diff --git "a/data_tmp/process_15/tokenized_finally.jsonl" "b/data_tmp/process_15/tokenized_finally.jsonl"
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-{"id": "2809.png", "formula": "\\begin{align*} X ( \\mathbf { x } ) = \\nabla H _ { X } ( \\mathbf { x } ) + u ( \\mathbf { x } ) , \\forall \\mathbf { x } \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} \\| e ^ { t L } \\tilde L _ \\beta e ^ { - t L } f \\| _ { X } & \\leq C _ L \\| \\tilde L _ \\beta e ^ { - t L } f \\| _ { X } \\\\ & \\leq C r \\sum _ { i = 1 } ^ { d } \\| \\partial _ { x _ i } e ^ { - t L } f \\| _ { X } \\\\ & \\leq C r \\sum _ { i = 1 } ^ { d } \\| e ^ { t L } \\partial _ { x _ i } e ^ { - t L } f \\| _ { X } . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} & \\nabla ^ { \\mathbb { S } } _ { u } ( \\omega \\otimes \\tau \\otimes s ) = ( \\nabla ^ { F } _ { X } \\omega ) \\otimes \\tau \\otimes s + \\omega \\otimes ( { \\mathcal L } _ { X } \\tau ) \\otimes s \\\\ & + \\frac { 1 } { 3 } ( \\iota _ { X } { \\mathcal H } ) \\wedge \\omega \\otimes \\tau \\otimes s + \\omega \\otimes \\tau \\otimes ( \\nabla ^ { 0 , \\mathcal S _ { \\mathcal G } } _ { X } s - \\frac { 1 } { 3 } ( \\mathrm { a d } _ { r } ) ( s ) ) . \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} R = 4 ( 2 n ) ^ 2 . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} L _ j : = X _ j - \\sqrt { - 1 } J X _ j , ~ ~ j = 1 , \\ldots , n \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} h \\left ( \\kappa \\right ) : = \\varepsilon ^ { 2 } d \\kappa ^ { 2 } + \\varepsilon \\kappa \\left ( d f _ { u _ { 0 } } + g _ { v _ { 0 } } \\right ) + \\det \\mathbb { J } . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} g _ m : [ - 1 , 1 ] \\rightarrow [ - l _ m , l _ m ] , t \\mapsto u = g _ m ( t ) , \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\int _ { \\R ^ { 2 | 2 } } F = \\int d x \\wedge d y \\circ \\partial _ \\xi \\ , \\partial _ \\eta \\ , F = \\int F \\ , d _ \\eta \\ , d _ \\xi \\ , d x \\ , d y . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\vec a = ( a _ 1 , a _ 2 , \\cdots , a _ { r } ) \\in \\Z ^ r , ~ ~ ~ ~ \\ \\ \\vec b = ( b _ 1 , b _ 2 , \\cdots , b _ s ) \\in \\Z ^ s \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\| \\mathcal { W } ^ { B D } [ y ; d _ 1 ] ( x , \\cdot ) - \\mathcal { W } ^ { B D } [ y ; d _ 2 ] ( x , \\cdot ) \\| _ { \\infty , t } \\le L \\ , \\| d _ 1 ( x , \\cdot ) - d _ 2 ( x , \\cdot ) \\| _ { \\infty , t } \\\\ \\| [ M ^ W d _ 1 ] ( x , \\cdot ) - [ M ^ W d _ 2 ] ( x , \\cdot ) ] \\| _ { \\infty , t } \\le L \\ , \\| d _ 1 ( x , \\cdot ) - d _ 2 ( x , \\cdot ) \\| _ { \\infty , t } \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} P _ \\alpha : = \\{ p ( \\beta ) : \\beta \\in C ^ * \\setminus \\{ \\alpha , \\alpha ^ c \\} \\} \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} V = \\sqrt { 2 } f ( \\theta _ 1 , \\theta _ 2 ) ( e ^ { i \\theta _ 1 } , e ^ { i \\theta _ 2 } ) . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\underset { k \\in \\mathbb { L } , n \\cdot k \\rightarrow \\pm \\infty } { \\lim } u ( \\cdot + k , t ) = p ^ { \\mp } ( \\cdot ) . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} g _ { 1 } = \\begin{cases} p \\tilde { f } | V ( p ) - \\frac { \\overline { a ( p ) } } { p } \\tilde { f } & p \\nmid N \\\\ p \\tilde { f } | V ( p ) - \\frac { \\overline { a ( p ) } } { p + 1 } \\tilde { f } & p | N , \\end{cases} \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} B ( k _ 1 , k _ 2 ; n , p _ { l _ 1 } , p _ { l _ 2 } ) : = { n \\choose k _ 1 , k _ 2 } p _ { l _ 1 } ^ { k _ 1 } p _ { l _ 2 } ^ { k _ 2 } ( 1 - p _ { l _ 1 } - p _ { l _ 2 } ) ^ { n - k _ 1 - k _ 2 } , \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} g \\ ! \\upharpoonleft _ { A } = h \\ ! \\upharpoonleft _ { A } g = h . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} A ( t ) = \\left ( \\begin{array} { c c } 0 & A _ 1 ( t ) \\\\ - A _ 1 ( t ) ^ T & O \\end{array} \\right ) \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\sum \\limits _ { b \\in B ( G , \\mu ) } \\mathrm { M a n t } _ { b , \\mu } ( \\mathrm { R e d } _ b ( \\pi ) ) = [ \\pi ] [ r _ { - \\mu } \\circ \\mathrm { L L } ( \\pi ) | _ { W _ { E _ { \\{ \\mu \\} _ G } } } \\otimes | \\cdot | ^ { \\mathrm { - \\sum _ { \\tau } p _ { \\tau } q _ { \\tau } / 2 } } ] . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\prod _ { m = 1 } ^ { \\infty } \\frac { \\left ( 1 - ( - q t ) ^ m \\right ) ^ 2 } { \\left ( 1 + q ( - q t ) ^ m \\right ) \\left ( 1 + q ^ { - 1 } ( - q t ) ^ m \\right ) } = 1 + \\sum _ { n = 1 } ^ { \\infty } r _ 2 \\left ( n , q \\right ) t ^ n , \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\chi _ { \\widetilde v } ( \\tau _ 1 ) = \\cdots = \\chi _ { \\widetilde v } ( \\tau _ { \\widetilde { v } } ) & = 1 \\\\ \\chi _ { \\widetilde v } ( \\tau _ { \\widetilde { v } + 1 } ) = \\cdots = \\chi _ { \\widetilde v } ( \\tau _ { r } ) & = - 1 . \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\sin ( \\log ( \\log | x | ^ { - 1 } ) ) , u _ 2 ( x ) = \\cos ( \\log ( \\log | x | ^ { - 1 } ) ) \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} & C _ { 2 n } ^ { \\lambda } ( t ) = \\frac { ( - 1 ) ^ n } { ( \\lambda + n ) B ( \\lambda , n + 1 ) } \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - n , n + \\lambda ; \\frac { 1 } { 2 } ; t ^ 2 \\Big ) ; \\\\ [ 4 p t ] & C _ { 2 n + 1 } ^ { \\lambda } ( t ) = \\frac { ( - 1 ) ^ n 2 t } { B ( \\lambda , n + 1 ) } \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - n , n + \\lambda + 1 ; \\frac { 3 } { 2 } ; t ^ 2 \\Big ) , \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\begin{aligned} { \\textbf { I } } _ l ( \\boldsymbol { \\psi } , { { \\bf { W } } } ) \\triangleq \\lim \\limits _ { M , N \\to + \\infty } \\frac { 1 } { M N } { \\textbf { I } _ S } ( \\boldsymbol { \\psi } , { { \\bf { W } } } ) . \\end{aligned} \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\partial _ t \\rho + u \\cdot \\nabla \\rho = 0 . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\delta ^ { 2 } V _ { 0 } ( \\delta ) = h ^ { 2 } , \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u _ \\varepsilon | ^ 2 d x ~ & = \\int _ \\Omega u _ \\varepsilon \\left ( \\lambda _ \\varepsilon H ( u _ \\varepsilon ) u _ \\varepsilon \\exp ( u _ \\varepsilon ^ 2 ) \\right ) d x \\ , , \\\\ & = \\int _ \\Omega U _ { \\varepsilon , x _ \\varepsilon } \\Delta U _ { \\varepsilon , x _ \\varepsilon } d x + o \\left ( \\check { \\zeta } _ \\varepsilon \\right ) \\ , . \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { y } = & p _ y ( z , u _ h ) - r _ y ( y , u _ h ) + d _ y ( t ) \\mathbf { - g _ y ( y , x ) + g _ x ( y , x ) - d _ b ( t ) } \\\\ \\dot { z } = & p _ z ( y , z , u _ h ) - r _ z ( y , z , u _ h ) + d _ z ( t ) \\\\ \\mathbf { \\dot { x } } = & \\mathbf { g _ y ( y , x ) - g _ x ( y , x ) - r _ x ( x ) + d _ x ( t ) + d _ b ( t ) } \\end{aligned} \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\underset { \\rho , n _ p } { \\max } & \\gamma _ { } , \\\\ & \\xi ^ { \\ast } \\geq 1 - \\epsilon , \\\\ & n _ p = 1 , 2 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} \\bar H ^ { 1 ^ * } _ { , k } = 0 . \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} v _ { 1 } ( 0 , \\lambda ) = v _ { 2 } ^ { \\prime } ( 0 , \\lambda ) = 1 , v _ { 2 } ( 0 , \\lambda ) = v _ { 1 } ^ { \\prime } ( 0 , \\lambda ) = 0 , \\ \\lambda \\in \\mathbb { C } . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} a _ { ( f ) } b - b _ { ( f ) } a = 2 \\pi i \\ , a ( 0 ) b \\pmod { V _ { ( g ) } V } , \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\sin ( \\log ( \\log | x | ^ { - 1 } ) ) , u _ 2 ( x ) = \\cos ( \\log ( \\log | x | ^ { - 1 } ) ) \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} & \\frac { d p ^ { \\pm } } { d t } - F ( t , p ^ { \\pm } ) = ( 0 , 0 , \\dots , 0 ) , \\ \\ \\frac { d \\varphi ^ { \\pm } } { d t } - D F ( t , p ^ { \\pm } ) \\varphi ^ { \\pm } = \\lambda _ { \\pm } \\varphi ^ { \\pm } , \\\\ & p ^ { \\pm } ( \\cdot + T ) = p ^ { \\pm } ( \\cdot ) , \\ \\varphi ^ { \\pm } ( \\cdot + T ) = \\varphi ^ { \\pm } ( \\cdot ) . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} 1 \\ , \\leq \\ , \\frac { 1 } { \\sqrt { 1 - t / d } } \\ , \\leq \\ , \\frac { 1 } { \\sqrt { 1 - \\tau / \\sqrt { d } - 1 / d } } \\ , = : \\ , \\kappa _ { \\tau } ( d ) \\ , \\xrightarrow [ d \\rightarrow \\infty ] { } \\ , 1 \\ , . \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} j ^ \\textnormal { c o h } ( t _ 1 , z ) = e ^ { \\frac { t _ 1 H } { z } } \\sum _ { d \\geq 0 } \\frac { e ^ { t _ 1 d } } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ L ^ H \\int \\alpha _ t \\frac { \\partial } { \\partial x } M ( d t , \\theta ^ H _ t ) \\right ] = 0 \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } \\log ( 1 - r \\cos ( w ) ) ~ d w & = 4 \\pi \\log \\frac { \\sqrt { 1 + r } + \\sqrt { 1 - r } } { 2 } . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} k _ { r + 1 } \\in \\{ 2 k _ 1 , \\dotso , r k _ 1 , ( r + 1 ) k _ 1 \\} = \\{ k _ 2 , \\dotso , k _ r , ( r + 1 ) k _ 1 \\} . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} c ^ 2 ( \\mu ) : = \\int c ^ 2 _ \\mu ( z ) d \\mu ( z ) . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} & h ( x ) \\\\ & \\le \\Bigl ( g _ 3 ( x ) + \\frac { 2 } { 6 ^ 4 } ( 3 ^ 4 x - 3 3 ) ( 7 - 2 ^ 4 x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 1 - 3 ^ 5 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 3 0 2 } { 3 ^ 6 } } \\\\ & + \\frac 1 { 1 0 \\cdot 6 ^ 5 } - \\eta = - \\frac { 1 3 2 5 0 6 6 1 0 8 5 5 2 8 9 7 4 2 1 9 0 5 0 8 2 0 5 7 5 4 6 8 7 6 6 8 8 3 } { 1 5 5 1 3 9 8 3 8 5 0 9 9 1 9 8 2 9 3 0 4 6 4 9 4 8 3 4 1 0 2 5 9 1 8 8 3 1 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} k _ t ( x ) = \\mathcal { F } _ k ^ { - 1 } ( e ^ { - t | . | ^ 2 } ) ( x ) = t ^ { - \\gamma _ k - d / 2 } e ^ { - | x | ^ 2 / t } \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} \\int _ { \\tilde { M } _ \\tau } e ^ { - \\frac { | x | ^ 2 } { 4 } } - \\int _ \\Sigma e ^ { - \\frac { | x | ^ 2 } { 4 } } & \\geq \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , \\Big [ - C u ^ 2 + \\frac { 1 } { C } \\ , | \\nabla ^ \\Sigma u | ^ 2 \\Big ] \\\\ & - C L ^ { - 1 } \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , u ^ 2 . \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} - \\log \\left ( \\prod _ { p \\leq X } \\frac { p - 1 } { p - q _ p } \\right ) & = - \\sum _ { p \\leq X } \\log \\left ( 1 - \\frac { p } { p + ( p - 1 ) X } \\right ) \\\\ & \\leq \\sum _ { p \\leq X } \\left ( \\frac { 1 } { X } + \\frac { 2 } { p X } \\right ) \\\\ & \\leq \\left ( 1 + \\mathcal { O } \\left ( \\frac { 1 } { ( \\log X ) ^ 2 } \\right ) \\right ) \\frac { 1 } { \\log X } \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} w _ \\varepsilon ( z ) = u _ \\varepsilon ( y _ \\varepsilon + \\nu _ \\varepsilon z ) . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} g ^ { - 1 } a g & = g ^ { - 1 } [ x _ 1 , y _ 1 ] [ x _ 2 , y _ 2 ] \\cdot \\cdot \\cdot [ x _ n , y _ n ] g \\\\ & = g ^ { - 1 } [ x _ 1 , y _ 1 ] e [ x _ 2 , y _ 2 ] e \\cdot \\cdot \\cdot e [ x _ n , y _ n ] g \\\\ & = g ^ { - 1 } [ x _ 1 , y _ 1 ] g g ^ { - 1 } [ x _ 2 , y _ 2 ] g \\cdot \\cdot \\cdot g ^ { - 1 } [ x _ n , y _ n ] g \\\\ & = [ g ^ { - 1 } x _ 1 g , g ^ { - 1 } y _ 1 g ] [ g ^ { - 1 } x _ 2 g , g ^ { - 1 } y _ 2 g ] \\cdot \\cdot \\cdot [ g ^ { - 1 } x _ n g , g ^ { - 1 } y _ n g ] \\in \\mathcal { G } ' . \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} f ^ { ' } ( t ) = 4 t ^ 3 - 6 S t ^ 2 - 1 2 S ( S - 1 ) t + 2 S ( 2 - 3 S ) ^ 2 , \\ \\ f ^ { '' } ( t ) = 1 2 ( t ^ 2 - S t - S ( S - 1 ) ) , \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} & \\lim _ { b \\to 0 } \\frac { ( 1 - \\alpha ^ 2 ) b } { g ( z _ - ) } = 1 \\lim _ { b \\to 0 } z _ - ( b ) = - \\infty \\ , , \\\\ & \\lim _ { b \\to 0 } \\frac { b g ( z _ + ) } { 1 - \\alpha ^ 2 } = \\frac 1 { 1 - \\alpha ^ 2 } > 1 \\lim _ { b \\to 0 } z _ + ( b ) = \\alpha \\ , . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} \\left \\| f _ \\lambda \\right \\| _ { \\dot { H } ^ s ( \\R ^ d ) } = \\lambda ^ { \\frac { d } { 2 } - s } \\left \\| f \\right \\| _ { \\dot { H } ^ s ( \\R ^ d ) } \\ , . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} R _ { i j k l } = \\sum _ { p } \\left ( h ^ { p ^ { \\ast } } _ { i k } h ^ { p ^ { \\ast } } _ { j l } - h ^ { p ^ { \\ast } } _ { i l } h ^ { p ^ { \\ast } } _ { j k } \\right ) , \\ \\ R _ { i k } = \\sum _ { p } H ^ { p ^ { \\ast } } h ^ { p ^ { \\ast } } _ { i k } - \\sum _ { p , j } h ^ { p ^ { \\ast } } _ { i j } h ^ { p ^ { \\ast } } _ { j k } . \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} I _ 0 = \\int _ 0 ^ { \\pi } \\dfrac { d \\theta } { ( a _ { 2 n } + \\cos { 2 n \\theta } ) ^ { 2 s } } = ( 2 \\rho ^ { 2 n } ) ^ { 2 s } \\dfrac { \\pi \\sum _ { m = 0 } ^ { 2 s - 1 } { 2 s - 1 \\choose m } { 4 s - m - 2 \\choose 2 s - 1 } \\left ( \\rho ^ { 4 n } - 1 \\right ) ^ m } { \\left ( \\rho ^ { 4 n } - 1 \\right ) ^ { 4 s - 1 } } \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\xi _ { \\pi } ^ { - 1 } = \\xi _ { \\pi ' } \\xi \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\nu ( \\partial T _ x ) = \\sum _ { y : y ^ - = x } \\nu ( \\partial T _ y ) \\ , . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{gather*} \\delta X ^ i = \\rho ( \\epsilon ) ^ i = \\rho ^ i _ a ( X ) \\epsilon ^ a , \\end{gather*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\psi _ 1 ' ( a ) & = \\xi '' ( a ) \\Big [ \\frac { ( 1 + z _ 1 + z _ 2 ) q } { z _ 1 } - a - \\frac { q \\xi ' ( q ) ( 1 + z _ 2 ) ( 1 + z _ 1 + z _ 2 ) } { z _ 1 [ ( 1 + z _ 2 ) \\xi ' ( q ) + z _ 1 \\xi ' ( a ) ] } \\Big ] \\\\ & = \\frac { \\xi '' ( a ) [ \\xi ' ( a ) [ ( 1 + z _ 1 + z _ 2 ) q - z _ 1 a ] - \\xi ' ( q ) ( 1 + z _ 2 ) a ] } { ( 1 + z _ 2 ) \\xi ' ( q ) + z _ 1 \\xi ' ( a ) } . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} m _ 0 ^ { ( r ) } = 0 , m _ { 2 n } ^ { ( r ) } = 2 ^ r m _ n ^ { ( r ) } , m _ { 2 n + 1 } ^ { ( r ) } = 2 ^ r m _ n ^ { ( r ) } + 1 . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\begin{array} { c } \\displaystyle \\int _ { \\Omega } p \\left | \\nabla u \\right | ^ { 2 } \\geq p _ { 0 } \\int _ { \\Omega } \\left | \\nabla u _ { 0 } \\right | ^ { 2 } - 2 \\pi p _ { 0 } \\left ( \\sum _ { i = 1 } ^ { m } d _ { i } ^ { 2 } \\right ) I \\left ( \\dfrac { R } { R _ { 0 } } \\right ) + \\\\ \\\\ \\displaystyle - 2 \\pi \\left ( 1 - a ^ { 2 } \\right ) p _ { 0 } \\sum _ { i \\neq j } \\left | d _ { i } \\right | \\left | d _ { j } \\right | \\log \\frac { R } { | a _ { i } - a _ { j } | } - C , \\end{array} \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} P _ { \\textrm { o u t , F } } ^ { \\textrm { I S A O C } } = & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N H F = 0 , F D F = 0 , N D F = 1 , N D N = 1 } \\right \\rbrace \\\\ & + \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace \\times \\left ( 1 - \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D F = 1 , N D N = 1 } \\right \\rbrace \\right ) . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} e _ N ( x ) & : = u ( x ) - I _ N ^ { \\lambda } u ( x ) = S ( t ) \\big ( \\breve U ( t ) - I _ N ^ G \\breve U ( t ) \\big ) : = S ( t ) \\breve e _ N ( t ) , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} A ^ { * n } A ^ n A ^ n = A ^ n A ^ { * n } A ^ n . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\{ \\varphi , h \\} = \\{ \\varphi , \\varphi + \\psi \\} \\in T ^ * T + I , \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} v _ 1 ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { r r } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) , \\sigma v _ 1 ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { r r } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ) , \\\\ v _ 2 ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { r r } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) , \\sigma v _ 2 ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { r r } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\nu _ 2 = [ \\nu _ 1 ; \\nu _ 2 ( \\phi _ 2 ) = \\nu ( \\phi _ 2 ) ] . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\theta ( J _ { n } ^ p ) = \\biggl ( { p \\over ( p + 1 ) ^ 2 } + o ( 1 ) \\biggr ) n ^ 2 . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} a ^ * : = v ^ { ( 1 ) } ( t ^ * ) = v ^ { ( 2 ) } ( t ^ * ) = a ^ * _ { } + a _ 0 ^ * \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\pi _ { \\mathcal { K } } \\big ( 3 \\mathcal { S } _ { 0 , 2 } ( V _ A ^ { \\perp } , W _ 2 ) + \\mathcal { S } _ { 0 , 3 } ( V _ A ^ { \\perp } , V _ A ^ { \\perp } , V _ A ^ { \\perp } ) \\big ) = - 4 V _ A ^ { \\perp } . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\mathbf { Z } ( p _ 1 , \\ldots , p _ r ) = \\{ ( x _ { j , 1 } , \\ldots , x _ { j , m } ) \\ : | \\ : 1 \\leq j \\leq L \\} . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} A ^ q A ^ { * } A = A ^ { * } A ^ { } A ^ q . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} T _ 0 ( x ) = \\log \\left ( 1 + | x | ^ 2 \\right ) \\ , . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} | \\Lambda ( f _ 1 , \\ldots , f _ { n + 1 } ) | \\lesssim \\| \\mathrm { M } ( f _ 1 , \\ldots , f _ { n + 1 } ) \\| _ { 1 } , \\mathrm { M } ( f _ 1 , \\ldots , f _ { n + 1 } ) ( x ) = \\sup _ { x \\in Q } \\prod _ { j = 1 } ^ { n + 1 } \\langle | f _ j | \\rangle _ Q . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\mathcal { S } _ 1 ^ + = \\mathcal { S } _ 1 ^ - = \\left \\{ s \\in \\mathbb { R } \\colon 0 < s < \\frac { \\eta + \\sqrt { 1 + \\eta ^ 2 } } { \\sigma ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} G ( m , 2 , a , \\epsilon , s ) \\leq 2 \\sum _ { k = 1 } ^ { ( { m ( 1 + \\varepsilon ) \\over 2 } ) ^ { { 1 \\over a } } } k ^ { - d s } ( m - k ^ a ) ^ { - { d s \\over a } } \\cdot N _ { m , a , \\epsilon } ( k ) , \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\overline { Q } _ { k , i } ( x ; q ) - \\overline { Q } _ { k , i - 1 } ( x ; q ) = ( x q ) ^ i \\overline { Q } _ { k , k - i } ( x q ; q ) + ( x q ) ^ { i - 1 } \\overline { Q } _ { k , k - i + 1 } ( x q ; q ) . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\lim _ { z \\to 1 ^ - } \\frac { { } _ 2 F _ 1 ( a , b ; c ; z ) } { ( 1 - z ) ^ { c - a - b } } = \\frac { \\Gamma ( c ) \\Gamma ( a + b - c ) } { \\Gamma ( a ) \\Gamma ( b ) } . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\frac { p } { m _ j q } = \\frac { p _ { n } } { q _ { n } } . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} { u _ n } ^ q \\simeq \\left ( \\prod _ { k = n _ 0 } ^ n \\mu _ q ( k ) \\right ) e _ q \\ , , \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} f _ { n } ( y _ { 1 } , \\ldots , y _ { n } ) = \\exp ( y _ { 1 } ) \\otimes \\cdots \\otimes \\exp ( { y _ { n } } ) , \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( e ( J _ b ) L J ( \\delta ^ { \\frac { 1 } { 2 } } _ { G , P _ b } I ^ { M _ b } _ { M _ S } ( \\rho ) ) ) = \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( I ^ G _ { M _ S } ( \\rho ) ) ) . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} 1 - \\frac 1 { \\beta _ 0 } : = \\frac { p } { q } + \\frac { 2 p } { r } < 1 \\ , , \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} \\bar p ^ n _ 3 + \\bar p ^ n _ 4 = \\bar p ^ { n + 1 } _ 3 + \\bar p ^ { n + 1 } _ 4 , L ^ { 3 / 2 } ( ( 0 , T ) \\times B _ { 2 ^ n } ) \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} ( D ^ s h ) ( x ) & = C ( n , s ) \\ , \\mathrm { P . V . } \\int _ { \\mathbb R ^ n } \\frac { h ( x + y ) - h ( x ) } { | y | ^ { n + s } } \\ , d y \\\\ & = C ( n , s ) \\lim _ { \\varepsilon \\searrow 0 } \\int _ { | y | \\ge \\varepsilon } \\frac { h ( x + y ) - h ( x ) } { | y | ^ { n + s } } \\ , d y , \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} | 1 - e ^ { \\mathbf { i } t } | = 2 \\sin \\left ( \\frac { t } { 2 } \\right ) . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} \\eta _ { i j } : = ( \\partial _ { t ^ i } , \\partial _ { t ^ j } ) = \\partial _ { t ^ r } \\partial _ { t ^ i } \\partial _ { t ^ j } \\mathbb { F } ( t ) \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\langle x , y \\rangle : = ( x , \\varphi ^ * ( y ) ) _ { R T } \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} \\frac { \\tilde \\lambda _ { j } } { \\tilde \\lambda _ { j - 1 } - \\tilde \\lambda _ j } \\leq \\frac { \\lambda _ { j - 1 } } { \\tilde \\lambda _ { j - 1 } - \\lambda _ { j - 1 } } = \\frac { 1 } { 1 - A } \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } \\leq 2 \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } , \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} f ( g ) = \\Phi \\Big ( \\frac { g } { t ^ N } \\Big ) \\cdot e \\Big ( g \\cdot ( X _ { i _ 1 } - X _ { i _ 2 } ) \\Big ) , \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\phi ( y + 1 ) \\psi ( y ) - \\phi ( y ) \\psi ( y + 1 ) = c . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\Delta _ K ( t ) f ( t ) f ( t ^ { - 1 } ) = \\Delta _ { K ' } ( t ) g ( t ) g ( t ^ { - 1 } ) \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\sum ^ { \\infty } _ { j = 1 } \\omega ( \\sigma ^ { j } ) \\leq C _ { b } . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} f ( T ) = \\psi ( T ) ^ { - 1 } ( f \\psi ) ( T ) = ( I + T ) ^ 2 T ^ { - 1 } ( f \\psi ) ( T ) . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} D _ f ( \\lambda ) = \\mu \\{ x \\in \\R ^ n : | f ( x ) | > \\lambda \\} = b _ n \\lambda ^ { - \\frac { n } { s } } \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} Y = \\ln X , \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} \\begin{aligned} { } \\| v - L _ \\nu \\| _ { L ^ { \\infty } ( B ( \\sigma ^ \\nu ) ) } & \\leq \\sigma ^ \\nu \\omega ( \\sigma ^ \\nu ) ~ ~ ~ ~ ~ ~ | b _ \\nu | \\leq C , \\\\ | a _ { \\nu + 1 } - a _ \\nu | & \\leq C \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) , ~ ~ ~ ~ | b _ { \\nu + 1 } - b _ \\nu | \\leq C \\omega ( \\sigma ^ \\nu ) . \\end{aligned} \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - \\hat h _ n x _ n ^ 2 ) + u _ { n + 1 } , x _ 0 \\in \\mathbb R . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { ( 0 , a ) ^ 2 } \\det \\nabla f _ n = - \\frac { a } { 2 } < 0 . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 + \\alpha ) } _ m ( z ) , z ^ j { \\bar z } ^ l \\rangle _ { \\alpha } = 0 \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } D ( x _ n , x ) = 0 . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} w _ j = p _ 1 ^ { ( j ) } \\dots p _ m ^ { ( j ) } \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} ( d S _ { l } ) _ { \\alpha \\sqcup h _ { 0 } } ( 0 \\sqcup \\gamma ) = ( d L _ { a } ) _ { S _ { l } ( h _ { 0 } ) } \\circ ( d S _ { l } ) _ { h _ { 0 } } ( \\gamma ) = ( d L _ { a } ) _ { g _ { 0 } } ( \\xi _ { 0 } ) = \\xi . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} & \\frac { \\partial \\gamma _ { } } { \\partial n _ p } = \\\\ & \\frac { \\lambda _ { a b } ^ 2 { \\rho ^ \\ast } ^ 2 \\left [ \\ ! - \\ ! \\lambda _ { a b } \\rho ^ \\ast n _ p ^ 2 - ( 2 \\sigma _ b ^ 2 + 2 \\lambda _ { a b } \\rho ^ \\ast ) n _ p + n \\sigma _ b ^ 2 + \\lambda _ { a b } \\rho ^ \\ast n \\right ] } { n \\ln 2 \\left [ \\sigma _ b ^ 3 + \\lambda _ { a b } \\rho ^ \\ast \\sigma _ b ( n _ p + 1 ) \\right ] ^ 2 } . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\| \\nabla u _ { x ^ * , \\rho _ 0 } \\| _ { L ^ { n , \\infty } ( B _ { \\theta _ 0 } ) } = \\| \\nabla u \\| _ { L ^ { n , \\infty } ( B _ { \\theta _ 0 \\rho _ 0 } ( x ^ * ) ) } \\\\ \\leq \\frac { 1 } { 2 } \\| \\nabla u _ { x ^ * , \\rho _ 0 } \\| _ { L ^ { n , \\infty } ( B _ 1 ) } = \\frac { 1 } { 2 } \\| \\nabla u \\| _ { L ^ { n , \\infty } ( B _ { \\rho _ 0 } ( x ^ * ) ) } . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} W _ w ( C ) = \\{ \\alpha \\in C : | \\alpha | = w \\} \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} I = I _ { 0 } + I _ { 1 } p + \\ldots + I _ { N - 1 } p ^ { N - 1 } , \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} w _ { i , p } ( z ) \\leq \\left ( 4 - \\gamma \\right ) \\log \\frac { 1 } { | z | } + \\widetilde C _ { \\gamma } , \\qquad \\forall i = 1 , \\ldots , k \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\Phi _ { S _ 0 } ( g x ) = g g _ 1 \\Phi _ { \\eta } ( g ^ { - 1 } _ 1 x ) = g \\Phi _ { \\eta _ 1 } ( x ) = g \\Phi _ { S _ 0 } ( x ) , \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} F _ { n + 1 } = \\left [ \\begin{array} { c c c c | c } & & F _ n & & 0 \\\\ \\hline 0 & \\cdots & 0 & U I _ { n + 1 } & C _ { n + 1 } \\end{array} \\right ] : { \\begin{array} { l l l } R ( - 1 ) ^ { \\binom { n + 2 } { 2 } } & \\xrightarrow { F _ n } & R ^ { \\binom { n + 1 } { 2 } } \\\\ \\ ; \\ ; \\bigoplus & \\searrow & \\ ; \\bigoplus \\\\ R ( - 1 ) ^ { n + 2 } & \\longrightarrow & R ^ { n + 1 } \\end{array} } \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\sigma ( \\theta ) = \\left ( ( R + r \\cos ( \\theta _ 1 ) ) \\cos ( \\theta _ 2 ) , ( R + r \\cos ( \\theta _ 1 ) ) \\sin ( \\theta _ 2 ) , r \\sin ( \\theta _ 1 ) \\right ) \\ ; , \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} \\Gamma ( k ) = \\begin{bmatrix} 0 & - 1 & - 2 & - 6 & - 4 \\\\ - 3 & - 4 & - 5 & - 9 & - 7 \\\\ - 1 0 & - 1 1 & - 1 2 & - 1 6 & - 1 4 \\\\ - 1 0 & - 1 1 & - 1 2 & - 1 6 & - 1 4 \\\\ - 6 & - 7 & - 8 & - 1 2 & - 1 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} f \\left ( x _ { o } - , u ^ { - } \\right ) = f \\left ( x _ { o } + , u ^ { + } \\right ) \\dot = \\bar f , \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\frac { 1 } { C } R ( \\alpha t ) e ^ { \\alpha t } \\leq \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } R _ k \\leq C R ( \\alpha t ) e ^ { \\alpha t } \\quad \\mbox { f o r a l l $ t \\geq t _ 0 $ } . \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) ( z , Q ) = \\lim _ { t \\to 0 } P _ { q ^ t , z } \\cdot \\varphi _ { q ^ t , z } ^ * \\widetilde { J ^ { K \\textnormal { t h } } } ( z , q ^ t , Q ) \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\frac { d } { d x } \\big ( h \\cdot g \\big ) = \\frac { d } { d x } h \\cdot g + h \\cdot \\frac { d } { d x } g \\in V + P ( V ) , \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} m _ { k , \\lambda + i \\gamma } ( \\xi ) = \\sum _ { m = 1 } ^ { \\infty } \\frac { e ^ { - i 2 \\pi m ^ k \\xi } } { m ^ { \\lambda + i \\gamma } } . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( x ) \\left ( \\mathcal { D } _ 0 ^ \\theta g \\right ) ( x ) \\ , d x = \\int _ 0 ^ 1 ( H _ 0 ^ { \\theta - 1 } \\varphi _ g ) ( x ) \\varphi _ f ( x ) \\ , d x . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} g ( s ) : = \\sum _ { n = 0 } ^ { \\infty } b ( n ) e ^ { - s n } , \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} | x _ n | < | x _ 0 | ^ { { 3 } ^ { n } } \\prod _ { i = 0 } ^ { n - 1 } h _ { n - 1 - i } ^ { { 3 } ^ { i } } , n \\in \\mathbb N . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} Q ( \\theta ) = \\int _ 0 ^ \\theta C _ \\vartheta ( z ) d z , \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} \\ddot { u } ( t ) - \\div ( A \\nabla u ( t ) ) = f ( t ) \\hbox { i n } \\Omega \\setminus \\Gamma ( t ) \\ , , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\begin{aligned} & a _ { 1 i } = \\sigma _ i , \\ a _ { 2 i } = \\sigma _ i , \\ a _ { 3 i } = \\mu _ i \\xi _ 1 - \\theta _ i \\xi _ 2 , \\ a _ { 4 i } = \\theta _ i \\xi _ 1 + \\mu _ i \\xi _ 2 , \\\\ & a _ { 5 i } = - \\mu _ i \\xi _ 1 + \\theta _ i \\xi _ 2 , \\ a _ { 6 i } = - \\theta _ i \\xi _ 1 - \\mu _ i \\xi _ 2 , \\ a _ { 7 i } = \\sigma _ i , \\ a _ { 8 i } = \\sigma _ i . \\end{aligned} \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} ( A \\otimes B ) _ { i , j } = \\bigoplus _ { 1 \\leq k \\leq n } a _ { i , k } \\otimes b _ { k , j } \\ = \\max _ { 1 \\leq k \\leq n } a _ { i , k } + b _ { k , j } . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} g _ { ( i ) } ( z , t ) = \\underbrace { ( f ( z , t ) - 1 ) } _ { \\sigma } \\cdot \\underbrace { \\left ( z \\cdot f ( z , 1 ) \\right ) } _ { 1 \\ominus \\tau } = z f ( z , 1 ) ( f ( z , t ) - 1 ) . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} Q _ 1 \\leq C _ 2 C _ { V , l } \\| \\gamma \\| _ { 1 - \\rm v a r } ^ { \\bar { l } } = C _ 2 C _ { V , l } \\| v \\| _ { \\textsc { C C } } ^ { \\bar { l } } , \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\bar { \\mathsf h } _ { i i } - \\mathsf h _ o \\rightarrow 0 \\ ; \\ ; i = 1 , \\ldots , \\mathbf k ^ * . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} & 1 + \\varpi ^ { l - 1 } X ^ { \\prime } + 2 ^ { - 1 } \\varpi ^ { 2 l - 2 } X ^ { \\prime 2 } \\pmod { \\frak { p } ^ r } \\\\ = & 1 + \\varpi ^ { - 1 } X + 2 ^ { - 1 } \\varpi ^ { 2 l - 2 } X ^ 2 + \\varpi ^ l T \\pmod { \\frak { p } ^ r } \\\\ = & ( 1 + \\varpi ^ { l - 1 } X + 2 ^ { - 1 } \\varpi ^ { 2 l - 2 } X ^ 2 ) ( 1 + \\varpi ^ l T ) \\pmod { \\frak { p } ^ r } , \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\partial _ r l _ { a b } = R _ { L a b L } - l _ { a c } l ^ c _ b \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} g _ { 1 } = x _ { 0 } ^ { n + 1 } - x _ { 3 } ^ { n } & = \\sum _ { \\mu = 1 } ^ { n - 1 } \\alpha _ { \\mu } \\left [ x _ { 2 } ^ { ( n + 1 ) - \\mu } x _ { 3 } ^ { \\mu } - x _ { 0 } ^ { ( n + 1 ) - \\mu } x _ { 1 } ^ { \\mu + 1 } \\right ] \\\\ & + \\sum _ { t = 0 } ^ { n } \\beta _ { t } \\left [ x _ { 1 } ^ { ( n + 1 ) - t } x _ { 3 } ^ { t } - x _ { 0 } ^ { n - t } x _ { 2 } ^ { t + 1 } \\right ] + \\gamma _ { 2 } ( x _ { 1 } x _ { 2 } - x _ { 0 } x _ { 3 } ) , \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\bigg \\| \\Big ( \\sum _ j ( f ^ j ) ^ s \\Big ) ^ \\frac 1 s \\bigg \\| _ { L ^ { q } ( v ) } \\le C ( [ \\vec v ] _ { A _ { \\vec q } } ) \\prod _ { i = 1 } ^ m \\bigg \\| \\Big ( \\sum _ j ( f _ i ^ j ) ^ { s _ i } \\Big ) ^ \\frac 1 { s _ i } \\bigg \\| _ { L ^ { q _ i } ( v _ i ) } \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} \\gamma _ { p , q } = \\frac { d } { 2 } - \\frac { d } { q } - \\frac { 2 s } { p } , \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} | | H f _ k | | ^ 2 _ { L ^ 2 ( 1 , + \\infty ) } & = \\int ^ { + \\infty } _ 1 \\Big | \\int ^ { 1 / 2 + 2 ^ { - k } } _ { 1 / 2 - 2 ^ { - k } } \\frac { f _ k ( y ) } { x - y } d y \\Big | ^ 2 d x \\\\ & \\leq \\int ^ { + \\infty } _ 1 \\frac { 1 } { | x - \\frac { 3 } { 4 } | ^ 2 } \\Big ( \\int ^ { 1 / 2 + 2 ^ { - k } } _ { 1 / 2 - 2 ^ { - k } } | f _ k ( y ) | d y \\Big ) ^ 2 d x \\\\ & \\leq 2 ^ { - k + 1 } \\int ^ { + \\infty } _ 1 \\frac { 1 } { | x - \\frac { 3 } { 4 } | ^ 2 } d x \\lesssim 2 ^ { - k } , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} { } _ { 2 } F _ { 1 } ( b , a ; a ; z ) = ( 1 - z ) ^ { - b } . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} | f ( x ) | \\lesssim C _ { 1 } \\chi _ { B ( y _ { 1 } , r ) } ( x ) + C _ { 2 } \\chi _ { B ( y _ { 2 } , r ) } ( x ) | y _ { 1 } - y _ { 2 } | = N r . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\left \\{ \\R ^ n \\setminus ( \\Omega _ 1 \\setminus G ) \\right \\} \\cap \\Omega _ 1 = \\left \\{ ( \\R ^ n \\setminus \\Omega _ 1 ) \\cup G \\right \\} \\cap \\Omega _ 1 = G \\cap \\Omega _ 1 \\neq \\emptyset . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} u _ x ( \\bar x ) = 0 , u _ { x x } ( \\bar x ) \\le 0 , \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} W ( z ) = \\frac { 1 } { 2 \\pi } | z | ^ n \\exp ( - \\alpha | z | ^ { 2 m } ) , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\overline { U } _ { 2 k , 2 a } ( x ; q ) = ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} g '' ( u ) = - \\xi '' ( u ) + \\frac 1 { \\nu ( [ u , 1 ] ) ^ 2 } . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\bigg ( \\frac { \\mathrm { d } } { \\mathrm { d } r } \\bigg ) ^ 2 \\bigg ( \\frac { 1 } { r } \\frac { \\mathrm { d } } { \\mathrm { d } r } \\bigg ) ^ { k - 1 } \\left ( r ^ { 2 k - 1 } \\phi ( r ) \\right ) = \\bigg ( \\frac { 1 } { r } \\frac { \\mathrm { d } } { \\mathrm { d } r } \\bigg ) ^ { k } \\left ( r ^ { 2 k } \\frac { \\mathrm { d } \\phi } { \\mathrm { d } r } ( r ) \\right ) \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\sum _ { k , p } \\partial _ { t ^ i } \\partial _ { t ^ j } \\partial _ { t ^ k } \\mathbb { F } ( t ) ~ \\eta ^ { k p } ~ \\partial _ { t ^ p } \\partial _ { t ^ q } \\partial _ { t ^ s } \\mathbb { F } ( t ) = \\sum _ { k , p } \\partial _ { t ^ s } \\partial _ { t ^ j } \\partial _ { t ^ k } \\mathbb { F } ( t ) ~ \\eta ^ { k p } ~ \\partial _ { t ^ p } \\partial _ { t ^ q } \\partial _ { t ^ i } \\mathbb { F } ( t ) , \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} f _ 1 ( a , b ) & = f _ 4 ( b ^ { - 1 } , a b ) f _ 3 ( a , b ) , \\\\ f _ 2 ( a , b ) & = - f _ 3 ( b ^ { - 1 } , a b ) f _ 4 ( b ^ { - 1 } , e _ \\lambda ) f _ 3 ( b , b ^ { - 1 } ) + f _ 4 ( b ^ { - 1 } , a b ) f _ 4 ( a , b ) . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\rho = ( n , n - 1 , \\dots , 2 , 1 ) \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} s ( z ) = \\begin{cases} 1 , & z \\in ( - 2 h , 0 ) , \\\\ - 1 , & x \\in ( 0 , h ) . \\end{cases} \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} G ( \\zeta ) = G ( \\zeta , 1 ) & = \\lim _ { z \\to 1 } \\frac { 2 n ^ 2 z ^ { \\frac { n - 1 } { 2 } } ( 1 - z ) ^ 2 - n ( n - 1 ) z ^ { \\frac { n - 1 } { 2 } } ( 1 - z ^ n ) ( 1 - z ) ^ 2 - 2 ( 1 - z ^ n ) ^ 2 } { 2 ( 1 - z ^ n ) ^ 2 ( 1 - z ) ^ 2 } \\\\ [ 5 p t ] & = \\frac { n ^ 2 - 1 } { 2 4 } , \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\gamma \\left \\{ y _ n , \\gamma y _ n , \\gamma ^ 2 y _ n , \\dots , \\gamma ^ { M - 1 } y _ n \\right \\} = \\left \\{ y _ n , \\gamma y _ n , \\gamma ^ 2 y _ n , \\dots , \\gamma ^ { M - 1 } y _ n \\right \\} , \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\begin{cases} y _ 1 = x _ 1 - \\phi ( x ' ) = : \\Phi _ 1 ( x ) , \\\\ y _ j = x _ j = : \\Phi _ j ( x ) ( j = 2 , \\ldots , n ) , \\end{cases} \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\Phi ( x , y ) = ( x ' , x _ m - \\psi ( x ' , y ) , y ) . \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\| x \\| _ p = \\| ( x _ i ) _ { i = 1 } ^ n \\| _ p = \\begin{cases} \\displaystyle \\Bigl ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\Bigr ) ^ { 1 / p } \\ , , & 0 < p < \\infty \\ , ; \\\\ \\displaystyle \\max _ { 1 \\leq i \\leq n } | x _ i | \\ , , & p = \\infty . \\end{cases} \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\prod _ { m = 1 } ^ { \\infty } \\frac { 1 - ( - t ) ^ m } { 1 + ( - t ) ^ m } = \\sum _ { n = - \\infty } ^ { \\infty } t ^ { n ^ 2 } , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\int _ { D } \\kappa ( y , x ) & \\nabla ( u ( y , x ) - u _ M ^ { N , h } ( y , x ) ) \\cdot \\nabla v ( x ) \\mathrm { d } x \\\\ & = \\int _ { D } ( \\kappa _ M ^ { N , h } ( y , x ) - \\kappa ( y , x ) ) \\nabla u _ M ^ { N , h } ( y , x ) \\cdot \\nabla v ( x ) \\mathrm { d } x \\forall v \\in V . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} u _ t ( t , x ) + f ( u ( t , x ) ) _ x = 0 \\qquad \\forall ( t , x ) \\in ( 0 , + \\infty ) \\times \\R \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} q ( x , t ) = q _ { s o l } ( x , t ) + q _ { p a r } ( x , t ) + \\mathcal { O } ( t ^ { - 1 } \\ln t ) . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} C _ { 2 n } ^ { ( \\alpha + 1 ) } ( z / c ) = \\frac { ( \\alpha + 1 ) _ n } { \\left ( \\frac 1 2 \\right ) _ n } P _ n ^ { ( \\alpha + \\frac 1 2 , - \\frac 1 2 ) } \\left ( 2 \\left ( \\frac { z } { c } \\right ) ^ 2 - 1 \\right ) \\ , \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} g ( u ) & = \\xi ( 1 ) - \\xi ( u ) - \\frac 1 { A _ 1 ^ 2 } \\log \\Big ( 1 + \\frac { A _ 1 ( q - u ) } { A _ 2 ( 1 - q ) + \\Delta } \\Big ) + \\frac { q - u } { A _ 1 [ A _ 1 q + A _ 2 ( 1 - q ) + \\Delta ] } \\\\ & - \\frac 1 { A _ 2 ^ 2 } \\log \\Big ( 1 + \\frac { A _ 2 ( 1 - q ) } { \\Delta } \\Big ) + \\frac { 1 - q } { A _ 2 ( A _ 2 ( 1 - q ) + \\Delta ) } - \\frac { ( 1 - q ) q } { [ A _ 2 ( 1 - q ) + \\Delta ] [ A _ 1 q + A _ 2 ( 1 - q ) + \\Delta ] } ; \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\frac { e ^ { - 2 \\pi \\sqrt { 3 } n / w } } { n / w } = \\int _ { n } ^ { \\infty } \\left ( \\frac { 1 } { x ( x / w ) } + \\frac { 2 \\pi \\sqrt { 3 } } { x } \\right ) e ^ { - 2 \\pi \\sqrt { 3 } ( x / w ) } \\ , d x \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\lim _ { V \\rightarrow \\left \\{ e \\right \\} } \\int _ { \\cup _ { \\gamma \\in \\Psi _ { k } } \\gamma ^ { - 1 } V } \\left \\vert \\varphi \\left ( x \\right ) \\right \\vert ^ { 2 } d x = 0 . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} c o v ( T _ { l _ 1 } , T _ { l _ 2 } ) = \\mathbb { E } T _ { l _ 1 } T _ { l _ 2 } - \\mathbb { E } T _ { l _ 1 } \\mathbb { E } T _ { l _ 2 } . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\int _ { \\R } \\left ( \\int _ { \\varphi ^ { - 1 } ( s ) } f ( \\ell ) { \\rm d } \\ell \\right ) { \\rm d } s = \\int _ \\Omega \\vert \\nabla \\varphi ( x ) \\vert f ( x ) { \\rm d } x . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\dfrac { \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } \\ln n } { \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } } = \\sum _ { p } \\dfrac { p ^ s } { p ^ s - 1 } \\dfrac { p ^ t } { p ^ s - 1 + p ^ t } \\ln { p } \\ , \\ , \\ , : = \\ , \\mathcal { S } ( s , \\ , t ) . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\begin{aligned} & ( D _ 0 ^ { k ' } u ) ( t ) = f ( t , u ( t ) ) , \\mbox { a . e . } t \\in [ 0 , 1 ] , \\\\ & ( I _ 0 ^ { k ' } u ) ( 0 ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } { \\Big [ } - 2 \\bar R _ { L \\underline L L \\underline L } - \\bar R i c ( L , \\underline L ) - \\frac { 1 } { 2 } \\bar R i c ( L , L ) { \\Big ] } \\tilde X ^ i d S ^ 2 = & - \\int _ { S ^ 2 } \\bar R i c ( e _ 0 , e _ j ) \\tilde { X } ^ j \\tilde { X } ^ i d S ^ 2 \\\\ = & - \\frac { 4 \\pi } { 3 } \\bar R i c ( e _ 0 , e _ i ) . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} ( \\hat a _ j , \\hat b _ j , \\hat c _ j ) = \\begin{cases} ( 1 2 0 , \\ 1 1 9 , \\ 1 6 9 ) & j = 1 \\\\ ( 2 h _ j - 1 , 2 h _ j ^ 2 - 2 h _ j , 2 h _ j ^ 2 - 2 h _ j + 1 ) h _ j : = 2 ^ { j - 2 } + 2 & j = 2 , \\dots , \\nu , \\end{cases} \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} d x _ t = \\sum _ { \\alpha = 1 } ^ d V _ \\alpha ( x _ t ) \\ , d w _ t ^ \\alpha , \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} & \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( B _ n ^ { ( 3 ) } ) \\\\ & \\ , \\ , \\ , \\leq \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log [ ( \\lfloor n - \\varepsilon f _ 2 ( n ) \\rfloor - \\lfloor n - \\varepsilon ' f _ 2 ( n ) \\rfloor ) P ( S _ n ( t _ n ) \\leq t _ n - a _ n ) ] \\\\ & \\ , \\ , \\ , = - H ( \\ell _ 2 \\varepsilon ) , \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\sum _ { n \\ge 1 } \\chi _ G ( n ) \\ , z ^ n \\ = \\ \\frac { \\chi ^ * _ G ( z ) } { ( 1 - z ) ^ { d + 1 } } \\ , . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\mathtt p \\leftrightarrow ( \\mathtt q + \\mathtt r ) = ( \\mathtt p \\leftrightarrow \\mathtt q ) + ( \\mathtt p \\leftrightarrow \\mathtt r ) , \\ ( \\mathtt q + \\mathtt r ) \\leftrightarrow \\mathtt p = ( \\mathtt q \\leftrightarrow \\mathtt p ) + ( \\mathtt r \\leftrightarrow \\mathtt p ) , \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 0 } \\right \\rbrace = e ^ { - \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } \\theta } { \\lambda _ { \\textrm { B F } } P _ { \\textrm { F } } } \\left ( 2 ^ { \\frac { R } { \\theta } } - 1 \\right ) } - e ^ { - \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } ( 1 - \\theta ) } { \\lambda _ { \\textrm { B F } } P _ { \\textrm { N } } } \\left ( 2 ^ { \\frac { R } { 1 - \\theta } } - 1 \\right ) } . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} \\Gamma _ { k m } ( x ) = \\left \\{ \\begin{array} { l } \\gamma _ { k m } ( x ) \\quad \\quad k \\neq m \\\\ - \\sum \\nolimits _ { j : j \\neq k } \\gamma _ { k j } ( x ) \\quad \\ , k = m , \\end{array} \\right . \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\bar { X } : = \\left ( \\cos \\left ( \\frac { ( i - j ) \\pi } { n - 1 } \\right ) \\right ) _ { i , j = 0 } ^ { n - 1 } , \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { L } } d \\sigma _ { \\Sigma , L } = d \\sigma _ \\Sigma + d \\overline { \\sigma _ \\Sigma } L ^ { - 1 } + O ( L ^ { - 2 } ) , ~ ~ { \\rm a s } ~ ~ L \\rightarrow + \\infty . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} d ( e _ x ) = 1 - x , d ( x ) = 0 x \\in X ; \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\underline { \\Delta } \\varphi & = \\frac { \\psi ^ { ' } \\underline \\Delta d } { \\rho } + \\frac { \\psi ^ { '' } \\left \\| \\overline { \\nabla } d \\right \\| ^ { 2 } } { \\rho } \\ge \\frac { c _ { 1 } } { \\rho } ( n - 1 ) \\sqrt { \\rho _ { 2 } } \\ ; \\coth ( \\sqrt { \\rho _ { 2 } } \\rho ) - \\frac { c _ { 2 } } { \\rho ^ { 2 } } , \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\int | d u | ^ { p - 2 } \\langle d u , d v \\rangle = \\int | d u | ^ p \\langle u , v \\rangle \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} u ( t ) = \\mathbf B _ \\alpha ^ { ( 0 ) } ( t ) \\overline { f } & + \\int _ 0 ^ t { \\bf P } _ { \\alpha } ( t - r ) G ( r , u ( r ) ) d r , \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{gather*} Q = \\rho _ a ^ i ( x ) q ^ a \\frac { \\partial } { \\partial x ^ i } - \\frac { 1 } { 2 } C ^ c _ { a b } ( x ) q ^ a q ^ b \\frac { \\partial } { \\partial q ^ c } , \\end{gather*}"}
-{"id": "9438.png", "formula": "\\begin{align*} a _ k \\leq C b ^ { - \\frac { s } { Q } } \\bigg ( \\sum _ j \\int _ { 2 B _ 0 } g _ j ^ p \\ , d \\mu \\bigg ) ^ { \\frac { s } { Q } } \\sum _ { n = k _ 0 } ^ { k - 1 } 2 ^ { n ( 1 - \\frac { s p } { Q } ) } + C ' r _ 0 ^ { s } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} r & = \\sum _ \\alpha E _ \\alpha \\wedge Y _ \\alpha \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { m } } _ H ( \\Sigma ) = \\sqrt { \\frac { | \\Sigma | } { 1 6 \\pi } } \\left ( 1 - \\frac { 1 } { 1 6 \\pi } \\int _ { \\Sigma } H ^ 2 \\ , d \\sigma + \\frac { | \\Sigma | } { 4 \\pi } \\right ) \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d u _ t & = & a \\Delta u _ t d t + \\sigma d X _ t \\medskip \\\\ u _ 0 & = & h _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} | \\lambda | ^ { 1 / 2 } \\| \\partial _ z ( \\alpha v ) \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } \\le C _ p | \\lambda | ^ { 1 / p } \\sup _ { x _ 0 ' \\in G } \\biggl ( | \\lambda | ^ { 1 / 2 } \\| \\partial _ z ( \\alpha v ) \\| _ { L ^ p ( C ( x _ 0 ' ; | \\lambda | ^ { - 1 / 2 } ) ) } + \\| \\nabla _ H \\partial _ z ( \\alpha v ) \\| _ { L ^ p ( C ( x _ 0 ' ; | \\lambda | ^ { - 1 / 2 } ) ) } \\biggr ) , \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} y = \\frac { 1 } { | W ^ { \\mathrm { r e l } } _ { M _ S } | } \\sum \\limits _ { \\sigma \\in W ^ { \\mathrm { r e l } } _ { M _ S } } \\sigma ( x ) \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} E [ e ^ { X t } ] = \\int _ 0 ^ \\infty e ^ { x t } e ^ { - t } d x = \\frac { 1 } { 1 - t } = \\frac { 1 } { 1 - t } e ^ { - t } e ^ t . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} Y [ a , z ] = Y ( e ^ { 2 \\pi i z L _ 0 } a , \\phi ( z ) ) . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } \\mathrm { I n d } ^ G _ { P _ b } ( \\mathrm { M a n t } _ { M _ b , b ' , \\mu _ b } ( e ( J _ b ) L J ( \\delta ^ { \\frac { 1 } { 2 } } _ { G , P _ b } \\otimes I ^ { M _ b } _ { M _ S } ( \\rho ) ) ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b - \\mu \\rangle } ] , \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\nu _ 1 = \\left [ \\nu _ 0 ; \\nu _ 1 ( x ) = - \\frac { 1 } { p } \\right ] . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\beta ( x ) = \\frac { f ' ( x ) } { f ' ( \\alpha ( x ) ) } . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} A _ { 3 } = \\begin{bmatrix} 0 & 2 & - 4 & \\varepsilon & \\varepsilon \\\\ - 5 & \\varepsilon & \\varepsilon & \\varepsilon & - 6 \\\\ \\varepsilon & \\varepsilon & \\varepsilon & - 4 & \\varepsilon \\\\ \\varepsilon & - 3 & \\varepsilon & \\varepsilon & \\varepsilon \\\\ - 2 & \\varepsilon & \\varepsilon & 2 & \\varepsilon \\end{bmatrix} . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\left ( ( n + 1 ) ^ d - \\chi _ G ( n + 1 ) \\right ) z ^ n \\ = \\ \\frac { h _ d \\ , z ^ d + h _ { d - 1 } \\ , z ^ { d - 1 } + \\dots + h _ 0 } { ( 1 - z ) ^ d } \\ , , \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf B } _ \\alpha ( t , T ) v : = \\sum _ { j = 1 } ^ \\infty \\frac { E _ { \\alpha , 1 } ( - \\lambda _ j t ^ \\alpha ) } { E _ { \\alpha , 1 } ( - \\lambda _ j T ^ \\alpha ) } \\langle v , \\varphi _ j \\rangle \\varphi _ j , { \\bf P } _ \\alpha ( t ) v : = \\sum _ { j = 1 } ^ \\infty t ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\lambda _ j t ^ \\alpha ) \\langle v , \\varphi _ j \\rangle \\varphi _ j . \\end{aligned} \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ { j } b _ { i j } e _ i : \\ j \\in [ d ] \\right \\} . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\begin{bmatrix} c _ 1 a _ 1 & c _ 1 a _ 2 & f _ 1 \\\\ c _ 2 a _ 1 & c _ 2 a _ 2 & f _ 2 \\\\ b _ 1 & b _ 2 & 0 \\end{bmatrix} \\in S U ( 3 ) . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} D W ( \\mathbb { I } ) = 0 . \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\mu _ { f , A } ^ \\star \\ge \\frac { \\Phi _ 0 ( \\bar A ) ^ 2 } { 4 } = \\dfrac { 1 } { 2 } , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } P \\left ( \\frac { E ( t ) } { t } \\leq x \\right ) = \\int _ 0 ^ { x } \\frac { C _ { \\alpha } } { y ^ { \\alpha } ( y + 1 ) } d y , ~ \\forall x > 0 , \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} \\frac 1 s : = \\sum _ { i = 1 } ^ m \\frac 1 { s _ i } = r \\sum _ { i = 1 } ^ m \\frac 1 { r _ i } = r \\left ( \\frac 1 r - \\frac 1 { r _ { m + 1 } } \\right ) > 1 - \\frac 1 { r _ { m + 1 } } = \\frac 1 { r _ { m + 1 } ' } . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} c _ 0 ( \\Gamma ) : = \\widehat { \\Z \\Gamma } \\otimes _ { \\Z } \\Q = \\Big \\{ \\sum _ { \\gamma \\in \\Gamma } x _ { \\gamma } \\gamma \\mid x _ { \\gamma } \\in \\Q _ p \\ ; \\ ; | x _ { \\gamma } | \\to 0 \\ ; \\ ; \\gamma \\to \\infty \\Big \\} \\ ; . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} ( d t ) _ { i j k l } & = t _ { j k l } - t _ { i k l } + t _ { i j l } - t _ { i j k } \\\\ & = [ g _ { i j } s _ { j k l } g _ { i j } ^ { - 1 } - s _ { i k l } + s _ { i j l } - s _ { i j k } ] \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\partial _ y \\int _ D G ( y , z ) f ( z ) \\ , d z = \\int _ D \\partial _ y \\ , G ( y , z ) f ( z ) \\ , d z \\ , , y \\in D \\ , . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{gather*} S p l _ X ( f , i d ) = \\{ p \\le X \\} , \\\\ S p l _ X ( f , i d , k _ 1 ) = \\{ p \\le X \\} , \\\\ S p l _ X ( f , i d , k _ 1 , L , R _ 1 ) = \\{ p \\le X \\mid - a + k _ 1 p \\equiv R _ 1 \\bmod L \\} , \\end{gather*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\sigma ( G ) : = \\begin{cases} 2 & G V _ 4 ( 2 ^ \\ast ) , \\\\ 1 & G V _ 4 ( 1 ^ \\ast ) , \\\\ 0 & G V _ 4 . \\end{cases} \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} F _ n \\left ( e _ { ( a _ 1 , a _ 2 , a _ 3 ) } \\right ) = \\epsilon _ 1 U e _ { ( a _ 1 - 1 , a _ 2 , a _ 3 ) } + \\epsilon _ 2 V e _ { ( a _ 1 , a _ 2 - 1 , a _ 3 ) } + \\epsilon _ 3 W e _ { ( a _ 1 , a _ 2 , a _ 3 - 1 ) } , \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\gamma ( E _ { i j } v _ t ) = ( \\varphi \\theta ) ( E _ { i j } ) \\gamma ( v _ t ) i \\rho j , t . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} \\mathbb { E } T _ l = \\sum _ { k \\geq 3 } \\mathbb { P } ( N _ l = k ) k D _ l ( k ) \\geq \\sum _ { \\frac { \\eta _ 1 n } { 2 N } \\leq k \\leq \\frac { 2 \\eta _ 2 n } { N } } \\mathbb { P } ( N _ l = k ) k D _ l ( k ) , \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} F = c \\circ P + \\langle v , \\cdot \\rangle , \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} t _ 2 ( ( B L \\iota ) ^ \\ast \\circ \\nu ) ( q _ 4 ) = 4 \\lambda s _ 1 t _ 2 . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} L _ { Z } \\psi = L _ { Z } g ( Z , Z ) = ( L _ Z g ) ( Z , Z ) + 2 g ( [ Z , Z ] , Z ) = 0 . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} x = A \\phi , y = A \\phi _ r , z = A ^ { \\tfrac { 1 } { 2 } } \\psi . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\delta = 3 & & K _ 1 = 1 & & K _ 2 = 2 & & C = 1 0 & & C ' = 1 1 \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\mathcal { O } _ { i , d } \\cap \\mathcal { O } \\neq \\emptyset , \\ \\ i = 1 , 2 \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} \\check H _ 0 ( t ) = \\int _ 0 ^ t \\frac { 1 } { 1 - \\Delta \\tilde H ( s ) } \\tilde H _ 0 ( \\d s ) \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N H F } } ^ { \\textrm { N O M A } } = \\frac { P _ \\textrm { F } } { P _ \\textrm { N } + { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } / { | h _ { \\textrm { B F } } | ^ 2 } } + \\frac { \\beta _ { \\textrm { N } } \\eta P _ \\textrm { B } | h _ { \\textrm { B N } } | ^ 2 | h _ { \\textrm { N F } } | ^ 2 } { d _ { \\textrm { B N } } ^ { \\alpha } d _ { \\textrm { N F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} L ( 0 ) = \\frac { - a ^ 2 \\left ( \\eta - \\eta _ 1 \\right ) } { a ^ 2 - 1 } , \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{gather*} \\boldsymbol { \\Sigma } _ { \\frac { n + 1 } { 2 } } : = \\mathrm { d i a g } \\left ( \\lambda _ { 3 } , \\lambda _ { 5 } , \\ldots , \\lambda _ { n } , \\lambda _ { n + 2 } \\right ) + \\lambda _ { 1 } \\mathbf { u } \\mathbf { u } ^ { \\top } \\\\ \\boldsymbol { \\Gamma } _ { \\frac { n - 1 } { 2 } } : = \\mathrm { d i a g } \\left ( \\lambda _ { 2 } , \\lambda _ { 4 } , \\ldots , \\lambda _ { n - 1 } \\right ) + \\lambda _ { n + 1 } \\mathbf { v } \\mathbf { v } ^ { \\top } \\end{gather*}"}
-{"id": "9638.png", "formula": "\\begin{align*} C _ { i } ( s ) = C \\left ( 1 , . . . , 1 , \\underset { i - t h \\ a r g u m e n t } { \\underbrace { s } } , 1 , . . . , 1 \\right ) = s . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\mathcal E _ { 0 , j } : = 2 \\frac { a _ { 0 , j , w _ j } } { \\phi ^ { 2 \\alpha _ j } } - 2 \\langle \\nabla a _ { 0 , j } , \\nabla u _ j \\rangle \\ \\mathcal E _ { 1 , j } : = ( m + 2 - p ) \\ , \\frac { a _ { 1 , j , w _ j } } { \\phi ^ { 2 \\alpha _ j } } - 2 \\langle \\nabla a _ { 1 , j } , \\nabla u _ j \\rangle . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} \\varphi _ n ( s ) : = \\begin{cases} 1 & \\mbox { i f } | s | < n \\ , , \\\\ \\displaystyle \\frac { 2 n - | s | } { n } & \\mbox { i f } n \\le | s | \\le 2 n \\ , , \\\\ 0 & \\mbox { i f } | s | > 2 n \\ , . \\end{cases} \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} [ M _ { S ' } , \\mu _ { S ' } ] ( I ^ G _ { M _ S } ( \\rho ) ) = ( \\mathrm { I n d } ^ G _ { P _ { S ' } } \\circ [ \\mu _ { S ' } ] ) ( \\delta ^ { \\frac { 1 } { 2 } } _ { P _ S } \\otimes J ^ G _ { P ^ { o p } _ { S ' } } I ^ G _ { M _ S } ( \\rho ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ { S ' } - \\mu \\rangle } ] \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\chi _ n ( t ) : = \\chi ( \\mathcal { P } ( n ) , t ) = \\frac { ( t ) _ n } { t } = \\sum \\limits _ { k = 1 } ^ n s ( n , k ) t ^ { k - 1 } . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\lambda _ { n } = \\inf \\{ \\norm { \\mathcal { R } - L } _ { \\mathcal { B } ( L ^ 2 ( D ) ) } : L \\in \\mathfrak { F } ( L ^ 2 ( D ) ) , { } ( L ) < n \\} \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\frac 1 q + \\frac 2 r < \\frac { 1 } { q } + \\frac { q - p } { p q } = \\frac 1 p < 1 \\ , , \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{gather*} \\langle \\imath _ { \\kappa } \\gamma ( x ) , \\eta \\rangle = \\langle \\gamma ( x ) , \\kappa \\otimes \\eta \\rangle = \\langle x , [ \\kappa , \\eta ] \\rangle \\\\ = \\langle x , a d _ { \\kappa } \\eta \\rangle = \\langle a d _ { \\kappa } ^ * x , \\eta \\rangle . \\end{gather*}"}
-{"id": "10001.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\underline { \\nu _ L } } \\psi \\left ( \\nu , L \\right ) = \\frac { \\alpha \\underline { \\nu _ L } + d } { r _ 0 L } > 0 , \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} I ( R ) = \\dfrac { 1 } { 2 } \\displaystyle \\int _ { \\frac { 1 } { R ^ { 2 } } } ^ { j \\left ( \\eta _ { 0 } \\right ) } \\frac { j ^ { - 1 } ( t ) } { t } d t \\ , \\ , \\ , \\ , \\forall R \\geq 1 . \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{gather*} \\eta ^ { ( k - 1 ) } ( e _ k , \\dots , e _ n ) = ( - 1 ) ^ k \\iota _ { \\rho ( e _ k ) } \\eta ^ { ( k ) } ( e _ { k + 1 } , \\dots , e _ n ) + \\operatorname { c y c l } ( e _ k , \\dots , e _ n ) , \\\\ \\iota _ { \\rho ( e _ k ) } \\eta ^ { ( k ) } ( e _ { k + 1 } , \\dots , e _ { k + m } , \\dots , e _ n ) + \\iota _ { \\rho ( e _ { k + m } ) } \\eta ^ { ( k ) } ( e _ { k + 1 } , \\dots , e _ k , \\dots , e _ n ) = 0 , \\\\ k = 1 , \\dots , n - 1 , m = 1 , \\dots , n - k , \\end{gather*}"}
-{"id": "9112.png", "formula": "\\begin{align*} & \\Delta u = F _ 1 + F _ 2 u \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\prod _ { p \\equiv 1 \\mod 3 } \\left ( 1 + \\frac { 2 } { p ^ s } \\right ) = \\prod _ { p \\neq 3 } \\left ( 1 + \\frac { 1 } { p ^ s } + \\frac { \\xi ( p ) } { p ^ s } \\right ) . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} I _ 1 + I _ 3 & = - u _ 0 ( x ) + \\frac { 1 } { 2 } ( 1 + t ) ^ { - \\frac { \\mu } { 2 } } \\big [ u _ 0 ( x + t ) + u _ 0 ( x - t ) \\big ] + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } } } \\int _ { x - t } ^ { x + t } u _ 0 ( y ) \\bigg [ - \\frac { \\partial E } { \\partial b } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\bigg ] _ { b = 0 } \\mathrm { d } y \\\\ & + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } } } \\int _ { x - t } ^ { x + t } \\mu \\ , u _ 0 ( y ) E ( t , x ; 0 , y ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } y . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} c _ { 1 } ^ { 2 } \\left [ f _ { 2 } ^ { 2 } f _ { 2 } ^ { \\prime \\prime } - f _ { 2 } \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } \\right ] f _ { 1 } ^ { 3 } + \\left [ f _ { 2 } ^ { \\prime \\prime } + 2 a c _ { 1 } f _ { 2 } ^ { \\prime } + a ^ { 2 } c _ { 1 } ^ { 2 } f _ { 2 } \\right ] f _ { 1 } = 0 , \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} y ( \\cdot , T ) = \\overline { y } ( \\cdot , T ) \\quad \\mbox { a n d } \\theta ( \\cdot , T ) = \\overline { \\theta } ( \\cdot , T ) \\quad \\mbox { i n } \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} 2 ^ { \\Phi ( n ) } \\leq | \\mathcal { Z } _ n | \\leq \\sum _ { 1 \\leq i \\leq \\Phi ( n ) } \\binom { n } { i } \\leq n ^ { ( 1 / 2 + o ( 1 ) ) \\sqrt { n } } . \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} Q '' ( x _ 3 ) & = Q ( x _ 3 ) - Q ( x _ 3 + d ' ) , \\\\ P '' ( x _ 2 , x _ 3 ) & = P ( x _ 2 , x _ 3 ) - P \\big ( x _ 2 + Q '' ( x _ 3 ) + Q ' ( x _ 3 + d ) , x _ 3 + d ' \\big ) . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 1 . 0 5 \\\\ 1 . 9 6 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c } - 0 . 6 7 0 & - 0 . 4 4 2 \\\\ - 0 . 4 4 2 & - 0 . 4 3 6 \\end{array} \\right ] , d = \\left [ \\begin{array} { r } 0 . 0 8 9 1 1 \\\\ - 0 . 2 3 1 5 \\ \\ , \\end{array} \\right ] . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} L = \\sum _ { i = 1 } ^ d X _ i ^ 2 . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\sum _ i 2 k _ i + \\textnormal { d e g } ( \\alpha _ i ) = 2 ( 1 - g ) ( \\textnormal { d i m } ( X ) - 3 ) + 2 n - 2 \\int _ X c _ 1 ( T X ) \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} I I I '' _ 1 & = \\frac 1 6 ( p + s - n - 3 ) ( p + s - n - 4 ) ( - 2 p + 4 s - n - 5 ) \\\\ I I I '' _ 2 & = \\sum _ { j = 1 } ^ { s - 2 } [ 2 j ^ 2 + ( 2 p - 3 n - 7 ) j - ( n + 2 ) ( p - n - 3 ) ] - ( n + 3 - s ) ( p + s - n - 4 ) \\\\ & = ( p - \\frac 3 2 n - \\frac 7 2 ) ( s - 1 ) ( s - 2 ) + ( s - 2 ) ( s - 1 ) ( \\frac 2 3 s - 1 ) \\\\ & - ( n + 2 ) ( p - 3 - n ) ( s - 2 ) - ( n + 3 - s ) ( p + s - 4 - n ) \\\\ & = ( s - 1 ) n ^ 2 - [ p s - p + \\frac 3 2 s ^ 2 - \\frac { 1 5 } 2 s + 6 ] n + ( p - \\frac 7 2 ) ( s - 1 ) ( s - 2 ) \\\\ & + ( s - 2 ) ( s - 1 ) ( \\frac 2 3 p s - 1 ) - 2 ( s - 2 ) ( p - 3 ) + ( p + s - 4 ) ( s - 3 ) . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} H ^ * _ { a , V } ( M ) : = \\frac { \\left \\{ w \\in \\Omega ^ * ( M , \\R ) \\ \\big | \\ \\imath _ V ( w ) = 0 \\ \\ \\ \\ d _ { a , V } ( w ) = 0 \\right \\} } { \\left \\{ d _ { a , V } ( z ) \\ \\big | \\ z \\in \\Omega ^ { * - 1 } ( M , \\R ) \\ \\ \\ \\ \\imath _ V ( z ) = 0 \\right \\} } . \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\mathrm { i n } ( v ) = \\sum _ { w \\in V } \\ , a _ { w , v } \\begin{cases} = 0 & v , \\\\ \\ge 1 & v , \\end{cases} \\mathrm { o u t } ( v ) = \\sum _ { w \\in V } \\ , a _ { v , w } = \\begin{cases} 0 & v = R , \\\\ 1 & . \\end{cases} \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\begin{aligned} m _ \\epsilon = e ^ { \\frac { | P + \\nabla u _ \\epsilon ( x ) | ^ 2 } { 2 } + V ( x , \\frac { x } { \\epsilon } ) - \\overline { H } _ \\epsilon ( P ) } , \\end{aligned} \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } w _ { i , p } = U \\ \\mbox { i n } \\ C ^ 2 _ { l o c } ( \\R ^ 2 ) , \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} \\Theta : \\mathcal { R } \\to \\widehat { \\mathcal { R } } , \\sum _ { i = 0 } ^ { n - 1 } a _ i x ^ i \\mapsto \\sum _ { i = 0 } ^ { n - 1 } \\sigma ^ { - i } ( a _ i ) x ^ { - i } . \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , g \\otimes \\chi \\right ) \\ll _ { g , \\varepsilon } M ^ { 1 / 2 - 1 / 8 + \\varepsilon } . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} b = \\delta \\ ( 1 - \\frac { 2 m } { r _ o } \\ ) \\ ( \\frac { H _ o ^ 2 } { 4 } - \\delta \\ ) ^ { - 1 } r _ o ^ { - 2 } , \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} \\gamma : \\mathrm { C l } ( E ) \\rightarrow \\mathrm { E n d } ( S ) , a \\mapsto \\gamma ( a ) : = \\gamma _ a , \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} ( B \\lambda _ { 2 k + 1 } ) ^ \\ast ( q _ 4 ) = q _ 4 - k t _ 2 ^ 2 . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} ( \\frak X T ) ( x , y ) : = \\int _ { \\R ^ 6 } \\frak X ( x _ 0 , y _ 0 ) T ( x _ 0 , x , y - y _ 0 ) \\dd x _ 0 \\dd y _ 0 . \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} u _ \\varepsilon ( x _ \\varepsilon ) = \\max _ \\Omega u _ \\varepsilon = \\gamma _ \\varepsilon \\to + \\infty \\ , , \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} \\mathbf R _ j = \\frac { p } { n } \\sum _ { i = 1 } ^ { n } \\mathbb E [ R _ { u } ^ { j , \\ , i } ] + ( 1 - p ) \\mathbb E [ R _ o ^ { \\ , j } ] , \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} F ( \\gamma I ) = 1 . \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol \\Sigma _ { \\textbf { x } } \\ ! = \\ ! \\varepsilon _ S ^ 2 \\cdot \\ ! \\ ! \\int _ 0 ^ \\infty \\ ! e ^ { \\mathbf { E } v } \\mathbf { F } e ^ { \\mathbf { E } ^ v } d v \\ ! = \\ ! \\frac { \\varepsilon _ S ^ 2 } { 2 \\varepsilon _ S \\ ! - \\ ! 1 } \\mathbf { I } _ S ( \\boldsymbol { \\psi } , \\widetilde { \\textbf { W } } ^ * ) ^ { \\ ! - 1 } \\ ! . \\ ! \\end{aligned} \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\varphi ( s ) = 1 , { \\rm \\ f o r \\ } s \\in [ - \\frac 3 4 , \\frac 3 4 ] , { \\rm \\ a n d \\ } \\varphi ( s ) = 0 , { \\rm \\ f o r \\ } \\vert s \\vert \\geq 1 . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\pi ( y ) = ( 1 + \\| y \\| ^ 2 ) ^ { \\frac { - 1 } 2 } ( y + e ) , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\left ( h ^ 2 \\Delta _ X + z n ( x ) \\right ) u = 0 & \\mbox { i n } & X , \\\\ u = f & \\mbox { o n } & \\partial X . \\end{array} \\right . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} Q _ n \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ n F ( \\mu _ { n , i } ) . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { X } Y = \\nabla _ { X } Y - \\frac { 1 } { 4 } \\{ ( \\nabla J ) X , J \\} Y - \\frac { 1 } { 2 } J ( \\nabla _ { X } J ) Y , \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\# V _ { \\ominus } \\leq n \\ , \\binom { b } { t } \\ , , \\quad \\# V _ { \\oplus } \\leq n \\ , . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\omega = M ^ W d , M ^ W \\in \\mathcal { W } ^ { N D } [ y _ h ] , \\Omega _ T . \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} d _ 4 ( n , n - 1 ) = \\begin{cases} 1 & n \\\\ 2 & n \\end{cases} \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} \\norm { - \\Lambda _ f ( y ) - \\tilde u } \\leq \\sum _ { i = 1 } ^ m \\alpha _ i ^ * \\norm { \\nabla f _ i ( y ) - \\nabla f _ i ( z ) } \\leq \\sum _ { i = 1 } ^ m \\alpha _ i ^ * L \\norm { y - z } = L \\norm { y - z } \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ G ( x ) - G ( y ) : x , y \\in E \\} = \\R ^ n . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 i } + \\bar h ^ { 1 ^ * } _ { 2 2 i } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 i } + \\bar h ^ { 2 ^ * } _ { 2 2 i } = 0 , \\\\ & ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) \\bar h ^ { 1 ^ * } _ { 1 1 i } = 0 . \\end{cases} \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} g ^ { ( j , { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c ) } : = \\mathcal { S } ^ { { \\mathcal { R } ( { \\zeta } , { \\tau _ 1 } ) } , { { \\tau _ 2 } - { \\tau _ 1 } } } \\left ( f ^ { ( j , { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c ) } \\right ) \\in \\mathcal { C } ^ { \\infty } ( \\overline { \\Omega ^ { ( { \\mathcal { R } ( { \\zeta } , { \\tau _ 1 } ) } ) } } ) = \\mathcal { C } ^ { \\infty } ( \\overline { { \\Omega } _ { \\zeta } ( { \\tau _ 1 } ) } ) \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} T ^ { * * } = ( T _ { \\rm r e g } ) ^ { * * } + ( T _ { \\rm s i n g } ) ^ { * * } . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} x ^ * = [ 0 . 5 4 5 2 1 8 8 1 3 3 8 8 \\ { - 1 . 4 6 4 4 1 0 1 8 9 7 9 2 } \\ { - 0 . 7 2 0 6 0 6 6 5 4 2 7 6 } \\ 1 . 1 7 8 1 4 4 2 6 5 5 9 2 \\ 0 . 7 9 4 0 6 5 1 0 8 2 4 3 \\ { - 0 . 4 6 5 7 9 4 1 1 9 4 4 8 } ] \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla F ( f ) ( t ) | _ p & \\leq t ^ { \\frac { 1 } { p } - \\frac { 3 } { 2 } + \\gamma } c \\mathcal { B } ^ 2 \\eta _ t \\max \\left \\{ 1 , \\frac { 4 } { \\theta _ i ^ 2 } , \\ i = 1 , 2 , . . . , N \\right \\} B \\left ( \\frac { 2 } { p } - \\frac { 3 } { 2 } + \\gamma , 1 - \\frac { 1 } { p } \\right ) \\| f \\| ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} b ( u - u _ { h } , u - u _ { h } ) & = \\int _ { \\Omega } \\boldsymbol { \\eta } \\boldsymbol { \\Pi } _ K ^ { k - 1 } \\nabla ( u - u _ h ) \\cdot \\boldsymbol { \\Pi } _ K ^ { k - 1 } \\nabla ( u - u _ h ) \\leq \\Vert \\boldsymbol { \\eta } \\Vert _ { \\infty } | | u - u _ h | | _ { 2 , \\Omega } ^ 2 \\leq C \\gamma _ h ^ 2 . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} J _ k ( x ) = \\int _ { ( \\R ^ n ) ^ { k - 1 } } J ( x - y _ 1 ) J ( y _ 1 - y _ 2 ) \\cdots J ( y _ { k - 1 } ) d y _ { k - 1 } \\cdots d y _ 1 , \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} v _ 1 & : = ( 1 - \\sigma - \\sigma \\mu + \\sigma ^ 2 \\mu ) / 3 \\ , ( = v _ 1 c _ \\chi ) , \\\\ v _ 2 & : = - ( \\sigma - \\sigma ^ 2 - \\mu + \\sigma \\mu ) / 3 \\ , ( = v _ 2 c _ \\chi = - v _ 1 \\sigma ) , \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} \\dim \\delta _ { \\beta , \\theta } & = \\left ( G ( O _ r ) : G ( O _ r , \\beta ) \\right ) \\cdot \\dim \\sigma _ { \\beta , \\theta } \\\\ & = \\sharp \\Omega \\cdot q ^ { ( r - 2 ) ( \\dim G - \\ , G ) / 2 } \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} g = B ( y _ { 2 } , r ) , h _ { 1 } = B ( y _ { 1 } , r ) h _ { 2 } = B ( x _ { 0 } , r ) . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} t _ k : = 2 b ^ { - 1 / Q } \\mu ( 2 B _ 0 \\setminus E _ { k - 1 } ) ^ { 1 / Q } . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} \\begin{aligned} & S \\leq \\bar H ^ 2 \\leq 3 S - 2 , \\\\ & S ( 1 - \\dfrac 1 2 S ) - ( S - \\bar H ^ 2 ) ^ 2 + \\dfrac 1 2 \\bar H ^ 4 - \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 + 2 \\bar \\lambda ^ 2 ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} \\sigma _ { \\mathbf { \\Delta } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) - \\sigma _ { \\mathbf { \\Delta } , \\mathbf { G } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) = \\sum _ { k = 0 } ^ { p } v a r _ { P } \\left [ t _ { k } \\left ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } , P \\right ) \\right ] \\geq 0 , \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } | X | ^ { 2 } = 2 - H ^ { 2 } - \\langle X , \\sum _ { i } \\langle X , e _ { i } \\rangle e _ { i } \\rangle = 2 - H ^ { 2 } - | X ^ { \\top } | ^ { 2 } = 2 - | X | ^ { 2 } . \\endaligned \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} L _ { n } ^ { ( \\delta ) } ( z ; q ) = \\frac { 1 } { q ^ n } L _ { n } ^ { ( \\delta ) } ( z q ; q ) - \\frac { 1 - q ^ { n + \\delta } } { q ^ n } L _ { n - 1 } ^ { ( \\delta ) } ( z q ; q ) . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\gamma = ( q + 1 ) \\beta \\geq \\beta , \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\xi ^ T A \\xi & = e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } \\xi ^ T \\big ( ( \\Lambda + \\nabla _ y \\widetilde { w } ) ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T + I \\big ) \\xi \\\\ & = e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } \\big | ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T \\xi \\big | ^ 2 + e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } | \\xi | ^ 2 \\geq e ^ { \\inf V } | \\xi | ^ 2 . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} j u : = u ^ * ( d \\theta ) = u ^ 1 d u ^ 2 - u ^ 2 d u ^ 1 . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\omega \\left ( ( f _ 1 , \\phi _ 1 ) , ( f _ 2 , \\phi _ 2 ) \\right ) \\ = \\ \\langle f _ 2 , \\phi _ 1 \\rangle - \\langle f _ 1 , \\phi _ 2 \\rangle \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\alpha _ 1 ^ + = \\max _ { 1 \\le i \\le n } \\alpha _ i \\ ; \\ ; \\alpha _ n ^ + = \\min _ { 1 \\le i \\le n } \\alpha _ i \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} A _ i \\cong B _ i \\mbox { f o r } i = 1 , 2 . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} I _ n ^ \\zeta ( a , b ) & : = \\int _ { \\{ a < r _ 1 < r _ 2 < \\cdots < r _ n < b \\} } \\Big [ ( r _ 2 - r _ 1 ) ( r _ 3 - r _ 2 ) \\cdots ( b - r _ n ) \\Big ] ^ \\zeta d r _ 1 d r _ 2 \\cdots d r _ n \\\\ & = \\frac { \\Gamma ( 1 + \\zeta ) ^ { n + 1 } ( b - a ) ^ { n ( 1 + \\zeta ) } } { \\Gamma \\big ( n ( 1 + \\zeta ) + 1 \\big ) } , \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\norm { T ( v ) } & = \\norm { T ( x + y ) } \\leq \\norm { T ( x ) } + \\norm { T ( y ) } \\\\ & \\leq \\norm { T } _ { L ( X , W ) } \\norm { x } _ X + \\norm { T } _ { L ( Y , W ) } \\norm { y } _ Y \\leq \\max \\{ \\norm { T } _ { L ( X , W ) } , \\norm { T } _ { L ( Y , W ) } \\} \\bigl ( \\norm { x } _ X + \\norm { y } _ Y \\bigr ) . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} f _ j ( - t , - z ) = f _ j ( t , z ) . \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\begin{cases} \\hat { y } : = \\rho _ 0 ^ { - 2 } ( L ^ * \\hat { u } - \\alpha _ 1 \\hat { z } ^ 1 { 1 } _ { \\mathcal { O } _ { 1 , d } } - \\alpha _ 2 \\hat { z } ^ 2 { 1 } _ { \\mathcal { O } _ { 2 , d } } ) \\ \\ & \\ \\ Q , \\\\ \\hat { p } ^ i : = \\rho _ 0 ^ { - 2 } ( L \\hat { z } ^ i + \\dfrac { 1 } { \\mu _ i } \\hat { u } { 1 } _ { \\mathcal { O } _ i } ) \\ \\ & \\ \\ Q , \\\\ \\hat { f } : = - \\rho _ 1 ^ { - 2 } \\ \\hat { u } \\ \\ & \\ \\ \\mathcal { O } \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\nu _ { \\beta , h } ^ + ( \\eta _ 0 & = + \\mid + ^ { - ( n + 1 ) } _ { - \\infty } \\omega ^ { - 1 } _ { - n } ) - \\nu _ { \\beta , h } ^ + ( \\eta _ 0 = + \\mid - ^ { - ( n + 1 ) } _ { - \\infty } \\omega ^ { - 1 } _ { - n } ) | \\\\ & \\leq \\lim _ { n \\rightarrow \\infty } | \\mu _ { \\beta , h } ^ { \\Lambda _ n , \\omega ^ + } ( \\eta _ { ( 0 , 0 ) } = + 1 ) - \\mu _ { \\beta , h } ^ { \\Lambda _ n , \\omega ^ - } ( \\eta _ { ( 0 , 0 ) } = + 1 ) | \\\\ & \\leq \\lim _ { n \\rightarrow \\infty } 1 0 n C e ^ { - c n } , \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} R _ { c , m } = \\min \\limits _ { j } \\{ \\log _ 2 ( 1 + { \\rm S I N R } _ { c , j } ^ { c , m } ) , j \\ge m \\} , \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\lambda \\int _ { \\O } u \\phi ~ d x = \\int _ { \\O } u ^ p \\phi ~ d x . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\dot { x } & = \\lambda - d x + r x \\left ( 1 - \\frac { x } { K } \\right ) - \\frac { \\beta x v } { \\alpha y + \\gamma x } , \\\\ \\dot { y } & = \\frac { \\beta x ( t - \\tau _ 1 ) v ( t - \\tau _ 1 ) } { \\alpha y ( t - \\tau _ 1 ) + \\gamma x ( t - \\tau _ 1 ) } e ^ { - \\alpha _ 1 \\tau _ 1 } - a y - p y z , \\\\ \\dot { v } & = k e ^ { - \\alpha _ 2 \\tau _ 2 } y ( t - \\tau _ 2 ) - u v , \\\\ \\dot { z } & = c y z - b z , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} w ^ \\beta _ + w ^ \\gamma _ - = ( \\theta ^ \\beta _ + \\theta ^ \\gamma _ - b _ { \\beta , \\gamma } + b _ { \\alpha + \\beta , \\alpha + \\gamma } ) e ^ { \\beta + \\gamma } + ( \\theta ^ \\beta _ + b _ { \\beta , \\alpha + \\gamma } + \\theta ^ \\gamma _ - b _ { \\alpha + \\beta , \\gamma } ) e ^ { \\alpha + \\beta + \\gamma } \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\Delta _ L = \\frac { 1 } { 2 } \\Delta _ 0 = - \\frac { 1 } { 2 } \\left ( \\frac { \\partial ^ 2 } { \\partial \\theta _ 1 ^ 2 } + \\frac { \\partial ^ 2 } { \\partial \\theta _ 2 ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} 2 \\cos ( \\theta ) c + b = 0 a + \\cos ( \\theta ) c + \\cos ( \\phi ) d = 0 . \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\alpha } ( K | E ) ~ = ~ \\max \\left \\{ m \\in \\mathbb { N } ~ | ~ \\exists ~ \\alpha \\mathrm { - p a c k i n g ~ o f } ~ K ~ \\mathrm { h a v i n g ~ s i z e } ~ m \\right \\} , \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} h _ { \\pi ^ { * } H } ( g ^ { n } ( y ) ) = h _ { A + E } ( \\widetilde { g } ^ { n } ( \\widetilde { y } ) ) . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} q _ { ( i - j ) m } : = \\omega ( W _ { \\Theta _ { i } \\xi - \\Theta _ { j } \\xi } ) , h \\sigma ( \\Theta _ { i } \\xi , \\Theta _ { j } \\xi ) = ( i - j ) \\phi _ { m , n } \\ , \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} \\mu ( P ) \\leq C _ 0 l ( { P } ) = 3 { C _ 0 } l ( Q ) / C _ 1 , \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} T ( f _ { 1 } , \\cdots , f _ { m } ) ( x ) = \\int _ { \\mathbb { R } ^ { m n } } K ( x , y _ { 1 } , \\cdots , y _ { m } ) \\Pi _ { j = 1 } ^ { m } f _ { j } ( y _ { j } ) d y _ { 1 } \\cdots d y _ { m } , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} R _ { a L b L } = & W _ { a L b L } \\\\ R _ { a L b \\underline { L } } = & W _ { a \\underline { L } b \\underline { L } } - \\kappa ^ 2 g _ { a b } \\\\ R _ { a L L \\underline { L } } = & W _ { a L L \\underline { L } } \\\\ R _ { L \\underline { L } L \\underline { L } } = & W _ { L \\underline { L } L \\underline { L } } - \\kappa ^ 2 . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} w ^ { \\theta } & = ( \\theta ^ { 2 } \\otimes \\theta \\otimes { \\rm i d } ) ( w ) \\\\ & = ( \\alpha ^ { 2 } \\beta x y z + \\beta ^ { 2 } \\gamma y z x + \\alpha \\gamma ^ { 2 } z x y ) - ( \\beta \\gamma ^ { 2 } z y x + \\alpha \\beta ^ { 2 } y x z + \\alpha ^ { 2 } \\gamma x z y ) . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\emptyset \\not = \\Lambda \\subsetneq \\{ 1 , 2 , \\cdots , m \\} \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\xi = \\xi _ { ( \\cdot , \\cdot ) } : \\mathfrak { g } \\mapsto \\mathfrak { g } \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} k _ { \\gamma } ^ { \\infty } = \\frac { \\sqrt { \\dot { \\gamma } _ 1 ^ 2 + \\dot { \\gamma } _ 2 ^ 2 } } { | \\gamma _ 1 | | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} e ^ { t \\Delta } ( f \\otimes g ) = e ^ { t \\Delta _ H } f \\otimes e ^ { t \\Delta _ z } g , f : G \\to \\R ^ 2 , g : ( - h , 0 ) \\to \\R , \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\Big [ ( 6 + t ) E ( U _ { m + 1 } , t ) + 2 E _ * ( U _ { m + 1 } , t ) \\Big ] \\\\ & = \\| U _ { m + 1 } ' \\| ^ 2 + \\| A ^ { 1 / 2 } U _ { m + 1 } \\| ^ 2 + 2 ( 6 + t ) ( U _ { m + 1 } ' , - U _ { m + 1 } ' - V _ { m } ' ) \\\\ & + 4 \\| U _ { m + 1 } ' \\| ^ 2 + 4 ( U _ { m + 1 } , - A U _ { m + 1 } - V _ { m } ' ) \\\\ & = - ( 7 + 2 t ) \\| U _ { m + 1 } ' \\| ^ 2 - 3 \\| A ^ { 1 / 2 } U _ { m + 1 } \\| ^ 2 - 2 ( 6 + t ) ( U _ { m + 1 } ' , V _ { m } ' ) + 4 ( U _ { m + 1 } , A V _ { m , 1 } ) \\\\ & = - ( 1 + t ) \\| U _ { m + 1 } ' \\| ^ 2 - \\| A ^ { 1 / 2 } U _ { m + 1 } \\| ^ 2 - ( 6 + t ) \\| V _ { m } ' \\| ^ 2 + 2 \\| A ^ { 1 / 2 } V _ { m , 1 } \\| ^ 2 . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal J ( t ) = \\frac { \\sigma } { 2 L } \\int \\varphi ' \\left ( u ^ 2 + \\frac 2 { p + 1 } u ^ { p + 1 } \\right ) + \\frac { 1 } { 2 L } \\int \\varphi ' ( v ^ 2 - v _ x ^ 2 ) . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} H ^ * ( f ) _ { ( c } = H ^ { - 1 } ( f _ { ( c } ) . \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta u _ \\varepsilon = \\lambda _ \\varepsilon u _ \\varepsilon H _ { N _ \\varepsilon } ( u _ \\varepsilon ) \\exp ( u _ \\varepsilon ^ 2 ) , u _ \\varepsilon > 0 \\Omega \\ , , \\\\ & u _ \\varepsilon = 0 \\partial \\Omega \\ , , \\end{cases} \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} f : = | \\{ j : ( \\ref { e q u : d f e a s j } ) \\} | = | \\{ j : ( \\ref { e q u : p f e a s j } ) \\} | . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\left ( \\frac { c _ 1 + \\lambda _ 1 } { w _ 1 } + \\cdots + \\frac { c _ { | \\mathcal { S } | } + \\lambda _ { | \\mathcal { S } | } } { w _ { | \\mathcal { S } | } } \\right ) \\frac { \\left ( \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } \\right ) ^ 2 } { B } \\\\ = \\left ( | \\mathcal { S } | - 1 \\right ) \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\left | \\widehat { J } _ k \\left ( \\frac { \\xi } { \\theta _ k } \\right ) \\right | & = \\left | \\widehat { J } \\left ( \\frac { \\xi } { \\theta _ k } \\right ) \\right | ^ { k } \\\\ & \\leq \\left ( 1 - D \\frac { | \\xi | ^ \\sigma } { k } \\right ) ^ { k } \\\\ & \\leq e ^ { - D | \\xi | ^ { \\sigma } } , \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} { C } _ S ^ { \\min } ( \\boldsymbol { \\psi } ) = & \\min _ { \\textbf { W } } { C } _ S ( \\boldsymbol { \\psi } , \\textbf { W } ) = { C } _ S ( \\boldsymbol { \\psi } , \\textbf { W } _ S ^ * ) . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\frac 1 s - \\frac 1 { p } = \\frac 1 { q } - \\frac 1 { p } = \\sum _ { i = 1 } ^ m \\Big ( \\frac 1 { q _ i } - \\frac 1 { p _ i } \\Big ) = \\frac 1 { q _ m } - \\frac 1 { p _ m } = \\frac 1 { s _ m } - \\frac 1 { p _ m } \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} l : = 2 \\left \\lfloor \\nu \\frac { N } { k } \\right \\rfloor \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 } \\alpha _ { i } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} f _ 1 = \\cos ( \\theta _ 1 + \\theta _ 2 ) \\quad f _ 2 = \\frac { 5 \\cos ( 2 \\theta _ 1 + 2 \\theta _ 2 ) - 3 } { 6 } \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} \\widehat { \\chi _ k } ( j ) = \\delta _ { j k } ( j , k \\in \\mathbb { Z } ) . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} g ( s ) : = \\sum _ { n = 0 } ^ { \\infty } b ( n ) e ^ { - s n } , \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\| v \\| _ * & \\leq 2 \\left ( \\| u _ 0 \\| _ { H ^ 1 } + \\| u _ 1 \\| _ { L ^ 2 } \\right ) + C T ( | \\rho _ 1 | + | \\rho _ 2 | ) + C T ( | \\rho _ 1 | + | \\rho _ 2 | ) e ^ { \\frac { 1 } { 8 \\pi } R ^ 2 } \\\\ & = \\frac 2 3 R + C T ( | \\rho _ 1 | + | \\rho _ 2 | ) + C T ( | \\rho _ 1 | + | \\rho _ 2 | ) e ^ { \\frac { 1 } { 8 \\pi } R ^ 2 } . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} D ( P | | Q ) = \\sup _ { g } ~ \\mathbb { E } _ P [ g ( X ) ] - \\log \\mathbb { E } _ Q [ \\exp { g ( X ) } ] , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} y = x + \\sum _ { \\alpha = 1 } ^ { d } \\int _ { 0 } ^ { | I | } V _ { \\alpha } ( z _ { t } ) d \\tilde { h } _ { t } ^ { \\alpha } . \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} [ y ^ { p ^ r } _ { 1 2 } , y ^ { p ^ s } _ { 2 3 } ] = [ x ^ { p ^ r } _ { 1 2 } , x ^ { p ^ s } _ { 2 3 } ] = x ^ { p ^ r } _ { 1 2 } ( 1 - x ^ { - p ^ r } _ { 1 2 } x ^ { p ^ s } _ { 2 3 } x ^ { p ^ r } _ { 1 2 } x ^ { - p ^ s } _ { 2 3 } ) x ^ { p ^ s } _ { 2 3 } . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} I I I ' & = 4 \\pi c - \\frac 1 2 c ^ { - 1 } W ( h a ) \\int Q _ { a , c } ( x ) ^ 2 \\ , d x \\\\ & + \\frac 3 4 c ^ { - 1 } h ^ 2 W '' ( h a ) \\int ( x - a ) ^ 2 Q _ { a , c } ( x ) ^ 2 \\ , d x + O ( h ^ 3 ) \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { H } _ 0 & : Y _ k ( t ) \\sim \\mathcal { N } ( \\mu _ 0 , \\sigma ^ 2 ) , \\\\ \\mathcal { H } _ 1 & : Y _ k ( t ) \\sim \\mathcal { N } ( \\mu _ 1 , \\sigma ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\tau _ 0 = \\inf \\{ \\tau ' \\mid \\exists \\tau \\in \\R , \\ u ( \\cdot , \\cdot + \\tau ) \\preceq v ( \\cdot , \\cdot ) \\preceq u ( \\cdot , \\cdot + \\tau + \\tau ' ) \\} \\leq \\tau _ 0 - \\eta < \\tau _ 0 . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} E _ { \\beta , \\gamma } : = \\sum _ { k = 0 } ^ { \\infty } \\frac { z ^ k } { \\Gamma ( \\beta k + \\gamma ) } . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\widehat { w } : = \\Big ( \\prod _ { i = 1 } ^ { m - 1 } w _ i ^ { \\frac 1 { p _ i } } \\Big ) ^ { \\varrho } \\in A _ { \\frac { 1 - r } { r } \\varrho } ; \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} T _ { \\varepsilon _ { n } } = \\left ( \\log \\frac { 1 } { \\varepsilon _ { n } } \\right ) ^ { - \\frac { 1 } { s _ { k } } } . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} \\gamma _ 1 { G } : = G \\gamma _ i { G } : = \\lbrack \\gamma _ { i - 1 } { G } , G \\rbrack i \\ge 2 , \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} P + \\sum _ { B \\in \\mathcal B ( \\Phi ) } y ^ - _ B B = \\sum _ { B \\in \\mathcal B ( \\Phi ) } y ^ + _ B B . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} & L ( t ) : = \\left ( \\begin{array} { c c c } \\frac { 1 } { \\sqrt { 1 - | s ( t ) | ^ 2 } } & 0 \\\\ 0 & 1 \\end{array} \\right ) \\ , , \\quad \\tilde \\Phi ( t , x ) : = L ( t ) R ( t ) ( x - r ( t ) ) \\ , , \\\\ & \\tilde \\Omega ( t ) : = \\tilde \\Phi ( t , \\Omega ) \\ , , \\quad \\tilde \\Gamma ( t ) : = \\tilde \\Phi ( t , \\Gamma ( t ) ) \\ , , \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} A - \\lambda = 0 . \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} W _ l ( x ; \\tau _ 1 , \\dots , \\tau _ n ) & : = \\sum _ { 0 \\le k \\le n } \\sum _ { 1 \\le i _ 1 < \\dots < i _ k \\le n } ( - 1 ) ^ k \\tilde { M } ( x - \\tau _ { i _ 1 } - \\dots - \\tau _ { i _ k } ) ^ l \\\\ & = \\sum _ { S } ( - 1 ) ^ { | S | } \\tilde { M } ( x - \\sum _ { i \\in S } \\tau _ { i } ) ^ l , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\varphi _ N ( T ) = \\varphi _ N ( \\Gamma ) \\exp \\left ( - ( \\Gamma - T ) \\right ) - \\exp ( T ) \\int _ T ^ \\Gamma \\exp ( - s ) \\frac { s ^ N } { N ! } d s \\ , . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} P : = \\frac { 2 } { 3 } \\sum _ { i = 1 } ^ { 3 } \\Pi _ { \\mathcal J _ { i } } \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } E [ u , u _ t ] ( t ) = & - \\int _ 0 ^ 1 g ( u ( x , t ) ) u _ t ( x , t ) ^ 2 \\ , d x \\\\ & - m ' ( t ) \\int _ 0 ^ 1 \\Big \\{ \\big [ 1 - g ( u ( x , t ) ) \\big ] u _ t ( x , t ) + f ( u ( x , t ) ) \\Big \\} \\ , d x , \\end{aligned} \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\begin{aligned} \\iint | \\omega _ t * f ( x ) | ^ 2 \\ , t ^ { d - 1 } d t \\ , d \\lambda ( x ) = & \\iint | \\widehat { \\omega _ r } * f ( x ) | ^ 2 \\ , r ^ { d - 1 } d r \\ , d \\lambda ( x ) \\\\ = & c _ d \\iint \\left | D _ x ^ { - \\frac { d - 1 } { 2 } } e ^ { - 2 \\pi i t \\sqrt { - \\Delta } } f ( x ) \\right | ^ 2 \\ , d t \\ , d \\lambda ( x ) \\\\ = & c _ d ' \\iint \\left | D _ x ^ { - \\frac { d - 2 } { 2 } } e ^ { 2 \\pi i t \\Delta } f ( x ) \\right | ^ 2 \\ , d t \\ , d \\lambda ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\mathit { S I N R } = \\frac { \\displaystyle \\max _ { i = 1 , 2 , \\ldots , n } h _ { i } \\rho _ { \\mathrm { o } } } { \\sigma ^ 2 + I } , \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} { \\mathbf { y } _ { b d } } = \\sqrt { \\rho _ d } \\widehat { h } _ { a b } \\mathbf { x } _ d + \\underbrace { \\sqrt { \\rho _ d } \\widetilde { h } _ { a b } \\mathbf { x } _ d + \\mathbf { n } _ b } _ { \\mathbf { n } _ b ' } , \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\mathsf { A } ^ { m + n } _ T ( x ) = \\mathsf { A } ^ m _ T ( T ^ n x ) \\mathsf { A } ^ n _ T ( x ) \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\gamma ( z , \\psi ^ D ) = 1 / { \\textstyle \\sqrt { \\| 2 D \\| } } . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} w = ( \\lambda _ 1 + y - \\Sigma ) ^ { - 1 / 2 } v = ( \\lambda _ 1 + y - \\Sigma ) ^ { - 1 } F . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\| d _ a \\eta \\| ^ 2 \\le C \\sum _ { i = 1 } ^ d \\| \\nabla _ { X _ i } \\eta \\| ^ 2 , \\eta \\in \\Gamma ( \\mathbb { M } , \\mathcal { E } ) , \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\mathcal T ( ( P _ 1 , \\leqq _ 1 ) ) \\times \\cdots \\times \\mathcal T ( ( P _ n , \\leqq _ n ) ) = \\mathcal T \\Bigl ( ( P _ 1 , \\leqq _ 1 ) \\times \\cdots \\times ( P _ n , \\leqq _ n ) \\Bigr ) \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} H ( \\mu , \\mathcal { D } _ n ) = - \\sum _ { E \\in \\mathcal { D } _ n } \\mu ( E ) \\log \\mu ( E ) , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\alpha _ { \\rm H J M } = \\frac { d } { d x } \\Psi ( - T \\sigma ) , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} \\psi _ r = & \\ \\phi ^ { - 1 } X \\psi = A \\phi ^ { - 1 } \\xi \\psi = A \\phi \\psi . \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D N = 0 } \\right \\rbrace = \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace + \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\sum _ { g \\in G } m _ g g = ( \\sum _ { g \\in G } m _ g ) \\delta + \\sum _ { \\chi ( \\ne 1 ) \\in \\Psi } M _ \\chi , \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} d _ 2 \\left ( 1 \\cdot a \\otimes b \\otimes m \\right ) = { } & a \\otimes b ( 0 ) m - a ( 0 ) b \\otimes m - b \\otimes a ( 0 ) m , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} \\theta _ { i } ( t ) = \\int _ { 0 } ^ { t } \\omega _ { i } ( W _ { s } ) \\circ d W _ { s } \\ , . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} T = T _ { \\rm r e g } + T _ { \\rm s i n g } , \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\xi _ k \\times ( \\zeta _ j , 2 b _ j + 1 ) ) = \\prod _ { i = 0 } ^ { 2 b _ j } L ^ S ( s + \\frac { 1 } { 2 } + b _ j - i , \\xi _ k \\times \\zeta _ j ) , \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } \\equiv [ n ] \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ { k } } { [ 2 k ] ^ 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( k , H ^ q ( C \\otimes _ k \\overline { k } , \\mu _ n ^ { \\otimes b } ) ) \\ , \\Rightarrow H ^ { p + q } ( C , \\mu _ n ^ { \\otimes b } ) \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ { 1 } ( u ) : = \\Delta _ { 1 } ( - b _ { 0 } \\left ( u + q \\right ) ) = - 2 A ^ { 2 } \\frac { \\sin \\frac { \\sqrt { u } } { 2 } } { \\sqrt { u } } \\left ( \\frac { A ^ { 2 } + 1 } { 2 A } - \\cos \\frac { \\sqrt { u } } { 2 } \\right ) . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} ( f , g ) \\circ ( f ' , g ' ) = ( f \\circ f ' , g \\circ g ' ) \\mbox { a n d } 1 _ { ( a , b ) } = ( 1 _ a , 1 _ b ) . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\epsilon ' & \\leq \\epsilon _ n + \\frac { C } { \\sqrt { n } } + \\frac { 1 } { n } + \\frac { 1 } { q ( n ) } \\\\ & = \\epsilon . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} \\| u \\| _ { \\textsc { C C } } < \\kappa _ { 2 } \\implies C _ { V , l } | I _ { 1 } | = C _ { V , l } \\| u _ 1 \\| _ { \\textsc { C C } } \\leq \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} w ( \\theta _ { M _ S } ( \\mu _ S ) ) = x = w ' ( \\theta _ { M _ S } ( \\mu _ S ) ) . \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\det ( \\nabla ^ 2 h + h { \\rm I d } ) = h ^ { p - 1 } ( \\| \\nabla h \\| ^ 2 + h ^ 2 ) ^ { \\frac { n - q } 2 } f \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\begin{aligned} d f ( T _ x ( \\mathcal O ) ) & \\subset H _ { f ( x ) } ( M ) , ~ ~ \\forall x \\in \\mathcal O ~ ~ & \\\\ J \\circ d f & = d f \\circ J _ { s t } , \\end{aligned} \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} f ^ { ( j ) } ( w ) ~ = ~ 0 \\qquad \\forall j \\in \\overline { 2 , p } \\qquad \\mathrm { a n d } f ^ { ( p + 1 ) } ( w ) ~ \\neq ~ 0 . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} I _ 4 ( x ) : = \\left \\{ \\begin{array} { l l } + \\infty & \\ { \\rm i f } \\ x \\in \\R _ - \\\\ \\lceil \\ell _ 4 x \\rceil & \\ { \\rm i f } \\ x \\geq 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} ( - z ; \\ , q ) _ j L _ { n } ^ { ( \\delta ) } ( z q ^ j ; \\ , q ) = \\sum _ { i = 0 } ^ { j } a _ { j , n + i } L _ { n + i } ^ { ( \\delta ) } ( z ; \\ , q ) . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} \\| u - u _ N \\| _ { H ^ s ( \\mathbb R ) } & \\le c \\big ( \\| \\tilde e _ N \\| _ { H ^ s ( \\mathbb R ) } + \\| I _ N ^ \\lambda f - f \\| _ { L ^ 2 ( \\mathbb R ) } \\big ) \\\\ & \\le c N ^ { s - m } | u | _ { { \\mathbb B } ^ m _ { \\lambda } ( \\mathbb R ) } + c N ^ { - k } | f | _ { { \\mathbb B } ^ { k } _ { \\lambda } ( \\mathbb R ) } . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) = - J ( x _ 0 ) . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { k } \\left ( \\hat { \\boldsymbol { \\psi } } _ { k } - \\boldsymbol { \\psi } \\right ) \\overset { d } { \\rightarrow } \\mathcal { N } \\left ( 0 , \\mathbf { I } _ S ( \\boldsymbol { \\psi } , \\widetilde { \\textbf { W } } ^ * ) ^ { - 1 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} { A ( \\gamma _ \\varepsilon ) } - 2 \\xi _ \\varepsilon = o \\left ( \\tilde { \\zeta } _ \\varepsilon \\right ) \\ , , \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\eta _ k ^ ( t ) = \\log \\ ! \\left ( \\frac { q _ 1 ( y _ k ( t ) ) } { q _ 0 ( y _ k ( t ) ) } \\right ) . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\Pr { 0 \\leq F \\leq \\sigma } = \\sigma , 0 \\leq \\sigma < 1 , \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & - \\phi ^ s _ t - ( ( D _ 1 a ( y ^ s _ x , t , x ) y _ x ^ s + a ( y _ x ^ s , t , x ) ) \\phi ^ s _ x ) _ { x } + D _ 1 F ( y ^ s , y ^ s _ x ) \\phi ^ s - ( D _ 2 F ( y ^ s , y ^ s _ x ) \\phi ^ 2 ) _ x = \\alpha _ 1 ( y ^ s - y _ { 1 , d } ) { 1 } _ { \\mathcal { O } _ { 1 , d } } \\ \\ Q , \\\\ & \\phi ^ s ( 0 , t ) = \\phi ^ s ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & \\phi ^ s ( T ) = 0 \\ \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} \\norm { i } { a } = a \\sigma ( a ) \\dots \\sigma ^ { i - 1 } ( a ) . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\frac 1 r : = \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { r _ i } ; \\qquad \\frac 1 { p _ { m + 1 } } : = 1 - \\frac 1 p ; \\qquad \\frac 1 { q _ { m + 1 } } : = 1 - \\frac 1 q , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\tilde x \\frac { \\lambda _ { j - 2 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } = \\tilde x \\Big ( 1 + \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } \\Big ) \\leq 2 \\tilde { x } \\frac { \\lambda _ { j - 1 } } { g _ { j - 1 } } \\leq 2 z _ 2 \\frac { \\lambda _ { j - 1 } } { g _ { j - 1 } } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} I _ 2 \\leq C _ 1 r _ n \\sqrt { 2 } \\frac { n } { N } \\exp \\left ( - C _ 2 \\frac { n } { N } \\right ) = C _ 1 \\sqrt { 2 } \\left ( r _ n \\sqrt { \\frac { n } { N } } \\right ) \\left ( \\sqrt { \\frac { n } { N } } \\exp \\left ( - C _ 2 \\frac { n } { N } \\right ) \\right ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} x _ { n , m } = \\begin{cases} & n < n _ 1 , \\\\ x ^ k _ { m \\ , ( n _ k ) } & n _ k \\leq n < n _ { k + 1 } 0 \\leq m < n ! . \\end{cases} \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\overline { g } = \\frac { u ^ 2 } { 1 + t ^ 2 } d t ^ 2 + t ^ 2 g ( t ) , \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\Phi ' ( t ) = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { t ^ 2 } { 2 } } ; \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} C _ { a , b } ( x , y , z ) : & = \\dfrac { a } { 2 } ( x ^ 2 + y ^ 2 + z ^ 2 ) + b x y ^ { \\lambda } , \\\\ H _ { c , d } ( x , y , z ) : & = \\dfrac { c } { 2 } ( x ^ 2 + y ^ 2 + z ^ 2 ) + d x y ^ { \\lambda } , ~ \\forall ( x , y , z ) \\in \\Omega , \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { m } } \\varphi ( x ) d x = \\int _ { q \\in N } \\mathrm { v o l } _ { N } ( d q ) \\int _ { x \\in F ^ { - 1 } ( q ) } \\frac { \\varphi ( x ) } { \\| F ^ * \\mathrm { v o l } _ { N } \\| } \\mathrm { v o l } _ { F ^ { - 1 } ( q ) } ( d x ) , \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\nu ( f ( a _ \\sigma ) ) = \\nu ( f ( a _ { \\rho _ f } ) ) \\mbox { f o r e v e r y } \\rho _ f \\leq \\sigma < \\lambda , \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} p _ { c , n } ( n + a ( c ) ) - a ( c ) p _ { c , n } = \\sum _ { k = 1 } ^ n a _ { c , k } p _ { c , n - k } \\ \\ \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} A ^ k A ^ { * k } A ^ { k + 1 } = A ^ { k + 1 } A ^ { * k } A ^ k . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} H ( x , y ) = G ( x , y ) + \\frac { \\log ( | x - y | ) } { 2 \\pi } \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} X _ * ( T ) = X ^ * ( \\widehat { T } ) \\xrightarrow { r e s } X ^ * ( Z ( \\widehat { M _ S } ) ^ { \\Gamma } ) \\to X ^ * ( Z ( \\widehat { M _ S } ) ^ { \\Gamma } ) _ { \\mathbb { Q } } \\cong \\mathfrak { A } _ { M _ S , \\mathbb { Q } } \\subset \\mathfrak { A } _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "4159.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": "8141.png", "formula": "\\begin{align*} E ( L _ 1 , \\ell _ \\nu , \\ell _ \\mu ) + E ( L _ 1 \\ell _ \\nu , \\ell _ { \\mu ' } ) = & D ( L _ 1 , \\ell _ \\nu , 2 \\ell _ \\mu ) + D ( L _ 1 , \\ell _ \\nu , 2 \\ell _ { \\mu ' } ) \\\\ & + L _ 1 - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ \\mu ) - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ { \\mu ' } ) . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\nu _ { \\beta , h } ^ { + } ( \\{ \\omega \\colon \\omega _ { \\Lambda } = \\eta _ { \\Lambda } \\} ) : = \\mu _ { \\beta , h } ^ { + } ( \\{ \\tilde { \\omega } \\colon \\tilde { \\omega } _ { \\Lambda \\times \\{ 0 \\} } = \\eta _ { \\Lambda } \\} ) \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} \\lambda _ { s , t } \\cdot \\left [ \\binom { n - 1 - a } { s - 1 } \\cdot \\N ( T _ { r - 1 } ( a ) , K ^ { ( r - 1 ) } _ { s , t } ) + \\binom { n - 1 - a } { t - 1 } \\cdot \\N ( T _ { r - 1 } ( a ) , K ^ { ( r - 1 ) } _ { s , s } ) \\right ] . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} D ^ 2 f ( G ) = f '' ( G ) D G \\otimes D G + f ' ( G ) D ^ 2 G . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} N = \\binom { h } { i } \\binom { i } { i + j - h } , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} ( \\rho v _ x ^ 2 ) _ t + ( \\rho u v _ x ^ 2 ) _ x = 2 v _ x u _ { x x } . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\sigma _ { \\Delta , \\mathbf { G } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) - \\sigma _ { \\Delta , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) = \\sum _ { a \\in \\mathcal { A } } c _ { \\mathbf { a } } ^ { 2 } E _ { P } \\left \\{ \\pi _ { \\mathbf { a } } ( \\mathbf { G } ; P ) v a r _ { P } ( Y \\mid \\mathbf { A } = \\mathbf { a } , \\mathbf { G } ) v a r _ { P } \\left [ \\frac { 1 } { \\pi _ { \\mathbf { a } } ( \\mathbf { G , B } ; P ) } \\mid \\mathbf { A = a } , \\mathbf { G } \\right ] \\right \\} \\mathbf { \\geq } 0 . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} & \\left ( \\int _ { \\Omega \\cap \\{ | \\eta | \\geq 2 | \\gamma | \\} } \\left | | \\eta \\gamma | ^ { \\alpha - Q } - | \\eta | ^ { \\alpha - Q } \\right | ^ { Q ' } d \\eta \\right ) ^ { 1 / Q ' } \\\\ \\leq & C \\left ( \\int _ { \\{ | \\eta | \\geq 2 | \\gamma | \\} } \\left ( | \\gamma | | \\eta | ^ { \\alpha - Q - 1 } \\right ) ^ { Q ' } d \\eta \\right ) ^ { 1 / Q ' } \\\\ \\leq & C | \\gamma | \\cdot | \\gamma | ^ { \\alpha - Q - 1 + \\frac Q { Q ' } } = C | \\gamma | ^ { \\alpha ' } , \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} ( \\mathbf A \\leftrightarrow \\mathbf B ) \\otimes ( \\mathbf A ' \\leftrightarrow \\mathbf B ' ) = ( \\mathbf A \\otimes \\mathbf A ' ) \\leftrightarrow ( \\mathbf B \\otimes \\mathbf B ' ) . \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} - \\left ( \\alpha \\varphi _ k + d \\psi _ k \\right ) '' = \\mu _ 1 \\left ( 1 - 2 u _ { 1 , k } \\right ) \\alpha \\varphi _ k + \\mu _ 2 \\left ( 1 - 2 u _ { 2 , k } \\right ) \\psi _ k + \\lambda _ { 1 , k } \\left ( \\alpha \\varphi _ k + \\psi _ k \\right ) . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\log P ( B _ n ^ { ( 4 ) } ) = - \\varepsilon . \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\max _ { 1 \\leq j \\leq n } f ( I _ j ^ n ) = \\int _ I s ( x ) d x ; \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} f ^ { q - 1 } ( \\xi ) = \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta + \\lambda \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha - 1 } } d \\eta , \\xi \\in \\overline \\Omega . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\inf _ { \\stackrel { v \\in H _ 0 ^ 1 ( \\Omega ) \\cap \\mathcal { V } _ 0 , \\ , v \\not = 0 } { v \\bot \\phi _ 1 , \\dots v \\bot \\tilde { \\phi } _ { m _ 0 } } } \\frac { Q _ u ( v ) } { ( v , v ) _ { L ^ 2 ( \\Omega ) } } \\ge 0 . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\Big \\| { \\bf B } _ \\alpha ( t , T ) f \\Big \\| _ { \\mathbb { H } ^ { \\nu } ( \\Omega ) } ^ 2 = \\sum _ { j = 1 } ^ \\infty \\left | \\frac { E _ { \\alpha , 1 } ( - \\lambda _ j t ^ \\alpha ) } { E _ { \\alpha , 1 } ( - \\lambda _ j T ^ \\alpha ) } \\right | ^ 2 f _ j ^ 2 \\le \\mathcal N _ 2 ^ 2 t ^ { - 2 \\alpha \\vartheta } \\norm { f } ^ 2 _ { \\mathbb { H } ^ { \\nu + ( 1 - \\vartheta ) } ( \\Omega ) } , \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} 0 = ( B \\iota ) ^ \\ast ( B p ) ^ \\ast ( p _ 1 ) = ( B \\iota ) ^ \\ast ( 2 q _ 4 + t _ 2 ^ 2 ) = 2 ( B \\iota ) ^ \\ast ( q _ 4 ) + 4 t _ 2 ^ 2 . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\begin{cases} X ^ { \\delta } ( 0 ) = ( x _ 0 , Z _ { 0 } ) \\\\ d x ^ { \\delta } ( t ) = b _ { i ^ { \\delta } ( t ) } ( t , x ^ { \\delta } ( t ) ) d t + \\sigma _ { i ^ { \\delta } ( t ) } ( t , x ^ { \\delta } ( t ) ) d W ( t ) + \\delta d ( \\displaystyle \\sum _ { 0 \\leq s \\leq t } \\mathbf { 1 } _ { \\{ x ^ { \\delta } ( s - ) = 0 \\} } ) \\\\ i ^ { \\delta } ( t ) = Z _ { \\displaystyle \\sum _ { 0 \\leq s \\leq t } \\mathbf { 1 } _ { \\{ x ^ { \\delta } ( s - ) = 0 \\} } } \\\\ \\end{cases} , ~ ~ \\mathbb { P } \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} \\frac { d } { d t } E \\xi _ t ( x ) & = - E \\xi _ t ( x ) + \\lambda \\sum _ { y : y \\sim x } \\Big [ ( b - 1 ) E \\xi _ t ( x ) + a E \\xi _ t ( y ) \\Big ] + \\Big ( 1 - 2 d \\lambda [ ( b - 1 ) + a ] \\Big ) E \\xi _ t ( x ) \\\\ & = - E \\xi _ t ( x ) + 2 d \\lambda ( a + b - 1 ) E \\xi _ t ( x ) + \\big ( 1 - 2 d \\lambda ( a + b - 1 ) \\big ) E \\xi _ t ( x ) = 0 . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} p \\ast q : = \\sum p _ { n _ 1 , \\dots , n _ r } \\cdot q _ { n _ 1 , \\dots , n _ r } T _ 1 ^ { n _ 1 } \\cdots T _ r ^ { n _ r } , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mathcal { I } u ( x ) = 0 , \\ , \\ , , \\\\ u = g , \\qquad \\ , \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ * \\leq ~ & 2 \\rho _ 1 \\int _ 0 ^ T \\left \\| \\left ( \\frac { e ^ { u _ 1 } } { \\int _ { M } e ^ { u _ 1 } } - \\frac { e ^ { u _ 2 } } { \\int _ { M } e ^ { u _ 2 } } \\right ) \\right \\| _ { L ^ 2 } \\mathrm { d } s \\\\ & + 2 \\rho _ 2 \\int _ 0 ^ T \\left \\| \\left ( \\frac { e ^ { - u _ 1 } } { \\int _ { M } e ^ { - u _ 1 } } - \\frac { e ^ { - u _ 2 } } { \\int _ { M } e ^ { - u _ 2 } } \\right ) \\right \\| _ { L ^ 2 } \\mathrm { d } s . \\end{aligned} \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} ( t _ 0 , \\dots , t _ N ) \\cdot [ z _ 0 : \\cdots : z _ N ] = [ t _ 0 z _ 0 : \\cdots : t _ N z _ N ] \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\imath _ \\xi ( F \\circ \\gamma ( x ) ) = - a d _ { \\imath _ { \\xi } \\gamma ( x ) } ^ * = - a d _ { a d _ { \\xi } ^ * x } ^ * & = - \\lambda ( x , \\xi ) . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} \\chi _ { P , a , e f f } ^ { 1 , d e s c } \\left ( \\mathbf { V } ; \\mathcal { G } \\right ) = \\frac { I _ { a } ( A ) } { \\pi _ { a } ( \\mathbf { O } _ { m i n } ; P ) } \\left \\{ Y - b _ { a } \\left ( \\mathbf { O ; } P \\right ) \\right \\} . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\lambda ( \\delta ) = \\sup \\P ( | X - Y | \\geq 1 - \\delta ) , \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s & R _ { 2 n } ^ \\lambda ( x ) = a _ n ^ \\lambda \\\\ & \\times \\sum _ { k = 0 } ^ { n } \\frac { ( - n ) _ { k } ( n + \\lambda ) _ { k } } { ( \\lambda + \\frac { 1 } { 2 } ) _ { k } \\ , k ! } A _ s ^ { k + \\frac { \\lambda + 1 } { 2 } } { } _ { 2 } F _ { 1 } \\Big ( s + k + \\frac { \\lambda + 1 } { 2 } , s + \\frac { 1 } { 2 } ; \\frac { 1 } { 2 } ; - x ^ 2 \\Big ) , \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\Phi ( t ) = t ^ p \\log ( e + t ) ^ { p - 1 + \\delta } , p \\in ( 1 , \\infty ) , \\ \\delta \\in ( 0 , \\infty ) \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } = \\sum _ { i , j , k , l , p } ( h _ { i j k l } ^ { p ^ { \\ast } } ) ^ { 2 } \\endaligned \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} \\| \\mathcal { A } _ { 1 } ( y , p ^ 1 , p ^ 2 , f ) ( 0 ) \\| _ { H _ 0 ^ 1 ( I ) } & = \\int _ I | ( y _ { x t } ( 0 ) - ( a ( y _ x ( 0 , t , 0 ) ) y _ x ( 0 ) ) _ { x x } - f _ x ( 0 ) { 1 } _ { \\mathcal { O } } + \\frac { 1 } { \\mu } p _ x ( 0 ) { 1 } _ { \\omega } | ^ 2 d x \\\\ & \\leq C ( \\| ( y , p ^ 1 , p ^ 2 , f ) \\| _ Y ^ 2 + \\| ( y , p ^ 1 , p ^ 2 , f ) \\| _ Y ^ 4 ) < + \\infty . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} \\omega _ { 3 } ( \\sigma ^ { l } ) : = \\frac { 1 } { \\tilde { \\delta } } \\sum ^ { l } _ { j = 0 } \\omega _ { 1 } ( \\sigma ^ { l - j } ) \\omega _ { 2 } ( \\sigma ^ { j } ) , \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\mathsf b ( z ; \\eta ) = \\bigl ( \\exp \\eta \\partial _ { \\tilde e _ 0 } \\bigr ) \\ , \\sum _ { r \\geq 0 } ( - z ) ^ r \\tilde { e } _ r , \\mathsf c ( z ; \\eta ) = \\exp \\bigl ( \\eta \\sum _ { r \\geq 0 } ( - z ) ^ r \\partial _ { \\tilde e _ r } ^ \\perp \\bigr ) \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\vert u ( x _ k ) \\vert \\leq \\bigg ( \\sum _ { i = 0 } ^ { k - k _ 0 - 1 } \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\bigg ) + \\vert u ( x _ { k _ 0 } ) \\vert \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} \\displaystyle C _ m : = \\left \\{ \\bigcap _ { n \\geq m } \\bigcap _ { j = 0 } ^ { c _ n - 1 } \\left \\{ Y _ { l } ^ { t _ n + j b _ n } - ( t _ n + j b _ n ) \\leq b _ n \\right \\} \\right \\} \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} ( A _ { n , 4 } V _ { n , 4 } + \\epsilon V _ { n , 4 } ) ( \\zeta , \\eta , y , x ) = & \\mathcal { S } _ 4 ( \\zeta , \\eta ) + \\eta [ \\delta y + x ] + ( A _ { n , 2 } V _ { n , 2 } + \\epsilon V _ { n , 2 } ) ( y , x ) - \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} x _ t \\overset { \\mathrm { D e f . } } { = } x _ 0 + \\int _ 0 ^ t \\vartheta _ s \\ , \\mathrm { d } { s } + 2 c _ H ^ { B , R } \\int _ 0 ^ t \\varphi _ s \\delta B _ s ^ { \\frac { H } { 2 } + \\frac { 1 } { 2 } } + \\int _ 0 ^ t \\psi _ s \\delta { R } ^ H _ s \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } { 1 \\over n _ k } { \\sum _ { j = 1 } ^ k \\log u _ j } = \\lim _ { k \\to \\infty } { \\sum _ { j = 1 } ^ k j ^ { 1 - \\epsilon } / a \\over k ^ { { 1 \\over \\alpha } ( 1 - \\epsilon ) } } = 0 . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} \\xi = \\left \\| \\boldsymbol { W } ^ m \\left ( \\boldsymbol { W } ^ \\top \\right ) ^ m \\right \\| _ , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} \\hat \\nabla _ W Z & = \\nabla _ W Z + W ( \\Upsilon ) \\ , Z + Z ( \\Upsilon ) \\ , W , \\\\ \\hat \\nabla _ { \\bar W } Z & = \\nabla _ { \\bar W } Z - h ( Z , \\bar W ) \\nabla ^ { 1 , 0 } \\Upsilon , \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} E = x \\frac { \\partial } { \\partial x } + y \\frac { \\partial } { \\partial y } + 2 t \\frac { \\partial } { \\partial t } . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} v _ n = \\lambda v ( T ^ n x ) , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} f _ \\Lambda ( b _ 1 , b _ 2 ) = ( \\Re { \\wp _ \\Lambda ( b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 ) } , \\Im \\wp _ \\Lambda ( b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 ) ) ; ~ ~ ( b _ 1 , b _ 2 ) \\in [ 0 , 1 ) ^ 2 , b _ 1 ^ 2 + b _ 2 ^ 2 \\neq 0 . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } d i s t _ { \\rho } ( f _ { n } , F _ { \\rho } ( T ) ) = 0 . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} { \\psi } _ n ^ { N , h } ( y ) : = \\frac { 1 } { \\sqrt { \\lambda _ n ^ { N , h } } } h ^ 2 \\sum \\limits _ { n = 1 } ^ { N } \\sum \\limits _ { K \\in \\mathcal { T } _ h } \\sum \\limits _ { x _ j \\in I _ { K } } \\log \\kappa ( x _ j , y _ n ) \\phi _ n ^ { N , h } ( x _ j ) L _ n ( y ) . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} - \\div _ y ( A \\nabla _ y v ) = \\div _ y \\phi . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\epsilon _ k = \\sum _ { i _ { k - 1 } + 1 \\leq i \\leq i _ k - 1 } y _ i ^ { - 2 / b } ( 1 \\leq k \\leq K + 1 ) . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\mathcal { X } _ { q ( t ) } = \\widetilde { \\mathcal { X } } \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } e ^ { - \\| x \\| _ p ^ p } d x & = { \\rm v o l } ( B _ p ^ n ) \\cdot \\int _ 0 ^ \\infty n t ^ { n - 1 } e ^ { - t ^ p } d t = { \\rm v o l } ( B _ p ^ n ) \\cdot \\frac { n } { p } \\cdot \\int _ 0 ^ \\infty s ^ { n / p - 1 } e ^ { - s } d s \\\\ & = \\frac { n \\ , { \\rm v o l } ( B _ p ^ n ) \\Gamma ( n / p ) } { p } = { \\rm v o l } ( B _ p ^ n ) \\Gamma ( 1 + n / p ) . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\begin{aligned} & | \\dot { c } - c h W ' ( h a ) | \\lesssim \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu h t } \\\\ & | \\dot { a } - c + W ( h a ) + \\frac 1 2 c ^ { - 2 } h ^ 2 W '' ( h a ) | \\lesssim \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu h t } \\end{aligned} \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\Delta ( - \\tau _ 0 ) = 4 \\exp ( - \\tau _ 0 ) \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\int _ K ( d d ^ c w ) ^ n \\leq \\lim _ { j \\to + \\infty } \\int _ \\Omega \\chi _ k ( d d ^ c \\max ( u , w + \\frac { 1 } { j } ) ) ^ n = \\int _ \\Omega \\chi _ k ( d d ^ c u ) ^ n , \\ \\forall k \\geq 1 . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\left ( f \\ast \\varphi \\right ) \\left ( x \\right ) = \\int _ { G } f \\left ( y \\right ) \\varphi \\left ( y ^ { - 1 } x \\right ) d y \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} e ^ { - \\rho \\cdot x } ( \\Delta + q ) e ^ { \\rho \\cdot x } r = - q \\Omega , \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\Pr { B _ k } { M _ k } & = \\sum _ { i = k + 1 } ^ \\infty \\Pr { s _ i = M _ k , s _ { k + 1 } , \\ldots , s _ { i - 1 } < M _ k } { M _ k } . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\sum _ { l \\in \\overline S } \\frac { R _ { i l } } { \\lambda _ 0 - \\omega ( l , l ) } u _ l = \\frac { R _ { i j } } { \\lambda _ 0 - \\omega ( j , j ) } u _ j = c _ 1 u _ i \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\tau _ 1 ^ 2 ( x ) = g _ 1 ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac 1 3 , \\tfrac 1 2 ) . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} B ( \\alpha + i y ) = O _ { \\alpha , f , M } ( y ^ M ) + \\sum _ { j = 0 } ^ { M - 1 } y ^ { j + \\frac 1 2 } \\begin{cases} a _ j ( \\alpha ) + b _ j ( \\alpha ) \\log { y } & \\nu = k = 0 , \\\\ a _ j ( \\alpha ) y ^ \\nu + b _ j ( \\alpha ) y ^ { - \\nu } & . \\end{cases} \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\frac { a _ { 3 , j , w _ j } } { \\phi ^ { 2 \\alpha _ j } } = a _ { 3 , j , z _ j } - 4 \\alpha _ j ( p - 2 ) \\frac { \\langle \\nabla u _ j , \\nabla \\phi \\rangle \\langle \\nabla u _ j , \\nabla z _ j \\rangle } { z _ j \\phi } + \\Upsilon _ j , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\left \\langle T _ k , ( \\tau ' ) ^ l , \\frac { \\alpha } { z + \\psi } \\right \\rangle _ { 0 , l + 2 , d } = \\sum _ { u \\geq 0 } \\frac { ( - 1 ) ^ u } { z ^ { u + 1 } } \\left \\langle \\psi ^ u \\alpha , T _ k , ( \\tau ' ) ^ l \\right \\rangle _ { 0 , l + 2 , d } \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} X = \\left [ \\begin{array} [ c ] { c c } Z & S \\\\ S ^ { t r } & Y \\end{array} \\right ] \\left ( Z , Y \\right ) \\in \\mathfrak { s o } \\left ( p \\right ) \\times \\mathfrak { s o } \\left ( q \\right ) . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} F ( n , k - 1 ) - F ( n , k ) = G ( n + 1 , k ) - G ( n , k ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} t ^ { p - l } \\int _ { B _ t } | \\nabla u | ^ p d x \\leq ( k t ) ^ { p - l } \\int _ { B _ { k t } } | \\nabla u | ^ p d x = t ^ { p - l } \\int _ { B _ t } | \\nabla u _ k | ^ p d x \\to 0 \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} \\alpha ^ { r q - s ' - a } \\beta ^ { b - r q + s ' + a } \\binom { b } { r q - s ' - a } \\ & = \\ \\alpha ^ { r q - s ' - a } \\beta ^ { b - r q + s ' + a } \\binom { b _ 0 q + b _ 1 } { b _ 0 q + q - s ' - a _ 1 } . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} L ( \\zeta ) = \\left [ \\begin{pmatrix} 1 + \\frac { n ^ 2 + \\frac { n } { 2 } } { \\zeta } & \\frac { - 2 \\pi \\i \\ n } { \\Gamma ( n + \\frac 1 2 ) \\ , n ! \\ \\zeta } \\\\ \\frac { - n ! \\ \\Gamma ( n + \\frac 3 2 ) } { 2 \\pi \\i \\ \\zeta } & 1 - \\frac { n ^ 2 + \\frac { n } { 2 } } { \\zeta } \\end{pmatrix} + \\mathrm { O } ( \\zeta ^ { - 2 } ) \\right ] \\zeta ^ { - n \\sigma _ 3 } , \\quad \\zeta \\to \\infty , \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} a ( x ) : = \\lim _ { y \\to x } \\frac { f ( x ) - f ( y ) } { ( x - y ) ^ { p ^ { n _ i } } } \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} m : = \\min \\{ k \\in \\mathbb { N } : \\ , \\alpha ( k ) \\ge n \\} \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\mathbb { E } [ Z _ i | Y _ i = y _ i ] = \\sum _ { w = 0 } ^ { y _ i } { { y _ i } \\choose { w } } \\delta ^ { y _ i - w } ( 1 - \\delta ) ^ { w } \\frac { g ( w ) } { q } \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} I _ { N } ^ { \\lambda } u ( x ) = \\sum _ { n = 0 } ^ { N } \\tilde { u } _ { n } ^ { \\lambda } R _ { n } ^ { \\lambda } ( x ) , { \\rm w h e r e } \\tilde { u } _ { n } ^ { \\lambda } = \\frac { 1 } { \\gamma _ { n } ^ { \\lambda } } \\sum _ { j = 0 } ^ { N } u ( x _ { j } ^ \\lambda ) R _ { n } ^ { \\lambda } ( x _ j ^ \\lambda ) \\omega _ { j } ^ \\lambda . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\partial _ { \\mathbb { H } } \\circ \\tilde { \\pi } = 6 ( \\mathrm { I d } _ { \\Lambda ^ { 3 } E ^ { * } } - P ) , \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\delta _ q Q ^ k = k Q ^ k = Q \\partial _ Q Q ^ k \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } ( \\beta \\varepsilon ) ^ { n / 2 } & \\int _ { \\mathbb { R } ^ n } e ^ { i 2 \\pi ( x , ( \\alpha / \\beta ) ^ { \\frac { 1 } { 2 } } ( \\beta \\varepsilon ) ^ { \\frac { 1 } { 2 } } \\eta + m ) } e ^ { - i 2 \\pi k x } a ( x , ( \\alpha \\varepsilon ) ^ { \\frac { 1 } { 2 } } \\eta + m ) e ^ { - \\pi \\varepsilon \\beta | x | ^ 2 } d x \\\\ & = \\int _ { \\mathbb { T } ^ n } e ^ { i 2 \\pi ( x , m ) - i 2 \\pi k x } a ( x , m ) d x . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} H ( x _ 0 ) \\xi = E \\xi , \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} G ( q , z ) : = \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ { k } } { ( 1 - q ^ { 2 k } z ) ^ 2 } . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} u ' ( x ) & = \\frac { 1 } { 1 - \\gamma } \\mathrm { e } ^ { - \\gamma x } x ^ { \\nu + 1 } \\big ( \\mathbf { L } _ \\nu ( x ) - \\gamma \\mathbf { L } _ { \\nu + 1 } ( x ) \\big ) - \\mathrm { e } ^ { - \\gamma x } x ^ { \\nu + 1 } \\mathbf { L } _ \\nu ( x ) \\\\ & = \\frac { 1 } { 1 - \\gamma } \\mathrm { e } ^ { - \\gamma x } x ^ { \\nu + 1 } \\big ( \\mathbf { L } _ \\nu ( x ) - \\mathbf { L } _ { \\nu + 1 } ( x ) \\big ) > 0 , \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\mathcal { A } f _ g ( x ) = g ( x ) - \\mathbb { E } g ( Z ) , \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} d Y _ t = X _ 0 ( t ) d t + \\sum _ { i = 1 } ^ d X _ i ( Y _ t ) \\circ d \\beta _ t ^ i , \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} d _ r ( C _ D ) & = n - \\max \\{ \\ | D \\cap H | : H \\in [ \\widetilde { D } , m - 1 - r ] _ p \\ \\} \\\\ & = p ^ { m - 1 } - 1 - ( p ^ { m - 1 - r } - 1 ) \\\\ & = p ^ { m - 1 } \\left ( 1 - \\frac { 1 } { p ^ r } \\right ) \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} & t ^ { \\frac { N + 2 A } { 2 } } u ( x , t ) \\\\ & = q _ * [ M _ { 0 , 1 } + o ( 1 ) ] U ( | x | ) - \\left [ \\frac { N + 2 A } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] t ^ { - 1 } U ( | x | ) F ( | x | ) + o ( t ^ { - 1 } ) \\\\ & = q _ * [ M _ { 0 , 1 } + o ( 1 ) ] U ( | x | ) + O ( t ^ { - 1 } ) \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} ( z \\partial _ z + \\mathfrak { E } _ { ( \\partial ) } + \\mu ) \\textbf { S } ( t , z ) ( \\alpha ) = \\left ( \\frac { ( \\alpha ) } { 2 } - \\frac { \\textnormal { d i m } _ \\mathbb { C } ( X ) } { 2 } \\right ) \\textbf { S } ( t , z ) ( \\alpha ) \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} d V _ t = \\left ( g _ 0 ' ( t ) + ( K ' * d Z ) _ t - K ( 0 ) \\lambda V _ t \\right ) d t + K ( 0 ) \\nu \\sqrt { V _ t ^ + } d W _ t , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\sigma _ 2 ( \\theta _ 1 , \\theta _ 2 ) = \\frac { 1 } { \\sin ( \\omega _ 1 ) } \\ , \\cos ( \\theta _ 1 ) \\sin ( \\theta _ 2 ) \\ ; , \\ ; \\ ; ( \\theta _ 1 , \\theta _ 2 ) \\in [ - \\omega _ 1 , \\omega _ 1 ] \\times [ 0 , \\omega _ 2 ] \\ ; . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\coprod _ { M _ S } \\mathfrak { A } ^ + _ { M _ S , \\mathbb { Q } } = \\overline { C } _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} & P _ { \\textrm { o u t , N } } ^ { \\textrm { C S A N C } } = \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace = \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D F = 0 } \\right \\rbrace + \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D F = 1 , N D N = 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\gamma _ \\varepsilon \\left ( \\gamma _ \\varepsilon - u _ \\varepsilon ( x _ \\varepsilon - \\mu _ \\varepsilon \\cdot ) \\right ) \\to T _ 0 : = \\log \\left ( 1 + | \\cdot | ^ 2 \\right ) C ^ { 1 , \\theta } _ { l o c } ( \\mathbb { R } ^ 2 ) \\ , , \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} h ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\gamma ( n ) \\log ( ( 1 - e ^ { - s n } ) ^ { - 1 } ) = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\phi ( m s ) , \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} b _ 2 ( y ) = ( M ( 1 _ { I _ 0 } ) ) ^ { - \\frac 1 2 } = \\sqrt { \\frac { y + 1 } 2 } , \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} u ^ \\varepsilon _ t + f ( x , u ^ \\varepsilon ) _ x ~ = ~ \\varepsilon \\ , u ^ \\varepsilon _ { x x } \\ , , u ^ \\varepsilon ( 0 , x ) = \\bar u ( x ) , \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ j } H e s s _ { \\widetilde { g } } h ( X _ i , X _ k ) = H e s s _ { g _ B } f ( X _ i , X _ k ) e ^ { l ( y _ 1 , \\ldots , y _ m ) } , \\ \\ \\ \\forall i , k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} T _ { i , c ^ \\dagger } ^ { \\dagger , - 1 } A _ c T _ { i , c } ^ { - 1 } = A _ c , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} Y _ 3 ^ { \\perp } = \\cos ( \\theta _ 1 + \\theta _ 2 ) ( J X _ 1 - J X _ 2 ) \\quad Y _ 4 ^ { \\perp } = \\sin ( \\theta _ 1 + \\theta _ 2 ) ( J X _ 1 - J X _ 2 ) . \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\gamma _ { { } _ { F , g } } & = 1 + \\tau \\int _ { - 1 } ^ 1 \\sqrt { F ( s ) } F '' ( s ) \\ , d s \\left ( \\int _ { - 1 } ^ 1 \\sqrt { F ( s ) } \\ , d s \\right ) ^ { - 1 } \\\\ & = 1 - \\tau \\int _ { - 1 } ^ 1 \\frac { F ' ( s ) ^ 2 } { 2 \\sqrt { F ( s ) } } \\ , d s \\left ( \\int _ { - 1 } ^ 1 \\sqrt { F ( s ) } \\ , d s \\right ) ^ { - 1 } < 1 , \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} Y _ { ( M _ S , \\mu _ S ) } = \\{ b \\in \\mathbf { B } ( G , \\mu ) : ( M _ S , \\mu _ S ) \\in \\mathcal { R } _ { G , b , \\mu } \\} . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} { \\rm s u p p } [ \\nu ] \\subset \\left \\{ y \\in R ^ m \\ \\Big | \\ E ( y ) = 0 \\right \\} . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\beta _ n ( a , q ) = \\sum _ { 0 \\le j \\le n } \\frac { \\alpha _ j ( a , q ) } { ( q ; q ) _ { n - j } ( a q ; q ) _ { n + j } } . \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} e = \\frac { 1 } { Q y _ \\mathrm { s m a x } } \\sum _ { j = m } ^ Q \\left | \\tilde { y } _ j - \\hat { y } _ j \\right | , y _ \\mathrm { s m a x } = \\frac { 1 } { 2 } \\left ( \\sqrt { L ^ 2 + H ^ 2 } - L \\right ) , \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { S _ { n } \\varphi ( x ) } { \\Phi ( n ) } = \\lim _ { k \\to \\infty } \\frac { S _ { n _ k } \\varphi ( x ) } { \\Phi ( n _ k ) } \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\frac { \\det J } { \\cosh t + \\frac 1 2 | \\tilde s | ^ 2 e ^ t } \\ , d t \\wedge d s ^ { 1 } \\wedge \\dots \\wedge d s ^ { n - 1 } = e ^ { ( n - 1 ) t } \\ , d t \\wedge d s ^ { 1 } \\wedge \\dots \\wedge d s ^ { n - 1 } . \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} X _ { ( ( T ^ \\ast Q ) _ \\mu , \\omega _ \\mu , h _ \\mu , f _ \\mu , u _ \\mu ) } \\cdot \\pi _ \\mu = T \\pi _ \\mu \\cdot X _ { ( T ^ \\ast Q , G , \\omega , H , F , u ) } \\cdot i _ \\mu . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} B _ i ( t ) U : = \\sigma _ { i } ( t ) \\mathbf { 1 } _ 3 \\cdot \\nabla U , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} d i v \\left [ V ^ 2 \\nabla \\sinh ^ { - 1 } \\frac { f d i v ( V ^ 2 \\nabla \\tau ) } { | H _ 0 | | H | } - f V ^ 4 \\nabla \\tau + V ^ 2 ( \\alpha _ { H _ 0 } - \\alpha _ H ) \\right ] = O ( 1 ) \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\mathbb { E } \\# \\{ \\textrm { c r i t i c a l p o i n t s o f $ f | _ { \\Gamma \\cap Z ( p ) } $ } \\} = \\frac { | \\Gamma | } { \\pi } \\frac { d ^ { \\frac { n - 1 } { 2 } } } { ( 2 \\pi ) ^ { \\frac { n - 2 } { 2 } } } \\cdot \\mathbb { E } | \\det Q _ { n - 2 } | + O ( d ^ { \\frac { n - 2 } { 2 } } ) . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} & \\lim _ { r \\to 0 } r ^ { - 5 } E ( \\Sigma _ r , Y _ r ( T _ 0 ) , T _ 0 ) = \\frac { 1 } { 9 0 } \\Big [ Q ( e _ 0 , e _ 0 , e _ 0 , A e _ 0 + C _ i e _ i ) + \\frac { \\sum _ { m , n } \\bar W _ { 0 m 0 n } ^ 2 } { 2 A } \\Big ] . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} \\mathcal { D } ( w ^ { \\theta } ) & = k \\langle x , y , z \\rangle / ( \\partial _ { x } w , \\partial _ { y } w , \\partial _ { z } w ) \\\\ & = k \\langle x , y , z \\rangle / ( \\alpha ^ { 2 } \\beta y z - \\alpha ^ { 2 } \\gamma z y , \\beta ^ { 2 } \\gamma z x - \\alpha \\beta ^ { 2 } x z , \\alpha \\gamma ^ { 2 } x y - \\beta \\gamma ^ { 2 } y x ) \\\\ & = k \\langle x , y , z \\rangle / ( \\beta y z - \\gamma z y , \\gamma z x - \\alpha x z , \\alpha x y - \\beta y x ) . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} \\lambda _ { s , t } \\cdot \\sum _ { a = \\lfloor \\beta n \\rfloor } ^ { \\lfloor \\frac { r - 1 } { r } n \\rfloor - 1 } \\Delta ( a ) = \\lambda _ { s , t } \\cdot \\sum _ { a = \\lfloor \\beta n \\rfloor } ^ { \\lfloor \\frac { r - 1 } { r } n \\rfloor - 1 } M ( a ) \\left [ H \\left ( \\frac { n - a } { a } \\right ) + o ( 1 ) \\right ] , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} C _ \\alpha ( p ) : = \\{ \\beta \\in C ^ * \\setminus \\{ \\alpha , \\alpha ^ c \\} : p ( \\beta ) = p \\} \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\gamma \\left ( \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) \\right ) ( z , Q ) = \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} e _ \\rho ( \\rho , \\theta ) = \\frac { 1 } { \\rho ^ 2 } ( p - \\theta p _ \\theta ( \\rho , \\theta ) ) . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} f ( x ) = g ( x ) ^ T \\ , G g ( x ) + c ^ T h ( x ) , \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} P _ { A _ 1 ^ n , F _ 1 ^ n } \\triangleq \\prod _ { i = 1 } ^ n P _ { F _ i | A _ i } P _ { A _ i } . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} \\lambda _ j ( \\Sigma ) \\geq \\sum _ { k = 1 } ^ d \\omega _ k ^ 2 \\langle u _ j , f _ k \\rangle ^ 2 + \\lambda _ d ( \\Gamma ) . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\| u \\| _ * = \\| u \\| _ { C _ T ( H ^ 1 ) } + \\| \\partial _ t u \\| _ { C _ T ( L ^ 2 ) } . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} g _ { i } ( x ) : = \\begin{cases} 1 , & x _ i < b _ i - \\dfrac { 1 } { \\lambda } \\\\ - \\lambda ( x _ i - b _ i ) , & b _ i - \\dfrac { 1 } { \\lambda } < x _ i < b _ i \\\\ 0 , & b _ i < x _ i . \\end{cases} \\\\ \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\phi ( s ) = \\int _ { s } ^ { s ^ { 1 / 2 } } F ( x / s ) e ^ { - x } d x + \\int _ { s ^ { 1 / 2 } } ^ { \\infty } F ( x / s ) e ^ { - x } d s . \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} p _ 1 ( V \\oplus \\xi _ \\mathbb { R } ^ { \\oplus ( - 2 k - 1 ) } ) = p _ 1 ( V ) - ( 2 k + 1 ) c _ 1 ( \\xi ) ^ 2 \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} K ( x , y _ 1 , y _ 2 ) = \\sum _ { m \\in \\mathbb Z } \\int _ { \\R ^ d } 2 ^ { 3 m d } \\phi _ s ( 2 ^ m ( u - x ) ) \\psi ( 2 ^ m ( u - y _ 1 ) ) \\psi ( 2 ^ m ( u - y _ 2 ) ) \\ , \\mathrm { d } u \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\begin{cases} f ( \\xi ) = c _ 1 [ ( a - N ) \\xi + c ] ^ { - \\frac { 1 } { a - N } } \\\\ h ( \\xi ) = c _ 2 [ ( a - N ) \\xi + c ] ^ { - \\frac { N } { a - N } } \\\\ \\varphi ( \\xi ) = c _ 3 [ ( a - N ) \\xi + c ] ^ { - \\frac { k } { a - N } } \\end{cases} , \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} k ^ { L , s } _ { \\gamma _ i , \\Sigma } = k ^ { \\infty , s } _ { \\gamma _ i , \\Sigma } + O ( L ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} s _ r ( \\widetilde { H } ) & = \\sum _ { j = 1 } ^ r \\sum _ { ( i , j ) \\in V ( \\widetilde { H } ) } \\left | d _ { \\widetilde { H } } ( ( i , j ) ) - \\frac { r ! m } { r n } \\right | \\\\ & = r \\sum _ { i \\in V ( H ) } \\left | ( r - 1 ) ! d _ { H } ( i ) - \\frac { r ! m } { r n } \\right | \\\\ & = r ! s ( H ) . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\Phi ( \\chi _ k ) = \\Phi ( \\chi _ k \\ast \\chi _ k ) = P ( \\chi _ k ) = \\sum _ { j = 0 } ^ { + \\infty } \\widehat { \\chi _ k } ( - j ) = \\sum _ { j = 0 } ^ { + \\infty } { \\delta _ { - j k } } = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} p _ { w , i } = \\begin{cases} \\mu _ { i } [ w ] \\cdot c _ { i } & w \\in \\mathcal { W } _ { i } \\\\ 2 ^ { - m \\epsilon ^ { - 1 } } \\cdot c _ { i } & \\end{cases} , \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} h = ( x _ 1 + \\sum _ { i = 0 } ^ { b } c _ i x _ 2 ^ i x _ 3 ^ { m - p i } , x _ 2 + c x _ 3 ^ p , x _ 3 ) , c _ b \\neq 0 , c \\neq 0 . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\tau ^ * ( f _ 1 ) ( h g h ^ { - 1 } ) = f _ 1 ( \\sigma ( h g ^ { - 1 } h ^ { - 1 } ) ) = f _ 1 ( \\sigma ( h ) \\sigma ( g ) ^ { - 1 } \\sigma ( h ) ^ { - 1 } ) = \\tau ^ * ( f _ 1 ) ( g ) \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\lambda _ j ^ { q } < \\infty . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\mathrm { I m } ( w ^ \\xi _ \\eta ( \\rho ) ) = \\mathrm { I m } & \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\triangle ^ + ( \\rho ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } \\\\ = - \\mathrm { I m } & \\sum _ { \\{ \\alpha , \\beta , \\epsilon \\} \\in \\triangle ^ - ( \\rho ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } \\beta ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & 0 \\\\ \\int _ { S ^ 2 } D \\beta ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & 0 . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\mathbf { y } _ \\mathcal { Q } = \\mathcal { Q } ( \\mathbf { y } ) = \\mathcal { Q } \\left ( \\mathfrak { R } \\{ \\mathbf { y } \\} \\right ) + j \\mathcal { Q } \\left ( \\mathfrak { I } \\{ \\mathbf { y } \\} \\right ) , \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\frac { d } { d t } y _ { j } ( t ) = - 2 \\pi \\Im \\Psi _ { D _ t } ( z _ { j } ( t ) , \\xi ( t ) ) , \\\\ \\frac { d } { d t } x _ { j } ( t ) = - 2 \\pi \\Re \\Psi _ { D _ t } ( z _ { j } ( t ) , \\xi ( t ) ) , \\\\ \\frac { d } { d t } x ^ { r } _ { j } ( t ) = - 2 \\pi \\Re \\Psi _ { D _ t } ( z ^ { r } _ { j } ( t ) , \\xi ( t ) ) . \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\int _ 0 ^ L \\mu _ 1 w \\alpha \\left ( z + t w \\right ) ^ { + } \\varphi = \\int _ 0 ^ L \\mu _ 1 w \\alpha z ^ { + } \\varphi . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} v = \\pi ^ 0 { v } \\rightarrow \\pi ^ 1 { v } \\rightarrow \\pi ^ 2 { v } \\rightarrow \\ldots \\rightarrow \\pi ^ d { v } = m _ e . \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} F = \\sum _ { I , J \\subset \\Lambda } F _ { I , J } ( x , y ) ( \\eta \\xi ) _ { I , J } , \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} g - \\frac { 1 } { \\sqrt { \\lambda } } O g = h , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\mathcal { I } ( \\omega ) = 8 \\pi ^ { 2 } \\int _ { M } | \\omega | ^ { 2 } \\phi _ { 0 } ^ { 2 } + 4 \\pi \\int _ { M } \\phi _ { 0 } \\omega \\cdot \\nabla g - 4 \\pi \\int _ { M } g \\omega \\cdot \\nabla \\phi _ { 0 } \\ , , \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\psi _ { P , \\mathbf { a } } \\left ( \\mathbf { B } ; \\mathcal { G } \\right ) = \\psi _ { P , \\mathbf { a } } \\left ( \\mathbf { G } , \\mathbf { B } ; \\mathcal { G } \\right ) + \\sum _ { k = 0 } ^ { p } r _ { k } \\left ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } ; s _ { \\mathbf { a , } k } ^ { \\ast } , P \\right ) \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} v o l ( \\hat { \\mathfrak D } _ n ) = \\frac { 1 } { n ! } v o l ( \\{ ( x _ 1 , \\dots , x _ n ) \\in [ 0 , 1 ) ^ n \\mid \\sum x _ i \\in \\mathbb { Z } \\} ) \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} t _ { m _ { n } } & = t _ n , \\\\ t _ { u _ { 2 n } } & = t _ { u _ { 2 n + 1 } } = 1 - t _ n , \\\\ b _ { m _ { n } } & = \\left \\{ \\begin{array} { l l } 1 & n = 0 n = 2 ^ k , \\ , k \\in \\mathbb { N } , \\\\ 0 & \\end{array} \\right . \\\\ b _ { u _ { n } } & = \\left \\{ \\begin{array} { l l } 1 & n = 0 , \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\Lambda [ C ( z , t ) ] = \\frac { z } { 1 - 2 z } \\left ( \\frac { t ( C ( z , 1 ) - C ( z , t ) ) } { 1 - t } \\right ) . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\partial = \\sum _ { i \\ge 1 } ( - 1 ) ^ { i - 1 } ( \\delta _ i ^ 0 - \\delta _ i ^ 1 ) . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} H = \\lambda f ( ( T ^ n x ) _ { 1 } ) \\delta _ { n n ' } + \\Delta , \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{gather*} H = \\frac { 1 } { 2 } g ^ { i j } ( x ) p _ i p _ j + \\beta ^ i ( x ) p _ i + V ( x ) . \\end{gather*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\lim _ { { \\varepsilon } \\to 0 } { \\mathbb { P } } \\left ( \\exp \\left ( { \\varepsilon } ^ { - 1 } ( V ( \\partial D ) - \\eta ) \\right ) \\leq \\tau ^ { \\varepsilon } _ x \\leq \\exp \\left ( { \\varepsilon } ^ { - 1 } ( V ( \\partial D ) + \\eta ) \\right ) \\right ) = 1 . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\mathcal { L } _ { k } ^ { \\epsilon , \\hat { v } _ k } \\psi _ k ( x ) = \\bigl \\langle \\bigtriangledown _ x \\psi _ k ( x ) , f _ k ( x , \\hat { v } _ k ( x ) ) \\bigr \\rangle + \\frac { \\epsilon } { 2 } \\operatorname { t r } \\bigl \\{ a _ k ( x ) \\bigtriangledown _ x ^ 2 \\psi _ k ( x ) \\bigr \\} . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\langle \\vec { x } , \\vec { x } \\rangle & = \\frac { 1 } { 2 } \\langle \\vec { a } , \\vec { a } \\rangle + c \\langle \\vec { a } , \\vec { b } \\rangle + \\frac { c ^ { 2 } } { 2 } \\langle \\vec { b } , \\vec { b } \\rangle . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\lambda _ { k } ^ { \\epsilon , v _ k } = - \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\sup _ { x \\in D } \\mathbb { P } _ { x , k } ^ { \\epsilon , v _ k } \\Bigl \\{ \\tau _ { k } ^ { \\epsilon , v _ k } > t \\Bigr \\} , \\ , \\ , \\ , x \\in D , \\ , \\ , \\ , k \\in \\{ 1 , 2 , \\ldots , n \\} \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} f _ 1 ( n ) : = 1 \\vee \\frac { g ( n ) a _ c ^ { ( n ) } } { n p _ n } , \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} \\P \\Biggl [ \\sum _ { i = 1 } ^ n X _ i \\geq C ' n \\Biggr ] & \\leq e ^ { - b ' n } . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} E [ \\rho , u ] ( t ) : = \\int _ \\Omega \\eta ^ * ( \\rho , m ) d x = \\int _ \\Omega \\left ( \\frac { 1 } { 2 } \\rho u ^ 2 + \\frac { a } { \\gamma - 1 } \\rho ^ \\gamma \\right ) d x . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} b _ I ^ { ( \\theta ) } = \\sum _ { K \\in \\mathcal { B } _ I } \\theta _ K h _ K , I \\in \\mathcal { D } _ { \\leq n } , \\ \\theta \\in \\{ \\pm 1 \\} . \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} - z '' = \\frac { \\mu _ 1 } { \\alpha } \\left ( \\alpha - z \\right ) z ^ + - \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d + z \\right ) z ^ - . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\begin{aligned} { { \\widetilde { \\textbf { { I } } } } _ { i { p _ 1 } } } \\triangleq \\left [ \\begin{matrix} { { \\widetilde { \\bf { A } } } ^ { - 1 } } & { \\bf { 0 } } \\\\ { \\bf { 0 } } & { \\bf { 0 } } \\end{matrix} \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} \\sigma ^ { r } = x ^ { o ( 1 ) } . \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\begin{cases} u _ { 1 } + \\lambda \\left [ f \\left ( x , u _ { 1 } \\right ) - \\varepsilon u _ { 1 , x } \\right ] _ { x } = w _ { 1 } , \\\\ u _ { 2 } + \\lambda \\left [ f \\left ( x , u _ { 2 } \\right ) - \\varepsilon u _ { 2 , x } \\right ] _ { x } = w _ { 2 } , \\end{cases} \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\mathcal { Q } : = \\{ r \\in { \\mathcal { C } ^ 2 ( \\mathbb { C } ^ n ; \\mathbb { R } ) \\colon { } } & d r \\neq { 0 } r \\\\ & r \\Omega ^ { ( r ) } , \\\\ & \\Omega ^ { ( r ) } \\} \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\left \\Vert \\left ( \\boldsymbol { E } _ { M } \\oplus \\boldsymbol { E } _ { M } \\right ) \\left ( \\boldsymbol { P } _ { M } \\oplus \\boldsymbol { P } _ { M } \\right ) \\left ( u \\oplus v \\right ) - \\left ( u \\oplus v \\right ) \\right \\Vert = 0 , \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\frac { 1 } { p } - \\frac { a - \\frac 1 2 } { n } = \\frac { 1 } { p _ 0 } - \\frac { a _ 0 - \\frac 1 2 } { n } . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} \\theta : = 1 - \\left ( 1 - \\alpha \\right ) ^ { n - 1 \\choose 2 } . \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { i = 1 } ^ { m + 1 } \\eta _ i G _ i ( x ) , y - x \\right \\rangle \\geq 0 \\ \\ ; \\forall y \\in K _ Q . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } e ^ { K } = W , \\\\ \\| K \\| \\leq \\frac { \\pi } { 2 } \\| \\mathbf { 1 } _ n - W \\| = \\frac { \\pi } { 2 } \\| Z - \\hat { W } \\| \\leq \\frac { \\pi } { 2 } \\frac { 4 } { \\pi } \\arcsin ( \\varepsilon / 2 ) = 2 \\arcsin ( \\varepsilon / 4 ) \\end{array} \\right . \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} \\begin{cases} t - \\mu ^ 2 \\left ( \\sum \\limits _ { i = 1 } ^ m \\cfrac { 2 t } { t ^ 2 - \\sigma _ i ^ 2 } + \\cfrac { n - m } { t } \\right ) = z _ o , \\\\ \\sigma _ i + 2 \\mu ^ 2 \\cfrac { \\sigma _ i } { t ^ 2 - \\sigma _ i ^ 2 } , = \\sigma _ i ^ o \\\\ t > | \\sigma _ i | , \\end{cases} \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\left [ F ' ( u _ 1 ) - F ' ( u _ 2 ) - F '' ( u _ 2 ) ( u _ 1 - u _ 2 ) \\right ] \\partial _ t u _ 2 = \\frac 1 2 F ''' ( \\tilde { u } _ { 1 2 } ) | u _ 1 - u _ 2 | ^ 2 \\partial _ t u _ 2 \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} 0 = { \\rm t r } ( \\bar { X } \\bar { Z } ) = ( n - 2 ) b , \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} P \\{ X \\in A , \\ , \\ , Y \\in B \\} = \\int _ B \\int _ A f ( x , y ) d x d y . \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\eta _ { K , M } ^ { ( 1 ) } = \\left ( { M N - { \\lambda _ { \\cal M } } } \\right ) G _ { K , M } ^ { ( 1 ) } + { \\lambda _ { \\cal M } } G _ { K , M } ^ { ( 2 ) } , \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} s - \\frac { \\epsilon + \\eta } { \\epsilon \\eta } = t - ( \\epsilon + \\eta ) \\ , , \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\alpha _ { \\ell } & = r _ 1 + \\frac { ( r _ 1 - r _ 2 ) \\frac { \\alpha _ 1 - r _ 1 } { \\alpha _ 1 - r _ 2 } q ^ { \\ell - 1 } } { 1 - \\frac { \\alpha _ 1 - r _ 1 } { \\alpha _ 1 - r _ 2 } q ^ { \\ell - 1 } } , \\\\ & = r _ 2 + \\frac { r _ 2 - r _ 1 } { \\frac { \\alpha _ 1 - r _ 1 } { \\alpha _ 1 - r _ 2 } q ^ { \\ell - 1 } - 1 } . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\sum ^ { \\infty } _ { j = 1 } \\omega _ { 1 } ( \\sigma ^ { j } ) \\leq \\frac { 1 } { - \\log \\sigma } \\sum ^ { \\infty } _ { j = 1 } \\int ^ { \\sigma ^ { j - 1 } } _ { \\sigma ^ { j } } \\frac { \\omega _ { 1 } ( t ) } { t } d t = \\int ^ { 1 } _ { 0 } \\frac { \\omega _ 1 ( t ) } { t } d t < \\infty . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\textnormal { v l i f t } ( u _ 1 ) - \\textnormal { v l i f t } ( \\varphi ^ \\ast u _ 2 \\varphi _ \\ast ) = - X _ { H _ 1 } - \\textnormal { v l i f t } ( F _ 1 ) + T \\varphi ^ \\ast ( X _ { H _ 2 } ) + \\textnormal { v l i f t } ( \\varphi ^ \\ast F _ 2 \\varphi _ \\ast ) \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ * \\leq ~ & 2 \\left ( \\| u _ 0 \\| _ { H ^ 1 } + \\| u _ 1 \\| _ { L ^ 2 } \\right ) + 2 \\rho _ 1 \\int _ 0 ^ T \\left \\| \\left ( \\frac { e ^ u } { \\int _ { M } e ^ u } - \\frac { 1 } { | M | } \\right ) \\right \\| _ { L ^ 2 } \\mathrm { d } s \\\\ & + 2 \\rho _ 2 \\int _ 0 ^ T \\left \\| \\left ( \\frac { e ^ { - u } } { \\int _ { M } e ^ { - u } } - \\frac { 1 } { | M | } \\right ) \\right \\| _ { L ^ 2 } \\mathrm { d } s . \\end{aligned} \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} f ( x ) = f _ 0 ( x ) + f _ 1 ( x ) q ( x ) + \\ldots + f _ n ( x ) q ^ n ( x ) \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} f & = \\big ( x _ 1 + P ( x _ 2 , x _ 3 ) , x _ 2 + Q ( x _ 3 ) , x _ 3 + d \\big ) \\in M _ \\alpha , \\\\ g & = \\big ( x _ 1 + P ' ( x _ 2 , x _ 3 ) , x _ 2 + Q ' ( x _ 3 ) , x _ 3 + d ' \\big ) \\in N _ \\alpha , \\\\ f ^ { - 1 } & = \\Big ( x _ 1 - P \\big ( x _ 2 - Q ( x _ 3 - d ) , x _ 3 - d \\big ) , x _ 2 - Q ( x _ 3 - d ) , x _ 3 - d \\Big ) . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} \\tag * { ( B ) } \\begin{cases} x ^ k > x , \\\\ x ^ 2 > 2 ^ { k } ( x + x ^ { 2 / k } ) . \\end{cases} \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} | | \\dot { \\gamma } | | _ { \\Sigma , L } = \\sqrt { ( \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 ) ^ 2 + ( \\frac { l _ L } { l } ) ^ 2 L \\omega ( \\dot { \\gamma } ( t ) ) ^ 2 } \\sim L ^ { \\frac { 1 } { 2 } } | \\omega ( \\dot { \\gamma } ( t ) ) | , ~ ~ ~ ~ { \\rm a s } ~ ~ ~ ~ L \\rightarrow + \\infty . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\psi _ { P , \\mathbf { a } } ( \\mathbf { Z } ; P ) \\equiv \\frac { I _ { \\mathbf { a } } ( \\mathbf { A } ) } { \\lambda _ { \\overline { \\mathbf { a } } _ { p } } ( \\mathbf { Z } ; P ) } \\lbrace Y - \\chi _ { \\mathbf { a } } ( P ; \\mathcal { G } ) \\rbrace - \\sum \\limits _ { k = 0 } ^ { p } g _ { k } ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { Z } } _ { k } ; P ) , \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\beta ( x _ 1 , \\ldots x _ n ) \\wp ^ { ( k ) } ( x _ i - x _ { n + 1 } ) = - \\tfrac { 1 } { k } \\partial _ { x _ { n + 1 } } \\left ( \\beta ( x _ 1 , \\ldots x _ n ) \\wp ^ { ( k - 1 ) } ( x _ i - x _ { n + 1 } ) \\right ) , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\eta _ k \\times ( \\delta _ i , 2 b _ i ) ) = \\prod _ { j = 0 } ^ { 2 b _ i - 1 } L ^ S ( s + b _ i - j , \\eta _ k \\times \\delta _ i ) . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} F ( t ) = t ^ { e _ { 0 } } F _ { 1 } ( t ) ^ { e _ { 1 } } \\cdots F _ { r } ( t ) ^ { e _ { r } } \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( B _ 3 ^ { ( n ) } ) = - H ( \\ell _ 2 \\varepsilon ) . \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\alpha ^ { - 1 } ( S ( x _ 3 ) , x _ 4 ) \\alpha ( S ( x _ 2 ) , x _ 5 ) S ( x _ 1 ) x _ 6 = S ( x _ 1 ) x _ 2 = \\epsilon ( x ) \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { 2 k , 2 a } ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ { 2 a } , q ^ { 4 k - 2 a } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\inf _ { \\textbf { v } \\in \\partial \\mathcal { B } ( x ) , t \\ge 0 } \\ ! \\left \\| \\textbf { v } \\ ! - \\ ! \\tilde { \\textbf { x } } ^ 0 ( t ) \\right \\| _ 2 \\ ! = \\ ! \\inf _ { \\textbf { v } \\in \\partial \\mathcal { B } ( x ) } \\left \\| \\textbf { v } - \\hat { \\textbf { x } } _ { } \\right \\| _ 2 > 0 . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s Q + Q - | Q | ^ { \\frac { 4 s } { d } } Q = 0 . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} r = \\varphi ( Y ) \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} j = \\bar N - 1 - i , i = \\bar N - 1 - j , i = 0 , \\dots , \\bar N - 1 , j = \\bar N - 1 , \\dots , 0 , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} s ( e _ { x _ 1 } e _ { x _ 2 } \\cdots e _ { x _ { n - 1 } } e _ { x _ n } ) & = ( - 1 ) ^ { n \\choose 2 } ( - e _ { x _ n } x _ n ^ { - 1 } ) ( - e _ { x _ { n - 1 } } x _ { n - 1 } ^ { - 1 } ) \\cdots ( - e _ { x _ 2 } x _ 2 ^ { - 1 } ) ( - e _ { x _ 1 } x _ 1 ^ { - 1 } ) \\\\ & = ( - 1 ) ^ { \\frac { n ( n + 1 ) } { 2 } } \\ , e _ { x _ n } \\ , e _ { x _ { n - 1 } ^ { x _ n } } \\cdots e _ { x _ 2 ^ { x _ 3 \\cdots x _ n } } \\ , e _ { x _ 1 ^ { x _ 2 \\cdots x _ n } } \\ , x _ n ^ { - 1 } \\cdots x _ 1 ^ { - 1 } \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\alpha ( y ' ) : = 1 + | \\nabla _ { y ^ \\prime } \\phi ( y ' ) | ^ 2 , \\beta _ j ( y ' ) : = - 2 \\partial _ { y _ j } \\phi ( y ^ \\prime ) , \\gamma ( y ' ) : = - \\Delta _ { y ' } \\phi ( y ^ \\prime ) , \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} f '' _ { \\varepsilon } ( s ) = & \\ \\int _ { - \\infty } ^ \\infty \\widetilde { f } '' ( s - \\varepsilon \\eta _ { \\delta } ( s ) t ) \\ , ( 1 - \\varepsilon \\eta _ \\delta ' ( s ) t ) ^ 2 \\phi ( t ) \\ , d t \\\\ & \\ - \\varepsilon \\int _ { - \\infty } ^ \\infty \\widetilde { f } ' ( s - \\varepsilon \\eta _ { \\delta } ( s ) t ) \\ , \\eta _ \\delta '' ( s ) t \\phi ( t ) \\ , d t . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} W ( T ) = \\sum _ { 1 \\leq m \\leq n } \\left [ W _ { n , m } \\right ] \\L ^ { - n d } T ^ n , \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} d \\left ( x _ 1 \\wedge \\ldots \\wedge x _ n \\right ) = \\sum _ { i < j } ( - 1 ) ^ { i + j } [ x _ i , x _ j ] \\wedge x _ 1 \\wedge \\ldots \\wedge \\widehat { x } _ i \\wedge \\ldots \\wedge \\widehat { x } _ j \\wedge \\ldots \\wedge x _ n . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\xi _ j ^ { ( i ) } \\buildrel \\Delta \\over = \\left \\{ \\begin{array} { l } \\hat \\psi _ j ^ { ( i - M ) } - \\hat \\psi _ j ^ { ( i - M - 1 ) } , \\ ; i \\ge M + 1 j \\in { \\cal A } , \\\\ 0 , , \\end{array} \\right . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} V ( \\delta _ { - } ) = V ( \\delta _ { + } ) = \\frac { h ^ { 2 } } { ( \\delta _ { + } - \\delta _ { - } ) ^ { 2 } } . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} f ( P ) A g ( P ) A & = \\sum _ { i = 1 } ^ n f ( \\lambda _ i ) E _ i A \\sum _ { j = 1 } ^ n g ( \\lambda _ j ) E _ j A = \\sum _ { i = 1 } ^ n f ( \\lambda _ i ) g ( \\lambda _ i ) E _ i A E _ i A \\\\ & + \\sum _ { i < j } \\left ( f ( \\lambda _ i ) g ( \\lambda _ j ) E _ i A E _ j A + f ( \\lambda _ j ) g ( \\lambda _ i ) E _ j A E _ i A \\right ) . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} a , b \\in A , \\ a b = b a = 0 \\ \\Rightarrow \\ P ( a + b ) = P ( a ) + P ( b ) . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\lambda ( \\theta \\alpha v ) - \\Delta ( \\theta \\alpha v ) & = - \\theta \\alpha \\nabla _ H \\pi - 2 \\nabla ( \\theta \\alpha ) \\cdot \\nabla v - ( \\Delta ( \\theta \\alpha ) ) v \\quad \\Omega ' , \\\\ \\partial _ z ( \\theta \\alpha v ) | _ { \\Gamma _ u ' \\cup \\Gamma _ b ' } & = 0 , \\theta \\alpha v \\Gamma _ l ' . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} h _ { n + 1 } ( . , t , x ) = \\widetilde { p _ D ( t - s , x , y ) h _ n ( . , s , y ) } . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\Delta _ z v : = \\partial _ z ^ 2 v , D ( \\Delta _ z ) = \\{ f \\in W ^ { 2 , p } ( - h , 0 ) : f ( - h ) = \\partial _ z f ( 0 ) = 0 \\} . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\varphi _ n ^ { ( k ) } ( z ) : = \\ _ 3 \\phi _ 2 \\left ( \\begin{matrix} q ^ { - n } , q ^ { \\gamma + k } , q ^ { - z } \\\\ b q , q ^ \\gamma \\end{matrix} ; q , - { q ^ { n + 1 } \\over c } \\right ) \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} { \\rm e v } ^ \\ast ( x ) = s _ 1 \\otimes \\sigma ^ \\ast ( x ) . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} V ( k ) = \\sum _ { m = 1 } ^ { n - 1 } ( U _ { k , n - 1 } ( m - a ) - U _ { k , n - 1 } ( m - 1 ) ) , \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} x ' ( t ) & = s - d x ( t ) - x ( t ) f ( v ( t ) ) , \\\\ y ' ( t ) & = x ( t ) f ( v ( t ) ) - a y ( t ) - p y ( t ) z ( t ) , \\\\ v ' ( t ) & = k y ( t ) - u v ( t ) , \\\\ z ' ( t ) & = c y ( t ) z ( t ) - b z ( t ) , \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\mathbf P _ { s o } ^ { \\ , i } = \\mathbb P \\left ( C _ s ^ { \\ , i } < r _ s \\right ) , \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} N ( p , q ) : = \\{ \\nu ^ { p ( \\beta + \\gamma ) } _ + , \\nu ^ { p ( \\beta + \\gamma ) } _ - & : \\beta \\in C _ \\alpha ' ( p ) , \\gamma \\in C _ \\alpha ' ( q ) , \\\\ & \\gamma \\neq \\beta , \\beta ^ c , \\alpha + \\beta , \\alpha + \\beta ^ c \\} . \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} h _ \\alpha ( B _ \\alpha \\phi ) ( x ) = | x | ^ 2 h _ \\alpha ( \\phi ) ( x ) , \\ , \\ , \\ , x \\in \\mathbb { R } ^ d _ + . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\overline { x } ^ { \\alpha } = \\overline { x } ^ { \\alpha } ( x ^ { 1 } , \\dots , x ^ { r } ) , \\ ; \\ ; \\ ; \\ ; \\overline { x } ^ { a } = \\overline { x } ^ { a } ( x ^ { r + 1 } , \\dots , x ^ { m } ) . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\overline { C } _ { \\mathbb { Q } } = \\{ x \\in \\mathfrak { A } _ { \\mathbb { Q } } : \\langle x , \\alpha \\rangle \\geq 0 , \\alpha \\in \\Delta \\} . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\frac { \\partial q _ a } { \\partial y _ 0 } = \\dots = \\frac { \\partial q _ a } { \\partial y _ 5 } = 0 , \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} \\theta ( g ) = \\psi _ { \\beta , \\rho } ( g ) & = \\tau \\left ( \\varpi ^ { - l } B ( X + 2 ^ { - 1 } \\varpi ^ { l - 1 } X ^ 2 , \\beta ) - 2 ^ { - 1 } \\varpi ^ { - 1 } B ( X , \\beta ) \\right ) \\cdot \\rho ( \\overline X ) \\\\ & = \\tau \\left ( \\varpi ^ { - l } B ( X , \\beta ) \\right ) \\cdot \\rho ( \\overline X ) . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} X ^ { ( j + 1 , H + h ) } = X _ A + \\sum _ { i = 0 } ^ { \\infty } ( X _ { B _ i } + X _ { C _ i } ) \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} f _ d ( q ) = \\frac { f _ 0 ( q ) } { \\prod _ { r = 1 } ^ d ( 1 - q ^ r ) ^ 3 } = \\frac { f _ 0 ( q ) } { ( q ; q ) _ d ^ 3 } \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\Lambda : = \\sup _ { i \\in \\mathbb { N } } \\int _ M ( 2 - p _ i ) | d u _ i | ^ { p _ i } d v _ g < \\infty . \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\left | \\tau _ s ( g f ) \\right | = \\left | \\tau _ s \\left ( g \\prod _ { u = 1 } ^ k f _ u \\right ) \\right | \\leq \\| g \\| _ { Y ( \\{ X _ { j } ^ s : j \\in \\mathcal J \\} ) } \\prod _ { u = 1 } ^ k \\| f _ u \\| _ { X _ { j _ u } ^ s } , \\| f _ u \\| _ { X _ { j _ u } ^ s } = 1 , u = 1 , \\ldots , k , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\alpha _ { f \\times g } ( x , y ) = \\max \\{ \\alpha _ { f } ( x ) , \\alpha _ { g } ( y ) \\} . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} J ( t ) = J _ h ( t ) = \\begin{cases} \\exp ( - 1 / t ^ h ) & t > 0 \\ , , \\\\ 0 & t \\leq 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} D h _ { { K } } ( u ) = D h _ { { K } } ( e + y ) = D v ( y ) + \\left ( \\langle D v ( y ) , y \\rangle - v ( y ) \\right ) \\cdot e . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} P ( S _ n ' ( c n ) \\leq \\lfloor n - p _ n ^ { - 1 } \\rfloor ) & = P ( n - S _ n ' ( c n ) \\geq n - \\lfloor n - p _ n ^ { - 1 } \\rfloor ) \\\\ & = P ( \\mathrm { B i n } ( n , 1 - \\pi _ n ( c n ) ) \\geq n - \\lfloor n - p _ n ^ { - 1 } \\rfloor ) \\\\ & = P ( \\mathrm { B i n } ( n , 1 - \\pi _ n ( c n ) ) \\geq \\lceil p _ n ^ { - 1 } \\rceil ) \\\\ & \\leq \\mathrm { e } ^ { - \\frac { \\lceil p _ n ^ { - 1 } \\rceil } { 2 } \\log \\left ( \\frac { \\lceil p _ n ^ { - 1 } \\rceil } { n ( 1 - \\pi _ n ( c n ) ) } \\right ) } . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} 0 = ( \\mathbb { L } - \\phi ) ( \\bar { A } _ t ; Z _ t , W _ t ) = \\inf _ { A _ s \\in \\mathbb { C } ^ { \\kappa , \\mu , \\mu _ 0 } } ( \\mathbb { L } - \\phi ) ( A _ s ; Z _ t , W _ t ) , \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\nabla ^ { \\frac { H } { 2 } } \\left ( \\int _ 0 ^ T g _ s \\delta R _ s ^ H \\right ) ( x ) & = \\int _ 0 ^ T ( \\nabla ^ \\frac { H } { 2 } g _ s ) ( x ) \\delta R _ s ^ H + 2 c _ { H } ^ { B , R } \\frac { \\mathrm { B } \\left ( \\frac { H } { 2 } , 1 - H \\right ) } { \\Gamma \\left ( \\frac { H } { 2 } \\right ) ^ 2 } \\int _ 0 ^ T g _ s | s - x | ^ { H - 1 } \\delta B _ s ^ { \\frac { H } { 2 } + \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} D F ( t , p ^ { \\pm } ) = \\left ( \\begin{array} { c c c c } f _ { 1 , u _ 1 } ( t , p ^ { \\pm } ) & f _ { 1 , u _ 2 } ( t , p ^ { \\pm } ) & \\cdots & f _ { 1 , u _ m } ( t , p ^ { \\pm } ) \\\\ f _ { 2 , u _ 1 } ( t , p ^ { \\pm } ) & f _ { 2 , u _ 2 } ( t , p ^ { \\pm } ) & \\cdots & f _ { 2 , u _ m } ( t , p ^ { \\pm } ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ f _ { m , u _ 1 } ( t , p ^ { \\pm } ) & f _ { m , u _ 2 } ( t , p ^ { \\pm } ) & \\cdots & f _ { m , u _ m } ( t , p ^ { \\pm } ) . \\end{array} \\right ) . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\chi _ { V _ v , \\sigma _ v } ( a ) = \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau _ v [ a ] \\otimes \\tau ' _ v ) \\cdot \\xi _ { \\tau _ v [ a ] } ( - 1 ) ^ { N - 1 } \\cdot \\xi _ { \\tau _ v ' } ( - 1 ) ^ { N [ a ] } ; \\\\ \\chi _ { V ' _ v , \\sigma ' _ v } ( a ' ) = \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau _ v \\otimes \\tau _ v ' [ a ' ] ) \\cdot \\xi _ { \\tau _ v } ( - 1 ) ^ { N [ a ' ] } \\cdot \\xi _ { \\tau _ v ' [ a ' ] } ( - 1 ) ^ { N } . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\frac { 1 } { ( k + j ) ! } \\sum _ { i = 0 } ^ k ( - 1 ) ^ i { k \\choose i } \\sum _ { h = 0 } ^ j ( - 1 ) ^ { j + h } { j \\choose h } M ( t + h - j - ( i + h ) a ) ^ { k + j } . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } r _ 2 \\left ( n , q \\right ) t ^ n \\right ) ^ 2 = 1 + \\sum _ { n = 1 } ^ { \\infty } r _ 4 \\left ( n , q \\right ) t ^ n \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\ ! x ^ { \\lambda } K _ { \\mu } ( a x ) \\cos ( b x ) d x & = { 2 ^ { \\lambda - 1 } a ^ { - \\lambda - 1 } \\Gamma \\Big ( \\frac { \\mu + \\lambda + 1 } { 2 } \\Big ) \\Gamma \\Big ( \\frac { 1 + \\lambda - \\mu } { 2 } \\Big ) } \\ , \\\\ & \\ ; \\ ; \\ ; \\ ; \\times { } _ { 2 } F _ { 1 } \\Big ( \\frac { \\mu + \\lambda + 1 } { 2 } , \\frac { 1 + \\lambda - \\mu } { 2 } ; \\ ! \\frac { 1 } { 2 } ; - \\frac { b ^ 2 } { a ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\lim _ { m \\to - \\infty } k ^ 2 \\ , m & = - \\ ( \\frac { H _ o ^ 2 } { 4 } - \\delta \\ ) \\frac { r _ o ^ 3 } { 2 } , \\\\ [ 0 . 5 e x ] \\lim _ { m \\to - \\infty } - \\frac { 2 m } { b r _ o ^ 3 } & = \\frac { 1 } { \\delta } \\ ( \\frac { H _ o ^ 2 } { 4 } - \\delta \\ ) , \\\\ [ 1 . 2 5 e x ] \\lim _ { m \\to - \\infty } b & = + \\infty . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} e _ { q , D } : = P ^ { - 1 } ( e _ { q , \\lambda _ i } ( Q ) ) P \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} \\P ( C ' _ { j , k } \\cap A ' ) & = \\P ( C _ { j , k } \\cap A ) , \\P ( C ' _ { j , k } \\cap ( A ' ) ^ c ) = \\P ( C _ { j , k } \\cap A ^ c ) , \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} f = \\prod _ { u = 1 } ^ k f _ u , \\| f _ u \\| _ { X _ { j _ u } ^ S } = 1 . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\det D \\pi ( y ) = ( 1 + \\| y \\| ^ 2 ) ^ { \\frac { - n } 2 } . \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t } = \\sum _ { \\alpha = 1 } ^ { d } V _ { \\alpha } ( X _ { t } ) d B _ { t } , & 0 \\leq t \\leq 1 , \\\\ X _ { 0 } = x , \\end{cases} \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} & ( \\rho s ) _ t + ( \\rho u s ) _ x - \\left ( \\frac { \\kappa \\theta _ x } { \\theta } \\right ) _ x = \\frac { \\kappa \\theta _ x ^ 2 } { \\theta ^ 2 } + \\frac { \\varepsilon u _ x ^ 2 } { \\theta } + \\frac { \\mu | \\mathbf { w } _ x | ^ 2 } { \\theta } + \\frac { \\nu | \\mathbf { h } | _ x ^ 2 } { \\theta } \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\mathbf { B } ( G ) = \\coprod _ { S \\subset \\Delta } \\mathbf { B } ( G ) _ { M _ S } \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - h _ n x _ n ^ 2 ) + \\rho _ { n + 1 } \\xi _ { n + 1 } , n \\in \\mathbb N , x _ 0 \\in \\mathbb R . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F D F } } ^ { \\textrm { N O M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { F } } \\right ) P _ \\textrm { F } } { \\left ( 1 - \\beta _ { \\textrm { F } } \\right ) P _ \\textrm { N } + { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } / { | h _ { \\textrm { B F } } | ^ 2 } } . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mu _ { 2 } ^ { \\prime } \\mu _ { 4 } - \\mu _ { 2 } \\mu _ { 4 } ^ { \\prime } = 0 , \\\\ \\mu _ { 2 } ^ { \\prime } \\mu _ { 3 } - \\mu _ { 2 } \\mu _ { 3 } ^ { \\prime } + \\mu _ { 1 } ^ { \\prime } \\mu _ { 4 } - \\mu _ { 1 } \\mu _ { 4 } ^ { \\prime } = 0 , \\\\ \\mu _ { 1 } ^ { \\prime } \\mu _ { 3 } - \\mu _ { 1 } \\mu _ { 3 } ^ { \\prime } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\liminf _ { p \\to 2 } ( 2 - p ) c _ p ( S ^ n ) = \\liminf _ { p \\to 2 } ( 2 - p ) E _ p ( u _ p ) > 0 . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} T M ^ k = N ^ k T . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\P [ A ] & \\leq \\frac { 4 } { b 2 ^ { j - 1 } \\mu } = O \\bigl ( 2 ^ { - j } \\bigr ) , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\left ( P _ q \\mathcal { X } ^ { K \\textnormal { t h , e q } } \\left ( q , \\varphi _ { q , z } ^ { - 1 } ( Q ) \\right ) \\right ) _ { 0 i } = Q ^ \\frac { \\lambda _ i } { z } \\sum _ { d \\geq 0 } Q ^ d \\prod _ { r = 1 } ^ d \\prod _ { j = 0 } ^ N \\frac { 1 } { ( \\lambda _ i - \\lambda _ j + r z ) } \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} B _ \\rho ( z , r ) ~ = ~ \\left \\{ y \\in E ~ | ~ \\rho ( z , y ) < r \\right \\} ; \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\det A _ n ( { \\textstyle \\frac 1 2 } ) = 2 ^ { - n } \\ , n ^ { ( n + 1 ) / 2 } . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} \\varphi ' ( \\Lambda _ 0 \\Lambda _ 1 \\dots \\Lambda _ n ) 1 = \\Lambda _ 0 \\Lambda _ 1 \\dots \\Lambda _ { n + 1 } \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\Phi _ { t } ( x ; B ) \\stackrel { { \\rm l a w } } { = } \\Phi _ { 1 } ( x ; \\varepsilon B ) , \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} K _ { m + 1 } ( t ) & \\le C _ 1 \\biggl ( \\lVert a _ 0 \\rVert _ { L ^ \\infty _ H L ^ p _ z } + K _ m ( t ) H _ m ( t ) + K _ m ( t ) ^ 2 + R t ^ { 1 / 2 } K _ m ( t ) \\biggr ) , \\\\ H _ { m + 1 } ( t ) & \\le C _ 1 \\biggl ( \\lVert a _ 0 \\rVert _ { L ^ \\infty _ H L ^ p _ z } + K _ m ( t ) H _ m ( t ) + R t ^ { 1 / 2 } H _ m ( t ) + R t ^ { 1 / 2 } K _ m ( t ) \\biggr ) . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } > \\varepsilon \\right ) = - H ( \\ell _ 2 \\varepsilon ) \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} D F ( x , p ^ { \\pm } ) = \\left ( \\begin{array} { c c c c } f _ { 1 , u _ 1 } ( x , p ^ { \\pm } ) & f _ { 1 , u _ 2 } ( x , p ^ { \\pm } ) & \\cdots & f _ { 1 , u _ m } ( x , p ^ { \\pm } ) \\\\ f _ { 2 , u _ 1 } ( x , p ^ { \\pm } ) & f _ { 2 , u _ 2 } ( x , p ^ { \\pm } ) & \\cdots & f _ { 2 , u _ m } ( x , p ^ { \\pm } ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ f _ { m , u _ 1 } ( x , p ^ { \\pm } ) & f _ { m , u _ 2 } ( x , p ^ { \\pm } ) & \\cdots & f _ { m , u _ m } ( x , p ^ { \\pm } ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} F _ { t o t } ( n + 1 ) : = \\bigcap _ { j = 1 } ^ { n + 1 } \\bigcap _ { i = 1 } ^ { N _ W } F _ j ( i ) . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} & g _ { i _ 0 } \\ , e _ { i _ 0 i _ 1 } \\ , g _ { i _ 1 } \\ , e _ { i _ 1 i _ 2 } \\ , g _ { i _ 2 } \\ , \\ldots \\ , g _ { i _ { k - 1 } } \\ , e _ { i _ { k - 1 } i _ k } \\ , g _ { i _ k } \\\\ & = e _ { i _ 0 i _ 1 } \\ , e _ { i _ 1 i _ 2 } \\ , \\ldots \\ , e _ { i _ { k - 1 } i _ k } \\\\ & = e _ { i j } . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} & \\frac { 1 } { m + 1 } ( B _ { m + 1 } ( a + 1 ) - B _ { m + 1 } ( a ) ) \\\\ = & \\frac { 1 } { m + 1 } \\left ( \\Delta _ { m + 1 } \\left ( \\frac { x e ^ { ( a + 1 ) x } } { e ^ x - 1 } \\right ) - \\Delta _ { m + 1 } \\left ( \\frac { x e ^ { a x } } { e ^ x - 1 } \\right ) \\right ) \\\\ = & \\frac { \\Delta _ { m + 1 } ( x e ^ { a x } ) } { m + 1 } \\\\ = & \\ , a ^ m , \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\lambda u - \\Delta u = G \\R ^ 3 , G : = \\chi _ r E ( \\partial _ z f ) - 2 ( \\nabla \\chi _ r ) \\cdot E ( \\nabla v ) - ( \\Delta \\chi _ r ) E v . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} & x _ 0 = d ( y _ 0 ) + a y _ 1 - b y _ 0 \\ \\ \\ \\Rightarrow \\ \\ \\ 0 = \\imath _ V ( x _ 0 ) = L ( y _ 0 ) + \\imath _ V ( a ) y _ 1 - \\imath _ V ( b ) y _ 0 \\ \\ \\ \\Rightarrow \\ \\ \\ y _ 1 = L ( y _ 0 ) . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} L = L ( 1 - \\hat h _ N L ) , \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} Y : = \\{ [ 0 ; \\gamma ] : \\gamma \\in \\{ 1 , 2 \\} ^ { \\mathbb { N } } \\textrm { n o t c o n t a i n i n g t h e s u b w o r d s i n } P \\} \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} u ( t ) & = V _ 0 ( t ) + u ( t ) - V _ 0 ( t ) \\\\ & = V _ 0 ( t ) + U _ 1 ' ( t ) \\\\ & = V _ 0 ( t ) + V _ 1 ( t ) ' + ( V _ 1 ( t ) - U _ 1 ( t ) ) ' \\\\ & = V _ 0 ( t ) + V _ 1 ' ( t ) + U _ 2 '' ( t ) \\\\ & \\ , \\ , \\ , \\vdots \\\\ & = \\sum _ { \\ell = 0 } ^ m \\frac { d ^ \\ell } { d t ^ \\ell } V _ { \\ell } ( t ) + \\frac { d ^ { m + 1 } } { d t ^ { m + 1 } } U _ { m + 1 } ( t ) . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\tilde V _ i ( K ) = \\frac { \\kappa _ { n } } { \\kappa _ i } \\int _ { G ( n , i ) } V _ i ( K \\cap H ) \\ , d H , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} & U _ { 2 k , 2 a } ( x ; q ) - U _ { 2 k , 2 a - 2 } ( x ; q ) \\\\ & = ( 1 + x q ) \\left [ ( x q ) ^ { 2 a } \\sum _ { h = 1 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a } \\sum _ { h = 0 } ^ { k - a - 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ] \\\\ & + ( 1 + x q ) \\left [ ( x q ) ^ { 2 a - 2 } \\sum _ { h = 1 } ^ { k - a + 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a - 2 } \\sum _ { h = 0 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ] . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} ( \\bar \\lambda _ 1 \\bar \\lambda _ 2 - \\bar \\lambda ^ 2 ) \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) = 0 . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ { y _ 0 } ( x ) \\leq \\frac { ( R + R _ 0 ) ^ 2 \\widetilde { M } _ { R } } { 2 } = \\frac { ( R + R _ 0 ) ^ 2 ( M _ R + 2 a ) } { 2 } \\textrm { f o r e v e r y } x \\in B ( 0 , R ) . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\mathbf { L } ^ { p , q } ( \\Omega ) = \\{ f \\ , \\mbox { i s a m e a s u r a b l e f u n c t i o n o n $ \\Omega $ } : \\ , \\| f \\| _ { \\mathbf { L } ^ { p , q } ( X ) } < \\infty \\} \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} p ( \\rho , \\theta ) = p _ e ( \\rho ) + \\theta p _ \\theta ( \\rho ) , \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } _ \\varphi } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\phi ) + \\ell _ { \\beta } ( \\phi ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = 0 , \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} \\bigg ( \\frac { \\partial z } { \\partial t } \\bigg ) ^ 2 - \\bigg ( \\frac { \\partial z } { \\partial x } \\bigg ) ^ 2 = \\frac { 4 ( 1 + b ) \\ , z ( 1 - z ) } { ( 1 + t ) \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] } . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\lim _ { | s | \\to \\infty } \\frac { f ( s , t ) } { f _ \\infty ( s , t ) } = 1 \\mbox { w i t h } f _ \\infty ( s , t ) : = | \\theta ' ( s ) | \\ , | t | \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} L ( \\boldsymbol r , \\boldsymbol { \\dot r } ) = \\frac { m } { 2 } | \\boldsymbol { \\dot r } | ^ 2 - V ( \\boldsymbol r ) . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} v _ y ( z ) = \\frac { z + ( 1 - | y | ) ^ { - 1 } y } { | z + ( 1 - | y | ) ^ { - 1 } y | } , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\nabla _ { X } s = 0 , \\forall X \\in \\overline { P } \\ , . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} f _ q ( Q ) = \\sum _ { d \\geq 0 } \\frac { 1 } { \\prod _ { r = 1 } ^ d \\left ( 1 - q ^ r \\right ) ^ { N + 1 } } Q ^ d \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} \\phi _ { \\omega + 2 } : = \\phi _ { \\omega + 1 } - y ^ { \\frac { 1 } { p } } = \\phi _ \\omega - 1 - y ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} f _ { i } ^ { J _ { 0 } + 1 } = \\alpha _ { i } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } + ( - 1 ) ^ { i } \\alpha ^ { J _ { 0 } } \\chi _ { B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } + 1 } r ) } , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { \\pi _ { F } ( x ) } \\sum _ { N ( \\mathfrak { p } ) \\le x } N _ { \\mathfrak { p } } ^ k ( Y ) = \\sum _ { m = 1 } ^ { | Y | } m ^ k \\frac { | G ( m ) | } { | G | } . \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} C _ { \\rm b d } = \\sup \\{ 2 { C _ { \\rm u n f } } , \\sqrt { \\frac { 1 } { 1 6 \\sqrt { \\lambda _ 0 } } } \\} , \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\int _ { 0 } ^ { L } \\mu _ 1 \\left ( \\frac { \\left ( z + t w \\right ) ^ { + } - z ^ { + } } { t } \\right ) \\left ( \\alpha - z \\right ) \\varphi = \\int _ { 0 } ^ { L } \\mu _ { 1 } w \\mathbf { 1 } _ { z > 0 } \\left ( \\alpha - z \\right ) \\varphi . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} U _ { l _ 0 } ( x , 0 ) & = U _ { l _ 0 } ( x _ 1 + c \\tau + \\sigma ^ * , x ' + \\rho , \\tau ) \\\\ & = \\cdots = U _ { l _ 0 } ( x _ 1 + c n \\tau + n \\sigma ^ * , x ' + n \\rho , n \\tau ) \\overset { n \\rightarrow \\infty } { \\longrightarrow } p ^ - _ { l _ 0 } \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\alpha _ j a _ j ^ n = \\sum _ { j = 1 } ^ m \\left ( \\beta _ j a _ j \\right ) ^ n , \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} B ( z ) = \\frac 1 { 2 \\pi i } \\int _ { \\Re ( s ) = \\frac 1 2 } X _ f ( s ) \\Lambda _ f ( s ) H _ f ( s , x / y ) y ^ { \\frac 1 2 - s } \\ , d s - \\sum _ \\rho \\Lambda _ f ' ( \\rho ) H _ f ( \\rho , x / y ) y ^ { \\frac 1 2 - \\rho } , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} R i c _ { \\widetilde { g } } - \\frac { r } { h } H e s s _ { \\widetilde { g } } h = \\rho \\widetilde { g } , \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} v = ( \\lambda - \\Delta _ p ) ^ { - 1 } ( f + B v ) , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil f _ 5 ( n ) \\varepsilon \\rceil ) = - \\gamma ^ { - 1 } \\lceil \\ell _ 5 \\varepsilon \\rceil . \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} u ( 0 , x ) = \\bar u ( x ) . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} p ( t , x , y ) = \\mathbb { E } \\left [ \\delta _ { y } \\left ( \\Phi _ { 1 } ( x ; \\varepsilon B ) \\right ) \\right ] & \\ge C \\varepsilon ^ { - N } \\cdot \\mathbb { E } \\left [ \\delta _ { 0 } \\left ( \\frac { \\Phi _ { 1 } ( x ; \\varepsilon B + h ) - \\Phi _ { 1 } ( x ; h ) } { \\varepsilon } \\right ) { \\rm e } ^ { - I \\left ( \\frac { h } { \\varepsilon } \\right ) } \\right ] . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\mathcal { U } _ i ^ F ( \\boldsymbol { d } , p , B ) = \\mathcal { S } _ i ( p , d _ i ) + \\mathcal { B } _ i ( B , \\boldsymbol { w } , \\boldsymbol { d } ) - \\mathcal { C } _ i ( d _ i ) , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\sum _ { N _ 1 \\geq N _ 2 \\geq \\cdots \\geq N _ { k - 1 } \\geq 0 } \\frac { q ^ { N _ 1 ^ 2 + N _ 2 ^ 2 + \\cdots + N _ { k - 1 } ^ 2 + N _ { a + 1 } + \\cdots + N _ { k - 1 } } } { ( q ) _ { N _ 1 - N _ 2 } \\cdots ( q ) _ { N _ { k - 2 } - N _ { k - 1 } } ( q ) _ { N _ { k - 1 } } } = \\frac { ( q ^ a , q ^ { 2 k + 1 - a } , q ^ { 2 k + 1 } ; q ^ { 2 k + 1 } ) _ \\infty } { ( q ) _ \\infty } . \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} a = \\sum _ { g \\in G } g = \\sum _ { g \\in G _ 2 } g . \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\langle \\nu _ K ( z ) , \\tilde { \\pi } _ K ( z ) \\rangle \\| z \\| ^ { - ( n - 1 ) } = \\langle \\nu _ K ( z ) , z \\rangle \\| z \\| ^ { - n } . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} & \\quad \\ \\| \\mathcal { V } [ v + \\eta ] - \\mathcal { V } [ v ] - \\mathcal { V } ^ { B D } [ v ; \\eta ] \\| _ { L ^ r ( 0 , T ) } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\le \\rho _ { v , p , r } ( \\| \\eta \\| _ { \\infty } ) ( \\| \\eta ' \\| _ { L ^ p ( 0 , T ) } + | \\eta ( 0 ) | ) . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ { i + m - 1 } H ( \\mu , \\mathcal { D } _ { i + 1 } | \\mathcal { D } _ { i } ) & = \\int H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) \\\\ & = \\int _ { x : \\ , H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) \\leq \\varepsilon } H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) + \\int _ { x : \\ , H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) > \\varepsilon } H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} & \\big | \\big \\{ x \\in B ( x _ 0 , r ) : | b ( x ) - b _ { B ( x _ 0 , r ) } | > \\lambda \\big \\} \\big | \\\\ & \\leq C _ 1 | B ( x _ 0 , r ) | \\exp \\bigg [ - \\bigg ( 1 + \\frac { r } { \\rho ( x _ 0 ) } \\bigg ) ^ { - \\theta ^ { \\ast } } \\frac { C _ 2 \\lambda } { \\| b \\| _ { \\mathrm { B M O } _ { \\rho , \\theta } } } \\bigg ] , \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ { 1 j } = ( X _ j + X _ j ^ \\ast ) / 2 , \\\\ X _ { 2 j } = ( X _ j - X _ j ^ \\ast ) / { ( 2 \\mathbf { i } ) } , \\end{array} \\right . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\partial _ z v = 0 \\Gamma _ u \\times ( 0 , \\infty ) , \\pi , v \\Gamma _ l \\times ( 0 , \\infty ) , v = 0 \\Gamma _ b \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\left ( \\frac { 2 H _ { 0 } } { f _ { 2 } } \\right ) \\frac { d } { d f _ { 1 } } \\left ( \\frac { 1 } { p _ { 1 } } \\right ) = \\left ( 1 + a ^ { 2 } \\right ) \\frac { d } { d f _ { 1 } } \\left ( \\frac { f _ { 1 } } { p _ { 1 } } \\right ) \\left ( \\frac { p _ { 2 } \\dot { p } _ { 2 } } { f _ { 2 } } \\right ) + \\ddot { p } _ { 1 } . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\beta _ \\varepsilon : = \\left [ \\tilde { \\gamma } _ \\varepsilon \\left ( u _ \\varepsilon ( z _ \\varepsilon ) - \\tilde { \\gamma } _ \\varepsilon \\right ) \\right ] \\to \\beta _ 0 \\in ( 0 , + \\infty ) \\ , , \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) = x q ( - x q ^ 3 ; q ^ 2 ) _ \\infty [ \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) - \\overline { Q } _ { k , a - 1 } ( x ^ 2 ; q ^ 2 ) ] . \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} w ( x ) = A W _ { \\rho , \\nu } \\left ( \\sqrt { A M } x \\right ) \\textup { a n d } \\rho = \\sqrt { A M } R . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\left [ \\left ( H + z Q \\partial _ Q \\right ) ^ { N + 1 } - Q \\right ] J ^ \\textnormal { c o h } ( z , Q ) = 0 \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\lambda ^ \\star _ n = \\frac { 1 } { 2 \\theta _ n } . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} n & \\geq \\dim \\biggl ( ( X _ 0 + X _ 1 ) + \\sum _ { i \\in I } X _ { i } \\biggr ) \\\\ & = \\dim ( X _ 0 + X _ 1 ) + \\sum _ { i \\in I } \\dim ( X _ { i } ) \\\\ & = \\dim ( X _ 0 ) + \\dim ( X _ 1 ) - \\dim ( X _ 0 \\cap X _ 1 ) + \\sum _ { i \\in I } \\dim ( X _ { i } ) \\\\ & = ( k + 1 ) d - \\dim ( X _ 0 \\cap X _ 1 ) . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} & - t ^ { \\frac { N + 2 A _ 1 } { 2 } } ( \\partial _ r ^ 2 u _ { 1 , 1 } ) ( x , t ) \\ge - C , \\\\ & - t ^ { \\frac { N + 2 A _ 1 } { 2 } } ( \\partial _ { \\theta _ \\alpha } \\partial _ { \\theta _ \\beta } u _ { 1 , 1 } ) ( x , t ) \\ge \\frac { q _ N M _ { 1 , 1 } } { 2 } U _ 1 ( r _ * ) \\delta _ { \\alpha \\beta } - \\epsilon , \\\\ & - t ^ { \\frac { N + 2 A _ 1 } { 2 } } ( \\partial _ r \\partial _ { \\theta _ j } u _ { 1 , 1 } ) ( x , t ) \\ge - C | \\theta _ \\alpha | U _ 1 ' ( r ) + O ( t ^ { - 1 } ) \\ge - \\epsilon , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} q _ { 4 n } = q _ { 4 n + 3 } = 0 , q _ { 4 n + 1 } = q _ { 4 n + 2 } = q _ { n + 1 } . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} \\varphi ( E _ { i j } ) \\gamma ( v _ t ) = a _ { i j } \\gamma ( E _ { i j } v _ t ) i \\rho j t . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\tau ( s _ 1 ) = t _ 2 , \\ \\ \\ \\ \\tau ( x _ 2 ) = \\mu _ 3 , \\ \\ \\ \\ \\tau ( \\mu _ 3 ) = q _ 4 . \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} 2 \\bigtriangledown ^ { ( \\textnormal { L C } ) } = \\varphi ^ * \\nabla _ { | q = - 1 } \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} [ L : K ] = e ( w / v ) \\ , f ( w / v ) & \\Longleftrightarrow [ L ^ { h } : K ^ { h } ] = e ( w ^ { h } / v ^ { h } ) \\ , f ( w ^ { h } / v ^ { h } ) . \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\psi _ n ( y ) = \\frac { 1 } { \\sqrt { \\lambda _ n } } \\int _ { D } \\log \\kappa ( y , x ) \\phi _ n ( x ) \\mathrm { d } x . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} C = 2 \\delta + 3 C ' = 2 \\delta + 4 . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} F ( \\alpha + i y ) + A ( \\alpha + i y ) = \\eta \\left ( i \\frac { | \\alpha + i y | } { \\alpha + i y } \\right ) ^ k \\overline { F } \\ ! \\left ( - \\frac 1 { N ( \\alpha + i y ) } \\right ) + B ( \\alpha + i y ) . \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} f ( t , X _ t ) - f ( 0 , X _ 0 ) = \\int _ 0 ^ t ( f _ t + b \\nabla f + \\frac { 1 } { 2 } \\Delta f ) ( s , X _ s ) d s + \\int _ 0 ^ t \\nabla f ( s , X _ s ) d B _ s \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} d _ 4 ( n , n - 2 ) = \\begin{cases} 3 & n = 3 , \\\\ 2 & n \\ge 4 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} a c - b d = 1 \\Longrightarrow e _ 2 c + d e _ 1 = 1 . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} \\int M ( d t , \\theta _ t ) = \\int \\mu ( t , \\theta _ t ) d B _ t . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\vert \\vert \\vert a \\vert \\vert \\vert \\le \\sum _ { j = 1 } ^ m \\vert \\vert \\vert a _ j ^ n \\vert \\vert \\vert \\le \\sum _ { j = 1 } ^ m M _ 2 \\left \\Vert a _ j \\right \\Vert ^ n , \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} A = \\left \\{ \\left ( J _ { \\lambda } w , \\frac { 1 } { \\lambda } \\left ( w - J _ { \\lambda } w \\right ) \\right ) : \\ w \\in D , \\ ; \\lambda > 0 \\right \\} \\subset D \\times \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} Z _ n ( t ) = a _ 0 + \\sum _ { ( x , y ) \\in \\Pi ^ + : \\ , y \\leq n } 1 _ { ( t _ 0 , t ] } ( x ) S ( x , y ) y ^ { - 1 / \\alpha ( Z _ n ( x _ - ) ) } ( t _ 0 \\leq t < t _ 1 ) . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\gamma & = P \\big ( S _ n \\neq O n \\geq 1 \\big | S _ 0 = O \\big ) \\\\ & = P \\big ( S _ n \\neq O n \\geq 0 \\big | S _ 0 = e _ 1 \\big ) = 1 - H ( e _ 1 ) . \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} G ' ( x ) = \\frac { 1 + x ^ { \\alpha + \\beta ^ 2 } - \\beta ( \\beta + 1 ) g ( 1 / x ) x ^ { \\beta ^ 2 } ( 1 + x ) } { x ^ { \\alpha } ( 1 + x ) } . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} H _ I ( q ) = \\sum _ { n = 0 } ^ \\infty \\dim ( R / I ) _ n q ^ n . \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} & h ( x ) \\\\ & \\le \\Bigl ( g _ { 9 - } ( x ) + \\frac { 2 } { 6 ^ { 1 0 } } ( 2 ^ { 1 0 } x - 4 2 6 ) ( 2 4 5 9 6 - 3 ^ { 1 0 } x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 4 1 6 9 } { 5 8 0 2 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ { 1 0 } } - \\eta \\\\ & = - \\frac { 1 1 8 9 9 4 1 8 9 6 7 2 6 3 8 0 2 8 3 4 4 6 1 4 5 5 7 9 3 9 0 6 3 0 8 3 1 1 7 3 1 } { 4 9 5 2 5 6 1 0 1 2 0 6 7 0 6 3 8 6 4 7 5 3 4 7 1 1 5 7 0 9 5 6 2 1 1 4 4 0 7 9 7 9 1 3 6 0 0 0 0 } < 0 , \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\Psi _ { ( 1 - \\gamma ) \\mu _ k } ( \\tilde { w } ^ { ( k + 1 ) } ) \\leq \\Psi _ { \\mu _ k } ( \\tilde { w } ^ { ( k + 1 ) } ) + \\sqrt { \\vartheta } \\gamma \\mu _ k . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\mathcal { W } [ y ] ( x , t ) = \\mathcal { V } [ y ( x , \\cdot ) ] ( t ) , ( x , t ) \\in \\Omega \\times [ 0 , T ] . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} n ! { \\| \\tilde { h } _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 & \\geq \\frac { 1 } { n ! } C _ 9 e ^ { - 2 \\mu _ 1 t } \\eta ( t ) ^ n t ^ { n \\big ( \\frac { \\alpha - \\beta } { \\alpha } \\big ) } C _ { \\alpha , \\beta , \\epsilon } ^ n \\int _ { [ 1 / 2 , 1 ] ^ n } \\prod _ { i = 1 } ^ n ( t _ { i + 1 } - t _ { i } ) ^ { - \\beta / \\alpha } d \\textbf { t } , \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\delta & \\leq \\| T ( e _ \\xi ) \\| = | T ( e _ \\xi ) ( x _ \\xi , y _ \\xi ) | = | R _ L ( e _ \\xi ) ( x _ \\xi , y _ \\xi ) | \\\\ & = | R ( e _ \\xi \\restriction _ { \\alpha \\times \\{ y _ \\xi \\} } ) ( x _ \\xi ) | \\leq \\| R ( e _ \\xi \\restriction _ { \\alpha \\times \\{ y _ \\xi \\} } ) \\| \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} [ l _ { 0 } , l _ { 0 } ] = [ l _ { 0 } , l _ { h } ] = [ l _ { 0 } , l _ { h } ' ] = [ l _ { h } , l _ { h } ' ] = 0 \\in \\mathfrak { g } ^ { ( l ) } , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { r } \\frac { \\mathrm { d } } { \\mathrm { d } r } \\bigg ) ^ { k - 1 } \\left ( r ^ { 2 k - 1 } \\phi ( r ) \\right ) & = \\sum _ { j = 0 } ^ { k - 1 } \\beta ^ { ( k ) } _ j r ^ { j + 1 } \\frac { \\mathrm { d } ^ j \\phi } { \\mathrm { d } r ^ j } ( r ) , \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} n & = \\frac { q - 1 } { p } + \\frac { 1 } { p } \\sum _ { y \\in \\mathbb { F } _ p ^ * } \\Omega ( y , 0 ) \\\\ & = \\frac { q - 1 } { p } + \\frac { q - 1 } { p ( p + 1 ) } \\sum _ { y \\in \\mathbb { F } _ p ^ * } \\eta _ { t ( y ) } ^ { ( d , q ) } , \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} ( d F ) _ { h _ { 0 } \\sqcup \\alpha } = J _ { 1 } ( F ( h _ { 0 } ) ; \\alpha ) \\circ ( d F ) _ { h _ { 0 } } . \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\int _ { \\Omega _ y } \\Big ( \\tfrac { 1 } { 2 } | \\psi _ { y } | ^ 2 + \\tfrac { 1 } { 4 } | \\psi | ^ 4 \\Big ) ( t ) d y = \\int _ { \\Omega _ y } \\Big ( \\tfrac { 1 } { 2 } | \\psi _ { 0 y } | ^ 2 + \\tfrac { 1 } { 4 } | \\psi _ 0 | ^ 4 \\Big ) d y \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} k ^ 2 \\ ( 1 - \\frac { 2 m } { r _ o } \\ ) = \\frac { ( H _ o ^ 2 - 4 k ^ 2 b ) r _ o ^ 2 } { 4 } . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} T _ n ( z / c ) = \\frac { 1 } { 2 } ( ( w / c ) ^ n + ( c / w ) ^ { n } ) n \\geq 0 , \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} d X ^ { 0 , u } _ x ( t ) = [ A X ^ { 0 , u } _ x ( t ) + B ( t , X ^ { 0 , u } _ x ( t ) ) + G ( t , X ^ { 0 , u } _ x ( t ) ) u ( t ) ] d t , \\ ; X ^ { 0 , u } _ x ( 0 ) = x . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} & \\nabla ^ L _ { X _ j } X _ j = 0 , ~ ~ ~ 1 \\leq j \\leq 2 , ~ ~ ~ \\nabla ^ L _ { X _ 1 } X _ 2 = \\frac { 1 } { 2 } X _ 3 , ~ ~ ~ \\nabla ^ L _ { X _ 2 } X _ 1 = - \\frac { 1 } { 2 } X _ 3 , \\\\ & \\nabla ^ L _ { X _ 1 } X _ 3 = - \\frac { L } { 2 } X _ 2 , ~ ~ \\nabla ^ L _ { X _ 3 } X _ 1 = - \\frac { L } { 2 } X _ 2 - X _ 3 , \\\\ & \\nabla ^ L _ { X _ 2 } X _ 3 = \\nabla ^ L _ { X _ 3 } X _ 2 = \\frac { L } { 2 } X _ 1 , ~ ~ \\nabla ^ L _ { X _ 3 } X _ 3 = L X _ 1 . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} B _ 2 ^ { ( n ) } : = \\bigcup _ { t = \\lfloor n - f _ 1 ( n ) \\rfloor } ^ { \\lfloor n - \\varepsilon f _ 4 ( n ) \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} . \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} a _ k \\leq C b ^ { - \\frac { s } { Q } } \\bigg ( \\int _ { 2 B _ 0 } \\Big ( \\sum _ { j } g _ j ^ q \\Big ) ^ { \\frac { p } { q } } \\ , d \\mu \\bigg ) ^ { \\frac { s } { Q } } \\sum _ { n = k _ 0 } ^ { k - 1 } 2 ^ { n ( 1 - \\frac { s p } { Q } ) } + C ' r _ 0 ^ { s } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\chi _ { P , a , e f f } ^ { 1 } ( \\mathbf { V } ; \\mathcal { G } ) = \\Pi \\left [ \\psi _ { P , a } ( \\mathbf { O } ; \\mathcal { G } ) | \\Lambda \\left ( P \\right ) \\right ] \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\frac { 1 } { p _ 1 } + \\cdots + \\frac { 1 } { p _ k } = k - 1 + \\frac { 1 } { r } . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} u _ { t x x } + \\frac { 1 } { 2 } \\left ( u ^ 2 \\right ) _ { x x x } = 2 \\omega u _ x + \\frac { 1 } { 2 } \\left ( u _ x ^ 2 \\right ) _ x \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\frac { 1 } { Q ( \\pi ( q ) ) Q ( \\pi ' ( q ) ) Q ( \\xi ) } \\sim _ { E _ q ( \\Pi ) E _ { q ' } ( \\Pi ' ) } \\ \\cfrac { P ^ { ( q ) } ( \\Pi , \\imath _ { v _ 0 } ) P ^ { ( n - q - 2 ) } ( \\Pi ' , \\imath _ { v _ 0 } ) } { P ^ { ( q + 1 ) } ( \\Pi , \\imath _ { v _ 0 } ) P ^ { ( n - q - 1 ) } ( \\Pi ' , \\imath _ { v _ 0 } ) } \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ \\infty e ^ { \\frac { - \\lambda ^ 2 - \\mu _ { \\eta , 1 } ^ 2 t ^ 2 + 2 \\lambda \\mu _ { \\eta , 1 } t } { 2 \\sigma _ { \\eta , 1 } ^ 2 \\xi t } } & = \\beta \\\\ \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ \\infty e ^ { \\frac { - \\upsilon ^ 2 - \\mu _ { \\eta , 0 } ^ 2 t ^ 2 + 2 \\upsilon \\mu _ { \\eta , 0 } t } { 2 \\sigma _ { \\eta , 0 } ^ 2 \\xi t } } & = \\alpha . \\end{aligned} \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\norm { w } ^ 2 & = \\norm { u } ^ 2 + 2 \\sum \\limits _ { i = 1 } ^ m \\cfrac { \\mu ^ 2 } { ( A _ i x - b _ i ) ^ 2 } ( A _ i u ) ^ 2 + \\norm { \\sum \\limits _ { i = 1 } ^ m \\cfrac { \\mu ^ 2 } { ( A _ i x - b _ i ) ^ 2 } ( A _ i u ) A _ i ^ T } ^ 2 \\\\ & \\geq \\max \\left \\{ \\norm { u } ^ 2 , \\sum \\limits _ { i = 1 } ^ m \\cfrac { \\mu ^ 2 } { ( A _ i x - b _ i ) ^ 2 } ( A _ i u ) ^ 2 \\right \\} . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\chi ' _ k ( r ) = \\frac { 4 } { \\rho _ { k + 1 } - \\rho _ { k } } \\chi ' ( s ) = O ( \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } ) \\chi ' ( s ) . \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} { \\mathbb { E } } \\left ( \\int _ { 0 } ^ { T } \\left \\langle u ( s ) , d w ( s ) \\right \\rangle _ { H } \\right ) ^ { 2 } = \\sum _ { k = 1 } ^ { \\infty } { \\mathbb { E } } \\int _ { 0 } ^ { T } \\left \\langle u ( s ) , e _ { k } \\right \\rangle _ { H } ^ { 2 } = { \\mathbb { E } } \\int _ { 0 } ^ { T } | u ( s ) | _ { H } ^ { 2 } d s . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} C ^ { ( 1 + \\alpha ) } _ n ( z ) = \\sum _ { j = 0 } ^ n \\kappa ^ n _ j ( \\alpha ) z ^ j \\ , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\textbf { G } _ \\tau ( S ^ { K \\textnormal { t h } } ( \\phi _ i ) , \\phi _ j ) = \\textbf { g } ( \\phi _ i , T ^ { K \\textnormal { t h } } ( \\phi _ j ) ) \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { U } _ i ^ F } { \\partial d _ i } = p + \\frac { B w _ i \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { \\left ( \\sum _ { j \\in \\mathcal { N } } w _ j d _ j \\right ) ^ 2 } - c _ i , \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } { 1 \\over n _ k } { \\sum _ { j = 1 } ^ k \\log u _ j } = \\lim _ { k \\to \\infty } { \\sum _ { j = 1 } ^ k { 1 - \\epsilon \\over c } \\log j \\over k ^ { { 1 \\over \\alpha } ( 1 - \\epsilon ) } } = 0 . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\bar A ( t ) } { t ^ 2 } \\le \\frac 4 { \\log 2 } \\lim _ { t \\rightarrow \\infty } \\int _ { t } ^ { 2 t } \\frac { \\bar A ( s ) } { s ^ 2 } \\frac { d s } { s } = 0 . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} s ( n , n - k ) = \\sum _ { j = 0 } ^ k ( - 1 ) ^ j \\binom { n + j - 1 } { k + j } \\binom { n + k } { k - j } S ( j + k , j ) . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} H _ { n } ( y ) _ { t } = y _ { 1 } + \\cdots y _ { i - 1 } + \\left ( \\int _ { 0 } ^ { t } \\eta _ { i } ( s ) d s \\right ) y _ { i } , \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\mathbf { D } _ h F = \\sum _ { i = 1 } ^ { m } \\frac { \\partial f } { \\partial x _ i } \\left ( B _ { t _ 1 } , \\ldots , B _ { t _ m } \\right ) \\ , h _ { t _ i } . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\Pi _ { s } ^ { t , t + \\tau , A _ t ; U , V } [ b ] : = \\breve { y } _ { s } ^ { t , t + \\tau , A _ t ; U , V } , ~ s \\in [ t , t + \\tau ] , \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\theta _ p ( u , x ) : = \\lim _ { r \\to 0 } \\theta _ p ( u , x , r ) . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\Delta = & E [ D ] + \\frac { E [ Y ^ 2 ] } { 2 E [ Y ] } \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} \\sigma _ { \\beta , \\theta } ( g h ) = \\theta ( g ) \\cdot \\psi _ { \\beta } ( h ) ( g \\in G _ { \\beta } ( O _ r ) , h \\in K _ l ( O _ r ) ) . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\frac { \\frac { n _ { 1 } } { r _ { 1 } } } { \\frac { n } { r } } \\geq \\frac { 1 - \\frac { k } { l n ^ { 2 } } } { 1 + \\frac { 1 } { l n _ { 1 } ^ { 2 } } } = 1 - \\frac { \\frac { k } { l n ^ { 2 } } + \\frac { 1 } { l n _ { 1 } ^ { 2 } } } { 1 + \\frac { 1 } { l n _ { 1 } ^ { 2 } } } . \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { | y _ \\varepsilon - x _ \\varepsilon | } { \\mu _ \\varepsilon } = + \\infty \\ , . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} A = \\bigl \\{ w ^ { - 1 } s t w \\mid w \\in W , s , t \\in S , 3 \\leq m _ { s t } \\leq \\infty \\bigr \\} . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\begin{cases} \\bar { u } _ { t t } - \\sum _ { j = 1 } ^ { n + 1 } \\bar { u } _ { x _ j x _ j } + \\frac { \\mu } { 1 + t } \\bar { u } _ t + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } \\bar { u } = \\bar { f } ( t , x , x _ { n + 1 } ) , & ( x , x _ { n + 1 } ) \\in \\mathbb { R } ^ { n + 1 } , \\ t > 0 , \\\\ \\bar { u } ( 0 , x , x _ { n + 1 } ) = \\bar { u } _ 0 ( x , x _ { n + 1 } ) , & ( x , x _ { n + 1 } ) \\in \\mathbb { R } ^ { n + 1 } , \\\\ \\bar { u } _ t ( 0 , x , x _ { n + 1 } ) = \\bar { u } _ 1 ( x , x _ { n + 1 } ) , & ( x , x _ { n + 1 } ) \\in \\mathbb { R } ^ { n + 1 } , \\end{cases} \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} W _ H ( c ( a ) ) & = n - \\frac { 1 } { p ^ 2 } \\sum _ { y \\in \\mathbb { F } _ p } \\sum _ { z \\in \\mathbb { F } _ p } \\sum _ { x \\in \\mathbb { F } _ q ^ * } \\chi _ m ( ( z + y a ) x ) \\\\ & = n - \\frac { 1 } { p ^ 2 } ( q - 1 + ( p ^ 2 - 1 ) ( - 1 ) ) \\\\ & = p ^ { m - 1 } - p ^ { m - 2 } \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} M _ { x , y } \\triangleq \\{ u \\in \\mathfrak { g } ^ { ( l ) } : \\| u \\| _ \\mathrm { H S } < r \\ \\mathrm { a n d } \\ x + F _ l ( u , x ) = y \\} \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla _ { \\partial _ { t _ j } } T _ k = \\left ( \\partial _ { t _ j } + \\frac { 1 } { z } T _ i \\circ _ { \\tau _ 2 } \\right ) T _ k , & & 1 \\leq j \\leq r \\\\ & \\nabla _ { \\partial _ z } T _ k = \\left ( \\partial _ z - \\frac { 1 } { z ^ 2 } c _ 1 ( \\textnormal { T } X ) \\circ _ { \\tau _ 2 } + \\frac { 1 } { z } \\left ( 1 - \\frac { 1 } { 2 } \\textnormal { d i m } ( X ) \\right ) \\right ) T _ k \\end{aligned} \\right . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} \\psi ^ * ( \\omega _ { Z _ Y } ) = \\textrm { p r } _ 1 ^ * ( \\omega _ { F ( Y ) } ) - \\textrm { p r } _ 2 ^ * ( \\omega _ { F ( Y ) } ) \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} 1 - \\beta _ \\varepsilon = O \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , . \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( m ) = \\sigma \\left ( W _ { m n } ( [ - \\omega _ 1 , \\omega _ 1 ] ) \\times W _ { m n } ( [ 0 , \\omega _ 2 ] ) \\right ) \\ ; , \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\tau = \\tau _ 1 \\boxplus \\tau _ 2 \\boxplus \\cdots \\boxplus \\tau _ r , \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} ( k - 1 ) g _ { \\lambda } ( k ) & \\geq ( k - 1 ) s _ { \\mu } ( 1 ^ { k - 1 } ) \\\\ & = s _ { 1 } ( 1 ^ { k - 1 } ) s _ { \\mu } ( 1 ^ { k - 1 } ) \\\\ & \\geq s _ { \\lambda } ( 1 ^ { k - 1 } ) + s _ { \\nu } ( 1 ^ { k - 1 } ) \\\\ & \\geq s _ { \\lambda } ( 1 ^ { k - 1 } ) + k - 1 . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} M _ r : = \\max \\{ { \\tau } ( \\beta _ { r , 1 } ) , { \\dots } , { \\tau } ( \\beta _ { r , m _ r } ) \\} \\in \\mathbb { Z } _ { > 0 } \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} | A | = \\sum _ { \\sigma \\in S _ m } { s g n ( \\sigma ) } \\mu _ { 1 \\sigma ( 1 ) } \\cdots \\mu _ { m \\sigma ( m ) } , \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} b ^ { ( \\theta ) } = \\sum _ { K \\in \\mathcal { B } } \\theta _ K h _ K , \\theta \\in \\{ \\pm 1 \\} ^ { \\mathcal { D } } . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} \\Delta \\circ \\sigma ^ \\ast = \\bar { { \\rm e v } } ^ \\ast : H ^ { n + 1 } ( X ) \\rightarrow H ^ { n + 1 } ( \\Sigma \\Omega X ) , \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} 1 ^ { T } s _ { 1 } & = \\mu / 3 \\lambda ( v _ { 1 } \\oplus v _ { 2 } ) , & 1 ^ { T } s _ { 2 } & = 1 , \\\\ s _ { 1 } ^ { - } 1 & = 3 / 2 , & s _ { 2 } ^ { - } 1 & = \\mu / 3 \\lambda ( v _ { 1 } \\oplus v _ { 2 } ) . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - \\hat h _ n x _ n ^ 2 ) + \\rho _ { n + 1 } \\xi _ { n + 1 } , n \\in \\mathbb N , x _ 0 \\in \\mathbb R , \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\# ( F _ 1 + F _ 2 ) ( n _ i ) \\leq 2 N ( F _ 1 + F _ 2 , 2 ^ { - n _ i } ) \\leq N ( F _ 1 , 2 ^ { - n _ i } ) ^ { 1 + \\delta } = \\# F _ 1 ( n _ i ) ^ { 1 + \\delta } . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} J ( \\tau ) : = J + \\tau P _ 1 L ^ * L P _ 1 \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\begin{cases} u _ t = v , \\\\ \\displaystyle \\tau v _ t = \\mathcal { L } ( u ) - g ( u ) v - \\int _ 0 ^ 1 f ( u ) \\ , d x - \\int _ 0 ^ 1 [ 1 - g ( u ) ] v \\ , d x , \\end{cases} \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} V ( G ) = V _ 1 \\dot { \\cup } \\cdots \\dot { \\cup } V _ r \\sum _ { i = 1 } ^ { r } e ( V _ i ) < \\eta n ^ 2 . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\Lambda _ g ( \\Omega ) : = \\max _ { u \\in H ^ 1 _ 0 : \\| u \\| _ { H ^ 1 _ 0 } ^ 2 \\le 4 \\pi } \\int _ \\Omega \\left ( ( 1 + g ( u ) ) ( 1 + u ^ 2 ) - ( 1 + g ( 0 ) ) \\right ) d x \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\frac { \\| e ^ { - t A } f \\| ^ 2 } { 2 } + \\frac { 2 ^ { n } } { n ! } \\int _ 0 ^ t s ^ { n } \\| A ^ { \\frac { n + 1 } { 2 } } e ^ { - s A } f \\| ^ 2 \\ , d s = \\frac { \\| f \\| ^ 2 } { 2 } . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\rho _ { i j } \\to 0 , \\frac { \\rho _ { i j } } { 2 \\mu ^ 2 } = \\cfrac { t ^ 2 - \\sigma _ i ^ 2 } { 2 \\mu ^ 2 } \\times \\cfrac { ( t ^ 2 - \\sigma _ j ^ 2 ) } { 2 \\mu ^ 2 } \\rightarrow \\frac { ( t ^ * ) ^ 2 } { ( \\zeta _ i - t ^ * ) ( \\zeta _ j - t ^ * ) } ; \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\int Q ^ 2 = 8 \\pi \\ , , \\int Q ^ 3 = 2 4 \\pi \\ , , \\int x ^ 2 Q ^ 2 = 8 \\pi \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } = - 1 + \\frac { 2 } { n } \\\\ x _ { j + 1 } = \\frac { 2 } { n } - \\frac { 1 } { x _ { j } } \\end{cases} \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u = \\lambda u H ( u ) \\exp ( u ^ 2 ) \\Omega \\ , , \\\\ u = 0 \\partial \\Omega \\ , , \\end{cases} \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\sum _ { i \\in Z } i ^ { - t } \\leq \\sum _ { i = 1 } ^ { \\# Z } i ^ { - t } \\leq \\int _ 0 ^ { \\# Z } x ^ { - t } d x = \\frac { 1 } { 1 - t } ( \\# Z ) ^ { - t + 1 } . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\sum _ { j \\leq N } \\eta ( j ) = \\lceil K ( N + 1 ) ^ { u } ( \\log ( N + 1 ) ) ^ { v } \\rceil + 1 ( \\forall N \\in \\mathbb { N } ) \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} q ^ \\mathbb { Z } : = \\left \\{ q ^ k \\ , \\middle | \\ , k \\in \\mathbb { Z } \\right \\} \\subset \\mathbb { C } & & q ^ \\mathbb { R } : = \\left \\{ q ^ k \\ , \\middle | \\ , k \\in \\mathbb { R } \\right \\} \\subset \\mathbb { C } \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} ( \\partial \\ < V \\ > ) \\llcorner \\Gamma ^ c = \\left ( \\sum c _ i \\eta ( \\sigma _ i ) \\right ) [ \\tau ] . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } f ( x ^ { k } ) = f ^ { * } . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\hat X _ j = \\partial _ { x _ j } + \\frac { y _ j } { 2 } \\partial _ z , \\hat Y _ j = \\partial _ { y _ j } - \\frac { x _ j } { 2 } \\partial _ z , \\hat Z = \\partial _ z . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} & \\frac 1 2 \\int _ { M } ( | \\partial _ t u ( t , \\cdot ) | ^ 2 + | \\nabla u ( t , \\cdot ) | ^ 2 ) \\\\ & = E ( u ( t , \\cdot ) ) + \\rho _ 1 \\log \\int _ { M } e ^ { u ( t , \\cdot ) - \\overline { u } ( t ) } + \\rho _ 2 \\log \\int _ { M } e ^ { - u ( t , \\cdot ) + \\overline { u } ( t ) } \\\\ & \\leq E ( u ( t , \\cdot ) ) + ( \\rho _ 2 - \\rho _ 1 ) \\overline { u } ( t ) + C \\\\ & = E ( u ( 0 , \\cdot ) ) + ( \\rho _ 2 - \\rho _ 1 ) \\overline { u } ( 0 ) + C \\\\ & \\leq C \\mbox { f o r a l l } t \\in [ 0 , T _ 0 ) , \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\eta _ a ^ { ( 3 ) } = & \\frac { 1 } { 4 } D \\beta _ a \\\\ \\eta _ a ^ { ( 4 ) } = & \\frac { 1 } { 1 0 } D ^ 2 \\beta _ a - \\frac { 1 } { 4 5 } \\alpha _ { a b } \\beta ^ b . \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} a ( \\alpha _ 1 - \\alpha _ 2 - \\alpha _ 5 + \\alpha _ 6 ) = b ( \\alpha _ 2 - \\alpha _ 3 - \\alpha _ 4 + \\alpha _ 5 ) , \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} \\aligned R _ { \\widetilde { g } } - | \\widetilde { K } | _ { \\widetilde { g } } ^ 2 + ( _ { \\widetilde { g } } \\widetilde { K } ) ^ 2 & = 0 \\\\ _ { \\widetilde { g } } \\widetilde { K } - d ( _ { \\widetilde { g } } \\widetilde { K } ) & = 0 , \\endaligned \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} ( 1 - \\partial _ x ^ 2 ) u _ t + \\left ( u + u ^ p \\right ) _ x = 0 , ( t , x ) \\in \\R \\times \\R , p = 2 , 3 , 4 , \\ldots \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} \\omega = \\mathcal { W } ^ { B D } [ y ; d ] \\Omega _ T \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} b _ { q , m } = \\frac { 1 } { 2 } \\log _ 2 \\Big ( \\frac { 6 \\rho _ { q , m } \\ln 2 } { \\lambda } \\Big ) . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} g _ 2 ( \\lambda _ { } ) = \\mathrm { e } ^ { - 4 \\pi \\lambda _ { } R ^ 2 } . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\Phi ( t , x ) & = \\Phi ( t , r ( t ) ) + D \\Phi ( t , r ( t ) ) ( x - r ( t ) ) + D ^ 2 \\Phi ( t , z ) ( x - r ( t ) ) \\cdot ( x - r ( t ) ) \\\\ & = \\tilde \\Phi ( t , x ) + D ^ 2 \\Phi ( t , z ) ( x - r ( t ) ) \\cdot ( x - r ( t ) ) \\ , . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } a ( n ) f ( n ) = A ( N ) f ( N ) - \\sum _ { n = 1 } ^ { N - 1 } A ( n ) ( f ( n + 1 ) - f ( n ) ) . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k \\log \\lambda _ j ( | G ( \\theta ) | ) \\leq \\sum _ { j = 1 } ^ k \\int _ { - \\infty } ^ { + \\infty } d t \\Big ( ( 1 - \\theta ) \\beta _ { 1 - \\theta } ( t ) \\log \\lambda _ j \\big ( | G ( i t ) | \\big ) + \\theta \\beta _ \\theta ( t ) \\log \\lambda _ j \\big ( | G ( 1 + i t ) | \\big ) \\Big ) . \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} u + \\lambda A ^ { \\varepsilon } u = w \\quad u + \\lambda \\left [ f ( x , u ) - \\varepsilon u _ { x } \\right ] _ { x } = w , \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\nabla _ H \\pi = - B v - ( 1 - \\mathbb { P } ) ( u \\cdot \\nabla ) v = - B v - ( 1 - Q ) \\overline { ( u \\cdot \\nabla ) v } \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} ( n _ i ) _ { 1 \\le i \\le 6 } = \\begin{cases} ( 3 3 , 3 9 , 4 4 , 5 4 , 5 7 , 5 9 ) r = 4 , \\\\ ( 3 7 , 4 3 , 4 8 , 5 8 , 6 1 , 6 3 ) r = 6 . \\end{cases} \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} T _ i \\bullet _ \\tau T _ j = \\sum _ k \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } ( \\tau , Q ) T ^ k \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\nabla _ { \\partial t _ i } S ^ \\textnormal { c o h } ( \\tau , z ) \\alpha & = 0 \\\\ \\nabla _ { z \\partial _ z } S ^ \\textnormal { c o h } ( \\tau , z ) \\alpha & = S ^ \\textnormal { c o h } ( \\tau , z ) \\left ( \\mu ( \\alpha ) - \\frac { 1 } { z } \\rho ( \\alpha ) \\right ) \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} a _ i = \\# \\{ j : \\deg P _ j = i \\} . \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} \\Re a ( z ^ { - \\beta } u , u ) = \\| u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 + \\Re z ^ { - \\beta } \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 \\geq \\Re z ^ { - \\beta } \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\bar { r } _ \\varepsilon } F _ \\varepsilon ( r ) r d r = o \\left ( \\frac { \\mu _ \\varepsilon ^ 2 } { \\gamma _ \\varepsilon ^ 4 } \\right ) \\ , . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} \\delta _ \\Omega ( z ) = \\inf \\{ \\norm { w - z } : w \\in \\partial \\Omega \\} . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} H ( 0 , 0 ) = 0 , \\ H _ u ( u , v ) < 0 \\ \\ { \\rm a n d } \\ \\ H _ v ( u , v ) > 0 \\ \\ { \\rm f o r } \\ \\ u > 0 , \\ v > 0 . \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} ( f \\diamond g ) ( z ) : = \\sum _ { j = 0 } ^ { q - 1 } \\left ( \\sum _ { k + l = j } a _ k ( z ) b _ l ( z ) \\right ) \\bar { z } ^ j . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\partial h _ K ( e ) = F ( K , e ) \\mbox { a n d } o , \\gamma w \\in F ( K , e ) . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\left [ L _ { J I } \\right ] _ { J , I \\in V ( \\mathcal { G } ) } = \\left [ A _ { J I } - \\gamma _ { I } \\delta _ { J I } \\right ] _ { J , I \\in V ( \\mathcal { G } ) } , \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} & F : H ^ l = H \\oplus H \\oplus \\dots \\oplus H \\to H \\oplus H \\oplus \\dots \\oplus H , \\\\ & F ( a _ 1 , \\dots , a _ l ) = ( a _ 1 + T ( a _ l ) ^ 3 , a _ 2 + T ( a _ 1 ) ^ 3 , \\dots , a _ l + T ( a _ { l - 1 } ) ^ 3 ) . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} | \\psi ( t ) - \\psi ( u ) | & = 1 - \\psi ( u ) = 1 - \\dfrac { b _ { 2 n } - u } { b _ { 2 n } - a _ { 2 n } } = \\dfrac { u - a _ { 2 n } } { b _ { 2 n } - a _ { 2 n } } \\\\ & < \\dfrac { u - t } { b _ { 2 n } - a _ { 2 n } } \\leq \\dfrac { t ^ { 1 / k } } { b _ { 2 n } - a _ { 2 n } } = \\dfrac { t ^ { 1 / k } } { ( a _ { 2 n } ) ^ { 2 / k } } \\\\ & < \\dfrac { ( a _ { 2 n } ) ^ { 1 / k } } { ( a _ { 2 n } ) ^ { 2 / k } } = \\dfrac { 1 } { ( a _ { 2 n } ) ^ { 1 / k } } < \\dfrac { 1 } { 2 ^ { 2 n } } < \\dfrac { 1 } { 2 ^ { 2 n - 1 } } . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} & \\mathcal H \\left ( G , L , \\mathcal A \\right ) = \\mathcal H \\left ( G , L , \\mathbb Q \\right ) \\otimes _ { \\mathbb Q } \\mathcal A \\\\ & \\mathcal H \\left ( G , \\mathcal A \\right ) = \\mathcal H \\left ( G , \\mathbb Q \\right ) \\otimes _ { \\mathbb Q } \\mathcal A . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\gamma _ { X _ { l _ { 0 } } ( t , x ) } = J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) \\cdot \\gamma _ { U _ { t } ^ { ( l _ { 0 } ) } } \\cdot J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) ^ { * } . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{gather*} f _ { i ' } ( \\varphi _ * M , 0 ) > f _ { i ' + 1 } ( \\varphi _ * M , 0 ) , \\\\ \\phi ( \\exp { ( - f _ i ( M , 0 ) ) } ) = \\exp { ( - f _ { i ' } ( \\varphi _ * M , 0 ) ) } . \\end{gather*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\delta _ \\epsilon = ( 1 + \\epsilon ) ^ { 1 / \\lceil \\log _ 2 ( N - 1 ) \\rceil } - 1 \\ge ( 2 ^ { \\lceil \\log _ 2 ( N - 1 ) \\rceil } - 1 ) \\epsilon \\ge \\frac { \\ln 2 } { \\lceil \\log _ 2 ( N - 1 ) \\rceil } \\epsilon . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\sum _ { l _ 1 , l _ 2 } c o v ( T _ { l _ 1 } , T _ { l _ 2 } ) \\leq N ^ 2 \\left ( C _ 2 \\frac { r _ n ^ 2 n ^ 2 } { N ^ 3 } \\right ) = C _ 2 \\frac { r _ n ^ 2 n ^ 2 } { N } . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\tilde { M } Z = \\tilde { Y } . \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) : = \\frac { 1 } { ( 2 t ) ^ n } \\ , \\int _ { x _ n - t } ^ { x _ n + t } \\cdots \\int _ { x _ 1 - t } ^ { x _ 1 + t } f ( \\tau _ 1 , \\ldots , \\tau _ n ) \\ , d \\tau _ 1 \\cdots d \\tau _ n \\ , , \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} s ( z , t ) = \\frac { z } { 1 - t z } + \\frac { t z ^ 3 } { ( 1 - 2 z ) ( 1 - t z ) ^ 2 } + \\frac { t ^ 2 z ^ 5 } { ( 1 - z ) ^ 2 ( 1 - t z ) ^ 2 ( 1 - ( 1 + t ) z ) } . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} & e _ { [ k ] } ^ a = \\exp \\{ \\exp \\{ \\dots \\{ a \\underbrace { \\} \\dots \\} } _ { k \\ , \\ , t i m e s } \\mbox { f o r e a c h } a \\in \\mathbb R , e _ { [ 0 ] } ^ a = 1 ; \\\\ & \\ln _ k b = \\ln [ \\ln [ \\dots [ \\ln b \\underbrace { ] \\dots ] ] } _ { k \\ , \\ , t i m e s } \\mbox { f o r e a c h } b \\ge e _ { [ k ] } ^ 1 , \\ln _ 0 b = b . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m a x i m i z e } & \\sum _ { i = 1 } ^ n H _ i ( \\hat { x } _ i ) \\\\ \\mbox { s u b j e c t t o } & x _ t \\in F _ t \\subseteq \\mathbb { R } _ + ^ n ~ \\forall t \\in [ m ] \\\\ & \\hat { c } _ i ^ T \\hat { x } _ i \\leq 1 ~ \\forall i \\in [ n ] \\end{array} \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} A _ p : = \\frac { u _ { p } ( y _ { j , p } ) } { p } \\int _ { B _ { \\frac { r } { \\varepsilon _ { j , p } } } ( 0 ) } H ( y _ { j , p } , y _ { j , p } + \\varepsilon _ { j , p } z ) \\left ( 1 + \\frac { w _ { j , p } ( z ) } { p } \\right ) ^ { p } d z = o _ p ( 1 ) . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} ( g \\cdot \\nu ) ( P ) : = \\nu ( g ^ * ( P ) ) . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\sum _ { k \\in \\Bbb Z } a _ k ( y ) { \\bf e } ^ { i k x } , \\psi ( x , y ) = \\sum _ { k \\in \\Bbb Z } b _ k ( y ) { \\bf e } ^ { i k x } , \\xi ( x ) = \\displaystyle \\sum _ { k \\in \\Bbb Z } d _ k { \\bf e } ^ { i k x } , \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\frac { \\langle f _ 2 , f _ 3 \\rangle } { | | f _ 2 | | | | f _ 3 | | } & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 0 } ^ { \\infty } \\beta _ j \\overline { \\gamma _ j } \\\\ & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 0 } ^ { \\infty } e ^ { i j \\eta _ 3 } \\sqrt { ( 1 - \\rho ) \\rho ^ j } e ^ { - i ( \\eta _ 1 + j \\eta _ 3 ) } \\sqrt { ( 1 - \\rho ) \\rho ^ j } \\\\ & = - e ^ { i \\theta _ 1 } \\frac { | a | } { 1 + | a | ^ 2 } ; \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log \\chi ( \\Gamma _ n , \\Z \\Gamma / f \\Z \\Gamma ) ) = n ^ { - N } \\sum _ { \\zeta \\in \\mu _ n ( \\C ) ^ N } \\log | f ( \\zeta ) | \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} b _ k ( n , n ) = \\frac { ( - 1 ) ^ n \\Gamma ( 1 + \\alpha + n + k ) \\Gamma ( 2 + \\alpha + n - k ) } { \\Gamma ( 1 + \\alpha ) \\Gamma ( 3 + \\alpha + 2 n ) } \\ , \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\mathbf { B = } \\left ( \\mathbf { B } _ { 0 } , \\dots , \\mathbf { B } _ { p } \\right ) \\mathbf { \\subset V \\backslash } \\left \\{ \\mathbf { A } , Y \\right \\} \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} N _ d = ( 3 d - 4 ) ! \\sum _ { d _ 1 + d _ 2 = d } N _ { d _ 1 } N _ { d _ 2 } \\left ( - d _ 1 ^ 3 d _ 2 \\frac { 1 } { ( 3 d _ 1 - 1 ) ! } \\frac { 1 } { ( 3 d _ 2 - 3 ) ! } + d _ 1 ^ 2 d _ 2 ^ 2 \\frac { 1 } { ( 3 d _ 1 - 2 ) ! } \\frac { 1 } { ( 3 d _ 2 - 2 ) ! } \\right ) \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} 0 & = \\lim _ { j \\rightarrow \\infty } \\{ v _ { l _ 0 } ( x _ j , t _ j ) - u _ { l _ 0 } ( x _ j , t _ j + \\tau _ * ) \\} \\\\ & = \\lim _ { j \\rightarrow \\infty } \\{ v _ { l _ 0 } ( x _ j - k _ j , t _ j - n \\cdot k _ j / c ) - u _ { l _ 0 } ( x _ j - k _ j , t _ j - n \\cdot k _ j / c + \\tau _ * ) \\} \\\\ & = v _ { l _ 0 } ( x _ * , t _ * ) - u _ { l _ 0 } ( x _ * , t _ * + \\tau _ * ) . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { E _ { \\rm e l } } ( w _ 0 ) + E _ { \\rm g } ( w _ 0 ) + E _ { \\rm s f } ( w _ 0 ) & \\le \\liminf _ h \\overline { E _ { \\rm e l } } ( w _ h ) + E _ { \\rm g } ( w _ h ) + E _ { \\rm s f } ( w _ h ) \\\\ & \\le \\liminf _ h E ( w _ h ) = \\inf _ W E = \\inf _ W \\overline { E _ { \\rm e l } } + E _ { \\rm g } + E _ { \\rm s f } . \\end{aligned} \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} S ( T _ 0 , T _ 1 , \\dots , T _ r , T _ { r + 1 } ) : = \\sum _ { ( n , \\alpha ) \\in \\mathbb N ^ { r + 1 } } \\sum _ { \\begin{smallmatrix} \\beta \\in \\mathbb N ^ m \\\\ \\theta ' ( \\beta , \\alpha ) \\end{smallmatrix} } \\sum _ { \\begin{smallmatrix} ( k _ j ) _ { j \\in I } \\in \\mathbb N _ { > 0 } ^ I \\\\ ( \\ref { e q 3 . 1 2 } ) _ { n , \\beta } \\end{smallmatrix} } T _ 0 ^ n T _ 1 ^ { \\alpha _ 1 } \\cdots T _ r ^ { \\alpha _ r } T _ { r + 1 } ^ { \\sum _ { j \\in I } k _ j \\nu _ j } . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} G ( t ) = \\prod _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } ( t - \\lambda _ { 2 j } ) + \\frac { 4 } { n + 3 } \\sum _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } \\sin ^ { 2 } \\left ( \\frac { 2 j \\pi } { n + 3 } \\right ) ( \\lambda _ { 2 j } - \\lambda _ { n + 1 } ) \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n - 1 } { 2 } } ( t - \\lambda _ { 2 m } ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} ( U f ) ( x ) = ( \\delta _ { + } - \\delta _ { - } ) ^ { - \\frac { 1 } { 2 } } f \\Big ( \\frac { x - \\delta _ { - } } { \\delta _ { + } - \\delta _ { - } } \\Big ) . \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} \\left | \\bigcup _ { k = 1 } ^ N \\Phi _ { N + 1 } - \\Phi _ k \\right | \\leq C | \\Phi _ { N + 1 } | \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} \\mathrm { C o v } ( X ( h ) ) \\equiv \\Gamma _ { \\Phi _ 1 ( x ; h ) } = \\langle D \\Phi _ 1 ( x ; h ) , D \\Phi _ 1 ( x ; h ) \\rangle _ \\mathcal { H } . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} ( \\tau _ z \\otimes \\mathrm { i d } ) ( u ^ { ( \\pi ) } ) = Q _ \\pi ^ { \\mathrm { i } z } u ^ { ( \\pi ) } Q _ \\pi ^ { - \\mathrm { i } z } , ( \\sigma _ z \\otimes \\mathrm { i d } ) ( u ^ { ( \\pi ) } ) = Q _ \\pi ^ { \\mathrm { i } z } u ^ { ( \\pi ) } Q _ \\pi ^ { \\mathrm { i } z } . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x ^ j _ n ) + v ^ l _ n ( x ) , \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} s = s _ n = \\frac { \\eta _ n } { \\sigma ^ 2 } \\in \\mathcal { S } _ n ^ + . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } - \\tilde { \\psi } _ { t } - \\Delta \\tilde { \\psi } = \\rho g _ 0 + \\rho \\varphi _ 3 - \\rho ' \\psi & & Q , \\\\ \\nabla \\tilde { \\psi } \\cdot n = 0 & & \\Sigma , \\\\ \\tilde { \\psi } ( \\cdot , T ) = 0 & & \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} x ' ( t ) & = s - d x ( t ) - \\beta x ( t ) v ( t ) , \\\\ y ' ( t ) & = \\beta x ( t ) v ( t ) - a y ( t ) - p y ( t ) z ( t ) , \\\\ v ' ( t ) & = k y ( t ) - u v ( t ) , \\\\ z ' ( t ) & = c y ( t ) z ( t ) - b z ( t ) . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} f ( \\lambda _ i ) g ( \\lambda _ i ) E _ i A ^ 2 & = f ( \\lambda _ i ) g ( \\lambda _ i ) E _ i A E _ 1 A + \\cdots + f ( \\lambda _ i ) g ( \\lambda _ i ) E _ i A E _ n A . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} \\xi ( x ) = \\frac 5 7 x ^ 3 + \\frac 2 7 x ^ { 1 6 } . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} h ^ * _ 0 + \\dots + h ^ * _ { j + 1 } \\le h ^ * _ d + \\dots + h ^ * _ { d - j } + \\binom { h ^ * _ 1 - h ^ * _ d + j + 1 } { j + 1 } \\ \\ 0 \\le j \\le \\tfrac d 2 - 1 \\ , . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} a ( P ) = \\sum _ { G \\in G ( P ) } ( - 1 ) ^ { | E ( G ) | } = \\prod _ { P _ i \\in P } ( - 1 ) ^ { | P _ i | - 1 } ( | P _ i | - 1 ) ! \\ , . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} [ 0 , + \\infty ) \\ni t \\mapsto w _ t : = \\textup { u s c } \\Big ( \\lim _ { C \\to + \\infty } w _ t ^ C \\Big ) \\in \\textup { P S H } ( X , \\theta ) \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} w f _ 1 + 2 z f _ 2 - 3 g _ 1 g _ 2 + 6 h = 0 \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} u + \\lambda f ( x , u ) _ { x } = w . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} P \\left ( \\max _ { x \\in V } Z _ { n _ 0 + l , x } \\in I _ { j _ l } ~ \\bigg \\vert ~ \\xi = y \\right ) = P \\Big ( \\max \\big \\{ V _ x ( y ) ~ ; ~ x \\in V , ~ x \\neq x _ { n _ 0 + l - 1 } \\big \\} \\in I _ { j _ l } \\Big ) , \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\mu ( [ Z , X ] ) = \\mu ( L _ Z X ) = L _ Z ( \\mu X ) - ( L _ Z \\mu ) X = 0 . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\begin{aligned} \\textbf { w } _ { k , i } = \\frac { 1 } { \\sqrt { M N } } \\textbf { a } \\left ( \\hat { \\textbf { x } } _ { k - 1 } + { \\boldsymbol { \\Delta } } _ { D I I , i } \\right ) , i = 1 , 2 , 3 , \\end{aligned} \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} f _ 1 ( s , t ) = t ^ { - 1 } , ~ f _ 2 ( s , t ) = t ^ { - 1 } s - t ^ { - 1 } . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} C _ { \\ell ( \\lambda _ 1 ) , \\xi _ 1 } = \\sum \\limits _ { ( \\Lambda ^ { \\prime } , A ^ { \\prime } , \\Xi ^ { \\prime } ) } \\left ( \\prod _ { j = 2 } ^ q m ( \\lambda _ j ) \\sum \\limits _ { ( d ^ j _ k ) _ { k = 1 } ^ { \\ell ( \\lambda _ j ) } } \\prod \\limits _ { k = 1 } ^ { \\ell ( \\lambda _ j ) } s ( b _ k ^ j , d _ k ^ j ) \\right ) . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ m X _ { i } ^ \\star ( a _ { i j } X _ j v _ \\ell ) = \\sum _ { i = 1 } ^ m X _ { i } ^ \\star F _ i ^ \\ell + g , \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} d ( e ^ { - C _ 1 A ^ n _ t } | Y _ t ^ n - Y _ t | ^ 2 ) & = e ^ { - C _ 1 A ^ n _ t } d ( | Y _ t ^ n - Y _ t | ^ 2 ) - C _ 1 e ^ { - C _ 1 A ^ n _ t } | Y _ t ^ n - Y _ t | ^ 2 d A ^ n _ t \\\\ & \\leq e ^ { - C _ 1 A ^ n _ t } [ C _ 1 \\norm { b _ n - b } _ { L ^ { q , 1 } _ t ( L ^ p _ x ) } ^ 2 d A ^ n _ t + C _ 1 \\norm { b _ n - b } _ { L ^ { q , 1 } _ t ( L ^ p _ x ) } ^ 2 d t + d M _ t ] . \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} L = \\max ( \\underset { m } \\max ( \\| { \\bf a } ^ { ( n ) } _ m \\| _ 2 ) , \\lambda ) \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} & n _ { k + 1 } : = n _ { k } + 2 \\lfloor n _ k ^ { 1 / 2 } \\rfloor , d _ k : = \\frac { n _ { k + 1 } - n _ { k } } { 2 } , \\\\ & \\rho _ { k } : = n _ k ^ { \\frac { 1 } { 1 + \\delta } } , \\rho _ { k j } : = \\rho _ { k } + j \\ , \\frac { \\rho _ { k + 1 } - \\rho _ { k } } { 4 } , j = 1 , \\dots , 4 , \\\\ & a _ 0 : = 1 , a _ { k + 1 } : = \\rho _ k ^ { 2 d _ k } a _ k . \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} f _ q ( Q ) = \\frac { 1 } { ( Q ; q ) _ \\infty } \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\alpha _ { \\rm H J M } ( h ) = \\sum _ { k = 1 } ^ m \\sigma ^ k ( h ) \\cdot T \\sigma ^ k ( h ) , \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} w _ { t t } - \\div D W ( D w + \\mathbb { I } ) + w = 0 \\quad \\mbox { i n } \\Omega _ e \\times ( 0 , T ) , i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) & = \\lim _ { \\epsilon \\rightarrow 0 } \\mathbb { E } _ { x , k } ^ { \\epsilon , \\hat { v } } \\Bigl \\{ \\psi _ { \\zeta ^ { \\epsilon } ( \\tau _ D ^ { \\epsilon , \\hat { v } } ) } ^ { \\epsilon , \\hat { v } } ( X ^ { \\epsilon , \\hat { v } } ( \\tau _ { D } ^ { \\epsilon , \\hat { v } } ) ) \\Bigr \\} \\\\ & = g _ { k _ 0 } ( \\bar { y } _ 0 ) \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\hat { u } ( \\xi ) = \\frac { \\hat { f } ( \\xi ) } { | \\xi | ^ { \\alpha } + \\rho } , \\xi \\in { \\mathbb R } ^ d . \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ { \\Omega \\backslash B _ { x _ \\varepsilon } ( \\rho _ \\varepsilon ) } \\Psi _ { N _ \\varepsilon } ( u _ \\varepsilon ) d y = | \\Omega | ( 1 + g ( 0 ) ) \\ , . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} I _ b ^ k \\left ( D _ b ^ { k ' } f \\right ) ( x ) = ( I _ b ^ k \\varphi ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} f \\left ( u ; t \\right ) = \\begin{pmatrix} t \\mu _ 1 ( 1 - u _ 1 ) u _ 1 + ( 1 - t ) \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ + - k \\omega u _ 1 u _ 2 \\\\ \\frac { 1 } { d } \\left ( t \\mu _ 2 ( 1 - u _ 2 ) u _ 2 + ( 1 - t ) \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d - ( \\alpha u _ { 1 } - d u _ 2 ) ^ - \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ - - \\alpha k \\omega u _ 1 u _ 2 \\right ) \\end{pmatrix} . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\chi _ { P ^ { \\ast } , a , e f f } ^ { 1 } \\left ( \\mathbf { V } ; \\mathcal { G } \\right ) & = q _ { \\mathcal { G } } \\left ( \\mathbf { W } ; P ^ { \\ast } \\right ) + h _ { \\mathcal { G } } ( A , \\mathbf { O } , \\mathbf { M } , Y ; P ^ { \\ast } ) \\\\ & = b _ { a } ( \\mathbf { O } ; P ^ { \\ast } ) - \\chi _ { a } ( P ^ { \\ast } ; \\mathcal { G } ) + g ( \\mathbf { W } ) + \\frac { I _ { a } ( A ) } { \\pi _ { a } ( \\mathbf { O } _ { m i n } ; P ^ { \\ast } ) } ( Y - b _ { a } ( \\mathbf { O } ; P ^ { \\ast } ) ) . \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P \\left ( | S _ n ( \\kappa _ n ( x ) ) - S _ n ( \\ell _ n ( x ) ) | > \\delta a _ c ^ { ( n ) } \\right ) = - \\infty . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} B _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon } + \\frac { S _ { 0 , \\varepsilon } } { \\gamma _ \\varepsilon ^ 3 } + \\frac { S _ { 1 , \\varepsilon } } { \\gamma _ \\varepsilon ^ 5 } + ( A ( \\gamma _ \\varepsilon ) - 2 \\xi _ \\varepsilon ) \\frac { S _ { 2 , \\varepsilon } } { \\gamma _ \\varepsilon } + \\frac { \\zeta _ \\varepsilon \\bar { w } _ \\varepsilon } { \\gamma _ \\varepsilon } \\ , . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} V = \\bigoplus _ { k = - r } ^ { r } V _ { k } \\mbox { a n d } { V ' } = \\bigoplus _ { k = - r - 1 } ^ { r + 1 } { V ' } _ { k } , \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\phi _ \\delta ( x ) : = \\begin{cases} 1 / \\sqrt { \\delta } & 0 \\le x < \\delta \\\\ 1 / \\sqrt { x } & \\delta \\le x \\end{cases} \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} Q _ t ( - \\varphi _ 0 ) \\geq Q _ t ( - \\varphi _ s ) \\stackrel { \\eqref { e q : h l 1 } } = - \\varphi _ { t + s } \\forall s \\in ( 0 , 1 ] \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} g _ { t } ( x ) = \\Delta _ t g _ 0 ( x ) + \\int _ 0 ^ { t } \\Delta _ { t - s } \\left ( - \\lambda K g _ s ( 0 ) \\right ) ( x ) d s + \\int _ 0 ^ t \\Delta _ { t - s } \\left ( K \\nu \\sqrt { g _ s ( 0 ) } \\right ) ( x ) d W _ s . \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\mathbb { I } ^ m _ n = \\left \\{ ( X _ 1 , \\ldots , X _ m ) \\in M _ n ^ m \\ : \\left | \\ : \\ : \\begin{array} { l } X _ j X _ k - X _ k X _ j = \\mathbf { 0 } _ n , \\\\ X _ j - X _ j ^ \\ast = \\mathbf { 0 } _ n , \\\\ \\| X _ j \\| \\leq 1 \\end{array} 1 \\leq j , k \\leq m \\right . \\right \\} \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\ddot { u } ( t ) - \\div ( A \\nabla u ( t ) ) = f ( t ) \\hbox { i n } \\Omega \\setminus \\Gamma ( t ) \\ , , \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} u ( x ) = e ^ { i \\theta } \\lambda ^ { \\frac { 2 s } { \\alpha } } g ( \\lambda x + x _ 0 ) , \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\mathrm { w d } ( \\mathcal { T } ) = \\max \\lbrace N _ 1 ( m _ { n - 1 } ) \\mid n _ \\ast + 1 \\le n \\le n _ \\ast + \\ell _ \\ast + \\ell \\rbrace . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} g _ { t _ 0 } ( t ) = g _ 0 ( { t _ 0 + t } ) - \\int _ 0 ^ { t _ 0 } K ( t + t _ 0 - s ) \\lambda V _ s d s + \\int _ 0 ^ { t _ 0 } K ( t + t _ 0 - s ) \\nu \\sqrt { V _ s } d W _ s , \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\overline { B _ r } ( X ^ { 2 ^ r } ) + X \\overline { B _ r } ( X ^ 2 ) - \\overline { B _ r } ( X ) = 0 . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\varphi ^ { \\pm } = { } ^ t ( \\varphi ^ { \\pm } _ 1 , \\varphi ^ { \\pm } _ 2 , \\cdots , \\varphi ^ { \\pm } _ m ) \\gg { } ^ t ( 0 , 0 , \\cdots , 0 ) , \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\mathcal { A C } = \\{ A ~ : \\Lambda ( A ) < \\infty \\} . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} B _ { W } ( z , w ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( z \\bar { w } ) ^ k } { \\Gamma ( 2 k + 1 ) } = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( z \\bar { w } ) ^ k } { ( 2 n ) ! } = \\cos ( i \\sqrt { z \\bar { w } } ) , \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} g _ { 2 } = x _ { 1 } x _ { 2 } - x _ { 0 } x _ { 3 } & = \\sum _ { \\mu = 1 } ^ { n - 1 } \\alpha _ { \\mu } \\left [ x _ { 2 } ^ { ( n + 1 ) - \\mu } x _ { 3 } ^ { \\mu } - x _ { 0 } ^ { ( n + 1 ) - \\mu } x _ { 1 } ^ { \\mu + 1 } \\right ] \\\\ & + \\sum _ { t = 0 } ^ { n } \\beta _ { t } \\left [ x _ { 1 } ^ { ( n + 1 ) - t } x _ { 3 } ^ { t } - x _ { 0 } ^ { n - t } x _ { 2 } ^ { t + 1 } \\right ] + \\gamma _ { 1 } ( x _ { 0 } ^ { n + 1 } - x _ { 3 } ^ { n } ) , \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} \\partial h _ K ( u ) = F ( K , u ) , \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} N _ 1 ( m _ { n - 1 } ) + \\sum _ { i = 2 } ^ { C _ 1 ( m _ { n - 2 } ) } \\ , N _ 1 ( v _ i ( m _ { n - 2 } ) ) + \\sum _ { i = 2 } ^ { C _ 1 ( m _ { n - 3 } ) } \\ , \\sum _ { j = 1 } ^ { C _ 1 ( v _ i ( m _ { n - 3 } ) ) } \\ , N _ 1 ( v _ { i , j } ( m _ { n - 3 } ) ) , \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} s - \\lim \\left ( e ^ { - t A _ k / n } e ^ { - t V / n } \\right ) ^ n = e ^ { - t L _ k } , t \\geq 0 . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\Pr { A _ k } = & \\ \\Pr { s _ i < k ^ 2 \\ \\forall i \\in \\{ 1 , \\ldots , k \\} } \\\\ = & \\ ( 1 - \\Pr { s _ 1 < k ^ 2 } ) ^ k \\\\ \\leq & \\ ( 1 - 4 c ( k ^ 2 ) ^ { - 1 / 4 } ) ^ k \\\\ \\leq & \\ e ^ { - 4 c k ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} S _ { k } = \\left | \\sum _ { n = 1 } ^ { L } e ^ { 2 \\pi i n k _ d \\omega } \\right | \\leq \\frac { 1 } { \\| k _ d \\omega \\| } \\leq C | k _ d | ^ { 2 } . \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} O _ a ^ { - 1 } & = O _ { a ^ { - 1 } } . \\\\ O _ { a b } & = O _ a \\cdot O _ b . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\widehat { A } ( m , n , b , \\epsilon ) : = \\left \\{ ( i _ 1 , \\dots , i _ n ) \\in \\mathbb { N } ^ n : \\ \\sum _ { k = 1 } ^ n e ^ { ( \\log i _ k ) ^ b } \\in [ m , m ( 1 + \\epsilon ) ] \\right \\} , \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} E [ e ^ { ( X - 1 ) t } ] = \\frac { 1 } { 1 - t } e ^ { - t } = \\sum _ { n = 0 } ^ \\infty d _ n \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { M _ b , b ' , \\mu _ b } ( \\rho _ 1 \\boxtimes . . . \\boxtimes \\rho _ k ) = \\boxtimes ^ k _ { i = 1 } \\mathrm { M a n t } _ { G _ i , b ' _ i , \\mu _ { b , i } } ( \\rho _ i ) , \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta \\gtrsim ( \\alpha t ) ^ \\beta e ^ { \\alpha t } \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} w ^ a ( x ) = \\frac { 1 } { a } w \\left ( \\frac { x } { a } \\right ) \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} & D ^ { i j } _ l ( t + T ) = D ^ { i j } _ l ( t ) , \\ q _ l ( t + T ) = q _ l ( t ) , \\ f _ l ( t + T , \\cdot ) = f _ l ( t , \\cdot ) \\\\ & { \\rm f o r } \\ \\ t \\in \\R \\ \\ ( i , j = 1 , 2 , \\cdots , N ; \\ l = 1 , 2 , \\cdots , m ) \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} | | x + i y | | _ { r _ 0 } = 1 . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\min _ { x } \\ ( \\max _ { k = 1 } ^ { m } w _ { k } ( \\max _ { i , j = 1 } ^ { n } a _ { i j } ^ { ( k ) } x _ { j } / x _ { i } ) ) = \\min _ { x } \\ ( \\max _ { i , j = 1 } ^ { n } \\max _ { k = 1 } ^ { m } ( w _ { k } a _ { i j } ^ { ( k ) } ) x _ { j } / x _ { i } ) . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\textbf { S } ( t , z ) ( \\alpha ) = \\alpha - \\sum _ { \\substack { d \\in H _ 2 ( X ; \\mathbb { Z } ) - \\{ 0 \\} \\\\ l \\geq 0 } } \\ , \\sum _ { k = 0 } ^ N \\frac { 1 } { l ! } \\left \\langle T _ k , \\tau ' , \\dots , \\tau ' , \\frac { \\alpha } { z + \\psi } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } T ^ k e ^ { \\tau _ 2 ( d ) } \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} \\varepsilon L _ { \\lambda } \\varphi \\left ( x \\right ) = \\varepsilon \\int \\limits _ { \\mathcal { K } _ { N } } \\left \\{ \\varphi ( y ) - \\lambda \\varphi \\left ( x \\right ) \\right \\} J \\left ( x , y \\right ) d y . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} 2 \\lambda \\int _ { \\O } u \\phi ~ d x & = \\int _ { u > t _ p } ( 2 \\lambda u ) \\phi ~ d x + 2 \\lambda \\int _ { u \\leq t _ p } u \\phi ~ d x \\\\ & \\leq \\int _ { \\Omega } u ^ p \\phi ~ d x + 2 \\lambda t _ p \\int _ { \\O } \\phi ~ d x = \\lambda \\int _ { \\O } u \\phi ~ d x + 2 \\lambda t _ p , \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\int _ { - l _ m } ^ { l _ m } w _ m ( u ) \\d u = \\int _ { - l _ m } ^ { l _ m } C _ m w ( g _ m ^ { - 1 } ( u ) ) \\d u = C _ m \\int _ { - 1 } ^ { 1 } w ( t ) | g _ m ' ( t ) | \\d t \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\phi _ n ^ { ( k ) } ( z ) = \\sum _ { j = 0 } ^ { k } A _ j \\frac { ( q ; q ) _ { n - j } } { ( q ^ { \\delta + 1 } ; q ) _ { n - j } } L _ { n - j } ^ { ( \\delta ) } ( z q ^ j ; \\ , q ) . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell \\beta ^ { j / b } } { d \\sum _ { j = 1 } ^ { \\ell + 1 } \\beta ^ { j / b } - \\beta ^ { ( \\ell + 1 ) / b } } = \\frac 1 { d \\beta ^ { 1 / b } - \\beta ^ { 1 / b } + 1 } . \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} 1 - \\langle C _ \\varphi ^ * f , f \\rangle = & \\langle 2 f _ 2 , f _ 1 + f _ 2 \\rangle \\\\ = & 2 | | f _ 2 | | ^ 2 + 2 \\langle f _ 2 , f _ 1 \\rangle , \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} w _ t = v - \\gamma w _ { \\nu _ { \\mathcal { N } } } \\quad \\mbox { o n } \\Gamma _ c \\times ( 0 , T ) , \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} I _ 1 = \\sum _ { \\frac { \\eta _ 1 n } { 2 N } \\leq k \\leq \\frac { 2 \\eta _ 2 n } { N } } \\mathbb { P } ( N _ l = k ) \\mathbb { E } ( T _ l | N _ l = k ) = \\sum _ { \\frac { \\eta _ 1 n } { 2 N } \\leq k \\leq \\frac { 2 \\eta _ 2 n } { N } } B ( k ; n , p _ l ) \\Delta ( k , q _ l ) \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\left [ \\bigcup _ { d _ 1 + d _ 2 = d } \\mathcal { Z } _ { d _ 1 , d _ 2 } \\right ] ^ \\textnormal { v i r } = \\sum _ { d _ 1 + d _ 2 = d } \\Delta ^ ! \\left [ \\overline { \\mathcal { M } } _ { g _ 1 , n _ 1 + 1 } ( X , d _ 1 ) \\right ] ^ \\textnormal { v i r } \\otimes \\left [ \\overline { \\mathcal { M } } _ { g _ 2 , n _ 2 + 1 } ( X , d _ 2 ) \\right ] ^ \\textnormal { v i r } \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} J _ { \\mathcal { M } , T } ( x ) & = h ( x ) x \\in X _ T \\\\ J _ { \\mathcal { M } , t } ( x ) & = \\inf _ { a \\in \\psi ( x ) } \\{ c ( x , a ) + \\int _ { X _ { t + 1 } } J _ { \\mathcal { M } , t + 1 } ( y ) Q _ { t } ( d y | x , a ) \\} \\\\ & x \\in X _ t , t \\in [ T - 1 ] \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ \\Omega \\rho \\mathbf { w } \\varphi _ t d x d s - \\int _ \\Omega \\rho _ 0 \\mathbf { w } _ 0 \\varphi | _ { t = 0 } d s - \\int _ 0 ^ t \\int _ \\Omega \\rho u \\mathbf { w } \\varphi _ x d x d s = - \\int _ 0 ^ t \\int _ \\Omega \\mu \\mathbf { w } _ x \\varphi _ x d x d s \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\log \\biggl ( \\frac { \\Gamma ( 1 + x / p ) ^ { 1 / x } } { x ^ { 1 / p } } \\biggr ) & = \\frac { 1 } { x } \\log \\Gamma ( 1 + x / p ) - \\frac { 1 } { p } \\log ( x ) \\\\ & \\ge \\frac { 1 } { x } \\log ( c _ p ) + \\frac { 1 } { x } \\sum _ { j = 0 } ^ { k - 1 } \\log ( x / p - j ) - \\frac { 1 } { p } \\log ( x ) \\\\ & \\ge \\frac { 1 } { x } \\log ( c _ p ) + \\frac { 1 } { x } \\int _ { x / p - k } ^ { x / p } \\log ( t ) d t - \\frac { 1 } { p } \\log ( x ) \\\\ & = \\frac { 1 } { x } \\log ( c _ p ) + \\frac { 1 } { p } \\log ( 1 / p ) - \\frac { 1 } { p } - \\frac { 1 } { x } f ( x / p - k ) , \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} { \\bf h } _ m ^ H = \\sqrt { \\frac { N } { L _ m } } \\sum \\limits _ { l = 1 } ^ { L _ m } \\alpha _ m ^ { ( l ) } { \\bf a } _ t ( \\theta ^ { ( l ) } ) ^ H , \\forall m = 1 , \\cdots , M \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\left ( g _ { i j } \\right ) = \\left ( g ^ { i j } \\right ) = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} J _ { n _ 1 } ( \\alpha _ 1 ) \\boxplus \\cdots \\boxplus J _ { n _ r } ( \\alpha _ r ) = \\begin{bmatrix} J _ { n _ 1 } ( \\alpha _ 1 ) & & \\\\ & \\ddots & \\\\ & & J _ { n _ r } ( \\alpha _ r ) \\end{bmatrix} \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} | l _ \\nu | \\leq \\sum ^ { \\nu } _ { j = 0 } \\theta \\omega ( \\sigma ^ { j } ) \\leq \\theta \\sum ^ { \\infty } _ { j = 0 } \\omega ( \\sigma ^ { j } ) \\leq C _ { b } \\theta . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\mbox { v e c } Q = ( I - J _ { \\infty } \\otimes J _ { \\infty } ) ^ { - 1 } \\mbox { v e c } V _ { \\infty } . \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} 2 H _ { 0 } = \\left ( 1 + a ^ { 2 } \\right ) f _ { 1 } p _ { 2 } \\dot { p } _ { 2 } + 2 a p _ { 1 } p _ { 2 } + f _ { 2 } p _ { 1 } \\dot { p } _ { 1 } , \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\textrm { c o n v } ( g ) ( x ) = \\sup \\{ \\varphi ( x ) \\ , : \\ , \\varphi \\textrm { i s c o n v e x , } \\varphi \\leq g \\} . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} ( x \\cdot a ) \\cdot g = ( x \\cdot g ) \\cdot ( a \\cdot g ) \\ ; \\ ; \\ ; \\ ; \\ ; \\mbox { f o r a n y } x \\in B , a \\in A , g \\in G . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left ( \\mathcal { Y } ^ { K \\textnormal { t h } } ( z , q , Q ) \\right ) _ { 0 i } = \\sum _ { \\substack { 0 \\leq a , b \\leq N \\\\ a + b = i } } \\frac { \\log ( Q ) ^ a } { z ^ a a ! } f _ b ( z , Q ) \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} d S _ t ^ { t _ 0 } & = S _ t ^ { t _ 0 } \\sqrt { V _ t ^ { t _ 0 } } d B _ t ^ { t _ 0 } , S _ 0 ^ { t _ 0 } = S _ { t _ 0 } , \\\\ V _ t ^ { t _ 0 } & = g _ { t _ 0 } ( t ) + \\int _ 0 ^ { t } K ( t - s ) \\left ( - \\lambda V _ s ^ { t _ 0 } d s + \\nu \\sqrt { V _ s ^ { t _ 0 } } d W _ s ^ { t _ 0 } \\right ) , \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} I _ 2 = ( p _ 1 + p _ 2 ) ^ 2 ( n - k _ 1 - k _ 2 ) \\leq n ( p _ 1 + p _ 2 ) ^ 2 \\leq \\frac { \\eta _ 2 ^ 2 n } { N ^ 2 } \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\widetilde { [ \\chi _ k , \\left . \\Delta \\right | _ { \\mathit { D } } ] u } = - \\widetilde { ( \\Delta \\chi _ k ) u } - 2 \\widetilde { \\nabla \\chi _ k \\cdot \\nabla u } = \\widetilde { ( \\Delta \\chi _ k ) u } - 2 \\nabla \\cdot ( \\widetilde { ( \\nabla \\chi _ k ) u } ) , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\lambda \\star \\lambda = \\begin{cases} 1 , \\lambda - \\frac { 1 } { 2 } & \\mbox { i f } | \\alpha | = 2 \\\\ 1 , \\lambda , \\lambda - \\frac { 1 } { 2 } & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} B ( c , d ) = \\int _ 0 ^ 1 u ^ { c - 1 } ( 1 - u ) ^ { d - 1 } d u , \\ \\ c > 0 , d > 0 . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { N } \\mathbb { E } T _ l ^ 2 \\leq N C _ 1 \\left ( r _ n \\sqrt { \\frac { n } { N } } \\right ) ^ 2 = C _ 1 r _ n ^ 2 n \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} h _ \\varepsilon \\geq \\theta _ \\varepsilon / 2 \\geq \\frac { 1 6 n \\varepsilon } { \\gamma } \\mbox { a n d } \\lim _ { \\varepsilon \\to 0 ^ + } \\frac { \\varepsilon } { h _ \\varepsilon } = 0 . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\lambda v _ 1 - \\Delta v _ 1 + \\nabla _ H \\pi _ 1 = f \\Omega , \\Delta _ H \\ , \\pi _ 1 = - h ^ { - 1 } \\mathrm { d i v } _ H ( \\partial _ z v | _ { \\Gamma _ b } ) G , \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} f _ n ( x ) = \\sup \\{ B _ \\Omega ( \\gamma ^ m y _ n , x ) : m = 0 , 1 , \\dots , M - 1 \\} \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\alpha = \\{ ( 1 , 2 , 4 ) , ( 3 ) \\} \\ , , \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\epsilon = 1 . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\tau ^ { \\leq i } \\mathrm { h o f i b } ( \\tau ^ { \\leq i } \\mathcal N ^ { \\geq i } A \\Omega \\{ i \\} / p ^ n \\xrightarrow { \\varphi _ i - 1 } \\tau ^ { \\leq i } A \\Omega \\{ i \\} / p ^ n ) = \\tau ^ { \\leq i } \\mathrm { h o f i b } ( \\tau ^ { \\leq i } A \\Omega / p ^ n \\xrightarrow { 1 - \\xi ^ i \\varphi ^ { - 1 } } \\tau ^ { \\leq i } A \\Omega / p ^ n ) \\ , \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} | e ^ { \\int _ 0 ^ t B _ i ( s ) d \\beta _ i } f | _ q = | e ^ { \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s } f | _ q = | f | _ q , \\ \\forall f \\in L ^ q ( \\mathbb { R } ^ d ) , \\ s \\in \\mathbb { R } , \\ t \\geq 0 , i = 1 , 2 , . . . , N ; \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} ( e ^ { \\frac { 1 } { 2 } \\ell _ \\nu } + 1 ) ^ { - 1 } = ( e ^ { \\frac { 1 } { 2 } ( \\ell _ \\nu + 2 \\ell _ \\mu ) } + 1 ) ^ { - 1 } + ( e ^ { \\frac { 1 } { 2 } ( \\ell _ \\nu + 2 \\ell _ { \\mu ' } ) } + 1 ) ^ { - 1 } + ( e ^ { \\frac { 1 } { 2 } ( \\ell _ \\mu + \\ell _ { \\mu ' } ) } - 1 ) ^ { - 1 } . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\rho ( x , 0 ) = 1 + 0 . 2 ( \\pi x ) , \\ \\ \\ v ( x , 0 ) = 1 , \\ \\ p ( x , 0 ) = 1 , \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\frac { 1 } { | F | } \\sum _ { f ' } U _ f ^ { - 1 } U _ f U _ { f '' } U _ { f ' } ^ { - 1 } U _ { f ' } = U _ { f '' } , \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\int _ { \\mathfrak { g } ^ { ( l ) } } f ( u ) \\rho _ { t } ( u ) d u & = \\int _ { \\mathfrak { g } ^ { ( l ) } } f ( u ) Q _ { t } ( d u ) = \\int _ { \\mathfrak { g } ^ { ( l ) } } f ( \\delta _ { t ^ { H } } u ) Q _ { 1 } ( d u ) \\\\ & = \\int _ { \\mathfrak { g } ^ { ( l ) } } f ( \\delta _ { t ^ { H } } u ) \\rho _ { 1 } ( u ) d u = \\int _ { \\mathfrak { g } ^ { ( l ) } } t ^ { - H \\nu } f ( u ) \\rho _ { 1 } ( \\delta _ { t ^ { - H } } u ) d u , \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} u _ p \\ = \\ \\frac { 1 } { \\sqrt { n } } \\left [ 1 , \\ e ^ { - 2 \\pi i p \\frac { 1 } { n } } , \\ldots , \\ e ^ { - 2 \\pi i p \\frac { n - 1 } { n } } \\right ] ^ T p \\ = \\ 0 , \\ldots , \\ n - 1 \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} D _ { n , \\alpha } : = \\left ( \\frac { \\pi ^ { n + 1 } } { 2 ^ { n - 1 } n ! } \\right ) ^ { ( Q - \\alpha ) / Q } \\frac { n ! \\Gamma ( \\alpha / 2 ) } { \\Gamma ^ 2 ( ( Q + \\alpha ) / 4 ) } . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} S p l ( f , \\sigma ) = ( \\cup _ { \\mu } M ( f , \\mu ) ) \\cup T _ \\sigma \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} & \\widetilde { J _ 0 } ( q , Q ) = \\sum _ { d \\in \\mathbb { Z } } f _ d ( q ) Q ^ d & & \\widetilde { J _ 1 } ( q , Q ) = \\sum _ { d \\in \\mathbb { Z } } g _ d ( q ) Q ^ d + \\ell _ q ( Q ) \\sum _ { d \\in \\mathbb { Z } } g _ d ( q ) Q ^ d \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { f _ 1 ( n ) } { n ( 1 - \\pi _ n ( n - p _ n ^ { - 1 } ) ) } & \\geq \\lim _ { n \\to \\infty } \\frac { f _ 1 ( n ) } { n ( n p _ n ) ^ { r - 1 } [ ( 1 - p _ n ) ^ { p _ n ^ { - 1 } } ] ^ { n p _ n - 1 } } \\\\ & \\geq \\mathrm { e } ^ { - 1 } \\lim _ { n \\to \\infty } \\frac { a _ c ^ { ( n ) } g ( n ) } { n ( n p _ n ) ^ { r } ( \\mathrm { e } ^ { - 1 } + \\delta ) ^ { n p _ n } } = + \\infty , \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} g _ { 1 } ( \\lambda _ { } ) = 1 - \\mathrm { e } ^ { - \\pi \\lambda _ { } R ^ 2 } . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} a _ s ( \\pi _ { N , \\lambda } ^ s u - u , v \\big ) = 0 , \\forall \\ , v \\in V _ N ^ \\lambda . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} \\frac { 1 } { | W ^ { \\mathrm { a b s } } _ { M _ S } | } \\sum \\limits _ { w \\in W ^ { \\mathrm { a b s } } _ { M _ S } } \\sum \\limits _ { \\gamma \\in \\Gamma } w ( \\gamma ( \\mu ) ) = \\frac { 1 } { | W ^ { \\mathrm { a b s } } _ { M _ S } | } \\sum \\limits _ { w \\in W ^ { \\mathrm { a b s } } _ { M _ S } } \\sum \\limits _ { \\gamma \\in \\Gamma } \\gamma ( w ) ( \\gamma ( \\mu ) ) \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} e _ k : = \\begin{cases} ( n - a ) ^ k , & \\mbox { i f } a > 1 , \\\\ ( n - a ) ^ { k - 1 } , & \\mbox { i f } a = 1 . \\end{cases} \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\nu ( t _ 2 ^ 2 ) = 2 s _ 1 t _ 2 , \\ \\ \\ \\nu ( q _ 4 ) = \\mu _ 3 + \\lambda s _ 1 t _ 2 , \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} \\frac { 1 } { L ( \\chi , s ) } = \\sum _ { h = 1 } ^ { \\infty } \\frac { \\mu ( h ) \\chi ( h ) } { h ^ s } . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\widetilde { I } [ x , \\Lambda ; \\widetilde { w } ] = \\inf _ { { w } \\in { W } ^ { 1 , p } _ \\# ( \\mathcal { Y } ^ d ) / \\mathbb { R } } \\widetilde { I } [ x , \\Lambda ; { w } ] , \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} H _ { a l g } ( \\phi , U ) = \\limsup _ { n \\to \\infty } \\frac { \\log [ T _ n ( \\phi , U ) : U ] } { n } . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} F ( x _ 1 , \\dots , x _ n ) : = \\max _ { 1 \\leq j \\leq n } f _ j ( ( x _ { j - 1 } , x _ { j } ) ) . \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } C _ { g , 4 \\pi ( 1 - \\varepsilon ) } ( \\Omega ) = C _ { g , 4 \\pi } ( \\Omega ) \\ , , \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} ( y , x x ' ) _ { R T } = ( \\Delta ( y ) , x ' \\otimes x ) _ { R T } , ( y y ' , x ) _ { R T } = ( y \\otimes y ' , \\Delta ( x ) ) _ { R T } \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} \\psi \\left ( \\overline { \\nu _ L } , L \\right ) = \\frac { \\alpha \\overline { \\nu _ L } + \\frac { d } { 2 } } { r _ 0 L } - \\Phi _ 2 \\left ( \\frac { 1 } { 2 } , L \\right ) . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\widehat { w } ( \\xi ) = \\frac { 1 } { L } \\frac { \\sin ^ 2 \\left ( L \\pi \\xi \\right ) } { \\left ( \\pi \\xi \\right ) ^ 2 } . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} - a d _ { x } ^ * S ( \\xi ) + S ( \\xi ) a d _ { x } ^ * + S ( a d _ { x } ^ * ( \\xi ) ) + a d _ { a d _ { \\xi } ^ * ( x ) } ^ * = 0 \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} ( \\lambda t _ \\alpha + \\mu e ^ { \\alpha } ) \\cdot e ^ { \\alpha ^ c } = 0 \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta u _ \\varepsilon = \\frac { \\lambda _ \\varepsilon } { 2 } \\Psi ' _ { N _ \\varepsilon } ( u _ \\varepsilon ) , u _ \\varepsilon > 0 \\Omega \\ , , \\\\ & u _ \\varepsilon = 0 \\partial \\Omega \\ , , \\end{cases} \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\mathbb { E } e ^ { q \\log J _ t ( \\mathcal { H } ) } \\leq \\sum _ { s = 0 } ^ { t - 1 } \\exp ( 4 q ^ 2 \\alpha _ { s , t } + q L _ s + q \\log q ) . \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\frac { 1 } { F ( z ) } = R ( z ) + \\sum _ n \\frac { 1 } { F ' ( t _ n ) } \\cdot \\bigg ( \\frac { 1 } { z - t _ n } + \\frac { 1 } { t _ n } + \\cdots + \\frac { z ^ { k - 1 } } { t _ n ^ k } \\bigg ) , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} \\langle \\vec { x } , \\vec { y } \\rangle = Q ( \\vec { x } + \\vec { y } ) - Q ( \\vec { x } ) - Q ( \\vec { y } ) . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} & \\| ( \\varphi - \\varphi _ D ) ( t ) \\| _ { V ^ * } ^ 2 + | \\varphi _ D ( t ) | ^ 2 + \\int _ 0 ^ t \\| \\nabla \\varphi \\| _ H ^ 2 \\ , \\d s \\\\ & \\lesssim \\left ( 1 + e ^ { T + N } ( T + N ) \\right ) \\left ( \\| \\varphi _ 0 - ( \\varphi _ 0 ) _ D \\| _ { V ^ * } ^ 2 + | ( \\varphi _ 0 ) _ D | ^ 2 + \\| \\sigma \\| _ { L ^ 2 ( 0 , \\tau ^ N ; H ) } ^ 2 + \\| u \\| _ { L ^ 2 ( 0 , \\tau ^ N ; V ^ * ) } ^ 2 \\right ) \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} f ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\| \\partial _ z ( \\alpha v ) \\| _ { L ^ p ( C ( x _ 0 ' ; | \\lambda | ^ { - 1 / 2 } ) ) } & \\le \\| \\partial _ z ( \\theta \\alpha v ) \\| _ { L ^ p ( \\Omega ' ) } , \\\\ \\| \\nabla \\partial _ z ( \\alpha v ) \\| _ { L ^ p ( C ( x _ 0 ' ; | \\lambda | ^ { - 1 / 2 } ) ) } & \\le \\| \\nabla \\partial _ z ( \\alpha v ) \\| _ { L ^ p ( C ( x _ 0 ' ; | \\lambda | ^ { - 1 / 2 } ) ) } . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big [ ( 6 + t ) E ( U _ 1 ; t ) \\Big ] & = \\| U _ 1 ' \\| ^ 2 + \\| A ^ { 1 / 2 } U _ 1 \\| ^ 2 + 2 ( 6 + t ) ( U _ 1 ' , - U _ 1 ' + A e ^ { - t A } ( u _ 0 + u _ 1 ) ) \\\\ & \\leq \\| U _ 1 ' \\| ^ 2 + \\| A ^ { 1 / 2 } U _ 1 \\| ^ 2 - ( 6 + t ) \\| U _ 1 ' \\| ^ 2 + ( 6 + t ) \\| A e ^ { - t A } ( u _ 0 + u _ 1 ) \\| ^ 2 \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\left \\langle T _ k , \\tau ' , \\dots , \\tau ' , \\frac { e ^ { - \\tau _ 2 / z } \\alpha } { z + \\psi } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } : = \\sum _ { n , m \\geq 0 } \\frac { ( - 1 ) ^ { n + m } } { z ^ { n + m + 1 } } \\left \\langle T _ k , \\tau ' , \\dots , \\tau ' , \\psi _ { l + 2 } ^ n \\tau _ 2 ^ m \\cup \\alpha \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} R i c _ { \\overline { g } } = \\frac { 1 } { \\varphi ^ 2 } \\left \\{ ( n - 2 ) \\varphi H e s s _ { g } \\varphi + [ \\varphi \\Delta _ g \\varphi - ( n - 1 ) | g r a d _ g \\varphi | ^ 2 ] g \\right \\} \\ , . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\gamma _ 1 \\geqslant \\theta ( w , x ) \\wedge ( \\exists b \\in K ) ( \\exists c \\in K ) ( v ( b ) = \\alpha ( w , x ) \\wedge ( \\forall \\gamma > \\gamma _ 1 ) ( ( c / v ) \\in A _ \\gamma ^ { w , x , b } ) ) . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\| T ( f _ 1 , \\dots , f _ m ) \\| _ { L ^ { q } ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { q _ i } ( v _ i ) } , \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} T [ u ] ( \\xi _ R , \\eta _ R ) = - [ \\xi _ R , \\eta _ R ] _ R + [ \\xi _ R , \\eta _ R ] = [ \\xi _ R , \\eta _ R ] _ N \\subset V _ N \\ , . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} P \\left ( \\ell _ q ( Q ) + 1 \\right ) = \\ell _ q ( Q ) ^ d + \\cdots + ( a _ 0 + \\cdots + a _ { d - 1 } + 1 ) = 0 \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\lim _ { h \\downarrow 0 } \\lim _ { L \\to \\infty } G _ { \\beta , h } ( 0 ) = \\infty . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} & f ( x ) \\equiv \\prod _ { i = 1 } ^ n ( x - r _ i ) \\bmod p , \\\\ & 0 \\le r _ 1 \\le r _ 2 \\le \\dots \\le r _ n < p . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} & \\inf _ { \\omega \\in O ( \\frac { n ( n - 1 ) } { 2 } ) } \\sup _ { B _ { 1 0 n H ( x _ 2 , t _ 2 ) ^ { - 1 } } ( x _ 2 ) } \\max _ \\alpha \\bigg | K _ \\alpha ^ { ( x _ 1 , t _ 1 ) } - \\sum _ { \\beta = 1 } ^ { \\frac { n ( n - 1 ) } { 2 } } \\omega _ { \\alpha \\beta } K _ \\beta ^ { ( x _ 2 , t _ 2 ) } \\bigg | \\\\ & \\leq C \\ , 2 ^ { - j } \\ , H ( x _ 2 , t _ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} w = u + \\mu ^ 2 \\nabla ^ 2 f ( x ) u = u + \\sum \\limits _ { i = 1 } ^ m \\cfrac { \\mu ^ 2 } { ( A _ i x - b _ i ) ^ 2 } ( A _ i u ) A _ i ^ T . \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} ( F _ f * G _ \\alpha ) _ { | [ 0 , 1 ] } = S _ 0 ^ \\alpha f \\ , \\mbox { i n } \\ , L ^ 1 ( [ 0 , 1 ] ; \\mathbb { R } ) . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\left \\| \\sum _ n f _ n \\right \\| ^ 2 _ { \\dot { H } ^ s ( \\R ^ d ) } \\geq \\limsup _ { N \\to \\infty } \\sum _ { n = 1 } ^ N \\left [ \\| f _ n \\| ^ 2 _ { \\dot { H } ^ s ( \\R ^ d ) } - \\frac { C _ d } { s } \\frac { 1 } { \\lambda _ n ^ { 2 s } } \\| f _ n \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 \\right ] \\ , . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\tilde \\Delta ( \\tilde \\Delta + 2 ) Y _ 0 ^ { ( 3 ) } = \\tilde \\nabla ^ a ( \\alpha _ H ^ { ( 2 ) } ) _ a + \\tilde \\nabla ^ a ( f ^ { ( 1 ) } \\tilde \\nabla _ a C _ i \\tilde X ^ i ) + \\frac { 1 } { 2 } \\tilde \\Delta ( f ^ { ( 1 ) } C _ i \\tilde X ^ i ) . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 ^ + } \\varphi ' ( \\delta ) = + \\infty . \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} I _ \\epsilon [ u ] = \\int _ { \\mathbb { T } ^ d } e ^ { \\frac { | P + \\nabla u ( x ) | ^ 2 } { 2 } + V ( x , \\frac { x } { \\epsilon } ) } d x . \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot c = \\frac { 1 } { 4 \\pi } \\partial _ a E ( Q _ { a , c } ) = c h W ' ( h a ) + \\frac 1 2 c ^ { - 1 } h ^ 3 W ''' ( h a ) + O ( h ^ 5 ) \\\\ & \\dot a = - \\frac { 1 } { 4 \\pi } \\partial _ c E ( Q _ { a , c } ) = c - W ( h a ) + \\frac 1 2 c ^ { - 2 } h ^ 2 W '' ( h a ) + O ( h ^ 4 ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} \\begin{array} { r l } & \\left ( h ( t ) - \\mu ( t , \\theta ^ H _ t ) \\right ) \\frac { \\partial } { \\partial x } \\mu ( t , \\theta ^ H _ t ) \\equiv 0 \\\\ \\Leftrightarrow & \\left ( \\sum _ { j = 1 } ^ n \\theta _ j ( t ) \\mathbb { E } [ \\frac { \\partial } { \\partial y _ j } q _ { \\gamma } ( \\Theta ) | \\mathcal { F } _ t ] - \\sum _ { i = 0 } ^ k \\tilde { a } _ i ( \\theta ^ H _ t ) ^ i \\right ) \\sum _ { i = 1 } ^ k i \\tilde { a } _ i ( \\theta ^ H _ t ) ^ { i - 1 } \\equiv 0 \\end{array} \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\textnormal { c h } \\left ( \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } } \\right ) \\right ) = J ^ \\textnormal { c o h } \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} p _ { c , n } : = ( \\rho _ { n + c _ i a } - \\lambda _ { c _ i a } ) ^ { - 1 } \\sum _ { k = 1 } ^ n a _ { c , k } p _ { c , n - k } \\ \\ \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\Theta ( a ) \\Theta ( z ) = \\theta ( a ) z = z \\theta \\sigma ^ { - 1 } \\theta \\theta ( a ) = z \\theta \\sigma ^ { - 1 } ( a ) = \\Theta ( z a ) \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\textrm { I M F } = \\sum _ { k = 0 } ^ { n - 1 } u _ k ( 1 - \\lambda _ k ) ^ { N _ 0 } \\sigma _ k = \\textrm { I D F T } \\left ( ( I - D ) ^ { N _ 0 } \\textrm { D F T } ( s ) \\right ) \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} V ( \\tilde { x } ' , \\tilde { x } _ m , \\tilde { y } ) = \\begin{cases} v ( \\tilde { x } ' , \\tilde { x } _ m , \\tilde { y } ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\tilde x _ m \\ge 0 , \\\\ \\sum ^ { k _ 0 + 1 } _ { i = 1 } c _ i v ( \\tilde { x } ' , - \\frac { \\tilde { x } _ m } { i } , \\tilde { y } ) \\ \\tilde { x } _ m < 0 , \\end{cases} \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} [ \\left ( z \\partial _ { t _ 1 } \\right ) ^ { N + 1 } - e ^ { t _ 1 } ] J = 0 \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} Z ( T ) = \\sum _ { \\emptyset \\not = I \\subseteq J , \\ h ( E _ I ^ { \\circ } ) \\subseteq X _ 0 ( g ) } ( \\L - 1 ) ^ { | I | - 1 } [ \\widetilde { E } _ I ^ { \\circ } ] S _ { I , \\varepsilon , \\ell } ( T ) , \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} w ( x ) = \\left \\{ \\begin{array} { c c } \\frac { 1 } { L } - \\frac { 1 } { L ^ 2 } | x | & \\textrm { f o r } \\ ; | x | \\leq L \\\\ 0 & \\textrm { o t h e r w i s e } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} m ^ { - 1 } = ( x _ { 1 } e _ { 1 } + y _ { 1 } e _ { 2 } , \\ , x _ { 1 } e _ 1 + h ^ { - 1 } _ { 1 } e _ { 1 } + y _ { 1 } e _ { 2 } ) \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\| E _ { p , \\epsilon } ' ( u ) \\| _ { ( W ^ { 1 , p } ) ^ * } = 0 , \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\Phi ( - g \\tilde \\sigma ) = \\Phi ( g \\sigma ) . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} B ^ 4 + X B ^ 2 + B = 0 . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} ( \\N _ X I ) ( Y ) = g ( Z , Y ) X - g ( X , Y ) Z . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} \\begin{array} { c } \\hat { \\mathfrak U } _ { i } ^ { ( n - 1 ) } \\\\ \\updownarrow \\\\ \\hat { \\mathfrak U } _ { i } ^ { ( n - 1 ) } \\end{array} = d \\hat { \\mathfrak U } _ { i } ^ { ( n - 1 ) } . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} ( a ; q ) _ 0 & : = 1 , \\\\ ( a ; q ) _ n & : = ( 1 - a ) ( 1 - a q ) \\cdots ( 1 - a q ^ { n - 1 } ) , \\\\ ( a ; q ) _ { \\infty } & : = \\lim _ { n \\to \\infty } ( a ; q ) _ n , | q | < 1 . \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} f ( X ) = g ( X ) + h ( X ) \\prod _ { \\alpha \\in S } ( X - \\alpha ) , \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\left ( I _ a ^ { k _ 3 } f \\right ) ( x ) = \\left ( \\left ( \\mathbb { K } _ 1 * \\mathbb { K } _ 2 \\right ) * f \\right ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} x _ { N + k } = x _ N e ^ { \\sum _ { i = 0 } ^ { k - 1 } \\ln \\left ( 1 - h _ { N + i } x _ { N + i } ^ 2 \\right ) } . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} f ( n ) : = \\left \\{ \\begin{array} { l l } 2 ^ { l + 1 } - 2 & \\mbox { i f } n = 2 ^ l - 1 \\\\ 2 ^ { l + 1 } - 1 & \\mbox { i f } 2 ^ l \\le n \\le 2 ^ { l + 1 } - 2 \\end{array} \\right . \\mbox { f o r a l l i n t e g e r s $ l \\ge 1 $ } . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} A _ + & = A _ 1 \\oplus A _ 0 \\oplus A _ \\lambda \\oplus A _ { \\lambda - \\frac { 1 } { 2 } } \\oplus A _ { \\nu ^ { 0 } _ + } \\oplus A _ { \\nu ^ { 0 } _ - } \\\\ A _ - & = A _ { \\nu ^ { 1 } _ + } \\oplus A _ { \\nu ^ { 1 } _ - } \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ m ( x , t ) + P ^ - \\big ( D ^ 2 u _ m ( x , t ) \\big ) + \\big \\langle \\mu , D u _ m ( x , t ) \\big \\rangle - r u _ m ( x , t ) = 0 , \\\\ \\partial _ t u _ m ( x , t ) + P ^ + \\big ( D ^ 2 u _ m ( x , t ) \\big ) + \\big \\langle \\mu , D u _ m ( x , t ) \\big \\rangle - r u _ m ( x , t ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} V ( u ) = \\frac { ( 1 - u ^ 2 ) ^ 2 } { 4 } , \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} u _ t + u _ { x x x x } + u _ { x x } + u \\ , u _ x = 0 \\textrm { f o r } 0 \\le x < L , \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\int _ 0 ^ T g _ s \\ , \\mathrm { d } ^ - R _ s ^ H = \\int _ 0 ^ T g _ s \\delta R _ s ^ H + 2 c _ H ^ { B , R } \\int _ 0 ^ T ( \\nabla ^ \\frac { H } { 2 } g _ s ) ( s ) \\delta { B } _ s ^ { \\frac { H } { 2 } + \\frac { 1 } { 2 } } + c _ H ^ R \\int _ 0 ^ T ( \\nabla ^ { \\frac { H } { 2 } , \\frac { H } { 2 } } g _ s ) ( s , s ) \\ , \\mathrm { d } { s } . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\bar { \\mathbb { L } } _ 1 ( A _ t ) + \\bar { \\mathcal { H } } ( A _ t , ( \\bar { \\mathbb { L } } _ 1 , \\partial _ x \\bar { \\mathbb { L } } _ 1 , \\partial _ { x x } \\bar { \\mathbb { L } } _ 1 ) ( A _ t ) ) \\geq \\frac { \\delta } { t ^ 2 } , ~ t \\in [ 0 , T ) , ~ A _ t \\in \\Lambda \\\\ \\bar { \\mathbb { L } } _ 1 ( A _ T ) = m ( A _ T ) - \\frac { \\delta } { T } , ~ A _ T \\in \\Lambda _ T . \\end{cases} \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} S _ k ( t ) = \\sum _ { j = 1 } ^ t \\boldsymbol { e } _ k ^ \\top \\boldsymbol { W } ^ { t + 1 - l } \\boldsymbol { \\eta } ( j ) , \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\lVert f \\rVert _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } = \\sup _ { x _ 0 ' \\in G } \\lVert f \\rVert _ { L ^ \\infty ( B ( x ' _ 0 , R ) ; L ^ p _ z ) } , R > 0 . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} \\frac { | D _ \\ell ( a _ 1 , \\ldots , a _ \\ell ) | } { | I _ \\ell ( a _ 1 , \\ldots , a _ \\ell ) | } \\ \\stackrel { { e } } { \\sim } \\ \\sum _ { i = s _ { \\ell + 1 } - t _ { \\ell + 1 } } ^ { s _ { \\ell + 1 } + t _ { \\ell + 1 } } i ^ { - d } \\ \\stackrel { { e } } { \\sim } \\ t _ { \\ell + 1 } s _ { \\ell + 1 } ^ { - d } . \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} | x _ { N + 1 } | = | x _ N | | 1 - h _ N x ^ 2 _ N | < | x _ N | . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\mathfrak { A } _ { M _ S , \\mathbb { Q } } : = \\{ x \\in \\mathfrak { A } _ { \\mathbb { Q } } : \\langle x , \\alpha \\rangle = 0 , \\alpha \\in S \\} , \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { \\pi ( x ) } \\sum _ { p \\le x } N _ { p } ^ k ( E [ \\ell ] ) = \\frac { \\ell ^ 4 - 2 \\ell ^ 3 - \\ell ^ 2 + 3 \\ell } { ( \\ell ^ 2 - \\ell ) ( \\ell ^ 2 - 1 ) } + \\ell ^ k \\frac { \\ell ^ 3 - 2 \\ell - 1 } { ( \\ell ^ 2 - \\ell ) ( \\ell ^ 2 - 1 ) } + \\ell ^ { 2 k } \\frac { 1 } { ( \\ell ^ 2 - \\ell ) ( \\ell ^ 2 - 1 ) } . \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\epsilon { } _ { \\eta } : = \\frac { 1 } { 2 } \\cdot \\min { \\{ 1 , { \\rho } ( R _ 0 , M _ 4 , M _ 5 ) , { \\rho } ( { \\eta } , M _ 4 , M _ 5 ) \\} } \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} Y _ 1 ^ { \\perp } & = \\cos ( \\theta _ 1 - \\theta _ 2 ) ( J X _ 1 - J X _ 2 ) = 2 J \\nabla ( \\sin ( \\theta _ 1 - \\theta _ 2 ) ) , \\\\ Y _ 2 ^ { \\perp } & = - \\sin ( \\theta _ 1 - \\theta _ 2 ) ( J X _ 1 - J X _ 2 ) = 2 J \\nabla ( \\cos ( \\theta _ 1 - \\theta _ 2 ) ) , \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\rho ( e , u , f ) = \\rho ( e , g , g ) \\rho ( g , u , h ) \\rho ( h , h , f ) \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} f ( X _ v ) = d _ G ( v ) \\mbox { f o r a l l } v \\in V ( G ) . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} v \\tau _ a = \\begin{cases} v & \\mbox { i f } v \\in A _ + \\\\ - v & \\mbox { i f } v \\in A _ - \\end{cases} \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} a _ k ( n , n ) = \\frac { ( - 1 ) ^ n \\Gamma ( 1 + \\alpha + n + k ) \\Gamma ( 1 + \\alpha + n - k ) } { \\Gamma ( 1 + \\alpha ) \\Gamma ( 2 n + \\alpha + 2 ) } \\ , \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\boxtimes ^ k _ { i = 1 } [ M _ { S _ i } , \\mu _ { S _ i } ] : \\mathrm { G r o t h } ( G ( \\mathbb { Q } _ p ) ) \\to \\mathrm { G r o t h } ( G ( \\mathbb { Q } _ p ) \\times W _ { E _ { \\{ \\mu _ S \\} _ { M _ S } } } ) \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { I \\in \\mathcal { D } } a _ I h _ I \\Big \\| _ { S L ^ \\infty } = \\Big \\| \\Bigl ( \\sum _ { I \\in \\mathcal { D } } a _ I ^ 2 h _ I ^ 2 \\Bigr ) ^ { 1 / 2 } \\Big \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} & \\left < Z _ \\alpha ( \\varphi ) ( t ) , x ^ \\star \\right > _ { E _ 1 , E _ 1 ^ \\star } = \\int _ 0 ^ t ( t - s ) ^ { - \\alpha } \\left < d w ( s ) , [ S ( t - s ) G ( s , \\varphi ( s ) ) ] ^ \\star x ^ \\star \\right > _ H \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\begin{aligned} \\L \\Theta ( X ( t ) ) & \\leq - \\frac { \\gamma } { 2 m } v ( t ) ^ 2 - \\frac { 1 } { m } \\sum _ { k = 1 } ^ N \\lambda _ k z _ k ( t ) ^ 2 - a _ 1 \\sum _ { k > N } k ^ { - 2 s } \\lambda _ k z _ k ( t ) ^ 2 + a , \\end{aligned} \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} S D : = \\frac { \\| s _ { m + 1 } - s _ { m } \\| _ 2 } { \\| s _ { m } \\| _ 2 } \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} N _ { \\alpha \\beta } ^ { \\gamma } = N ^ { \\alpha } _ { \\gamma \\overline { \\beta } } = N ^ { \\beta } _ { \\overline { \\alpha } \\gamma } \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} & ( n - 1 ) ! V ( k ) \\\\ = & \\sum _ { 0 \\le h \\le n \\atop 1 \\le l \\le n - 1 } ( - 1 ) ^ { n + k + h } \\sum _ { m = 1 } ^ { n - 1 } { k \\choose n - h - m + l } { n - k \\choose m - l } M ( l - h a ) ^ { n - 1 } . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\Delta u ( x ) \\Delta \\varphi ( x ) = \\int _ { \\Omega } \\Delta ^ 2 u ( x ) \\varphi ( x ) d x . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} x - \\mu ^ 2 \\sum \\limits _ { i = 1 } ^ m \\cfrac { 1 } { A _ i x - b _ i } A _ i ^ T = z \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} \\sin ( 2 \\hat \\theta _ 2 ) = 2 \\sin ( \\hat \\theta _ 2 ) \\cos ( \\hat \\theta _ 2 ) = 2 \\cdot \\frac { 5 } { 1 3 } \\cdot \\frac { 1 2 } { 1 3 } = \\frac { 1 2 0 } { 1 6 9 } = \\sin ( \\hat \\theta _ 1 ) . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u _ t ( x ) = - ( - \\Delta ) ^ { \\alpha / 2 } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot { F } ( t , x ) , \\ \\ \\ x \\in D , \\ \\ t > 0 , \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\lim _ { \\substack { \\rho \\to 0 , \\\\ \\frac { m } { \\rho } } = } \\eta ( \\rho , m ) = 0 . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\partial _ { t } g _ { i j } ( x , t ) = - 2 R i c _ { i j } ( x , t ) , \\ ; \\ ; \\ ; ( x , t ) \\in M \\times [ 0 , T ] , \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\varphi ( y ) = \\displaystyle \\sum _ { s ; z _ { i + 1 } \\in \\partial I ( z _ i ^ \\prime ) } G _ { I ( y ) } ( k , z _ 1 ) G _ { I ( z _ 1 ^ \\prime ) } ( z _ 1 ^ \\prime , z _ 2 ) \\cdots G _ { I ( z _ s ^ \\prime ) } ( z _ s ^ \\prime , z _ { s + 1 } ) \\varphi ( z _ { s + 1 } ^ \\prime ) , \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ { x _ j } d y = 0 . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} \\| T ( f _ 1 , f _ 2 , \\dots , f _ m ) \\| _ { L ^ s ( w ) } \\lesssim \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { s _ i } \\left ( w _ i \\right ) } , \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} f ( z ) = z g ( z ) = \\max ( z / R ) \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} ( x + y ) ^ { - n } = \\sum _ { k \\geq 0 } \\binom { - n } { k } _ q x ^ k y ^ { - n - k } . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\langle \\hat u _ j , E u _ k \\rangle & = \\sum _ { l \\geq 1 } \\langle \\hat u _ j , u _ l \\rangle \\langle u _ l , E u _ k \\rangle \\\\ & = \\langle \\hat u _ j , u _ j \\rangle \\langle u _ j , E u _ k \\rangle + \\sum _ { l \\neq j } \\langle \\hat u _ j , u _ l \\rangle \\langle u _ l , E u _ k \\rangle \\\\ & = \\langle \\hat u _ j , u _ j \\rangle \\sqrt { \\lambda _ k \\lambda _ j } \\bar \\eta _ { k j } + \\sum _ { l \\neq j } \\langle \\hat u _ j , u _ l \\rangle \\sqrt { \\lambda _ k \\lambda _ l } \\bar \\eta _ { k l } \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\rho ( \\ T f _ { n } - w ) \\leq \\lim _ { n \\rightarrow \\infty } \\rho ( \\ f _ { n } - w ) = \\ m . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} ( D _ a ^ { k ' } f ) ( x ) = \\varphi ( x ) \\omega ^ { - 1 } ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} ( \\beta _ 1 - t _ 1 ) ( \\beta _ 2 - t _ 2 ) = ( a - \\sigma ^ 2 s ) ^ 2 \\beta _ 1 < 0 . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} J = \\begin{bmatrix} A _ { 1 \\times 1 } & B _ { 1 \\times n - 1 } \\\\ C _ { n - 1 \\times 1 } & D _ { n - 1 \\times n - 1 } \\end{bmatrix} \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} I _ 2 & = ( v , \\delta _ { \\varepsilon , x _ 0 ' } | v | ^ { p - 2 } v ^ * ) _ { \\Omega ' } = ( v , \\lambda ^ * w - \\Delta w + \\nabla _ H \\Pi ) _ { \\Omega ' } = ( \\lambda v - \\Delta v + \\nabla _ H \\pi , w ) _ { \\Omega ' } = ( \\partial _ z f , w ) _ { \\Omega ' } \\\\ & = - ( f , \\partial _ z w ) _ { \\Omega ' } , \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} ( D _ { E _ \\pm } \\mathcal { J } ) E _ \\mp = 0 , ( D _ { E _ \\pm } \\mathcal { J } ) E _ \\pm \\subset E _ \\pm . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\lambda g - O ^ 2 g = f , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\mathbb { P } _ l ^ \\eta ( t , x , A ) = \\int _ { \\frak { g } ^ { ( l ) } } \\eta ( u ) { \\bf 1 } _ { \\{ F _ l ( u , x ) \\in A \\} } \\rho _ t ( u ) d u . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} l ( G ) = \\max \\{ \\O ( q - 1 ) + f , \\O ( q + 1 ) + 1 \\} , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\bigg \\langle \\int _ { - \\infty } ^ { t } | \\nabla | ^ { 2 \\sigma } e ^ { i ( t - s ) \\sqrt { 1 - \\Delta } } P _ k ^ 2 F ( \\cdot , s ) d s , G ( x , t ) \\bigg \\rangle _ { L _ { x , t } ^ 2 } = \\sum _ { j \\ge 0 } B _ j ( F , G ) , \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\psi ( x ) : = x ^ 2 e ^ { x } \\ , , 0 \\le x \\ , . \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\mathcal { V } _ n = \\{ U \\in \\mathbb { R P } ^ 1 \\colon d ( U , V ) < \\tfrac { 1 } { n } V \\in X _ u \\} \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\sigma ( t , \\theta , \\phi ) = ( r ( t , \\theta ) \\cos ( \\phi ) , r ( t , \\theta ) \\sin ( \\phi ) , z ( t , \\theta ) ) \\ ; , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq t \\leq T } \\| { \\cal I } _ { a , m } ^ { - 1 } \\frac { \\partial } { \\partial t } P _ t ^ { m } f \\| _ { \\infty } = \\sup _ { 0 \\leq t \\leq T } \\| { \\cal I } _ { a , m } ^ { - 1 } G _ m P _ t ^ { m } f \\| _ { \\infty } = \\sup _ { 0 \\leq t \\leq T } \\| { \\cal I } _ { a , m } ^ { - 1 } P _ t ^ { m } G _ m f \\| _ { \\infty } < \\infty . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} & C _ { 2 n } ^ { \\lambda } ( t ) = a _ n ^ { \\lambda } \\ , \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - n , n + \\lambda ; \\lambda + \\frac { 1 } { 2 } ; 1 - t ^ 2 \\Big ) , \\\\ [ 4 p t ] & C _ { 2 n + 1 } ^ { \\lambda } ( t ) = b _ n ^ { \\lambda } \\ , t \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - n , n + \\lambda + 1 ; \\lambda + \\frac { 1 } { 2 } ; 1 - t ^ 2 \\Big ) , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} U '' + \\frac { N - 1 } { r } U ' - V ( r ) U = 0 \\quad \\mbox { i n } \\quad ( 0 , \\infty ) \\quad \\mbox { w i t h } \\lim _ { r \\to 0 } r ^ { - A ^ + ( \\lambda _ 1 ) } U ( r ) = 1 . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\Lambda _ i = q ^ { - \\frac { \\lambda _ i } { z } } \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} v _ { p _ { \\infty } } ( ( r _ { k } - A ) ) = 1 + \\sum _ { i = 1 } ^ k \\frac { n - 1 } { ( n - a ) ^ i } . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\Xi ( \\mathrm { e } _ { [ \\alpha ] } ) = V _ { [ \\alpha ] } , \\ \\ \\ \\forall \\alpha \\in \\mathcal { A } _ 1 ( l ) . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} u _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon } - \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon ^ 3 } - ( A ( \\gamma _ \\varepsilon ) - 2 \\xi _ \\varepsilon ) \\frac { t _ \\varepsilon } { 2 \\gamma _ \\varepsilon } + o \\left ( \\frac { t _ \\varepsilon \\tilde { \\zeta } _ \\varepsilon } { \\gamma _ \\varepsilon } \\right ) \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} K [ \\ ! [ t ] \\ ! ] _ 0 & = \\left \\{ \\sum _ { i \\in \\mathbb { N } } a _ i t ^ i \\in K [ \\ ! [ t ] \\ ! ] ; a _ i \\in K , \\ \\sup _ { i \\in \\mathbb { N } } { | a _ i | } < \\infty \\right \\} = \\mathcal { O } _ K [ \\ ! [ t ] \\ ! ] \\otimes _ { \\mathcal { O } _ K } K , \\\\ K \\{ t \\} & = \\left \\{ \\sum _ { i \\in \\mathbb { N } } a _ i t ^ i \\in K [ \\ ! [ t ] \\ ! ] ; a _ i \\in K , \\lim _ { i \\to \\infty } { | a _ i | \\eta ^ i } = 0 \\ ( \\eta \\in ( 0 , 1 ) ) \\right \\} . \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } ( 2 - p _ i ) \\| S _ i - S _ i ^ s - S _ i ^ h \\| _ { L ^ 1 } = 0 . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} ( 1 - p _ 1 - p _ 2 ) ^ { n - k _ 1 - k _ 2 } \\geq e ^ { - ( p _ 1 + p _ 2 ) ( n - k _ 1 - k _ 2 ) - ( p _ 1 + p _ 2 ) ^ 2 ( n - k _ 1 - k _ 2 ) } = e ^ { - n p _ 1 } e ^ { - n p _ 2 } e ^ { I _ { 1 } - I _ 2 } , \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} 0 \\le 1 - \\prod _ { k = 1 } ^ { m } ( 1 - x _ k ) \\le \\sum _ { k = 1 } ^ { m } x _ k \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} P ^ 2 < M ^ { 1 - \\delta } L . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} & M = \\max _ { x \\in \\Omega } \\mathcal { H } _ x ( x ) \\ , , \\\\ & K _ \\Omega = \\{ y \\in \\Omega \\mathcal { H } _ y ( y ) = M \\} \\\\ & S = \\max _ { z \\in K _ \\Omega } \\int _ { \\Omega } G _ z ( y ) F ( 4 \\pi G _ z ( y ) ) d y \\ , , \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ \\infty e ^ { \\frac { - \\lambda ^ 2 - \\mu _ { \\eta ^ , 1 } ^ 2 t ^ 2 + 2 \\lambda \\mu _ { \\eta ^ , 1 } t } { 2 \\sigma _ { \\eta ^ , 1 } ^ 2 \\xi t } } & = \\beta \\\\ \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ \\infty e ^ { \\frac { - \\upsilon ^ 2 - \\mu _ { \\eta ^ , 0 } ^ 2 t ^ 2 + 2 \\upsilon \\mu _ { \\eta ^ , 0 } t } { 2 \\sigma _ { \\eta ^ , 0 } ^ 2 \\xi t } } & = \\alpha . \\end{aligned} \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} w _ t = v - \\gamma w _ { \\nu _ \\Lambda } \\quad \\mbox { o n } \\Gamma _ c \\times ( 0 , T ) , \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\mathcal { B } _ { N } = \\{ \\omega \\in \\mathbb { T } \\mid ( \\omega , T ^ { j } x _ { 0 } ) \\in { \\rm P r o j } _ { \\mathbb { T } ^ { d + 1 } } S _ N , \\ \\exists | j | \\sim N _ 2 \\} . \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\| { \\sup _ { R > 0 } } ^ + A _ { R } ( x ) \\| _ 2 \\lesssim \\| G ^ \\alpha ( x ) \\| ^ \\frac { 1 } { 2 } _ 2 \\| G _ * ^ \\alpha ( x ) \\| ^ \\frac { 1 } { 2 } _ 2 = \\| G ^ \\alpha ( x ) \\| _ 2 \\lesssim \\| x \\| _ 2 . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\frac { | \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) | ^ 2 } { 2 } + V ( x , y ) = \\ln \\widetilde { m } _ 1 ( x , y ) + \\widehat { H } ( x , \\widetilde { \\Lambda } ) . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\Delta _ f ( s ) + \\bigl ( \\psi _ f ' ( s ) - \\psi _ { \\bar { f } } ' ( 1 - s ) \\bigr ) \\Lambda _ f ( s ) = \\eta \\epsilon ^ { 1 - k } N ^ { \\frac 1 2 - s } \\Delta _ { \\bar { f } } ( 1 - s ) , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} C _ { j ( \\lambda , \\lambda ' ) \\cap j ( \\lambda ' , \\lambda '' ) } = C _ { j ( \\lambda , \\lambda ' ) } \\cap C _ { j ( \\lambda ' , \\lambda '' ) } \\subset C _ { j ( \\lambda , \\lambda '' ) } , \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { D } _ { 0 1 } } { \\partial \\eta } = \\frac { n ^ 2 ( n \\eta - n _ d ) ( n ^ 2 \\rho ^ 2 \\eta ^ 2 - n ^ 2 \\rho ^ 2 \\eta + n _ d n _ p \\sigma _ w ^ 4 ) } { ( n \\eta \\rho + n _ d \\sigma _ w ^ 2 ) ^ 2 [ ( 1 - \\eta ) n \\rho + n _ p \\sigma _ w ^ 2 ] ^ 2 } . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\pi { G } ) = \\gamma _ 1 { G } / \\gamma _ c { G } > \\gamma _ 2 ( \\pi { G } ) = \\gamma _ 2 { G } / \\gamma _ c { G } > \\ldots > \\gamma _ { c - 1 } ( \\pi { G } ) = \\gamma _ { c - 1 } { G } / \\gamma _ c { G } > \\gamma _ c ( \\pi { G } ) = 1 , \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} G _ D ( x , y ) = \\int _ 0 ^ \\infty p _ D ( t , x , y ) d t , x , y \\in \\R \\ , \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } L _ n & = \\lim _ { n \\rightarrow \\infty } R _ n \\\\ & = \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\pi } \\log ( 1 + a ^ 2 - 2 a \\cos ( w ) ) ~ d w \\\\ & = 2 \\log a , \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} r _ { R } ( n ) = \\sum _ { z } \\sum _ { w } \\# \\{ ( x , y ) \\in \\Z ^ { 2 } : R ( x , y , z , w ) = n \\} . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } \\mathcal { L } \\sum _ { i , j , k , p } ( \\bar h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } = \\sum _ { i , j , k , l , p } ( \\bar h _ { i j k l } ^ { p ^ { \\ast } } ) ^ { 2 } > 0 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} k + p ( T ( 0 , x _ 0 ) ) - p ( x _ 0 ) > 0 . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} d \\geq 3 & : \\phantom { \\ell _ \\nu \\ll } \\ \\ell _ \\nu \\ll \\ell \\ll R \\ll L \\quad { \\rm a n d } \\tau \\ll \\tau _ \\ell , \\\\ d = 2 & : \\ell _ \\nu \\ll \\ell _ f \\ll \\ell \\ll R \\ll L { \\rm a n d } \\tau \\ll \\tau _ \\ell , \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\mathcal { S } _ { n } ^ { \\prime } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { k } \\sin ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n } } , \\\\ \\mathcal { C } _ { n } ^ { \\prime } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { k } \\cos ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n + 1 } } , \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\dfrac { 1 } { [ T _ n ( z ) ] ^ { 2 s } } = \\sum _ { k = 0 } ^ { + \\infty } \\beta ^ { ( s ) } _ { n , k } u ^ { - 2 n s - k } , \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\hat { H } ( t , x , y ) = \\int _ { \\mathcal { G } } H _ { \\chi } ( t , x , y ) \\ , d \\chi \\ , , \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } f ( w _ n ( x , y ) ) \\phi ( x , y ) d y d x = \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } f ( w ( x , y ) ) \\phi ( x , y ) d y d x . \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} s ( g ( x ) ) = \\chi ( g ) s ( x ) \\ , , \\forall x \\in \\hat { M } , \\ g \\in G \\ , . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\dim F \\ = \\ \\dim P - \\dim B \\ \\ge \\ n - 1 - ( n - c ) \\ = \\ c - 1 \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\frac { X ( s ) ^ { \\perp } } { 2 } & = \\frac { 1 } { 2 } ( X ( s ) - g ^ { i j } \\langle X ( s ) , X _ i ( s ) \\rangle X _ j ( s ) ) \\\\ & = \\sqrt { 2 } \\left ( \\frac { 1 } { 2 } + \\frac { s } { 2 } f - s ^ 2 \\left ( \\left ( \\frac { \\partial f } { \\partial \\theta _ 1 } \\right ) ^ 2 + \\left ( \\frac { \\partial f } { \\partial \\theta _ 2 } \\right ) ^ 2 \\right ) \\right ) ( e ^ { i \\theta _ 1 } , e ^ { i \\theta _ 2 } ) + T _ 2 ( s ) , \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} f ( z ) = \\left \\langle f , B _ z ^ { D , \\mu } \\right \\rangle _ { \\mu } . \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } W P W ^ \\ast = W _ P ^ \\ast P W _ P \\\\ \\| \\mathbf { 1 } _ n - W _ P \\| \\leq \\sqrt { 2 } \\| W P W ^ \\ast - P \\| \\end{array} \\right . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} U _ * = 0 \\ \\ { \\rm a n d } \\ \\ \\underset { u \\rightarrow U _ * } { \\lim } \\zeta ( u ) = 0 . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} - \\int _ { \\Sigma } \\overline { q } ^ 2 d \\overline { \\sigma _ \\Sigma } + \\int _ { \\Sigma } A d \\sigma _ \\Sigma + \\sum _ { i = 1 } ^ n \\int _ { \\gamma _ i } k ^ { \\infty , s } _ { \\gamma _ i , \\Sigma } d \\overline { s } = 0 . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{gather*} D \\iota _ { \\rho } \\widetilde { h } = 0 . \\end{gather*}"}
-{"id": "992.png", "formula": "\\begin{align*} & \\mathfrak { R } ( f , \\sigma , \\{ k _ j \\} , L ) = \\\\ & \\left \\{ \\{ R _ i \\} \\in [ 0 , L - 1 ] ^ n \\left | \\begin{array} { l } R _ 1 + R _ 4 \\equiv R _ 2 + R _ 3 \\bmod L ( R _ 1 + R _ 4 , L ) = 1 , \\\\ R _ 1 + R _ 4 = 1 \\mathbb { Q } ( \\zeta _ 8 ) \\cap \\mathbb { Q } ( \\zeta _ L ) \\end{array} \\right . \\right \\} . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} \\binom { a + b } { a } \\alpha _ L ^ a ( - 1 ) ^ b \\sum _ { \\substack { \\beta _ 1 , \\dots , \\beta _ { q - 1 - a - b } \\in \\mathbb { F } _ q \\\\ , \\neq \\beta _ L } } ^ { } \\prod _ { j = 1 } ^ { q - 1 - a - b } \\beta _ j . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\int _ 0 ^ x t ^ \\nu \\mathbf { L } _ \\nu ( t ) \\ , \\mathrm { d } t = \\frac { x ^ { 2 \\nu + 2 } } { \\sqrt { \\pi } 2 ^ { \\nu + 1 } ( \\nu + 1 ) \\Gamma ( \\nu + \\frac { 3 } { 2 } ) } { } _ 2 F _ 3 \\bigg ( 1 , \\nu + 1 ; \\frac { 3 } { 2 } , \\nu + \\frac { 3 } { 2 } , \\nu + 2 ; \\frac { x ^ 2 } { 4 } \\bigg ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\P [ \\sigma _ 0 < \\sigma _ { J + 1 } V _ k = 1 ] = \\frac { 1 } { 2 ( J + 1 - k ) k } . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} K = w _ { x x } w _ { y y } - w _ { x y } ^ { 2 } , H = \\frac { w _ { x x } + w _ { y y } } { 2 } . \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} f ( \\rho ^ \\varepsilon , u ^ \\varepsilon ) \\rightharpoonup \\overline { f ( \\rho , u ) } ( x , t ) : = \\langle \\nu _ { x , t } , f ( \\rho , u ) \\rangle . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} U _ { l _ 0 } ( x , 0 ) & = U _ { l _ 0 } ( x _ 1 - c \\tau - \\sigma ^ * , x ' - \\rho , - \\tau ) \\\\ & = \\cdots = U _ { l _ 0 } ( x _ 1 - c n \\tau - n \\sigma ^ * , x ' - n \\rho , - n \\tau ) \\overset { n \\rightarrow \\infty } { \\longrightarrow } p ^ + _ { l _ 0 } \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} \\Phi _ 1 ( Y _ j ) = X _ j = \\Psi ^ { - 1 } ( Y _ j ) , \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } ( \\alpha _ 1 \\sqrt { k } ) ^ { 1 / k } = \\lim _ { k \\to \\infty } ( \\alpha _ 2 k ^ 2 ) ^ { 1 / k } = 1 \\quad \\therefore \\ , ( k L ( k ) ) ^ { 1 / k } \\to 1 , \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} p _ 1 ( T M ) _ { G } - ( 2 k + 1 ) c _ 1 ( \\xi ) ^ 2 _ { G } = p _ 1 ( T M - \\xi ^ { \\oplus ( 2 k + 1 ) } ) _ { G } = n \\cdot \\pi ^ \\ast q \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\langle \\rho _ { M _ b } , \\mu ' \\rangle + \\langle \\frac { \\det ( A d _ { N _ b } ( M _ b ) ) | _ T } { 2 } , \\mu ' \\rangle = \\langle \\rho _ G , \\mu ' \\rangle , \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} R \\left ( f ( z , 1 ) , f _ t ( z , 1 ) , f \\left ( z , \\frac { 1 } { 1 - z } \\right ) , z , t _ 1 ( z ) \\right ) & = 0 , \\\\ R \\left ( f ( z , 1 ) , f _ t ( z , 1 ) , f \\left ( z , \\frac { 1 } { 1 - z } \\right ) , z , t _ 2 ( z ) \\right ) & = 0 , \\\\ R \\left ( f ( z , 1 ) , f _ t ( z , 1 ) , f \\left ( z , \\frac { 1 } { 1 - z } \\right ) , z , t _ 3 ( z ) \\right ) & = 0 . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\min x _ 1 ^ 2 - x _ 2 + 2 \\cos x _ 2 x _ 2 = 0 . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} B _ \\epsilon = \\{ z \\in \\Omega : B _ \\Omega ( z _ 0 , z ) \\leq \\epsilon \\} , \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { \\gamma } x _ { \\gamma } \\gamma \\Big \\| = \\max _ { \\gamma } | x _ { \\gamma } | \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} q + i + \\ell m + k ' - k = 0 . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} 1 < p _ 1 , p _ 2 , r _ 1 , r _ 2 \\leq \\infty , \\ \\frac 1 2 < q _ 3 < \\infty , \\ \\textstyle \\frac { 1 } { q _ 3 } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } = \\frac { 1 } { r _ 1 } + \\frac { 1 } { r _ 2 } . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} E ( u ; t ) = \\| u ' ( t ) \\| ^ 2 + \\| A ^ { 1 / 2 } u ( t ) \\| ^ 2 . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} ( a _ { i j } ) _ { i \\rho j } \\cdot \\ , ^ { g ^ { - 1 } } A ^ { g ^ { - 1 } } = \\ , ^ { g ^ { - 1 } } \\large ( ( a _ { \\tilde { g } ( i ) \\tilde { g } ( j ) } ) _ { i \\rho j } \\cdot A \\large ) ^ { g ^ { - 1 } } \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} d _ H ( v , K ^ { ( r ) } _ { s , t } ) \\geq \\delta ( H , K ^ { ( r ) } _ { s , t } ) \\geq \\delta _ \\ell = F _ { r , s , t } \\left ( \\left \\lfloor \\frac { r - 1 } { r } \\ell \\right \\rfloor , \\ell \\right ) \\geq \\Omega ( \\ell ^ { ( r - 1 ) s + t - 1 } ) . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\Phi ( t ) : = \\P ( t ) \\circ \\Psi ( t ) \\circ \\L \\circ \\C \\ , , \\Phi ( t ) : \\overline { \\Omega } \\to \\overline { \\Omega } ^ { ( 4 ) } \\ , , \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\partial _ t E = \\nabla \\wedge B - j , \\quad \\nabla \\cdot E = \\rho , \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} P _ { \\textrm { i s o u t } } ^ { \\textrm { I S A O C } } = & \\left ( 1 - \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 1 } \\right \\rbrace \\right ) \\times \\left ( 1 - \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D F = 1 , N D N = 1 } \\right \\rbrace \\right ) . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} H = \\int M ( d t , \\theta ^ H _ t ) + L ^ H \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\overline { g } = \\frac { u ^ 2 } { 1 + t ^ 2 } d t ^ 2 + t ^ 2 g ( t ) . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\{ \\mathfrak q : u ( \\mathfrak q ) = \\mathfrak p _ 1 \\leftrightarrow \\mathfrak p _ 2 , \\mathtt { O u t } ( \\mathfrak q ) = I _ 1 I _ 2 , \\mbox { a n d } \\mathtt { I n } ( \\mathfrak q ) = J _ 1 J _ 2 \\} \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\sum _ { n < R ( n ) ^ { { \\mathcal { S } } / { \\mathcal { T } } } } \\dfrac { R ( n ) ^ { t } } { n ^ s } \\ln \\dfrac { R ( n ) ^ { \\mathcal { S } } } { n ^ { \\mathcal { T } } } = \\sum _ { n > R ( n ) ^ { { \\mathcal { S } } / { \\mathcal { T } } } } \\dfrac { R ( n ) ^ { t } } { n ^ s } \\ln \\dfrac { n ^ { \\mathcal { T } } } { R ( n ) ^ { \\mathcal { S } } } . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} L ( \\varepsilon ) : = \\max \\left \\{ \\dfrac { \\| d \\| _ 1 } { \\varepsilon } , \\sqrt { \\dfrac { n ( \\rho ( B ) ) + \\varepsilon } { a _ k } } \\right \\} , \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\int _ D | \\C ^ \\epsilon _ \\mu f ( z ) | ^ 2 d A ( z ) & = \\int _ D \\Big | \\int _ { | z - w | < \\epsilon } \\frac { f ( w ) } { z - w } d A ( w ) \\Big | ^ 2 d A ( z ) \\\\ & = \\int _ D \\Big | \\int ^ { 2 \\pi } _ 0 \\int ^ \\epsilon _ 0 { e ^ { - i \\theta } } f ( z + r e ^ { i \\theta } ) \\chi _ D ( z + r e ^ { i \\theta } ) d r d \\theta \\Big | ^ 2 d A ( z ) \\\\ & \\leq \\int _ D \\int ^ { 2 \\pi } _ 0 \\int ^ \\epsilon _ 0 | f ( z + r e ^ { i \\theta } ) \\chi _ D ( z + r e ^ { i \\theta } ) | ^ 2 d r d \\theta d A ( z ) \\\\ & \\leq 2 \\pi \\epsilon | | f | | ^ 2 _ { L ^ 2 ( d A ) } , \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\int _ { x > 0 } x \\theta ( z , t ) d z = \\int _ { x < 0 } | x | \\theta ( z , t ) d z \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} A _ { ( 0 , 0 ) } & = A _ 1 \\oplus A _ 0 \\oplus A _ { \\lambda - \\frac { 1 } { 2 } } \\oplus A _ { \\nu ^ { 0 } _ + } \\oplus A _ { \\nu ^ { 0 } _ - } \\\\ A _ { ( 1 , 0 ) } & = A _ { \\nu ^ { 1 } _ + } \\\\ A _ { ( 0 , 1 ) } & = A _ { \\nu ^ { 1 } _ - } \\\\ A _ { ( 1 , 1 ) } & = A _ { \\lambda } \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\delta _ k ( x _ 2 ) = \\mu _ 1 ( V ) - ( 2 k + 1 ) s c \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} C _ { \\textrm { F } } ^ { \\textrm { N O M A } } = \\begin{cases} \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 \\left ( 2 ^ R - 1 \\right ) } { P _ { \\textrm { N } } } , & \\textrm { i f } k > 2 ^ R , \\\\ \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 \\left ( 2 ^ R - 1 \\right ) } { P _ { \\textrm { F } } - P _ { \\textrm { N } } \\left ( 2 ^ R - 1 \\right ) } , & \\textrm { i f } k \\le 2 ^ R . \\end{cases} \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} W _ n = \\coprod _ { 1 \\leq m \\leq n } W _ { n , m } , \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{gather*} J _ { N } ( x , y ) = \\\\ p ^ { N } { \\textstyle \\sum \\limits _ { J \\in G _ { N } ^ { 0 } } } { \\textstyle \\sum \\limits _ { K \\in G _ { N } ^ { 0 } } } { \\textstyle \\sum \\limits _ { J _ { j } \\in G _ { J } ^ { M } } } { \\textstyle \\sum \\limits _ { K _ { j } \\in G _ { K } ^ { M } } } A _ { J K } \\Omega \\left ( p ^ { M } \\left \\vert x - J _ { j } \\right \\vert _ { p } \\right ) \\Omega \\left ( p ^ { M } \\left \\vert y - K _ { j } \\right \\vert _ { p } \\right ) \\end{gather*}"}
-{"id": "8308.png", "formula": "\\begin{align*} C _ s ^ { \\ , i } = \\frac { \\beta } { n } \\left [ \\log \\left ( \\frac { 1 + \\mathrm { S I N R } _ { A _ i , B } } { 1 + \\mathrm { S I N R } _ { A _ i , E } } \\right ) \\right ] ^ + , \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } D \\rho \\tilde X ^ i d S ^ 2 = \\frac { 1 } { 2 0 } \\int _ { S ^ 2 } \\tilde \\nabla ^ a \\tilde \\nabla ^ b D \\alpha _ { a b } \\tilde X ^ i d S ^ 2 = - \\frac { 1 } { 2 0 } \\int _ { S ^ 2 } D \\alpha _ { a b } \\tilde \\sigma ^ { a b } \\tilde X ^ i d S ^ 2 = 0 . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} B _ { W } ( z , w ) & = 2 m \\alpha ^ { \\tfrac { 3 - 2 m + n } { 2 m } } \\sum _ { k = 0 } ^ { \\infty } \\alpha ^ { k / m } \\frac { ( z \\bar { w } ) ^ k } { \\Gamma \\left ( \\frac { 2 k + 2 + n } { 2 m } \\right ) } \\\\ & = 2 m \\alpha ^ { \\tfrac { 3 - 2 m + n } { 2 m } } \\sum _ { k = 0 } ^ { \\infty } \\frac { \\big ( \\alpha ^ { 1 / m } ( z \\bar { w } ) \\big ) ^ k } { \\Gamma \\left ( \\tfrac { k } { m } + \\tfrac { 2 + n } { 2 m } \\right ) } . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty ( m _ k - P _ k ) = \\sum _ { k = 0 } ^ \\infty \\mu _ k a _ k \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} F ( n , k ) & = ( - 1 ) ^ n ( 3 n + 2 k + 1 ) \\frac { ( \\frac { 1 } { 2 } ) _ n ( \\frac { 1 } { 2 } + k ) _ n ^ 2 ( \\frac { 1 } { 2 } ) _ k } { ( 1 ) _ n ^ 2 ( 1 + 2 k ) _ n ( 1 ) _ k } 2 ^ { 3 n } , \\\\ [ 5 p t ] G ( n , k ) & = ( - 1 ) ^ n \\frac { ( \\frac { 1 } { 2 } ) _ n ( \\frac { 1 } { 2 } + k ) _ { n - 1 } ^ 2 ( \\frac { 1 } { 2 } ) _ k } { ( 1 ) _ { n - 1 } ^ 2 ( 1 + 2 k ) _ { n - 1 } ( 1 ) _ k } 2 ^ { 3 n - 2 } . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} | A ^ { p + m } | ^ 2 & = A ^ { * m } | A ^ { p } | ^ 2 A ^ m \\leq A ^ { * m } | A ^ { m + p } | ^ \\frac { 2 p } { p + m } A ^ m \\\\ & = f _ { p + m , p } ( \\frac { p } { p + m } ) \\leq f _ { p + m , p } ( 1 ) \\\\ & = | A | ^ \\frac { 2 ( p + m ) } { p + 2 m } , \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} x ( \\alpha ) = \\frac 1 2 \\left ( \\sqrt { \\alpha ( \\alpha + 2 ) + ( 4 \\cos ^ 2 ( \\frac { \\theta } { 3 } ) - 1 ) ^ 2 } - ( 1 + \\alpha ) \\right ) . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { F } } { \\partial x } = \\left ( \\begin{array} { c c } \\frac { \\partial y } { \\partial x _ { 1 } } & \\frac { \\partial y } { \\partial x _ { 2 } } \\\\ 0 & \\mathrm { I } _ { m - n } \\end{array} \\right ) . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\widehat { U } _ k ( ( T _ { m _ N } x ) \\rtimes \\lambda ( g ) ) & = U ( T _ { m _ N } ( x ) ) \\otimes \\lambda ( g ) = \\left ( m U ( x ) \\right ) \\otimes \\lambda ( g ) \\\\ & = ( m \\otimes \\mathrm { i d } ) \\widehat { U } _ k ( x \\rtimes \\lambda ( g ) ) . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\sigma _ 0 & = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ { V _ k } \\tau _ k ( j ) . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\norm { \\exp ( \\gamma ( \\lambda ) \\ ! \\ ! \\sum \\limits _ { k = \\min ( m , j ) } ^ { \\max ( m , j ) - 1 } \\phi _ \\delta ( \\abs { A _ k } ) ) G _ { m j } ( \\lambda ) } \\le C \\ , . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\begin{cases} \\hat { a } _ { 1 1 } ( t ) = n \\\\ \\hat { a } _ { 1 , l _ k - 1 } ( t ) = \\hat { a } _ { l _ k - 1 , 1 } ( t ) = ( s - k ) t _ { s - k } t \\ ( 0 \\leq k \\leq s - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\frac { ( - i ) ^ { | \\beta | } } { | \\beta | ! } \\partial _ y ^ \\beta \\left ( e ^ { i \\varphi / h } \\right ) = e ^ { i \\varphi / h } \\sum _ { k = 0 } ^ { | \\beta | } h ^ { - k } G _ { k } ^ { ( \\beta ) } ( \\varphi ) \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 = \\dfrac { \\bar H ^ 2 + 2 S + \\sqrt { ( 4 S - 3 \\bar H ^ 2 ) \\bar H ^ 2 } } 4 . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} F ( x ) = \\textrm { c o n v } \\left ( x \\mapsto \\inf _ { y \\in E ^ { * } } \\{ t _ y + \\langle \\xi _ y , x - y \\rangle + \\frac { 1 } { 2 } \\left ( A _ { k ( y ) } + 4 \\| G \\| _ { \\infty } + 1 \\right ) | P ( x - y ) | ^ 2 \\} \\right ) \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\Phi ( n ) \\over n } = \\infty . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} g ( t ) = \\prod _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } ( t - \\lambda _ { 2 j } ) + ( t - \\lambda _ { n + 1 } ) \\sum _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } \\frac { \\sin ^ { 2 } \\left ( \\frac { 2 j \\pi } { n + 3 } \\right ) } { \\sin ^ { 2 } \\left [ \\frac { ( n + 1 ) \\pi } { n + 3 } \\right ] } \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n - 1 } { 2 } } ( t - \\lambda _ { 2 m } ) . \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar h _ { 1 1 2 2 } ^ { 2 ^ { * } } = \\bar h _ { 2 2 1 1 } ^ { 2 ^ { * } } , \\ \\bar h _ { 1 1 1 2 } ^ { 1 ^ { * } } = \\bar h _ { 1 1 2 1 } ^ { 1 ^ { * } } , \\\\ & \\bar h _ { 1 1 1 2 } ^ { 2 ^ { * } } - \\bar h _ { 1 1 2 1 } ^ { 2 ^ { * } } = ( \\bar \\lambda _ 1 - 2 \\bar \\lambda _ 2 ) \\bar K , \\\\ & \\bar h _ { 2 2 1 2 } ^ { 2 ^ { * } } - \\bar h _ { 2 2 2 1 } ^ { 2 ^ { * } } = 3 \\bar \\lambda _ 2 \\bar K . \\end{aligned} \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} u ' _ 0 ( x \\pm ( t - b ) ) = \\mp \\frac { \\partial } { \\partial b } \\big ( u _ 0 ( x \\pm ( t - b ) ) \\big ) . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\left \\langle \\psi _ 1 ^ { k _ 1 } \\alpha _ 1 , \\dots , \\psi _ { n - 1 } ^ { k _ { n - 1 } } \\alpha _ { n - 1 } , \\psi _ n ^ 1 \\right \\rangle ^ \\textnormal { c o h } _ { g , n , d } = ( 2 g - 2 + n ) \\left \\langle \\psi _ 1 ^ { k _ 1 } \\alpha _ 1 , \\dots , \\psi _ { n - 1 } ^ { k _ { n - 1 } } \\alpha _ { n - 1 } \\right \\rangle ^ \\textnormal { c o h } _ { g , n - 1 , d } \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} X _ 1 ( s ) & = \\sqrt { 2 } ( 1 + s f ) ( i e ^ { i \\theta _ 1 } , 0 ) + \\sqrt { 2 } s \\textstyle \\frac { \\partial f } { \\partial \\theta _ 1 } ( e ^ { i \\theta _ 1 } , e ^ { i \\theta _ 2 } ) \\\\ X _ 2 ( s ) & = \\sqrt { 2 } ( 1 + s f ) ( 0 , i e ^ { i \\theta _ 2 } ) + \\sqrt { 2 } s \\textstyle \\frac { \\partial f } { \\partial \\theta _ 2 } ( e ^ { i \\theta _ 1 } , e ^ { i \\theta _ 2 } ) . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} f _ 0 = f _ a + f _ b + f _ c , \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} p ^ + = ( p ^ + _ 1 , p ^ + _ 2 , \\cdots , p ^ + _ m ) \\gg p ^ - = ( p ^ - _ 1 , p ^ - _ 2 , \\cdots , p ^ - _ m ) , \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\avg { | D H | ^ 2 } & = \\frac { 1 } { | \\Lambda | } \\sum _ { j , l } \\beta _ { j l } \\avg { ( x _ j + z _ j ) ( x _ l + z _ l ) } ( 1 - e ^ { i p \\cdot ( j - l ) } ) + \\frac { h } { | \\Lambda | } \\sum _ j \\avg { x _ j + z _ j } \\\\ & \\leq \\frac { 1 } { | \\Lambda | } \\sum _ { j , l } \\beta _ { j l } ( 1 + ( n + 1 ) \\avg { y _ 0 ^ 2 } ) ( 1 - \\cos ( p \\cdot ( j - l ) ) ) + h \\avg { z _ 0 } . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} D \\Psi ( 0 ) \\zeta = \\partial _ y \\psi \\big | _ { y = \\rho _ s } - \\mu ( \\bar \\sigma - \\tilde \\sigma ) \\zeta \\qquad \\mbox { f o r } \\ ; \\ ; \\zeta \\in h ^ { 4 + \\alpha } ( \\Bbb S ) . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\overline { U } _ { 2 k + 1 , 2 a } ( n ) q ^ n = \\sum _ { n = 0 } ^ \\infty \\overline { U } _ { 2 k + 1 , 2 a - 1 } ( n ) q ^ n = \\frac { ( - q ^ 2 ; q ^ 2 ) ^ 2 _ \\infty ( q ^ { 2 a } , q ^ { 4 k - 2 a + 2 } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\chi ( \\xi ) = [ \\rho ^ { 2 \\vartheta } - ( u - \\xi ) ^ 2 ] _ + ^ { \\Lambda } . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} V _ { \\bar { f } } ^ { ( - ) ^ a } & \\ ! \\left ( \\frac { | \\beta n u | } { 1 + u ^ 2 } \\right ) \\cos ^ { ( a ) } \\ ! \\left ( \\frac { 2 \\pi \\beta n } { 1 + u ^ 2 } \\right ) \\\\ & = \\sum _ { j + 2 l < 2 l _ 0 } \\frac { ( - 2 \\pi ) ^ j } { j ! l ! } ( V _ { \\bar { f } } ^ { ( - ) ^ a } ) ^ { ( l ) } ( | \\beta n u | ) \\cos ^ { ( j + a ) } ( 2 \\pi \\beta n ) u ^ l \\left ( \\frac { \\beta n u ^ 2 } { 1 + u ^ 2 } \\right ) ^ { j + l } + O _ { \\alpha , \\sigma , l _ 0 } \\bigl ( | n u | ^ { - \\sigma } u ^ { 2 l _ 0 } \\bigr ) . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} \\| \\nu _ \\varepsilon ^ { 2 / p } w _ \\varepsilon \\| _ { L ^ p ( \\tilde { \\Omega } _ { R , \\varepsilon } ) } = O ( 1 ) \\ , , \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\phi _ m = ( \\zeta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ s , 1 ) \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} u ' _ 1 u ' _ 2 \\dots & = ( u '' _ 1 + u _ 1 ''' ) ( u '' _ 2 + u _ 2 ''' ) ( u _ 3 '' + u _ 3 ''' ) \\dots ( u _ m '' + u _ m ''' ) \\\\ & = u '' _ 1 u '' _ 2 \\dots u _ m '' + \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} & \\left | \\zeta _ { v , t } ^ { \\mathcal { M } } ( x , a ) - \\zeta _ { v , t } ^ { \\mathcal { M } } ( y , b ) \\right | \\\\ & = \\left | \\int _ { - \\infty } ^ { \\infty } [ v ( x + A _ { \\cdot , t + 1 } w ) - v ( y + A _ { \\cdot , t + 1 } w ) ] \\phi ( w ) d w \\right | \\\\ & \\le L _ { v } \\int _ { - \\infty } ^ { \\infty } \\left | ( x + A _ { \\cdot , t + 1 } w ) - ( y + A _ { \\cdot , t + 1 } w ) \\right | \\phi ( w ) d w \\\\ & = L _ { v } | x - y | \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} R _ 2 ( y _ 0 , x _ 0 , x _ 1 ) = x _ 0 ^ { 1 4 } ( x _ 0 - 1 ) ^ 9 ( 9 x _ 0 - 1 ) S _ 2 ( y _ 0 , x _ 0 , x _ 1 ) S _ 3 ( y _ 0 , x _ 0 , x _ 1 ) , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ + } \\frac { V ( K + \\varepsilon L ) - V ( K ) } { \\varepsilon } = \\int _ { S ^ { n - 1 } } h _ L \\ , d S ( K , \\cdot ) \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} n ( x ) = f ( x , y , v ) v = \\frac { a \\varphi _ { 1 } ( y ) } { G _ { 1 } } = \\frac { a u v } { k G _ { 1 } G _ { 2 } } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} \\Lambda : = \\{ l \\in \\{ 1 , 2 , \\cdots , m \\} \\mid \\zeta _ { l } : = u _ { l } - v _ { l } \\not > 0 \\ \\ { \\rm o n } \\ \\ \\R ^ N \\times \\{ t _ 0 \\} \\} \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} \\operatorname { E } \\left ( \\boldsymbol { Y } \\right ) & = \\mu _ 1 \\cdot \\boldsymbol { \\pi } , \\\\ \\operatorname { C o v } \\left ( \\boldsymbol { Y } \\right ) & = \\mu _ 1 \\cdot \\textrm { d i a g } ( \\boldsymbol { \\pi } ) + ( \\mu _ 2 - \\mu _ 1 ^ 2 ) \\cdot \\boldsymbol { \\pi } \\boldsymbol { \\pi } ^ t , \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\partial _ t \\mathbb { L } ( A _ t ; Z _ t ^ u , W _ t ^ u ) = \\lim _ { \\delta t \\downarrow 0 } \\frac { \\mathbb { L } ( A _ { t , \\delta t } ; Z _ { t , \\delta t } , W _ { t , \\delta t } ) - \\mathbb { L } ( A _ t ; Z _ t , W _ t ) } { \\delta t } , \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} f _ { \\lambda ' } ( b _ 1 , b _ 2 ) & = \\lambda / X _ { \\lambda } ( b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 ) , z = b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 \\in U _ 2 . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\nu ( q _ 4 ) = \\mu _ 3 . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\norm { \\mathcal { R } _ N } _ { \\mathcal { B } ( L ^ 2 ( D ) , H ^ s ( D ) ) } \\leq \\norm { \\log \\kappa } _ { L ^ { \\infty } ( \\Omega , L ^ 2 ( D ) ) } \\norm { \\log \\kappa } _ { L ^ { \\infty } ( \\Omega , H ^ s ( D ) ) } . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} w _ { k } ( t , i ) & = W \\left ( \\frac { \\eta _ k ( t ) - \\hat { \\eta } _ k ^ ( t , i ) } { \\hat { \\sigma } ( \\boldsymbol { \\eta } _ k ( t ) ) } \\right ) \\\\ \\hat { \\eta } _ k ^ ( t , i + 1 ) & = \\frac { \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } w _ l ( i ) \\eta _ l ( t , i ) } { \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } w _ l ( i ) } \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} ( x \\cdot g ) \\cdot a = ( x \\cdot ( g \\cdot a ) ) \\cdot g \\ ; \\ ; \\ ; \\ ; \\ ; \\mbox { f o r a n y } x \\in B , a \\in A , g \\in G . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} T _ m = \\lim _ { a \\to \\infty } \\frac { 1 } { 2 a } \\int _ { - a } ^ { a } \\sigma _ { r } \\circ T _ { \\varphi } \\circ \\sigma _ { - r } d r , \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} R _ { n , s } ( f ) \\equiv R _ { n , s } ( f T _ n ) = \\frac { 1 } { 2 \\pi i } \\oint _ { \\Gamma } K _ { n , s } ( z ) f ( z ) d z , \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} & H ^ 1 _ D ( \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) ) : = \\{ v \\in H ^ 1 ( \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) ) \\ : \\ v = 0 \\ \\hbox { o n } \\partial _ D \\Omega ^ { ( 4 ) } \\} \\ , , \\\\ & { \\mathcal H } : = \\{ v \\in H ^ 2 ( \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) ) \\ : \\eqref { b c - v } \\ \\hbox { h o l d t r u e } \\} \\oplus \\{ k \\zeta S \\ : \\ k \\in \\mathbb R \\} \\ , , \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} F \\left ( { \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } } \\right ) = 1 - S \\ge q \\left ( \\theta \\right ) - { \\epsilon _ 2 } = \\frac { 1 } { 2 } q \\left ( \\theta \\right ) \\ge { \\epsilon ^ * } , \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} f _ { U } ^ { \\sharp } ( x ) = \\mathrm { e s s \\ , s u p } _ { y \\in x U } | f ( y ) | . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\alpha = \\max \\{ | K | : K \\in \\mathcal { B } _ I , \\ I \\in \\mathcal { D } _ { \\leq n } \\} . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\theta _ { \\min } \\triangleq \\min _ { w \\in [ - \\pi , \\pi ] } ~ \\frac { \\sigma ^ 2 } { g ( w ) } = \\frac { \\sigma ^ 2 } { ( a + 1 ) ^ 2 } \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} E [ g ( X ) ] = \\int _ { - \\infty } ^ \\infty g ( x ) f ( x ) d x , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } ^ { 2 } & \\leq C _ { H } \\int _ { 0 } ^ { 1 } d t \\int _ { t } ^ { 1 } ( s - t ) ^ { - \\frac { 1 } { 2 } - H } | \\varphi ( s ) | ^ { 2 } d s = C _ { H } \\int _ { 0 } ^ { 1 } | \\varphi ( s ) | ^ { 2 } d s \\int _ { 0 } ^ { s } ( s - t ) ^ { - \\frac { 1 } { 2 } - H } d t \\\\ & = C _ { H } \\int _ { 0 } ^ { 1 } s ^ { \\frac { 1 } { 2 } - H } | \\varphi ( s ) | ^ { 2 } d s \\leq C _ { H } \\| \\varphi \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } ^ { 2 } = C _ { H } \\| f \\| _ { \\mathcal { H } } ^ { 2 } , \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} - D ^ 2 \\log \\abs { \\det D F \\circ G } ( v , v ) & = t r ( A ^ 2 ) \\geq \\\\ & \\geq \\frac { 1 } { n - m } ( t r A ) ^ 2 \\geq \\frac 1 { N - m } ( D \\rho ( v ) - t r A ) ^ 2 - \\frac { ( D \\rho ( v ) ) ^ 2 } { N - n } . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\nabla f _ 1 ( y _ t ) - \\nabla f _ 2 ( y _ t ) = y _ t - ( 1 , 0 ) = ( - \\sin ^ 2 t , \\sin t \\cos t ) = \\sin t ( - \\sin t , \\cos t ) \\perp \\nabla f _ 1 ( y _ t ) \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} H = - \\Delta - 5 W ^ { 4 } ( x ) \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\alpha _ { f } ( x ) = 1 \\iff \\# \\ O _ { f } ( x ) < \\infty \\iff x \\in X ( \\overline { \\mathbb Q } ) _ { \\rm t o r s } \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} q _ { k } ^ { - 1 } a _ { l k } ^ { - 1 } p _ { l } = \\Delta . \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( m ) = \\sigma \\left ( X _ { m n } ( [ 0 , 1 ] ) \\times W _ { m n } ( [ \\omega _ 1 , \\omega _ 2 ] ) \\times W _ { m n } ( [ \\phi _ 1 , \\phi _ 2 ] ) \\right ) \\ ; , \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\tau ^ { \\Gamma } _ p ( M ) = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log _ p \\chi ( \\Gamma _ n , M ) \\ ; \\Q _ p \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} G _ N ^ { ( y ) } ( s ) = \\frac { ( b ) _ n } { ( a + b ) _ n } \\frac { ( - n ) _ y ( a ) _ y } { ( - b - n + 1 ) _ y } { } _ 2 F _ 1 \\{ ( - n + y , a + y ) ; - b - n + 1 + y ; s \\} , \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i , 1 } & = w _ { i } ^ { \\ast } + v _ { 1 } ^ { \\ast } = w _ { i } ^ { \\ast } , \\\\ \\Gamma ( k ) _ { 1 , j } & = w _ { 1 } ^ { \\ast } + v _ { j } ^ { \\ast } = v _ { j } ^ { \\ast } , \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} ( \\sigma _ \\lambda ^ \\phi - w ) ( \\sigma _ \\lambda ^ \\phi - 1 ) _ + = ( \\sigma _ \\lambda ^ \\phi - 1 ) ( \\sigma _ \\lambda ^ \\phi - 1 ) _ + + ( 1 - w ) ( \\sigma _ \\lambda ^ \\phi - 1 ) _ + \\geq 0 \\quad \\Omega \\times Q \\ , . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} \\int _ { \\mathcal { H } _ { - s } } \\ ! \\ ! \\ ! \\ ! \\L ^ R \\psi ( X ) \\mu ( d X ) = \\int _ { \\mathcal { H } _ { - s } } \\ ! \\ ! \\ ! \\ ! v \\Phi ' ( x ) \\left ( 1 - \\theta ^ R ( x ) \\right ) \\psi ( X ) \\mu ( d X ) . \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} & h _ { N _ 1 + 2 } x _ { N _ 1 + 2 } ^ 2 < 2 h _ { N _ 1 + 2 } \\left [ x _ { N _ 1 + 1 } ^ 2 + u _ { N _ 1 + 2 } ^ 2 \\right ] = 2 e ^ { - 9 } \\left [ x _ { N _ 1 + 1 } ^ 2 + 1 \\right ] \\\\ & = 4 e ^ { - 9 } \\approx 0 . 0 0 0 4 9 < 1 . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} Q ( x ) A ( x ) Q ^ T ( x ) = I \\ , , \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} A \\bigl ( ( T _ { a _ { 0 } } \\circ f _ { 0 } ) \\times \\cdots \\times ( T _ { a _ { r } } \\circ f _ { r } ) \\bigr ) & = A ( T _ { a _ { 0 } } \\circ f _ { 0 } ) \\cup \\cdots \\cup A ( T _ { a _ { r } } \\circ f _ { r } ) \\\\ & = A ( f _ { 0 } ) \\cup \\cdots \\cup A ( f _ { r } ) = A ( f ) . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} u ( 0 , x ) = f ( x ) , x \\in \\Omega , \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} | \\psi ( t ) - \\psi ( u ) | & = 1 - \\psi ( u ) = 1 - \\dfrac { u - a _ { 2 n - 1 } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } = \\dfrac { b _ { 2 n - 1 } - u } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\\\ & \\leq \\dfrac { ( b _ { 2 n - 1 } ) ^ { 1 / k } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } < \\dfrac { ( b _ { 2 n - 1 } ) ^ { 1 / k } } { 2 ^ { 2 n - 1 } ( b _ { 2 n - 1 } ) ^ { 1 / k } } < \\dfrac { 1 } { 2 ^ { 2 n - 1 } } . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} S _ I ( T _ 0 , T _ 1 , \\dots , T _ r ) : = \\sum _ { ( n , \\alpha ) \\in \\mathbb N ^ { r + 1 } } \\sum _ { \\begin{smallmatrix} \\beta \\in \\mathbb N ^ m \\\\ \\theta ' ( \\beta , \\alpha ) \\end{smallmatrix} } \\sum _ { \\begin{smallmatrix} ( k _ j ) _ { j \\in I } \\in \\mathbb N _ { > 0 } ^ I \\\\ ( \\ref { e q 3 . 1 2 } ) _ { n , \\beta } \\end{smallmatrix} } \\L ^ { - \\sum _ { j \\in I } k _ j \\nu _ j } T _ 0 ^ n T _ 1 ^ { \\alpha _ 1 } \\cdots T _ r ^ { \\alpha _ r } . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N D F } } ^ { \\textrm { N O M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { N } } \\right ) P _ \\textrm { F } } { \\left ( 1 - \\beta _ { \\textrm { N } } \\right ) P _ \\textrm { N } + { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } / { | h _ { \\textrm { B N } } | ^ 2 } } , \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} T ( f ^ { \\sharp } ) ~ = ~ \\varphi _ { f ^ { \\sharp } } \\qquad \\forall f ^ { \\sharp } \\in \\mathcal { F } ^ { \\sharp } _ { [ N _ 1 , h _ 2 ] } ~ , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} f ( k ) : = r _ - ( k ) - r _ + ( k ) , \\bar f ( \\bar k ) = - f ( k ) . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} { \\bf E } | \\sum _ { i = 1 } ^ n X _ i | ^ s \\le a _ s \\ \\left [ \\sum _ { r = 0 } ^ { n - 1 } \\alpha ^ { 1 - s / v } ( r ) \\ ( r + 1 ) ^ { s / 2 - 1 } \\right ] \\ \\left ( \\ \\sum _ { i = 1 } ^ n { \\bf E } ^ { 2 / v } | X _ i | ^ v \\ \\right ) ^ { s / 2 } , \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} u _ t ( x ) = \\big ( \\mathcal { G } u _ 0 \\big ) _ t ( x ) + \\xi \\int _ 0 ^ t \\int _ { D } p _ D ( t - s , x , y ) u _ s ( y ) F ( \\delta s , \\delta y ) . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} N ( X , T ) = \\# X ( \\Q , T ) . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + F \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) & \\mathbb R ^ n . \\end{cases} \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} \\sum _ { j = a } ^ \\infty \\sum _ { l = a } ^ \\infty e ^ { - { \\frac { 1 } { 2 } } | j - l | } \\left ( \\int _ { j } ^ { j + 1 } \\theta \\right ) \\left ( \\int _ { l } ^ { l + 1 } \\theta \\right ) \\le \\left ( \\sum _ { j = a } ^ \\infty \\int _ j ^ { j + 1 } \\theta \\right ) ^ 2 = \\left ( \\int _ a ^ \\infty \\theta \\right ) ^ 2 \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\begin{aligned} 1 _ { x L } \\star 1 _ { y L } ( z ) & = \\int _ G 1 _ { x L } ( z t ^ { - 1 } ) 1 _ { y L } ( t ) d t \\\\ & = \\int _ { y L } 1 _ { x L } ( z t ^ { - 1 } ) d t = v o l \\left ( L x ^ { - 1 } z \\cap y L \\right ) \\\\ & = M ( x , y , z ) \\cdot v o l \\left ( L \\cap y L y ^ { - 1 } \\right ) \\in \\mathbb Q , \\end{aligned} \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} \\partial _ { t } z - \\partial _ { x x } z = f \\left ( z , x \\right ) \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} { \\rm J } _ k & = \\mathbb { E } \\left \\lbrace \\left \\| \\textbf { h } ' _ k - \\hat { \\textbf { h } ' } _ k \\right \\| ^ 2 \\right \\rbrace = { \\rm t r } \\left \\lbrace \\textbf { R } ' _ k - \\hat { \\textbf { R } ' } _ k \\right \\rbrace , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} U _ k ( r ) = r ^ k + \\Gamma _ k ( r ) \\quad \\mbox { i f } \\quad \\lambda _ 1 = 0 , U _ k ( r ) = \\Gamma _ k ( r ) \\quad \\mbox { i f } \\quad \\lambda _ 1 > 0 . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { c _ i z _ i } { z _ i + 1 } < 0 . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\frac { ( 2 n - 1 ) ! } { n ! } = ( n + 1 ) ( n + 2 ) \\cdots ( n + n - 1 ) \\equiv ( n - 1 ) ! \\pmod { n } , \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\| u _ \\varepsilon \\| _ { H ^ 1 _ 0 } ^ 2 = 4 \\pi \\left ( 1 + I ( \\gamma _ \\varepsilon ) + o \\left ( \\gamma _ \\varepsilon ^ { - 4 } + | A ( \\gamma _ \\varepsilon ) | + \\gamma _ \\varepsilon ^ { - 3 } | B ( \\gamma _ \\varepsilon ) | \\right ) \\right ) \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} S _ \\phi ( n , n ' ) = \\hat { \\phi } ( n - n ' ) , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} J ( f ) = \\int _ G f ( g ) j ( g ) d g . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} R _ { j , d } = q _ { j \\mathtt { N } } \\sum _ { l = 1 } ^ { d } e ^ { \\i \\frac { 2 \\pi } { d } l j } = q _ { j \\mathtt { N } } \\sum _ { l = 1 } ^ d r ^ l = q _ { j \\mathtt { N } } \\frac { r } { 1 - r } ( 1 - r ^ { d } ) \\ , , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} L _ { k , p , q } \\leq C _ { H , l _ { 0 } } | I _ { k } | \\cdot \\left ( \\frac { \\sum _ { j = 1 } ^ k | I _ { j } | } { | I _ { 1 } | } \\right ) ^ { 2 H - 1 } \\cdot \\left ( \\sum _ { l = 1 } ^ { k - 1 } | I _ { l } | ^ { \\frac { 1 } { 2 } - H } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} u _ { t t } ( t , x ) - \\Delta u ( t , x ) + 2 a \\ , ( - \\Delta ) ^ \\theta u _ t ( t , x ) = 0 \\ , , t \\geq 0 , \\ , \\ , x \\in \\R ^ n , \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} & h = - \\sum _ { I = 1 } ^ { q } ( \\omega ^ { I } ) ^ { 2 } + \\sum _ { A = q + 1 } ^ { q + p } ( \\omega ^ { A } ) ^ { 2 } , \\ ; \\ ; h = g _ { i j } d x ^ { i } d x ^ { j } , \\ ; \\ ; \\mathrm { r a n k } | g _ { i j } | = p + q < m , \\\\ & g _ { i j } = g ( \\partial _ { i } , \\partial _ { j } ) = - \\sum _ { I = 1 } ^ { q } \\omega ^ { I } _ { i } \\omega ^ { I } _ { j } + \\sum _ { A = q + 1 } ^ { q + p } \\omega ^ { A } _ { i } \\omega ^ { A } _ { j } , \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 1 } ^ { s } \\lambda _ { j _ l } b _ { 1 , j _ l } \\right | \\leq r . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} A = \\bigoplus _ { \\lambda \\in \\mathcal { F } } A _ \\lambda \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} X _ u & = \\overline { \\{ u ( A ) : A \\in \\SS ( A ) \\} } , \\\\ X _ s & = \\overline { \\{ s ( A ) : A \\in \\SS ( A ) \\} } . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ k = \\sum _ { l = 0 } ^ N G ^ { k l } \\frac { 1 } { 2 } \\left ( \\partial _ { t _ j } G _ { i l } + \\partial _ { t _ i } G _ { j l } - \\partial _ { t _ l } G _ { i j } \\right ) \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} | | \\C _ { \\mu } \\chi _ Q | | ^ 2 _ { L ^ 2 ( \\mu \\llcorner Q ) } = \\frac { \\pi ^ 2 } { 3 } \\int _ Q \\theta _ { \\mu } ( z ) ^ 2 d \\mu ( z ) + \\frac { 1 } { 6 } c ^ 2 ( \\mu \\llcorner Q ) . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} I = \\sum _ { u \\in \\Phi _ { \\mathrm { i u } } } g _ u p _ u d _ u ^ { - \\alpha } , \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} f | _ { k } \\gamma = ( c z + d ) ^ { - k } f \\left ( \\frac { a z + b } { c z + d } \\right ) . \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { A ( G ) } = \\sum _ { [ \\pi ] \\in \\widehat { G } } d _ \\pi \\bigl \\Vert \\widehat { f } ( \\pi ) \\bigr \\Vert _ 1 \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( H - \\int M ( d t , \\theta ^ H _ t ) \\right ) \\int \\frac { \\partial } { \\partial x } M ( d t , \\theta ^ H _ t ) \\right ] = 0 . \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} \\begin{cases} - u _ 1 '' = \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ + - k \\omega u _ 1 u _ 2 \\\\ - d u _ 2 '' = \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d - ( \\alpha u _ { 1 } - d u _ 2 ) ^ - \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ - - \\alpha k \\omega u _ 1 u _ 2 \\end{cases} \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 1 } + \\bar h ^ { 2 ^ * } _ { 2 2 1 } = 0 \\\\ & 3 \\bar \\lambda _ 2 \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} Q _ 1 T _ 2 v = T _ 1 P _ 1 v v \\in V _ 2 . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} q _ L = \\begin{cases} w ^ { + 1 } _ { p , \\alpha } \\oplus q _ { \\alpha , \\beta } & \\left ( \\frac { - \\beta } { p } \\right ) = + 1 \\\\ w ^ { - 1 } _ { p , \\alpha } \\oplus q _ { \\alpha , \\beta } & \\left ( \\frac { - \\beta } { p } \\right ) = - 1 \\\\ \\end{cases} \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\forall \\mathcal { O } \\in \\mathcal { B } ( \\mathbb { R } ^ d ) , \\forall z \\in \\mathbb { R } ^ d , \\ : \\forall t > 0 , \\ : P _ { n , t } ( z , \\mathcal { O } ) \\triangleq \\mathbb { P } ( Z _ t ^ n \\in \\mathcal { O } \\vert Z _ 0 ^ n = z ) \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F H N } } ^ { \\textrm { N O M A } } = \\frac { P _ \\textrm { N } { | h _ { \\textrm { B N } } | ^ 2 } } { { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } } + \\frac { \\beta _ { \\textrm { F } } \\eta P _ \\textrm { B } | h _ { \\textrm { B F } } | ^ 2 | h _ { \\textrm { N F } } | ^ 2 } { d _ { \\textrm { B F } } ^ { \\alpha } d _ { \\textrm { N F } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} T S P C ^ { ( P ) } _ n \\geq V ^ { ( P ) } _ n = \\sum _ { l = 1 } ^ { N } T _ l ^ { ( P ) } . \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\alpha ( i ) \\stackrel { d e f } { = } \\sup _ n \\max _ { k \\in [ 1 , n ] } \\alpha ( M _ 1 ^ k , M _ { k + i } ^ n ) . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\left ( b c + q ^ { - z } \\right ) M _ n \\left ( q ^ { - z } ; \\ , b , \\ , { c \\over q } ; q \\right ) = { c ( b q ^ { n + 1 } - 1 ) \\over q ^ { n + 1 } } M _ { n + 1 } \\left ( q ^ { - z } ; \\ , b , \\ , c ; q \\right ) + { q ^ { n + 1 } + c \\over q ^ { n + 1 } } M _ n \\left ( q ^ { - z } ; \\ , b , \\ , c ; q \\right ) . \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} & \\Delta ( f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ { 2 - n } - g ) \\\\ & = - \\lambda f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ n ( | \\nabla \\bar b _ i | ^ 2 - g ^ { \\frac { 2 n - 2 } { n - 2 } } ) \\\\ & + \\lambda f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ n g \\frac { \\left ( f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ { 2 - n } \\right ) ^ { \\frac { n } { n - 2 } } - g ^ { \\frac { n } { n - 2 } } } { f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ { 2 - n } - g } ( f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ { 2 - n } - g ) . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} X = \\left ( \\begin{array} { c c c c c c } 0 & 0 & 0 & \\ldots & 0 & X ^ { [ r ] } \\\\ X ^ { [ 1 ] } & 0 & 0 & \\ldots & 0 & 0 \\\\ 0 & X ^ { [ 2 ] } & 0 & \\ldots & 0 & 0 \\\\ \\vdots & { } & { } & \\vdots & { } & \\vdots \\\\ 0 & 0 & 0 & \\ldots & X ^ { [ r - 1 ] } & 0 \\end{array} \\right ) \\ . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} - \\div _ x \\big ( \\widetilde { m } _ 0 ( x ) \\widetilde { m } _ 1 ( x , y ) ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) \\big ) = 0 . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } m ^ { \\flat } ( x ) = \\infty , \\ , \\ , \\ , \\textrm { a n d } \\ , \\ , \\ , m ^ { \\flat } ( x ) \\geq 0 \\textrm { f o r a l l } x \\in X , \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\widehat { G } ( m , N , b , \\varepsilon , s ) \\leq 2 \\sum _ { k = 1 } ^ { e ^ { ( \\log ( m ( 1 + \\varepsilon ) / 2 ) ^ { 1 / b } } } k ^ { - d s } \\sum _ { j = 0 } ^ 2 \\widehat { G } ( ( m - e ^ { ( \\log k ) ^ b } ) ( 1 + \\epsilon ) ^ j , N - 1 , b , \\varepsilon , s ) . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial } { \\partial t } u _ t ( x ) = - ( - \\Delta ) ^ { \\alpha / 2 } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot { F } ( t , x ) \\ \\ \\ x \\in D , \\ \\ t > 0 , \\\\ u _ t ( x ) = 0 \\ \\ \\ x \\in D ^ c \\end{cases} \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { y } & = p _ y ( y , z , u _ h ) - r _ y ( y , z , u _ h ) + u _ b + d _ y ( t ) \\\\ \\dot { z } & = p _ z ( y , z , u _ h ) - r _ z ( y , z , u _ h ) + d _ z ( t ) \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} \\check h _ { \\check \\nu ^ C } = \\bigg \\lfloor \\frac 1 2 ( 1 + \\sqrt { 2 C - 1 } ) \\bigg \\rfloor . \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\dim \\Pi \\mu = \\min \\{ 1 , \\frac { h _ { \\mu } } { - \\chi _ { \\mu } } \\} \\ : . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} F ( T ) = \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\eta T ) ^ { k } } { k ! } , G ( T ) = \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\tilde \\eta T ) ^ { k } } { k ! } \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ' ( p _ n ^ { - 1 } ) \\leq \\lfloor c n \\rfloor ) & \\leq - c H ( 1 / 2 ) \\lim _ { n \\to \\infty } \\frac { n } { a _ c ^ { ( n ) } } = - \\infty . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 + \\alpha ) } _ 1 ( z / c ) , 1 \\rangle _ { \\alpha } = \\kappa ^ 1 _ 1 ( \\alpha ) \\langle z / c , 1 \\rangle _ { \\alpha } = 0 \\ , \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\gamma ( 0 ) & = \\sum _ { j = 1 } ^ { n } \\alpha _ j \\gamma ( j ) + \\sigma ^ 2 _ \\epsilon , \\\\ \\gamma ( h ) & = \\sum _ { j = 1 } ^ { n } \\alpha _ j \\gamma ( h - j ) \\textrm { f o r } h > 0 . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\begin{aligned} { } & | u ( p ) - \\mathbb { L } ( p ) | \\leq | u ( p ) - L _ { \\nu } ( p ) | + | L _ { \\nu } ( p ) - \\mathbb { L } ( p ) | \\\\ & \\leq \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) + \\sum ^ { \\infty } _ { j = 0 } | L _ { \\nu + j } - L _ { \\nu + j + 1 } | _ { \\Omega \\cap B ( \\sigma ^ { \\nu } ) } ~ ~ ~ ~ ~ ~ \\leq \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) + C _ { b } \\sigma ^ { \\nu } \\sum ^ { \\infty } _ { j = \\nu } \\omega ( \\sigma ^ { j } ) . \\end{aligned} \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N - 1 } f ( k ) \\leq \\int _ { 1 } ^ N f ( x ) d x . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} Q \\eta ( x , y ) = P _ y \\eta ( x ) = e ^ { - y \\sqrt { - \\mathcal { L } } } \\eta ( x ) , \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\psi : = g ( Z , Z ) = g ( W , W ) , \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\min \\limits _ { \\sum \\limits _ { i = 1 } ^ n c _ i ^ 2 = 1 } f ( c _ 1 , \\cdots , c _ n ) = f ( \\boldsymbol { c ^ * } ) & = \\mu ^ * \\forall 1 \\leq i \\leq n : ( \\lambda _ i - \\mu ^ { * } ) c _ i ^ * = \\frac { 1 } { N ^ { 1 - \\delta } } \\sum \\limits _ { j = 1 } ^ { n } c _ j ^ { * } . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\nu _ { f \\sigma , \\alpha } = \\nu _ { f , \\sigma ( \\alpha ) } . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} ( \\phi _ i ( X ( k \\Phi ) ) ) = r k \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{gather*} ( k _ j ^ { \\prime } , k _ j ) : = \\begin{cases} ( 0 , 0 ) & ~ m _ j = 0 \\\\ ( m _ j , 0 ) & ~ m _ j > 0 \\\\ ( 0 , - m _ j ) & ~ m _ j < 0 . \\end{cases} \\end{gather*}"}
-{"id": "9941.png", "formula": "\\begin{align*} \\Re z ^ \\beta \\| \\hat u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 = - \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 + \\Re \\int _ \\Omega \\hat f \\overline { \\hat u } d x \\leq \\| \\hat f \\| _ { L ^ 2 ( \\Omega ) } \\| \\hat u \\| _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\epsilon = & \\norm { \\sigma ( 0 ) - \\sigma ( T _ 0 ) } \\leq \\int _ 0 ^ { T _ 0 } \\norm { \\sigma ^ \\prime ( t ) } d t \\leq C \\int _ 0 ^ { T _ 0 } \\delta _ \\Omega ( \\sigma ( t ) ) ^ { 1 / 2 } d t \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\log P ( B _ 1 ^ { ( n ) } ) & = \\lim _ { n \\to \\infty } \\frac { a _ c ^ { ( n ) } } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) \\\\ & = - J ( x _ 0 ) \\lim _ { n \\to \\infty } \\frac { a _ c ^ { ( n ) } } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } = - \\infty . \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} e _ k ( i d : \\ell ^ n _ p ( \\mathbb { C } ) \\rightarrow \\ell ^ n _ q ( \\mathbb { C } ) ) \\sim \\begin{cases} 1 & 1 \\leq k \\leq \\log _ 2 ( 2 n ) , \\\\ \\displaystyle \\biggl ( \\frac { \\log _ 2 ( 1 + 2 n / k ) } { k } \\biggr ) ^ { \\frac { 1 } { p } - \\frac { 1 } { q } } & \\log _ 2 ( 2 n ) \\leq k \\leq 2 n , \\\\ [ 1 0 p t ] \\displaystyle 2 ^ { - \\frac { k - 1 } { 2 n } } n ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } & 2 n \\leq k . \\end{cases} \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} \\left | u _ \\varepsilon ( y ) - \\frac { 4 \\pi G _ { x _ \\varepsilon } ( y ) } { \\gamma _ \\varepsilon } \\right | = o \\left ( \\frac { G _ { x _ \\varepsilon } ( y ) } { \\gamma _ \\varepsilon } \\right ) \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\rho _ B ( X , Y ) = & \\ \\tfrac { 1 } { 2 } \\sum _ { i = 1 } ^ 4 \\Omega ( X , Y , e _ i , J e _ i ) \\\\ = & \\ \\Omega ( X , Y , e _ 1 , e _ 2 ) + \\Omega ( X , Y , Z , W ) \\\\ = & \\ \\Omega ( X , Y , e _ 1 , e _ 2 ) , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\left [ \\frac { u ( x ) - w ( x ) } { \\lambda } \\phi ( x ) - f ( x , u ) \\phi ' ( x ) - \\varepsilon u ( x ) \\phi '' ( x ) \\right ] \\ ; d x = 0 . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} u ( x , y ) = ( u _ 1 , u _ 2 ) = k * \\theta = \\int _ S k ( x - \\xi _ 1 , y - \\xi _ 2 ) \\theta ( \\xi _ 1 , \\xi _ 2 ) d \\xi , \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} \\begin{cases} ( v , u , \\mathbf { w } , \\mathbf { h } , \\theta , \\psi ) | _ { t = 0 } = ( v _ 0 , u _ 0 , \\mathbf { w } _ 0 , \\mathbf { h } _ 0 , \\theta _ 0 , \\psi _ 0 ) ( y ) , & y \\in \\Omega , \\\\ ( u , \\mathbf { w } , \\mathbf { h } , \\theta _ y , \\psi ) | _ { \\partial \\Omega } = 0 . \\end{cases} \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} \\tau _ 5 ( x ) = g _ 5 ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 1 0 1 } { 3 ^ 5 } , \\tfrac { 8 8 } { 2 1 1 } ) . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\Xi _ K = \\nu _ K ^ { - 1 } \\left ( N ( K , o ) \\cap S ^ { n - 1 } \\right ) . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} \\alpha ' = \\frac { 1 } { 2 } ( \\alpha + \\min \\{ \\frac { \\tau } { q - 1 } , 1 \\} ) , \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } ( I - W ) ^ m = U Z U ^ T \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\widetilde Z _ H \\coloneqq \\{ ( \\delta , p ) \\in \\widetilde Y \\times Z \\mid i ( p ) = \\delta ( 0 ) \\} , \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\chi _ 1 ( s ) & = \\sum _ { m \\leq s ^ { - 1 / 2 } } - \\phi ' ( m s ) \\\\ & = K u \\Gamma ( u + 1 ) \\sum _ { m \\leq s ^ { - 1 / 2 } } ( m s ) ^ { - u - 1 } ( 1 + O ( ( m s ) ^ { \\epsilon / 2 } ) ) \\\\ & = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } \\left ( \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { 1 } { m ^ { u + 1 } } + O ( s ^ { \\epsilon / 2 } ) \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { 1 } { m ^ { u + 1 - \\epsilon / 2 } } \\right ) \\\\ & = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} E _ { P } \\left \\{ E _ { P } \\left [ Y \\mid A = a , \\mathbf { O } _ { m i n } \\right ] \\right \\} = E _ { P } \\left [ \\frac { I _ { a } ( A ) Y } { \\pi _ { a } ( \\mathbf { O } _ { m i n } , P ) } \\right ] = E _ { P } \\left [ \\frac { I _ { a } ( A ) Y } { \\pi _ { a } ( \\mathbf { O } , P ) } \\right ] = \\chi _ { a } ( P ; \\mathcal { G } ) \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\left ( \\mathcal { X } ^ { K \\textnormal { t h , e q } } \\left ( q , \\varphi _ { q , z } ^ { - 1 } ( Q ) \\right ) P _ { q , z } \\right ) _ { l i } = \\left ( \\delta _ q \\right ) ^ l \\Lambda _ i ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { 1 } { z ^ { d ( N + 1 ) } } \\frac { ( 1 - q ) ^ { d ( N + 1 ) } Q ^ d } { \\left ( q \\Lambda _ 0 \\Lambda _ i ^ { - 1 } , \\dots q , \\dots , q \\Lambda _ N \\Lambda _ i ^ { - 1 } ; q \\right ) _ d } \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} t ^ { 1 / 2 } \\lVert \\nabla \\tilde { V } _ m ( t ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } \\le C _ 2 \\biggl ( & 2 H _ m ( t ) \\tilde { K } _ { m - 1 } ( t ) + 2 \\tilde { H } _ { m - 1 } ( t ) K _ { m - 1 } ( t ) + K _ m ( t ) \\tilde { K } _ { m - 1 } ( t ) \\\\ & + K _ { m - 1 } ( t ) \\tilde { K } _ { m - 1 } ( t ) + 2 R t ^ { 1 / 2 } \\tilde { K } _ { m - 1 } \\biggr ) \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} \\binom { 2 n - 1 } { n - 1 } = \\prod _ { d | n } \\binom { 2 d - 1 } { d - 1 } ' , \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{gather*} \\Phi _ a = \\rho _ a ^ i ( x ) p _ i + \\alpha _ a ( x ) , \\end{gather*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x } & & x ^ { - } A x . \\end{aligned} \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\int Q _ { a , c } ^ 2 = c \\int Q ^ 2 = 8 \\pi c \\ , , E _ 0 ( Q _ { a , c } ) = c ^ 2 E _ 0 ( Q ) = - 2 \\pi c ^ 2 \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} [ \\xi , \\eta ] _ 0 = [ \\xi , \\eta ] ' + u ^ { - 1 } T [ u ] ( \\xi , \\eta ) \\ , , \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\frac { \\varepsilon ^ 2 } { d ( z _ \\varepsilon , \\partial \\Omega ) ^ 2 } = o \\left ( { \\left ( \\log \\frac { 1 } { \\varepsilon } \\right ) ^ { - 1 } } \\right ) \\ , . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\| \\delta ^ { k + 1 } ( \\tau _ V ^ t ( V ) ) \\| & \\leq \\| \\delta _ V ^ { k + 1 } ( V ) \\| + \\sum _ { l = 0 } ^ k \\sum _ { j = 0 } ^ l { l \\choose j } 2 \\| \\delta _ V ^ j ( V ) \\| \\| \\delta _ V ^ { l - j } \\delta ^ { k - l } ( \\tau _ V ^ t ( V ) ) \\| . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} ( ( B L \\iota ) ^ \\ast \\circ \\nu ) ( q _ 4 ) = ( B L \\iota ) ^ \\ast ( \\mu _ 3 ) + \\lambda ( B L \\iota ) ^ \\ast ( s _ 1 t _ 2 ) . \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} \\inf | X | ^ 2 = \\lim _ { m \\rightarrow \\infty } H ^ 2 ( p _ m ) \\leq 2 , \\lim _ { m \\rightarrow \\infty } | \\nabla ^ { \\perp } \\vec H | ^ 2 ( p _ m ) = 0 . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} t _ \\varepsilon ( \\rho _ \\varepsilon ' ) = ( 1 - \\varepsilon _ 0 ' ) \\gamma _ \\varepsilon ^ 2 \\ , , \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 0 } ^ { q - 1 } a _ j ( z ) \\overline z ^ j , a _ j \\in A _ 1 ( K ) . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} E [ e ^ { k X t } ] = \\int _ 0 ^ \\infty e ^ { k x t } e ^ { - x } d t = \\left ( \\frac { 1 } { 1 - k t } e ^ { - t } \\right ) e ^ t . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} ( M _ { S _ 1 } , \\mu _ { S _ 1 } ) = ( M _ { S ^ 0 } , \\mu _ { S ^ 0 } ) \\leq . . . \\leq ( M _ { S ^ k } , \\mu _ { S ^ k } ) = ( M _ { S ' _ 1 } , \\mu _ { S ' _ 1 } ) \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} g _ 2 ( a , b ) & = - g _ 1 ( b ^ { - 1 } , b ) + g _ 1 ( b ^ { - 1 } , b ) g _ 1 ( a , a ^ { - 1 } ) \\\\ & = - g _ 1 ( a , b ) + g _ 1 ( a , b ) g _ 1 ( a , a ^ { - 1 } ) \\\\ & = - g _ 1 ( a , b ) + g _ 1 ( a , a ^ { - 1 } b ) , \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} u _ { t } + f ( x , u ) _ { x } = 0 , f ( x , u ) ~ = ~ \\alpha ( x ) \\ , u ( 1 - u ) \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} L = \\lim _ { n \\to \\infty } x _ { n } > x _ N e ^ { - 2 x _ N ^ 2 S } > 0 . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\mathbb { E } _ 0 T ^ { ( P ) } _ l = I ^ { ( P ) } _ 1 + I ^ { ( P ) } _ 2 \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} w _ i ^ { \\frac { \\theta _ i } { p _ i } } = w _ i ^ { \\frac { \\theta _ i } { q _ i } } \\in A _ { \\frac { 1 - r } { r } \\theta _ i } , \\mbox { w h e r e } \\frac 1 { \\theta _ i } : = \\frac { 1 - r } r - \\frac 1 { \\widetilde { \\delta } _ i } ; \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} U _ t + F ( U ) _ x = 0 \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} S : = \\sum _ { k = 1 } ^ \\infty f ( k ) \\leq n ! \\Big ( \\frac { 1 } { ( s - 1 ) ^ { n + 1 } } + \\frac { 1 } { \\sqrt { 2 \\pi n } ( s - 1 ) ^ n } \\Big ) . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} & \\left ( \\int _ { \\Omega \\cap \\{ | \\eta | \\leq 2 | \\gamma | \\} } \\left | | \\eta \\gamma | ^ { \\alpha - Q } - | \\eta | ^ { \\alpha - Q } \\right | ^ { Q ' } d \\eta \\right ) ^ { 1 / Q ' } \\\\ \\leq & C \\left ( \\int _ { \\{ | \\eta | \\leq B | \\gamma | \\} } | \\eta | ^ { ( \\alpha - Q ) Q ' } d \\eta \\right ) ^ { 1 / Q ' } \\\\ \\leq & C | \\gamma | ^ { \\alpha - Q + \\frac Q { Q ' } } = C | \\gamma | ^ { \\alpha ' } . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} | G _ { I ( \\hat { y } ) } ( \\hat { y } , x _ i ) | \\leq e ^ { - \\tau | \\hat { y } - x _ i | } , \\ ; i = 1 , 2 , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\mu ( J X ) = & \\ \\mu ^ Z ( J X ) \\xi _ 1 + \\mu ^ W ( J X ) \\xi _ 2 \\\\ = & \\ \\psi ^ { - 1 } g ( Z , J X ) \\xi _ 1 + \\psi ^ { - 1 } g ( W , J X ) \\xi _ 2 \\\\ = & \\ - \\psi ^ { - 1 } g ( W , X ) \\xi _ 1 + \\psi ^ { - 1 } g ( Z , X ) \\xi _ 2 \\\\ = & \\ \\mu ^ W ( X ) \\xi _ 1 - \\mu ^ Z ( X ) \\xi _ 2 \\\\ = & \\ J _ { \\mathfrak t ^ 2 } \\mu ( X ) . \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} ( x , t ) \\mapsto \\Psi ( x , t ) : = & - \\varphi ( x , y _ j ) - \\frac j 2 ( t - s _ j ) ^ 2 + \\varphi ( x _ j , y _ j ) + \\frac j 2 ( t _ j - s _ j ) ^ 2 \\\\ & + u ( x _ j , t _ j ) \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\textbf { g } ( \\nabla _ { \\partial t _ k } T _ i , T _ j ) + \\textbf { g } ( T _ i , \\nabla _ { \\partial t _ k } T _ j ) = \\frac { 1 } { - z } \\textbf { g } ( T _ k , T _ i \\bullet T _ j ) + \\frac { 1 } { z } \\textbf { g } ( T _ i , T _ j \\bullet T _ j ) = 0 = \\partial _ { t _ k } \\textbf { g } ( T _ i , T _ j ) \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} H e s s _ { \\widetilde { g } } h ( X _ i , Y _ j ) = 0 , \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} | H _ 0 | = & \\frac { 2 } { r } + r h _ 0 ^ { ( 1 ) } + r ^ 2 h _ 0 ^ { ( 2 ) } + r ^ 3 h _ 0 ^ { ( 3 ) } + O ( r ^ 4 ) \\\\ \\alpha _ { H _ 0 } = & r ^ 2 \\alpha _ { H _ 0 } ^ { ( 2 ) } + r ^ 3 \\alpha _ { H _ 0 } ^ { ( 3 ) } + r ^ 4 \\alpha _ { H _ 0 } ^ { ( 4 ) } + O ( r ^ { 5 } ) . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} & \\int _ { \\Sigma _ r } \\alpha _ { H } ( V ^ 2 \\nabla \\tau ) d \\Sigma _ r \\\\ = & \\int _ { \\Sigma _ r } ( \\alpha _ H ) \\left [ A \\nabla Y ^ 0 + ( C _ i Y ^ 4 + D _ p \\epsilon _ { p q i } Y ^ q ) \\nabla Y ^ i \\right ] d \\Sigma _ r + O ( r ^ 6 ) \\\\ = & \\int _ { \\Sigma _ r } ( \\alpha _ H ) \\left [ A \\nabla Y ^ 0 + ( C _ i ( 1 + \\frac { r ^ 2 } { 2 } ) + D _ p \\epsilon _ { p q i } Y ^ q ) \\nabla Y ^ i \\right ] d \\Sigma _ r + O ( r ^ 6 ) \\\\ \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} T _ n = \\min \\{ t \\in \\mathbb N : \\ , \\ , A _ n ( t ) = t \\} = \\min \\{ t \\in \\mathbb N : \\ , \\ , A _ n ( t ) \\leq t \\} \\quad A _ n ^ * = T _ n , \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\widehat { \\Z \\Gamma } = \\Big \\{ \\sum _ { \\gamma \\in \\Gamma } x _ { \\gamma } \\gamma \\mid x _ { \\gamma } \\in \\Z _ p \\ ; \\ ; | x _ { \\gamma } | \\to 0 \\ ; \\ ; \\gamma \\to \\infty \\Big \\} \\ ; . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} V : = v ^ { \\frac { r _ m } { q } } \\widehat { w } ^ { - \\frac { r _ m } { \\widetilde { \\delta } _ { m + 1 } } } \\in A _ { \\frac { q _ m } { r _ m } , \\frac { \\widetilde { \\delta } _ { m + 1 } } { r _ m } } ( \\widehat { w } ) . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} e _ k = { ( \\mu \\tilde \\sigma - \\nu ) d _ k \\over k [ \\coth k ( \\rho _ s - \\eta _ s ) + \\tanh k \\eta _ s ] } \\mbox { f o r } \\ ; \\ k \\neq 0 , \\ ; k \\in \\Bbb Z . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} L \\overset { \\Delta } { = } \\underset { \\mathbf { v } _ 1 \\ne \\mathbf { v } _ 2 } { \\sup } \\frac { \\left \\| \\mathbf { f } _ { \\boldsymbol { \\psi } } ( \\mathbf { v } _ 1 ) - \\mathbf { f } _ { \\boldsymbol { \\psi } } ( \\mathbf { v } _ 2 ) \\right \\| _ 2 } { \\left \\| \\mathbf { v } _ 1 - \\mathbf { v } _ 2 \\right \\| _ 2 } . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } y _ { 1 } ( x , \\lambda ) \\\\ y _ { 2 } ( x , \\lambda ) \\end{array} \\right ) = \\left ( \\begin{array} { c c } b _ { 1 1 } ( \\lambda ) & b _ { 1 2 } ( \\lambda ) \\\\ b _ { 2 1 } ( \\lambda ) & b _ { 2 2 } ( \\lambda ) \\end{array} \\right ) \\left ( \\begin{array} { c } \\phi _ { 1 } ( x , \\lambda ) \\\\ \\phi _ { 2 } ( x , \\lambda ) \\end{array} \\right ) \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\gamma _ { a b } ^ c = \\tilde \\gamma _ { a b } ^ c + r ^ 2 \\gamma _ { a b } ^ { ( 2 ) c } + O ( r ^ 3 ) \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} & X _ k = L ^ { p _ k ^ S } ( M _ S , \\mu _ s ; L ^ { p _ k ^ { S - 1 } } ( M _ { S - 1 } , \\mu _ { S - 1 } ; \\cdots L ^ { p _ k ^ { 1 } } ( M _ { 1 } , \\mu _ { 1 } ; L ^ { p _ k ^ { 0 } } ( \\mathcal M ) ) \\cdots ) , k = 1 , \\ldots , n , \\\\ & Y _ { n + 1 } = L ^ { q _ { n + 1 } ^ S } ( M _ S , \\mu _ S ; L ^ { q _ { n + 1 } ^ { S - 1 } } ( M _ { S - 1 } , \\mu _ { S - 1 } ; \\cdots L ^ { q _ { n + 1 } ^ { 1 } } ( M _ { 1 } , \\mu _ { 1 } ; L ^ { q _ { n + 1 } ^ { 0 } } ( \\mathcal M ) ) \\cdots ) . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} & \\sum _ { I _ { n _ k } ( a _ 1 , \\ldots , a _ { n _ k } ) \\cap F \\neq \\emptyset } | I _ { n _ k } ( a _ 1 , \\ldots , a _ { n _ k } ) | ^ { s } \\\\ \\leq & K _ 2 ^ { s n _ k } \\prod _ { j = 1 } ^ k G ( m ( j ) , n ( j ) , a , 1 / 3 , s ) \\\\ \\leq & K _ 2 ^ { s n _ k } C _ 1 ^ k C _ 2 ^ { n _ k - k - 1 } 3 ^ { - k } \\prod _ { j = 1 } ^ k m ( j ) ^ { { 1 - d s \\over a } } . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & ( A r _ t + \\alpha ( r _ t ) ) d t + \\sigma ( r _ { t - } ) d X _ t \\medskip \\\\ r _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\left ( T _ i \\bullet _ \\tau T _ j \\right ) _ { | Q = 0 } = T _ i \\cup T _ j \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} L R _ 0 & = \\left \\{ ( l _ 1 , \\dots , l _ n ) \\in \\mathbb { Q } ^ n \\mid \\left ( \\begin{array} { c } { l } _ 1 \\\\ \\vdots \\\\ { l } _ r \\end{array} \\right ) = - T \\left ( \\begin{array} { c } { l } _ { r + 1 } \\\\ \\vdots \\\\ { l } _ n \\end{array} \\right ) \\right \\} \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} R = a _ { 1 } x ^ { 2 } + a _ { 2 } ( y + m _ { 1 2 } x ) ^ { 2 } + a _ { 3 } ( z + m _ { 1 3 } x + m _ { 2 3 } y ) ^ { 3 } + a _ { 4 } ( w + m _ { 1 2 } x + m _ { 1 3 } y + m _ { 1 4 } z ) ^ { 2 } , \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} A _ { K , m , \\xi } ( t ) = \\tilde V _ m ( K \\cap \\{ \\xi ^ \\perp + t \\xi \\} ) & = \\frac { 1 } { n - 1 } \\int _ { S ^ { n - 1 } \\cap \\xi ^ \\perp } \\rho _ { K - t \\xi } ^ m ( \\theta ) \\ , d \\theta \\\\ & = \\frac { \\kappa _ { n - 1 } } { \\kappa _ m } \\int _ { G ( \\xi ^ \\perp , m ) } V _ m ( ( K - t \\xi ) \\cap H ) \\ , d H , \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} e _ { x _ 1 } \\cdots e _ { x _ { i - 1 } } x _ i e _ { x _ { i + 1 } } \\cdots e _ { x _ n } = x _ i e _ { x _ 1 ^ { x _ i } } \\cdots e _ { x _ { i - 1 } ^ { x _ i } } e _ { x _ { i + 1 } } \\cdots e _ { x _ n } , \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\xi _ \\varepsilon = \\frac { \\gamma _ \\varepsilon ^ { 2 ( N _ \\varepsilon - 1 ) } } { \\varphi _ { N _ \\varepsilon - 1 } ( \\gamma _ \\varepsilon ^ 2 ) ( N _ \\varepsilon - 1 ) ! } \\ , , \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} 0 \\leq s ' _ j = [ N , Q ( E _ j ) ] - [ N , E _ j ] = [ d , Q ( E _ j ) ] - [ N , E _ j ] , \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\begin{cases} x ' = \\left [ \\frac { ( a ^ 2 - b ) - N ^ 2 r - ( 2 k + a ) N r } { N r } \\right ] x ^ 2 \\\\ x ' = [ a - N r ] x ^ 2 \\end{cases} . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} A _ i = \\textstyle \\sum _ { j , k \\in I } \\big ( \\tfrac 1 2 k _ { j k } D _ { c _ { j i } } D _ { c _ { k i } } + c _ { j k } D _ { c _ { j i } } D _ { e _ { k i } } + \\tfrac 1 2 e _ { j k } D _ { e _ { j i } } D _ { e _ { k i } } \\big ) \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} \\dot z = F ( z ) , \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\epsilon } F _ n ( t , \\epsilon ) | _ { \\epsilon = 0 } = ( - 2 ) \\mathbb { E } \\left [ \\int _ 0 ^ { t \\wedge \\tau _ n } \\left ( h _ s - \\mu ( s , \\theta ^ H _ s ) \\right ) \\frac { \\partial } { \\partial x } \\mu ( s , \\theta ^ H _ s ) d [ B ] _ s \\right ] = 0 . \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\omega ( z ; q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } , \\nu ( z ; q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( - z q ; q ^ 2 ) _ { n + 1 } } , \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\iota _ \\partial \\omega ^ 2 = \\omega - \\sum _ { k = 1 } ^ j c _ k \\cdot d ( \\partial ^ { k - 1 } f ) - \\sum _ i d _ i f \\cdot d e _ i , \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\mathcal { F } ( r ) = \\mathcal { F } _ 0 ( r ) \\dot { \\cup } \\left ( \\dot { \\bigcup } _ { j = 1 } ^ t \\ , \\mathcal { T } _ j ^ r \\right ) \\# \\mathcal { F } _ 0 ( r ) = s \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} h _ p = \\tau ^ { \\Gamma } _ p ( M ) \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} T ( h \\otimes \\chi _ { \\{ y _ 0 \\} } ) ( \\xi , y _ 0 ) & = \\int ( h \\otimes \\chi _ { \\{ y _ 0 \\} } ) d T ^ * ( \\delta _ { ( \\xi , y _ 0 ) } ) \\\\ & = \\sum _ { \\eta \\in \\alpha } h ( \\eta ) \\cdot T ^ * ( \\delta _ { ( \\xi , y _ 0 ) } ) ( \\{ ( \\eta , y _ 0 ) \\} ) = \\sum _ { \\eta \\in \\alpha } r ( \\eta , \\xi ) \\cdot h ( \\eta ) . \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 1 ( x ) + \\frac { 2 } { 6 ^ 2 } ( 3 ^ 2 x - 4 ) ( 2 - 2 ^ 2 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 4 1 } { 9 0 } } + \\frac 1 { 1 0 \\cdot 6 ^ 2 } - \\eta \\\\ & = - \\frac { 1 5 9 6 7 7 3 7 5 6 0 2 7 9 3 7 8 6 6 2 0 8 4 5 8 3 4 8 3 7 1 9 5 9 1 } { 2 9 5 5 7 2 0 1 3 6 2 2 0 0 8 5 1 4 9 0 7 5 0 1 6 8 3 0 7 0 9 0 9 9 0 0 0 } < 0 , \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} \\phi _ { n + 1 } = x - \\sum _ { i = 0 } ^ n y ^ { - \\frac { 1 } { p ^ i } } \\in K [ x ] . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\int | d u | ^ { p - 2 } \\langle j u , d \\psi \\rangle = 0 \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} | \\mathcal { C } _ 0 | \\leq \\sum _ { i = 0 } ^ { \\frac { \\sqrt { n } } { \\log n } } \\binom { n } { i } + \\sum _ { i = 0 } ^ { \\frac { 1 6 \\sqrt { n } } { \\log n } } \\binom { n } { i } \\leq 2 ^ { 1 6 \\sqrt { n } + 1 } . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} l ( G ) = \\max \\{ \\O ( q - 1 ) + f , \\O ( q + 1 ) + 1 , l ( { \\rm L } _ { 2 } ( q ^ { 1 / r } ) ) + 1 + \\delta _ { 2 , r } \\ , : \\ , r \\in \\pi ( f ) \\} , \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} E [ g ] ( y ) = \\tilde { g } ( y ) : = g _ { D , \\mathrm { o d d } } ( y ) + g _ { N , \\mathrm { e v e n } } ( y ) , y \\in \\mathbb R ^ n , \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} s ( y ) : = c - \\sum _ { i = 1 } ^ m y _ i a _ i , \\ \\ \\ \\ \\ \\ \\ S ( y ) : = C - \\sum _ { i = 1 } ^ m y _ i A _ i . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} x = \\sinh t - \\frac 1 2 | \\tilde s | ^ { 2 } e ^ t , y ^ i = e ^ { t } s ^ i , z = \\cosh t + \\frac 1 2 | \\tilde s | ^ 2 e ^ t , \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} { \\rm C r i t } ( f ) = \\{ x \\in X \\colon d f _ x = 0 \\} \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\nabla _ { \\ ! x } L ( x , \\lambda ) = a ; \\\\ \\lambda \\in \\mathcal { N } _ { \\mathcal { K } } ( g ( x ) - b ) \\end{array} \\right . \\Longleftrightarrow \\left \\{ \\begin{array} { l l } \\nabla \\ ! f ( x ) + \\nabla \\ ! g ( x ) \\lambda = a ; \\\\ g ( x ) - b = \\Pi _ { \\mathcal { K } } ( g ( x ) - b + \\lambda ) \\end{array} \\right . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} g ( x ) = \\left ( H _ 0 ^ { \\theta - 1 } \\varphi _ g \\right ) ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\delta _ { n , k } \\coloneqq \\psi _ { n , k } \\lor ( x \\to \\bigvee _ { i = 1 } ^ { n + 1 } y _ { i } ) \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} p = \\sum _ { i = 1 } ^ r a _ i \\textbf { Q } ^ { \\lambda _ i } \\mbox { w i t h } \\nu \\left ( a _ i \\textbf { Q } ^ { \\lambda _ i } \\right ) \\geq \\nu ( p ) , \\mbox { f o r e v e r y } i , 1 \\leq i \\leq r , \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} | \\mathcal { W } [ y _ h ] - \\mathcal { W } [ y + d ] | ( x , t ) \\le L \\sup _ { s \\le t } | y _ h - y - d | ( x , s ) = L \\sup _ { s \\le t } | r _ h ( x , s ) | . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} \\{ a _ i , a _ j \\} = 0 , \\{ a ^ \\ast _ i , a ^ \\ast _ j \\} = 0 , \\{ a _ i , a ^ \\ast _ j \\} = \\delta _ { i j } , \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\varphi ^ { \\pm } = { } ^ t ( \\varphi ^ { \\pm } _ 1 , \\varphi ^ { \\pm } _ 2 , \\cdots , \\varphi ^ { \\pm } _ m ) \\gg { } ^ t ( 0 , 0 , \\cdots , 0 ) , \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} [ f , D ^ { - \\frac 1 2 } \\nabla ] = [ \\langle x \\rangle ^ a , R D ^ \\frac 1 2 ] = [ \\langle x \\rangle ^ a , R ] D ^ \\frac 1 2 + R [ \\langle x \\rangle ^ a , D ^ \\frac 1 2 ] , \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} \\Phi _ j ( t ) = \\frac { 1 } { 2 } \\Bigl ( \\sqrt { 1 + 4 p _ j p _ { - j } \\ , t ^ 2 } - 1 \\Bigr ) \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} Q ( x , t ) : = \\begin{pmatrix} 0 & q ( x , t ) \\\\ - \\bar q ( x , t ) & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} N ( 0 , 0 ) & = \\nu ^ { 0 } _ + , \\nu ^ { 0 } _ - \\\\ N ( 1 , 0 ) & = \\nu ^ { 1 } _ + , \\nu ^ { 1 } _ - . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\check \\nu ^ C = 2 + \\lfloor \\log _ 2 ( \\check h _ { \\check \\nu ^ C } - 2 ) \\rfloor \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\frac { q ^ 2 } { z _ 1 ^ 2 } \\log ( 1 + \\frac { z _ 1 } { 1 + z _ 2 } ) - \\frac { q ^ 2 } { z _ 1 ( 1 + z _ 1 + z _ 2 ) } & = \\xi ( q ) \\Delta ^ 2 , \\\\ \\frac { ( 1 - q ) q } { ( 1 + z _ 2 ) ( 1 + z _ 1 + z _ 2 ) } + \\frac { ( 1 - q ) ^ 2 } { z _ 2 ^ 2 } \\log ( 1 + z _ 2 ) - \\frac { ( 1 - q ) ^ 2 } { z _ 2 ( 1 + z _ 2 ) } & = [ \\xi ( 1 ) - \\xi ( q ) ] \\Delta ^ 2 , \\\\ \\frac { q } { ( 1 + z _ 2 ) ( 1 + z _ 1 + z _ 2 ) } + \\frac { 1 - q } { 1 + z _ 2 } & = \\xi ' ( 1 ) \\Delta ^ 2 . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} V ( \\partial D ) = V ( \\bar { D } ^ c ) \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\begin{aligned} p _ i & = ( 1 - \\varepsilon ) p _ i ^ 0 + \\varepsilon h _ i \\\\ & = p _ i ^ { \\prime } + \\varepsilon h _ i , \\end{aligned} \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} d ( n ) = d ( n - 1 ) \\max \\{ d ' ( n ) , \\max _ { 1 \\leq m \\leq n - 1 } d ( m ) d ( n - m ) \\} , \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} { e _ i } ^ 2 = \\left ( U _ { 1 1 } - U ^ i _ { 1 1 } \\right ) ^ 2 + \\left ( U _ { 1 2 } - U ^ i _ { 1 2 } \\right ) ^ 2 + \\ldots + \\left ( U _ { m n } - U ^ i _ { m n } \\right ) ^ 2 . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta ( z , q ) = z ^ { - 1 } - \\sum _ { k = 0 } ^ \\infty G _ { 2 k + 2 } z ^ { 2 k + 1 } . \\end{aligned} \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\sum _ { k = N + 1 } ^ { \\infty } f ( k ) \\leq \\int _ { N } ^ \\infty f ( x ) d x . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} h , _ { y _ j } = f e ^ { l ( y _ 1 , \\ldots , y _ m ) } , \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\hbox { P f } \\left ( \\sum \\limits _ { i = 1 } ^ p \\lambda _ i W _ i \\right ) = 0 \\in \\mathbb { K } ( z ) \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} z = z ( t , x ; b , y ) \\doteq \\frac { ( t - b ) ^ 2 - ( y - x ) ^ 2 } { ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 } . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} P ^ { ( j ) } ( 1 ) + 4 \\Biggl [ \\frac { ( - 1 ) ^ { j - 1 } ( j - 1 ) ! } { 2 ^ { j / 2 } } \\sin ( j \\pi / 4 ) + \\frac { ( - 1 ) ^ { j } ( j - 1 ) ! } { 2 ^ { ( j - 1 ) / 2 } } \\sin ( ( j - 1 ) \\pi / 4 ) \\Biggr ] = 0 . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N D N } } ^ { \\textrm { N O M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { N } } \\right ) P _ \\textrm { N } | h _ { \\textrm { B N } } | ^ 2 } { { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } } . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\tilde x \\frac { \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\lambda _ 1 } \\sum _ { l > 1 } \\sqrt { \\lambda _ l } \\langle \\tilde u _ 1 , u _ l \\rangle = \\Big ( 1 - \\tilde x \\frac { \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\lambda _ 1 } \\Big ) \\sqrt { \\lambda _ 1 } \\langle \\tilde u _ 1 , u _ 1 \\rangle , \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\Delta _ \\pi ( z ) = \\left ( D ^ { ( \\pi ( 1 ) ) } ( z ) , \\dots , D ^ { ( \\pi ( n ) ) } ( z ) \\right ) . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} u ( x , y , z ) = \\chi _ r ( x , y , z ) ( E v ) ( x , y , z ) \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} A \\circ \\varphi ( M ) \\circ A = \\varphi ( A d ( M ) ^ t ) \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = \\binom { n - 1 } { k - 1 } _ q + q ^ k \\binom { n - 1 } { k } _ q , \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x ^ j _ n ) + v ^ l _ n ( x ) , \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\max _ { x \\in S } \\ ( \\max _ { i = 1 } ^ { n } x _ { i } ) ( \\max _ { j = 1 } ^ { n } ( 1 / x _ { j } ) ) , \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} I _ { N } ^ { \\lambda } u ( x ) = S ( t ) \\ , I _ { N } ^ { G } \\Big \\{ \\frac { U ( t ) } { S ( t ) } \\Big \\} = S ( t ) \\ , I _ { N } ^ { G } \\breve U ( t ) . \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} & \\nu _ { \\beta , h } ^ { + , \\pm } ( \\cdot ) : = \\lim _ { n \\rightarrow \\infty } \\lim _ { l \\rightarrow \\infty } \\nu _ { \\beta , h } ^ + \\left ( \\cdot \\mid \\eta ^ { - n } _ { - ( n + l ) } = \\pm ^ { - n } _ { - ( n + l ) } \\right ) \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} H _ 0 ^ \\alpha ( S _ 0 ^ \\alpha f ) ( x ) = S _ 0 ^ \\alpha ( H _ 0 ^ \\alpha f ) ( x ) = \\int _ 0 ^ x f ( y ) \\ , d y , \\mbox { a . e . } x \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\hat { \\breve { h } } _ { i , k k ' } [ \\breve { \\nu } ] & \\triangleq \\begin{cases} \\hat { h } _ { i , k ' } [ \\breve { \\nu } + \\breve { d } _ { i , k } ] , & \\frac { P | \\hat { h } _ { i , k ' } [ \\breve { \\nu } + \\breve { d } _ { i , k } ] | ^ 2 } { \\sigma _ { m _ i } ^ 2 + \\varepsilon _ i ^ 2 } \\geq \\eta , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} a & = \\inf \\left \\{ x \\le x _ { o } : \\ u \\left ( \\xi \\right ) < v \\left ( \\xi \\right ) \\xi \\in \\left ] x , x _ { o } \\right ] \\right \\} , \\\\ b & = \\sup \\left \\{ x \\ge x _ { o } : \\ u \\left ( \\xi \\right ) \\le v \\left ( \\xi \\right ) \\xi \\in \\left [ x _ { o } , x \\right ] \\right \\} . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} E = E _ D + E _ { D T } + E _ T + E _ S \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} h ( x _ 1 , \\ldots , x _ n , y _ 1 , \\ldots , y _ m ) = f ( x _ 1 , \\ldots , x _ n ) \\int e ^ { l ( y _ 1 , \\ldots , y _ m ) } d y _ j + m ( \\hat { y _ j } ) . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} w \\left ( \\frac { 1 } { X } \\right ) + \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\psi _ n ( \\alpha ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } \\sum _ { \\alpha } \\frac { \\nu ( \\alpha ) \\psi _ n ( \\alpha ) } { N ( \\alpha ) ^ s } X ^ s \\mathfrak { w } ( s ) d s . \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\Omega } p \\left | \\nabla u \\right | ^ { 2 } \\geq & \\ , 2 \\pi p _ { 0 } \\left ( \\sum _ { i = 1 } ^ { m } d _ { i } ^ { 2 } \\right ) \\left ( \\log \\dfrac { R } { R _ { 0 } } - I \\left ( \\dfrac { R } { R _ { 0 } } \\right ) \\right ) \\\\ & + 2 \\pi p _ { 0 } \\displaystyle \\sum _ { i \\neq j } \\left ( - \\left ( 1 - a ^ { 2 } \\right ) \\left | d _ { i } \\right | \\left | d _ { j } \\right | + d _ { i } d _ { j } \\right ) \\log \\frac { R } { | a _ { i } - a _ { j } | } - C , \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} I ' _ { k , \\lambda + i \\gamma } f ( x ) = \\int \\limits _ { 1 } ^ { \\infty } \\frac { f ( x - y ^ k ) } { y ^ { \\lambda + i \\gamma } } d y . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} z \\frac { d F ( z ; c ) } { d z } = A ( z ; c ) F ( z ; c ) \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m - 1 } ( - 1 ) ^ k [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ; q ) _ k ^ 3 } = \\sum _ { n = 0 } ^ { m - 1 } F ( n , 0 ) . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathcal { P } _ 1 , \\ldots , \\mathcal { P } _ s ) = \\{ P _ { 1 , j _ 1 } P _ { 2 , j _ 2 } \\cdots P _ { s , j _ s } | P _ { k , j _ k } \\in \\mathcal { P } _ k \\} \\backslash \\{ \\mathbf { 0 } \\} , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( Z _ \\varepsilon ) = \\frac 1 { 2 ^ { n - 1 } n ^ { n - 1 } } \\ , \\mathcal { H } ^ { n - 1 } ( E _ \\varepsilon ) . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} ( 2 k + 3 ) \\lambda _ j - 2 k q \\sim _ E \\begin{cases} 2 \\lambda _ 1 + \\lambda _ 2 + 2 q - q _ 1 - q _ 2 & j = 1 , \\\\ \\lambda _ { j - 1 } - \\lambda _ j + \\lambda _ { j + 1 } + q + q ' _ 3 & j \\leq n - 2 , \\\\ \\lambda _ { j - 1 } + \\lambda _ j + \\lambda _ { j + 1 } + q - q _ 3 & j \\geq 3 , \\\\ \\lambda _ { n - 1 } - 2 \\lambda _ n + q ' _ 1 + q ' _ 2 + 2 q & j = n . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\rho + \\div ( u \\rho ) = 0 \\\\ \\rho ( 0 , \\cdot ) = \\bar \\rho \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} t \\sum _ { j = 0 } ^ { n } a _ { j } \\bigl ( t ^ { 2 } - 1 \\bigr ) ^ { j } \\leq f \\bigl ( t ^ { 2 } - 1 \\bigr ) \\sum _ { j = 0 } ^ { m } b _ { j } \\bigl ( t ^ { 2 } - 1 \\bigr ) ^ { j } , t \\geq 1 \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} S _ n ( t ) \\stackrel { d e f } { = } n ^ { - 1 / 2 } \\sum _ { i = 1 } ^ n \\xi _ i ( t ) , \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ y ( x ) = \\varphi _ y ( x ) + a | x - z | ^ 2 . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} | g r a d _ { g _ B } h | ^ 2 = | g r a d _ { \\overline { g } } h | ^ 2 = \\varepsilon _ { i _ 0 } \\varphi ^ 2 ( h ' ) ^ 2 . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} K ' * d Z = ( K ' * L ) ( 0 ) ( V - g _ 0 ) + d ( K ' * L ) * ( V - g _ 0 ) . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} T _ { \\rm r e g } = ( T _ { \\rm o p } ) _ { \\rm r e g } , T _ { \\rm s i n g } = ( T _ { \\rm o p } ) _ { \\rm s i n g } + T _ { \\rm m u l } . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\langle \\kappa + v _ 1 , \\alpha \\rangle = 0 \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 1 } ^ k x _ j ^ q \\Big ) ^ { \\frac 1 { q } } \\leq 1 6 \\left ( 1 + V _ { \\Phi , h } ( f ) \\right ) \\max _ { 1 \\leq m \\leq k } m ^ { \\frac 1 { q } } \\Phi _ m ^ { - 1 } \\big ( h ( k ) \\big ) . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} 1 \\cdot V _ { ( \\wp ) } V \\otimes V + 1 \\cdot V \\otimes V _ { ( \\wp ) } V + \\wp ( x ) \\cdot V \\otimes V . \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} a \\otimes b ( 0 ) m - a ( 0 ) b \\otimes m - b \\otimes a ( 0 ) m = \\frac { 1 } { 2 \\pi i } \\left ( a \\otimes [ b , m ] - [ a , b ] \\otimes m - b \\otimes [ a , m ] \\right ) , \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\mathbf R = \\frac { 1 } { \\ln 2 } \\left ( \\frac { p \\Delta ( \\emptyset ) } { \\frac { \\gamma _ { A , D ^ { r } } } { \\gamma _ { D ^ { t } , D ^ { r } } } - 1 } - ( 1 - p ) ( 1 - \\beta ) \\Psi \\left ( \\frac { 1 } { \\gamma _ { D ^ { t } , D ^ { r } } } \\right ) \\right ) , \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} H ^ { \\circ _ { t _ 1 } ( N + 1 ) } = e ^ { t _ 1 } \\in S Q H ^ * \\left ( \\mathbb { P } ^ N \\right ) \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} ( z Q \\partial _ Q ) ^ { N + 1 } Q ^ { \\frac { H } { z } } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } = ( H + d z ) ^ { N + 1 } Q ^ { \\frac { H } { z } } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } = Q ^ { \\frac { H } { z } } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ { d - 1 } \\left ( H + r z \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} a _ { k _ 0 } \\leq C ' r _ 0 ^ s 2 ^ { k _ 0 } \\leq C '' b ^ { - 1 / p } \\left ( \\sum _ { j = - \\infty } ^ { l - 2 } 2 ^ { s ' j p } \\left ( \\int _ { 2 B _ 0 } g _ j ^ p \\ , d \\mu \\right ) ^ { \\frac { q } { p } } \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\bar H ^ { 1 ^ * } _ { , k } = 0 . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} & p : = \\min ( \\P ( A \\cap C _ { k _ 1 , j _ 1 } ) , \\P ( A \\cap C _ { k _ 2 , j _ 2 } ) ) > 0 . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\{ h , 0 \\} = \\{ \\varphi , \\varphi ' \\} + \\{ \\psi , \\psi ' \\} , \\{ \\varphi , \\varphi ' \\} \\in T , \\{ \\psi , \\psi ' \\} \\in T ^ \\perp = J T ^ * , \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} d ( t ) : = d _ { \\theta ( t ) } \\le C ( t + 1 ) ^ { \\frac 1 3 } \\log ^ 2 ( 2 + t ) \\mbox { f o r a n y } t > 0 \\ , , \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} - \\phi ' ( s ) & = \\int _ { 1 } ^ { \\infty } F _ 1 ( y ) s e ^ { - s y } d y \\\\ & = \\int _ { s } ^ { \\infty } F ( x / s ) e ^ { - x } d x \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { + \\infty } \\bigl \\vert a ( k ) \\widehat { f } ( k ) ^ 2 \\bigr \\vert \\le \\left \\Vert a \\right \\Vert _ r \\bigl \\Vert \\widehat { f } ^ { \\ 2 } \\bigr \\Vert _ s = \\left \\Vert a \\right \\Vert _ r \\bigl \\Vert \\widehat { f } \\ , \\bigr \\Vert _ q ^ 2 \\le \\left \\Vert a \\right \\Vert _ r \\left \\Vert f \\right \\Vert _ p ^ 2 . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\sigma _ { a b } = & r ^ 2 \\tilde \\sigma _ { a b } + r ^ 4 \\sigma _ { a b } ^ { ( 4 ) } + r ^ 5 \\sigma _ { a b } ^ { ( 5 ) } + O ( r ^ 6 ) \\\\ | H | = & \\frac { 2 } { r } + r h ^ { ( 1 ) } + r ^ 2 h ^ { ( 2 ) } + r ^ 3 h ^ { ( 3 ) } + O ( r ^ 4 ) \\\\ \\alpha _ H = & r ^ 2 \\alpha _ H ^ { ( 2 ) } + r ^ 3 \\alpha _ H ^ { ( 3 ) } + r ^ 4 \\alpha _ H ^ { ( 4 ) } + O ( r ^ { 5 } ) . \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} ( \\tilde { \\nabla } ^ 2 u _ { 1 , i } ) ( x , t ) = o ( t ^ { - \\frac { N + 2 A _ 1 } { 2 } } ) \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\alpha ^ { ( 1 ) } S ( \\alpha ^ { ( 2 ) } ) S ( \\beta ^ { ( 1 ) } ) \\beta ^ { ( 2 ) } = 1 . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} f ^ { ( j , { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c ) } : = \\begin{cases} \\overline { \\partial } \\Big ( \\chi { } ( \\cdot { } , { \\zeta } ) \\cdot { } c _ j \\Big ) & C _ { \\zeta } ( { \\tau _ 2 } ) \\\\ 0 & { \\Omega } _ { \\zeta } ( { \\tau _ 2 } ) \\setminus { } C _ { \\zeta } ( { \\tau _ 2 } ) \\end{cases} \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle \\mu _ { g _ s } ( \\hat x , t ) & = & \\displaystyle g _ s ( \\hat x ) + \\frac { t ^ 2 } { 6 } \\ , \\sum _ { i = 1 } ^ { k - 1 } { ( g _ s ) } _ { i i } ( \\hat x ) + \\left ( \\frac { 1 } { 1 2 0 } \\sum _ { i = 1 } ^ { k - 1 } { ( g _ s ) } _ { i i i i } + \\frac { 1 } { 3 6 } \\ , \\sum _ { \\substack { i , j = 1 \\\\ j > i } } ^ { k - 1 } { ( g _ s ) } _ { i i j j } \\right ) t ^ 4 \\ , , \\end{array} \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\theta _ n = \\theta . \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) & \\leq - ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) \\sup _ { \\varepsilon > x } \\inf _ { y \\in \\left ( - \\infty , \\frac { \\varepsilon } { h ( x ) } \\right ] } H ( y ) \\\\ & = - ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) \\inf _ { y \\in \\left ( - \\infty , \\frac { x } { h ( x ) } \\right ] } H ( y ) \\\\ & = - J ( x ) , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\left \\| \\varrho _ { P _ 0 } ( u ) \\ , u - \\frac { z _ i ( t ) } { \\langle u _ i , u \\rangle } \\ , u \\right \\| = \\frac { | z _ i ( 0 ) - z _ i ( t ) | } { \\langle u _ i , u \\rangle } \\leq \\frac { R Z } { r } \\cdot | t | . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} k _ { \\gamma , \\Sigma } ^ { \\infty } = \\frac { | \\overline { p } \\dot { \\gamma } _ 1 + \\overline { q } \\dot { \\gamma } _ 2 | } { | \\gamma _ 1 | | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\| v _ \\varepsilon \\| _ { L ^ \\infty ( B _ { x _ \\varepsilon } ( r _ \\varepsilon ) ) } = O \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , , \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} w ( X - \\beta ) = w ( X - \\alpha + \\alpha - \\beta ) = \\begin{cases} w ( X - \\alpha ) , & v ( \\alpha - \\beta ) \\geq w ( X - \\alpha ) \\\\ v ( \\alpha - \\beta ) , & v ( \\alpha - \\beta ) < w ( X - \\alpha ) \\end{cases} \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} F _ 0 '' ( r ) & = \\left \\{ ( 1 - N ) r ^ { - N } [ U ( r ) ] ^ { - 2 } - 2 r ^ { 1 - N } [ U ( r ) ] ^ { - 3 } U ' ( r ) \\right \\} \\int _ 0 ^ r \\tau ^ { N - 1 } U ( \\tau ) ^ 2 \\ , d \\tau + 1 \\\\ & \\to \\frac { 1 - N } { N } + 1 = \\frac { 1 } { N } \\quad \\mbox { a s } r \\to 0 . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\mathcal { C } _ 1 & = \\overline { \\{ O \\big ( C _ 2 ( \\alpha ) \\big ) \\} _ { \\alpha \\in \\mathbb { C } } } = \\{ \\mathbb { C } ^ { n + 1 } , B , C _ 1 , C _ 2 ( \\alpha ) , C _ 3 \\} , \\\\ \\mathcal { C } _ 2 & = \\overline { O ( D _ { n + 1 } ) } = \\{ \\mathbb { C } ^ { n + 1 } , B , C _ 1 , D _ 3 , \\ldots , D _ { n + 1 } \\} . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} N _ d = \\left \\langle H ^ 2 , \\dots , H ^ 2 \\right \\rangle ^ \\textnormal { c o h } _ { 0 , 3 d - 1 , d [ l ] } \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} ( \\lambda t _ \\alpha + \\mu e ^ { \\alpha } ) \\cdot ( x e ^ \\beta + e ^ { \\alpha + \\beta } ) & = \\big ( \\lambda x a \\vert \\alpha \\cap \\beta \\vert + \\mu b _ { \\alpha , \\beta } \\big ) e ^ \\beta \\\\ & + \\big ( \\lambda a \\vert \\alpha \\cap ( \\alpha + \\beta ) \\vert + \\mu x b _ { \\alpha , \\beta } \\big ) e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\left | \\widehat { J } _ k \\left ( \\frac { \\xi } { \\theta _ k } \\right ) \\right | & \\leq ( 1 - \\delta ) ^ k \\leq ( k L ( k ) ) ^ { - n / \\sigma } = \\theta _ k ^ { - n } , \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\Phi ( \\lambda ) : = \\frac { u ( p + \\lambda ( s - p ) ) - u ( p ) } { \\lambda } , \\qquad u ( z ) : = z \\log ( z ) . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} a _ { s } = a + 2 l n s + r l s ^ { 2 } . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{gather*} \\begin{pmatrix} 1 _ t & - A ^ { - 1 } B ) \\\\ 0 & 1 _ { n - t } \\end{pmatrix} \\begin{pmatrix} A ^ { - 1 } & 0 \\\\ 0 & ( C - { } ^ t B A ^ { - 1 } B ) ^ { - 1 } \\end{pmatrix} \\begin{pmatrix} 1 _ t & 0 \\\\ - { } ^ t ( A ^ { - 1 } B ) & 1 _ { n - t } \\end{pmatrix} \\\\ = \\begin{pmatrix} & \\\\ & ( C - { } ^ t B A ^ { - 1 } B ) ^ { - 1 } \\end{pmatrix} , \\end{gather*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ \\Omega ( 1 + g ( u _ \\varepsilon ) ) \\exp ( u _ \\varepsilon ^ 2 ) d x = | \\Omega | ( 1 + g ( 0 ) ) + \\pi \\exp ( 1 + M ) \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\dim _ F M _ { ( 2 a , b , c ) } & = \\dfrac { 2 ^ n n ! } { 4 ^ a a ! 2 ^ { b + c } ( b + c ) ! } \\times \\binom { b + c } { b } \\\\ & = \\dfrac { 2 ^ { 2 a } \\times ( 2 a ) ! } { 4 ^ a a ! } \\times \\dfrac { n ! } { ( 2 a ) ! b ! c ! } \\\\ & = \\dim _ F F S \\times [ C _ 2 \\wr S _ n : C _ 2 \\wr S _ { ( 2 a , b , c ) } ] . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} p ( z ) = \\lim _ { j \\to \\infty } p _ j ( z ) \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } d _ \\mathcal { H } ( 2 ( F _ i \\cap [ 0 , 1 ] ^ d ) , 2 E ) = 0 . \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} V ^ { ( P ) } _ n : = \\sum _ { l = 1 } ^ { N } T ^ { ( P ) } _ l , \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sharp ( \\{ I _ j ^ n \\} \\cap J ) - 1 } { n - 1 } = \\frac { \\int _ J s ( x ) d x } { \\int _ I s ( x ) d x } , \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = \\sum \\nolimits _ { k = 1 } ^ n \\omega _ k ( x ( t ) ) f _ k \\bigl ( x ( t ) , \\hat { v } _ k ^ { 0 } ( x ( t ) ) \\bigr ) \\\\ & \\triangleq f _ { \\rm a v } ^ { \\hat { v } } \\bigl ( x ( t ) \\bigr ) , x ( 0 ) = x _ 0 \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left ( 1 - q ^ d \\right ) ^ 3 h _ { d } = h _ { d - 1 } \\\\ & \\left ( 1 - q ^ d \\right ) ^ 3 g _ d + \\sum _ { k = 0 } ^ 3 \\binom { 3 } { k } ( - q ^ d ) ^ k k h _ { d } = g _ { d - 1 } \\end{aligned} \\right . \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} ( e _ 1 \\wedge v ) \\delta _ { k + 1 } = v - e _ 1 \\wedge ( v \\delta _ k ) = v , \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} V _ x ( y ) \\sim \\begin{cases} \\displaystyle \\frac { E \\big ( y _ { n _ 0 + l - 1 , x } \\big ) } { y _ { n _ 0 + l - 1 , x } } , & \\mbox { i f } y _ { n _ 0 + l - 1 , x } \\geq t ^ { \\ast } , \\\\ \\displaystyle \\frac { E \\big ( y _ { n _ 0 + l - 1 , x } + t ^ { \\ast } \\big ) } { y _ { n _ 0 + l - 1 , x } + t ^ { \\ast } } + \\frac { t ^ { \\ast } } { \\Tilde { t } } , & \\mbox { i f } y _ { n _ 0 + l - 1 , x } < t ^ { \\ast } . \\end{cases} \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} R ( t , s ) = \\int _ { 0 } ^ { t \\wedge s } K ( t , r ) K ( s , r ) d r , \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} & ( n - 1 ) ! \\lim _ { X \\to \\infty } \\frac { \\# \\{ p \\in S p l _ X ( f ) \\mid r _ i / p < a \\} } { \\# S p l _ X ( f ) } = \\\\ & \\sum _ { 0 \\le h \\le n \\atop 1 \\le l \\le n - 1 } \\sum _ { k = i } ^ n ( - 1 ) ^ { h + k + n } { n \\choose k } \\sum _ { m = 1 } ^ { n - 1 } { k \\choose n - h - m + l } { n - k \\choose m - l } M ( l - h a ) ^ { n - 1 } , \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} d _ { \\max } = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } \\frac { \\sigma ^ 2 } { g ( w ) } ~ d w . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\sigma _ { a b } = & r ^ 2 \\tilde \\sigma _ { a b } + r ^ 4 \\tilde \\sigma ^ { ( 4 ) } _ { a b } + O ( r ^ 5 ) \\\\ | H | = & \\frac { 2 } { r } + h ^ { ( 1 ) } r + O ( r ^ 2 ) \\\\ ( \\alpha _ H ) _ a = & ( \\alpha _ H ^ { ( 2 ) } ) _ a r ^ 2 + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\mathbb { C } ^ { \\kappa , \\mu , \\mu _ 0 } ( \\Lambda ^ n ) : = \\{ A _ t \\in \\Lambda ^ n ~ : ~ [ \\ ! [ A _ t ] \\ ! ] _ { \\kappa } \\leq \\mu , ~ \\| A _ t \\| _ { \\infty } \\leq \\mu _ 0 \\} . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} h _ p : = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log _ p \\chi ( \\Gamma _ n , M ) \\ ; . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to \\infty } d ( x _ n , x _ { n + 1 } ) = 0 . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\mathbb C \\otimes T ( \\mathcal S ) = T ^ { 1 , 0 } ( \\mathcal S ) \\oplus T ^ { 0 , 1 } ( \\mathcal S ) , \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\sum _ { \\{ \\hat { \\alpha } , \\hat { \\beta } \\} \\in \\mathcal { S } _ { ( \\hat { d \\varphi } , \\iota ) } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\hat { \\alpha } } ( \\hat { \\phi } ) + \\ell _ { \\hat { \\beta } } ( \\hat { \\phi } ) ) } + ( - 1 ) ^ { \\hat { \\alpha } \\cdot \\hat { \\beta } } \\right ) ^ { - 1 } = 0 . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} X = \\begin{pmatrix} J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) \\\\ \\delta _ q J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) \\\\ \\vdots \\\\ \\left ( \\delta _ q \\right ) ^ N J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) \\end{pmatrix} \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} D _ l ( k ) = \\mathbb { E } d ( Z _ k , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = \\left [ \\overline { q } \\dot { \\gamma } _ 3 - \\frac { \\sqrt { 2 } } { 2 } \\overline { p } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) \\right ] e _ 1 - \\frac { l _ L } { l } L ^ { \\frac { 1 } { 2 } } \\omega ( \\dot { \\gamma } ( t ) ) e _ 2 . \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} 1 _ n = \\overbrace { 1 _ 1 \\otimes \\dots \\otimes 1 _ 1 } ^ { n \\mbox { \\footnotesize t i m e s } } \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} \\nu _ 3 : = \\left [ \\nu _ 2 ; \\nu _ 3 ( \\phi _ 3 ) = - \\frac { 1 } { p ^ 3 } \\right ] \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} x \\nu & = \\lambda x a \\vert \\alpha \\cap \\beta \\vert + \\mu b _ { \\alpha , \\beta } \\\\ \\nu & = \\lambda a \\vert \\alpha \\cap ( \\alpha + \\beta ) \\vert + \\mu x b _ { \\alpha , \\beta } \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} q ^ { * } : = \\inf \\{ q \\geq q _ { 0 } : q \\in U , \\tilde \\zeta ( q ) = z _ { 2 } ^ { b } ( q ) \\} < \\ 8 . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} & R ^ L ( X _ 1 , X _ 2 ) X _ 1 = \\frac { 3 } { 4 } L X _ 2 + X _ 3 , ~ ~ ~ R ^ L ( X _ 1 , X _ 2 ) X _ 2 = - \\frac { 3 } { 4 } L X _ 1 , ~ ~ ~ R ^ L ( X _ 1 , X _ 2 ) X _ 3 = - L X _ 1 , ~ ~ ~ \\\\ & R ^ L ( X _ 1 , X _ 3 ) X _ 1 = L X _ 2 + \\frac { 3 } { 4 } L X _ 3 , ~ ~ ~ R ^ L ( X _ 1 , X _ 3 ) X _ 2 = - L X _ 1 , ~ ~ ~ R ^ L ( X _ 1 , X _ 3 ) X _ 3 = ( \\frac { L ^ 2 } { 4 } - L ) X _ 1 , ~ ~ ~ \\\\ & R ^ L ( X _ 2 , X _ 3 ) X _ 1 = 0 , ~ ~ ~ R ^ L ( X _ 2 , X _ 3 ) X _ 2 = - \\frac { L } { 4 } X _ 3 , ~ ~ ~ R ^ L ( X _ 2 , X _ 3 ) X _ 3 = \\frac { L ^ 2 } { 4 } X _ 2 . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\textnormal { v d i m } _ \\mathbb { C } \\left ( \\overline { \\mathcal { M } } _ { g , n } ( X , d ) \\right ) = ( 1 - g ) ( \\textnormal { d i m } ( X ) - 3 ) + n - \\int _ X c _ 1 ( T X ) \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} { \\rm A L G } & : = \\sum _ { i = 1 } ^ n H _ i ( \\omega _ { i , m } ( K ) ) \\\\ { \\rm P _ { g s e q } } & : = \\sum _ { i = 1 } ^ n \\big ( H _ i ( \\omega _ { i , m } ( K ) ) + G _ i ( \\hat { c } _ i ^ T \\omega _ { i , m } ( K ) ) \\big ) \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\mathcal { T } _ { t , k } ^ { \\epsilon , v _ k } \\psi ( x ) = \\mathbb { E } _ { x , k } ^ { \\epsilon , v _ k } \\left \\{ \\psi \\bigl ( X _ { k } ^ { \\epsilon , v _ k } ( t ) \\bigr ) ; \\ , \\tau _ { k } ^ { \\epsilon , v _ k } > t \\right \\} , \\ , \\ , \\ , x \\in D , \\ , \\ , \\ , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\int _ { \\bigcup _ { \\gamma \\in \\Psi _ { k } } \\gamma ^ { - 1 } V } \\left \\vert \\varphi \\left ( x \\right ) \\right \\vert ^ { 2 } d x \\leq \\int _ { G } \\left \\vert \\varphi \\left ( x \\right ) \\right \\vert ^ { 2 } d x = \\left \\Vert \\varphi \\right \\Vert ^ { 2 } < \\infty . \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} \\gamma ( c ) & = \\mu ( c ) \\ : u & & \\\\ & = q ( c , c \\wedge m _ \\mu ) \\iota ( c \\wedge m _ \\mu , m _ \\mu ) \\ : u & & \\ref { r m k i n v c o n e } \\\\ & = q ( c , c \\wedge m _ \\gamma ) \\iota ( c \\wedge m _ \\gamma , m _ \\gamma ) \\gamma ( m _ \\mu ) & & m _ \\gamma = m _ \\mu . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} y ( t , \\Phi ( t , x ) ) = y _ 0 ( x ) , \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} r _ n ^ 2 = \\frac { M \\log { n } + c _ n } { n } s _ n ^ 2 = \\frac { 2 M \\log { n } + d _ n } { n } \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} Q \\partial _ Q \\widetilde { f } ( Q ) = Q \\widetilde { f } ( Q ) \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} \\int _ \\Omega J d x & = \\int _ \\Omega ( v u _ x ( \\rho u ) _ x - u v _ x ( \\rho u ) _ x + v _ x ( u ( \\rho u ) _ x + \\rho u u _ x ) ) d x \\\\ & = \\int _ \\Omega ( v u _ x ( \\rho u ) _ x + \\rho u v _ x u _ x ) d x = \\int _ \\Omega u _ x ^ 2 d x . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} ( B \\phi ) ^ \\ast ( p _ 1 ) = p _ 1 - ( 2 k + 1 ) t _ 2 ^ 2 , \\ \\ ( B \\lambda _ { 2 k + 1 } ) ^ \\ast ( q _ 4 ) = q _ 4 - k t _ 2 ^ 2 . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} y _ l = t _ l \\big ( u ( z _ l ) - u ( x _ l ) \\big ) z _ l \\in \\mathcal { S } _ { x _ l } 0 < t _ l < \\gamma _ { k , \\rho } ( x _ k , y _ l ) + 1 / l . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\alpha _ { i } = - \\nu ( f _ i ) \\ge - \\nu ( x _ { j _ 1 } ^ { m _ { j _ 1 } } \\dots x _ { j _ r } ^ { m _ { j _ r } } ) = \\sum _ { k = 1 } ^ r m _ { j _ k } \\alpha _ { j _ k } . \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} 0 \\leq W _ t ( x , y ) \\leq K _ t ( x , y ) = \\frac { 1 } { ( 2 t ) ^ { \\gamma _ k + d / 2 } c _ k } e ^ { - ( | x | ^ 2 + | y | ^ 2 ) / 4 t } E _ k ( \\frac { x } { \\sqrt { 2 t } } , \\frac { y } { \\sqrt { 2 t } } ) . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb R ^ n } g ( x ) + \\sup _ { x , y \\in \\mathbb R ^ n , x \\not = y } \\frac { | g ( x ) - g ( y ) | } { | x - y | } < L _ g \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} ( T _ { \\rm r e g } ) ^ { * * } = ( ( T ^ { * * } ) _ { \\rm r e g } ) ^ { * * } . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\sigma ( \\phi ( t , z ) ) = \\phi ( t , \\sigma ( z ) ) , \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\lvert \\nabla K _ \\lambda ( x ) \\rvert \\ , d x \\le \\int _ 0 ^ \\infty ( 1 + \\lvert \\lambda \\rvert ^ { 1 / 2 } r ) e ^ { - \\lvert \\lambda \\rvert ^ { 1 / 2 } \\cos ( \\psi / 2 ) r } \\ , d r . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\frac { d } { d t } \\log b ( u ( t ) ) = s ( u ( t ) ) g ( \\Psi _ { u ( t ) } ( v ) ) ^ { - \\frac { 2 } { n - 2 } } = w ( u ( t ) ) . \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} F ( x ) = \\lim _ { k \\to \\infty } \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ { i } ^ { ( k ) } g ( x _ { i } ^ { ( k ) } ) . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} k p - ( n - k ) p ^ 2 \\geq k p - n p ^ 2 \\geq \\frac { \\eta _ 1 n } { 2 N } \\frac { \\eta _ 1 } { N } - n \\left ( \\frac { \\eta _ 2 } { N } \\right ) ^ 2 = - \\eta \\frac { n } { N ^ 2 } \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} w ' ( t ) & = V _ m ' ( t ) + U _ { m + 1 } '' ( t ) \\\\ & = - A U _ { m + 1 } ( t ) - U _ { m + 1 } ' ( t ) . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} | f | _ { 2 , \\kappa ; \\Lambda ^ n } & : = | f | _ { \\kappa ; \\Lambda ^ n } + | \\partial _ t f | _ { \\alpha ; \\Lambda ^ n } + | \\partial _ x f | _ { \\kappa ; \\Lambda ^ n } + | \\partial _ { x x } f | _ { \\kappa ; \\Lambda ^ n } . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\nabla f ( x _ 0 ) + \\frac { t _ 0 ^ 2 } { 6 } \\left ( \\sum _ { i = 1 } ^ n \\nabla f _ { i i } \\right ) ( x _ 0 ) = 0 \\ , . \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} p _ { w } = \\begin{cases} \\mu [ w ] \\cdot c & w \\in \\mathcal { W } \\\\ 2 ^ { - m \\epsilon ^ { - 1 } } \\cdot c & \\end{cases} , \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} p _ { l } ^ { \\eta } ( t , x , y ) & \\geq \\int _ { M _ { x , y } \\cap W } \\eta ( u ) K ( u , x ) \\rho _ { t } ( u ) m _ { x , y } ( d u ) \\\\ & = \\int _ { M _ { x , x } \\cap V _ { x , h } } \\eta ( S _ { x , y } ^ { - 1 } v ) K ( S _ { x , y } ^ { - 1 } v , x ) \\rho _ { t } ( S _ { x , y } ^ { - 1 } v ) \\ , m _ { x , y } \\circ S _ { x , y } ^ { - 1 } ( d v ) . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\zeta ( z , q = 0 ) & = 2 \\pi i \\frac { e ^ { 2 \\pi i z } } { e ^ { 2 \\pi i z } - 1 } - \\pi i = f ( z ) - \\pi i \\\\ \\wp ( z , q = 0 ) & = ( 2 \\pi i ) ^ 2 \\frac { e ^ { 2 \\pi i z } } { ( e ^ { 2 \\pi i z } - 1 ) ^ 2 } = g ( z ) . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} [ W _ j , \\sqrt { - L } ] f ( 0 ) = \\lim _ { t \\to 0 } H _ t [ W _ j , \\sqrt { - L } ] f ( 0 ) . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} ( \\partial _ { t } & - \\underline { \\Delta } ) \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } \\\\ & = - 2 f ^ { 2 } _ { i j } + 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } ) - 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\ , S c a l ' ) - 4 R i c ' _ { i j } \\overline { \\nabla } _ { i } f \\overline { \\nabla } _ { j } f . \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\log \\frac { 1 } { \\mu _ \\varepsilon ^ 2 } = \\gamma _ \\varepsilon ^ 2 ( 1 + o ( 1 ) ) \\ , , \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} Q _ { \\lambda } ( { \\bf x } ) = X _ { \\lambda _ 1 } ^ + X _ { \\lambda _ 2 } ^ + \\cdots X _ { \\lambda _ l } ^ + \\cdot 1 = [ z ^ \\lambda ] X ^ + ( z _ 1 ) X ^ + ( z _ 2 ) \\cdots X ^ + ( z _ l ) \\cdot 1 \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} H ^ * ( X ; \\C ) \\cong S R ( X ) \\Big / \\Big ( \\sum _ { j = 1 } ^ N \\nu _ j Z _ j \\Big ) , \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} P f ' = 0 \\mbox { f o r a l l } \\{ f , f ' \\} \\in T . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\vert u ( x ) \\vert = \\lim _ { i \\rightarrow \\infty } \\vert u ( x ) - u ( y _ i ) \\vert \\leq C ' r _ 0 ^ { s - s ' } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} V _ { e , n } & = { \\bf 1 } _ { \\zeta _ t ( e ) = 1 \\ ; t \\in [ n T , ( n + 1 ) T ) } , \\\\ U _ { x , n } & = { \\bf 1 } _ { \\mathcal { R } ^ { x } \\cap [ n T , ( n + 1 ) T ) = \\emptyset } . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} h + g = \\phi _ 1 ( y _ 1 ) + T ( y _ 2 + z _ 2 ) . \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} I = \\frac 1 { A _ 1 } [ \\xi ' ( q ) - \\xi ' ( b ) ] \\frac 1 { A _ 1 \\xi ' ( s ) + \\frac 1 { | \\nu | } } \\Big | _ { s = 0 } ^ q = \\frac { [ \\xi ' ( b ) - \\xi ' ( q ) ] | \\nu | \\xi ' ( q ) } { A _ 1 \\xi ' ( q ) + \\frac 1 { | \\nu | } } , \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} S ( x ' , a , \\Delta ) = \\left \\{ y \\in \\mathbb { R } ^ 2 \\colon \\lvert y - x ' \\rvert \\in [ a , a + \\Delta ] \\right \\} . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\mathbb { Z D } _ n ^ m ( p _ 1 , \\ldots , p _ r ) = \\{ ( X _ 1 , \\ldots , X _ m ) \\in \\mathbb { D } ^ m _ n \\ : | p _ j ( X _ 1 , \\ldots , X _ m ) = \\mathbf { 0 } _ n , 1 \\leq j \\leq r \\} \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\infty } ^ 1 = X _ 1 ( \\overline { p } ) + X _ 2 ( \\overline { q } ) . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} E ( \\Phi ) = \\left \\{ x \\in ( 0 , 1 ) : \\lim _ { n \\to \\infty } \\frac { S _ n ( x ) } { \\Phi ( n ) } = 1 \\right \\} , \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} A _ { k } ( \\theta ) = \\left ( \\begin{array} { c c } P _ k ( \\theta ) & - P _ { k - 1 } ( \\theta + \\alpha ) \\\\ P _ { k - 1 } ( \\theta ) & - P _ { k - 2 } ( \\theta + \\alpha ) \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} f \\circ g = \\sum _ { j = 0 } ^ { p - 1 } ( - 1 ) ^ { ( p - 1 - j ) ( q - 1 ) } f \\underset { ( j ) } { \\circ } g \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\beta _ { g , a } ^ { \\prime } : = C ^ { - 1 } \\beta _ { g , a } ( C ) \\in U _ E / U _ E ^ { m + 1 } , \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\gamma = 1 6 \\sigma ^ 4 ( 2 a \\eta - a ^ 2 + 1 ) ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} - ( 2 \\pi i ) ^ 2 q \\frac { d \\zeta } { d q } = \\zeta \\wp + \\frac { 1 } { 2 } \\wp ' . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} N < s _ { i _ { s , 1 } } < s _ { i _ { s , 2 } } < \\cdots < s _ { i _ { s , l ( s ) } } = \\max \\{ s _ 1 , \\ldots , s _ m \\} . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\theta ( G ) = \\min \\max _ { v \\in V } { 1 \\over \\langle \\mathbf { x } _ v , \\mathbf { h } \\rangle ^ 2 } \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\lambda _ \\varepsilon = \\frac { 4 } { \\gamma _ \\varepsilon ^ 2 \\exp \\left ( 1 + M + \\frac { \\gamma _ \\varepsilon ^ 2 ( A ( \\gamma _ \\varepsilon ) - 2 \\xi _ \\varepsilon ) } { 2 } + o ( \\tilde { \\zeta } _ \\varepsilon \\gamma _ \\varepsilon ^ 2 ) \\right ) } \\ , , \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\begin{aligned} k _ { i i } & = 0 & k _ { i j } - k _ { j i } & = 0 & \\textstyle \\sum _ { i \\in I } k _ { i j } & = 0 \\\\ c _ { i i } & = 0 & & & \\textstyle \\sum _ { i \\in I } c _ { i j } & = 0 \\\\ e _ { i i } & = 0 & e _ { i j } - e _ { j i } & = 0 \\end{aligned} \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} \\bar \\omega = \\sum \\bar { a } _ i \\cdot d \\bar { b } _ i . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\xi ^ 0 & \\ge | x _ 1 | \\\\ \\eta ^ 0 & \\ge | x _ 2 | \\\\ \\xi ^ j & = \\ \\ \\ \\ \\cos ( \\theta _ j ) \\xi ^ { j - 1 } + \\sin ( \\theta _ j ) \\eta ^ { j - 1 } \\ j = 1 , \\dots , \\nu \\\\ \\eta ^ j & \\ge | - \\sin ( \\theta _ j ) \\xi ^ { j - 1 } + \\cos ( \\theta _ j ) \\eta ^ { j - 1 } | j = 1 , \\dots , \\nu \\\\ \\xi ^ \\nu & \\le x _ 3 \\\\ \\eta ^ \\nu & \\le \\tan ( \\theta _ \\nu ) \\xi ^ \\nu \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\widetilde { J _ i } \\left ( q , \\varphi ^ { - 1 } ( Q ) \\right ) = \\sum _ { \\substack { a + b = i \\\\ 0 \\leq a , b \\leq N } } ( - 1 ) ^ a \\binom { \\ell _ q \\left ( \\left ( \\frac { 1 - q } { z } \\right ) ^ { N + 1 } Q \\right ) } { a } J _ b \\left ( q , \\left ( \\frac { 1 - q } { z } \\right ) ^ { N + 1 } Q \\right ) \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = 0 , u ( 0 , x ) = \\begin{cases} u _ l = \\frac { 1 } { 2 } , & x \\le 0 , \\\\ u _ r = \\frac 1 2 , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} H ( u _ 1 , u _ 2 ; t ) = 0 \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} P = - h ^ { 2 } \\Delta + V , \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} w : = ( w ^ 1 _ 1 , w ^ 2 , \\dots , w ^ N ) \\in L ^ p ( [ 0 , L ^ 1 ] ; \\R ^ 3 ) \\times \\prod _ { i = 2 } ^ N ( ( L ^ p ( [ 0 , L ^ i ] ; \\R ^ 3 ) \\times \\R ^ 3 \\times \\R ^ 3 \\times \\R ^ 3 ) = : V . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} m ( f ) : = \\int _ 0 ^ 1 \\log | f ( e ^ { 2 \\pi i t } ) | \\ , { \\rm d } t \\ , . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi _ i + R \\varphi _ i = & - \\frac { n - 1 } { n } \\tau ^ { 2 a } \\varphi _ i ^ { N - 1 } + \\left ( | \\sigma + L W _ i | ^ { 2 } + k _ i ^ { 2 } \\right ) \\varphi _ i ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ { * } L W _ i = & \\frac { n - 1 } { n } \\varphi _ i ^ { N } d \\tau ^ a . \\endaligned \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} - \\phi ' ( s ) = \\frac { K u \\Gamma ( u + 2 ) } { u + 1 } s ^ { - u - 1 } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} g _ j ( \\lambda ) = 0 , j = 1 , 2 , \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{gather*} p _ 1 + p _ 2 + \\dots + p _ l + q _ 1 + q _ 2 + \\dots + q _ l \\leq n ; \\\\ p _ i = 0 \\ \\Rightarrow \\ q _ i = 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j = 1 . \\end{gather*}"}
-{"id": "1206.png", "formula": "\\begin{align*} & u ( x ) = U ( t ( x ) ) , \\breve u ( x ) = \\frac { u ( x ) } { s ( x ) } = \\frac { U ( t ) } { S ( t ) } = \\breve U ( t ) , \\ ; \\ ; \\ ; { \\rm w h e r e } \\\\ & s ( x ) : = \\frac { 1 } { ( 1 + x ^ 2 ) ^ { ( \\lambda + 1 ) / 2 } } = ( 1 - t ^ 2 ) ^ { ( \\lambda + 1 ) / 2 } : = S ( t ) . \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\ker d _ x f = \\textrm { s p a n } \\{ v _ 2 ( x ) , \\ldots , v _ { n - 1 } ( x ) \\} . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\ll \\min \\limits _ { i = 2 , \\cdots , I } \\{ \\lambda _ { k _ i } - \\lambda _ { k _ { i - 1 } } \\} . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} & \\widetilde { S ^ { K \\textnormal { t h } } } = S ^ { K \\textnormal { t h } } \\left . \\right . _ \\circ \\left . \\right . \\prod _ { j = 1 } ^ r \\phi _ j ^ { \\ell _ q ( Q _ j ) } & & \\widetilde { T ^ { K \\textnormal { t h } } } = \\prod _ { j = 1 } ^ r \\phi _ j ^ { - \\ell _ q ( Q _ j ) } \\left . \\right . _ \\circ \\left . \\right . T ^ { K \\textnormal { t h } } \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} ( 1 , 0 ) ^ { \\perp } & = - \\cos \\theta _ 1 J E _ 1 = - \\sqrt { 2 } J \\nabla ( \\sin \\theta _ 1 ) , & ( i , 0 ) ^ { \\perp } & = - \\sin \\theta _ 1 J E _ 1 = \\sqrt { 2 } J \\nabla ( \\cos \\theta _ 1 ) , \\\\ ( 0 , 1 ) ^ { \\perp } & = - \\cos \\theta _ 2 J E _ 2 = - \\sqrt { 2 } J \\nabla ( \\sin \\theta _ 2 ) , & ( 0 , i ) ^ { \\perp } & = - \\sin \\theta _ 2 J E _ 2 = \\sqrt { 2 } J \\nabla ( \\cos \\theta _ 2 ) , \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} 0 \\ge \\int _ { \\Omega } | \\nabla w _ 1 ^ e | ^ 2 \\ : d x - \\int _ { \\Omega } V _ { e s } ( x ) ( w _ 1 ^ e ) ^ 2 \\ : d x = Q _ { e s } ( w _ 1 ^ e ) = Q _ u ( w _ 1 ^ e ) , \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & \\gamma ( r _ { t - } ) d N _ t \\medskip \\\\ r _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\Delta = t ^ { a _ 0 } - t ^ { a _ 1 } + t ^ { a _ 2 } . . . . + t ^ { a _ n } \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\log \\phi \\Big ( \\Big | \\prod _ { j = 1 } ^ m A _ j ^ \\theta \\Big | ^ \\frac { p } { \\theta } \\Big ) \\leq \\int _ { - \\infty } ^ { + \\infty } d t \\beta _ \\theta ( t ) \\log \\phi \\Big ( \\Big | \\prod _ { j = 1 } ^ m A _ j ^ { 1 + i t } \\Big | ^ p \\Big ) . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\| f _ { u } \\| _ { E _ { u } ^ s } = 1 , 1 \\leq u \\leq k . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - F _ m ^ + \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n . \\end{cases} \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} & \\| \\varphi - \\varphi _ D \\| _ { C ^ 0 ( [ 0 , t ] ; V ^ * ) ) } + \\| \\varphi _ D \\| + \\| \\nabla \\varphi \\| _ { L ^ 2 ( 0 , t ; H ) } ^ 2 + \\norm { \\sigma } _ { C ^ 0 ( [ 0 , t ] ; H ) \\cap L ^ 2 ( 0 , T ; V ) } = 0 \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} j = f V ^ 2 d \\tau - d [ \\sinh ^ { - 1 } ( \\frac { f d i v ( V ^ 2 \\nabla \\tau ) } { | H _ 0 | | H | } ) ] - \\alpha _ { H _ 0 } + \\alpha _ { H } . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} r = \\psi + Y \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\sigma _ t [ h , h ' ] : = \\frac 1 n \\sum _ { i = 1 } ^ n h _ i h _ i ' \\ , p _ { i , t } ( 1 - p _ { i , t } ) . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\infty } | a _ { i , j } | q _ i \\leq \\alpha p _ j ~ \\mbox { f o r a l l } ~ j \\geq 1 \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\gamma \\ast \\delta : = ( { ( g _ \\alpha h _ { \\ell } ) } _ { \\alpha \\in L } ; k ) \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\mathcal { T } _ { \\mathcal { K } } ( g ( \\overline { x } ) ) \\ni g ( x ^ k ) - b ^ k - g ( \\overline { x } ) = g ' ( \\overline { x } ) ( x ^ k - \\overline { x } ) + o ( t _ k ) , \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} g _ 1 ' ( u ) = \\frac { q [ \\xi ' ( u ) ( q + q z _ 1 + q z _ 2 - u z _ 1 ) - ( 1 + z _ 2 ) \\xi ' ( q ) u ] } { ( 1 + z _ 2 ) ( 1 + z _ 1 + z _ 2 ) [ q ( 1 + z _ 1 + z _ 2 ) - u z _ 1 ] \\xi ' ( q ) } , \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} A _ I & = c _ { 1 2 } e _ { 3 1 } + c _ { 2 3 } e _ { 1 2 } + c _ { 3 1 } e _ { 2 3 } \\\\ M _ I & = ( A _ I ) ^ 2 \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} s _ 0 = \\frac { 1 - r } { 1 2 \\sqrt { d } } . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} C = \\sum _ { i = 1 } ^ n \\nabla ^ 2 f _ { i i } = 2 4 \\ , { \\rm d i a g } ( a ) \\ , . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) , u _ t ( x , 0 ) = u _ 1 ( x ) , x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} U _ 1 = - X ^ { \\perp } = J X _ 1 + J X _ 2 . \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\dfrac { - \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } \\ln n } { \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } } & = \\sum _ { p } \\dfrac { p ^ { s } \\ln p ( p ^ s - 1 ) - ( p ^ s - 1 + p ^ t ) p ^ s \\ln p } { ( p ^ s - 1 ) ( p ^ s - 1 + p ^ t ) } \\\\ & = \\sum _ { p } \\dfrac { ( p ^ s - 1 - p ^ s + 1 - p ^ t ) p ^ s \\ln p } { ( p ^ s - 1 ) ( p ^ s - 1 + p ^ t ) } \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\left \\{ \\frac { n r _ i } { p } \\right \\} < \\left \\{ \\frac { m r _ i } { p } \\right \\} ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} U _ { N } = \\left [ \\begin{array} [ c ] { c c c } 0 & N ^ { 2 } & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] , V _ { N , \\ell } = \\left [ \\begin{array} [ c ] { c c c } 0 & N ^ { 2 } & \\frac { \\ell } { 2 } \\\\ 0 & 0 & \\frac { \\ell } { N ^ { 2 } } \\\\ 0 & 0 & 0 \\end{array} \\right ] , \\ell = 1 , \\cdots , N . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} w _ 0 ( x ) = \\begin{cases} \\exp ( - ( 1 - x ^ 2 ) ^ { - 1 } ) , & \\mbox { $ | x | < 1 $ } \\\\ 0 & \\mbox { $ | x | \\geq 1 $ } , \\end{cases} \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} | \\hat \\lambda _ j ^ { ( n ) } - \\lambda _ j ^ { ( n ) } | / g _ j ^ { ( n ) } \\geq \\frac { \\hat \\lambda _ j ^ { ( n ) } - \\lambda _ j ^ { ( n ) } } { \\lambda _ { j - 1 } ^ { ( n ) } - \\lambda _ j ^ { ( n ) } } = 1 - \\frac { \\lambda _ { j - 1 } ^ { ( n ) } - \\hat \\lambda _ j ^ { ( n ) } } { \\lambda _ { j - 1 } ^ { ( n ) } - \\lambda _ j ^ { ( n ) } } , \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} E ( \\prod _ { x , y } s _ { x , y } ^ { N ^ { ( 1 ) } _ { x , y } } \\prod _ x e ^ { - \\sum _ { x } \\chi _ { x } \\hat { \\mathcal { L } } _ 1 ^ x } ) = E ( e ^ { - \\frac { 1 } { 2 } \\sum _ { x , y } C _ { x , y } ( s _ { x , y } - 1 ) \\varphi _ x \\bar \\varphi _ y - \\frac { 1 } { 2 } \\sum \\chi _ { x } \\phi _ x \\bar \\phi _ x } ) . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\parallel v \\parallel _ { H ^ { s + 2 } ( \\Omega _ f ) } + \\parallel q \\parallel _ { H ^ { s + 1 } ( \\Omega _ f ) } \\leqslant C \\parallel v _ t \\parallel _ { H ^ { s } ( \\Omega _ f ) } + C \\parallel \\omega _ { \\nu _ { \\Lambda } } \\parallel _ { H ^ { s + \\frac { 1 } { 2 } } ( \\Gamma _ c ) } , s = 0 , 1 \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} A _ { m n } ( x ) = \\phi _ { m - n } ( T ^ { m } x ) + \\overline { \\phi _ { n - m } ( T ^ { n } x ) } , m \\neq n , \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} f = f ^ { ( 1 ) } r + O ( r ^ 2 ) \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} h ( x ) & \\le g _ { 6 + } ( x ) \\Bigr | _ { x = \\frac { 8 5 8 } { 2 0 5 9 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 2 4 5 7 3 4 8 3 0 2 4 3 2 6 8 3 5 4 9 4 1 0 8 2 6 8 5 8 5 3 5 6 1 5 4 3 6 6 1 } { 2 0 0 4 9 1 5 0 9 7 4 1 1 5 9 4 0 4 9 2 6 1 7 1 2 2 2 8 5 5 5 4 3 0 6 7 8 0 1 9 0 4 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{gather*} Q ^ 2 = 0 . \\end{gather*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\check { H } ( z ; \\eta ) = \\exp \\bigl ( \\sum _ { m > 0 } z ^ m ( x _ m - \\frac { \\eta } { z } ( - 1 ) ^ m \\tilde { e } _ { m - 1 } ) \\bigr ) \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\{ p , z \\} = 2 i ( \\gamma ^ \\prime ) ^ \\alpha ( \\gamma ^ \\prime ) ^ \\beta A _ { \\alpha \\beta } \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\phi \\big ( | G _ X ( i t ) | ^ \\frac { 2 } { s } \\big ) = & \\ \\phi \\Big ( \\big ( X _ 2 ^ { \\frac { r s ( 1 + i t ) } { 2 } } C _ 2 ^ { - \\frac { r s ( 1 + i t ) } { 2 } } M ^ * M C _ 2 ^ { - \\frac { r s ( 1 - i t ) } { 2 } } X _ 2 ^ { \\frac { r s ( 1 - i t ) } { 2 } } \\big ) ^ \\frac { 1 } { s } \\Big ) \\\\ = & \\ \\phi \\Big ( \\big ( M C _ 2 ^ { - \\frac { r s ( 1 - i t ) } { 2 } } X _ 2 ^ { r s } C _ 2 ^ { - \\frac { r s ( 1 + i t ) } { 2 } } M ^ * \\big ) ^ \\frac { 1 } { s } \\Big ) . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} E ( z ; \\eta ) = \\sum _ { m \\geq 0 } z ^ m ( e _ m + \\eta \\tilde { e } _ m ) , \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} \\int _ a ^ { r _ 3 } ( r _ 2 - a ) ^ { 1 + \\zeta } ( r _ 3 - r _ 2 ) ^ \\zeta d r _ 2 = & \\int _ 0 ^ { r _ 3 - a } r _ 2 ^ { 1 + \\zeta } ( r _ 3 - a - r _ 2 ) ^ \\zeta d r _ 2 \\\\ = & ( r _ 3 - a ) ^ { 2 ( 1 + \\zeta ) } B \\Big ( ( 1 + \\zeta ) + 1 , \\zeta + 1 \\Big ) , \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} & h ( x ) \\le g _ { 1 0 - } ( x ) \\Bigr | _ { x = \\frac { 7 2 9 3 5 } { 1 7 5 0 9 9 } } + \\frac 1 { 1 0 \\cdot 6 ^ { 1 0 } } - \\eta \\\\ & = - \\frac { 9 2 7 0 2 3 5 2 3 6 5 4 1 1 5 2 9 6 1 4 4 8 0 1 9 8 0 2 4 7 7 9 4 3 0 8 0 0 9 6 5 4 0 9 } { 1 6 9 1 2 1 2 3 2 7 2 1 6 4 3 4 0 3 7 4 2 7 3 9 3 2 6 1 8 3 4 2 8 9 7 8 7 3 3 5 2 2 3 2 6 9 6 5 7 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} f = - \\sum a _ i \\partial b _ i , \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} V ( x ) = W ( h x ) \\ , , W \\in C _ c ^ \\infty ( \\mathbb { R } ) 0 < h \\ll 1 \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} V a r ( X ) = E [ ( X - E [ X ] ) ^ 2 ] = E [ X ^ 2 ] - ( E [ X ] ) ^ 2 . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\mathcal L ( \\mathcal A ) = \\bigcup _ { m , n } \\bigcup _ { u \\in I \\cup Q ^ { n } \\atop v \\in J \\cup Q ^ { m } } \\left \\{ \\mathfrak x \\in \\mathcal F ( \\mathcal X ) \\left | \\begin{array} { c } \\mathrm { O U T } _ { u } \\\\ \\updownarrow \\\\ \\mu _ { \\Gamma } ( \\mathfrak x ) \\\\ \\updownarrow \\\\ \\mathrm { I N } _ { v } \\end{array} = 1 \\right . \\right \\} . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\mathbb { S } : = \\mathcal { S } \\otimes L , L : = | \\mathrm { d e t } \\ , T ^ * M | ^ \\frac 1 2 . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} e _ A - e _ B = \\sum _ { i = 1 } ^ k \\gamma _ i ( e _ { C _ i } - e _ B ) . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} a ^ { ( j ) } _ \\bullet & = ( a _ 0 , \\ldots , a _ { j - 2 } , a _ { j } , a _ j \\ ! + \\ ! 1 , \\ldots , a _ { j + m } \\ ! + \\ ! 1 , \\ldots , a _ r ) . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\tau _ 9 ( x ) = g _ { 9 - } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 7 9 8 5 } { 1 9 1 7 1 } , c ) . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} S _ n ' ( t ) : = S _ n ( t ) + \\mathrm { B i n } ( a _ n , \\pi _ n ( t ) ) \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} c _ 2 4 ^ { - ( n + 1 ) } = g _ { n + 1 } ( t _ { { n + 1 } } ) = M _ n ( g _ n ) ( t _ { n + 1 } ) \\ge c _ 2 4 ^ { - 2 n } \\int _ { 0 } ^ { t _ { n + 1 } } g ^ 2 _ n ( \\tau ) d t \\ge c _ 2 4 ^ { - 2 n } \\int _ { t _ n } ^ { t _ { n + 1 } } c _ 2 ^ 2 4 ^ { - 2 n } d t . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} U _ { T X } = U + 2 U U _ { X X } + U _ X ^ 2 . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\hat { \\textbf { V } } _ k = \\left [ \\textbf { a } \\left ( \\textbf { x } \\right ) , j \\textbf { a } \\left ( \\textbf { x } \\right ) , \\beta \\frac { \\partial \\textbf { a } \\left ( \\textbf { x } \\right ) } { \\partial x _ 1 } , \\beta \\frac { \\partial \\textbf { a } \\left ( \\textbf { x } \\right ) } { \\partial x _ 2 } \\right ] \\Bigg | _ { { \\boldsymbol { \\psi } } = \\hat { \\boldsymbol { \\psi } } _ { k - 1 } } . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\varphi _ y ( x ) = F ( x ) - F ( y ) - \\langle \\nabla F ( y ) , x - y \\rangle \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\| Q \\{ f , f ' \\} \\| _ { \\overline { T } } ^ { 2 } = \\| \\{ 0 , P f ' \\} \\| _ { \\overline { T } } ^ { 2 } = \\| P f ' \\| ^ { 2 } . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\mathrm { R e l } ^ { G , \\mu } _ { M _ S , b } = \\coprod _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } \\mathrm { R e l } ^ { M _ b , \\mu _ b } _ { M _ S , b ' } . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { \\gamma \\in \\Lambda } \\left ( \\prod _ { l = 1 } ^ { m } p _ { j _ l } \\right ) = \\sum _ { \\gamma \\in \\Lambda } \\mathbb { P } \\left ( \\Tilde { Y } _ 1 = a _ { j _ 1 } , \\ldots , \\Tilde { Y } _ m = a _ { j _ m } \\right ) \\leq \\mathbb { P } \\left ( \\prod _ { l = 1 } ^ m \\Tilde { Y _ l } \\geq 1 \\right ) , \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\left ( 1 - q ^ d \\right ) ^ 3 h _ d a _ d + \\sum _ { k = 0 } ^ 3 \\binom { 3 } { k } ( - q ^ d ) ^ k k h _ { d } = h _ { d - 1 } a _ { d - 1 } \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} H = \\int M ( d s , \\theta ^ H _ s ) + L ^ H , \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} q _ { \\mathcal { G } } \\left ( \\mathbf { W } ; P ^ { \\ast } \\right ) = b _ { a } ( \\mathbf { O } ; P ^ { \\ast } ) - \\chi _ { a } ( P ^ { \\ast } ; \\mathcal { G } ) + g ( \\mathbf { W } ) , \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ l \\left \\Vert a _ j \\right \\Vert ^ n < \\left \\Vert a \\right \\Vert _ { \\mathcal { P } _ n } + \\varepsilon / 2 , \\sum _ { j = 1 } ^ m \\left \\Vert b _ j \\right \\Vert ^ n < \\left \\Vert b \\right \\Vert _ { \\mathcal { P } _ n } + \\varepsilon / 2 . \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} T _ { z } \\coloneqq \\{ a \\in Y \\colon f ( a ) = z \\} . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\vec H _ { h } ^ l = \\{ \\vec z _ h \\in L ^ 2 ( \\Omega ) \\mid \\vec z _ h { } _ { | _ { K } } \\circ T _ K \\in \\mathbb { Q } _ l ^ d \\ , , \\ ; \\vec z _ h { } _ { | \\partial \\Omega } = \\vec 0 \\} \\ , . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\lambda _ \\varepsilon = \\frac { 4 + o ( 1 ) } { \\gamma _ \\varepsilon ^ 2 \\exp ( 1 + M ) } \\ , , \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} ( K ^ { - 1 } h ) _ { t } = C _ { H } \\cdot t ^ { H - \\frac { 1 } { 2 } } D _ { 0 ^ { + } } ^ { H - \\frac { 1 } { 2 } } \\left ( s ^ { \\frac { 1 } { 2 } - H } \\dot { h } _ { s } \\right ) ( t ) . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\mathbf { G } _ { j } \\mathbf { \\perp \\ ! \\ ! \\ ! \\perp } _ { \\mathcal { G } } \\left [ \\overline { \\mathbf { B } } _ { j - 1 } \\mathbf { \\backslash } \\overline { \\mathbf { G } } _ { j - 1 } \\right ] \\mathbf { | } \\overline { \\mathbf { G } } _ { j - 1 } , \\overline { \\mathbf { A } } _ { j - 1 } j = 1 , \\dots , p . \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} \\frac { t } { \\log ( 1 + t ) } = \\sum _ { n = 0 } ^ \\infty b _ n \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 3 , 8 , 9 ] ) . \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} a ( u , v ) = \\int _ \\Omega z ^ \\beta u \\overline v d x + \\int _ \\Omega \\nabla u \\cdot \\nabla \\overline v d x . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} d z = \\frac { \\mathrm { v o l } _ { F ^ { - 1 } ( y ) } } { \\sqrt { \\det \\left ( \\left ( \\langle \\partial _ { i } z , \\partial _ { j } z \\rangle \\right ) _ { n + 1 \\leqslant i , j \\leqslant m } \\right ) } } \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} \\sum _ { i \\in [ k ] } w ( F _ i ^ + ) \\alpha _ i - w ( F _ i ^ - ) \\alpha _ i = 0 \\mod L _ { \\R } ( A ) . \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\delta ^ 1 ( \\tau _ V ^ t ( V ) ) & = \\tau _ V ^ t ( \\delta ( V ) ) - \\i [ V , \\tau _ V ^ t ( V ) ] \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} F ( \\sigma ( z ) ) = - J _ \\sigma ( z ) \\cdot F ( z ) , \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ n z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( y q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ n } { ( y q ; q ^ 2 ) _ { n + 1 } } \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} g _ D ( y _ 1 , y ' ) = \\sum _ { k = 1 } ^ \\infty b _ k ( y ' ) \\sin ( k y _ 1 ) , \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot H ^ s ( \\mathcal { H } ) } = \\| \\mathcal { H } ^ { \\frac { s } { 2 } } f \\| _ { L ^ 2 ( \\Omega ) } \\| f \\| _ { H ^ s ( \\mathcal { H } ) } = \\| ( I + \\mathcal { H } ) ^ { \\frac { s } { 2 } } f \\| _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = \\dot { \\gamma } _ 3 X _ 1 + \\frac { \\sqrt { 2 } } { 2 } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) X _ 2 + \\omega ( \\dot { \\gamma } ( t ) ) X _ 3 . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} B = \\left [ \\begin{array} { c c } - 2 & - 0 . 3 5 \\\\ - 0 . 3 5 & - 4 \\end{array} \\right ] , \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} h = ( x _ 1 , x _ 2 + c x _ 3 ^ p , x _ 3 ) , c \\neq 0 . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\nu ^ p _ \\epsilon \\star \\nu ^ p _ \\iota = \\begin{cases} N ( p , p ) \\cup \\{ 1 , 0 , \\lambda - \\frac { 1 } { 2 } \\} & \\mbox { i f } \\epsilon = \\iota \\\\ N ( p , p ) \\cup \\{ \\lambda \\} & \\mbox { i f } \\epsilon = - \\iota \\end{cases} \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\langle X ( s ) , X _ 1 ( s ) \\rangle & = 4 s ( 1 + s f ) \\textstyle \\frac { \\partial f } { \\partial \\theta _ 1 } , \\\\ \\langle X ( s ) , X _ 2 ( s ) \\rangle & = 4 s ( 1 + s f ) \\textstyle \\frac { \\partial f } { \\partial \\theta _ 2 } . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} b _ { n , ( j + 1 ) } = \\min \\Big ( c _ 2 4 ^ { - 2 ( n + j ) } b _ { n , j } \\cdot ( b _ { n , j } + e ^ { - { \\frac { 1 } { 2 } } 4 ^ { ( n + j ) } } ) \\cdot c _ 6 4 ^ { 3 n } , c _ 2 4 ^ { - ( n + j + 1 ) } \\Big ) \\mbox { f o r } j \\ge 0 . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} m ( f _ i ) = \\int _ { T ^ N } \\log | f _ i | \\ , d \\mu \\ ; . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} & F \\big ( ( x , t ) , u ( x , t ) , D u ( x , t ) , D ^ 2 u ( x , t ) \\big ) \\\\ & : = \\triangle _ { p ( x , t ) } ^ N u ( x , t ) + \\sum _ { i = 1 } ^ n \\mu _ i \\frac { \\partial u } { \\partial x _ i } ( x , t ) - r u ( x , t ) \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} w ^ \\beta _ + w ^ \\beta _ - = - c _ \\beta t _ \\beta + c _ { \\alpha + \\beta } t _ { \\alpha + \\beta } - 2 \\xi _ \\beta b _ { \\beta , \\alpha + \\beta } e ^ \\alpha \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha _ 2 , \\beta _ 2 \\} \\in \\mathcal { S } _ 2 ( S ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha _ 2 } ( \\rho ) + \\ell _ { \\beta _ 2 } ( \\rho ) ) } + 1 \\right ) ^ { - 1 } = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\underline { \\Delta } G & = t \\left ( 2 f _ { i j } ^ { 2 } + 2 f _ { i } f _ { j j i } + 2 R i c ' _ { i j } f _ { i } f _ { j } + 2 \\alpha R i c ' _ { i j } f _ { i j } - \\alpha \\partial _ { t } ( \\underline { \\Delta } f ) \\right ) \\\\ & \\geq t \\left [ \\left ( 2 f _ { i j } ^ { 2 } + \\alpha R i c ' _ { i j } f _ { i j } \\right ) + 2 f _ { j } f _ { j j i } - 2 \\rho _ { 1 } \\left \\| \\overline { \\nabla } f \\right \\| ^ { 2 } - \\alpha \\partial _ { t } ( \\underline { \\Delta } f ) \\right ] , \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} P ( B _ 2 ^ { ( n ) } ) & \\leq P ( S _ n ( n - f _ 1 ( n ) ) \\leq \\lfloor n - \\varepsilon ' f _ 2 ( n ) \\rfloor - a _ n ) \\\\ & \\leq P ( S _ n ' ( n - f _ 1 ( n ) ) \\leq \\lfloor n - \\varepsilon ' f _ 2 ( n ) \\rfloor ) \\\\ & = P ( \\mathrm { B i n } ( n , 1 - \\pi _ n ( n - f _ 1 ( n ) ) ) \\geq n - \\lfloor n - \\varepsilon ' f _ 2 ( n ) \\rfloor ) \\\\ & = P ( \\mathrm { B i n } ( n , 1 - \\pi _ n ( n - f _ 1 ( n ) ) ) \\geq \\lceil \\varepsilon ' f _ 2 ( n ) \\rceil ) , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 3 ^ { ( n ) } ) = - \\infty . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} Q _ \\lambda ( \\omega ) g ^ { q _ \\alpha - 1 } ( \\varsigma ) = \\int _ { \\Omega _ { q _ \\alpha } } \\frac { g ( \\eta ) } { | \\eta ^ { - 1 } \\varsigma | ^ { Q - \\alpha } } d \\eta , \\ \\ \\ \\ g ( 0 ) = 1 . \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\begin{aligned} { } \\sum ^ { \\infty } _ { j = 1 } \\omega _ { 3 } ( \\sigma ^ { j } ) & = \\frac { 1 } { \\tilde { \\delta } } \\sum ^ { \\infty } _ { j = 1 } \\sum ^ { j } _ { i = 0 } \\omega _ { 1 } ( \\sigma ^ { j - i } ) \\omega _ { 2 } ( \\sigma ^ { i } ) \\\\ & = \\frac { 1 } { \\tilde { \\delta } } \\Big ( \\sum ^ { \\infty } _ { j = 0 } \\omega _ { 1 } { ( \\sigma ^ { j } ) } \\Big ) \\Big ( \\sum ^ { \\infty } _ { j = 1 } \\omega _ { 2 } ( \\sigma ^ { j } ) \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} & \\varphi = t T ( t , \\varphi ) , \\\\ & F _ { \\kappa , 1 } ( t , \\varphi ) , \\ , F _ { \\kappa , 2 } ( t , \\varphi ) \\leq 0 , \\\\ & \\big ( F _ { \\kappa , 1 } F _ { \\kappa , 2 } \\big ) ( t , \\varphi ) = 0 . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } t ^ { 2 H } \\log p ( t , x , y ) = - \\frac { 1 } { 2 } d ( x , y ) ^ { 2 } . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\theta _ { M _ S } ( \\mu _ S ) - \\theta _ { M _ { S ' } } ( \\mu _ { S ' } ) = \\frac { \\theta _ { M _ S } ( \\mu _ S ) - \\sigma _ { \\alpha } ( \\theta _ { M _ S } ( \\mu _ S ) ) } { 2 } = \\frac { 1 } { 2 } \\langle \\theta _ { M _ S } ( \\mu _ S ) , \\alpha \\rangle \\check { \\alpha } . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} R _ { N } ( x , x ' ) : = \\sum \\limits _ { n = 1 } ^ { N } \\omega _ n \\log \\kappa ( { y } _ n , x ) \\log \\kappa ( { y } _ n , x ' ) . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} V _ { N } ^ { \\lambda } & = \\big \\{ \\phi ( x ) : \\phi ( x ) = S ( t ) P ( t ) , \\ ; \\ ; \\forall P \\in { \\mathcal P } _ N \\big \\} \\\\ & = \\big \\{ R _ { n } ^ { \\lambda } \\left ( x \\right ) : n = 0 , 1 , . . . , N \\big \\} . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} i \\frac { \\partial \\Psi ( t , x ) } { \\partial t } = a \\frac { \\partial ^ { 2 } \\Psi ( t , x ) } { \\partial x ^ { 2 } } + i b \\frac { \\partial \\Psi ( t , x ) } { \\partial x } + V ( t , x ) \\Psi ( t , x ) , x \\in \\mathbb { T } , \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} \\varPhi ( x ) = \\varPhi ( t _ 0 , \\cdots , t _ s ; x ) = \\sum _ { k = 0 } ^ { s } h _ { s - k } ( t _ 0 , \\cdots , t _ s ) x ^ { s - k } \\in E _ 1 [ x ] \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} A ( M ( \\sum _ { g \\in G } g ^ { - 1 } ( \\alpha ) g ) ) = ( m _ { i , j } ) ( \\sum _ { g \\in G } g ^ { - 1 } ( \\alpha ) A ( g ) ) . \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} E ~ = ~ \\R ^ d \\qquad \\mathrm { a n d } \\rho ( x , y ) ~ = ~ \\| x - y \\| \\qquad \\forall x , y \\in \\R ^ d . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} F _ m - F _ { m - 1 } = \\left ( \\tilde { U } _ { m - 1 } \\cdot \\nabla \\right ) V _ m + \\left ( U _ { m - 1 } \\cdot \\nabla \\right ) \\tilde { V } _ { m - 1 } + \\left ( \\tilde { U } _ { m - 1 } \\cdot \\nabla \\right ) v _ { } + \\left ( U _ { } \\cdot \\nabla \\right ) \\tilde { V } _ { m - 1 } \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} ( 2 \\mu c _ \\alpha t _ \\alpha - e ^ \\alpha ) ( \\theta ^ \\beta _ \\epsilon e ^ \\beta + e ^ { \\alpha + \\beta } ) = - b _ { \\alpha , \\beta } e ^ \\beta + ( - 2 \\mu c _ \\alpha - b _ { \\alpha , \\beta } \\theta ^ \\beta _ \\epsilon ) e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} l ( k e r ( f ) ) = l ( \\sum _ { s = 1 } ^ { d i m T } t _ s H ^ * _ { T } ( X ) ) = \\sum _ { s = 1 } ^ { d i m T } \\underbrace { ( t _ s , \\cdots , t _ s ) } _ l ( H ^ * _ { T } ( X ) ) . \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} x ^ { s } = \\sum _ { j = 0 } ^ { \\infty } { \\frac { s } { s + t j } \\binom { s + t j } { j } z ^ { j } } , \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} c _ { n , j } \\le 4 ^ { - ( n + 2 ^ { j - j _ 0 } ) } = 4 ^ { - ( n + ( 2 ^ { - j _ 0 } ) 2 ^ { j } ) } . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\mathcal E _ { 0 , j } ( x ) = 4 \\alpha _ j a _ { 0 , j } \\frac { \\langle \\nabla u _ j , \\nabla \\phi \\rangle } { \\phi } \\mathcal E _ { 1 , j } ( x ) = 2 \\left ( \\mathcal Q _ j - \\frac { h _ j ' ( u _ j ) } { h _ j ( u _ j ) } z _ j \\right ) a _ { 1 , j } . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} c _ { { } _ F } : = \\displaystyle \\int _ { - 1 } ^ 1 \\sqrt { 2 F ( s ) } \\ , d s \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} t \\frac { d w _ { * } } { d t } = - w _ { * } \\left ( 1 + t \\left ( \\frac { 2 t } { 1 + t ^ { 2 } } + \\frac { t \\left ( | M | ^ { * } \\right ) ^ { 2 } } { 2 } \\right ) \\right ) + \\frac { ( R _ { g ( t ) } - t ^ { 2 } \\overline { R } ) _ { * } } { 2 ( 1 + t ) ^ { 2 } } . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} { \\epsilon _ 1 } \\buildrel \\Delta \\over = \\frac { 1 } { 2 } \\left ( { 1 - q \\left ( \\theta \\right ) } \\right ) > 0 { \\epsilon _ 2 } \\buildrel \\Delta \\over = \\frac { 1 } { 2 } q \\left ( \\theta \\right ) > 0 . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} K = \\frac { w _ { y y } w _ { z z } - w _ { y z } ^ { 2 } } { w _ { z } ^ { 4 } } , H = \\frac { w _ { z } ^ { 2 } w _ { y y } - 2 w _ { y } w _ { z } w _ { y z } + \\left ( 1 + w _ { y } ^ { 2 } \\right ) w _ { z z } } { 2 w _ { z } ^ { 3 } } , \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} \\int _ { \\mathcal { H } _ { - s } } \\ ! \\ ! \\ ! \\ ! I _ { k , 1 } ( X ) \\mu ( d X ) = 0 , \\qquad \\int _ { \\mathcal { H } _ { - s } } \\ ! \\ ! \\ ! \\ ! I _ { k , 2 } ( X ) \\mu ( d X ) = 0 , \\ k \\geq 1 . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} { \\rm d e t } ( I I ^ L _ 1 ) = - \\frac { L } { 4 } - \\langle e _ 1 , \\nabla _ H ( \\frac { X _ 3 u } { | \\nabla _ H u | } ) \\rangle + \\frac { 1 } { 2 } ( \\overline { q } ^ 2 - \\overline { p } ^ 2 ) + O ( L ^ { - \\frac { 1 } { 2 } } ) ~ ~ { \\rm a s } ~ ~ L \\rightarrow + \\infty . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\frac { \\lambda _ 1 ^ { ( n ) } } { \\lambda _ 1 ^ { ( n ) } - \\lambda _ { 2 } ^ { ( n ) } } & = 1 + \\frac { \\lambda _ { 2 } ^ { ( n ) } } { \\lambda _ 1 ^ { ( n ) } - \\lambda _ { 2 } ^ { ( n ) } } \\leq 1 + \\sum _ { k > 1 } \\frac { \\lambda _ k ^ { ( n ) } } { \\lambda _ 1 ^ { ( n ) } - \\lambda _ k ^ { ( n ) } } , \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} ( A ^ { * m } | A ^ p | ^ { 2 k } A ^ m ) ^ \\frac { m } { p k + m } & = ( U _ m ^ * | A ^ { * m } | | A ^ p | ^ { 2 k } | A ^ { * m } | U _ m ) ^ \\frac { m } { p k + m } \\\\ & = U _ m ^ * ( | A ^ { * m } | | A ^ p | ^ { 2 k } | A ^ { * m } | ) ^ \\frac { m } { p k + m } U _ m . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\frac { \\partial ^ 3 \\mathcal { F } } { \\partial t _ 2 ^ 3 } + \\frac { \\partial ^ 3 \\mathcal { F } } { \\partial t _ 1 ^ 3 } \\frac { \\partial ^ 3 \\mathcal { F } } { \\partial t _ 1 \\partial t _ 2 ^ 2 } = \\left ( \\frac { \\partial ^ 3 \\mathcal { F } } { \\partial t _ 1 ^ 2 \\partial t _ 2 } \\right ) ^ 2 \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\L u = f \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} v = 0 \\hbox { o n } \\partial _ D \\Omega ^ { ( 4 ) } \\ , , \\partial _ W v = 0 \\hbox { o n } \\partial _ N \\Omega ^ { ( 4 ) } \\cup \\Gamma ^ { ( 4 ) } ( t _ 0 ) \\ , , \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} \\begin{array} { l l } U ( Q _ { ( \\lambda , t , u ) } ) & \\cong U ( R _ { \\omega } ) ( Q _ { ( \\lambda , t , u ) } ) \\\\ & = \\bigoplus _ { ( \\tilde \\lambda , \\tilde { t } , \\tilde { u } ) } N _ { ( \\lambda , t , u ) } ( R _ { \\omega } ) ( Q _ { ( \\lambda , t , u ) } ) \\boxtimes W _ { ( \\lambda , t , u ) } , \\end{array} \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} V _ 0 ( t ) = e ^ { t A } a _ 0 , V _ { m + 1 } ( t ) = e ^ { t A } a _ 0 + \\int _ 0 ^ t e ^ { ( t - s ) A } F _ m ( s ) \\ , d s \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} & \\dot X ( s ) = f ( X ( s ) ) + h ^ 2 g ( X , s ) \\\\ & \\dot { \\bar X } ( s ) = f ( \\bar X ( s ) ) \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\delta _ { 0 } ( X ( h ) ) { \\rm e } ^ { - I \\left ( \\frac { h } { \\varepsilon } \\right ) } \\right ] = \\mathbb { E } \\left [ \\delta _ { 0 } ( X ( h ) ) { \\rm e } ^ { - G _ { 1 } ^ { \\varepsilon } } \\right ] \\cdot \\mathbb { E } \\left [ { \\rm e } ^ { - G _ { 2 } ^ { \\varepsilon } } \\right ] . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\gamma _ \\textnormal { e q } \\left ( \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } \\right ) \\right ) ( z , Q ) = J ^ { \\textnormal { c o h } , \\textnormal { e q } } ( z , Q ) \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\chi ( S ) + \\sum _ { i = 1 } ^ n k _ i < 0 , k _ i = \\textsf { \\emph { o r d } } ( p _ i ) . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} I d _ 0 = \\begin{array} { c } I d _ 0 \\leftrightarrow \\varepsilon \\\\ \\updownarrow \\\\ \\varepsilon \\leftrightarrow I d _ 0 \\end{array} = \\begin{array} { c } I d _ 0 \\\\ \\updownarrow \\\\ \\varepsilon \\end{array} \\leftrightarrow \\begin{array} { c } \\varepsilon \\\\ \\updownarrow \\\\ I d _ 0 \\end{array} = \\varepsilon \\leftrightarrow \\varepsilon = \\varepsilon \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} T = ( I - P ) T \\Leftrightarrow P T = 0 , \\mbox { a n d } T = P T \\Leftrightarrow ( I - P ) T = 0 . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d u _ t & = & \\big ( - \\frac { \\Delta } { 2 } + \\langle x , \\nabla \\rangle \\big ) u _ t d t + \\sigma ( u _ { t - } ) d X _ t \\medskip \\\\ u _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ d \\frac { ( x _ i - y _ i ) ^ 2 } { \\abs { x - y } ^ { 2 } } = \\sum _ { i = 1 } ^ d \\frac { ( x _ i - y _ i ) } { \\abs { x - y } ^ { d + 1 } } ( x _ i - y _ i ) \\abs { x - y } ^ { d - 1 } , \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} \\widehat { w ^ a } ( \\xi ) = \\int _ { - \\infty } ^ { + \\infty } \\frac { 1 } { a } w \\left ( \\frac { x } { a } \\right ) e ^ { - i \\xi \\frac { x } { a } a 2 \\pi } \\d x = \\widehat { w } ( a \\xi ) \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} c = c _ { p , p ' } = 1 - 6 \\frac { ( p - p ' ) ^ 2 } { p p ' } . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} u _ t ( x ) = \\big ( \\mathcal { G } u _ 0 \\big ) _ t ( x ) + \\xi \\int _ 0 ^ t \\int _ { D } p _ D ( t - s , x , y ) \\sigma \\big ( u _ s ( y ) \\big ) F ( d y , d s ) , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\int _ { M _ a ( \\R ^ d ) } \\ < f , \\nu ^ m \\ > { \\tilde { \\Q } } _ { \\xi , t } ( \\mu _ 0 , d \\nu ) = \\int _ { M _ 0 ( \\R ^ d ) } \\ < { \\cal I } _ { a , m } ^ { - 1 } f , T _ { I _ a } ( \\nu ) ^ m \\ > { \\Q } _ t ( T _ { I _ a } ( \\mu _ 0 ) , d [ T _ { I _ a } ( \\nu ) ] ) \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\psi \\varphi = F ( \\ ( B \\rtimes ( h \\ltimes ( b _ { i j } ) _ { i \\rho j } ) ) \\cdot ( A \\rtimes ( g \\ltimes ( a _ { i j } ) _ { i \\rho j } ) ) \\ ) \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} r _ \\alpha ^ \\lambda = ( v _ \\lambda , Q _ \\alpha u ) , \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} D F ( p ^ { \\pm } ) = \\left ( \\begin{array} { c c c c } f _ { 1 , u _ 1 } ( p ^ { \\pm } ) & f _ { 1 , u _ 2 } ( p ^ { \\pm } ) & \\cdots & f _ { 1 , u _ m } ( p ^ { \\pm } ) \\\\ f _ { 2 , u _ 1 } ( p ^ { \\pm } ) & f _ { 2 , u _ 2 } ( p ^ { \\pm } ) & \\cdots & f _ { 2 , u _ m } ( p ^ { \\pm } ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ f _ { m , u _ 1 } ( p ^ { \\pm } ) & f _ { m , u _ 2 } ( p ^ { \\pm } ) & \\cdots & f _ { m , u _ m } ( p ^ { \\pm } ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\left | \\nabla \\left ( e ^ { \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } f \\right ) \\right | _ q \\leq | \\nabla f | _ q , \\ \\forall f \\in W ^ { 1 , q } ( \\mathbb { R } ^ d ) , \\ t \\geq 0 , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) ' } ) & \\leq \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log \\left ( \\sum _ { t = a _ n } ^ { \\lfloor \\kappa _ n ( x _ 0 ) \\rfloor } P ( S _ n ( t ) \\leq t - a _ n ) \\right ) \\\\ & \\leq \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log ( ( \\lfloor \\kappa _ n ( x _ 0 ) \\rfloor - a _ n + 1 ) P ( S _ n ( t _ n ) \\leq t _ n - a _ n ) ) \\\\ & \\leq - J ( x _ 0 ) . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\hat { H } ( t , x , y ) = \\sum _ { j = 0 } ^ { \\infty } \\int _ { \\mathcal { G } } e ^ { - \\lambda _ { \\chi , j } t } s _ { \\chi , j } ( x ) \\overline { s _ { \\chi , j } ( y ) } d \\chi \\ , . \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} & \\triangle _ { p ( x , t ) } ^ N u ( x , t ) \\\\ & : = \\bigg ( \\frac { p ( x , t ) - 2 } { | D u ( x , t ) | ^ 2 } \\bigg ) \\sum _ { i , j = 1 } ^ n \\frac { \\partial ^ 2 u } { \\partial x _ i \\partial x _ j } ( x , t ) \\frac { \\partial u } { \\partial x _ i } ( x , t ) \\frac { \\partial u } { \\partial x _ j } ( x , t ) + \\sum _ { i = 1 } ^ n \\frac { \\partial ^ 2 u } { \\partial x _ i ^ 2 } ( x , t ) \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} I _ 1 = C \\int _ { \\Sigma _ { R } ^ C ( \\zeta ) } f _ \\epsilon ^ { \\frac { 2 Q } { Q + \\alpha } } ( \\xi ) d \\xi = O ( \\frac { R } \\epsilon ) ^ { - Q } , \\quad \\quad \\quad \\epsilon \\to 0 . \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} & \\overline { Q } _ { k , a } ( x ; q ) = \\sum _ { i = 1 } ^ a ( x q ) ^ i \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) \\right ] \\\\ & \\quad + \\sum _ { i = 0 } ^ { a - 1 } ( x q ) ^ i \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) \\right ] . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} K ( \\lambda , \\tilde { f } ) = \\inf \\left \\{ \\| g _ D \\| _ { L ^ 2 ( \\mathbb R ^ n ) } + \\lambda \\| g _ N \\| _ { \\dot H ^ 2 ( \\mathbb R ^ n ) } : \\tilde { f } = g _ D + g _ N \\in L ^ 2 ( \\mathbb R ^ n ) + \\dot H ^ 2 ( \\mathbb R ^ n ) \\right \\} . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} B _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { \\tau _ \\varepsilon } { \\gamma _ \\varepsilon } \\ , . \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} q \\text - \\Omega ^ \\bullet _ { \\tilde { A } / A _ { \\inf } } = \\tilde { A } \\to \\bigoplus _ { i = 1 } ^ d \\tilde { A } \\to \\ldots \\to \\tilde { A } \\to 0 \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\Xi _ { P } ^ { \\vec { p } } ( \\phi , \\gamma ) = \\mathrm { C S } ^ { \\vec { p } } ( A _ { \\phi } ^ { \\gamma } ) \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\bold { D } = \\left [ \\begin{array} { c c | c c | c c c c } 2 & 3 & 2 & 2 & 2 & 2 & 2 & 2 \\\\ \\hline 6 & 7 & 5 & 5 & 8 & 8 & 8 & 8 \\\\ \\hline 9 & 1 0 & 9 & 9 & 1 1 & 1 1 & 1 1 & 1 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p ( I , L ^ q ) } : = \\Big ( \\int _ I \\Big ( \\int _ { \\R ^ d } | u ( t , x ) | ^ q d x \\Big ) ^ { \\frac { p } { q } } \\Big ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { | H | } { | H _ 0 | } = 1 . \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} { \\mathrm { R e } ( e ^ { - i \\alpha } w ) } = & \\zeta _ 1 \\left ( | | f _ 1 | | ^ 2 - 2 | | f _ 2 | | | | f _ 3 | | \\delta _ 1 \\cos { ( \\theta _ 1 - \\frac { \\pi } { 3 } ) } \\right ) \\\\ & + \\zeta _ 2 \\left ( | | f _ 2 | | ^ 2 - 2 | | f _ 1 | | | | f _ 3 | | \\delta _ 2 \\cos { ( \\theta _ 2 - \\frac { \\pi } { 3 } ) } \\right ) \\\\ & + \\zeta _ 3 \\left ( | | f _ 3 | | ^ 2 - 2 | | f _ 1 | | | | f _ 2 | | \\delta _ 3 \\cos { ( \\theta _ 3 - \\frac { \\pi } { 3 } ) } \\right ) , \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\dot x & = n ( x ) - f ( x ( t ) , y ( t ) , v ( t ) ) v ( t ) , \\\\ \\dot y & = \\int ^ { \\infty } _ { 0 } f _ { 1 } ( \\tau ) f ( x ( t - \\tau ) , y ( t - \\tau ) , v ( t - \\tau ) ) v ( t - \\tau ) e ^ { - \\alpha _ 1 \\tau } d \\tau - a \\varphi _ { 1 } ( y ( t ) ) - p w ( y ( t ) , z ( t ) ) , \\\\ \\dot v & = k \\int ^ { \\infty } _ { 0 } f _ { 2 } ( \\tau ) e ^ { - \\alpha _ { 2 } \\tau } \\varphi _ { 1 } ( y ( t - \\tau ) ) d \\tau - u v ( t ) , \\\\ \\dot z & = c \\int ^ { \\infty } _ { 0 } f _ { 3 } ( \\tau ) w ( y ( t - \\tau ) , z ( t - \\tau ) ) d \\tau - b \\varphi _ { 2 } ( z ( t ) ) . \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{gather*} \\mathfrak { R } ( f , \\sigma , \\{ k _ j \\} , L ) : = \\{ \\{ R _ i \\} \\in [ 0 , L - 1 ] ^ n \\mid ( C _ 1 ) \\} . \\end{gather*}"}
-{"id": "4545.png", "formula": "\\begin{align*} u ( n + 1 ) + u ( n - 1 ) + V ( n ) u ( n ) = E u ( n ) \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} u _ 0 ^ { \\lambda } ( x ) = \\lambda u _ 0 ( \\lambda x ) . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} \\varphi ( \\eta ) : = \\frac { 1 } { ( 1 + \\eta ^ 2 ) ^ { 3 / 2 + \\kappa } } , \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\ell : = \\left \\lfloor \\frac { n + d + 1 } { 2 } \\right \\rfloor = \\frac { n + d + 1 - \\delta } { 2 } \\delta _ { \\ell } : = \\begin{cases} 0 & \\\\ 1 & . \\end{cases} \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} f = \\sum _ { k = 1 } ^ \\infty g _ k \\ast h _ k \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\widetilde { J _ i } ( q , Q ) = \\sum _ { \\substack { a + b = i \\\\ 0 \\leq a , b \\leq N } } ( - 1 ) ^ a \\binom { \\ell _ q ( Q ) } { a } J _ b ( q , Q ) \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c c c } 0 . 7 & 0 . 2 & 0 . 1 & 0 . 5 & 1 . 0 \\\\ 0 . 3 & 0 . 6 & 0 . 2 & 0 . 8 & 0 . 3 \\\\ 0 . 5 & 0 . 7 & 0 . 9 & 1 . 0 & 0 . 5 \\\\ 0 . 1 & 0 . 1 & 0 . 3 & 0 . 8 & 0 . 3 \\\\ 0 . 8 & 0 . 2 & 0 . 9 & 0 . 3 & 0 . 2 \\end{array} \\right ) , \\mbox { w i t h } \\rho ( A ) = 2 . 4 0 3 1 . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\lambda _ u ( A _ i A _ j ) = - \\lambda _ u ( A _ i ) \\lambda _ u ( A _ j ) \\quad \\quad \\lambda _ u ( A _ i A _ j ^ 2 ) = - \\lambda _ u ( A _ i ) \\lambda _ u ( A _ j ) ^ 2 . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} g : = f \\prod _ { i = 1 } ^ m f _ i \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\langle X , Y \\rangle = I m ( t r a c e ( X Y ) ) \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\mu _ { x , y , w } = \\frac { \\delta _ x - \\delta _ y } { d ( x , y ) } w \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} A _ t = \\bigoplus _ { \\lambda \\in \\mathcal { F } _ t } A _ \\lambda ( a ) \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} & E [ ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } ( U _ 1 + U _ 2 + \\cdots + U _ { k - 1 } + 1 ) } ] = ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } E [ ( 1 + \\lambda t ) ^ { \\frac { U _ 1 + \\cdots + U _ { k - 1 } } { \\lambda } } ] \\\\ & = ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } \\Big ( \\frac { 1 } { \\lambda } \\log ( 1 + \\lambda t ) \\Big ) ^ { - ( k - 1 ) } \\Big ( ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 \\Big ) ^ { k - 1 } . \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\sum _ { v \\in S } \\sum _ { i = 1 } ^ d c _ { v , i } \\mathbf { a } _ { v , i } + \\sum _ { \\mathbf { x } \\in \\cal { B } } d _ \\mathbf { x } \\mathbf { x } = 0 , \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} \\alpha ^ N _ X : C ^ N X & \\rightarrow \\Omega ^ N \\Sigma ^ N X \\\\ \\left [ d _ { \\underline { n } } , x ^ { \\underline { n } } \\right ] & \\mapsto \\left ( t \\mapsto \\begin{cases} [ x ^ i , d _ i ^ { - 1 } ( t ) ] , & t \\in d _ i ( I ^ N ) \\\\ \\ast , & t \\not \\in d _ { \\underline n } \\left ( \\coprod _ { \\underline n } I ^ N \\right ) \\end{cases} \\right ) \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} 1 - t ^ 2 = \\frac 1 { 1 + x ^ 2 } , \\frac { d x } { d t } = \\frac { 1 } { ( 1 - t ^ 2 ) ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\phi ( s ) & = K \\Gamma ( u + 1 ) s ^ { - u } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 + 1 / 2 } ) + O ( s ^ { - u + 1 / 2 } ) \\\\ & = K \\Gamma ( u + 1 ) s ^ { - u } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} \\frac { N } { ( \\log N ) ^ { \\beta } } \\# \\Big \\{ q \\leq N / ( \\log N ) ^ { \\beta } : ~ \\| m _ { j } q \\alpha \\| \\leq \\frac { 1 } { N } \\Big \\} = o ( N ) \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} F ( \\varphi ) = \\int _ { \\mathbb { R } _ + \\times \\mathbb { R } ^ { d } } \\varphi ( t , x ) F ( d t , d x ) , \\ \\ \\varphi \\in \\mathcal { H } . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} U ^ { ( P ) } _ l = U ^ { ( P ) } _ l ( n ) : = \\left \\{ \\frac { \\eta _ 1 n } { 2 N } \\leq N ^ { ( P ) } _ l \\leq \\frac { 2 \\eta _ 2 n } { N } \\right \\} , \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\| u _ \\varepsilon \\| ^ 2 _ { H ^ 1 _ 0 } = \\alpha _ \\varepsilon \\ , , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\| [ T , \\textbf { b } ] _ \\alpha ( f _ 1 , f _ 2 , \\dots , f _ m ) \\| _ { L ^ q ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\| b _ j \\| ^ { \\alpha _ i } _ { { \\rm B M O } } \\| f _ i \\| _ { L ^ { q _ i } \\left ( v _ i \\right ) } , \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} R _ { i j k l } = \\left \\langle R ( e _ j , e _ k ) e _ l , e _ i \\right \\rangle , \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} w _ t \\geq P [ \\psi _ { \\tau } ] ( \\phi ) + t \\tau = \\psi _ { \\tau } + t \\tau , \\ \\forall \\tau \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} 0 = & \\ X ( e ^ { - f } \\psi ) = e ^ { - f } \\left ( X \\psi - A \\xi \\psi \\right ) , \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} C _ { n } ^ { ( 1 + \\alpha ) } ( 1 ) = \\frac { \\Gamma ( 2 + 2 \\alpha + n ) } { \\Gamma ( 2 + 2 \\alpha ) \\Gamma ( n + 1 ) } \\ , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} Q _ 2 : = \\sum _ { j = 1 } ^ N \\chi _ j ( P _ j ) v _ j v _ j ^ T . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} ( D y _ 0 \\cdot n ) _ { t g } + ( f ( y _ 0 ) ) _ { t g } = 0 \\mbox { o n } \\partial \\Omega \\quad \\mbox { a n d } | ( y _ 0 , \\theta _ 0 ) - ( 0 , \\overline { \\theta } _ 0 ) \\| _ { [ H ^ 3 ( \\Omega ) ^ N \\cap W ] \\times H ^ 1 ( \\Omega ) } \\leq \\delta , \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} k ( k + s ^ 2 - 4 s + 2 ) = 0 , \\ - s ^ 3 + 4 s ^ 2 - 5 s + 2 = 0 \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\mathrm { P } ^ 1 { } _ \\# \\Xi = m _ * \\star \\varkappa , \\ \\ \\mathrm { P } ^ 2 { } _ \\# \\Xi = \\nu _ * \\star \\vartheta . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} | \\sin \\pi ( 2 \\theta + x _ 0 ^ \\prime \\alpha ) | = e ^ { - \\eta ^ \\prime | \\ell | } . \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\varepsilon } \\Big ( \\mathcal { G } _ { h , N _ { 1 } } ~ \\big | ~ { \\bf { L ^ 1 } } ( [ 0 , L ] , E ) \\Big ) ~ \\geq ~ { \\mathrm { C a r d } \\left ( \\mathcal { G } _ { h , N _ { 1 } } \\right ) \\over \\mathrm { C a r d } \\left ( \\mathcal { I } _ { \\tilde { \\delta } } ( 2 \\varepsilon ) \\right ) } ~ \\geq ~ { 2 ^ { N _ 1 ( { \\bf \\tilde { p } } + 2 ) } \\over 2 ^ { N _ 1 ( 2 + { \\bf \\tilde { p } } / 2 ) } } ~ = ~ 2 ^ { N _ 1 { \\bf \\tilde { p } } / 2 } ~ . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} w ^ \\beta _ \\epsilon w ^ \\beta _ \\epsilon & = ( \\theta ^ \\beta _ \\epsilon ) ^ 2 c _ \\beta t _ \\beta + c _ { \\alpha + \\beta } t _ { \\alpha + \\beta } + 2 \\theta ^ \\beta _ \\epsilon b _ { \\beta , \\alpha + \\gamma } e ^ \\alpha \\\\ & = \\left ( ( \\theta ^ \\beta _ \\epsilon ) ^ 2 c _ \\beta + c _ { \\alpha + \\beta } \\right ) t _ \\beta + c _ { \\alpha + \\beta } t _ \\alpha + 2 \\theta ^ \\beta _ \\epsilon b _ { \\beta , \\alpha + \\gamma } e ^ \\alpha \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\nabla ^ \\frac { H } { 2 } \\left ( \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\nabla ^ \\frac { H } { 2 } \\psi _ u ( u ) \\right ) ( u ) = \\frac { \\partial ^ 2 f } { \\partial x } ( s , x _ s ) \\nabla ^ \\frac { H } { 2 } x _ s ( u ) \\nabla ^ \\frac { H } { 2 } \\psi _ u ( u ) + \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\nabla ^ { \\frac { H } { 2 } , \\frac { H } { 2 } } \\psi _ u ( u , u ) \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} \\Delta g ( X , t ) & = n ^ n \\prod _ { \\substack { c \\in \\overline { \\C ( t ) } , \\\\ \\frac { \\partial g } { \\partial t } ( c , t ) = 0 } } g ( c , t ) ^ { m _ c } \\\\ & = n ^ n g ( 0 , t ) ^ { n - 2 } g ( \\frac { n - 1 } { n } t , t ) \\\\ & = n ^ n ( - t ) ^ { n - 2 } \\left ( \\left ( - \\frac { 1 } { n } t \\right ) \\left ( \\frac { n - 1 } { n } t \\right ) ^ { n - 1 } - t \\right ) \\\\ & = n ^ n ( - t ) ^ { n - 1 } \\left ( \\left ( \\frac { 1 } { n } \\right ) \\left ( \\frac { n - 1 } { n } t \\right ) ^ { n - 1 } + 1 \\right ) \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} g ( t ) = \\prod _ { \\substack { j = 1 } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 j } ) + ( t - \\lambda _ { n + 2 } ) \\sum _ { j = 1 } ^ { \\frac { n } { 2 } } \\frac { \\sin ^ { 2 } \\left ( \\frac { 2 j \\pi } { n + 3 } \\right ) } { \\sin ^ { 2 } \\left ( \\frac { \\pi } { n + 3 } \\right ) } \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 m } ) . \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\xi ^ { \\ast } = 1 - \\frac { \\gamma ( n , \\frac { n \\tau ^ { \\ast } } { \\sigma _ w ^ 2 } ) } { \\Gamma ( n ) } + \\frac { \\gamma \\left ( n , \\frac { n \\tau ^ { \\ast } } { \\rho + \\sigma _ w ^ 2 } \\right ) } { \\Gamma ( n ) } , \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = 0 , \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} A & = \\{ j \\in [ n ] \\ ; : \\ ; b _ j = b _ { n + j } = * \\} , \\\\ B & = \\{ j \\in [ n ] \\ ; : \\ ; b _ j = 0 , \\ , b _ { n + j } = * \\} , \\\\ C & = \\{ j \\in [ n ] \\ ; : \\ ; b _ j = * , \\ , b _ { n + j } = 0 \\} , \\\\ D & = \\{ j \\in [ n ] \\ ; : \\ ; b _ j = b _ { n + j } = 0 \\} . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} o s c \\Big ( y _ 0 , \\frac { R } { 4 } , v \\Big ) = & \\sup _ { B ( y _ 0 , \\frac { R } { 4 } ) } ( \\psi + z ) - \\inf _ { B ( y _ 0 , \\frac { R } { 4 } ) } ( \\psi + z ) \\\\ \\leq & \\sup _ { B ( y _ 0 , \\frac { R } { 4 } ) } \\psi + \\sup _ { B ( y _ 0 , \\frac { R } { 4 } ) } z - \\inf _ { B ( y _ 0 , \\frac { R } { 4 } ) } \\psi - \\inf _ { B ( y _ 0 , \\frac { R } { 4 } ) } z . \\\\ \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\| T ( f _ 1 , f _ 2 , \\dots , f _ m ) \\| _ { L ^ p ( w ) } \\lesssim \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { p _ i } \\left ( w _ i \\right ) } , \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\int _ { B _ 0 } \\vert u \\vert ^ { p ^ * } \\ , d \\mu \\leq \\sum _ { k = - \\infty } ^ { \\infty } a _ k ^ { p ^ * } \\mu ( B _ 0 \\cap ( E _ k \\setminus E _ { k - 1 } ) ) . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { g } U _ b = & - \\sigma _ y \\bar { y } Y + \\sigma _ x \\bar { y } X - \\hat d _ b D _ b \\\\ s \\bar { y } X = & \\sigma _ y \\bar { y } Y - \\sigma _ x \\bar { y } X - a _ { x x } \\bar { y } X + \\hat d _ x D _ x + \\hat d _ b D _ b \\\\ \\end{aligned} \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { P r ( f , \\sigma ) = 0 , v o l ( \\mathfrak { D } ( f , \\sigma ) ) > 0 } { { v o l } ( D _ { i , a } \\cap { \\mathfrak { D } } ( f , \\sigma ) ) } . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} p ( v , \\theta ) = p _ e ( v ) + \\theta p _ \\theta ( v ) , \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\overline { H } _ \\epsilon ( P ) = \\ln I _ \\epsilon [ u _ \\epsilon ] \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\partial _ t B = - \\nabla \\wedge E , \\quad \\nabla \\cdot B = 0 . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\delta V ( X ) : = \\int _ { A _ m ( M ) } \\langle S , \\nabla X \\rangle d V ( S ) . \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\beta _ { \\ell } & = t _ 1 + \\frac { ( t _ 1 - t _ 2 ) \\frac { \\beta _ 1 - t _ 1 } { \\beta _ 1 - t _ 2 } p ^ { \\ell - 1 } } { 1 - \\frac { \\beta _ 1 - t _ 1 } { \\beta _ 1 - t _ 2 } p ^ { \\ell - 1 } } , \\\\ & = t _ 2 + \\frac { t _ 2 - t _ 1 } { \\frac { \\beta _ 1 - t _ 1 } { \\beta _ 1 - t _ 2 } p ^ { \\ell - 1 } - 1 } . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\Lambda _ { a _ 1 } ^ { \\Omega , \\Gamma } ( f ) = \\Lambda _ { a _ 2 } ^ { \\Omega , \\Gamma } ( f ) \\Gamma , \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} { } \\left \\langle X + \\left ( \\dfrac { \\sqrt { \\left \\langle Y _ { \\mathfrak { m } } , Y _ { \\mathfrak { m } } \\right \\rangle } - \\left \\langle X , Y _ { \\mathfrak { m } } \\right \\rangle } { \\left \\langle Y _ { \\mathfrak { m } } , Y _ { \\mathfrak { m } } \\right \\rangle } \\right ) Y _ { \\mathfrak { m } } , \\left [ Y , Z \\right ] _ { \\mathfrak { m } } \\right \\rangle = 0 . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} B ( k ; n , p ) \\geq \\frac { 1 } { k ! } ( n - k ) ^ { k } p ^ { k } e ^ { - p ( n - k ) - p ^ 2 ( n - k ) } = P o i ( k ; n p ) \\left ( 1 - \\frac { k } { n } \\right ) ^ { k } e ^ { k p - ( n - k ) p ^ 2 } \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} j = j ( d , n , \\mu ) = \\max \\{ i \\colon H _ i \\leq n - 1 \\} , \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\displaystyle \\mathcal K ( \\rho ) = - ( 1 + \\rho _ x ^ 2 ) ^ { - { 3 \\over 2 } } \\rho _ { x x } . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} & \\inf _ { \\omega \\in O ( \\frac { n ( n - 1 ) } { 2 } ) } \\sup _ { B _ { 1 / C } ( x _ 2 ) } \\max _ \\alpha \\bigg | K _ \\alpha ^ { ( x _ 1 , t _ 1 ) } - \\sum _ { \\beta = 1 } ^ { \\frac { n ( n - 1 ) } { 2 } } \\omega _ { \\alpha \\beta } K _ \\beta ^ { ( x _ 2 , t _ 2 ) } \\bigg | \\\\ & \\leq C \\ , 2 ^ { - j + \\frac { j } { 4 0 0 } } , \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} k ^ 2 & = \\frac { H _ o ^ 2 r _ o ^ 2 } { 4 } \\ ( 1 + b r _ o ^ 2 - \\frac { 2 m } { r _ o } \\ ) ^ { - 1 } , \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\pi } \\dfrac { a } { c ^ { 2 s } } \\ , d \\theta = \\sum _ { k = 0 } ^ s { 2 s + 1 \\choose k } ^ 2 \\rho ^ { 4 n k } I _ 0 + 2 \\sum _ { l = 1 } ^ { s } \\rho ^ { 2 n l } \\sum _ { i = 0 } ^ { s - l } { 2 s + 1 \\choose i } { 2 s + 1 \\choose i + l } \\rho ^ { 4 n i } I _ l , \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\psi _ 1 ( C ) : = \\min _ { m \\in \\mathbb { Z } , n \\in \\mathbb { Z } } \\bigg \\{ \\frac { m ^ 2 + n ^ 2 } { 2 m n } : \\ m ^ 2 + n ^ 2 \\le C , \\ m - 1 \\ge n \\ge 1 \\bigg \\} . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { c _ i z _ i } { z _ i + 1 } < \\sum _ { i = 1 } ^ n c _ i z _ i = 0 . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } \\buildrel \\Delta \\over = \\arg \\mathop { \\max } \\limits _ \\theta { f _ 0 } \\left ( \\theta \\right ) , \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\bar { \\check { y } } _ { \\mathrm { d } , i , k } [ n ] & = \\bar { h } _ { i , k k } [ 0 ] x _ { \\mathrm { d } , k } [ n ] + \\sum _ { \\nu \\in \\Delta \\setminus \\{ 0 \\} } \\bar { h } _ { i , k k } [ \\nu ] x _ { \\mathrm { d } , k } [ n - \\nu ] \\\\ & \\quad + \\sum _ { k ' \\in \\mathcal { K } \\setminus \\{ k \\} } \\sum _ { \\nu \\in \\Delta } \\bar { h } _ { i , k k ' } [ \\nu ] x _ { \\mathrm { d } , k ' } [ n - \\nu ] \\\\ & \\quad + \\bar { z } _ { \\mathrm { d } , i , k } [ n ] + \\bar { e } _ { \\mathrm { d } , i , k } [ n ] , \\ , i \\in \\mathcal { I } , k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} d ( e ^ { - \\eta t } \\Theta ( X ( t ) ) ) = - \\eta e ^ { - \\eta t } \\Theta ( X ( t ) ) \\ , d t + e ^ { - \\eta t } \\mathcal { L } \\Theta ( X ( t ) ) \\ , d t + d M _ \\eta ( t ) \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\begin{array} { l } \\lim _ { t \\to 0 } \\mu _ f ( x , t ) = f ( x ) \\ , , \\ \\ \\lim _ { t \\to 0 } \\nabla _ x \\mu _ f ( x , t ) = \\nabla f ( x ) \\ , , \\ \\ \\lim _ { t \\to 0 } \\ , \\nabla _ { x x } \\mu _ f ( x , t ) = \\nabla ^ 2 f ( x ) \\ , , \\\\ \\\\ \\lim _ { t \\to 0 } \\ , \\nabla _ { t x } \\mu _ f ( x , t ) = 0 \\ , . \\end{array} \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} \\Phi _ n ^ { ( k ) } ( z ) : = \\ _ 3 \\phi _ 2 \\left ( \\begin{matrix} q ^ { - n } , q ^ { \\gamma + k } , a b q ^ { n + 1 } \\\\ a q , q ^ \\gamma \\end{matrix} ; q , q z \\right ) , \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} X _ j ^ s = L ^ { p _ j ^ { s } } ( M _ s , \\mu _ s ; X _ j ^ { s - 1 } ) , s = 1 , \\ldots , S , \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\bar \\eta _ { K , M } ^ { ( 3 ) } \\buildrel \\Delta \\over = \\sum \\limits _ { j \\in { \\cal A } } { \\mathop { \\max } \\limits _ { 1 \\le { k _ j } \\le K } \\sum \\limits _ { i = { k _ j } } ^ K { \\hat \\phi _ { j i } ^ { \\left ( 4 \\right ) } } } = \\sum \\limits _ { j \\in { \\cal A } } { \\hat \\psi _ j ^ { ( K ) } } = \\sum \\limits _ { j \\in { \\cal A } } { \\sum \\limits _ { l = 1 } ^ { M + K } { \\xi _ j ^ { ( l ) } } } \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi + R \\varphi + \\frac { n - 1 } { n } \\tau ^ { 2 } \\varphi ^ { N - 1 } = \\frac { w ^ { 2 } } { \\varphi ^ { N + 1 } } , \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} 8 \\pi ^ 2 q \\frac { d \\zeta } { d q } = 2 \\zeta \\wp + \\wp ' , \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} X _ 1 = \\partial _ { x _ 3 } , ~ ~ X _ 2 = \\frac { 1 } { \\sqrt { 2 } } ( - e ^ { x _ 3 } \\partial _ { x _ 1 } + e ^ { - x _ 3 } \\partial _ { x _ 2 } ) , ~ ~ X _ 3 = - \\frac { 1 } { \\sqrt { 2 } } ( e ^ { x _ 3 } \\partial _ { x _ 1 } + e ^ { - x _ 3 } \\partial _ { x _ 2 } ) . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} [ e , \\alpha , f \\rangle \\ : [ f , \\beta , g \\rangle = [ e , \\alpha { \\downharpoonright } h \\cdot h { \\downharpoonleft } \\beta , g \\rangle . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} y _ k = \\frac { P } { N _ { \\rm M S } } \\textbf { h } _ k ^ { \\rm H } \\textbf { p } _ k x _ k + \\frac { P } { N _ { \\rm M S } } \\sum \\limits _ { k ' \\neq k } \\textit { \\textbf { h } } _ k ^ { \\rm H } \\textbf { F } \\textbf { p } _ { k ' } x _ { k ' } + n _ k , \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} f ( z , t ) = \\frac { 1 } { 1 - z t } + \\frac { f ^ { \\not \\ominus } ( z , t ) } { ( 1 - z t ) ( 1 - z ) } . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} G = \\log \\frac { \\xi '' ( 1 ) } { \\xi ' ( 1 ) } - \\frac { [ \\xi '' ( 1 ) - \\xi ' ( 1 ) ] [ \\xi '' ( 1 ) - \\xi ' ( 1 ) + \\xi ' ( 1 ) ^ 2 ] } { \\xi '' ( 1 ) \\xi ' ( 1 ) ^ 2 } \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} & \\begin{aligned} \\mu _ { \\eta ^ , i } & = E _ i \\{ \\eta ^ \\} \\\\ & = \\int _ { \\Omega } p _ { \\eta ^ , i } ( x ) x \\ ; d x \\\\ & = \\int _ { C _ 0 } ^ { C _ 1 } \\left ( A _ { 0 , i } \\delta ( x - C _ 0 ) + A _ { 1 , i } \\delta ( x - C _ 1 ) + A _ { 2 , i } \\right ) x \\ ; d x \\\\ & = A _ { 0 , i } C _ 0 + A _ { 1 , i } C _ 1 + \\frac { 1 } { 2 } \\left [ A _ { 2 , i } x ^ 2 \\right ] _ { C _ 0 } ^ { C _ 1 } \\\\ & = A _ { 0 , i } C _ 0 + A _ { 1 , i } C _ 1 + A _ { 2 , i } \\frac { C _ 1 ^ 2 - C _ 0 ^ 2 } { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} u ( x , t ) = \\phi ( n \\cdot x - \\theta _ 0 ) , \\ \\ v ( x , t ) = \\psi ( n \\cdot x - \\theta _ 0 ) . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} { \\rm E n d s } ( S ) = { \\rm E n d s } _ { \\infty } ( S ) = \\mathcal { B } \\sqcup \\mathcal { U } , \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} \\widehat { I } [ u _ 0 ] = \\overline { I } [ u _ 0 , u _ 1 ] = \\lim \\limits _ { \\epsilon \\rightarrow 0 } I _ \\epsilon [ u _ \\epsilon ] \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} c = c _ { p , p ' } = 1 - 6 \\frac { ( p - p ' ) ^ 2 } { p p ' } . \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq \\varepsilon \\right ) = - H ( \\ell _ 2 \\varepsilon ) . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} [ X ] ^ { < \\omega } & = , \\\\ [ f ] ^ { < \\omega } ( a ) & = \\{ f ( x ) \\ , | \\ , x \\in a \\} , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} & \\frac { | a | ^ 2 | \\beta _ 0 | ^ 2 } { 1 + | a | ^ 2 } + \\frac { \\sum _ { j = 0 } ^ { \\infty } \\left | \\overline { a } \\beta _ j + a \\beta _ { j + 1 } \\right | ^ 2 } { ( 1 + | a | ^ 2 ) ^ 2 } \\\\ < & \\frac { | a | ^ 2 | \\beta _ 0 | ^ 2 } { 1 + | a | ^ 2 } + \\frac { 2 \\sum _ { j = 1 } ^ { \\infty } | a | ^ 2 | \\beta _ j | ^ 2 } { ( 1 + | a | ^ 2 ) ^ 2 } + \\frac { 2 \\sum _ { j = 0 } ^ { \\infty } | a | ^ 2 | \\beta _ j | ^ 2 } { ( 1 + | a | ^ 2 ) ^ 2 } \\\\ \\leqslant & \\frac { 4 | a | ^ 2 } { ( 1 + | a | ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} T ^ - ( \\rho ) f ( x ) = ( 4 \\pi \\rho ) ^ { - 1 } V ( x ) \\int _ { | x - y | = \\rho } f ( y ) \\ , d y \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} | \\det B | = 2 ^ { - n } | \\det A | \\le 2 ^ { - n } ( n + 1 ) ^ { ( n + 1 ) / 2 } , \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} F _ m ^ - \\big ( ( x , t ) , \\xi , \\nu , M \\big ) & = \\inf _ { ( a , c ) \\in \\mathcal H _ m } \\sup _ { ( b , d ) \\in \\mathcal H _ m } \\Phi \\big ( a , b , c , d ; ( x , t ) , \\nu , M \\big ) + r \\xi , \\\\ F _ m ^ + \\big ( ( x , t ) , \\xi , \\nu , M \\big ) & = \\sup _ { ( b , d ) \\in \\mathcal H _ m } \\inf _ { ( a , c ) \\in \\mathcal H _ m } \\Phi \\big ( a , b , c , d ; ( x , t ) , \\nu , M \\big ) + r \\xi \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} x _ 1 ^ 2 + ( a _ 2 x _ 2 ^ 2 + \\ldots + a _ n x _ n ^ 2 ) & = x _ 1 , \\\\ 2 a _ k x _ k x _ 1 & = x _ k , \\quad k = 2 , \\ldots , n . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} h _ { a l g } ( \\widetilde { \\phi } ) = h _ { a l g } ( \\widetilde { \\phi } \\upharpoonright _ { H G ' / G ' } ) + h _ { a l g } ( \\overline { \\widetilde { \\phi } } ) , \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} j = \\frac { P } { \\int _ { 0 } ^ { 1 } \\int _ 0 ^ 1 \\frac { 1 } { m ( x , y ) } d y d x } , \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} w ( x ) = \\begin{cases} 0 . 5 & ( x < - 1 ) , \\\\ 0 . 2 5 & ( - 1 < x < 0 ) , \\\\ 1 & ( 0 < x < 1 ) , \\\\ 0 . 5 & ( x > 1 ) , \\end{cases} \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\phi _ { m , k } ( t ) \\ = \\ \\max ( 0 , \\min ( 1 , m \\cdot t + k ) ) . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} c _ k = \\frac { 1 } { k ^ { 1 + \\alpha \\beta } } , \\ \\lambda _ k = \\frac { 1 } { k ^ \\beta } . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} & = \\# ( e ) - \\# ( f ) , \\\\ & = \\# ( e ) + \\# ( h ) + \\# ( f ) , \\\\ & = \\sum _ { m \\in \\Z _ { \\geq 0 } } m ( \\# ( e _ m ) + \\# ( h _ m ) + \\# ( f _ m ) ) . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\mathcal { S } _ i ( p , d _ i ) = p d _ i . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} d i v ( | j u | ^ { p - 2 } j u ) = 0 \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} \\gamma & = ( \\{ i _ { a + 1 } , \\overline { i _ { a + 1 } } \\} , \\ldots , \\{ i _ { a + b } , \\overline { i _ { a + b } } \\} ) \\\\ \\delta & = ( [ i _ { a + b + 1 } , \\overline { i _ { a + b + 1 } } ] , \\ldots , [ i _ n , \\overline { i _ n } ] ) , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} f _ { T } : = f | _ { T } & \\colon T \\longrightarrow T \\\\ g & \\colon A \\longrightarrow A \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} f ( \\omega ) = - \\frac { 1 } { m } 1 _ { \\mathcal { W } } ( \\omega | _ { m } ) \\log \\mu [ \\omega | _ { m } ] , \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} \\hat \\nu _ \\delta = \\left \\lceil \\log _ 2 \\left ( - 6 + 2 \\sqrt { 1 + 2 / \\delta } \\right ) \\right \\rceil , \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = \\frac { 1 } { 2 } \\left ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 \\right ) + \\sum _ { d > 0 } \\sum _ { n \\geq 0 } \\sum _ { \\alpha _ 1 + 3 d - 1 = n } N _ d \\frac { ( d \\ , t _ 1 ) ^ { \\alpha _ 1 } } { { \\alpha _ 1 } ! } \\frac { t _ 2 ^ { 3 d - 1 } } { ( 3 d - 1 ) ! } Q ^ d \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} z ^ { - \\mu } z ^ \\rho ( T _ i ) = \\exp ( - \\mu \\log ( z ) ) \\exp ( \\rho \\log ( z ) ) \\cdot T _ i \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} ( 2 k - 1 ) ! ! = \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + k } \\binom { 2 k } { k + j } S ( j + k , j ) , \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\sum _ { i = 1 } ^ { m } \\frac { 1 } { r _ { i } } . \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} | a ( u , u ) | = | z | ^ \\beta | a ( z ^ { - \\beta } u , u ) | \\geq \\left \\{ \\begin{array} { c c } \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 & \\Re z ^ { - \\beta } > 0 , \\\\ | z | ^ \\beta | \\Im z ^ { - \\beta } | \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 & . \\end{array} \\right . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} ( a - 1 ) & \\equiv \\left ( \\sum \\limits _ { k = 1 } ^ { a - 3 } m _ { k } k + ( a - 3 ) m _ { a - 3 } + m _ { a - 2 } ( a - 2 ) \\right ) \\ , ( \\mathrm { m o d } \\ , a ) \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ t ( x ) = & - 2 F _ t ( x ) + \\lambda \\sum _ { y : y \\sim O } \\Big ( ( b - 1 ) F _ t ( 0 ) + a E \\big [ \\xi _ t ( y ) \\xi _ t ( x ) \\big ] \\Big ) \\\\ & + \\lambda \\sum _ { y : y \\sim x } \\Big ( ( b - 1 ) F _ t ( 0 ) + a F _ t ( y ) \\Big ) + 2 \\Big ( 1 - 2 d \\lambda ( a + b - 1 ) \\Big ) F _ t ( x ) \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\mathcal { W } _ j ( x ) : = h _ j ' ( u _ j ) z _ j f ( x , z _ j ) + 2 \\theta a _ { 0 , j } ( x ) \\ , a _ { 1 , j } ( x ) \\mbox { f o r e v e r y } x \\in \\omega . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} I _ { N , M } \\risingdotseq \\cup _ { K = 1 } ^ { M - 1 } S _ { \\tau ( K ) } \\quad S _ { \\tau ( K ) } : = \\left ( \\frac { K - \\tau ( K ) } { M } , \\frac { K } { M } \\right ) . \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} t > t _ 0 : = \\sqrt { \\dfrac { | \\lambda _ n ( B ) | } { 2 \\ , a _ k } } \\ , . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\sup _ { ( b , d ) \\in \\mathcal H _ m } & \\Phi \\big ( a ^ 1 , b , c _ 1 , d , D u _ m ( x , t ) , D ^ 2 u _ m ( x , t ) \\big ) + r u _ m ( x , t ) \\\\ & \\leq \\partial _ t u _ m ( x , t ) + \\frac { 1 } { k } , \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla _ { \\partial _ { t _ i } } T _ j = \\left ( \\partial _ { t _ i } + \\frac { 1 } { z } T _ i \\bullet _ \\tau \\right ) T _ j , & & 0 \\leq i \\leq N \\\\ & \\nabla _ { \\partial _ z } T _ j = \\left ( \\partial _ z - \\frac { 1 } { z ^ 2 } \\mathfrak { E } \\bullet _ \\tau + \\frac { 1 } { z } \\frac { \\textnormal { d e g } _ { H ^ * ( X ) } } { 2 } \\right ) T _ j \\end{aligned} \\right . \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\eta _ n = \\sqrt { \\frac { \\log \\log n } { n } } . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\mathcal { G } _ { h , N _ { 1 } } ~ = ~ \\left \\{ g _ { \\delta } = \\sum _ { i = 0 } ^ { N _ 1 - 1 } \\delta _ i \\cdot \\chi _ { I _ i } ~ \\Big | ~ \\delta \\in \\Delta _ { h , N _ { 1 } } \\right \\} . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\big ( a ^ 2 ( l ) , c _ 2 ( l ) \\big ) = \\begin{cases} ( a ^ 1 , c _ 1 ) , & ~ l \\in E _ 1 , \\\\ z ^ 2 \\big ( X ( t _ 1 ) \\big ) , & ~ l \\in E _ 2 . \\end{cases} \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\tau ( t ; g ) = \\langle 0 | \\mathrm e ^ { H ( t ) } g | 0 \\rangle , g \\in \\mathrm { G L } ( \\infty ) \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\chi _ { \\varepsilon , R } = 4 \\pi \\Lambda _ { \\varepsilon , R } G _ { x _ \\varepsilon } \\ , , \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 q ^ { - { k + 1 \\choose 2 } } } { ( q ; q ) _ k ^ 2 ( q ^ 2 ; q ^ 2 ) _ k } & \\equiv [ n ] q ^ { \\frac { 1 - n } { 2 } } + \\frac { ( n ^ 2 - 1 ) ( 1 - q ) ^ 2 } { 2 4 } [ n ] ^ 3 q ^ { \\frac { 1 - n } { 2 } } \\pmod { [ n ] \\Phi _ n ( q ) ^ 3 } . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\mathcal { A C } _ m & : = \\{ A \\in \\mathcal { A C } : \\Lambda ( A ) \\leq m \\} , \\\\ \\mathcal { S } _ m & : = \\{ S \\in \\mathcal { S } : \\Lambda ( S ) \\leq m \\} , \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} i _ A ( M _ 1 , M _ 2 ; A ) = 0 \\ ; \\dim M _ 1 + \\dim M _ 2 < \\dim A \\ ; . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} c _ k ^ - = ( A ^ - ) ^ { n ^ k - 1 } n _ { 2 } ^ { n ^ k } ( - 1 ) ^ { ( n - a ) n ^ { k - 1 } } . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} X _ { i } = f ^ { e _ { 0 } } F _ { 1 } ( f ) ^ { e _ { 1 } } \\cdots F _ { i - 1 } ( f ) ^ { e _ { i - 1 } } F _ { i + 1 } ( f ) ^ { e _ { i + 1 } } \\cdots F _ { r } ( f ) ^ { e _ { r } } ( X ) . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} f ( r ) = \\cosh \\left ( r \\sqrt { \\frac { \\lambda } { n - 2 } } \\right ) . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\norm { w ^ { ( k ) } + \\triangle \\hat { w } - w ^ * } = \\tau _ k \\norm { w ^ { ( k ) } - w ^ * } , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} G _ { \\boldsymbol { Y } } ( \\boldsymbol { s } ) = \\operatorname { E } \\left \\lbrace ( \\boldsymbol { \\pi } ^ t \\boldsymbol { s } ) ^ { \\vert \\boldsymbol { Y } \\vert } \\right \\rbrace = G _ { \\vert \\boldsymbol { Y } \\vert } \\left ( \\boldsymbol { \\pi } ^ t \\boldsymbol { s } \\right ) , \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} d ( x , x + F _ { l } ( u , x ) ) ^ { 2 } \\leq C _ { H , l } \\lim _ { m \\to \\infty } \\left ( \\sum _ { k = 1 } ^ { m } | I _ { k } | \\right ) ^ { 2 H } \\left ( \\sum _ { k = 1 } ^ { m } k ^ { 2 H + 1 } | I _ { k } | ^ { 2 ( 1 - H ) } \\right ) . \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\beta _ { 1 } & = \\{ k ( m + d ) \\mid 0 \\leq k \\leq q + 1 \\} , \\\\ \\beta _ { 2 } & = \\{ k ( m + d ) + ( q + 1 ) m + ( q + e - 1 ) d \\mid 0 \\leq k \\leq q + 1 \\} , \\\\ \\beta _ { 3 } & = \\{ ( q + 1 ) m + ( q + i ) d \\mid 2 \\leq i \\leq e - 2 \\} . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} A ( 2 ) = \\frac { 3 \\sqrt 3 } { 4 } < \\frac 4 3 = h ( 2 ) . \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} | \\mathbf { p } S _ i | = P _ i [ 3 ] \\oplus \\bigoplus _ { \\rho \\in Q _ 1 : t ( \\rho ) = i } P ^ \\rho _ { s ( \\rho ) } [ 2 ] \\oplus \\bigoplus _ { \\tau \\in Q _ 1 : s ( \\tau ) = i } P ^ \\tau _ { t ( \\tau ) } [ 1 ] \\oplus P _ i \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} F _ n = \\sum _ k ( - 1 ) ^ k \\binom { n } { \\lfloor \\frac { n - 1 - 5 k } { 2 } \\rfloor } \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} C ( l , h ) = \\begin{cases} \\displaystyle \\frac { - ( - 1 ) ^ { n - l } } { ( n - 1 ) ! } { n - 2 \\choose l - 1 } & h = 0 , \\\\ [ 4 m m ] \\displaystyle \\frac { ( - 1 ) ^ { n - l } } { ( n - 1 ) ! } { n - 2 \\choose l - 1 } & h = 1 , \\\\ [ 4 m m ] 0 & h \\ge 2 . \\end{cases} \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} v = K _ \\lambda \\ast f \\hbox { a n d } w = \\partial _ j K _ \\lambda \\ast f \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} ( 1 & - \\frac { 2 7 } { 6 4 L ^ 3 } ) ( x ^ 2 + y ^ 2 ) ^ 3 + \\frac { 2 7 } { 6 4 L ^ 3 } ( x ^ 3 - 3 x y ^ 2 ) ^ 2 \\\\ & - \\frac { 9 } { 4 L } ( x ^ 2 + y ^ 2 ) ( x ^ 3 - 3 x y ^ 2 ) + ( \\frac { 2 7 } { 1 6 L ^ 2 } - L ) ( x ^ 2 + y ^ 2 ) ^ 2 \\\\ & + ( 2 - \\frac { 2 7 } { 6 4 L ^ 3 } ) ( x ^ 3 - 3 x y ^ 2 ) - \\frac { 9 } { 8 L } ( x ^ 2 + y ^ 2 ) + \\frac { 2 7 } { 2 5 6 L ^ 3 } = 0 . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ n z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( y q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ n q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} \\sup \\bigg \\{ \\int _ X \\langle u , d \\mu \\rangle | u \\colon X \\to \\mathbb { R } ^ m 1 \\bigg \\} = \\norm { \\mu } _ { \\mathcal { W } _ 1 ( X , \\mathbb { R } ^ m ) } . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\xi ^ { 2 n } n ! { \\| h _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 < \\infty . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} K = - 1 + \\frac { 1 } { 4 } | H _ 0 | ^ 2 + O ( r ^ 2 ) \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot c = \\frac { 1 } { 4 \\pi } \\partial _ a E ( Q _ { a , c } ) \\\\ & \\dot a = - \\frac { 1 } { 4 \\pi } \\partial _ c E ( Q _ { a , c } ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\tau _ \\epsilon ( i ) & = \\begin{cases} ( \\delta + \\epsilon ) - i & \\mbox { f o r $ \\min ( i , ( \\delta + \\epsilon ) - i ) $ o d d } \\\\ i & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} \\sigma ( 3 ) = 6 . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\underline { b } _ { k } \\in \\Lambda = p ^ { - 1 } ( p _ 0 ) \\ , \\ , \\ , k = 1 , . . , N _ { 2 } . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\| u _ \\varepsilon \\| _ { L ^ { p ' } } = o \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } + \\| u _ \\varepsilon \\| _ { L ^ { p ' } } \\right ) + O \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} Q _ { 2 , k } = \\int _ { 0 } ^ { 1 } \\left | \\sum _ { l = 1 } ^ { k - 1 } \\int _ { 0 } ^ { 1 } \\frac { \\frac { \\dot { h } _ { k } ( u ) } { | I _ { k } | } - \\left ( \\frac { a _ { k } + u | I _ { k } | } { a _ { l } + v | I _ { l } | } \\right ) ^ { H - \\frac { 1 } { 2 } } \\cdot \\frac { \\dot { h } _ { l } ( v ) } { | I _ { l } | } } { ( a _ { k } + u | I _ { k } | - a _ { l } - v | I _ { l } | ) ^ { H + \\frac { 1 } { 2 } } } | I _ { l } | d v \\right | ^ { 2 } | I _ { k } | d u . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\norm { \\Lambda _ f ( y _ t ) - \\Lambda _ f ( z _ t ) } = \\norm { ( 1 , 0 ) - y _ t } = \\sin t , \\norm { y _ t - z _ t } = ( \\sin t ) ^ 2 . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { m - 1 } [ 3 n ] \\frac { ( 1 - q ) ^ 2 ( q ; q ^ 2 ) _ { n } ( q ^ { m + 2 } ; q ^ 2 ) _ { n - 1 } ^ 2 q ^ { m - { n + 1 \\choose 2 } - \\frac { ( 2 n + 1 ) ( m - 1 ) } { 2 } } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } \\equiv 0 \\pmod { \\Phi _ m ( q ) } , \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} T _ j ( \\varphi \\psi ) = T _ j ( \\varphi ) \\psi + \\varphi T _ j ( \\psi ) , j = 1 , 2 . . . d \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} 1 - \\langle \\hat { u } _ j ^ { ( n ) } , u _ j ^ { ( n ) } \\rangle = 1 - | \\langle \\hat { u } _ j ^ { ( n ) } , u _ j ^ { ( n ) } \\rangle | \\leq 1 - \\langle \\hat { u } _ j ^ { ( n ) } , u _ j ^ { ( n ) } \\rangle ^ 2 = \\frac { 1 } { 2 } \\| \\hat P _ j ^ { ( n ) } - P _ j ^ { ( n ) } \\| _ 2 ^ 2 \\xrightarrow { \\P } 0 . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} | I _ 1 ( x ' ) | = \\left \\vert \\int _ { \\Omega ' } \\big ( | v ( x ' , z ) | ^ p - | v ( y ' , z ) | ^ p \\big ) \\delta _ { \\varepsilon , x ' _ 0 } ( y ' ) \\ , d y ' d z \\right \\vert \\le C _ { \\theta , p } | \\lambda | ^ { - 1 / 2 } \\| f \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } \\| v \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } ^ { p - 1 } . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} P _ 3 ( x _ 2 , x _ 3 ) = x _ 2 ^ a ( x _ 2 + d x _ 3 ^ p ) ^ b R _ 3 ( x _ 2 , x _ 3 ) \\\\ P _ 1 ( x _ 2 , x _ 3 ) = x _ 2 ^ { a ' } ( x _ 2 - d x _ 3 ^ p ) ^ { b ' } R _ 1 ( x _ 2 , x _ 3 ) \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} d s ^ { 2 } = - d u d v + \\frac { 1 } { 4 } ( u - v ) ^ { 2 } ( d \\theta ^ { 2 } + \\sin ^ { 2 } \\theta d \\phi ^ { 2 } ) , \\ ; \\ ; \\ ; - \\infty < u , v < \\infty . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega ( A \\sigma : \\tau + u \\cdot \\div \\tau ) + ( \\div \\sigma - g ) \\cdot v \\ d x & = 0 , & & \\tau \\in H ( \\div ; \\mathbb { S } ) , & v \\in L ^ 2 ( \\mathbb { V } ) . \\end{aligned} \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\left | \\begin{array} { r l } & L ^ * u = 0 + \\alpha _ 1 z ^ 1 { 1 } _ { \\mathcal { O } _ { 1 , d } } + \\alpha _ 2 z ^ 2 { 1 } _ { \\mathcal { O } _ { 2 , d } } \\ \\ Q , \\\\ & L z ^ i = 0 - \\dfrac { 1 } { \\mu _ i } u { 1 } _ { \\mathcal { O } _ i } \\ \\ Q , \\\\ & u ( 0 , t ) = u ( L , t ) = 0 , \\ \\ z ^ i ( 0 , t ) = z ^ i ( L , t ) = 0 , \\ \\ \\ ( 0 , T ) , \\\\ & u ( T ) = u ^ { T } , \\ \\ z ^ i ( 0 ) = 0 , \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\bold { D } = \\left [ \\begin{array} { c c | c c | c c c c } 2 & 3 & 2 & 2 & 2 & 2 & 2 & 2 \\\\ \\hline 6 & 7 & 5 & 5 & 8 & 8 & 8 & 8 \\\\ \\hline 9 & 1 0 & 9 & 9 & 1 1 & 1 1 & 1 1 & 1 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\langle D _ 1 J _ 1 ( f ; v ^ 1 + s w ^ 1 , v ^ 2 ) - D _ 1 J _ 1 ( f ; v ^ 1 , v ^ 2 ) , w ^ 2 \\rangle & = \\alpha _ 1 \\iint _ { \\mathcal { O } _ { 1 , d } \\times ( 0 , T ) } ( y ^ s - y _ { 1 , d } ) z ^ s d x d t \\\\ & - \\alpha _ 1 \\iint _ { \\mathcal { O } _ { 1 , d } \\times ( 0 , T ) } ( y - y _ { 1 , d } ) z \\ d x d t \\\\ & + s \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } w ^ 1 w ^ 2 d x d t . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\Delta ^ 2 u = Q ( x , u , \\nabla u ) , x \\in \\Omega \\subset \\mathbb { R } ^ 4 , \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = \\sup \\{ H _ { a l g } ( \\phi , U ) : U \\in \\mathcal C ( G ) \\} . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} ( L _ Z I ) ( X ) = & \\ L _ Z ( I X ) - I ( L _ Z X ) \\\\ = & \\ \\N _ Z ( I X ) - \\N _ { I X } Z - I ( [ Z , X ] ) \\\\ = & \\ ( \\N _ Z I ) X + I ( \\N _ Z X ) - I ^ 2 X - I ( [ Z , X ] ) \\\\ = & \\ g ( Z , X ) Z - g ( Z , X ) Z + I ( \\N _ X Z ) + X \\\\ = & \\ 0 , \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} \\rho _ B ( X , Y ) = \\tfrac { 1 } { 2 } \\Omega ( X , Y , e _ i , J e _ i ) \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} R _ o ^ { \\ , j } = \\frac { ( 1 - \\beta ) } { | \\mathcal D _ o | } \\log \\left ( 1 + \\mathrm { S N R } _ { D _ { j } ^ { t } , D _ j ^ { r } } \\right ) . \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} & \\delta _ + ( k , \\xi ) = \\delta _ - ( k , \\xi ) ( 1 + | r ( k ) | ^ 2 ) , \\ k \\in ( - \\infty , k _ 0 = - \\xi ) , \\\\ & \\lim \\limits _ { k \\to \\infty } \\delta ( k ) = 1 , \\\\ & \\delta ( k ) \\mbox { i s a n a l y t i c i n } k \\in \\mathbb { C } \\setminus ( - \\infty , k _ 0 ] . \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\mathcal { D } _ { i , j } ^ { L , \\alpha , \\lambda } = ( - \\Delta ) ^ { \\alpha / 2 } l _ { j } ( x _ i ^ \\lambda ) = \\sum _ { k = 0 } ^ { N - 1 } b _ { k } ^ { j } ( - \\Delta ) ^ { \\alpha / 2 } R _ { j } ^ { \\lambda } ( x _ { i } ^ \\lambda ) , \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} a _ t & : = \\nabla f ( x _ t ) + \\nabla g ( x _ t ) ( \\lambda _ t + b _ t ) \\\\ & = \\nabla f ( \\overline { x } ) + t \\nabla ^ 2 f ( \\overline { x } ) \\xi ^ * + ( \\nabla g ( \\overline { x } ) + t \\nabla ( \\nabla g ( \\cdot ) ) ( \\overline { x } ) \\xi ^ * + o ( t ) ) ( \\overline { \\lambda } + t v ^ * + b _ t ) + o ( t ) \\\\ & = t \\nabla _ { x x } ^ 2 L ( \\overline { x } , \\overline { \\lambda } ) \\xi ^ * + t \\nabla g ( \\overline { x } ) v ^ * + o ( t ) = o ( t ) \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} ( ( h _ i ) _ { 1 \\leq i \\leq n } \\leftarrow ( \\psi _ \\alpha ) _ { \\alpha \\in \\mathcal { C } } ) \\leftarrow ( \\sigma _ t ) _ { 1 \\leq t \\leq q } = ( ( h _ i ) _ { 1 \\leq i \\leq n } \\leftarrow ( \\sigma _ t ) _ { 1 \\leq t \\leq q } ) \\leftarrow ( \\psi _ \\alpha ) _ { \\alpha \\in \\mathcal { C } } \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq \\varepsilon \\right ) & = \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( B _ 4 ^ { ( n ) } \\cap B _ 5 ^ { ( n ) } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( B _ 5 ^ { ( n ) } ) = - H ( \\ell _ 2 \\varepsilon ) , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { f } ( x ) = & \\limsup _ { n \\to \\infty } h _ { H _ { X } } ( f ^ { n } ( x ) ) ^ { 1 / n } = \\limsup _ { n \\to \\infty } h _ { H _ { X } } ( \\pi ( g ^ { n } ( y ) ) ) ^ { 1 / n } \\\\ & \\leq \\limsup _ { n \\to \\infty } h _ { H _ { Y } } ( g ^ { n } ( y ) ) ^ { 1 / n } = \\overline { \\alpha } _ { g } ( y ) . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} B H ( f , g ) ( x ) = \\ , \\int _ { \\mathbb R } f ( x - t ) g ( x + t ) \\frac { d t } { t } . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} - \\nu ( h _ i h _ { i + 1 } \\dots h _ n ) = - \\nu ( h _ i ) - \\dots - \\nu ( h _ n ) \\ge \\alpha ^ + _ i + \\dots + \\alpha ^ + _ n \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} \\Delta _ m ( f ( x ) ) & : = \\frac { d ^ m f ( x ) } { d x ^ m } | _ { x = 0 } , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} K \\left ( \\mathbb { P } ^ N \\right ) = \\mathbb { Z } [ P , P ^ { - 1 } ] \\left / \\left ( \\left ( 1 - P ^ { - 1 } \\right ) ^ { N + 1 } \\right ) \\right . \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} 4 d N ^ 2 N '^ 2 = O ( d ) ^ { 4 k + 3 } . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} v ^ i = - \\dfrac { 1 } { \\mu _ i } p ^ i 1 _ { \\mathcal { O } _ i } . \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} ( T ^ n x ) _ { i } = x _ { i } + n x _ { i + 1 } + \\cdots + \\binom { n } { d - i } x _ { d } + \\binom { n } { d - i + 1 } \\omega , i = 1 , \\ldots , d , x = ( x _ 1 , \\ldots , x _ d ) . \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} b ( t , x ) = - \\beta \\frac { x } { | x | ^ 2 } \\ 1 _ { x \\neq 0 } , \\beta > \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\| z \\| _ { L ^ { \\infty } _ x ( \\Omega _ { T } ) } = \\sup _ { t \\le T } \\| z ( \\cdot , t ) \\| _ { L ^ { \\infty } _ x } & \\le \\left ( \\bigl \\| z _ 0 \\bigr \\| _ { L ^ { \\infty } _ x } + \\int _ 0 ^ T \\bigl \\| f ( \\cdot , s ) \\bigr \\| _ { L ^ { \\infty } _ x } d s \\right ) e ^ { L T } , \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} T _ { \\geq j } ( y ) = \\sum _ { k \\geq j } ( \\lambda _ j + y - \\lambda _ k ) ^ { - 1 / 2 } P _ k , T _ { \\leq j } ( y ) = \\sum _ { k \\leq j } ( \\lambda _ k + y - \\lambda _ j ) ^ { - 1 / 2 } P _ k . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{gather*} D \\gamma = 0 , \\end{gather*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\tau _ { \\varphi _ p } ^ r \\Big ( \\sum _ { i = 1 } ^ n \\xi _ i \\Big ) \\le \\sum _ { i = 1 } ^ n \\tau _ { \\varphi _ p } ^ r ( \\xi _ i ) ; \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} & V _ { j , j ' } : = V _ j ^ { \\perp } \\cap V _ { j ' } , \\ \\ E _ { r _ i } ^ { j . j ' } : = D _ { r _ i } ^ j \\times V _ { j , j ' } \\ \\ \\subset \\ \\ V _ { j ' } , \\\\ & E _ { r _ i } ^ j : = D _ { r _ i } ^ j \\times V _ j ^ { \\perp } \\ \\ \\subset \\ \\ H . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\nu _ 1 ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( z q ; q ^ 2 ) _ n ( - q ) ^ n . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} c _ j x _ j = - 1 , \\ \\ \\ a _ { 1 j } x _ j \\le 0 , \\ \\ \\ X _ { k k } = 0 \\ \\ \\forall \\ k \\ne j , \\ \\ \\ X _ { j j } - x _ j = 0 \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\gamma _ 1 \\geqslant \\delta ( w ) \\wedge ( \\exists b \\in K ) ( \\exists c \\in \\mathcal { O } _ K ) ( v ( b ) = \\alpha ( w ) \\wedge ( \\forall \\gamma < \\gamma _ 1 ) ( ( c / v ) \\in A _ \\gamma ^ { w , b } ) ) . \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} v ( t ) = \\int _ { 0 } ^ { t } S ( t - s ) [ B ( s , v ( s ) + \\Psi ( s ) ) ] d s , \\ ; t \\in \\lbrack 0 , T ] . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( \\sum _ { j = 1 } ^ { V _ i } Z _ { i j } \\right ) ^ 2 \\right ] = & \\frac { \\mathbb { E } \\left [ Z ^ 2 \\right ] } { \\epsilon _ { k , n } ( \\delta ) } + \\frac { 2 - 2 \\epsilon _ { k , n } ( \\delta ) } { \\epsilon _ { k , n } ^ 2 ( \\delta ) } \\mathbb { E } \\left [ Z \\right ] ^ 2 \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = q ^ { n - k } \\binom { n - 1 } { k - 1 } _ q + \\binom { n - 1 } { k } _ q . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} G \\left ( \\frac { m + 1 } { 2 } , k \\right ) & = - \\frac { ( 1 + q ^ { \\frac { m - 1 } { 2 } + 2 k } ) ( q ; q ^ 2 ) _ { ( m + 1 ) / 2 } ( q ^ { 2 k + 1 } ; q ^ 2 ) _ { ( m - 1 ) / 2 } ^ 2 q ^ { - \\frac { m ^ 2 - 1 } { 8 } - p k } } { ( 1 - q ) ( q ; q ) _ { ( m - 1 ) / 2 } ^ 3 ( - q ; q ) _ { ( m - 1 ) / 2 } } \\\\ [ 5 p t ] & \\equiv 0 \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } , \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} 0 = \\bigg \\langle \\sum _ { v \\in S } \\sum _ { i = 1 } ^ d c _ { v , i } \\mathbf { a } _ { v , i } + \\sum _ { \\mathbf { x } \\in \\cal { B } } d _ \\mathbf { x } \\mathbf { x } , \\mathbf { b } _ { u , j } \\bigg \\rangle = c _ { u , j } . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} B = M _ 1 ( t _ 1 x + t _ 0 ) = \\left [ \\begin{array} { c } t _ 1 ^ 2 \\end{array} \\right ] \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\left | \\Gamma ^ { - 1 } ( t ) [ K ( \\Gamma ( t ) f ) \\cdot \\nabla ] ( \\Gamma ( t ) f ) \\right | _ q \\leq \\mathcal { B } ^ 2 e ^ { \\sum _ { i = 1 } ^ N \\left [ \\beta _ i ( t ) \\theta _ i - \\frac { t } { 2 } \\theta _ i ^ 2 \\right ] } | f | _ { p } | \\nabla f | _ { p } , \\ \\forall f \\in W ^ { 1 , p } ( \\mathbb { R } ^ 2 ) . \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} \\Phi _ 2 \\left ( \\frac { 1 } { 2 } , L ^ \\star \\right ) L ^ \\star = \\frac { 1 } { r _ 0 } \\max \\left ( \\alpha + \\frac { d } { 2 } , \\frac { \\alpha } { 2 } + d \\right ) . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} V = \\bigoplus _ { i = 0 } ^ n V ( i ) , \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} [ y ^ { p ^ r } _ { 1 2 } , y ^ { p ^ s } _ { 2 1 } ] = [ x ^ { p ^ r } _ { 1 2 } , x ^ { p ^ s } _ { 2 1 } ] = x ^ { p ^ r } _ { 1 2 } ( 1 - x ^ { - p ^ r } _ { 1 2 } x ^ { p ^ s } _ { 2 1 } x ^ { p ^ r } _ { 1 2 } x ^ { - p ^ s } _ { 2 1 } ) x ^ { p ^ s } _ { 2 1 } , \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } - \\mathcal { L } ^ { 0 , \\hat { v } } \\psi _ k ^ { 0 , \\hat { v } _ k } ( x ) = \\lambda ^ { 0 , \\hat { v } _ k } \\ , \\psi _ k ^ { 0 , \\hat { v } _ k } ( x ) D \\\\ \\psi _ k ^ { 0 , \\hat { v } _ k } ( x ) = 0 \\partial D , k = 1 , 2 , \\ldots , n \\end{array} \\right \\} , \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} f = ( c _ 1 x _ 1 + t _ 1 , c _ 2 x _ 2 + t _ 2 , \\dots ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} \\nu \\cdot \\alpha ( x , y ) = \\nu ( x _ 1 ) \\nu ( x _ 2 ) \\alpha ( x _ 2 , y _ 2 ) \\nu ^ { - 1 } ( x _ 3 y _ 3 ) \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} v ^ n ( t , y ) : = v _ 0 ( y ) + \\int _ 0 ^ t u _ y ^ n ( y , s ) d s , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} y _ { \\textrm { B N } } = \\left ( \\sqrt { P _ { \\textrm { N } } } x _ { \\textrm { N } } + \\sqrt { P _ { \\textrm { F } } } x _ { \\textrm { F } } \\right ) \\sqrt { d _ { \\textrm { B N } } ^ { - \\alpha } } h _ { \\textrm { B N } } + n _ { \\textrm { N } } , \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} x _ { n + 1 } \\le x _ n ( 1 - h _ n x _ n ^ 2 ) \\le x _ 1 \\prod _ { i = 1 } ^ n ( 1 - h _ i x _ i ^ 2 ) . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} T _ { \\rm o p } = ( T _ { \\rm o p } ) _ { \\rm r e g } + ( T _ { \\rm o p } ) _ { \\rm s i n g } , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} \\frac { f _ { 1 } f _ { 1 } ^ { \\prime \\prime } } { \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } } = c _ { 1 } = - \\frac { \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } } { f _ { 2 } f _ { 2 } ^ { \\prime \\prime } } , \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} V _ f = X _ f + \\hbar ^ { - 1 } L \\frac { \\partial } { \\partial \\phi } \\ , , \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} H _ { K , M } ^ { \\left ( A \\right ) } \\buildrel \\Delta \\over = \\eta _ { K , M } ^ { ( 1 ) } + \\eta _ { K , M } ^ { ( 2 ) } + \\mathop { \\max } \\limits _ { 1 \\le k \\le K } \\eta _ { K , M } ^ { ( 4 ) } . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} U = \\left \\{ v \\in X _ { \\infty } ; a < v \\left ( x \\right ) < b x \\in \\mathcal { K } _ { N } \\right \\} . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\begin{cases} u '' = \\lambda u & e \\\\ \\sum _ { e \\ni v } u ' _ e ( v ) = \\lambda q u ( v ) , & v , \\end{cases} \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} b ( u , v , w ) = - b ( u , w , v ) . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\| ( \\hat { Q } _ r - Q _ r ) E R _ r \\| _ 2 & = \\sqrt { \\sum _ { s \\neq r } \\frac { \\| \\hat { Q } _ r - Q _ r ) E Q _ s \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ s ) ^ 2 } } \\\\ & \\leq \\sqrt { \\sum _ { \\substack { s < r _ 0 : \\\\ s \\neq r } } \\frac { \\| \\hat { Q } _ r - Q _ r ) E Q _ s \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ s ) ^ 2 } + \\frac { \\| \\hat { Q } _ r - Q _ r ) E Q _ { \\geq r _ 0 } \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ { r _ 0 } ) ^ 2 } } . \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} P _ X ^ { } ( x ) = \\sum _ { m = m _ 0 } ^ { m _ 1 } \\frac { p _ m ^ { } } { x \\ln b } \\ , \\chi _ m ^ { } ( x ) , 0 < x < \\infty , \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { n - 1 } { 2 } } ( - 1 ) ^ k [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ; q ) _ k ^ 3 } \\equiv [ n ] q ^ { \\frac { ( n - 1 ) ^ 2 } { 4 } } ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\pmod { [ n ] \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} f ^ { q - 1 } ( \\xi ) = \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta , \\xi \\in \\overline { \\Omega } . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\tilde { p } _ { m a x } = \\max _ { x \\in \\mathcal { A } _ n } p ( x ) \\ ; , \\ ; \\ ; \\tilde { p } _ { m i n } = \\min _ { x \\in \\mathcal { A } _ n } p ( x ) \\ ; , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ t = \\Big ( \\frac { d } { d t } F _ t ( x ) \\Big ) _ { x \\in Z ^ d } = G _ \\lambda F _ t , \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} h ^ { p ^ { \\ast } } _ { i j k l } - h ^ { p ^ { \\ast } } _ { i j l k } = \\sum _ m h ^ { p ^ { \\ast } } _ { m j } R _ { m i k l } + \\sum _ m h ^ { p ^ { \\ast } } _ { i m } R _ { m j k l } + \\sum _ { q } h ^ { q ^ { \\ast } } _ { i j } R _ { { q ^ { \\ast } p ^ { \\ast } } k l } . \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\xi ( 1 ) = \\sum _ { p } c _ p ^ 2 = 1 . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} E _ { P } \\left [ T _ { P , a , \\mathcal { G } } | Y , _ { \\mathcal { G } } \\left ( Y \\right ) \\right ] = \\frac { I _ { a } ( A ) Y } { \\pi _ { a } ( \\mathbf { O } _ { m i n } ; P ) } . \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} | D ^ 2 F | : = \\sqrt { \\sum _ { l , j , k = 1 } ^ m f ^ 2 _ { l , u _ j u _ k } } . \\end{align*}"}
-{"id": "10030.png", "formula": "\\begin{align*} R _ \\alpha f ( x ) = \\frac { d } { d x } B _ \\alpha ^ { - 1 / 2 } f ( x ) , x \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} 1 + \\frac { p - 1 } { p } \\left [ \\sum _ { i = 1 } ^ { a _ { 1 } } p ^ { \\max \\{ 2 i - a _ { 3 } , 0 \\} } + \\sum _ { i = a _ { 1 } + 1 } ^ { a _ { 2 } } p ^ { \\max \\{ - i , i - a _ { 3 } \\} } + \\sum _ { i = a _ { 2 } + 1 } ^ { a _ { 3 } } p ^ { \\max \\{ - 2 i , - a _ { 3 } \\} } \\right ] . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\zeta } ^ { ( { \\tau _ 1 } , { \\tau _ 2 } ) } ( c ) : = \\begin{cases} - g _ { { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c } + ( \\chi { } ( \\cdot { } , { \\zeta } ) - 1 ) \\cdot { } c & A _ { \\zeta } ( { \\tau _ 1 } ) \\cap { } C _ { \\zeta } ( { \\tau _ 2 } ) \\\\ - g _ { { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c } & A _ { \\zeta } ( { \\tau _ 1 } ) \\setminus { } C _ { \\zeta } ( { \\tau _ 2 } ) \\end{cases} \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} F _ 1 ( N ) & = \\sum _ { n = 1 } ^ { N } n \\gamma ( n ) \\\\ & = N F ( N ) - \\sum _ { n = 1 } ^ { N - 1 } F ( n ) \\\\ & = N ( 1 + O ( N ^ { - \\epsilon } ) ) K N ^ u - \\sum _ { n = 1 } ^ { N - 1 } K n ^ u ( 1 + O ( n ^ { - \\epsilon } ) ) \\\\ & = \\frac { K u } { u + 1 } ( 1 + O ( N ^ { - \\epsilon } ) ) N ^ { u + 1 } . \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} W = e _ 1 K ^ { \\alpha } F \\oplus e _ g K ^ { \\nu ( \\alpha ) } F \\oplus e _ { g ^ 2 } K ^ { \\alpha } F \\oplus e _ { g ^ 3 } K ^ { \\nu ( \\alpha ) } F , \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} g ( t ) = t ^ 3 e ^ { - t } . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} T _ { 2 , k } & : = \\sum _ { 0 \\le K _ i \\le M - 1 ( i > k ) , \\atop \\tau ' _ j \\in \\Sigma ' ( j \\le k ) } B _ { \\tilde { n } } \\left ( \\left \\{ \\frac { \\sum _ { i > k } K _ i + \\sum _ { j \\le k } \\tau ' _ j } { M } \\right \\} \\right ) , \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\Delta _ L ^ { \\perp } H = 0 = \\Delta _ L ^ { \\perp } X ^ { \\perp } . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\gamma ( t ) = \\left \\{ \\begin{array} { l l } t e ^ { i \\theta } , & \\hbox { $ t \\geq 0 $ } \\\\ - t e ^ { - i \\theta } , & \\hbox { $ t \\leq 0 $ } \\end{array} \\right . \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} B _ 1 ^ { ( n ) } : = \\bigcup _ { t = a _ n } ^ { \\lfloor n - f _ 1 ( n ) \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} , \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} ( n - 2 ) f h \\varphi '' - r f \\varphi h '' - m h \\varphi f '' - 2 m h \\varphi ' f ' - 2 r f \\varphi ' h ' = 0 . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} H ^ { - 1 } ( A ^ \\bullet ( q = 0 ) ) \\cong H ^ { - 1 } ( Q ^ \\bullet ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\sum _ { k = \\nu } ^ { n } \\sqrt { k } & = \\frac 2 3 \\sqrt { n + 1 } \\left ( n + \\frac 1 4 \\right ) - \\frac 2 3 \\sqrt \\nu \\left ( \\nu - \\frac 3 4 \\right ) - \\widehat { \\delta } _ { \\nu , n } \\\\ & = n A ( n ) - \\frac 2 3 \\sqrt \\nu \\left ( \\nu - \\frac 3 4 \\right ) - \\widehat { \\delta } _ { \\nu , n } . \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\begin{aligned} | I _ 2 ( t ) | & \\le \\sum _ { h , k } | \\partial ^ 2 _ { h k } S ( \\Phi ( t ) ) - \\partial ^ 2 _ { h k } \\tilde S ( t , \\tilde \\Phi ( t ) ) | | \\partial _ j \\tilde \\Phi _ k ( t ) | | \\partial _ i \\tilde \\Phi _ h ( t ) | \\\\ & + \\sum _ { h , k } | \\partial ^ 2 _ { h k } S ( \\Phi ( t ) ) | | \\partial _ j \\Phi _ k ( t ) \\partial _ i \\Phi _ k ( t ) - \\partial _ j \\tilde \\Phi _ k ( t ) \\partial _ i \\tilde \\Phi _ h ( t ) | \\ , . \\end{aligned} \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( \\partial _ { t } - \\underline { \\Delta } + S c a l ' ) u ( x , t ) = 0 , \\\\ { \\partial _ { t } } g ' ( x , t ) = 2 R i c ' ( x , t ) \\end{array} \\right . \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\left \\{ \\begin{array} { l } ( \\partial _ { t } - \\underline { \\Delta } + S c a l ' ) u ( x , t ) = 0 , \\\\ { \\partial _ { t } } g ' ( x , t ) = - 2 R i c ' ( x , t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} C _ 1 = 2 \\cdot 3 ^ { \\frac { d s } a } C _ 4 ( a ) , \\ C _ 2 = 6 \\cdot 3 ^ { d s - 1 \\over a } \\cdot \\zeta ( d s ) . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } m n \\log ( \\theta ) = \\lim _ { n \\to \\infty } m n \\log ( 1 - ( 1 - \\alpha ) ^ { n - 1 \\choose 2 } ) . \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\partial _ { y _ j } \\partial _ { y _ k } g _ { D , \\mathrm { o d d } } ( y ) = ( \\partial _ { y _ j } \\partial _ { y _ k } g _ D ) _ { \\mathrm { o d d } } ( y ) , \\partial _ { y _ j } \\partial _ { y _ k } g _ { N , \\mathrm { e v e n } } ( y ) = ( \\partial _ { y _ j } \\partial _ { y _ k } g _ N ) _ { \\mathrm { e v e n } } ( y ) , \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\overline { \\xi } \\frac { 1 } { \\sqrt { | D ^ { c } | } } \\sum _ { \\beta \\in D ^ { c * } } \\xi _ { \\beta } \\# \\{ \\vec { v } \\in \\bigcup L + \\beta : \\beta \\in D ^ { c * } , S ( \\vec { v } ) = n / w \\} , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} 0 \\le \\hat \\tau _ i < 1 , \\sum _ { i = 1 } ^ m \\hat A _ i = 1 . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} & \\iota : B _ n ( G ) \\to N _ n ^ { \\rm { c y } } ( G ) , \\ \\ \\ \\iota : G ^ n \\to G ^ { n + 1 } , \\\\ & \\iota ( g _ 1 , \\ , g _ 2 , \\ , \\ldots \\ , , g _ n ) = \\big ( ( g _ 1 g _ 2 \\ldots g _ n ) ^ { - 1 } , \\ , g _ 1 , \\ , g _ 2 , \\ , \\ldots \\ , , g _ n \\big ) . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{gather*} P r ( f , \\sigma ) : = \\lim _ { X \\to \\infty } \\frac { \\# S p l _ X ( f , \\sigma ) } { \\# S p l _ X ( f ) } , \\\\ P r ( f , \\sigma , \\{ k _ j \\} ) : = \\lim _ { X \\to \\infty } \\frac { \\# S p l _ X ( f , \\sigma , \\{ k _ j \\} ) } { \\# S p l _ X ( f , \\sigma ) } , \\\\ P r ( f , \\sigma , \\{ k _ j \\} , L , \\{ R _ i \\} ) : = \\lim _ { X \\to \\infty } \\frac { \\# S p l _ X ( f , \\sigma , \\{ k _ j \\} , L , \\{ R _ i \\} ) } { \\# S p l _ X ( f , \\sigma , \\{ k _ j \\} ) } , \\end{gather*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\Pi ( \\underline { a } ) = \\lim _ { n \\to \\infty } f _ { a _ 1 } \\circ \\cdots \\circ f _ { a _ n } ( 1 ) . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\gamma _ i = \\inf \\left ( - \\nu ( h _ i h _ { i + 1 } \\dots h _ n ) \\right ) - \\alpha ^ + _ { i + 1 } - \\dots - \\alpha ^ + _ n . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\tilde { x } _ \\varepsilon \\to \\tilde { x } , ~ ~ | \\tilde { x } | = 1 \\ , , \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\chi _ i = \\sum _ { j = 1 } ^ { m } \\left ( \\left ( 1 - \\psi _ { i - j + 1 } \\right ) ^ 2 + { \\psi _ { i - j + 1 } } ^ 2 \\right ) . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} y = c _ 0 x \\cdot \\frac { \\log x \\log _ 3 x } { \\log _ 2 x } , \\end{align*}"}
-{"id": "10034.png", "formula": "\\begin{align*} t \\partial _ t P _ t ^ \\alpha f ( x ) & = - \\int _ { \\mathbb R _ + ^ d } | y | t \\ e ^ { - | y | t } \\phi _ y ^ \\alpha ( x ) h _ \\alpha ( f ) ( y ) y ^ { 2 \\alpha } d y \\\\ & = - h _ \\alpha ( | \\cdot | t \\ e ^ { - | \\cdot | \\ t } h _ \\alpha ( f ) ( \\cdot ) ) ( x ) , \\ , \\ , \\ , x \\in \\mathbb { R } _ + ^ d \\ , \\ , \\ , a n d \\ , \\ , \\ , t > 0 , \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{gather*} \\operatorname { a d } ^ * _ { e _ 1 } \\mu ( e _ 2 ) = \\mu ( [ e _ 1 , e _ 2 ] ) . \\end{gather*}"}
-{"id": "1732.png", "formula": "\\begin{align*} y ' _ { i } - x ' _ k \\geq y ' _ { j } - x ' _ k = \\frac { \\alpha } { q _ j } + y _ { j } - \\frac { \\alpha } { p _ k } - x _ { k } \\geq y _ { j } - x _ { k } \\geq 1 - \\delta . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} { \\rm m s } _ t ( { F _ { \\cdot } } ( x ) ) = | v _ t | ( F _ t ( x ) ) { \\rm a . e . } \\ t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} L ( s , \\phi _ n \\times \\psi _ m ) = \\prod _ { \\substack { 1 \\leq i \\leq l \\\\ 1 \\leq j \\leq s } } L ( s , \\zeta _ i \\times \\eta _ j ) \\prod _ { \\substack { 1 \\leq i \\leq s \\\\ 1 \\leq j \\leq k } } L ( s + \\frac { 1 } { 2 } , \\eta _ i \\times \\xi _ j ) L ( s - \\frac { 1 } { 2 } , \\eta _ i \\times \\xi _ j ) . \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\mathcal { H } ^ { m } ( B _ { r } ( x ) \\cap M ) - \\mathcal { H } ^ { m } ( B _ { r } ^ { g } ( x ) ) = O ( r ^ { m + 2 } ) . \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} Q _ { B G K } ( f ) = \\frac { \\rho _ f ( 1 - \\eta \\rho _ f ) } { \\tau } ( F _ f - f ) , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} B _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon } + o \\left ( \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon } \\right ) \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} R ( x , x ) = \\sum \\limits _ { n = 1 } ^ { \\infty } \\lambda _ n | \\phi _ n ( x ) | ^ 2 \\quad \\mbox { a n d } \\| \\log \\kappa - \\log \\kappa _ M \\| ^ 2 _ { L ^ 2 ( \\Omega , C ( { D } ) ) } \\leq \\sum \\limits _ { n > M } \\lambda _ n \\| { \\phi _ n } \\| _ { C ( D ) } ^ 2 . \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} \\mathcal { T } _ { G , b , \\mu } : = \\mathcal { T } ^ { - 1 } ( b ) \\cap \\mathcal { S D } _ { \\mu } \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} s _ r ( H ( k , k , \\ldots , k ) ) & = k \\sum _ { i \\in [ r ] } \\sum _ { j \\in V _ i } \\left | d _ { H ( k , \\ldots , k ) } ( j ) - \\frac { k ^ r m } { k n _ i } \\right | \\\\ & = k ^ r \\sum _ { i \\in [ r ] } \\sum _ { j \\in V _ i } \\left | d _ { H } ( j ) - \\frac { m } { n _ i } \\right | \\\\ & = k ^ r s _ r ( H ) , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} f \\left ( \\sum _ { i = 1 } ^ { N - 1 } D ( u _ i , u _ { i + 1 } ) \\right ) \\leq f ( \\varepsilon ) . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} r _ 2 ( n , q ) = 4 d _ { 1 , 4 } ( n , q ) - 4 d _ { 3 , 4 } ( n , q ) \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\sqrt { \\lambda _ k } \\langle \\tilde u _ 1 , u _ k \\rangle = \\tilde x \\frac { \\lambda _ k } { \\tilde \\lambda _ 1 - \\lambda _ k } \\big ( \\sqrt { \\lambda _ 1 } \\langle \\tilde u _ 1 , u _ 1 \\rangle + \\sum _ { l > 1 } \\sqrt { \\lambda _ l } \\langle \\tilde u _ 1 , u _ l \\rangle \\big ) , \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\partial _ Q \\widetilde { f } ( Q ) = \\frac { 1 } { ( Q - 1 ) ( Q - i ) ( Q + 1 ) } \\widetilde { f } ( Q ) \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 + } t ^ { 1 / 2 } \\lVert \\nabla e ^ { t A } v \\rVert _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } = 0 . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} I _ m ( \\psi ) : = s ^ { m - 4 } \\lambda ^ { m - 3 } \\iint _ Q e ^ { - 2 s \\sigma } ( \\xi ) ^ { m - 4 } ( | \\psi _ t | ^ 2 + | \\psi _ { x x } | ^ 2 ) d x d t + L _ m ( \\psi ) , \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\Delta _ T = \\int _ { 0 } ^ { T } \\left ( t - u ( t ) \\right ) d t \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} C _ { p _ l } ^ k & = \\left ( \\frac { ( k / ( k - 1 ) ) ^ { ( k - 1 ) / k } } { k ^ { 1 / k } } \\right ) ^ { k / 2 } \\\\ & = \\frac { 1 } { k ^ { 1 / 2 } } \\left ( 1 + \\frac { 1 } { k - 1 } \\right ) ^ { ( k - 1 ) / 2 } \\\\ & \\leq \\frac { \\sqrt { e } } { k ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} & \\sup _ { x _ 1 ^ 2 + \\hdots + x _ n ^ 2 \\leq r _ 0 ^ 2 , t = t _ 0 } H \\ , \\langle \\omega , \\nu \\rangle ^ { - 1 } \\\\ & \\leq \\max \\Big \\{ \\sup _ { x _ 1 ^ 2 + \\hdots + x _ n ^ 2 = r _ 0 ^ 2 , t _ j \\leq t \\leq t _ 0 } H \\ , \\langle \\omega , \\nu \\rangle ^ { - 1 } , \\ , \\sup _ { x _ 1 ^ 2 + \\hdots + x _ n ^ 2 \\leq r _ 0 ^ 2 , t = t _ j } H \\ , \\langle \\omega , \\nu \\rangle ^ { - 1 } \\Big \\} . \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} S \\setminus C = \\bigcup _ { g \\in G } ( S _ { g } \\setminus C _ { g } ) , \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 4 \\\\ 1 \\\\ 8 \\\\ 4 \\\\ 6 \\\\ 7 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c c c c c } 4 & 0 & 0 & 3 & 0 & 3 \\\\ 0 & 0 & 0 & 6 & 6 & 0 \\\\ 0 & 0 & 5 & 0 & 3 & 6 \\\\ 3 & 6 & 0 & 4 & 4 & 3 \\\\ 0 & 6 & 3 & 4 & 4 & 5 \\\\ 3 & 0 & 6 & 3 & 5 & 2 \\end{array} \\right ] , d = \\left [ \\begin{array} { c } 8 \\\\ 7 \\\\ 7 \\\\ 8 \\\\ 6 \\\\ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} p _ { w } = \\begin{cases} \\mu [ w ] \\cdot c & w \\in \\mathcal { W } \\\\ 2 ^ { - m \\epsilon ^ { - 1 } } \\cdot c & \\end{cases} \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} 0 = \\widehat \\alpha ( 0 ) = \\widehat \\alpha _ \\Gamma ( 0 ) = \\widehat \\beta ( 0 ) = \\widehat \\beta _ \\Gamma ( 0 ) \\ , , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} f ( D ( x , y ) ) \\leq f \\left ( \\sum _ { i = 1 } ^ { N - 1 } D ( u _ i , u _ { i + 1 } ) \\right ) + \\alpha . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\mathcal { L } _ I ( s ) = \\exp \\left ( - 2 \\pi \\lambda ^ { ( n > 0 ) } _ { } s ^ { 2 / \\alpha } \\mathbb { E } _ p \\left [ p ^ { 2 / \\alpha } \\right ] \\int ^ \\infty _ { ( s \\rho _ { \\mathrm { o } } ) ^ { \\frac { - 1 } { \\alpha } } } \\frac { y } { y ^ \\alpha + 1 } \\ , \\mathrm { d } y \\right ) , \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} \\Delta u + \\lambda u = 0 , u = 0 \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} v = 0 \\quad \\mbox { o n } \\quad \\Gamma _ f \\times ( 0 , T ) . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} & \\ \\phi \\big ( | G _ X ( 1 + i t ) | ^ 2 \\big ) \\\\ = & \\ \\phi \\big ( G _ X ( 1 + i t ) ^ * G _ X ( 1 + i t ) \\big ) \\\\ = & \\ \\phi \\big ( | M | ^ \\frac { ( 1 - i t ) } { s } Q ^ * C ^ { - \\frac { r ( 1 - i t ) } { 2 } } X ^ r C ^ { - \\frac { r ( 1 + i t ) } { 2 } } Q | M | ^ \\frac { ( 1 + i t ) } { s } \\big ) \\\\ = & \\ \\phi \\big ( | M | ^ \\frac { 1 } { s } Q ^ * C ^ { - \\frac { r ( 1 - i t ) } { 2 } } X ^ r C ^ { - \\frac { r ( 1 + i t ) } { 2 } } Q | M | ^ \\frac { 1 } { s } \\big ) \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\nu ( f ) = \\lim _ { \\alpha \\in A } \\{ \\nu _ \\alpha ( f ) \\} : = \\sup _ { \\alpha \\in A } \\{ \\nu _ \\alpha ( f ) \\} . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d u _ t & = & - \\big ( \\langle x , \\partial ^ 2 \\rangle + \\langle 1 - x , \\nabla \\rangle \\big ) u _ t d t + \\sigma ( u _ { t - } ) d X _ t \\medskip \\\\ u _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\tau ( \\beta ) ^ { - 1 } b \\tau ( b ) \\tau ( \\beta ) = b ' \\tau ( b ' ) \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\iint _ Q & \\rho _ 0 ^ { - 2 } | L ^ * u - \\alpha _ 1 z ^ 1 { 1 } _ { \\mathcal { O } _ { 1 , d } } - \\alpha _ 2 z ^ 2 { 1 } _ { \\mathcal { O } _ { 2 , d } } | ^ 2 d x d t \\\\ & + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\rho _ 0 ^ { - 2 } | L z ^ i + \\dfrac { 1 } { \\mu _ i } u { 1 } _ { \\mathcal { O } _ i } | ^ 2 d x d t + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\rho _ 1 ^ { - 2 } | u | ^ 2 d x d t = 0 . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} H e s s _ { \\widetilde { g } } h ( X _ i , Y _ j ) = h , _ { x _ i y _ j } - \\frac { f , _ { x _ i } } { f } h , _ { y _ j } . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} ( t _ 0 , \\dots , t _ N ) \\cdot v = \\chi ( f ( t _ 0 , \\dots , t _ N ) , v ) = \\chi ( t _ 0 \\cdots t _ N , v ) \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} K _ { n , s } ( z ) = \\sum _ { k = 0 } ^ { + \\infty } \\omega _ { n , k } ^ { ( s ) } u ^ { - ( 2 s + 1 ) n - k - 1 } , \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} ( \\partial _ t \\varphi ) ^ 2 + \\partial _ { y _ 1 } \\varphi - \\rho ^ 2 + g _ M ( m _ 0 , \\varphi ) = t ^ { M } R _ M \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\det ( \\nabla ^ 2 h _ K + h _ K \\ , { \\rm I d } ) = \\mbox { $ \\frac 1 n $ } \\ , h _ K ^ { p - 1 } \\left ( \\| \\nabla h _ K \\| ^ 2 + h _ K ^ 2 \\right ) ^ { \\frac { n - q } 2 } \\cdot f . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\frac 1 q - \\frac 1 { q _ 0 } = \\frac 1 r - \\frac 1 { r _ 0 } = \\frac 1 p - \\frac 1 { p _ 0 } , \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} h ^ { p ^ { \\ast } } _ { i j } = h ^ { p ^ { \\ast } } _ { j i } = h ^ { i ^ { \\ast } } _ { p j } , \\ \\ \\ i , j , p . \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\langle \\nabla F ( x ) , x \\rangle = \\sum _ t \\mathbb { E } _ { S \\sim x } [ f ( S \\cup \\{ t \\} ) - f ( S \\setminus \\{ t \\} ) ] x _ t \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\eta _ t : = e ^ { \\sup _ { 0 \\leq s \\leq t } \\sum _ { i = 1 } ^ N [ \\beta _ i ( s ) \\theta _ i - \\frac { s } { 4 } \\theta _ i ^ 2 ] } , \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\mathcal { O } ^ { } _ { \\overline { \\mathcal { M } } _ { g , n } ( X , 0 ) } = \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\bigwedge \\nolimits ^ i \\left ( p r _ 1 ^ * R ^ 1 \\pi _ * \\mathcal { O } _ \\mathcal { C } \\otimes \\textnormal { e v } ^ * T _ X \\right ) ^ \\vee \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} \\partial _ t ^ { \\beta } u - \\Delta u = f , u ^ { ( j ) } ( 0 ) = 0 , \\ j = 0 , \\dots , m - 1 , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 0 } ~ ~ \\Big | ~ ~ y _ 0 + \\int _ { 0 } ^ { \\cdot } b _ { j ^ { \\delta } ( s ) } ( s , y ^ { \\delta } ( s ) ) d s + \\int _ { 0 } ^ { \\cdot } \\sigma _ { j ^ { \\delta } ( s ) } ( s , y ^ { \\delta } ( s ) ) d B ^ { \\delta } ( s ) ~ ~ \\\\ & - ~ ~ \\Big ( y _ 0 + \\int _ { 0 } ^ { \\cdot } b _ { j ( s ) } ( s , y ( s ) ) d s + \\int _ { 0 } ^ { \\cdot } \\sigma _ { j ( s ) } ( s , y ( s ) ) d B ( s ) ~ ~ \\Big ) ~ ~ \\Big | _ { ( 0 , T ) } ~ ~ = ~ ~ 0 . \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} u ( x ) \\leq \\left \\{ \\begin{aligned} & \\max _ { \\partial B _ { r _ 0 } ( 0 ) } u + c _ 1 \\log ( r _ 0 / | x | ) & & k = \\ell ; & \\\\ & \\max _ { \\partial B _ { r _ 0 } ( 0 ) } u + c _ 1 | x | ^ { 1 - \\frac { \\ell } { k } } & & k < \\ell . & \\end{aligned} \\right . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\eta = \\eta ( R ) : = R + ( R + R _ 0 ) \\sqrt { \\widetilde { M _ R } / 2 a } = R + ( R + R _ 0 ) \\sqrt { 1 + M _ R / 2 a } . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} b ( w _ m ) = \\lim _ { n \\rightarrow \\infty } B _ \\Omega ( w _ m , y _ n ) - B _ \\Omega ( z _ 0 , y _ n ) \\geq B _ \\Omega ( w _ m , z _ 0 ) - R . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} V _ i & = \\textstyle \\sum _ { a , b , j } ( 2 k _ { j i } c _ { a b } P _ { a b j i } + k _ { a b } c _ { i j } D _ { e _ { i j } } Q _ { a b } ) \\\\ W _ i & = \\textstyle \\sum _ { a , b , c , d , j } c _ { a b } c _ { c d } c _ { i j } D _ { e _ { i j } } P _ { a b c d } \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} \\zeta _ { l _ 0 } ( \\cdot , t ) = 0 \\ \\ { \\rm f o r } \\ \\ t \\in [ 0 , t _ 0 ] . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} { I } _ { N } ^ \\lambda f ( x ) = \\sum _ { n \\in \\Upsilon _ { \\ ! N } } \\tilde { f } _ { n } \\ , R _ { n } ^ \\lambda ( x ) , \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\prod _ v \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau _ v [ a ] \\otimes \\tau ' _ v ) = + 1 . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\left ( 1 - q ^ d \\right ) ^ 3 f _ d ( q ) = f _ { d - 1 } ( q ) \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} A = \\sum g _ i [ \\sigma _ i ] \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathbf { P A } _ { q } ( K ) } = \\norm { f } _ \\infty = \\sup _ { z \\in K } \\abs { f ( z ) } = \\| f \\| _ K . \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} f ( u , v ) + \\langle G ( u , v ) , ( x - u , y - v ) \\rangle = \\begin{cases} - x & \\textrm { i f } ( u , v ) \\in E _ 1 , u < 0 \\\\ x & \\textrm { i f } ( u , v ) \\in E _ 1 , u > 0 \\\\ - x & \\textrm { i f } ( u , v ) \\in E _ 2 , u < 0 , 1 \\leq v \\leq j + 1 \\\\ x & \\textrm { i f } ( u , v ) \\in E _ 2 , u > 0 , 1 \\leq v \\leq j + 1 \\\\ 2 ( y - j - 1 ) & \\textrm { i f } ( u , v ) \\in E _ 2 , v > j + 1 , \\end{cases} \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\eta \\langle \\Delta \\big ( | d u | ^ { p - 2 } d u \\big ) , d u \\rangle d \\mu & = - \\int _ { \\Omega } \\eta \\langle ( d d ^ * + d ^ * d ) \\big ( | d u | ^ { p - 2 } d u \\big ) , d u \\rangle d \\mu \\\\ & = - \\int _ { \\Omega } \\langle d ^ * \\big ( \\eta d \\big ( | d u | ^ { p - 2 } d u \\big ) \\big ) , d u \\rangle d \\mu \\\\ & = - \\int _ { \\Omega } \\langle \\eta d \\big ( | d u | ^ { p - 2 } d u \\big ) , d ( d u ) \\rangle d \\mu = 0 . \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} B = \\{ | X - Y | \\geq 1 - \\delta \\} . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} U _ { 2 k + 1 , 2 a + 1 } ( x ; q ) = ( - x q ; q ^ 2 ) _ \\infty \\overline { Q } _ { k + \\frac { 1 } { 2 } , a + \\frac { 1 } { 2 } } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} ( ( I - P ) T ) ^ * = T ^ * ( I - P ) = ( ( I - P ) T ^ { * * } ) ^ * . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} d ( ( e _ x ^ 2 - e _ y ^ 2 ) b ) & = d ( e _ x ^ 2 - e _ y ^ 2 ) b + ( e _ x ^ 2 - e _ y ^ 2 ) d ( b ) = ( e _ x ^ 2 - e _ y ^ 2 ) d ( b ) . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } \\varepsilon _ { i , p } \\left ( : = \\left [ p u _ { p } ( y _ { i , p } ) ^ { p - 1 } \\right ] ^ { - 1 / 2 } \\right ) = 0 \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} u ( t ) = u ^ R ( t ) + \\zeta ( t ) k ( t ) S ( \\Phi ( t ) ) \\ , , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\| P _ h u - u _ h \\| _ 0 ^ 2 = b ( \\tau , P _ h u - u _ h ) = b ( \\Pi _ h \\tau , P _ h u - u _ h ) . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\gamma _ n = e ^ { - n ^ 2 } , \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\Delta _ { M D S - B E } = \\frac { n } { \\epsilon _ n } - \\frac { n } { 2 } + \\frac { k \\epsilon _ { k + 1 , n + 1 } ( q ) } { q \\epsilon _ { k , n } ( q ) } \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\mu _ { k _ i } = \\mathbf { w } _ { k _ i } ^ H \\mathbf { A } ( \\mathbf { H } ) _ { k _ { i } } \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla ( e ^ { s B _ i ( t ) } f ) | _ q = | \\nabla f | _ q , \\ \\forall f \\in W ^ { 1 , q } ( \\mathbb { R } ^ d ) , \\ s \\in \\mathbb { R } , \\ t \\geq 0 , i = 1 , 2 , . . . , N . \\end{aligned} \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} E [ X | Y = y ] = \\int _ { - \\infty } ^ \\infty x f _ { X | Y } ( x | y ) d x , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} F _ 1 ( N ) & = \\sum _ { n = 1 } ^ { N } n \\gamma ( n ) \\\\ & = N F ( N ) - \\sum _ { n = 1 } ^ { N - 1 } F ( n ) \\\\ & = N ( 1 + O ( N ^ { - \\epsilon } ) ) K N ^ u - \\sum _ { n = 1 } ^ { N - 1 } K n ^ u ( 1 + O ( n ^ { - \\epsilon } ) ) \\\\ & = \\frac { K u } { u + 1 } ( 1 + O ( N ^ { - \\epsilon } ) ) N ^ { u + 1 } . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { h } ^ { ( j ) } : = \\frac { \\lambda _ { h } ^ { ( j ) } L } { \\pi ^ 2 } , \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\left \\| { \\cal T } ^ { ( k ) } - \\sum _ { j = 0 } ^ { s } h ^ j { \\cal T } _ j ^ { ( k ) } \\right \\| \\lesssim h ^ { s + 1 } | { \\rm I m } \\ , z | ^ { - ( 3 s + 2 - k ) / 2 } . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} \\nu _ { n + 1 } : = \\left [ \\nu _ n ; \\nu _ { n + 1 } ( \\phi _ { n + 1 } ) = - \\frac { 1 } { p ^ { n + 1 } } \\right ] . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} W _ 1 ( \\mu _ { 1 } , \\mu _ { 2 } ) = \\inf _ { \\pi } \\sum _ { y \\in V } \\sum _ { x \\in V } d ( x , y ) \\pi ( x , y ) , \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} f ( x ) = g ( x ) ^ T G g ( x ) + c ^ T h ( x ) \\ , , \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} n _ K = \\# \\left \\{ ( i , j ) \\in C _ F ( r ) \\times C _ F ( r ) \\colon D ( i , j ) \\cap K \\neq \\emptyset \\right \\} . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} f ^ { - 1 } ( f ( \\mathbb { C } ^ { m } ) \\cap \\mathcal { D } ) = \\coprod _ { i \\in I } U _ { i } \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} T = \\frac { \\rho + \\sigma _ w ^ 2 } { 2 n } ( \\chi _ { 2 n _ p } ^ 2 + \\chi _ { 2 n _ d } ^ 2 ) = \\frac { \\rho + \\sigma _ w ^ 2 } { 2 n } \\chi _ { 2 n } ^ 2 , \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\tilde { a } _ { i j } ( p ) = a _ { i j } ( \\delta _ { \\sigma ^ { \\nu } } p ) , \\ , \\ , \\ , \\tilde { f } _ { i } ( p ) = f _ { i } ( \\delta _ { \\sigma ^ { \\nu } } p ) , \\ , \\ , \\ , \\tilde { g } ( p ) = g ( \\delta _ { \\sigma ^ { \\nu } } p ) , \\ , \\ , \\ , \\tilde { L } _ { \\nu } ( p ) = L _ { \\nu } ( \\delta _ { \\sigma ^ { \\nu } } p ) . \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\left \\langle f , g \\right \\rangle _ { \\mu } : = \\int _ { D } f ( \\zeta ) \\overline { g ( \\zeta ) } \\mu ( \\zeta ) d A ( \\zeta ) . \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\Gamma & = 4 \\sigma ^ 4 [ ( - a + \\eta ) ^ 2 - 1 ] s ^ 2 + \\\\ & 4 \\sigma ^ 2 [ ( - a + \\eta ) ( a ^ 2 - 1 ) + 2 \\eta ] s + ( a ^ 2 - 1 ) ^ 2 . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ d \\frac { ( x _ i - y _ i ) ^ 2 } { \\abs { x - y } ^ { 2 } } = \\sum _ { i = 1 } ^ d \\frac { ( x _ i - y _ i ) } { \\abs { x - y } ^ { d + 1 } } ( x _ i - y _ i ) \\abs { x - y } ^ { d - 1 } , \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell j ^ { \\alpha / b } } { d \\sum _ { j = 1 } ^ { \\ell + 1 } j ^ { \\alpha / b } - ( \\ell + 1 ) ^ { \\alpha / b } } = \\frac 1 d . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\big ( j ( \\lambda , \\lambda ' ) \\cap j ( \\lambda ' , \\lambda '' ) \\big ) \\cap j ( \\lambda , \\lambda '' ) = j ( \\lambda , \\lambda ' ) \\cap j ( \\lambda ' , \\lambda '' ) . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} - h ' ( s ) & = K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u - 1 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 1 } ) + O ( s ^ { - 1 } ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u - 1 } , \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} \\norm { \\exp \\left ( \\gamma \\sum _ { m = 1 } ^ { j - 1 } \\phi _ \\delta ( \\norm { A _ m } _ { B ( \\mathcal { K } ) } ) \\right ) G _ { j k } ( \\lambda ) \\exp \\left ( - \\gamma \\sum _ { m = 1 } ^ { k - 1 } \\phi _ \\delta ( \\norm { A _ m } _ { B ( \\mathcal { K } ) } ) \\right ) } _ { B ( \\mathcal { K } ) } \\le C \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\frac { \\prod _ { \\substack { 1 \\leq i \\leq l \\\\ 1 \\leq j \\leq s } } L ( s + \\frac { 1 } { 2 } , \\zeta _ i \\times \\eta _ j ) \\prod _ { \\substack { 1 \\leq i \\leq s \\\\ 1 \\leq j \\leq k } } L ( s + 1 , \\eta _ i \\times \\xi _ j ) L ( s , \\eta _ i \\times \\xi _ j ) } { \\prod _ { \\substack { 1 \\leq i \\leq l \\\\ 1 \\leq j \\leq k } } L ( s + \\frac { 1 } { 2 } , \\zeta _ i \\times \\xi _ j ) \\prod _ { j = 1 } ^ { k } L ( s + 1 , \\xi _ j \\times \\xi _ j ) L ( s , \\xi _ j , \\rho ) } \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} \\mathfrak d ( h ) ~ = ~ \\min _ { a \\in [ - M , M - h ] } \\left ( \\inf _ { g \\in \\mathcal { A } _ { [ a , a + h ] } } \\| f - g \\| _ { { \\bf L } ^ { \\infty } ( [ a , a + h ] ) } \\right ) \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\chi ( \\Gamma _ n , M ) : = \\prod _ i | H _ i ( \\Gamma _ n , M ) | ^ { ( - 1 ) ^ i } \\in \\Q ^ { \\times } \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\ln ( 1 + x ) \\cdot \\sum _ { j = 0 } ^ { m } b _ { j } x ^ { j } \\geq \\sum _ { j = 0 } ^ { n } a _ { j } x ^ { j } \\geq \\frac { f ( x ) } { \\sqrt { x + 1 } } \\sum _ { j = 0 } ^ { m } b _ { j } x ^ { j } , x \\in ( - 1 , 0 ] . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} Q \\partial _ Q \\widetilde { f } ( Q ) = \\frac { 1 } { Q } f ( Q ) \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } = \\prod _ { j = 1 } ^ r \\phi _ j ^ { - \\ell _ q ( Q _ j ) } \\left . \\right . _ \\circ \\left . \\right . J ^ { K \\textnormal { t h } } \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\Phi ^ { ( 2 ) } _ x ( x , t , k ) = ( Q ( x , t ) - \\i k \\sigma _ 3 ) \\Phi ^ { ( 2 ) } ( x , t , k ) , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\tau ( \\beta ) \\beta = U D U ^ { - 1 } \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} \\max _ { \\overline { \\Omega } } u _ { \\epsilon } & \\ge u _ { \\epsilon } ( x ) = \\max _ { \\overline { \\Omega } } u + \\epsilon \\phi ( x ) \\\\ & \\ge \\max _ { \\overline { \\Omega } } u + \\epsilon \\inf _ { \\overline { \\Omega } } \\phi \\\\ & > \\max _ { \\partial \\Omega } u + \\epsilon \\max _ { \\Omega } \\phi \\\\ & \\ge \\max _ { \\partial \\Omega } u _ { \\epsilon } , \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} Z _ { t o t } ( n + 1 ) : = \\bigcap _ { j = 1 } ^ { n + 1 } \\bigcap _ { i = 1 } ^ { N _ W } Z _ j ( i ) , \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\AA ' _ { a b } = r ^ 3 \\AA _ { a b } ^ { ' ( 3 ) } + O ( r ^ 4 ) \\\\ \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi + R \\varphi = & - \\frac { n - 1 } { n } \\tau ^ { 2 a } \\varphi ^ { N - 1 } + \\left ( | \\sigma + L W | ^ { 2 } + k ^ { 2 } \\right ) \\varphi ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ { * } L W = & \\frac { n - 1 } { n } \\varphi ^ { N } d \\tau ^ a . \\endaligned \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} c _ 1 ( E ) = - \\frac { \\sqrt { - 1 } } { 2 \\pi } [ \\imath _ { \\kappa } \\gamma ] . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } - \\Delta u = u ^ p & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { o n } \\partial \\Omega \\\\ u > 0 & \\mbox { i n } \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} x _ { m } = \\frac { \\psi ( \\tau x ' , \\tau ^ { 2 } y _ { 2 } , \\cdots , \\tau ^ { k } y _ { k } ) } { \\tau } ~ ~ ~ ~ ~ \\overline { x } _ { m } = \\frac { \\psi ( \\tau \\overline { x } ' , \\tau ^ { 2 } \\overline { y } _ { 2 } , \\cdots \\tau ^ { k } \\overline { y } _ { k } ) } { \\tau } . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} \\chi ( \\Gamma _ n , M ) = \\prod _ i | H _ i ( \\Gamma _ n , M ) | ^ { ( - 1 ) ^ i } \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} \\begin{cases} w _ { t t } - ( 1 + t ) ^ { 2 \\ell } \\Delta w = g ( t , x ) , & x \\in \\mathbb { R } ^ n , t > 0 , \\\\ w ( 0 , x ) = w _ 0 ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ w _ t ( 0 , x ) = w _ 1 ( x ) , & x \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} B ( z ; \\eta ) & = H ( z ) \\ , \\mathsf b ( z ; \\eta ) \\ , E ^ { \\perp } ( - z ^ { - 1 } ) \\\\ C ( z ; \\eta ) & = H ( z ) \\ , \\mathsf c ( z ; \\eta ) \\ , E ^ { \\perp } ( - z ^ { - 1 } ) \\\\ \\bar { B } ( z ; \\eta ) & = V _ + ( z ) \\ , H ( z ; \\eta ) \\ , E ^ { \\perp } ( - z ^ { - 1 } ) \\\\ \\bar { C } ( z ; \\eta ) & = H ( z ) \\ , \\bar { \\mathsf c } ( z ; \\eta ) \\ , E ^ { \\perp } ( - z ^ { - 1 } ) \\ , V _ - ( z ) . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial X _ f } { \\partial z } & = \\frac { 1 } { 2 } f ( x _ 1 ) f ' _ 2 ( x _ 2 ) + \\frac i 2 f ' ( x _ 1 ) f _ 2 ( x _ 2 ) , \\\\ \\frac { \\partial X _ f } { \\partial \\bar { z } } & = - \\frac { 1 } { 2 } f ( x _ 1 ) f ' _ 2 ( x _ 2 ) + \\frac i 2 f ' ( x _ 1 ) f _ 2 ( x _ 2 ) , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} G _ \\tau ( \\phi _ 0 \\star _ \\tau \\phi _ j , \\phi _ k ) = \\partial _ { t _ 0 } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } ( \\tau , Q ) = G _ \\tau ( \\phi _ j , \\phi _ k ) \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\left | { \\hat \\eta _ { K , M , j } ^ { ( 2 ) } - \\eta _ { K , M } ^ { ( 2 ) } } \\right | = 0 , . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\left ( \\sum ^ 3 _ { i = 0 } b _ i \\right ) ^ 2 - 2 b _ 0 b _ 1 \\geq ( 1 - 4 \\varepsilon ) ( b _ 0 ^ 2 + b _ 1 ^ 2 ) - \\frac { 1 } { \\varepsilon } b _ 2 ^ 2 - \\frac { 1 } { \\varepsilon } b _ 3 ^ 2 . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k \\lambda _ i Q ( x _ i ) \\right \\| \\leq C \\cdot \\left \\| \\sum _ { i = 1 } ^ k \\lambda _ i P ( x _ i ) \\right \\| \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} L = \\theta ( X _ f ) - f \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( H _ { i j k l } ( t _ n ) e _ { k l } ( u _ n ) ) _ { , j } + f _ i = 0 \\Omega , \\\\ \\\\ H _ { i j k l } ( t _ n ) e _ { k l } ( u _ n ) n _ j - \\hat { f } _ i = 0 , \\Gamma _ t , \\ ; \\forall i \\in \\{ 1 , 2 , 3 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} u _ 1 \\overline { u _ 2 } = - 1 . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\mathbb { E } _ { t } [ \\langle \\zeta _ { t + 1 } , h \\rangle f ( \\langle \\zeta _ { t + 1 } , h \\rangle ) ] = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } h _ { i } \\{ \\mathbb { E } _ { t } [ X _ { i , t + 1 } f ( \\langle \\zeta _ { t + 1 } , h \\rangle ) ] - p _ { i , t + 1 } \\mathbb { E } _ { t } f ( \\langle \\zeta _ { t + 1 } , h \\rangle ) \\} , \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ 1 } \\mathcal { K } ^ { \\Sigma _ 1 , \\infty } d \\sigma _ { \\Sigma _ 1 } + \\sum _ { i = 1 } ^ n \\int _ { \\gamma _ i } k ^ { \\infty , s } _ { \\gamma _ i , \\Sigma _ 1 } d { s } = 0 . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\log N ( \\alpha ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } \\sum _ { \\alpha } \\frac { \\nu ( \\alpha ) \\log N ( \\alpha ) } { N ( \\alpha ) ^ s } X ^ s \\mathfrak { w } ( s ) d s = - \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } \\frac { \\partial I ( s ) } { \\partial s } X ^ s \\mathfrak { w } ( s ) d s . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} v = 0 \\quad \\mbox { o n } \\quad \\Gamma _ f \\times ( 0 , T ) . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 1 \\\\ 1 \\\\ 1 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c c } - 2 & - 0 . 3 5 & - 0 . 3 5 \\\\ - 0 . 3 5 & - 4 & - 0 . 3 5 \\\\ - 0 . 3 5 & - 0 . 3 5 & - 6 \\end{array} \\right ] , d = \\left [ \\begin{array} { c } 0 . 2 \\\\ 0 . 2 \\\\ 0 . 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\mathfrak { C } = \\begin{cases} - \\chi ( n r _ 2 \\overline { \\beta } ) & \\chi , \\\\ \\chi ( \\overline { n } r _ 2 \\beta ) ( M - 1 ) & \\chi . \\end{cases} \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\lambda ( \\delta ) = \\frac { 2 \\delta } { 1 + \\delta } . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} q _ 0 T _ d = T _ { d - 1 } \\cdots T _ 1 ( X _ 1 T _ 0 ^ { - 1 } ) T _ 1 ^ { - 1 } \\cdots T _ { d - 1 } ^ { - 1 } . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\mathbb { P } \\Bigl \\{ \\zeta ^ { \\epsilon } ( t + \\triangle ) = m \\ , \\bigl \\vert \\ , \\zeta ^ { \\epsilon } ( t ) = k , X _ t ^ { \\epsilon , \\hat { v } } = x \\Bigr \\} = \\gamma _ { k m } ( x ) \\triangle + o ( \\triangle ) \\ , \\ , \\ , \\ , \\ , \\ , \\triangle \\downarrow 0 , \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} D ( \\Pi _ { i } \\nu _ { i } , q ) = \\min \\{ 1 , \\frac { \\log \\Vert p _ { i } \\Vert _ { q } ^ { q } } { ( q - 1 ) \\log r _ { i } ^ { m } } \\} \\ge \\min \\{ 1 , \\frac { h _ { i } - \\delta } { - \\log r _ { i } } \\} \\ : . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\mathbf { 1 } _ { [ 0 , t ] } ( s ) d h ( s ) = h ( t ) = \\int _ { 0 } ^ { t } K ( t , s ) \\varphi ( s ) d s = \\int _ { 0 } ^ { 1 } \\left ( \\mathcal { K } ^ { * } \\mathbf { 1 } _ { [ 0 , t ] } \\right ) ( s ) \\varphi ( s ) d s . \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\tilde { u } ( x , t , \\o ) = \\sum _ { j = 1 } ^ m u ( x _ j , t , \\o ) \\phi _ j ( x ) , \\tilde { f } ( x ) = \\sum _ { j = 1 } ^ m f ( x _ j ) \\phi _ j ( x ) . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} v = \\mathcal V ( \\rho ) + \\mathcal W ( \\rho ) . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} \\langle D _ 1 J _ 1 ( f ; v ^ 1 + s w ^ 1 , v ^ 2 ) , w ^ 2 \\rangle & = \\alpha _ 1 \\iint _ { \\mathcal { O } _ { 1 , d } \\times ( 0 , T ) } ( y ^ s - y _ { 1 , d } ) z ^ s d x d t \\\\ & + \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } ( v ^ 1 + s w ^ 1 ) w ^ 2 d x d t \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } ( q , Q ) = P ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\left ( q P ^ { - 1 } ; q \\right ) _ d ^ { N + 1 } } \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} u _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon } + \\frac { S _ { 0 , \\varepsilon } } { \\gamma _ \\varepsilon ^ 3 } + \\frac { S _ { 1 , \\varepsilon } } { \\gamma _ \\varepsilon ^ 5 } + ( A ( \\gamma _ \\varepsilon ) - 2 \\xi _ \\varepsilon ) \\frac { S _ { 2 , \\varepsilon } } { \\gamma _ \\varepsilon } + o \\left ( t _ \\varepsilon \\frac { \\zeta _ \\varepsilon } { \\gamma _ \\varepsilon } \\right ) \\ , , \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} g ( \\delta ) = 2 Q \\delta - L _ 0 T \\left ( \\cosh \\delta - 1 \\right ) / 2 . \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\sqrt { - 1 } \\mathfrak { t } = \\mathfrak { n } _ { + } ^ { \\perp } . \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i , j } = \\Gamma ( k ) _ { i , 1 } + \\Gamma ( k ) _ { 1 , j } . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\tilde { N } \\left ( \\frac { a } { x _ K ^ n } \\right ) = \\frac { N ( a ) } { N ( x _ K ) ^ n } . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} k ^ { ( 1 ) } = h _ 0 ^ { ( 1 ) } - 1 . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} ( F , F _ 1 , \\dots , F _ m ) = \\bigg ( \\Big ( \\sum _ j ( f ^ j ) ^ { s } \\Big ) ^ \\frac 1 { s } , \\Big ( \\sum _ j ( f _ 1 ^ j ) ^ { s _ 1 } \\Big ) ^ \\frac 1 { s _ 1 } , \\dots , \\Big ( \\sum _ j ( f _ m ^ j ) ^ { s _ m } \\Big ) ^ \\frac 1 { s _ m } \\bigg ) \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ n ( t , x ) = I + \\nabla u _ n ( t , \\Phi _ n ^ { - 1 } ( t , x ) ) , \\tilde { \\sigma } ( t , x ) = I + \\nabla u ( t , \\Phi ^ { - 1 } ( t , x ) ) , \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} C ( l , 1 ) = & \\frac { - 1 } { n ! } \\sum _ { 0 \\le k \\le n } ( - 1 ) ^ { 1 + k + n } \\ , k \\ , { n \\choose k } \\sum _ { 1 \\le q \\le l } { k \\choose q - 1 } { n - k \\choose n - q } . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} & - f ( a e _ x ) g ( a x e _ y ) + f ( a e _ y ) ( g ( a e _ x ) - g ( a e _ y ) + g ( a x e _ y ) ) \\\\ & \\qquad - g ( a e _ x ) ( f ( a e _ x y ) - f ( a e _ x ) + f ( a e _ y ) ) + g ( a e _ y ) f ( a e _ { x } y ) \\\\ & = ( f ( a e _ y ) - f ( a e _ x ) ) g ( a x e _ y ) - h ( a e _ y ) + h ( a e _ x ) + ( g ( a e _ y ) - g ( a e _ x ) ) f ( a e _ { x } y ) \\\\ & = ( f ( a x e _ y ) - f ( a e _ x y ) ) g ( a x e _ y ) - h ( a e _ y ) + h ( a e _ x ) + ( g ( a x e _ y ) - g ( a e _ x y ) ) f ( a e _ { x } y ) \\\\ & = h ( a x e _ y ) - h ( a e _ y ) + h ( a e _ x ) - h ( a e _ { x } y ) = ( d ^ * h ) ( a e _ x e _ y ) , \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} r _ { \\textrm { F } } = { R } / { \\log \\left ( 1 + \\frac { P _ { \\textrm { B } } \\lambda _ { \\textrm { B F } } } { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } \\right ) } . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\iint _ Q { \\rho } _ 5 ^ 2 | y _ { t t } | ^ 2 d x d t & + \\underset { [ 0 , T ] } { } \\left ( { \\rho } _ 5 ^ 2 ( t ) \\| y _ { x t } ( t ) \\| ^ 2 \\right ) \\leq C \\left ( \\| y _ { x t } ( 0 ) \\| ^ 2 + \\iint _ Q { \\rho } _ 3 ^ 2 | G _ { t } | ^ 2 d x d t \\right . \\\\ & + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } { \\rho } _ 1 ^ 2 | f | ^ 2 d x d t + \\iint _ Q { \\rho } _ 0 ^ 2 | y | ^ 2 d x d t + \\left . \\sum _ { i = 1 } ^ 2 \\iint _ Q { \\rho } _ 0 ^ 2 | p ^ i | ^ 2 d x d t \\right ) . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\mathfrak p \\leftrightarrow \\mathfrak q = \\begin{array} { c } \\mathfrak q \\\\ \\updownarrow \\\\ \\mathfrak p \\end{array} = \\mathfrak q \\leftrightarrow \\mathfrak p . \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\Sigma _ { \\theta } ^ 2 & = \\sup _ { 0 \\le a \\le 1 } \\sigma _ { \\theta } ^ 2 ( a ) = \\sup _ { 0 \\le a \\le 1 } \\biggl ( V ( a , a ) + 2 \\sum _ { k = 1 } ^ \\infty \\frac { V ( \\langle p ^ k a \\rangle , \\langle q ^ k a \\rangle ) } { p ^ k q ^ k } \\biggr ) , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} D ( P _ X | | P _ Y ) = \\frac { \\sigma _ X ^ 2 + ( \\mu _ X - \\mu _ Y ) ^ 2 } { 2 \\sigma _ Y ^ 2 } - \\frac { 1 } { 2 } \\log \\frac { \\sigma _ X ^ 2 } { \\sigma _ Y ^ 2 } - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} c _ i \\ne c _ k , \\ \\textrm { i f } \\ a _ { i k } = 1 , \\ \\forall k \\in \\{ 1 , 2 , \\ldots , i - 1 \\} \\ . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} A _ i = \\int _ { \\mathbb { R } ^ 2 } \\Delta S _ i d x \\ , . \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} \\begin{cases} - w '' = M \\left ( A - w \\right ) w & \\textup { i n } \\left ( - R , R \\right ) \\\\ w \\left ( \\pm R \\right ) = \\nu A . \\end{cases} \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n p _ n ( 1 - \\pi _ n ( c n ) ) } = + \\infty . \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} \\gamma & = ( \\{ 2 a + 1 , \\overline { 2 a + 1 } \\} , \\ldots , \\{ 2 a + b , \\overline { 2 a + b } \\} ) \\\\ \\delta & = ( [ 2 a + b + 1 , \\overline { 2 a + b + 1 } ] , \\ldots , [ n , \\overline { n } ] ) , \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 ^ + } \\| ( F _ f * G _ \\alpha ) _ { | [ 0 , 1 ] } - f \\| _ { L ^ 1 ( [ 0 , 1 ] ; \\mathbb { R } ) } = 0 . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial ^ 2 } { \\partial t ^ 2 } - \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\mu } { 1 + t } \\frac { \\partial } { \\partial t } + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } \\right ) E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) = 0 . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\varPhi _ n ^ { ( k ) } ( z ) : = \\ _ 3 \\phi _ 2 \\left ( \\begin{matrix} q ^ { - n } , q ^ { \\gamma + k } , z ^ { - 1 } \\\\ 0 , q ^ \\gamma \\end{matrix} ; q , { q z \\over a } \\right ) . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} \\P \\Biggl [ 1 + \\sum _ { i = 1 } ^ k \\tau _ i > C k \\Biggr ] & \\leq \\exp \\biggl [ - k \\biggl ( \\frac { C ( 1 - e ^ { - \\beta } ) } { 2 } - 1 \\biggr ) \\biggr ] \\leq \\exp \\biggl ( - \\frac { C ( 1 - e ^ { - \\beta } ) } { 2 } + 1 \\biggr ) . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\max \\{ l _ \\varepsilon ( y ) - v ( y ) : \\ , y \\in \\widetilde { \\Omega } _ \\varepsilon \\} = \\max \\{ l _ \\varepsilon ( y ) - v ( y ) : \\ , y \\in \\Omega _ \\varepsilon \\} = l _ \\varepsilon ( o ) - v ( o ) = \\varepsilon . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\bigg \\| \\Big ( \\sum _ j ( f ^ j ) ^ s \\Big ) ^ \\frac 1 s \\bigg \\| _ { L ^ { q } ( v ) } \\le C ( [ \\vec v ] _ { A _ { \\vec q , \\vec r } } ) \\prod _ { i = 1 } ^ m \\bigg \\| \\Big ( \\sum _ j ( f _ i ^ j ) ^ { s _ i } \\Big ) ^ \\frac 1 { s _ i } \\bigg \\| _ { L ^ { q _ i } ( v _ i ) } \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} J ( \\langle x _ 1 , \\dots , x _ n \\rangle ) < _ { D ^ G ( X ) } D ^ { \\mu ^ G } _ X ( x ) \\qquad . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } ( E ( L _ 1 , \\ell _ \\nu , \\ell _ \\mu ) + E ( L _ 1 , \\ell _ \\nu , \\ell _ { \\mu ' } ) ) = 1 - \\frac { \\sinh \\frac { \\ell _ \\nu } { 2 } } { \\cosh \\frac { \\ell _ \\nu } { 2 } + 1 } = 2 ( e ^ { \\frac { 1 } { 2 } \\ell _ \\nu } + 1 ) ^ { - 1 } . \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} F ( u ) & = f ( g _ 1 ( u ) , \\dots , g _ n ( u ) ) = \\\\ & = \\sum _ { w \\in \\left ( \\mathbb { F } _ 2 \\right ) ^ n } \\lambda _ f ( w ) g _ 1 ( u ) ^ { w _ 1 } \\cdots g _ n ( u ) ^ { w _ n } \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} F _ i ^ \\ell \\overset { d e f } { = } f _ i - \\sum _ { j = 1 } ^ m a _ { i j } X _ j L _ { p _ \\ell } . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\Gamma _ { [ T , L , M , f ] } ~ = ~ 2 ^ { 2 p _ f + 1 } \\left [ 3 \\log _ 2 5 + \\log _ 2 ( 5 e ) \\right ] \\tilde { \\gamma } _ { [ L , M ] } \\left ( L + T \\cdot f ' _ M \\right ) ^ { p _ f } \\left ( 1 + { 1 \\over T } \\right ) . \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} S _ 0 ( f ) & = \\frac { P _ { \\rm R } } { B } , \\\\ G ( \\cos \\theta _ { \\rm c } \\cdot f ) & = \\frac { 1 } { | \\cos \\theta _ { \\rm c } | \\log \\sqrt { \\frac { f _ { \\rm m a x } } { f _ { \\rm m i n } } } } , \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\sigma ( M ) \\cap \\sigma ( M _ { \\overline S \\overline S } ) = \\varnothing , \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\widetilde { u } | _ { B _ 1 } = u | _ { B _ 1 } - L , \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\chi ( \\Gamma _ n , X ) = \\prod _ { i \\in I } | X ^ { \\Gamma _ n } _ { f _ i } | \\ ; . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} d = 3 \\begin{cases} u \\approx n ^ { \\prime } ( p + ( k - 1 ) q ) \\\\ v \\approx n ^ { \\prime } ( 2 q + ( k - 2 ) r ) \\end{cases} \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} f _ { T } : = f | _ { T } & \\colon T \\longrightarrow T \\\\ g & \\colon A \\longrightarrow A \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} x \\cdot ( y \\otimes z ) = a d _ x ( y ) \\otimes z + y \\otimes a d _ x ( z ) , \\\\ x \\cdot ( y \\otimes T ) = a d _ x ( y ) \\otimes T + y \\otimes [ - a d _ { x } ^ * , T ] \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\nabla ^ N _ { \\partial _ a } L & = - l _ a ^ c \\partial _ c - \\eta _ a L \\\\ \\nabla ^ N _ { \\partial _ a } \\partial _ b & = \\gamma _ { a b } ^ c \\partial _ c - l _ { a b } \\underline L - n _ { a b } L \\\\ \\nabla ^ N _ { \\partial _ a } \\underline L & = - n _ a ^ c \\partial _ c + \\eta _ a \\underline L , \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} & \\frac { d } { d t } X ( t ) = C _ 1 ( 1 + X ( t ) ^ { q _ 1 } ) \\\\ & X ( 0 ) = C _ 2 , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} u ( x , 0 ) = \\sin ( 0 . 5 \\ , \\pi \\ , \\ , x ) + \\sin ( 0 . 8 5 \\ , \\pi \\ , x ) + 0 . 2 \\ , v , v = \\mathcal { N } ( 0 , 1 ) . \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} - \\Delta w _ j = \\left ( \\Theta _ j - \\Delta z _ j \\right ) \\phi ^ { 2 \\alpha _ j } = \\left ( \\Theta _ j - 2 | D ^ 2 u _ j | ^ 2 - 2 \\sum ^ 3 _ { i = 0 } \\langle \\nabla a _ { i , j } , \\nabla u _ j \\rangle \\right ) \\phi ^ { 2 \\alpha _ j } . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\| D _ y ^ { 1 / 2 } [ \\chi ( h y ) \\cdot w ] \\| _ { L _ y ^ 2 } \\lesssim \\begin{aligned} [ t ] & \\| D _ y ^ { 1 / 2 } [ \\chi ( h y ) ] \\| _ { L _ y ^ \\infty } \\| w \\| _ { L _ y ^ 2 } + \\| \\chi ( h y ) \\| _ { L _ y ^ \\infty } \\| D _ y ^ { 1 / 2 } w \\| _ { L _ y ^ 2 } \\\\ & + \\| D _ y ^ { 1 / 2 } [ \\chi ( h y ) ] \\| _ { L _ y ^ 2 } \\| D _ y ^ { 1 / 2 } w \\| _ { L _ y ^ 2 } \\end{aligned} \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\sigma ( a , b , x , y ) & = ( \\omega a , \\omega ^ { - 1 } b , x , y ) , \\\\ \\tau ( a , b , x , y ) & = ( b , a , - b ^ 3 x + ( 1 + a b + a ^ 2 b ^ 2 ) y , ( 1 - a b ) x + a ^ 3 y ) . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} E _ { P _ { \\alpha } } \\left [ b _ { a } ^ { 2 } \\left ( \\mathbf { O } ; P _ { \\alpha } \\right ) \\right ] = E _ { P _ { \\alpha } } \\left [ \\left \\{ O _ { 1 } + O _ { 2 } + \\alpha O _ { 1 } O _ { 2 } \\right \\} ^ { 2 } \\right ] = E _ { P _ { \\alpha } } \\left [ O _ { 1 } ^ { 2 } + O _ { 2 } ^ { 2 } + \\alpha ^ { 2 } ( O _ { 1 } O _ { 2 } ) ^ { 2 } \\right ] = 2 + \\alpha ^ { 2 } . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n F ( \\mu _ { n , i } ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } F \\left ( g ( w ) \\right ) ~ d w , \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} i ^ \\ast \\circ \\nu ( q _ 4 ) = \\sigma ^ \\ast ( q _ 4 ) = \\mu _ 3 , \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\max \\left \\{ { { X _ M } , { Y _ M } } \\right \\} = \\max \\left \\{ { \\mathop { \\lim } \\limits _ { M \\to \\infty } { X _ M } , \\mathop { \\lim } \\limits _ { M \\to \\infty } { Y _ M } } \\right \\} . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} H ^ { 0 } ( A ^ \\bullet ( q = 0 ) ) \\rightarrow H ^ { 0 } ( Q ^ \\bullet ) \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\left | { \\hat \\eta _ { K , M , j } ^ { ( 2 ) } - \\eta _ { K , M } ^ { ( 2 ) } } \\right | = 0 \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} \\varnothing & \\neq \\big ( W \\smallsetminus \\{ x \\} \\big ) \\cap K ^ { ( \\alpha ) } \\\\ & = \\big ( W \\smallsetminus \\{ x \\} \\big ) \\cap K ^ { ( \\alpha ) } \\cap F \\\\ & = \\big ( W \\smallsetminus \\{ x \\} \\big ) \\cap ( K \\cap F ) ^ { ( \\alpha ) } \\\\ & \\subseteq \\big ( V \\smallsetminus \\{ x \\} \\big ) \\cap ( K \\cap F ) ^ { ( \\alpha ) } . \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} ( f ^ { \\circ k } ) ' ( X ) = f ' ( X ) \\prod _ { 0 < i < k } f ' ( f ^ { \\circ i } ( X ) ) \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} a _ { i } ^ { J _ { 0 } + 1 } = \\frac { f _ { i } ^ { J _ { 0 } + 1 } } { \\| f _ { i } ^ { J _ { 0 } + 1 } \\| _ { L ^ { \\infty } } } | B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } + 1 } r ) | ^ { - 1 / p } , \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} | \\mathbb { Z } ^ d / L _ { \\Gamma } | = | \\mathbb { Z } ^ d / T _ { \\Gamma } | \\leq | P _ { \\Gamma } | | \\mathbb { Z } ^ d / \\Gamma | \\leq 2 ^ { d } d ! x = O _ d ( x ) . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} W ( \\mathcal N ) = \\bigcap _ { A \\in \\mathcal N _ { n - 2 } } \\ker _ { \\mathbb Z } \\phi ^ A \\cap \\mathbb N _ { 0 } ^ m \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} \\limsup _ { v \\rightarrow \\infty } \\lambda _ 0 ( v , p ) \\leq \\lim _ { v \\rightarrow \\infty } \\hat \\lambda ( v , p ) \\ ; = \\ ; \\bar { \\lambda } / p , \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} \\lambda _ l ^ k = \\lambda _ { l } \\lambda _ { l + 1 } \\cdots \\lambda _ { l + k - 1 } , l \\in \\mathbb { Z } , \\ , k \\in \\{ m , n \\} . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } \\left ( 2 ^ { R } - 1 \\right ) ( 1 + k ) } { \\lambda _ { \\textrm { B F } } \\left ( k - \\left ( 2 ^ { \\textrm { R } } - 1 \\right ) \\right ) } \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} u _ 1 = - e ^ { - 3 } \\approx - 0 . 0 4 9 8 , u _ n = 0 \\quad n \\geq 2 . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} | \\langle \\C _ { \\mu } \\varphi _ { Q ' } , \\varphi _ { Q '' } \\rangle | \\leq | | \\C _ { \\mu } \\varphi _ { Q ' } | | _ { L ^ 2 ( \\mu ) } | | \\varphi _ { Q '' } | | _ { L ^ 2 ( \\mu ) } = | | \\C _ { \\mu } \\varphi _ { Q ' } | | _ { L ^ 2 ( \\mu ) } . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} v _ 1 = ( 2 , \\ldots , 2 , 1 ) , v _ 2 = ( 2 , \\ldots , 2 , 1 , 1 ) , \\ldots , v _ n = ( 1 , \\dots , 1 , 1 ) . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) = f ( x ) + \\frac { t ^ 2 } { 6 } \\ , \\sum _ { i = 1 } ^ n f _ { i i } ( x ) + \\left ( \\frac { 1 } { 1 2 0 } \\sum _ { i = 1 } ^ n f _ { i i i i } + \\frac { 1 } { 3 6 } \\ , \\sum _ { \\substack { i , j = 1 \\\\ j > i } } ^ n f _ { i i j j } \\right ) t ^ 4 \\ , , \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\left \\Vert g _ k \\right \\Vert _ p \\left \\Vert h _ k \\right \\Vert _ q < \\infty . \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m + 1 } } \\cdot L : L ] = [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m + 1 } } : k ] . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} K ( z , w ) = \\frac { \\Im ( z - w ) } { | z - w | ^ 2 } \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} S _ { k } = \\Big \\lvert \\sum _ { n = 1 } ^ { L } e ^ { 2 \\pi i \\langle T ^ n x , k \\rangle } \\Big \\lvert , 0 < | k | < \\frac { 1 } { \\epsilon } , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\mathbf { y } _ w = [ y _ w ^ 1 , . . . , y _ w ^ { n _ p } , \\underbrace { y _ w ^ { n _ p + 1 } . . . , y _ w ^ { n } } _ { n _ d } ] , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\Delta _ H \\pi = \\mathrm { d i v } _ H F \\quad G , \\pi | _ { \\partial G } : , \\int _ G \\pi \\ , d x ' = 0 , \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\Delta & = 4 \\sigma ^ 4 [ ( a + \\eta ) ^ 2 - 1 ] s ^ 2 + \\\\ & 4 \\sigma ^ 2 [ ( a + \\eta ) ( a ^ 2 - 1 ) + 2 \\eta ] s + ( a ^ 2 - 1 ) ^ 2 . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\mathbb { P } ( N _ l = k ) = B ( k ; n , p _ { l } ) : = { n \\choose k } p _ l ^ { k } ( 1 - p _ l ) ^ { n - k } , \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { \\| F [ u + h ] - F [ u ] - F ^ { B D } [ u ; h ] \\| } { \\| h \\| } = 0 , \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} ( - x ; q ) _ n = ( 1 + x ) ( 1 + x q ) \\cdots ( 1 + x q ^ { n - 1 } ) . \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} g ( t ) : = \\left \\{ \\begin{aligned} & { \\tilde { g } ( t ) } & & { 0 \\leq t \\leq \\tau } \\\\ & { h \\left ( t - \\tau \\right ) } & & { \\tau < t \\leq \\tau + S _ k } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} p * f ( t , x ) : = \\int _ 0 ^ \\infty \\int _ { \\R ^ d } p ( s , y ) f ( t - s , x - y ) d y d s \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\mathcal { H } ^ 2 = \\left \\{ f ( s ) = \\sum _ { n = 1 } ^ \\infty a _ n n ^ { - s } : \\ , \\Vert f \\Vert ^ 2 = \\sum _ { n = 1 } ^ \\infty | a _ n | ^ 2 < \\infty \\right \\} . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} A _ n = { q ^ { n + \\gamma - 1 } ( 1 - q ^ n ) ( 1 - a q ^ n ) ( 1 - a b q ^ { n - \\gamma + 1 } ) \\over ( 1 - q ^ { n + \\gamma } ) ( 1 - a b q ^ { 2 n } ) ( 1 - a b q ^ { 2 n + 1 } ) } . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = \\sup \\{ H _ { a l g } ( \\phi , U ) : U \\in \\mathcal B ( G ) \\} . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} \\frac { f _ { 2 } ^ { \\prime \\prime } } { f _ { 2 } ^ { \\prime } } \\left ( \\mu _ { 1 } + \\mu _ { 2 } \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } \\right ) = \\left ( \\mu _ { 3 } \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } + \\mu _ { 4 } \\left ( \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } \\right ) ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} a _ { n + 1 } = \\frac { c ( n + 1 ) } { 2 ( n + \\alpha + 1 ) } \\sqrt { \\frac { h _ { n + 1 } } { h _ { n } } } \\ , b _ n = \\frac { c ( n + 2 \\alpha + 1 ) } { 2 ( n + \\alpha + 1 ) } \\sqrt { \\frac { h _ { n - 1 } } { h _ { n } } } \\ . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} f ( y ) = t ( n - 2 ) / s _ 1 \\mbox { f o r a l l } y \\in U _ x ^ 1 . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\left \\langle T _ 1 ^ { ( \\alpha _ 1 ) } , T _ 2 ^ { ( \\alpha _ 2 ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , \\alpha _ 1 + \\alpha _ 2 , d [ l ] } = d ^ { \\alpha _ 1 } \\left \\langle T _ 2 ^ { ( \\alpha _ 2 ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , \\alpha _ 2 , d [ l ] } \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} \\sigma ^ k & \\in \\bigoplus _ { n \\in I } \\langle x \\mapsto \\sin ( n x ) \\rangle , k = 1 , \\ldots , m . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\partial _ \\xi \\textbf { g } ( s _ 1 , s _ 2 ) = \\textbf { g } ( \\nabla _ \\xi s _ 1 , s _ 2 ) + \\textbf { g } ( s _ 1 , \\nabla _ \\xi s _ 2 ) \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} 3 | | \\vec a | | ^ 2 + b _ 1 ^ 2 + b _ 2 ^ 2 + \\cdots + b _ { 2 n } ^ 2 = 2 n - 2 + 2 4 \\alpha ( X ) \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\mathrm { S O } \\left ( p , q \\right ) = \\left \\{ M \\in G L \\left ( p + q , \\mathbb { R } \\right ) : M ^ { t r } J \\left ( p , q \\right ) M = J \\left ( p , q \\right ) \\right \\} \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} M H ( \\mu ; - ) : = \\{ c \\in v \\mathcal { C } : \\mu ( c ) \\} . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} z ^ { * } \\Gamma _ { \\Phi _ { 1 } ( x ; h ) } z & \\geq C _ { H } \\| z ^ { * } D \\Phi _ { 1 } ( x ; h ) \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } ^ { 2 } \\\\ & = C _ { H } \\int _ { 0 } ^ { 1 } z ^ { * } J _ { 1 } J _ { t } ^ { - 1 } V ( \\Phi _ { t } ) V ^ { * } ( \\Phi _ { t } ) ( J _ { t } ^ { - 1 } ) ^ { * } J _ { 1 } ^ { * } z d t . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\langle D ^ 2 _ 1 J _ 1 ( f ; v ^ 1 , v ^ 2 ) , ( w ^ 1 , w ^ 2 ) \\rangle & = \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } \\eta w ^ 2 d x d t + \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } w ^ 1 w ^ 2 d x d t . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} j = \\widetilde { m } ( \\Lambda + \\nabla _ y \\widetilde { w } ) \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - h _ n x _ n ^ 2 ) , x _ 0 \\in \\mathbb R , n \\in \\mathbb N . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\theta ^ \\beta _ \\epsilon = \\frac { \\mu c _ { \\alpha + \\beta } } { 2 \\lambda b _ { \\beta , \\alpha + \\gamma } } \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\forall x \\in \\mathbb { R } ^ d , \\ F ( x ) = C ( F _ 1 ( x ) , . . . , F _ d ( x ) ) . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\| x \\| _ K = \\min \\{ a \\ge 0 : x \\in a K \\} , x \\in \\mathbb R ^ n . \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} | X ^ { 0 , u } _ x - X ^ 0 _ x | _ { C ( [ 0 , T _ 0 ] : E ) } \\leq \\kappa _ 3 | Y ^ { 0 , u } _ x - Y ^ 0 _ x | _ { C ( [ 0 , T _ 0 ] : E ) } = \\kappa _ 3 | Y ^ { 0 , u } _ x | _ { C ( [ 0 , T _ 0 ] : E ) } \\leq \\kappa _ 4 T _ 0 ^ { \\frac { 1 } { p } } | u | _ { L ^ 2 ( [ 0 , T _ 0 ] : H ) } \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\tilde q \\left ( { \\theta , { \\mu _ j } } \\right ) \\buildrel \\Delta \\over = F \\left ( { \\tau - \\theta - { \\mu _ j } } \\right ) , \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} \\begin{aligned} & | \\dot { c } - c h W ' ( h a ) | \\lesssim \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu h t } \\\\ & | \\dot { a } - c + W ( h a ) + \\frac 1 2 c ^ { - 2 } h ^ 2 W '' ( h a ) | \\lesssim \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu h t } \\end{aligned} \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} d ( x , y ) \\leq \\sum _ { i = 1 } ^ { N - 1 } d ( u _ i , u _ { i + 1 } ) . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} v ( ( f _ { w } ( u ) - f _ { w } ( z ) ) ^ t ) = v ( ( u - z ) ^ s \\cdot b ^ t ) . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} w ( z , \\bar { z } ) = z ^ Q / | z | ^ { Q - 1 } . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} S = \\bigcup _ { j } \\bigcap _ { l \\in L _ { j } } \\left \\{ \\mathbb { R } ^ { n } \\Big \\lvert P _ l s _ { j l } 0 \\right \\} , \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} h '' ( s ) & = K u \\Gamma ( u + 2 ) s ^ { - u - 2 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 2 } ) + O ( s ^ { - 2 } ) \\\\ & = ( 1 + O ( s ^ { \\eta } ) ) K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) s ^ { - u - 2 } . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\frac 1 { \\varrho } : = \\frac 1 { r _ m } - \\frac 1 { r _ { m + 1 } ' } + \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { p _ i } = \\frac 1 { \\delta _ m } + \\frac 1 { \\delta _ { m + 1 } } > 0 , \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} y _ k ( t ) = \\mu ^ \\circ + x ( t ) \\sigma ^ \\circ , \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} 0 \\le \\lim _ { n \\to \\infty } - m n \\log ( 1 - \\beta ^ { n - 1 \\choose 2 } ) \\le \\lim _ { n \\to \\infty } - n ^ { q + 1 } \\log ( 1 - \\beta ^ n ) = 0 , \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} d ( p ^ { \\star } , p _ 0 ) \\leq C _ 0 ( d ( p ^ { \\star } , p ) + d ( p , p _ 0 ) ) \\leq C _ 0 ( d ( p ^ { \\star } , p ) + 2 d ( p ^ { \\star } , p ) ) = 3 C _ 0 d ( p , p ^ { \\star } ) . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} r _ 1 : = \\begin{cases} \\min \\{ | a _ 1 | , | a _ 2 | \\} & \\mbox { i f } a _ 1 a _ 2 \\ge 0 \\ , , \\\\ 0 & \\mbox { i f } a _ 1 a _ 2 \\le 0 \\ , , \\end{cases} r _ 2 : = \\max \\{ | a _ 1 | , | a _ 2 | \\} \\ , . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} Q ^ h : = \\sum _ \\alpha X _ 0 ^ { e - \\sum _ i \\alpha _ i } X _ 1 ^ { \\alpha _ 1 } \\cdots X _ k ^ { \\alpha _ k } , \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} & f ^ * _ 1 \\le f ^ * _ { 2 } \\le \\dots \\le f ^ * _ { \\lfloor \\frac { \\xi } { 2 } \\rfloor + 1 } \\\\ & f ^ * _ { j } \\le f ^ * _ { \\xi + 1 - j } \\ \\ 1 \\le j \\le \\tfrac \\xi 2 \\\\ & t ^ * _ 1 \\le t ^ * _ { 2 } \\le \\dots \\le t ^ * _ { \\lfloor \\frac { d - c } { 2 } \\rfloor + 1 } \\\\ & t ^ * _ { j } \\le t ^ * _ { d - c + 1 - j } \\ \\ 1 \\le j \\le \\tfrac { d - c } 2 \\ , . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\widehat { \\rho } _ t \\psi ( m ) = \\psi ( \\rho _ t m ) \\exp \\left ( - \\frac { i } { \\hbar } \\int \\limits ^ { t } _ 0 L ( \\rho _ { t ' } m ) d t ' \\right ) \\ , . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 2 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = - i ( i R _ { 2 1 } + R _ { 2 4 } ) = R _ { 2 1 } - i R _ { 2 4 } = 0 ; \\\\ \\frac { R _ { 3 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 3 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = R _ { 3 1 } - i R _ { 3 4 } = 0 . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} h = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log \\chi ( \\Gamma _ n , X ) \\ ; \\Gamma _ n \\to 0 \\ ; . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\mathcal { C } _ { a , k } ( z ) = \\sum _ n a _ n \\bigg ( \\frac { 1 } { z - t _ n } + \\frac { 1 } { t _ n } + + \\dots + \\frac { z ^ { k - 1 } } { t _ n ^ k } \\bigg ) = z ^ k \\sum _ n \\frac { a _ n } { t _ n ^ k ( z - t _ n ) } \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} g _ i ( x , q ) = \\sum _ { j = 1 } ^ r h _ { i , j } ( x , q ) g _ j ( x q ^ { e ( i , j ) } , q ) , \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta u _ \\varepsilon = \\lambda _ \\varepsilon u _ \\varepsilon H ( u _ \\varepsilon ) \\exp ( u _ \\varepsilon ^ 2 ) , u _ \\varepsilon > 0 \\Omega \\ , , \\\\ & u _ \\varepsilon = 0 \\partial \\Omega \\ , , \\end{cases} \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\theta ^ N _ X : C ^ { N } \\Omega ^ N X & \\rightarrow \\Omega ^ N X \\\\ \\left [ d _ { \\underline n } , \\gamma ^ { \\underline n } \\right ] & \\mapsto \\left ( t \\mapsto \\begin{cases} \\gamma ^ i ( d _ i ^ { - 1 } ( t ) ) , & t \\in d _ i ( I ^ N ) \\\\ \\ast _ c , & t \\not \\in d _ { \\underline n } \\left ( \\coprod _ { \\underline n } I ^ N \\right ) \\end{cases} \\right ) \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\varphi _ f ( x ) = ( \\mathcal { D } _ 1 ^ \\theta f ) ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } [ \\mathrm { I n d } ^ G _ { P _ b } ( \\delta ^ { \\frac { 1 } { 2 } } _ { G , P _ b } \\otimes I ^ { M _ b } _ { M _ S } ( \\rho ) ) ] \\left [ \\bigoplus _ { ( M _ S , \\mu _ S ) \\in \\mathrm { R e l } ^ { M _ b , \\mu _ b } _ { M _ S , b ' } } r _ { - \\mu _ S } \\circ L L ( I ^ { M _ b } _ { M _ S } ( \\rho ) ) | _ { W _ { E _ { \\{ \\mu _ S \\} _ { M _ S } } } } | \\cdot | ^ { S } \\right ] , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } W ^ n ( b , T ) = W ( b , T ) . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} \\lambda _ 1 ( J ) = \\min _ { \\begin{subarray} { c } u \\in H ^ { 1 } _ 0 ( J ) \\\\ u \\neq 0 \\end{subarray} } \\bigg \\{ \\frac { \\int _ { J } p ( x ) u ' ( x ) ^ 2 d x + \\int _ J q ( x ) u ( x ) ^ 2 d x } { \\int _ { J } w ( x ) u ( x ) ^ 2 d x } \\bigg \\} , \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} m _ { k , 0 } = \\frac { 1 } { k ! } \\left ( ( r _ k - z n _ k ) n _ 0 ^ { - ( k + 2 ) / 2 } \\psi \\right ) \\circ \\kappa , k \\ge 1 . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\vert \\nabla ( w _ i ) ^ 2 ( x ) \\vert = 2 \\left \\vert w _ i ( x ) \\right \\vert . \\left \\vert \\nabla \\left \\vert w _ i ( x ) \\right \\vert \\right \\vert \\ , \\leq 2 \\vert \\nabla u \\vert \\sqrt { 4 \\lambda _ 0 ^ { - 1 } V ( u ( x ) ) } \\leq 4 \\sqrt { \\lambda _ 0 } ^ { - 1 } J ( u ) ( x ) , \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} 0 & = s _ k ^ 2 \\left ( \\lambda \\mathcal { L } _ 0 ( W _ k ) + \\frac { 1 } { 2 } \\mathcal { S } _ { 0 , 2 } ( U _ k , U _ k ) \\right ) \\\\ & + u _ k \\mathcal { L } _ 0 ( W _ k ) + \\lambda s _ k ^ 3 \\mathcal { S } _ { 0 , 2 } ( U _ k , W _ k ) + \\frac { s _ k ^ 3 } { 6 } \\mathcal { S } _ { 0 , 3 } ( U _ k , U _ k , U _ k ) + o ( s _ k ^ 3 ) \\\\ & = \\lambda s _ k ^ 2 \\mathcal { L } _ 0 ( E _ k ) + u _ k \\mathcal { L } _ 0 ( W _ k ) \\\\ & + \\frac { s _ k ^ 3 } { 6 } ( 6 \\lambda \\mathcal { S } _ { 0 , 2 } ( U _ k , W _ k ) + \\mathcal { S } _ { 0 , 3 } ( U _ k , U _ k , U _ k ) ) + o ( s _ k ^ 3 ) . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} T _ { \\rm r e g } = ( I - P ) T , T _ { \\rm s i n g } = P T , \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\nabla _ { X _ i } ^ { \\perp } ( J X _ j ) = 0 \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\delta _ n = \\min _ { \\gamma \\in \\Gamma \\setminus \\{ 1 \\} } K _ \\Omega ( y _ n , \\gamma y _ n ) . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} G _ { \\{ 0 \\} ^ c } ( x , y ) = K ( x ) + K ( y ) - K ( y - x ) . \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\hat { y } ^ i _ t - ( ( D _ 1 a ( y _ x , t , x ) y _ x + a ( y _ x , t , x ) ) \\hat { y ^ i } _ x ) _ { x } + D _ 1 F ( y , y _ x ) \\hat { y } ^ i + D _ 2 F ( y , y _ x ) \\hat { y } ^ i _ x = \\hat { v } ^ i 1 _ { \\mathcal { O } } \\ \\ Q , \\\\ & \\hat { y } ^ i ( 0 , t ) = \\hat { y } ^ i ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & \\hat { y } ^ i ( 0 ) = 0 \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} P ^ { ( \\theta ) } ( f ) = \\sum _ { I \\in \\mathcal { D } } \\frac { \\langle f , b _ I ^ { ( \\theta ) } \\rangle } { \\| b _ I \\| _ 2 ^ 2 } b _ I ^ { ( \\theta ) } . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\mathcal A = \\bigoplus _ { k = 0 } ^ \\infty V ^ { \\otimes k } . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} \\nabla _ { X } \\mu = { \\mathcal L } _ { X } \\mu - \\frac { 1 } { r } \\mathrm { t r a c e } \\ , ( \\nabla X ) \\mu , \\ \\forall \\mu \\in \\Gamma ( | \\mathrm { d e t } \\ , T ^ { * } M | ^ { 1 / r } ) , \\ X \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } f ( \\| x \\| _ p ) d x & = - \\int _ { \\R ^ n } \\int _ { \\| x \\| _ p } ^ \\infty f ' ( t ) d t d x = - \\int _ 0 ^ \\infty \\biggl ( \\int _ { x : \\| x \\| _ p \\le t } 1 \\ , d x \\biggr ) f ' ( t ) d t \\\\ & = - \\int _ 0 ^ \\infty { \\rm v o l } ( t B _ p ^ n ) f ' ( t ) d t = - { \\rm v o l } ( B _ p ^ n ) \\int _ 0 ^ \\infty t ^ n f ' ( t ) d t \\\\ & = { \\rm v o l } ( B _ p ^ n ) \\cdot \\int _ 0 ^ \\infty n t ^ { n - 1 } f ( t ) d t . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} ( 0 , 1 ) \\ni \\theta \\mapsto \\beta ( \\theta ; a , b ) : = \\frac { \\Gamma ( a + b ) } { \\Gamma ( a ) \\Gamma ( b ) } \\theta ^ { a - 1 } ( 1 - \\theta ) ^ { b - 1 } \\ . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\partial _ { \\alpha } g _ { a b } = 0 , \\ ; \\ ; \\ ; \\ ; \\forall a , b \\in \\{ r + 1 , \\dots , m \\} , \\ ; \\ ; \\ ; \\ ; \\alpha \\in \\{ 1 , \\dots , r \\} , \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d \\Phi } { d t } ( t ; x ) = u ( t , \\Phi ( t ; x ) ) , \\\\ \\Phi ( 0 ; x ) = x . \\end{cases} \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} - 2 a = \\frac { 1 + a ^ { 2 } } { c _ { 1 } } \\left ( \\frac { f _ { 2 } ^ { \\prime \\prime } } { f _ { 2 } ^ { \\prime } } \\right ) + c _ { 1 } \\left ( \\frac { f _ { 2 } } { f _ { 2 } ^ { \\prime } } \\right ) , \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { \\infty } & \\le \\left \\| u ( t ) - h \\mathbf { 1 } \\right \\| _ { \\infty } + \\left | h \\right | \\le \\sqrt { L } \\| u _ x ( t ) \\| _ 2 + | h | = L \\sqrt { | 4 \\omega h | } + | h | , \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\rho _ n = \\left ( \\frac { \\beta } { n + e } \\right ) ^ { 3 ^ n } , \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 = 0 , \\ \\ \\bar \\lambda = 0 , \\ \\ \\bar \\lambda _ 1 \\bar \\lambda _ 2 \\neq 0 , \\\\ & S < \\sup H ^ 2 = \\bar H ^ 2 \\leq 3 S - 2 \\ \\ \\ \\ S < \\sup H ^ 2 < \\dfrac 4 3 S , \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\mathcal { S } : = \\bigcap _ { i = 1 } ^ I \\mathcal { B } _ { R _ i } ( u _ i ) , \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} \\begin{aligned} L ( s ) & = L _ b ( s ) + L _ h ( s ) \\\\ L _ b ( s ) & = G _ y C _ b \\sigma _ y \\bar { y } , L _ h ( s ) = G _ y G _ h C _ h \\\\ S ( s ) & = \\frac { 1 } { 1 + L ( s ) } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\hat \\mu _ N ^ 2 + \\hat \\varepsilon _ { q _ N , N } ^ 2 - \\bar m _ { q _ N , N } ^ 2 = 0 . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} r = \\sum _ { \\alpha \\in \\Delta _ + } E _ { \\alpha } \\wedge E _ { - \\alpha } \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} L ( s ) = \\exp \\left ( ( \\ln s ) ^ { \\mu _ 1 } \\cdots ( \\ln _ m s ) ^ { \\mu _ m } \\right ) , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\int _ \\Omega \\rho d x = \\int _ \\Omega \\rho _ 0 d x , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} d \\theta _ t ^ 1 = F _ 1 ( \\theta _ t ^ 1 , \\hdots , \\theta _ t ^ d ) d t , d \\theta _ t ^ i = F _ i ( \\theta _ t ^ { i - 1 } , \\hdots , \\theta _ t ^ d ) d t , 2 \\leq i \\leq d \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\hat { \\phi } _ R ( \\xi ) = \\left \\{ \\begin{array} { c l c } 1 & & 1 / R \\leq | \\xi | \\leq R , \\\\ 0 & & | \\xi | \\leq 1 / 2 R \\vee | \\xi | \\geq 2 R . \\end{array} \\right . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} w _ { k , l } = \\begin{cases} \\frac { 1 } { | \\mathcal { N } _ k \\cup \\{ k \\} | } & , l \\in \\mathcal { N } _ k \\cup \\{ k \\} \\\\ 0 & , \\end{cases} . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} s ' = \\begin{cases} 0 , & 0 < \\eta \\leq \\eta _ 1 \\\\ s ^ \\star , & \\eta _ 1 < \\eta < \\eta _ 2 \\\\ \\frac { 2 \\eta } { \\sigma ^ 2 } , & \\eta \\geq \\eta _ 2 , \\end{cases} \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} f ( S \\cup \\{ t \\} ) - f ( S \\setminus \\{ t \\} ) & = \\big ( f ( S \\cup \\{ t \\} ) - f ( S ) \\big ) \\\\ & + \\big ( f ( S ) - f ( S \\setminus \\{ t \\} ) \\big ) \\\\ & \\geq ( 1 - \\kappa _ { f } ) f ( \\{ t \\} ) \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\begin{cases} z _ t - \\Delta z = 0 , & \\Omega _ T \\\\ \\mathcal { B } [ z ] = 0 , & \\Gamma _ T , \\\\ z ( \\cdot , 0 ) = z _ 0 \\in L ^ { \\infty } ( \\Omega ) , & \\Omega , \\end{cases} \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 = - R _ { 1 2 } = - 1 = u _ 1 ; \\quad \\frac { R _ { 3 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 = - R _ { 3 2 } = 0 ; \\quad \\frac { R _ { 4 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 = 0 . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} l _ m = \\sup \\{ l : N _ 2 + 2 + k _ l \\le m \\} . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] = X _ 3 , ~ ~ [ X _ 2 , X _ 3 ] = 0 , ~ ~ [ X _ 1 , X _ 3 ] = X _ 2 . \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} ( \\check a _ j , \\check b _ j , \\check c _ j ) = \\begin{cases} ( 1 2 0 , \\ 1 1 9 , \\ 1 6 9 ) & j = 1 \\\\ ( 2 \\check h _ j - 1 , 2 \\check h _ j ^ 2 - 2 \\check h _ j , 2 \\check h _ j ^ 2 - 2 \\check h _ j + 1 ) & j = 2 , \\dots , \\check \\nu ^ C \\end{cases} \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} e _ { 1 ^ { * } } = \\frac { { \\vec H } } { | { \\vec H } | } , \\ \\ H ^ { 1 ^ { * } } = H = | { \\vec H } | , \\ \\ H ^ { 2 ^ { * } } = h _ { 1 1 } ^ { 2 ^ { * } } + h _ { 2 2 } ^ { 2 ^ { * } } = 0 . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\Phi _ { S _ 0 } ( g x ) = \\Phi _ { \\eta _ 2 } ( g x ) = g _ 2 \\Phi _ { \\eta } ( g ^ { - 1 } _ 2 g x ) = g _ 2 \\Phi _ { \\eta } ( g ^ { - 1 } _ 2 g g _ 1 g ^ { - 1 } _ 1 x ) = g _ 2 g ^ { - 1 } _ 2 g g _ 1 \\Phi _ { \\eta } ( g ^ { - 1 } _ 1 x ) , \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\eta _ k \\times ( \\eta _ j , 2 a _ j + 1 ) ) = \\prod _ { i = 0 } ^ { 2 a _ j } L ^ S ( s + \\frac { 1 } { 2 } + a _ j - i , \\eta _ k \\times \\eta _ j ) , \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i \\leq l , ~ 1 \\leq j \\leq k ' } L ( s + \\frac { 1 } { 2 } , \\zeta _ i \\times \\xi ' _ j ) = \\frac { \\prod _ { \\substack { 1 \\leq i \\leq l \\\\ 1 \\leq j \\leq s } } L ( s + \\frac { 1 } { 2 } , \\zeta _ i \\times \\eta _ j ) } { \\prod _ { \\substack { 1 \\leq i \\leq l \\\\ 1 \\leq j \\leq k } } L ( s + \\frac { 1 } { 2 } , \\zeta _ i \\times \\xi _ j ) } \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} \\mu = \\| V \\| + | \\bar { h } | ^ 2 d v o l _ g , \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} F ( X ) = ( 0 , - \\Phi ' ( x ) / m , 0 , 0 , \\dots ) ^ T . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\gamma _ \\varepsilon : = \\max _ { y \\in \\Omega } u _ \\varepsilon \\to + \\infty \\ , . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\nabla \\overline { p } = \\overline { \\theta } e _ { N } , \\overline { \\theta } _ t - \\Delta \\overline { \\theta } = 0 & & Q , \\\\ \\nabla \\overline \\theta \\cdot n = 0 & & \\Sigma , \\\\ \\overline { \\theta } ( \\cdot , 0 ) = \\overline { \\theta } _ 0 ( \\cdot ) & & \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} g _ { 1 1 } & = \\langle X _ 1 ( s ) , X _ 1 ( s ) \\rangle = 2 ( 1 + s f ) ^ 2 + 4 s ^ 2 \\big ( \\textstyle \\frac { \\partial f } { \\partial \\theta _ 1 } \\big ) ^ 2 , \\\\ g _ { 1 2 } & = \\langle X _ 1 ( s ) , X _ 2 ( s ) \\rangle = 4 s ^ 2 \\textstyle \\frac { \\partial f } { \\partial \\theta _ 1 } \\frac { \\partial f } { \\partial \\theta _ 2 } , \\\\ g _ { 2 2 } & = \\langle X _ 2 ( s ) , X _ 2 ( s ) \\rangle = 2 ( 1 + s f ) ^ 2 + 4 s ^ 2 \\big ( \\textstyle \\frac { \\partial f } { \\partial \\theta _ 2 } \\big ) ^ 2 . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\| \\tilde { h } ^ { ( m ) } \\| _ { \\bar { { \\cal H } } ( [ 0 , a _ { m + 1 } ] ) } ^ { 2 } & = C _ { H } \\cdot \\int _ { 0 } ^ { a _ { m + 1 } } \\left | t ^ { H - \\frac { 1 } { 2 } } \\left ( t ^ { 1 - 2 H } \\dot { \\tilde { h } } ^ { ( m ) } ( t ) \\right . \\right . \\\\ & \\ \\ \\ \\left . \\left . + \\left ( H - \\frac { 1 } { 2 } \\right ) \\int _ { 0 } ^ { t } \\frac { t ^ { \\frac { 1 } { 2 } - H } \\dot { \\tilde { h } } ^ { ( m ) } ( t ) - s ^ { \\frac { 1 } { 2 } - H } \\dot { \\tilde { h } } ^ { ( m ) } ( s ) } { ( t - s ) ^ { H + \\frac { 1 } { 2 } } } d s \\right ) \\right | ^ { 2 } d t . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\norm { ( t , \\sigma ) - ( t ^ * , \\sigma ^ * ) } & = \\norm { p _ \\mu ( z _ o , \\sigma ^ o ) - \\Pi _ { K _ { m , n } } ( z _ o ^ * , \\zeta ) } \\\\ & \\leq \\norm { p _ \\mu ( z _ o , \\sigma ^ o ) - p _ 0 ( z _ o , \\sigma ^ o ) } + \\norm { \\Pi _ { K _ { m , n } } ( z _ o , \\sigma ^ o ) - \\Pi _ { K _ { m , n } } ( z _ o ^ * , \\zeta ) } \\\\ & = O ( \\norm { ( z _ o - z _ o ^ * , \\sigma ^ o - \\zeta , \\mu ) } . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} & | f ( \\xi \\eta ) + f ( \\xi \\eta ^ { - 1 } ) - 2 f ( \\xi ) | \\\\ = & | f ( \\xi + ( \\xi \\eta - \\xi ) ) + f ( \\xi - ( \\xi \\eta - \\xi ) ) - 2 f ( \\xi ) | \\\\ \\leq & C \\| \\xi \\eta - \\xi \\| \\leq C \\| \\eta \\| \\leq C | \\eta | , \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} b _ c ^ { ( n ) } = b _ c ^ { ( n ) ' } ( 1 + o ( 1 ) ) , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} \\mathcal { G } ( p , r , R ) = \\mathcal { G } _ 0 ( p , r , R ) \\dot { \\cup } \\left ( \\dot { \\bigcup } _ { j = 1 } ^ { \\tilde { t } } \\ , \\mathcal { T } _ j ^ r \\right ) \\# \\mathcal { G } _ 0 ( p , r , R ) = \\tilde { s } . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mu _ { \\beta , h } ^ + ( \\cdot \\mid \\eta = - 1 ( - \\infty , - n ] \\times \\{ 0 \\} ) = \\mu _ { \\beta , h } ^ + ( \\cdot ) . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} h _ { i j k } ^ { p ^ { \\ast } } = h _ { i k j } ^ { p ^ { \\ast } } = h ^ { i ^ { \\ast } } _ { p j k } . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d U = \\Delta U d t + ( X \\cdot \\nabla ) U d t + \\sum _ { i = 1 } ^ N ( B _ i ( t ) + \\theta _ i I ) U d \\beta _ i ( 0 , \\infty ) \\times \\mathbb { R } ^ 2 , \\\\ U ( 0 , \\xi ) = U _ 0 ( \\xi ) = ( c u r l \\ x ) ( \\xi ) , \\ \\xi \\in \\mathbb { R } ^ 2 , \\end{array} \\right . \\ \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} S ( y ) : = I m ( \\sqrt { y _ 1 + i y _ 2 } ) \\ , . \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} f _ 1 ( a , a ^ { - 1 } ) f _ 1 ( b , b ^ { - 1 } ) = f _ 1 ( a b , b ^ { - 1 } a ^ { - 1 } ) . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} G _ D \\ , \\phi ( x ) = \\int _ \\R G _ D ( x , y ) \\phi ( y ) d y , x \\in \\R \\ , . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\left \\| | x _ \\varepsilon - \\cdot | | \\nabla u _ \\varepsilon | \\right \\| _ { L ^ \\infty ( B _ { x _ \\varepsilon } ( r _ \\varepsilon ) ) } = O \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , , \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} k _ n = M ( 1 - \\frac { 1 } { 2 ^ n } ) . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\| [ \\langle x \\rangle ^ a , D ^ { - \\frac 1 2 } \\nabla ] f \\| _ { L ^ q ( \\mathbb R ^ n ) } & \\le \\| [ \\langle x \\rangle ^ a , R ] D ^ \\frac 1 2 f \\| _ { L ^ q ( \\mathbb R ^ n ) } + \\| R [ \\langle x \\rangle ^ a , D ^ \\frac 1 2 ] f \\| _ { L ^ q ( \\mathbb R ^ n ) } . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} | \\psi ( t ) - \\psi ( u ) | & = \\dfrac { | t - u | } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\leq \\dfrac { t ^ { 1 / k } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } < \\dfrac { 1 } { 2 ^ { 2 n - 1 } } . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\lim _ { z \\to 1 ^ - } \\frac { { } _ 2 F _ 1 ( a , b ; a + b ; z ) } { - \\ln ( 1 - z ) } = \\frac { \\Gamma ( a + b ) } { \\Gamma ( a ) \\Gamma ( b ) } . \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! S _ 0 ( f ) \\ ! = \\ ! \\frac { N _ 0 } { \\alpha _ { \\rm c } } \\ ! \\left ( \\ ! \\frac { 1 } { \\frac { \\mu } { 2 } \\int \\limits _ 0 ^ { \\pi } G ( \\cos \\theta \\cdot f ) \\sin \\theta { \\rm d } \\theta } - \\frac { 1 } { G ( \\cos \\theta _ { \\rm c } \\cdot f ) } \\ ! \\right ) _ { \\ ! \\ ! \\ ! \\ ! + } \\ ! \\ ! , \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} h _ j ( T _ \\varepsilon ) - h _ { j - 1 } ( T _ \\varepsilon ) = \\varepsilon / \\rho , \\textrm { f o r s o m e } j \\in \\{ 1 , \\dots , N + 2 \\} . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} m ^ { \\alpha + 2 } \\leq \\sum _ { j = 1 } ^ \\infty \\| V ^ j \\| _ { L ^ { \\alpha + 2 } } ^ { \\alpha + 2 } . \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} I _ 0 = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ { L _ 0 } x ^ { - \\alpha } e ^ { - t x } d x , \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} c _ s ( z ) = z ^ { 2 s } \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\chi ^ { ( 1 ) k } _ n = \\exp _ p ( k h _ p ) ^ { a _ n } u ^ { k a _ n } _ n \\ ; n \\ ; . \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} [ L : K ] & = \\sum _ { i \\leq s } \\sum _ { j \\leq r _ { i } } e ( u _ { i , j } / v ) \\ , f ( u _ { i , j } / v ) . \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} I I _ 1 = \\| \\langle x \\rangle ^ { a } ( \\widetilde { ( \\nabla \\chi _ k ) u } ) \\| _ { L ^ 2 ( \\mathbb R ; \\dot H ^ \\frac 1 2 ( \\mathbb { R } ^ n ) ) } \\le C \\| \\langle x \\rangle ^ { a } ( \\nabla \\chi _ k ) u \\| _ { L ^ 2 ( \\mathbb R ; H ^ \\frac 1 2 ( - \\left . \\Delta \\right | _ { \\mathit { D } } ) ) } . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\mathbb { P } _ { x , k } ^ { \\epsilon , \\hat { v } } \\Bigl \\{ \\tau _ D ^ { \\epsilon , \\hat { v } } < \\infty \\Bigr \\} = 1 , \\ , \\ , \\ , \\ , \\ , x \\in \\hat { D } , \\ , \\ k \\in \\{ 1 , 2 , \\ldots , n \\} . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\big ( T _ { e } ( c ) \\big ) _ { e ' } = \\begin{cases} c _ { e ' } - 2 \\cdot q _ { e , e ' } & e ' = ( u , w ) \\in A _ { v } u \\neq s \\\\ c _ { e ' } - 2 \\cdot q _ { e , e ' } + c _ { e } & e ' = ( u , w ) \\in A _ { v } u = s \\\\ \\end{cases} , \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{gather*} \\# \\mathfrak { R } ( f , \\sigma , k , L ) = L ^ { n - 1 } [ \\mathbb { Q } ( \\zeta _ { L / d } ) : \\mathbb { Q } ( f ) \\cap \\mathbb { Q } ( \\zeta _ { L / d } ) ] \\quad ( d : = ( k , L ) ) . \\end{gather*}"}
-{"id": "1380.png", "formula": "\\begin{gather*} \\tilde \\Phi _ a = M ^ b _ a \\Phi _ b \\end{gather*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\left [ ( - \\lambda _ 0 + z Q \\partial _ Q ) \\cdots ( - \\lambda _ N + z Q \\partial _ Q ) - Q \\right ] J ^ { \\textnormal { c o h } , \\textnormal { e q } } ( z , Q ) = 0 \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ 1 \\left ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 \\right ) ^ T \\nabla _ y \\left ( \\nabla _ { \\widetilde { \\Lambda } } \\widetilde { u } _ 1 \\right ) d y = 0 . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} H ^ { - 1 } ( A ^ \\bullet ( q = 0 ) ) = 0 . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 0 . 4 & 0 . 4 & 0 . 1 \\\\ 0 . 5 & 0 . 3 & 0 . 3 \\\\ 0 . 1 & 0 . 1 & 0 . 5 \\end{array} \\right ) , \\mbox { w i t h } \\rho ( A ) = 0 . 8 9 6 0 . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} X = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} , Y = \\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix} , Z = \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} , \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} m _ 1 ^ { - \\alpha } & < 2 \\epsilon \\left ( \\frac { ( m _ 1 + 1 ) ^ { - \\alpha } } { 2 \\epsilon } + 1 \\right ) \\\\ & = ( m _ 1 + 1 ) ^ { - \\alpha } + 2 \\epsilon \\\\ & \\leq m _ 1 ^ { - \\alpha } \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} h ( \\xi ) = \\frac { 1 } { \\lambda } \\int _ 0 ^ \\infty e ^ { - s } f \\left ( - \\frac { 1 } { \\sqrt { \\lambda } } s \\mathbf { 1 } _ d + \\xi \\right ) d s , \\ \\xi \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} S ^ * ( \\delta _ { ( \\xi , y ) } ) ( \\{ ( \\eta , y ) \\} ) & = T ^ * ( \\delta _ { ( \\xi , y ) } ) ( \\{ ( \\eta , y ) \\} ) - R _ L ^ * ( \\delta _ { ( \\xi , y ) } ) ( \\{ ( \\eta , y ) \\} ) \\\\ & = r ( \\eta , \\xi ) - r ( \\eta , \\xi ) = 0 \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\begin{aligned} \\| e ^ { u ( s , \\cdot ) } \\| _ { L ^ 2 } ^ 2 = ~ & \\int _ { M } e ^ { 2 u ( s , \\cdot ) } = \\int _ { M } e ^ { 2 ( u ( s , \\cdot ) - \\overline { u } ) } e ^ { 2 I } \\\\ \\leq ~ & C \\exp \\left ( \\frac { 1 } { 4 \\pi } \\int _ { M } | \\nabla u ( s , \\cdot ) | ^ 2 \\right ) e ^ { 2 I } \\leq C e ^ { 2 I } e ^ { \\frac { 1 } { 4 \\pi } R ^ 2 } , \\end{aligned} \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\ddot { v } ( t ) - \\div ( \\widetilde { A } ( t ) \\nabla v ( t ) ) + l . o . t . = \\widetilde { f } ( t ) \\hbox { i n } \\widetilde \\Omega \\setminus \\widetilde { \\Gamma } _ 0 \\ , , \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\phi \\Big ( \\big ( K _ N ^ * A ^ \\frac { r } { N } K _ N \\big ) ^ { N } \\Big ) = & \\ \\lim _ { N \\rightarrow + \\infty } \\phi \\left ( \\Big ( \\exp \\big ( \\frac { 1 } { 2 N } H \\big ) \\exp \\big ( \\frac { r } { N } \\log A \\big ) \\exp \\big ( \\frac { 1 } { 2 N } H \\big ) \\Big ) ^ N \\right ) \\\\ = & \\ \\phi \\big ( \\exp ( H + r \\log A ) \\big ) . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} h _ { w , y } ( \\gamma ) = r \\gamma + \\alpha ( w , x ) . \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} u ^ { \\epsilon } ( x ) : = \\sup _ { y \\in \\mathbb R ^ d } \\left \\{ u ( y ) - \\frac { | y - x | ^ 2 } { 2 \\epsilon } \\right \\} = u ( x ^ \\epsilon ) - \\frac { | x ^ \\epsilon - x | ^ 2 } { 2 \\epsilon } . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big [ ( 3 + t ) E ( u ; t ) \\Big ] & = \\| u ' \\| ^ 2 + \\| A ^ { 1 / 2 } u \\| ^ 2 - 2 ( 3 + t ) \\| u ' \\| ^ 2 , \\\\ \\frac { d } { d t } E _ * ( u ; t ) & = 2 \\| u ' \\| ^ 2 + 2 ( u , u '' + u ' ) \\\\ & = 2 \\| u ' \\| ^ 2 - 2 \\| A ^ { 1 / 2 } u \\| ^ 2 . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { \\mathrm { G L } _ { n _ 1 } , ( 1 ^ 2 , 0 ^ { n _ 1 - 2 } ) } ( L J ( \\rho _ 1 ( n _ 2 / 2 ) ) ) = [ \\rho _ 1 ( n _ 2 / 2 ) ] [ r _ { ( - 1 ^ 2 , 0 ^ { n _ 1 - 2 } ) } \\circ L L ( \\rho _ 1 ( n _ 2 / 2 ) ) \\otimes | \\cdot | ^ { 2 - n _ 1 } ] , \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\eta \\cdot \\xi = \\xi \\cdot \\zeta = \\zeta \\cdot \\eta = 0 , | \\eta | ^ 2 = | \\xi | ^ 2 + | \\zeta | ^ 2 . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} q _ 1 ( z ) & = \\frac { q ^ { n + \\gamma } } { ( 1 - q ^ \\gamma ) ( 1 - z q b ) } \\left [ ( q ^ { - n - \\gamma } - 1 ) ( 1 - z q b ) + \\frac { b ( q ^ n - 1 ) ( a q ^ n - 1 ) } { ( a b q ^ { 2 n } - 1 ) } \\right ] \\\\ \\ q _ 0 ( z ) & = \\frac { q ^ { \\gamma } ( q ^ { n } - 1 ) ( b q ^ n - 1 ) } { ( 1 - z q b ) ( 1 - q ^ \\gamma ) ( a b q ^ { 2 n } - 1 ) } . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\tau _ 0 : = \\inf \\{ \\tau ' > 0 \\mid \\exists \\tau \\in \\R , \\ \\widetilde { \\phi } ( \\cdot - \\tau ) \\preceq \\phi ( \\cdot ) \\preceq \\widetilde { \\phi } ( \\cdot - \\tau - \\tau ' ) \\} \\ ( \\in [ 0 , b - a ] ) \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} ( \\Delta _ h K * d Z ) & = ( \\Delta _ h K * L ) ( 0 ) ( V - g _ 0 ) + d ( \\Delta _ h K * L ) * ( V - g _ 0 ) . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ s } \\ln \\dfrac { R ( n ) ^ { \\mathcal { S } } } { n ^ { \\mathcal { T } } } = 0 . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} u = \\exp ( l _ 0 + l _ h ) \\ \\Longrightarrow \\ v = \\exp ( l _ 0 ) \\in G ^ { ( l - 1 ) } . \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\nu ( x _ 1 ) \\nu ( y _ 1 ) \\alpha ' ( x _ 2 , y _ 2 ) \\nu ^ { - 1 } ( x _ 3 y _ 3 ) = \\alpha ( x , y ) \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} f ^ \\delta ( x , u ) \\ ; \\dot = \\ ; \\frac { ( u - 1 + H ^ \\delta ( x ) ) ^ 2 } { 2 } . \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\P ( | X _ n - X | \\leq 1 / n ) = 1 . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} a _ n \\geq { \\rm c } _ 0 ^ n \\left ( a _ 0 - \\underset { k = 0 } { \\overset { n } \\sum } \\frac { 1 } { { \\rm c } _ 0 ^ { k + 1 } } f _ k \\right ) { \\rm \\ f o r \\ } n \\in \\N ^ * n \\leq n _ \\star . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} n \\pi _ n ( x a _ c ^ { ( n ) } ) & \\sim _ e n \\frac { x ^ { r } } { r ! } ( p _ n a _ c ^ { ( n ) } ) ^ { r } \\\\ & = x ^ { r } ( a _ c ^ { ( n ) } ) ^ { r - 1 } \\frac { n p _ n ^ { r } } { r ! } a _ c ^ { ( n ) } \\\\ & = x ^ { r } \\left ( 1 - \\frac { 1 } { r } \\right ) ^ { r - 1 } \\frac { ( r - 1 ) ! } { n p _ n ^ { r } } \\frac { n p _ n ^ { r } } { r ! } a _ c ^ { ( n ) } \\\\ & = \\frac { 1 } { r } \\left ( 1 - \\frac { 1 } { r } \\right ) ^ { r - 1 } x ^ { r } a _ c ^ { ( n ) } . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F H N } } ^ { \\textrm { N O M A } } = \\frac { P _ \\textrm { N } } { P _ \\textrm { F } + { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } / { | h _ { \\textrm { B N } } | ^ 2 } } + \\frac { \\beta _ { \\textrm { F } } \\eta P _ \\textrm { B } | h _ { \\textrm { B F } } | ^ 2 | h _ { \\textrm { N F } } | ^ 2 } { d _ { \\textrm { B F } } ^ { \\alpha } d _ { \\textrm { N F } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } u ( x ) v ( x ) \\ , d x = \\sum _ { j = 0 } ^ N u ( x _ j ^ \\lambda ) v ( x _ j ^ \\lambda ) \\omega _ j ^ \\lambda , \\forall \\ , u \\cdot v \\in V _ { 2 N + 1 } ^ \\lambda . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { t } ( - 1 ) ^ { i } { m + s - 1 + i \\choose 2 s - 1 } { 2 s + 1 \\choose i } = ( - 1 ) ^ { t } \\dfrac { s ( 2 m + 2 s + 2 ) - t } { ( m + s ) ( m + s + 1 ) } { m + s + t \\choose 2 s } { 2 s \\choose t } . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} e _ { \\pm } : = \\lambda t _ \\alpha \\pm \\mu e ^ { \\alpha } , \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} L R ( \\chi ) : = \\{ M \\in \\mathbb { Q } [ G ] \\mid M = M C ( \\chi ) , M ( \\sum _ { g \\in G } g ( \\alpha ) g ^ { - 1 } ) = 0 \\} . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} X = X ^ 0 \\supseteq X ^ { 1 } \\supseteq \\cdots \\supseteq X ^ { \\dim X } \\supseteq X ^ { \\dim X + 1 } = \\emptyset . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\gamma _ { X _ { l } ( t , x ) } = J F _ { l } ( U _ { t } ^ { ( l ) } , x ) \\cdot \\gamma _ { U _ { t } ^ { ( l ) } } \\cdot J F _ { l } ( U _ { t } ^ { ( l ) } , x ) ^ { * } . \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} k ^ 2 = \\dfrac { H _ o ^ 2 r _ o ^ 2 } { 4 } \\ ( 1 + b r _ o ^ 2 - \\dfrac { 2 m } { r _ o } \\ ) ^ { - 1 } \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c } \\frac { \\partial u } { \\partial t } ( x , t ) = f \\left ( u , v \\right ) \\\\ \\\\ \\frac { \\partial v } { \\partial t } ( x , t ) = g \\left ( u , v \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} \\mu ^ Z ( J X ) \\xi _ 1 + \\mu ^ W ( J X ) \\xi _ 2 = \\mu ( J X ) = J _ { \\mathfrak t ^ 2 } \\mu ( X ) = J _ { \\mathfrak t ^ 2 } ( \\mu ^ Z ( X ) \\xi _ 1 + \\mu ^ W ( X ) \\xi _ 2 ) = - \\mu ^ W ( X ) \\xi _ 1 + \\mu ^ Z ( X ) \\xi _ 2 , \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} E ^ 2 _ { d , q } ( X / F , \\Z / n \\Z ( b ) ) = H ^ { d - q } _ { u r } ( F ( X ) , \\Z / n \\Z ( d - b ) ) . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} ( \\rho ^ { u } ) \\mathtt { C } \\mathtt { S } ( \\phi ) & = ( \\mu _ { S _ 1 e } \\ast \\rho ( e , u , f ) ) \\mathtt { C } \\mathtt { S } ( \\phi ) \\\\ & = \\mu _ { v \\mathtt { C } ( \\phi ) ( S _ 1 e ) } \\ast \\mathtt { C } ( \\phi ) ( \\rho ( e , u , f ) ) \\\\ & = \\mu _ { v \\Phi ( S _ 1 e ) } \\ast \\Phi ( \\rho ( e , u , f ) ) \\\\ & = \\mu _ { S _ 2 e \\phi } \\ast \\rho ( e \\phi , u \\phi , f \\phi ) \\\\ & = \\rho ^ { e \\phi } \\ast \\rho ( e \\phi , u \\phi , f \\phi ) \\\\ & = \\rho ^ { e \\phi u \\phi } = \\rho ^ { e u \\phi } = \\rho ^ { u \\phi } . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 1 \\\\ 2 \\\\ 1 \\\\ 6 \\\\ 2 \\\\ 1 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c c c c c } 4 & 1 & 4 & 2 & 4 & 4 \\\\ 1 & 1 & 4 & 0 & 1 & 7 \\\\ 4 & 4 & 4 & 6 & 6 & 7 \\\\ 2 & 0 & 6 & 6 & 7 & 9 \\\\ 4 & 1 & 6 & 7 & 3 & 0 \\\\ 4 & 7 & 7 & 9 & 0 & 3 \\end{array} \\right ] , d = \\left [ \\begin{array} { c } 8 \\\\ 7 \\\\ 6 \\\\ 4 \\\\ 7 \\\\ 6 \\end{array} \\right ] . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\mathcal E _ { 2 , j } ( x ) = 2 \\sigma \\left [ \\left ( 1 - 2 \\frac { \\langle x , \\nabla u _ j \\rangle ^ 2 } { | x | ^ 2 z _ j } \\right ) \\phi - \\alpha _ j \\langle x , \\nabla \\phi \\rangle \\right ] \\frac { z _ j } { | x | ^ 2 \\phi } . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} \\textbf { h } _ k = \\textbf { F } ^ { \\rm H } \\textit { \\textbf { h } } _ k . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n } } { ( z q ^ { n + 1 } ; q ) _ { n + 2 } ( z q ^ { 2 n + 4 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } p _ { \\omega } ( m , n ) z ^ m q ^ n , \\\\ [ 6 p t ] \\sum _ { n = 0 } ^ { \\infty } q ^ { n } ( - z q ^ { n + 1 } ; q ) _ { n } ( - z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } p _ { \\nu } ( m , n ) z ^ m q ^ n . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\mathcal { S } _ { 0 } ( z ) & = - < z > , \\\\ C _ { 1 } ( z ) & = < z > ^ { 2 } - \\frac { 1 } { 1 2 } , \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} u '' + \\lambda u = 0 , u ( 0 ) = u ( \\pi ) = 0 \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} p ^ { \\lambda } ( x , t ) = \\lambda ^ 2 p ( \\lambda x , \\lambda ^ 2 t ) , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} P _ \\beta : = \\{ y \\in R \\mid \\nu ( y ) \\geq \\beta \\} \\mbox { a n d } P _ \\beta ^ + : = \\{ y \\in R \\mid \\nu ( y ) > \\beta \\} . \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\alpha / 2 } u ( x ) : = C _ { d , \\alpha } { \\rm p . v . } \\int _ { \\mathbb { R } ^ d } \\dfrac { u ( x ) - u ( y ) } { \\lvert x - y \\rvert ^ { d + \\alpha } } d y \\ ; \\ ; \\textmd { w i t h } \\ ; \\ ; C _ { d , \\alpha } : = \\dfrac { { \\alpha } 2 ^ { \\alpha - 1 } \\Gamma \\Big ( \\dfrac { \\alpha + d } { 2 } \\Big ) } { \\pi ^ { d / 2 } \\Gamma \\Big ( \\dfrac { 2 - \\alpha } { 2 } \\Big ) } . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } \\circ \\mathrm { R e d } _ b = [ \\mathcal { M } _ { G , b , \\mu } ] . \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = \\frac { 1 } { 2 } ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 ) + \\sum _ { d = 1 } ^ \\infty N _ d \\frac { t _ 2 ^ { 3 d - 1 } } { ( 3 d - 1 ) ! } e ^ { d t _ 1 } Q ^ d \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\max _ { y \\in \\Omega _ \\varepsilon } | y - x _ \\varepsilon | ^ 2 | \\Delta u _ \\varepsilon ( y ) | u _ \\varepsilon ( y ) = | y _ \\varepsilon - x _ \\varepsilon | ^ 2 | \\Delta u _ \\varepsilon ( y _ \\varepsilon ) | u _ \\varepsilon ( y _ \\varepsilon ) \\to + \\infty \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathbf A \\hat \\oplus \\mathbf A ' \\\\ \\updownarrow \\\\ \\mathbf B \\hat \\oplus \\mathbf B ' \\\\ \\end{array} = \\left ( \\begin{array} { c } \\mathbf A \\\\ \\updownarrow \\\\ \\mathbf B \\end{array} \\right ) \\hat \\oplus \\left ( \\begin{array} { c } \\mathbf A ' \\\\ \\updownarrow \\\\ \\mathbf B ' \\end{array} \\right ) . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace = 1 - e ^ { - \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } \\left ( 2 ^ { R } - 1 \\right ) } { \\lambda _ { \\textrm { B F } } \\left ( P _ { \\textrm { F } } - P _ { \\textrm { N } } \\left ( 2 ^ { \\textrm { R } } - 1 \\right ) \\right ) } } . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} v o l ( D _ { 1 , a } \\cap \\hat { \\mathfrak D } _ n ) / v o l ( { \\hat { \\mathfrak D } } _ n ) = \\sum _ { 0 \\le h \\le n , \\atop 1 \\le l \\le n - 1 } C _ 1 ( l , h ) M ( l - h a ) ^ { n - 1 } , \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ( x , t ) + \\triangle _ { p ( x , t ) } ^ N u ( x , t ) + \\sum _ { i = 1 } ^ n \\mu _ i \\frac { \\partial u } { \\partial x _ i } ( x , t ) = r u ( x , t ) ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n \\end{cases} \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} ( \\forall \\ , j \\in A ) \\ ; & w _ j = w _ { n + j } = 1 ; \\\\ ( \\forall \\ , j \\in B ) \\ ; & w _ j = 0 , \\ , w _ { n + j } = 1 ; \\\\ ( \\forall \\ , j \\in C ) \\ ; & w _ j = 1 , \\ , w _ { n + j } = 0 ; \\\\ ( \\forall \\ , j \\in D ) \\ ; & w _ j = w _ { n + j } = 0 . \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\mathfrak p = \\begin{array} { c } ( \\mathfrak p _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak p _ i ) \\\\ \\updownarrow \\\\ ( \\mathfrak q _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak q _ j ) \\end{array} \\leftrightarrow \\begin{array} { c } ( \\mathfrak p _ { i + 1 } \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak p _ k ) \\\\ \\updownarrow \\\\ ( \\mathfrak q _ { j + 1 } \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak q _ \\ell ) \\end{array} \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} B _ { \\delta , \\gamma } ( x , y , t ) : = \\delta ( | x | ^ 2 + | y | ^ 2 ) + \\gamma t ^ { - 1 } . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\widehat { J } ( \\xi ) = 1 - A | \\xi | ^ 2 \\ln ( 1 / | \\xi | ) + o ( | \\xi | ^ 2 \\ln ( 1 / | x | ) ) \\quad \\mbox { a s $ | \\xi | \\to 0 $ } . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} w ^ { \\alpha + \\beta } _ \\pm = - \\theta ^ \\beta _ \\mp w ^ \\beta _ \\pm \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\log P ( \\mathrm { B i n } ( \\lfloor m _ n \\rfloor , q _ n ) \\geq r _ n ) = r _ n \\log \\left ( \\frac { m _ n q _ n } { r _ n } \\right ) ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} S _ k ( t ) & = \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } w _ { k l } \\left ( S _ l ( t - 1 ) \\right ) + \\hat { \\eta } _ k ( t ) , \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} & \\Lambda _ { T _ 0 } : L ^ \\infty _ c \\otimes X _ 1 \\times \\cdots \\times L ^ \\infty _ c \\otimes X _ { n + 1 } \\to \\C , \\\\ & \\Lambda _ { T _ 0 } ( f _ 1 , \\ldots , f _ { n + 1 } ) = \\sum _ { a _ 1 , \\ldots , a _ { n + 1 } } \\langle T _ 0 ( f _ { 1 , a _ 1 } , \\ldots , f _ { n , a _ n } ) , f _ { n + 1 , a _ { n + 1 } } \\rangle \\tau \\Big ( \\prod _ { j = 1 } ^ { n + 1 } e _ { j , a _ j } \\Big ) , \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x _ n ^ j ) + v _ n ^ l ( x ) , \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\frac { \\sigma ' ( z ) } { \\sigma ( z ) } = \\zeta ( z ) + G _ 2 z . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} ( 1 ) = 2 \\pi p _ { 0 } \\left ( d _ { k } ^ { 2 } - d _ { k } \\right ) \\log \\frac { 1 } { c _ { n } } + \\frac { \\pi } { 2 } \\alpha _ { k } \\left ( 1 - \\alpha \\right ) \\left ( c _ { n } ^ { s _ { k } } - 1 \\right ) \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\deg ( f _ a ) & \\le \\Big ( \\Big \\lfloor \\frac { k ' } { r } \\Big \\rfloor - 1 \\Big ) ( r + 1 ) + ( r - 1 ) \\\\ & = k ' + \\Big \\lceil \\frac { k ' } { r } \\Big \\rceil - 2 . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\mbox { $ \\vec { p } = ( p _ 1 , \\dots , p _ m ) $ e x t r a p o l a t e s t o $ \\vec { t } = ( t _ 1 , t _ 2 , \\dots , t _ m ) = ( q _ 1 , p _ 2 , \\dots , p _ m ) $ } . \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\Gamma ( k ) = \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } \\\\ \\Gamma ( k ) _ { 2 , 1 } \\\\ \\vdots \\\\ \\Gamma ( k ) _ { n , 1 } \\end{bmatrix} \\otimes \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } & \\Gamma ( k ) _ { 1 , 2 } & \\ldots & \\Gamma ( k ) _ { 1 , n } \\end{bmatrix} . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\ < V \\ > = \\sum _ { i = 1 } ^ k c _ i \\ < \\sigma _ i \\ > . \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} \\lambda _ { 1 , k } = \\lambda _ { 1 , \\textup { $ L $ - p e r } } \\left ( - \\left ( \\begin{matrix} \\frac { \\mathrm { d } ^ { 2 } } { \\mathrm { d } x ^ { 2 } } + \\mu _ { 1 } \\left ( 1 - 2 u _ { 1 , k } \\right ) - k \\omega u _ { 2 , k } & k \\omega u _ { 1 , k } \\\\ \\alpha k \\omega u _ { 2 , k } & d \\frac { \\mathrm { d } ^ { 2 } } { \\mathrm { d } x ^ { 2 } } + \\mu _ { 2 } \\left ( 1 - 2 u _ { 2 , k } \\right ) - \\alpha k \\omega u _ { 1 , k } \\end{matrix} \\right ) \\right ) \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\mu ( X _ 0 = \\omega _ 0 \\vert X _ { - \\infty } ^ { - 1 } = \\omega _ { - \\infty } ^ { - 1 } , X _ { 1 } ^ { \\infty } = \\omega _ { 1 } ^ { \\infty } ) = \\gamma ( \\omega _ { - \\infty } ^ { - 1 } , \\omega _ 0 , \\omega _ { 1 } ^ { \\infty } ) , \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\overline { A } _ { k , i } ( n ) = \\overline { B } _ { k , i } ( n ) . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\delta ^ { * } ( t ) = \\frac { 1 } { t ( 1 + t ^ { 2 } ) } \\int _ { 1 } ^ { t } \\frac { \\left ( R _ { g ( s ) } - s ^ { 2 } \\overline { R } \\right ) ^ { * } } { 2 } \\exp \\left ( - \\int _ { s } ^ { t } \\frac { \\tau ( | M | _ { * } ) ^ { 2 } } { 2 } d \\tau \\right ) d s \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} v _ x p _ x = - a \\gamma \\rho ^ { \\gamma - 3 } \\rho _ x ^ 2 - \\delta \\frac { \\rho _ x } { \\rho ^ 2 } \\theta _ x p _ \\theta ( \\rho ) - \\delta \\frac { \\rho _ x ^ 2 } { \\rho ^ 2 } \\theta p _ \\theta ' ( \\rho ) . \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\mathbf { W } = \\mathbf { C } _ { \\mathbf { y } _ { \\mathcal { Q } } } ^ { - 1 } \\mathbf { C } _ { \\mathbf { y } _ { \\mathcal { Q } } \\mathbf { x } } , \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} S _ { l } ( \\gamma ) = S _ { l } ( \\alpha ) \\otimes S _ { l } ( \\beta ) = S _ { l } ( \\alpha ) \\otimes w = u , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} & I _ { \\mu } ( x ) = { \\rm i } ^ { - \\mu } J _ { \\mu } ( { \\rm i } x ) , K _ { \\mu } ( x ) = \\frac { \\pi } { 2 } \\frac { I _ { - \\mu } ( x ) - I _ { \\mu } \\left ( x \\right ) } { \\sin ( \\mu \\pi ) } , \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} e ^ { - 2 K t / 3 } \\bigg ( \\underbrace { \\frac { \\Delta u _ t } { u _ t } - | \\nabla \\log u _ t | ^ 2 } _ { = \\Delta \\log u _ t } \\bigg ) \\geq ( 1 - e ^ { - 2 K t / 3 } ) | \\nabla \\log u _ t | ^ 2 - \\frac { N K } { 3 } \\frac { e ^ { - 4 K t / 3 } } { 1 - e ^ { - 2 K t / 3 } } \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\ < V \\ > = \\sum _ { i = 1 } ^ k c _ i \\eta ( \\sigma _ i ) [ \\sigma _ i ] , \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} \\binom { \\ell _ q ( Q ) } { k } = \\frac { 1 } { k ! } \\prod _ { r = 0 } ^ { k - 1 } ( \\ell _ q ( Q ) - r ) \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} E [ g ( X ) h ( Y ) ] = E [ g ( X ) ] E [ h ( Y ) ] , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} \\phi \\big ( ( K ^ * X ^ { r s } K ) ^ { \\frac { 1 } { s } } \\big ) = & \\ \\phi \\big ( ( M ^ * C ^ { - \\frac { r s } { 2 } } X ^ { r s } C ^ { - \\frac { r s } { 2 } } M ) ^ \\frac { 1 } { s } \\big ) \\\\ = & \\ \\phi \\big ( ( | M | Q ^ * C ^ { - \\frac { r s } { 2 } } X ^ \\frac { r s } { 2 } X ^ \\frac { r s } { 2 } C ^ { - \\frac { r s } { 2 } } Q | M | ) ^ \\frac { 1 } { s } \\big ) \\\\ = & \\ \\phi \\big ( | G _ X ( s ) | ^ \\frac { 2 } { s } \\big ) . \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\{ p , w \\} = \\bigl ( \\{ p , z \\} - \\{ w , z \\} \\bigr ) \\left ( \\frac { d z } { d w } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 f ( t ) T _ n ( t ) \\ , \\frac { d t } { \\sqrt { 1 - t ^ 2 } } = \\sum _ { \\nu = 1 } ^ n \\sum _ { i = 0 } ^ { 2 s - 1 } A _ { i , \\nu } f ^ { ( i ) } ( \\xi _ \\nu ) + R _ { n , s } ( f ) \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} B _ { \\lambda } ( x ^ { \\mu } , x ^ { \\nu } ) & = \\dfrac { x ^ { \\mu } y ^ { \\nu } - x ^ { \\nu } y ^ { \\mu } } { x - y } \\\\ & = \\sum _ { k = 1 } ^ { \\mu - \\nu } x ^ { \\mu - k } y ^ { \\nu + k - 1 } = \\sum _ { k = 1 } ^ { \\mu - \\nu } x ^ { \\lambda - ( \\lambda - \\mu + k ) } y ^ { \\lambda - ( \\lambda - \\nu - k + 1 ) } , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\bold { F } _ { x , y } = \\left [ \\begin{array} { c | c | c | c | c } \\bold { 0 } & \\bold { A } _ { x , y ; 1 , 1 } & \\dots & \\bold { 0 } & \\bold { A } _ { x , y ; 1 , p _ y } \\\\ \\hline \\vdots & \\ddots & \\ddots & \\vdots & \\vdots \\\\ \\hline \\bold { 0 } & \\bold { A } _ { x , y ; p _ x , 1 } & \\dots & \\bold { 0 } & \\bold { A } _ { x , y ; p _ x , p _ y } \\\\ \\end{array} \\right ] . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\nu _ { \\omega + n + 1 } : = \\left [ \\nu _ { \\omega + n } ; \\nu _ { \\omega + n + 1 } ( \\phi _ { \\omega + n } ) = \\frac { 1 + \\ldots + p ^ n } { p ^ n } \\right ] . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\pi ( T ) = \\beta _ 2 ( \\varphi ( T ) - \\varphi _ T ) + \\frac { \\beta _ 3 } 2 \\ , , \\rho ( T ) = 0 \\ , . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) : = \\frac { 1 } { 2 t } \\ , \\int _ { x - t } ^ { x + t } f ( \\tau ) \\ , d \\tau \\ , . \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ n a _ { i _ 0 \\ell } \\neq \\sum _ { \\ell = 1 } ^ n a _ { j _ 0 \\ell } \\ ; \\ ; \\mbox { f o r s o m e } \\ ; \\ ; i _ 0 , j _ 0 \\in \\{ 1 , 2 , \\dots , n \\} ; \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\alpha \\coloneqq \\left [ \\alpha _ { 1 } , \\alpha _ { 2 } , \\ldots \\right ] = \\frac { 1 } { \\alpha _ { 1 } + \\frac { 1 } { \\alpha _ { 2 } + \\frac { 1 } { \\ddots } } } \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} \\mathcal { M } _ { G , b , \\mu } = \\sum \\limits _ { ( M _ S , \\mu _ S ) \\in \\mathcal { R } _ { G , b , \\mu } } ( - 1 ) ^ { L _ { M _ S , M _ b } } ( M _ S , \\mu _ S ) \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\mathfrak p \\leftrightarrow ( \\mathfrak p ' \\leftrightarrow \\mathfrak p '' ) = ( \\mathfrak p \\leftrightarrow \\mathfrak p ' ) \\leftrightarrow \\mathfrak p '' . \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\displaystyle \\mathbb { E } [ \\log ( Y ) ] = M C _ { \\alpha } ^ { M } \\int _ { 0 } ^ { \\infty } \\log ( x ) \\frac { g ( x ) ^ { M - 1 } } { x ^ { \\alpha } ( x + 1 ) } d x . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\big | \\Re \\langle \\C _ \\mu \\chi _ { Q ' } , \\chi _ { Q '' } \\rangle \\big | = \\Big | \\Re \\int _ { Q '' } \\C _ \\mu \\chi _ { Q ' } ( z ) d \\mu ( z ) \\Big | = \\int _ { Q '' } \\int _ { Q ' } \\frac { \\Re ( z - w ) } { | z - w | ^ 2 } d \\mu ( w ) d \\mu ( z ) . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\frac { h , _ { x _ i y _ j } } { h , _ { y _ j } } = \\frac { f , _ { x _ i } } { f } , \\ \\ \\ \\forall i = 1 , \\ldots , n , j = 1 , \\ldots , m . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { \\pi ( x ) } \\sum _ { \\substack { p \\leq x \\\\ p \\textrm { i s i n e r t o r r a m i f i e s i n } K } } N _ p ^ k ( E [ \\ell ] ) = \\frac { 1 } { 2 } M _ k ( \\ell ) = \\frac { \\ell - 2 } { 2 ( \\ell - 1 ) } + \\ell ^ { k } \\frac { 1 } { 2 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\frac { \\partial \\sigma _ { t + 1 } } { \\partial m } ( z ) & = s _ t ( z ) [ 1 - s _ t ( z ) ] \\frac { \\partial \\pi _ { t } } { \\partial m } ( z ) + c \\left [ C _ { t } \\left ( z \\right ) \\right ] ( 1 - c [ C _ { t } ( z ) ] ) \\left ( 1 - \\frac { \\partial \\pi _ { t } } { \\partial m } ( z ) \\right ) \\\\ & + ( s _ t ( z ) - c [ C _ { t } ( z ) ] ) ^ { 2 } \\cdot \\frac { \\partial \\sigma _ { t } } { \\partial m } ( z ) . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = \\eta \\epsilon ^ { 1 - k } N ^ { \\frac 1 2 - s } \\Lambda _ { \\bar { f } } ( 1 - s ) , \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} & \\overline { U } _ { 2 k + 1 , 2 a } ( n ) - \\overline { U } _ { 2 k + 1 , 2 a - 2 } ( n ) \\\\ & = ( x q ) ^ { 2 a } U _ { 2 k + 1 , 2 k - 2 a + 1 } ( x q ; q ) + ( x q ) ^ { 2 a - 2 } U _ { 2 k + 1 , 2 k - 2 a + 3 } ( x q ; q ) , \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\sup _ n \\ | | \\ n ^ { - 1 / 2 } \\ \\sum _ { i = 1 } ^ n X _ i | | _ { s , \\Omega } \\le Z [ \\alpha ] ( s , v ) \\ Y [ \\{ X _ i \\} ] ( v ) , \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\frac { 1 } { 8 \\pi } \\int _ { \\Sigma _ r } ( h _ 0 ^ { ( 1 ) } - h ^ { ( 1 ) } ) d S ^ 2 = & \\frac { 4 \\pi } { 3 } T ( e _ 0 , e _ 0 ) \\\\ \\frac { 1 } { 8 \\pi } \\int _ { \\Sigma _ r } \\tilde X ^ i \\tilde \\nabla ^ a ( \\alpha _ H ^ { ( 2 ) } - \\alpha _ { H _ 0 } ^ { ( 2 ) } ) _ a d S ^ 2 = & \\frac { 4 \\pi } { 3 } T ( e _ 0 , e _ i ) \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} f ( x ) = \\left ( H _ 1 ^ { \\theta - 1 } \\varphi _ f \\right ) ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} - \\Lambda _ f ( y ) = \\sum _ { i = 1 } ^ m \\alpha ^ * _ i \\nabla f _ i ( y ) \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ( N ) } \\mathrm { R e } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left | \\lambda _ g ( n ) \\right | W \\left ( \\frac { n } { X } \\right ) \\ll _ { g } X . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\{ f , C g \\} = \\{ f , C ( I - P ) g + C P g \\} \\in S , \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\frac { I ( R ) } { \\log R } = 0 . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\nu ( x , y , z ) & : = \\dfrac { 1 } { y ^ { \\lambda - 1 } } , ~ \\forall ( x , y , z ) \\in \\Omega : = \\{ ( x , y , z ) \\in \\mathbb { R } ^ { 3 } : y \\neq 0 \\} , \\\\ H ( x , y , z ) & : = x y ^ { \\lambda } , ~ \\forall ( x , y , z ) \\in \\Omega , \\\\ C ( x , y , z ) & : = \\dfrac { 1 } { 2 } \\left ( x ^ 2 + y ^ 2 + z ^ 2 \\right ) , ~ \\forall ( x , y , z ) \\in \\Omega . \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} H ( u _ 1 , u _ 2 ; 0 ) = 0 \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} | k _ a ' | \\lesssim \\sum _ { j = - \\infty } ^ \\infty \\sum _ { l = a } ^ \\infty e ^ { - { \\frac { 1 } { 2 } } | j - l | } \\left ( \\int _ { j } ^ { j + 1 } \\theta \\right ) \\left ( \\int _ { l } ^ { l + 1 } \\theta \\right ) . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\Delta _ { P _ { \\alpha } } \\left ( \\mathbf { O } \\right ) = \\Pi \\left [ \\psi _ { P _ { \\alpha } , a } ( \\mathbf { O } ; \\mathcal { G } ) | \\Lambda \\left ( P _ { \\alpha } \\right ) ^ { \\perp } \\right ] \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} \\lambda _ k ( \\gamma ) = k ^ 3 \\tanh k \\rho _ s \\ , \\big ( \\gamma - \\gamma _ k \\big ) \\mbox { f o r } \\ ; \\ ; k \\neq 0 , \\ ; k \\in \\Bbb Z . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\varphi ( 0 ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\varphi ( 1 ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\frac { 1 } { \\lambda _ 1 ( I _ j ) ^ { 1 / r } \\mathcal L ( I _ j ) } \\chi _ { I _ j } ( x ) \\leq C { \\beta ^ { 1 / r + 1 / t } } \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} E _ i \\left \\{ \\boldsymbol { \\eta } ( j ) \\boldsymbol { \\eta } ( l ) \\right \\} = \\begin{cases} \\mu _ { \\eta , i } ^ 2 \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top & , j \\neq l \\\\ \\sigma ^ 2 _ { \\eta , i } \\boldsymbol { I } + \\mu _ { \\eta , i } ^ 2 \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top & , j = l \\end{cases} , \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\rho \\left ( u _ { n } , u _ { n + 1 } \\right ) = 0 . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} \\int _ { M } u ( t , \\cdot ) = \\int _ { M } u _ 0 \\mbox { f o r a l l } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} x _ { N + 1 } = x _ N ( 1 - h _ N x ^ 2 _ N ) = - x _ N , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\langle \\xi + r + X , \\eta + s + y \\rangle = \\frac { 1 } { 2 } ( \\xi ( Y ) + \\eta ( X ) ) + ( r , s ) ^ { \\mathcal G } , \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} h ^ { n } = T _ { b } \\circ T _ { f ^ { n } ( b ) - b } \\circ f ^ { n } \\circ T _ { - b } = T _ { f ^ { n } ( b ) } \\circ f ^ { n } \\circ T _ { - b } . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\begin{cases} y ' = \\frac { ( a ^ 2 - b ) } { r } x ^ 2 - y ^ 2 - ( 2 k + a ) x y \\\\ x ' = a x ^ 2 - r x y \\end{cases} \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\sum _ { S } ( - 1 ) ^ { | S | } ( c - \\sum _ { i \\in S } \\tau _ i ) ^ { l } = \\ , \\left \\{ \\begin{array} { c l l } 0 & & 0 \\le l \\le n - 1 , \\\\ n ! \\prod _ i \\tau _ i & & l = n , \\end{array} \\right . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\begin{cases} v _ { \\tau \\tau } - \\mathrm { e } ^ { 2 \\tau } \\Delta v - \\frac { \\delta } { 4 } v = \\mathrm { e } ^ { \\frac { \\mu + 3 } { 2 } \\tau } f ( \\mathrm { e } ^ \\tau - 1 , x ) , & x \\in \\mathbb { R } ^ n , \\ \\tau > 0 , \\\\ v ( 0 , x ) = u _ 0 ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ v _ \\tau ( 0 , x ) = \\frac { \\mu - 1 } { 2 } u _ 0 ( x ) + u _ 1 ( x ) , & x \\in \\mathbb { R } ^ n . \\end{cases} \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\Pi _ { x , y } \\triangleq \\left \\{ h \\in \\bar { \\mathcal { H } } : \\Phi _ { 1 } ( x ; h ) = y \\right \\} \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} E \\leq C \\sigma ^ { \\nu \\alpha } = C W _ { 2 } ( \\sigma ^ { 2 \\nu } ) \\leq C W _ { 2 } ( \\sigma ^ { \\nu } ) ~ ~ ~ ~ W _ { 2 } , \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} X = X ^ { \\perp } = - J X _ 1 - J X _ 2 . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} ( f \\circ g ) \\otimes ( f ' \\circ g ' ) = ( f \\otimes f ' ) \\circ ( g \\otimes g ' ) \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u \\geq P \\quad , \\\\ u ( x ) = P ( x ) , \\\\ P ( y ) = \\widetilde { A } + \\widetilde { B } ( y - x ) + \\widetilde { C } \\left ( - \\frac { 1 } { 2 } | y - x | ^ 2 \\right ) \\end{array} \\right . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\varepsilon _ N = \\sigma \\sqrt { 1 / N } , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\beta ( x , t ) : = \\frac { 2 } { 5 } \\overline { \\sigma } ( x , t ) , \\ \\ \\beta ^ * ( t ) : = \\underset { x \\in \\Omega } { } \\ { \\beta } ( x , t ) , \\ \\ \\hat { \\beta } ( t ) : = \\underset { x \\in \\Omega } { } \\ { \\beta } ( x , t ) \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\left ( \\frac { t } { e ^ t - 1 } \\right ) ^ r e ^ { x t } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( r ) } ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 9 , 1 0 ] ) . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} { \\cal I } _ { a , J _ t } ^ { - 1 } Y _ t = \\left ( { \\cal I } _ { a , J _ t } ^ { - 1 } P ^ { J _ { \\tau _ k } } _ { t - \\tau _ k } { \\cal I } _ { a , J _ { \\tau _ { k - 1 } } } \\right ) \\left ( { \\cal I } _ { a , J _ { \\tau _ { k - 1 } } } ^ { - 1 } S _ k { \\cal I } _ { a , J _ { \\tau _ { k - 2 } } } \\right ) \\cdots \\left ( { \\cal I } _ { a , J _ { \\tau _ { 1 } } } ^ { - 1 } S _ 1 { \\cal I } _ { a , J _ { \\tau _ { 0 } } } \\right ) \\left ( { \\cal I } _ { a , J _ { 0 } } ^ { - 1 } P ^ { J _ 0 } _ { \\tau _ 1 } Y _ 0 \\right ) , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\mathcal { V } _ { t + 1 } [ h _ 1 , h _ 2 ] = \\sigma _ { t + 1 } [ h _ 1 , h _ 2 ] + \\mathcal { V } _ t [ \\mathcal { J } _ t h _ 1 , \\mathcal { J } _ t h _ 2 ] , t \\geq 0 , \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} f _ 0 ( x ) = \\frac 1 { \\Gamma ( 1 - \\alpha ) } \\left ( \\frac { L _ 0 } { 2 } \\right ) ^ { 1 - \\alpha } e ^ { - t ( x + 1 ) L _ 0 / 2 } , w ( x ) = ( 1 + x ) ^ { - \\alpha } \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} & u ( \\cdot , t ) \\ll v ( \\cdot , t ) \\ \\ { \\rm f o r } \\ \\ t \\in [ 0 , t _ 0 ) , \\\\ & u ( \\cdot , t _ 0 ) \\preceq v ( \\cdot , t _ 0 ) , \\ u ( \\cdot , t _ 0 ) \\not \\ll v ( \\cdot , t _ 0 ) . \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} ( T _ { \\rm o p } ) _ { \\rm r e g } = ( I - P _ r ) T _ { \\rm o p } = ( I - P _ r ) ( I - Q _ m ) T = ( I - P ) T = T _ { \\rm r e g } . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} & \\partial _ r \\varphi _ { n , k } ( r , l ) = - \\int _ r ^ l \\left ( \\frac { s _ k ( \\tau ) } { s _ k ( r ) } \\right ) ^ { n - 1 } d \\tau , \\\\ & \\partial _ r ^ 2 \\varphi _ { n , k } ( r , l ) = 1 + ( n - 1 ) \\int _ r ^ l \\left ( \\frac { s _ k ( \\tau ) } { s _ k ( r ) } \\right ) ^ { n - 1 } \\left ( \\frac { s _ k ' ( r ) } { s _ k ( r ) } \\right ) d \\tau , \\\\ & \\partial _ r ^ 2 \\varphi _ { n , k } ( r , l ) + \\frac { ( n - 1 ) s _ k ' ( r ) } { s _ k ( r ) } \\partial _ r \\varphi _ { n , k } ( r , l ) = 1 . \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\bigg \\| \\Big ( \\sum _ j | T ( f _ 1 ^ j , \\dots , f _ m ^ j ) | ^ s \\Big ) ^ \\frac 1 s \\bigg \\| _ { L ^ { q } ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\bigg \\| \\Big ( \\sum _ j | f _ i ^ j | ^ { s _ i } \\Big ) ^ \\frac 1 { s _ i } \\bigg \\| _ { L ^ { q _ i } ( v _ i ) } , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} x _ 1 = \\underbrace { x _ 0 ( 1 - h _ 0 x _ 0 ^ 2 ) + \\frac { u _ 1 } { 2 } } _ { = 0 } + \\frac { u _ 1 } { 2 } < 0 . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} g ( r , y ) = h ( r , y ) \\gamma ( y ) . \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} | \\theta ( t ) - \\theta ( u ) | & = \\theta ( u ) = \\dfrac { ( 2 n + 1 ) ( u - a _ { 2 n + 1 } ) } { b _ { 2 n + 1 } - a _ { 2 n + 1 } } = \\dfrac { ( 2 n + 1 ) ( u - a _ { 2 n + 1 } ) } { ( a _ { 2 n + 1 } ) ^ { 2 / k } } \\\\ & < \\dfrac { ( 2 n + 1 ) t ^ { 1 / k } } { ( a _ { 2 n + 1 } ) ^ { 2 / k } } < \\dfrac { ( 2 n + 1 ) ( a _ { 2 n + 1 } ) ^ { 1 / k } } { ( a _ { 2 n + 1 } ) ^ { 2 / k } } = \\dfrac { 2 n + 1 } { ( a _ { 2 n + 1 } ) ^ { 1 / k } } \\\\ & < \\dfrac { 2 n + 1 } { 2 ^ { 2 n + 1 } } = \\ell ( 2 n + 1 ) . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} c _ 0 ( \\Gamma ) ^ { \\times } = p ^ { \\Z } \\mu _ { p - 1 } \\Gamma U ^ 1 _ { \\Gamma } \\ ; \\Gamma = \\Z ^ N \\ ; . \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ' ( c n ) \\leq \\lfloor n - p _ n ^ { - 1 } \\rfloor ) = - \\infty . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\begin{aligned} d x ( t ) & = v ( t ) \\ , d t , \\\\ m \\ , d v ( t ) & = \\big ( - \\gamma v ( t ) - \\Phi ' ( x ( t ) ) - \\sum _ { k \\geq 1 } \\sqrt { c _ k } z _ k ( t ) \\big ) \\ , d t + \\sqrt { 2 \\gamma } \\ , d W _ 0 ( t ) , \\\\ d z _ k ( t ) & = \\left ( - \\lambda _ k z _ k ( t ) + \\sqrt { c _ k } v ( t ) \\right ) \\ , d t + \\sqrt { 2 \\lambda _ k } \\ , d W _ k ( t ) , k \\geq 1 , \\end{aligned} \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\eta = \\epsilon = \\frac 1 2 \\left ( \\frac { t - s } { 2 } + \\left [ \\left ( \\frac { t - s } { 2 } \\right ) ^ 2 + 4 \\right ] ^ { 1 / 2 } \\right ) \\ , . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} X _ { j _ 0 } = Y \\left ( \\left \\{ X _ { j } : j \\in \\mathcal J _ { m } \\setminus \\{ j _ 0 \\} \\right \\} \\right ) . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} d \\left ( f ( x , y ) \\cdot a \\otimes b \\otimes m + f ( y , x ) \\cdot b \\otimes a \\otimes m \\right ) = 0 . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} F ( p ^ { \\pm } ) = ( 0 , 0 , \\cdots , 0 ) , \\ \\ D F ( p ^ { \\pm } ) \\varphi ^ { \\pm } = - \\lambda _ { \\pm } \\varphi ^ { \\pm } . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} \\mathcal U : = & \\Big \\{ ( u , w ) \\in L ^ 2 ( \\Omega ; L ^ 2 ( 0 , T ; H ) ) ^ 2 0 \\leq u , w \\leq 1 \\Omega \\times ( 0 , T ) \\times D \\Big \\} \\ , , \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} m _ \\epsilon ( x ) = F ^ { - 1 } _ \\epsilon \\Big ( \\overline { H } _ \\epsilon ( P ) - V \\Big ( x , \\frac { x } { \\epsilon } \\Big ) \\Big ) , \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 4 { R _ 4 ^ 4 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 4 { R _ 4 ^ 4 } ) = 5 0 , \\mathrm { I C } ( \\mathcal { T } ^ 4 { R _ 4 ^ 4 } ) = 8 9 . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { \\sigma \\in S _ n } { { v o l } ( D _ { i , a } \\cap { \\mathfrak { D } } ( f , \\sigma ) ) } \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\Delta _ N } \\left ( \\boldsymbol { \\pi } \\right ) \\underset { N } { \\wedge } \\mathcal { B } _ { n } \\left ( p \\right ) = \\mathcal { M } _ { \\blacktriangle _ { n } } \\left ( p \\cdot \\boldsymbol { \\pi } \\right ) , \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} \\Delta u _ { x ^ * , \\rho _ 0 } ( y ) = \\widetilde { Q } ( y , u _ { x ^ * , \\rho _ 0 } , \\nabla u _ { x ^ * , \\rho _ 0 } ) \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} P _ { \\tilde { X } _ i } ( x ) = \\frac { { { x - 1 } \\choose { k - 1 } } ( 1 - \\delta ) ^ k \\delta ^ { x - k } } { \\epsilon _ { k , n } ( \\delta ) } , \\ \\ x = k , k + 1 , \\ldots , n \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = 0 , \\mbox { w h e r e } f ( x , u ) = \\begin{cases} \\displaystyle f _ l ( u ) = \\frac 1 2 ( u - 1 ) ^ 2 , & x \\le 0 , \\\\ [ 2 m m ] \\displaystyle f _ r ( u ) = \\frac 1 2 { u ^ 2 } , & x > 0 . \\end{cases} \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\sum _ { Q \\in \\mathcal { S } } | Q | \\prod _ { m = 1 } ^ { n + 1 } \\langle | g _ m | \\rangle _ Q \\lesssim _ { \\eta _ 1 , \\eta _ 2 } \\sum _ { Q \\in \\mathcal { U } } | Q | \\prod _ { m = 1 } ^ { n + 1 } \\langle | g _ m | \\rangle _ Q . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } \\right ) ( z , Q ) = \\sum _ { i = 0 } ^ N \\left ( Q ^ \\frac { \\lambda _ i } { z } \\sum _ { d \\geq 0 } Q ^ d \\prod _ { r = 1 } ^ d \\prod _ { j = 0 } ^ N \\frac { 1 } { ( \\lambda _ i - \\lambda _ j + r z ) } \\right ) \\Psi _ i \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} a ( w , x ) : = \\lim _ { y \\to x } \\frac { f _ w ( x ) - f _ w ( y ) } { ( x - y ) ^ { p ^ { n _ i } } } \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} \\phi ( x ) = \\eta ( \\rho _ 0 ^ { - 1 } ( x - x _ 0 ) ) x \\in \\mathbb R ^ N . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} A _ t f : = Q ^ V f ( Y _ 0 , B _ 0 ) - 2 \\int _ 0 ^ { t \\wedge \\tau } V ( Y _ s ) Q ^ V f ( Y _ s , B _ s ) d s . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} \\left ( \\phi _ i \\star _ \\tau \\phi _ j \\right ) _ { | Q = 0 } = \\phi _ i \\otimes \\phi _ j \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} \\mathcal { B ' } = \\mathcal { B } \\cup \\{ a ( \\lambda ) b ( \\lambda ' ) - a ( \\lambda ' ) b ( \\lambda ) , \\lambda ' \\in \\mathcal { A } \\setminus \\{ \\lambda \\} \\} , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\partial _ x \\int _ B G ( x , z ) P _ { B ^ c } ( y , z ) d z & = \\partial _ x \\int _ B G ( x , z ) P _ 1 ( y , z ) d z + \\partial _ x \\int _ B G ( x , z ) P _ 2 ( y , z ) d z \\\\ & = \\int _ B \\partial _ x G ( x , z ) P _ 1 ( y , z ) d z + \\int _ B \\partial _ x G ( x , z ) P _ 2 ( y , z ) d z \\\\ & = \\int _ B \\partial _ x G ( x , z ) P _ { A } ( y , z ) d z , \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} R _ { n } ^ { \\lambda } \\left ( x \\right ) = S ( t ) C _ { n } ^ { \\lambda } ( t ) , S ( t ) : = \\sqrt { \\omega _ \\lambda ( t ) \\frac { d x } { d t } } = ( 1 - t ^ 2 ) ^ { \\frac { \\lambda + 1 } 2 } , \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\{ \\gamma _ i , \\gamma _ j \\} = 2 \\eta _ { i j } , \\eta = ( \\underbrace { + 1 , \\dots , + 1 } _ { p } , \\underbrace { - 1 , \\dots , - 1 } _ { q } ) \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\varphi _ n ^ { ( k ) } ( z ) = \\sum _ { i = 0 } ^ { k } u _ { n - i } ( z ) M _ { n - i } \\left ( q ^ { - z } ; b , c ; q \\right ) , \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} \\textbf { P } = \\eta \\textbf { K } \\hat { \\textbf { H } } , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} W _ l ( x ; \\tau _ 1 , \\dots , \\tau _ n ) = 0 { } ^ \\exists \\tau _ i = 0 . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} V ( x ) = b R ( \\mu x + y _ 0 ) , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} f ( x ) = x + \\alpha x ^ 3 . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} A ^ t W + W A + W ' + \\lambda W = 0 . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\langle D J _ 1 ( f ; v ^ 1 , v ^ 2 ) , w ^ 2 \\rangle = \\alpha _ 1 \\iint _ { { \\mathcal { O } } _ { 1 , d } \\times ( 0 , T ) } ( y - y _ { 1 , d } ) z \\ d x d t + \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } v ^ 1 w ^ 2 d x d t \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} d & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\min ( \\theta _ n , ~ \\sigma _ { n , i } ^ 2 ) , \\\\ d _ n & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\min ( \\theta , ~ \\sigma _ { n , i } ^ 2 ) , \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\int _ { T , S , h _ 0 } [ \\langle h y ^ h _ t , h \\varphi _ t \\rangle - D W ( \\nabla _ h y ^ h ) : \\nabla \\varphi P _ h + h \\sqrt { e ^ h } \\langle f , \\varphi \\rangle ] \\det F ( \\frac { s h } { h _ 0 } ) d s d x d t = 0 , \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} t \\partial P _ t = - 2 \\int ^ \\infty _ 0 t ^ 2 s ^ 2 \\varphi ^ \\prime ( s ) \\partial M _ { t ^ 2 s } d s = - 2 \\int ^ \\infty _ 0 s \\varphi ^ \\prime ( s ) ( t ^ 2 s \\partial M _ { t ^ 2 s } ) d s \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\bigcup _ { n = 1 } ^ { \\infty } F _ { n } ( ( \\mathbb { R } ^ { d } ) ^ { n } ) \\backslash E \\subseteq { \\cal P } . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} h _ z ( \\gamma ) = h _ x ( \\gamma ) = h _ y ( \\gamma ) , \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} | \\sin \\pi \\hat \\theta _ j | = \\min _ { \\ell \\in B _ j } | \\sin \\pi ( 2 \\theta + ( \\ell + i ) \\alpha ) | . \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} K _ 1 = \\infty & & K _ 2 = 0 & & C = C _ 1 = 2 \\delta + 1 . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} d ( \\nabla Y ^ n _ t ) = [ \\nabla ^ 2 u _ n ( t , \\Phi _ n ^ { - 1 } ( t , Y ^ n _ t ) ) \\nabla \\Phi _ n ^ { - 1 } ( t , Y ^ n _ t ) \\nabla Y ^ n _ t ] d B _ t . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} d ( \\varphi _ { w _ { 1 } , i } , \\varphi _ { w _ { 2 } , i } ) \\ge c ^ { n _ { j } } i = 1 , 2 , \\ : j \\ge 1 w _ { 1 } , w _ { 2 } \\in \\Lambda _ { i } ^ { n _ { j } } w _ { 1 } \\ne w _ { 2 } \\ : . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} ( 1 + \\epsilon ) \\Phi ( n ) - ( 1 - \\epsilon ) \\Phi ( n - 1 ) = ( 1 + \\epsilon ) e ^ { n ^ \\alpha } - ( 1 - \\epsilon ) e ^ { ( n - 1 ) ^ \\alpha } \\leq ( 1 + \\epsilon ) e ^ { n ^ \\alpha } , \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{gather*} \\mathbf { F } _ { \\frac { n + 1 } { 2 } } : = \\left [ \\frac { ( \\lambda _ { 2 k + 1 } - \\lambda _ { 1 } ) \\sin \\left [ \\frac { ( 2 k + 1 ) \\pi } { n + 3 } \\right ] } { \\lambda _ { 2 k + 1 } - \\phi _ { \\ell } } \\right ] _ { k , \\ell } \\\\ \\mathbf { G } _ { \\frac { n - 1 } { 2 } } : = \\left [ \\frac { ( \\lambda _ { 2 k } - \\lambda _ { n + 1 } ) \\sin \\left ( \\frac { 2 k \\pi } { n + 3 } \\right ) } { \\lambda _ { 2 k } - \\psi _ { \\ell } } \\right ] _ { k , \\ell } \\end{gather*}"}
-{"id": "3059.png", "formula": "\\begin{align*} P _ q \\cdot [ A _ q ] : = \\left ( q ^ { Q \\partial _ Q } P _ q \\right ) A _ q P _ q ^ { - 1 } \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} h ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\gamma ( n ) \\log ( ( 1 - e ^ { - s n } ) ^ { - 1 } ) = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\phi ( m s ) , \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { 0 \\le t \\le T } | J _ { 1 , n } ( t ) | _ E = 0 . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} | \\nabla u _ n ( t , X _ t ^ n ) - \\nabla u ( t , X _ t & ) | = | ( \\nabla u _ n ( t , X _ t ^ n ) - \\nabla u _ n ( t , X _ t ) ) + ( \\nabla u _ n ( t , X _ t ) - \\nabla u ( t , X _ t ) ) | \\\\ & \\leq \\bar { C } ( | \\nabla u _ n ( t , X _ t ^ n ) - \\nabla u _ n ( t , X _ t ) | + \\norm { b _ n - b } _ { L ^ { q , 1 } _ t ( L ^ p _ x ) } ) . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\frac { \\partial w _ m } { \\partial y _ 1 ^ { p ^ s } } = i _ 1 \\sum _ { i _ 1 = 1 } ^ { \\alpha _ 1 } ( y _ 1 ^ { p ^ s } ) ^ { i _ 1 - 1 } v _ { i _ 1 } ( y _ 2 ^ { p ^ s } , y _ 3 ^ { p ^ s } , \\cdots , y _ 8 ^ { p ^ s } ) = 0 . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\mathbb { C } [ \\ ! [ Q ] \\ ! ] = \\left \\{ \\sum _ { d \\in H _ 2 ( X ; \\mathbb { Z } ) } f _ d Q ^ d \\ , \\middle | \\ , f _ d \\in \\mathbb { C } \\right \\} \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} & \\begin{cases} \\partial _ t \\mathbb { U } ( A _ t ) + \\inf _ { u \\in \\mathrm { U } } \\sup _ { v \\in \\mathrm { V } } \\mathcal { H } ( A _ t , u , v , ( \\mathbb { U } , \\partial _ x \\mathbb { U } , \\partial _ { x x } \\mathbb { U } ) ( A _ t ) ) = 0 , \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ t \\in [ 0 , T ) , ~ A _ t \\in \\Lambda \\\\ \\mathbb { U } ( A _ T ) = m ( A _ T ) , ~ A _ T \\in \\Lambda _ T . \\end{cases} \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} ( a \\vee b ) ^ { - 1 } ( a \\vee b ) = ( a ^ { - 1 } a ) \\vee ( b ^ { - 1 } b ) , \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\small { W _ { t + 1 } ( x , y ) = q _ t W _ t ( x , y ) + W ( x , y ) [ 1 - W _ t ( x , y ) ] f \\left ( \\frac { d _ { W _ t } ( x ) + d _ { W _ t } ( y ) } { 2 } \\right ) . } \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} \\mathcal U ( U _ c , U _ d ) = w _ c \\ln U _ c + w _ d \\ln U _ d , \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} p = p _ 0 + p _ 1 q + \\ldots + p _ s q ^ s \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} k = a \\left ( 2 ^ R - 1 \\right ) + b , \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} V ( t , x ) = \\kappa \\left ( \\sum _ { n \\in \\mathbb { Z } } \\delta ( t - n ) \\right ) \\cos ( 2 \\pi x ) , \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} & S _ { 2 , \\lambda } ( n , k ) + ( n - 1 ) \\lambda S _ { 2 , \\lambda } ( n - 1 , k ) \\\\ & = \\sum _ { m = k - 1 } ^ { n - 1 } { m \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , m ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { m - k + 1 } ] . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\frac { d ^ 2 R } { d t ^ 2 } ( s ) = t r \\Big ( A _ s ^ { - 1 } \\frac { d ^ 2 A _ t } { d t ^ 2 } ( s ) \\Big ) - t r \\Bigg ( \\Big ( A _ s ^ { - 1 } \\frac { d A _ t } { d t } ( s ) \\Big ) ^ 2 \\Bigg ) . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} U _ l = U _ l ( n ) : = \\left \\{ \\frac { \\eta _ 1 n } { 2 N } \\leq N _ l \\leq \\frac { 2 \\eta _ 2 n } { N } \\right \\} , \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} ( t _ i - t _ j ) ( t _ i - t _ k ) = t _ i + \\delta _ { j k } t _ j \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} R i c _ g + H e s s _ g f - \\frac { 1 } { r } d f \\otimes d f = \\rho g , \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\Sigma _ \\epsilon = \\Sigma - \\frac { 1 } { 2 r } F \\otimes F \\leq \\Sigma - \\frac { 1 } { 2 r } F \\otimes F + \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ^ 2 \\Big ) F \\otimes F = \\tilde \\Sigma , \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\partial _ t \\rho _ n + \\div ( u _ n \\rho _ n ) = 0 \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} f _ { ( c } \\ = _ { d e f } \\ f ^ { - 1 } ( c , \\infty ) \\mbox { a n d } f _ { [ c } \\ = _ { d e f } \\ f ^ { - 1 } [ c , \\infty ) , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} M _ b = M _ 1 \\times . . . \\times M _ k \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} | \\gamma _ { i } ^ { k } | & = \\| f _ { i } ^ { k } \\| _ { L ^ { \\infty } } | B ( y _ { i } , 2 ^ { k } r ) | ^ { 1 / p } \\\\ & \\leq \\frac { | B ( y _ { i } , r ) | } { | B ( y _ { i } , 2 ^ { k - 1 } r ) | } \\| f _ { i } \\| _ { L ^ { \\infty } } | B ( y _ { i } , 2 ^ { k } r ) | ^ { 1 / p } \\\\ & \\lesssim C _ { i } ( 2 ^ { ( k - 1 ) } ) ^ { n } ( 2 ^ { k n } | B ( y _ { i } , r ) | ) ^ { 1 / p } \\\\ & \\lesssim C _ { i } 2 ^ { k n ( \\frac { 1 } { p } - 1 ) } | B ( y _ { i } , r ) | ^ { 1 / p } . \\\\ \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a ( n ) z ^ n = \\prod _ { j = 1 } ^ { \\infty } ( 1 - z ^ j ) ^ { - \\eta ( j ) } . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} g ^ { ( i ) } ( t ) = g _ \\infty ^ { ( i ) } ( t ) ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} \\liminf _ { k \\to + \\infty } \\lambda _ { 1 , k } = \\lambda _ 1 > 0 \\textup { a n d } \\lim _ { k \\to + \\infty } Z _ k = Z \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\tilde { \\mathcal { A } } _ { F S } = \\begin{pmatrix} A _ q & - A _ q \\mathbb { P } S \\\\ 0 & 0 \\end{pmatrix} , B _ { F S } = \\begin{pmatrix} 0 & 0 \\\\ K ^ { - 1 } C _ 1 & K ^ { - 1 } C _ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} x _ { \\pi ( i _ 1 ) } > x _ { \\pi ( i _ 2 ) } > \\ldots > x _ { \\pi ( i _ { j - 1 } ) } > x _ { \\pi ( i _ j ) } = x _ { \\pi ( i _ { j + 1 } ) } > \\ldots > x _ { \\pi ( i _ k ) } \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} e ( y ) = c ( y , q _ 0 ) + c ( y , q _ 1 ) - c ( q _ 0 , q _ 1 ) . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ { k } } { [ 2 k ] ^ 2 } \\equiv \\frac { ( n ^ 2 - 1 ) ( 1 - q ) ^ 2 } { 2 4 } \\pmod { \\Phi _ n ( q ) } . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\delta _ N = 2 C \\frac { n } { N } - \\log { D } - \\log { N } - \\frac { 1 } { 2 } \\log { n } . \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} 0 & = \\langle A \\varpi , \\varpi \\rangle = B [ \\varpi , \\varpi ] = \\int _ { \\mathcal { Y } ^ d } e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + \\lambda V } \\big ( \\big ( ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T \\nabla _ y \\varpi \\big ) ^ 2 + | \\nabla _ y \\varpi | ^ 2 \\big ) d y . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} a b = \\left ( a b ^ { \\frac { 2 - p } { p } } \\right ) b ^ { 2 \\frac { p - 1 } { p } } \\le \\frac { 1 } { p } \\left ( a b ^ { \\frac { 2 - p } { p } } \\right ) ^ p + \\frac { 1 } { q } b ^ { 2 \\frac { p - 1 } { p } q } = \\frac { 1 } { p } a ^ p b ^ { 2 - p } + \\left ( 1 - \\frac { 1 } { p } \\right ) b ^ 2 . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} A ( n ) = \\sum _ { k \\in \\mathbb { Z } } \\binom { n } { k } ^ 2 \\binom { n + k } { k } ^ 2 , \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} g _ { ( a ) } ( z , t ) = \\Phi [ f ( z , t ) ] = s ( z , t ) \\left ( \\frac { f ( z , 1 ) - t f ( z , t ) } { 1 - t } \\right ) . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} & | H _ 0 | \\sqrt { V ^ 2 + V ^ 4 | \\nabla \\tau | ^ 2 + \\frac { ( d i v V ^ 2 \\nabla \\tau ) ^ 2 } { | H _ 0 | ^ 2 } } \\\\ = & A \\Big [ \\frac { 2 } { r } + \\frac { 2 A B _ i \\tilde { X } ^ i } { A ^ 2 } + r ( h _ 0 ^ { ( 1 ) } + \\frac { A ^ 2 + ( C _ i \\tilde X ^ i ) ^ 2 + g _ 1 + \\frac { g _ 2 } { 4 } - h _ 0 ^ { ( 1 ) } \\sum _ { i j } C _ i C _ j \\tilde { X } ^ i \\tilde { X } ^ j } { A ^ 2 } ) \\Big ] + O ( r ^ 2 ) . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\delta _ \\lambda ( \\xi ) = ( \\lambda z , \\lambda ^ 2 t ) \\in \\Omega \\quad \\forall \\lambda \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} S _ { p , p - k } & = \\frac { ( p + 1 + k ) ! } { p ! ( p - k ) ! } \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + 1 } \\frac { 1 } { ( k + j ) ! ( k - j ) ! ( p + 1 + j ) } S ( j + k , j ) \\\\ & = \\frac { p + 1 } { ( 2 k ) ! ( p - k ) ! } \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + 1 } \\binom { 2 k } { k + j } \\frac { C ( p , k ) } { p + 1 + j } S ( j + k , j ) \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} z C ^ { ( 1 + \\alpha ) } _ n ( z ) = \\frac { n + 1 } { 2 ( n + \\alpha + 1 ) } C ^ { ( 1 + \\alpha ) } _ { n + 1 } ( z ) + \\frac { n + 2 \\alpha + 1 } { 2 ( n + \\alpha + 1 ) } C ^ { ( 1 + \\alpha ) } _ { n - 1 } ( z ) , \\ \\ \\ n = 1 , 2 , 3 , \\ldots \\ . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} H e s s _ { \\overline { g } } f ( X _ { i } , X _ { j } ) = f _ { , x _ i x _ j } - \\sum _ k \\overline { \\Gamma } _ { i j } ^ k f _ { , x _ k } , \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} T _ d : = q _ 0 ^ { - 1 } X _ d T _ { d - 1 } ^ { - 1 } \\cdots T _ 1 ^ { - 1 } T _ 0 ^ { - 1 } T _ 1 ^ { - 1 } \\cdots T _ { d - 1 } ^ { - 1 } . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} T _ 1 + T _ 2 = \\{ \\ , \\{ f , h + k \\} : \\ , \\{ f , h \\} \\in T _ 1 , \\ , \\ , \\{ f , k \\} \\in T _ 2 \\ , \\} . \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\big ( ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { m } \\big ) _ { y _ i } \\widetilde { w } _ { x _ j } d y = 0 . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} ( A ^ { * m } | A ^ n | ^ \\frac { 2 n } { n } A ^ m ) ^ \\frac { m } { m + n } = ( A ^ { * m } A ^ { * n } A ^ n A ^ m ) ^ \\frac { m } { m + n } = | A ^ { m + n } | ^ \\frac { 2 m } { m + n } = | A | ^ { 2 m } = | A ^ m | ^ 2 \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\Delta _ L ^ { \\perp } = - \\nabla _ { E _ 1 } ^ { \\perp } \\nabla _ { E _ 1 } ^ { \\perp } - \\nabla _ { E _ 2 } ^ { \\perp } \\nabla _ { E _ 2 } ^ { \\perp } . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\mathbb { E } \\langle \\zeta _ 2 , h \\rangle = n ^ { 1 / 2 } \\bar { h } ( q _ 2 - 2 q _ 1 ) ^ 2 p _ 0 ^ 2 ( 1 - p _ 0 ) ( q _ 1 + q _ 2 - 2 p _ 0 q _ 1 ) \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} a _ i = 0 & \\Rightarrow b _ i = 0 , \\ , \\ , c _ i = * , \\\\ a _ i = 1 & \\Rightarrow b _ i = * , \\ , \\ , c _ i = 0 , \\\\ a _ i = * & \\Rightarrow b _ i = * , \\ , \\ , c _ i = * . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} O _ { I , I ' } = \\bigl \\{ \\theta \\in \\{ \\pm 1 \\} ^ { \\mathcal { D } } : | \\langle T b _ I ^ { ( \\theta ) } , b _ { I ' } ^ { ( \\theta ) } \\rangle | > \\eta _ 0 \\bigr \\} , I , I ' \\in \\mathcal { D } _ { \\leq n } , \\ I \\neq I ' \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{gather*} \\left ( \\frac { F ( x ) ^ p - ( 1 + \\lambda x ) G ( f ) } { \\lambda ^ p } \\cdot f - ( 1 + \\lambda x ) \\right ) \\cdot ( 1 / y ) ^ p + \\frac { p f } { \\lambda ^ { p - 1 } } F ( x ) ^ { p - 1 } \\cdot ( 1 / y ) ^ { p - 1 } \\\\ \\qquad + \\cdots + \\frac { p f } { \\lambda } F ( x ) \\cdot ( 1 / y ) + f = 0 \\end{gather*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\breve { y } _ s ^ { t , t + \\tau , A _ t ; U , V } & = b + \\int _ s ^ { t + \\tau } l ( X _ r ^ { t , A _ t ; U , V } , \\breve { y } _ r ^ { t , t + \\tau , A _ t ; U , V } , \\breve { q } _ r ^ { t , t + \\tau , A _ t ; U , V } , U _ r , V _ r ) \\dd r \\\\ & ~ ~ ~ - \\int _ s ^ { t + \\tau } \\breve { q } _ r ^ { t , t + \\tau , A _ t ; U , V } \\dd B _ r , ~ s \\in [ t , t + \\tau ] . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\nabla \\ ! g ( \\overline { x } ) \\Delta \\lambda = 0 , \\\\ \\Pi _ { \\mathcal { K } } ' ( g ( \\overline { x } ) + \\ ! \\overline { \\lambda } ; \\Delta \\lambda ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} x = \\sum _ { k = 1 } ^ { m } w _ { k } y _ { k } , \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} B [ v , \\varpi ] = \\int _ { \\mathcal { Y } ^ d } e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + \\lambda V } \\big ( ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T \\nabla _ y v ( \\Lambda + \\nabla _ y \\widetilde { w } ) + \\nabla _ y v ) \\big ) ^ T \\nabla _ y \\varpi d y . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} B _ { j k } = \\left \\{ \\begin{array} [ c ] { c } 1 \\left ( j , k \\right ) \\in I \\\\ 0 \\end{array} \\right . Y _ { j k } = \\left \\{ \\begin{array} [ c ] { c } 1 \\left ( j , k \\right ) \\in J \\\\ - 1 \\left ( j , k \\right ) = \\left ( p , 1 \\right ) \\\\ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\lvert K _ \\lambda ( x ) \\rvert \\ , d x = \\int _ 0 ^ \\infty r e ^ { - \\lvert \\lambda \\rvert ^ { 1 / 2 } \\cos ( \\psi / 2 ) r } \\ , d r \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} t _ \\varepsilon ( x ) = \\log \\left ( 1 + \\frac { | x - x _ \\varepsilon | ^ 2 } { \\mu _ \\varepsilon ^ 2 } \\right ) \\ , . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} d \\Gamma ( t ) \\phi = \\Gamma ( t ) \\left [ \\sum _ { i = 1 } ^ N ( B _ i ( t ) + \\theta _ i ) \\phi d \\beta _ i ( t ) \\right ] . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} P ( Q ) = \\begin{pmatrix} Q ^ { - 1 } & 0 \\\\ 0 & Q ^ { 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} \\rho _ 0 ( s ) : = \\lim _ { ( \\xi _ 1 , \\xi _ 2 ) \\to ( 0 , 0 ) } \\rho ( s , \\xi _ 1 , \\xi _ 2 ) \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} F = d \\mu = F ^ Z \\xi _ 1 + F ^ W \\xi _ 2 . \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} e ^ { \\alpha ^ c } ( \\theta ^ \\beta _ \\pm e ^ \\beta + e ^ { \\alpha + \\beta } ) & = b _ { \\alpha ^ c , \\beta } \\theta ^ \\beta _ \\pm e ^ { \\alpha ^ c + \\beta } + b _ { \\alpha ^ c , \\alpha + \\beta } e ^ { \\beta ^ c } \\\\ & = b _ { \\alpha ^ c , \\beta } ( \\theta ^ { \\alpha ^ c + \\beta } _ \\pm e ^ { \\alpha + \\beta ^ c } + e ^ { \\beta ^ c } ) \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\langle x \\mapsto x ^ j \\exp ( \\mu x ) \\cos ( \\nu x ) , x \\mapsto x ^ j \\exp ( \\mu x ) \\sin ( \\nu x ) : j = 0 , \\ldots , n - 1 \\rangle . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} h ( x ) & \\le g _ 3 ( x ) \\Bigr | _ { x = \\frac { 8 7 } { 2 1 1 } } + \\frac 1 { 1 0 \\cdot 6 ^ 3 } - \\eta \\\\ & = - \\frac { 4 0 8 7 5 6 2 1 2 8 8 4 6 7 2 6 8 0 8 4 0 9 6 6 5 1 4 9 1 7 1 7 0 1 9 8 3 } { 2 9 2 4 2 5 8 1 3 7 4 3 6 7 6 4 6 8 7 1 5 4 8 6 2 7 6 2 6 6 6 6 6 2 1 4 6 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} x = \\frac { t } { \\sqrt { 1 - t ^ 2 } } \\quad \\ ; { \\rm o r } \\quad \\ ; t = \\frac { x } { \\sqrt { 1 + x ^ 2 } } , t \\in I , \\ ; \\ ; x \\in { \\mathbb R } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 } \\alpha _ { i } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } & = \\alpha _ { 1 } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } + \\alpha _ { 2 } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } - \\alpha ^ { J _ { 0 } } \\chi _ { B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } + 1 } ) } + \\alpha ^ { J _ { 0 } } \\chi _ { B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } + 1 } ) } \\\\ & = f _ { 1 } ^ { J _ { 0 } + 1 } + f _ { 2 } ^ { J _ { 0 } + 1 } \\\\ & = \\sum _ { i = 1 } ^ { 2 } f _ { i } ^ { J _ { 0 } + 1 } . \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\nabla f _ 1 ( z _ t ) - \\nabla f _ 2 ( z _ t ) = z _ t - ( 1 , 0 ) = ( 0 , \\cos t \\sin t ) \\perp \\nabla f _ 2 ( z _ t ) . \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\sigma ( x , y ) = \\left ( - \\frac { b y } c , - \\frac { c x } b \\right ) . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} \\mathrm { m o d } _ p ( d \\varphi , \\iota ; l ) = \\left ( \\frac { 2 } { t } \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ^ + _ { ( d \\varphi , \\iota ) } } + \\frac { 1 } { t } \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ^ 0 _ { ( d \\varphi , \\iota ) } } \\right ) \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\phi ) + \\ell _ { \\beta } ( \\phi ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } - \\frac { k _ l \\cdot \\mathrm { g c d } ( s , t ) } { t } . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} A ^ { * ( p + q ) } A ^ { p } A ^ { * p } A ^ { p + q } & = A ^ { * q } ( A ^ { * p } A ^ { p } ) ^ 2 A ^ q = A ^ { * q } ( A ^ { * } A ) ^ { 2 p } A ^ q \\\\ & = A ^ { * q } ( A ^ { * 2 p } A ^ { 2 p } ) A ^ q = A ^ { * 2 p + q } A ^ { 2 p + q } = ( A ^ * A ) ^ { 2 p + q } . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} \\lambda = - \\mathbb { R } ' ( A _ 1 ^ n , B _ 1 ^ n , d ) . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} \\pi ^ T ( X ) : = X - \\eta ( X ) Z . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} Z ( a ) = \\begin{pmatrix} \\tau _ 1 ( a ( z _ 1 ) ) & & \\\\ & \\tau _ 2 ( a ( z _ 2 ) ) & \\\\ & & \\ddots \\end{pmatrix} \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\epsilon _ { k , n } ( \\delta ) = \\sum _ { x = k } ^ { n } { { x - 1 } \\choose { k - 1 } } ( 1 - \\delta ) ^ k \\delta ^ { x - k } \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\nu _ \\gamma ( a _ 0 + a _ 1 x + \\ldots + a _ n x ^ n ) : = \\min _ { 0 \\leq i \\leq n } \\{ \\nu ( a _ i ) + i \\gamma \\} \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} w _ { 1 , 2 } = \\frac { q ^ 2 \\pm [ q ^ 2 - \\frac { 2 q \\xi ( q ) } { \\xi ' ( q ) } ] } { \\frac { 2 [ q \\xi ' ( q ) - \\xi ( q ) ] ( 1 - q ) } { \\xi ' ( 1 ) - \\xi ' ( q ) } } > 0 . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} f ( x ) = 0 . 5 \\| P x - q \\| ^ { 2 } , \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\{ x ' = 0 \\in \\R ^ k \\} , k \\ge 2 , \\mbox { w h e r e } x ' : = ( x _ 1 , . . , x _ k ) , x '' : = ( x _ { k + 1 } , . . . , x _ d ) . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} ( \\l _ { h } ^ { ( i ) } - \\l ) b _ h ( u _ { h } , u _ { h } ) & = a ( u - u _ { h } , u - u _ { h } ) - \\l b ( u - u _ { h } , u - u _ { h } ) \\\\ & \\quad + ( a _ { h } ( u _ { h } , u _ { h } ) - a ( u _ { h } , u _ { h } ) ) + \\l \\left [ b ( u _ { h } , u _ { h } ) - b _ { h } ( u _ { h } , u _ { h } ) \\right ] . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\rho ( i ) & = \\begin{cases} 2 i & \\mbox { $ i \\leq \\delta / 2 $ } \\\\ 2 ( \\delta - i ) + 1 & \\mbox { $ i > \\delta / 2 $ } \\end{cases} & \\rho ^ { - 1 } ( i ) & = \\begin{cases} i / 2 & \\mbox { $ i $ e v e n } \\\\ \\delta - \\frac { i - 1 } { 2 } & \\mbox { $ i $ o d d } \\end{cases} \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} s ( t ) = \\frac { b } { 2 } \\ln \\left ( b - t \\right ) - \\frac { a } { 2 } \\ln \\left ( t - a \\right ) , \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) = P ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\left ( q \\Lambda _ 0 P ^ { - 1 } , \\dots , q \\Lambda _ N P ^ { - 1 } ; q \\right ) _ d } \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} \\phi _ n = ( \\eta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ s , 1 ) . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} C ' _ { k _ 1 , j _ 1 } & = C _ { k _ 1 , j _ 1 } \\setminus A _ 1 , C ' _ { k _ 2 , j _ 2 } = C _ { k _ 2 , j _ 2 } \\setminus A _ 2 , \\\\ C ' _ { k _ 1 , j _ 2 } & = C _ { k _ 1 , j _ 2 } \\cup A _ 1 , C ' _ { k _ 2 , j _ 1 } = C _ { k _ 2 , j _ 1 } \\cup A _ 2 . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\Psi _ 1 ( k ) & = ( \\Delta s ) ^ \\beta \\int _ 0 ^ 1 \\mu ( k + \\mu ) ^ { \\beta - 1 } d \\mu , \\\\ \\Psi _ 2 ( k ) & = ( \\Delta s ) ^ \\beta \\int _ 0 ^ 1 ( 1 - \\mu ) ( k + \\mu ) ^ { \\beta - 1 } d \\mu , \\beta = 1 - \\alpha , \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } W _ { k , a } ( n ) q ^ n = & \\frac { ( - q ^ 3 ; q ^ 2 ) _ \\infty ( q ^ { a + 1 } , q ^ { 2 k + 1 - a } , q ^ { 2 k + 2 } ; q ^ { 2 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\\\ & + \\frac { q ( - q ^ 3 ; q ^ 2 ) _ \\infty ( q ^ { a - 1 } , q ^ { 2 k + 3 - a } , q ^ { 2 k + 2 } ; q ^ { 2 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} u = \\{ u _ m \\} _ { m = 1 } ^ \\infty \\ , , u _ m \\in \\mathcal { K } _ m \\ , , \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\sigma = 1 _ { I _ 0 } , w = M ( \\sigma ) ^ { - 1 } , \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\norm { a } \\le \\sum _ { j = 1 } ^ m \\left \\Vert S _ n \\left ( a _ { 1 , j } , \\dotsc , a _ { n , j } \\right ) \\right \\Vert \\le \\sum _ { j = 1 } ^ m \\left \\Vert a _ { 1 , j } \\right \\Vert \\dotsb \\left \\Vert a _ { n , j } \\right \\Vert , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} b _ I ^ { ( \\theta ) } = \\sum _ { K \\in \\mathcal { B } _ I } \\theta _ K h _ K , I \\in \\mathcal { D } . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} d d ^ c u _ { j , k _ j } = 0 \\ \\Omega ' \\backslash ( \\overline { \\mathbb B } ( a _ j , r _ j ) \\cap \\{ \\varphi _ j \\geq - 1 \\} ) . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} { \\bf { W } } = { \\bf { I } } - \\frac { 2 \\left ( { { \\bf { D } } - { \\bf { A } } } \\right ) } { \\sigma _ 1 ^ 2 \\left ( { { \\bf { D } } - { \\bf { A } } } \\right ) + \\sigma _ { N - 1 } ^ 2 \\left ( { { \\bf { D } } - { \\bf { A } } } \\right ) } , \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} \\mathbf P _ { s o } ^ { \\ , i } = \\mathbb P \\left [ \\frac { 1 + \\frac { \\mathrm { S N R } _ { A _ i , B } } { I _ B + 1 } } { 1 + \\frac { \\mathrm { S N R } _ { A _ i , E } } { I _ E + 1 } } < 2 ^ { \\frac { n r _ s } { \\beta } } \\right ] . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 1 } \\right \\rbrace = \\exp \\left ( - \\frac { C _ { \\textrm { F } } ^ { \\textrm { O F D M A } } } { \\lambda _ { \\textrm { B F } } } \\right ) . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} w [ \\varphi ] ( t , x ) = \\frac { 1 } { ( n - 2 ) ! ! } \\frac { \\partial } { \\partial t } \\ , \\Omega _ t [ \\varphi ] ( x ) . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\widetilde { J _ 1 } ( q , Q ) = \\left ( \\sum _ { d \\geq 0 } \\frac { Q ^ d } { ( q ; q ) _ d ^ 3 } \\right ) \\left ( \\ell _ q ( Q ) + \\sum _ { k = 1 } ^ d \\frac { - 3 q ^ k } { 1 - q ^ k } \\right ) \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\mu ( n ) = \\begin{cases} ( - 1 ) ^ r , & n = p _ 1 \\cdots p _ r , \\mbox { f o r d i s t i n c t p r i m e n u m b e r s $ p _ i $ } \\\\ 0 , & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} \\begin{aligned} F ( f ) ( t ) : = \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\Gamma ^ { - 1 } ( s ) [ K ( \\Gamma ( s ) f ( s ) ) \\cdot \\nabla ] ( \\Gamma ( s ) f ( s ) ) d s - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\Gamma ^ { - 1 } ( s ) ( \\Gamma y ( s ) \\cdot \\nabla ) ( K ( \\Gamma y ( s ) ) d s , \\ t \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} | \\lambda - \\lambda _ h ^ { ( j ) } | \\leq C h ^ { 2 \\min \\{ s , k - 1 \\} } \\forall j = 1 , \\ldots , m . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} w ^ + : = \\mathcal W ^ + ( \\rho , \\zeta ) \\in h ^ { 4 + \\alpha } ( \\Bbb E _ { \\mathcal S ( \\rho ) } ) \\mbox { a n d } w ^ - : = \\mathcal W ^ - ( \\rho , \\zeta ) \\in h ^ { 4 + \\alpha } ( \\Bbb D _ { \\mathcal S ( \\rho ) } ) . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} | \\nabla f | _ x - V ( x ) = 0 . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} v _ { t t } ( t , x ) - \\Delta v ( t , x ) - \\Delta v _ { t t } ( t , x ) + \\Delta ^ 2 v ( t , x ) + ( - \\Delta ) ^ { \\theta } v _ t ( t , x ) = 0 , t \\geq 0 , \\ , \\ , \\ , x \\in \\R ^ n \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} 1 - 2 z & = \\frac { ( t + b + 2 ) ^ 2 - 2 ( t - b ) ^ 2 + ( y - x ) ^ 2 } { ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 } . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} M ( \\mathbb { A } / F , n ) = \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { \\pi _ F ( x ) } \\sum _ { N ( \\mathfrak { p } ) \\leq x } N _ { \\mathfrak { p } } ( \\mathbb { A } [ n ] ) , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} m _ N ( g ) = \\frac { | K _ N \\cap g K _ N | } { | K _ N | } . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\delta _ q Q ^ k = \\frac { q ^ k Q ^ k - Q ^ k } { q - 1 } = \\left ( 1 + q + \\cdots + q ^ { k - 1 } \\right ) Q ^ k \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} L ^ { ( 3 ) } & = \\mathbb { E } \\Bigl [ \\int _ t ^ { t + \\tau } l ( X _ r ^ { t , A _ t ; Z _ t \\otimes \\alpha ^ \\epsilon ( v ) , W _ t \\otimes v } , y _ r ^ { t , t + \\tau , A _ t ; Z _ t \\otimes \\alpha ^ \\epsilon ( v ) , W _ t \\otimes v } , \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ q _ r ^ { t , t + \\tau , A _ t ; Z _ t \\otimes \\alpha ^ \\epsilon ( v ) , W _ t \\otimes v } , ( Z _ t \\otimes \\alpha ^ \\epsilon ( v ) ) _ { r } , ( W _ t \\otimes v ) _ { r } ) \\dd r \\bigl | \\mathcal { F } _ t \\Bigr ] . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\iota _ \\partial \\left ( \\omega - \\sum _ { k = 1 } ^ j c _ k \\cdot d ( \\partial ^ { k - 1 } f ) \\right ) = c _ 0 f . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\psi _ n ^ { N } ( y ) = \\frac { 1 } { \\sqrt { \\lambda _ n ^ { N } } } \\int _ { D } \\log \\kappa ( y , x ) \\phi _ n ^ { N } ( x ) \\mathrm { d } x . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\frac { y ^ m q ^ { m ^ 2 + m } } { ( y q ; q ^ 2 ) _ { m + 1 } } \\cdot \\frac { z ^ m q ^ { m ^ 2 + m } } { ( z q ; q ^ 2 ) _ { m + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\infty } b _ \\omega ( \\ell , m , n ) z ^ \\ell y ^ m q ^ n = \\sum _ { m = 0 } ^ { \\infty } \\frac { y ^ m q ^ m } { ( z q ; q ^ 2 ) _ { m + 1 } } . \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\gamma _ * : = \\sup _ { k \\neq 0 } \\{ \\gamma _ k \\} . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\| \\hat Q _ r Q _ { \\geq r _ 0 } \\| _ 2 ^ 2 = \\sum _ { s \\geq r _ 0 } \\| \\hat Q _ r Q _ s \\| _ 2 ^ 2 & \\leq \\sum _ { s \\geq r _ 0 } \\frac { 4 \\| \\hat Q _ r E Q _ s \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ s ) ^ 2 } \\leq \\sum _ { s \\geq r _ 0 } \\frac { 4 \\| \\hat Q _ r E Q _ s \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ { r _ 0 } ) ^ 2 } = \\frac { 4 \\| \\hat Q _ r E Q _ { \\geq r _ 0 } \\| _ 2 ^ 2 } { ( \\mu _ r - \\mu _ { r _ 0 } ) ^ 2 } , \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} ( z \\partial _ z + \\mathfrak { E } _ { ( \\partial ) } + \\mu ) \\ , _ \\circ \\ , S ^ \\textnormal { c o h } ( \\tau , z ) ( e ^ { \\tau _ 2 / z } \\alpha ) = S ^ \\textnormal { c o h } ( \\tau , z ) \\ , _ \\circ \\ , e ^ { \\tau _ 2 / z } \\ , _ \\circ \\ , ( z \\partial z + \\mathfrak { E } _ { ( \\partial ) } + \\mu ) ( \\alpha ) \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} \\Delta _ { P _ { \\alpha } } \\left ( \\mathbf { O } \\right ) = b _ { a } \\left ( \\mathbf { O } ; P _ { \\alpha } \\right ) - E _ { P _ { \\alpha } } \\left [ b _ { a } \\left ( \\mathbf { O } ; P _ { \\alpha } \\right ) | O _ { 1 } \\right ] - E _ { P _ { \\alpha } } \\left [ b _ { a } \\left ( \\mathbf { O } ; P _ { \\alpha } \\right ) | O _ { 2 } \\right ] + E _ { P _ { \\alpha } } \\left [ b _ { a } \\left ( \\mathbf { O } ; P _ { \\alpha } \\right ) \\right ] = \\alpha O _ { 1 } O _ { 2 } \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} C _ l ^ { ( 1 + \\alpha ) } ( 1 ) = \\frac { \\Gamma ( 2 + 2 \\alpha + l ) } { \\Gamma ( 2 + 2 \\alpha ) l ! } \\ , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} d X ^ { \\epsilon , \\hat { v } } ( t ) = f _ { \\zeta _ t ^ { \\epsilon } } \\bigl ( X ^ { \\epsilon , \\hat { v } } ( t ) , \\hat { v } _ { { \\zeta _ t ^ { \\epsilon } } } ^ { \\epsilon } ( X ^ { \\epsilon , \\hat { v } } ( t ) ) \\bigr ) d t + \\sqrt { \\epsilon } \\sigma _ { { \\zeta _ t ^ { \\epsilon } } } \\bigl ( X ^ { \\epsilon , \\hat { v } } ( t ) \\bigr ) d W ( t ) , \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} A _ m ( M ) : = \\{ S \\in E n d ( T M ) \\mid S = S ^ * , - n I \\leq S \\leq I , t r ( S ) = m \\} \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} w _ \\theta ( \\xi _ t ) & = \\sum _ { v \\in \\xi _ t } \\theta ^ { - L ( v ) } . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} w _ 0 & = \\frac { 1 } { \\Delta s } \\int _ { s _ 0 } ^ { s _ 1 } ( s _ 1 - s ) w ( s ) d s = \\Psi _ 2 ( 0 ) , \\\\ w _ K & = \\frac { 1 } { \\Delta s } \\int _ { s _ { K - 1 } } ^ { s _ K } ( s - s _ { K - 1 } ) w ( s ) d s = \\Psi _ 1 ( K - 1 ) , \\\\ w _ k & = \\frac { 1 } { \\Delta s } \\int _ { s _ { k - 1 } } ^ { s _ k } ( s - s _ { k - 1 } ) w ( s ) d s + \\frac { 1 } { \\Delta s } \\int _ { s _ k } ^ { s _ { k + 1 } } ( s _ { k + 1 } - s ) w ( s ) d s = \\Psi _ 1 ( k - 1 ) + \\Psi _ 2 ( k ) . \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\mathcal { O } _ { e ^ { | M | } _ { - } } : & = \\left \\{ ( x , y , z ) \\in \\Omega \\mid x ^ 2 + y ^ 2 + z ^ 2 = ( \\lambda + 1 ) M ^ 2 , ~ y < 0 \\right \\} , \\\\ \\mathcal { O } _ { e ^ { | M | } _ { + } } : & = \\left \\{ ( x , y , z ) \\in \\Omega \\mid x ^ 2 + y ^ 2 + z ^ 2 = ( \\lambda + 1 ) M ^ 2 , ~ y > 0 \\right \\} . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { L ( \\lambda s ) } { L ( s ) } = 1 \\quad \\mbox { f o r a l l $ \\lambda > 0 $ } . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} | \\tilde { J } _ { s } ( x ) - J _ { s } ( x ) | & \\le ( | | J _ T | | _ w + | | \\tilde { J } _ T | | _ w ) \\theta . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} X ( \\Q , T ) = \\{ P \\in X ( \\Q ) | H ( P ) \\leq T \\} . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\xi \\ ; \\stackrel { \\textnormal { d e f } } { = } \\ ; \\textstyle \\sum _ { j , j ' \\in J } k _ { j j ' } = 0 \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} \\sum _ { 0 \\le K \\le q } ( - 1 ) ^ { q + K } ( q - K ) { h \\choose K } = ( - 1 ) ^ q \\sum _ { 0 \\le K \\le h } ( - 1 ) ^ { K } ( q - K ) { h \\choose K } = 0 , \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} J _ k = J \\ast \\cdots \\ast J , \\quad \\mbox { t h e $ ( k - 1 ) $ - t i m e s c o n v o l u t i o n } . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} d & | Y ^ 1 _ s - Y ^ 2 _ s | ^ r \\\\ & = \\frac { r ( r - 1 ) } { 2 } \\big ( [ \\tilde { \\sigma } ( s , Y ^ 1 _ s ) - \\tilde { \\sigma } ( s , Y ^ 2 _ s ) ] [ \\tilde { \\sigma } ( s , Y ^ 1 _ s ) - \\tilde { \\sigma } ( s , Y ^ 2 _ s ) ] ^ T \\big ) | Y ^ 1 _ s - Y ^ 2 _ s | ^ { r - 2 } d s + d M _ s \\\\ & \\leq \\frac { r ( r - 1 ) } { 2 } | \\tilde { \\sigma } ( s , Y ^ 1 _ s ) - \\tilde { \\sigma } ( s , Y ^ 2 _ s ) | ^ 2 | Y ^ 1 _ s - Y ^ 2 _ s | ^ { r - 2 } d s + d M _ s \\\\ & = | Y ^ 1 _ s - Y ^ 2 _ s | ^ r d A _ s + d M _ s \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } p ( z , \\mu ) + \\mu ^ 2 \\nabla f ( p ( z , \\mu ) ) = z & \\mbox { w h e n } \\ ; \\mu > 0 \\\\ p ( z , \\mu ) = \\Pi _ X ( z ) & \\mbox { w h e n } \\ ; \\mu = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} d \\left ( T \\{ x _ n ^ i \\} , T \\{ x _ n ^ j \\} \\right ) & = d \\left ( \\{ x _ n ^ { i + 1 1 } \\} , \\{ x _ n ^ { j + 1 1 } \\} \\right ) \\\\ & = \\frac { 1 0 } { i + j + 2 2 } + 1 \\\\ & < \\frac { 1 0 } { i + j } + 1 \\\\ & = d \\left ( \\{ x _ n ^ i \\} , \\{ x _ n ^ j \\} \\right ) . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} J ^ { \\textnormal { c o h } , \\textnormal { e q } } ( z , Q ) = \\sum _ { i = 0 } ^ N J ^ { \\textnormal { c o h } , \\textnormal { e q } } _ { | H = \\lambda _ i } ( z , Q ) L a g _ i \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\int _ { B _ 0 } \\vert u \\vert ^ { p ' } \\ , d \\mu \\leq \\sum _ { k = - \\infty } ^ { \\infty } a _ k ^ { p ' } \\mu ( B _ 0 \\cap ( E _ k \\setminus E _ { k - 1 } ) ) . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} u _ { \\zeta , x _ 0 } ^ \\nu ( x ) : = \\begin{cases} \\zeta & \\ \\langle x - x _ 0 , \\nu \\rangle \\geq 0 , \\\\ 0 & \\ \\langle x - x _ 0 , \\nu \\rangle < 0 , \\end{cases} \\qquad u _ { M , x _ 0 } ( x ) : = M ( x - x _ 0 ) , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} ( e ^ { t G _ \\lambda } K ) ( x ) = \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { t G _ \\lambda } ( x , y ) K ( y ) = \\sum _ { y \\in \\mathbb { Z } ^ d } G ^ 0 _ \\lambda ( x , y ) K ( y ) = K ( x ) \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} h ( s ) & = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\phi ( m s ) \\\\ & = \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { \\phi ( m s ) } { m } + \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } + \\sum _ { m > s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } \\\\ & = : \\psi _ 1 ( s ) + \\psi _ 2 ( s ) + \\psi _ 3 ( s ) . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} K _ 1 = K _ 2 = \\frac { \\delta + 1 } { 2 } . \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\int _ A \\L ^ { - \\ell } d \\tilde \\mu : = \\sum _ { n \\in \\mathbb Z } \\tilde \\mu ( \\ell ^ { - 1 } ( n ) ) \\L ^ { - n } \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} 1 + 3 b u _ { m , b } ^ 2 - u _ { m , b } '^ 2 - 2 u _ { m , b } u _ { m , b } '' = 0 , \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\eth ( \\hat { \\Psi } _ t ( Y _ 1 , \\ldots , Y _ m ) , ( Y _ 1 , \\ldots , Y _ m ) ) = \\max _ { 1 \\leq j \\leq m } \\| \\Psi _ t ( Y _ j ) - Y _ j \\| \\leq \\varepsilon , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} * 1 ^ p = * \\ ! * \\ ! * 0 , 0 * ^ p = 0 \\ ! * \\ ! * * . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\mathbb { S } = \\left ( \\Lambda F ^ { * } \\otimes | \\mathrm { d e t } \\ , ( \\mathrm { A n n } \\ , F ) | ^ { 1 / 2 } \\right ) \\hat { \\otimes } \\mathcal { S } _ { \\mathcal G } \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} d s ^ 2 = ( d x ^ n ) ^ 2 + ( x ^ n ) ^ 2 H ( x ^ \\alpha , d x ^ \\alpha ) , \\ , \\ , \\ , \\alpha = 1 , . . . , n - 1 , \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} \\int _ { D } | K _ { \\epsilon } ( z , w ) | ^ 2 d A ( z ) \\leq \\frac { A ( D ) } { \\epsilon ^ 2 } = \\frac { \\pi } { \\epsilon ^ 2 } , \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 1 } ^ n x _ j ^ q z _ j \\Big ) ^ { \\frac 1 { q } } \\leq \\sum _ { j = 1 } ^ n x _ j y _ j \\max _ { 1 \\leq k \\leq n } \\Big ( \\sum _ { j = 1 } ^ k z _ j \\Big ) ^ { \\frac 1 { q } } \\Big ( \\sum _ { j = 1 } ^ k y _ j \\Big ) ^ { - 1 } , \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} p ^ + ( t ) = ( p ^ + _ 1 ( t ) , p ^ + _ 2 ( t ) , \\cdots , p ^ + _ m ( t ) ) \\gg p ^ - ( t ) = ( p ^ - _ 1 ( t ) , p ^ - _ 2 ( t ) , \\cdots , p ^ - _ m ( t ) ) \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\mu _ i ( D _ { 1 / 2 } ( 0 ) \\setminus U _ i ) = 0 , \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} \\hat \\rho ( k ; \\hat \\theta ) = \\sum _ { i = 1 } ^ m \\hat A _ i \\ , \\hat \\tau _ i ^ k \\textrm { w h e r e } \\hat \\theta = [ \\hat A _ 1 , \\hat A _ 2 , \\dots , \\hat A _ m , \\hat \\tau _ 1 , \\hat \\tau _ 2 , \\dots , \\hat \\tau _ m ] , \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\nu _ q ( f ) : = \\min _ { 0 \\leq i \\leq n } \\{ \\nu ( f _ i q ^ i ) \\} \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} X ^ { * } _ { i } ( \\tilde { a } _ { i j } X _ { j } \\tilde { L } _ { \\nu } ) = \\sum ^ { m } _ { i j = 1 } X ^ { * } _ { i } \\big ( ( \\tilde { a } _ { i j } - \\tilde { a } ^ { 0 } _ { i j } ) X _ { j } \\tilde { L } _ { \\nu } \\big ) \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } F ' ( T _ k u , A ) & = F ' ( u , A ) , \\\\ \\lim _ { k \\to + \\infty } F '' ( T _ k u , A ) & = F '' ( u , A ) . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\nu _ { x , t } [ \\mathbb { H } \\setminus ( \\mathcal { H } \\cup V ) ] = 0 , \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} P \\cdot [ A ] : = P ^ { - 1 } A P + P ^ { - 1 } \\textnormal { d } P \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\rm { g r } _ \\nu ( R ) : = \\bigoplus _ { \\beta \\in \\nu R } P _ \\beta / P _ \\beta ^ + . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { 2 k , 2 a } ( n ) q ^ n = ( - q ; q ^ 2 ) _ \\infty \\sum _ { n \\geq 0 } \\overline { B } _ { k , a } ( n ) q ^ { 2 n } , \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} g _ { k , U _ { 2 } } ( U _ { 1 } ) : = \\min ( h _ { k - 1 , U _ { 2 } } ( U _ { 1 } ) , h _ { k - 1 , U _ { 2 } } ( U _ { 2 } \\setminus U _ { 1 } ) ) , \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{gather*} ( \\overline { \\nabla } _ V W - \\overline { \\nabla } _ W V - [ V , W ] ) ^ K = \\overline T _ { I J } { } ^ K V ^ I W ^ J , \\\\ ( \\overline { \\nabla } _ V \\overline { \\nabla } _ W Y - \\overline { \\nabla } _ W \\overline { \\nabla } _ V Y - \\overline { \\nabla } _ { [ V , W ] } Y ) ^ J = \\overline R _ I { } ^ J { } _ { K L } Y ^ I V ^ K W ^ L \\end{gather*}"}
-{"id": "3717.png", "formula": "\\begin{align*} \\Omega _ { i n } ( 0 ) = \\{ x \\mid ( u _ 0 ( x ) , v _ 0 ( x ) ) \\in \\Delta _ 1 \\} , \\ \\ \\Omega _ { o u t } ( 0 ) = \\{ x \\mid ( u _ 0 ( x ) , v _ 0 ( x ) ) \\in \\Delta _ 2 \\} . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\widehat { \\alpha _ \\lambda ^ { - 1 } } ( \\alpha _ \\lambda ( u _ { 0 \\lambda } ' ) ) \\leq \\widehat \\alpha _ \\lambda ( u _ { 0 \\lambda } ' ) + \\widehat { \\alpha _ \\lambda ^ { - 1 } } ( \\alpha _ \\lambda ( u _ { 0 \\lambda } ' ) ) = \\alpha _ \\lambda ( u _ { 0 \\lambda } ' ) u _ { 0 \\lambda } ' \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} S ^ \\textnormal { c o h } ( \\tau , z ) ( \\alpha ) = e ^ { - \\tau _ 2 / z } \\alpha - \\sum _ { \\substack { d \\in H _ 2 ( X ; \\mathbb { Z } ) - \\{ 0 \\} \\\\ l \\geq 0 } } \\ , \\sum _ { k = 0 } ^ N \\frac { 1 } { l ! } \\left \\langle T _ k , \\tau ' , \\dots , \\tau ' , \\frac { e ^ { - \\tau _ 2 / z } \\alpha } { z + \\psi } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } T ^ k e ^ { \\tau _ 2 ( d ) } \\in Q H ( X ) \\otimes \\mathbb { C } ( \\ ! ( z ) \\ ! ) \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} a = S _ n \\left ( z _ 1 , \\dotsc , z _ { n - 1 } , a _ { n - 1 } \\right ) \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\Lambda ( S ) : = \\sup _ { A \\in \\mathcal { A C } } \\Lambda ( S ( A ) ) \\in [ 0 , \\infty ] . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\mathcal { U } _ n = \\bigcup _ { A \\in \\mathcal { S } ( \\mathsf { A } ) } A \\mathcal { V } _ n \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} u \\otimes v + v \\otimes u + 2 \\langle u , v \\rangle 1 = 0 . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} \\mathbf { c } ^ { T } E _ { P } \\left [ v a r _ { P } \\left [ \\mathbf { \\psi } _ { P } \\left ( \\mathbf { G , B } ; \\mathcal { G } \\right ) | \\mathbf { A } , Y , \\mathbf { G } \\right ] \\right ] \\mathbf { c = } \\sum _ { \\mathbf { a } \\in \\mathcal { A } } c _ { \\mathbf { a } } ^ { 2 } E _ { P } \\left \\{ \\pi _ { \\mathbf { a } } ( \\mathbf { G } ; P ) v a r _ { P } ( Y \\mid \\mathbf { A } = \\mathbf { a } , \\mathbf { G } ) v a r _ { P } \\left [ \\frac { 1 } { \\pi _ { \\mathbf { a } } ( \\mathbf { G , B } ; P ) } \\mid \\mathbf { A = a } , \\mathbf { G } \\right ] \\right \\} . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} A _ n = \\left \\{ \\sup _ { x , y \\in K } | M ( x ) - M ( y ) | \\leq n \\right \\} \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} w _ + = \\frac 1 { \\sqrt 2 } \\begin{pmatrix} - 1 \\\\ \\lambda _ - \\end{pmatrix} \\ , , w _ - = \\frac 1 { \\sqrt 2 } \\begin{pmatrix} - \\lambda _ - \\\\ 1 \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} d _ c = \\theta _ { \\min } = \\frac { \\sigma ^ 2 } { ( a + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\rho ( x , q ) = \\sup _ { p \\in \\mathbb { R } ^ d } \\bigl [ q \\cdot p - \\lambda ( x , p , 0 ) \\bigr ] , x , q \\in \\mathbb { R } ^ d \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\| f \\| _ { W ^ { 1 , d } } ^ d = \\int _ { B ( 0 , 1 ) } | \\nabla f ( x ) | ^ d \\ , d x , \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} K = \\frac { f _ { 1 } f _ { 2 } f _ { 1 } ^ { \\prime \\prime } f _ { 2 } ^ { \\prime \\prime } - \\left ( f _ { 1 } ^ { \\prime } f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } } { \\left ( a f _ { 1 } ^ { \\prime } f _ { 2 } + f _ { 1 } f _ { 2 } ^ { \\prime } \\right ) ^ { 4 } } , \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} F ( r ) : = \\int _ 0 ^ r s ^ { 1 - N } [ U ( s ) ] ^ { - 2 } \\left ( \\int _ 0 ^ s \\tau ^ { N - 1 } U ( \\tau ) ^ 2 \\ , d \\tau \\right ) \\ , d s . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { R ( \\lambda s ) } { R ( s ) } = \\lambda ^ \\beta \\quad \\mbox { f o r a l l $ \\lambda > 0 $ } . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\partial _ t \\eta = - \\dot a \\partial _ a Q _ { a , c } - \\dot c \\partial _ c Q _ { a , c } + J E ' ( Q _ { a , c } ) + J E '' ( Q _ { a , c } ) \\eta - \\frac 1 2 \\partial _ x ( \\eta ^ 2 ) \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{gather*} 0 = A \\phi _ r - \\frac { \\phi _ { r r } } { \\phi } - \\psi ^ 2 , \\psi _ r = - A \\phi \\psi , \\end{gather*}"}
-{"id": "3307.png", "formula": "\\begin{align*} c ( 0 ) = \\sum _ { i > 0 } b _ { S , i } ^ 2 \\le \\frac { \\delta ^ 2 \\lvert \\textbf { s } \\rvert ^ 2 } { 3 2 C _ { \\mathbf { I } } C _ e ^ 2 \\sigma _ z ^ 2 } . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} \\dim _ { H } \\zeta = \\inf \\{ \\dim _ { H } E \\ : : \\ : E \\subset \\mathbb { R } \\zeta ( E ) > 0 \\} \\ : . \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\phi ( n + 1 ) + \\phi ( n - 1 ) + V ( n ) u ( n ) = E \\phi ( n ) \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} & \\overline { \\alpha } _ { f } ( x ) = \\limsup _ { n \\to \\infty } \\max \\{ 1 , h _ { H } ( f ^ { n } ( x ) ) \\} ^ { 1 / n } , \\\\ & \\underline { \\alpha } _ { f } ( x ) = \\liminf _ { n \\to \\infty } \\max \\{ 1 , h _ { H } ( f ^ { n } ( x ) ) \\} ^ { 1 / n } . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\gamma ( t _ i - s _ i ) d s _ i = & \\int _ { t _ i - t } ^ { t _ i } \\gamma ( r ) d r \\\\ \\leq & \\int _ { - t } ^ 0 \\gamma ( r ) d r + \\int _ 0 ^ t \\gamma ( r ) d r . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} D ^ { ( 1 ) } : = D - \\frac { 1 } { 2 } \\mathcal J _ { 2 } D \\mathcal J _ { 2 } \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} - A _ n u = \\bar { g } \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\mathcal { L } _ j [ w _ j ] = \\left ( \\mathcal X _ j - 2 | D ^ 2 u _ j | ^ 2 + \\mathcal E _ { 0 , j } + \\mathcal E _ { 1 , j } \\right ) \\phi ^ { 2 \\alpha _ j } \\omega , \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\nu _ n ( z ) = \\min \\left ( n , g ( z ) \\right ) \\cdot \\mu ( z ) \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} E ( Y ) = 0 . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} ( \\widetilde { f } ^ { n } ) ^ { * } = \\begin{pmatrix} ( g ^ { n } ) ^ { * } & h _ { n } ^ { * } \\\\ 0 & ( f _ { T } ^ { n } ) ^ { * } \\end{pmatrix} . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} [ u , \\mathcal J v ] _ { + } - \\mathcal J [ u , v ] _ { + } = 0 , \\ \\forall u \\in \\Gamma ( E _ - ) , \\ v \\in \\Gamma ( E _ + ) . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} q _ i : = q ^ { d _ i } , [ n ] _ i : = \\frac { q _ i ^ n - q _ i ^ { - n } } { q _ i - q _ i ^ { - 1 } } \\mbox { a n d } [ n ] _ i ! : = [ 1 ] _ i \\ldots [ k ] _ i . \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} - u '' = \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - v ^ + \\right ) v ^ + + \\frac { k \\omega } { d } u ( v - \\alpha u ) . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ { i + m - 1 } H ( \\mu , \\mathcal { D } _ { i + 1 } | \\mathcal { D } _ { i } ) \\leq \\varepsilon \\mu ( \\{ x \\colon H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) \\leq \\varepsilon \\} ) + m \\log 2 \\left ( \\mu ( \\{ x \\colon H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) > \\varepsilon \\} ) \\right ) \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\alpha / 2 } u ( x ) + \\rho u ( x ) = f ( x ) , & x \\in { \\mathbb R } ^ d , \\\\ u ( x ) = 0 , & \\lvert x \\rvert \\to \\infty , \\end{cases} \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} ( ( T _ { a } \\circ f ) ^ { n } ) ^ { * } = ( T _ { b } \\circ f ^ { n } ) ^ { * } = ( f ^ { n } ) ^ { * } \\circ T _ { b } ^ { * } = ( f ^ { n } ) ^ { * } \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\begin{aligned} \\inf _ { \\tau \\in \\ker ( b , H ( \\div ; \\mathbb { S } ) ; L ^ 2 ( \\mathbb { V } ) ) } a ( \\tau , \\tau ) & \\geq \\alpha \\| \\tau \\| _ { \\div } ^ 2 , \\\\ \\inf _ { v \\in L ^ 2 ( \\mathbb { V } ) } \\sup _ { \\tau \\in H ( \\div ; \\mathbb { S } ) } b ( \\tau , v ) & \\geq \\beta \\| \\tau \\| _ { \\div } \\| v \\| _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} A = \\langle \\alpha ( g , h ) \\rangle / \\langle \\alpha ( g , h ) \\alpha ( g h , k ) \\alpha ^ { - 1 } ( h , k ) \\alpha ^ { - 1 } ( g , h k ) \\rangle \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} X _ { \\lambda } ^ { ( n ) } ( z ) / n ! = \\frac 1 { 2 \\pi i } \\oint \\frac { X _ \\lambda ( w ) d w } { ( z - w ) ^ { n + 1 } } \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} H _ 4 = ( 1 , \\dots , 1 , - 1 \\ , | \\ , 1 , \\dots , 1 , - 1 ) . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} ( \\lambda - G _ 1 ) u = 0 , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} ( d i v _ { \\sigma } \\eta ) ^ { ( 0 ) } = & \\frac { 1 } { 3 } \\tilde { \\nabla } ^ a ( \\beta _ a ) = \\rho \\\\ ( d i v _ { \\sigma } \\eta ) ^ { ( 1 ) } = & \\frac { 1 } { 4 } \\tilde { \\nabla } ^ a D \\beta _ a = D \\rho \\\\ ( d i v _ { \\sigma } \\eta ) ^ { ( 2 ) } = & \\frac { 1 } { 1 0 } \\tilde { \\nabla } ^ a ( D ^ 2 \\beta _ a ) - \\frac { 2 } { 1 5 } \\tilde { \\nabla } ^ a ( \\alpha _ { a b } \\beta ^ b ) \\\\ = & \\frac { 1 } { 2 } D ^ 2 \\rho + \\frac { 1 } { 1 5 } ( | \\alpha | ^ 2 - 8 | \\beta | ^ 2 ) . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\omega ^ \\xi _ \\eta ( \\rho ) : = \\psi _ { \\rho } ( x _ 0 ) - \\psi _ { \\rho } ( y _ 0 ) = x ( \\rho ) - y ( \\rho ) \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} \\gamma _ { v } ( s _ { + } { \\otimes } s _ { - } ) = \\gamma _ { v _ { + } } ( s _ { + } ) { \\otimes } s _ { - } + ( - 1 ) ^ { | s _ { + } | } s _ { + } { \\otimes } \\gamma _ { v _ { - } } ( s _ { - } ) , \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} M & = \\sum _ { \\sigma \\in G a l ( \\chi ) } M c _ { \\sigma ( \\chi ) } \\\\ & = \\sum _ \\sigma \\sigma ( m ) c _ { \\sigma ( \\chi ) } \\\\ & = | G | ^ { - 1 } \\sum _ g t r _ { \\mathbb { Q } ( \\chi ) / \\mathbb { Q } } ( m \\overline { \\chi ( g ) } ) g , \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } | 1 - \\langle C _ \\varphi ^ * f , f \\rangle | ^ 2 - \\frac { 1 } { 4 } | 1 + \\langle C _ \\varphi ^ * f , f \\rangle | ^ 2 = & ( | | f _ 2 | | ^ 2 - | | f _ 1 | | ^ 2 ) | | f | | ^ 2 \\\\ = & | | f _ 2 | | ^ 2 - | | f _ 1 | | ^ 2 . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} ( \\lambda t _ \\alpha + \\mu e ^ { \\alpha } ) \\cdot ( t _ i - t _ j ) = \\lambda t _ i + \\mu a e ^ \\alpha - \\lambda t _ j - \\mu a e ^ \\alpha = \\lambda ( t _ i - t _ j ) \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\lambda = h ( \\alpha _ 1 ^ 2 , \\ldots , \\alpha _ n ^ 2 ) , \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} \\left [ 2 \\kappa + n \\rho ^ \\ast \\lambda _ { a b } \\right ] ^ 2 - \\left [ 2 \\sqrt { \\kappa ( \\kappa + n \\rho ^ \\ast \\lambda _ { a b } ) } \\right ] ^ 2 = ( n \\rho ^ \\ast \\lambda _ { a b } ) ^ 2 > 0 , \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} Q _ 0 ( n ) : = \\mathrm { C a r d } \\left \\{ \\{ \\gamma _ 1 , \\gamma _ 2 \\} \\in \\mathcal { S } \\mid \\norm { \\{ \\gamma _ 1 , \\gamma _ 2 \\} } \\leq n \\right \\} . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} | I _ { n _ k } | \\geq K _ 3 ^ { 2 k - 1 } K _ 1 ^ k \\prod _ { j = 1 } ^ k | I _ { n _ j - n _ { j - 1 } - 1 } ( a _ { n _ { j - 1 } + 1 } , \\cdots , a _ { n _ j - 1 } ) | \\cdot a _ { n _ j } ^ { - d } , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\left [ \\Lambda _ { \\lambda } ^ { w } \\left ( u \\right ) \\right ] ( x ) = \\int _ { \\mathbb { R } } H \\left ( \\lambda \\varepsilon , x - y \\right ) w ( y ) \\ ; d y - \\int _ { \\mathbb { R } } \\lambda H _ { x } \\left ( \\lambda \\varepsilon , x - y \\right ) f \\left ( y , u ( y ) \\right ) \\ ; d y . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} f ( t ) = \\prod _ { \\substack { j = 1 } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 j + 1 } ) + ( t - \\lambda _ { 1 } ) \\sum _ { j = 1 } ^ { \\frac { n } { 2 } } \\frac { \\sin ^ { 2 } \\left [ \\frac { ( 2 j + 1 ) \\pi } { n + 3 } \\right ] } { \\sin ^ { 2 } \\left ( \\frac { \\pi } { n + 3 } \\right ) } \\prod _ { \\substack { \\substack { m = 1 \\\\ m \\neq j } } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 m + 1 } ) \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} s ( H ) { - } s ( H ' ) & { = } \\left | d _ i { - } \\frac { r m } { n } \\right | { + } \\left | d _ j { - } \\frac { r m } { n } \\right | { - } \\left | ( d _ i { + } 1 ) { - } \\frac { r m } { n } \\right | { - } \\left | ( d _ j { - } 1 ) { - } \\frac { r m } { n } \\right | \\\\ & { = } \\left ( \\frac { r m } { n } { - } d _ i \\right ) \\ ! \\ ! + \\ ! \\ ! \\left ( d _ j { - } \\frac { r m } { n } \\right ) \\ ! \\ ! - \\ ! \\ ! \\left ( \\frac { r m } { n } { - } ( d _ i { + } 1 ) \\right ) \\ ! \\ ! - \\ ! \\ ! \\left ( ( d _ j { - } 1 ) { - } \\frac { r m } { n } \\right ) \\ ! = \\ ! 2 . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\forall \\psi \\in \\mathcal C ^ 2 ( \\mathbb { R } ^ d ) , A _ n \\psi ( z ) \\triangleq \\frac { 1 } { 2 } \\partial _ { z _ 1 z _ 1 } \\psi ( z ) + \\sum _ { i = 1 } ^ d F _ { n , i } ( z ) \\partial _ { z _ i } \\psi ( z ) , z \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\mathcal { C } ^ L ( p , B , \\boldsymbol { d } ) = p \\sum _ { i \\in \\mathcal { N } } d _ i + B . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\overline { \\nabla } { \\phi } = \\frac { 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\ , \\overline { \\nabla } { f } ) } { ( 1 - f ) ^ { 2 } } + \\frac { 2 \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } \\overline { \\nabla } { f } } { ( 1 - f ) ^ { 3 } } = \\frac { 2 f _ { i } f _ { i j } } { ( 1 - f ) ^ { 2 } } + \\frac { 2 f _ { i } ^ { 2 } f _ { j } } { ( 1 - f ) ^ { 3 } } . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\begin{aligned} & S \\geq \\inf H ^ 2 = \\bar H ^ 2 , \\\\ & S ( 1 - \\dfrac 1 2 S ) - ( S - \\bar H ^ 2 ) ^ 2 + \\dfrac 1 2 \\bar H ^ 4 - \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 + 2 \\bar \\lambda ^ 2 ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} I ( X _ d ) = I _ { + } ( \\eta ) = \\sum _ { n \\geq 2 } ( I _ { + } ( \\eta ) _ { n } ) = \\sum _ { i = 2 } ^ { m } ( I _ { + } ( \\eta ) _ { i } ) \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} D _ 0 ( t ) = \\frac { 1 } { C } \\parallel D v ( t ) \\parallel ^ 2 _ { L ^ 2 } + \\gamma \\parallel w _ { \\nu _ { \\mathcal { N } } } ( t ) \\parallel ^ 2 _ { L ^ 2 ( \\Gamma _ c ) } \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\mathbf C _ s ^ { \\ , i } = \\mathbb E [ C _ s ^ { \\ , i } ] , \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} q _ S = \\begin{cases} \\left ( w ^ { + 1 } _ { p , 1 } \\right ) ^ { \\oplus k } & s _ { ( + ) } - s _ { ( - ) } \\equiv k ( 1 - p ) 8 \\\\ \\left ( w ^ { + 1 } _ { p , 1 } \\right ) ^ { \\oplus k - 1 } \\oplus w ^ { - 1 } _ { p , 1 } & s _ { ( + ) } - s _ { ( - ) } \\equiv k ( 1 - p ) + 4 8 \\\\ \\end{cases} \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} I I I ' & = 4 \\pi c - \\frac 1 2 c ^ { - 1 } W ( h a ) \\int _ { | x - a | < h ^ { - 1 } } Q _ { a , c } ( x ) ^ 2 \\ , d x \\\\ & + \\frac 1 4 c ^ { - 1 } h ^ 2 W '' ( h a ) \\int _ { | x - a | < h ^ { - 1 } } ( x - a ) ^ 2 Q _ { a , c } ( x ) ^ 2 \\ , d x + O ( h ^ 3 ) \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} \\left | \\exp ( y ) - \\sum _ { j = 0 } ^ { N } \\frac { y ^ j } { j ! } \\right | \\le \\frac { | y | ^ { N + 1 } } { ( N + 1 ) ! } \\exp ( | y | ) \\ , , \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\begin{aligned} \\underset { [ 0 , T ] } { } ( { \\rho } _ 4 ^ 2 ( t ) \\| y _ t ( t ) \\| ^ 2 ) & + \\iint _ Q { \\rho } _ 4 ^ 2 | y _ { x t } | ^ 2 d x d t \\leq C \\left ( \\| y _ t ( 0 ) \\| ^ 2 + \\iint _ Q \\rho _ 3 ^ 2 | G _ t | ^ 2 d x d t \\right . \\\\ & + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } { \\rho } _ 1 ^ 2 | f | ^ 2 d x d t + \\iint _ Q { \\rho } _ 0 ^ 2 | y | ^ 2 d x d t + \\left . \\sum _ { i = 1 } ^ 2 \\iint _ Q { \\rho } _ 0 ^ 2 | p ^ i | ^ 2 d x d t \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} | D ^ 2 F | : = \\sqrt { \\sum _ { l = 1 } ^ m \\sum _ { i , j = 1 } ^ N f _ { l , u _ i u _ j } ^ 2 } , \\ \\ | \\varphi | ^ 2 : = \\sum _ { l = 1 } ^ m \\varphi ^ 2 . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} B ^ { \\alpha + \\beta } _ R & = - C _ { \\alpha , \\beta } R ^ { - 2 ( \\alpha + \\beta ) } \\int ^ R _ 0 \\left ( \\int ^ t _ 0 B ^ \\alpha _ r d r \\right ) \\frac { d } { d t } [ ( R ^ 2 - t ^ 2 ) ^ { \\beta - 1 } t ^ { 2 \\alpha + 1 } ] d t \\\\ & = C _ { \\alpha , \\beta } \\int ^ 1 _ 0 \\varphi ( t ) M ^ \\alpha _ { R t } d t , \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} R ( s ) = s ^ \\beta L ( s ) , \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d v _ t & = & \\frac { 1 } { \\tau } \\big ( \\lambda ^ 2 \\frac { d ^ 2 } { d x ^ 2 } v _ t - v _ t \\big ) d t + \\sigma d X _ t \\medskip \\\\ v _ 0 & = & h _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} A _ \\gamma ^ { w , b } : = \\{ [ f _ w ( x ) ^ { t } x ^ { - s } b ^ { - t } ] / v : \\gamma < \\delta ( w ) , v ( x ) = \\gamma \\} . \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} d S _ K = h _ K ^ { p - 1 } \\| D h _ K \\| _ Q ^ { q - n } f \\ , d \\mathcal { H } ^ { n - 1 } \\mbox { \\ \\ o n $ S ^ { n - 1 } $ } . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} r _ i \\le p _ i , i = 1 , \\dots , m ; \\quad \\mbox { a n d } r _ { m + 1 } ' > p , \\mbox { w h e r e } \\frac 1 p : = \\frac 1 { p _ 1 } + \\dots + \\frac 1 { p _ { m } } . \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} ( I - P ) T _ { \\rm o p } = ( I - P ) ( I - P _ m ) T = ( I - P ) T . \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} A = - \\frac { \\kappa } { 2 } \\frac { d ^ 2 } { d x ^ 2 } - \\frac { d } { d x } \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\textstyle \\sum _ { j \\in I } c _ { i j } D _ { e _ { i j } } P = 0 \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\avg { e ^ { t _ j + t _ l } } = 1 + \\avg { y _ j y _ l } \\leq 1 + \\avg { y _ 0 ^ 2 } . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} T _ j ^ { ( D ) } = \\min \\left \\{ { K : H _ { K , M , j } ^ { \\left ( D \\right ) } \\ge { h } } \\right \\} . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} \\begin{cases} \\eta _ { \\rho \\rho } - \\frac { p _ e ' ( \\rho ) } { \\rho ^ 2 } \\eta _ { u u } = 0 , & \\rho > 0 \\\\ \\eta | _ { \\rho = 0 } = 0 . & \\end{cases} \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\tilde { h } \\triangleq \\sqcup _ { m = 1 } ^ { \\infty } \\tilde { h } _ { m } : I \\rightarrow \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} f ( S ) = 2 ( S - 1 ) ( S - 2 ) S ^ 2 \\leq 0 , \\ \\ f ( 3 S - 2 ) = 2 ( 3 S - 2 ) ^ 2 ( S - 1 ) ( S - 2 ) \\leq 0 , \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} \\mathcal { Z } _ p : = \\left \\{ f : \\ t ^ { 1 - \\frac { 3 } { 2 p } } f \\in C _ b ( [ 0 , \\infty ) ; L ^ p ( \\mathbb { R } ^ 3 ) ) , \\ t ^ { \\frac { 3 } { 2 } \\left ( 1 - \\frac { 1 } { p } \\right ) } \\partial _ j f \\in C _ b ( [ 0 , \\infty ) ; L ^ p ( \\mathbb { R } ^ 3 ) ) , \\ j = 1 , 2 , 3 \\right \\} . \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} 1 - P ( \\mathrm { B i n } ( \\lfloor m _ n \\rfloor , q _ n ) \\geq k ) = ( 1 - q _ n ) ^ { m _ n } \\frac { ( q _ n m _ n ) ^ { k - 1 } } { ( k - 1 ) ! } ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} \\bar { R } ( \\omega \\otimes \\tau \\otimes s ) = \\frac { 1 } { 2 } \\sum _ { i , j , k } ( R ( X _ { i } , X _ { j } ) , r _ { k } ) ^ { \\mathcal G } ( \\alpha _ { i } \\wedge \\alpha _ { j } \\wedge \\omega ) \\otimes \\tau \\otimes \\tilde { r } _ { k } s , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - \\log b _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) = - \\infty \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} p ( x , y ) = x y \\ , . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ln | S ( i \\omega ) | d \\omega = \\pi \\sum _ k R e ( p _ k ) - \\frac { \\pi } { 2 } \\sigma _ y \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} H : = \\bigcup _ { k = 1 } ^ \\infty { \\rm B o x } ( x _ k , 2 ^ { - k - 1 } ) . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} M ( r ) : = \\{ z \\in { \\mathbb { C } ^ n } \\colon { } \\exists { x \\in { M } } | x - z | < r \\} \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} h _ 1 ( u ) & = \\xi ' ( u ) ( q + q z _ 1 + q z _ 2 - u z _ 1 ) - ( 1 + z _ 2 ) \\xi ' ( q ) u , \\\\ h _ 2 ( u ) & = [ \\xi ' ( u ) - \\xi ' ( q ) ] ( 1 + z _ 2 - q - u z _ 2 ) - ( u - q ) ( \\xi ' ( 1 ) - \\xi ' ( q ) ) . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} B = \\sqrt { d ( d - 1 ) } Q _ { n - 2 } \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} & H ( U ( - \\infty ) , V ( - \\infty ) ) = H ( R _ 1 , 0 ) = - \\zeta ( R _ 1 ) < 0 , \\\\ & H ( U ( + \\infty ) , V ( + \\infty ) ) = H ( 0 , R _ 2 ) = R _ 2 > 0 , \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\hat { \\textbf { H } } = \\left [ \\hat { \\textbf { h } } _ 1 \\ , \\hat { \\textbf { h } } _ 2 \\cdots \\hat { \\textbf { h } } _ { N _ { \\rm M S } } \\right ] , \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} | \\lambda _ { r - k } | b _ { r - k , r - k } & \\leq r + \\sum _ { j = r - k + 1 } ^ { r } | \\lambda _ j | b _ { j , r - k } \\\\ & \\leq r + \\left ( \\sum _ { j = r - k + 1 } ^ { r } | \\lambda _ j | \\right ) ( b _ { r - k , r - k } - 1 ) \\\\ & \\leq r + ( 1 + 1 + 2 + \\ldots + 2 ^ { k - 2 } ) r ( b _ { r - k , r - k } - 1 ) \\\\ & = r + 2 ^ { k - 1 } r ( b _ { r - k , r - k } - 1 ) \\\\ & \\leq 2 ^ { k - 1 } r b _ { r - k , r - k } , \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} \\omega ' ( u ) = ( \\cos 2 \\theta - i \\sin 2 \\theta ) \\omega ( u ) = \\exp ( - 2 i \\theta ) \\omega ( u ) . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 2 ( x ) + \\frac { 2 } { 6 ^ 3 } ( 2 ^ 3 x - 3 ) ( 1 1 - 3 ^ 3 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 6 0 7 } { 1 5 1 2 } } + \\frac 1 { 1 0 \\cdot 6 ^ 3 } - \\eta \\\\ & = - \\frac { 1 1 5 3 0 5 3 7 9 4 2 0 4 1 4 3 8 9 4 1 0 4 6 0 0 0 2 6 5 5 9 3 5 8 2 7 } { 2 1 2 8 1 1 8 4 9 8 0 7 8 4 6 1 3 0 7 3 3 4 0 1 2 1 1 8 1 1 0 5 5 1 2 8 0 0 0 } < 0 , \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\Gamma = G \\ltimes \\prod _ { g \\in G } H , \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} X = X ^ 0 \\supseteq X ^ 1 \\supseteq \\cdots \\supseteq X ^ { \\dim X } \\supseteq X ^ { \\dim X + 1 } = \\emptyset . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - \\log b _ c ^ { ( n ) } } \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) = - \\lceil \\ell _ 4 \\varepsilon \\rceil . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} w _ 1 ^ { - e } = w _ 2 ^ e w _ 2 ^ { - e } = w _ 1 ^ e \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\leq \\varepsilon \\right ) = P ( A _ n ^ * \\geq n - \\varepsilon f _ 1 ( n ) ) = P ( A _ n ( t ) > t , \\quad \\forall t = a _ n , \\ldots , \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor ) . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} | u _ { n + 1 } | = \\left | \\left [ \\frac { \\beta _ 1 ( e - 1 ) } { e } \\right ] ^ { 3 ^ n } \\times \\left [ \\frac { \\beta } { \\beta _ 1 } \\right ] ^ { 3 ^ n } \\xi _ n \\right | \\le \\left [ \\frac { \\beta _ 1 ( e - 1 ) } { e } \\right ] ^ { 3 ^ n } . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } K _ { D _ k , \\mu _ k } = K _ { D , \\mu } \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\left | \\tau \\left ( e \\prod _ { \\ell = 1 } ^ { m } e _ { \\sigma ( \\ell ) } \\right ) \\right | \\leq | e | _ Z \\prod _ { j = 1 } ^ m | e _ j | _ { X _ j } , \\forall \\sigma \\in \\Sigma ( m ) , \\ , e _ j \\in X _ j , \\ , j = 1 , \\ldots , m , \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\frac { | P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , \\frac { x } { \\epsilon } ) | ^ 2 } { 2 } + V \\Big ( x , \\frac { x } { \\epsilon } \\Big ) = \\ln \\widetilde { m } _ 0 ( x ) + \\ln \\widetilde { m } _ 1 \\Big ( x , \\frac { x } { \\epsilon } \\Big ) + \\overline { H } . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\kappa _ { \\min } = \\frac { 1 + \\alpha } { 1 - \\alpha } \\ , . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} \\frac { \\partial n _ p } { \\partial \\rho ^ \\ast } = - \\frac { \\sigma _ b ^ 2 \\left [ 2 \\kappa + n \\rho ^ \\ast \\lambda _ { a b } - 2 \\sqrt { \\kappa ( \\kappa + n \\rho ^ \\ast \\lambda _ { a b } ) } \\right ] } { 2 \\lambda _ { a b } { \\rho ^ \\ast } ^ 2 \\sqrt { \\kappa ( \\kappa + n \\rho ^ \\ast \\lambda _ { a b } ) } } . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\lim _ { j \\in J } \\frac { | e ^ { - 1 } g E F _ j \\setminus F _ j | } { | F _ j | } = 0 . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} { \\mathcal X } _ { 0 } \\stackrel { \\mathrm { d e f } } { = } \\| ( u _ 0 , \\tau _ 0 ) \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac { n } { 2 } - 1 } } + \\| u _ 0 \\| ^ h _ { \\dot B ^ { \\frac n p - 1 } _ { p , 1 } } + \\| \\tau _ 0 \\| ^ h _ { \\dot B ^ { \\frac n p } _ { p , 1 } } \\leq c _ 0 , \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} & T ^ { \\tilde { D } } ( u , v , w ) - T ^ { D ^ { ( 1 ) } } ( u , v , w ) = \\\\ & - \\frac { 1 } { 8 } \\sum _ { ( u , v , w ) \\ , \\mathrm { c y c l i c } } \\left ( \\eta ( \\mathcal J v , u , w ) + \\eta ( w , u , \\mathcal J v ) - \\eta ( v , u , \\mathcal J w ) - \\eta ( \\mathcal J w , u , v ) \\right ) \\\\ & = - \\frac { 1 } { 4 } \\sum _ { ( u , v , w ) \\ , \\mathrm { c y c l i c } } \\left ( \\eta ( \\mathcal J u , w , v ) + \\eta ( u , v , \\mathcal J w ) \\right ) , \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\rho ( e , e , f ) \\rho ( f , f e , e ) = \\rho ( e , e ( f e ) , e ) = \\rho ( e , ( e f ) e , e ) = \\rho ( e , e , e ) = 1 _ { S e } . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\gamma \\left ( \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) ( z , Q ) \\right ) = \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} \\left ( c _ { 2 } - c _ { 1 } \\right ) \\left ( \\mu _ { 1 } ^ { \\prime } \\mu _ { 2 } - \\mu _ { 1 } \\mu _ { 2 } ^ { \\prime } \\right ) = 0 . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} \\lambda _ { k } ^ { \\epsilon , v _ k } = - \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\mathbb { P } _ { x , k } ^ { \\epsilon , v _ k } \\Bigl \\{ \\tau _ { k } ^ { \\epsilon , v _ k } > t \\Bigr \\} , \\ , \\ , \\ , x \\in D , \\ , \\ , \\ , k \\in \\{ 1 , 2 , \\ldots , n \\} . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} Z _ { n + 1 , x } = \\begin{cases} \\displaystyle \\frac { X _ { n + 1 , x } } { W _ { n , x } } , & \\mbox { i f } W _ { n , x } \\geq t ^ { \\ast } , \\\\ \\displaystyle \\frac { X _ { n + 1 , x } - t ^ { \\ast } } { W _ { n , x } + t ^ { \\ast } } + \\frac { t ^ { \\ast } } { \\Tilde { t } } , & \\mbox { i f } W _ { n , x } < t ^ { \\ast } . \\end{cases} \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\exp ( f _ n ) - 1 = \\sum \\limits _ { i = 1 } ^ { \\infty } \\frac { f _ n ^ i } { i ! } , a . e . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\lambda _ \\varepsilon H ( \\gamma _ \\varepsilon ) \\mu _ \\varepsilon ^ 2 \\gamma _ \\varepsilon ^ 2 \\varphi _ { N _ \\varepsilon - 1 } ( \\gamma _ \\varepsilon ^ 2 ) = 4 \\ , , \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} d \\mu _ m = \\frac 1 { \\widetilde { V } _ q ( P _ m ) } \\ , d \\widetilde { C } _ { p , q } ( P _ m , \\cdot ) = \\frac { h _ { P _ m } ^ { - p } } { \\widetilde { V } _ q ( P _ m ) } \\ , d \\widetilde { C } _ { q } ( P _ m , \\cdot ) \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} 2 H = \\left ( 1 + a ^ { 2 } \\right ) f _ { 1 } f _ { 2 } ^ { \\prime \\prime } + 2 a f _ { 1 } ^ { \\prime } f _ { 2 } ^ { \\prime } + f _ { 1 } ^ { \\prime \\prime } f _ { 2 } . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} f ^ { \\sharp } ( s ) ~ = ~ a _ { f , i } \\quad \\forall s \\in I _ i , ~ i \\in \\overline { 0 , N _ 1 - 1 } \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} f ^ { ( n ) } ( t _ 0 , \\cdots , t _ s , \\xi ; x ) = \\sum _ { k = 0 } ^ n z _ k ( t _ 0 , \\cdots , t _ s ) x ^ k \\in E _ 1 [ x ] \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} & \\Omega ^ { ( 2 ) } : = \\Lambda ( \\Omega ^ { ( 1 ) } ) \\ , , \\ \\Gamma ^ { ( 2 ) } : = \\L ( \\Gamma ^ { ( 1 ) } ) \\ , , \\ \\Gamma ^ { ( 2 ) } ( t ) : = \\L ( \\Gamma ^ { ( 1 ) } ( t ) ) \\ , , \\\\ & A ^ { ( 2 ) } ( x ) : = [ D \\L \\ , A ^ { ( 1 ) } \\ , D \\L ^ T ] ( \\L ^ { - 1 } ( x ) ) \\ , . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\nu _ { \\omega + 3 } : = \\left [ \\nu _ { \\omega + 2 } ; \\nu _ { \\omega + 3 } ( \\phi _ { \\omega + 2 } ) = \\frac { 1 + p } { p } \\right ] . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\displaystyle \\alpha _ i \\iint _ { \\mathcal { O } _ { i , d } \\times ( 0 , T ) } ( y - y _ { i , d } ) \\hat { y } ^ i \\ d x d t + \\mu _ i \\iint _ { \\mathcal { O } _ i \\times ( 0 , T ) } v ^ i \\hat { v } ^ i \\ d x d t = 0 , \\\\ & \\forall \\hat { v } ^ i \\in L ^ 2 ( \\mathcal { O } _ i \\times ( 0 , T ) ) , \\ \\ { v } ^ i \\in L ^ 2 ( \\mathcal { O } _ i \\times ( 0 , T ) ) , \\ \\ i = 1 , 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} d _ i ^ { n e } = \\frac { B \\left ( | \\mathcal { S } | - 1 \\right ) } { w _ i \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } \\left ( 1 - \\frac { c _ i + \\lambda _ i } { w _ i } \\frac { \\left ( | \\mathcal { S } | - 1 \\right ) } { \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } \\right ) . \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\bar { b } _ { i j } ( t _ 0 , \\cdots , t _ s ) & = b _ { i j } ( t _ 0 , \\cdots , t _ s ) - \\dfrac { ( s - i + 1 ) ( s - j + 1 ) } { n } t _ { s - i + 1 } t _ { s - j + 1 } \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} \\langle \\tilde { D } _ { u } v , w \\rangle = \\langle D ^ { ( 1 ) } _ { u } v , w \\rangle & - \\frac { 1 } { 8 } \\left ( \\eta ( \\mathcal J v , u , w ) + \\eta ( w , u , \\mathcal J v ) \\right ) \\\\ & + \\frac { 1 } { 8 } \\left ( \\eta ( v , u , \\mathcal J w ) + \\eta ( \\mathcal J w , u , v ) \\right ) , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\left ( \\sum ^ { \\infty } _ { n = 1 } f ( x _ { n } ) e _ { n } , h \\right ) = \\left ( f , k \\right ) \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} \\begin{cases} - \\overline { W } '' = \\left ( 1 - \\overline { W } \\right ) \\overline { W } & \\textup { i n } \\left ( 0 , + \\infty \\right ) \\\\ \\overline { W } \\left ( 0 \\right ) = \\nu \\\\ \\nu < \\overline { W } < 1 & \\textup { i n } \\left ( 0 , + \\infty \\right ) \\end{cases} \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\nu ' : = [ \\nu ; \\nu ' ( x ) = \\gamma ] . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} M ( \\lambda _ 0 ) \\begin{pmatrix} v \\\\ u \\end{pmatrix} = \\begin{pmatrix} M _ { \\overline S \\overline S } ( \\lambda _ 0 ) & M _ { \\overline S S } ( \\lambda _ 0 ) \\\\ M _ { S \\overline S } ( \\lambda _ 0 ) & M _ { S S } ( \\lambda _ 0 ) \\end{pmatrix} \\begin{pmatrix} v \\\\ u \\end{pmatrix} = \\lambda _ 0 \\begin{pmatrix} v \\\\ u \\end{pmatrix} . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n / 2 } \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } \\varepsilon _ j \\leq \\sum _ { j = 1 } ^ { n / 2 } \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } \\leq e ^ { ( n / 2 ) ^ \\alpha } , \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} 0 = \\mathcal { S } ( V _ k ) & = s _ k \\mathcal { L } _ { 0 } ( U _ k ) + t _ k \\mathcal { L } _ 0 ( W _ k ) + \\frac { s _ k ^ 2 } { 2 } \\mathcal { S } _ { 0 , 2 } ( U _ k , U _ k ) + s _ k t _ k \\mathcal { S } _ { 0 , 2 } ( U _ k , W _ k ) \\\\ & + \\frac { t _ k ^ 2 } { 2 } \\mathcal { S } _ { 0 , 2 } ( W _ k , W _ k ) + O ( s _ k ^ 3 , s _ k t _ k ^ 2 , s _ k ^ 2 t _ k , t _ k ^ 3 ) . \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} \\mathcal Z _ j ( x ) : = \\ 4 \\alpha _ j a _ { 0 , j } \\frac { \\langle \\nabla u _ j , \\nabla \\phi \\rangle } { \\phi } + 2 \\mathcal Q _ j ( x ) a _ { 1 , j } . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} c _ 0 + \\sum _ { k = 1 } ^ { \\frac { n - 1 } { 2 } } c _ { k } \\left ( e ^ { \\frac { 2 \\pi i j } { n } k } - e ^ { \\frac { 2 \\pi i j } { n } k } e ^ { 2 \\pi i j } \\right ) \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} j _ { \\ell } = M ( i _ { \\ell - 1 } ) + 1 \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n } } { ( q / z ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = - \\infty } ^ { \\infty } b ^ 2 _ \\omega ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { - n } q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\Psi : \\bigcup _ { d _ 1 + d _ 2 = d } \\mathcal { Z } _ { d _ 1 , d _ 2 } \\to \\mathcal { Z } _ d \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\in O \\right ) & \\geq \\liminf _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } > \\varepsilon \\right ) \\\\ & = - J ( x _ 0 ) = - \\inf _ { x \\in O } I _ 1 ( x ) . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\phi ( x y ) = & \\ \\phi [ \\exp \\big ( \\tau \\log ( ( t x ) ^ \\frac { 1 } { \\tau } ) + ( 1 - \\tau ) \\log ( ( \\frac { y } { t } ) ^ \\frac { 1 } { 1 - \\tau } ) \\big ) ] \\\\ \\leq & \\ \\tau \\phi [ \\exp \\log ( ( t x ) ^ \\frac { 1 } { \\tau } ) ] + ( 1 - \\tau ) \\phi [ \\exp \\log ( ( \\frac { y } { t } ) ^ \\frac { 1 } { 1 - \\tau } ) ] \\\\ = & \\ \\tau t ^ \\frac { 1 } { \\tau } \\phi ( x ^ \\frac { 1 } { \\tau } ) + \\frac { ( 1 - \\tau ) } { t ^ \\frac { 1 } { 1 - \\tau } } \\phi ( y ^ \\frac { 1 } { 1 - \\tau } ) . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } _ { ( d \\varphi , \\iota ) } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\phi ) + \\ell _ { \\beta } ( \\phi ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = 0 , \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\Delta ( \\varphi \\circ f ) = \\varphi '' \\circ f | \\d f | ^ 2 + \\varphi ' \\circ f \\Delta f \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} P _ { q , z } \\cdot \\varphi _ { q , z } ^ * \\widetilde { J ^ { K \\textnormal { t h } } } ( z , q , Q ) = \\sum _ { i = 0 } ^ N \\left ( 1 - P ^ { - 1 } \\right ) ^ i a _ { 0 i } \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\left \\langle \\psi _ 1 ^ n T _ a , T _ j , \\tau ^ { ( k ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , k + 2 , d } = \\sum _ { \\substack { u + v = k \\\\ x + y = u } } \\frac { k ! } { v ! x ! y ! } ( \\tau _ 2 ( d ) ) ^ x \\left \\langle \\psi _ 1 ^ { n - y } ( T _ a \\cup \\tau _ 2 ^ y ) , T _ j , \\tau '^ { ( v ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , v + 2 , d } \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\cos \\ ! \\left ( \\frac { 2 \\pi n } q \\right ) & = 1 - \\frac { q } { q - 1 } \\chi _ 0 ( n ) + \\frac { \\sqrt { q } } { q - 1 } \\sum _ { \\substack { \\chi \\pmod * { q } \\\\ \\chi ( - 1 ) = 1 \\\\ \\chi \\neq \\chi _ 0 } } \\overline { \\epsilon _ \\chi } \\chi ( n ) , \\\\ \\sin \\ ! \\left ( \\frac { 2 \\pi n } q \\right ) & = \\frac { \\sqrt { q } } { q - 1 } \\sum _ { \\substack { \\chi \\pmod * { q } \\\\ \\chi ( - 1 ) = - 1 } } \\overline { \\epsilon _ \\chi } \\chi ( n ) , \\\\ \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} e _ \\pm : = \\lambda t _ \\alpha \\pm \\mu e ^ \\alpha \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\tilde { \\Sigma } = \\Sigma + \\tilde x ( F \\otimes F ) , \\tilde x = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\Big ( f _ i ^ 2 - \\frac { 1 } { 2 r } \\Big ) . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\| y \\| _ { L ^ 2 ( Q ) } ^ 2 \\leq C ( \\underset { [ 0 , T ] } { } \\| y _ x ( t ) \\| ^ 2 + \\| y _ { x x } \\| _ { L ^ 2 ( Q ) } ^ 2 ) \\leq C ( \\| f \\| _ { L ^ 2 ( \\mathcal { O } \\times ( 0 , T ) ) } ^ 2 + \\| y _ 0 \\| ^ 2 + \\sum _ { i = 1 } ^ 2 \\frac { 1 } { \\mu } \\| \\phi ^ i \\| _ { L ^ 2 ( \\mathcal { O } _ i \\times ( 0 , T ) ) } ^ 2 ) \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} ( A ^ - n _ 2 , A ^ + a ( n - a ) ) = 1 , \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\sum _ { \\alpha _ 0 + \\alpha _ 1 + \\alpha _ 2 = 3 } \\frac { t _ 0 ^ { \\alpha _ 0 } } { \\alpha _ 0 ! } \\cdots \\frac { t _ 2 ^ { \\alpha _ 2 } } { \\alpha _ 2 ! } \\int _ { \\mathbb { P } ^ 2 } H ^ { \\alpha _ 1 + 2 \\alpha _ 2 } = \\frac { 1 } { 2 } \\left ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 \\right ) \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} x _ i = - x _ { 2 s + 1 - i } = \\dfrac { 1 } { i - ( i + 1 ) \\xi _ 1 } , \\ i = 1 , . . . , s . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} W _ { L a L \\underline L } & = r \\beta _ a + r ^ 2 D \\beta _ a + \\frac { 1 } { 2 } r ^ 3 D ^ 2 \\beta _ a + O ( r ^ 4 ) \\\\ W _ { L \\underline L L \\underline L } & = \\rho + r D \\rho + r ^ 2 [ \\frac { 1 } { 2 } D ^ 2 \\rho - \\frac { 1 } { 3 } | \\beta | ^ 2 ] + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} \\sum _ { j \\in I } k _ j N _ j ( f ) = n , \\ \\ \\sum _ { j \\in I } k _ j N _ j ( f _ i ) = \\beta _ i , \\ 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i , j } & = \\Gamma ( k ) _ { i , 1 } \\otimes \\Gamma ( k ) _ { 1 , j } \\\\ & = \\Gamma ( k ) _ { i , 1 } + \\Gamma ( k ) _ { 1 , j } . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} | I _ { m } | = \\| u _ { m } \\| _ { \\textsc { C C } } \\leq C _ { V , l } \\| u _ { m - 1 } \\| ^ { 1 + \\frac { 1 } { l } } = C _ { V , l } | I _ { m - 1 } | ^ { 1 + \\frac { 1 } { l } } . \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\lambda : = \\min \\{ 1 , p _ { } - 1 \\} \\Lambda : = \\max \\{ 1 , p _ { } - 1 \\} . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\mathcal { L } _ 2 [ w _ 2 ] ( x ^ * ) = \\mathcal { A } _ 2 [ w _ 2 ] ( x ^ * ) = - ( \\Delta w _ 2 ) ( x ^ * ) - ( p - 2 ) \\frac { \\langle ( D ^ 2 w _ 2 ) ( \\nabla u _ 2 ) , \\nabla u _ 2 \\rangle } { | \\nabla u _ 2 | ^ 2 } ( x ^ * ) \\geq 0 . \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} \\ \\alpha = 0 , \\ h = - n , \\ n = 1 . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { \\partial u _ { J } } { \\partial t } = f ( u _ { J } , v _ { J } ) + \\varepsilon { \\textstyle \\sum \\limits _ { I \\in G _ { N } ^ { 0 } } } L _ { J I } u _ { I } \\\\ \\\\ \\frac { \\partial v _ { J } } { \\partial t } = g ( u _ { J } , v _ { J } ) + \\varepsilon d { \\textstyle \\sum \\limits _ { I \\in G _ { N } ^ { 0 } } } L _ { J I } v _ { I } , \\end{array} \\right . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\dfrac { p ^ { t } _ { n } \\ln p _ { n } } { p ^ { s } _ { n } - 1 + p ^ { t } _ { n } } & = \\dfrac { \\ln p _ { n } } { p _ { n } ^ { s - t } - { p ^ { - t } _ { n } } + 1 } \\\\ & < \\dfrac { \\ln p _ { n } } { p _ { n } ^ { s - t } } \\\\ & < \\dfrac { p _ { n } - 1 } { p _ { n } ^ { s - t } } \\ , \\ , \\ , \\ , ( \\mbox { u s e o f } \\ , \\ln x < 1 - x , \\ , x > 0 ) \\\\ & < \\dfrac { 1 } { p _ { n } ^ { s - t - 1 } } \\\\ & < \\dfrac { 1 } { n ^ { s - t - 1 } } \\ , \\ , \\ , \\ , ( \\mbox { u s e o f } \\ , p _ { n } > n ) . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} k ^ L _ { \\gamma , \\Sigma } : = \\sqrt { \\frac { | | \\nabla ^ { \\Sigma , L } _ { \\dot { \\gamma } } { \\dot { \\gamma } } | | _ { \\Sigma , L } ^ 2 } { | | \\dot { \\gamma } | | ^ 4 _ { \\Sigma , L } } - \\frac { \\langle \\nabla ^ { \\Sigma , L } _ { \\dot { \\gamma } } { \\dot { \\gamma } } , \\dot { \\gamma } \\rangle ^ 2 _ { \\Sigma , L } } { | | \\dot { \\gamma } | | ^ 6 _ { \\Sigma , L } } } . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} & \\| ( I _ { N } ^ { \\lambda } u - u ) ' \\| _ { L ^ 2 ( \\mathbb R ) } \\le c \\big ( \\| I _ N ^ G \\breve U - \\breve U \\| _ { L ^ 2 _ { \\omega _ \\lambda } ( I ) } + \\| \\sqrt { 1 - t ^ 2 } ( I _ N ^ G \\breve U - \\breve U ) ' \\| _ { L ^ 2 _ { \\omega _ \\lambda } ( I ) } \\big ) . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\| y ^ \\prime _ v \\| _ 1 = \\tau ( y ^ \\prime _ v ) = \\frac { 2 ^ { v + 1 } } { A _ j } \\int _ { 2 ^ v } ^ { 2 ^ { v + 1 } } \\left \\| \\left ( \\frac { \\partial T _ { s , j } ( x _ { v + j } ) } { \\partial s } \\right ) ^ * \\right \\| _ 2 ^ 2 d s + 2 \\sum _ { k = 0 } ^ { A _ j - 1 } \\| T _ { { \\gamma _ k } , j } ( x _ { v + j } ) ^ * \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} S ^ { ( - 1 ) } = I _ { j - 1 } \\cup I _ j , \\ \\ S ^ { ( 0 ) } = I _ { j - 1 } \\cup I _ j \\cup I _ { j + 1 } , \\ \\ S ^ { ( 1 ) } = I _ j \\cup I _ { j + 1 } . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} \\overline { Q } _ { k , 0 } ( m , n ) = 0 , k \\geq 1 , \\ m , n \\geq 0 , \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) ( z , Q ) = \\sum _ { i = 0 } ^ N \\left ( 1 - P ^ { - 1 } \\right ) ^ i \\sum _ { \\substack { 0 \\leq a , b \\leq N \\\\ a + b = i } } \\frac { 1 } { a ! } \\left ( \\frac { \\log ( Q ) } { z } \\right ) ^ a g _ b ( z , Q ) \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} d F \\circ J = J \\circ d F \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} t ^ { n - 1 } P _ n ( t ^ { - 1 } ) = \\sum \\limits _ { F \\in L ( \\mathcal { B } _ n ) } \\chi ( ( \\mathcal { B } _ n ) _ F , t ) P ( ( \\mathcal { B } _ n ) ^ F , t ) . \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} P ^ { ( j ) } ( 1 ) + 2 ( - 1 ) ^ { j + 1 } ( j - 2 ) ! = 0 \\ \\ \\mbox { f o r a l l } \\ \\ j = 1 , 2 , 3 , 4 , \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} \\partial _ t z - \\partial _ { x x } z = g _ L ( x , z ) , \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} u _ { 0 } ( z ) = \\left ( \\frac { z - a _ { 1 } } { \\left | z - a _ { 1 } \\right | } \\right ) ^ { d _ { 1 } } \\left ( \\frac { z - a _ { 2 } } { \\left | z - a _ { 2 } \\right | } \\right ) ^ { d _ { 2 } } . . . . \\left ( \\frac { z - a _ { m } } { \\left | z - a _ { m } \\right | } \\right ) ^ { d _ { m } } . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & ( \\frac { d } { d \\xi } r _ t + \\alpha _ { \\rm H J M } ( r _ t ) ) d t + \\gamma ( r _ { t - } ) d X _ t \\medskip \\\\ r _ 0 & = & h _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} u ( x , t ) = \\widetilde { \\phi } ( x _ 1 - c t ) \\ \\ { \\rm w i t h } \\ \\ \\widetilde { \\phi } ( - \\infty ) = p ^ + , \\ \\widetilde { \\phi } ( + \\infty ) = p ^ - . \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} J _ { \\rho _ 1 , \\rho _ 2 } ( u ) = \\frac 1 2 \\int _ { M } | \\nabla u | ^ 2 - \\rho _ 1 \\log \\int _ { M } e ^ { u - \\overline { u } } - \\rho _ 2 \\log \\int _ { M } e ^ { - u + \\overline { u } } . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\begin{array} { r l l } \\partial _ t v - \\Delta v + ( u \\cdot \\nabla ) v + \\nabla _ H \\pi & = 0 & \\Omega \\times ( 0 , \\infty ) , \\\\ \\partial _ z \\pi & = 0 & \\Omega \\times ( 0 , \\infty ) , \\\\ _ H \\overline { v } & = 0 & G \\times ( 0 , \\infty ) , \\\\ v ( 0 ) & = a & \\Omega , \\end{array} \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\sum \\limits _ { ( M _ { S ' } , \\mu _ { S ' } ) \\in X } \\ , \\ \\sum \\limits _ { b \\in Y _ { ( M _ { S ' } , \\mu _ { S ' } ) } } ( - 1 ) ^ { L _ { M _ { S ' } , M _ b } } = 0 . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} Z _ \\nu ( p _ 1 , \\ldots , p _ r ) = \\bigcup _ { \\mathbf { x } \\in \\mathbf { Z } ( p _ 1 , \\ldots , p _ r ) } B ( \\mathbf { x } , \\nu ) \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} m = 2 n + d - 1 \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} { } r _ { - \\mu } | _ { \\widehat { M _ S } } = \\bigoplus _ { ( M _ S , \\mu _ S ) \\in \\mathcal { C } _ G , \\mu _ S \\sim _ G \\mu } r _ { - \\mu _ S } | _ { \\widehat { M _ S } } . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\bar { u } _ T ^ { i + j } = u _ \\zeta ^ { e _ n } ( i + j ) = u _ T ^ { i - T j ' + j } . \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} - \\left ( \\kappa W _ { \\rho , \\nu } \\right ) '' = \\left ( 1 - W _ { \\rho , \\nu } \\right ) \\kappa W _ { \\rho , \\nu } \\geq \\left ( 1 - \\kappa W _ { \\rho , \\nu } \\right ) \\kappa W _ { \\rho , \\nu } \\textup { i n } \\left ( - \\rho , \\rho \\right ) . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\beta ^ { * } _ n ( a q , b , q ) = \\frac { ( 1 - b ) } { 1 - b q ^ n } \\beta _ n ( a , q ) . \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} \\lim _ { L _ 1 , L _ j \\to 0 } \\tfrac { 1 } { L _ 1 } R ( L _ 1 , L _ j , z ) & = 2 ( e ^ { \\frac { 1 } { 2 } ( 0 + z ) } + 1 ) ^ { - 1 } , \\\\ \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } D ( L _ 1 , y , z ) & = \\tfrac { 1 } { L _ 1 } \\left ( R ( L _ 1 , y , z ) + R ( L _ 1 , z , y ) - L _ 1 \\right ) \\\\ & = 2 ( e ^ { \\frac { 1 } { 2 } ( y + z ) } + 1 ) ^ { - 1 } . \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\overline { Q } _ { k , i } ( x ; q ) = H _ { k , i } ( - 1 / q ; x q ; q ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n x ^ { k n } q ^ { k n ^ 2 + k n - i n } ( 1 - x ^ i q ^ { ( 2 n + 1 ) i } ) ( - x q ^ { n + 1 } ) _ { \\infty } ( - q ) _ n } { ( q ) _ n ( x q ^ { n + 1 } ) _ \\infty } , \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\frac { \\rho } { z } z ^ { - \\mu } = z ^ { - \\mu } \\rho \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} G : = \\nabla \\mathcal { L } \\cdot ( \\nabla \\mathcal { L } ) ^ \\top = \\begin{pmatrix} f ^ 2 & 0 \\\\ 0 & 1 \\end{pmatrix} \\qquad \\mbox { w i t h } f ( s , t ) : = \\sqrt { 1 + \\theta ' ( s ) ^ 2 \\ , t ^ 2 } \\ , . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\psi ( t , w ) = \\bigl ( s + t , ( \\psi ) w \\bigr ) , \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\Theta ( a z ) = \\Theta ( z \\sigma ( a ) ) = z \\theta \\sigma ^ { - 1 } \\sigma ( a ) = z \\theta ( a ) = \\Theta ( z ) \\Theta ( a ) . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\binom { n } { k } = ( - 1 ) ^ k \\operatorname { s g n } ( k ) \\binom { k - n - 1 } { k } , \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} ( 2 \\mu c _ \\alpha t _ \\alpha - e ^ \\alpha ) ( \\theta ^ \\beta _ \\pm e ^ \\beta + e ^ { \\alpha + \\beta } ) & = ( 2 \\mu c _ \\alpha \\theta ^ \\beta _ \\pm a | \\alpha \\cap \\beta | - b _ { \\alpha , \\alpha + \\beta } ) e ^ \\beta \\\\ & + ( 2 \\mu c _ \\alpha a | \\alpha \\cap ( \\alpha + \\beta ) | - b _ { \\alpha , \\beta } \\theta ^ \\beta _ \\pm ) e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} E ( Q _ { a , c } ) = - 2 \\pi c ^ 2 + \\frac 1 2 \\int V Q _ { a , c } ^ 2 \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} h f _ { s _ 1 } ( w _ 1 ) \\cdots f _ { s _ { i _ 1 - 1 } } ( w _ { i _ 1 - 1 } ) = f _ { t _ 1 } ( v _ 1 ) \\cdots f _ { t _ { j _ 1 - 1 } } ( v _ { j _ 1 - 1 } ) . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\norm { \\chi _ k ' } _ { L ^ \\infty ( [ \\rho _ k , \\rho _ { k + 1 } ] ) } = \\norm { \\tilde \\chi _ k ' } _ { L ^ \\infty ( [ \\rho _ k , \\rho _ { k + 1 } ] ) } = O ( \\rho _ k ^ { - \\frac { 1 - \\delta } { 2 } } ) . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} f = \\sum _ { k = 1 } ^ { M } \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { k } \\Pi _ { l } ( g _ { j } ^ { k } , h _ { j , 1 } ^ { k } , h _ { j , 2 } ^ { k } ) + E _ { M } , \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\sup _ n | | n ^ { - 1 / 2 } \\sum _ { i = 1 } ^ n X _ i | | L ( s , \\Omega ) \\le K _ N [ \\beta ] ( s ) \\ \\sup _ i | | X _ i | | L ( s , \\Omega ) , \\ s \\in [ 2 , \\infty ) , \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i x _ j } \\widetilde { m } _ { x _ j } d y + \\int _ { \\mathcal { Y } ^ d } V _ { x _ j } \\widetilde { m } _ { x _ j } d y = \\int _ { \\mathcal { Y } ^ d } \\frac { 1 } { \\widetilde { m } } \\widetilde { m } ^ 2 _ { x _ j } d y . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} T _ 0 = A ( y ^ 0 \\frac { \\partial } { \\partial y ^ 4 } - y ^ 4 \\frac { \\partial } { \\partial y ^ 0 } ) - B _ i ( y ^ 0 \\frac { \\partial } { \\partial y ^ i } + y ^ i \\frac { \\partial } { \\partial y ^ 0 } ) - C _ j ( y ^ 4 \\frac { \\partial } { \\partial y ^ j } + y ^ j \\frac { \\partial } { \\partial y ^ 4 } ) + D _ p \\epsilon _ { p q r } y ^ q \\frac { \\partial } { \\partial y ^ r } \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\omega _ { i , t } ( k ) & : = \\big [ [ \\tilde { x } _ 1 ( K ) ] _ i , \\dots , [ \\tilde { x } _ { t - 1 } ( K ) ] _ i , [ \\tilde { x } _ t ( k ) ] _ i , \\underbrace { 0 , \\dots , 0 } _ { m - t ~ } \\big ] ^ T \\\\ [ d _ t ( k ) ] _ i & : = \\nabla _ t H _ i \\big ( \\omega _ { i , t } ( k ) \\big ) + c _ { i , t } G ' _ i \\big ( \\hat { c } _ i ^ T \\omega _ { i , t } ( k ) \\big ) \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} u ( t _ { 0 } , x ) = \\varphi ( x ) , x \\in \\Sigma , \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\dot { x } & = n ( x ) - h ( x , v ) , \\\\ \\dot { y } & = \\int _ 0 ^ { \\infty } f _ 1 ( \\tau ) h ( x ( t - \\tau ) , v ( t - \\tau ) ) d \\tau - a g _ 1 ( y ) - p w ( y , z ) , \\\\ \\dot { v } & = k \\int _ 0 ^ { \\infty } f _ 2 ( \\tau ) g _ 1 ( y ( t - \\tau ) ) d \\tau , \\\\ \\dot { z } & = c \\int _ 0 ^ { \\infty } f _ 3 ( \\tau ) w ( y ( t - \\tau ) , z ( t - \\tau ) ) d \\tau - b g _ 3 ( z ) . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} & \\overline { U } _ { 2 k , 2 a } ( x ; q ) - \\overline { U } _ { 2 k , 2 a - 2 } ( x ; q ) \\\\ & = ( x q ) ^ { 2 a } U _ { 2 k , 2 k - 2 a } ( x q ; q ) + ( x q ) ^ { 2 a - 2 } U _ { 2 k , 2 k - 2 a + 2 } ( x q ; q ) \\\\ & = ( x q ) ^ { 2 a } ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , k - a } ( x ^ 2 q ^ 2 ; q ^ 2 ) + ( x q ) ^ { 2 a - 2 } ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , k - a + 1 } ( x ^ 2 q ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} L ^ S ( \\tfrac { 1 } { 2 } , B C ( \\pi ( q ) ) \\otimes B C ( \\pi ' ( q ) ) ) = L ^ S ( \\tfrac { 1 } { 2 } , \\Pi ^ { { \\sf v } } \\otimes \\Pi '^ { { \\sf v } } ) = L ^ S ( \\tfrac { 1 } { 2 } , \\Pi ^ { c } \\otimes \\Pi '^ { c } ) = L ^ S ( \\tfrac { 1 } { 2 } , \\Pi \\otimes \\Pi ' ) . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ z } = \\partial _ z + T , \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} v = e ^ { \\chi _ 0 t } u , \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} [ X \\times _ k \\mathbb A _ k ^ n \\to S , \\sigma ] = [ X \\times _ k \\mathbb A _ k ^ n \\to S , \\sigma ' ] \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\max \\left \\{ { { X _ M } , { Y _ M } } \\right \\} = \\frac { 1 } { 2 } \\left ( { { X _ M } + { Y _ M } + \\left | { { X _ M } - { Y _ M } } \\right | } \\right ) \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - h _ n x _ n ^ 2 ) + \\rho _ { n + 1 } \\xi _ { n + 1 } , n \\in \\mathbb N , x _ 0 \\in \\mathbb R . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\tilde { f } ( x ) : = \\tilde { g } ( \\Phi ( x ) ) , x \\in U . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} { \\frak W } ^ { \\rm u n } _ t ( W _ 0 ) \\left ( y , k \\right ) : = W ^ { \\rm u n } ( t , y , k ) , \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\delta _ \\varepsilon ( x ' ) : = \\frac 1 { \\varepsilon ^ 2 } \\delta \\left ( \\frac { x ' } \\varepsilon \\right ) , \\delta _ { \\varepsilon , x _ 0 ' } ( x ' ) : = \\delta _ \\varepsilon ( x ' - x _ 0 ' ) . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} y _ { \\textrm { N F } } = \\sqrt { \\beta _ { \\textrm { N } } \\eta P _ \\textrm { B } d _ { \\textrm { B N } } ^ { - \\alpha } d _ { \\textrm { N F } } ^ { - \\alpha } | h _ { \\textrm { B N } } | ^ 2 } h _ { \\textrm { N F } } x _ { \\textrm { F } } + n _ { \\textrm { F } } , \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} K ( y , y ' ) & = - \\frac { \\partial ^ 2 } { \\partial y \\partial y ' } \\left ( \\frac { \\chi ( y ) - \\chi ( y ' ) } { y - y ' } \\right ) \\\\ & = \\frac { 2 ( \\chi ( y ) - \\chi ( y ' ) ) - ( \\chi ' ( y ) + \\chi ' ( y ' ) ) ( y - y ' ) } { ( y - y ' ) ^ 3 } \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F ( n , 0 ) \\equiv \\sum _ { n = 0 } ^ { m - 1 } F ( n , 1 ) \\equiv \\cdots \\equiv \\sum _ { n = 0 } ^ { m - 1 } F \\left ( n , \\frac { m - 1 } { 2 } \\right ) \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} S _ 2 ( \\overline { f ( z , t ) } , z , t ) = 0 . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\bar \\eta _ { j k } = \\frac { \\langle u _ j , E u _ k \\rangle } { \\sqrt { \\lambda _ j \\lambda _ k } } . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\Lambda _ i \\widetilde { \\delta } _ { i j } \\widetilde { m } _ { \\Lambda _ j } d y = 0 . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} \\Phi \\left ( \\sum _ i g [ \\sigma _ i ] \\right ) = 0 . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} & \\norm { \\sum \\limits _ { n = 1 } ^ { M } \\Big ( \\sqrt { \\lambda _ n ^ { N } } \\phi _ n ^ { N } \\psi _ n - \\sqrt { \\lambda _ n ^ { N , h } } \\phi _ n ^ { N , h } \\psi _ n \\Big ) } _ { L ^ 2 ( \\Omega , C ( D ) ) } \\lesssim M ^ { \\frac { 3 s } { d } + 2 } N ^ { - 1 / 2 } h ^ { s } + M ^ { \\frac { s } { d } + \\frac { 3 } { 2 } } h ^ { s } . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} I _ { i } ~ = ~ [ i h _ 1 , ( i + 1 ) h _ 1 \\big ) \\quad \\forall i \\in \\overline { 0 , N _ 1 - 2 } ~ ; \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} { \\rm d e g } \\ , v _ i = ( { \\rm d e g } \\ , w _ i ) \\sigma _ t \\ ; \\ ; \\ ; \\mbox { f o r a n y } i \\in \\alpha \\mbox { w i t h } \\alpha \\in \\mathcal { C } ^ t \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} \\lambda w - \\Delta w = \\partial _ i f \\Omega . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} & E _ { j } ^ 1 = L ^ { p _ { j } ^ 1 } ( M _ 1 , \\mu _ 1 ) , \\ , \\\\ & E _ { j } ^ s = L ^ { p _ { j } ^ s } ( M _ s , \\mu _ s ; E _ { j } ^ { s - 1 } ) , s = 2 , \\ldots , S , \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} ( A ^ - n _ 2 , A ^ + c _ k ^ + ( c _ k - 1 ) ^ + ) = 1 , \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} F = \\textrm { c o n v } \\left ( x \\mapsto \\inf _ { y \\in E ^ { * } } \\left \\{ t _ y + \\langle \\xi _ y , x - y \\rangle + \\varphi _ { y } ( P ( x ) ) + a | P ( x - y ) | ^ 2 \\right \\} \\right ) \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 1 } ^ s | f ( I _ j ) | ^ q \\Big ) ^ { \\frac 1 { q } } \\leq 1 6 \\left ( 1 + V _ { \\Phi , h } ( f ) \\right ) \\max _ { 1 \\leq m \\leq s } m ^ { \\frac 1 { q } } \\Phi _ m ^ { - 1 } \\big ( h ( s ) \\big ) . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F \\left ( n , \\frac { m - 1 } { 2 } \\right ) = F \\left ( m - 1 , \\frac { m - 1 } { 2 } \\right ) & = [ 2 m - 1 ] \\frac { ( q ; q ^ 2 ) _ { m - 1 } ( q ; q ^ 2 ) _ { ( m - 1 ) / 2 } } { ( q ; q ) _ { m - 1 } ( q ^ 2 ; q ^ 2 ) _ { ( m - 1 ) / 2 } } \\\\ [ 5 p t ] & = { 2 m - 1 \\brack m - 1 } { m - 1 \\brack \\frac { m - 1 } { 2 } } _ { q ^ 2 } \\frac { [ m ] } { ( - q ; q ) _ { m - 1 } ^ 2 } . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\mathcal I _ { t _ 0 } ( t ) : = \\frac 1 2 \\int \\varphi \\left ( \\frac { x + \\sigma t _ 0 - \\tilde \\sigma ( t _ 0 - t ) } { L } \\right ) \\left ( u ^ 2 + u _ x ^ 2 \\right ) ( t , x ) d x , \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\| \\mathcal { L } _ { \\mathcal { A } _ n } \\| = \\sup _ { f \\in C ( K ) , f \\neq 0 } \\frac { \\| \\mathcal { L } _ { \\mathcal { A } _ n } f \\| _ K } { \\| f \\| _ K } \\leq c \\ , \\sqrt { c a r d ( \\mathcal { A } _ n ) } \\ ; , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\sup _ { | k | \\leq \\varepsilon | K | } | | U ( K + k ) | | = \\sup _ { | k | \\leq \\frac { 3 \\varsigma _ 1 } { 4 \\ln \\lambda } | K | } | | U ( K + k ) | | . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} U _ n ( k ) - U _ n ( k - 1 ) = \\frac { 1 } { n ! } \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i { n + 1 \\choose i } ( k - i ) ^ n = \\frac { A ( n , k ) } { n ! } , \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\| S _ { 1 / N } - T _ { m _ { N } } : V N ( \\Gamma ) \\to V N ( \\Gamma ) \\| & \\leq \\sum _ { r = N ^ 2 + 1 } ^ \\infty \\| e ^ { - \\frac { r } { N } } p _ r \\| \\lesssim \\beta \\sum _ { r = N ^ 2 + 1 } ^ \\infty \\frac { N ^ 6 } { r ^ 6 } ( r + 1 ) \\\\ & \\lesssim \\beta \\frac { 1 } { ( N + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} k _ 2 : = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } \\sup _ { x \\in \\R ^ d } \\sup _ { 0 < \\tau \\leq T } \\ < \\varphi _ { \\tau } ( x - y ) , \\mu _ 0 ( d y ) \\ > . \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\log | R ( \\chi _ 0 ) | ^ 2 & = \\log \\prod _ { p \\leq X } ( 1 - q _ p ) ^ { - 2 } \\\\ & = 2 \\sum _ { p \\leq X } ( \\log X - \\log p ) \\\\ & = 2 ( 1 + o ( 1 ) ) \\frac { X } { \\log X } , \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\langle \\Delta ^ { l _ j } _ K f _ j , h ^ 0 _ { Q _ j } \\rangle = \\langle f _ j , h _ { Q _ j ^ { ( k _ j - l _ j ) } } \\rangle \\langle h _ { Q _ j ^ { ( k _ j - l _ j ) } } , h ^ 0 _ { Q _ j } \\rangle , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} ( d S _ { l } ) _ { h } ( H _ { n } ( w ) ) & = \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } S _ { l } ( H _ { n } ( y ) + \\varepsilon \\cdot H _ { n } ( w ) ) = \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } S _ { l } ( H _ { n } ( y + \\varepsilon \\cdot w ) ) \\\\ & = \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } f _ { n } ( y + \\varepsilon \\cdot w ) = ( d f _ { n } ) _ { y } ( w ) = \\xi . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} 1 - \\gamma & \\geq P \\big ( S _ 2 = O \\big | S _ 0 = O \\big ) + P \\big ( S _ 4 = O , S _ 2 \\neq O \\big | S _ 0 = O \\big ) \\\\ & = \\frac { 4 d ^ 2 + 4 d - 3 } { 8 d ^ 3 } > \\frac { 1 } { 2 d - 1 } \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} J ( x ) \\sim \\frac { 1 } { | x | ^ { n + 2 + \\varepsilon } } \\quad \\varepsilon > 0 \\ , \\ , ( \\sigma = 2 ) , J ( x ) \\sim \\frac { 1 } { | x | ^ \\alpha } n < \\alpha < n + 2 \\ , \\ , ( \\sigma = \\alpha - n \\in ( 0 , 2 ) ) , \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} N _ { C _ 2 \\wr S _ { n } } ( R _ r ) = N _ { C _ 2 \\wr S _ { r p } } ( R _ r ) \\times C _ 2 \\wr S _ { \\{ r p + 1 , \\ldots , n \\} } , \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\widehat { \\Z \\Gamma } = \\varprojlim _ i \\Z \\Gamma / p ^ i \\Z \\Gamma \\longrightarrow \\varprojlim _ i \\Z \\Gamma ^ { ( n ) } / p ^ i \\Z \\Gamma ^ { ( n ) } = \\Z _ p \\Gamma ^ { ( n ) } \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\left \\Vert { f } \\right \\Vert _ M : = \\sup _ { x \\in { } M } { \\left \\Vert { f ( x ) } \\right \\Vert } \\in \\mathbb { R } _ { \\geq { } 0 } \\cup \\{ \\infty \\} \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} \\| u _ x ( t ) \\| _ { \\infty } & = \\Theta \\left ( \\frac { 1 } { T ^ { ( 2 ) } - t } \\right ) , & \\| u _ { x x } ( t ) \\| _ { \\infty } & = \\Theta \\left ( \\frac { 1 } { ( T ^ { ( \\infty ) } - t ) ^ 2 } \\right ) . & \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} S = \\{ ( u , v ) \\in \\R _ + \\times \\R _ + \\mid H ( u , v ) = 0 \\} , \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} g ( x ) & = \\chi _ { B ( y _ { 2 } , r ) } ( x ) \\\\ h _ { 1 } ( x ) & = \\chi _ { B ( y _ { 1 } , r ) } ( x ) \\\\ h _ { 2 } ( x ) & = \\frac { a ( x ) } { { T _ { 2 } } ^ { * } ( h _ { 1 } , g ) ( x _ { 0 } ) } \\\\ \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} | \\Lambda _ { P _ 1 , P _ 2 } ( f _ 0 , f _ 1 , f _ 2 ) | \\leq q ^ { \\delta _ 1 } \\min _ { i = 1 , 2 } \\| f _ i \\| _ { U ^ 1 } ^ { 1 / 2 } + O _ { P _ 1 , P _ 2 } ( q ^ { \\delta _ 1 - 1 / 4 } + q ^ { - \\delta _ 2 } + q ^ { 3 \\delta _ 3 / 4 - \\delta _ 4 / 4 } + q ^ { \\delta _ 3 - 1 / 4 } ) \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} M ^ m Z = Y , \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\lambda \\in L ^ p ( \\mathcal M ) ^ * \\mapsto B _ \\lambda \\in L ^ { p ' } ( \\mathcal M ) , \\lambda ( A ) = \\tau ( B _ \\lambda A ) \\quad \\forall A \\in L ^ p ( \\mathcal M ) . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\hat \\eta _ { K , M , j } ^ { ( 3 ) } \\buildrel \\Delta \\over = { \\bf { e } } _ j ^ T \\sum \\limits _ { l = 1 } ^ { M + K } { { { { \\bf { \\bar W } } } ^ { Q \\left ( { M + K - l + 1 } \\right ) } } { { \\boldsymbol { \\xi } } ^ { ( l ) } } } = { \\bf { e } } _ j ^ T \\sum \\limits _ { l = 1 } ^ { K } { { { { \\bf { \\bar W } } } ^ { Q \\left ( { K - l + 1 } \\right ) } } { { \\boldsymbol { \\xi } } ^ { ( l + M ) } } } . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\chi ( k , \\xi ) = ( k + N ) ^ { \\i \\nu } \\cdot \\exp [ \\frac { 1 } { 2 \\pi \\i } \\int \\limits _ { - N } ^ { k _ 0 } \\frac { \\ln \\frac { 1 + | r ( s ) | ^ 2 } { 1 + | r ( k _ 0 ) | ^ 2 } \\ \\d s } { s - k } + \\frac { 1 } { 2 \\pi \\i } \\int \\limits _ { - \\infty } ^ { - N } \\frac { \\ln ( 1 + | r ( s ) | ^ 2 ) \\ \\d s } { s - k } ] . \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} ( f , g ) _ { H } = \\int _ { \\R ^ N } f g \\rho \\ , d x . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\begin{aligned} f ^ { \\prime } ( x ) = \\dfrac { d f ( x ) } { d x } & = \\dfrac 3 2 S - \\dfrac 3 2 x - \\dfrac { \\sqrt { ( 4 S - 3 x ) x } } 4 - \\dfrac { x ( 2 S - 3 x ) } { 4 \\sqrt { ( 4 S - 3 x ) x } } \\\\ & = \\dfrac 3 2 ( S - x ) - \\dfrac { 3 x ( S - x ) } { 2 \\sqrt { ( 4 S - 3 x ) x } } \\\\ & = \\dfrac 3 2 ( S - x ) \\bigl ( 1 - \\dfrac { x } { \\sqrt { ( 4 S - 3 x ) x } } \\bigl ) > 0 \\end{aligned} \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} h _ 2 ' ( u _ 2 ) + 2 \\theta ( p - 1 ) h _ 2 ( u _ 2 ) = \\left ( 2 \\theta ( p - 1 ) - k \\right ) h _ 2 ( u _ 2 ) > 0 . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\begin{array} { r c l } { \\partial } K \\backslash \\Xi _ K & = & \\{ z \\in { \\partial } K : \\ , h _ K ( u ) > 0 \\mbox { a t s o m e } u \\in N ( K , z ) \\} , { a n d } \\\\ { \\partial } K \\backslash \\Xi _ K & \\mbox { i s } & C ^ 1 . \\end{array} \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} r \\partial _ r ( r ^ { - 1 } n _ { a b } ) = & r ^ 2 [ \\bar W _ { L a b \\underline L } + \\kappa ^ 2 \\tilde \\sigma _ { a b } - \\frac { 1 } { 6 } \\bar W _ { L a b L } + \\tilde \\nabla _ a \\eta ^ { ( 2 ) } _ b ] + O ( r ^ 3 ) \\\\ = & r ^ 2 ( \\rho + \\kappa ^ 2 ) \\tilde \\sigma _ { a b } + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\sum _ { \\beta = 1 } ^ { \\frac { n ( n - 1 ) } { 2 } } \\omega _ { \\alpha \\beta } ^ { ( l ) } \\ , K _ \\beta ^ { ( 2 , l ) } ( x ) & = S ^ { ( 2 , l ) } ( U ^ { ( l ) } ) ^ { - 1 } J _ \\alpha ^ { ( 1 , l ) } U ^ { ( l ) } ( S ^ { ( 2 , l ) } ) ^ { - 1 } ( x - q ^ { ( 2 , l ) } ) \\\\ & = S ^ { ( 1 , l ) } \\exp ( \\sigma ^ { ( l ) } ) J _ \\alpha ^ { ( 1 , l ) } \\exp ( - \\sigma ^ { ( l ) } ) ( S ^ { ( 1 , l ) } ) ^ { - 1 } ( x - q ^ { ( 2 , l ) } ) . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = \\frac { ( q ; q ) _ n } { ( q ; q ) _ k ( q ; q ) _ { n - k } } , \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} | \\ ! | \\varphi ^ { \\sharp } F | \\ ! | ' _ { K , k } & \\le \\widetilde C _ { \\varphi , \\varphi ( K ) , k } | \\ ! | F | \\ ! | ' _ { \\varphi ( K ) , k } , \\\\ \\widetilde C _ { \\varphi , \\varphi ( K ) , k } & = \\sup _ { \\alpha , | \\alpha | \\le k } \\left ( \\sum _ { \\beta \\le \\alpha } L ^ { k - | \\alpha | } \\ : C _ { \\varphi , \\varphi ( K ) , | \\alpha | } \\ : \\operatorname { d i a m } ( \\varphi ( K ) ) ^ { | \\alpha | - | \\beta | } \\right ) . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} W ( s , t ) : = \\frac { 1 } { t } - \\frac { \\partial _ t f ( s , t ) } { f ( s , t ) } = \\frac { 1 } { t \\ , \\big [ 1 + \\theta ' ( s ) ^ 2 \\ , t ^ 2 \\big ] } \\ , . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} d X _ t = - b X _ t ^ 3 d t + g ( t ) d W _ t , n \\in \\mathbb N , X _ 0 \\in \\mathbb R , \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} \\sup _ { m \\geq 1 } \\| \\dot { \\tilde { h } } ^ { ( m ) } \\| _ { \\infty ; [ 0 , a _ { m + 1 } ] } = \\sup _ { m \\geq 1 } \\| \\dot { \\tilde { h } } _ { m } \\| _ { \\infty ; I _ { m } } = \\sup _ { m \\geq 1 } \\frac { 1 } { | I _ { m } | } \\cdot \\| \\dot { { h } } _ { m } \\| _ { \\infty ; [ 0 , 1 ] } . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { 2 } \\left \\{ \\mathit { T r } \\mathbb { J } \\pm \\sqrt { \\left ( \\mathit { T r } \\mathbb { J } \\right ) ^ { 2 } - 4 \\det \\mathbb { J } } \\right \\} . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} [ \\kappa + v , x ] = ( a d _ v - a d _ { \\kappa } ^ * ) x . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} \\textbf { g } \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ i ) , S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ j ) \\right ) = \\textbf { g } ( T _ i , T _ j ) \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} F \\left ( { \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } } \\right ) = 1 - \\underbrace { \\frac { 1 } { { \\left ( { M + K } \\right ) N } } \\sum \\limits _ { j \\in { \\cal N } } { \\left ( { \\sum \\limits _ { m = 1 } ^ M { u _ j ^ { ( m ) } } + \\sum \\limits _ { i = 1 } ^ K { u _ j ^ { ( i ) } } } \\right ) } } _ { \\buildrel \\Delta \\over = S } \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 ( x , y ) = u _ 1 ( x , 0 ) + \\int _ { 0 } ^ { y } \\frac { j } { m ( x , s ) } d s - y \\int _ 0 ^ 1 \\frac { j } { m ( x , z ) } d z . \\end{aligned} \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} I I '' & = \\sum _ { n = p - 2 } ^ { 2 p - 5 } q ^ n \\sum _ { j = 1 } ^ { 2 p - 4 - n } ( n - 2 p + 2 j + 3 ) j + \\sum _ { n = 0 } ^ { p - 3 } q ^ n \\sum _ { j = 1 } ^ { n + 1 } ( 2 j - n - 1 ) j \\\\ & = \\sum _ { n = p - 2 } ^ { 2 p - 5 } q ^ n \\frac 1 6 ( 2 p - n - 3 ) ( 2 p - n - 4 ) ( 2 p - n - 5 ) + \\sum _ { n = 0 } ^ { p - 3 } q ^ n \\frac 1 6 ( n + 1 ) ( n + 2 ) ( n + 3 ) \\\\ & \\ge 0 . \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } | 1 - \\langle C _ \\varphi ^ * f , f \\rangle | ^ 2 = | | f _ 2 | | ^ 4 + | | f _ 1 | | ^ 2 | | f _ 2 | | ^ 2 \\delta ^ 2 + 2 | | f _ 1 | | \\cdot | | f _ 2 | | ^ 3 \\delta \\cos \\theta \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} m ( k ) \\leq C \\sum _ { p = k } ^ { \\infty } \\frac { 1 } { | | A _ p | | ^ 2 } . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} p ( \\rho , \\theta ) = a \\rho ^ \\gamma + \\delta \\theta p _ \\theta ( \\rho ) , \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\phi ( x ) \\sin ( 2 \\pi \\alpha n x ) x ^ { \\frac 1 2 - s } \\ , d x = \\sum _ { j = 0 } ^ { m - 1 } \\frac { \\sin ^ { ( j + 1 ) } ( 2 \\pi \\alpha n ) } { ( 2 \\pi \\alpha n ) ^ { j + 1 } } \\phi _ j ( 1 , s ) + \\int _ 1 ^ \\infty \\frac { \\sin ^ { ( m ) } ( 2 \\pi \\alpha n x ) } { ( 2 \\pi \\alpha n ) ^ m } \\phi _ k ( x , s ) x ^ { \\frac 1 2 - m - s } \\ , d x . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} J ^ G _ { P ^ { o p } _ b } I ^ G _ { M _ S } ( \\rho ) = \\sum \\limits _ { w \\in W ^ { M _ S , M _ b } } I ^ { M _ b } _ { M ' _ b } ( w ( J ^ { M _ S } _ { { P ' } ^ { o p } _ S } ( \\rho ) ) ) , \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { l o } ( \\pi { G } ) = \\mathrm { l o } ( G / \\gamma _ c { G } ) = \\mathrm { l o } ( G ) - \\mathrm { l o } ( \\gamma _ c { G } ) = \\mathrm { l o } ( G ) - 1 , \\mathrm { c l } ( \\pi { G } ) = \\mathrm { c l } ( G / \\gamma _ c { G } ) = \\mathrm { c l } ( G ) - 1 , \\\\ \\mathrm { c c } ( \\pi { G } ) = \\mathrm { l o } ( \\pi { G } ) - \\mathrm { c l } ( \\pi { G } ) = \\mathrm { l o } ( G ) - 1 - ( \\mathrm { c l } ( G ) - 1 ) = \\mathrm { l o } ( G ) - \\mathrm { c l } ( G ) = \\mathrm { c c } ( G ) . \\end{aligned} \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} W _ \\star ( b , T ) \\triangleq 1 - \\frac 4 \\pi \\sum _ { k = 0 } ^ { + \\infty } \\frac { ( - 1 ) ^ k } { 2 k + 1 } \\exp \\left ( - \\frac { ( 2 k + 1 ) ^ 2 \\pi ^ 2 T } { 8 b ^ 2 } \\right ) , \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{gather*} S p l ( f , \\sigma ) = S p l ( f , \\sigma \\nu ) , \\ , \\\\ S p l ( f , \\sigma , \\{ k _ j \\} ) = S p l ( f , \\sigma \\nu , \\{ k _ j ' \\} ) , \\ , \\\\ S p l ( f , \\sigma , \\{ k _ j \\} , L , \\{ R _ j \\} ) = S p l ( f , \\sigma \\nu , \\{ k _ j ' \\} , L , \\{ R _ j \\} ) , \\end{gather*}"}
-{"id": "225.png", "formula": "\\begin{align*} [ u , v ] = D _ { u } v - D _ { v } u + ( D u ) ^ { * } v - T ^ { D } ( u , v ) \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} & U _ { 2 k , 2 a } ( x ; q ) \\\\ & = ( 1 + x q ) \\bigg [ \\sum _ { i = 1 } ^ a ( x q ) ^ { 2 i } \\left ( \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ) \\\\ & + \\sum _ { i = 1 } ^ { a } ( x q ) ^ { 2 i - 2 } \\left ( \\sum _ { h = 1 } ^ { k - i + 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ) \\bigg ] . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} e ^ { x } = 1 + x + \\sum \\frac { x ^ { k } } { k ! } \\leq 1 + x + \\sum _ { k \\geq 2 } x ^ { k } \\leq 1 + 2 x \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\frac { 1 } { F ( z ) } = \\sum _ n \\frac { 1 } { F ' ( t _ n ) ( z - t _ n ) } , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\varphi ( \\sigma ( M ) \\cdot A d ( M ) ^ t ) = \\varphi ( \\sigma ( M ) ) \\circ A \\circ \\varphi ( M ) \\circ A \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\nu _ 1 ( a _ 0 + a _ 1 x + \\ldots + a _ r x ^ r ) = \\min \\{ \\nu _ 0 ( a _ i ) + i \\gamma _ 0 \\} . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\Big ( ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 \\Big ) ^ k = \\sum _ { n = k } ^ \\infty S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ n } { n ! } , \\ , \\ , ( k \\geq 0 ) . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\Psi ( Q ) : = \\{ f \\in K [ x ] \\mid f \\mbox { i s m o n i c } , \\nu _ Q ( f ) < \\nu ( f ) \\mbox { a n d } \\alpha ( Q ) = \\deg ( f ) \\} . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} ( L + \\pi ( X _ t ) ) ^ { n + 1 } \\Omega = L ^ n \\pi ( X _ t ) \\Omega + Q _ t \\Omega \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\sum _ { \\tau ' \\in \\Sigma ' } e ^ { \\frac { \\tau ' } { M } x } & = \\sum _ { A _ 1 = 0 } ^ { N - 1 } e ^ { \\frac { A _ 1 } { N } x } + \\sum _ { A _ 2 = 0 } ^ { M - N - 1 } e ^ { \\frac { A _ 2 } { M - N } x } \\\\ & = \\frac { e ^ x - 1 } { e ^ { \\frac { x } { N } } - 1 } + \\frac { e ^ x - 1 } { e ^ { \\frac { x } { M - N } } - 1 } . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} \\frac { \\pi } { 3 } = \\theta _ { 1 4 } = \\theta _ { 1 2 } + \\theta _ { 1 3 } + ( \\theta _ { 3 4 } - \\theta _ { 1 2 } ) . \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( x ) ) = h _ { H } ( \\pi ( g ^ { n } ( y ) ) ) = h _ { \\pi ^ { * } H } ( g ^ { n } ( y ) ) . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} \\mathrm { m o d } _ q ( d \\varphi ) = - \\lim _ { \\tau \\to \\infty } \\sum _ { [ \\{ \\alpha , \\beta ; \\epsilon \\} ] \\in \\vec { \\triangle } ^ { \\eta ^ - _ i } _ { \\xi ^ - _ i } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho _ \\tau ) + \\ell _ { \\beta } ( \\rho _ \\tau ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ n _ S ( x ) : x \\in S \\} = Y = \\textrm { s p a n } \\{ n _ S ( x ) - n _ S ( y ) : x , y \\in S \\} = \\textrm { s p a n } \\{ n _ S ( x ) : x \\in S \\cap Y \\} . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\P ( \\tilde { \\mathcal { E } } ^ c ) = \\P \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ^ 2 > \\frac { 1 } { r } \\Big ) \\leq e ^ { - \\frac { 3 n } { 1 6 r ^ 2 } } . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\lim _ { C \\to + \\infty } \\int _ X ( u ^ C - v ^ C ) \\theta _ { u ^ C } ^ k \\wedge \\theta _ { v ^ C } ^ { n - k } = \\int _ X ( u - v ) \\theta _ { u } ^ k \\wedge \\theta _ { v } ^ { n - k } . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} 0 < a < \\frac { \\pi } { \\ln 1 0 } = 1 . 3 6 4 \\cdots . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} | \\Lambda _ U ( g _ 1 , \\dots , g _ { n + 1 } ) | \\lesssim \\prod _ { j = 1 } ^ { n + 1 } \\| g _ j \\| _ { L ^ { p _ j } ( X _ j ) } \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\nu _ 1 \\left ( a _ 0 + a _ 1 x + \\ldots + a _ n x ^ n \\right ) : = \\min _ { 0 \\leq i \\leq n } \\left \\{ \\nu _ 0 ( a _ i ) - \\frac { i } { p } \\right \\} . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} [ D ^ s , f ] g & = D ^ s ( f g ) - f D ^ s g \\\\ & = C ( n , s ) \\ , \\mathrm { P . V . } \\int _ { \\mathbb R ^ n } \\frac { f ( x + y ) - f ( x ) } { | y | ^ \\alpha } \\cdot \\frac { g ( x + y ) } { | y | ^ { n + s - \\alpha } } \\ , d y , \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d U = \\Delta U d t + [ K ( U ) \\cdot \\nabla ] U d t - ( U \\cdot \\nabla ) K ( U ) d t + \\sum _ { i = 1 } ^ N ( B _ i ( t ) + \\theta _ i I ) U d \\beta _ i ( 0 , \\infty ) \\times \\mathbb { R } ^ 3 , \\\\ U ( 0 , \\xi ) = U _ 0 ( \\xi ) = ( c u r l \\ x ) ( \\xi ) , \\ \\xi \\in \\mathbb { R } ^ 3 . \\end{array} \\right . \\ \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} ( \\omega _ { P } ^ { p } ) _ { A } ( a , b ) & = r ( r - 1 ) { \\textstyle \\int \\nolimits _ { M } } p ( a , b , F _ { A } , \\overset { ( r - 2 ) } { \\ldots } , F _ { A } ) , \\\\ ( \\mu _ { P } ^ { p } ) _ { A } ( X ) & = - r { \\textstyle \\int \\nolimits _ { M } } p ( v _ { A } ( X ) , F _ { A } , \\overset { ( r - 1 ) } { \\ldots } , F _ { A } ) \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} k \\left \\{ 0 , ( 1 - r ) \\right \\} = \\left \\{ n ( 1 - r ) \\colon n = 0 , 1 , \\ldots , k \\right \\} \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} [ g _ 1 , g _ 2 ] = a _ 1 a _ 1 ^ { p ^ s n _ 1 } \\ , a _ 2 a _ 2 ^ { p ^ s n _ 2 } \\ , a _ 1 ^ { - 1 } a _ 1 ^ { - p ^ s n _ 2 } \\ , a _ 2 ^ { - 1 } a _ 2 ^ { - p ^ s n _ 3 } , \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} t = a \\int _ { t _ { 0 } } ^ { t ^ { * } } \\exp \\left ( \\int _ { s _ { 0 } } ^ { s } \\widetilde { h } ^ { * } ( t ^ { * } ) d t ^ { * } \\right ) d s + b , \\ ; \\ ; \\ ; \\ ; a , b \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} X ^ n = 1 \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} b ( ( \\hat { u } , \\hat { z } ^ 1 , \\hat { z } ^ 2 ) , ( u , z ^ 1 , z ^ 2 ) ) = < \\mathbb { G } , ( u , z ^ 1 , z ^ 2 ) > , \\ \\ \\forall ( u , z ^ 1 , z ^ 2 ) \\in \\mathcal { X } . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} E [ e ^ { ( k X - 1 ) t } ] = \\frac { 1 } { 1 - k t } e ^ { - t } = \\sum _ { n = 0 } ^ \\infty d _ n ( k ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\delta _ { k } ^ { ( \\ell ) } = \\begin{cases} 2 ^ { 1 - 2 s \\ell } \\gamma \\beta & k = 1 \\\\ 2 ^ { 2 \\ell - 4 s ^ 2 } \\gamma \\beta ^ 2 & k = 2 \\\\ 2 ^ { 1 + 2 \\ell - 4 s ^ 2 } \\gamma \\beta ^ 2 & k = 3 \\\\ 2 ^ { - 2 s \\ell } \\gamma \\beta & k = 4 \\end{cases} . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\eta : = \\sum _ { i = 0 } ^ \\infty y ^ { - \\frac { 1 } { p ^ i } } \\in \\overline { K } , \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} p _ c = \\frac { n + 2 } { n - 2 } n \\geq 3 p _ c = \\infty n = 2 . \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} 1 / ( n p _ n ) \\to 0 \\quad p _ n = o ( n ^ { - 1 / r } ) , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} Z = \\{ x _ { k n + m } \\colon m \\in \\omega \\} \\cup \\{ y _ { k n + m } \\colon m \\in \\omega \\} \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} Q _ { 3 } \\leq C _ { H , l _ { 0 } } \\sum _ { k = 1 } ^ { m } | I _ { k } | ^ { 2 ( 1 - H ) } . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} | P _ n - P _ { n , q } | = O \\Big ( \\frac { 1 } { q } \\Big ) ; \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} & \\frac { u e ^ { - 2 \\pi i k n \\theta } } { 2 k \\pi } \\exp \\left \\{ - \\frac { n \\theta ( 1 - \\theta ) } { 2 } ( u - 2 k \\pi ) ^ 2 \\right \\} \\cdot \\left [ 1 + \\frac { 1 - 2 \\theta } { 6 } n \\theta ( 1 - \\theta ) ( i u - 2 \\pi k i ) ^ 3 \\right ] \\\\ & = : \\hat { \\delta } _ { n , k } ( u ; \\theta ) , \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} S _ i ^ \\alpha = I _ i ^ { k ' _ \\alpha } , i = 0 , 1 . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} C _ { j } ^ { ( 1 + \\alpha ) } ( \\cos \\theta ) = \\sum _ { l = 0 } ^ j \\frac { ( 1 + \\alpha ) _ l ( 1 + \\alpha ) _ { j - l } } { l ! ( j - l ) ! } \\cos ( ( j - 2 l ) \\theta ) \\ , \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\begin{cases} \\dd y _ s ^ { t , A _ t ; U , V } = - l ( X _ s ^ { t , A _ t ; U , V } , y _ s ^ { t , A _ t ; U , V } , q _ s ^ { t , A _ t ; U , V } , U _ s , V _ s ) \\dd s \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + q _ s ^ { t , A _ t ; U , V } \\dd B _ s , ~ s \\in [ t , T ) \\\\ y _ T ^ { t , A _ t ; U , V } = m ( X _ T ^ { t , A _ t ; U , V } ) , \\end{cases} \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} \\sigma ( t , \\gamma ) = [ z _ { j , i } , \\theta ^ l ] , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { S _ p ( G ) } = \\left \\Vert f \\right \\Vert _ 1 + \\left ( \\sum _ { [ \\pi ] \\in \\widehat { G } } d _ \\pi \\bigl \\Vert \\widehat { f } ( \\pi ) \\bigr \\Vert _ { S ^ p ( H _ \\pi ) } ^ p \\right ) ^ { 1 / p } \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} = \\frac { 1 } { | W ^ { \\mathrm { a b s } } _ { M _ S } | } \\sum \\limits _ { w \\in W ^ { \\mathrm { a b s } } _ { M _ S } } \\sum \\limits _ { \\gamma \\in \\Gamma } \\gamma ( w ( \\mu ) ) = \\frac { 1 } { | W ^ { \\mathrm { a b s } } _ { M _ S } | } \\sum \\limits _ { \\gamma \\in \\Gamma } \\sum \\limits _ { w \\in W ^ { \\mathrm { a b s } } _ { M _ S } } \\gamma ( w ( \\mu ) ) . \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( A ^ { \\varepsilon } \\right ) = \\left \\{ u \\in \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) : \\ \\left [ f ( x , u ) - \\varepsilon u _ { x } \\right ] _ { x } \\in \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) \\right \\} . \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} w _ m ( B \\cap M _ S ) = B ' \\cap M _ S . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\zeta ( z + 1 , q ) = \\zeta ( z , q ) , \\zeta ( z + \\tau , q ) = \\zeta ( z , q ) - 2 \\pi i . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + F \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n , \\end{cases} \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} t ^ { \\alpha - 1 } = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ \\infty x ^ { - \\alpha } e ^ { - t x } d x , \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} F ( t ) = F _ { 0 } ( t ) ^ { e _ { 0 } } F _ { 1 } ( t ) ^ { e _ { 1 } } \\cdots F _ { r } ( t ) ^ { e _ { r } } \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( \\rho ) ) = \\sum \\limits _ { ( M _ S , \\mu _ S ) \\in \\mathcal { R } _ { G , b , \\mu } } ( - 1 ) ^ { L _ { M _ S , M _ b } } [ M _ S , \\mu _ S ] ( \\rho ) , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\eta _ { m u } ^ \\zeta ( \\rho , \\rho u ) = \\int _ { - 1 } ^ 1 \\zeta '' ( u + \\rho ^ \\vartheta s ) [ 1 - s ^ 2 ] _ + ^ \\Lambda d s , \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} f \\left ( \\sum _ { i = n } ^ { m - 1 } D ( x _ i , x _ { i + 1 } ) \\right ) \\leq f \\left ( \\frac { k ^ n } { 1 - k } D ( x _ 0 , x _ 1 ) \\right ) < f ( \\varepsilon ) - \\alpha , m > n \\geq N . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) = \\sum _ { i = 0 } ^ N H ^ i \\sum _ { \\substack { 0 \\leq a , b \\leq N \\\\ a + b = i } } \\frac { 1 } { a ! } \\left ( \\frac { \\log ( Q ) } { z } \\right ) ^ a g _ b ( z , Q ) \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} L = X ^ 2 + Y ^ 2 . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} F ( p ^ { \\pm } ) = ( 0 , 0 ) , \\ \\ D F ( p ^ { \\pm } ) \\varphi ^ { \\pm } = - \\lambda _ { \\pm } \\varphi ^ { \\pm } , \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} L ^ 1 \\Gamma \\twoheadrightarrow L ^ 1 G = \\C G c _ 0 ( \\Gamma ) \\longrightarrow c _ 0 ( G ) = \\Q _ p G \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } \\mathcal I ( t ) = & ~ \\frac { \\sigma } { 2 L } \\int \\varphi ' ( u ^ 2 + u _ x ^ 2 ) + \\int \\varphi ( u u _ t + u _ x u _ { t x } ) \\\\ = & ~ \\frac { \\sigma } { 2 L } \\int \\varphi ' ( u ^ 2 + u _ x ^ 2 ) + \\int \\varphi u ( u _ t - u _ { t x x } ) - \\frac { 1 } { L } \\int \\varphi ' u u _ { t x } . \\end{aligned} \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} G _ \\tau ( \\phi _ i \\star _ \\tau \\phi _ j , \\phi _ k ) = \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } ( \\tau , Q ) \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b ^ 3 _ \\nu ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( - q / z ; q ^ 2 ) _ n ( z q ) ^ n . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\epsilon \\eta = ( 1 + k ) \\epsilon = \\frac { t - s } { 2 } + \\left [ \\frac { ( t - s ) ^ 2 } { 4 } + \\frac { ( k + 1 ) ^ 2 } { k } \\right ] ^ { 1 / 2 } \\ , , \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} R _ 1 = \\log _ 2 \\left ( 1 + \\frac { P _ 1 | h _ 1 | ^ 2 } { 1 + P _ 2 | h _ 1 | ^ 2 } \\right ) , R _ 2 = \\log _ 2 ( 1 + { P _ 2 | h _ 2 | ^ 2 } ) , \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\tau ( A + \\lambda B ) = \\tau ( A ) + \\lambda \\tau ( B ) , \\forall A , B \\in \\mathcal M _ + , \\lambda > 0 \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} I ^ i _ m ( \\psi ) : = s ^ { m - 4 } \\lambda ^ { m - 3 } \\iint _ Q e ^ { - 2 s \\sigma _ i } ( \\xi _ i ) ^ { m - 4 } ( | \\psi _ t | ^ 2 + | \\psi _ { x x } | ^ 2 ) d x d t + L ^ i _ m ( \\psi ) , \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} d ( x _ { n + 1 } , x _ n ) & = d ( x _ { m + 1 } , x _ m ) \\\\ & < d ( x _ m , x _ { m - 1 } ) \\\\ & \\hdots \\\\ & \\hdots \\\\ & < d ( x _ { n + 1 } , x _ n ) \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} & \\binom { ( s - 4 ) / 2 } { ( k - 2 ) / 2 } = \\frac { s - k } { s - 2 } \\binom { ( s - 2 ) / 2 } { ( k - 2 ) / 2 } , \\ \\binom { ( s - 2 ) / 2 } { k / 2 } = \\frac { s - k } { k } \\binom { ( s - 2 ) / 2 } { ( k - 2 ) / 2 } , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} H _ { k , i } ( a ; x ; q ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { x ^ { k n } q ^ { k n ^ 2 + n - i n } a ^ n ( 1 - x ^ i q ^ { 2 n i } ) ( a x q ^ { n + 1 } ) _ { \\infty } ( 1 / a ) _ n } { ( q ) _ n ( x q ^ n ) _ \\infty } . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} g _ 0 ( t ) = V _ 0 + \\int _ 0 ^ { t } K ( s ) \\lambda \\theta d s , t \\geq 0 , \\mbox { f o r s o m e } V _ 0 , \\theta \\geq 0 , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} g _ n = f _ n + h _ n ^ \\beta K \\Big ( \\frac { \\cdot - x _ 0 } { h _ n } \\Big ) , \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} C _ \\delta : = 2 \\hat h ^ 2 - 2 \\hat h + 1 , \\hat h : = 2 ^ { \\hat \\nu _ \\delta - 2 } + 2 . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\left \\Vert f ^ { * n } - g ^ { * n } \\right \\Vert _ { \\mathcal { P } _ n } \\le \\frac { n ^ n } { n ! } \\sum _ { j = 1 } ^ { n } \\binom { n } { j } \\left \\Vert g \\right \\Vert _ { A } ^ { n - j } \\left \\Vert f - g \\right \\Vert _ { A } ^ { j } . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\Gamma _ { \\Phi _ { 1 } ( x ; h ) } = C _ { H } \\sum _ { \\alpha = 1 } ^ { d } \\int _ { [ 0 , 1 ] ^ { 2 } } J _ { 1 } J _ { s } ^ { - 1 } V _ { \\alpha } ( \\Phi _ { s } ) V _ { \\alpha } ^ { * } ( \\Phi _ { t } ) ( J _ { t } ^ { - 1 } ) ^ { * } J _ { 1 } ^ { * } | t - s | ^ { 2 H - 2 } d s d t , \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} M ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) & = \\sum _ { k , l } m _ { k , l } v _ { l , k } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) \\\\ & = \\sum _ { k , l } m _ { k , l } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) E _ { k , l } \\\\ & = ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) ( m _ { i , j } ) , \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} \\bold { G } = \\left [ \\begin{array} { c | c | c | c } \\bold { F } _ { 1 , 1 } & \\bold { F } _ { 1 , 2 } & \\dots & \\bold { F } _ { 1 , p _ 0 } \\\\ \\hline \\bold { F } _ { 2 , 1 } & \\bold { F } _ { 2 , 2 } & \\dots & \\bold { F } _ { 2 , p _ 0 } \\\\ \\hline \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\hline \\bold { F } _ { p _ 0 , 1 } & \\bold { F } _ { p _ 0 , 2 } & \\dots & \\bold { F } _ { p _ 0 , p _ 0 } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\mathring { C } _ I = \\bigoplus _ { k \\geq 0 } \\mathring { C } _ I ^ { - k } , \\mathring { C } _ I ^ { - k } = \\bigoplus _ { [ R ] \\in Q ^ * ( I , k + 1 ) } \\mathring { C } _ { I , [ R ] } . \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} \\begin{cases} - W '' = \\left ( 1 - W \\right ) W & \\textup { i n } \\left ( - \\rho , \\rho \\right ) \\\\ W \\left ( \\pm \\rho \\right ) = \\nu . \\end{cases} \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\mathrm { i } \\partial _ t u ( t , x ) + \\Delta u ( t , x ) = 0 \\mathbb R \\times \\Omega \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\Delta = \\sum _ { i = 1 } ^ d ( \\nabla _ { e _ i } \\nabla _ { e _ i } - \\nabla _ { \\nabla _ { e _ i } e _ i } ) . \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to \\infty } \\frac { N ( \\xi ) } { \\xi ^ { 1 / 2 } } = \\frac { 1 } { \\pi } \\int _ I \\sqrt { \\frac { w ( x ) } { p ( x ) } } d x . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} ( x q ) ^ { 2 a - 2 } \\sum _ { h = 1 } ^ { k - a + 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a - 2 } \\sum _ { h = 0 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\alpha _ i = \\sum _ { j \\neq i } m _ j \\alpha _ j . \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\left ( \\mathcal { D } _ 0 ^ \\theta f \\right ) ( x ) = \\left ( D _ 0 ^ { k ' _ { \\theta - 1 } } f \\right ) ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } H ^ { 2 } = & | \\nabla ^ { \\perp } { \\vec H } | ^ { 2 } + H ^ { 2 } - \\sum _ { i , j , p , q } H ^ { p ^ { \\ast } } h _ { i j } ^ { p ^ { \\ast } } \\cdot H ^ { q ^ { \\ast } } h _ { i j } ^ { q ^ { \\ast } } . \\endaligned \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\begin{cases} u '' ( t ) + A u ( t ) + u ' ( t ) = 0 , & t > 0 , \\\\ ( u , u ' ) ( 0 ) = ( u _ 0 , u _ 1 ) , \\end{cases} \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} r = \\frac { 1 } { 2 } \\sum h _ i \\otimes h ^ i + \\sum _ { \\alpha \\in \\Delta _ + } E _ { \\alpha } \\otimes E _ { - \\alpha } \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\# H _ m = { n \\choose 2 m } 2 ^ { 2 m } . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} H _ { \\nu } ( \\Delta , \\Z \\Gamma / f _ i \\Z \\Gamma ) = H _ { \\nu } ( \\Z G \\xrightarrow { f _ i } \\Z G ) \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} v _ { r ' } ( X ) = v _ { r ' } ( ( P - \\widetilde { P } ) R _ i T ^ i + ( P - \\widetilde { P } ) ( Q - R _ i T ^ i ) ) = v _ { r ' } ( ( P - \\widetilde { P } ) R _ i T ^ i ) = v _ { r ' } ( P - \\widetilde { P } ) + r ' i + v ( R _ i ) . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} a _ l & : = \\max \\{ h : h \\in E _ { p , 1 } ^ l , 1 \\le p \\le m \\} , a ' _ l : = \\min \\{ h : h \\in E _ { p , 1 } ^ l , 1 \\le p \\le m \\} , \\\\ e _ l & : = \\max \\{ \\delta _ p ^ l - a _ l \\cdot t _ { p , 1 } ^ l : 1 \\le p \\le m \\mbox { f o r w h i c h } a _ l \\in E _ { p , 1 } ^ l \\} \\mbox { a n d } \\\\ e ' _ l & : = \\min \\{ \\delta _ p ^ l - a _ l \\cdot t _ { p , 1 } ^ l : 1 \\le p \\le m \\mbox { f o r w h i c h } a _ l \\in E _ { p , 1 } ^ l \\} . \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} J _ 0 ( r ) = \\sqrt { \\frac { 2 } { \\pi r } } \\cos \\Big ( r - \\frac { \\pi } { 4 } \\Big ) + O ( r ^ { - 3 / 2 } ) , r \\to \\infty , \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\langle P , \\delta ^ k ( u ) \\rangle _ { L ^ 2 ( \\Omega ) } = \\langle u ; D ^ k P \\rangle _ { L ^ 2 ( \\Omega ; L ^ 2 ( \\mathbb { R } ^ k ) ) } \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\sup _ n \\ | | n ^ { - 1 / 2 } \\sum _ { i = 1 } ^ n X _ i | | _ { s , \\Omega } \\le Z [ \\alpha ] ( s , v ) \\cdot | | X | | _ { v , \\Omega } , \\ v > s . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\overline { r } ( n ; \\theta ) + 1 } \\mathfrak { J } _ k ^ { ( 1 ) } ( n ; \\theta ) \\leq \\frac { \\lambda _ { 1 2 } } { n \\theta ( 1 - \\theta ) } + \\frac { \\lambda _ { 1 3 } } { [ n \\theta ( 1 - \\theta ) ] ^ { 1 + \\varepsilon ( \\gamma ) } } \\log \\left ( \\frac { n } { \\theta ( 1 - \\theta ) } \\right ) \\ . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\| e _ N \\| _ { L ^ 2 ( \\mathbb R ) } = \\| \\breve e _ N \\| _ { L ^ 2 _ { \\omega _ \\lambda } ( I ) } \\le c N ^ { - m } \\| \\partial _ t ^ m \\breve U \\| _ { L ^ 2 _ { \\omega _ { \\lambda + m } } ( I ) } = c N ^ { - m } | u | _ { { \\mathbb B } ^ m _ { \\lambda } ( \\mathbb R ) } . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} ( x - \\gamma _ i ) N _ i = x ^ n - 1 \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\delta ^ { k + 1 } ( \\tau _ V ^ t ( V ) ) & = \\tau _ V ^ t ( \\delta _ V ^ { k + 1 } ( V ) ) - \\i \\sum _ { l = 0 } ^ k \\delta _ V ^ l [ V , \\delta ^ { k - l } ( \\tau _ V ^ t ( V ) ) ] \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} A ^ t W + W A + W ' + \\lambda W = 0 . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\Phi _ { x y } ^ { \\epsilon , \\delta } ( w ) : = h _ p ( y - w ) [ \\partial _ p Q ^ { \\lambda } _ { \\epsilon } ( x - w ) - \\partial _ p Q ^ { \\lambda } _ { \\delta } ( x - w ) ] . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} | x _ { n + 1 } | ^ { \\frac 1 { 3 ^ { n + 1 } } } & \\le e ^ { - ( n + 1 ) } | x _ 0 | + \\frac { \\beta ( e - 1 ) } { e } [ e ^ { - n } + e ^ { - n + 1 } + \\dots + e ^ { - 1 } + 1 ] \\\\ & = e ^ { - ( n + 1 ) } | x _ 0 | + \\frac { \\beta ( e - 1 ) } { e } \\frac { 1 - e ^ { - n - 1 } } { 1 - e ^ { - 1 } } \\le e ^ { - ( n + 1 ) } | x _ 0 | + \\beta . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} | T ( \\chi _ V ) ( z ) | & = | \\int \\chi _ V d T ^ * ( \\delta _ { z } ) | = | T ^ * ( \\delta _ { z } ) ( V ) | \\\\ & \\geq | T ^ * ( \\delta _ { z } ) | ( \\{ w _ 0 \\} ) - | T ^ * ( \\delta _ { z } ) | ( V \\setminus \\{ w _ 0 \\} ) \\geq \\frac { \\epsilon } { 2 } \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\nabla _ { z \\partial _ { t _ 1 } } & = z \\partial _ { t _ 1 } + H \\circ _ { \\tau _ 2 } \\\\ \\nabla _ { z \\partial _ z } & = z \\partial _ z - \\frac { 1 } { z } ( N + 1 ) H \\circ _ { \\tau _ 2 } + \\left ( 1 - \\frac { N + 1 } { 2 } \\right ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\tilde { \\psi } _ n ^ { N , h } ( y ) : = \\frac { 1 } { \\sqrt { \\lambda _ n ^ { N , h } } } \\int _ { D } \\log \\kappa ( y , x ) \\phi _ n ^ { N , h } ( x ) \\mathrm { d } x . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} T ^ 2 - T & = ( R _ L + S ) \\circ ( R _ L + S ) - ( R _ L + S ) \\\\ & = R _ L ^ 2 - R _ L + R _ L \\circ S + S \\circ R _ L + S ^ 2 - S \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} \\overline { W } _ { \\rho , \\nu } ( x ) = W _ { \\rho , \\nu } \\left ( x - \\rho \\right ) \\textup { f o r } x \\in [ 0 , 2 \\rho ] . \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\nabla \\ ! f ( \\overline { x } ) + \\nabla g ( \\overline { x } ) \\widehat { \\lambda } = 0 . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} b ( \\alpha ) = \\begin{cases} \\frac { \\alpha ( \\cos ( \\frac { \\theta } { 3 } ) + \\cos ( \\theta ) ) } { ( 1 + \\alpha ) ( \\cos ( \\frac { \\theta } { 3 } ) + \\cos ( \\theta ) ) - 2 x ( \\alpha ) \\cos ( \\frac { \\theta } { 3 } ) } , & \\theta \\in [ 0 , \\pi ] , \\ \\theta \\ne \\frac { 3 \\pi } { 4 } , \\\\ \\frac { \\alpha ( 1 + \\alpha + x ( \\alpha ) ) } { ( 1 + \\alpha ) ( 1 + \\alpha + x ( \\alpha ) ) - 2 \\cos ^ 2 ( \\theta ) } , & \\theta = \\frac { 3 \\pi } { 4 } . \\end{cases} \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { p _ i } = 1 \\qquad \\mbox { a n d } \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { \\delta _ i } = \\frac 1 { r } - 1 = \\frac { 1 - r } { r } . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{gather*} \\boldsymbol { L } _ { J K } u ( x ) = \\left ( A _ { J K } ( u ; K + p ^ { N } \\mathbb { Z } _ { p } ) \\right ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) \\\\ - A _ { J K } u ( x ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) . \\end{gather*}"}
-{"id": "3775.png", "formula": "\\begin{align*} \\sigma ^ * : = \\inf \\{ \\sigma \\mid w ^ { \\sigma ' } \\preceq u \\ \\ { \\rm h o l d s \\ f o r \\ a l l } \\ \\ \\sigma ' \\geq \\sigma \\} . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} F _ { \\kappa , 1 } ( t , \\varphi ) = \\frac { \\| \\psi \\| _ { L ^ \\infty } } { \\max \\big \\{ \\| \\varphi \\| _ { L ^ \\infty } , \\ , 1 \\big \\} } - \\kappa = 0 . \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} & \\dot s ( t ) + \\frac { s ( t ) ^ 2 } { n - 1 } - \\frac { 2 s ( t ) } { n - 2 } \\frac { \\left < \\nabla g ( \\Psi _ t ( v ) ) , \\dot \\Psi _ t ( v ) \\right > } { g ( \\Psi _ t ( v ) ) } - \\epsilon g ( \\Psi _ t ( v ) ) ^ { \\frac { 2 n - 2 } { n - 2 } } \\leq 0 . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} { P r } _ D ( f , \\sigma ) : & = \\lim _ { X \\to \\infty } \\displaystyle \\frac { \\# \\{ p \\in { S p l } _ X ( f , \\sigma ) \\mid ( r _ 1 / p , \\dots , r _ n / p ) \\in D \\} } { \\# { S p l } _ X ( f , \\sigma ) } \\\\ & = \\displaystyle \\frac { { v o l } ( { D } \\cap { \\mathfrak { D } } ( f , \\sigma ) ) } { { v o l } ( { \\mathfrak { D } } ( f , \\sigma ) ) } . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{gather*} \\mathbf { R } _ { n } ^ { - 1 } \\mathbf { x } = - \\frac { \\mathbf { x } + \\mathbf { y } } { 2 \\theta } , \\\\ \\mathbf { R } _ { n } ^ { - 1 } \\mathbf { y } = - \\frac { \\mathbf { x } - \\mathbf { y } } { 2 \\theta } . \\end{gather*}"}
-{"id": "6945.png", "formula": "\\begin{align*} & \\mathbb { E } [ \\ln ( 1 + \\mathit { S I N R } ) ] = \\\\ & \\ \\mathbb { E } _ N { \\left [ \\sum _ { k = 1 } ^ { n } \\binom { n } { k } ( - 1 ) ^ { k + 1 } \\int _ 0 ^ \\infty \\frac { \\mathrm { e } ^ { - ( 1 - f _ { N _ i } ( 0 ) ) \\sqrt { k x } \\arctan ( \\sqrt { k x } ) } \\ , \\mathrm { d } x } { x + 1 } \\right ] } . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} & = \\frac { 2 \\nu + n + 1 } { 2 ( \\nu + n + 1 ) } t ^ { \\nu } \\mathbf { L } _ { \\nu + n } ( t ) + \\frac { n + 1 } { 2 ( \\nu + n + 1 ) } t ^ { \\nu } \\mathbf { L } _ { \\nu + n + 2 } ( t ) \\\\ & \\quad + \\frac { n + 1 } { \\nu + n + 1 } \\frac { t ^ { 2 \\nu + n + 1 } } { \\sqrt { \\pi } 2 ^ { \\nu + n + 2 } \\Gamma ( \\nu + n + \\frac { 5 } { 2 } ) } . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} & \\frac { 1 } { r } \\int _ { r - t } ^ { r + t } \\widetilde { \\Omega } _ s [ u _ 0 ] ( x ) K _ 0 ( t , r ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s \\\\ & = \\int _ { 0 } ^ { t } \\frac { 1 } { r } \\Big [ \\Omega _ { s + r } [ u _ 0 ] ( x ) - \\Omega _ { s - r } [ u _ 0 ] ( x ) \\Big ] K _ 0 ( t , r ; s + r ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s \\xrightarrow [ r \\to 0 ] { } 2 \\int _ { 0 } ^ { t } \\frac { \\partial } { \\partial s } \\ , \\Omega _ { s } [ u _ 0 ] ( x ) K _ 0 ( t , 0 ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s . \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} R ( \\chi ) = \\sum _ { n = 1 } ^ \\infty q _ n \\chi ( n ) . \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} c _ k = A ^ { n - 1 } c _ { k - 1 } ^ a ( c _ { k - 1 } - 1 ) ^ { n - a } + 1 . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} i _ { s , 1 } = & \\ \\min \\{ j \\in \\{ 1 \\ldots , m \\} \\ | \\ s _ j > N \\} , \\\\ i _ { s , 2 } = & \\ \\min \\{ j > i _ { s , 1 } \\ | \\ s _ j \\geq s _ { i _ { s , 1 } } \\} , \\\\ & \\vdots \\\\ i _ { s , l ( s ) } = & \\ \\min \\{ j > i _ { s , l ( s ) - 1 } \\ | \\ s _ j \\geq s _ { i _ { s , l ( s ) - 1 } } \\} . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} p _ e ( \\rho ) = a \\rho ^ \\gamma , \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} D _ { 0 ; 1 ^ n , k } ( q ) = S _ { n , k } ( q ) . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} S \\odot S ' = \\sum _ { w } ( \\mu ( \\mathrm { c i r c u i t } ( w ) ) \\mu ' ( \\mathrm { c i r c u i t } ( w ) ) w = \\sum _ { w } ( \\mu \\hat \\odot \\mu ' ( \\mathrm { c i r c u i t } ( w ) ) w . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} K ( S { \\frak C } _ F ( H ' ) ) & = \\lim _ l \\ K ( S { \\frak C } _ { F ^ l } ( W ' _ l ) ) \\cong \\lim _ l \\ K ( S { \\frak C } ( W ' _ l ) ) \\\\ & = K ( S { \\frak C } ( H ' ) ) \\cong K ( S ) \\cong { \\mathbb Z } \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} \\begin{cases} \\widehat { H } ( x , P + \\nabla \\widetilde { u } _ 0 ( x ) ) = \\ln \\widetilde { m } _ 0 ( x ) + \\overline { H } , \\\\ - \\div _ x \\big ( \\widetilde { m } _ 0 ( x ) \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ 1 ( x , y ) ( P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) d y \\big ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\overline { k } \\otimes _ k F & \\simeq \\overline { k } \\otimes _ k k [ T ] / k [ T ] P \\\\ & \\simeq \\overline { k } / \\overline { k } [ T ] P \\\\ & = \\overline { k } [ T ] / \\overline { k } [ T ] ( T - \\alpha _ 1 ) ( T - \\alpha _ 2 ) \\cdots ( T - \\alpha _ u ) \\\\ & \\simeq \\oplus _ { i = 1 } ^ u \\ \\overline { k } [ T ] / \\overline { k } [ T ] ( T - \\alpha _ i ) \\\\ & \\simeq \\oplus _ { i = 1 } ^ u \\ \\overline { k } . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\frac { \\N ( R , K _ m ) } { k ^ m } \\geq \\binom { r } { m } \\left ( \\frac 1 r \\right ) ^ m - 3 \\delta . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} c _ { x _ 0 , R } ( t ) : = p ( x , t ) - \\hat p _ { x _ 0 , R } ( x , t ) \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} u _ \\varepsilon = B _ \\varepsilon + v _ \\varepsilon \\ , . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} x _ { 1 } = 2 u _ 1 + \\underbrace { x _ 0 ( 1 - h _ 0 x _ 0 ^ 2 ) - u _ 1 } _ { = 0 } = 2 u _ 1 = 2 e ^ { - 3 } > 0 . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} | x _ { N _ 1 + 2 + k } | \\le | x _ { N _ 1 + 1 } | + \\sum _ { i = 1 } ^ { k } | u _ { N _ 1 + 2 + i } | \\mbox { a n d } x ^ 2 _ { N _ 1 + 2 + k } h _ { N _ 1 + 2 + k } < 1 , \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\phi _ R * v ^ l _ n \\| _ { L ^ \\infty } = \\sup _ { x _ n } \\limsup _ { n \\rightarrow \\infty } | \\phi _ R * v ^ l _ n ( x _ n ) | . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\Theta ( x ) : = \\left \\{ \\begin{array} { l l } 1 & \\ ; x \\geq 0 \\ , \\\\ & \\\\ 0 & \\mbox { f o r } \\ ; x < 0 \\ . \\\\ \\end{array} \\right . \\\\ \\ \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ r \\left ( \\frac { m } { n _ i } + 1 \\right ) & \\leq \\sum _ { i = 0 } ^ r \\binom { r } { i } \\left ( \\frac { m } { \\sqrt [ r ] { n _ 1 n _ 2 \\cdots n _ r } } \\right ) ^ { r - i } \\left ( \\frac { n } { r } \\right ) ^ { \\frac { ( r - 1 ) i } { r } } \\\\ & = \\left ( \\frac { m } { \\sqrt [ r ] { n _ 1 n _ 2 \\cdots n _ r } } + \\left ( \\frac { n } { r } \\right ) ^ { 1 - \\frac { 1 } { r } } \\right ) ^ r . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} F = \\Phi ^ { - 1 } J \\Phi - J \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\biggl | 2 \\sum _ { n = N } ^ \\infty \\frac { 1 } { 6 ^ n } \\frac { d } { d x } V \\bigl ( \\langle 2 ^ n x \\rangle , \\langle 3 ^ n x \\rangle \\bigr ) \\biggr | \\le \\frac { 1 } { 2 ^ { N - 2 } } \\quad \\hbox { a . e . } \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { S } } _ { n } ( z ) & = \\mathcal { S } _ { n } ( z - 1 / 2 ) , \\\\ \\widetilde { \\mathcal { C } } _ { n } ( z ) & = \\mathcal { C } _ { n } ( z - 1 / 2 ) , \\\\ \\widetilde { \\mathcal { S } } _ { n } ^ { \\prime } ( z ) & = \\mathcal { S } _ { n } ^ { \\prime } ( z - 1 / 2 ) , \\\\ \\widetilde { \\mathcal { C } } _ { n } ^ { \\prime } ( z ) & = \\mathcal { C } _ { n } ^ { \\prime } ( z - 1 / 2 ) . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} \\nabla f ( x ^ k ) & = \\nabla f ( \\overline { x } ) + \\nabla ^ 2 f ( \\overline { x } ) ( x ^ k - \\overline { x } ) + o ( t _ k ) , \\\\ g ' ( x ^ k ) & = g ' ( \\overline { x } ) + g '' ( \\overline { x } ) ( x ^ k - \\overline { x } ) + o ( t _ k ) . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} b ( z ) a = e ^ { z T } a ( - z ) b , b { ( n ) } a = - \\sum _ { j \\in \\Z _ { \\geq 0 } } ( - 1 ) ^ { n + j } T ^ { ( j ) } ( a { ( n + j ) } b ) , \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} x ' _ k = x ' _ { k + 1 } y ' _ j = y ' _ { j + 1 } x ' _ k = x ' _ { k - 1 } y ' _ j = y ' _ { j - 1 } \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde b _ 1 + \\ ( 1 - \\frac { a _ 1 } { 2 \\i B } \\ ) \\frac { H _ d } { z _ d ' } = 0 , \\\\ \\widetilde a _ 1 - \\frac { b _ 1 } { 2 \\i B } \\cdot \\frac { H _ d } { z _ d ' } = 0 ; \\end{cases} \\begin{cases} b _ 1 + ( 1 + \\frac { \\widetilde a _ 1 } { 2 \\i B } ) \\frac { H } { z ' } = 0 , \\\\ a _ 1 + \\frac { \\widetilde b _ 1 } { 2 \\i B } \\cdot \\frac { H } { z ' } = 0 , \\end{cases} \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} V _ C = \\bigoplus \\limits _ { ( M _ S , \\mu _ S ) \\in C } [ M _ S , \\mu _ S ] ( \\rho ) , \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} \\| T ( f _ 1 , f _ 2 , \\dots , f _ m ) \\| _ { L ^ s ( w ) } \\lesssim \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { s _ i } \\left ( w _ i \\right ) } \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} f ( \\xi ) = c _ 1 g ( \\xi ) = c _ 2 H ( \\delta _ r ( \\zeta ^ { - 1 } \\xi ) , \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} V _ t = g _ 0 ( t ) + \\int _ 0 ^ t K ( t - s ) \\left ( - \\lambda V _ s d s + \\nu \\sqrt { V _ s ^ + } d W _ s \\right ) , \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { 1 } { \\sqrt { L } } \\int _ { \\gamma } d s _ L = \\int _ a ^ b d s . \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} a d _ { \\kappa + v } = a d _ { \\kappa + v _ 1 } + a d _ { v _ 2 } , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\begin{cases} ( n - 2 ) f h \\varphi '' - r f \\varphi h '' - m h \\varphi f '' - 2 m h \\varphi ' f ' - 2 r f \\varphi ' h ' = 0 \\\\ \\rho = \\lambda _ F = 0 \\end{cases} \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} M ( x ) = \\sum _ { i = 0 } ^ k a _ i x ^ i p ^ { ( i ) } _ { \\{ \\delta \\} } ( \\Theta ) . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\tilde x \\sum _ { k = 1 } ^ r \\frac { \\lambda _ k } { \\tilde \\lambda _ j - \\lambda _ k } = 1 . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } A ^ * W ( A X _ 0 - I ) = 0 , \\\\ B ^ * W ( B X _ 0 - I ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} p _ i = \\frac { r _ i } { r } , 1 \\le i \\le m , \\qquad \\mbox { a n d } \\frac 1 r = \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { r _ i } > 1 , \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} E _ { P } \\left [ T _ { P , \\mathcal { G } } | Y , _ { \\mathcal { G } } \\left ( Y \\right ) \\right ] = \\frac { I _ { a } ( A ) } { \\pi _ { a } \\left ( \\mathbf { O } _ { \\min } ; P \\right ) } Y + g \\left ( A , \\mathbf { O } , \\mathbf { M } ; P \\right ) , \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\frac { d } { d z } H ( U , V ) = ( H _ u ( U , V ) , H _ v ( U , V ) ) \\cdot ( U ' , V ' ) > 0 . \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} p = \\sum _ { i = 1 } ^ r a _ i \\textbf { Q } ^ { \\lambda _ i } \\mbox { w i t h } \\nu \\left ( a _ i \\textbf { Q } ^ { \\lambda _ i } \\right ) \\geq \\nu ( p ) , \\mbox { f o r e v e r y } i , 1 \\leq i \\leq r , \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\| T ( h _ n ) \\| = \\| R _ L ( h _ n ) + S ( h _ n ) \\| \\leq \\| R _ L ( h _ n ) \\| + \\| S ( h _ n ) \\| \\leq 2 \\| R ( f _ n ) \\| + \\frac { 1 } { n } . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\chi _ k ( t ) : = \\sqrt { \\frac { 2 } { a _ 2 - a _ 1 } } \\ , \\sin \\big ( \\sqrt { E _ k } \\ , ( t - a _ 1 ) \\big ) \\ , . \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} n = 0 , h - f = 0 . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\| \\tilde T _ j F \\| \\Big \\| \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i \\tilde T _ j \\epsilon _ i \\Big \\| & \\leq C \\sqrt { \\frac { 1 } { n } \\frac { \\tilde \\lambda _ j } { \\tilde g _ j } \\operatorname { t r } ( \\tilde T _ j \\Sigma _ \\epsilon \\tilde T _ j ) } \\bigvee \\sqrt { \\frac { t } { n } \\frac { \\tilde \\lambda _ j } { \\tilde g _ j } \\| \\tilde T _ j \\Sigma _ \\epsilon \\tilde T _ j \\| _ { \\infty } } . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} P _ { W _ i } ( w ) = { { w - 1 } \\choose { n - 1 } } p ^ n ( 1 - p ) ^ { w - n } , \\ \\ w = n , n + 1 , \\ldots \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} v a r _ { P } \\left [ \\psi _ { P , \\mathbf { a } } \\left ( \\mathbf { B } ; \\mathcal { G } \\right ) \\right ] = v a r _ { P } \\left [ \\psi _ { P , \\mathbf { a } } \\left ( \\mathbf { G } , \\mathbf { B } ; \\mathcal { G } \\right ) \\right ] + \\sum _ { k = 0 } ^ { p } v a r _ { P } \\left [ r _ { k } \\left ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } ; s _ { \\mathbf { a , } k } ^ { \\ast } , P \\right ) \\right ] . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} \\mathcal { B } = \\dot { \\bigcup } _ { n = e } ^ { e + d } \\ , \\mathrm { L y r } _ n { \\mathcal { B } } \\# \\mathcal { B } = \\sum _ { n = e } ^ { e + d } \\ , \\# \\mathrm { L y r } _ n { \\mathcal { B } } . \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\| \\tilde { x } - x \\| \\leq q ^ { - N } \\Longleftrightarrow | \\tilde { r } _ i f _ i - r _ i \\tilde { f } _ i | _ \\infty \\leq q ^ { \\deg f _ i + \\deg \\tilde { f } _ i - N } \\mbox { f o r a l l } i = 1 , . . . , n . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} P ^ { ( \\theta ) } f = \\sum _ { I \\in \\mathcal { D } _ { \\leq n } } \\frac { \\langle f , b _ I ^ { ( \\theta ) } \\rangle } { \\| b _ I ^ { ( \\theta ) } \\| _ 2 ^ 2 } b _ I ^ { ( \\theta ) } , f \\in W _ N . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} \\| [ T , \\textbf { b } ] _ \\alpha ( f _ 1 , f _ 2 , \\dots , f _ m ) \\| _ { L ^ s ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\| b _ j \\| ^ { \\alpha _ i } _ { { \\rm B M O } } \\| f _ i \\| _ { L ^ { s _ i } \\left ( v _ i \\right ) } , \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} | F | _ { K , k } : = \\sup _ { x \\in K \\atop | \\alpha | \\leq k } | F _ \\alpha ( x ) | \\ : . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\| \\gamma \\| _ { \\bar { \\mathcal { H } } } = \\| \\varphi \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } \\leq C _ { H , V } . \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} K ( t ) = \\int _ 0 ^ { \\infty } e ^ { - x t } \\mu ( d x ) , t > 0 . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\tilde P ^ \\nu ( \\theta ) : = \\{ x \\in \\mathbb { R } ^ 3 : \\ \\exists ( \\xi ^ j , \\eta ^ j ) _ { j = 0 } ^ \\nu : \\ \\eqref { e q : s t e p 0 x 1 } - \\eqref { e q : s t e p 0 x 2 } , \\eqref { e q : s t e p x 3 } , \\eqref { e q : o u t e r R e f o r m } \\} . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\Psi _ i = \\prod _ { j \\neq i } \\frac { 1 - \\Lambda _ j P ^ { - 1 } } { 1 - \\Lambda _ j \\Lambda _ i ^ { - 1 } } \\in K _ { T ^ { N + 1 } } \\left ( \\mathbb { P } ^ N \\right ) \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\inf _ { \\stackrel { v \\in H _ 0 ^ 1 ( \\Omega ) \\setminus \\{ 0 \\} } { v \\bot \\phi _ 1 , \\dots v \\bot \\phi _ j } } \\frac { Q _ u ( v ) } { ( v , v ) _ { L ^ 2 ( \\Omega ) } } = \\lambda _ { j + 1 } \\ge 0 . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} g ^ { \\flat } ( x ) = m ^ { \\flat } ( x ) = f ^ { \\flat } ( x ) \\textrm { f o r a l l } x \\in E ^ { \\flat } . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ n : = \\lambda _ n \\lambda ^ { - 1 } , \\tilde { x } _ n : = \\lambda _ n \\lambda ^ { - 1 } x _ 0 + x _ n , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\frac { q _ 1 + p _ 2 q _ 2 } { q _ 1 + q _ 2 } - \\frac { p _ 2 q _ 2 } { q _ 2 + q _ 3 } = \\frac { q _ 1 } { q _ 1 + q _ 2 } \\leq \\frac { q _ 1 + q _ 3 } { q _ 1 + q _ 2 + q _ 3 } = \\P ( ( G \\cap H ^ c ) \\cup ( H \\cap G ^ c ) \\mid G \\cup H ) . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} W _ m ^ * = W _ { - m } \\ , , m \\in \\Z ^ { 2 g } \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p m _ i \\alpha _ i = m \\quad ( m _ i , m \\in \\mathbb Z ) . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\nu ( f ) = 5 , \\ \\nu ( \\partial _ 1 f ) \\geq 3 , \\ \\nu ( \\partial _ 2 f ) = 1 \\mbox { a n d } \\nu ( \\partial _ 3 f ) = 0 , \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} A ( x , \\ell ) : = \\pi e ^ { e - 2 } \\times \\begin{cases} \\ell / \\log \\left ( ( e - 1 ) \\right ) , & \\ell > x , \\\\ x , & \\ell \\leq x . \\end{cases} \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\mathcal { L } w _ 0 = \\alpha w _ 0 + \\nu Q \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} v _ t = 0 , \\quad \\mbox { o n } \\Gamma _ f \\times ( 0 , T ) . \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\tilde x \\sum _ { k = 1 , k \\neq j - 1 } ^ r \\frac { \\lambda _ k } { \\lambda _ { j - 1 } - \\lambda _ k } \\geq L + 1 \\qquad \\tilde x \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 1 } - \\lambda _ j } \\leq \\ell . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} | \\nabla ^ { \\perp } \\vec H | ^ 2 = \\sum _ { i , p } ( H ^ { p ^ { \\ast } } _ { , i } ) ^ 2 , \\ \\ \\ \\ \\Delta ^ \\perp H ^ { p ^ { \\ast } } = \\sum _ i H ^ { p ^ { \\ast } } _ { , i i } . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} 0 \\to H ^ 2 ( M ) \\cong \\Z \\to H ^ 2 ( U ) \\oplus H ^ 2 ( V ) \\cong \\Z \\oplus \\Z \\oplus \\Z / 2 \\Z \\to H ^ 2 ( U \\cap V ) \\cong \\Z \\to H ^ 3 ( M ) = 0 , \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} f _ i ( x + t v ) < f _ i ( x ) i = 1 , \\dots , m \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} \\mathfrak { H } _ j : = \\left ( \\begin{array} { c c c c c } H _ { \\nu } & \\cdots & H _ 0 & & 0 \\\\ & \\ddots & & \\ddots & \\\\ 0 & & H _ { \\nu } & \\cdots & H _ 0 \\\\ \\end{array} \\right ) \\in \\mathbb F ^ { ( j + 1 ) ( n - k ) \\times ( \\nu + j + 1 ) n } , \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} \\alpha ( Q ) : = \\min \\{ \\deg ( f ) \\mid \\nu _ Q ( f ) < \\nu ( f ) \\} , \\mbox { a n d } \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} p _ 0 = \\min \\left \\lbrace p ( x ) : x \\in \\overline { G } \\right \\rbrace \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} u = \\int _ { G } V _ { \\eta } u ( x ) \\pi ( x ) \\eta \\ , d x \\ , \\ , . \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { r e s } ( \\alpha ) } = w _ 1 \\circ . . \\circ w _ n . \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\int _ { B } | \\nabla v | ^ { p - 2 } \\Big ( h _ { i j } ( v ) \\frac { \\partial v ^ i } { \\partial x _ \\alpha } \\frac { \\partial \\Phi ^ j } { \\partial x _ \\beta } + \\frac { 1 } { 2 } \\big ( \\frac { \\partial h _ { i j } } { \\partial x _ k } \\circ v \\big ) \\frac { \\partial v ^ i } { \\partial x _ \\alpha } \\frac { \\partial v ^ j } { \\partial x _ \\beta } \\Phi ^ k \\Big ) \\delta ^ { \\alpha \\beta } d x = 0 \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\tilde J : \\tilde { \\mathcal U } \\to [ 0 , + \\infty ) \\ , , \\tilde J ( u , w ) : = J ( S ( u , w ) , u , w ) \\ , , ( u , w ) \\in \\tilde { \\mathcal U } \\ , . \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\mathbf P _ { o } = 1 - p \\Omega ( \\emptyset , 1 ) - ( 1 - p ) e ^ { - \\frac { 2 ^ { \\frac { r _ t } { 1 - \\beta } } - 1 } { \\gamma _ { D ^ { t } , D ^ { r } } } } , \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\pi _ \\mu ^ \\ast \\omega _ \\mu = i _ \\mu ^ \\ast \\omega . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} a ( u - u _ { h } , u - u _ { h } ) & - \\l b ( u - u _ { h } , u - u _ { h } ) = a ( u _ { h } , u _ { h } ) - \\l b ( u _ { h } , u _ { h } ) \\\\ & = a ( u _ { h } , u _ { h } ) - a _ { h } ( u _ { h } , u _ { h } ) + \\l _ { h } ^ { ( j ) } b _ { h } ( u _ { h } , u _ { h } ) - \\l b ( u _ { h } , u _ { h } ) \\\\ & = a ( u _ { h } , u _ { h } ) - a _ { h } ( u _ { h } , u _ { h } ) + ( \\l _ { h } ^ { ( j ) } - \\l ) b _ { h } ( u _ { h } , u _ { h } ) + \\l [ b _ h ( u _ { h } , u _ { h } ) - b ( u _ { h } , u _ { h } ) ] , \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} A ( a ) : = \\sup _ { k \\ge 0 } \\bigg \\{ \\sup _ { ( \\varphi , W ) \\in \\mathcal { A } _ a ( k ) } \\| \\varphi \\| _ { L ^ \\infty } \\bigg \\} < + \\infty \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} Q _ { \\lambda , q } ( \\Omega ) g _ q ^ { q - 1 } ( \\varsigma ) = \\int _ { \\Omega _ q } \\frac { g _ q ( \\eta ) } { | \\eta ^ { - 1 } \\varsigma | ^ { Q - \\alpha } } d \\eta + \\lambda \\int _ \\Omega \\frac { f _ q ( \\xi _ q ) ^ { 1 - q } \\cdot f _ q ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha - 1 } } d \\eta \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} x - \\sigma _ { \\mathrm { r e s } ( \\alpha ) } ( x ) & = ( x - \\sigma _ { \\beta } ( x ) ) + \\sigma _ { \\beta } ( x - \\sigma _ { \\alpha } ( x ) ) + ( \\sigma _ { \\beta } \\circ \\sigma _ { \\alpha } ) ( x - \\sigma _ { \\beta } ( x ) ) \\\\ & = b \\beta + a ( \\alpha + \\beta ) + b \\alpha \\\\ & = ( a + b ) ( \\alpha + \\beta ) , \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} M _ { \\tau } ^ f = e ^ { \\int _ 0 ^ { \\tau } V ( Y _ v ) d v } Q ^ V f ( Y _ { \\tau } , 0 ) = e ^ { \\int _ 0 ^ { \\tau } V ( Y _ u ) d u } f ( Y _ { \\tau } ) . \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\to 0 } S ^ { \\sigma } _ t u ~ = ~ S _ t u \\qquad \\qquad \\forall u \\in \\overline D , \\ ; \\forall t \\ge 0 , \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n ( - 1 ) ^ k P ( k ) { n \\choose k } = c _ n ( - 1 ) ^ n \\ , n ! . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} & \\ \\tau \\phi \\big ( \\exp ( H + \\sum _ { j = 1 } ^ m p _ j \\log A _ j ) \\big ) + ( 1 - \\tau ) \\phi \\big ( \\exp ( H + \\sum _ { j = 1 } ^ m p _ j \\log B _ j ) \\big ) \\\\ \\leq & \\ \\sum _ { j = 1 } ^ m \\frac { p _ j } { r } \\Big ( \\tau \\phi \\big ( \\exp ( L + r \\log A _ j - r \\log C _ j ) \\big ) + ( 1 - \\tau ) \\phi \\big ( \\exp ( L + r \\log B _ j - r \\log C _ j ) \\big ) \\Big ) \\\\ \\leq & \\ \\sum _ { j = 1 } ^ m \\frac { p _ j } { r } \\phi \\big ( \\exp ( L ) \\big ) \\\\ = & \\ \\phi \\big ( \\exp ( H + \\sum _ { j = 1 } ^ m p _ j \\log C _ j ) \\big ) , \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\beta _ n = \\sum _ { \\alpha \\in \\Phi ^ + } c _ { \\alpha } \\cdot X _ { \\alpha } \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} - \\div _ y ( A \\nabla _ y \\psi ) = \\div _ y \\phi , \\ \\forall y \\in B ( y _ 0 , 2 R ) \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} y ( 0 , \\lambda ) = h , y ^ { \\prime } ( 0 , \\lambda ) = k , \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} \\aligned \\| W _ { t _ i , \\varphi _ i } \\| _ { W ^ { 2 , p } } \\le & c _ 2 t _ i ^ { - N } \\big \\| \\varphi _ i ^ { N } d \\tau ^ { a _ 0 / t _ i } \\big \\| _ { L ^ p } \\\\ \\le & a _ 0 t _ i ^ { - N - 1 } \\big \\| \\tau \\| _ { L ^ \\infty } ^ { \\frac { a _ 0 - t _ i } { t _ i } } \\| \\varphi _ i ^ N d \\tau \\big \\| _ { L ^ p } \\to 0 . \\endaligned \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} P _ 1 E ( \\{ \\lambda \\} ) u = 0 \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} & \\| I _ { N } ^ { \\lambda } u - u \\| _ { L ^ 2 ( \\mathbb R ) } = \\| I _ N ^ G \\breve U - \\breve U \\| _ { L ^ 2 _ { \\omega _ \\lambda } ( I ) } , \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\dim _ H B ( \\{ s _ n \\} , \\{ t _ n \\} , 1 ) \\geq \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { i = 1 } ^ \\ell \\log t _ i } { d \\sum _ { i = 1 } ^ { \\ell + 1 } \\log s _ i - \\log t _ { \\ell + 1 } } . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} \\| \\tilde T _ j ( \\hat \\Sigma - \\tilde { \\Sigma } ) \\tilde T _ j \\| _ { \\infty } \\leq \\| \\tilde T _ j ( \\hat { \\Sigma } _ \\epsilon - \\Sigma _ \\epsilon ) \\tilde T _ j \\| _ { \\infty } + 2 \\| \\tilde T _ j F \\| \\Big \\| \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i \\tilde T _ j \\epsilon _ i \\Big \\| . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} \\mu = \\mu _ x \\times \\mu _ v \\times \\prod _ { k \\geq 1 } \\nu . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\nabla ^ N _ { L } \\partial _ a = & - l _ a ^ c \\partial _ c - \\eta _ a L \\\\ \\nabla ^ N _ { L } \\underline L = & - \\eta ^ b \\partial _ b . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N F ( x _ n ) = \\int _ 0 ^ 1 F ( x ) d x \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\lambda \\frac { \\frac { ( | \\mathcal { N } | - 1 ) } { a | \\mathcal { N } | } } { \\frac { ( | \\mathcal { N } | - 1 ) } { a | \\mathcal { N } | } \\hat { B } ^ * + 1 } - p \\frac { ( | \\mathcal { N } | - 1 ) } { a | \\mathcal { N } | } - 1 = 0 , \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\widehat { { L } } _ 3 ( t ) : = \\int _ 0 ^ t ( t - r ) ^ { \\alpha ( 1 - \\mu ) - 1 } \\left ( r ^ { - \\alpha \\vartheta } + { \\mathcal R } ^ { s } r ^ { - ( 1 + s ) \\alpha \\vartheta } \\right ) { { L } } _ 3 ( r ) d r . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\| D \\varphi _ { y } ( x ) \\| = 2 A _ { k ( y ) } . \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\tilde { p } ^ { ( \\alpha ) } _ n ( z ) : = \\frac { n ! c ^ n } { 2 ^ n ( 1 + \\alpha ) _ { n } } C _ { n } ^ { ( 1 + \\alpha ) } ( z / c ) \\ . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial x } + 2 y \\frac { \\partial } { \\partial t } , Y = \\frac { \\partial } { \\partial y } - 2 x \\frac { \\partial } { \\partial t } , T = \\frac { \\partial } { \\partial t } , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & A r _ t d t \\medskip \\\\ r _ 0 & = & h _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} & B ( \\gamma ) = \\frac { 1 + g ( 0 ) } { \\gamma } + \\frac { c ( a + 1 ) } { 2 } \\gamma ^ { - a } \\left ( \\log \\gamma \\right ) ^ { - b } F ( t ) = t ^ { \\min ( a , 1 ) } \\ , , \\\\ & A ( \\gamma ) = c ' \\times \\begin{cases} a ' \\gamma ^ { - ( a ' + 2 ) } ( \\log \\gamma ) ^ { - b ' } a ' > 0 \\ , , \\\\ b ' \\gamma ^ { - 2 } ( \\log \\gamma ) ^ { - ( b ' + 1 ) } a ' = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\beta _ { 2 , j } : = ( 1 - 3 \\alpha _ j c ' \\varepsilon ) M - 6 \\alpha _ j c ' ( 1 + | p - 2 | ) \\varepsilon - 3 | p - 2 | / 2 . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} u _ j : = u _ j ( x ; \\epsilon ) , j = 1 , 2 , \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { x ^ k } { ( k ! ) ^ \\nu } \\leq C _ 1 e ^ { c _ 1 x ^ { 1 / \\nu } } , \\ \\ x > 0 , \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} ( \\bar { g } ( u / v ) - \\bar { g } ( z / v ) ) ^ t = ( u / v - z / v ) ^ s . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} y _ i = \\sum _ { j = 1 } ^ { m } \\left ( \\left ( 1 - \\psi _ { i - j + 1 } \\right ) U \\left [ j , k ^ \\mathrm { L } _ { i - j + 1 } \\right ] + \\psi _ { i - j + 1 } U \\left [ j , k ^ \\mathrm { R } _ { i - j + 1 } \\right ] \\right ) , \\psi _ i = b _ i - k _ i ^ \\mathrm { L } . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} H _ D ( X ) = \\ker D = \\ker D ^ 2 . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\frac { \\Psi ' _ { N _ \\varepsilon } ( B _ \\varepsilon ) } { 2 } = B _ \\varepsilon H ( \\gamma _ \\varepsilon ) \\left [ \\left ( 1 + L ^ H _ \\varepsilon \\right ) \\left ( 1 + \\varphi _ { \\tilde { N } _ \\varepsilon } ( B _ \\varepsilon ^ 2 ) \\right ) + L ^ g _ \\varepsilon \\left ( \\frac { B _ \\varepsilon ^ { 2 N _ \\varepsilon } } { N _ \\varepsilon ! } - B _ \\varepsilon ^ 2 \\right ) \\right ] \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} I _ a ^ k \\left ( I _ a ^ { k ' } \\left ( f - I _ a ^ k \\omega ^ { - 1 } \\varphi \\right ) \\right ) ( x ) = I _ a ^ k \\left ( ( I _ a ^ { k ' } f ) ( a ) \\right ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\tau _ l = \\inf \\{ 0 \\leq t \\leq T ^ r _ 1 \\ | \\ | \\nabla Y ^ n _ t | > l \\} , \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} 0 = \\pi _ { \\mathcal { K } } ( 6 \\lambda \\mathcal { S } _ { 0 , 2 } ( U _ k , W _ k ) + \\mathcal { S } _ { 0 , 3 } ( U _ k , U _ k , U _ k ) ) + o ( 1 ) . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} | H _ 0 | = \\frac { 2 } { r } + ( 2 W _ 0 + 1 ) r + O ( r ^ 2 ) \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} w _ n ^ { ( r ) } - n = \\left | \\left \\{ k \\in \\mathbb { N } : m _ k ^ { ( r ) } < w _ n ^ { ( r ) } \\right \\} \\right | . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} T = A [ y _ 1 , \\ldots , y _ c ] = A _ 0 [ x _ 1 , \\ldots , x _ d , y _ 1 , \\ldots , y _ c ] , \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} & - \\div _ y \\Big ( e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } \\big ( ( \\Lambda + \\nabla _ y \\widetilde { w } ) ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T + I \\big ) \\nabla _ y v \\Big ) \\\\ & \\quad = \\div _ y \\Big ( e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } V _ { y _ l } ( \\Lambda + \\nabla _ y \\widetilde { w } ) \\Big ) , \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} k _ { \\gamma , \\Sigma _ 1 } ^ { \\infty , s } = \\frac { \\overline { p } { \\dot { \\gamma } _ 3 } + \\frac { \\sqrt { 2 } } { 2 } \\overline { q } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) } { | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} \\widetilde \\rho \\left ( \\gamma _ { \\frak { h } } ( v , g ) \\right ) & = \\rho \\left ( \\gamma _ { \\frak g } ( v ^ { \\prime } , g ) \\right ) = \\overline \\tau \\left ( \\langle v ^ { \\prime } , v _ g \\rangle _ { \\beta } \\right ) \\\\ & = \\overline \\tau \\left ( \\langle v , v _ g \\rangle _ { \\beta } \\right ) \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} H _ { n _ i } ( \\mu ) = \\frac { 1 } { n _ i \\log 2 } \\log \\# F _ 1 ( n _ i ) \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} T ^ { * * } = ( T ^ { * * } ) _ { \\rm r e g } + ( T ^ { * * } ) _ { \\rm s i n g } . \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b ^ 2 _ \\nu ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( - q ; q ^ 2 ) _ n ( z q ) ^ n . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\nu \\stackrel { \\mathrm { d e f } } { = } \\lambda + 2 \\mu , I ( a ) \\stackrel { \\mathrm { d e f } } { = } \\frac { a } { 1 + a } , k ( a ) \\stackrel { \\mathrm { d e f } } { = } - \\frac { P ' ( 1 + a ) } { 1 + a } + P ' ( 1 ) \\hbox { w i t h $ P ' ( 1 ) = 1 $ } , \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} W _ 1 \\ ( \\mu , \\nu \\ ) = \\min _ q \\sum _ { \\ ( x ' , y ' \\ ) \\in G \\times G } q ( x ' , y ' ) d ( x ' , y ' ) \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} ( \\mathcal { S } v ) ( { y } ) = \\int _ { D } \\log \\kappa ( y , x ) v ( x ) \\mathrm { d } x , \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} L _ i \\cap f ^ { - 1 } ( 0 ) \\setminus V = \\varnothing \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\overline { T } _ { 2 k + 1 , 2 a } ( x ; q ) : = ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k + \\frac { 1 } { 2 } , a } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} h _ \\alpha ( B _ \\alpha f ) ( x ) = x ^ 2 h _ \\alpha ( f ) ( x ) , x \\in ( 0 , \\infty ) \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\sum _ { m \\equiv b \\mod { q } } w ( m ) = \\frac { 1 } { q } \\sum _ { m \\in \\mathbf { Z } } \\widehat { w } \\left ( \\frac { m } { q } \\right ) e _ q ( b m ) , \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\textrm { i n } _ \\nu ( y ) : = y + P _ { \\nu ( y ) } ^ + \\in P _ { \\nu ( y ) } / P _ { \\nu ( y ) } ^ + \\subset \\textrm { g r } _ \\nu ( R ) . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\tilde { \\delta } } ( 2 \\varepsilon ) ~ = ~ \\Big \\lbrace \\delta \\in \\Delta _ { h , N _ { 1 } } ~ \\Big | ~ \\rho _ { \\bf { L } ^ 1 } ( g _ \\delta , g _ { \\tilde { \\delta } } ) \\leq 2 \\varepsilon \\Big \\rbrace , \\eta ( \\delta , \\tilde { \\delta } ) ~ = ~ \\left ( \\left \\lbrace i \\in \\overline { 0 , N _ 1 - 1 } ~ \\big | ~ \\delta _ i \\neq \\tilde { \\delta } _ i \\right \\rbrace \\right ) . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\kappa _ { f } = 1 - \\min _ { j : f ( j ) \\neq 0 } \\frac { f ( j | ( V \\setminus j ) ) } { f ( j ) } = 1 - \\min _ { S \\subset V \\setminus \\{ j \\} , f ( j ) \\neq 0 } \\frac { f ( j | S ) } { f ( j ) } \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d U = \\Delta U d t + ( X \\cdot \\nabla ) U d t - ( U \\cdot \\nabla ) X d t + \\sum _ { i = 1 } ^ N ( B _ i ( t ) + \\theta _ i I ) U d \\beta _ i ( 0 , \\infty ) \\times \\mathbb { R } ^ 3 , \\\\ U ( 0 , \\xi ) = U _ 0 ( \\xi ) = ( c u r l \\ x ) ( \\xi ) , \\ \\xi \\in \\mathbb { R } ^ 3 , \\end{array} \\right . \\ \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\xi & = \\textstyle \\sum _ { j , j ' \\in J } k _ { j j ' } = \\textstyle \\sum _ { k , k ' \\in K } k _ { k k ' } \\\\ \\Xi & = \\textstyle \\frac { 1 } { | J | ( | J | - 1 ) } \\sum _ { j , j ' \\in J } D _ { k _ { j j ' } } \\textstyle - \\frac { 2 } { | J | | K | } \\sum _ { j \\in J , k \\in K } D _ { k _ { j k } } + \\frac { 1 } { | K | ( | K | - 1 ) } \\sum _ { k , k ' \\in K } D _ { k _ { k k ' } } \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\alpha _ { i } \\ge \\sum _ { k = 1 } ^ r m _ { j _ k } \\alpha _ { j _ k } , \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\left ( \\mathcal { X } ^ { K \\textnormal { t h , e q } } \\left ( q , \\varphi _ { q , z } ^ { - 1 } ( Q ) \\right ) P _ { q , z } \\right ) _ { 0 i } = \\Lambda _ i ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { 1 } { z ^ { d ( N + 1 ) } } \\frac { ( 1 - q ) ^ { d ( N + 1 ) } Q ^ d } { \\left ( q \\Lambda _ 0 \\Lambda _ i ^ { - 1 } , \\dots q , \\dots , q \\Lambda _ N \\Lambda _ i ^ { - 1 } ; q \\right ) _ d } \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\rho ^ { ( k ) } ( x ) : = \\sup _ { j \\geq k } \\rho _ j ( x ) . \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} B ' = \\big ( \\bigcup _ { x \\in A \\cap C _ i } B ' _ { x } \\big ) \\cup \\big ( \\bigcup _ { y \\in A \\setminus C _ i } B ' _ { y } \\big ) \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} \\overline { T } = T ^ { * * } , \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\begin{cases} \\dot X ( t , x ) = u ( t , X ( t , x ) ) \\\\ X ( 0 , x ) = x \\ , . \\end{cases} \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} ( u _ 0 ) _ x ( x ) + ( u _ 1 ) _ y ( x , y ) = \\frac { j } { m ( x , y ) } - P . \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} P \\left ( \\{ Z _ { n _ 0 + l } \\in I _ { j _ l } \\} \\cap A _ l \\right ) \\leq \\int _ { \\psi ^ { - 1 } ( \\mathcal { R } _ l ) } C p _ { j _ l } d \\tilde F ( y ) = C p _ { j _ l } P ( \\xi \\in \\psi ^ { - 1 } ( \\mathcal { R } _ l ) ) = C p _ { j _ l } P ( A _ l ) . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\log B ( n ) = ( 1 + O ( n ^ { - \\kappa } ) ) \\tfrac { 1 } { u } ( K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) ) ^ { \\frac { 1 } { u + 1 } } ( u + 1 ) ^ { \\frac { u } { u + 1 } } n ^ { \\frac { u } { u + 1 } } \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\sqrt { \\lambda _ k } \\langle \\tilde u _ j , u _ k \\rangle = \\tilde x \\frac { \\lambda _ k } { \\tilde \\lambda _ j - \\lambda _ k } \\big ( \\sqrt { \\lambda _ j } \\langle \\tilde u _ j , u _ j \\rangle + \\sum _ { l \\neq j } \\sqrt { \\lambda _ l } \\langle \\tilde u _ j , u _ l \\rangle \\big ) , \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\langle E _ K f _ j , h ^ 0 _ { Q _ j } \\rangle = \\langle f _ j , h ^ 0 _ K \\rangle \\langle h _ K ^ 0 , h _ { Q _ j } ^ 0 \\rangle \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} h ' ( x ) = \\frac 1 2 ( \\xi '' ( x ) ( 1 - x ) - [ \\xi ' ( 1 ) - \\xi ' ( x ) ] ) . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\partial _ { y _ j } g _ { D , \\mathrm { o d d } } ( y ) = ( \\partial _ { y _ j } g _ D ) _ { \\mathrm { o d d } } ( y ) , \\partial _ { y _ j } g _ { N , \\mathrm { e v e n } } ( y ) = ( \\partial _ { y _ j } g _ N ) _ { \\mathrm { e v e n } } ( y ) , \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} ( d d ^ c w ) ^ n \\leq ( d d ^ c u ) ^ n \\Omega \\cap \\{ w = u \\} . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} G = \\cup _ z z L \\ \\ , \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} A = ( A _ { i , j } ) _ { i , j \\in \\{ 0 , \\dots , N \\} } \\in \\textnormal { M } _ { N + 1 } ( V ) \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\lambda [ z _ 0 , z _ 1 , \\cdots , z _ { 2 n } ] = [ \\lambda ^ { a _ 0 } z _ 0 , \\lambda ^ { a _ 1 } z _ 1 , \\cdots , \\lambda ^ { a _ { 2 n } } z _ { 2 n } ] . \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\frac { d } { d t } R _ L ( t ) = 2 \\phi ( L ) t ^ { - 2 / 3 } \\log t \\left ( \\frac { 1 } { 3 } \\log t + 2 \\right ) . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} J _ { \\nu } ( x ) = e ^ { i x } W _ { \\nu } ( x ) + e ^ { - i x } \\overline { W _ { \\nu } ( x ) } . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { i = 1 } ^ m \\xi _ i F _ i ( x ) , y - x \\right \\rangle \\geq 0 \\ \\ \\forall y \\in K _ P . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} \\psi _ { \\beta } ( g _ r ^ { - 1 } h g _ r ) = \\psi _ { ( g ) \\beta } ( h ) ( h \\in K _ l ( O _ r ) ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} H = \\int \\theta ^ H _ t d B _ t + L ^ H \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} f _ { 2 } \\left ( u _ { 2 } \\right ) = c _ { 2 } \\cos \\left ( \\frac { c _ { 1 } } { 1 + a ^ { 2 } } u _ { 2 } \\right ) + c _ { 3 } \\sin \\left ( \\frac { c _ { 1 } } { 1 + a ^ { 2 } } u _ { 2 } \\right ) . \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} H ^ { \\bullet } ( G , b , \\mu ) [ \\rho ] = \\sum \\limits _ { ( M _ S , \\mu _ S ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ S , b _ S } } ( \\mathrm { I n d } ^ G _ { P _ S } H ^ { \\bullet } ( M _ S , b _ S , \\mu _ S ) [ \\rho ] ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ S \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] , \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\langle \\hat u _ j , u _ k \\rangle = \\frac { \\langle u _ j , E u _ k \\rangle } { \\lambda _ j - \\lambda _ k } + \\frac { \\langle \\hat u _ j - u _ j , E u _ k \\rangle } { \\lambda _ j - \\lambda _ k } + \\frac { \\lambda _ j - \\hat \\lambda _ j } { \\lambda _ j - \\lambda _ k } \\langle \\hat u _ j , u _ k \\rangle , \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\lambda _ { k } = - 2 a \\cos \\left ( \\tfrac { n k \\pi } { n + 3 } \\right ) - 2 b \\cos \\left [ \\tfrac { ( n + 1 ) k \\pi } { n + 3 } \\right ] - 2 c \\cos \\left [ \\tfrac { ( n + 2 ) k \\pi } { n + 3 } \\right ] - d \\cos \\left ( k \\pi \\right ) , k = 1 , 2 , \\ldots , n + 2 . \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} & y _ { i + 1 } = \\left ( 2 y _ i - y _ { i - 1 } \\left ( 1 - 2 \\zeta \\omega \\frac { \\Delta t } { 2 } \\right ) + f \\left ( y _ i , x _ i \\right ) \\Delta t ^ 2 \\right ) \\left ( 1 + 2 \\zeta \\omega \\frac { \\Delta t } { 2 } \\right ) ^ { - 1 } , \\\\ & y _ 1 = y _ 0 + \\Delta t v _ 0 \\left ( 1 - 2 \\zeta \\omega \\frac { \\Delta t } { 2 } \\right ) + f \\left ( y _ 0 , x _ 0 \\right ) \\frac { \\Delta t ^ 2 } { 2 } , \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} H _ n = K \\langle g , x \\rangle / ( g ^ n - 1 , x ^ n , g x g ^ { - 1 } - \\zeta x ) \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} L _ { 2 ( n ) } ( q , b , q ) = \\frac { \\beta _ n ( 1 , q ) } { 1 - b ^ 2 q ^ { 2 n } } . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} & [ C o n f _ 3 ( B ) ] = [ B ^ 3 ] - [ x = y ] - [ x = z ] - [ z = x ] + 3 [ x = y = z ] - [ x = y = z ] = \\\\ & = [ B _ { \\{ \\{ 1 \\} , \\{ 2 \\} , \\{ 3 \\} \\} } ] - [ B _ { \\{ \\{ 1 , 2 \\} , \\{ 3 \\} \\} } ] - [ B _ { \\{ \\{ 1 , 3 \\} , \\{ 2 \\} \\} } ] - [ B _ { \\{ \\{ 1 \\} , \\{ 2 , 3 \\} \\} } ] + 2 [ B _ { \\{ \\{ 1 , 2 , 3 \\} \\} } ] \\ , . \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} \\Phi _ 1 \\left ( \\nu _ L , L \\right ) = \\Phi _ 2 \\left ( - \\frac { \\delta \\left ( \\nu _ L , L \\right ) } { d } , L \\right ) . \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} \\sum \\limits _ { q \\le Q } \\sum \\limits _ { \\substack { a = 1 \\\\ ( a , q ) = 1 } } ^ { q ^ k } \\left | \\sum \\limits _ { M < n \\le M + N } a _ n e \\left ( n \\cdot \\frac { a } { q ^ k } \\right ) \\right | ^ 2 \\ll \\left ( N + Q ^ { k + 1 } \\right ) \\sum \\limits _ { M < n \\le M + N } | a _ n | ^ 2 . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\delta : = \\left \\lfloor \\frac { n + 1 } { 2 } \\right \\rfloor - \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor , \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} & \\mathbb { H } ( A _ t , Z _ t , W _ t , r , y , p , P ) \\\\ & : = \\mathbb { H } ^ - ( A _ t , Z _ t , W _ t , r , y , p , P ) = \\mathbb { H } ^ + ( A _ t , Z _ t , W _ t , r , y , p , P ) . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\hat \\Phi ( t , k ) = \\left [ \\begin{array} { c } \\hat \\phi _ + ( t , k ) \\\\ \\\\ \\hat \\phi _ - ( t , k ) , \\end{array} \\right ] , \\hat \\Psi ( k ) = \\left [ \\begin{array} { c } \\hat \\psi _ + ( k ) \\\\ \\\\ \\hat \\psi _ - ( k ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} f = \\sum _ { [ \\pi ] \\in \\widehat { G } } d _ \\pi f \\ast \\chi _ \\pi . \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { r r r r r } 0 . 6 4 7 0 & 0 . 1 7 2 0 & - 0 . 7 4 9 0 & 0 . 7 2 8 0 & 0 . 7 1 7 0 \\\\ - 0 . 3 5 4 0 & - 0 . 0 6 2 0 & - 0 . 9 3 6 0 & - 0 . 7 7 3 0 & - 0 . 7 7 8 0 \\\\ 0 . 0 4 6 0 & 1 . 1 9 9 0 & - 1 . 2 6 9 0 & 0 . 8 3 7 0 & 0 . 3 1 6 0 \\\\ - 0 . 7 9 3 0 & 0 . 8 0 2 0 & 0 . 4 9 8 0 & - 1 . 1 2 8 0 & 1 . 4 0 7 0 \\\\ - 1 . 5 5 1 0 & 1 . 0 5 3 0 & 2 . 7 8 9 0 & - 1 . 4 2 5 0 & 0 . 4 0 1 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\sum _ { n \\ge 1 } \\phi _ G ( n ) \\ , z ^ n \\ = \\ \\frac { \\phi ^ * _ G ( z ) } { ( 1 - z ) ^ { \\xi + 1 } } \\ , , \\sum _ { n \\ge 1 } f _ G ( n ) \\ , z ^ n \\ = \\ \\frac { f ^ * _ G ( z ) } { ( 1 - z ) ^ { \\xi + 1 } } \\ , , \\sum _ { n \\ge 1 } t _ G ( n ) \\ , z ^ n \\ = \\ \\frac { t ^ * _ G ( z ) } { ( 1 - z ) ^ { d - c + 1 } } \\ , , \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} m _ 1 = m _ { \\sigma ^ 0 } , \\ , m _ 2 = m _ { \\sigma } , \\ , m _ 3 = m _ { \\sigma ^ 2 } , \\ , m _ 4 = m _ { \\mu } , \\ , m _ 5 = m _ { \\sigma \\mu } , \\ , m _ 6 = m _ { \\sigma ^ 2 \\mu } . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ \\lambda \\in [ 0 , 1 ] : \\exists \\widetilde { w } \\in C ^ { 2 , \\alpha } _ \\# ( \\mathcal { Y } ^ d ) \\ \\ F ( \\widetilde { w } , \\lambda , x , \\Lambda ) = 0 \\} . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} \\beta ( v | \\mathbf { h } | ^ 2 ) _ t + 2 \\nu v | \\mathbf { h } _ x | ^ 2 + \\beta ( v u | \\mathbf { h } | ^ 2 ) _ x - ( 2 \\nu v \\mathbf { h } \\cdot \\mathbf { h } _ x ) _ x + 2 \\beta v \\mathbf { h } \\cdot \\mathbf { w } _ x = - \\nu v _ x ( | \\mathbf { h } | ^ 2 ) _ x . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} t _ { k + 1 } : = \\inf \\bigg \\{ t \\in [ t _ k , 1 ] : f ( t ) \\notin \\bigcup _ { i = 0 } ^ k U _ { g _ { \\pi ( i ) } } \\bigg \\} . \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} \\gamma = \\gamma _ 0 \\cup \\gamma _ 1 ^ { - 1 } \\cup \\gamma _ 2 \\cup \\gamma ^ { - 1 } _ 3 \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} F _ 1 & = y ^ 2 z + y z ^ 2 - x ^ 3 - x ^ 2 z + 7 x z ^ 2 - 5 z ^ 3 \\\\ F _ 2 & = x ^ 2 z - y ^ 3 + 2 6 z ^ 3 \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} h _ \\lambda ^ { * n } \\ast f = \\sum _ { j = 1 } ^ m \\bigl ( h _ \\lambda \\ast f _ j \\bigr ) ^ { * n } . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} ( A - \\lambda ) ^ n = 0 , \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} T _ { \\lambda _ { 0 , r } + 1 } & \\cdots T _ { d - 1 } T _ d T _ { d - 1 } \\cdots T _ { \\lambda _ { 0 , r } + k } \\\\ & = q _ 0 ^ { - 1 } T _ { \\lambda _ { 0 , r } + 1 } \\cdots T _ { d - 1 } ( T _ { d - 1 } \\cdots T _ 1 X _ 1 T _ 0 ^ { - 1 } T _ 1 ^ { - 1 } \\cdots T _ { d - 1 } ^ { - 1 } ) T _ { d - 1 } \\cdots T _ { \\lambda _ { 0 , r } + k } \\\\ & = q _ 0 ^ { - 1 } T _ { \\lambda _ { 0 , r } + 1 } \\cdots T _ { d - 1 } T _ { d - 1 } \\cdots T _ 1 X _ 1 T _ 0 ^ { - 1 } T _ 1 ^ { - 1 } \\cdots T _ { \\lambda _ { 0 , r } + k - 1 } ^ { - 1 } . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\tilde a _ s ( u _ N , v _ N ) = ( I _ N ^ \\lambda f , v _ N ) , \\forall v _ N \\in V _ N ^ \\lambda . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\real ( u z ) = | z | t \\cos ( \\mathrm { A r g } ( z ) + { \\beta _ p } ) = - | z | t \\sin ( \\mathrm { A r g } ( z ) - \\pi / 2 p ) , \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} - \\phi ' ( s ) = \\frac { K u \\Gamma ( u + 2 ) } { u + 1 } s ^ { - u - 1 } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( H - \\int M ( d s , \\theta ^ H _ s ) \\right ) \\int \\frac { \\partial } { \\partial x } M ( d s , \\theta ^ H _ s ) \\right ] = 0 , \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\alpha _ n ( 1 , q ) = 2 ( - 1 ) ^ n q ^ { n ^ 2 } , \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { W } _ { k , a } ( n ) q ^ n = \\frac { ( - q ^ 2 ; q ^ 2 ) _ \\infty ( q ^ { a + 1 } , q ^ { 2 k + 1 - a } , q ^ { 2 k + 1 } ; q ^ { 2 k + 1 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} Z _ m ( x _ i ) = a _ 0 + \\sum _ { 0 < j \\leq i } S ( x _ j , y _ j ) y _ j ^ { - 1 / \\alpha ( Z _ m ( x _ { j - 1 } ) ) } , Z _ n ( x _ i ) = a _ 0 + \\sum _ { k : 0 < i _ k \\leq i } S ( x _ { i _ k } , y _ { i _ k } ) y _ { i _ k } ^ { - 1 / \\alpha ( Z _ n ( x _ { { i _ k } - 1 } ) ) } . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\partial ^ 2 _ { y _ 1 } g _ { D , \\mathrm { o d d } } ( y ) = ( \\partial ^ 2 _ { y _ 1 } g _ D ) _ { \\mathrm { o d d } } ( y ) , \\partial ^ 2 _ { y _ 1 } g _ { N , \\mathrm { e v e n } } ( y ) = ( \\partial ^ 2 _ { y _ 1 } g _ N ) _ { \\mathrm { e v e n } } ( y ) , \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} U ( \\xi ) = U ( z , t ) = ( ( 1 + | z | ^ 2 ) ^ 2 + t ^ 2 ) ^ { - ( Q - 2 ) / 4 } . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\int _ { B _ { d } ( x , t ^ { H } ) } p _ { l } ^ { \\eta , \\psi } ( t , x , y ) d y \\geq \\mathbb { P } \\left ( \\| U _ { t } \\| _ { \\textsc { C C } } \\leq \\gamma t ^ { H } \\right ) = \\mathbb { P } \\left ( \\| \\delta _ { t ^ { H } } ^ { - 1 } U _ { t } \\| _ { \\textsc { C C } } \\leq \\gamma \\right ) . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} p \\sigma + \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 = & - \\rho u ^ 2 \\sigma + \\varepsilon u _ x \\sigma + \\alpha g ' ( 1 / \\rho ) h ( | \\psi | ^ 2 ) \\sigma - \\left ( \\int _ b ^ x \\rho u \\sigma d \\xi \\right ) _ t \\\\ & + \\int _ b ^ x \\left [ \\left ( \\rho u ^ 2 + p + \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 - \\alpha g ' ( 1 / \\rho ) h ( | \\psi | ^ 2 ) \\right ) \\sigma _ x - \\varepsilon u _ x \\sigma _ x \\right ] d \\xi . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} F ( z ) + A ( z ) = \\eta \\left ( i \\frac { | z | } { z } \\right ) ^ k \\overline { F } \\ ! \\left ( - \\frac 1 { N z } \\right ) + B ( z ) , \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\partial _ t v - \\Delta v + \\nabla _ H \\pi = f , \\div _ H \\overline { v } = 0 , v ( 0 ) = a \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} ( 2 \\pi i ) ^ 2 q \\frac { d \\wp } { d q } = \\zeta \\wp ' - \\wp ^ 2 + \\frac { 1 } { 2 } \\wp '' . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} L _ { \\nu + 1 } ( p ) : = L _ { \\nu } ( p ) + \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) L ( \\delta _ { \\sigma ^ { - \\nu } } ( p ) ) , ~ ~ ~ ~ ~ p \\in \\Omega \\cap B ( \\sigma ^ { \\nu + 1 } ) . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\mathrm { h o f i b } ( \\mathcal N ^ { \\geq i } A \\Omega \\{ i \\} / p ^ n \\xrightarrow { \\varphi _ i - 1 } A \\Omega \\{ i \\} / p ^ n ) = \\tau ^ { \\leq i } R \\nu _ \\ast \\mathbb Z / p ^ n \\mathbb Z ( i ) = \\tau ^ { \\leq i } R \\psi _ \\ast \\mathbb Z / p ^ n \\mathbb Z ( i ) \\ , \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\alpha _ 1 | \\xi | ^ 2 \\leq \\sum _ { i , j = 1 } ^ N D ^ { i j } _ l ( x ) \\xi _ i \\xi _ j \\leq \\alpha _ 2 | \\xi | ^ 2 \\ \\ { \\rm f o r } \\ \\ x \\in \\R ^ N , \\ \\xi \\in \\R ^ N \\ ( l = 1 , 2 , \\cdots , m ) , \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\mathrm { I } ( u ) : = \\frac { 1 } { n + 1 } \\sum _ { k = 0 } ^ n \\int _ X ( u - V _ \\theta ) \\theta _ u ^ k \\wedge \\theta _ { V _ \\theta } ^ { n - k } . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} \\chi _ f ( G ) : = \\inf _ d { \\chi ( G \\ltimes K _ d ) \\over d } . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} I ^ { ( P ) } _ 1 = \\sum _ { \\frac { \\eta _ 1 n } { 2 N } \\leq k \\leq \\frac { 2 \\eta _ 2 n } { N } } \\Delta ( k , q _ l ) P o i ( k ; n p _ l ) , \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} R _ j g ( x ) : = & \\ , c _ n \\ , \\mathrm { P . V . } \\int _ { \\mathbb R ^ n } \\frac { x _ j - y _ j } { | x - y | ^ { n + 1 } } g ( y ) \\ , d y \\\\ = & \\ , c _ n \\lim _ { \\varepsilon \\searrow 0 } \\int _ { | x - y | \\ge \\varepsilon } \\frac { x _ j - y _ j } { | x - y | ^ { n + 1 } } g ( y ) \\ , d y \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} K ( x ) = \\sqrt { \\frac { h ( x ) } { k ( x ) } } \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} v ( x ) = - \\int _ { \\R ^ n } \\Phi ( x - y ) Q ( y , \\widetilde { u } ( y ) + L , \\nabla \\widetilde { u } ( y ) ) d y \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} K _ 1 & = \\int _ { \\Gamma _ \\gamma } | F ( z ) | \\left | \\frac { d z } { z } \\right | = 2 \\int _ 0 ^ \\infty | t e ^ { - i \\gamma } e ^ { - t ( \\cos ( \\gamma ) - i \\sin ( \\gamma ) ) } | \\frac { d t } { t } \\\\ & = 2 \\int _ 0 ^ \\infty e ^ { - t \\cos ( \\gamma ) } d t \\lesssim \\frac { 1 } { \\cos ( \\gamma ) } , \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty U _ { 2 k + 1 , 2 a + 1 } ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ { 2 a + 1 } , q ^ { 4 k + 1 - 2 a } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\mathcal { I } ^ { G , \\mu } _ { M _ { S _ 2 } , b _ { S _ 2 } } = \\{ ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\in \\mathcal { C } _ { M _ { S _ 2 } } : ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\in \\mathcal { I } ^ { M _ { S _ 1 } , \\mu _ { S _ 1 } } _ { M _ { S _ 2 } , b _ { S _ 2 } } \\ , \\ \\ , \\ ( M _ { S _ 1 } , \\mu _ { S _ 1 } ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ { S _ 1 } , b _ { S _ 1 } } \\} . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\int _ { 2 B _ 0 } \\Big ( \\sum _ { j \\leq l - 2 } 2 ^ { s ' j p } g _ j ^ p \\Big ) \\ , d \\mu \\approx \\sum _ { k = - \\infty } ^ { \\infty } 2 ^ { k p } \\mu ( E _ k \\setminus E _ { k - 1 } ) . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} T ^ { D } ( u , v ) = D _ { u } v - D _ { v } u - [ u , v ] + ( D u ) ^ { * } v , \\ \\forall u , v \\in \\Gamma ( E ) , \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} ( \\mathbf A \\leftrightarrow \\mathbf B ) \\odot ( \\mathbf A ' \\leftrightarrow \\mathbf B ' ) = ( \\mathbf A \\odot \\mathbf A ' ) \\leftrightarrow ( \\mathbf B \\odot \\mathbf B ' ) . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\begin{aligned} \\textbf { I } _ S \\left ( \\hat { \\boldsymbol { \\psi } } _ { k - 1 } , \\textbf { W } _ k \\right ) ^ { - 1 } = \\frac { { { \\sigma _ z ^ 2 } } } { { 2 { { \\lvert \\textbf { s } \\rvert } ^ 2 } } } \\left \\{ { { { \\widetilde { \\bf { I } } } _ { i { p _ 1 } } } + { { \\widetilde { \\bf { I } } } _ { i { p _ 2 } } } \\left ( \\hat { \\beta } _ { k - 1 } \\right ) } \\right \\} , \\end{aligned} \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} S ( G , r ) : = \\{ x \\in G : d ( x , \\partial G ) \\geq r \\} , \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\frac { \\P ( G \\cap H ) } { \\P ( G \\cup H ) } & = \\P ( G \\cap H \\mid G \\cup H ) = 1 - \\P ( ( G \\cap H ^ c ) \\cup ( H \\cap G ^ c ) \\mid G \\cup H ) \\\\ & \\leq 1 - ( \\P ( A \\mid G ) - \\P ( A \\mid H ) ) \\leq \\delta , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 = \\dfrac { \\bar H ^ 2 + 2 S - \\sqrt { ( 4 S - 3 \\bar H ^ 2 ) \\bar H ^ 2 } } 4 . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} M _ { ( H ) } : = \\{ x \\in M \\mid G _ x H \\} \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} b \\in L ^ q ( [ 0 , T ] , L ^ p _ x ) , \\frac { 2 } { q } + \\frac { d } { p } = 1 , \\ 1 < p , q < \\infty . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\in \\Lambda } \\left \\Vert h _ \\lambda ^ { * n } \\ast f - f \\right \\Vert _ A = 0 . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} ( X + 1 ) P + X ^ 3 + X ^ 2 + X = 0 . \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\dot { x } & = \\lambda - d x + r x \\left ( 1 - \\frac { x } { K } \\right ) - \\frac { \\beta x v } { ( 1 + y ) ( 1 + v ) } , \\\\ \\dot { y } & = \\frac { \\beta x ( t - \\tau _ 1 ) v ( t - \\tau _ 1 ) } { ( 1 + y ( t - \\tau _ 1 ) ) ( 1 + v ( t - \\tau _ 1 ) ) } e ^ { - \\alpha _ 1 \\tau _ 1 } - a y - p y z , \\\\ \\dot { v } & = k e ^ { - \\alpha _ 2 \\tau _ 2 } y ( t - \\tau _ 2 ) - u v , \\\\ \\dot { z } & = c y z - b z , \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} - \\frac { n - 1 } { n } t _ 1 ^ 2 \\cdot \\binom { ( s - 2 ) / 2 } { ( k - 1 ) / 2 } \\{ ( - 1 ) \\cdot ( s - 1 ) ^ 2 t _ 1 ^ 2 \\} ^ { ( k - 1 ) / 2 } \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\mathbb { R } ( A _ 1 ^ n , B _ 1 ^ { \\star n } , d ) = \\R _ { A _ 1 ^ n } ( d ) , \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\psi _ i ^ b ( \\{ z ^ j \\} _ { j \\in \\Z ^ n } ) \\geq c _ 2 \\sum _ { k = 1 } ^ n | D _ 1 ^ k z ( 0 ) | ^ p ; \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} G ( m , N , a , \\varepsilon , s ) \\leq 2 \\sum _ { k = 1 } ^ { ( { m ( 1 + \\epsilon ) \\over 2 } ) ^ { { 1 \\over a } } } k ^ { - d s } \\sum _ { j = 0 } ^ 2 G \\big ( ( m - k ^ a ) ( 1 + \\epsilon ) ^ j , N - 1 , a , \\varepsilon , s \\big ) . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} r _ i ( P ^ { - ( r - 1 ) } \\mathcal { A } ( H ) P ) & = \\sum _ { i _ 2 , \\ldots , i _ r = 1 } ^ n p _ i ^ { - ( r - 1 ) } a _ { i i _ 2 \\cdots i _ r } p _ { i _ 2 } \\cdots p _ { i _ r } \\\\ & = \\sum _ { \\{ i , i _ 2 , \\ldots , i _ r \\} \\in E ( H ) } p _ i ^ { - ( r - 1 ) } p _ { i _ 2 } \\cdots p _ { i _ r } . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { \\infty } a r ^ t = \\frac { a } { 1 - r } , \\ | r | < 1 , \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} \\begin{aligned} E ( A , \\Pi _ h \\xi , \\Pi _ h \\zeta ) \\leq C h ^ 2 ( & \\| A \\| _ { 2 , \\infty } \\| \\Pi _ h \\xi \\| _ 0 \\| \\Pi _ h \\zeta \\| _ 0 + \\\\ & \\| A \\| _ { 1 , \\infty } ( \\| \\xi \\| _ 1 \\| \\Pi _ h \\zeta \\| _ 0 + \\| \\Pi _ h \\xi \\| _ 0 \\| \\zeta \\| _ 1 ) + \\\\ & \\| A \\| _ 0 \\| \\xi \\| _ 1 \\| \\zeta \\| _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\frac { \\lambda _ \\varepsilon } { 2 } \\int _ { B _ { x _ \\varepsilon } ( A _ \\varepsilon \\mu _ \\varepsilon ) } \\Psi _ { N _ \\varepsilon } ( u _ \\varepsilon ) d y = \\frac { 4 \\pi + o ( 1 ) } { \\gamma _ \\varepsilon ^ 2 } \\ , . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} x _ j ^ { \\left ( k \\right ) } = \\theta + n _ j ^ { \\left ( k \\right ) } , \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} ( \\dot B ^ { s _ 0 } _ { p , q _ 0 } ( \\mathcal { H } ) , \\dot B ^ { s _ 1 } _ { p , q _ 1 } ( \\mathcal { H } ) ) _ { \\theta , q } = \\dot B ^ s _ { p , q } ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) & = \\log P ( \\mathrm { B i n } ( n - a _ n , 1 - \\pi _ n ( n - h ( n ) ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) \\\\ & \\sim _ e \\varepsilon f _ 4 ( n ) \\log ( b _ c ^ { ( n ) ' } / f _ 4 ( n ) ) . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\P ( G _ k \\cap A ^ c ) = \\P ( C _ { k , j + 1 } \\cap A ^ c ) . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\overline { Q } _ { k , i } ( 0 , 0 ) = 1 , k \\geq i \\geq 1 , \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b _ { F , r } ( n ) z ^ n = \\prod _ { j = 1 } ^ { \\infty } ( 1 - z ^ j ) ^ { - \\gamma _ { F , r } ( j ) } . \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} T V ( f _ h , [ 0 , L ] ) ~ \\leq ~ { h \\over \\Psi ( h ) } \\cdot V ~ = : ~ V _ h ~ . \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\mu _ { \\beta , 0 } ^ + ( A _ n \\mid \\eta = - 1 ( - \\infty , 0 ) \\times \\{ 0 \\} ) > 0 . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\nu _ 2 : = [ \\nu _ 1 ; \\nu _ 2 ( \\phi _ 1 ) = \\gamma _ 2 ] . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} S _ { k } \\leq C | k _ { d - j } | ^ { \\frac { 1 } { 2 ^ { j } } } L ^ { 1 - \\frac { 1 } { 2 ^ { j + 1 } } + } , k _ 1 = \\cdots = k _ { d - j - 1 } = 0 , k _ { d - j } \\neq 0 , 2 \\leq j \\leq d - 1 . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} X _ { ( T ^ \\ast Q , G , \\omega , H , F , u ) } = X _ H + \\textnormal { v l i f t } ( F ) + \\textnormal { v l i f t } ( u ) . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\lambda _ { \\phi _ t } ( i ) = 0 i \\ne 0 \\mod 2 ^ e . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} \\omega : = \\mathcal G \\cap \\{ x \\in B _ { \\rho _ 0 / 2 } ( x _ 0 ) : \\ ( x - x _ 0 ) / \\rho _ 0 \\in \\mathrm { I n t } \\ , ( \\mathrm { S u p p } ( \\eta ) ) \\} . \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\begin{aligned} Y ( s ) = & \\frac { \\hat { p } } { \\bar { y } } \\hat G _ y ( s ) \\hat G _ h ( s ) U _ h ( s ) + \\frac { \\hat { g } } { \\bar { y } } \\hat G _ y ( s ) U _ b ( s ) \\\\ & + \\hat G _ y ( s ) \\hat G _ d ( s ) \\frac { \\hat d _ { y z } } { \\bar { y } } D _ { y z } ( s ) \\\\ \\end{aligned} \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( H _ T - \\int _ 0 ^ T M ( d s , \\theta ^ H _ s ) \\right ) ^ 2 \\right ] = \\mathbb { E } [ ( \\lambda ^ H _ T ) ^ 2 ] \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} E [ X ] = \\int _ { - \\infty } ^ \\infty x f ( x ) d x . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\int _ 0 ^ { r _ \\varepsilon } | ( \\Delta \\bar { w } _ \\varepsilon ) | r d r = & O \\left ( \\| \\bar { w } ' _ \\varepsilon \\| _ { L ^ \\infty ( [ 0 , r _ \\varepsilon ] ) } \\int _ 0 ^ { r _ \\varepsilon / \\mu _ \\varepsilon } \\frac { \\mu _ \\varepsilon r ^ 2 d r } { ( 1 + r ^ 2 ) ^ { 1 + \\varepsilon _ 0 + o ( 1 ) } } \\right ) \\\\ & ~ ~ ~ ~ ~ + o \\left ( \\int _ 0 ^ { r _ \\varepsilon / \\mu _ \\varepsilon } \\frac { r d r } { ( 1 + r ^ 2 ) ^ { 1 + \\varepsilon _ 0 + o ( 1 ) } } \\right ) \\ , . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} C _ { i j k } = \\frac { r } { n + 1 } ( h _ { i j } C _ k + h _ { k i } C _ j + h _ { j k } C _ i ) + \\frac { t } { C ^ 2 } C _ i C _ j C _ k , \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\left | \\Big | \\sum _ { j = 1 } ^ l a _ j \\Big | ^ { \\alpha + 2 } - \\sum _ { j = 1 } ^ l | a _ j | ^ { \\alpha + 2 } \\right | \\leq C \\sum _ { j \\ne k } | a _ j | | a _ k | ^ { \\alpha + 1 } , \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} f ( x ) = \\sup \\{ d ( x , k ) : k \\in K \\} \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} x _ { 1 , n } - c t _ n \\in [ - 2 C , 2 C ] , \\ ( x ' _ { n } , t _ n ) \\in \\R ^ { N - 1 } \\times \\R \\ \\ ( n = 1 , 2 , 3 , \\cdots ) \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} f ( t , x _ t ) - f ( 0 , x _ 0 ) = \\int _ 0 ^ t \\frac { \\partial f } { \\partial s } ( s , x _ s ) \\ , \\mathrm { d } { s } + \\int _ 0 ^ t \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\ , \\mathrm { d } ^ { - } x _ s \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} g _ q ( W ) = \\frac { ( q \\alpha _ 1 ( q ) ^ { - 1 } W , q ) _ \\infty ( q \\alpha _ 2 ( q ) ^ { - 1 } W , q ) _ \\infty ( q \\alpha _ 3 ( q ) ^ { - 1 } W , q ) _ \\infty } { ( q W , q ) _ \\infty ( i q W , q ) _ \\infty ( - q W , q ) _ \\infty } \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} J = J ( x - y ) , \\quad \\int _ { \\R ^ n } | J ( x ) | d x = 1 . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} M : = { \\rm o r d e r } ( \\gamma ) < \\infty . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} & \\pi _ { a _ { 0 } , a _ { 1 } } \\left ( \\mathbf { Z } _ { 0 } , \\mathbf { Z } _ { 1 } ; P \\right ) \\equiv P \\left [ A _ { 1 } = a _ { 1 } | A _ { 0 } = a _ { 0 } , \\mathbf { Z } _ { 0 } , \\mathbf { Z } _ { 1 } \\right ] , \\\\ & \\pi _ { a _ { 0 } } \\left ( \\mathbf { Z } _ { 0 } ; P \\right ) \\equiv P \\left [ A _ { 0 } = a _ { 0 } | \\mathbf { Z } _ { 0 } \\right ] . \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} ( K \\varphi ) ( t ) = \\int _ { 0 } ^ { t } K ( t , s ) \\varphi ( s ) d s , \\ \\ \\ \\varphi \\in L ^ { 2 } ( [ 0 , 1 ] ) . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} F _ \\lambda ( x ) : = \\begin{cases} \\frac \\lambda 2 \\int _ \\Omega | x | ^ 2 + \\int _ \\Omega | \\nabla x | ^ 2 \\quad & x \\in V \\ , , \\\\ + \\infty \\quad & x \\in H \\setminus V \\ , . \\end{cases} \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } E _ { p , \\epsilon } ( v _ j ) = c _ { p , \\epsilon } ( M ) , \\lim _ { j \\to \\infty } \\| E _ { p , \\epsilon } ' ( u ) \\| _ { ( W ^ { 1 , p } ) ^ * } = 0 , \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} E _ H ^ { } \\lvert \\tau \\rvert ^ { 1 / 2 } \\nabla _ H e ^ { \\tau \\Delta _ H } Q f = \\lvert \\tau \\rvert ^ { 1 / 2 } \\nabla _ H e ^ { \\tau \\Delta _ H } E _ H ^ { } Q f = \\lvert \\tau \\rvert ^ { 1 / 2 } \\nabla _ H e ^ { \\tau \\Delta _ H } Q _ { \\R ^ 2 } E _ H ^ { } f . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} x = \\sinh t - e ^ t ( \\frac 1 2 s ^ 2 + & \\bar \\psi \\psi ) , y = e ^ t s , z = \\cosh t + e ^ t ( \\frac 1 2 s ^ 2 + \\bar \\psi \\psi ) , \\\\ & \\xi = e ^ t \\bar \\psi , \\eta = e ^ t \\psi . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} D _ { \\phi } ( \\bf x , { \\bf x ' } ) = \\phi ( \\bf x ) - \\phi ( { \\bf x ' } ) - \\nabla \\phi ( { \\bf x ' } ) ^ { T } ( \\bf x - { \\bf x ' } ) \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} s _ { n + 1 } & = d ( x _ { n + 1 } , z ) \\\\ & = d ( T x _ n , T z ) \\\\ & < d ( x _ n , z ) = s _ n \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} H = v _ n \\delta _ { n n ' } + \\Delta , \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} \\sum _ { p } h _ { j k } ^ { p ^ { \\ast } } \\mathcal { L } H ^ { p ^ { \\ast } } = \\sum _ { p } h _ { j k } ^ { p ^ { \\ast } } H ^ { p ^ { \\ast } } - \\sum _ { i , l , p , q } h _ { j k } ^ { p ^ { \\ast } } h _ { i l } ^ { p ^ { \\ast } } h _ { i l } ^ { q ^ { \\ast } } H ^ { q ^ { \\ast } } , \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\int _ { 2 B _ 0 } \\Big ( \\sum _ j g _ j ^ q \\Big ) ^ { \\frac { p } { q } } \\ , d \\mu \\approx \\sum _ { k = - \\infty } ^ { \\infty } 2 ^ { k p } \\mu ( E _ k \\setminus E _ { k - 1 } ) . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} u _ \\varepsilon = \\gamma _ \\varepsilon - \\frac { ( 1 + o ( 1 ) ) t _ \\varepsilon } { \\gamma _ \\varepsilon } \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\mathfrak q ' \\\\ \\updownarrow \\\\ \\mathfrak q '' \\end{array} : u ( \\mathfrak q ' ) = \\mathfrak p ' , u ( \\mathfrak q '' ) = \\mathfrak p '' , \\mathtt { I n } ( \\mathfrak q '' ) = J , \\mathtt { I n } ( \\mathfrak q ' ) = \\mathtt { O u t } ( \\mathfrak q '' ) , \\mathtt { O u t } ( \\mathfrak q ' ) = I \\right \\} \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} q _ { 1 j } = q _ { 2 j } = - 1 , \\ j = 1 , . . . , n . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} M ' _ { \\varphi _ R } ( t ) & = - \\int _ 0 ^ \\infty m ^ s \\int \\Delta ^ 2 \\varphi _ R | u _ m ( t ) | ^ 2 d x d m + 4 \\sum _ { j , k = 1 } ^ d \\int _ 0 ^ \\infty m ^ s \\int \\partial ^ 2 _ { j k } \\varphi _ R \\partial _ j \\overline { u } _ m ( t ) \\partial _ k u _ m ( t ) d x d m \\\\ & \\mathrel { \\phantom { = - \\int _ 0 ^ \\infty m ^ s \\int \\Delta ^ 2 \\varphi _ R | u _ m ( t ) | ^ 2 d x d m } } - \\frac { 2 \\alpha } { \\alpha + 2 } \\int \\Delta \\varphi _ R | u ( t ) | ^ { \\alpha + 2 } d x , \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\int _ { S ^ { n - 1 } } g ( u ) \\ , d \\widetilde { C } _ q ( K _ m , u ) = \\int _ { S ^ { n - 1 } } g ( u ) \\ , d \\widetilde { C } _ q ( K , u ) \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} A ^ t W + W A + W ' = 0 . \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\mathcal { X } ( \\lambda , a , b ) : = \\big \\{ x \\in \\mathbb { X } \\ | \\ ( x , \\lambda ) \\in \\mathcal { S } _ { \\rm K K T } ( a , b ) \\big \\} . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} & \\sum _ { m = k - 1 } ^ { n - 1 } { m \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , m ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { m - k + 1 } ] \\\\ & = { n - 1 \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , n - 1 ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { n - k } \\\\ & + \\sum _ { m = k - 1 } ^ { n - 2 } { m \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , m ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { m - k + 1 } ] . \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} | x _ { m + 1 } - y | = | \\Phi _ 1 ( x _ m , h _ m ) - x _ m - F ( S _ { l _ 0 } ( h _ m ) , x _ m ) | . \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} S _ { 1 } = \\sum _ { p = 1 } ^ { X } \\frac { 1 } { p ^ { \\sigma } } \\left ( \\phi ( q ) \\sum _ { \\substack { m , n , p m = n \\bmod q } } q _ m q _ n \\right ) . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\bar { \\mathsf h } _ { i j } \\rightarrow \\bar { w } ^ * _ { i j } \\ ; \\ ; i , j = \\mathbf k ^ * + 1 , \\ldots , m , i \\ne j . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} B _ R & = C _ { \\alpha , \\beta } R ^ { - 2 ( \\alpha + \\beta ) } \\int ^ R _ 0 ( R ^ 2 - t ^ 2 ) ^ { \\beta - 1 } t ^ { 2 \\alpha + 1 } ( 1 - R ^ { - 2 } t ^ 2 ) B ^ { \\alpha + 1 } _ t d t \\\\ & = C _ { \\alpha , \\beta } R ^ { - 2 ( \\alpha + \\beta ) - 2 } \\int ^ R _ 0 ( R ^ 2 - t ^ 2 ) ^ { \\beta } t ^ { 2 \\alpha + 1 } B ^ { \\alpha + 1 } _ t d t \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 2 } ^ n F _ { } ( \\mu _ { n , i } ) + \\frac { 1 } { n } F _ { } ( \\mu _ { n , 1 } ) , \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} \\mathbb { P } _ { x , k } ^ { \\epsilon , \\hat { v } } \\Bigl \\{ \\zeta ^ { \\epsilon } ( \\tau _ D ^ { \\epsilon , \\hat { v } } ) = k _ 0 \\ , \\bigl \\vert \\tau _ D ^ { \\epsilon , \\hat { v } } < \\infty \\Bigr \\} = 1 \\rightarrow 0 \\epsilon \\rightarrow 0 , \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} \\theta _ { \\pm } : = - \\lim _ { n \\rightarrow \\infty } n ^ { - 1 } \\log \\nu _ { \\beta , 0 } ^ { + } ( \\eta = \\mp 1 [ - n , - 1 ] ) , \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} ( 1 - \\lambda ) ^ 2 = \\tfrac { 1 } { 4 } \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\left ( G ( O _ r ) : G ( O _ r , \\beta ) \\right ) & = \\frac { | G ( O _ { l ^ { \\prime } } ) | } { | G _ { \\beta } ( O _ { l ^ { \\prime } } ) | } \\\\ & = \\sharp \\Omega \\cdot q ^ { ( l ^ { \\prime } - 1 ) ( \\dim G - \\ , G ) } , \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} ( D _ { a ^ { + } } ^ { \\alpha } f ) ( t ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left ( \\frac { f ( t ) } { ( t - a ) ^ { \\alpha } } + \\alpha \\int _ { a } ^ { t } \\frac { f ( t ) - f ( s ) } { ( t - s ) ^ { \\alpha + 1 } } d s \\right ) , \\ \\ \\ t \\in [ a , b ] . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} A ^ G = \\{ x \\in A | \\mu ( x ) = x \\otimes 1 \\in A \\otimes _ k R ) \\} \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\tau ( \\beta ) \\beta b \\tau ( b ) = b \\tau ( b ) \\tau ( \\beta ) \\beta \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} X _ * ( T ) ^ { \\Gamma } = \\mathfrak { A } . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} ( d F ) _ { h _ { 0 } \\sqcup \\alpha } ( l \\sqcup 0 ) = ( { \\rm I d } + J _ { 1 } ( F ( h _ { 0 } ) ; \\alpha ) ) \\circ ( d F ) _ { h _ { 0 } } ( l ) , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - \\log b _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) & = \\lim _ { n \\to \\infty } \\frac { a _ c ^ { ( n ) } } { - \\log b _ c ^ { ( n ) } } \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) \\\\ & = - J ( x _ 0 ) \\lim _ { n \\to \\infty } \\frac { a _ c ^ { ( n ) } } { - \\log b _ c ^ { ( n ) } } = - \\infty . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\rho ( i ) & = \\begin{cases} 2 i & \\mbox { $ i \\leq \\delta / 2 $ } \\\\ 2 ( \\delta - i ) + 1 & \\mbox { $ i > \\delta / 2 $ } \\end{cases} & \\rho ^ { - 1 } ( i ) & = \\begin{cases} i / 2 & \\mbox { $ i $ e v e n } \\\\ \\delta - \\frac { i - 1 } { 2 } & \\mbox { $ i $ o d d } \\end{cases} \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { 2 k , 2 a - 1 } ( n ) q ^ n = & \\frac { ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a } , q ^ { 4 k - 2 a } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\\\ & + \\frac { x q ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a - 2 } , q ^ { 4 k - 2 a + 2 } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} \\| { \\bf w } \\| _ { 1 , 2 } : = \\sum \\limits _ { g } \\| { \\bf w } _ g \\| _ 2 \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\Lambda _ { a _ 1 } ( f ) = \\Lambda _ { a _ 2 } ( f ) \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} h ( x ) & \\le g _ { 8 - } ( x ) \\Bigr | _ { x = \\frac { 7 9 8 5 } { 1 9 1 7 1 } } + \\frac 1 { 1 0 \\cdot 6 ^ 8 } - \\eta \\\\ & = - \\frac { 4 7 6 8 9 0 6 9 4 4 2 9 4 3 7 5 0 6 6 3 0 2 3 5 9 0 7 4 8 2 4 5 5 9 2 1 8 9 } { 1 5 5 9 9 4 9 8 4 6 2 2 2 3 9 6 9 3 5 6 8 6 9 8 1 2 4 1 6 7 7 0 3 7 8 7 3 5 3 9 4 4 9 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\alpha ' ( t ) = \\sum _ { i = 1 } ^ m b _ i ( t ) X _ i ( \\alpha ( t ) ) . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\Delta _ n \\left ( \\frac { x e ^ { t x } } { e ^ x - 1 } \\right ) = B _ n ( t ) = \\sum _ { i = 0 } ^ n \\binom { n } { i } B _ { n - i } ( 0 ) t ^ i , B _ m ( 0 ) = \\Delta _ m \\left ( \\frac { x } { e ^ x - 1 } \\right ) . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} A _ { n + 1 } = \\frac { 1 } { ( n + 1 ) ! } \\sum _ { \\sigma \\in S _ { n + 1 } } ( - 1 ) ^ { \\sigma } \\sigma \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\beta _ { j _ { t } } = A O _ { j _ { t - 1 } } ^ { \\ast } \\tau _ { t } \\rho _ { j _ { t - 1 } } \\lambda _ { j _ { t - 1 } } \\sigma \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\theta _ \\mu ( \\xi ) = \\mu ( \\xi ) \\ , . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\theta _ { M _ S } ( \\mu ) = \\frac { 1 } { | W ^ { \\mathrm { r e l } } _ { M _ S } | } \\sum \\limits _ { \\sigma \\in W ^ { \\mathrm { r e l } } _ { M _ S } } \\sigma ( \\mu ^ { \\Gamma } ) . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\asymp \\sum _ { j = u _ k + 1 } ^ \\infty j ^ { - d } \\asymp u _ k ^ { - d + 1 } . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} u ( t ) = \\sum _ { \\ell = 0 } ^ m \\frac { d ^ \\ell } { d t ^ \\ell } V _ { \\ell } ( t ) + \\frac { d ^ { m + 1 } } { d t ^ { m + 1 } } U _ { m + 1 } ( t ) . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\hat { \\Psi } _ 0 ( Y _ 1 , \\ldots , Y _ m ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\hat { \\Psi } _ 1 ( Y _ 1 , \\ldots , Y _ m ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\nu [ w _ { 1 } . . . w _ { l } ] = p _ { w _ { 1 } } \\cdot . . . \\cdot p _ { w _ { l } } w _ { 1 } , . . . , w _ { l } \\in \\Lambda ^ { m } \\ : . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} U _ { l _ 0 } ( x , t ) - U _ { l _ 0 } ( x _ 1 + c \\tau + \\sigma ^ * , x ' + \\rho , t + \\tau ) & = U _ { l _ 0 } ( x , t ) - W ^ { \\sigma ^ * } _ { l _ 0 } ( x , t ) \\\\ & = z _ { l _ 0 } ( x , t ) = 0 . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\mu u _ v \\phi ( v ) + z u _ v \\phi ( v ) = 0 \\quad { v \\in \\Gamma . } \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\begin{cases} \\eta ^ \\zeta ( \\rho , \\rho u ) = \\rho \\int _ { - 1 } ^ 1 \\zeta ( u + \\rho ^ { \\vartheta } s ) [ 1 - s ^ 2 ] _ + ^ \\Lambda d s , & \\\\ q ^ \\zeta ( \\rho , \\rho u ) = \\rho \\int _ { - 1 } ^ 1 ( u + \\vartheta \\rho ^ \\vartheta s ) \\zeta ( u + \\rho ^ { \\vartheta } s ) [ 1 - s ^ 2 ] _ + ^ \\Lambda d s . & \\end{cases} \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\phi ( y _ 1 + y _ 2 ) = \\phi _ 1 ( y _ 1 ) + T y _ 2 \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle \\mu _ { g _ s } ( \\hat x , t ) & = & \\displaystyle \\frac { 1 } { ( 2 t ) ^ { k - 1 } } \\ , \\int _ { x _ { k - 1 } - t } ^ { x _ { k - 1 } + t } \\cdots \\int _ { x _ 1 - t } ^ { x _ 1 + t } g _ s ( \\tau _ 1 , \\ldots , \\tau _ { k - 1 } ) \\ , d \\tau _ 1 \\cdots d \\tau _ { k - 1 } \\\\ & & \\\\ & = & \\displaystyle \\frac { 1 } { ( 2 t ) ^ { k - 1 } } \\ , \\int _ { x _ { k - 1 } - t } ^ { x _ { k - 1 } + t } \\cdots \\int _ { x _ 1 - t } ^ { x _ 1 + t } f ( \\tau _ 1 , \\ldots , \\tau _ { k - 1 } , s ) \\ , d \\tau _ 1 \\cdots d \\tau _ { k - 1 } \\end{array} \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq n } f _ j ( I _ j ) = f _ { j _ 2 } ( I _ { j _ 2 } ) \\leq f _ { j _ 2 } ( \\widehat { I } _ { j _ 2 } ) \\leq \\max _ { 1 \\leq j \\leq n } f _ j ( \\widehat I _ j ) . \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} v _ { p _ { \\infty } } ( \\epsilon _ 0 ) = 0 v _ { p _ { \\infty } } ( \\epsilon _ i ) = \\frac { v _ { p _ { \\infty } } ( \\epsilon _ { i - 1 } ) } { n - a } + \\frac { n - 1 } { n - a } i > 1 . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\mu ( S e ) & = q ( S e , S e \\wedge m _ { \\mu } ) \\mu ( S e \\wedge m _ { \\mu } ) & & \\\\ & = q ( S e , S e \\wedge m _ { \\mu } ) \\iota ( S e \\wedge m _ { \\mu } , m _ { \\mu } ) \\mu ( m _ { \\mu } ) & & S e \\wedge m _ { \\mu } \\le m _ { \\mu } \\\\ & = \\rho ( e , e f , e f ) \\rho ( e f , e f , f ) \\rho ( f , f , f ) & & m _ { \\mu } = \\{ S f \\} \\\\ & = \\rho ( e , e f , f ) & & \\\\ & = \\rho ^ f ( S e ) . & & \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\left ( \\Delta _ X + \\lambda ^ 2 n ( x ) \\right ) u = 0 & \\mbox { i n } & X , \\\\ u = f & \\mbox { o n } & \\partial X , \\end{array} \\right . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} L ( x , \\varphi , \\dot x , \\dot \\varphi ) = \\frac { m } { 2 } \\Big ( r ^ 2 \\dot \\varphi ^ 2 + \\dot x ^ 2 \\Big ) - m g x . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} { \\rm S p a n } \\{ V _ { [ \\alpha ] } ( x ) : \\alpha \\in { \\cal A } _ { 1 } ( l ) \\} = { \\rm S p a n } \\{ V _ { \\mu } ( x ) : 1 \\leq \\mu \\leq m _ { l } \\} , \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\langle D _ 1 J _ 1 ( f ; v ^ 1 + s w ^ 1 , v ^ 2 ) - D _ 1 J _ 1 ( f ; v ^ 1 , v ^ 2 ) , w ^ 2 \\rangle & = \\iint _ { \\mathcal { O } _ { 1 } \\times ( 0 , T ) } ( \\phi ^ s - \\phi ) w ^ 2 d x d t + s \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } w ^ 1 w ^ 2 d x d t . \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} \\bold { T } _ { x , i } = \\left [ \\begin{array} { c | c } \\bold { A } _ { x , x ; i , i } & \\begin{array} { c | c | c | c } \\bold { B } _ { x , x ; i } & \\bold { E } _ { x , 1 ; i } & \\dots & \\bold { E } _ { x , p _ 0 ; i } \\end{array} \\\\ \\hline \\begin{array} { c } \\bold { U } _ { x , i } \\\\ \\hline \\bold { V } _ { x , i } \\end{array} & \\bold { Z } _ { x , i } \\end{array} \\right ] , \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} f _ i ( x ) \\frac { \\partial g ^ { i j } } { \\partial y ^ k } = 0 , \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} U _ m ^ - ( x , t ) & = \\inf _ { S \\in \\mathcal { S } _ m } \\sup _ { A \\in \\mathcal { A C } _ m } \\mathbb E \\Big [ e ^ { - r ( T - t ) } g \\big ( X ( T ) \\big ) \\Big ] , \\\\ U _ m ^ + ( x , t ) & = \\sup _ { S \\in \\mathcal { S } _ m } \\inf _ { A \\in \\mathcal { A C } _ m } \\mathbb E \\Big [ e ^ { - r ( T - t ) } g \\big ( X ( T ) \\big ) \\Big ] \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\hat { E } ( L _ 1 , \\ell _ \\mu , \\ell _ { \\mu ' } ) : = \\tfrac { 1 } { L _ 1 } ( L _ 1 - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ \\mu ) - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ { \\mu ' } ) ) \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { x ^ k } { ( k ! ) ^ \\upsilon } \\geq c _ 1 e ^ { c _ 2 x ^ { 1 / \\upsilon } } , \\ \\ x > 0 , \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m i n i m i z e } & \\sum _ { t = 1 } ^ m \\sigma _ { F _ t } \\big ( \\begin{bmatrix} y _ { 1 , t } - z _ 1 c _ { 1 , t } \\\\ \\vdots \\\\ y _ { n , t } - z _ n c _ { n , t } \\end{bmatrix} \\big ) - \\sum _ { i = 1 } ^ n H _ i ^ * ( \\hat { y } _ i ) + \\sum _ { i = 1 } ^ n z _ i \\\\ \\mbox { s u b j e c t t o } & z _ i \\geq 0 ~ \\forall i \\in [ n ] \\\\ \\end{array} \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} M = \\begin{pmatrix} c _ { 0 , 0 } & c _ { 0 , 1 } & c _ { 0 , 2 } & c _ { 0 , 3 } & \\dots \\\\ c _ { 1 , 0 } & c _ { 1 , 1 } & c _ { 1 , 2 } & c _ { 1 , 3 } & \\dots \\\\ 0 & c _ { 2 , 1 } & c _ { 2 , 2 } & c _ { 2 , 3 } & \\dots \\\\ 0 & 0 & c _ { 3 , 2 } & c _ { 3 , 3 } & \\dots \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\ddots \\end{pmatrix} \\ . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\beta , 0 } ^ { + } ( \\{ [ - n , - 1 ] \\leftrightarrow \\infty \\} ^ c ) = \\mathbb { P } _ { \\beta ^ * , 0 } ^ { * } ( O D C [ - n , - 1 ] ) , \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\Lambda = \\Big \\{ a \\in w ( S _ p ^ i ) \\big | \\alpha _ m ( w ^ { - 1 } ( a ) ) > 1 / j \\Big \\} \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} X = \\{ | x _ 1 - x _ 2 | , \\dots , | x _ 1 - x _ Z | \\} . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} v _ { g h } = v _ h \\sigma _ g ^ { - 1 } + v _ g \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ G ( y ) + \\nabla \\psi ( y ) - G ( z ) - \\nabla \\psi ( z ) : y , z \\in E \\} = \\R ^ n ; \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\Delta ^ 2 u = Q _ 1 ( x , u , \\nabla u , \\nabla ^ 2 u ) + \\nabla \\cdot Q _ 2 ( x , u , \\nabla u , \\nabla ^ 2 u ) \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} S ( k ) : = \\begin{pmatrix} a ( k ) & - \\bar b ( k ) \\\\ b ( k ) & \\bar a ( k ) \\end{pmatrix} = \\Psi ^ { - 1 } ( 0 , 0 , k ) \\Phi ^ { \\R { p } } ( 0 , 0 , k ) = \\Phi ^ { \\R { p } } ( 0 , 0 , k ) = N ( 0 , 0 , k ) . \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} w ^ { \\ast } = \\begin{bmatrix} 0 \\\\ - 3 \\\\ - 1 0 \\\\ - 1 0 \\\\ - 6 \\end{bmatrix} , \\ ; v ^ { \\ast } = \\begin{bmatrix} 0 \\\\ - 1 \\\\ - 2 \\\\ - 6 \\\\ - 4 \\end{bmatrix} . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} \\left ( \\langle f _ 2 , f _ 3 \\rangle , \\langle f _ 3 , f _ 1 \\rangle , \\langle f _ 1 , f _ 2 \\rangle \\right ) = ( \\delta _ 1 e ^ { i \\theta _ 1 } | | f _ 2 | | | | f _ 3 | | , \\delta _ 2 e ^ { i \\theta _ 2 } | | f _ 3 | | | | f _ 1 | | , \\delta _ 3 e ^ { i \\theta _ 3 } | | f _ 1 | | | | f _ 2 | | ) . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} T ^ { i j k } = T ( \\tilde { e } _ i , \\tilde { e } _ j , \\tilde { e } _ k ) = \\epsilon _ i \\epsilon _ j \\epsilon _ k T ( e _ i , e _ j , e _ k ) = - \\epsilon _ i \\epsilon _ j \\epsilon _ k \\langle [ e _ i , e _ j ] , e _ k \\rangle , \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\alpha = \\alpha _ { T } : { M ' } _ { X ' } \\longrightarrow M _ { X } \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\rho ) = 0 . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} g - \\frac { 1 } { \\lambda } O ^ 2 g = \\frac { 1 } { \\lambda } f . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} \\alpha : = \\partial \\widehat \\alpha \\ , , \\alpha _ \\Gamma : = \\partial \\widehat \\alpha _ \\Gamma \\ , , \\beta : = \\partial \\widehat \\beta \\ , , \\beta _ \\Gamma : = \\partial \\widehat \\beta _ \\Gamma \\ , . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\mathcal { S } _ { ( d \\varphi , \\iota ) } \\times \\{ \\pm \\} } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\phi ) + \\ell _ { \\beta } ( \\phi ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = t \\ ; \\mathrm { m o d } _ q ( d \\varphi ) - t \\ ; \\mathrm { m o d } _ q ( d \\varphi ) = 0 . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k \\lambda _ j ( ( B ^ \\frac { p } { 2 } A ^ p B ^ \\frac { p } { 2 } ) ^ \\frac { 1 } { p } ) = & \\ \\lambda _ 1 \\Big ( \\big ( ( \\wedge ^ { k } B ) ^ \\frac { p } { 2 } ( \\wedge ^ { k } A ) ^ p ( \\wedge ^ { k } B ) ^ \\frac { p } { 2 } \\big ) ^ \\frac { 1 } { p } \\Big ) \\\\ \\geq & \\ \\lambda _ 1 \\Big ( ( \\wedge ^ { k } B ) ^ \\frac { 1 } { 2 } ( \\wedge ^ { k } A ) ( \\wedge ^ { k } B ) ^ \\frac { 1 } { 2 } \\Big ) = \\prod _ { j = 1 } ^ k \\lambda _ j ( B ^ \\frac { 1 } { 2 } A B ^ \\frac { 1 } { 2 } ) . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} 2 \\mu c _ \\alpha a | \\alpha \\cap \\beta | + b _ { \\alpha , \\beta } \\theta ^ \\beta _ \\mp = 2 \\mu c _ \\alpha a | \\alpha \\cap ( \\alpha + \\beta ) | - b _ { \\alpha , \\beta } \\theta ^ \\beta _ \\pm \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} H ( B _ \\varepsilon ) = H ( \\gamma _ \\varepsilon ) \\left ( 1 + L ^ H _ \\varepsilon \\right ) ( 1 + g ( B _ \\varepsilon ) ) = H ( \\gamma _ \\varepsilon ) \\left ( 1 + L ^ H _ \\varepsilon + L ^ g _ \\varepsilon \\right ) \\ , . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} & \\mathcal { S } _ n : = _ \\mathbb { C } \\{ s i n ( k \\pi y ) : k = 1 , . . . , n \\} , & \\mathcal { C } _ n : = _ \\mathbb { C } \\{ c o s ( j \\pi y ) : j = 0 , 1 , . . . , n \\} , \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} B ^ 0 = 1 \\cdot V _ { ( \\wp ) } V \\subset 1 \\cdot V = A ^ 0 , \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} Z _ t \\overset { \\mathrm { D e f . } } { = } \\int _ 0 ^ t \\psi _ r \\delta R _ r ^ H . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\langle \\imath _ { \\xi } \\gamma ( x ) , \\eta \\rangle & = \\langle x , [ \\xi , \\eta ] \\rangle \\\\ & = \\langle [ x , \\xi ] , \\eta \\rangle \\qquad \\\\ & = \\langle - a d _ x ^ * \\xi + a d _ { \\xi } ^ * x , \\eta \\rangle \\\\ & = \\langle a d _ { \\xi } ^ * x , \\eta \\rangle . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} c = \\alpha ^ { d _ 1 + d _ 2 } \\ , \\beta ^ { d _ 3 } \\ ; , \\ ; \\ ; c a r d ( \\mathcal { A } _ n ) \\leq N _ 1 ^ { d _ 1 } \\ , N _ 2 ^ { d _ 2 + d _ 3 } \\ ; , \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\lim _ { n \\to 0 } \\left [ \\frac { \\beta } { \\beta _ 1 } \\right ] ^ { 3 ^ n } \\ln ^ { \\frac 1 2 + \\zeta } n = 0 , \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} f ( u _ { 0 } , v _ { 0 } ) = g ( u _ { 0 } , v _ { 0 } ) = 0 u _ { 0 } v _ { 0 } \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} f _ { 1 , v } = \\partial _ { v } f _ 1 < 0 , \\ f _ { 2 , u } < 0 \\ \\ { \\rm i n } \\ \\ ( p _ 1 ^ { - } , p _ 1 ^ { + } ) \\times ( p _ 2 ^ { + } , p _ 2 ^ { - } ) . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ n _ S ( x ) : x \\in S \\} = \\textrm { s p a n } \\{ n _ S ( x ) : x \\in S \\cap Y \\} = \\textrm { s p a n } \\{ n _ { S \\cap Y } ( x ) : x \\in S \\cap Y \\} = Y . \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\begin{aligned} S ( 1 - \\dfrac 3 2 S ) + \\dfrac 3 2 S \\bar H ^ 2 - \\dfrac 3 4 \\bar H ^ 4 + \\dfrac { \\bar H ^ 2 \\sqrt { ( 4 S - 3 \\bar H ^ 2 ) \\bar H ^ 2 } } 4 = 0 . \\end{aligned} \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\frac { R _ { 1 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = i R _ { 1 3 } = 0 ; \\quad \\frac { R _ { 2 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = i R _ { 2 3 } = i = u _ 2 ; \\quad \\frac { R _ { 4 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = 2 u _ 4 . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} A _ { \\epsilon } : = I + ( p - 2 ) \\gamma _ { \\epsilon } ^ { - 2 } d \\varphi _ { \\epsilon } \\otimes d \\varphi _ { \\epsilon } . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\imath \\{ f , f ' \\} = f , \\{ f , f ' \\} \\in T ^ { * * } . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} f _ i ( n ) = ( g _ n + ( y ^ r \\frac { \\partial a } { \\partial x } ) ^ n ) f _ i ( 0 ) \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\bold { F } _ { x , x } = \\left [ \\begin{array} { c | c | c | c | c } \\bold { I } _ { k _ { x , 1 } } & \\bold { A } _ { x , x ; 1 , 1 } & \\dots & \\bold { 0 } & \\bold { A } _ { x , x ; 1 , p _ x } \\\\ \\hline \\vdots & \\ddots & \\ddots & \\vdots & \\vdots \\\\ \\hline \\bold { 0 } & \\bold { A } _ { x , x ; p _ x , 1 } & \\dots & \\bold { I } _ { k _ { x , p _ x } } & \\bold { A } _ { x , x ; p _ x , p _ x } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} E ( w ) : = E _ { \\rm e l } ( w ) + E _ { \\rm g } ( w ) + E _ { \\rm s f } ( w ) , w \\in W . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\varepsilon _ k \\alpha _ k ^ 2 [ f h \\varphi \\varphi '' - ( n - 1 ) f h ( \\varphi ' ) ^ 2 + m h \\varphi \\varphi ' f ' + r f \\varphi \\varphi ' h ' ] = \\rho f h . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ { x _ j } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { x _ j y _ i } d y + \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } \\widetilde { w } _ { y _ i x _ j } \\widetilde { w } _ { x _ j y _ i } d y = 0 . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} j ! ! \\doteq \\begin{cases} j ( j - 2 ) \\cdots 1 & \\mbox { i f } \\ \\ j \\ \\ \\mbox { i s o d d } , \\\\ j ( j - 2 ) \\cdots 2 & \\mbox { i f } \\ \\ j \\ \\ \\mbox { i s e v e n } . \\end{cases} \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\partial _ { t _ i } \\textbf { G } _ \\tau ( s _ 1 , s _ 2 ) = \\textbf { G } _ \\tau \\left ( \\nabla _ { \\partial t _ i } s _ 1 , s _ 2 \\right ) + \\textbf { G } _ \\tau \\left ( s _ 1 , \\nabla _ { \\partial t _ i } s _ 2 \\right ) \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} B _ 1 ^ { ( n ) } : = \\bigcup _ { t = a _ n } ^ { \\lfloor K a _ c ^ { ( n ) } \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} , \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} R ^ T = - \\xi ^ { - 1 } X X \\log \\xi . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\begin{aligned} - 1 6 L \\mathcal Q ( t ) \\geq & ~ { } \\frac 1 2 \\tilde \\sigma \\int | \\varphi ' | f ^ 2 + 3 \\tilde \\sigma \\int | \\varphi ' | f _ x ^ 2 + 2 \\tilde \\sigma \\int | \\varphi ' | f _ { x x } ^ 2 + \\frac 1 4 \\tilde \\sigma \\int | \\varphi ' | f _ { x x x } ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} - \\beta + \\sum _ { i = 0 } ^ { s - 2 } \\delta _ 3 ^ { ( s - i ) } & = - \\beta \\left ( 1 - \\gamma \\beta \\sum _ { i = 0 } ^ { s - 2 } 2 ^ { 1 + 2 ( s - i ) - 4 s ^ 2 } \\right ) \\\\ & < - \\beta \\left ( 1 - 2 ^ { - 4 s ^ 2 + 2 s + 2 } \\right ) \\\\ & \\leq - \\beta \\left ( 1 - 2 ^ { - 1 0 } \\right ) , \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} k _ { \\eta } ( \\tau ) : = \\left \\{ \\begin{array} { l l l } 1 & \\hbox { i f } 0 \\leq \\tau < \\eta / 2 \\\\ \\Big ( 2 \\frac { \\tau } { \\eta } - 2 \\Big ) ^ 2 \\Big ( 4 \\frac { \\tau } { \\eta } - 1 \\Big ) & \\hbox { i f } \\eta / 2 \\leq \\tau < \\eta \\\\ 0 & \\hbox { i f } \\tau \\geq \\eta \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} \\delta _ { f } = \\max \\{ \\delta _ { g } , \\delta _ { f _ { T } } \\} = \\max \\{ \\rho ( F ) ^ { 2 } , \\rho ( F ) \\} = \\rho ( F ) ^ { 2 } = \\delta _ { g } \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} A _ \\lambda \\partial _ t ( u _ \\lambda , v _ \\lambda ) + B _ \\lambda ( u _ \\lambda , v _ \\lambda ) = ( g _ \\lambda , g _ { \\Gamma \\lambda } ) - ( T _ \\lambda \\pi ( u _ \\lambda ) , T _ \\lambda \\pi _ \\Gamma ( v _ \\lambda ) ) \\ , , ( u _ \\lambda , v _ \\lambda ) ( 0 ) = ( u _ 0 , u _ { 0 | \\Gamma } ) \\ , . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} ( a ) = ( 1 ) + ( 2 ) \\leq - 2 \\pi p _ { 0 } I \\left ( \\frac { 1 } { \\varepsilon _ { n } } \\left ( \\log \\frac { 1 } { \\varepsilon _ { n } } \\right ) ^ { - \\frac { 1 } { s _ { k } } } \\right ) - 2 \\pi p _ { 0 } \\frac { 1 } { s _ { k } } \\log \\log \\frac { 1 } { \\varepsilon _ { n } } + 2 \\pi p _ { 0 } \\log \\frac { 1 } { \\varepsilon _ { n } } + O ( 1 ) . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} E ( \\prod _ i N ^ { ( \\frac { 1 } { 2 } ) } _ { \\{ x _ i , y _ i \\} } \\prod _ l ( N ^ { ( \\frac { 1 } { 2 } ) } _ { z _ l } + 1 ) ) = E ( \\prod _ i C _ { ( x _ i , y _ i ) } { \\varphi ^ { \\mathbb { R } } _ { x _ i } \\varphi ^ { \\mathbb { R } } _ { y _ i } } \\prod _ l \\frac { 1 } { 2 } \\lambda _ { z _ l } ( { \\varphi ^ { \\mathbb { R } } _ { z _ l } } ) ^ 2 ) . \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} ( \\sqrt { \\delta } - 1 ) ^ 2 \\left ( ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\right ) ^ { - 1 } \\bigg ( 1 - \\frac { t + b + 2 } { 1 + t } \\bigg ) & = - ( \\sqrt { \\delta } - 1 ) ^ 2 \\frac { ( 1 + b ) } { ( 1 + t ) \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] } \\\\ & = - \\frac { 4 ( 1 + b ) } { ( 1 + t ) \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] } \\big ( \\tfrac { \\sqrt { \\delta } - 1 } { 2 } \\big ) ^ 2 . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} w ( t , x ) & = \\frac { 1 } { 2 ^ { \\sqrt { \\delta } } } \\int _ 0 ^ t \\int _ { x - t + b } ^ { x + t - b } \\left [ u _ 0 '' ( y ) + b u _ 1 '' ( y ) - \\left ( \\frac { \\nu ^ 2 } { ( 1 + b ) ^ 2 } u _ 0 ( y ) + \\frac { \\mu } { 1 + b } u _ 1 ( y ) + \\frac { \\nu ^ 2 \\ , b } { ( 1 + b ) ^ 2 } u _ 1 ( y ) \\right ) \\right ] E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } y \\ , \\mathrm { d } b \\\\ & \\doteq I _ 1 + I _ 2 + I _ 3 + I _ 4 + I _ 5 . \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\nabla _ { z \\partial _ z } S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) = \\left ( z \\partial _ z + \\mathfrak { E } _ { ( \\partial ) } + \\mu \\right ) S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} h _ 0 h _ 1 h _ 2 h _ 3 = ( x _ 1 + P ( x _ 2 , x _ 3 ) , x _ 2 + Q ( x _ 3 ) , x _ 3 ) \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} | \\tilde { \\mathcal { L } } ^ d _ { \\leq x } | & = \\frac { 1 } { 2 ^ { d - 1 } d ! } ( 1 + O _ { d } ( x ^ { - 1 + O ( 1 / \\log \\log x ) } ) ) | \\mathcal { L } ^ d _ { \\leq x } | \\\\ & = ( 1 + O _ { d } ( x ^ { - 1 + O ( 1 / \\log \\log x ) } ) ) \\frac { \\prod _ { i = 2 } ^ { d } \\zeta ( i ) } { 2 ^ { d - 1 } d ! d } x ^ d \\\\ & = ( 1 + O _ { d } ( x ^ { - 1 + O ( 1 / \\log \\log x ) } ) ) c _ d x ^ d , \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} v ( f _ w ( x ) ) = h _ { w } ( \\gamma ) = r _ i \\gamma + \\alpha ( w ) . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\Delta _ 0 \\Delta _ 1 \\Delta _ 2 \\hdots = x _ 1 \\varphi ( \\Delta _ 0 \\Delta _ 1 \\Delta _ 2 \\dots ) . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\phi _ { \\omega , j } = \\frac { s _ { \\chi _ { \\omega } , j } } { \\sigma _ { \\omega } } \\ , , j \\geq 0 \\ , . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\bar f ( k , \\eta ) : = \\frac { 1 } { 2 } \\left [ f \\left ( k + \\frac { \\eta } { 2 } \\right ) + f \\left ( k - \\frac { \\eta } { 2 } \\right ) \\right ] \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\wp ( x ) \\sum _ { j \\in \\Z _ { \\geq 0 } } \\frac { 1 } { j + 1 } ( - T ) ^ { ( j ) } ( a _ { ( x ^ { j + 1 } \\wp ' ) } b ) \\otimes m = \\wp ( x ) \\int \\left \\{ a _ { ( \\wp ' ) } b \\right \\} \\otimes m . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} \\pi ^ T _ * J X = I \\pi _ * X . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} 4 K = ( H _ 0 ' ) ^ 2 - 4 - 2 | \\AA ' | ^ 2 . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( F ) = 0 \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} V _ f ^ { \\pm } ( y ) = \\pi ^ { - \\frac { k } 2 } \\left ( W _ { \\frac { k } 2 , \\nu } ( 4 \\pi y ) \\pm \\epsilon \\nu ^ k W _ { - \\frac { k } 2 , \\nu } ( 4 \\pi y ) \\right ) = \\begin{cases} 4 \\sqrt { y } K _ { \\nu } ( 2 \\pi y ) & k = 0 \\epsilon = \\pm 1 , \\\\ 0 & k = 0 \\epsilon = \\mp 1 , \\\\ 4 y K _ { \\nu \\pm \\frac { \\epsilon } 2 } ( 2 \\pi y ) & k = 1 . \\end{cases} \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} N _ \\varepsilon = 1 \\varepsilon \\ , . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } { 1 \\over n _ k } { \\sum _ { j = 1 } ^ k \\log u _ j } = \\lim _ { k \\to \\infty } { \\sum _ { j = 1 } ^ k j ^ { 1 - \\epsilon \\over b } \\over k ^ { { 1 \\over \\alpha } ( 1 - \\epsilon ) } } = 0 . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\alpha = \\frac 1 d - \\frac 1 2 \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} \\widetilde { \\Sigma } ( \\nu , n ) : = \\frac { 1 } { n } \\left ( n A ( n ) + \\nu \\Sigma ( \\nu ) - \\nu A ( \\nu ) \\right ) , \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\begin{cases} w _ t - \\frac { 1 } { 2 } \\Delta w = - ( f + b \\cdot \\nabla u ) , 0 \\leq t \\leq T , \\\\ w ( 0 , x ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} 1 - h ( z ) = 1 - \\frac { c A } { 2 } | w + c | + \\frac { c B } { 2 } \\Re ( w + c ) = 1 - \\tilde { j } ( w ) \\ , \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} H ^ 2 ( P _ { f _ \\theta } , P _ g ) = 2 - 2 \\int _ 0 ^ 1 \\sqrt { g ( x - \\theta ) g ( x ) } d x = 4 0 \\theta ^ 2 ( 1 - 2 \\theta + 8 \\theta ^ 3 / 5 ) . \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\rho _ { n , s } ( z ) = \\int _ { - 1 } ^ 1 \\dfrac { \\omega ( t ) } { z - t } \\ , T _ n ^ { 2 s + 1 } ( t ) d t . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} I I _ 1 = \\| \\langle x \\rangle ^ { a } ( \\widetilde { ( \\nabla \\chi _ k ) u } ) \\| _ { L ^ 2 ( \\mathbb R ; \\dot H ^ \\frac 1 2 ( \\mathbb { R } ^ n ) ) } \\lesssim \\| f \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} ( \\nabla ^ j \\varphi _ R ) \\subset \\left \\{ \\begin{array} { c l } \\{ | x | \\leq 2 R \\} & j = 1 , 2 , \\\\ \\{ R \\leq | x | \\leq 2 R \\} & j = 3 , 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\Phi \\left ( \\cap _ { i = 1 } ^ k A _ i \\right ) ( \\leqq \\times \\cdots \\times \\leqq ) \\Phi \\left ( \\cap _ { i = 1 } ^ k A ' _ i \\right ) & \\Longleftrightarrow \\pi ( \\ell _ i ( A _ i ) ) \\leqq \\pi ( \\ell _ i ( A ' _ i ) ) ( \\forall i ) \\\\ & \\Longleftrightarrow \\cap _ { i = 1 } ^ k A _ i \\subset \\overline { \\cap _ { i = 1 } ^ k A ' _ i } . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} { } _ { 2 } F _ { 1 } ( a , b ; c ; x ) = ( 1 - x ) ^ { - a } \\ , { } _ { 2 } F _ { 1 } \\Big ( a , c - b ; c ; \\frac x { x - 1 } \\Big ) , \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} Y ' = \\{ ( ( A _ g ) , m , ( T _ { g } ) ) | \\forall i \\ , A _ { g _ i } = e _ { i i } \\} . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} \\mathcal I _ { \\alpha } f ( x ) & : = { \\mathcal L } ^ { - { \\alpha } / 2 } f ( x ) \\\\ & = \\int _ 0 ^ { \\infty } e ^ { - t \\mathcal L } f ( x ) \\ , t ^ { \\alpha / 2 - 1 } d t . \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} & \\frac { d } { d t } | v ( t ) | _ { H ^ { - 1 } } ^ 2 = 2 \\left < v ' ( t ) , ( - A ) ^ { - 1 } v ( t ) \\right > _ H \\\\ & = - 2 | v ( t ) | _ H ^ 2 - 2 b ( \\mathcal { M } ( \\Phi ) ( t ) , \\mathcal { M } ( \\Phi ) ( t ) , ( - A ) ^ { - 1 } v ( t ) ) + 2 b ( \\mathcal { M } ( \\Psi ) ( t ) , \\mathcal { M } ( \\Psi ) ( t ) , ( - A ) ^ { - 1 } v ( t ) ) . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\prod _ { l = 1 } ^ m \\Tilde { Y _ l } \\geq 1 \\right ) = \\mathbb { P } \\left ( \\prod _ { l = 1 } ^ m \\Tilde { Y _ l } ^ { \\theta } \\geq 1 \\right ) \\leq \\mathbb { E } \\left [ \\Tilde { Y } ^ { \\theta } \\right ] ^ { m } \\leq \\left ( \\frac { \\mu } { C } \\right ) ^ { m } . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\hat Q _ r ( \\hat \\Sigma - \\mu _ r I ) \\hat Q _ r - Q _ r E Q _ r = ( \\hat { Q } _ r E \\hat { Q } _ r - Q _ r E Q _ r ) + \\hat { Q } _ r ( \\Sigma - \\lambda _ r I ) \\hat { Q } _ r . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\lambda _ { t } ^ { ( l _ { 0 } ) } = \\inf _ { \\eta \\in S ^ { N - 1 } } \\langle \\eta , M _ { X _ { l _ { 0 } } ( t , x ) } \\eta \\rangle _ { \\mathbb { R } ^ { N } } \\geq c _ { V , l } \\mu _ { t } ^ { ( l _ { 0 } ) } , \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\Q ^ { \\times } _ p = \\mu _ { p - 1 } \\times ( 1 + p \\Z _ p ) \\times p ^ { \\Z } \\ ; . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} w ^ { \\sigma } ( x , t ) : = u ( x _ 1 + c \\tau + \\sigma , x ' + \\rho , t + \\tau ) \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) = x q ( - x q ^ 3 ; q ^ 2 ) _ \\infty [ \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) - \\overline { Q } _ { k , a - 1 } ( x ^ 2 ; q ^ 2 ) ] . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} & \\P ( G \\cap H ) \\leq \\delta \\P ( G \\cup H ) = \\delta ( \\P ( G ) + \\P ( H ) ) - \\delta \\P ( G \\cap H ) , \\\\ & ( 1 + \\delta ) \\P ( G \\cap H ) \\leq \\delta ( \\P ( G ) + \\P ( H ) ) . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\mathrm { i } \\partial _ t u ( t , x ) + \\Delta u ( t , x ) = F ( t , x ) \\mathbb R \\times \\mathbb R ^ n \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\alpha _ { i } ^ { k } = \\frac { | B ( y _ { i } , r ) | } { | B ( y _ { i } , 2 ^ { k } r ) | } \\langle f _ { i } \\rangle _ { B ( y _ { i } , r ) } , \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\lambda _ j = c _ 0 + c _ { n - 1 } \\omega _ j + \\ldots + c _ 1 \\omega _ j ^ { n - 1 } , \\textrm { f o r } j = 0 , \\ldots , \\ n - 1 \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\mathcal { F } ^ { \\Psi } _ { \\left [ 2 \\ell _ { [ L , M , T ] } , M , \\gamma _ { [ L , M , T ] } \\right ] } ~ = ~ \\Big \\{ g \\in B V ^ { \\Psi } \\Big ( \\left [ 0 , 2 \\ell _ { [ L , M , T ] } \\right ] , [ - M , M ] \\Big ) ~ \\big | ~ T V ^ { \\Psi } ( g , [ 0 , 2 \\ell _ { [ L , M , T ] } ] ) \\leq \\gamma _ { [ L , M , T ] } \\Big \\} \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\left ( 6 - c _ { 1 } \\right ) \\omega _ { 3 } ^ { 2 } \\omega _ { 3 } ^ { \\prime } - c _ { 1 } \\omega _ { 3 } \\omega _ { 3 } ^ { \\prime \\prime } \\omega _ { 4 } + 4 c _ { 1 } \\left ( \\omega _ { 3 } ^ { \\prime } \\right ) ^ { 2 } \\omega _ { 4 } = 0 . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } - \\Delta U = e ^ U \\quad \\mbox { i n } \\R ^ 2 \\\\ \\int _ { \\R ^ 2 } e ^ U d x = 8 \\pi . \\end{array} \\right . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} h , _ { x _ i y _ j } - \\frac { f , _ { x _ i } } { f } h , _ { y _ j } = 0 . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s \\ , \\varepsilon _ s ^ 2 = Q \\textrm { f o r f i n i t e } Q . \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} J _ { \\mu } ( x ) = \\sum _ { m = 0 } ^ \\infty \\ , \\frac { ( - 1 ) ^ m } { m ! \\ , \\Gamma ( m + \\mu + 1 ) } \\left ( \\frac { x } { 2 } \\right ) ^ { 2 m + \\mu } . \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} N ( G ) \\coloneqq \\left \\{ g \\in G : g d _ 1 ( g ) = 1 \\right \\} . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\alpha _ k ( x ) = \\sup \\Big \\{ \\epsilon \\geq 0 \\big | \\exists _ { V \\in G _ { n , k } } \\exists _ { W \\in G _ { m , k } } \\exists _ { T \\in O ( V , W ) } & \\forall _ { y \\in ( x + V ) \\cap B ( x , \\epsilon ) } \\\\ & u ( x ) - u ( y ) = T ( x - y ) \\Big \\} , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} ( u ^ * , v ^ * ) : = \\Big ( \\frac { b _ 2 R _ 1 - b _ 1 R _ 2 } { a _ 1 b _ 2 - a _ 2 b _ 1 } , \\frac { a _ 1 R _ 2 - a _ 2 R _ 1 } { a _ 1 b _ 2 - a _ 2 b _ 1 } \\Big ) \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\# ( F _ 1 + F _ 2 ) ( n _ i ) \\leq 2 N ( F _ 1 + F _ 2 , 2 ^ { - n _ i } ) \\leq N ( F _ 1 , 2 ^ { - n _ i } ) ^ { 1 + \\delta } = \\# F _ 1 ( n _ i ) ^ { 1 + \\delta } . \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} \\Gamma ( k ) = \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } \\\\ \\Gamma ( k ) _ { 2 , 1 } \\\\ \\vdots \\\\ \\Gamma ( k ) _ { n , 1 } \\end{bmatrix} \\otimes \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } & \\Gamma ( k ) _ { 1 , 2 } & \\ldots & \\Gamma ( k ) _ { 1 , n } \\end{bmatrix} . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\lambda \\star \\nu ^ p _ \\pm = \\begin{cases} \\emptyset & \\mbox { i f } p = ( 0 , | \\alpha | ) \\\\ \\nu ^ p _ \\mp & \\mbox { i f } \\xi _ \\beta = 0 \\mbox { w h e r e } p ( \\beta ) = p \\\\ \\nu ^ p _ + , \\nu ^ p _ - & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - t ) ( 1 - 2 t ) \\cdots ( 1 - k t ) } = \\frac { 1 } { k ! } \\sum _ { l = 0 } ^ k { k \\choose l } ( - 1 ) ^ { k - l } l ^ k \\frac { 1 } { 1 - l t } . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} A _ { K , M } ^ { ( 1 , 1 ) } \\buildrel \\Delta \\over = \\max \\left \\{ { \\frac { 1 } { { \\left | { 1 - \\frac { 1 } { { M N + K { N _ { \\cal S } } } } \\left ( { { \\lambda _ { \\cal M } } + { \\lambda _ { \\cal S } } } \\right ) } \\right | } } , \\frac { 1 } { { \\left | { 1 - \\frac { 1 } { { M N + K { N _ { \\cal S } } } } \\left ( { \\hat \\lambda _ { { \\cal M } , j } ^ { ( K ) } + \\hat \\lambda _ { { \\cal S } , j } ^ { ( K ) } } \\right ) } \\right | } } } \\right \\} . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} Q + \\sum _ { i = 1 } ^ r y ^ - _ i B _ i = \\sum _ { i = 1 } ^ r y ^ + _ i B _ i , \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} h \\Delta _ { g _ B } h + ( r - 1 ) | g r a d _ { g _ B } h | ^ 2 + \\rho h ^ 2 = \\mu , \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\mu _ { i j } = 1 , \\sum _ { j = 1 } ^ n \\mu _ { i j } = \\sum _ { j = 1 } ^ n \\mu _ { j i } , \\textrm { f o r a l l } i \\in [ n ] , \\ ; \\mu \\in \\R _ + ^ { n \\times n } . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} \\Xi : t ^ k \\mapsto & \\frac { t z } { 1 - t z } \\left ( t ^ { k - 1 } + \\frac { t ^ { k - 2 } } { 1 - z } + \\frac { t ^ { k - 3 } } { ( 1 - z ) ^ 2 } + \\cdots + \\frac { t } { ( 1 - z ) ^ { k - 2 } } + \\frac { 1 } { ( 1 - z ) ^ { k - 1 } } \\right ) \\\\ = & \\left ( \\frac { t z } { 1 - t z } \\right ) \\left ( \\frac { t ^ k - \\left ( \\frac { 1 } { 1 - z } \\right ) ^ k } { t - \\frac { 1 } { 1 - z } } \\right ) . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\begin{aligned} & w _ d = \\sum _ { i _ 1 = 0 } ^ { p - 1 } \\cdots \\sum _ { i _ 8 = 0 } ^ { p - 1 } ( y _ 1 ^ { p ^ s } ) ^ { i _ 1 } ( y _ 2 ^ { p ^ s } ) ^ { i _ 2 } \\cdots ( y _ 8 ^ { p ^ s } ) ^ { i _ 8 } h _ { i _ 1 i _ 2 \\cdots i _ 8 } ( y _ 1 ^ { p ^ { s + 1 } } , y _ 2 ^ { p ^ { s + 1 } } , \\cdots , y _ 8 ^ { p ^ { s + 1 } } ) \\\\ & \\in \\mathbb { F } _ p [ y _ 1 ^ { p ^ s } , y _ 2 ^ { p ^ s } , \\cdots , y _ 8 ^ { p ^ s } ] \\backslash \\mathbb { F } _ p [ y _ 1 ^ { p ^ { s + 1 } } , y _ 2 ^ { p ^ { s + 1 } } , \\cdots , y _ 8 ^ { p ^ { s + 1 } } ] , \\end{aligned} \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\nu = \\delta _ Y , \\ Y = \\left < \\nu ; \\widetilde { y } \\right > \\in { \\rm D o m } [ E ] , \\ E ( Y ) > 0 , \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\xi ( x ) = \\frac 5 6 x ^ 3 + \\frac 1 6 x ^ { 1 6 } . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} ( ( A - \\lambda ) ( A - \\overline { \\lambda } ) ) ^ n = 0 . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} D ^ { \\mathcal S _ { + } } _ { e } ( v \\eta _ { + } ) = ( D ^ { + } _ { e } v ) \\eta _ { + } + v D ^ { \\mathcal S _ { + } } _ { e } \\eta _ { + } , \\ e \\in \\Gamma ( E ) , \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} & \\# \\{ p \\in S p l _ X ( f ) \\mid \\left \\{ \\frac { m r } { p } \\right \\} < \\left \\{ \\frac { n r } { p } \\right \\} ^ \\exists r \\in \\mathbb { Z } f ( r ) \\equiv 0 \\bmod p \\} \\\\ = & \\# S p l _ X ( f ) - \\# \\{ p \\in S p l _ X ( f ) \\mid \\left \\{ \\frac { n r } { p } \\right \\} \\le \\left \\{ \\frac { m r } { p } \\right \\} ^ \\forall r f ( r ) \\equiv 0 \\bmod p \\} , \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} T ' = T + \\alpha , \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = \\frac { 1 } { 2 } ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 ) + \\sum _ { d = 1 } ^ \\infty N _ d \\frac { t _ 2 ^ { 3 d - 1 } } { ( 3 d - 1 ) ! } e ^ { d t _ 1 } Q ^ d \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} j ^ { K \\textnormal { t h } } ( q , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( 1 - q ^ r P ^ { - 1 } \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} \\chi ( \\Gamma , X ) = \\prod _ i | H ^ i ( \\Gamma , X ) | ^ { ( - 1 ) ^ i } \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\textbf { S } ( t , z ) ( z \\partial z + \\mathfrak { E } _ { ( \\partial ) } + \\mu ) \\alpha = \\textbf { S } ( t , z ) \\left ( \\frac { ( \\alpha ) } { 2 } - \\frac { \\textnormal { d i m } _ \\mathbb { C } ( X ) } { 2 } \\right ) \\alpha = \\left ( \\frac { \\alpha } { 2 } - \\frac { \\textnormal { d i m } _ \\mathbb { C } ( X ) } { 2 } \\right ) S ( t , z ) ( \\alpha ) \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } M x - A ^ T \\lambda + q = 0 , \\\\ \\lambda ^ T ( A x - b ) = 0 , \\ \\lambda \\geq 0 , \\ A x \\geq b . \\end{array} \\right . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} z Q \\partial _ Q \\left ( Q ^ { \\frac { H } { z } } \\right ) = H Q ^ \\frac { H } { z } \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} & D ^ { i j } _ l ( t ) \\ \\ ( i , j = 1 , 2 , \\cdots , N ) , \\ \\ q _ l ( t ) \\in \\R ^ N \\ \\ ( l = 1 , 2 , \\cdots , m ) , \\\\ & F ( t , u _ 1 , \\cdots , u _ m ) = ( f _ 1 , f _ 2 , \\cdots , f _ m ) \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} v _ l = \\sum _ { k \\in S } \\frac { \\omega ( l , k ) } { \\lambda _ 0 - \\omega ( l , l ) } v _ k + \\sum _ { l _ 1 \\in \\overline S , l _ 1 \\neq l } \\frac { \\omega ( l , l _ 1 ) } { \\lambda _ 0 - \\omega ( l , l ) } v _ { l _ 1 } - \\frac { u _ l } { \\lambda _ 0 - \\omega ( l , l ) } . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\displaystyle X _ { n + 1 , x } = \\begin{cases} 0 , & \\mbox { i f } x = x _ n , \\\\ E _ x ( S _ n ) , & \\mbox { i f } x \\neq x _ n \\textrm { a n d } W _ { n , x } \\geq t ^ { \\ast } , \\\\ E _ x ( S _ n + t ^ { \\ast } ) + t ^ { \\ast } , & \\mbox { i f } x \\neq x _ n \\textrm { a n d } W _ { n , x } < t ^ { \\ast } . \\end{cases} \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} | \\left ( e _ { n _ 0 } , h \\right ) | = | \\left ( f _ m , k \\right ) | \\leq \\| f _ m \\| _ { L ^ { 2 } [ 0 , \\infty ) } \\| \\ , \\| k \\| _ { L ^ { 2 } [ 0 , \\infty ) } \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\overline { \\Gamma } _ { i j } ^ k = 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\overline { \\Gamma } _ { i j } ^ i = - \\frac { \\varphi _ { , x _ { j } } } { \\varphi } \\ \\ \\ \\ \\ \\ \\ \\overline { \\Gamma } _ { i i } ^ k = \\varepsilon _ { i } \\varepsilon _ { k } \\frac { \\varphi _ { , x _ { k } } } { \\varphi } \\ \\ \\ \\ \\ \\ \\ \\overline { \\Gamma } _ { i i } ^ i = - \\frac { \\varphi _ { , x _ { j } } } { \\varphi } . \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\sigma _ { h } ( x ) \\ , C ^ h _ { i j } = 0 , \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} D ^ \\xi _ X \\circ D ^ { \\mu ^ { T \\circ S } } _ X = D ^ { \\mu ^ S } _ X . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} ( 3 + t ) E ( u ; t ) + E _ * ( u ; t ) = ( 1 + t ) E ( u , t ) + 2 \\| A ^ { 1 / 2 } u \\| ^ 2 + 2 \\left \\| u ' + \\frac { 1 } { 2 } u \\right \\| ^ 2 + \\frac { 1 } { 2 } \\| u \\| ^ 2 , \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} w ( x ) = \\begin{cases} 0 . 5 & ( | x | > 1 ) , \\\\ 1 & ( - 1 < x < 1 ) , \\end{cases} \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} e \\bigg ( \\sum _ { i = - \\infty } ^ n a _ i t ^ i \\bigg ) = E ( a _ { - 1 } ) . \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\int \\mathcal { E } d x + \\frac { d } { d t } \\int \\left ( \\frac { 1 } { 2 } | \\psi _ y ( t , y ) | ^ 2 + \\frac { 1 } { 4 } | \\psi ( t , y ) | ^ 4 + \\alpha g ( v ( t , y ) ) h ( | \\psi ( t , y ) | ^ 2 ) \\right ) d y = 0 . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} p _ m ^ { } \\geq 0 , ~ ~ ~ ~ m = m _ 0 , m _ 0 + 1 , \\ldots , m _ 1 , \\sum _ { m = m _ 0 } ^ { m _ 1 } p _ m = 1 . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\varphi _ { f ^ { \\sharp } } ( s ) ~ = ~ \\varphi _ { f ^ { \\sharp } } ( t _ i ) ~ \\in ~ \\left \\{ 0 , 1 , 2 , \\dots , \\Gamma _ { [ N _ 1 , h _ 2 ] } - 1 \\right \\} \\quad \\forall s \\in I _ i , ~ i \\in \\overline { 0 , N _ 1 - 1 } ~ . \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} \\begin{array} { l } h \\Delta _ { g _ B } h + ( r - 1 ) | g r a d _ { g _ B } h | ^ 2 + \\rho h ^ 2 \\\\ = h \\Big \\{ \\varphi ^ 2 \\varepsilon _ { i _ 0 } \\left [ h '' - ( n - 2 ) \\frac { \\varphi ' h ' } { \\varphi } \\right ] + m \\varepsilon _ { i _ 0 } \\varphi ^ 2 \\frac { h ' f ' } { f } \\Big \\} + ( r - 1 ) \\varepsilon _ { i _ 0 } \\varphi ^ 2 ( h ' ) ^ 2 + \\rho h ^ 2 . \\end{array} \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\textrm { I M F } _ 1 = \\lim \\limits _ { m \\rightarrow \\infty } { \\mathcal { M } ^ m ( s ) ( x ) } = \\int _ { - \\infty } ^ { \\infty } \\widehat { s } ( \\xi ) \\chi _ { \\{ \\widehat { w } ( \\xi ) = 0 \\} } e ^ { 2 \\pi i \\xi x } \\textrm { d } \\xi \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\nu _ { \\omega + 2 } : = \\left [ \\nu _ { \\omega + 1 } ; \\nu _ { \\omega + 2 } ( \\phi _ { \\omega + 1 } ) = \\frac { 1 } { p } \\right ] . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } \\frac { \\dim \\ker Q ( h ) } { \\dim V _ h ( 1 ) } = r . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} { \\alpha } _ { a b } & = \\bar W _ { a L b L } \\ & \\underline { \\alpha } _ { a b } = & \\bar W _ { a \\underline { L } b \\underline { L } } \\\\ \\beta _ a & = \\bar W _ { a L \\underline { L } L } \\ & \\underline { \\beta } _ a = & \\bar W _ { a \\underline { L } \\underline { L } L } \\\\ \\rho & = \\bar W _ { \\underline { L } L \\underline { L } L } \\ & \\sigma = & \\epsilon ^ { a b } \\bar W _ { a b \\underline { L } L } . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\overline { r _ 1 } & < \\underset { t } { \\min } \\Big ( \\frac { b _ 1 ( t ) } { b _ 2 ( t ) } \\Big ) \\overline { r _ 2 } , \\ \\ \\overline { r _ 2 } < \\underset { t } { \\min } \\Big ( \\frac { a _ 2 ( t ) } { a _ 1 ( t ) } \\Big ) \\overline { r _ 1 } , \\\\ \\overline { r _ 1 } + \\overline { r _ 2 } & > \\underset { t } { \\max } \\Big ( \\frac { a _ 2 ( t ) } { a _ 1 ( t ) } \\Big ) \\overline { r _ 1 } , \\ \\ \\overline { r _ 1 } + \\overline { r _ 2 } > \\underset { t } { \\max } \\Big ( \\frac { b _ 1 ( t ) } { b _ 2 ( t ) } \\Big ) . \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\bar { { \\cal H } } _ { 0 } \\triangleq \\bigcup _ { n = 1 } ^ { \\infty } H _ { n } \\left ( ( \\mathbb { R } ^ { d } ) ^ { n } \\right ) \\subseteq \\bar { \\cal H } . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} T ^ { * * } = ( T ^ { * * } ) _ { \\rm r e g } + ( T ^ { * * } ) _ { \\rm s i n g } , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} m ( u ) = x = 0 & \\Leftrightarrow \\frac { L _ { X } ^ { \\prime } ( u ) } { L _ { X } ( u ) } = 0 \\\\ & \\Leftrightarrow L _ { X } ^ { \\prime } ( u ) = 0 \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\sum _ i \\partial _ a Y _ i \\partial _ b Y _ i - \\partial _ a Y _ 0 \\partial _ b Y _ 0 - \\partial _ a Y _ 4 \\partial _ b Y _ 4 = \\sigma _ { a b } . \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} ( \\partial _ { t } - \\underline { \\Delta } ) ( t F ) & = F + t ( \\partial _ { t } - \\underline { \\Delta } ) ( \\varphi \\phi ) \\\\ & = F - 2 t g ' ( \\overline { \\nabla } \\varphi , \\overline { \\nabla } \\phi ) + t \\varphi ( \\partial _ { t } - \\underline { \\Delta } ) \\phi + t \\phi ( \\partial _ { t } - \\underline { \\Delta } ) \\varphi . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} I = J \\sqcup K \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} K _ { n , s } ( z ) = \\dfrac { \\rho _ { n , s } ( z ) } { T _ n ^ { 2 s } ( z ) } , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\eta ( x ) = A d _ x ( r ) - r \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} T _ { 2 n + 1 } ( x ) = x W _ n ( 2 x ^ 2 - 1 ) \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} X ^ { 0 , u } _ x = \\mathcal { M } \\left ( S ( \\cdot ) x + \\int _ 0 ^ \\cdot S ( \\cdot - s ) G ( s , X ^ { 0 , u } _ x ( s ) ) u ( s ) d s \\right ) = \\mathcal { M } \\left ( S ( \\cdot ) x + \\mathcal { L } ( X ^ { 0 , u } _ x ) u \\right ) \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} & m _ 1 \\alpha _ 1 + m _ 2 \\alpha _ 2 + m _ 3 \\alpha _ 3 + m _ 4 \\alpha _ 4 = m , \\\\ & m _ 2 \\alpha _ 1 + m _ 1 \\alpha _ 2 + m _ 4 \\alpha _ 3 + m _ 3 \\alpha _ 4 = m , \\\\ & m _ 3 \\alpha _ 1 + m _ 4 \\alpha _ 2 + m _ 1 \\alpha _ 3 + m _ 2 \\alpha _ 4 = m , \\\\ & m _ 4 \\alpha _ 1 + m _ 3 \\alpha _ 2 + m _ 2 \\alpha _ 3 + m _ 1 \\alpha _ 4 = m . \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\delta _ { k _ \\alpha , k ' _ \\alpha } ( x , y ) = \\delta _ { k ' _ \\alpha , k _ \\alpha } ( x , y ) = 1 , ( x , y ) \\in \\Omega . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\mathbb P _ { N } ( \\eta , A ) : = \\mathbb P ( \\exists \\ ; \\sigma ^ { 1 } , \\sigma ^ { 2 } \\in \\mathcal L ( \\eta ) , R _ { 1 , 2 } \\in A ) . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} g _ { N , \\mathrm { e v e n } } ( y _ 1 , y ' ) = \\frac { a _ 0 } { 2 } + \\sum _ { k = 1 } ^ \\infty a _ k ( y ' ) \\cos ( k y _ 1 ) \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\tilde \\nabla ^ a D \\beta _ a = & 4 D \\rho \\\\ \\tilde \\nabla ^ a D ^ 2 \\beta _ a = & 5 D ^ 2 \\rho \\\\ \\tilde \\nabla ^ a ( D \\alpha _ { a b } ) = & 5 D \\beta _ b \\\\ \\tilde \\nabla ^ a ( D ^ 2 \\alpha _ { a b } ) = & 6 D ^ 2 \\beta _ b \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\chi _ m ^ { } ( x ) = \\begin{cases} 1 , & ~ ~ x \\in [ \\ , b ^ m , \\ , b ^ { m + 1 } \\ , ) , \\\\ 0 , & ~ ~ { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\| \\rho _ 3 f \\| _ { L ^ 2 ( 0 , T ; H _ 0 ^ 1 ( I ) ) } ^ 2 + \\| ( \\rho _ 3 ) _ t f \\| _ { L ^ 2 ( Q ) } ^ 2 \\leq & C \\ b ( ( \\hat { u } , \\hat { z } ^ 1 , \\hat { z } ^ 2 ) , ( \\hat { u } , \\hat { z } ^ 1 , \\hat { z } ^ 2 ) ) \\\\ : = & C \\left ( \\iint _ Q \\rho _ 0 ^ 2 | \\hat { y } | ^ 2 d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\rho _ 0 ^ 2 | \\hat { p } ^ i | ^ 2 d x d t \\right . \\\\ & + \\left . \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\rho _ 1 ^ 2 | \\hat { f } | ^ 2 d x d t \\right ) \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} c ^ 2 _ p ( x ) \\approx \\sum _ { k = 0 } ^ \\infty \\theta ^ 2 _ k \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 1 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = i ( R _ { 1 2 } i + R _ { 1 3 } ) = - R _ { 1 2 } + i R _ { 1 3 } = - 1 - 1 = - 2 = 2 u _ 1 ; \\\\ \\frac { R _ { 4 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 4 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = - R _ { 4 2 } + i R _ { 4 3 } = 0 - 2 i = - 2 i = 2 u _ 4 . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\begin{cases} { \\bf d } & : = ~ { \\bf d } ( E ) ~ ~ \\mathrm { t h e ~ d o u b l i n g ~ d i m e n s i o n ~ o f ~ } E , \\cr { \\bf p } & : = ~ { \\bf p } ( E ) ~ ~ \\mathrm { t h e ~ p a c k i n g ~ d i m e n s i o n ~ o f ~ } E . \\end{cases} \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\bar { m } ^ 2 _ s = \\frac { 1 } { p _ s } \\sum _ { \\ell \\in L _ s } m ^ 2 _ { \\ell , \\ell + s } ; \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\max _ { i \\in I _ { k } } \\left | b _ { k } - y _ { i } \\right | = R _ n \\sim | \\log \\varepsilon _ { n } | ^ { - \\frac { 1 } { s _ { k } } } \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\gamma _ 2 ( \\pi { G } ) = \\gamma _ 2 { G } / \\gamma _ c { G } \\pi { G } / \\gamma _ 2 ( \\pi { G } ) = ( G / \\gamma _ c { G } ) / ( \\gamma _ 2 { G } / \\gamma _ c { G } ) \\simeq G / \\gamma _ 2 { G } , \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F D F } } ^ { \\textrm { O F D M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { F } } \\right ) P _ \\textrm { F } { | h _ { \\textrm { B F } } | ^ 2 } } { { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } \\theta } , \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} L ( ( \\theta _ k / | \\xi | ) ^ \\gamma ) \\geq L ( \\theta _ k ^ \\gamma ) = L ( k L ( k ) ) , \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} K _ { i j } : = \\begin{cases} 1 i + j = n + 1 , \\\\ 0 , \\end{cases} \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} b _ { n + 1 } \\in ( A - c _ 1 ) \\cap \\cdots \\cap ( A - c _ n ) \\cap L = ( A - e _ { \\sigma ( j _ 1 ) } ) \\cap \\cdots \\cap ( A - e _ { \\sigma ( j _ n ) } ) \\cap L \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\varphi _ k ( v ) = ( a _ { k } + v | I _ { k } | ) ^ { \\frac { 1 } { 2 } - H } \\dot { \\bar { h } } _ { k } ( v ) . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} \\hat { p } ( t , x , \\xi ) = \\mathbb { E } \\left [ e ^ { 2 \\pi i \\langle \\xi , X _ t ^ x \\rangle } \\right ] , \\quad \\hat { p } _ l ^ \\eta ( t , x , \\xi ) = \\mathbb { E } \\left [ \\eta ( U _ t ) e ^ { 2 \\pi i \\langle \\xi , X _ l ( t , x ) \\rangle } \\right ] . \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\frac { L ' \\left ( \\chi _ { \\alpha } , \\frac { 1 } { 2 } + \\left ( c - \\frac { 1 } { 2 } + i t \\right ) \\right ) } { L \\left ( \\chi _ { \\alpha } , \\frac { 1 } { 2 } + \\left ( c - \\frac { 1 } { 2 } + i t \\right ) \\right ) } = \\frac { \\partial } { \\partial \\nu ' } R _ w ( \\nu ' ; \\nu ) \\Biggr | _ { \\nu ' = \\nu = c - 1 / 2 + i t } . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\mathcal { K } ^ { L } ( e _ 1 , e _ 2 ) = \\left [ \\frac { 1 } { 4 } ( \\overline { p } \\frac { l } { l _ L } ) ^ 2 - \\frac { 3 } { 4 } ( \\overline { q } \\frac { l } { l _ L } ) ^ 2 \\right ] L - \\frac { 3 } { 4 } L \\overline { r _ L } ^ 2 + 2 \\frac { l } { l _ L } \\overline { q } L ^ { \\frac { 1 } { 2 } } \\overline { r _ L } . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\overline { Q } _ { k , a } ( x ; q ) & = \\overline { Q } _ { k , a - 2 } ( x ; q ) + ( x q ) ^ { a - 1 } \\overline { Q } _ { k , k - a + 1 } ( x q ; q ) + ( x q ) ^ { a - 2 } \\overline { Q } _ { k , k - a + 2 } ( x q ; q ) \\\\ & \\qquad + ( x q ) ^ a \\overline { Q } _ { k , k - a } ( x q ; q ) + ( x q ) ^ { a - 1 } \\overline { Q } _ { k , k - a + 1 } ( x q ; q ) \\\\ & \\quad = \\overline { Q } _ { k , 0 } ( x ; q ) + \\sum _ { i = 1 } ^ a ( x q ) ^ i \\overline { Q } _ { k , k - i } ( x q ; q ) + \\sum _ { i = 0 } ^ { a - 1 } ( x q ) ^ i \\overline { Q } _ { k , k - i } ( x q ; q ) . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\ln | | A _ k | | } { | k | } = \\ln | \\lambda | . \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\frac { p _ 0 } { p _ { N + 1 } } = \\frac { 1 } { \\Lambda _ 0 \\cdots \\Lambda _ N } \\left ( - \\frac { Q } { ( q - 1 ) ^ { N + 1 } } + \\prod _ { j = 0 } ^ N \\frac { 1 - \\Lambda _ j } { 1 - q } \\right ) \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\gamma ( x ) = \\alpha ( x _ 1 , S ( x _ 2 ) ) . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) = - H ( \\ell _ 2 x ) . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\chi _ { P , a , e f f } ^ { 1 } \\left ( \\mathbf { V } ; \\mathcal { G } \\right ) = \\psi _ { P , a } \\left ( \\mathbf { O } ; \\mathcal { G } \\right ) P \\in \\mathcal { M } ( \\mathcal { G } ) . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\Delta _ { R C - B E } = \\frac { k } { q } \\left ( \\frac { 3 } { 2 } + \\frac { 1 - q } { k } \\right ) \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\int _ { r \\le | x | } \\theta ( z , t ) d z \\le \\frac { 1 } { r } \\int _ { r \\le | x | } { | x | } \\theta ( z , t ) d z \\le \\frac { C ( d _ { \\theta _ 0 } , \\| \\theta _ 0 \\| _ { L ^ \\infty { ( S ) } } ) } { r } \\mbox { f o r a n y } r > 0 \\mbox { f o r a n y } t \\ge 0 . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\mathbf { M } ( g ) ( V ^ { ( 0 ) } ) = \\lim _ { r \\rightarrow \\infty } \\tilde { \\mathfrak { m } } _ H ( S _ r ) \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} P _ i ( \\theta ) = A _ i \\cos ( \\theta ) + B _ i \\sin ( \\theta ) + C _ i \\ ; , \\ ; \\ ; i = 1 , 2 \\ ; , \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { d ( y _ \\varepsilon , \\partial \\Omega ) } { \\nu _ \\varepsilon } = + \\infty . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\varphi ( 0 ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\varphi ( 1 ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\eth ( \\varphi ( t ) , ( Y _ 1 , \\ldots , Y _ m ) ) \\leq \\varepsilon , \\end{array} \\right . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} f ( x , t ) : = | x | ^ { - \\tau - \\sigma } t ^ { \\frac { m + 2 - p } { 2 } } \\ \\mathrm { f o r } \\ x \\in \\mathbb R ^ N \\setminus \\{ 0 \\} \\ \\mathrm { a n d } \\ h _ j ( t ) : = \\left \\{ \\begin{aligned} & t ^ q & & \\mathrm { i f } \\ j = 1 , & \\\\ & - C _ 0 ^ { - k } e ^ { - k t } & & \\mathrm { i f } \\ j = 2 . & \\end{aligned} \\right . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} I _ h \\Big ( \\lambda _ n ^ { N , h } \\phi _ { n } ^ { N , h } - \\mathcal { R } _ N \\phi _ { n } ^ { N , h } \\Big ) = 0 . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial x _ { 1 } } { \\partial y } = \\left ( \\frac { \\partial y } { \\partial x _ { 1 } } \\right ) ^ { - 1 } , \\\\ \\frac { \\partial x _ { 1 } } { \\partial z } = - \\left ( \\frac { \\partial y } { \\partial x _ { 1 } } \\right ) ^ { - 1 } \\cdot \\frac { \\partial y } { \\partial x _ { 2 } } , \\\\ \\frac { \\partial x _ { 2 } } { \\partial y } = 0 , \\\\ \\frac { \\partial x _ { 2 } } { \\partial z } = \\mathrm { I } _ { m - n } . \\end{cases} \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} P \\left ( Y _ j = y \\right ) = \\frac { \\pi _ j ^ y } { y ! } G _ { \\vert \\boldsymbol { Y } \\vert } ^ { ( y ) } ( 1 - \\pi _ j ) , \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} C _ \\delta = 2 \\hat h ^ 2 - 2 \\hat h + 1 = 2 \\left ( \\hat h - \\frac 1 2 \\right ) ^ 2 + \\frac 1 2 \\le 2 \\left ( - \\frac 3 2 + \\sqrt { 1 + \\frac { 2 } \\delta } \\right ) ^ 2 + \\frac 1 2 = \\frac 4 \\delta + 7 - 6 \\sqrt { 1 + \\frac 2 \\delta } . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} p _ { } = \\inf _ { y \\in \\mathbb R ^ n \\times [ 0 , T ] } p ( y ) > 1 ~ p _ { } = \\sup _ { y \\in \\mathbb R ^ n \\times [ 0 , T ] } p ( y ) < \\infty . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\mathbf { w } _ { s , t } ^ { n } = \\int _ { s < u _ { 1 } < \\cdots < u _ { n } < t } d w _ { u _ { 1 } } \\otimes \\cdots \\otimes d w _ { u _ { n } } . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} \\frac { 1 + \\alpha } { \\pi } \\int _ 0 ^ 1 \\d r \\ , r ^ { 2 p + 1 } \\frac { r ^ { 2 l } } { 2 ^ { 2 l } } ( 1 - r ^ 2 ) ^ { \\alpha } = \\frac { \\Gamma ( 2 + \\alpha ) \\Gamma ( 1 + p + l ) } { 2 ^ { 2 l + 1 } \\pi \\Gamma ( 2 + \\alpha + p + l ) } . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} \\sigma ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) A ( \\sigma ) , \\mu ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) A ( \\mu ) , \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\mathbf { y } ^ { m } _ { \\mathcal { Q } _ p } = \\mathcal { Q } \\left ( \\mathbf { X } ^ H _ p \\mathbf { h } ^ { m } + \\mathbf { n } _ p ^ { m } \\right ) , \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\gamma ' = \\sigma ( \\gamma ) ^ { - 1 } . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} H _ n = \\begin{bmatrix} H _ { n - 1 } & H _ { n - 1 } \\\\ H _ { n - 1 } & - H _ { n - 1 } \\\\ \\end{bmatrix} = H _ 1 \\otimes H _ { n - 1 } = H _ 1 ^ { \\otimes n } , \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} \\mathbf C _ s ^ { \\ , i } = \\frac { a _ i \\beta + b _ i } { n \\ln 2 } , \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\beta ' _ n ( a ^ 2 , q ^ 2 ) = \\frac { q ^ { - n } } { ( - a ; q ) _ { 2 n } } \\sum _ { n \\ge j \\ge 0 } \\frac { ( - 1 ) ^ { n - j } q ^ { ( n - j ) ^ 2 - ( n - j ) } } { ( q ^ 2 ; q ^ 2 ) _ { n - j } } \\beta _ j ( a , q ) . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\# \\{ n = 1 , \\ldots , M | T ^ n x \\in \\Omega _ { N } ( E ) \\} < M ^ { 1 - \\delta } , \\delta > 0 , \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} I I I '' & = \\sum _ { n = p - 2 } ^ { p + s - 5 } q ^ n I I I '' _ 1 + \\sum _ { n = s - 2 } ^ { p - 3 } q ^ n I I I '' _ 2 + \\sum _ { n = 0 } ^ { s - 3 } q ^ n \\frac 1 6 ( n + 1 ) ( n + 2 ) ( n + 3 ) , \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} C _ 1 b _ n = N \\left ( C _ 1 r _ n \\sqrt { \\frac { n } { N } } \\right ) \\leq \\mathbb { E } V _ n \\leq N \\left ( C _ 2 r _ n \\sqrt { \\frac { n } { N } } \\right ) = C _ 2 b _ n \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & y ^ s _ t - ( a ( y _ x ^ s , t , x ) y ^ s _ x ) _ x + F ( y ^ s , y _ x ^ s ) = f { 1 } _ \\mathcal { O } + ( v ^ 1 + s w ^ 1 ) { 1 } _ { \\mathcal { O } _ 1 } + v ^ 2 1 _ { \\mathcal { O } _ 1 } \\ \\ Q , \\\\ & y ^ s ( 0 , t ) = y ^ s ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & y ^ s ( 0 ) = y ^ { 0 } \\ \\ \\ \\ I , \\end{array} \\right . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} M ( f ; P ) = \\sup _ { t > 0 } | ( P _ { t } \\ast f ) ( x ) | \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\int _ 0 ^ t e ^ { - ( t - s ) A } \\Big ( \\frac { d } { d t } V _ { m } ^ { ( 2 ) } ( s ) \\Big ) \\ , d s & = ( - 1 ) ^ { m } \\sum _ { k = 1 } ^ { m } \\begin{pmatrix} m \\\\ k \\end{pmatrix} \\frac { ( - t A ) ^ { k } } { k ! } e ^ { - t A } u _ 1 \\\\ & = - V _ { m + 1 } ^ { ( 1 ) } ( t ) + ( - 1 ) ^ { m + 1 } e ^ { - t A } u _ 1 . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} T _ 1 = ( I - Q ) T T _ 2 = Q T . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} P ( B _ 2 ^ { ( n ) } ) & \\leq \\sum _ { j = \\lceil K \\rceil } ^ { J _ n - 1 } P \\left ( \\bigcup _ { t \\in [ \\theta _ j ^ { ( n ) } , \\theta _ { j + 1 } ^ { ( n ) } ] \\cap \\mathbb N } \\{ S _ n ' ( t ) \\leq t \\} \\right ) \\leq \\sum _ { j = \\lceil K \\rceil } ^ { J _ n - 1 } P \\left ( S _ n ' ( \\theta _ j ^ { ( n ) } ) \\leq \\theta _ { j + 1 } ^ { ( n ) } \\right ) , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} ( J - \\lambda I ) v _ \\tau = - \\tau P _ 1 L ^ * L P _ 1 v _ \\tau \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\sigma ^ 2 ( c ) & = V ( c , c ) + 2 \\sum _ { k = 1 } ^ { 3 6 } \\sum _ { j = 0 } ^ \\infty \\frac 1 { 6 ^ { k + 3 6 j } } V \\bigl ( \\langle 2 ^ { k + 3 6 j } c \\rangle , \\langle 3 ^ { k + 3 6 j } c \\rangle \\bigr ) \\\\ & = V ( c , c ) + \\frac { 2 \\cdot 6 ^ { 3 6 } } { 6 ^ { 3 6 } - 1 } \\sum _ { k = 1 } ^ { 3 6 } \\frac 1 { 6 ^ { k } } V \\bigl ( \\langle 2 ^ { k } c \\rangle , \\langle 3 ^ { k } c \\rangle \\bigr ) \\\\ & = \\frac { 1 2 2 2 6 8 5 8 0 7 0 5 0 4 6 7 5 5 8 6 4 5 7 8 2 5 4 7 1 6 3 4 9 2 } { 4 5 6 1 2 9 6 5 0 6 5 1 2 4 7 7 0 8 1 9 0 5 8 9 0 1 7 0 8 4 7 3 7 5 } . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} ( X ; S _ { l - 1 } , T _ { l - 1 } ) ^ { g _ r } = \\left ( ( \\overline g ) ^ { - 1 } X ; ( ( g ) ^ { - 1 } S ) _ { l - 1 } , ( ( g ) ^ { - 1 } T + \\nu ( X , g ) ) _ { l - 1 } \\right ) . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\lambda & = \\lambda ^ 2 + \\mu ^ 2 c \\frac { l + 2 } { 2 } , \\\\ \\mu & = 2 \\lambda \\mu a d + \\mu ^ { 2 } b l . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} A ( m , n , a , \\epsilon ) : = \\left \\{ ( i _ 1 , \\dots , i _ n ) \\in \\mathbb { N } ^ n : \\ \\sum _ { k = 1 } ^ n i _ k ^ a \\in [ m , m ( 1 + \\epsilon ) ] \\right \\} . \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} f ( n ) \\coloneqq \\begin{cases} ( - 1 ) ^ n , & n \\in \\left [ N _ k , \\big \\lfloor \\tfrac { N _ { k + 1 } } { 2 } \\big \\rfloor \\right ) \\\\ \\\\ - ( - 1 ) ^ { n } , & n \\in \\left [ \\big \\lfloor \\tfrac { N _ { k + 1 } } { 2 } \\big \\rfloor , N _ { k + 1 } \\right ) \\end{cases} \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} d = 4 \\begin{cases} u \\approx ( n ^ { \\prime } ) ^ 2 \\big ( \\frac { 1 } { 2 } p _ 1 + ( k - 1 ) p _ 2 + \\frac { k - 1 } { 2 } p _ 3 + \\binom { k - 1 } { 2 } p _ 4 \\big ) \\\\ v \\approx ( n ^ { \\prime } ) ^ 2 \\big ( p _ 2 + p _ 3 + \\frac { 5 ( k - 2 ) } { 2 } p _ 4 + \\binom { k - 2 } { 2 } p _ 5 \\big ) \\end{cases} \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} H e s s _ { \\widetilde { g } } h ( X , Y ) = 0 , \\ \\ \\ \\ \\forall X \\in \\mathcal { L } ( B ) , Y \\in \\mathcal { L } ( F ) , \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\left ( \\frac { 1 - q } { z } \\right ) ^ i \\sum _ { \\substack { a + b = i \\\\ 0 \\leq a , b \\leq N } } ( - 1 ) ^ a \\binom { \\ell _ q \\left ( Q \\right ) } { a } J _ b \\left ( q , \\left ( \\frac { 1 - q } { z } \\right ) ^ { N + 1 } Q \\right ) \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} { \\bf a } _ t ( \\theta ) = [ 1 , \\exp ^ { i k d \\sin ( \\theta ) } , \\cdots , \\exp ^ { i k d ( N - 1 ) \\sin ( \\theta ) } ] \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\rho \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } _ { | P = \\Lambda _ i } ( q , Q ) \\right ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { ( q ; q ) _ d ^ { N + 1 } } = \\widetilde { J _ 0 } ( q , Q ) \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} ( \\alpha \\cdot A ) ^ 2 = ( A \\cdot A ) \\mathbb { I } _ N , A \\in \\C ^ n . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} d ( e _ { x _ 1 } \\cdots e _ { x _ n } ) = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } e _ { x _ 1 } \\cdots e _ { x _ { i - 1 } } d ( e _ { x _ i } ) e _ { x _ { i + 1 } } \\cdots e _ { x _ n } \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} A \\Omega _ A \\to L \\eta _ \\xi \\varphi _ \\ast A \\Omega _ A = \\varphi _ \\ast L \\eta _ { \\tilde \\xi } A \\Omega _ A \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} \\bigcap _ { \\delta > 0 } \\overline { g _ t ( F _ { t + \\delta } \\setminus F _ t ) } = \\{ \\xi ( t ) \\} \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\begin{array} { l } \\Delta v = ( 1 - v ) ^ p \\Omega ' : = M ^ { \\frac { p - 1 } { 2 } } \\Omega \\subseteq B _ { M ^ { ( p - 1 ) / 2 } } \\\\ v \\geq 0 v ( 0 ) = 0 . \\end{array} \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\mathcal { B } _ { [ 0 , 1 ) } = \\mathcal { D } _ { m _ 0 } . \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} E ( u ( t , \\cdot ) ) = E ( u ( 0 , \\cdot ) ) \\mbox { f o r a l l } t \\in [ 0 , T _ 0 ) . \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} A _ \\gamma : = \\{ x \\in K : v ( x ) = \\gamma \\wedge v ( f ( x ) ) = h ( \\gamma ) \\} . \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\begin{cases} \\frac { | \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) | ^ 2 } { 2 } + V ( x , y ) = \\ln \\widetilde { m } _ 1 ( x , y ) + \\widehat { H } ( x , \\widetilde { \\Lambda } ) , \\\\ - \\div _ y \\big ( \\widetilde { m } _ 1 ( x , y ) ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) \\big ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} L ^ q ( U ' ; C ( U _ 3 ) ) : = \\{ f \\in L ^ q _ H L ^ \\infty _ z ( U ) : f ( x ' , \\cdot ) \\in C ( U _ 3 ) x ' \\in U ' \\} , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} z _ i = \\sum _ { j = 1 } ^ { m } g _ j \\left ( x _ { i - j + 1 } , y _ { i - j + 1 } \\right ) , \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} K _ { 0 } \\underset { \\omega _ { 1 } } { \\underbrace { \\left ( \\frac { 1 } { f _ { 1 } f _ { 1 } ^ { \\prime \\prime } } \\right ) ^ { \\prime } } } \\left ( \\frac { 1 } { f _ { 2 } ^ { \\prime } } \\right ) ^ { 2 } + \\underset { \\omega _ { 2 } } { \\underbrace { \\left ( \\frac { \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } } { f _ { 1 } f _ { 1 } ^ { \\prime \\prime } } \\right ) ^ { \\prime } } } = 0 . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} p _ f ~ : = ~ \\max _ { w \\in I _ f } p _ w ~ . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} a _ { v , j } : = a ( \\mu _ { \\imath _ v } , j ) : = - \\mu _ { \\imath _ v , n - j + 1 } + \\tfrac { n + 1 } { 2 } - j \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} G _ x ( y ) = \\frac { 1 } { 4 \\pi } \\left ( \\log \\frac { 1 } { | x - y | ^ 2 } + \\mathcal { H } _ x ( y ) \\right ) \\ , , \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\Psi _ f ( ( g _ 1 , g _ 2 , \\cdots , g _ n ) ) & = e ( f _ 1 g _ 1 + f _ 2 g _ 2 + \\cdots , f _ n g _ n ) \\\\ & = \\prod _ { i = 1 } ^ n e ( f _ i g _ i ) \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ i ) = \\sum _ { \\substack { n \\geq 0 \\\\ d \\in H _ 2 ( X ; \\mathbb { Z } ) } } \\frac { 1 } { n ! } \\langle \\tau , \\dots , \\tau \\rangle ^ \\textnormal { c o h } _ { 0 , n , d } \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} F ( c _ 1 , \\cdots , c _ n ; \\mu ) : = f ( c _ 1 , \\cdots , c _ n ) - \\mu ( \\sum \\limits _ { i = 1 } ^ n c _ i ^ 2 - 1 ) \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\widetilde { b } ( x , \\widetilde { \\Lambda } ) = \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ 1 ( x , y ) ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) d y \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} g _ q ( \\varsigma ) = \\mu _ q ^ { \\frac { \\alpha } { 2 - q } } f _ q ( \\xi _ q \\delta _ { \\mu _ q } ( \\varsigma ) ) , \\ , \\ \\ , \\ , \\mbox { f o r } \\ , \\ , \\varsigma \\in \\Omega _ q . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\chi ( \\Gamma , M ) = \\prod _ i | H _ i ( \\Gamma , M ) | ^ { ( - 1 ) ^ i } \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} G _ \\tau ( \\phi _ i \\star _ \\tau \\phi _ j , \\phi _ k ) = G _ \\tau ( \\phi _ i , \\phi _ j \\star _ \\tau \\phi _ k ) \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} - \\Phi _ 1 \\left ( \\frac { 1 } { 2 } , L _ 0 \\right ) r _ 0 L _ 0 + \\frac { \\alpha } { 2 } = - d . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} H ^ { p ^ { \\ast } } _ { , i } = \\sum _ k h ^ { p ^ { \\ast } } _ { i k } \\langle X , e _ k \\rangle , i , p = 1 , 2 . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} \\Phi _ n ^ { ( k ) } ( z ) = \\sum _ { i = 0 } ^ { k } q _ { n - i } ( z ) p _ { n - i } ( z ; a , b | q ) , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} P ( z ) = \\{ x \\in \\R ^ n : \\ , \\langle x , u _ i \\rangle \\leq z _ i \\mbox { f o r } i = 1 , \\ldots , k \\} . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} f ( p ^ { a _ { 3 } } ) = 1 + \\sum _ { i = 1 } ^ { a _ { 1 } } \\frac { p ^ { 3 i } p ^ { \\min \\{ i - 1 , a _ { 3 } - i - 1 \\} } ( p - 1 ) } { p ^ { 2 i } p ^ { \\min \\{ 2 i , 2 a _ { 3 } - 2 i \\} } } + \\sum _ { i = a _ { 1 } + 1 } ^ { a _ { 2 } } \\frac { p ^ { 2 i } p ^ { \\min \\{ i - 1 , a _ { 3 } - i - 1 \\} } ( p - 1 ) } { p ^ { 2 i } p ^ { \\min \\{ 2 i , 2 a _ { 3 } - 2 i \\} } } + \\sum _ { i = a _ { 2 } + 1 } ^ { a _ { 3 } } \\frac { p ^ { i } p ^ { \\min \\{ i - 1 , a _ { 3 } - i - 1 \\} } ( p - 1 ) } { p ^ { 2 i } p ^ { \\min \\{ 2 i , 2 a _ { 3 } - 2 i \\} } } . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\textbf { h } ' _ k = \\textbf { B } _ { k } ^ { \\rm H } \\textit { \\textbf { h } } _ k , \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { \\log ( 1 - \\beta ^ { 1 / x } ) } { ( 1 / x ) ^ q } = 0 , \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\mathfrak { A } ^ + _ { M _ S , \\mathbb { Q } } = \\{ x \\in \\mathfrak { A } _ { \\mathbb { Q } } : \\langle x , \\alpha \\rangle = 0 , \\alpha \\in S , \\langle x , \\alpha \\rangle > 0 , \\alpha \\in \\Delta \\setminus S \\} , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} A _ 1 = q \\widetilde { m } ^ { q - 1 } \\widetilde { m } _ { y _ j } \\widetilde { m } _ { y _ i } K ^ i K ^ j = q \\widetilde { m } ^ { q - 1 } \\big | ( \\nabla _ y \\widetilde { m } ) ^ T K \\big | ^ 2 , \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\int _ \\Omega ( - \\psi ) d \\mu & = \\sum _ { j \\geq 1 } \\int _ \\Omega ( 1 + \\gamma _ \\Omega ) d d ^ c u _ { j , k _ j } \\\\ & = \\sum _ { j \\geq 1 } \\int _ { \\Omega \\cap \\{ \\gamma _ \\Omega < - 1 + \\frac { 1 } { 2 ^ { j } } \\} } ( 1 + \\gamma _ \\Omega ) d d ^ c u _ { j , k _ j } \\\\ & \\leq \\sum _ { j \\geq 1 } \\frac { 1 } { 2 ^ { j } } \\int _ { \\Omega ' } d d ^ c u _ { j , k _ j } \\leq 1 . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} | \\nabla \\tilde { w } _ \\varepsilon ( \\tilde { z } _ \\varepsilon ) | = \\frac { | \\nabla u _ \\varepsilon ( z _ \\varepsilon ) | } { | \\nabla u _ \\varepsilon ( y _ \\varepsilon ) | } \\le \\frac { u _ \\varepsilon ( y _ \\varepsilon ) } { u _ \\varepsilon ( z _ \\varepsilon ) } \\frac { 1 } { | \\tilde { x } _ \\varepsilon - \\tilde { z } _ \\varepsilon | } \\le \\frac { 1 + o ( 1 ) } { | \\tilde { x } _ \\varepsilon - \\tilde { z } _ \\varepsilon | } \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} r : = \\lfloor \\log ^ { \\eta _ 0 } x \\rfloor , \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} U _ { t o t } = U _ { t o t } ( n ) : = \\bigcap _ { l = 1 } ^ { N } U _ l \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\partial _ t ( \\varphi ^ 1 - \\varphi ^ 2 ) - \\Delta ( \\mu ^ 1 - \\mu ^ 2 ) & = \\mathcal P ( \\sigma ^ 1 - \\sigma ^ 2 ) h ( \\varphi ^ 1 ) + ( \\mathcal P \\sigma ^ 1 - a - \\alpha u ) \\left ( h ( \\varphi ^ 1 ) - h ( \\varphi ^ 2 ) \\right ) \\\\ \\mu ^ 1 - \\mu ^ 2 & = - \\Delta ( \\varphi ^ 1 - \\varphi ^ 2 ) + \\psi ' ( \\varphi ^ 1 ) - \\psi ' ( \\varphi ^ 2 ) . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\Theta ^ n _ \\mu ( Q ) : = \\frac { \\mu ( Q ) } { l ( Q ) ^ n } \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\theta _ q ( Q ) = \\sum _ { d \\in \\mathbb { Z } } q ^ \\frac { d ( d - 1 ) } { 2 } Q ^ d \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - u _ { x x } + \\frac { \\mu } { 1 + t } u _ t + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } u = f ( t , x ) , & x \\in \\mathbb { R } , \\ t > 0 , \\\\ u ( 0 , x ) = 0 , & x \\in \\mathbb { R } , \\\\ u _ t ( 0 , x ) = 0 , & x \\in \\mathbb { R } , \\end{cases} \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} \\lVert V \\rVert _ { \\mathcal { S } ( T ) } = \\max \\left \\{ \\sup _ { 0 < t < T } \\lVert V ( t ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } , \\sup _ { 0 < t < T } t ^ { 1 / 2 } \\lVert \\nabla V ( t ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } \\right \\} \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\rho ( z ) = C & z \\in b \\Delta _ 1 = \\bigcup _ { a = 1 } ^ k b T _ { a , 1 } \\medskip \\\\ \\rho ( z ) = 1 & z \\in b \\Delta _ 2 = \\bigcup _ { a = 1 } ^ k b T _ { a , 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\leq | | 2 \\theta + ( 2 \\sum _ { i = 0 } ^ { s } ( - 1 ) ^ { i + 1 } b _ { { j _ i } } ^ \\prime ) \\alpha + ( - 1 ) ^ { s + 1 } k \\alpha | | _ { \\R / \\Z } + 2 \\sum _ { i = 0 } ^ { s } e ^ { - ( \\varsigma + \\varepsilon ) | K _ { j _ i } | } \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { p , 1 } ^ { \\gamma _ 3 } } \\le C ( 1 + t ) ^ { - \\frac { ( \\frac n 2 - \\sigma - 1 ) ( 1 - \\theta _ { 4 } ) } { 2 } } = C ( 1 + t ) ^ { - \\frac { n } { 2 } ( \\frac 1 2 - \\frac 1 p ) - \\frac { \\gamma _ 3 - \\sigma - 1 } { 2 } } . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} F ( 0 ) = 0 ; F ; \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} f ^ I _ { \\phi , \\theta } = \\begin{pmatrix} \\cos ( \\phi ) \\\\ \\sin ( \\phi ) e ^ { i \\theta } \\end{pmatrix} \\in \\R \\times \\C \\ , , \\phi , \\theta \\in [ 0 , 2 \\pi ) \\ , , \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} t ^ { \\frac { s } { n } } f ^ { * * } ( t ) = \\frac { n } { n - s } ( b _ n ) ^ { \\frac { s } { n } } , \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\kappa _ { t } = \\theta _ { t } Z _ { u = t , 1 } ^ { \\ast } Y \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} \\langle H _ t , K \\rangle = \\langle h ( - t + \\cdot ) , K \\rangle + \\int _ 0 ^ t F ( \\psi _ 1 , \\psi _ 2 ) ( t - x ) K ( x ) d x = \\langle h , K ( t + \\cdot ) \\rangle + \\psi _ 2 ( t ) . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\mathcal { E } ( \\tilde { L } ' _ \\tau ) \\geq \\mathcal { E } ( L ) = \\frac { 2 \\pi } { e } . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} { A ( \\gamma _ \\varepsilon ) } = o \\left ( \\frac { 1 } { \\gamma _ \\varepsilon ^ 2 } \\right ) \\ , , \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\eta _ \\varepsilon = I _ { \\bar { x } _ 0 } ( \\gamma _ \\varepsilon ) - I _ { x _ \\varepsilon } ( \\gamma _ \\varepsilon ) + o ( \\check { \\zeta } _ \\varepsilon ) \\ , , \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} v a r _ k ( P ) : = \\sup _ { \\omega _ { - \\infty } ^ { - 1 } , \\sigma _ { - \\infty } ^ { - 1 } \\in \\Omega < 0 , \\atop \\omega _ { - k } ^ { - 1 } = \\sigma _ { - k } ^ { - 1 } } \\sup _ { a \\in A } | P ( a \\mid \\omega _ { - 1 } ^ { - \\infty } ) - P ( a \\mid \\sigma _ { - 1 } ^ { - \\infty } ) | \\rightarrow 0 k \\rightarrow \\infty . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { p ^ r - 1 } { 2 } } \\frac { ( 6 k + 1 ) ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 4 ^ k } \\equiv ( - 1 ) ^ \\frac { p - 1 } { 2 } p \\sum _ { k = 0 } ^ { \\frac { p ^ { r - 1 } - 1 } { 2 } } \\frac { ( 6 k + 1 ) ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 4 ^ k } \\pmod { p ^ { 4 r } } \\quad . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\begin{aligned} D _ { k _ { i i } } & = 0 & D _ { k _ { i j } } & = D _ { k _ { j i } } & \\textstyle \\sum _ i D _ { k _ { i j } } & = 0 \\\\ D _ { c _ { i i } } & = 0 & & & \\textstyle \\sum _ i D _ { c _ { i j } } & = 0 \\\\ D _ { e _ { i i } } & = 0 & D _ { e _ { i j } } & = D _ { e _ { j i } } & & \\end{aligned} \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\delta _ 0 ^ s ( \\ell ) + \\ell \\delta _ 1 ^ s ( \\ell ) + \\frac { \\ell ^ 2 } { 2 ( \\ell ^ 2 - 1 ) } = \\frac { 1 } { 2 } d _ K ( \\ell ) . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\left [ b ^ 2 l ^ 2 + 2 a ^ { 2 } c d ^ 2 ( l + 2 ) \\right ] \\mu ^ { 2 } + 2 b l ( a d - 1 ) \\ , \\mu + 1 - 2 a d = 0 \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} M ^ { m + 1 } = \\begin{bmatrix} G ^ 1 & M ^ m \\\\ G ^ 2 & M ^ m \\\\ \\vdots & \\vdots \\\\ G ^ n & M ^ m \\end{bmatrix} . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\mu = \\lim _ { i \\to \\infty } ( 2 - p _ i ) | \\alpha _ i | ^ { p _ i } d v _ g + | \\bar { h } | ^ 2 d v _ g . \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} K ( \\partial _ t ) f = \\left ( \\frac { d } { d t } \\right ) ^ m K _ m ( \\partial _ t ) f , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} B _ { { \\bf L } ^ 1 ( I , E ) } ( \\varphi , r ) ~ = ~ \\left \\{ g \\in \\mathbf { L } ^ { 1 } ( I , E ) ~ \\big | ~ \\rho _ { { \\bf L } ^ 1 } ( \\varphi , g ) < r \\right \\} ; \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} f _ { r } \\left ( u ( 0 ) \\right ) - f _ { l } \\left ( u ( 0 ) \\right ) = \\varepsilon \\left ( u _ { x } ( 0 + ) - u _ { x } ( 0 - ) \\right ) . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} ( M _ n h ) ( t ) = c _ 2 \\min \\left ( 4 ^ { - 2 n } \\int _ 0 ^ t h ( \\tau ) \\cdot ( h ( \\tau ) + e ^ { - { \\frac { 1 } { 2 } } 4 ^ n } ) d \\tau , 4 ^ { - ( n + 1 ) } \\right ) \\mbox { f o r } t \\ge 0 \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} T _ q = \\sum _ { r \\mod { q } } | T _ q ( r ) | . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} c ( x , y ) = \\inf _ { \\gamma \\in \\Gamma } \\int _ 0 ^ T V ( \\gamma ( t ) ) \\ , d t , \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { r , \\alpha } ( \\bar O ) } : = \\left \\{ \\begin{array} { l l } \\sum _ { j = 0 } ^ { r } [ u ] _ { j , 0 , O } , & \\hbox { i f } \\ , \\alpha = 0 ; \\\\ \\| u \\| _ { C ^ { r , 0 } ( \\bar O ) } + [ u ] _ { r , \\alpha ; O } , & \\hbox { i f } \\ , \\alpha > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} r _ \\varepsilon | \\bar { w } ' _ \\varepsilon ( r _ \\varepsilon ) | = & O \\left ( \\| \\bar { w } ' _ \\varepsilon \\| _ { L ^ \\infty ( [ 0 , r _ \\varepsilon ] ) } \\frac { { \\mu _ \\varepsilon ( r _ \\varepsilon / \\mu _ \\varepsilon ) ^ 3 } } { 1 + ( r _ \\varepsilon / \\mu _ \\varepsilon ) ^ 3 } \\right ) + o \\left ( \\frac { ( r _ \\varepsilon / \\mu _ \\varepsilon ) ^ 2 } { 1 + ( r _ \\varepsilon / \\mu _ \\varepsilon ) ^ 2 } \\right ) \\ , . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} s = \\sum \\limits _ { k = 1 } ^ { a - 4 } m _ { k } ( a + k d ) + m _ { a - 3 } b . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} d _ 1 ( v , u ) \\geq d _ 1 ( v , \\max ( v , P ( u , \\varphi ) ) ) \\geq d _ 1 ( P ( u , \\varphi ) , P ( P ( u , \\varphi ) , v ) ) = d _ 1 ( P ( u , \\varphi ) , P ( v , \\varphi ) ) . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} H _ { ( n , s ) , ( n ' , s ' ) } ( \\omega , \\theta ) = \\lambda v _ s ( \\theta + n \\omega ) \\delta _ { n n ' } \\delta _ { s s ' } + \\Delta , \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} \\partial _ { t _ i } \\mathcal { F } ( \\tau , Q ) = \\sum _ { \\substack { d \\in H _ 2 ( X ; \\mathbb { Z } ) \\\\ n \\geq 0 } } \\frac { 1 } { n ! } \\langle T _ i , \\tau , \\dots , \\tau \\rangle ^ \\textnormal { c o h } _ { 0 , n , d } Q ^ d \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} A _ { E , m , \\xi } ( t ) & = \\frac { \\kappa _ { n - 1 } } { \\kappa _ m } \\int _ { G ( \\xi ^ \\perp , m ) } V _ m ( ( E - t \\xi ) \\cap H ) \\ , d H \\\\ & = \\frac { \\kappa _ { n - 1 } } { \\kappa _ m } \\int _ { G ( \\xi ^ \\perp , m ) } V _ m ( E \\cap ( H \\vee \\xi ) \\cap \\{ \\xi ^ \\perp + t \\xi \\} ) \\ , d H , \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} \\sum _ { x \\in X _ u } f ( x ) \\geq \\sum _ { x \\in X _ u } ( r - d _ { x , u } ) = r | X _ u | - \\sum _ { x \\in X _ u } d _ { x , u } = d _ G ( u ) - \\sum _ { x \\in X _ u } d _ { x , u } \\geq d _ B ( u ) . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\begin{aligned} & | \\dot { C } - C W ' ( A ) | \\leq \\kappa ^ 2 h ^ 2 e ^ { 2 \\mu S } \\\\ & | \\dot { A } - C + W ( A ) + \\frac 1 2 C ^ { - 2 } h ^ 2 W '' ( A ) | \\leq \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu S } \\end{aligned} \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\bar L _ j : = X _ j + \\sqrt { - 1 } J X _ j , ~ ~ j = 1 , \\ldots , n \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} ( A ) _ { i k } = \\Phi ( \\boldsymbol { x } _ i , \\boldsymbol { x } _ k ) , i , k = 1 , \\ldots , N , \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} x ^ * = [ - 0 . 6 5 4 6 6 4 1 7 1 6 0 3 \\ { - 1 . 8 6 9 5 1 6 0 0 7 1 1 5 } \\ { - 0 . 3 6 8 1 3 5 0 7 1 9 8 2 } \\ 0 . 8 1 9 0 8 6 6 4 6 3 2 4 \\ 0 . 7 7 5 6 2 2 3 1 6 9 6 4 \\ { - 0 . 5 3 1 3 2 2 7 9 0 2 0 7 } ] \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} v a r _ { P _ { \\alpha } } \\left [ \\Delta _ { P _ { \\alpha } } \\left ( \\mathbf { O } \\right ) \\right ] = \\alpha ^ { 2 } E _ { P _ { \\alpha } } \\left [ O _ { 1 } ^ { 2 } O _ { 2 } ^ { 2 } \\right ] = \\alpha ^ { 2 } . \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} t ^ { \\frac { N } { 2 } } u ( x , t ) & = q _ * [ M _ { 0 , 1 } + o ( 1 ) ] U ( | x | ) - q _ * \\left [ \\frac { N } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] t ^ { - 1 } ( U F _ 0 ) ( | x | ) \\\\ & + t ^ { - 2 } O ( | x | ^ 4 U ( | x | ) ) + \\sum ^ N _ { i = 1 } [ M _ { 1 , i } + o ( 1 ) ] t ^ { - 1 } U _ 1 ( | x | ) Q _ { 1 , i } \\left ( \\frac { x } { | x | } \\right ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + t ^ { - 2 } U _ 1 ( | x | ) O ( | x | ^ 2 ) + o ( t ^ { - 1 } ) \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\Sigma ( n ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sqrt { k } , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\omega _ 2 ( V \\oplus \\xi _ { \\mathbb { R } } ^ { \\oplus ( - 2 k - 1 ) } ) = \\omega _ 2 ( M ) - ( 2 k + 1 ) c _ 1 ( \\xi ) ~ { \\rm m o d } ~ 2 = 0 , \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\widetilde { V } _ q ( K ) = \\frac 1 n \\int _ { S ^ { n - 1 } } \\varrho ^ q _ K ( u ) \\ , \\mathcal { H } ^ { n - 1 } ( d u ) , \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} ( u ( x , t ) , v ( x , t ) ) = ( \\phi _ 1 ( n \\cdot x - c t - \\theta _ 0 ) , \\phi _ 2 ( n \\cdot x - c t - \\theta _ 0 ) ) . \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x ~ = ~ \\varepsilon \\ , u _ { x x } , \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} - \\det A ( 0 ) = 2 1 6 4 0 9 2 5 4 8 3 1 0 2 5 > - \\det A ( \\pm 1 ) = 2 1 5 0 5 5 7 8 2 1 1 7 3 7 6 . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} d { s } : = | \\omega ( \\dot { \\gamma } ( t ) ) | d t , ~ ~ ~ ~ d \\overline { s } : = \\frac { 1 } { 2 } \\frac { 1 } { | \\omega ( \\dot { \\gamma } ( t ) ) | } \\left ( \\frac { \\dot { \\gamma } _ 1 ^ 2 } { \\gamma _ 1 ^ 2 } + \\dot { \\gamma } _ 3 ^ 2 \\right ) d t . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\rho = ( n - 1 , n - 2 , \\dots , 1 , 0 ) \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\epsilon , \\hat { v } } \\psi _ k ( x ) = \\mathcal { L } _ k ^ { \\epsilon , \\hat { v } _ k } \\psi _ k ( x ) + \\sum \\nolimits _ { j = 1 } ^ n \\gamma _ { k j } ( x ) \\bigl [ \\psi _ j ( x ) - \\psi _ k ( x ) \\bigr ] , \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} m _ e > \\frac { f ( b ) } { 2 } \\ ( 1 + f ( b ) ^ 2 - ( f ' ( b ) ) ^ 2 \\ ) = \\frac { f ( b ) } { 2 } ( f ( b ) ^ 2 - u _ { m _ e } ( s _ 0 ) ^ 2 ) + \\frac { f ( b ) } { u _ { m _ e } ( s _ 0 ) } \\ , m _ e , \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} P ( \\boldsymbol { Y } = \\boldsymbol { y } ) = P ( \\vert \\boldsymbol { Y } \\vert = \\vert \\boldsymbol { y } \\vert ) \\binom { \\vert \\boldsymbol { y } \\vert } { \\boldsymbol { y } } \\prod _ { j = 1 } ^ J \\pi _ j ^ { y _ j } . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\| \\bar { J } _ t \\| _ q \\leq 2 q \\sum _ { s = 0 } ^ { t - 1 } ( 2 n ^ { - 1 } + 3 \\beta _ s \\sqrt { \\pi q } + \\gamma _ s ) e ^ { 4 q \\alpha _ { s , t } } . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} n ! { \\| \\tilde { h } _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 \\geq \\frac { 1 } { n ! } C _ 5 e ^ { - 2 \\mu _ 1 t } \\int _ { [ 0 , t ] ^ { 2 n } } \\prod _ { i = 1 } ^ n \\gamma ( t _ i - s _ i ) \\prod _ { i = 1 } ^ n \\Big ( t + t _ { i + 1 } - t _ { i } \\Big ) ^ { - \\beta / \\alpha } d \\textbf { t } d \\textbf { s } . \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} p ( x _ 0 ) < 0 . \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\bar { u } _ T : = \\begin{cases} u _ T ^ { i - T j ' } & \\ i \\in Z _ 1 ( T j ' + T Q ) \\ \\ j ' \\in \\Z ^ { n - 1 } \\times \\{ 0 \\} , \\\\ u _ \\zeta ^ { e _ n } ( i ) & \\ \\Z ^ n . \\end{cases} \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} { \\rm { A u t } } ( E / L ^ { G } ( U ) ) = G . \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\max _ { \\chi \\not = \\chi _ 0 } | L ( 1 , \\chi ) | \\leq ( 1 + o ( 1 ) ) \\frac { \\log q } { 3 } . \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\frac 1 { \\varrho } : = \\frac 1 { r _ m } - \\frac 1 { r _ { m + 1 } ' } + \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { p _ i } = \\frac 1 { r _ m } - \\frac 1 { r _ { m + 1 } ' } + \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { q _ i } = \\frac 1 { \\delta _ m } + \\frac 1 { \\delta _ { m + 1 } } = \\frac 1 { \\widetilde { \\delta } _ m } + \\frac 1 { \\widetilde { \\delta } _ { m + 1 } } \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} P _ { \\beta , h } ^ { + , \\pm } ( \\pm \\mid \\omega ^ { - 1 } _ { - \\infty } ) : = \\lim _ { n \\rightarrow \\infty } \\lim _ { l \\rightarrow \\infty } \\nu _ { \\beta , h } ^ + \\left ( \\eta _ 0 = \\pm \\mid \\eta ^ { - 1 } _ { - ( n + l ) } = \\pm ^ { - ( n + 1 ) } _ { - ( n + l ) } \\omega ^ { - 1 } _ { - n } \\right ) \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} ( a _ { i j } \\cdot q ) _ { i j } , \\ { \\rm w h e r e } \\ q \\ \\ Z _ { n + 1 } , \\ { \\rm a n d } \\ ( a _ { i j } ) = a ( \\theta _ n ^ i ) ; \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\begin{bmatrix} \\xi _ 1 x _ 1 ^ 3 - ( 1 - \\xi _ 1 ) \\\\ ( 1 - \\xi _ 1 ) x _ 2 ^ 2 - \\xi _ 1 \\end{bmatrix} + \\begin{bmatrix} - \\lambda \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { k \\to + \\infty } \\frac { k } { M N } \\mathbb { E } \\left [ { \\left \\| \\hat { \\textbf { h } } _ k - \\textbf { h } \\right \\| } _ 2 ^ 2 \\bigg | \\hat { \\boldsymbol { \\psi } } _ k \\to \\boldsymbol { \\psi } \\right ] = { C } _ S ^ { \\min } ( \\boldsymbol { \\psi } ) . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\hat { Q } _ r = \\sum _ { j \\in \\mathcal { I } _ r } \\hat { P } _ j . \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } - \\mathcal { L } _ k ^ { \\epsilon , v _ k } \\ , \\psi _ k ^ { \\epsilon , v _ k } ( x ) = \\lambda _ k ^ { \\epsilon , v _ k } \\ , \\psi _ k ^ { \\epsilon , v _ k } ( x ) D \\\\ \\psi _ k ^ { \\epsilon , v _ k } ( x ) = 0 \\partial D , k = 1 , 2 , \\ldots , n \\end{array} \\right \\} , \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} & \\sum _ { n \\geq 0 } U _ { 2 k + 1 , 2 a } ( n ) q ^ n = U _ { 2 k + 1 , 2 a } ( 1 ; q ) \\\\ & = \\frac { ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a + 1 } , q ^ { 4 k - 2 a + 1 } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } + \\frac { x q ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a - 1 } , q ^ { 4 k - 2 a + 3 } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\eta \\langle \\Delta \\big ( | d u | ^ { p - 2 } d u \\big ) , d u \\rangle d \\mu = 0 . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} B ^ h ( x ) \\ , C _ { h i j } = 0 . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\sum _ { 0 \\le K \\le M - 1 } e ^ { \\frac { K } { M } x } = \\frac { e ^ x - 1 } { e ^ { \\frac { x } { M } } - 1 } \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} [ 1 - ( 1 - \\mu ) p _ { j , \\infty } ] \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ n \\frac { r _ { i j } ( 1 - p _ { i , \\infty } ) } { 1 - r _ { i j } p _ { j , \\infty } } < \\mu , \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\check { g } ( t ) : = \\sup _ { \\tau \\in \\Bbb R } ( g ( \\tau ) + t \\tau ) , \\ t \\geq 0 . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} r _ { - \\mu } | _ { \\widehat { M } } = [ r _ { ( - 1 ^ 2 , 0 ^ { n _ 1 - 2 } ) } \\boxtimes r _ { ( 0 ^ { n _ 2 } ) } ] \\oplus [ r _ { ( - 1 , 0 ^ { n _ 1 - 1 } ) } \\boxtimes r _ { ( - 1 , 0 ^ { n _ 2 - 1 } ) } ] \\oplus [ r _ { ( 0 ^ { n _ 1 } ) } \\boxtimes r _ { ( - 1 ^ 2 , 0 ^ { n _ 2 - 2 } ) } ] , \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} H ^ 0 ( \\mathring { C } _ I ^ 0 \\otimes _ { \\C [ I ] } D _ I ^ \\bullet ) = M . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } f \\left ( x , t \\right ) = \\int \\limits _ { \\mathbb { Q } _ { p } } q \\left ( \\left \\vert x - y \\right \\vert _ { p } \\right ) \\left \\{ f ( x , t ) - f ( y , t ) \\right \\} d y , \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } \\left ( 2 ^ { R } - 1 \\right ) ( 1 + k ) \\left ( k - 2 ^ { \\textrm { R } } \\right ) } { \\lambda _ { \\textrm { B F } } \\left ( k - \\left ( 2 ^ { \\textrm { R } } - 1 \\right ) \\right ) } \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{gather*} m _ j : = \\begin{cases} \\frac { c _ i | d _ { 1 j _ 1 } d _ { 2 j _ 2 } \\dots d _ { r j _ r } | } { d _ { i j _ i } } & ~ j = j _ i ~ ( 1 \\leq i \\leq r ) \\\\ 0 & , \\end{cases} \\end{gather*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\underset { { y } _ { k } \\in \\R ^ { n } , z _ { k } \\in \\R } { \\max } & \\{ \\ ; b ^ { \\mathrm T } { y } _ { k } + z _ { k } \\ ; | \\ ; B ^ { \\mathrm T } { y } _ { k } + I _ { k } z _ { k } \\leq { q } _ { k } \\ ; \\} , \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} P _ q ( Q ) = \\frac { \\theta _ q ( - Q ) \\theta _ q ( i Q ) \\theta _ q ( Q ) } { \\theta _ q \\left ( - \\alpha _ 1 ( q ) Q \\right ) \\theta _ q \\left ( ( - \\alpha _ 2 ( q ) Q \\right ) \\theta _ q \\left ( ( - \\alpha _ 3 ( q ) Q \\right ) } \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} L _ { Z } \\mu = L _ W \\mu = 0 . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\ddot { u } ( t ) - \\Delta u ( t ) = f ( t ) \\Omega \\setminus \\Gamma ( t ) \\ , , \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} X _ { i } = F _ { 0 } ( f ) ^ { e _ { 0 } } \\cdots F _ { i - 1 } ( f ) ^ { e _ { i - 1 } } F _ { i + 1 } ( f ) ^ { e _ { i + 1 } } \\cdots F _ { r } ( f ) ^ { e _ { r } } ( X ) . \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} X _ i & = \\textstyle \\sum _ { j \\in I } ( k _ { j i } D _ { c _ { j i } } + c _ { i j } D _ { e _ { i j } } ) \\\\ Y _ i & = \\textstyle \\sum _ { j \\in I } ( c _ { j i } D _ { c _ { j i } } + e _ { i j } D _ { e _ { i j } } ) - h \\\\ Z & = \\textstyle \\sum _ { i , j \\in I } ( k _ { i j } D _ { k _ { i j } } + c _ { i j } D _ { c _ { i j } } ) - s \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} f & = \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 1 } a _ { j } ^ { 1 } \\\\ & = \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 1 } \\Pi _ { l } ( g _ { j } ^ { 1 } , h _ { j , 1 } ^ { 1 } , h _ { j , 2 } ^ { 1 } ) + \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 1 } \\big ( a _ { j } ^ { 1 } - \\Pi _ { l } ( g _ { j } ^ { 1 } , h _ { j , 1 } ^ { 1 } , h _ { j , 2 } ^ { 1 } ) \\big ) \\\\ & = : M _ { 1 } + E _ { 1 } . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\frac { f _ { 1 1 } f _ { 1 2 } } { f _ { 2 1 } f _ { 2 2 } } = \\frac { g _ { 1 1 } g _ { 1 2 } } { g _ { 2 1 } g _ { 2 2 } } < 0 . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} F ( n , k ) & = \\frac { [ n + 1 ] } { 2 } { 2 n + 1 \\brack n } \\frac { ( 1 + q ^ { 2 k - n - 1 } ) q ^ { - { n - 2 k + 1 \\choose 2 } } } { [ 2 k ] ^ 2 { n \\brack k } _ { q ^ 2 } ^ 2 } , \\\\ [ 5 p t ] G ( n , k ) & = - \\frac { [ 3 n - 2 k + 5 ] } { 2 } { 2 n + 1 \\brack n } \\frac { ( 1 + q ^ { n + 1 } ) q ^ { - { n - 2 k + 3 \\choose 2 } } } { [ 2 k ] ^ 2 { n + 1 \\brack k } _ { q ^ 2 } ^ 2 } . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Z _ j P _ j = P _ j Z _ j \\\\ \\| W - Z _ j \\| = \\| \\mathbf { 1 } _ n - W _ { P _ j } \\| \\leq \\sqrt { 2 } \\| W P _ j W ^ \\ast - P _ j \\| \\end{array} \\right . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} A ^ { ( 4 ) } ( t , 0 ) = I \\ , . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = \\varepsilon \\sum _ Y q ^ { \\sigma ( Y ) - k ( k - 1 ) / 2 } , \\varepsilon = \\pm 1 , \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} ( \\mathbf A + \\mathbf A ' ) \\hat \\oplus ( \\mathbf B + \\mathbf B ' ) = \\mathbf A \\hat \\oplus \\mathbf B + \\mathbf A \\hat \\oplus \\mathbf B ' + \\mathbf A ' \\hat \\oplus \\mathbf B + \\mathbf A ' \\hat \\oplus \\mathbf B ' \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} y _ { \\textrm { B F } } = \\left ( \\sqrt { P _ { \\textrm { N } } } x _ { \\textrm { N } } + \\sqrt { P _ { \\textrm { F } } } x _ { \\textrm { F } } \\right ) \\sqrt { d _ { \\textrm { B F } } ^ { - \\alpha } } h _ { \\textrm { B F } } + n _ { \\textrm { F } } , \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\nu ^ { 0 } _ \\epsilon \\star \\nu ^ { 0 } _ \\iota = 1 , 0 , \\lambda - \\tfrac { 1 } { 2 } , \\nu ^ { 0 } _ + , \\nu ^ { 0 } _ - \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} \\quad { \\sigma } _ 1 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , { \\sigma } _ 2 = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\quad { \\sigma } _ 3 = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} n _ { S \\cap Y } ( P ( x ) ) = n _ S ( x ) \\textrm { f o r e v e r y } x \\in S , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\int _ { J _ \\varepsilon } F _ \\varepsilon ( r ) r d r \\lesssim \\int _ { \\{ r \\le \\rho _ \\varepsilon , t _ \\varepsilon \\ge \\gamma _ \\varepsilon \\} } \\exp \\left ( - ( 1 + \\varepsilon _ 0 + o ( 1 ) ) t _ \\varepsilon ( r ) \\right ) r d r = o \\left ( \\frac { \\mu _ \\varepsilon ^ 2 } { \\gamma _ \\varepsilon ^ 4 } \\right ) \\ , . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } \\sqrt { p } u _ { p } = 0 \\ \\mbox { i n $ C ^ 2 _ { l o c } ( \\bar \\Omega \\setminus \\mathcal { S } ) $ } \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot B ^ s _ { p , q } ( \\mathcal { H } ) } : = \\begin{cases} \\displaystyle \\left \\{ \\sum _ { j = - \\infty } ^ \\infty \\left ( 2 ^ { s j } \\| \\phi _ j ( \\sqrt { \\mathcal { H } } ) f \\| _ { L ^ p ( \\Omega ) } \\right ) ^ q \\right \\} ^ \\frac { 1 } { q } & , \\\\ \\displaystyle \\sup _ { j \\in \\mathbb Z } 2 ^ { s j } \\| \\phi _ j ( \\sqrt { \\mathcal { H } } ) f \\| _ { L ^ p ( \\Omega ) } & , \\end{cases} \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} [ F : K ] & = e ( w ^ { 0 } / v ^ { 0 } ) \\ , f ( w ^ { 0 } / v ^ { 0 } ) . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\frac { d { f _ 2 ( q , z _ 2 ) } } { d q } = [ - \\xi ' ( 1 ) - \\xi ' ( q ) - ( 1 - q ) \\xi '' ( q ) ] \\Big ( \\frac { 1 + z _ 2 } { z _ 2 ^ 2 } \\log ( 1 + z _ 2 ) - \\frac 1 { z _ 2 } \\Big ) + \\xi '' ( q ) ( 1 - q ) \\\\ - ( 1 - q ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] \\Big ( \\frac { 2 + z _ 2 } { z _ 2 ^ 3 } \\log ( 1 + z _ 2 ) - \\frac 2 { z _ 2 ^ 2 } \\Big ) \\frac { d z _ 2 } { d q } = 0 . \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} L _ { \\alpha + \\hat { e } _ i } : = & \\ [ L _ \\alpha , A _ i ] , \\quad \\mbox { w h e r e } \\substack { \\hat { e } _ i = \\\\ \\ } \\substack { ( 0 , \\ldots , 1 , \\ldots , 0 ) \\\\ i } \\mbox { f o r } i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } { 1 \\over n _ k } { \\sum _ { j = 1 } ^ k \\log u _ j } = 0 . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} v o l ( L ' ) \\cdot \\sum _ { x ' } f ( x ' ) x ' . v & = v o l ( L ' ) \\cdot \\sum _ { x ' } \\frac { 1 } { \\left [ L ' : L \\cap L ' \\right ] } \\left ( \\sum _ { l ' _ 1 } f ( x ' l ' _ 1 ) \\ x ' l ' _ 1 . v \\right ) \\\\ & = v o l ( L \\cap L ' ) \\cdot \\sum _ { x ' } \\sum _ { l ' _ 1 } f ( x ' l ' _ 1 ) \\ x ' l ' _ 1 . v \\\\ & = v o l ( L \\cap L ' ) \\cdot \\sum _ { x } \\sum _ { l _ 1 } f ( x l _ 1 ) \\ x l _ 1 . v \\\\ & = v o l ( L ) \\cdot \\sum _ { x } f ( x ) \\ x . v . \\\\ \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} d ( v _ \\ell , v _ { \\ell + 1 } ) & = 2 \\mbox { f o r $ 0 \\leq \\ell < k $ } \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} M ( \\sum _ { g \\in G } g ( \\alpha ) g ^ { - 1 } ) V = 0 . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} [ b , \\mathcal I _ { \\alpha } ] f ( x ) : = b ( x ) \\cdot \\mathcal I _ { \\alpha } f ( x ) - \\mathcal I _ { \\alpha } ( b f ) ( x ) , x \\in \\mathbb R ^ d . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\boldsymbol { w } ( x , t ) = \\sum \\limits _ { r n j } \\boldsymbol { a } _ { r n j } e ^ { \\lambda t } \\Psi _ { r n j } + \\sum \\limits _ { I \\in G _ { N } ^ { 0 } } \\boldsymbol { b } _ { I } \\varphi _ { I } \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} ( y ( g ) , \\gamma , \\delta ) { \\tau _ { i _ j } } & = \\sum _ { E \\subseteq \\{ i _ 1 , \\ldots , i _ s \\} } ( \\prod _ { e \\in E } y _ e ) ( g ^ { \\tau _ E \\tau _ { i _ j } } , \\gamma , \\delta ) \\\\ & = y _ j \\sum _ { E \\subseteq \\{ i _ 1 , \\ldots , i _ s \\} } ( \\prod _ { e \\in E } y _ e y _ j ) ( g ^ { \\tau _ E \\tau _ { i _ j } } , \\gamma , \\delta ) = y _ j ( y ( g ) , \\gamma , \\delta ) . \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} \\begin{cases} d X _ { x } ^ { \\varepsilon } ( t ) = [ A X _ { x } ^ { \\varepsilon } ( t ) + B ( t , X _ { x } ^ { \\varepsilon } ( t ) ) ] d t + \\sqrt { { \\varepsilon } } G ( t , X _ { x } ^ { \\varepsilon } ( t ) ) d w ( t ) , \\\\ X _ { x } ^ { \\varepsilon } ( 0 ) = x \\in E . \\end{cases} \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\gamma _ { } = \\frac { n - n _ p } { n } \\times \\gamma , \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ k \\epsilon _ j d _ j \\Big \\| _ { L ^ p ( X ) } \\lesssim \\Big \\| \\sum _ { j = 1 } ^ k d _ j \\Big \\| _ { L ^ p ( X ) } . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\delta _ { e } = \\delta _ { e } \\ast \\eta ^ { * } \\ast \\eta = \\eta ^ { * } \\ast \\eta ~ , \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} R _ k = R ( k ) = k ^ { \\beta } L ( k ) \\quad ( \\beta \\in \\R ) \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} f \\wedge g \\ = \\ f \\mbox { a n d } f \\vee g \\ = \\ g . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} D _ { u } ^ { ( 1 ) } : = D _ { u } - \\frac { 1 } { 2 } \\mathcal J _ { 2 } D _ { u } \\mathcal J _ { 2 } \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} & \\int _ { x _ i \\ge 0 , \\sum x _ i \\le 1 } e ( \\sum _ { i = 1 } ^ m ( \\alpha _ i - \\alpha _ { m + 1 } ) x _ i ) d x _ 1 \\dots d x _ m \\\\ = & \\frac { 1 } { ( 2 \\pi i ) ^ m } \\sum _ { J = 1 } ^ u \\frac { e ( a _ J - \\alpha _ { m + 1 } ) } { \\prod \\limits _ { l \\notin C _ J } ( a _ J - \\alpha _ l ) } \\Bigl \\{ \\sum _ { n _ 0 , n _ d \\ge 0 \\ , ( d \\not \\in C _ J ) , \\atop n _ 0 + \\sum \\limits _ { d \\not \\in C _ J } n _ d = \\# C _ J - 1 } \\frac { ( 2 \\pi i ) ^ { n _ 0 } } { \\prod _ { d \\not \\in C _ J } { ( \\alpha _ d - a _ J ) ^ { n _ d } } } \\Bigr \\} , \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\gamma _ 1 { G } > \\gamma _ 2 { G } > \\ldots > \\gamma _ c { G } > \\gamma _ { c + 1 } { G } , \\gamma _ i { G } = 1 i \\ge c + 1 , \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\left | { \\eta _ { K , M } ^ { ( 3 ) } } \\right | = 0 , \\ ; \\ ; . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\lambda _ z \\circ \\Psi _ { ( h ' , v ) } = \\lambda _ z \\circ \\Psi _ { ( h ' h ^ { - 1 } , z ) } \\circ \\Psi _ { ( h , v ) } = \\lambda _ z \\circ \\Psi _ { ( h , v ) } , \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} N _ \\pm = \\frac { r ( k + a ) \\pm \\sqrt { r ^ 2 ( k + a ) ^ 2 - r ( r - 1 ) ( a ^ 2 - b ) } } { r ( r - 1 ) } , \\ \\ \\ \\ \\ r ( k + a ) ^ 2 \\geq ( r - 1 ) ( a ^ 2 - b ) . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } \\equiv \\frac { ( n ^ 2 - 1 ) ( 1 - q ) ^ 2 } { 2 4 } [ n ] \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\Omega _ r [ u ] ( t , x ) & \\doteq \\bigg ( \\frac { 1 } { r } \\frac { \\partial } { \\partial r } \\bigg ) ^ { k - 1 } \\Big ( r ^ { 2 k - 1 } I _ r [ u ] ( t , x ) \\Big ) , \\\\ \\Omega _ r [ u _ j ] ( x ) & \\doteq \\bigg ( \\frac { 1 } { r } \\frac { \\partial } { \\partial r } \\bigg ) ^ { k - 1 } \\Big ( r ^ { 2 k - 1 } I _ r [ u _ j ] ( x ) \\Big ) \\mbox { f o r } \\ \\ j = 0 , 1 , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} \\mathbf { M } ( \\overline { g } ) ( V ^ { ( 0 ) } ) = \\tilde { \\mathfrak { m } } _ H ( \\Sigma ) \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\hat { c } _ { e } : = \\begin{cases} c _ { e } - c _ { f } & e v , \\\\ c _ { e } + c _ { f } & e v , \\\\ c _ { e } & . \\end{cases} \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ w \\in \\Lambda ^ { m } \\ : : \\ : 2 ^ { - m ( h + \\delta ) } \\le \\mu [ w ] \\le 2 ^ { - m ( h - \\delta ) } | r _ { w } | \\ge 2 ^ { m ( \\chi - \\delta ) } \\} \\ : . \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} t _ { k + 1 } > t _ k f ( t _ { k + 1 } ) \\notin \\bigcup _ { i = 0 } ^ k U _ { g _ { \\pi ( i ) } } . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\left | { H _ { K , M } ^ { \\left ( A \\right ) } - H _ { K , M } ^ { \\left ( G \\right ) } } \\right | = 0 \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\lambda ^ * \\alpha w - \\Delta ( \\alpha w ) & = \\alpha F - \\alpha ( \\nabla _ H \\Pi ) - 2 ( \\partial _ z \\alpha ) ( \\partial _ z w ) - ( \\partial _ z ^ 2 \\alpha ) w , \\\\ \\partial _ z ( \\alpha w ) & = 0 \\quad \\Gamma _ u ' \\cup \\Gamma _ b ' , \\alpha w \\Gamma _ l ' \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { U } _ { 2 k , 2 a } ( n ) q ^ n = ( - q ^ 2 ; q ^ 2 ) _ \\infty \\sum _ { n \\geq 0 } \\overline { B } _ { k , a } ( n ) q ^ { 2 n } . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\tilde \\tau _ 1 & = \\frac { ( 2 \\phi _ 2 ^ 2 - \\phi _ 1 ^ 2 ) - ( 2 \\phi _ 2 ^ 1 - \\phi _ 1 ^ 1 ) } { 2 \\omega _ 2 - \\omega _ 1 } \\\\ \\mbox { a n d } \\tilde \\tau _ 2 & = \\frac { ( \\phi _ 2 ^ 2 - 2 \\phi _ 1 ^ 2 ) - ( \\phi _ 2 ^ 1 - 2 \\phi _ 1 ^ 1 ) } { \\omega _ 2 - 2 \\omega _ 1 } . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\nabla _ { i } | X | ^ { 2 } = 2 \\langle X , e _ { i } \\rangle , \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\left | \\frac { M ( x + h ) - M ( x ) } { h } - \\frac { d } { d x } M ( x ) \\right | = \\left | \\frac { d } { d x } M ( \\alpha ( x , h ) ) - \\frac { d } { d x } M ( x ) \\right | < K | h | ^ \\delta , \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} F ( t ) = \\frac { j ^ 2 } { 2 t ^ 2 } - \\ln t . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} H _ n ( \\mu ) = \\frac { 1 } { n \\log 2 } H ( \\mu , \\mathcal { D } _ n ) . \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} J _ 2 ( t , \\delta ) & \\leq 2 \\| f \\| t \\Pr ( Y _ 1 \\wedge V _ 2 > b _ 0 ( t ) \\min \\{ ( \\eta - \\delta ) , \\delta \\} ) \\leq C _ { 2 } t ^ { \\frac { - \\alpha + \\epsilon } { \\alpha _ 0 } } \\stackrel { t \\to \\infty } { \\to } 0 . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} - \\frac { 1 } { n } \\log \\epsilon ^ \\star ( n , d , M ) = \\min _ { P _ { \\hat { X } } } ~ D ( P _ { \\hat { X } } | | P _ { X } ) + O \\left ( \\frac { \\log n } { n } \\right ) , \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} I _ l = \\int _ 0 ^ { \\pi } \\dfrac { \\cos { 2 n l \\theta } \\ , d \\theta } { ( a _ { 2 n } + \\cos { 2 n \\theta } ) ^ { 2 s } } = ( - 1 ) ^ l ( 2 \\rho ^ { 2 n } ) ^ { 2 s } \\dfrac { \\pi \\sum _ { m = 0 } ^ { 2 s - 1 } { 2 s + l - 1 \\choose m } { 4 s - m - 2 \\choose 2 s - 1 } \\left ( \\rho ^ { 4 n } - 1 \\right ) ^ m } { \\rho ^ { 2 n l } \\left ( \\rho ^ { 4 n } - 1 \\right ) ^ { 4 s - 1 } } . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} & E [ e ^ { ( U _ 1 + U _ 2 + \\cdots + U _ k ) ( e ^ t - 1 ) } ] = \\sum _ { m = 0 } ^ \\infty E [ ( U _ 1 + \\cdots + U _ k ) ^ m ] \\frac { ( e ^ t - 1 ) ^ m } { m ! } \\\\ & = \\sum _ { m = 0 } ^ \\infty E [ ( U _ 1 + \\cdots + U _ k ) ^ m ] \\sum _ { n = m } ^ \\infty S _ 2 ( n , m ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n S _ 2 ( n , m ) E [ ( U _ 1 + \\cdots + U _ k ) ^ m ] \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\sum _ j \\frac { \\partial f } { \\partial y ^ j } ( g . z ) \\frac { \\partial \\Phi ^ j _ g } { \\partial x ^ i } ( z ) = \\frac { \\partial f } { \\partial x ^ i } ( z ) . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} d _ a \\mathcal { L } \\eta = \\mathcal { L } d _ a \\eta , \\eta \\in \\Gamma ( \\mathbb { M } , \\mathcal { E } ) , \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\| I _ \\alpha f \\| _ { L ^ { p _ \\alpha } ( \\mathbb { H } ^ n ) } = \\| I _ \\alpha f _ \\epsilon \\| _ { L ^ { p _ \\alpha } ( \\mathbb { H } ^ n ) } , \\quad \\| f \\| _ { L ^ { q _ \\alpha } ( \\mathbb { H } ^ n ) } = \\| f _ \\epsilon \\| _ { L ^ { q _ \\alpha } ( \\mathbb { H } ^ n ) } , \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } f ( T ^ n x ) \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} i \\partial _ { t } \\psi - \\Delta \\psi = \\pm | \\psi | ^ { \\frac { 4 } { d } } \\psi , \\psi ( 0 ) = \\psi _ { 0 } \\in L ^ { 2 } \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} g _ + = \\exp ( x A _ + ) \\exp ( \\psi _ + d _ + ) \\exp ( \\psi _ - d _ - ) . \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} g _ { m , b } = d s ^ 2 + u _ { m , b } ( s ) ^ 2 g _ * , \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\tilde { z } _ * = ( 0 , \\cdots , 0 , \\tilde { z } _ { * , N } ) = ( 0 , \\cdots , 0 , \\theta _ 0 ) . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sharp ( \\{ I _ j ^ n \\} \\cap J ) } { n } = \\frac { \\int _ J s ( x ) d x } { \\int _ I s ( x ) d x } . \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\varphi \\left ( x _ 1 , \\ldots , x _ n \\right ) = \\frac { 1 } { n ! \\ , 2 ^ n } \\sum _ { \\epsilon _ 1 , \\ldots , \\epsilon _ n = \\pm 1 } \\epsilon _ { 1 } \\cdots \\epsilon _ { n } P \\left ( \\epsilon _ 1 x _ 1 + \\cdots + \\epsilon _ { n } x _ { n } \\right ) \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\begin{aligned} & 0 = S ( 1 - \\dfrac 3 2 S ) + 2 \\bar H ^ 2 S - \\dfrac 1 2 \\bar H ^ 4 - \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 ) \\\\ & = - ( S - \\bar H ^ 2 ) ^ 2 - \\dfrac 1 2 \\bar H ^ 2 ( \\bar \\lambda _ 1 - \\bar \\lambda _ 2 ) ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} H _ { - 1 } ( K _ \\bullet ^ A , \\iota _ T ) = 0 . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} ( \\varepsilon \\log v ) _ { y t } = u _ t + \\left ( p + \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 - \\alpha g ' ( v ) h ( | \\mathbf { w } | ^ 2 ) \\right ) _ y . \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\frac { d } { d x } g = \\frac { d } { d x } \\int _ 0 ^ { \\bullet } f ( y ) d y = f = \\big ( f - f ( 0 ) \\big ) + f ( 0 ) = \\int _ 0 ^ { \\bullet } f ' ( y ) d y + f ( 0 ) \\in T V + \\langle 1 \\rangle . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} \\psi _ n = ( \\eta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ k , 1 ) \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\left ( 1 - P ^ { - 1 } \\right ) ^ { N + 1 } - Q = 0 \\in Q K ( \\mathbb { P } ^ N ) \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} x ^ p + \\frac { p } { \\lambda } x ^ { p - 1 } + \\frac { p ( p - 1 ) } { 2 \\lambda } x ^ { p - 2 } + \\cdots + \\frac { p } { \\lambda ^ { p - 1 } } x - f = 0 \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} f _ 1 ( a , b ) & = f _ 1 ( b ^ { - 1 } , b ) , \\\\ f _ 2 ( a , b ) & = - f _ 1 ( b , b ) + f _ 1 ( b ^ { - 1 } , b ) f _ 1 ( a , a ^ { - 1 } ) . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} G _ { i j } = G _ \\tau ( \\phi _ i , \\phi _ j ) = \\partial _ { t _ 0 } \\partial _ { t _ i } \\partial _ { t _ j } \\mathcal { F } \\in \\mathbb { Z } [ \\ ! [ t _ 0 , \\dots , t _ N ] \\ ! ] \\otimes \\mathbb { C } [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} V ^ 4 | \\nabla \\tau | ^ 2 = & ( C _ i Y ^ 4 + B _ i Y ^ 0 + D _ p \\epsilon _ { p q i } Y ^ q ) ( C _ j Y ^ 4 + B _ j Y ^ 0 + D _ m \\epsilon _ { m n j } Y ^ n ) ( \\delta ^ { i j } - \\tilde { X } ^ i \\tilde { X } ^ j ) \\\\ & + r ^ 2 \\left [ \\sigma ^ { ( 0 ) a b } ( C _ i \\tilde { X } _ a ^ i ) ( C _ j \\tilde { X } _ b ^ j ) + 2 ( C _ i \\tilde \\nabla \\tilde Y ^ i ) ( C _ j \\tilde \\nabla Y _ j ^ { ( 3 ) } + A \\tilde \\nabla Y _ 0 ^ { ( 3 ) } ) \\right ] + O ( r ^ { 3 } ) , \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} a + b = \\sum _ { j = 1 } ^ l a _ j ^ n + \\sum _ { j = 1 } ^ m b _ j ^ n \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} & w \\big ( \\big \\{ x \\in B ( x _ 0 , r ) : | b ( x ) - b _ { B ( x _ 0 , r ) } | > \\lambda \\big \\} \\big ) \\\\ & \\leq C _ 1 w \\big ( B ( x _ 0 , r ) \\big ) \\exp \\bigg [ - \\bigg ( 1 + \\frac { r } { \\rho ( x _ 0 ) } \\bigg ) ^ { - \\theta ^ { \\ast } } \\frac { C _ 2 \\lambda } { \\| b \\| _ { \\mathrm { B M O } _ { \\rho , \\theta } } } \\bigg ] \\left ( 1 + \\frac { r } { \\rho ( x _ 0 ) } \\right ) ^ { \\eta } , \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} H = \\int M ( d t , \\theta ^ H _ t ) + L ^ H \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\frac { d } { d t } f _ { i } ( t ) = - \\sum _ { j } Q _ { i , j } \\left \\{ f _ { i } ( t ) - f _ { j } ( t ) \\right \\} , \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} s \\sum _ { j \\le k } \\binom { s } { j } 4 ^ { j } d ^ { O ( k ) } = s ^ { k + 1 } d ^ { O ( k ) } \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} Y ' = \\{ ( m , T , A ) | w _ { i , j } = w _ { 1 , 0 } ^ i w _ { n - 1 , 1 } ^ j , A _ { \\xi } ( w _ { n - 1 , 1 } ) = 1 , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} & x _ { i } = y _ { 0 , i } , x _ N = y _ { K , 1 } & i & = 1 , \\dots , N - 1 \\\\ & ( y _ { k , 2 i - 1 } , y _ { k , 2 i } , y _ { k + 1 , i } ) \\in L ^ 3 \\ & k & = 0 , \\dots , K - 1 , i = 1 , \\dots , 2 ^ { K - k - 1 } . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\hat \\psi _ j ^ { ( K ) } = \\mathop { \\max } \\limits _ { 1 \\le { k _ j } \\le K } \\sum \\limits _ { i = { k _ j } } ^ K { \\hat \\phi _ { j i } ^ { \\left ( 4 \\right ) } } = \\max \\left \\{ { \\hat \\psi _ j ^ { ( K - 1 ) } , 0 } \\right \\} + \\hat \\phi _ { j K } ^ { \\left ( 4 \\right ) } \\ ; \\ ; \\ ; \\ ; \\hat \\psi _ j ^ { ( 0 ) } = 0 , \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} G = G _ 0 > G _ 1 > \\cdots > G _ t = 1 \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\nabla q ( \\mathbf { U } ) = \\nabla \\eta ( \\mathbf { U } ) \\nabla F ( \\mathbf { U } ) . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} M = C ( I ) \\oplus E \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} H _ { \\chi } ( t , x , y ) = \\sum _ { j = 0 } ^ { \\infty } e ^ { - \\lambda _ { \\chi , j } t } s _ { \\chi , j } ( x ) \\overline { s _ { \\chi , j } ( y ) } \\ , . \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\nabla _ { z \\partial _ z } S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) = S ^ \\textnormal { c o h } ( \\tau , z ) \\left ( \\mu ( \\alpha ) - \\frac { 1 } { z } \\rho ( T _ a ) \\right ) \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} \\hat \\Sigma - \\tilde { \\Sigma } = \\hat \\Sigma _ \\epsilon - \\Sigma _ \\epsilon + F \\otimes \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i \\epsilon _ i \\Big ) + \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i \\epsilon _ i \\Big ) \\otimes F . \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\rho ) = \\sum \\limits _ { i , j \\geq 0 } ( - 1 ) ^ { i + j } \\varinjlim \\limits _ { U \\subset G ( \\mathbb { Q } _ p ) } \\mathrm { E x t } ^ i _ { J _ b ( \\mathbb { Q _ p } ) } ( H ^ j _ c ( \\mathbb { M } ^ { r i g } _ { b , \\mu , U } \\times \\overline { \\hat { \\mathbb { Q } ^ { u r } _ p } } , \\overline { \\mathbb { Q } _ l } ) , \\rho ) ( - \\mathrm { d i m } \\mathbb { M } ^ { r i g } _ { b , \\mu , U } ) . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} a ^ k = \\nabla ^ 2 f ( \\overline { x } ) ( x ^ k - \\overline { x } ) + \\nabla g ( \\overline { x } ) ( \\lambda ^ k - \\overline { \\lambda } ) + \\big [ g '' ( \\overline { x } ) ( x ^ k - \\overline { x } ) ] ^ * \\lambda ^ k + o ( t _ k ) . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} \\underset { \\bar { { \\bf w } } \\in \\mathcal { W } } \\min ~ & \\frac { 1 } { 2 } \\| \\bar { { \\bf w } } - \\bar { { \\bf w } } ' \\| _ 2 ^ 2 , \\\\ \\underset { { { \\bf s } } \\in \\mathcal { S } } \\min ~ & \\frac { 1 } { 2 } \\| { { \\bf s } } - { { \\bf s } } ' \\| _ 2 ^ 2 , \\\\ \\underset { { { \\bf y } } \\in \\bigtriangleup _ M } \\min ~ & \\sum \\limits _ { m = 1 } ^ { M } { y } _ m \\log \\frac { { y } _ m } { { y } ' _ m } - \\sum \\limits _ { m = 1 } ^ { M } ( { y } _ m - { y } ' _ m ) \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} Q _ { t } = Q _ { 1 } \\circ \\delta _ { t ^ { H } } ^ { - 1 } . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} ( x ^ { \\nu } J _ { \\nu } ( x ) ) ' = x ^ { \\nu } J _ { \\nu - 1 } ( x ) . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} h _ T ( z ) \\ : = \\ ( 1 - z ) ^ { d + 1 } \\ , f _ T \\left ( \\frac { z } { 1 - z } \\right ) . \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} D _ a ^ { k ' } ( I _ a ^ k f ) ( x ) = f ( x ) , \\mbox { a . e . } x \\in [ a , b ] \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\begin{cases} u _ { 1 } + \\lambda \\left [ f \\left ( x , u _ { 1 } \\right ) - \\varepsilon u _ { 1 , x } \\right ] _ { x } \\le w _ { 1 } , \\\\ u _ { 2 } + \\lambda \\left [ f \\left ( x , u _ { 2 } \\right ) - \\varepsilon u _ { 2 , x } \\right ] _ { x } \\ge w _ { 2 } . \\end{cases} \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} z _ * = \\tilde { z } _ * \\ \\ { \\rm o r } \\ \\ \\partial B _ { | \\tilde { z } _ * - z _ * | } ( z _ * ) \\ \\ { \\rm a n d } \\ \\ H \\ \\ { \\rm i n t e r s e c t \\ a t } \\ \\ \\tilde { z } _ * . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{gather*} R _ + : = \\{ z \\in \\C ; \\ ; x _ j < \\Re z < x ^ r _ j , \\ ; y _ j < \\Im z < y _ j + \\delta \\} , \\\\ R _ - : = \\{ z \\in \\C ; \\ ; x _ j < \\Re z < x ^ r _ j , \\ ; y _ j - \\delta < \\Im z < y _ j \\} , \\end{gather*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\left | e ^ { \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } f \\right | _ q \\leq | f | _ q , \\ \\forall f \\in L ^ q ( \\mathbb { R } ^ d ) , \\ t \\geq 0 , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} Y _ { x } ^ { \\varepsilon } ( t ) = \\sqrt { { \\varepsilon } } \\int _ { 0 } ^ { t } S ( t - s ) G ( s , \\mathcal { M } ( S ( \\cdot ) x + Y _ { x } ^ { \\varepsilon } ) ( s ) ) d w ( s ) . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} E E f - E ( \\log \\xi ) E f = 0 . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\Psi ^ { - 1 } \\left ( V \\over N _ { f , h } - 1 \\right ) \\cdot { N _ { f , h } - 1 \\over V } ~ \\leq ~ \\Psi ^ { - 1 } \\left ( \\Psi ( h ) \\right ) \\cdot { 1 \\over \\Psi ( h ) } ~ = ~ { h \\over \\Psi ( h ) } \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\begin{aligned} G _ a ^ n ( z , w ) : = \\langle x , y , s _ 2 , \\ldots , s _ { n - 1 } \\mid s _ 2 = \\lbrack y , x \\rbrack , \\ ( \\forall _ { i = 3 } ^ n ) \\ s _ i = \\lbrack s _ { i - 1 } , x \\rbrack , \\ s _ n = 1 , \\ \\lbrack y , s _ 2 \\rbrack = s _ { n - 1 } ^ a , \\\\ ( \\forall _ { i = 3 } ^ { n - 1 } ) \\ \\lbrack y , s _ i \\rbrack = 1 , \\ x ^ 3 = s _ { n - 1 } ^ w , \\ y ^ 3 s _ 2 ^ 3 s _ 3 = s _ { n - 1 } ^ z , \\ ( \\forall _ { i = 2 } ^ { n - 3 } ) \\ s _ i ^ 3 s _ { i + 1 } ^ 3 s _ { i + 2 } = 1 , \\ s _ { n - 2 } ^ 3 = s _ { n - 1 } ^ 3 = 1 \\ \\rangle , \\end{aligned} \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} q _ j ( x ) = \\frac { 1 } { 2 } \\gamma _ j x _ 1 ^ 2 . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ G ( x ) - G ( y ) : x , y \\in E \\} = \\R ^ n . \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\beta b \\tau ( b ) \\beta ^ { - 1 } = b ' \\tau ( b ' ) , \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} F _ n ( 1 ) = { a q ^ { 2 n - 1 } ( 1 - b q ^ { n + 1 } ) ( 1 - a b q ^ { n + 1 } ) \\over ( 1 - a b q ^ { 2 n } ) ( 1 - a b q ^ { 2 n + 1 } ) } . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F ( n , 0 ) \\equiv \\sum _ { n = 0 } ^ { m - 1 } F ( n , 1 ) \\equiv \\cdots \\equiv \\sum _ { n = 0 } ^ { m - 1 } F \\left ( n , \\frac { m - 1 } { 2 } \\right ) \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} A _ V ( \\xi _ 1 , \\xi _ 2 ) = ( 2 \\xi _ 1 + \\xi _ 2 , - \\xi _ 1 + 2 \\xi _ 2 ) \\quad J _ { A _ V } ( \\xi _ 1 , \\xi _ 2 ) = \\frac { 1 } { 1 0 } ( 3 \\xi _ 1 - \\xi _ 2 , \\xi _ 1 + 3 \\xi _ 2 ) . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\begin{array} { c } \\displaystyle R i c _ { g _ B } = \\rho g _ B + \\frac { r } { h } H e s s _ { g _ B } h , \\\\ R i c _ { g _ F } = \\mu g _ F , \\\\ h \\Delta _ { g _ B } h + ( r - 1 ) | g r a d _ { g _ B } h | ^ 2 + \\rho h ^ 2 = \\mu , \\end{array} \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} B _ n ( X ) = 0 \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} = ( \\mathrm { I n d } ^ G _ { P _ { S ' } } \\circ [ \\mu _ { S ' } ] ) ( \\bigoplus _ { w \\in W _ { \\rho } } \\delta ^ { \\frac { 1 } { 2 } } _ { P _ { S ' } } \\otimes I ^ { M _ { S ' } } _ { w ( M _ S ) } ( w ( \\rho ) ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ { S ' } - \\mu \\rangle } ] , \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} 2 \\sigma ^ 2 x ^ 2 + [ a ^ 2 + 2 \\sigma ^ 2 s ( - a + \\eta ) - 1 ] x + \\beta _ 1 = 0 . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} T _ { \\rm s } = ( I - P ) T = \\{ \\{ f , ( I - P ) f ' \\} : \\ , \\{ f , f ' \\} \\in T \\} . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} u _ { t } + f \\left ( x , u \\right ) _ { x } = 0 \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ t ( x ) = \\begin{cases} - 4 a d \\lambda F _ t ( x ) + 2 a \\lambda \\sum _ { y : y \\sim x } F _ t ( y ) & x \\neq O , \\\\ \\big [ 1 - 4 d \\lambda ( a + b - 1 ) + 2 d \\lambda ( b ^ 2 - 1 ) + 2 d a ^ 2 \\lambda \\big ] F _ t ( O ) + 4 a b d \\lambda F _ t ( e _ 1 ) & x = O . \\end{cases} \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\pi ( u ) \\langle v , w \\rangle = \\langle D _ u v , w \\rangle + \\langle v , D _ u w \\rangle \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\begin{array} { r l } \\Delta _ { g _ B } h & = \\Delta _ { \\overline { g } } h + m \\frac { \\overline { g } ( g r a d _ { \\overline { g } } h , g r a d _ { \\overline { g } } f ) } { f } \\\\ & = \\varphi ^ 2 \\varepsilon _ { i _ 0 } \\left [ h '' - ( n - 2 ) \\frac { \\varphi ' h ' } { \\varphi } \\right ] + m \\varepsilon _ { i _ 0 } \\varphi ^ 2 \\frac { h ' f ' } { f } , \\end{array} \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} d Y _ t = X _ 0 ( t ) d t + \\sum _ { i = 1 } ^ d X _ i ( Y _ t ) \\circ d \\beta _ t ^ i , \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} [ y _ k ^ { p ^ r } , w _ d ] _ { \\circ } = w _ m \\cdot ( \\delta _ k ( r ) ) _ { \\circ } \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\zeta ( x ) = \\displaystyle \\sum _ { k \\in \\Bbb Z } c _ k { \\bf e } ^ { i k x } \\in C ^ \\infty ( \\Bbb S ) , \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} y ( T ) = 0 \\ \\ \\ I . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} & B _ k \\bullet Y _ k = - 1 \\\\ & D _ k \\bullet Y _ k = 0 \\\\ & I \\bullet Y _ k = 0 . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\mu \\left ( T _ j \\right ) = \\frac { 1 } { 2 } \\left ( \\textnormal { d e g } _ { H ^ * ( X ) } ( T _ j ) - \\textnormal { d i m } _ \\mathbb { C } ( X ) \\right ) T _ j \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\sigma x = ( x _ { \\sigma ^ { - 1 } ( 1 ) } , x _ { \\sigma ^ { - 1 } ( 2 ) } , \\ldots x _ { \\sigma ^ { - 1 } ( n ) } ) . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} F ( z , w ) : = \\sum _ { j = 0 } ^ { q - 1 } a _ j ( z ) w ^ j , ( z , w ) \\in \\Omega \\times \\C . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} M _ { X } = \\prod _ { j = 1 } ^ { l } G ( V ^ { \\gamma , t _ { j } } _ { t _ { j } - 1 } ) \\times G ( U ) \\cong \\prod _ { j = 1 } ^ { l } G ( ( V ' ) ^ { \\gamma ' , t _ { j } + 1 } _ { t _ { j } } ) \\times G ( U ) . \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} P _ { \\lambda _ 1 } ( \\eta _ s ( O ) = 1 ) \\geq P _ { \\lambda _ 2 } ( \\eta _ t ( O ) = 1 ) . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} { \\mathcal { M } _ { 2 ^ { - ( 2 + 2 / { \\bf \\tilde { p } } ) } \\cdot h } ( B _ { \\rho } ( x , h ) | E ) } ~ \\geq ~ \\left ( { h \\over 2 \\cdot 2 ^ { - ( 2 + 2 / { \\bf \\tilde { p } } ) } \\cdot h } \\right ) ^ { \\bf { \\tilde { p } } } ~ = ~ 2 ^ { { \\bf \\tilde { p } } + 2 } \\qquad \\forall h > 0 ~ . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 3 ( x ) + \\frac { 2 } { 6 ^ 4 } ( 3 ^ 4 x - 3 4 ) ( 7 - 2 ^ 4 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 8 } { 1 9 } } + \\frac 1 { 1 0 \\cdot 6 ^ 4 } - \\eta \\\\ & = - \\frac { 6 3 8 5 5 1 9 3 6 2 9 7 4 5 3 9 8 3 9 6 3 8 0 1 1 8 8 1 4 2 2 6 3 } { 3 9 4 0 9 6 0 1 8 1 6 2 6 7 8 0 1 9 8 7 6 6 6 8 9 1 0 7 6 1 2 1 3 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} \\partial _ t g + L g = Q _ t ( g ) . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi \\upharpoonright _ { H } ) = h _ { a l g } ( \\phi \\upharpoonright _ { H ' } ) + h _ { a l g } ( \\widetilde { \\phi \\upharpoonright _ { H } } ) , \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} d _ 0 > 0 , 1 - 3 \\alpha _ j c ' \\varepsilon > 0 , \\mbox { a n d } d _ { 2 , j } > 0 \\mbox { f o r } j = 1 , 2 . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} R _ E ^ { \\nabla } ( l _ 1 , l _ 2 ) = \\nabla _ { l _ 1 } \\nabla _ { l _ 2 } - \\nabla _ { l _ 2 } \\nabla _ { l _ 1 } - \\nabla _ { [ l _ 1 , l _ 2 ] } \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\int _ { x \\in S } f ( x ) d x = 1 . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} E ( \\Sigma , Y , T _ 0 ) = ( A e + C _ i p ^ i ) r ^ 3 + O ( r ^ 4 ) \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } x _ { N _ 1 + 2 + k _ l } = L . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} 4 \\int _ { S ^ 2 } \\beta ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & \\int _ { S ^ 2 } \\tilde \\nabla ^ a \\alpha _ { a b } ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = 0 \\\\ 5 \\int _ { S ^ 2 } D \\beta ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & \\int _ { S ^ 2 } \\tilde \\nabla ^ a D \\alpha _ { a b } ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = 0 . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} 0 & - 1 \\\\ A & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} g _ k ( i ) = \\begin{cases} 1 ; \\textrm { i f } \\ , \\lambda _ k ( i ) \\ \\geq \\ \\delta \\\\ 0 ; \\textrm { o t h e r w i s e } \\\\ \\end{cases} , \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} S ( \\xi ) = - a d _ { r ( \\xi ) } ^ * \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\Lambda [ w ] : = \\bigcup _ { i = 1 } ^ N \\Lambda ^ i [ w ] . \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} v ^ l _ n = \\phi _ R * v ^ l _ n + ( \\delta - \\phi _ R ) * v ^ l _ n , \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} N _ { \\mathcal J } ( u , v , w ) = & \\langle N _ { \\mathcal J } ( u , v ) , w \\rangle = \\eta ( \\mathcal J u , v , w ) - \\eta ( \\mathcal J v , u , w ) + \\eta ( w , u , \\mathcal J v ) \\\\ & - \\eta ( v , u , \\mathcal J w ) + \\eta ( \\mathcal J w , u , v ) + \\eta ( u , v , \\mathcal J w ) . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i x _ j \\Lambda _ l } \\widetilde { m } d y = 0 . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} S _ n ( x ; \\tau _ 1 , \\dots , \\tau _ n ) : = \\{ ( x _ 1 , \\dots , x _ n ) \\mid 0 \\le x _ i \\le \\tau _ i , \\sum _ { i = 1 } ^ n x _ i = x \\} \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} Y _ 1 \\leq 2 \\sum _ { j \\ge 1 } \\frac { 1 } { j ^ { \\pi _ q ( j ) + 1 } ( \\pi _ q ( j ) + 1 ) ! } \\le 2 \\sum _ { j \\ge 1 } \\frac { 1 } { ( \\pi _ q ( j ) + 1 ) ! } = O \\Big ( \\frac { 1 } { q } \\Big ) . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\mathcal { R } ^ { n } _ { \\epsilon , \\mu } f ( z ) : = \\int _ { | z - w | > \\epsilon } \\frac { x - y } { | x - y | ^ { n + 1 } } f ( y ) d \\mu ( y ) . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\phi _ { i } ( \\gamma _ { i } ( 1 ) ) = \\gamma _ { i + 1 } ( 0 ) i = 1 , \\ldots , n - 1 \\phi _ { n } ( \\gamma _ { n } ( 1 ) ) = \\gamma _ { 1 } ( 0 ) \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} P ( p , q , r ) : = ( \\underbrace { 0 , \\dots , 0 } _ { p } , \\underbrace { \\epsilon , \\dots , \\epsilon } _ { q } , \\underbrace { 1 , \\dots , 1 } _ { r } ) \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} f ( I ) = f ( I ) \\cap \\bigg ( \\bigcup _ { k = 0 } ^ n U _ { g _ k } \\bigg ) . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ k = \\frac { 1 } { 2 } \\sum _ { l = 0 } ^ N G ^ { k l } \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} d ^ { - 1 } + \\sum _ { k = 1 } ^ { \\min ( H , t ) - 1 } \\mu d ^ k d ^ { - k - 1 } \\leq \\frac { 1 + t \\mu } { d } . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} f _ 1 ( q , z _ 2 ) : = - \\frac { [ q \\xi ' ( q ) - \\xi ( q ) ] ( 1 - q ) ( 1 + z _ 2 ) } { \\xi ' ( 1 ) - \\xi ' ( q ) } - q ^ 2 \\log \\frac { q [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { ( 1 + z _ 2 ) \\xi ' ( q ) ( 1 - q ) } + q ^ 2 - \\frac { 2 \\xi ( q ) q } { \\xi ' ( q ) } \\\\ + \\frac { \\xi ( q ) q ^ 2 [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { ( 1 + z _ 2 ) \\xi ' ( q ) ^ 2 ( 1 - q ) } & = 0 , \\\\ f _ 2 ( q , z _ 2 ) : = ( 1 - q ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] \\Big ( \\frac { 1 + z _ 2 } { z _ 2 ^ 2 } \\log ( 1 + z _ 2 ) - \\frac 1 { z _ 2 } \\Big ) + \\xi ' ( q ) ( 1 - q ) - 1 + \\xi ( q ) & = 0 . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} S _ { n , k } ( q ) = S _ { n - 1 , k - 1 } ( q ) + [ k ] _ q S _ { n - 1 , k } ( q ) \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\sup _ { ( b , d ) \\in \\mathcal H _ m } \\bigg ( & \\Phi \\big ( a ^ { 2 , i } , b , c _ { 2 , i } , d , D u _ m ( y , t _ 1 ) , D ^ 2 u _ m ( y , t _ 1 ) \\big ) + r u _ m ( y , t _ 1 ) \\bigg ) \\\\ & \\leq \\partial _ t u _ m ( y , t _ 1 ) + \\frac { 1 } { k } \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} v _ n ( x + x ^ { j _ 0 } _ n ) = V ^ { j _ 0 } ( x ) + \\sum _ { 1 \\leq j \\leq l \\atop j \\ne j _ 0 } V ^ j ( x + x _ n ^ { j _ 0 } - x ^ j _ n ) + \\tilde { v } ^ l _ n ( x ) , \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\delta ( K _ { G / G _ i } ( X ^ { G _ i } ) ) = 0 . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} d X _ { k } ^ { \\epsilon , u _ k } ( t ) = f _ k \\bigl ( X _ { k } ^ { \\epsilon , u _ k } ( t ) , u _ { k } ^ { \\epsilon } ( t ) \\bigr ) d t + \\sqrt { \\epsilon } \\sigma _ k \\bigl ( X _ { k } ^ { \\epsilon , u _ k } ( t ) \\bigr ) d W ( t ) , \\\\ X _ { k } ^ { \\epsilon , u _ k } ( 0 ) = x _ { 0 } , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} N _ 1 = \\lceil m n + 1 \\rceil \\ ; , \\ ; \\ ; N _ 2 = \\lceil 2 m n + 1 \\rceil \\ ; . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} X ( s ) & = \\sqrt { 2 } \\big ( 1 + s f \\big ) ( e ^ { i \\theta _ 1 } , e ^ { i \\theta _ 2 } ) \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\gamma _ i : = \\inf \\left ( - \\nu ( h _ i h _ { i + 1 } \\dots h _ n ) \\right ) - \\sum _ { j = i + 1 } ^ n \\gamma _ { j } , \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { \\partial L } { \\partial \\dot z ^ i } ( t , z ( t ) , \\dot z ( t ) ) - \\frac { \\partial L } { \\partial z ^ i } ( t , z ( t ) , \\dot z ( t ) ) = \\alpha ^ j ( t ) \\frac { \\partial f _ j } { \\partial z ^ i } ( t , z ( t ) ) \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} [ X _ i , A _ j ] & = - ( 1 - \\delta _ { i j } ) \\tfrac { 1 } { | I | - 1 } D _ { e _ { i j } } X _ j \\\\ [ X _ i , B _ j ] & = - 2 \\delta _ { i j } A _ i - ( 1 - \\delta _ { i j } ) ( \\textstyle \\frac { 2 } { | I | - 2 } D _ { c _ { j i } } X _ j - D _ { c _ { i j } } X _ i - D _ { e _ { i j } } Y _ i + \\frac { 1 } { | I | - 1 } D _ { e _ { i j } } ( Y _ j + Z ) ) \\\\ [ X _ i , C _ j ] & = - \\delta _ { i j } B _ i + \\tfrac { 1 } { | I | } B _ i + ( 1 - \\delta _ { i j } ) ( D _ { k _ { i j } } X _ i + D _ { c _ { j i } } Y _ i - \\tfrac { 1 } { | I | - 2 } D _ { c _ { j i } } Z ) \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( 1 + g ( U _ { \\varepsilon , x _ \\varepsilon } ) \\right ) \\exp \\left ( U _ { \\varepsilon , x _ \\varepsilon } ^ 2 \\right ) d y ~ & ~ = \\int _ \\Omega \\left ( 1 + g ( u _ \\varepsilon ) \\right ) \\exp \\left ( u _ \\varepsilon ^ 2 \\right ) d y + o \\left ( \\gamma _ \\varepsilon ^ 2 \\check { \\zeta } _ \\varepsilon \\right ) \\ , , \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} L ( { \\cal D } _ j ) \\geq \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { q } l ( Y _ k , { \\cal D } _ j ) \\geq q \\cdot A w _ n \\geq 2 A ^ 3 \\epsilon _ 1 n ^ { 2 / 3 } w _ n . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} ( d i v V ^ 2 \\nabla \\tau ) ^ 2 = & 4 \\sum _ { i j } ( C _ i Y ^ 4 + B _ i Y ^ 0 + D _ p \\epsilon _ { p q i } Y ^ q ) ( C _ j Y ^ 4 + B _ j Y ^ 0 + D _ m \\epsilon _ { m n j } Y ^ n ) ( \\tilde { X } ^ i \\tilde { X } ^ j ) r ^ { - 2 } \\\\ & + 4 ( C _ i \\tilde { X } ^ i ) ( C _ j \\tilde { \\sigma } ^ { a b } \\gamma _ { a b } ^ { ( 2 ) c } \\tilde { X } _ c ^ j ) - 4 ( C _ i \\tilde { X } ^ i ) ( C _ j \\tilde \\Delta Y _ j ^ { ( 3 ) } + A \\tilde \\Delta Y _ 0 ^ { ( 3 ) } ) + O ( r ) , \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\tilde { D } ^ { \\prime } _ { u } v = D _ { u } v - \\frac { 1 } { 4 } \\{ ( D \\mathcal J ) u , \\mathcal J \\} v - \\frac { 1 } { 2 } \\mathcal J ( D _ { u } \\mathcal J ) v . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\frac { 1 } { \\widetilde { m } } \\widetilde { m } ^ 2 _ { x _ j } d y + \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } \\widetilde { w } _ { y _ i x _ j } ^ 2 d y = \\int _ { \\mathcal { Y } ^ d } V _ { x _ j } \\widetilde { m } _ { x _ j } d y \\leq \\int _ { \\mathcal { Y } ^ d } \\frac { 1 } { 2 \\widetilde { m } } \\widetilde { m } _ { x _ j } ^ 2 d y + \\int _ { \\mathcal { Y } ^ d } \\frac { V _ { x _ j } ^ 2 } { 2 } \\widetilde { m } d y . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\omega \\left ( ( W _ m + W _ { ( 0 , 0 ) } ) ^ * ( W _ m + W _ { ( 0 , 0 ) } ) \\right ) = 2 + \\omega \\left ( W _ { m } ^ * \\right ) + \\omega \\left ( W _ m \\right ) \\in [ 0 , \\infty ) \\ , . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} - 2 t f ( 1 - f ) ^ { - 1 } \\left \\| \\overline { \\nabla } f \\right \\| \\left \\| \\overline { \\nabla } \\varphi \\right \\| \\phi & \\leq t \\phi \\left ( \\frac { \\left \\| \\overline { \\nabla } f \\right \\| ^ { 2 } } { 1 - f } \\varphi + \\frac { \\left \\| \\overline { \\nabla } \\varphi \\right \\| ^ { 2 } } { \\varphi } \\frac { f ^ { 2 } } { 1 - f } \\right ) \\\\ & = t ( 1 - f ) F ^ { 2 } + t F \\frac { c _ { 2 } } { \\rho ^ { 2 } } \\frac { f ^ { 2 } } { 1 - f } . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\lambda ^ * \\beta w - \\Delta ( \\beta w ) & = \\beta F - \\beta \\nabla _ H \\Pi - 2 ( \\partial _ z \\beta ) ( \\partial _ z w ) - ( \\partial _ z ^ 2 \\beta ) w \\quad \\Omega ' , \\\\ \\partial _ z ( \\beta w ) & = 0 \\Gamma _ u ' , \\beta w = 0 \\Gamma _ b ' , \\partial _ z ( \\beta w ) \\Gamma _ l ' . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\varphi ( 0 ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\varphi ( 1 ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} ( T ^ * T ^ { * * } ) _ { \\rm { s } } ^ { 1 / 2 } = \\int _ 0 ^ \\infty \\lambda \\ , d E _ \\lambda \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) = \\sigma ^ 2 f ' ( x ) - ( x - \\mu ) f ( x ) \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq n } f _ j ( I _ j ) = f _ { j _ 1 } ( I _ { j _ 1 } ) \\leq f _ { j _ 1 } ( \\widehat { I } _ { j _ 1 } ) \\leq \\max _ { 1 \\leq j \\leq n } f _ j ( \\widehat I _ j ) \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\beta _ { \\textrm { N } } = \\max \\left \\lbrace 1 - \\frac { C _ { \\textrm { N } } ^ { \\textrm { N O M A } } } { | h _ { \\textrm { B N } } | ^ 2 } , 0 \\right \\rbrace , \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} T _ n ( \\phi , F ) = T _ m ( \\phi , F ) \\phi ^ { k } ( T _ m ( \\phi , F ) ) \\leq T _ m ( \\phi , F ) \\cdot T _ k ( \\phi , F _ 2 ) , \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\varepsilon _ k \\alpha _ k ^ 2 [ - f h \\varphi ^ 2 f '' + ( n - 2 ) f h \\varphi \\varphi ' f ' - ( m - 1 ) h \\varphi ^ 2 ( f ' ) ^ 2 - r f \\varphi ^ 2 f ' h ' ] = h [ \\rho f ^ 2 - \\lambda _ F ] . \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 2 } \\right ) ^ { \\alpha ^ { m - 1 } - 1 } , \\ \\quad \\ \\sum _ { m = 1 } ^ { \\infty } m ^ { 2 H + 1 } \\left ( \\frac { 1 } { 2 } \\right ) ^ { 2 ( 1 - H ) ( \\alpha ^ { m - 1 } - 1 ) } \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} d ( g _ { 1 } , g _ { 2 } ) = \\left \\{ \\begin{array} { l l l } | a _ { 1 } - a _ { 2 } | & & r _ { 1 } = r _ { 2 } \\\\ \\infty & & r _ { 1 } \\neq r _ { 2 } \\end{array} \\right . . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} ( B \\rtimes ( h \\ltimes ( b _ { i j } ) _ { i \\rho j } ) ) \\cdot ( A \\rtimes ( g \\ltimes ( a _ { i j } ) _ { i \\rho j } ) ) = ( B \\ ; \\ ; ^ { h ^ { - 1 } } ( ( b _ { i j } ) _ { i \\rho j } \\cdot A ) ^ { h ^ { - 1 } } ) \\rtimes ( h g \\ltimes ( b _ { \\tilde { g } ( i ) \\tilde { g } ( j ) } a _ { i j } ) _ { i \\rho j } ) . \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} w '' = \\left ( \\sum _ { j = 0 } ^ 3 k _ j \\wp ( z - \\omega _ j / 2 ) + \\lambda \\right ) w , k _ j = \\alpha _ j ^ 2 - 1 / 4 . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\sum \\limits _ { q \\le Q } \\sum \\limits _ { \\substack { a = 1 \\\\ ( a , q ) = 1 } } ^ { q ^ k } \\left | \\sum \\limits _ { M < n \\le M + N } a _ n e \\left ( n \\cdot \\frac { a } { q ^ k } \\right ) \\right | ^ 2 \\ll _ { \\varepsilon } ( Q N ) ^ { \\varepsilon } \\left ( N + Q ^ { k + 1 } \\right ) \\cdot \\sum \\limits _ { M < n \\le M + N } | a _ n | ^ 2 \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} u _ \\lambda ( t , x ) : = \\lambda ^ { \\frac { 2 s } { \\alpha } } u ( \\lambda ^ { 2 s } t , \\lambda x ) , \\lambda > 0 . \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} | T ( f , g ) | \\lesssim \\sum _ { i = 1 } ^ { 3 ^ n } T _ { \\mathcal { S } _ i } ( f , g ) , \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} M ^ { v } ( n , 1 ) = ( M n , 1 ) M ^ { v } ( n , - 1 ) = ( M n + v , - 1 ) . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} & 0 = - \\partial _ x F _ \\lambda - k ( x , \\lambda \\ , M _ \\lambda ) F _ \\lambda = \\mathcal { L } _ { \\lambda M _ \\lambda } F _ \\lambda , \\\\ & F _ \\lambda ( 0 ) = M _ \\lambda , \\langle F _ \\lambda \\rangle = 1 . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} & \\Omega ^ { ( 1 ) } : = \\Omega \\ , , \\ \\Gamma ^ { ( 1 ) } : = \\C ( \\Gamma ) \\ , , \\ \\Gamma ^ { ( 1 ) } ( t ) : = \\C ( \\Gamma ( t ) ) \\ , , \\\\ & A ^ { ( 1 ) } ( x ) : = [ D \\C \\ , A \\ , D \\C ^ T ] ( \\C ^ { - 1 } ( x ) ) \\ , . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\mathcal { X } ( t ) + \\int ^ t _ { 0 } \\mathcal { X } ( \\tau ) d \\tau \\leqslant C _ 0 \\sum _ { j = 1 } ^ { m } \\int ^ t _ { 0 } \\mathcal { X } ( \\tau ) ^ { \\alpha _ j } d \\tau + C _ 0 \\sum _ { k = 1 } ^ { n } \\mathcal { X } ( t ) ^ { \\beta _ k } + C _ 0 \\sum _ { k = 1 } ^ { n } \\mathcal { X } ( 0 ) ^ { \\beta _ k } + C _ 0 \\mathcal { X } ( 0 ) , \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\sum _ { \\{ \\rho _ 1 , \\rho _ 2 : | \\rho _ 1 - \\rho _ 2 | \\ge 4 \\sqrt { n } \\} } & C _ { \\rho _ 1 , \\rho _ 2 } ( j ) \\| F _ { \\rho _ 1 } \\| _ { L _ { y , s } ^ 2 ( | ( y , s ) | ^ { \\alpha p } ) } \\| G _ { \\rho _ 2 } \\| _ { L _ { x , t } ^ 2 ( | ( x , t ) | ^ { \\alpha p } ) } \\\\ & = \\sum _ { \\{ \\rho _ 1 , \\rho _ 2 : | \\rho _ 1 - \\rho _ 2 | \\ge 4 \\sqrt { n } , | \\rho _ 1 | \\ge 1 \\} } + \\sum _ { \\{ \\rho _ 1 , \\rho _ 2 : | \\rho _ 1 - \\rho _ 2 | \\ge 4 \\sqrt { n } , | \\rho _ 1 | < 1 \\} } . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal J } ( T ^ { D } ) = \\frac { 1 } { 4 } N _ { \\mathcal J } \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\mathcal { L } _ { X _ { h e t } } L ( x , y , z ) = - 2 \\varepsilon x ^ 2 y ^ { 2 ( 2 \\lambda - 1 ) } \\left [ ( y ^ 2 - \\lambda x ^ 2 ) ^ 2 + y ^ 2 z ^ 2 + \\lambda ^ 2 x ^ 2 z ^ 2 \\right ] \\leq 0 , ~ \\forall ( x , y , z ) \\in \\mathbb { R } ^ 3 . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} ( \\rho ( x , 0 ) , u ( x , 0 ) ) = ( \\rho _ 0 ( x ) , u _ 0 ( x ) ) . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = \\sup \\{ H _ { a l g } ( \\phi , F ) : F \\ \\ G \\} . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\delta _ k F _ k + F _ { k + 1 } \\delta _ { k + 1 } = n \\ , \\mathrm { i d } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & \\big ( \\frac { d } { d x } r _ t + \\alpha _ { \\rm H J M } \\big ) d t + \\sigma d X _ t \\medskip \\\\ r _ 0 & = & h _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} \\nabla f ( x ^ k ) + \\nabla g ( x ^ k ) \\lambda ^ k = a ^ k \\ \\ { \\rm a n d } \\ \\ \\lambda ^ k \\in \\mathcal { N } _ { \\mathcal { K } } ( g ( x ^ k ) - b ^ k ) . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\kappa ^ n _ j ( \\alpha ) = \\frac { \\Gamma ( \\alpha + 1 + ( n + j ) / 2 ) } { \\Gamma ( \\alpha + 1 ) \\Gamma ( 1 + ( n + j ) / 2 ) } \\kappa ^ n _ j ( 0 ) \\ , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} d ( v _ \\ell , v _ { \\ell + 1 } ) & = 2 \\mbox { $ ( \\ell < k ) $ } \\mbox { a n d } d ( b , v _ \\ell ) = k \\mbox { $ ( \\ell \\leq k ) $ } \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} g _ { - 1 } ( q ) = h _ 0 ( q ) \\sum _ { k = 0 } ^ 3 \\binom { 3 } { k } ( - 1 ) ^ k k = 0 \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} & W _ { n - 1 } \\cdot \\exp \\left ( \\alpha _ 1 U _ { n - 1 } ^ 2 \\right ) \\\\ = & W _ { n - 2 } \\cdot \\exp \\Bigg ( \\alpha _ 1 \\left ( Z _ { n - 1 } + \\left ( a + \\frac { s } { 2 \\alpha _ 1 } \\right ) U _ { n - 2 } \\right ) ^ 2 + \\\\ & \\left ( a ^ 2 \\alpha _ 1 - s \\eta \\right ) U _ { n - 2 } ^ 2 - \\alpha _ 1 \\left ( a + \\frac { s } { 2 \\alpha _ 1 } \\right ) ^ 2 U _ { n - 2 } ^ 2 \\Bigg ) . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} { } _ { 2 } F _ { 1 } \\left ( a , b ; c ; x \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( a \\right ) _ { k } \\left ( b \\right ) _ { k } } { \\left ( c \\right ) _ { k } } \\frac { x ^ { k } } { k ! } , { \\rm f o r } \\ ; \\ ; | x | < 1 , \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { A ( G ) } = \\inf \\bigl \\{ \\left \\Vert g \\right \\Vert _ 2 \\left \\Vert h \\right \\Vert _ 2 : f = g \\ast h , \\ g , h \\in L ^ 2 ( G ) \\bigr \\} \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\chi _ 0 = \\| J \\| _ { L ^ 1 } , \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} P _ { V ^ { ( n ) } } ( v ) = \\epsilon _ { k , n } ( \\delta ) ( 1 - \\epsilon _ { k , n } ( \\delta ) ) ^ { v - 1 } , \\ \\ v = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = h _ { a l g } ( \\phi \\upharpoonright _ { G ' \\cap H } ) + h _ { a l g } ( \\widetilde { \\phi \\upharpoonright _ { G ' } } ) + h _ { a l g } ( \\widetilde { \\phi } \\upharpoonright _ { H G ' / G ' } ) + h _ { a l g } ( \\overline { \\widetilde { \\phi } } ) . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} R _ L ( F ) ( x , y ) & = R _ L ( \\sum _ { i = 1 } ^ { m } f _ i \\otimes g _ i ) ( x , y ) = R ( \\sum _ { i = 1 } ^ { m } ( f _ i \\otimes g _ i ) \\restriction _ { [ 0 , \\alpha ] \\times { y } } ) ( x ) \\\\ & = \\sum _ { i = 1 } ^ { m } R ( f _ i ) ( x ) \\cdot g _ i ( y ) = \\sum _ { i = 1 } ^ { m } ( R ( f _ i ) \\otimes g _ i ) ( x , y ) = G ( x , y ) . \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\begin{cases} d Z _ { t } ^ { \\varepsilon } = \\sum _ { \\alpha = 1 } ^ { d } \\varepsilon \\cdot W _ { \\alpha } ( Z _ { t } ^ { \\varepsilon } ) d B _ { t } ^ { \\alpha } , \\\\ Z _ { 0 } ^ { \\varepsilon } = 0 . \\end{cases} \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} \\left \\langle \\psi ^ n T _ a , T _ j , \\tau ^ { ( k ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , k + 2 , d } = \\sum _ { u + v = k } \\frac { k ! } { u ! v ! } \\left \\langle \\psi ^ n T _ a , T _ j , \\tau _ 2 ^ { ( u ) } , \\tau '^ { ( v ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , k + 2 , d } \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} w _ m ( u ) = \\frac { w \\left ( \\frac { u } { l _ m } \\right ) } { l _ m } \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\sum _ { k } h ^ { p ^ { \\ast } } _ { i j k } \\omega _ k = d h ^ { p ^ { \\ast } } _ { i j } + \\sum _ k h ^ { p ^ { \\ast } } _ { i k } \\omega _ { k j } + \\sum _ k h ^ { p ^ { \\ast } } _ { k j } \\omega _ { k i } + \\sum _ { q } h ^ { q ^ { \\ast } } _ { i j } \\omega _ { p ^ { \\ast } q ^ { \\ast } } , \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} | | v _ \\ell | | _ { L ^ { \\infty } ( B ( p _ 3 , a t ) ) } \\leq C t W ( t ) , \\ \\ \\ \\ \\ell = 1 , 2 . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} T _ { \\rm s i n g } = ( P _ r + Q _ m ) T = P _ r T + T _ { \\rm m u l } = ( T _ { \\rm o p } ) _ { \\rm s i n g } + T _ { \\rm m u l } . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\frac { 1 } { \\displaystyle \\max _ { \\{ I _ j \\} \\in \\mathcal C _ n ( I ) } \\min _ { 1 \\leq j \\leq n } f _ j ( I _ j ) } = \\min _ { \\{ I _ j \\} \\in \\mathcal C _ n ( I ) } \\max _ { 1 \\leq j \\leq n } \\frac { 1 } { f _ j ( I _ j ) } , \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} z ( x , t ) & : = U ( x , t ) - W ^ { \\sigma ^ * } ( x , t ) \\\\ & : = U ( x , t ) - U ( x _ 1 + c \\tau + \\sigma ^ * , x ' + \\rho , t + \\tau ) \\succeq ( 0 , 0 , \\cdots , 0 ) , \\\\ z _ { l _ 0 } ( x _ { 1 , \\infty } , & 0 , 0 ) = 0 \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\nu ( \\frac { p _ 1 ( V ) - ( 2 k + 1 ) c ^ 2 } { 2 } ) = \\mu _ 1 ( V ) - ( 2 k + 1 ) s c . \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{gather*} S p l _ X ( f , \\sigma , \\{ k _ j \\} , L , \\{ R _ i \\} ) \\ ! : = \\{ p \\in S p l _ X ( f , \\sigma , \\{ k _ j \\} ) \\mid r _ i \\equiv R _ i \\bmod L \\ , ( 1 \\le { } ^ \\forall i \\le n ) \\} \\end{gather*}"}
-{"id": "821.png", "formula": "\\begin{align*} f ( t x _ 1 + ( 1 - t ) x _ 3 ) = t f ( x _ 1 ) + ( 1 - t ) f ( x _ 3 ) . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} h ( s ( y ) ) + y s ( y ) & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & + ( 1 + O ( y ^ { - \\kappa } ) ) ( K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } ( u + 1 ) y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) \\tfrac { 1 } { u } ( K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } ( u + 1 ) ^ { u / ( u + 1 ) } y ^ { u / ( u + 1 ) } , \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\psi + R \\psi = & - \\frac { n - 1 } { n } \\tau ^ { 2 a _ 0 / t } \\psi ^ { N - 1 } \\\\ & + \\bigg [ | \\sigma + L W | ^ 2 + \\bigg ( 2 \\max \\big \\{ \\| \\varphi _ 0 \\| _ { L ^ \\infty } , \\ , 2 \\big \\} - \\| \\varphi \\| _ { L ^ \\infty } \\bigg ) _ + \\bigg ( \\| \\sigma + L W _ 0 \\| ^ 2 _ { L ^ \\infty } + k _ 0 ^ 2 \\bigg ) \\bigg ] \\psi ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ * L W = & \\frac { n - 1 } { n } \\psi ^ N d \\tau ^ { a _ 0 / t } . \\endaligned \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} & \\frac 1 2 \\norm { \\varphi ( t ) } _ H ^ 2 + \\int _ { Q _ t } | \\Delta \\varphi | ^ 2 - \\int _ { Q _ t } \\left ( \\psi ' ( \\varphi _ 1 ) - \\psi ' ( \\varphi _ 2 ) \\right ) \\Delta \\varphi \\\\ & = \\frac 1 2 \\norm { \\varphi _ 0 } _ H ^ 2 + \\int _ { Q _ t } \\left ( \\mathcal P \\sigma - \\alpha u \\right ) h ( \\varphi _ 1 ) \\varphi + \\int _ { Q _ t } ( \\mathcal P \\sigma _ 2 - a - \\alpha u _ 2 ) \\left ( h ( \\varphi _ 1 ) - h ( \\varphi _ 2 ) \\right ) \\varphi \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} g ( x , y ) = \\lim _ { n \\to \\infty } R _ L ( f _ n ) ( x , y ) = \\lim _ { n \\to \\infty } R ( f _ n \\restriction _ { [ 0 , \\alpha ] \\times \\{ y \\} } ) ( x ) = 0 . \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\delta _ q Q ^ k = Q \\partial _ Q Q ^ k \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\frac { p _ i } { p _ { N + 1 } } = \\frac { 1 } { ( - z ) ^ { N + 1 - i } } \\sum _ { 0 \\leq j _ 1 < \\cdots < j _ { N + 1 - i } \\leq N } \\lambda _ { j _ 1 } \\cdots \\lambda _ { j _ { N + 1 - i } } \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} x _ j ^ \\lambda = \\frac { t _ j ^ \\lambda } { \\sqrt { 1 - ( t _ j ^ \\lambda ) ^ 2 } } , \\omega _ j ^ \\lambda = ( 1 + ( t _ j ^ \\lambda ) ^ 2 ) ^ { - \\lambda } \\rho _ j ^ \\lambda , 0 \\le j \\le N . \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} \\left ( 1 + a ^ { 2 } \\right ) \\left ( \\frac { f _ { 1 } } { f _ { 1 } ^ { \\prime } } \\right ) ^ { \\prime } f _ { 2 } ^ { \\prime \\prime } + \\left ( \\frac { f _ { 1 } ^ { \\prime \\prime } } { f _ { 1 } ^ { \\prime } } \\right ) ^ { \\prime } f _ { 2 } = 0 . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} | x _ 1 | = \\left | h _ 0 x _ 0 ^ 3 \\left ( 1 - \\frac 1 { h _ 0 x _ 0 ^ 2 } \\right ) \\right | = h _ 0 | x _ 0 | ^ 3 \\left | 1 - \\frac 1 { h _ 0 x _ 0 ^ 2 } \\right | < h _ 0 | x _ 0 | ^ 3 . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} L _ { o c } ^ \\diamond ( X , Z ) : = L _ { o c } ( X , Z ) _ + - L _ { o c } ( X , Z ) _ + . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} M ^ m _ { i q } = \\begin{cases} 1 , & q = n \\left ( j - 1 \\right ) + K ^ i _ { m - j + 1 } j \\in \\left \\lbrace 1 , 2 , \\ldots , m \\right \\rbrace \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} H _ u ( U _ 0 ( & - \\theta _ 0 , x _ * ) , V _ 0 ( - \\theta _ 0 , x _ * ) ) \\ , U _ 0 ' ( - \\theta _ 0 , x _ * ) \\\\ & + H _ v ( U _ 0 ( - \\theta _ 0 , x _ * ) , V _ 0 ( - \\theta _ 0 , x _ * ) ) V ' _ 0 ( - \\theta _ 0 , x _ * ) \\\\ = & \\lim _ { j \\rightarrow \\infty } \\{ H _ u ( w _ 1 ^ j ( 0 , 0 ) , w _ 2 ^ j ( 0 , 0 ) ) \\ , n ( p ( x _ j , t _ j ) , t _ j ) \\cdot \\nabla w _ 1 ^ j ( 0 , 0 ) \\\\ & + H _ v ( w _ 1 ^ j ( 0 , 0 ) , w _ 2 ^ j ( 0 , 0 ) ) \\ , n ( p ( x _ j , t _ j ) , t _ j ) \\cdot \\nabla w _ 2 ^ j ( 0 , 0 ) \\} = 0 . \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} c Q _ { a , c } - H Q _ { a , c } ' - \\frac 1 2 Q ^ 2 _ { a , c } = 0 \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\left [ ( - \\lambda _ 0 + z Q \\partial _ Q ) \\cdots ( - \\lambda _ N + z Q \\partial _ Q ) - Q \\right ] f ( z , Q ) = 0 \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} f _ { X | Y } ( x | y ) = \\frac { f ( x , y ) } { f _ Y ( y ) } . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { m - 1 } { 2 } } [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 q ^ { - { k + 1 \\choose 2 } } } { ( q ; q ) _ k ^ 2 ( q ^ 2 ; q ^ 2 ) _ k } = \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , 0 ) . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big [ ( 3 + t ) E ( u ; t ) + E _ * ( u ; t ) \\Big ] = - ( 3 + 2 t ) \\| u ' \\| ^ 2 - \\| A ^ { 1 / 2 } u \\| ^ 2 . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\widehat { \\bf F } \\cdot \\widehat { \\bf W } = \\sum _ { \\iota = \\pm } \\left ( \\widehat { F } _ \\iota \\widehat { W } ^ \\star _ \\iota + \\widehat { H } _ \\iota \\widehat { Y } ^ \\star _ \\iota \\right ) . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - y ^ { \\prime \\prime } ( x ) = \\lambda y ( x ) , x \\in ( 0 , 1 ) , \\\\ y ( 0 ) = y ( a ) = y ( 1 ) , a \\in ( 0 , 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i x _ j x _ l } \\widetilde { m } d y = - \\int _ { \\mathcal { Y } ^ d } \\big ( \\widetilde { m } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\big ) _ { y _ i } \\widetilde { w } _ { x _ j x _ l } d y = 0 . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} V o l _ { \\gamma C } ( \\gamma C \\cap \\Theta ) = \\int _ { \\gamma C \\cap \\Theta } \\omega \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} \\eta ( \\sigma _ i ) = \\eta ( \\tau ) \\wedge \\nu _ i , \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} \\Sigma _ { 3 / 2 } & = \\sigma _ { 3 / 2 } \\Bigl ( \\frac { 2 7 7 } { 3 ^ 6 - 2 ^ 6 } \\Bigr ) = \\frac { 2 } { 6 6 5 } \\sqrt { \\frac { 3 0 5 6 7 1 4 5 1 7 6 2 6 1 6 8 8 9 6 6 1 4 4 5 6 3 6 7 9 0 8 7 3 } { 1 0 3 1 4 4 2 4 7 9 8 4 9 0 5 3 5 5 4 6 1 7 1 9 4 9 0 5 5 } } . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\hat h _ n = \\left \\{ \\begin{array} { c c } h _ n , & n < N _ 4 , \\\\ \\frac 1 { x ^ 2 _ { N _ 4 } } , & n \\ge N _ 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{gather*} g ( v _ { i , 1 } , \\dots , v _ { i , \\chi ( 1 ) } ) = ( v _ { i , 1 } , \\dots , v _ { i , \\chi ( 1 ) } ) A ( g ) { } ^ \\forall g \\in G , \\\\ v _ { i , j } v _ { k , l } = \\delta _ { i , l } v _ { k , j } 1 \\le i , j , k , l \\le \\chi ( 1 ) . \\end{gather*}"}
-{"id": "3271.png", "formula": "\\begin{align*} \\mathbf { H } ( t ) \\mathbf { U } = \\mathbf { f } , \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} P _ { \\textrm { o u t , F } } ^ { \\textrm { I S A N C } } = P _ { \\textrm { o u t , F } } ^ { \\textrm { C S A N C } } = & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace \\times \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace \\\\ & + \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N H F = 0 , F D F = 0 , N D F = 1 , N D N = 1 } \\right \\rbrace . \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} \\ell = \\frac { 1 } { 4 \\| { v ^ + } ' \\| _ { L ^ \\infty } } > 0 \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} & \\left ( a \\otimes b m - a b \\otimes m + b \\otimes m a \\right ) + \\left ( - a \\otimes m b + b a \\otimes m - b \\otimes a m \\right ) \\\\ = { } & d _ { } ( a \\otimes b \\otimes m ) - d _ { } ( b \\otimes a \\otimes m ) . \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} f = ( x _ 1 + x _ 2 ^ q , x _ 2 , x _ 3 ) g = ( x _ 1 + x _ 3 ^ m , x _ 2 , x _ 3 ) \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\int _ A \\L ^ { - \\ell } d \\tilde \\mu ^ { \\hat { G } } : = \\sum _ { n \\in \\mathbb N } \\tilde \\mu ^ { \\hat { G } } \\left ( \\ell ^ { - 1 } ( n ) \\right ) \\L ^ { - n } , \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} \\frac { \\langle v , S v \\rangle } { \\langle v , v \\rangle } = \\tilde x \\langle v , v \\rangle + \\frac { \\langle w , ( \\hat \\Sigma - \\tilde \\Sigma ) w \\rangle } { \\langle v , v \\rangle } > \\frac { \\tilde x } { 2 } \\frac { \\sqrt { n } } { z } + \\frac { \\langle w , ( \\hat \\Sigma - \\tilde \\Sigma ) w \\rangle } { \\langle v , v \\rangle } . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} 2 \\sum _ { k = 0 } ^ { A _ j - 1 } \\| T _ { { \\gamma _ k } , j } ( x _ { v + j } ) \\| _ 2 ^ 2 & = \\sum _ { k = 0 } ^ { A _ j - 1 } \\int _ { \\Omega } | m _ { { \\gamma _ k } } ( \\omega ) | ^ 2 \\left | \\eta _ j \\left ( \\frac { f ( \\omega ) } { { \\gamma _ k } } \\right ) \\right | ^ 2 | U _ { v + j } ( x ) ( \\omega ) | ^ 2 d \\mu ( \\omega ) \\\\ & \\leq 2 A _ j a _ j ^ 2 \\| x _ { v + j } \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\| u ' ( t ) \\| ^ 2 + \\| A ^ { 1 / 2 } u \\| ^ 2 + 2 \\int _ 0 ^ t \\| u ' ( s ) \\| ^ 2 \\ , d s = \\| u _ 1 \\| ^ 2 + \\| A ^ { 1 / 2 } u _ 0 \\| ^ 2 . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} V & , \\\\ \\Pi _ U \\alpha & \\\\ \\sigma ^ k ( \\mathbb { M } ) & \\subset V , k = 1 , \\ldots , m , \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m - 1 } [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 q ^ { - { k + 1 \\choose 2 } } } { ( q ; q ) _ k ^ 2 ( q ^ 2 ; q ^ 2 ) _ k } = \\sum _ { n = 0 } ^ { m - 1 } F ( n , 0 ) . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| \\bar { \\boldsymbol { \\psi } } ( t _ { k + l } ) - \\tilde { \\boldsymbol { \\psi } } ^ k ( t _ { k + l } ) \\right \\| _ 2 \\\\ \\le & L \\sum _ { i = 1 } ^ { l } b _ { S , k + i } \\left \\| \\bar { \\boldsymbol { \\psi } } ( t _ { k + i - 1 } ) - \\tilde { \\boldsymbol { \\psi } } ^ k ( t _ { k + i - 1 } ) \\right \\| _ 2 + C , \\end{aligned} \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\beta _ { \\textrm { F } } = \\max \\left \\lbrace 1 - \\frac { C _ { \\textrm { F } } ^ { \\textrm { O F D M A } } } { | h _ { \\textrm { B F } } | ^ 2 } , 0 \\right \\rbrace , \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} 6 u ( x ) = u ( x , x , x ) . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\mbox { v a r } ( Y _ i ) = & \\frac { m p } { 1 \\ ! - \\ ! p ^ 2 } + 2 \\frac { m p ^ 2 } { ( 1 \\ ! - \\ ! p ) ( 1 \\ ! - \\ ! p ^ 2 ) } = \\frac { m p ( 1 \\ ! + \\ ! p ) } { ( 1 \\ ! - \\ ! p ) ( 1 \\ ! - \\ ! p ^ 2 ) } \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} n & = \\sum _ { x \\in \\mathbb { F } _ q ^ * } \\left ( \\frac { 1 } { p } \\sum _ { y \\in \\mathbb { F } _ p } \\zeta _ p ^ { y ^ m _ 1 ( x ^ d ) } \\right ) \\\\ & = \\frac { 1 } { p } \\sum _ { y \\in \\mathbb { F } _ p } \\sum _ { x \\in \\mathbb { F } _ q ^ * } \\chi _ m ( y x ^ d ) \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\mu ( \\{ - M , M \\} ) = 0 \\ , , \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla U _ { \\varepsilon , z _ \\varepsilon } | ^ 2 d x = 4 \\pi \\left ( 1 + I _ { z _ \\varepsilon } ( \\gamma _ \\varepsilon ) + o \\left ( \\check { \\zeta } _ \\varepsilon \\right ) \\right ) \\ , , \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} v ^ T \\nabla _ { x x } \\mu _ f ( x , t ) v = v ^ T \\nabla ^ 2 f ( x ) v + \\frac { t ^ 2 } { 6 } v ^ T C v = v ^ T \\nabla ^ 2 f ( x ) v + \\frac { a \\ , t ^ 2 } { 6 } \\ , . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\mu \\left ( [ - M + 1 , M - 1 ] ^ c \\right ) = 0 \\ , , \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\mathcal { F } ( V ( s ) ) & = s \\mathcal { F } _ 1 ( V _ 1 ) + \\frac { s ^ 2 } { 2 } \\big ( \\mathcal { F } _ 1 ( V _ 2 ) + \\mathcal { F } _ 2 ( V _ 1 , V _ 1 ) \\big ) + \\frac { s ^ 3 } { 3 ! } \\big ( \\mathcal { F } _ 1 ( V _ 3 ) + 3 \\mathcal { F } _ 2 ( V _ 1 , V _ 2 ) + \\mathcal { F } _ 3 ( V _ 1 , V _ 1 , V _ 1 ) \\big ) \\\\ & + \\ldots + \\frac { s ^ n } { n ! } \\big ( \\mathcal { F } _ 1 ( V _ n ) + n \\mathcal { F } _ 2 ( V _ 1 , V _ { n - 1 } ) + \\ldots + \\mathcal { F } _ n ( V _ 1 , \\ldots , V _ 1 ) \\big ) + \\ldots \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} \\psi _ { P , a } \\left ( \\mathbf { O } \\left ( A , Y ; \\mathcal { G } \\right ) ; \\mathcal { G } \\right ) = \\chi _ { P , a , e f f } ^ { 1 } ( \\mathbf { V } ; \\mathcal { G } ) = q _ { \\mathcal { G } } \\left ( \\mathbf { W } ; P \\right ) + h _ { \\mathcal { G } } \\left ( A , \\mathbf { O , M , } Y ; P \\right ) P \\in \\mathcal { M ( G ) } . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\hat { p } _ j ( X _ j ) = \\hat { p } _ j ( \\Lambda ( X _ j ) ) = \\mathbf { 0 } _ n = \\hat { p } _ j ( \\Lambda ( Y _ j ) ) = \\hat { p } _ j ( Y _ j ) \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} \\beta _ \\theta ( t ) = \\frac { \\sin ( \\pi \\theta ) } { 2 \\theta \\big ( \\cosh ( \\pi t ) + \\cos ( \\pi \\theta ) \\big ) } , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} z ^ { * } \\Gamma _ { \\Phi _ { 1 } ( x ; h ) } z = C _ { H } \\int _ { [ 0 , 1 ] ^ { 2 } } \\langle \\xi _ { s } , \\xi _ { t } \\rangle _ { \\mathbb { R } ^ { d } } | t - s | ^ { 2 H - 2 } d s d t , \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\mu ( \\cdot ) : = \\lim _ { n \\rightarrow \\infty } \\mu _ { \\beta , h } ^ + ( \\cdot \\mid \\eta = - 1 ( - \\infty , - n ] \\times \\{ 0 \\} ) . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} - 2 a \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } = c _ { 1 } \\left ( 1 + a ^ { 2 } \\right ) \\frac { f _ { 1 } } { f _ { 1 } ^ { \\prime } } + \\frac { f _ { 1 } ^ { \\prime \\prime } } { f _ { 1 } ^ { \\prime } } . \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} ( w _ p - 1 ) / p ^ 3 = S _ { p , 2 } / p + p S _ { p , 4 } + p ^ 3 S _ { p , 6 } + \\cdots + p ^ { p - 6 } S _ { p , p - 3 } + p ^ { p - 4 } S _ { p , p - 1 } . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\underline { \\Delta } F ( x _ { 0 } , t _ { 0 } ) = 0 , \\ ; \\ ; \\ ; \\partial _ { t } F ( x _ { 0 } , t _ { 0 } ) \\ge 0 \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\underline { \\Delta } F ( x _ { 0 } , t _ { 0 } ) \\le 0 . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\lim _ { C \\to + \\infty } \\int _ X ( u ^ C - V _ { \\theta } ) \\theta _ { u ^ C } ^ k \\wedge \\theta _ { v ^ C } ^ { n - k } = \\int _ X ( u - V _ { \\theta } ) \\theta _ { u } ^ k \\wedge \\theta _ { v } ^ { n - k } . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} E = E ( \\lambda ) : = \\bigcap ^ \\infty _ { n = 1 } E _ n . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} f _ n : B ( 0 , 1 ) \\to \\R ^ d , f _ n ( x ) = \\frac { 2 } { n } \\frac { x - a _ n } { | x - a _ n | ^ 2 } , a _ n = ( 1 + 1 / n , 0 , \\ldots , 0 ) . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 5 { R _ 3 ^ 5 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 5 { R _ 3 ^ 5 } ) = 4 0 , \\mathrm { I C } ( \\mathcal { T } ^ 5 { R _ 3 ^ 5 } ) = 4 0 . \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N H F } } ^ { \\textrm { O F D M A } } = \\frac { P _ \\textrm { F } { | h _ { \\textrm { B F } } | ^ 2 } } { { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } \\theta } + \\frac { \\beta _ { \\textrm { N } } \\eta P _ \\textrm { B } | h _ { \\textrm { B N } } | ^ 2 | h _ { \\textrm { N F } } | ^ 2 } { d _ { \\textrm { B N } } ^ { \\alpha } d _ { \\textrm { N F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 \\theta } . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\sum _ { t _ n \\in \\mathcal { Z } _ { A _ 2 } } \\frac { G _ 2 ( t _ n ) } { A _ 2 ' ( t _ n ) ( z - t _ n ) } = \\frac { G _ 2 ( z ) } { A _ 2 ( z ) } - G _ 2 ( z ) \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} V = - ( N - 1 ) k ( x ) \\kappa - \\frac { \\partial } { \\partial n } k ( x ) - \\frac { 2 k ( x ) ( C + 1 ) } { K ( x ) } \\frac { \\partial } { \\partial n } K ( x ) . \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} 2 q _ 4 = p _ 1 \\in H ^ \\ast ( B S p i n ( n ) ; \\mathbb { Z } ) , \\ \\ \\ \\ 2 q _ 4 + t _ 2 ^ 2 = p _ 1 \\in H ^ \\ast ( B S p i n ^ c ( n ) ; \\mathbb { Z } ) . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} \\Delta = \\Delta _ 0 \\supset \\Delta _ 1 \\supset \\dots \\supset \\Delta _ m = \\emptyset \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\nabla \\phi _ t = \\nabla \\phi _ { t ' } \\mu _ t \\ae \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 9 \\\\ 2 \\\\ 6 \\\\ 4 \\\\ 8 \\\\ 7 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c c c c c } 4 & 4 & 9 & 3 & 4 & 1 \\\\ 4 & 3 & 7 & 9 & 9 & 2 \\\\ 9 & 7 & 4 & 7 & 6 & 6 \\\\ 3 & 9 & 7 & 4 & 2 & 6 \\\\ 4 & 9 & 6 & 2 & 8 & 3 \\\\ 1 & 2 & 6 & 6 & 3 & 5 \\end{array} \\right ] , d = \\left [ \\begin{array} { c } 2 \\\\ 6 \\\\ 5 \\\\ 0 \\\\ 0 \\\\ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} | x _ { N + k } | \\le | x _ N | + \\sum _ { i = 1 } ^ k | u _ { N + i } | , \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\frac { \\eta _ 1 n } { N } \\leq \\mathbb { E } _ 0 N ^ { ( P ) } _ l = n p _ l \\leq \\frac { \\eta _ 2 n } { N } \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\textit { \\textbf { h } } _ k ( f ) = \\sum \\limits _ { p = 1 } ^ { N _ { \\rm S } + 1 } \\alpha _ { k , p } \\ , \\textbf { a } \\left ( \\theta _ { k , p } \\right ) \\mathrm { e } ^ { - j 2 \\pi \\tau _ { k , p } f } , \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} & T \\left ( f _ 1 , f _ 2 , \\ldots , f _ n \\right ) ( x ) = \\sum e _ { 1 , j } T ( f _ { 1 , j } , f _ 2 , \\ldots , f _ n ) ( x ) , x \\in \\R ^ d , \\\\ & f _ 1 = \\sum e _ { 1 , j } f _ { 1 , j } , f _ { 1 , j } \\in L ^ \\infty _ c ( \\R ^ d ) , \\ , e _ { 1 , j } \\in X . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\overline { \\nabla } ( \\varphi G ) ( x _ { 0 } , t _ { 0 } ) = 0 , \\ ; \\ ; \\ ; \\frac { \\partial } { \\partial t } ( \\varphi G ) ( x _ { 0 } , t _ { 0 } ) \\geq 0 , \\ ; \\ ; \\ ; \\underline { \\Delta } ( \\varphi G ) ( x _ { 0 } , t _ { 0 } ) \\leq 0 , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} R : = \\epsilon _ m \\log N \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} \\int _ D p _ n ( z ) \\overline { p _ m ( z ) } w ( z ) \\ , \\d { A } ( z ) = \\delta _ { n , m } \\ . \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\begin{aligned} g ( s ) : = - \\frac { \\left ( \\frac { F ' ( s ) } { s ^ 2 } \\right ) ' F ' ( s ) } { \\left ( \\frac { F '^ 2 ( s ) } { s ^ 2 } \\right ) ^ 2 } , \\ \\ g ( p _ N ) > 0 . \\end{aligned} \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\int _ { | x | > R } | u ( t ) | ^ { \\alpha + 2 } d x = \\sum _ { j = 0 } ^ \\infty \\int _ { 2 ^ j R < | x | \\leq 2 ^ { j + 1 } R } | u ( t ) | ^ { \\alpha + 2 } d x , \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} & ( \\delta f ) ( a _ 1 , \\ , a _ 2 , \\ , \\ldots \\ , , a _ { n + 1 } ) = a _ 1 f ( a _ 2 , \\ , \\ldots \\ , , a _ { n + 1 } ) \\ , + \\\\ & \\Big ( \\sum _ { i = 1 } ^ n ( - 1 ) ^ i f ( a _ 1 , \\ , a _ 2 , \\ , \\ldots \\ , , a _ i a _ { i + 1 } , \\ , \\ldots \\ , , a _ n ) \\Big ) + ( - 1 ) ^ { n + 1 } f ( a _ 1 , \\ , a _ 2 , \\ , \\ldots \\ , , a _ n ) a _ { n + 1 } . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} \\Phi ( x ^ * , 0 ) - \\langle \\lambda , 0 \\rangle + r \\sigma ( 0 ) = f ( x ^ * ) = f ^ * \\forall r \\ge 0 , \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\Psi \\varphi ( x ) d x = - \\int _ { \\R ^ d } \\phi ( 0 , t , x ) \\nabla \\varphi ( x ) d x \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} f _ { l , u _ j } ( v ( \\cdot , t _ 0 ) ) = 0 \\ \\ { \\rm f o r } \\ \\ l \\in \\Lambda , \\ j \\not \\in \\Lambda . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} \\frac { c _ i + \\lambda _ i } { w _ i B } \\left ( \\frac { B \\left ( | \\mathcal { S } | - 1 \\right ) } { \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } \\right ) ^ 2 = \\frac { B \\left ( | \\mathcal { S } | - 1 \\right ) } { \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } - w _ i d _ i ^ { n e } , \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} Y [ a , z ] = R ^ { - 1 } Y ( R a , z ) R . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} d ( x _ n , x _ { n + k } ) & \\leq s \\{ d ( x _ n , x _ { n + 1 } ) + d ( x _ { n + 1 } , x _ { n + 2 } ) + \\hdots + d ( x _ { n + v } , x _ { n + k } ) \\} \\\\ & < s \\{ d ( x _ 0 , x _ 1 ) + d ( x _ 0 , x _ 1 ) + \\hdots + d ( x _ { n + v - 1 } , x _ { n + k - 1 } ) \\} \\\\ & \\hdots \\\\ & \\hdots \\\\ & < s \\{ v d ( x _ 0 , x _ 1 ) + d ( x _ 0 , x _ { k - v } ) \\} \\\\ & < s \\{ v d ( x _ 0 , x _ 1 ) + M \\} \\\\ & = M _ 1 , \\mbox { s a y } . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\left | \\tau ( \\sum _ n B ^ \\delta _ { R _ n } ( x ) y _ n ) \\right | & = | C _ { \\alpha , \\beta } | \\left | \\tau ( \\sum _ n \\left ( \\int ^ 1 _ 0 \\varphi ( t ) M ^ \\alpha _ { R _ n t } ( x ) d t \\right ) y _ n ) \\right | \\\\ & \\leq | C _ { \\alpha , \\beta } | \\int ^ 1 _ 0 | \\varphi ( t ) | \\left | \\tau ( \\sum _ n M ^ \\alpha _ { R _ n t } ( x ) y _ n ) \\right | d t \\\\ & \\lesssim \\| { \\sup _ { R > 0 } } ^ + M ^ \\alpha _ { R } ( x ) \\| _ 2 . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\langle T ( u ) , \\zeta \\rangle : = \\int _ M j u \\wedge d \\zeta . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} ( B L \\lambda _ { 2 k + 1 } ) ^ \\ast ( \\mu _ 3 ) = \\mu _ 3 - ( 2 k + 1 ) s _ 1 t _ 2 = s _ k ( x _ 2 ) . \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\chi _ G ( n ) \\ = \\sum _ { \\Pi \\in A ( G ) } \\Omega _ \\Pi ^ \\circ ( n ) \\ , . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} & T \\left ( f _ 1 , \\ldots , f _ n \\right ) ( x ) = \\sum _ { j _ 1 , \\ldots , j _ n } T ( f _ { 1 , j _ 1 } , \\ldots , f _ { n , j _ n } ) ( x ) \\prod _ { k = 1 } ^ n e _ { k , j _ k } , x \\in \\R ^ d , \\\\ & f _ { k } = \\sum _ { j _ k } e _ { k , j _ k } f _ { k , j _ k } , f _ { k , j _ k } \\in L ^ \\infty _ c ( \\R ^ d ) , \\ , e _ { k , j _ k } \\in X _ k . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\int _ { \\Gamma } \\overline { \\nabla u } \\nabla \\phi \\ , d t + ( z + \\mu ) \\int _ { \\Gamma } \\overline { u } \\phi \\ , d t = \\int _ { \\Gamma } \\overline f \\phi \\ , d t \\qquad \\forall \\phi \\in \\bigoplus _ { e \\in \\Gamma } H ^ 1 _ 0 ( e ) . \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\biggl | 2 \\sum _ { n = N } ^ \\infty \\frac { 1 } { 6 ^ n } V \\bigl ( \\langle 2 ^ n x \\rangle , \\langle 3 ^ n x \\rangle \\bigr ) \\biggr | & \\le \\frac { 1 } { 1 0 \\cdot 6 ^ { N - 1 } } . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\hat { \\lambda } _ j - \\lambda _ j - \\lambda _ j \\bar { \\eta } _ { j j } & = \\langle \\hat u _ j , \\hat \\Sigma \\hat u _ j \\rangle - \\lambda _ j \\langle \\hat u _ j , \\hat u _ j \\rangle - \\langle u _ j , E u _ j \\rangle \\\\ & = \\langle \\hat u _ j , E \\hat u _ j \\rangle - \\langle u _ j , E u _ j \\rangle + \\langle \\hat u _ j , ( \\Sigma - \\lambda _ j I ) \\hat u _ j \\rangle \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} c ( x , \\eta ) = \\begin{cases} 1 & \\eta ( x ) = 1 , \\\\ \\lambda \\sum _ { y : y \\sim x } \\eta ( y ) & \\eta ( x ) = 0 \\end{cases} \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\Xi ( \\tau \\otimes \\tau ) = 0 , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} Q ( u ^ { \\ast } ) & = \\sup _ { 0 < u \\le \\lambda } ( u x - \\log L _ { X } ( u ) ) \\\\ & = - \\log L _ { X } ( u ^ { \\ast } ) \\\\ & = I _ { p _ 1 p _ 2 } \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} d _ r ( C _ D ) = p ^ { m - 1 } \\left ( 1 - \\frac { 1 } { p ^ r } \\right ) , \\ 1 \\le r \\le m - 1 . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 2 \\xi _ 1 - 1 \\\\ 1 - 2 \\xi _ 1 & 0 \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ x _ 2 \\end{bmatrix} + \\begin{bmatrix} 0 \\\\ - 1 \\end{bmatrix} + \\begin{bmatrix} 2 \\lambda x _ 1 \\\\ - \\lambda \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\frac { d } { d t } | v ( t ) | _ H ^ 2 & = 2 \\left < v ( t ) , v ' ( t ) \\right > _ H \\\\ & = - 2 | v ( t ) | _ { H ^ 1 } ^ 2 - 2 b ( \\mathcal { M } ( \\Phi ) ( t ) , \\mathcal { M } ( \\Phi ) ( t ) , v ( t ) ) + 2 b ( \\mathcal { M } ( \\Psi ) ( t ) , \\mathcal { M } ( \\Psi ) ( t ) , v ( t ) ) . \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} A _ i \\tilde Q ( x , y ) = \\tilde Q ( A _ i x , y ) + \\tilde Q ( x , A _ i y ) \\quad \\mbox { f o r a l l } x , y \\in Y , \\ i = 1 , \\ldots , N \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} F = \\prod _ { u = 1 } ^ k B _ u ^ s ( F ) , \\| B _ u ^ s ( F ) \\| _ { E _ { u } ^ s } = 1 , \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } ( \\mathrm { I n d } ^ G _ { P _ b } \\circ \\otimes ^ k _ { i = 1 } ( \\mathrm { M a n t } _ { M _ { b _ i } , b ' _ i , \\mu _ { b i } } \\circ \\mathrm { R e d } _ { b ' _ i } ) \\circ ( \\delta _ { P _ b } \\otimes \\mathrm { J a c } ^ G _ { P ^ { o p } _ b } ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} ( A _ { n , 3 } V _ { n , 3 } + \\epsilon V _ { n , 3 } ) ( \\eta , y , x ) \\leq K _ 3 - K _ { 3 , \\eta } \\ ; \\eta ^ 2 - K _ { 3 , y } \\ ; y ^ 2 - K _ { 3 , x } \\ ; x ^ 2 + \\delta | y | + | x | \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} A _ { 1 } = \\begin{bmatrix} 0 & - 1 & - 2 & \\varepsilon & \\varepsilon \\\\ - 3 & \\varepsilon & \\varepsilon & \\varepsilon & - 3 \\\\ \\varepsilon & \\varepsilon & \\varepsilon & - 4 & \\varepsilon \\\\ \\varepsilon & - 5 & \\varepsilon & \\varepsilon & \\varepsilon \\\\ - 6 & \\varepsilon & \\varepsilon & - 5 & \\varepsilon \\end{bmatrix} , \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} & m _ 1 \\alpha _ 1 + m _ 2 \\alpha _ 2 + m _ 3 \\alpha _ 3 = m - \\sum _ { i > 3 } m _ i \\alpha _ i , \\\\ & m _ 3 \\alpha _ 1 + m _ 1 \\alpha _ 2 + m _ 2 \\alpha _ 3 = m - \\sum _ { i > 3 } m _ i \\alpha _ i , \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} \\mathcal K ( \\epsilon \\zeta ) = - \\epsilon \\zeta _ { x x } + O ( \\epsilon ^ 2 ) . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 ( \\varPi _ N ^ \\lambda \\varPhi ( t ) - \\varPhi ( t ) ) \\varPsi ( t ) \\omega _ \\lambda ( t ) \\ , d t = 0 , \\forall \\ , \\varPsi \\in { \\mathcal P } _ N . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\phi = 2 B ( x + 4 A t ) + ( 2 n + \\frac 3 2 ) \\ln t + \\ln \\ ( \\frac { 2 \\pi \\ | \\delta ( E _ 0 , A ) | ^ 2 } { n ! \\Gamma ( n + \\frac 3 2 ) } \\cdot \\frac { ( 1 6 B ^ 2 ) ^ { 2 n + \\frac 3 2 } } { | \\hat \\phi ( E _ 0 ) | \\sqrt { 2 B } } \\ ) \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = \\varepsilon u _ { x x } , \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} k _ { \\gamma , \\Sigma } ^ { \\infty , s } = \\frac { \\overline { p } \\dot { \\gamma } _ 1 + \\overline { q } \\dot { \\gamma } _ 2 } { \\gamma _ 1 | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} y \\le 0 , \\ \\ S ( y ) \\succeq 0 , \\ \\ S ( y ) _ { j j } = 0 , \\ \\ s ( y ) _ j = 0 . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) \\Phi ( n ) - ( 1 + \\epsilon ) \\Phi ( n - 1 ) = ( 1 - \\epsilon ) e ^ { n ^ \\alpha } - ( 1 + \\epsilon ) e ^ { ( n - 1 ) ^ \\alpha } \\geq ( 1 - 2 \\epsilon ) e ^ { n ^ \\alpha } . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} X _ { ( ( T ^ \\ast Q ) _ { \\mathcal { O } _ \\mu } , \\omega _ { \\mathcal { O } _ \\mu } , h _ { \\mathcal { O } _ \\mu } , f _ { \\mathcal { O } _ \\mu } , u _ { \\mathcal { O } _ \\mu } ) } \\cdot \\pi _ { \\mathcal { O } _ \\mu } = T \\pi _ { \\mathcal { O } _ \\mu } \\cdot X _ { ( T ^ \\ast Q , G , \\omega , H , F , u ) } \\cdot i _ { \\mathcal { O } _ \\mu } . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\nu _ { \\omega + 1 } : = \\left [ \\{ \\nu _ n \\} _ { n \\in \\N } , \\nu _ { \\omega + 1 } ( \\phi _ \\omega ) = \\gamma \\right ] . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { j = 0 } ^ N \\left ( \\sum _ { k = 0 } ^ { M } c _ { j k } \\underbrace { ( z \\diamond _ q \\cdots \\diamond _ q z ) } _ { k - \\mbox { t i m e s } } \\right ) \\underbrace { ( \\bar { z } \\diamond _ q \\cdots \\diamond _ q \\bar { z } ) } _ { j - \\mbox { t i m e s } } . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\phi _ \\tau ^ { ( 2 ) } = ( \\tau _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\tau _ r , 2 ) , \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} & \\left | u _ 0 ( x , t ) \\right | \\le C t ^ { - \\frac { N + 2 A } { 2 } } U ( | x | ) \\le \\nu t ^ { - \\frac { N + 2 A } { 2 } } , \\\\ & \\left | u _ { 1 , i } ( x , t ) \\right | \\le \\nu t ^ { - \\frac { N + 2 A } { 2 } } , | R _ 2 ( x , t ) | \\le \\nu t ^ { - \\frac { N + 2 A } { 2 } } , \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} { { \\pmb \\circ } } _ { O _ 1 O _ 2 O _ 4 } ( { { \\pmb \\circ } } _ { O _ 2 O _ 3 O _ 4 } \\otimes 1 ) = { { \\pmb \\circ } } _ { O _ 1 O _ 3 O _ 4 } ( 1 \\otimes { { \\pmb \\circ } } _ { O _ 1 O _ 2 O _ 3 } ) \\alpha . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\nu _ { b _ 1 } = ( { ( 2 / n _ 1 ) } ^ { n _ 1 } , 0 ^ { n _ 2 } ) , \\nu _ { b _ 2 } = ( { ( 2 / n _ 2 ) } ^ { n _ 2 } , 0 ^ { n _ 1 } ) . \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\mathcal { E } : = \\rho \\left ( e + \\frac { 1 } { 2 } u ^ 2 + \\frac { 1 } { 2 } | \\mathbf { w } | ^ 2 \\right ) + \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 , \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} s ( v , \\theta ) = \\int _ 1 ^ \\theta \\frac { C _ \\vartheta ( z ) } { z } d z - \\int _ v ^ 1 p _ \\theta ( z ) d z . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} q _ \\lambda ( { \\bf x } ) = q _ { \\lambda _ 1 } ( { \\bf x } ) q _ { \\lambda _ 2 } ( { \\bf x } ) \\cdots q _ { \\lambda _ l } ( { \\bf x } ) \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} z ^ * - \\bar { z } ^ * = \\lim \\limits _ { ( z , \\mu ) \\rightarrow ( z ^ * , 0 ) } ( x - z ) = \\lim \\limits _ { ( z , \\mu ) \\rightarrow ( z ^ * , 0 ) } \\sum \\limits _ { i \\in \\mathcal { I } ^ * } \\cfrac { \\mu ^ 2 } { A _ i x - b _ i } ( - A _ i ^ T ) \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\bf P } ( | | S _ n ( \\cdot ) | | L ( p , T ) | > u ) = { \\bf P } ( | | S ( t ) | | L ( p , T ) > u ) , \\ u > 0 . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { z } & < \\zeta > ^ { 2 n + 1 } d \\zeta = \\frac { 1 } { 2 n + 2 } < z > ^ { 2 n + 2 } , \\\\ \\int _ { 0 } ^ { z } & < \\zeta > ^ { 2 n } d \\zeta = \\frac { 1 } { 2 n + 1 } [ < z > ^ { 2 n + 1 } + \\frac { 1 } { 2 ^ { 2 n } } ( z - < z > ) ] . \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} t _ 0 : = \\sup \\{ t ' > 0 \\mid u ( \\cdot , t ) \\ll v ( \\cdot , t ) \\ \\ { \\rm f o r \\ a l l } \\ \\ t \\in [ 0 , t ' ] \\} \\in ( 0 , \\infty ) \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\begin{aligned} & \\max _ { x } & & 1 ^ { T } A x ( A x ) ^ { - } 1 . \\end{aligned} \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} 1 < p _ 1 , \\ldots , p _ { m } < \\infty , \\sum _ { j = 1 } ^ { m } \\textstyle \\frac { 1 } { p _ j } = 1 . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\Big | \\mu _ { \\beta , h } ^ { \\Lambda , \\sigma } ( \\eta _ { ( 0 , 0 ) } = + 1 ) - \\mu _ { \\beta , h } ^ { \\Lambda , \\omega } ( \\eta _ { ( 0 , 0 ) } = + 1 ) \\Big | \\leq C e ^ { - c \\norm { x } _ 1 } . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} g . \\big ( T _ x ( Z \\cap S ) \\big ) = T _ y ( Z \\cap S ) , \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} M \\ = \\ ( I - A ^ T ) ( I - A ) \\ ; N \\ = \\ ( I - A ) ( I - A ^ T ) \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\begin{aligned} u _ b = - \\sigma _ y y + \\sigma _ x x - d _ b \\end{aligned} \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} \\lim _ { | x _ 0 | \\to \\infty } \\int _ 0 ^ T \\int _ { B _ 1 ( x _ 0 ) } | u | ^ 3 + | p - c _ { x _ 0 , 1 } ( t ) | ^ { 3 / 2 } \\ , d x \\ , d t = 0 \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} S _ k ( t ) & = \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } w _ { k l } \\left ( S _ l ( t - 1 ) + \\eta _ l ( t ) \\right ) , \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} & U _ k ( r ) \\thicksim r ^ { A ^ + ( \\lambda _ 1 + \\omega _ k ) } \\quad \\mbox { a s } r \\to + 0 , \\\\ & U _ k ( r ) \\thicksim c _ k \\ , r ^ { A ^ + ( \\lambda _ 2 + \\omega _ k ) } , U _ k ' ( r ) = O \\left ( r ^ { A ^ + ( \\lambda _ 2 + \\omega _ k ) - 1 } \\right ) \\quad \\mbox { a s } r \\to \\infty , \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ k \\lambda _ i P ( x _ i ) = \\sum \\limits _ { j = 1 } ^ l \\alpha _ j P ( x _ j ) & \\Longrightarrow \\left \\| \\sum _ { i = 1 } ^ k \\lambda _ i P ( x _ i ) - \\sum _ { j = 1 } ^ l \\alpha _ j P ( x _ j ) \\right \\| = 0 \\\\ & \\stackrel { ( \\ref { 4 a a s t } ) } { \\Longrightarrow } \\left \\| \\sum _ { i = 1 } ^ k \\lambda _ i Q ( x _ i ) - \\sum _ { j = 1 } ^ l \\alpha _ j Q ( x _ j ) \\right \\| = 0 \\\\ & \\Longrightarrow \\sum _ { i = 1 } ^ k \\lambda _ i Q ( x _ i ) = \\sum _ { j = 1 } ^ l \\alpha _ j Q ( x _ j ) . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\Psi _ N ( t ) = ( 1 + g _ N ( t ) ) \\exp ( t ^ 2 ) \\ , . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\{ \\emph { d i s t } ^ - ( x , \\Gamma ) < \\epsilon \\} : = \\bigcup _ { \\sigma \\in [ 0 , \\ell ] } B _ \\epsilon ( \\gamma ( \\sigma ) ) \\cap \\{ x \\in \\Omega \\ : \\ x \\cdot ( \\gamma ' ( \\sigma ) ) ^ { \\perp } < 0 \\} \\ , . \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} X ^ { * } _ { i } ( \\tilde { f } _ { i } - \\tilde { f } _ { i } ( e ) ) = X ^ { * } _ { i } \\tilde { f } _ { i } , \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\| t \\circ P \\circ Q \\| _ { \\mathcal { Q } ^ { i n j } } & = \\| I _ H \\circ t \\circ P \\circ Q \\| _ { \\mathcal { Q } } = \\| s \\circ I _ G \\circ P \\circ Q \\| _ { \\mathcal { Q } } \\leq \\| s \\| \\cdot \\| I _ G \\circ P \\| _ { \\mathcal { Q } } \\cdot \\| Q \\| ^ m \\\\ & = \\| I _ H \\circ t \\| \\cdot \\| I _ G \\circ P \\| _ { \\mathcal { Q } } \\cdot \\| Q \\| ^ m \\leq \\| t \\| \\cdot \\| P \\| _ { \\mathcal { Q } ^ { i n j } } \\cdot \\| Q \\| ^ m . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } ( \\mathrm { I n d } ^ G _ { P _ b } \\circ [ \\mathcal { M } _ { M _ b , b ' , \\mu _ b } ] \\circ ( \\delta _ { P _ b } \\otimes \\mathrm { J a c } ^ G _ { P ^ { o p } _ b } ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\bar U '' ( t ) = \\frac { \\lambda } { n - 2 } \\bar U ( t ) \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\det ( \\nabla ^ 2 h ( u ) + h ( u ) { \\rm I d } ) = h ^ { p - 1 } ( u ) \\| \\nabla h ( u ) + h ( u ) \\ , u \\| _ Q ^ { n - q } f . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} T \\mbox { i s s i n g u l a r } \\Leftrightarrow T = T _ { \\rm s i n g } \\Leftrightarrow T _ { \\rm r e g } = 0 . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} S _ { l } ( \\alpha \\sqcup ( h _ { 0 } + t \\cdot \\gamma ) ) = a \\otimes S _ { l } ( h _ { 0 } + t \\cdot \\gamma ) . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\tilde { \\delta } = \\min \\Big \\{ \\frac { \\delta } { 2 C _ { b } \\tilde { C } \\theta } , ~ ~ \\frac { \\delta } { C _ { b } 2 m ^ { 2 } \\theta } \\Big \\} , \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } f _ { 1 } ( \\tau ) e ^ { - \\alpha _ 1 \\tau } x ' ( t - \\tau ) d \\tau + y ' ( t ) = \\int ^ { \\infty } _ { 0 } f _ { 1 } ( \\tau ) e ^ { - \\alpha _ 1 \\tau } n ( x ( t - \\tau ) ) d \\tau - a \\varphi _ { 1 } ( y ) \\leq M _ { 1 } G _ { 1 } - a k _ { 1 } y , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\int _ { B ( y _ 0 , 2 R ) \\cap \\{ \\psi > k \\} } ( \\nabla _ y \\psi ) ^ T A ^ T \\nabla _ y \\psi d y = \\int _ { B ( y _ 0 , 2 R ) \\cap \\{ \\psi > k \\} } - \\phi ^ T \\nabla _ y \\psi d y . \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\textbf { Q } : = \\{ x - a _ \\rho \\mid \\rho < \\lambda \\} \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\widehat { w } : = \\Big ( \\prod _ { i = 1 } ^ { m - 1 } w _ i ^ { \\frac 1 { p _ i } } \\Big ) ^ { \\varrho } \\mbox { a n d } W : = w ^ { \\frac { r _ m } { p } } \\widehat { w } ^ { - \\frac { r _ m } { \\delta _ { m + 1 } } } = w _ m ^ { \\frac { r _ m } { p _ m } } \\widehat { w } ^ { \\frac { r _ m } { \\delta _ m } } \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} d i v \\alpha _ { H _ 0 } = \\Delta \\frac { \\Delta Y ^ 0 } { | H _ 0 | } + d i v ( | H _ 0 | \\nabla Y ^ 0 ) - \\frac { 1 } { 2 } ( ( \\frac { 1 } { 2 } | H _ 0 | + r ) \\sigma ^ { a b } - \\AA ^ { a b } ) ) \\nabla _ a \\nabla _ b Y ^ 0 + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} u ( x , t ) \\succeq w ^ 0 ( x , t ) = u ( x _ 1 + c \\tau , x ' + \\rho , t + \\tau ) . \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} A & = \\{ i \\in V ( H ^ * ) \\ , | \\ , d _ { H ^ * } ( i ) = d \\} , \\\\ B & = \\{ i \\in V ( H ^ * ) \\ , | \\ , d _ { H ^ * } ( i ) = d + 1 \\} , \\\\ C & = \\{ i \\in V ( H ^ * ) \\ , | \\ , d _ { H ^ * } ( i ) \\geq d + 2 \\} , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} s I I I _ 1 '' + p I V '' _ 1 & = \\frac 1 6 ( p + s - n - 3 ) ( p + s - n - 4 ) [ 3 ( p - s ) ^ 2 + ( p + s ) ( p + s - 5 - n ) ] > 0 \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } I ( r ) } { \\mathrm { d } r } & = \\int _ { - \\pi } ^ { \\pi } \\frac { \\partial } { \\partial r } \\log ( 1 - r \\cos ( w ) ) ~ d w \\\\ & = - 2 \\cdot \\int _ { 0 } ^ { \\pi } \\frac { \\cos w } { 1 - r \\cos w } ~ d w . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} n _ { s } = n + r s = n _ { s - 1 } + r , \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} H ^ 0 ( \\mathring { C } _ I ^ 0 \\otimes _ { \\C [ I ] } D _ I ^ \\bullet ) = M . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} \\begin{aligned} { } & | \\nabla \\psi _ { \\tau } ( x ' , y ) - \\nabla \\psi _ { \\tau } ( \\bar { x ' } , \\bar { y } ) | \\\\ & \\leq ( 1 + \\tau + \\tau ^ { 2 } + \\cdots + \\tau ^ { k - 1 } ) \\omega _ { \\nabla \\psi } ( \\tau | x ' - \\bar { x ' } | + \\cdots + \\tau ^ { k } | y _ { k } - \\bar { y _ { k } } | ) \\rightarrow 0 , \\end{aligned} \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\int _ 0 ^ x t ^ \\nu \\mathbf { L } _ { \\nu + n } ( t ) \\ , \\mathrm { d } t \\sim \\int _ 0 ^ x \\frac { t ^ { 2 \\nu + n + 1 } } { \\sqrt { \\pi } 2 ^ { \\nu + n } \\Gamma ( \\nu + n + \\frac { 3 } { 2 } ) } \\ , \\mathrm { d } t = \\frac { x ^ { 2 \\nu + n + 2 } } { 2 ^ { \\nu + n } ( 2 \\nu + n + 2 ) \\Gamma ( \\nu + n + \\frac { 3 } { 2 } ) } , \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} & x \\left ( t \\right ) = \\left ( k - 1 \\right ) \\Delta x \\quad \\mbox { f o r } \\left ( j - 1 \\right ) \\Delta \\tau < t \\leq j \\Delta \\tau , \\\\ & k \\in \\left \\lbrace 1 , 2 , \\ldots , n \\right \\rbrace , j \\in \\left \\lbrace 1 , 2 , \\ldots , Q \\right \\rbrace , Q = \\operatorname { r o u n d } \\left ( t _ \\mathrm { m a x } / \\Delta \\tau \\right ) . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} A _ { \\rho } ( \\alpha ) & = \\{ f \\in C _ 0 ( \\alpha \\times L ) : f ( \\eta , y ) = 0 ( \\eta , y ) \\in \\alpha \\times ( L \\setminus L _ { \\rho } ) \\} , \\\\ B _ { \\rho } ( \\alpha ) & = \\{ f \\in C _ 0 ( \\alpha \\times L ) : f ( \\eta , y ) = 0 ( \\eta , y ) \\in \\alpha \\times L _ { \\rho } \\} \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} x _ { m } = \\psi _ { \\tau } ( x ' , y ) = \\psi _ { \\tau } ( x ' , y _ { 2 } , \\cdots y _ { k } ) : = \\frac { 1 } { \\tau } \\psi ( \\tau x ' , \\tau ^ { 2 } y _ { 2 } , \\cdots \\tau ^ { k } y _ { k } ) . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\min _ { ( U _ { 1 } , \\ldots , U _ { k } ) \\in P _ { k , V } } \\sum _ { i = 1 } ^ { k } f ( U _ { i } ) = \\min _ { \\emptyset \\subset U \\subset V } g _ { k , V } ( U ) , \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} T ^ * = J T ^ \\perp = ( J T ) ^ { \\perp } , \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} T = T _ { 1 } + T _ { 2 } \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} u \\in \\bigcap _ { k = 0 } ^ { n + 1 } C ^ { k } ( [ 0 , \\infty ) ; D ( A ^ { \\frac { n - k } { 2 } } ) ) \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} P \\cdot \\left [ B ( Q ) \\right ] = B ( 0 ) \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} N _ { \\Sigma ( W ) } ( T ) = \\left \\{ \\gamma \\in \\Gamma \\cap \\Sigma ( W ) \\mathop { | } H ( \\gamma ) \\leq T \\right \\} . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} f ( z ) = z + b _ 0 + \\sum _ { n = 1 } ^ { \\infty } b _ n z ^ { - n } . \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\lim _ { | \\mathbf { z } | \\to \\infty } \\frac { \\zeta ( \\mathbf { z } ) } { \\Phi ( \\mathbf { z } ) } = 0 . \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} R i c _ g - \\frac { r } { h } H e s s _ g h = \\rho g . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} g _ { D , \\mathrm { o d d } } ( y _ 1 , y ^ \\prime ) : = \\begin{cases} g _ D ( y _ 1 , y ^ \\prime ) & \\\\ - g _ D ( - y _ 1 , y ^ \\prime ) & \\end{cases} \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} f _ p ^ { - 1 } = f _ { \\frac { p } { p - 1 } } . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 \\phi + \\ldots + f _ n \\phi ^ n \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} \\begin{array} { r c l } v _ { R L T _ 1 ' } : = & \\underset { x \\in \\mathbb { R } ^ { m } , X \\in \\mathcal { S } ^ { m } } { \\min } & \\langle Q , X \\rangle \\\\ & & ( x , X ) \\in \\mathcal { F } . \\end{array} \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\alpha \\in \\ \\begin{cases} W ^ { 1 - \\frac { 1 } { \\frac { 3 } { 2 } + \\varepsilon } , \\frac { 3 } { 2 } + \\varepsilon } ( \\partial \\Omega _ F ( 0 ) ) & \\ 1 < p \\le \\frac { 3 } { 2 } \\\\ W ^ { 1 - \\frac { 1 } { p } , p } ( \\partial \\Omega _ F ( 0 ) ) & \\ p > \\frac { 3 } { 2 } \\end{cases} \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\mathcal { H } ^ { m } ( B _ { r } ( x ) \\cap M ) = \\omega _ { m } r ^ { m } \\left ( 1 + \\frac { 2 \\lVert I I ( x ) \\rVert - \\lVert H ( x ) \\rVert } { 8 ( m + 2 ) } r ^ { 2 } + O ( r ^ { 3 } ) \\right ) , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ k [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ; q ) _ k ^ 3 } \\equiv [ n ] q ^ { \\frac { ( n - 1 ) ^ 2 } { 4 } } ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\pmod { [ n ] \\Phi _ n ( q ) ^ 2 } , \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} I = ( I \\cap R _ + ) - ( I \\cap R _ + ) \\mbox { a n d } s \\in R _ + , r \\in I \\cap R _ + , s \\leq r \\Rightarrow s \\in I . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} 0 = A ^ t W + W ' + \\lambda W + W A . \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} y _ t = y _ 0 + \\int _ 0 ^ t \\tilde { \\vartheta } _ s \\ , \\mathrm { d } { s } + 2 c _ H ^ { B , R } \\int _ 0 ^ t \\tilde { \\varphi } _ s \\delta B _ s ^ { \\frac { H } { 2 } + \\frac { 1 } { 2 } } + \\int _ 0 ^ t \\tilde { \\psi } _ s \\delta R ^ H _ s \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\begin{aligned} I I I ' & = 4 \\pi c - c ^ { - 1 } \\int W ( h x ) Q _ { a , c } ( x ) ^ 2 \\ , d x + \\frac 1 2 c ^ { - 1 } \\int \\partial _ x [ W ( h x ) ( x - a ) ] Q _ { a , c } ( x ) ^ 2 \\ , d x \\\\ & = 4 \\pi c + c ^ { - 1 } \\int [ - \\frac 1 2 W ( h x ) + \\frac 1 2 h W ' ( h x ) ( x - a ) ] Q _ { a , c } ( x ) ^ 2 \\ , d x \\end{aligned} \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} A ^ { * n } A ^ n = ( A ^ * A ) ^ n \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} & [ x _ 1 , x _ 2 ] = x _ 3 , [ x _ 2 , x _ 3 ] = 0 , [ x _ 3 , x _ 1 ] = 0 ; \\\\ & [ \\xi ^ 1 , \\xi ^ 2 ] = \\xi ^ 2 , [ \\xi ^ 2 , \\xi ^ 3 ] = 0 , [ \\xi ^ 3 , \\xi ^ 1 ] = - \\xi ^ 3 ; \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\Theta ( b x ^ j ) \\Theta ( a x ^ i ) = \\sigma ^ { - j } ( b ) x ^ { - j } \\sigma ^ { - i } ( a ) x ^ { - i } = \\sigma ^ { - j } ( b ) \\sigma ^ { - j - i } ( a ) x ^ { - i - j } . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} [ h , L _ 1 ] = L _ 1 , [ h , f ] = - f , [ L _ 1 , f ] = 2 h . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\Xi [ D ( z , t ) ] = \\frac { t z } { 1 - t z } \\left ( \\frac { D ( z , t ) - D \\left ( z , \\frac { 1 } { 1 - z } \\right ) } { t - \\frac { 1 } { 1 - z } } \\right ) . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} E ( u ) = \\frac 1 2 \\| D ^ { 1 / 2 } u \\| _ { L ^ 2 } ^ 2 - \\frac 1 6 \\int u ^ 3 + \\frac 1 2 \\int V u ^ 2 \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Q } _ { p } } \\Psi _ { r n j } \\left ( x \\right ) d x = 0 . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} [ L : K ] & = \\sum _ { w \\supseteq v } e ( w / v ) f ( w / v ) , \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} a ( u - u _ { h } , u - u _ { h } ) = | u - u _ h | _ { 2 , \\O } ^ 2 \\le C \\gamma _ h ^ 2 , \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} G ( x ) = ( w ( z ) , u ( x ) - u ( z ) ) , \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} g \\cdot [ f _ 1 , \\ldots , f _ r ] : = [ f _ 1 \\circ g ^ { - 1 } , \\ldots , f _ r \\circ g ^ { - 1 } ] . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\eta _ n = \\omega \\left ( \\frac { 1 } { \\sqrt { n } } \\right ) . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} \\begin{aligned} [ D _ { k _ { i j } } , c _ { a b } ] & = 0 & [ D _ { c _ { i j } } , k _ { a b } ] & = 0 & [ D _ { e _ { i j } } , k _ { a b } ] & = 0 \\\\ [ D _ { k _ { i j } } , e _ { a b } ] & = 0 & [ D _ { c _ { i j } } , e _ { a b } ] & = 0 & [ D _ { e _ { i j } } , c _ { a b } ] & = 0 \\end{aligned} \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\mathcal { A } _ t f ( x ) & = \\sigma _ { t + 1 } ^ 2 [ h ] \\cdot f ' ( x ) - \\{ x - \\langle \\zeta _ t , D _ t h \\rangle \\} f ( x ) , \\\\ \\tilde { \\mathcal { A } } _ t f ( x ) & = \\sigma _ { t + 1 } ^ 2 [ h ] \\cdot f ' ( x ) - \\{ x - \\mathbb { E } _ t \\langle \\zeta _ { t + 1 } , h \\rangle \\} f ( x ) . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\left ( \\mathcal { D } _ 0 ^ \\theta f \\right ) ( x ) = f ( 0 ) \\left [ \\frac { 1 } { \\theta - 1 } E _ 1 \\left ( \\frac { x } { \\theta - 1 } \\right ) \\right ] + \\int _ 0 ^ x \\frac { 1 } { \\theta - 1 } E _ 1 \\left ( \\frac { x - y } { \\theta - 1 } \\right ) \\frac { d f } { d y } ( y ) \\ , d y , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} 0 = \\frac { d } { d t } \\Big | _ { t = 0 } E _ p ( u _ t ) = \\int _ { \\Omega } \\langle | \\nabla u | ^ { p - 2 } \\nabla u , \\nabla \\psi \\rangle d \\mu , \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\tau _ 6 ( x ) & = g _ { 6 - } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 1 0 1 } { 3 ^ 5 } , c ) , \\\\ \\tau _ 6 ( x ) & = g _ { 6 + } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , c , \\tfrac { 3 0 4 } { 3 ^ 6 } ) . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } J _ { \\lambda } ^ { \\varepsilon } w = J _ { \\lambda } w , \\quad w \\in D , \\ ; \\lambda > 0 , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} K _ t ( x , y ) = \\tau _ x ( k _ t ) ( - y ) = t ^ { - \\gamma _ k - d / 2 } e ^ { - ( | x | ^ 2 + | y | ^ 2 ) / t } E ( 2 x / t , y ) . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} \\C ^ \\epsilon _ \\mu f ( z ) : = \\C _ \\mu f ( z ) - \\C _ { \\epsilon , \\mu } f ( z ) . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} | \\nabla ( \\Gamma ( t ) f ) | _ q \\leq \\mathcal { B } e ^ { \\sum _ { i = 1 } ^ N \\left [ \\beta _ i ( t ) \\theta _ i - \\frac { t } { 2 } \\theta _ i ^ 2 \\right ] } \\left | \\nabla f \\right | _ q , | \\nabla ( \\Gamma ^ { - 1 } ( t ) f ) | _ q \\leq e ^ { \\sum _ { i = 1 } ^ N \\left [ - \\beta _ i ( t ) \\theta _ i + \\frac { t } { 2 } \\theta _ i ^ 2 \\right ] } \\left | \\nabla f \\right | _ q \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\Delta u + V u = E u \\R ^ n , \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\mu ( t , \\infty ) = L ( t ) t ^ { - \\alpha } , ~ t > 0 , \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\Sigma ( W ) \\cap \\Gamma = \\Sigma ' ( W ) \\cap \\Gamma . \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} & \\pi : N _ n ^ { c y } ( G ) \\to B _ n ( G ) , \\ \\ \\ \\pi : G ^ { n + 1 } \\to G ^ n , \\\\ & \\pi ( g _ 0 , \\ , g _ 1 , \\ , g _ 2 , \\ , \\ldots \\ , , g _ n ) = ( g _ 1 , \\ , g _ 2 , \\ , \\ldots \\ , , g _ n ) . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\begin{cases} p _ \\theta \\in C [ 0 , \\infty ) \\cap C ^ 1 ( 0 , \\infty ) , & p _ \\theta ( 0 ) = 0 \\\\ p _ \\theta \\rho \\in [ 0 , \\infty ] \\\\ p _ \\theta ( \\rho ) \\leq p _ 0 ( 1 + \\rho ^ \\Gamma ) , & \\rho \\geq 0 , \\end{cases} \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ { 1 / 6 } } \\leq w _ n : = \\frac { 1 + c _ n } { n ^ { 1 / 6 } } \\leq \\frac { 2 } { n ^ { 1 / 6 } } , \\ ; \\ ; \\ ; A : = \\left ( \\frac { 3 } { \\epsilon _ 1 } \\right ) ^ { \\frac { 1 } { 3 } } \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\Phi _ 1 ( x ) = \\begin{cases} 1 , & \\ | x | _ \\infty \\leq 1 / q , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} x = ( x _ 1 , x _ 2 , x _ 3 ) = ( r \\cos ( \\phi ) , r \\sin ( \\phi ) , z ) \\ ; , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\xi _ \\varepsilon = o \\left ( \\frac { 1 } { \\gamma _ \\varepsilon ^ 4 } \\right ) \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} \\mathbf P _ { s o } ^ { \\ , i } = 1 - \\mathbb E \\left [ \\frac { e ^ { - \\frac { ( 2 ^ { \\frac { n r _ s } { \\beta } } - 1 ) ( I _ B + 1 ) } { \\gamma _ { A _ i , B } } } } { \\frac { \\gamma _ { A _ i , E } } { \\gamma _ { A _ i , B } } \\frac { I _ B + 1 } { I _ E + 1 } 2 ^ { \\frac { n r _ s } { \\beta } } + 1 } \\right ] , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} L _ { M , \\lambda } = L _ { \\lambda } \\mid _ { X _ { M } } . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} - h ''' ( s ) & = K u \\Gamma ( u + 3 ) s ^ { - u - 3 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 3 } ) + O ( s ^ { - 3 } ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K u \\Gamma ( u + 3 ) \\zeta ( u + 1 ) s ^ { - u - 3 } . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} \\nu \\left ( \\left \\lbrace ( g ^ 1 , g ^ 2 ) \\in ( C ( [ 0 , T ] ; H ) \\cap L ^ 2 ( 0 , T ; V ) ) ^ 2 : g ^ 1 = g ^ 2 \\right \\rbrace \\right ) = 1 . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\omega ( y , z ; q ) = z ^ { - 2 } \\bar { \\omega } ( \\sqrt { y } q / \\sqrt { z } , \\sqrt { z } q ; q ) . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ m h _ { i j k l m } ^ { p ^ { \\ast } } \\omega _ m & = d h _ { i j k l } ^ { p ^ { \\ast } } + \\sum _ m h _ { m j k l } ^ { p ^ { \\ast } } \\omega _ { m i } + \\sum _ m h _ { i m k l } ^ { p ^ { \\ast } } \\omega _ { m j } + \\sum _ m h _ { i j m l } ^ { p ^ { \\ast } } \\omega _ { m k } \\\\ & \\ \\ + \\sum _ m h _ { i j k m } ^ { p ^ { \\ast } } \\omega _ { m l } + \\sum _ { m } h _ { i j k l } ^ { m ^ { \\ast } } \\omega _ { m ^ { \\ast } p ^ { \\ast } } \\end{aligned} \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\sup _ { n \\geq 1 } \\P ( \\sqrt { n } | \\hat \\lambda _ j ^ { ( n ) } - \\lambda _ j ^ { ( n ) } | / \\lambda _ j ^ { ( n ) } > R ) = 0 . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} u _ t = - \\Delta \\left ( \\Delta u + f ( u ) \\right ) . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} \\mathcal F : = \\{ Q \\in \\textbf { Q } \\mid \\lambda _ i ( Q ) \\neq 0 \\mbox { f o r s o m e } i , 1 \\leq i \\leq n \\} . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} g _ 4 ( a b , c ) = f _ 1 ( e _ { \\lambda } , b ^ { - 1 } a ^ { - 1 } ) = f _ 1 ( e _ { \\lambda } , a ^ { - 1 } ) f _ 1 ( e _ { \\lambda } , b ^ { - 1 } ) = g _ 4 ( a , b c ) g _ 4 ( b , c ) , \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} M ( x ) = \\int \\mu ( s , x ) d B _ s \\frac { \\partial } { \\partial x } M ( x ) = \\int \\frac { \\partial } { \\partial x } \\mu ( s , x ) d B _ s . \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} & C \\bullet X = - 1 \\\\ & D _ i \\bullet Y _ i \\le 0 \\ \\forall \\ i = 1 , \\ldots , m \\\\ & I \\bullet X + \\textstyle { \\sum _ { i = 1 } ^ m } I \\bullet Y _ i \\le 0 \\\\ & X , Y _ i . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{gather*} D \\alpha ^ { \\prime } = \\gamma , \\end{gather*}"}
-{"id": "4380.png", "formula": "\\begin{align*} u ^ h \\mid _ { \\xi = 0 } = \\bar { w } ^ h , \\partial _ { \\xi } u ^ h \\mid _ { \\xi = 0 } = \\hat { w } ^ h , \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} & , \\\\ & f _ 3 ( a b , c ) f _ 3 ( a , b ) = f _ 3 ( a , b c ) , \\\\ & f _ 3 ( a b , c ) f _ 4 ( a , b ) = f _ 4 ( a , b c ) f _ 3 ( b , c ) , \\\\ & f _ 4 ( a b , c ) = f _ 4 ( a , b c ) f _ 4 ( b , c ) . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} f ^ \\frac { Q - \\alpha } { Q + \\alpha } ( \\xi ) = \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { n - \\alpha } } d \\eta , \\xi \\in \\overline \\Omega . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\mathcal { D } ( \\alpha _ j ; \\varepsilon ) = \\{ z \\in \\C \\mid | z - \\alpha _ j | < \\epsilon \\} \\ ( j = 1 , \\cdots , k ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} | f ( a ( t ) , t ) | = \\left ( \\int _ \\Omega \\rho _ 0 d x \\right ) ^ { - k } \\left ( \\int _ \\Omega \\rho ( x , t ) | f ( x , t ) | ^ { 1 / k } d x \\right ) ^ k . \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} R i c _ { \\widetilde { g } } ( X _ { i } , X _ { j } ) = \\frac { ( n - 2 ) \\varphi _ { , x _ { i } x _ { j } } } { \\varphi } - \\frac { m } { f } \\left [ f _ { , x _ i x _ j } + \\frac { \\varphi _ { , x _ j } } { \\varphi } f _ { , x _ i } + \\frac { \\varphi _ { , x _ i } } { \\varphi } f _ { , x _ j } \\right ] , \\ \\forall \\ i \\neq j \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\lambda ' ( \\delta ) = \\sup \\P ( | X - Y | \\geq 1 - \\delta ) , \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 - \\theta } { r } + \\frac { \\theta } { s } . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\textup { E u l e r } \\left ( t \\otimes \\left ( \\textup { H o d g e } \\oplus \\textup { H o d g e } ^ \\vee \\right ) \\right ) = t ^ { 2 g } \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} Q ^ N ( \\theta ) : = \\{ x \\in \\mathbb { R } ^ { n } : \\exists y _ { k , i } : \\eqref { e q : t o w e r L i n e a r } , ( y _ { k , 2 i - 1 } , y _ { k , 2 i } , y _ { k + 1 , i } ) \\in \\tilde P ^ { \\nu _ \\delta } ( \\theta ) \\ k = 0 , \\dots , K - 1 , i = 1 , \\dots , 2 ^ { K - k - 1 } \\} , \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & ( B r _ t + \\beta ( r _ t ) ) d t + \\sigma ( r _ { t - } ) d X _ t \\medskip \\\\ r _ 0 & = & h _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\in O \\right ) & \\geq - \\inf _ { x \\in O \\cap ( 0 , \\ell _ 2 ^ { - 1 } ) } H ( \\ell _ 2 x ) \\wedge \\inf _ { z \\in O \\cap ( \\ell _ 2 ^ { - 1 } , + \\infty ) } H ( \\ell _ 2 z ) \\\\ & = - \\inf _ { y \\in O \\cap [ 0 , \\infty ) } H ( \\ell _ 2 y ) = - \\inf _ { y \\in O } I _ 2 ( y ) , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} Z & = ( \\pi \\times \\pi ) _ * \\left [ ( \\widetilde { \\varphi } \\times \\widetilde { \\varphi } ) ^ { - 1 } ( \\Delta _ { \\mathbb { P } ^ 2 } ) - \\Delta _ { \\widetilde { A } } - \\sum _ { ( i , j ) \\in I ^ 2 } \\mu _ { i , j } \\cdot E _ i \\times E _ j \\right ] \\\\ & = ( \\pi \\times \\pi ) _ * ( \\widetilde { \\varphi } \\times \\widetilde { \\varphi } ) ^ { * } ( \\Delta _ { \\mathbb { P } ^ 2 } ) - \\Delta _ { A } - \\sum _ { ( i , j ) \\in I ^ 2 } \\mu _ { i , j } \\cdot D _ i \\times D _ j , \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\omega ( X _ { \\pm } ) = X _ { \\mp } , \\omega ( Y ) = Y , X _ { \\pm } = A _ { \\pm } , \\ ; a _ { \\pm } , \\ ; b _ { \\pm } , Y = N , \\ ; F \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} A & \\le \\sum _ { x \\in \\Gamma } \\left | \\langle \\mathbf { 1 } _ { \\{ y \\} } , \\lambda _ { G } ( x ) \\varphi \\rangle \\right | ^ { 2 } \\\\ & = \\sum _ { x \\in y ^ { - 1 } \\Gamma } | \\varphi ( x ) | ^ { 2 } \\\\ & \\le \\sum _ { x \\not \\in U } | \\varphi ( x ) | ^ { 2 } < A \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} | Z _ j | = \\vert \\{ x _ 1 , x _ 2 \\} ^ { \\Delta _ j } \\vert = 2 ^ { | \\Delta _ j | } . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} a ^ { i + 1 } _ q = a ^ i _ q + \\frac { \\alpha } { m } \\left ( y _ i - \\hat { y } _ i \\right ) = a ^ i _ q - \\frac { \\alpha } { m } \\left ( \\sum _ { j = 1 } ^ m a ^ i _ j - \\sum _ { j = 1 } ^ m a _ j \\right ) , q \\in \\left \\lbrace 1 , 2 , \\ldots , m \\right \\rbrace . \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} i \\pi _ q ( i ) = q ^ i + \\sum _ { 1 \\neq d \\mid i } \\mu ( d ) q ^ { i / d } . \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} & \\lim _ { j \\rightarrow \\infty } u ^ j ( \\cdot , 0 ) = u ( \\cdot , 0 ) , \\ \\lim _ { j \\rightarrow \\infty } v ^ j ( \\cdot , 0 ) = v ( \\cdot , 0 ) , \\\\ & p ^ - \\ll u ^ j ( \\cdot , 0 ) \\ll v ^ j ( \\cdot , 0 ) \\ll p ^ + \\ \\ { \\rm f o r } \\ \\ j \\in \\N . \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\alpha _ 1 | \\xi | ^ 2 \\leq \\sum _ { i , j = 1 } ^ N D ^ { i j } _ l ( t ) \\xi _ i \\xi _ j \\leq \\alpha _ 2 | \\xi | ^ 2 \\ \\ { \\rm f o r } \\ \\ t \\in \\R , \\ \\xi \\in \\R ^ N \\ ( l = 1 , 2 , \\cdots , m ) , \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} F _ 1 & : = - \\theta ( \\partial _ z \\alpha ) ( \\nabla _ H \\pi ) - ( \\Delta _ H \\theta ) ( \\partial _ z \\alpha ) v - ( \\Delta _ H \\theta ) \\alpha ( \\partial _ z v ) , \\\\ F _ 2 & : = - 2 ( \\nabla _ H \\theta ) \\alpha \\cdot ( \\nabla _ H v ) - 2 \\theta ( \\partial _ z \\alpha ) ( \\partial _ z v ) - \\theta ( \\partial _ z ^ 2 \\alpha ) v . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\frac { \\sum _ { i = 1 } ^ { n s } \\xi _ i - s \\theta L _ n ( \\theta ) } { \\sigma } \\Rightarrow { \\sf N } ( 0 , 1 ) . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} f _ { 1 } ( q , z _ { 2 } ) = f _ { 2 } ( q , z _ { 2 } ) = 0 , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} & H ( { \\mu } ; { c } ) : = \\{ \\mu \\ast f ^ \\circ : f \\in \\mathcal { C } ( c _ { \\mu } , c ) \\} \\\\ & H ( { \\mu } ; { g } ) \\colon H ( { \\mu } ; { c } ) \\to H ( { \\mu } ; { d } ) \\mu \\ast f ^ \\circ \\mapsto \\mu \\ast ( f g ) ^ \\circ . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} ( x q ) ^ { 2 a - 1 } \\sum _ { h = 1 } ^ { k - a + 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a - 1 } \\sum _ { h = 0 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d y } { d t } = & \\Delta y ( t ) + \\Gamma ^ { - 1 } ( t ) [ K ( \\Gamma ( t ) y ( t ) ) \\cdot \\nabla ] ( \\Gamma ( t ) y ( t ) ) - \\Gamma ^ { - 1 } ( t ) ( \\Gamma y ( t ) \\cdot \\nabla ) ( K ( \\Gamma y ( t ) ) ; \\ y ( 0 ) = U _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 = - i R _ { 2 1 } = 0 ; \\quad \\frac { R _ { 3 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 = 0 ; \\quad \\frac { R _ { 4 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 = i = - u _ 4 . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m + 1 } } : k ] = [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m } } : k ] [ E _ { t _ { m + 1 } } : k ] . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } \\alpha _ H ^ { ( 2 ) } ( \\tilde \\nabla \\tilde X ^ i ) = & 0 \\\\ \\int _ { S ^ 2 } \\alpha _ H ^ { ( 2 ) } ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & 0 \\\\ \\int _ { S ^ 2 } \\alpha _ H ^ { ( 3 ) } ( \\epsilon _ { p q i } \\tilde X ^ q \\tilde \\nabla \\tilde X ^ i ) = & 0 . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} M = \\bigoplus _ { \\chi \\in Z ( H ) } M _ { \\chi } , \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} a _ n ^ { \\lambda } = \\frac { \\left ( \\lambda \\right ) _ { n } } { \\left ( 1 \\right ) _ { n } } \\frac { \\left ( \\lambda + \\frac { 1 } { 2 } \\right ) _ { n } } { \\left ( \\frac { 1 } { 2 } \\right ) _ { n } } , b _ n ^ { \\lambda } = \\frac { 2 \\lambda \\left ( \\lambda + 1 \\right ) _ { n } } { \\left ( 1 \\right ) _ { n } } \\frac { \\left ( \\lambda + \\frac { 1 } { 2 } \\right ) _ { n } } { \\left ( \\frac { 3 } { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} A _ { 2 } = \\begin{bmatrix} 0 & - 4 & - 3 & \\varepsilon & \\varepsilon \\\\ - 4 & \\varepsilon & \\varepsilon & \\varepsilon & - 3 \\\\ \\varepsilon & \\varepsilon & \\varepsilon & - 2 & \\varepsilon \\\\ \\varepsilon & - 1 & \\varepsilon & \\varepsilon & \\varepsilon \\\\ - 1 & \\varepsilon & \\varepsilon & 1 & \\varepsilon \\end{bmatrix} , \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\cfrac { t ^ 2 - \\sigma _ i ^ 2 } { \\mu ^ 2 } = \\cfrac { 2 \\sigma _ i } { \\sigma _ i ^ o - \\sigma _ i } \\rightarrow \\cfrac { 2 t ^ * } { \\zeta _ i - t ^ * } \\ ; \\ ; i = 1 , \\ldots , \\mathbf k ^ * , \\ ; \\\\ \\cfrac { \\mu ^ 2 } { t ^ 2 - \\sigma _ i ^ 2 } \\rightarrow \\cfrac { 0 } { ( t ^ * ) ^ 2 - ( \\zeta _ i ) ^ 2 } = 0 \\ ; \\ , i = \\mathbf k ^ * + 1 , \\ldots , m . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} ( x q ) ^ { 2 a } \\sum _ { h = 1 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a } \\sum _ { h = 0 } ^ { k - a - 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} \\| \\tilde { h } \\| _ { \\bar { \\mathcal { H } } ( [ 0 , T _ { 2 } ] ) } = \\left ( \\frac { T _ { 1 } } { T _ { 2 } } \\right ) ^ { H } \\| h \\| _ { \\bar { { \\cal H } } ( [ 0 , T _ { 1 } ] ) } . \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\begin{aligned} q _ 0 & = \\min \\left \\{ p _ 0 ^ { \\prime \\prime } , \\max \\left \\{ c _ 0 ( \\nu q _ 0 + q _ 1 ) , p _ 0 ^ { \\prime } \\right \\} \\right \\} \\\\ q _ 1 & = \\min \\left \\{ p _ 1 ^ { \\prime \\prime } , \\max \\left \\{ c _ 1 ( q _ 0 + \\nu q _ 1 ) , p _ 1 ^ { \\prime } \\right \\} \\right \\} \\end{aligned} \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log b _ c ^ { ( n ) } } { - n p _ n } & = \\lim _ { n \\to \\infty } \\frac { \\log n + ( r - 1 ) \\log ( n p _ n ) - n p _ n } { - n p _ n } = - M ' + 1 > 0 . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} \\sum _ { N \\leq k < t } \\frac { ( \\alpha t ) ^ k } { k ! } k ^ { \\beta + \\varepsilon } \\sim \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ { \\beta + \\varepsilon } \\gtrsim ( \\alpha t ) ^ { \\beta + \\varepsilon } e ^ { \\alpha t } \\quad \\mbox { a s $ t \\to \\infty $ , b y T h e o r e m \\ref { T h m : K u m m e r } } . \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\hat { r } _ \\mathrm { l e n s } = \\bigg ( 1 - \\frac { T _ \\mathrm { a } + T _ \\mathrm { p } + 2 d _ \\mathrm { m a x } } { T _ \\mathrm { f } } \\bigg ) \\sum _ { k \\in \\mathcal { K } } \\log _ 2 ( 1 + \\hat { \\gamma } _ k ) \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} & k ( k + s ^ 2 - 4 s + 2 ) - s ^ 3 + 4 s ^ 2 - 5 s + 2 \\\\ & \\leq ( s - 2 ) \\{ ( s - 2 ) + s ^ 2 - 4 s + 2 \\} - s ^ 3 + 4 s ^ 2 - 5 s + 2 \\\\ & = - s ^ 2 + s + 2 \\\\ & = - ( s + 1 ) ( s - 2 ) < 0 , \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\| \\exp ( u _ \\varepsilon ^ 2 ) \\| _ { L ^ p ( B _ { x _ \\varepsilon } ( \\rho _ \\varepsilon ' / 2 ) ^ c ) } = O ( 1 ) \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} d M _ \\eta ( t ) & = e ^ { - \\eta t } \\frac { \\sqrt { 2 \\gamma } } { m } v ( t ) \\ , d W _ 0 ( t ) + \\frac { e ^ { - \\eta t } } { m } \\sum _ { k = 1 } ^ N \\sqrt { 2 \\lambda _ k } z _ k ( t ) \\ , d W _ k ( t ) \\\\ & + e ^ { - \\eta t } \\sum _ { k > N } \\sqrt { 2 \\lambda _ k } k ^ { - 2 s } z _ k ( t ) \\ , d W _ k ( t ) . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} L _ { k , 2 , 3 } & = \\int _ { J _ { 2 } } \\left | \\sum _ { l = 1 } ^ { k - 1 } \\int _ { J _ { 3 } } \\frac { \\left | \\dot { \\bar { h } } _ { k } ( u ) \\right | | I _ { l } | d v } { ( ( 1 - v ) | I _ { l } | + | I _ { l + 1 } | + \\cdots + | I _ { k - 1 } | + u | I _ { k } | ) ^ { H + \\frac { 1 } { 2 } } } \\right | ^ { 2 } | I _ { k } | d u \\\\ & \\leq C _ { H , l _ { 0 } } \\left | \\sum _ { l = 1 } ^ { k - 1 } \\int _ { J _ { 3 } } \\frac { | I _ { l } | d v } { ( ( 1 - v ) | I _ { l } | + | I _ { l + 1 } | + \\cdots + | I _ { k } | ) ^ { H + \\frac { 1 } { 2 } } } \\right | ^ { 2 } \\cdot | I _ { k } | , \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\tau _ { D } ^ { \\epsilon , \\hat { v } } = \\min \\Bigl \\{ t > 0 \\ , \\bigl \\vert \\ , X ^ { \\epsilon , \\hat { v } } ( t ) \\notin D \\Bigr \\} . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} x _ 0 ~ = ~ 0 , x _ { i + 1 } ~ = ~ \\sup \\left \\{ x \\in ( x _ i , L ) ~ \\big | ~ \\rho ( f ( y ) , f ( x _ i ) ) \\in [ 0 , h ] \\quad \\forall y \\in ( x _ i , x ] \\right \\} \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\partial _ { \\tau } V ( x ) = A ( x ) \\tau \\quad \\mbox { a n d } A ( x ) ^ T = - A ( x ) \\quad \\forall x \\in S , \\mbox { a . e . } , \\quad \\tau \\in S _ x . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} & \\bar { \\check { y } } _ { \\mathrm { d } , i , k } [ n ] \\\\ & = \\sum _ { k ' \\in \\mathcal { K } } \\sum _ { \\ell ' \\in \\mathcal { L } _ { m _ i , k ' } } h _ { i , k ' , \\ell ' } x _ { \\mathrm { d } , k } \\big [ n - \\big ( d _ { m _ i , k ' , \\ell ' } - d _ { m _ i , k , \\ell ^ \\star _ { i , k } } \\big ) \\big ] \\\\ & \\quad + z _ { \\mathrm { d } , i } \\big [ n + d _ { m _ i , k , \\ell ^ \\star _ { i , k } } \\big ] + e _ { \\mathrm { d } , i } \\big [ n + d _ { m _ i , k , \\ell ^ \\star _ { i , k } } \\big ] . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} [ \\vec w ] _ { A _ { \\vec p , \\vec r } } \\le \\Big [ w ^ { \\frac { \\delta _ { m + 1 } } { p } } \\Big ] _ { A _ { \\frac { 1 - r } { r } \\delta _ { m + 1 } } } ^ { \\frac 1 { \\delta _ { m + 1 } } } \\prod _ { i = 1 } ^ { m } \\Big [ w _ i ^ { \\frac { \\theta _ i } { p _ i } } \\Big ] _ { A _ { \\frac { 1 - r } { r } \\theta _ i } } ^ { \\frac 1 { \\theta _ i } } \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\P ( C ' _ { j , k } \\cap A ' ) & = \\P ( C _ { j , k } \\cap A ) , \\P ( C ' _ { j , k } \\cap ( A ' ) ^ c ) = \\P ( C _ { j , k } \\cap A ^ c ) , \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} 2 D _ { I J K } = \\overline { \\nabla } _ I g _ { J K } + \\overline { \\nabla } _ J g _ { K I } - \\overline { \\nabla } _ K g _ { I J } - \\overline T _ { I J K } + \\overline T _ { J K I } - \\overline T _ { K I J } . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\overline { v _ L } = \\left | \\begin{array} { c c c } X _ 1 , & X _ 2 , & \\widetilde { X _ 3 } \\\\ \\frac { ( f _ 1 ) _ { u _ 1 } } { f _ 1 } , & ( f _ 3 ) _ { u _ 1 } , & \\sqrt { L } \\left [ \\frac { ( f _ 2 ) _ { u _ 1 } } { f _ 1 } - ( f _ 3 ) _ { u _ 1 } \\right ] \\\\ \\frac { ( f _ 1 ) _ { u _ 2 } } { f _ 1 } , & ( f _ 3 ) _ { u _ 2 } , & \\sqrt { L } \\left [ \\frac { ( f _ 2 ) _ { u _ 2 } } { f _ 1 } - ( f _ 3 ) _ { u _ 2 } \\right ] \\\\ \\end{array} \\right | . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\frac { j ^ 2 _ \\epsilon } { 2 m ^ 2 _ \\epsilon ( x ) } + V \\Big ( x , \\frac { x } { \\epsilon } \\Big ) = \\ln m _ \\epsilon ( x ) + \\overline { H } _ \\epsilon ( P ) , \\ \\ \\mathbb { T } . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ n a _ i x _ i ^ 4 + x ^ T B x + d ^ T x \\ , , \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} \\rho = ( n - 1 , n - 2 , \\dots , 1 , 0 ) , \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} W : = \\big \\{ w \\in V : \\big \\} . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} F _ { \\kappa , 1 } ( t , \\varphi ) = \\frac { \\| \\psi \\| _ { L ^ \\infty } } { \\max \\big \\{ \\| \\varphi \\| _ { L ^ \\infty } , \\ , 1 \\big \\} } - \\kappa = 0 . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} \\left ( \\frac { | 1 + i u | } { 1 + i u } \\right ) ^ k ( 1 + u ^ 2 ) ^ { - j - l } & = ( 1 - i u ) ^ k ( 1 + u ^ 2 ) ^ { - j - l - \\frac { k } 2 } = \\sum _ { m = 0 } ^ \\infty b _ { j , k , l , m } ( - i u ) ^ m \\\\ & = \\sum _ { m = 0 } ^ { m _ 0 - 1 } b _ { j , k , l , m } ( - i u ) ^ m + O \\ ! \\left ( \\sum _ { m = m _ 0 } ^ \\infty 2 ^ { j + l + \\frac { m } 2 } | u | ^ m \\right ) \\\\ & = \\sum _ { m = 0 } ^ { m _ 0 - 1 } b _ { j , k , l , m } ( - i u ) ^ m + O _ { j , l , m _ 0 } ( | u | ^ { m _ 0 } ) . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\hat { x } ^ * = [ - 1 . 3 5 0 3 9 1 4 5 9 \\ { - 1 . 4 8 3 1 5 0 3 3 2 } \\ { - 1 . 3 6 9 0 0 6 7 7 2 } \\ { - 1 . 1 0 5 9 4 1 1 8 } \\ 1 . 5 4 3 5 3 0 2 4 \\ 2 . 3 3 0 8 8 4 1 2 ] \\ , , \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} 0 = & - \\div _ x \\bigg ( \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ 0 ( x ) \\widetilde { m } _ 1 ( x , y ) ( P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) d y \\bigg ) \\\\ = & - \\div _ x \\bigg ( \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } ( x , y ) ( P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) d y \\bigg ) . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\beta _ { \\mathcal { M } _ h ^ v } : = \\min _ { \\mathcal { T } _ h ^ w \\sim \\mathcal { M } _ h ^ v } \\beta _ w > 0 . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\phi _ 1 ^ 2 - \\phi _ 1 ^ 1 & = \\omega _ 1 \\tau _ 1 , \\\\ \\phi _ 2 ^ 2 - \\phi _ 2 ^ 1 & = \\omega _ 2 \\tau _ 2 . \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\widetilde { \\Delta f } ( x ) = \\Delta \\tilde { f } ( x ) , x \\in U . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\widehat { J } ( \\xi ) = 1 - A | \\xi | ^ \\sigma ( \\ln ( 1 / | \\xi | ) ^ \\mu + o ( | \\xi | ^ \\sigma ( \\ln ( 1 / | \\xi | ) ^ \\mu ) \\quad \\mbox { a s $ | \\xi | \\to 0 $ } . \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\psi ( C ) = \\frac { \\check c _ { \\check \\nu ^ C } } { \\check b _ { \\check \\nu ^ C } } , \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\inf V = 0 . \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\textrm { c o n v } ( g ) ( x ) = \\inf \\left \\lbrace \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ { j } g ( x _ j ) \\ , : \\ , \\lambda _ j \\geq 0 , \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j = 1 , \\ , x = \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j x _ j \\right \\rbrace \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} x ^ p - x & = \\overline f \\\\ y ^ p - y - c ( x ^ p , - x ) & = \\overline f ^ { - 1 } \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} - \\Delta u + W \\cdot \\nabla u + V u = \\lambda u \\R ^ n , \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} J _ L ( \\dot { \\gamma } ) = \\frac { l _ L } { l } L ^ { \\frac { 1 } { 2 } } \\omega ( \\dot { \\gamma } ( t ) ) e _ 1 + \\left [ \\overline { q } \\dot { \\gamma } _ 3 - \\frac { \\sqrt { 2 } } { 2 } \\overline { p } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) \\right ] e _ 2 . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla e ^ { \\int _ 0 ^ t B _ i ( s ) d \\beta _ i } f | _ q = | \\nabla e ^ { \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s } f | _ q = | \\nabla f | _ q , \\ \\forall f \\in W ^ { 1 , q } ( \\mathbb { R } ^ d ) , \\ s \\in \\mathbb { R } , \\ t \\geq 0 , i = 1 , 2 , . . . , N . \\end{aligned} \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} p ( \\beta ) : = \\left ( | \\alpha \\cap \\beta | , | \\alpha \\cap ( \\alpha + \\beta ) | \\right ) = ( | \\alpha \\cap \\beta | , | \\alpha \\cap \\beta ^ c | ) \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\overline { U } _ { 2 k + 1 , 2 a } ( x ; q ) = \\overline { U } _ { 2 k + 1 , 2 a - 1 } ( x ; q ) = ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k + \\frac { 1 } { 2 } , a } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} r ( ( t , 1 ) ) ( ( 1 , \\gamma _ i ) v ) & = ( t , \\gamma _ i ) v \\\\ & = ( 1 , \\gamma _ i ) ( \\gamma ^ { - 1 } _ i ( t ) , 1 ) v \\\\ & = ( 1 , \\gamma _ i ) r _ { - \\mu _ S } ( ( \\gamma ^ { - 1 } _ i ( t ) , 1 ) ) ( v ) \\\\ & = ( 1 , \\gamma _ i ) \\mu ' ( \\gamma ^ { - 1 } _ i ( t ) ) v \\\\ & = \\gamma _ i ( \\mu ' ) ( t ) ( 1 , \\gamma _ i ) v . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\sum _ { i < j } \\mbox { c o v } ( E _ i , E _ j ) = & \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = i + 1 } ^ { \\infty } m p ^ j ( 1 - p ^ i ) \\\\ = & \\frac { m p ^ 2 } { ( 1 - p ) ( 1 - p ^ 2 ) } \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} X ^ { i + 1 } = X ^ i + \\frac { B _ p - A _ p X ^ i } { \\left | A _ p \\right | ^ 2 } { A _ p } ^ \\mathrm { T } , \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\frac { \\wp ' ( u ) } { \\wp ( u ) - \\wp ( v ) } & = \\frac { d } { d u } \\left [ \\log { \\sigma ( u + v ) } + \\log { \\sigma ( u - v ) } - 2 \\log { \\sigma ( u ) } \\right ] \\\\ & = \\zeta ( u + v ) + \\zeta ( u - v ) - 2 \\zeta ( u ) . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\sup _ { t \\in \\mathbb { R } } \\frac { t ^ { 2 N } } { N ! } \\exp ( - t ^ 2 ) = \\frac { N ^ N } { N ! } \\exp ( - N ) \\underset { N \\to + \\infty } { = } \\frac { 1 + o ( 1 ) } { \\sqrt { 2 \\pi N } } \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} P ( f ) = \\sum _ { k = 0 } ^ { + \\infty } \\widehat { f \\ast f } ( - k ) = \\sum _ { k = 0 } ^ { + \\infty } \\widehat { f } ( - k ) ^ 2 ( f \\in L ^ \\infty ( \\mathbb { T } ) ) . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} \\mathcal { F } _ 1 ( V _ 2 ) + \\mathcal { F } _ 2 ( V _ 1 , V _ 1 ) = 0 . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} ( z \\partial _ z + \\mathfrak { E } _ { ( \\partial ) } + \\mu ) _ \\circ \\textbf { S } ( t , z ) ( \\alpha ) = \\left ( \\frac { ( \\alpha ) } { 2 } - \\frac { \\textnormal { d i m } _ \\mathbb { C } ( X ) } { 2 } \\right ) \\textbf { S } ( t , z ) ( \\alpha ) \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 = i R _ { 1 2 } = i = u _ 1 ; \\quad \\frac { R _ { 3 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 = i R _ { 3 2 } = 0 ; \\quad \\frac { R _ { 4 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 = 0 . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} N _ { \\mathcal J } ( u , v ) & = ( D _ { \\mathcal J u } \\mathcal J ) v - ( D _ { \\mathcal J v } \\mathcal J ) u - \\mathcal J ( D _ { u } \\mathcal J ) v \\\\ & + \\mathcal J ( D _ { v } \\mathcal J ) u + ( D ( \\mathcal J u ) ) ^ { * } \\mathcal J v - ( D u ) ^ { * } v \\\\ & - \\mathcal J ( D ( \\mathcal J u ) ) ^ { * } v - \\mathcal J ( D u ) ^ { * } \\mathcal J v . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} \\mu ^ Z ( J X ) = - \\mu ^ W ( X ) , \\mu ^ W ( J X ) = \\mu ^ Z ( X ) . \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} \\inf \\{ | z _ k ( t ) - & \\sigma ' | \\mid t \\in \\mathbb { R } , \\sigma ' \\in \\sigma , k \\in \\mathbb { N } \\} \\\\ & = \\inf \\{ | z _ k ( t ) - \\sigma ' | \\mid t \\in [ 0 , \\omega ] , \\sigma ' \\in \\sigma , k \\in \\mathbb { N } \\} > 0 . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} u ( x , 0 ) = \\overline f ( x ) , x \\in \\Omega , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\alpha ( x _ 1 , S ( x _ 2 ) ) \\gamma ^ { - 1 } ( x _ 3 ) = \\gamma ( x _ 1 ) \\gamma ^ { - 1 } ( x _ 2 ) = \\epsilon ( x ) \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} u _ \\varepsilon ( z _ \\varepsilon ) = \\frac { \\lambda _ \\varepsilon } { 2 } \\int _ \\Omega G _ { z _ \\varepsilon } ( y ) \\Psi ' _ { N _ \\varepsilon } ( u _ \\varepsilon ( y ) ) ~ d y \\ , . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} u ' ( x ) = \\frac { w - u } { \\lambda f _ u ( x , u ) } . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} F ( t ) = \\prod _ { j = 1 } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 j + 1 } ) + \\frac { 4 } { n + 3 } \\sum _ { j = 1 } ^ { \\frac { n } { 2 } } \\sin ^ { 2 } \\left [ \\frac { ( 2 j + 1 ) \\pi } { n + 3 } \\right ] ( \\lambda _ { 2 j + 1 } - \\lambda _ { 1 } ) \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 m + 1 } ) . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } p _ j ( \\Lambda ^ { ( k ) } ( X _ 1 , \\ldots , X _ m ) ) = p _ j ( \\Lambda ^ { ( k ) } ( X _ 1 ) , \\ldots , \\Lambda ^ { ( k ) } ( X _ m ) ) = 0 , \\\\ p _ j ( \\Lambda ^ { ( k ) } ( Y _ 1 , \\ldots , Y _ m ) ) = p _ j ( \\Lambda ^ { ( k ) } ( Y _ 1 ) , \\ldots , \\Lambda ^ { ( k ) } ( Y _ m ) ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla F ( f ) ( t ) | _ p \\leq & c \\mathcal { B } ^ 2 \\eta _ t \\max \\left \\{ 1 , \\frac { 4 } { \\theta _ i ^ 2 } , \\ i = 1 , 2 , . . . , N \\right \\} \\int _ 0 ^ t ( t - s ) ^ { - \\frac { 1 } { p } } s ^ { - \\gamma } | f ( s ) | _ p | \\nabla f ( s ) | _ p d s . \\end{aligned} \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} & T _ { 2 k + 1 , 2 a + 1 } ( x ; q ) - T _ { 2 k + 1 , 2 a - 1 } ( x ; q ) \\\\ & = ( 1 + x q ) [ ( x q ) ^ { 2 a + 1 } \\overline { T } _ { 2 k + 1 , 2 k - 2 a } ( x q ; q ) + ( x q ) ^ { 2 a - 1 } \\overline { T } _ { 2 k + 1 , 2 k - 2 a + 2 } ( x q ; q ) ] . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} d _ i ^ * = \\chi _ i ( p ) T _ i . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} d ( p , p _ 0 ) = | p ^ { - 1 } \\circ p _ 0 | . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} C = \\frac { C _ { \\mathbf { f } } L } { 2 } \\sum _ { i = 1 } ^ { k _ T } b _ { S , k + i } ^ 2 + \\underset { 1 \\le l \\le k _ T } { \\sup } \\left \\| \\boldsymbol { \\xi } _ { k + l } - \\boldsymbol { \\xi } _ { k } \\right \\| _ 2 , \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} J _ { p _ o } ( q ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ - 2 y _ o & 2 x _ o & 1 \\end{pmatrix} ; \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\langle u _ 0 , d \\mu \\rangle = \\sup \\Big \\{ \\int _ { \\mathbb { R } ^ n } \\langle u , d \\mu \\rangle \\big | u \\in \\mathcal { F } \\Big \\} \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\nabla _ { x x } \\mu _ f ( x ( t ) , t ) \\ , \\dot { x } ( t ) + \\nabla _ { t x } \\mu _ { f } ( x ( t ) , t ) = 0 \\ , , \\hbox { f o r a . e . \\ } t \\in ( 0 , t _ 0 ] \\ , , \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} \\tau _ j ( Q ) : = I f _ j - I _ Q f _ j \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\norm { \\exp ( \\gamma ( \\lambda ) \\sum _ { k = 1 } ^ { m - 1 } \\phi _ \\delta ( \\abs { A _ k } ) ) u _ m } _ { \\mathcal { K } } \\le \\widetilde { C } \\ , . \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\left [ A , Y _ { 1 } \\right ] = Y _ { 1 } + \\beta Y _ { 2 } \\left [ A , Y _ { 2 } \\right ] = - \\beta Y _ { 1 } + Y _ { 2 } \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\begin{aligned} c _ 0 & : = \\Delta x + ( 1 + \\Delta x ) ( y + \\Delta \\lambda ) = 1 + ( 1 + \\Delta x ) ( 1 + y + \\Delta \\lambda ) \\\\ c _ 1 & : = x + \\lambda ( 1 + \\Delta x ) . \\end{aligned} \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} & \\int _ a ^ 1 \\max ( t - w , 0 ) ^ m d w \\\\ = & \\int _ { t - a } ^ { t - 1 } \\max ( W , 0 ) ^ m ( - d W ) \\\\ = & - \\int _ { - \\infty } ^ { t - 1 } \\max ( W , 0 ) ^ m d W + \\int _ { - \\infty } ^ { t - a } \\max ( W , 0 ) ^ m d W \\\\ = & - \\frac { 1 } { m + 1 } M ( t - 1 ) ^ { m + 1 } + \\frac { 1 } { m + 1 } M ( t - a ) ^ { m + 1 } . \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} G ^ { - 1 } ( t _ 0 , t _ k , t _ { n - k } ) = \\left ( \\sqrt { f _ 0 } , \\frac { f _ k } { \\sqrt { f _ 0 } } , \\frac { f _ { n - k } } { \\sqrt { f _ 0 } } \\right ) , \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\{ z ^ u ( x ) , z ^ v ( y ) \\} _ \\lambda ^ \\Q : = { \\partial z ^ u ( x ) \\over \\partial ( b ^ { i } _ I ) ^ { ( k ) } } \\partial _ x ^ k \\Big ( { \\partial z ^ v ( y ) \\over \\partial ( b _ { J } ^ { j } ) ^ { ( l ) } } \\partial _ y ^ n \\big ( \\{ b _ { I } ^ { i } ( x ) , b _ { J } ^ j ( y ) \\} _ \\lambda \\big ) \\Big ) \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} a _ { ( \\zeta ) } b = a _ { ( f - \\pi i ) } b = a b - \\frac { 1 } { 2 } ( a b - b a ) = \\frac { 1 } { 2 } ( a b + b a ) . \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} A ^ { * 2 q } A ^ { 2 q } & = A ^ { * q } A ^ { q * } A ^ q A ^ q = A ^ { * q } ( A ^ { * } A ) ^ q A ^ q \\\\ & = A ^ { * q } A ^ q ( A ^ { * } A ) ^ q = ( A ^ { * } A ) ^ { 2 q } \\\\ A ^ { * s q + 1 } A ^ { s q + 1 } & = A ^ { * s q } A ^ { * } A A ^ { s q } = A ^ { * s q } A ^ { s q } A ^ { * } A = ( A ^ { * } A ) ^ { s q + 1 } , \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\varphi _ { y } ( P ( y ) ) = 0 , \\nabla \\varphi _ { y } ( P ( y ) ) = 0 ; \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} f _ j ( t , z ) = 0 , j = 1 , \\ldots , l < m + n , f _ j \\in C ^ 2 ( \\mathbb { R } ^ { m + n + 1 } ) . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} P _ X ( \\mathbf { x } ) = \\prod _ { i = 0 } ^ { p - 2 } \\bigl ( \\langle \\mathbf { 1 } _ X , \\mathbf { x } \\rangle - i \\bigr ) , \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} T ^ { D } ( u , v , w ) - T ^ { D } ( \\mathcal J u , v , \\mathcal J w ) - T ^ { D } ( u , \\mathcal J v , \\mathcal J w ) - T ^ { D } ( \\mathcal J u , \\mathcal J v , w ) = N _ { \\mathcal J } ( u , v , w ) , \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\lim _ { \\varepsilon \\to 0 } \\int _ { \\Omega \\backslash B _ { x _ \\varepsilon } ( R \\mu _ \\varepsilon ) } ( \\Delta u _ \\varepsilon ( y ) ) ^ + u _ \\varepsilon ~ d y = 0 \\ , . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} P _ N = \\psi ( Y _ N ) \\ , , \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} 1 = 2 a d \\lambda + b l \\mu \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 4 { R _ 5 ^ 4 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 4 { R _ 5 ^ 4 } ) = 3 0 , \\mathrm { I C } ( \\mathcal { T } ^ 4 { R _ 5 ^ 4 } ) = 5 7 . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} d Z _ t ^ 1 = F _ 1 ( Z _ t ^ 1 , \\hdots , Z _ t ^ d ) d t + d W _ t , d Z _ t ^ i = F _ i ( Z _ t ^ { i - 1 } , \\hdots , Z _ t ^ d ) d t , 2 \\leq i \\leq d , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} S ( t ) _ 1 = \\displaystyle Q _ { n } ( 1 ; 1 / \\sqrt { n } ) \\prod _ { k = 0 } ^ { s - 1 } R _ n ( 1 , l _ k - 1 ; - a _ { 1 , l _ k - 1 } ( t ) / \\sqrt { n } ) . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\tilde T _ { \\leq j } ( y ) = \\sum _ { k \\leq j } ( \\tilde \\lambda _ k + y - \\tilde \\lambda _ j ) ^ { - 1 / 2 } \\tilde P _ k \\quad \\tilde T _ j = | \\tilde R _ j | ^ { 1 / 2 } + \\tilde g _ j ^ { - 1 / 2 } \\tilde P _ j , \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\overline { I } [ u , w ] = \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } e ^ { \\frac { | P + \\nabla u ( x ) + \\nabla _ y w ( x , y ) | ^ 2 } { 2 } + V ( x , y ) } d y d x . \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} a _ i ( x ) F _ i ( x ) + b _ i ( x ) f _ i ( x ) = 1 . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} j _ \\epsilon = m _ \\epsilon ( P + ( u _ \\epsilon ) _ x ) . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} h _ t ( r , z ) = \\sum _ { n = { - \\infty } } ^ { + \\infty } \\sum _ { k = 0 } ^ { + \\infty } ( 2 k + | n | + 1 ) e ^ { - ( 4 k ( k + | n | + 1 ) + 2 | n | ) t } e ^ { i n z } ( \\cos r ) ^ { | n | } P _ k ^ { 0 , | n | } ( \\cos 2 r ) . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} g ( x , y ) = \\lim _ { n \\to \\infty } P ( h _ n ) ( x , y ) = \\lim _ { n \\to \\infty } \\left ( R ( h _ n \\restriction _ { [ 0 , \\alpha ] \\times \\{ y \\} } ) ( x ) + S ( h _ n ) ( x , y ) \\right ) = 0 . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} H = - \\frac { X ^ { \\perp } } { 2 } , \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\prod _ { m = 1 } ^ { \\infty } \\frac { \\left ( 1 - ( - t ) ^ m \\right ) ^ 2 } { \\left ( 1 + ( - t ) ^ m \\right ) ^ 2 } = 1 + \\sum _ { n = 1 } ^ { \\infty } r _ 2 ( n ) t ^ n . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} y ( t ) = G ( y ( t ) ) : = e ^ { t \\Delta } U _ 0 + F ( y ) ( t ) , \\ t \\geq 0 , \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\Omega ( [ 0 0 0 ] ) & = \\{ ( 3 2 1 0 ) \\} \\\\ \\Omega ( [ 1 0 0 ] ) & = \\{ ( 2 3 1 0 ) , ( 2 1 3 0 ) , ( 2 1 0 3 ) \\} \\\\ \\Omega ( [ 0 1 0 ] ) & = \\{ ( 3 1 2 0 ) , ( 1 3 2 0 ) , ( 1 3 0 2 ) , ( 3 1 0 2 ) , ( 1 0 3 2 ) \\} \\\\ \\Omega ( [ 0 0 1 ] ) & = \\{ ( 3 2 0 1 ) , ( 3 0 2 1 ) , ( 0 3 2 1 ) \\} \\\\ \\Omega ( [ 1 1 0 ] ) & = \\{ ( 1 2 0 3 ) , ( 1 0 2 3 ) , ( 1 2 3 0 ) \\} \\\\ \\Omega ( [ 1 0 1 ] ) & = \\{ ( 2 0 1 3 ) , ( 2 0 3 1 ) , ( 0 2 1 3 ) , ( 0 2 3 1 ) , ( 2 3 0 1 ) \\} \\\\ \\Omega ( [ 0 1 1 ] ) & = \\{ ( 3 0 1 2 ) , ( 0 3 1 2 ) , ( 0 1 3 2 ) \\} \\\\ \\Omega ( [ 1 1 1 ] ) & = \\{ ( 0 1 2 3 ) \\} \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} f _ n ( 1 ) = a _ n , \\ f _ n ( 2 ) = a _ n ^ { - 1 } , \\ f _ n ( 3 ) = g _ n , \\ f _ n ( 4 ) = g _ n ^ { - 1 } , \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} w _ 1 ( t ) & = e ^ { - t A } \\Big [ - u _ 1 \\Big ] - \\int _ 0 ^ t e ^ { - ( t - s ) A } V _ { 0 } ' ( s ) \\ , d s \\\\ & = - e ^ { - t A } u _ 1 - \\int _ 0 ^ t e ^ { - ( t - s ) A } ( - A e ^ { s A } v _ 0 ) \\ , d s \\\\ & = - e ^ { - t A } u _ 1 + t A e ^ { - t A } v _ 0 \\\\ & = V _ 1 ^ { ( 1 ) } ( t ) + V _ 1 ^ { ( 2 ) } ( t ) . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\left \\Vert \\exp \\left [ \\begin{array} [ c ] { c c c } 0 & x _ { 1 } & x _ { 3 } \\\\ 0 & 0 & x _ { 2 } \\\\ 0 & 0 & 0 \\end{array} \\right ] \\right \\Vert = \\left ( \\left ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } \\right ) ^ { 2 } + x _ { 3 } ^ { 4 } \\right ) ^ { 1 / 4 } . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} P ^ \\nu ( \\theta ) : = \\{ x \\in \\mathbb { R } ^ 3 : \\ \\exists ( \\xi ^ j , \\eta ^ j ) _ { j = 0 } ^ \\nu : \\ \\eqref { e q : o u t e r } \\} . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} V ^ 2 = & - \\langle \\frac { \\partial } { \\partial t } , \\frac { \\partial } { \\partial t } \\rangle \\\\ V ^ 2 \\nabla \\tau = & - ( \\frac { \\partial } { \\partial t } ) ^ \\top , \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} d x & = \\frac { 1 } { \\| F ^ { * } d y \\| } d y \\wedge \\mathrm { v o l } _ { F ^ { - 1 } ( y ) } \\\\ & = \\frac { 1 } { \\| F ^ { * } \\mathrm { v o l } _ { N } \\| } \\mathrm { v o l } _ { N } \\wedge \\mathrm { v o l } _ { F ^ { - 1 } ( y ) } \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\delta ^ n ( A B ) = \\sum _ { k = 0 } ^ n { n \\choose k } \\delta ^ { n - k } ( A ) \\delta ^ k ( B ) . \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} \\P \\Big ( \\Big \\{ \\frac { \\hat \\lambda _ j - \\tilde \\lambda _ j } { \\lambda _ { j - 1 } - \\lambda _ j } \\geq - \\frac { \\delta } { 2 } \\Big \\} \\cap \\tilde { \\mathcal { E } } _ { z _ 1 , z _ 2 } \\Big ) \\geq 1 - e ^ { - t } , \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\| h \\| _ { \\bar { { \\cal H } } } = \\| K ^ { - 1 } h \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } . \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\P ( C ' _ { k , j } \\cap B ' ) = \\P ( C ' _ { k , j } ) = \\P ( C _ { k , j } ) + \\P ( C _ { k + 1 , j } ) \\geq \\P ( C _ { k , j } \\cap B ) + \\P ( C _ { k + 1 , j } \\cap B ) . \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\hat \\rho ( k ) = \\exp ( - \\alpha _ f \\ , k ) , \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\langle \\nabla _ { X _ 1 ( s ) } X _ 1 ( s ) , X _ 1 ( s ) \\rangle & = - 2 s \\sin ( \\theta _ 1 + \\theta _ 2 ) + s ^ 2 \\sin ( 2 \\theta _ 1 + 2 \\theta _ 2 ) , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\alpha ' _ n ( a ^ 2 , q ^ 2 ) = \\frac { ( 1 + a q ^ { 2 n } ) } { ( 1 + a ) q ^ n } \\alpha _ n ( a , q ) , \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} V _ { ( r ) } = V _ { - r } \\oplus V _ { r } \\oplus V _ { ( r - 1 ) } . \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} \\Sigma _ { p / q } = \\sigma _ { p / q } \\Bigl ( \\frac { n } { p ^ k - q ^ k } \\Bigr ) \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} k ^ L _ { \\gamma } : = \\sqrt { \\frac { | | \\nabla ^ L _ { \\dot { \\gamma } } { \\dot { \\gamma } } | | _ L ^ 2 } { | | \\dot { \\gamma } | | ^ 4 _ L } - \\frac { \\langle \\nabla ^ L _ { \\dot { \\gamma } } { \\dot { \\gamma } } , \\dot { \\gamma } \\rangle ^ 2 _ L } { | | \\dot { \\gamma } | | ^ 6 _ L } } . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\norm { \\mu } _ { \\mathcal { W } _ 1 ( X , \\mathbb { R } ^ m ) } = 0 \\mu = 0 . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} & \\int _ { x _ i \\ge 0 , \\sum x _ i \\le 1 } e ( \\sum _ { i = 1 } ^ m ( \\alpha _ i - \\alpha _ { m + 1 } ) ) d x _ 1 \\dots d x _ m \\\\ = & \\lim _ { \\epsilon \\to 0 } \\int _ { x _ i \\ge 0 , \\sum x _ i \\le 1 } e ( \\sum _ { i = 1 } ^ m ( \\beta _ i - \\beta _ { m + 1 } ) ) d x _ 1 \\dots d x _ m \\\\ & = \\frac { ( - 1 ) ^ m } { ( 2 \\pi i ) ^ m } \\lim _ { \\epsilon \\to 0 } \\frac { N ( \\epsilon ) } { D ( \\epsilon ) } , \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} v = e ^ { i \\varphi } \\bar { v } \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\pi _ n ( \\theta _ j ^ { ( n ) } ) H \\left ( \\frac { \\theta _ { j + 1 } ^ { ( n ) } } { n \\pi _ n ( \\theta _ j ^ { ( n ) } ) } \\right ) & \\geq \\pi _ n ( \\theta _ j ^ { ( n ) } ) H ( x _ K ) \\geq \\mathrm { e } ^ { - 3 } ( 1 - r ^ { - 1 } ) ^ { r - 1 } \\frac { ( K ^ { r / \\lceil K \\rceil } ) ^ j } { n r } H ( x _ K ) a _ c ^ { ( n ) } , \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} ( l - \\lambda _ { 0 } I ) u = \\eta \\in \\ker ( ( L - \\lambda _ { 0 } I ) ^ { m _ { 0 } } ) \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\mu ( X ) = 0 . \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\max _ { 1 \\leq j \\leq n } f ( I _ j ^ n ) = \\int _ I s ( x ) d x . \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\sigma _ \\ell \\in \\bigotimes _ { i = 1 } ^ { d _ 1 } \\mathbb { P } _ 1 ( I _ i ) \\otimes \\bigotimes _ { j = 1 } ^ { d _ 2 + d _ 3 } \\mathbb { T } _ 1 ( \\Theta _ j ) \\ ; \\ ; , \\ ; \\ ; 1 \\leq \\ell \\leq d \\ ; , \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} u ( t , x ) = u ^ R ( t , x ) + k ( t ) \\zeta ( t , x ) \\overline { S } ( t , x ) \\ , , \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial ^ 2 } { \\partial b ^ 2 } - \\frac { \\partial ^ 2 } { \\partial y ^ 2 } - \\frac { \\mu } { 1 + b } \\frac { \\partial } { \\partial b } + \\frac { \\mu + \\nu ^ 2 } { ( 1 + t ) ^ 2 } \\right ) E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) = 0 . \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\theta ( z , \\tau ) & = - i \\cdot \\sum _ { n \\in \\Z } ( - 1 ) ^ n q ^ { ( n + 1 / 2 ) ^ 2 / 2 } e ^ { 2 \\pi i ( n + 1 / 2 ) z } , \\\\ \\theta _ 1 ( z , \\tau ) & = \\sum _ { n \\in \\Z } q ^ { ( n + 1 / 2 ) ^ 2 / 2 } e ^ { 2 \\pi i ( n + 1 / 2 ) z } , \\\\ \\theta _ 2 ( z , \\tau ) & = \\sum _ { n \\in \\Z } ( - 1 ) ^ n q ^ { n ^ 2 / 2 } e ^ { 2 \\pi i n z } , \\\\ \\theta _ 3 ( z , \\tau ) & = \\sum _ { n \\in \\Z } q ^ { n ^ 2 / 2 } e ^ { 2 \\pi i n z } . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\lambda _ { k } ^ { \\epsilon , v _ k } = \\inf \\Bigl \\{ \\operatorname { R e } \\bigl \\{ \\lambda \\bigr \\} \\ , \\bigl \\vert \\ , \\lambda \\in \\operatorname { S p } \\bigl \\{ \\mathcal { L } _ { k } ^ { \\epsilon , v _ k } \\bigr \\} \\Bigr \\} , \\ , \\ , \\ , k = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} \\begin{aligned} Y = & \\frac { \\hat d _ { y z } } { \\bar { y } } S \\hat G _ y \\hat G _ d D _ { y z } + \\frac { \\hat d _ x } { \\bar { y } } S \\hat G _ y ( 1 - \\hat C _ b ) D _ x - \\frac { \\hat d _ b } { \\bar { y } } S \\hat G _ y \\hat C _ b D _ b \\\\ S = & \\frac { 1 } { 1 + \\hat G _ y \\hat C _ b \\sigma _ y + \\hat G _ y \\hat G _ h \\hat C _ h } \\end{aligned} \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\frac { 2 } { p } + \\frac { n } { q } = \\frac { n } { 2 } - s \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\min x _ 1 ^ 2 - | x _ 2 | x _ 2 = 0 \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\mathcal { P } _ i = \\left \\{ P _ i \\ \\Big | \\ p _ i ^ { \\prime } \\leq p _ i \\leq p _ i ^ { \\prime \\prime } \\right \\} , \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\begin{cases} \\sum _ { i , j = 1 } ^ m X _ i ^ \\star ( a _ { i j } X _ j u ) = \\sum _ { i = 1 } ^ m X _ i ^ \\star f _ i + g , \\\\ u = h \\ \\ \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} \\lim _ { { \\varepsilon } \\rightarrow 0 } \\sup _ { x \\in K } \\left \\vert F ( { \\varepsilon } , x ) - F ( 0 , x ) \\right \\vert = 0 . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\overline { Y } _ N = \\sum _ { j = 1 } ^ { N ^ 2 } a _ j \\mathbb { 1 } _ { \\{ Y \\in I _ j \\} } . \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\left | \\frac { d } { d t } \\gamma ( u ( t ) ) \\right | \\leq \\frac { \\left | \\dot \\gamma ( u ( t ) ) \\right | } { g ( \\gamma ( u ( t ) ) ) ^ { \\frac { 2 } { n - 2 } } } = g ( \\gamma ( u ( t ) ) ) ^ { \\frac { n - 3 } { n - 2 } } \\leq K ^ { \\frac { n - 3 } { n - 2 } } . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\frac { d } { d t } g _ { t } ( z ) = - 2 \\pi \\Psi _ { D _ t } ( g _ { t } ( z ) , \\xi ( t ) ) , g _ { 0 } ( z ) = z \\in D , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} \\Big | \\sum _ { k = 1 } ^ K e _ { 1 , k } e _ { 2 } \\Big | _ { Y _ 3 } \\le \\Big | \\sum _ { k = 1 } ^ K e _ { 1 , k } \\Big | _ { X _ 1 } | e _ { 2 } | _ { X _ { 2 } } , \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { k _ { \\gamma } ^ { L } } { \\sqrt { L } } = \\frac { | \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) | } { \\left ( \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } \\right ) ^ 2 + \\dot { \\gamma } _ 3 ^ 2 } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) = 0 ~ ~ a n d ~ ~ \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) \\neq 0 . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} \\frac { \\lambda _ { j - 1 } - \\tilde \\lambda _ j } { \\lambda _ { j - 1 } - \\lambda _ j } \\leq \\delta / 2 . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} \\phi ( L ) : = 1 6 \\left ( \\frac { L - \\frac 1 3 } { c _ 4 \\log 4 } \\right ) ^ 2 / c _ 5 ^ { \\frac 1 3 } , c _ { 7 } : = c _ 3 \\cdot c _ 5 ^ { \\frac 1 3 } , L : = \\frac 1 3 + ( c _ 4 \\log 4 ) A \\ , , \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} [ I ^ G _ { w ( M _ S ) } ( w ( \\rho ) ) ] = [ I ^ G _ { M _ S } ( \\rho ) ] , \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} x _ \\lambda \\mapsto M _ { \\lambda } : = M _ { ( 0 ^ { \\lambda _ 0 } , \\ldots , r + 1 ^ { \\lambda _ { r + 1 } } ) } = v _ 0 ^ { \\otimes \\lambda _ 0 } \\otimes \\cdots \\otimes v _ { r + 1 } ^ { \\otimes \\lambda _ { r + 1 } } \\in \\mathbb { V } ^ { \\otimes d } , \\forall \\lambda \\in \\Lambda _ { n , d } . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { n \\in \\Psi _ M } | f _ j ( n ) - f ( n ) | ^ 2 & \\leq \\sum _ { n \\in \\Psi _ M } | f _ j ( n ) - f _ M ( n ) | ^ 2 + \\sum _ { n \\in \\zeta _ M } | f _ M ( n ) - f ( n ) | ^ 2 \\\\ & \\leq \\frac { 2 | \\Psi _ M | } { j } + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { n \\in \\Xi _ i } | f _ M ( n ) - f _ i ( n ) | ^ 2 \\\\ & \\leq \\frac { 2 | \\Psi _ M | } { j } + \\frac { | \\Psi _ M | } { M } \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} R \\left ( G G \\right ) \\coloneqq R \\left ( G G , \\alpha , s , N \\right ) \\coloneqq \\frac { 2 } { N } \\sum _ { d \\in \\mathcal { D } _ { N } \\left ( G G \\right ) } \\mathrm { r } \\left ( d \\right ) I _ { s , N } \\left ( d \\alpha \\right ) , I _ { s , N } \\left ( x \\right ) \\coloneqq \\begin{cases} 1 & \\left \\Vert x \\right \\Vert \\leq s / N \\\\ 0 & \\mathrm { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\xi _ V F = 0 , \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} E _ { R } = \\Big \\{ \\sup _ { t \\geq 0 } \\frac { e ^ { - \\lambda t / 2 q } \\Phi ( x ( t ) ) } { m } - \\Theta ( X _ 0 ) - \\frac { 2 q a } { \\lambda } < R \\Big \\} , \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} l ^ { a b } n _ { a b } = & l _ { a b } \\sigma ^ { a c } \\sigma ^ { b d } n _ { c d } \\\\ = & ( l _ { a b } + \\frac { \\sigma _ { a b } } { r } ) \\sigma ^ { a c } \\sigma ^ { b d } ( n _ { c d } - \\frac { \\sigma _ { c d } } { 2 r } ) + \\frac { 1 } { 2 } r ^ { - 1 } \\sigma ^ { a b } l _ { a b } - r ^ { - 1 } \\sigma ^ { a b } n _ { a b } + r ^ { - 2 } . \\\\ \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} \\begin{aligned} & \\Big \\lVert \\sum _ { i = 1 } ^ k t _ i u ( z _ i ) - u ( z _ l ) \\Big \\rVert ^ 2 = \\Big \\lVert \\sum _ { i = 1 } ^ k t _ i ( u ( z _ i ) - u ( z _ l ) ) \\Big \\rVert ^ 2 = \\\\ & = \\sum _ { i , j = 1 } ^ k t _ i t _ j \\langle u ( z _ i ) - u ( z _ l ) , u ( z _ j ) - u ( z _ l ) \\rangle = \\sum _ { i , j = 1 } ^ k t _ i t _ j \\langle z _ i - z _ l , z _ j - z _ l \\rangle = \\\\ & = \\Big \\lVert \\sum _ { i = 1 } ^ k t _ i z _ i - z _ l \\Big \\rVert ^ 2 = \\norm { z - z _ l } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\bar Q ^ N ( \\hat \\theta ) : = \\{ x \\in \\mathbb { R } ^ { n } : \\exists y _ { k , i } : \\eqref { e q : t o w e r L i n e a r } , ( y _ { k , 2 i - 1 } , y _ { k , 2 i } , y _ { k + 1 , i } ) \\in \\tilde P ^ { \\hat \\nu _ { \\delta _ \\epsilon } } ( \\hat \\theta ) \\ k = 0 , \\dots , K - 1 , i = 1 , \\dots , 2 ^ { K - k - 1 } \\} , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} F ( x ) = \\textrm { c o n v } \\left ( x \\mapsto \\inf _ { y \\in E } \\{ f ( y ) + \\langle G ( y ) , x - y \\rangle + \\frac { 1 } { 2 } \\left ( A _ { k ( y ) } + 4 \\| G \\| _ { \\infty } + 1 \\right ) | x - y | ^ 2 \\} \\right ) , \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} 1 - m _ N ( g ) = \\frac { | g S ^ N \\backslash S ^ N | } { | S ^ N | } \\leq \\frac { | S ^ { N + | g | } \\backslash S ^ N | } { | S ^ N | } \\leq \\beta \\frac { \\sum _ { i = N + 1 } ^ { N + | g | } i ^ { d - 1 } } { N ^ d } \\lesssim \\beta \\frac { | g | } { N } , g \\in \\Gamma . \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} G ( x ) & = f ( x , y ( x ) , v ( x ) ) - \\frac { n ( x ) } { v ( x ) } , \\\\ & = f \\left ( x , \\phi ( x ) , \\frac { n ( x ) k G _ { 1 } G _ { 2 } } { a u } \\right ) - \\frac { a u } { k G _ 1 G _ 2 } , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\begin{cases} \\widehat { H } ( x , P + \\nabla \\widetilde { u } _ 0 ( x ) ) = \\ln \\widetilde { m } _ 0 ( x ) + \\overline { H } , \\\\ - \\div \\big ( \\widetilde { m } _ 0 ( x ) \\widetilde { b } ( x , P + \\nabla \\widetilde { u } _ 0 ( x ) ) \\big ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} { A } _ n = \\left ( \\begin{matrix} 2 & - 1 & 0 & \\cdots & 0 \\\\ - 1 & 2 & - 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & - 1 & 2 & - 1 \\\\ 0 & \\cdots & 0 & - 1 & 2 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) = g _ { k _ 0 } ( \\bar { y } _ 0 ) , \\ , \\ , 1 \\le k \\le n , \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} f _ { a } ( x ) = \\sum _ { i = 0 } ^ { r - 1 } f _ i ( x ) x ^ i + h _ { B } ( x ) \\sum _ { m = 0 } ^ { r - t - 1 } b _ m x ^ m , \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} { \\rm m e s } _ { d + 1 } S < \\eta = e ^ { - B ^ { \\sigma } } , \\quad \\log B \\ll \\log M \\ll \\log \\frac { 1 } { \\eta } . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} R _ 1 { } ^ 2 { } _ { 1 \\bar 1 } = - 4 \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\mathcal { L } = \\mathcal { F } + \\nabla _ { 0 } + \\sum _ { i = 1 } ^ d \\nabla _ { i } ^ 2 , \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} \\tau \\big ( e \\otimes e ^ * + f ^ * \\otimes f \\big ) = \\langle f ^ * , e \\rangle + \\langle e ^ * , f \\rangle \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} & e _ i ^ * : = e _ i , f _ i ^ * : = f _ i , ( q ^ h ) ^ * : = q ^ { - h } , \\mbox { a n d } \\\\ & \\varphi ( e _ i ) : = f _ i , \\varphi ( f _ i ) : = e _ i , \\varphi ( q ^ h ) : = q ^ h \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\iota _ \\partial \\bar { \\omega } ^ 2 = \\bar { \\omega } , \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\begin{aligned} \\Gamma _ i ^ v & : = \\{ f \\in \\Delta _ 2 ( \\mathcal { T } _ h ^ v ) : v \\in \\overline { f } \\} , & & \\Gamma _ o ^ v & : = \\Delta _ 2 ( \\mathcal { T } _ h ^ v ) \\backslash \\Gamma _ i ^ v , \\end{aligned} \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} F ( n , k ) & = ( - 1 ) ^ { n } [ 3 n - 2 k + 1 ] { 2 n - 2 k \\brack n } \\frac { ( q ; q ^ 2 ) _ { n } ( q ; q ^ 2 ) _ { n - k } } { ( q ; q ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { n - k } } , \\\\ [ 5 p t ] G ( n , k ) & = ( - 1 ) ^ { n + 1 } [ n ] { 2 n - 2 k \\brack n - 1 } \\frac { ( q ; q ^ 2 ) _ { n } ( q ; q ^ 2 ) _ { n - k } q ^ { n + 1 - 2 k } } { ( q ; q ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { n - k } } . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} i ^ G _ { M _ { S _ 1 } } ( [ M _ { S _ 2 } , \\mu _ { S _ 2 } ] ) ( \\pi ) = ( \\mathrm { I n d } ^ G _ { P _ { S _ 1 } } \\circ [ M _ { S _ 2 } , \\mu _ { S _ 2 } ] \\circ ( \\delta _ { P _ { S _ 1 } } \\otimes \\mathrm { J a c } ^ G _ { P ^ { o p } _ { S _ 1 } } ) ) ( \\pi ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ { S _ 1 } \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] , \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} T ( t ) = S ( t ) + \\int _ 0 ^ t S ( t - s ) \\tilde { B } T ( s ) \\ , d s ( t \\geq 0 ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} Z = \\{ x _ { n + m } \\colon m \\in \\omega \\} \\cup \\{ y _ { n + m } \\colon m \\in \\omega \\} \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\min \\{ \\phi , \\psi \\} = \\frac { \\phi + \\psi - | \\phi - \\psi | } { 2 } \\max \\{ \\phi , \\psi \\} = \\frac { \\phi + \\psi + | \\phi - \\psi | } { 2 } \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} | \\Im a ( z ^ { - \\beta } u , u ) | = | \\Im z ^ { - \\beta } | \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} F ( \\widetilde { w } , \\lambda , x , \\Lambda ) = \\div \\Big ( e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + \\lambda V ( x , y ) } ( \\Lambda + \\nabla _ y \\widetilde { w } ) \\Big ) . \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} A _ + = A _ 1 \\oplus A _ 0 A _ - = A _ { \\lambda - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} A _ p ( n ) = \\mathbf { C } ^ H _ { \\mathbf { y } ^ m _ p ( n ) \\mathbf { y } ^ m _ { \\mathcal { Q } _ p } ( n ) } \\mathbf { C } ^ { - 1 } _ { \\mathbf { y } ^ m _ p ( n ) } = \\sqrt { \\frac { 2 } { \\pi } } \\left ( \\mathbf { x } _ p ( n ) ^ H \\mathbf { x } _ p ( n ) + \\sigma _ n ^ 2 \\right ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta & \\gtrsim ( \\alpha t ) ^ N \\sum _ { k = 0 } ^ \\infty \\frac { k ! } { ( 1 + N - \\beta ) ( 2 + N - \\beta ) \\cdots ( k + N - \\beta ) } \\frac { ( \\alpha t ) ^ k } { k ! } \\\\ & = ( \\alpha t ) ^ N \\sum _ { k = 0 } ^ \\infty \\frac { ( 1 ) _ k } { ( 1 + N - \\beta ) _ k } \\frac { ( \\alpha t ) ^ k } { k ! } \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\Phi = \\{ \\varphi _ { \\lambda } ( x ) = r _ { \\lambda } x + a _ { \\lambda } \\} _ { \\lambda \\in \\Lambda } \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} B _ 0 = - \\overline { \\phi _ { \\omega } } \\partial _ \\nu \\phi _ { \\omega } + \\phi _ { \\omega } f \\partial _ \\nu \\phi _ { 0 } - 2 \\pi i ( \\phi _ { 0 } + \\epsilon ) \\phi _ { \\omega } f \\omega \\cdot \\nu \\ , . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} & g ( 0 ) = \\bar { A } , g \\ge g ( 0 ) [ 1 / R , R ] | g | \\le C \\mathbb { R } \\ , . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 4 ( x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 2 - 3 ^ 5 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 7 8 } { 6 6 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 5 } - \\eta \\\\ & = - \\frac { 5 0 9 1 9 0 5 4 6 8 4 5 3 4 7 6 6 7 4 8 0 1 5 9 2 8 4 3 4 5 9 9 4 9 } { 7 0 9 3 7 2 8 3 2 6 9 2 8 2 0 4 3 5 7 7 8 0 0 4 0 3 9 3 7 0 1 8 3 7 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} E [ e ^ { ( X _ 1 + X _ 2 + \\cdots + X _ k ) t } ] & = \\left ( \\frac { - t } { ( 1 - t ) \\log ( 1 - t ) } \\right ) ^ k = \\left ( \\frac { - t } { \\log ( 1 - t ) } \\right ) ^ k ( 1 - t ) ^ { - k } \\\\ & = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( n - k + 1 ) } ( - k + 1 ) ( - 1 ) ^ n \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) = \\frac { 1 } { 2 t } \\int _ { x _ k - t } ^ { x _ k + t } \\left ( \\frac { 1 } { ( 2 t ) ^ { k - 1 } } \\ , \\int _ { x _ { k - 1 } - t } ^ { x _ { k - 1 } + t } \\cdots \\int _ { x _ 1 - t } ^ { x _ 1 + t } f ( \\tau _ 1 , \\ldots , \\tau _ { k - 1 } , \\tau _ k ) \\ , d \\tau _ 1 \\cdots d \\tau _ { k - 1 } \\right ) d \\tau _ k \\ , . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} W _ 2 ( \\mu _ { t _ 1 ' } , \\mu _ { t _ 0 ' } ) = \\frac { t _ 1 ' - t _ 0 ' } { t _ 1 - t _ 0 } W _ 2 ( \\mu _ { t _ 1 } , \\mu _ { t _ 0 } ) \\qquad \\forall \\ , [ t _ 0 ' , t _ 1 ' ] \\subset [ t _ 0 , t _ 1 ] \\subset ( 0 , 1 ) . \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} \\zeta _ { v , t } ^ { \\mathcal { M } } ( x , a ) = \\int _ { X _ { t + 1 } } v ( y ) Q _ { t } ( d y | x , a ) . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } [ I ^ G _ { M _ S } ( \\rho ) ] \\left [ \\bigoplus _ { ( M _ S , \\mu _ S ) \\in \\mathrm { R e l } ^ { M _ b , \\mu _ b } _ { M _ S , b ' } } r _ { - \\mu _ S } \\circ L L ( I ^ { G } _ M ( \\rho ) ) | _ { W _ { E _ { \\{ \\mu _ S \\} _ { M _ S } } } } | \\cdot | ^ { - \\langle \\rho _ G , \\mu \\rangle } \\right ] . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} ( \\xi ^ j , \\eta ^ j ) = \\begin{cases} ( | x _ 1 | , | x _ 2 | ) & j = 0 \\\\ ( \\cos ( \\theta _ j ) \\xi ^ { j - 1 } + \\sin ( \\theta _ j ) \\eta ^ { j - 1 } , | - \\sin ( \\theta _ j ) \\xi ^ { j - 1 } + \\cos ( \\theta _ j ) \\eta ^ { j - 1 } | ) & j = 1 , \\dots , \\nu . \\end{cases} \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} P _ \\mu ^ \\sharp ( z _ o , z ) = P _ \\mu ( - z _ o , - z ) + ( z _ o , z ) . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} & A ( t _ 0 , \\cdots , t _ s , t ) = ( a _ { i j } ( t _ 0 , \\cdots , t _ s , t ) ) _ { 1 \\leq i , j \\leq n } = M _ n ( f ( t ; x ) ) \\in \\mathrm { S y m } _ n ( E _ 2 ) , \\\\ & B ( t _ 0 , \\cdots , t _ s ) = ( b _ { i j } ( t _ 0 , \\cdots , t _ s ) ) _ { 1 \\leq i , j \\leq s } = M _ s ( g ( x ) ) \\in \\mathrm { S y m } _ s ( E _ 1 ) . \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} \\left < \\tilde { L } , x ^ \\star \\right > _ { E _ 1 , E _ 1 ^ \\star } = \\int _ 0 ^ t \\left < ( t - s ) ^ { - \\alpha } S ( t - s ) G ( s , \\varphi ( s ) ) u ( s ) , x ^ \\star \\right > _ { E _ 1 , E _ 1 ^ \\star } d s , \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\nabla _ { E } E = \\widetilde { h } E , \\ ; \\ ; \\nabla _ { E } W _ { 1 } = - k _ { 1 } E + k _ { 3 } W _ { 2 } , \\ ; \\ ; \\nabla _ { E } W _ { 2 } = - k _ { 2 } E - k _ { 3 } W _ { 1 } , \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\delta _ H \\bigl ( A ( s _ 0 + s ) , A ( s _ 0 ) \\bigr ) = \\delta _ H \\bigl ( A _ { s _ 0 } A ( s ) , A _ { s _ 0 } A ( 0 ) \\bigr ) = \\delta _ H \\bigl ( A ( s ) , A ( 0 ) \\bigr ) \\rightarrow 0 \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\langle \\partial _ { i } z , \\partial _ { j } z \\rangle & = \\sum _ { \\alpha , \\beta = 1 } ^ { m } \\frac { \\partial x ^ { \\alpha } } { \\partial z ^ { i } } \\frac { \\partial x ^ { \\beta } } { \\partial z ^ { j } } \\langle \\partial _ { \\alpha } x , \\partial _ { \\beta } x \\rangle \\\\ & = \\sum _ { \\alpha = 1 } ^ { m } \\frac { \\partial x ^ { \\alpha } } { \\partial z ^ { i } } \\frac { \\partial x ^ { \\alpha } } { \\partial z ^ { j } } . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { V } _ { U } ( d ) = \\mathbb { V } _ Z ( d ) , & d \\in ( 0 , d _ c ] , \\\\ \\mathbb { V } _ { U } ( d ) < \\mathbb { V } _ Z ( d ) , & d \\in ( d _ c , \\sigma ^ 2 ) . \\end{cases} \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\Theta ( f ) = \\sum _ { i = 0 } ^ { n - 1 } \\sigma ^ { n - i } ( a _ i ) u x ^ { n - i } = a _ 0 + \\sum _ { j = 1 } ^ { n - 1 } u \\sigma ^ j ( a _ { n - j } ) x ^ j . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} w = s _ { i _ 1 } \\ldots s _ { i _ N } \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\Sigma _ { s , u } [ h _ s , h _ u ] = \\begin{cases} \\mathcal { V } _ s [ h _ s , \\mathcal { J } _ s \\circ \\cdots \\mathcal { J } _ { u - 1 } h _ u ] , & s \\leq u \\\\ \\Sigma _ { u , s } [ h _ u , h _ s ] , & s > u , \\\\ \\end{cases} \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\textrm { R a t e } _ k = \\left ( 1 - \\frac { b ' } { T } \\right ) \\log ( 1 + \\textrm { S I N R } _ k ) . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{gather*} A _ { 0 } : = y _ { 1 } y _ { 2 } - y _ { 3 } ^ 2 , A _ { 1 } : = y _ { 0 } y _ { 2 } - y _ { 4 } ^ 2 , A _ { 2 } : = y _ { 0 } y _ { 1 } - y _ { 5 } ^ 2 , \\\\ A _ { 3 } : = y _ { 4 } y _ { 5 } - y _ { 0 } y _ { 3 } , A _ { 4 } : = y _ { 3 } y _ { 5 } - y _ { 1 } y _ { 4 } , A _ { 5 } : = y _ { 3 } y _ { 4 } - y _ { 2 } y _ { 5 } , \\end{gather*}"}
-{"id": "9275.png", "formula": "\\begin{align*} w _ a w _ b w _ a w _ c \\cdot X + ( ) = 0 \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} | \\phi ( - 1 ) | , | \\phi ( 0 ) | \\leq \\sum _ { i = 1 , 2 } e ^ { \\varepsilon a _ n } | \\varphi ( x _ i ^ { \\prime } ) | e ^ { - | x _ i | \\ln \\lambda } , \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} g ( x + h ) & + g ( x - h ) - 2 g ( x ) \\leq f ( y ) + \\langle G ( y ) , x + h - y \\rangle + \\varphi _ y ( x + h ) \\\\ & + f ( y ) + \\langle G ( y ) , x - h - y \\rangle + \\varphi _ y ( x - h ) \\\\ & - 2 \\left ( f ( y ) + \\langle G ( y ) , x - y \\rangle + \\varphi _ y ( x ) \\right ) + 2 \\varepsilon \\\\ & = \\varphi _ y ( x + h ) + \\varphi _ y ( x - h ) - 2 \\varphi _ y ( x ) + 2 \\varepsilon \\\\ & \\leq C _ R | h | ^ 2 + 2 \\varepsilon , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} S ^ \\textnormal { c o h } ( \\tau , z ) ^ { - 1 } ( T _ a ) = \\sum _ { j = 0 } ^ N \\textbf { g } \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ j ) , T _ a \\right ) T ^ j \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\beta v _ t | \\mathbf { h } | ^ 2 + \\beta v _ x u | \\mathbf { h } | ^ 2 - \\beta v u _ x | \\mathbf { h } | ^ 2 = 0 . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} 0 < \\delta < \\min \\Big \\{ \\frac { p ^ + _ l - p ^ - _ l } { \\max \\{ \\varphi ^ - _ l , \\varphi ^ + _ l \\} } \\mid l = 1 , 2 , \\cdots , m \\Big \\} \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\begin{alignedat} { 2 } & \\# ( h _ { m + 1 } ) + \\# ( f _ { m } ) + \\# ( f _ { m + 1 } ) = k + 1 , \\quad & & \\# ( e _ { m + 1 } ) + \\# ( h _ { m + 1 } ) + \\# ( f _ { m } ) = k + 1 , \\\\ & \\# ( e _ { m + 1 } ) + \\# ( h _ { m } ) + \\# ( f _ { m } ) = k + 1 , \\quad & & \\# ( e _ { m } ) + \\# ( e _ { m + 1 } ) + \\# ( h _ { m } ) = k + 1 , \\end{alignedat} \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} Y ( y ) = \\textstyle \\frac { | J | - 1 } { | I | - 2 } D _ { k _ { j \\bullet } } ( k _ { a b } ) = \\textstyle \\frac { | J | - 1 } { | I | - 2 } ( 1 - \\delta _ { a b } ) ( - \\frac { 1 } { | J | - 1 } ( \\delta _ { j a } + \\delta _ { j b } ) + \\frac { 2 } { | J | ( | J | - 1 ) } ) \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} N _ 1 = 2 ^ { - d } \\frac { n _ { K _ 0 } } { N ( F , r ) } \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} w : = \\lim _ { k \\to \\infty } \\frac { \\nabla g ( \\overline { x } ) ( \\lambda ^ k - \\overline { \\lambda } ) } { t _ k } = - \\nabla _ { x x } ^ 2 L ( \\overline { x } , \\overline { \\lambda } ) \\xi \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} ( m - k ^ a ) ( 1 + \\epsilon ) ^ j \\geq { m \\over 3 } , \\ \\ j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} E ( \\{ \\lambda \\} ) P _ 1 L ^ * L P _ 1 v _ \\tau = 0 \\quad \\tau > 0 \\ , . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} D ( x , y ) > 0 \\implies D ( x , y ) \\leq 3 \\sum _ { i = 1 } ^ { N - 1 } D ( u _ i , u _ { i + 1 } ) , \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\Theta [ A ( z , t ) ] = \\frac { t ( A ( z , 1 ) - A ( z , t ) ) } { 1 - t } . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} d ( \\delta _ { \\lambda } \\Gamma _ { t } ) & = \\sum _ { \\alpha = 1 } ^ { d } \\left ( ( \\delta _ { \\lambda } ) _ { * } \\tilde { W } _ { \\alpha } \\right ) ( \\delta _ { \\lambda } \\Gamma _ { t } ) d B _ { t } ^ { \\alpha } = \\lambda \\sum _ { \\alpha = 1 } ^ { d } \\tilde { W } _ { \\alpha } ( \\delta _ { \\lambda } \\Gamma _ { t } ) d B _ { t } ^ { \\alpha } . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} [ T _ n : U ] = [ T _ n : T _ n \\cap ( U H ) ] [ T _ n \\cap ( U H ) : U ] , \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} \\mu _ { i } & = \\sigma _ { i } ^ { 2 } \\sigma _ { i + 1 } , \\\\ \\nu _ { i } & = \\sigma _ { 3 } ^ { - 1 } \\sigma _ { 4 } ^ { - 1 } \\cdots \\sigma _ { i + 1 } ^ { - 1 } , \\\\ \\omega _ { i } & = \\mu _ { i + 1 } \\mu _ { i } \\cdots \\mu _ { 2 } , \\\\ \\xi _ { j , i } & = \\nu _ { j } \\nu _ { i } \\nu _ { i - 1 } \\cdots \\nu _ { 2 } . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\sigma _ { m + 1 } = w ^ { \\frac { r _ { m + 1 } ' } { r _ { m + 1 } ' - p } } = w ^ { ( p ' - 1 ) \\frac { r } { 1 - r } } ; \\sigma _ i = w _ i ^ { \\frac { r _ i } { r _ i - p _ i } } = w _ i ^ { - \\frac { r } { 1 - r } } , 1 \\le i \\le m , \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} \\tilde { u } ( p ) : = \\frac { ( u - L _ { \\nu } ) ( \\delta _ { \\sigma ^ { \\nu } } ( p ) ) } { \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) } , ~ ~ ~ ~ p \\in \\tilde { \\Omega } \\cap B ( 1 ) , \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\phi ( s , t , x ) : = \\Phi ^ { - 1 } ( t , \\psi ( s , t , \\Phi ( s , x ) ) ) \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} B = A _ 1 \\xi ' ( q ) + A _ 2 [ \\xi ' ( 1 ) - \\xi ' ( q ) ] + \\Delta \\xi '' ( 1 ) + \\frac 1 { A _ 1 q + A _ 2 ( 1 - q ) + \\Delta } \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\textbf { G } _ \\tau \\left ( S ^ { K \\textnormal { t h } } ( \\phi _ i ) , S ^ { K \\textnormal { t h } } ( \\phi _ j ) \\right ) = \\textbf { g } ( \\phi _ i , \\phi _ j ) \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} ( X + \\xi ) \\cdot \\alpha : = \\iota _ { X } \\alpha + \\xi \\wedge \\alpha , \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} l ( G ) = \\O ( q ^ 2 - 1 ) + 3 f + 1 - \\O ( ( 3 , q + 1 ) ) \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\tilde { \\ell } ( g _ 1 , g _ 2 ) \\tilde { \\ell } ( g _ 1 g _ 2 ) = [ f _ 1 \\tilde { \\ell } ( g _ 1 , g _ 2 g _ 3 ) { f _ 1 } ^ { - 1 } ] \\tilde { \\ell } ( g _ 2 , g _ 3 ) \\times c ( g _ 1 , g _ 2 , g _ 3 ) \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 4 ( x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 1 - 3 ^ 5 x ) \\\\ & + \\frac { 2 } { 6 ^ 6 } ( 2 ^ 6 x - 2 6 ) ( 3 0 3 - 3 ^ 6 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 7 } { 6 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 1 9 0 6 3 4 9 5 2 1 5 4 4 4 9 3 8 7 3 2 6 1 5 4 9 9 7 1 9 2 0 3 4 9 } { 4 2 5 6 2 3 6 9 9 6 1 5 6 9 2 2 6 1 4 6 6 8 0 2 4 2 3 6 2 2 1 1 0 2 5 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} u ( t , x ) & = ( 1 + t ) ^ { - \\frac { \\mu } { 2 } } w [ u _ 0 ] ( t , x ) + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } - 1 } } \\int _ 0 ^ t w [ u _ 0 ] ( s , x ) K _ 0 ( t , 0 ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } - 1 } } \\int _ 0 ^ t w [ u _ 1 + \\mu \\ , u _ 0 ] ( s , x ) K _ 1 ( t , 0 ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s \\\\ & + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } - 1 } } \\int _ 0 ^ t \\int _ 0 ^ { t - b } w [ f ] ( s , x ; b ) E ( t , 0 ; b , s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s \\ , \\mathrm { d } b , \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\P ( H _ { j + 1 } \\cap A ) = \\P ( C _ { k , j + 1 } \\cap A ) . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} & H _ t = H _ v u _ y + \\frac { \\kappa } { v } \\theta _ t , \\\\ & H _ { t y } = H _ v u _ { y y } + H _ { v v } u _ y v _ y + \\left ( \\frac { 1 } { v } \\right ) _ v \\kappa \\theta _ t v _ y + \\left ( \\frac { \\kappa } { v } \\theta _ y \\right ) _ t . \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\alpha _ { j , k } = \\frac { 1 } { j ! k ! } \\left . \\frac { d ^ k } { d r ^ k } \\phi _ j \\right | _ { r = 0 } , \\phi _ j ( r ) = \\left ( \\frac { 1 } { 2 } + \\sqrt { \\frac { 1 } { 4 } - r } \\right ) ^ { - 2 j } , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\delta ( g _ 3 , g _ 2 ) : = b _ 2 \\delta _ 1 ^ 2 - b _ 1 \\delta _ 1 \\delta _ 2 + b _ 0 \\delta _ 2 ^ 2 , \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} { ( - b c q ^ { z - k + 1 } ; q ) _ k \\over q ^ { k z } } \\varphi _ n ^ { ( k ) } ( z ) = \\sum _ { i = 0 } ^ { k } v _ { n - i } ( z ) M _ { n - i } \\left ( q ^ { - z } ; b , c ; q \\right ) , \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} | a _ h ( u _ h , u _ h ) - a ( u _ h , u _ h ) | & = \\Big | \\sum \\limits _ { K \\in \\mathcal { T } _ h } \\Big \\{ a _ { h , K } ( u _ h - u _ { \\pi } , u _ h ) - a _ { K } ( u _ h - u _ { \\pi } , u _ h ) \\Big \\} \\Big | \\\\ & \\leq \\sum \\limits _ { K \\in \\mathcal { T } _ h } ( 1 + \\alpha _ 2 ) a _ K ( u _ h - u _ { \\pi } , u _ h - u _ { \\pi } ) \\\\ & \\leq C \\sum \\limits _ { K \\in \\mathcal { T } _ h } | u _ h - u _ { \\pi } | _ { 2 , K } ^ 2 . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} u _ \\epsilon ( x ) = \\int _ 0 ^ x \\frac { j _ \\epsilon } { m _ \\epsilon ( s ) } d s - P x + P - \\int _ 0 ^ 1 \\int _ 0 ^ z \\frac { j _ \\epsilon } { m _ \\epsilon ( s ) } d s d z . \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} \\delta \\left ( \\underline { \\nu _ L } , L \\right ) = - d \\textup { a n d } \\delta \\left ( \\overline { \\nu _ L } , L \\right ) = - \\frac { d } { 2 } . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} s _ { m + 1 } = ( I - W _ m ) s _ m \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\psi _ m = ( \\zeta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ l , 1 ) \\boxplus ( \\xi _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\xi _ k , 2 ) , \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} & | \\det ( V ) | \\int _ { x _ i \\ge 0 , \\sum x _ i \\le 1 } d x _ 1 \\dots d x _ m \\\\ = & | \\det ( V ) | v o l ( \\Delta ) \\\\ = & \\frac { | \\det ( V ) | } { m ! } \\sum _ { \\sigma \\in S _ m } v o l ( \\sigma ( \\Delta ) ) \\\\ = & \\frac { | \\det ( V ) | } { m ! } \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n - 1 } f ( \\sigma ^ { k m - j } \\omega ) + \\frac { 1 } { \\epsilon n } \\sum _ { k = 1 } ^ { n - 1 } 1 _ { \\{ ( \\sigma ^ { k m - j } \\omega ) | _ { m } \\notin \\mathcal { W } \\} } \\le h + 2 \\delta ( 1 + \\epsilon ^ { - 1 } ) \\ : . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} { e _ { i + 1 } } ^ 2 = { e _ i } ^ 2 - \\frac { 2 \\alpha - \\alpha ^ 2 } { m } \\left ( \\hat { y } _ i - y _ i \\right ) ^ 2 . \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} T \\circ \\sigma _ t = \\sigma _ t \\circ T , t \\in \\mathbb R . \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\mu _ f ( B ) = \\int _ B \\psi _ f \\dd \\nu _ f \\mu _ h ( B ' ) = \\int _ { B ' } \\phi _ h \\dd \\nu _ h \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\Gamma ( 1 + x / p ) = \\Gamma ( x / p - k + 1 ) \\prod _ { j = 0 } ^ { k - 1 } ( x / p - j ) \\le C _ p ( x / p ) ^ k , \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} J ^ \\textnormal { c o h } ( z , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} \\pi _ n ( p _ n ^ { - 1 } ) & = P ( \\mathrm { B i n } ( \\lfloor p _ n ^ { - 1 } \\rfloor , p _ n ) \\geq r ) = P ( \\mathrm { P o } ( \\lfloor p _ n ^ { - 1 } \\rfloor p _ n ) \\geq r ) + O ( p _ n ) \\geq 2 c , \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} ( 1 - z q b ) p _ n ( z ; a , \\ , b q | q ) = { b q ^ { n + 1 } ( a q ^ { n + 1 } - 1 ) \\over a b q ^ { 2 n + 2 } - 1 } p _ { n + 1 } ( z ; a , b | q ) + { b q ^ { n + 1 } - 1 \\over a b q ^ { 2 n + 2 } - 1 } p _ { n } ( z ; a , b | q ) \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} \\langle \\rho _ G , \\mu _ { S _ 2 } \\rangle - \\langle \\rho _ G , \\mu \\rangle = ( \\langle \\rho _ G , \\mu _ { S _ 1 } \\rangle - \\langle \\rho _ G , \\mu \\rangle ) + ( \\langle \\rho _ { M _ { S _ 1 } } , \\mu _ { S _ 2 } \\rangle - \\langle \\rho _ { M _ { S _ 1 } } , \\mu _ { S _ 1 } \\rangle ) . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} 4 \\sum _ { j , k = 1 } ^ \\infty \\int _ 0 ^ \\infty m ^ s \\int _ { | x | > R } \\partial ^ 2 _ { j k } \\varphi _ R \\partial _ j \\overline { u } _ m ( t ) \\partial _ k u _ m ( t ) d x d m = 4 \\int _ 0 ^ \\infty m ^ s \\int _ { | x | > R } \\varphi '' _ R | \\nabla u _ m ( t ) | ^ 2 d x d m . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} C _ p = \\left ( \\frac { p ^ { 1 / p } } { q ^ { 1 / q } } \\right ) ^ { 1 / 2 } , \\quad \\frac { 1 } { p } + \\frac { 1 } { q } = 1 . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\mathrm { h o f i b } ( \\mathcal N ^ { \\geq i } A \\Omega \\{ i \\} / p ^ n \\xrightarrow { \\varphi _ i - 1 } A \\Omega \\{ i \\} / p ^ n ) = \\tau ^ { \\leq i } \\mathrm { h o f i b } ( \\tau ^ { \\leq i } \\mathcal N ^ { \\geq i } A \\Omega \\{ i \\} / p ^ n \\xrightarrow { \\varphi _ i - 1 } \\tau ^ { \\leq i } A \\Omega \\{ i \\} / p ^ n ) \\ . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} \\left ( d N \\times \\mathbb { R } \\right ) / ( x , t ) \\sim ( \\iota ( x ) , - t ) = ( d N \\times [ 0 , \\infty ) ) / ( x , 0 ) \\sim ( \\iota ( x ) , 0 ) . \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\begin{array} { l l } f ^ { * } = \\underset { y } { \\max } \\{ b ^ { \\mathrm T } y \\ ; | \\ ; B ^ { \\mathrm T } y \\leq C \\alpha ^ { * } + \\gamma ^ { * } \\} . \\\\ \\end{array} \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} \\lim _ { J \\downarrow x } \\frac { 1 } { f ( J ) \\mathcal L ( J ) } = s ( x ) , . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} G \\left ( \\sqrt { f _ 0 } , \\frac { f _ k } { \\sqrt { f _ 0 } } , \\frac { f _ { n - k } } { \\sqrt { f _ 0 } } \\right ) = ( t _ 0 , t _ k , t _ { n - k } ) . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} V ( x , y ) = \\sum _ { i = 1 } ^ { d } V _ i ( x , y _ i ) . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} \\psi _ i ^ b ( \\{ z ^ j \\} _ { j \\in \\Z ^ n } ) = \\psi _ i ^ b ( \\{ w ^ j \\} _ { j \\in \\Z ^ n } ) ; \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} \\mu _ { n } = \\frac { 1 } { n - 1 } \\sum _ { 1 \\leq j \\leq n - 1 } \\delta _ { x _ j ^ n } , \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} & - q _ * ^ { - 1 } t ^ { \\frac { N + 2 A } { 2 } + 1 } ( \\nabla ^ 2 u _ 0 ) ( x , t ) \\\\ & = - t [ M _ { 0 , 1 } + o ( 1 ) ] \\nabla ^ 2 U ( | x | ) + \\left [ \\frac { N + 2 A } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] \\nabla ^ 2 [ U F _ 0 ] ( | x | ) + O ( t ^ { - 1 } ) \\\\ & \\ge \\left [ \\frac { N + 2 A } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] \\nabla ^ 2 [ U F _ 0 ] ( | x | ) + O ( t ^ { - 1 } ) \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} G = \\prod _ { \\substack { q \\leq p : \\\\ } } G _ { p , q } , \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} G _ \\lambda : H \\to H \\ , , G _ \\lambda x : = \\lambda x - \\Delta x \\ , , x \\in D ( G _ \\lambda ) : = W _ { \\bf n } \\ , , \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} L _ { j ( \\lambda , \\lambda ' ) } \\cap L _ { j ( \\lambda ' , \\lambda '' ) } = L _ { j ( \\lambda , \\lambda ' ) \\cap j ( \\lambda ' , \\lambda '' ) } \\subset L _ { j ( \\lambda , \\lambda '' ) } , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} B v = A v - \\Pi _ U A v = \\Pi _ V A v \\in V , \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\mathrm { k e r } ( D ) = \\mathrm { r a n } ( D ^ * ) ^ { \\bot } , L ^ p _ { \\mathrm { p o t } } ( \\Omega ) = \\overline { \\mathrm { r a n } ( D ) } = \\mathrm { k e r } ( D ^ * ) ^ { \\bot } . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} 0 < \\lambda _ m ^ * | \\xi | ^ 2 \\leq \\sum _ { k , l = 1 } ^ m \\sum _ { p , q = 1 } ^ d \\Gamma _ { p q } ^ { k l } ( \\cdot ) \\xi _ p ^ { ( k ) } \\xi _ q ^ { ( l ) } \\leq \\Lambda _ m ^ * | \\xi | ^ 2 < \\infty . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\| d _ a \\eta \\| ^ 2 \\le C \\sum _ { i = 1 } ^ d \\| \\nabla _ { X _ i } \\eta \\| ^ 2 , \\eta \\in \\Gamma ( \\mathbb { M } , \\mathcal { E } ) , \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\| \\mathcal { S } _ { 0 , 2 } ( U , U ) \\| _ { C ^ { 0 , \\alpha } } \\geq \\delta , \\quad \\pi _ { \\mathcal { K } } ( \\mathcal { S } _ { 0 , 2 } ( U , U ) ) = 0 , \\| \\pi _ { \\mathcal { K } } ( \\mathcal { S } _ { 0 , 3 } ( U , U , U ) ) \\| _ { C ^ { 0 , \\alpha } } \\geq \\delta , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} y _ 0 ( x ) = \\int _ 0 ^ x \\rho _ 0 ( z ) d z . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } f ( s ) d h ( s ) = \\int _ { 0 } ^ { 1 } \\mathcal { K } ^ { * } f ( s ) \\varphi ( s ) d s \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} D _ { i , a } : = \\{ ( x _ 1 , \\dots , x _ n ) \\in [ 0 , 1 ) ^ n \\mid x _ i \\le a \\} , \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} f ^ { - 1 } g f = f ^ { - 1 } & \\Big ( x _ 1 + P ( x _ 2 , x _ 3 ) + P ' \\big ( x _ 2 + Q ( x _ 3 ) , x _ 3 + d \\big ) , x _ 2 + Q ( x _ 3 ) + Q ' ( x _ 3 + d ) , x _ 3 + d + d ' \\Big ) \\\\ = & \\Big ( x _ 1 + P '' ( x _ 2 , x _ 3 ) + P ' \\big ( x _ 2 + Q ( x _ 3 ) , x _ 3 + d \\big ) , x _ 2 + Q '' ( x _ 3 ) + Q ' ( x _ 3 + d ) , x _ 3 + d ' \\Big ) \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { \\infty } | a _ { i , j } | p _ j \\leq \\beta q _ i ~ \\mbox { f o r a l l } ~ i \\geq 0 . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} | k _ { a b } | = 0 | c _ { a b } | = \\delta _ { b i } | e _ { a b } | = \\delta _ { a i } + \\delta _ { b i } \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\chi ( \\Gamma _ n , M ) ^ { ( 1 ) } = \\exp _ p ( h _ p ) ^ { ( \\Gamma : \\Gamma _ n ) } u ^ { ( \\Gamma : \\Gamma _ n ) } _ n \\ ; . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\Gamma ( x , & y ) = P ( x ^ 2 + y ^ 2 ) ^ 3 + Q ( x ^ 3 - 3 x y ^ 2 ) ^ 2 - \\frac { 9 } { 4 L } ( x ^ 2 + y ^ 2 ) ( x ^ 3 - 3 x y ^ 2 ) \\\\ & + ( \\frac { 2 7 } { 1 6 L ^ 2 } - L ) ( x ^ 2 + y ^ 2 ) ^ 2 + ( 2 - \\frac { 2 7 } { 6 4 L ^ 3 } ) ( x ^ 3 - 3 x y ^ 2 ) - \\frac { 9 } { 8 L } ( x ^ 2 + y ^ 2 ) + \\frac { 2 7 } { 2 5 6 L ^ 3 } , \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} G ( \\rho , \\zeta ) = \\partial _ n \\mathcal W ^ + ( \\rho , \\zeta ) \\Big | _ { J _ { \\mathcal S ( \\rho ) } } - \\partial _ n \\mathcal W ^ - ( \\rho , \\zeta ) \\Big | _ { J _ { \\mathcal S ( \\rho ) } } \\mbox { f o r } \\ ; \\ ; \\rho \\in \\mathcal O _ { \\delta _ 1 } , \\ ; \\ ; \\zeta \\in h ^ { 4 + \\alpha } ( \\Bbb S ) . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\Phi ^ \\rho _ * u = u \\circ \\Phi _ \\rho ^ { - 1 } \\mbox { f o r } \\ ; \\ ; u \\in C ( \\Omega _ s ) , \\qquad \\ ; \\ ; \\Phi ^ * _ \\rho v = v \\circ \\Phi _ \\rho \\mbox { f o r } \\ ; \\ ; v \\in C ( \\Omega _ \\rho ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\underset { \\textbf { W } _ k } { \\max } & \\left \\{ \\ ! \\underset { \\hat { \\textbf { x } } _ k } { \\max } \\sum \\limits _ { l = 1 } ^ k \\left [ \\ , p _ { D I } \\left ( \\textbf { y } _ l \\big | { \\textbf { x } } , \\textbf { W } _ { l } \\right ) \\bigg | _ { \\textbf { x } = \\hat { \\textbf { x } } _ k } \\right ] \\right \\} \\\\ ~ & \\eqref { e q _ o b s e r v a t i o n _ v e c t o r } , \\eqref { e q _ F c } , \\eqref { e q _ s v } , \\eqref { e q _ F e D I } , \\eqref { e q _ c o n s t r a n t 1 D I } . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\theta _ 1 = \\biggr ( 1 - \\sum _ { \\alpha = 2 } ^ N \\theta _ \\alpha ^ 2 \\biggr ) ^ { 1 / 2 } , \\frac { \\partial \\theta _ 1 } { \\partial \\theta _ \\alpha } = - \\theta _ 1 ^ { - 1 } \\theta _ \\alpha , \\frac { \\partial ^ 2 \\theta _ 1 } { \\partial \\theta _ \\alpha \\partial \\theta _ \\beta } = - \\delta _ { \\alpha \\beta } \\theta _ 1 ^ { - 1 } - \\theta _ 1 ^ { - 3 } \\theta _ \\alpha \\theta _ \\beta , \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\mathcal { B } '' _ n : = \\left \\lbrace \\dfrac { X ^ n Y ^ n Z ^ n } { m _ i } : 1 \\le i \\le { n + 2 \\choose 2 } \\right \\rbrace . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\int _ { - h } ^ 0 | v ( x ' , z ) | ^ p \\ , d z = \\int _ { - h } ^ 0 \\int _ { G ' } \\big ( | v ( x ' , z ) | ^ p - | v ( y ' , z ) | ^ p \\big ) \\delta _ { \\varepsilon , x ' _ 0 } ( y ' ) \\ , d y ' d z + ( v , \\delta _ { \\varepsilon , x _ 0 ' } | v | ^ { p - 2 } v ^ * ) _ { \\Omega ' } = : I _ 1 ( x ' ) + I _ 2 , \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} F ( t ) = \\prod _ { j = 1 } ^ { \\frac { n + 1 } { 2 } } ( t - \\lambda _ { 2 j + 1 } ) + \\frac { 4 } { n + 3 } \\sum _ { j = 1 } ^ { \\frac { n + 1 } { 2 } } \\sin ^ { 2 } \\left [ \\frac { ( 2 j + 1 ) \\pi } { n + 3 } \\right ] ( \\lambda _ { 2 j + 1 } - \\lambda _ { 1 } ) \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n + 1 } { 2 } } ( t - \\lambda _ { 2 m + 1 } ) \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\begin{cases} d \\Gamma _ { t } = \\sum _ { \\alpha = 1 } ^ { d } \\tilde { W } _ { \\alpha } ( \\Gamma _ { t } ) d B _ { t } ^ { \\alpha } , \\\\ \\Gamma _ { 0 } = \\mathbf { 1 } . \\end{cases} \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\begin{aligned} q _ 0 & = \\max \\left \\{ c _ 0 q _ 1 , p _ 0 ^ { \\prime } \\right \\} \\\\ q _ 1 & = \\max \\left \\{ c _ 1 q _ 0 , p _ 1 ^ { \\prime } \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\zeta _ \\varepsilon = \\max \\left ( { \\frac { 1 } { \\gamma _ \\varepsilon ^ 4 } , | A ( \\gamma _ \\varepsilon ) | , \\xi _ \\varepsilon } \\right ) \\ , . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} u _ 2 ( x ) : = - \\log ( C _ 0 u _ 1 ( x ) ) \\mathrm { f o r \\ e v e r y } \\ | x | \\in ( \\rho _ 0 / 2 , 3 \\rho _ 0 / 2 ) . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\lvert \\beta \\left ( \\textbf { x } \\right ) \\rvert ^ 2 \\right ] = \\lvert \\eta \\left ( \\textbf { x } \\right ) \\rvert ^ 2 \\left ( \\sigma _ { \\beta } ^ c \\right ) ^ 2 \\triangleq \\sigma _ { \\beta } ^ 2 . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} \\zeta \\lambda & = 4 \\mu ^ 2 c _ \\alpha ^ 2 + c _ \\alpha \\\\ \\zeta \\mu & = - 4 \\mu c _ \\alpha | \\alpha | a \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} F \\cap [ 0 , \\infty ) = ( F \\cap [ 0 , 1 ] ) \\cup \\left ( \\bigcup _ { i \\geq 0 } F \\cap [ 2 ^ i , 2 ^ { i + 1 } ] \\right ) = \\bigcup _ { i \\geq - 1 } F _ i , \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\beta _ { p } ( Q ; m ) = \\lim _ { k \\to \\infty } \\frac { \\# \\{ \\vec { x } \\in ( \\Z / p ^ { k } \\Z ) ^ { r } : Q ( \\vec { x } ) \\equiv m \\pmod { p ^ { k } } } { p ^ { ( r - 1 ) k } } \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\Phi _ 1 ^ X ( s ) & : = \\Phi \\big ( a ^ 1 , b ( s ) , c _ 1 , d ( s ) ; ( X ( s ) , s ) , D u _ m ( X ( s ) , s ) , D ^ 2 u _ m ( X ( s ) , s ) \\big ) , \\\\ \\Phi _ 1 ^ x ( s ) & : = \\Phi \\big ( a ^ 1 , b ( s ) , c _ 1 , d ( s ) ; ( x , s ) , D u _ m ( x , s ) , D ^ 2 u _ m ( x , s ) \\big ) . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} \\| ( G , G _ 1 , G _ 2 , y _ 0 ) \\| _ Z ^ 2 & : = \\| \\rho G \\| ^ 2 _ { L ^ 2 ( Q ) } + \\| \\rho _ 3 G _ { t } \\| ^ 2 _ { L ^ 2 ( Q ) } + \\| G ( 0 ) \\| _ { H _ 0 ^ 1 ( I ) } ^ 2 + \\| \\rho G _ 1 \\| ^ 2 _ { L ^ 2 ( Q ) } \\\\ & + \\| \\rho G _ 2 \\| ^ 2 _ { L ^ 2 ( Q ) } + \\| y _ 0 \\| ^ 2 _ { H ^ 3 ( I ) } . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace = 1 - \\exp \\left ( - \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ { 2 } ( 1 - \\theta ) } { \\lambda _ { \\textrm { B N } } P _ { \\textrm { N } } } \\left ( 2 ^ { \\frac { R } { 1 - \\theta } } - 1 \\right ) \\right ) . \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} T _ j f ( z ) : = \\int _ { \\Delta _ j ( z ) } \\frac { f ( w ) } { z - w } d \\mu ( w ) \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} c = \\sqrt { \\sup _ { t \\geq 0 } m _ + ( t ) } \\ , , \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\textbf { g } \\left ( \\Phi _ 1 ( q ) , \\Phi _ 2 ( q ) \\right ) = g \\left ( \\Phi _ 1 ( q ^ { - 1 } ) , \\Phi _ 2 ( q ) \\right ) & & \\textbf { G } _ \\tau \\left ( \\Phi _ 1 ( q ) , \\Phi _ 2 ( q ) \\right ) = G _ \\tau \\left ( \\Phi _ 1 ( q ^ { - 1 } ) , \\Phi _ 2 ( q ) \\right ) \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\Pr { B _ k } { M _ k } & = \\Pr { s _ { ( k ) } = M _ k } { M _ k } \\\\ & = \\sum _ { i = k + 1 } ^ \\infty \\Pr { s _ i = M _ k , ( k ) = i } { M _ k } . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\left ( \\Delta + \\lambda ^ 2 n _ 1 ( x ) \\right ) u _ 1 = 0 & \\mbox { i n } & \\Omega , \\\\ \\left ( \\Delta + \\lambda ^ 2 n _ 2 ( x ) \\right ) u _ 2 = 0 & \\mbox { i n } & \\Omega , \\\\ u _ 1 = u _ 2 , \\ , \\ , \\ , \\partial _ \\nu u _ 1 = \\partial _ \\nu u _ 2 & \\mbox { o n } & \\Gamma , \\end{array} \\right . \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\mathcal C _ { x } \\cap \\mathcal C _ { y } = \\emptyset , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} v \\Delta _ S ( S e ) : = H ( \\lambda ^ e ; - ) \\quad \\Delta _ S ( \\rho ( e , u , f ) ) : = \\eta _ { \\lambda ^ e } \\mathbb { R } ( S ) ( \\lambda ( f , u , e ) , - ) \\eta _ { \\lambda ^ f } ^ { - 1 } . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} Y _ x : = ( { \\frak p } _ x - { \\frak p } _ { x + 1 } ) \\partial _ { { \\frak p } _ { x - 1 } } + ( { \\frak p } _ { x + 1 } - { \\frak p } _ { x - 1 } ) \\partial _ { { \\frak p } _ { x } } + ( { \\frak p } _ { x - 1 } - { \\frak p } _ { x } ) \\partial _ { { \\frak p } _ { x + 1 } } \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} L = - \\sum _ { i = 1 } ^ d X _ i ^ * X _ i + V , \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} f ( x ) = ( I _ b ^ k \\varphi ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} P ^ { - \\ell _ q ( Q ) } = \\left ( 1 - \\left ( 1 - P ^ { - 1 } \\right ) \\right ) ^ { \\ell _ q ( Q ) } = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\binom { \\ell _ q ( Q ) } { k } \\left ( 1 - P ^ { - 1 } \\right ) ^ k \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} 4 \\pi e \\left . \\frac { \\partial ^ 2 F } { \\partial s ^ 2 } \\right | _ { s = 0 } = \\langle V , ( \\Delta _ L ^ { \\perp } - 1 ) V \\rangle _ { L ^ 2 } = 2 \\langle f _ 1 , ( \\Delta _ L - 1 ) f _ 1 \\rangle _ { L ^ 2 } + 2 \\langle f _ 2 , ( \\Delta _ L - 1 ) f _ 2 \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} - H Q _ x + Q - \\frac 1 2 Q ^ 2 = 0 \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} I = \\left \\{ \\left ( 1 , p + 1 \\right ) , \\left ( p + 1 , 1 \\right ) \\right \\} J = \\left \\{ \\left ( 1 , p \\right ) , \\left ( p , p + 1 \\right ) , \\left ( p + 1 , p \\right ) \\right \\} \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} T _ r | _ { V _ { ( r - 1 ) } \\oplus V _ { - r } } = 0 , \\ \\ T _ r : V _ r \\rightarrow V ' _ { r + 1 } \\ , . \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} | \\mathcal { L } _ 1 \\cap \\mathcal { R } | = | \\mathcal { L } _ 2 \\cap \\mathcal { R } | . \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} | \\mathcal { C } _ { \\ell } | & \\leq 2 ^ { 1 4 2 \\sqrt { n } } \\cdot \\binom { \\log \\log n } { \\ell - 1 } 2 ^ { 2 5 \\sqrt { n } } \\leq 2 ^ { 1 6 7 \\sqrt { n } } \\log n . \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} [ T , \\textbf { b } ] _ \\alpha = [ \\cdots [ [ T , \\textbf { b } ] _ { \\alpha _ 1 e _ 1 } , \\textbf { b } ] _ { \\alpha _ 2 e _ 2 } \\cdots , \\textbf { b } ] _ { \\alpha _ m e _ m } . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\int _ \\delta ^ { 1 - \\delta } h '' ( t ) \\ , \\d t = h ' ( 1 - \\delta ) - h ' ( \\delta ) \\leq \\frac { h ( 1 ) } { 1 - \\delta } + \\frac { h ( 0 ) } { \\delta } . \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} u ' ( x ) = \\frac { w - u } { \\lambda f ' ( u ) } = \\frac { w - u } { 2 \\lambda ( 0 . 5 - u ) } . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} I I I ' & = \\sum _ { i = 0 } ^ { s - 2 } ( q ^ { s - 2 } - q ^ i ) \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { j = 0 } ^ { k - 1 } q ^ { j } + \\sum _ { i = 0 } ^ { s - 2 } q ^ i \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { j = 0 } ^ { k - 1 } ( q ^ { j } - q ^ { j + p - 1 - k } ) \\\\ & = ( q - 1 ) \\sum _ { i = 0 } ^ { s - 3 } \\sum _ { k = 0 } ^ { s - 3 - i } q ^ { i + k } \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { j = 0 } ^ { k - 1 } q ^ { j } + ( 1 - q ) \\sum _ { i = 0 } ^ { s - 2 } q ^ i \\sum _ { k = 1 } ^ { p - 2 } \\sum _ { j = 0 } ^ { k - 1 } q ^ j \\sum _ { i = 0 } ^ { p - 2 - k } q ^ { i } . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} B _ i ( t ) f : = \\sigma _ { i } ( t ) \\mathbf { 1 } _ 2 \\cdot \\nabla f , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} \\rho _ 1 ( S _ 1 ) = \\rho _ 1 ( S _ 2 ) = 0 \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ \\xi _ y - \\xi _ z : y , z \\in E ^ { * } \\} = X ; \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} e ^ { - z H _ k } ( f ) = \\sum _ { n \\in \\mathbb { N } ^ d } e ^ { - z ( 2 | n | + \\gamma _ k + d ) } \\langle f , h _ n ^ k \\rangle \\ ; h _ n ^ k . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\log \\kappa ( y , x ) = \\sum \\limits _ { n = 1 } ^ { \\infty } \\sqrt { \\lambda _ n } \\phi _ n ( x ) \\psi _ n ( y ) , \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} f = P _ 1 \\cdots P _ k , \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\big ( q \\widetilde { m } ^ { q - 1 } | ( \\nabla _ y \\widetilde { m } ) ^ T K | ^ 2 + \\widetilde { m } ^ { q + 1 } | \\nabla _ y ^ 2 \\widetilde { w } | ^ 2 + ( q + 1 ) \\widetilde { m } ^ q ( \\nabla _ y \\widetilde { m } ) ^ T \\nabla _ y ^ 2 \\widetilde { w } ( \\Lambda + \\nabla _ y \\widetilde { w } ) \\big ) d y = 0 . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} C _ m : = \\sup _ { t \\in ( a _ 1 , a _ 2 ) } \\frac { | \\psi _ 1 ^ m ( t ) | } { | t | } \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\Delta u _ 1 & = - \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 , \\\\ \\Delta u _ 2 & = - \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} x _ { m } = \\psi _ { \\sigma ^ { \\nu } } ( x ' , y _ { 2 } , \\cdots , y _ { k } ) = \\frac { \\psi ( \\sigma ^ { \\nu } x ' , \\sigma ^ { 2 \\nu } y _ { 2 } , \\cdots , \\sigma ^ { k \\nu } y _ { k } ) } { \\sigma ^ { \\nu } } . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} g _ j ^ { - 1 } ( 0 ) \\cap O _ j \\setminus A = \\varnothing . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} | | x | | _ p + | | y | | _ p \\leq 2 ^ { \\frac { 1 } { q } } \\Big ( | | x | | ^ p _ p + | | y | | ^ p _ p \\Big ) ^ { \\frac { 1 } { p } } = 2 ^ { \\frac { 1 } { q } } \\Big ( \\sum _ { j = 1 } ^ { n } ( | x _ j | ^ p + | y _ j | ^ p ) \\Big ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} \\begin{aligned} { } & \\frac { | d ( p , p ^ { \\star } ) | ^ { \\alpha } } { \\delta ( p ) ^ { \\alpha } } [ W ( \\delta ( p ) ) ] \\leq W ( d ( p , p ^ { \\star } ) ) . \\end{aligned} \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} g ( m ( T ) ) \\geq \\omega _ k ( s , \\mu ) - \\varepsilon \\sum _ { l = 0 } ^ k 2 ^ { - l } . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\overline { R } = - 6 + O ( t ^ { - 5 } ) \\geq - 6 . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\nabla _ X ( f ) = X ( f ) - \\frac { i } { \\hbar } \\theta ( X ) f \\ , , \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\check { F } _ t : = h ( F _ { \\tilde { a } ^ { - 1 } ( 2 t ) } ) , \\ \\check { g } _ t : = \\tilde { g } _ { \\tilde { a } ^ { - 1 } ( 2 t ) } , \\ \\check { \\xi } ( t ) : = h _ { \\tilde { a } ^ { - 1 } ( 2 t ) } ( \\xi ( \\tilde { a } ^ { - 1 } ( 2 t ) ) ) . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P _ n = e ^ { - 1 } . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} \\frac { Q _ 1 u ( x ) - Q _ 1 u ( x _ 0 ) - T _ 1 P _ 1 ( x - x _ 0 ) } { \\norm { x - x _ 0 } } = ( Q _ 1 T _ 2 - T _ 1 P _ 1 ) \\bigg ( \\frac { x - x _ 0 } { \\norm { x - x _ 0 } } \\bigg ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} \\pi _ { N } ^ { \\lambda } u ( x ) = \\sum _ { n = 0 } ^ N \\hat u _ n ^ \\lambda R _ n ^ \\lambda ( x ) , \\quad \\ ; \\ ; \\hat u _ n ^ \\lambda = \\frac 1 { \\gamma _ n ^ \\lambda } \\int _ { \\mathbb R } u ( x ) R _ n ^ \\lambda ( x ) d x . \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} [ L _ m , L _ n ] = ( m - n ) L _ { m + n } + \\frac { m ^ 3 - m } { 1 2 } \\delta _ { m , - n } C . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} \\gamma ( x ) = \\sum _ { i = 1 } ^ p \\gamma _ i ( x ) e ^ i , \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } x _ { 1 } ^ { ( k ) } : = x _ 1 \\in B ( 0 , R ' ) , \\ , \\ , \\ , \\lim _ { k \\to \\infty } \\lambda _ { 1 } ^ { ( k ) } : = \\lambda _ 1 \\in [ \\tfrac { 1 } { n + 1 } , 1 ] . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\phi \\big ( ( X _ 2 ^ \\frac { q s } { 2 } K ^ * X _ 1 ^ { p s } K X _ 2 ^ \\frac { q s } { 2 } ) ^ { \\frac { 1 } { s } } \\big ) = & \\ \\phi \\big ( ( X _ 2 ^ \\frac { q s } { 2 } C _ 2 ^ { - \\frac { q s } { 2 } } M ^ * C _ 1 ^ { - \\frac { p s } { 2 } } X _ 1 ^ \\frac { p s } { 2 } X _ 1 ^ \\frac { p s } { 2 } C _ 1 ^ { - \\frac { p s } { 2 } } M C _ 2 ^ { - \\frac { q s } { 2 } } X _ 2 ^ \\frac { q s } { 2 } ) ^ \\frac { 1 } { s } \\big ) \\\\ = & \\ \\phi \\big ( | G _ X \\big ( \\frac { p } { r } \\big ) | ^ \\frac { 2 } { s } \\big ) . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} 2 ^ d \\lceil 2 ^ { d n } / n \\rceil - \\lceil 2 ^ { d ( n + 1 ) } / ( n + 1 ) \\rceil \\leq 2 ^ { d ( n + 1 ) } \\left ( \\frac { 1 } { n } - \\frac { 1 } { n + 1 } \\right ) + 1 = \\frac { 2 ^ { d ( n + 1 ) } } { n ( n + 1 ) } + 1 \\leq \\frac { 1 } { 2 } 2 ^ d \\lceil 2 ^ { d n } / n \\rceil , \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} 1 + \\langle C _ \\varphi ^ * f , f \\rangle = & \\langle 2 f _ 1 , f _ 1 + f _ 2 \\rangle \\\\ = & 2 | | f _ 1 | | ^ 2 + 2 \\langle f _ 1 , f _ 2 \\rangle . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { W ( L ^ { \\infty } , L ^ { p } ) } = \\Vert f _ { U } ^ { \\sharp } \\Vert _ { p } \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 0 ( x ) + \\frac { 2 } { 6 } 2 x ( 1 - 3 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 5 } { 1 8 } } + \\frac 1 { 6 0 } - \\eta \\\\ & = - \\frac { 4 9 0 3 6 6 0 3 9 3 4 5 8 0 5 5 2 6 9 3 3 3 3 2 9 2 5 7 4 7 3 7 4 3 } { 2 4 6 3 1 0 0 1 1 3 5 1 6 7 3 7 6 2 4 2 2 9 1 8 0 6 9 2 2 5 7 5 8 2 5 0 } < 0 , \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} { \\rm K } _ { \\rm e x t } ( \\mathcal U , \\delta ) = \\frac { { \\rm K } _ { \\rm p o t } ( M _ 0 ) \\delta } { 8 C ( \\mathcal U , \\delta ) } . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} n _ p = \\frac { - \\kappa + \\sqrt { \\kappa ( \\kappa + n \\lambda _ { a b } \\rho ^ \\ast ) } } { \\lambda _ { a b } \\rho ^ \\ast } . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\left \\Vert \\alpha a \\right \\Vert _ { \\mathcal { P } _ n } \\le \\sum _ { j = 1 } ^ m \\left \\Vert \\beta a _ j \\right \\Vert ^ n = \\sum _ { j = 1 } ^ m \\left \\vert \\alpha \\right \\vert \\left \\Vert a _ j \\right \\Vert ^ n , \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} F ( t , q ) & = \\frac { 1 } { 2 } \\left ( \\frac { 1 + t } { 1 - t } \\right ) ^ { q / 2 } \\left ( \\vphantom { \\frac { t + 1 } { t - 1 } } ( 1 - t ^ 2 ) ^ { - 1 / 2 } + ( 1 - t ^ 2 ) ^ { 1 / 2 } \\right ) \\\\ & = \\frac { 1 } { 2 } \\left ( \\vphantom { \\frac { t + 1 } { t - 1 } } ( 1 + t ) ^ { \\frac { q - 1 } { 2 } } ( 1 - t ) ^ { - \\frac { q + 1 } { 2 } } + ( 1 + t ) ^ { \\frac { q + 1 } { 2 } } ( 1 - t ) ^ { - \\frac { q - 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 1 } + \\bar h ^ { 2 ^ * } _ { 2 2 1 } = 0 \\\\ & 3 \\bar \\lambda _ 2 \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} G ( t ) = \\prod _ { j = 1 } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 j } ) + \\frac { 4 } { n + 3 } \\sum _ { j = 1 } ^ { \\frac { n } { 2 } } \\sin ^ { 2 } \\left ( \\frac { 2 j \\pi } { n + 3 } \\right ) ( \\lambda _ { 2 j } - \\lambda _ { n + 2 } ) \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n } { 2 } } ( t - \\lambda _ { 2 m } ) . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} ( D \\Phi _ { t } ^ { i } ( x ; h ) ) _ { s } = \\left ( J _ { t } ( x ; h ) J _ { s } ^ { - 1 } ( x ; h ) V ( \\Phi _ { s } ( x ; h ) ) \\right ) ^ { i } \\mathbf { 1 } _ { [ 0 , t ] } ( s ) , \\ \\ \\ 1 \\leq i \\leq N . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\partial _ t V ( t ) + \\mathcal A ( t ) V ( t ) = { G } ( t ) \\ , , \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\N ^ T ( J \\N f ) ^ { \\flat } ( Y , Z ) + \\N ^ T ( J \\N f ) ^ { \\flat } ( Z , Y ) = \\N ^ T \\N ^ T f ( J Y , Z ) + \\N ^ T \\N ^ T f ( J Z , Y ) \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\mathcal { F } = - \\sum _ { i j k l } R _ { i j k l } a _ i ^ \\ast a _ j a _ k ^ \\ast a _ l \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\| \\tilde { f } \\| _ { \\dot H ^ s ( \\mathbb R ^ n ) } \\le C \\| \\tilde { f } \\| _ { ( L ^ 2 ( \\mathbb R ^ n ) , \\dot H ^ 2 ( \\mathbb R ^ n ) ) _ { \\frac { s } { 2 } , 2 } } = C \\left \\{ \\int _ 0 ^ \\infty ( \\lambda ^ { - \\frac { s } { 2 } } K ( \\lambda , \\tilde { f } ) ) ^ 2 \\frac { d \\lambda } { \\lambda } \\right \\} ^ \\frac 1 2 , \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} f = \\sum _ { k = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { k } \\Pi _ { l } ( g _ { j } ^ { k } , h _ { j , 1 } ^ { k } , h _ { j , 2 } ^ { k } ) . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} h ^ { - 1 } ( A ) = S ^ 1 \\times A = S ^ 1 \\times ( [ - 1 , 1 ] \\times S ^ 1 ) = ( S ^ 1 \\times [ - 1 , 1 ] ) \\times S ^ 1 , \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} R : = \\{ j \\in \\mathbb N : \\ ; 1 \\leq j \\leq ( n - 1 ) , \\ ; x _ j > \\widehat x _ j \\} . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p } ( w ) } \\le C ( [ \\vec w ] _ { A _ { \\vec p } } ) \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { p _ i } ( w _ i ) } \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} \\nabla ^ I W ^ { - } _ { I J K L } = C ^ { - } _ { J K L } , \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\Theta : \\mathcal { R } \\to \\mathcal { R } , \\sum _ { i = 0 } ^ { n - 1 } a _ i x ^ i \\mapsto \\sum _ { i = 0 } ^ { n - 1 } \\sigma ^ { - i } ( a _ i ) x ^ { - i } = a _ 0 + \\sum _ { j = 1 } ^ { n - 1 } \\sigma ^ j ( a _ { n - j } ) x ^ j , \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} R _ { u , w } : = ( B _ - u B _ + ) / B _ + \\cap ( B _ + w B _ + ) / B _ + . \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\alpha _ \\varepsilon = 4 \\pi u _ \\varepsilon ( I _ { \\alpha _ \\varepsilon } ^ { g _ { N _ \\varepsilon } } ( \\Omega ) ) \\ , , \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} 0 = \\frac { 1 } { a } ( a ^ 2 - d ^ 2 ) , \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} s ( \\rho , \\theta ) = \\int _ 1 ^ \\theta \\frac { C _ \\vartheta ( z ) } { z } d z - \\int _ 1 ^ \\rho \\frac { p _ \\theta ( z ) } { z ^ 2 } d z . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r d _ i t ^ i \\partial _ { t ^ i } \\mathbb { F } ( t ) = \\left ( 3 - d \\right ) \\mathbb { F } ( t ) ; ~ ~ ~ ~ d _ { r } = 1 . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\Lambda : = \\left \\{ a _ { i , \\alpha } , b _ { \\alpha , \\beta } , c _ { i , \\alpha } \\in \\mathbb { F } : i = 1 , \\dots , n , \\alpha , \\beta \\in C ^ * \\right \\} . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} B _ { \\rho } \\left ( x _ i , { h _ 2 \\over 2 } \\right ) \\bigcap B _ { \\rho } \\left ( x _ j , { h _ 2 \\over 2 } \\right ) ~ = ~ \\emptyset \\quad \\forall i \\neq j \\in \\overline { 1 , { \\bf M } _ { h _ 2 } } ~ . \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\lim _ { n } \\left ( \\log \\dfrac { 1 } { \\varepsilon _ { n } } \\right ) ^ { - 1 } I \\left ( \\dfrac { 1 } { \\varepsilon _ { n } ^ { 1 - \\alpha } } \\right ) = 0 , \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} f _ 1 ( x ) ^ n = f _ 2 ( x ) ^ n . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\mathfrak { H } : = \\left ( \\begin{array} { c c c c c } H _ { \\nu } & \\cdots & H _ 0 & & 0 \\\\ & \\ddots & & \\ddots & \\\\ 0 & & H _ { \\nu } & \\cdots & H _ 0 \\end{array} \\right ) \\in \\mathbb F ^ { ( L + 1 ) ( n - k ) \\times ( \\nu + L + 1 ) n } \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} \\Delta _ 1 : = \\{ ( \\xi , \\eta ) \\in \\R _ + \\times \\R _ + \\mid \\lim _ { \\tau \\rightarrow \\infty } ( u ( \\tau ; \\xi , \\eta ) , v ( \\tau ; \\xi , \\eta ) ) = ( R _ 1 , 0 ) \\} , \\\\ \\Delta _ 2 : = \\{ ( \\xi , \\eta ) \\in \\R _ + \\times \\R _ + \\mid \\lim _ { \\tau \\rightarrow \\infty } ( u ( \\tau ; \\xi , \\eta ) , v ( \\tau ; \\xi , \\eta ) ) = ( 0 , R _ 2 ) \\} . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ m a ^ { 0 } _ { i j } X _ i X _ j U _ 0 = 0 \\ \\ \\ B ( 4 / 5 ) \\cap \\{ x _ m > 0 \\} , \\ \\ \\ \\ \\ U _ 0 = 0 \\ \\ \\ B ( 4 / 5 ) \\cap \\{ x _ m = 0 \\} . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} \\nabla _ { K Z } ' : = d - \\sum _ { s \\in S } c _ s \\frac { d \\alpha _ s } { \\alpha _ s } s = d - \\sum _ { s \\in S } c _ s d ( \\log \\alpha _ s ) s \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} d _ 2 \\left ( \\wp ( x ) \\cdot a \\otimes b \\otimes m \\right ) = { } & - b \\otimes a _ { ( \\wp ) } m \\\\ & + \\wp ( x ) \\left ( a \\otimes b { ( 0 ) } m + b ( 0 ) a \\otimes m \\right ) , \\\\ \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} ( | \\xi | ^ { \\alpha } + \\rho ) \\hat { u } ( \\xi ) = \\hat { f } ( \\xi ) , \\xi \\in { \\mathbb R } ^ d , \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\L \\Theta & = - \\frac { \\gamma } { m } v ^ 2 - \\frac { 1 } { m } \\sum _ { k = 1 } ^ N \\lambda _ k z _ k ^ 2 - \\sum _ { k > N } \\lambda _ k k ^ { - 2 s } z _ k ^ 2 - \\frac { 1 } { m } \\sum _ { k > N } \\sqrt { c _ k } z _ k v \\\\ & \\qquad + \\sum _ { k > N } \\sqrt { c _ k } k ^ { - 2 s } z _ k v + \\frac { \\gamma } { m ^ 2 } + \\frac { 1 } { m } \\sum _ { k = 1 } ^ N \\lambda _ k + \\sum _ { k > N } \\lambda _ k k ^ { - 2 s } . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} C = C _ 1 = 2 \\delta + 1 & & C ' = C _ 0 = 2 \\delta + 2 . \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} a = \\lambda _ { 1 } a _ { 1 } + \\cdots + \\lambda _ { d _ { k } } a _ { d _ { k } } \\in \\mathcal { L } _ { k } \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} v _ p ( c _ { k + 1 } ^ - ) = v _ p ( ( A ^ - ) ^ { n - 1 } ( c _ k ^ - ) ^ n ) , \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} Q _ { 2 , k } \\leq C _ { H } \\int _ { 0 } ^ { 1 } \\left | \\sum _ { l = 1 } ^ { k - 1 } \\int _ { 0 } ^ { 1 } \\frac { \\left | \\dot { \\bar { h } } _ { k } ( u ) \\right | + \\left ( \\frac { \\sum _ { j _ 1 = 1 } ^ k | I _ { j _ 1 } | } { \\sum _ { j _ 2 = 1 } ^ { l - 1 } | I _ { j _ 2 } | } \\right ) ^ { H - \\frac { 1 } { 2 } } \\cdot \\left | \\dot { \\bar { h } } _ { l } ( v ) \\right | } { | q _ { k , l } ( u , v ) | ^ { H + \\frac { 1 } { 2 } } } | I _ { l } | d v \\right | ^ { 2 } | I _ { k } | d u . \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} \\varphi \\in L ^ 2 ( { \\bf R } ^ N , e ^ { | x | ^ 2 / 4 } \\ , d x ) , M ( \\varphi ) : = \\int _ { { \\bf R } ^ N } \\varphi ( y ) U ( | y | ) \\ , d y > 0 . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} \\Lambda _ \\alpha = \\{ f \\in C ( \\mathbb R ^ n ) : | f ( x ) - f ( y ) | \\le C | x - y | ^ \\alpha \\} \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\textbf { w } _ { k , i } = \\frac { 1 } { \\sqrt { M N } } \\textbf { a } \\left ( \\boldsymbol { \\omega } _ { k , i } \\right ) , \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} H _ { K , M , j } ^ { \\left ( D \\right ) } \\buildrel \\Delta \\over = \\hat \\eta _ { K , M , j } ^ { ( 1 ) } + \\hat \\eta _ { K , M , j } ^ { ( 2 ) } + \\hat \\eta _ { K , M , j } ^ { ( 3 ) } . \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } P _ { q ^ t } ( Q ) = ( - Q ) ^ \\frac { 1 + i } { 4 } ( - i Q ) ^ { - \\frac { 1 } { 2 } } Q ^ \\frac { 1 - i } { 4 } \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} S ( \\tau \\otimes \\tau ) = 0 . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} R _ { 0 } = R ( \\bar { x } , 0 , 0 ) = \\frac { k G _ { 1 } G _ { 2 } f ( \\bar { x } , 0 , 0 ) } { a u } . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n } } { ( q / z ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = - \\infty } ^ { \\infty } b ^ { 2 ' } _ \\omega ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ n } { ( q / z ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\displaystyle \\beta ^ { ( s ) } _ { n , k } = \\left \\{ \\begin{array} { l l } \\displaystyle 2 ^ { 2 s } ( - 1 ) ^ j { { j + 2 s - 1 } \\choose { 2 s - 1 } } , & k = 2 j n , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} q _ { k , l } ( u , v ) = ( 1 - v ) | I _ l | + | I _ { l + 1 } | + \\cdots + | I _ { k - 1 } | + u | I _ k | . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} { \\tilde { \\Q } } _ t ( \\mu , \\Gamma ) : = { \\Q } _ t ( T _ { I _ a } ( \\mu ) , T _ { I _ a } ( \\Gamma ) ) \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\Lambda \\cap P ^ { a _ i } = \\langle \\lambda _ i , \\ldots , \\lambda _ r \\rangle , \\Lambda \\cap Q ^ { b _ i } = \\langle \\lambda _ { \\sigma ^ { - 1 } ( i ) } , \\ldots , \\lambda _ { \\sigma ^ { - 1 } ( r ) } \\rangle . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} n _ { g . C , w } ( w _ 0 ) = g . C \\cap p _ { w } ^ { - 1 } ( w _ { 0 } ) . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} X = \\begin{cases} L ^ 2 ( \\mathbb R ; L ^ { \\frac { 2 n } { n - 2 } } ( \\mathbb R ^ n ) ) & n \\ge 3 \\\\ L ^ p ( \\mathbb R ; L ^ q ( \\mathbb R ^ 2 ) ) & n = 2 , \\end{cases} \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\frac { 1 } { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } | f ( n ) | ^ 2 = \\frac { 1 } { N } | \\sqrt { N } | ^ 2 = 1 \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} A ( n ) = \\sum _ { k = 0 } ^ n \\binom { n } { k } ^ 2 \\binom { n + k } { k } ^ 2 . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\{ f , ( I - P ) f ' \\} = \\{ f , f ' \\} - \\{ 0 , P f ' \\} , \\{ f , f ' \\} \\in T \\subset T ^ { * * } , \\{ 0 , P f ' \\} \\in T ^ { * * } , \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } e ^ { i \\psi ( 2 ^ k \\xi ) } \\varphi ( \\xi ) d \\xi & = \\int _ { \\mathbb { R } ^ n } e ^ { i 2 ^ k ( x - y ) \\cdot \\xi + i 2 ^ k ( t - s ) \\sqrt { 2 ^ { - 2 k } + | \\xi | ^ 2 } } \\varphi ( \\xi ) d \\xi \\\\ & \\leq C ( 1 + 2 ^ k | ( x - y , t - s ) | ) ^ { - \\frac { n } { 2 } } \\\\ & \\leq C 2 ^ { - \\frac { n } { 2 } ( k + j ) } \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} A _ { j } \\mathbf { \\perp \\ ! \\ ! \\ ! \\perp } _ { \\mathcal { G } } \\overline { \\mathbf { G } } _ { j } \\mathbf { | } \\overline { \\mathbf { B } } _ { j } , \\overline { \\mathbf { A } } _ { j - 1 } j = 0 , \\dots , p , \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} ~ D ^ { u } _ { ( n , 2 i ) } = 0 ~ ~ ( n , i ) \\geq ( n _ u , i _ u ) ~ ~ D ^ { u } _ { ( n , 2 i ) } \\neq 0 ~ ~ ( n , i ) \\geq ( n _ u , i _ u ) . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} M ( a , b , s ) : = \\sum _ { k = 0 } ^ \\infty \\frac { ( a ) _ k } { ( b ) _ k } \\frac { s ^ k } { k ! } , \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\mathcal L ^ 1 = \\mathrm { s p a n } \\{ e \\otimes e ^ * + f ^ * \\otimes f , \\ , e , f \\in X , \\ , e ^ * , f ^ * \\in X ^ * \\} ; \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\begin{aligned} { } \\bigg | \\psi ( \\tau \\overline { x ' } , \\tau ^ { 2 } y _ { 2 } \\cdots , \\tau ^ { k } y _ { k } ) - \\psi ( \\tau \\overline { x ' } , \\tau ^ { 2 } \\overline { y } _ { 2 } \\cdots , \\tau ^ { k } \\overline { y } _ { k } ) \\bigg | \\leq C _ { 3 } \\tilde { \\delta } \\tau ^ { 1 + \\alpha } d ( p , \\overline { p } ) \\leq C _ { 3 } \\tau \\omega ( { \\tau } ) \\tilde { \\delta } d ( p , \\overline { p } ) . \\end{aligned} \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\hat { S } = \\begin{bmatrix} S ^ { \\star } _ 1 & 0 & \\cdots & 0 \\\\ 0 & S ^ { \\star } _ 2 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & S ^ { \\star } _ { \\ell } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\mathbb { A } ^ { \\left ( M \\right ) } = \\left [ \\begin{array} [ c ] { c c c c c } \\mathbb { A } _ { N ; M } & & & & \\\\ & \\ddots & & & \\\\ & & \\mathbb { A } _ { N ; M } & & \\\\ & & & \\ddots & \\\\ & & & & \\mathbb { A } _ { N ; M } \\end{array} \\right ] _ { p ^ { M - N } \\times p ^ { M - N } } , \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\| V \\| _ { \\dot { H } ^ s } \\leq \\liminf _ { n \\rightarrow \\infty } \\| v _ n \\| _ { \\dot { H } ^ s } = \\| Q \\| _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\beta ' = - w _ d \\frac { \\sum _ { i = 1 } ^ { n } b _ i } { \\sum _ { i = 1 } ^ { n } a _ i } - w _ c \\frac { \\sum _ { j = 1 } ^ { m } ( v _ j - s _ j ) } { \\sum _ { j = 1 } ^ { m } ( u _ j + s _ j ) } . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ i - \\mu ^ { * } } = N ^ { 1 - \\delta } \\frac { c _ i ^ * } { \\sum \\limits _ { j = 1 } ^ { n } c _ j ^ { * } } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\varphi _ { y } ( P ( y ) ) = 0 , \\nabla \\varphi _ { y } ( P ( y ) ) = 0 ; \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\textbf { R } ' _ k & = \\mathbb { E } \\left \\lbrace \\textbf { h } ' _ k \\left ( { \\textbf { h } ' } _ k \\right ) ^ { \\textrm { H } } \\right \\rbrace = \\mathbb { E } \\left \\lbrace \\textbf { B } _ { k } ^ { \\rm H } \\textit { \\textbf { h } } _ k \\textit { \\textbf { h } } _ k ^ { \\rm H } \\textbf { B } _ { k } \\right \\rbrace = \\textbf { B } _ { k } ^ { \\rm H } \\textbf { R } _ k \\textbf { B } _ { k } , \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} | K _ { n , s } ( z ) | ^ 2 = \\dfrac { \\pi ^ 2 } { 2 ^ { 2 s - 1 } \\rho ^ { 2 ( 2 s + 1 ) n } } \\cdot \\dfrac { a } { b c ^ { 2 s } } . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} - n t \\sum _ { k = 0 } ^ { s } t _ { s - k } M _ n ( x ^ { n - 1 } , x ^ { s - k } ) = - n t \\sum _ { k = 1 } ^ { s } t _ { s - k } M _ n ( x ^ { n - 1 } , x ^ { s - k } ) , \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} a ( n ) = \\sum _ { j = 1 } ^ J c _ j e ^ { 2 \\pi i \\theta _ j n } \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} \\omega ( z ; q ) & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n } } { ( q ; q ^ 2 ) _ { n + 1 } } , \\\\ [ 6 p t ] \\nu ( z ; q ) & = \\sum _ { n = 0 } ^ { \\infty } ( q / z ; q ^ 2 ) _ n ( - z q ) ^ n , \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\mathfrak { A } : = X _ * ( A ) . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi \\upharpoonright _ { G ' \\cap H } ) = h _ { a l g } ( \\phi \\upharpoonright _ { H ' } ) + h _ { a l g } ( \\xi ) , \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\left [ \\mathbf { S } _ { n + 2 } \\right ] _ { k , \\ell } : = \\sqrt { \\frac { 2 } { n + 3 } } \\sin \\left ( \\frac { k \\ell \\pi } { n + 3 } \\right ) . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau [ a ] \\otimes \\tau ' ) = \\prod _ v \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau _ v [ a ] \\otimes \\tau ' _ v ) = + 1 ; \\\\ \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau \\otimes \\tau ' [ a ' ] ) = \\prod _ v \\varepsilon ( \\tfrac { 1 } { 2 } , \\tau _ v \\otimes \\tau ' _ v [ a ' ] ) = + 1 . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{gather*} [ \\rho ( e _ 1 ) , \\rho ( e _ 2 ) ] = \\rho ( [ e _ 1 , e _ 2 ] ) , \\end{gather*}"}
-{"id": "9962.png", "formula": "\\begin{align*} S _ 2 & \\geq \\phi ( q ) \\exp \\left ( ( 1 + o ( 1 ) ) ( 2 - \\log 4 ) \\frac { X } { \\log X } \\right ) \\\\ & = \\exp \\left ( ( 1 + o ( 1 ) ) \\left ( \\left ( 1 + \\frac { 2 - \\log 4 } { B } \\right ) \\log q \\right ) \\right ) , \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\frac { d \\varphi _ { k } } { d u } = \\frac { \\ddot { \\bar { h } } _ { k } ( u ) } { ( a _ { k } + u | I _ { k } | ) ^ { H - \\frac { 1 } { 2 } } } + \\left ( \\frac { 1 } { 2 } - H \\right ) \\cdot \\frac { | I _ { k } | \\ , \\dot { \\bar { h } } _ { k } ( u ) } { ( a _ { k } + u | I _ { k } | ) ^ { H + \\frac { 1 } { 2 } } } . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} ( m _ { i , j } ) \\left ( \\sum _ { g \\in G } g ^ { - 1 } ( \\alpha ) A ( g ) \\right ) = 0 ^ { ( \\chi ( 1 ) ) } . \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\inf _ { w \\in W } \\overline { E _ { \\rm e l } } ( w ) + E _ { \\rm g } ( w ) + E _ { \\rm s f } ( w ) = \\inf _ { w \\in W } E ( w ) . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\bar f ( s ) = \\int _ s ^ 1 f ( r ) \\xi '' ( r ) d r f ( r ) = \\int _ 0 ^ r \\frac { \\xi '' ( t ) d t } { ( B - \\hat \\nu ( s ) ) ^ 2 } - r . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\textbf { W } _ k = & \\textbf { F } _ { k } ^ c \\left ( \\hat { \\boldsymbol { \\psi } } _ 0 , \\textbf { W } _ { 1 } , \\cdots , \\textbf { W } _ { k - 1 } , \\textbf { y } _ 1 , \\cdots , \\textbf { y } _ { k - 1 } \\right ) \\\\ \\hat { \\boldsymbol { \\psi } } _ k = & \\textbf { F } _ { k } ^ e \\left ( \\hat { \\boldsymbol { \\psi } } _ 0 , \\textbf { W } _ { 1 } , \\cdots , \\textbf { W } _ { k } , \\textbf { y } _ 1 , \\cdots , \\textbf { y } _ { k } \\right ) , \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} \\int _ a ^ { r _ 2 } ( r _ 2 - r _ 1 ) ^ \\zeta d r _ 1 = \\frac { ( r _ 2 - a ) ^ { 1 + \\zeta } } { 1 + \\zeta } . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\omega _ { n , k } ^ { ( s ) } = \\sum _ { j = 0 } ^ k \\beta _ { n , j } ^ { ( s ) } \\gamma _ { n , k - j } ^ { ( s ) } . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } R _ k \\asymp R ( \\alpha t ) e ^ { \\alpha t } \\quad \\mbox { a s $ t \\to \\infty $ } . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } f ( x ^ { k } ) = \\mu \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\widetilde { m } ( x , \\Lambda , y ) = e ^ { \\frac { \\left | \\Lambda + \\nabla _ y w ( x , \\Lambda , y ) \\right | ^ 2 } { 2 } + V ( x , y ) } . \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { \\delta } _ { i j } \\widetilde { m } _ { \\Lambda _ j } d y + \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i \\Lambda _ j } \\widetilde { m } _ { \\Lambda _ j } d y = \\int _ { \\mathcal { Y } ^ d } \\frac { \\widetilde { m } _ { \\Lambda _ j } ^ 2 } { \\widetilde { m } } d y . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } = \\frac { [ n + 1 ] } { 2 } { 2 n + 1 \\brack n } \\sum _ { k = 1 } ^ { n } \\frac { ( 1 + q ^ { 2 k - n - 1 } ) q ^ { - { n - 2 k + 1 \\choose 2 } } } { [ 2 k ] ^ 2 { n \\brack k } _ { q ^ 2 } ^ 2 } . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\iint _ Q & \\hat { y } ( L ^ * u - \\alpha _ 1 z ^ 1 { 1 } _ { \\mathcal { O } _ { 1 , d } } - \\alpha _ 2 z ^ 2 { 1 } _ { \\mathcal { O } _ { 2 , d } } ) d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\hat { p } ^ i ( L z ^ i + \\dfrac { 1 } { \\mu _ i } u { 1 } _ { \\mathcal { O } _ i } ) d x d t \\\\ = & ( y _ 0 , u ( 0 ) ) + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\hat { y } u \\ d x d t + \\iint _ Q ( G u { G } _ 1 z ^ 1 + { G } _ { 2 } z ^ 2 ) d x d t \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\pi _ { l , l _ { 0 } } = \\left ( \\begin{array} { c c } { \\rm I d } _ { \\mathfrak { g } ^ { ( l _ { 0 } ) } } & 0 \\\\ 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} E _ { P } \\left [ b _ { \\mathbf { a } } ( \\mathbf { G } ; P ) \\right ] = E _ { P } \\left [ b _ { \\mathbf { a } } ( \\mathbf { G , B } ; P ) \\right ] = \\chi _ { \\mathbf { a } } ( P ; \\mathcal { G } ) , \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} Z _ P P = P Z _ P \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} E [ ( k X - 1 ) ^ n ] = d _ n ( k ) , \\ , \\ , ( n \\geq 0 ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\sqrt { - L } f ( \\mathbf x ) = - \\frac { 1 } { 2 \\sqrt { \\pi } } \\int _ 0 ^ { \\infty } t ^ { - 3 / 2 } ( H _ t f ( \\mathbf x ) - f ( \\mathbf x ) ) d t . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\begin{cases} w '' + A w + w ' = A F , t > 0 , \\\\ ( w , w ' ) ( 0 ) = ( \\tilde { u } _ 0 , \\tilde { u } _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E [ \\mathrm { B i n } ( n - a _ n , \\Pi _ n ( x , \\eta ) ) ] / b _ c ^ { ( n ) } = 0 . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\exp \\left \\{ - \\sum _ { i = 0 } ^ { n } \\frac 1 { \\left ( i + e _ { [ k ] } ^ 1 \\right ) \\ln \\left ( i + e _ { [ k ] } ^ 1 \\right ) \\dots \\ln _ k \\left ( i + e _ { [ k ] } ^ 1 \\right ) } \\right \\} \\le \\frac { 1 } { \\ln _ { k } ( n + 1 + e _ { [ k ] } ^ 1 ) } . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} m _ { \\tau , j , k , l } ( \\xi ) = \\begin{cases} & \\lvert \\tau \\rvert ^ { 1 / 2 } \\xi _ l \\left ( \\delta _ { j , k } - \\frac { \\xi _ j \\xi _ k } { \\lvert \\xi \\rvert ^ 2 } \\right ) e ^ { - \\tau \\lvert \\xi \\rvert ^ 2 } , \\xi \\in \\R ^ 2 \\setminus \\{ 0 \\} , \\\\ & 0 , \\xi = 0 , \\end{cases} \\hbox { f o r } 1 \\le j , k , l \\le 2 . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\partial _ r ( \\sigma ^ { a b } n _ { a b } ) = R i c ( L , \\underline L ) + R _ { L \\underline L L \\underline L } + l ^ { a b } n _ { a b } + d i v _ { \\sigma } \\eta - \\eta _ a \\eta ^ a . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\mathbf { A } = \\mathbf { C } _ { \\mathbf { y y } _ \\mathcal { Q } } ^ H \\mathbf { C } _ { \\mathbf { y } } ^ { - 1 } = \\sqrt { \\frac { 2 } { \\pi } } \\left ( \\mathbf { C } _ { \\mathbf { y } } \\right ) ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\alpha _ 2 & = 2 \\alpha _ 3 ; & \\alpha _ 1 & = 3 \\alpha _ 3 ; & \\alpha _ 1 & = \\alpha _ 2 + \\alpha _ 3 . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\omega ^ \\xi _ \\eta ( \\rho ) = w ^ \\xi _ \\eta ( \\rho ) : = \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\vec { \\mathcal { \\triangle } } ^ \\xi _ \\eta } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} & L ^ 2 \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } u ^ 2 + L \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , u ^ 2 \\\\ & \\leq C \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , | \\nabla ^ \\Sigma u | ^ 2 + C L ^ 2 \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq \\frac { L } { 2 } \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , u ^ 2 . \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d y } { d t } = & \\Delta y ( t ) + \\Gamma ^ { - 1 } ( t ) [ K ( \\Gamma ( t ) y ( t ) ) \\cdot \\nabla ] ( \\Gamma ( t ) y ( t ) ) , \\ t > 0 ; \\ y ( 0 ) = U _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\zeta _ l \\times ( \\xi _ i , 2 a _ i ) ) = \\prod _ { j = 0 } ^ { 2 a _ i - 1 } L ^ S ( s + a _ i - j , \\zeta _ l \\times \\xi _ i ) . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} & ( \\gamma ^ { ( 1 ) } ) ' ( s ^ { ( 1 ) } ( t ) ) = \\frac { A ^ { - 1 / 2 } ( r ( t ) ) \\gamma ' ( s ( t ) ) } { | A ^ { - 1 / 2 } ( r ( t ) ) \\gamma ' ( s ( t ) ) | } \\ , , n ^ { ( 1 ) } ( s ^ { ( 1 ) } ( t ) ) = \\frac { A ^ { 1 / 2 } ( r ( t ) ) n ( s ( t ) ) } { | A ^ { 1 / 2 } ( r ( t ) ) n ( s ( t ) ) | } \\ , \\\\ & \\dot s ^ { ( 1 ) } ( t ) = | A ^ { - 1 / 2 } ( r ( t ) ) \\gamma ' ( s ( t ) ) | \\dot s ( t ) \\ , , D \\chi ( r ( t ) ) = A ^ { - 1 / 2 } ( r ( t ) ) \\ , , \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\begin{aligned} Y ( s ) = & \\frac { \\hat { p } } { \\bar { y } } \\hat G _ y \\hat G _ h U _ h + \\frac { \\hat { g } } { \\bar { y } } \\hat G _ y U _ b + \\frac { \\hat d _ { y z } } { \\bar { y } } \\hat G _ y \\hat G _ d D _ { y z } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} ( \\sigma \\otimes \\sigma ) ( r ) + r = ( \\sigma \\otimes 1 ) r - ( 1 \\otimes \\sigma ) r \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} A ( - n ) = A ( n - 1 ) . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\frac { 1 } { p ' } = \\frac { 1 } { p } + \\frac { \\alpha } { m } , \\frac { 1 } { q ' } = \\frac { 1 } { q } + \\frac { \\alpha } { n } . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\sum _ { \\omega _ 0 \\in A } P \\bigl ( \\omega _ 0 \\bigm | \\omega _ { - \\infty } ^ { - 1 } \\bigr ) \\ ; = \\ ; 1 \\quad \\forall \\omega _ { - \\infty } ^ { - 1 } \\in \\Omega _ { < 0 } . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} & F _ * \\big ( ( x _ 0 , t _ 0 ) , \\phi ( x _ 0 , t _ 0 ) , D \\phi ( x _ 0 , t _ 0 ) , D ^ 2 \\phi ( x _ 0 , t _ 0 ) \\big ) \\\\ & = F ^ * \\big ( ( x _ 0 , t _ 0 ) , \\phi ( x _ 0 , t _ 0 ) , D \\phi ( x _ 0 , t _ 0 ) , D ^ 2 \\phi ( x _ 0 , t _ 0 ) \\big ) \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} W _ N ( i , j ) = \\frac 1 { M _ N } \\sum _ { k = 1 } ^ { M _ N } \\overline { Z _ { k , i } } Z _ { k , j } \\ , . \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\boldsymbol { \\nu } _ { k , i } = \\beta \\left ( \\textbf { x } _ k \\right ) \\textbf { w } _ { k , i } ^ \\textbf { a } ( \\textbf { x } _ k ) \\textbf { s } + \\boldsymbol { \\zeta } _ { k , i } . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\langle \\eta , \\gamma _ { X _ { l } ( t , x ) } \\eta \\rangle _ { \\mathbb { R } ^ { N } } & = \\left \\Vert D \\left ( \\langle \\eta , X _ { l } ( t , x ) \\rangle _ { \\mathbb { R } ^ { N } } \\right ) \\right \\Vert _ { \\bar { { \\cal H } } } ^ { 2 } \\\\ & = \\left \\Vert D \\left ( \\langle \\eta , X _ { l _ { 0 } } ( t , x ) \\rangle _ { \\mathbb { R } ^ { N } } \\right ) + D \\left ( \\langle \\eta , R _ { t } \\rangle _ { \\mathbb { R } ^ { N } } \\right ) \\right \\Vert _ { \\bar { { \\cal H } } } ^ { 2 } . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { q } ( v ) } \\le C ( [ \\vec v ] _ { A _ { \\vec q } } ) \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { q _ i } ( v _ i ) } \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\frac { R _ { 1 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = - i R _ { 1 4 } = 0 ; \\quad \\frac { R _ { 2 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = 0 ; \\quad \\frac { R _ { 3 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = - i = u _ 3 . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\nabla _ { ( 1 - q ) \\partial _ { t _ i } } \\left . \\right . _ \\circ \\left . \\right . S ^ { K \\textnormal { t h } } = S ^ { K \\textnormal { t h } } \\left . \\right . _ \\circ \\left . \\right . ( 1 - q ) \\partial _ { t _ i } & & T ^ { K \\textnormal { t h } } \\left . \\right . _ \\circ \\left . \\right . \\nabla _ { ( 1 - q ) \\partial _ { t _ i } } = ( 1 - q ) \\partial _ { t _ i } \\left . \\right . _ \\circ \\left . \\right . T ^ { K \\textnormal { t h } } \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} \\eta _ { k _ i } ^ 2 = \\sigma _ x ^ 2 ( \\mu _ { k _ i } - \\mu _ { k _ i } ^ 2 ) . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} S _ n ( x ) = \\sum _ { j = 1 } ^ n a _ j ( x ) = \\sum _ { j = 1 } ^ n a _ 1 ( T ^ { j - 1 } x ) \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\log B ( n ) = ( 1 + O ( n ^ { - \\kappa } ) ) \\tfrac { 1 } { u } ( K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) ) ^ { \\frac { 1 } { u + 1 } } ( u + 1 ) ^ { \\frac { u } { u + 1 } } n ^ { \\frac { u } { u + 1 } } \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\gamma \\in \\Delta } x _ { \\gamma } \\Big | = | x _ e | \\neq 0 \\ ; . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\nu _ { 2 \\omega + 1 } : = \\left [ \\{ \\nu _ { \\omega + n } \\} _ { n \\in \\N } , \\nu _ { 2 \\omega + 1 } ( \\phi _ { 2 \\omega } ) = \\gamma ' \\right ] . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\frac { 4 } { 3 } < p < 2 , \\ r = \\frac { 2 p } { 2 - p } , \\ q = \\frac { 2 p } { 4 - p } > 1 . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} A B - q B A = I \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} f _ 1 = f _ 0 ^ { - 1 } f _ 1 & \\sim _ \\nu h _ 0 , & f _ 1 ^ { - 1 } f _ 2 & \\sim _ \\nu h _ 1 , \\\\ f _ 2 ^ { - 1 } f _ 3 & \\sim _ \\nu h _ 2 , & f _ 3 ^ { - 1 } = f _ 3 ^ { - 1 } f _ 0 & \\sim _ \\nu h _ 3 , \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to + \\infty } B ( \\gamma ) = 0 \\ , , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} 1 _ { x L } \\star 1 _ { y L } = \\sum _ z M ( x , y , z ) \\cdot v o l \\left ( L \\cap y L y ^ { - 1 } \\right ) \\cdot 1 _ { z L } . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} I _ 2 = - \\int \\varphi ( u u _ x + p u ^ p u _ x ) = \\frac { 1 } { L } \\int \\varphi ' \\left ( \\frac 1 2 u ^ 2 + \\frac { p } { p + 1 } u ^ { p + 1 } \\right ) . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} & T ^ { \\nabla ^ { E } } ( \\omega \\otimes \\tau \\otimes s ) = - 8 ( { \\mathcal H } \\wedge \\omega ) \\otimes \\tau \\otimes s \\ + ( - 1 ) ^ { | \\omega | } \\omega \\otimes \\tau \\otimes C s \\\\ & + 2 ( - 1 ) ^ { | \\omega | + 1 } \\sum _ { i , j , k } ( R ( X _ { i } , X _ { j } ) , r _ { k } ) ^ { \\mathcal G } ( \\alpha _ { i } \\wedge \\alpha _ { j } \\wedge \\omega ) \\otimes \\tau \\otimes { \\tilde { r } _ { k } } s . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} h ( z ) = h ( i L ) + \\int _ { \\Gamma _ z } h ' ( w ) \\ , d w , z \\in D \\cup \\bigcup _ j \\partial _ p C _ j . \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} f _ 1 ( L ) \\sim f _ 2 ( L ) , ~ ~ a s ~ ~ L \\rightarrow + \\infty \\Leftrightarrow { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { f _ 1 ( L ) } { f _ 2 ( L ) } = 1 . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { 1 } \\log ( x ) \\frac { g ( x ) ^ { M - 1 } } { x ^ { \\alpha } ( x + 1 ) } d x = - \\int _ { 1 } ^ { \\infty } \\log ( x ) \\frac { g ( 1 / x ) ^ { M - 1 } } { x ^ { 1 - \\alpha } ( x + 1 ) } d x . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} V ( y ) = \\inf \\Bigl \\{ I _ { 0 T } ( \\varphi ) \\ , \\bigl \\vert \\ , \\varphi ( 0 ) = x _ 0 \\in D , \\ , \\ , T > 0 , \\ , \\ , \\varphi ( T ) = y \\ , \\ , \\ , \\ , \\ , \\ , y \\in \\partial D \\Bigr \\} , \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\sin \\Big ( ( 2 - n ) \\log | x | \\Big ) , u _ 2 ( x ) = \\cos \\Big ( ( 2 - n ) \\log | x | \\Big ) \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} B _ \\varepsilon ^ k = o ( 1 ) \\times \\frac { B _ \\varepsilon ^ { 2 N _ \\varepsilon } } { N _ \\varepsilon ! } \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\frac 1 { \\eta _ i } : = \\frac 1 { \\delta _ i } \\Big ( \\sum _ { j = 1 } ^ { m - 1 } \\frac 1 { \\delta _ j } \\Big ) ^ { - 1 } = \\frac 1 { \\delta _ i } \\Big ( \\sum _ { j = 1 } ^ { m + 1 } \\frac 1 { \\delta _ j } - \\frac 1 { \\delta _ m } - \\frac 1 { \\delta _ { m + 1 } } \\Big ) ^ { - 1 } = \\frac 1 { \\delta _ i } \\Big ( \\frac { 1 - r } { r } - \\frac 1 { \\varrho } \\Big ) ^ { - 1 } \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\chi _ 2 ( s ) & = \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } - \\phi ' ( m s ) \\\\ & \\leq O ( s ^ { - u - 1 } ) \\sum _ { s ^ { - 1 / 2 } < m < s ^ { - 1 } } \\frac { 1 } { m ^ { u + 1 } } \\\\ & = O ( s ^ { - u - 1 } ) O ( s ^ { u / 2 } ) \\\\ & = O ( s ^ { - u / 2 - 1 } ) , \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\sigma ^ 2 f _ g ' ( x ) - ( x - \\mu ) f _ g ( x ) = g ( x ) - \\frac 1 { \\sigma } \\int _ \\mathbb { R } g ( x ) \\phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\mathrm { d } x , \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\psi ( t ) = \\begin{cases} 0 , & \\mbox { i f $ b _ { 2 n - 2 } \\leq t < a _ { 2 n - 1 } $ } , \\\\ \\dfrac { t - a _ { 2 n - 1 } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } , & \\mbox { i f $ a _ { 2 n - 1 } \\leq t < b _ { 2 n - 1 } $ } , \\\\ 1 , & \\mbox { i f $ b _ { 2 n - 1 } \\leq t < a _ { 2 n } $ , } \\\\ \\dfrac { b _ { 2 n } - t } { b _ { 2 n } - a _ { 2 n } } , & \\mbox { i f $ a _ { 2 n } \\leq t < b _ { 2 n } $ } . \\end{cases} \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| f _ n \\| _ { W ^ { 1 , d } } ^ d = d ^ { d / 2 } c , \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) \\leq & \\liminf _ { \\ell \\to \\infty } { \\sum _ { j = 1 } ^ \\ell \\log 3 \\varepsilon e ^ { \\beta ^ j / a } / a \\over d \\sum _ { j = 1 } ^ { \\ell + 1 } \\log e ^ { \\beta ^ j / a } - \\log 3 \\varepsilon e ^ { \\beta ^ { \\ell + 1 } / a } / a } \\\\ = & \\liminf _ { \\ell \\to \\infty } { \\sum _ { j = 1 } ^ \\ell { \\beta ^ j / a } \\over d \\sum _ { j = 1 } ^ { \\ell + 1 } { \\beta ^ j / a } - { \\beta ^ { \\ell + 1 } / a } } \\\\ = & { 1 \\over d \\beta - \\beta + 1 } . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} d X _ t = b ( t , X _ t ) d t + \\sigma ( t , X _ t ) d B _ t , X _ 0 = x , \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} F ( t ) = t ^ { e _ { 0 } } F _ { 1 } ( t ) ^ { e _ { 1 } } \\cdots F _ { r } ( t ) ^ { e _ { r } } \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\gamma _ { } = \\frac { \\lambda _ { a b } ^ 2 { \\rho ^ \\ast } ^ 2 n _ p } { \\sigma _ b ^ 2 \\left [ \\sigma _ b ^ 2 + \\lambda _ { a b } \\rho ^ \\ast ( n _ p + 1 ) \\right ] } , \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k i a _ i ( \\theta ^ H _ t ) ^ { i - 1 } = 0 . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\int _ D \\tilde { p } _ n ( z ) \\overline { \\tilde { p } _ m ( z ) } w ( z ) \\ , \\d { A } ( z ) = \\delta _ { n , m } \\tilde { h } _ n \\ . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} E _ { P } \\left [ T _ { P , a , \\mathcal { G } } | M _ { k } , _ { \\mathcal { G } } \\left ( M _ { k } \\right ) \\right ] = \\frac { I _ { a } ( A ) } { \\pi _ { a } ( \\mathbf { O } _ { m i n } ; P ) } E _ { P } \\left [ Y | M _ { k } , _ { \\mathcal { G } } \\left ( M _ { k } \\right ) \\right ] . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} p _ { l } ^ { \\eta , \\psi } ( t , x , y ) & = \\int _ { M _ { x , y } } \\eta ( u ) \\psi \\left ( \\frac { \\Lambda \\| u \\| _ { \\textsc { C C } } } { t ^ { H } } \\right ) K ( u , x ) \\rho _ { t } ( u ) m _ { x , y } ( d u ) \\\\ & = \\int _ { M _ { x , y } \\cap W } \\eta ( u ) \\psi \\left ( \\frac { \\Lambda \\| u \\| _ { \\textsc { C C } } } { t ^ { H } } \\right ) K ( u , x ) \\rho _ { t } ( u ) m _ { x , y } ( d u ) , \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\aligned a \\sqrt { \\frac { n - 1 } { n } } \\int _ M \\big ( | L W _ \\infty | + k _ \\infty \\big ) \\bigg ( \\frac { d \\tau } { \\tau } \\bigg ) ^ 2 d v = & - \\frac { 1 } { 2 } \\int _ M \\bigg \\langle L W _ \\infty , \\ , L \\bigg ( \\frac { d \\tau } { \\tau } \\bigg ) \\bigg \\rangle d v \\\\ \\leq & \\frac { 1 } { 2 } \\int _ M | L W _ \\infty | \\bigg | L \\bigg ( \\frac { d \\tau } { \\tau } \\bigg ) \\bigg | d v \\\\ \\leq & \\frac { c } { 2 } \\int _ M | L W _ \\infty | \\bigg ( \\frac { d \\tau } { \\tau } \\bigg ) ^ 2 d v . \\endaligned \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\liminf _ { \\varepsilon \\to 0 } \\underset { : = \\delta _ \\varepsilon \\in ( 0 , 1 ) } { \\underbrace { \\frac { \\varphi _ { N _ \\varepsilon } \\left ( \\gamma _ \\varepsilon ^ 2 \\right ) } { \\exp \\left ( \\gamma _ \\varepsilon ^ 2 \\right ) } } } > 0 \\ , , \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} v \\mathbb { L } ( S ) = L ^ \\circ : = L \\cup \\{ 0 \\} \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} \\omega ' ( ( f _ 2 , \\phi _ 2 ) , ( f _ 1 , \\phi _ 1 ) ) \\ & = \\ \\langle A ( f _ 1 ) , \\phi _ 2 \\rangle + \\langle B ( \\phi _ 1 ) , \\phi _ 2 \\rangle + \\langle f _ 2 , D ( \\phi _ 1 ) \\rangle \\\\ & = \\ \\langle f _ 1 , { ^ t A } ( \\phi _ 2 ) \\rangle + \\langle { ^ t B } ( \\phi _ 2 ) , \\phi _ 1 \\rangle + \\langle { ^ t D } ( f _ 2 ) , \\phi _ 1 \\rangle . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\zeta _ l \\times ( \\zeta _ j , 2 b _ j + 1 ) ) = \\prod _ { i = 0 } ^ { 2 b _ j } L ^ S ( s + \\frac { 1 } { 2 } + b _ j - i , \\zeta _ l \\times \\zeta _ j ) , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} u v = \\dot { T } _ u v + \\dot { T } _ v u + \\dot { R } ( u , v ) = \\dot { T } _ u v + \\dot { T } ' _ v u , \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\Phi '' ( x ) \\right | & \\leq c ( \\Phi ( x ) ^ { q _ 1 } + 1 ) , \\\\ \\quad \\Phi ( ( 1 - t ) x + t y ) & \\leq c ( \\Phi ( x ) ^ { q _ 2 } + \\Phi ( | x - y | ) ^ { q _ 2 } + 1 ) , \\end{aligned} \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\left [ \\left ( 1 - \\Lambda _ 0 + ( 1 - q ) \\Lambda _ 0 \\delta _ q \\right ) \\cdots \\left ( 1 - \\Lambda _ N + ( 1 - q ) \\Lambda _ N \\delta _ q \\right ) - Q \\right ] J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) = 0 \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ * ( U _ 1 ; t ) & = 2 \\| U _ 1 ' \\| ^ 2 + 2 ( U _ 1 , - A U _ 1 + A e ^ { - t A } ( u _ 0 + u _ 1 ) ) \\\\ & = 2 \\| U _ 1 ' \\| ^ 2 - 2 \\| A ^ { 1 / 2 } U _ 1 \\| ^ 2 + 2 ( U _ 1 , A e ^ { - t A } ( u _ 0 + u _ 1 ) ) \\\\ & \\leq 2 \\| U _ 1 ' \\| ^ 2 - \\| A ^ { 1 / 2 } U _ 1 \\| ^ 2 + \\| A ^ { 1 / 2 } e ^ { - t A } ( u _ 0 + u _ 1 ) \\| ^ 2 \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( 0 ) ) \\leq \\lfloor \\kappa _ n ( 0 ) \\rfloor - a _ n ) & \\geq \\liminf _ { n \\to \\infty } \\frac { n - a _ n } { a _ c ^ { ( n ) } } \\log ( 1 - \\pi _ n ( \\kappa _ n ( 0 ) ) ) \\\\ & = \\liminf _ { n \\to \\infty } \\left ( - \\frac { n } { a _ c ^ { ( n ) } } \\pi _ n ( \\kappa _ n ( 0 ) ) + \\frac { n } { a _ c ^ { ( n ) } } o ( \\pi _ n ( \\kappa _ n ( 0 ) ) ) \\right ) \\\\ & = - r ^ { - 1 } ( 1 - r ^ { - 1 } ) ^ { r - 1 } \\alpha ^ r = - J ( 0 ) . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty f ( x ) \\ , \\d x = h \\sum _ { m = - \\infty } ^ \\infty f ( m h ) . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} T _ { 2 , 0 } & = \\sum _ { 0 \\le K _ i \\le M - 1 , \\atop ( i \\ge 1 ) } B _ { \\tilde { n } } \\left ( \\left \\{ \\frac { \\sum _ { i \\ge 1 } K _ i } { M } \\right \\} \\right ) = B _ { \\tilde { n } } ( 0 ) . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} + \\infty & > \\int _ { \\mathbb B ( a , r ) } d d ^ c w \\geq \\int _ { \\mathbb B ( a , r ) \\cap \\Omega } d \\mu \\\\ & \\geq \\sum _ { j \\geq j _ 0 } \\int _ { \\overline { \\mathbb B } ( a _ j , r _ j ) \\cap \\{ \\varphi _ j \\geq - 1 \\} } d d ^ c u _ { j , k _ j } \\\\ & = \\sum _ { j \\geq j _ 0 } \\int _ { \\Omega ' } d d ^ c u _ { j , k _ j } \\geq \\sum _ { j \\geq j _ 0 } \\frac { 1 } { 2 } = + \\infty . \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} & \\mathcal { S } : C ^ { 2 , \\alpha } ( \\mathcal { U } ) \\to C ^ { 0 , \\alpha } ( N L ) \\\\ & \\mathcal { S } ( V ) = - X _ V ^ * \\left ( H ( X + V ) + \\frac { ( X + V ) ^ { \\perp _ V } } { 2 } \\right ) ^ { \\perp } . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} \\lim _ { \\eta _ n \\rightarrow 0 } r _ 1 = - \\frac { a ^ 2 - 1 } { 2 \\sigma ^ 2 } . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ \\xi _ y - \\xi _ z : y , z \\in E ^ { * } \\} = X , \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\frac { d } { d t } x ( t ) = \\int _ { U \\times V } f ( t , x ( t ) , m ( t ) , u , v ) \\eta ( t , d ( u , v ) ) , \\ \\ x ( s ) = y . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} C _ { 2 l } ^ { ( 1 + \\alpha ) } ( x ) = \\frac { \\Gamma ( 2 l + 1 + \\alpha ) 2 ^ { 2 l } } { \\Gamma ( 1 + \\alpha ) ( 2 l ) ! } x ^ { 2 l } + O ( x ^ { 2 l - 2 } ) \\ . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} Z _ t ^ { u } [ s ] : = \\begin{cases} z _ s , & ~ s \\in [ 0 , t ) \\\\ u & ~ s = t , \\end{cases} ~ ~ W _ t ^ { v } [ s ] : = \\begin{cases} w _ s , & ~ s \\in [ 0 , t ) \\\\ v & ~ s = t . \\end{cases} \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\lambda _ { q ^ t } ^ { \\ell _ { q ^ t } ( Q ) } = e ^ { \\mu \\log ( q ^ t ) \\ell _ q ( Q ) } \\sim _ { t \\to 0 } e ^ { \\mu ( q ^ t - 1 ) \\ell _ { q ^ t } ( Q ) } \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} V ( x ) = V _ { 0 } \\left ( \\vert x \\vert \\theta ( \\widehat { x } ) \\right ) , \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} T _ 2 ( t , c ) & = \\begin{cases} ( 1 - c ) & \\textup { i f } \\frac { 1 } { 2 } \\leq t \\leq 1 , 0 \\leq c < \\frac { 1 } { 2 } \\\\ t & \\textup { o t h e r w i s e } \\end{cases} \\\\ C _ 2 ( t , c ) & = \\begin{cases} ( 1 - t ) & \\textup { i f } 0 \\leq t \\leq \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\leq c \\leq 1 \\\\ c & \\textup { o t h e r w i s e } \\end{cases} , \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} \\lim _ { \\ell \\rightarrow \\infty } \\alpha _ { \\ell } = r _ 1 . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B _ 1 ^ { ( n ) } ) = - J ( x _ 0 ) , \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} ( \\partial _ r \\partial _ { \\theta _ \\alpha } u _ 0 ) ( x , t ) = ( \\partial _ { \\theta _ \\alpha } \\partial _ { \\theta _ \\beta } u _ 0 ) ( x , t ) = 0 \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} \\mathbb { E } \\# \\{ F = 0 \\} \\cap W & = \\int _ { W } \\mathbb { E } \\left \\{ | \\det J ( x ) | \\ , \\bigg | \\ , F ( x ) = 0 \\right \\} \\rho _ { F ( x ) } ( 0 ) \\omega ( x ) d x \\\\ & = \\int _ { W } \\rho ( x ) \\omega ( x ) d x . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} f ^ { ' } ( t ) = 4 t ^ 3 - 6 S t ^ 2 - 1 2 S ( S - 1 ) t + 2 S ( 2 - 3 S ) ^ 2 , \\ \\ f ^ { '' } ( t ) = 1 2 ( t ^ 2 - S t - S ( S - 1 ) ) , \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} K ( t ) = \\sum _ { k \\geq 1 } c _ k e ^ { - \\lambda _ k t } . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\widetilde \\Sigma ( n ) = 2 1 0 8 . 1 8 5 2 6 4 8 7 2 4 2 8 5 \\ldots \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\gamma _ 1 \\frac { r _ n } { \\sqrt { k } } \\leq D _ l ( k ) \\leq \\gamma _ 2 \\frac { r _ n } { \\sqrt { k } } \\mathbb { E } \\left ( d ^ 2 ( Y _ { k } , \\{ Y _ u \\} _ { 1 \\leq u \\leq k - 1 } ) | N _ l = k \\right ) \\leq \\gamma _ 3 \\frac { r _ n ^ 2 } { k } . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} X _ j = \\partial _ { x _ j } - \\frac { y _ j } { 2 } \\partial _ z , Y _ j = \\partial _ { y _ j } + \\frac { x _ j } { 2 } \\partial _ z , Z = \\partial _ z , \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} \\delta < \\min \\Big \\{ \\frac { \\lambda _ + \\varphi ^ + _ l ( t ) } { a _ + ( t ) } , \\frac { \\lambda _ - \\varphi ^ - _ l ( t ) } { a _ - ( t ) } \\mid t \\in \\R , \\ l = 1 , 2 , \\cdots , m \\Big \\} , \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} u _ { t x x } = \\mathcal { A } ( u ( t ) ) u , \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} I [ w ] = \\int ^ T _ 0 \\int _ { \\Omega _ e } [ \\frac { 1 } { 2 } \\mid w _ t \\mid ^ 2 - W ( D w ) + | w | ^ 2 ] d x d \\tau , \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\mathcal { B } = \\bigcap _ { N _ { 0 } } \\bigcup _ { N > N _ { 0 } } \\mathcal { B } _ { N } , \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { p ^ r - 1 } { 2 } } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 } ( 3 k + 1 ) ( - 1 ) ^ k 2 ^ { 3 k } & \\equiv p ^ r ( - 1 ) ^ { \\frac { p - 1 } { 2 } } \\pmod { p ^ { r + 2 } } \\quad . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} D _ { I J } { } ^ K = D _ { J I } { } ^ K + \\overline T _ { J I } { } ^ K . \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} ( x _ 1 , \\varphi ^ * ( x _ 2 ) ) _ { R T } = ( x _ 1 , x _ 2 ) _ K . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} E _ { 1 } & = \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 2 } a _ { j } ^ { 2 } \\\\ & = \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 2 } \\Pi _ { l } ( g _ { j } ^ { 2 } , h _ { j , 1 } ^ { 2 } , h _ { j , 2 } ^ { 2 } ) + \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { 2 } ( a _ { j } ^ { 2 } - \\Pi _ { l } ( g _ { j } ^ { 2 } , h _ { j , 1 } ^ { 2 } , h _ { j , 2 } ^ { 2 } ) \\\\ & = : M _ { 2 } + E _ { 2 } , \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} & \\Delta S _ 0 - 8 \\exp ( - 2 T _ 0 ) S _ 0 = 4 \\exp ( - 2 T _ 0 ) \\left ( T _ 0 ^ 2 - T _ 0 \\right ) \\ , , \\\\ & \\Delta S _ 1 - 8 \\exp ( - 2 T _ 0 ) S _ 1 = 4 \\exp ( - 2 T _ 0 ) \\left ( S _ 0 + 2 S _ 0 ^ 2 - 4 T _ 0 S _ 0 + 2 S _ 0 T _ 0 ^ 2 - T _ 0 ^ 3 + \\frac { T _ 0 ^ 4 } { 2 } \\right ) \\ , , \\\\ & \\Delta S _ 2 - 8 \\exp ( - 2 T _ 0 ) S _ 2 = 4 \\exp ( - 2 T _ 0 ) T _ 0 \\ , , \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\varphi _ { n } ( \\pi ) = \\frac { \\sum _ { \\alpha , \\beta \\in K _ { n } } N _ { \\bar { \\alpha } \\beta } ^ { \\pi } d _ \\alpha d _ \\beta } { d _ \\pi ( \\sum _ { \\xi \\in K _ { n } } d _ \\xi ^ 2 ) } , \\pi \\in \\mathrm { I r r } ( \\mathbb G ) . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} L _ u ( D _ e u ) = 0 , L _ u ( D _ { e e } u ) \\le 0 \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} \\langle \\nabla ^ { \\Sigma , L } _ { \\dot { \\gamma } } { \\dot { \\gamma } } , \\dot { \\gamma } \\rangle _ { \\Sigma , L } & = ( \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 ) \\cdot \\left ( \\overline { q } \\frac { \\ddot { \\gamma } _ 1 \\gamma _ 1 - ( \\dot { \\gamma } _ 1 ) ^ 2 } { \\gamma ^ 2 _ 1 } - \\overline { p } \\ddot { \\gamma } _ 3 \\right ) . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\Delta _ \\pi ( z ) = \\left ( D ( z ) , \\dot D ( z ) , \\dots , D ^ { ( n - 1 ) } ( z ) \\right ) , \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} { \\lim \\limits _ { M , N \\to + \\infty } } { C } _ S ^ { \\min } ( \\boldsymbol { \\psi } ) = \\mathop { \\lim } \\limits _ { M , N \\to + \\infty } C _ S ( \\boldsymbol { \\psi } , \\textbf { W } ^ * ) \\\\ \\overset { ( g ) } { \\ge } \\mathop { \\lim } \\limits _ { M , N \\to + \\infty } C _ S ( \\boldsymbol { \\psi } , \\widetilde { \\textbf { W } } _ S ^ * ) , \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { U } _ i ^ F } { \\partial d _ i } \\big | _ { d _ i = d _ i ^ { n e } } = \\frac { 1 } { w _ i } \\left ( \\frac { \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } { | \\mathcal { S } | - 1 } - \\frac { c _ i + \\lambda _ i } { w _ i } \\right ) > 0 , \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\left [ \\frac { B ( z , 1 ) - B ( z , t ) } { 1 - t } \\right ] _ { t = 1 } = B _ t ( z , 1 ) , \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} \\textnormal { c h } ( \\Lambda _ i ) = \\textnormal { c h } ( f _ K ( q ) ) ^ { - \\frac { \\lambda _ i } { g _ H ( z ) } } \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\langle b _ { i , j } ^ { \\epsilon _ k } , g \\rangle = \\langle b _ { i , j } , g \\rangle , \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} \\nabla ^ { E } _ { \\xi + r + X } ( \\eta + s + Y ) : = ( \\nabla ^ { F } _ { X } \\eta - \\frac { 1 } { 3 } { \\mathcal H } ( X , Y , \\cdot ) , \\nabla _ { X } s + \\frac { 2 } { 3 } [ r , s ] ^ { \\mathcal G } , \\nabla _ { X } ^ { F } Y ) \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\exp ( - \\gamma ( \\lambda ) \\sum _ { k = 1 } ^ { m - 1 } \\phi _ \\delta ( \\norm { A _ k } _ { B ( \\mathcal { K } ) } ) ) & \\asymp \\exp \\left [ - ( 1 - \\epsilon _ 0 ) \\sqrt { - \\lambda } \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { k ^ { \\alpha / 2 } } \\right ] \\\\ & \\asymp \\exp \\left [ - ( 1 - \\epsilon _ 0 ) \\frac { \\sqrt { - \\lambda } } { 1 - \\frac { \\alpha } { 2 } } n ^ { 1 - \\alpha / 2 } \\right ] \\ , . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} p ^ * = c , ~ B ^ * = 0 . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} { \\rm d e t } ( I I ^ L ) = - \\frac { L } { 4 } - \\langle e _ 1 , \\nabla _ H ( \\frac { X _ 3 u } { | \\nabla _ H u | } ) \\rangle - \\overline { p } [ X _ 1 ( \\overline { p } ) + X _ 2 ( \\overline { q } ) ] + O ( L ^ { - \\frac { 1 } { 2 } } ) ~ ~ { \\rm a s } ~ ~ L \\rightarrow + \\infty . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\sigma ^ \\ast ( t _ 2 ) = s _ 1 , \\ \\ \\ \\ \\sigma ^ \\ast ( \\mu _ 3 ) = x _ 2 , \\ \\ \\ \\ \\sigma ^ \\ast ( q _ 4 ) = \\mu _ 3 . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} B _ n = \\{ z \\in \\Omega : B _ \\Omega ( z , y _ n ) < \\delta _ n / 2 \\} . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} y _ { t t } - ( D _ 2 a ( 0 , t , x ) y _ x + a ( 0 , t , x ) y _ { x t } ) _ x + D _ 1 F ( 0 , 0 ) y _ t + D _ 2 F ( 0 , 0 ) y _ { x t } = f _ t 1 _ { \\mathcal { O } } - \\frac { 1 } { \\mu _ 1 } p ^ 1 _ t 1 _ { \\mathcal { O } _ 1 } - \\frac { 1 } { \\mu _ 2 } p ^ 2 _ t 1 _ { \\mathcal { O } _ 2 } + G _ { t } . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} \\theta _ p ( u , x , r ) : = r ^ { p - n } \\int _ { B _ r ( x ) } | d u | ^ p . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\varphi ' ( w ) = - \\frac { [ q \\xi ' ( q ) - \\xi ( q ) ] ( 1 - q ) } { \\xi ' ( 1 ) - \\xi ' ( q ) } + \\frac { q ^ 2 } w - \\frac { \\xi ( q ) q ^ 2 [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { \\xi ' ( q ) ^ 2 ( 1 - q ) w ^ 2 } . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ 2 h ) = 1 & \\\\ h = u & \\end{cases} \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} \\sum ^ { k _ 0 + 1 } _ { i = 1 } c _ i ( - 1 / i ) ^ m = 1 , m = 0 , 1 , \\dots , k _ 0 , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} \\varphi ( s ) = c _ 1 + c _ { 2 } 2 ^ { - s } \\hbox { \\ w i t h } \\ { \\rm R e } \\ , c _ 1 \\geq \\frac { 1 } { 2 } + | c _ 2 | . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\psi ( f ^ { - 1 } ( 0 ) \\setminus V ) = \\psi ( f ^ { - 1 } ( 0 ) ) \\setminus \\psi ( V ) = \\Sigma \\cap g ^ { - 1 } ( 0 ) \\setminus A . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} N = d i m ( \\mathbb { P } _ n ^ d ( K ) ) = d i m ( \\mathbb { P } _ n ^ d ( \\mathcal { M } ) ) = { n + d \\choose d } - { n - k + d \\choose d } \\ ; , \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\sum _ { j \\le k } \\binom { s } { j } 4 ^ { j } d ^ { O ( k ) } = s ^ { k } d ^ { O ( k ) } . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} N ( g \\alpha ) = N ( \\alpha ) . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m } } \\cdot L : L ] = [ E _ { t _ 1 } \\cdot \\cdots \\cdot E _ { t _ { m } } : k ] \\ \\ \\hbox { \\rm e t } \\ \\ [ E _ { t _ { m + 1 } } \\cdot L : L ] = [ E _ { t _ { m + 1 } } : k ] . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\frac { \\wp ' ( u ) - \\wp ' ( v ) } { \\wp ( u ) - \\wp ( v ) } = 2 \\zeta ( u + v ) - 2 \\zeta ( u ) - 2 \\zeta ( v ) , \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\frac { \\partial { l o g \\ , p _ { D I } ( \\textbf { y } _ k | \\textbf { x } , \\textbf { W } _ k ) } } { \\partial { x _ p } } \\ ! = \\ ! - \\frac { 1 } { \\lvert \\boldsymbol { \\Sigma } _ { \\textbf { y } , k } \\rvert } \\frac { \\partial \\lvert \\boldsymbol { \\Sigma } _ { \\textbf { y } , k } \\rvert } { \\partial { x } _ p } \\ ! - \\ ! \\textbf { y } _ k ^ \\frac { \\partial \\boldsymbol { \\Sigma } _ { \\textbf { y } , k } ^ { - 1 } } { \\partial { x } _ p } \\textbf { y } _ k , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\mu _ { D ( x , n ) } = \\mu _ { x , n } = \\frac { 1 } { \\mu ( D ( x , n ) ) } \\mu \\vert _ { D ( x , n ) } \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\omega _ { \\mathcal { O } _ \\mu } ^ + ( \\nu ) ( \\xi _ { \\mathfrak { g } ^ \\ast } ( \\nu ) , \\eta _ { \\mathfrak { g } ^ \\ast } ( \\nu ) ) = < \\nu , [ \\xi , \\eta ] > , \\ ; \\ ; \\forall \\ ; \\nu \\in \\mathcal { O } _ \\mu , \\ ; \\xi , \\eta \\in \\mathfrak { g } . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} E ( A , \\Pi _ h \\xi , \\zeta _ h ) = E ( ( I - P _ 0 ) A , \\Pi _ h \\xi , \\zeta _ h ) + E ( P _ 0 A , ( I - P _ 0 ) \\Pi _ h \\xi , \\zeta _ h ) + E ( P _ 0 A , P _ 0 \\Pi _ h \\xi , \\zeta _ h ) . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\int _ 0 ^ { n + 1 } f ( x ) d x > \\sum _ { i = 1 } ^ { n } f ( i ) > \\int _ 1 ^ { n + 1 } f ( x ) d x > \\sum _ { i = 2 } ^ { n + 1 } f ( i ) . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} w ^ { \\eta ^ - _ i } _ { \\xi ^ - _ i } ( \\rho _ \\tau ) = \\sum _ { [ \\{ \\alpha , \\beta ; \\epsilon \\} ] \\in \\vec { \\triangle } ^ { \\eta ^ - _ i } _ { \\xi ^ - _ i } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho _ \\tau ) + \\ell _ { \\beta } ( \\rho _ \\tau ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\begin{aligned} - 1 6 L \\mathcal Q ( t ) \\geq & ~ { } \\tilde \\sigma \\int g ^ 2 + 4 \\tilde \\sigma \\int g _ x ^ 2 + 3 \\tilde \\sigma \\int g _ { x x } ^ 2 + \\frac 1 2 \\tilde \\sigma \\int g _ { x x x } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} & \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } \\left ( L _ 1 - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ \\mu ) - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ { \\mu ' } ) \\right ) \\\\ & = 1 - \\left ( \\frac { \\cosh \\frac { \\ell _ \\nu } { 2 } + e ^ { - \\ell _ \\mu } } { \\cosh \\frac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu } + \\frac { \\cosh \\frac { \\ell _ \\nu } { 2 } + e ^ { - \\ell _ { \\mu ' } } } { \\cosh \\frac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ { \\mu ' } } \\right ) . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\exp \\left ( X \\right ) \\exp \\left ( Y \\right ) = \\exp \\left ( X + Y + \\frac { 1 } { 2 } \\left [ X , Y \\right ] \\right ) = \\exp \\left [ \\begin{array} [ c ] { c c c } 0 & x _ { 1 } + y _ { 1 } & x _ { 3 } + y _ { 3 } + \\frac { x _ { 1 } y _ { 2 } - x _ { 2 } y _ { 1 } } { 2 } \\\\ 0 & 0 & x _ { 2 } + y _ { 2 } \\\\ 0 & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} & ( f \\smile g ) ( a e _ x e _ y ) = - f ( a e _ x ) g ( a x e _ y ) + f ( a e _ y ) g ( a e _ { x } y ) , \\\\ & ( f \\smile g + g \\smile f ) ( a e _ x e _ y ) = \\\\ & - f ( a e _ x ) g ( a x e _ y ) + f ( a e _ y ) g ( a e _ { x } y ) - g ( a e _ x ) f ( a x e _ y ) + g ( a e _ y ) f ( a e _ { x } y ) . \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\textrm { I M F } _ 1 = \\lim _ { m \\rightarrow \\infty } ( I - W ) ^ m s = U Z U ^ T s \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\| \\nabla ^ 2 u _ { 1 , i } \\| _ { L ^ \\infty ( B ( 0 , \\eta _ 1 ) ) } + \\| \\nabla ^ 2 R _ 2 \\| _ { L ^ \\infty ( B ( 0 , \\eta _ 1 ) ) } = O ( t ^ { - \\frac { N + 2 A _ 1 } { 2 } } ) \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} Q ^ V f ( x , y ) = P _ y f ( x ) = e ^ { - y \\sqrt { - L } } f ( x ) , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\chi _ { P , a , e f f } ^ { 1 , n o n - d e s c } \\left ( \\mathbf { V } ; \\mathcal { G } \\right ) = b _ { a } \\left ( \\mathbf { O ; } P \\right ) - \\chi _ { a } \\left ( P ; \\mathcal { G } \\right ) . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} \\binom { a } { b } \\equiv \\prod _ { i = 0 } ^ { \\ell - 1 } \\binom { a _ i } { b _ i } \\mod p \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - \\Delta u + \\frac { \\mu } { 1 + t } u _ t + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } u = f ( t , x ) , & x \\in \\mathbb { R } ^ n , \\ t > 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ u _ t ( 0 , x ) = u _ 1 ( x ) , & x \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} v ^ { ( \\ell ) } : = v _ 1 ^ { ( \\ell ) } = v _ 2 ^ { ( \\ell ) } \\Omega \\ell = 1 , 2 , \\cdots , N + 1 , \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} & \\{ ( k , l ) \\mid k , l \\ne j , \\alpha _ k \\ne \\alpha _ l \\} \\\\ = & \\{ ( k , l ) \\mid k , l \\not \\in C _ J , \\alpha _ k \\ne \\alpha _ l \\} \\\\ & \\cup \\{ ( k , l ) \\mid k \\not \\in C _ J , l \\in C _ J , l \\ne j \\} \\\\ & \\cup \\{ ( k , l ) \\mid k \\in C _ J , k \\ne j , l \\not \\in C _ J \\} , \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\frac { v o l ( D _ { i , a } \\cap { \\hat { \\mathfrak { D } } _ n } ) } { v o l ( \\hat { \\mathfrak { D } } _ n ) } = \\sum _ { k = i } ^ n { n \\choose k } V ( k ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\left [ ( z Q \\partial _ Q ) ^ { N + 1 } - Q \\right ] \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) = H ^ { N + 1 } Q ^ \\frac { H } { z } \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} \\begin{aligned} P \\left ( \\hat { \\textbf { x } } _ k \\in \\mathcal { I } , \\forall k \\ge 0 \\right ) \\ge 1 - 8 e ^ { - \\frac { R \\lvert \\textbf { s } \\rvert ^ 2 } { \\epsilon _ S ^ 2 \\sigma _ z ^ 2 } } . \\end{aligned} \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\varphi _ N ( t ) = \\sum _ { k = N + 1 } ^ { + \\infty } \\frac { t ^ k } { k ! } \\ , . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\| T ( f _ 1 , \\dots , f _ m ) \\| _ { L ^ p ( w ) } \\le \\zeta ^ { - 1 } ( 1 - r ) ^ { - ( m + 1 ) } [ \\vec { w } ] _ { A _ { \\vec { p } , \\vec { r } } } ^ { \\frac 1 { 1 - r } } \\prod _ { i = 1 } ^ { m + 1 } \\| f _ i \\| _ { L ^ { p _ i } ( w _ i ) } , \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 2 } ^ { e - 1 } t _ { i } ( ( q + 1 ) m + ( q + i ) d ) \\equiv ( j - x _ { 1 } ) ( m + d ) + ( q + 1 ) m + ( q + e - 1 ) d ( \\textrm { m o d } \\ , m ) . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} f ( z ) = \\frac { 1 } { 1 - z } + \\frac { f ^ { \\not \\ominus } ( z ) } { ( 1 - z ) ^ 2 } . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\frac { \\Psi _ { N _ \\varepsilon } ' ( u _ \\varepsilon ) } { 2 } ~ & = u _ \\varepsilon H ( u _ \\varepsilon ) \\varphi _ { N _ \\varepsilon } ( u _ \\varepsilon ^ 2 ) + u _ \\varepsilon ( 1 + g ( u _ \\varepsilon ) ) \\frac { u _ \\varepsilon ^ { 2 N _ \\varepsilon } } { N _ \\varepsilon ! } + O \\left ( \\gamma _ \\varepsilon ^ 3 \\right ) \\\\ & \\le ( 1 + o ( 1 ) ) \\gamma _ \\varepsilon \\varphi _ { N _ \\varepsilon - 1 } ( \\gamma _ \\varepsilon ^ 2 ) \\ , . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ N \\sum _ { l \\geq 0 } \\frac { 1 } { v ! } e ^ { \\tau _ 2 ( d ) } \\left \\langle \\frac { e ^ { - \\tau _ 2 / z } } { z + \\psi } , T _ k , ( \\tau ' ) ^ v \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } T ^ j \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} d ( z , z _ 1 ) & = d ( T z , T z _ 1 ) \\\\ & < d ( z , z _ 1 ) , \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\norm { u _ m } _ { \\mathcal { K } } & = \\norm { P _ m u } _ { \\mathcal { H } } \\\\ & = \\norm { P _ m \\tau ( J ( \\tau ) - \\lambda I ) ^ { - 1 } P _ 1 L ^ * L P _ 1 u } _ { \\mathcal { H } } \\\\ & \\le \\tau \\norm { P _ m ( J ( \\tau ) - \\lambda I ) ^ { - 1 } P _ 1 } _ { B ( \\mathcal { H } ) } \\norm { P _ 1 u } _ \\mathcal { H } \\\\ & \\le C \\exp ( - \\gamma ( \\lambda ) \\sum _ { k = 1 } ^ { m - 1 } \\phi _ \\delta ( \\norm { A _ k } _ { B ( \\mathcal { K } ) } ) ) \\norm { P _ 1 u } _ \\mathcal { H } \\ , , \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = 0 } ^ { \\infty } { ' } \\alpha _ k T _ k ( z ) , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} n _ 0 = \\begin{cases} ( n - s + 1 ) / 2 , & \\\\ ( n - s + 2 ) / 2 , & . \\end{cases} \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} h _ 0 & = ( x _ 1 , x _ 2 + d ' x _ 3 ^ p , x _ 3 ) , & h _ 1 & = ( x _ 1 + P _ 1 ( x _ 2 , x _ 3 ) , x _ 2 , x _ 3 ) , \\\\ h _ 2 & = ( x _ 1 , x _ 2 + d x _ 3 ^ p , x _ 3 ) , & h _ 3 & = ( x _ 1 + P _ 3 ( x _ 2 , x _ 3 ) , x _ 2 , x _ 3 ) , \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} | H _ 0 | = & \\frac { 2 } { r } + h _ 0 ^ { ( 1 ) } r + O ( r ^ 2 ) \\\\ ( \\alpha _ { H _ 0 } ) _ a = & ( \\alpha ^ { ( 2 ) } _ { H _ 0 } ) _ a r ^ 2 + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} T ( x _ { 1 } , x _ { 2 } ) = ( x _ { 1 } + x _ { 2 } , x _ { 2 } + \\omega ) , \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} \\mathcal { S } _ \\infty ^ { + } = \\left ( 0 , ~ \\frac { 2 \\eta } { \\sigma ^ 2 } \\right ] , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} & p _ i = G \\sqrt { \\Delta t } w _ i , i \\geq 1 , \\\\ & q _ i = \\sum _ { j = 1 } ^ i p _ j , \\\\ & x _ i = \\begin{cases} q _ i - \\left \\lfloor q _ i \\right \\rfloor , & \\left \\lfloor q _ i \\right \\rfloor \\\\ 1 - q _ i + \\left \\lfloor q _ i \\right \\rfloor , & \\left \\lfloor q _ i \\right \\rfloor \\end{cases} \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} \\begin{aligned} & | - c ^ { - 1 } \\dot { c } - h W ' ( h a ) | \\lesssim \\kappa ^ 2 h ^ 3 e ^ { 2 \\mu h t } \\\\ & | \\dot { a } - c + W ( h a ) | \\lesssim \\kappa ^ 2 h ^ 2 e ^ { 2 \\mu h t } \\end{aligned} \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} & \\frac { \\partial E } { \\partial x } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\Big | _ { y = x \\pm ( t - b ) } = \\pm 2 ^ { \\sqrt { \\delta } - 1 } ( 1 + b ) ^ { \\frac { \\mu } { 2 } - 1 } ( 1 + t ) ^ { - \\frac { \\mu } { 2 } - 1 } \\bigg [ 2 ^ { - 3 } ( 1 - \\sqrt { \\delta } ) ^ 2 ( t - b ) + 2 ^ { - 2 } ( \\sqrt { \\delta } - 1 ) ( t - b ) \\bigg ] . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} D ^ L _ v \\mu : = \\mathcal { L } _ { \\pi ( v ) } \\mu - \\frac 1 2 \\mathrm { d i v } _ D ( v ) \\mu , \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { - i \\infty } ^ { i \\infty } \\frac { ( \\frac { 1 } { 2 } ) _ s ^ 3 } { ( 1 ) _ s ^ 3 } \\Gamma ( - s ) ( 3 s + 1 ) 2 ^ { 3 s } d s = \\frac { 1 } { \\pi } . \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} & \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < \\frac { 1 } { 2 } \\} \\\\ = & \\{ ( 0 , 0 , 1 , 1 ) + y _ 1 ( \\frac { 1 } { \\sqrt { 2 } } , 0 , 0 , - \\frac { 1 } { \\sqrt { 2 } } ) + y _ 2 ( 0 , \\frac { 1 } { \\sqrt { 2 } } , - \\frac { 1 } { \\sqrt { 2 } } , 0 ) \\mid 0 \\le y _ 1 \\le y _ 2 < \\frac { 1 } { \\sqrt { 2 } } \\} , \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\norm { \\Lambda _ f ( y ) - \\Lambda _ f ( z ) } = \\norm { v _ y - v _ z } \\leq \\sqrt { 2 L M } \\ ; \\norm { y - z } ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\partial _ { t _ 0 } \\partial _ { t _ a } \\partial _ { t _ b } \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = g _ { a b } \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { \\partial u _ { J } } { \\partial t } = f ( u _ { J } , v _ { J } ) + \\varepsilon { \\textstyle \\sum \\limits _ { I \\in G _ { N } ^ { 0 } } } A _ { J I } \\left \\{ u _ { I } - u _ { J } \\right \\} \\\\ \\\\ \\frac { \\partial v _ { J } } { \\partial t } = g ( u _ { J } , v _ { J } ) + \\varepsilon d { \\textstyle \\sum \\limits _ { I \\in G _ { N } ^ { 0 } } } A _ { J I } \\left \\{ v _ { I } - v _ { J } \\right \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} ( d F ) _ { h _ { 0 } \\sqcup \\alpha } ( \\gamma \\sqcup 0 ) = J _ { 1 } ( F ( h _ { 0 } ) ; \\alpha ) \\circ ( d F ) _ { h _ { 0 } } ( \\gamma ) , \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} C _ b \\sigma _ y = \\sigma _ y \\frac { s + a _ { x x } } { s + \\sigma _ x + a _ { x x } } \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\widetilde { q } ( X ) = q ( X ) = \\dim B < \\dim B + \\dim B _ 0 \\le q ( B \\times B _ 0 \\times Z _ 0 ) \\le \\widetilde { q } ( X ) , \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} \\mathcal { S } _ 0 ( s V _ A ^ { \\perp } ) = \\frac { 2 s ^ 2 ( 1 + 5 \\cos ( 2 \\theta _ 1 + 2 \\theta _ 2 ) ) - 1 9 s ^ 3 \\cos ( 3 \\theta _ 1 + 3 \\theta _ 2 ) } { 8 } ( J X _ 1 + J X _ 2 ) + \\frac { s ^ 3 } { 8 } V _ A ^ { \\perp } + O ( s ^ 4 ) . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} x = ( \\mu ^ { - 1 } B ) ^ { \\ast } u , B = \\bigoplus _ { k = 1 } ^ { m } w _ { k } A _ { k } , \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} Z [ \\alpha ] ( s , v ) : = \\left \\{ a _ s \\ \\left [ \\ \\sum _ { r = 0 } ^ { \\infty } \\alpha ^ { 1 - s / v } ( r ) \\ ( r + 1 ) ^ { s / 2 - 1 } \\ \\right ] \\ \\right \\} ^ { 1 / s } , \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} f ( N ) \\leq f ( e ^ { n / s } ) = \\frac { ( i / s ) ^ i } { e ^ i } \\leq \\frac { i ! } { \\sqrt { 2 \\pi i } s ^ i } \\leq \\frac { i ! } { \\sqrt { 2 \\pi } ( s - 1 ) ^ i } \\cdot \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} M = \\max _ { i = 1 , \\dots , m , \\ ; x \\in W } \\norm { \\nabla f _ i ( x ) } . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\bar { \\mathsf h } _ { i , m + j } \\rightarrow \\begin{cases} \\cfrac { t ^ * } { \\zeta _ i } \\bar { w } ^ * _ { i , m + j } & \\mbox { i f } \\ ; i = 1 , \\ldots , \\mathbf k ^ * \\\\ \\bar { w } ^ * _ { i , m + j } & \\mbox { i f } \\ ; i = \\mathbf k ^ * + 1 , \\ldots , m . \\end{cases} \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} \\int _ { \\partial T } \\varphi \\ , d \\nu = \\sum _ { x \\in \\tau } \\varphi _ x \\ , \\biggl ( \\nu ( \\partial T _ x ) - \\sum _ { y \\in S _ { \\tau } ( x ) } \\nu ( \\partial T _ y ) \\biggr ) . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } p _ j ( \\varphi ( t ) ) = \\mathbf { 0 } _ n , \\ : \\ : 1 \\leq j \\leq r , \\\\ \\eth ( \\varphi ( t ) , ( Y _ 1 , \\ldots , Y _ m ) ) \\leq \\varepsilon , \\end{array} \\right . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} v _ 1 ( x ) & = - \\int _ { \\R ^ n } \\Psi ( x - y ) Q _ 1 ( y , \\widetilde { u } ( y ) + L , \\nabla \\widetilde { u } ( y ) , \\nabla ^ 2 \\widetilde { u } ( y ) ) d y \\\\ v _ 2 ( x ) & = \\int _ { \\R ^ n } \\nabla \\Psi ( x - y ) Q _ 2 ( y , \\widetilde { u } ( y ) + L , \\nabla \\widetilde { u } ( y ) , \\nabla ^ 2 \\widetilde { u } ( y ) ) d y \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\psi _ { E , \\alpha } ( z ) = \\psi ( \\Delta ( \\alpha _ 1 z _ 0 + \\alpha _ 0 z _ 1 ) ) . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} \\frac { \\partial \\boldsymbol { w } } { \\partial t } ( x , t ) = \\mathbb { J } \\boldsymbol { w } \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\mathcal { W } = \\C \\setminus \\{ \\pm i \\ , t : t \\ge 1 / \\kappa \\} \\ , . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\frac { 1 } { r \\ ! - \\ ! 1 } \\rho ^ { \\frac { r } { r - 1 } } \\ ! + \\ ! \\left ( \\frac { r m } { n } \\right ) ^ { \\frac { r } { r - 1 } } \\ ! & \\ ! = \\frac { 1 } { r \\ ! - \\ ! 1 } \\Bigg [ \\rho ^ { \\frac { r } { r - 1 } } + \\underbrace { \\left ( \\frac { r m } { n } \\right ) ^ { \\frac { r } { r - 1 } } + \\cdots + \\left ( \\frac { r m } { n } \\right ) ^ { \\frac { r } { r - 1 } } } _ { r - 1 } \\Bigg ] \\\\ & \\geq \\frac { r ^ 2 m } { ( r - 1 ) n } \\rho ^ { \\frac { 1 } { r - 1 } } . \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} w ( \\bar x ) - u ( \\bar x ) = \\lambda \\left [ ( f _ r ) ' ( u ( \\bar x ) ) u _ x ( \\bar x ) - \\varepsilon u _ { x x } ( \\bar x ) \\right ] \\ge 0 . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} \\mu _ { \\eta , 1 } & = \\frac { \\mu _ 0 ^ 2 + \\mu _ 1 ^ 2 - 2 \\mu _ 0 \\mu _ 1 + \\sigma _ 1 ^ 2 - \\sigma _ 0 ^ 2 } { 2 \\sigma _ 0 ^ 2 } + \\log \\ ! \\left ( \\frac { \\sigma _ 0 } { \\sigma _ 1 } \\right ) \\\\ \\sigma _ { \\eta , 1 } ^ 2 & = \\frac { 1 } { 2 } \\left ( 1 + \\frac { \\sigma _ 1 ^ 4 } { \\sigma _ 0 ^ 4 } \\right ) + \\left ( \\mu _ 0 - \\mu _ 1 \\right ) ^ 2 \\frac { \\sigma _ 1 ^ 2 } { \\sigma _ 0 ^ 4 } - \\frac { \\sigma _ 1 ^ 2 } { \\sigma _ 0 ^ 2 } . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\left ( \\ref { c l a i m : d i f i c i l } \\right ) u = 1 \\exists Z _ { u = 1 , 2 } ^ { \\ast } \\mathbf { Z } _ { m i n } \\backslash \\left [ \\mathbf { O } _ { m i n } \\cup \\left \\{ Z _ { u = 1 , 1 } ^ { \\ast } \\right \\} \\right ] Z _ { u = 1 , 2 } ^ { \\ast } Z _ { u = 1 , 1 } ^ { \\ast } \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} m \\ , \\dot { v } ( t ) = - \\Phi ' ( x ( t ) ) - \\gamma v ( t ) - \\int _ { - \\infty } ^ t K ( t - s ) v ( s ) \\ , d s + \\sqrt { 2 \\gamma } \\ , \\dot { W } ( t ) + F ( t ) . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\theta _ T ( \\mu _ S ) = \\theta _ T ( \\gamma ( \\mu _ S ) ) , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} \\left | \\frac { ( A ^ + ) ^ { n - 1 } ( c _ k ^ + ) ^ a ( ( c _ k - 1 ) ^ + ) ^ { n - a } } { ( A ^ - ) ^ { n - 1 } ( c _ k ^ - ) ^ n } \\right | = \\left | A ^ { n - 1 } c _ { k } ^ a ( c _ { k } - 1 ) ^ { n - a } \\right | = | c _ { k + 1 } - 1 | > 2 > 1 , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} c _ S ( F _ 1 ) = ( - 1 , 0 ) , c _ S ( F _ 2 ) = ( - 1 , 1 ) , c _ S ( F _ 3 ) = ( 0 , 1 ) , c _ S ( F _ 4 ) = ( 1 , 1 ) , \\\\ c _ S ( F _ 5 ) = ( 1 , 0 ) , c _ S ( F _ 6 ) = ( 1 , - 1 ) , c _ S ( F _ 7 ) = ( - 1 , 0 ) , c _ S ( F _ 8 ) = ( - 1 , - 1 ) , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\mathbf { Y } _ { \\mathcal { Q } _ p } = \\mathcal { Q } \\left ( \\mathbf { Y } _ p \\right ) = \\mathcal { Q } \\left ( \\mathbf { H } \\mathbf { X } _ p + \\mathbf { N } _ p \\right ) , \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} P _ e ( \\rho ) = \\frac { a } { \\gamma - 1 } \\rho ^ { \\gamma - 1 } . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n m _ { j , i } r _ { \\sigma ( i ) } = \\sum _ { i = 1 } ^ n m _ { j , \\sigma ^ { - 1 } ( i ) } r _ { i } \\equiv m _ j \\bmod p \\quad ( 1 \\le { } ^ \\forall j \\le t ) , \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\int _ { M } g _ s \\delta R _ s ^ H \\overset { \\textnormal { D e f . } } { = } c _ H ^ R \\ , \\ , \\left ( \\delta ^ 2 \\circ I _ { - , \\mathrm { t r } } ^ { \\frac { H } { 2 } , \\frac { H } { 2 } } \\right ) ( \\textbf { 1 } _ M g ) . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\lambda _ n ( H _ { \\mu } ( x , t ) ) \\ge \\theta _ L + \\frac { t ^ 2 \\ , \\lambda _ n ( C ) } { 6 } = - | \\theta _ L | + \\frac { t ^ 2 \\ , \\lambda _ n ( C ) } { 6 } \\ , . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} v a r ( T S P _ { n + 1 } ) = \\mathbb { E } \\left ( \\sum _ { j = 1 } ^ { n + 1 } D _ j \\right ) ^ 2 = \\sum _ { j = 1 } ^ { n + 1 } \\mathbb { E } D _ j ^ 2 . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} k ^ 2 = \\ ( \\dfrac { H _ o ^ 2 } { 4 } - \\delta \\ ) r _ o ^ 2 \\ ( 1 - \\dfrac { 2 m } { r _ o } \\ ) ^ { - 1 } \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\mathbb { Z I } _ { n , \\epsilon } ^ m ( p _ 1 , \\ldots , p _ r ) = \\{ ( X _ 1 , \\ldots , X _ m ) \\in \\mathbb { I } ^ m _ n \\ : | \\ : \\ : \\| p _ j ( X _ 1 , \\ldots , X _ m ) \\| \\leq \\epsilon , 1 \\leq j \\leq r \\} \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} e _ A - e _ B = \\sum _ { i = 1 } ^ k \\gamma _ i ( e _ { C _ i } - e _ B ) \\qquad A \\cap B \\subseteq C _ i \\subseteq A \\cup B . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\partial _ { t } \\phi = \\frac { \\partial } { \\partial { t } } \\left ( \\frac { \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } } { ( 1 - f ) ^ { 2 } } \\right ) = \\frac { \\partial _ { t } \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } } { ( 1 - f ) ^ { 2 } } + \\frac { 2 \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } \\partial _ { t } f } { ( 1 - f ) ^ { 3 } } . \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} F ( n + 1 , k ) - F ( n , k ) = G ( n , k + 1 ) - G ( n , k ) . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} C _ 5 \\sqrt { n } \\leq b _ n = r _ n \\sqrt { n N } \\leq C _ 6 \\sqrt { n } \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { p - 1 } { 2 } } ( - 1 ) ^ k q ^ { k ^ 2 } [ 4 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 3 } \\equiv [ p ] q ^ { \\frac { ( p - 1 ) ^ 2 } { 4 } } ( - 1 ) ^ { \\frac { p - 1 } { 2 } } \\pmod { [ p ] ^ 3 } \\quad , \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal Q ( t ) = & ~ { } - \\frac { 1 } { 2 L } b \\int \\varphi ' f ^ 2 - \\frac { 3 } { 2 L } ( 1 + b ) \\int \\varphi ' f _ x ^ 2 - \\frac { 1 } { 2 L } ( 4 + 3 b ) \\int \\varphi ' f _ { x x } ^ 2 - \\frac { 1 } { 2 L } ( 1 + b ) \\int \\varphi ' f _ { x x x } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\frac { \\left \\| \\overline { \\nabla } \\varphi \\right \\| ^ { 2 } } { \\varphi } = \\frac { \\left \\| \\psi ' \\right \\| ^ { 2 } \\left \\| \\overline { \\nabla } d \\right \\| ^ { 2 } } { \\rho ^ { 2 } \\varphi } \\leq \\frac { c _ { 2 } } { \\rho ^ { 2 } } , \\quad \\frac { \\partial \\varphi } { \\partial t } \\leq c _ { 2 } \\rho _ { 1 } , \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\omega ( \\sigma ^ { l } ) : = \\max \\{ \\omega _ { 3 } ( \\sigma ^ { l } ) , \\sigma ^ { l \\alpha } \\} . \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} s ( z ) = \\sum \\limits ^ { 2 s } _ { n = 0 } a _ n z ^ n \\ , . \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix} - u _ { 1 } '' ( x ) = \\mu _ { 1 } ( x ) \\left ( 1 - u _ { 1 } ( x ) \\right ) u _ { 1 } ( x ) - k \\omega ( x ) u _ { 1 } ( x ) u _ { 2 } ( x ) \\\\ - d u _ { 2 } '' ( x ) = \\mu _ { 2 } ( x ) \\left ( 1 - u _ { 2 } ( x ) \\right ) u _ { 2 } ( x ) - \\alpha k \\omega ( x ) u _ { 1 } ( x ) u _ { 2 } ( x ) \\end{matrix} \\right . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\frac { g _ i } { t _ i } \\cdot \\widetilde { C } _ q ( P ( z _ 0 ) , \\{ u _ i \\} ) = 0 . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } \\Big \\{ \\frac { x } { \\left ( 1 + x ^ 2 \\right ) ^ { \\gamma } } \\Big \\} = ( 2 s + 1 ) A _ s ^ \\gamma \\ , \\frac { x } { ( 1 + x ^ 2 ) ^ { s + \\gamma } } \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - s , s + \\gamma ; \\frac { 3 } { 2 } ; \\frac { x ^ 2 } { 1 + x ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} m \\ , d v ( t ) = - \\gamma v ( t ) - \\Phi ' ( x ( t ) ) d t + \\sqrt { 2 \\gamma } d W ( t ) , \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } b _ { 0 } y ^ { \\prime \\prime } ( x ) + b _ { 1 } y ^ { \\prime } ( x ) = \\lambda y ( x ) , x \\in ( 0 , 1 ) , \\\\ y ( 0 ) = y ( \\frac { 1 } { 2 } ) = y ( 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{gather*} C _ { a b } ^ e C _ { c e } ^ d + \\rho _ { a } ^ j \\partial _ j C _ { b c } ^ d + \\operatorname { c y c l } ( a b c ) = \\sigma ^ { d } _ { a b c } , \\end{gather*}"}
-{"id": "7888.png", "formula": "\\begin{align*} g ( 0 ) = 0 , g ( q ) = 0 , g ' ( q ) = 0 , \\\\ \\xi ' ( 1 ) = \\int _ 0 ^ 1 \\frac { d r } { \\nu ( ( r , 1 ] ) ^ 2 } , g ( u ) \\ge 0 , u \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} S _ { i , \\varepsilon } ( x ) = S _ i \\left ( \\frac { | x - x _ \\varepsilon | } { \\mu _ \\varepsilon } \\right ) \\ , , \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\rho ( f _ { n } - w ) = m \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} E [ S _ n ' ( \\theta _ j ^ { ( n ) } ) ] = n \\pi _ n ( \\theta _ j ^ { ( n ) } ) & \\geq \\mathrm { e } ^ { - 3 } ( 1 - r ^ { - 1 } ) ^ { r - 1 } \\frac { K ^ { r - 1 } } { r } \\theta _ j ^ { ( n ) } > K ^ { 1 / \\lceil K \\rceil } \\theta _ j ^ { ( n ) } = \\theta _ { j + 1 } ^ { ( n ) } . \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} L ( t , z , \\dot z ) = \\frac { 1 } { 2 } g _ { i j } \\dot z ^ i \\dot z ^ j + a _ i \\dot z ^ i - V , z = ( z ^ 1 , \\ldots , z ^ { m + n } ) . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\hat k = \\underset { 1 \\leq k \\leq { \\frac { N } { 2 } } } { } \\ \\frac { 1 } { ( N - k - 1 ) ^ 2 } \\sum _ { i = k + 1 } ^ N ( X _ i - \\bar { X } _ { k , N } ) ^ 2 \\ \\ \\ \\ \\bar { X } _ { k , N } = \\frac { 1 } { N - k } \\sum _ { i = k + 1 } ^ N X _ i . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} & \\norm { { \\kappa - \\kappa _ M ^ { N , h } } } _ { L ^ p ( \\Omega , C ( D ) ) } \\\\ & \\leq \\norm { { \\log \\kappa ( y , x ) - \\log \\kappa _ M ^ { N , h } ( y , x ) } } _ { L ^ 2 ( \\Omega , C ( D ) ) } \\Big ( \\| { \\kappa } \\| _ { L ^ q ( \\Omega , C ( D ) ) } + \\| { \\kappa _ M ^ { N , h } } \\| _ { L ^ q ( \\Omega , C ( D ) ) } \\Big ) , \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} f ( x , u ) = \\frac { ( u - 1 + H ( x ) ) ^ 2 } { 2 } , \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{gather*} [ \\rho ( e _ 1 ) , \\rho ( e _ 2 ) ] = \\rho ( [ e _ 1 , e _ 2 ] ) . \\end{gather*}"}
-{"id": "1258.png", "formula": "\\begin{align*} g ( y _ 1 , y ' ) = \\frac { a _ 0 } { 2 } + \\sum _ { k = 1 } ^ \\infty a _ k ( y ' ) \\cos ( k y _ 1 ) + \\sum _ { k = 1 } ^ \\infty b _ k ( y ' ) \\sin ( k y _ 1 ) \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} L ( E ) = \\lim _ { k \\rightarrow \\infty } \\frac { 1 } { k } \\int _ { \\mathbb { R } / \\mathbb { Z } } \\ln \\| A _ k ( \\theta ) \\| d \\theta . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} & \\sum _ { 1 - l \\le q \\le n - 1 - l } { k \\choose n - h - q } { n - k \\choose q } \\\\ = \\ , & \\sum _ { 0 \\le q \\le \\min ( n - 1 - l , n - h ) } { k \\choose n - h - q } { n - k \\choose q } \\\\ = \\ , & { n \\choose n - h } - \\sum _ { \\min ( n - 1 - l , n - h ) + 1 \\le q \\le n - h } { k \\choose n - h - q } { n - k \\choose q } \\\\ = \\ , & { n \\choose h } - \\sum _ { h \\le q \\le \\max ( l , h - 1 ) } { k \\choose q - h } { n - k \\choose n - q } , \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} ( 1 - \\eta _ \\varepsilon ) = \\frac { 4 \\pi ( 1 - \\varepsilon ) } { \\| U _ { \\varepsilon , \\bar { x } _ 0 } \\| _ { H ^ 1 _ 0 } ^ 2 } \\ , . \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to \\infty } d ( x _ n , x _ { n + 2 } ) = 0 . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} & u _ { k , i } ( x , t ) : = [ e ^ { - t L _ V } \\varphi ^ { 1 , i } ] ( | x | ) = [ e ^ { - t L _ V ^ k } \\phi ^ { k , i } ] ( | x | ) \\ , Q _ { k , i } \\left ( \\frac { x } { | x | } \\right ) , \\\\ & u ( x , t ) = u _ 0 ( x , t ) + \\sum _ { k = 1 } ^ { m - 1 } \\sum _ { i = 1 } ^ { \\ell _ k } u _ { k , i } ( x , t ) + R _ m ( x , t ) , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} \\eta ( u , v , w ) : = \\langle ( D _ { u } \\mathcal J ) v , w \\rangle . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 5 ( x ) + \\frac { 2 } { 6 ^ 6 } ( 3 ^ 6 x - 3 0 4 ) ( 2 7 - 2 ^ 6 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 8 8 } { 2 1 1 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 5 6 4 1 1 0 5 4 1 4 7 3 3 8 6 5 5 8 5 2 2 2 2 2 7 9 1 5 3 7 1 4 1 2 5 7 0 3 } { 6 3 1 6 3 9 7 5 7 6 8 6 3 4 1 1 7 2 4 2 5 4 5 0 3 5 6 7 3 5 9 9 9 0 2 3 5 7 9 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} \\Phi ( u ) ( t ) : = e ^ { - i t ( - \\Delta ) ^ s } u _ 0 + i \\int _ 0 ^ t e ^ { - i ( t - \\tau ) ( - \\Delta ) ^ s } | u ( \\tau ) | ^ \\alpha u ( \\tau ) d \\tau \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} R _ 1 & = \\Delta u _ 1 + \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 , \\\\ R _ 2 & = \\Delta u _ 2 + \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\sup _ { M \\in F ^ { N D } ( u + h ) } \\frac { \\| F [ u + h ] - F [ u ] - M h \\| } { \\| h \\| } = 0 . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\Psi _ { x y } ^ \\epsilon ( w _ 1 , w _ 2 ) : = h _ p ( y - w _ 1 ) \\partial _ p Q ^ { \\lambda } _ { \\epsilon } ( x - w _ 1 ) \\times h _ p ( y - w _ 2 ) \\partial _ p Q ^ { \\lambda } _ { \\epsilon } ( x - w _ 2 ) . \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} \\vert u ( x _ k ) \\vert \\leq \\bigg ( \\sum _ { i = 0 } ^ { k - k _ 0 - 1 } \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\bigg ) + \\vert u ( x _ { k _ 0 } ) \\vert \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} J ^ { \\textnormal { c o h } , \\textnormal { e q } } _ { | H = \\lambda _ i } ( z , Q ) = Q ^ \\frac { \\lambda _ i } { z } \\sum _ { d \\geq 0 } Q ^ d \\prod _ { r = 1 } ^ d \\prod _ { j = 0 } ^ N \\frac { 1 } { ( \\lambda _ i - \\lambda _ j + r z ) } \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} w _ p = \\binom { 2 p - 1 } { p - 1 } = \\prod _ { k = 1 } ^ { p - 1 } \\left ( \\frac { p } { k } + 1 \\right ) = \\sum _ { k = 0 } ^ { p - 1 } p ^ k S _ { p - 1 , k } \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} B _ r ^ { 2 ^ r } + X B _ r ^ 2 + B _ r = 0 . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} g = \\frac { 1 } { \\lambda } \\int _ 0 ^ \\infty e ^ { - \\sigma } \\left [ \\int _ 0 ^ \\infty e ^ { - s } f \\left ( \\frac { 1 } { \\sqrt { \\lambda } } ( \\sigma - s ) \\mathbf { 1 } _ d + \\xi \\right ) d s \\right ] d \\sigma \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} \\varphi _ { y } ( y ) = 0 , \\nabla \\varphi _ { y } ( y ) = 0 ; \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} l ( G ) = \\O ( | B | ) + 1 = \\O ( q ^ 2 - 1 ) + 3 f + 1 - \\O ( ( 3 , q + 1 ) ) \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} H ( \\omega _ 1 , \\omega _ 2 ; \\theta _ 1 , \\theta _ 2 ) = \\lambda v ( \\theta _ 1 + n _ 1 \\omega _ 1 , \\theta _ 2 + n _ 2 \\omega _ 2 ) + \\Delta . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} Q _ { u } = Q ^ h ( g _ 0 , g _ 1 , \\ldots , g _ k ) . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} H \\left ( \\lambda , x \\right ) = \\frac { 1 } { 2 \\sqrt { \\lambda } } e ^ { - \\frac { \\left | x \\right | } { \\sqrt { \\lambda } } } . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} H = v ( x + n \\omega ) \\delta _ { n n ' } + S _ \\phi , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\begin{cases} v ( z ) z ' - c ( a - r z ) e ^ { \\int v ( z ) d z } = 0 , \\ \\ \\ \\ \\ c > 0 , \\\\ x = c e ^ { \\int v ( z ) d z } \\end{cases} \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\Delta = \\lim _ { T \\rightarrow \\infty } \\frac { \\Delta _ T } { T } = \\lim _ { i \\rightarrow \\infty } \\frac { \\sum _ { j = 1 } ^ { i } Q _ j } { \\sum _ { j = 1 } ^ { i } T _ j } \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\bar A = \\bigcup _ { i = 1 } ^ { m } { A _ i } . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} h + \\frac { 1 } { \\sqrt { \\lambda } } O h = \\frac { 1 } { \\lambda } f , \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\dim _ { P } ^ { * } \\theta \\le \\min \\{ 1 , \\dim ( \\Pi _ { 1 } \\mu _ { 1 } \\times \\Pi _ { 2 } \\mu _ { 2 } ) \\} = \\beta \\ : . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta \\exp \\left ( ( \\ln k ) ^ { \\mu _ 1 } \\cdots ( \\ln _ m k ) ^ { \\mu _ m } \\right ) \\asymp ( \\alpha t ) ^ \\beta \\exp \\left ( ( \\ln ( \\alpha t ) ) ^ { \\mu _ 1 } \\cdots ( \\ln _ m ( \\alpha t ) ) ^ { \\mu _ m } \\right ) \\quad \\mbox { a s $ t \\to \\infty $ } , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} - \\div _ y ( A \\nabla _ y z ) = 0 , \\ \\forall y \\in B ( y _ 0 , 2 R ) \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\phi ( 0 ) ~ = ~ 0 \\quad \\mathrm { a n d } \\phi ( s ) ~ = ~ \\phi ( t _ i ) \\quad \\forall i \\in \\overline { 0 , N _ 1 - 1 } , ~ s \\in I _ i ~ . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} \\Q ^ { \\times } _ 2 = \\mu _ 2 \\times ( 1 + 4 \\Z _ 2 ) \\times 2 ^ { \\Z } \\ ; p = 2 \\ ; . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - t } e ^ { - t } & = \\left ( \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! } t ^ k \\right ) \\left ( \\sum _ { m = 0 } ^ \\infty t ^ m \\right ) = \\sum _ { n = 0 } ^ \\infty \\left ( n ! \\sum _ { k = 0 } ^ n \\frac { ( - 1 ) ^ k } { k ! } \\right ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty d _ n \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 3 , 4 , 7 ] ) . \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} T _ a + \\sum _ { n = 0 } ^ \\infty \\sum _ j \\frac { ( - 1 ) ^ { n + 1 } } { ( n + 1 ) ! } \\frac { 1 } { z ^ { n + 1 } } \\left ( \\int _ X T _ a \\cup \\tau _ 2 ^ { n + 1 } \\cup T _ j \\right ) T ^ j = e ^ { - \\tau _ 2 / z } T _ a \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { L } \\left ( \\psi ' \\right ) ^ { 2 } & = \\int _ { 0 } ^ { L } \\left [ \\mu _ { 1 } \\left ( \\alpha - 2 v \\right ) \\mathbf { 1 } _ { v > 0 } + \\mu _ { 2 } \\left ( d + 2 v \\right ) \\mathbf { 1 } _ { v < 0 } \\right ] \\psi ^ { 2 } + \\lambda \\\\ & = M _ 1 \\int _ { \\{ \\mu _ 1 > 0 \\} \\cap \\{ v > 0 \\} } \\left ( \\alpha - 2 v \\right ) \\psi ^ { 2 } + M _ 2 \\int _ { \\{ \\mu _ 2 > 0 \\} \\cap \\{ v < 0 \\} } \\left ( d + 2 v \\right ) \\psi ^ { 2 } + \\lambda . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} g _ n ( z ) = c _ 0 \\prod _ { k = 1 } ^ m ( z - z _ { k , n } ) . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} K _ 0 ( S { \\frak C } _ F ( { \\bf H } ) \\rtimes { \\mathbb Z } ) & \\cong K _ 0 ( S { \\frak C } ( { \\bf H } ) \\rtimes { \\mathbb Z } ) \\\\ & \\cong K _ 1 ( C ^ * ( { \\mathbb Z } ) ) = K ^ 1 ( S ^ 1 ) \\cong { \\mathbb Z } . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} \\left ( \\ref { c l a i m : d i f i c i l } \\right ) u = t \\exists Z _ { u = t , 2 } ^ { \\ast } \\mathbf { Z } _ { m i n } \\backslash \\left [ \\mathbf { O } _ { m i n } \\cup \\left \\{ Z _ { u = t , 1 } ^ { \\ast } \\right \\} \\right ] Z _ { u = t , 2 } ^ { \\ast } Z _ { u = t , 1 } ^ { \\ast } \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } h _ { y _ j } h _ { y _ j } d y + \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } \\widetilde { w } _ { y _ i y _ j } \\widetilde { w } _ { y _ i y _ j } d y = \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } h _ { y _ j } V _ { y _ j } d y . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} w ^ \\xi _ \\eta ( \\rho ) & : = \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\vec { \\mathcal { \\triangle } } ^ \\xi _ \\eta } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } \\\\ & = 1 - \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\vec { \\mathcal { \\triangle } } ^ \\eta _ \\xi } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} b ' _ { q , m } = \\max \\Big \\{ \\frac { 1 } { 2 } \\log _ 2 \\Big ( \\frac { 6 \\rho _ { q , m } \\ln 2 } { \\lambda ^ \\star } \\Big ) , 0 \\Big \\} , \\ , q \\in \\mathcal { Q } , \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} A ^ { * s } A ^ s = ( A ^ * A ) ^ s \\qquad s \\in S , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} ( 1 + g _ N ( t ) ) \\exp ( t ^ 2 ) = ( 1 + g ( t ) ) ( 1 + t ^ 2 ) + ( 1 + g ( t ) ) \\left ( \\sum _ { k = N + 1 } ^ { + \\infty } \\frac { t ^ { 2 k } } { k ! } \\right ) \\ , , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\mathrm { w d } ( \\mathcal { T } ) : = \\sup \\lbrace \\# \\mathrm { L y r } _ n { \\mathcal { T } } \\mid n \\ge n _ \\ast \\rbrace . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\forall \\ , r > 0 , \\ , \\exists \\ , ( x _ i ) _ { i = 1 , \\cdots , n } \\subset A : \\ , A \\subset \\bigcup _ { i = 1 , \\cdots , n } B ( x _ i , r ) . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} T \\mbox { i s r e g u l a r } \\Leftrightarrow T = T _ { \\rm r e g } \\Leftrightarrow T _ { \\rm s i n g } = 0 , \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\binom { n } { k } = \\binom { n - 1 } { k - 1 } + \\binom { n - 1 } { k } \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} g ( x _ { ( k + 1 ) n + i } ) = y _ { ( k + 1 ) n + i } . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} { \\rm A u t } ( G ) \\cong \\prod _ { i = 1 } ^ m { \\rm A u t } ( T _ i ^ { k _ i } ) \\cong \\prod _ { i = 1 } ^ m { \\rm A u t } ( T _ i ) \\wr S _ { k _ i } . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} y _ 0 y _ 1 y _ 2 + 2 y _ 3 y _ 4 y _ 5 - y _ 0 y _ 3 ^ 2 - y _ 1 y _ 4 ^ 2 - y _ 2 y _ 5 ^ 2 = 0 . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} g \\left ( R _ { n } \\right ) = 2 \\pi d _ { k } ^ { 2 } p _ { 0 } I \\left ( \\frac { 1 } { R _ { n } } \\right ) + 2 \\pi d _ { k } p _ { 0 } I \\left ( \\frac { R _ { n } } { \\varepsilon _ { n } ^ { \\alpha } } \\right ) + 2 \\pi \\left ( p _ { 0 } + \\alpha _ { k } \\frac { R _ { n } ^ { s _ { k } } } { 4 } \\right ) I \\left ( \\frac { 1 } { \\varepsilon _ { n } ^ { 1 - \\alpha } } \\right ) + C _ { 4 } . \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} v ( 0 , x ) = v _ 0 ( x ) , v _ t ( 0 , x ) = v _ 1 ( x ) . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} U _ 0 ( x ' , x _ m , y ) = \\begin{cases} U _ 0 ( x ' , x _ m , y ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ m \\geq 0 , \\\\ \\sum ^ 3 _ { i = 1 } c _ i U _ 0 ( x ' , - x _ m / i , y ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ m < 0 , \\end{cases} \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} N ( p , q ) : = \\{ \\nu ^ { p ( \\beta + \\gamma ) } _ + , \\nu ^ { p ( \\beta + \\gamma ) } _ - : \\beta \\in C _ \\alpha ' ( p ) , \\gamma \\in C _ \\alpha ' ( q ) , \\gamma \\neq \\beta , \\alpha + \\beta , \\beta ^ c , \\alpha + \\beta ^ c \\} \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\| f _ u \\| _ { X _ u ^ s } = \\| B _ u ^ s ( F ) \\| _ { E _ { u } ^ s } = 1 , \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } w ' ( t ) = - A U _ { m + 1 } ( 0 ) - U _ { m + 1 } ' ( 0 ) = ( - 1 ) ^ m u _ 1 \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} c = \\max \\big ( C , 4 , 1 + ( 1 + \\sqrt { 8 / a } ) ^ 2 \\| K \\| _ \\infty \\big ) \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\mu ( c ) & = \\gamma ( c ) \\ : u ^ { - 1 } & & \\\\ & = q ( c , c \\wedge m _ \\gamma ) \\iota ( c \\wedge m _ \\gamma , m _ \\gamma ) \\gamma ( m _ \\gamma ) ( \\gamma ( m _ \\gamma ) ) ^ { - 1 } & & \\ref { r m k i n v c o n e } \\\\ & = q ( c , c \\wedge m _ \\gamma ) \\iota ( c \\wedge m _ \\gamma , m _ \\gamma ) . & & \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} | I ' | = r \\Big \\lfloor \\frac { k - 1 } { r } \\Big \\rfloor + s - 1 . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\begin{cases} \\frac { | P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) | ^ 2 } { 2 } + V ( x , y ) = \\ln \\widetilde { m } _ 1 ( x , y ) + \\widehat { H } ( x , P + \\nabla \\widetilde { u } _ 0 ( x ) ) , \\\\ - \\div _ y \\big ( \\widetilde { m } _ 1 ( x , y ) ( P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) \\big ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\alpha : = x ^ { \\frac { \\log _ 3 x } { 4 \\log _ 2 x } } . \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} u _ { t x } + \\frac { 1 } { 2 } \\left ( u ^ 2 \\right ) _ { x x } = \\gamma u , \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} f _ { k , u _ l } ( t , u ) = \\partial _ { u _ l } f _ k \\geq 0 \\ \\ ( k \\not = l ) \\ \\ { \\rm f o r \\ e a c h } \\ \\ u \\in ( p ^ - ( t ) , p ^ + ( t ) ) , \\ t \\in \\R , \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\Phi _ n ^ { ( 1 ) } ( z ) = q _ 1 ( z ) p _ n ( z ; a , b | q ) + q _ 0 ( z ) p _ { n - 1 } ( z ; a , b | q ) , \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} m _ i = \\lim _ { \\delta \\rightarrow 0 } \\lim _ { p \\rightarrow + \\infty } \\max _ { \\overline { B _ { \\delta } ( x _ i ) } } u _ { p } \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} G ( T _ { p i j } ^ { 0 , \\rho } ) = \\Big \\{ ( a , b ) \\in \\mathbb { R } ^ { n - m } \\times \\mathbb { R } ^ m \\big | a \\in \\Lambda , \\gamma _ { m , \\rho } ( v ( a ) , b ) < 1 \\Big \\} . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\phi _ n ^ { ( 1 ) } ( z ) = \\frac { ( 1 - q ^ { n + \\gamma } ) } { ( 1 - q ^ { \\gamma } ) } \\frac { ( q ; q ) _ n } { ( q ^ { \\delta + 1 } ; q ) _ n } L _ n ^ { ( \\delta ) } ( z ; q ) + \\frac { q ^ { \\gamma } ( q ^ { n } - 1 ) } { ( 1 - q ^ { \\gamma } ) } \\ \\frac { ( q ; q ) _ { n - 1 } } { ( q ^ { \\delta + 1 } ; q ) _ { n - 1 } } L _ { n - 1 } ^ { ( \\delta ) } ( z q ; q ) . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\tau u _ { t t } + \\left \\{ u - \\tau f ( u ) + \\tau \\lambda _ f \\right \\} _ t = \\Delta u + f ( u ) - \\lambda _ f . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\mathcal S = \\{ S ~ : \\Lambda ( S ) < \\infty \\} . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\frac { ( 1 - q ) ^ d } { ( q ; q ) _ d } = \\frac { 1 } { d ! } \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} R _ k = R ( k ) \\quad \\mbox { w h e r e $ R ( s ) $ i s a r e g u l a r v a r y i n g f u n c t i o n } . \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\mu \\cdot \\nu : = \\begin{cases} 0 , & \\quad \\quad ~ \\mu = 0 ~ ~ \\nu = 0 , \\\\ 0 , & \\quad r ( \\mu ) \\neq s ( \\nu ) , \\\\ \\mu \\nu , & \\quad r ( \\mu ) = s ( \\nu ) . \\end{cases} \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } g ( u ) \\ , d \\widetilde { C } _ { q } ( \\varphi K , \\varphi Q , u ) = \\int _ { S ^ { n - 1 } } g \\left ( \\frac { \\varphi ^ { - t } u } { \\| \\varphi ^ { - t } u \\| } \\right ) d \\widetilde { C } _ { q } ( K , Q , u ) . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} I _ 1 = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { k _ 1 } l \\left ( v _ j , { \\cal C } _ { t o t } \\right ) I _ 2 = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { k _ 2 } l \\left ( u _ j , { \\cal C } _ { t o t } \\right ) . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} f ( \\xi ) = \\int _ \\Omega \\frac { f ^ { p - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta + \\lambda \\int _ \\Omega \\frac { f ^ { p - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha - 1 } } d \\eta , \\xi \\in \\overline \\Omega , \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\nabla _ i = \\nabla _ { X _ i } , 0 \\le i \\le d , \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } f d \\mu _ { \\mathcal { S } } = 0 \\nu \\mathcal { S } \\in C C ( \\mathbb { R } ^ n ) , \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\Phi ~ = ~ \\sup _ { \\varphi \\in \\mathcal { G } } \\varphi \\quad \\mathrm { w i t h } \\mathcal { G } ~ : = ~ \\{ \\varphi : [ 0 , + \\infty ) \\to [ 0 , + \\infty ) ~ | ~ \\varphi ~ \\mathrm { i s ~ c o n v e x } , ~ ~ \\varphi ( 0 ) = 0 , ~ ~ \\varphi \\leq \\mathfrak d \\} . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\dd { } { t } g ( t ) & = 2 ( - K ( g ( t ) ) + r _ o ^ { - 1 } ) g ( t ) , \\\\ g ( 0 ) & = g _ o , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi _ * + R \\varphi _ * = & - \\frac { n - 1 } { n } \\tau ^ { 2 a _ 0 } \\varphi _ * ^ { N - 1 } \\\\ & + \\bigg [ | \\sigma + L W _ * | ^ 2 + \\bigg ( 2 \\max \\big \\{ \\| \\varphi _ 0 \\| _ { L ^ \\infty } , \\ , 2 \\big \\} - \\| \\varphi _ * \\| _ { L ^ \\infty } \\bigg ) _ + \\bigg ( \\| \\sigma + L W _ 0 \\| ^ 2 _ { L ^ \\infty } + k _ 0 ^ 2 \\bigg ) \\bigg ] \\varphi _ * ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ * L W _ * = & \\frac { n - 1 } { n } \\varphi _ * ^ N d \\tau ^ { a _ 0 } . \\endaligned \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} H _ 2 = ( 1 , \\dots , 1 , - 1 \\ , | \\ , 1 , \\dots , 1 ) . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} | b ( \\sigma - \\sigma _ h , v ) | = | b ( \\sigma - \\sigma _ h , v - P _ h v ) | \\leq C h ^ 2 \\| \\div \\sigma \\| _ 1 \\| v \\| _ 1 . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} d _ { k } > 0 \\ , \\ , \\ , k = 1 , . . , N _ { 2 } \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ s X _ i . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\| \\tilde { v } ^ l \\| _ { L ^ { \\alpha + 2 } } \\leq \\limsup _ { n \\rightarrow \\infty } \\| \\tilde { v } ^ l _ n \\| _ { L ^ { \\alpha + 2 } } = \\limsup _ { n \\rightarrow \\infty } \\| v ^ l _ n \\| _ { L ^ { \\alpha + 2 } } \\rightarrow 0 , l \\rightarrow \\infty . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 0 . 6 & 0 . 4 & 0 . 1 \\\\ 0 . 5 & 0 . 5 & 0 . 3 \\\\ 0 . 1 & 0 . 1 & 0 . 7 \\end{array} \\right ) , \\mbox { w i t h } \\rho ( A ) = 1 . 0 9 6 0 . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} H ( t ) = \\frac { E ( t ) } { 1 - \\nu ^ 2 } \\left \\{ \\begin{array} { l c r } 1 & \\nu & 0 \\\\ \\nu & 1 & 0 \\\\ 0 & 0 & \\frac { 1 } { 2 } ( 1 - \\nu ) \\end{array} \\right \\} \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\pi _ n ( t ) : = P ( Y ^ { ( n ) } _ 1 \\leq t ) = P ( \\mathrm { B i n } ( t , p _ n ) \\geq r ) , t \\in \\mathbb N \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & z ^ s _ t - ( ( D _ 1 a ( y _ x ^ s , t , x ) y _ x ^ s + a ( y _ x ^ s , t , x ) ) z ^ s _ x ) _ { x } + D _ 1 F ( y ^ s , y ^ s _ x ) z ^ s + D _ 2 F ( y ^ s , y ^ s _ x ) z _ x ^ s = w ^ 2 { 1 } _ { \\mathcal { O } _ 1 } \\ \\ Q , \\\\ & z ^ s ( 0 , t ) = z ^ s ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & z ^ s ( 0 ) = 0 \\ \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} v _ x ( | \\mathbf { h } | ^ 2 ) _ x = - \\frac { \\beta } { \\nu } ( v | \\mathbf { h } | ^ 2 ) _ t - 2 v | \\mathbf { h } _ x | ^ 2 + \\frac { \\beta } { \\nu } ( v u | \\mathbf { h } | ^ 2 ) _ x + ( 2 v \\mathbf { h } \\cdot \\mathbf { h } _ x ) _ x - \\frac { 2 \\beta } { \\nu } v \\mathbf { h } \\cdot \\mathbf { w } _ x . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} W _ { t } = \\beta _ { t } + \\int _ { 0 } ^ { t } u ( W _ { s } ) \\ , d L _ { s } \\ , . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\textnormal { c h } ( f _ K ( q ) ) ^ { - \\frac { \\lambda _ i } { g _ H ( z ) } } = 1 + \\frac { \\lambda _ i } { \\lambda _ 0 + \\cdots + \\lambda _ N } c _ 1 ( \\Lambda _ 0 \\otimes \\cdots \\otimes \\Lambda _ N ) + \\cdots = 1 + \\lambda _ i + \\cdots = \\textnormal { c h } ( \\lambda _ i ) \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} & \\frac { s a _ r - t ( a _ 1 + 1 ) } { a _ 1 + 1 - s } ( a _ r - s ) _ { q } = \\frac { ( s \\hat \\alpha \\hat { x } - q \\hat { x } - q s - t ) ( \\hat { x } + \\hat \\alpha \\hat { x } ) _ q } { \\hat { x } + 1 } = H _ 1 ( \\hat { x } , \\hat \\alpha ) \\cdot p ( \\hat { x } ) , \\\\ & \\frac { t a _ r - s ( a _ 1 + 1 ) } { a _ 1 + 1 - t } ( a _ 1 - s ) _ { q } = \\frac { ( t \\hat \\alpha \\hat { x } + q \\hat { x } + q s - s ) ( \\hat { x } ) _ q } { \\hat { x } + 1 - q } = H _ 2 ( \\hat { x } , \\hat \\alpha ) \\cdot p ( \\hat { x } ) . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} F ^ Z = \\gamma ( r ) d r \\wedge d \\theta . \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} B _ q = P _ q \\cdot [ A _ q ] \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\sum _ i H ^ { p ^ { \\ast } } _ { , i } \\omega _ i = d H ^ { p ^ { \\ast } } + \\sum _ { q } H ^ { q ^ { \\ast } } \\omega _ { q ^ { \\ast } p ^ { \\ast } } , \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { - \\ln | | U ( k ) | | } { | k | } = \\ln | \\lambda | , \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\hat { f } ( \\phi ) = \\phi ( f ) . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\theta _ i ( t ) = \\int _ { 0 } ^ { t } \\omega _ i ( W _ s ) \\ , \\circ d \\beta _ s \\ , . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\sum _ k a ^ f _ { i , k } a ^ g _ { k , j } = a ^ { f g } _ { i , j } i , j = 1 , \\ldots n . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} | T | _ { \\mathrm { o p } , G } & = \\sup _ { g \\in G , g \\neq 0 } \\{ | T ( g ) | / | g | \\} , \\\\ | T | _ { \\mathrm { s p } , G } & = \\lim _ { s \\to \\infty } | T ^ s | ^ { 1 / s } _ { \\mathrm { o p } , G } . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\mathcal { F } ( \\tau , Q ) = \\sum _ { d , n \\geq 0 } \\frac { 1 } { n ! } \\langle \\tau ' , \\dots , \\tau ' \\rangle ^ \\textnormal { c o h } _ { 0 , n , d } e ^ { \\tau _ 2 ( d ) } Q ^ d \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} ( \\alpha _ 1 - r _ 1 ) ( \\alpha _ 1 - r _ 2 ) = ( a + \\sigma ^ 2 s ) ^ 2 \\alpha _ 1 . \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} Q _ i ( u ) : = 1 + \\dots + u ^ { i - 1 } = \\prod _ { d \\mid i , d > 1 } \\Phi _ d ( u ) , \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} R _ { u } ^ { j , \\ , i } = \\frac { \\beta } { n } \\log \\left ( 1 + \\mathrm { S I N R } _ { D _ { j } ^ { t } , D _ { j } ^ { r } } \\right ) . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} J ^ { K \\textnormal { t h } , \\textnormal { e q } } ( q , Q ) = \\sum _ { i = 0 } ^ N J ^ { K \\textnormal { t h } , \\textnormal { e q } } _ { | P = \\Lambda _ i } ( q , Q ) \\Psi _ i \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m + 1 } \\sigma _ i ^ { \\frac { r } { r _ i } } = w ^ { \\frac { r } { p ( 1 - r ) } } \\prod _ { i = 1 } ^ { m } w _ i ^ { - \\frac { r } { p _ i ( 1 - r ) } } = 1 \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} \\left \\| u ( t ) - \\sum _ { \\ell = 0 } ^ m \\overline { u } _ \\ell ( t ) \\right \\| & \\leq C ( 1 + t ) ^ { - m - 1 / 2 } . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\langle \\sigma , \\varpi \\rangle _ { \\mathcal { H } } = \\langle v _ 0 , \\varpi \\rangle _ { \\mathcal { H } ^ 0 } . \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} u = Q _ { a , c } + \\eta \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & 3 \\bar \\lambda _ 2 \\bar h ^ { 1 ^ { \\ast } } _ { 2 2 k } + 4 \\bar \\lambda \\bar h ^ { 2 ^ { \\ast } } _ { 1 1 k } = 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} \\mathcal S ( v _ 1 ) & = \\mathcal S ( v ) \\cap \\{ 1 , 2 , \\ldots , r p \\} \\\\ \\mathcal T ( v _ 1 ) & = \\mathcal T ( v ) \\cap \\{ 1 , 2 , \\ldots , r p \\} \\\\ \\mathcal U ( v _ 1 ) & = \\mathcal U ( v ) \\cap \\{ 1 , 2 , \\ldots , r p \\} . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\gamma ( ( - y + T y ) P , Q ) = \\gamma ( P , y Q ) \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\phi _ j ^ { - \\ell _ q ( Q _ j ) } = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\binom { \\ell _ q ( Q _ j ) } { k } \\left ( 1 - \\phi _ j ^ { - 1 } \\right ) ^ k = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\left ( \\frac { 1 } { k ! } \\prod _ { r = 0 } ^ { k - 1 } ( \\ell _ q ( Q ) - r ) \\right ) \\left ( 1 - \\phi _ j ^ { - 1 } \\right ) ^ k \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\{ i \\} = \\bigcap _ { j = 1 } ^ k \\beta _ j \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} & 1 + f \\left ( \\sum _ { i = 1 } ^ { N - 1 } D ( u _ i , u _ { i + 1 } ) \\right ) - f ( D ( x , y ) ) \\\\ & = 1 - \\frac { 1 } { \\displaystyle \\sum _ { i = 1 : N - 1 , \\ , u _ { i + 1 } \\neq u _ i } \\exp \\left ( | u _ { i + 1 } - u _ i | \\right ) } + \\frac { 1 } { \\exp \\left ( | x - y | \\right ) } \\\\ & \\geq 1 - 1 + \\frac { 1 } { \\exp \\left ( | x - y | \\right ) } \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m _ r } | \\lambda _ k ( \\hat Q _ r ( \\hat \\Sigma - \\mu _ r I ) \\hat Q _ r ) - \\lambda _ k ( Q _ r E Q _ r ) | \\leq \\| \\hat Q _ r ( \\hat \\Sigma - \\mu _ r I ) \\hat Q _ r - Q _ r E Q _ r \\| _ 1 . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} f \\circ h = u \\prod _ { j \\in I } y _ j ^ { N _ j ( f ) } , \\ \\ f _ i \\circ h = u _ i \\prod _ { j \\in I } y _ j ^ { N _ j ( f _ i ) } , \\ 1 \\leq i \\leq m , \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} W = \\left \\{ ( z _ { 0 } , g , w _ { 0 } ) \\in U _ { i } \\cap B ( 0 , R _ { i } ) \\times G ( \\mathbb { R } ) \\times \\mathbb { C } \\mathop { | } p _ { g , w } \\circ f ( z _ { 0 } ) = w _ { 0 } \\right \\} . \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log _ p | X ^ { \\Gamma _ n } | = ( 3 n ) ^ { - 1 } \\log _ p | X ^ { \\Gamma _ n } | = ( 3 n ) ^ { - 1 } \\log _ p 4 \\ ; . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} K ( x , y ) = K ( y , x ) \\mbox { f o r a . e . ~ } x , y \\in \\R ^ n \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} d \\left ( \\wp ( x ) \\cdot a \\otimes b \\otimes m \\right ) = - d \\left ( \\wp ( y ) \\cdot b \\otimes a \\otimes m \\right ) . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} \\mu ( x _ 0 ) & = x _ 0 x _ { r - 1 } , \\\\ \\mu ( x _ 1 ) & = x _ 1 x _ 0 x _ { r - 1 } , \\\\ & \\ ; \\ ; \\vdots \\\\ \\mu ( x _ { r - 1 } ) & = x _ { r - 1 } \\dots x _ 1 x _ 0 x _ { r - 1 } . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} f \\ast h \\ast f ( \\eta ) = \\sum _ { \\alpha _ 1 \\alpha _ 2 \\alpha _ 3 = \\eta } f ( \\alpha _ 1 ) h ( \\alpha _ 2 ) f ( \\alpha _ 3 ) . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\wp ( u ) - \\wp ( v ) = - \\frac { \\sigma ( u + v ) \\sigma ( u - v ) } { \\sigma ^ 2 ( u ) \\sigma ^ 2 ( v ) } . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} C ( s ) = F ( F _ 1 ^ { - 1 } ( s _ 1 + 0 ) , F _ 2 ^ { - 1 } ( s _ 2 + 0 ) , \\cdots , F _ d ^ { - 1 } ( s _ d + 0 ) ) . \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\frac { r m } { n } & = \\frac { 1 } { n } \\Bigg ( \\sum _ { i \\in A } d _ { H ^ * } ( i ) + \\sum _ { i \\in B } d _ { H ^ * } ( i ) + \\sum _ { i \\in C } d _ { H ^ * } ( i ) \\Bigg ) \\\\ & = \\frac 1 n \\Bigg ( k d + s ( d + 1 ) + \\sum _ { i \\in C } d _ { H ^ * } ( i ) \\Bigg ) \\\\ & = \\frac 1 n [ k d + s ( d + 1 ) + ( n - k - s ) ( d + 1 ) + \\ell ] \\\\ & = d + 1 + \\frac { \\ell - k } { n } . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\tilde { r } _ \\mathrm { u p a } = \\sum _ { k \\in \\mathcal { K } } \\frac { 1 } { N + \\mu } \\sum _ { n \\in \\mathcal { N } } \\log _ 2 ( 1 + \\tilde { \\gamma } _ { k , n } ) \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} d _ { 3 , j } > 0 , \\mbox { w h e r e $ d _ { 3 , j } $ i s g i v e n b y ( \\ref { d c 1 } ) f o r $ j = 1 , 2 $ } . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} ( \\rho , u , v , p ) ( x , y , 0 ) = \\left \\{ \\begin{array} { l l } ( 1 , 0 . 1 , 0 . 1 , 1 ) , & 0 . 5 < x < 1 , 0 . 5 < y < 1 , \\\\ ( 0 . 5 1 9 7 , - 0 . 6 2 5 9 , 0 . 1 , 0 . 4 ) , & 0 < x < 0 . 5 , 0 . 5 < y < 1 , \\\\ ( 0 . 8 , 0 . 1 , 0 . 1 , 0 . 4 ) , & 0 < x < 0 . 5 , 0 < y < 0 . 5 , \\\\ ( 0 . 5 1 9 7 , 0 . 1 , - 0 . 6 2 5 9 , 0 . 4 ) , & 0 . 5 < x < 1 , 0 < y < 0 . 5 . \\end{array} \\right . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\Psi ( X ) : = \\frac { 1 } { m } \\Phi ( x ) + \\frac { 1 } { 2 } v ^ 2 + \\frac { 1 } { 2 } \\sum _ { k \\geq 1 } k ^ { - 2 s } z _ k ^ 2 . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\lambda _ 1 = \\left ( \\frac { j _ { 0 , 1 } } { r _ 2 } \\right ) ^ 2 \\ , , \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} f _ { u _ { 0 } } ^ { 2 } d _ { c } ^ { 2 } + 2 \\left ( 2 f _ { v _ { 0 } } g _ { u _ { 0 } } - f _ { u _ { 0 } } g _ { v _ { 0 } } \\right ) d _ { c } + g _ { v _ { 0 } } ^ { 2 } = 0 . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} b ( \\gamma ^ { - n } z _ 0 ) = b ( z _ 0 ) = 0 \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} B ( y ) & \\geq \\exp ( h ( s ( y ) ) + y s ( y ) - 3 0 s ( y ) C ( y ) ^ { 1 / 2 } - 1 0 0 ) \\\\ & = \\exp ( h ( s ( y ) ) + y s ( y ) - O ( y ^ { u / ( 2 u + 2 ) } ) ) . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } P r ( f , \\sigma , k ) & = \\lim _ { X \\to \\infty } \\frac { \\# \\{ p \\in S p l _ X ( f ) \\mid ( \\sum _ { i = 1 } ^ n r _ i + a _ { n - 1 } ) / p = k \\} } { \\# S p l _ X ( f ) } \\\\ & = E _ n ( k ) , \\\\ v o l ( \\hat { \\frak { D } } _ n ) & = \\frac { \\sqrt { n } } { n ! } , \\\\ v o l ( \\frak { D } ( f , \\sigma , k ) ) & = P r ( f , \\sigma , k ) v o l ( \\hat { \\frak { D } } _ n ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\gamma _ 0 = \\min { 1 , \\frac { \\beta } { 2 C _ { p , \\eta } \\left ( 1 + \\mathrm { d i a m } ( \\Omega _ S ( 0 ) ) \\right ) } } C _ { p , \\eta } = \\left ( \\frac { 1 } { p ' \\eta } \\right ) ^ { 1 / p ' } . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} - q _ * ^ { - 1 } t ^ { \\frac { N + 2 A } { 2 } } ( \\nabla ^ 2 u _ 0 ) ( x , t ) = - [ M _ { 0 , 1 } + o ( 1 ) ] ( \\nabla ^ 2 U ) ( | x | ) + O ( t ^ { - 1 } ) \\ge - \\frac { M _ { 0 , 1 } U '' ( 0 ) } { 2 } I _ N - \\epsilon I _ N \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} ( M , G ) = ( M / G ) - ( M ^ G ) - 1 , \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} & \\int _ \\Omega f ^ { p - 1 } ( \\xi ) E f ( \\xi ) d \\xi = \\int _ \\Omega E ( \\frac { u ^ p ( \\xi ) } p ) d \\xi = \\frac 1 p \\int _ { \\partial \\Omega } u ^ p ( \\xi ) E \\cdot \\nu d S - \\frac Q p \\int _ \\Omega u ^ p ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} & \\ddot B ( t ) + 2 \\dot B ( t ) A ( t ) + B ( t ) \\dot A ( t ) + B ( t ) A ( t ) ^ 2 = - B ( t ) R ( t ) + B ( t ) W ( t ) \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\sum ^ { m } _ { i j = 1 } X ^ { * } _ { i } X _ { j } ( \\tilde { a } ^ { 0 } _ { i j } \\tilde { L } _ { \\nu } ) = 0 , \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} p _ { r } ( t ) & = p ( t ) = \\int _ { \\mathbf { R } } g ( r , y ) e ( - t y ) \\ , d y \\ll _ j ( r | t | ) ^ { - j } . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\frac { d R } { d t } ( s ) = t r \\Big ( A _ s ^ { - 1 } \\frac { d A _ t } { d t } ( s ) \\Big ) . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} [ X _ j , Y _ k ] = \\delta _ { j k } Z , [ X _ j , Z ] = 0 , [ Y _ k , Z ] = 0 . \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} W ^ c _ { 2 k + 1 , \\vec a , \\vec b } ( M ^ { 4 m + 2 } ) : = \\int _ { M ^ { 4 m + 2 } } \\mathcal { W } ^ c _ { 2 k + 1 , \\vec a , \\vec b } ( M ^ { 4 m + 2 } ) . \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} R _ { \\rm e n d - f i r e } = \\int \\limits _ { f _ { \\rm m i n } } ^ { f _ { \\rm m a x } } \\left ( \\log _ 2 \\mu - \\log _ 2 \\log \\left ( \\frac { f _ { \\rm m a x } } { \\sqrt { f ( 2 f _ { \\rm c } - f ) } } \\right ) \\right ) _ + { \\rm d } f . \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} R ( \\chi ) = \\prod _ { p \\leq Y } \\left ( 1 - q _ p \\chi ( p ) \\right ) ^ { - 1 } = \\prod _ { p \\leq Y } \\left ( 1 - \\frac { \\chi ( p ) } { 2 } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} \\begin{cases} u _ t + f ( v , u ) _ x ~ = ~ 0 \\ , , \\\\ v _ t + g ( v ) _ x ~ = ~ 0 , \\end{cases} \\begin{cases} u ( 0 , x ) = u _ 0 ( x ) , \\\\ v ( 0 , x ) = v _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\leq e ^ { - \\frac { 3 } { 4 } \\varsigma | ( - 1 ) ^ { s + 1 } K _ { j _ { s + 1 } } + 2 \\sum _ { i = 0 } ^ { s } ( - 1 ) ^ { s + i + 1 } b _ { { j _ i } } ^ \\prime | } . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} \\tilde { I } _ { x } ( \\varphi ) \\doteq \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } | u ( s ) | _ { H } ^ { 2 } d s : \\varphi = Y _ { x } ^ { 0 , u } \\right \\} \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\mathcal { T } ^ { r , s } _ { m ( \\cdot ) , \\nu ( \\cdot ) } & ( x ' , x '' , \\xi ' , \\zeta ' , \\xi '' , \\zeta '' ) \\\\ & \\triangleq ( e _ r ( \\mathrm { t r a j } ^ { s } _ { m ( \\cdot ) } ( x ' , \\xi ' \\zeta ' ) ) , e _ r ( \\mathrm { t r a j } ^ { s } _ { \\nu ( \\cdot ) } ( x '' , \\xi '' \\zeta '' ) ) ) . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\| \\nabla ^ j \\varphi _ R \\| _ { L ^ \\infty } \\lesssim R ^ { 2 - j } , j = 0 , \\cdots , 4 , \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\bar \\lambda _ 1 \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) + \\bar \\lambda \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) = 0 , \\\\ & \\bar \\lambda \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) + \\bar \\lambda _ 2 \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) = 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} S ( t , x ) = \\left \\{ \\begin{array} { l l } \\big ( 1 , T ( t , x ) \\big ) & \\textrm { i f $ F _ i ( t , x ) \\leq 0 $ f o r a l l $ 1 \\le i \\le l $ } , \\\\ ( 0 , 0 ) & \\textrm { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} A _ + = A _ 1 \\oplus A _ 0 \\oplus A _ { \\lambda - \\frac { 1 } { 2 } } A _ - = A _ { \\nu ^ p _ + } \\oplus A _ { \\nu ^ p _ - } . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} d \\left ( \\exp U _ { N } , \\exp V _ { N , \\ell } \\right ) = \\frac { \\ell } { N ^ { 2 } } \\leq \\frac { 1 } { N } . \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} \\abs { \\frac { z _ 1 - t } { t z _ 1 - 1 } - r } ^ 2 & = \\abs { ( t - r ) + \\frac { ( 1 - t ^ 2 ) z _ 1 } { t z _ 1 - 1 } } ^ 2 \\leq ( t - r ) ^ 2 + \\frac { 2 ( t - r ) ( 1 - t ^ 2 ) } { \\abs { t z _ 1 - 1 } } \\abs { z _ 1 } + \\frac { ( 1 - t ^ 2 ) ^ 2 } { \\abs { t z _ 1 - 1 } ^ 2 } \\abs { z _ 1 } ^ 2 \\\\ & \\leq ( t - r ) ^ 2 + 4 ( 1 - t ) \\abs { z _ 1 } + 1 6 ( 1 - t ) \\abs { z _ 1 } ^ 2 \\\\ & \\leq ( t - r ) ^ 2 + 4 ( 1 - t ) \\abs { z _ 1 } + 8 ( 1 - t ) \\abs { z _ 1 } \\\\ & \\leq ( t - r ) ^ 2 + 1 2 ( 1 - t ) \\abs { z _ 1 } . \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} \\int _ M | d u | ^ p d i v ( X ) - p | d u | ^ { p - 2 } \\langle d u ^ * d u , \\nabla X \\rangle = 0 \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} R = 2 \\alpha + 4 \\beta \\log \\frac { 4 C M } { \\epsilon } . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} p _ n ( z ; \\ , a , \\ , b | q ) = { 1 - a b q ^ { n + 1 } \\over 1 - a b q ^ { 2 n + 1 } } p _ { n } ( z ; \\ , a , \\ , b q | q ) + { a b q ^ { n + 1 } ( 1 - q ^ n ) \\over 1 - a b q ^ { 2 n + 1 } } p _ { n - 1 } ( z ; \\ , a , \\ , b q | q ) . \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} { } \\theta _ { M _ { S ' } } ( \\mu _ { S } ) = \\frac { 1 } { | W ^ { \\mathrm { r e l } } _ { M _ { S ' } } | } \\sum \\limits _ { \\sigma \\in W ^ { \\mathrm { r e l } } _ { M _ { S ' } } } \\sigma ( \\theta _ { M _ S } ( \\mu _ S ) ) . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} \\theta _ { M _ { S ' } } ( \\mu _ { S ' } ) = \\theta _ { M _ { S ' } } ( \\mu _ S ) = \\frac { 1 } { | W _ { M _ { S ' } } | } \\sum \\limits _ { \\sigma \\in W _ { M _ { S ' } } } \\sigma ( \\theta _ { M _ S } ( \\mu _ S ) ) , \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { N } { M _ N } = \\lambda \\in ( 0 , \\infty ) \\ , . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} ( x , t ) \\mapsto ( M ^ W d ) ( x , t ) : = [ M ^ W ( x ) d ( x ) ] ( t ) \\\\ \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} T ^ { K \\textnormal { t h } } = \\left ( S ^ { K \\textnormal { t h } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & y _ t - ( a ( y _ x , t , x ) y _ x ) _ x + F ( y , y _ x ) = f 1 _ \\mathcal { O } + v ^ 1 1 _ { \\mathcal { O } _ 1 } + v ^ 2 1 _ { \\mathcal { O } _ 2 } \\ \\ Q , \\\\ & y ( 0 , t ) = y ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & y ( 0 ) = y _ { 0 } \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\widetilde { I } \\left ( R , c \\right ) = \\sup \\left \\{ \\int _ { 1 } ^ { R } \\frac { 1 - f ^ { 2 } } { r } d r + 4 \\int _ { 1 } ^ { R } \\frac { \\left ( 1 - f ^ { 2 } \\right ) ^ { 2 } } { r } d r : \\int _ { 1 } ^ { R } J \\left ( 1 - f ^ { 2 } \\right ) r d r \\leq c \\right \\} . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} e ^ { i t \\Phi ( H ) } P _ { c } ( H ) = W e ^ { i t \\Phi ( H _ { 0 } ) } W ^ { * } \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\Sigma _ { p / q } = \\sqrt { \\begin{aligned} & \\frac { ( | p | q ) ^ { I } + 1 } { ( | p | q ) ^ { I } - 1 } v \\Bigl ( \\frac { | p | - q - 1 } { 2 ( | p | - q ) } \\Bigr ) \\\\ & + \\frac { 2 ( | p | q ) ^ { I } } { ( | p | q ) ^ { I } - 1 } \\sum _ { m = 1 } ^ { I - 1 } \\frac 1 { ( | p | q ) ^ { m } } v \\Bigl ( q ^ m \\frac { | p | - q - 1 } { 2 ( | p | - q ) } \\Bigr ) \\end{aligned} } \\quad , \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\textit { a n d } \\bold { E } _ { x , y ; i } = \\left [ \\begin{array} { c | c | c } \\bold { E } _ { x , y ; i ; 1 } & \\dots & \\bold { E } _ { x , y ; i ; p _ y } \\end{array} \\right ] , \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} \\xi _ { | \\alpha \\cap \\beta | } : = \\frac { \\lambda a } { 2 \\mu b _ { \\alpha , \\beta } } ( | \\alpha | - 2 | \\alpha \\cap \\beta | ) = \\frac { 1 } { 4 \\mu b _ { \\alpha , \\beta } } \\left ( 1 - \\frac { 2 \\vert \\alpha \\cap \\beta \\vert } { \\vert \\alpha \\vert } \\right ) \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\partial _ t \\theta + \\div ( v \\theta ) = 0 \\ , . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} g ( \\lambda ) & = B ^ \\frac { - 1 } { 2 } ( B ^ \\frac { 1 } { 2 } A ^ \\lambda B ^ \\frac { 1 } { 2 } ) ^ \\frac { \\delta + m } { p k \\lambda + m } B ^ \\frac { - 1 } { 2 } \\\\ & = | A ^ { * m } | ^ { - 1 } ( | A ^ { * m } | | A ^ p | ^ { 2 k \\lambda } | A ^ { * m } | ) ^ \\frac { \\delta + m } { p k \\lambda + m } | A ^ { * m } | ^ { - 1 } \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} f _ { i } ^ { k } = \\alpha _ { i } ^ { k - 1 } \\chi _ { B ( y _ { i } , 2 ^ { k - 1 } r ) } - \\alpha _ { i } ^ { k } \\chi _ { B ( y _ { i } , 2 ^ { k } r ) } , \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} e ^ { \\int _ 0 ^ t V ( Y _ v ) d v } Q ^ V f ( Y _ t , B _ t ) = & Q ^ V f ( Y _ 0 , B _ 0 ) + \\sum _ { i = 1 } ^ d \\int _ 0 ^ \\tau e ^ { \\int _ 0 ^ s V ( Y _ v ) d v } X _ i Q ^ V f ( Y _ s , B _ s ) d \\beta _ s ^ i \\\\ & + \\int _ 0 ^ \\tau e ^ { \\int _ 0 ^ s V ( Y _ v ) d v } \\partial _ y Q ^ V f ( Y _ s , B _ s ) d B _ s . \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} m ^ { - n } = ( x _ { n } e _ { 1 } + y _ { n } e _ { 2 } , \\ , x _ { n } e _ 1 + h ^ { - 1 } _ { n } e _ { 1 } + y _ { n } e _ { 2 } ) , \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} F ( 0 , x _ n ) = \\inf _ { u \\in \\mathcal { S } ^ N } \\left [ \\frac { 1 } { 2 } \\int _ 0 ^ T | u ( s ) | _ H ^ 2 d s + h ( Y ^ { 0 , u } _ { x _ n } ) \\right ] > \\frac { 1 } { 2 } \\int _ 0 ^ T | u _ n ( s ) | _ H ^ 2 d s + h ( Y ^ { 0 , u _ n } _ { x _ n } ) - \\frac { 1 } { n } . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} v ( q ) & = v ( o ) + ( X v ) ( o ) x + ( Y v ) ( o ) y + ( T v ) ( o ) t \\\\ & + \\frac { 1 } { 2 } ( X ^ 2 v ) ( o ) x ^ 2 + \\frac { 1 } { 2 } ( Y ^ 2 v ) ( o ) y ^ 2 + [ 2 ( T v ) ( o ) + ( X Y v ) ( o ) ] x y + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\left [ \\frac { \\beta ( e - 1 ) } { e } \\right ] ^ { 3 ^ n } = \\left [ \\frac { \\beta _ 1 ( e - 1 ) } { e } \\right ] ^ { 3 ^ n } \\times \\left [ \\frac { \\beta } { \\beta _ 1 } \\right ] ^ { 3 ^ n } , \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\begin{cases} U _ 1 '' + A U _ 1 + U _ 1 ' = A e ^ { t A } ( u _ 0 + u _ 1 ) , t > 0 , \\\\ ( U _ 1 , U _ 1 ' ) ( 0 ) = ( 0 , - u _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\eta _ a = \\frac { r ^ 2 } { 3 } \\beta _ a + \\frac { r ^ 3 } { 4 } D \\beta _ a + r ^ 4 [ \\frac { 1 } { 1 0 } D ^ 2 \\beta _ a - \\frac { 1 } { 4 5 } \\alpha _ { a b } \\beta ^ b ] + O ( r ^ 5 ) , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\frac { \\varphi _ { \\tilde { N } _ \\varepsilon } ( B _ \\varepsilon ^ 2 ) } { \\varphi _ { \\tilde { N } _ \\varepsilon } ( \\gamma _ \\varepsilon ^ 2 ) } = \\exp ( B _ \\varepsilon ^ 2 - \\gamma _ \\varepsilon ^ 2 ) - F _ \\varepsilon \\ , , \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } \\frac { g ( A _ i ) } { \\prod _ { j = 1 \\atop j \\ne i } ^ { m } ( A _ i - A _ j ) } = c _ { m - 1 } . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\ ; \\ ; \\ ; \\mathcal A ( \\rho ) u : = \\Phi _ { \\rho } ^ * \\Delta ( \\Phi _ * ^ { \\rho } u ) , \\qquad \\ ; \\mathcal B ( \\rho ) u : = \\langle \\nabla ( \\Phi _ * ^ \\rho u ) | _ { \\Gamma _ \\rho } , { \\bf n } _ \\rho \\rangle \\qquad \\ ; \\mbox { f o r } \\ ; \\ ; u \\in H ^ 2 ( \\Omega _ s ) , \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\tau _ { k } ^ { \\epsilon , u _ k } = \\min \\bigl \\{ t > 0 \\ , \\bigl \\vert \\ , X _ { k } ^ { \\epsilon , u _ k } ( t ) \\notin D \\bigr \\} . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\chi ( \\Delta , M ) = \\prod _ { i \\in I } \\chi ( \\Delta , \\Z \\Gamma / f _ i \\Z \\Gamma ) \\ ; \\chi ( \\Delta , \\Z \\Gamma / f _ i \\Z \\Gamma ) = | \\Z G / f _ i \\Z G | \\ ; . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} { \\mathbf { y } _ { b } } = h _ { a b } \\mathbf { x } + \\mathbf { n } _ b . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} w _ 0 = \\nu ( \\mathcal { L } - \\alpha ) ^ { - 1 } Q \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} A ' ( x ) = \\frac { 4 x ^ 2 - x - 2 } { 1 2 x ^ 2 \\sqrt { x + 1 } } > 0 , \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} w ^ { \\xi ^ + _ i } _ { \\eta ^ + _ i } ( \\rho _ \\tau ) & = \\sum _ { [ \\{ \\alpha , \\beta ; \\epsilon \\} ] \\in \\vec { \\triangle } ^ { \\xi ^ + _ i } _ { \\eta ^ + _ i } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho _ \\tau ) + \\ell _ { \\beta } ( \\rho _ \\tau ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} { \\rm C o r r } _ { \\rho , \\Phi } ^ { f , g } ( A , B ) & : = \\Phi \\left ( f ( \\rho ) g ( \\rho ) A ^ * B \\right ) - \\Phi \\left ( f ( \\rho ) A ^ * g ( \\rho ) B \\right ) \\\\ { \\rm I } _ { \\rho , \\Phi } ^ { f , g } ( A ) & : = { \\rm C o r r } _ { \\rho , \\Phi } ^ { f , g } ( A , A ) \\ , , \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\dot { x } _ { t } = \\sum _ { \\alpha = 1 } ^ { d } \\lambda ^ { \\alpha } ( t ) V _ { \\alpha } ( x _ { t } ) , \\ \\ \\ 0 \\leq t \\leq 1 . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} L ( { \\cal C } ) = \\sum _ { i = 1 } ^ { t - 1 } d ( Y _ i , Y _ { i + 1 } ) + d ( Y _ { t } , Y _ 1 ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { t } l ( Y _ i , { \\cal C } ) , \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} | K ( y _ { 0 } , y _ { 1 } , \\cdots , y _ { m } ) | \\leq \\frac { A } { ( \\sum _ { k , l = 0 } ^ { m } | y _ { k } - y _ { l } | ) ^ { m n } } , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\nu ( d ) & = \\mu \\ast e ( d ) = \\mu ( d ) \\ : e \\\\ & = q ( d , d \\wedge c _ \\mu ) \\ : \\iota ( d \\wedge c _ \\mu , c _ \\mu ) \\ : q ( c _ \\mu , d ) & & e = q ( c _ \\mu , d ) \\\\ & = q ( d , d ) \\ : \\iota ( d , c _ \\mu ) \\ : q ( c _ \\mu , d ) & & d \\wedge c _ \\mu = d \\\\ & = 1 _ d & & \\iota ( d , c _ \\mu ) \\ : q ( c _ \\mu , d ) = 1 _ d . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} \\lim _ { \\eta _ n \\rightarrow 0 } ~ q = \\frac { 1 } { a ^ 2 } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} V _ t = g _ 0 ( t ) + \\int _ 0 ^ t K ( t - s ) ( - \\lambda V _ s d s + \\nu \\sqrt { V _ s } d W _ s ) { . } \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} r E ' ( r ) & = ( n - p + O ( r ) ) E ( r ) + ( p r + O ( r ^ 2 ) ) A \\\\ & \\geq ( n - p + O ( r ) ) E ( r ) + \\frac { p + O ( r ) } { K } E ( r ) \\\\ & = ( n - p + \\frac { p } { K } + O ( r ) ) E ( r ) \\\\ & \\geq ( n - p + p \\gamma ) E ( r ) \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} \\langle C _ \\varphi ^ * f , f \\rangle = \\langle f _ 1 - f _ 2 , f _ 1 + f _ 2 \\rangle . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} v ^ { ( \\ell ) } : = v _ 1 ^ { ( \\ell ) } = v _ 2 ^ { ( \\ell ) } \\Omega \\setminus \\overline { D } , \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} c _ { \\gamma ( c , d ) } = c H ( \\gamma ( c , d ) ; - ) = \\Gamma ( d ) . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} 1 = \\tilde x \\sum _ { k = 1 } ^ r \\frac { \\lambda _ k } { \\tilde \\lambda _ 1 - \\lambda _ k } & = \\frac { \\tilde x } { A } \\frac { \\lambda _ 1 } { \\lambda _ 1 - \\lambda _ 2 } + \\tilde x \\sum _ { k = 2 } ^ r \\frac { \\lambda _ k } { \\lambda _ 1 - \\lambda _ k + A ( \\lambda _ 1 - \\lambda _ 2 ) } \\\\ & \\geq \\frac { \\tilde x } { 1 + A } \\Big ( \\frac { \\lambda _ 1 } { \\lambda _ 1 - \\lambda _ 2 } + \\sum _ { k = 2 } ^ r \\frac { \\lambda _ k } { \\lambda _ 1 - \\lambda _ k } \\Big ) . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} f ^ { \\not \\ominus } ( z , t ) = g _ { ( i ) } ( z , t ) + g _ { ( i i ) } ( z , t ) , \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { \\dot { H } ^ s } = \\frac { 2 } { \\alpha + 2 } \\| u \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\widetilde { \\chi _ k u } ( t ) = e ^ { i t \\Delta } ( \\widetilde { \\chi _ k f } ) + \\mathrm { i } \\int _ 0 ^ t e ^ { \\mathrm { i } ( t - \\tau ) \\Delta } ( \\widetilde { [ \\chi _ k , \\mathcal { H } ] u } ) ( \\tau ) \\ , d \\tau \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} A : = \\{ x \\in X \\setminus F : \\neg ( \\exists \\gamma \\in \\Gamma _ K ) ( \\gamma \\geqslant \\gamma _ 3 ( x ) \\wedge ( \\forall y , z \\in B ^ \\circ ( x , \\gamma ) \\setminus \\{ x \\} ) ( g _ x ( y ) = g _ x ( z ) ) \\} , \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial z } = \\left ( \\begin{array} { c } - \\left ( \\frac { \\partial y } { \\partial x _ { 1 } } \\right ) ^ { - 1 } \\cdot \\frac { \\partial y } { \\partial x _ { 2 } } \\\\ \\mathrm { I } _ { m - n } \\end{array} \\right ) , \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x \\sin ( \\mu x ) { } _ { 2 } F _ { 1 } \\Big ( a , b ; \\frac { 3 } { 2 } ; - c ^ 2 x ^ { 2 } \\Big ) d x = 2 ^ { - a - b + 1 } \\pi c ^ { - a - b } \\mu ^ { a + b - 2 } \\frac { K _ { a - b } ( { \\mu } / { c } ) } { \\Gamma ( a ) \\Gamma ( b ) } . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) & \\leq \\limsup _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P \\left ( \\frac { S _ n ( \\kappa _ n ( x ) ) } { ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) a _ c ^ { ( n ) } } \\leq \\frac { \\varepsilon } { h ( x ) } \\right ) \\\\ & \\leq - ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) \\inf _ { y \\in \\left ( - \\infty , \\frac { \\varepsilon } { h ( x ) } \\right ] } H ( y ) . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} Z = \\left [ \\begin{array} { c c } B ^ { \\beta } & 0 \\\\ 0 & 0 \\end{array} \\right ] , U = \\left [ \\begin{array} { c c } Q & R \\\\ S & - Q ^ * \\end{array} \\right ] , V = \\left [ \\begin{array} { c c } Q & - R \\\\ S & Q ^ * \\end{array} \\right ] , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( I ^ G _ { M _ S } ( \\rho ) ) ) = [ I ^ G _ { M _ S } ( \\rho ) ] \\left [ \\bigoplus _ { ( M _ S , \\mu _ S ) \\in \\mathrm { R e l } ^ { G , \\mu } _ { M _ S , b } } r _ { - \\mu _ S } \\circ L L ( \\rho ) | _ { W _ { \\{ \\mu _ S \\} _ { M _ S } } } | \\cdot | ^ { - \\langle \\rho _ G , \\mu \\rangle } \\right ] . \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} f ( \\lambda _ i ( C ) ) = & \\ f \\Big ( \\sum _ { j = 1 } ^ n D _ { i j } \\big ( \\tau \\lambda _ j ( A ) + ( 1 - \\tau ) \\lambda _ j ( B ) \\big ) \\Big ) \\\\ \\geq & \\ \\sum _ { j = 1 } ^ n D _ { i j } \\big ( \\tau f ( \\lambda _ j ( A ) ) + ( 1 - \\tau ) f ( \\lambda _ j ( B ) ) \\big ) = x _ i , 1 \\leq i \\leq n , \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} w ( \\theta _ { M _ S } ( \\mu _ S ) ) = \\theta _ { w ( M _ S ) } ( w ( \\mu _ S ) ) . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} & \\ \\tau \\phi \\big ( \\exp ( L + r \\log A _ j - r \\log C _ j ) \\big ) + ( 1 - \\tau ) \\phi \\big ( \\exp ( L + r \\log B _ j - r \\log C _ j ) \\big ) \\\\ \\leq & \\ \\phi \\big ( \\exp ( L + r \\log ( \\tau A _ j + ( 1 - \\tau ) B _ j ) - r \\log C _ j ) \\big ) \\\\ = & \\ \\phi \\big ( \\exp ( L ) \\big ) . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} M _ t ^ f = e ^ { \\int _ 0 ^ { t \\wedge \\tau } V ( Y _ s ) d s } Q ^ V f ( Y _ { t \\wedge \\tau } , B _ { t \\wedge \\tau } ) . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} A _ { \\inf } \\{ 1 \\} \\otimes _ { A _ { \\inf } , \\tilde \\theta _ r } W _ r ( R ) = ( \\ker \\tilde \\theta _ r ) / ( \\ker \\tilde \\theta _ r ) ^ 2 \\ . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} u ( 1 , \\cdot ) = \\varphi ( \\cdot ) . \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} | x _ { N + k } | & = | x _ N | e ^ { \\sum _ { i = 0 } ^ { k - 1 } \\ln \\left ( 1 - h _ { N + i } x _ { N + i } ^ 2 \\right ) } < | x _ N | e ^ { \\sum _ { i = 0 } ^ { k - 1 } \\left ( - h _ { N + i } x _ { N + i } ^ 2 \\right ) } \\\\ & < | x _ N | e ^ { - L ^ 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } } . \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 3 ( n ) } \\in O \\right ) & \\geq \\mathrm { e } ^ { ( \\eta + x - \\delta ) f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } - \\mathrm { e } ^ { ( x + \\delta - \\eta ) f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\\\ & = \\mathrm { e } ^ { ( x - \\delta + \\eta ) f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\left ( 1 - \\mathrm { e } ^ { 2 ( \\delta - \\eta ) f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\right ) , \\quad \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\Delta _ { M D S - S T } = \\frac { \\mathbb { E } [ Q _ i ] } { \\mathbb { E } [ T _ i ] } \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\nu = \\sum _ n | A ' ( t _ n ) | ^ { - 2 } \\mu _ n ^ { - 1 } \\delta _ { t _ n } . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} N _ d = \\sum _ { \\substack { d _ 1 + d _ 2 = d \\\\ d _ 1 , d _ 2 > 0 } } N _ { d _ 1 } N _ { d _ 2 } \\left ( \\binom { 3 d - 4 } { 3 d _ 1 - 2 } d _ 1 ^ 2 d _ 2 ^ 2 - \\binom { 3 d - 4 } { 3 d _ 1 - 1 } d _ 1 ^ 3 d _ 2 \\right ) \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } - r ^ { - 5 } \\int _ { \\Sigma _ r } \\alpha _ { H } ( V ^ 2 \\nabla \\tau ) d \\Sigma _ r = \\int _ { S ^ 2 } \\left [ \\frac { 4 A } { 3 } W _ 0 ^ 2 + \\frac { 2 } { 3 } C _ i W _ i W _ 0 - ( C _ i \\tilde { X } ^ i ) | \\beta | ^ 2 \\right ] d S ^ 2 . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\partial _ x \\left ( \\frac { \\partial _ x u } { \\sqrt { 1 + | \\partial _ x u | ^ 2 + | \\partial _ y u | ^ 2 } } \\right ) + \\partial _ y \\left ( \\frac { \\partial _ y u } { \\sqrt { 1 + | \\partial _ x u | ^ 2 + | \\partial _ y u | ^ 2 } } \\right ) = 0 , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} 0 = ( \\{ h , h ' \\} , \\{ 0 , t ' \\} ) _ { \\overline { T } } = ( h ' , t ' ) , \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\dim _ H B ( \\{ s _ n \\} , \\{ t _ n \\} , N ) = \\liminf _ { \\ell \\to \\infty } { \\sum _ { i = 1 } ^ \\ell \\log t _ i \\over d \\sum _ { i = 1 } ^ { \\ell + 1 } \\log s _ i - \\log t _ { \\ell + 1 } } . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} x = x _ k : = \\frac { \\sqrt { 1 + 4 k ^ 2 } + 1 - 2 k } { 2 } = \\frac 1 2 + \\frac { 1 } { 8 k } + O ( k ^ { - 3 } ) , \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} r v ( f _ w ( y ) - f _ w ( z ) ) = r v ( y - z ) c ( w , x ) . \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} U _ + : = U \\cap \\Omega , U _ - : = U \\setminus \\overline { U _ + } , V _ { \\pm } : = \\Phi ( U _ \\pm ) . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\overline { y } _ t - ( a ( \\overline { y } _ x , t , x ) \\overline { y } _ x ) _ x + F ( \\overline { y } , \\overline { y } _ x ) = 0 \\ \\ Q , \\\\ & \\overline { y } ( 0 , t ) = \\overline { y } ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & \\overline { y } ( 0 ) = \\overline { y } _ { 0 } \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} E [ ( 1 + \\lambda t ) ^ { \\frac { U _ 1 } { \\lambda } } ] & = \\int _ 0 ^ 1 ( 1 + \\lambda t ) ^ { \\frac { u _ 1 } { \\lambda } } f ( u _ 1 ) d u _ 1 = \\int _ 0 ^ 1 e ^ { \\frac { u _ 1 } { \\lambda } \\log ( 1 + \\lambda t ) } d u _ 1 \\\\ & = \\frac { 1 } { \\frac { 1 } { \\lambda } \\log ( 1 + \\lambda t ) } \\big ( ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 \\big ) . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } \\Big \\{ \\frac { 1 } { \\left ( 1 + x ^ 2 \\right ) ^ { \\gamma } } \\Big \\} = \\frac { A _ s ^ { \\gamma } } { ( 1 + x ^ 2 ) ^ { s + \\gamma } } \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - s , s + \\gamma ; \\frac { 1 } { 2 } ; \\frac { x ^ 2 } { 1 + x ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\varphi ( x ) = - G _ { [ x _ 1 , x _ 2 ] } ( x _ 1 , x ) \\varphi ( x _ 1 - 1 ) - G _ { [ x _ 1 , x _ 2 ] } ( x , x _ 2 ) \\varphi ( x _ 2 + 1 ) , \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B _ n ( t ) \\frac { x ^ n } { n ! } & : = \\frac { x e ^ { t x } } { e ^ x - 1 } = \\left ( \\sum _ { k = 0 } ^ \\infty \\frac { B _ k ( 0 ) } { k ! } x ^ k \\right ) \\left ( \\sum _ { m = 0 } ^ \\infty \\frac { ( t x ) ^ m } { m ! } \\right ) , \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tilde J ( u _ n , w _ n ) = \\inf _ { ( v , z ) \\in \\mathcal U } \\tilde J ( v , z ) \\ , . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\widetilde { J _ 0 } ( q , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { ( q ; q ) _ d ^ 3 } \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\sigma ^ { a b } n _ { a b } = & \\frac { 1 } { r } + \\frac { r } { 2 } [ ( \\rho + 2 \\kappa ^ 2 + ( d i v _ { \\sigma } \\eta ) ^ { ( 0 ) } ] + \\frac { r ^ 2 } { 3 } [ D \\rho + ( d i v _ { \\sigma } \\eta ) ^ { ( 1 ) } ] + \\frac { r ^ 3 } { 4 } \\Big [ \\frac { 1 } { 2 } ( D ^ 2 \\rho - \\frac { 2 } { 3 } | \\beta | ^ 2 ) \\\\ & + ( d i v _ { \\sigma } \\eta ) ^ { ( 2 ) } - \\frac { 1 } { 9 } | \\beta | ^ 2 + \\frac { 1 } { 1 5 } | \\alpha | ^ 2 \\Big ] + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\{ x \\in X ( \\overline { \\mathbb Q } ) \\mid q ( x ) = 0 \\} = Y ( \\overline { \\mathbb Q } ) + X ( \\overline { \\mathbb Q } ) _ { \\rm t o r s } . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & ( A r _ t + \\alpha ( r _ t ) ) d t + \\sigma ( r _ t ) d W _ t + \\gamma ( r _ { t - } ) d X _ t \\medskip \\\\ r _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} \\psi _ j ^ { ( K ) } = \\mathop { \\max } \\limits _ { 1 \\le { k _ j } \\le K } \\sum \\limits _ { i = { k _ j } } ^ K { \\phi _ { j i } ^ { \\left ( 4 \\right ) } } = \\max \\bigg \\{ { \\mathop { \\max } \\limits _ { 1 \\le { k _ j } \\le K - 1 } \\sum \\limits _ { i = { k _ j } } ^ K { \\phi _ { j i } ^ { \\left ( 4 \\right ) } } , \\phi _ { j K } ^ { \\left ( 4 \\right ) } } \\bigg \\} = \\max \\left \\{ { \\psi _ j ^ { ( K - 1 ) } , 0 } \\right \\} + \\phi _ { j K } ^ { \\left ( 4 \\right ) } \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} { \\Psi _ i } _ { | P = \\Lambda _ j } = \\delta _ { i , j } \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\tilde x \\Big ( \\sum _ { k = 2 } ^ r \\frac { \\lambda _ k } { \\lambda _ 1 - \\lambda _ k } + \\frac { \\lambda _ 1 } { \\lambda _ 1 - \\lambda _ 2 } \\Big ) \\geq L + 1 . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\sum \\limits _ { F \\in L ( \\mathcal { B } _ n ) } \\left [ \\prod \\limits _ { k = 1 } ^ { \\ell ( F ) } \\chi _ { b _ k } ( t ) \\right ] P _ { \\ell ( F ) } ( t ) \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\int _ { M } \\phi _ { 0 } \\omega \\cdot \\nabla \\phi _ { 0 } = \\frac { 1 } { 2 } \\int _ { M } \\omega \\cdot \\nabla \\phi _ { 0 } ^ { 2 } = 0 \\ , . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\left | R _ { n , s } ( f ) \\right | \\leq 2 \\pi \\left ( \\max _ { z \\in \\mathcal { E } _ { \\rho } } | f ( z ) | \\right ) \\dfrac { \\sum _ { k = 0 } ^ s ( - 1 ) ^ { k } { 2 s + 1 \\choose s - k } \\rho ^ { 2 n ( s - k ) } } { \\rho ^ n ( \\rho ^ { 2 n } - 1 ) ^ { 2 s } } \\ , . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} m ( x , t ) = & \\lim \\limits _ { k \\rightarrow \\infty } k ( M ( x , t , k ) - I ) \\\\ = & \\lim \\limits _ { k \\rightarrow \\infty } k \\ ( { M } ^ { ( 2 ) } ( x , t , k ) - I \\ ) \\ ( { G } ^ { ( 1 ) } ( x , t , k ) \\ ) ^ { - 1 } \\delta ^ { \\sigma _ 3 } ( k , \\xi ) \\\\ + & k \\ ( \\ ( \\hat { G } ^ { ( 1 ) } ( x , t , k ) \\ ) ^ { - 1 } \\delta ^ { \\sigma _ 3 } ( k , \\xi ) - I \\ ) \\\\ = & \\lim \\limits _ { k \\rightarrow \\infty } k \\ ( { M } ^ { ( 2 ) } ( x , t , k ) - I \\ ) = { m } ^ { ( 2 ) } ( x , t ) \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{gather*} S p l _ X ( f , \\sigma ) : = \\{ p \\in S p l _ X ( f ) \\mid \\sum _ { i = 1 } ^ n m _ { j , i } r _ { \\sigma ( i ) } \\equiv m _ j \\bmod p \\ , ( 1 \\le { } ^ \\forall j \\le t ) \\} , \\\\ S p l _ X ( f , \\sigma , \\{ k _ j \\} ) : = \\{ p \\in S p l _ X ( f , \\sigma ) \\mid \\sum _ { i = 1 } ^ n m _ { j , i } r _ { \\sigma ( i ) } = m _ j + k _ j p \\ , ( 1 \\le { } ^ \\forall j \\le t ) \\} , \\end{gather*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { k , a } ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ a , q ^ { 2 k - a } , q ^ { 2 k } ; q ^ { 2 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} f ( z ) = \\eta \\left ( i \\frac { | z | } { z } \\right ) ^ k \\bar { f } \\ ! \\left ( - \\frac 1 { N z } \\right ) \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} p ( \\Pi ' ) \\sim \\prod _ { i = 1 } ^ r p ( { \\Pi ' _ i } ) \\cdot \\prod _ { i < j } L ^ S ( 1 , \\Pi ' _ i \\times ( \\Pi ' _ j ) ^ { { \\sf v } } ) . \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } p _ n \\int _ \\Omega | \\nabla u _ { p _ n } ( x ) | ^ 2 \\ , d x = 8 \\pi e \\cdot k . \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\partial ^ \\beta ( f \\circ \\varphi ) = \\sum _ { \\lambda \\in \\Lambda _ { n , m } ( \\beta ) } \\frac { \\beta ! } { \\lambda ! } \\left ( \\partial ^ { \\sum _ { \\alpha } \\lambda _ \\alpha } f \\right ) \\circ \\varphi \\prod _ { \\alpha } \\frac { \\left ( \\partial ^ \\alpha \\varphi \\right ) ^ { \\lambda _ \\alpha } } { ( \\alpha ! ) ^ { \\sum _ i \\lambda _ { i , \\alpha } } } \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} \\| z \\| _ { L ^ \\infty ( \\Omega _ T ) } \\le C _ 2 ( T ) \\left ( \\int _ 0 ^ T \\| f \\| _ { L ^ { \\infty } _ x } ( s ) \\ , d s + \\| z _ 0 \\| _ { L ^ { \\infty } ( \\Omega ) } \\right ) . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} Y _ i ^ { t } = \\begin{cases} t , & \\mbox { i f } i = 0 \\\\ Y _ { i - 1 } ^ { t } + E _ { e _ i } ( Y _ { i - 1 } ^ { t } ) , & \\mbox { i f } 1 \\leq i \\leq l . \\end{cases} \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} f \\ast h \\ast g ( \\eta ) = \\sum _ { \\alpha _ 1 \\alpha _ 2 \\alpha _ 3 = \\eta } f ( \\alpha _ 1 ) h ( \\alpha _ 2 ) g ( \\alpha _ 3 ) . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} - \\frac { 2 R i c ' _ { i j } f _ { i } f _ { j } } { ( 1 - f ) ^ { 2 } } + \\frac { 2 f _ { j } f _ { j j i } } { ( 1 - f ) ^ { 2 } } - \\frac { 2 f _ { i } f _ { i j j } } { ( 1 - f ) ^ { 2 } } = - \\frac { 4 R i c ' _ { i j } f _ { i } f _ { j } } { ( 1 - f ) ^ { 2 } } , \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} ( \\lambda t _ \\alpha + \\mu e ^ { \\alpha } ) \\cdot t _ i = 0 \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} z _ o ^ * + \\sum \\limits _ { i = 1 } ^ n | z _ i ^ * | & = \\lim ( z _ o + \\sum \\limits _ { i = 1 } ^ n | z _ i | ) = \\lim ( t + \\sum \\limits _ { i = 1 } ^ n | x _ i | - \\mu ^ 2 \\sum \\limits _ { i = 1 } ^ n \\cfrac { 2 } { t + | x _ i | } \\\\ & \\leq \\lim ( t + \\sum \\limits _ { i = 1 } ^ n | x _ i | ) = t ^ * + \\sum \\limits _ { i = 1 } ^ n | x _ i ^ * | = 0 . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } d _ n = d , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} \\Omega \\cap U = \\{ x \\in U : x _ 1 > \\phi ( x ' ) \\} , \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} T ( f _ 1 , f _ 2 , \\dots , f _ m ) ( x ) = \\int _ { \\mathbb R ^ { n m } } K ( x , y _ 1 , \\dots , y _ m ) f _ 1 ( y _ 1 ) \\cdots f _ m ( y _ m ) d y _ 1 \\dots d y _ m , \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\kappa ( b _ { S _ 1 } ) = \\mu _ { S _ 2 } | _ { Z ( \\widehat { M _ { S _ 1 } } ) ^ { \\Gamma } } . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} \\nu ( b _ { S _ 1 } ) = \\nu ( b _ { S _ 2 } ) \\preceq \\theta _ T ( \\mu _ { S _ 2 } ) \\preceq \\theta _ T ( \\mu _ { S _ 1 } ) , \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\lambda x & = c _ { \\alpha + \\beta } \\\\ \\mu x & = 2 \\theta ^ \\beta _ \\epsilon b _ { \\beta , \\alpha + \\gamma } \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} \\mathcal { S } _ { n } ( z ) & = 2 \\int _ { 0 } ^ { z } \\mathcal { C } _ { n } ( \\zeta ) d \\zeta , \\\\ \\mathcal { C } _ { n + 1 } ( z ) & = - 2 \\int _ { 0 } ^ { z } \\mathcal { S } _ { n } ( \\zeta ) d \\zeta - \\frac { ( 2 ^ { 2 n + 1 } - 1 ) } { ( 2 n + 2 ) ! } | B _ { 2 n + 2 } | , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} A _ 2 = \\widetilde { m } ^ { q + 1 } K ^ i _ { y _ j } K ^ j _ { y _ i } = \\widetilde { m } ^ { q + 1 } \\widetilde { w } _ { y _ i y _ j } \\widetilde { w } _ { y _ j y _ i } = \\widetilde { m } ^ { q + 1 } | \\nabla _ y ^ 2 \\widetilde { w } | ^ 2 , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} ( e _ { i } ) : = ( X _ { i } , r _ { j } , \\alpha _ { k } ) , \\ ( \\tilde { e } _ { i } ) : = ( 2 \\alpha _ { i } , \\tilde { r } _ { j } , 2 X _ { k } ) \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} k = O ( \\rho _ k ^ { \\frac { 1 + \\delta } { 2 } } ) , \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\mu = 0 , \\quad \\mbox { o r } \\quad \\mu = 1 , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\begin{aligned} a ( \\sigma , \\tau ) & : = \\int _ { \\Omega } A \\sigma : \\tau \\ d x , & b ( \\tau , v ) & : = \\int _ { \\Omega } \\div \\tau \\cdot v \\ d x . \\end{aligned} \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} & \\liminf _ { n \\rightarrow \\infty } P ( n , a , \\eta ) \\\\ = ~ & \\liminf _ { n \\rightarrow \\infty } \\min \\left \\{ P ^ + ( n , a , \\eta ) , P ^ - ( n , a , \\eta ) \\right \\} \\\\ \\geq ~ & I ^ - ( a , \\eta ) . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} b ( w _ m ) = \\lim _ { n \\rightarrow \\infty } K _ \\Omega ( w _ m , y _ n ) - K _ \\Omega ( z _ 0 , y _ n ) \\geq K _ \\Omega ( w _ m , z _ 0 ) - R . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} ( A ^ { ( 4 ) } ) ^ T ( t , x ) n ( x ) = W ( x ) \\ , , \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\Omega ^ i : = \\{ ( s , \\zeta _ 1 , \\zeta _ 2 ) \\in \\R ^ 3 : s \\in [ 0 , L ^ i ] , \\ , ( \\zeta _ 1 , \\zeta _ 2 ) \\in \\mathcal A ^ i ( s ) \\} , \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\textbf { w } _ { k , i } = \\frac { 1 } { \\sqrt { M N } } \\textbf { a } \\left ( \\hat { \\textbf { x } } _ { k - 1 } + \\boldsymbol { \\Delta } _ { i } \\right ) , \\ , i = 1 , \\cdots , q . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} S ( f _ { \\lambda } ) ( \\xi , y _ { \\lambda } ) = \\int f _ { \\lambda } d S ^ * ( \\delta _ { ( \\xi , y _ { \\lambda } ) } ) = \\sum _ { \\substack { \\eta \\in \\alpha \\\\ x \\in L } } f _ { \\lambda } ( \\eta , x ) S ^ * ( \\delta _ { ( \\xi , y _ { \\lambda } ) } ) ( \\{ ( \\eta , x ) \\} ) \\neq 0 , \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\int \\varphi \\left ( \\frac { x - \\beta t - \\gamma } { L } \\right ) ( u ^ 2 + u _ x ^ 2 ) ( \\delta , x ) d x = 0 , \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} W _ n W _ m = e ^ { \\i h \\sigma ( n , m ) } W _ { n + m } \\ , , n , m \\in \\Z ^ { 2 g } \\ , . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\mathcal { K } ( \\theta ) : = \\int _ 0 ^ \\theta \\kappa ( z ) d z . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\mathcal { E } _ \\rho = \\left \\{ z \\in \\mathbb { C } \\ , \\left | \\right . \\ z = \\frac 1 2 \\left ( u + u ^ { - 1 } \\right ) , \\ 0 \\le \\theta \\le 2 \\pi \\right \\} , u = \\rho \\ , e ^ { i \\theta } . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} E [ X ] = E [ E [ X | Y ] ] = \\int _ { - \\infty } ^ \\infty E [ X | Y = y ] f _ Y ( y ) d y . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D F = 1 , N D N = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ { 2 } \\left ( 2 ^ { R } - 1 \\right ) ( 1 + k ) \\left ( k - 2 ^ { \\textrm { R } } \\right ) } { \\lambda _ { \\textrm { B N } } \\left ( k - \\left ( 2 ^ { \\textrm { R } } - 1 \\right ) \\right ) } \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} R i c _ { \\widetilde { g } } ( Y _ { i } , Y _ { j } ) = \\gamma _ { i j } g _ { F } ( Y _ { i } , Y _ { j } ) \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{gather*} X ( 0 ) = X , \\frac { \\partial X ( s ) } { \\partial s } = V ( s ) , \\end{gather*}"}
-{"id": "3217.png", "formula": "\\begin{align*} f ( t ) = \\prod _ { j = 1 } ^ { \\frac { n + 1 } { 2 } } ( t - \\lambda _ { 2 j + 1 } ) + ( t - \\lambda _ { 1 } ) \\sum _ { j = 1 } ^ { \\frac { n + 1 } { 2 } } \\frac { \\sin ^ { 2 } \\left [ \\frac { ( 2 j + 1 ) \\pi } { n + 3 } \\right ] } { \\sin ^ { 2 } \\left ( \\frac { \\pi } { n + 3 } \\right ) } \\prod _ { \\substack { m = 1 \\\\ m \\neq j } } ^ { \\frac { n + 1 } { 2 } } ( t - \\lambda _ { 2 m + 1 } ) \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\langle A ^ * \\xi , v \\rangle & = \\big \\langle \\xi , T \\big ( \\tilde { \\alpha } ( t _ 0 + s , \\langle \\zeta , h _ 0 - \\phi ( t _ 0 ) + v \\rangle ) - \\tilde { \\alpha } ( t _ 0 + s , \\langle \\zeta , h _ 0 - \\phi ( t _ 0 ) \\rangle ) \\big ) \\big \\rangle \\\\ & - \\langle \\xi , \\alpha ( h _ 0 + v ) - \\alpha ( h _ 0 ) \\rangle . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} f _ { k , u _ l } ( x , u ) = \\partial _ { u _ l } f _ k \\geq 0 \\ \\ ( k \\not = l ) \\ \\ { \\rm f o r \\ e a c h } \\ \\ u \\in ( p ^ { - } ( x ) , p ^ { + } ( x ) ) , \\ x \\in \\R ^ N , \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} & \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } \\left ( E ( L _ 1 , \\ell _ \\nu , \\ell _ \\mu ) + E ( L _ 1 , \\ell _ \\nu , \\ell _ { \\mu ' } ) \\right ) \\\\ & = \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } ( R ( L _ 1 , 2 \\ell _ \\mu , \\ell _ \\nu ) + R ( L _ 1 , 2 \\ell _ { \\mu ' } , \\ell _ \\nu ) - L _ 1 ) \\\\ & = 1 - \\frac { \\sinh \\frac { \\ell _ \\nu } { 2 } ( 2 \\cosh \\frac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu + \\cosh \\ell _ { \\mu ' } ) } { ( \\cosh \\frac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu ) ( \\cosh \\frac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ { \\mu ' } ) } . \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} z ^ \\beta \\hat u - \\Delta \\hat u = \\hat f , | \\arg ( z ) | < \\min ( \\pi , ( \\pi - \\delta ) / \\beta ) , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} \\mathcal { D } _ N = \\{ I \\in \\mathcal { D } : | I | = 2 ^ { - N } \\} \\qquad \\mathcal { D } _ { \\leq N } = \\bigcup _ { n = 0 } ^ N \\mathcal { D } _ n . \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\begin{cases} f _ \\pm ( \\xi ) = c _ 1 [ ( a - r N _ \\pm ) \\xi + c ] ^ { - \\frac { 1 } { a - r N _ \\pm } } \\\\ h _ \\pm ( \\xi ) = c _ 2 [ ( a - r N _ \\pm ) \\xi + c ] ^ { - \\frac { N _ \\pm } { a - r N _ \\pm } } \\\\ \\varphi _ \\pm ( \\xi ) = c _ 3 [ ( a - r N _ \\pm ) \\xi + c ] ^ { - \\frac { k } { a - r N _ \\pm } } \\end{cases} , \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\sigma ( 4 ) = 3 . \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} 1 / ( n p _ n ) \\to 0 \\quad p _ n = o ( n ^ { - 1 / r } ) , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} x \\cdot y = \\alpha ( x _ 1 , y _ 1 ) \\alpha ^ { - 1 } ( x _ 3 , y _ 3 ) x _ 2 y _ 2 . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} ( \\Delta + q ) v _ 1 = ( \\Delta + q ) v _ 2 = 0 \\Omega \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\tilde { \\check { y } } _ { \\mathrm { d } , i } [ n ] & = \\sum _ { k \\in \\mathcal { K } } \\tilde { h } _ { i , k } [ n ] \\tilde { x } _ { \\mathrm { d } , k } [ n ] + \\tilde { z } _ { \\mathrm { d } , i } [ n ] + \\tilde { e } _ { \\mathrm { d } , i } [ n ] , \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} a _ { i } ^ { k } = \\frac { f _ { i } ^ { k } } { \\| f _ { i } ^ { k } \\| _ { L ^ { \\infty } } } | B ( y _ { i } , 2 ^ { k } r ) | ^ { - 1 / p } . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ { t _ e } \\vec q _ h \\cdot ( \\partial _ { t _ e } \\lambda _ { v _ e } \\vec q _ h ( v ) + \\partial _ { t _ e } \\lambda _ v \\vec q _ h ( v _ e ) ) \\\\ = \\ & \\partial _ { t _ e } \\vec q _ h \\cdot ( \\partial _ { t _ e } ( \\lambda _ v + \\lambda _ { v _ e } ) \\vec q _ h ( ( v + v _ e ) / 2 ) + \\frac { h _ e } { 2 } \\partial _ { t _ e } ( \\lambda _ v - \\lambda _ { v _ e } ) \\partial _ { t _ e } \\vec q _ h ) . \\end{aligned} \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} a ( w ) : = \\lim _ { x \\to \\infty } \\frac { f _ w ( x ) } { x ^ r } . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\sum _ { d _ 1 + d _ 2 = d } N _ { d _ 1 } N _ { d _ 2 } d _ 1 ^ 2 d _ 2 ^ 2 \\frac { 1 } { ( 3 d _ 1 - 2 ) ! } \\frac { 1 } { ( 3 d _ 2 - 2 ) ! } \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\widetilde { J _ 1 } ( q , Q ) = \\sum _ { d \\in \\mathbb { Z } } g _ d ( q ) Q ^ d + \\ell _ q ( Q ) \\sum _ { d \\in \\mathbb { Z } } h _ d ( q ) Q ^ d \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\max \\{ u , v \\} = \\frac { 1 } { 2 } ( u + v + | u - v | ) \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} K [ x ] / ( p ) = K ( a ) \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} x _ { m + 1 } = x + \\sum _ { \\alpha = 1 } ^ { d } \\int _ { 0 } ^ { a _ { m + 1 } } V _ { \\alpha } ( z _ { t } ) d \\tilde { h } _ { t } ^ { \\alpha } . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\in C \\cap ( 0 , \\ell _ 1 ^ { - 1 } ) \\right ) & \\leq P \\left ( \\kappa _ 1 < \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\leq \\kappa _ 2 \\right ) \\\\ & = P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } > \\kappa _ 1 \\right ) - P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } > \\kappa _ 2 \\right ) . \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} S _ { n , n - k } = S _ { n - 1 , n - k } + \\frac { S _ { n - 1 , n - ( k + 1 ) } } { n } . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\gamma = \\frac { \\lambda _ { a b } ^ 2 \\rho n \\eta ( 1 - \\eta ) } { ( n - n _ p ) \\sigma _ b ^ 2 \\left [ \\lambda _ { a b } ( 1 - \\eta ) + \\frac { \\sigma _ b ^ 2 } { \\rho n } + \\frac { \\lambda _ { a b } \\eta } { n - n _ p } \\right ] } . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} R _ L ( h ) ( \\xi , y ) = \\left \\{ \\begin{array} { r l } 0 & y \\notin \\{ y _ 0 , y _ 1 \\} , \\\\ R ( f ) ( \\xi ) & y = y _ 0 \\\\ - R ( f ) ( \\xi ) & y = y _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\frac 1 p = \\frac 1 { p _ 1 } + \\frac 1 { p _ 2 } < \\frac 1 { r _ 1 } + \\frac 1 { r _ 2 } \\le \\frac 1 { \\min \\{ r _ 1 , 2 \\} } + \\frac 1 { \\min \\{ r _ 2 , 2 \\} } < 2 - \\frac 1 { \\min \\{ r _ 3 , 2 \\} } \\le \\frac 3 2 . \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} H ( x _ 0 ) ( m , m ) = v ( T ^ { m } x _ 0 ) , \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} [ u ] _ { r , \\alpha ; O } : = \\left \\{ \\begin{array} { l l } \\sup _ { x \\in O , | j | = r } | \\partial ^ { j } u ( x ) | , & \\hbox { i f } \\ , \\alpha = 0 ; \\\\ \\sup _ { x , y \\in O , x \\not = y , | j | = r } \\frac { | \\partial ^ { j } u ( x ) - \\partial ^ { j } u ( y ) | } { | x - y | ^ { \\alpha } } , & \\hbox { i f } \\ , \\alpha > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\langle y , u _ i \\rangle \\leq h _ { P ( \\tilde { z } _ t ) } ( u _ i ) - 2 r \\mbox { \\ f o r $ i = 2 , \\ldots , k $ } \\mbox { \\ a n d \\ } P ( z _ t ) \\subset R B ^ n . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} H ( \\mu , \\mathcal { D } _ n ) \\geq \\frac { ( 1 - \\varepsilon ) n } { m } ( 1 - \\varepsilon ) ^ 2 m \\log 2 = ( 1 - \\varepsilon ) ^ 3 n \\log 2 \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} J _ { \\tilde { \\rho } _ 1 , \\tilde { \\rho } _ 2 } ( u ) = \\frac 1 2 \\int _ { M } | \\nabla u | ^ 2 - \\tilde { \\rho } _ 1 \\int _ { M } e ^ { u - \\overline { u } } - \\tilde { \\rho } _ 2 \\int _ { M } e ^ { - u + \\overline { u } } . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} & \\nabla ^ { F ^ { * } + F } _ { u } ( \\eta + Y ) = \\nabla ^ { F } _ { X } \\eta - \\frac { 1 } { 3 } { \\mathcal H } ( X , Y , \\cdot ) + \\nabla ^ { F } _ { X } Y , \\\\ & \\nabla ^ { \\mathcal G } _ { u } ( s ) : = \\nabla _ { X } s + \\frac { 2 } { 3 } \\mathrm { a d } _ { r } ( s ) \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\frac { d } { d s } | V | ^ 2 = 2 V \\dot V = 2 V \\cdot ( f ( X ) - f ( \\bar X ) ) + 2 h ^ 2 V \\cdot g ( X , s ) \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\sum _ j \\langle X , e _ j \\rangle ^ 2 ( p _ m ) = 0 \\ \\ \\ \\ 2 - \\inf | X | ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} P ( \\mathrm { B i n } ( \\lfloor m _ n \\rfloor , q _ n ) \\geq k ) = \\frac { ( q _ n m _ n ) ^ { k } } { k ! } ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} I I ^ L = \\left ( \\begin{array} { c c } \\langle \\nabla ^ { L } _ { e _ 1 } v _ L , e _ 1 ) \\rangle _ { L } , & \\langle \\nabla ^ { L } _ { e _ 1 } v _ L , e _ 2 ) \\rangle _ { L } \\\\ \\langle \\nabla ^ { L } _ { e _ 2 } v _ L , e _ 1 ) \\rangle _ { L } , & \\langle \\nabla ^ { L } _ { e _ 2 } v _ L , e _ 2 ) \\rangle _ { L } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} \\widetilde { \\phi } ( z , t + T ) \\equiv \\widetilde { \\phi } ( z , t ) , \\ \\widetilde { \\phi } _ z \\gg ( 0 , 0 , \\cdots , 0 ) , \\ \\widetilde { \\phi } ( \\pm \\infty , \\cdot ) = p ^ { \\mp } ( \\cdot ) . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} D ( P _ { v } \\Pi \\nu , q ) = \\min \\{ 1 , \\frac { \\log \\Vert p \\Vert _ { q } ^ { q } } { ( q - 1 ) \\log r ^ { m } } \\} \\ge \\min \\{ 1 , \\frac { h - \\delta } { - \\log r } \\} , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } | 1 + \\langle C _ \\varphi ^ * f , f \\rangle | ^ 2 = | | f _ 1 | | ^ 4 + | | f _ 1 | | ^ 2 | | f _ 2 | | ^ 2 \\delta ^ 2 + 2 | | f _ 1 | | ^ 3 | | f _ 2 | | \\delta \\cos \\theta . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} K _ { T ^ { N + 1 } } \\left ( \\mathbb { P } ^ N \\right ) = \\mathbb { Z } [ \\Lambda _ 0 ^ { \\pm 1 } , \\dots , \\Lambda _ N ^ { \\pm 1 } ] [ P ^ { \\pm 1 } ] \\left / \\left ( \\left ( 1 - \\Lambda _ 0 P ^ { - 1 } \\right ) \\cdots \\left ( 1 - \\Lambda _ N P ^ { - 1 } \\right ) \\right ) \\right . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { a = \\pm 1 } \\bigg | P \\Big ( a \\big | \\sigma _ { - \\infty } ^ { - ( k + 1 ) } \\omega _ { - k } ^ { - 1 } \\Big ) - P \\Big ( a \\big | \\tau _ { - \\infty } ^ { - ( k + 1 ) } \\omega _ { - k } ^ { - 1 } \\Big ) \\bigg | ^ 2 < \\infty \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} V ( n ) = \\frac { 1 } { ( n - 1 ) ! } \\sum _ { 0 \\le i \\le n , \\atop 1 \\le k \\le n - 1 } ( - 1 ) ^ i { n \\choose i } M ( k - i a ) ^ { n - 1 } . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\rho _ { { \\bf L } ^ 1 } ( \\tilde { g } , g ) ~ = ~ \\int _ { [ a , b ] \\backslash \\mathcal { D } _ g } \\rho ( \\tilde { g } ( x ) , g ( x ) ) d x ~ = ~ 0 ~ . \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\varphi ( d _ q \\log ( T _ a ) ) = \\tilde \\xi d _ q \\log ( T _ a ) \\ . \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\omega ' ( ( f _ 1 , \\phi _ 1 ) , ( f _ 2 , \\phi _ 2 ) ) \\ = \\ \\langle A ( f _ 2 ) , \\phi _ 1 \\rangle - \\langle A ( f _ 1 ) , ( \\phi _ 2 ) \\rangle + \\langle B ( \\phi _ 2 ) , \\phi _ 1 \\rangle \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} P ( N ) & = \\{ D \\in \\mathbb { F } _ q [ t ] : D | D | < N \\} \\\\ g ( N ) & = \\{ D \\in P ( N ) : L ( 1 / 2 , \\chi _ D ) = 0 \\} \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} V ^ { ( i ) } ( t ) : = v ^ { ( i ) } ( t ^ * + t ) - v ^ * _ { } ( t ^ * ) , i = 1 , 2 . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\{ t _ i : i = 1 , \\dots , n \\} \\cup \\{ e ^ { \\alpha } : \\alpha \\in C ^ * \\} , \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} | \\Gamma ( t ) f | _ q \\leq \\mathcal { B } e ^ { \\sum _ { i = 1 } ^ N \\left [ \\beta _ i ( t ) \\theta _ i - \\frac { t } { 2 } \\theta _ i ^ 2 \\right ] } \\left | f \\right | _ q | \\Gamma ^ { - 1 } ( t ) f | _ q \\leq e ^ { \\sum _ { i = 1 } ^ N \\left [ - \\beta _ i ( t ) \\theta _ i + \\frac { t } { 2 } \\theta _ i ^ 2 \\right ] } \\left | f \\right | _ q \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\cos ( \\mu x ) { } _ { 2 } F _ { 1 } \\Big ( a , b ; \\frac { 1 } { 2 } ; - c ^ 2 x ^ { 2 } \\Big ) d x = 2 ^ { - a - b + 1 } \\pi c ^ { - a - b } \\mu ^ { a + b - 1 } \\frac { K _ { a - b } ( { \\mu } / { c } ) } { \\Gamma ( a ) \\Gamma ( b ) } , \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} & G ( m , 2 , a , \\epsilon , s ) \\\\ \\leq & 2 \\sum _ { k = 1 } ^ { ( { m ( 1 + \\epsilon ) \\over 2 } ) ^ { { 1 \\over a } } } k ^ { - d s } ( { m \\over 3 } ) ^ { - { d s \\over a } } \\cdot 2 \\varepsilon m ^ { { 1 \\over a } } \\cdot C _ 4 ( a ) \\\\ \\leq & 4 \\cdot \\zeta ( d s ) \\cdot 3 ^ { \\frac { d s } a } \\cdot C _ 4 ( a ) \\cdot \\varepsilon m ^ { \\frac { 1 - d s } a } . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} f = f _ { 1 } + f _ { 2 } f _ { i } \\subseteq B ( y _ { i } , r ) i = 1 , 2 . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} = - \\langle \\rho _ G , \\mu \\rangle . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\theta _ p ( v , x _ 1 ) \\leq \\theta _ p ( v , 0 ) = \\theta _ p ( u , x ) . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} I ^ h ( y ^ h ) = \\int _ { S ^ { h _ 0 } } W ( \\nabla _ h y ^ h ) d z \\leqslant C e ^ h , \\quad \\lim _ { h \\rightarrow 0 } \\frac { e ^ h } { h ^ 2 } = 0 . \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} b \\in L ^ { q , 1 } ( [ 0 , T ] , L ^ p _ x ) \\frac { 2 } { q } + \\frac { d } { p } = 1 , \\ 1 < p , q < \\infty \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\rho _ s = \\eta _ s + \\ln ( \\bar \\sigma + \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } ) - \\ln \\hat \\sigma . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} \\omega _ k ( x ) > 0 , \\sum \\nolimits _ { k = 1 } ^ { n } \\omega _ k ( x ) = 1 \\bar { \\omega } ( x ) \\Gamma ( x ) = 0 , \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} u _ 1 = - 2 e ^ { - 3 } \\approx - 0 . 0 9 9 6 , u _ n < 0 , \\quad n \\geq 2 , \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\O ^ { 1 , i } _ 2 ( s ) = ( \\eta _ i + 1 ) s ^ i , ~ \\Gamma ^ { 1 j } _ { 2 k } ( s ) = \\eta _ j \\delta ^ j _ k . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\frac { d } { d x } P ( s , x ) = \\frac { x ^ { s - 1 } e ^ { - x } } { \\Gamma ( s ) } , s > 0 , \\ , x > 0 . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\Phi ( f ) = \\int _ \\mathbb { T } f ( z ) \\ , d \\mu ( z ) ( f \\in C ( \\mathbb { T } ) ) . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} C _ I = \\bigoplus _ { k \\geq 1 } C _ I ^ { - k } C _ I ^ { - k } = \\bigoplus _ { [ R ] \\in Q ( I , k ) } C _ { I , [ R ] } . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\hat { \\Psi } _ 0 ( Y _ 1 , \\ldots , Y _ m ) = ( { \\Psi } _ 0 ( Y _ 1 ) , \\ldots , { \\Psi } _ 0 ( Y _ m ) ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\hat { \\Psi } _ 1 ( Y _ 1 , \\ldots , Y _ m ) = ( { \\Psi } _ 1 ( Y _ 1 ) , \\ldots , { \\Psi } _ 1 ( Y _ m ) ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} r _ 2 \\left ( n \\right ) = 4 d _ { 1 , 4 } ( n ) - 4 d _ { 3 , 4 } ( n ) , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} | x _ { n + 1 } | & \\le | x _ n ( 1 - h _ n x _ n ^ 2 ) | + | u _ { n + 1 } | \\\\ & \\le h _ n | x _ n | ^ 3 \\left | 1 - \\frac 1 { h _ n x _ n ^ 2 } \\right | + | u _ { n + 1 } | \\\\ & \\le h _ n | x _ n | ^ 3 + | u _ { n + 1 } | . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{gather*} w _ 0 = ( n ( n - 1 ) ( n - 2 ) \\dots 2 1 2 \\dots ( n - 1 ) n ) \\dots ( 3 2 1 2 3 ) ( 2 1 2 ) ( 1 ) , \\\\ w ' _ 0 = ( ( n - 1 ) ( n - 2 ) ( n - 3 ) \\dots 2 1 2 \\dots ( n - 2 ) ( n - 1 ) ) \\dots ( 3 2 1 2 3 ) ( 2 1 2 ) ( 1 ) , \\\\ w _ 0 = ( n ( n - 1 ) ( n - 2 ) \\dots 2 1 2 \\dots ( n - 1 ) n ) w ' _ 0 . \\end{gather*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\liminf _ { t \\to \\infty } \\frac { \\mu \\left ( \\{ s \\in [ 0 , t ] \\colon d ( T _ s f , T _ s g ) < \\delta \\} \\right ) } { t } = 0 \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} p = \\frac { 4 s ( \\alpha + 2 ) } { \\alpha ( d - 2 s ) } , q = \\frac { d ( \\alpha + 2 ) } { d + \\alpha s } . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ 2 v ( r , t ; x ) - \\partial _ r ^ 2 v ( r , t ; x ) + \\frac { \\mu } { 1 + t } \\partial _ t v ( r , t ; x ) + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } v ( r , t ; x ) = \\Omega _ r [ f ] ( t , x ) , t > 0 , r > 0 , \\\\ v ( r , 0 ; x ) = \\Omega _ r [ u _ 0 ] ( x ) , \\ \\ \\ , r > 0 , \\\\ \\partial _ t v ( r , 0 ; x ) = \\Omega _ r [ u _ 1 ] ( x ) , r > 0 , \\\\ v ( 0 , t ; x ) = 0 , \\ \\ , t \\geqslant 0 . \\end{cases} \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\eta = \\sqrt { \\frac { { N _ { \\rm M S } } } { \\textrm { t r } \\left \\lbrace \\hat { \\textbf { H } } ^ { \\rm H } \\textbf { K } ^ { \\rm H } \\textbf { F } ^ { \\rm H } \\textbf { F } \\textbf { K } \\hat { \\textbf { H } } \\right \\rbrace } } . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} I _ 0 ^ 2 ( j , | \\gamma | ) : = \\int _ { \\R ^ n } \\dfrac { | \\partial _ t ^ j ( \\lambda _ { + } e ^ { \\lambda _ { - } t } - \\lambda _ { - } e ^ { \\lambda _ { + } t } ) | ^ 2 } { | \\lambda _ { + } - \\lambda _ { - } | ^ 2 } \\ , | \\xi | ^ { 2 | \\gamma | } \\chi ( \\xi ) ^ 2 d \\xi \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\mathrm { I C } ( \\mathcal { F } ( r ) ) = \\begin{cases} 1 9 & r = 1 , \\\\ 7 3 9 = 5 1 5 + 2 2 4 & r = 4 , \\\\ 5 0 1 = 2 0 7 + 2 9 4 & r \\ge 5 , \\\\ 5 8 1 = 3 5 7 + 2 2 4 & r \\ge 6 . \\end{cases} \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} & N _ { \\mathcal J } ( u , v , w ) + N _ { \\mathcal J } ( u , w , v ) \\\\ & = \\pi ( \\mathcal J u ) \\langle \\mathcal J v , w \\rangle - \\pi ( u ) \\langle v , w \\rangle + \\pi ( \\mathcal J u ) \\langle v , \\mathcal J w \\rangle + \\pi ( u ) \\langle \\mathcal J v , \\mathcal J w \\rangle = 0 . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} I ( \\varphi ) = \\sum _ { k = 1 } ^ K \\int _ { s _ { k - 1 } } ^ { s _ k } w ( s ) \\varphi ( s ) d s \\approx \\sum _ { k = 1 } ^ K \\int _ { s _ { k - 1 } } ^ { s _ k } w ( s ) \\varphi _ K ( s ) d s = \\sum _ { k = 1 } ^ K w _ k \\varphi ( s _ k ) , \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} N _ 2 = 2 ^ { - d + 1 } \\frac { N ^ 2 _ 1 } { N _ 1 N ( D ( F ) , r ) } \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} K _ { 0 } \\left ( \\frac { 1 } { f _ { 1 } f _ { 1 } ^ { \\prime \\prime } } \\right ) \\left ( \\frac { 1 } { f _ { 2 } ^ { \\prime } } \\right ) ^ { 2 } = \\frac { f _ { 2 } f _ { 2 } ^ { \\prime \\prime } } { \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } } - \\frac { \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } } { f _ { 1 } f _ { 1 } ^ { \\prime \\prime } } . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\overline { u } _ \\ell ( t ) & = \\frac { 1 } { 2 } \\sum _ { j = 0 } ^ \\ell \\alpha _ { j , k } ( - t ) ^ j ( - \\Delta ) ^ { j + \\ell } e ^ { t \\Delta } u _ 0 \\\\ & \\quad + \\sum _ { 0 \\leq k _ 1 + k _ 2 \\leq \\ell } \\alpha _ { \\ell - k _ 1 - k _ 2 , k _ 1 } \\beta _ { k _ 2 } ( - t ) ^ { \\ell - k _ 1 - k _ 2 } ( - \\Delta ) ^ { 2 \\ell - k _ 1 - k _ 2 } e ^ { t \\Delta } \\left ( \\frac { 1 } { 2 } u _ 0 + u _ 1 \\right ) , \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\cdots = p _ { - 2 } ^ { } = p _ { - 1 } ^ { } = p _ 0 ^ { } = p _ { 1 } ^ { } = p _ 2 ^ { } = \\cdots , \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} u _ \\epsilon ( x ) = \\int _ 0 ^ x \\frac { j _ \\epsilon } { m _ \\epsilon ( s ) } d s - P x + P - \\int _ 0 ^ 1 \\int _ 0 ^ z \\frac { j _ \\epsilon } { m _ \\epsilon ( s ) } d s d z . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\varepsilon _ i ( x ) \\equiv a _ i ( - x ^ { 2 } ) F _ i ( - x ^ { 2 } ) = 1 - b _ i ( - x ^ { 2 } ) f _ i ( - x ^ { 2 } ) \\ ( { \\rm m o d } \\ x ^ { 2 n } + 1 ) . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} E ^ { k _ j } _ K f _ j = \\sum _ { l _ j = 0 } ^ { k _ j - 1 } \\Delta ^ { l _ j } _ K f _ j + E _ K f _ j . \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} \\mathbf { G } _ { j } \\mathbf { \\perp \\ ! \\ ! \\ ! \\perp } _ { \\mathcal { G } } \\overline { \\mathbf { B } } _ { j - 1 } \\mathbf { | } \\overline { \\mathbf { G } } _ { j - 1 } , \\overline { \\mathbf { A } } _ { j - 1 } j = 1 , \\dots , p . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { A _ p ( G ) } = \\inf \\biggl \\{ \\sum _ { k = 1 } ^ \\infty \\left \\Vert g _ k \\right \\Vert _ p \\left \\Vert h _ k \\right \\Vert _ q : f = \\sum _ { k = 1 } ^ \\infty g _ k \\ast h _ k \\biggr \\} \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\boldsymbol { w } = \\left [ \\begin{array} [ c ] { c } w _ { 1 } \\\\ \\\\ w _ { 2 } \\end{array} \\right ] = \\left [ \\begin{array} [ c ] { c } u - u _ { 0 } \\\\ \\\\ v - v _ { 0 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } L ( s ) = \\begin{cases} \\infty & \\mbox { i f $ L $ i s i n c r e a s i n g } , \\\\ 0 & \\mbox { i f $ L $ i s d e c r e a s i n g } . \\end{cases} \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x ^ { \\lambda } K _ { \\mu } ( a x ) \\sin ( b x ) d x & = \\frac { { 2 ^ { \\lambda } b \\ , \\Gamma \\big ( \\frac { 2 + \\mu + \\lambda } { 2 } \\big ) \\Gamma \\big ( \\frac { 2 + \\lambda - \\mu } { 2 } \\big ) } } { a ^ { 2 + \\lambda } } \\\\ & \\ ; \\ ; \\ ; \\ ; \\times { } _ { 2 } F _ { 1 } \\Big ( \\frac { 2 + \\mu + \\lambda } { 2 } , \\frac { 2 + \\lambda - \\mu } { 2 } ; \\frac { 3 } { 2 } ; - \\frac { b ^ 2 } { a ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\left [ \\nabla _ { ( 1 - q ) \\partial _ { t _ i } } , \\nabla _ { ( 1 - q ) \\partial _ { t _ j } } \\right ] = 0 \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} E ( \\prod _ { x , y } s _ { x , y } ^ { N ^ { ( \\frac { 1 } { 2 } ) } _ { x , y } } \\prod _ x e ^ { - \\sum _ { x } \\chi _ { x } \\hat { \\mathcal { L } } _ { \\frac { 1 } { 2 } } ^ x } ) = E ( e ^ { - \\frac { 1 } { 2 } \\sum _ { x , y } C _ { x , y } ( s _ { x , y } - 1 ) \\varphi ^ { \\mathbb { R } } _ x \\varphi ^ { \\mathbb { R } } _ y - \\frac { 1 } { 2 } \\sum \\chi _ { x } ( { \\varphi ^ { \\mathbb { R } } _ x } ) ^ 2 } ) . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} d ( x , y ) & = | I | ^ { H } d _ { | I | } ( x , y ) \\leq | I | ^ { H } \\| \\tilde { h } \\| _ { \\bar { { \\cal H } } ( [ 0 , | I | ] ) } \\\\ & = \\lim _ { m \\rightarrow \\infty } \\left ( \\left ( \\sum _ { k = 1 } ^ { m } | I _ { k } | \\right ) ^ { H } \\| \\tilde { h } ^ { ( m ) } \\| _ { \\bar { { \\cal H } } ( [ 0 , a _ { m + 1 } ] ) } \\right ) , \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} & \\inf _ { \\omega \\in O ( \\frac { n ( n - 1 ) } { 2 } ) } \\sup _ { B _ { 2 ^ { j / 2 0 } } ( x _ 2 ) } \\max _ \\alpha \\bigg | K _ \\alpha ^ { ( x _ 1 , t _ 1 ) } - \\sum _ { \\beta = 1 } ^ { \\frac { n ( n - 1 ) } { 2 } } \\omega _ { \\alpha \\beta } K _ \\beta ^ { ( x _ 2 , t _ 2 ) } \\bigg | \\\\ & \\leq C \\ , 2 ^ { - j + \\frac { j } { 1 0 } } . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} d _ n = n ! \\sum _ { k = 0 } ^ n \\frac { ( - 1 ) ^ k } { k ! } , \\ , \\ , ( n \\geq 0 ) , ( \\textnormal { s e e } \\ , \\ , [ 7 ] ) . \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\int _ I a ( 0 , t , x ) \\rho _ 3 ^ 2 | y _ { x x } | ^ 2 d x & \\leq - \\int _ I D _ 3 a ( 0 , t , x ) \\rho _ 3 ^ 2 y _ x y _ { x x } d x + \\epsilon \\int _ I \\rho _ 3 ^ 2 | y _ { x x } | ^ 2 d x \\\\ & + C _ \\epsilon \\left ( \\int _ I \\rho _ 3 ^ 2 | G | ^ 2 d x + \\int _ \\mathcal { O } \\rho _ 3 ^ 2 | f | ^ 2 d x + \\int _ I \\rho _ 3 ^ 2 | y _ t | ^ 2 d x \\right . \\\\ & + \\left . \\int _ I \\rho _ 3 ^ 2 | y | ^ 2 d x + \\int _ I \\rho _ 3 ^ 2 | y _ x | ^ 2 d x + \\sum _ { i = 1 } ^ 2 \\int _ I \\rho _ 3 ^ 2 | p ^ i | ^ 2 d x \\right ) . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\iota _ \\partial \\left ( \\omega - \\sum _ { k = 1 } ^ j c _ k \\cdot d ( \\partial ^ { k - 1 } f ) - \\sum _ i d _ i f \\cdot d e _ i \\right ) = 0 . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\norm { T h } ^ p & \\ , = \\ , \\int _ 0 ^ \\infty | h ( x + t ) | ^ p c ^ x \\ , d x \\\\ & \\ , = \\ , c ^ { - t } \\int _ 0 ^ \\infty | h ( x + t ) | ^ p c ^ { x + t } \\ , d x \\\\ & \\ , = \\ , c ^ { - t } \\int _ t ^ \\infty | h ( x ) | ^ p c ^ x \\ , d x \\\\ & \\ , = \\ , c ^ { - t } \\norm { h } ^ p . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\int _ 0 ^ t B _ i ^ 2 ( s ) d s = \\int _ 0 ^ t \\sigma ^ 2 _ i ( s ) d s \\ O ^ 2 . \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\frac { \\partial ^ + ( g ^ + _ t ) ' ( z ) } { \\partial t } = - 2 \\pi ( \\eta ^ + _ s ( g ^ + _ s ( z ) , \\xi ( s ) ) ) ' . \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} | \\nabla u | ^ 2 & = \\frac { | \\nabla f | ^ 2 } { f ^ 2 } = \\frac { 1 } { | x | ^ 2 f ^ 2 } , \\\\ | \\nabla ^ 2 u | ^ 2 & \\leq C \\left ( \\frac { | \\nabla f | ^ 4 } { f ^ 4 } + \\frac { | \\nabla ^ 2 f | ^ 2 } { f ^ 2 } \\right ) \\leq C \\left ( \\frac { 1 } { | x | ^ 4 f ^ 4 } + \\frac { 1 } { | x | ^ 4 f ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} E ( \\Sigma , Y , T _ 0 ) = - \\frac { 1 } { 8 \\pi } \\int _ \\Sigma \\left [ \\langle T _ 0 , T _ 0 \\rangle f + j ( T _ 0 ^ \\top ) \\right ] d \\Sigma \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} \\left ( k ( k + s ^ 2 - 4 s + 2 ) - s ^ 3 + 4 s ^ 2 - 5 s + 2 \\right ) n - k ( k + s ^ 2 - 4 s + 2 ) = 0 , \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} R _ w ( \\nu ' , \\nu ) & = \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\frac { L \\left ( \\chi _ { \\alpha } , \\frac { 1 } { 2 } + \\nu ' \\right ) } { L \\left ( \\chi _ { \\alpha } , \\frac { 1 } { 2 } + \\nu \\right ) } \\\\ & = W ^ * ( X ) \\left ( \\frac { \\zeta \\left ( \\frac { 3 } { 2 } + 3 \\nu ' \\right ) } { \\zeta \\left ( \\frac { 3 } { 2 } + 2 \\nu ' + \\nu \\right ) } A ( \\nu ' ; \\nu ) \\right ) + O \\left ( X ^ { 1 / 2 + \\epsilon } \\right ) , \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} f _ { u _ 2 } = \\frac { ( f _ 1 ) _ { u _ 2 } } { f _ 1 } X _ 1 + ( f _ 3 ) _ { u _ 2 } X _ 2 + \\sqrt { L } \\left [ \\frac { ( f _ 2 ) _ { u _ 2 } } { f _ 1 } - ( f _ 3 ) _ { u _ 2 } \\right ] \\widetilde { X _ 3 } . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} \\frac { R _ { 3 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 3 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 = - i ( i R _ { 3 1 } - R _ { 3 2 } ) = R _ { 3 1 } + i R _ { 3 2 } = 0 ; \\\\ \\frac { R _ { 4 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 4 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 = R _ { 4 1 } + i R _ { 4 2 } = - 1 - 1 = - 2 = - 2 u _ 4 . \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} \\lim _ { { \\varepsilon } \\to 0 } { \\varepsilon } \\log { \\mathbb { E } } \\tau ^ { \\varepsilon } _ x = V ( \\partial D ) \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} H ^ * ( p t ) = k [ t _ 1 , \\cdots , t _ n ] \\cong S ( \\mathfrak { t } ^ * ) . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} \\Psi _ { r n j } \\left ( x \\right ) = p ^ { \\frac { - r } { 2 } } \\chi _ { p } \\left ( p ^ { r - 1 } j x \\right ) \\Omega \\left ( \\left \\vert p ^ { r } x - n \\right \\vert _ { p } \\right ) , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\tilde { D } : = D ^ { ( 1 ) } - \\frac { 1 } { 6 } \\tilde { \\pi } ( T ^ { D ^ { ( 1 ) } } ) , \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} \\frac { \\tilde \\lambda _ j } { \\tilde \\lambda _ j - \\tilde \\lambda _ { j + 1 } } = 1 + \\frac { \\tilde \\lambda _ { j + 1 } } { \\tilde \\lambda _ j - \\tilde \\lambda _ { j + 1 } } \\leq 1 + 2 \\frac { \\lambda _ j } { \\lambda _ { j - 1 } - \\lambda _ { j } } \\leq 2 \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 1 } - \\lambda _ { j } } , \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} \\frac { d \\zeta } { d u } ( u ) = \\frac { g ( u , \\zeta ( u ) ) } { f ( u , \\zeta ( u ) ) } . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} W _ m = \\left [ w _ m ( x _ i - x _ j ) \\cdot \\frac { 1 } { n } \\right ] _ { i , \\ j = 0 } ^ { n - 1 } = \\left [ \\frac { w ( g _ { m } ^ { - 1 } ( x _ i - x _ j ) ) } { \\sum _ { z _ r = - 1 } ^ { 1 } w ( z _ r ) | g ' _ { m } ( z _ r ) | \\Delta z _ r } \\cdot \\frac { 1 } { n } \\right ] _ { i , \\ j = 0 } ^ { n - 1 } \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\sum _ { y < n \\leq x } a _ n h ( n ) = A ( x ) h ( x ) - A ( y ) h ( y ) - \\int _ y ^ x A ( t ) h ' ( t ) d t , \\ , \\ , \\ , \\ , \\ , \\ , A ( t ) : = \\sum _ { n \\leq t } a _ n , \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } | \\mathcal { M } ( \\Phi _ { n } ) - \\mathcal { M } ( \\Phi ) | _ { L ^ { p } ( [ 0 , T ] : E ) } = 0 . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} I V '' _ 1 & = \\frac 1 6 ( p + s - n - 3 ) ( p + s - n - 4 ) ( 4 p - 2 s - n - 5 ) \\\\ I V '' _ 2 & = \\frac 1 6 s ( s - 1 ) ( 3 n - 2 s + 7 ) . \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} \\eta _ \\Gamma + \\partial _ { \\bf n } u + \\xi _ \\Gamma + \\pi _ \\Gamma ( v ) = g _ \\Gamma \\ , , \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} g ( y , \\xi ) = f ( y , \\xi ) + \\delta | \\xi | ^ p . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\dot { \\overline { S } } ( t , x ) & = \\bigg [ \\frac { - \\dot { s } ( t ) ( 1 - | \\dot { s } ( t ) | ^ 2 ) + \\dot { s } ( t ) \\ddot { s } ( t ) ( x _ 1 - ( s ( t ) - s ( 0 ) ) ) } { ( 1 - | \\dot { s } ( t ) | ^ 2 ) ^ { 3 / 2 } } \\bigg ] { \\partial _ 1 } S ( \\Phi _ 1 ( t , x ) , x _ 2 ) \\\\ & = \\dot { \\Phi } _ 1 ( t , x ) \\sqrt { 1 - | \\dot s ( t ) | ^ 2 } { \\partial _ 1 } \\bar S ( t , x ) \\ , . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\N ( H , K ^ { ( r ) } _ { s , t } ) & = \\N ( G , K ^ { ( r ) } _ { s , t } ) - \\sum _ { i = \\ell + 1 } ^ n d _ { G _ i } ( v _ i , K ^ { ( r ) } _ { s , t } ) \\\\ & \\geq \\N ( T _ r ( n ) , K ^ { ( r ) } _ { s , t } ) - \\sum _ { i = \\ell + 1 } ^ n \\delta _ i + ( n - \\ell ) = \\N ( T _ r ( \\ell ) , K ^ { ( r ) } _ { s , t } ) + ( n - \\ell ) . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\Vert f \\Vert ' _ { \\ell ^ { p , \\infty } } : = \\sup _ { E \\subset \\mathbb { Z } ^ n , \\ , 1 \\leq \\mu ( E ) < \\infty } \\mu ( E ) ^ { \\frac { 1 } { p } - \\frac { 1 } { r } } ( \\sum _ { n ' \\in E } | f ( n ' ) | ^ r ) ^ { \\frac { 1 } { r } } \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} y _ t = x y - z ^ 2 \\leq x y = \\tfrac { 1 } { 2 } ( x ^ 2 ) _ t . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} W : = \\big \\{ w \\in L ^ p ( [ 0 , L ] ; \\R ^ 3 ) : \\big \\} . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} \\nabla ^ 2 f ( x ) = 1 2 \\ , { \\rm d i a g } ( a _ 1 x _ 1 ^ 2 , \\ldots , a _ n x _ n ^ 2 ) + 2 B \\ , , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} L L ( I ^ G _ { M _ S } ( \\rho ) ) = L L ( \\rho ) . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} F ( x + i y ) = \\sum _ { n = 1 } ^ \\infty \\frac { c _ f ( n ) } { \\sqrt { n } } \\bigl ( V _ f ^ + ( n y ) \\cos ( 2 \\pi n x ) + i V _ f ^ - ( n y ) \\sin ( 2 \\pi n x ) \\bigr ) . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\mathbf I ( M ) \\hat \\oplus \\mathbf I ( N ) = \\mathbf I ( M + N ) . \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\mathfrak Q _ { \\star , r } ( x _ 0 ) = \\mathfrak S _ { \\star , r } ( x _ 0 ) \\cup \\S ^ 2 ( x _ 0 , r ) \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\rho _ { t } ( u ) = t ^ { - H \\nu } \\rho _ { 1 } ( \\delta ^ { - 1 } _ { t ^ { H } } u ) , \\ \\ \\ \\ ( u , t ) \\in \\mathfrak { g } ^ { ( l ) } \\times ( 0 , 1 ) , \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} | B | = \\frac { q ^ { r ( 2 r + 1 ) } ( q ^ 2 - 1 ) ^ r } { ( 2 r + 1 , q + 1 ) } . \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} H ^ 0 ( \\mathring { C } _ I ^ { p } \\otimes _ { \\C [ I ] } D _ I ^ \\bullet ) = \\mathring { C } _ I ^ { p } \\otimes _ { \\C [ I ] } \\C . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} B ^ * = \\left ( \\lambda - \\frac { 1 } { \\sum _ { i \\in \\mathcal { S } } H _ i } \\right ) ^ + , \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\widetilde { C } _ 1 M _ I = \\widetilde { C } _ 2 M _ I = \\widetilde { C } _ 3 M _ I = \\cdot e _ { 1 2 } e _ { 2 3 } e _ { 3 1 } \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } N _ p ^ k ( E [ \\ell ] ) = \\sum _ { \\substack { p \\leq x \\\\ p \\textrm { s p l i t s i n } K } } N _ p ^ k ( E [ \\ell ] ) + \\sum _ { \\substack { p \\leq x \\\\ p \\textrm { i s i n e r t o r r a m i f i e s i n } K } } N _ p ^ k ( E [ \\ell ] ) . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\nu _ { \\beta , h } ^ { \\pm } ( \\{ \\omega \\colon \\omega _ { \\Lambda } = \\eta _ { \\Lambda } \\} ) = \\nu _ { \\beta , h } ^ { \\pm } ( \\{ \\omega \\colon \\omega _ { T ^ { - 1 } \\Lambda } = \\eta _ { T ^ { - 1 } \\Lambda } \\} ) \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} u _ t = \\Delta u + f ( u ) , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\| \\psi _ n - \\psi _ n ^ N \\| _ { L ^ 2 ( \\Omega ) } \\lesssim \\frac { 1 } { \\sqrt { \\lambda _ n ^ { N } } } N ^ { - 1 / 2 } \\quad n = 1 , \\cdots , M . \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} ( i + 1 ) g = ( x ) ( \\sigma _ 1 \\sigma _ 2 ) ^ i , \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\left ( z Q \\partial _ Q \\right ) ^ { N + 1 } J ^ \\textnormal { c o h } ( z , Q ) = Q J ^ \\textnormal { c o h } ( z , Q ) \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\hat \\varepsilon ( T ) = \\sigma \\sqrt { 2 \\ , \\tau _ f / T } , \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\partial _ Q \\widetilde { f } ( Q ) = \\frac { 1 } { ( Q - 1 ) ( Q - i ) ( Q + 1 ) } \\widetilde { f } ( Q ) \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} R _ { E } ^ { \\nabla } ( x , \\xi ) = - a d _ { x } ^ * S ( \\xi ) + S ( \\xi ) a d _ { x } ^ * + S ( a d _ { x } ^ * ( \\xi ) ) + a d _ { a d _ { \\xi } ^ * ( x ) } ^ * . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} \\Pi \\sigma ^ { j } \\nu ( B ( x , \\eta ) ) = \\sum _ { u \\in \\Lambda ^ { j } } \\Pi \\sigma ^ { j } ( \\nu | _ { [ u ] } ) ( B ( x , \\eta ) ) \\le | \\Lambda | ^ { m } \\eta ^ { ( 1 - \\delta ) \\alpha ' } < \\eta ^ { ( 1 - \\delta ) \\alpha } , \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} 4 \\sum _ { j , k = 1 } ^ \\infty \\int _ 0 ^ \\infty & m ^ s \\int _ { | x | > R } \\partial ^ 2 _ { j k } \\varphi _ R \\partial _ j \\overline { u } _ m ( t ) \\partial _ k u _ m ( t ) d x d m \\\\ & = 8 s \\| u ( t ) \\| ^ 2 _ { \\dot { H } ^ s ( | x | > R ) } - 4 \\int _ 0 ^ \\infty m ^ s \\int _ { | x | > R } ( 2 - \\varphi '' _ R ) | \\nabla u _ m ( t ) | ^ 2 d x d m \\\\ & \\leq 8 s \\| u ( t ) \\| ^ 2 _ { \\dot { H } ^ s ( | x | > R ) } . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} ( s , ( \\psi ) ) ( s \\ , ' , ( \\psi ' ) ) = ( s + s \\ , ' , ( \\psi ) ( \\psi ' ) ) . \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\begin{cases} w '' + A w + w ' = - V _ { m } '' = - A ^ 2 V _ { m , 2 } , t > 0 , \\\\ ( w , w ' ) ( 0 ) = ( ( - 1 ) ^ { m + 1 } u _ 1 , - A U _ { m + 1 } ( 0 ) - V _ { m } ' ( 0 ) ) = ( ( - 1 ) ^ { m + 1 } u _ 1 , A V _ { m , 1 } ( 0 ) ) \\end{cases} \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} p ( x ) : = ( 3 \\pi ^ 2 \\rho ( x ) ) ^ { 1 / 3 } , \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} ( 2 a - c - a x + b x ) { } _ { 2 } F _ { 1 } & ( a , b ; c ; x ) + ( c - a ) { } _ { 2 } F _ { 1 } ( a - 1 , b ; c ; x ) \\\\ & + a ( x - 1 ) { } _ { 2 } F _ { 1 } ( a + 1 , b ; c ; x ) = 0 . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\mathcal { T } _ * Y _ \\alpha ^ L & = 0 \\\\ \\mathcal { T } _ * Y _ \\alpha ^ R & = Y _ \\alpha ^ R - Y _ \\alpha ^ L \\\\ \\mathcal { T } _ * E _ { \\alpha } ^ R & = E _ { \\alpha } ^ R + E _ { - \\alpha } ^ L \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} u _ { \\overline S } = - ( M _ { \\overline S \\overline S } ( \\lambda _ 0 ) - \\lambda _ 0 I ) ^ { - 1 } M _ { \\overline S S } ( \\lambda _ 0 ) u _ S . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\left | \\frac { d \\varphi _ { k } } { d u } \\right | \\leq \\frac { C _ { H , l _ { 0 } } } { ( \\sum _ { j = 1 } ^ k | I _ { j } | ) ^ { H - \\frac { 1 } { 2 } } } . \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\mathcal { S ' } _ \\varepsilon ( \\sigma ) v = \\int _ D s _ \\varepsilon ( \\sigma ) v , \\left [ \\mathcal { S '' } _ \\varepsilon ( \\sigma ) v \\right ] = s ' _ \\varepsilon ( \\sigma ( x ) ) v ( x ) . \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\frac { \\log | \\det A | } { n } = \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\log | f ( e ^ { 2 \\pi i j / n } ) | \\ , . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} ( g \\circ f ) ^ { * } H & = { \\pi _ { X } } _ { * } \\widetilde { f } ^ { * } \\widetilde { g } ^ { * } H \\\\ f ^ { * } ( g ^ { * } H ) & = { \\pi _ { X } } _ { * } \\widetilde { f } ^ { * } \\pi _ { Y } ^ { * } { \\pi _ { Y } } _ { * } \\widetilde { g } ^ { * } H . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 \\\\ & ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) \\bar h ^ { 1 ^ * } _ { 1 1 k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} ( n , k , d ) = ( 1 4 , 6 , 7 ) , ( 1 5 , 7 , 7 ) , ( 1 7 , 6 , 9 ) , ( 1 7 , 7 , 8 ) , ( 1 9 , 7 , 9 ) , ( 2 0 , 7 , 1 0 ) . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} k _ { \\gamma } ^ { \\infty } = \\frac { \\sqrt { \\frac { 1 } { 2 } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) ^ 2 + \\dot { \\gamma } _ 3 ^ 2 } } { | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} G L R & : = \\{ M \\in \\mathbb { Q } [ G ] \\mid M [ \\alpha ] \\in \\mathbb { Q } \\} . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\alpha / 2 } u ( x ) + \\rho u ( x ) = f ( x ) , & x \\in \\mathbb { R } , \\\\ u ( x ) = 0 , & | x | \\to \\infty , \\end{cases} \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} 0 , & \\mbox { s e } x \\leq 0 , \\\\ \\int _ { 0 } ^ { x } \\frac { 1 } { t ^ { \\alpha } ( 1 + t ) } d t , & \\mbox { s e } x > 0 . \\end{cases} \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} \\lambda ^ { 2 } - \\left \\{ \\left ( 1 + d \\right ) \\varepsilon \\kappa + \\mathit { T r } \\mathbb { J } \\right \\} \\lambda + h \\left ( \\kappa \\right ) = 0 , \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} Y ^ { 0 , u } _ x ( t ) = \\int _ 0 ^ t S ( t - s ) G ( s , \\mathcal { M } ( S ( \\cdot ) x + Y ^ { 0 , u } _ x ) ( s ) ) u ( s ) d s , \\ ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\| y _ v \\| _ 1 = \\tau ( y _ v ) = \\frac { 2 ^ { v + 1 } } { A _ j } \\int _ { 2 ^ v } ^ { 2 ^ { v + 1 } } \\left \\| \\frac { \\partial T _ { s , j } ( x _ { v + j } ) } { \\partial s } \\right \\| _ 2 ^ 2 d s + 2 \\sum _ { k = 0 } ^ { A _ j - 1 } \\| T _ { { \\gamma _ k } , j } ( x _ { v + j } ) \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\sigma _ { \\Delta , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) - \\sigma _ { \\Delta , \\mathbf { G } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) = \\mathbf { c } ^ { T } v a r _ { P } \\left ( \\mathbf { Q } \\right ) \\mathbf { c \\geq } 0 \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\left \\langle u ( s ) , d w ( s ) \\right \\rangle _ { H } \\doteq \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { T } \\left \\langle u ( s ) , e _ { k } \\right \\rangle _ { H } d \\beta _ { k } ( s ) . \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\mathcal U : = & \\Big \\{ ( u , w ) \\in L ^ 2 ( \\Omega ; L ^ 2 ( 0 , T ; H ) ) ^ 2 0 \\leq u , w \\leq 1 \\Omega \\times ( 0 , T ) \\times D \\Big \\} \\ , , \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} n + \\alpha ( n - f ) = 0 . \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } } { \\lambda _ { \\textrm { B F } } } \\left ( \\frac { ( 1 - \\theta ) } { ( 1 - \\rho ) } \\left ( 2 ^ { \\frac { R } { 1 - \\theta } } - 1 \\right ) - \\frac { \\theta } { \\rho } \\left ( 2 ^ { \\frac { R } { \\theta } } - 1 \\right ) \\right ) \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} L _ { W } g ^ T ( Y , Z ) = \\N ^ T W ^ { \\flat } ( Y , Z ) + \\N ^ T W ^ { \\flat } ( Z , Y ) . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ \\infty \\ln | S ( i \\omega ) | w ( z , \\omega ) d \\omega = & \\pi \\ln \\prod _ k \\left | \\frac { p _ k + z } { p _ k - z } \\right | - \\pi \\ln | 1 + L _ b ( z ) | \\end{aligned} \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathbf { L } ^ { p , q } ( \\Omega ) } = \\left \\{ \\begin{aligned} & \\left ( \\int _ 0 ^ \\infty \\left ( t ^ { \\frac { 1 } { p } } f ^ { * * } ( t ) \\right ) ^ q \\frac { d t } { t } \\right ) ^ { \\frac { 1 } { q } } , & 1 & \\leq p < \\infty , 1 \\leq q < \\infty , \\\\ & \\sup _ { t > 0 } t ^ { \\frac { 1 } { p } } f ^ { * * } ( t ) , & 1 & \\leq p < \\infty , q = \\infty , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\psi _ 2 ' ( b ) & = \\xi '' ( b ) q - b \\xi '' ( b ) + \\frac { \\xi '' ( b ) ( 1 - q ) } { z _ 2 } + \\xi '' ( b ) ( 1 - q ) - \\frac { \\xi '' ( b ) ( 1 - q ) ( 1 + z _ 2 ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { z _ 2 [ \\xi ' ( 1 ) - \\xi ' ( q ) + ( \\xi ' ( b ) - \\xi ' ( q ) ) z _ 2 ] } \\\\ & = \\frac { \\xi '' ( b ) [ ( q - b ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] + [ \\xi ' ( b ) - \\xi ' ( q ) ] ( 1 - q + ( 1 - b ) z _ 2 ) ] } { \\xi ' ( 1 ) - \\xi ' ( q ) + [ \\xi ' ( b ) - \\xi ' ( q ) ] z _ 2 } . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} K ( n ) = \\frac { 1 } { n } \\sum _ { d | n } 2 ^ { n / d } \\phi ( d ) \\simeq { 2 ^ n } / { n } , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\eta _ { v , e } \\ = \\ \\begin{cases} 1 & \\\\ - 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\chi _ G ( n ) \\ = \\ \\chi ^ * _ 0 \\binom { n + d } d + \\chi ^ * _ 1 \\binom { n + d - 1 } d + \\dots + \\chi ^ * _ d \\binom n d \\ , . \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} T ^ W ( x ) : = \\inf _ { u \\in S } \\{ T ( u ) + K \\ , d ( x , u ) \\} , x \\in M , \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} [ X , Y ] = 2 Z , [ Y , Z ] = 2 X , [ Z , X ] = 2 Y . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\| { \\bf x } \\| ^ { \\star } = \\underset { \\| { \\bf s } \\| \\leq 1 } { \\max } { \\bf s } ^ T { \\bf x } \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\| b ^ { ( \\theta ) } \\| _ { H ^ p } = \\Big | \\bigcup \\mathcal { B } \\Big | ^ { 1 / p } , \\| b ^ { ( \\theta ) } \\| _ { ( H ^ p ) ^ * } = \\Big | \\bigcup \\mathcal { B } \\Big | ^ { 1 / { p ' } } \\qquad \\| b ^ { ( \\theta ) } \\| _ { S L ^ \\infty } = 1 . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} L ^ { 2 } ( \\mathcal { K } _ { N } , \\mathbb { C } ) = X _ { N } \\otimes \\mathbb { C } \\mathbb { \\oplus } \\mathcal { L } _ { 0 } ^ { 2 } ( \\mathcal { K } _ { N } , \\mathbb { C } ) , \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x } & & \\max _ { i , j = 1 } ^ { n } \\max _ { k = 1 } ^ { m } ( w _ { k } a _ { i j } ^ { ( k ) } ) x _ { j } / x _ { i } . \\end{aligned} \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} [ C o n f _ k ( B ) \\to M ^ k ] = \\sum _ { P \\in X _ k } a ( P ) [ B _ P \\to M ^ k ] \\ , , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\norm { ( T _ t f ) \\chi _ { { } _ { [ 0 , a ] } } } ^ p & = \\int _ 0 ^ a | f ( x + t ) | ^ p \\ , v ( x ) \\ , d x \\\\ & = \\int _ 0 ^ a | f ( x + t ) | ^ p \\ , v ( x + t ) \\frac { v ( x ) } { v ( x + t ) } \\ , d x \\\\ & \\leq \\frac { B _ a } { c } \\int _ 0 ^ a | f ( x + t ) | ^ p \\ , v ( x + t ) \\ , d x \\\\ & \\leq \\frac { B _ a } { c } \\int _ t ^ { t + a } | f ( x ) | ^ p \\ , v ( x ) \\ , d x \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\nabla u _ 0 + \\nabla _ y u _ 1 = \\nabla \\overline { u } _ 0 + \\nabla _ y \\overline { u } _ 1 . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} P _ { \\vert \\boldsymbol { Y } \\vert = n } \\left ( \\boldsymbol { Y } = \\boldsymbol { y } \\right ) = \\binom { n } { \\boldsymbol { y } } \\prod _ { j = 1 } ^ J \\left ( \\frac { \\theta _ j } { \\vert \\boldsymbol { \\theta } \\vert } \\right ) ^ { y _ j } \\cdot \\mathbb { 1 } _ { \\Delta _ { n } } ( \\boldsymbol { y } ) , \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} \\i Q _ t - Q _ { x x } \\sigma _ 3 + 2 Q ^ 3 \\sigma _ 3 = 0 . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\mathrm { i n } ( v ) = \\sum _ { w \\in V } \\ , a _ { w , v } \\mathrm { o u t } ( v ) = \\sum _ { w \\in V } \\ , a _ { v , w } \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} ( h _ 0 ^ { ( 1 ) } - h ^ { ( 1 ) } ) = & \\rho \\\\ ( h _ 0 ^ { ( 2 ) } - h ^ { ( 2 ) } ) = & D \\rho \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\nu ^ p _ \\pm : = \\frac { 1 } { 4 } + \\mu b _ { \\alpha , \\beta } ( \\theta ^ \\beta _ \\pm + \\xi _ \\beta ) \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s \\ , \\hat \\varepsilon _ { s , N } ^ 2 = \\hat Q _ { N } , \\textrm { f o r f i n i t e } \\hat Q _ N . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} W ( s ) : = W _ { 1 } ( s ) + W _ { 2 } ( s ) + W _ { 3 } ( s ) + W _ { 4 } ( s ) , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} 0 < \\delta < \\min \\Big \\{ \\frac { p ^ + _ l - p ^ - _ l } { \\max \\{ \\varphi _ l ^ + , \\varphi ^ - _ l \\} } \\mid l = 1 , 2 , \\cdots , m \\Big \\} \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\mathbb { E } [ Y _ i ^ 2 ] = \\frac { m p ( 1 + p ) } { ( 1 - p ) ( 1 - p ^ 2 ) } + \\frac { m ^ 2 p ^ 2 } { ( 1 - p ) ^ 2 } \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } P _ 0 ( h , \\mu ) u ( t , y ) = 0 & \\mbox { i n } & ( 0 , 1 ) \\times Y , \\\\ u ( 0 , y ) = \\phi ( { \\cal D } _ { y _ 1 } ) f & \\mbox { o n } & Y , \\end{array} \\right . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\begin{aligned} & \\dfrac 1 2 \\lim _ { m \\to \\infty } \\mathcal L \\sum _ { i , j , k , p } ( h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 ( p _ m ) \\\\ & = \\bar H ^ 2 \\biggl [ - \\dfrac { \\bar H ^ 4 } 2 + \\dfrac { 3 \\bar H ^ 2 } 2 - S + \\dfrac { ( \\bar H ^ 2 - 1 ) \\sqrt { ( 4 S - 3 \\bar H ^ 2 ) \\bar H ^ 2 } } 2 \\biggl ] . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} - \\left ( \\int _ { \\Sigma } \\overline { q } ^ 2 d \\sigma _ \\Sigma \\right ) L + \\left ( - \\int _ { \\Sigma } \\overline { q } ^ 2 d \\overline { \\sigma _ \\Sigma } + \\int _ { \\Sigma } A d \\sigma _ \\Sigma + \\sum _ { i = 1 } ^ n \\int _ { \\gamma _ i } k ^ { \\infty , s } _ { \\gamma _ i , \\Sigma } d \\overline { s } \\right ) + O ( L ^ { - \\frac { 1 } { 2 } } ) = 2 \\pi \\frac { \\chi ( \\Sigma ) } { \\sqrt { L } } . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} c = \\frac { { v o l } ( { \\mathfrak { D } } ( f , \\sigma ) ) } { P r ( f , \\sigma ) } \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} \\varphi ( s ) = c _ { 0 } s + \\sum _ { n = 1 } ^ \\infty c _ n n ^ { - s } = : c _ { 0 } s + \\psi ( s ) , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} d _ 4 ( n , n - 3 ) = \\begin{cases} 3 & 4 \\le n \\le 1 8 , \\\\ 2 & n \\ge 1 9 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} \\psi _ n = ( \\eta _ 1 , 2 a _ 1 + 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ k , 2 a _ k + 1 ) \\boxplus ( \\delta _ 1 , 2 b _ 1 ) \\boxplus \\cdots \\boxplus ( \\delta _ l , 2 b _ l ) \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell j ( 1 / c - 1 ) \\log \\beta } { d \\sum _ { j = 1 } ^ { \\ell + 1 } j / c \\log \\beta - ( \\ell + 1 ) ( 1 / c - 1 ) \\log \\beta } = \\frac { 1 - c } d . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} c _ j \\xi ^ j & = \\ \\ \\ \\ \\ b _ j \\xi ^ { j - 1 } + a _ j \\eta ^ { j - 1 } \\ j = 1 , \\dots , \\nu \\\\ c _ j \\eta ^ j & \\ge | - a _ j \\xi ^ { j - 1 } + b _ j \\eta ^ { j - 1 } | j = 1 , \\dots , \\nu \\\\ b _ \\nu \\eta ^ \\nu & \\le a _ \\nu \\xi ^ \\nu , \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} 2 \\sqrt { K } = \\frac { 2 } { r } + k ^ { ( 1 ) } r + k ^ { ( 2 ) } r ^ 2 + k ^ { ( 3 ) } r ^ 3 + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } \\| u _ p \\| _ { \\infty } = \\sqrt { e } . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} a ( w ) : = \\lim _ { x \\to \\infty } \\frac { f _ w ( x ) } { x ^ { n _ i / p ^ { m _ i } } } \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} A ( a ) : = \\sum _ { j = 0 } ^ { n - 1 } a _ j J ^ j , \\ B ( a ) : = \\sum _ { j = 1 } ^ { n - 1 } a _ j \\left ( J ^ T \\right ) ^ { n - j } . \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} \\frac { \\eta _ 1 } { N } \\leq p _ l : = \\frac { \\int _ { S _ l } f ( x ) d x } { \\int _ { \\cup _ j S _ j } f ( x ) d x } \\leq \\frac { \\eta _ 2 } { N } , \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} v ( t w ) = \\gamma \\ , t + o ( t ) . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} d _ { \\infty } ( A _ t , B _ { t ^ \\prime } ) : = | t - t ^ \\prime | + \\| A _ { t , t ^ \\prime - t } - B _ { t ^ \\prime } \\| _ { \\infty } , \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\frac { 1 } { 1 2 0 } \\sum _ { i = 1 } ^ 1 f _ { i i i i } + \\frac { 1 } { 3 6 } \\ , \\sum _ { \\substack { i , j = 1 \\\\ j > i } } ^ 1 f _ { i i j j } = \\frac { 1 } { 1 2 0 } f ^ { ( 4 ) } ( x ) = \\frac { a _ 4 } { 5 } \\ , , \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u _ \\varepsilon | ^ 2 d x = 4 \\pi \\left ( 1 + I _ { x _ \\varepsilon } ( \\gamma _ \\varepsilon ) + o \\left ( \\check { \\zeta } _ \\varepsilon \\right ) \\right ) \\ , , \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} A _ n ( t ) = a _ n + S _ n ( t ) , { t \\le T _ n } . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} - \\phi ' ( s ) & = \\int _ { 1 } ^ { \\infty } F _ 1 ( y ) s e ^ { - s y } d y \\\\ & = \\int _ { s } ^ { \\infty } F ( x / s ) e ^ { - x } d x \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} a _ 1 = 0 & \\Rightarrow b _ 1 = 0 , \\ , \\ , c _ 1 = * , \\\\ a _ 2 = * & \\Rightarrow b _ 2 = * , \\ , \\ , c _ 2 = * . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} I _ 1 ( t ) + I _ 2 ( t ) & \\leq \\frac { 2 m ^ 2 } { \\gamma } \\Big [ \\left ( 3 \\lambda - \\frac { \\gamma } { m } \\right ) ^ 2 C _ 2 ^ 2 + 4 \\lambda ^ 4 C _ 1 ^ 2 \\Big ] e ^ { - 2 \\lambda t } = C _ 3 e ^ { - 2 \\lambda t } , \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} d x = \\frac { 1 } { \\det \\left ( \\frac { \\partial \\overline { F } } { \\partial x } \\right ) } d y \\wedge d z = \\frac { 1 } { \\det \\left ( \\frac { \\partial y } { \\partial x _ { 1 } } \\right ) } d y \\wedge d z . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} ( \\rho _ \\varepsilon ' ) ^ 2 = \\exp ( - \\varepsilon ' _ 0 ( 1 + o ( 1 ) ) \\gamma _ \\varepsilon ^ 2 ) \\ , . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\left \\{ x \\ : \\ \\| x \\| _ 2 \\le \\rho _ 1 , \\ \\| x \\| _ \\infty \\le \\rho _ 2 \\right \\} = \\left \\{ x \\ : \\ x ^ T x \\le \\rho _ 1 ^ 2 , \\ x _ j ^ 2 \\le \\rho _ 2 ^ 2 \\ \\forall \\ j \\right \\} \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s W - | W | ^ { \\frac { 4 s } { d - 2 s } } W = 0 , \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 & = - S ( 1 - \\dfrac 1 2 S ) + ( S - \\sup H ^ 2 ) ^ 2 - \\dfrac 1 2 ( \\sup H ^ 2 ) ^ 2 + \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 ) \\\\ & = ( \\frac 5 6 S - 1 ) S . \\end{aligned} \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t ^ \\beta u - \\Delta u & = f , & & ( x , t ) \\in \\Omega \\times [ 0 , T ] , \\\\ u ^ { ( k ) } ( x , 0 ) & = 0 , & & x \\in \\Omega , \\ ; k = 0 , \\dots , m - 1 , \\end{aligned} \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell ( 1 / c - 1 ) \\gamma ^ j } { d \\sum _ { j = 1 } ^ { \\ell + 1 } 1 / c \\gamma ^ j - ( 1 / c - 1 ) \\gamma ^ { \\ell + 1 } } = \\frac { 1 - c } { d \\gamma - ( 1 - c ) ( \\gamma - 1 ) } . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\phi _ 1 ^ { N } : = \\sum \\limits _ { i = 1 } ^ { \\infty } c _ i \\phi _ i + v \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\nabla _ x H ( x _ i , x _ i ) + \\sum _ { i \\neq \\ell } \\nabla _ x G ( x _ i , x _ \\ell ) = 0 . \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{gather*} \\{ \\hat \\theta _ N , \\hat \\sigma _ N , \\hat \\mu _ N \\} = \\textrm { a r g m i n } \\ , f ( \\hat \\theta , \\hat \\sigma , \\hat \\mu ) = \\sum _ { s = 1 } ^ { q _ N } \\big [ g _ { s } ( \\hat \\theta , \\hat \\sigma , \\hat \\mu ) \\big ] ^ 2 \\textrm { w h e r e } \\\\ g _ { s } ( \\hat \\theta , \\hat \\sigma , \\hat \\mu ) = \\hat \\mu ^ 2 + \\frac { \\hat \\sigma ^ 2 } { s } \\Big [ 1 + 2 \\ , \\sum \\limits _ { k = 1 } ^ { s - 1 } \\big ( 1 - \\frac { k } { s } \\big ) \\ , \\hat \\rho ( k ; \\hat \\theta ) \\Big ] - \\bar { m } _ s ^ 2 , \\end{gather*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\widetilde { m } ^ 2 } + \\frac { 1 } { j ^ 2 } V = \\frac { 1 } { j ^ 2 } \\ln \\widetilde { m } + \\frac { 1 } { 2 } + \\frac { 1 } { j ^ 2 } \\widetilde { H } _ 1 . \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\Lambda = \\left \\lbrace b _ { 1 } , . . , b _ { N } \\right \\rbrace \\subset G \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\mathcal { I } ( \\omega ) = 8 \\pi ^ { 2 } \\int _ { M } \\abs { \\omega } ^ { 2 } \\phi _ { 0 } ^ { 2 } + 8 \\pi \\int _ { M } \\phi _ { 0 } \\omega \\cdot \\nabla g _ \\omega \\ , , \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } \\frac { e ^ h } { h ^ 4 } = \\kappa < \\infty . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} A _ + & = A _ 1 \\oplus A _ 0 \\oplus A _ \\lambda \\oplus A _ { \\lambda - \\frac { 1 } { 2 } } \\oplus \\bigoplus _ { m \\in w t ( D _ + ) } A _ { \\nu ^ { ( m , | \\alpha | - m ) } _ \\pm } \\\\ A _ - & = \\bigoplus _ { m \\in w t ( D ) - w t ( D _ + ) } A _ { \\nu ^ { ( m , | \\alpha | - m ) } _ \\pm } \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} \\left ( \\mu _ 2 ^ \\star \\right ) _ { | \\left [ 0 , 1 \\right ] } = M _ 2 \\mathbf { 1 } _ { \\left [ r _ 1 + r _ 0 , r _ 1 + r _ 0 + 2 r _ 2 \\right ] } \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} T _ 1 = ( I - Q ) T T _ 2 = Q T . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\mathcal { G } : = \\bigg \\{ \\bigg ( \\Big ( f \\widehat { w } ^ { - \\frac 1 { r _ { m + 1 } ' } } \\Big ) ^ { r _ m } , \\Big ( \\Big ( \\prod _ { i = 1 } ^ { m - 1 } \\| f _ i \\| _ { L ^ { p _ i } ( w _ i ) } \\Big ) f _ m \\widehat { w } ^ { - \\frac 1 { r _ m } } \\Big ) ^ { r _ m } \\bigg ) : ( f , f _ 1 , \\dots , f _ m ) \\in \\mathcal { F } \\bigg \\} , \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\frac { 1 } { \\beta - \\alpha } , & \\ , \\ , \\alpha < x < \\beta , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} \\varphi : = \\exists u , v , s , t , b \\in \\N [ & x ^ 2 - ( a ^ 2 - 1 ) y ^ 2 = 1 \\ \\wedge \\\\ & u ^ 2 - ( a ^ 2 - 1 ) v ^ 2 = 1 \\ \\wedge \\\\ & s ^ 2 - ( b ^ 2 - 1 ) t ^ 2 = 1 \\ \\wedge \\\\ & b > 1 \\ \\wedge \\ b \\equiv 1 \\pmod { 4 y } \\ \\wedge \\ b \\equiv a \\pmod u \\ \\wedge \\\\ & v > 0 \\ \\wedge \\ y ^ 2 \\mid v \\ \\wedge \\\\ & s \\equiv x \\pmod u \\ \\wedge \\ t \\equiv k \\pmod { 4 y } ] . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( \\mu ^ t ) = \\rho _ { \\mu ^ t } = \\psi _ { \\mu ^ t _ \\# } \\circ \\rho _ 0 \\circ \\psi _ { \\mu ^ t _ \\# } ^ { - 1 } . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\mathrm { m o d } _ q ( \\varphi ) = \\lim _ { \\tau \\to \\infty } \\left ( \\mathrm { F i x } ^ + ( \\rho _ \\tau ( d \\varphi _ * \\gamma ) ) - \\mathrm { F i x } ^ + ( \\rho _ \\tau ( \\gamma ) ) \\right ) . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\mathcal L = \\bigcup _ { m , n } \\bigcup _ { u \\in I \\cup Q ^ { n } \\atop v \\in J \\cup Q ^ { m } } \\left \\{ \\mathfrak g \\in \\mathcal F ( \\mathcal X ) \\left | \\begin{array} { c } \\mathrm { O U T } _ { u } \\\\ \\updownarrow \\\\ \\mu ( \\mathfrak g ) \\\\ \\updownarrow \\\\ \\mathrm { I N } _ { v } \\end{array} = 1 \\right . \\right \\} . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} A ^ * A = \\left [ \\begin{array} { c c } { A ^ \\prime } ^ * A ^ \\prime + R ^ * R & 0 \\\\ 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\Vert t _ m f \\Vert _ { \\ell ^ { p , \\infty } } : = \\sup _ { \\alpha > 0 } \\alpha \\cdot \\mu \\{ n : | t _ m f ( n ) | > \\alpha \\} ^ { \\frac { 1 } { p } } \\leq C \\Vert f \\Vert _ { \\ell ^ 1 } \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} A \\Omega _ A ^ a = L \\eta _ \\mu R \\Gamma ( \\mathrm { S p f } ( A ) _ C , \\mathbb A _ { \\inf } ) ^ a \\to \\varprojlim ( L \\eta _ \\xi \\varphi _ \\ast ) ^ { \\circ r } R \\Gamma ( \\mathrm { S p f } ( A ) _ C , \\mathbb A _ { \\inf } ) ^ a \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n ^ 2 = L ^ 2 . \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} X = \\| \\mathbb { P } _ { \\mathcal { M } } - \\mathbb { P } _ { S _ n } \\| _ { 1 } . \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\Gamma ( t ) : = e ^ { \\sum _ { i = 1 } ^ N \\left [ \\int _ 0 ^ t B _ i ( s ) d \\beta _ i + \\theta _ i \\beta _ i - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s - \\frac { 1 } { 2 } \\theta _ i ^ 2 t - \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s \\right ] } \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} C _ { k , j } \\cup C _ { k + 1 , j } \\subset B \\P ( ( C _ { k , j } \\cup C _ { k + 1 , j } ) \\cap B ) = 0 . \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\mathbf C _ s = \\frac { 1 } { \\ln 2 } \\Big ( p \\ , \\Lambda ( D ) + \\left ( 1 - p \\right ) \\beta \\Lambda ( \\emptyset ) \\Big ) , \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\nabla _ { \\partial t _ i } s ^ \\textnormal { c o h } ( \\tau _ 2 , z ) = 0 \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} { \\epsilon ^ * } \\buildrel \\Delta \\over = \\min \\left \\{ { { \\epsilon _ 1 } , { \\epsilon _ 2 } } \\right \\} = \\frac { 1 } { 2 } \\min \\left \\{ { q \\left ( \\theta \\right ) , 1 - q \\left ( \\theta \\right ) } \\right \\} > 0 . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\chi ( y ) = \\left \\{ \\begin{array} { l l } 1 , & \\mbox { f o r } | y | \\le 1 / 2 , \\\\ 0 , & \\mbox { f o r } | y | \\ge 1 , \\\\ \\mbox { b e l o n g s t o } [ 0 , 1 ] , & \\mbox { i f o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { 1 } { ( 1 + 2 \\lambda \\beta _ i ^ 2 ) ^ 4 } \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} \\mathbb { P } _ 0 ( N ^ { ( P ) } _ l = k ) = P o i ( k ; n p _ l ) : = e ^ { - n p _ l } \\frac { ( n p _ l ) ^ { k } } { k ! } , \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} \\left ( u , v \\right ) \\in B \\quad u , \\ ; v \\in \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) v = f \\left ( x , u \\right ) _ { x } , \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} P ( N ) & = \\{ D \\in \\mathbb { F } _ q [ t ] : D | D | < N \\} \\\\ g ( N ) & = \\{ D \\in P ( N ) : L ( 1 / 2 , \\chi _ D ) = 0 \\} . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\begin{aligned} \\L \\Psi ( X ) & = - \\frac { \\gamma } { m } v ^ 2 - \\sum _ { k \\geq 1 } \\lambda _ k k ^ { - 2 s } z _ k ^ 2 \\\\ & - \\frac { 1 } { m } \\sum _ { k \\geq 1 } \\sqrt { c _ k } z _ k v + \\sum _ { k \\geq 1 } \\sqrt { c _ k } k ^ { - 2 s } z _ k v + \\frac { \\gamma } { m ^ 2 } + \\sum _ { k \\geq 1 } k ^ { - 2 s } \\lambda _ k . \\end{aligned} \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} V _ + ( t , y , k ) : = { \\frak W } ( t ) { V } _ { 0 , + } ( y , k ) - \\int _ 0 ^ t { \\frak W } ( s ) F ( y , k ) d s . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ X | u _ j - u | ( \\theta _ u ^ n + \\theta _ { u _ j } ^ n ) = \\lim _ { j \\to + \\infty } \\int _ X ( u _ j - u ) ( \\theta _ u ^ n + \\theta _ { u _ j } ^ n ) = 0 . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} U : = e ^ { \\sum _ { i = 1 } ^ N \\left [ \\int _ 0 ^ t B _ i ( s ) d \\beta _ i + \\theta _ i \\beta _ i - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s - \\frac { 1 } { 2 } \\theta _ i ^ 2 t - \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s \\right ] } y . \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} \\forall x , y , z \\in H : \\alpha ( x _ 1 , y _ 1 ) \\alpha ( x _ 2 y _ 2 , z ) = \\alpha ( y _ 1 , z _ 1 ) \\alpha ( x , y _ 2 z _ 2 ) . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} u _ \\varepsilon = ( 1 + o ( 1 ) ) \\frac { 4 \\pi G _ { x _ \\varepsilon } + o ( 1 ) } { \\gamma _ \\varepsilon } \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\delta < \\min \\Big \\{ \\frac { \\lambda _ + \\varphi ^ + _ l ( x ) } { a _ + ( x ) } , \\frac { \\lambda _ - \\varphi ^ - _ l ( x ) } { a _ - ( x ) } \\mid x \\in \\R ^ N , \\ l = 1 , 2 , \\cdots , m \\Big \\} , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} q _ N ^ { - 1 } ( \\tilde { \\nabla } ^ 2 u _ { 1 , i } ) ( x , t ) = [ M _ { 1 , i } + o ( 1 ) ] t ^ { - \\frac { N + 2 A _ 1 } { 2 } } \\tilde { \\nabla } ^ 2 [ U _ 1 ( | x | ) \\theta _ i ] + O ( t ^ { - \\frac { N + 2 A _ 1 } { 2 } - 1 } ) \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\omega ( q ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ^ 2 } , \\nu ( q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( - q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} a _ { d } - a _ { d - 1 } = \\frac { 1 } { \\left ( 1 - q ^ d \\right ) ^ 3 } \\left ( - 3 q ^ d + 6 q ^ { 2 d } - 3 q ^ { 3 d } \\right ) = \\frac { - 3 q ^ d } { 1 - q ^ d } \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} \\Theta ( a x ^ i b x ^ j ) = \\Theta ( a \\sigma ^ i ( b ) x ^ { i + j } ) = \\sigma ^ { - i - j } ( a \\sigma ^ i ( b ) ) x ^ { - i - j } = \\sigma ^ { - i - j } ( a ) \\sigma ^ { - j } ( b ) x ^ { - i - j } , \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\sigma _ { i } \\cdot \\mathbf { x } & = t _ { 1 } t _ { 2 } \\cdots t _ { i - 1 } t _ { i + 1 } ( t _ { i + 1 } ^ { - 1 } t _ { i } t _ { i + 1 } ) t _ { i + 2 } \\cdots t _ { k } , \\\\ \\sigma _ { i } ^ { - 1 } \\cdot \\mathbf { x } & = t _ { 1 } t _ { 2 } \\cdots t _ { i - 1 } ( t _ { i } t _ { i + 1 } t _ { i } ^ { - 1 } ) t _ { i } t _ { i + 2 } \\cdots t _ { k } . \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} Q ( \\eta ) : = \\big ( - 1 / 2 , 1 / 2 \\big ) ^ { n - 1 } \\times \\big ( - \\eta / 2 , \\eta / 2 \\big ) . \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} 0 = \\frac { 1 } { 2 } \\mathcal { L } S = \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } + S ( 1 - \\frac { 3 } { 2 } S ) + 2 H ^ { 2 } S - \\frac { 1 } { 2 } H ^ { 4 } - \\sum _ { j , k , p , q } H ^ { p ^ { \\ast } } h _ { j k } ^ { p ^ { \\ast } } H ^ { q ^ { \\ast } } h _ { j k } ^ { q ^ { \\ast } } . \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} b _ { \\beta , \\alpha + \\beta ^ c } = b _ { \\alpha + \\beta , \\alpha + \\beta ^ c } = b _ { \\alpha + \\beta , \\beta ^ c } \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\infty } = { \\rm l i m } _ { L \\rightarrow + \\infty } \\mathbb { } \\mathcal { H } _ L = X _ 1 ( \\overline { p } ) + X _ 2 ( \\overline { q } ) - \\overline { p } . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} e _ k ( i d : \\ell _ p ^ n ( \\R ) \\to \\ell _ q ^ n ( \\R ) ) \\le \\tau & \\le \\frac { C _ { p , q } } { [ 2 ^ { ( k - 1 ) \\bar p / n } V _ n ( p , q ) ^ { - \\bar p / n } ] ^ { 1 / \\bar p } } \\le C ' _ { p , q } 2 ^ { - \\frac { k - 1 } { n } } n ^ { 1 / q - 1 / p } , \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{gather*} [ \\rho ( e _ 1 ) , \\rho ( e _ 2 ) ] = \\rho ( [ e _ 1 , e _ 2 ] ) , \\\\ [ e _ 1 , f e _ 2 ] = f [ e _ 1 , e _ 2 ] + \\rho ( e _ 1 ) f \\cdot e _ 2 . \\end{gather*}"}
-{"id": "4544.png", "formula": "\\begin{align*} | | U ( x _ 0 ) | | = | | U ^ { \\phi } ( x _ 0 ) | | \\leq e ^ { - ( \\ln \\lambda - \\varepsilon ) | x _ 0 | } e ^ { \\eta \\ell } . \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} d ( x _ m , x _ n ) & \\leq c d ( x _ m , x _ { n + 1 } ) + c d ( x _ { n + 1 } , x _ n ) \\\\ & \\leq c ^ 2 d ( x _ m , x _ { n + 2 } ) + c ^ 2 d ( x _ { n + 2 } , x _ { n + 1 } ) + c d ( x _ { n + 1 } , x _ n ) \\\\ & \\leq \\cdots \\\\ & \\leq c ^ { m - ( n + 1 ) } d ( x _ m , x _ { m - 1 } ) + \\sum _ { k = 1 } ^ { m - ( n + 1 ) } c ^ k d ( x _ { n + k } , x _ { n + k - 1 } ) \\\\ & \\leq \\Big [ c ^ { m - ( n + 1 ) } \\theta ^ { m - 1 } + \\sum _ { k = 1 } ^ { m - ( n + 1 ) } c ^ k \\theta ^ { n + k - 1 } \\Big ] d ( x _ 1 , x _ 0 ) \\\\ & < ( ( c \\theta ) ^ { m - 1 } c ^ { - n } + ( 1 - c \\theta ) ^ { - 1 } c ^ { - ( n - 1 ) } ) d ( x _ 1 , x _ 0 ) . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} m _ 0 = 0 , m _ { 2 n } = 4 m _ n , m _ { 2 n + 1 } = 4 m _ n + 1 \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\begin{aligned} a ( \\sigma , \\tau ) + \\{ b ( \\tau , u ) + c ( \\tau , p ) \\} + \\{ b ( \\sigma , v ) + c ( \\sigma , q ) \\} & = ( g , v ) , & & ( \\tau , v , q ) \\in \\Sigma \\times V \\times Q . \\end{aligned} \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\frac { d } { d t } V _ { m } ^ { ( 1 ) } ( t ) & = ( - 1 ) ^ { m } ( - A ) \\left ( \\sum _ { j = 0 } ^ { m } \\begin{pmatrix} 1 + ( m - 1 ) \\\\ 1 + ( j - 1 ) \\end{pmatrix} \\frac { ( - t A ) ^ { j } } { j ! } e ^ { - t A } v _ 0 \\right ) , \\\\ \\frac { d } { d t } V _ { m } ^ { ( 2 ) } ( t ) & = ( - 1 ) ^ { m } ( - A ) \\left ( \\sum _ { k = 0 } ^ { m - 1 } \\begin{pmatrix} 1 + ( m - 1 ) \\\\ 1 + k \\end{pmatrix} \\frac { ( - t A ) ^ { k } } { k ! } e ^ { - t A } u _ 1 \\right ) . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} \\left ( I _ a ^ { k _ i } f \\right ) ( x ) = \\left ( \\mathbb { K } _ i * f \\right ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] , \\ , \\ , i = 1 , 2 , \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\Psi ( t , x ) = x - ( s ^ { ( 1 ) } ( t ) - s ^ { ( 1 ) } ( t _ 0 ) ) e _ 1 \\hbox { a n d } \\Psi ^ { - 1 } ( t , x ) = x + ( s ^ { ( 1 ) } ( t ) - s ^ { ( 1 ) } ( t _ 0 ) ) e _ 1 \\ , , \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} A ( \\nu ' ; \\nu ) & = \\frac { \\zeta \\left ( \\frac { 3 } { 2 } + 2 \\nu ' + \\nu \\right ) } { \\zeta \\left ( \\frac { 3 } { 2 } + 3 \\nu ' \\right ) } \\prod _ p \\left ( 1 + a ( p ) \\frac { 1 - p ^ { \\nu ' - \\nu } } { p ^ { 3 / 2 + 3 \\nu ' } - 1 } \\right ) \\\\ & = \\prod _ { p \\equiv 1 \\mod 3 } \\left ( 1 + O \\big ( p ^ { - 5 / 2 - 2 \\nu ' - \\nu } + p ^ { - 5 / 2 - 3 \\nu ' } + p ^ { - 3 - 4 \\nu ' - 2 \\nu } + p ^ { - 2 } \\big ) \\right ) . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} & \\frac { 1 } { k ! } \\sum _ { l = 0 } ^ k { k \\choose l } ( - 1 ) ^ { k - l } l ^ k \\frac { 1 } { 1 - l t } = \\sum _ { m = 0 } ^ \\infty \\frac { 1 } { k ! } \\sum _ { l = 0 } ^ k { k \\choose l } ( - 1 ) ^ { k - l } l ^ { m + k } t ^ m \\\\ & = \\sum _ { m = 0 } ^ \\infty \\left ( \\frac { 1 } { k ! } \\Delta ^ k 0 ^ { m + k } \\right ) t ^ m = \\sum _ { m = 0 } ^ \\infty S _ 2 ( m + k , k ) t ^ m , \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} F ^ { K \\textnormal { t h } } = U \\times \\mathbb { P } ^ 1 \\times K ( X ) \\to U \\times \\mathbb { P } ^ 1 \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} z \\in \\Delta _ { n - 1 } , \\bar { A } z = \\bar { u } \\Rightarrow \\| x - z \\| _ 1 > \\frac { 2 \\| \\bar { A } x - \\bar { u } \\| _ v } { \\Phi _ v ( \\bar { A } ) } . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} p ( s + t , x , y ) = \\int _ { \\mathbb { R } ^ { N } } p ( s , x , z ) p ( t , z , y ) d z \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} { w } _ t ^ D = \\sup _ { \\tau \\in \\mathbb { R } } \\left ( P ( \\phi , \\psi _ \\tau + D ) + t \\tau \\right ) , \\ t \\geq 0 . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} P _ S ^ { } ( s ) = \\frac { 1 } { s \\ , \\ln b } , s \\in [ 1 , \\ , b \\ , ) . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { \\dot { H } ^ s ( \\R ^ d ) } = \\int _ { \\R ^ d } | \\xi | ^ { 2 s } | \\hat { f } ( \\xi ) | ^ 2 \\ , d \\xi < \\infty \\ , . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} & F \\left ( x _ { i } , \\dots , x _ { i - m + 1 } \\right ) = F _ 1 \\left ( x _ { i } \\right ) + \\ldots + F _ m \\left ( x _ { i - m + 1 } \\right ) , \\\\ & F _ j \\left ( x _ { i - j + 1 } \\right ) = U \\left [ j , \\frac { x _ \\mathrm { m a x } - n x _ \\mathrm { m i n } + \\left ( n - 1 \\right ) x _ { i - j + 1 } } { x _ \\mathrm { m a x } - x _ \\mathrm { m i n } } \\right ] , \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} u ( t , x ) : = - ( c + 1 ) ^ { 1 / ( p - 1 ) } Q \\left ( \\sqrt { \\frac { c + 1 } { c } } ( x + c t ) \\right ) , \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} g ( \\bot ) = y _ { k n } \\leq ^ { \\boldsymbol { X } _ { n } ^ { \\infty } } x _ { ( k + 2 ) n } . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\nu ( y , z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n ( y q ) ^ n . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} \\norm { \\nabla P } _ { L ^ { q , \\infty } _ t ( L ^ p _ x ) } & \\leq \\norm { \\norm { \\frac { | x | } { 2 t ^ { ( d + 2 ) / 2 } } e ^ { - | x | ^ 2 / 4 t } } _ { L ^ p _ x } } _ { L ^ { q , \\infty } _ t } = C \\norm { \\frac { t ^ { ( p + d ) / 2 p } } { 2 t ^ { ( d + 2 ) / 2 } } } _ { L ^ { q , \\infty } _ t } = C \\norm { \\frac { 1 } { 2 } t ^ { - 1 / q } } _ { L ^ { q , \\infty } _ t } < \\infty . \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} | \\Im z ^ \\beta | \\| \\hat u \\| _ { L ^ 2 ( \\Omega ) } ^ 2 = \\left | \\Im \\int _ \\Omega \\hat f \\overline { \\hat u } d x \\right | \\leq \\| \\hat f \\| _ { L ^ 2 ( \\Omega ) } \\| \\hat u \\| _ { L ^ 2 ( \\Omega ) } \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} I ( X _ { d } ) = \\sum _ { n \\geq 2 } ( I _ { + } ( \\eta ) _ { n } ) . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\overline { \\textrm { I M F } } _ 1 = ( I - W ) ^ { N _ 0 } s = U ( I - D ) ^ { N _ 0 } U ^ T s \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} I _ a ^ k \\left ( I _ a ^ { k ' } \\left ( f - I _ a ^ k \\omega ^ { - 1 } \\varphi \\right ) \\right ) ( x ) = \\int _ a ^ x \\left ( f - I _ a ^ k \\omega ^ { - 1 } \\varphi \\right ) ( y ) \\omega ( y ) \\ , d y , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} f ( x ) = x ^ n + a _ { n - 1 } x ^ { n - 1 } + \\dots + a _ 0 \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} \\mathfrak { D } ( f , i d ) & = \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < 1 / 2 \\} , \\\\ D _ 1 & = \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < 1 / 2 , x _ 1 \\le A \\} , \\\\ D _ 2 & = \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < \\min ( 1 / 2 , A ) \\} , \\\\ D _ 3 & = \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < 1 / 2 , 1 - x _ 2 \\le A \\} , \\\\ D _ 4 & = \\{ ( x _ 1 , x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\mid 0 \\le x _ 1 \\le x _ 2 < 1 / 2 , 1 - x _ 1 \\le A \\} , \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} F ( \\widetilde { w } _ n , \\lambda _ n , x , \\Lambda ) = 0 . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} & \\sum _ { n \\geq 0 } U _ { 2 k , 2 a - 1 } ( n ) q ^ n = U _ { 2 k , 2 a - 1 } ( 1 ; q ) \\\\ & = \\frac { ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a } , q ^ { 4 k - 2 a } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } + \\frac { x q ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a - 2 } , q ^ { 4 k - 2 a + 2 } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} D ( x , y , z ) = R ( z , y , z ) + R ( x , z , y ) - x , \\ ; E ( x , y , z ) = R ( x , 2 z , y ) - \\tfrac { x } { 2 } \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} p \\in [ 2 , \\infty ] , q \\in [ 2 , \\infty ) , ( p , q ) \\ne \\left ( 2 , \\frac { 4 d - 2 } { 2 d - 3 } \\right ) , \\frac { 2 s } { p } + \\frac { d } { q } = \\frac { d } { 2 } . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\partial _ t ( u _ \\lambda ) _ \\Omega + \\lambda ( \\mu _ \\lambda ) _ \\Omega = 0 \\ , , \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} - z '' ( x ) = \\mu _ { 1 } ( x ) \\left ( 1 - \\frac { 1 } { \\alpha } z ( x ) \\right ) z ^ { + } ( x ) - \\frac { 1 } { d } \\mu _ { 2 } ( x ) \\left ( 1 + \\frac { 1 } { d } z ( x ) \\right ) z ^ { - } ( x ) . \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} V I I - I V ' = 0 \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( z q ^ n ; q ) _ { n + 1 } ( z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n + 1 } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } , \\\\ [ 6 p t ] \\sum _ { n = 0 } ^ { \\infty } q ^ n ( - z q ^ { n + 1 } ; q ) _ { n } ( - z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\gamma _ \\textnormal { e q } \\left ( \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } \\right ) ( z , Q ) \\right ) = J ^ { \\textnormal { c o h , e q } } ( z , Q ) \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} w _ n \\equiv 1 \\pmod { n ^ 2 } \\iff & n \\\\ & n = p ^ 2 p \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} p _ { \\tau , h } ^ { k } { } _ { | I _ n } ( t ) = \\sum _ { j = 0 } ^ r P ^ { j , k } _ { n , h } \\varphi _ { n , j } ( t ) \\ , , \\vec z _ { \\tau , h } ^ { k } { } _ { | I _ n } ( t ) = \\sum _ { j = 0 } ^ r \\vec Z ^ { j , k } _ { n , h } \\varphi _ { n , j } ( t ) \\ , , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} \\mathcal { B } _ i ( B , \\boldsymbol { w } , \\boldsymbol { d } ) = \\frac { w _ i d _ i } { \\sum _ { j \\in \\mathcal { N } } w _ j d _ j } B . \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\widehat { \\sigma } = \\widehat { \\sigma _ U \\sigma _ { \\tau ^ h } } = \\theta \\sigma _ { \\tau ^ h } ^ { - 1 } \\sigma _ U ^ { - 1 } \\theta = \\theta \\sigma _ { \\tau ^ h } ^ { - 1 } \\theta \\theta \\sigma _ U ^ { - 1 } \\theta = \\widehat { \\sigma _ { \\tau ^ h } } \\widehat { \\sigma _ U } , \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 = R _ { 2 1 } = 0 ; \\quad \\frac { R _ { 3 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 = R _ { 3 1 } = 0 ; \\quad \\frac { R _ { 4 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 = - 1 = - u _ 4 . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} & e _ { q , \\lambda _ q } ( Q ) = \\frac { \\theta _ q ( Q ) } { \\theta _ q ( \\lambda _ q Q ) } & & \\lambda _ q ^ { \\ell _ q ( Q ) } = \\sum _ { k \\geq 0 } \\frac { 1 } { k ! } \\left ( \\log ( \\lambda _ q ) \\ell _ q ( Q ) \\right ) ^ k \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} X ^ * = \\left ( \\begin{array} { r r r r r } 0 & 0 . 1 7 2 0 & 0 & 0 . 7 2 8 0 & 0 . 7 1 7 0 \\\\ 0 & - 0 . 0 6 2 0 & 0 & 0 & 0 \\\\ 0 & 1 . 1 9 9 0 & - 1 . 3 4 4 4 & 0 . 1 1 1 4 & 0 \\\\ 0 & 0 . 8 0 2 0 & 0 . 4 9 1 0 & - 1 . 1 9 5 6 & 1 . 2 7 7 9 \\\\ 0 & 1 . 0 5 3 0 & 2 . 7 5 3 5 & 0 & - 0 . 2 5 2 5 \\end{array} \\right ) \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} A _ M = t ^ M B _ M + h ^ M C _ M \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\Pr { A _ k \\ } = 0 . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} \\bar { \\mathsf h } _ { i j } \\to \\cfrac { \\rho _ { i j } ^ * } { ( \\rho _ { i j } ^ * + ( t ^ * ) ^ 2 ) ^ 2 - ( t ^ * ) ^ 2 ( \\zeta _ j ) ^ 2 } ( \\bar { w } ^ * _ { i j } ( \\rho _ { i j } ^ * + ( t ^ * ) ^ 2 ) - \\bar { w } ^ * _ { j i } t ^ * \\zeta _ j ) = \\cfrac { t ^ * \\zeta _ i - \\zeta _ j ^ 2 } { \\zeta _ i ^ 2 - \\zeta _ j ^ 2 } \\bar { w } ^ * _ { i j } - \\cfrac { \\zeta _ j ( \\zeta _ i - t ^ * ) } { \\zeta _ i ^ 2 - \\zeta _ j ^ 2 } \\bar { w } ^ * _ { j i } , \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} h ( a b ) f _ 4 ( a , b ) & = f _ 3 ( a b , b ^ { - 1 } a ^ { - 1 } ) f _ 4 ( a , b ) f _ 3 ( b , b ^ { - 1 } ) ^ { - 1 } f _ 3 ( b , b ^ { - 1 } ) \\\\ & = f _ 4 ( a , a ^ { - 1 } ) f _ 3 ( b , b ^ { - 1 } a ^ { - 1 } ) f _ 3 ( e _ \\lambda , b ) f _ 3 ( b , b ^ { - 1 } ) \\\\ & = f _ 4 ( a , a ^ { - 1 } ) f _ 3 ( e _ \\lambda , a ^ { - 1 } ) f _ 3 ( b , b ^ { - 1 } ) \\\\ & = f _ 1 ( e _ \\lambda , a ^ { - 1 } ) f _ 3 ( b , b ^ { - 1 } ) \\\\ & = g _ 4 ( a , b ) h ( b ) , \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\O _ { \\lambda } = \\left \\{ z \\in \\C \\colon \\frac { z + z ^ * } 2 = \\lambda , \\frac { z - z ^ * } 2 = 0 \\right \\} \\ , . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) ( \\bar h _ { 1 1 1 2 } ^ { 2 ^ { * } } - \\bar h _ { 1 1 2 1 } ^ { 2 ^ { * } } ) = - S + ( 3 \\bar \\lambda _ 2 ^ 3 + \\bar \\lambda _ 1 ^ 3 ) \\bar H \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\lambda u - \\Delta u = F \\R ^ 3 , F : = \\chi _ r E f - 2 ( \\nabla \\chi _ r ) \\cdot E ( \\nabla v ) - ( \\Delta \\chi _ r ) E v . \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\sin \\Big ( ( 4 - n ) \\log | x | \\Big ) , u _ 2 ( x ) = \\cos \\Big ( ( 4 - n ) \\log | x | \\Big ) \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} n ! { \\| h _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 & \\leq \\frac { 1 } { n ! } C _ 1 e ^ { - ( 2 - \\delta ) \\mu _ 1 t } \\\\ & \\times \\int _ { [ 0 , t ] ^ { 2 n } } \\prod _ { i = 1 } ^ n \\gamma ( t _ i - s _ i ) \\prod _ { i = 1 } ^ n \\Big ( t _ { \\tau ( i + 1 ) } + s _ { \\iota ( i + 1 ) } - ( t _ { \\tau ( i ) } + s _ { \\iota ( i ) } ) \\Big ) ^ { - \\beta / \\alpha } d { \\bf t } d { \\bf s } . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} ( d S _ { l } ) _ { \\alpha \\sqcup h _ { 0 } } ( 0 \\sqcup \\gamma ) = ( d L _ { a } ) _ { S _ l ( h _ 0 ) } \\circ ( d S _ { l } ) _ { h _ { 0 } } ( \\gamma ) = ( d L _ { a } ) _ { g _ { 0 } } ( \\xi _ { 0 } ) = \\xi , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\ln \\left [ F ( \\bar N ) \\right ] & = \\ln h _ { \\bar N } + 2 \\cdot 3 ^ { \\bar N } \\ln | x _ 0 | + \\sum _ { j = 0 } ^ { \\bar N - 1 } \\ln h _ { j } ^ { 2 \\cdot 3 ^ { \\bar N - 1 - j } } \\\\ & = \\ln h _ { \\bar N } + 2 \\cdot 3 ^ { \\bar N } \\ln | x _ 0 | + \\frac { 2 } { 3 } \\cdot \\sum _ { j = 0 } ^ { \\bar N - 1 } 3 ^ { \\bar N - j } \\ln h _ { j } . \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } f ( n ) e ^ { 2 \\pi i n \\theta } = 0 \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} E ( \\prod _ i N ^ { ( 1 ) } _ { ( x _ i , y _ i ) } \\prod _ l ( N ^ { ( 1 ) } _ { z _ l } + 1 ) ) = E ( \\prod _ i \\dfrac { 1 } { 2 } C _ { ( x _ i , y _ i ) } { \\varphi _ { x _ i } \\bar { \\varphi } _ { y _ i } } \\prod _ l \\frac { 1 } { 2 } { \\lambda _ { z _ l } \\varphi _ { z _ l } \\bar { \\varphi } _ { z _ l } } ) . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} u ( t ) = \\sum _ { \\ell = 0 } ^ m ( - A ) ^ \\ell V _ { \\ell , \\ell } ( t ) + \\frac { d ^ { m + 1 } } { d t ^ { m + 1 } } U _ { m + 1 } ( t ) . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} R _ 1 = R ( \\hat { x } , \\hat { y } , \\hat { v } ) , \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} M ^ { j } & = \\{ m ^ { j } _ { i } = \\left ( ( 2 i - 1 ) e _ { 1 } + j e _ { 2 } , \\ , 2 i e _ { 1 } + j e _ { 2 } \\right ) : \\forall i \\in \\mathbb { N } \\} , \\\\ M & = \\{ m _ i = ( ( 4 i - 1 ) e _ 1 , 4 i e _ 1 ) : \\forall i \\in \\mathbb { N } \\} . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} g _ 1 y _ 1 & = h _ 1 , \\\\ x _ 2 g _ 2 y _ 2 & = h _ 2 , \\\\ & \\vdots \\\\ x _ { m - 1 } g _ { m - 1 } y _ { m - 1 } & = h _ { m - 1 } , \\\\ x _ m g _ m & = h _ m . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} \\sum \\limits _ { b \\in B ( G , \\mu ) } \\mathcal { M } _ { G , b , \\mu } = ( G , \\mu ) , \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} { \\mathbb B } ^ m _ { \\lambda } ( \\mathbb R ) = \\big \\{ u : u \\ , \\ , \\ , \\ , \\| u \\| _ { { \\mathbb B } ^ m _ { \\lambda } ( \\mathbb R ) } < \\infty \\big \\} , \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\Delta F ( a , b ) = \\sum _ { \\varepsilon \\in \\{ 0 , 1 \\} ^ d } ( - 1 ) ^ { s ( \\varepsilon ) } F ( b + \\varepsilon \\ast ( a - b ) ) \\geq 0 , \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} ( y - z ) ^ { 2 } & - n p \\rho _ { 1 } y - n p \\frac { \\sqrt { c _ { 2 } } } { \\rho } y ^ { \\frac { 1 } { 2 } } ( y - \\alpha z ) \\\\ & = \\frac { 1 } { \\alpha ^ { 2 } } ( y - \\alpha z ) ^ { 2 } + \\left ( \\frac { \\alpha - 1 } { \\alpha ^ { 2 } } y ^ { 2 } - n p \\rho _ { 1 } y \\right ) \\\\ & + \\left ( \\frac { 2 ( \\alpha - 1 ) } { \\alpha ^ { 2 } } y - n p \\frac { \\sqrt { c _ { 2 } } } { \\rho } y ^ { \\frac { 1 } { 2 } } \\right ) ( y - \\alpha z ) , \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\eta u - \\varepsilon \\boldsymbol { L } u = h \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} T = R _ L + S \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} - \\Big \\langle V ( 0 ) , \\frac { \\partial } { \\partial s } \\Big ( H ( s ) + \\frac { X ( s ) } { 2 } & \\Big ) \\Big | _ { s = 0 } \\Big \\rangle _ { L ^ 2 } = \\left \\langle V ( 0 ) , ( \\Delta _ L ^ { \\perp } - 1 ) V ( 0 ) \\right \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} Z ( t ) = a _ 0 + \\sum _ { ( x , y ) \\in \\Pi ^ + } 1 _ { ( t _ 0 , t ] } ( x ) S ( x , y ) y ^ { - 1 / \\alpha ( Z ( x _ - ) ) } ( t _ 0 \\leq t < t _ 1 ) . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\overline { \\dim } _ B S & = \\inf \\left \\{ \\delta \\ , \\Big | \\ , \\epsilon ^ { \\delta } N _ \\epsilon ( S ) \\to 0 \\right \\} . \\\\ \\bigg ( \\underline \\dim _ B S & = \\sup \\left \\{ \\delta \\ , \\Big | \\ , \\epsilon ^ { \\delta } N _ \\epsilon ( S ) \\to + \\infty \\right \\} \\bigg ) . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} f _ 2 ( q , + \\ 8 ) = \\xi ' ( q ) ( 1 - q ) - 1 + \\xi ( q ) < 0 . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} f _ \\epsilon ( \\xi ) = \\epsilon ^ { - \\frac { Q + \\alpha } 2 } H ( \\delta _ { \\epsilon ^ { - 1 } } ( \\zeta ^ { - 1 } \\xi ) ) , \\forall \\epsilon > 0 . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} h _ y : = v _ y \\circ f . \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} \\mathbb { E } _ p \\left [ p ^ { 2 / \\alpha } \\right ] = \\frac { \\rho _ { \\mathrm { o } } ^ { 2 / \\alpha } \\gamma \\left ( 2 , \\pi \\lambda _ { } R ^ 2 \\right ) } { \\pi \\lambda _ { } \\left ( 1 - \\mathrm { e } ^ { - \\pi \\lambda _ { } R ^ 2 } \\right ) } . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } > \\varepsilon \\right ) & = 1 - P ( A _ n ( t ) > t , \\quad \\forall t = a _ n , \\ldots , \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor ) \\\\ & = P \\left ( \\bigcup _ { t = a _ n } ^ { \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } \\{ A _ n ( t ) - t \\leq 0 \\} \\right ) \\\\ & = P \\left ( \\bigcup _ { t = a _ n } ^ { \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} \\right ) , \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} | u _ n | = | \\rho _ n \\xi _ n | \\le | \\rho _ n | . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\Lambda _ f ( y _ t ) = - \\nabla f _ 1 ( y _ t ) = - y _ t , \\Lambda _ f ( z _ t ) = - \\nabla f _ 2 ( z _ t ) = - ( 1 , 0 ) . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\sigma f + L f = h . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} t _ \\varepsilon ( \\rho _ { \\varepsilon } ) = ( 1 - \\varepsilon _ 0 ) \\gamma _ \\varepsilon ^ 2 \\ , . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} K _ 1 = \\delta & & K _ 2 = \\delta & & C = 3 \\delta + 1 & & C ' = C + 1 \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\Omega ( m , n ) \\times \\Omega ( m + n , g ) & = \\Omega ( m , n ) \\times \\big ( \\Omega ( m , g ) \\times \\Omega ( n , g ) \\big ) = \\\\ & = \\big ( \\Omega ( m , n ) \\times \\Omega ( m , g ) \\big ) \\times \\Omega ( n , g ) = \\Omega ( m , n + g ) \\times \\Omega ( n , g ) \\ , . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} f ( x ' ) = \\gamma > f ^ { * } . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} h ( x ) & \\le g _ 1 ( x ) \\Bigr | _ { x = \\frac { 3 } { 2 ^ 3 } } + \\frac 1 { 1 0 \\cdot 6 } - \\eta \\\\ & = - \\frac { 5 7 7 8 5 9 0 3 2 6 7 6 3 4 2 1 7 4 8 3 1 4 5 6 2 4 7 8 8 5 2 2 3 9 } { 8 7 5 7 6 8 9 2 9 2 5 0 3 9 5 5 9 9 7 2 5 9 3 0 9 1 2 8 0 2 6 9 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\widehat { u } _ { N } ( \\xi _ { j } ^ { \\lambda } ) = \\frac { \\widehat { { I } _ { N } ^ \\lambda f } ( \\xi _ { j } ^ { \\lambda } ) } { | \\xi _ { j } ^ { \\lambda } | ^ 2 + \\rho } , j \\in \\Upsilon _ { \\ ! N } , \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\mathcal H \\left ( G , L , \\mathcal A \\right ) = \\epsilon _ L \\mathcal H \\left ( G , \\mathcal A \\right ) \\epsilon _ L \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} h ^ { n } = T _ { b } \\circ ( T _ { a } \\circ f ) ^ { n } \\circ T _ { - b } = T _ { b } \\circ T _ { a + f ( a ) + \\cdots + f ^ { n - 1 } ( a ) } \\circ f ^ { n } \\circ T _ { - b } . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\mathcal B \\otimes _ { \\mathcal A } V ^ L = \\mathcal B \\otimes _ { \\mathcal A } W = W _ { \\mathcal B } \\simeq W _ 1 \\oplus W _ 2 \\oplus \\cdots \\oplus W _ t \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} X _ { \\dot { w } } ^ m ( 1 ) = \\coprod _ { g \\in G ( F ) / I _ F } g . X _ { \\dot { w } } ^ m ( 1 ) _ { P _ { 1 / 2 } } , X _ w ( 1 ) = \\coprod _ { g \\in G ( F ) / I _ F } g . X _ w ( 1 ) _ { P _ { 1 / 2 } } , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} | \\Lambda _ { T _ 0 } ( f _ 1 , \\dots , f _ { n + 1 } ) | \\lesssim \\prod _ { j = 1 } ^ { n + 1 } \\| f _ j \\| _ { L ^ { p _ j } ( X _ j ) } . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} R ( X , Y ) Z = \\nabla _ X \\nabla _ Y - \\nabla _ Y \\nabla _ X - \\nabla _ { [ X , Y ] } . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} \\tilde { u } _ \\varepsilon ( z ) = u _ \\varepsilon ( y _ \\varepsilon ) \\left ( u _ \\varepsilon ( z _ \\varepsilon ) - u _ \\varepsilon ( y _ \\varepsilon ) \\right ) \\ , , \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\gamma ( r ) = C _ H | r | ^ { 2 H - 2 } , \\ \\ \\ f o r \\ \\ H \\in \\big ( 1 / 2 , 1 \\big ) \\ \\ C _ H = H ( 2 H - 1 ) . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\mathbf { s } \\mathbf { s } _ \\mathcal { Q } } = \\sqrt { \\frac { 2 } { \\pi } } \\mathbf { K } \\mathbf { C } _ { \\mathbf { s } } , \\mbox { w h e r e } \\mathbf { K } = \\left ( \\mathbf { C } _ { \\mathbf { s } } \\right ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} \\omega ( W _ m ) = \\begin{cases} 1 & m = \\ 0 \\\\ p ^ { ( m ) } \\in \\C & \\end{cases} \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} S \\circ T ( ( a _ { \\xi } ) _ { \\xi < \\omega _ 1 } ) = \\left ( T ( ( a _ { \\xi } ) _ { \\xi < \\omega _ 1 } ) ( z _ \\xi ) \\right ) _ { \\xi < \\omega _ 1 } = \\left ( a _ { \\xi } \\cdot \\chi _ { A _ \\xi } ( z _ \\xi ) \\right ) _ { \\xi < \\omega _ 1 } = ( a _ { \\xi } ) _ { \\xi < \\omega _ 1 } \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta u _ \\varepsilon = \\tilde { \\lambda } _ \\varepsilon u _ \\varepsilon ^ { 2 N _ \\varepsilon + 1 } , u _ \\varepsilon > 0 \\Omega \\ , , \\\\ & u _ \\varepsilon = 0 \\partial \\Omega \\ , , \\\\ & N _ \\varepsilon \\to + \\infty \\ , , \\end{cases} \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left ( \\frac { q ^ t - 1 } { z } \\right ) ^ a \\binom { \\ell _ { q ^ t } \\left ( Q \\right ) } { a } = \\frac { 1 } { a ! } \\left ( \\frac { \\log ( Q ) } { z } \\right ) ^ a \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} P _ n = \\mathbb { P } _ { S _ n } ( \\Omega _ 0 ) \\mbox { a n d } P _ { n , q } = \\mathbb { P } _ { \\mathcal { M } } ( \\Omega _ 0 ) . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} \\prod _ { i = 2 } ^ 4 p ^ { \\frac { k + a _ i } { 2 } } \\epsilon _ { p ^ { k - a _ i } } \\left ( \\frac { a A _ i ' } { p ^ { k - a _ i } } \\right ) e _ { p ^ { k - a _ i } } ( - \\overline { 4 a A _ i ' } c _ i '^ 2 ) , \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} I _ { N } ^ { G } U ( t ) = \\sum _ { n = 0 } ^ { N } \\tilde { U } _ { n } ^ { \\lambda } C _ { n } ^ { \\lambda } ( t ) , { \\rm w h e r e } \\tilde { U } _ { n } ^ { \\lambda } = \\frac { 1 } { \\gamma _ { n } ^ { \\lambda } } \\sum _ { j = 0 } ^ { N } U ( t _ { j } ^ \\lambda ) C _ { n } ^ { \\lambda } ( t _ j ^ \\lambda ) \\rho _ { j } ^ \\lambda . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} c _ 1 ( H ) \\overset { \\textnormal { D e f . } } { = } \\frac { 4 ( 2 H - 1 ) } { H + 1 } , c _ 2 ( H ) \\overset { \\textnormal { D e f . } } { = } \\frac { 8 ( 2 H - 1 ) } { H + 1 } \\sqrt { \\frac { H ( 2 H - 1 ) } { 2 } } . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} \\dot { \\tilde { a } } _ t = h _ t ' ( \\xi ( t ) ) ^ 2 \\dot { a } _ t , t \\in [ 0 , t _ 0 ) . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\sup _ { N \\in \\mathbb { N } , 0 = x _ 0 < x _ 1 < . . . < x _ N = L } \\sum _ { i = 0 } ^ { N - 1 } \\Psi \\left ( \\rho \\left ( g ( x _ { i } ) , g ( x _ { i + 1 } \\right ) ) \\right ) ~ \\leq ~ V . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\norm { \\nabla f _ i ( y ) - \\nabla f _ i ( z ) } \\leq L \\norm { y - z } \\forall y , z \\in W , \\ ; i = 1 , \\dots , m . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} ( t - r ) ^ 2 - ( 1 - r ) ^ 2 = ( 2 r - 1 - t ) ( 1 - t ) \\leq ( r - 1 ) ( 1 - t ) \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} ( y - z ) ^ { 2 } & = \\frac { 1 } { \\alpha ^ { 2 } } ( y - \\alpha z ) ^ { 2 } + \\frac { \\alpha - 1 } { \\alpha ^ { 2 } } y ^ { 2 } + \\frac { 2 ( \\alpha - 1 ) } { \\alpha ^ { 2 } } y ( y - \\alpha z ) , \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { H } _ 0 & : Y _ k ( t ) \\sim \\mathcal { N } ( \\mu _ 0 , \\sigma _ 0 ^ 2 ) \\\\ \\mathcal { H } _ 1 & : Y _ k ( t ) \\sim \\mathcal { N } ( \\mu _ 1 , \\sigma _ 1 ^ 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\mathcal { H } _ { ( + , 0 ) } ^ { ( 0 , 0 , \\pm | M | ) } ( t ) & = : \\left ( x ( t ) , 0 , \\tilde { z } ( t ) \\right ) , \\\\ \\mathcal { H } _ { ( - , 0 ) } ^ { ( 0 , 0 , \\pm | M | ) } ( t ) & = : \\left ( - x ( t ) , 0 , \\tilde { z } ( t ) \\right ) , \\\\ \\mathcal { H } _ { ( 0 , + ) } ^ { ( 0 , 0 , \\pm | M | ) } ( t ) & = : \\left ( 0 , y ( t ) , z ( t ) \\right ) , \\\\ \\mathcal { H } _ { ( 0 , - ) } ^ { ( 0 , 0 , \\pm | M | ) } ( t ) & = : \\left ( 0 , - y ( t ) , z ( t ) \\right ) , ~ ~ t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\underline { p } : = \\min \\{ p _ - , \\ 1 \\} . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\frac { | P + \\nabla \\widetilde { u } _ 0 ( x ) + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) | ^ 2 } { 2 } + V ( x , y ) = \\ln \\widetilde { m } ( x , y ) + \\overline { H } . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\Sigma , L } ( e _ 1 , e _ 2 ) = - \\langle R ^ { \\Sigma , L } ( e _ 1 , e _ 2 ) e _ 1 , e _ 2 \\rangle _ { \\Sigma , L } , ~ ~ ~ ~ \\mathcal { K } ^ { L } ( e _ 1 , e _ 2 ) = - \\langle R ^ { L } ( e _ 1 , e _ 2 ) e _ 1 , e _ 2 \\rangle _ L . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} F & = \\{ \\sigma _ 0 \\leq 4 ( 1 0 ) ^ 5 n \\log n / \\beta \\} , \\\\ E & = \\{ \\sigma _ 0 < \\sigma _ { J + 1 } \\} . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta B _ { \\varepsilon } = \\frac { \\lambda _ \\varepsilon } { 2 } \\Psi ' _ { N _ \\varepsilon } ( B _ \\varepsilon ) \\ , , \\\\ & B _ \\varepsilon ( x _ \\varepsilon ) = \\gamma _ \\varepsilon \\ , , \\end{cases} \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\lambda ^ { 2 } - \\left ( \\mathit { T r } \\mathbb { J } \\right ) \\lambda + \\det \\mathbb { J } = 0 . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} U = B ( x _ 0 , \\delta ) = \\{ x \\in R ^ n : | x - x _ 0 | < \\delta \\} \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\nabla \\psi ( x ) = \\lambda ( x ) x , \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} \\P [ A _ 1 ^ c ] + \\P [ A _ 2 ^ c ] + \\P [ A _ 3 ^ c ] = O ( 1 ) e ^ { - \\Omega ( c d n ) } . \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} \\sum _ { i \\in Z _ 1 ( T j ' + T \\overline { Q } ) } \\psi _ i ^ s ( \\{ \\bar { u } _ T ^ { i + j } \\} _ { j \\in \\Z ^ n } ) = \\sum _ { i \\in Z _ 1 ( T j ' + T \\overline { Q } ) } \\psi _ i ^ s ( \\{ u _ T ^ { i - T j ' + j } \\} _ { j \\in \\Z ^ n } ) = \\sum _ { i \\in Z _ 1 ( T \\overline { Q } ) } \\psi _ i ^ s ( \\{ u _ T ^ { i + j } \\} _ { j \\in \\Z ^ n } ) . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} S ( z _ k ) & = \\int _ 0 ^ { \\omega } p _ k ( t , \\dot z _ k - \\dot z _ * ) d t \\ge \\int _ 0 ^ { \\omega } L ( t , z _ k , \\dot z _ * ) d t \\\\ & + \\int _ 0 ^ { \\omega } \\frac { \\partial L } { \\partial \\dot z ^ i } ( t , z _ k , \\dot z _ * ) ( \\dot z _ k ^ i - \\dot z _ * ^ i ) d t . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\mathcal { G } : = \\Big \\{ \\widetilde { Q } \\in \\mathcal { D } : \\mu ( \\widetilde { Q } ) \\geq \\frac { l ( Q ) } { C ' _ 1 } \\Big \\} . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} L ( s ) = c ( s ) \\exp \\left ( \\int _ { s _ 0 } ^ s \\frac { \\varepsilon ( \\tau ) } { \\tau } d \\tau \\right ) . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} d ( e _ x b ) = ( 1 - x ) b - e _ x d ( b ) . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { k \\geq 1 } \\int _ { \\mathcal { H } _ { - s } } \\Big | \\lambda _ k \\frac { \\partial ^ 2 \\psi ( X ) } { \\partial z _ k ^ 2 } \\Big | \\mu ( d X ) & = \\int _ { \\mathcal { H } _ { - s } } \\Big | \\sum _ { k \\geq 1 } \\lambda _ k k ^ { - 2 s } k ^ { 2 s } \\frac { \\partial ^ 2 \\psi ( X ) } { \\partial z _ k ^ 2 } \\Big | \\mu ( d X ) \\\\ & \\leq \\| D ^ 2 \\psi \\| _ \\infty \\int _ { \\mathcal { H } _ { - s } } \\sum _ { k \\geq 1 } \\lambda _ k k ^ { - 2 s } \\mu ( d X ) < \\infty . \\end{aligned} \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} d & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i ^ 2 + \\beta _ i ^ 2 ( 1 + 2 \\lambda \\beta _ i ^ 2 ) } { ( 1 + 2 \\lambda \\beta _ i ^ 2 ) ^ 2 } , \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\widetilde { H } _ { \\Lambda _ j } = \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { \\delta } _ { i j } \\widetilde { m } d y + \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i \\Lambda _ j } \\widetilde { m } d y . \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\mathbf P _ { s o } ^ { \\ , i } = 1 - \\sum _ { \\mathbf D _ u \\in 2 ^ { \\mathcal D } } \\varepsilon ^ { | \\mathbf D _ u | } \\ , \\left [ \\vartheta ^ { m - | \\mathbf D _ u | } \\Theta _ { i } ( \\mathbf D _ u , 1 ) + \\left ( \\left ( 1 \\ ! - \\ ! \\varepsilon \\right ) ^ { m - | \\mathbf D _ u | } \\ ! - \\ ! \\vartheta ^ { m - | \\mathbf D _ u | } \\right ) \\Theta _ { i } ( \\mathbf D _ u , \\beta ) \\right ] , \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\rho ( \\nu _ { f , [ \\alpha ] } ) = \\rho ( \\nu _ { f , [ \\sigma ( \\alpha ^ + ) ] } ) = \\rho ( \\nu _ { f \\sigma , [ \\alpha ^ + ] } ) = [ \\sigma ( \\alpha ^ + ) ] = [ \\alpha ] , \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} P ^ 2 L \\ll N ^ { 1 - \\varepsilon } . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\Phi _ 1 & = W _ { l ^ 1 } ( 2 , - 3 , 1 ) = \\{ ( 2 , - 3 , 1 ) , ( 1 , - 3 , 2 ) \\} , \\\\ \\Phi _ 2 & = W _ { l ^ 2 } ( - 1 , - 2 , 3 ) = \\{ ( - 1 , - 2 , 3 ) , ( 1 , - 3 , 2 ) \\} \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} m ^ { * } = c ^ { * } \\circ P + \\langle v , \\cdot \\rangle . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} h = 2 \\ , \\tau _ f / c \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} F ( x ) = F ^ { \\flat } ( P ( x ) ) + \\langle v , x \\rangle . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} S _ { l _ { 0 } } ( h _ { m } ) = S _ { l _ { 0 } } ( \\| u _ { m } \\| _ { } \\cdot \\bar { h } _ { m } ) = \\delta _ { \\| u _ { m } \\| _ { } } ( S _ { l _ { 0 } } ( \\bar { h } _ { m } ) ) = \\delta _ { \\| u _ { m } \\| _ { } } ( \\overline { u } _ { m } ) = u _ { m } . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} y _ m : = x _ { m + \\mathcal N ( \\omega ) } , u _ { m + 1 } : = \\rho _ { m + \\mathcal N ( \\omega ) } \\xi _ { m + \\mathcal N ( \\omega ) } ( \\omega ) , { \\rm h } _ m : = h _ { m + \\mathcal N ( \\omega ) } , \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\omega ( \\sigma ^ { j } ) \\leq \\frac { 1 } { \\tilde { \\delta } } \\sum ^ { j / 2 } _ { l = 0 } \\omega _ { 1 } ( \\sigma ^ { j - l } ) \\omega _ { 2 } ( \\sigma ^ { l } ) + \\frac { 1 } { \\tilde { \\delta } } \\sum ^ { j } _ { l = j / 2 } \\omega _ { 1 } ( \\sigma ^ { j - l } ) \\omega _ { 2 } ( \\sigma ^ { l } ) + \\sigma ^ { j \\alpha } . \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\varphi _ y ( x ) = c ( x ) - c ( P ( y ) ) - \\langle \\nabla c ( P ( y ) ) , x - P ( y ) \\rangle , \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\frac { \\tilde \\lambda _ { j } - \\lambda _ j } { \\lambda _ { j - 1 } - \\tilde \\lambda _ j } = \\frac { 1 - A } { A } \\geq \\frac { L - \\ell } { \\ell } , \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 1 { R _ 1 ^ 1 } ) = 1 , \\mathrm { w d } ( \\mathcal { T } ^ 1 { R _ 1 ^ 1 } ) = 7 , \\mathrm { I C } ( \\mathcal { T } ^ 1 { R _ 1 ^ 1 } ) = 1 9 . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m + 1 } v _ j \\lambda ( B ( x _ j , \\epsilon _ k ) \\cap N _ { i k } ) = 0 . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} f ( x ) = \\sup \\{ d ( x , k ) : k \\in K \\} \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\frac { f ' } { f } ( \\xi ) = x ( \\xi ) , \\ \\ \\ \\ \\ \\frac { \\varphi ' } { \\varphi } ( \\xi ) = k x ( \\xi ) , \\ \\ \\ \\ \\ \\frac { h ' } { h } ( \\xi ) = x ( \\xi ) z ( \\xi ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 5 { P _ { 1 1 } } ) = 1 , \\mathrm { w d } ( \\mathcal { T } ^ 5 { P _ { 1 1 } } ) = 4 0 , \\mathrm { I C } ( \\mathcal { T } ^ 5 { P _ { 1 1 } } ) = 9 4 . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} \\partial _ b : = \\frac { 1 } { b ! } \\frac { \\partial ^ b } { \\partial x ^ b } . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } h ( ( f \\times g ) ^ { n } ( x , y ) ) ^ { 1 / n } = \\lim _ { n \\to \\infty } ( h _ { H _ { X } } ( f ^ { n } ( x ) ) + h _ { H _ { Y } } ( g ^ { n } ( y ) ) ) ^ { 1 / n } \\\\ & = \\max \\{ \\lim _ { n \\to \\infty } h _ { H _ { X } } ( f ^ { n } ( x ) ) ^ { 1 / n } , \\lim _ { n \\to \\infty } h _ { H _ { Y } } ( g ^ { n } ( y ) ) ^ { 1 / n } \\} = \\max \\{ \\alpha _ { f } ( x ) , \\alpha _ { g } ( y ) \\} . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\begin{array} { c l } \\underset { x \\in \\mathbb { R } ^ { m } , X \\in \\mathcal { S } ^ { m } } { \\min } & \\langle Q , X \\rangle \\\\ & ( x , X ) \\in \\mathcal { F } \\\\ & \\langle Q _ { i } , X \\rangle = \\langle c _ { i } , x \\rangle i = 1 , \\ldots , k . \\end{array} \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\underline { \\Delta } d ( x , p ) = \\begin{cases} ( n - 1 ) \\sqrt { k } \\cot ( \\sqrt { k } \\rho ) , & k > 0 , \\\\ ( n - 1 ) \\rho ^ { - 1 } , & k = 0 , \\\\ ( n - 1 ) \\sqrt { | k | } \\coth ( \\sqrt { k } \\rho ) , & k < 0 . \\end{cases} \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} v : = \\sum _ n u _ n \\ , , \\theta : = \\sum _ n \\rho _ n \\ , , \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} S ( N ) = S ^ \\star ( N ) + O \\left ( ( P M L ) ^ { \\varepsilon } \\left ( \\frac { N L } { P M ^ { 1 / 2 } } + \\frac { M } { P } + \\frac { N } { L } \\right ) \\right ) \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} \\log \\rho ( A ) \\ge \\sum _ { i , j = 1 } ^ n \\mu _ { i j } \\log \\frac { a _ { i j } \\sum _ { k = 1 } ^ n \\mu _ { i k } } { \\mu _ { i j } } . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } ( q ( t ) - 1 ) \\ell _ { q ( t ) } ( - Q ) = \\log ( Q ) \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} 0 = & \\ A X \\xi - X ^ 2 \\log \\xi - \\psi ^ 2 \\xi . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} x _ i = \\sum _ { k = 1 } ^ { n } \\alpha _ k \\ , x _ { i - k } + \\epsilon _ i ; \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x } & & 1 ^ { T } x x ^ { - } 1 , \\\\ & & & x = ( \\lambda ^ { - 1 } A ) ^ { \\ast } u , u > 0 . \\end{aligned} \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} & \\langle \\eta , \\gamma _ { X _ { l _ { 0 } } ( t , x ) } \\eta \\rangle _ { \\mathbb { R } ^ { N } } \\\\ & = \\eta ^ { * } \\cdot J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) \\cdot \\gamma _ { U _ { t } ^ { ( l _ { 0 } ) } } \\cdot J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) ^ { * } \\eta \\\\ & \\geq \\mu _ { t } ^ { ( l _ { 0 } ) } \\cdot \\left ( \\eta ^ { * } \\cdot J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) \\cdot J F _ { l _ { 0 } } ( U _ { t } ^ { ( l _ { 0 } ) } , x ) ^ { * } \\cdot \\eta \\right ) , \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} x = a r \\cos ( \\theta ) \\ , \\ \\ y = b r \\sin ( \\theta ) \\ , \\ \\ \\mbox { w i t h } \\ \\ r \\in [ 0 , 1 ) , \\ ; \\theta \\in [ 0 , 2 \\pi ] \\ . \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\Delta u + a ( x , u ) = 0 \\Omega \\subset \\R ^ n , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} k _ { \\gamma , \\Sigma _ 1 } ^ { \\infty } = \\frac { \\sqrt { \\frac { 1 } { 2 } \\overline { q } ^ 2 \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) ^ 2 + \\overline { p } ^ 2 \\dot { \\gamma _ 3 } ^ 2 } } { | \\omega ( \\dot { \\gamma } ( t ) ) | } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) \\neq 0 , \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} & { \\rm { M o r } } \\ , ( i , \\ , j ) = \\begin{cases} e _ { i j } , & i \\preccurlyeq j , \\ \\ i \\neq j , \\\\ \\emptyset , & i \\npreceq j , \\end{cases} \\\\ & { \\rm { M o r } } \\ , ( i , \\ , i ) = G _ i , \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} \\bigg \\{ ( t , x ) \\in K ~ : ~ F _ c ( x ) = 0 \\bigg \\} = \\emptyset \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N \\left ( 1 - \\frac { \\textnormal { d e g } ( T _ k ) } { 2 } \\right ) a _ i + \\sum _ { j = 1 } ^ r \\omega _ j T _ j ( d ) = - \\textnormal { d i m } _ \\mathbb { C } ( X ) + \\frac { T _ k } { 2 } + \\frac { \\alpha } { 2 } + u + 1 \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\alpha _ { f } ( x ) = \\lim _ { n \\to \\infty } \\max \\{ h _ { X } ( f ^ { n } ( x ) ) , 1 \\} ^ { 1 / n } \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\mathcal { R } ( x - \\gamma ' ) = \\mathcal { R } \\Theta ( x - \\gamma ) . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\overline { R } _ { u , w } : = \\mathrm { C l } _ { ( B _ + w B _ + ) / B _ + } ( R _ { u , w } ) \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\mathcal { B } _ n = \\mathcal { B } _ n ( m ) = \\left ( X _ { m n } ( I _ 1 ) \\times \\dots \\times X _ { m n } ( I _ { d _ 1 } ) \\right ) \\times \\left ( W _ { m n } ( \\Theta _ 1 ) \\times \\dots \\times W _ { m n } ( \\Theta _ { d _ 2 + d _ 3 } ) \\right ) \\ ; . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} M ( g , \\mu ) & = \\{ p \\in S p l ( g ) \\mid \\beta _ i \\equiv R _ { \\mu ( i ) } \\bmod \\mathfrak { p } \\ , ( 1 \\le { } ^ \\forall i \\le n ) { } ^ \\exists \\mathfrak { p } \\mid p \\} \\\\ & = \\left \\{ \\begin{array} { c c c } M ( f , \\mu ) & & \\delta = 1 , \\\\ M ( f , { \\mu ' } ) & & \\delta = - 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) = ( 1 + b ) ^ { \\mu } ( 1 + t ) ^ { - \\mu } E ( b , y ; t , x ; \\mu , \\nu ^ 2 ) . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} [ T , \\textbf { b } ] _ { k e _ j } = [ \\cdots [ [ T , \\textbf { b } ] _ { e _ j } , \\textbf { b } ] _ { e _ j } \\cdots , \\textbf { b } ] _ { e _ j } , \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\le x \\\\ p \\textrm { s p l i t s i n } K } } N _ { p } ^ k ( E [ \\ell ] ) = 1 ^ k \\cdot \\pi _ E ^ s ( x ; \\ell , 0 ) + \\ell ^ k \\cdot \\pi _ E ^ s ( x ; \\ell , 1 ) + \\ell ^ { 2 k } \\cdot \\pi _ E ^ s ( x ; \\ell , 2 ) . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\lim _ { q \\to \\infty } S ( q ) = 0 . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\cosh [ \\ln ( R ^ 2 ) ] = \\frac 1 2 ( R ^ 2 + R ^ { - 2 } ) = \\frac { a ^ 2 + b ^ 2 } { a ^ 2 - b ^ 2 } \\ , \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} A _ { } \\leq \\sum _ { j = 1 } ^ \\infty \\| V ^ j \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} R _ L ( f ) ( \\xi , y ) & = R ( f \\restriction _ { [ 0 , \\alpha ] \\times \\{ y \\} } ) ( \\xi ) \\\\ & = T ( f \\restriction _ { [ 0 , \\alpha ] \\times \\{ y \\} } \\otimes \\chi _ { \\{ y _ 0 \\} } ) ( \\xi , y _ 0 ) = \\sum _ { \\eta \\in \\alpha } r ( \\eta , \\xi ) \\cdot f ( \\eta , y ) . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} D ^ { \\mathcal S _ { + } } _ { e } \\eta _ { + } = \\lambda ( e ) \\eta _ { + } , \\ \\forall e \\in \\Gamma ( E ) , \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} f ( 1 ) = 0 , \\bar f ( q ) = 0 , \\bar f ( 0 ) = 0 , \\\\ \\bar f ( u ) \\ge 0 u \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = h _ { a l g } ( \\bar \\phi ) + h _ { a l g } ( \\phi \\restriction _ H ) , \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\nu \\left \\{ R ^ m \\setminus { \\rm D o m } [ E ] \\right \\} = 0 . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } _ { | P = \\Lambda _ i } \\right ) _ { | \\Lambda _ 0 = \\cdots = \\Lambda _ N = 1 } ( q , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { ( q ; q ) _ d ^ { N + 1 } } \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} = \\Sigma _ { + } + \\Sigma _ - + ( s q _ { n } - 1 ) \\ln 2 , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\| \\sqrt { f _ n } - \\sqrt { g _ n } \\| _ 2 ^ 2 = \\int \\frac { ( f _ n - g _ n ) ^ 2 } { ( \\sqrt { f _ n } + \\sqrt { g _ n } ) ^ 2 } \\asymp \\frac { h _ n ^ { 2 \\beta + 1 } } { f _ n ( x _ 0 ) + h _ n ^ \\beta } . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{gather*} { } ^ E \\nabla g = 0 , \\end{gather*}"}
-{"id": "2734.png", "formula": "\\begin{align*} \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D F = 1 , N D N = 1 } \\right \\rbrace = \\exp \\left ( - \\frac { C _ { \\textrm { N } } ^ { \\textrm { O F D M A } } } { \\lambda _ { \\textrm { B N } } } \\right ) . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\Pr { B _ k } { M _ k } & = \\sum _ { i = k + 1 } ^ \\infty \\Pr { s _ i = M _ k } { M _ k } \\prod _ { n = 1 } ^ { i - ( k + 1 ) } \\Pr { s _ { k + n } < M _ k } { M _ k } \\\\ & = \\sum _ { i = k + 1 } ^ \\infty \\frac { c } { M _ k ^ { 5 / 4 } } \\Pr { s _ { k + 1 } < M _ k } { M _ k } ^ { i - ( k + 1 ) } . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} L _ f = \\lambda _ { \\max } ( Q ) = \\lambda _ { \\max } ( Q ^ { 1 / 2 } ) ^ 2 , \\ ; \\ ; \\mu _ f = \\lambda _ { \\min } ( Q ) = \\lambda _ { \\min } ( Q ^ { 1 / 2 } ) ^ 2 . \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\frac 1 r : = \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { r _ i } , \\frac 1 { p _ { m + 1 } } : = 1 - \\frac 1 p , \\qquad \\mbox { a n d } \\frac 1 { \\delta _ i } = \\frac 1 { r _ i } - \\frac 1 { p _ i } , i = 1 , \\dots , m + 1 . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { p , 1 } ^ { \\gamma _ 3 } } \\le & C \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\gamma _ 3 + \\frac n 2 - \\frac n p } } \\le C \\big ( \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { 2 , \\infty } ^ { \\sigma + 1 } } \\big ) ^ { \\theta _ { 4 } } \\big ( \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 } } \\big ) ^ { 1 - \\theta _ { 4 } } , \\theta _ { 4 } = \\frac { \\frac n p - \\gamma _ 3 } { \\frac n 2 - 1 - \\sigma } \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} \\tau _ { 1 0 } ( x ) = g _ { 1 0 - } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 2 4 5 9 6 } { 3 ^ { 1 0 } } , c ) . \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\frac { D } { d t } d \\Psi _ { ( t , v ) } ( w _ i ) = \\sum _ { j = 0 } ^ { n - 1 } \\left ( \\dot B _ { i j } ( t ) + \\sum _ { k = 0 } ^ { n - 1 } B _ { i k } ( t ) A _ { k j } ( t ) \\right ) v _ j ( t ) . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} | x _ { N _ 1 + 2 + k _ { l _ { m } + 1 } } | \\le | x _ { N _ 1 + 2 + m } | + \\sum ^ { k _ { l _ m + 1 } } _ { i = m } | u _ { N _ 1 + 2 + i } | \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 4 { P _ 9 } ) = 1 , \\mathrm { w d } ( \\mathcal { T } ^ 4 { P _ 9 } ) = 3 0 , \\mathrm { I C } ( \\mathcal { T } ^ 4 { P _ 9 } ) = 7 8 . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} B ^ s _ { q , p } ( D ) : = \\left ( W ^ { m _ 1 , q } ( D ) , W ^ { m _ 2 , q } ( D ) \\right ) _ { \\theta , p } . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} d _ 0 ^ { } \\in \\{ 1 , 2 , \\ldots , 9 \\} , d _ i ^ { } \\in \\{ 0 , 1 , 2 , \\ldots , 9 \\} , ~ ~ i = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 2 } ^ { e - 1 } t _ { i } ( ( q + 1 ) m + ( q + i ) d ) \\equiv j ( m + d ) + & ( q + 1 ) m + ( q + e - 1 ) d ( \\textrm { m o d } \\ , m ) . \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} \\vec { w } : = ( w _ 1 , \\dots , w _ m ) : = ( v _ 1 e _ { b _ 1 } , \\dots , v _ m e _ { b _ m } ) : = ( v _ 1 e ^ { - ( z _ 1 ) s _ 1 b _ 1 } , \\dots , v _ m e ^ { - ( z _ m ) s _ m b _ m } ) . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} \\partial _ t z - \\partial _ { x x } z = \\mu _ L ( x ) ( 1 - z ^ 2 ) z . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\Delta _ { \\tilde { n } } ( f g ^ { \\tilde { n } } ) & = \\sum _ { k _ 1 + \\dots + k _ { \\tilde { n } + 1 } = \\tilde { n } , \\atop { } ^ \\forall k _ i \\ge 0 } \\frac { \\tilde { n } ! } { k _ 1 ! \\dots k _ { \\tilde { n } + 1 } ! } \\Delta _ { k _ 1 } ( f ) \\Delta _ { k _ 2 } ( g ) \\dots \\Delta _ { k _ { \\tilde { n } + 1 } } ( g ) \\\\ & = \\tilde { n } ! \\ , \\Delta _ 1 ( g ) ^ { \\tilde { n } } = \\frac { \\tilde { n } ! } { 2 ^ { \\tilde { n } } } , \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta } D ( L _ 1 , \\ell _ { \\alpha } , \\ell _ { \\beta } ) + \\sum _ { j = 2 } ^ n \\sum _ \\gamma R ( L _ 1 , L _ j , \\ell _ \\gamma ) + \\sum _ { \\mu , \\nu } E ( L _ 1 , \\ell _ \\nu , \\ell _ \\mu ) = L _ 1 \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } \\right ) ^ 2 \\Omega _ r [ u ] ( t , x ) + \\frac { \\mu } { 1 + t } \\left ( \\frac { \\partial } { \\partial t } \\right ) \\ , \\Omega _ r [ u ] ( t , x ) + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } \\ , \\Omega _ r [ u ] ( t , x ) = \\Omega _ r [ \\Delta u ] ( t , x ) + \\Omega _ r [ f ] ( t , x ) . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\tau _ 7 ( x ) & = g _ { 7 - } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 8 5 7 } { 2 0 5 9 } , c ) , \\\\ \\tau _ 7 ( x ) & = g _ { 7 + } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , c , \\tfrac { 9 1 1 } { 3 ^ 7 } ) . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} r \\partial _ r ( r ^ { - 1 } n _ { a b } ) = & R _ { L a b \\underline L } - ( l _ b ^ c + r ^ { - 1 } \\delta _ b ^ c ) n _ { a c } + \\nabla _ a \\eta _ b - \\eta _ a \\eta _ b \\\\ = & W _ { L a b \\underline L } + \\sigma _ { a b } - ( l _ b ^ c + r ^ { - 1 } \\delta _ b ^ c ) n _ { a c } + \\nabla _ a \\eta _ b - \\eta _ a \\eta _ b . \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} 2 \\mu c _ \\alpha \\theta ^ \\beta _ \\pm a | \\alpha \\cap \\beta | - b _ { \\alpha , \\beta } = \\theta ^ \\beta _ \\pm \\left ( 2 \\mu c _ \\alpha a | \\alpha \\cap ( \\alpha + \\beta ) | - b _ { \\alpha , \\beta } \\theta ^ \\beta _ \\pm \\right ) \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - F _ m ^ + \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n . \\end{cases} \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} f ( x , t ) : = | \\mathcal { W } [ y + d ] - \\mathcal { W } [ y ] - \\omega | ( x , t ) \\ge 0 . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\omega _ i \\subset \\subset \\mathcal { O } _ { i , d } \\cap \\tilde { \\mathcal { O } } , \\ i = 1 , 2 \\ \\ \\omega _ 1 \\cap \\omega _ 2 \\neq \\emptyset . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} f _ { d + 1 } = \\left ( - q \\right ) ^ { 1 + s - r } \\frac { ( 1 - a _ 0 q ^ d ) \\cdots ( 1 - a _ r q ^ d ) } { ( 1 - q ^ { d + 1 } ) ( 1 - b _ 0 q ^ d ) \\cdots ( 1 - b _ s q ^ d ) } \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\displaystyle p _ { N , \\rho , j } = \\left \\{ \\begin{array} { r c } P ( Y \\in I _ j ) + \\rho , & \\mbox { i f } j \\leq N ^ 2 , \\\\ M a ^ { \\log ( N ) + j - N ^ 2 - 2 } , & \\mbox { i f } j > N ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} U ( x ) = a Q ( \\lambda x + x _ 0 ) , \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\tilde \\Sigma = \\sum _ { j = 1 } ^ r \\lambda _ j ( u _ j \\otimes u _ j ) + \\tilde x ( F \\otimes F ) \\tilde x > 0 F = \\sum _ { j = 1 } ^ r \\sqrt { \\lambda } _ j u _ j \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\left ( \\frac { t } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 } \\right ) ^ r ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty \\beta _ { n , \\lambda } ^ { ( r ) } ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 1 , 2 ] ) . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ { t _ e } \\vec q _ h \\cdot ( \\vec q _ h ( ( v + v _ e ) / 2 ) + \\frac { h _ e } { 2 } ( \\partial _ { t _ e } ( \\lambda _ v - \\lambda _ { v _ e } ) \\partial _ { t _ e } \\vec q _ h ) ) = & | \\partial _ { t _ e } \\vec q _ h | ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) \\Phi ( n ) \\leq \\sum _ { k = 1 } ^ n \\varphi ( a _ k ) \\leq ( 1 + \\epsilon ) \\Phi ( n ) . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} ( \\sigma ^ { - 1 } \\circ \\mathcal { I } _ g \\circ \\sigma ) ( x ) & = \\sigma ^ { - 1 } ( \\mathcal { I } _ g ( \\sigma ( x ) ) ) \\\\ & = \\sigma ^ { - 1 } ( g \\sigma ( x ) g ^ { - 1 } ) \\\\ & = \\sigma ^ { - 1 } ( g ) x \\sigma ^ { - 1 } ( g ^ { - 1 } ) \\\\ & = \\sigma ^ { - 1 } ( g ) x ( \\sigma ^ { - 1 } ( g ) ) ^ { - 1 } \\\\ & = \\mathcal { I } _ { \\sigma ^ { - 1 } ( g ) } ( x ) . \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\theta _ q ( Q ) = \\sum _ { d \\in \\mathbb { Z } } q ^ \\frac { d ( d - 1 ) } { 2 } Q ^ d \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] = X _ 3 , ~ ~ [ X _ 2 , X _ 3 ] = 0 , ~ ~ [ X _ 1 , X _ 3 ] = X _ 3 . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\Pi \\omega = \\underset { n \\rightarrow \\infty } { \\lim } \\varphi _ { \\omega | _ { n } } ( 0 ) \\omega \\in \\Omega . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m ( - 1 ) ^ k { n \\choose k } = ( - 1 ) ^ m { n - 1 \\choose m } . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} H ^ { ( i ) } ( t ) & \\le C _ 1 \\left ( \\lVert a ^ * _ 0 \\rVert _ { L ^ \\infty _ H L ^ p _ z } + K ^ { ( i ) } ( t ) H ^ { ( i ) } ( t ) + R ^ * t ^ { 1 / 2 } H ^ { ( i ) } ( t ) + R ^ * t ^ { 1 / 2 } K ^ { ( i ) } ( t ) \\right ) , \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\min f _ 0 ( x ) G ( x ) \\preceq 0 , h ( x ) = 0 , \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( n + 1 , k ) = k S _ { 2 , \\lambda } ( n , k ) + S _ { 2 , \\lambda } ( n , k - 1 ) - n \\lambda S _ { 2 , \\lambda } ( n , k ) , \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { 1 } { ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) a _ c ^ { ( n ) } } \\log P \\left ( \\frac { S _ n ( \\kappa _ n ( x ) ) } { ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) a _ c ^ { ( n ) } } < \\frac { \\varepsilon } { h ( x ) } \\right ) & \\geq - \\inf _ { y \\in \\left ( - \\infty , \\frac { \\varepsilon } { h ( x ) } \\right ) } H ( y ) \\\\ & = - H \\left ( \\frac { \\varepsilon } { h ( x ) } \\right ) , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} { \\cal C } _ { t o t } = ( v _ 1 , { \\cal E } _ 1 , v _ 2 , { \\cal E } _ 2 , \\ldots , { \\cal E } _ { k _ 1 - 1 } , v _ { k _ 1 } , { \\cal E } _ { k _ 1 } , v _ 1 = v _ { k _ 1 + 1 } ) , \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\widetilde { V } _ f ^ \\pm ( s ) = \\int _ 0 ^ \\infty V _ f ^ \\pm ( y ) y ^ { s - \\frac 1 2 } \\frac { d y } y = \\begin{cases} \\gamma _ f ^ \\pm ( s ) & k = 1 \\epsilon = \\pm 1 , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} L \\cdot F ( L ) \\cdot F ^ 2 ( L ) = w \\cdot M \\cdot F ( M ) . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } a ( g ) = \\left \\langle v , \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } M ( g ) w \\right \\rangle \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} \\sum _ { g \\in G } g ( M [ \\alpha ] ) g ^ { - 1 } & = \\sum _ { g \\in G } ( \\sum _ { h \\in G } m _ h g h ( \\alpha ) ) ( g h h ^ { - 1 } ) ^ { - 1 } \\\\ & = ( \\sum _ { h \\in G } m _ h h ) ( \\sum _ { g \\in G } g ( \\alpha ) g ^ { - 1 } ) . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\sum _ { w \\in \\Lambda ^ { m } } p _ { w } ^ { q } | r _ { w } | ^ { - \\tau } = 1 \\ : . \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\langle 1 , 1 \\rangle _ K = 1 \\mbox { a n d } \\langle f _ i x , y \\rangle _ K = \\langle x , e ' _ i ( y ) \\rangle _ K \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F ( n , k - 1 ) - \\sum _ { n = 0 } ^ { m - 1 } F ( n , k ) = G ( m , k ) . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\int _ \\Omega z _ { 0 \\lambda } u _ { 0 \\lambda } ' = - \\lambda \\int _ \\Omega z _ { 0 \\lambda } \\mu _ { 0 \\lambda } - \\int _ \\Omega \\nabla z _ { 0 \\lambda } \\cdot \\nabla \\mu _ { 0 \\lambda } \\leq \\frac \\lambda 2 \\int _ \\Omega | \\mu _ { 0 \\lambda } | ^ 2 + \\frac 1 2 \\int _ \\Omega | \\nabla \\mu _ { 0 \\lambda } | ^ 2 + \\norm { z _ { 0 \\lambda } } _ V ^ 2 \\ , , \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\Sigma ( n ) = \\frac { 1 } { n } \\left ( \\nu \\Sigma ( \\nu ) + \\sum _ { k = \\nu + 1 } ^ n \\sqrt { k } \\right ) . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\boldsymbol { L } \\varphi \\left ( x \\right ) = { \\textstyle \\int \\limits _ { \\mathcal { K } _ { N } } } \\left \\{ \\varphi \\left ( y \\right ) - \\varphi \\left ( x \\right ) \\right \\} J _ { N } \\left ( x , y \\right ) d y \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} H ( x ) = \\frac { 1 } { 2 d } \\sum _ { y : y \\sim x } H ( y ) \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} h _ { i + 1 } - h _ i < h _ { j + 1 } - h _ j , j \\neq i , j = 0 , \\dots , N + 1 , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} Y ^ 0 = r ^ 3 Y _ 0 ^ { ( 3 ) } + r ^ 4 Y _ 0 ^ { ( 4 ) } + O ( r ^ 5 ) . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\langle C _ \\varphi ^ * f , f \\rangle = & \\langle f _ 1 + \\mu f _ 2 + \\mu ^ 2 f _ 3 , f _ 1 + f _ 2 + f _ 3 \\rangle \\\\ = & | | f _ 1 | | ^ 2 - 2 | | f _ 2 | | | | f _ 3 | | \\delta _ 1 \\cos { ( \\theta _ 1 - \\frac { \\pi } { 3 } ) } \\\\ & + \\mu \\left ( | | f _ 2 | | ^ 2 - 2 | | f _ 1 | | | | f _ 3 | | \\delta _ 2 \\cos { ( \\theta _ 2 - \\frac { \\pi } { 3 } ) } \\right ) \\\\ & + \\mu ^ 2 \\left ( | | f _ 3 | | ^ 2 - 2 | | f _ 1 | | | | f _ 2 | | \\delta _ 3 \\cos { ( \\theta _ 3 - \\frac { \\pi } { 3 } ) } \\right ) . \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{gather*} \\{ \\Phi _ a , \\Phi _ b \\} = C ^ c _ { a b } \\Phi _ c , \\end{gather*}"}
-{"id": "7151.png", "formula": "\\begin{align*} \\bar H ^ { 1 ^ * } _ { , k } = 0 . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} \\Phi _ v ( \\bar { A } ) \\le \\max \\{ \\lambda : \\exists y , z \\in \\Delta _ { n - 1 } , I ( y ) \\subseteq I , I ( z ) \\subseteq J , \\bar { A } ( y - z ) = \\lambda d \\} . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} ( \\mu \\hat \\oplus \\mu ' ) \\left ( \\begin{array} { c } \\mathfrak c \\\\ \\updownarrow \\\\ \\mathfrak c ' \\end{array} \\right ) = \\mu \\left ( \\begin{array} { c } \\mathfrak c \\\\ \\updownarrow \\\\ \\mathfrak c ' \\end{array} \\right ) \\hat \\oplus \\mu ' \\left ( \\begin{array} { c } \\mathfrak c \\\\ \\updownarrow \\\\ \\mathfrak c ' \\end{array} \\right ) . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\sum \\limits _ { ( M _ S , \\mu _ S ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ S , b _ S } } i ^ G _ { M _ S } ( \\mathcal { M } _ { M _ S , b _ S , \\mu _ S } ) = \\mathcal { M } _ { G , b , \\mu } . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} \\| f \\| _ \\infty = \\inf _ { f ' \\sim f } | f ' | _ \\infty , \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} J [ v ] : = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla v | ^ 2 ~ d x v \\in H _ 0 ^ 1 ( \\O ) \\int _ { \\O } | v | ^ { p + 1 } ~ d x = 1 , \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} D ^ { ( P ) } _ l ( k ) = \\mathbb { E } d ( Z _ k , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) = D _ l ( k ) , \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } a _ { N 0 } ( z , q , Q ) = \\frac { 1 } { z ^ { N + 1 } } \\left ( ( - 1 ) ^ { N + 1 } \\lambda _ 0 \\cdots \\lambda _ N - Q \\right ) \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\psi _ f ' ( s ) - \\psi _ { \\bar { f } } ' ( 1 - s ) = \\psi ' ( s + \\nu ) + \\psi ' ( s - \\nu ) - X _ f ( s ) . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } [ 3 k + 1 ] { 2 k \\brack k } ^ 3 \\frac { ( - q ; q ) _ n ^ 4 } { ( - q ; q ) _ k ^ 4 } q ^ { - { k + 1 \\choose 2 } } \\equiv 0 \\pmod { ( 1 + q ^ n ) ^ 2 [ 2 n + 1 ] { 2 n \\brack n } } . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\N ( G , K _ m ) \\geq \\N ( T _ r ( n ) , K _ m ) - \\delta n ^ m \\geq \\binom { r } { m } \\left ( \\frac n r \\right ) ^ m - 2 \\delta n ^ m , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\vert \\mathcal { B } \\vert = 1 + ( n - \\vert \\alpha \\vert ) + 1 + ( \\vert \\alpha \\vert - 1 ) + 1 + 2 \\cdot ( \\frac { \\vert C ^ * \\vert - 2 } { 2 } ) = n + \\vert C ^ * \\vert = \\dim ( A ) \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\mathcal { C } : = \\big \\{ ( t , x ) \\in [ 0 , 1 ] \\times X \\quad \\| x \\| \\leq C \\big \\} . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\lim _ { x \\to x _ 0 } \\frac { Q _ 1 u ( x ) - Q _ 1 u ( x _ 0 ) - T _ 1 P _ 1 ( x - x _ 0 ) } { \\norm { x - x _ 0 } } = 0 . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} e _ k ( T ) : = \\inf \\Bigr \\{ r > 0 : \\exists y _ 1 , \\dots , y _ { 2 ^ { k - 1 } } \\in Y \\ \\ T ( B _ X ) \\subset \\bigcup _ { j = 1 } ^ { 2 ^ { k - 1 } } ( y _ j + r B _ Y ) \\Bigl \\} . \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\sigma ( z ) = z \\prod _ { \\omega \\in \\Lambda _ \\tau \\backslash 0 } \\left ( 1 - \\frac { z } { \\omega } \\right ) e ^ { z / \\omega + \\frac { 1 } { 2 } ( z / \\omega ) ^ 2 } , \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\limsup _ { | n ' | \\rightarrow \\infty } \\sup _ { \\xi \\in \\mathbb { T } ^ n } | a ( \\xi , n ' ) | = 0 . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} c _ { p , \\epsilon } ( M ) : = \\inf _ { h \\in \\Gamma _ p ( M ) } \\max _ { y \\in D _ 1 ^ 2 } E _ { p , \\epsilon } ( h _ y ) , \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} h ^ { * } ( x ) : = \\sup _ { v \\in \\R ^ n } \\{ \\langle v , x \\rangle - h ( v ) \\} \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\delta _ l \\times ( \\eta _ j , 2 a _ j + 1 ) ) = \\prod _ { i = 0 } ^ { 2 a _ j } L ^ S ( s + \\frac { 1 } { 2 } + a _ j - i , \\delta _ l \\times \\eta _ j ) , \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} T _ { 1 } T _ { 2 } = \\{ \\{ f , T _ { 1 } \\varphi \\} : \\ , \\{ f , \\varphi \\} \\in T _ { 2 } \\} , \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} t _ G ( n ) \\ = \\sum _ { \\Pi \\in A ( G ) } u _ \\Pi ( n ) \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} | x _ { N _ 1 + 2 + m } | \\le | x _ { N _ 1 + 2 + m - 1 } | + | u _ { N _ 1 + 2 + m } | \\le | x _ { N _ 1 + 2 + k _ { l _ m } } | + \\sum _ { i = k _ { l _ m } } ^ { m } | u _ { N _ 1 + 2 + i } | , \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\psi ( C ) : = \\min _ { a \\in \\mathbb { Z } , b \\in \\mathbb { Z } , c \\in \\mathbb { Z } } \\bigg \\{ \\frac { c } { b } : \\ a ^ 2 + b ^ 2 = c ^ 2 , c \\le C , \\ a , b , c \\ge 1 \\bigg \\} . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} { \\mathfrak K } = A ( y ^ 0 \\frac { \\partial } { \\partial y ^ 4 } - y ^ 4 \\frac { \\partial } { \\partial y ^ 0 } ) - B _ i ( y ^ 0 \\frac { \\partial } { \\partial y ^ i } + y ^ i \\frac { \\partial } { \\partial y ^ 0 } ) - C _ j ( y ^ 4 \\frac { \\partial } { \\partial y ^ j } + y ^ j \\frac { \\partial } { \\partial y ^ 4 } ) + D _ p \\epsilon _ { p q r } y ^ q \\frac { \\partial } { \\partial y ^ r } . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} R _ 1 & = \\Delta u _ 1 + \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 , \\\\ R _ 2 & = \\Delta u _ 2 + \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} \\sum _ { j \\le k } \\binom { s } { j } 4 ^ { j } . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} \\beta \\leq \\min _ { 1 \\leq i \\leq n } \\left \\{ \\nu \\left ( a _ i \\textbf Q ^ { \\lambda _ i } \\right ) \\right \\} = \\min _ { 1 \\leq i \\leq n } \\left \\{ \\nu _ Q \\left ( a _ i \\textbf Q ^ { \\lambda _ i } \\right ) \\right \\} \\leq \\nu _ Q ( p ) \\leq \\nu ( p ) = \\beta . \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} { n \\choose k _ 1 , k _ 2 } = \\frac { 1 } { k _ 1 ! k _ 2 ! } n ( n - 1 ) \\ldots ( n - k _ 1 - k _ 2 + 1 ) \\leq \\frac { n ^ { k _ 1 + k _ 2 } } { k _ 1 ! k _ 2 ! } \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} & t ^ { \\frac { N + 2 A _ k } { 2 } } \\partial _ r ^ \\ell \\frac { [ e ^ { - t L _ V ^ k } \\phi ^ { k , i } ] ( | x | ) } { U _ k ( | x | ) } \\\\ & = \\left [ M _ { k , i } + o ( 1 ) \\right ] \\delta _ { 0 \\ell } - \\left [ \\frac { N + 2 A _ k } { 2 } M _ { k , i } + o ( 1 ) \\right ] t ^ { - 1 } ( \\partial _ r ^ \\ell F _ k ) ( | x | ) + t ^ { - 2 } O ( | x | ^ { 4 - \\ell } ) \\\\ & = \\left [ M _ { k , i } + o ( 1 ) \\right ] \\delta _ { 0 \\ell } + O ( t ^ { - 1 } | x | ^ { 2 - \\ell } ) \\quad \\mbox { a s } t \\to \\infty , \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} ( \\mathcal { D } _ 0 ^ \\theta g ) ( x ) = \\varphi _ g ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\Sigma ( n ) > A ( n ) - \\frac { 1 } { 6 n } - \\frac { 1 } { 1 6 n } = A ( n ) - \\frac { 1 1 } { 4 8 n } . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} F ( \\lambda ) = \\left ( \\int _ 0 ^ \\infty \\frac { \\lambda ^ { t - 1 } } { \\Gamma ( t ) } \\ , d t \\right ) e ^ { - \\lambda } , \\lambda > 0 . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} \\lambda : = \\frac { 1 } { 2 a _ \\alpha \\vert \\alpha \\vert } \\ \\ \\mu ^ 2 : = \\frac { \\lambda - \\lambda ^ 2 } { c _ { \\alpha } } . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} \\left . \\frac { \\partial U _ { t } ^ { u } } { \\partial u } \\right | _ { u = 0 } \\cdot v \\neq { \\rm e } ^ { - U _ { t } ^ { 0 } } \\otimes v \\otimes { \\rm e } ^ { U _ { t } ^ { 0 } } , \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} T \\colon L ^ { p _ 1 } ( \\R ^ d ; X _ 1 ) \\times & L ^ { p _ 2 } ( \\R ^ d ; X _ 2 ) \\times \\prod _ { k = 3 } ^ n L ^ { p _ j } ( \\R ^ d ) \\to L ^ { q _ { n + 1 } } ( \\R ^ d ; Y _ 3 ) , \\\\ & { 1 < p _ k \\leq \\infty , \\ , \\frac { 1 } { n } < q _ { n + 1 } < \\infty } , \\ , { \\textstyle \\frac { 1 } { q _ { n + 1 } } } = \\sum _ { k = 1 } ^ n { \\textstyle \\frac { 1 } { p _ k } } . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} Q ^ { \\lambda } _ { \\epsilon } ( x - \\cdot ) : = \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda s } q _ { s + \\epsilon } ^ { 1 } ( x - \\cdot ) d s = e ^ { \\lambda \\epsilon } \\int _ { \\epsilon } ^ { \\infty } e ^ { - \\lambda t } q _ t ^ { 1 } ( x - \\cdot ) d t \\in C _ b ^ { \\infty } ( \\R ^ d ) . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\sup _ { m \\geq 1 } \\| \\dot { \\tilde { h } } ^ { ( m ) } \\| _ { \\infty ; [ 0 , a _ { m + 1 } ] } = \\sup _ { m \\geq 1 } \\left \\{ \\| \\dot { \\bar { h } } _ m \\| _ { \\infty ; [ 0 , 1 ] } \\right \\} \\leq C _ { l _ 0 } , \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} \\Xi _ { P } ^ { \\vec { p } } ( \\phi , \\gamma ) = \\mathrm { C S } ^ { \\vec { p } } ( A _ { \\phi } ^ { \\gamma } ) \\in \\mathbb { R } / \\mathbb { Z } . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} \\begin{cases} u _ \\epsilon ( x ) = \\widetilde { u } _ 0 ( x ) + \\epsilon \\widetilde { u } _ 1 ( x , \\frac { x } { \\epsilon } ) , \\\\ m _ \\epsilon ( x ) = \\widetilde { m } _ 0 ( x ) ( \\widetilde { m } _ 1 ( x , \\frac { x } { \\epsilon } ) + \\epsilon m _ 2 ( x , \\frac { x } { \\epsilon } ) ) \\end{cases} \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\Sigma , L } ( e _ 1 , e _ 2 ) = \\mathcal { K } ^ { L } ( e _ 1 , e _ 2 ) + { \\rm d e t } ( I I ^ L ) . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\Xi ( x , y ) & = \\left | w _ - ( x ) - w _ - ( y ) \\right | ^ 2 - \\left | \\left ( w _ - ( x ) - w _ - ( y ) \\right ) + \\eta ( y ) w _ - ( y ) \\right | ^ 2 \\\\ & \\ge - 2 \\eta ( y ) w _ - ( x ) w _ - ( y ) . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\mathcal { A } v ( x ) = - \\sum _ { m = 1 } ^ N \\frac { \\partial } { \\partial x _ m } \\left ( \\sum _ { n = 1 } ^ N a _ { m n } ( x ) \\frac { \\partial } { \\partial x _ n } v ( x ) \\right ) + q ( x ) v ( x ) , x \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\bar { F } _ { \\mathit { S I N R } } ( \\theta ) & = \\mathbb { E } _ N { \\left [ \\ , \\mathbb { P } ( \\ , \\mathit { S I N R } > \\theta \\mid n \\ , ) \\ , \\right ] } \\\\ & = \\sum _ { n = 0 } ^ \\infty f _ N ( n ) \\mathbb { P } ( \\ , \\mathit { S I N R } > \\theta \\mid n \\ , ) , \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} V _ A ^ { \\perp } & = - \\cos ( \\theta _ 1 + \\theta _ 2 ) ( J X _ 1 + J X _ 2 ) = - 2 J \\nabla ( \\sin ( \\theta _ 1 + \\theta _ 2 ) ) , \\\\ V _ B ^ { \\perp } & = - \\sin ( \\theta _ 1 + \\theta _ 2 ) ( J X _ 1 + J X _ 2 ) = 2 J \\nabla ( \\cos ( \\theta _ 1 + \\theta _ 2 ) ) , \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) & : = \\{ \\gamma \\in { \\bf O R } : \\alpha < \\gamma < \\beta \\} , \\\\ [ \\alpha , \\beta ) & : = \\{ \\gamma \\in { \\bf O R } : \\alpha \\leq \\gamma < \\beta \\} . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\mathbb { E } [ Z _ i ^ 2 | Y _ i \\ ! = \\ ! y _ i ] \\ ! \\ ! = \\ ! \\ ! \\ ! \\sum _ { w = 0 } ^ { y _ i } \\ ! \\ ! { { y _ i } \\choose { w } } \\delta ^ { y _ i \\ ! - \\ ! w } ( 1 \\ ! \\ ! - \\ ! \\ ! \\delta ) ^ { \\ ! w } \\frac { g ( w ) ( g ( w ) \\ ! + \\ ! ( 1 \\ ! \\ ! - \\ ! q ) ) } { q ^ 2 } \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} P _ { \\hat { F } _ 1 ^ { \\star n } | \\hat { X } _ 1 ^ n = \\hat { x } _ 1 ^ n } = \\prod _ { i = 1 } ^ n P _ { \\hat { F } _ i ^ { \\star } | \\hat { X } _ i = \\hat { x } _ i } , \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\vert u ( x _ k ) \\vert & \\leq \\sum _ { i = 0 } ^ { k - k _ 0 - 1 } 2 ^ { m _ 0 ( s - s ' ) - \\frac { ( k - i - 1 ) p ( s - s ' ) } { Q } } \\bigg ( \\sum _ { j = - \\infty } ^ { m _ 0 - ( k - i - 1 ) p / Q } 2 ^ { j s ' p } \\Big [ g _ j ( x _ { k - i } ) + g _ j ( x _ { k - i - 1 } ) \\Big ] ^ p \\bigg ) ^ { 1 / p } \\\\ & \\qquad + \\vert u ( x _ { k _ 0 } ) \\vert \\\\ & \\leq C 2 ^ { m _ 0 ( s - s ' ) } \\sum _ { i = 0 } ^ { k - k _ 0 - 1 } 2 ^ { - \\frac { ( k - i - 1 ) p ( s - s ' ) } { Q } } 2 ^ { k - i } + \\vert u ( x _ { k _ 0 } ) \\vert . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\psi _ n = ( \\eta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ k , 1 ) \\boxplus ( \\delta _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\delta _ l , 2 ) . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} d i v ( A _ { \\epsilon } \\nabla ( \\gamma _ { \\epsilon } ^ p ) ) = p \\gamma _ { \\epsilon } ^ { p - 2 } [ \\langle A _ { \\epsilon } , H e s s ( \\varphi _ { \\epsilon } ) ^ 2 \\rangle + R i c ( d \\varphi _ { \\epsilon } , d \\varphi _ { \\epsilon } ) ] , \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\chi ( k _ 0 , \\xi ) & = \\lim \\limits _ { k \\to k _ 0 } ( k - k _ 0 ) ^ { \\i \\nu } \\cdot \\exp \\left [ \\frac { 1 } { 2 \\pi \\i } \\int \\limits _ { - \\infty } ^ { k _ 0 } \\frac { \\ln ( 1 + | r ( s ) | ^ 2 ) \\ \\d s } { s - k } \\right ] \\\\ & = ( k + N ) ^ { \\i \\nu } \\cdot \\exp \\left [ \\frac { 1 } { 2 \\pi \\i } \\int \\limits _ { - N } ^ { k _ 0 } \\frac { \\ln \\frac { 1 + | r ( s ) | ^ 2 } { 1 + | r ( k _ 0 ) | ^ 2 } \\ \\d s } { s - k _ 0 } + \\frac { 1 } { 2 \\pi \\i } \\int \\limits _ { - \\infty } ^ { - N } \\frac { \\ln ( 1 + | r ( s ) | ^ 2 ) \\ \\d s } { s - k } \\right ] \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} u _ { p } ( y _ { i , p } ) : = \\max _ { \\overline { B _ { 2 r } ( x _ i ) } } u _ { p } \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} { \\alpha \\circledast \\beta } ( s ) : = \\left \\{ \\begin{array} { l } \\alpha ( 2 s ) , \\ : \\ : 0 \\leq s \\leq \\frac { 1 } { 2 } , \\\\ \\beta ( 2 s - 1 ) , \\ : \\ : \\frac { 1 } { 2 } \\leq s \\leq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} | H | = \\frac { 2 } { r } + ( W _ 0 + 1 ) r + O ( r ^ 2 ) . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ { 6 + } ( x ) + \\frac 2 { 6 ^ 7 } ( 3 ^ 7 x - 9 1 1 ) ( 5 4 - 2 ^ 7 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 6 2 7 } { 6 3 0 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 7 } - \\eta \\\\ & = - \\frac { 2 9 6 1 5 2 8 6 4 1 8 0 2 8 3 6 2 6 3 4 3 1 9 8 6 7 0 5 3 1 3 9 6 7 2 1 } { 3 5 7 8 0 7 6 5 6 8 1 0 2 5 8 6 2 7 8 0 6 4 2 5 2 3 7 4 5 8 3 2 0 6 8 8 5 4 4 0 0 0 } < 0 , \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} q _ { { \\cal X } _ n } = \\dfrac { 1 } { 2 } \\min _ { i \\neq k } \\left \\| \\boldsymbol { x } _ i - \\boldsymbol { x } _ k \\right \\| _ 2 . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} \\mathrm { r e s } ( x - \\sigma _ { \\mathrm { r e s } ( \\alpha ) } ( x ) ) = 2 ( a _ 1 + b _ 1 + . . . + a _ n + b _ n ) \\mathrm { r e s } ( \\alpha ) . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} h '' ( s ) & = K u \\Gamma ( u + 2 ) s ^ { - u - 2 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 2 } ) + O ( s ^ { - 2 } ) \\\\ & = ( 1 + O ( s ^ { \\eta } ) ) K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) s ^ { - u - 2 } . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\abs { u _ k ( z ) } = O ( a _ k r ^ { - n _ k } ) , \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} \\mathcal { U } = \\{ w \\in \\Lambda ^ { n m } \\ : : \\ : [ w ] \\cap \\Pi ^ { - 1 } ( B ( x , 2 ^ { n m \\chi } ) ) \\ne \\emptyset \\mu _ { 0 } [ w ] > 0 \\} \\ : . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{gather*} \\omega = \\omega _ { \\rm c a n } + B , \\end{gather*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d y } { d t } = & \\Delta y ( t ) + \\Gamma ^ { - 1 } ( t ) [ K ( \\Gamma ( t ) y ( t ) ) \\cdot \\nabla ] ( \\Gamma ( t ) y ( t ) ) , \\ t > 0 ; \\ y ( 0 ) = U _ 0 = c u r l \\ x . \\end{aligned} \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} F ( x + h ) - F ( x ) \\leq \\lambda _ { 1 } ^ { ( k ) } \\left ( g \\left ( x _ { 1 } ^ { ( k ) } + \\frac { h } { \\lambda _ { 1 } ^ { ( k ) } } \\right ) - g ( x _ { 1 } ^ { ( k ) } ) \\right ) + \\left ( \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ { i } ^ { ( k ) } g ( x _ { i } ^ { ( k ) } ) - F ( x ) \\right ) , \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = \\lambda x z \\\\ \\dot y = - y z \\\\ \\dot z = y ^ 2 - \\lambda x ^ 2 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} \\mu _ { \\beta , h } ^ { \\Lambda , \\sigma } ( \\{ \\omega _ { \\Lambda } \\} ) : = \\frac { 1 } { Z _ { \\beta , h } ^ { \\Lambda , \\sigma } } e ^ { - H _ { \\beta , h } ^ { \\Lambda , \\sigma } ( \\omega _ { \\Lambda } ) } , Z _ { \\beta , h } ^ { \\Lambda , \\sigma } : = \\sum _ { \\omega _ { \\Lambda } \\in \\tilde { \\Omega } _ { \\Lambda } } e ^ { - H _ { \\beta , h } ^ { \\Lambda , \\sigma } ( \\omega _ { \\Lambda } ) } , \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\| f _ 1 \\ast \\cdots \\ast f _ k \\| _ { L ^ r } \\leq \\left ( \\prod _ { l = 1 } ^ k C _ { p _ l } \\right ) ^ n \\| f _ 1 \\| _ { L ^ { p _ 1 } } \\cdots \\| f _ k \\| _ { L ^ { p _ k } } \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ) : = \\sup \\lbrace \\mathrm { d p } ( \\mathcal { B } ( n ) ) \\mid n \\ge n _ \\ast \\rbrace . \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\nu _ \\varepsilon ^ { 2 / p } u _ \\varepsilon ( y _ \\varepsilon ) = O ( 1 ) \\ , , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\liminf _ { n \\rightarrow \\infty } \\P \\Big ( \\frac { \\lambda _ { j - 1 } ^ { ( n ) } - \\hat \\lambda _ j ^ { ( n ) } } { \\lambda _ { j - 1 } ^ { ( n ) } - \\lambda _ j ^ { ( n ) } } \\leq \\delta \\Big ) > 0 . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} - C \\ln q _ n \\leq \\sum _ { k = 0 , k \\neq k _ 0 } ^ { q _ n - 1 } \\ln | \\sin \\pi ( x + k \\alpha ) | + ( q _ n - 1 ) \\ln 2 \\leq C \\ln q _ n . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} I _ a ^ k \\left ( D _ a ^ { k ' } f \\right ) ( x ) = ( I _ a ^ k \\varphi ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\| u _ \\lambda ( 0 ) \\| _ { \\dot { H } ^ \\gamma } = \\lambda ^ { \\gamma + \\frac { 2 s } { \\alpha } - \\frac { d } { 2 } } \\| u _ 0 \\| _ { \\dot { H } ^ \\gamma } . \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} C = \\sup _ { k > \\bar k , x \\in \\R } \\left ( \\left | \\mu _ 1 ( 1 - 2 u _ { 1 , k } ) \\right | + \\left | \\mu _ 2 ( 1 - 2 u _ { 2 , k } ) \\right | \\right ) . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\mu _ { n , i } = 1 . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} 4 \\delta ^ { 2 } V ( \\delta ) = h ^ { 2 } . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\mathcal { L } ^ { 0 , \\hat { v } } \\psi _ k ^ { 0 , \\hat { v } } ( x ) & = \\bigl \\langle \\bigtriangledown _ x \\psi _ k ^ { 0 , \\hat { v } } ( x ) , f _ k ( x , \\hat { v } _ k ( x ) ) \\bigr \\rangle + \\sum \\nolimits _ { j = 1 } ^ n \\gamma _ { k j } ( x ) \\bigl [ \\psi _ j ^ { 0 , \\hat { v } } ( x ) - \\psi _ k ^ { 0 , \\hat { v } } ( x ) \\bigr ] \\\\ & = 0 , x \\in D , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} z ( P _ n ) & = \\sum _ { l = 0 } ^ { r - 1 } \\Big ( ( a _ l - a _ { l + 1 } ) ( p ^ l + \\dots + p ^ l ) + \\dots a _ r p ^ r \\Big ) + a _ r p ^ r \\\\ & = \\sum _ { l = 0 } ^ { r - 1 } \\Big ( ( l + 1 ) ( a _ l - a _ { l + 1 } ) p ^ l \\Big ) + ( r + 1 ) a _ r p ^ r \\\\ & = a _ 0 - a _ 1 + \\left ( \\sum _ { l = 1 } ^ { r - 1 } a _ l ( l + 1 ) p ^ l \\right ) - \\left ( \\sum _ { l = 1 } ^ { r - 1 } a _ { l + 1 } ( l + 1 ) p ^ l \\right ) + ( r + 1 ) a _ r p ^ r \\\\ & = a _ 0 + \\sum _ { l = 1 } ^ { r } a _ l \\Big ( ( l + 1 ) p ^ l - l p ^ { l - 1 } \\Big ) , \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} u V _ N = 0 \\mathrm { a n d } u V = V _ R \\ , . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 1 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = - i ( R _ { 1 2 } ( - 1 ) - i R _ { 1 3 } ) = i R _ { 1 2 } - R _ { 1 3 } = i + i = 2 i = 2 u _ 1 ; \\\\ \\frac { R _ { 4 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 4 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = i R _ { 4 2 } - R _ { 4 3 } = 0 - ( - 2 ) = 2 = 2 u _ 4 . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { k , a } ( n ) q ^ n = \\frac { ( - q ^ 2 ; q ) _ \\infty ( q ^ { a + 1 } , q ^ { 2 k - a - 1 } , q ^ { 2 k } ; q ^ { 2 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } + \\frac { x q ( - q ^ 2 ; q ) _ \\infty ( q ^ { a - 1 } , q ^ { 2 k - a + 1 } , q ^ { 2 k } ; q ^ { 2 k } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} H _ 2 = ( 1 , \\dots , 1 , - 1 \\ , | \\ , 1 , \\dots , 1 ) . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} \\left ( J + F - \\lambda I \\right ) ^ { - 1 } = ( J + F - K \\widetilde { P } _ M - \\lambda I ) ^ { - 1 } \\left [ I + K \\widetilde { P } _ M ( J + F - K \\widetilde { P } _ M - \\lambda I ) ^ { - 1 } \\right ] ^ { - 1 } \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 1 ( x ) + \\frac { 2 } { 6 ^ 2 } ( 3 ^ 2 x - 3 ) ( 2 - 2 ^ 2 x ) + \\frac { 2 } { 6 ^ 3 } ( 2 ^ 3 x - 3 ) ( 1 1 - 3 ^ 3 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 } { 5 } } \\\\ & + \\frac 1 { 1 0 \\cdot 6 ^ 3 } - \\eta = - \\frac { 4 0 6 2 3 8 6 4 9 3 4 0 7 9 0 2 7 7 7 5 3 1 0 8 2 3 3 8 9 1 0 7 } { 7 2 9 8 0 7 4 4 1 0 4 1 9 9 6 3 3 3 1 0 4 9 4 2 4 2 7 3 3 5 5 8 0 0 0 } < 0 , \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\mathbf { L } _ { \\nu - 1 } ( x ) - \\mathbf { L } _ { \\nu + 1 } ( x ) & = \\frac { 2 \\nu } { x } \\mathbf { L } _ { \\nu } ( x ) + \\frac { \\big ( \\frac { 1 } { 2 } x \\big ) ^ \\nu } { \\sqrt { \\pi } \\Gamma ( \\nu + \\frac { 3 } { 2 } ) } , \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } x } \\big ( x ^ { \\nu } \\mathbf { L } _ { \\nu } ( x ) \\big ) & = x ^ { \\nu } \\mathbf { L } _ { \\nu - 1 } ( x ) . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} Z ( \\alpha ) = : \\begin{pmatrix} a ( \\alpha ) & c ( \\alpha ) & 0 & d ( \\alpha ) \\cr c ( \\alpha ) & b ( \\alpha ) & c ( \\alpha ) & 0 \\cr 0 & c ( \\alpha ) & b ( \\alpha ) & c ( \\alpha ) \\cr d ( \\alpha ) & 0 & c ( \\alpha ) & a ( \\alpha ) \\end{pmatrix} , \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\phi _ n ^ { ( k ) } ( z ) : = \\ _ 2 \\phi _ 2 \\left ( \\begin{matrix} q ^ { - n } , q ^ { \\gamma + k } \\\\ q ^ { \\delta + 1 } , q ^ \\gamma \\end{matrix} ; q , - q ^ { n + \\delta + 1 } z \\right ) , \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} e ^ { - t A _ k } f ( x ) = = k _ t * _ k f = \\int _ { \\mathbb { R } ^ d } K _ t ( x , y ) f ( y ) w _ k ( y ) d y \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} l ( G ) = \\O ( | B | ) + 2 = 2 \\O ( q - 1 ) + 3 f + 2 - \\O ( ( 3 , q - 1 ) ) \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} b _ { \\overline { \\mathbf { a } } _ { k } } \\left ( \\overline { \\mathbf { B } } _ { k } , \\overline { \\mathbf { G } } _ { k } \\mathbf { ; } P \\right ) = b _ { \\overline { \\mathbf { a } } _ { k } } \\left ( \\overline { \\mathbf { G } } _ { k } \\mathbf { ; } P \\right ) . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} ( ( \\mathrm { L i e } \\ , \\theta ) \\omega ) ( \\theta _ 1 , \\hdots , \\theta _ n ) = \\theta ( \\omega ( \\theta _ 1 , \\hdots , \\theta _ n ) ) - \\sum _ { i = 1 } ^ n \\omega ( \\theta _ 1 , \\hdots , [ \\theta , \\theta _ i ] , \\hdots , \\theta _ n ) \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} x = x _ { - k } p ^ { - k } + x _ { - k + 1 } p ^ { - k + 1 } + \\ldots + x _ { 0 } + x _ { 1 } p + \\ldots , x _ { - k } \\neq 0 \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} \\frac { c _ { k + 1 } ^ - } { | c _ { k + 1 } ^ - | } = \\left ( \\frac { c _ { k } ^ - } { | c _ { k } ^ - | } \\right ) ^ n . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\phi = \\frac { \\theta _ 1 + \\theta _ 2 } { 2 } \\quad \\quad \\rho = \\frac { \\theta _ 1 - \\theta _ 2 } { 2 } . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\begin{bmatrix} a ^ 3 & - a ^ 2 \\bar { b } & a \\bar { b } ^ 2 & - \\bar { b } ^ 3 \\\\ 3 a ^ 2 b & a ( 3 | a | ^ 2 - 2 ) & \\bar { b } ( 1 - 3 | a | ^ 2 ) & 3 \\bar { a } \\bar { b } ^ 2 \\\\ 3 a b ^ 2 & b ( 3 | a | ^ 2 - 1 ) & \\bar { a } ( 3 | a | ^ 2 - 2 ) & - 3 \\bar { a } ^ 2 \\bar { b } \\\\ b ^ 3 & \\bar { a } b ^ 2 & \\bar { a } ^ 2 b & \\bar { a } ^ 3 \\end{bmatrix} . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} S _ { I , \\varepsilon , \\ell } ( T ) : = \\sum _ { ( n , \\alpha ) \\in \\Delta \\cap \\mathbb N ^ { r + 1 } } \\sum _ { \\begin{smallmatrix} \\beta \\in \\mathbb N ^ m \\\\ \\theta ' ( \\beta , \\alpha ) \\end{smallmatrix} } \\sum _ { \\begin{smallmatrix} ( k _ j ) _ { j \\in I } \\in \\mathbb N _ { > 0 } ^ I \\\\ ( \\ref { e q 3 . 1 2 } ) _ { n , \\beta } \\end{smallmatrix} } \\L ^ { - \\sum _ { j \\in I } k _ j \\nu _ j } \\L ^ { - \\varepsilon ( n , \\alpha ) } T ^ { \\ell ( n , \\alpha ) } . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} G / B _ + = \\coprod _ { w \\in W } ( B _ + w B _ + ) / B _ + = \\coprod _ { u \\in W } ( B _ - u B _ + ) / B _ + \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} & \\alpha ( h _ 0 + v ) - \\alpha ( h _ 0 ) \\\\ & = A v - T \\big ( \\tilde { \\alpha } ( t _ 0 + s , \\langle \\zeta , h _ 0 - \\phi ( t _ 0 ) + v \\rangle ) - \\tilde { \\alpha } ( t _ 0 + s , \\langle \\zeta , h _ 0 - \\phi ( t _ 0 ) \\rangle ) \\big ) \\in V , \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} - h ' ( s ) & = \\sum _ { m = 1 } ^ { \\infty } - \\phi ' ( m s ) \\\\ & = \\sum _ { m \\leq s ^ { - 1 / 2 } } - \\phi ' ( m s ) + \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } - \\phi ' ( m s ) + \\sum _ { m > s ^ { - 1 } } - \\phi ' ( m s ) \\\\ & = : \\chi _ 1 ( s ) + \\chi _ 2 ( s ) + \\chi _ 3 ( s ) . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\bar { \\boldsymbol { \\psi } } ( t _ { k + l } ) \\ ! = \\ ! \\bar { \\boldsymbol { \\psi } } ( t _ k ) \\ ! + \\ ! \\ ! \\sum _ { i = 1 } ^ { l } \\ ! b _ { S , k + i } \\mathbf { f } _ { \\boldsymbol { \\psi } } \\left ( \\bar { \\boldsymbol { \\psi } } ( t _ { n + i \\ ! - \\ ! 1 } ) \\right ) \\ ! + \\ ! ( \\boldsymbol { \\xi } _ { k + l } \\ ! - \\ ! \\boldsymbol { \\xi } _ { k } ) \\ ! , \\ ! \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} u ( x , t ) = \\phi ( n \\cdot x - c t - \\theta _ 0 , t ) \\ \\ { \\rm f o r \\ a l l } \\ \\ ( x , t ) \\in \\R ^ N \\times \\R . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} d _ i ^ \\circ = \\sqrt { \\frac { B \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { w _ i ( c _ i + \\lambda _ i ) } } - \\frac { \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { w _ i } . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\mathcal { L } Q _ x = 0 \\ , \\mathcal { L } ( Q + x Q ' ) = - Q \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\Sigma ' ( W ) = \\left \\{ g \\in G ( \\mathbb { R } ) \\mathop { | } V \\cap g ^ { - 1 } . \\mathcal { F } \\neq \\emptyset \\right \\} . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\sup _ { \\psi \\in C _ 0 ( \\bar { D } ) , \\ , \\Vert \\psi \\Vert < 1 } \\left \\Vert \\mathcal { T } _ { t , k } ^ { \\epsilon , v _ k } \\psi ( x ) \\right \\Vert = \\sup _ { x \\in D } \\mathbb { P } _ { x , k } ^ { \\epsilon , v _ k } \\Bigl \\{ \\tau _ { k } ^ { \\epsilon , v _ k } > t \\Bigr \\} , \\ , \\ , \\ , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\textstyle \\sum _ { j \\in J } c _ { j \\bullet } = 0 \\textstyle \\sum _ { k \\in K } c _ { k \\bullet } = 0 \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} N = \\sum _ { \\mu \\in M _ + } \\lambda _ { \\mu } T ^ \\mu \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} W = W _ { 1 } \\circ C \\circ W _ { 2 } \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} \\sigma ^ { \\alpha } \\omega ( \\sigma ^ { \\nu } ) = \\sigma ^ { \\alpha ( 1 + \\nu ) } \\leq \\omega ( \\sigma ^ { 1 + \\nu } ) . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} W _ m = \\left [ \\begin{array} { c c c c } c _ 0 & c _ { n - 1 } & \\ldots & c _ 1 \\\\ c _ { 1 } & c _ 0 & \\ldots & c _ 2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { n - 1 } & c _ { n - 2 } & \\ldots & c _ 0 \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} G ( \\zeta ^ m ) = \\sum _ { k = 1 } ^ { n - 1 } \\frac { \\zeta ^ { m k } } { ( 1 - \\zeta ^ { 2 m k } ) ^ 2 } = \\sum _ { k = 1 } ^ { n - 1 } \\frac { \\zeta ^ { k } } { ( 1 - \\zeta ^ { 2 k } ) ^ 2 } = G ( \\zeta ) , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\chi _ { \\mu } = \\sum _ { \\lambda \\in \\Lambda } \\mu [ \\lambda ] \\log | r _ { \\lambda } | \\ : . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} F ( h ) ( x ) = F ( f ) ( b ) + F ( f ) ( x - b ) = 0 . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} ( 1 + t ) ^ 3 \\| U _ { m + 1 } '' ( t ) \\| ^ 2 = ( 1 + t ) ^ 3 \\| w ' ( t ) \\| ^ 2 \\in L ^ 1 ( ( 0 , \\infty ) ) . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\mathcal { L } _ { k } ^ { \\epsilon , v _ k } \\psi ( x ) = \\bigl \\langle \\bigtriangledown _ x \\psi ( x ) , f _ k ( x , v _ k ^ { \\epsilon } ( x ) ) \\bigr \\rangle + \\frac { \\epsilon } { 2 } \\operatorname { t r } \\bigl \\{ a _ k ( x ) \\bigtriangledown _ x ^ 2 \\psi ( x ) \\bigr \\} , \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\theta _ \\varepsilon \\ , w \\in \\Omega _ \\varepsilon \\mbox { \\ a n d \\ } \\lim _ { \\varepsilon \\to 0 ^ + } \\varepsilon / \\theta _ \\varepsilon = 0 . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} \\P \\bigl [ \\bigr ] & \\leq \\sum _ { i = 1 } ^ k e ^ { - \\beta ( k - i + 1 ) } \\leq \\frac { e ^ { - \\beta } } { 1 - e ^ { - \\beta } } , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\P ( A \\mid G ) - \\P ( A \\mid H ) = \\frac { p _ 1 q _ 1 + p _ 2 q _ 2 } { q _ 1 + q _ 2 } - \\frac { p _ 2 q _ 2 + p _ 3 q _ 3 } { q _ 2 + q _ 3 } \\leq \\frac { q _ 1 + p _ 2 q _ 2 } { q _ 1 + q _ 2 } - \\frac { p _ 2 q _ 2 } { q _ 2 + q _ 3 } . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} \\left | f _ 0 ^ { ( r - q ) } ( x ) \\prod _ { s = 1 } ^ q f _ 0 ^ { ( s - 1 ) } ( x ) ^ { m _ s } \\right | & \\leq R ^ \\frac { r - q } { \\beta } f _ 0 ( x ) ^ \\frac { \\beta - ( r - q ) } { \\beta } \\prod _ { s = 1 } ^ q R ^ \\frac { ( s - 1 ) m _ s } { \\beta } f _ 0 ( x ) ^ \\frac { ( \\beta - s + 1 ) m _ s } { \\beta } \\\\ & = R ^ \\frac { r - M _ q } { \\beta } f _ 0 ( x ) ^ \\frac { \\beta - r + ( \\beta + 1 ) M _ q } { \\beta } . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} c _ k = M ^ 2 \\ , y _ { i _ k } ^ { - 2 / ( a , b ) } ( 1 \\leq k \\leq K ) \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} \\kappa ^ n _ j ( \\alpha ) = \\left \\{ \\begin{array} { l l } \\frac { \\displaystyle ( - 1 ) ^ { ( n - j ) / 2 } 2 ^ j \\Gamma ( \\alpha + 1 + ( n + j ) / 2 ) } { \\displaystyle \\Gamma ( \\alpha + 1 ) \\Gamma ( j + 1 ) \\Gamma ( 1 + ( n - j ) / 2 ) } , & \\mbox { f o r } \\ \\ n - j \\ { \\rm e v e n } , \\\\ 0 , & \\mbox { f o r } \\ \\ n - j \\ { \\rm o d d } . \\end{array} \\right . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} & ( n - 1 ) ! \\lim _ { X \\to \\infty } \\frac { \\# \\{ p \\in S p l _ X ( f ) \\mid r _ i / p < a \\} } { \\# S p l _ X ( f ) } \\\\ = & \\sum _ { 0 \\le h \\le n \\atop 1 \\le l \\le n - 1 } \\sum _ { k = i } ^ n ( - 1 ) ^ { h + k + n } { n \\choose k } \\sum _ { m = 1 } ^ { n - 1 } { k \\choose n - h - m + l } { n - k \\choose m - l } M ( l - h a ) ^ { n - 1 } , \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } e _ { x _ 1 } \\cdots e _ { x _ { i - 1 } } ( 1 - x _ i ) e _ { x _ { i + 1 } } \\cdots e _ { x _ n } \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\textstyle \\sum _ j k _ { j i } P _ { j i } + \\sum _ { a , b , j } c _ { a b } c _ { i j } D _ { e _ { i j } } P _ { a b } = 0 \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} f ( x + i y ) = \\sum _ { n = 1 } ^ \\infty \\frac { \\lambda _ f ( n ) } { \\sqrt { n } } \\bigl ( V _ f ^ + ( n y ) \\cos ( 2 \\pi n x ) + i V _ f ^ - ( n y ) \\sin ( 2 \\pi n x ) \\bigr ) , \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} x ' ( t ) & = s - d x ( t ) - \\frac { \\beta x ( t ) v ( t ) } { 1 + a x ( t ) + b v ( t ) } , \\\\ y ' ( t ) & = \\frac { \\beta x ( t ) v ( t ) } { 1 + a x ( t ) + b v ( t ) } - a y ( t ) - p y ( t ) z ( t ) , \\\\ v ' ( t ) & = k y ( t ) - u v ( t ) , \\\\ z ' ( t ) & = c y ( t - \\tau ) z ( t - \\tau ) - b z ( t ) . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} X _ { \\dot { w } } ^ J ( b ) : = \\{ g J \\in G ( E ) / J \\colon g ^ { - 1 } b \\tau ( g ) \\in J \\dot { w } J \\} \\subseteq G ( E ) / J . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} Y ( X _ { j _ 1 } , \\dots , X _ { j _ k } ) = \\begin{cases} X & 1 \\notin \\{ j _ 1 \\ldots , j _ k \\} , \\ , 2 \\in \\{ j _ 1 \\ldots , j _ k \\} , \\\\ X ^ * & 1 \\in \\{ j _ 1 \\ldots , j _ k \\} , \\ , 2 \\not \\in \\{ j _ 1 \\ldots , j _ k \\} , \\\\ \\mathbb C & \\{ 1 , 2 \\} \\subset \\{ j _ 1 \\ldots , j _ k \\} \\ ; \\textrm { o r } \\ ; \\{ 1 , 2 \\} \\cap \\{ j _ 1 \\ldots , j _ k \\} = \\varnothing . \\end{cases} \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} l _ { a b } + r ^ { - a } \\sigma _ { a b } = & - \\frac { 1 } { 3 } r ^ 3 \\alpha _ { a b } + O ( r ^ 4 ) \\\\ n _ { c d } - \\frac { 1 } { 2 } r ^ { - 1 } \\sigma _ { c d } = & r ^ 3 ( \\frac { 1 } { 2 } ( \\rho + \\kappa ^ 2 ) \\tilde \\sigma _ { a b } - \\frac { 1 } { 6 } \\alpha _ { a b } ) + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\frac { \\eta _ 1 n } { N } \\leq \\mathbb { E } N _ l = n p _ l \\leq \\frac { \\eta _ 2 n } { N } \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} 0 < \\theta _ p ( v _ 1 , 0 , r ) = \\theta _ p ( v _ 1 , 0 ) \\leq \\theta _ p ( u , x ) r > 0 . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{gather*} H _ 0 = \\{ 0 , 3 \\} \\quad K _ 0 = \\{ 0 , 3 , 6 \\} , H _ 1 = \\{ 1 , 4 \\} \\quad K _ 1 = \\{ 1 , 4 , 7 \\} , \\\\ H _ 2 = \\{ 2 , 5 \\} \\quad K _ 2 = \\{ 2 , 5 , 8 \\} . \\end{gather*}"}
-{"id": "1658.png", "formula": "\\begin{align*} \\int _ \\Omega f v _ 1 v _ 2 \\ , d x = 0 \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\Vert \\Phi ( a ) \\Vert \\le \\sum _ { j = 1 } ^ m \\left \\Vert P ( a _ j ) \\right \\Vert \\le \\sum _ { j = 1 } ^ m \\left \\Vert P \\right \\Vert \\left \\Vert a _ j \\right \\Vert ^ n = \\Vert P \\Vert \\sum _ { j = 1 } ^ m \\left \\Vert a _ j \\right \\Vert ^ n . \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ { i + m - 1 } H ( \\mu , \\mathcal { D } _ { i + 1 } | \\mathcal { D } _ { i } ) & = \\int H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) \\\\ & = \\int _ { x : H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) \\leq m ( 1 - \\varepsilon ) } H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) + \\int _ { x : H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) > m ( 1 - \\varepsilon ) } H ( \\mu ^ { x , i } , \\mathcal { D } _ m ) d \\mu ( x ) . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\mathcal { L } = \\Delta - \\mathcal { F } , \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} p _ t ^ { ( 2 ) } ( x , y ) = ( 4 \\pi t ) ^ { - d / 2 } e ^ { - \\abs { x - y } / 4 t } . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} m _ { \\ell , \\ell + s } = \\frac { 1 } { s } \\sum _ { i = \\ell + 1 } ^ { \\ell + s } x _ i \\textrm { f o r } \\ell = 0 , 1 , \\ldots , N - s , \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} 0 \\longrightarrow R ( n - 1 ) ^ n \\xrightarrow { C ^ t _ n } R ( m + n - 1 ) ^ { n + 1 } \\xrightarrow { E _ n : = \\left [ - \\delta _ 1 \\ ; \\delta _ 2 \\ ; \\cdots \\ ; ( - 1 ) ^ { n + 1 } \\delta _ { n + 1 } \\right ] } R ( m n + m + n - 1 ) . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\tau ^ { \\Gamma } _ p ( M ) = \\tau ^ { \\Gamma } _ p ( M ' ) + \\tau ^ { \\Gamma } _ p ( M '' ) \\ ; . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| y ^ { k } - x ^ { k } \\| = 0 . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} v o l ( p r ( Y ) ) & = \\int _ { ( y _ 1 , \\dots , y _ T ) \\in p r ( Y ) } 1 d y _ 1 \\dots d y _ T \\\\ & = | \\det U _ 1 | \\int _ { ( x _ 1 , \\dots , x _ T ) \\in X } 1 d x _ 1 \\dots d x _ T \\\\ & = | \\det U _ 1 | v o l ( X ) = | \\det U _ 1 | | \\det C | ^ { - 1 } v o l ( Y ) . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} \\xi _ 1 = ( x _ 1 , . . . , x _ m ) , \\ \\xi _ 2 = ( x _ { 2 , 1 } , . . . , x _ { 2 , m _ 2 } ) , . . . , \\xi _ k = ( x _ { k , 1 } , . . . , x _ { k , m _ k } ) . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\mu ( \\{ x \\in \\mathbb { R } ^ n | P ^ { \\perp } x \\in A \\} ) = 0 A \\subset V ^ { \\perp } . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} & P _ e ( v ) = \\frac { a } { \\gamma - 1 } v ^ { 1 - \\gamma } , & Q ( \\theta ) = \\int _ 0 ^ \\theta C _ \\vartheta ( z ) d z . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} ( x ) _ { n , \\lambda } = \\sum _ { l = 0 } ^ n S _ { 1 , \\lambda } ( n , l ) x ^ l , \\ , \\ , ( n \\geq 0 ) , ( \\textnormal { s e e } \\ , \\ , [ 5 , 6 ] ) . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} \\int _ { 0 } ^ t \\gamma \\big ( t _ { 1 } - s _ { 1 } \\big ) d s _ { 1 } = & \\int _ { t _ { 1 } - t } ^ { t _ 1 } \\gamma ( r ) d r \\\\ \\geq & \\int _ 0 ^ { t _ 1 } \\gamma ( r ) d r , \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} J ^ k ( J ^ T ) ^ l & = \\left ( \\sum _ { j = 1 } ^ { n - k } e _ { j + k } e _ j ^ T \\right ) \\left ( \\sum _ { j = 1 } ^ { n - l } e _ j e _ { j + l } ^ T \\right ) , \\\\ & = \\sum _ { j = 1 } ^ { n - \\max \\ \\{ k , l \\} } ( e _ { j + k } e _ j ^ T ) ( e _ j e _ { j + l } ^ T ) , \\\\ & = \\sum _ { j = 1 } ^ { n - \\max \\ \\{ k , l \\} } e _ { j + k } e _ { j + l } ^ T . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} \\Gamma = \\langle a _ j \\ , , j \\in I \\mid a _ j ^ { - 1 } = a _ { - j } \\ ; \\ ; j \\in I \\rangle \\ , . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} ( \\cup _ { i = \\frac { q + 5 } { 4 } } ^ { \\frac { q + 1 } { 4 } + t } \\mathbb { C } _ { b } ) \\cap - q ( \\cup _ { i = 0 } ^ { \\frac { q + 1 } { 4 } } \\mathbb { C } _ { b } ) & = \\mathbb { C } _ { \\frac { ( q - 1 ) ^ 2 } { 4 } - m ( q + 1 ) } , \\\\ ( \\cup _ { i = 0 } ^ { \\frac { q + 1 } { 4 } } \\mathbb { C } _ { b } ) \\cap - q ( \\cup _ { i = \\frac { q + 5 } { 4 } } ^ { \\frac { q + 1 } { 4 } + t } \\mathbb { C } _ { b } ) & = \\mathbb { C } _ { s - 2 m ( q + 1 ) } , \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} A ( x ) = \\frac { 2 } { 3 } \\sqrt { x + 1 } \\left ( 1 + \\frac { 1 } { 4 x } \\right ) , \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\overline { U } _ { 2 k , 2 a } ( x ; q ) - \\overline { U } _ { 2 k , 2 a - 2 } ( x ; q ) = ( - x q ^ 2 ; q ^ 2 ) _ \\infty ( \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) - \\overline { Q } _ { k , a - 1 } ( x ^ 2 ; q ^ 2 ) ) . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} v _ { t t } ( t , x ) + ( - \\Delta ) ^ \\delta v _ { t t } ( t , x ) + ( - \\Delta ) ^ \\alpha v ( t , x ) + ( - \\Delta ) ^ { \\theta } v _ t ( t , x ) = 0 , t \\geq 0 , \\ , \\ , \\ , \\ , x \\in \\R ^ n \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} - \\frac { 2 } { ( 1 - f ) ^ { 2 } } \\left \\{ f ^ { 2 } _ { i j } + \\frac { 2 f _ { i } f _ { i j } f _ { j } } { ( 1 - f ) } + \\frac { f ^ { 2 } _ { i } f ^ { 2 } _ { j } } { ( 1 - f ) ^ { 2 } } \\right \\} = - \\frac { 2 } { ( 1 - f ) ^ { 2 } } \\Big ( f _ { i j } + \\frac { f _ { i } f _ { j } } { 1 - f } \\Big ) ^ { 2 } . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\mathrm { C a r d } \\left ( \\Big \\lbrace \\delta \\in \\Delta _ { h , N _ { 1 } } ~ \\Big | ~ \\eta ( \\delta , \\tilde { \\delta } ) = r \\Big \\rbrace \\right ) ~ = ~ { { N _ { 1 } } \\choose { r } } \\cdot \\left ( 2 ^ { { \\bf \\tilde { p } } + 2 } - 1 \\right ) ^ { r } . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} L = \\sum _ { i = 1 } ^ d X _ i ^ 2 + X _ 0 + V , \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\Phi ( x ) : & = \\begin{cases} 1 , & \\ | x | _ \\infty \\leq 1 , \\\\ 0 , & \\end{cases} \\\\ & = \\begin{cases} 1 , & \\ | x / t | _ \\infty \\leq 1 / q , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} Q _ { \\lambda , q } ( \\Omega ) f ^ { q - 1 } ( \\xi ) = \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta + \\lambda \\int _ \\Omega \\frac { f ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha - 1 } } d \\eta , \\xi \\in \\overline \\Omega \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} - \\div _ y \\big ( \\widetilde { m } _ 1 ( x , y ) ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) \\big ) = 0 . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} \\frac { \\rho _ { k + 1 } } { \\rho _ k } = O ( 1 + 4 \\rho _ k ^ { - \\frac { 1 + \\delta } { 2 } } ) = O ( 1 ) , \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\psi _ m = ( \\zeta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ l , 1 ) \\boxplus ( \\xi _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\xi _ k , 2 ) . \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\nabla ^ { F , \\ , \\mathrm { s p i n } } _ { u } : = \\nabla ^ { F } _ { X } + \\frac { 1 } { 3 } ( \\iota _ { X } { \\mathcal H } ) \\wedge \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\mathit { S I N R } > \\theta \\mid n ) \\\\ & \\ = \\sum _ { k = 1 } ^ { n } \\binom { n } { k } ( - 1 ) ^ { k + 1 } \\exp \\left ( - ( 1 - f _ N ( 0 ) ) \\sqrt { \\theta _ k } \\arctan \\sqrt { \\theta _ k } \\right ) . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} J ^ k ( J ^ T ) ^ l - J ^ { n - l } ( J ^ T ) ^ { n - k } = 0 \\ \\iff \\ l = n - k . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} p ( T ) \\ast q ( T ) : = \\sum _ { n \\geq 1 } p _ n \\cdot q _ n T ^ n \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\mathcal { Z } _ { \\zeta } ^ { ( { \\tau _ 1 } , { \\tau _ 2 } ) } ( c ) : = \\begin{cases} - g _ { { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c } + \\chi { } ( \\cdot { } , { \\zeta } ) \\cdot { } c & B _ { \\zeta } ( { \\tau _ 1 } ) \\cap { } C _ { \\zeta } ( { \\tau _ 2 } ) \\\\ - g _ { { \\zeta } , { \\tau _ 1 } , { \\tau _ 2 } , c } & B _ { \\zeta } ( { \\tau _ 1 } ) \\setminus { } C _ { \\zeta } ( { \\tau _ 2 } ) \\end{cases} \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} | T _ n ( \\phi , F ) | = | H \\cap T _ n ( \\phi , F ) | \\cdot | \\pi ( T _ n ( \\phi , F ) ) | \\leq m \\cdot | \\pi ( T _ n ( \\phi , F ) ) | . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} d \\Psi _ { ( t , v ) } ( w _ i ) = \\sum _ { j = 0 } ^ { n - 1 } B _ { i j } ( t ) v _ j ( t ) , \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} H _ 1 = \\begin{bmatrix} 1 & 1 \\\\ 1 & - 1 \\end{bmatrix} , H _ 2 = \\begin{bmatrix} H _ 1 & H _ 1 \\\\ H _ 1 & - H _ 1 \\end{bmatrix} = \\begin{bmatrix} 1 & 1 & 1 & 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 1 & 1 & - 1 & - 1 \\\\ 1 & - 1 & - 1 & 1 \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\mathcal { A } ( y , p ^ 1 , p ^ 2 , f ) : = ( \\mathcal { A } _ 1 ( y , p ^ 1 , p ^ 2 , f ) , \\mathcal { A } _ 2 ( y , p ^ 1 , p ^ 2 , f ) , \\mathcal { A } _ 3 ( y , p ^ 1 , p ^ 2 , f ) , \\mathcal { A } _ 4 ( y , p ^ 1 , p ^ 2 , f ) ) \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} F ( \\overline { x } ) & = \\left ( \\hat p ( 0 ) , ( d \\hat p v _ 1 ) ( 0 ) , \\ldots , ( d \\hat p v _ { n - 1 } ) ( 0 ) \\right ) \\\\ & = ( p _ { d , 0 , \\ldots , 0 } , p _ { d - 1 , 0 , 1 0 , \\ldots , 0 } , \\ldots , p _ { d - 1 , 0 , \\ldots , 0 , 1 , 0 , \\ldots 0 } ) \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t ( t - s ) ^ { 1 - \\gamma - \\frac { 1 } { \\alpha } } | M ( y ( s ) ) | _ \\alpha d s & \\leq C \\eta _ \\infty \\int _ 0 ^ t ( t - s ) ^ { 1 - \\gamma - \\frac { 4 - p } { 2 p } } \\cdot s ^ { \\frac { 2 } { p } - \\frac { 5 } { 2 } + 2 \\gamma - \\gamma } d s \\ \\| y \\| ^ 2 \\\\ & = C \\eta _ \\infty B \\left ( \\frac { 2 } { p } - \\frac { 5 } { 2 } + \\gamma + 1 , 1 - \\gamma - \\frac { 4 - p } { 2 p } + 1 \\right ) \\| y \\| ^ 2 , \\ \\forall t \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} b _ n ( z ) = K _ \\Omega ( z , y _ n ) - K _ \\Omega ( y _ n , z _ 0 ) . \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} g ' ( \\delta ) = 2 Q - L _ 0 T \\sinh \\delta / 2 , g '' ( \\delta ) = - L _ 0 T \\cosh \\delta / 2 < 0 , \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\frac { \\partial ^ 3 \\mathcal { F } } { \\partial t _ 2 ^ 3 } = \\sum _ { d = 2 } ^ \\infty N _ d e ^ { d t _ 1 } \\frac { 1 } { ( 3 d - 4 ) ! } t _ 2 ^ { 3 d - 4 } \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} ( f _ 1 ( a ) + \\lambda ) ^ q = ( f _ 1 ( a + \\lambda ) ) ^ q = ( f _ 2 ( a + \\lambda ) ) ^ q = ( f _ 2 ( a ) + \\lambda ) ^ q , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\hat \\Sigma & = \\{ i _ 1 i _ 2 \\cdots \\in \\Sigma : i _ n \\in \\{ 1 , \\ldots , M \\} n \\in \\N \\} , \\\\ \\Upsilon & = \\{ i _ 1 i _ 2 \\cdots \\in \\Sigma : n _ 0 \\in \\N i _ n \\in \\{ M + 1 , \\ldots , N \\} n > n _ 0 \\} . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\bigcup _ { n = 1 } ^ \\infty S ( G , 2 ^ { - n - 1 } ) = G \\setminus \\partial G . \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} a _ { ( f ( x ) ) } b + b _ { ( f ( - x ) ) } a = \\sum _ { j \\in \\Z _ { \\geq 0 } } \\frac { ( - 1 ) ^ j } { ( j + 1 ) ! } T ^ { j + 1 } \\left ( a _ { ( x ^ { j + 1 } f ( x ) ) } b \\right ) , . \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} \\mathcal X _ j = \\Theta _ j - M \\frac { \\langle \\nabla z _ j , \\nabla w _ j \\rangle } { z _ j \\phi ^ { 2 \\alpha _ j } } + \\mathcal E _ { 2 , j } + \\mathcal E _ { 3 , j } . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} X _ j = L ^ { p _ j ^ S } ( M _ S , \\mu _ S ; L ^ { p _ j ^ { S - 1 } } ( M _ { S - 1 } , \\mu _ { S - 1 } ; \\cdots L ^ { p _ j ^ { 1 } } ( M _ { 1 } , \\mu _ { 1 } ; L ^ { p _ j ^ { 0 } } ( \\mathcal M ) ) \\cdots ) \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\int _ { \\tilde { \\Omega } _ { R , \\varepsilon } } | \\nabla w _ \\varepsilon | ^ 2 d z = O ( 1 ) \\ , . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\psi _ 1 ( t ) = \\begin{cases} J _ 0 \\big ( \\sqrt { \\lambda _ 1 } t \\big ) & \\mbox { i f } t \\in ( a _ 1 , 0 ) \\ , , \\\\ 0 & \\mbox { i f } t \\in ( 0 , a _ 2 ) \\ , . \\end{cases} \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} ( ( B L \\iota ) ^ \\ast \\circ \\nu ) ( q _ 4 ) = ( \\nu \\circ ( B \\iota ) ^ \\ast ) ( q _ 4 ) . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\varrho _ { K _ m } ( u ) = \\varrho _ { K } ( u ) \\mbox { \\ f o r $ u \\in S ^ { n - 1 } \\backslash \\partial N ( K , o ) ^ * $ } , \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} j ^ { K \\textnormal { t h } } ( q , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( 1 - q ^ r P ^ { - 1 } \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} S L ^ \\infty = \\{ f \\in L ^ 2 : \\| f \\| _ { S L ^ \\infty } < \\infty \\} , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} ( A ^ { ( 4 ) } ( t , x ) \\xi ) \\cdot \\xi & \\geq \\frac 1 2 \\| D P ^ { T } ( t , y ) \\xi \\| ^ 2 - \\frac { m } { d ^ 4 } \\xi _ 1 ^ 2 = \\frac { 1 } { 2 d ^ 4 } \\left [ ( p ^ 2 + q ^ 2 - 2 m ) \\xi _ 1 ^ 2 + 2 q d ^ 2 \\xi _ 1 \\xi _ 2 + d ^ 4 \\xi _ 2 ^ 2 \\right ] \\\\ & \\geq \\frac { 1 } { 2 } \\left [ p ^ 2 - \\left ( \\frac { 1 } { \\varepsilon } - 1 \\right ) q ^ 2 - 2 | m | \\right ] \\xi _ 1 ^ 2 + \\frac { 1 } { 2 } ( 1 - \\varepsilon ) \\xi _ 2 ^ 2 \\ , , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\delta _ { * } ( t ) & \\geq 1 - \\frac { C _ 6 } { t ^ 3 } , \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} K ( \\partial _ t ) f : = \\int _ 0 ^ t k ( t - \\tau ) f ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} P ( i , j ) = \\begin{cases} \\frac { 1 } { \\sum \\limits ^ { n } _ { k = 1 } G ( k , j ) } , & \\mbox { i f } G ( i , j ) = 1 , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} \\rho _ { { \\bf L } ^ 1 } ( f , g ) ~ : = ~ ~ \\int _ { I } \\rho ( f ( t ) , g ( t ) ) d t ~ < ~ + \\infty \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\Delta ( k , q _ l ) = \\mathbb { E } ( T _ l | N _ l = k ) = \\int _ { S _ l } T S P ( z _ 1 , \\ldots , z _ k ; S _ l ) \\frac { f ( z _ 1 ) } { q _ l } \\ldots \\frac { f ( z _ k ) } { q _ l } d z _ 1 \\ldots d z _ k \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} G ( q ) : = \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ { k } } { ( 1 - q ^ { 2 k } ) ^ 2 } \\equiv \\frac { n ^ 2 - 1 } { 2 4 } \\pmod { \\Phi _ n ( q ) } . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\sigma _ { \\theta , \\rho } ( g h ) = \\theta ( g ) \\cdot U _ { \\psi } ( g ) \\circ \\pi _ { \\psi } ( h ) \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\nu _ \\infty ( f ) : = \\sup \\{ \\nu _ i ( f ) \\} . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\mathbf { y } _ \\mathcal { Q } } = \\frac { 2 } { \\pi } \\left ( ^ { - 1 } \\left ( \\mathbf { K } \\mathfrak { R } \\{ \\mathbf { C } _ { \\mathbf { y } } \\} \\mathbf { K } \\right ) + j ^ { - 1 } \\left ( \\mathbf { K } \\mathfrak { I } \\{ \\mathbf { C } _ { \\mathbf { y } } \\} \\mathbf { K } \\right ) \\right ) \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} b ( \\gamma ^ { - 1 } z ) = \\lim _ { n \\rightarrow \\infty } K _ \\Omega ( \\gamma ^ { - 1 } z , y _ n ) - K _ \\Omega ( y _ n , z _ 0 ) = \\lim _ { n \\rightarrow \\infty } K _ \\Omega ( z , \\gamma y _ n ) - K _ \\Omega ( y _ n , z _ 0 ) = b ( z ) \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} \\Phi _ { 1 } ( x ; ( h _ { 0 } + \\varepsilon \\cdot l ) \\sqcup \\alpha ) = \\Phi _ { 1 } ( x ; h _ { 0 } + \\varepsilon \\cdot l ) + \\Phi _ { 1 } ( \\Phi _ { 1 } ( x ; h _ { 0 } + \\varepsilon \\cdot l ) ; \\alpha ) , \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\varphi ( t ) & = C _ { H } t ^ { H - \\frac { 1 } { 2 } } \\left ( D _ { 0 ^ { + } } ^ { H - \\frac { 1 } { 2 } } \\left ( s ^ { \\frac { 1 } { 2 } - H } \\dot { \\gamma } _ { s } \\right ) \\right ) ( t ) . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} [ L ^ { p _ 1 } , L ^ { p _ 2 } ] _ { \\theta , q } = L ^ { p , q } , \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} & l : = \\sqrt { p ^ 2 + q ^ 2 } , ~ ~ ~ ~ l _ L : = \\sqrt { p ^ 2 + q ^ 2 + r ^ 2 } , ~ ~ ~ ~ \\overline { p } : = \\frac { p } { l } , \\\\ & \\overline { q } : = \\frac { q } { l } , ~ ~ ~ ~ \\overline { p _ L } : = \\frac { p } { l _ L } , ~ ~ ~ ~ \\overline { q _ L } : = \\frac { q } { l _ L } , ~ ~ ~ ~ \\overline { r _ L } : = \\frac { r } { l _ L } . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\mathcal { R } _ { G , b , \\mu } = \\{ ( M _ { S _ 1 } , \\mu _ { S _ 1 } ) \\in \\mathcal { C } _ G : ( M _ { S _ 1 } , \\mu _ { S _ 1 } ) \\leq ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\ , \\ \\ , \\ ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\in \\mathcal { T } _ { G , b , \\mu } \\} . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} J _ 1 ( t , \\delta ) & \\leq 2 \\| f \\| t \\Pr ( Y _ 1 \\wedge Y _ 2 > b _ 0 ( t ) \\delta ) \\\\ & = 2 \\| f \\| t \\ , \\Pr ( Y _ 1 > b _ 0 ( t ) \\delta ) \\Pr ( Y _ 2 > b _ 0 ( t ) \\delta ) \\leq C _ { 1 } t ^ { \\frac { \\alpha _ 0 - \\alpha - \\alpha ^ * + \\epsilon } { \\alpha _ 0 } } \\stackrel { t \\to \\infty } { \\to } 0 . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{gather*} ( u + v ) * w = u * w + v * w \\\\ w * ( u + v ) = w * u + w * v \\\\ ( s u ) * v = u * ( s v ) = s ( u * v ) . \\end{gather*}"}
-{"id": "2019.png", "formula": "\\begin{align*} | | \\dot { \\gamma } | | _ { \\Sigma _ 1 , L } ^ 2 = \\left [ \\overline { q } { \\dot { \\gamma } _ 3 } - \\frac { \\sqrt { 2 } } { 2 } \\overline { p } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) \\right ] ^ 2 , \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} & \\angle ( { y _ { k , i } } ) - \\angle ( { y _ { k , 1 } } ) \\\\ = & \\pi \\left [ { \\frac { M - 1 } { M } ( \\omega _ { k , 1 1 } - \\omega _ { k , i 1 } ) + \\frac { N - 1 } { N } { ( \\omega _ { k , 1 2 } - \\omega _ { k , i 2 } ) } } \\right ] , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 u _ 1 ( x ) \\ , d x = 0 , \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } R _ k \\asymp R ( \\alpha t ) e ^ { \\alpha t } \\quad \\mbox { a s $ t \\to \\infty $ } . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\left \\| N ( h , \\mu ) { \\rm O p } _ h ( \\chi ) U ^ { - 1 } { \\rm O p } _ h ( \\phi \\varrho ^ k ) U { \\rm O p } _ h ( \\psi ) - \\sum _ { j = 0 } ^ { s } h ^ j { \\cal T } _ j ^ { ( k ) } \\right \\| \\lesssim h ^ { s + 1 } | { \\rm I m } \\ , z | ^ { - ( 3 s + 2 - k ) / 2 } . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} x _ { N + k } > x _ N e ^ { - 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } x _ { N + i } ^ 2 } \\ge x _ N e ^ { - 2 x _ N ^ 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } } > x _ N e ^ { - 2 x _ N ^ 2 S } > 0 . \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\lim _ { { \\varepsilon } \\to 0 } { \\mathbb { P } } ( X ^ { \\varepsilon } _ x ( \\tau ^ { \\varepsilon } _ x ) \\in N ) = 0 . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{gather*} a ^ { } ( \\phi ) = a ( \\sqrt { 1 + \\rho } \\ , \\phi \\oplus 0 ) + a ^ * ( 0 \\oplus \\sqrt { \\rho } \\ , \\overline { \\phi } ) . \\end{gather*}"}
-{"id": "1025.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { d - 1 } B _ { \\tilde { n } } \\left ( \\frac { j } { d } \\right ) = \\frac { B _ { \\tilde { n } } ( 0 ) } { d ^ { \\tilde { n } - 1 } } , \\sum _ { 0 \\le j _ i \\le d _ i - 1 , \\atop ( i = 1 , 2 ) } B _ { \\tilde { n } } \\left ( \\left \\{ \\frac { j _ 1 } { d _ 1 } + \\frac { j _ 2 } { d _ 2 } \\right \\} \\right ) = \\frac { B _ { \\tilde { n } } ( 0 ) } { ( d _ 1 d _ 2 ) ^ { \\tilde { n } - 1 } } . \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} a ( u , v ) : = \\int _ { \\O } D ^ 2 u : \\ , D ^ 2 v , b ( u , v ) : = \\int _ { \\O } ( \\boldsymbol { \\eta } \\nabla u ) \\cdot \\nabla v , \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\frac { R _ { 3 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 3 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 = i ( - R _ { 3 1 } + i R _ { 3 2 } ) = - i R _ { 3 1 } - R _ { 3 2 } = 0 ; \\\\ \\frac { R _ { 4 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 4 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 = - i R _ { 4 1 } - R _ { 4 2 } = i + i = 2 i = - 2 u _ 4 . \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 ) } _ m ( z / c ) , z ^ j \\rangle _ 0 = 0 \\ . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\phi ( s ) = \\int _ { s } ^ { s ^ { 1 / 2 } } F ( x / s ) e ^ { - x } d x + \\int _ { s ^ { 1 / 2 } } ^ { \\infty } F ( x / s ) e ^ { - x } d s . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} L ( s , f \\otimes \\chi ) = \\prod _ { p } \\left ( 1 - \\frac { \\chi ( p ) \\lambda ( p ) } { p ^ s } + \\frac { \\chi ^ 2 ( p ) } { p ^ { 2 s } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} S [ u + h ] = S [ u ] + d + o ( \\| h \\| _ { X _ S } ) \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\P ( C ' _ { k _ 1 , j _ 1 } ) & = \\P ( C _ { k _ 1 , j _ 1 } ) - p , \\\\ \\P ( C ' _ { k _ 2 , j _ 2 } ) & = \\P ( C _ { k _ 2 , j _ 2 } ) - p , \\\\ \\P ( C ' _ { k _ 1 , j _ 2 } ) & = \\P ( C _ { k _ 1 , j _ 2 } ) + p , \\\\ \\P ( C ' _ { k _ 2 , j _ 1 } ) & = \\P ( C _ { k _ 2 , j _ 1 } ) + p , \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} \\sigma ( t , \\theta ) = t \\ , P _ 1 ( \\theta ) + ( 1 - t ) \\ , P _ 2 ( \\theta ) \\ ; , \\ ; \\ ; t \\in [ 0 , 1 ] \\ ; , \\ ; \\ ; \\theta \\in [ \\omega _ 1 , \\omega _ 2 ] \\ ; , \\ ; \\ ; \\omega _ 2 - \\omega _ 1 < 2 \\pi \\ ; , \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\beta _ { \\textrm { N } } = \\max \\left \\lbrace 1 - \\frac { C _ { \\textrm { N } } ^ { \\textrm { O F D M A } } } { | h _ { \\textrm { B N } } | ^ 2 } , 0 \\right \\rbrace , \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } P _ \\lambda \\big ( \\eta _ t ( O ) = 1 \\big ) & = \\lim _ { t \\rightarrow + \\infty } P _ \\lambda \\big ( \\xi _ t ( O ) > 0 \\big ) \\\\ & \\geq \\limsup _ { t \\rightarrow + \\infty } \\frac { ( E \\xi _ t ( O ) ) ^ 2 } { E ( \\xi _ t ^ 2 ( O ) ) } = \\limsup _ { t \\rightarrow + \\infty } \\frac { 1 } { E ( \\xi _ t ^ 2 ( O ) ) } \\\\ & \\geq \\frac { \\inf _ { x \\in \\mathbb { Z } ^ d } K ( x ) } { K ( O ) } > 0 . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\Phi _ m : = \\begin{cases} \\exp \\left ( - \\gamma \\sum _ { k = 1 } ^ { m - 1 } \\phi _ \\delta ( \\abs { A _ k } ) \\right ) \\ , , & m \\le N \\ , , \\\\ \\exp \\left ( - \\gamma \\sum _ { k = 1 } ^ { N - 1 } \\phi _ \\delta ( \\abs { A _ k } ) \\right ) \\ , , & m > N \\ , , \\end{cases} \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} w ( M _ S ) \\subset w ( Z _ G ( \\theta _ { M _ S } ( \\mu _ S ) ) ) = Z _ G ( x ) = M _ { S ' } . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\nu _ { \\beta , 0 } ^ + ( \\eta = - 1 [ - n , - 1 ] ) \\leq 1 / 2 ^ n , n . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} & \\d { A } _ { \\alpha } ( z ) \\ , = \\ , ( 1 + \\alpha ) ( 1 - h ( z ) ) ^ { \\alpha } \\d { A } ( z ) \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} N _ d = \\sum _ { d _ 1 + d _ 2 = d } N _ { d _ 1 } N _ { d _ 2 } \\left ( \\binom { 3 d - 4 } { 3 d _ 1 - 2 } d _ 1 ^ 2 d _ 2 ^ 2 - \\binom { 3 d - 4 } { 3 d _ 1 - 1 } d _ 1 ^ 3 d _ 2 \\right ) \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\prod _ { p \\equiv 1 \\mod 3 } \\left ( 1 + \\frac { 2 } { p ^ s } \\right ) = \\zeta ( s ) L ( \\xi , s ) \\zeta ( 2 s ) ^ { - 2 } L ( \\xi , 2 s ) ^ { - 1 } \\zeta ( 3 s ) L ( \\xi , 3 s ) J ( s ) = : I ( s ) , \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\widehat { C } = 2 \\cdot 3 ^ { d s } \\zeta ( d s ) \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} & \\frac { \\partial E } { \\partial t } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\Big | _ { y = x \\pm ( t - b ) } \\\\ & = 2 ^ { \\sqrt { \\delta } - 1 } ( 1 + b ) ^ { \\frac { \\mu } { 2 } - 1 } ( 1 + t ) ^ { - \\frac { \\mu } { 2 } - 1 } \\bigg [ 2 ^ { - 3 } ( 1 - \\sqrt { \\delta } ) ^ 2 ( t - b ) + \\big ( - \\tfrac { \\mu } { 2 } + \\tfrac { 1 - \\sqrt { \\delta } } { 2 } \\big ) ( 1 + b ) + 2 ^ { - 2 } ( \\sqrt { \\delta } - 1 ) ( t + b + 2 ) \\bigg ] . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\ell _ q ( Q ) = \\frac { - Q \\theta _ q ' ( Q ) } { \\theta _ q ( Q ) } \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\tau _ \\epsilon ( i ) & = \\begin{cases} ( \\delta + \\epsilon ) - i & \\mbox { $ i $ o d d } \\\\ i & \\mbox { $ i $ e v e n } \\end{cases} \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\mathfrak { E } = c _ 1 ( T X ) + \\sum _ i \\left ( 1 - \\frac { 1 } { 2 } \\textnormal { d e g } _ { H ^ * ( X ) } ( T _ i ) \\right ) t _ i T _ i \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\mathcal A ( \\rho ) z = z \\mbox { i n } \\ ; \\ ; \\Bbb E _ \\eta , z = 0 \\mbox { o n } \\ ; \\ ; \\Gamma _ s , \\partial _ y z = - \\partial _ { y y } \\ , \\mathcal U ( \\rho , \\eta ) \\zeta \\mbox { o n } \\ ; \\ ; J _ \\eta . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 - 2 } } \\le & C \\big ( \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , \\infty } ^ { \\sigma - 1 } } \\big ) ^ { \\theta _ { 1 } } \\big ( \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 } } \\big ) ^ { 1 - \\theta _ { 1 } } , \\theta _ 1 = \\frac { 4 } { n - 2 \\sigma + 2 } \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\begin{aligned} & P \\left ( \\underset { 0 \\le i \\le k } { \\sup } \\left | M _ i \\right | > \\eta \\right ) \\le 2 \\exp \\left \\{ - \\frac { \\eta ^ 2 } { 2 \\operatorname { V a r } \\left [ M _ k \\right ] } \\right \\} , \\end{aligned} \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\frac { w } { c } = 2 \\left ( \\frac { z } { c } \\right ) ^ 2 - 1 \\ . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\Delta _ t = \\int _ 0 ^ t \\mathbf { 1 } _ { \\{ | X _ s | = P _ Y \\} } Y _ s \\hbox { d } s . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\left [ U ' _ n \\right ] \\L ^ { - n ( d _ 1 + d _ 2 ) } T ^ n = ( \\L ^ { d _ 2 } - 1 ) \\sum _ { 1 \\leq m \\leq n } \\L ^ { - \\left ( n d _ 1 + ( d _ 2 - d _ 1 ) m \\right ) } T ^ n + \\L ^ { d _ 1 } \\sum _ { n \\geq 1 } \\L ^ { - n d _ 2 } T ^ n . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} \\Theta ( a x ^ k ) = \\Theta ( a u ^ i x ^ j ) = u ^ i \\sigma ^ { - j } ( a ) x ^ { - j } = \\sigma ^ { - k } ( a ) x ^ { - k } , \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} h ( s ) & = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\phi ( m s ) \\\\ & = \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { \\phi ( m s ) } { m } + \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } + \\sum _ { m > s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } \\\\ & = : \\psi _ 1 ( s ) + \\psi _ 2 ( s ) + \\psi _ 3 ( s ) . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} C _ n ^ { ( 1 + \\alpha ) } ( z ) = \\frac { ( 2 + 2 \\alpha ) _ n } { ( \\alpha + \\frac 3 2 ) _ n } P _ n ^ { ( \\alpha + \\frac 1 2 , \\alpha + \\frac 1 2 ) } ( z ) \\ , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\mu ( B ( x , \\tilde { \\tilde { r } } ) \\cap \\Omega ) = \\frac { 1 } { 2 } \\mu ( B ( x , \\tilde { r } ) \\cap \\Omega ) = \\frac { 1 } { 4 } \\mu ( B ( x , r ) \\cap \\Omega ) . \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} | \\bar { w } ' _ \\varepsilon ( s _ \\varepsilon ) | = \\| \\bar { w } ' _ \\varepsilon \\| _ { L ^ \\infty ( [ 0 , \\bar { r } _ \\varepsilon ] ) } \\limsup _ { \\varepsilon \\to 0 } \\mu _ \\varepsilon | \\bar { w } ' _ \\varepsilon ( s _ \\varepsilon ) | > 0 \\ , . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} ( g z , x ( g ) ) = ( g z , g ( g ^ { - 1 } x ( g ) ) ) \\in U . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} r & = N ^ { - \\frac { 1 } { \\delta } } \\\\ \\beta & = \\frac 1 \\delta - 1 \\\\ \\delta & = d \\alpha + d - 1 . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ { x x } ^ 2 L ( \\overline { x } , \\overline { \\lambda } ) \\xi + \\nabla g ( \\overline { x } ) v = 0 , \\\\ g ' ( \\overline { x } ) \\xi - \\Pi _ { \\mathcal { K } } ' ( g ( \\overline { x } ) \\ ! + \\ ! \\overline { \\lambda } ; g ' ( \\overline { x } ) \\xi + v ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} \\mathrm { P r } _ { \\mathcal G } [ r _ { 1 } , r _ { 2 } ] = [ r _ { 1 } , r _ { 2 } ] ^ { \\mathcal G } , \\ r _ { i } \\in \\Gamma ( \\mathcal G ) , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\theta & = \\frac { d + 1 \\pm \\sqrt { ( d + 1 ) ^ 2 - 4 ( 1 + \\mu ) d } } { 2 } . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} h = ( x _ 1 + \\sum _ { i = a } ^ { b } c _ i x _ 2 ^ i x _ 3 ^ { m - p i } , x _ 2 , x _ 3 ) , c _ a \\neq 0 , c _ b \\neq 0 . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\langle X , Y \\rangle = I m ( t r a c e ( X Y ) ) \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( m ) = \\sigma \\left ( W _ { m n } ( [ \\omega _ 1 , \\omega _ 2 ] ) \\times W _ { m n } ( [ \\omega _ 3 , \\omega _ 4 ] ) \\right ) \\ ; , \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} F _ { i } = \\frac { \\tilde { f } _ { i } - \\tilde { f } _ { i } ( e ) - \\sum ^ { m } _ { j = 1 } ( \\tilde { a } _ { i j } - \\tilde { a ^ { 0 } _ { i j } } ) X _ { j } \\tilde { L } _ { \\nu } } { \\omega ( \\sigma ^ { \\nu } ) } . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} W _ N : = \\frac 1 { M _ N } X _ N ^ * X _ N \\ , . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} \\chi _ k \\ast \\chi _ k = \\chi _ k ( k \\in \\mathbb { Z } ) \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} | e | _ { Y ( X _ 1 , \\ldots , X _ m ) } = \\sup \\left \\{ \\left | \\tau \\left ( e \\prod _ { \\ell = 1 } ^ { m } e _ { \\sigma ( \\ell ) } \\right ) \\right | : \\sigma \\in \\Sigma ( m ) , | e _ j | _ { X _ j } = 1 , j = 1 , \\ldots , m \\right \\} \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\bigcup _ { n = 1 } ^ { \\infty } F _ { n } ( ( \\mathbb { R } ^ { d } ) ^ { n } ) \\backslash E = \\mathbb { R } ^ { N } \\backslash E . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} f ( x ) = ( x _ 1 ^ 2 - 1 ) ^ 2 + ( x _ 2 ^ 2 - 2 ) ^ 2 + ( x _ 3 ^ 2 - 3 ) ^ 2 - 0 . 7 \\ , ( x _ 1 x _ 2 + x _ 1 x _ 3 + x _ 2 x _ 3 ) + 0 . 2 \\ , ( x _ 1 + x _ 2 + x _ 3 ) \\ , . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\begin{aligned} | D ^ 2 u _ j | ^ 2 - 2 \\theta a _ { 0 , j } a _ { 1 , j } \\geq & \\ \\left [ ( 1 - 4 \\varepsilon ) ( a _ { 0 , j } ^ 2 + a _ { 1 , j } ^ 2 ) - \\frac { \\sigma ^ 2 z _ j } { \\varepsilon | x | ^ 2 } - \\frac { ( p - 2 ) ^ 2 } { 4 \\varepsilon } \\frac { | \\nabla z _ j | ^ 2 } { z _ j } \\right ] \\theta . \\end{aligned} \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\mu ^ { D ( x , n ) } = \\mu ^ { x , n } = T ^ { D ( x , n ) } \\mu _ { D ( x , n ) } . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} M _ a M _ \\lambda = M _ \\lambda M _ { \\lambda ( a ) } \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} u _ 1 = ( 1 , 0 , . . . , 0 , - 1 ) , u _ 2 = ( 0 , 1 , . . . , - 1 , 0 ) , . . . , u _ s = ( 0 , . . . , 1 , - 1 , . . . , 0 ) . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} v ^ { ( \\ell ) } : = v _ 1 ^ { ( \\ell ) } = v _ 2 ^ { ( \\ell ) } \\Omega \\ell = 1 , 2 , \\cdots , N + 1 . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\rho ' = ( n , \\dots , n - p + 1 ; r , \\dots , 1 ; - ( r + 1 ) , \\dots , - ( n - p ) ) . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } \\Big \\{ \\frac { 1 } { \\left ( 1 + x ^ 2 \\right ) ^ { \\gamma } } \\Big \\} = A _ s ^ { \\gamma } \\ , { } _ { 2 } F _ { 1 } \\Big ( s + \\gamma , s + \\frac { 1 } { 2 } ; \\frac { 1 } { 2 } ; - x ^ 2 \\Big ) , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} [ Y _ i , A _ j ] & = - 2 \\delta _ { i j } A _ j & [ Z , A _ j ] & = 0 \\\\ [ Y _ i , B _ j ] & = - \\delta _ { i j } B _ j & [ Z , B _ j ] & = - B _ j \\\\ [ Y _ i , C _ j ] & = 0 & [ Z , C _ j ] & = - 2 C _ j \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} v a r _ { P } \\left [ \\psi _ { P , \\mathbf { a } } ( \\mathbf { G , B } ; \\mathcal { G } ) \\right ] = E _ { P } \\left [ v a r _ { P } ( \\psi _ { P , \\mathbf { a } } ( \\mathbf { G , B } ; \\mathcal { G } ) \\mid \\mathbf { A } , Y , \\mathbf { G } ) \\right ] + v a r _ { P } \\left [ E _ { P } ( \\psi _ { P , \\mathbf { a } } ( \\mathbf { G , B } ; \\mathcal { G } ) \\mid \\mathbf { A } , Y , \\mathbf { G } ) \\right ] . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} f _ { N _ 2 } ( n ) = \\frac { ( \\lambda _ { } \\pi R ^ 2 ) ^ n } { n ! } \\mathrm { e } ^ { - \\lambda _ { } \\pi R ^ 2 } , \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} [ \\kappa + v _ 1 , v _ 2 ] = [ \\kappa + v _ 1 + v _ 2 , v _ 2 ] = [ \\kappa + v , v _ 2 ] = 0 . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\| \\hat { Q } _ r - Q _ r \\| _ 2 ^ 2 = 2 \\langle \\hat Q _ r , I - Q _ r \\rangle & = \\sum _ { \\substack { s < r _ 0 : \\\\ s \\neq r } } 2 \\langle \\hat Q _ r , Q _ s \\rangle + 2 \\langle \\hat Q _ r , Q _ { \\geq r _ 0 } \\rangle \\\\ & = \\sum _ { \\substack { s < r _ 0 : \\\\ s \\neq r } } 2 \\| \\hat { Q } _ r Q _ s \\| _ 2 ^ 2 + 2 \\| \\hat { Q } _ r Q _ { \\geq r _ 0 } \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} \\widetilde J ( f ) = \\int _ { G } f ( g ) j ( g ) d g . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} E _ { \\varepsilon _ { n } } \\left ( U _ { \\varepsilon _ { n } } ^ { j , k } , B _ { \\lambda \\varepsilon _ { n } } \\left ( x _ { j } \\right ) \\right ) = O ( 1 ) . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\beta ( B _ 1 , B _ 2 ) \\stackrel { d e f } { = } \\sup _ { { \\bf P } ( A _ 1 \\in B _ 1 ) \\ { \\bf P } ( A _ 2 \\in B _ 2 ) > 0 } \\left | \\ \\frac { { \\bf P } ( A _ 1 \\cap A _ 2 ) - { \\bf P } ( A _ 1 ) { \\bf P } ( A _ 2 ) } { { \\bf P } ( A _ 1 ) { \\bf P } ( A _ 2 ) } \\ \\right | . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in M _ + } \\lambda _ { \\mu } T ^ \\mu = 0 \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} L \\Omega = 0 \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\| A \\| _ { q , r } = \\begin{cases} \\left [ \\sum _ { i = 1 } ^ m ( \\sum _ { j = 1 } ^ n | a _ { i j } | ^ q ) ^ { r / q } \\right ] ^ { 1 / r } & \\mbox { i f $ q \\geq r $ } \\\\ \\left [ \\sum _ { j = 1 } ^ n ( \\sum _ { i = 1 } ^ m | a _ { i j } | ^ r ) ^ { q / r } \\right ] ^ { 1 / q } & \\mbox { i f $ r > q $ , } \\end{cases} \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\tag * { ( C ) } \\begin{cases} ( x - b _ { n - 1 } ) ^ k > 2 ^ { ( n - 1 ) k } x , \\\\ x ^ 2 > 2 ^ { n k } ( x + x ^ { 2 / k } ) . \\end{cases} \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\norm { f g } _ { L ^ 1 _ { t , x } } & = \\int _ { \\R } \\int _ { \\R ^ d } | f | ( t , x ) | g | ( t , x ) d x d t \\\\ & \\leq C \\int _ { \\R } \\norm { f ( t , \\cdot ) } _ { L ^ { p _ 1 , s _ 1 } _ x } \\norm { g ( t , \\cdot ) } _ { L ^ { p _ 2 , s _ 2 } _ x } d t \\leq C \\norm { f } _ { L ^ { q _ 1 , r _ 1 } _ t ( L ^ { p _ 1 , s _ 1 } _ x ) } \\norm { g } _ { L ^ { q _ 2 , r _ 2 } _ t ( L ^ { p _ 2 , s _ 2 } _ x ) } . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\bar W _ { L a b \\underline L } & = \\frac { 1 } { 2 } \\tilde \\sigma _ { a b } \\rho + \\frac { 1 } { 4 } \\epsilon _ { a b } \\sigma \\\\ \\bar W _ { a b c L } & = - \\epsilon _ { a b } \\epsilon _ { c d } \\beta ^ d \\\\ \\bar W _ { a b c \\underline L } & = \\epsilon _ { a b } \\epsilon _ { c d } \\underline { \\beta } ^ d \\\\ \\bar W _ { a b \\underline { L } L } & = \\frac { 1 } { 2 } \\epsilon _ { a b } \\sigma . \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\frac { q _ 1 + p _ 2 q _ 2 } { q _ 1 + q _ 2 } - \\frac { p _ 2 q _ 2 } { q _ 2 + q _ 3 } = \\frac { q _ 1 + q _ 2 } { q _ 1 + q _ 2 } - \\frac { q _ 2 } { q _ 2 + q _ 3 } \\leq \\frac { q _ 1 + q _ 3 } { q _ 1 + q _ 2 + q _ 3 } = \\P ( ( G \\cap H ^ c ) \\cup ( H \\cap G ^ c ) \\mid G \\cup H ) . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\hat \\lambda _ { { \\cal A } , j } ^ { ( K ) } \\buildrel \\Delta \\over = \\frac { 1 } { { \\sum \\limits _ { l = 1 } ^ K { { \\alpha ^ { K - l } } } } } \\sum \\limits _ { l = 1 } ^ K { { \\alpha ^ { K - l } } u _ j ^ { ( l ) } } = \\frac { { 1 - \\alpha } } { { 1 - { \\alpha ^ K } } } \\sum \\limits _ { l = 1 } ^ K { { \\alpha ^ { K - l } } u _ j ^ { ( l ) } } \\ ; \\ ; \\alpha < 1 \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} \\left ( \\alpha \\varphi _ k + \\psi _ k \\right ) \\to \\left ( \\mathbf { 1 } _ { v > 0 } + \\frac { 1 } { d } \\mathbf { 1 } _ { v < 0 } \\right ) Z = \\sigma ( v ) Z \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} X _ i ( A _ j f ) & = 0 & ( Y _ i + 2 \\delta _ { i j } ) ( A _ j f ) & = 0 & Z ( A _ j f ) & = 0 \\intertext { I f $ A _ j f = 0 $ f o r a l l $ j $ t h e n } X _ i ( B _ j f ) & = 0 & ( Y _ i + \\delta _ { i j } ) ( B _ j f ) & = 0 & ( Z + 1 ) ( B _ j f ) & = 0 \\intertext { I f $ B _ j f = 0 $ f o r a l l $ j $ t h e n } X _ i ( C _ j f ) & = 0 & Y _ i ( C _ j f ) & = 0 & ( Z + 2 ) ( C _ j f ) & = 0 \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} C _ { + } ( \\mathbb { R } ^ n ) = \\Bigl \\{ \\mu = ( \\mu _ 1 , \\mu _ 2 , \\ldots , \\mu _ n ) \\ , \\bigl \\vert \\ , \\mu \\in C ( \\mathbb { R } ^ n ) , \\mu _ k ( 0 ) = 0 , \\ , \\ , \\ , \\ , 1 \\le k \\le n , & \\\\ \\mu _ k ( t ) \\ , \\ , \\ , \\ , \\sum \\nolimits _ { k = 1 } ^ n \\mu _ k ( t ) = t , \\ , \\ , t \\in [ 0 , T ] \\Bigr \\} . & \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\hat O \\le \\hat h + c _ 0 | x '' | ^ 2 + 3 \\delta = : g \\mbox { i n } B _ { 1 / 2 } . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{gather*} \\boldsymbol { L } _ { J K } \\left ( \\boldsymbol { P } _ { M } u ( x ) \\right ) = \\left ( A _ { J K } ( \\boldsymbol { P } _ { M } u ; K + p ^ { N } \\mathbb { Z } _ { p } ) \\right ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) \\\\ - A _ { J K } \\left ( \\boldsymbol { P } _ { M } u \\left ( x \\right ) \\right ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) . \\end{gather*}"}
-{"id": "8496.png", "formula": "\\begin{align*} A _ + & = A _ 1 \\oplus A _ 0 \\oplus A _ \\lambda \\oplus A _ { \\lambda - \\frac { 1 } { 2 } } \\oplus \\bigoplus _ { m \\in w t ( D _ + ) } A _ { \\nu ^ { ( m , | \\alpha | - m ) } _ \\pm } \\\\ A _ - & = \\bigoplus _ { m \\in w t ( D ) - w t ( D _ + ) } A _ { \\nu ^ { ( m , | \\alpha | - m ) } _ \\pm } \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} 0 < \\delta < \\min \\Big \\{ \\frac { p ^ + _ l ( x ) - p ^ - _ l ( x ) } { \\max \\{ \\varphi ^ + _ l ( x ) , \\varphi ^ - _ l ( x ) \\} } \\mid x \\in \\R ^ N , \\ l = 1 , 2 , \\cdots , m \\Big \\} \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { k = 0 } ^ \\infty a _ k T _ k ( x ) \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\hat \\Sigma = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n X _ i \\otimes X _ i , \\hat \\Sigma _ \\epsilon = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\epsilon _ i \\otimes \\epsilon _ i , \\Sigma _ \\epsilon = \\Sigma - \\frac { 1 } { 2 r } F \\otimes F , \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\binom { - k - 1 } { - n - 1 } _ q = \\sum _ { Y \\in D ( n , k ) } q ^ { \\sigma ( Y ) + k - n ( n + 1 ) / 2 } , \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} ( Z _ { t } \\otimes u ) [ s ] & : = \\begin{cases} z _ s , & ~ s \\in [ 0 , t ) \\\\ u _ s , & ~ s \\in [ t , T ] , \\\\ \\end{cases} ~ ~ ~ ( W _ { t } \\otimes v ) [ s ] : = \\begin{cases} w _ s , & ~ s \\in [ 0 , t ) \\\\ v _ s , & ~ s \\in [ t , T ] , \\\\ \\end{cases} \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} Z _ k = \\alpha \\varphi _ k + d \\psi _ k . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} & \\int _ { S ^ 2 } ( h _ 0 ^ { ( 3 ) } - h ^ { ( 3 ) } ) d S ^ 2 \\\\ = & \\frac { 1 } { 2 } \\int _ { S ^ 2 } | \\AA ^ { ' ( 3 ) } | ^ 2 _ { \\tilde { \\sigma } } d S ^ 2 - \\frac { 1 } { 4 } \\int _ { S ^ 2 } ( \\tilde { \\Delta } Y _ 0 ^ { ( 3 ) } ) ^ 2 d S ^ 2 - \\int _ { S ^ 2 } \\tilde X ^ i \\tilde { \\Delta } ( Y _ i ^ { ( 5 ) } - Y _ i ^ { ' ( 5 ) } ) d S ^ 2 + \\int _ { S ^ 2 } ( k ^ { ( 3 ) } - \\frac { 1 } { 4 } - h ^ { ( 3 ) } ) d S ^ 2 . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\lambda _ i = c _ 1 ( \\Lambda _ i ) \\in H ^ 2 _ { T ^ { N + 1 } } ( \\textnormal { p t } ) = H ^ 2 ( B T ^ { N + 1 } ) & & z = - c _ 1 ( q ) \\in H ^ 2 _ { T } ( \\textnormal { p t } ) \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - q ^ r P ^ { - 1 } } = \\frac { 1 } { ( 1 - q ^ r ) + q ^ r ( 1 - P ^ { - 1 } ) } = \\frac { 1 } { 1 - q ^ r } \\sum _ { m = 0 } ^ { N } \\left ( \\frac { q ^ r } { 1 - q ^ r } ( 1 - P ^ { - 1 } ) \\right ) ^ m \\in K \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } ( \\ ! ( q ) \\ ! ) \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} ( A _ { n , 3 } V _ { n , 3 } + \\epsilon V _ { n , 3 } ) ( \\eta , y , x ) = \\mathcal { S } _ 3 ( \\eta ) + \\eta [ \\delta y + x ] + ( A _ { n , 2 } V _ { n , 2 } + \\epsilon V _ { n , 2 } ) ( y , x ) - \\frac { \\delta } { 2 } , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\mu ( X ) = \\mu ^ Z ( X ) \\xi _ 1 + \\mu ^ W ( X ) \\xi _ 2 , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & y _ t - ( b ( y _ x , t ) x ^ { \\alpha } y _ x ) _ x + F ( y , y _ x ) = f 1 _ \\mathcal { O } + v ^ 1 1 _ { \\mathcal { O } _ 1 } + v ^ 2 1 _ { \\mathcal { O } _ 2 } , \\\\ & y ( 1 , t ) = 0 \\ \\ \\begin{cases} y ( 0 , t ) = 0 \\ \\ & \\ 0 \\leq \\alpha < 1 , \\\\ ( x ^ { \\alpha } y _ x ) ( 0 , t ) = 0 \\ \\ & \\ 1 \\leq \\alpha < 2 , \\end{cases} \\\\ & y ( 0 ) = y _ { 0 } . \\end{array} \\right . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} 0 = ( \\mathbb { L } - \\phi ) ( \\bar { A } _ t ; Z _ t , W _ t ) = \\sup _ { A _ s \\in \\mathbb { C } ^ { \\kappa , \\mu , \\mu _ 0 } } ( \\mathbb { L } - \\phi ) ( A _ s ; Z _ t , W _ t ) , \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\omega _ \\mu ( \\xi , \\eta ) = \\mu \\left ( [ \\xi , \\eta ] \\right ) \\ , , \\forall \\xi , \\eta \\in \\mathfrak { g } \\ , , \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} ( K \\cap F ) ^ { ( \\alpha ) } = K ^ { ( \\alpha ) } \\cap F . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\mathfrak { D } ( f , & ( 1 , 3 , 6 ) ) \\\\ = & \\left \\{ ( x _ 1 , x _ 2 , x _ 1 + x _ 2 , 1 - x _ 1 - x _ 2 , 1 - x _ 2 , 1 - x _ 1 ) \\left | \\begin{array} { l } 0 \\le x _ 1 \\le x _ 2 , \\\\ x _ 1 + x _ 2 \\le 1 / 2 \\end{array} \\right . \\right \\} \\\\ = \\ , & \\{ C + y _ 1 \\delta _ 1 + y _ 2 \\delta _ 2 \\mid y _ 2 / \\sqrt { 3 } \\le y _ 1 \\le \\sqrt { 3 } y _ 2 , \\sqrt { 3 } y _ 1 + y _ 2 \\le \\sqrt { 3 } \\} , \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} ( \\alpha \\bar u _ 1 - d \\bar u _ 2 ) ^ + = \\alpha \\bar u _ 1 \\textup { a n d } ( \\alpha \\bar u _ 1 - d \\bar u _ 2 ) ^ - = d \\bar u _ 2 . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\partial _ t \\varphi - \\Delta \\mu & = \\left ( \\mathcal P \\sigma - \\alpha u \\right ) h ( \\varphi _ 1 ) + ( \\mathcal P \\sigma _ 2 - a - \\alpha u _ 2 ) \\left ( h ( \\varphi _ 1 ) - h ( \\varphi _ 2 ) \\right ) \\\\ \\mu & = - A \\Delta \\varphi + B ( \\psi ' ( \\varphi _ 1 ) - \\psi ' ( \\varphi _ 2 ) ) , \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\tilde { \\alpha } ( t , z ) & : = \\langle A ^ * \\zeta , \\phi ( t ) + T z \\rangle + \\langle \\zeta , \\alpha ( \\phi ( t ) + T z ) \\rangle , \\\\ \\tilde { \\sigma } ( t , z ) & : = \\langle \\zeta , \\sigma ( \\phi ( t ) + T z ) \\rangle . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\eta _ { } \\ ! \\left ( \\ ! \\theta \\ ! , \\phi \\ ! \\right ) \\ ! = \\ ! - \\ ! \\min \\ ! { \\left \\{ \\ ! - \\ ! \\left ( \\ ! \\eta _ { } \\left ( \\ ! \\theta , \\phi \\ ! = \\ ! \\frac { \\pi } { 2 } \\ ! \\right ) \\ ! + \\ ! \\eta _ { } \\left ( \\theta \\ ! = \\ ! 0 , \\phi \\right ) \\ ! \\right ) \\ ! , \\eta _ { \\max } \\ ! \\right \\} } \\ ! . \\ ! \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big [ t ^ { k + 1 } \\| A ^ { \\frac { k + 1 } { 2 } } w ( t ) \\| ^ 2 \\Big ] & = ( k + 1 ) t ^ { k } \\| A ^ { \\frac { k + 1 } { 2 } } w ( t ) \\| ^ 2 - 2 t ^ { k + 1 } \\| A ^ { \\frac { k + 2 } { 2 } } w ( t ) \\| ^ 2 \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} d _ t ( K ) & \\preceq d _ t ( k - 1 ) \\\\ d _ t ( K ) & \\preceq \\frac { 1 } { K } \\sum _ { k = 1 } ^ K d _ t ( k - 1 ) \\\\ x ^ T d _ t ( K ) & \\leq \\frac { 1 } { K } x ^ T \\sum _ { k = 1 } ^ K d _ t ( k - 1 ) \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 2 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = i ( - R _ { 2 1 } + R _ { 2 3 } ) = - i R _ { 2 1 } + i R _ { 2 3 } = 0 + i = i = u _ 2 ; \\\\ \\frac { R _ { 4 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 4 3 } } { - i - \\omega ( 3 , 3 ) } u _ 3 = - i R _ { 4 1 } + i R _ { 4 3 } = i - 2 i = - i = u _ 4 . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar H ^ 6 - 3 \\bar H ^ 4 S + 3 \\bar H ^ 2 S ^ 2 - 3 S ^ 2 + 2 S \\\\ & \\geq \\dfrac { ( \\bar H ^ 2 - S ) ^ 2 } 4 \\bigl [ ( 6 + \\dfrac 3 4 ) S - ( 4 + \\dfrac 3 { 8 } ) \\bar H ^ 2 - \\dfrac { 3 S ( 3 S - 2 ) } { 8 \\bar H ^ 2 } \\bigl ] . \\end{aligned} \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\mathcal { F } ( u , v ) : = \\big \\{ \\lambda \\in \\mathbb { Y } \\ ! : \\lambda + u \\in \\mathcal { N } _ { \\mathcal { K } } ( \\overline { y } ) , \\ , \\lambda + v \\in \\mathcal { H } ( \\overline { \\eta } ) \\big \\} . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\psi _ { \\pm } = \\frac { 1 } { \\sqrt { 2 } } ( \\psi \\pm \\zeta \\theta ) . \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\Delta _ \\sigma = \\bigcup _ { a = 1 } ^ k T _ { a , \\sigma } , \\sigma = 1 , 2 , \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( | S _ n ( \\kappa _ n ( x ) ) - S _ n ( \\ell _ n ( x ) ) | > \\delta b _ c ^ { ( n ) } \\right ) = - \\infty \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ s | | 2 \\theta + K _ { j _ i } \\alpha | | _ { \\R / \\Z } \\leq \\sum _ { i = 0 } ^ s e ^ { - ( \\varsigma + \\varepsilon ) | K _ { j _ i } | } \\leq 2 e ^ { - ( \\varsigma + \\varepsilon ) | K _ { j _ s } | } . \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | x _ { n + 1 } | = \\lim _ { n \\to \\infty } | x _ n | \\lim _ { n \\to \\infty } | 1 - h _ n x _ n ^ 2 | = 0 , \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} \\mathcal { A } ( b _ 1 ^ 0 ) = b _ 2 ^ 0 \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\ll _ { \\varepsilon } M ^ { 1 / 4 - 1 / 1 6 + \\varepsilon } . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\omega ' ( ( f _ 1 , \\phi _ 1 ) , ( f _ 2 , \\phi _ 2 ) ) \\ = \\ \\langle A ( f _ 2 ) , \\phi _ 1 \\rangle + \\langle B ( \\phi _ 2 ) , \\phi _ 1 \\rangle + \\langle f _ 1 , D ( \\phi _ 2 ) \\rangle \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\frac { \\gamma \\left ( n , \\frac { n ( \\rho + \\sigma _ w ^ 2 ) } { \\rho } \\ln ( 1 + \\frac { \\rho } { \\sigma _ w ^ 2 } ) \\right ) } { \\Gamma ( n ) } - \\frac { \\gamma \\left ( n , \\frac { n \\sigma _ w ^ 2 } { \\rho } \\ln ( 1 + \\frac { \\rho } { \\sigma _ w ^ 2 } ) \\right ) } { \\Gamma ( n ) } = \\epsilon , \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\Phi _ n & : = N ( - n ) ^ { { n + 2 \\choose 2 } } \\xrightarrow { \\ ; \\ ; F _ n \\ ; \\ ; } N ( - n + 1 ) ^ { { n + 1 \\choose 2 } } \\mbox { a n d } \\\\ \\Psi _ n & : = N ( n - 1 ) ^ { { n + 1 \\choose 2 } } \\xrightarrow { \\ ; \\ ; F ^ t _ n \\ ; \\ ; } N ( n ) ^ { { n + 2 \\choose 2 } } \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\mu = \\mu _ 1 + \\cdots + \\mu _ k , \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} C : = \\max \\{ C _ { r _ 0 } C _ { s _ 0 ^ \\prime } , C _ { r _ 1 } C _ { s _ 1 ^ \\prime } \\} \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\mathcal { L } _ j [ w _ j ] = - \\Delta w _ j - M \\frac { \\langle \\nabla z _ j , \\nabla w _ j \\rangle } { z _ j } + 2 a _ { 0 , j , w _ j } + ( m + 2 - p ) a _ { 1 , j , w _ j } + a _ { 2 , w _ j } + a _ { 3 , j , w _ j } . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\psi _ 2 ( s ) & = \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } \\\\ & \\leq \\phi ( s ^ { 1 / 2 } ) \\sum _ { m = 1 } ^ { s ^ { - 1 } } \\frac { 1 } { m } \\\\ & = O ( s ^ { - u / 2 } \\log ( 1 / s ) ) . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} \\kappa ( x , y ) = \\int \\limits _ D \\left | b ( z ) \\frac { G ( x , z ) \\partial _ z G ( z , y ) } { G ( x , y ) } \\right | d z , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} \\mu \\bigg ( 1 - \\frac { m } { 1 - \\beta \\omega ( \\mu , s + 3 \\mu ) } \\bigg ) - \\eta = 0 , \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq z + \\delta \\right ) & - P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq z - \\delta \\right ) \\geq \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( z - \\delta ) ) + \\eta ) b _ c ^ { ( n ) } } - \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( z + \\delta ) ) - \\eta ) b _ c ^ { ( n ) } } \\\\ & = \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( z - \\delta ) ) + \\eta ) b _ c ^ { ( n ) } } \\left ( 1 - \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( z + \\delta ) ) - H ( \\ell _ 2 ( z - \\delta ) ) - 2 \\eta ) b _ c ^ { ( n ) } } \\right ) \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} - u _ i \\cdot u _ j = \\cosh ( t _ i - t _ j ) + \\frac 1 2 | \\tilde s _ i - \\tilde s _ j | ^ { 2 } e ^ { t _ i + t _ j } , z _ i = \\cosh t _ i + \\frac 1 2 | \\tilde s _ i | ^ { 2 } e ^ { t _ i } . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\lambda - \\tfrac { 1 } { 2 } \\star \\lambda - \\tfrac { 1 } { 2 } = \\begin{cases} 1 & \\mbox { i f } a = - \\frac { 1 } { | \\alpha | } \\\\ \\lambda - \\tfrac { 1 } { 2 } & \\mbox { i f } a = \\frac { 1 } { | \\alpha | } \\\\ 1 , \\lambda - \\tfrac { 1 } { 2 } & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} u _ { t t } - \\Delta u - u ^ { 5 } = 0 \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\sigma _ { \\mathbf { a } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) - \\sigma _ { \\mathbf { a } , \\mathbf { G } , \\mathbf { B } } ^ { 2 } \\left ( P \\right ) = E _ { P } \\left [ \\left \\{ \\frac { 1 } { \\pi _ { \\mathbf { a } } \\left ( \\mathbf { B } ; P \\right ) } - 1 \\right \\} v a r _ { P } \\left [ \\left . b _ { \\mathbf { a } } ( \\mathbf { G , B } ; P ) \\right \\vert \\mathbf { B } \\right ] \\right ] \\geq 0 . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} L ( { \\cal D } _ j ) = T S P ( X _ 1 , \\ldots , X _ { j - 1 } , X _ { j + 1 } , \\ldots , X _ { n + 1 } ; S ) , \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} t _ k : = 2 b ^ { - 1 / Q } \\mu ( 2 B _ 0 \\setminus E _ { k - 1 } ) ^ { 1 / Q } . \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil \\varepsilon f _ 5 ( n ) \\rceil ) = - \\ell _ 5 ' \\varepsilon . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} | f * g | ( t , x ) \\leq \\int _ { \\R } \\int _ { \\R ^ d } | f ( s , y ) g ( t - s , x - y ) | d y d s = \\norm { f ( \\cdot , \\cdot ) g ( t - \\cdot , x - \\cdot ) } _ { L ^ 1 _ t ( L ^ 1 _ x ) } . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} - \\frac { \\partial ^ 2 \\mathcal U } { \\partial p ^ 2 } = \\frac { w _ c } { \\left ( p + \\frac { 1 } { 1 / \\hat { \\mu } ( \\beta ) - 1 } \\right ) ^ 2 } + \\frac { w _ d } { \\left ( p + \\frac { 1 } { 1 / \\hat { \\nu } ( \\beta ) - 1 } \\right ) ^ 2 } > 0 . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} F _ 3 : = - \\theta \\beta ( \\nabla _ H \\pi ) - 2 ( \\nabla _ H \\theta ) \\beta \\cdot ( \\nabla _ H v ) - 2 \\theta ( \\partial _ z \\beta ) ( \\partial _ z v ) - ( \\Delta _ H \\theta ) \\beta v - 2 \\theta ( \\partial _ z ^ 2 \\beta ) v . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\begin{aligned} e _ x & = ( 1 , 0 , 0 , 0 , \\dots ) , \\\\ e _ v & = ( 0 , 1 , 0 , 0 , \\dots ) , \\\\ e _ k & = ( 0 , 0 , 0 , \\dots , k ^ s , 0 , \\dots ) , \\ , k \\geq 1 . \\end{aligned} \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( q ^ { n } ; q ) _ { n + 1 } ( q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n + 1 } } { ( q ; q ^ 2 ) ^ 2 _ { n + 1 } } , \\\\ [ 6 p t ] \\sum _ { n = 0 } ^ { \\infty } q ^ n ( - q ^ { n + 1 } ; q ) _ n ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} - h ' ( s ) & = \\sum _ { m = 1 } ^ { \\infty } - \\phi ' ( m s ) \\\\ & = \\sum _ { m \\leq s ^ { - 1 / 2 } } - \\phi ' ( m s ) + \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } - \\phi ' ( m s ) + \\sum _ { m > s ^ { - 1 } } - \\phi ' ( m s ) \\\\ & = : \\chi _ 1 ( s ) + \\chi _ 2 ( s ) + \\chi _ 3 ( s ) . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} J _ L ( e _ 1 ) : = e _ 2 ; ~ ~ ~ ~ J _ L ( e _ 2 ) : = - e _ 1 . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} w = \\begin{bmatrix} 0 \\\\ - 4 \\\\ - 1 3 \\\\ - 9 \\\\ - 6 \\end{bmatrix} , \\ ; v = \\begin{bmatrix} 0 \\\\ - 5 \\\\ - 4 \\\\ - 8 \\\\ - 1 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} X _ { ( ( T ^ \\ast Q ) _ \\mu , \\omega _ \\mu , h _ \\mu , f _ \\mu , u _ \\mu ) } = X _ { h _ \\mu } + \\textnormal { v l i f t } ( f _ \\mu ) + \\textnormal { v l i f t } ( u _ \\mu ) , \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} l ( A _ n ) = \\left \\lfloor \\frac { 3 n - 1 } { 2 } \\right \\rfloor - b _ n - 1 , \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} a = \\sum _ { j = 1 } ^ m a _ j ^ n = \\sum _ { j = 1 } ^ m S _ n ( a _ j , \\dotsc , a _ j ) \\in \\mathcal { S } _ n ( A ) . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} E _ v ( t ) = \\frac { 1 } { 2 } \\left \\{ | | v _ t ( t ) | | ^ 2 + | | ( - \\Delta ) ^ { \\frac { \\delta } { 2 } } v _ t ( t ) | | ^ 2 + | | ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } v ( t ) | | ^ 2 \\right \\} . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\tau _ 4 ( x ) = g _ 4 ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 2 7 } { 6 5 } , \\tfrac { 3 4 } { 3 ^ 4 } ) . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\frac 1 { \\theta _ i } : = \\frac { 1 - r } r - \\frac 1 { \\delta _ i } = \\sum _ { j = 1 } ^ { m + 1 } \\frac 1 { \\delta _ j } - \\frac 1 { \\delta _ i } > 0 , \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\bar { H } _ m = R _ { m + 1 } \\hat { H } ^ { ( h ) } _ m R ^ { - 1 } _ m . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\xi ^ j \\leftarrow \\frac { 1 } { \\prod _ { j ' = 1 } ^ j c _ { j ' } } \\xi ^ j \\eta ^ j \\leftarrow \\frac { 1 } { \\prod _ { j ' = 1 } ^ j c _ { j ' } } \\eta ^ j , \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\mathrm { h o l } _ { \\phi } ^ { \\Theta _ { P } } ( \\gamma ) = - \\mathrm { C S } ( A _ { \\phi } ^ { \\gamma } ) \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) = f ( x ) + 2 \\ , t ^ 2 \\ , \\left ( x _ 1 ^ 2 + x _ 2 ^ 2 - 1 \\right ) + \\frac { 2 } { 5 } \\ , t ^ 4 \\ , . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( \\pi ) ) = \\mathrm { M a n t } _ { G , b , \\mu } ( L J ( \\rho _ 1 ( n _ 2 / 2 ) ) \\boxtimes L J ( \\rho _ 2 ( - n _ 1 / 2 ) ) ) \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} V ^ 2 = & - \\langle T _ 0 , T _ 0 \\rangle = ( A ^ 2 - \\sum _ i C _ i ^ 2 ) + O ( r ^ 2 ) \\\\ V ^ 2 \\nabla _ a \\tau = & - ( T _ 0 ^ T ) _ a = r C _ i \\tilde \\nabla _ a \\tilde X ^ i + O ( r ^ 2 ) . \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } V ( \\gamma _ \\rho , N ) = V ( O , N ) . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} v _ { t t } = 0 , \\quad \\mbox { o n } \\Gamma _ f \\times ( 0 , T ) . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} H ^ { p ^ { \\ast } } = - \\langle X , e _ { p ^ { \\ast } } \\rangle , \\ \\ p = 1 , 2 . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} c _ { 1 } ^ { 2 } \\left ( f _ { 1 } f _ { 1 } ^ { \\prime \\prime } - \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } \\right ) - K _ { 0 } \\left ( a f _ { 1 } ^ { \\prime } + c _ { 1 } f _ { 1 } \\right ) ^ { 4 } f _ { 2 } ^ { 2 } = 0 , \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} d _ i ^ \\circ = \\sqrt { \\frac { B \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { w _ i ( c _ i - p ) } } - \\frac { \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { w _ i } . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} n = \\frac { k ( k + s ^ 2 - 4 s + 2 ) } { k ( k + s ^ 2 - 4 s + 2 ) - s ^ 3 + 4 s ^ 2 - 5 s + 2 } . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} u _ t ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\xi ^ n I _ n \\Big ( h _ n ( . , t , x ) \\Big ) , \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} | \\P ( A _ 2 \\geq 0 | f _ i , i \\leq n ) - 1 / 2 | \\leq C \\sum _ { i = 1 } ^ n \\frac { \\sigma ^ 6 + | f _ i | ^ 3 \\sigma ^ 3 } { n ^ { 3 / 2 } \\sigma ^ 6 } . \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\| \\hat P _ j - \\tilde P _ j \\| _ 2 \\leq 2 \\tilde g _ j ^ { - 1 } \\| \\hat \\Sigma - \\tilde \\Sigma \\| _ \\infty , \\tilde g _ j = \\min ( \\tilde \\lambda _ { j - 1 } - \\tilde \\lambda _ j , \\tilde \\lambda _ j - \\tilde \\lambda _ { j + 1 } ) , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} \\frac { \\partial G } { \\partial \\overline { m } } = - \\frac { 1 } { ( \\overline { m } + 1 ) ^ 3 } - \\epsilon \\frac { 1 } { \\overline { m } + 1 } . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\sigma ( [ x , y ] ) & = \\sigma ( x ^ { - 1 } y ^ { - 1 } x y ) \\\\ & = \\sigma ( x ^ { - 1 } ) \\sigma ( y ^ { - 1 } ) \\sigma ( x ) \\sigma ( y ) \\\\ & = \\sigma ( x ^ { - 1 } ) \\sigma ( x ) \\sigma ( y ^ { - 1 } ) \\sigma ( y ) \\\\ & = \\sigma ( e ) \\in \\mathcal { A } _ 0 . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n } } { ( z q ^ { n + 1 } ; q ) _ { n + 2 } ( z q ^ { 2 n + 4 } ; q ^ 2 ) _ { \\infty } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} v : = \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ - 1 \\end{bmatrix} , \\ u : = \\begin{bmatrix} 1 \\\\ - 1 \\\\ 1 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} Y ^ n _ t = \\Phi _ n ( t , X ^ n _ t ) , Y _ t = \\Phi ( t , X _ t ) , \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} \\zeta _ { l _ 0 , t } & - D _ { l _ 0 } \\Delta \\zeta _ { l _ 0 } = f _ { l _ 0 } ( v ) - f _ { l _ 0 } ( u ) \\\\ & \\geq f _ { l _ 0 } ( \\cdots , u _ { l _ 0 - 1 } , v _ { l _ 0 } , u _ { l _ 0 + 1 } , \\cdots ) - f _ { l _ 0 } ( u ) \\\\ & \\geq - M \\zeta _ { l _ 0 } \\ \\ \\ ( M : = \\sup _ { w \\in [ p ^ - , p ^ + ] } | D F ( w ) | ) . \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} \\alpha _ 1 & = 1 + \\frac { 1 + i } { 4 } ( q - 1 ) + o ( q - 1 ) = 1 \\left ( 1 + \\frac { 1 + i } { 4 } ( q - 1 ) + o ( q - 1 ) \\right ) \\\\ \\alpha _ 2 & = - i + \\frac { i } { 2 } ( q - 1 ) + o ( q - 1 ) = - i \\left ( 1 - \\frac { 1 } { 2 } ( q - 1 ) + o ( q - 1 ) \\right ) \\\\ \\alpha _ 3 & = - 1 - \\frac { 1 - i } { 4 } ( q - 1 ) + o ( q - 1 ) = - 1 \\left ( 1 + \\frac { 1 - i } { 4 } ( q - 1 ) + o ( q - 1 ) \\right ) \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} ( I - Q _ r ) \\hat { Q } _ r & = R _ r ( \\hat \\Sigma - \\mu _ r I + \\Sigma - \\hat \\Sigma ) \\hat { Q } _ r \\\\ & = R _ r ( \\hat \\Sigma - \\mu _ r I ) \\hat { Q } _ r - R _ r E Q _ r - R _ r E ( \\hat { Q } _ r - Q _ r ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\widetilde { \\ell _ { q , z } } ( Q ) = \\ell _ q \\left ( \\left ( \\frac { 1 - q } { z } \\right ) ^ { N + 1 } Q \\right ) \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} - \\phi ' ( s ) = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} & T _ { 2 k + 1 , 2 a + 1 } ( x ; q ) - T _ { 2 k + 1 , 2 a - 1 } ( x ; q ) \\\\ & = ( - x q ; q ^ 2 ) _ \\infty [ \\overline { Q } _ { k + \\frac { 1 } { 2 } , a + \\frac { 1 } { 2 } } ( x ^ 2 ; q ^ 2 ) - \\overline { Q } _ { k + \\frac { 1 } { 2 } , a - \\frac { 1 } { 2 } } ( x ^ 2 ; q ^ 2 ) ] \\\\ & = ( - x q ; q ^ 2 ) _ \\infty [ ( x q ) ^ { 2 a + 1 } \\overline { Q } _ { k + \\frac { 1 } { 2 } , k - a } ( x ^ 2 q ^ 2 ; q ^ 2 ) + ( x q ) ^ { 2 a - 1 } \\overline { Q } _ { k + \\frac { 1 } { 2 } , k - a + 1 } ( x ^ 2 q ^ 2 ; q ^ 2 ) ] . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\mu _ v ( \\omega ) = \\mathcal { H } ^ n \\Big ( N _ v \\big ( \\omega \\big ) \\Big ) . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} ( e ( t ) + y ( t ) ) ' & \\leq M _ { 1 } G _ { 1 } - a k _ { 1 } y \\\\ & = M _ { 1 } G _ { 1 } + M _ { 1 } G _ { 1 } - M _ { 1 } G _ { 1 } - a k _ { 1 } y \\\\ & \\leq 2 M _ { 1 } G _ { 1 } - \\frac { M _ { 1 } } { \\bar { x } } e ( t ) - a k _ { 1 } y \\\\ & \\leq 2 M _ { 1 } G _ { 1 } - \\bar { \\mu } ( e ( t ) + y ( t ) ) \\textnormal { w h e r e } \\bar { \\mu } = \\min \\left \\lbrace \\frac { M _ { 1 } } { \\bar { x } } , a k _ { 1 } \\right \\rbrace , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} I _ a ^ k \\left ( D _ a ^ { k ' } f \\right ) ( x ) = f ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\left ( z \\frac { \\partial } { \\partial z } + \\mu - \\frac { \\rho } { z } \\right ) ( z ^ { - \\mu } z ^ \\rho \\alpha ) & = z ^ { - \\mu } \\rho z ^ \\rho - \\frac { \\rho } { z } z ^ { - \\mu } z ^ \\rho = 0 \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\lambda _ j = a _ 0 + a _ 1 \\omega ^ j + \\cdots + a _ { n - 1 } \\omega ^ { ( n - 1 ) j } , \\ ; 0 \\le j < n , \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} e _ k ( i d : \\ell ^ n _ p ( \\mathbb { R } ) \\rightarrow \\ell ^ n _ q ( \\mathbb { R } ) ) \\sim \\begin{cases} 1 & 1 \\leq k \\leq \\log _ 2 n , \\\\ \\displaystyle \\biggl ( \\frac { \\log _ 2 ( 1 + n / k ) } { k } \\biggr ) ^ { \\frac { 1 } { p } - \\frac { 1 } { q } } & \\log _ 2 n \\leq k \\leq n , \\\\ [ 1 0 p t ] \\displaystyle 2 ^ { - \\frac { k - 1 } { n } } n ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } & n \\leq k . \\end{cases} \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\beta _ { j , j '' } ( a _ { i , j } ) & = \\bar { \\beta } _ { j , j '' } ( a _ { i , j } ) = \\bar { \\beta } _ { j ' , j '' } \\circ \\bar { \\beta } _ { j , j ' } ( a _ { i , j } ) \\\\ & = \\bar { \\beta } _ { j ' , j '' } \\circ \\beta _ { j , j ' } ( a _ { i , j } ) = \\beta _ { j ' , j '' } \\circ \\beta _ { j , j ' } ( a _ { i , j } ) . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} b _ { n + 1 } \\in A c _ 1 ^ { - 1 } \\cap \\cdots \\cap A c _ n ^ { - 1 } \\cap L = A e _ { \\sigma ( j _ 1 ) } ^ { - 1 } \\cap \\cdots \\cap A c _ { \\sigma ( j _ n ) } ^ { - 1 } \\cap L \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 2 ( x ) + \\frac { 2 } { 6 ^ 3 } ( 2 ^ 3 x - 3 ) ( 1 2 - 3 ^ 3 x ) + \\frac { 2 } { 6 ^ 4 } ( 3 ^ 4 x - 3 4 ) ( 7 - 2 ^ 4 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 8 } { 1 9 } } \\\\ & + \\frac 1 { 1 0 \\cdot 6 ^ 4 } - \\eta \\\\ & = - \\frac { 6 3 8 5 5 1 9 3 6 2 9 7 4 5 3 9 8 3 9 6 3 8 0 1 1 8 8 1 4 2 2 6 3 } { 3 9 4 0 9 6 0 1 8 1 6 2 6 7 8 0 1 9 8 7 6 6 6 8 9 1 0 7 6 1 2 1 3 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\mu _ { 1 } \\left ( u _ { 2 } \\right ) + 2 a \\mu _ { 2 } \\left ( u _ { 1 } \\right ) \\mu _ { 3 } \\left ( u _ { 2 } \\right ) + a ^ { 2 } \\mu _ { 4 } \\left ( u _ { 2 } \\right ) = 0 , \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} \\boldsymbol { P } _ { M } \\varphi \\left ( x \\right ) = { \\textstyle \\sum \\limits _ { I _ { j } \\in G _ { N } ^ { M } } } \\varphi \\left ( I _ { j } \\right ) \\Omega \\left ( p ^ { M } \\left \\vert x - I _ { j } \\right \\vert _ { p } \\right ) . \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} J ( x ) = \\frac { 1 } { | x | ^ n \\{ 1 + ( \\ln | x | ) ^ 2 \\} } \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} h _ 1 ' ( u ) = \\xi '' ( u ) [ q + ( q - u ) z _ 1 + q z _ 2 ] - z _ 1 \\xi ' ( u ) - ( 1 + z _ 2 ) \\xi ' ( q ) . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} b _ { 0 } ( x ) y ^ { \\prime \\prime } ( x ) + b _ { 1 } ( x ) y ^ { \\prime } ( x ) = \\lambda y ( x ) \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\P ( w ^ e _ n = 0 \\ , | \\ , w ^ e _ { n - 1 } , \\dotsc , w ^ e _ 0 ) \\geq \\delta \\coloneqq e ^ { - p v T } \\ ! \\left [ \\tfrac { e ^ { - v T } + ( 1 - p ) ( 1 - e ^ { - v T } ) - e ^ { - p v T } } { 1 - e ^ { - p v T } } \\right ] , \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} Q _ { 0 , 1 } \\left ( \\frac { x } { | x | } \\right ) = q _ * : = | { \\bf S } ^ { N - 1 } | ^ { - \\frac { 1 } { 2 } } , Q _ { 1 , i } \\left ( \\frac { x } { | x | } \\right ) = q _ N \\frac { x _ i } { | x | } \\quad \\mbox { w i t h } q _ N = N ^ { \\frac { 1 } { 2 } } q _ * , \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\langle \\nabla u _ j , \\nabla w _ j \\rangle = \\left ( 2 \\alpha _ j z _ j \\frac { \\langle \\nabla u _ j , \\nabla \\phi \\rangle } { \\phi } + \\langle \\nabla z _ j , \\nabla u _ j \\rangle \\right ) \\phi ^ { 2 \\alpha _ j } \\mbox { i n } \\omega . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} B _ \\Omega ( \\gamma _ n y _ n , y _ n ) = \\delta _ n . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{gather*} p _ 1 + p _ 2 + \\dots + p _ l + q _ 1 + q _ 2 + \\dots + q _ l \\leq n ; \\\\ p _ i = 0 \\ \\Rightarrow \\ q _ i = 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j = 1 . \\end{gather*}"}
-{"id": "626.png", "formula": "\\begin{gather*} \\boldsymbol { P } _ { M } \\left ( \\boldsymbol { L } _ { J K } u ( x ) \\right ) = \\left ( A _ { J K } ( u ; K + p ^ { N } \\mathbb { Z } _ { p } ) \\right ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) \\\\ - A _ { J K } \\boldsymbol { P } _ { M } u ( x ) \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) . \\end{gather*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\mathcal { I } _ g ( x ) & = g x y g ^ { - 1 } \\\\ & = g x d ( g ) y g ^ { - 1 } \\\\ & = g x g ^ { - 1 } g y g ^ { - 1 } \\\\ & = \\mathcal { I } _ g ( x ) \\mathcal { I } _ g ( y ) . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = ( - 1 ) ^ k \\operatorname { s g n } ( k ) q ^ { \\frac { 1 } { 2 } k ( 2 n - k + 1 ) } \\binom { k - n - 1 } { k } _ q . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\mathcal { L } = \\mathcal { F } + \\nabla _ { 0 } + \\sum _ { i = 1 } ^ d \\nabla _ { i } ^ 2 , \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\left [ ( z \\partial _ { t _ 1 } ) ^ { N + 1 } - e ^ { t _ 1 } \\right ] j ^ \\textnormal { c o h } ( t _ 1 , z ) = 0 \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} Q _ \\lambda ( \\Omega ) g ^ { q _ \\alpha - 1 } ( \\varsigma ) = \\int _ { \\mathbb { H } ^ n } \\frac { g ( \\eta ) } { | \\eta ^ { - 1 } \\varsigma | ^ { Q - \\alpha } } d \\eta , \\ \\ \\ \\ g ( 0 ) = 1 . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} d X _ { k } ^ { \\epsilon , v _ k } ( t ) = f _ k \\bigl ( X _ { k } ^ { \\epsilon , v _ k } ( t ) , v _ { k } ^ { \\epsilon } \\bigl ( X _ { k } ^ { \\epsilon , v _ k } ( t ) \\bigr ) \\bigr ) d t + \\sqrt { \\epsilon } \\sigma _ k \\bigl ( X _ { k } ^ { \\epsilon , v _ k } ( t ) \\bigr ) d W ( t ) , \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} f = \\sum _ { N \\in \\N } 1 _ { \\{ 2 ^ N \\} } \\sqrt { N } \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} a _ { k _ 0 } \\leq C ' r _ 0 ^ s 2 ^ { k _ 0 } \\leq C '' b ^ { - 1 / p } \\left ( \\sum _ { j = - \\infty } ^ { \\infty } \\int _ { 2 B _ 0 } g _ j ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\int _ I \\rho _ 3 ^ 2 | y _ { x x } | ^ 2 d x & \\leq C \\left ( \\int _ I \\rho ^ 2 | G | ^ 2 d x + \\int _ \\mathcal { O } \\rho _ 1 ^ 2 | f | ^ 2 d x + \\int _ I \\rho _ 3 ^ 2 | y _ t | ^ 2 d x \\right . \\\\ & + \\left . \\int _ I \\rho _ 0 ^ 2 | y | ^ 2 d x + \\int _ I \\rho _ 2 ^ 2 | y _ x | ^ 2 d x + \\sum _ { i = 1 } ^ 2 \\int _ I \\rho _ 0 ^ 2 | p ^ i | ^ 2 d x \\right ) . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} H = \\lambda v ( T ^ n x ) + \\Delta . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} S p i n ^ c ( n ) = ( S p i n ( n ) \\times S ^ 1 ) / \\{ \\pm 1 \\} , \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} F : = \\frac { a f _ 0 ( x _ 0 ) ^ { \\frac { \\beta + 1 } { \\beta } } } { 4 R ^ { \\frac 1 { \\beta } } } \\leq \\frac { 1 } { 1 6 n } \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} J ^ { \\textnormal { c o h } , \\textnormal { e q } } ( z , Q ) = \\sum _ { i = 0 } ^ N J ^ { \\textnormal { c o h } , \\textnormal { e q } } _ { | H = \\lambda _ i } ( z , Q ) L a g _ i \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} b = \\frac 1 a ( a ^ 2 + c ^ 2 - d ^ 2 ) , \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\varphi ' ( \\delta ) = \\frac { 1 } { \\sqrt { \\delta } } - \\frac { 1 } { ( 1 + \\delta ) \\sqrt { ( 1 + \\delta ) ^ 2 - 1 } } = \\frac { 1 } { \\sqrt { \\delta } } \\left ( 1 - \\frac { 1 } { ( 1 + \\delta ) \\sqrt { 2 + \\delta } } \\right ) , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} t _ n = \\frac { 1 } { C _ 1 ^ 2 } \\frac { \\log n } { 4 } , \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\textbf { a } ( \\textbf { x } _ k ) = \\textbf { a } _ 1 \\left ( x _ { k , 1 } \\right ) \\otimes \\textbf { a } _ 2 \\left ( x _ { k , 2 } \\right ) \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\beta : = \\left [ \\begin{matrix} b _ 1 ^ 1 & \\cdots & b _ 1 ^ n \\\\ \\vdots & & \\vdots \\\\ b _ q ^ 1 & \\cdots & b _ q ^ n \\end{matrix} \\right ] \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} \\mathcal { D } ( w ) & = k \\langle x , y , z \\rangle / ( \\partial _ { x } w , \\partial _ { y } w , \\partial _ { z } w ) \\\\ & = k \\langle x , y , z \\rangle / ( y z - z y , z x - x z , x y - y x ) = k [ x , y , z ] . \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} E _ { P } \\left \\{ v a r _ { P } \\left [ \\psi _ { P , \\mathbf { a } } ( \\mathbf { G , B } ; \\mathcal { G } ) \\mid \\mathbf { A } , Y , \\mathbf { G } \\right ] \\right \\} = E _ { P } \\left \\{ \\pi _ { \\mathbf { a } } ( \\mathbf { G } ; P ) v a r _ { P } ( Y \\mid \\mathbf { A } = \\mathbf { a } , \\mathbf { G } ) v a r _ { P } \\left [ \\frac { 1 } { \\pi _ { \\mathbf { a } } ( \\mathbf { G , B } ; P ) } \\mid \\mathbf { A = a } , \\mathbf { G } \\right ] \\right \\} \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} X _ 1 = x _ 1 \\partial _ { x _ 1 } , ~ ~ X _ 2 = x _ 1 \\partial _ { x _ 2 } + \\partial _ { x _ 3 } , ~ ~ X _ 3 = x _ 1 \\partial _ { x _ 2 } . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} p ^ + ( x ) = ( p ^ + _ 1 ( x ) , p ^ + _ 2 ( x ) , \\cdots , p ^ + _ m ( x ) ) \\gg p ^ - ( x ) = ( p ^ - _ 1 ( x ) , p ^ - _ 2 ( x ) , \\cdots , p ^ - _ m ( x ) ) \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} \\langle b _ 2 \\rangle _ { I _ k } = \\frac { \\sqrt 2 } { 3 k } \\big [ ( k + 1 ) ^ { \\frac 3 2 } - 2 \\sqrt 2 \\big ] \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} \\nabla _ x \\mu _ f ( x , t ) = 0 \\ , , \\forall t \\in ( 0 , t _ 0 ] \\ , . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} U _ i = a U _ { i - 1 } + Z _ i , \\quad \\forall i \\geq 1 , \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} ( V _ y - u ) _ t = \\left ( p + \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 - \\alpha g ' ( v ) h ( | \\psi | ^ 2 ) \\right ) _ y . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d U = \\Delta U d t + ( K ( U ) \\cdot \\nabla ) U d t + \\sum _ { i = 1 } ^ N ( B _ i ( t ) + \\theta _ i ) U d \\beta _ i ( 0 , \\infty ) \\times \\mathbb { R } ^ 2 , \\\\ U ( 0 , \\xi ) = U _ 0 ( \\xi ) , \\ \\xi \\in \\mathbb { R } ^ 2 . \\end{array} \\right . \\ \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\frac { 2 H _ { 0 } } { f _ { 2 } p _ { 1 } } = \\left ( 1 + a ^ { 2 } \\right ) \\left ( \\frac { f _ { 1 } } { p _ { 1 } } \\right ) \\left ( \\frac { p _ { 2 } \\dot { p } _ { 2 } } { f _ { 2 } } \\right ) + 2 a \\frac { p _ { 2 } } { f _ { 2 } } + \\dot { p } _ { 1 } . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\widehat { \\lambda } = ( 2 m \\pi ) ^ { 2 } = ( \\frac { 2 n \\pi } { a } ) ^ { 2 } = ( \\frac { 2 l \\pi } { 1 - a } ) ^ { 2 } , m , n , l \\in \\mathbb { N } : = \\left \\{ 1 , 2 , \\ldots \\right \\} . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\varprojlim _ i \\ ! ^ 1 M _ { p ^ i } = 0 \\ ; . \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} - H Q _ { a , c } ' + c Q _ { a , c } - \\frac 1 2 Q _ { a , c } ^ 2 = 0 \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\Sigma ( n ) = A ( n ) - \\frac { 1 } { 6 n } - \\frac { \\delta _ { 1 , n } } { 2 4 n } , \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } a ( n ) f ( n ) = A ( N ) f ( N ) - \\sum _ { n = 1 } ^ { N - 1 } A ( n ) ( f ( n + 1 ) - f ( n ) ) . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} \\hat S ( t , x ) : = I m \\left ( \\sqrt { \\frac { A ^ { - 1 } ( r ( t ) ) \\left ( x - r ( t ) \\right ) \\cdot \\gamma ' ( s ( t ) ) } { c _ { A , \\gamma ' } ( t ) \\sqrt { 1 - | c _ { A , \\gamma ' } ( t ) | ^ 2 | \\dot s ( t ) | ^ 2 } } + i \\frac { \\left ( x - r ( t ) \\right ) \\cdot n ( s ( t ) ) } { c _ { A , n } ( t ) } } \\right ) \\ , , \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} | B | = \\frac { q ^ { r ( 2 r - 1 ) } ( q ^ 2 - 1 ) ^ r } { ( 2 r , q + 1 ) } . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} u ( t , x ) = \\hat u ^ { R } _ i ( t , x ) + k _ i ( t ) \\hat S ( t , x ) \\ , , \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { 1 } { \\sqrt { L } } d s _ L = d s , ~ ~ ~ ~ { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { 1 } { \\sqrt { L } } d \\sigma _ { \\Sigma _ 1 , L } = d \\sigma _ { \\Sigma _ 1 } . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} T _ { n + 1 } ( z ( w ) / c ) = \\frac { 1 } { 2 } ( ( w / c ) ^ { n + 1 } + ( c / w ) ^ { n + 1 } ) \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} A _ { \\varepsilon } = \\{ x \\in [ - 1 , 1 ] : \\exists \\ ; y \\in A | x - y | < \\varepsilon \\} . \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} D ( f _ 1 , \\dots , f _ r ) \\longmapsto \\bigcap _ { i = 1 } ^ r M _ { f _ i } \\subseteq \\Gamma \\ , . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\overline { \\bf J } u ( t ) : = { \\bf B } _ { \\alpha } ( t , T ) f & + \\int _ 0 ^ t { \\bf P } _ { \\alpha } ( t - r ) G ( r , u ( r ) ) d r - \\int _ 0 ^ { T } { \\bf B } _ { \\alpha } ( t , T ) { \\bf P } _ { \\alpha } ( T - r ) G ( r , u ( r ) ) d r . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} T _ i \\circ _ { \\tau _ 2 } T _ j = \\left ( T _ i \\bullet _ \\tau T _ j \\right ) _ { | \\tau ' = 0 } \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} H ^ 1 ( G _ { \\beta } ( O _ r ) , \\Bbb C ^ { \\times } ) & = ( G _ { \\beta } ( O _ r ) , \\Bbb C ^ { \\times } ) , \\\\ H ^ 1 ( K _ 1 ( O _ r ) \\cap G _ { \\beta } ( O _ r ) , \\Bbb C ^ { \\times } ) ^ { G _ { \\beta } ( \\Bbb F ) } & = ( K _ 1 ( O _ r ) \\cap G _ { \\beta } ( O _ r ) , \\Bbb C ^ { \\times } ) \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} & \\norm { \\sum \\limits _ { n = 1 } ^ { M } \\Big ( \\sqrt { \\lambda _ n } \\phi _ n \\psi _ n - \\sqrt { \\lambda _ n ^ { N } } \\phi _ n ^ { N } \\psi _ n \\Big ) } _ { L ^ 2 ( \\Omega , C ( D ) ) } \\lesssim M ^ { \\frac { s } { d } + 1 } N ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\mu _ \\# ( z ) = \\left \\{ \\begin{array} { l l } \\mu ( z ) , & z \\in \\mathbb { H } ; \\\\ \\mu ( A _ 0 \\cdot z ) , & z \\in \\overline { \\mathbb { H } } ; \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} m = \\arg \\min \\limits _ { k \\in \\{ 1 , 2 , \\dots , M _ { i - 1 } \\} - \\mathcal { C } ^ { \\rm C } _ i } q ' _ k , \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} v ( x _ 0 : x _ 1 : x _ 2 ) = ( x _ 0 ^ 2 : x _ 1 ^ 2 : x _ 2 ^ 2 : x _ 1 x _ 2 : x _ 0 x _ 2 : x _ 0 x _ 1 ) . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\Theta _ { i } ( \\mathbf D _ u , \\beta ) = \\int _ { 0 } ^ { \\infty } \\ ! \\ ! \\ ! \\int _ { 0 } ^ { \\infty } \\ ! \\ ! \\ ! \\frac { e ^ { - \\frac { ( 2 ^ { \\frac { n r _ s } { \\beta } } - 1 ) ( x + 1 ) } { \\gamma _ { A _ i , B } } } } { \\frac { \\gamma _ { A _ i , E } } { \\gamma _ { A _ i , B } } \\frac { x + 1 } { y + 1 } 2 ^ { \\frac { n r _ s } { \\beta } } + 1 } \\mathbf f _ { \\mathbf D _ u } ^ i ( x ) \\mathbf g _ { \\mathbf D _ u } ^ i ( y ) \\mathrm { d } x \\mathrm { d } y . \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} L ^ 2 < | x _ N | e ^ { - L ^ 2 \\sum _ { j = 1 } ^ { \\infty } h _ j } = 0 , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} V _ 2 = \\mathrm { s p a n } \\{ x - x _ 1 | x \\in \\mathcal { S } _ 2 \\} \\subset \\mathrm { s p a n } \\{ x - x _ 0 | x \\in \\mathcal { S } _ 2 \\} \\subset V _ 2 . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} r _ { n , s } ( f ; t ) = f ( t ) - \\sum _ { \\nu = 1 } ^ n \\sum _ { i = 0 } ^ { 2 s - 1 } \\ell _ { i , \\nu } ( t ) f ^ { ( i ) } ( x _ { \\nu } ) = \\dfrac { 1 } { 2 \\pi i } \\oint _ { \\Gamma } \\dfrac { f ( z ) \\Omega _ { n , s } ( t ) } { ( z - t ) \\Omega _ { n , s } ( z ) } \\ , d z , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\mathfrak { E } _ { ( \\partial ) } : = \\sum _ { i = 1 } ^ r \\omega _ i \\partial _ { t _ i } + \\sum _ { k = 0 } ^ N \\left ( 1 - \\frac { T _ k } { 2 } t _ k \\right ) \\partial _ { t _ k } \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} V ^ 2 = - \\langle T _ 0 , T _ 0 \\rangle = ( A Y ^ 0 + C _ i Y ^ i ) ^ 2 + ( A Y ^ 4 + B _ i Y ^ i ) ^ 2 - \\sum _ i ( C _ i Y ^ 4 + B _ i Y ^ 0 + D _ p \\epsilon _ { p q i } Y ^ q ) ^ 2 \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 5 { R _ 4 ^ 5 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 5 { R _ 4 ^ 5 } ) = 4 5 , \\mathrm { I C } ( \\mathcal { T } ^ 5 { R _ 4 ^ 5 } ) = 8 5 . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} H _ n ( \\rho ) : = \\sum _ { \\substack { \\{ \\gamma _ 1 , \\gamma _ 2 \\} _ i \\\\ i \\leq n } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\gamma _ 1 } ( \\rho ) + \\ell _ { \\gamma _ 2 } ( \\rho ) ) } + ( - 1 ) ^ { \\gamma _ 1 \\cdot \\gamma _ 2 } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} A _ f ( w _ j ) = \\sum a ^ f _ { i , j } w _ i . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} L & = \\frac { \\partial X } { \\partial r } = \\frac { \\partial } { \\partial x ^ 0 } + \\tilde { X } ^ i ( u ^ a ) \\frac { \\partial } { \\partial x ^ i } + O ( r ) \\\\ \\partial _ a & = \\frac { \\partial X } { \\partial u ^ a } = r \\frac { \\partial \\tilde { X } ^ i } { \\partial u ^ a } \\frac { \\partial } { \\partial x ^ i } + O ( r ^ 2 ) , \\ , \\ , a = 1 , 2 . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} \\begin{cases} C _ \\vartheta \\in C ^ 1 [ 0 , \\infty ) , & \\inf _ { z \\in [ 0 , \\infty ) } C _ \\vartheta ( z ) > 0 \\\\ e _ 1 ( 1 + \\theta ^ r ) \\leq C _ \\vartheta ( \\theta ) \\leq e _ 2 ( 1 + \\theta ^ r ) , & \\end{cases} \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} \\Pr { s _ i \\geq n } = \\sum _ { m = n } ^ \\infty p ( m ) = c \\sum _ { m = n } ^ \\infty m ^ { - 5 / 4 } \\geq c \\int _ { n } ^ \\infty x ^ { - 5 / 4 } \\dd x = 4 c n ^ { - 1 / 4 } . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} t _ n = \\hat { t } _ 1 + \\sum _ { j = n _ 0 } ^ { n } [ j ^ { \\epsilon } - c _ j b _ j ] . \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} ( u ^ n , \\mathbf { w } ^ n , \\mathbf { h } ^ n , \\theta ^ n , \\psi ^ n ) | _ { t = 0 } = ( u _ 0 ^ n , \\mathbf { w } _ 0 ^ n , \\mathbf { h } _ 0 ^ n , \\theta _ 0 ^ n , \\psi _ 0 ^ n ) , \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\gamma = \\max \\{ \\| z \\| : z \\in \\Xi _ K \\} > 0 \\mbox { \\ a n d \\ } \\gamma w \\in \\Xi _ K . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} \\small \\textrm { O v e r h e a d R e d u c t i o n } = \\frac { \\textrm { \\# o f r e s o u r c e s : n o n - o r t h o g o n a l t r a i n i n g } } { \\textrm { \\# o f r e s o u r c e s : o r t h o g o n a l t r a i n i n g } } , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} \\zeta ( q ) = \\tilde \\zeta ( q ) \\ \\ q \\ge q _ 0 , q \\in U . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} X _ { ( ( T ^ \\ast Q ) _ { \\mathcal { O } _ \\mu } , \\omega _ { \\mathcal { O } _ \\mu } , h _ { \\mathcal { O } _ \\mu } , f _ { \\mathcal { O } _ \\mu } , u _ { \\mathcal { O } _ \\mu } ) } = X _ { h _ { \\mathcal { O } _ \\mu } } + \\textnormal { v l i f t } ( f _ { \\mathcal { O } _ \\mu } ) + \\textnormal { v l i f t } ( u _ { \\mathcal { O } _ \\mu } ) , \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} E _ 2 = \\frac { 1 } { 3 2 \\pi ^ 2 } \\int | d \\psi | ^ 2 \\ , d ^ 3 x , E _ 4 = \\frac { 1 } { 1 2 8 \\pi ^ 2 } \\int | \\psi ^ * \\omega | ^ 2 \\ , d ^ 3 x . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} \\lim _ { X \\to \\infty } \\frac { \\sum _ { p \\in S p l _ X ( f ) } \\# \\{ i \\mid 0 \\le r _ i / p \\le a \\} } { n \\# S p l _ X ( f ) } = a . \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} a _ k \\leq \\tilde { C } b ^ { - 1 / p } \\left ( \\sum _ { j = - \\infty } ^ { l - 2 } 2 ^ { s ' j p } \\left ( \\int _ { 2 B _ 0 } g _ j ^ p \\ , d \\mu \\right ) ^ { \\frac { q } { p } } \\right ) ^ { \\frac { 1 } { p } } ( k - k _ 0 ) . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} ( z b q ; q ) _ k \\Phi _ n ^ { ( k ) } ( z ) = \\sum _ { i = 0 } ^ { k } r _ { n - i } ( z ) p _ { n - i } ( z ; a , b | q ) , \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\abs { u _ k ( x ) } = O ( a _ k r ^ { - n _ k } ) , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} u _ { m , 0 } ' ( s ) = \\sqrt { 1 - \\dfrac { 2 m } { u _ { m , 0 } ( s ) } } < \\sqrt { 1 - \\dfrac { 2 m } { r _ o } } , \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\frac { p _ { \\geq 2 r + 2 } ( n ) } { p ( n ) } = \\Theta _ r ( n ^ { - ( 2 r + 1 ) / 2 } ) . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} X _ { \\rm A n d e r s o n } = \\left ( \\begin{array} { r r r r r } - 0 . 0 5 9 0 & 0 . 1 7 0 0 & 0 . 0 0 3 0 & 0 . 6 6 5 0 & 0 . 6 5 5 2 \\\\ 0 & - 0 . 1 7 3 0 & 0 . 0 3 0 0 & 0 & 0 \\\\ 0 & 1 . 1 8 0 0 & - 1 . 3 1 6 0 & 0 . 0 0 8 0 & 0 \\\\ 0 & 0 . 8 0 1 0 & 0 . 4 9 5 0 & - 1 . 1 7 8 0 & 1 . 3 5 7 0 \\\\ 0 & 1 . 0 4 0 0 & 2 . 7 5 6 0 & 0 & - 0 . 1 8 3 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\beta _ { \\textrm { F } } = \\max \\left \\lbrace 1 - \\frac { C _ { \\textrm { F } } ^ { \\textrm { N O M A } } } { | h _ { \\textrm { B F } } | ^ 2 } , 0 \\right \\rbrace , \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} y = S u \\ , , S : L ^ 2 ( \\Omega _ T ) \\to H ^ 1 ( 0 , T ; L ^ 2 ( \\Omega ) ) \\cap L ^ \\infty ( 0 , T ; V ) \\ , , \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} P _ 0 ( n ) : = \\mathrm { C a r d } \\left \\{ \\gamma d N \\mid \\norm { \\{ \\gamma \\} } \\leq n \\right \\} . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} & \\begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\end{bmatrix} , \\\\ & \\begin{bmatrix} 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} E [ ( 1 + t ) ^ U ] & = \\int _ 0 ^ 1 ( 1 + t ) ^ u p ( u ) d u = \\int _ 0 ^ 1 ( 1 + t ) ^ x d x = \\frac { t } { \\log ( 1 + t ) } \\\\ & = \\sum _ { n = 0 } ^ \\infty b _ n \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} \\| \\tilde { h } \\| _ { \\bar { { \\cal H } } ( [ 0 , T _ { 2 } ] ) } = \\| K ^ { - 1 } \\tilde { h } \\| _ { L ^ { 2 } ( [ 0 , T _ { 2 } ] ) } = \\left ( \\frac { T _ { 1 } } { T _ { 2 } } \\right ) ^ { H } \\| K ^ { - 1 } h \\| _ { L ^ { 2 } ( [ 0 , T _ { 1 } ] ) } = \\left ( \\frac { T _ { 1 } } { T _ { 2 } } \\right ) ^ { H } \\| h \\| _ { \\bar { { \\cal H } } ( [ 0 , T _ { 1 } ] ) } , \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\mathbb { E } f _ \\varepsilon ( Z _ t ) = \\varepsilon ^ 3 + c _ H ^ R \\int _ 0 ^ t \\mathbb { E } \\left [ \\psi _ s \\left ( f '' _ \\varepsilon ( Z _ s ) ( \\nabla ^ { \\frac { H } { 2 } , \\frac { H } { 2 } } Z _ s ) ( s , s ) + f ''' _ \\varepsilon ( Z _ s ) [ ( \\nabla ^ \\frac { H } { 2 } Z _ s ) ( s ) ] ^ 2 \\right ) \\right ] \\ , \\mathrm { d } { s } \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} m ^ { \\alpha + 2 } = \\frac { \\alpha + 2 } { 2 } \\| Q \\| ^ 2 _ { \\dot { H } ^ s } , M ^ 2 = \\| Q \\| ^ 2 _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\begin{array} { r c l } T _ 1 & = & \\displaystyle \\frac { 1 } { 2 t } \\int _ { x _ k - t } ^ { x _ k + t } f ( \\hat x , s ) d s = \\displaystyle \\frac { 1 } { 2 t } \\int _ { x _ k - t } ^ { x _ k + t } h ( s ) d s = \\mu _ h ( x _ k , t ) . \\\\ \\end{array} \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} \\left ( \\langle f _ 2 , f _ 3 \\rangle , \\langle f _ 3 , f _ 1 \\rangle , \\langle f _ 1 , f _ 2 \\rangle \\right ) = ( \\delta _ 1 e ^ { i \\theta _ 1 } , \\delta _ 2 e ^ { i \\theta _ 2 } , \\delta _ 3 e ^ { i \\theta _ 3 } ) . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} & C \\bullet X + \\epsilon \\textstyle { \\sum _ i } B _ i \\bullet Y _ i = - 1 \\\\ & D _ i \\bullet Y _ i \\le 0 \\ \\forall \\ i \\\\ & I \\bullet X + \\textstyle { \\sum _ i } I \\bullet Y _ i \\le 0 \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\nu ( ( r , 1 ] ) = \\begin{cases} A _ 1 ( q - r ) + A _ 2 ( 1 - q ) + \\Delta , & r \\le q , \\\\ A _ 2 ( 1 - r ) + \\Delta , & q < r \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\int _ { 0 } ^ t \\gamma ( t _ { 2 } - s _ { 2 } ) d s _ { 2 } = & \\int _ { t _ { 2 } - t } ^ { t _ { 2 } } \\gamma ( r ) d r \\\\ \\geq & \\int _ 0 ^ { t _ 2 } \\gamma ( r ) d r , \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} - \\mathbf { D } ^ 2 = \\Delta - \\frac { s } { 4 } , \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} A _ \\lambda ( \\partial _ t u _ \\lambda , \\partial _ t v _ \\lambda ) + B _ \\lambda ( u _ \\lambda , v _ \\lambda ) = ( g _ \\lambda , g _ { \\Gamma \\lambda } ) - ( T _ \\lambda \\pi ( u _ \\lambda ) , T _ \\lambda \\pi _ \\Gamma ( v _ \\lambda ) ) \\ , , \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\dot { z } _ { t } = y - x = \\left ( V V ^ { * } ( V V ^ { * } ) ^ { - 1 } \\right ) ( z _ { t } ) \\cdot ( y - x ) = V ( z _ { t } ) \\dot { h } _ { t } , \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} x ^ * = [ 0 . 7 0 7 4 2 3 2 3 7 4 8 3 \\ 1 . 2 3 9 5 1 4 8 5 0 4 0 0 \\ 1 . 2 6 0 3 8 1 2 1 9 5 9 4 \\ 1 . 0 8 2 0 7 8 2 0 5 4 8 8 \\ { - 1 . 6 4 4 0 2 4 0 0 6 2 3 6 } \\ { - 2 . 3 5 1 7 1 2 4 0 9 9 3 8 } ] \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} S ^ { K \\textnormal { t h } } ( \\phi _ i ) & = \\sum _ { \\alpha , \\beta = 0 } ^ N \\overline { S _ { i \\alpha } } G ^ { \\alpha \\beta } \\phi _ \\beta \\\\ T ^ { K \\textnormal { t h } } ( \\phi _ i ) & = \\sum _ { \\alpha , \\beta = 0 } ^ N S _ { i \\alpha } g ^ { \\alpha \\beta } \\phi _ \\beta \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} d s ^ 2 = \\frac { 1 + u _ t ^ 2 } { \\sqrt { 1 + u _ t ^ 2 + u _ x ^ 2 } } ( d t ^ 2 + d y ^ 2 ) + \\frac { 1 + u _ x ^ 2 } { \\sqrt { 1 + u _ t ^ 2 + u _ x ^ 2 } } ( d x ^ 2 + d z ^ 2 ) + \\frac { 2 u _ t u _ x } { \\sqrt { 1 + u _ t ^ 2 + u _ x ^ 2 } } ( d t d x + d y d z ) , \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\zeta \\left ( \\sum _ { w \\in G } n _ w x _ w \\wedge \\sum _ { v \\in F } m _ v x _ v \\right ) = \\sum _ { w \\in G } n _ w x _ w + \\sum _ { v \\in F } m _ v x _ v \\ , . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\imath _ { \\xi } \\gamma ( x ) = a d _ { \\xi } ^ * x . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} L R ( \\chi ) = \\{ M C ( \\chi ) \\mid M \\in G L R \\} \\ , ( = G L R \\cdot C ( \\chi ) ) , \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\tilde { \\zeta } _ \\varepsilon = \\max \\left ( \\frac { 1 } { \\gamma _ \\varepsilon ^ 2 } , | A ( \\gamma _ \\varepsilon ) | , \\xi _ \\varepsilon \\right ) \\ , . \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} l = \\lim _ { \\gamma \\to + \\infty } \\frac { \\gamma ^ { - 4 } + A ( \\gamma ) / 2 + 4 \\gamma ^ { - 3 } \\exp ( - 1 - M ) B ( \\gamma ) S } { \\gamma ^ { - 4 } + | A ( \\gamma ) | + \\gamma ^ { - 3 } | B ( \\gamma ) | } \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\to 0 } J _ \\lambda ^ { \\sigma } u ~ = ~ J _ \\lambda u \\forall u \\in \\overline D \\ , , \\ ; \\forall \\lambda \\in ] 0 , \\lambda _ { 0 } ] , \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\tau ( A B ) = \\tau ( B A ) . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\sup _ { w \\in V _ z } | E _ w ( f ) | = \\sup _ { w \\in V _ z } | f ( w ) | < \\infty \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} T _ l : = T S P ( Y _ 1 , \\ldots , Y _ { N _ l } ; S _ l ) \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\left [ { \\hat \\theta _ { { \\rm { M L E } } } ^ { ( a ) } , \\{ { { \\hat \\mu } _ { } ^ { ( j ) } } \\} } \\right ] \\buildrel \\Delta \\over = \\arg \\mathop { \\max } \\limits _ { \\theta , \\{ { \\mu _ j } \\ge b \\} } { f _ 1 } \\left ( { \\theta , \\{ { \\mu _ j } \\} } \\right ) , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} | \\lambda | ^ { 1 / 2 } \\| \\partial _ z ( \\beta v ) \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } & \\le C _ { \\theta , p , \\lambda _ 0 } \\left ( \\eta ^ { 2 / p - 1 } + \\eta ^ { 3 / p } | \\lambda | ^ { - 1 / 2 p } \\big ( 1 + | \\log ( \\eta | \\lambda | ^ { - 1 / 2 } ) | \\big ) \\right ) \\| f \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } \\\\ & + C _ { \\theta , p } \\eta ^ { 2 / p - 1 } | \\lambda | ^ { 1 / 2 } \\| \\partial _ z v \\| _ { L ^ \\infty _ H L ^ p _ z ( \\Omega ) } . \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\beta _ i ( x , t ) : = \\frac { 2 } { 5 } \\overline { \\sigma _ i } ( x , t ) , \\ \\ \\beta ^ * _ i ( t ) : = \\underset { x \\in \\Omega } { } \\ { \\beta } _ i ( x , t ) , \\ \\ \\hat { \\beta } _ i ( t ) : = \\underset { x \\in \\Omega } { } \\ { \\beta } _ i ( x , t ) . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} f _ q ( Q ) = \\frac { ( Q , q ) _ \\infty ( - i Q , q ) _ \\infty ( - Q , q ) _ \\infty } { ( \\alpha _ 1 ( q ) Q , q ) _ \\infty ( \\alpha _ 2 ( q ) Q , q ) _ \\infty ( \\alpha _ 3 ( q ) Q , q ) _ \\infty } \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\lambda _ 1 | \\boldsymbol { \\xi } | ^ 2 = \\left ( \\min _ { 1 \\leq j \\leq N } \\lambda _ j \\right ) | \\boldsymbol { \\xi } | ^ 2 \\leq \\langle ( D ^ 2 w _ 2 ) ( x ^ * ) \\ , \\boldsymbol { \\xi } , \\boldsymbol { \\xi } \\rangle \\leq \\left ( \\max _ { 1 \\leq j \\leq N } \\lambda _ j \\right ) | \\boldsymbol { \\xi } | ^ 2 = \\lambda _ N | \\boldsymbol { \\xi } | ^ 2 \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial t } = \\frac { 1 } { t ( 1 + t ^ { 2 } ) } \\left [ \\frac { 3 } { 2 } u \\nabla u \\cdot \\nabla w + \\frac { 1 } { 2 w } \\right . & \\Delta w + \\frac { 1 } { 2 } ( R _ { g ( t ) } - t { } ^ { 2 } \\overline { R } ) \\\\ & \\left . - w \\left ( 1 + 3 t ^ { 2 } + \\frac { t ^ { 2 } ( 1 + t ^ { 2 } ) | M | ^ { 2 } } { 2 } \\right ) \\right ] . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} t > \\sqrt { \\dfrac { 6 \\ , | \\theta _ L | } { \\lambda _ n ( C ) } } = : t _ 0 > 0 . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\lim _ { m \\rightarrow \\infty } H ^ 2 ( p _ m ) = \\sup H ^ 2 = \\bar H ^ 2 , \\lim _ { m \\rightarrow \\infty } | \\nabla H ^ 2 ( p _ m ) | = 0 , \\\\ & 0 \\geq \\lim _ { m \\rightarrow \\infty } | \\nabla ^ { \\perp } \\vec H | ^ 2 ( p _ m ) - \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 + \\dfrac 1 2 ( \\bar H ^ 2 - S ) ( \\bar H ^ 2 - 3 S + 2 ) , \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} f g = \\dot { T } _ f g + \\dot { T } _ g f + \\dot { R } ( f , g ) . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\log : U ^ 1 \\longrightarrow A \\ ; , \\ ; \\log u = - \\sum ^ { \\infty } _ { \\nu = 1 } \\frac { ( 1 - u ) ^ { \\nu } } { \\nu } \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} ( \\pi \\oplus \\tau ) ( i ) = \\begin{cases} \\pi ( i ) & \\mbox { f o r $ 1 \\le i \\le k $ } , \\\\ \\tau ( i - k ) + k & \\mbox { f o r $ k + 1 \\le i \\le k + \\ell $ } . \\end{cases} \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} S \\ast T ( \\rho ) f = \\int _ \\R S ( \\rho - \\sigma ) T ( \\sigma ) f \\ , d \\sigma \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\partial _ { x _ 1 } = - \\frac { \\sqrt { 2 } } { 2 } e ^ { - x _ 3 } ( X _ 2 + X _ 3 ) , ~ ~ \\partial _ { x _ 2 } = \\frac { \\sqrt { 2 } } { 2 } e ^ { x _ 3 } ( X _ 2 - X _ 3 ) , ~ ~ \\partial _ { x _ 3 } = X _ 1 , \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} { } \\sum \\limits _ { ( M _ { S ' } , \\mu _ { S ' } ) \\in X \\cap \\mathcal { R } _ { G , b , \\mu } } ( - 1 ) ^ { L _ { M _ { S ' } , M _ b } } = 0 . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\Omega = \\R ^ 2 , \\ ; \\ ; f _ 1 ( r , s ) = \\dfrac { r ^ 2 + s ^ 2 } 2 , \\ ; \\ ; f _ 2 ( r , s ) = r , \\ ; \\ ; \\ ; \\ ; f ( x ) = ( f _ 1 ( x ) , f _ 2 ( x ) ) \\ ; \\ ; x \\in \\R ^ 2 , \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} \\gamma = \\sup _ { 0 \\leq \\Re z \\leq b } \\frac { 1 } { \\Re z } \\log | r ( z ) | = \\max \\left \\{ 1 , \\frac { 1 } { b } \\sup _ { \\Re z = b } \\log | r ( z ) | \\right \\} , \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} \\sum _ { v \\in A _ { 0 } } t _ { \\ell _ 2 } ( v ) \\leq 2 \\frac { \\sqrt { n } } { \\log ^ 3 n } | A _ { 0 } | = 2 \\frac { \\sqrt { n } } { \\log ^ 3 n } | C _ 1 | . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} v F ( c ) & : = \\widehat { \\mathcal { C } } \\mu & & c \\in v \\mathcal { C } , \\\\ F ( f ) & : = \\rho ( \\mu , \\mu \\ast f ^ \\circ , \\nu ) & & f \\in \\mathcal { C } ( c , d ) , \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} ( \\alpha \\otimes \\alpha ) \\phi ( \\phi \\otimes 1 ) = \\phi ( 1 \\otimes \\phi ) \\alpha , \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} \\max _ { x \\in \\left [ 0 , L \\right ] } \\left ( \\alpha \\varphi _ k + d \\psi _ k \\right ) ( x ) = 1 . \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\widehat { H } ( x , \\widetilde { \\Lambda } ) = \\ln \\widetilde { m } _ 0 ( x ) + \\overline { H } . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} \\arg q _ { s o l } = \\pi n - 2 \\ ( A x + 2 t ( A ^ 2 - B ^ 2 ) \\ ) - \\arg \\hat \\phi ( E _ 0 ) + 2 \\arg \\delta ( E _ 0 , A ) , \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} S ^ { \\star } = Q \\begin{bmatrix} S ^ { \\star } _ 1 & 0 & \\cdots & 0 \\\\ 0 & S ^ { \\star } _ 2 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & S ^ { \\star } _ { \\ell } \\\\ \\end{bmatrix} Q ^ T . \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathfrak p \\\\ \\updownarrow \\\\ \\left ( \\begin{array} { c } \\mathfrak q \\\\ \\updownarrow \\\\ \\mathfrak r \\end{array} \\right ) \\end{array} = \\begin{array} { c } \\left ( \\begin{array} { c } \\mathfrak p \\\\ \\updownarrow \\\\ \\mathfrak q \\end{array} \\right ) \\\\ \\updownarrow \\\\ \\mathfrak r . \\end{array} \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} ( \\rho Q ( \\theta ) ) _ t + ( \\rho Q ( \\theta ) \\mathbf { u } ) _ x + \\theta p _ \\theta ( \\rho ) u _ x = ( \\kappa \\theta _ x ) _ x + \\varepsilon u _ x ^ 2 + \\mu | \\mathbf { w } _ x | ^ 2 + \\nu | \\mathbf { h } _ x | ^ 2 . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi + R \\varphi = & - \\frac { n - 1 } { n } t ^ 2 \\tau ^ { 2 a } \\varphi ^ { N - 1 } + \\left ( | \\sigma + L W | ^ { 2 } + k ^ { 2 } \\right ) \\varphi ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ { * } L W = & \\frac { n - 1 } { n } t \\varphi ^ { N } d \\tau ^ a . \\endaligned \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} f g ^ h = \\dot { T } _ f { g ^ h } + \\dot { T } _ { g ^ h } f + \\dot { R } ( f , g ^ h ) . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} Y _ { I , I ' } ( \\theta ) & = \\langle T b _ I ^ { ( \\theta ) } , b _ { I ' } ^ { ( \\theta ) } \\rangle , & I , I ' \\in \\mathcal { D } _ { \\leq n } , \\ I \\neq I ' , \\ \\theta \\in \\{ \\pm 1 \\} , & \\\\ Z _ I ( \\theta ) & = \\langle T b _ I ^ { ( \\theta ) } , b _ I ^ { ( \\theta ) } \\rangle - \\sum _ { K \\in \\mathcal { B } _ I } \\langle T h _ K , h _ K \\rangle , & I \\in \\mathcal { D } _ { \\leq n } , \\ \\theta \\in \\{ \\pm 1 \\} . & \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} k \\int _ 0 ^ L \\omega u _ 1 u _ 2 = \\int _ 0 ^ L t \\mu _ 1 ( 1 - u _ 1 ) u _ 1 + \\int _ 0 ^ L ( 1 - t ) \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ + . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a _ { 1 , 1 } & a _ { 1 , 2 } & a _ { 1 , 3 } & \\varepsilon & \\varepsilon \\\\ a _ { 2 , 1 } & \\varepsilon & \\varepsilon & \\varepsilon & a _ { 2 , 5 } \\\\ \\varepsilon & \\varepsilon & \\varepsilon & a _ { 3 , 4 } & \\varepsilon \\\\ \\varepsilon & a _ { 4 , 2 } & \\varepsilon & \\varepsilon & \\varepsilon \\\\ a _ { 5 , 1 } & \\varepsilon & \\varepsilon & a _ { 5 , 4 } & \\varepsilon \\end{bmatrix} , \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar \\lambda _ 1 \\bar h ^ { 1 ^ * } _ { 1 1 k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} a ( k ) = & N _ { 1 1 } ( 0 , 0 , k ) = \\bar N _ { 2 2 } ( 0 , 0 , \\bar k ) = \\frac { \\varkappa ( k ) + \\varkappa ^ { - 1 } ( k ) } { 2 } \\frac { F ^ 0 _ 1 ( k ) } { F ^ 0 _ 1 ( \\infty ) } , \\\\ b ( k ) = & N _ { 2 1 } ( 0 , 0 , k ) = - \\bar N _ { 1 2 } ( 0 , 0 , \\bar k ) = \\frac { \\varkappa ( k ) - \\varkappa ^ { - 1 } ( k ) } { 2 } \\frac { F ^ 0 _ 2 ( k ) } { H ^ 0 _ 2 ( \\infty ) } , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\theta ^ \\beta _ \\pm = - \\xi _ \\beta \\pm \\sqrt { ( \\xi _ \\beta ) ^ 2 + 1 } \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\mathbb { Z I } _ n ^ m ( p _ 1 , \\ldots , p _ r ) = \\{ ( X _ 1 , \\ldots , X _ m ) \\in \\mathbb { I } ^ m _ n \\ : | p _ j ( X _ 1 , \\ldots , X _ m ) = \\mathbf { 0 } _ n , 1 \\leq j \\leq r \\} \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sqrt { p _ n } u _ { p _ n } = 0 \\ \\mbox { i n $ C ^ 2 _ { l o c } ( \\bar \\Omega \\setminus \\mathcal { S } ) $ . } \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k e ^ { ( 1 - d s ) ( \\log m ( j ) ) ^ { 1 / b } } \\stackrel { { e } } { \\sim } e ^ { ( 1 - d s ) { b \\over b + 1 } k ^ { b + 1 \\over b } } \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} & - \\frac { 4 f _ { i } f _ { i j } f _ { j } } { ( 1 - f ) ^ { 3 } } - \\frac { 4 f ^ { 2 } _ { i } f ^ { 2 } _ { j } } { ( 1 - f ) ^ { 4 } } + \\frac { 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } ) } { ( 1 - f ) ^ { 2 } } \\\\ & = \\frac { - 2 } { 1 - f } g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } { \\phi } ) + 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } { \\phi } ) = \\frac { - 2 } { 1 - f } g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } { \\phi } ) . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} g ( y _ 1 , y ' ) = \\frac { a _ 0 } { 2 } + \\sum _ { k = 1 } ^ \\infty a _ k ( y ' ) \\cos ( k y _ 1 ) + \\sum _ { k = 1 } ^ \\infty b _ k ( y ' ) \\sin ( k y _ 1 ) \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} z = e ^ { i u } \\ , . \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} 0 < \\delta < \\min \\Big \\{ \\frac { p ^ + _ l ( t ) - p ^ - _ l ( t ) } { \\max \\{ \\varphi ^ + _ l ( t ) , \\varphi ^ - _ l ( t ) \\} } \\mid t \\in \\R , \\ l = 1 , 2 , \\cdots , m \\Big \\} \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} ( - z ; q ) _ k \\ , \\phi _ n ^ { ( k ) } ( z ) = \\sum _ { i = 0 } ^ { k } g _ { k - i } ( z ) L _ { n - i } ^ { ( \\delta ) } ( z ; q ) , \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} u _ 1 ' ( x ) = - \\int _ { x _ 0 } ^ { x } \\left [ t \\mu _ 1 ( 1 - u _ 1 ) u _ 1 + ( 1 - t ) \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ] + k \\int _ { x _ 0 } ^ { x } \\omega u _ 1 u _ 2 \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\phi _ m ( a ) = \\sum _ { k = - m } ^ m a ( k ) \\widehat { h } ( k ) ( a \\in \\ell ^ r ( \\mathbb { Z } ) , \\ m \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = \\sum _ { d \\geq 0 } \\sum _ { n = 0 } ^ \\infty \\sum _ { \\alpha _ 0 + \\alpha _ 1 + \\alpha _ 2 = n } \\frac { t _ 0 ^ { \\alpha _ 0 } } { \\alpha _ 0 ! } \\frac { t _ 1 ^ { \\alpha _ 1 } } { \\alpha _ 1 ! } \\frac { t _ 2 ^ { \\alpha _ 2 } } { \\alpha _ 2 ! } \\left \\langle T _ 0 ^ { ( \\alpha _ 0 ) } , T _ 1 ^ { ( \\alpha _ 1 ) } , T _ 2 ^ { ( \\alpha _ 2 ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , n , d [ l ] } \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} g _ i ( X ) & : = f ( A ( 1 + X ) ) - r _ { i - 1 } \\\\ & = A ^ { n } ( 1 + X ) ^ a X ^ { n - a } + ( A - r _ { i - 1 } ) \\\\ & = A ^ { n } ( 1 + X ) ^ a X ^ { n - a } + \\epsilon _ { i - 1 } A . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} \\varepsilon \\leftrightarrow \\mathfrak p = \\mathfrak p \\leftrightarrow \\varepsilon = \\mathfrak p . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\mathcal N : \\{ y \\in V ^ * : \\ ; y _ \\Omega = 0 \\} \\to \\{ y \\in V : \\ ; y _ \\Omega = 0 \\} \\ , , \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} u \\mapsto U _ { t } ^ { u } = \\log ( { \\rm e } ^ { u } \\otimes { \\rm e } ^ { U _ { t } ^ { 0 } } ) . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} k _ T \\overset { \\Delta } { = } \\inf \\left \\{ i \\in \\mathbb { Z } : t _ { k + i } \\ge t _ k + T \\right \\} . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} \\Delta _ { e _ k } = \\Delta ' _ { e _ k } + \\Delta ' _ { e _ { k - 1 } } . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} ( n - r k ) ( r + 1 ) = ( ( r + 1 ) k - r k ) ( r + 1 ) = k ( r + 1 ) = n . \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\mathbb { E } [ Y _ i ] = & \\sum _ { i = 1 } ^ { \\infty } \\mathbb { E } \\left [ E _ i \\right ] = \\frac { m p } { 1 - p } \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} F ( n , k - 1 ) - F ( n , k ) = G ( n + 1 , k ) - G ( n , k ) . \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} w ( x ) = \\begin{cases} 0 . 5 & ( x < - 1 ) , \\\\ 0 . 4 & ( - 1 < x < 0 ) , \\\\ 0 . 7 & ( 0 < x < 1 ) , \\\\ 0 . 5 & ( x > 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} F ( \\pm 1 ) = F ' ( \\pm 1 ) = 0 , F '' ( \\pm 1 ) > 0 , F ( u ) > 0 , \\ ; \\ , \\forall \\ , u \\neq \\pm 1 . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d u _ t & = & \\big ( \\langle v , \\nabla \\rangle u _ t + \\alpha \\big ) d t + \\sigma ( u _ { t - } ) d X _ t \\medskip \\\\ u _ 0 & = & h _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} - G ' _ i ( u ) + G _ i ( u ) & = - G ' _ i ( u ) + \\gamma _ i G _ i ( u ) - ( \\gamma _ i - 1 ) G _ i ( u ) \\\\ & = \\frac { L _ i \\gamma _ i } { e - 1 } u - ( \\gamma _ i - 1 ) G _ i ( u ) \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { \\partial f \\left ( x , t \\right ) } { \\partial t } = \\varepsilon \\boldsymbol { L } f \\left ( x , t \\right ) x \\in \\mathcal { K } _ { N } t > 0 \\\\ \\\\ f \\left ( x , 0 \\right ) = f _ { 0 } ( x ) \\in { \\large X } _ { \\infty } . \\end{array} \\right . \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} \\frac { K _ { 0 } } { c _ { 3 } ^ { 2 } \\left ( c f _ { 1 } f _ { 1 } ^ { \\prime \\prime } - \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } \\right ) } = \\frac { f _ { 2 } ^ { 2 c } } { \\left ( a f _ { 1 } ^ { \\prime } f _ { 2 } + c _ { 3 } f _ { 1 } f _ { 2 } ^ { c } \\right ) ^ { 4 } } . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { \\ell ^ { p , \\infty } } : = \\sup _ { \\alpha > 0 } \\alpha \\mu \\{ n ' : | f ( n ' ) | > \\alpha \\} ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\mathbf { W } = \\arg \\min _ { \\mathbf { W ' } } E \\left [ \\left \\vert \\left \\vert \\mathbf { x } - \\mathbf { W ' } ^ { H } \\mathbf { y } _ { \\mathcal { Q } } \\right \\vert \\right \\vert ^ 2 _ 2 \\right ] . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} S \\setminus \\bigcup \\limits _ { g \\in G } \\partial S ^ { ' } _ { g } = \\bigcup \\limits _ { g \\in G } S ^ { ' } _ { g } \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\| v \\| _ p : = \\left ( \\sum _ { k = 1 } ^ K \\left | v _ k \\right | ^ p \\Delta x \\right ) ^ { \\frac { 1 } { p } } \\ ( 1 \\le p < \\infty ) , \\| v \\| _ { \\infty } : = \\max _ { k \\in \\mathbb { Z } } \\left | v _ k \\right | . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} W ( T ) & = \\sum _ { 1 \\leq m \\leq n } \\left [ W _ { n , m } \\right ] \\L ^ { - n d } T ^ n \\\\ & = \\L ^ d \\sum _ { 1 \\leq m \\leq n } \\left [ V _ { n , m } \\right ] \\L ^ { - ( n + 1 ) ( d - 1 ) } T ^ n - \\L ^ d \\sum _ { 1 \\leq m \\leq n } \\left [ V _ { n , m } \\right ] \\L ^ { - ( n + 1 ) ( d - 1 ) } \\L ^ { - m } T ^ n \\\\ & = \\L ^ d \\sum _ { 1 \\leq m \\leq n } \\tilde { \\mu } \\left ( A _ { n , m } \\right ) T ^ n - \\L ^ d \\sum _ { 1 \\leq m \\leq n } \\tilde { \\mu } \\left ( A _ { n , m } \\right ) \\L ^ { - m } T ^ n . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } p ( t , x , z ) p ( s , y , z ) d z = p ( t + s , x , y ) \\ \\ \\ x , y \\in \\mathbb { R } ^ d \\ \\ \\ s , t > 0 . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} & T _ 0 X _ 1 ^ { - 1 } T _ 0 = q _ 0 ^ { - 1 } q _ 1 X _ 1 + ( q _ 0 ^ { - 1 } q _ 1 - 1 ) T _ 0 , \\\\ & T _ i X _ i T _ i = X _ { i + 1 } ~ ( i \\neq 0 ) , T _ i X _ j = X _ j T _ i ~ ( j \\neq i , i + 1 ) . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} F ( x ' ) : = \\lVert f ( x ' , \\cdot ) \\rVert _ { L ^ p ( \\R ) } , G ( x ' ) : = \\lVert g ( x ' , \\cdot ) \\rVert _ { L ^ 1 ( \\R ) } , \\psi ( x ' ) : = \\lVert ( g \\ast f ) ( x ' , \\cdot ) \\rVert _ { L ^ p ( \\R ) } \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} G ( f ) = \\frac { B N _ 0 } { \\alpha _ { \\rm c } P _ { \\rm R } } \\left ( \\frac { \\mu } { \\log \\left ( \\frac { f _ { \\rm m a x } } { \\sqrt { f ( 2 f _ { \\rm c } - f ) } } \\right ) } - 1 \\right ) _ + , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } d i s t _ { W ^ { 1 , p } } ( v _ j , h ^ j ( D _ 1 ^ 2 ) ) = 0 . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} Q ^ \\lambda _ \\alpha ( t ) = b ( t ) ^ { - 1 } \\exp ( t \\nabla ^ \\pm H ( g _ 0 ) ) Q ^ \\lambda _ \\alpha k _ - ( t ) . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} U _ i & = \\max _ { t \\in [ m ] } \\frac { \\sup _ { x \\in \\mathbb { R } ^ m : ~ \\hat { c } _ i ^ T x = 1 } \\nabla _ t H _ i ( x ) } { c _ { i , t } } \\\\ L _ i & = \\min _ { t \\in [ m ] } \\frac { \\inf _ { x \\in \\mathbb { R } ^ m : ~ \\hat { c } _ i ^ T x \\leq 1 } \\nabla _ t H _ i ( x ) } { c _ { i , t } } \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} E _ { M } = \\sum _ { k = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } ^ { k } ( a _ { j } ^ { k } - \\Pi _ { l } ( g _ { j } ^ { k } , h _ { j , 1 } ^ { k } , h _ { j , 2 } ^ { k } ) ) , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} Z _ { i j } = \\left \\lceil \\frac { W _ i } { n } \\right \\rceil n \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} \\mu _ { 1 } f _ { 2 } ^ { 2 } f _ { 2 } ^ { \\prime \\prime } + \\mu _ { 2 } f _ { 2 } f _ { 2 } ^ { \\prime } f _ { 2 } ^ { \\prime \\prime } - \\mu _ { 3 } f _ { 2 } \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } - \\mu _ { 4 } \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 3 } = 0 , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} & ( w _ 1 ( \\tilde { z } _ * , 0 ) , w _ 2 ( \\tilde { z } _ * , 0 ) ) \\in S , \\\\ & ( w _ 1 ( z , 0 ) , w _ 2 ( z , 0 ) ) \\in \\Delta _ 2 \\cup S \\ \\ { \\rm f o r } \\ \\ z \\ \\ { \\rm w i t h } \\ \\ | z - z _ * | \\leq | \\tilde { z } _ * - z _ * | . \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } H ^ \\delta ( x ) = H ( x ) \\mbox { p o i n t w i s e } ~ \\forall x \\in \\mathbb { R } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\overline { H } ( P ) = \\ln \\overline { I } [ u _ 0 , u _ 1 ] . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { L } v ' \\varphi ' = \\int _ { 0 } ^ { L } \\left ( \\mu _ { 1 } \\left ( \\alpha - v \\right ) v ^ { + } - \\mu _ { 2 } \\left ( d + v \\right ) v ^ { - } \\right ) \\varphi . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} { } \\sum \\limits _ { b \\in Y _ { ( M _ S , \\mu _ S ) } } ( - 1 ) ^ { L _ { M _ S , M _ b } } = 0 . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\partial _ \\pi F & = \\{ \\alpha \\in F : \\exists \\beta \\in F ^ c , \\beta \\in \\alpha \\otimes \\pi \\} \\cup \\{ \\beta \\in F ^ c : \\exists \\alpha \\in F , \\alpha \\in \\beta \\otimes \\pi \\} . \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} H ( u ; t ) = u - L \\left ( u + f ( u ; t ) \\right ) , \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq x + \\delta \\right ) & - P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq x - \\delta \\right ) \\geq \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( x + \\delta ) ) + \\eta ) b _ c ^ { ( n ) } } - \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( x - \\delta ) ) - \\eta ) b _ c ^ { ( n ) } } \\\\ & = \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( x + \\delta ) ) + \\eta ) b _ c ^ { ( n ) } } \\left ( 1 - \\mathrm { e } ^ { - ( H ( \\ell _ 2 ( x - \\delta ) ) - H ( \\ell _ 2 ( x + \\delta ) ) - 2 \\eta ) b _ c ^ { ( n ) } } \\right ) . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} m : = \\sup _ { \\phi \\in \\mathcal { G } _ n } \\| \\phi \\| < \\infty \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} q _ i \\in \\{ 0 , 1 \\} \\mbox { a n d } r _ i \\leq r - 1 , \\mbox { f o r a l l } i = 1 , \\ldots , L . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\frac { 1 } { s } = \\sum _ { j = 1 } ^ m \\frac 1 { s _ j } = \\sum _ { j = 1 } ^ { i } \\frac 1 { q _ j } + \\sum _ { j = i + 1 } ^ m \\frac 1 { p _ j } \\ge \\sum _ { j = 1 } ^ m \\frac 1 { q _ j } = \\frac 1 q > \\frac 1 { r _ { m + 1 } ' } . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\begin{array} { r l } L _ { n , s } ( \\mathcal { E } _ { \\rho } ) = & \\displaystyle \\dfrac { 1 } { 2 \\pi \\sqrt { 2 } } \\int _ { 0 } ^ { 2 \\pi } | K _ { n , s } ( z ) | \\sqrt { a _ 2 - \\cos { 2 \\theta } } \\ , d \\theta \\\\ = & \\displaystyle \\dfrac { 1 } { 2 \\sqrt { 2 } } \\int _ { 0 } ^ { 2 \\pi } \\dfrac { \\sqrt { a } } { \\rho ^ { ( 2 s + 1 ) n } 2 ^ { s - 1 / 2 } c ^ s } \\ , d \\theta = \\dfrac { 1 } { 2 ^ s \\rho ^ { ( 2 s + 1 ) n } } \\int _ 0 ^ { \\pi } \\dfrac { \\sqrt { a } } { c ^ s } \\ , d \\theta . \\\\ \\end{array} \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\Psi _ * \\left [ \\bigcup _ { d _ 1 + d _ 2 = d } \\mathcal { Z } _ { d _ 1 , d _ 2 } \\right ] ^ \\textnormal { v i r } = g ^ ! \\left [ \\overline { \\mathcal { M } } _ { g , n } ( X , d ) \\right ] ^ \\textnormal { v i r } \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i , j } = \\Gamma ( k ) _ { i , 1 } + \\Gamma ( k ) _ { 1 , j } . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} u _ n ( t , x ) = \\frac { \\lambda _ n } { \\tau _ n } \\ , u \\left ( \\frac { t } { \\tau _ n } , \\frac { x } { \\lambda _ n } \\right ) \\ , , \\rho _ n ( t , x ) = \\gamma _ n \\ , \\rho \\left ( \\frac { t } { \\tau _ n } , \\frac { x } { \\lambda _ n } \\right ) \\ , . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\frac { \\langle f _ 1 , f _ 2 \\rangle } { | | f _ 1 | | | | f _ 2 | | } & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 1 } ^ { \\infty } \\alpha _ j \\overline { \\beta _ j } \\\\ & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 1 } ^ { \\infty } e ^ { i ( \\eta _ 2 + ( j - 1 ) \\eta _ 3 ) } \\sqrt { ( 1 - \\rho ) \\rho ^ { j - 1 } } e ^ { - i j \\eta _ 3 } \\sqrt { ( 1 - \\rho ) \\rho ^ j } \\\\ & = - e ^ { i \\theta _ 3 } \\frac { | a | } { 1 + | a | ^ 2 } \\rho ^ { 1 / 2 } . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} L = M = 0 , & F ( L ) = F ( M ) = 0 \\\\ F ^ 2 ( L ) = M = 0 , & L = F ( M ) = 0 \\\\ F ( L ) = M = 0 , & F ^ 2 ( L ) = F ( M ) = 0 . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\mu ^ { Z } ( X ) : = \\psi ^ { - 1 } g ( Z , X ) , \\mu ^ W ( X ) : = \\psi ^ { - 1 } g ( W , X ) . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\mathrm { e p i } ( \\omega _ * ) = \\bigcap _ { k = 0 } ^ \\infty \\mathrm { e p i } ( \\omega _ k ) = \\bigcap _ { l = 1 } ^ \\infty \\mathrm { e p i } ( \\omega _ { k _ l - 1 } ) . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\nu ^ { 1 } _ \\epsilon \\star \\nu ^ { 1 } _ \\iota = \\begin{cases} 1 , 0 , \\lambda , \\lambda - \\frac { 1 } { 2 } , \\nu ^ { 0 } _ + , \\nu ^ { 0 } _ - & \\mbox { i f } n \\geq 5 \\\\ 1 , 0 , \\lambda , \\lambda - \\frac { 1 } { 2 } & \\mbox { i f } n = m = 3 , 4 \\end{cases} \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} & \\varphi = t T ( t , \\varphi ) , \\\\ & F _ { \\kappa , 1 } ( t , \\varphi ) , \\ , F _ { \\kappa , 2 } ( t , \\varphi ) \\leq 0 , \\\\ & \\big ( F _ { \\kappa , 1 } F _ { \\kappa , 2 } \\big ) ( t , \\varphi ) = 0 . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{gather*} S = S _ g + S _ h + S _ { \\eta } , \\end{gather*}"}
-{"id": "7502.png", "formula": "\\begin{align*} \\Omega ( t _ 0 , \\cdots , t _ s , \\xi ; x ) = \\sum _ { k = 0 } ^ n y _ k ( t _ 0 , \\cdots , t _ s ) x ^ k \\in E _ 1 [ x ] \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } e _ { x _ 1 } \\cdots e _ { x _ { i - 1 } } e _ { x _ { i + 1 } } \\cdots e _ { x _ n } - \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } e _ { x _ 1 } \\cdots e _ { x _ { i - 1 } } x _ i e _ { x _ { i + 1 } } \\cdots e _ { x _ n } . \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} \\log L ( \\sigma , \\chi ) = \\sum _ { n = 2 } ^ X \\frac { \\Lambda ( n ) \\chi ( n ) } { n ^ { \\sigma } \\log n } + \\mathcal { O } \\left ( \\frac { 1 } { ( \\log q ) ^ { 1 / 4 } } \\right ) , \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} F ( z ) = \\alpha z ^ { - 1 } + o ( z ) , z \\in \\mathbb { C } \\setminus D , \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} ( N _ 1 / C _ 1 , N _ 2 / C _ 2 ) = \\begin{cases} ( 2 1 / 1 , 1 5 1 / 2 1 ) & j \\ge 4 \\\\ ( 3 0 / 1 , 2 9 5 / 3 7 ) & j \\ge 5 \\\\ \\end{cases} \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} P _ { X _ i } ( x ) = { { x - 1 } \\choose { k - 1 } } ( 1 - \\delta ) ^ k \\delta ^ { x - k } , \\ \\ x = k , k + 1 , \\ldots \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} 0 \\star \\nu ^ p _ \\pm = \\begin{cases} \\emptyset & \\mbox { i f } \\ 1 \\notin C \\mbox { a n d f o r a l l } \\beta \\in C _ \\alpha ( p ) , \\alpha \\cap \\beta = \\beta \\\\ \\nu ^ p _ \\pm & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} D ' _ n : = D _ n \\otimes _ A A / ( y , z ) = \\begin{bmatrix} O _ n & O _ n \\\\ C _ n & O _ n \\end{bmatrix} \\mbox { f o r e v e r y } n \\ge 1 . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\left ( \\Phi \\left ( \\sum _ { k = - m } ^ m \\widehat { f } ( k ) \\chi _ k \\right ) \\right ) = \\left ( \\sum _ { k = - m } ^ m \\widehat { f } ( k ) \\Phi ( \\chi _ k ) \\right ) = \\left ( \\sum _ { k = - m } ^ m \\widehat { f } ( k ) a ( k ) \\right ) \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} j ^ \\textnormal { c o h } ( t _ 1 , z ) = e ^ { \\frac { t _ 1 H } { z } } \\sum _ { d \\geq 0 } \\frac { e ^ { t _ 1 d } } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\left [ \\mathcal { A } _ j , \\nabla _ { \\partial _ { t _ i } } \\right ] = 0 \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} a T ( p ) - b \\sqrt { T ( p ) } \\geq - b ^ 2 / ( 4 a ) = : C ( A ) \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} B _ { \\mathtt A } = \\sum _ { w \\in A ^ { * } } \\mu ( \\mathrm { c i r c u i t } ( w ) ) w . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} \\int _ { U ( 2 s ) \\setminus U ( s ) } | d u | ^ p \\leq \\frac { E _ p ( u ) } { K } \\leq \\frac { \\Lambda } { 2 - p } \\cdot \\frac { 2 - p } { p - 1 } = \\frac { \\Lambda } { p - 1 } . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d Y _ t & = & \\big ( \\frac { \\kappa } { 2 } \\frac { d ^ 2 } { d x ^ 2 } Y _ t + \\frac { d } { d x } Y _ t \\big ) d t + \\sigma d X _ t \\medskip \\\\ Y _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} \\begin{cases} u _ t + f ( v , u ) _ x ~ = ~ \\varepsilon u _ { x x } \\ , , \\\\ v _ t + g ( v ) _ x ~ = ~ 0 , \\end{cases} \\begin{cases} u ( 0 , x ) = u _ 0 ( x ) , \\\\ v ( 0 , x ) = v _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} L ( { \\cal C } _ j ) = \\min _ { \\cal C } L ( { \\cal C } ) = : T S P ( x _ 1 , \\ldots , x _ j ; R ) , \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} f ( x _ { i } ) = x _ { i } f ( y _ { i } ) = g ( x _ { i } ) = g ( y _ { i } ) = y _ { i } \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ 3 ( \\lambda - \\zeta _ j ) - \\Delta ^ 2 \\cdot \\sum _ { j = 1 } ^ 3 ( \\lambda ^ 3 - \\zeta _ j ^ 3 ) - 2 \\Delta ^ 3 \\cdot \\prod _ { j = 1 } ^ 3 \\sqrt { \\lambda ^ 2 + \\lambda \\zeta _ j + \\zeta _ j ^ 2 } = 0 . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} v ( x ) : = \\frac { 1 } { ( 1 + x ^ 2 ) ^ { \\gamma } } = { } _ { 2 } F _ { 1 } \\Big ( \\gamma , \\frac { 1 } { 2 } ; \\frac { 1 } { 2 } ; - x ^ { 2 } \\Big ) . \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\tau _ n = \\inf \\left \\{ t > 0 ; \\sup _ { \\{ \\phi \\} , | \\phi _ s | < 1 } \\left | \\int _ 0 ^ t \\frac { \\partial } { \\partial x } M ( d s , \\theta ^ H _ s + \\phi _ s ) - \\int _ 0 ^ t \\frac { \\partial } { \\partial x } M ( d s , \\theta ^ H _ s ) \\right | \\geq n \\right \\} , \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} = \\prod _ { j = 1 } ^ 3 ( \\lambda - \\zeta _ j ) - \\sum _ { j = 1 } ^ 3 ( \\lambda ^ 3 - \\zeta _ j ^ 3 ) \\delta _ j ^ 2 - 2 \\prod _ { j = 1 } ^ 3 \\sqrt { \\lambda ^ 2 + \\lambda \\zeta _ j + \\zeta _ j ^ 2 } \\cdot \\delta _ j . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\leq \\sum _ { j = - \\infty } ^ { m _ 0 - ( k - i - 1 ) p / Q } 2 ^ { j s } \\Big [ g _ j ( x _ { k - i } ) + g _ j ( x _ { k - ( i + 1 ) } ) \\Big ] , \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} F ( z ) + \\langle \\nabla G ( z ) , x - z \\rangle \\leq F ( x ) = F ( y ) + \\langle \\nabla F ( y ) , x - y \\rangle + \\varphi _ y ( x ) \\textrm { f o r a l l } x , y , z \\in \\R ^ n , \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} P ( S _ n ' ( n - p _ n ^ { - 1 } ) \\leq \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor ) & = P ( \\mathrm { B i n } ( n , 1 - \\pi _ n ( n - p _ n ^ { - 1 } ) ) \\geq \\lceil \\varepsilon f _ 1 ( n ) \\rceil ) . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} \\mathcal { D M } _ { \\Delta _ N } \\left ( \\boldsymbol { \\alpha } \\right ) \\underset { N } { \\wedge } \\beta \\mathcal { B } _ { n } \\left ( \\vert \\boldsymbol { \\alpha } \\vert , b \\right ) = \\mathcal { D M } _ { \\blacktriangle _ { n } } \\left ( \\boldsymbol { \\alpha } , b \\right ) . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\psi _ m = ( \\zeta _ 1 , 2 b _ 1 + 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ l , 2 b _ l + 1 ) \\boxplus ( \\xi _ 1 , 2 a _ 1 ) \\boxplus \\cdots \\boxplus ( \\xi _ k , 2 a _ k ) . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\Gamma _ t = S _ l ( B ) _ { 0 , t } = 1 + \\sum _ { i = 1 } ^ l \\int _ { 0 < t _ 1 < \\cdots < t _ i < t } d B _ { t _ 1 } \\otimes \\cdots \\otimes d B _ { t _ i } . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\begin{cases} \\dd y _ s ^ { t , \\bar { x } ; u , v } = - \\bigl [ l _ 1 ( x _ s ^ { t , \\bar { x } ; u , v } , y _ s ^ { t , \\bar { x } ; u , v } , q _ s ^ { t , \\bar { x } ; u , v } ) + l _ 2 ( u _ s , u _ { s - r } , v _ s , v _ { s - r } ) \\bigr ] \\dd s \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + q _ s ^ { t , \\bar { x } ; u , v } \\dd B _ s , ~ s \\in [ t , T ) \\\\ y _ T ^ { t , \\bar { x } _ t ; u , v } = m ( x _ T ^ { t , \\bar { x } ; u , v } ) . \\end{cases} \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} B ^ \\delta _ R = C _ { \\alpha , \\beta } R ^ { - 2 \\delta } \\int ^ R _ 0 ( R ^ 2 - t ^ 2 ) ^ { \\beta - 1 } t ^ { 2 \\alpha + 1 } B ^ \\alpha _ t d t , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} | H _ 0 ' | ^ 2 = - ( \\Delta Y ' _ 4 ) ^ 2 + \\sum _ { i = 1 } ^ 3 ( \\Delta ( Y _ i ) ' ) ^ 2 + 4 . \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} f ( n ) : = \\left \\{ \\begin{array} { l l } 2 ^ { l + 1 } - 2 & \\mbox { i f } n = 2 ^ l - 1 \\\\ 2 ^ { l + 1 } - 1 & \\mbox { i f } 2 ^ l \\le n \\le 2 ^ { l + 1 } - 2 \\end{array} \\right . \\mbox { f o r a l l i n t e g e r s $ l \\ge 1 $ } . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\Delta & _ { R C - S T } = \\frac { \\mathbb { E } [ Q _ i ] } { m + \\mathbb { E } [ Y _ i + Z _ i ] } \\\\ = & \\frac { \\mathbb { E } \\left [ \\left ( m + Y _ i \\ ! + \\ ! Z _ i \\right ) ^ 2 \\ ! + \\ ! 2 \\left ( m \\ ! + \\ ! Y _ i \\ ! + \\ ! Z _ i \\right ) \\left ( Y _ { i - 1 } \\ ! + \\ ! Z _ { i - 1 } \\right ) \\right ] } { 2 m + 2 \\mathbb { E } [ Y _ i + Z _ i ] } \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } ( k ^ { ( 3 ) } - \\frac { 1 } { 4 } - h ^ { ( 3 ) } ) d S ^ 2 = - \\frac { 3 } { 4 } \\int _ { S ^ 2 } W _ 0 ^ 2 d S ^ 2 - \\frac { 1 } { 6 0 } \\int _ { S ^ 2 } | \\alpha | ^ 2 d S ^ 2 + \\frac { 1 1 } { 4 5 } \\int _ { S ^ 2 } | \\beta | ^ 2 d S ^ 2 . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} M _ { \\tau } ^ \\eta = \\mathcal { M } _ { \\tau } \\theta _ { \\tau } \\alpha ( Y _ { \\tau } ) . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} m _ 2 ( z , t ) = 1 - z - 2 t + 2 t z + t ^ 2 + t ^ 2 z - 3 t ^ 2 z ^ 2 - 2 t ^ 3 z + 2 t ^ 2 z ^ 3 + 2 t ^ 3 z ^ 2 + t ^ 4 z ^ 2 - t ^ 4 z ^ 3 . \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} \\mathbb { P } ( s ( u , W , v ) > 2 \\mu _ { u v } ) < \\exp \\left ( - 2 \\frac { \\mu _ { u v } ^ 2 } { 3 s ( u , I , v ) } \\right ) = \\exp \\left ( - 2 \\frac { \\mu _ { u v } } { 3 p ^ 2 } \\right ) < \\exp \\left ( - \\frac { 1 6 \\sqrt { n } } { 3 \\log ^ 4 n } \\right ) . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} M _ { S \\overline S } ( \\lambda _ 0 ) ( M _ { \\overline S \\overline S } ( \\lambda _ 0 ) - \\lambda _ 0 I ) ^ { - 1 } u _ { \\overline S } = c u _ S . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} & D ^ { i j } _ l ( x + k ) = D ^ { i j } _ l ( x ) , \\ q _ l ( x + k ) = q _ l ( x ) , \\ f _ l ( x + k , \\cdot ) = f _ l ( x , \\cdot ) \\\\ & { \\rm f o r } \\ \\ x \\in \\R ^ N , \\ k \\in \\mathbb { L } : = L _ 1 \\Z \\times L _ 2 \\Z \\times \\cdots \\times L _ N \\Z \\\\ & ( i , j = 1 , 2 , \\cdots , N ; \\ l = 1 , 2 , \\cdots , m ) , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\textbf { h } _ k = \\beta \\left ( \\textbf { x } _ k \\right ) \\textbf { a } ( \\textbf { x } _ k ) . \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} R _ 0 = R ( \\bar { x } , 0 , 0 ) > R ( \\bar { x } , y , v ) > R ( x , y , v ) , \\forall x \\in [ 0 , \\bar { x } ) , y , v > 0 . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} V _ n : = \\sum _ { l = 1 } ^ { N } T _ l , \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} S _ { n - i } = \\# \\{ l \\mid a _ { i } > - y _ { l } + \\tfrac { 1 } { 2 } > a _ { i + 1 } \\} . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\bar H _ { , k } ^ { p ^ { \\ast } } = \\bar h ^ { p ^ { \\ast } } _ { 1 1 k } + \\bar h ^ { p ^ { \\ast } } _ { 2 2 k } = 0 , \\ \\ k , p = 1 , 2 . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\left | e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } f \\right | _ q \\leq \\mathcal { B } | f | _ q , \\ \\forall f \\in e ^ { \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } [ L ^ q ( \\mathbb { R } ^ d ) ] , \\ t \\geq 0 , \\ i = 1 , 2 , . . . , N , \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\vert u ( x ) \\vert = \\lim _ { i \\rightarrow \\infty } \\vert u ( x ) - u ( y _ i ) \\vert \\leq C ' r _ 0 ^ { s } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} ( a \\vee b ) ^ { - 1 } d c ^ { - 1 } & \\geq ( a \\vee b ) ^ { - 1 } a c c ^ { - 1 } = a ^ { - 1 } a c c ^ { - 1 } . \\\\ ( a \\vee b ) ^ { - 1 } d c ^ { - 1 } & \\geq ( a \\vee b ) ^ { - 1 } b c c ^ { - 1 } = b ^ { - 1 } b c c ^ { - 1 } . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\max _ { 0 \\le x \\le 1 } \\det A _ n ( x ) = \\det A _ n ( x _ k ) = \\det A _ n ( { \\textstyle \\frac 1 2 } ) \\left ( 1 + \\frac { 1 } { 8 k n } + O ( k ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\psi _ { \\tau } ( x ' , y ) \\rightarrow \\langle \\nabla _ { x ' } \\psi ( 0 , 0 ) , x ' \\rangle = 0 ~ ~ ~ ~ ~ ~ ~ ~ \\tau \\rightarrow 0 , \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\iint _ Q & \\hat { y } b d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\hat { p } ^ i b _ i d x d t \\\\ = & ( y _ 0 , u ( 0 ) ) + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\hat { y } u \\ d x d t + \\iint _ Q ( G u + { G } _ 1 z ^ 1 + { G } _ { 2 } z ^ 2 ) d x d t \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} u ( t , x ) : = ( c - 1 ) ^ { 1 / ( p - 1 ) } Q \\left ( \\sqrt { \\frac { c - 1 } { c } } ( x - c t ) \\right ) , Q ( s ) : = \\left ( \\frac { p + 1 } { 2 \\cosh ^ 2 ( \\frac { p - 1 } { 2 } s ) } \\right ) ^ { 1 / ( p - 1 ) } , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\frac { 2 N } { N - 4 \\nu } \\ge \\frac { 2 N } { N - 4 \\left ( \\frac { N } { 4 } - \\frac { \\varrho } { s } \\right ) } = \\frac { 2 N } { \\left ( 4 \\frac { \\varrho } { s } \\right ) } \\ge \\frac { 2 N } { \\left ( 4 \\frac { N } { 8 } \\right ) } = 4 , \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} \\partial _ { y _ 1 } g _ { D , \\mathrm { o d d } } ( y ) = ( \\partial _ { y _ 1 } g _ D ) _ { \\mathrm { e v e n } } ( y ) , \\partial _ { y _ 1 } g _ { N , \\mathrm { e v e n } } ( y ) = ( \\partial _ { y _ 1 } g _ N ) _ { \\mathrm { o d d } } ( y ) , \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { S } } _ { n } ^ { \\prime } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { \\sin ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n } } , \\\\ \\widetilde { \\mathcal { C } } _ { n } ^ { \\prime } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { \\cos ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n + 1 } } , \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} c = \\frac 1 { \\sqrt { 1 - \\alpha ^ 2 } } \\alpha : = \\Big | \\Big \\langle { \\frac { v _ 1 } { \\| v _ 1 \\| } } \\ , , { \\frac { v _ 2 } { \\| v _ 2 \\| } } \\Big \\rangle \\Big | \\ , . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} g \\circ f = h \\circ f g = h . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} G _ i ( x ) : = \\left \\{ \\begin{array} { l l } F _ i ( x ) & \\hbox { i f } \\ \\ i = 1 , . . . , m , \\\\ F _ { m } ( x ) & \\hbox { i f } \\ \\ i = m + 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} g _ q ( Q ) = \\sum _ { d \\geq 0 } \\frac { ( 1 - q ) ^ d } { ( q ; q ) _ d } Q ^ d \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} C _ { \\textrm { N } } ^ { \\textrm { N O M A } } = \\begin{cases} \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 \\left ( 2 ^ R - 1 \\right ) } { P _ { \\textrm { N } } } , & \\textrm { i f } k > 2 ^ R , \\\\ \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 \\left ( 2 ^ R - 1 \\right ) } { P _ { \\textrm { F } } - P _ { \\textrm { N } } \\left ( 2 ^ R - 1 \\right ) } , & \\textrm { i f } k \\le 2 ^ R , \\end{cases} \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} v _ 1 ( x ) = e ^ { \\rho _ 1 \\cdot x } ( 1 + r _ 1 ) , v _ 2 ( x ) = e ^ { \\rho _ 2 \\cdot x } ( 1 + r _ 2 ) , \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\norm { P ( a ) } = \\norm { \\Phi \\left ( a ^ n \\right ) } \\le \\left \\Vert \\Phi \\right \\Vert \\left \\Vert a ^ n \\right \\Vert _ { \\mathcal { P } _ n } \\le \\left \\Vert \\Phi \\right \\Vert \\left \\Vert a \\right \\Vert ^ n . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} - ( t _ 1 - t _ 0 ) \\varphi _ { t _ 1 } = \\big ( ( t _ 1 - t _ 0 ) \\varphi _ { t _ 0 } \\big ) ^ c , \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} Y _ i = \\sum _ { i = 1 } ^ { \\infty } E _ i \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\Gamma = \\left \\{ \\exp ( - V _ { M , \\ell } ) : M \\in \\mathbb { N } , \\ell = 1 , \\cdots , M \\right \\} . \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\eta ( u , v , w ) = - \\eta ( u , w , v ) , \\ \\eta ( u , \\mathcal J v , w ) = \\eta ( u , v , \\mathcal J w ) . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { u \\wedge v } ( u - x ) ^ { \\alpha - 1 } ( v - x ) ^ { \\alpha - 1 } \\ , \\mathrm { d } { x } = \\mathrm { B } ( \\alpha , 1 - 2 \\alpha ) | u - v | ^ { 2 \\alpha - 1 } , u \\neq v , \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} | G ^ \\alpha _ R ( x ) | ^ 2 & = \\left | \\frac { 1 } { R } \\int ^ R _ 0 ( B ^ { \\alpha + 1 } _ r ( x ) - B ^ { \\alpha } _ r ( x ) ) d r \\right | ^ 2 \\\\ & \\leq \\int ^ R _ 0 | B ^ { \\alpha + 1 } _ r ( x ) - B ^ { \\alpha } _ r ( x ) | ^ 2 \\frac { d r } { R } \\\\ & \\leq \\int ^ \\infty _ 0 | B ^ { \\alpha + 1 } _ r ( x ) - B ^ { \\alpha } _ r ( x ) | ^ 2 \\frac { d r } { r } = : ( G ^ \\alpha ( x ) ) ^ 2 . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} \\partial _ r \\eta _ a = R _ { L a L \\underline L } + l _ a ^ b \\eta _ b \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} ( X + 1 ) Q ^ 4 + X ^ 3 Q = 0 . \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } U _ { 2 k + 1 , 2 a } ( n ) q ^ n = & \\frac { ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a + 1 } , q ^ { 4 k - 2 a + 1 } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\\\ & + \\frac { x q ( - q ^ 2 ; q ) _ \\infty ( q ^ { 2 a - 1 } , q ^ { 4 k - 2 a + 3 } , q ^ { 4 k + 2 } ; q ^ { 4 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} I \\left ( R , c \\right ) = \\sup \\left \\{ \\int _ { 1 } ^ { R } \\frac { 1 - f ^ { 2 } } { r } d r : \\int _ { 1 } ^ { R } J \\left ( 1 - f ^ { 2 } \\right ) r d r \\leq c \\right \\} \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\C _ { \\mu } f ( z ) : = \\lim _ { \\epsilon \\to 0 } \\C _ { \\epsilon , \\mu } f ( z ) \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ t ( O ) = & - F _ t ( O ) + \\lambda \\sum _ { y : y \\sim O } 2 a b F _ t ( y ) + 2 d \\lambda ( b ^ 2 - 1 ) F _ t ( O ) + \\lambda \\sum _ { y : y \\sim O } a ^ 2 E \\big [ \\xi _ t ^ 2 ( y ) \\big ] \\\\ & + 2 \\big ( 1 - 2 d \\lambda ( a + b - 1 ) \\big ) F _ t ( O ) . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} q _ 1 ( V ) = g ^ \\ast ( q _ 4 ) : H ^ 4 ( M ) \\leftarrow H ^ 4 ( B S p i n ^ c ( n ) ) \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} w : = ( \\kappa _ 1 , \\kappa _ 2 , \\omega ) \\in L ^ p ( [ 0 , L ] ; \\R ^ 3 ) . \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} D ^ { \\mu ^ { T \\circ S } } _ X = \\zeta ^ { T , S } _ X \\circ D ^ { \\mu ^ T } _ { D ^ S ( X ) } \\circ D ^ { \\mu ^ S } _ X . \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , 0 ) \\equiv \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , 1 ) \\equiv \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , 2 ) \\equiv \\cdots \\equiv \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F \\left ( n , \\frac { m - 1 } { 2 } \\right ) \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } . \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\delta < \\frac { \\min \\{ \\lambda _ { + } \\varphi _ l ^ { + } , \\ , \\lambda _ { - } \\varphi _ l ^ { - } \\mid l = 1 , 2 , \\cdots , m \\} } { \\max \\Big \\{ 1 , \\ , \\underset { u \\in [ p ^ - , p ^ + ] } { \\sup } | D ^ 2 F ( u ) | \\Big \\} } , \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} f ^ { \\flat } ( z ) = c ^ { * } ( z ) , G ^ { \\flat } ( z ) = \\xi _ { y } - v , \\textrm { w h e r e } y \\in P ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} \\mu _ { n , 1 } = \\left ( \\prod _ { i = 2 } ^ n \\mu _ { n , i } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\Phi ( f ) = \\int _ \\mathbb { T } f ( z ) h ( z ) \\ , d z ( f \\in L ^ p ( \\mathbb { T } ) ) . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} T _ k & = \\inf \\left \\{ t | S _ k ( t ) \\notin \\left [ \\lambda , \\upsilon \\right ] \\right \\} , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} \\dim \\Pi \\mu _ { 0 } = \\dim \\Pi \\mu < \\beta - \\epsilon \\ : . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} ( x q ) ^ { 2 a + 1 } \\sum _ { h = 0 } ^ { k - a - 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} F _ { k + 1 } ( x ) & = \\frac { c - a } { a ( 1 + x ^ 2 ) } \\ , F _ { k - 1 } ( x ) + \\frac { ( 2 a - c ) + ( a - b ) x ^ 2 } { a ( 1 + x ^ 2 ) } F _ k ( x ) , \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} \\{ \\emph { d i s t } ^ + ( x , \\Gamma ) < \\epsilon \\} : = \\bigcup _ { \\sigma \\in [ 0 , \\ell ] } B _ \\epsilon ( \\gamma ( \\sigma ) ) \\cap \\{ x \\in \\Omega \\ : \\ x \\cdot ( \\gamma ' ( \\sigma ) ) ^ { \\perp } > 0 \\} \\ , . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} \\tilde { \\mathcal { U } } ^ L = \\lambda \\ln \\left ( 1 + | \\mathcal { N } | T \\right ) - p | \\mathcal { N } | T - \\frac { a | \\mathcal { N } | ^ 2 T } { | \\mathcal { N } | - 1 } . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} \\mu ( \\{ a \\} ) = \\begin{cases} \\frac { 1 } { \\# A } & a \\in A \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\int _ y \\chi ( h y ) \\cdot H \\partial _ y w \\cdot \\partial _ y w \\ , d y = \\iint _ { y , y ' } K ( y , y ' ) w ( y ) w ( y ' ) \\ , d y \\ , d y ' \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} H ( z ) = ( s ^ 2 ( r - 2 ) + s t ) z - ( s ^ 2 - s ) + s t ( r - 1 ) ^ q z ^ { q + 1 } - ( t ^ 2 - t ) ( r - 1 ) ^ { q - 1 } z ^ q . \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} E _ 0 ( w _ 0 ) = \\min _ W E _ 0 . \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} T ^ { ( P ) } _ l : = T S P ( Y _ 1 , \\ldots , Y _ { N ^ { ( P ) } _ l } ; S _ l ) \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} \\mathbf P _ { s o } ^ { \\ , i } \\ ! = \\ ! 1 \\ ! - \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\mathbf D _ u \\in 2 ^ { \\mathcal D } } \\ ! \\ ! \\ ! \\underbrace { \\mathbb E \\ ! \\ ! \\left [ \\frac { e ^ { - \\frac { ( 2 ^ { \\frac { n r _ s } { \\beta } } - 1 ) ( I _ B + 1 ) } { \\gamma _ { A _ i , B } } } } { \\frac { \\gamma _ { A _ i , E } } { \\gamma _ { A _ i , B } } \\frac { I _ B + 1 } { I _ E + 1 } 2 ^ { \\frac { n r _ s } { \\beta } } \\ ! + \\ ! 1 } \\Bigg | \\mathbf D _ u \\right ] \\ ! \\mathbb P \\ ! \\ ! \\left ( \\mathbf D _ u \\right ) } _ { T } , \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} y _ i = \\sum _ { j = 1 } ^ { m } g _ j \\left ( x _ { i - j + 1 } \\right ) , i \\in \\mathbb { N } , \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\mathcal { L } H ^ { p ^ { \\ast } } = \\sum _ { k } H _ { , k k } ^ { p ^ { \\ast } } - \\langle X , \\sum _ { k } H _ { , k } ^ { p ^ { \\ast } } e _ { k } \\rangle = H ^ { p ^ { \\ast } } - \\sum _ { i , j , q } h _ { i j } ^ { p ^ { \\ast } } h _ { i j } ^ { q ^ { \\ast } } H ^ { q ^ { \\ast } } . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} g ( u ) = \\xi ( 1 ) - \\xi ( u ) - \\frac { q ( 1 - u ) } { [ A _ 2 ( 1 - q ) + \\Delta ] [ A _ 1 q + A _ 2 ( 1 - q ) + \\Delta ] } \\\\ + \\frac { 1 - u } { A _ 2 [ A _ 2 ( 1 - q ) + \\Delta ] } - \\frac 1 { A _ 2 ^ 2 } \\log \\Big ( 1 + \\frac { A _ 2 ( 1 - u ) } { \\Delta } \\Big ) . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} & A _ 1 ( z ) = ( 1 - z b q ) ( 1 - z b q ^ 2 ) ( q ^ { - n } - q ^ \\gamma ) ( q ^ { - n } - q ^ { \\gamma + 1 } ) ( a b q ^ { 2 n } - 1 ) ( a b q ^ { 2 n - 1 } - 1 ) , \\\\ & B _ 1 ( z ) = b q ^ { n + \\gamma } ( 1 - z b q ^ 2 ) ( 1 + q ) ( a q ^ n - 1 ) ( 1 - q ^ { - n } ) ( q ^ { - n } - q ^ \\gamma ) ( a b q ^ { 2 n - 1 } - 1 ) , \\\\ & \\\\ & C _ 1 ( z ) = b ^ 2 q ^ { 2 n + 2 \\gamma } ( a q ^ n - 1 ) ( a q ^ { n - 1 } - 1 ) ( 1 - q ^ { - n } ) ( 1 - q ^ { - n + 1 } ) . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\langle u , f \\rangle d \\mu = \\sup \\Big \\{ \\int _ { \\mathbb { R } ^ n } \\langle v , f \\rangle d \\mu \\big | v \\colon \\mathbb { R } ^ n \\to \\mathbb { R } ^ m 1 \\Big \\} . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} 0 \\ge \\log \\left ( \\sum _ { w \\in \\Lambda ^ { m } } p _ { w } ^ { q } \\rho _ { 2 } ^ { - m \\tau } \\right ) = \\log \\Vert p \\Vert _ { q } ^ { q } - m \\tau \\log \\rho _ { 2 } \\ : . \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} Q _ 1 & = 1 + \\frac { - 1 } { 2 ( 1 - i ) } ( q - 1 ) + o ( q - 1 ) \\\\ Q _ i & = i + \\frac { i } { 2 } ( q - 1 ) + o ( q - 1 ) \\\\ Q _ { - 1 } & = - 1 + \\frac { 1 } { 2 ( 1 + i ) } ( q - 1 ) + o ( q - 1 ) \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} 0 \\le x _ i \\le f _ { q - k + i } , \\mbox { f o r } i = 1 , \\cdots , k , \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\gamma _ { p , q } = \\frac { d } { 2 } - \\frac { d } { q } - \\frac { 2 s } { p } . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} f ^ * ( t ) = \\inf \\{ \\lambda \\geq 0 : D _ f ( \\lambda ) \\leq t \\} = \\left ( \\frac { b _ n } { t } \\right ) ^ { \\frac { s } { n } } . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} u _ { t } + \\left [ u \\left ( 1 - u \\right ) \\right ] _ { x } = 0 . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\hat { \\Psi } _ 0 ( Y _ 1 , \\ldots , Y _ m ) = ( X _ 1 , \\ldots , X _ m ) , \\\\ \\hat { \\Psi } _ 1 ( Y _ 1 , \\ldots , Y _ m ) = ( Y _ 1 , \\ldots , Y _ m ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} i + \\ell ( d + m ) - e + 1 = 0 . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} c _ { n + 1 } = e _ { \\sigma ( j _ { n + 1 } ) } \\in ( A - b _ 1 ) \\cap \\cdots \\cap ( A - b _ { n + 1 } ) \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} \\mathbf { D } ^ k _ { h _ 1 , \\ldots , h _ k } F = \\mathbf { D } _ { h _ 1 } \\ldots \\mathbf { D } _ { h _ k } F . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} d i v \\alpha _ { H _ 0 } = \\frac { 1 } { 2 } \\tilde \\Delta ( \\tilde \\Delta + 2 ) Y _ 0 ^ { ( 3 ) } \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\eta ^ \\# ( \\rho , m ) _ t + q ^ \\# ( \\rho , m ) _ x = \\left ( - \\frac { \\beta } { 2 } | \\mathbf { h } | ^ 2 + \\alpha g ' ( 1 / \\rho ) h ( | \\psi | ^ 2 ) + \\varepsilon u _ x \\right ) _ x \\eta _ m ^ \\# ( \\rho , m ) . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\left ( \\Gamma \\cap x W \\right ) ^ { - 1 } = \\Gamma ^ { - 1 } \\cap x ^ { - 1 } W ^ { - 1 } . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\mathbb { E } _ 0 T ^ { ( P ) } _ { l _ 1 } T ^ { ( P ) } _ { l _ 2 } = J ^ { ( P ) } _ 1 + J ^ { ( P ) } _ 2 , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} ( 2 \\phi _ 2 ^ 2 - \\phi _ 1 ^ 2 ) - ( 2 \\phi _ 2 ^ 1 - \\phi _ 1 ^ 1 ) = ( 2 \\omega _ 2 - \\omega _ 1 ) \\tau . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} m ( \\pi ) & = \\lim _ { a \\to \\infty } \\frac { 1 } { 2 a } \\int _ { - a } ^ { a } Q _ \\pi ^ { i r } \\varphi ( \\pi ) Q _ \\pi ^ { - i r } d r = \\lim _ { a \\to \\infty } \\frac { 1 } { 2 a } \\int _ { - a } ^ { a } Q _ \\pi ^ { i r } [ ( \\Phi \\otimes \\mathrm { i d } ) ( u ^ { ( \\pi ) } ) ] Q _ \\pi ^ { - i r } d r \\\\ & = \\lim _ { a \\to \\infty } \\frac { 1 } { 2 a } \\int _ { - a } ^ { a } ( \\Phi \\circ \\tau _ r \\otimes \\mathrm { i d } ) ( u ^ { ( \\pi ) } ) d r = ( \\Psi \\otimes \\mathrm { i d } ) ( u ^ { ( \\pi ) } ) . \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\omega _ { \\alpha ^ { \\ast } } = 0 1 \\leq \\alpha \\leq n \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} g _ j = G _ j \\circ \\phi , \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots x _ { n + 1 } ) = { } & c ( x _ 1 , \\ldots x _ n ) + \\sum _ { i = 0 } ^ n \\alpha _ { i } ( x _ 1 , \\ldots x _ n ) \\zeta ( x _ { n + 1 } - x _ i ) \\\\ & + \\sum _ { k \\in \\Z _ { \\geq 0 } } \\sum _ { i = 0 } ^ n \\beta _ { i , k } ( x _ 1 , \\ldots x _ n ) \\wp ^ { ( k ) } ( x _ i - x _ { n + 1 } ) \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} x _ 1 = x _ 0 ( 1 - h _ 0 x _ 0 ^ 2 ) = - h _ 0 x _ 0 ^ 3 \\left ( 1 - \\frac 1 { h _ 0 x _ 0 ^ 2 } \\right ) , \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} H _ i ( \\Gamma , M ) ^ { \\ast } = H ^ i ( \\Gamma , X ) \\ ; i \\ge 0 \\ ; . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\boldsymbol { \\psi } } ( t ) \\ ! = \\ ! \\bar { \\boldsymbol { \\psi } } ( t _ k ) \\ ! + \\ ! \\frac { ( t \\ ! - \\ ! t _ k ) } { b _ { S , k + 1 } } \\left [ \\bar { \\boldsymbol { \\psi } } ( t _ { k + 1 } ) \\ ! - \\ ! \\bar { \\boldsymbol { \\psi } } ( t _ k ) \\right ] , t \\ ! \\in \\ ! [ t _ k , t _ { k + 1 } ] . \\end{aligned} \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} f = \\eta f = \\eta f _ 1 + \\eta f _ 2 . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} l ( \\mu ) = \\begin{cases} 0 , & \\quad \\mu = v \\in E ^ 0 , \\\\ n , & \\quad \\mu = e _ 1 e _ 2 \\dots e _ n . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\begin{aligned} A = & \\bar \\lambda _ 2 \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) - \\bar \\lambda \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) \\\\ = & \\bar H \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) \\\\ = & - \\dfrac { H \\bar \\lambda } { \\bar \\lambda _ 1 } \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) . \\end{aligned} \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} r ^ { \\epsilon , \\hat { v } } ( t ) & = \\bigl ( r _ { 1 } ^ { \\epsilon , \\hat { v } } ( t ) , r _ { 2 } ^ { \\epsilon , \\hat { v } } ( t ) , \\ldots , r _ { n } ^ { \\epsilon , \\hat { v } } ( t ) \\bigr ) \\\\ & = \\biggl ( \\int _ 0 ^ t \\chi _ 1 \\bigl ( \\zeta ^ { \\epsilon } ( s ) \\bigr ) d s , \\int _ 0 ^ t \\chi _ 2 \\bigl ( \\zeta ^ { \\epsilon } ( s ) \\bigr ) d s , \\ldots , \\int _ 0 ^ t \\chi _ n \\bigl ( \\zeta ^ { \\epsilon } ( s ) \\bigr ) d s \\biggr ) , \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} Y ^ { { \\varepsilon } , u } _ x ( t ) = & \\sqrt { { \\varepsilon } } \\int _ 0 ^ t S ( t - s ) G ( s , \\mathcal { M } ( Y ^ { { \\varepsilon } , u } _ x + S ( \\cdot ) x ) ( s ) ) d w ( s ) \\\\ & + \\int _ 0 ^ t S ( t - s ) G ( s , \\mathcal { M } ( Y ^ { { \\varepsilon } , u } _ x + S ( \\cdot ) x ) ( s ) ) u ( s ) d s . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\Delta = \\lim _ { i \\rightarrow \\infty } \\frac { \\frac { 1 } { i } \\sum _ { j = 1 } ^ { i } Q _ j } { \\frac { 1 } { i } \\sum _ { j = 1 } ^ { i } T _ j } = \\frac { \\mathbb { E } [ Q ] } { \\mathbb { E } [ T ] } \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ { 6 + } ( x ) + \\frac 2 { 6 ^ 7 } ( 3 ^ 7 x - 9 1 1 ) ( 5 4 - 2 ^ 7 x ) \\\\ & \\qquad \\qquad + \\frac { 2 } { 6 ^ 8 } ( 3 ^ 8 x - 2 7 3 3 ) ( 1 0 7 - 2 ^ 8 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 9 1 1 } { 3 ^ 7 } } + \\frac 1 { 1 0 \\cdot 6 ^ 8 } - \\eta \\\\ & = - \\frac { 2 9 0 3 2 4 4 1 3 7 5 7 1 8 6 0 2 6 7 2 3 3 3 3 3 5 5 7 8 0 2 3 4 3 7 0 1 } { 1 1 1 7 0 0 6 8 3 7 2 7 1 4 2 2 7 7 0 9 9 3 4 7 6 2 8 0 5 5 3 8 6 6 1 5 5 8 4 6 4 0 0 0 } < 0 , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} | A ^ { n } | U _ n ^ * ( A ^ * A ) ^ m U _ n | A ^ { n } | = ( A ^ * A ) ^ { m + n } . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} \\mathcal Z ( \\mathcal { H } ) : = \\left \\{ f \\in L ^ 1 ( \\Omega ) \\cap \\mathcal D ( \\mathcal { H } ) : \\sup _ { j \\in \\mathbb Z } 2 ^ { M | j | } \\big \\| \\phi _ j \\big ( \\sqrt { \\mathcal { H } } \\big ) f \\big \\| _ { L ^ 1 ( \\Omega ) } < \\infty M \\in \\mathbb N \\right \\} . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\mathrm { I C } ( \\mathcal { T } ) : = \\left ( \\sum _ { n = n _ \\ast } ^ { p _ \\ast - 1 } \\ , \\# \\mathcal { B } ( n ) \\right ) + \\left ( \\sum _ { n = p _ \\ast } ^ { p _ \\ast + \\ell - 1 } \\ , \\# \\mathcal { B } ( n ) \\right ) , \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} \\chi ' _ k ( r ) \\leq O ( \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } ) \\chi ' ( ( \\log \\rho _ k ) ^ { - 3 / 2 } ) = O ( \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } ) ( \\log \\rho _ k ) ^ { 3 } e ^ { - ( \\log \\rho _ k ) ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\mathcal { H } _ G = \\overline { \\rm S p a n } \\ , \\Big \\{ \\frac { G } { z - \\lambda } : G ( \\lambda ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} p \\circ \\sigma \\in \\bigotimes _ { i = 1 } ^ { d _ 1 } \\mathbb { P } _ n ( I _ i ) \\otimes \\bigotimes _ { j = 1 } ^ { d _ 2 + d _ 3 } \\mathbb { T } _ n ( \\Theta _ j ) \\ ; . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\Phi _ v ( \\bar { A } ) \\le \\max \\{ \\lambda : \\exists y , z \\in \\Delta _ { n - 1 } , I ( y ) \\subseteq I ( x ) , \\bar { A } ( y - z ) = \\lambda d \\} . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} L a g _ i = \\prod _ { j \\neq i } \\frac { H - \\lambda _ j } { \\lambda _ i - \\lambda _ j } \\in H _ { T ^ { N + 1 } } ^ * \\left ( \\mathbb { P } ^ N ; \\mathbb { Q } \\right ) \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} \\delta _ 1 = \\frac { 1 } { 3 } \\min _ { \\mathbf { x } , \\mathbf { y } \\in \\mathbf { Z } ( p _ 1 , \\ldots , p _ r ) } \\{ \\| \\mathbf { x } - \\mathbf { y } \\| _ 2 \\ : | \\ : \\mathbf { x } \\neq \\mathbf { y } \\} > 0 , \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} \\tau _ 0 : = \\inf \\{ \\tau ' > 0 \\mid \\exists \\tau \\in \\R , \\ \\widetilde { \\phi } ( \\cdot - \\tau ) \\preceq \\phi ( \\cdot ) \\preceq \\widetilde { \\phi } ( \\cdot - \\tau - \\tau ' ) \\} \\leq \\tau _ 0 - \\eta < \\tau _ 0 . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\| q \\alpha \\| = \\frac { \\| m _ j q \\alpha \\| } { m _ j } \\leq \\frac { 1 } { m _ j Q ( \\log Q ) ^ { 3 / 4 + \\varepsilon } } \\approx \\frac { 1 } { Q ( \\log Q ) ^ { 1 + \\varepsilon } } , 1 \\leq q \\leq Q . \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\varphi _ 0 & = ( 1 - P ^ { - 1 } ) ^ N \\\\ \\varphi _ i & = P ^ { - ( N - i ) } ( 1 - P ^ { - 1 } ) & & \\textnormal { i f } i \\neq 0 \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\nu ' ( f ) : = \\min _ { 0 \\leq i \\leq n } \\{ \\nu ( f _ i ) + i \\gamma \\} . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\tilde { D } _ { u } : = D ^ { ( 1 ) } _ { u } + \\eta _ { u } = D _ { u } - \\frac { 1 } { 2 } \\mathcal J _ { 2 } D _ { u } \\mathcal J _ { 2 } + \\eta _ { u } , \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\psi ' ( \\tau ) & \\ , = \\ , - \\frac { \\tau _ 0 } { \\tau ^ 2 } ( u + \\tau _ 0 ) - 1 \\\\ [ \\alpha _ 0 - \\tau _ 0 \\leq u ] & \\ , \\leq \\ , - \\frac { \\tau _ 0 } { \\tau ^ 2 } \\ , \\alpha _ 0 - 1 \\\\ [ \\alpha _ 0 \\geq - \\tau _ 0 ] & \\ , \\leq \\ , \\frac { \\tau _ 0 ^ 2 } { \\tau ^ 2 } - 1 \\\\ [ \\tau \\geq \\tau _ 0 ] & \\ , \\leq \\ , 0 \\ , . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} C _ p \\geq f ( \\overline { x } , \\overline { y } ) = 2 ^ { \\frac { 1 } { q } } . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} K ^ * _ { D _ { \\varepsilon } } ( z , \\varepsilon e ^ { i \\theta } ) = 2 K ^ * _ D ( z , 0 ) ( 1 + c ( z , \\theta ) \\varepsilon ) \\sin \\theta \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} a _ { \\varepsilon } ( x ) = \\int _ { - \\infty } ^ { x } \\lambda _ { \\varepsilon } \\left ( \\xi \\right ) \\ ; d \\xi . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\phi _ 1 ( q ) : = \\xi '' ( q ) [ ( 1 - q ) \\xi ' ( 1 ) - 1 + \\xi ( q ) ] - \\frac { \\xi ' ( 1 ) - \\xi ' ( q ) } { 1 - q } [ 1 - \\xi ( q ) - \\xi ' ( q ) ( 1 - q ) ] < 0 . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\mathcal { O } _ { 1 , d } = \\mathcal { O } _ { 2 , d } \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} & \\{ p \\in S p l ( f ) \\mid g _ i ( r _ { \\mu ( 1 ) } ) \\equiv r _ { \\mu ( i ) } \\bmod p \\ , \\ , ( 1 \\le { } ^ \\forall i \\le n ) \\} \\\\ = & \\left \\{ p \\in S p l ( f ) \\left | \\{ g _ { \\mu ^ { - 1 } ( 1 ) } ( r _ { \\mu ( 1 ) } ) / p \\} \\le \\dots \\le \\{ g _ { \\mu ^ { - 1 } ( n ) } ( r _ { \\mu ( 1 ) } ) / p \\} \\right . \\right \\} \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\tau _ * E _ { \\alpha } ^ L = E _ { - \\alpha } ^ R \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\Delta ^ 2 ( u - v _ 1 - v _ 2 ) = 0 , \\mbox { i n $ B _ { 1 } $ } , \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\aligned & f ( t ) = t ^ 4 - 2 S t ^ 3 - 6 S ( S - 1 ) t ^ 2 + 2 S ( 2 - 3 S ) ^ 2 t - ( 2 - 3 S ) ^ 2 S ^ 2 , \\endaligned \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { m \\rightarrow \\infty } | \\nabla ^ { \\perp } \\vec H | ^ 2 & = \\sum _ { i , p } ( \\bar H ^ { p ^ * } _ { , i } ) ^ 2 = ( \\bar h _ { 1 1 2 } ^ { 2 ^ * } + \\bar h _ { 2 2 2 } ^ { 2 ^ * } ) ^ 2 = A ^ 2 \\end{aligned} \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\begin{cases} w _ { m + 1 } ' + A w _ { m + 1 } = - V _ { m } ' , t > 0 , \\\\ w _ { m + 1 } ( 0 ) = ( - 1 ) ^ { m + 1 } u _ 1 . \\end{cases} \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta \\mu = 0 \\ , , \\mu = \\alpha ( \\partial _ t u ) - \\Delta u + \\beta ( u ) + \\pi ( u ) - g ( 0 , T ) \\times \\Omega \\ , , \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} k _ a ( t ) : = \\int _ { x > a } ( x - a ) ^ 2 \\theta ( z , t ) d z . \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\ell _ q ( Q _ j ) & = \\frac { - Q _ j \\theta _ q ' ( Q _ j ) } { \\theta _ q ( Q _ j ) } \\\\ \\theta _ q ( Q _ j ) & = \\prod _ { r \\geq 0 } ( 1 - q ^ { r + 1 } ) ( 1 + q ^ r Q _ j ) \\left ( 1 + \\frac { q ^ { r + 1 } } { Q _ j } \\right ) \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} \\pi _ { \\mathcal { K } } \\circ \\mathcal { S } _ { 0 } ( s U ) & = \\frac { s ^ 3 } { 8 } ( a ^ 2 + b ^ 2 + 1 8 c ^ 2 + 1 8 d ^ 2 ) ( a V _ A ^ { \\perp } + b V _ B ^ { \\perp } ) \\\\ & \\quad + \\frac { s ^ 3 } { 8 } ( 1 8 a ^ 2 + 1 8 b ^ 2 + c ^ 2 + d ^ 2 ) ( c V _ C ^ { \\perp } + d V _ D ^ { \\perp } ) + O ( s ^ 4 ) . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\det D ^ 2 v = f \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} n ( x _ { 1 } ) - f ( x _ { 1 } , y _ { 1 } , v _ { 1 } ) v _ { 1 } & = 0 , \\\\ f ( x _ { 1 } , y _ { 1 } , v _ { 1 } ) v _ { 1 } & = \\frac { a \\varphi _ { 1 } ( y _ { 1 } ) } { G _ { 1 } } , \\\\ \\frac { u } { k G _ { 2 } } & = \\frac { \\varphi _ { 1 } ( y _ { 1 } ) } { v _ { 1 } } . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} y \\le 0 , \\ \\ S ( y ) \\succeq 0 , \\ \\ S ( y ) _ { j j } = s ( y ) _ j = 0 \\ \\forall \\ j \\in J \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} P ( y _ 0 , y _ 1 , y _ 2 , y _ 3 , x _ 0 , x _ 1 ) = K ( x _ 0 , x _ 1 ) y _ 0 + R ( y _ 1 , y _ 2 , y _ 3 , x _ 0 , x _ 1 ) \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\theta _ R ( x ) = \\begin{cases} 1 & | x | \\leq R \\\\ 0 & | x | \\geq R + 1 \\end{cases} \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} g ( f ( z ) ) - z & = \\{ g ( f ( z ) ) - f _ { n _ k } ^ { - 1 } ( f ( z ) ) \\} + \\{ f _ { n _ k } ^ { - 1 } ( f ( z ) ) - f _ { n _ k } ^ { - 1 } ( f _ { n _ k } ( z ) ) \\} \\\\ & \\to 0 k \\to \\infty . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} A _ 0 ( r ) = r N ^ 0 _ r = \\sup _ { x _ 0 \\in \\R ^ 3 } \\int _ { B _ r ( x _ 0 ) } | u _ 0 | ^ 2 \\ , d x , \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} e _ { q , U } : = \\sum _ { d = 0 } ^ \\infty \\frac { 1 } { k ! } ( \\ell _ q ( Q ) N ) ^ d \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} \\Psi ( x ) = \\left \\{ \\begin{aligned} & c _ 4 \\log | x | , & n & = 4 \\\\ & c _ n \\frac { 1 } { | x | ^ { n - 4 } } , & n & \\geq 5 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} \\widetilde { u } | _ { B _ 1 } = u | _ { B _ 1 } - L \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\N ( T _ r ( n ) , K ^ { ( r ) } _ { s , t } ) & \\leq \\N ( G , K ^ { ( r ) } _ { s , t } ) = \\N ( H , K ^ { ( r ) } _ { s , t } ) + \\sum _ { i = \\ell + 1 } ^ n d _ { G _ i } ( v _ i , K ^ { ( r ) } _ { s , t } ) \\\\ & \\leq \\N ( T _ r ( \\ell ) , K ^ { ( r ) } _ { s , t } ) + \\sum _ { i = \\ell + 1 } ^ n \\delta _ i + \\mu \\cdot o ( \\ell ^ { ( r - 1 ) s + t } ) - ( n - \\ell ) \\\\ & = \\N ( T _ r ( n ) , K ^ { ( r ) } _ { s , t } ) + \\mu \\cdot o ( \\ell ^ { ( r - 1 ) s + t } ) - ( n - \\ell ) . \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) = \\mathbb { E } _ { x , k } ^ { \\epsilon , \\hat { v } } \\Bigl \\{ \\psi _ { \\zeta ^ { \\epsilon } ( \\tau _ D ^ { \\epsilon , \\hat { v } } ) } ^ { \\epsilon , \\hat { v } } ( X ^ { \\epsilon , \\hat { v } } ( \\tau _ { D } ^ { \\epsilon , \\hat { v } } ) ) \\Bigr \\} , \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} h ^ * \\mathcal { G } _ { \\varphi } ( M ' , \\varphi _ * D ' ) & \\cong \\mathcal { G } _ { \\Phi } ( h ^ * M ' , h ^ * \\varphi _ * D ' ) = \\mathcal { G } _ { \\Phi } ( ( h ^ * \\varphi _ * M ) _ { > \\phi ( s ) } , h ^ * \\varphi _ * D ' ) \\\\ & = \\mathcal { G } _ { \\Phi } ( ( \\Phi _ * H ^ * M ) _ { > \\phi ( s ) } , \\Phi _ * H ^ * D ' ) = \\Phi _ * ( ( H ^ * M ) _ { > s } , H ^ * D ' ) , \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} Z _ { G _ { \\alpha } } ( T ) = Z _ { G _ { \\mathrm { r e s } ( \\alpha ) } } ( T ) = T \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{align*} \\begin{cases} - u _ 1 '' = t \\mu _ 1 ( 1 - u _ 1 ) u _ 1 + ( 1 - t ) \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - ( \\alpha u _ { 1 } - d u _ 2 ) ^ + \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ + - k \\omega u _ 1 u _ 2 \\\\ - d u _ 2 '' = t \\mu _ 2 ( 1 - u _ 2 ) u _ 2 + ( 1 - t ) \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d - ( \\alpha u _ { 1 } - d u _ 2 ) ^ - \\right ) ( \\alpha u _ { 1 } - d u _ 2 ) ^ - - \\alpha k \\omega u _ 1 u _ 2 \\end{cases} \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\mathbb C \\otimes T ( \\mathcal S ) = H ^ { 1 , 0 } ( \\mathcal S ) \\oplus H ^ { 0 , 1 } ( \\mathcal S ) \\oplus \\left ( \\mathbb C \\otimes \\mathcal V ( \\mathcal S ) \\right ) . \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\left < d w ( s ) , [ S ( t - s ) G ( s , \\varphi ( s ) ) ] ^ { \\star } x ^ \\star \\right > _ { H } = \\left < \\tilde { Z } ( t ) , x ^ \\star \\right > _ { E , E ^ \\star } . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} \\cdots = \\Pr { b ^ { - 1 } \\leq X < 1 } = \\Pr { 1 \\leq X < b } = \\Pr { b \\leq X < b ^ 2 } = \\cdots . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} N ( X , 2 r ) ^ { 1 + \\delta } < N ( X - X , 2 r ) \\le 4 N ( D , 4 \\times 2 r ) \\le 4 C _ \\varepsilon \\left ( \\frac { r \\sqrt { 2 i - 1 } } { 4 \\times 2 r } \\right ) ^ { s + \\varepsilon } = 4 C _ \\varepsilon \\left ( \\frac { \\sqrt { 2 i - 1 } } { 8 } \\right ) ^ { s + \\varepsilon } \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} \\left [ ( z \\partial _ { t _ 1 } ) ^ { N + 1 } - e ^ { t _ 1 } \\right ] j ^ \\textnormal { c o h } ( t _ 1 , z ) = 0 \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\bold { D } = \\left [ \\begin{array} { c | c } 4 & 3 \\\\ \\hline 7 & 6 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} a _ { j ( 2 n + 1 ) } = 1 , \\ , a _ { ( n + j ) ( 2 n + 1 ) } = i , \\ , a _ { ( 2 n + 1 ) j } = - 1 , \\ , a _ { ( 2 n + 1 ) ( n + j ) } = - i ; 0 . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\sum _ { n = M + 1 } ^ { M + q } \\min \\left ( L , \\frac { 1 } { \\| n \\omega \\| } \\right ) \\lesssim L + q \\log q . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} t _ n \\le t _ 1 + \\sum _ { k = 1 } ^ n 4 ^ { 3 k - 4 } / ( c ^ 2 _ 2 ) \\le t _ 1 + 4 ^ { 3 n - 3 } / ( c ^ 2 _ 2 ) \\ , . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\displaystyle X _ { 1 , x } = \\left \\{ \\begin{array} { r c } T _ 1 ^ { v _ 0 } , & \\mbox { s e } x = v _ 0 , \\\\ 0 , & \\mbox { s e } x \\neq v _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\dfrac { \\| h ( x ) \\| } { \\| g ( x ) \\| } = \\dfrac { \\| x ^ T B x + d ^ T x \\| } { \\| g ( x ) \\| } \\le \\rho ( B ) \\dfrac { \\| x \\| ^ 2 } { \\| g ( x ) \\| } + \\| d \\| _ { 1 } \\ , \\dfrac { \\| x \\| _ { \\infty } } { \\| g ( x ) \\| } . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\| \\chi _ k u \\| _ { L ^ 2 ( \\mathbb R ; L ^ \\frac { 2 n } { n - 2 } ( \\Omega ) ) } \\le C \\| f \\| _ { L ^ 2 ( \\Omega ) } , k = 1 , \\cdots , N . \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} X _ 1 ^ { ( 1 , 1 ) } = \\left ( \\begin{array} { c c c c } 0 & 0 . 4 2 0 4 & 0 . 1 7 5 9 & 0 . 6 7 7 0 \\\\ 0 . 7 3 4 3 & 0 . 3 7 9 6 & 0 . 1 7 9 7 & 0 \\\\ 0 . 0 2 7 4 & 0 & 0 . 5 7 9 1 & 0 . 0 0 6 9 \\\\ 0 . 1 3 3 4 & 0 . 0 5 8 0 & 0 . 6 7 1 9 & 0 . 6 4 0 3 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\begin{array} { l l } d _ Q ( 1 4 , 6 ) = 6 7 , & d _ Q ( 1 5 , 7 ) = 6 7 , \\\\ d _ Q ( 1 7 , 6 ) = 8 9 , & d _ Q ( 1 7 , 7 ) = 7 8 , \\\\ d _ Q ( 1 9 , 7 ) = 8 9 , & d _ Q ( 2 0 , 7 ) = 9 1 0 . \\end{array} \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} X ( \\alpha ) = : \\begin{pmatrix} 1 + \\alpha & \\cos ( \\frac { \\theta } { 3 } ) & x ( \\alpha ) & \\cos ( \\theta ) \\cr \\cos ( \\frac { \\theta } { 3 } ) & 1 + \\alpha & \\cos ( \\frac { \\theta } { 3 } ) & x ( \\alpha ) \\cr x ( \\alpha ) & \\cos ( \\frac { \\theta } { 3 } ) & 1 + \\alpha & \\cos ( \\frac { \\theta } { 3 } ) \\cr \\cos ( \\theta ) & x ( \\alpha ) & \\cos ( \\frac { \\theta } { 3 } ) & 1 + \\alpha \\end{pmatrix} . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\frac { 1 } { t } = \\sum _ { i = 1 } ^ m \\frac 1 { t _ i } = \\frac { 1 } { q _ 1 } + \\sum _ { i = 2 } ^ m \\frac 1 { p _ i } \\ge \\sum _ { i = 1 } ^ m \\frac { 1 } { q _ i } = \\frac 1 q > \\frac 1 { r _ { m + 1 } ' } , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} m ^ { \\alpha + 2 } = \\frac { \\alpha + 2 } { 2 } \\| g \\| ^ 2 _ { \\dot { H } ^ s } , M ^ 2 = \\| g \\| ^ 2 _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} - u _ i \\cdot u _ j & = \\cosh ( t _ i - t _ j ) + \\frac 1 2 ( s _ i - s _ j ) ^ 2 e ^ { t _ i + t _ j } + ( \\bar \\psi _ i - \\bar \\psi _ j ) ( \\psi _ i - \\psi _ j ) e ^ { t _ i + t _ j } \\\\ z _ i & = \\cosh t _ i + ( \\frac 1 2 s _ i ^ 2 + \\bar \\psi _ i \\psi _ i ) e ^ { t _ i } . \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} X _ { \\tau + h } = g ( \\tau + h ) + ( K * d Z ) _ { \\tau + h } = g ( \\tau + h ) + ( \\Delta _ h K * d Z ) _ \\tau + \\int _ 0 ^ h K ( h - s ) d Z _ { \\tau + s } , h \\geq 0 . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\begin{aligned} N _ { f ( \\xi , x ) } = \\begin{cases} \\gamma + 1 & \\\\ \\gamma & \\\\ \\gamma + 2 & . \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\phi ( \\langle e , h \\rangle ) = h \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\epsilon = 0 . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\# \\mathcal { B } ( e ) = N _ 1 ( m _ e ) + \\sum _ { i = 2 } ^ { C _ 1 ( m _ e ) } \\ , \\left ( N _ 1 ( v _ i ) + \\sum _ { j = 1 } ^ { C _ 1 ( v _ i ) } \\ , N _ 1 ( v _ { i , j } ) \\right ) . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} g ( x ) : = \\inf _ { y \\in E } \\left \\{ f ( y ) + \\langle G ( y ) , x - y \\rangle + \\varphi _ { y } ( x ) \\right \\} \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} s _ { 1 } & = \\phi ^ { 1 } ( \\zeta ) = \\rho ( \\zeta ) , s _ { 2 } = \\phi ^ { 2 } ( \\zeta ) = I m ( \\rho _ { \\zeta } \\cdot ( \\zeta - \\zeta ^ { \\ast } ) ) , \\\\ \\left ( t _ { 2 k - 1 } , t _ { 2 k } \\right ) & = \\left ( \\phi ^ { 2 k - 1 } ( z , \\zeta ) , \\phi ^ { 2 k } ( z , \\zeta ) \\right ) = \\left ( R e ( \\zeta _ { k } - \\zeta ^ { \\ast } _ { k } ) , \\ , I m ( \\zeta _ { k } - \\zeta ^ { \\ast } _ { k } ) \\right ) , k = 2 , \\dots , n . \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\mathbf { C l } ( V ) = \\mathbf { C l } ^ - ( V ) \\oplus \\mathbf { C l } ^ + ( V ) , \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} b ( v ) & = \\int _ 0 ^ \\omega \\dot \\gamma ^ j ( t ) \\frac { d } { d t } \\Big ( \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z _ * ) v ^ k ( t ) \\Big ) d t \\\\ & + \\int _ 0 ^ \\omega \\gamma ^ j ( t ) \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z _ * ) v ^ k ( t ) d t \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ { \\alpha / 2 } u ( x ) + \\rho u ( x ) = f ( x ) , x \\in \\mathbb { R } ^ d ; \\\\ & u ( x ) \\to 0 , { \\rm a s } \\ ; \\ ; | x | \\to \\infty , \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} w ^ { \\sigma } ( x , t ) : = u ( x + \\rho , t + \\tau - \\sigma ) \\preceq u ( x , t ) \\ \\ { \\rm f o r \\ a l l } \\ \\ x \\in \\R ^ N , \\ y \\in \\R . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\phi ( x ) \\geq & \\ \\phi [ \\tau f ( \\lambda ( A ) ) + ( 1 - \\tau ) f ( \\lambda ( B ) ) ] \\\\ \\geq & \\ \\tau \\phi [ f ( \\lambda ( A ) ) ] + ( 1 - \\tau ) \\phi [ f ( \\lambda ( B ) ) ] \\\\ = & \\ \\tau \\phi [ f ( A ) ] + ( 1 - \\tau ) \\phi [ f ( B ) ] . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\mu ( I _ { n _ k } ) = \\prod _ { j = 1 } ^ k \\nu ( I _ { n _ j - n _ { j - 1 } - 1 } ( a _ { n _ { j - 1 } + 1 } , \\cdots , a _ { n _ j - 1 } ) ) . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} | | \\gamma F _ i ( m ) - F _ i ( \\gamma m ) | | & \\leq | | \\gamma F ( m ) - \\gamma F _ i ( m ) | | + | | \\gamma F ( m ) - F _ i ( \\gamma m ) | | \\\\ & = | | F ( m ) - F _ i ( m ) | | + | | F ( \\gamma m ) - F _ i ( \\gamma m ) | | \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} P ( T > t + s | T > s ) = \\frac { P ( T > t + s ) } { P ( T > s ) } \\geq \\frac { P ( T > 2 s ) } { P ( T > s ) } = \\frac { L ( 2 s ) } { L ( s ) } \\left ( \\frac { 1 } { 2 } \\right ) ^ { \\alpha } . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\nabla _ i = \\nabla _ { X _ i } , 0 \\le i \\le d , \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} 2 \\sigma ^ 2 x ^ 2 + [ a ^ 2 + 2 \\sigma ^ 2 s ( a + \\eta ) - 1 ] x + \\alpha _ 1 = 0 . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ \\int _ 0 ^ t { \\bigl \\vert u _ k ^ { \\epsilon } ( t ) \\bigr \\vert ^ 2 } d t \\right \\} < \\infty , \\forall t > 0 , k = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} q _ { \\infty } ^ { ( 1 ) } = \\frac { ( - 1 ) ^ { n } 2 B \\exp [ 2 \\i \\varphi ( t ) - \\i \\arg \\hat \\phi ( E _ 0 ) + 2 \\i \\arg \\delta ( E _ 0 , \\xi ) ] } { \\cosh \\left [ 2 B ( x + 4 A t ) + ( 2 n + \\frac 3 2 ) \\ln t + \\ln \\ ( \\frac { 2 \\pi } { n ! \\Gamma ( n + \\frac 3 2 ) } \\cdot \\frac { | \\delta ( E _ 0 ) | ^ 2 ( 1 6 B ^ 2 ) ^ { 2 n + \\frac 3 2 } } { | \\hat \\phi ( E _ 0 ) | \\sqrt { 2 B } } \\ ) \\right ] } + \\mathcal { O } \\ ( \\frac { \\ln t } { t } \\ ) , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( u ; t ) & = 2 ( u ' , u '' ) + 2 ( A ^ { 1 / 2 } u ' , A ^ { 1 / 2 } u ) \\\\ & = 2 ( u ' , u '' + A u ) \\\\ & = - 2 \\| u ' \\| ^ 2 . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\theta _ { Q } ( z ) = \\sum _ { n = 0 } ^ { \\infty } r _ { Q } ( n ) q ^ { n } , q = e ^ { 2 \\pi i z } \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\mu _ \\varepsilon \\| \\bar { w } ' _ \\varepsilon \\| _ { L ^ \\infty ( [ 0 , \\bar { r } _ \\varepsilon ] ) } = o ( 1 ) \\ , . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} T _ w : = T _ { i _ 1 } \\cdots T _ { i _ l } \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} Y = \\bar { V } ^ { I } + \\bar { V } ^ { I I } + \\bar { V } ^ { I I I ' } + Y _ { I I } + \\bar { \\bar { V } } ^ { I } + \\bar { \\bar { V } } ^ { I I } + \\bar { \\bar { V } } ^ { I I I } \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} [ L ^ { q _ 1 } ( [ 0 , T ] , L ^ p _ x ) , L ^ { q _ 2 } ( [ 0 , T ] , L ^ p _ x ) ] _ { \\theta , 1 } = L ^ { q , 1 } ( [ 0 , T ] , L ^ p _ x ) , \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { - f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 3 ( n ) } \\in C \\right ) \\leq - m = - \\inf _ { x \\in C } I _ 3 ( x ) , \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\rho ) = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } \\mathrm { I n d } ^ G _ { P _ b } ( \\boxtimes ^ k _ { i = 1 } \\mathrm { M a n t } _ { M _ b , b ' _ i , \\mu _ { b , i } } ( \\rho _ i ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] . \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} u ( t ) = \\mathcal { I } ^ { \\alpha } [ g ] ( t ) , \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} 0 = & \\ \\rho _ B ^ { 1 , 1 } ( J Z , Z ) - \\tfrac { 1 } { 2 } L _ { \\theta ^ { \\sharp } } g ( Z , Z ) + \\tfrac { 1 } { 2 } L _ { \\N f } g ( Z , Z ) \\\\ = & \\ \\N ^ 2 f ( Z , Z ) \\\\ = & \\ Z ( Z f ) - ( \\N _ Z Z ) f \\\\ = & \\ Z ( Z f ) , \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} x _ { s , t } \\sim \\sum _ { k = 1 } ^ { \\infty } \\sum _ { i _ { 1 } , \\ldots , i _ { k } = 1 } ^ { d } V _ { ( i _ { 1 } , \\ldots , i _ { k } ) } ( x _ { s } ) \\int _ { s < u _ { 1 } < \\cdots < u _ { k } < t } d w _ { u _ { 1 } } ^ { i _ { 1 } } \\cdots d w _ { u _ { k } } ^ { i _ { k } } , \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} p ^ + = ( p ^ + _ 1 , p ^ + _ 2 ) , \\ p ^ - = ( p ^ - _ 1 , p ^ - _ 2 ) \\ \\ ( p ^ - _ 1 < p ^ + _ 1 , \\ p ^ + _ 2 < p ^ - _ 2 ) , \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 + \\alpha ) } _ { k + 1 } ( z / c ) , z ^ j { \\bar z } ^ l \\rangle _ { \\alpha } = \\sum _ { p = 0 } ^ { k + 1 } \\kappa ^ { k + 1 } _ p ( \\alpha ) \\langle ( z / c ) ^ p , z ^ j { \\bar z } ^ l \\rangle _ { \\alpha } = 0 \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\Psi _ k ^ \\prime & : = \\bigl \\{ ( \\bar { A } _ { t + \\tau } , Z _ { t } , W _ { t } ) \\in \\Psi _ { t , t + \\tau } ^ { A _ t ; Z _ t , W _ t } ~ : ~ \\hat { d } ( ( \\bar { A } _ { t + \\tau } , Z _ { t } , W _ { t } ) , ( \\bar { A } _ { t + \\tau } ^ k , \\bar { Z } _ { t } ^ { k } , \\bar { W } _ { t } ^ { k } ) ) < \\epsilon \\bigr \\} . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} & C \\bullet X + c _ j x _ j = - 1 \\\\ & A _ i \\bullet X + a _ { i j } x _ j \\le 0 \\ \\ \\forall \\ i = 1 , \\ldots , m \\\\ & X , \\ X _ { k k } \\ge 0 \\ \\ \\forall \\ k \\ne j \\\\ & X _ { j j } , \\ x _ j \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} y _ t = \\cos t \\ ; ( \\cos t , \\sin t ) , z _ t = ( 1 , \\cos t \\ , \\sin t ) . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\begin{aligned} u _ 0 ( x ) = \\int _ { 0 } ^ { x } \\int _ { 0 } ^ { 1 } \\frac { j } { m ( s , y ) } d y d s - P x + P - \\int _ 0 ^ 1 \\int _ { 0 } ^ { z } \\int _ { 0 } ^ { 1 } \\frac { j } { m ( s , y ) } d y d s d z , \\end{aligned} \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\int _ \\Omega v ( t , y ) d y = \\int _ \\Omega v _ 0 ( y ) d y , \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} \\lim _ { C \\to + \\infty } C \\int _ { \\{ u \\leq V _ { \\theta } - C \\} } \\theta _ { u ^ C } ^ k \\wedge \\theta _ { v ^ C } ^ { n - k } = 0 , \\ \\ \\lim _ { C \\to + \\infty } C \\int _ { \\{ v \\leq V _ { \\theta } - C \\} } \\theta _ { u ^ C } ^ k \\wedge \\theta _ { v ^ C } ^ { n - k } = 0 , \\ \\forall k . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} F = \\textrm { c o n v } \\left ( x \\mapsto \\inf _ { y \\in E } \\left \\{ f ( y ) + \\langle G ( y ) , x - y \\rangle + \\frac { M } { 2 } | x - y | ^ 2 \\right \\} \\right ) \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} W _ 0 = & \\tilde X ^ i \\tilde X ^ j \\bar W _ { 0 i 0 j } = \\rho \\\\ W _ i = & \\tilde X ^ j \\tilde X ^ k \\bar W _ { 0 k i j } = \\frac { 1 } { 2 } ( \\beta ^ b - 2 \\underline { \\beta } ^ b ) \\tilde { \\nabla } _ b \\tilde { X } ^ i \\\\ P _ k = & \\frac { 1 } { 1 5 } \\bar W _ { 0 i 0 k } \\tilde X ^ i - \\frac { 1 } { 6 } W _ 0 \\tilde X ^ k = - \\frac { 1 } { 3 0 } ( \\beta ^ a + 2 \\underline { \\beta } ^ a ) \\tilde { \\nabla } _ a \\tilde { X } ^ k - \\frac { 1 } { 1 0 } \\rho \\tilde { X } ^ k \\\\ \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\langle \\tilde u _ j , u _ { j - 1 } \\rangle = \\sqrt { \\frac { \\lambda _ { j - 1 } } { \\lambda _ j } } \\frac { \\tilde \\lambda _ j - \\lambda _ j } { \\tilde \\lambda _ { j } - \\lambda _ { j - 1 } } \\langle \\tilde u _ j , u _ j \\rangle . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\mu _ { f } : = \\left ( \\mu ^ Z + d ^ c f _ 1 + d f _ 2 \\right ) \\otimes Z + \\left ( \\mu ^ W + d ^ c f _ 2 - d f _ 1 \\right ) \\otimes W \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\mathcal { S } _ { 0 , 2 } ( V , V ) & = - \\left ( f \\left ( f + 2 \\frac { \\partial ^ 2 f } { \\partial \\theta _ 1 ^ 2 } + 2 \\frac { \\partial ^ 2 f } { \\partial \\theta _ 2 ^ 2 } \\right ) \\right ) ( J X _ 1 + J X _ 2 ) \\\\ & - 2 \\left ( \\frac { \\partial f } { \\partial \\theta _ 2 } \\right ) ^ 2 J X _ 1 - 2 \\left ( \\frac { \\partial f } { \\partial \\theta _ 1 } \\right ) ^ 2 J X _ 2 . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} d _ i ^ { n e } = \\sqrt { \\frac { B \\sum _ { j \\in \\mathcal { S } / \\{ i \\} } w _ j d _ j ^ { n e } } { w _ i ( c _ i + \\lambda _ i ) } } - \\frac { \\sum _ { j \\in \\mathcal { S } / \\{ i \\} } w _ j d _ j ^ { n e } } { w _ i } , \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} \\sin ( \\theta _ j ) = \\frac { a _ j } { c _ j } \\cos ( \\theta _ j ) = \\frac { b _ j } { c _ j } , \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} ( \\delta _ { \\lambda } ) _ { * } \\tilde { W } = \\lambda \\cdot \\tilde { W } . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\Psi ' _ N ( t ) ~ & = 2 t H ( t ) \\left ( 1 + t ^ 2 + \\varphi _ N ( t ^ 2 ) \\right ) + 2 t ( 1 + g ( t ) ) \\left ( \\frac { t ^ { 2 N } } { N ! } - t ^ 2 \\right ) \\\\ & = 2 t H ( t ) \\varphi _ N ( t ^ 2 ) + 2 t \\left ( 1 + \\frac { t ^ { 2 N } } { N ! } \\right ) ( 1 + g ( t ) ) + g ' ( t ) ( 1 + t ^ 2 ) \\ , . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\lim _ { a \\rightarrow q } \\frac { ( q ^ 2 / a ; q ) _ n } { ( a ; q ) _ n } = \\operatorname { s g n } ( n ) . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} B ( \\{ s _ n \\} , \\{ t _ n \\} , 1 ) = & \\bigcap _ { \\ell = 1 } ^ \\infty \\bigcup _ { \\substack { a _ 1 , \\ldots , a _ \\ell \\\\ a _ i \\in [ s _ i - t _ i , s _ i + t _ i ] } } I _ \\ell ( a _ 1 , \\ldots , a _ \\ell ) \\\\ = & \\bigcap _ { \\ell = 1 } ^ \\infty \\bigcup _ { \\substack { a _ 1 , \\ldots , a _ \\ell \\\\ a _ i \\in [ s _ i - t _ i , s _ i + t _ i ] } } D _ \\ell ( a _ 1 , \\ldots , a _ \\ell ) . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} C \\left ( u , v \\right ) \\coloneqq \\sum _ { \\substack { n _ { 1 } , n _ { 2 } \\in \\mathbb { Z } \\setminus \\left \\{ 0 \\right \\} , \\\\ n _ { 1 } u = n _ { 2 } v } } c _ { n _ { 1 } } c _ { n _ { 2 } } . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\abs { \\phi } _ { \\infty } = \\max _ { i , j } \\abs { \\phi _ { i , j } } . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} k ^ { L , s } _ { \\gamma , \\Sigma } : = \\frac { \\langle \\nabla ^ { \\Sigma , L } _ { \\dot { \\gamma } } { \\dot { \\gamma } } , J _ L ( \\dot { \\gamma } ) \\rangle _ { \\Sigma , L } } { | | \\dot { \\gamma } | | ^ 3 _ { \\Sigma , L } } , \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} U _ { 2 k + 1 , 1 } ( x ; q ) = x ^ 2 q ^ 2 \\overline { U } _ { 2 k + 1 , 2 k } ( x q ; q ) + \\overline { U } _ { 2 k + 1 , 2 k + 2 } ( x q ; q ) . \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\psi ^ { ( \\nu _ 1 , \\widetilde { \\nu _ 1 } ) } + \\cdots + \\psi ^ { ( \\nu _ t , \\widetilde { \\nu _ t } ) } = \\sum _ { ( \\lambda ' , \\mu ' ) } \\chi ^ { ( \\lambda ' , \\mu ' ) } , \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} ( E _ 1 * F ) \\left ( \\frac { x - y } { \\alpha } \\right ) = 1 . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } X _ { q ( t ) } ( Q ) = \\widetilde { X } ( Q ) \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} \\frac { 1 } { | \\Psi _ N | } \\sum _ { n \\in \\Psi _ N } | f ( n ) | ^ 2 \\ge \\sum _ { k = 1 } ^ K \\frac { 1 } { | \\Psi _ N | } \\sum _ { n \\in \\Psi _ N } | 1 _ { \\{ a _ k < | f | \\le b _ k \\} } ( n ) f ( n ) | ^ 2 \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} f ( x , t ) : = \\big [ q ^ \\# ( \\rho , m ) + \\big ( \\tfrac { \\beta } { 2 } | \\mathbf { h } | ^ 2 - \\alpha g ' ( 1 / \\rho ) h ( | \\psi \\circ Y | ^ 2 ) - \\varepsilon u _ x \\big ) \\eta _ m ^ \\# ( \\rho , m ) \\big ] ( x , t ) . \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\alpha ^ { J _ { 0 } } = \\frac { | B ( y _ { 1 } , r ) | } { | B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } } r ) | } \\langle f _ { 1 } \\rangle _ { B ( y _ { 1 } , r ) } . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\lambda ( X \\setminus S ) = 0 , \\mu ( S ) = 0 . \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\sigma _ { \\mathbf { n } _ b ' } ^ 2 = \\frac { 1 } { n _ d } \\mathbb { E } \\left [ \\mathbf { n } _ b ^ H \\mathbf { n } _ b ' \\right ] = \\sigma _ b ^ 2 + \\rho _ d \\sigma _ { \\widetilde { h } _ { a b } } ^ 2 . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} ( \\mathcal { F } ( \\mathcal { F } f ) ) ( \\xi ) = f ( - \\xi ) , \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} S _ { l } ( \\alpha \\sqcup ( h _ { 0 } + \\varepsilon \\cdot \\gamma ) ) = a \\otimes S _ { l } ( h _ { 0 } + \\varepsilon \\cdot \\gamma ) . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\gamma _ k = 0 \\mbox { a n d } \\lim _ { k \\to \\infty } k ^ 3 \\tanh k \\rho _ s \\gamma _ k = \\mu ( \\bar \\sigma - \\tilde \\sigma ) > 0 . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} B \\ , d W ( t ) = \\begin{pmatrix} 0 & 0 & 0 & \\dots \\\\ \\frac { \\sqrt { 2 \\gamma } } { m } & 0 & 0 & \\cdots \\\\ 0 & \\sqrt { 2 \\lambda _ 1 } & 0 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix} \\begin{pmatrix} d W _ 0 \\\\ d W _ 1 \\\\ d W _ 2 \\\\ \\vdots \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ \\frac { \\sqrt { 2 \\gamma } } { m } d W _ 0 ( t ) \\\\ \\sqrt { 2 \\lambda _ 1 } d W _ 1 ( t ) \\\\ \\sqrt { 2 \\lambda _ 2 } d W _ 2 ( t ) \\\\ \\vdots \\end{pmatrix} . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\forall x \\in [ l e p ( G ) , u e p ( G ) ] , \\ G ^ { - 1 } ( G ( x ) + 0 ) = x . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} p _ i = - \\delta _ { i , 0 } Q + ( 1 - q ) ^ i \\sum _ { 0 \\leq j _ 1 < \\cdots < j _ i \\leq N } \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ i } \\prod _ { k \\in \\{ 0 , \\dots , N \\} - \\{ j _ 1 , \\dots , j _ i \\} } ( 1 - \\Lambda _ k ) \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} f ( a _ { i } ) = a _ { i } f ( b _ { i } ) = g ( a _ { i } ) = g ( b _ { i } ) = b _ { i } f ( c _ { i } ) = g ( c _ { i } ) = c _ { i } . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } 9 \\\\ 7 \\\\ 1 \\\\ 4 \\\\ 9 \\\\ 9 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c c c c c } 8 & 0 & 1 & 3 & 9 & 9 \\\\ 0 & 0 & 9 & 5 & 2 & 6 \\\\ 1 & 9 & 4 & 1 & 1 & 8 \\\\ 3 & 5 & 1 & 0 & 8 & 0 \\\\ 9 & 2 & 1 & 8 & 2 & 1 \\\\ 9 & 6 & 8 & 0 & 1 & 8 \\end{array} \\right ] , d = \\left [ \\begin{array} { c } 5 \\\\ 8 \\\\ 6 \\\\ 9 \\\\ 9 \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} G ( m , k ) & = - \\frac { ( 1 + q ^ { m + 2 k - 1 } ) ( q ; q ^ 2 ) _ { m } ( q ^ { 2 k + 1 } ; q ^ 2 ) _ { m - 1 } ^ 2 q ^ { - { m \\choose 2 } - ( 2 m - 1 ) k } } { ( 1 - q ) ( q ; q ) _ { m - 1 } ^ 3 ( - q ; q ) _ { m - 1 } } \\equiv 0 \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } , \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} f _ 1 ( a , b ) + f _ 2 ( a , b ) = f _ 1 ( a , a ^ { - 1 } b ) . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\mathrm { R e } \\left ( e ^ z + ( - 1 ) ^ k \\right ) ^ { - 1 } = \\mathrm { R e } \\left ( e ^ { \\bar { z } } + ( - 1 ) ^ k \\right ) ^ { - 1 } & = \\tfrac { 1 } { 2 } \\left ( \\left ( e ^ z + ( - 1 ) ^ k \\right ) ^ { - 1 } + \\left ( e ^ { \\bar { z } } + ( - 1 ) ^ k \\right ) ^ { - 1 } \\right ) \\\\ & = \\frac { e ^ x \\cos | y | + ( - 1 ) ^ k } { e ^ { 2 x } + 2 ( - 1 ) ^ k e ^ x \\cos | y | + 1 } z = x + y i . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} J _ L ( \\dot { \\gamma } ) = \\frac { l _ L } { l } L ^ { \\frac { 1 } { 2 } } \\omega ( \\dot { \\gamma } ( t ) ) e _ 1 + ( \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 ) e _ 2 . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} x ^ { \\frac { 2 \\alpha - 1 } { M - 1 } } \\geq x ^ { ( 1 - \\alpha ) ( 2 - \\alpha ) } = x ^ { \\beta ( \\beta + 1 ) } . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\frac { d } { d x } \\sigma ^ 2 ( x ) & \\ge \\frac { d } { d x } g _ { 1 1 - } ( x ) - \\frac 1 { 2 ^ { 1 0 } } = \\frac { 2 7 1 5 7 1 9 5 } { 1 4 1 7 1 7 6 } - 4 6 x - \\frac 1 { 2 ^ { 1 0 } } \\\\ & \\ge \\frac { 2 7 1 5 7 1 9 5 } { 1 4 1 7 1 7 6 } - 4 6 c - \\frac 1 { 2 ^ { 1 0 } } = \\frac { 1 2 2 5 9 1 8 6 9 } { 1 2 0 6 3 0 0 2 1 1 2 0 } > 0 \\quad \\hbox { a . e . } \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { n p _ n f _ 4 ( n ) } { a _ c ^ { ( n ) } } = 0 , \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 0 , t _ 1 , t _ 2 , Q ) = \\frac { 1 } { 2 } ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 ) + \\sum _ { d = 1 } ^ \\infty N _ d \\frac { t _ 2 ^ { 3 d - 1 } } { ( 3 d - 1 ) ! } e ^ { d t _ 1 } Q ^ d \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} u _ m ( x , t ) & = U _ m ^ - ( x , t ) \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} c _ m : = \\sqrt { \\frac { 4 \\pi } { C ' m } } , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} J _ t ( t , A _ t ; U , V ) & = \\mathbb { E } \\Bigl [ \\int _ t ^ T l ( X _ s ^ { t , A _ t ; U , V } , U _ s , V _ s ) \\dd s + m ( X _ T ^ { t , A _ t ; U , V } ) \\bigl | \\mathcal { F } _ t \\Bigr ] . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\mathbf { y } } = E \\left [ \\left ( \\mathbf { H } \\mathbf { x } + \\mathbf { n } \\right ) \\left ( \\mathbf { H } \\mathbf { x } + \\mathbf { n } \\right ) ^ H \\right ] = \\sigma _ x ^ 2 \\mathbf { H H } ^ H + \\sigma _ n ^ 2 \\mathbf { I } _ M . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} | D ^ 2 F | : = \\sqrt { \\sum _ { l = 1 } ^ m \\sum _ { i , j = 1 } ^ N f _ { l , u _ i u _ j } ^ 2 } , \\ \\ | \\varphi | ^ 2 : = \\sum _ { l = 1 } ^ m \\varphi _ l ^ 2 . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\| e _ 1 \\| _ { X _ 1 } \\sim \\sup _ { | e _ 2 | _ { X _ 2 } = | e _ 3 | _ { X _ 3 } = 1 } | \\tau ( e _ 1 e _ 2 e _ 3 ) | \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} \\mathcal { I } ^ { \\alpha } [ f ] ( t ) = \\partial _ t ^ { - \\alpha } f ( t ) , t > 0 , \\ ; \\alpha \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} T S P C _ n ^ { ( P ) } \\leq V _ n ^ { ( P ) } + 2 ( N - 1 ) ( s _ n + 8 r _ n ) = \\sum _ { l = 1 } ^ { N } T ^ { ( P ) } _ l + 2 ( N - 1 ) ( s _ n + 8 r _ n ) . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\langle \\mathbf { m } _ { u , s } \\otimes \\bar { \\mathbf { m } } _ { u , t } , \\mathbf { m } _ { v , s ' } \\otimes \\bar { \\mathbf { m } } _ { v , t ' } \\rangle = \\begin{cases} 1 & \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p } ( w ) } \\le C ( [ \\vec w ] _ { A _ { \\vec p , \\vec r } } ) \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { p _ i } ( w _ i ) } \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} y ( \\xi ) = x ( \\xi ) z ( \\xi ) , \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\underset { v \\rightarrow + \\infty } { \\lim } \\ \\ \\underset { k \\ge 1 } { \\sup } \\ \\ \\int \\limits _ { \\left \\| \\hat { z } _ k \\right \\| _ 2 > v } \\left \\| \\hat { \\mathbf { z } } _ k \\right \\| _ 2 ^ 2 p ( \\hat { \\mathbf { z } } _ k ) d \\hat { \\mathbf { z } } _ k = 0 . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\begin{cases} \\dd x _ s ^ { t , A _ t ; U , V } = f ( X _ s ^ { t , A _ t ; U , V } , U _ s , V _ s ) \\dd s + \\sigma ( X _ s ^ { t , A _ t ; U , V } , U _ s , V _ s ) \\dd B _ s , ~ s \\in ( t , T ] \\\\ X _ t ^ { t , A _ t ; U , V } = A _ t \\in \\Lambda _ t ^ n , \\end{cases} \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} | \\delta _ \\lambda ( p ) | = \\lambda | p | , \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\binom { n } { k } _ q = \\binom { n - 1 } { k - 1 } _ q + q ^ k \\binom { n - 1 } { k } _ q , \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} 0 = \\xi _ \\beta \\left ( 2 \\lambda - 1 \\right ) \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { r e s } ( \\alpha ) } = \\sigma _ { \\alpha _ n } \\circ . . . \\circ \\sigma _ { \\alpha _ 1 } \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} g ( \\xi _ 1 ) = \\frac { 1 } { 1 - 2 \\xi _ 1 } u _ 1 + \\frac { 1 } { 2 - 3 \\xi _ 1 } u _ 2 + \\cdots + \\frac { 1 } { s - ( s + 1 ) \\xi _ 1 } u _ s , \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} \\mathcal { N } : = p ^ { \\ast } \\mathcal { E } ^ { a } \\otimes \\mathcal { R } _ 1 \\oplus \\mathcal { R } _ 2 \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\int ( \\mu ( s , x ) - \\nu ( s , x ) ) ^ 2 d [ B ] _ s \\right ] = \\mathbb { E } [ ( M ( x ) - M ( x ) ) ^ 2 ] = 0 . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { U } _ { 2 k , 2 a } ( n ) q ^ n = \\frac { ( - q ^ 2 ; q ^ 2 ) ^ 2 _ \\infty ( q ^ { 2 a } , q ^ { 4 k - 2 a } , q ^ { 4 k } ; q ^ { 4 k } ) _ \\infty } { ( q ^ { 2 } ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} \\begin{gathered} \\| u \\| _ { L ^ p ( B _ 1 ) } ^ p + ( 1 - s ) [ u ] _ { W ^ { s , p } ( B _ 1 ) } ^ p \\le M , \\\\ | B _ 1 \\cap \\{ u \\le 0 \\} | \\ge \\gamma | B _ 1 | \\mbox { a n d } | B _ 1 \\cap \\{ u \\ge 1 \\} | \\ge \\gamma | B _ 1 | . \\end{gathered} \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\{ \\varphi _ { \\lambda , i } ( x ) = x / n _ { i } + \\lambda t _ { i } / n _ { i } \\} _ { \\lambda = 0 } ^ { n _ { i } - 1 } , \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal U } { \\partial \\beta } = \\frac { w _ c } { \\beta + \\frac { \\sum _ { i = 1 } ^ { n } b _ i } { \\sum _ { i = 1 } ^ { n } a _ i } } + \\frac { w _ d } { \\beta + \\frac { \\sum _ { j = 1 } ^ { m } ( v _ j - s _ j ) } { \\sum _ { j = 1 } ^ { m } ( u _ j + s _ j ) } } . \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} \\prod _ { \\beta \\neq \\beta _ L } ^ { } ( \\beta - \\beta _ L ) = \\prod _ { \\beta \\in \\mathbb { F } _ q ^ * } ^ { } \\beta = 1 . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\dot B ^ s _ { p , q } ( \\mathcal { H } ) : = \\{ f \\in \\mathcal Z ' ( \\mathcal { H } ) : \\| f \\| _ { \\dot B ^ s _ { p , q } ( \\mathcal { H } ) } < \\infty \\} \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} x = \\lim _ { n \\to \\infty } T _ { a _ 1 } \\circ \\cdots \\circ T _ { a _ n } ( 1 ) = \\frac { \\displaystyle 1 } { \\displaystyle a _ 1 + \\frac { \\displaystyle 1 } { \\displaystyle a _ 2 + \\frac { \\displaystyle 1 } { \\displaystyle a _ 3 + \\ddots } } } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} g ( x ) = g ( t _ 0 , \\cdots , t _ s ; x ) , \\ f ( t ; x ) = f ^ { ( n ) } ( t _ 0 , \\cdots , t _ s , t ; x ) \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\hat l _ { a b } = - \\frac { 1 } { 3 } r ^ 3 \\alpha _ { a b } + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} w '' + \\left ( \\sum _ { j = 0 } ^ 2 \\frac { 1 - \\alpha _ j } { z - a _ j } \\right ) w ' + \\frac { A z - q } { ( z - a _ 0 ) ( z - a _ 1 ) ( z - a _ 2 ) } w = 0 , \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\underbrace { \\tfrac 1 2 n ( n - 3 ) } _ { k } + \\underbrace { n ( n - 2 ) } _ { c } + \\underbrace { \\tfrac 1 2 n ( n - 1 ) } _ { e } = 2 n ( n - 2 ) \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} \\max \\Big \\{ f ( x , y ) : \\ , g ( x , y ) = 1 \\} = C _ p , \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\hat T ( y ) ( w ) : = \\displaystyle \\sup _ { x \\in S } \\{ r _ x ( w ) - K d ( x , y ) \\} , w \\in \\Omega \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} f _ n : ( 0 , a ) ^ 2 \\to \\R ^ 2 , f _ n ( x , y ) : = \\frac { 1 } { \\sqrt { n } } ( 1 - y ) ^ n ( \\sin n x , \\cos n x ) , \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} & L \\left ( \\frac { 2 \\eta } { \\sigma ^ 2 } \\right ) = \\\\ & \\begin{cases} \\frac { - 2 ( 2 \\eta - a ) ( \\eta - \\eta _ 2 ) ( \\eta - \\eta _ 2 ' ) } { ( a - 2 \\eta ) ^ 2 - 1 } , & \\eta \\in \\left ( 0 , \\frac { a - 1 } { 2 } \\right ) \\cup \\left ( \\frac { a + 1 } { 2 } , + \\infty \\right ) \\\\ \\frac { \\eta } { 1 - ( a - 2 \\eta ) ^ 2 } , & \\eta \\in \\left ( \\frac { a - 1 } { 2 } , \\frac { a + 1 } { 2 } \\right ) , \\end{cases} \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathbf E ( M , p , q ; [ r _ 1 , \\dots , r _ p ] , [ s _ 1 , \\dots , s _ q ] ) \\odot \\mathbf E ( N , p , q ; [ k _ 1 , \\dots , k _ p ] , [ \\ell _ 1 , \\dots , \\ell _ q ] ) = \\\\ \\mathbf E ( M N , p , q ; [ ( k _ 1 - 1 ) M + r _ 1 , \\dots , ( k _ p - 1 ) M + r _ p ] , [ ( \\ell _ 1 - 1 ) M + s _ 1 , \\dots , ( \\ell _ q - 1 ) M + s _ q ) ] . \\end{array} \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} a _ 1 = 1 & \\Rightarrow b _ 1 = * , \\ , c _ 1 = 0 , \\\\ a _ 2 = 0 & \\Rightarrow b _ 2 = 0 , \\ , c _ 2 = * , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} e _ k ( i d : \\ell ^ n _ p ( \\mathbb { R } ) \\rightarrow \\ell ^ n _ q ( \\mathbb { R } ) ) \\sim \\begin{cases} 1 & 1 \\leq k \\leq \\log _ 2 n , \\\\ \\displaystyle \\biggl ( \\frac { \\log _ 2 ( 1 + n / k ) } { k } \\biggr ) ^ { \\frac { 1 } { p } - \\frac { 1 } { q } } & \\log _ 2 n \\leq k \\leq n , \\\\ [ 1 0 p t ] \\displaystyle 2 ^ { - \\frac { k - 1 } { n } } n ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } & n \\leq k . \\end{cases} \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} I _ 1 ^ 2 ( j , | \\gamma | ) : = \\int _ { \\R ^ n } \\dfrac { | \\partial _ t ^ j ( e ^ { \\lambda _ { + } t } - e ^ { \\lambda _ { - } t } ) | ^ 2 } { | \\lambda _ { + } - \\lambda _ { - } | ^ 2 } \\ , | \\xi | ^ { 2 | \\gamma | } \\chi ( \\xi ) ^ 2 d \\xi \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} B _ { n } ( f _ { 1 } , f _ { 2 } ) : & = \\frac { f _ { 1 } ( x ) f _ { 2 } ( y ) - f _ { 1 } ( y ) f _ { 2 } ( x ) } { x - y } = \\sum _ { i , j = 1 } ^ { n } \\alpha _ { i j } x ^ { n - i } y ^ { n - j } \\in F [ x , y ] , \\\\ M _ { n } ( f _ { 1 } , f _ { 2 } ) : & = ( \\alpha _ { i j } ) _ { 1 \\leq i , j \\leq n } . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\ell ( g _ 1 , g _ 2 ) \\ell ( g _ 1 g _ 2 , g _ 3 ) = \\ , \\ , ^ { g _ 1 } \\ell ( g _ 2 , g _ 3 ) \\ell ( g _ 1 , g _ 2 g _ 3 ) . \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\varepsilon ^ { \\frac { n } { 2 } } \\int _ { \\mathbb { R } ^ n } T ( P w _ { \\varepsilon \\alpha } ) \\overline { Q ( x ) } w _ { \\varepsilon \\beta } d x = ( \\pi / \\beta ) ^ { n / 2 } \\int _ { \\mathbb { T } ^ n } ( A P ) ( x ) \\overline { Q ( x ) } d x . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} ( u ( y ) - u ( x ) ) w ( y ) & = w ( x ) \\nabla u ( x ) \\cdot ( y - x ) + \\frac { w ( x ) } { 2 } \\nabla ^ 2 u ( x ) \\cdot ( y - x , y - x ) \\\\ & + [ \\nabla u ( x ) \\cdot ( y - x ) ] [ \\nabla w ( x ) \\cdot ( y - x ) ] + E ( r ) r ^ 2 . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} m ( x ) : = \\sup _ { z \\in E } \\lbrace f ( z ) + \\langle G ( z ) , x - z \\rangle \\rbrace \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} L ( \\mathtt A ) = \\sum _ { t : \\delta ^ { * } ( t ) \\cap F \\neq \\emptyset } t . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} ( L \\lambda _ { 2 k + 1 } ) ^ \\ast ( x _ 2 ) = x _ 2 . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ n \\frac { 1 } { \\lambda _ i - \\mu ^ * } & = N ^ { 1 - \\delta } . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} \\mathcal { L } f _ 0 = \\mu f _ 0 + \\nu Q + \\omega x Q \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\Gamma \\cap x W = \\Gamma \\cap \\left ( x W x ^ { - 1 } \\right ) x = \\Gamma \\cap W x ~ , \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} \\delta _ 0 ^ s ( \\ell ) + \\ell \\delta _ 1 ^ s ( \\ell ) + \\frac { \\ell ^ 2 } { 2 ( \\ell ^ 2 - 1 ) } = \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { \\pi ( x ) } \\sum _ { \\substack { p \\le x \\\\ p \\textrm { s p l i t s i n } K } } N _ { p } ( E [ \\ell ] ) . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\begin{aligned} & \\max _ { x } & & q ^ { - } x ( A x ) ^ { - } p . \\end{aligned} \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( J - \\lambda I ) \\Phi = J + F - \\lambda I \\ , . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} Z _ \\nu ( [ a , b ] ) = \\left \\{ \\mathcal { A } ( \\eta _ j ) \\right \\} \\subset ( a , b ) \\ ; , \\ ; \\ ; \\eta _ j = \\cos \\left ( \\frac { ( 2 j + 1 ) \\pi } { 2 ( \\nu + 1 ) } \\right ) \\ ; , \\ ; \\ ; 0 \\leq j \\leq \\nu \\ ; , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\sqrt { - L } f ( \\mathbf x ) = \\int _ 0 ^ { 1 } t ^ { - 3 / 2 } \\int _ 0 ^ t L H _ s f ( \\mathbf x ) d s d t + \\int _ { 1 } ^ { \\infty } t ^ { - 3 / 2 } ( H _ t f ( \\mathbf x ) - f ( \\mathbf x ) ) d t . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\omega = \\mathcal { W } ^ { B D } [ y ; d ] \\Omega _ T \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} 0 = \\zeta _ { l _ 0 , t } & - D _ { l _ 0 } \\Delta \\zeta _ { l _ 0 } = f _ { l _ 0 } ( v ) - f _ { l _ 0 } ( u ) \\\\ & > f _ { l _ 0 } ( \\cdots , v _ { j _ 0 - 1 } , u _ { j _ 0 } , v _ { j _ 0 + 1 } , \\cdots ) - f _ { l _ 0 } ( u ) \\\\ & \\geq f _ { l _ 0 } ( \\cdots , u _ { l _ 0 - 1 } , v _ { l _ 0 } , u _ { l _ 0 + 1 } , \\cdots ) - f _ { l _ 0 } ( u ) \\\\ & \\geq - M \\zeta _ { l _ 0 } = 0 \\ \\ { \\rm a t } \\ \\ ( x , t ) = ( x _ 0 , t _ 0 ) . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\theta ( v , \\nabla v ) \\geq a _ { \\kappa } ( 2 R ) h _ { i j } ( v ) \\nabla v ^ i \\cdot \\nabla v ^ j = a _ { \\kappa } ( 2 R ) | \\nabla v | ^ 2 , \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\imath _ V ( X + \\nabla _ A ( Y ) ) = \\imath _ V \\left ( X + d Y + A Y ' - A ' Y \\right ) = Y ' + \\imath _ V ( A ) Y ' - \\imath _ V ( A ' ) Y = 0 . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{gather*} p _ 1 + p _ 2 + \\dots + p _ l + r = p ; \\\\ q _ 1 + q _ 2 + \\dots + q _ l + s = q ; \\\\ p _ i = 0 \\ \\Rightarrow \\ q _ i = 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j = 1 ; \\\\ r = p \\Rightarrow \\ s = q - 1 ; s = q \\Rightarrow \\ r = p - 1 ; \\\\ r = 0 \\Rightarrow \\ s \\leq 1 ; s = 0 \\Rightarrow \\ r \\leq 1 ; \\\\ r > 0 \\Rightarrow \\ n _ p = 0 ; s > 0 \\Rightarrow \\ n _ q = 0 ; \\\\ n _ p = 1 \\Rightarrow \\ r = 0 , p _ 1 \\geq 1 ; n _ q = 1 \\Rightarrow \\ s = 0 , q _ 1 \\geq 1 . \\end{gather*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\dot { x } & = s - d x - f ( x , y , v ) v , \\\\ \\dot { y } & = f ( x , y , v ) v - a y - p y z , \\\\ \\dot { v } & = k y - u v , \\\\ \\dot { z } & = c y z - b z , \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} 0 < x _ 0 = \\frac { ( 1 - c _ 2 ) - \\sqrt { ( 1 - c _ 2 ) ^ 2 - 4 c _ 1 a _ 0 } } { 2 c _ 1 } = \\frac { 1 } { 2 c _ 1 } \\frac { 4 c _ 1 a _ 0 } { ( 1 - c _ 2 ) + \\sqrt { ( 1 - c _ 2 ) ^ 2 - 4 c _ 1 a _ 0 } } < \\frac { 2 } { 1 - c _ 2 } a _ 0 , \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } { 2 k \\choose k } \\frac { 1 } { k } \\equiv 0 \\pmod { p ^ 2 } \\quad . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} Z & : = F \\times ( L ^ 2 ( \\rho ^ 2 ; Q ) ) ^ 2 \\times ( H ^ 3 ( I ) \\cap H ^ 1 _ 0 ( I ) ) \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\begin{cases} \\frac { | P + \\nabla u _ \\epsilon ( x ) | ^ 2 } { 2 } + V ( x , \\frac { x } { \\epsilon } ) = \\ln ( m _ \\epsilon ( x ) ) + \\overline { H } _ \\epsilon ( P ) , \\ & \\ \\mathbb { T } ^ d , \\\\ - \\div ( m _ \\epsilon ( x ) ( P + \\nabla u _ \\epsilon ( x ) ) ) = 0 , \\ & \\ \\mathbb { T } ^ d , \\\\ \\int _ { \\mathbb { T } ^ d } u _ \\epsilon ( x ) d x = 0 , \\ , \\int _ { \\mathbb { T } ^ d } m _ \\epsilon ( x ) d x = 1 . \\end{cases} \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} \\phi ( x ) : = \\sum \\limits _ { k = 1 } ^ K \\frac { \\kappa _ k } { | x - R _ k | } . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { ( q ^ 2 ; q ^ 2 ) _ n q ^ { 2 n ^ 2 } } { ( - q ^ 2 ; q ^ 2 ) _ n ( q ^ 2 ; q ^ 4 ) _ n ( 1 - q ^ { 2 n + 1 } ) } = \\frac { 1 } { 2 } \\sum _ { n \\ge 0 } q ^ { 3 n ^ 2 / 2 } ( 1 + q ^ { 2 n + 1 } ) ( 1 + q ^ { n + 1 / 2 } ) \\sum _ { | j | \\le n } q ^ { j ^ 2 / 2 } \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\underbrace { ( z \\diamond _ q \\cdots \\diamond _ q z ) } _ { k - \\mbox { t i m e s } } = z ^ k , \\underbrace { ( \\bar { z } \\diamond _ q \\cdots \\diamond _ q \\bar { z } ) } _ { j - \\mbox { t i m e s } } = \\bar { z } ^ j \\mbox { m o d } \\ , \\bar { z } ^ q \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} K = \\sup _ { 1 \\leq t < \\infty } \\left \\{ - \\int _ { 1 } ^ { t } \\frac { \\left ( R _ { g ( s ) } - s ^ { 2 } \\overline { R } \\right ) _ { * } } { 4 } \\exp \\left ( \\int _ { 1 } ^ { s } \\frac { \\tau ( | M | ^ { * } ) ^ { 2 } } { 2 } d \\tau \\right ) d s \\right \\} < \\infty . \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} I I I & = - \\frac 1 2 h \\int W ' ( h x ) Q _ { a , c } ( x ) ^ 2 \\ , d x \\\\ & = - \\frac { 1 } { 2 } h \\int ( W ' ( h a ) + h W '' ( h a ) ( x - a ) + O ( h ^ 2 ) ( x - a ) ^ 2 ) Q _ { a , c } ^ 2 ( x ) d x \\\\ & = - \\frac { 1 } { 2 } h W ' ( h a ) \\int Q _ { a , c } ^ 2 ( x ) d x + O ( h ^ 3 ) \\\\ & = - 4 \\pi c h W ' ( h a ) + O ( h ^ 3 ) \\ . \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\mu \\left ( [ - ( M - 1 ) , M - 1 ] \\right ) = 1 \\ , . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} t \\in I = I _ 1 \\times \\dots \\times I _ { d _ 1 } \\ ; , \\ ; \\ ; \\theta \\in \\Theta = \\Theta _ 1 \\times \\dots \\times \\Theta _ { d _ 2 + d _ 3 } \\ ; , \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\left ( \\mathcal { D } _ 1 ^ \\theta f \\right ) ( x ) = \\left ( D _ 1 ^ { k ' _ { \\theta - 1 } } f \\right ) ( x ) , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\Delta ^ 2 u _ 1 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } - \\frac { 2 0 u _ 1 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 , \\\\ \\Delta ^ 2 u _ 2 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } - \\frac { 2 0 u _ 2 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 , \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\sqrt { g ( x - \\theta ) g ( x ) } d x = \\int _ \\theta ^ { 1 / 2 } \\sqrt { g ( x - \\theta ) g ( x ) } d x = 1 - 2 0 \\theta ^ 2 ( 1 - 2 \\theta + 8 \\theta ^ 3 / 5 ) , \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} V _ n ( x ; \\tau _ 1 , \\dots , \\tau _ n ) : = \\{ ( x _ 1 , \\dots , x _ n ) \\mid 0 \\le x _ i \\le \\tau _ i , \\sum _ { i = 1 } ^ n x _ i \\le x \\} \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\bar { r } _ \\varepsilon = \\sup \\left \\{ r > 0 | \\bar { w } _ \\varepsilon | \\le \\bar { \\delta } t _ \\varepsilon [ 0 , r ] \\right \\} \\ , . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} 4 \\pi \\int _ { M } ( \\phi _ { 0 } + \\epsilon ) f _ { \\epsilon } \\omega \\cdot \\nabla ( \\phi _ { 0 } + \\epsilon ) & = 2 \\pi \\int _ { M } f _ { \\epsilon } \\omega \\cdot \\nabla ( \\phi _ { 0 } + \\epsilon ) ^ { 2 } \\\\ & = - 2 \\pi \\int _ { M } ( \\phi _ { 0 } + \\epsilon ) ^ { 2 } \\nabla f _ { \\epsilon } \\cdot \\omega \\ , . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} H & = \\nabla _ { E _ 1 } E _ 1 + \\nabla _ { E _ 2 } E _ 2 = \\frac { 1 } { 2 } \\nabla _ { X _ 1 } X _ 1 + \\frac { 1 } { 2 } \\nabla _ { X _ 2 } X _ 2 \\\\ & = - \\frac { \\sqrt { 2 } } { 2 } ( e ^ { i \\theta _ 1 } , 0 ) - \\frac { \\sqrt { 2 } } { 2 } ( 0 , e ^ { i \\theta _ 2 } ) = \\frac { 1 } { 2 } ( J X _ 1 + J X _ 2 ) = - \\frac { X ^ { \\perp } } { 2 } , \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} & \\frac { \\partial L } { \\partial \\dot z ^ k } ( t , z _ * , \\dot z _ * ) + \\int _ t ^ { \\omega } \\frac { \\partial L } { \\partial z ^ k } ( s , z _ * , \\dot z _ * ) d s \\\\ & = \\dot \\gamma ^ j ( t ) \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z _ * ) + \\int _ t ^ \\omega \\dot \\gamma ^ j ( s ) \\frac { d } { d s } \\Big ( \\frac { \\partial f _ j } { \\partial z ^ k } ( s , z _ * ) \\Big ) d s + \\lambda _ k . \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\mathbf { D } = \\sum _ { i = 1 } ^ d c ( e ^ * _ i ) \\nabla _ { e _ i } , \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} \\dot { x } ( t ) = - [ \\nabla _ { x x } \\mu _ f ( x ( t ) , t ) ] ^ { - 1 } \\nabla _ { t x } \\mu _ f ( x ( t ) , t ) \\ , , \\quad \\mbox { f o r a . e . \\ } t \\in ( 0 , t _ 0 ] \\ , , \\quad \\mbox { w i t h } x ( t _ 0 ) = x _ 0 \\ , , \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} [ u , \\mathcal J v ] _ { - } - \\mathcal J [ u , v ] _ { - } = 0 , \\ \\forall u \\in \\Gamma ( E _ + ) , \\ v \\in \\Gamma ( E _ - ) . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} A u = f \\left ( x , u \\right ) _ { x } \\in \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) u \\in \\mathcal { D } \\left ( A \\right ) . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} \\int _ 0 ^ { { \\rho } _ \\varepsilon } \\exp ( - \\gamma _ \\varepsilon ^ 2 ) r d r = o \\left ( \\frac { \\mu _ \\varepsilon ^ 2 } { \\gamma _ \\varepsilon ^ 4 } \\right ) \\ , . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\Xi ( x , y ) & = \\left | w _ - ( x ) - w _ - ( y ) \\right | ^ 2 - \\left | ( 1 - \\eta ( x ) ) \\left ( w _ - ( x ) - w _ - ( y ) \\right ) - \\left ( \\eta ( x ) - \\eta ( y ) \\right ) w _ - ( y ) \\right | ^ 2 \\\\ & \\ge \\left ( 1 - 2 ( 1 - \\eta ( x ) ) ^ 2 \\right ) \\left | w _ - ( x ) - w _ - ( y ) \\right | ^ 2 - 2 \\left | \\eta ( x ) - \\eta ( y ) \\right | ^ 2 w _ - ( y ) ^ 2 . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\det \\left ( \\lambda \\mathbb { I } - \\mathbb { J } - \\varepsilon \\kappa \\mathbb { D } \\right ) = 0 , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} U _ { 2 k , 2 a } ( x ; q ) = ( - x q ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\tilde { \\pi } : \\Lambda ^ { 3 } E ^ { * } \\rightarrow E ^ { * } \\otimes \\Lambda ^ { 1 , 1 } _ { \\mathbb { H } } E ^ { * } , \\ ( \\tilde { \\pi } \\alpha ) ( u , v , w ) : = \\alpha ( u , v , w ) + \\sum _ { i = 1 } ^ { 3 } \\alpha ( u , \\mathcal J _ { i } v , \\mathcal J _ { i } w ) . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} u _ { l , t t } \\geq & \\sum _ { i , j = 1 } ^ N D _ l ^ { i j } ( x ) u _ { l , t x _ i x _ j } + q _ l ( x ) \\cdot \\nabla u _ { l , t } + f _ { l , u _ l } ( x , u _ 1 , \\cdots , u _ m ) u _ { l , t } \\\\ & { \\rm f o r } \\ \\ x \\in \\R ^ N , \\ t \\in \\R \\ \\ ( l = 1 , 2 , \\cdots , m ) , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} K _ 1 = K _ 2 = \\frac { \\delta + 1 } { 2 } . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} & ( \\tilde { n } - 1 ) ! M ^ { { \\tilde { n } } - 1 } \\frac { v o l ( I _ { N , M } ^ { \\tilde { n } } \\cap \\hat { \\mathfrak { D } } _ { \\tilde { n } } ) } { v o l ( \\hat { \\mathfrak { D } } _ { \\tilde { n } } ) } \\\\ = \\ , & \\sum _ { S } ( - 1 ) ^ { | S | } \\sum _ { 0 \\le K _ i \\le M - 1 \\atop ( 1 \\le i \\le \\tilde { n } ) } \\sum _ { l = 0 } ^ { \\tilde { n } } \\tilde { M } \\left ( \\sum _ { i \\not \\in S } K _ i + \\sum _ { j \\in S } ( K _ j - \\tau _ j ) - M l \\right ) ^ { \\tilde { n } - 1 } , \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} - t ^ { \\frac { N } { 2 } + 1 } ( \\partial _ r \\partial _ { \\theta _ \\alpha } u _ { 1 , 1 } ) ( x , t ) = O ( | \\theta _ \\alpha | ) + O ( t ^ { - 1 } ) \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} v _ n ( x ) = \\sum _ { j = 1 } ^ l V ^ j ( x - x _ n ^ j ) + v ^ l _ n ( x ) , \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\Sigma _ 1 , \\infty } ( e _ 1 , e _ 2 ) = - \\langle e _ 1 , \\nabla _ H ( \\frac { X _ 3 u } { | \\nabla _ H u | } ) \\rangle - \\frac { ( X _ 3 u ) ^ 2 } { l ^ 2 } . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} c _ { \\delta ( c , d ) } = d H ( \\delta ( c , d ) ; - ) = \\Delta ( c ) . \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} A _ { K , M } ^ { ( 1 , 2 ) } \\buildrel \\Delta \\over = \\max \\left \\{ { \\frac { 1 } { { \\left | { 1 - \\frac { 1 } { { M N + K N } } \\left ( { { \\lambda _ { \\cal M } } + { \\lambda _ { \\cal N } } } \\right ) } \\right | } } , \\frac { 1 } { { \\left | { 1 - \\frac { 1 } { { M N + K N } } \\left ( { \\hat \\lambda _ { { \\cal M } , j } ^ { ( K ) } + \\hat \\lambda _ { { \\cal N } , j } ^ { ( K ) } } \\right ) } \\right | } } } \\right \\} , \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} \\mathfrak { D } = \\Delta \\cap { \\mathcal { S } } , \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} g ( x ) = m ( x ) = f ( x ) \\textrm { f o r a l l } x \\in E . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} R _ X = \\C [ x _ 1 , \\ldots , x _ m ] \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} \\frac { \\partial ( \\L ^ { - 1 } ) } { \\partial \\sigma } ( \\sigma , \\tau ) = ( \\gamma ^ { ( 1 ) } ) ' ( \\sigma _ 0 + \\sigma ) + \\tau ( n ^ { ( 1 ) } ) ^ \\prime ( \\sigma _ 0 + \\sigma ) \\ , , \\frac { \\partial ( \\L ^ { - 1 } ) } { \\partial \\tau } ( \\sigma , \\tau ) = n ^ { ( 1 ) } ( \\sigma _ 0 + \\sigma ) \\ , , \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\frac { 1 } { s } = \\sum _ { j = 1 } ^ m \\frac 1 { s _ j } = \\sum _ { j = 1 } ^ { i } \\frac 1 { q _ j } + \\sum _ { j = i + 1 } ^ m \\frac 1 { p _ j } > \\sum _ { j = 1 } ^ m \\frac 1 { p _ j } = \\frac 1 p > \\frac 1 { r _ { m + 1 } ' } . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\mathcal { L } Q = - \\frac 1 2 Q ^ 2 \\ , , \\mathcal { L } Q ^ 2 = - 2 Q - Q ^ 2 \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} S = \\int _ \\Omega G _ { \\bar { x } } ( y ) F ( 4 \\pi G _ { \\bar { x } } ( y ) ) ~ d y \\ , , \\frac { \\gamma _ \\varepsilon ^ { - 3 } B ( \\gamma _ \\varepsilon ) } { \\gamma _ \\varepsilon ^ { - 4 } + | A ( \\gamma _ \\varepsilon ) | } \\not \\to 0 \\ , , \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\phi _ 1 ^ 2 - \\phi _ 1 ^ 1 & = \\omega _ 1 \\tau , \\\\ \\phi _ 2 ^ 2 - \\phi _ 2 ^ 1 & = \\omega _ 2 \\tau . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} \\omega _ { f } ( s ) = \\sup _ { d ( p , \\overline { p } ) \\leq s } | f ( p ) - f ( \\overline { p } ) | . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\frac { v a r _ { P _ { \\alpha } } \\left [ \\chi _ { P _ { \\alpha } , a , e f f } ^ { 1 } ( \\mathbf { V } ; \\mathcal { G } ) \\right ] } { v a r _ { P _ { \\alpha } } \\left [ \\psi _ { P _ { \\alpha } , a } ( \\mathbf { O } ; \\mathcal { G } ) \\right ] } = 1 - \\frac { v a r _ { P _ { \\alpha } } \\left [ \\Delta _ { P _ { \\alpha } } \\left ( \\mathbf { O } \\right ) \\right ] } { v a r _ { P _ { \\alpha } } \\left [ \\psi _ { P _ { \\alpha } , a } ( \\mathbf { O } ; \\mathcal { G } ) \\right ] } . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\gamma = \\bigl ( \\sigma _ { N - 1 } ^ { - 1 } \\sigma _ { N - 2 } ^ { - 1 } \\cdots \\sigma _ { 1 } ^ { - 1 } \\bigr ) ^ { N } \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} I _ 0 = \\frac 1 { \\Gamma ( 1 - \\alpha ) } \\left ( \\frac { L _ 0 } { 2 } \\right ) ^ { 1 - \\alpha } \\int _ { - 1 } ^ 1 e ^ { - t ( y + 1 ) L _ 0 / 2 } ( y + 1 ) ^ { - \\alpha } d y . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\mathfrak { g } ^ { ( l ) } \\ni v \\mapsto \\left ( \\left . \\frac { \\partial U _ { t } ^ { u } } { \\partial u } \\right | _ { u = 0 } \\right ) ^ { - 1 } \\cdot v = \\left . \\frac { d } { d \\eta } \\right | _ { \\eta = 0 } \\log \\left ( { \\rm e } ^ { U _ { t } ^ { 0 } + \\eta v } \\otimes { \\rm e } ^ { - U _ { t } ^ { 0 } } \\right ) \\in \\mathfrak { g } ^ { ( l ) } . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} y _ 0 > 0 , \\tilde { y } = u ^ * \\begin{pmatrix} - D i a g ( \\tilde { y } _ 1 , \\ldots , \\tilde { y } _ r ) & 0 \\\\ 0 & 0 \\end{pmatrix} ( v ^ * ) ^ T , \\sum _ { i = 1 } ^ r \\tilde { y } _ i = y _ 0 , \\tilde { y } _ i > 0 , \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\dot { \\gamma } = ( \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 ) e _ 1 - \\frac { l _ L } { l } L ^ { \\frac { 1 } { 2 } } \\omega ( \\dot { \\gamma } ( t ) ) e _ 2 . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} I _ a ^ { k _ 1 } \\left ( I _ a ^ { k _ 2 } f \\right ) ( x ) = \\left ( I _ a ^ { k _ 3 } f \\right ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] , \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} \\begin{aligned} | \\partial _ j \\Phi _ k ( t , x ) & \\partial _ i \\Phi _ h ( t , x ) - \\partial _ j \\tilde \\Phi _ k ( t , x ) \\partial _ i \\tilde \\Phi _ h ( t , x ) | \\le \\frac { 2 } { c _ 1 ^ 2 } \\Vert D \\Lambda \\Vert _ \\infty \\Vert D ^ 2 \\Lambda \\Vert _ \\infty | x - r ( t ) | \\ , , \\end{aligned} \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\lambda _ { k } ^ { \\epsilon , v _ k } = - \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\mathbb { P } _ { x , k } ^ { \\epsilon , v _ k } \\left \\{ \\tau _ { k } ^ { \\epsilon , v _ k } > t \\right \\} , \\ , \\ , \\ , x \\in D , \\ , \\ , \\ , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} H ( \\varphi ) = \\mathcal { L } _ { k , d i s t } ( \\varphi ) \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\log N ( F , 2 ^ { - n } ) = \\log \\# F = H ( \\mu , \\mathcal { D } _ n ) \\le 5 \\varepsilon n \\log 2 . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} u ( t , x ) = u ( 0 , x ) = \\begin{cases} 1 , & x < 0 , \\\\ 0 , & x > 0 , \\end{cases} , \\forall t \\ge 0 . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} [ a t ^ m , b t ^ n ] = [ a , b ] t ^ { m + n } + m ( a , b ) \\delta _ { m , - n } K . \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} | \\partial _ j e ^ { t \\Delta } g | _ \\beta \\leq c t ^ { \\frac { 1 } { \\beta } - \\frac { 1 } { \\alpha } - \\frac { 1 } { 2 } } | g | _ \\alpha , \\ u \\in L ^ \\alpha ( \\mathbb { R } ^ 2 ) , \\ j = 1 , 2 . \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} x ^ { - k } = u ^ i x ^ { - j } \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} v \\mathbb { R } ( S ) & : = \\{ e S : e \\in E ( S ) \\} , \\\\ \\mathbb { R } ( S ) ( e S , f S ) & : = \\{ \\lambda ( e , u , f ) : u \\in f S e \\} , \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\R ( A _ 1 ^ n , B _ 1 ^ n , d ) & = - \\lambda d ~ + \\\\ & \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ n \\log ( 1 + 2 \\lambda \\beta _ i ^ 2 ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\lambda \\alpha _ i ^ 2 } { 1 + 2 \\lambda \\beta _ i ^ 2 } \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} T : = \\left [ T _ { j k } \\right ] _ { n \\times n } , T _ { j k } : = \\mathcal L ( L _ j , L _ k ) . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} d \\left ( T \\{ x _ n ^ i \\} , T \\{ x _ n ^ j \\} \\right ) & = d \\left ( \\{ x _ n ^ { i + 1 1 } \\} , \\{ x _ n ^ { j + 1 1 } \\} \\right ) \\\\ & = \\frac { 1 0 } { i + j + 2 2 } + 1 \\\\ & < \\frac { 1 0 } { i + j } + 1 \\\\ & = d \\left ( \\{ x _ n ^ i \\} , \\{ x _ n ^ j \\} \\right ) . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } = \\frac { B \\left ( | \\mathcal { S } | - 1 \\right ) } { \\sum _ { j \\in \\mathcal { S } } \\frac { c _ j + \\lambda _ j } { w _ j } } . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} N _ W = \\left ( \\frac { 1 } { 2 A w _ n } \\right ) ^ 2 \\geq D n ^ { 1 / 3 } \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} w _ m ( u ) = C _ m w ( g _ m ^ { - 1 } ( u ) ) = \\frac { w ( g _ m ^ { - 1 } ( u ) ) } { \\int _ { - 1 } ^ { 1 } w ( t ) | g _ m ' ( t ) | \\d t } \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( B _ 5 ^ { ( n ) } ) = - H ( \\ell _ 2 \\varepsilon ) \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\chi _ { P , a , e f f } ^ { 1 } \\left ( \\mathbf { V } ; \\mathcal { G } \\right ) = \\psi _ { P , a } \\left [ \\mathbf { O } \\left ( A , Y , \\mathcal { G } \\right ) ; \\mathcal { G } \\right ] P \\in \\mathcal { M } \\left ( \\mathcal { G } \\right ) \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\overline { C ^ \\infty _ c ( \\R ^ 3 ) } ^ { \\norm { \\cdot } _ { L ^ \\infty _ H L ^ \\infty _ z } } = C _ 0 ( \\R ^ 3 ) , \\overline { C _ { } ^ \\infty ( \\overline { \\Omega } ) } ^ { \\norm { \\cdot } _ { L ^ \\infty _ H L ^ \\infty _ z } } = C _ ( [ 0 , 1 ] ^ 2 ; C [ - h , 0 ] ) . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - F _ m ^ - \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n \\end{cases} \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} W L = ( L - 4 i Z + 4 ) W . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\mathbb { P } ( T S P C _ n \\geq C _ 3 b _ n ) \\geq 1 - D \\sqrt { n } e ^ { - 2 C _ 2 N } = 1 - e ^ { - \\alpha _ N } , \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} ( T ^ * T ) _ { \\rm { s } } ^ { 1 / 2 } = \\int _ 0 ^ \\infty \\lambda \\ , d E _ \\lambda , \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} m _ \\epsilon ( x ) = F ^ { - 1 } _ \\epsilon \\Big ( \\overline { H } _ \\epsilon ( P ) - V \\Big ( x , \\frac { x } { \\epsilon } \\Big ) \\Big ) . \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\dim \\theta = \\min \\{ 1 , \\frac { h _ { 1 } } { - \\log r _ { 1 } } + \\frac { h _ { 2 } } { - \\log r _ { 2 } } \\} \\ : . \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} u ( t ) = { \\bf B } _ \\alpha ( t , T ) f & + \\int _ 0 ^ t { \\bf P } _ \\alpha ( t - r ) G ( r , u ( r ) ) d r - \\int _ 0 ^ { T } { \\bf B } _ \\alpha ( t , T ) { \\bf P } _ { \\alpha } ( T - r ) G ( r , u ( r ) ) d r \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} R _ { i j k l } = K ( \\delta _ { i k } \\delta _ { j l } - \\delta _ { i l } \\delta _ { j k } ) = R _ { i ^ { \\ast } j ^ { \\ast } k l } , \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\sum _ { k = N + 1 } ^ { \\infty } f ( k ) \\leq \\int _ { N } ^ \\infty f ( x ) d x . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\sum _ { j } \\binom { 5 m } { 5 j } - \\sum _ { j } \\binom { 5 m } { 5 j + 1 } = ( - 1 ) ^ { m } F _ { 5 m - 1 } \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\Delta _ j ( z ) : = Q _ j ( z ) \\setminus Q _ { j + 1 } ( z ) . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} 1 = T _ { 2 k + 1 , 2 a + 1 } ( 0 ; q ) = T _ { 2 k + 1 , 2 a + 1 } ( x ; 0 ) = \\overline { T } _ { 2 k + 1 , 2 a } ( 0 ; q ) = \\overline { T } _ { 2 k + 1 , 2 a } ( x ; 0 ) . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\left \\Vert \\varepsilon \\boldsymbol { L } _ { M } \\boldsymbol { P } _ { M } u - \\boldsymbol { P } _ { M } \\varepsilon \\boldsymbol { L } u \\right \\Vert _ { \\infty } = 0 \\lim _ { M \\rightarrow \\infty } \\left \\Vert d \\varepsilon \\boldsymbol { L } _ { M } \\boldsymbol { P } _ { M } u - \\boldsymbol { P } _ { M } \\varepsilon d \\boldsymbol { L } u \\right \\Vert _ { \\infty } = 0 . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left [ \\sqrt { f _ { \\theta + h } } - \\sqrt { f _ \\theta } - \\tfrac { 1 } { 2 } h \\dot { \\ell } _ \\theta \\sqrt { f _ \\theta } \\right ] ^ 2 = 9 6 0 h ^ 4 , \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} | \\mathsf Y \\rangle = \\psi _ { r _ 1 } ^ { \\phantom * } \\cdots \\psi _ { r _ n } ^ { \\phantom * } \\psi _ { s _ 1 } ^ * \\cdots \\psi _ { s _ n } ^ * \\ , | 0 \\rangle , \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} V ( n ) = & \\frac { 1 } { ( n - 1 ) ! } \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ i { n - 1 \\choose i } \\{ M ( k - i a ) ^ { n - 1 } - M ( k - ( i + 1 ) a ) ^ { n - 1 } \\} \\\\ = & \\frac { 1 } { ( n - 1 ) ! } \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ i { n - 1 \\choose i } M ( k - i a ) ^ { n - 1 } \\\\ + & \\frac { 1 } { ( n - 1 ) ! } \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i { n - 1 \\choose i - 1 } M ( k - i a ) ^ { n - 1 } \\\\ = & \\frac { 1 } { ( n - 1 ) ! } \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ i { n \\choose i } M ( k - i a ) ^ { n - 1 } . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ { \\Lambda _ j } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { \\Lambda _ j y _ i } d y + \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } ( \\widetilde { \\delta } _ { i j } + \\widetilde { w } _ { y _ i \\Lambda _ j } ) \\widetilde { w } _ { \\Lambda _ j y _ i } d y = 0 . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\mathbb { V } _ U ( d ) = \\limsup _ { n \\rightarrow \\infty } ~ \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ n \\min \\left [ 1 , ~ \\left ( \\frac { \\sigma _ { n , i } ^ 2 } { \\theta _ n } \\right ) ^ 2 \\right ] . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} d ( u , v ) : = \\| u - v \\| _ { L ^ \\infty ( I , L ^ 2 ) } + \\| u - v \\| _ { L ^ p ( I , L ^ q ) } , \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} \\mathfrak { H } : = \\left ( \\begin{array} { c c c c c c c } H _ { 2 } & H _ 1 & H _ 0 & & & & \\\\ & H _ 2 & H _ 1 & H _ 0 & & & \\\\ & & H _ 2 & H _ 1 & H _ 0 & & \\\\ & & & H _ 2 & H _ 1 & H _ 0 & \\\\ & & & & H _ 2 & H _ 1 & H _ 0 \\end{array} \\right ) \\in \\mathbb F ^ { 5 \\times 1 4 } . \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\pi _ { \\mathbf { a } } \\left ( \\mathbf { G , B } ; P \\right ) = \\pi _ { \\mathbf { a } } ( \\mathbf { B } ; P ) . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\d ( \\varphi \\circ f ) = \\varphi ' \\circ f , \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} | | U ( K + k _ 0 ) | | = \\sup _ { | k | \\leq \\varepsilon | K | } | | U ( K + k ) | | = \\sup _ { | k | \\leq \\frac { 3 \\varsigma _ 1 } { 4 \\ln \\lambda } | K | } | | U ( K + k ) | | . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\vert u ( x ) \\vert = \\lim _ { i \\rightarrow \\infty } \\vert u ( x ) - u ( y _ i ) \\vert \\leq C ' r _ 0 ^ { s } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\sigma _ \\mathrm { e s s } ( - \\Delta _ D ^ \\Omega ) = [ \\lambda _ 1 , \\infty ) \\ , , \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} P ( S _ n ( n - x f _ 2 ( n ) ) + a _ n > \\lfloor n - \\varepsilon f _ 2 ( n ) \\rfloor ) & = P ( n - a _ n - S _ n ( n - x f _ 2 ( n ) ) < \\lceil \\varepsilon f _ 2 ( n ) \\rceil ) \\\\ & = P \\left ( \\frac { n - a _ n - S _ n ( n - x f _ 2 ( n ) ) } { f _ 2 ( n ) } \\leq \\varepsilon \\right ) . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j } ( d '' _ { i } - d _ i ) \\ge j , \\mbox { } j = 1 , \\cdots , s . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} B ( \\{ s _ n \\} , \\{ t _ n \\} , N ) = \\bigcup _ { a _ 1 \\cdots a _ { N - 1 } \\in \\mathbb { N } ^ { N - 1 } } f _ { a _ 1 } \\circ \\cdots \\circ f _ { a _ { N - 1 } } ( B ( \\{ s _ { n + N - 1 } \\} , \\{ t _ { n + N - 1 } \\} , 1 ) ) \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\gamma = d s ^ 2 + u ( s ) ^ 2 g _ * \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} A : = \\bigl ( C _ 1 \\bigr ) _ 2 \\times \\dotsb \\times \\bigl ( C _ k \\bigr ) _ 2 = \\{ 0 , c _ 1 \\} \\times \\dotsb \\times \\{ 0 , c _ k \\} . \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\{ H , J _ i \\} = 0 , \\ \\ i = 1 , \\dots , 2 n - k . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\sup _ { z \\in B ( t , \\rho / 2 ) } h ( z ) \\leq \\sup _ { z \\in E _ { \\rho } } h ( z ) \\leq h ( z _ { * } ) = : c ( \\rho ) \\ , z _ { * } \\in \\partial E _ { \\rho } \\ . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , k - 1 ) - \\sum _ { n = 0 } ^ { \\frac { m - 1 } { 2 } } F ( n , k ) = G \\left ( \\frac { m + 1 } { 2 } , k \\right ) - G ( 0 , k ) = G \\left ( \\frac { m + 1 } { 2 } , k \\right ) . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\mathcal { L } _ j [ w _ j ] = \\left [ \\mathcal X _ j - 2 | D ^ 2 u _ j | ^ 2 + 4 \\alpha _ j a _ { 0 , j } \\frac { \\langle \\nabla u _ j , \\nabla \\phi \\rangle } { \\phi } + 2 \\left ( \\mathcal Q _ j - \\frac { h _ j ' ( u _ j ) } { h _ j ( u _ j ) } z _ j \\right ) a _ { 1 , j } \\right ] \\phi ^ { 2 \\alpha _ j } , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} A = \\phi ^ { - 1 } ( A ) U , \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} \\begin{aligned} G _ { \\rm m a x , U L A , 2 8 ~ G H z } = \\frac { 1 } { | \\cos 6 0 ^ \\circ | \\log \\sqrt { \\frac { 2 8 . 3 5 } { 2 7 . 5 } } } \\approx 2 1 . 2 { \\rm d B } . \\end{aligned} \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} - 2 \\langle \\nabla a _ { 3 , j } , \\nabla u _ j \\rangle = - a _ { 3 , j , z _ j } + ( p - 2 ) \\left ( \\frac { 1 } { 2 } - \\frac { \\langle \\nabla z _ j , \\nabla u _ j \\rangle ^ 2 } { z _ j | \\nabla z _ j | ^ 2 } \\right ) \\frac { | \\nabla z _ j | ^ 2 } { z _ j } . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\frac { 1 } { ( 3 d - 4 ) ! } N _ d + \\sum _ { d _ 1 + d _ 2 = d } N _ { d _ 1 } N _ { d _ 2 } d _ 1 ^ 3 d _ 2 \\frac { 1 } { ( 3 d _ 1 - 1 ) ! } \\frac { 1 } { ( 3 d _ 2 - 3 ) ! } \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\mathrm { R e l } ^ { G , \\mu } _ { M _ S , b } = \\coprod _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } \\mathrm { R e l } ^ { M _ b , \\mu _ b } _ { M _ S , b ' } . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\eta _ s = { \\mu \\over \\nu } \\big ( \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } - \\tilde \\sigma \\ln ( \\bar \\sigma + \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } ) + \\tilde \\sigma \\ln \\hat \\sigma \\big ) , \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{gather*} C _ { a b } ^ e C _ { c e } ^ d + \\rho _ { a } ^ j \\partial _ j C _ { b c } ^ d + \\operatorname { c y c l } ( a b c ) = 0 . \\end{gather*}"}
-{"id": "9947.png", "formula": "\\begin{align*} \\frac { n } { \\log ^ 4 n } \\geq | R | = \\frac 1 2 \\sum _ { v \\in I } s _ { I , I } ( v ) = \\frac 1 4 \\sum _ { v \\in I } s _ { I } ( v ) \\geq \\frac 1 4 \\sum _ { v \\in I _ h } \\frac { \\sqrt { n } } { \\log ^ 4 n } \\geq \\frac { n } { 4 \\log ^ 5 n } , \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\begin{aligned} S ( 1 - \\dfrac 1 2 S ) - ( S - \\bar H ^ 2 ) ^ 2 + \\dfrac 1 2 ( \\bar H ^ 2 ) ^ 2 - \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} \\frac { 2 N ( 1 + s ) } { N - 4 \\sigma } \\ , & = \\frac { 2 N ( 1 + s ) } { N - 4 ( \\nu - \\mu ) } \\le \\frac { 2 N ( 1 + s ) } { N - 4 \\nu + 4 s \\left ( \\frac { N } { 4 } - \\nu \\right ) } = \\frac { 2 N } { N - 4 \\nu } . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} w _ { \\nu _ { \\mathcal { N } } } = ( \\mathbf { a } : \\mathbf { a } ^ { \\mathbf { T } } : D v ) \\nu - q ( \\mathbf { a } \\nu ) \\quad \\mbox { o n } \\quad \\Gamma _ c \\times ( 0 , T ) , \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { k _ { \\gamma , \\Sigma } ^ { L } } { \\sqrt { L } } = \\frac { | \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) | } { \\left ( \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 \\right ) ^ 2 } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) = 0 ~ ~ a n d ~ ~ \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) \\neq 0 . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\nabla g ( \\overline { x } ) \\nabla \\Xi ( \\overline { y } ) \\frac { \\mu ^ k - \\overline { \\mu } } { t _ k } = w - \\nabla g ( \\overline { x } ) \\nabla \\big ( \\nabla \\Xi ( \\cdot ) \\overline { \\mu } \\big ) ( \\overline { y } ) ( g ' ( \\overline { x } ) \\xi ) . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\| W _ { j , j } - V _ j \\| = \\| R _ j - P _ j \\| \\leq \\| R _ j ^ 2 - P _ j \\| , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\frac { \\partial x _ { 1 } } { \\partial y } & \\frac { \\partial x _ { 1 } } { \\partial z } \\\\ \\frac { \\partial x _ { 2 } } { \\partial y } & \\frac { \\partial x _ { 2 } } { \\partial z } \\end{array} \\right ) \\cdot \\left ( \\begin{array} { c c } \\frac { \\partial y } { \\partial x _ { 1 } } & \\frac { \\partial y } { \\partial x _ { 2 } } \\\\ 0 & \\mathrm { I } _ { m - n } \\end{array} \\right ) = \\mathrm { I } _ { m } . \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} { \\rm d i v } \\ ( | x | ^ { \\sigma } | \\nabla u | ^ { p - 2 } \\nabla u \\ ) = | x | ^ { - \\tau } u ^ q | \\nabla u | ^ m \\mathrm { i n } \\ \\Omega ^ * : = \\Omega \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} S ^ { \\alpha } ( x ) = \\gamma ( x _ 1 ) S ( x _ 2 ) \\gamma ^ { - 1 } ( x _ 3 ) \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } y _ { i , p } = x _ i , \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} Q ^ { ( i ) } ( M , \\imath ) : = Q _ { 0 } ( M , \\imath ) Q _ { 1 } ( M , \\imath ) \\cdots Q _ { i } ( M , \\imath ) , \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} g ( t ) = \\begin{cases} g _ 0 ( t ) : = g ( 0 ) + c t ^ { a + 1 } \\log ( 1 / t ) ^ { - b } ( 0 , 1 / R ' ] \\ , , \\\\ g _ \\infty ( t ) : = c ' t ^ { - a ' } ( \\log t ) ^ { - b ' } [ R ' , + \\infty ) \\ , , \\end{cases} \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\varphi \\in L ^ p _ { \\mathrm { i n v } } ( \\Omega ) \\Leftrightarrow U _ { h e _ i } U _ { y } \\varphi = U _ y \\varphi y \\in \\R ^ d , h \\in \\R , i = 1 , . . . , d . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { u _ k - 1 } | I _ { n _ k } [ a _ 1 , \\cdots , a _ { n _ k - 1 } , j ] | \\asymp \\left ( \\sum _ { j = 1 } ^ { u _ k - 1 } | I _ 1 ( j ) | \\right ) \\cdot I _ { n _ k - 1 } [ a _ 1 , \\cdots , a _ { n _ k - 1 } ] , \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\tilde { u } = m _ { j _ 2 } x - m _ { j _ 1 } y \\qquad \\mathrm { w h e r e } \\quad \\tilde { u } \\coloneqq u - \\min \\{ P _ { A } ( j _ 2 ) \\} + \\min \\{ P _ { A } ( j _ 1 ) \\} , \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} e ( \\rho , \\theta ) = P _ e ( \\rho ) + Q ( \\theta ) , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} A ^ \\pm ( \\lambda ) : = \\frac { - ( N - 2 ) \\pm \\sqrt { ( N - 2 ) ^ 2 + 4 \\lambda } } { 2 } . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\bar h ^ { p ^ { \\ast } } _ { 1 1 j } + \\bar h ^ { p ^ { \\ast } } _ { 2 2 j } = 0 , \\ \\ \\ j , p = 1 , 2 \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} N _ { \\mathcal J } ( u , v ) : = [ \\mathcal J u , \\mathcal J v ] - [ u , v ] - \\mathcal J ( [ \\mathcal J u , v ] + [ u , \\mathcal J v ] ) \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} D \\psi = \\frac { 1 } { 2 } \\sum _ { 1 \\le i < j \\le d } \\langle \\psi ( e _ i ) , e _ j \\rangle e _ i e _ j \\in \\mathbf { C l } ( V ) , \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\binom { k - n - 1 } { k } _ q = \\sum _ { Y \\in C ( n , k ) } q ^ { \\sigma ( Y ) } , \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\Theta = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} , a , b , c , d \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\left ( \\left \\lfloor \\frac { n } { e p ^ l } \\right \\rfloor - \\left \\lfloor \\frac { n } { e p ^ { l + 1 } } \\right \\rfloor \\right ) & = a _ l + a _ { l + 1 } p + \\dots + a _ r p ^ { r - l } - ( a _ { l + 1 } + a _ { l + 2 } p + \\dots + a _ r p ^ { r - l - 1 } ) \\\\ & = ( a _ l - a _ { l + 1 } ) + ( a _ { l + 1 } - a _ { l + 2 } ) p + \\dots ( a _ { r - 1 } - a _ r ) p ^ { r - l - 1 } + a _ r p ^ { r - l } . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\Phi _ 0 \\left ( h _ \\lambda ^ { * n } \\ast f \\right ) = \\sum _ { j = 1 } ^ m \\Phi _ 0 \\bigl ( \\left ( h _ \\lambda \\ast f _ j \\right ) ^ { * n } \\bigr ) = \\sum _ { j = 1 } ^ m P \\left ( h _ \\lambda \\ast f _ j \\right ) , \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\in \\Lambda } \\varphi \\left ( f , h _ \\lambda , \\dotsc , h _ \\lambda \\right ) = \\sum _ { [ \\pi ] \\in \\mathcal { F } } \\varphi \\left ( d _ \\pi f \\ast \\chi _ \\pi , d _ \\pi \\chi _ \\pi , \\dotsc , d _ \\pi \\chi _ \\pi \\right ) = \\Phi _ 0 ( f ) . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} & U _ + ( \\tau ) : = \\| P _ + \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\\\ & U _ 0 ( \\tau ) : = \\| P _ 0 \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\\\ & U _ - ( \\tau ) : = \\| P _ - \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} f _ { k , u _ l } = \\partial _ { u _ l } f _ k \\geq 0 \\ \\ ( k \\not = l ) \\ \\ { \\rm i n } \\ \\ ( p ^ - , p ^ + ) : = ( p _ 1 ^ { - } , p _ 1 ^ { + } ) \\times ( p _ 2 ^ { - } , p _ 2 ^ { + } ) \\times \\cdots \\times ( p _ m ^ { - } , p _ m ^ { + } ) , \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} a _ { i j } = \\begin{cases} 1 & \\textrm { i f $ \\exists k \\in \\mathcal { N } _ { \\rm U } , $ s . t . , $ \\textbf { b } _ i , \\textbf { b } _ j \\in \\mathcal { B } _ k $ } \\\\ 0 & \\textrm { o t h e r w i s e } \\end{cases} , \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} R _ { i , j } ( G , S , \\lambda ) = \\sum _ { p = 1 } ^ { m + 1 } R _ { i , j } ^ { ( p ) } ( G , S , \\lambda ) , \\forall i , j \\in S . \\\\ R _ { i , j } ( G , S , \\lambda ) = \\sum _ { p = 1 } ^ { m - 1 } R _ { i , j } ^ { ( p ) } ( G , S , \\lambda ) , \\forall i , j \\in \\overline { S } , i \\neq j . \\\\ R _ { i , i } ( G , S , \\lambda ) = w ( i , i ) , \\forall i \\in \\overline { S } . \\\\ R _ { i , j } ( G , S , \\lambda ) = \\sum _ { p = 1 } ^ m R _ { i , j } ^ { ( p ) } ( G , S , \\lambda ) , \\forall i \\in S , j \\in \\overline { S } i \\in \\overline { S } , j \\in S . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} \\mathbb { E } T _ { l _ 1 } T _ { l _ 2 } = J _ 1 + J _ 2 , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n m _ { j , i } r _ { \\sigma ( i ) } = \\sum _ { i = 1 } ^ n m _ { j , \\sigma ^ { - 1 } ( i ) } r _ { i } = m _ j + k _ j p \\quad ( 1 \\le { } ^ \\forall j \\le t ) . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { 2 \\beta _ i ^ 2 ( 2 \\alpha _ i ^ 2 + 1 + 2 \\lambda \\beta _ i ^ 2 ) } { ( 1 + 2 \\lambda \\beta _ i ^ 2 ) ^ 3 } \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\overline u ' \\phi ' \\ , d t + ( \\mu + z ) \\int _ 0 ^ 1 \\overline u \\phi \\ , d t = \\int _ 0 ^ 1 \\overline f \\phi \\ , d t , \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} P ( S _ n ( \\kappa _ n ( 0 ) ) \\leq \\lfloor \\kappa _ n ( 0 ) \\rfloor - a _ n ) & = P \\left ( \\frac { S _ n ( \\kappa _ n ( 0 ) ) } { a _ c ^ { ( n ) } } \\leq o ( 1 ) \\right ) \\\\ & \\geq P \\left ( S _ n ( \\kappa _ n ( 0 ) ) = 0 \\right ) = ( 1 - \\pi _ n ( \\kappa _ n ( 0 ) ) ) ^ { n - a _ n } . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ( x ) = \\max _ { y \\in D } \\{ \\langle G ( x ) , x - y \\rangle - 0 . 5 \\alpha \\| x - y \\| ^ { 2 } \\} ; \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} \\wedge V ^ \\ast = \\bigoplus _ { k \\ge 0 } \\wedge ^ k V ^ \\ast , \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\tau _ \\epsilon ( i ) & = \\begin{cases} ( \\delta + \\epsilon ) - i & \\mbox { f o r $ \\min ( i , ( \\delta + \\epsilon ) - i ) $ o d d } \\\\ i & \\mbox { o t h e r w i s e } \\end{cases} \\\\ \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} p ( n ) = ( 1 + O ( 1 / \\sqrt { n } ) ) \\frac { 1 } { 4 n \\sqrt { 3 } } \\exp ( \\pi \\sqrt { 2 n / 3 } ) . \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} \\gamma ' & = \\{ \\{ i _ { a + 1 } , \\overline { i _ { a + 1 } } \\} , \\ldots , \\{ i _ { a + b } , \\overline { i _ { a + b } } \\} \\} \\\\ \\delta ' & = \\{ [ i _ { a + b + 1 } , \\overline { i _ { a + b + 1 } } ] , \\ldots , [ i _ n , \\overline { i _ n } ] \\} . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} R ( d u ) = \\sum _ i \\langle _ M ( d u ( e _ i ) ) , d u ( e _ i ) \\rangle - \\sum _ { i , j } \\langle R ^ N \\big ( d u ( e _ i ) , d u ( e _ j ) \\big ) d u ( e _ i ) , d u ( e _ j ) \\rangle \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} \\| h _ { 2 } \\| _ { L ^ { r _ { 2 } } ( \\mathbb { R } ^ { n } ) } = \\frac { \\| a \\| _ { L ^ { r _ { 2 } } ( \\mathbb { R } ^ { n } ) } } { | T _ { 2 } ^ { * } ( h _ { 1 } , g ) ( x _ { 0 } ) | } \\lesssim \\frac { \\| a \\| _ { L ^ { \\infty } } | B ( x _ { 0 } , r ) | ^ { 1 / r _ { 2 } } } { C N ^ { 2 n } } \\lesssim \\frac { r ^ { - n / p } r ^ { n / r _ { 2 } } } { C N ^ { 2 n } } , \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} f _ j ( x ) = f _ 0 ( x ) \\left ( 1 - \\gamma F + \\gamma K \\left ( \\frac { F _ 0 ( x ) - F _ 0 ( x _ { j - 1 } ) } { F } \\right ) \\right ) , j \\in \\{ 1 , 2 \\} , \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} f ( t ) = \\alpha + \\beta t + \\int _ { - 1 } ^ { 1 } \\frac { t ^ 2 } { 1 - t \\lambda } d \\mu ( \\lambda ) , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\Delta u ( x ) + a ( x , u ) = 0 . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - \\Delta u + \\frac { \\mu } { 1 + t } u _ t + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } u = | u | ^ p , & x \\in \\mathbb { R } ^ n , \\ t > 0 , \\\\ u ( 0 , x ) = \\varepsilon u _ 0 ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ u _ t ( 0 , x ) = \\varepsilon u _ 1 ( x ) , & x \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} w | _ { \\partial X } = { \\rm O p } _ h ( \\chi ) U ^ { - 1 } { \\rm O p } _ h ( \\phi \\varrho ^ k ) U { \\rm O p } _ h ( \\psi ) f \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} { \\phi ( x ) ^ { - 1 } + \\phi ( y ) ^ { - 1 } \\over | x - y | } = \\sum \\limits _ { k = 1 } ^ K \\kappa _ k \\frac { | x - R _ k | + | y - R _ k | } { | x - y | } g _ k ( x ) g _ k ( y ) \\ge \\sum \\limits _ { k = 1 } ^ K \\kappa _ k g _ k ( x ) g _ k ( y ) . \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} C _ { g , \\alpha } ( \\Omega ) : = \\sup _ { u \\in H ^ 1 _ 0 : { \\| u \\| _ { H ^ 1 _ 0 } ^ 2 \\le \\alpha } } \\int _ { \\Omega } ( 1 + g ( u ) ) \\exp ( u ^ 2 ) d x \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\tilde { \\rho } _ { y } [ h ] : = \\bigl \\| { \\rho _ y } \\bigl ( x , \\| h \\| _ { L ^ 1 ( 0 , T ; L ^ \\infty ( \\Omega ) ) } \\bigr ) \\bigr \\| ^ 2 _ { { L ^ { 2 p ' } } ( \\Omega ) } \\xrightarrow { h \\to 0 } 0 \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} T = T _ { \\rm o p } + T _ { \\rm m u l } . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\chi _ 1 ( s ) & = \\sum _ { m \\leq s ^ { - 1 / 2 } } - \\phi ' ( m s ) \\\\ & = K u \\Gamma ( u + 1 ) \\sum _ { m \\leq s ^ { - 1 / 2 } } ( m s ) ^ { - u - 1 } ( 1 + O ( ( m s ) ^ { \\epsilon / 2 } ) ) \\\\ & = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } \\left ( \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { 1 } { m ^ { u + 1 } } + O ( s ^ { \\epsilon / 2 } ) \\sum _ { m \\leq s ^ { - 1 / 2 } } \\frac { 1 } { m ^ { u + 1 - \\epsilon / 2 } } \\right ) \\\\ & = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) , \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\frac { R _ { 1 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = R _ { 1 4 } = 0 ; \\quad \\frac { R _ { 2 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = 0 ; \\quad \\frac { R _ { 3 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = 1 = u _ 3 . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\partial _ { t } ( \\left \\| \\overline { \\nabla } f \\right \\| ^ { 2 } ) & = 2 R i c ' _ { i j } \\overline { \\nabla } _ { i } f \\overline { \\nabla } _ { j } f + 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\ , \\underline { \\Delta } { f } ) + 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\left \\| \\overline { \\nabla } { f } \\right \\| ^ { 2 } ) \\\\ & - 2 g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } \\ , { S c a l ' } ) , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\tilde { K } _ m ( t ) : = \\sup _ { 0 < s < t } s ^ { 1 / 2 } \\lVert \\nabla \\tilde { V } _ m ( t ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } , \\tilde { H } _ m ( t ) : = \\sup _ { 0 < s < t } \\lVert \\tilde { V } _ m ( t ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} x . \\left ( f . v \\right ) \\overset { d e f } { = } \\left ( l _ x f \\right ) . v , \\ \\ f \\in \\mathcal H ( G , \\mathcal A ) , \\ \\ v \\in V , \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\bar h _ { 2 2 1 1 } ^ { 1 ^ { * } } = 0 . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} g ( t ) = k _ + ( t ) ^ { - 1 } g _ 0 k _ - ( t ) , \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\widetilde { \\hat { \\Delta } g } = \\hat { \\Delta } \\tilde { g } . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } D ( x _ { n _ k } , x ) = 0 , \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} z _ i \\abs { z } ^ { d - 1 } = \\sum _ { k \\in \\Z ^ d } a _ i ( k ) e ^ { i 2 \\pi k \\cdot z } , \\qquad \\forall z \\in Q ( 0 , \\frac 1 4 ) , \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} I ^ + \\cup I ^ - = I \\qquad \\inf I ^ + < \\inf I ^ - . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n ( 3 k + 1 ) { 2 k \\choose k } ^ 3 1 6 ^ { n - k } & \\equiv 0 \\mod { 4 ( 2 n + 1 ) { 2 n \\choose n } } , \\\\ [ 5 p t ] \\sum _ { k = 0 } ^ n ( 3 k + 1 ) { 2 k \\choose k } ^ 3 ( - 8 ) ^ { n - k } & \\equiv 0 \\mod { 4 ( 2 n + 1 ) { 2 n \\choose n } } , \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 ) } _ m ( z / c ) , z ^ j { \\bar z } ^ l \\rangle _ 0 = \\langle z ^ l C ^ { ( 1 ) } _ m ( z / c ) , z ^ j \\rangle _ 0 = 0 \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} h = \\varphi + \\psi , \\varphi ' + \\psi ' = 0 , \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\Delta w - \\lambda w = \\partial _ j f \\R ^ 3 , f \\in C ^ \\infty _ c ( \\R ^ 3 ) . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} f _ q ( Q ) = Q ^ k \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} ( i + ( k ' - k ) - e + 1 ) d + ( k ' - k ) m = 0 . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} & m _ 1 \\alpha _ 1 + m _ 2 \\alpha _ 2 + m _ 3 \\alpha _ 3 + m _ 4 \\alpha _ 4 = m , \\\\ & m _ 1 \\alpha _ 2 + m _ 2 \\alpha _ 1 + m _ 3 \\alpha _ 3 + m _ 4 \\alpha _ 4 = m , \\\\ & m _ 1 \\alpha _ 1 + m _ 2 \\alpha _ 2 + m _ 3 \\alpha _ 4 + m _ 4 \\alpha _ 3 = m , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} | \\nabla _ y G _ x ( y ) | \\le \\frac { C } { | x - y | } \\ , , 0 < G _ x ( y ) \\le \\frac { 1 } { 2 \\pi } \\log \\frac { C } { | x - y | } \\ , , \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} - \\hat { \\sigma } ( z ) \\varphi '' - \\hat { \\sigma } ( z ) f _ { 1 } \\left [ z \\right ] \\varphi = \\lambda \\varphi . \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} - q _ * ^ { - 1 } t ^ { \\frac { N + 2 A } { 2 } } ( \\partial _ r ^ 2 u _ 0 ) ( x , t ) = - [ M _ { 0 , 1 } + o ( 1 ) ] ( \\partial _ r ^ 2 U ) ( | x | ) + O ( t ^ { - 1 } ) \\ge - \\frac { M _ { 0 , 1 } U '' ( r _ * ) } { 2 } \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} p _ i = 0 \\ \\Rightarrow \\ q _ i = 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j = 1 . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} D _ \\ell ( a _ 1 , \\dots , a _ \\ell ) : = \\big \\{ x \\in I _ \\ell ( a _ 1 , \\dots , a _ \\ell ) : a _ { \\ell + 1 } ( x ) \\in [ s _ { \\ell + 1 } - t _ { \\ell + 1 } , s _ { \\ell + 1 } + t _ { \\ell + 1 } ] \\big \\} . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\alpha _ { l } = \\sqrt { \\lambda _ l } | \\langle \\hat u _ j , u _ l \\rangle | \\qquad \\forall l \\neq j , \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} 4 \\lambda ^ 2 - 4 \\lambda - 3 = 0 \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\theta _ i ( t ) = \\int _ { 0 } ^ { t } \\omega _ i ( W _ s ) \\ , \\circ d \\beta _ s \\ , . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} H ( \\mu , \\mathcal { D } _ n ) = H ( \\mu , \\mathcal { D } _ 0 ) + \\sum _ { i = 0 } ^ { n - 1 } ( H ( \\mu , \\mathcal { D } _ { i + 1 } ) - H ( \\mu , \\mathcal { D } _ { i } ) ) . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} L _ - : & = \\frac { l _ 1 } 2 + \\sum _ { i = 1 } ^ { N / 2 } l _ { 2 i + 1 } , & L _ + & = \\sum _ { i = 1 } ^ { N / 2 } l _ { 2 i } + \\frac { l _ { N + 2 } } { 2 } , & \\mbox { i f } \\ , N \\mbox { i s e v e n } , \\\\ L _ - : & = \\frac { l _ 1 } 2 + \\sum _ { i = 1 } ^ { ( N - 1 ) / 2 } l _ { 2 i + 1 } + \\frac { l _ { N + 2 } } { 2 } , & L _ + & = \\sum _ { i = 1 } ^ { ( N + 1 ) / 2 } l _ { 2 i } , & \\mbox { i f } \\ , N \\mbox { i s o d d } , \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} & f _ l ( v ) - f _ l ( u ) \\geq - M \\ , ( v _ l - u _ l ) \\ \\ ( l = 1 , 2 , \\cdots , m ) \\\\ & \\ \\ { \\rm f o r \\ a n y } \\ \\ u , \\ , v \\in [ p ^ - , p ^ + ] \\ \\ { \\rm w i t h } \\ \\ u \\preceq v \\ \\ \\ \\ ( M : = \\max _ { w \\in [ p ^ - , p ^ + ] } | D F ( w ) | ) . \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\{ \\psi \\in \\Sigma ( A ) \\colon \\abs { \\phi ( f _ k ) - \\psi ( f _ k ) } < \\epsilon , k = 1 , \\ldots , r \\} , \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\left ( f _ { 1 } ^ { \\prime } f _ { 2 } \\right ) ^ { 2 } f _ { 1 } f _ { 2 } ^ { \\prime \\prime } - 2 \\left ( f _ { 1 } ^ { \\prime } f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } f _ { 1 } f _ { 2 } + \\left ( f _ { 1 } f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } f _ { 2 } f _ { 1 } ^ { \\prime \\prime } + f _ { 1 } f _ { 2 } ^ { \\prime \\prime } + 2 a f _ { 1 } ^ { \\prime } f _ { 2 } ^ { \\prime } + a ^ { 2 } f _ { 1 } ^ { \\prime \\prime } f _ { 2 } = 0 . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} a = v _ 1 . . . v _ { 2 k } , 1 \\le k \\le \\frac { d } { 2 } , v _ i \\in V , \\parallel v _ i \\parallel = 1 . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\alpha & \\in \\left \\langle { \\frac 1 d - 1 , \\frac 1 d } \\right \\rangle , \\\\ \\delta & = d \\alpha + d - 1 . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} & f ( x ) a \\otimes b ( 0 ) m - \\sum _ { j \\in \\Z _ { \\geq 0 } } \\partial ^ { ( j ) } f ( x ) a ( j ) b \\otimes m - b \\otimes a _ { ( f ) } m , \\\\ & a \\otimes b _ { ( f ) } m - f ( x ) a ( 0 ) b \\otimes m - f ( x ) b \\otimes a ( 0 ) m , \\\\ & \\sum _ { j \\in \\Z _ { \\geq 0 } } ( - 1 ) ^ j \\partial ^ { ( j ) } f ( x ) a \\otimes b ( j ) m - a _ { ( f ) } b \\otimes m - \\sum _ { j \\in \\Z _ { \\geq 0 } } \\partial ^ { ( j ) } f ( - x ) b \\otimes a ( j ) m . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } ( \\mathrm { I n d } ^ G _ { P _ b } \\circ \\otimes ^ k _ { i = 1 } \\mathrm { M a n t } _ { M _ { b _ i } , b ' _ i , \\mu _ { b i } } \\circ \\mathrm { R e d } _ b ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} W _ \\nu ( [ u , v ] ) = \\psi _ \\omega ( Z _ { 2 \\nu } ( [ - 1 , 1 ] ) ) + \\frac { u + v } { 2 } \\subset ( u , v ) \\ ; , \\ ; \\ ; \\omega = \\frac { v - u } { 2 } \\leq \\pi \\ ; , \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t } = \\sum _ { \\alpha = 1 } ^ { d } V _ { \\alpha } ( X _ { t } ) d B _ { t } ^ { \\alpha } , & 0 \\leq t \\leq 1 , \\\\ X _ { 0 } = x \\in \\mathbb { R } ^ { N } . \\end{cases} \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\C ( x _ S | x _ T , { F } _ \\ell ) = \\C ( x _ S , { F } _ \\ell | x _ T ) - \\C ( { F } _ \\ell | x _ T ) + O ( \\log ( \\ell n ) ) . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } ^ { q - 1 } | \\nabla _ y \\widetilde { m } | ^ 2 d y \\leq \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } ^ { q + 1 } | \\nabla _ y V | ^ 2 d y \\leq \\sup _ { x , y } | \\nabla _ y V | ^ 2 \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } ^ { q + 1 } d y . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { H } \\mathbf { x } + \\mathbf { n } = \\sum _ { k = 1 } ^ { K } \\mathbf { h } _ k x _ k + \\mathbf { n } , \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} ( \\tilde { N } _ 1 / \\tilde { C } _ 1 , \\tilde { N } _ 2 / \\tilde { C } _ 2 ) = \\begin{cases} ( 1 0 / 1 , 1 5 / 8 ) & j \\ge 4 \\\\ ( 1 2 / 1 , 2 7 / 1 4 ) & j \\ge 3 \\\\ \\end{cases} \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\wp ( z , q ) = z ^ { - 2 } + \\sum _ { k = 0 } ^ \\infty ( 2 k + 1 ) G _ { 2 k + 2 } z ^ { 2 k } , \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\Delta z _ j = 2 | D ^ 2 u _ j | ^ 2 + 2 \\sum ^ 3 _ { i = 0 } \\langle \\nabla a _ { i , j } , \\nabla u _ j \\rangle \\mbox { f o r a l l } x \\in \\omega . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} s = \\# \\mathcal { F } _ 0 ( r ) = 2 0 7 \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\liminf _ { n \\rightarrow \\infty } \\P ( | \\hat \\lambda _ j ^ { ( n ) } - \\lambda _ { j } ^ { ( n ) } | / g _ { j } ^ { ( n ) } \\geq 1 - \\delta ) > 0 . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} K : = \\{ x \\in \\mathbb { R } ^ { m } \\ ; | \\ ; B x = b , ~ x \\in \\{ 0 , 1 \\} ^ { m } \\} , \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { + \\infty } \\bigl \\vert \\widehat { f } ( - k ) \\bigr \\vert ^ 2 \\le \\sum _ { k = - \\infty } ^ { + \\infty } \\bigl \\vert \\widehat { f } ( k ) \\bigr \\vert ^ 2 \\le \\Vert f \\Vert _ \\infty ^ 2 . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} \\displaystyle \\mathbb { E } [ \\log ( Y ) ] = M C _ { \\alpha } ^ { M } \\int _ { 1 } ^ { \\infty } \\frac { \\log ( x ) } { x + 1 } \\left [ \\frac { g ( x ) ^ { M - 1 } } { x ^ { \\alpha } } - \\frac { g ( 1 / x ) ^ { M - 1 } } { x ^ { 1 - \\alpha } } \\right ] d x . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\zeta ( z , q ) & = 2 \\pi i \\left ( \\frac { e ^ { 2 \\pi i z } } { e ^ { 2 \\pi i z } - 1 } - \\frac { 1 } { 2 } - \\sum _ { n = 1 } ^ \\infty \\frac { q ^ n } { 1 - q ^ n } \\left [ e ^ { 2 \\pi i n z } - e ^ { - 2 \\pi i n z } \\right ] \\right ) \\\\ \\wp ( z , q ) & = ( 2 \\pi i ) ^ 2 \\left ( \\frac { e ^ { 2 \\pi i z } } { ( e ^ { 2 \\pi i z } - 1 ) ^ 2 } + \\sum _ { n = 1 } ^ \\infty \\frac { n q ^ n } { 1 - q ^ n } \\left [ e ^ { 2 \\pi i n z } + e ^ { - 2 \\pi i n z } \\right ] \\right ) , \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\int { \\{ a _ { ( f ) } b \\} } = \\sum _ { j \\in \\Z _ { \\geq 0 } } \\frac { ( - 1 ) ^ j } { ( j + 1 ) ! } T ^ { j } ( a _ { ( x ^ { j + 1 } f ( x ) ) } b ) . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} A _ e ( v ) = - \\frac 1 2 [ A _ e , v ] . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\hat { \\eta } ( t , x , q , \\beta ) = \\sup _ { p \\in \\mathbb { R } ^ d , \\ , \\alpha \\in \\mathbb { R } ^ n } \\bigl [ q \\cdot p + \\beta \\cdot \\alpha - \\hat { \\lambda } ( t , x , p , \\alpha ) \\bigr ] , \\ , \\ , \\ , x , q \\in \\mathbb { R } ^ d , \\ , \\ , \\ , \\beta \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} \\tau _ { r } ( \\lambda _ { } ) = ( 1 - f _ N ( 0 ) ) \\int _ 0 ^ \\infty \\frac { \\mathrm { e } ^ { - \\frac { x \\sigma ^ 2 } { \\rho _ { \\mathrm { o } } } } } { x + 1 } \\mathcal { L } _ I \\left ( \\frac { x } { \\rho _ { \\mathrm { o } } } \\right ) \\mathrm { d } x , \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} B - \\hat \\nu _ m ( s ) = \\begin{cases} A _ 1 \\xi ' ( a ) + A _ 1 [ \\xi ' ( s ) - \\xi ' ( a ) ] m + \\frac 1 { | \\nu | } , & 0 \\le s \\le a , \\\\ A _ 1 \\xi ' ( s ) + \\frac 1 { | \\nu | } , & a \\le s \\le q , \\\\ A _ 1 \\xi ' ( q ) + A _ 2 [ \\xi ' ( s ) - \\xi ' ( q ) ] + \\frac 1 { | \\nu | } , & q \\le s \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} T ^ { \\tilde { D } } ( u , v , w ) = \\frac { 1 } { 4 } \\sum _ { ( u , v , w ) \\ ; \\mathrm { c y c l i c } } \\left ( \\eta ( u , v , \\mathcal J w ) + \\eta ( \\mathcal J u , v , w ) \\right ) , \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} a _ { ( \\wp \\zeta ) } b \\otimes m + \\frac { 1 } { 2 } a _ { ( \\wp ' ) } b \\otimes m - \\sum _ { j \\geq 1 } \\partial ^ { ( j ) } \\zeta ( x ) a _ { ( x ^ { j } \\wp ) } b \\otimes m \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} d v ( t ) = [ A v ( t ) + B ( t , v ( t ) + \\Psi ( t ) ) ] d t , \\ \\ \\ \\ v ( 0 ) = 0 , \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} \\int _ { \\Gamma } \\overline { \\nabla u } \\nabla \\phi \\ , d t + ( z + \\mu ) \\int _ { \\Gamma } \\overline { u } \\phi \\ , d t = \\int _ { \\Gamma } \\overline f \\phi \\ , d t \\qquad \\forall \\phi \\in H ^ 1 ( \\Gamma ) , \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\omega _ { i } & = \\sigma _ { 1 } ^ { - 1 } \\sigma _ { 2 } ^ { - 2 } \\sigma _ { 3 } \\sigma _ { 4 } \\cdots \\sigma _ { i + 1 } \\\\ \\tau _ { i , k } & = \\sigma _ { 1 } ^ { 2 } \\sigma _ { 2 } \\cdots \\sigma _ { k - 1 } \\sigma _ { k } ^ { - 2 } \\sigma _ { k + 1 } \\sigma _ { k + 2 } \\cdots \\sigma _ { i + k - 1 } . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} { \\cal L } ^ { * } f = { G } _ { m } f + \\frac { \\gamma \\sigma ^ 2 } { 2 } \\sum _ { i , j = 1 , \\ , i \\not = j } ^ m ( \\Phi _ { i j } ^ m f - f ) . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\Delta ^ 2 u _ 1 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } + \\frac { 4 u _ 2 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 + ( R _ 2 - R _ 1 ) | \\nabla u | ^ 2 , \\\\ \\Delta ^ 2 u _ 2 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } - \\frac { 4 u _ 1 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 - ( R _ 1 + R _ 2 ) | \\nabla u | ^ 2 , \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} G _ { k } ^ { ( \\beta ) } ( \\varphi ) = \\sum _ { \\nu = k } ^ { k M } t ^ \\nu \\Theta _ \\nu ^ { ( k , \\beta ) } ( m _ { 1 , 0 } , . . . , m _ { \\nu - k , 0 } ) \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigr ( g _ 2 ( x ) + \\frac { 2 } { 6 ^ 3 } ( 2 ^ 3 x - 3 ) ( 1 2 - 3 ^ 3 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 8 } { 6 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 3 } - \\eta \\\\ & = - \\frac { 6 2 1 4 5 5 7 2 5 3 1 0 0 1 9 5 9 6 8 2 8 6 6 0 9 0 5 8 7 8 2 6 5 3 7 } { 2 5 6 1 6 2 4 1 1 8 0 5 7 4 0 7 1 2 9 1 9 8 3 4 7 9 1 9 9 4 7 8 8 5 8 0 0 0 } < 0 , \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} T ( f , g ) ( x ) = \\int _ { \\mathbb R ^ n } \\int _ { \\mathbb R ^ n } K ( x , y , z ) f ( y ) g ( z ) \\ , \\d y \\d z \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} A ( h t ) = h a ( t ) \\ , , C ( h t ) = c ( t ) \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\frac { \\tilde \\lambda _ { j - 1 } } { \\tilde \\lambda _ { j - 1 } - \\tilde \\lambda _ j } = 1 + \\frac { \\tilde \\lambda _ { j } } { \\tilde \\lambda _ { j - 1 } - \\tilde \\lambda _ j } \\leq 1 + 2 \\frac { \\lambda _ { j - 1 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } \\leq 2 \\frac { \\lambda _ { j - 2 } } { \\lambda _ { j - 2 } - \\lambda _ { j - 1 } } . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\delta _ { f } = \\lim _ { n \\to \\infty } ( ( f ^ { n } ) ^ { * } H \\cdot H ^ { \\dim X - 1 } ) ^ { 1 / n } . \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\int _ 0 ^ { t } \\gamma ( t _ { n } - s _ { n } ) d s _ { n } & = \\int _ { t _ { n } - t } ^ { t _ { n } } \\gamma ( r ) d r \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\in \\Lambda } \\left \\Vert h _ \\lambda \\ast f - f \\right \\Vert _ A = 0 , \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\langle V _ i , f \\rangle : = \\int _ M ( 2 - p _ i ) | \\alpha _ i | ^ { p _ i } f ( I - 2 | \\alpha _ i | ^ { - 2 } \\alpha _ i \\otimes \\alpha _ i ) f \\in C ^ 0 ( A _ { n - 2 } ( M ) ) . \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} \\int _ a ^ b \\tilde { M } ( t - w ) ^ m d w = \\frac { 1 } { m + 1 } \\left ( \\tilde { M } ( t - a ) ^ { m + 1 } - \\tilde { M } ( t - b ) ) ^ { m + 1 } \\right ) . \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} = [ \\mathcal { M } _ { G , b , \\mu } ] . \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} \\frac { d } { d t } u + A u ~ \\ni ~ 0 \\ , , \\qquad u ( 0 ) = \\bar u \\ , . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} & \\beta _ 1 : C _ 0 ( { \\mathbb R } ) \\to S { \\frak C } _ l ( W ' ) , \\beta _ 1 ( f ) = f ( X \\hat { \\otimes } 1 + 1 \\hat { \\otimes } C _ l ) , \\\\ & \\beta _ 2 : C _ 0 ( { \\mathbb R } ) \\to S { \\frak C } ( W ' ) , \\beta _ 2 ( f ) = f ( X \\hat { \\otimes } 1 + 1 \\hat { \\otimes } C ) \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} b _ { \\alpha , \\beta } \\left ( \\theta ^ \\beta _ \\pm + \\theta ^ \\beta _ \\mp \\right ) & = 2 \\mu c _ \\alpha a \\left ( | \\alpha \\cap ( \\alpha + \\beta ) | - | \\alpha \\cap \\beta | \\right ) \\\\ - 2 b _ { \\alpha , \\beta } \\xi _ \\beta & = 2 \\mu c _ \\alpha a \\left ( | \\alpha | - 2 | \\alpha \\cap \\beta | \\right ) \\\\ & = \\frac { 2 \\mu ^ 2 b _ { \\alpha , \\beta } c _ \\alpha } { \\lambda } \\xi _ \\beta \\\\ & = 2 ( 1 - \\lambda ) b _ { \\alpha , \\beta } \\xi _ \\beta \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\{ f , ( I - Q ) g + Q g ' \\} = \\{ f , g \\} - \\{ 0 , Q ( g - g ' ) \\} \\in T , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\alpha ^ { * } _ n ( a q , b , q ) = \\frac { ( 1 - a q ^ { 2 n + 1 } ) ( a q / b ; q ) _ n ( - b ) ^ n q ^ { n ( n - 1 ) / 2 } } { ( 1 - a q ) ( b q ) _ n } \\sum _ { n \\ge j \\ge 0 } \\frac { ( b ) _ j } { ( a q / b ) _ j } ( - b ) ^ { - j } q ^ { - j ( j - 1 ) / 2 } \\alpha _ j ( a , q ) \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\frac { \\log \\mu ( B _ r ( x ) ) } { \\log r } = \\frac { \\log \\mu ( B _ { r _ \\ell } ( x ) ) } { \\log r _ \\ell } = \\frac { \\log \\mu ( D _ \\ell ( a _ 1 ( x ) , \\ldots , a _ \\ell ( x ) ) ) } { \\log r _ \\ell } . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} g x g = y = g h x g h \\implies & \\ ( x g ) = h ( x g ) h \\\\ \\implies & \\ ( x g ) ^ { - 1 } h ^ { - 1 } ( x g ) = h \\\\ \\implies & \\ h = ( x g ) ^ { - 1 } h ^ { - 1 } ( x g ) \\\\ & = ( x g ) ^ { - 1 } [ ( x g ) ^ { - 1 } h ^ { - 1 } ( x g ) ] ^ { - 1 } ( x g ) \\\\ & = ( x g ) ^ { - 2 } h ( x g ) ^ 2 \\\\ \\implies & \\ \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\mu _ { \\beta , 0 } ^ { + } \\big ( \\eta ( n , 0 ) = 1 \\mid \\eta = - 1 \\partial \\Gamma \\big ) \\leq \\mu _ { \\beta , 0 } ^ { - } \\big ( \\eta ( n , 0 ) = 1 \\big ) , \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\limsup _ { i \\to \\infty } d i s t ( ( 2 - p _ i ) \\Sigma _ { \\ell = 1 } ^ { m _ 0 } \\theta _ { p _ i } ( u _ i , x ^ i _ { \\ell } , r _ i ) , 2 \\pi \\mathbb { Z } ) = 0 . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k a _ i ( \\theta _ t ^ H ) ^ i = \\frac { \\sum _ { j = 1 } ^ n \\theta _ j ( t ) \\mathbb { E } [ \\frac { \\partial } { \\partial y _ j } q _ { \\gamma } ( \\Theta ) | \\mathcal { F } _ t ] } { \\sum _ { j = 1 } ^ n \\theta _ j ( t ) \\mathbb { E } [ \\frac { \\partial } { \\partial y _ j } p _ { \\delta } ( \\Theta ) | \\mathcal { F } _ t ] } \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { x \\in H ( u ( t ) ) } \\left | \\ , x - \\min \\Pi \\ , \\frac { \\Xi ( \\varphi ) } { | \\Xi ( \\varphi ) | } \\ , \\right | = 0 . \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\frac { r _ \\alpha ^ \\lambda ( t ) } { r _ \\beta ^ \\lambda ( t ) } = \\exp \\left ( t ( f ( h _ \\alpha ) - f ( h _ \\beta ) ) \\right ) \\frac { r _ \\alpha ^ \\lambda } { r _ \\beta ^ \\lambda } . \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\mathfrak { E } _ { ( \\partial ) } e ^ { \\tau _ 2 ( d ) } = \\left ( \\sum _ { i = 1 } ^ r \\omega _ i \\partial _ { t _ i } \\right ) e ^ { \\tau _ 2 ( d ) } = \\sum _ { i = 1 } ^ r \\omega _ i T _ i ( d ) e ^ { \\tau _ 2 ( d ) } \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\P ( A \\cap G _ { 1 } ) = 0 . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { M } e ^ { u ( t _ k , \\cdot ) } = + \\infty \\mbox { o r } \\lim _ { k \\to \\infty } \\int _ { M } e ^ { - u ( t _ k , \\cdot ) } = + \\infty . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\begin{aligned} \\underset { t \\in \\left [ t _ k , t _ { k + k _ T } \\right ] } { \\sup } \\ ! \\left \\| \\bar { \\boldsymbol { \\psi } } ( t ) \\ ! - \\ ! \\tilde { \\boldsymbol { \\psi } } ^ k ( t ) \\ ! \\right \\| _ 2 \\ ! \\le \\ ! \\frac { C _ { \\mathbf { f } } b _ { S , k + 1 } } { 2 } \\ ! + \\ ! C e ^ { L ( T + b _ { S , 1 } ) } \\ ! , \\ ! \\end{aligned} \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} u _ { \\gamma } ( x ) = \\begin{cases} \\phi ( x ) & x \\le - 1 , \\\\ 1 & - 1 < x \\le - \\gamma , \\\\ \\frac { 1 } { 2 } \\left ( 1 - \\frac { x } { \\gamma } \\right ) & x \\in \\left ] - \\gamma , \\gamma \\right [ , \\\\ 0 & x \\ge \\gamma . \\end{cases} \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} \\mathbf { \\dot { C } } _ { 0 } & = \\left . \\tfrac { d } { d t } \\right \\vert _ { t = 0 } ( W _ { t } ^ { \\mathbf { P } } ) ^ { \\ast } ( \\mathrm { p r } ^ { \\ast } \\mathbf { A ) } = L _ { Y } ( \\mathrm { p r } ^ { \\ast } \\mathbf { A ) } ) = - s L _ { \\overline { X } } ( \\mathrm { p r } ^ { \\ast } \\mathbf { A ) } ) - ( \\mathbb { \\iota } _ { \\overline { X } } \\mathrm { p r } ^ { \\ast } \\mathbf { A ) } ) d s \\\\ & = - \\mathrm { p r } ^ { \\ast } ( v _ { \\mathbf { A } } ( X ) ) d s . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\mathcal { M } \\left ( \\mathcal { G } ^ { 1 } , \\mathbf { V } ^ { 2 } \\right ) = \\mathcal { M } \\left ( \\mathcal { G } ^ { 2 } \\right ) \\mathcal { M } \\left ( \\mathcal { G } ^ { 2 } , \\mathbf { V } ^ { 3 } \\right ) = \\mathcal { M } \\left ( \\mathcal { G } ^ { 3 } \\right ) \\Rightarrow \\mathcal { M } \\left ( \\mathcal { G } ^ { 1 } , \\mathbf { V } ^ { 3 } \\right ) = \\mathcal { M } \\left ( \\mathcal { G } ^ { 3 } \\right ) \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} \\frac { d \\tilde { \\boldsymbol { \\psi } } ^ k ( t ) } { d t } = \\mathbf { f } _ { \\boldsymbol { \\psi } } \\left ( \\tilde { \\boldsymbol { \\psi } } ^ k ( t ) \\right ) , \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } D ( x _ n , x ) = 0 , x \\in X . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} P _ 1 ( \\Lambda ) = P ( \\{ \\omega \\} \\cap \\tilde { A } _ m \\cap \\tilde { B } _ m ) > 0 . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} Y _ { x } ^ { \\varepsilon } ( t ) = \\sqrt { { \\varepsilon } } \\int _ { 0 } ^ { t } S ( t - s ) G ( s , \\mathcal { M } ( Y _ { x } ^ { \\varepsilon } + S ( \\cdot ) x ) ( s ) ) d w ( s ) , \\ ; t \\in \\lbrack 0 , T ] . \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} I _ { 1 } = ( p _ 1 + p _ 2 ) ( k _ 1 + k _ 2 ) \\geq 0 \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} d _ a = \\sum _ { i = 1 } ^ d a _ i \\nabla _ { X _ i } , \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} N _ t = \\int _ 0 ^ { t \\wedge \\tau } A _ i ( \\nabla , \\partial _ y ) ^ { \\mathrm T } Q f ( Y _ s , B _ s ) ( d \\beta _ s , d B _ s ) \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} z _ 2 ^ s ( q ) & = \\frac { q \\xi ( q ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { \\xi ' ( q ) [ q \\xi ' ( q ) - \\xi ( q ) ] ( 1 - q ) } - 1 , \\\\ z _ 2 ^ b ( q ) & = \\frac { q [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { ( 1 - q ) \\xi ' ( q ) } - 1 . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} \\delta ( r _ 1 , r _ 2 ) = \\sum _ { j = 1 } ^ { \\infty } { 2 ^ { - j } \\cdot \\frac { | r _ 2 - r _ 1 | _ { 2 , B _ j } } { | r _ 2 - r _ 1 | _ { 2 , B _ j } + 1 } } \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} R ( - n ) ^ { \\binom { n + 2 } { 2 } } \\xrightarrow { \\ ; F _ n \\ ; } R ( - n + 1 ) ^ { \\binom { n + 1 } { 2 } } , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} J ^ h ( u ) = I ^ h ( u ) - \\frac { 1 } { h } \\int _ { S ^ h } \\langle f ^ h , u ( z ) \\rangle d z , \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} u _ t - { \\rm d i v } \\ , ( A ( x ) \\nabla u ) = F ( x , u ) , \\ \\ x \\in \\R ^ N , \\ t \\in \\R , \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} [ y _ k ^ { p ^ r } , w _ m ] _ { \\circ } = w _ m \\cdot ( \\delta _ k ( r ) ) _ { \\circ } \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} A ( f ) = \\{ 1 , \\delta _ { f _ { T } } , \\delta _ { g } \\} = \\begin{cases} \\{ 1 , \\rho ( F ) \\} \\\\ \\{ 1 , \\rho ( F ) ^ { 2 } \\} \\\\ \\{ 1 , \\rho ( F ) , \\rho ( F ) ^ { 2 } \\} . \\end{cases} \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} ( L _ { Z } \\mu ^ Z ) ( X ) = & \\ \\psi ^ { - 1 } \\left ( L _ Z ( \\mu ^ Z ( X ) ) - \\mu ^ Z ( L _ Z X ) \\right ) \\\\ = & \\ \\psi ^ { - 1 } ( L _ Z g ( Z , X ) - g ( Z , [ Z , X ] ) ) \\\\ = & \\ 0 . \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} b _ 1 y _ 1 + b _ 2 y _ 2 + b _ 3 y _ 3 & = - 1 \\\\ d _ 1 y _ 1 + d _ 2 y _ 2 + d _ 3 y _ 3 & = 0 \\\\ y _ 1 + y _ 2 + y _ 3 & = 0 \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} b _ { { j _ 0 } , { j _ 1 } , \\cdots , { j _ s } } = \\sum _ { i = 0 } ^ s ( - 1 ) ^ i K _ { j _ i } + ( - 1 ) ^ { i - s } b _ { { j _ i } } ^ \\prime \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} v a r _ { P _ { \\alpha } } \\left [ \\psi _ { P , a } ( \\mathbf { V } ; \\mathcal { G } ) \\right ] = E _ { P _ { \\alpha } } \\left [ \\frac { I _ { a } ( A ) } { \\pi _ { a } ^ { 2 } ( \\mathbf { O } _ { m i n } ; P _ { \\alpha } ) } v a r _ { P _ { \\alpha } } ( Y \\mid A = a , \\mathbf { O } ) \\right ] + 2 + \\alpha ^ { 2 } . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} E [ ( 1 + t ) ^ U ] & = E [ e ^ { U \\log ( 1 + t ) } ] = \\sum _ { k = 0 } ^ \\infty E [ U ^ k ] \\frac { 1 } { k ! } \\log ^ k ( 1 + t ) \\\\ & = \\sum _ { k = 0 } ^ \\infty E [ U ^ k ] \\sum _ { n = k } ^ \\infty S _ 1 ( n , k ) \\frac { t ^ n } { n ! } = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { k = 0 } ^ n E [ U ^ k ] S _ 1 ( n , k ) \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 2 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = i ( - R _ { 1 2 } - i R _ { 1 4 } ) = - i R _ { 2 1 } + R _ { 2 4 } = 0 ; \\\\ \\frac { R _ { 3 1 } } { - i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 3 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = - i R _ { 3 1 } + R _ { 3 4 } = - 1 + 1 = 0 . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} { { \\bf y } } = \\left \\{ \\begin{array} { l r } { { \\bf y } } ' , & { { \\bf y } } ' \\in \\bigtriangleup _ M \\\\ \\frac { { { \\bf y } } ' } { \\lVert { { \\bf y } } ^ { ' } \\rVert _ 1 } , & \\end{array} \\right \\} \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} f = \\sum _ { [ \\pi ] \\in \\widehat { G } } d _ \\pi f \\ast \\chi _ \\pi . \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} u ( x , t ) = \\sin ^ 3 \\left ( \\tfrac 3 2 \\pi t \\right ) \\cos \\left ( \\tfrac 1 2 \\pi x _ 1 \\right ) \\cos \\left ( \\tfrac 1 2 \\pi x _ 2 \\right ) . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\max _ j \\| \\nabla u _ j ( t _ k , \\cdot ) \\| _ { L ^ 2 } = + \\infty , \\lim _ { k \\to + \\infty } \\max _ j \\int _ { M } e ^ { u _ j ( t _ k , \\cdot ) } = + \\infty . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} f = M _ { 1 } + M _ { 2 } + E _ { 2 } = \\sum _ { k = 1 } ^ { 2 } M _ { k } + E _ { 2 } . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\delta _ { f } = \\max \\{ \\delta _ { g } , \\delta _ { f _ { T } } \\} \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} B - \\hat \\nu _ m ( s ) = \\begin{cases} A _ 1 \\xi ' ( s ) + A _ 2 [ \\xi ' ( q ) - \\xi ' ( b ) ] m + A _ 2 [ \\xi ' ( b ) - \\xi ' ( q ) ] + \\frac 1 { | \\nu | } , & 0 \\le s \\le q , \\\\ A _ 1 \\xi ' ( q ) + A _ 2 [ \\xi ' ( b ) - \\xi ' ( q ) ] + A _ 2 [ \\xi ' ( s ) - \\xi ' ( b ) ] m + \\frac 1 { | \\nu | } , & q \\le s \\le b , \\\\ A _ 1 \\xi ' ( q ) + A _ 2 [ \\xi ' ( s ) - \\xi ' ( q ) ] + \\frac 1 { | \\nu | } , & b \\le s \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} & \\nabla ^ L _ { X _ j } X _ j = 0 , ~ ~ ~ 1 \\leq j \\leq 3 , ~ ~ ~ \\nabla ^ L _ { X _ 1 } X _ 2 = \\frac { L - 1 } { 2 L } X _ 3 , ~ ~ ~ \\nabla ^ L _ { X _ 2 } X _ 1 = \\frac { - L - 1 } { 2 L } X _ 3 , \\\\ & \\nabla ^ L _ { X _ 1 } X _ 3 = \\frac { 1 - L } { 2 } X _ 2 , ~ ~ \\nabla ^ L _ { X _ 3 } X _ 1 = \\frac { - 1 - L } { 2 } X _ 2 , \\nabla ^ L _ { X _ 2 } X _ 3 = \\nabla ^ L _ { X _ 3 } X _ 2 = \\frac { 1 + L } { 2 } X _ 1 . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} f ( x _ { i } ) = x _ { i } f ( y _ { i } ) = g ( x _ { i } ) = g ( y _ { i } ) = y _ { i } . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} c _ { n + 1 } = e _ { \\sigma ( j _ { n + 1 } ) } \\in b _ 1 ^ { - 1 } A \\cap \\cdots \\cap b _ { n + 1 } ^ { - 1 } A \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\widetilde { C } _ { q } ( K , Q , S ^ { n - 1 } \\cap N ( K , o ) ) = 0 \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} 2 ^ { m _ 0 } > \\frac { 2 ^ { 6 ( n + 2 ) } \\Gamma ^ 4 } { \\eta _ 0 ^ 4 } , \\qquad \\eta _ 0 = \\frac { \\eta \\delta } { ( 1 + \\eta ) 2 ^ { 3 ( n + 2 ) } } . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ \\Omega \\chi ( d d ^ c \\max ( u , w + \\frac { 1 } { j } ) ) ^ n = \\int _ \\Omega \\chi ( d d ^ c u ) ^ n \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} \\nu ( t _ 2 ) = s _ 1 . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\bar N } 3 ^ { - j } \\ln h ^ { - 1 } _ { j } > \\ln | x _ 0 | ^ 3 . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} P _ { q , z } \\cdot \\varphi _ { q , z } ^ * J ^ { K \\textnormal { t h , e q } } ( t , z , Q ) = \\sum _ { i = 0 } ^ N \\left ( \\Lambda _ i ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { 1 } { z ^ { d ( N + 1 ) } } \\frac { ( 1 - q ) ^ { d ( N + 1 ) } Q ^ d } { \\left ( q \\Lambda _ 0 \\Lambda _ i ^ { - 1 } , \\dots q , \\dots , q \\Lambda _ N \\Lambda _ i ^ { - 1 } ; q \\right ) _ d } \\right ) \\Psi _ i \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\alpha } ( K | E ) ~ = ~ \\min \\left \\{ n \\in \\mathbb { N } ~ | ~ \\exists ~ \\alpha \\mathrm { - c o v e r i n g ~ o f } ~ K ~ \\mathrm { h a v i n g ~ s i z e } ~ n \\right \\} \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\Omega _ \\Pi ^ \\circ ( n ) \\ : = \\ \\left | \\left \\{ \\varphi \\in [ n ] ^ \\Pi : \\ , a \\prec b \\ \\Longrightarrow \\varphi ( a ) < \\varphi ( b ) \\right \\} \\right | . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} N = 1 9 ( n + 2 ) + \\bigl \\lfloor 4 \\log _ 2 ( \\Gamma / \\delta ) + 4 \\log _ 2 \\bigl ( 1 + \\eta ^ { - 1 } \\bigr ) \\bigr \\rfloor . \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} w _ { n } \\coloneqq \\bigvee _ { i = 0 } ^ { n } \\big ( x _ { i } \\to \\bigvee _ { j \\ne i } x _ { j } \\big ) . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\mathcal { L } w _ 0 = \\mu w _ 0 + \\nu Q \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} t ^ * : = \\inf \\{ t \\in [ 0 , \\infty ) : v ^ { ( 1 ) } ( t ) \\neq v ^ { ( 2 ) } ( t ) \\} . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} f ( t ) = \\alpha + \\beta t + \\int _ 0 ^ \\infty \\Big ( \\frac { \\lambda } { 1 + \\lambda ^ 2 } - \\frac { 1 } { t + \\lambda } \\Big ) d \\mu ( \\lambda ) , \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} y \\leq ^ { \\boldsymbol { Y } } x _ { ( t + 1 ) n } ' f ( x _ { ( t + 1 ) n } ' ) = x _ { ( t + 1 ) n } . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} f _ { u _ 1 } = ( f _ 1 ) _ { u _ 1 } \\partial _ { x _ 1 } + ( f _ 2 ) _ { u _ 1 } \\partial _ { x _ 2 } + ( f _ 3 ) _ { u _ 1 } \\partial _ { x _ 3 } = \\frac { ( f _ 1 ) _ { u _ 1 } } { f _ 1 } X _ 1 + ( f _ 3 ) _ { u _ 1 } X _ 2 + \\sqrt { L } \\left [ \\frac { ( f _ 2 ) _ { u _ 1 } } { f _ 1 } - ( f _ 3 ) _ { u _ 1 } \\right ] \\widetilde { X _ 3 } , \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} E _ { \\varepsilon } \\left ( u , D \\right ) = \\int _ { D } p | \\nabla u | ^ { 2 } + \\dfrac { 1 } { \\varepsilon ^ { 2 } } \\int _ { D } J \\left ( 1 - | u | ^ { 2 } \\right ) \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} | e ^ { s B _ i ( t ) } f | _ q = | f | _ q , \\ \\forall f \\in L ^ q ( \\mathbb { R } ^ d ) , \\ s \\in \\mathbb { R } , \\ t \\geq 0 , i = 1 , 2 , . . . , N ; \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} C _ 1 ( z ) & = \\frac { \\partial } { \\partial \\nu ' } \\left [ \\prod _ p \\left ( 1 + a ( p ) \\frac { 1 - p ^ { \\nu ' - \\nu } } { p ^ { 3 / 2 + 3 \\nu ' } - 1 } \\right ) \\right ] _ { \\nu ' = \\nu = z } \\\\ & = \\prod _ p \\left ( 1 + a ( p ) \\frac { 1 - 1 } { p ^ { 3 / 2 + 3 z } - 1 } \\right ) \\cdot \\sum _ q \\left ( 1 + a ( q ) \\frac { 1 - 1 } { q ^ { 3 / 2 + 3 z } - 1 } \\right ) ^ { - 1 } B ( q , z ) \\\\ & = \\sum _ p \\frac { - a ( p ) \\log p } { p ^ { 3 / 2 + 3 z } - 1 } , \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} - \\left ( \\alpha \\varphi _ k + d \\psi _ k \\right ) '' = \\mu _ 1 ( 1 - 2 u _ { 1 , k } ) \\alpha \\varphi _ k + \\mu _ 2 ( 1 - 2 u _ { 2 , k } ) \\psi _ k + \\lambda _ { 1 , k } \\left ( \\alpha \\varphi _ k + \\psi _ k \\right ) , \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} \\pi { G } : = G / \\gamma _ c { G } c \\ge 2 \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta \\big ( | d u | ^ { 2 ( p - 1 ) } \\big ) & = | d u | ^ { p - 1 } \\Delta ( | d u | ^ { p - 1 } ) + \\Big | \\nabla | d u | ^ { p - 1 } \\Big | ^ 2 \\\\ & \\leq | d u | ^ { p - 1 } \\Delta ( | d u | ^ { p - 1 } ) + \\Big | \\nabla \\big ( | d u | ^ { p - 2 } d u \\big ) \\Big | ^ 2 . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\# \\theta _ 1 = 3 , \\ , \\# \\theta _ 2 = 1 \\ , . \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} I _ { z _ \\varepsilon } ( \\gamma _ \\varepsilon ) = \\gamma _ \\varepsilon ^ { - 4 } + \\frac { A ( \\gamma _ \\varepsilon ) } { 2 } + \\frac { 4 B ( \\gamma _ \\varepsilon ) } { \\gamma _ \\varepsilon ^ 3 \\exp ( 1 + \\mathcal { H } _ { z _ \\varepsilon } ( z _ \\varepsilon ) ) } \\int _ { \\Omega } G _ { z _ \\varepsilon } ( y ) F ( 4 \\pi G _ { z _ \\varepsilon } ( y ) ) ~ d y \\ , , \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{gather*} D \\mu = \\gamma . \\end{gather*}"}
-{"id": "7478.png", "formula": "\\begin{align*} g ( t _ 0 , \\cdots , t _ s ; x ) & = \\sum _ { k = 0 } ^ { s } t _ { s - k } x ^ { s - k } , \\\\ f ^ { ( n ) } ( t _ 0 , \\cdots , t _ s , t ; x ) & = x ^ n + t \\cdot g ( t _ 0 , \\cdots , t _ s ; x ) \\\\ \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 4 ( n ) } > \\varepsilon \\right ) = - \\varepsilon . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} P ( f ) = \\sum _ { [ \\pi ] \\in \\widehat { G } } P \\left ( d _ \\pi f \\ast \\chi _ \\pi \\right ) \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\left ( S ^ \\textnormal { c o h } \\right ) ^ { - 1 } ( \\tau , z ) ( T _ a ) = \\sum _ { j = 0 } ^ N \\textbf { g } \\left ( e ^ { \\tau _ 2 / z } T _ j + \\sum _ { \\substack { d \\in H _ 2 ( X ; \\mathbb { Z } ) - \\{ 0 \\} \\\\ l \\geq 0 } } \\sum _ k \\frac { 1 } { l ! } e ^ { \\tau _ 2 ( d ) } \\left \\langle \\frac { e ^ { \\tau _ 2 / z } T _ j } { - z + \\psi } , T _ k , \\tau ' , \\dots , \\tau ' \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } T ^ k , T _ a \\right ) T ^ j \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} L ( a _ 0 ) = d ( \\imath _ V ( a _ 0 ) ) + \\imath _ V ( d ( a _ 0 ) ) = \\imath _ V ( a _ 1 a _ 0 ) = a _ 1 ( V ) a _ 0 - a _ 0 ( V ) a _ 1 = a _ 1 . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { W } \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top = \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top \\\\ \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top \\boldsymbol { W } ^ \\top = \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top \\\\ \\boldsymbol { W } \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top \\boldsymbol { W } ^ \\top = \\boldsymbol { 1 } \\boldsymbol { 1 } ^ \\top \\end{aligned} \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\Gamma ( x , 0 ) = & ( x - \\frac { 3 } { 4 L } ) ^ 3 ( x ^ 3 - L x - \\frac { 1 } { 4 } ) \\\\ = & x ^ 6 - \\frac { 9 } { 4 L } x ^ 5 + ( \\frac { 2 7 } { 1 6 L ^ 2 } - L ) x ^ 4 + ( 2 - \\frac { 2 7 } { 6 4 L ^ 3 } ) x ^ 3 - \\frac { 9 } { 8 L } x ^ 2 + \\frac { 2 7 } { 2 5 6 L ^ 3 } . \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\check { \\zeta } _ \\varepsilon = \\max \\left ( \\frac { 1 } { \\gamma _ \\varepsilon ^ 4 } , | A ( \\gamma _ \\varepsilon ) | , \\frac { | B ( \\gamma _ \\varepsilon ) | } { \\gamma _ \\varepsilon ^ 3 } \\right ) \\ , . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} H _ { X } ( \\mathbf { x } ) = \\int _ { 0 } ^ { 1 } \\langle X ( t \\mathbf { x } ) , \\mathbf { x } \\rangle \\mathrm { d } t , \\forall \\mathbf { x } \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\exp _ p ( a _ n k h _ p + a _ n k \\varepsilon _ n ) = \\exp _ p ( k h _ p ) ^ { a _ n } \\exp _ p ( \\varepsilon _ n ) ^ { k a _ n } \\ ; . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\begin{cases} ( n - 2 ) f h \\varphi '' - r f \\varphi h '' - m h \\varphi f '' - 2 m h \\varphi ' f ' - 2 r f \\varphi ' h ' = 0 \\\\ \\rho = \\lambda _ F = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\bigcap _ { n = 1 } ^ \\infty E ( G , d 2 ^ { - n } ) = G \\cup \\partial G , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} \\left \\| | f | \\ast ( \\varphi ^ { * } ) _ { U } ^ { \\sharp } \\right \\| _ { 2 } \\le \\| f \\| _ { 1 } \\| ( \\varphi ^ { * } ) _ { U } ^ { \\sharp } \\| _ { 2 } = \\| f \\| _ { 1 } \\| \\varphi ^ { * } \\| _ { W ( L ^ { \\infty } , L ^ { 2 } ) } ~ , \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\chi _ { A ^ { \\prime } } ( u ) = \\chi _ { A } ( u ) + { \\textstyle \\int _ { u } } T p ( A ^ { \\prime } , A ) , \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\frac { \\lambda e ^ { - \\lambda x } ( \\lambda x ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } , & \\ , \\ , x \\geq 0 , \\\\ 0 , & \\ , \\ , x < 0 , \\end{cases} \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} \\mathrm { S I N R } _ { D _ { j } ^ { t } , D _ j ^ { r } } \\ ! = \\ ! \\frac { \\mathrm { S N R } _ { D _ { j } ^ { t } , D _ j ^ { r } } } { \\mathrm { S N R } _ { A _ i , D _ { j } ^ { r } } \\ ! + \\ ! \\underset { D _ k \\in \\mathcal D _ u ^ i \\backslash \\{ D _ j \\} } { \\sum } \\mathrm { S N R } _ { D _ { k } ^ { \\ , t } , D _ { j } ^ { r } } \\ ! + \\ ! 1 } . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\partial _ t W ^ { \\rm u n } ( t , y , k ) + \\bar \\omega ' ( k ) \\partial _ y W ^ { \\rm u n } ( t , y , k ) = - 2 \\gamma _ 0 R ( k ) W ^ { \\rm u n } ( t , y , k ) , y \\not = 0 , \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\omega } & f ( u _ k ) \\dot u _ k v d t \\\\ & = \\int _ 0 ^ { \\omega } \\big ( f ( u _ k ) - f ( u ) \\big ) \\dot u _ k v d t + \\int _ 0 ^ { \\omega } f ( u ) \\dot u _ k v d t . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} f _ { j _ 1 } ( I _ { j _ 1 } ) = f _ { j _ 2 } ( I _ { j _ 2 } ) , \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal { K } } ( g ( x _ t ) + \\lambda _ t ) & = \\Pi _ { \\mathcal { K } } \\big ( g ( \\overline { x } ) + t g ' ( \\overline { x } ) \\xi ^ * + \\overline { \\lambda } + t v ^ * + o ( t ) \\big ) \\\\ & = \\Pi _ { \\mathcal { K } } ( g ( \\overline { x } ) + \\overline { \\lambda } ) + t \\Pi _ { \\mathcal { K } } ' ( g ( \\overline { x } ) \\ ! + \\ ! \\overline { \\lambda } ; g ' ( \\overline { x } ) \\xi ^ * + v ^ * ) + o ( t ) \\\\ & = g ( \\overline { x } ) + t g ' ( \\overline { x } ) \\xi ^ * + o ( t ) , \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} T ^ { \\tilde { D } } = T ^ { D ^ { ( 1 ) } } - \\frac { 1 } { 6 } ( \\partial _ { \\mathbb { H } } \\circ \\tilde { \\pi } ) ( T ^ { D ^ { ( 1 ) } } ) = P ( T ^ { D ^ { ( 1 ) } } ) = \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { 3 } N _ { \\mathcal J _ { i } } , \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\Gamma ( k ) = \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } \\\\ \\Gamma ( k ) _ { 2 , 1 } \\\\ \\vdots \\\\ \\Gamma ( k ) _ { n , 1 } \\end{bmatrix} \\otimes \\begin{bmatrix} \\Gamma ( k ) _ { 1 , 1 } \\\\ \\Gamma ( k ) _ { 1 , 2 } \\\\ \\vdots \\\\ \\Gamma ( k ) _ { 1 , n } \\end{bmatrix} ^ { \\top } \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} B _ { W } ( z , w ) = 2 m \\alpha ^ { \\tfrac { 3 - 2 m + n } { 2 m } } E _ { \\tfrac { 1 } { m } , \\tfrac { 2 + n } { 2 m } } \\big ( \\alpha ^ { 1 / m } ( z \\bar { w } ) \\big ) . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} H ( t ) = \\dfrac { 2 E ' ( t ) } { v ( t , \\cdot ) E ( t ) } . \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } = \\frac { \\partial } { \\partial x ^ { i } } a _ { i } ( x , s , u , D u ) - a ( x , s , u , D u ) \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\eta _ n ( t ) = | | ( u ^ n , \\mathbf { w } ^ n , \\mathbf { h } ^ n , \\theta ^ n , \\psi ^ n ) ( t ) | | _ { H ^ 1 ( \\Omega ) } ^ 2 + \\int _ 0 ^ t | | ( u _ { y y } ^ n , \\mathbf { w } _ { y y } ^ n , \\mathbf { h } _ { y y } ^ n , \\theta _ { y y } ^ n ) ( s ) | | _ { L ^ 2 ( \\Omega ) } ^ 2 d s . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} ( \\R _ + \\times \\R _ + ) \\backslash S = \\Delta _ 1 \\cup \\Delta _ 2 , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} H ^ { p ^ { \\ast } } _ { , i } = \\sum _ k h ^ { p ^ { \\ast } } _ { i k } \\langle X , e _ k \\rangle , i , p = 1 , 2 . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} X _ { ( T ^ \\ast Q , \\omega , H , F , u ) } = X _ H + \\textnormal { v l i f t } ( F ) + \\textnormal { v l i f t } ( u ) . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} p \\left ( \\mathbf { V } \\right ) = \\prod \\limits _ { k = 1 } ^ { s } p _ { k } \\left ( V _ { k } | _ { \\mathcal { G } } \\left ( V _ { k } \\right ) \\right ) \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ s R _ { 2 n } ^ \\lambda ( x ) = a _ n ^ \\lambda \\ , \\sum _ { k = 0 } ^ { n } \\frac { ( - n ) _ { k } ( n + \\lambda ) _ { k } } { ( \\lambda + \\frac { 1 } { 2 } ) _ { k } k ! } ( - \\Delta ) ^ s \\bigg \\{ \\frac { 1 } { \\left ( 1 + x ^ 2 \\right ) ^ { k + \\frac { \\lambda + 1 } { 2 } } } \\bigg \\} , \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\frac 1 2 \\left ( \\frac { r } { c } + \\frac { c } { r } \\right ) = \\frac { a } { c } \\ \\ \\mbox { a n d } \\ \\ \\frac 1 2 \\left ( \\frac { r } { c } - \\frac { c } { r } \\right ) = \\frac { b } { c } \\ . \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\tau _ { i + 1 } = \\tau _ { i } \\omega _ { i } , \\sigma _ { i + 1 } = \\sigma _ { i } / \\omega _ { i } , \\omega _ i = 1 / \\sqrt { 1 + 2 \\tau _ i \\tilde { \\gamma } _ G } \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} C ^ { - 1 } ( \\{ M ^ 2 / 2 \\} ) \\cap U _ { 0 } & = C ^ { - 1 } ( \\{ M ^ 2 / 2 \\} ) \\cap \\left ( \\{ ( x , y , z ) \\in \\mathbb { R } ^ 3 : x = 0 \\} \\cup \\{ ( x , y , z ) \\in \\mathbb { R } ^ 3 : y = 0 \\} \\right ) \\\\ & = \\mathcal { E C } ^ { - 1 } ( \\{ ( 0 , M ^ 2 / 2 ) \\} ) \\subset C ^ { - 1 } ( \\{ M ^ 2 / 2 \\} ) \\cap U _ { \\beta } , ~ \\forall \\beta \\in ( 0 , \\beta _ 0 ) , \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { H } ( x , P + \\nabla u _ 0 ( x ) ) = \\ln ( m _ 0 ( x ) ) + \\overline { H } ( P ) , \\ & \\ \\mathbb { T } ^ d , \\\\ - \\div \\big ( m _ 0 ( x ) D _ \\Lambda \\widetilde { H } ( x , P + \\nabla u _ 0 ( x ) \\big ) = 0 , \\ & \\ \\mathbb { T } ^ d , \\\\ \\int _ { \\mathbb { T } ^ d } m _ 0 d x = 1 . \\end{cases} \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\sqrt { 2 } \\{ \\tilde { M } ( x ) - \\tilde { M } ( x - \\tau _ 1 ) - \\tilde { M } ( x - \\tau _ 2 ) + \\tilde { M } ( x - \\tau _ 1 - \\tau _ 2 ) \\} = \\sqrt { 2 } U _ 2 ' ( x ) . \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\mathrm { r e s } ( x - \\sigma _ { \\mathrm { r e s } ( \\alpha ) } ( x ) ) = ( a _ 1 + . . . + a _ n ) \\mathrm { r e s } ( \\alpha ) \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\bar { \\sigma } _ { A } ^ { \\ast } ( \\mathbf { A } ) = A \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\Omega _ { \\widetilde { s } , { s } , { \\eta } } : = \\left ( { \\Omega ^ { ( s ) } } \\setminus { W _ s } \\right ) \\cup \\{ x \\in { W _ s } \\colon \\widetilde { s } ( x ) < \\eta \\} \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\begin{aligned} d x ( t ) & = v ( t ) \\ , d t , \\\\ m \\ , d v ( t ) & = \\Big ( - \\gamma v ( t ) - \\Phi ' ( x ( t ) ) - \\sum _ { k = 1 } ^ N \\sqrt { c _ i } z _ i ( t ) \\Big ) d t + \\sqrt { 2 \\gamma } \\ , d W _ 0 ( t ) , \\\\ d z _ k ( t ) & = ( - \\lambda _ k z _ k ( t ) + \\sqrt { c _ k } v ( t ) ) \\ , d t + \\sqrt { 2 \\lambda _ k } \\ , d W _ k ( t ) , 1 \\leq k \\leq N , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} & U _ { 2 k + 1 , 2 a + 1 } ( x ; q ) - U _ { 2 k + 1 , 2 a - 1 } ( x ; q ) \\\\ & = ( 1 + x q ) ( x q ) ^ { 2 a + 1 } \\overline { U } _ { 2 k + 1 , 2 k - 2 a } ( x q ; q ) + ( 1 + x q ) ( x q ) ^ { 2 a - 1 } \\overline { U } _ { 2 k + 1 , 2 k - 2 a + 2 } ( x q ; q ) , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\frac { \\langle f _ 3 , f _ 1 \\rangle } { | | f _ 3 | | | | f _ 1 | | } & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 0 } ^ { \\infty } \\gamma _ j \\overline { \\alpha _ { j + 1 } } \\\\ & = - \\frac { \\overline { a } } { 1 + | a | ^ 2 } \\sum _ { j = 0 } ^ { \\infty } e ^ { i ( \\eta _ 1 + j \\eta _ 3 ) } \\sqrt { ( 1 - \\rho ) \\rho ^ j } e ^ { - i ( \\eta _ 2 + j \\eta _ 3 ) } \\sqrt { ( 1 - \\rho ) \\rho ^ j } \\\\ & = - e ^ { i \\theta _ 2 } \\frac { | a | } { 1 + | a | ^ 2 } ; \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\mathcal { E L } ( L ) ( q , \\dot { q } , \\ddot { q } ) = F ( q , \\dot { q } ) + u ( q , \\dot { q } ) , \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} \\mathcal { N _ { \\alpha } } \\left ( B _ \\rho ( z _ 0 , 2 ^ { i + 1 } \\alpha ) ~ \\Big | ~ E \\right ) ~ \\leq ~ \\sum _ { j = 1 } ^ { 2 ^ { { \\bf d } ( E ) } } \\mathcal { N _ { \\alpha } } \\left ( B _ \\rho ( x _ j , 2 ^ { i } \\alpha ) ~ \\Big | ~ E \\right ) ~ \\leq ~ 2 ^ { { \\bf d } ( E ) } \\cdot 2 ^ { i { \\bf d } ( E ) } ~ = ~ 2 ^ { ( i + 1 ) { \\bf d } ( E ) } . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\wp ( x ) \\cdot \\left ( a _ { ( x \\wp ' ) } b - \\frac { 1 } { 2 } T ( a _ { ( x ^ 2 \\wp ' ) } b ) \\right ) \\otimes m = \\wp ( x ) \\cdot \\left ( - 2 a ( - 2 ) b + T ( a ( - 1 ) b ) \\right ) \\otimes m . \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\parallel v \\parallel _ { H ^ { s + 2 } ( \\Omega _ f ) } + \\parallel q \\parallel _ { H ^ { s + 1 } ( \\Omega _ f ) } \\leqslant C \\parallel v _ t \\parallel _ { H ^ { s } ( \\Omega _ f ) } + C \\parallel w _ { \\nu _ { \\mathcal { N } } } \\parallel _ { H ^ { s + \\frac { 1 } { 2 } } ( \\Gamma _ c ) } , s = 0 , 1 , 2 \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\nabla _ { ( 1 - q ) \\partial _ { t _ i } } = ( 1 - q ) \\partial _ { t _ i } + \\phi _ i \\star _ \\tau \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} { P r } _ D ( f , \\sigma , \\{ k _ j \\} ) : & = \\lim _ { X \\to \\infty } \\displaystyle \\frac { \\# \\{ p \\in { S p l } _ X ( f , \\sigma , \\{ k _ j \\} ) \\mid ( r _ 1 / p , \\dots , r _ n / p ) \\in D \\} } { \\# { S p l } _ X ( f , \\sigma , \\{ k _ j \\} ) } \\\\ & = \\displaystyle \\frac { { v o l } ( { D } \\cap { \\mathfrak { D } } ( f , \\sigma , \\{ k _ j \\} ) ) } { { v o l } ( { \\mathfrak { D } } ( f , \\sigma , \\{ k _ j \\} ) ) } . \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\deg ( q ) \\leq \\deg ( p ) \\mbox { a n d } \\nu ( p ) = \\nu _ q ( p ) . \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla _ { \\partial _ { t _ i } } T _ j = \\left ( \\partial _ { t _ i } + \\frac { 1 } { z } T _ i \\bullet _ \\tau \\right ) T _ j , & & 0 \\leq i \\leq N \\\\ & \\nabla _ { \\partial _ z } T _ j = \\left ( \\partial _ z - \\frac { 1 } { z ^ 2 } \\mathfrak { E } \\bullet _ \\tau + \\frac { 1 } { z } \\mu \\right ) T _ j \\end{aligned} \\right . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} c _ 1 & = 2 \\log ( a + 1 ) + \\frac { a \\pi } { a ^ 2 - 1 } , \\\\ c _ 2 & = 2 \\log \\frac { a } { a ^ 2 - 1 } + \\frac { 2 a \\pi } { a ^ 2 - 1 } . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} ( r , \\theta , z ) \\to \\exp ( r \\cos \\theta X + r \\sin \\theta Y ) \\exp ( z Z ) = \\begin{pmatrix} \\cos ( r ) e ^ { i z } & \\sin ( r ) e ^ { i ( \\theta - z ) } \\\\ - \\sin ( r ) e ^ { - i ( \\theta - z ) } & \\cos ( r ) e ^ { i z } \\end{pmatrix} \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} C _ { i j k } = \\frac { 1 } { n + 1 } ( h _ { i j } \\ , C _ k + h _ { k i } \\ , C _ j + h _ { j k } \\ , C _ i ) . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} 2 Q - L _ 0 T \\sinh \\delta / 2 = 0 \\implies \\delta = \\sinh ^ { - 1 } \\left ( \\frac { 4 Q } { T L _ 0 } \\right ) . \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} M [ \\alpha ] = 0 \\ , \\ , M \\in L R ( \\chi ) . \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} ( x + y ) ^ n = \\sum _ { k = 0 } ^ n \\binom { n } { k } _ q x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\int _ { B _ { x _ \\varepsilon } ( R \\mu _ \\varepsilon ) } | \\nabla u _ \\varepsilon | ^ 2 d y = \\frac { 4 \\pi } { \\gamma _ \\varepsilon ^ 2 } \\left ( \\log ( 1 + R ^ 2 ) - \\frac { R ^ 2 } { 1 + R ^ 2 } + o ( 1 ) \\right ) \\ , , \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} a _ { E } ( n ) = \\frac { \\pi ^ { 2 } n } { \\sqrt { D ( Q ) } } \\prod _ { p ~ } \\beta _ { p } ( Q ; n ) . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\phi _ \\zeta = ( \\zeta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ l , 1 ) \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} ( A _ 1 , A _ 2 , \\dots , A _ m ) \\ \\longmapsto \\ \\phi \\big ( \\exp \\big ( H + \\sum _ { j = 1 } ^ m p _ j \\log A _ j \\big ) \\big ) \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} u _ 2 & = 0 , u _ 1 = { \\rm i } \\nu \\langle K , h \\rangle , f _ 1 ( t ) = - { \\rm i } \\nu \\langle K , h ' ( - t + \\cdot ) \\rangle , \\\\ \\psi _ 1 ( t ) & = u _ 1 + ( 1 * f _ 1 ) ( t ) = { \\rm i } \\nu \\langle K ( t + \\cdot ) , h \\rangle , \\\\ f _ 2 ( t ) & = { \\rm i } ( \\frac { \\nu } 2 - \\lambda ) \\langle K ( t + \\cdot ) , h \\rangle , \\psi _ 2 = K * F ( \\psi _ 1 , \\psi _ 2 ) , \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\int _ { t - a } ^ { t - b } \\tilde { M } ( W ) ^ m ( - d W ) & = - \\left ( \\int _ { - \\infty } ^ { t - b } \\tilde { M } ( W ) ^ m d W - \\int _ { - \\infty } ^ { t - a } \\tilde { M } ( W ) ^ m d W \\right ) \\\\ & = \\int _ { 0 } ^ { t - a } \\tilde { M } ( W ) ^ m d W - \\int _ { 0 } ^ { t - b } \\tilde { M } ( W ) ^ m d W \\\\ & = \\frac { 1 } { m + 1 } \\left ( \\tilde { M } ( t - a ) ^ { m + 1 } - \\tilde { M } ( t - b ) ^ { m + 1 } \\right ) . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\int _ { D _ r ( a ) } ( 2 - p ) | d u | ^ p = r \\int _ { \\partial D _ r ( a ) } ( | d u | ^ p - p | d u | ^ { p - 2 } | d u ( \\frac { z - a } { | z - a | } ) | ^ 2 \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\langle | b _ 1 - \\langle b _ 1 \\rangle _ { J _ k } | \\rangle _ { A , J _ k } \\langle | b _ 2 - \\langle b _ 2 \\rangle _ { J _ k } | \\rangle _ { B , J _ k } & = \\langle | b _ 1 | \\rangle _ { A , J _ k } \\langle | b _ 2 | \\rangle _ { B , J _ k } \\\\ & \\simeq k ^ { - \\frac 1 2 } \\log ( e + k ) ^ { \\frac { 1 + \\alpha } 2 } k ^ { \\frac 1 2 } \\log ( e + k ) ^ { - \\frac 1 2 } \\log \\log ( e ^ e + k ) ^ { - \\frac 3 4 } \\\\ & \\overset { k \\rightarrow \\infty } { \\rightarrow } \\infty . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { \\bar C } = \\bar C W ' ( \\bar A ) \\\\ & \\dot { \\bar A } = \\bar C - W ( \\bar A ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\psi = ( \\tau _ 1 , b _ 1 ) \\boxplus \\cdots \\boxplus ( \\tau _ r , b _ r ) , \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\exp ( t \\nabla ^ \\pm H ( g ) ) = b ( t ) k _ \\pm ( t ) ^ { - 1 } , \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} K _ G ( X ) = K _ 0 ' ( ( X \\times T ) / G ) . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\aligned & \\frac { 1 } { 2 } \\mathcal { L } \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } \\\\ = & H ^ { 2 } \\bigl [ H ^ { 2 } - 2 S + \\frac { 1 } { 2 } H ^ { 4 } - H ^ 2 \\lambda ^ 2 - K H ^ { 2 } - K ^ { 2 } \\bigl ] \\\\ & + H ^ { 2 } ( S + 2 - \\frac { 3 } { 2 } H ^ { 2 } - \\lambda _ { 1 } ^ { 2 } - \\lambda _ { 2 } ^ { 2 } - 2 \\lambda ^ 2 ) ( \\lambda _ { 1 } ^ { 2 } + \\lambda _ { 2 } ^ { 2 } + 2 \\lambda ^ 2 ) . \\endaligned \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} \\mu \\left ( T _ j \\right ) = \\frac { 1 } { 2 } \\left ( \\textnormal { d e g } _ { H ^ * ( X ) } ( T _ j ) - \\textnormal { d i m } _ \\mathbb { C } ( X ) \\right ) T _ j \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} \\theta _ { M _ { S ' } } ( \\mu _ { S ' } ) = \\theta _ { M _ { S ' } } ( \\mu _ { S } ) = \\frac { \\theta _ { M _ S } ( \\mu _ S ) + \\sigma _ { \\alpha } ( \\theta _ { M _ S } ( \\mu _ S ) ) } { 2 } , \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} L ^ p ( \\mathcal M ) = \\left \\{ A \\in \\mathcal A : \\tau \\left ( ( A ^ \\star A ) ) ^ { \\frac p 2 } \\right ) < \\infty \\right \\} ; \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\Vert t _ m f \\Vert _ { \\ell ^ { p , \\infty } } = \\Vert k \\ast f \\Vert _ { \\ell ^ { p , \\infty } } \\leq \\Vert k \\Vert _ { \\ell ^ { p , \\infty } } \\Vert f \\Vert _ { \\ell ^ 1 } \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} \\mathbf { M } ( \\overline { g } ) ( V ^ { ( 0 ) } ) = \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { 4 \\pi } \\int _ { \\Sigma } \\frac { t ( 1 + t ^ 2 ) } { 2 } ( 1 - u ^ { - 2 } ) d \\sigma . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\frac { m - 1 } { 2 } } ( 1 + q ^ n + q ^ { 2 n } ) \\frac { ( 1 - q ) ( q ; q ^ 2 ) _ { n } ( q ^ { 2 } ; q ^ 2 ) _ { n - 1 } q ^ { - { n \\choose 2 } - \\frac { m - 1 } { 2 } } } { ( 1 + q ^ n ) ( q ; q ) _ { n } ^ 2 } \\\\ [ 5 p t ] & = \\sum _ { n = 1 } ^ { \\frac { m - 1 } { 2 } } \\frac { 1 + q ^ n + q ^ { 2 n } } { ( 1 + q ^ n ) [ 2 n ] } { 2 n \\brack n } q ^ { - { n \\choose 2 } - \\frac { m - 1 } { 2 } } \\equiv 0 \\pmod { \\Phi _ m ( q ) } , \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} f _ i ( x ^ * ) = & \\int _ { - \\frac { 1 } { 2 a _ { i , T } \\lambda } } ^ { \\frac { 1 } { 2 a _ { i , T } \\lambda } } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left ( \\dfrac { - w ^ 2 } { 2 } \\right ) d w \\le \\int _ { - \\frac { 1 } { 2 a _ { i , T } \\lambda } } ^ { \\frac { 1 } { 2 a _ { i , T } \\lambda } } d w \\\\ = & \\frac { 1 } { | a _ { i , T } | \\lambda } \\le \\frac { 1 } { \\min _ { 1 \\le i \\le m } | a _ { i , T } | \\lambda } . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} ( P ^ { - ( r - 1 ) } \\mathcal { A } ( H ) P ) _ { i _ 1 i _ 2 \\cdots i _ r } = p _ { i _ 1 } ^ { - ( r - 1 ) } a _ { i _ 1 i _ 2 \\cdots i _ r } p _ { i _ 2 } \\cdots p _ { i _ r } . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\beta ( h ) = \\tilde { u } + v + \\Pi _ V \\beta ( h ) \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} B ^ { ( \\theta ) } f = \\sum _ { I \\in \\mathcal { D } } \\frac { \\langle f , h _ I \\rangle } { \\| h _ I \\| _ 2 ^ 2 } b _ I ^ { ( \\theta ) } \\qquad A ^ { ( \\theta ) } f = \\sum _ { I \\in \\mathcal { D } } \\frac { \\langle f , b _ I ^ { ( \\theta ) } \\rangle } { \\| b _ I ^ { ( \\theta ) } \\| _ 2 ^ 2 } h _ I \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} G ( m , n , a , \\epsilon , s ) = \\sum _ { i _ 1 \\cdots i _ n \\in A ( m , n , a , \\epsilon ) } \\prod _ { k = 1 } ^ n i _ k ^ { - d s } . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} f _ { \\lambda ' } ( b _ 1 , b _ 2 ) = X _ \\lambda ( b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 ) , z = b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 \\in U _ 1 \\cup \\{ r \\omega _ 1 ; r \\in ( 0 , 1 ) \\} . \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { k _ { \\gamma , \\Sigma _ 1 } ^ { L } } { \\sqrt { L } } = \\frac { | \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) | } { \\left [ \\overline { q } { \\dot { \\gamma } _ 3 } - \\frac { \\sqrt { 2 } } { 2 } \\overline { p } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) \\right ] ^ 2 } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) = 0 ~ ~ a n d ~ ~ \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) \\neq 0 . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\left ( \\int _ \\R ( g \\ast f ) ( x ' , x _ 3 ) ^ p \\ , d x _ 3 \\right ) ^ { 1 / p } & = \\left ( \\int _ \\R \\left ( \\int _ { \\R ^ 2 } \\left ( \\int _ { \\R } f ( x ' - y ' , x _ 3 - y _ 3 ) g ( y ' , y _ 3 ) \\ , d y _ 3 \\right ) \\ , d y ' \\right ) ^ p d x _ 3 \\right ) ^ { 1 / p } \\\\ & \\le \\int _ { \\R ^ 2 } \\left ( \\int _ \\R \\left ( \\int _ \\R f ( x ' - y ' , x _ 3 - y _ 3 ) g ( y ' , y _ 3 ) \\ , d y _ 3 \\right ) ^ p \\ , d x _ 3 \\right ) ^ { 1 / p } \\ , d y ' \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} P _ { Z _ { i j } } ( z ) & = \\sum _ { w = z - n + 1 } ^ { z } P _ { W _ i } ( w ) , \\ \\ z = n , 2 n , \\ldots \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} \\mathbb { E } d ( x , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) = \\int _ { 0 } ^ { \\infty } \\mathbb { P } ( d ( x , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) \\geq r ) d r . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} & \\widehat W _ { + } ^ \\star ( \\eta , k ) = \\widehat W _ { + } ( - \\eta , k ) , \\widehat Y _ { + } ( \\eta , k ) = \\widehat Y _ { + } ( \\eta , - k ) , \\\\ & \\widehat W _ { - } ( \\eta , k ) = \\widehat W _ { + } ( \\eta , - k ) , \\widehat Y _ { - } ( \\eta , k ) = \\widehat Y _ { + } ^ \\star ( - \\eta , k ) . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} G _ { k } = ( 2 \\pi i ) ^ { k } \\left ( - \\frac { B _ { k } } { k ! } + \\frac { 2 } { ( k - 1 ) ! } \\sum _ { n = 1 } ^ \\infty n ^ { k - 1 } \\frac { q ^ n } { 1 - q ^ n } \\right ) \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} \\dim _ { P } ^ { * } \\zeta = \\inf \\{ s \\ge 0 \\ : : \\ : \\underset { \\eta \\downarrow 0 } { \\limsup } \\ : \\frac { \\log \\zeta ( B ( x , \\eta ) ) } { \\log \\eta } \\le s \\zeta x \\in \\mathbb { R } \\} \\ : . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} - f ( e _ x ) g ( x e _ y ) + f ( e _ y ) g ( y e _ { x ^ y } ) = - f ( e _ x ) g ( e _ y ) + f ( e _ y ) g ( e _ { x ^ y } ) . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { k = 1 : k \\neq j - 1 } ^ { r _ n } \\frac { \\lambda _ k ^ { ( n ) } } { \\lambda _ { j - 1 } ^ { ( n ) } - \\lambda _ k ^ { ( n ) } } \\geq \\epsilon \\qquad \\forall n \\geq 1 . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\frac 1 { \\theta _ i } : = \\frac { 1 - r } r - \\frac 1 { \\delta _ i } = \\left ( \\sum _ { j = 1 } ^ { m + 1 } \\frac 1 { \\delta _ j } \\right ) - \\frac 1 { \\delta _ i } > 0 . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\begin{cases} v _ i = v _ i ^ S \\qquad i \\in S _ 0 = S \\\\ \\displaystyle v _ l + \\frac { u _ l } { \\lambda _ 0 - \\omega ( l , l ) } = \\frac { 1 } { \\lambda _ 0 - \\omega ( l , l ) } \\sum _ { j \\in S _ { k - 1 } } \\omega ( l , j ) v _ j \\qquad l \\in S _ k \\setminus S _ { k - 1 } \\end{cases} \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\dot S _ { j _ 0 + 1 } ( u v ) = \\dot { T } _ { u } \\dot S _ { j _ 0 + 1 } v + \\dot S _ { j _ 0 + 1 } \\bigl ( \\dot { T } _ { v } u + \\dot { R } ( v , u ) \\bigr ) + [ \\dot S _ { j _ 0 + 1 } , \\dot { T } _ { u } ] v . \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} S _ { 1 , \\lambda } ( n + 1 , k ) = S _ { 1 , \\lambda } ( n , k - 1 ) - n \\lambda S _ { 1 , \\lambda } ( n , k ) , \\ , \\ , ( 1 \\leq k \\leq n ) , ( \\textnormal { s e e } \\ , \\ , [ 6 ] ) . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\Omega } p \\left | \\nabla u \\right | ^ { 2 } \\geq 2 \\pi p _ { 0 } \\left ( \\sum _ { i = 1 } ^ { m } d _ { i } ^ { 2 } \\right ) \\left ( \\log \\dfrac { R } { R _ { 0 } } - I \\left ( \\dfrac { R } { R _ { 0 } } \\right ) \\right ) + 2 \\pi p _ { 0 } a ^ { 2 } \\displaystyle \\sum _ { i \\neq j } d _ { i } d _ { j } \\log \\frac { R } { | a _ { i } - a _ { j } | } - C , \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( H - \\int M ( d s , \\theta ^ H _ s ) \\right ) \\int \\frac { \\partial } { \\partial x } M ( d s , \\theta ^ H _ s ) \\right ] = 0 . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\sum _ { j = n / 2 } ^ { n } \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } \\varepsilon _ j \\leq \\alpha n ^ { \\alpha - 1 } e ^ { n ^ \\alpha } \\sum _ { j = 1 } ^ { n / 2 } \\varepsilon _ j = o ( e ^ { n ^ \\alpha } ) . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} | A _ R ( x ) | & = | C _ { \\alpha , \\beta } | R ^ { - 2 ( \\alpha + \\beta ) } \\Big | \\int ^ R _ 0 ( R ^ 2 - t ^ 2 ) ^ { \\beta - 1 } t ^ { 2 \\alpha + 1 } ( B ^ \\alpha _ t ( x ) - B ^ { \\alpha + 1 } _ t ( x ) ) d t \\Big | \\\\ & \\leq | C _ { \\alpha , \\beta } | R ^ { - 2 ( \\alpha + \\beta ) } \\Big ( \\int ^ R _ 0 | ( R ^ 2 - t ^ 2 ) ^ { \\beta - 1 } t ^ { 2 \\alpha + 1 } | ^ 2 d t \\Big ) ^ { \\frac 1 2 } \\\\ & \\quad \\times \\Big ( \\int ^ R _ 0 | B ^ \\alpha _ t ( x ) - B ^ { \\alpha + 1 } _ t ( x ) | ^ 2 d t \\Big ) ^ { \\frac 1 2 } \\lesssim G ^ \\alpha ( x ) , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\left \\| f _ \\lambda \\right \\| _ { \\dot { W } ^ { s , p } ( \\R ^ d ) } = \\lambda ^ { \\frac { d } { p } - s } \\left \\| f \\right \\| _ { \\dot { W } ^ { s , p } ( \\R ^ d ) } \\ , . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} S _ 2 \\left ( \\left [ \\overline { f ( z , t ) } \\right ] _ { t = 1 } , z , 1 \\right ) = ( z - 1 ) ^ 3 \\left ( \\left [ \\overline { f ( z , t ) } \\right ] _ { t = 1 } \\right ) ^ 3 P _ 2 \\left ( \\left [ \\overline { f ( z , t ) } \\right ] _ { t = 1 } , z \\right ) = 0 , \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\tilde { K _ 1 } = 1 & & \\tilde { K _ 2 } = \\delta - 1 & & \\tilde { C } = 2 \\delta + 3 & & \\tilde { C ' } = 2 \\delta + 4 \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} A ( V ) = U ( V ) _ 0 / ( U ( V ) \\cdot U ( V ) _ { > 0 } ) _ 0 . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} a \\otimes b \\mapsto a ( 0 ) b = \\frac { 1 } { 2 \\pi i } \\left ( a b - b a \\right ) , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\chi : = F \\circ p \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\Psi [ B ( z , t ) ] = \\frac { t ^ 2 z ^ 4 } { ( 1 - z ) ^ 2 ( 1 - t z ) ( 1 - ( 1 + t ) z ) } \\left ( \\frac { B ( z , 1 ) - B ( z , t ) } { 1 - t } \\right ) . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\mathbb { P } ( U ^ c _ { t o t } ( n ) ) \\leq \\sum _ { l = 1 } ^ { N } \\mathbb { P } ( U ^ c _ l ) \\leq N \\exp \\left ( - C \\frac { n } { N } \\right ) , \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} H _ { g } ^ { 1 } ( G , \\mathbf { C } ) = \\left \\{ u \\in H ^ { 1 } ( G , \\mathbf { C } ) : u = g \\partial G \\right \\} . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} ( E F ) _ \\alpha : = \\sum _ { \\beta + \\gamma = \\alpha \\atop \\beta , \\gamma \\in \\N ^ n } E _ \\beta F _ \\gamma , | \\alpha | \\leq m . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} { \\cal R } \\left ( \\boldsymbol { x } _ i \\right ) = f _ i , i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} & x _ i ^ 2 \\le z _ i x _ N \\ & i & = 1 , \\dots , N - 1 \\\\ & \\sum _ { i = 1 } ^ { N - 1 } z _ { i } \\le x _ N \\\\ & x _ N \\ge 0 , z _ i \\ge 0 \\ & i & = 1 , \\dots , N - 1 , \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} T ( f ) ( x ) = \\lim _ { j \\rightarrow \\infty } T _ { n _ j } ( f ) ( x ) ; a . e . \\ ; x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\left ( 6 - c _ { 1 } \\right ) \\frac { \\omega _ { 3 } } { \\omega _ { 4 } } - c _ { 1 } \\frac { \\omega _ { 3 } ^ { \\prime \\prime } } { \\omega _ { 3 } ^ { \\prime } } + 4 c _ { 1 } \\frac { \\omega _ { 3 } ^ { \\prime } } { \\omega _ { 3 } } = 0 . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\nabla \\varphi _ { 0 } ( z ) = \\sum _ { i = 1 } ^ { m } d _ { i } \\frac { V _ { i } ( z ) } { \\left | z - a _ { i } \\right | } , \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} y _ t - \\Delta y = u + \\mathcal { W } [ y ] , \\mathcal { W } [ y ] ( x , t ) = \\mathcal { V } [ y ( x , \\cdot ) ] ( t ) . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\left [ \\partial _ { t _ i } , \\phi _ j \\star _ \\tau \\right ] \\phi _ k = 0 \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\boldsymbol { \\nu } ( \\partial T _ x ) = q ^ { ( n ) } ( o , x ) \\ , , \\ ; n = | x | \\ , . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} A _ i X : = \\left ( \\begin{array} { c } a _ { 1 i } \\partial _ 1 X _ 1 + a _ { 2 i } \\partial _ 2 X _ 1 + a _ { 3 i } \\partial _ 1 X _ 2 + a _ { 4 i } \\partial _ 2 X _ 2 \\\\ a _ { 5 i } \\partial _ 1 X _ 1 + a _ { 6 i } \\partial _ 2 X _ 1 + a _ { 7 i } \\partial _ 1 X _ 2 + a _ { 8 i } \\partial _ 2 X _ 2 \\end{array} \\right ) , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} \\rho _ { n , s } ( z ) = \\sum _ { k = 0 } ^ { + \\infty } \\gamma _ { n , k } ^ { ( s ) } u ^ { - n - k - 1 } , \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\log _ p \\chi ^ { ( 1 ) k } _ n = k \\log _ p \\chi ^ { ( 1 ) } _ n = k \\log _ p \\chi _ n = a _ n k h _ p + a _ n k \\varepsilon _ n \\ ; . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} d _ \\mathcal { H } ( A , B ) = \\inf \\left \\{ \\varepsilon \\ge 0 \\colon A \\subseteq [ B ] _ \\varepsilon B \\subseteq [ A ] _ \\varepsilon \\right \\} \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\min _ { \\alpha \\in [ 0 , \\alpha _ { \\max } ] } \\ , a \\alpha + \\frac { b } { 2 } \\alpha ^ 2 = \\left \\{ \\begin{array} { l l } - \\frac { a ^ 2 } { 2 b } & \\ ; \\alpha _ { \\max } > - \\frac { a } { b } \\\\ \\alpha _ { \\max } \\left ( a + \\frac { b } { 2 } \\alpha _ { \\max } \\right ) \\le \\frac { a } { 2 } \\alpha _ { \\max } & \\ ; \\alpha _ { \\max } \\le - \\frac { a } { b } . \\end{array} \\right . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\mathcal { C } _ i ( d _ i ) = c _ i d _ i , \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} s ^ \\textnormal { c o h } ( \\tau _ 2 , z ) = \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) \\right ) _ { | \\tau ' = 0 } \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} G \\backslash E = \\left ( \\bigcup _ { n = 1 } ^ { \\infty } f _ { n } ( ( \\mathbb { R } ^ { d } ) ^ { n } ) \\right ) \\backslash E \\neq \\emptyset , \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} { T _ G } = \\min \\left \\{ { K : \\mathop { \\max } \\limits _ { 1 \\le k \\le K } \\Lambda _ G ^ { ( k , K ) } \\ge h } \\right \\} , \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\Upsilon ( h ) : = - \\sigma \\big ( \\overline { \\lambda } , \\mathcal { T } _ { \\mathcal { K } } ^ 2 ( \\overline { y } , h ) \\big ) = \\langle \\overline { \\mu } , \\Xi '' ( \\overline { y } ) ( h , h ) \\rangle \\forall h \\in \\mathcal { C } _ { \\mathcal { K } } ( \\overline { y } , \\overline { \\lambda } ) . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b _ \\omega ^ { 1 ' } ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ n } { ( q ; q ^ 2 ) _ { n + 1 } } , \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\begin{cases} p _ \\theta \\in C ( 0 , \\infty ) \\cap C ^ 1 ( 0 , \\infty ) , & \\lim _ { v \\to \\infty } p _ \\theta ( v ) = 0 \\\\ p _ \\theta v \\in ( 0 , \\infty ) \\\\ p _ \\theta ( v ) \\leq p _ 0 ( 1 + v ^ { - \\Gamma } ) , & \\rho \\geq 0 , \\end{cases} \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\Delta ^ 2 = \\frac { 1 - q } { ( 1 + z _ 2 ) ( \\xi ' ( 1 ) - \\xi ' ( q ) ) } = \\frac { q } { \\xi ' ( q ) ( 1 + z _ 2 ) ( 1 + z _ 1 + z _ 2 ) } . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} d _ { } ( a \\otimes b \\otimes m ) & = b \\otimes m a - a b \\otimes m + a \\otimes b m \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\begin{cases} d U _ { t } ^ { ( l ) } = \\sum _ { \\alpha = 1 } ^ { d } W _ { \\alpha } ( U _ { t } ^ { ( l ) } ) d B _ { t } ^ { \\alpha } , \\\\ U _ { 0 } ^ { ( l ) } = 0 \\end{cases} \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} b _ \\alpha b _ \\beta = \\sum _ { \\gamma \\in \\Gamma } n _ { \\alpha , \\beta } ^ \\gamma b _ \\gamma \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} X _ p ( \\vec { a } ) = \\big ( 1 + O _ { \\leq } ( \\log ^ { - 3 } x ) \\big ) \\cdot \\sigma ^ r . \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} k : = m + q - p + 1 \\ \\mathrm { a n d } \\ \\ell : = q + 1 - \\sigma - \\tau . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} \\begin{array} { c } I d _ m \\\\ \\updownarrow \\\\ \\mathfrak p \\end{array} = \\begin{array} { c } \\mathfrak p \\\\ \\updownarrow \\\\ I d _ n \\end{array} = \\mathfrak p \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\biggl ( \\frac { 1 + 4 \\mu } { . 9 6 d } \\biggr ) ^ { h } & \\leq \\exp \\Biggl [ \\biggl ( - \\log \\Bigl ( \\frac { d } { 1 + 4 \\mu } \\Bigr ) + \\log \\Bigl ( \\frac { 2 5 } { 2 4 } \\Bigr ) \\biggr ) \\frac { C \\bigl ( j + \\log ( 1 + \\mu ) \\bigr ) } { \\log \\bigl ( \\frac { d } { 1 + 4 \\mu } \\bigr ) } \\Biggr ] \\\\ & \\leq \\exp \\biggl [ - . 9 C \\bigl ( j + \\log ( 1 + \\mu ) \\bigr ) \\biggr ] = e ^ { - . 9 C j } ( 1 + \\mu ) ^ { - . 9 C } . \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\sum _ { k \\in \\Z } \\zeta ( x - k e _ 1 ) \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} \\widetilde { K } ( z ) = F _ { c ^ \\dagger } ( z ) ^ { \\dagger , - 1 } \\widetilde { K } ( x _ 0 ) F _ c ( z ) ^ { - 1 } , \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\begin{cases} w _ { t t } - \\Delta w = 0 , & x \\in \\mathbb { R } ^ n , \\ t > 0 , \\\\ w ( 0 , x ) = \\varphi ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ w _ t ( 0 , x ) = 0 , & x \\in \\mathbb { R } ^ n . \\end{cases} \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} \\mathcal { C } = \\bigcup _ { \\ell = 0 } ^ { \\log \\log n + 1 } \\mathcal { C } _ { \\ell } . \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} b _ { i j } = \\max _ { k = 1 } ^ { m } w _ { k } a _ { i j } ^ { ( k ) } , \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\begin{aligned} G ( v ) = \\frac { 1 } { v } \\exp \\left [ - \\frac { C } { v } \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\sum _ { n , i \\geq 0 } \\lambda _ A ( C _ j ) _ { ( n , i ) } z ^ i \\omega ^ n = \\frac { P _ { C _ j } ( z , \\omega ) } { ( 1 - z ^ 2 ) ^ c ( 1 - \\omega ) ^ r } ~ ~ ~ ~ j = 1 , 2 , 3 , 4 , \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} \\Phi _ 2 ( \\nu , L ) = \\Phi ( d , M _ 2 , \\nu , r _ 2 L ) , \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\gamma ^ { - 1 } ( x _ 1 y _ 1 ) \\alpha ^ { - 1 } ( x _ 2 , y _ 2 ) = \\alpha ( S ( y _ 1 ) , S ( x _ 1 ) ) \\gamma ^ { - 1 } ( x _ 2 ) \\gamma ^ { - 1 } ( y _ 2 ) \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\lim _ { | s | \\to \\infty } \\frac { \\theta '' ( s ) } { \\theta ' ( s ) ^ 2 } = 0 \\mbox { a n d } \\lim _ { | s | \\to \\infty } \\frac { \\theta ''' ( s ) } { \\theta ' ( s ) ^ 3 } = 0 \\ , . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} T _ m : H _ \\varphi \\to H _ \\varphi , U ( T _ m x ) = ( m ( i ) ( U x ) ( i ) ) _ { i \\in I } , x \\in H _ \\varphi . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} \\underset { [ 0 , T ] } { } ( \\rho _ 2 ^ 2 \\| y ( t ) \\| ^ 2 ) + \\iint _ Q \\rho _ 2 ^ 2 | y _ x | ^ 2 d x d t & \\leq C \\left ( \\| y _ 0 \\| ^ 2 + \\iint _ Q \\rho ^ 2 | G | ^ 2 d x d t + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\rho _ 1 ^ 2 | f | ^ 2 d x d t \\right . \\\\ & + \\left . \\iint _ Q \\rho _ 0 ^ 2 | y | ^ 2 d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\rho _ 0 ^ 2 | p ^ i | ^ 2 d x d t \\right ) . \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\int _ { 2 B _ 0 } \\Big ( \\sum _ j g _ j ^ p \\Big ) \\ , d \\mu \\approx \\sum _ { k = - \\infty } ^ { \\infty } 2 ^ { k p } \\mu ( E _ k \\setminus E _ { k - 1 } ) . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\nabla u ( t , x ) = \\int _ 0 ^ t \\int _ { \\R ^ d } \\nabla ( \\frac { 1 } { s ^ { d / 2 } } e ^ { - | y | ^ 2 / 4 s } ) \\cdot f ( t - s , x - y ) d y d s . \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} \\max _ { y \\in \\Omega _ \\varepsilon } | y - x _ \\varepsilon | | \\nabla u _ \\varepsilon ( y ) | u _ \\varepsilon ( y ) = | y _ \\varepsilon - x _ \\varepsilon | | \\nabla u _ \\varepsilon ( y _ \\varepsilon ) | u _ \\varepsilon ( y _ \\varepsilon ) : = C _ \\varepsilon \\to + \\infty \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} W = \\phi _ { m - n } ( T ^ m x ) , \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} | \\Gamma ( \\eta ) | \\sim | \\log | \\eta | | \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\begin{cases} x ' = \\left [ \\frac { ( a ^ 2 - b ) - N ^ 2 - ( 2 k + a ) N } { N } \\right ] x ^ 2 \\\\ x ' = [ a - N ] x ^ 2 \\end{cases} . \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} N = 1 9 ( n + 2 ) + \\bigl \\lfloor 4 \\log _ 2 ( \\Gamma / \\delta ) + 4 \\log _ 2 \\bigl ( 1 + \\eta ^ { - 1 } \\bigr ) \\bigr \\rfloor . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} C _ m = \\frac { 1 } { \\int _ { - 1 } ^ { 1 } w ( t ) | g _ m ' ( t ) | \\d t } \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\tau _ { 1 } = \\theta _ { 1 } A Z _ { u = 1 , 1 } ^ { \\ast } \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { 2 n } q ^ { n ^ 2 + n } } { ( - z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } ( - 1 ) ^ m b ^ 4 _ \\nu ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( z q ; q ^ 2 ) _ n ( - z q ) ^ n , \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} h _ { i _ 1 i _ 2 \\cdots i _ 8 } ( y _ 1 ^ { p ^ { s + 1 } } , y _ 2 ^ { p ^ { s + 1 } } , \\cdots , y _ 8 ^ { p ^ { s + 1 } } ) = w _ m h _ { i _ 1 i _ 2 \\cdots i _ 8 } ^ { * } , \\ i _ 1 \\geq 1 \\ \\ i _ 2 \\geq 1 \\cdots \\ \\ i _ 8 \\geq 1 . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} M ( v _ i , \\sigma v _ i ) = ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { r r } c _ 1 & c _ 3 \\\\ c _ 2 & c _ 4 \\end{array} \\right ) \\ , ( = ( v _ i , \\sigma v _ i ) C , ) . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\mu ^ a ( x ) = \\frac { s ( x ) } { \\int _ I s ( t ) d t } . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} p _ i = 0 \\ \\Rightarrow \\ q _ i = 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j = 1 . \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} H _ \\mu ( x , y ) = \\begin{pmatrix} x - p _ \\mu ( x - y ) \\\\ F ( x ) - y \\end{pmatrix} . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ 3 ( \\lambda - \\zeta _ j ) - \\Delta ^ 2 \\cdot \\sum _ { j = 1 } ^ 3 ( \\lambda ^ 3 - \\zeta _ j ^ 3 ) - 2 \\Delta ^ 3 \\cdot \\prod _ { j = 1 } ^ 3 \\sqrt { \\lambda ^ 2 + \\lambda \\zeta _ j + \\zeta _ j ^ 2 } . \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\| \\tilde { a } _ { i j } - \\tilde { a } ^ { 0 } _ { i j } \\| _ { L ^ { \\infty } ( \\tilde { \\Omega } \\cap B ( 1 ) ) } = \\| a _ { i j } - a ^ { 0 } _ { i j } \\| _ { L ^ { \\infty } ( \\Omega \\cap B ( \\sigma ^ { \\nu } ) ) } \\leq \\omega _ { a _ { i j } } ( \\sigma ^ { \\nu } ) \\leq \\omega _ { A } ( \\sigma ^ { \\nu } ) . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} \\cos ^ 3 \\alpha - \\frac { 3 } { 4 } \\cos \\alpha = \\Lambda _ 0 ^ 3 - L \\Lambda _ 0 , \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} J _ { N } ( x , y ) = p ^ { N } { \\textstyle \\sum \\limits _ { J \\in G _ { N } ^ { 0 } } } { \\textstyle \\sum \\limits _ { K \\in G _ { N } ^ { 0 } } } A _ { J K } \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) \\Omega \\left ( p ^ { N } \\left \\vert y - K \\right \\vert _ { p } \\right ) \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} G ( \\cos \\theta _ { \\rm c } \\cdot f ) = \\frac { N _ 0 } { \\alpha _ { \\rm c } } \\left ( \\ ! \\left ( \\frac { \\mu } { 2 } \\ ! \\int \\limits _ { \\frac { \\cos \\theta _ { \\rm c } \\cdot f } { f _ { \\rm m a x } } } ^ { \\frac { \\cos \\theta _ { \\rm c } \\cdot f } { f _ { \\rm m i n } } } \\frac { S _ 0 ( \\frac { \\cos \\theta _ { \\rm c } \\cdot f } { u } ) } { u } { \\rm d } u \\right ) ^ { \\ ! \\ ! - 1 } \\ ! \\ ! \\ ! - \\frac { 1 } { S _ 0 ( f ) } \\right ) _ { \\ ! + } \\ ! \\ ! . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\kappa _ { 1 } = \\theta _ { 1 } Z _ { u = 1 , 1 } ^ { \\ast } Y . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{gather*} \\mathbf { R } _ { n } ^ { - 1 } \\mathbf { x } = - \\frac { \\mathbf { z } + \\mathbf { w } } { 2 \\theta } , \\\\ \\mathbf { R } _ { n } ^ { - 1 } \\mathbf { y } = - \\frac { \\mathbf { z } - \\mathbf { w } } { 2 \\eta } \\end{gather*}"}
-{"id": "6564.png", "formula": "\\begin{align*} N _ d ( x ) = ( 1 + O ( ( \\log x ) / x ) ) \\frac { x ^ d } { d } \\prod _ { i = 2 } ^ { d } \\zeta ( i ) . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} & \\lim _ { L _ 1 \\to 0 } \\tfrac { 1 } { L _ 1 } ( L _ 1 - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ \\mu ) - R ( L _ 1 , \\ell _ \\nu , 2 \\ell _ { \\mu ' } ) ) \\\\ & = \\frac { e ^ { - \\frac { \\ell _ { \\mu } } { 2 } } + e ^ { - \\frac { \\ell _ { \\mu ' } } { 2 } } } { \\sinh \\frac { \\ell _ \\mu } { 2 } + \\sinh \\frac { \\ell _ { \\mu ' } } { 2 } } = 2 ( e ^ { \\frac { 1 } { 2 } ( \\ell _ \\mu + \\ell _ { \\mu ' } ) } - 1 ) ^ { - 1 } . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} Q _ { 1 , k } & = \\int _ { I _ k } t ^ { 1 - 2 H } \\left | \\dot { \\tilde { h } } _ { k } ( t ) \\right | ^ { 2 } d t = \\int _ { I _ { k } } t ^ { 1 - 2 H } \\left | \\frac { 1 } { | I _ { k } | } \\dot { h } _ { k } \\left ( \\frac { t - a _ { k } } { a _ { k + 1 } - a _ { k } } \\right ) \\right | ^ { 2 } d t . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } B _ { q ( t ) } = \\Tilde { B } \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ^ k & \\in \\bigoplus _ { \\lambda \\in I } \\langle x \\mapsto x ^ j \\exp ( \\lambda x ) : j = 0 , \\ldots , p \\rangle \\\\ & \\oplus \\bigoplus _ { ( \\mu , \\nu ) \\in J } \\langle x \\mapsto x ^ j \\exp ( \\mu x ) \\cos ( \\nu x ) , \\\\ & \\qquad \\qquad \\quad \\ , \\ , \\ , x \\mapsto x ^ j \\exp ( \\mu x ) \\sin ( \\nu x ) : j = 0 , \\ldots , p \\rangle \\end{aligned} \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} z _ { l _ 0 , t } - D _ { l _ 0 } \\Delta z _ { l _ 0 } = f _ { l _ 0 } ( U ) - f _ { l _ 0 } ( W ^ { \\sigma ^ * } ) \\geq - M z _ { l _ 0 } \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { p ^ r - 1 } { 2 } } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 } ( 3 k + 1 ) 2 ^ { 2 k } & \\equiv p ^ r \\pmod { p ^ { r + 2 } } , \\\\ [ 5 p t ] \\sum _ { k = 0 } ^ { p ^ r - 1 } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 } ( 3 k + 1 ) 2 ^ { 2 k } & \\equiv p ^ r \\pmod { p ^ { r + 2 } } . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( B ^ { ( n ) } ) = - \\infty . \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} v ( f _ w ( x ) ^ { t } ) = v ( x ^ { s } b ^ { t } ) . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} Z _ t = e ^ { \\int _ 0 ^ { t } V _ s d s } \\int _ 0 ^ t e ^ { - \\int _ 0 ^ { s } V _ v d v } d Y _ s , \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } \\Big \\{ \\frac { x } { \\left ( 1 + x ^ 2 \\right ) ^ { \\gamma } } \\Big \\} = ( 2 s + 1 ) A _ s ^ \\gamma \\ , \\ , x \\ , { } _ { 2 } F _ { 1 } \\Big ( s + \\gamma , s + \\frac { 3 } { 2 } \\ , ; \\frac { 3 } { 2 } ; - x ^ 2 \\Big ) , \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} w \\varpi _ i ( g \\cdot t ^ { - \\lambda _ w } ) = w \\varpi _ i ( g ) \\cdot w \\varpi _ i ( t ^ { - \\lambda _ w } ) = w \\varpi _ i ( g ) \\cdot t ^ { - ( w \\varpi _ i , \\ \\lambda _ w ) } \\\\ = w \\varpi _ i ( g ) t ^ { - A _ { w \\varpi _ i } } = \\gamma ( g ) z ^ { - A _ \\gamma } \\gamma = w \\varpi _ i . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} f ( N ) \\leq f ( e ^ { n / s } ) = \\frac { ( n / s ) ^ n } { e ^ n } \\leq \\frac { n ! } { \\sqrt { 2 \\pi n } s ^ n } \\leq \\frac { n ! } { \\sqrt { 2 \\pi n } ( s - 1 ) ^ n } \\cdot \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} ( \\lambda _ g ) _ { g \\in G } \\cdot ( \\alpha ( g , h ) ) _ { g , h \\in G } = ( \\alpha ( g , h ) \\lambda _ g \\lambda _ h \\lambda _ { g h } ^ { - 1 } ) _ { g , h \\in G } . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbb { P } _ { x _ 0 , k } ^ { \\epsilon , \\hat { v } } \\bigl \\{ \\vert X ^ { \\epsilon , \\hat { v } } ( \\tau _ { D } ^ { \\epsilon , \\hat { v } } ) - \\bar { y } _ 0 \\vert > \\delta \\bigr \\} = 0 , \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{gather*} \\mathbf { F } _ { \\frac { n } { 2 } } : = \\left [ \\frac { ( \\lambda _ { 2 k + 1 } - \\lambda _ { 1 } ) \\sin \\left [ \\frac { ( 2 k + 1 ) \\pi } { n + 3 } \\right ] } { \\lambda _ { 2 k + 1 } - \\phi _ { \\ell } } \\right ] _ { k , \\ell } \\\\ \\mathbf { G } _ { \\frac { n } { 2 } } : = \\left [ \\frac { ( \\lambda _ { 2 k } - \\lambda _ { n + 2 } ) \\sin \\left ( \\frac { 2 k \\pi } { n + 3 } \\right ) } { \\lambda _ { 2 k } - \\psi _ { \\ell } } \\right ] _ { k , \\ell } \\end{gather*}"}
-{"id": "1553.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi \\upharpoonright _ { G ' } ) = h _ { a l g } ( \\phi \\upharpoonright _ { G ' \\cap H } ) + h _ { a l g } ( \\widetilde { \\phi \\upharpoonright _ { G ' } } ) , \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} H _ i ( \\Gamma _ n , M ) = H _ i ( ( \\Z \\Gamma ^ { ( n ) } ) ^ r \\xrightarrow { f ^ { ( n ) } } ( \\Z \\Gamma ^ { ( n ) } ) ^ r ) \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} h _ d ( q ) = \\frac { h _ 0 ( q ) } { ( q ; q ) _ d ^ 3 } \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} \\frac { \\partial f _ j } { \\partial t } ( t , z _ * ( t ) ) + \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z _ * ( t ) ) \\dot z ^ k _ * ( t ) = 0 . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} H = \\int M ( d t , \\theta ^ H _ t ) + L ^ H \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\nu [ w _ { 1 } . . . w _ { l } ] = p _ { w _ { 1 } } \\cdot . . . \\cdot p _ { w _ { l } } w _ { 1 } , . . . , w _ { l } \\in \\Lambda ^ { m } \\ : . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} M _ { \\mathcal { R } } ( f ) = \\begin{pmatrix} a _ 0 & a _ 1 & \\dots & a _ { n - 1 } \\\\ u \\sigma ( a _ { n - 1 } ) & \\sigma ( a _ 0 ) & \\dots & \\sigma ( a _ { n - 2 } ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ u \\sigma ^ { n - 1 } ( a _ 1 ) & u \\sigma ^ { n - 1 } ( a _ 2 ) & \\dots & \\sigma ^ { n - 1 } ( a _ 0 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\max _ { x \\in [ - 1 , 1 ] } \\max _ { i = 1 , \\cdots , k + 1 } \\prod _ { j = 1 , j \\neq i } ^ { k + 1 } \\frac { | x - \\cos 2 \\pi \\theta _ j | } { | \\cos 2 \\pi \\theta _ i - \\cos 2 \\pi \\theta _ j | } < e ^ { k \\epsilon } . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\rho ) = [ \\mathcal { M } _ { G , b , \\mu } ] ( J L ( \\rho ) ) = [ J L ( \\rho ) ] [ r _ { - \\mu } \\circ L L ( \\rho ) | \\cdot | ^ { - \\langle \\rho _ G , \\mu \\rangle } ] . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\frac { n - 1 } { 2 } } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } \\equiv 0 \\pmod { \\Phi _ n ( q ) } , \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\begin{aligned} & 0 = S ( 1 - \\dfrac 3 2 S ) + 2 \\bar H ^ 2 S - \\dfrac 1 2 \\bar H ^ 4 - \\bar H ^ 2 ( \\bar \\lambda _ 1 ^ 2 + \\bar \\lambda _ 2 ^ 2 ) \\\\ & = - ( S - \\bar H ^ 2 ) ^ 2 - \\dfrac 1 2 \\bar H ^ 2 ( \\bar \\lambda _ 1 - \\bar \\lambda _ 2 ) ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} K _ 1 = 1 & & K _ 2 = \\delta & & C = 2 \\delta + 2 & & C ' = 2 \\delta + 3 . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\psi _ 2 ( C ) : = \\min _ { m \\in \\mathbb { Z } , n \\in \\mathbb { Z } } \\bigg \\{ \\frac { m ^ 2 + n ^ 2 } { m ^ 2 - n ^ 2 } : \\ m ^ 2 + n ^ 2 \\le C , \\ m - 1 \\ge n \\ge 1 \\bigg \\} . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mu _ { \\beta , 0 } ^ + ( \\cdot \\mid \\eta = - 1 [ ( - N ( n ) , - n ) \\cup ( n , N ( n ) ] \\times \\{ 0 \\} ) = \\mu _ { \\beta , 0 } ^ - ( \\cdot ) . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} \\bar { { \\bf Q } } _ m = \\begin{bmatrix} { \\rm I \\ ! R e } \\{ { \\bf Q } _ m \\} & - { \\rm I \\ ! I m } \\{ { \\bf Q } _ m \\} \\\\ { \\rm I \\ ! I m } \\{ { \\bf Q } _ m \\} & { \\rm I \\ ! R e } \\{ { \\bf Q } _ m \\} \\end{bmatrix} , \\forall \\ ; m \\in [ M ] \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} P _ e ( \\rho ) = \\frac { a } { \\gamma - 1 } \\rho ^ { \\gamma - 1 } , \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} a _ s ( u , v ) = \\big ( ( - \\Delta ) ^ { s / 2 } u , ( - \\Delta ) ^ { s / 2 } v \\big ) + ( u , v ) . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\frac { d ^ v } { d t ^ v } m _ t ( | g | ) = \\int _ { - 1 } ^ { 1 } \\frac { d ^ v } { d t ^ v } \\left ( \\frac { y } { t } + e ^ { - \\frac { 2 } { t } } \\right ) ^ { | g | } d \\nu ( y ) . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { \\partial u _ { J } } { \\partial t } = f ( u _ { J } , v _ { J } ) + \\varepsilon { \\textstyle \\sum \\limits _ { I } } L _ { J I } u _ { I } \\\\ \\\\ \\frac { \\partial v _ { J } } { \\partial t } = g ( u _ { J } , v _ { J } ) + \\varepsilon d { \\textstyle \\sum \\limits _ { I } } L _ { J I } v _ { I } . \\end{array} \\right . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\frac 1 { p } = r \\sum _ { i = 1 } ^ { m } \\frac 1 { r _ i } = 1 - \\frac { r } { r _ { m + 1 } } = \\frac 1 { ( \\frac { r _ { m + 1 } } { r } ) ' } . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} v a r ( V _ n ) \\leq C \\frac { r _ n ^ 2 n ^ 2 } { N } = C b _ n ^ 2 \\left ( \\frac { n } { N ^ 2 } \\right ) \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} H _ 1 ^ \\alpha ( S _ 1 ^ \\alpha f ) ( x ) = S _ 1 ^ \\alpha ( H _ 1 ^ \\alpha f ) ( x ) = \\int _ x ^ 1 f ( y ) \\ , d y , \\mbox { a . e . } x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } = - \\dfrac { 3 S } 2 | \\nabla ^ { \\perp } { \\vec H } | ^ { 2 } = - \\dfrac { 3 } { 5 } S ^ 2 ( 3 S - 2 ) . \\endaligned \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\begin{cases} w _ { t t } - \\Delta w = 0 , & x \\in \\mathbb { R } ^ n , \\ t > 0 , \\\\ w ( 0 , x ) = f ( b , x ) , & x \\in \\mathbb { R } ^ n , \\\\ w _ t ( 0 , x ) = 0 , & x \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} f _ { i \\lambda } ^ { - 1 } f _ { i \\mu } = f _ { j \\lambda } ^ { - 1 } f _ { j \\mu } , \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\Theta _ { i } \\xi - \\Theta _ { j } \\xi = \\begin{pmatrix} ( 1 + i m ) \\xi _ 2 \\\\ n \\xi _ 2 \\end{pmatrix} - \\begin{pmatrix} ( 1 + j m ) \\xi _ 2 \\\\ n \\xi _ 2 \\end{pmatrix} = \\begin{pmatrix} ( i - j ) m \\xi _ 2 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\omega ( y , z ; q ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ n q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ n } { ( y q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} ( w _ 1 ( z , \\tau ) , w _ 2 ( z , \\tau ) ) = ( U _ 0 ( z _ N - \\theta _ 0 , x _ * ) , V _ 0 ( z _ N - \\theta _ 0 , x _ * ) ) . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} u ( x , T ) = f ( x ) , x \\in \\Omega , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} F ( z , w ) = \\sum _ { j = 0 } ^ { q - 1 } a _ j ( z ) w ^ j , ( z , w ) \\in K \\times \\C \\subset \\C ^ 2 , \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} U _ m = \\left \\{ z + f _ m ( z ) w : \\ , z \\in B ( \\varrho ) \\backslash B ( \\delta ) \\right \\} \\subset ( { \\partial } K _ m ) \\cap ( { \\rm i n t } B ^ n ) , \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} M ( u ( t ) ) & = \\int | u ( t , x ) | ^ 2 d x , \\\\ E ( u ( t ) ) & = \\frac { 1 } { 2 } \\int | ( - \\Delta ) ^ { s / 2 } u ( t , x ) | ^ 2 d x - \\frac { 1 } { \\alpha + 2 } \\int | u ( t , x ) | ^ { \\alpha + 2 } d x . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} g ( u ) & = \\int _ u ^ 1 \\bigg ( \\xi ' ( s ) - \\int _ 0 ^ s \\frac { d r } { \\nu ( ( r , 1 ] ) ^ 2 } \\bigg ) d s . \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} f _ q ( Q ) = \\sum _ { d \\geq 0 } \\frac { 1 } { ( q ; q ) _ d } Q ^ d \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} ( d d ^ c \\max ( \\sum _ { k = 1 } ^ j \\psi _ k , a _ j ( j + 1 ) \\psi ) ) ^ n & \\geq 1 _ { \\Omega \\cap \\{ \\psi < - \\frac { 1 } { j + 1 } \\} } ( d d ^ c ( \\sum _ { k = 1 } ^ j \\psi _ k ) ) ^ n \\\\ & \\geq \\sum _ { k = 1 } ^ j 1 _ { \\Omega \\cap \\{ - \\frac { 1 } { k } \\leq \\psi < - \\frac { 1 } { k + 1 } \\} } ( d d ^ c \\psi _ k ) ^ n \\geq \\sum _ { k = 1 } ^ j \\min ( f _ k , j ) ( d d ^ c \\psi _ k ) ^ n \\ Q B ( \\Omega ) . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} L _ d ( \\zeta ) = \\left [ \\begin{pmatrix} 1 - \\frac { n ^ 2 + \\frac { n } { 2 } } { \\zeta _ d } & \\frac { - n ! \\ \\Gamma ( n + \\frac 3 2 ) } { 2 \\pi \\i \\ \\zeta _ d } \\\\ \\frac { - 2 \\pi \\i \\ n } { \\Gamma ( n + \\frac 1 2 ) \\ , n ! \\ \\zeta _ d } & 1 + \\frac { n ^ 2 + \\frac { n } { 2 } } { \\zeta _ d } \\end{pmatrix} + \\mathrm { O } ( \\zeta _ d ^ { - 2 } ) \\right ] \\zeta _ d ^ { n \\sigma _ 3 } , \\quad \\zeta _ d \\to \\infty . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} R a n ( e ^ { - L _ k } ) \\subset D ( L _ k ^ { 1 / 2 } ) = D ( q _ k ) . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} s p ( i , \\Pi ^ { c } ; \\Pi '^ { c } , \\imath _ v ) = s p ( i , \\Pi ; \\Pi ' , \\bar { \\imath } _ v ) = s p ( n - i , \\Pi ; \\Pi ' , \\imath _ { v } ) . \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } + \\nu ' , \\chi \\right ) = \\sum _ { n \\leq x } \\frac { \\chi ( n ) } { n ^ { 1 / 2 + \\nu ' } } + \\epsilon ( \\chi ) \\left ( \\frac { q } { \\pi } \\right ) ^ { - \\nu ' } \\frac { \\Gamma \\left ( \\frac { 1 } { 4 } - \\frac { \\nu ' } { 2 } \\right ) } { \\Gamma \\left ( \\frac { 1 } { 4 } + \\frac { \\nu ' } { 2 } \\right ) } \\sum _ { n \\leq y } \\frac { \\overline { \\chi } ( n ) } { n ^ { 1 / 2 - \\nu ' } } + , \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ r \\frac { \\tilde \\lambda _ k } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ k } + \\frac { \\tilde \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ 2 } & \\leq 1 + 2 \\sum _ { k = 1 } ^ r \\frac { \\lambda _ k } { \\tilde \\lambda _ 1 - \\lambda _ k } , \\frac { \\tilde \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ 2 } \\leq 1 + \\frac { \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\lambda _ 1 } . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} g ( \\tau + h ) + ( \\Delta _ h K * d Z ) _ \\tau & = g ( \\tau + h ) + ( \\Delta _ h K * L ) ( 0 ) ( X - g ) ( \\tau ) \\\\ & + ( d ( \\Delta _ h K * L ) * X ) _ { \\tau } - ( d ( \\Delta _ h K * L ) * g ) ( \\tau ) \\\\ & \\geq g ( \\tau + h ) - ( d ( \\Delta _ h K * L ) * g ) ( \\tau ) - ( \\Delta _ h K * L ) ( 0 ) g ( \\tau ) , \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\bar \\lambda _ 1 \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) + \\bar \\lambda \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) = 0 , \\\\ & \\bar \\lambda \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 1 \\rangle ( p _ m ) + \\bar \\lambda _ 2 \\lim _ { m \\rightarrow \\infty } \\langle X , e _ 2 \\rangle ( p _ m ) = 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} \\partial _ t ^ \\beta u : = \\mathcal { I } ^ { m - \\beta } [ \\partial ^ m _ t u ] ( t ) = \\frac { 1 } { \\Gamma ( 1 - m + \\beta ) } \\int _ 0 ^ t ( t - s ) ^ { m - \\beta - 1 } \\partial ^ { m } _ t u ( s ) \\ , d s . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\P ( | Y _ n - Y | \\leq 1 / n ) = 1 . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\sigma ^ \\ast _ i ( \\mu _ 3 ) = x _ 2 , \\ \\ \\sigma ^ \\ast _ i ( s _ 1 t _ 2 ) = 0 \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\frac { t ^ 2 } { 6 } \\ , \\sum _ { i = 1 } ^ 1 f _ { i i } ( x ) = \\frac { t ^ 2 } { 6 } \\ , f ^ { '' } ( x ) , \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} S _ 2 & \\geq \\phi ( q ) \\sum _ m q _ m ^ 2 \\\\ & = \\phi ( q ) \\prod _ { p \\leq X } ( 1 - q _ p ^ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal { L } u ( y , \\cdot ) & = f , x \\in D , \\\\ u ( y , \\cdot ) & = 0 , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\frac { \\pi } { n - 1 } & \\equiv \\mathrm { A r g } ( r ^ n ) \\mod 2 \\pi \\\\ & \\equiv \\mathrm { A r g } ( t _ 0 ) + \\mathrm { A r g } ( r ^ { n - 1 } + 1 ) \\mod 2 \\pi \\\\ & \\equiv \\frac { \\pi } { n - 1 } + \\mathrm { A r g } ( r ^ { n - 1 } + 1 ) \\mod 2 \\pi . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} y _ i = y _ i \\left ( X ^ i \\right ) , X ^ i = \\begin{bmatrix} x _ { i - m + 1 } & \\ldots & x _ { i - 1 } & x _ i \\end{bmatrix} , \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} f _ i ( n ) = ( g _ n + ( y ^ r \\frac { \\partial a } { \\partial x } ) ^ n ) f _ i ( 0 ) \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} \\mathcal S ( v _ 2 ) & = \\mathcal S ( v ) \\backslash \\mathcal S ( v _ 1 ) \\\\ \\mathcal T ( v _ 2 ) & = \\mathcal T ( v ) \\backslash \\mathcal T ( v _ 1 ) \\\\ \\mathcal U ( v _ 2 ) & = \\mathcal U ( v ) \\backslash \\mathcal U ( v _ 1 ) . \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} \\mathbb { P } _ l ^ \\eta ( t , x , A ) = \\mathbb { E } \\left [ \\eta ( U _ { t } ) { \\bf 1 } _ { \\{ F _ l ( U _ t , x ) \\in A \\} } \\right ] , \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\langle \\tilde u _ 1 , u _ 2 \\rangle & = \\sqrt { \\frac { \\lambda _ 2 } { \\lambda _ 1 } } \\frac { \\tilde \\lambda _ 1 - \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\lambda _ 2 } \\langle \\tilde u _ 1 , u _ 1 \\rangle . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\int _ { G ( \\mathrm { i n t } T _ { p i j } ) } \\phi ( F ( x ) ) J _ n F ( x ) d \\lambda ( x ) = \\int _ { \\mathrm { i n t } T _ { p i j } } \\phi ( z ) d \\lambda ( z ) . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} w _ \\pm ^ \\beta : = \\theta ^ \\beta _ \\pm e ^ \\beta + e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} A ( n ) \\le A ( \\alpha ( m ) ) < \\frac 2 3 \\sqrt { \\alpha ( m ) + 2 } = m + 1 . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} e _ 0 = \\int _ S \\int _ S \\theta ( z , t ) \\theta ( \\xi , t ) \\Gamma ( z - \\xi ) d \\xi d z = \\sum _ { n \\in \\mathbb { Z } } \\sum _ { m \\in \\mathbb { Z } } \\int _ { n < x < n + 1 } \\int _ { m < \\xi _ 1 < m + 1 } \\theta ( z , t ) \\theta ( \\xi , t ) \\Gamma ( z - \\xi ) d \\xi d z . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\mu \\left ( f ( z _ 1 , z _ 2 ) a d z _ 1 \\boxtimes b d z _ 2 \\right ) = f ( z _ 1 , z _ 2 ) d z _ 1 \\boxtimes a ( z _ 1 - z _ 2 ) b d z _ 2 \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} J _ 3 ( t , \\delta ) & \\leq 2 \\| f \\| t \\Pr ( V _ 1 \\wedge Y _ 2 > b _ 0 ( t ) \\min \\{ ( \\eta - \\delta ) , \\delta \\} ) \\leq C _ { 3 } t ^ { \\frac { - \\alpha ^ { * } + \\epsilon } { \\alpha _ 0 } } \\stackrel { t \\to \\infty } { \\to } 0 . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\texttt { c } _ { i j } = 2 \\delta _ { i j } - \\delta _ { i , j + 1 } - \\delta _ { i , j - 1 } . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} 2 i \\partial _ t \\varphi \\partial _ t a _ j + i \\partial _ { y _ 1 } a _ j + i \\partial _ t ^ 2 \\varphi a _ j + \\partial _ t ^ 2 a _ { j - 1 } - E _ { j } ^ { ( M ) } = t ^ { M } Q _ M ^ { ( j ) } , 0 \\le j \\le M , \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\begin{array} { r l l l } \\lambda u _ 1 - \\Delta u _ 1 & = \\partial _ z G _ 1 + _ H G _ 2 & \\R ^ 3 , & G _ 1 : = \\chi _ r s E f , G _ 2 : = - 2 ( \\nabla \\chi _ r ) E v , \\\\ \\lambda u _ 2 - \\Delta u _ 2 & = G _ 3 & \\R ^ 3 , & G _ 3 : = - ( \\partial _ z \\chi _ r ) s E f + ( \\Delta \\chi _ r ) E v . \\end{array} \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} Y _ t - Y _ 0 = \\Phi ( t , X _ t ) - \\Phi ( t , X _ 0 ) & = \\int _ 0 ^ t \\nabla u ( s , X _ s ) d B _ s + B _ t = \\int _ 0 ^ t \\nabla u ( s , \\Phi ^ { - 1 } ( s , Y _ s ) ) d B _ s + B _ t . \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} u ( t ) = \\left ( I _ 0 ^ k f ( \\cdot , u ( \\cdot ) ) \\right ) ( t ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} \\epsilon ( f ) = \\max _ { b \\in \\N } \\left \\{ \\frac { \\nu ( f ) - \\nu ( \\partial _ b f ) } { b } \\right \\} . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\phi ( s ) & = K \\Gamma ( u + 1 ) s ^ { - u } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 + 1 / 2 } ) + O ( s ^ { - u + 1 / 2 } ) \\\\ & = K \\Gamma ( u + 1 ) s ^ { - u } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} g _ { i j } & = \\left \\{ \\begin{aligned} & 1 & & \\textnormal { i f } i + j \\leq N \\\\ & 0 & & \\textnormal { o t h e r w i s e } \\end{aligned} \\right . \\\\ \\left ( G _ { i j } \\right ) _ { | t = 0 } & = g _ { i j } + \\frac { Q } { 1 - Q } \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} U ^ 1 = 1 + p A = \\{ b \\in B \\mid \\| 1 - b \\| < 1 \\} \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} | \\nabla | ^ { 2 \\sigma } & e ^ { i ( t - s ) \\sqrt { 1 - \\Delta } } P _ k ^ 2 F _ { \\rho _ 1 } ( \\cdot , s ) \\\\ & = \\int _ { \\mathbb { R } ^ n } \\bigg ( \\int _ { \\mathbb { R } ^ n } e ^ { i ( x - y ) \\cdot \\xi + i ( t - s ) \\sqrt { 1 + | \\xi | ^ 2 } } | \\xi | ^ { 2 \\sigma } \\phi ( 2 ^ { - k } | \\xi | ) ^ 2 d \\xi \\bigg ) F _ { \\rho _ 1 } ( y , s ) d y \\\\ & : = \\int _ { \\mathbb { R } ^ n } I _ k ( x - y , t - s ) F _ { \\rho _ 1 } ( y , s ) d y \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\angle ( { y _ { k , i } } ) \\ ! = \\ ! \\angle { \\beta \\left ( \\textbf { x } _ k \\right ) } { \\rm { - } } \\pi \\ ! \\left [ \\ ! { \\frac { M \\ ! - \\ ! 1 } { M } ( \\omega _ { k , i 1 } \\ ! - \\ ! x _ { k , 1 } ) \\ ! + \\ ! \\frac { N \\ ! - \\ ! 1 } { N } { ( \\omega _ { k , i 2 } \\ ! - \\ ! x _ { k , 2 } ) } } \\ ! \\right ] \\ ! . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\sigma ^ { a b } l _ { a b } = - \\frac { 2 } { r } + \\frac { 1 } { 4 5 } r ^ 3 | \\alpha | ^ 2 + O ( r ^ 4 ) \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} w ( 0 ) = V _ m ( 0 ) + U _ { m + 1 } ' ( 0 ) = ( - 1 ) ^ m u _ 1 + ( - 1 ) ^ { m + 1 } u _ 1 = 0 \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\alpha _ { g } ( \\pi ( x ) ) = \\begin{cases} 1 \\\\ \\delta _ { g } = \\delta _ { f } \\end{cases} \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} p ( t , x , y ) = ( 4 \\pi t ) ^ { - d / 2 } e ^ { - \\frac { | x - y | ^ 2 } { 4 t } } \\ \\ \\ \\ x , y \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} B _ \\Omega ( z , w ) & = B _ \\Omega ( z , \\sigma ( \\tau ) ) + B _ \\Omega ( \\sigma ( \\tau ) , w ) \\geq B _ \\Omega ( z , z _ 0 ) + B _ \\Omega ( z _ 0 , w ) - 2 B _ \\Omega ( z _ 0 , \\sigma ( \\tau ) ) \\\\ & \\geq B _ \\Omega ( z , z _ 0 ) + B _ \\Omega ( z _ 0 , w ) - R \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} H \\phi = E \\phi , | \\phi ( k ) | \\leq \\hat { C } ( 1 + | k | ) . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\left \\langle ( T _ a \\cup \\tau _ 2 ^ n ) , T _ j , \\tau _ 2 , \\right \\rangle ^ \\textnormal { c o h } _ { 0 , 3 , 0 } = \\int _ X T _ a \\cup T _ j \\cup \\tau _ 2 ^ { n + 1 } \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} u ( t , x ) = \\hat u ^ { R } _ 1 ( t , x ) + k _ 1 ( t ) \\hat S ( t , x ) = \\hat u ^ { R } _ 2 ( t , x ) + k _ 2 ( t ) \\hat S ( t , x ) \\ , , \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\alpha ( 1 ) = 2 , \\ , \\alpha ( 2 ) = 4 , \\ , \\alpha ( 3 ) = 3 , \\ , \\alpha ( 4 ) = 1 \\ , , \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} _ r \\phi _ s ( q ^ { - n } , q ^ { \\alpha _ 2 + 1 } ; z ) = a _ n ^ { \\alpha } \\ _ r \\phi _ s ( q ^ { - n + 1 } , q ^ { \\alpha _ 2 } ; z ) + b _ n ^ { \\alpha } \\ _ r \\phi _ s ( q ^ { - n } , q ^ { \\alpha _ 2 } ; z ) . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} 0 < d < \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( \\alpha _ i ^ 2 + \\beta _ i ^ 2 ) , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} B _ j ( F , G ) = \\sum _ { \\ell \\in \\mathbb { Z } } \\int _ { 2 ^ j \\ell } ^ { 2 ^ j ( \\ell + 1 ) } \\int _ { t - I _ j } \\Big \\langle | \\nabla | ^ { 2 \\sigma } e ^ { i ( t - s ) \\sqrt { 1 - \\Delta } } P _ k ^ 2 F ( \\cdot , s ) , G ( x , t ) \\Big \\rangle _ { L _ x ^ 2 } d s d t . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x _ j ^ { \\left ( k \\right ) } = \\theta + n _ j ^ { \\left ( k \\right ) } , k < { t _ a } \\\\ x _ j ^ { \\left ( k \\right ) } = \\theta + { \\mu _ j } + n _ j ^ { \\left ( k \\right ) } , k \\ge { t _ a } , \\end{array} \\right . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\Vert t _ m f \\Vert _ { \\ell ^ { p } } = \\Vert k \\ast f \\Vert _ { \\ell ^ { p } } \\leq \\Vert k \\Vert _ { \\ell ^ { p } } \\Vert f \\Vert _ { \\ell ^ 1 } \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } A _ { K , M } ^ { ( 1 , 2 ) } = \\frac { 1 } { { q \\left ( \\theta \\right ) } } . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } w ( p , n ) = ( 1 + O ( \\frac { 1 } { \\log ^ { 1 0 } _ 2 x } ) ) \\tau \\frac { x } { \\log ^ r x } \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F \\left ( n , \\frac { m - 1 } { 2 } \\right ) = \\sum _ { n = 0 } ^ { m - 1 } [ 3 n + m ] \\frac { ( q ; q ^ 2 ) _ { n } ( q ^ { m } ; q ^ 2 ) _ { n } ^ 2 q ^ { - { n + 1 \\choose 2 } - \\frac { ( 2 n + 1 ) ( m - 1 ) } { 2 } } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } . \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\begin{array} { l } ( n - 2 ) f h \\varphi , _ { x _ i x _ j } - r f \\varphi h , _ { x _ i x _ j } - m h \\varphi f , _ { x _ i x _ j } - m h \\varphi , _ { x _ i } f , _ { x _ j } - m h \\varphi , _ { x _ j } f , _ { x _ i } - r f \\varphi , _ { x _ i } h , _ { x _ j } \\\\ - r f \\varphi , _ { x _ j } h , _ { x _ i } = 0 , \\ \\ \\ \\forall i , j = 1 , \\ldots , n , \\ i \\neq j , \\end{array} \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\delta _ k = \\nu _ k , \\ k = 1 , . . . , K . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} & g \\in C ^ 1 ( \\mathbb { R } ) \\ , , \\quad \\lim _ { s \\to + \\infty } g ( s ) = 0 \\ , , g ( t ) > - 1 g ( t ) = g ( - t ) t \\ , . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} z ^ \\mu \\frac { \\rho } { z } z ^ { - \\mu } & = e ^ { ( \\mu \\log z ) } \\frac { \\rho } { z } = \\sum _ k \\frac { 1 } { k ! } ( ( \\mu \\log z ) ) ^ k \\frac { \\rho } { z } \\\\ & = \\sum _ k \\frac { 1 } { k ! } ( \\log z ) ^ k \\frac { 1 } { z } [ \\underbrace { \\mu , [ \\mu , [ \\cdots , [ \\mu } _ { } , \\rho ] \\cdots ] ] = \\sum _ k \\frac { 1 } { k ! } ( \\log z ) ^ k \\frac { 1 } { z } \\rho = \\rho \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\i [ L , \\pi ( a ^ \\sharp ( \\phi ) ) ] \\Psi = \\pi ( \\delta ( a ^ \\sharp ( \\phi ) ) ) \\Psi = \\pi ( a ^ \\sharp ( \\i h _ 0 \\phi ) ) \\Psi , \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} G ( \\zeta ^ m ) = \\frac { n ^ 2 - 1 } { 2 4 } \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} \\phi _ { m , n } : = h m n \\xi _ 2 ^ 2 . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} \\gamma & = ( \\{ i _ { a + 1 } , \\overline { i _ { a + 1 } } \\} , \\ldots , \\{ i _ { a + b } , \\overline { i _ { a + b } } \\} ) \\\\ \\delta & = ( [ i _ { a + b + 1 } , \\overline { i _ { a + b + 1 } } ] , \\ldots , [ i _ n , \\overline { i _ n } ] ) . \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} E = \\frac { V } { 3 2 \\pi ^ 2 } \\int _ M | \\psi ^ * \\omega | ^ 2 \\ , \\eta , Q = \\frac { 1 } { 4 \\pi } \\int _ M \\psi ^ * \\omega . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\rho ( B ) \\dfrac { \\| x \\| ^ 2 } { \\| g ( x ) \\| } \\le \\rho ( B ) \\ , \\dfrac { { n } \\| x \\| _ { \\infty } ^ 2 } { \\| x \\| _ { \\infty } ^ 2 } = n \\rho ( B ) , \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} \\begin{cases} z ' = \\frac { a ^ 2 - b } { r } x - 2 ( k + a ) x z + ( r - 1 ) x z ^ 2 \\\\ x ' = ( a - r z ) x ^ 2 \\end{cases} . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} \\nabla _ { \\gamma ^ \\prime } \\gamma ^ \\prime = u \\gamma ^ \\prime , \\nabla _ { \\bar \\gamma ^ \\prime } \\gamma ^ \\prime = 0 \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\langle x \\mapsto x ^ j \\exp ( \\lambda x ) : j = 0 , \\ldots , n - 1 \\rangle . \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} \\frac 1 2 \\norm { x _ n ^ \\gamma ( t ) } _ H ^ 2 + A \\int _ { Q _ t } | \\Delta x _ n ^ \\gamma | ^ 2 & = B \\int _ { Q _ t } \\psi '' _ n ( \\varphi ) x _ n ^ \\gamma \\Delta x _ n ^ \\gamma \\\\ & + \\int _ { Q _ t } \\left [ h ( \\varphi ) \\mathcal P z _ n ^ \\gamma + h ' ( \\varphi ) x _ n ^ \\gamma ( \\mathcal P \\sigma - a - \\alpha u ) + \\gamma _ 1 \\right ] x _ n ^ \\gamma \\ , . \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} V : = \\mathbb { R } ^ { 3 } \\setminus \\{ \\{ ( x , y , z ) \\in \\mathbb { R } ^ { 3 } : y = 0 \\} & \\cup \\{ ( M , - M \\sqrt { \\lambda } , 0 ) : M \\in \\mathbb { R } \\} \\\\ & \\cup \\{ ( M , M \\sqrt { \\lambda } , 0 ) : M \\in \\mathbb { R } \\} \\cup \\{ ( 0 , 0 , M ) : M \\in \\mathbb { R } \\} \\} . \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} \\partial _ t \\theta + u \\cdot \\nabla \\theta = 0 , \\theta | _ { t = 0 } = \\theta _ 0 . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\mathcal { W } ^ { B D } [ y ; d ] ( x , t ) = \\mathcal { V } ^ { B D } [ y ( x ) , d ( x ) ] ( t ) . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\log \\kappa ( y , x ) : = e ^ { - | x - 1 / 2 | \\times \\| y \\| _ { \\ell _ 1 } } x \\in [ 0 , 1 ] y \\in \\mathbb { R } ^ { d ' } . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} H _ { \\beta , h } ^ { \\Lambda , \\sigma } ( \\omega _ { \\Lambda } ) : = - \\beta \\left [ \\sum _ { \\{ x , y \\} \\subset \\Lambda \\atop x \\sim y } \\omega _ x \\omega _ y + \\sum _ { x \\in \\Lambda , y \\notin \\Lambda \\atop x \\sim y } \\omega _ x \\sigma _ y \\right ] + h \\sum _ { x \\in \\Lambda } \\omega _ x . \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} \\gamma & = ( \\{ i _ { a + 1 } , \\overline { i _ { a + 1 } } \\} , \\ldots , \\{ i _ { a + b } , \\overline { i _ { a + b } } \\} ) \\\\ \\delta & = ( [ i _ { a + b + 1 } , \\overline { i _ { a + b + 1 } } ] , \\ldots , [ i _ n , \\overline { i _ n } ] ) , \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } \\big \\{ x v ( x ) \\big \\} & = \\frac { 2 ^ { 2 s + 1 } \\Gamma ( s + \\gamma ) \\Gamma ( s + 3 / 2 ) } { \\sqrt { \\pi } \\Gamma ( \\gamma ) } \\ , x \\ , { } _ { 2 } F _ { 1 } \\Big ( s + \\gamma , s + \\frac { 3 } { 2 } \\ , ; \\frac { 3 } { 2 } ; - x ^ 2 \\Big ) . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\frac { c _ 1 + \\lambda _ 1 } { w _ 1 B } \\left ( \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } \\right ) ^ 2 & = \\sum _ { j \\in \\mathcal { S } / \\{ 1 \\} } w _ j d _ j ^ { n e } , \\\\ & ~ ~ \\vdots \\\\ \\frac { c _ { | \\mathcal { S } | } + \\lambda _ { | \\mathcal { S } | } } { w _ { | \\mathcal { S } | } B } \\left ( \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } \\right ) ^ 2 & = \\sum _ { j \\in \\mathcal { S } / \\{ | \\mathcal { S } | \\} } w _ j d _ j ^ { n e } , \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\forall f \\in \\mathcal C ^ 2 ( \\mathbb { R } ^ d ) , \\ : \\int _ { \\mathbb { R } ^ d } A _ n f ( z ) m _ n ( z ) \\d z = 0 . \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\bigg \\| \\Big ( \\sum _ j | [ T , \\textbf { b } ] _ \\alpha ( f _ 1 ^ j , f _ 2 ^ j , \\dots , f _ m ^ j ) | ^ s \\Big ) ^ \\frac 1 s \\bigg \\| _ { L ^ { q } ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\| b _ j \\| ^ { \\alpha _ i } _ { { \\rm B M O } } \\bigg \\| \\Big ( \\sum _ j | f _ i ^ j | ^ { s _ i } \\Big ) ^ \\frac 1 { s _ i } \\bigg \\| _ { L ^ { q _ i } ( v _ i ) } , \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} f ( x ) = f _ 1 ( x _ 1 ) + \\ldots + f _ n ( x _ n ) \\ , , \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} D _ n : = \\left [ \\begin{array} { c | c } E _ n & \\\\ \\hline F _ n & E _ n \\end{array} \\right ] \\mbox { o f o r d e r } \\ ; 2 { n + 1 \\choose 2 } \\times 2 { n + 2 \\choose 2 } . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\int _ { ( \\R ^ { 2 | 2 } ) ^ \\Lambda } F = F _ { \\varnothing , \\varnothing } ( 0 , 0 ) . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\bigoplus _ { j = 1 } ^ { n } a _ { l j } x _ { j } = a _ { l k } x _ { k } \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\lim _ { m \\rightarrow \\infty } H ^ 2 ( p _ m ) = \\inf H ^ 2 = \\bar H ^ 2 , \\lim _ { m \\rightarrow \\infty } | \\nabla H ^ 2 ( p _ m ) | = 0 , \\\\ & \\lim _ { m \\rightarrow \\infty } | \\nabla ^ { \\perp } \\vec H | ^ 2 ( p _ m ) - \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 + \\dfrac 1 2 ( \\bar H ^ 2 - S ) ( \\bar H ^ 2 - 3 S + 2 ) \\geq 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} x = a r \\cos ( \\theta ) \\ , \\ \\ y = b r \\sin ( \\theta ) \\ , \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} g _ q ( Q ) = f _ q \\left ( ( 1 - q ) Q \\right ) = \\frac { 1 } { ( ( 1 - q ) Q ; q ) _ \\infty } \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} T ( u ) ( \\mu ) = \\int _ { X } \\langle u , d \\mu \\rangle \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} 3 \\div \\ , [ w ^ { i } G ( x ) \\nabla w ^ i ] - 3 ( w ^ i ) ^ { 2 } = 3 ( w ^ i w ^ i _ { t } ) _ t + 3 [ \\mid \\nabla _ { g } w ^ i \\mid _ { g } ^ { 2 } - ( w ^ i _ t ) ^ 2 ] . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\cosh ( \\rho _ s - \\eta _ s ) = { \\bar \\sigma \\over \\hat \\sigma } . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} a \\cdot ( g \\cdot x ) = g \\cdot ( ( a \\cdot g ) \\cdot x ) \\ ; \\ ; a \\in A , \\ g \\in G , \\ x \\in B . \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} I _ { x } ( \\varphi ) \\doteq \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } | u ( s ) | _ { H } ^ { 2 } d s : X _ { x } ^ { 0 , u } = \\varphi \\right \\} , \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} A _ 2 = A _ 3 D _ 2 \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\mathcal L ( V _ { \\l } , V _ { \\l } ) \\\\ & = \\sum _ { j = 1 } ^ n | b _ { \\l } ^ j | ^ 2 d _ j , \\l = 1 , \\ldots , q . \\end{aligned} \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} v ^ 1 _ n ( x + x ^ 2 _ n ) = v ^ 1 _ n ( x + ( x ^ 2 _ n - x ^ 1 _ n ) + x ^ 1 _ n ) , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\Theta ( X ; s , N ) = \\frac { 1 } { m } \\Phi ( x ) + \\frac { 1 } { 2 } v ^ 2 + \\frac { 1 } { 2 m } \\sum _ { k = 1 } ^ N z _ k ^ 2 + \\frac { 1 } { 2 } \\sum _ { k > N } k ^ { - 2 s } z _ k ^ 2 . \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\mbox { r a n k } ( D B , D A B , \\dots , D A ^ { n - 2 } B ) = n - 1 . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} - \\frac { 4 f _ { i } f _ { i j } f _ { j } } { ( 1 - f ) ^ { 3 } } - \\frac { 4 f ^ { 2 } _ { i } f ^ { 2 } _ { j } } { ( 1 - f ) ^ { 4 } } = \\frac { - 2 } { 1 - f } f _ { j } \\left ( \\frac { 2 f _ { i } f _ { i j } } { ( 1 - f ) ^ { 2 } } + \\frac { 2 f ^ { 2 } _ { i } f _ { j } } { ( 1 - f ) ^ { 3 } } \\right ) = \\frac { - 2 } { 1 - f } g ' ( \\overline { \\nabla } { f } , \\overline { \\nabla } { \\phi } ) . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega ( A \\sigma : \\tau + \\div \\tau \\cdot u + \\tau : p ) \\ d x & = 0 , & & \\tau \\in H ( \\div ; \\mathbb { M } ) , \\\\ \\int _ \\Omega \\div \\sigma \\cdot v \\ d x & = \\int _ { \\Omega } g \\cdot v \\ d x , & & v \\in L ^ 2 ( \\mathbb { V } ) , \\\\ \\int _ \\Omega \\sigma : q \\ d x & = 0 , & & q \\in L ^ 2 ( \\mathbb { K } ) . \\end{aligned} \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\dot x _ { k } ^ { 0 , u _ k } ( t ) = f _ k \\bigl ( x _ { k } ^ { 0 , u _ k } ( t ) , u _ { k } ^ { 0 } ( t ) \\bigr ) , x _ { k } ^ { 0 , u _ k } ( 0 ) = x , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} H _ { N } ( t ) = \\frac { \\Psi ' _ { N } ( t ) \\exp ( - t ^ 2 ) } { 2 t } \\ , . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\begin{aligned} & { \\rho } _ { i } \\leq C \\rho _ { i - 1 } \\ \\ \\forall i \\in \\{ \\ 1 , 2 , 3 , 4 , 5 \\} , \\\\ & { \\rho } _ 0 \\leq C \\rho \\leq C \\rho _ 5 ^ 2 \\\\ & | \\rho _ i \\rho _ { i , t } | \\leq C | \\rho _ { i - 1 } | ^ 2 , \\ \\ \\forall i \\in \\{ \\ 2 , 3 , 4 , 5 \\} \\ . \\end{aligned} \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} A ( t _ 0 , \\cdots , t _ s , t ) = A ( t ) = ( a _ { i j } ( t ) ) _ { 1 \\leq i , j \\leq n } , \\ B ( t _ 0 , \\cdots , t _ s ) = B = ( b _ { i j } ) _ { 1 \\leq i , j \\leq s } \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} \\inf _ { s > 0 } ~ \\mathbb { E } \\left [ W _ n \\right ] = \\inf _ { s \\in \\mathcal { S } _ n ^ + } ~ \\mathbb { E } \\left [ W _ n \\right ] . \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} { \\rm A L G } & \\geq \\sum _ { t = 1 } ^ m \\sigma _ { F _ t } ( d _ t ( K ) ) - \\sum _ { i = 1 } ^ n G _ i ( \\hat { c } _ i ^ T \\omega _ { i , m } ( K ) ) - \\frac { L m \\lambda ^ 2 } { K } \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} X ( \\alpha ) \\begin{bmatrix} c ( \\alpha ) \\\\ b ( \\alpha ) \\\\ c ( \\alpha ) \\\\ 0 \\end{bmatrix} = \\alpha \\begin{bmatrix} 0 \\\\ \\alpha \\\\ 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} - \\Delta _ g u = \\rho \\left ( \\frac { e ^ u } { \\int _ { M } e ^ u } - \\frac { 1 } { | M | } \\right ) , \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} \\varphi \\mapsto - \\int _ { 0 } ^ { L } \\left ( \\mu _ { 1 } \\alpha w ^ { + } - \\mu _ { 2 } d w ^ { - } \\right ) \\mathbf { 1 } _ { z = 0 } \\varphi , \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} N _ R ^ 0 : = \\sup _ { x _ 0 \\in \\R ^ 3 } \\frac 1 R \\int _ { B _ R ( x _ 0 ) } | u _ 0 | ^ 2 \\ , d x . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { L } _ k ^ { \\epsilon , \\hat { v } } \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) + \\sum \\nolimits _ { j = 1 } ^ n \\gamma _ { k j } ( x ) \\bigl [ \\psi _ j ^ { \\epsilon , \\hat { v } } ( x ) - \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) \\bigr ] = 0 , x \\in D , \\\\ \\psi _ k ^ { \\epsilon , \\hat { v } } ( x ) \\vert _ { \\partial D } = g _ k ( x ) , k = 1 , 2 , \\ldots , n , \\end{array} \\right . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} & \\langle x ^ * - x , y - x ^ * \\rangle + \\langle x - x ^ * , z - x \\rangle + \\frac { 1 } { 2 } \\big ( | y - x ^ * | ^ 2 + | z - x | ^ 2 \\big ) - \\langle y - x ^ * , z - x \\rangle \\\\ & = \\frac { 1 } { 2 } \\big ( | y - z | ^ 2 + | x - x ^ * | ^ 2 \\big ) \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} & f _ 3 ( a , e _ \\lambda ) = 1 = f _ 4 ( e _ \\lambda , a ) , \\\\ & f _ 3 ( a , b ) ^ { - 1 } = f _ 3 ( a b , b ^ { - 1 } ) , \\\\ & f _ 4 ( a , b ) ^ { - 1 } = f _ 4 ( a ^ { - 1 } , a b ) \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} \\sigma ( \\alpha ) = ( \\alpha _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , \\alpha _ { \\sigma ^ { - 1 } ( n ) } ) . \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} & : S { \\frak C } _ l ( H ' ) \\to S { \\frak C } ( H ' ) , \\\\ & ( x ) = \\lim _ a \\ \\beta _ 2 ( 1 \\hat { \\otimes } _ 1 ( x _ a ) ) \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} f _ i ( x , q ) = \\sum _ { j = 1 } ^ r h _ { i , j } ( x , q ) f _ j ( x q ^ { e ( i , j ) } , q ) , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} ( r _ 1 , r _ 2 ) A _ 1 + ( r _ 1 , r _ 2 ) B A _ 2 = 0 \\mod \\R ^ 2 . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} - 2 ( \\nabla \\chi _ r ) \\cdot E ( \\nabla v ) - ( \\Delta \\chi _ r ) E v = - 2 ( \\nabla \\chi _ r E v ) + ( \\Delta \\chi _ r ) E v , \\chi _ r E ( \\partial _ z f ) = \\partial _ z ( \\chi _ r s E f ) - ( \\partial _ z \\chi _ r ) s E f \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} \\Theta ( G ) : = \\sup _ n \\alpha ( G ^ { \\boxtimes n } ) ^ { 1 / n } = \\lim _ { n \\to \\infty } \\alpha ( G ^ { \\boxtimes n } ) ^ { 1 / n } , \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\bar \\eta _ { k l } = \\frac { \\langle u _ k , E u _ l \\rangle } { \\sqrt { \\lambda _ k \\lambda _ l } } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\langle u _ k , X _ i \\rangle } { \\sqrt { \\lambda _ k } } \\frac { \\langle u _ l , X _ i \\rangle } { \\sqrt { \\lambda _ l } } - \\delta _ { k , l } , k , l \\geq 1 , \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\nu ( y , z ; q ) = \\bar { \\nu } ( i q / \\sqrt { z } , \\sqrt { y z } q ; q ) \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} | x _ { k + 1 } | < h _ k | x _ k | ^ 3 < h _ k | x _ 0 | ^ { 3 ^ { k + 1 } } \\prod _ { i = 0 } ^ { k - 1 } h _ { k - 1 - i } ^ { { 3 } ^ { i + 1 } } = | x _ 0 | ^ { 3 ^ { k + 1 } } \\prod _ { i = - 1 } ^ { k - 1 } h _ { k - 1 - i } ^ { { 3 } ^ { i + 1 } } , \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta _ g u = \\rho _ 1 \\left ( \\frac { e ^ u } { \\int _ { M } e ^ u } - \\frac { 1 } { | M | } \\right ) - \\rho _ 2 \\left ( \\frac { e ^ { - u } } { \\int _ { M } e ^ { - u } } - \\frac { 1 } { | M | } \\right ) \\mbox { o n } M , \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} \\frac { c _ i + \\lambda _ i } { w _ i B } \\left ( \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } \\right ) ^ 2 = \\sum _ { j \\in \\mathcal { S } / \\{ i \\} } w _ j d _ j ^ { n e } . \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\Vert \\phi _ m \\Vert = \\left ( \\sum _ { k = - m } ^ m \\bigl \\vert \\widehat { h } ( k ) \\bigr \\vert ^ s \\right ) ^ { 1 / s } ( m \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} & - n ! C ( l , h ) \\\\ = & \\sum _ { h \\le q \\le \\max ( l , h - 1 ) } ( - 1 ) ^ { h + n } { n \\choose h } { n - h \\choose n - q } \\sum _ { 0 \\le k \\le n } ( - 1 ) ^ k k { h \\choose q - k } \\\\ = & \\sum _ { h \\le q \\le \\max ( l , h - 1 ) } ( - 1 ) ^ { h + n } { n \\choose h } { n - h \\choose n - q } \\sum _ { 0 \\le K \\le q } ( - 1 ) ^ { q + K } ( q - K ) { h \\choose K } \\\\ = & \\ , 0 , \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\begin{cases} V _ { m + 1 } ' + A V _ { m + 1 } = - V _ { m } ' , t > 0 , \\\\ V _ { m + 1 } ( 0 ) = ( - 1 ) ^ { m + 1 } u _ 1 . \\end{cases} \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} f _ G ( n ) \\ = \\sum _ { \\Pi \\in T ( G ) } p _ \\Pi ( n ) \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} n F = F + F + \\cdots + F = \\left \\{ a _ 1 + a _ 2 + \\cdots + a _ n \\ , \\vert \\ , a _ i \\in F , \\forall \\ , i \\in \\left \\{ 1 , 2 , \\ldots , n \\right \\} \\right \\} ( n \\geq 2 ) \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} A ( s + s ' ) = A ( s ) A ( s ' ) = A ( s ' ) A ( s ) , A ( - s ) = A ( s ) ^ { - 1 } , A ( s ) = A _ s A ( 0 ) \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} \\mathcal X _ j ( x ) \\leq \\beta _ { 1 , j } \\frac { z _ j } { | x | ^ 2 \\phi } - \\beta _ { 2 , j } \\ , \\frac { | \\nabla z _ j | ^ 2 } { z _ j } x \\in \\omega \\ \\mbox { a n d } j = 1 , 2 . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} & - \\nabla \\cdot \\left ( 2 \\mu \\vec \\varepsilon ( \\vec u ) + \\lambda \\nabla \\cdot \\vec u \\vec I - b \\ , p \\vec I \\right ) = \\rho _ b \\vec g \\ , , \\\\ [ 0 e x ] & \\partial _ t \\Big ( \\frac { 1 } { M } p + \\nabla \\cdot ( b \\vec u ) \\Big ) + \\nabla \\cdot \\vec q = f \\ , , \\vec q = - \\vec K \\nabla p \\ , , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} a _ { 0 , j } = ( j - 1 ) ( p - 1 ) z _ j , a _ { 1 , j } ( x ) = h _ j ( u _ j ) f ( x , z _ j ) , a _ { 3 , j } = - \\frac { ( p - 2 ) } { 2 } \\frac { \\langle \\nabla z _ j , \\nabla u _ j \\rangle } { z _ j } . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} | B _ C ( p , R ) | = \\omega _ C R ^ Q , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ | B ( p , R ) | = \\omega R ^ Q . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\frac { t ^ { - \\frac { N + 2 A _ k } { 2 } } U _ k ( | x | ) } { t ^ { - \\frac { N + 2 A } { 2 } } U ( | x | ) } \\le C t ^ { A - A _ k } ( 1 + | x | ) ^ { A _ k - A } \\le C t ^ { - \\frac { A _ k - A } { 2 } } = o ( 1 ) \\quad \\mbox { a s } t \\to \\infty \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} 1 / ( n p _ n ) \\to 0 \\quad p _ n = o ( n ^ { - 1 / r } ) , \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} \\theta _ { i } ( t ) = \\sum _ { l = 0 } ^ { \\infty } \\int _ { \\tau _ { l } \\wedge t } ^ { \\tau _ { l + 1 } \\wedge t } \\omega _ { i } ( W _ { s } ) \\circ W _ { s } . \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} L a g _ i = \\prod _ { j \\neq i } \\frac { H - \\lambda _ j } { \\lambda _ i - \\lambda _ j } \\in H _ { T ^ { N + 1 } } ^ * \\left ( \\mathbb { P } ^ N ; \\mathbb { Q } \\right ) \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\lambda _ { q ^ t } ^ { \\ell _ { q ^ t } ( Q ) } = e ^ { \\mu \\log ( Q ) } = Q ^ \\mu \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} E ( u , t ) + 2 \\int _ 0 ^ t \\| u ' ( s ) \\| ^ 2 \\ , d s = E ( u , 0 ) = \\| u _ 1 \\| ^ 2 + \\| A ^ { 1 / 2 } u _ 0 \\| ^ 2 , \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { | ( \\Phi _ N g ) \\ , \\triangle \\ , \\Phi _ N | } { | \\Phi _ N | } = 0 = \\lim _ { N \\to \\infty } \\frac { | \\Phi _ N \\ , \\triangle \\ , ( g \\Phi _ N ) | } { | \\Phi _ N | } \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{gather*} a _ { k - 1 } = z _ k a _ k , \\\\ \\left \\Vert a _ { k - 1 } - a _ { k } \\right \\Vert < \\frac { \\varepsilon } { n } , \\end{gather*}"}
-{"id": "1184.png", "formula": "\\begin{align*} R _ { n } ^ { \\lambda } \\left ( x \\right ) : = \\big ( 1 + x ^ 2 \\big ) ^ { - \\frac { \\lambda + 1 } { 2 } } C _ { n } ^ { \\lambda } \\Big ( \\frac { x } { \\sqrt { 1 + x ^ 2 } } \\Big ) , x \\in { \\mathbb R } , \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\frac { \\Psi ^ { - 1 } ( s ) } { s } ~ = ~ \\frac { \\Psi ^ { - 1 } ( s ) - \\Psi ^ { - 1 } ( 0 ) } { s } ~ > ~ \\frac { \\Psi ^ { - 1 } ( r ) } { r } \\qquad \\forall 0 < s < r \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\widetilde { \\textbf { W } } _ S ^ * = \\mathop { \\arg \\min } \\limits _ { \\textbf { W } } \\left ( \\mathop { \\lim } \\limits _ { M , N \\to + \\infty } C _ S ( \\boldsymbol { \\psi } , \\textbf { W } ) \\right ) . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} \\sum _ l h ^ { p ^ { \\ast } } _ { i j k l } \\omega _ l = d h ^ { p ^ { \\ast } } _ { i j k } + \\sum _ l h ^ { p ^ { \\ast } } _ { l j k } \\omega _ { l i } + \\sum _ l h ^ { p ^ { \\ast } } _ { i l k } \\omega _ { l j } + \\sum _ l h ^ { p ^ { \\ast } } _ { i j l } \\omega _ { l k } + \\sum _ { q } h ^ { q ^ { \\ast } } _ { i j k } \\omega _ { { q ^ { \\ast } p ^ { \\ast } } } , \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\mathcal { L } ( v ) = \\div \\Big ( e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + \\lambda V ( x , y ) } \\big ( ( \\Lambda + \\nabla _ y \\widetilde { w } ) ^ T \\nabla _ y v ( \\Lambda + \\nabla _ y \\widetilde { w } ) + \\nabla _ y v \\big ) \\Big ) . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} \\mathbb { V } _ Z ( d ) = \\frac { 1 } { 2 } , \\quad \\forall d \\in ( 0 , \\sigma ^ 2 ) . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} A _ q ( X ) \\cap C ^ { q - 1 } ( X ) = \\mathbf { P A } _ q ( X ) . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\big ( ( A + B ) ^ 2 - C ^ 2 \\big ) \\big ( ( A - B ) ^ 2 - C ^ 2 \\big ) & = \\big ( ( A + C ) ^ 2 - B ^ 2 \\big ) \\big ( ( A - C ) ^ 2 - B ^ 2 \\big ) = \\big ( ( B + C ) ^ 2 - A ^ 2 \\big ) \\big ( ( B + C ) ^ 2 - A ^ 2 \\big ) \\\\ & = A ^ 4 + B ^ 4 + C ^ 4 - 2 A ^ 2 B ^ 2 - 2 A ^ 2 C ^ 2 - 2 B ^ 2 C ^ 2 \\mbox { f o r a n y } \\ \\ A , B , C \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\overline { T } _ { 2 k , 2 a } ( x ; q ) & = ( 1 + x q ) \\sum _ { i = 1 } ^ a ( x q ) ^ { 2 i } \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ h \\overline { T } _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ h \\overline { T } _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ] \\\\ & \\quad + ( 1 + x q ) \\sum _ { i = 0 } ^ { a - 1 } ( x q ) ^ { 2 i } \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ h \\overline { T } _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ h \\overline { T } _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\right ] . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} & \\int _ { J _ { 3 } } \\frac { | I _ { l } | d v } { ( ( 1 - v ) | I _ { l } | + | I _ { l + 1 } | + \\cdots + | I _ { k } | ) ^ { H + \\frac { 1 } { 2 } } } \\\\ = & \\frac { 1 } { \\left ( H - \\frac { 1 } { 2 } \\right ) } \\left ( \\frac { 1 } { ( | I _ { l + 1 } | + \\cdots + | I _ { k } | ) ^ { H - \\frac { 1 } { 2 } } } - \\frac { 1 } { \\left ( \\frac { 1 } { 3 } | I _ { l } | + | I _ { l + 1 } | + \\cdots + | I _ { k } | \\right ) ^ { H - \\frac { 1 } { 2 } } } \\right ) \\leq \\frac { C _ { H } } { | I _ { k } | ^ { H - \\frac { 1 } { 2 } } } , \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} V _ n ( z / c ) = \\frac { 1 + 2 n } { P _ n ^ { ( 1 / 2 , - 1 / 2 ) } ( 1 ) } P _ n ^ { ( 1 / 2 , - 1 / 2 ) } ( z / c ) \\ . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} T _ n ( z ) = \\left ( u ^ n + u ^ { - n } \\right ) / { 2 } , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} T = T _ 1 + T _ 2 , \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} E ( A _ N ) = \\# \\{ n _ 1 , n _ 2 , n _ 3 , n _ 4 \\leq N : ~ a _ { n _ 1 } + a _ { n _ 2 } = a _ { n _ 3 } + a _ { n _ 4 } \\} . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} Q ^ * B ^ s Q = B ^ { s } s = \\alpha , \\beta . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\| a ^ \\sharp ( \\phi _ k ) \\ldots a ^ \\sharp ( \\phi _ 1 ) \\Omega \\| \\leq \\sqrt { ( k + 1 ) ! } \\prod _ { i = 1 } ^ k \\| \\phi _ i \\| \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} F ( t ) = t ^ { e _ { 0 } } F _ { 1 } ( t ) ^ { e _ { 1 } } \\cdots F _ { r } ( t ) ^ { e _ { r } } \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} L _ { m + 1 } = Q _ { m + 1 } R _ { m + 1 } , \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} 0 < \\eta < \\frac { 1 } { 2 } { \\min } \\left ( \\underset { i \\neq j } { \\min } \\left | b _ { i } - b _ { j } \\right | , \\underset { i = 1 , . . , N _ { 2 } } { \\min } d i s t \\left ( b _ { i } , \\partial G \\right ) \\right ) . \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} B = B _ { 1 } + B _ { 2 } , \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\rho _ { i j } \\rightarrow \\cfrac { t ^ * ( ( t ^ * ) ^ 2 - ( \\zeta _ j ) ^ 2 ) } { \\zeta _ i - t ^ * } = \\rho _ { i j } ^ * ; \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} L R ( \\chi ) & = \\{ \\sum _ g t r _ { \\mathbb { Q } ( \\zeta _ l ) / \\mathbb { Q } } ( \\ , m \\ , \\overline { \\chi ( g ) } \\ , ) g \\mid m \\in \\mathbb { Q } ( \\zeta _ l ) \\} \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} F ( n ) = \\left \\{ a : [ a , b ) \\in \\mathcal { D } _ n [ a , b ) \\cap F \\neq \\emptyset \\right \\} . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} u ( x , t + \\tau _ j ) \\preceq v ( x , t ) \\preceq u ( x , t + \\tau _ j + \\tau _ j ' ) \\ \\ { \\rm f o r } \\ \\ ( x , t ) \\in \\R ^ N \\times \\R \\ \\ ( j = 1 , 2 , \\cdots ) . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} w ( r , s , t ) = \\cot ( r ) \\exp [ i n ( s - t ) ] , \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\mu _ { k } & = \\sigma _ { 1 } ^ { 2 } \\sigma _ { 2 } \\sigma _ { 3 } \\cdots \\sigma _ { k - 2 } \\sigma _ { k - 1 } ^ { 2 } , \\\\ \\alpha _ { p , k , j } & = \\mu _ { k } ^ { j } \\beta _ { p , k } . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} X _ { ( T ^ \\ast Q , G , \\omega , H , F , u ) } = X _ H + \\textnormal { v l i f t } ( F ) + \\textnormal { v l i f t } ( u ) . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} u _ i ( n + 1 ) + u _ i ( n - 1 ) + V ( n ) u _ i ( n ) = E u _ i ( n ) . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} X _ d T _ { d - 1 } ^ { - 1 } \\cdots T _ 1 ^ { - 1 } T _ 0 ^ { - 1 } = T _ { d - 1 } \\cdots T _ 1 ( X _ 1 T _ 0 ^ { - 1 } ) . \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} T : = L P _ 1 E ( \\{ \\lambda \\} ) \\ , , \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} & ( 1 + o ( 1 ) ) \\exp ( \\tfrac { 1 } { 4 } ( 1 - d ^ 2 ) ) \\frac { ( d n ) ! } { ( d n / 2 ) ! 2 ^ { ( d n ) / 2 } ( d ! ) ^ n } \\\\ & = ( 1 + o ( 1 ) ) \\sqrt { 2 } \\exp ( \\tfrac { 1 } { 4 } ( 1 - d ^ 2 ) ) \\left ( \\frac { d ^ d n ^ d } { e ^ d ( d ! ) ^ 2 } \\right ) ^ { n / 2 } \\\\ & = \\exp ( ( 1 + o ( 1 ) ) \\tfrac { 1 } { 2 } d n \\log n ) , \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\tilde { g } ( b ) ~ = ~ g ( b ) , \\tilde { g } ( x ) ~ : = ~ g ( x + ) \\qquad \\forall x \\in [ a , b ) \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\nu ^ { 1 } _ \\epsilon \\star \\nu ^ { 1 } _ \\iota = \\begin{cases} 1 , 0 , \\lambda - \\frac { 1 } { 2 } , \\nu ^ { 0 } _ + , \\nu ^ { 0 } _ - & \\mbox { i f } \\epsilon = \\iota \\mbox { a n d } n \\geq 5 \\\\ 1 , 0 , \\lambda - \\frac { 1 } { 2 } & \\mbox { i f } \\epsilon = \\iota \\mbox { a n d } n = m = 3 , 4 \\\\ \\lambda & \\mbox { i f } \\epsilon = - \\iota \\end{cases} \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\mathcal L _ 1 [ w ] : = \\mathcal A _ 1 [ w ] + \\left ( m + 2 - p \\right ) a _ { 1 , 1 } \\frac { \\langle \\nabla u _ 1 , \\nabla w \\rangle } { z _ 1 } - M \\frac { \\langle \\nabla z _ 1 , \\nabla w \\rangle } { z _ 1 } . \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} J _ { | Q } = I _ { | Q } , J ( Z ) = W . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} T f ( z ) = \\int _ { \\mathbb { C } } K ( z , w ) f ( w ) d \\mu ( w ) , \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { D } p _ D ( t - s , x , y ) u _ s ( y ) F ( \\delta s , \\delta y ) = & \\sum _ { n = 0 } ^ \\infty \\int _ 0 ^ t \\int _ { D } I _ n \\big ( p _ D ( t - s , x , y ) h _ n ( . , s , y ) \\big ) F ( \\delta s , \\delta y ) \\\\ = & \\sum _ { n = 0 } ^ \\infty I _ { n + 1 } \\Big ( \\widetilde { p _ D ( t - s , x , y ) h _ n ( . , s , y ) } \\Big ) , \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\mathbf { B } ( G , \\mu ) : = \\{ b \\in \\mathbf { B } ( G ) : \\nu ( b ) \\preceq \\theta _ T ( \\mu ) , \\kappa ( b ) = \\mu | _ { Z ( \\widehat { G } ) ^ { \\Gamma } } \\} . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k q ^ { ( k - j ) ( n - j ) } \\binom { n } { j } _ q \\binom { m } { k - j } _ q = \\binom { n + m } { k } _ q . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} D ( F ) = \\{ | x - y | \\colon x , y \\in F \\} . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} A ( f ) = \\{ 1 \\} \\cup A _ { 1 } \\cup \\cdots \\cup A _ { r } . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} t ( \\triangle , A ) = \\frac { 1 } { v _ n ^ 3 } \\sum _ { i , j , k = 1 } ^ { v _ n } A _ { i j } A _ { j k } A _ { i k } , \\partial _ { i j } t ( \\triangle , A ) = \\frac { 3 } { v _ n ^ 3 } \\sum _ { k = 1 } ^ { v _ n } A _ { i k } A _ { j k } . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} \\partial ^ 2 _ { j i } w ( t ) & = \\sum _ h ( \\partial _ h S ( \\Phi ( t ) ) \\partial ^ 2 _ { j i } \\Phi _ h ( t ) - \\partial _ h \\tilde S ( t , \\tilde \\Phi ( t ) ) \\partial ^ 2 _ { j i } \\tilde \\Phi _ h ( t ) ) \\\\ & + \\sum _ { h , k } ( \\partial ^ 2 _ { h k } S ( \\Phi ( t ) ) \\partial _ j \\Phi _ k ( t ) \\partial _ i \\Phi _ h ( t ) - \\partial ^ 2 _ { h k } \\tilde S ( t , \\tilde \\Phi ( t ) ) \\partial _ j \\tilde \\Phi _ k ( t ) \\partial _ i \\tilde \\Phi _ h ( t ) ) = : I _ 1 ( t ) + I _ 2 ( t ) \\ , , \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} \\Phi _ 1 ( \\nu , L ) = \\Phi ( \\alpha , M _ 1 , \\nu , r _ 1 L ) , \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} ( k - 1 ) a _ l = a _ { l + 1 } + \\cdots + a _ { l + k - 1 } , l \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\mathbb { C } [ \\ ! [ Q ] \\ ! ] : = \\left \\{ \\sum _ { d \\in H _ 2 ( X ; \\mathbb { Z } ) } f _ d Q ^ d \\middle | f _ d \\in \\mathbb { C } \\right \\} \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\delta _ { f } = \\delta _ { \\widetilde { f } } = \\lim _ { n \\to \\infty } \\max \\{ \\rho ( ( g ^ { n } ) ^ { * } | _ { N ^ { 1 } ( A ) } ) , \\rho ( ( f _ { T } ^ { n } ) ^ { * } | _ { N ^ { 1 } ( \\overline { T } ) } ) \\} ^ { 1 / n } = \\max \\{ \\delta _ { g } , \\delta _ { f _ { T } } \\} . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\geq | | 2 \\theta + ( 2 \\sum _ { i = 0 } ^ { s } ( - 1 ) ^ { i + 1 } b _ { { j _ i } } ^ \\prime ) \\alpha + ( - 1 ) ^ { s + 1 } k \\alpha | | _ { \\R / \\Z } - 2 \\sum _ { i = 0 } ^ { s } e ^ { - ( \\varsigma + \\varepsilon ) | K _ { j _ i } | } \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\alpha = ( \\underbrace { \\gamma _ 1 , \\dots , \\gamma _ 1 } _ { m _ 1 } , \\dots , \\underbrace { \\gamma _ r , \\dots , \\gamma _ r } _ { m _ r } ) \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\alpha _ t = \\exp \\Big [ \\int _ 0 ^ t b ( s , X _ s ) d X _ s - \\frac { 1 } { 2 } \\int _ 0 ^ t | b ( s , X _ s ) | ^ 2 d s \\Big ] \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} F ( n , k ) & = [ 3 n + 2 k + 1 ] \\frac { ( q ; q ^ 2 ) _ { n } ( q ^ { 2 k + 1 } ; q ^ 2 ) _ { n } ^ 2 q ^ { - { n + 1 \\choose 2 } - ( 2 n + 1 ) k } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } , \\\\ [ 5 p t ] G ( n , k ) & = - \\frac { ( 1 + q ^ { n + 2 k - 1 } ) ( q ; q ^ 2 ) _ { n } ( q ^ { 2 k + 1 } ; q ^ 2 ) _ { n - 1 } ^ 2 q ^ { - { n \\choose 2 } - ( 2 n - 1 ) k } } { ( 1 - q ) ( q ; q ) _ { n - 1 } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } , \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} ( a _ 1 - \\i B ) ^ 2 = - B ^ 2 \\ ( 1 - \\frac { | q ^ { ( 1 ) } _ { \\infty } | ^ 2 } { 4 B ^ 2 } \\ ) , ( \\tilde b _ 1 ) ^ 2 = \\frac { - \\ ( q ^ { ( 1 ) } _ { \\infty } \\ ) ^ 2 } { 4 } . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\Gamma _ 0 : = \\{ x \\in \\Omega \\mid ( u _ 0 ( x ) , v _ 0 ( x ) ) \\in S \\} \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} \\C ^ N _ \\mu f ( w ) : = \\sum _ { j = 0 } ^ { N - 1 } T _ j f ( w ) \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} 2 H _ { 0 } c _ { 1 } ^ { 2 } = \\frac { f _ { 2 } ^ { \\prime \\prime } } { \\left ( f _ { 2 } ^ { \\prime } \\right ) ^ { 3 } } . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} Q ^ \\lambda _ \\alpha ( t ) = k _ - ( t ) ^ { - 1 } Q ^ \\lambda _ \\alpha k _ - ( t ) , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } a _ { j } \\bigl ( t ^ { 2 } - 1 \\bigr ) ^ { j } \\geq 2 \\ln t \\sum _ { j = 0 } ^ { m } b _ { j } \\bigl ( t ^ { 2 } - 1 \\bigr ) ^ { j } , t \\geq 1 . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\chi ( \\Gamma _ n , M ) = \\prod _ i | H _ i ( \\Gamma _ n , M ) | ^ { ( - 1 ) ^ i } \\in \\Q ^ { \\times } \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\frac { \\mathcal { H } ^ { m } ( B _ { r } ^ { g } ( x ) ) } { \\mathcal { H } ^ { m } ( B _ { r } ( x ) \\cap M ) } = \\frac { \\omega _ { m } r ^ { m } ( 1 + O ( r ^ { 2 } ) ) } { \\omega _ { m } r ^ { m } ( 1 + O ( r ^ { 2 } ) ) } \\to 1 , r \\to 0 ^ { + } , \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\| \\tau \\| _ { L ^ \\infty } ^ { 2 a _ 0 / t _ i } \\to 0 \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ K \\delta _ k = \\sum \\limits _ { k = 1 } ^ K \\nu _ k = 1 . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} S = \\big \\{ ( f _ 1 ) \\times \\cdots \\times ( f _ n ) \\ : \\ f \\in \\tilde S \\big \\} . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} a _ { 1 3 } = 1 , \\ , a _ { 2 3 } = i , \\ , a _ { 3 1 } = - 1 , \\ , a _ { 3 2 } = - i ; 0 . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} e _ N ( x ) & : = u ( x ) - \\pi _ N ^ { \\lambda } u ( x ) = \\sum _ { n = N + 1 } ^ \\infty \\hat u _ n R _ n ^ \\lambda ( x ) = S ( t ) \\sum _ { n = N + 1 } ^ \\infty \\widehat { \\breve U } _ n C _ n ^ \\lambda ( t ) \\\\ & = S ( t ) \\big ( \\breve U ( t ) - \\varPi _ N ^ \\lambda \\breve U ( t ) \\big ) : = S ( t ) \\breve e _ N ( t ) . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\gamma \\triangleq & \\frac { \\rho _ d \\sigma _ { \\widehat { h } _ { a b } } ^ 2 } { \\sigma _ b ^ 2 + \\rho _ d \\sigma _ { \\widetilde { h } _ { a b } } ^ 2 } = & \\frac { \\rho _ d \\lambda _ { a b } ^ 2 n _ p \\rho _ p } { \\sigma _ b ^ 4 + \\rho _ d \\lambda _ { a b } \\sigma _ b ^ 2 + \\sigma _ b ^ 2 \\lambda _ { a b } n _ p \\rho _ p } . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\bar f ( s ) & = s \\xi ' ( s ) - \\xi ' ( s ) + 1 - \\xi ( s ) + \\frac { [ \\xi ' ( 1 ) - \\xi ' ( s ) ] ( 1 - q ) } { z _ 2 } \\\\ & \\ - \\frac { ( 1 - q ) ( 1 + z _ 2 ) ( \\xi ' ( 1 ) - \\xi ' ( q ) ) } { z _ 2 ^ 2 } \\log \\frac { 1 + z _ 2 } { 1 + \\frac { z _ 2 ( \\xi ' ( s ) - \\xi ' ( q ) ) } { \\xi ' ( 1 ) - \\xi ' ( q ) } } . \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\Phi ^ { ( 2 ) } _ t ( x , t , k ) = ( Q _ 0 ( x , t ) + k Q _ 1 ( x , t ) - 2 \\i k ^ 2 \\sigma _ 3 ) \\Phi ^ { ( 2 ) } ( x , t , k ) . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\| V \\| _ { \\dot { H } ^ s } = \\| Q \\| _ { \\dot { H } ^ s } = \\lim _ { n \\rightarrow \\infty } \\| v _ n \\| _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} E = \\sum _ i d _ i t ^ i { \\partial _ { t ^ i } } ; ~ ~ ~ e = { \\partial _ { t ^ { r } } } . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} ( y ^ * , x ^ * ) _ { R T } = ( y , x ) _ { R T } \\mbox { f o r } y \\in U ^ { \\leq 0 } , x \\in U ^ { \\geq 0 } , \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} d ( m \\otimes \\xi _ 1 \\wedge \\cdots \\wedge \\xi _ k ) = { } & \\sum _ i ( - 1 ) ^ i ( m \\xi _ i ) \\otimes \\xi _ 1 \\wedge \\cdots \\wedge \\widehat { \\xi } _ i \\wedge \\cdots \\wedge \\xi _ k \\\\ & + \\sum _ { i < j } ( - 1 ) ^ { i + j } m \\otimes [ \\xi _ i , \\xi _ j ] \\wedge \\xi _ 1 \\wedge \\cdots \\wedge \\widehat { \\xi } _ i \\wedge \\cdots \\wedge \\widehat { \\xi } _ j \\wedge \\cdots \\wedge \\xi _ k . \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} x \\in C , \\ ; f ( x ) \\leq f ( \\bar x ) \\Rightarrow f ( x ) = f ( \\bar x ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\begin{aligned} ( a _ { i j } , i = 0 , & \\dots , r - 1 , j = 1 , \\dots , S _ { k ' \\ ! , r } ( i ) ) \\\\ & ( b _ m , m = 0 , \\dots , s - 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { C } = C W ' ( A ) + \\frac 1 2 C ^ { - 1 } h ^ 2 W ''' ( A ) + O ( h ^ 4 ) \\\\ & \\dot { A } = C - W ( A ) + \\frac 1 2 C ^ { - 2 } h ^ 2 W '' ( A ) + O ( h ^ 4 ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} W Y _ j W ^ * = \\hat { W } ^ * Y _ j \\hat { W } = \\Psi ^ { - 1 } ( Y _ j ) = X _ j , \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\mathcal { S } : = S \\otimes | \\mathrm { d e t } \\ , S ^ * | ^ { \\frac { 1 } { r _ S } } , \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\sum _ { \\{ \\gamma \\} \\in \\mathcal { S } _ 1 ' ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } \\ell _ { \\gamma } } + 1 \\right ) ^ { - 1 } + \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } _ 2 ' ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } + \\ell _ { \\beta } ) } + 1 \\right ) ^ { - 1 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} x _ m + F ( u _ m , x _ m ) = y , \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\omega _ { 1 } = c _ { 5 } \\omega _ { 3 } ^ { 4 } \\omega _ { 4 } ^ { \\frac { 6 } { c _ { 1 } } } . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} F ( y , s ) = \\sum _ { \\rho \\in \\mathbb { Z } ^ n } F _ { \\rho } ( y , s ) \\textnormal { a n d } G ( x , t ) = \\sum _ { \\rho \\in \\mathbb { Z } ^ n } G _ { \\rho } ( x , t ) , \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\Gamma ( t ) : = e ^ { \\sum _ { i = 1 } ^ N \\left [ \\int _ 0 ^ t B _ i ( s ) d \\beta _ i + \\beta _ i \\theta _ i - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s - \\frac { 1 } { 2 } \\theta _ i ^ 2 t - \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s \\right ] } \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} X Y = Y X = \\begin{cases} e , & \\quad X \\cap Y = \\{ e \\} , \\\\ 0 , & \\quad . \\\\ \\end{cases} \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} D _ { v } ( e _ { k } ) & = - \\epsilon _ { k } \\sum _ { j } \\omega _ { j k } ( v ) e _ { j } = 2 \\sum _ { j < p } \\omega _ { p j } ( v ) ( \\epsilon _ { p } e _ { p } ^ { * } \\otimes e _ { j } - \\epsilon _ { j } e _ { j } ^ { * } \\otimes e _ { p } ) ( e _ { k } ) \\\\ & = 2 \\sum _ { j < p } \\omega _ { p j } ( v ) ( e _ { p } \\wedge e _ { j } ) ( e _ { k } ) , \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\nabla _ { z \\partial t _ i } S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) = 0 \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\Tilde { Y } = \\sum _ { j \\geq 1 } a _ j \\mathbb { 1 } _ { I _ j } . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} y _ i = l _ { t , u , i } ( y _ 3 , y _ 5 ) , i \\in \\{ 0 , 1 , 2 , 4 \\} , \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} x \\leq y \\Longleftrightarrow & y = \\top ( x , y \\in X _ { n } n \\in \\omega x \\leq ^ { \\mathbb { X } _ { n } } y ) \\\\ & ( x \\in X _ { n } y \\in X _ { m } n , m \\in \\omega n < m ) . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\int _ { A } \\langle u , d \\mu \\rangle = \\int _ { \\mathbb { R } ^ n \\times \\mathbb { R } ^ n } \\mathbf { 1 } _ I ( x , y ) \\Big \\langle \\mathbf { 1 } _ A ( x ) u ( x ) - \\mathbf { 1 } _ A ( y ) u ( y ) , d \\pi ( x , y ) \\Big \\rangle . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} | \\varphi ( k - 1 ) | , | \\varphi ( k ) | \\leq \\sum _ { i = 1 , 2 } e ^ { ( t + \\varepsilon ) k } | \\varphi ( x _ i ^ { \\prime } ) | e ^ { - | k - x _ i | \\ln \\lambda } , \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - u _ { x x } + \\frac { \\mu } { 1 + t } u _ t + \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } u = 0 , & x \\in \\mathbb { R } , \\ t > 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in \\mathbb { R } , \\\\ u _ t ( 0 , x ) = u _ 1 ( x ) , & x \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { y } { z } = \\frac { x y - z ^ 2 } { z } - \\frac { y x z } { z ^ 2 } = - z < - z _ 0 . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\frac { z ( w ) } { c } = \\sqrt { \\frac { w + c } { 2 c } } \\ , \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\gamma _ { U _ { t } ^ { ( l ) } } = \\left ( \\begin{array} { c c } \\gamma _ { U _ { t } ^ { ( l _ { 0 } ) } } & P \\\\ Q & R \\end{array} \\right ) , \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\mu ( B ( x , r _ j ) \\cap \\Omega ) = 2 ^ { - j } \\mu ( B ( x , r ) \\cap \\Omega ) . \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} - z '' = \\mu _ { 1 } \\left ( \\alpha - z \\right ) z ^ { + } - \\mu _ { 2 } \\left ( d + z \\right ) z ^ { - } . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\theta _ { 1 } A Y \\theta _ { 1 } \\mathbf { Z } _ { m i n } \\backslash Z _ { u = 1 , 1 } ^ { \\ast } \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} { \\mathfrak { D } } ( f , \\sigma ) = \\cup _ { \\{ k _ j \\} \\in \\mathbb { Z } ^ t } \\mathfrak { D } ( f , \\sigma , \\{ k _ j \\} ) \\quad , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} f _ h ( x ) ~ = ~ \\begin{cases} f ( x _ i ) & \\qquad \\forall x \\in [ x _ i , x _ { i + 1 } ) ~ , ~ i \\in \\overline { 0 , N _ { f , h } - 2 } \\cr \\cr f \\left ( x _ { N _ { f , h } - 1 } \\right ) & \\qquad \\forall x \\in \\left [ x _ { N _ { f , h } - 1 } , L ] \\right . . \\end{cases} \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} f = g \\circ \\psi \\colon X \\to \\c , \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\frac { R _ { 1 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = - R _ { 1 3 } = 0 ; \\quad \\frac { R _ { 2 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = - R _ { 2 3 } = - 1 = u _ 2 ; \\quad \\frac { R _ { 4 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = 2 u _ 4 . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} f _ F ( z ) = z + \\frac { c } { z } + o ( z ^ { - 1 } ) , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} T ( \\chi _ { V \\times U } ) ( \\xi , z ) & = \\lim _ { i \\to \\infty } T ( \\chi _ { V \\times U } ) ( \\xi , x _ { \\lambda _ { i } } ) \\leq \\frac { 2 p + q } { 3 } \\\\ & < \\frac { 2 q + p } { 3 } \\leq \\lim _ { i \\to \\infty } T ( \\chi _ { V \\times U } ) ( \\xi , y _ { \\lambda _ { i } } ) = T ( \\chi _ { V \\times U } ) ( \\xi , z ) , \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} Y ( X _ 1 , \\ldots , X _ m ) = \\left \\{ e \\in \\mathcal A : e \\prod _ { \\ell = 1 } ^ { m } e _ { \\sigma ( \\ell ) } \\in \\mathcal L ^ 1 \\ , \\forall \\sigma \\in \\Sigma ( m ) , \\ , e _ j \\in X _ j , \\ , j = 1 , \\ldots , m \\right \\} \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} h f ( n ' ) : = \\sum _ { m \\neq n ' } \\frac { f ( m ) } { n ' - m } . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\partial _ { t } g ' ( x , t ) = - 2 R i c ' ( x , t ) , \\ ; \\ ; \\ ; \\ ; ( x , t ) \\in M ' \\times ( 0 , T ] , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\underset { [ 0 , T ] } { } ( \\rho _ 3 ^ 2 ( t ) \\| y _ x ( t ) \\| ^ 2 ) + \\iint _ Q \\rho _ 3 ^ 2 | y _ t | ^ 2 d x d t & \\leq C \\left ( \\| y _ 0 \\| _ { H _ 0 ^ 1 ( I ) } ^ 2 + \\iint _ Q \\rho ^ 2 | G | ^ 2 d x d t + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\rho _ 1 ^ 2 | f | ^ 2 d x d t \\right . \\\\ & + \\left . \\iint _ Q \\rho _ 0 ^ 2 | y | ^ 2 d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\rho _ 0 ^ 2 | p ^ i | ^ 2 d x d t \\right ) . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} Z _ f ( T ) = \\sum _ { \\emptyset \\not = I \\subseteq J } ( \\mathbb { L } - 1 ) ^ { | I | - 1 } [ \\widetilde { E } _ I ^ { \\circ } ] \\prod _ { i \\in I } \\frac { \\L ^ { - \\nu _ i } T ^ { N _ i } } { 1 - \\L ^ { - \\nu _ i } T ^ { N _ i } } . \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} & M _ s ( g ( x ) ) = \\sum _ { \\ell = 0 } ^ s \\sum _ { m = 1 } ^ s m t _ { \\ell } t _ m M _ s ( x ^ { \\ell } , x ^ { m - 1 } ) , \\\\ & M _ n ( g ( x ) ) = \\sum _ { \\ell = 0 } ^ s \\sum _ { m = 1 } ^ s m t _ { \\ell } t _ m M _ n ( x ^ { \\ell } , x ^ { m - 1 } ) , \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\check { y } _ { q , m } [ n ] = y _ { q , m } [ n ] + e _ { q , m } [ n ] , \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\min \\ M _ { \\rm t r } , \\ { \\rm s . \\ t . } & \\ c _ i \\neq c _ j , \\ \\textrm { i f } \\ a _ { i j } = 1 , \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} A _ j \\boldsymbol { \\alpha } _ j = \\boldsymbol { p } _ j , \\textrm { a n d } A _ j \\boldsymbol { \\beta } _ j = \\boldsymbol { q } _ j . \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\nu _ \\omega : = \\displaystyle \\lim _ { n \\rightarrow \\infty } \\nu _ n \\mbox { o n } K [ x ] . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\Gamma ^ T \\nabla _ y \\widetilde { w } + \\epsilon \\bigg ( \\frac { | \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V - \\ln \\widetilde { m } - \\widetilde { H } _ 1 \\bigg ) = 0 . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c c } \\pi ^ 0 I _ { k _ 0 } & M _ { 0 , 1 } & M _ { 0 , 2 } & M _ { 0 , 3 } & M _ { 0 , 4 } \\cr 0 & \\pi I _ { k _ 1 } & \\pi M _ { 1 , 2 } & \\pi M _ { 1 , 3 } & \\pi M _ { 1 , 4 } \\cr 0 & 0 & \\pi ^ 2 I _ { k _ 2 } & \\pi ^ 2 M _ { 2 , 3 } & \\pi ^ 2 M _ { 2 , 4 } \\cr 0 & 0 & 0 & \\pi ^ 3 I _ { k _ 3 } & \\pi ^ 3 M _ { 3 , 4 } \\end{array} \\right ) U \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} i _ * ( M ) = i _ \\bullet ( M ) \\otimes _ \\C \\C [ \\partial _ { y _ { k + 1 } } , \\ldots , \\partial _ { y _ { \\dim ( Y ) } } ] \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - h _ n x _ n ^ 2 ) + \\rho _ { n + 1 } \\xi _ { n + 1 } , n \\in \\mathbb N , x _ 0 \\in \\mathbb R , \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} L _ { k , p , q } \\triangleq \\int _ { J _ { p } } \\left | \\sum _ { l = 1 } ^ { k - 1 } \\int _ { J _ { q } } \\frac { \\left | \\dot { \\bar { h } } _ { k } ( u ) \\right | + \\left ( \\frac { | I _ { 1 } | + \\cdots + | I _ { k } | } { | I _ { 1 } | + \\cdots + | I _ { l - 1 } | } \\right ) ^ { H - \\frac { 1 } { 2 } } \\cdot \\left | \\dot { \\bar { h } } _ { l } ( v ) \\right | } { | q _ { k , l } ( u , v ) | ^ { H + \\frac { 1 } { 2 } } } | I _ { l } | d v \\right | ^ { 2 } | I _ { k } | d u , \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} \\limsup _ { t \\to 0 + } t ^ { 1 / 2 } \\lVert \\nabla e ^ { t A } ( a _ 1 - a _ { } ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } = 0 , t ^ { 1 / 2 } \\lVert \\nabla e ^ { t A } a _ 2 \\rVert _ { L ^ \\infty _ H L ^ p _ z } \\le C _ 4 \\lVert a _ 2 \\rVert _ { L ^ \\infty _ H L ^ p _ z } , t > 0 , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\begin{pmatrix} B _ 1 & A _ 1 & 0 & 0 & \\cdots \\\\ A _ 1 ^ * & B _ 2 & A _ 2 & 0 & \\\\ 0 & A _ 2 ^ * & B _ 3 & A _ 3 & \\ddots \\\\ 0 & 0 & A _ 3 ^ * & B _ 4 & \\ddots \\\\ \\vdots & & \\ddots & \\ddots & \\ddots \\end{pmatrix} \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\Delta u ^ i = - | \\nabla u | ^ 2 u ^ i , i = 1 , \\cdots , K . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\tilde { a } ^ { 0 } _ { i j } = \\tilde { a } _ { i j } ( e ) = a _ { i j } ( \\delta _ { \\sigma ^ { \\nu } } e ) = a _ { i j } ( e ) . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} \\# \\mathrm { L y r } _ n { \\mathcal { T } } = \\sum _ { i = M } ^ { n } \\ , \\# \\mathrm { L y r } _ n { \\mathcal { B } ( i ) } , M : = \\max ( n _ \\ast , n - d ) . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} f ( x ) = ( I _ a ^ k \\varphi ) ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} I ( \\theta ) = \\int _ { \\theta } ^ { \\theta + 1 / 2 } \\frac { g ' ( x - \\theta ) ^ 2 } { g ( x - \\theta ) } d x = \\int _ 0 ^ 1 \\frac { g ' ( x ) ^ 2 } { g ( x ) } d x = 9 6 0 \\int _ 0 ^ { 1 / 2 } ( 1 - 4 x ) ^ 2 d x = 1 6 0 , \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} C _ { i j } ^ k ( t ) : = \\sum _ p \\eta ^ { k p } \\partial _ { t ^ p } \\partial _ { t ^ i } \\partial _ { t ^ j } \\mathbb { F } ( t ) \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} \\frac { 1 } { r } \\Big ( \\widetilde { \\Omega } _ { r + t } [ u _ 0 ] ( x ) + \\widetilde { \\Omega } _ { r - t } [ u _ 0 ] ( x ) \\Big ) & = \\frac { 1 } { r } \\Big ( \\Omega _ { r + t } [ u _ 0 ] ( x ) - \\Omega _ { t - r } [ u _ 0 ] ( x ) \\Big ) \\xrightarrow [ r \\to 0 ] { } 2 \\frac { \\partial } { \\partial t } \\ , \\Omega _ { t } [ u _ 0 ] ( x ) . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\Sigma = \\sum _ { k = 1 } ^ d \\omega _ k ^ 2 f _ k f _ k ^ { \\top } + \\Gamma , \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} ( u _ 1 \\cdot \\nabla ) v _ 2 = \\nabla \\cdot ( u _ 1 \\otimes v _ 2 ) = \\nabla _ H \\cdot ( v _ 1 \\otimes v _ 2 ) + \\partial _ z ( w _ 1 v _ 2 ) \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} \\tau ^ { \\Gamma } _ p ( M ) = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log _ p \\chi ( \\Gamma _ n , M ) \\ ; \\Q _ p \\ ; . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\langle \\hat u _ j , ( E - \\lambda _ j I ) \\hat u _ j \\rangle = \\sum _ { k \\neq j } \\langle \\hat u _ j , u _ k \\rangle \\langle ( \\Sigma - \\lambda _ j I ) \\hat u _ j , u _ k \\rangle = \\sum _ { k \\neq j } ( \\lambda _ k - \\lambda _ j ) \\langle \\hat u _ j , u _ k \\rangle ^ 2 . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} h = \\tfrac { 1 } { 2 } d \\eta \\left ( \\cdot , I \\cdot \\right ) + \\eta \\otimes \\eta + d t \\otimes d t . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} X _ N ( i , j ) : = Z _ { i , j } , \\ , 1 \\le i \\le M _ N , \\ , 1 \\le j \\le N \\ , . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { U } _ { k , a } ( n ) q ^ n = \\sum _ { n \\geq 0 } \\overline { U } _ { k , a - 1 } ( n ) q ^ n = \\frac { ( - q ^ 2 ; q ^ 2 ) ^ 2 _ \\infty ( q ^ { a } , q ^ { 2 k - a } , q ^ { 2 k } ; q ^ { 2 k } ) _ \\infty } { ( q ^ { 2 } ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} I ( z ) & = \\int _ { H ^ { + } } F ( s ) J _ { k - 1 - \\tau _ 0 } ( z , s ) \\ , d V ( s ) \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} U _ t ( x ) = e ^ { - x t } U _ 0 ( x ) + \\int _ 0 ^ t e ^ { - x ( t - s ) } ( - \\lambda V _ s d s + \\nu \\sqrt { V _ s } d W _ s ) , t \\geq 0 . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { f ( n ) \\over g ( n ) } = 1 . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} G = G _ 0 > G _ 1 > \\cdots > G _ { t - 1 } > G _ t = 1 , \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} \\rho ( H ) = \\rho ( P ^ { - ( r - 1 ) } \\mathcal { A } ( H ) P ) \\leq \\max _ { 1 \\leq i \\leq n } \\left \\{ r _ i ( P ^ { - ( r - 1 ) } \\mathcal { A } ( H ) P ) \\right \\} . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} ( \\ell - 1 ) e + ( 2 \\ell + 1 ) q + \\ell + i + j - k + 1 = 0 . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { L } } d \\sigma _ { \\Sigma , L } = \\frac { 1 } { \\sqrt { L } } e _ 1 ^ { \\star } \\wedge e _ 2 ^ { \\star } = \\frac { l } { l _ L } ( \\overline { p } \\omega _ 2 - \\overline { q } \\omega _ 1 ) \\wedge \\omega + \\frac { 1 } { \\sqrt { L } } \\overline { r _ L } \\omega _ 1 \\wedge \\omega _ 2 . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\eta _ { K , M } ^ { ( 2 ) } = \\left ( { K { N _ { \\cal S } } - { \\lambda _ { \\cal S } } } \\right ) G _ { K , M } ^ { ( 1 ) } + { \\lambda _ { \\cal S } } G _ { K , M } ^ { ( 2 ) } , \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} { } N ( x , t , s ) - \\mathcal { K } [ N ] ( x , t , s ) = F ( x , t , s ) . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} & \\left ( \\ref { c l a i m : d i f i c i l } \\right ) u = 1 \\exists Z _ { u = 1 , 3 } ^ { \\ast } \\mathbf { Z } _ { m i n } \\backslash \\left [ \\mathbf { O } _ { m i n } \\cup \\left \\{ Z _ { u = 1 , 1 } ^ { \\ast } , Z _ { u = 1 , 2 } ^ { \\ast } \\right \\} \\right ] Z _ { u = 1 , 3 } ^ { \\ast } \\\\ & Z _ { u = 1 , 2 } ^ { \\ast } \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\Psi _ i = \\prod _ { j \\neq i } \\frac { 1 - \\Lambda _ j P ^ { - 1 } } { 1 - \\Lambda _ j \\Lambda _ i ^ { - 1 } } \\in K _ { T ^ { N + 1 } } \\left ( \\mathbb { P } ^ N \\right ) \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} E = \\kappa \\int | \\psi ^ * \\omega | ^ 2 \\ , \\sqrt { g } \\ , d ^ 3 x = \\kappa \\int F _ { \\mu \\nu } F ^ { \\mu \\nu } \\ , \\sqrt { g } \\ , d ^ 3 x , \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} D ^ { ( 1 ) } _ { u } : = D _ { u } - \\frac { 1 } { 2 } \\mathcal J D _ { u } \\mathcal J \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} A _ s ^ \\gamma : = \\frac { 2 ^ { 2 s } \\Gamma ( s + \\gamma ) \\Gamma ( s + \\frac { 1 } { 2 } ) } { \\sqrt { \\pi } \\Gamma ( \\gamma ) } . \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\mathcal { N } _ r ( S i n g ( u ) \\cap B _ 2 ( x ) ) \\subset \\bigcup _ { i = 1 } ^ m B _ { 5 r } ( x _ j ) . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} \\nabla ^ { \\Sigma , L } _ { \\dot { \\gamma } } \\dot { \\gamma } = \\langle \\nabla ^ L _ { \\dot { \\gamma } } \\dot { \\gamma } , e _ 1 \\rangle _ L e _ 1 + \\langle \\nabla ^ L _ { \\dot { \\gamma } } \\dot { \\gamma } , e _ 2 \\rangle _ L e _ 2 , \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } ( q , Q ) = \\sum _ { i = 0 } ^ N \\widetilde { J _ i } ( q , Q ) \\left ( 1 - P ^ { - 1 } \\right ) ^ i \\in K \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } ( q ) [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\psi _ r \\ , \\mapsto \\begin{cases} B _ r ^ { ( 0 ) } r \\in \\mathbb Z \\\\ B _ { r - \\frac 1 2 } ^ { ( 1 ) } r \\in \\mathbb Z + \\tfrac 1 2 \\end{cases} \\psi _ s ^ * \\ , \\mapsto \\begin{cases} K _ s ^ { \\perp ( 0 ) } s \\in \\mathbb Z \\\\ K _ { s + \\frac 1 2 } ^ { \\perp ( 1 ) } s \\in \\mathbb Z + \\tfrac 1 2 \\end{cases} \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\left ( \\mu _ \\lambda , G \\right ) _ H = \\left ( \\mu _ \\lambda - ( \\mu _ \\lambda ) _ D , G \\right ) _ H + | D | ( \\mu _ \\lambda ) _ D G _ D \\lesssim \\norm { \\nabla \\mu _ \\lambda } _ H \\norm { G } _ { V ^ * } + ( \\mu _ \\lambda ) _ D \\norm { G } _ { V ^ * } \\ , , \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} m [ \\sum _ { i = 1 } ^ n v _ i ] = p [ \\sum _ { i = 1 } ^ n v _ i ] \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\langle D ^ 2 _ 1 J _ 1 ( f ; v ^ 1 , v ^ 2 ) , ( w ^ 1 , w ^ 1 ) \\rangle & = \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } \\eta w ^ 1 d x d t + \\mu _ 1 \\iint _ { \\mathcal { O } _ 1 \\times ( 0 , T ) } | w ^ 1 | ^ 2 d x d t . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} d \\alpha _ { \\phi } = \\phi _ { M } ^ { \\ast } \\lambda - \\lambda . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} \\begin{aligned} \\mu \\bar h _ { 2 2 2 } ^ { 2 ^ * } = & \\bigl \\{ 1 2 \\bar \\lambda ^ 2 + ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) ^ 2 \\bigl \\} A , \\\\ \\bar h _ { 2 2 2 } ^ { 1 ^ * } = & - \\bar h _ { 2 1 1 } ^ { 1 ^ * } , \\\\ \\bar h _ { 2 2 1 } ^ { 1 ^ * } = & - \\bar h _ { 1 1 1 } ^ { 1 ^ * } , \\\\ \\mu \\bar h _ { 2 1 1 } ^ { 1 ^ * } = & \\bar \\lambda ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) A , \\\\ \\mu \\bar h _ { 1 1 1 } ^ { 1 ^ * } = & - 4 \\bar \\lambda ^ 2 A \\\\ \\end{aligned} \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\dim _ H B ( \\{ s _ n \\} , \\{ t _ n \\} , N ) = \\dim _ H B ( \\{ s _ { n + N - 1 } \\} , \\{ t _ { n + N - 1 } \\} , 1 ) . \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\overline { Q } _ { k , i } ( m , n ) = 0 , m n < 0 . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} P = p ( I ( \\xi _ 1 ) , I ( \\xi _ 2 ) , \\ldots , I ( \\xi _ n ) ) \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F D N } } ^ { \\textrm { N O M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { F } } \\right ) P _ \\textrm { N } | h _ { \\textrm { B F } } | ^ 2 } { { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } } . \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} V \\subset h ^ { - 1 } ( 0 ) \\theta | _ U = d h . \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\lambda _ p \\ = \\ \\sum _ { q = 0 } ^ { n - 1 } c _ { 1 q } e ^ { - 2 \\pi i p \\frac { q } { n } } p \\ = \\ 0 , \\ldots , \\ n - 1 \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} d ( p _ 0 , p _ 0 ^ { \\star } ) \\leq C _ 0 ( d ( p _ 0 , p ^ { \\star } ) + d ( p ^ { \\star } , p _ 0 ^ { \\star } ) ) = C _ 0 ( d ( p _ 0 , p ^ { \\star } ) + \\delta ( p ^ { \\star } ) ) \\leq 6 C _ 0 ^ 2 d ( p , p ^ { \\star } ) . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\mathrm { m o d } _ p ( \\rho ) = & \\mathrm { I m } \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\triangle ^ + ( \\rho ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } \\\\ = - & \\mathrm { I m } \\sum _ { \\{ \\alpha , \\beta ; \\epsilon \\} \\in \\triangle ^ - ( \\rho ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { L } } d s _ L = d s + d \\overline { s } L ^ { - 1 } + O ( L ^ { - 2 } ) ~ ~ { \\rm a s } ~ ~ L \\rightarrow + \\infty . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\langle a , b \\rangle \\in S \\Longleftrightarrow & a = b a , b \\notin Z ' . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\nu ( p ) \\leq \\nu ( p _ i ) + i \\nu ( q ) \\leq \\nu \\left ( a _ { i j } \\textbf { Q } ^ { \\lambda _ { i j } } \\right ) + i \\nu ( q ) = \\nu \\left ( a _ { i j } \\textbf { Q } ^ { \\lambda ' _ { i j } } \\right ) , \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} \\dot H ^ s ( \\mathcal { H } ) : = \\dot { B } ^ s _ { 2 , 2 } ( \\mathcal { H } ) H ^ s ( \\mathcal { H } ) : = B ^ s _ { 2 , 2 } ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} L _ { n } ^ { ( \\delta ) } ( z , q ) = q ^ { - n } { L } _ { n } ^ { ( \\delta + 1 ) } ( z , q ) - q ^ { - n } { L } _ { n - 1 } ^ { ( \\delta + 1 ) } ( z , q ) . \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} h ( x ) = \\frac 2 3 \\sqrt { x + 2 } , \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} A ^ k ( A ^ { * } A ) ^ { k } = ( A A ^ { * } ) ^ k A ^ { k } . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} d \\mapsto \\gamma _ d : = \\sup \\{ \\gamma \\in \\Gamma _ K ^ { > \\theta ( w , x ) } \\cup \\{ + \\infty \\} : d \\in A _ \\gamma ^ { w , x , b } \\} . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} \\alpha ( x ) = f ^ { - 1 } \\big ( - f ( x ) \\big ) , \\beta ( x ) = \\frac { f ' ( x ) } { f ' ( \\alpha ( x ) ) } . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\mathcal D ( - \\left . \\Delta \\right | _ { \\mathit { D } } ) = H ^ 2 ( \\Omega ) \\cap H ^ 1 _ 0 ( \\Omega ) , \\mathcal D ( - \\left . \\Delta \\right | _ { \\mathit { N } } ) = \\left \\{ u \\in H ^ 2 ( \\Omega ) : \\left . \\partial _ n u \\right | _ { \\partial \\Omega } = 0 \\right \\} \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} = \\sum \\limits _ { b \\in \\mathbf { B } ( G , \\mu ) } \\mathrm { M a n t } _ { G , b , \\mu } ( e ( J _ b ) L J ( \\delta ^ { \\frac { 1 } { 2 } } _ { P _ b } \\otimes J ^ G _ { P ^ { o p } _ b } I ^ G _ { M _ S } ( \\rho ) ) ) . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} O \\Bigl ( \\textstyle \\sum _ { k = J + 1 } ^ { n - 1 } I _ k / \\beta \\Bigr ) , \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\mathcal { H } = I _ { 1 ^ { - } } ^ { 1 / 2 - H } ( L ^ { 2 } ( [ 0 , 1 ] ) ) . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\omega _ { 3 } ^ { \\prime } = c _ { 5 } \\omega _ { 3 } ^ { 4 } \\omega _ { 4 } ^ { \\frac { 6 - c _ { 1 } } { c _ { 1 } } } , c _ { 5 } \\in \\mathbb { R } , c _ { 5 } \\neq 0 , \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\langle x , \\partial ^ 2 u \\rangle + \\langle 1 - x , \\nabla u \\rangle + \\lambda u = 0 \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\sum _ { j = 1 } ^ { V _ i } Z _ { i j } \\right ] = & \\frac { \\mathbb { E } \\left [ Z \\right ] } { \\epsilon _ { k , n } ( \\delta ) } \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\epsilon _ m ^ { \\frac { n } { 2 } } \\int _ { \\mathbb { R } ^ n } e ^ { - \\epsilon _ m | x | ^ 2 } f ( x ) g _ { m } ( x ) d x = \\int _ { \\mathbb { T } ^ n } f ( x ) g ( x ) d x \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} & V ^ 2 + V ^ 4 | \\nabla \\tau | ^ 2 + \\frac { ( d i v V ^ 2 \\nabla \\tau ) ^ 2 } { | H _ 0 | ^ 2 } \\\\ = & A ^ 2 + 2 A B _ i \\tilde { X } ^ i r \\\\ & + r ^ 2 \\left [ ( B _ i \\tilde { X } ^ i ) ^ 2 + A ^ 2 + ( C _ i \\tilde { X } ^ i ) ^ 2 + g _ 1 + \\frac { g _ 2 } { 4 } - h _ 0 ^ { ( 1 ) } \\sum _ { i j } C _ i C _ j \\tilde { X } ^ i \\tilde { X } ^ j \\right ] , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\gamma ( \\lambda ) : = \\sqrt { \\delta } \\psi ^ { - 1 } \\left ( \\frac { ( b - \\Re \\lambda ) ( 1 - \\epsilon ) } { \\delta } \\right ) \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} 0 = b _ { \\alpha + \\beta , \\gamma } ( \\theta ^ \\beta _ + + \\theta ^ \\gamma _ - ) = - 2 \\xi _ \\beta b _ { \\alpha + \\beta , \\gamma } \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} \\kappa ^ { \\star } = \\inf \\left \\{ \\kappa > 1 \\ | \\ \\kappa W _ { \\rho ' , \\nu } \\geq W _ { \\rho , \\nu } \\textup { i n } \\left ( - \\rho , \\rho \\right ) \\right \\} . \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { \\pi n \\theta ( 1 - \\theta ) } \\right ) ^ { 1 / 2 } \\sum _ { r = 2 } ^ { + \\infty } \\frac { 1 + 6 \\pi ^ 3 r ^ 3 n \\theta ( 1 - \\theta ) } { r } e ^ { - \\pi ^ 2 n \\theta ( 1 - \\theta ) r ^ 2 } \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\rho ( t , \\cdot ) \\| _ { \\dot { H } ^ { s } ( \\R ^ d ) } & \\geq \\| \\rho ( t , \\cdot ) \\| ^ 2 _ { L ^ 2 ( \\R ^ d ) } \\| \\rho ( t , \\cdot ) \\| ^ { - 1 } _ { \\dot { H } ^ { - s } ( \\R ^ d ) } \\geq \\| \\rho ( 0 , \\cdot ) \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 \\hat { C } _ s ^ { - 1 } \\exp ( s c t ) \\\\ & = C _ s \\exp ( s c t ) \\end{aligned} \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left \\| \\hat { \\mathbf { z } } _ k \\right \\| _ 2 ^ 2 \\right ] = \\operatorname { t r } \\left \\{ \\mathbf { I } _ S ( \\hat { \\boldsymbol { \\psi } } _ { k \\ ! - \\ ! 1 } , \\ ! \\mathbf { W } _ k ) ^ { - 1 } \\right \\} < + \\infty , \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | S ( \\cdot ) x _ n - S ( \\cdot ) x ^ { \\star \\star } | _ { C ( [ t _ 1 , T ] : E ) } = 0 \\lim _ { n \\to \\infty } | S ( \\cdot ) x _ n - S ( \\cdot ) x ^ { \\star \\star } | _ { L ^ p ( [ 0 , T ] : E ) } = 0 . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} \\biguplus _ { n \\in \\omega } K _ n \\uplus \\{ x \\} & = \\biguplus _ { n \\in \\omega } ( K \\cap F _ n ) \\uplus \\{ x \\} \\\\ & = \\left ( K \\cap \\biguplus _ { n \\in \\omega } F _ n \\right ) \\uplus \\{ x \\} \\\\ & = K \\cap \\left ( \\biguplus _ { n \\in \\omega } F _ n \\uplus \\{ x \\} \\right ) \\\\ & = K \\cap E = K . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} h _ n \\le e ^ { - \\frac { 3 ^ { n + 1 } } { \\left ( n + e _ { [ k ] } ^ 1 \\right ) \\ln \\left ( n + e _ { [ k ] } ^ 1 \\right ) \\dots \\ln _ k \\left ( n + e _ { [ k ] } ^ 1 \\right ) } } < e ^ { - 3 ^ { \\frac n 2 + 1 } } \\le e ^ { - 3 ^ { 2 } } = e ^ { - 9 } . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\| y ( x , \\cdot ) \\| _ { \\infty , t } = \\sup _ { 0 \\le s \\le t } | y ( x , s ) | , \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} H ( \\xi ) = H ( z , t ) = ( ( 1 + | z | ^ 2 ) ^ 2 + t ^ 2 ) ^ { - ( Q + \\alpha ) / 4 } . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} C _ { \\rho _ 1 , \\rho _ 2 } ( j ) = \\begin{cases} ( 2 ^ j | \\rho _ 1 - \\rho _ 2 | ) ^ { - 1 0 n } | \\rho _ 1 - \\rho _ 2 | \\geq 4 \\sqrt { n } , \\\\ 2 ^ { - n j / 2 } | \\rho _ 1 - \\rho _ 2 | < 4 \\sqrt { n } , \\end{cases} \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { H } ( A _ t , Z _ t , W _ t , ( \\partial _ t \\mathbb { G } ^ { u , v } , \\mathbb { G } , \\partial _ x \\mathbb { G } , \\partial _ { x x } \\mathbb { G } ) ( A _ t ; Z _ t , W _ t ) ) \\\\ \\mathbb { G } ( A _ T ; Z _ T , W _ T ) = m ( A _ T ) , ~ ( A _ T , Z _ T , W _ T ) \\in \\Lambda _ T \\times \\hat { \\Lambda } _ T . \\end{cases} \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} T _ { \\rm r e g } = ( I - P ) T _ { \\rm o p } . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\tilde { X } \\tilde { W } = \\tilde { W } \\tilde { X } , \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\delta : = 1 - 2 \\epsilon . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} \\lambda \\left ( V _ { B } \\right ) = \\biggl \\{ \\begin{array} { l } 1 \\quad \\mathrm { i f } ~ \\eqref { e q : s e r i e s i n v o l v i n g t h e c o m p a r i s o n o f g r o w t h i n B o r e l - B e r n s t e i n t h e o r e m } \\ , \\mathrm { ~ c o n v e r g e s , } \\\\ 0 \\quad \\mathrm { i f } ~ \\eqref { e q : s e r i e s i n v o l v i n g t h e c o m p a r i s o n o f g r o w t h i n B o r e l - B e r n s t e i n t h e o r e m } \\ , \\mathrm { ~ d i v e r g e s . } \\end{array} \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } ( X _ 1 ) _ n ( X _ 2 ) _ n ( a q / X _ 1 X _ 2 ) ^ n \\beta _ n ( a , q ) = \\frac { ( a q / X _ 1 ) _ { \\infty } ( a q / X _ 2 ) _ { \\infty } } { ( a q ) _ { \\infty } ( a q / X _ 1 X _ 2 ) _ { \\infty } } \\sum _ { n \\ge 0 } \\frac { ( X _ 1 ) _ n ( X _ 2 ) _ n ( a q / X _ 1 X _ 2 ) ^ n \\alpha _ n ( a , q ) } { ( a q / X _ 1 ) _ n ( a q / X _ 2 ) _ n } . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} \\frac { \\left ( \\lambda x + 1 \\right ) ^ { p } - 1 } { \\lambda ^ { p } } & = f \\\\ \\frac { 1 } { \\lambda ^ { p } } \\left [ \\frac { \\left ( \\lambda y + F ( x ) \\right ) ^ { p } } { \\lambda x + 1 } - G ( f ) \\right ] & = \\frac { 1 } { f } \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} U _ { \\varepsilon _ { n } } ( x ) = U _ { \\varepsilon _ { n } } ^ { k } ( x ) = \\left ( \\dfrac { x - \\bar { b } _ { k } } { | x - \\bar { b } _ { k } | } \\right ) ^ { d _ { k } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , B _ { T _ { \\varepsilon _ { n } } , \\eta _ { 0 } } ( \\bar { b } _ { k } ) . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\log \\phi \\Big ( \\Big ( \\exp \\big ( \\sum _ { j = 1 } ^ m A _ j \\big ) \\Big ) ^ p \\Big ) \\leq \\int _ { - \\infty } ^ { + \\infty } d t \\beta _ 0 ( t ) \\log \\phi \\Big ( \\Big | \\prod _ { j = 1 } ^ m \\exp \\big ( ( 1 + i t ) A _ j \\big ) \\Big | ^ p \\Big ) . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} & \\lim _ { i \\to \\infty } \\ \\sup _ { m \\in D _ { r _ { i _ 0 } } ' } \\ | | F ( m ) - F _ i ( m ) | | = 0 ( * _ 1 ) \\\\ & \\lim _ { i \\to \\infty } \\ \\ | | \\ ( l - l _ i ) | W _ { i _ 0 } ' \\ | | = 0 ( * _ 2 ) . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} & \\int _ { \\Delta _ \\tau \\cap \\{ | x _ { N + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , | \\langle \\omega , \\nu _ { } \\rangle | \\\\ & \\leq C L ^ { - 1 } \\int _ { \\Delta _ \\tau \\cap \\{ | x _ { n + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , | x _ 1 ^ 2 + \\hdots + x _ n ^ 2 - 2 ( n - 1 ) | \\\\ & \\leq C L ^ { - 1 } \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , u ^ 2 . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} \\Gamma ^ T \\nabla _ y \\widetilde { m } + \\epsilon \\big ( \\nabla _ y \\widetilde { m } ^ T \\nabla _ y \\widetilde { w } + \\widetilde { m } \\Delta _ y \\widetilde { w } \\big ) = 0 . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} R _ { 2 n } ^ { \\lambda } ( x ) & = \\frac { a _ n ^ \\lambda } { ( 1 + x ^ 2 ) ^ { \\frac { \\lambda + 1 } { 2 } } } \\ , { } _ { 2 } F _ { 1 } \\Big ( \\ ! - n , n + \\lambda ; \\lambda + \\frac { 1 } { 2 } ; \\frac { 1 } { 1 + x ^ 2 } \\Big ) \\\\ [ 2 p t ] & = a _ n ^ \\lambda \\ , \\sum _ { k = 0 } ^ { n } \\frac { \\left ( - n \\right ) _ { k } \\left ( n + \\lambda \\right ) _ { k } } { \\left ( \\lambda + \\frac { 1 } { 2 } \\right ) _ { k } k ! } \\frac { 1 } { \\left ( 1 + x ^ 2 \\right ) ^ { k + \\frac { \\lambda + 1 } { 2 } } } , \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} y ( t ) = \\tfrac { 1 } { 2 } x ^ 2 + C , \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\check h _ { j - 1 } : = \\bigg \\lceil \\frac { \\check h _ j + 1 } { 2 } \\bigg \\rceil , \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\mathbb { Z D } _ { n , \\epsilon } ^ m ( p _ 1 , \\ldots , p _ r ) = \\{ ( X _ 1 , \\ldots , X _ m ) \\in \\mathbb { D } ^ m _ n \\ : | \\ : \\ : \\| p _ j ( X _ 1 , \\ldots , X _ m ) \\| \\leq \\epsilon , 1 \\leq j \\leq r \\} \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} d x ( t ) = \\sigma _ { i ( t ) } ( x ( t ) ) d W ( t ) + b _ { i ( t ) } ( x ( t ) ) d t + d l ( t ) \\ ; , ~ ~ 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} D A = \\widetilde A D . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\Omega f ( \\xi ) = & \\sum _ { x \\in \\mathbb { Z } ^ d } \\big [ f ( \\xi ^ { x , 0 } ) - f ( \\xi ) \\big ] + \\sum _ { x \\in \\mathbb { Z } ^ d } \\sum _ { y : y \\sim x } \\lambda \\big [ f ( \\xi _ { a , b } ^ { x , y } ) - f ( \\xi ) \\big ] \\\\ & + \\sum _ { x \\in \\mathbb { Z } ^ d } f _ x ^ { \\prime } ( \\xi ) \\Big ( 1 - 2 d \\lambda [ ( b - 1 ) + a ] \\Big ) \\xi ( x ) \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} L _ { k , p , q } \\leq C _ { H , l _ { 0 } } | I _ { k } | \\cdot \\left ( \\sum _ { l = 1 } ^ { k - 1 } \\frac { | I _ { l } | } { ( | I _ { l } | + \\cdots + | I _ { k - 1 } | ) ^ { H + \\frac { 1 } { 2 } } } \\cdot \\left ( \\frac { | I _ { 1 } | + \\cdots + | I _ { k } | } { | I _ { 1 } | + \\cdots + | I _ { l - 1 } | } \\right ) ^ { H - \\frac { 1 } { 2 } } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} \\| \\xi _ 1 \\xi _ 2 \\| _ { L ^ p ( \\mathcal M ) } \\leq \\| \\xi _ 1 \\| _ { L ^ { p _ 1 } ( \\mathcal M ) } \\| \\xi _ 2 \\| _ { L ^ { p _ 2 } ( \\mathcal M ) } , \\textstyle \\frac { 1 } { p } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} K _ m ( t ) : = \\sup _ { 0 < s < t } s ^ { 1 / 2 } \\lVert \\nabla V _ m ( s ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } , H _ m ( t ) : = \\sup _ { 0 < s < t } \\lVert V _ m ( s ) \\rVert _ { L ^ \\infty _ H L ^ p _ z } \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} D _ l ( k ) = \\frac { 1 } { \\int _ { S _ l } f ( x ) d x } \\int _ { S _ l } \\mathbb { E } d ( x , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) f ( x ) d x , \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} Q : = \\frac { ( \\delta _ { + } - \\delta _ { - } ) ^ { 2 } } { h ^ { 2 } } U ^ { - 1 } P U = - \\Delta + W _ { h } ( x ) , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} C ^ * C & = Q ^ * B ^ { \\beta } Q = B ^ { \\beta } , \\\\ C ^ { * 2 } C ^ 2 & = Q ^ * B ^ \\frac { \\beta } { 2 } C ^ * C B ^ \\frac { \\beta } { 2 } Q = Q ^ * B ^ { 2 \\beta } Q = B ^ { 2 \\beta } = ( C ^ * C ) ^ 2 , \\\\ C ^ { * 3 } C ^ 3 & = C ^ { * } ( C ^ { * 2 } C ^ 2 ) C = Q ^ * B ^ \\frac { \\beta } { 2 } B ^ { 2 \\beta } B ^ \\frac { \\beta } { 2 } Q \\\\ & = Q ^ * B ^ { 3 \\beta } Q = B ^ { 3 \\beta } = ( C ^ * C ) ^ 3 . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{gather*} \\alpha : k ^ 6 \\to S _ 3 , ( y _ 0 , \\dots , y _ 5 ) \\mapsto \\begin{pmatrix} y _ 0 & y _ 5 & y _ 4 \\\\ y _ 5 & y _ 1 & y _ 3 \\\\ y _ 4 & y _ 3 & y _ 2 \\end{pmatrix} . \\end{gather*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sharp ( Z _ n \\cap J ) } { n } = \\frac { \\int _ J \\sqrt { w ( x ) / p ( x ) } d x } { \\int _ I \\sqrt { w ( x ) / p ( x ) } d x } . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\frac { z } { \\sqrt { n } } \\sum _ { k \\geq 1 } \\frac { \\lambda _ k } { \\lambda _ 1 + y - \\lambda _ k } = 1 , \\frac { z } { \\sqrt { n } } \\sum _ { k > r } \\frac { \\lambda _ k } { \\lambda _ 1 - \\lambda _ k } < 1 / 2 . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} g _ N ( y _ 1 , y ' ) = \\frac { a _ 0 } { 2 } + \\sum _ { k = 1 } ^ \\infty a _ k ( y ' ) \\cos ( k y _ 1 ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} & \\ \\tau \\phi \\big ( ( K ^ * A ^ { r s } K ) ^ { \\frac { 1 } { s } } \\big ) + ( 1 - \\tau ) \\phi \\big ( ( K ^ * B ^ { r s } K ) ^ { \\frac { 1 } { s } } \\big ) \\\\ = & \\ \\tau \\phi \\big ( | G _ A ( s ) | ^ \\frac { 2 } { s } \\big ) + ( 1 - \\tau ) \\phi \\big ( | G _ B ( s ) | ^ \\frac { 2 } { s } \\big ) \\\\ \\leq & \\ \\phi \\big ( ( M ^ * M ) ^ \\frac { 1 } { s } \\big ) \\\\ = & \\ \\phi \\big ( ( K ^ * C ^ { r s } K ) ^ { \\frac { 1 } { s } } \\big ) . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} f ( u , v ) = g ( u , v ) = 0 . \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} x _ 0 = ( - 0 . 1 0 5 0 0 6 6 2 8 3 3 5 0 8 , \\ - 0 . 3 8 0 9 4 3 6 3 0 9 4 0 6 1 ) \\ , . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} Z ^ { h _ 0 } : = \\langle \\zeta , r ^ { h _ 0 } - \\phi ( t _ 0 + \\bullet ) \\rangle , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\mathcal { L } f = \\Delta f - \\langle X , \\nabla f \\rangle , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\| D ^ 2 _ { x } \\left ( \\psi ( x ) - f ( y ) - G ( y ) , x - y \\rangle \\right ) \\| = \\| D ^ { 2 } \\psi ( x ) \\| \\leq \\sup _ { z \\in B ( 0 , R ) } \\| D ^ 2 \\psi ( z ) \\| . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\bar { V } ^ { I } & = \\sum _ { i = 1 } ^ { n ^ a } \\bar { U } ^ { I } _ i , & & \\bar { U } ^ { I } = X ^ { ( 2 ) } _ { n : n } \\\\ \\bar { V } ^ { I I } & = \\sum _ { i = 1 } ^ { n ^ { b - a } } \\bar { U } ^ { I I } _ i , & & \\bar { U } ^ { I I } = ( X ^ { ( 1 ) } _ { 1 : n ^ { 2 a } } ) _ { n ^ { 1 \\ ! - \\ ! a } : n ^ { 1 \\ ! - \\ ! a } } \\\\ \\bar { V } ^ { I I I ' } & = \\sum _ { i = 1 } ^ { n ^ { b - a } } \\bar { U } ^ { I I I ' } _ i , & & \\bar { U } ^ { I I I ' } = X ^ { ( 2 ) } _ { n : n } \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} A _ { j } \\mathbf { \\perp \\ ! \\ ! \\ ! \\perp } _ { \\mathcal { G } } \\left [ \\overline { \\mathbf { G } } _ { j } \\mathbf { \\backslash } \\overline { \\mathbf { B } } _ { j } \\right ] \\mathbf { | } \\overline { \\mathbf { B } } _ { j } , \\overline { \\mathbf { A } } _ { j - 1 } j = 0 , \\dots , p \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} u ( t + \\lambda ) ~ = ~ J _ \\lambda u ( t ) \\ , . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\gamma \\cdot \\delta : = \\gamma \\ast ( \\delta ( c _ { \\gamma } ) ) ^ \\circ \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\mathcal { K } _ { x x } = ( \\rho Q ( \\theta ) ) _ t + ( \\rho u Q ( \\theta ) ) _ x + \\delta \\theta p _ \\theta ( \\rho ) u _ x - \\varepsilon u _ x ^ 2 - \\mu | w _ x | ^ 2 - \\nu | \\mathbf { h } _ x | ^ 2 . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\Theta ^ { n , * } _ { \\mu } ( z ) : = \\limsup _ { l ( Q ) \\to 0 } \\Theta ^ n _ \\mu ( Q ) , \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} 1 + z _ 1 + z _ 2 = \\frac { q [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { \\xi ' ( q ) ( 1 - q ) } . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\varphi ^ * ( e _ i ) = f _ i , \\varphi ^ * ( f _ i ) = e _ i \\mbox { a n d } \\varphi ^ * ( q ^ h ) = q ^ { - h } . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\Lambda _ { k } \\left ( P \\right ) \\equiv \\left \\{ q _ { k } \\left ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } \\right ) : E _ { P } \\left [ q _ { k } \\left ( \\overline { \\mathbf { A } } _ { k } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } \\right ) | \\overline { \\mathbf { A } } _ { k - 1 } , \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } \\right ] = 0 \\right \\} . \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} d X ^ R ( t ) = \\left [ L X ^ R ( t ) + F \\left ( X ^ R ( t ) \\right ) \\theta _ R \\left ( x ^ R ( t ) \\right ) \\right ] d t + B \\ , d W ( t ) , X ^ R ( 0 ) = X _ 0 . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} v _ { i , j } v _ { k , l } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) & = v _ { i , j } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) E _ { l , k } \\\\ & = ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) E _ { j , i } E _ { l , k } \\\\ & = \\delta _ { i , l } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) E _ { j , k } \\\\ & = \\delta _ { i , l } v _ { k , j } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\beta ( \\alpha ) = \\limsup _ { k \\to \\infty } - \\frac { \\ln | | k \\alpha | | _ { \\R / \\Z } } { | k | } \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} r _ \\alpha ^ \\lambda ( t ) = h _ \\lambda ( t ) ^ { - 1 } e ^ { t f ( h _ \\alpha ) } r _ \\alpha ^ \\lambda , \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\abs { \\mathbb V ( x ) } = O ( \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } ) s ^ { - 2 } \\leq C ( \\log \\rho _ k ) ^ 3 \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } \\leq C ( \\log \\abs { x } ) ^ 3 \\abs { x } ^ { - \\epsilon } , \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} & ( \\cosh \\tfrac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu ) ( \\cosh \\tfrac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ | { \\mu ' } ) \\\\ & = ( 1 + \\cosh \\tfrac { \\ell _ \\nu } { 2 } ) ( 2 \\cosh \\tfrac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu + \\cosh \\ell _ { \\mu ' } ) . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} ( u _ 0 ) _ x = \\int _ { 0 } ^ { 1 } \\frac { j } { m ( \\cdot , y ) } d y - P . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} \\widehat { \\Gamma } = & \\bigcup \\ : \\{ \\Gamma ( c , d ) : ( c , d ) \\in v \\mathcal { C } \\times v \\mathcal { D } \\} \\\\ \\widehat { \\Delta } = & \\bigcup \\ : \\{ \\Delta ( c , d ) : ( c , d ) \\in v \\mathcal { C } \\times v \\mathcal { D } \\} \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} T _ { 1 , k } & = \\Delta _ { \\tilde { n } } \\left ( \\frac { x e ^ x } { e ^ x - 1 } \\cdot \\left ( \\frac { e ^ x - 1 } { e ^ { \\frac { x } { M } } - 1 } \\right ) ^ { \\tilde { n } - k } \\cdot \\left ( \\frac { e ^ x - 1 } { e ^ { \\frac { x } { N } } - 1 } + \\frac { e ^ x - 1 } { e ^ { \\frac { x } { M - N } } - 1 } \\right ) ^ k \\right ) , \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\Delta ( \\lambda ) : = \\det \\left ( \\begin{array} { c c } \\int _ { 0 } ^ { 1 } y _ { 1 } ( x , \\lambda ) \\mathrm { d } \\nu _ { 0 } ( x ) - 1 & \\int _ { 0 } ^ { 1 } y _ { 2 } ( x , \\lambda ) \\mathrm { d } \\nu _ { 0 } ( x ) \\\\ \\int _ { 0 } ^ { 1 } y _ { 1 } ( x , \\lambda ) \\mathrm { d } \\nu _ { 1 } ( x ) - y _ { 1 } ( 1 , \\lambda ) & \\int _ { 0 } ^ { 1 } y _ { 2 } ( x , \\lambda ) \\mathrm { d } \\nu _ { 1 } ( x ) - y _ { 2 } ( 1 , \\lambda ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\frac { p _ k ( z ) } { z - 1 } = \\frac { d } { d z } \\frac { z ^ { k + 1 } - 1 } { z - 1 } \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} | | \\beta ( F ^ * _ i & ( \\alpha _ i ) ) - F ^ * _ { i + 1 } ( \\alpha _ { i + 1 } ) | | = | | ( F ^ 0 _ { i + 1 } ) ^ * ( \\beta ( \\alpha _ i ) ) - F ^ * _ { i + 1 } ( \\alpha _ { i + 1 } ) | | \\\\ & | | ( F ^ 0 _ { i + 1 } ) ^ * ( \\beta ( \\alpha _ i ) ) - F _ { i + 1 } ^ * ( \\beta ( \\alpha _ i ) ) + | | F _ { i + 1 } ^ * ( \\beta ( \\alpha _ i ) ) - F ^ * _ { i + 1 } ( \\alpha _ { i + 1 } ) | | \\\\ & \\leq \\delta _ i | | \\beta ( \\alpha _ i ) | | + | | \\beta ( \\alpha _ i ) - \\alpha _ { i + 1 } | | \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} v _ t + v _ x u = v u _ x . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\left | \\left ( \\frac { \\partial } { \\partial t } \\right ) ^ k \\left ( \\frac { \\partial ^ { | \\beta | } } { \\partial x ^ \\beta } \\right ) ( u - 1 ) \\right | \\leq \\frac { C } { t ^ 3 } \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} ( T _ { \\rm o p } ) _ { \\rm s i n g } = P _ r T _ { \\rm o p } = P _ r ( I - Q _ m ) T = P _ r T . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} \\phi _ i \\star _ \\tau \\phi _ j = \\sum _ { k = 0 } ^ N 2 \\Gamma _ { i j } ^ k \\phi _ k \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - v ^ { \\prime \\prime } ( x ) + q v ( x ) = - \\frac { \\lambda } { b _ { 0 } } v ( x ) , x \\in ( 0 , 1 ) , \\\\ v ( 0 ) = A v ( \\frac { 1 } { 2 } ) = A ^ { 2 } v ( 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} \\sum _ { v \\in A _ 0 } t _ { \\ell _ 2 } ( v ) = \\sum _ { u _ i \\in T _ { \\ell _ 2 } } d _ { H _ 1 [ A _ i ] } ( u _ i ) \\geq \\left | T _ { \\ell _ 2 } \\right | \\frac { | C _ 1 | | U _ 1 | ^ 2 } { 2 4 n } . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} C _ 0 ( L _ { \\rho } ) & \\cong \\{ g \\in C _ 0 ( L ) : g ( y ) = 0 y \\in L \\setminus L _ { \\rho } \\} \\\\ C _ 0 ( L \\setminus L _ { \\rho } ) & \\cong \\{ g \\in C _ 0 ( L ) : g ( y ) = 0 y \\in L _ { \\rho } \\} \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} u _ x ( 0 , t ) = u _ x ( 1 , t ) = 0 , t > 0 , \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} \\eta _ k & = \\frac { \\partial L } { \\partial \\dot z ^ k } ( t , z _ * , \\dot z _ * ) + \\int _ t ^ { \\omega } \\frac { \\partial L } { \\partial z ^ k } ( s , z _ * , \\dot z _ * ) d s \\\\ & - \\dot \\gamma ^ j ( t ) \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z _ * ) - \\int _ t ^ \\omega \\dot \\gamma ^ j ( s ) \\frac { d } { d s } \\Big ( \\frac { \\partial f _ j } { \\partial z ^ k } ( s , z _ * ) \\Big ) d s - \\lambda _ k \\in Y . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\P \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ^ 2 - \\frac { 1 } { 2 r } \\geq t \\Big ) \\leq \\exp \\Big ( - \\frac { n t ^ 2 } { 1 + \\frac { 2 r t } { 3 } } \\Big ) . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\Lambda _ G ^ { ( k , K ) } = \\mathop { \\sup } \\limits _ { \\theta , \\{ { \\mu _ j } \\} } { f _ 1 } \\left ( { \\theta , \\{ { \\mu _ j } \\} } \\right ) - \\mathop { \\sup } \\limits _ \\theta { f _ 0 } \\left ( { \\theta } \\right ) \\ge \\mathop { \\sup } \\limits _ { \\theta } { f _ 1 } \\left ( { \\theta , \\{ 0 \\} } \\right ) - \\mathop { \\sup } \\limits _ \\theta { f _ 0 } \\left ( { \\theta } \\right ) = 0 \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} & \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\leq \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - a _ n - S _ n ( \\kappa _ n ( x ) ) } { f _ 2 ( n ) } \\geq x ( 1 - \\delta ) \\right ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\leq - \\inf _ { [ x ( 1 - \\delta ) , + \\infty ) } H ( \\ell _ 2 y ) = - H ( \\ell _ 2 x ( 1 - \\delta ) ) \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ z \\in \\mathbb { C } : 0 \\leq \\mathrm { R e } ( z ) \\leq 1 \\} . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} & \\sigma ^ { \\ast } \\mathcal { L } \\cong \\sigma ^ { \\ast } ( \\mathcal { O } _ { E } ( 3 o _ { E } ) ) \\cong \\mathcal { O } _ { E } ( \\widetilde { \\sigma } ( 3 o _ { E } ) ) = \\mathcal { O } _ { E } ( 3 q ) , \\\\ & ( \\sigma ^ { 2 } ) ^ { \\ast } \\mathcal { L } \\cong ( \\sigma ^ { 2 } ) ^ { \\ast } ( \\mathcal { O } _ { E } ( 3 o _ { E } ) ) \\cong \\mathcal { O } _ { E } ( \\widetilde { \\sigma ^ { 2 } } ( 3 o _ { E } ) ) = \\mathcal { O } _ { E } ( 3 r ) , \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} ( \\mu ^ 0 \\hat \\oplus \\mu ^ 1 ) ( \\mathfrak c ) = \\mu ^ 0 ( \\mathfrak c ) \\hat \\oplus \\mu ^ 1 ( \\mathfrak c ) . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x } & & q ^ { - } x x ^ { - } p , \\\\ & & & A x \\leq x . \\end{aligned} \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} f ( t , x ) = x ^ n + t \\cdot g ( x ) \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} p ( t , x , y ) = \\mathbb { E } \\left [ \\delta _ { y } \\left ( \\Phi _ { 1 } ( x ; \\varepsilon B ) \\right ) \\right ] . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\max _ { 1 \\leq j \\leq n } f ( I _ j ^ n ) = \\frac { 1 } { \\int _ I s ( x ) d x } . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\mbox { r a n k } ( B , A B , \\dots , A ^ { n - 1 } B ) = n . \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } x _ i = l , \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\tau m '' ( t ) + m ' ( t ) = 0 , m ( 0 ) = \\int _ 0 ^ 1 u _ 0 ( x ) \\ , d x , m ' ( 0 ) = \\int _ 0 ^ 1 u _ 1 ( x ) \\ , d x , \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} X _ { x } ^ { \\varepsilon } = \\mathcal { M } \\left ( S ( \\cdot ) x + \\sqrt { { \\varepsilon } } \\int _ { 0 } ^ { \\cdot } S ( \\cdot - s ) G ( s , X _ { x } ^ { \\varepsilon } ( s ) ) d w ( s ) \\right ) = \\mathcal { M } \\left ( S ( \\cdot ) x + \\sqrt { { \\varepsilon } } Z ( X _ { x } ^ { \\varepsilon } ) \\right ) , \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} d ( T x , T y ) & = \\frac { | ( x + 2 0 ) - ( y + 2 0 ) | } { 1 0 } \\\\ & = \\frac { | x - y | } { 1 0 } \\\\ \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 + } \\left ( \\lim _ { \\varepsilon \\to 0 } u ^ \\varepsilon ( t , \\cdot ) \\right ) ~ \\not = ~ \\bar u \\ , . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} W ( h x ) & = W ( h a ) + h W ' ( h a ) ( x - a ) + \\frac 1 2 h ^ 2 W '' ( h a ) ( x - a ) ^ 2 \\\\ & + \\frac 1 6 h ^ 3 W ''' ( h a ) ( x - a ) ^ 3 + \\frac 1 { 2 4 } h ^ 4 ( x - a ) ^ 4 \\nu _ { h , a } ^ 0 ( x ) \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} \\mbox { v a r } ( Y _ i ) = & \\sum _ { i = 1 } ^ { \\infty } \\mbox { v a r } ( E _ i ) + 2 \\sum _ { i < j } ^ { \\infty } \\mbox { c o v } ( E _ i , E _ j ) \\\\ = & \\frac { m p } { 1 - p ^ 2 } + 2 \\sum _ { i < j } \\mbox { c o v } ( E _ i , E _ j ) \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\dot { p } _ { 1 } = \\frac { c _ { 3 } + c _ { 4 } p _ { 1 } ^ { 2 } } { c _ { 1 } + c _ { 2 } p _ { 1 } ^ { 2 } } . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } = [ n + 1 ] { 2 n + 1 \\brack n } \\sum _ { k = 1 } ^ { n } \\frac { q ^ { - { n - 2 k + 1 \\choose 2 } } } { [ 2 k ] ^ 2 { n \\brack k } _ { q ^ 2 } ^ 2 } . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} u ( 0 , x ) = \\begin{cases} 1 , & x < 0 , \\\\ 0 , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} f _ T ( z ) \\ = \\ f _ { T _ \\mu } ( z ) \\ , ( 1 + z ) \\ , . \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} c _ 0 = \\sum d _ i \\partial e _ i \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} d _ 1 & : = s _ 1 + j - i + 1 \\\\ d _ 2 & : = s _ 2 + j - i + 1 . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} 1 ^ { T } s _ { 1 } & = 1 , & 1 ^ { T } s _ { 2 } & = \\mu / 4 \\lambda , & 1 ^ { T } s _ { 3 } & = 1 , \\\\ s _ { 1 } ^ { - } 1 & = \\mu / 3 \\lambda , & s _ { 2 } ^ { - } 1 & = 1 , & s _ { 3 } ^ { - } 1 & = \\mu / 3 \\lambda , \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} K = \\cup _ { \\lambda \\in \\Lambda } \\varphi _ { \\lambda } ( K ) \\ : . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} K _ 1 ( c _ 0 ( \\Z ^ N ) ) / R ^ { \\times } = K _ T ( c _ 0 ( \\Z ^ N ) ) \\ ; . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} X _ { i } = F _ { 0 } ( f ) ^ { e _ { 0 } } \\cdots F _ { i - 1 } ( f ) ^ { e _ { i - 1 } } F _ { i + 1 } ( f ) ^ { e _ { i + 1 } } \\cdots F _ { r } ( f ) ^ { e _ { r } } ( X ) \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } S = & \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } + S ( 1 - \\frac { 3 } { 2 } S ) + 2 H ^ { 2 } S - \\frac { 1 } { 2 } H ^ { 4 } - \\sum _ { j , k , p , q } H ^ { p ^ { \\ast } } h _ { j k } ^ { p ^ { \\ast } } H ^ { q ^ { \\ast } } h _ { j k } ^ { q ^ { \\ast } } . \\endaligned \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} f \\circ \\Phi _ g = f \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\nu ^ { 1 } _ { \\epsilon } \\star \\nu ^ { 0 } _ { \\iota } = \\nu ^ { 1 } _ { \\epsilon } \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\widehat { G } ( m , 2 , b , \\epsilon , s ) \\leq 2 \\sum _ { k = 1 } ^ { e ^ { ( \\log ( m ( 1 + \\varepsilon ) / 2 ) ^ { 1 / b } } } k ^ { - d s } e ^ { - d s ( \\log ( m - e ^ { ( \\log k ) ^ b } ) ) ^ { 1 / b } } \\cdot \\widehat { N } _ { m , b , \\epsilon } ( k ) , \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { \\partial u _ { J } } { \\partial t } = f ( u _ { J } , v _ { J } ) \\\\ \\\\ \\frac { \\partial v _ { J } } { \\partial t } = g ( u _ { J } , v _ { J } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\boldsymbol { \\varsigma } _ { k } \\ ! \\triangleq \\ ! \\textbf { I } _ S \\left ( \\hat { \\boldsymbol { \\psi } } _ { k - 1 } , \\textbf { W } _ k \\right ) ^ \\frac { \\partial \\ , p _ S \\left ( \\textbf { y } _ k | \\boldsymbol { \\psi } , \\textbf { W } _ k \\right ) } { \\partial \\boldsymbol { \\psi } } \\Bigg | _ { { \\boldsymbol { \\psi } } \\ ! = \\ ! \\hat { \\boldsymbol { \\psi } } _ { k \\ ! - \\ ! 1 } } . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} I _ \\epsilon [ u _ \\epsilon ] = \\inf _ { \\overset { u \\in { C } ^ 1 ( \\mathbb { T } ^ d ) } { \\int _ { \\mathbb { T } ^ d } u ( x ) d x = 0 } } I _ \\epsilon [ u ] , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} \\{ f , g \\} = \\{ f , ( I - Q ) g + Q g \\} , \\{ f , ( I - Q ) g \\} \\in ( I - Q ) T , \\{ f , Q g \\} \\in Q T ; \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} ( ( a , u ) ( b , v ) ) ( c , w ) & = ( a b , f _ 3 ( a , b ) u + f _ 4 ( a , b ) v ) ( c , w ) \\\\ & = ( a b c , f _ 3 ( a b , c ) ( f _ 3 ( a , b ) u + f _ 4 ( a , b ) v ) + f _ 4 ( a b , c ) w ) , \\\\ ( a , u ) ( ( b , v ) ( c , w ) ) & = ( a , u ) ( b c , f _ 3 ( b , c ) v + f _ 4 ( b , c ) w ) \\\\ & = ( a b c , f _ 3 ( a , b c ) u + f _ 4 ( a , b c ) ( f _ 3 ( b , c ) v + f _ 4 ( b , c ) w ) ) . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} E = E ^ - \\oplus E ^ + \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar H ^ 6 - 3 \\bar H ^ 4 S + 3 \\bar H ^ 2 S ^ 2 - 3 S ^ 2 + 2 S \\\\ & \\geq \\dfrac { ( \\bar H ^ 2 - S ) ^ 2 } 4 \\bigl [ ( 5 - \\dfrac { 7 } { 8 0 } ) S - ( 4 - \\dfrac 3 { 8 } ) \\bar H ^ 2 + \\dfrac 5 { 8 } \\bigl ] . \\end{aligned} \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\textbf { G } _ \\tau ( S ^ { K \\textnormal { t h } } ( \\phi _ i ) , \\phi _ j ) = S _ { i j } & & \\textbf { g } ( \\phi _ i , T ^ { K \\textnormal { t h } } ( \\phi _ j ) ) = S _ { i j } \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} a _ j = a _ j ( \\rho ) = \\dfrac { 1 } { 2 } ( \\rho ^ j + \\rho ^ { - j } ) , \\ j \\in \\mathbb { N } ( \\rho > 1 ) , \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } > \\varepsilon \\right ) & = P \\left ( \\bigcup _ { t = a _ n } ^ { \\lfloor n - \\varepsilon f _ 2 ( n ) \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} \\right ) . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\nu _ { x , t } = \\delta _ { ( \\rho ( x , t ) , \\rho ( x , t ) u ( x , t ) ) } ( x , t ) . \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} H _ { n _ i } ( \\nu ) = \\frac { 1 } { n _ i \\log 2 } \\log \\# F _ 2 ( n _ i ) . \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} n \\log ( n ) - n \\le \\log ( n ! ) = \\log ( \\Gamma ( 1 + n ) ) \\le n \\log ( n ) , \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\leq \\sum _ { j = - \\infty } ^ { m _ 0 - ( k - i - 1 ) p / Q } 2 ^ { j s } \\Big [ g _ j ( x _ { k - i } ) + g _ j ( x _ { k - ( i + 1 ) } ) \\Big ] , \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} T S P C _ n = L ( { \\cal C } _ n ) : = \\min _ { { \\cal C } } L ( { \\cal C } ) , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} x _ { n + 1 } \\geq x _ n ( 1 - h _ n x _ n ^ 2 ) \\geq x _ 1 \\prod _ { i = 1 } ^ { n } ( 1 - h _ i x _ i ^ 2 ) . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\{ f , f ' \\} = \\{ f , f _ { 1 } ' + f _ { 2 } ' \\} , \\mbox { w h e r e } \\{ f , f _ { 1 } ' \\} \\in T _ { 1 } , \\ , \\ , \\{ f , f _ { 2 } ' \\} \\in T _ { 2 } , \\ , \\ , f _ { 1 } ' \\perp f _ { 2 } ' . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\| g \\| _ { L ^ { q } ( \\mathbb { R } ^ n ) } = | B ( y _ { 2 } , r ) | ^ { 1 / q } \\approx r ^ { n / q } , \\\\ \\| h _ { 1 } \\| _ { L ^ { r _ { 1 } } ( \\mathbb { R } ^ n ) } = | B ( y _ { 1 } , r ) | ^ { 1 / r _ { 1 } } \\approx r ^ { n / r _ { 1 } } , \\\\ \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} e ( v , \\theta ) = P _ e ( v ) + Q ( \\theta ) , \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} ( J u , u ) & = \\sum _ { n = 1 } ^ \\infty \\left ( \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} v _ { n - 1 } + \\begin{pmatrix} s & 0 \\\\ 0 & t \\end{pmatrix} v _ n + \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} v _ { n + 1 } , v _ n \\right ) _ { \\mathcal { K } } \\\\ & + \\sum _ { n = 1 } ^ \\infty \\left ( O ( n ^ { \\alpha - 1 } ) u _ { n - 1 } + O ( n ^ { \\alpha - 1 } ) u _ { n + 1 } , u _ n \\right ) _ { \\mathcal { K } } \\ , . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & ( \\bar h ^ { 1 ^ * } _ { 1 1 } - 3 \\bar h ^ { 1 ^ * } _ { 2 2 } ) \\bar h ^ { 1 ^ * } _ { 1 1 k } + 4 \\bar h ^ { 2 ^ { \\ast } } _ { 1 1 } \\bar h ^ { 2 ^ * } _ { 1 1 k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} \\frac { S _ 1 } { S _ 2 } & \\geq \\sum _ { k = 1 } ^ \\infty a _ k q _ k \\\\ & = \\prod _ { p \\leq X } \\left ( 1 - q _ p p ^ { - 1 } \\right ) ^ { - 1 } \\\\ & = \\left ( \\prod _ { p \\leq X } ( 1 - p ^ { - 1 } ) ^ { - 1 } \\right ) \\left ( \\prod _ { p \\leq X } \\frac { p - 1 } { p - q _ p } \\right ) . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m a x i m i z e } & \\sum _ { i = 1 } ^ n H _ i ( \\hat { x } _ i ) \\\\ \\mbox { s u b j e c t t o } & x _ t \\in F _ t \\subseteq \\mathbb { R } _ + ^ n ~ \\forall t \\in [ m ] \\\\ & \\hat { c } _ i ^ T \\hat { x } _ i \\leq 1 ~ \\forall i \\in [ n ] \\end{array} \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} x _ { n + 1 } = g ( x _ n ) , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} d \\hat { \\Gamma } ^ \\lambda _ { t } = \\lambda \\sum _ { \\alpha = 1 } ^ { d } \\tilde { W } _ { \\alpha } ( \\Gamma ^ \\lambda _ { t } ) d B ^ { \\alpha } _ t . \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} u _ t + f ^ { \\delta _ n } ( x , u ) _ x = { \\varepsilon _ n } u _ { x x } . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} ( 1 + z ) L _ n ^ { ( \\delta ) } ( z q ; \\ , q ) = \\frac { q ^ { n + 1 } - 1 } { q ^ { n + \\delta + 1 } } L _ { n + 1 } ^ { ( \\delta ) } ( z ; \\ , q ) + \\frac { 1 } { q ^ { n + \\delta + 1 } } L _ { n } ^ { ( \\delta ) } ( z ; \\ , q ) \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} E \\{ y ^ 2 \\} & = \\mu _ y ^ 2 + \\sigma _ y ^ 2 E \\{ y ^ 3 \\} = \\mu _ y ^ 3 + 3 \\mu _ y \\sigma _ y ^ 2 E \\{ y ^ 4 \\} = \\mu _ y ^ 4 + 6 \\mu _ y ^ 2 \\sigma _ y ^ 2 + 3 \\sigma _ y ^ 4 . \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ j } H e s s _ { \\widetilde { g } } h ( X _ i , X _ k ) = 0 , \\ \\ \\ \\forall i , k = 1 , \\ldots n . \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} E [ X _ { k : n } ^ 2 ] = & \\frac { 1 } { \\lambda ^ 2 } \\left ( ( H _ n - H _ { n - k } ) ^ 2 + G _ { n } - G _ { n - k } \\right ) \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\partial _ r ( \\sigma ^ { a b } l _ { a b } ) = \\frac { 1 } { 2 } ( \\sigma ^ { a b } l _ { a b } ) ^ 2 + \\hat { l } _ a ^ b \\hat { l } _ b ^ a + R i c ( L , L ) \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\sum ^ \\infty 3 ^ { - j } \\ln h ^ { - 1 } _ { j } = \\infty , \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} W ^ { M _ S } = \\{ w \\in W ^ { \\mathrm { r e l } } : w ( M _ S \\cap B ) \\subset B \\} , \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\frac { f _ { 2 } ^ { \\prime \\prime } } { f _ { 2 } ^ { \\prime } } = \\frac { \\mu _ { 3 } \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } + \\mu _ { 4 } \\left ( \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } \\right ) ^ { 2 } } { \\mu _ { 1 } + \\mu _ { 2 } \\frac { f _ { 2 } ^ { \\prime } } { f _ { 2 } } } . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\begin{aligned} { \\textbf { I } _ S } \\left ( \\boldsymbol { \\psi } , { { \\textbf { W } } } \\right ) ^ { - 1 } = \\frac { { { \\sigma _ z ^ 2 } } } { { 2 { { \\lvert \\textbf { s } \\rvert } ^ 2 } } } \\left \\{ { { { \\bf { I } } _ { i { p _ 1 } } } + { { \\bf { I } } _ { i { p _ 2 } } } \\left ( \\beta \\right ) } \\right \\} , \\end{aligned} \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\varepsilon ^ { n / 2 } \\int _ { \\mathbb { R } ^ n } T ( P w _ { \\varepsilon \\alpha } ) \\overline { Q ( x ) } w _ { \\varepsilon \\beta } ( x ) d x = ( \\pi / \\beta ) ^ { n / 2 } \\int _ { \\mathbb { T } ^ n } e ^ { i 2 \\pi ( x , m ) - i 2 \\pi k x } a ( x , m ) d x . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} \\left [ ( \\delta _ \\lambda ) _ * \\tilde { W } \\right ] ( \\delta _ \\lambda u ) = \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } \\delta _ \\lambda ( u \\otimes \\exp ( \\varepsilon \\cdot \\tilde { W } ( \\mathbf { 1 } ) ) ) . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} S _ { I , \\varepsilon , \\ell } ( T ) = \\sum _ { \\delta \\in C \\cap \\mathbb N ^ N } \\L ^ { - \\omega ' ( \\delta ) } T ^ { \\omega ( \\delta ) } . \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} u ( x , t ) = v ( x , t + \\theta _ 0 ) . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} | X _ 1 | ^ 2 = | X _ 2 | ^ 2 = 2 \\quad \\langle X _ 1 , X _ 2 \\rangle = 0 , \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} S _ l ( h _ 0 ) = g _ 0 , S _ l ( \\alpha ) = a , \\quad \\ \\ \\xi _ 0 = d L _ { a ^ { - 1 } } ( \\xi ) . \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} w _ t \\leq \\sup _ { \\tau \\in \\mathbb { R } } \\left ( P [ \\psi _ \\tau ] ( \\phi ) + t \\tau \\right ) = \\sup _ { \\tau \\in \\mathbb { R } } \\left ( \\psi _ \\tau + t \\tau \\right ) . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\| \\tilde { V } \\| = \\| V \\| \\delta \\tilde { V } = \\delta V . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\tfrac { 1 } { L _ 1 } \\left ( E ( L _ 1 , \\ell _ \\nu , \\ell _ \\mu ) + E ( L _ 1 , \\ell _ \\nu , \\ell _ { \\mu ' } ) \\right ) & = \\tfrac { 1 } { L _ 1 } \\left ( \\hat { R } ( L _ 1 ^ A , L _ 1 ^ B , \\ell _ \\nu ) L _ 1 ^ A + \\hat { R } ( L _ 1 ^ B , L _ 1 ^ A , \\ell _ \\nu ) L _ 1 ^ B \\right ) \\\\ & < \\tfrac { 1 } { L _ 1 } ( 6 e ^ { - \\frac { 1 } { 2 } \\ell _ \\nu } L _ 1 ^ A + 6 e ^ { - \\frac { 1 } { 2 } \\ell _ \\nu } L _ 1 ^ B ) = 6 e ^ { - \\frac { 1 } { 2 } \\ell _ \\nu } . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\dim \\Xi _ { \\chi } = \\sharp X _ w ( 1 ) _ { P _ { 1 / 2 } } = ( q - 1 ) q ^ { n + d - 1 } . \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} B _ 2 ^ { ( n ) } : = \\bigcup _ { t = \\lceil K a _ c ^ { ( n ) } \\rceil } ^ { \\lfloor p _ n ^ { - 1 } \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} , \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\xi _ t ( \\widehat { \\rho } _ t s ( m ) ) = s ( \\rho _ t m ) \\ , . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\lambda \\in ( X _ j ^ s ) ^ * \\mapsto g _ \\lambda \\in Y _ j ^ s \\lambda ( f ) = \\tau _ s ( g _ \\lambda f ) , f \\in X _ j ^ s . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\det ( D ^ 2 v ( y ) ) = v ( y ) ^ { p - 1 } \\left \\| D v ( y ) + \\left ( \\langle D v ( y ) , y \\rangle - v ( y ) \\right ) \\cdot e \\right \\| _ Q ^ { n - q } \\ , g ( y ) \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\Delta v ^ { ( 0 ) } + q ( x ) v ^ { ( 0 ) } = 0 \\Omega \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} d i v ( \\gamma _ { \\epsilon } ^ { p - 2 } d \\varphi _ { \\epsilon } ) = 0 , \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} E [ e ^ { ( X _ 1 + 2 X _ 2 + \\cdots + k x _ k - k ) t } ] = \\sum _ { n = 0 } ^ \\infty E [ ( X _ 1 + 2 X _ 2 + \\cdots + k X _ k - k ) ^ n ] \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} C _ { I , [ R ] } = \\Bigl ( \\Gamma _ R \\otimes V [ 1 ] ^ { \\otimes R } \\Bigr ) \\otimes _ { \\C [ R ] } \\C [ I ] = \\frac { \\Gamma _ R \\otimes V [ 1 ] ^ { \\otimes R } \\otimes \\C [ I ] } { \\left \\langle - ( \\partial _ { z _ r } + T _ r ) + \\sum _ { i \\in I _ r } \\lambda _ { i } \\right \\rangle _ { r \\in R } } . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} [ T , \\textbf { b } ] _ { e _ j } ( f _ 1 , f _ 2 , \\dots , f _ m ) = b _ j T ( f _ 1 , \\dots , f _ j , \\dots f _ m ) - T ( f _ 1 , \\dots , b _ j f _ j , \\dots f _ m ) ; \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} E _ { P _ { \\alpha } } \\left [ \\frac { I _ { a } ( A ) } { \\pi _ { a } ^ { 2 } ( \\mathbf { O } _ { m i n } ; P _ { \\alpha } ) } v a r _ { P _ { \\alpha } } ( Y \\mid A = a , \\mathbf { O } ) \\right ] \\leq C E _ { P _ { \\alpha } } \\left [ \\frac { I _ { a } ( A ) } { \\pi _ { a } ^ { 2 } ( \\mathbf { O } _ { m i n } ; P _ { \\alpha } ) } \\right ] \\leq C ^ { 3 } . \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } & \\left ( { \\rho } _ 4 ^ 2 ( t ) \\| y _ t ( t ) \\| ^ 2 d x \\right ) + \\int _ I a ( 0 , t , x ) { \\rho } _ 4 ^ 2 | y _ { x t } | ^ 2 d x \\leq - \\int _ I D _ 2 a ( 0 , t , x ) \\rho _ 4 ^ 2 y _ x y _ { x t } d x + \\int _ I \\rho _ 4 \\rho _ { 4 , t } | y _ t | ^ 2 d x \\\\ & + C \\left ( \\int _ I { \\rho } _ 4 ^ 2 | G _ { t } | ^ 2 d x + \\int _ \\mathcal { O } { \\rho } _ 4 ^ 2 | f _ t | ^ 2 d x + \\int _ I { \\rho } _ 4 ^ 2 | y _ t | ^ 2 d x + \\sum _ { i = 1 } ^ 2 \\int _ I { \\rho } _ 4 ^ 2 | p ^ i _ t | ^ 2 d x \\right ) . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\frac { d \\mathcal { M } _ t } { d t } = \\mathcal { M } _ t \\theta _ t \\mathcal { F } \\theta _ t ^ { - 1 } , \\mathcal { M } _ 0 = \\mathbf { I d } . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ' ( t ) + A u ( t ) & = f ( t ) , \\ t \\ge 0 \\\\ u ( 0 ) & = u _ 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\mathcal { M } _ { m } ( s _ m ) = s _ m - \\mathcal { L } _ m ( s _ m ) = s _ { m + 1 } \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} & U _ + ( \\tau ) : = \\| P _ + \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\\\ & U _ 0 ( \\tau ) : = \\| P _ 0 \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\\\ & U _ - ( \\tau ) : = \\| P _ - \\hat { u } ( \\cdot , \\tau ) \\| _ { \\mathcal { H } } ^ 2 , \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\lambda \\norm { G _ \\lambda ^ { - 1 } ( y ) } _ H ^ 2 + \\norm { \\nabla G _ \\lambda ^ { - 1 } ( y ) } _ H ^ 2 \\leq C \\norm { y } _ { V ^ * } ^ 2 \\qquad \\forall \\ , y \\in H : \\ ; y _ \\Omega = 0 \\ , , \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\avg { ( x _ j + z _ j ) ( x _ l + z _ l ) } = \\avg { x _ j x _ l + z _ j z _ l } \\leq \\avg { x _ 0 ^ 2 } + \\avg { z _ 0 ^ 2 } = 1 + ( n + 1 ) \\avg { y _ 0 ^ 2 } . \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\iota _ \\partial \\omega = \\sum a _ i \\partial b _ i = \\sum _ { k = 0 } ^ j c _ k \\partial ^ k f , \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } F ( n , 0 ) - \\sum _ { n = 0 } ^ { N } F ( n , N ) = \\sum _ { k = 1 } ^ { N } G \\left ( N + 1 , k \\right ) . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\eta ( x , q , \\beta ) & = - \\sup _ { \\beta \\in \\mathbb { R } ^ n } \\bigl [ - 0 \\cdot \\beta - \\eta ( x , q , \\beta ) \\bigr ] \\\\ & = \\sup _ { p \\in \\mathbb { R } ^ d } \\bigl [ q \\cdot p - \\lambda ( x , p , 0 ) \\bigr ] \\\\ & = \\rho ( x , q ) . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\alpha _ t = \\sqrt { m _ t \\mu _ t } \\| \\hat Q _ r Q _ t \\| _ 2 \\quad \\forall t < r _ 0 , t \\neq r , \\quad \\alpha _ { r _ 0 } = \\sqrt { \\operatorname { t r } _ { \\geq r _ 0 } ( \\Sigma ) } \\| \\hat Q _ r Q _ { \\geq r _ 0 } \\| _ 2 , \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} J ^ k ( J ^ T ) ^ l - J ^ { n - l } ( J ^ T ) ^ { n - k } = \\sum _ { j = 1 } ^ { n - l } e _ { j + k } e _ { j + l } ^ T - \\sum _ { j = 1 } ^ k e _ { j + n - l } e _ { j + n - k } ^ T . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\mathbf { E } = \\mathbf { D } _ 1 \\cdot \\varepsilon _ S + \\frac { 1 } { 2 } = \\left [ \\begin{smallmatrix} \\frac { 1 } { 2 } - \\varepsilon _ S & 0 & 0 & 0 \\\\ 0 & \\frac { 1 } { 2 } - \\varepsilon _ S & 0 & 0 \\\\ 0 & 0 & \\frac { 1 } { 2 } - \\varepsilon _ S & 0 \\\\ 0 & 0 & 0 & \\frac { 1 } { 2 } - \\varepsilon _ S & \\end{smallmatrix} \\right ] \\prec 0 , \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } ( q , Q ) = \\sum _ { i = 0 } ^ N \\widetilde { J _ i } ( q , Q ) \\left ( 1 - P ^ { - 1 } \\right ) ^ i \\in K \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } ( q ) [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} & n + \\alpha ( h + 2 n - f ) = 0 , \\\\ & n - \\alpha ( h + f ) = 0 , \\\\ & n ( h + n - 1 ) + \\alpha ( h + f ) = 0 , \\\\ & \\alpha ( n - 1 ) ( h + n ) = 0 . \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} R _ L ^ 2 ( h ) ( \\xi , y ) = \\left \\{ \\begin{array} { r l } 0 & y \\notin \\{ y _ 0 , y _ 1 \\} , \\\\ R ^ 2 ( f ) ( \\xi ) & y = y _ 0 , \\\\ - R ^ 2 ( f ) ( \\xi ) & y = y _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} u ( x , t ) = \\cos ( t \\sqrt { 1 - \\Delta } ) f + \\frac { \\sin ( t \\sqrt { 1 - \\Delta } ) } { \\sqrt { 1 - \\Delta } } g . \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} t ^ { \\alpha - 1 } = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ L x ^ { - \\alpha } e ^ { - t x } d x + \\tau ( L ) , \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} [ O : \\phi ^ { - 1 } O ] = [ O : ( \\phi ^ { - 1 } O ) ( O \\cap H ) ] \\cdot [ ( \\phi ^ { - 1 } O ) ( O \\cap H ) : \\phi ^ { - 1 } O ] . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} g ( T _ i \\bullet _ \\tau T _ j , T _ k ) = g ( T _ i , T _ j \\bullet _ \\tau T _ k ) = \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } ( \\tau , Q ) \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} A ( f ) & = A ( f _ { 1 } ) \\cup \\cdots \\cup A ( f _ { r } ) \\\\ & = \\{ 1 \\} \\cup A _ { 1 } \\cup \\cdots \\cup A _ { r } . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } \\int _ { B _ { \\frac { r } { \\varepsilon _ { j , p } } } ( 0 ) } \\left ( 1 + \\frac { w _ { j , p } ( z ) } { p } \\right ) ^ { p } d z = \\int _ { \\R ^ 2 } e ^ { U } \\overset { \\eqref { L i o u v i l l e E q u a t i o n I N T R O } } { = } 8 \\pi , \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} \\phi _ { M \\times U ( 1 ) } ^ { \\ast } \\Theta & = \\phi _ { M \\times U ( 1 ) } ^ { \\ast } \\vartheta - 2 \\pi i \\phi _ { M } ^ { \\ast } \\lambda = ( \\vartheta + 2 \\pi i d \\alpha _ { \\phi } ) - 2 \\pi i \\phi _ { M } ^ { \\ast } \\lambda \\\\ & = ( \\vartheta + 2 \\pi i ( \\phi _ { M } ^ { \\ast } \\lambda - \\lambda ) ) - 2 \\pi i \\phi _ { M } ^ { \\ast } \\lambda = \\Theta . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| \\nabla u ( t _ k , \\cdot ) \\| _ { L ^ 2 } = + \\infty . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\norm { T _ t f } ^ p & \\ , = \\ , \\int _ 0 ^ \\infty | f ( x + t ) | ^ p v ( x ) \\ , d x \\\\ & \\ , = \\ , \\int _ 0 ^ \\infty | f ( x + t ) | ^ p v ( x + t ) \\frac { v ( x ) } { v ( x + t ) } \\ , d x \\\\ & \\ , \\leq \\ , B \\int _ 0 ^ \\infty | f ( x + t ) | ^ p v ( x + t ) \\ , d x \\\\ & \\ , = \\ , B \\int _ { t } ^ \\infty | f ( x ) | ^ p v ( x ) \\ , d x \\\\ & \\ , \\leq \\ , B \\int _ { 0 } ^ \\infty | f ( x ) | ^ p v ( x ) \\ , d x \\\\ & \\ , = \\ , B \\norm { f } ^ p . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} x \\leq y \\Longleftrightarrow & x = y ( x = a _ { 1 } y \\in \\{ b _ { 2 } , \\dots , b _ { n } \\} ) \\\\ & ( x = a _ { m } m > 1 y \\in \\{ b _ { 1 } , b _ { m } \\} ) . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} x _ { j , s } ( p ) = \\langle \\xi _ j ( p ) , e _ { j , s } \\rangle , s = 1 , \\ldots , m _ j , \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\gamma _ { i } ^ { k } = \\begin{cases} \\| f _ { i } ^ { k } \\| _ { L ^ { \\infty } } | B ( y _ { i } , 2 ^ { k } r ) | ^ { 1 / p } & k = 1 , \\cdots , J _ { 0 } \\\\ \\| f _ { i } ^ { k } \\| _ { L ^ { \\infty } } | B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } + 1 } r ) | ^ { 1 / p } & k = J _ { 0 } + 1 . \\end{cases} \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta ( \\ln k ) ^ \\mu \\asymp ( \\alpha t ) ^ \\beta ( \\ln ( \\alpha t ) ) ^ \\mu e ^ { \\alpha t } \\quad \\mbox { a s $ t \\to \\infty $ } , \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ { x x } ^ 2 L ( \\overline { x } , \\overline { \\lambda } ) \\xi = 0 , \\\\ g ' ( \\overline { x } ) \\xi - \\Pi _ { \\mathcal { K } } ' ( g ( \\overline { x } ) \\ ! + \\ ! \\overline { \\lambda } ; g ' ( \\overline { x } ) \\xi ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\Vert k _ { k , \\lambda + i \\gamma } \\Vert _ { \\ell ^ { p , \\infty } } : = \\sup _ { \\alpha > 0 } \\alpha \\mu \\{ s \\in \\mathbb { Z } : | k _ { k , \\lambda + i \\gamma } ( s ) | > \\alpha \\} ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\hat \\Phi ( x ) = q ^ n \\hat \\Phi _ 1 ( t x ) = \\begin{cases} q , & \\ | x | _ \\infty \\leq q ^ { - 2 } , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} G L R = \\mathbb { Q } \\delta \\oplus ( V \\cap G L R ) . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} u ( t , x ) = \\lim _ { r \\to 0 } I _ r [ u ] ( t , x ) = \\lim _ { r \\to 0 } \\frac { 1 } { \\beta ^ { ( k ) } _ 0 r } \\Omega _ r [ u ] ( t , x ) = \\frac { 1 } { ( n - 2 ) ! ! } \\lim _ { r \\to 0 } \\frac { \\widetilde { v } ( r , t ; x ) } { r } . \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { x \\in H ( u ( t ) ) } \\left | \\ , | x | - \\min \\Pi \\ , \\right | = 0 . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} T = \\{ U ^ { j } z \\ : : \\ : 0 \\le j < m \\} \\ : . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} g _ { D , \\mathrm { o d d } } ( y _ 1 , y ' ) = \\sum _ { k = 1 } ^ \\infty b _ k ( y ' ) \\sin ( k y _ 1 ) , \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} N _ { \\mathcal J } ( u , v , w ) = \\sum _ { ( u , v , w ) \\ ; \\mathrm { c y c l i c } } \\left ( \\eta ( u , v , \\mathcal J w ) + \\eta ( \\mathcal J u , v , w ) \\right ) . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\frac 1 s - \\frac 1 { p } = \\frac 1 { \\tau } - \\frac 1 { \\delta _ { m + 1 } } = \\frac 1 { s _ m } - \\frac 1 { p _ m } , \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\frac { 1 } { H - r z } = \\frac { - 1 } { r z } \\sum _ { m \\geq 0 } \\left ( \\frac { H } { r z } \\right ) ^ m = \\frac { - 1 } { r z } \\sum _ { m = 0 } ^ N \\left ( \\frac { H } { r z } \\right ) ^ m \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} \\boldsymbol { L } _ { N } \\varphi \\left ( x \\right ) = { \\textstyle \\int \\limits _ { \\mathcal { K } _ { N } } } \\left ( \\varphi \\left ( y \\right ) - \\varphi \\left ( x \\right ) \\right ) J _ { N } ( x , y ) d y , \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\exp _ p \\log _ p \\chi ^ { ( 1 ) k } _ n = \\chi ^ { ( 1 ) k } _ n \\ ; . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} { } \\boxtimes ^ k _ { i = 1 } [ \\mathcal { M } _ { G _ i , b _ i , \\mu _ i } ] = [ \\mathcal { M } _ { G , b , \\mu } ] . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} K _ 1 = \\left \\lfloor \\frac { \\delta + \\epsilon } { 2 } \\right \\rfloor & & K _ 2 = \\left \\lceil \\frac { \\delta + \\epsilon } { 2 } \\right \\rceil & & C = 2 ( \\delta + \\epsilon ) + 1 & & C ' = C + 1 . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} S _ { n , n - k } = P _ { n , k } / n ! , k = 0 , 1 , 2 , \\dots , n , \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} E ( y ) = 0 \\ \\mbox { w h e n e v e r } \\ y \\in { \\rm D o m } [ E ] \\cap \\partial { \\rm D o m } [ E ] . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} \\widetilde { \\boldsymbol { \\eta } } _ 2 : = \\begin{pmatrix} 1 - \\alpha \\frac { y } { L } & 0 \\\\ 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} \\beta : = \\nu ( x - a ) = \\nu _ { x - b } ( x - a ) = \\min \\{ \\nu ( x - b ) , \\nu ( b - a ) \\} . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} T _ { l } ^ { * } ( f _ { 1 } , \\cdots , f _ { m } ) ( x ) = \\int _ { \\mathbb { R } ^ { m n } } K ( y _ { l } , y _ { 1 } , \\cdots , y _ { l - 1 } , x , y _ { l + 1 } , \\cdots , y _ { m } ) \\Pi _ { j = 1 } ^ { m } f _ { j } ( y _ { j } ) d y _ { 1 } \\cdots d y _ { m } . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} & \\bigg ( 2 \\frac { \\partial E } { \\partial b } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\mp 2 \\frac { \\partial E } { \\partial y } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) - \\frac { \\mu } { 1 + b } E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\bigg ) _ { y = x \\pm ( t - b ) } \\\\ & = 2 ^ { \\sqrt { \\delta } - 1 } ( 1 + t ) ^ { - \\frac { \\mu } { 2 } - 1 } ( 1 + b ) ^ { \\frac { \\mu } { 2 } - 1 } \\Big [ 2 \\big ( \\tfrac { \\mu } { 2 } + \\tfrac { 1 - \\sqrt { \\delta } } { 2 } \\big ) ( 1 + t ) + 2 ^ { - 1 } ( \\sqrt { \\delta } - 1 ) ( 2 t + 2 ) - \\mu ( 1 + t ) \\Big ] = 0 . \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} d ( u , v ) = \\delta - d ( u ' , v ) u , v \\in \\Gamma . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{gather*} q _ { ( i - j ) m } = \\omega ( W _ { \\Theta _ { i } \\xi - \\Theta _ { j } \\xi } ) = \\omega ( W _ { ( ( i - j ) \\mathtt { N } \\xi _ 2 , 0 ) } ) \\\\ ( i - j ) \\phi _ { m , n } = h \\sigma ( \\Theta _ { j } \\xi , \\Theta _ { i } \\xi ) = ( i - j ) l h \\mathtt { N } \\xi _ 2 ^ 2 = ( i - j ) l \\frac { 2 \\pi } { d } + ( i - j ) l \\frac { \\epsilon } { 4 d ^ 2 } \\ , . \\end{gather*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\Xi _ { P } ^ { \\vec { p } } ( \\phi , \\gamma ) = \\xi _ { \\mathbf { A } _ { \\phi } } ^ { \\vec { p } } ( f _ { \\phi } ^ { \\gamma } ) . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\lambda v _ 2 - \\Delta v _ 2 + \\nabla _ H \\pi _ 2 = 0 \\Omega , \\Delta _ H \\ , \\pi _ 2 = \\mathrm { d i v } _ H \\ , \\bar f G , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} & n R _ { n } ^ { \\lambda } ( x ) = \\frac { 2 x } { \\sqrt { 1 + x ^ 2 } } ( n + \\lambda - 1 ) R _ { n - 1 } ^ { \\lambda } ( x ) - ( n + 2 \\lambda - 2 ) R _ { n - 2 } ^ { \\lambda } ( x ) , \\ ; \\ ; \\ ; n \\geq 2 ; \\\\ [ 2 p t ] & R _ { 0 } ^ { \\lambda } ( x ) = \\frac { 1 } { ( 1 + x ^ 2 ) ^ { \\frac { \\lambda + 1 } { 2 } } } , R _ { 1 } ^ { \\lambda } ( x ) = \\frac { 2 \\lambda x } { ( 1 + x ^ 2 ) ^ { 1 + \\frac { \\lambda } { 2 } } } . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\rho _ d n _ d = \\eta \\rho n , ~ \\rho _ p n _ p = ( 1 - \\eta ) \\rho n , ~ 0 < \\eta < 1 , \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} 3 | | \\vec a | | ^ 2 + b _ 1 ^ 2 + b _ 2 ^ 2 + \\cdots + b _ { 2 m } ^ 2 = 2 k - 2 , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\lambda ( \\lambda I _ 9 - \\tilde { \\mathcal { A } } _ { F S } ) ^ { - 1 } = \\begin{pmatrix} \\lambda ( \\lambda I _ 3 - A _ q ) ^ { - 1 } & - \\lambda ( \\lambda I _ 3 - A _ q ) ^ { - 1 } \\mathbb { P } S + \\mathbb { P } S \\\\ 0 & I \\end{pmatrix} . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\lim _ { | s | \\to \\infty } | \\theta ' ( s ) | = \\infty \\ , . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} P ( \\boldsymbol { Y } = \\boldsymbol { y } ) = \\frac { P ( \\vert \\boldsymbol { Y } \\vert = \\vert \\boldsymbol { y } \\vert ) } { \\binom { \\vert \\boldsymbol { y } \\vert + \\vert \\boldsymbol { \\alpha } \\vert - 1 } { \\vert \\boldsymbol { y } \\vert } } \\prod _ { j = 1 } ^ J \\binom { y _ j + \\alpha _ j - 1 } { y _ j } . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\nabla ^ { \\mathcal G , \\ , \\mathrm { s p i n } } _ { u } : = \\nabla ^ { 0 , S _ { \\mathcal G } } _ { X } - \\frac { 1 } { 3 } \\mathrm { a d } _ { r } \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} f _ { \\mu , \\lambda } ( x ) : = \\mu f ( \\lambda x ) , \\mu , \\lambda > 0 . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\frac { \\kappa } { 2 } u '' + u ' + \\lambda u = 0 , u ( 0 ) = u ( 1 ) = 0 \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} g _ { m , b } = \\ ( 1 + b r ^ 2 - \\frac { 2 m } { r } \\ ) ^ { - 1 } d r ^ 2 + r ^ 2 g _ * , \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} ( n , k , d ) = ( 1 4 , 6 , 7 ) , ( 1 5 , 7 , 7 ) , ( 1 7 , 6 , 9 ) , ( 1 7 , 7 , 8 ) , ( 1 9 , 7 , 9 ) , ( 2 0 , 7 , 1 0 ) . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} K _ \\Omega ( \\gamma _ n y _ n , y _ n ) = \\delta _ n . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} ( g t ) ^ { - 1 } \\mathcal { H } _ { r ( g t ) } ( g t ) & = t ^ { - 1 } g ^ { - 1 } \\mathcal { H } _ { r ( g ) } g t \\\\ & = t ^ { - 1 } \\mathcal { H } _ { d ( g ) } t \\\\ & = t ^ { - 1 } \\mathcal { H } _ { r ( t ) } t \\\\ & = \\mathcal { H } _ { d ( t ) } \\\\ & = \\mathcal { H } _ { d ( g t ) } . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} A ( g ) = \\sum _ h c _ { j , h } \\times ( A ( g h ) ) . \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\sum _ { k = \\nu } ^ { n } \\sqrt { k } = n A ( n ) - \\frac 2 3 \\sqrt { \\nu } \\left ( \\nu - \\frac 3 4 \\right ) - \\frac { \\delta _ { \\nu , n } } { 2 4 } , \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\mathbb { E } d ^ 2 ( Z _ k , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) = \\frac { 1 } { \\int _ { S _ l } f ( x ) d x } \\int _ { S _ l } \\mathbb { E } d ^ 2 ( x , \\{ Z _ j \\} _ { 1 \\leq j \\leq k - 1 } ) f ( x ) d x \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} & U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) \\\\ & = ( x q ) ^ { 2 a + 1 } ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\sum _ { h = 1 } ^ { k - a } [ ( x q ^ 2 ) ^ { 2 h } \\overline { Q } _ { k , k - h } ( x ^ 2 q ^ 4 ; q ) + ( x q ^ 2 ) ^ { 2 h - 2 } \\overline { Q } _ { k , k - h + 1 } ( x ^ 2 q ^ 4 ; q ) ] \\\\ & + ( x q ) ^ { 2 a - 1 } ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\sum _ { h = 1 } ^ { k - a + 1 } [ ( x q ^ 2 ) ^ { 2 h } \\overline { Q } _ { k , k - h } ( x ^ 2 q ^ 4 ; q ) + ( x q ^ 2 ) ^ { 2 h - 2 } \\overline { Q } _ { k , k - h + 1 } ( x ^ 2 q ^ 4 ; q ) ] . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} ( D ^ 2 u ( x ) ) y = x y , \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\Psi ( \\rho ) : = \\mathcal B ( \\rho ) \\mathcal V ( \\rho ) + \\mathcal B ( \\rho ) \\mathcal W ( \\rho ) \\qquad \\mbox { f o r } \\ ; \\ ; \\rho \\in \\mathcal O _ \\delta . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\boldsymbol { L } \\Psi _ { r n j } \\left ( x \\right ) = - \\gamma _ { I } \\Psi _ { r n j } \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\mathcal { P } ^ - ( X ) = \\lambda \\sum _ { \\lambda _ i > 0 } \\lambda _ i + \\Lambda \\sum _ { \\lambda _ i < 0 } \\lambda _ i , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} & & M _ { j , R } : = \\sup \\left \\{ \\textrm { L i p } \\left ( ( \\varphi _ { ( j , u , v ) } ) _ { | _ { B ( 0 , R ) } } \\right ) : ( u , v ) \\in E \\cap B ( 0 , R ) \\right \\} \\\\ & & = \\sup \\left \\{ 2 A _ { j , u , v } : ( u , v ) \\in E \\cap B ( 0 , R ) \\right \\} \\leq \\max \\left \\{ e ^ { R } , \\tfrac { 1 0 } { 4 } \\right \\} < \\infty . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\log P ( B _ 1 ^ { ( n ) } ) = - \\infty \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\sum _ { \\tau \\subset \\sigma } w ( \\sigma ) c _ { \\Sigma , \\tau } ( \\sigma , x ) = 0 \\mod L _ { \\Z } ( \\tau ) ~ x \\in \\Z ^ n . \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\imath _ { \\kappa } \\gamma ( x ) = a d _ v ( x ) \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} V _ C ^ { \\perp } = - \\cos ( \\theta _ 1 - \\theta _ 2 ) ( J X _ 1 + J X _ 2 ) \\quad V _ D ^ { \\perp } = \\sin ( \\theta _ 1 - \\theta _ 2 ) ( J X _ 1 + J X _ 2 ) , \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ \\int ( \\mu ( s , x ) - \\mu ( s , y ) ) ^ 2 d [ B ] s \\right ] = \\mathbb { E } \\left [ ( M ( x ) - M ( y ) ) ^ 2 \\right ] \\\\ & \\leq \\mathbb { E } [ ( M ( x ) - M ^ n ( x ) ) ^ 2 ] + \\mathbb { E } [ ( M ^ n ( x ) - M ^ n ( y ) ) ^ 2 ] \\\\ & + \\mathbb { E } [ ( M ^ n ( y ) - M ( y ) ) ^ 2 ] . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\langle \\kappa , t \\rangle = t r a c e ( a d _ t ) = \\sum _ { \\alpha \\in \\Delta _ { + } } \\langle \\alpha , t \\rangle = \\langle \\sum _ { \\alpha \\in \\Delta _ { + } } \\alpha , t \\rangle , \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} d ( T x , T y ) & = \\frac { | 1 0 - ( y + 2 0 ) | } { 1 0 } \\\\ & = \\frac { y + 1 0 } { 1 0 } \\\\ & \\leq 2 . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} I _ n ( a _ 1 , \\cdots , a _ n ) = f _ { a _ 1 } \\circ \\cdots \\circ f _ { a _ n } ( [ 0 , 1 ] ) \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} ( T _ w + q _ { s _ i } T _ { w s _ i } ) T _ i = T _ { w s _ i } ( T _ i + q _ { s _ i } ) T _ i = p _ { s _ i } T _ { w s _ i } ( T _ i + q _ { s _ i } ) = p _ { s _ i } ( T _ w + q _ { s _ i } T _ { w s _ i } ) , \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} c _ s : = \\sqrt { \\frac { \\sin \\pi s } { \\pi } } . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\rho ( e , u , f ) = \\theta \\sigma \\iota = \\rho ( e , g , g ) \\rho ( g , v , h ) \\rho ( h , h , f ) . \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} P _ t f = \\frac { t } { 2 \\sqrt { \\pi } } \\int _ 0 ^ { \\infty } e ^ { - \\frac { t ^ 2 } { 4 s } } H _ s f \\frac { d s } { s ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} d \\Gamma ^ \\lambda _ { t } = \\sum _ { \\alpha = 1 } ^ { d } \\tilde { W } _ { \\alpha } ( \\Gamma ^ \\lambda _ { t } ) d ( B _ { t } ^ { \\alpha , \\lambda } ) = \\lambda \\sum _ { \\alpha = 1 } ^ { d } \\tilde { W } _ { \\alpha } ( \\Gamma ^ \\lambda _ { t } ) d \\left ( \\lambda ^ { - 1 } B ^ { \\alpha , \\lambda } _ { t } \\right ) _ t . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} a _ k \\leq C b ^ { - \\frac { s - s ' } { Q } } \\bigg ( \\int _ { 2 B _ 0 } \\Big ( \\sum _ { j \\leq l - 2 } 2 ^ { s ' j p } g _ j ^ p \\Big ) \\ , d \\mu \\bigg ) ^ { \\frac { s - s ' } { Q } } \\sum _ { n = k _ 0 } ^ { k - 1 } 2 ^ { n \\left ( 1 - ( s - s ' ) \\frac { p } { Q } \\right ) } + C ' r _ 0 ^ { s - s ' } 2 ^ { k _ 0 } . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} f ( t , x _ t ) = f ( 0 , x _ 0 ) + \\int _ 0 ^ t \\left [ \\frac { \\partial f } { \\partial s } ( s , x _ s ) + \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\vartheta _ s + \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\varphi _ s D ^ \\phi _ s x _ s \\right ] \\ , \\mathrm { d } { s } + \\int _ 0 ^ t \\frac { \\partial f } { \\partial x } ( s , x _ s ) \\varphi _ s \\delta B _ s ^ { H } . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta _ { \\hat { g } } \\hat { \\varphi } + R _ { \\hat { g } } \\hat { \\varphi } + \\frac { n - 1 } { n } \\hat { \\tau } ^ { 2 } \\hat { \\varphi } ^ { N - 1 } = \\frac { \\hat { w } ^ { 2 } } { \\hat { \\varphi } ^ { N + 1 } } . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} T _ { 2 , k } & = \\frac { B _ { \\tilde { n } } ( 0 ) } { M ^ { k } ( M - N ) ^ { \\tilde { n } - k } } + \\sum _ { l = 1 } ^ { k - 1 } \\binom { k } { l } \\frac { B _ { \\tilde { n } } ( 0 ) } { M ^ { k } N ^ { \\tilde { n } - l } ( M - N ) ^ { \\tilde { n } - k + l } } \\\\ & + \\frac { B _ { \\tilde { n } } ( 0 ) } { M ^ k N ^ { \\tilde { n } - k } } \\\\ & = \\biggl ( \\frac { 1 } { M ^ { k } ( M - N ) ^ { \\tilde { n } - k } } + \\frac { M ^ k - N ^ k - ( M - N ) ^ k } { M ^ { k } ( N ( M - N ) ) ^ { \\tilde { n } } } + \\frac { 1 } { M ^ { k } N ^ { \\tilde { n } - k } } \\biggr ) B _ { \\tilde { n } } ( 0 ) \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\widehat { f } ( s ) = - i \\hbar \\nabla _ { X _ f } s + f s \\ , , \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} | n \\cdot x _ j - c t _ j | \\leq 2 C \\ \\ ( j \\in \\N ) \\ \\ { \\rm a n d } \\ \\ \\lim _ { j \\rightarrow \\infty } \\{ v _ { l _ 0 } ( x _ j , t _ j ) - u _ { l _ 0 } ( x _ j , t _ j + \\tau _ * ) \\} = 0 . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\begin{array} { r c l } f ( x ) & = & \\xi ( g ( x ) , h ( x ) ) \\ge \\lambda \\| g ( x ) \\| ^ 2 - \\| c \\| \\ , \\| h ( x ) \\| \\\\ & & \\\\ & = & \\| g ( x ) \\| \\left ( \\lambda \\| g ( x ) \\| - \\| c \\| \\ , \\dfrac { \\| h ( x ) \\| } { \\| g ( x ) \\| } \\right ) \\\\ & & \\\\ & \\ge & \\| g ( x ) \\| \\left ( \\lambda \\| g ( x ) \\| - R \\| c \\| \\right ) \\ , , \\end{array} \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } F ( n , k - 1 ) - \\sum _ { n = 0 } ^ { N } F ( n , k ) = G \\left ( N + 1 , k \\right ) . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} H _ N ^ + ( t , x ) & = C \\sum _ { f ' ( N ) \\leq m \\leq f ' ( 2 N ) } ( m - x ) ^ { \\frac { 2 - \\alpha } { 2 \\alpha - 2 } } e \\left ( t x _ m ^ { \\alpha } + ( x - m ) x _ m \\right ) + O ( N ^ { 1 - \\frac \\alpha 2 } ) \\\\ & = C \\sum _ { f ' ( N ) \\leq m \\leq f ' ( 2 N ) } ( m - x ) ^ { \\frac { r } 2 - 1 } e \\left ( c _ { t , r } ( m - x ) ^ { r } \\right ) + O ( N ^ { \\frac 1 2 } ) , \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} v : = M ^ { \\varepsilon } . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} P _ { \\textrm { s o u t } } ^ { \\textrm { C S A N C } } = & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N H F = 0 , F D F = 0 , N D F = 1 , N D N = 1 } \\right \\rbrace + \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\tau ( g \\otimes s ) = \\tau ( g ) \\otimes s + g \\sum _ { i = 1 } ^ { \\dim ( Y ) } \\tau ( y _ i \\circ f ) \\otimes \\frac { \\partial s } { \\partial { y _ i } } . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} P _ n = 1 + n t - ( 2 n + 1 ) t ^ 2 + n t ^ 3 + t ^ 4 . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} [ M _ S , \\mu _ S ] = ( I ^ G _ { M _ S } \\otimes \\delta ^ { - \\frac { 1 } { 2 } } _ { P _ S } \\circ [ \\mu _ S ] \\circ ( \\delta ^ { \\frac { 1 } { 2 } } _ { P _ S } \\otimes J ^ G _ { P ^ { o p } _ S } ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ S - \\mu \\rangle } ] . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} - \\frac { \\partial w _ m } { \\partial y _ 1 ^ { p ^ s } } y _ 4 ^ { p ^ { r + s + 1 } - p ^ r } = w _ m U _ 6 ^ 4 ( y _ 1 , y _ 2 , \\cdots , y _ 8 ) \\neq 0 . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} e ^ { n ^ \\alpha } - 2 \\sum _ { j = 1 } ^ n \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } \\varepsilon _ j \\leq \\sum _ { j = 1 } ^ n \\varphi ( a _ j ( x ) ) \\leq e ^ { n ^ \\alpha } - 2 \\sum _ { j = 1 } ^ n \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } \\varepsilon _ j . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} Q = J _ { \\infty } Q J _ { \\infty } ^ \\top + V _ { \\infty } . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\mathbf { O } & = \\begin{pmatrix} 1 & \\xi \\end{pmatrix} \\begin{pmatrix} S _ { 0 0 } & S _ { 0 1 } & S _ { 0 2 } \\\\ S _ { 1 0 } & S _ { 1 1 } & S _ { 1 2 } \\end{pmatrix} \\begin{pmatrix} 1 \\\\ \\Xi \\\\ \\Xi ^ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\| \\nabla u ( t _ k , \\cdot ) \\| _ { L ^ 2 } = + \\infty , \\lim _ { k \\to + \\infty } \\max \\left \\{ \\int _ { M } e ^ { u ( t _ k , \\cdot ) } , \\int _ { M } e ^ { - u ( t _ k , \\cdot ) } \\right \\} = + \\infty . \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} I _ 1 \\geq \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { k _ 1 } l \\left ( v _ j , { \\cal C } _ { 1 } \\right ) = L ( { \\cal C } _ 1 ) \\geq L \\left ( { \\cal Q } _ 1 \\right ) , \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} L ^ { ' } = \\{ l ^ { ' } _ { i } = ( ( 2 i + 1 ) e _ { 2 } , \\ , e _ { 1 } + ( 2 i + 1 ) e _ { 2 } ) : \\forall i \\in \\mathbb { N } \\} , \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\begin{cases} U _ 1 '' + A U _ 1 + U _ 1 ' = A e ^ { t A } ( u _ 0 + u _ 1 ) , t > 0 , \\\\ ( U _ 1 , U _ 1 ' ) ( 0 ) = ( 0 , - u _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\big | ( \\Phi _ N + m ) \\triangle \\Phi _ N \\big | } { | \\Phi _ N | } = 0 \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} L ' ( t ) & = \\frac { c G _ { 3 } } { p } G _ { 1 } f ( x , y , v ) v - \\frac { a c G _ { 3 } } { p } \\varphi _ { 1 } ( y ) - c G _ { 3 } \\varphi _ { 1 } ( y ) \\varphi _ { 2 } ( z ) + c G _ { 3 } \\varphi _ { 1 } ( y ) \\varphi _ { 2 } ( z ) - b \\varphi _ { 2 } ( z ) \\\\ & \\leq \\frac { c G _ { 3 } G _ { 1 } } { p } M _ { 3 } - \\frac { a c G _ { 3 } } { p } k _ { 1 } y - b k _ { 2 } z \\\\ & \\leq \\frac { c G _ { 3 } G _ { 1 } } { p } M _ { 3 } - \\tilde { \\mu } \\left ( \\frac { c G _ { 3 } } { p } y + z \\right ) \\\\ & = \\frac { c G _ { 3 } G _ { 1 } } { p } M _ { 3 } - \\tilde { \\mu } L ( t ) , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal { L } _ { M } ^ { N , h } u _ M ^ { N , h } ( y , \\cdot ) & = f , x \\in D , \\\\ u _ M ^ { N , h } ( y , \\cdot ) & = 0 , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} d _ a \\mathcal { L } \\eta = \\mathcal { L } d _ a \\eta , \\eta \\in \\Gamma ( \\mathbb { M } , \\mathcal { E } ) , \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\begin{cases} \\kappa \\in C ^ 2 ( [ 0 , \\infty ) ) & \\\\ k _ 1 ( 1 + \\theta ^ q ) \\leq \\kappa ( \\theta ) \\leq k _ 2 ( 1 + \\theta ^ q ) , & \\theta \\geq 0 \\\\ \\kappa _ \\theta ( \\theta ) \\leq k _ 2 ( 1 + \\theta ^ { q ' } ) , & \\theta \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} & \\lim _ { h \\to 0 } \\mathbb { E } \\left [ \\left ( \\frac { M ( x + h ) - M ( x ) } { h } - \\frac { d } { d x } M ( x ) \\right ) ^ 2 \\right ] \\\\ & = \\lim _ { h \\to 0 } \\mathbb { E } \\left [ \\left ( \\frac { d } { d x } M ( \\alpha ( x , h ) ) - \\frac { d } { d x } M ( x ) \\right ) ^ 2 \\right ] \\\\ & \\leq \\lim _ { h \\to 0 } \\mathbb { E } [ K ^ 2 | h | ^ { 2 \\delta } ] = 0 . \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( A \\right ) = \\left \\{ u \\in D : \\ w \\in D , \\ ; \\lambda > 0 u = J _ { \\lambda } w \\right \\} . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\frac { p _ 1 ( V ) - ( 2 k + 1 ) c _ 1 ( \\xi ) ^ 2 } { 2 } = 0 . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F D N } } ^ { \\textrm { O F D M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { F } } \\right ) P _ \\textrm { N } { | h _ { \\textrm { B F } } | ^ 2 } } { { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 } \\left ( 1 - \\theta \\right ) } . \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\frac { \\Pr ( \\max ( Z _ 1 , Z _ 2 ) > x t , Z _ 2 > t ) } { \\Pr ( \\max ( Z _ 1 , Z _ 2 ) > t , Z _ 2 > t ) } & \\leq \\frac { \\Pr ( Z _ 1 + Z _ 2 > x t , Z _ 2 > t ) } { \\Pr ( Z _ 2 > t ) } \\\\ & = \\frac { \\Pr ( Z _ 1 + Z _ 2 > x t , Z _ 2 > t ) } { \\Pr ( Z _ 1 + Z _ 2 > t , Z _ 2 > t ) } . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } ( 1 - \\varepsilon _ j ) \\leq \\sum _ { j = 1 } ^ n \\varphi ( a _ j ( x ) ) \\leq \\sum _ { j = 1 } ^ n \\alpha j ^ { \\alpha - 1 } e ^ { j ^ \\alpha } ( 1 + \\varepsilon _ j ) , \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) & = P ( S _ n ( \\kappa _ n ( x ) ) \\leq ( ( 1 - r ^ { - 1 } ) ^ { - 1 } x + o ( 1 ) ) a _ c ^ { ( n ) } ) \\\\ & = P \\left ( \\frac { S _ n ( \\kappa _ n ( x ) ) } { h ( x ) a _ c ^ { ( n ) } } \\leq \\frac { ( ( 1 - r ^ { - 1 } ) ^ { - 1 } x + o ( 1 ) ) } { h ( x ) } \\right ) \\\\ & = P \\left ( \\frac { S _ n ( \\kappa _ n ( x ) ) } { ( 1 - r ^ { - 1 } ) ^ { - 1 } h ( x ) a _ c ^ { ( n ) } } \\leq \\frac { ( x + o ( 1 ) ) } { h ( x ) } \\right ) . \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\left ( \\widetilde { \\mathcal { X } ^ { K \\textnormal { t h } } } \\right ) _ { 0 i } ( q , Q ) = \\sum _ { \\substack { a + b = i \\\\ 0 \\leq a , b \\leq N } } ( - 1 ) ^ a \\binom { \\ell _ q \\left ( Q \\right ) } { a } J _ b \\left ( q , \\left ( \\frac { 1 - q } { z } \\right ) ^ { N + 1 } Q \\right ) \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} ( - 1 ) ^ k s ( n , n - k ) & = \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + k } \\binom { n + j - 1 } { k + j } \\binom { n + k } { k - j } S ( j + k , j ) \\\\ & = \\frac { ( n + k ) ! } { ( 2 k ) ! ( n - 1 - k ) ! } \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + k } \\binom { 2 k } { k + j } \\frac { S ( j + k , j ) } { n + j } \\\\ & = \\frac { 1 } { ( 2 k ) ! } \\sum _ { j = 0 } ^ k ( - 1 ) ^ { j + k } \\binom { 2 k } { k + j } \\frac { D ( n , k ) } { n + j } S ( j + k , j ) \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\frac { \\partial f ^ { \\left ( N \\right ) } \\left ( x , t \\right ) } { \\partial t } = \\varepsilon ^ { \\prime } \\boldsymbol { L } _ { N , \\lambda } f ^ { \\left ( N \\right ) } \\left ( x , t \\right ) \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} E _ { P } \\left [ \\left . \\frac { 1 } { \\lambda _ { \\overline { \\mathbf { a } } _ { 0 } } \\left ( \\overline { \\mathbf { G } } _ { 0 } , \\overline { \\mathbf { B } } _ { 0 } \\mathbf { ; } P \\right ) } \\right \\vert \\overline { \\mathbf { G } } _ { 0 } \\mathbf { , } \\overline { \\mathbf { A } } _ { 0 } \\mathbf { = } \\overline { \\mathbf { a } } _ { 0 } \\right ] = \\frac { 1 } { \\pi _ { a _ { 0 } } \\left ( \\overline { \\mathbf { G } } _ { 0 } \\mathbf { ; } P \\right ) } . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\xi ^ \\epsilon & : = \\inf \\{ r > t ~ : ~ [ \\ ! [ X _ r ^ { t , \\bar { A } _ t ^ \\epsilon ; Z _ t \\otimes \\alpha ( v ) , W _ t \\otimes v } ] \\ ! ] _ { \\kappa } > \\mu \\} \\\\ & ~ ~ ~ \\wedge \\inf \\{ r > t ~ : ~ \\| X _ r ^ { t , \\bar { A } _ t ^ \\epsilon ; Z _ t \\otimes \\alpha ( v ) , W _ t \\otimes v } \\| _ \\infty > \\mu _ 0 \\} . \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} & f > 0 , \\ g > 0 \\ \\ { \\rm o n } \\ \\ ( 0 , u ^ * ) \\times ( 0 , v ^ * ) , \\\\ & f < 0 , \\ g < 0 \\ \\ { \\rm o n } \\ \\ ( u ^ * , \\infty ) \\times ( v ^ * , \\infty ) , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} \\inf _ { u \\in \\psi ( x ) } \\sup _ { a \\in \\psi ( x ) } d _ A ( u , a ) \\lor \\inf _ { a \\in \\psi ( x ) } \\sup _ { u \\in \\psi ( x ) } d _ A ( u , a ) \\\\ = d _ H ( \\psi ( x ) , \\psi ( x ) ) = 0 . \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 0 } ^ { q - 1 } a _ j ( z ) \\bar { z } ^ j , z \\in \\Omega , \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } ( q , Q ) : = P ^ { - \\ell _ q ( Q ) } J ^ { K \\textnormal { t h } } ( q , Q ) = P ^ { - \\ell _ q ( Q ) } \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\left ( q P ^ { - 1 } ; q \\right ) _ d ^ { N + 1 } } \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} H ^ { + } = [ 0 , 1 ] \\times [ - 1 , 1 ] ^ { 2 n - 1 } . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} d = \\sum _ i a _ i ^ * \\nabla _ { e _ i } . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\mathcal { F } ( \\tau , Q ) = \\sum _ { d , n \\geq 0 } \\frac { 1 } { n ! } \\langle \\tau , \\dots , \\tau \\rangle ^ { K \\textnormal { t h } } _ { 0 , n , d } Q ^ d \\in \\mathbb { Z } [ \\ ! [ t _ 0 , \\dots , t _ N ] \\ ! ] \\otimes \\mathbb { C } [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} s ( D , v ) : = \\lambda t _ D + \\mu \\sum _ { \\alpha \\in D ^ * } \\left ( - 1 \\right ) ^ { \\left ( v , \\alpha \\right ) } e ^ { \\alpha } , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} k _ { r + 2 } \\in \\{ 2 k _ 1 , \\dotso , r k _ 1 , ( r + 1 ) k _ 1 , k _ { r + 1 } + k _ 1 \\} = \\{ k _ 2 , \\dotso , k _ r , k _ r + k _ 1 , n \\} . \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} \\| A \\| _ { L ^ p ( \\mathcal M ) } = \\left [ \\tau \\left ( \\left ( A ^ \\star A \\right ) ^ { \\frac p 2 } \\right ) \\right ] ^ { \\frac 1 p } , 1 \\leq p < \\infty . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\partial _ r n _ { a b } = R _ { L a b \\underline L } - l _ b ^ c n _ { a c } + \\nabla _ a \\eta _ b - \\eta _ a \\eta _ b \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} B _ { W } ( z , w ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { W _ k } ( z \\bar { w } ) ^ k , \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} & i / ( p - 1 ) - v _ p ( i ! ) = \\sum _ { k = 1 } ^ r \\{ ( i / p ^ k ) - \\lfloor i / p ^ k \\rfloor \\} + \\sum _ { k = r + 1 } ^ { \\infty } i / p ^ k \\le \\sum _ { k = 1 } ^ r ( ( p - 1 ) + \\dots + ( p - 1 ) p ^ { k - 1 } ) / p ^ k + \\sum _ { k = r + 1 } ^ { \\infty } i / p ^ k \\\\ = & r - ( p ^ r - 1 ) / p ^ r ( p - 1 ) + i / p ^ r ( p - 1 ) \\le r - ( p ^ r - 1 ) / p ^ r ( p - 1 ) + ( p ^ { r + 1 } - 1 ) / p ^ r ( p - 1 ) = r + 1 . \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ m X _ i ^ \\star ( a _ { i j , \\tau } X _ j v ) = \\sum _ { i = 1 } ^ m X _ i ^ \\star f _ { i , \\tau } + g _ { \\tau } ~ ~ ~ ~ ~ ~ ~ B ( 1 ) , \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} \\sum _ { i , j , k , l , p } ( h _ { i j k l } ^ { p ^ { \\ast } } ) ^ { 2 } = \\sum _ { i , j , k , l } ( h _ { i j k l } ^ { 1 ^ { \\ast } } ) ^ { 2 } + \\sum _ { i , j , k , l } ( h _ { i j k l } ^ { 2 ^ { \\ast } } ) ^ { 2 } , \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} C ' _ { j , k } \\cap A ' & = \\{ \\omega ' _ { j , k , 1 } \\} , C ' _ { j , k } \\cap ( A ' ) ^ c = \\{ \\omega ' _ { j , k , 2 } \\} , \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} \\nu ' : = [ \\nu ; \\nu ' ( \\phi ) = \\gamma ] . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\alpha = \\alpha ( m ) = \\frac { 1 } { \\cos ( \\pi / ( 2 m ) ) } \\ ; , \\ ; \\ ; \\beta = \\beta ( m ) = \\frac { m } { m - 1 } \\ ; , \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} H = - \\frac { ( X - x _ 0 ) ^ { \\perp } } { 2 t _ 0 } \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} ( ( a , u ) ( b , v ) ) ( c , w ) & = ( a b , u + f _ 1 ( a , a ^ { - 1 } ) v ) ( c , w ) \\\\ & = ( a b c , u + f _ 1 ( a , a ^ { - 1 } ) v + f _ 1 ( a b , b ^ { - 1 } a ^ { - 1 } ) w ) , \\\\ ( a , u ) ( ( b , v ) ( c , w ) ) & = ( a , u ) ( b c , v + f _ 1 ( b , b ^ { - 1 } ) w ) \\\\ & = ( a b c , u + f _ 1 ( a , a ^ { - 1 } ) ( v + f _ 1 ( b , b ^ { - 1 } ) w ) ) . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\delta _ \\Omega ( \\sigma ( \\tau ) ) = \\max \\{ \\delta _ \\Omega ( \\sigma ( t ) ) : t \\in [ 0 , T _ 0 ] \\} . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\nabla ^ { \\mathcal G , \\ , \\mathrm { s p i n } } _ { u } = \\nabla ^ { 0 , \\mathcal S _ { \\mathcal G } } _ { X } - \\frac { 1 } { 3 } \\mathrm { a d } _ { r } \\quad \\mbox { o n } \\mathcal S _ { \\mathcal G } . \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} e ( H ^ U ( A ) ) = | P ' | = | P | - \\binom { | A \\cap U | } { 2 } > \\frac { | A | ^ 2 | U | ^ 2 } { 6 n } - \\frac { | A | \\cdot | U | } { 2 } \\geq \\frac { | A | ^ 2 | U | ^ 2 } { 1 2 n } , \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\binom { n } { k } _ { a , q } = \\binom { n - 1 } { k - 1 } _ { a , q } + a q ^ { k - 1 } \\frac { 1 - q ^ { n - k } } { 1 - a q ^ { n - k - 1 } } \\binom { n - 1 } { k } _ { a , q } \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\varepsilon } ( K | E ) ~ = ~ \\log _ 2 \\mathcal { N } _ { \\varepsilon } ( K | E ) . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\mathbf { B = } \\left ( \\mathbf { B } _ { 0 } , \\dots , \\mathbf { B } _ { p } \\right ) \\mathbf { \\subset V \\backslash } \\left \\{ \\mathbf { A } , Y \\right \\} \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\varphi _ \\delta ( s ) = 1 { \\rm \\ f o r \\ } s \\in [ 0 , r - \\delta ) , \\ \\ , \\vert \\varphi ' ( s ) \\vert \\leq 2 \\delta { \\rm \\ f o r \\ } s \\in [ r - \\delta , r ] { \\rm \\ a n d \\ } \\varphi ( r ) = 0 . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } ^ 2 { x + k \\choose 2 n + 1 } = \\frac { 1 } { ( 4 n + 2 ) { 2 n \\choose n } } \\sum _ { k = 0 } ^ { n } ( 2 x - 3 k ) { x \\choose k } ^ 2 { 2 k \\choose k } . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\lim _ { n } d _ { \\mathcal A } ( f ( t ) , f _ n ( t ) ) = 0 a . e . \\ \\ t \\in \\Omega _ S . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\sigma ( t , \\theta ) = ( t \\cos ( \\theta ) , t \\sin ( \\theta ) ) \\ ; , \\ ; \\ ; ( t , \\theta ) \\in [ 0 , 1 ] \\times [ - \\omega , \\omega ] \\ ; , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\left | \\nabla \\left ( e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } f \\right ) \\right | _ q \\leq \\mathcal { B } | \\nabla f | _ q , \\ \\forall f \\in e ^ { \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } [ W ^ { 1 , q } ( \\mathbb { R } ^ d ) ] , \\ t \\geq 0 , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\Delta ( u - v ) = 0 , \\mbox { i n } \\ , \\ , B _ { 1 } . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} ( \\lambda _ g ) \\cdot a = \\prod _ { i = 1 } ^ n \\lambda _ { g _ i } ^ { \\psi _ i ( a ) } a \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} u _ t + \\frac { 1 } { 2 } \\Delta u - b \\cdot \\nabla u - \\lambda u = b \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} { h } _ { a b } = \\widehat { h } _ { a b } + \\widetilde { h } _ { a b } , \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} Q _ { m _ { 1 } , \\ldots , m _ { i + 1 } } ^ { ( j ) } = Q _ { m _ { 1 } , \\ldots , m _ { i } } \\cap \\left [ x _ { d - i + 1 } = \\frac { i + 1 } { j - i } x _ { d - i } - \\frac { i + 1 } { j - i } \\left ( x _ { d - i } ^ { 0 } + \\cdots + \\frac { 1 } { i + 1 } \\binom { j + 1 } { i } x _ { d } ^ { 0 } + m _ { i + 1 } \\right ) \\right ] , \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\hat { D } ( x , y , z ) = \\frac { 2 } { x } \\left ( 1 + \\frac { 2 \\sinh \\frac { x } { 2 } } { e ^ { \\frac { x } { 2 } } + e ^ { \\frac { y + z } { 2 } } } \\right ) \\leq \\frac { 4 \\sinh \\frac { x } { 2 } } { x ( e ^ { \\frac { x } { 2 } } + e ^ { \\frac { y + z } { 2 } } ) } . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} W [ \\alpha ] ( s , v ) \\stackrel { d e f } { = } Z ^ s [ \\alpha ] ( s , v ) \\ \\sup _ i \\left \\{ \\ \\int _ T { \\bf E } | U ^ { - 1 } \\ \\xi _ i ( t ) | ^ v \\ \\mu ( d t ) \\ \\right \\} < \\infty , \\ 1 \\le s < v . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\begin{alignedat} { 2 } & \\# ( f _ { m + 1 } ) + \\# ( h _ { m + 1 } ) + \\# ( f _ { m } ) \\leq k , \\quad & & \\# ( h _ { m + 1 } ) + \\# ( e _ { m + 1 } ) + \\# ( f _ { m } ) \\leq k , \\\\ & \\# ( e _ { m + 1 } ) + \\# ( f _ { m } ) + \\# ( h _ { m } ) \\leq k , \\quad & & \\# ( e _ { m + 1 } ) + \\# ( h _ { m } ) + \\# ( e _ { m } ) \\leq k . \\end{alignedat} \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} & S _ { 2 , \\lambda } ( n + 1 , k ) + n \\lambda S _ { 2 , \\lambda } ( n , k ) \\\\ & = \\sum _ { m = k - 1 } ^ n { m \\choose k - 1 } S _ { 1 , \\lambda } ( n , m ) E [ ( U _ 1 + \\cdots + U _ k + 1 ) ^ { m - k + 1 } ] . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\nabla ^ { \\mathbb { S } } : = \\nabla ^ { F , \\ , \\mathrm { s p i n } } \\otimes \\nabla ^ { { \\mathcal G } , \\ , \\mathrm { s p i n } } \\otimes ( \\nabla ^ { E } ) ^ { L } \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} w _ 1 = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\ , , w _ 2 = \\begin{pmatrix} \\alpha \\\\ \\sqrt { 1 - \\alpha ^ 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} E _ { m , n } = \\frac { p - p ^ { - 1 } } { 2 \\log { p } } Q , \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} \\N ( T _ r ( n ) , K ^ { ( r ) } _ { s , t } ) - \\N ( T _ r ( n - 1 ) , K ^ { ( r ) } _ { s , t } ) = F _ { r , s , t } \\left ( \\left \\lfloor \\frac { r - 1 } { r } n \\right \\rfloor , n \\right ) . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\in C \\right ) = P \\left ( \\frac { n - A _ n ^ * } { f _ 1 ( n ) } \\in C \\cap ( 0 , \\ell _ 1 ^ { - 1 } + \\delta ) \\right ) . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\mathrm { F } \\Phi ( \\mathrm { Y } ) \\cong \\Phi \\Psi ( \\mathrm { F } \\Phi ( \\mathrm { Y } ) ) = \\Phi ( \\mathrm { F } \\Psi \\Phi ( \\mathrm { Y } ) ) \\to \\Phi ( \\mathrm { F } \\ , \\mathrm { Y } ) \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} \\Delta = \\sum _ { i = 1 } ^ d ( \\nabla _ { e _ i } \\nabla _ { e _ i } - \\nabla _ { \\nabla _ { e _ i } e _ i } ) , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} \\dim V _ h ( \\beta ) = c _ n \\beta ^ n h ^ { - n } + o ( h ^ { - n } ) \\quad h \\to 0 , c _ n : = \\frac { 1 } { 2 ^ n \\cdot n ! } . \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\overline { \\varphi _ { t } } = \\zeta _ { t } \\varphi _ { - t } \\ , , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\int _ 0 ^ T g _ s \\ , \\mathrm { d } ^ { - } B _ s ^ H = \\int _ 0 ^ T g _ s \\delta B _ s ^ H + c _ H ^ B \\int _ 0 ^ T ( \\nabla ^ { H - \\frac { 1 } { 2 } } g _ s ) ( s ) \\ , \\mathrm { d } { s } . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} w ( \\gamma _ 1 ^ 2 ) = t _ { 1 1 1 2 } ^ { 3 / 4 } t _ { 2 1 1 1 } ^ { 1 / 4 } \\enspace . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} R _ { L a b L } = & r ^ 2 \\bar R _ { L a b L } + O ( r ^ 3 ) \\\\ R _ { L a L \\underline L } = & r \\bar R _ { L a L \\underline L } + O ( r ^ 2 ) \\\\ R _ { L \\underline L L \\underline L } = & \\bar R _ { L \\underline L L \\underline L } + O ( r ) , \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} u _ i \\cdot u _ i = - 1 , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} a ^ { 2 } & = 1 , & x ^ { 2 p } & = 1 , & z ^ { 2 } & = { a - 1 } , \\\\ a x & = x a , & a z & = z a , & x z & = - z x . \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\overline { B } ( X ^ 4 ) + X \\overline { B } ( X ^ 2 ) - \\overline { B } ( X ) = 0 . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} ( u _ 1 , u _ 2 ) = ( u , - v ) , \\ f _ 1 ( u _ 1 , u _ 2 ) = f ( u _ 1 , - u _ 2 ) , \\ f _ 2 ( u _ 1 , u _ 2 ) = - g ( u _ 1 , - u _ 2 ) . \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} J _ 1 ( f ; v ^ 1 , v ^ 2 ) = \\underset { \\hat { v } ^ 1 } { } \\ J _ 1 ( f ; \\hat { v } ^ 1 , v ^ 2 ) , \\ \\ J _ 2 ( f ; v ^ 1 , v ^ 2 ) = \\underset { \\hat { v } ^ 2 } { } \\ J _ 2 ( f ; { v } ^ 1 , \\hat { v } ^ 2 ) . \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} & \\sum _ { r , s \\in \\mathbb Z } ( - 1 ) ^ { n + k + s } { k \\choose n + r - s - m } \\ ! \\ ! \\left \\{ \\ ! { n - 1 - k \\choose m - r } \\ ! + \\ ! { n - 1 - k \\choose m - r - 1 } \\right \\} \\ ! M ( r - s a ) ^ { n - 1 } \\\\ & = \\ ! \\ ! \\sum _ { r , s \\in \\mathbb { Z } } ( - 1 ) ^ { n + k + s } { k \\choose n + r - s - m } { n - k \\choose m - r } M ( r - s a ) ^ { n - 1 } , \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} D _ { \\gamma , \\mu } : = f ^ { - 1 } ( v ^ { - 1 } ( ] - \\infty , \\mu ] ) ) \\cap ( v ^ { - 1 } ( \\gamma ) ) , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} | u ( p ) - \\mathbb { L } ( p ) | \\leq C \\sigma ^ { \\nu } W ( \\sigma ^ { \\nu } ) = C | p | W ( | p | ) , \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{gather*} \\alpha _ 1 = 3 . 1 3 7 8 , \\ \\alpha _ 2 = - 3 . 9 7 8 9 , \\ \\alpha _ 3 = 2 . 6 7 8 8 , \\ \\alpha _ 4 = - 1 . 0 4 0 1 , \\ \\alpha _ 5 = 0 . 2 1 3 9 , \\ \\alpha _ 6 = - 0 . 0 1 3 3 . \\end{gather*}"}
-{"id": "7979.png", "formula": "\\begin{align*} d _ i ^ { n e } = T , ~ \\forall i . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\big | [ b , \\mathcal I _ { \\alpha } ] f _ 2 ( x ) \\big | & \\leq \\big | b ( x ) - b _ B \\big | \\int _ { \\mathbb R ^ d } \\big | \\mathcal K _ { \\alpha } ( x , y ) f _ 2 ( y ) \\big | \\ , d y + \\int _ { \\mathbb R ^ d } \\big | b ( y ) - b _ B \\big | \\big | \\mathcal K _ { \\alpha } ( x , y ) f _ 2 ( y ) \\big | \\ , d y \\\\ & : = \\xi ( x ) + \\zeta ( x ) . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} P ( T > t + s | T > s ) \\geq P ( T > t + s ) \\geq P ( T > 2 t ) = \\frac { L ( 2 t ) } { ( 2 t ) ^ { \\alpha } } \\geq \\frac { 1 } { t ^ { \\alpha + \\epsilon } } , \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\gamma ^ * ( \\alpha _ i ) ( m ' ) & = \\gamma ^ * ( ( F _ i ) ^ * ( u _ { i } ) ) ( m ' ) = \\gamma u _ i ( F _ i ( \\gamma ^ { - 1 } ( m ' ) ) ) \\\\ & = \\gamma \\beta ( u _ { i _ 0 } ) ( F _ i ( \\gamma ^ { - 1 } ( m ' ) ) ) = \\beta ( \\gamma u _ { i _ 0 } ) ( F _ i ( \\gamma ^ { - 1 } ( m ' ) ) ) \\\\ & = \\beta ( \\gamma u _ { i _ 0 } \\gamma ^ { - 1 } ) ( \\gamma F _ i ( \\gamma ^ { - 1 } ( m ' ) ) ) = ( \\gamma ^ * F _ i ) ^ * ( \\gamma ^ * ( u _ i ) ) ( m ' ) . \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} p = \\sum _ { i = 1 } ^ r a _ i \\textbf { Q } ^ { \\lambda _ i } \\mbox { w i t h } \\nu \\left ( a _ i \\textbf { Q } ^ { \\lambda _ i } \\right ) \\geq \\beta , \\mbox { f o r e v e r y } i , 1 \\leq i \\leq r , \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\small { c ( l , \\sigma , f , e _ g , g ( 1 ) , \\ldots g ( l ) ) = T r _ W ( m ^ { l } L _ { \\sigma } T ( g ( 1 ) , g ( 2 ) , \\ldots g ( l ) ) A _ { e _ g } ) . } \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty F ( \\alpha + i y ) y ^ { s - \\frac 1 2 } \\frac { d y } { y } = \\sum _ { a \\in \\{ 0 , 1 \\} } i ^ { - a } \\Delta _ f \\bigl ( s , \\alpha , \\cos ^ { ( a ) } \\bigr ) \\begin{cases} 1 & k = 1 ( - 1 ) ^ a = \\epsilon , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} u _ { t } + f ( x , u ) _ { x } = \\varepsilon u _ { x x } , u ( 0 , x ) = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\frac 1 { \\delta _ i } = \\frac 1 { r _ i } - \\frac 1 { p _ i } , \\frac 1 { \\widetilde { \\delta } _ i } = \\frac 1 { r _ i } - \\frac 1 { q _ i } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} & c _ 1 \\cdots c _ l + c _ { l + 1 } \\cdots c _ { 2 l } + \\cdots + c _ { ( n - 1 ) l + 1 } \\cdots c _ { l n } \\\\ = \\ , & \\sum _ { i = 0 } ^ { n - 1 } ( 1 0 ^ l s _ i - s _ { i + 1 } ) / b \\\\ = \\ , & \\frac { 1 0 ^ l - 1 } { L } \\cdot \\frac { \\sum _ { i = 0 } ^ { n - 1 } s _ i } { B } , \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\frac { w _ p - 1 } { p ^ 3 } & = \\frac { p + 1 } { ( 2 p - 4 ) ! ( p - 1 ) ! } ( a _ { 2 p - 7 } p ^ { 2 p - 7 } + \\cdots + a _ 2 p ^ 2 + a _ 1 p + a _ 0 ) \\\\ & = \\frac { p + 1 } { ( 2 p - 4 ) ! ( p - 1 ) ! } W ( p ) , a _ i \\in \\Z . \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} w ^ \\beta _ + w ^ \\gamma _ - = ( \\theta ^ \\beta _ + \\theta ^ \\gamma _ - b _ { \\beta , \\gamma } + b _ { \\alpha + \\beta , \\alpha + \\gamma } ) e ^ { \\beta + \\gamma } + ( \\theta ^ \\beta _ + b _ { \\beta , \\alpha + \\gamma } + \\theta ^ \\gamma _ - b _ { \\alpha + \\beta , \\gamma } ) e ^ { \\alpha + \\beta + \\gamma } \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} 0 * ^ p = 0 \\ ! * \\ ! * * , * 1 ^ p = * \\ ! * \\ ! * 0 , \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\dot { u } = \\nu ( u ) \\left ( \\nabla H ( u ) \\times \\nabla C ( u ) \\right ) , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\overline { U } _ { k , a } ( n ) = \\overline { U } _ { k , a - 1 } ( n ) . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\left \\Vert a \\right \\Vert \\le \\sum _ { j = 1 } ^ m \\left \\Vert a _ j ^ n \\right \\Vert \\le \\sum _ { j = 1 } ^ m \\left \\Vert a _ j \\right \\Vert ^ n , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} w ( \\gamma ^ 1 _ 2 ) = t _ { 2 2 2 2 } \\enspace . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} E _ { \\varepsilon } \\left ( u \\right ) = \\int _ { G } p \\left \\vert \\nabla u \\right \\vert ^ { 2 } + \\frac { 1 } { \\varepsilon ^ { 2 } } \\int _ { G } J \\left ( 1 - \\left \\vert u \\right \\vert ^ { 2 } \\right ) \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} E ^ 1 _ { r , q } ( X ) = \\bigoplus _ { x \\in X _ r } H _ { r + q } ( x ) \\Rightarrow H _ { r + q } ( X ) , \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} h _ 2 ' ( u ) = \\xi '' ( u ) ( 1 + z _ 2 - q - u z _ 2 ) - z _ 2 [ \\xi ' ( u ) - \\xi ' ( q ) ] - [ \\xi ' ( 1 ) - \\xi ' ( q ) ] . \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} & { n \\choose k } E [ ( U _ 1 + \\cdots + U _ k ) ^ { n - k } ] S _ { 1 , \\lambda } ( n , n ) \\\\ & \\quad - { n - 1 \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , n - 1 ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { n - k } \\\\ & = - \\sum _ { m = k - 1 } ^ { n - 1 } E [ ( U _ 1 + \\cdots + U _ k ) ^ { m - k } ] { m \\choose k } S _ { 1 , \\lambda } ( n - 1 , m - 1 ) \\\\ & + \\sum _ { m = k - 1 } ^ { n - 2 } { m \\choose k - 1 } S _ { 1 , \\lambda } ( n - 1 , m ) E [ ( U _ 1 + \\cdots + U _ { k - 1 } + 1 ) ^ { m - k + 1 } ] . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} \\int _ { A \\times A } [ 0 _ A \\times A ] \\cdot \\eta \\wedge \\overline { \\eta } = 4 . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\mathbf P _ o ^ { j } = \\frac { p } { n } \\sum _ { i = 1 } ^ { n } \\mathbb P ( R _ { u } ^ { j , \\ , i } < r _ t ) + ( 1 - p ) \\mathbb P ( R _ o ^ { \\ , j } < r _ t ) , \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathcal { S } } w _ j d _ j ^ { n e } = \\sqrt { \\frac { w _ i B \\sum _ { j \\in \\mathcal { S } / \\{ i \\} } w _ j d _ j ^ { n e } } { ( c _ i + \\lambda _ i ) } } . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} V _ { k } ^ { - } = o _ { \\varepsilon \\to 0 } ( 1 ) + o _ { h \\to 0 } ( 1 ) . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} h _ { k , U _ { 2 } } ( U _ { 1 } ) = \\begin{cases} f ( U _ { 1 } ) + \\min _ { \\emptyset \\subset U ' \\subset U _ { 2 } \\setminus U _ { 1 } } g _ { k , U _ { 2 } \\setminus U _ { 1 } } ( U ' ) & | U _ { 2 } \\setminus U _ { 1 } | > 1 \\\\ \\infty & { } \\end{cases} \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} [ y _ k ^ { p ^ r } , W ] = W \\cdot \\delta _ k ( r ) \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\| \\varphi _ * \\| _ { L ^ { \\infty } } \\leq A ( a _ 0 ) = \\| \\varphi _ 0 \\| _ { L ^ \\infty } , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} & ( \\alpha _ 1 - \\alpha _ 2 ) ( m _ 1 - m _ 2 ) + ( \\alpha _ 3 - \\alpha _ 4 ) ( m _ 3 - m _ 4 ) = 0 , \\\\ & ( \\alpha _ 3 - \\alpha _ 4 ) ( m _ 1 - m _ 2 ) + ( \\alpha _ 1 - \\alpha _ 2 ) ( m _ 3 - m _ 4 ) = 0 , \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\mathcal { I } = \\Big ( x _ 1 - | x _ 1 - \\hat { x } _ { 1 , } | - \\delta , ~ x _ 1 + | x _ 1 - \\hat { x } _ { 1 , } | + \\delta \\Big ) \\times \\Big ( x _ 2 - | x _ 2 - \\hat { x } _ { 2 , } | - \\delta , ~ x _ 2 + | x _ 2 - \\hat { x } _ { 2 , } | + \\delta \\Big ) \\subset \\mathcal { B } ( \\textbf { x } ) . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} Q = \\left ( \\begin{matrix} B \\\\ a ^ { \\mathrm T } \\end{matrix} \\right ) ^ { \\mathrm T } \\left ( \\begin{matrix} Y \\\\ y ^ { \\mathrm T } \\end{matrix} \\right ) \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} u ( t , x ) = u ^ { R } ( t , x ) + k ( t ) \\zeta ( t , x ) S ( \\Phi ( t , x ) ) \\ , , \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} Q ( x ) = \\frac { 4 } { 1 + x ^ 2 } \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} ( x q ) ^ { 2 a + 1 } \\sum _ { h = 1 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} v _ L = \\overline { p _ L } X _ 1 + \\overline { q _ L } X _ 2 + \\overline { r _ L } \\widetilde { X _ 3 } , ~ ~ ~ ~ e _ 1 = \\overline { q } X _ 1 - \\overline { p } X _ 2 , ~ ~ ~ ~ e _ 2 = \\overline { r _ L } ~ ~ \\overline { p } X _ 1 + \\overline { r _ L } ~ ~ \\overline { q } X _ 2 - \\frac { l } { l _ L } \\widetilde { X _ 3 } , \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} F ( x ) = ( \\hat { p } ( x ) , d _ x \\hat p v _ 2 ( x ) , \\ldots , d _ x \\hat p v _ { n - 1 } ( x ) ) . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} M _ \\triangledown ( f ) ( x ) : = \\sup _ { t \\in ( 0 , \\infty ) , | y - x | \\le 3 ( | x _ 0 | + 1 ) t } [ | F ( y , t ) | + | G ( y , t ) | ] \\in W X , \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} H = - | G | ^ { - 1 / 2 } \\partial _ i | G | ^ { 1 / 2 } G ^ { i j } \\partial _ j = - f ^ { - 1 } \\partial _ s f ^ { - 1 } \\partial _ s - f ^ { - 1 } \\partial _ t f \\partial _ t \\qquad \\mbox { w i t h } | G | : = \\det ( G ) = f ^ 2 . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} Z _ 1 & : = \\partial _ { z _ 1 } + i \\left ( \\bar z _ 2 + 2 z _ 1 \\bar z _ 1 ^ 2 \\right ) \\partial _ t , \\\\ Z _ 2 & : = \\partial _ { z _ 2 } + i \\bar z _ 1 \\partial _ t , \\\\ Z _ 3 & : = \\partial _ { z _ 3 } + i \\bar z _ 4 \\partial _ t , \\\\ Z _ 4 & : = \\partial _ { z _ 4 } + i \\bar z _ 3 \\partial _ t . \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} H _ 1 = ( 1 , \\dots , 1 \\ , | \\ , 1 , \\dots , 1 ) . \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} P _ q ( X ) \\cap C ^ { q - 1 } ( X ) = \\mathbf { P } _ q ( X ) . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} M _ R : = \\sup \\left \\{ \\frac { | \\nabla \\varphi _ { y } ( x ) - \\nabla \\varphi _ { y } ( z ) | } { | x - z | } \\ , : x , z \\in B ( 0 , R ) , x \\neq z , \\ , y \\in E \\cap B ( 0 , R ) \\right \\} < \\infty \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\beta ( i ) \\stackrel { d e f } { = } \\sup _ n \\max _ { k \\in [ 1 , n ] } \\beta ( M _ 1 ^ k , M _ { k + i } ^ n ) . \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} T _ t ^ \\alpha f ( x ) & = \\int _ { \\mathbb R _ + ^ d } e ^ { - | z | ^ 2 t } \\phi _ z ^ \\alpha ( x ) h _ \\alpha f ( z ) z ^ { 2 \\alpha } d z \\\\ & = \\int _ { \\mathbb R _ + ^ d } \\left ( \\int _ { \\mathbb R _ + ^ d } e ^ { - | z | ^ 2 t } \\phi _ z ^ \\alpha ( x ) \\phi _ z ^ \\alpha ( y ) z ^ { 2 \\alpha } d z \\right ) f ( y ) y ^ { 2 \\alpha } d y \\\\ & = \\int _ { \\mathbb R ^ d _ + } \\mathcal { W } _ t ^ \\alpha ( x , y ) f ( y ) y ^ { 2 \\alpha } d y , \\ , \\ , \\ , x \\in \\mathbb { R } _ + ^ d , \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} \\aligned \\nabla _ { j } \\nabla _ { i } | X | ^ { 2 } = & 2 \\langle e _ { i } , e _ { j } \\rangle + 2 \\langle X , X _ { i j } \\rangle \\\\ = & 2 \\delta _ { i j } + 2 \\langle X , \\sum _ { p } h _ { i j } ^ { p ^ { \\ast } } e _ { p ^ { \\ast } } \\rangle \\\\ = & 2 \\delta _ { i j } - 2 \\sum _ { p } h _ { i j } ^ { p ^ { \\ast } } H ^ { p ^ { \\ast } } , \\endaligned \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\bar { u } _ T ^ { i + j } = u _ T ^ { i - T j ' + j } \\quad \\ j \\in Z _ 1 ( L Q ) . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} ( Q ; q ) _ 0 & = 1 \\\\ ( Q ; q ) _ d & = \\prod _ { r = 0 } ^ { d - 1 } ( 1 - q ^ r Q ) \\\\ ( Q ; q ) _ \\infty & = \\prod _ { r \\geq 0 } ( 1 - q ^ r Q ) \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} d _ { \\mathbf { p } S _ i } = \\left ( \\begin{smallmatrix} d _ { P _ i [ 3 ] } & 0 & 0 & 0 \\\\ \\rho & d _ { P _ { s ( \\rho ) } [ 2 ] } & 0 & 0 \\\\ - \\tau ^ \\ast & - \\partial _ { \\rho \\tau } w & d _ { P _ { t ( \\tau ) } [ 1 ] } & 0 \\\\ e _ i ^ \\ast & \\rho ^ \\ast & \\tau & d _ { P _ i } \\end{smallmatrix} \\right ) \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\partial _ \\tau w _ g ^ n ( y , \\tau ) - \\frac { 1 } { 2 } \\partial _ { y y } w _ g ^ n ( y , \\tau ) + ( y + a _ n ' ( y ) ) \\partial _ y w _ g ^ n ( y , \\tau ) = g ( y , \\tau ) , \\ : \\forall ( y , \\tau ) \\in \\mathbb { R } \\times [ 0 , \\infty ) , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\pi _ L ( X ) F ( g ) & = \\left . \\frac { d } { d \\tau } F ( e ^ { - \\tau X } g ) \\right | _ { \\tau = 0 } , \\\\ \\pi _ R ( X ) F ( g ) & = \\left . \\frac { d } { d \\tau } F ( g e ^ { \\tau X } ) \\right | _ { \\tau = 0 } , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\langle \\imath _ { \\kappa } \\gamma ( x ) , \\eta \\rangle = \\langle a d _ v ( x ) , \\eta \\rangle \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ n \\lambda ^ { n - m } S _ 1 ( n , m ) \\frac { 1 } { k ! } \\Delta ^ k 0 ^ m = \\begin{cases} S _ { 2 , \\lambda } ( n , k ) , & \\ , \\ , n \\geq k , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n x _ i \\beta _ i & = ( \\beta _ 1 , \\dots , \\beta _ r ) \\left \\{ \\left ( \\begin{array} { c } { x } _ 1 \\\\ \\vdots \\\\ { x } _ r \\end{array} \\right ) + T \\left ( \\begin{array} { c } { x } _ { r + 1 } \\\\ \\vdots \\\\ { x } _ n \\end{array} \\right ) \\right \\} ( x _ i \\in \\mathbb { R } ) , \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} q \\theta _ q ' ( q Q ) = \\frac { 1 } { Q } \\theta _ q ' ( Q ) - \\frac { 1 } { Q ^ 2 } \\theta _ q ( Q ) \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} F _ { \\chi , \\theta } ( s ) & = \\sum _ { \\substack { \\beta \\mid \\theta \\\\ ( \\beta , \\kappa ) = 1 } } \\frac { \\mu ( \\beta ) \\chi ( \\beta ) \\lambda ( \\theta / \\beta ) } { \\beta ^ s } \\sum _ { ( n , \\kappa ) = 1 } \\frac { \\chi ( n ) \\lambda ( n ) } { n ^ s } \\\\ & = P ( \\chi , \\theta , \\kappa ) L ( s , f \\otimes \\chi ) , \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} & \\varphi ( a _ n ( x ) ) = S _ n \\varphi ( x ) - S _ { n - 1 } \\varphi ( x ) \\\\ \\in & \\Big ( \\Phi ( n ) ( 1 - \\varepsilon ) - \\Phi ( n - 1 ) ( 1 + \\varepsilon ) , \\ \\Phi ( n ) ( 1 + \\varepsilon ) - \\Phi ( n - 1 ) ( 1 - \\varepsilon ) \\Big ) \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\gamma u \\times H _ { e f f } + L _ 1 \\frac { 1 } { | u | ^ 2 } ( u \\cdot H _ { e f f } ) u - L _ 2 \\frac { 1 } { | u | ^ 2 } u \\times ( u \\times H _ { e f f } ) \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\{ ( w , x ) \\in X : ( \\forall \\gamma > \\theta ( w , x ) ) ( h _ { w , x } ( \\gamma ) = r _ i \\gamma + \\alpha ( w , x ) ) \\} . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} I n d ( u , t ) = \\left \\{ \\begin{array} { l l } 0 , & ( u , t ) \\in A ^ * , \\\\ + \\infty , & , \\end{array} \\right . \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m a x i m i z e } & \\sum _ { i = 1 } ^ n \\big ( H _ i ( \\hat { x } _ i ) + G _ i ( \\hat { c } _ i ^ T \\hat { x } _ i ) \\big ) \\\\ \\mbox { s u b j e c t t o } & x _ t \\in F _ t \\subseteq \\mathbb { R } ^ n ~ \\forall t \\in [ m ] \\\\ \\end{array} \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} & \\liminf _ { n \\to \\infty } \\frac { 1 } { - f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 3 ( n ) } \\in O \\right ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\geq \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 3 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 3 ( n ) ) } \\log \\left ( 1 - P \\left ( \\frac { n - A _ n ^ * } { f _ 3 ( n ) } > \\varepsilon \\right ) \\right ) = 0 = - \\inf _ { x \\in O } I _ 3 ( x ) , \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} \\partial _ { Q } : E ^ { * } \\otimes \\mathrm { a d } _ { Q } ( E ) \\rightarrow \\Lambda ^ { 3 } E ^ { * } , \\ ( \\partial _ { Q } \\eta ) ( u , v , w ) : = \\eta ( u , v , w ) + \\eta ( w , u , v ) + \\eta ( v , w , u ) \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} f = & \\frac { h _ 0 ^ { ( 1 ) } - h ^ { ( 1 ) } } { A } r + O ( r ^ 2 ) \\\\ V ^ 2 = & A ^ 2 - \\sum _ i C _ i ^ 2 + O ( r ) \\\\ V ^ 2 \\nabla _ a \\tau = & r C _ i \\tilde \\nabla _ a \\tilde X ^ i + O ( r ^ 2 ) . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} a ( e ( t ) , h ( t ) , f ( t ) ) = \\sum _ { k \\in \\Z _ { \\geq 0 } } [ a ( e , h , f ) ] _ k \\frac { t ^ k } { k ! } , \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} f _ k ( x ) : = 2 ^ { ( k - 1 ) / 2 } \\big ( \\chi _ { [ 1 / 2 - 2 ^ { - k } , 1 / 2 ] } ( x ) - \\chi _ { [ 1 / 2 , 1 / 2 + 2 ^ { - k } ] } ( x ) \\big ) . \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial b } ( t , x ; b , y ) & = \\frac { 4 ( 1 + t ) \\big [ ( y - x ) ^ 2 - ( t - b ) ( t + b + 2 ) \\big ] } { \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] ^ 2 } , \\\\ \\frac { \\partial z } { \\partial y } ( t , x ; b , y ) & = - \\frac { 8 ( y - x ) ( 1 + t ) ( 1 + b ) } { \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] ^ 2 } , \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\overline { q } ^ 2 d \\sigma _ \\Sigma = 0 , \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } k ^ { 2 H + 1 } | I _ { k } | ^ { 2 ( 1 - H ) } \\leq ( C _ { V , l } | I _ { 1 } | ) ^ { 2 ( 1 - H ) } \\cdot \\left ( \\sum _ { k = 1 } ^ { m } k ^ { 2 H + 1 } ( C _ { V , l } | I _ { 1 } | ) ^ { 2 ( 1 - H ) ( \\alpha ^ { k - 1 } - 1 ) } \\right ) . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) = - \\varepsilon . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\eta ( x , q , \\beta ) = \\sup _ { p \\in \\mathbb { R } ^ d , \\ , \\alpha \\in \\mathbb { R } ^ n } \\bigl [ q \\cdot p + \\beta \\cdot \\alpha - \\lambda ( x , p , \\alpha ) \\bigr ] , x , q \\in \\mathbb { R } ^ d , \\ , \\ , \\beta \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( 1 + | \\xi | ^ { 2 \\delta } ) \\ , \\widehat { v } _ { t t } ( t , \\xi ) + | \\xi | ^ { 2 \\theta } \\ , \\widehat { v } _ t ( t , \\xi ) + | \\xi | ^ { 2 \\alpha } \\ , \\widehat { v } ( t , \\xi ) = 0 , t \\geq 0 , \\ , \\ , \\ , \\ , \\xi \\in \\R ^ n \\\\ \\widehat { v } ( 0 , \\xi ) = \\widehat { v } _ 0 ( \\xi ) , \\widehat { v } _ t ( 0 , \\xi ) = \\widehat { v } _ 1 ( \\xi ) , \\xi \\in \\R ^ n . \\end{array} \\right . \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = 0 , f ( x , u ) = \\begin{cases} f _ l ( u ) = u ( 1 - u ) & ( x < 0 ) , \\\\ f _ r ( u ) = 2 u ( 1 - u ) & ( x > 0 ) . \\end{cases} \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} \\frac { ( A ^ + ) ^ { n - 1 } a ^ a ( n - a ) ^ { n - a } } { ( A ^ - ) ^ { n - 1 } n ^ n } = ( A \\left ( \\frac { a } { n } \\right ) ^ { \\frac { a } { n - 1 } } \\left | \\frac { a } { n } - 1 \\right | ^ { \\frac { n - a } { n - 1 } } ) ^ { n - 1 } > 2 \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\tau _ 3 ( x ) = g _ 3 ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 1 1 } { 3 ^ 3 } , \\tfrac { 8 } { 1 9 } ) . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} & | \\nabla \\psi '' ( \\iota \\varphi _ 1 + ( 1 - \\iota ) \\varphi _ 2 ) | = | \\psi ''' ( \\iota \\varphi _ 1 + ( 1 - \\iota ) \\varphi _ 2 ) ( \\iota \\nabla \\varphi _ 1 + ( 1 - \\iota ) \\nabla \\varphi _ 2 ) | \\\\ & \\lesssim ( 1 + | \\varphi _ 1 | + | \\varphi _ 2 | ) ( | \\nabla \\varphi _ 1 | + | \\nabla \\varphi _ 2 | ) \\\\ & \\lesssim 1 + | \\varphi _ 1 | ^ 2 + | \\varphi _ 2 | ^ 2 + | \\nabla \\varphi _ 1 | ^ 2 + | \\nabla \\varphi _ 2 | ^ 2 \\ , , \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} d \\widetilde { C } _ { p , q } ( K , Q , \\cdot ) = h _ K ^ { - p } d \\widetilde { C } _ q ( K , \\cdot ) . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} P ( Y = 0 ) & = \\omega , \\\\ P ( Y = y ) & = \\frac { ( 1 - \\omega ) p ^ y } { - y \\ln ( 1 - p ) } , \\ ; y \\geq 1 . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} S ' ( q ) = - \\frac { 3 } { 2 } \\frac { \\frac { 5 } { 3 } P ( q ) - P ' ( q ) q } { q ^ 2 } < 0 . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} A _ i X : = \\sigma _ i \\mathbf { 1 } _ 3 \\cdot \\nabla X + \\theta _ i X , \\ i = 1 , 2 , . . . , N , \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} = [ I ^ { M _ { S ' } } _ { w ( M _ S ) } ( w ( \\rho ) ) ] [ r _ { - \\mu _ { S ' } } \\circ L L ( w ( \\rho ) ) \\otimes | \\cdot | ^ { - \\langle \\rho _ { M _ { S ' } } , \\mu _ { S ' } \\rangle } ] \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} \\min _ { x } \\ ( \\max _ { i , j = 1 } ^ { n } a _ { i j } x _ { j } / x _ { i } ) . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\left ( \\partial ^ { p - 1 } ( \\sigma ) \\right ) _ { i _ 0 , \\dots , i _ p } = \\sum _ { k = 0 } ^ { p } ( - 1 ) ^ k \\sigma _ { i _ 0 , \\dots , \\widehat { i _ k } , \\dots , i _ p } \\restriction _ { U _ { i _ 0 , \\dots , i _ p } } \\ , . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} \\varphi _ R ( x ) = \\varphi _ R ( r ) : = R ^ 2 \\chi ( r / R ) , | x | = r . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} v ( t , x ) = \\cos \\left ( t \\sqrt { - \\Delta _ g } \\right ) u _ 0 + \\frac { \\sin ( t \\sqrt { - \\Delta _ g } ) } { \\sqrt { - \\Delta _ g } } \\ , u _ 1 + \\int _ 0 ^ t \\frac { \\sin \\bigr ( ( t - s ) \\sqrt { - \\Delta _ g } \\bigr ) } { \\sqrt { - \\Delta _ g } } \\ , f ( s ) \\ , \\mathrm { d } s . \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\begin{aligned} F ( f ) ( t ) : & = \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\Gamma ^ { - 1 } ( s ) [ K ( \\Gamma ( s ) f ( s ) ) \\cdot \\nabla ] ( \\Gamma ( s ) f ( s ) ) d s , \\ t \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } A _ { K , m , \\xi } ^ { ( q ) } ( 0 ) H _ l ( \\xi ) \\ , d \\xi & = \\frac { m } { ( m - q ) ( n - 1 ) } \\int _ { S ^ { n - 1 } } I _ q \\left ( \\rho _ K ^ { m - q } \\right ) ( \\xi ) H _ l ( \\xi ) \\ , d \\xi \\\\ & = \\frac { m } { ( m - q ) ( n - 1 ) } \\int _ { S ^ { n - 1 } } \\rho _ K ^ { m - q } ( \\xi ) I _ q \\left ( H _ l \\right ) ( \\xi ) \\ , d \\xi \\\\ & = \\frac { m } { ( m - q ) ( n - 1 ) } \\lambda _ l ( q ) \\int _ { S ^ { n - 1 } } \\rho _ K ^ { m - q } ( \\xi ) H _ l ( \\xi ) \\ , d \\xi . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\mathcal { Z } _ N = \\int _ { D ^ N } \\left | \\Delta _ N ( z ) \\right | ^ 2 \\prod _ { i = 1 } ^ N w ( z _ i ) \\d { A } ( z _ i ) = N ! \\prod _ { j = 0 } ^ { N - 1 } \\tilde { h } _ j \\ . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} E ( \\xi _ t ^ 2 ( O ) ) = F _ t ( O ) \\leq \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { t G _ \\lambda } ( O , y ) \\frac { K ( y ) } { \\inf _ { x \\in \\mathbb { Z } ^ d } K ( x ) } = \\frac { K ( O ) } { \\inf _ { x \\in \\mathbb { Z } ^ d } K ( x ) } \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\lim _ { m \\rightarrow \\infty } H ^ 2 ( p _ m ) = \\sup H ^ 2 = \\bar H ^ 2 , \\lim _ { m \\rightarrow \\infty } | \\nabla H ^ 2 ( p _ m ) | = 0 , \\\\ & 0 \\geq \\lim _ { m \\rightarrow \\infty } | \\nabla ^ { \\perp } \\vec H | ^ 2 ( p _ m ) - \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 + \\dfrac 1 2 ( \\bar H ^ 2 - S ) ( \\bar H ^ 2 - 3 S + 2 ) . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} { { \\hat \\mu } _ { } ^ { ( j ) } } = \\max \\left \\{ { { { \\tilde \\mu } _ j } , b } \\right \\} , \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} P _ \\sharp { \\rm O p } _ h ( \\chi ) = { \\rm O p } _ h ( \\chi ) \\widetilde P _ \\sharp + { \\cal O } ( h ^ \\infty ) \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} & \\frac { 1 } { \\sqrt { n } } \\Big ( \\sum _ { k = 2 } ^ r \\frac { \\tilde \\lambda _ k } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ k } + \\frac { \\tilde \\lambda _ 1 } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ 2 } \\Big ) \\leq \\frac { 2 \\tilde x } { z } \\sum _ { k = 1 } ^ r \\frac { \\lambda _ k } { \\tilde \\lambda _ 1 - \\lambda _ k } + \\frac { 1 } { \\sqrt { n } } = \\frac { 2 } { z } + \\frac { 1 } { \\sqrt { n } } . \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} p + q + r = n ; p = 0 \\ , \\Rightarrow \\ , q = 1 ; q = 0 \\ , \\Rightarrow \\ , p = 1 . \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} \\overline { Q } _ { k , a } ( x ; q ) & = \\sum _ { i = 1 } ^ a ( x q ) ^ i \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) \\right ] \\\\ & \\quad + \\sum _ { i = 0 } ^ { a - 1 } ( x q ) ^ i \\left [ \\sum _ { h = 1 } ^ { k - i } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) + \\sum _ { h = 0 } ^ { k - i - 1 } ( x q ^ 2 ) ^ h \\overline { Q } _ { k , k - h } ( x q ^ 2 ; q ) \\right ] . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} D ( t ) = \\frac { 1 } { C } \\parallel \\nabla v ( t ) \\parallel ^ 2 _ { L ^ 2 } + \\gamma \\parallel w _ { \\nu _ { \\Lambda } } ( t ) \\parallel ^ 2 _ { L ^ 2 ( \\Gamma _ c ) } \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} ( t _ i ^ p - \\alpha t ^ p ) ^ { \\frac 1 p } > t _ i - c _ 0 t ^ p > \\varrho _ 0 / 2 \\mbox { \\ f o r $ i = m + 1 , \\ldots , k $ } . \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} g _ { 2 } = p ^ { 2 } \\tilde { f } | V ( p ^ { 2 } ) - \\overline { a ( p ) } \\tilde { f } | V ( p ) + \\frac { \\overline { \\chi ( p ) } } { p } \\tilde { f } \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} z ( z - A ) ^ { - 1 } = - \\int _ 0 ^ \\infty z e ^ { t z } S _ t d t = - \\int _ { \\Gamma _ { \\beta _ p } ^ + } z e ^ { u z } S _ u d u . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} B _ t = X _ t - \\int _ 0 ^ t b ( s , X _ s ) d s - x \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - \\log b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 4 ( n ) } > \\varepsilon \\right ) = - \\lceil \\ell _ 4 \\varepsilon \\rceil ; \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b _ \\omega ^ 1 ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\dot { y } & = A _ { y y } \\Delta y + A _ { y z } \\Delta z - \\sigma _ y \\Delta y + \\sigma _ x \\Delta x + B _ { y h } \\Delta u _ h + d _ y - d _ b \\\\ \\Delta \\dot { z } & = A _ { z y } \\Delta y + A _ { z z } \\Delta z + B _ { z h } \\Delta u _ h + d _ z \\\\ \\Delta \\dot { x } & = \\sigma _ y \\Delta y - ( \\sigma _ x + a _ { x x } ) \\Delta x + d _ x + d _ b \\end{aligned} \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} & \\sup _ { ( x , t ) \\in \\Omega ^ { ( j ) } , \\ , \\bar { t } - 2 ^ { \\frac { j } { 1 0 0 } } \\leq t \\leq \\bar { t } } | f _ \\alpha ^ { ( j ) } ( x , t ) | \\\\ & \\leq \\max \\bigg \\{ \\sup _ { ( x , t ) \\in \\partial \\Omega ^ { ( j ) } , \\ , \\bar { t } - 2 ^ { \\frac { j } { 1 0 0 } } \\leq t \\leq \\bar { t } } | f _ \\alpha ^ { ( j ) } ( x , t ) | , \\sup _ { ( x , t ) \\in \\Omega ^ { ( j ) } , \\ , t = \\bar { t } - 2 ^ { \\frac { j } { 1 0 0 } } } | f _ \\alpha ^ { ( j ) } ( x , t ) | \\bigg \\} \\\\ & \\leq C \\ , 2 ^ { - \\frac { j } { 4 } } \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} \\tilde { \\mu } ( A _ { n , \\alpha } ) = \\sum _ { \\beta \\in \\mathbb N ^ m , \\theta ' ( \\beta , \\alpha ) } \\tilde { \\mu } ( D _ { n , \\beta } ) \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} & p _ { - + } = \\pi ( \\{ ( x , y ) | x < 0 , y > 0 \\} ) , p _ { 0 + } = \\pi ( \\{ ( 0 , y ) | y > 0 \\} ) , p _ { + + } = \\pi ( \\{ ( x , y ) | x > 0 , y > 0 \\} ) \\\\ & p _ { - 0 } \\ , \\ , = \\pi ( \\{ ( x , 0 ) | x < 0 \\} ) , \\ , \\ , \\ , \\ , p _ { 0 0 } \\ , \\ , = \\pi ( \\{ ( 0 , 0 ) \\} ) , \\ , \\ , \\ , \\ , p _ { + 0 } \\ , \\ , = \\pi ( \\{ ( x , 0 ) | x > 0 \\} ) , \\\\ & \\ , p _ { - - } = \\pi ( \\{ ( x , y ) | x < 0 , y < 0 \\} ) , p _ { 0 - } = \\pi ( \\{ ( 0 , y ) | y < 0 \\} ) , p _ { + - } = \\pi ( \\{ ( x , y ) | x > 0 , y < 0 \\} ) . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} H = \\left ( \\begin{matrix} \\nu & \\sigma ( \\nu ) & \\cdots & \\sigma ^ { n - \\delta } ( \\nu ) \\\\ \\sigma ( \\nu ) & \\sigma ^ 2 ( \\nu ) & \\cdots & \\sigma ^ { n - \\delta + 1 } ( \\nu ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\sigma ^ { n - 1 } ( \\nu ) & \\nu & \\cdots & \\sigma ^ { n - \\delta - 1 } ( \\nu ) \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\Big ( \\log ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } \\Big ) ^ k = \\sum _ { n = k } ^ \\infty S _ { 1 , \\lambda } ( n , k ) \\frac { t ^ n } { n ! } , \\ , \\ , ( \\lambda \\in \\mathbb { R } ) . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} h _ n ( t _ 1 , x _ 1 , . . . , t _ n , x _ n , t , x ) = \\frac { 1 } { n ! } \\prod _ { i = 1 } ^ n p _ D ( t _ { \\tau ( i + 1 ) } - t _ { \\tau ( i ) } , x _ { \\tau ( i + 1 ) } , x _ { \\tau ( i ) } ) ( \\mathcal { G } u _ 0 ) _ { t _ { \\tau ( 1 ) } } ( x _ { \\tau ( 1 ) } ) , \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\mbox { t h e r e g u l a r i z e d e n e r g y } e _ 0 : = \\int _ S \\theta _ 0 ( z ) \\Psi _ 0 ( z ) d z = \\int _ { S } \\int _ { S } \\theta _ 0 ( z ) \\theta _ 0 ( \\xi ) \\Gamma ( z - \\xi ) d \\xi d z . \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} \\lim _ { { \\varepsilon } \\to 0 } \\sup _ { x \\in K } \\left | { \\varepsilon } \\log { \\mathbb { E } } \\left ( \\exp ( - { \\varepsilon } ^ { - 1 } h ( Y ^ { \\varepsilon } _ x ) ) \\right ) + \\inf _ { \\varphi \\in C ( [ 0 , T ] : E ) } \\{ h ( \\varphi ) + \\tilde { I } _ { x } ( \\varphi ) \\} \\right | = 0 . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} R _ { S ' } ( \\lambda ) = \\begin{pmatrix} 0 & 1 / \\lambda ^ 2 \\\\ - 1 & - 2 / \\lambda \\end{pmatrix} , \\det ( R _ { S ' } ( \\lambda ) - \\lambda I ) = \\frac { ( \\lambda ^ 2 + 1 ) ^ 2 } { \\lambda ^ 2 } , \\sigma ( R _ { S ' } ( \\lambda ) ) = \\{ i , i , - i , - i \\} ; \\\\ R _ { S ' } ( i ) = \\begin{pmatrix} 0 & - 1 \\\\ - 1 & 2 i \\end{pmatrix} , \\det ( R _ { S ' } ( i ) - \\lambda I ) = ( \\lambda - i ) ^ 2 , \\sigma ( R _ { S ' } ( i ) ) = \\{ i , i \\} ; \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} g ( 0 ) & = 1 , \\\\ g ( \\alpha + 1 ) & = ( \\alpha + 1 ) \\cdot 2 , \\\\ g ( \\lambda ) & = \\textstyle \\sup _ { \\alpha < \\lambda } g ( \\alpha ) \\quad . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} F ( C , y ) = \\partial h _ C ( y ) = \\{ z \\in \\R ^ n : h _ C ( x ) \\geq h _ C ( y ) + \\langle z , x - y \\rangle \\} . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} ( K - s _ i K s _ i ) ( z ) = 2 P _ { i , c ^ \\dagger } ( z ) ^ { \\dagger , - 1 } ( K _ i ^ { 1 , - 1 } + K _ i ^ { - 1 , 1 } ) P _ { i , c } ( z ) ^ { - 1 } + z _ 1 ^ { 1 - 2 | c _ i | } R _ i ( z ) \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} u _ { \\varepsilon _ { n } } \\rightarrow u _ { * } = e ^ { i \\phi } \\prod _ { j = 1 } ^ { N } ( \\dfrac { z - b _ { j } } { | z - b _ { j } | } ) ^ { d _ j } \\ , \\ , \\ , \\ , i n \\ , C ^ { 1 , \\alpha } _ { l o c } \\left ( \\overline { G } \\setminus \\left \\lbrace b _ { 1 } , . . . , b _ { N } \\right \\rbrace \\right ) \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} { x _ { T } } & = { x _ { T - 1 } } + { A } _ { \\cdot , T } \\epsilon _ { T - 1 } = { x _ { T - 2 } } + { A } _ { \\cdot , T - 1 } \\epsilon _ { T - 2 } + { A } _ { \\cdot , T } \\epsilon _ { T - 1 } \\\\ & \\cdots \\cdots = { A } _ { \\cdot , 1 } \\epsilon _ { 0 } + . . . . + { A } _ { \\cdot , T } \\epsilon _ { T - 1 } = A { z } \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} & \\left < L x ^ i , y ^ \\star \\right > _ { F , F ^ \\star } = \\left < x ^ i , L ^ \\star y ^ \\star \\right > _ { E , E ^ \\star } = \\left < L ^ { \\star } y ^ { \\star } , J _ E ( x ^ i ) \\right > _ { E ^ \\star , E ^ { \\star \\star } } \\\\ & \\to \\left < L ^ { \\star } y ^ \\star , x ^ { \\star \\star } \\right > _ { E ^ \\star , E ^ { \\star \\star } } = \\left < y ^ \\star , L ^ { \\star \\star } x ^ { \\star \\star } \\right > _ { F ^ \\star , F ^ { \\star \\star } } . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\| \\hat u _ j - u _ j \\| ^ 2 = \\sum _ { k \\geq 1 } \\langle \\hat u _ j - u _ j , u _ k \\rangle ^ 2 = \\sum _ { k \\neq j } \\langle \\hat u _ j , u _ k \\rangle ^ 2 + ( 1 - \\langle \\hat u _ j , u _ j \\rangle ) ^ 2 . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} & \\norm { x _ 1 - x _ 0 } ^ 2 + \\norm { x _ 2 - x _ 0 } ^ 2 - 2 \\langle T _ 1 ( x _ 1 - x _ 0 ) , T _ 2 ( x _ 2 - x _ 0 ) \\rangle = \\norm { u ( x _ 1 ) - u ( x _ 2 ) } ^ 2 \\leq \\\\ & \\leq \\norm { x _ 1 - x _ 2 } ^ 2 = \\norm { x _ 1 - x _ 0 } ^ 2 + \\norm { x _ 2 - x _ 0 } ^ 2 - 2 \\langle x _ 1 - x _ 0 , x _ 2 - x _ 0 \\rangle . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} W _ k = \\int _ { 0 } ^ { \\infty } r ^ { 2 k + 1 + n } \\exp ( - \\alpha r ^ { 2 m } ) = \\frac { 1 } { 2 m } \\alpha ^ { \\tfrac { 2 m - 2 k - n - 3 } { 2 m } } \\cdot \\Gamma \\left ( \\frac { 2 k + 2 + n } { 2 m } \\right ) . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\widetilde { \\mathbf { S } } _ { n } ( z ) & = \\mathbf { C } _ { n } ^ { \\prime } ( z - 1 / 4 ) \\\\ \\widetilde { \\mathbf { C } } _ { n } ( z ) & = \\mathbf { S } _ { n } ^ { \\prime } ( z + 1 / 4 ) , \\\\ \\widetilde { \\mathbf { S } } _ { n } ^ { \\prime } ( z ) & = \\mathbf { C } _ { n } ( z - 1 / 4 ) , \\\\ \\widetilde { \\mathbf { C } } _ { n } ^ { \\prime } ( z ) & = \\mathbf { S } _ { n } ( z + 1 / 4 ) . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} Q T _ { 2 } = \\{ \\ , \\{ f , Q f ' \\} : \\ , \\{ f , f ' \\} \\in T _ { 2 } \\ , \\} = T _ { 2 } . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\hat h = 2 ^ { \\hat \\nu _ \\delta - 2 } + 2 = 2 ^ { \\left \\lceil \\log _ 2 \\left ( - 6 + 2 \\sqrt { 1 + \\frac 2 \\delta } \\right ) \\right \\rceil - 2 } + 2 \\le - 1 + \\sqrt { 1 + \\frac { 2 } \\delta } . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P _ n = \\lim _ { n \\to \\infty } P _ { n , q } = e ^ { - 1 } , \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\int _ { \\Omega ' } d d ^ c u _ { j , k } = \\int _ { \\Omega ' } d d ^ c u _ { j } = 1 . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} & - q _ * \\frac { N } { 2 } M _ { 0 , 1 } U ( | x | ) F _ 0 ( | x | ) + \\sum ^ N _ { i = 1 } M _ { 1 , i } U _ 1 ( | x | ) Q _ { 1 , i } \\left ( \\frac { x } { | x | } \\right ) \\\\ & = \\frac { q _ * ^ 2 } { 2 ^ N \\Gamma ( N / 2 ) } \\biggr [ - \\frac { N } { c _ * ^ 2 } M ( \\varphi ) + \\frac { 1 } { c _ 1 ^ 2 } \\biggr ( \\frac { x } { | x | } \\cdot \\Xi ( \\varphi ) \\biggr ) U _ 1 ( | x | ) \\biggr ] . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } i \\partial _ t u - ( - \\Delta ) ^ s u & = & - | u | ^ { \\alpha } u , [ 0 , + \\infty ) \\times \\R ^ d , \\\\ u ( 0 ) & = & u _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} - h ' ( s ) & = K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u - 1 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 1 } ) + O ( s ^ { - 1 } ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u - 1 } , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\gamma _ 1 : = \\nu ( \\phi _ 1 ) > \\nu _ 1 ( \\phi _ 1 ) . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } | I _ { k } | \\leq C _ { V , l } | I _ { 1 } | \\cdot \\left ( \\sum _ { k = 1 } ^ { m } ( C _ { V , l } | I _ { 1 } | ) ^ { \\alpha ^ { k - 1 } - 1 } \\right ) \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} H ( x , p ) = | p | _ x - V ( x ) . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} ( ( h _ i ) _ { 1 \\leq i \\leq n } \\leftarrow ( \\sigma _ t ) _ { 1 \\leq t \\leq q } ) \\leftarrow g = ( ( h _ i ) _ { 1 \\leq i \\leq n } \\leftarrow g ) \\leftarrow ( ( \\sigma _ t ) _ { 1 \\leq t \\leq q } \\leftarrow g ) \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Lambda _ n ( \\theta ) } { v ( n ) } & = \\lim _ { n \\to \\infty } \\frac { \\log ( 1 + \\pi ( n ) ( \\mathrm { e } ^ \\theta - 1 ) ) } { \\pi ( n ) } \\lim _ { n \\to \\infty } \\frac { n - a _ n } { n } \\\\ & = \\lim _ { n \\to \\infty } \\frac { \\mathrm { e } ^ \\theta - 1 } { 1 + \\pi ( n ) ( \\mathrm { e } ^ \\theta - 1 ) } \\\\ & = \\mathrm { e } ^ { \\theta } - 1 , \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\sigma ^ { a b } n _ { a b } = \\frac { 1 } { r } + r ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 1 ) } + r ^ 2 ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 2 ) } + r ^ 3 ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 3 ) } + O ( r ^ 4 ) \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\mathrm { m o d } _ q ( d \\varphi ) = \\lim _ { \\tau \\to \\infty } \\sum _ { [ \\{ \\alpha , \\beta ; \\epsilon \\} ] \\in \\vec { \\triangle } ^ { \\xi ^ + _ i } _ { \\eta ^ + _ i } } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho _ \\tau ) + \\ell _ { \\beta } ( \\rho _ \\tau ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} & D _ { i , 0 } \\cap { \\mathfrak { D } } ( f , \\sigma ) \\\\ = & \\{ ( x _ 1 , \\dots , x _ n ) \\mid 0 \\le x _ 1 \\le \\dots \\le x _ n < 1 , x _ i = 0 , \\sum _ { k = 1 } ^ n m _ { j , k } x _ { \\sigma ( k ) } \\in \\mathbb { Z } \\ , ( { } ^ \\forall j ) \\} \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\overline { \\epsilon _ i } = \\left \\{ \\begin{array} { l l } \\pm 1 , \\ \\ d ^ i \\ \\ \\epsilon \\\\ 0 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} | K ( y _ { 0 } , \\cdots , y _ { j } , \\cdots , y _ { m } ) - K ( y _ { 0 } , \\cdots , y ' _ { j } , \\cdots , y _ { m } ) | \\leq \\frac { A | y _ { j } - y _ { j ' } | ^ { \\epsilon } } { ( \\sum _ { k , l = 0 } ^ { m } | y _ { k } - y _ { l } | ) ^ { m n + \\epsilon } } , \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\P ( C '' _ { k , j } ) > \\P ( C '' _ { k , j } \\cap ( A '' ) ^ c ) = 0 \\P ( C '' _ { k , j } ) > \\P ( C '' _ { k , j } \\cap A '' ) = 0 , \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} K = \\frac { 1 } { r ^ 2 } + \\frac { 1 } { 6 } \\tilde \\nabla ^ a \\tilde \\nabla ^ b \\alpha _ { a b } + O ( r ) . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { k _ { \\gamma , \\Sigma } ^ { L , s } } { \\sqrt { L } } = \\frac { ( - \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } + \\overline { p } \\dot { \\gamma } _ 3 ) \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) } { | \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 | ^ 3 } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) = 0 ~ ~ a n d ~ ~ \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) \\neq 0 . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} P ( Y _ j = y _ j ) & = \\sum _ { \\boldsymbol { y } _ { - j } } P ( \\boldsymbol { Y } = \\boldsymbol { y } ) , \\\\ & = \\sum _ { n \\geq y _ j } P ( \\vert \\boldsymbol { Y } \\vert = n ) \\sum _ { \\boldsymbol { y } _ { - j } } P _ { \\vert \\boldsymbol { Y } \\vert = n } ( \\boldsymbol { Y } = \\boldsymbol { y } ) , \\\\ P ( Y _ j = y _ j ) & = \\sum _ { n \\geq y _ j } P ( \\vert \\boldsymbol { Y } \\vert = n ) P _ { \\vert \\boldsymbol { Y } \\vert = n } ( Y _ j = y _ j ) , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} P _ D ( x , z ) = \\int _ D G ( x , y ) \\nu ( z - y ) d y , x \\in D , \\ , z \\in \\overline { D } ^ c . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} T = T _ { \\rm o p } + T _ { \\rm m u l } , \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\theta _ { Q } ( z ) = E ( z ) + C ( z ) \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} R _ r = \\sum _ { s \\neq r } \\frac { 1 } { \\mu _ s - \\mu _ r } Q _ s . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} X ( t ) = \\sum _ { i = 0 } ^ 3 V _ i ( t ) + \\epsilon _ 1 ( \\| D v \\| ^ 2 _ { L ^ 2 } + \\| D v _ t \\| ^ 2 _ { L ^ 2 } + \\| D v _ { t t } \\| ^ 2 _ { L ^ 2 } ) \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\Omega _ r \\left [ \\frac { \\mu } { 1 + t } u _ { t } \\right ] ( t , x ) & = \\frac { \\mu } { 1 + t } \\bigg ( \\frac { \\partial } { \\partial t } \\bigg ) \\ , \\Omega _ r [ u ] ( t , x ) , \\\\ \\Omega _ r \\left [ \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } u \\right ] ( t , x ) & = \\frac { \\nu ^ 2 } { ( 1 + t ) ^ 2 } \\ , \\Omega _ r [ u ] ( t , x ) . \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\textrm { r a n k } ( D B , \\widetilde A D B , \\dots , \\widetilde A ^ { n - 2 } D B ) = \\textrm { r a n k } ( D B , D A B , \\dots , D A ^ { n - 2 } B ) = n - 1 . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} ( - 1 ) ^ { n + 1 } & \\sum _ { r = 1 } ^ { n + 1 } \\frac { ( - 4 ) ^ r r ! ^ 2 } { ( 2 r ) ! } { { - ( s + t - ( - 1 ) ^ a \\epsilon \\nu + 1 ) / 2 } \\choose { r - 1 } } { { - ( s + t + ( - 1 ) ^ a \\epsilon \\nu ) / 2 } \\choose { r } } { { s + t - 1 } \\choose n + 1 - r } , \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} B ( x , y ) = \\frac { \\Gamma ( x ) \\Gamma ( y ) } { \\Gamma ( x + y ) } . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} P _ { \\vert \\boldsymbol { Y } _ { \\mathcal { I } } \\vert = n } ( \\boldsymbol { Y } _ { \\mathcal { I } } = \\boldsymbol { y } _ { \\mathcal { I } } ) = \\frac { 1 } { a _ { \\vert \\boldsymbol { \\theta } _ { \\mathcal { I } } \\vert } ( n ) } \\prod _ { j \\in \\mathcal { I } } a _ { \\theta _ j } ( y _ j ) \\mathbb { 1 } _ { \\Delta _ n } ( \\boldsymbol { y } _ { \\mathcal { I } } ) , \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} y _ { m + 1 } = y _ m ( 1 - { \\rm h } _ m y _ m ^ 2 ) + u _ { m + 1 } , m \\in \\mathbb N , y _ 0 = x _ { \\mathcal N ( \\omega ) } . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\hat { \\eta } _ k ^ ( t ) = \\mathrm { a r g } \\min _ { \\eta } \\prod _ { { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } } \\left [ m ^ { 2 } + \\left ( \\eta _ l ( t ) - \\eta \\right ) ^ { 2 } \\right ] \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} B ^ { ( \\theta ) } f & = \\sum _ { I \\in \\mathcal { D } _ { \\leq n } } \\frac { \\langle f , h _ I \\rangle } { \\| h _ I \\| _ 2 ^ 2 } b _ I ^ { ( \\theta ) } , & & f \\in W _ n , \\\\ A ^ { ( \\theta ) } f & = \\sum _ { I \\in \\mathcal { D } _ { \\leq n } } \\frac { \\langle f , b _ I ^ { ( \\theta ) } \\rangle } { \\| b _ I ^ { ( \\theta ) } \\| _ 2 ^ 2 } h _ I , & & f \\in W _ N \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} U _ { 2 k + 1 , 2 a } ( x ; q ) = U _ { 2 k + 1 , 2 a + 1 } ( x ; q ) - U ^ { 2 a + 1 } _ { 2 k + 1 , 2 a + 1 } ( x ; q ) \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} J ^ { K \\textnormal { t h } } ( q , Q ) = \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\left ( q P ^ { - 1 } ; q \\right ) _ d ^ { N + 1 } } \\in K \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } ( q ) [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} r _ \\varepsilon = \\sup \\left \\{ r \\in ( 0 , \\rho _ \\varepsilon ] | \\bar { u } _ \\varepsilon - B _ \\varepsilon | \\le \\frac { 1 } { \\gamma _ \\varepsilon } B _ { x _ \\varepsilon } ( r ) \\right \\} \\ , . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} R = R \\left ( G G \\right ) + R \\left ( A G \\right ) + R \\left ( A A \\textsubscript { d i f f } \\right ) + R \\left ( A A \\textsubscript { d i f f } \\right ) . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} 1 0 ^ p = * 0 0 \\ ! * . \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} \\varphi ( z _ 1 , \\dots , z _ d ) = \\left ( \\frac { z _ 1 - t } { t z _ 1 - 1 } , \\frac { ( 1 - t ^ 2 ) ^ { 1 / 2 } } { t z _ 1 - 1 } z _ 2 , \\dots , \\frac { ( 1 - t ^ 2 ) ^ { 1 / 2 } } { t z _ 1 - 1 } z _ d \\right ) . \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\lambda _ d = c _ 0 + 2 \\sum _ { k = 1 } ^ { \\frac { n - 1 } { 2 } } c _ { k } \\cos \\left ( \\frac { 2 \\pi d k } { n } \\right ) , \\textrm { f o r } d \\in \\left \\{ 1 , \\ 2 , \\ldots , \\ n - 1 \\right \\} \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} \\partial _ r \\sigma _ { a b } = - 2 l _ { a b } \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\hat { \\tilde { r } } _ \\mathrm { u p a } = \\Big ( 1 - \\frac { \\tau _ \\mathrm { a } + \\tau _ \\mathrm { p } } { \\tau _ \\mathrm { f } } \\Big ) \\frac { 1 } { N + \\mu } \\sum _ { k \\in \\mathcal { K } } \\sum _ { n \\in \\mathcal { N } } \\log _ 2 ( 1 + \\hat { \\tilde { \\gamma } } _ { k , n } ) , \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} \\eta ' \\left ( u ^ { \\varepsilon } \\right ) \\lambda f \\left ( x , u ^ { \\varepsilon } \\right ) _ { x } - \\lambda \\varepsilon \\eta ' \\left ( u ^ { \\varepsilon } \\right ) u ^ { \\varepsilon } _ { x x } + \\eta ' \\left ( u ^ { \\varepsilon } \\right ) \\left [ u ^ { \\varepsilon } - w \\right ] = 0 . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} \\varphi _ { y } ( y ) = 0 , \\nabla \\varphi _ { y } ( y ) = 0 ; \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r e ( V _ i ) < \\varepsilon n ^ 2 . \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\begin{aligned} E _ i \\{ S _ k ( t ) \\} & = \\sum _ { j = 1 } ^ t \\boldsymbol { e } _ k ^ \\top \\boldsymbol { W } ^ { t + 1 - j } E _ i \\{ \\boldsymbol { \\eta } ( j ) \\} \\\\ & = \\mu _ { \\eta , i } \\sum _ { j = 1 } ^ t \\boldsymbol { e } _ k ^ \\top \\boldsymbol { W } ^ { t + 1 - j } \\boldsymbol { 1 } \\\\ & = \\mu _ { \\eta , i } t , \\end{aligned} \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} \\widetilde { H } ( x , \\Lambda ) = \\ln \\int _ { \\mathcal { Y } ^ d } e ^ { \\frac { \\left | \\Lambda + \\nabla _ y ( x , y ) \\widetilde { w } \\right | ^ 2 } { 2 } + V ( x , y ) } d y \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( m ) = \\sigma \\left ( W _ { m n } ( [ \\omega _ 1 , \\omega _ 2 ] ) \\times W _ { m n } ( [ \\omega _ 3 , \\omega _ 4 ] ) \\times W _ { m n } ( [ \\phi _ 1 , \\phi _ 2 ] ) \\right ) \\ ; , \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} & F ^ * : S { \\frak C } ( E ) = C _ 0 ( E , C l ( E ) ) \\to S { \\frak C } ( E ' ) \\\\ & F ^ * ( h ) ( v ) = \\bar { l } ^ { - 1 } ( h ( F ( v ) ) ) . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} [ m ] ^ 2 \\sum _ { n = 1 } ^ { \\frac { m - 1 } { 2 } } [ 3 n ] \\frac { ( 1 - q ) ^ 2 ( q ; q ^ 2 ) _ { n } ( q ^ { m + 2 } ; q ^ 2 ) _ { n - 1 } ^ 2 q ^ { m - { n + 1 \\choose 2 } - \\frac { ( 2 n + 1 ) ( m - 1 ) } { 2 } } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } , \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } D ( x _ n , x ^ * ) = 0 . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\alpha ( u , v , w ) = \\sum _ { ( u , v , w ) \\ ; \\mathrm { c y c l i c } } \\langle A _ u v , w \\rangle . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\P ( \\{ X _ n \\notin [ k / n , ( k + 1 ) / n ) \\} \\cap \\{ k / n \\leq X < ( k + 1 ) / n \\} ) = 0 , \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} L ( s ) = \\exp \\left ( ( \\ln s ) ^ { 1 / 3 } ( \\cos ( \\ln s ) ) ^ { 1 / 3 } \\right ) . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\beta _ { 1 , j } : = \\overline { c } _ j + 3 \\alpha _ j c ' ( M + 2 + 2 | p - 2 | ) / ( 4 \\varepsilon ) . \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} \\begin{aligned} { { \\textbf { { I } } } _ { i { p _ 1 } } } \\triangleq \\left [ \\begin{matrix} { { \\bf { A } } ^ { - 1 } } & { \\bf { 0 } } \\\\ { \\bf { 0 } } & { \\bf { 0 } } \\end{matrix} \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ 2 H ^ i ( B , \\Z ) \\otimes H ^ { 4 - i } ( S , \\Z ) \\xrightarrow { \\sim } H ^ 4 ( X , \\Z ) , \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} m \\tau T _ { e _ f } ( U _ { f '' } ) = \\frac { 1 } { | F | } \\sum _ { f ' } U _ f ^ { - 1 } U _ { f ' } ^ { - 1 } U _ f U _ { f '' } U _ { f ' } . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} ( f ' , f ) = \\| h \\| ^ 2 \\geq 0 , \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} u ^ { \\lambda } ( x , t ) = \\lambda u ( \\lambda x , \\lambda ^ 2 t ) , \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} \\delta _ 0 ^ s ( \\ell ) + \\delta _ 1 ^ s ( \\ell ) + \\frac { 1 } { 2 ( \\ell ^ 2 - 1 ) } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} M ( 1 - \\delta ) & = \\sum _ g m _ { g } ( g - \\delta ) \\\\ & = \\sum _ { g \\ne 1 } ( m _ { g } - m _ { 1 } ) ( g - \\delta ) . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} h = \\sum _ { i \\in I } h _ { f _ i } \\ ; h _ f = \\ ; \\ ; X _ f : = ( \\Z \\Gamma / f \\Z \\Gamma ) ^ { \\ast } \\ ; . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} d & = \\inf \\left \\{ \\delta \\ , \\Big | \\ , \\lim _ { \\epsilon \\to 0 } A _ \\epsilon ^ { n , \\delta } ( S ) = + \\infty \\right \\} \\\\ \\bigg ( d & = \\sup \\left \\{ \\delta \\ , \\Big | \\ , \\lim _ { \\epsilon \\to 0 } A _ \\epsilon ^ { n , \\delta } ( S ) = 0 \\right \\} \\bigg ) . \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} d ( a _ 0 \\otimes \\ldots \\otimes a _ n \\otimes m ) = { } & a _ 1 \\otimes \\ldots \\otimes a _ n \\otimes m a _ 0 \\\\ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { i } a _ 0 \\otimes \\ldots \\otimes a _ { i - 2 } \\otimes a _ { i - 1 } a _ i \\otimes a _ { i + 1 } \\otimes \\ldots \\otimes a _ n \\otimes m \\\\ & - ( - 1 ) ^ n a _ 0 \\otimes a _ 1 \\otimes \\ldots \\otimes a _ { n - 1 } \\otimes a _ { n } m . \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ N \\textbf { g } \\left ( e ^ { \\tau _ 2 / z } T _ j + \\sum _ { \\substack { d \\in H _ 2 ( X ; \\mathbb { Z } ) - \\{ 0 \\} \\\\ l \\geq 0 } } \\sum _ k \\frac { 1 } { l ! } e ^ { \\tau _ 2 ( d ) } \\left \\langle \\frac { e ^ { \\tau _ 2 / z } T _ j } { - z + \\psi } , T _ k , \\tau ' , \\dots , \\tau ' \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 2 , d } T ^ k , T _ a \\right ) T ^ j \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} p = 0 \\ , \\Rightarrow \\ , q = 1 ; q = 0 \\ , \\Rightarrow \\ , p = 1 \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\beta _ \\varepsilon : = \\frac { \\tilde { N } _ \\varepsilon } { \\gamma _ \\varepsilon ^ 2 } \\lim _ { \\varepsilon \\to 0 } \\beta _ \\varepsilon = \\beta _ 0 \\in [ 0 , 1 ] \\ , , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} E _ \\Omega = \\{ ( c , d ) \\in v \\mathcal { C } \\times v \\mathcal { D } : c \\in M \\Gamma ( d ) \\} \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } u _ { 1 } - \\partial _ { x x } u _ { 1 } = \\mu _ { 1 } ( x ) \\left ( a _ { 1 } - u _ { 1 } \\right ) u _ { 1 } - k \\omega ( x ) u _ { 1 } u _ { 2 } \\\\ \\partial _ { t } u _ { 2 } - d \\partial _ { x x } u _ { 2 } = \\mu _ { 2 } ( x ) \\left ( a _ { 2 } - u _ { 2 } \\right ) u _ { 2 } - \\alpha k \\omega ( x ) u _ { 1 } u _ { 2 } \\end{cases} \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} ( L + X _ t ) ^ n X _ t = \\sum _ { A _ i = L , X _ t } A _ 1 A _ 2 \\cdots A _ n X _ t . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} V = \\sqrt { 1 + r ^ 2 } + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\mu ^ { \\Gamma } = \\frac { 1 } { [ \\Gamma : \\Gamma _ { \\mu } ] } \\sum \\limits _ { \\gamma \\in \\Gamma / \\Gamma _ { \\mu } } \\gamma ( \\mu ) \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} ( a ; q ) _ n = ( - a ) ^ n q ^ { n ( n - 1 ) / 2 } ( q ^ { - n + 1 } / a ; q ) _ n . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} = \\overline { } \\frac { ~ _ { } ( V ^ n ) } { } . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\det J & = ( \\det D ) \\det { ( A - B D ^ { - 1 } C ) } \\\\ & = e ^ { ( n - 1 ) t } \\left ( \\cosh { t } - \\frac { 1 } { 2 } | \\tilde { s } ^ 2 | e ^ { t } - \\sum _ { i = 1 } ^ { n - 1 } ( - s _ i e ^ { t } ) e ^ { - t } ( s _ i e ^ { t } ) \\right ) \\\\ & = e ^ { ( n - 1 ) t } ( \\cosh { t } + \\frac { 1 } { 2 } | \\tilde { s } | ^ 2 e ^ { t } ) , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} K _ { T ^ { N + 1 } } ( \\textnormal { p t } ) = \\mathbb { Z } [ \\Lambda _ 0 ^ { \\pm 1 } , \\cdots , \\Lambda _ N ^ { \\pm 1 } ] \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} b _ n ( z ) = B _ \\Omega ( z , y _ n ) - B _ \\Omega ( y _ n , z _ 0 ) . \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\overline { u } _ 0 ( t ) & = e ^ { - t A } v _ 0 , \\\\ \\overline { u } _ \\ell ( t ) & = A ^ \\ell \\left ( \\sum _ { j = 0 } ^ \\ell \\begin{pmatrix} 2 \\ell - 1 \\\\ \\ell + j - 1 \\end{pmatrix} \\frac { ( - t A ) ^ j } { j ! } e ^ { - t A } v _ 0 + \\sum _ { k = 0 } ^ { \\ell - 1 } \\begin{pmatrix} 2 \\ell - 1 \\\\ \\ell + k \\end{pmatrix} \\frac { ( - t A ) ^ k } { k ! } e ^ { - t A } u _ 1 \\right ) . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} A _ \\gamma ^ { w , x , b } : = \\{ [ ( f _ { w } ( u ) - f _ { w } ( z ) ) ^ t ( u - z ) ^ { - s } b ^ { - t } ] / v : u , z \\in B ^ \\circ ( x , \\theta ( w , x ) ) , v ( u - z ) = \\gamma \\} , \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} g _ { ( b ) } ( z , t ) = \\frac { t ^ 2 z ^ 4 } { ( 1 - z ) ^ 2 ( 1 - t z ) ( 1 - ( 1 + t ) z ) } \\left ( \\frac { f _ t ( z , 1 ) } { 1 - t } - \\frac { t ( f ( z , 1 ) - f ( z , t ) ) } { ( 1 - t ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\mathcal { O } _ k : = \\bigcup _ { s \\in \\mathcal { Q } \\colon \\widehat { M } _ s \\leq { k } } { B ( s , { d _ s } ) } \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\mathfrak p \\\\ \\updownarrow \\\\ \\mathfrak q \\end{array} \\right ) \\leftrightarrow \\left ( \\begin{array} { c } \\mathfrak p ' \\\\ \\updownarrow \\\\ \\mathfrak q ' \\end{array} \\right ) = \\begin{array} { c } ( \\mathfrak p \\leftrightarrow \\mathfrak p ' ) \\\\ \\updownarrow \\\\ ( \\mathfrak q \\leftrightarrow \\mathfrak q ' ) . \\end{array} \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} \\bar { y } _ s ^ { t , \\bar { A } _ t ^ \\epsilon ; Z _ t \\otimes u ^ { \\epsilon ^ \\prime } , W _ t ^ \\otimes v } & : = \\phi ( X _ s ^ { t , \\bar { A } _ t ^ \\epsilon ; Z _ t \\otimes u ^ { \\epsilon ^ \\prime } , W _ t \\otimes v } ; ( Z _ t \\otimes u ^ \\epsilon ) _ s , ( W _ t \\otimes v ) _ s ) \\\\ & ~ ~ ~ - y _ s ^ { t , \\bar { A } _ t ^ \\epsilon ; Z _ t \\otimes u ^ { \\epsilon ^ \\prime } , W _ t \\otimes v } \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\int _ { - r } ^ { r } ( u ( x ) - k ) _ - \\left \\{ \\int _ \\R \\frac { ( u ( y ) - k ) _ + ^ { p - 1 } } { | x - y | ^ { 1 + s p } } \\ , d y \\right \\} d x & \\ge \\int _ { - r } ^ { 0 } k \\left \\{ \\int _ { 0 } ^ { x + r } \\frac { ( 1 - k ) ^ { p - 1 } } { ( y - x ) ^ { 1 + s p } } \\ , d y \\right \\} d x \\\\ & = \\frac { r ^ { 1 - s p } k ( 1 - k ) ^ { p - 1 } } { 1 - s p } . \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\upsilon _ \\theta ^ { \\left ( 2 \\right ) } = \\frac { 1 } { 2 } \\left [ { 1 + q \\left ( \\theta \\right ) } \\right ] \\ln \\frac { { 1 + q \\left ( \\theta \\right ) } } { { q \\left ( \\theta \\right ) } } - \\ln 2 > 0 . \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda ^ * w - \\Delta w + \\nabla _ H \\Pi & = \\delta _ { \\varepsilon , x _ 0 ' } | v | ^ { p - 2 } v ^ * \\quad \\Omega ' , \\\\ \\partial _ z \\Pi & = 0 \\quad \\Omega ' , \\\\ \\mathrm { d i v } _ H \\ , \\bar w & = 0 \\quad G ' , \\\\ \\partial _ z w | _ { \\Gamma _ u ' } = 0 , w | _ { \\Gamma _ b ' } & = 0 , w , \\Pi \\Gamma _ l ' , \\end{aligned} \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { a _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) + a _ n - \\lfloor \\kappa _ n ( x ) \\rfloor \\leq 0 ) = - J ( x ) , x \\geq 0 . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} L _ { \\tilde { J } _ { t } } = [ L _ c + L _ q L _ { \\tilde { J } _ { t + 1 } } ] [ 1 + L _ { \\psi } ] \\\\ L _ { \\tilde { J } _ { T } } = L _ { \\tilde { h } _ \\lambda } + L _ h . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} \\Delta ( f ( X , B ^ { - 1 } ( 1 + D t _ 0 ) ) & = n ^ n \\prod _ { c \\in \\overline { \\Q } : h ' ( c , t _ 0 ) = 0 } f ( c , B ^ { - 1 } ( 1 + D t _ 0 ) ) ^ { m _ c } \\\\ & = n ^ n ( B ^ { - 1 } ( 1 + D t _ 0 ) ) ^ { n - 1 } \\left ( ( B ^ { - 1 } ( 1 + D t _ 0 ) ) ^ { n - 1 } \\left ( \\frac { 1 } { n } - 1 \\right ) + 1 \\right ) \\\\ & \\equiv B ^ { 1 - n } \\mod l , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} H _ j : = \\int | T ( \\omega _ j ) - T ( \\omega _ { j - 1 } ) | f ( y _ j ) \\ldots f ( y _ n ) d y _ { j } \\ldots d y _ { n + 1 } \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} f : \\Omega \\to \\R ^ m , f ( x ) = ( f _ 1 ( x ) , \\dots , f _ m ( x ) ) . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} ( L _ X g ) ( Y , Z ) = X g ( Y , Z ) - g ( L _ X Y , Z ) - g ( Y , L _ X Z ) . \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} f ^ * ( t ) = \\inf \\{ \\lambda \\geq 0 : D _ f ( \\lambda ) \\leq t \\} . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} J F _ l ( 0 , x ) \\cdot J F _ l ( 0 , x ) ^ { * } = \\sum _ { \\mu = 1 } ^ { m _ { l } } V _ { \\mu } ( x ) \\otimes V _ { \\mu } ( x ) . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} f = ( c _ 1 x _ 1 + t _ 1 , c _ 2 x _ 2 + t _ 2 , \\dots ) . \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\int _ { \\Omega } f v _ 1 \\cdots v _ m \\ , d x = \\sum _ { j = 1 } ^ { 2 ^ m } \\int _ { \\Omega } c _ j f w _ 1 ^ { ( j ) } \\cdots w _ m ^ { ( j ) } \\ , d x \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\mathfrak p = \\begin{array} { c } ( \\mathfrak p _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak p _ k ) \\\\ \\updownarrow \\\\ ( \\mathfrak q _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak q _ \\ell ) \\end{array} \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} \\left ( \\bigotimes { } ^ { } _ { r \\in R } L _ r \\right ) _ { X ^ I } = \\bigoplus _ { \\pi : I \\rightarrow R } j ^ { [ I / R ] } _ * { j ^ { [ I / R ] } } ^ * \\left ( \\boxtimes \\left ( L _ r \\right ) _ { X ^ { \\pi ^ { - 1 } ( r ) } } \\right ) . \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\Delta _ L ^ { \\perp } V = ( \\Delta _ L f _ 1 ) J X _ 1 + ( \\Delta _ L f _ 2 ) J X _ 2 , \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} g _ { i j } ( 0 ) s _ { j k l } g _ { i j } ^ { - 1 } ( 0 ) - s _ { i k l } + s _ { i j l } - s _ { i j k } = 0 \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} f \\ \\geq \\ \\hat { f } \\ = _ { d e f } \\ \\bigvee _ { c \\in I } \\ f _ c . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} P _ N = \\psi ( Y _ N ) , \\ , N \\ge 1 \\ , . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} t ( 1 + t ^ { 2 } ) \\frac { \\partial u } { \\partial t } = \\frac { u ^ { 2 } \\Delta _ { g ( t ) } u } { 2 } & - \\frac { u ^ { 3 } } { 4 } ( R _ { g ( t ) } - t ^ { 2 } \\overline { R } ) \\\\ & + u \\left ( \\frac { 1 + 3 t ^ { 2 } } { 2 } + \\frac { t ^ { 2 } ( 1 + t ^ { 2 } ) | M | _ { g ( t ) } ^ { 2 } } { 4 } \\right ) . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} { } \\triangle ^ { 1 / 2 } u = W ' ( u ) , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} p _ 1 ( T M ) _ { S ^ 1 } - ( 2 k + 1 ) c _ 1 ( \\xi ) ^ 2 _ { S ^ 1 } = p _ 1 ( T M - \\xi ^ { \\oplus ( 2 k + 1 ) } ) _ { S ^ 1 } = n \\cdot \\pi ^ \\ast u ^ 2 , \\ \\ n > 0 . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\hat { \\phi } ( n + 1 ) + \\hat { \\phi } ( n - 1 ) + \\hat { V } ( n ) \\tilde { \\phi } ( n ) = E \\hat { \\phi } ( n ) . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} | Y _ t ^ n - Y _ t | & = | X _ t ^ n + u _ n ( t , X _ t ^ n ) - X _ t - u ( t , X _ t ) | \\\\ & \\geq | X ^ n _ t + u ( t , X ^ n _ t ) - X _ t - u ( t , X _ t ) | - | u _ n ( t , X ^ n _ t ) - u ( t , X ^ n _ t ) | \\\\ & \\geq \\frac { 1 } { 2 } | X ^ n _ t - X _ t | - \\norm { u _ n - u } _ { L ^ \\infty } \\geq \\frac { 1 } { 2 } | X ^ n _ t - X _ t | - \\bar { C } \\norm { b _ n - b } _ { L ^ { q , 1 } _ t ( L ^ p _ x ) } \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} Q ( n ) : = \\ln \\beta - \\sum _ { i = 1 } ^ { k + 1 } \\ln _ { i } \\left ( n + e ^ 1 _ { [ k ] } \\right ) + \\left ( \\prod _ { i = 0 } ^ { k } \\ln _ { i } \\left ( n + e ^ 1 _ { [ k ] } \\right ) \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} v \\mapsto \\begin{cases} v & \\mbox { i f } v \\in A _ + \\\\ - v & \\mbox { i f } v \\in A _ - \\end{cases} \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} [ D _ { c _ { j \\bullet } } , c _ { j ' \\bullet } ] & = \\delta _ { j j ' } - \\tfrac { 1 } { | J | } & [ D _ { c _ { k \\bullet } } , c _ { k ' \\bullet } ] & = \\delta _ { k k ' } - \\tfrac { 1 } { | K | } \\\\ [ D _ { e _ { j \\bullet } } , e _ { j ' \\bullet } ] & = \\delta _ { j j ' } & [ D _ { e _ { k \\bullet } } , e _ { k ' \\bullet } ] & = \\delta _ { k k ' } \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{gather*} \\{ \\Phi ^ { \\prime } _ a , \\Phi ^ { \\prime } _ b \\} = C _ { a b } ^ c \\Phi ^ { \\prime } _ c , \\\\ \\{ H , \\Phi ^ { \\prime } _ a \\} = \\lambda _ a ^ { \\prime b } \\Phi ^ { \\prime } _ b , \\end{gather*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\begin{aligned} { } \\big | \\psi ( \\tau x ' , \\tau ^ { 2 } y _ { 2 } \\cdots , \\tau ^ { k } y _ { k } ) - \\psi ( \\tau \\overline { x } ' , \\tau ^ { 2 } y _ { 2 } \\cdots , \\tau ^ { k } y _ { k } ) \\big | & \\leq \\tilde { \\delta } \\tau | x ' - \\overline { x } ' | \\omega _ { \\nabla \\psi } ( \\tau ) \\\\ & \\leq C _ { 2 } \\tilde { \\delta } \\tau \\omega _ { \\nabla \\psi } ( \\tau ) d ( p , \\overline { p } ) \\leq C _ { 2 } \\tilde { \\delta } \\tau \\omega ( \\tau ) d ( p , \\overline { p } ) \\end{aligned} \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} \\frac { R _ { 2 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 2 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = - i ( i R _ { 2 1 } - i R _ { 2 3 } ) = R _ { 2 1 } - R _ { 2 3 } = - 1 = u _ 2 ; \\\\ \\frac { R _ { 4 1 } } { i - \\omega ( 1 , 1 ) } u _ 1 + \\frac { R _ { 4 3 } } { i - \\omega ( 3 , 3 ) } u _ 3 = R _ { 4 1 } - R _ { 4 3 } = - 1 - ( - 2 ) = 1 = u _ 4 . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\psi _ { j , k } : = P ( \\varphi _ j , \\varphi _ { j + 1 } , \\dots , \\varphi _ { k } ) , \\ j \\in \\mathbb { N } , k \\geq j . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} n = 2 r + 2 . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\asymp \\sum _ { j = 1 } ^ { u _ k - 1 } j ^ { - d } \\asymp 1 . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} 1 ^ { T } x x ^ { - } 1 = ( \\max _ { i = 1 } ^ { n } x _ { i } ) ( \\max _ { j = 1 } ^ { n } ( 1 / x _ { j } ) ) , \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} | | f | | ^ 2 = & | | f _ 1 | | ^ 2 + | | f _ 2 | | ^ 2 + | | f _ 3 | | ^ 2 + 2 | | f _ 1 | | | | f _ 2 | | \\delta _ 3 \\cos { \\theta _ 3 } \\\\ & + | | f _ 2 | | | | f _ 3 | | \\delta _ 1 \\cos { \\theta _ 1 } + | | f _ 1 | | | | f _ 3 | | \\delta _ 2 \\cos { \\theta _ 2 } . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\det A _ n ( x ) = ( x + 2 k ) ( x ^ 2 - x - k ) ^ { 2 k } . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\Xi ( x , y ) & = \\eta ( x ) w _ - ( x ) \\big ( w _ - ( x ) + 2 w _ + ( y ) + ( 1 - \\eta ( x ) ) w _ - ( x ) \\big ) \\\\ & \\ge \\eta ( x ) \\left ( \\vphantom { \\big ( } \\left | w _ - ( x ) - w _ - ( y ) \\right | ^ 2 + 2 w _ - ( x ) w _ + ( y ) \\right ) . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} r _ 1 & = \\frac { - [ a ^ 2 + 2 \\sigma ^ 2 ( a + \\eta ) s - 1 ] - \\sqrt { \\Delta } } { 4 \\sigma ^ 2 } , \\\\ r _ 2 & = \\frac { - [ a ^ 2 + 2 \\sigma ^ 2 ( a + \\eta ) s - 1 ] + \\sqrt { \\Delta } } { 4 \\sigma ^ 2 } , \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\frac 1 { \\theta _ { m + 1 } } : = \\frac { 1 - r } r - \\frac 1 { \\delta _ { m + 1 } } = \\sum _ { j = 1 } ^ { m + 1 } \\frac 1 { \\delta _ j } - \\frac 1 { \\delta _ { m + 1 } } > 0 , \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} P ( B _ 2 ^ { ( n ) } ) & \\le \\mathrm { e } ^ { - \\mathrm { e } ^ { - 3 } r ^ { - 1 } ( 1 - r ^ { - 1 } ) ^ { r - 1 } K ^ { r } H ( x _ K ) a _ c ^ { ( n ) } } \\frac { 1 } { 1 - \\mathrm { e } ^ { - \\mathrm { e } ^ { - 3 } ( 1 - r ^ { - 1 } ) ^ { r - 1 } K ^ { r } ( \\lceil K \\rceil ) ^ { - 1 } H ( x _ K ) ( \\log K ) a _ c ^ { ( n ) } } } . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 0 } a _ m z ^ m = z ^ { 2 s } \\sum \\limits _ { n = 0 } \\tilde { a } _ n z ^ { - n } a \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} d ( e ^ { - A _ s } | Y ^ 1 _ s - Y ^ 2 _ s | ^ r ) & = - e ^ { - A _ s } | Y ^ 1 _ s - Y ^ 2 _ s | ^ r d A _ s + e ^ { - A _ s } d | Y ^ 1 _ s - Y ^ 2 _ s | ^ r \\leq e ^ { - A _ s } d M _ s . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} R i c _ { \\widetilde { g } } - \\frac { r } { h } H e s s _ { \\widetilde { g } } h = \\rho \\widetilde { g } , \\ \\ \\ \\ \\ r \\in \\mathbb { R } ^ * _ + , \\ \\ \\ \\ \\ \\rho \\in \\mathbb { R } . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} | k _ { a b } | = 2 | c _ { a b } | = 1 | e _ { a b } | = 0 \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\widetilde { H } _ { \\Lambda _ l x _ j } = \\int _ { \\mathcal { Y } ^ d } \\frac { \\widetilde { m } _ { x _ j } \\widetilde { m } _ { \\Lambda _ l } } { \\widetilde { m } } d y + \\int _ { \\mathcal { Y } ^ d } ( \\widetilde { \\delta } _ { i l } + \\widetilde { w } _ { y _ i \\Lambda _ l } ) \\widetilde { w } _ { y _ i x _ j } \\widetilde { m } d y + \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i x _ j \\Lambda _ l } \\widetilde { m } d y . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} p _ 0 & = - Q + \\prod _ { i = 0 } ^ N \\left ( 1 - \\Lambda _ i \\right ) \\\\ p _ { N + 1 } & = ( 1 - q ) ^ { N + 1 } \\Lambda _ 0 \\cdots \\Lambda _ N \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} h ( a b ) f _ 3 ( a , b ) = g _ 3 ( a , b ) h ( a ) , \\\\ h ( a b ) f _ 4 ( a , b ) = g _ 4 ( a , b ) h ( b ) . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} s _ { k } ( t ) = \\frac { \\sinh ( k \\ , t ) } { k } \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} s = \\sum \\limits _ { k = 1 } ^ { a - 4 } m _ { k } ( a + k d ) + m _ { a - 3 } b + m _ { a - 2 } ( b + d ) \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\nabla _ { X _ 1 } X _ 1 = J X _ 1 , \\nabla _ { X _ 2 } X _ 2 = J X _ 2 , \\nabla _ { X _ 1 } X _ 2 = \\nabla _ { X _ 2 } X _ 1 = 0 . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\left ( \\langle f _ 2 , f _ 3 \\rangle , \\langle f _ 3 , f _ 1 \\rangle , \\langle f _ 1 , f _ 2 \\rangle \\right ) = ( \\delta _ 1 e ^ { i \\theta _ 1 } | | f _ 2 | | | | f _ 3 | | , \\delta _ 2 e ^ { i \\theta _ 2 } | | f _ 3 | | | | f _ 1 | | , \\delta _ 3 e ^ { i \\theta _ 3 } | | f _ 1 | | | | f _ 2 | | ) . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} A ^ p = \\left [ \\begin{array} { c c } A ^ { \\prime p } & 0 \\\\ 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} d & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\hat { \\alpha } _ i ^ 2 + \\beta _ i ^ 2 ( 1 + 2 \\hat { \\lambda } \\beta _ i ^ 2 ) } { ( 1 + 2 \\hat { \\lambda } \\beta _ i ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\begin{cases} \\chi _ { \\rho \\rho } - \\frac { p _ e ' ( \\rho ) } { \\rho ^ 2 } \\chi _ { u u } = 0 , & \\\\ \\chi ( 0 , u ; s ) = 0 , & \\chi _ \\rho ( 0 , u ; s ) = \\delta _ { u = s } . \\end{cases} \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} X _ * ( T ) = X ^ * ( \\widehat { T } ) \\xrightarrow { r e s } X ^ * ( Z ( \\widehat { M _ S } ) ^ { \\Gamma } ) \\to X ^ * ( Z ( \\widehat { M _ S } ) ^ { \\Gamma } ) _ { \\mathbb { Q } } \\cong \\mathfrak { A } _ { M _ S , \\mathbb { Q } } \\subset \\mathfrak { A } _ { \\mathbb { Q } } \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\tilde a _ s ( u , v ) : = \\big ( ( - \\Delta ) ^ { s / 2 } u , ( - \\Delta ) ^ { s / 2 } v \\big ) + \\rho ( u , v ) = ( f , v ) , \\forall v \\in H ^ s ( \\mathbb R ) . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\overline { \\rho } \\left ( \\overline { \\pi } \\left ( \\frac { z - \\alpha } { c } \\right ) \\right ) = \\overline { \\pi } \\left ( \\frac { w - \\alpha } { c } \\right ) & \\iff \\overline { \\pi } \\left ( \\rho \\left ( \\frac { z - \\alpha } { c } \\right ) \\right ) = \\overline { \\pi } \\left ( \\frac { w - \\alpha } { c } \\right ) \\\\ & \\iff \\rho \\left ( \\frac { z - \\alpha } { c } \\right ) - \\frac { w - \\alpha } { c } \\in M _ U . \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} \\dfrac { \\partial X ( p , t ) } { \\partial t } = \\vec H ( p , t ) , ( p , t ) \\in M \\times [ 0 , T ) , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\lambda ( x , \\xi ) = a d _ { x } ^ * \\cdot S ( \\xi ) - S ( \\xi ) \\cdot a d _ { x } ^ * - S ( a d _ { x } ^ * ( \\xi ) ) , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { 2 n } q ^ { n ^ 2 + n } } { ( z q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b ^ 4 _ \\nu ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n ( z q ) ^ n . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\begin{aligned} & ( i , a , \\ell ) \\ : ( j , b , k ) = \\begin{cases} ( i , a p _ { \\ell j } b , k ) & p _ { \\ell j } \\neq 0 , \\\\ 0 & p _ { \\ell j } = 0 , \\end{cases} \\\\ & ( i , a , \\ell ) \\ : 0 = 0 \\ : ( i , a , \\ell ) = 0 \\ : 0 = 0 . \\end{aligned} \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} \\lim _ { X \\to \\infty } \\frac { \\sum _ { p \\in S p l _ X ( f ) } \\# \\{ i \\mid r _ i / p \\le a , 1 \\le i \\le n \\} } { n \\cdot \\# S p l _ X ( f ) } = a \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\gamma _ { d , r , R } ( n ) = O _ { d , R } ( x ^ { d - 1 + O ( 1 / \\log \\log n ) } ) \\forall x \\geq 1 . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} ( i _ 1 , g _ 1 , \\lambda _ 1 ) \\dots ( i _ m , g _ m , \\lambda _ m ) = ( j _ 1 , h _ 1 , \\mu _ 1 ) \\dots ( j _ m , h _ m , \\mu _ m ) \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\hat Q _ r - Q _ r = \\hat Q _ r ( I - Q _ r ) + ( I - Q _ r ) \\hat Q _ r + Q _ r ( \\hat Q _ r - Q _ r ) Q _ r - ( I - Q _ r ) \\hat Q _ r ( I - Q _ r ) . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\tau _ 2 ( x ) = g _ 2 ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac 2 5 , \\tfrac 4 { 3 ^ 2 } ) . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\int _ { \\gamma _ i } \\omega _ j = \\delta _ { i , j } \\ , . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} P \\in \\mathcal { Q } ^ { i n j } ( ^ m E ; F ) \\Longleftrightarrow I _ F \\circ P \\in \\mathcal { Q } ( ^ m E ; \\ell _ \\infty ( B _ { F ^ * } ) { \\rm ~ ~ a n d ~ ~ } \\| P \\| _ { \\mathcal { Q } ^ { i n j } } : = \\| I _ F \\circ P \\| _ { \\mathcal { Q } } . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\phi _ n = ( \\eta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ s , 1 ) , \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\tilde { \\omega } _ { 2 } ( \\sigma ) = \\max \\{ \\sigma ^ { \\alpha } , \\omega _ { f } ( \\sigma ) \\} . \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} d _ G ( v ^ * ) = f ( X _ { v ^ * } ) = f ^ 1 ( X _ { v ^ * } ) + f ^ 2 ( X _ { v ^ * } ) = d _ { G ^ 1 } ( v ^ * ) + d _ { G ^ 2 } ( v ^ * ) . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} & \\sum _ { I _ { n _ k } ( a _ 1 , \\ldots , a _ { n _ k } ) \\cap F \\neq \\emptyset } | I _ { n _ k } ( a _ 1 , \\ldots , a _ { n _ k } ) | ^ { s } \\\\ \\leq & K _ 2 ^ { s n _ k } \\prod _ { j = 1 } ^ k \\widehat { G } ( m ( j ) , n ( j ) , b , 1 / 3 , s ) \\\\ \\leq & K _ 2 ^ { s n _ k } \\cdot 6 ^ k \\cdot \\widehat { C } ^ { n _ k - k - 1 } \\cdot 3 ^ { - k } \\prod _ { j = 1 } ^ k e ^ { ( 1 - d s ) ( \\log m ( j ) ) ^ { 1 / b } } . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\frac { 1 } { t } = \\frac { \\| \\psi \\| _ { L ^ \\infty } } { \\| \\varphi \\| _ { L ^ \\infty } } = \\kappa . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} \\widetilde \\psi _ 0 \\left ( ( X ; S _ { l - 1 } , T _ { l - 1 } ) ^ { g _ r ^ { - 1 } } \\right ) & = \\tau \\left ( \\varpi ^ { - l } B ( \\lambda ( X ) + \\varpi T , ( g ) \\beta ) \\right ) \\\\ & = \\tau \\left ( \\varpi ^ { - l } B ( \\lambda ( X ) + \\varpi T , \\beta ) \\right ) \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} & \\mu _ { \\beta , 0 } ^ { + } \\big ( \\eta ( n , 0 ) = 1 \\mid \\eta = - 1 ( - \\infty , 0 ) \\times \\{ 0 \\} \\big ) > \\mu _ { \\beta , 0 } ^ { - } \\big ( \\eta ( n , 0 ) = 1 \\big ) , \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} A _ { t , \\delta t } [ s ] : = \\begin{cases} a _ s & ~ s \\in [ 0 , t ) \\\\ a _ t & ~ s \\in [ t , t + \\delta t ] \\end{cases} , ~ A _ { t } ^ { ( h ) } [ s ] : = \\begin{cases} a _ s & ~ s \\in [ 0 , t ) \\\\ a _ t + h & ~ s = t . \\end{cases} \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\# \\mathcal { B } ( e ) = N _ 1 ( m _ e ) . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} t \\cdot \\mathrm { m o d } _ q ( d \\varphi ) = \\left ( 2 \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ^ + _ { ( d \\varphi , \\iota ) } } + \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ^ 0 _ { ( d \\varphi , \\iota ) } } \\right ) \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\phi ) + \\ell _ { \\beta } ( \\phi ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ n ( x _ i ^ 2 - i ) ^ 2 + \\sum _ { \\substack { i , j = 1 \\\\ j > i } } ^ n b _ { i j } \\ , x _ i x _ j + \\ , \\sum _ { i = 1 } ^ n d _ i \\ , x _ i \\ , , \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\mathbb { P } ( Z _ 1 \\in A ) = \\frac { \\int _ { A \\cap S _ l } f ( x ) d x } { \\int _ { S _ l } f ( x ) d x } . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( m ) = \\sigma \\left ( X _ { m n } ( [ 0 , 1 ] ) \\times W _ { m n } ( [ \\omega _ 1 , \\omega _ 2 ] ) \\right ) \\ ; , \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} \\aligned \\frac { 1 } { 2 } \\mathcal { L } \\sum _ { i , j , k , p } ( h _ { i j k } ^ { p ^ { \\ast } } ) ^ { 2 } = \\sum _ { i , j , k , l , p } ( h _ { i j k l } ^ { p ^ { \\ast } } ) ^ { 2 } . \\endaligned \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\mathrm { L y r } _ n { \\mathcal { T } } : = \\lbrace v \\in \\mathcal { T } \\mid \\mathrm { l o } ( v ) = n \\rbrace , \\mathrm { L y r } _ n { \\mathcal { B } } : = \\lbrace v \\in \\mathcal { B } \\mid \\mathrm { l o } ( v ) = n \\rbrace , \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\left < \\nu ; E ( \\widetilde { y } ) \\right > = 0 \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\prod _ { m = 1 } ^ { \\infty } \\frac { \\left ( 1 - t ^ m \\right ) ^ 2 } { \\left ( 1 - q t ^ m \\right ) \\left ( 1 - q ^ { - 1 } t ^ m \\right ) } = 1 + \\left ( q ^ { 1 / 2 } - q ^ { - 1 / 2 } \\right ) \\sum _ { d = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { \\infty } t ^ { d k } \\left ( q ^ { d - k / 2 } - q ^ { - d + k / 2 } \\right ) . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} c _ n : = \\frac { \\Gamma ( ( n + 1 ) / 2 ) } { \\pi ^ { ( n + 1 ) / 2 } } . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\P ( C _ { k , j } \\cap ( A ) ^ c ) = 0 \\P ( C _ { k , j } \\cap A ) = 0 \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{gather*} d _ k = n _ k ^ { 1 / 2 } + O ( 1 ) , \\\\ \\rho _ { k + 1 } - \\rho _ { k } = n _ { k + 1 } ^ { \\frac { 1 } { 1 + \\delta } } - n _ { k } ^ { \\frac { 1 } { 1 + \\delta } } = \\frac { 1 } { 1 + \\delta } n _ k ^ { - \\frac { \\delta } { 1 + \\delta } } ( 2 n _ k ^ { 1 / 2 } + O ( 1 ) ) = \\frac { 2 } { 1 + \\delta } \\rho _ k ^ { \\frac { 1 - \\delta } { 2 } } + O ( \\rho _ k ^ { - \\delta } ) . \\end{gather*}"}
-{"id": "3344.png", "formula": "\\begin{align*} \\| h \\| _ { W ^ { 1 , 2 } } & \\geq \\frac { \\int _ { 0 } ^ { 1 } f ( s ) d h ( s ) } { \\| f \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } } = \\frac { \\| h \\| _ { \\bar { \\mathcal { H } } } \\| f \\| _ { \\mathcal { H } } } { \\| f \\| _ { L ^ { 2 } ( [ 0 , 1 ] ) } } \\geq C _ { H } \\| h \\| _ { \\bar { \\mathcal { H } } } , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} I _ \\alpha f _ j ( \\xi ) & = I _ \\alpha ^ 1 f _ j ( \\xi ) + I _ \\alpha ^ 2 f _ j ( \\xi ) \\\\ & : = f _ j * | \\xi | ^ { \\alpha - Q } \\chi _ { \\{ | \\xi | > \\rho \\} } + f _ j * | \\xi | ^ { \\alpha - Q } \\chi _ { \\{ | \\xi | < \\rho \\} } . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} - \\div _ x \\bigg ( \\widetilde { m } _ 0 ( x ) \\int _ { \\mathcal { Y } ^ d } ( \\widetilde { m } _ 1 ( x , y ) ( \\widetilde { \\Lambda } + \\nabla _ y \\widetilde { u } _ 1 ( x , y ) ) ) d y \\bigg ) = 0 . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ { x _ j x _ l } d y = 0 . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} ( W _ m v ) ( n ) : = e ^ { \\i h { \\sigma ( m , n ) } } v ( n + m ) . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} & \\partial _ t f = - \\partial _ x f - k ( x , \\lambda \\ , m ( t ) ) f = : \\mathcal { L } _ { \\lambda m ( t ) } f , \\\\ & f ( t , 0 ) = p ( t ) , \\ \\ f ( 0 , x ) = f _ 0 ( x ) , \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\tilde { D } _ { u } = D ^ { ( 1 ) } _ { u } - \\frac { 1 } { 4 } \\{ A _ { u } ^ { \\mathrm { s y m } } , \\mathcal J \\} \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} & h ^ * _ d \\le h ^ * _ { d - 1 } \\le \\dots \\le h ^ * _ { \\lfloor \\frac { d + 1 } { 2 } \\rfloor } \\\\ & h ^ * _ { j + 1 } \\ge h ^ * _ { d - j } \\ \\ 0 \\le j \\le \\tfrac d 2 - 1 \\\\ & h ^ * _ j \\le \\binom { h ^ * _ 1 + j - 1 } { j } \\ \\ 0 \\le j \\le d \\ , . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( H _ { i j k l } ( t ) e _ { k l } ( u ) ) _ { , j } + f _ i = 0 \\Omega , \\\\ \\\\ H _ { i j k l } ( t ) e _ { k l } ( u ) n _ j - \\hat { f } _ i = 0 , \\Gamma _ t , \\ ; \\forall i \\in \\{ 1 , 2 , 3 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} Q _ { m _ { 1 } , m _ { 2 } } ^ { ( j ) } = Q _ { m _ { 1 } } \\cap \\left [ x _ d = \\frac { 2 } { j - 1 } x _ { d - 1 } - \\frac { 2 } { j - 1 } \\left ( x _ { d - 1 } ^ { 0 } + \\frac { j + 1 } { 2 } x _ d ^ { 0 } + m _ { 2 } \\right ) \\right ] , | m _ { 2 } | < M . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} T M = \\mathrm { R a d } \\ ; T M \\bot S ( T M ) = T C \\bot S ( T M ) , \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\Sigma _ { n } \\coloneqq \\{ \\delta _ { n , k } \\thickapprox 1 \\colon k = 1 , \\dots , k _ n \\} . \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} \\sharp \\{ \\gamma ^ { \\prime } \\in \\Gamma : \\gamma ^ { \\prime } V \\cap \\gamma _ { 0 } V \\not = \\emptyset \\} \\ge m + 1 ~ , \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\tau _ n = \\frac { 1 } { n ^ 3 } \\lambda _ n = e ^ { - n } \\ , , \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} H \\frac { R ^ { ( 1 - s ) p } } { ( R - r ) ^ p } \\| ( u - k ) _ - \\| _ { L ^ p ( x _ 0 - R , x _ 0 + R ) } ^ p = H \\frac { R ^ { ( 1 - s ) p } } { ( R - r ) ^ p } k ^ p \\int _ { x _ 0 - R } ^ 0 d x \\ge H ( R - | x _ 0 | ) ^ { 1 - s p } k ^ p . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} g _ N ( x ) = \\frac { f ( x ) } { \\int _ { \\cup _ { 1 \\leq j \\leq N } S _ j } f _ j ( x ) d x } \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} \\bar { u } _ \\varepsilon ( z ) = \\frac { 1 } { 2 \\pi | x _ \\varepsilon - z | } \\int _ { \\partial B _ { x _ \\varepsilon } ( | x _ \\varepsilon - z | ) } u _ \\varepsilon ~ d \\sigma \\ , , \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ n z ^ n q ^ { n ^ 2 + n } } { ( y q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\infty } b _ \\nu ( \\ell , m , n ) z ^ \\ell y ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n ( y q ) ^ n . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} & m C \\left ( C _ k ( \\C ^ n ) \\right ) = \\sum _ { P \\in X } \\prod _ { P _ k \\in P } \\left [ ( - 1 ) ^ { | P _ k | - 1 } ( | P _ k | - 1 ) ! \\sum _ { i = 1 } ^ n \\left ( \\prod _ { j \\neq i } \\frac { 1 + y \\frac { \\alpha _ i } { \\alpha _ j } } { 1 - \\frac { \\alpha _ i } { \\alpha _ j } } \\prod _ { j = 1 } ^ n \\prod _ { a \\in P _ k } \\psi _ { i , j } ( \\beta _ a \\alpha _ j ) \\right ) \\right ] \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} e ( \\tau ) = - 8 \\pi ^ 2 \\sum _ { n = 1 } ^ \\infty \\frac { n \\ , q ^ { n / 2 } } { 1 - q ^ { n } } . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\Psi ( s ) ~ \\leq ~ { t - s \\over t } \\cdot \\Psi ( 0 ) + { s \\over t } \\cdot \\Psi ( t ) ~ = ~ { s \\over t } \\cdot \\Psi ( t ) ~ < ~ \\Psi ( t ) \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\Big \\| \\mathop { { \\sup _ { i \\in I } } ^ { + } } x _ { i } \\Big \\| _ { p } = \\sup _ { J } \\Big \\| \\mathop { { \\sup _ { i \\in J } } ^ { + } } x _ { i } \\Big \\| _ { p } . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} c _ p ( M ) : = \\sup _ { \\epsilon > 0 } c _ { p , \\epsilon } ( M ) = \\lim _ { \\epsilon \\to 0 } c _ { p , \\epsilon } ( M ) . \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} J \\bigl ( \\C [ e , h , f ] \\bigr ) = \\C [ e _ 0 , h _ 0 , f _ 0 , e _ 1 , h _ 1 , \\ldots ] \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\Omega _ \\epsilon } \\partial _ y ( K ( x ) \\phi ( x ) ) d x = 0 \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} L ( x , \\phi , \\dot x , \\dot \\phi ) = \\frac { 1 } { 2 } \\Big ( m x ^ 2 + J \\Big ) \\dot \\phi ^ 2 + \\frac { 1 } { 2 } m \\dot x ^ 2 - m g x \\sin \\phi . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} y ' _ { j } - x ' _ n \\geq y ' _ { j } - x ' _ k = \\frac { \\alpha } { q _ j } + y _ { j } - \\frac { \\alpha } { p _ k } - x _ { k } \\geq y _ { j } - x _ { k } \\geq 1 - \\delta . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ l \\frac { \\alpha _ s ( y _ i ) } { \\alpha _ s } ( 1 - s ) ( x _ i ) = \\frac { 1 } { \\alpha _ s } \\left ( \\sum _ { i = 1 } ^ l \\alpha _ s ( y _ i ) ( x _ i - s ( x _ i ) s ) \\right ) = \\frac { 1 } { \\alpha _ s } ( \\alpha _ s - s ( \\alpha _ s ) s ) = 1 + s \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} [ X + \\xi , Y + \\eta ] = [ X , Y ] + \\mathcal { L } _ X \\eta - i _ { Y } d \\xi + H ( X , Y , \\cdot ) , \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\sup _ { r \\leq r _ 0 } f _ t ( r , t _ 0 ) & \\leq \\max \\Big \\{ \\sup _ { t _ j \\leq t \\leq t _ 0 } f _ t ( r _ 0 , t ) , \\ , \\sup _ { r \\leq r _ 0 } f _ t ( r , t _ j ) \\Big \\} \\\\ & = \\max \\Big \\{ f _ t ( r _ 0 , t _ 0 ) , \\ , \\sup _ { r \\leq r _ 0 } f _ t ( r , t _ j ) \\Big \\} . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} & \\lambda _ 1 \\leq \\tilde \\lambda _ 1 \\qquad \\lambda _ j \\leq \\tilde \\lambda _ j \\leq \\lambda _ { j - 1 } j = 2 , \\dots , r . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\psi _ 2 ( s ) & = \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } \\frac { \\phi ( m s ) } { m } \\\\ & \\leq \\phi ( s ^ { 1 / 2 } ) \\sum _ { m = 1 } ^ { s ^ { - 1 } } \\frac { 1 } { m } \\\\ & = O ( s ^ { - u / 2 } \\log ( 1 / s ) ) . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} ( U ^ { \\leq 0 } ) _ { \\geq l } : = ( U ^ { \\leq 0 } ) _ l \\oplus ( U ^ { \\leq 0 } ) _ { l + 1 } \\oplus \\ldots \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} r ( l ( n ) ) \\le \\frac { 9 l ( n ) ^ { 2 } } { 2 n } ( 1 + o ( 1 ) ) = \\frac { n ^ { 5 } } { 8 } ( 1 - 2 c _ { n } + c _ { n } ^ { 2 } ) ( 1 + o ( 1 ) ) , \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} & w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( 1 , \\{ 2 , 3 \\} ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( 2 , \\{ 1 , 3 \\} ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( 3 , \\{ 1 , 2 \\} ) } ) = 0 \\\\ & w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( \\{ 2 , 3 \\} , 1 ) } ) = 1 , w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( \\{ 1 , 3 \\} , 2 ) } ) = 1 , w ^ \\uparrow _ { \\{ 1 , 2 , 3 \\} } ( C _ { ( \\{ 1 , 2 \\} , 3 ) } ) = 1 \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} d \\left ( T \\{ x _ n ^ i \\} , T \\{ x _ n ^ j \\} \\right ) & = d \\left ( \\{ x _ n ^ { i + 1 1 } \\} , \\{ x _ n ^ { j + 1 1 } \\} \\right ) \\\\ & = \\frac { 1 0 } { i + j + 2 2 } + 1 \\\\ & < \\frac { 1 0 0 } { i + j } + 1 \\\\ & = d \\left ( \\{ x _ n ^ i \\} , \\{ x _ n ^ j \\} \\right ) . \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\Omega \\left ( p ^ { N } \\left \\vert x - J \\right \\vert _ { p } \\right ) = { \\textstyle \\sum \\limits _ { J _ { j } \\in G _ { J } ^ { M } } } \\Omega \\left ( p ^ { N } \\left \\vert x - J _ { j } \\right \\vert _ { p } \\right ) M \\geq N . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} F _ { \\gamma } ( x , r ) = \\begin{cases} f _ 0 ( x ) + r d ( 0 , G ( x ) ) ^ { \\gamma } , & d ( 0 , G ( x ) ) < \\tau , \\\\ + \\infty , & \\end{cases} \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\varkappa ( v ) _ i : = \\begin{cases} j & \\ker ( T _ i ) = u _ j / v ^ \\prime j \\in \\lbrace 1 , \\ldots , 4 \\rbrace , \\\\ 0 & \\ker ( T _ i ) = v / v ^ \\prime . \\end{cases} \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} | \\sin \\pi ( 2 \\theta + \\alpha k ) | = e ^ { - t | k | } . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} d \\eta ( X , Y ) = & \\ X \\eta ( Y ) - Y \\eta ( X ) - \\eta [ X , Y ] \\\\ = & \\ X g ( Z , Y ) - Y g ( Z , X ) - g ( Z , [ X , Y ] ) \\\\ = & \\ g ( \\N _ X Z , Y ) + g ( Z , \\N _ X Y ) - g ( \\N _ Y Z , X ) - g ( Z , \\N _ Y X ) - g ( Z , [ X , Y ] ) \\\\ = & \\ g ( I X , Y ) - g ( I Y , X ) \\\\ = & \\ 2 g ( I X , Y ) . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - f _ 4 ( n ) \\log ( b _ c ^ { ( n ) } / f _ 4 ( n ) ) } \\log P ( B _ 2 ^ { ( n ) } ) = - \\varepsilon . \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} & T \\left ( f _ 1 , f _ 2 , \\ldots , f _ n \\right ) ( x ) = \\sum _ { j _ 1 , j _ 2 } e _ { 1 , j _ 1 } e _ { 2 , j _ 2 } T ( f _ { 1 , j _ 1 } , f _ { 2 , j _ 2 } , f _ 3 , \\ldots , f _ n ) ( x ) , x \\in \\R ^ d , \\\\ & f _ 1 = \\sum _ { j _ 1 } e _ { 1 , j _ 1 } f _ { 1 , j _ 1 } , \\ , f _ 2 = \\sum _ { j _ 2 } e _ { 2 , j _ 2 } f _ { 2 , j _ 2 } f _ { 1 , j } , f _ { 2 , j } \\in L ^ \\infty _ c ( \\R ^ d ) , \\ , e _ { 1 , j _ 1 } \\in X _ 1 , \\ , e _ { 2 , j _ 2 } \\in X _ 2 , \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} \\Phi ( x ) = \\left \\{ \\begin{aligned} & - \\frac { 1 } { 2 \\pi } \\log | x | , & n & = 2 \\\\ & \\frac { 1 } { n ( n - 2 ) b _ n } \\frac { 1 } { | x | ^ { n - 2 } } , & n & \\geq 3 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} & \\frac { 1 } { r } \\int _ { r - t } ^ { r + t } \\widetilde { \\Omega } _ s [ u _ 1 + \\mu u _ 0 ] ( x ) K _ 1 ( t , r ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s \\xrightarrow [ r \\to 0 ] { } 2 \\int _ { 0 } ^ { t } \\frac { \\partial } { \\partial s } \\ , \\Omega _ { s } [ u _ 1 + \\mu u _ 0 ] ( x ) K _ 1 ( t , 0 ; s ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } s . \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} h _ { f _ i } = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log | X ^ { \\Gamma _ n } _ { f _ i } | \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} \\ddot { u } ( t ) - \\Delta u ( t ) = f ( t ) \\Omega \\setminus \\Gamma ( t ) \\ , , \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( S _ n ( n - x f _ 2 ( n ) ) + a _ n > \\lfloor n - \\varepsilon f _ 2 ( n ) \\rfloor ) = - H ( \\ell _ 2 \\varepsilon ) . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\dim _ H E _ M = 1 . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} H _ { T ^ { N + 1 } } ^ * \\left ( \\mathbb { P } ^ N ; \\mathbb { Q } \\right ) = \\mathbb { Q } [ \\lambda _ 0 , \\dots , \\lambda _ N ] [ H ] \\left / \\left ( ( H - \\lambda _ 0 ) \\cdots ( H - \\lambda _ N ) \\right ) \\right . \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} & U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) \\\\ & = ( x q ) ^ { 2 a + 1 } \\sum _ { h = 1 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a + 1 } \\sum _ { h = 0 } ^ { k - a - 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) \\\\ & + ( x q ) ^ { 2 a - 1 } \\sum _ { h = 1 } ^ { k - a + 1 } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) + ( x q ) ^ { 2 a - 1 } \\sum _ { h = 0 } ^ { k - a } ( x q ^ 2 ) ^ { 2 h } U _ { 2 k , 2 k - 2 h } ( x q ^ 2 ; q ) . \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\nu _ \\varepsilon ^ 2 | \\Delta u _ \\varepsilon ( y _ \\varepsilon ) | u _ \\varepsilon ( y _ \\varepsilon ) = 1 \\ , . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\mathcal { F } _ 0 ( r ) = \\mathcal { F } ( r ) \\setminus \\left ( \\dot { \\bigcup } _ { i = 1 } ^ t \\ , \\mathcal { T } ^ r _ i \\right ) t \\ge 0 . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} | x _ { n + 1 } | ^ { \\frac 1 { 3 ^ { n + 1 } } } & < | x _ 0 | \\prod _ { i = 0 } ^ { n } h _ { i } ^ { \\frac 1 { 3 ^ { i + 1 } } } + \\sum _ { i = 1 } ^ { n + 1 } \\prod _ { j = i } ^ { n } h _ { j } ^ { \\frac 1 { 3 ^ { j + 1 } } } | u _ { i } | ^ { \\frac 1 { 3 ^ { i } } } \\\\ & = \\prod _ { i = 0 } ^ { n } h _ { i } ^ { \\frac 1 { 3 ^ { i + 1 } } } \\left [ | x _ 0 | + \\sum _ { i = 1 } ^ { n + 1 } \\prod _ { j = 0 } ^ { i - 1 } h _ { j } ^ { - \\frac 1 { 3 ^ { j + 1 } } } | u _ { i } | ^ { \\frac 1 { 3 ^ { i } } } \\right ] . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\langle \\hat u _ j - u _ j , E u _ k \\rangle & = \\sum _ { l \\geq 1 } \\langle \\hat u _ j - u _ j , u _ l \\rangle \\langle u _ l , E u _ k \\rangle \\\\ & = \\sqrt { \\lambda _ j \\lambda _ k } \\langle \\hat u _ j - u _ j , u _ j \\rangle \\bar \\eta _ { j k } + \\sum _ { l \\neq j } \\sqrt { \\lambda _ k \\lambda _ l } \\langle \\hat u _ j , u _ l \\rangle \\bar \\eta _ { k l } \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} x _ m ( t ) = \\cos ( \\omega _ 1 t + \\phi _ 1 ^ m ) + \\cos ( \\omega _ 2 t + \\phi _ 2 ^ m ) . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} i _ { \\mathcal { O } _ \\mu } ^ \\ast \\omega = \\pi _ { \\mathcal { O } _ { \\mu } } ^ \\ast \\omega _ { \\mathcal { O } _ \\mu } + \\mathbf { J } _ { \\mathcal { O } _ \\mu } ^ \\ast \\omega _ { \\mathcal { O } _ \\mu } ^ + , \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} t _ 0 > \\sqrt { \\dfrac { | \\lambda _ 3 ( B ) | } { 2 } } = \\sqrt { 3 . 0 4 9 8 0 2 } \\ , . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\liminf _ { r \\to 0 } \\frac { \\log \\mu ( B _ r ( x ) ) } { \\log r } = \\liminf _ { \\ell \\to \\infty } \\frac { \\log \\mu ( B _ { r _ \\ell } ( x ) ) } { \\log r _ \\ell } . \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\sum _ { j } ( - 1 ) ^ j \\binom { 2 m + 1 } { 2 j } = \\begin{cases} ( - 1 ) ^ { m / 2 } 2 ^ m & \\\\ ( - 1 ) ^ { ( m + 1 ) / 2 } 2 ^ m & \\end{cases} \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ m X _ i ^ \\star ( a _ { i j } X _ j u ) = \\sum _ { i = 1 } ^ m X _ i ^ \\star f _ i + g \\ \\ \\ ~ ~ ~ B ( \\tau ) , \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} & W _ l ( x ; \\tau _ 1 , \\dots , \\tau _ { n } ) \\\\ = \\ , & W _ l ( x ; \\tau _ 1 , \\dots , \\tau _ { n - 1 } ) - W _ l ( x - \\tau _ n ; \\tau _ 1 , \\dots , \\tau _ { n - 1 } ) . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\binom { \\ell _ q \\left ( Q \\right ) } { a } = \\frac { 1 } { a ! } \\prod _ { r = 0 } ^ { a - 1 } ( \\ell _ q ( Q ) - r ) = \\frac { 1 } { a ! } \\ell _ q ( Q ) ^ a + \\cdots \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ + } \\frac { \\widetilde { V } _ q ( K + \\varepsilon L ) - \\widetilde { V } _ q ( K ) } { \\varepsilon } = \\int _ { S ^ { n - 1 } } \\frac { h _ L } { h _ K } \\ , d \\widetilde { C } _ q ( K , \\cdot ) . \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} y _ { j } = y _ { j } ^ { \\varepsilon _ { n } } \\rightarrow l _ { j } \\in \\overline { G } , \\ , j = 1 , . . , N _ { 1 } . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\tilde { \\mathcal { E } } _ { z _ 1 } & = \\Big \\{ \\tilde x \\geq \\frac { z _ 1 } { \\sqrt { n } } \\Big \\} , \\qquad \\tilde { \\mathcal { E } } _ { z _ 1 , z _ 2 } = \\Big \\{ \\frac { z _ 1 } { \\sqrt { n } } \\leq \\tilde x \\leq \\frac { z _ 2 } { \\sqrt { n } } \\Big \\} , \\tilde { \\mathcal { E } } = \\Big \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ^ 2 \\leq \\frac { 1 } { r } \\Big \\} . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\mathcal L ( V , V ) \\\\ & = \\sum _ { j = 1 } ^ n | b ^ j | ^ 2 d _ j . \\end{aligned} \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\| T ( f _ 1 , \\dots , f _ n ) \\| _ { L ^ { q _ { n + 1 } } ( \\R ^ d ) } \\lesssim \\prod _ { m = 1 } ^ n \\| f _ m \\| _ { L ^ { p _ m } ( \\R ^ d ) } \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\phi ( x ) \\cos ( 2 \\pi \\alpha n x ) x ^ { \\frac 1 2 - s } \\ , d x = \\sum _ { j = 0 } ^ { m - 1 } \\frac { \\cos ^ { ( j + 1 ) } ( 2 \\pi \\alpha n ) } { ( 2 \\pi \\alpha n ) ^ { j + 1 } } \\phi _ j ( 1 , s ) + \\int _ 1 ^ \\infty \\frac { \\cos ^ { ( m ) } ( 2 \\pi \\alpha n x ) } { ( 2 \\pi \\alpha n ) ^ m } \\phi _ k ( x , s ) x ^ { \\frac 1 2 - m - s } \\ , d x \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} - \\div _ y \\Big ( e ^ { \\frac { | \\Lambda + \\nabla _ y \\widetilde { w } | ^ 2 } { 2 } + V } ( \\Lambda + \\nabla _ y \\widetilde { w } ) \\Big ) = 0 . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} & \\widetilde { p _ D ( t - s , x , y ) h _ n ( . , s , y ) } = \\frac { 1 } { n + 1 } \\Big [ p _ D ( t - s , x , y ) h _ n ( s _ 1 , x _ 1 , s _ 2 , x _ 2 , \\cdots , s _ n , x _ n , s , y ) \\\\ + & \\sum _ { i = 1 } ^ n p _ D ( t - s _ j , x , y _ j ) h _ n ( s _ 1 , x _ 1 , s _ 2 , x _ 2 , \\cdots , s _ { j - 1 } , x _ { j - 1 } , s , y , s _ { j + 1 } , x _ { j + 1 } , \\cdots , s _ n , x _ n , s _ j , x _ j ) \\Big ] . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\tilde { b } _ { q , m } = \\begin{cases} \\lfloor b ' _ { q , m } \\rfloor & b ' _ { q , m } - \\lfloor b ' _ { q , m } \\rfloor \\leq \\beta \\\\ \\lceil b ' _ { q , m } \\rceil & b ' _ { q , m } - \\lfloor b ' _ { q , m } \\rfloor > \\beta \\end{cases} , \\ , q \\in \\mathcal { Q } ' _ m , \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} Z ( T _ 0 , T _ 1 , \\dots , T _ r ) = \\sum _ { \\emptyset \\not = I \\subseteq J , \\ h ( E _ I ^ { \\circ } ) \\subseteq X _ 0 ( g ) } ( \\L - 1 ) ^ { | I | - 1 } [ \\widetilde { E } _ I ^ { \\circ } ] S _ I ( T _ 0 , T _ 1 , \\dots , T _ r ) , \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} [ T a ] ( z ) = \\partial _ z a ( z ) , [ T a ] ( n ) = - n a ( n - 1 ) , \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\psi _ m = ( \\zeta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\zeta _ l , 1 ) \\boxplus ( \\xi _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\xi _ k , 2 ) \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\lim _ h \\mathcal H ^ 2 ( S _ h \\cap U _ { \\varepsilon } ( \\gamma ) ) = 0 . \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} h _ { a l g } ( \\phi ) = h _ { a l g } ( \\phi \\upharpoonright _ { G ' } ) + h _ { a l g } ( \\widetilde { \\phi } ) , \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} u _ n : = \\rho _ { n } \\xi _ { n } \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\widehat { J } ( \\xi ) = 1 - A | \\xi | ^ \\sigma ( \\ln 1 / | \\xi | ) ^ { \\mu _ 1 } ( \\ln _ 2 1 / | \\xi | ) ^ { \\mu _ 2 } \\cdots ( \\ln _ m 1 / | \\xi | ) ^ { \\mu _ m } + l . o . t \\quad \\mbox { a s $ \\xi \\to 0 $ } , \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} \\{ t _ 0 , t _ 1 , t _ { n - 2 } \\} = \\{ 1 , \\cos \\theta , \\cos \\theta _ { n - 2 } \\} . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} g _ 0 ( t ) = c _ 1 , g _ { n + 1 } : = M _ n ( g _ n ) \\mbox { f o r a l l } t \\ge 0 . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} [ r _ { - \\mu ' } \\circ L L ( w ( \\rho ) ) | _ { W _ { E _ { \\{ \\mu ' \\} _ { w ( M _ S ) } } } } ] = [ r _ { - w ^ { - 1 } ( \\mu ' ) } \\circ L L ( \\rho ) | _ { W _ { E _ { \\{ w ^ { - 1 } ( \\mu ' ) \\} _ { M _ S } } } } ] . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\bold { T } _ x = \\left [ \\begin{array} { c | c } \\bold { A } _ { x , x } & \\begin{array} { c | c | c } \\bold { B } _ { x , 1 } & \\dots & \\bold { B } _ { x , p } \\end{array} \\\\ \\hline \\bold { U } _ x & \\bold { Z } _ { x } \\end{array} \\right ] , \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} f _ i ( x ) ^ 2 = 2 g _ i ( x ) \\ { \\rm i n } \\ \\mathcal { K } _ i \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\begin{cases} \\eta ^ \\zeta ( \\rho , \\rho u ) = \\int _ \\mathbb { R } \\chi ( \\rho ; s - u ) \\zeta ( s ) d s , & \\\\ q ^ \\zeta ( \\rho , \\rho u ) = \\int _ \\mathbb { R } ( \\vartheta s + ( 1 + \\vartheta ) u ) \\chi ( \\rho ; s - u ) \\zeta ( s ) d s , & \\vartheta = \\frac { \\gamma - 1 } { 2 } , \\end{cases} \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n ( 1 - h _ n x _ n ^ 2 ) + u _ { n + 1 } , x _ 0 \\in \\mathbb R . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} l ( G ) = \\max \\{ \\O ( q - 1 ) + f , \\O ( q + 1 ) + 1 , l ( { \\rm L } _ { 2 } ( p ) ) + 1 + \\delta _ { 2 , f } \\} , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} [ \\xi , \\eta ] ' & : = [ u \\xi , \\eta ] + [ \\xi , u \\eta ] - u [ \\xi , \\eta ] \\\\ T [ u ] ( \\eta , \\xi ) & = [ u \\xi , u \\eta ] - u [ \\xi , \\eta ] ' \\ , . \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} { { \\tilde \\mu } _ j } \\buildrel \\Delta \\over = \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( a ) } - { F ^ { - 1 } } \\left [ { 1 - \\frac { 1 } { { K - k + 1 } } \\sum \\limits _ { i = k } ^ K { u _ j ^ { ( i ) } } } \\right ] . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} \\mathcal { S } _ { n } ^ { \\prime } ( z ) & = \\frac { 1 } { \\pi ^ { 2 n } } \\operatorname { I m } [ L i _ { 2 n } ( - \\exp ( - 2 \\pi i z ) ) ] , n > 0 , \\\\ \\mathcal { C } _ { n } ^ { \\prime } ( z ) & = \\frac { 1 } { \\pi ^ { 2 n + 1 } } \\operatorname { R e } [ L i _ { 2 n + 1 } ( - \\exp ( - 2 \\pi i z ) ) ] , \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} C _ { 2 n + 1 } ^ { ( 1 ) } ( x ) = x C _ n ^ { ( 1 ) } ( 2 x ^ 2 - 1 ) \\ . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } f _ { q ^ t } = \\widetilde { f } \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\frac { 1 - r } { r } = \\sum _ { i = 1 } ^ { m + 1 } \\frac 1 { \\delta _ i } = \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { \\delta _ i } + \\frac 1 { \\varrho } > \\frac 1 { \\varrho } . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\omega _ { f } ( s ) = \\sup _ { d ( p , \\overline { p } ) \\leq s } | f ( p ) - f ( \\overline { p } ) | . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } D ( x _ n , x ) = 0 , \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} h ' _ 1 & = \\frac { 1 } { 4 ( h _ 2 - h _ 1 ) } \\left ( \\alpha ^ 3 - \\alpha ^ 1 \\right ) , \\\\ h ' _ 2 & = \\frac { 1 } { 4 ( h _ 2 - h _ 1 ) } \\left ( \\alpha ^ 3 - \\alpha ^ 1 \\right ) + \\frac { 1 } { 4 ( h _ 3 - h _ 2 ) } \\left ( \\alpha ^ 4 - \\alpha ^ 2 \\right ) , \\\\ h ' _ 3 & = \\frac { 1 } { 4 ( h _ 3 - h _ 2 ) } \\left ( \\alpha ^ 4 - \\alpha ^ 2 \\right ) . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{gather*} \\mu _ a = - \\eta _ i \\rho ^ i _ a , \\\\ \\rho ^ j _ a b _ { j i } + \\rho ^ j _ a \\partial _ j \\eta _ i + \\eta _ j \\partial _ i \\rho ^ j _ a + \\Gamma ^ b _ { a i } \\mu _ b = 0 , \\\\ \\rho ^ i _ a \\partial _ i \\mu _ b - C ^ c _ { a b } \\mu _ c - \\rho ^ i _ b \\Gamma ^ c _ { a i } \\mu _ c = 0 . \\end{gather*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\widetilde { J ^ { K \\textnormal { t h } } } ( q , Q ) = \\sum _ { i = 0 } ^ N \\widetilde { J _ i } ( q , Q ) \\left ( 1 - P ^ { - 1 } \\right ) ^ i \\in K \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } ( q ) [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\mathcal { Q } _ { 2 \\rho , T } : = \\{ ( x , t ) \\in M ' \\times ( 0 , T ] : d ( x , y , t ) \\leq 2 \\rho \\} , \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} p _ 1 ( M ) _ G - ( 2 k + 1 ) c _ 1 ( \\xi ) ^ 2 _ G = \\alpha \\cdot \\pi ^ * q , \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} ( T x ) _ { i } = x _ { i } + x _ { i + 1 } , 1 \\leq i \\leq d - 1 , \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\sum _ { m = 1 } ^ r \\ \\| \\ , s _ m \\ , X ^ { ( m ) } \\ , - \\ , A ^ { ( m ) } \\ , \\| ^ 2 \\to \\min \\\\ s _ 1 \\cdots s _ r \\ \\le \\ 1 \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} N _ 1 ( m _ { n - 1 } ) + \\sum _ { i = 2 } ^ { C _ 1 ( m _ { n - 2 } ) } \\ , N _ 1 ( v _ i ( m _ { n - 2 } ) ) , \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} T ^ { ( n ) } = T ^ { ( n ) } _ { \\geq 1 } ( y _ n ) = \\sum _ { k \\geq 1 } \\frac { 1 } { \\sqrt { \\lambda _ 1 ^ { ( n ) } + y _ n - \\lambda _ k ^ { ( n ) } } } P _ k ^ { ( n ) } , y _ n = \\sqrt { \\frac { \\log n } { n } } \\lambda _ 1 ^ { ( n ) } . \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - F _ m ^ - \\big ( ( x , t ) , u , D u , D ^ 2 u \\big ) = 0 ~ ~ & \\mathbb R ^ n \\times ( 0 , T ) , \\\\ u ( x , T ) = g ( x ) ~ ~ & \\mathbb R ^ n , \\end{cases} \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( T S P C _ n \\leq C _ 1 b _ n \\right ) \\geq 1 - D N \\sqrt { n } \\exp \\left ( - 2 C \\frac { n } { N } \\right ) = 1 - e ^ { - \\delta _ N } , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\P ( C ' _ { k _ 1 , j _ 1 } \\cap B ' ) + \\P ( C ' _ { k _ 1 , j _ 2 } \\cap B ' ) & = \\P ( C ' _ { k _ 1 , j _ 1 } ) + \\P ( C ' _ { k _ 1 , j _ 2 } ) = \\P ( C _ { k _ 1 , j _ 1 } ) - p + \\P ( C _ { k _ 1 , j _ 2 } ) + p \\\\ & = \\P ( C _ { k _ 1 , j _ 1 } ) + \\P ( C _ { k _ 1 , j _ 2 } ) = \\P ( C _ { k _ 1 , j _ 1 } \\cap B ) + \\P ( C _ { k _ 1 , j _ 2 } \\cap B ) , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} C \\bullet X + \\epsilon \\sum _ { i = 1 } ^ m I \\bullet Y = - 1 , \\ \\ \\ D _ i \\bullet Y _ i \\le 0 , \\ \\ \\ I \\bullet X + \\sum _ { i = 1 } ^ m I \\bullet Y _ i \\le 0 \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} x ^ n - 1 = f _ 1 ( x ) f _ 2 ( x ) \\ldots f _ r ( x ) , \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\left \\{ 0 \\right \\} = \\mathfrak { g } _ { \\mathbb { C } } ^ { \\left ( 0 \\right ) } \\subset \\mathfrak { g } _ { \\mathbb { C } } ^ { \\left ( 1 \\right ) } \\cdots \\subset \\mathfrak { g } _ { \\mathbb { C } } ^ { \\left ( n - 1 \\right ) } \\subset \\mathfrak { g } _ { \\mathbb { C } } ^ { \\left ( n \\right ) } = \\mathfrak { g } _ { \\mathbb { C } } . \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} = \\oplus ^ k _ { i = 1 } [ \\pi _ i ] [ r _ { - \\mu _ { S ' } } \\circ L L ( w ( \\rho ) ) \\otimes | \\cdot | ^ { - \\langle \\rho _ { M _ { S ' } } , \\mu _ { S ' } \\rangle } ] \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\begin{cases} U _ { m + 1 } '' + A U _ { m + 1 } + U _ { m + 1 } ' = - V _ { m } ' , t > 0 , \\\\ ( U _ { m + 1 } , U _ { m + 1 } ' ) ( 0 ) = ( 0 , ( - 1 ) ^ { m + 1 } u _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\| h \\| : = \\bigg ( | h ( 0 ) | ^ 2 + \\int _ { \\mathbb { R } _ + } | h ' ( x ) | ^ 2 w ( x ) d x \\bigg ) ^ { 1 / 2 } < \\infty \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} P _ { \\textrm { i s o u t , o p t } } ^ { \\textrm { I S A N C } } = & \\left ( 1 - \\exp \\left ( - \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ { 2 } } { \\lambda _ { \\textrm { B N } } P _ { \\textrm { B } } } \\left ( 2 ^ { 2 R } - 1 \\right ) \\right ) \\right ) \\times \\left ( 1 - \\exp \\left ( - \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } } { \\lambda _ { \\textrm { B F } } P _ { \\textrm { B } } } \\left ( 2 ^ { 2 R } - 1 \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\hat { \\sigma } _ ( \\boldsymbol { \\eta } _ k ( t ) ) = 1 . 4 8 3 \\cdot \\left ( | \\boldsymbol { \\eta } _ k ( t ) - \\hat { \\eta } _ k ^ ( t ) | \\right ) , \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\cfrac { | I ^ { c a n } ( f , f ' ) | ^ { 2 } } { \\ < f , f \\ > \\ \\ < f ' , f ' \\ > } = ( * ) \\frac { L ( \\tfrac { 1 } { 2 } , \\Pi \\otimes \\Pi ' ) } { L ( 1 , \\Pi , { \\rm A s } ^ { ( - 1 ) ^ n } ) L ( 1 , \\Pi ' , { \\rm A s } ^ { ( - 1 ) ^ { n - 1 } } ) } . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\bar \\rho \\big ( X ( t , \\cdot ) ^ { - 1 } ( x ) \\big ) \\ , , \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} G / B _ + = \\coprod _ { \\substack { u \\leq w \\\\ u , w \\in W } } R _ { u , w } . \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} \\mu _ { u v } = p ^ 2 s ( u , I , v ) \\leq p ^ 2 \\frac { \\sqrt { n } } { \\log ^ 3 n } = \\frac { 4 \\sqrt { n } } { \\log ^ 4 n } . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\nu _ 0 = [ \\nu ; \\nu _ 0 ( x ) = \\nu ( x ) ] . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\textbf { V } = \\left [ \\textbf { V } _ 1 , \\beta \\textbf { V } _ 2 \\right ] , \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\psi _ n = ( \\eta _ 1 , 1 ) \\boxplus \\cdots \\boxplus ( \\eta _ k , 1 ) \\boxplus ( \\delta _ 1 , 2 ) \\boxplus \\cdots \\boxplus ( \\delta _ l , 2 ) . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} & n C _ { n } ^ { \\lambda } ( t ) = 2 t ( n + \\lambda - 1 ) C _ { n - 1 } ^ { \\lambda } ( t ) - ( n + 2 \\lambda - 2 ) C _ { n - 2 } ^ { \\lambda } ( t ) , \\ ; \\ ; n \\geq 2 , \\\\ & C _ { 0 } ^ { \\lambda } ( t ) = 1 , C _ { n } ^ { \\lambda } ( t ) = 2 \\lambda t . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\tilde D : = \\bigcap _ { k = 0 } ^ { m - 1 } ( e ^ { 2 \\pi i k / m } \\cdot D ) , \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\lim _ { \\tau ' \\to 0 } \\textbf { g } \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) , S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ b ) \\right ) = \\textbf { g } \\left ( e ^ { - \\tau _ 2 / z } T _ a , e ^ { - \\tau _ 2 / z } T _ b \\right ) = g ( T _ a , T _ b ) \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} \\frac { \\partial ^ + g _ t ( z ) } { \\partial a _ t } = - \\pi \\Psi _ { D _ t } ( g _ t ( z ) , \\xi ( t ) ) , g _ 0 ( z ) = z , t \\in [ 0 , t _ 0 ) . \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} \\frac 1 { \\xi ' ( 1 ) } = \\frac { 1 + z } { z ^ 2 } \\log ( 1 + z ) - \\frac 1 z . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} f _ i \\circ N _ A \\circ \\mu = f _ i | _ { A ^ G } \\circ N \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} ( \\mathcal { A } x ) _ i = \\sum _ { i _ 2 , \\ldots , i _ r = 1 } ^ n a _ { i i _ 2 \\cdots i _ r } x _ { i _ 2 } \\cdots x _ { i _ r } , ~ ~ i \\in [ n ] . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} \\gamma \\left ( \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) ( z , Q ) \\right ) = \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} P ( f ) = \\Phi _ 0 \\left ( f ^ { * n } \\right ) \\left ( f \\in \\mathcal { T } ( G ) \\right ) . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\norm { u _ \\lambda } _ V ^ 2 = \\norm { \\nabla u _ \\lambda } ^ 2 _ H + \\norm { u _ \\lambda } ^ 2 _ H \\ , , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left ( \\frac { 1 - q ^ t } { z } \\right ) ^ a \\binom { \\ell _ { q ^ t } \\left ( Q \\right ) } { a } = ( - 1 ) ^ a \\frac { \\log ( Q ) ^ a } { z ^ a } \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} T _ i = n V _ i + \\sum _ { j = 1 } ^ { V _ i } Z _ { i j } \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\mathbb { T } _ n ( [ u , v ] ) = s p a n ( 1 , c o s ( \\theta ) , \\sin ( \\theta ) , \\dots , \\cos ( n \\theta ) , \\sin ( n \\theta ) ) \\ ; , \\ ; \\ ; \\theta \\in [ u , v ] \\ ; , \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\mathcal L _ j [ w ] : = \\mathcal A _ j [ w ] - M \\frac { \\langle \\nabla z _ j , \\nabla w \\rangle } { z _ j } + 2 a _ { 0 , j , w } + \\left ( m + 2 - p \\right ) a _ { 1 , j , w } , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\mathcal { C } ^ { m , \\delta } ( K ) = \\left \\{ f : \\mathbb { R } \\rightarrow \\mathbb { R } : \\left | \\frac { \\partial ^ m } { \\partial x ^ m } f ( x ) - \\frac { \\partial ^ m } { \\partial x ^ m } f ( y ) \\right | \\leq K | x - y | ^ \\delta \\right \\} \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} \\mathbf { T } ( V ) = \\mathbb { R } \\oplus V \\oplus ( V \\otimes V ) \\oplus \\cdots \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} H _ 1 = ( 1 , \\dots , 1 \\ , | \\ , 1 , \\dots , 1 ) . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar \\lambda _ 1 \\bar h ^ { 1 ^ { \\ast } } _ { 1 1 k } + 4 \\bar \\lambda \\bar h ^ { 2 ^ { \\ast } } _ { 1 1 k } = 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} - \\lambda \\sigma ( z ) \\varphi - \\varphi '' = f _ { 1 } \\left [ z \\right ] \\varphi , \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} { { T ^ M } } ( x ) : = \\sup _ { u \\in S } \\{ T ( u ) - K \\ , d ( x , u ) \\} , x \\in M , \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\widehat { G } ( m , n , b , \\epsilon , s ) = \\sum _ { i _ 1 \\cdots i _ n \\in \\widehat { A } ( m , n , b , \\epsilon ) } \\prod _ { k = 1 } ^ n i _ k ^ { - d s } . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\lambda ( x ) = - x \\cdot S , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ r } ( \\alpha _ H ) \\left [ ( \\frac { r ^ 2 C _ i } { 2 } + D _ p \\epsilon _ { p q i } Y ^ q ) \\nabla Y ^ i \\right ] d \\Sigma _ r = O ( r ^ 6 ) \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} P \\left ( f ( z , t ) , f ( z , 1 ) , f _ t ( z , 1 ) , f \\left ( z , \\frac { 1 } { 1 - z } \\right ) , z , t \\right ) = 0 . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\Delta Q : = V - V \\circ \\Xi ^ t . \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} F _ { \\{ 0 \\} } ( n + 1 ) & = \\sum _ { \\substack { B \\subseteq \\Z \\\\ B } } \\P ( N _ \\leq n + 1 , \\ , Z _ { 1 } = B \\ , | \\ , Z _ 0 = \\{ 0 \\} ) \\\\ & = \\sum _ { \\substack { B \\subseteq \\Z \\\\ B } } \\P ( N _ \\leq n , \\ , | \\ , Z _ { 0 } = B ) \\P ( Z _ 1 = B \\ , | \\ , Z _ 0 = \\{ 0 \\} ) \\\\ & \\geq \\sum _ { \\substack { B \\subseteq \\Z \\\\ B } } \\big ( F _ { \\{ 0 \\} } ( n ) \\big ) ^ { | B | } \\P ( Z _ 1 = B \\ , | \\ , Z _ 0 = \\{ 0 \\} ) = g ( F _ { \\{ 0 \\} } ( n ) , \\epsilon ) . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\lVert f \\rVert _ { L ^ q _ H L ^ p _ z ( U ) } : = \\begin{cases} & \\left ( \\int _ { U ' } \\lVert f ( x ' , \\cdot ) \\rVert ^ q _ { L ^ p ( U _ 3 ) } \\ , d x ' \\right ) ^ { 1 / q } , q \\in [ 1 , \\infty ) , \\\\ & _ { x ' \\in U ' } \\lVert f ( x ' , \\cdot ) \\rVert _ { L ^ p ( U _ 3 ) } , q = \\infty . \\end{cases} \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\begin{aligned} a _ h ( \\sigma _ h , \\tau _ h ) + c _ h ( \\sigma _ h , q _ h ) + c _ h ( \\tau _ h , p _ h ) + b ( \\sigma _ h , v _ h ) + b ( \\tau _ h , u _ h ) & = ( g , v _ h ) , & & ( \\tau _ h , v _ h , q _ h ) \\in \\Sigma _ h \\times V _ h \\times \\tilde Q _ h . \\end{aligned} \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\hat \\nu ( s ) = \\int _ s ^ 1 \\xi '' ( r ) \\nu ( d r ) , \\ \\ s \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} C _ { k , j } \\cup C _ { k + 1 , j } \\subset B \\P ( ( C _ { k , j } \\cup C _ { k + 1 , j } ) \\cap B ) = 0 . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - t ^ { - 1 } \\big ( t \\ , \\psi _ 1 ' ( t ) \\big ) ' & = \\lambda _ 1 \\psi _ 1 ( t ) \\ , , & & t \\in ( a _ 1 , 0 ) \\cup ( 0 , a _ 2 ) \\ , , \\\\ \\psi _ 1 ( a _ 1 ) = \\psi _ 1 ( a _ 2 ) & = 0 \\ , , \\\\ \\psi _ 1 ' ( 0 ^ - ) = \\psi _ 1 ' ( 0 ^ + ) & = 0 \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} c / v \\mapsto \\gamma _ { c } : = \\inf \\{ \\gamma \\in \\Gamma _ K ^ { < \\delta ( w ) } \\cup \\{ - \\infty \\} : ( c / v ) \\in A _ \\gamma ^ { w , b } \\} . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ N g ( \\alpha , T _ j ) = \\sum _ { j , k } \\alpha _ k g _ { k j } T ^ j = \\sum _ k \\alpha _ k T _ k = \\alpha \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} ( \\nu \\circ ( B \\iota ) ^ \\ast ) ( q _ 4 ) = - 2 \\nu ( t _ 2 ^ 2 ) = - 4 s _ 1 t _ 2 \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} v \\mathbb { R } ( S ) = I ^ \\circ : = I \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} P ( f ) = \\sum _ { k = - \\infty } ^ { + \\infty } a ( k ) \\widehat { f \\ast f } ( k ) = \\sum _ { k = - \\infty } ^ { + \\infty } a ( k ) \\widehat { f } ( k ) ^ 2 ( f \\in L ^ p ( \\mathbb { T } ) ) . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\begin{cases} v _ { \\tau \\tau } - ( 1 + \\tau ) ^ { 2 \\ell } \\Delta _ y v = \\frac { 1 } { \\delta } ( 1 + \\tau ) ^ { \\frac { \\mu - 1 + \\sqrt { \\delta } } { 2 } ( \\ell + 1 ) + 2 \\ell } f ( ( 1 + \\tau ) ^ { \\ell + 1 } - 1 , ( \\ell + 1 ) y ) , & y \\in \\mathbb { R } ^ n , \\ \\tau > 0 , \\\\ v ( 0 , y ) = u _ 0 ( y ) , & y \\in \\mathbb { R } ^ n , \\\\ v _ \\tau ( 0 , y ) = \\frac { \\mu - 1 + \\sqrt { \\delta } } { 2 } ( \\ell + 1 ) u _ 0 ( ( \\ell + 1 ) y ) + ( \\ell + 1 ) u _ 1 ( ( \\ell + 1 ) y ) , & y \\in \\mathbb { R } ^ n . \\end{cases} \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} - \\frac 1 { 2 q } + \\left ( 1 - \\frac 1 s \\right ) \\frac { p } { p - 2 } = - \\frac 1 2 + \\frac 1 { 2 p } + \\left ( 1 - \\frac 1 s \\right ) \\frac { p } { p - 2 } \\le - \\frac 1 4 + \\frac 1 { 2 p } < 0 \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\alpha _ i ' = \\min ( 2 , \\alpha _ i / \\alpha _ { \\min } ) , \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big [ \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( 2 t ) ^ { k + 1 } } { ( k + 1 ) ! } \\| A ^ { \\frac { k + 1 } { 2 } } w ( t ) \\| ^ 2 \\Big ] & = 2 \\sum _ { k = 0 } ^ { n - 1 } \\left ( \\frac { ( 2 t ) ^ { k } } { k ! } \\| A ^ { \\frac { k + 1 } { 2 } } w ( t ) \\| ^ 2 - \\frac { ( 2 t ) ^ { k + 1 } } { ( k + 1 ) ! } \\| A ^ { \\frac { k + 2 } { 2 } } w ( t ) \\| ^ 2 \\right ) \\\\ & = \\| A ^ { 1 / 2 } w ( t ) \\| ^ 2 - \\frac { ( 2 t ) ^ { n } } { n ! } \\| A ^ { \\frac { n + 1 } { 2 } } w ( t ) \\| ^ 2 . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} h _ 1 ( 1 ) = ( 1 + z _ 2 ) [ \\xi ' ( 1 ) q - \\xi ' ( q ) + \\xi ' ( 1 ) ( 1 - q ) ] - \\frac { \\xi ' ( 1 ) q [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } { \\xi ' ( q ) } . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} A ( t _ 0 , \\cdots , t _ s , \\xi ; x ) = M _ n ( f ^ { ( n ) } ( t _ 0 , \\cdots , t _ s , \\xi ; x ) , f ^ { ( n ) } ( t _ 0 , \\cdots , t _ s , \\xi ; x ) ^ { \\prime } ) . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} 0 = & \\ \\psi _ r - f _ r \\psi = \\psi _ r - A \\phi \\psi , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} I _ j = \\left ( \\frac { j - 1 } { n } , \\frac { j } { n } \\right ) \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\bar h ^ { 1 ^ { \\ast } } _ { 1 1 } \\bar h ^ { 1 ^ { \\ast } } _ { 1 1 k } + 3 \\bar h ^ { 1 ^ { \\ast } } _ { 1 2 } \\bar h ^ { 1 ^ { \\ast } } _ { 1 2 k } + 3 \\bar h ^ { 2 ^ { \\ast } } _ { 1 2 } \\bar h ^ { 2 ^ { \\ast } } _ { 1 2 k } + \\bar h ^ { 2 ^ { \\ast } } _ { 2 2 } \\bar h ^ { 2 ^ { \\ast } } _ { 2 2 k } = 0 , \\ \\ \\ k = 1 , 2 . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\psi _ 2 = \\int _ 0 ^ { \\infty } \\chi _ 2 ( { \\cdot } , x ) \\mu ( d x ) , \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar H ^ 6 - 3 \\bar H ^ 4 S + 3 \\bar H ^ 2 S ^ 2 - 3 S ^ 2 + 2 S \\\\ & \\geq \\dfrac { ( \\bar H ^ 2 - S ) ^ 2 } 4 \\bigl [ ( 5 - \\dfrac { 3 } { 3 2 } ) S - ( 4 - \\dfrac 3 { 8 } ) \\bar H ^ 2 + \\dfrac 9 { 1 6 } \\bigl ] . \\end{aligned} \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} \\langle [ u , v ] , w \\rangle = - T ^ { D } ( u , v , w ) + \\langle D _ { u } v - D _ { v } u , w \\rangle + \\langle D _ { w } u , v \\rangle . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} h _ { a l g } ( \\widetilde { \\phi \\upharpoonright _ { H } } ) = h _ { a l g } ( \\xi ) + h _ { a l g } ( \\varphi ) , \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} C = ( c _ { i j } ) , c _ { i j } = \\frac 1 2 \\ , ( d _ { i j } + d _ { j i } ) \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\log \\varepsilon _ { j , p } = - \\frac { ( p - 1 ) } { 2 } \\log u _ { p } ( y _ { j , p } ) - \\frac { 1 } { 2 } \\log p , \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\partial _ { x _ 1 } = \\frac { 1 } { x _ 1 } X _ 1 , ~ ~ \\partial _ { x _ 2 } = \\frac { 1 } { x _ 1 } X _ 3 , ~ ~ \\partial _ { x _ 3 } = X _ 2 - X _ 3 , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\lambda _ { k } ^ { \\epsilon , \\hat { v } _ k } = - \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\mathbb { P } _ { x , k } ^ { \\epsilon , \\hat { v } _ k } \\bigl \\{ \\tau _ { k } ^ { \\epsilon , \\hat { v } _ k } > t \\bigr \\} , \\ , \\ , \\ , x \\in D , \\ , \\ , \\ , k \\in \\{ 1 , 2 , \\ldots , n \\} . \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\overline { U } _ { 2 k , 2 a } ( x ; q ) = ( - x q ^ 2 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} E \\left [ \\ne ^ { T t } \\right ] = \\left ( \\frac { ( \\theta \\ne ^ { t } ) _ n } { ( \\theta ) _ n } \\right ) ^ s = \\prod _ { j = 1 } ^ n \\left ( \\frac { \\theta } { \\theta + j - 1 } \\ne ^ t + \\frac { j - 1 } { \\theta + j - 1 } \\right ) ^ s . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} p _ j = \\frac { p _ { N , \\rho , j } } { C } . \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\left ( \\langle \\partial _ { i } z , \\partial _ { j } z \\rangle \\right ) _ { n + 1 \\leqslant i , j \\leqslant m } = \\left ( \\frac { \\partial x } { \\partial z } \\right ) ^ { * } \\cdot \\frac { \\partial x } { \\partial z } . \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} W ^ c _ { 2 k + 1 , \\vec a , \\vec b } ( M ^ { 4 m } ) : = \\int _ { M ^ { 4 m } } \\mathcal { W } ^ c _ { 2 k + 1 , \\vec a , \\vec b } ( M ^ { 4 m } ) , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\tau ( A A ^ * ) = \\tau ( A ^ * A ) \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} q ( x , 0 ) = q _ 0 ( x ) = \\begin{cases} 0 , & x \\ge 0 \\\\ q _ p ( x ) , & x < 0 , \\end{cases} \\\\ \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d X - \\Delta X d t + ( X \\cdot \\nabla ) X d t = \\sum _ { i = 1 } ^ N A _ i ( X ) d \\beta _ i ( t ) + \\nabla \\pi d t ( 0 , \\infty ) \\times \\mathbb { R } ^ d , \\\\ \\nabla \\cdot X = 0 ( 0 , \\infty ) \\times \\mathbb { R } ^ d , \\\\ X ( 0 ) = x \\left ( L ^ p ( \\mathbb { R } ^ d \\right ) ^ d , \\end{array} \\right . \\ \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\dim \\Xi _ { \\chi } = ( q - 1 ) q ^ { \\frac { m + d - 1 } { 2 } } . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\begin{cases} t - \\mu ^ 2 ( 2 t \\mathbf M ^ { - 1 } ) & = t ^ o \\\\ z _ 1 + \\mu ^ 2 ( 2 z _ 1 \\mathbf M ^ { - 1 } ) & = z _ 1 ^ o \\\\ z _ 2 + \\mu ^ 2 ( 2 z _ 2 \\mathbf M ^ { - 1 } ) & = z _ 2 ^ o . \\end{cases} \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\Phi ( t ) = \\mathbb { E } [ e ^ { t \\log ( Y ) } ] = \\mathbb { E } [ Y ^ { t } ] . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} \\varepsilon U ' = f ( x , U ) . \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} f _ p ( x ) = \\frac { x } { p x + 1 - p } . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} A ^ 2 - B ^ 2 = \\frac { 2 B } { c ^ 2 } \\ \\ \\mbox { a n d } \\ \\ \\frac { B c ^ 2 } { 2 } + 1 = \\frac { ( a ^ 2 + b ^ 2 ) ^ 2 } { 4 a ^ 2 b ^ 2 } \\ . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} e ( t ) = \\frac { 1 } { L } \\int _ 0 ^ { L } u ^ 2 ( x , t ) \\ , d x . \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } f ( g ) a ( g ) = 0 \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} A _ j = \\left ( \\begin{array} { c c c c } \\Phi ( \\boldsymbol { x } ^ j _ 1 , \\boldsymbol { x } ^ j _ 1 ) & \\cdots & \\Phi ( \\boldsymbol { x } ^ j _ 1 , \\boldsymbol { x } _ { N _ j } ^ j ) \\\\ \\vdots & \\ddots & \\vdots \\\\ \\Phi ( \\boldsymbol { x } ^ j _ { N _ j } , \\boldsymbol { x } ^ j _ 1 ) & \\cdots & \\Phi ( \\boldsymbol { x } ^ j _ { N _ j } , \\boldsymbol { x } ^ j _ { N _ j } ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} H ^ { - 1 } ( A ^ \\bullet ( q = 0 ) ) = 0 . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} Q = \\{ a _ { n + m } \\colon m \\in \\omega \\} \\cup \\{ b _ { n + m } \\colon m \\in \\omega \\} \\cup \\{ c _ { n + m } \\colon m \\in \\omega \\} \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log b _ c ^ { ( n ) } } { \\log b _ c ^ { ( n ) ' } } & = \\lim _ { n \\to \\infty } \\frac { \\log n + ( r - 1 ) \\log ( n p _ n ) - n p _ n } { \\log n + ( r - 1 ) \\log ( n p _ n ) + n \\log ( 1 - p _ n ) } \\\\ & = \\lim _ { n \\to \\infty } \\frac { \\log n / ( n p _ n ) + ( r - 1 ) \\log ( n p _ n ) / ( n p _ n ) - 1 } { \\log n / ( n p _ n ) + ( r - 1 ) \\log ( n p _ n ) / ( n p _ n ) + \\log ( 1 - p _ n ) / p _ n } = 1 \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\Gamma _ A : = \\bigcup _ { \\alpha \\in A } \\Gamma _ { \\nu _ \\alpha } \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} \\sum _ { i \\in [ k ] } w ( F _ i ^ + ) c _ { \\Sigma , A } ( F _ i ^ + , x ) + w ( F _ i ^ - ) c _ { \\Sigma , A } ( F _ i ^ - , x ) = 0 \\mod L _ { \\R } ( A ) ~ x \\in \\R ^ n \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } D ( x _ n , x ) = 0 , \\ , x \\in X \\implies D ( x , y ) \\leq \\limsup _ { n \\to + \\infty } D ( x _ n , y ) , \\ , y \\in X . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} f & = \\frac { \\sqrt { | H _ 0 | ^ 2 + \\frac { d i v ( V ^ 2 \\nabla \\tau ) ^ 2 } { V ^ 2 + V ^ 4 | \\nabla \\tau | ^ 2 } } - \\sqrt { | H | ^ 2 + \\frac { d i v ( V ^ 2 \\nabla \\tau ) ^ 2 } { V ^ 2 + V ^ 4 | \\nabla \\tau | ^ 2 } } } { V \\sqrt { 1 + V ^ 2 | \\nabla \\tau | ^ 2 } } . \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\P \\left ( A ^ c \\cap C _ { k + 1 , j + 1 } \\right ) & = 0 , \\\\ \\P \\left ( A ^ c \\cap \\bigcup _ { j _ 1 \\ne j , j + 1 } C _ { k , j _ 1 } \\right ) & = 0 . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} g ( \\xi ) = \\int _ \\Omega \\frac { g ^ { q ' - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta , \\xi \\in \\overline { \\Omega } \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\partial ^ 2 _ { j k } = \\left ( \\delta _ { j k } - \\frac { x _ j x _ k } { r ^ 2 } \\right ) \\frac { \\partial _ r } { r } + \\frac { x _ j x _ k } { r ^ 2 } \\partial ^ 2 _ r , \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} g = d r ^ 2 + \\phi ^ 2 ( r ) d \\theta ^ 2 , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} M & = u ( 0 ) = \\int _ { \\Omega } G ( y ) u ^ p ( y ) ~ d y = \\frac { 1 } { 2 \\pi } \\int _ { \\Omega } \\log ( 1 / | y | ) \\ , u ^ p ~ d y - \\int _ { \\Omega } g u ^ p d y \\\\ & \\geq \\frac { 1 } { 2 \\pi } \\int _ { \\Omega } \\log ( 1 / | y | ) \\ , u ^ p ~ d y - C \\int _ { \\Omega } u ^ p ~ d y \\\\ & \\geq \\frac { 1 } { 2 \\pi } \\int _ { \\Omega } \\log ( 1 / | y | ) \\ , u ^ p ~ d y - C ' , \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\langle v _ 1 , v _ 2 \\rangle = \\langle T _ 1 v _ 2 , T _ 2 v _ 2 \\rangle . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} z \\frac { \\partial } { \\partial z } ( z ^ { - \\mu } z ^ \\rho \\alpha ) = - \\mu z ^ { - \\mu } z ^ \\rho + z ^ { - \\mu } \\rho z ^ \\rho \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} X = X ^ 0 \\supseteq X ^ 1 \\supseteq \\cdots \\supseteq X ^ { \\dim X } \\supseteq X ^ { \\dim X + 1 } = \\emptyset . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } \\right ) ( z , Q ) = \\lim _ { t \\to 0 } P _ { q ^ t , z } \\cdot \\varphi _ { q ^ t , z } ^ * J ^ { K \\textnormal { t h , e q } } ( q ^ t , Q ) \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\Delta t : = t - \\tau \\le \\frac { 3 } { 4 } \\left ( \\frac { L ^ 2 } { 4 } + L + 2 \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} f ( 0 ) & = 1 , \\\\ f ( \\alpha + 1 ) & = f ( \\alpha ) + 1 + \\alpha , \\\\ f ( \\lambda ) & = \\textstyle \\sup _ { \\alpha < \\lambda } f ( \\alpha ) \\qquad , \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} I = \\int _ a ^ b f ( x ) \\ , \\d x . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} M _ { 0 , 1 } > 0 , M _ { 1 , i } = \\frac { c _ { A _ 1 } ^ 2 } { c _ 1 ^ 2 } \\Xi _ i ( \\varphi ) = \\frac { c _ { A _ 1 } ^ 2 } { c _ 1 ^ 2 } | \\Xi ( \\varphi ) | \\delta _ { 1 , i } , i = 1 , \\dots , N , \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} E : = \\left \\{ ( j , k ) : [ \\tilde A _ i ] _ { j k } \\ne 0 i \\in \\{ 0 , 1 , \\ldots m \\} \\right \\} \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ H f = f ^ { 2 ^ * - 1 } + g ( \\xi , f ) , & u > 0 , \\Omega , \\\\ u = 0 & \\quad \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} A = ( g _ { j - 1 } | _ \\Sigma ) ^ { - 1 } ( 0 ) \\cap U _ j . \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\widetilde { \\textbf { h } ' } _ k = \\sqrt { P _ { \\rm t r } } \\textbf { B } _ { k } ^ { \\rm H } \\textit { \\textbf { h } } _ k + \\sqrt { P _ { \\rm t r } } \\textbf { C } _ k \\textbf { F } ^ { \\rm H } \\textit { \\textbf { h } } _ k + \\textbf { n } _ k , \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell j ^ \\alpha / a } { d \\sum _ { j = 1 } ^ { \\ell + 1 } j ^ \\alpha / a - ( \\ell + 1 ) ^ \\alpha / a } = \\frac 1 d . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} P ^ { - \\ell _ q ( Q ) } = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\binom { \\ell _ q ( Q ) } { k } \\left ( 1 - P ^ { - 1 } \\right ) ^ k \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq \\varepsilon \\right ) & = P ( A _ n ( t ) > t , \\quad \\forall t = a _ n , \\ldots , \\lfloor n - \\varepsilon f _ 2 ( n ) \\rfloor ) \\\\ & = P ( B _ 4 ^ { ( n ) } \\cap B _ 5 ^ { ( n ) } ) , \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} e r ( g _ 1 ) & = \\| g _ 1 - \\hat g _ 1 \\| _ { L ^ 2 [ 0 , T ] } \\approx \\Big ( \\frac { \\Delta t } { 2 } \\sum _ { i = 0 } ^ N p _ i \\left | g _ 1 ( t _ i ) - \\hat g _ 1 ( t _ i ) \\right | ^ 2 \\Big ) ^ \\frac { 1 } { 2 } , \\\\ e r ( | g _ 2 | ) & = \\big \\| | g _ 2 | - | \\hat g _ 2 | \\big \\| _ { L ^ 2 [ 0 , T ] } \\approx \\Big ( \\frac { \\Delta t } { 2 } \\sum _ { i = 0 } ^ N p _ i \\left ( | g _ 2 ( t _ i ) | - | \\hat g _ 2 ( t _ i ) | \\right ) ^ 2 \\Big ) ^ \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} g _ { ( b ) } ( z , t ) = \\Psi [ \\Theta [ f ( z , t ) ] ] . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\kappa _ { t } = ( 1 + \\alpha _ t + n \\Gamma _ { t } + n ^ { 1 / 2 } \\delta _ t ) \\sum _ { s = 0 } ^ { t - 1 } [ 1 + \\psi _ s \\sqrt { n } ( 1 + \\psi _ s \\sqrt { n } ) ] t e ^ { 1 6 \\alpha _ { s , t } } . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} B B ^ * e _ x = 0 , B B ^ * e _ v = \\frac { 2 \\gamma } { m ^ 2 } e _ v , \\qquad B B ^ * e _ k = 2 \\lambda _ k k ^ { - 2 s } e _ k , \\ k \\geq 1 . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} r _ h = y _ h - y - d \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\leq \\sum _ { j = - \\infty } ^ { m _ 0 - ( k - i - 1 ) p / Q } 2 ^ { j s } \\Big [ g _ j ( x _ { k - i } ) + g _ j ( x _ { k - ( i + 1 ) } ) \\Big ] , \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} D _ \\zeta G ( 0 , p _ 0 ) \\xi = \\sum _ { k \\in \\Bbb Z } \\tau _ k \\xi _ k e ^ { i k x } , \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} C \\bullet X + c _ j x _ j & = - 1 \\\\ \\pm I \\bullet X + a _ { i j } x _ j & \\le 0 \\ \\ \\ \\ \\forall \\ i A _ i = \\pm I \\\\ a _ { i j } x _ j & \\le 0 \\ \\ \\ \\ \\forall \\ i A _ i = 0 \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} \\Big | \\sum _ { k = 1 } ^ K \\prod _ { j = 1 } ^ { n - 1 } e _ { j , k } e _ { n } \\Big | _ { { Y ( } X _ { n + 1 } ) } \\le \\Big | \\sum _ { k = 1 } ^ K \\prod _ { j = 1 } ^ { n - 1 } e _ { j , k } \\Big | _ { Y ( Y ( X _ 1 , \\dots , X _ { n - 1 } ) ) } | e _ { n } | _ { X _ { n } } \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { \\boldsymbol { \\psi } } ^ k ( t ) & = \\bar { \\boldsymbol { \\psi } } ( t _ k ) + \\int _ { t _ k } ^ t \\mathbf { f } _ { \\boldsymbol { \\psi } } \\left ( \\tilde { \\boldsymbol { \\psi } } ^ k ( v ) \\right ) d v , t \\ge t _ k . \\end{aligned} \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ n a _ { i \\ell } = \\sum _ { \\ell = 1 } ^ n a _ { j \\ell } \\ ; \\ ; \\mbox { f o r a l l } \\ ; \\ ; i , j \\in \\{ 1 , 2 , \\dots , n \\} ; \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} H _ j = I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\langle \\tilde { v } , v \\rangle _ { - \\gamma , \\gamma } = \\langle \\tilde { v } , v \\rangle , \\textrm { f o r } \\tilde { v } \\in L ^ 2 ( \\Omega ) , v \\in \\mathbb H ^ { \\gamma } ( \\Omega ) . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } { { v o l } ( { \\mathfrak { D } } ( f , \\sigma ) ) } = { v o l } ( [ 0 , 1 ) ^ n \\cap { \\mathfrak { D } } _ f ) . \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} m _ \\varphi : = \\inf _ { \\textup { A m p } ( \\{ \\theta \\} ) } \\frac { \\varphi _ a - \\varphi _ b } { a - b } = \\inf _ { \\textup { A m p } ( \\{ \\theta \\} ) } \\frac { \\varphi _ c - \\varphi _ d } { c - d } , \\ \\ M _ \\varphi : = \\sup _ { \\textup { A m p } ( \\{ \\theta \\} ) } \\frac { \\varphi _ a - \\varphi _ b } { a - b } = \\sup _ { \\textup { A m p } ( \\{ \\theta \\} ) } \\frac { \\varphi _ c - \\varphi _ d } { c - d } . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} K ( z , t ) & = ( 1 - 2 z ) ( 1 - z ) ( 1 - z - t + t ^ 2 z - t ^ 2 z ^ 2 ) \\\\ & \\qquad ( 1 - z - 2 t + 2 t z + t ^ 2 + t ^ 2 z - 3 t ^ 2 z ^ 2 - 2 t ^ 3 z + 2 t ^ 2 z ^ 3 + 2 t ^ 3 z ^ 2 + t ^ 4 z ^ 2 - t ^ 4 z ^ 3 ) . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} S _ { k } ( \\gamma ) = \\exp ( \\lambda _ { 1 } a _ { 1 } ) \\otimes \\cdots \\otimes \\exp ( \\lambda _ { d _ { k } } a _ { d _ { k } } ) = \\exp ( a ) = u . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} X ( C ^ 2 + 1 ) + ( C ^ 4 + C ) = 0 . \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} u _ t + f ^ \\delta ( x , u ) _ x = 0 \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} - \\Delta _ g u = \\rho _ 1 \\left ( \\frac { e ^ u } { \\int _ { M } e ^ u } - \\frac { 1 } { | M | } \\right ) - \\rho _ 2 \\left ( \\frac { e ^ { - u } } { \\int _ { M } e ^ { - u } } - \\frac { 1 } { | M | } \\right ) . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\frac { L ( e ^ { n + 1 } t ) } { L ( t ) } = \\frac { L ( e ^ { n + 1 } t ) } { L ( e ^ n t ) } \\frac { L ( e ^ { n } t ) } { L ( t ) } = \\frac { L ( e s ) } { L ( s ) } \\frac { L ( e ^ { n } t ) } { L ( t ) } . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} B ( \\{ s _ n \\} , \\{ t _ n \\} , N ) : = \\big \\{ x \\in ( 0 , 1 ) : s _ n - t _ n \\leq { a _ n ( x ) } \\leq s _ n + t _ n , \\ \\forall n \\geq N \\big \\} . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} { \\mathcal P } : = \\{ ( p , q , r ) \\in { \\mathbb R } _ { + + } \\times { \\mathbb R } ^ 2 : p + q z ^ k + r z ^ { n - k } { \\rm \\ h a s \\ a l l \\ r o o t s \\ s a t i s f y \\ } | z | > 1 \\} . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\sup _ { \\alpha > 0 } \\alpha \\mu \\{ s \\in \\mathbb { Z } : | k _ { k , \\lambda + i \\gamma } ( s ) | > \\alpha \\} ^ { \\frac { 1 } { p } } & = \\sup _ { 0 < \\alpha \\leq 1 } \\mu \\{ m ^ { k } \\in \\mathbb { Z } : m > 0 \\ , \\ , \\textnormal { a n d } \\ , \\ , m < \\frac { 1 } { \\alpha ^ { \\frac { 1 } { \\lambda } } } \\} \\\\ & = \\sup _ { 0 < \\alpha \\leq 1 } \\alpha [ \\frac { 1 } { \\alpha ^ { \\frac { 1 } { \\lambda } } } ] ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} F ( z ) = g ( z ) + \\frac { \\alpha } { z } + o \\Big ( \\frac { 1 } { z } \\Big ) , z \\in D , \\ | z | \\to \\infty . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { ( q ^ 2 ; q ^ 2 ) _ n q ^ { n ^ 2 } } { ( - q ^ 2 ; q ^ 2 ) _ n ( q ; q ^ 2 ) _ { n + 1 } } = \\frac { 1 } { 2 } \\sum _ { n \\ge 0 } q ^ { n ^ 2 / 2 } ( 1 + q ^ { 2 n + 1 } ) ( 1 + q ^ { n + 1 / 2 } ) \\sum _ { | j | \\le n } q ^ { j ^ 2 / 2 } \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\det ( \\rho _ { [ l ] } ( \\tilde { \\Theta } ) ) & = c _ l \\cdot ( t - \\theta ) ^ { s _ l } c _ l \\in \\bar { K } ^ \\times , s _ l \\geq 1 , \\\\ \\sigma \\bigl ( \\rho _ { [ l ] } ( R ) \\bigr ) & = \\rho _ { [ l ] } ( R ) \\cdot \\rho _ { [ l ] } ( \\tilde { \\Theta } ) . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} ( ( - \\Delta ) ^ { \\alpha / 2 } + x ^ 2 ) u ( x ) = \\lambda u ( x ) , x \\in { \\mathbb R } . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} \\gamma _ { w } \\gamma _ { v } \\gamma _ { e } = - \\gamma _ { w } \\gamma _ { e } \\gamma _ { v } + 2 \\langle v , e \\rangle \\gamma _ { w } . \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} & \\int _ { x _ i \\ge 0 , \\sum x _ i \\le 1 } e ( \\sum _ { i = 1 } ^ m A _ i x _ i ) d x _ 1 \\dots d x _ m \\\\ = \\ , & \\frac { 1 } { ( 2 \\pi i ) ^ m } \\left \\{ \\sum _ { i = 1 } ^ { m } \\frac { e ( A _ i ) } { A _ i \\prod _ { j = 1 \\atop j \\ne i } ^ { m } ( A _ i - A _ j ) } + \\frac { 1 } { \\prod _ { j = 1 } ^ m ( - A _ j ) } \\right \\} , \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} & \\Psi _ 0 ( v ) = x , \\frac { d } { d t } \\Psi _ t ( v ) \\Big | _ { t = 0 } = v , \\\\ & \\frac { D ^ 2 } { d t ^ 2 } \\Psi _ t ( v ) = \\frac { 1 } { 2 } \\nabla V ^ 2 ( \\Psi _ t ( v ) ) , \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} | g | _ H = \\min \\left \\{ \\sum _ { i = 1 } ^ k | n _ i | \\ , | g _ i | : g = \\sum _ { i = 1 } ^ k n _ i g _ i \\right \\} , \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\jmath _ { U _ 1 ^ n } \\left ( u _ 1 ^ n , d \\right ) & = \\sum _ { i = 1 } ^ n \\frac { \\min ( \\theta _ n , \\sigma _ { n , i } ^ 2 ) } { 2 \\theta _ n } \\left ( \\frac { x _ i ^ 2 } { \\sigma _ { n , i } ^ 2 } - 1 \\right ) + \\\\ & \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\log \\frac { \\max ( \\theta _ n , \\sigma _ { n , i } ^ 2 ) } { \\theta _ n } , \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\widetilde X _ 1 = \\left ( \\begin{array} { c c c c c } 0 . 4 3 4 9 & 0 . 1 4 0 6 & 0 . 0 6 5 2 & 0 . 4 9 1 2 & 0 . 9 3 4 5 \\\\ 0 & 0 . 3 7 5 1 & 0 . 0 6 8 2 & 0 . 7 6 6 8 & 0 . 0 5 1 8 \\\\ 0 & 0 . 3 3 8 3 & 0 . 6 8 8 1 & 0 . 9 4 6 6 & 0 . 1 0 0 9 \\\\ 0 & 0 & 0 & 0 . 5 9 1 7 & 0 \\\\ 0 . 2 9 8 9 & 0 . 0 8 7 8 & 0 . 8 3 4 3 & 0 . 2 8 3 4 & 0 . 0 7 6 2 \\end{array} \\right ) , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { Q ( \\pi ( q ) ) Q ( \\pi '' ( q ) ) } & \\sim _ { E ( \\pi ( q ) ) E ( \\pi ' ( q ) ) } ( 2 \\pi i ) ^ { d n ( n + 1 ) / 2 } \\frac { L ^ S ( \\tfrac { 1 } { 2 } , \\Pi \\otimes \\Pi ' ) } { L ^ S ( 1 , \\Pi , { \\rm A s } ^ { ( - 1 ) ^ n } ) L ^ S ( 1 , \\Pi ' , { \\rm A s } ^ { ( - 1 ) ^ { n - 1 } } ) } . \\end{aligned} \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} f ( x ) \\leq y \\in Y , y = f ( z ) z \\geq x . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} U ( x ) = a Q ( \\lambda x + x _ 0 ) , \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} [ \\vec w ] _ { A _ { \\vec p , \\vec r } } \\le [ W ] _ { A _ { \\frac { p _ m } { r _ m } , \\frac { \\delta _ { m + 1 } } { r _ m } } ( \\widehat { w } ) } ^ { \\frac 1 { \\delta _ { m + 1 } } } [ \\widehat { w } ] _ { A _ { \\frac { 1 - r } { r } \\varrho } } ^ { \\frac 1 { \\varrho } } \\prod _ { i = 1 } ^ { m - 1 } \\Big [ w _ i ^ { \\frac { \\theta _ i } { p _ i } } \\Big ] _ { A _ { \\frac { 1 - r } { r } \\theta _ i } } ^ { \\frac 1 { \\theta _ i } } . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} - \\frac { n - 1 } { n } t _ 1 ^ 2 \\cdot \\binom { ( s - 3 ) / 2 } { ( k - 2 ) / 2 } \\{ ( - 1 ) \\cdot ( s - 1 ) ^ 2 t _ 1 ^ 2 \\} ^ { ( k - 2 ) / 2 } \\cdot ( s - 1 ) t _ 1 \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\left [ ( z Q \\partial _ Q ) ^ { N + 1 } - Q \\right ] \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) = 0 \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to \\infty } ( 1 + \\alpha ) ^ { - \\frac { n } { 2 } } C _ n ^ { ( 1 + \\alpha ) } \\left ( ( 1 + \\alpha ) ^ { - \\frac { 1 } { 2 } } x \\right ) = H _ n ( x ) / n ! \\ , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} - \\int _ { S } x ( - \\Psi _ y \\theta _ x + \\Psi _ x \\theta _ y ) d z = - \\int _ { S } \\Psi _ y \\theta d z = \\int _ { S \\times S } \\frac { - \\sin ( y - \\xi _ 2 ) } { 2 ( \\cosh ( x - \\xi _ 1 ) - \\cos ( y - \\xi _ 2 ) ) } \\theta ( z , t ) \\theta ( \\xi , t ) d z d \\xi = 0 \\ , , \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} | \\psi ( t ) - \\psi ( u ) | & = 1 - \\psi ( t ) = 1 - \\dfrac { t - a _ { 2 n - 1 } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } = \\dfrac { b _ { 2 n - 1 } - t } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\\\ & \\leq \\dfrac { u - t } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\leq \\dfrac { t ^ { 1 / k } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } < \\dfrac { 1 } { 2 ^ { 2 n - 1 } } . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} k _ i ^ \\mathrm { L } = \\left \\lfloor b _ i \\right \\rfloor , k _ i ^ \\mathrm { R } = \\left \\lceil b _ i \\right \\rceil , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\sigma _ { a b } = \\tilde { \\sigma } _ { a b } r ^ 2 + O ( r ^ 3 ) , \\ , \\ , l _ { a b } = - \\tilde { \\sigma } _ { a b } r + O ( r ^ 2 ) , \\ , \\ , n _ { a b } = \\frac { 1 } { 2 } \\tilde { \\sigma } _ { a b } r + O ( r ^ 2 ) , \\ , \\ , \\eta _ { a } = \\frac { 1 } { 3 } { \\beta } _ a r ^ 2 + O ( r ^ 3 ) \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\# \\mathcal { B } ( e ) = N _ 1 ( m _ e ) + \\sum _ { i = 2 } ^ { C _ 1 ( m _ e ) } \\ , N _ 1 ( v _ i ) . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} v _ { G L } \\leq v _ { G G L } \\leq v _ { L B B ' } = v _ { R L T _ 1 ' } = v _ { R L T _ 1 } \\leq v _ { L B B ^ { * } } . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} & h _ n \\le \\exp \\left \\{ { - \\frac { 3 ^ { n + 1 } } { \\left ( n + e _ { [ k ] } ^ 1 \\right ) \\ln \\left ( n + e _ { [ k ] } ^ 1 \\right ) \\dots \\ln _ k \\left ( n + e _ { [ k ] } ^ 1 \\right ) } } \\right \\} , \\\\ & ( h _ n ) _ { n \\in \\mathbb N } \\mbox { i s a d e c r e a s i n g s e q u e n c e } , \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} v _ p ( c _ 1 ^ - ) = v _ p ( ( A ^ - ) ^ { n - 1 } n _ 2 ^ n ) , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\Phi ( \\chi _ k ) = \\Phi ( \\chi _ k \\ast \\chi _ k ) = P ( \\chi _ k ) = \\sum _ { j = - \\infty } ^ { + \\infty } a ( j ) \\widehat { \\chi _ k } ( j ) = \\sum _ { j = - \\infty } ^ { + \\infty } a ( j ) { \\delta _ { j k } } = a ( k ) \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} d _ F ( g , \\sigma , g ( 1 ) , \\ldots g ( l ) ) = \\begin{cases} T r _ { W _ 1 } ( A _ g m ^ { l } L _ { \\sigma } T ( e _ { g ( 1 ) } , \\ldots , e _ { g ( l ) } ) g , g _ 1 , \\ldots g _ { l } \\in F \\\\ 0 \\end{cases} \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\mbox { c o v } ( E _ j , E _ i ) = & \\mathbb { E } [ E _ j E _ i ] - \\mathbb { E } [ E _ j ] \\mathbb { E } [ E _ i ] = m p ^ j ( 1 - p ^ i ) \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} J ( u ) = \\vert \\nabla u \\vert \\sqrt { V ( u ) } . \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} \\nu _ { \\eta ' } ( a _ 0 + \\ldots + a _ n x ^ n ) : = \\sum _ { i = 1 } ^ r \\nu _ t ( \\eta ' - \\eta _ i ) \\mbox { w h e r e } \\iota ( a _ 0 ) + \\ldots + \\iota ( a _ n ) x ^ n = \\prod _ { i = 1 } ^ r ( x - \\eta _ i ) . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} f ^ { \\uparrow } ( x ; p ) = f ' ( x ; p ) . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { | y _ \\varepsilon - x _ \\varepsilon | } { \\nu _ \\varepsilon } = + \\infty \\ , , \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} U ( t ) : = e ^ { \\sum _ { i = 1 } ^ N \\left [ \\int _ 0 ^ t B _ i ( s ) d \\beta _ i + \\beta _ i \\theta _ i - \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s - \\frac { 1 } { 2 } \\theta _ i ^ 2 t - \\theta _ i \\int _ 0 ^ t B _ i ( s ) d s \\right ] } y ( t ) . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} J _ X = Q J _ Y = Q ^ T . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N D F } } ^ { \\textrm { O F D M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { N } } \\right ) P _ \\textrm { F } { | h _ { \\textrm { B N } } | ^ 2 } } { { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } \\theta } , \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} & J _ { E ^ { ( 1 ) } } = G \\begin{pmatrix} 1 + \\mathrm { O } ( t ^ { - 1 } ) & \\mathrm { O } ( t ^ { - 2 \\gamma + 2 n - 1 } ) \\\\ \\mathrm { O } ( t ^ { 2 \\gamma - 2 n - 2 } ) & 1 + \\mathrm { O } ( t ^ { - 1 } ) \\end{pmatrix} G ^ { - 1 } , k \\in \\partial C , \\\\ & J _ { E ^ { ( 1 ) } } = G \\begin{pmatrix} 1 + \\mathrm { O } ( t ^ { - 1 } ) & \\mathrm { O } ( t ^ { 2 \\gamma - 2 n - 2 } ) \\\\ \\mathrm { O } ( t ^ { - 2 \\gamma + 2 n - 1 } ) & 1 + \\mathrm { O } ( t ^ { - 1 } ) \\end{pmatrix} G ^ { - 1 } , k \\in \\partial C _ d . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} \\hat { u } _ j ( t , \\xi ) = \\sum _ { k = 0 } ^ \\infty \\sum _ { m = 0 } ^ \\infty \\sum _ { n = 0 } ^ \\infty \\hat { u } _ { j k m n } ( t , \\xi ) , \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} A _ 0 ( r ) = r N ^ 0 _ r = \\sup _ { x _ 0 \\in \\R ^ 3 } \\int _ { B _ r ( x _ 0 ) } | u _ 0 | ^ 2 \\ , d x , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\lambda ^ 3 - \\frac { 3 \\lambda } { 4 } - & \\frac { 1 } { 4 } \\cos { 3 \\alpha } - \\left ( 3 \\lambda ^ 3 - \\frac { 3 } { 4 } \\cos 3 \\alpha \\right ) \\cdot \\Delta ^ 2 = \\\\ & 2 \\sqrt { \\lambda ^ 6 + \\frac { 3 \\lambda ^ 4 } { 4 } - \\frac { \\lambda ^ 3 } { 2 } \\cos 3 \\alpha + \\frac { 9 \\lambda ^ 2 } { 1 6 } - \\frac { 3 \\lambda } { 1 6 } \\cos 3 \\alpha + \\frac { 1 } { 1 6 } \\cos ^ 2 3 \\alpha } \\cdot \\Delta ^ 3 . \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\frac { c _ 1 ^ - } { | c _ 1 ^ - | } = ( - 1 ) ^ { n - a } . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\bar { u } _ \\varepsilon ( r _ { \\varepsilon } ) = B _ { \\varepsilon } ( r _ { \\varepsilon } ) + o \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , , \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} L ( \\boldsymbol r , \\boldsymbol { \\dot r } ) = \\frac { m } { 2 } \\Big ( | \\boldsymbol { \\dot r } | ^ 2 + \\big ( \\nabla f ( \\boldsymbol r ) , \\boldsymbol { \\dot r } \\big ) ^ 2 \\Big ) - m g f ( \\boldsymbol r ) . \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\begin{aligned} - 1 6 L \\mathcal Q ( t ) = & ~ { } 2 ( 9 + \\tilde \\sigma ) \\int | \\varphi ' | f ^ 2 + ( - 3 + 5 \\tilde \\sigma ) \\int | \\varphi ' | f _ x ^ 2 \\\\ & ~ { } + 4 ( - 1 + \\tilde \\sigma ) \\int | \\varphi ' | f _ { x x } ^ 2 + ( 1 + \\tilde \\sigma ) \\int | \\varphi ' | f _ { x x x } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} \\| \\tilde { h } \\| _ { \\bar { \\mathcal { H } } ( [ 0 , T _ 2 ] ) } = \\| K ^ { - 1 } \\tilde { h } \\| _ { L ^ 2 ( [ 0 , T _ 2 ] ) } . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} ( \\tilde { \\nabla } ^ 2 R _ 2 ) ( x , t ) = O ( t ^ { - \\frac { N + 2 A _ 2 } { 2 } } ) \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} P _ { ( y , \\eta ) } = T _ { ( y , \\eta ) } P T _ { ( y , \\eta ) } ^ { - 1 } , T _ { ( y , \\eta ) } f ( x ) = e ^ { { \\frac { i } { h } } \\langle x , \\eta \\rangle } f ( x - y ) . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\begin{cases} b _ { 1 } = x _ { 1 } + r \\left ( 1 - \\frac { 1 } { x _ { 2 } } \\right ) \\\\ b _ { j + 1 } = \\frac { 2 } { n } - \\frac { 1 } { b _ { j } } \\end{cases} \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} K _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) & + K _ \\Omega ( z _ 0 , y _ n ) - K _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , y _ n ) \\\\ & = K _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) + K _ \\Omega ( z _ 0 , y _ n ) - K _ \\Omega ( z _ 0 , \\gamma _ n y _ n ) \\\\ & \\geq K _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) - K _ \\Omega ( \\gamma _ n y _ n , y _ n ) \\rightarrow \\infty . \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } p _ n u _ { p _ n } ( x ) = 8 \\pi \\sqrt { e } \\sum _ { i = 1 } ^ k G ( x , x _ i ) \\ \\mbox { i n } C ^ 2 _ { l o c } ( \\bar \\Omega \\setminus \\mathcal S ) \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} B _ n ( t ) = e ^ { - t } \\sum _ { k = 0 } ^ \\infty \\frac { t ^ k } { k ! } k ^ n \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\Gamma ( k - 1 ) / \\Gamma ( k ) \\cong \\bigoplus _ { \\substack { \\pi \\in \\overline { F } \\\\ f ^ { \\circ ( k - 1 ) } ( \\pi ) = 0 } } S _ { X _ \\pi } \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{gather*} \\# \\mathfrak { R } ( f , i d , k _ 1 , L ) = \\begin{cases} 1 & a \\le 0 , \\\\ \\varphi ( L ) & a > 0 . \\end{cases} \\end{gather*}"}
-{"id": "6940.png", "formula": "\\begin{align*} f _ { N _ 1 } ( n ) = \\frac { ( \\lambda _ { } / c \\lambda _ { } ) ^ n } { c B ( n + 1 , c - 1 ) ( \\lambda _ { } / c \\lambda _ { } + 1 ) ^ { n + c } } , \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} B _ \\rho ( z _ 0 , 2 ^ { i + 1 } \\alpha ) ~ \\subseteq ~ \\bigcup ^ { 2 ^ { { \\bf d } ( E ) } } _ { j = 1 } B _ \\rho ( x _ j , 2 ^ { i } \\alpha ) \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} j ( w ) = \\frac { a } { b ^ 2 } | c + w | - \\frac { c } { b ^ 2 } \\Re { ( c + w ) } \\ . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} X ^ { h } ( x ) \\ , C _ { h i j } = 0 , { X ^ i } _ { | _ j } = - \\delta ^ i _ j . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} & \\sum _ { j \\in \\Z _ { \\geq 0 } } \\binom { m } { j } \\left ( a ( n + j ) b \\right ) ( m + k - j ) c \\\\ & \\phantom { . . . . . . . . . . . . . } = \\sum _ { j \\in \\Z _ { \\geq 0 } } ( - 1 ) ^ j \\binom { n } { j } \\left [ a ( m + n - j ) b ( k + j ) c - ( - 1 ) ^ n b ( n + k - j ) a ( m + j ) c \\right ] \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} { n _ p ^ \\ast } = \\left \\{ \\begin{aligned} & n _ p ^ { \\mathrm { c e i l } } , & ~ \\gamma _ { } ^ { \\mathrm { c e i l } } \\geq \\gamma _ { } ^ { \\mathrm { f l o o r } } , \\\\ & n _ p ^ { \\mathrm { f l o o r } } , & ~ \\gamma _ { } ^ { \\mathrm { c e i l } } < \\gamma _ { } ^ { \\mathrm { f l o o r } } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} ( \\alpha _ { H _ 0 } ) _ a = \\frac { r ^ 2 } { 2 } \\tilde \\nabla _ a ( \\tilde \\Delta + 2 ) Y _ 0 ^ { ( 3 ) } + \\frac { r ^ 3 } { 2 } \\tilde \\nabla _ a ( \\tilde \\Delta + 2 ) Y _ 0 ^ { ( 4 ) } + O ( r ^ 4 ) \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} F ( \\sigma ( z ) ) = J _ \\sigma ( z ) \\cdot F ( z ) . \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} \\dim _ K \\frak { g } _ { \\beta } ( K ) = \\dim \\frak { g } _ { \\beta } { \\otimes } _ O K \\geq \\dim G _ { \\beta } { \\otimes } _ O K \\geq ( G ) . \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i , j , k , l } a _ { i k } d _ { j l } x _ { i j } x _ { k l } : ~ X = ( x _ { i j } ) , ~ X \\in \\Pi _ n \\right \\} , \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} ( 2 \\mu c _ \\alpha t _ \\alpha - e ^ \\alpha ) ( 2 \\mu c _ \\alpha t _ \\alpha - e ^ \\alpha ) = ( 4 \\mu ^ 2 c _ \\alpha ^ 2 + c _ \\alpha ) t _ \\alpha - 4 \\mu c _ \\alpha | \\alpha | a e ^ \\alpha \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} z ( P _ \\tau ) & = a _ 0 + \\sum _ { i = 1 } ^ t \\Big ( a _ i ( i + 1 ) p ^ i - a _ i i p ^ { i - 1 } \\Big ) \\\\ & = \\sum _ { i = 0 } ^ t a _ i p ^ i + \\sum _ { i = 1 } ^ t a _ i i p ^ { i - 1 } ( p - 1 ) . \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\| X [ s _ 1 , \\ldots , s _ r ] \\ , - \\ , A \\| \\ \\to \\ \\min \\\\ s _ 1 \\cdots s _ r \\ = \\ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} & \\left < u , \\tilde { L } ^ \\alpha _ t ( \\varphi ) x ^ \\star \\right > _ { L ^ 2 ( [ 0 , t ] : H ) } = \\int _ 0 ^ t \\left < u ( s ) , ( t - s ) ^ { - \\alpha } [ S ( t - s ) G ( s , \\varphi ( s ) ) ] ^ \\star x ^ \\star \\right > _ H d s \\\\ & = \\int _ 0 ^ t \\left < ( t - s ) ^ { - \\alpha } S ( t - s ) G ( s , \\varphi ( s ) ) u ( s ) , x ^ \\star \\right > _ { E _ 1 , E _ 1 ^ \\star } d s = \\left < L ^ \\alpha _ t ( \\varphi ) u , x ^ \\star \\right > _ { E _ 1 , E _ 1 ^ \\star } . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\mathring { M } _ h ^ v : = \\{ \\mu _ h \\in M _ h : \\mu _ h ( x ) = 0 , \\ x \\in \\Omega \\backslash \\overline { \\mathcal { T } _ h ^ v } ; ( \\Xi \\mu _ h ) n = 0 , f \\in \\Gamma _ o ^ v \\} . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} A + 1 \\le A ^ + ( \\lambda _ 2 ) + 1 < A ^ + ( \\lambda _ 2 + \\omega _ 1 ) = A _ 1 . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} l ( G ) = \\O ( q - 1 ) + 2 f + 1 \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\omega _ { 1 } = c _ { 6 } \\left ( a \\omega _ { 3 } - c _ { 4 } \\omega _ { 4 } \\right ) ^ { 4 } , c _ { 6 } \\in \\mathbb { R } , c _ { 6 } \\neq 0 . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} a = \\sum _ { j = 1 } ^ l a _ j ^ n , b = \\sum _ { j = 1 } ^ m b _ j ^ n , \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} J _ 4 & \\leq C \\cdot w ^ q ( B ) ^ { { ( 1 / q - \\kappa / p ) } } \\sum _ { k = 1 } ^ \\infty \\left ( 1 + \\frac { r } { \\rho ( x _ 0 ) } \\right ) ^ { N \\cdot \\frac { N _ 0 } { N _ 0 + 1 } } \\left ( 1 + \\frac { 2 ^ { k + 1 } r } { \\rho ( x _ 0 ) } \\right ) ^ { - N } \\\\ & \\times \\frac { 1 } { | 2 ^ { k + 1 } B | ^ { 1 - ( \\alpha / d ) } } \\int _ { 2 ^ { k + 1 } B } \\big | b ( y ) - b _ B \\big | \\big | f ( y ) \\big | \\ , d y . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\pi \\varepsilon ^ 2 = \\mathcal H ^ 2 ( B _ \\varepsilon ) \\le \\mathcal H ^ 2 ( \\hat { \\pi } ( S _ h \\cap U _ \\varepsilon ( \\gamma ) ) \\le ( { \\rm L i p } \\ , \\hat { \\pi } ) ^ 2 \\mathcal H ^ 2 ( S _ h \\cap U _ \\varepsilon ( \\gamma ) ) \\ , . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} D ( k ) = D _ { l } ( k ) : = \\mathbb { E } \\left ( d ( Y _ { k } , \\{ Y _ u \\} _ { 1 \\leq u \\leq k - 1 } ) | N _ l = k \\right ) , \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\int _ A \\L ^ { - \\ell } d \\tilde \\mu ^ { G _ i } & = \\sum _ { n } [ A _ n ] \\L ^ { - ( N + 1 ) d - n } = \\sum _ { n , e } [ A _ { n , e } ] \\L ^ { - ( N + 1 ) d - n } \\\\ & = \\sum _ { n , e } [ \\tilde { A } _ { n , e } ] \\L ^ { - ( N + 1 ) d - ( n + e ) } = \\sum _ { m } \\left ( \\sum _ { n + e = m } [ \\tilde { A } _ { n , e } ] \\right ) \\L ^ { - ( N + 1 ) d - m } \\\\ & \\qquad = \\sum _ { m } \\left [ \\pi _ N \\left ( \\tilde { \\ell } ^ { - 1 } ( m ) \\right ) \\right ] \\L ^ { - ( N + 1 ) d - m } = \\int _ { h ^ { - 1 } ( A ) } \\L ^ { - \\tilde { \\ell } } d \\tilde \\mu ^ { G _ i } , \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} \\lim _ { L _ 1 \\to 0 } \\frac { R ( L _ 1 , y , z ) } { L _ 1 } = \\frac { \\cosh \\frac { y } { 2 } + e ^ { - \\frac { z } { 2 } } } { \\cosh \\frac { y } { 2 } + \\cosh \\frac { z } { 2 } } , \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} h ' ( s ( y ) ) = - y . \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\dim _ { H } \\zeta = \\sup \\{ s \\ge 0 \\ : : \\ : \\underset { \\eta \\downarrow 0 } { \\liminf } \\ : \\frac { \\log \\zeta ( B ( x , \\eta ) ) } { \\log \\eta } \\ge s \\zeta x \\in \\mathbb { R } \\} \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} J ' _ 1 ( f ; v ^ 1 , v ^ 2 ) ( \\hat { v } ^ 1 , 0 ) = 0 , \\ \\ \\ \\forall \\hat { v } ^ 1 \\in L ^ 2 ( \\mathcal { O } _ 1 \\times ( 0 , T ) ) \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ { 2 } \\big ( \\sum _ { k = 1 } ^ { J _ { 0 } } \\| f _ { i } ^ { k } \\| _ { L ^ { \\infty } } | B ( y _ { i } , 2 ^ { k } r ) | ^ { 1 / p } a _ { i } ^ { k } \\big ) + \\sum _ { i = 1 } ^ { 2 } \\alpha _ { i } ^ { J _ { 0 } } \\chi _ { B ( y _ { i } , 2 ^ { J _ { 0 } } r ) } . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} c ^ { \\lambda } ( z ) = e ^ { i \\lambda z } , \\ , z \\in \\C \\ , . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\rho ( g , u , h ) \\rho ( h , u ' , g ) = 1 _ { S g } = \\rho ( g , g , g ) \\ \\ \\rho ( h , u ' , g ) \\rho ( g , u , h ) = 1 _ { S h } = \\rho ( h , h , h ) . \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} E ^ { ( 1 ) } = \\mathbf { 1 } + \\mathcal { C } [ \\mathbf { 1 } - J _ { E ^ { ( 1 ) } } ] + \\mathcal { C } [ ( E ^ { ( 1 ) } _ + - \\mathbf { 1 } ) \\cdot ( \\mathbf { 1 } - J _ { E ^ { ( 1 ) } } ) ] , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } I ( r ) } { \\mathrm { d } r } = \\frac { 2 \\pi } { r } - \\frac { 2 \\pi } { r \\sqrt { 1 - r ^ 2 } } . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\left [ \\mu , \\rho \\right ] ( T _ a ) = \\frac { ( T _ a ) + 2 } { 2 } \\rho ( T _ a ) - \\rho \\left ( \\frac { ( T _ a ) } { 2 } T _ a \\right ) = \\rho ( T _ a ) \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\widetilde { A } ( m , n , k , a , \\epsilon ) : = \\left \\{ ( i _ 2 , \\dots , i _ n ) \\in \\mathbb { N } ^ n : \\ \\sum _ { k = 2 } ^ n i _ k ^ a \\in [ m - k ^ a , m ( 1 + \\epsilon ) - k ^ a ] \\right \\} . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} M ( Q , N ) = \\max _ { x \\in S _ { Q } } \\# \\Big \\{ \\tilde { x } \\in S _ { Q } \\mathrel \\Big | \\| \\tilde { x } - x \\| \\leq q ^ { - N } \\Big \\} . \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} p _ m ^ { } = \\Pr { b ^ m \\leq X < b ^ { m + 1 } } , m = 0 , \\pm 1 , \\pm 2 , \\ldots , \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\epsilon \\cdot | F | & \\leq | \\sum _ { \\gamma \\in F } \\mathrm { s i g n } ( T ( g _ \\gamma ) ( x ) ) \\cdot T ( g _ \\gamma ) ( x ) | = | T ( \\sum _ { \\gamma \\in F } \\mathrm { s i g n } ( T ( g _ \\gamma ) ( x ) ) \\cdot g _ \\gamma ) ( x ) | \\\\ & \\leq \\| T ( \\sum _ { \\gamma \\in F } \\mathrm { s i g n } ( T ( g _ \\gamma ) ( x ) ) \\cdot g _ \\gamma ) \\| \\leq \\| T \\| \\| \\sum _ { \\gamma \\in F } \\mathrm { s i g n } ( T ( g _ \\gamma ) ( x ) ) \\cdot g _ \\gamma \\| \\leq \\| T \\| \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} \\left ( f \\ast \\chi _ \\pi \\right ) \\ast \\left ( f \\ast \\chi _ { \\pi ' } \\right ) = f \\ast f \\ast \\chi _ \\pi \\ast \\chi _ { \\pi ' } = 0 , \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} \\bar { \\mathsf h } _ { i j } & = \\cfrac { \\frac { \\rho _ { i j } } { 2 \\mu ^ 2 } } { \\rho _ { i j } \\frac { \\rho _ { i j } } { 2 \\mu ^ 2 } + 2 t ^ 2 \\frac { \\rho _ { i j } } { 2 \\mu ^ 2 } + t ^ 2 \\frac { t ^ 2 - \\sigma _ i ^ 2 } { 2 \\mu ^ 2 } + \\sigma _ i ^ 2 \\frac { t ^ 2 - \\sigma _ j ^ 2 } { 2 \\mu ^ 2 } } ( \\bar { w } _ { i j } ( \\rho _ { i j } + t ^ 2 ) - \\bar { w } _ { j i } \\sigma _ i \\sigma _ j ) \\to \\frac { t ^ * } { \\zeta _ i + \\zeta _ j } ( \\bar { w } ^ * _ { i j } - \\bar { w } ^ * _ { j i } ) . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} \\cosh \\tfrac { \\ell _ \\nu } { 2 } + e ^ { - \\ell _ \\mu } & = 2 \\sinh ( \\ell _ { \\mu ' } ) \\left ( - e ^ { - \\frac { \\ell _ \\mu } { 2 } } + \\sinh \\tfrac { \\ell _ { \\mu ' } } { 2 } \\right ) , \\\\ \\cosh \\tfrac { \\ell _ \\nu } { 2 } + \\cosh \\ell _ \\mu & = 2 \\sinh ( \\ell _ { \\mu ' } ) \\left ( \\sinh \\tfrac { \\ell _ \\mu } { 2 } + \\sinh \\tfrac { \\ell _ { \\mu ' } } { 2 } \\right ) . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\delta _ { * } ( t ) = \\frac { 1 } { t ( 1 + t ^ { 2 } ) } \\int _ { 1 } ^ { t } \\frac { \\left ( R _ { g ( s ) } - s ^ { 2 } \\overline { R } \\right ) _ { * } } { 2 } \\exp \\left ( - \\int _ { s } ^ { t } \\frac { \\tau ( | M | ^ { * } ) ^ { 2 } } { 2 } d \\tau \\right ) d s \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} | | \\sum _ { i = 1 } ^ n X _ i | | _ { s , \\Omega } \\le Z [ \\alpha ] ( s , v ) \\cdot \\sqrt { \\sum _ { i = 1 } ^ n | | X _ i | | _ v ^ 2 } . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} w ( \\beta , \\lambda ) : = w ( i _ 0 , i _ 1 ) \\prod _ { l = 1 } ^ { p - 1 } \\frac { w ( i _ l , i _ { l + 1 } ) } { \\lambda - w ( i _ l , i _ l ) } . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\sum _ { k = \\nu + 1 } ^ n \\sqrt { k } = n A ( n ) - \\nu A ( \\nu ) - \\frac { \\delta _ { \\nu + 1 , n } } { 2 4 } . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} w ^ { ( k \\ell ) } : = w _ 1 ^ { ( k \\ell ) } = w _ 2 ^ { ( k \\ell ) } \\Omega . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} 0 = \\int _ { \\mathcal { Y } ^ d } ( \\widetilde { m } h _ { y _ j } K ^ i + \\widetilde { m } K ^ i _ { y _ j } ) \\widetilde { w } _ { y _ i y _ j } d y . \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} X _ { n } & = \\{ \\xi \\in [ 0 , \\bar { x } ] : ( n ( x ) - n ( \\xi ) ) ( x - \\xi ) < 0 \\textnormal { f o r } x \\neq \\xi , x \\in [ 0 , \\bar { x } ] \\} , \\\\ X _ { f } ( y , v ) & = \\{ \\xi \\in [ 0 , \\bar { x } ] : ( f ( x , y , v ) - f ( \\xi , y , v ) ) ( x - \\xi ) < 0 \\textnormal { f o r } x \\neq \\xi , x \\in [ 0 , \\bar { x } ] \\} , \\\\ X & = \\cap _ { y , v \\in ( 0 , \\bar { y } ) \\times ( 0 , \\bar { v } ) } X _ { f } ( y , v ) \\cap X _ { n } . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} I _ 1 = J _ 1 + J _ 2 + J _ 3 + J _ 4 = J _ 1 + \\frac { 1 } { 2 ^ { \\sqrt { \\delta } } } \\int _ { x - t } ^ { x + t } u _ 0 ( y ) \\bigg [ \\frac { \\partial E } { \\partial b } ( t , x ; b , y ; \\mu , \\nu ^ 2 ) - \\frac { \\mu } { 1 + b } E ( t , x ; b , y ; \\mu , \\nu ^ 2 ) \\bigg ] _ { b = 0 } ^ { b = t - | x - y | } \\ , \\mathrm { d } y - I _ 3 . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} Q : = \\frac { \\delta ^ { 2 } } { h ^ { 2 } } U ^ { - 1 } P U = - \\Delta + W _ { h } ( x ) , \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\mathcal { H } _ { z _ \\varepsilon , \\varepsilon } ( y ) = \\mathcal { H } _ { z _ \\varepsilon } ( y ) + O \\left ( \\frac { \\varepsilon ^ 2 } { d ( z _ \\varepsilon , \\partial \\Omega ) ^ 2 } \\right ) y \\in \\Omega \\ , , \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} & \\zeta ( \\cdot , t ) \\gg ( 0 , 0 , \\cdots , 0 ) \\ \\ { \\rm f o r } \\ \\ t \\in [ 0 , t _ 0 ) , \\ \\ \\zeta ( \\cdot , t _ 0 ) \\succeq ( 0 , 0 , \\cdots , 0 ) , \\\\ & \\exists l _ 0 \\in \\{ 1 , 2 , \\cdots , m \\} , \\ , x _ 0 \\in \\R ^ N , \\ \\zeta _ { l _ 0 } ( x _ 0 , t _ 0 ) = 0 . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} & x _ i ^ \\mathrm { S } = x _ i + x _ \\mathrm { s m a x } \\sigma v _ i , \\\\ & y _ i ^ \\mathrm { S } = y _ i + y _ \\mathrm { s m a x } \\sigma w _ i , \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\begin{aligned} \\| B ^ { ( \\theta ) } f \\| _ { W } & \\leq \\| f \\| _ { W } , & & f \\in W , \\\\ \\| A ^ { ( \\theta ) } f \\| _ { W } & \\leq \\| f \\| _ { W } , & & f \\in W . \\end{aligned} \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} A _ { i } = \\int _ { \\Omega } \\frac { 1 - \\rho ^ { 2 } } { \\left | z - a _ { i } \\right | ^ { 2 } } d z \\leq \\int _ { B _ { R } ( 0 ) \\setminus B _ { R _ { 0 } } ( a _ { i } ) } \\frac { 1 - \\rho ^ { 2 } } { \\left | z - a _ { i } \\right | ^ { 2 } } d z \\leq 2 \\pi I \\left ( \\dfrac { \\delta _ { i } } { R _ { 0 } } \\right ) \\leq 2 \\pi I \\left ( \\dfrac { R } { R _ { 0 } } \\right ) + C \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\sigma ^ { ( 0 ) a b } = - \\frac { 1 } { 3 } \\alpha ^ { a b } \\tilde { \\sigma } ^ { a b } \\gamma _ { a b } ^ { ( 2 ) c } = - \\frac { 4 } { 3 } \\beta ^ c . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} L ( \\mathtt A ) = \\sum _ { t } \\mu ( \\mathrm { c i r c u i t } ( t ) ) t . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} ( { A ^ \\prime } ^ * A ^ \\prime + R ^ * R ) ^ { k + 1 } = { A ^ \\prime } ^ * ( { A ^ \\prime } ^ * A ^ \\prime + R ^ * R ) ^ { k } A ^ \\prime . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\overline { W } _ { k , a } ( n ) = \\overline { W } _ { k , a - 1 } ( n ) . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} X ( t ) = ( x ( t ) , v ( t ) , z _ 1 ( t ) , z _ 2 ( t ) , \\ldots ) , \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} z \\mapsto [ u : v : w ] = [ \\wp ( z ) : \\wp ' ( z ) : 1 ] , 0 \\mapsto [ 0 : 1 : 0 ] , \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\frac { \\hat { \\lambda } _ j ^ { ( n ) } - \\lambda _ k ^ { ( n ) } } { \\sqrt { \\lambda _ j ^ { ( n ) } \\lambda _ k ^ { ( n ) } } } \\langle \\hat { u } _ j ^ { ( n ) } , u _ k ^ { ( n ) } \\rangle & = \\bar { \\eta } _ { k j } ^ { ( n ) } \\langle \\hat { u } _ j ^ { ( n ) } , u _ j ^ { ( n ) } \\rangle + \\sum _ { l \\neq j } \\bar { \\eta } _ { k l } ^ { ( n ) } \\sqrt { \\frac { \\lambda _ l ^ { ( n ) } } { \\lambda _ j ^ { ( n ) } } } \\langle \\hat { u } _ j ^ { ( n ) } , u _ l ^ { ( n ) } \\rangle . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} C _ { } = \\frac { 2 s + d } { d } \\| Q \\| ^ { - \\frac { 4 s } { d } } _ { L ^ 2 } , \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\frac { d } { d t } \\tilde { x } ( t ) = \\int _ { U \\times V } \\tilde { f } ( t , \\tilde { x } ( t ) , m ( t ) , u , v ) \\eta ( d ( u , v ) ) , \\ \\ \\tilde { x } ( s ) = y ' . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} \\nu _ { P } ( \\xi ) = m _ { 1 } \\delta _ { 0 } + m _ { 2 } \\delta _ { q } + \\Delta _ { P } \\delta _ { 1 } . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\abs { t - \\tau } & = B _ \\Omega ( \\sigma ( t ) , \\sigma ( \\tau ) ) \\leq B _ \\Omega ( \\sigma ( t ) , z _ 0 ) + B _ \\Omega ( z _ 0 , \\sigma ( \\tau ) ) \\\\ & \\leq 2 \\alpha + \\beta \\log \\frac { 1 } { \\delta _ \\Omega ( \\sigma ( t ) ) \\delta _ \\Omega ( \\sigma ( \\tau ) ) } \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} F \\left ( \\frac 1 2 , - \\alpha ; \\frac 3 2 ; 1 \\right ) = \\frac { \\sqrt { \\pi } \\Gamma ( 1 + \\alpha ) } { 2 \\Gamma ( \\alpha + 3 / 2 ) } \\ , \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} E [ X _ { k : n } ] = & \\frac { 1 } { \\lambda } ( H _ n - H _ { n - k } ) \\\\ V a r [ X _ { k : n } ] = & \\frac { 1 } { \\lambda ^ 2 } ( G _ { n } - G _ { n - k } ) \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\beta = A Y \\tau _ { 1 } \\rho \\lambda \\alpha ^ { \\ast } . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\left ( z \\frac { \\partial } { \\partial z } + \\mu - \\frac { \\rho } { z } \\right ) ( z ^ { - \\mu } z ^ \\rho \\alpha ) = 0 \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} 0 \\longrightarrow R ( - 2 n - 1 ) ^ { \\binom { n + 1 } { 2 } } & \\xrightarrow { \\ ; F _ n ^ t \\ ; } R ( - 2 n ) ^ { \\binom { n + 2 } { 2 } } \\\\ & \\xrightarrow { \\ ; G _ n \\ ; } R ( - n ) ^ { \\binom { n + 2 } { 2 } } \\xrightarrow { \\ ; F _ n \\ ; } R ( - n + 1 ) ^ { \\binom { n + 1 } { 2 } } \\longrightarrow 0 \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} C = 2 \\sqrt { 2 } m N \\max _ { 1 \\leq k \\leq m } C _ k = 6 \\sqrt { 2 } m N \\frac { \\max _ { 1 \\leq k \\leq m } r _ k ( r _ k - 1 ) } { \\min _ { 1 \\leq k \\leq m } \\min _ { 1 \\leq j , l \\leq r _ k , j \\neq l } | \\alpha _ { k , j } - \\alpha _ { k , l } | } \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} u ^ 3 - 3 u v ^ 2 - 4 L u ^ 2 w - 4 L v ^ 2 w + 4 w ^ 3 = 0 . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} p _ S ( \\textbf { y } _ k | \\boldsymbol { \\psi } , \\textbf { W } _ k ) = { \\frac { 1 } { \\pi ^ { 3 } \\sigma _ z ^ { 6 } } e ^ { - \\frac { { \\left \\| \\textbf { y } _ k - \\lvert \\textbf { s } \\rvert \\beta \\textbf { W } _ k ^ \\textbf { a } ( \\textbf { x } ) \\right \\| } _ 2 ^ 2 } { \\sigma _ z ^ 2 } } } . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\frac { s _ { \\lambda } ( 1 ^ { \\ell ( \\lambda ) + 1 } ) } { \\ell ( \\lambda ) + 1 } & \\geq \\frac { s _ { \\mu } ( 1 ^ { \\ell ( \\lambda ) + 1 } ) } { \\ell ( \\lambda ) + 1 } \\\\ & \\geq \\frac { s _ { \\mu } ( 1 ^ { \\ell ( \\lambda ) } ) } { \\ell ( \\lambda ) } + 1 \\\\ & = \\frac { s _ { \\lambda } ( 1 ^ { \\ell ( \\lambda ) } ) } { \\ell ( \\lambda ) } + 1 . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { N D N } } ^ { \\textrm { O F D M A } } = \\frac { \\left ( 1 - \\beta _ { \\textrm { N } } \\right ) P _ \\textrm { N } { | h _ { \\textrm { B N } } | ^ 2 } } { { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } \\left ( 1 - \\theta \\right ) } , \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\frac { | \\cdot - x _ \\varepsilon | } { \\gamma _ \\varepsilon \\rho _ \\varepsilon } = o \\left ( \\frac { t _ \\varepsilon } { \\gamma _ \\varepsilon ^ 5 } \\right ) \\tilde { \\Omega } _ \\varepsilon : = \\{ y t _ \\varepsilon ( y ) \\le \\gamma _ \\varepsilon \\} \\ , , \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} \\phi ( U ) = U ^ \\sqsubset . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\nu ( q _ 4 ) = \\mu _ 3 - s _ 1 t _ 2 , \\ \\ \\nu ( t _ 2 ^ 2 ) = 2 s _ 1 t _ 2 , \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} s _ 1 \\left ( z \\right ) = c _ s \\left ( z \\right ) s _ 2 \\left ( \\frac 1 z \\right ) , z \\in U _ 1 \\cap U _ 2 \\ , . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } - d i v ( p \\nabla u _ { \\varepsilon } ) = \\dfrac { 1 } { \\varepsilon ^ { 2 } } j ( 1 - | u _ { \\varepsilon } | ^ { 2 } ) u _ { \\varepsilon } & $ i n $ \\ , G \\\\ u _ { \\varepsilon } = g & $ o n $ \\ , \\partial G , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} a _ r ^ \\epsilon : = \\begin{cases} a _ r , & ~ | a _ r - a _ t | \\leq ( \\mu - \\epsilon ) | r - t | ^ { \\kappa } \\\\ a _ t + ( \\mu - \\epsilon ) ( t - r ) ^ \\kappa \\frac { a _ r - a _ t } { | a _ r - a _ t | } , & ~ | a _ r - a _ t | \\geq ( \\mu - \\epsilon ) | r - t | ^ { \\kappa } . \\end{cases} \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} u : [ 0 , \\omega ] \\to \\Sigma , u ( t ) = ( u ^ 1 , \\ldots , u ^ { m + n } ) ( t ) , u ^ i \\in H ^ 1 [ 0 , \\omega ] \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} C _ \\vartheta ( \\theta ) \\theta _ t + \\theta p _ \\theta u _ y = \\left ( \\frac { \\kappa \\theta _ y } { v } \\right ) _ y + \\frac { \\varepsilon u _ y ^ 2 } { v } + \\frac { \\mu | \\mathbf { w } _ y | ^ 2 } { v } + \\frac { \\nu | \\mathbf { h } _ y | ^ 2 } { v } , \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} [ \\xi , \\eta ] _ 0 = T [ u ] ( \\xi , \\eta ) \\ , . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} E ~ \\subseteq ~ \\bigcup _ { i = 1 } ^ { { \\bf N } _ { h _ 2 } } B _ \\rho ( a _ i , h _ 2 ) ~ , \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } { 2 k \\choose k } \\frac { 1 } { k } = \\frac { N + 1 } { 3 } { 2 N + 1 \\choose N } \\sum _ { k = 1 } ^ { N } \\frac { 1 } { k ^ 2 { N \\choose k } ^ 2 } . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\nu ( f ) = 6 , \\ \\nu ( \\partial _ 1 f ) = 3 , \\ \\nu ( \\partial _ 2 f ) = 1 \\mbox { a n d } \\nu ( \\partial _ 3 f ) = 0 , \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} \\lambda v - \\Delta v = f \\Omega , \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} l ( G ) = \\max \\{ \\O ( q - 1 ) + f , \\O ( q + 1 ) + 1 \\} . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\in C \\right ) & \\leq ( - H ( \\ell _ 2 M ) ) \\vee ( - H ( \\ell _ 2 m ) ) \\\\ & = - \\inf _ { y \\in C \\cap [ 0 , \\ell _ 2 ^ { - 1 } ) } I _ 2 ( y ) \\vee - \\inf _ { y \\in C \\cap ( \\ell _ 2 ^ { - 1 } , \\infty ) } I _ 2 ( y ) \\\\ & = - \\inf _ { y \\in C } I _ 2 ( y ) , \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} { V ' } _ { ( r + 1 ) } = { V ' } _ { r + 1 } \\oplus { V ' } _ { - r - 1 } \\oplus { V ' } _ { ( r ) } , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\xi _ k \\times ( \\xi _ i , 2 a _ i ) ) = \\prod _ { j = 0 } ^ { 2 a _ i - 1 } L ^ S ( s + a _ i - j , \\xi _ k \\times \\xi _ i ) . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\lim _ { | s | \\to \\infty } | \\theta ' ( s ) | = 0 \\Longrightarrow \\sigma _ \\mathrm { e s s } ( H ) = [ E _ 1 , \\infty ) \\ , . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} f \\left ( R _ { n } \\right ) = 2 \\pi p _ { 0 } d _ { k } \\log \\frac { 1 } { \\varepsilon _ { n } } + 2 \\pi p _ { 0 } \\left ( d _ { k } ^ { 2 } - d _ { k } \\right ) \\log \\frac { 1 } { R _ { n } } + \\frac { \\pi } { 2 } \\alpha _ { k } \\left ( 1 - \\alpha \\right ) R _ { n } ^ { s _ { k } } \\log \\frac { 1 } { \\varepsilon _ { n } } \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\Psi _ n ( s , t ) : = \\varphi _ n ( s ) \\ , \\psi _ 1 ( t ) \\ , , \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\left \\lVert \\sum _ { i = 1 } ^ n X _ i \\right \\rVert _ q \\leq 2 q \\left ( 1 + \\left \\lVert \\sum _ { i = 1 } ^ n \\mathbb { E } [ X _ i \\vert \\mathcal { F } ] \\right \\rVert _ q \\right ) . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\psi _ i ^ s ( \\{ z ^ j \\} _ { j \\in \\Z ^ n } ) = \\psi _ i ^ s ( \\{ w ^ j \\} _ { j \\in \\Z ^ n } ) . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} | \\psi ( t ) - \\psi ( u ) | & = \\psi ( t ) = \\dfrac { t - a _ { 2 n - 1 } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\leq \\dfrac { t - u } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\\\ & \\leq \\dfrac { t ^ { 1 / k } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } < \\dfrac { ( b _ { 2 n - 1 } ) ^ { 1 / k } } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } \\\\ & < \\dfrac { ( b _ { 2 n - 1 } ) ^ { 1 / k } } { 2 ^ { 2 n - 1 } ( b _ { 2 n - 1 } ) ^ { 1 / k } } < \\dfrac { 1 } { 2 ^ { 2 n - 1 } } . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { L } } d s _ L = \\frac { 1 } { \\sqrt { L } } \\sqrt { \\frac { \\dot { \\gamma } _ 1 ^ 2 } { \\gamma _ 1 ^ 2 } + \\dot { \\gamma } _ 3 ^ 2 } d t . \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} \\Delta _ 1 ( \\Gamma ) = \\Delta ( \\Gamma ) + 1 \\leq \\frac { 2 \\sqrt { n } } { \\log ^ 3 n } . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} K \\langle \\alpha / t , t ] \\ ! ] _ 0 & = \\left \\{ \\sum _ { i \\in \\mathbb { Z } } a _ i t ^ i ; a _ i \\in K , \\lim _ { i \\to - \\infty } { | a _ i | \\alpha ^ i } = 0 , \\ \\sup _ { i \\in \\mathbb { Z } } { | a _ i | } < \\infty \\right \\} , \\\\ K \\langle \\alpha / t , t \\} & = \\left \\{ \\sum _ { i \\in \\mathbb { Z } } a _ i t ^ i ; a _ i \\in K , \\lim _ { i \\to \\pm \\infty } { | a _ i | \\eta ^ i } = 0 \\ ( \\eta \\in ( \\alpha , 1 ) ) \\right \\} . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} \\mu = [ \\mu _ { i j } ] , \\ ; \\mu _ { i j } = \\frac { 1 } { \\rho ( A ) } w _ i a _ { i j } u _ j \\textrm { f o r } i , j \\in [ n ] , \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} a ^ { 2 } = 1 , a s _ { \\pm } = s _ { \\pm } a , s ^ { 2 } _ { \\pm } = 1 , ( s _ { + } s _ { - } ) ^ { p } = 1 . \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} \\partial _ { t } \\left ( \\sigma ( z ) z \\right ) - \\partial _ { x x } z = f \\left ( z , x \\right ) , \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\sigma ^ k ( x + t v ) = \\xi ^ k ( x ) \\cdot h ^ k ( t ) \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\partial _ { t _ i } \\textbf { g } \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) , S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ b ) \\right ) = 0 \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} j u = d \\varphi + d ^ * \\xi + h \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\partial _ \\nu G _ { x _ \\varepsilon } = - \\frac { 1 } { 2 \\pi R \\mu _ \\varepsilon } + O \\left ( \\frac { 1 } { d ( x _ \\varepsilon , \\partial \\Omega ) } \\right ) \\partial B _ { x _ \\varepsilon } ( R \\mu _ \\varepsilon ) \\ , . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} & \\left ( \\ref { c l a i m : d i f i c i l } \\right ) u = t \\exists Z _ { u = t , 3 } ^ { \\ast } \\mathbf { Z } _ { m i n } \\backslash \\left [ \\mathbf { O } _ { m i n } \\cup \\left \\{ Z _ { u = t , 1 } ^ { \\ast } , Z _ { u = t , 2 } ^ { \\ast } \\right \\} \\right ] Z _ { u = t , 3 } ^ { \\ast } \\\\ & Z _ { u = t , 2 } ^ { \\ast } . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} \\Phi ( g \\sigma ) = \\Phi ( g \\sigma ' ) + \\Phi ( g \\sigma '' ) \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} H _ i ^ \\alpha = I _ i ^ { k _ \\alpha } , i = 0 , 1 . \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} C _ { n , i } = \\sum \\limits _ { \\substack { \\lambda _ 1 \\vdash n \\\\ \\alpha _ 1 , \\xi _ 1 } } \\sum \\limits _ { ( \\Lambda ^ { \\prime } , A ^ { \\prime } , \\Xi ^ { \\prime } ) } \\left ( \\prod _ { j = 1 } ^ q m ( \\lambda _ j ) \\sum \\limits _ { ( d ^ j _ k ) } \\prod \\limits _ { k = 1 } ^ { \\ell ( \\lambda _ j ) } s ( b _ k ^ j , d _ k ^ j ) \\right ) . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} v \\Gamma _ S ( e S ) : = H ( \\rho ^ e ; - ) \\quad \\Gamma _ S ( \\lambda ( e , u , f ) ) : = \\eta _ { \\rho ^ e } \\mathbb { L } ( S ) ( \\rho ( f , u , e ) , - ) \\eta _ { \\rho ^ f } ^ { - 1 } \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\Psi _ { x y } ^ { \\epsilon , \\delta } ( w _ 1 , w _ 2 ) : = \\Phi _ { x y } ^ { \\epsilon , \\delta } ( w _ 1 ) \\times \\Phi _ { x y } ^ { \\epsilon , \\delta } ( w _ 2 ) \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\det ( \\nabla ^ 2 h + h \\ , { \\rm I d } ) = \\mbox { $ \\frac 1 n $ } \\ , h ^ { p - 1 } \\left ( \\| \\nabla h \\| ^ 2 + h ^ 2 \\right ) ^ { \\frac { n - q } 2 } \\cdot f . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\sup _ { y \\in G } \\sharp \\left ( \\Gamma ^ { - 1 } \\cap y W ^ { - 1 } \\right ) = \\sup _ { x \\in G } \\sharp \\left ( \\Gamma ^ { - 1 } \\cap x ^ { - 1 } W ^ { - 1 } \\right ) = \\sup _ { x \\in G } \\sharp \\left ( \\Gamma \\cap x W \\right ) < \\infty \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} v = \\sum _ { i = 1 } ^ t \\ w _ i \\ \\ w _ i \\in W _ i . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\lambda = \\bigoplus _ { 1 \\leq k \\leq n } \\bigoplus _ { 1 \\leq i _ { 1 } , \\ldots , i _ { k } \\leq n } ( a _ { i _ { 1 } i _ { 2 } } a _ { i _ { 2 } i _ { 3 } } \\cdots a _ { i _ { k } i _ { 1 } } ) ^ { 1 / k } = \\mathop \\mathrm { t r } A \\oplus \\cdots \\oplus \\mathop \\mathrm { t r } \\nolimits ^ { 1 / n } ( A ^ { n } ) . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} S _ 3 ( \\overline { f ^ { \\not \\ominus } ( z , t ) } , z , t ) = 0 . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} B ( y ) & \\geq \\exp ( h ( s ( y ) ) + y s ( y ) - 3 0 s ( y ) C ( y ) ^ { 1 / 2 } - 1 0 0 ) \\\\ & = \\exp ( h ( s ( y ) ) + y s ( y ) - O ( y ^ { u / ( 2 u + 2 ) } ) ) . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} \\mathfrak { A } _ { M _ S , \\mathbb { Q } } \\to X _ * ( T ) _ { \\mathbb { Q } } = X ^ * ( \\widehat { T } ) _ { \\mathbb { Q } } \\xrightarrow { r e s } X ^ * ( Z ( \\widehat { M _ S } ) ^ { \\Gamma } ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} M ( f , \\mu ) : = \\{ p \\in S p l ( f ) \\mid \\alpha _ i \\equiv r _ { \\mu ( i ) } \\bmod \\mathfrak p \\ , \\ , \\ , ( 1 \\le { } ^ \\forall i \\le n ) { } ^ \\exists \\mathfrak p \\ , | \\ , p \\} , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\ < \\int _ { \\R ^ d } \\phi ( x - y ) M ( d y d s ) \\ > = \\gamma \\sigma ^ 2 \\int _ { \\R ^ d } \\phi ( x ) ^ 2 d x d s . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} a _ k \\leq \\tilde { C } b ^ { - 1 / p } \\left ( \\sum _ { j = - \\infty } ^ { \\infty } \\int _ { 2 B _ 0 } g _ j ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } ( k - k _ 0 ) \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} f ( x ) = \\bigoplus _ { a \\in A } ( v _ a \\odot x ^ { \\odot a } ) \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} l _ { a b } = & \\langle \\nabla ^ N _ { \\partial _ a } \\partial _ b , L \\rangle \\\\ n _ { a b } = & \\langle \\nabla ^ N _ { \\partial _ a } \\partial _ b , \\underline L \\rangle \\\\ \\eta _ a = & \\langle \\nabla ^ N _ { L } \\partial _ a , \\underline L \\rangle \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} e _ 1 ^ { \\star } = \\overline { q } \\omega _ 1 - \\overline { p } \\omega _ 2 , ~ ~ ~ ~ e _ 2 ^ { \\star } = \\overline { r _ L } ~ \\overline { p } \\omega _ 1 + \\overline { r _ L } ~ \\overline { q } \\omega _ 2 - \\frac { l } { l _ L } L ^ { \\frac { 1 } { 2 } } \\omega . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} U _ { \\varepsilon , z } ( z ) = \\gamma _ \\varepsilon \\ , . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} F ( \\bar N ) : = h _ { \\bar N } \\left | | x _ 0 | ^ { { 3 } ^ { \\bar N } } \\prod _ { j = 0 } ^ { \\bar N - 1 } h _ { j } ^ { { 3 } ^ { \\bar N - 1 - j } } \\right | ^ 2 . \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\left \\Vert a + b \\right \\Vert _ { \\mathcal { P } _ n } \\le \\sum _ { j = 1 } ^ l \\left \\Vert a _ j \\right \\Vert ^ n + \\sum _ { j = 1 } ^ m \\left \\Vert b _ j \\right \\Vert ^ n \\le \\left \\Vert a \\right \\Vert _ { \\mathcal { P } _ n } + \\left \\Vert b \\right \\Vert _ { \\mathcal { P } _ n } + \\varepsilon , \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\nabla _ { z \\partial t _ i } S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) = 0 \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} h ( s ( y ) ) & = ( 1 + O ( ( s ( y ) ) ^ { \\kappa } ) ) \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ( y ) ^ { - u } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ( K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} E _ { P } \\left [ \\left . \\frac { 1 } { \\lambda _ { \\overline { \\mathbf { a } } _ { k } } \\left ( \\overline { \\mathbf { G } } _ { k } , \\overline { \\mathbf { B } } _ { k } \\mathbf { ; } P \\right ) } \\right \\vert \\overline { \\mathbf { G } } _ { k } \\mathbf { , } \\overline { \\mathbf { A } } _ { k } \\mathbf { = } \\overline { \\mathbf { a } } _ { k } \\right ] = \\frac { 1 } { \\lambda _ { \\overline { \\mathbf { a } } _ { k } } \\left ( \\overline { \\mathbf { G } } _ { k } \\mathbf { ; } P \\right ) } . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} y _ { V , v } ( z ) = \\begin{pmatrix} ( z / \\bar { z } ) I _ { r _ v } & 0 \\\\ 0 & I _ { s _ v } \\end{pmatrix} \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} \\abs { \\mathbb V ( x ) } \\leq & C \\abs { \\chi ' _ k ( r ) n _ k ^ { \\frac 7 4 } } \\leq C \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } ( \\log \\rho _ k ) ^ { 3 } e ^ { - ( \\log \\rho _ k ) ^ { 3 / 2 } } n _ k ^ \\frac 7 4 \\\\ \\leq & C ( \\log \\rho _ k ) ^ { 3 } \\rho _ { k } ^ { - \\frac { 1 - \\delta } { 2 } } \\leq C ( \\log \\abs { x } ) ^ 3 \\abs { x } ^ { - \\epsilon } , \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} f V ( 1 + 2 V ^ 2 | \\nabla \\tau | ^ 2 ) - 2 V \\nabla \\tau \\nabla \\sinh ^ { - 1 } \\frac { f d i v ( V ^ 2 \\nabla \\tau ) } { | H _ 0 | | H | } + ( \\alpha _ { H } - \\alpha _ { H _ 0 } ) ( 2 V \\nabla \\tau ) = O ( r ) . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\xi ^ { 2 n } n ! { \\| h _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 \\leq C _ 2 e ^ { - ( 2 - \\delta ) \\mu _ 1 t } \\sum _ { n \\geq 0 } \\frac { \\Big ( C _ 1 \\xi ^ { 2 } \\kappa \\Big ) ^ n t ^ { n ( 1 - \\beta / \\alpha ) } } { ( n ! ) ^ { 1 - \\beta / \\alpha } } . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} \\pi _ n ( c n ) & = P ( \\mathrm { B i n } ( \\lfloor c n \\rfloor , p _ n ) \\geq r ) \\geq 1 - \\mathrm { e } ^ { - \\lfloor c n \\rfloor p _ n H ( r / ( \\lfloor c n \\rfloor p _ n ) ) } ; \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\mu _ f ( x , t ) = f ( x ) + \\frac { t ^ 2 } { 6 } \\ , f '' ( x ) + \\frac { a _ 4 \\ , t ^ 4 } { 5 } = f ( x ) + \\frac { t ^ 2 } { 6 } \\ , f '' ( x ) + \\frac { t ^ 4 } { 1 2 0 } f ^ { ( 4 ) } ( x ) \\ , . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ w \\in \\Lambda ^ { m } \\ : : \\ : 2 ^ { - m ( h + \\delta ) } \\le \\mu [ w ] \\le 2 ^ { - m ( h - \\delta ) } \\} , \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} B _ s \\gamma ( \\phi ) & = 2 ( \\cosh s \\cos \\phi + \\sinh s \\sin \\phi ) + 2 i ( \\sinh s \\cos \\phi + \\cosh s \\sin \\phi ) \\\\ & = 2 ( \\cosh s e ^ { i \\phi } + i \\sinh s e ^ { - i \\phi } ) . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} L ( { \\cal C } _ { f i n } ) \\leq \\sum _ { i = 1 } ^ { N } T _ l + 2 ( N - 1 ) ( s _ n + 8 r _ n ) . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } [ \\hat { H } _ j , \\hat { H } _ k ] = [ \\hat { H } ' _ j , \\hat { H } ' _ k ] = \\mathbf { 0 } _ n \\\\ \\Lambda ( \\Phi ( \\hat { H } _ j ) ) = \\Lambda ( \\hat { H } ' _ j ) , \\\\ p _ k ( \\hat { H } _ 1 , \\ldots , \\hat { H } _ m ) = p _ k ( \\hat { H } ' _ 1 , \\ldots , \\hat { H } ' _ m ) = \\mathbf { 0 } _ n , \\\\ \\eth ( \\mathbf { H } , \\hat { \\mathbf { H } } ) < \\delta _ 3 / 2 , \\\\ \\eth ( \\mathbf { H } ' , \\hat { \\mathbf { H } } ' ) < \\delta _ 3 / 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} ( 1 + g ( t ) ) = \\frac { 1 } { 1 + t ^ 2 + \\varphi _ { N _ \\varepsilon } ( t ^ 2 ) } \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\delta ( f ) : = \\max \\{ \\nu ( x - a ) \\mid a \\mbox { i s a r o o t o f } f \\} . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N H F = 0 , F D F = 0 , N D F = 1 , N D N = 1 } \\right \\rbrace \\to \\int _ { 0 } ^ { \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ 2 \\left ( 2 ^ R - 1 \\right ) } { P _ { \\textrm { F } } - P _ { \\textrm { N } } \\left ( 2 ^ R - 1 \\right ) } } \\frac { 1 } { \\lambda _ { \\textrm { B F } } } \\ln \\left ( 2 \\psi _ { 1 } ( x ) ^ { - 1 } \\right ) \\frac { 1 } { 2 } \\psi _ { 1 } ( x ) ^ { 2 } d x . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\cosh \\tfrac { L _ 1 } { 2 } + \\cosh \\tfrac { \\ell _ \\nu } { 2 } = 2 \\sinh \\tfrac { \\ell _ \\mu } { 2 } \\ ; \\sinh \\tfrac { \\ell _ { \\mu ' } } { 2 } L _ 1 0 , \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } \\frac { [ 3 k ] } { [ 2 k ] ^ 2 } { 2 k \\brack k } q ^ { - { k \\choose 2 } } = [ n ] { 2 n - 1 \\brack n - 1 } \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ { - { n - 2 k \\choose 2 } } } { [ 2 k ] ^ 2 { n - 1 \\brack k } _ { q ^ 2 } ^ 2 } . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} G _ { \\boldsymbol { Y } } ( \\boldsymbol { s } ) = \\sum _ { \\boldsymbol { y } \\in \\mathbb { N } ^ J } \\Gamma \\left ( \\vert \\boldsymbol { y } \\vert + 1 \\right ) P \\left ( \\vert \\boldsymbol { Y } \\vert = \\vert \\boldsymbol { y } \\vert \\right ) \\frac { \\prod _ { j = 1 } ^ J ( \\alpha _ j ) _ { y _ j } } { ( \\vert \\boldsymbol { \\alpha } \\vert ) _ { \\vert \\boldsymbol { y } \\vert } } \\prod _ { j = 1 } ^ J \\frac { s _ j ^ { y _ j } } { y _ j ! } . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} & u \\preceq v \\ \\ { \\rm i f } \\ \\ u ( x ) \\preceq v ( x ) \\ \\ { \\rm f o r \\ a l l } \\ \\ x \\in \\R ^ l , \\\\ & u \\prec v \\ \\ { \\rm i f } \\ \\ u \\preceq v \\ \\ { \\rm a n d } \\ \\ u \\not = v , \\\\ & u \\ll v \\ \\ { \\rm i f } \\ \\ u ( x ) \\ll v ( x ) \\ \\ { \\rm f o r \\ a l l } \\ \\ x \\in \\R ^ l . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\sigma _ { r } \\circ T _ { m } \\circ \\sigma _ { - r } = T _ { Q ^ { i r } m Q ^ { - i r } } , r \\in \\mathbb R , Q = \\oplus _ \\pi Q _ \\pi . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} ( d _ 1 ^ { n e } , d _ 2 ^ { n e } ) = ( T _ 1 , T _ 2 ) . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\Big \\vert D _ { x _ j } ( \\frac { 1 } { t ^ { d / 2 } } e ^ { - | x | ^ 2 / 4 t } ) \\Big \\vert = \\Big \\vert \\frac { x _ j } { 2 t } \\frac { 1 } { t ^ { d / 2 } } e ^ { - | x | ^ 2 / 4 t } \\Big \\vert \\leq \\frac { | x | } { 2 t ^ { ( d + 2 ) / 2 } } e ^ { - | x | ^ 2 / 4 t } . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} m : = \\sup _ { \\phi \\in \\mathcal { G } _ { \\gamma } } \\| \\phi \\| < \\infty . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} T = T _ { \\rm r e g } + T _ { \\rm s i n g } . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\overline { C ^ \\infty _ c ( \\R ^ 3 ) } ^ { \\norm { \\cdot } _ { L ^ \\infty _ H L ^ p _ z } } = C _ 0 ( \\R ^ 2 ; L ^ p ( \\R ) ) , \\overline { C ^ \\infty _ c ( \\R ^ 3 ) } ^ { \\norm { \\cdot } _ { L ^ q _ H L ^ \\infty _ z } } = L ^ q ( \\R ^ 2 ; C _ 0 ( \\R ) ) , \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} d d ^ c v _ { j } \\leq d d ^ c u _ { j } = 1 _ { \\{ a _ j \\} } d d ^ c u _ { j } = 1 _ { \\{ a _ j \\} } d d ^ c v _ { j } \\leq d d ^ c v _ { j } \\ \\Omega ' . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} d S _ t & = S _ t \\sqrt { V _ t } d B _ t , S _ 0 > 0 , \\\\ V _ t & = g _ 0 ( t ) + \\int _ 0 ^ t K ( t - s ) \\left ( - \\lambda V _ s d s + \\nu \\sqrt { V _ s } d W _ s \\right ) , \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } | u _ { N _ 1 + 2 + i } | < \\sum _ { i = 1 } ^ { \\infty } | u _ { N _ 1 + 2 + i } | \\le \\sum _ { n = 4 } ^ { \\infty } \\left ( \\frac \\beta 4 \\right ) ^ { n } = \\frac { \\left ( \\frac \\beta 4 \\right ) ^ { 4 } } { 1 - \\frac \\beta 4 } < \\frac { 4 \\left ( \\frac 1 4 \\right ) ^ { 4 } } { 4 - \\beta } < \\frac 1 { 4 ^ 3 \\times 3 } < 1 . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} \\epsilon = \\frac { \\sigma ^ { 1 + \\alpha } } { 2 } . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } 3 ^ { - j } \\ln h ^ { - 1 } _ { j } = \\infty . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} R _ i ^ j = { d - 1 \\over 2 } \\delta _ i ^ j + { \\nabla _ 1 } _ i E ^ j \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} 0 & \\le \\overbrace { b _ 1 = \\dots = b _ 1 } ^ { d _ 1 } < \\dots < \\overbrace { b _ { k _ 2 } = \\dots = b _ { k _ 2 } } ^ { d _ { k _ 2 } } \\\\ & < r _ 1 \\le \\dots \\le r _ l \\\\ & < \\underbrace { p + a _ 1 = \\dots = p + a _ 1 } _ { c _ 1 } < \\dots < \\underbrace { p + a _ { k _ 1 } = \\dots = p + a _ { k _ 1 } } _ { c _ { k _ 1 } } < p , \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\lambda _ l ( k ) & = ( - 1 ) ^ k \\frac { d ^ k } { d z ^ k } \\left ( P ^ n _ l ( z ) ( 1 - z ^ 2 ) ^ { ( m - 2 ) / 2 } \\right ) \\Big | _ { z = 0 } \\\\ & = ( - 1 ) ^ k \\sum _ { j = 0 } ^ k \\frac { d ^ { k - j } } { d z ^ { k - j } } P ^ n _ l ( z ) \\Big | _ { z = 0 } \\ , \\frac { d ^ { j } } { d z ^ { j } } ( 1 - z ^ 2 ) ^ { ( m - 2 ) / 2 } \\Big | _ { z = 0 } . \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\frac { d } { { d c } } { \\varphi _ X } \\left ( c \\right ) + { \\epsilon _ 2 } { \\varphi _ X } \\left ( c \\right ) = 0 . \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\mathcal { D } _ n = \\left \\{ D _ n ( k ) = [ k 2 ^ { - n } , ( k + 1 ) 2 ^ { - n } ) : k \\in \\mathbb { N } , 0 \\leq k \\leq 2 ^ n - 1 \\right \\} . \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} K _ p = \\left \\lbrace [ \\alpha ] ; \\ , p \\ge \\frac { \\alpha _ 1 ^ + } { \\alpha _ n ^ + } \\right \\rbrace . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} h ( b _ 1 b _ 2 ) = h ( b _ 1 ) \\Delta ( b _ 2 ) + ( - 1 ) ^ { | b _ 1 | } ( \\tau \\Delta ) ( b _ 1 ) h ( b _ 2 ) . \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\Delta u = Q ( x , u , \\nabla u ) , \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} x ' ( t ) & = s - d x ( t ) - \\frac { \\beta x ( t ) v ( t ) } { ( 1 + a x ( t ) ) ( 1 + b v ( t ) ) } , \\\\ y ' ( t ) & = \\frac { \\beta x ( t - \\tau ) v ( t ) e ^ { - s \\tau } } { ( 1 + a x ( t - \\tau ) ) ( 1 + b v ( t - \\tau ) ) } - a y ( t ) - p y ( t ) z ( t ) , \\\\ v ' ( t ) & = k y ( t ) - u v ( t ) , \\\\ z ' ( t ) & = c y ( t ) z ( t ) - b z ( t ) . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{gather*} \\langle \\alpha _ 1 , \\dots , \\alpha _ { \\ell } , x _ 1 , y _ 1 , \\dots , x _ { g _ 0 } , y _ { g _ 0 } \\ , | \\ , \\alpha _ 1 ^ { n _ 1 } = \\cdots = \\alpha _ { \\ell } ^ { n _ l } = 1 , \\ , \\prod _ { i = 1 } ^ { \\ell } \\alpha _ i = \\prod _ { j = 1 } ^ { g _ 0 } [ x _ j , y _ j ] \\ , \\rangle . \\end{gather*}"}
-{"id": "6925.png", "formula": "\\begin{align*} l ( G ) = \\max \\{ 9 , \\O ( q + 1 ) + l ( { \\rm L } _ { 2 } ( q ) ) + 1 , \\O ( q ^ 2 - q + 1 ) + 1 \\} . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( J L ( \\rho ) ) ) = \\mathrm { M a n t } _ { G , b , \\mu } ( \\rho ) , \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\frac { p _ i } { p _ { N + 1 } } = \\frac { - \\delta _ { i , 0 } Q + ( 1 - q ) ^ i \\sum _ { 0 \\leq j _ 1 < \\cdots < j _ i \\leq N } \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ i } \\prod _ { k \\in \\{ 0 , \\dots , N \\} - \\{ j _ 1 , \\dots , j _ i \\} } ( 1 - \\Lambda _ k ) } { ( 1 - q ) ^ { N + 1 } \\Lambda _ 0 \\cdots \\Lambda _ N } \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} P _ D ( x , z ) = \\int _ B G ( x , y ) \\nu ( | y - z | ) d y + \\int _ { \\tilde { D } } G ( x , y ) \\nu ( | y - z | ) d y = : \\mathrm { I } + \\mathrm { I I } . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} \\mathcal G ( t , r _ 1 , r _ 2 , l ) & = \\frac { F ( r _ 1 + t ) - F ( r _ 2 ) } { l \\ , f ^ n ( r _ 1 + t ) } \\\\ & + \\frac { 1 } { l \\ , f ^ n ( r _ 1 + t ) } \\int _ 0 ^ l \\int _ s ^ l f ' ( F ^ { - 1 } ( \\bar U ( s ' ) ) \\ , d s ' \\ , d s . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\frac { d \\psi ( t ) } { d t } & = & A \\psi ( t ) + \\Pi _ U \\alpha ( \\psi ( t ) ) \\medskip \\\\ \\psi ( 0 ) & = & u _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} & \\int _ { \\Sigma _ r } f ( V ^ 2 + V ^ 4 | \\nabla \\tau | ^ 2 ) + ( d i v V ^ 2 \\nabla \\tau ) \\sinh ^ { - 1 } ( \\frac { f d i v V ^ 2 \\nabla \\tau } { | H | | H _ 0 | } ) d \\Sigma _ r \\\\ & - \\int _ { \\Sigma _ r } \\alpha _ { H _ 0 } ( V ^ 2 \\nabla \\tau ) d \\Sigma _ r + \\int _ { \\Sigma _ r } \\alpha _ { H } ( V ^ 2 \\nabla \\tau ) d \\Sigma _ r . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} { \\bf H } _ \\alpha ~ : = ~ \\log _ 2 { \\bf N } _ \\alpha \\quad \\mathrm { a n d } { \\bf K } _ { \\alpha } ~ : = ~ \\log _ 2 { \\bf M } _ { \\alpha } \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\begin{aligned} & 0 \\leq W _ { h } ( x ) \\leq m ( h ) & & x \\in ( 1 - \\varepsilon ) \\Omega , \\\\ & W _ { h } ( x ) \\geq M ( h ) & & x \\notin ( 1 + \\varepsilon ) \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\varphi _ { A _ { \\inf } \\{ 1 \\} } = \\pi _ 2 \\varphi : A _ { \\inf } \\{ 1 \\} [ \\tfrac 1 \\xi ] \\cong A _ { \\inf } \\{ 1 \\} [ \\tfrac 1 { \\tilde \\xi } ] \\ , \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} & w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( 1 , \\{ 2 , 3 \\} ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( 2 , \\{ 1 , 3 \\} ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( 3 , \\{ 1 , 2 \\} ) } ) = 1 \\\\ & w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( \\{ 2 , 3 \\} , 1 ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( \\{ 1 , 3 \\} , 2 ) } ) = 0 , w ^ \\uparrow _ { \\{ 1 , 2 \\} } ( C _ { ( \\{ 1 , 2 \\} , 3 ) } ) = 1 \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} B _ n = \\{ z \\in \\Omega : K _ \\Omega ( z , y _ n ) < \\delta _ n / 2 \\} . \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} z _ k ( t ) = e ^ { - \\lambda _ k t } z _ k ( 0 ) + \\sqrt { c _ k } \\int _ 0 ^ t e ^ { - \\lambda _ k ( t - s ) } v ( s ) \\ , d s + \\sqrt { 2 \\lambda _ k } \\int _ 0 ^ t e ^ { - \\lambda _ k ( t - s ) } \\ , d W _ k ( s ) . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} \\tilde { p } ^ { ( \\alpha ) } _ n ( z ) = \\frac { n ! c ^ n } { 2 ^ n ( 1 + \\alpha ) _ { n } } C _ { n } ^ { ( 1 + \\alpha ) } ( z / c ) \\ , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Phi ( f ( \\rho ) A ^ * A ) & \\Phi ( f ( \\rho ) A ^ * B ) & \\Phi ( f ( \\rho ) g ( \\rho ) A ^ * ) & 0 \\\\ \\Phi ( f ( \\rho ) B ^ * A ) & \\Phi ( f ( \\rho ) B ^ * B ) & \\Phi ( f ( \\rho ) g ( \\rho ) B ^ * ) & 0 \\\\ \\Phi ( f ( \\rho ) g ( \\rho ) A ) & \\Phi ( f ( \\rho ) g ( \\rho ) B ) & \\Phi \\left ( f ( \\rho ) g ( \\rho ) ^ 2 \\right ) & 0 \\\\ 0 & 0 & 0 & \\Phi \\left ( f ( \\rho ) g ( \\rho ) ^ 2 \\right ) \\end{bmatrix} \\geq 0 . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\tilde J ( u , w ) = \\inf _ { ( v , z ) \\in \\mathcal U } \\tilde J ( v , z ) \\ , . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\psi _ 1 & = u _ 1 + 1 * f _ 1 , \\\\ \\psi _ 2 & = u _ 2 K + K * F ( \\psi _ 1 , \\psi _ 2 ) , \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ { 2 } \\sum _ { k = 1 } ^ { J _ { 0 } + 1 } \\gamma _ { i } ^ { k } a _ { i } ^ { k } , \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} J \\C [ x ] = \\C [ x _ 0 , x _ 1 , x _ 2 , \\ldots ] \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} t _ 1 & = \\frac { - [ a ^ 2 + 2 \\sigma ^ 2 s ( - a + \\eta ) - 1 ] - \\sqrt { \\Gamma } } { 4 \\sigma ^ 2 } , \\\\ t _ 2 & = \\frac { - [ a ^ 2 + 2 \\sigma ^ 2 s ( - a + \\eta ) - 1 ] + \\sqrt { \\Gamma } } { 4 \\sigma ^ 2 } , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} ( \\mathcal { K } ^ { * } f _ { \\beta } ) ( t ) = C _ { H } \\frac { \\Gamma \\left ( \\beta + \\frac { 3 } { 2 } - H \\right ) } { \\Gamma ( \\beta + 1 ) } t ^ { \\frac { 1 } { 2 } - H } ( 1 - t ) ^ { \\beta } . \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } g _ q ( Q ) = e ^ Q \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} A : = \\sum _ { j = 0 } ^ s a _ j J ^ { k _ j } , \\ B : = \\sum _ { j = 1 } ^ s a _ j \\left ( J ^ T \\right ) ^ { n - k _ j } . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} g ( t ) = \\frac 5 4 a + b \\int _ 0 ^ t ( g ( s ) + g ( s ) ^ m ) d s , 0 < t < T _ 1 . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} D _ i = \\{ ( x _ 1 , x _ 2 ) \\mid 0 \\le x _ 1 \\le x _ 2 \\le \\frac { 1 } { 2 } , \\{ g _ 1 ( x _ i ) \\} < \\{ g _ 2 ( x _ i ) \\} \\} \\quad ( i = 1 , 2 ) , \\\\ D _ i = \\{ ( x _ 3 , x _ 4 ) \\mid \\frac { 1 } { 2 } \\le x _ 3 \\le x _ 4 \\le 1 , \\{ g _ 1 ( x _ i ) \\} < \\{ g _ 2 ( x _ i ) \\} \\} \\quad ( i = 3 , 4 ) , \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} L ^ S ( s + \\frac { 1 } { 2 } , \\delta _ l \\times ( \\delta _ i , 2 b _ i ) ) = \\prod _ { j = 0 } ^ { 2 b _ i - 1 } L ^ S ( s + b _ i - j , \\delta _ l \\times \\delta _ i ) . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} & \\sigma _ 1 = [ 1 , 2 , 3 , 5 , 4 , 6 , 7 , 8 ] , \\sigma _ 2 = [ 1 , 2 , 4 , 3 , 6 , 5 , 7 , 8 ] , \\sigma _ 3 = [ 1 , 4 , 6 , 2 , 7 , 3 , 5 , 8 ] , \\\\ & \\sigma _ 4 = [ 2 , 1 , 3 , 4 , 5 , 6 , 8 , 7 ] , \\sigma _ 5 = [ 2 , 1 , 3 , 5 , 4 , 6 , 8 , 7 ] , \\sigma _ 6 = [ 2 , 1 , 4 , 3 , 6 , 5 , 8 , 7 ] , \\\\ & \\sigma _ 7 = [ 3 , 1 , 4 , 2 , 7 , 5 , 8 , 6 ] . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\det \\mathcal { P } ( k ) & = \\det \\begin{pmatrix} I _ 1 ( k + 9 ) & I _ 2 ( k + 9 ) & { } & { } \\\\ I _ 3 ( k + 9 ) & I _ 4 ( k + 9 ) & { } & { } \\\\ { } & { } & I _ 5 ( k + 9 ) & { } \\\\ { } & { } & { } & I _ 6 ( k + 9 ) \\end{pmatrix} \\\\ & = ( k + 1 ) ( k + 3 ) ( k + 4 ) ( k + 5 ) ( k + 9 ) ^ 2 ( k + 1 0 ) ( k + 1 1 ) \\\\ & \\neq 0 \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\mbox { t h e h o r i z o n t a l c e n t e r o f m a s s } h _ 0 : = \\int _ S x \\theta _ 0 ( z ) d z , \\mbox { a n d } \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} 0 & = \\mathcal { L } _ 0 ( E _ k ) + \\left ( \\frac { s _ k ^ 2 } { 2 \\lambda s _ k ^ 2 + 2 u _ k } - \\frac { 1 } { 2 \\lambda } \\right ) \\mathcal { S } _ { 0 , 2 } ( U _ k , U _ k ) + O ( s _ k ) \\\\ & = \\mathcal { L } _ 0 ( E _ k ) - \\frac { u _ k } { 2 \\lambda ^ 2 s _ k ^ 2 } \\mathcal { S } _ { 0 , 2 } ( U _ k , U _ k ) + O \\left ( s _ k , \\frac { u _ k ^ 2 } { s _ k ^ 4 } \\right ) . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} & a _ j = U \\left [ j , k _ { i - j + 1 } \\right ] , \\\\ & a ^ i _ j = U ^ i \\left [ j , k _ { i - j + 1 } \\right ] , \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\alpha _ { \\rm H J M } = - \\gamma \\cdot \\Psi ' \\bigg ( - \\int _ 0 ^ { \\bullet } \\gamma ( \\xi ) d \\xi \\bigg ) , \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} I ( f ) = \\widetilde I ( f ) . \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( 1 - \\rho ^ { 2 } \\right ) \\left | \\nabla \\varphi _ { 0 } \\right | ^ { 2 } \\leq 2 \\pi \\left ( \\sum _ { i = 1 } ^ { m } d _ { i } ^ 2 \\right ) I \\left ( \\frac { R } { R _ { 0 } } \\right ) + 2 \\pi ( 1 - a ^ { 2 } ) \\sum _ { i \\neq j } \\left | d _ { i } \\right | \\left | d _ { j } \\right | \\log \\dfrac { R } { \\left | a _ { i } - a _ { j } \\right | } + C . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\mathcal { I } _ g ( g ^ { - 1 } m g ) & = g ( g ^ { - 1 } m g ) g ^ { - 1 } \\\\ & = g g ^ { - 1 } m g g ^ { - 1 } \\\\ & = r ( g ) m r ( g ) \\\\ & = m . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\theta ( k ) = \\theta ( k ; \\xi ) = 2 k ^ 2 + 4 \\xi k \\ \\xi = \\frac { x } { 4 t } . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} \\Pi _ U \\alpha ( h _ 0 + v ) - \\Pi _ U \\alpha ( h _ 0 ) = 0 , \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} D _ i \\cdot D _ j \\le \\frac { 1 } { r ( r - 1 ) } \\left ( 2 \\pi - 2 + 2 r \\right ) = \\frac { 1 } { r ( r - 1 ) } ( 2 \\pi - 2 ) + \\frac { 2 } { r - 1 } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} { } _ { 2 } F _ { 1 } ( - n , b ; c ; x ) = \\frac { ( c - b ) _ { n } } { ( c ) _ { n } } { } _ { 2 } F _ { 1 } ( - n , b ; b - c - n + 1 ; 1 - x ) . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} p _ i : = - \\delta _ { i , 0 } Q + ( 1 - q ) ^ i \\sum _ { 0 \\leq j _ 1 < \\cdots < j _ i \\leq N } \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ i } \\prod _ { k \\in \\{ 0 , \\dots , N \\} - \\{ j _ 1 , \\dots , j _ i \\} } ( 1 - \\Lambda _ k ) \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\sigma ( t , \\theta ) = ( \\cos ( \\theta ) , t \\sin ( \\theta ) ) \\ ; , \\ ; \\ ; ( t , \\theta ) \\in [ - 1 , 1 ] \\times [ - \\omega , \\omega ] \\ ; . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\Delta u + a _ j ( x , u ) = 0 \\Omega , \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} = [ \\pi ] [ ( r _ { ( - 1 ^ 2 , 0 ^ { n _ 2 - 2 } ) } \\boxtimes r _ { ( 0 ^ { n _ 1 } ) } ) \\circ ( L L ( \\rho _ 2 ) \\oplus L L ( \\rho _ 1 ) ) \\otimes | \\cdot | ^ { 2 - n _ 1 - n _ 2 } ] . \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} J ( t , A _ t ; Z _ { t } \\otimes u , W _ { t } \\otimes v ) = y _ t ^ { t , A _ t ; Z _ { t } \\otimes u , W _ { t } \\otimes v } . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\left ( \\underline { \\Delta } - \\partial _ { t } \\right ) u ( x , t ) = 0 , \\ ; \\ ; \\ ; \\ ; ( x , t ) \\in M ' \\times ( 0 , T ] . \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} \\left \\langle Z , x ^ { \\star } \\right \\rangle _ { E , E ^ { \\star } } = \\left \\langle \\int _ { 0 } ^ { t } \\Phi ( s ) d w ( s ) , x ^ { \\star } \\right \\rangle _ { E , E ^ { \\star } } = \\int _ { 0 } ^ { t } \\left \\langle d w ( s ) , \\Phi ^ { \\star } ( s ) x ^ { \\star } \\right \\rangle _ { H } . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{gather*} \\mathrm { F l u x } ( \\{ l _ t \\} ) = [ \\sigma _ { \\psi } ] \\in H ^ 1 ( M : \\mathbb { R } ) \\end{gather*}"}
-{"id": "1860.png", "formula": "\\begin{align*} \\mathbb { S } \\Omega = \\{ ( \\gamma , \\delta ) \\in \\widehat { \\Gamma } \\times \\widehat { \\Delta } : ( \\gamma , \\delta ) \\ : \\} . \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\theta ( t ) = \\begin{cases} 0 , & \\mbox { i f $ b _ { 2 n - 2 } \\leq t < a _ { 2 n - 1 } $ } , \\\\ \\dfrac { ( 2 n - 1 ) ( t - a _ { 2 n - 1 } ) } { b _ { 2 n - 1 } - a _ { 2 n - 1 } } , & \\mbox { i f $ a _ { 2 n - 1 } \\leq t < b _ { 2 n - 1 } $ } , \\\\ 2 n - 1 , & \\mbox { i f $ b _ { 2 n - 1 } \\leq t < a _ { 2 n } $ , } \\\\ \\dfrac { ( 2 n - 1 ) ( b _ { 2 n } - t ) } { b _ { 2 n } - a _ { 2 n } } , & \\mbox { i f $ a _ { 2 n } \\leq t < b _ { 2 n } $ } . \\end{cases} \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} 2 \\pi - 2 & = ( K _ X + D ) \\cdot D \\\\ & = \\sum _ { i = 1 } ^ r ( K _ X + D _ i ) \\cdot D _ i + 2 \\sum _ { i < j } D _ i \\cdot D _ j \\\\ & = \\sum _ { i = 1 } ^ r ( 2 p _ a ( D _ i ) - 2 ) + 2 \\sum _ { i < j } D _ i \\cdot D _ j . \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} f ( z ) & = 2 \\pi i \\cdot \\frac { e ^ { 2 \\pi i z } } { e ^ { 2 \\pi i z } - 1 } = z ^ { - 1 } + \\pi i - \\frac { \\pi ^ 2 } { 3 } z - \\frac { \\pi ^ 4 } { 4 5 } z ^ 3 \\cdots , \\\\ g ( z ) & = ( 2 \\pi i ) ^ 2 \\cdot \\frac { e ^ { 2 \\pi i z } } { ( e ^ { 2 \\pi i z } - 1 ) ^ 2 } = z ^ { - 2 } + \\frac { \\pi ^ 2 } { 3 } + \\frac { \\pi ^ 4 } { 1 5 } z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} H ( x ) : = 1 - x + x \\log x , x \\in \\R _ + , H ( 0 ) : = 1 , H ( x ) : = + \\infty , x \\in \\R _ - : = ( - \\infty , 0 ) . \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\partial ^ { \\uparrow } f ( x ) = { \\rm c o n v } \\left \\{ \\lim \\limits _ { y \\rightarrow x } \\nabla f ( y ) \\ : \\ y \\in X _ { f } , \\ y \\notin S \\right \\} , \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} N _ 4 = \\inf \\{ n \\in \\mathbb N : x _ n ^ 2 h _ n < 1 \\} , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} \\widetilde { H } _ { \\Lambda _ j \\Lambda _ l } = \\int _ { \\mathcal { Y } ^ d } ( \\widetilde { \\delta } _ { i l } + \\widetilde { w } _ { y _ i \\Lambda _ l } ) ( \\widetilde { \\delta } _ { i j } + \\widetilde { w } _ { y _ i \\Lambda _ j } ) \\widetilde { m } d y + \\int _ { \\mathcal { Y } ^ d } \\frac { \\widetilde { m } _ { \\Lambda _ l } \\widetilde { m } _ { \\Lambda _ j } } { \\widetilde { m } } d y . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\mathbb { E } T _ l = I _ 1 + I _ 2 \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} a _ { t , i } = 1 + \\sum _ { j = t } ^ { i - 1 } a _ { t , j } \\cdot \\binom { i } { j } \\mod 2 . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} { F _ { \\min } } \\buildrel \\Delta \\over = \\min \\left \\{ { F \\left ( { \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } } \\right ) , 1 - F \\left ( { \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } } \\right ) } \\right \\} \\ge { \\epsilon ^ * } . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} & C _ { k , i } \\subset B , C _ { k + 1 , i } \\subset B ^ c , \\quad \\P ( C _ { k , i } ) = 0 . \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} \\left ( \\partial _ { t _ j } \\partial _ { t _ k } \\partial _ { t _ \\alpha } \\mathcal { F } ( \\tau , Q ) \\right ) \\left ( \\partial _ { t _ i } G ^ { \\alpha \\beta } \\right ) = \\left ( \\partial _ { t _ j } \\partial _ { t _ k } \\partial _ { t _ \\alpha } \\mathcal { F } ( \\tau , Q ) \\right ) \\sum _ { \\alpha ' , \\beta ' } \\left ( \\partial _ { t _ i } G _ { \\alpha ' \\beta ' } \\right ) G ^ { \\alpha \\alpha ' } G ^ { \\beta \\beta ' } \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\mathfrak c = \\begin{array} { c } \\mathfrak p _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak p _ k \\\\ \\updownarrow \\\\ \\mathfrak q _ 1 \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak q _ \\ell \\end{array} , \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} M = \\Big ( \\widetilde { C } _ \\Lambda | E ( 0 ) | ^ { \\frac { \\theta } { 2 } - 1 } 2 ^ { \\frac { \\theta ^ 2 } { \\theta - 2 } } \\Big ) ^ { \\frac { 1 } { \\theta } } . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} g ' | _ { D ' } ( x ) : = l _ D ( x ) + w ^ \\uparrow _ g ( \\sigma ) d _ { f , \\sigma } ( D ' , x ) + c , \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} \\Phi ( g [ \\sigma _ i ] ) = | g | \\int _ { \\sigma _ i } \\Omega . \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} & \\Delta f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ { 2 - n } = - \\lambda f ( G ^ { - 1 } ( \\bar b _ i ) ) ^ n | \\nabla \\bar b _ i | ^ 2 \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\mathfrak { H } _ j ^ { ( e ) } [ v ^ { ( e ) } _ t , \\hdots , v ^ { ( e ) } _ { t + j } ] = - \\mathfrak { H } _ j ^ { ( r ) } [ v ^ { ( r ) } _ t , \\hdots , v ^ { ( r ) } _ { t + j } ] \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( T S P _ n \\leq 5 \\sqrt { n } \\right ) = 1 \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} u _ \\varepsilon ( x _ \\varepsilon + \\mu _ \\varepsilon \\cdot ) = \\gamma _ \\varepsilon - \\frac { \\tau _ \\varepsilon } { \\gamma _ \\varepsilon } \\ , . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathbf A \\\\ \\updownarrow \\\\ \\mathbf B \\end{array} \\odot \\begin{array} { c } \\mathbf A ' \\\\ \\updownarrow \\\\ \\mathbf B ' \\end{array} = \\begin{array} { c } \\mathbf A \\odot \\mathbf A ' \\\\ \\updownarrow \\\\ \\mathbf B \\odot \\mathbf B ' \\end{array} . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\dim \\Xi _ { \\chi } = ( q - 1 ) q ^ { \\frac { m - 1 } { 2 } } . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ H f = f ^ { \\frac { Q + 2 } { Q - 2 } } & \\Omega , \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ u = 0 \\quad & \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} ( M _ { i } ) _ { e , f } : = \\begin{cases} - 1 & e v _ { i } , \\\\ 1 & e v _ { i } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} 0 = - \\sum _ { i , j , p } \\bar h _ { i j l } ^ { p ^ { * } } \\bar h _ { i j k } ^ { p ^ { * } } = \\sum _ { i , j , p } \\bar h _ { i j } ^ { p ^ { * } } \\bar h _ { i j k l } ^ { p ^ { * } } = \\bar h _ { 1 1 } ^ { 1 ^ { * } } \\bar h _ { 1 1 k l } ^ { 1 ^ { * } } + 3 \\bar h _ { 1 2 } ^ { 2 ^ { * } } \\bar h _ { 1 2 k l } ^ { 2 ^ { * } } = \\bar \\lambda _ 1 \\bar h _ { 1 1 k l } ^ { 1 ^ { * } } + 3 \\bar \\lambda _ 2 \\bar h _ { 2 2 k l } ^ { 1 ^ { * } } . \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\zeta _ { ( 0 , 0 ) , m } & = 1 \\otimes x _ m , \\zeta _ { ( 1 , 0 ) , \\mu } = \\gamma _ 1 \\otimes \\xi _ { \\mu } , \\\\ \\zeta _ { ( 0 , 1 ) , \\mu } & = \\gamma _ 2 \\otimes \\xi _ { \\mu } , \\zeta _ { ( 1 , 1 ) , m } = \\gamma _ 1 \\gamma _ 2 \\otimes x _ m , \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { H } _ 0 & : Y _ k ( t ) \\sim \\mathcal { N } ( 0 , \\sigma _ n ^ 2 ) , \\\\ \\mathcal { H } _ 1 & : Y _ k ( t ) \\sim \\mathcal { N } ( 0 , \\sigma _ x ^ 2 + \\sigma _ n ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} V _ t = g _ 0 ( t ) + \\langle 1 , U _ t \\rangle _ { \\mu } , g _ { t _ 0 } ( t ) = g _ 0 ( t _ 0 + t ) + \\langle e ^ { - t ( \\cdot ) } , U _ t \\rangle _ { \\mu } , t , t _ 0 \\geq 0 , \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\beta ( C _ { x , y , \\lambda } , z _ { x , y , \\lambda } + z _ { u , y , \\lambda } ) = i ( t _ { \\gamma ( y , \\lambda ) } , r _ { \\gamma ( y , \\lambda ) } ) . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\left ( q \\Lambda _ 0 P ^ { - 1 } , \\dots , q \\Lambda _ N P ^ { - 1 } ; q \\right ) _ d = \\prod _ { i = 0 } ^ N \\left ( q \\Lambda _ i P ^ { - 1 } ; q \\right ) _ d \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} \\frac { 1 } { s } = \\sum _ { i = 1 } ^ m \\frac 1 { s _ i } = \\frac { 1 } { q _ 1 } + \\frac { 1 } { q _ 2 } + \\sum _ { i = 3 } ^ m \\frac 1 { p _ i } \\ge \\sum _ { i = 1 } ^ m \\frac { 1 } { q _ i } = \\frac 1 q > \\frac 1 { r _ { m + 1 } ' } , \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } x _ { N + i } ^ 2 < x _ N ^ 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } < x _ N ^ 2 S . \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} S _ { i j } = g _ { i j } + \\sum _ { \\substack { n \\geq 0 \\\\ d \\in H _ 2 ( X ; \\mathbb { Z } ) } } \\frac { 1 } { n ! } \\left \\langle \\phi _ i , \\tau , \\dots , \\tau , \\frac { \\phi _ j } { 1 - q L } \\right \\rangle ^ { K \\textnormal { t h } } _ { 0 , n + 2 , d } Q ^ d \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} n & = \\frac { 1 } { p } \\sum _ { y \\in \\mathbb { F } _ p } \\sum _ { x \\in \\mathbb { F } _ q ^ * } \\chi _ m ( y x ) \\\\ & = \\frac { 1 } { p } ( q - 1 + ( p - 1 ) ( - 1 ) ) \\\\ & = p ^ { m - 1 } - 1 . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} i ^ * [ Z ] = i ^ * [ Z _ 0 ] + i ^ * [ Z _ 1 ] = i ^ * [ Z _ 0 ] + i ^ * \\circ i _ * ( y ) = i ^ * [ Z _ 0 ] . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\pi _ E ( x ; \\ell , i ) = \\# \\{ p \\le x ; ~ N _ p ( E [ \\ell ] ) = \\ell ^ i \\} . \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\| z \\| ^ { \\ell } _ { \\dot B ^ { s } _ { p , r } } \\stackrel { \\mathrm { d e f } } { = } \\| z ^ { \\ell } \\| _ { \\dot B ^ { s } _ { p , r } } \\ \\hbox { a n d } \\ \\| z \\| ^ { h } _ { \\dot B ^ { s } _ { p , r } } \\stackrel { \\mathrm { d e f } } { = } \\| z ^ { h } \\| _ { \\dot B ^ { s } _ { p , r } } . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} { \\rm v o l } ( B _ p ^ n ) \\Gamma ( 1 + n / p ) & = \\int _ { \\R ^ n } e ^ { - \\| x \\| _ p ^ p } d x = \\int _ { \\R ^ n } e ^ { - | x _ 1 | ^ p - \\dots - | x _ n | ^ p } d x = \\biggl ( 2 \\int _ 0 ^ \\infty e ^ { - t ^ p } d t \\biggr ) ^ n \\\\ & = 2 ^ n \\biggl ( \\frac { 1 } { p } \\int _ 0 ^ \\infty s ^ { 1 / p - 1 } e ^ { - s } d s \\biggr ) ^ n = 2 ^ n \\biggl ( \\frac { \\Gamma ( 1 / p ) } { p } \\biggr ) ^ n = 2 ^ n \\Gamma ( 1 + 1 / p ) ^ n . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} \\left ( u , v \\right ) \\in A ^ { \\varepsilon } \\quad u , \\ ; v \\in \\L { 1 } \\left ( \\mathbb { R } , \\mathbb { R } \\right ) \\left [ f ( x , u ) - \\varepsilon u _ { x } \\right ] _ { x } = v . \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} f = V _ { \\eta } ^ { * } V _ { \\eta } f = f \\ast \\eta ^ { * } \\ast \\eta ~ , \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} A _ 1 = D _ 1 A _ 2 \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} Y ( a , z ) = R ( f ) Y ( R ( f _ z ) ^ { - 1 } a , f ( z ) ) R ( f ) ^ { - 1 } . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\begin{aligned} m ( x , y ) = F ^ { - 1 } ( \\overline { H } ( P ) - V ( x , y ) ) . \\end{aligned} \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\widetilde { G } = G - \\frac { \\alpha ^ { 2 } n p } { 2 ( \\alpha - 1 ) } t \\rho _ { 1 } - \\frac { \\alpha n } { 2 } t ( \\rho _ { 1 } + \\rho _ { 2 } ) \\sqrt { p q } , \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} n - \\lfloor n - y \\rfloor = n + \\lceil y - n \\rceil = n + \\lceil y \\rceil - n = \\lceil y \\rceil . \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta \\asymp ( \\alpha t ) ^ \\beta e ^ { \\alpha t } \\quad \\mbox { a s $ t \\to \\infty $ } . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 5 { R _ 2 ^ 5 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 5 { R _ 2 ^ 5 } ) = 7 5 , \\mathrm { I C } ( \\mathcal { T } ^ 5 { R _ 2 ^ 5 } ) = 7 5 . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} R _ E ^ \\nabla ( a , \\bar { l } ) = \\nabla _ { a } \\nabla _ { l } - \\nabla _ { l } \\nabla _ { a } - \\nabla _ { [ a , l ] } \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\frac { A _ n } { F _ n ( 1 ) } = \\frac { \\left ( 1 - q ^ n \\right ) \\left ( 1 - a q ^ n \\right ) \\left ( 1 - a b q ^ { n - \\gamma + 1 } \\right ) } { a q ^ { n - \\gamma } \\left ( 1 - b q ^ { n + 1 } \\right ) \\left ( 1 - q ^ { \\gamma + n } \\right ) \\left ( 1 - a b q ^ { n + 1 } \\right ) } \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\begin{aligned} & \\| G ( f ) - G ( \\bar { f } ) \\| \\\\ & = \\left \\| \\int _ 0 ^ \\cdot e ^ { ( \\cdot - s ) \\Delta } \\left ( \\Gamma ^ { - 1 } ( s ) [ K ( \\Gamma ( s ) f ( s ) \\cdot \\nabla ] ( \\Gamma ( s ) f ( s ) ) - \\Gamma ^ { - 1 } ( s ) [ K ( \\Gamma ( s ) \\bar { f } ( s ) \\cdot \\nabla ] ( \\Gamma ( s ) \\bar { f } ( s ) ) \\right ) d s \\right \\| \\\\ & \\leq \\mathcal { C } _ 1 \\eta _ \\infty R ^ * \\| f - \\bar { f } \\| , \\end{aligned} \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} U ^ { - 1 } [ U \\xi , U \\eta ] & = U ^ { - 1 } \\left \\{ \\lambda ^ 2 [ \\xi , \\eta ] + \\lambda \\left ( [ u \\xi , \\eta ] + [ \\xi , u \\eta ] \\right ) + [ u \\xi , u \\eta ] \\right \\} \\ , . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\langle X ( \\mathbf { x } ) , \\mathbf { x } \\rangle = \\langle \\nabla H _ X ( \\mathbf { x } ) , \\mathbf { x } \\rangle , \\forall \\mathbf { x } \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar \\lambda _ 1 \\bar h ^ { 1 ^ { \\ast } } _ { 1 1 k } + 3 \\bar \\lambda \\bar h ^ { 1 ^ { \\ast } } _ { 1 2 k } + 3 \\bar \\lambda _ 2 \\bar h ^ { 2 ^ { \\ast } } _ { 1 2 k } - \\bar \\lambda \\bar h ^ { 2 ^ { \\ast } } _ { 2 2 k } = 0 . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} \\mathbf { y } _ \\mathcal { Q } = \\mathbf { A y } + \\mathbf { n } _ q , \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\bigg \\| \\bigg ( \\sum _ j \\bigg ( \\sum _ k ( f ^ { j k } ) ^ s \\bigg ) ^ { \\frac { t } { s } } \\bigg ) ^ { \\frac { 1 } { t } } \\bigg \\| _ { L ^ p ( v ) } \\lesssim \\prod _ { i = 1 } ^ m \\bigg \\| \\bigg ( \\sum _ j \\bigg ( \\sum _ k ( f _ i ^ { j k } ) ^ { s _ i } \\bigg ) ^ { \\frac { t _ i } { s _ i } } \\bigg ) ^ { \\frac { 1 } { t _ i } } \\bigg \\| _ { L ^ { q _ i } ( v _ i ) } . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{gather*} \\Psi : \\mathcal { N } ( \\Delta ) \\longrightarrow \\mathcal { N } ( M _ 0 ) \\\\ \\Psi ( \\Delta ) = M _ 0 \\\\ \\pi ( \\Psi ( q , q ) ) = q \\end{gather*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\mu ( f ) ( x _ 1 , \\dots , x _ n ) = ( 1 + x _ 1 ) \\cdots ( 1 + x _ n ) \\cdot f \\left ( \\frac { x _ 1 } { 1 + x _ 1 } , \\dots , \\frac { x _ n } { 1 + x _ n } \\right ) , \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\gamma ' ( n ) = \\lfloor x / h _ 1 \\rfloor \\forall x > 0 . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\nu _ 2 : = \\left [ \\nu _ 1 ; \\nu _ 2 ( \\phi _ 2 ) = - \\frac { 1 } { p ^ 2 } \\right ] . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} [ E _ 2 ^ t , E _ 3 ^ t , \\ldots , E _ { n + 1 } ^ t ] & = [ e _ 2 , t e _ 3 , e _ 4 , \\ldots , e _ { n + 1 } ] = t e _ 1 = E _ 1 ^ t , \\\\ [ E _ 1 ^ t , E _ 3 ^ t , \\ldots , E _ { n + 1 } ^ t ] & = [ t e _ 1 , t e _ 3 , e _ 4 , \\ldots , e _ { n + 1 } ] = t ^ 2 e _ 2 = t ^ 2 E _ 2 ^ t , \\\\ [ E _ 1 ^ t , \\ldots , E _ { i - 1 } ^ t , E _ { i + 1 } ^ t , \\ldots , E _ { n + 1 } ^ t ] & = [ t e _ { 1 } , e _ 2 , \\ldots , e _ { i - 1 } , e _ { i + 1 } , \\ldots , e _ { n + 1 } ] = 0 \\mbox { f o r } i > 2 . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} ( L \\rho ) ^ \\ast \\circ \\nu = \\nu \\circ \\rho ^ \\ast : H ^ 3 ( M ) \\rightarrow H ^ 2 ( L S ( \\xi ) ) . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} \\textbf { h } _ k = \\eta \\left ( \\theta _ k , \\phi _ k \\right ) \\beta _ k ^ c \\textbf { a } ( \\textbf { x } _ k ) , \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{gather*} \\big ( C _ { a b } ^ e C _ { c e } ^ d + \\partial _ j C _ { a b } ^ d \\rho _ c ^ j + \\operatorname { c y c l } ( a b c ) \\big ) { \\rho ^ i _ d } = 0 . \\end{gather*}"}
-{"id": "7438.png", "formula": "\\begin{align*} & \\Psi _ { \\mu _ k } ( w ^ { ( k ) } + \\lambda \\triangle \\tilde { w } ^ { ( k ) } ) \\\\ & \\leq ( 1 - \\lambda ) \\Psi _ { \\mu _ k } ( w ^ { ( k ) } ) + \\varpi ( \\lambda \\Psi _ { \\mu _ k } ( w ^ { ( k ) } ) ) + \\frac 1 4 \\lambda ^ 2 C ^ 2 ( 1 + \\theta _ 1 ) ^ 2 \\beta \\Psi _ { \\mu _ k } ( w ^ { ( k ) } ) + \\lambda \\sqrt 2 \\theta _ 1 \\Psi _ { \\mu _ k } ( w ^ { ( k ) } ) . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} \\Big < \\nu ; E ( \\widetilde { y } ) - E ( Y ) - \\Lambda \\cdot ( \\widetilde { y } - Y ) \\Big > = 0 \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} a _ { N 0 } ( z , q , Q ) = \\frac { 1 } { \\Lambda _ 0 \\cdots \\Lambda _ N } \\left ( - \\frac { Q } { z ^ { N + 1 } } + \\prod _ { j = 0 } ^ N \\frac { 1 - \\Lambda _ j } { 1 - q } \\right ) \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} U ^ { ( \\theta ) } f = \\sum _ { I \\in \\mathcal { D } _ { \\leq n } } \\frac { \\langle f , b _ I ^ { ( \\theta ) } \\rangle } { \\langle T b _ I ^ { ( \\theta ) } , b _ I ^ { ( \\theta ) } \\rangle } b _ I ^ { ( \\theta ) } , f \\in W _ N . \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} T _ 0 = A ( y ^ 0 \\frac { \\partial } { \\partial y ^ 4 } - y ^ 4 \\frac { \\partial } { \\partial y ^ 0 } ) - B _ i ( y ^ 0 \\frac { \\partial } { \\partial y ^ i } + y ^ i \\frac { \\partial } { \\partial y ^ 0 } ) - C _ j ( y ^ 4 \\frac { \\partial } { \\partial y ^ j } + y ^ j \\frac { \\partial } { \\partial y ^ 4 } ) + D _ p \\epsilon _ { p q r } y ^ q \\frac { \\partial } { \\partial y ^ r } . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} \\frac { d } { d s } | _ { s = 0 } \\frak { H } _ 2 = \\int ( 1 + r ^ 2 ) \\delta \\tau \\left [ d i v ( | H _ 0 | \\nabla Y ^ 0 ) + \\Delta \\frac { \\Delta Y ^ 0 } { | H _ 0 | } - d i v \\alpha _ { H _ 0 } \\right ] . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} T ( l , h ) : = & \\sum _ { k = 1 } ^ n ( - 1 ) ^ { h + k + n } { n \\choose k } \\left ( { n \\choose h } - \\sum _ { h \\le q \\le \\max ( l , h - 1 ) } { k \\choose q - h } { n - k \\choose n - q } \\right ) \\\\ = & - ( - 1 ) ^ { h + n } { n \\choose h } - \\sum _ { k = 1 } ^ n ( - 1 ) ^ { h + k + n } { n \\choose k } \\sum _ { q = h } ^ { \\max ( l , h - 1 ) } { k \\choose q - h } { n - k \\choose n - q } , \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} h \\mathcal { R } = \\bigcap _ { i = 0 } ^ { k - 1 } ( x - \\sigma ^ i ( \\gamma ) ) \\mathcal { R } = \\bigcap _ { i = 0 } ^ { k - 1 } \\sigma ^ i \\left ( ( x - \\gamma ) \\mathcal { R } \\right ) . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} W _ { i } = S _ { 1 } \\times V ' \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\wp ( z + 1 , q ) = \\wp ( z + \\tau , q ) = \\wp ( z , q ) , \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) = Q ^ { \\frac { H } { z } } \\sum _ { d \\geq 0 } \\frac { Q ^ d } { \\prod _ { r = 1 } ^ d \\left ( H + r z \\right ) ^ { N + 1 } } \\in H \\left ( \\mathbb { P } ^ N \\right ) \\otimes \\mathbb { C } [ z , z ^ { - 1 } ] [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} ( a ; q ) _ n ( a q ^ n ; q ) _ k = ( a ; q ) _ { n + k } . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\sigma ( t , \\theta _ 1 , \\theta _ 2 ) = t \\left ( \\cos ( \\theta _ 1 ) \\cos ( \\theta _ 2 ) , \\cos ( \\theta _ 1 ) \\sin ( \\theta _ 2 ) , \\sin ( \\theta _ 1 ) \\right ) + ( 1 - t ) V \\ ; , \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} & \\left ( a b \\otimes m - b \\otimes m a - a \\otimes b m \\right ) + \\left ( b a \\otimes m - b \\otimes a m - a \\otimes m b \\right ) \\\\ = { } & - d _ { } ( a \\otimes b \\otimes m ) - d _ { } ( b \\otimes a \\otimes m ) . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} \\bar { B } = \\left [ \\begin{array} { c } t _ 1 ^ 2 - \\dfrac { 1 } { n } t _ 1 ^ 2 \\end{array} \\right ] = \\left [ \\begin{array} { c } \\dfrac { n - 1 } { n } t _ 1 ^ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\mathcal { U } _ { j _ { 1 } , j _ { 2 } } = \\{ ( w _ { 1 } , w _ { 2 } ) \\in \\mathcal { U } \\ : : \\ : \\sigma _ { i } ^ { j _ { i } } \\nu _ { i } [ w _ { i } ] \\ge \\exp ( - n _ { i } m ( h _ { i } + c _ { 2 } \\delta / \\epsilon ) ) i = 1 , 2 \\} \\ : . \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} 0 & < \\int _ { a } ^ { x _ { o } } \\left [ v - u \\right ] \\ ; d x \\le \\int _ { a } ^ { b } \\left [ v - u \\right ] \\ ; d x = \\lambda \\int _ { a } ^ { b } \\left [ f ( x , u ) - f ( x , v ) \\right ] _ { x } \\ ; d x \\\\ & = \\lambda \\left [ f \\left ( b - , u ( b - ) \\right ) - f \\left ( b - , v ( b - ) \\right ) \\right ] - \\lambda \\left [ f \\left ( a + , u ( a + ) \\right ) - f \\left ( a + , v ( a + ) \\right ) \\right ] . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} g ( T _ i \\bullet _ \\tau T _ j , T _ k ) = \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ k } \\mathcal { F } \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} r _ { y , 2 y , \\dots , ( k - 1 ) y } ( [ N ] ) = o _ { k } ( N ) , \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} e _ { 1 ^ { * } } = \\frac { { \\vec H } } { | { \\vec H } | } , \\ \\ H ^ { 1 ^ { * } } = H = | { \\vec H } | , \\ \\ H ^ { 2 ^ { * } } = h _ { 1 1 } ^ { 2 ^ { * } } + h _ { 2 2 } ^ { 2 ^ { * } } = 0 . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} ( \\mathcal { K } ^ { * } f ) ( t ) = \\begin{cases} C _ { H } \\cdot t ^ { \\frac { 1 } { 2 } - H } \\cdot \\left ( I _ { 1 ^ { - } } ^ { H - \\frac { 1 } { 2 } } \\left ( s ^ { H - \\frac { 1 } { 2 } } f ( s ) \\right ) \\right ) ( t ) , & H > \\frac { 1 } { 2 } ; \\\\ C _ { H } \\cdot t ^ { \\frac { 1 } { 2 } - H } \\cdot \\left ( D _ { 1 ^ { - } } ^ { \\frac { 1 } { 2 } - H } \\left ( s ^ { H - \\frac { 1 } { 2 } } f ( s ) \\right ) \\right ) ( t ) , & H \\leq \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { m - 1 } { 2 } } ( - 1 ) ^ k [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ; q ) _ k ^ 3 } \\equiv 0 \\pmod { \\Phi _ m ( q ) } . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} u _ 1 = e ^ { - 3 } \\approx 0 . 0 4 9 8 , u _ n > 0 , \\quad n \\geq 2 , \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} ( \\underline { \\Delta } - \\partial _ { t } ) ( t \\varphi G ) = 2 t \\overline { \\nabla } \\varphi \\overline { \\nabla } G - \\varphi G + t \\varphi ( \\underline { \\Delta } - \\partial _ { t } ) G + t G ( \\underline { \\Delta } - \\partial _ { t } ) \\varphi . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} X _ 2 = \\left ( \\begin{array} { c c c c c } 0 & 0 . 4 2 0 4 & 0 . 1 7 5 9 & 0 . 6 7 7 0 & 0 . 3 \\\\ 0 . 7 3 4 3 & 0 . 3 7 9 6 & 0 . 1 7 9 7 & 0 & 0 . 5 \\\\ 0 . 0 2 7 4 & 0 & 0 . 5 7 9 1 & 0 . 0 0 6 9 & 0 . 8 \\\\ 0 . 1 3 3 4 & 0 . 0 5 8 0 & 0 . 6 7 1 9 & 0 . 6 4 0 3 & 1 \\\\ 0 & 0 & 0 & 0 & 0 . 8 \\end{array} \\right ) , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\phi ( \\mathcal { H } ) : = \\int _ 0 ^ \\infty \\phi ( \\lambda ) \\ , d E _ { \\mathcal { H } } ( \\lambda ) \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} B _ r ( x _ i ) \\cap B _ r ( x _ j ) = \\varnothing i \\neq j , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} v ( \\phi ( s _ \\nu ) ) & = v ( c ) + \\sum _ { \\alpha \\in S } \\epsilon _ i u ( s _ \\nu - \\alpha ) + \\sum _ { \\alpha \\in \\Omega _ { \\phi } \\setminus S } \\epsilon _ i u ( s _ \\nu - \\alpha ) = \\lambda \\delta + \\gamma \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} u _ \\varepsilon ( z _ \\varepsilon ) = \\frac { 4 \\pi G _ { z _ \\varepsilon } ( x _ \\varepsilon ) } { \\gamma _ \\varepsilon } \\left ( 1 + \\frac { 1 } { \\gamma _ \\varepsilon ^ 2 } + \\frac { A ( \\gamma _ \\varepsilon ) - 2 \\xi _ \\varepsilon } { 2 } + o ( \\tilde { \\zeta } _ \\varepsilon ) \\right ) \\\\ + o \\left ( \\frac { 1 } { \\gamma _ \\varepsilon } \\right ) \\ , . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\| Q _ r ( \\hat Q _ r - Q _ r ) Q _ r \\| _ 2 \\leq \\| Q _ r ( \\hat Q _ r - Q _ r ) Q _ r \\| _ 1 = \\operatorname { t r } ( Q _ r - Q _ r \\hat Q _ r Q _ r ) = \\| \\hat Q _ r - Q _ r \\| _ 2 ^ 2 / 2 , \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} - Z _ k '' = \\left [ \\mu _ 1 \\left ( 1 - 2 \\frac { v ^ + } { \\alpha } \\right ) + \\frac { 1 } { d } \\mu _ 2 \\left ( 1 + 2 \\frac { v ^ - } { d } \\right ) \\right ] Z _ k + \\lambda _ { 1 , k } \\sigma ( v ) Z _ k + o _ k ( 1 ) \\\\ \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\dim _ H E _ \\varphi ( \\Phi ) = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { j = 1 } ^ \\ell \\alpha ( 1 / c - 1 ) \\log j } { d \\sum _ { j = 1 } ^ { \\ell + 1 } \\alpha / c \\log j - \\alpha ( 1 / c - 1 ) \\log ( \\ell + 1 ) } = \\frac { 1 - c } d . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\tilde { D } _ { u } v = D _ { u } v - \\frac { 1 } { 4 } \\{ A _ { u } ^ { \\mathrm { s y m } } , \\mathcal J \\} v - \\frac { 1 } { 2 } \\mathcal J ( D _ { u } \\mathcal J ) v , \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\lambda ^ * \\partial _ z ( \\alpha w ) - \\Delta ( \\partial _ z ( \\alpha w ) ) & = \\partial _ z \\left [ \\alpha F - 2 ( \\partial _ z \\alpha ) ( \\partial _ z w ) - ( \\partial _ z ^ 2 \\alpha ) w \\right ] - ( \\partial _ z \\alpha ) ( \\nabla _ H \\Pi ) \\quad \\Omega ' , \\\\ \\partial _ z ( \\alpha w ) & = 0 \\quad \\Gamma _ u ' \\cup \\Gamma _ b ' , \\partial _ z ( \\alpha w ) \\Gamma _ l ' . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\textrm { c o n v } ( g ) = g ^ { * * } , \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} P ( \\vert \\boldsymbol { Y } _ { \\mathcal { I } } \\vert = n ) = \\sum _ { k \\geq n } \\frac { a _ { \\vert \\boldsymbol { \\theta } _ { \\mathcal { I } } \\vert } ( n ) a _ { \\vert \\boldsymbol { \\theta } _ { - \\mathcal { I } } \\vert } ( k - n ) } { a _ { \\vert \\boldsymbol { \\theta } \\vert } ( k ) } P ( \\vert \\boldsymbol { Y } \\vert = k ) \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\bar { K } = \\{ x \\geq 0 : { B } x = { b } \\} . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} \\log \\lambda _ { g } ^ { \\left ( p , q \\right ) } \\left ( f \\right ) = \\underset { \\sigma \\rightarrow + \\infty } { \\underline { \\lim } } \\left [ \\log ^ { \\left [ p + 1 \\right ] } M _ { g } ^ { - 1 } \\left ( \\sigma \\right ) - \\log ^ { \\left [ q + 1 \\right ] } M _ { f } ^ { - 1 } ( \\sigma ) \\right ] . \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} u ( t , x ) = \\hat u ^ { R } ( t , x ) + k ( t ) \\hat S ( t , x ) \\ , , \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} I _ 2 + I _ 4 + I _ 5 = \\frac { 1 } { 2 ^ { \\sqrt { \\delta } } } \\int _ { x - t } ^ { x + t } u _ 1 ( y ) E ( t , x ; 0 , y ; \\mu , \\nu ^ 2 ) \\ , \\mathrm { d } y - t u _ 1 ( x ) . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\mathbb { H } _ n = \\mathbb { G } _ { n + 1 } ( - m + 1 ) \\quad \\mbox { a n d } d _ n ^ { \\mathbb { H } } = ( - 1 ) d _ { n + 1 } ^ { \\mathbb { G } } \\mbox { f o r e v e r y } n ; \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} X = \\sum _ { k = 1 } ^ d \\omega _ k F _ { k } f _ k + \\epsilon , \\epsilon = \\Gamma ^ { 1 / 2 } Y . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\| \\tilde { h } ^ { ( m ) } \\| _ { \\bar { { \\cal H } } ( [ 0 , a _ { m + 1 } ] ) } ^ { 2 } & \\leq C _ { H } ( Q _ { 1 } + Q _ { 2 } + Q _ { 3 } ) \\leq C _ { H , l _ { 0 } } \\sum _ { k = 1 } ^ { m } k ^ { 2 H + 1 } | I _ { k } | ^ { 2 ( 1 - H ) } . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\beta _ { k } = \\frac { 1 } { k ! } \\left . \\frac { d ^ k } { d r ^ k } \\psi \\right | _ { r = 0 } , \\psi ( r ) = \\frac { 1 } { 2 } \\left ( \\frac { 1 } { 4 } - r \\right ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\hat { p } _ k ( x _ k ) = \\prod _ { y \\in \\breve { Z } _ k } ( x _ k - y ) , \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\| V ^ j \\| ^ 2 _ { \\dot { H } ^ s } \\leq \\limsup _ { n \\rightarrow \\infty } \\| v _ n \\| ^ 2 _ { \\dot { H } ^ s } \\leq M ^ 2 . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\{ f ^ { - 1 } ( U ) \\colon f \\in F , U \\mbox { o p e n i n } T \\} , f ^ { - 1 } ( U ) : = \\{ y \\in Y \\colon f ( y ) \\in U \\} . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} g ^ { 1 1 } & = \\frac { 1 } { 2 } - s f + \\frac { s ^ 2 } { 2 } \\big ( 3 f ^ 2 - 2 \\textstyle ( \\frac { \\partial f } { \\partial \\theta _ 1 } ) ^ 2 \\big ) + O ( s ^ 3 ) , \\\\ g ^ { 1 2 } & = - s ^ 2 \\textstyle \\frac { \\partial f } { \\partial \\theta _ 1 } \\frac { \\partial f } { \\partial \\theta _ 2 } + O ( s ^ 3 ) , \\\\ g ^ { 2 2 } & = \\frac { 1 } { 2 } - s f + \\frac { s ^ 2 } { 2 } \\big ( 3 f ^ 2 - 2 \\textstyle ( \\frac { \\partial f } { \\partial \\theta _ 2 } ) ^ 2 \\big ) + O ( s ^ 3 ) . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\mathcal { H } _ { z _ \\varepsilon } ( z _ \\varepsilon + \\varepsilon z ) = \\mathcal { H } _ { z _ \\varepsilon } ( z _ \\varepsilon ) + O \\left ( \\frac { \\varepsilon | z | } { d ( z _ \\varepsilon , \\partial \\Omega ) } \\right ) \\ , , \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\delta _ { T _ { a } \\circ f } = \\lim _ { n \\to \\infty } \\left \\| ( ( T _ { a } \\circ f ) ^ { n } ) ^ { * } \\right \\| ^ { 1 / n } = \\lim _ { n \\to \\infty } \\left \\| ( f ^ { n } ) ^ { * } \\right \\| ^ { 1 / n } = \\delta _ { f } \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\sigma ( \\phi ( t , z ) ) = \\phi ( - t , \\sigma ( z ) ) , \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { + \\infty } \\vert \\widehat { h } ( k ) \\vert ^ s = + \\infty . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} \\frac { \\hat z ^ { \\prime \\prime } } { \\hat z ^ \\prime } = 2 \\Upsilon ^ \\prime , \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} | z - \\omega | = | \\omega _ 1 | | c _ 1 + c _ 2 \\tau | \\geq | \\omega _ 1 | | c _ 2 | | \\Im ( \\tau ) | \\geq | \\omega _ 2 | / 6 0 . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\angle _ m ( - \\sigma ^ \\prime ( T / 2 ) , \\gamma ^ \\prime ( 0 ) ) + \\angle _ m ( \\sigma ^ \\prime ( T / 2 ) , \\gamma ^ \\prime ( 0 ) ) = \\pi , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\left | \\nabla u \\right | ^ { 2 } = \\left | \\nabla \\rho \\right | ^ { 2 } + \\rho ^ { 2 } \\left | \\nabla \\varphi _ { 0 } + \\nabla \\psi \\right | ^ { 2 } . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} b \\tau ( b ) = U \\Lambda U ^ { - 1 } \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\frac { \\lambda _ \\varepsilon } { 2 } \\int _ \\Omega u _ \\varepsilon \\left [ \\Psi _ { N _ \\varepsilon } ' ( u _ \\varepsilon ) \\right ] ^ + d y = 4 \\pi + o ( 1 ) \\ , , \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} & \\frac { 1 } { d ^ 4 } \\geq 1 \\ , , \\\\ & p = d + \\frac { y _ 1 ^ 2 } { | y | } ( \\alpha - c _ 1 ) | k ' _ \\eta ( | y | ) | \\geq d \\geq c _ 1 \\ , , \\\\ & q ^ 2 = ( \\alpha - c _ 1 ) ^ 2 \\frac { y _ 1 ^ 2 y _ 2 ^ 2 } { | y | ^ 2 } ( k ' _ \\eta ( | y | ) ) ^ 2 \\leq 9 ( 1 - c _ 1 ) ^ 2 \\ , , \\\\ & | m | \\leq \\frac { 4 2 c _ 2 ( 1 - c _ 1 ^ 2 ) } { c _ 1 } \\eta + \\frac { c _ 2 ^ 2 ( 1 - c _ 1 ^ 2 ) } { c _ 1 ^ 2 } \\eta ^ 2 \\ , . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} & b _ { a _ { 0 } , a _ { 1 } } \\left ( \\mathbf { Z } _ { 0 } , \\mathbf { Z } _ { 1 } ; P \\right ) \\equiv E _ { P } \\left [ Y | A _ { 0 } = a _ { 0 } , A _ { 1 } = a _ { 1 } , \\mathbf { Z } _ { 0 } , \\mathbf { Z } _ { 1 } \\right ] , \\\\ & b _ { a _ { 0 } } \\left ( \\mathbf { Z } _ { 0 } ; P \\right ) \\equiv E _ { P } \\left [ b _ { a _ { 0 } , a _ { 1 } } \\left ( \\mathbf { Z } _ { 0 } , \\mathbf { Z } _ { 1 } ; P \\right ) | A _ { 0 } = a , \\mathbf { Z } _ { 0 } \\right ] \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} E ( \\left < \\nu ; \\widetilde { y } \\right > ) = \\left < \\nu ; E ( \\widetilde { y } ) \\right > < \\infty . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} 2 \\sqrt { K } = \\frac { 2 } { r } + k ^ { ( 1 ) } r + O ( r ^ 2 ) \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} & \\zeta ( u ^ * ) = v ^ * , \\ \\ H ( u , v ) = v - \\zeta ( u ) \\ \\ { \\rm f o r } \\ \\ u \\geq 0 , \\ v \\geq 0 \\ \\ { \\rm o r } \\\\ & \\zeta ( v ^ * ) = u ^ * , \\ \\ H ( u , v ) = \\zeta ( v ) - u \\ \\ { \\rm f o r } \\ \\ u \\geq 0 , \\ v \\geq 0 . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} | g | ( x , t ) = | \\omega + h | ( x , t ) \\le L \\sup _ { s \\le t } | d ( x , s ) | + | h ( x , t ) | \\ , . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} & \\quad \\ \\| \\mathcal { V } [ v + \\eta ] - \\mathcal { V } [ v ] - M \\eta \\| _ { L ^ r ( 0 , T ) } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\le \\rho _ { v , p , r } ( \\| \\eta \\| _ { \\infty } ) ( \\| \\eta ' \\| _ { L ^ p ( 0 , T ) } + | \\eta ( 0 ) | ) . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\sigma y = \\bigg ( 1 + O \\bigg ( \\frac { 1 } { \\log ^ { 1 0 } _ { 2 } x } \\bigg ) \\bigg ) \\cdot 8 0 c x \\log _ 2 x \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} d _ r ( C _ D ) = \\begin{cases} \\frac { p ^ s ( 2 p ^ s + 1 - p ) } { p + 1 } \\left ( 1 - \\frac { 1 } { p ^ r } \\right ) , & 1 \\le r \\le s , \\\\ \\frac { 2 ( q - 1 ) } { p + 1 } + 1 - p ^ { m - r } , & s \\le r \\le m . \\end{cases} \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} & \\lim _ { \\varepsilon \\to 0 } \\frac { \\gamma _ \\varepsilon ^ 2 - \\tilde { N } _ \\varepsilon } { \\sqrt { \\tilde { N } _ \\varepsilon } } = + \\infty \\ , , \\\\ & \\frac { \\gamma _ \\varepsilon ^ 2 - \\tilde { N } _ \\varepsilon } { \\sqrt { \\tilde { N } _ \\varepsilon } } = O ( 1 ) \\ , . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} 0 = & \\ \\int _ M Z ( Z f ) e ^ { - f } d V _ g \\\\ = & \\ \\int _ M \\left \\{ Z \\left ( e ^ { - f } Z f \\right ) + e ^ { - f } ( Z f ) ^ 2 \\right \\} d V _ g \\\\ = & \\ \\int _ M L _ Z \\left ( e ^ { - f } Z f d V _ g \\right ) + \\int _ M ( Z f ) ^ 2 e ^ { - f } d V _ g \\\\ = & \\ \\int _ M ( Z f ) ^ 2 e ^ { - f } d V _ g . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} f _ j ( - t , - z ) = - f _ j ( t , z ) \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} & T : \\prod _ { k = 1 } ^ n L ^ { p _ k } ( \\R ^ d ; X _ k ) \\to L ^ { q _ { n + 1 } } ( \\R ^ d ; Y _ { n + 1 } ) , { 1 < p _ k \\leq \\infty , \\ , \\frac { 1 } { n } < q _ { n + 1 } < \\infty } , \\ , { \\textstyle \\frac { 1 } { q _ { n + 1 } } } = \\sum _ { k = 1 } ^ n { \\textstyle \\frac { 1 } { p _ k } } , \\\\ & T : \\prod _ { k = 1 } ^ n L ^ { 1 } ( \\R ^ d ; X _ k ) \\to L ^ { \\frac { 1 } { n } , \\infty } ( \\R ^ d ; Y _ { n + 1 } ) . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} p _ i = 0 \\ \\Rightarrow \\ q _ i \\leq 1 ; q _ j = 0 \\ \\Rightarrow \\ p _ j \\leq 1 . \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} W ( \\Omega ) \\coloneqq \\{ ( x , y ) \\in A ^ G \\times A ^ G : x \\vert _ \\Omega = y \\vert _ \\Omega \\} , \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} T ^ { \\tilde { D } } ( u , v , w ) = T ^ { D } ( u , v , w ) + \\eta ( u , v , w ) + \\eta ( w , u , v ) + \\eta ( v , w , u ) . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} ( I - A ) ^ { - 1 } \\ = \\ \\sum _ { k = 0 } ^ { \\infty } A ^ k \\ \\ge 0 \\ ; ( I - A ^ T ) ^ { - 1 } \\ = \\ \\sum _ { k = 0 } ^ { \\infty } ( A ^ T ) ^ k \\ \\ge 0 \\ \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} v ( 0 , x ) = v _ 0 ( x ) , v _ t ( 0 , x ) = v _ 1 ( x ) , \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\boldsymbol { w } \\left ( t ; \\lambda \\right ) = e ^ { \\lambda t } \\boldsymbol { w } _ { 0 } . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\lambda _ 1 ^ { r , t } ( J ) : = \\min _ { \\begin{subarray} { c } u \\in W ^ { 1 , r } _ 0 ( D ) \\\\ u \\neq 0 \\end{subarray} } \\bigg \\{ \\frac { \\int _ { J } p ( x ) u ' ( x ) ^ r d x + \\int _ J q ( x ) u ( x ) ^ r d x } { ( \\int _ { J } w ( x ) u ( x ) ^ t d x ) ^ { { r } / { t } } } \\bigg \\} . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} T _ n = \\sup _ { \\frac { 1 } { 4 ^ { n + 1 } } \\leq r \\leq \\frac { 1 } { 4 ^ n } } \\frac { \\operatorname { o s c } ( y _ 0 , r , v ) } { r ^ \\beta } . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\frac { D ^ 2 } { d t ^ 2 } \\gamma = \\nabla \\log V ( \\gamma ) - \\left < \\nabla \\log V , \\dot \\gamma \\right > \\dot \\gamma . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} \\begin{array} { r c l } f ( x ) & > & \\| g ( x ) \\| \\left ( \\lambda \\| g ( x ) \\| - R \\| c \\| \\right ) > 0 = f ( 0 ) \\ , , \\end{array} \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\mathcal { F } _ 1 ( V _ 3 ) + 3 \\mathcal { F } _ 2 ( V _ 1 , V _ 2 ) + \\mathcal { F } _ 3 ( V _ 1 , V _ 1 , V _ 1 ) = 0 . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} N _ { G _ { \\mathrm { r e s } ( \\alpha ) } } ( A ) ( k ) / Z _ { G _ { \\mathrm { r e s } ( \\alpha ) } } ( A ) ( k ) = N _ { G _ { \\mathrm { r e s } ( \\alpha ) } } ( T ) ( k ) / Z _ { G _ { \\mathrm { r e s } ( \\alpha ) } } ( T ) ( k ) . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i \\Lambda _ j \\Lambda _ l } \\widetilde { m } d y = 0 . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} v _ 1 ( x ) : = \\exp ( ( \\eta + i \\xi ) \\cdot x ) , v _ 2 ( x ) : = \\exp ( ( - \\eta + i \\xi ) \\cdot x ) , \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} E [ X ^ n ] = \\int _ { - \\infty } ^ \\infty x ^ n f ( x ) d x , \\ , \\ , ( n \\geq 1 ) . \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R } \\left ( \\int _ { \\vert u \\vert ^ { - 1 } ( s ) } \\vert \\nabla u \\vert ( \\ell ) { \\rm d } \\ell \\right ) { \\rm d } s & = \\int _ \\Omega \\vert \\nabla u ( x ) \\vert . \\vert \\nabla \\vert u \\vert \\vert { \\rm d } x , \\\\ & \\leq \\int _ \\Omega \\vert \\nabla u ( x ) \\vert ^ 2 { \\rm d } x . \\end{aligned} \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} b _ { \\beta , \\alpha + \\gamma } ( \\theta ^ \\beta _ + + \\theta ^ \\beta _ - ) = - 2 b _ { \\beta , \\alpha + \\gamma } \\xi _ \\beta \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\partial v ( y ) = \\partial h _ K ( y + e ) | e ^ \\bot = F ( K , y + e ) | e ^ \\bot = F ( K , \\pi ( y ) ) | e ^ \\bot . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} A \\sigma ^ k ( x + t v ) = \\xi ^ k ( x ) \\cdot ( h ^ k ) ' ( t ) , k = 1 , \\ldots , m , \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{gather*} D \\alpha ^ { \\prime } = \\gamma + ( \\tau ^ { \\prime t } \\otimes g _ { \\flat } ) \\rho , \\end{gather*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\mathbf { G = } \\left ( \\mathbf { G } _ { 0 } , \\dots , \\mathbf { G } _ { p } \\right ) \\mathbf { \\subset V \\backslash } \\left \\{ \\mathbf { A } , Y \\right \\} \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} T ^ { \\tilde { D } } ( u , v , w ) = \\frac { 1 } { 4 } N _ { \\mathcal J } ( u , v , w ) . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\delta _ n = \\min _ { \\gamma \\in \\Gamma \\setminus \\{ 1 \\} } B _ \\Omega ( y _ n , \\gamma y _ n ) . \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} | B _ 1 | \\bar { f } ( M ) & \\geq \\liminf _ { h \\to + \\infty } G _ { \\sigma _ h } ( w _ h , B _ 1 ) - 2 \\eta \\geq G ( u _ M , B _ 1 ) - 2 \\eta = | B _ 1 | f _ { \\rm h o m } ( M ) - 2 \\eta \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } = \\sum \\limits _ { ( M _ b , \\mu ' ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } \\mathrm { I n d } ^ G _ { P _ b } ( \\otimes ^ k _ { i = 1 } \\mathrm { M a n t } _ { M _ { b ' _ i } , b ' _ i , \\mu ' _ i } ) , \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} y _ { 2 } = \\varphi ^ { - 1 } _ { 1 } \\left ( \\frac { b } { c G _ { 3 } } \\right ) . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} { T _ A } = \\min \\left \\{ { K : H _ { K , M } ^ { ( A ) } \\ge h } \\right \\} . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\textbf { Q } ^ \\lambda : = \\prod _ { q \\in \\textbf { Q } } q ^ { \\lambda ( q ) } \\in K [ x ] . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\nabla _ { z \\partial _ z } S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) = S ^ \\textnormal { c o h } ( \\tau , z ) \\left ( \\mu - \\frac { \\rho } { z } \\right ) ( T _ a ) \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} t _ k : = 2 b ^ { - 1 / Q } \\mu ( 2 B _ 0 \\setminus E _ { k - 1 } ) ^ { 1 / Q } . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\mathbb { D } ^ m _ n = \\left \\{ ( X _ 1 , \\ldots , X _ m ) \\in M _ n ^ m \\ : \\left | \\ : \\ : \\begin{array} { l } X _ j X _ k - X _ k X _ j = \\mathbf { 0 } _ n , \\\\ X _ j X _ j ^ \\ast - X _ j ^ \\ast X _ j = \\mathbf { 0 } _ n , \\\\ \\| X _ j \\| \\leq 1 \\end{array} 1 \\leq j , k \\leq m \\right . \\right \\} \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} q ( \\beta , C ) & \\leq \\P \\bigl [ \\bigr ] + \\P \\Biggl [ 1 + \\sum _ { i = 1 } ^ k \\tau _ i > C k \\Biggr ] . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} v \\varPsi ( c ) & : = \\widetilde { \\mathcal { C } } \\mu & & c \\in v \\mathcal { C } , \\\\ \\varPsi ( f ) & : = \\rho ( \\mu , \\mu \\ast f ^ \\circ , \\nu ) & & f \\in \\mathcal { C } ( c , d ) , \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d r _ t & = & \\big ( \\frac { d } { d x } r _ t + \\alpha _ { \\rm H J M } ( r _ t ) \\big ) d t + \\sigma ( r _ { t } ) d W _ t \\medskip \\\\ r _ 0 & = & h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} s _ { n + 1 } & = d ( x _ { n + 1 } , x _ { n + 2 } ) \\\\ & < d ( x _ n , x _ { n + 1 } ) \\\\ & = s _ n \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} c _ U ( g , h ) = c _ { \\beta , \\rho } ( \\overline g , \\overline h ) \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} M ' _ { X ' } = M _ { X , X ' } \\times L ' , M _ { X , X ' } = \\prod _ { j = 1 } ^ { l } G ( ( V ' ) ^ { \\gamma ' , t _ { j } + 1 } _ { t _ { j } } ) \\times G ( U _ 1 ) . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\int _ a ^ b ( I _ a ^ k f ) ( x ) g ( x ) \\omega ( x ) \\ , d x = \\int _ a ^ b I _ b ^ k ( g ) ( x ) f ( x ) \\omega ( x ) \\ , d x . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} w ^ { ( k \\ell ) } : = w _ 1 ^ { ( k \\ell ) } = w _ 2 ^ { ( k \\ell ) } \\Omega . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} A _ { I J } = a ^ \\ast _ { i _ 1 } \\cdots a ^ \\ast _ { i _ k } a _ { j _ 1 } \\cdots a _ { j _ l } . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} f ( x ) = ( x _ 1 ^ 2 - 1 ) ^ 2 + ( x _ 2 ^ 2 - 2 ) ^ 2 - 0 . 7 \\ , x _ 1 x _ 2 + 0 . 2 \\ , x _ 1 + 0 . 3 \\ , x _ 2 \\ , . \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} d u \\circ \\delta _ \\lambda ^ { - 1 } = \\lambda ^ { - \\nu } d u , \\quad \\nu \\triangleq \\sum _ { k = 1 } ^ { l } k \\dim ( \\mathcal { L } _ k ) . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\gamma _ { _ { \\mathcal { G } } } : = \\max _ { I \\in G _ { N } ^ { 0 } } \\gamma _ { I } . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} \\overline { Q } _ { k , i } ( m , n ) = 0 , m n , \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} { n \\choose k _ 1 , k _ 2 } \\geq \\frac { 1 } { k _ 1 ! k _ 2 ! } ( n - k _ 1 - k _ 2 ) ^ { k _ 1 + k _ 2 } = \\frac { n ^ { k _ 1 + k _ 2 } } { k _ 1 ! k _ 2 ! } \\left ( 1 - \\frac { k _ 1 + k _ 2 } { n } \\right ) ^ { k _ 1 + k _ 2 } . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} d | Y ^ n _ t - Y _ t | ^ 2 = [ ( \\nabla u _ n ( t , X ^ n _ t ) - \\nabla u ( t , X _ t ) ) ( \\nabla u _ n ( t , X ^ n _ t ) - \\nabla u ( t , X _ t ) ) ^ T ] d t + d M _ t \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} \\bar { X } = \\begin{pmatrix} 1 & \\frac 1 2 & - \\frac 1 2 & - 1 \\cr \\frac 1 2 & 1 & \\frac 1 2 & - \\frac 1 2 \\cr - \\frac 1 2 & \\frac 1 2 & 1 & \\frac 1 2 \\cr - 1 & - \\frac 1 2 & \\frac 1 2 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\tau _ 8 ( x ) = g _ { 8 - } ( x ) \\quad \\hbox { f o r } x \\in [ \\ , \\tfrac { 2 0 2 } { 4 8 5 } , c ) . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} Q _ { 3 , k } & = | I _ { k } | ^ { 2 ( 1 - H ) } \\int _ { 0 } ^ { 1 } \\left | ( a _ { k } + v _ 1 | I _ { k } | ) ^ { H - \\frac { 1 } { 2 } } \\cdot \\int _ { 0 } ^ { v _ 1 } \\frac { \\varphi _ k ( v _ 1 ) - \\varphi _ k ( v _ 2 ) } { \\left ( v _ 1 - v _ 2 \\right ) ^ { H + \\frac { 1 } { 2 } } } d v _ 2 \\right | ^ { 2 } d v _ 1 , \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta \\big ( | d u | ^ { 2 ( p - 1 ) } \\big ) = \\langle \\Delta \\big ( | d u | ^ { p - 2 } d u \\big ) , | d u | ^ { p - 2 } d u \\rangle + \\Big | \\nabla \\big ( | d u | ^ { p - 2 } d u \\big ) \\Big | ^ 2 + | d u | ^ { 2 ( p - 2 ) } R ( d u ) , \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\sum _ { j \\le k } \\binom { s } { j } 4 ^ { j } O ( d ) ^ { k } , \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} f ( u _ { i } ^ { 0 } / v ^ { 0 } ) & = \\sum _ { j \\leq r _ { i } } e ( \\bar { u } _ { i , j } / \\bar { v } ) \\ , f ( \\bar { u } _ { i , j } / \\bar { v } ) , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\mathrm { i n t } T _ { p i j } = \\bigcup _ { \\rho \\in \\mathbb { N } } T _ { p i j } ^ { 0 , \\rho } \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} \\tau _ { t } = \\theta _ { t } A Z _ { u = t , 1 } ^ { \\ast } \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} ( K \\circ v _ j ) f _ 0 ^ { ( r ) } + \\sum _ { q = 1 } ^ r \\binom { r } { q } \\sum c _ { m _ 1 , \\ldots , m _ q } \\frac { K ^ { ( M _ q ) } \\circ v _ j } { F ^ { M _ q } } f _ 0 ^ { ( r - q ) } \\prod _ { s = 1 } ^ q \\big ( f _ 0 ^ { ( s - 1 ) } \\big ) ^ { m _ s } , \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\varepsilon ^ 2 _ { q , m } \\triangleq \\mathbb { E } [ | e _ { q , m } [ n ] | ^ 2 ] = 3 \\rho _ { q , m } / 2 ^ { 2 b _ { q , m } } , b _ { q , m } \\in \\mathbb { Z } _ { + + } . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ m ( y q ) ^ m = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } \\sum _ { \\ell = 0 } ^ { \\infty } b _ { \\nu } ( \\ell , m , n ) z ^ \\ell y ^ m q ^ n . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\mathrm { R a d } \\ ; T _ { x } M = \\left \\{ E _ { x } \\in \\mathrm { R a d } \\ ; T _ { x } M , g ( E _ { x } , X ) = 0 , \\ ; X \\in T _ { x } M \\right \\} . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} X ^ 3 ( D ^ 2 + D ) + D ^ 4 = 0 . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\int _ \\Omega f v _ 1 v _ 2 \\ , d x = 0 \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} f _ 1 ( ( a , x ) , ( b , y ) ) = f ( b ) ^ { - 1 } , ~ f _ 2 ( ( a , x ) , ( b , y ) ) = f ( b ) ^ { - 1 } ( f ( a ) - 1 ) . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} L ^ S ( \\tfrac { 1 } { 2 } , \\Pi \\times \\Pi ' ) \\sim _ { E ( \\Pi ) E ( \\Pi ' ) } ( 2 \\pi i ) ^ { d n n ' / 2 } \\prod \\limits _ { \\imath \\in \\Sigma } \\left ( \\prod \\limits _ { i = 0 } ^ { n } P ^ { ( i ) } ( \\Pi , \\imath ) ^ { s p ( i , \\Pi ; \\Pi ' , \\imath ) } \\prod \\limits _ { j = 0 } ^ { n ' } P ^ { ( j ) } ( \\Pi ' , \\imath ) ^ { s p ( j , \\Pi ' ; \\Pi , \\imath ) } \\right ) . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\| d \\| _ 1 \\ , \\dfrac { \\| x \\| _ { \\infty } } { \\| g ( x ) \\| } \\le \\| d \\| _ 1 \\ , \\dfrac { \\| x \\| _ { \\infty } } { \\| x \\| _ { \\infty } ^ 2 } = \\dfrac { \\| d \\| _ 1 } { \\| x \\| _ { \\infty } } < \\varepsilon , \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\mathbb { T } ^ d } f ( w _ { n , \\epsilon } ( x ) ) \\phi _ \\epsilon ( x ) d x = \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } f ( w _ n ( x , y ) ) \\phi ( x , y ) d y d x . \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} T ^ { \\nabla ^ { E } } ( u , v , w ) = & - { \\mathcal H } ( X , Y , Z ) - ( R ( X , Y ) , t ) ^ { \\mathcal G } - ( R ( Y , Z ) , r ) ^ { \\mathcal G } - ( R ( Z , X ) , s ) ^ { \\mathcal G } \\\\ & + ( [ r , s ] ^ { \\mathcal G } , t ) ^ { \\mathcal G } , \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\tau _ \\varepsilon = t _ \\varepsilon ( 1 + o ( 1 ) ) \\ , . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} f ( x - y ) : = \\int _ 0 ^ { \\infty } \\frac { e ^ { - \\lambda r } } { \\sqrt { r } } g _ r ( x - y ) d r \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} & | \\langle T ( f _ 1 , \\ldots , f _ n ) , f _ { n + 1 } \\rangle | \\lesssim \\left \\| \\mathrm { M } \\left ( | f _ 1 | _ { X _ 1 } , \\ldots , | f _ n | _ { X _ n } , | f _ { n + 1 } | _ { Y _ { n + 1 } ^ * } \\right ) \\right \\| _ { 1 } , \\\\ & \\mathrm { M } ( g _ 1 , \\ldots , g _ { n + 1 } ) ( x ) \\coloneqq \\sup _ { x \\in Q } \\prod _ { j = 1 } ^ { n + 1 } \\langle | g _ j | \\rangle _ Q , \\langle g \\rangle _ Q { \\coloneqq } \\frac { 1 } { | Q | } \\int _ Q g . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} D \\Psi ( 0 ) \\zeta = \\sum _ { k \\in \\Bbb Z } \\lambda _ k ( \\gamma ) c _ { k } { \\bf e } ^ { i k x } , \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\eta _ a = \\frac { r ^ 2 } { 3 } \\beta _ a + \\frac { r ^ 3 } { 4 } D \\beta _ a + r ^ 4 [ \\frac { 1 } { 1 0 } D ^ 2 \\beta _ a - \\frac { 1 } { 4 5 } \\alpha _ { a b } \\beta ^ b ] + O ( r ^ 5 ) , \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} f ( u , v ) : = ( R _ 1 - a _ 1 u - b _ 1 v ) u , \\ \\ g ( u , v ) : = ( R _ 2 - a _ 2 u - b _ 2 v ) v . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} X _ { 1 } = \\left [ \\begin{array} [ c ] { c c c } 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] , X _ { 2 } = \\left [ \\begin{array} [ c ] { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ] , X _ { 3 } = \\left [ \\begin{array} [ c ] { c c c } 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\chi _ 2 ( s ) & = \\sum _ { s ^ { - 1 / 2 } < m \\leq s ^ { - 1 } } - \\phi ' ( m s ) \\\\ & \\leq O ( s ^ { - u - 1 } ) \\sum _ { s ^ { - 1 / 2 } < m < s ^ { - 1 } } \\frac { 1 } { m ^ { u + 1 } } \\\\ & = O ( s ^ { - u - 1 } ) O ( s ^ { u / 2 } ) \\\\ & = O ( s ^ { - u / 2 - 1 } ) , \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} V _ { C _ i } = \\bigoplus \\limits _ { ( M _ S , \\mu _ S ) \\in C _ i } [ M _ S , \\mu _ S ] ( \\rho ) , \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\int _ { \\tau _ { l } \\wedge t } ^ { \\tau _ { l + 1 } \\wedge t } \\omega _ { i } ( W _ { s } ) \\circ W _ { s } = \\int _ { \\tau _ { l } \\wedge t } ^ { \\tau _ { l + 1 } \\wedge t } \\omega _ { i } ( W _ { s } ) \\circ \\beta _ { s } . \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\int _ { \\mathbb { T } ^ d } w _ \\epsilon ( x ) \\psi \\Big ( x , \\frac { x } { \\epsilon } \\Big ) d x = \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } w ( x , y ) \\psi ( x , y ) d y d x . \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} & u _ k \\in C ^ { \\infty } ( \\{ z \\in \\R ^ 2 \\colon \\rho _ k \\leq \\abs { z } \\leq \\rho _ { k + 1 } \\} ; \\C ^ 2 ) , \\\\ & V _ k \\in C ^ \\infty ( \\{ z \\in \\R ^ 2 \\colon \\rho _ k \\leq \\abs { z } \\leq \\rho _ { k + 1 } \\} ; \\C ^ { 2 \\times 2 } ) , \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\Xi _ K = \\{ x \\in \\partial K : \\ , \\mbox { t h e r e e x i s t s e x t e r i o r n o r m a l $ u \\in S ^ { n - 1 } $ a t $ x $ w i t h $ h _ K ( u ) = 0 $ } \\} , \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} U _ 2 = J X _ 1 - J X _ 2 , \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\inf _ { \\mathcal { A } _ c } \\| f \\| _ { W ^ { 1 , d } } ^ d = \\inf _ { \\mathcal { B } _ c } \\| f \\| _ { W ^ { 1 , d } } ^ d = d ^ { d / 2 } c , \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\tau ( A ^ * ) = \\overline { \\tau ( A ) } , \\tau ( A B ) = \\tau ( B A ) , \\forall A , B \\in \\mathcal S . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\lim _ { j \\in J } \\frac { | g F _ j \\setminus F _ j | } { | F _ j | } = 0 . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\partial _ Q X ( Q ) = \\frac { 1 } { Q } B ( Q ) X ( Q ) \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\textbf { g } \\left ( S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ a ) , S ^ \\textnormal { c o h } ( \\tau , z ) ( T _ b ) \\right ) = g ( T _ a , T _ b ) \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} l ( \\ss ( G ) ) = \\sum _ { i = 1 } ^ m l ( T _ i ) . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} | \\sin \\pi ( 2 \\theta + x _ 0 \\alpha ) | = e ^ { - \\eta | \\ell | } . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\widehat { f } ( \\pi ) = \\int _ G f ( t ) \\pi ( t ^ { - 1 } ) \\ , d t \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} K _ 1 = 1 , K _ 2 = \\delta , \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} P [ \\psi ] ( \\phi ) : = \\textup { u s c } \\Big ( \\lim _ { C \\to + \\infty } P ( \\psi + C , \\phi ) \\Big ) . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\| J _ k \\| _ { L ^ \\infty } \\leq R _ k : = C k ^ { - n / 2 } C = C ( n , J ) > 0 \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\mathrm { m o d } _ q ( \\hat { d \\varphi } ) = \\tfrac { t } { \\mathrm { g c d } ( s , t ) } \\cdot \\mathrm { m o d } _ q ( d \\varphi ; l ) + k _ l . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\chi _ k \\in C _ 0 ^ \\infty ( U _ k ) , \\chi _ k \\geq 0 \\sum _ { k = 1 } ^ N \\chi _ k ( x ) = 1 x \\in K . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\Phi ( \\chi _ k ) = \\Phi ( \\chi _ k \\ast \\chi _ k ) = P ( \\chi _ k ) = \\sum _ { j = - \\infty } ^ { + \\infty } \\widehat { \\chi _ k } ( j ) = \\sum _ { j = - \\infty } ^ { + \\infty } { \\delta _ { j k } } = 1 , \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} \\frac 1 { \\eta _ i } : = \\frac 1 { \\delta _ i } \\Big ( \\sum _ { j = 1 } ^ { m } \\frac 1 { \\delta _ j } \\Big ) ^ { - 1 } = \\frac 1 { \\delta _ i } \\Big ( \\sum _ { j = 1 } ^ { m + 1 } \\frac 1 { \\delta _ j } - \\frac 1 { \\delta _ { m + 1 } } \\Big ) ^ { - 1 } = \\frac 1 { \\delta _ i } \\Big ( \\frac { 1 - r } { r } - \\frac 1 { \\delta _ { m + 1 } } \\Big ) ^ { - 1 } \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\mathcal { B } _ I \\cap \\mathcal { B } _ { I ' } & = \\emptyset , & & I , I ' \\in \\mathcal { D } _ { \\leq n } , I \\neq I ' \\\\ K \\cap K ' & = \\emptyset , & & K , K ' \\in \\mathcal { B } _ I , K \\neq K ' , \\ I \\in \\mathcal { D } _ { \\leq n } . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\limsup _ { \\epsilon \\rightarrow 0 } \\bigg | \\int _ { \\mathbb { T } ^ d } f ( w _ \\epsilon ( x ) ) \\phi _ \\epsilon ( x ) d x - \\int _ { \\mathbb { T } ^ d } f ( w _ { n , \\epsilon } ( x ) ) \\phi _ \\epsilon ( x ) d x \\bigg | = 0 . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J } \\rho _ { j } ( - \\Delta ) ^ { \\alpha _ { j } / 2 } u ( x ) = f ( x ) , x \\in \\mathbb { R } ; u ( x ) \\to 0 , \\ ; \\ ; { \\rm a s } \\ ; \\ ; | x | \\to \\infty . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} U _ { k , r + 1 } ( t ) = \\int _ { x _ { r + 1 } = a } ^ 1 ( \\int _ { D } d x _ 1 , \\dots d x _ r ) d x _ { r + 1 } \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} ( & \\varphi _ { X , Y } ^ { \\sharp } F ) _ \\beta = \\sum _ { \\lambda \\in \\Lambda _ { n , m } ( \\beta ) } \\frac { \\beta ! } { \\lambda ! } F _ { \\sum _ { \\alpha } \\lambda _ \\alpha } \\circ \\varphi \\prod _ { \\alpha } \\frac { \\left ( \\partial ^ \\alpha \\varphi \\right ) ^ { \\lambda _ \\alpha } } { ( \\alpha ! ) ^ { \\sum _ i \\lambda _ { i , \\alpha } } } , \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} ( i = 1 ) & \\ , \\ , 0 . 2 5 0 3 0 , 0 . 2 4 9 6 0 , 0 . 2 4 9 5 3 , 0 . 2 5 0 5 6 , \\\\ ( i = 2 ) & \\ , \\ , 0 . 2 5 1 1 8 , 0 . 2 4 7 1 1 , 0 . 2 4 9 5 6 , 0 . 2 5 2 1 6 , \\\\ ( i = 3 ) & \\ , \\ , 0 . 2 5 0 1 3 , 0 . 2 5 1 0 3 , 0 . 2 4 8 3 1 , 0 . 2 5 0 5 3 , \\\\ ( i = 4 ) & \\ , \\ , 0 . 2 4 9 1 4 , 0 . 2 5 0 4 1 , 0 . 2 5 0 2 0 , 0 . 2 5 0 2 5 , \\\\ ( i = 5 ) & \\ , \\ , 0 . 2 4 9 4 1 , 0 . 2 4 9 5 6 , 0 . 2 5 0 8 1 , 0 . 2 5 0 2 3 , \\\\ ( i = 6 ) & \\ , \\ , 0 . 2 4 9 5 0 , 0 . 2 5 0 7 2 , 0 . 2 4 9 2 4 , 0 . 2 5 0 5 4 , \\\\ ( i = 7 ) & \\ , \\ , 0 . 2 5 0 3 9 , 0 . 2 4 8 6 0 , 0 . 2 5 1 5 6 , 0 . 2 4 9 4 6 . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\Omega _ j \\cap \\Lambda \\Omega _ k = \\varnothing \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} A _ \\delta & = e ^ \\gamma \\left ( \\log _ 2 q + \\log _ 3 q - C - \\delta + \\frac { 1 } { 2 \\sqrt { \\log _ 2 q } } \\right ) , \\\\ A _ { \\delta } ' & = e ^ \\gamma \\left ( \\log _ 2 q + \\log _ 3 q - C - \\delta \\right ) . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { ( 1 ) _ k ^ 3 } ( 3 k + 1 ) 2 ^ { 2 k } \\ \\ \\frac { - 2 i } { \\pi } , \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { ( 1 ) _ k ^ 3 } ( 3 k + 1 ) ( - 1 ) ^ k 2 ^ { 3 k } \\ \\ \\frac { 1 } { \\pi } \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} r _ 0 : = \\eta \\ , \\lambda _ 0 ^ { - 1 / 2 } < \\min \\{ 1 / 8 , h / 4 \\} . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} [ L _ m , b ( n ) ] = \\sum _ { j \\geq 0 } \\binom { m + 1 } { j } [ L _ { j - 1 } b ] ( m + n + 1 - j ) \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } = \\tau - { F ^ { - 1 } } \\left [ { 1 - \\frac { 1 } { { \\left ( { M + K } \\right ) N } } \\sum \\limits _ { j \\in { \\cal N } } { \\left ( { \\sum \\limits _ { m = 1 } ^ M { { \\check { u } } _ j ^ { ( m ) } } + \\sum \\limits _ { i = 1 } ^ K { u _ j ^ { ( i ) } } } \\right ) } } \\right ] \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} v ( q ) & = v ( o ) + ( X u ) ( p _ o ) x + ( Y u ) ( p _ o ) y + ( T u ) ( p _ o ) t \\\\ & + \\frac { 1 } { 2 } ( X ^ 2 u ) ( p _ o ) x ^ 2 + \\frac { 1 } { 2 } ( Y ^ 2 u ) ( p _ o ) y ^ 2 + [ 2 ( T u ) ( p _ o ) + ( X Y u ) ( p _ o ) ] x y + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} D ( ( - \\Delta _ p ) ^ \\vartheta ) = [ L ^ p ( \\Omega ) , D ( \\Delta _ p ) ] _ \\vartheta \\subset W ^ { 2 \\vartheta , p } ( \\Omega ) , \\vartheta \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} Q _ { m _ 1 } ^ { ( j ) } = \\left [ \\omega = \\frac { x _ d } { j } \\right ] - \\frac { m _ { 1 } + x _ d ^ { 0 } } { j } e _ 0 , | m _ { 1 } | < M . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} w _ { m + 1 } ( t ) = e ^ { - t A } \\Big [ ( - 1 ) ^ { m + 1 } u _ 1 \\Big ] - \\int _ 0 ^ t e ^ { - ( t - s ) A } V _ { m } ' ( s ) \\ , d s . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} z _ \\pm = \\frac 1 { 2 \\alpha b } \\Big ( b - \\frac 1 b \\pm \\sqrt { \\big ( b - \\frac 1 b \\big ) ^ 2 + 4 \\alpha ^ 2 } \\Big ) \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta z + x z & = f \\Omega , \\\\ \\partial _ \\nu z & = 0 \\quad \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} H ^ { p ^ \\ast } _ { , i } = \\nabla _ { i } H ^ { p ^ { \\ast } } = - \\nabla _ { i } \\langle X , e _ { p ^ { \\ast } } \\rangle = \\sum _ { j } h _ { i j } ^ { p ^ { \\ast } } \\langle X , e _ { j } \\rangle . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\liminf _ { \\ell \\to \\infty } \\frac { \\log \\mu ( B _ { r _ \\ell } ( x ) ) } { \\log r _ \\ell } = \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { i = 1 } ^ \\ell \\log t _ i } { d \\sum _ { i = 1 } ^ { \\ell + 1 } \\log s _ i - \\log t _ { \\ell + 1 } } . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} & \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , | \\nabla ^ \\Sigma u | ^ 2 \\\\ & \\leq C \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | \\leq L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } u ^ 2 + C L ^ { - 1 } \\int _ { \\Sigma \\cap \\{ | x _ { n + 1 } | = L \\} } e ^ { - \\frac { | x | ^ 2 } { 4 } } \\ , u ^ 2 . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} D _ j = \\mathbb { E } ( T S P _ { n + 1 } | { \\cal F } _ j ) - \\mathbb { E } ( T S P _ { n + 1 } | { \\cal F } _ { j - 1 } ) , \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\sigma ^ { a b } l _ { a b } = - \\frac { 2 } { r } + \\frac { r ^ 3 } { 4 5 } | \\alpha | ^ 2 + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} l = 2 | D ^ * | - | D | . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s Q + Q - | Q | ^ \\alpha Q = 0 , \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\delta _ { f } = \\max \\{ \\delta _ { g } , \\delta _ { f _ { T } } \\} . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\partial _ a \\tilde X ^ i \\partial _ b Y _ i ^ { ( 3 ) } + \\partial _ b \\tilde X ^ i \\partial _ a Y _ i ^ { ( 3 ) } = - \\frac { 1 } { 3 } \\bar R _ { L a b L } = \\frac { 1 } { 3 } \\alpha _ { a b } - \\frac { 1 } { 6 } \\bar R i c ( L , L ) \\tilde \\sigma _ { a b } \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} d ( e _ { x _ 1 } \\cdots e _ { x _ n } ) & = - e _ { x _ 1 } d ( e _ { x _ 2 } \\cdots e _ { x _ n } ) . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} V _ { \\eta } u ( x ) = \\langle u , \\pi ( x ) \\eta \\rangle ~ . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha _ 1 , \\beta _ 1 \\} \\in \\mathcal { S } _ 1 ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha _ 1 } ( \\rho ) + \\ell _ { \\beta _ 1 } ( \\rho ) ) } - 1 \\right ) ^ { - 1 } + \\sum _ { \\{ \\alpha _ 2 , \\beta _ 2 \\} \\in \\mathcal { S } _ 2 ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha _ 2 } ( \\rho ) + \\ell _ { \\beta _ 2 } ( \\rho ) ) } + 1 \\right ) ^ { - 1 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} w ( \\gamma ^ 1 _ 1 ) = t _ { 1 1 1 1 } \\enspace . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\int _ { \\Omega ^ i } ( 1 - \\zeta _ 1 k ^ i _ 2 ( s ) + \\zeta _ 2 k ^ i _ 1 ( s ) ) \\ , d s d \\zeta _ 1 d \\zeta _ 2 \\leq | \\Lambda ^ i [ w ] | , \\bigcap _ { i = 1 } ^ N { \\rm i n t } ( \\Lambda ^ i [ w ] ) = \\emptyset . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} & v _ r ^ { \\alpha } ( Q _ { l + 1 } - R _ i T ^ i ) \\ge \\inf { \\{ v _ r ^ { \\alpha } ( Q _ l - R _ i T ^ i ) , v _ r ^ { \\alpha } ( Y _ l ) \\} } \\\\ \\ge & \\inf { \\{ c _ 0 ^ { \\alpha } + r i + v ^ { \\alpha } ( R _ i ) , c _ l ^ { \\alpha } + r i + v ^ { \\alpha } ( R _ i ) \\} } = c _ 0 ^ { \\alpha } + r i + v ^ { \\alpha } ( R _ i ) . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\Psi _ { N _ \\varepsilon } ( u _ \\varepsilon ) = ( 1 + o ( 1 ) ) \\varphi _ { N _ \\varepsilon - 1 } ( u _ \\varepsilon ^ 2 ) \\Psi _ { N _ \\varepsilon } ' ( u _ \\varepsilon ) = 2 ( 1 + o ( 1 ) ) ~ u _ \\varepsilon \\varphi _ { N _ \\varepsilon - 1 } ( u _ \\varepsilon ^ 2 ) \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} f = \\prod _ { u = 1 } ^ k f _ u , f _ u ( t ) = B _ u ^ s ( F ) ( t ) B _ u ( A ( t ) ) , \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\eta ( Z ) = 1 , d \\eta ( Z , X ) = 0 . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\max _ { \\partial \\Omega } u = \\max _ { \\overline { \\Omega } } u , \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} = [ \\pi ] [ r _ { ( - 1 , 0 ^ { n _ 1 - 1 } ) } \\boxtimes r _ { ( - 1 , 0 ^ { n _ 2 - 1 } ) } \\circ ( L L ( \\rho _ 1 ) + L L ( \\rho _ 2 ) ) | \\cdot | ^ { 1 - n _ 2 } ] . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\frac { p _ i } { p _ { N + 1 } } = \\frac { 1 } { \\Lambda _ 0 \\cdots \\Lambda _ N } \\sum _ { 0 \\leq j _ 1 < \\cdots < j _ i \\leq N } \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ i } \\prod _ { k \\in \\{ 0 , \\dots , N \\} - \\{ j _ 1 , \\dots , j _ i \\} } \\frac { 1 - \\Lambda _ k } { 1 - q } \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { p , 1 } ^ { \\gamma _ 2 } } \\le & C \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\gamma _ 2 + \\frac n 2 - \\frac n p } } \\le C \\big ( \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , \\infty } ^ { \\sigma - 1 } } \\big ) ^ { \\theta _ { 3 } } \\big ( \\| \\eta \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 - 2 } } \\big ) ^ { 1 - \\theta _ { 3 } } , \\theta _ { 3 } = \\frac { \\frac n p - \\gamma _ 2 - 2 } { \\frac n 2 - 1 - \\sigma } \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\mathcal D ( \\phi ( \\mathcal { H } ) ) = \\left \\{ f \\in L ^ 2 ( \\Omega ) : \\int ^ \\infty _ { 0 } | \\phi ( \\lambda ) | ^ 2 d \\| E _ { \\mathcal { H } } ( \\lambda ) f \\| _ { L ^ 2 ( \\Omega ) } ^ 2 < \\infty \\right \\} , \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\begin{aligned} \\operatorname { C o v } \\left ( \\hat { \\textbf { x } } _ k \\right ) - \\left ( \\sum \\limits _ { l = 1 } ^ k \\textbf { I } _ { D I } \\left ( \\textbf { x } , { { \\bf { W } } _ { l } } \\right ) \\right ) \\succeq \\textbf { 0 } , \\end{aligned} \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} a _ { c , \\lambda } = \\frac { p - q } { \\dim K Z ( L _ c ( \\lambda ) ) } \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} a = \\sum _ { i \\ge 0 } a _ i { \\bf t } ^ i , c = \\sum _ { i \\ge 0 } c _ i { \\bf t } ^ i , b = \\sum _ { i \\ge 0 } b _ i { \\bf t } ^ i , d = \\sum _ { i \\ge 0 } d _ i { \\bf t } ^ i , \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 = 0 , \\ \\ \\bar \\lambda = 0 , \\ \\ \\bar \\lambda _ 1 \\bar \\lambda _ 2 \\neq 0 , \\\\ & S < \\sup H ^ 2 = \\bar H ^ 2 \\leq 3 S - 2 \\ \\ \\ \\ S < \\sup H ^ 2 < \\dfrac 4 3 S , \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\widetilde { C } _ q \\left ( K , N ( K , o ) \\cap S ^ { n - 1 } \\right ) = \\widetilde { C } _ { q } \\left ( K , \\{ u \\in S ^ { n - 1 } : \\ , h _ K ( u ) = 0 \\} \\right ) = 0 . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} h & \\overbrace { f _ { s _ 1 } ( w _ 1 ) \\cdots f _ { s _ { i _ { l ( s ) } - 1 } } ( w _ { i _ { l ( s ) } - 1 } ) } ^ { b _ 1 } f _ { s _ { i _ { l ( s ) } } } ( w _ { i _ { l ( s ) } } ) \\overbrace { f _ { s _ { i _ { l ( s ) } + 1 } } ( w _ { i _ { l ( s ) } + 1 } ) \\cdots f _ { s _ m } ( w _ m ) } ^ { b _ 2 } \\\\ = & \\ \\underbrace { f _ { t _ 1 } ( v _ 1 ) \\cdots f _ { t _ { j _ { l ( t ) } - 1 } } ( v _ { j _ { l ( t ) } - 1 } ) } _ { c _ 1 } f _ { t _ { j _ { l ( t ) } } } ( v _ { j _ { l ( t ) } } ) \\underbrace { f _ { t _ { j _ { l ( t ) } + 1 } } ( v _ { j _ { l ( t ) } + 1 } ) \\cdots f _ { t _ m } ( v _ m ) } _ { c _ 2 } . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} \\sum _ { l \\in \\overline S } \\frac { \\omega ( i , l ) } { \\lambda _ 0 - \\omega ( l , l ) } u _ l = c u _ i , \\forall i \\in S . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 = 0 , \\ \\ \\bar \\lambda = 0 , \\ \\ \\bar \\lambda _ 1 \\bar \\lambda _ 2 \\neq 0 , \\\\ & S < \\sup H ^ 2 = \\bar H ^ 2 \\leq 3 S - 2 \\ \\ \\ \\ S < \\sup H ^ 2 < \\dfrac 4 3 S \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} D H _ n ( X ; q , t ) = \\nabla e _ n . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} ( F _ { i + 1 } ^ 0 ) ^ * \\circ \\beta ( f \\hat { \\otimes } h ) & = ( l \\oplus F _ i ) ^ * \\circ \\beta ( f ) \\hat { \\otimes } h = l ^ * ( \\beta ( f ) ) \\hat { \\otimes } F _ i ^ * ( h ) \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\sigma ( k ) = k & \\mbox { i f $ k $ i s e v e n } \\\\ \\sigma ( k ) = \\delta + \\epsilon - k & \\mbox { i f $ k $ i s o d d } \\\\ \\sigma ( \\delta + \\epsilon - k ) = k & \\mbox { i f $ k $ i s o d d } \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\tau _ 0 : = \\{ \\tau ' \\mid \\exists \\tau \\in \\R , \\ & u ( x , t + \\tau ) \\preceq v ( x , t ) \\preceq u ( x , t + \\tau + \\tau ' ) \\\\ & ( ( x , t ) \\in \\R ^ N \\times \\R ) \\} \\ ( \\in [ 0 , b - a ] ) \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} A _ { } = \\frac { \\alpha + 2 } { 2 } S _ { } ^ { - \\alpha } , B _ { } = \\frac { \\alpha + 2 } { 2 } L _ { } ^ { - \\alpha } . \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\nu _ 2 = [ \\nu _ 1 ; \\nu _ 2 ( \\phi _ 1 ) = \\nu ( \\phi _ 1 ) ] . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\begin{cases} d x _ { t } = \\sum _ { \\alpha = 1 } ^ { d } V _ { \\alpha } ( x _ { t } ) d h _ { t } ^ { \\alpha } , & 0 \\leq t \\leq 1 , \\\\ x _ { 0 } = x , \\end{cases} \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\| p \\| _ K = \\| p \\circ \\sigma \\| _ { I \\times \\Theta } \\leq \\alpha ^ { d _ 1 + d _ 2 } \\ , \\beta ^ { d _ 3 } \\| p \\circ \\sigma \\| _ { \\mathcal { B } _ n } = \\alpha ^ { d _ 1 + d _ 2 } \\ , \\beta ^ { d _ 3 } \\| p \\| _ { \\sigma ( \\mathcal { B } _ n ) } \\ ; , \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} K _ G ( X ) = K _ 0 ' ( ( X \\times T ) / G ) \\to K _ 0 ' ( X \\times T ) \\simeq K _ 0 ' ( X ) . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} L R _ 0 & = \\{ ( l _ 1 , \\dots , l _ n ) \\in \\mathbb { Q } ^ n \\mid \\sum _ { i = 1 } ^ n l _ i \\beta _ i = 0 \\} \\\\ & = \\left \\{ ( l _ 1 , \\dots , l _ n ) \\in \\mathbb { Q } ^ n \\mid \\left ( \\begin{array} { c } { l } _ 1 \\\\ \\vdots \\\\ { l } _ r \\end{array} \\right ) = - T \\left ( \\begin{array} { c } { l } _ { r + 1 } \\\\ \\vdots \\\\ { l } _ n \\end{array} \\right ) \\right \\} , \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\alpha _ { \\zeta } & : = { { { } \\hat { \\alpha } _ { \\zeta } } } _ { \\big | { A _ { \\zeta } ( 2 { \\tau } ) } } \\\\ \\beta _ { \\zeta } & : = { { { } \\hat { \\beta } _ { \\zeta } } } _ { \\big | { B _ { \\zeta } ( 2 { \\tau } ) } } \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} \\int _ 0 ^ T f ^ 2 ( x , t ) \\ , d t & = \\int _ 0 ^ T | \\mathcal { W } [ y + d ] - \\mathcal { W } [ y ] - \\omega | ^ 2 ( x , t ) \\ , d t \\\\ & \\le \\rho ^ 2 _ { y ( x , \\cdot ) } ( \\| d ( x , \\cdot ) \\| _ { \\infty , T } ) \\cdot \\| d _ t ( x , \\cdot ) \\| ^ 2 _ { L ^ p ( 0 , T ) } , \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} k \\le n - | U | \\le n - \\frac { n } { 1 + r - \\frac { s t } { n } } = n \\frac { r - \\frac { s t } n } { r - \\frac { s t } n + 1 } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\mathcal A _ { a , b } ^ { ( x , t ) } : = \\frac { 1 } { 2 } \\big ( p ( x , t ) - 2 \\big ) ( a \\otimes a + b \\otimes b ) + I . \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} a _ c ^ { ( n ) } \\to + \\infty , a _ c ^ { ( n ) } / n \\to 0 , p _ n a _ c ^ { ( n ) } \\to 0 \\quad b _ c ^ { ( n ) } = o \\left ( \\frac { a _ c ^ { ( n ) } } { n p _ n } \\right ) . \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\Phi _ n ( q ) : = \\prod _ { \\substack { 1 \\leqslant k \\leqslant n \\\\ \\gcd ( n , k ) = 1 } } ( q - \\zeta ^ k ) , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} x _ { j } ^ { \\lambda } = ( x _ { j _ 1 } ^ \\lambda , \\cdots , x _ { j _ d } ^ \\lambda ) , j \\in \\Upsilon _ { \\ ! N } ; R _ { n } ^ \\lambda ( x ) = \\prod _ { i = 1 } ^ d R _ { n _ i } ^ \\lambda ( x _ i ) . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\sigma _ s ' ( \\rho _ s ) = { \\bar \\sigma \\cosh ( \\rho _ s - \\eta _ s ) - \\hat \\sigma \\over \\sinh ( \\rho _ s - \\eta _ s ) } = \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } P ^ + ( n , a , \\eta ) \\geq \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { 2 n } \\sum _ { \\ell = 1 } ^ { n - 1 } \\log ( 1 - 2 \\sigma ^ 2 \\alpha _ { \\ell } ) . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ \\xi _ y - \\xi _ z : y , z \\in E ^ { * } \\} = X ; \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} g = \\tfrac { 1 } { 2 } d \\eta \\left ( \\cdot , I \\cdot \\right ) + \\eta \\otimes \\eta . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} h = \\tau ^ { \\Gamma } _ { ( 2 ) } ( M ) \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\| \\pi _ { N , \\lambda } ^ { s } u - u \\| _ { H ^ s ( \\mathbb R ) } = \\inf _ { \\phi \\in V _ N ^ \\lambda } \\| \\phi - u \\| _ { H ^ s ( \\mathbb R ) } . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} P ( a ) = \\Phi \\left ( a ^ n \\right ) ( a \\in A ) \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} ( \\mathfrak { F } , \\mathfrak { F } _ 1 , \\dots , \\mathfrak { F } _ m ) = \\bigg ( \\Big ( \\sum _ j ( F ^ j ) ^ { t } \\Big ) ^ \\frac 1 { t } , \\Big ( \\sum _ j ( F _ 1 ^ j ) ^ { t _ 1 } \\Big ) ^ \\frac 1 { t _ 1 } , \\dots , \\Big ( \\sum _ j ( F _ m ^ j ) ^ { t _ m } \\Big ) ^ \\frac 1 { t _ m } \\bigg ) \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\varphi ( x ) = \\varphi _ { { \\bf m } _ q } ( x ) = F ^ { - M _ q } ( K ^ { ( M _ q ) } \\circ v _ j ) ( x ) f _ 0 ^ { ( r - q ) } ( x ) \\prod _ { s = 1 } ^ q f _ 0 ^ { ( s - 1 ) } ( x ) ^ { m _ s } , \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} d ( y x ^ y - x y ) = d ( y ) x ^ y + y d ( x ^ y ) - d ( x ) y - x d ( y ) = 0 + 0 - 0 - 0 = 0 . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\Phi \\varPsi ( \\mathcal { C } _ 2 ) ( f ) = \\varPsi ( \\mathcal { C } _ 2 ) ( \\Phi ( f ) ) = \\rho ( \\mu , \\mu \\ast ( \\Phi ( f ) ) ^ \\circ , \\nu ) \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} D = \\sum _ i b _ i \\otimes d _ i . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} \\left ( x , y , z \\right ) \\longmapsto \\left ( \\tilde { x } , \\tilde { y } , \\tilde { z } \\right ) : \\left \\{ \\begin{array} { l } \\tilde { x } = \\theta _ { 1 } + x \\cos \\theta - y \\sin \\theta , \\\\ \\tilde { y } = \\theta _ { 2 } + x \\sin \\theta + y \\cos \\theta , \\\\ \\tilde { z } = \\theta _ { 3 } + \\theta _ { 4 } x + \\theta _ { 5 } y + z , \\end{array} \\right . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} \\mathcal { M } _ { G , b , \\mu } = \\mathcal { M } _ { G _ 1 , b _ 1 , \\mu _ 1 } \\otimes . . . \\otimes \\mathcal { M } _ { G _ k , b _ k , \\mu _ k } . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} T _ { C , 2 } ( b _ 1 , b _ 2 ) \\ge \\lim _ { k \\rightarrow \\infty } c ( 1 - k ^ { - 1 } ) k ^ { \\frac 1 2 } \\langle 1 _ { ( 0 , 1 ) } \\rangle _ { C , I _ k } = \\lim _ { k \\rightarrow \\infty } c ( 1 - k ^ { - 1 } ) \\frac { k ^ { \\frac 1 2 } } { C ^ { - 1 } ( k ) } = \\infty . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta _ g u = \\rho \\left ( \\frac { e ^ u } { \\int _ { \\S ^ 2 } e ^ u } - \\frac { 1 } { 4 \\pi } \\right ) \\mbox { o n } \\S ^ 2 , \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} X = \\pm S ( X ) \\cdot 1 0 ^ M , 1 \\leq S ( X ) < 1 0 , M \\in { \\mathbb Z } , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\P ( C ' _ { k , j } ) > \\P ( C ' _ { k , j } \\cap ( A ' ) ^ c ) = 0 \\P ( C ' _ { k , j } ) > \\P ( C ' _ { k , j } \\cap A ' ) = 0 . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} | f | _ { \\mathcal { H } ^ \\beta } = \\max _ { 1 \\leq j < \\beta } \\| | f ^ { ( j ) } | ^ \\beta / | f | ^ { \\beta - j } \\| _ \\infty ^ { 1 / j } = \\max _ { 1 \\leq j < \\beta } \\left ( \\sup _ { x \\in [ 0 , 1 ] } \\frac { | f ^ { ( j ) } ( x ) | ^ \\beta } { | f ( x ) | ^ { \\beta - j } } \\right ) ^ { 1 / j } \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\langle \\zeta _ t ^ { ( s ) } , h \\rangle = \\langle \\zeta _ { t - s } , D _ { t - s , t } h \\rangle + \\sum _ { r = t - s + 1 } ^ t \\sigma _ r [ D _ { r , t } h ] \\cdot z _ r , \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\pi _ R ( A _ + ) & = \\frac { \\partial } { \\partial x } , \\\\ \\pi _ R ( d _ + ) & = \\frac { \\partial } { \\partial \\psi _ + } + 2 \\psi _ - \\frac { \\partial } { \\partial x } , \\\\ \\pi _ R ( d _ - ) & = \\frac { \\partial } { \\partial \\psi _ - } \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\mathbb Z / p ^ n \\mathbb Z ( i ) = \\mathrm { h o f i b } ( \\varphi _ i - 1 : \\mathcal N ^ { \\geq i } A \\Omega \\{ i \\} \\to A \\Omega \\{ i \\} ) / p ^ n \\ . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\P [ A _ 2 ^ c ] & \\leq \\sum _ { k = J + 1 } ^ { n - 1 } O ( 1 ) \\exp \\Bigl ( - \\Omega \\bigl ( d ^ { k - J } n \\log n / \\beta \\bigr ) \\Bigr ) = O ( 1 ) e ^ { - \\Omega ( d n \\log n / \\beta ) } . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\varphi _ { y } ( y ) = 0 , \\ , \\ , \\ , \\nabla \\varphi _ { y } ( y ) = 0 , \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} = \\sum \\limits _ { ( M _ b , \\mu _ b ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ b , b ' } } ( \\mathrm { I n d } ^ G _ { P _ b } \\circ \\otimes ^ k _ { i = 1 } [ \\mathcal { M } _ { M _ { b _ i } , b ' _ i , \\mu _ { b i } } ] \\circ ( \\delta _ { P _ b } \\otimes \\mathrm { J a c } ^ G _ { P ^ { o p } _ b } ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ b \\rangle - \\langle \\rho _ G , \\mu \\rangle } ] , \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} X _ \\nu ( [ a , b ] ) = \\left \\{ \\mathcal { A } ( \\xi _ j ) \\right \\} \\subset [ a , b ] \\ ; , \\ ; \\ ; \\xi _ j = \\cos { ( j \\pi / \\nu ) } \\ ; , \\ ; 0 \\leq j \\leq \\nu \\ ; , \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\frac { \\tilde \\lambda _ { 1 } - \\lambda _ 1 } { \\tilde \\lambda _ { 1 } - \\lambda _ 2 } & = \\frac { A } { A + 1 } \\geq \\frac { L } { L + 1 } , \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\Big ( \\big | \\nabla _ y \\widetilde { m } ^ { \\frac { 1 } { 2 } } \\big | ^ 2 + \\big ( \\widetilde { m } ^ { \\frac { 1 } { 2 } } \\big ) ^ 2 \\Big ) d y = & \\int _ { \\mathcal { Y } ^ d } \\Big ( \\frac { 1 } { 4 } \\widetilde { m } | \\nabla _ y h | ^ 2 + \\widetilde { m } \\Big ) d y \\leq C . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} \\mathcal { T } ( R ) = \\dot { \\cup } _ { r = e - 1 } ^ \\infty \\ , \\mathcal { G } ( p , r , R ) \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} T [ u ] ( \\xi , \\eta ) : = u ^ 2 [ \\xi , \\eta ] - u \\left ( [ u \\xi , \\eta ] + [ \\xi , u \\eta ] \\right ) + [ u \\xi , u \\eta ] \\ , , \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} m : = \\sup _ { \\phi \\in \\mathcal { G } _ { n - 1 } } \\| \\phi \\| < \\infty \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} 2 ^ \\star : = \\left \\{ \\begin{array} { c l } \\frac { 4 s } { d - 2 s } & d > 2 s , \\\\ \\infty & d \\leq 2 s . \\end{array} \\right . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} x \\iff \\mathrm { r a n g e } ( D f ( x ) ) \\cap ( - \\R _ { + + } ^ m ) = \\emptyset . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( \\widehat F ) = 0 \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} X _ { j } ^ 0 = L ^ { p _ j ^ 0 } ( \\mathcal M ) . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} f ( x ) & : = a ( x ) - \\pi _ { l } ( g , h _ { 1 } , h _ { 2 } ) ( x ) \\\\ & = a ( x ) - ( h _ { 2 } T _ { 2 } ^ { * } ( h _ { 1 } , g ) ( x ) - g T ( h _ { 1 } , h _ { 2 } ) ( x ) ) \\\\ & = \\underbrace { a ( x ) \\Big ( \\frac { T _ { 2 } ^ { * } ( h _ { 1 } , g ) ( x _ { 0 } ) - T _ { 2 } ^ { * } ( h _ { 1 } , g ) ( x ) } { T _ { 2 } ^ { * } ( h _ { 1 } , g ) ( x _ { 0 } ) } \\Big ) } _ { W _ { 1 } ( x ) } - \\underbrace { g ( x ) T ( h _ { 1 } , h _ { 2 } ) ( x ) } _ { W _ { 2 } ( x ) } , \\\\ \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\hat S ( t , x ) : = I m \\left ( \\sqrt { \\frac { \\left ( x - r ( t ) \\right ) \\cdot \\gamma ' ( s ( t ) ) } { \\sqrt { 1 - | \\dot s ( t ) | ^ 2 } } + i \\left ( x - r ( t ) \\right ) \\cdot n ( s ( t ) ) } \\right ) \\ , , \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } ( \\Lambda _ i + \\widetilde { w } _ { y _ i } ) \\widetilde { w } _ { y _ i \\Lambda _ j } \\widetilde { m } d y = 0 . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} C = \\bigoplus _ { i = 1 } ^ r C _ i \\mbox { w h e r e } C _ i \\mbox { a l l h a v e l e n g t h } m \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} T ( f _ \\xi ) ( x , y ) & = \\int f _ \\xi d T ^ * ( \\delta _ { ( x , y ) } ) = \\int _ { \\alpha \\times L _ { \\xi } } f _ \\xi d T ^ * ( \\delta _ { ( x , y ) } ) = 0 , \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\alpha ( t ) & : = \\sqrt { 1 - | \\dot { s } ^ { ( 1 ) } ( t ) | ^ 2 } \\ , , \\\\ d ( t , x ) & : = \\alpha ( t ) k _ { \\eta } ( | x | ) + ( 1 - k _ \\eta ( | x | ) ) { c _ 1 } \\ , , \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\mathcal { D } _ { g } ~ : = ~ \\left \\{ x \\in [ a , b ] ~ \\big | ~ g ( x + ) = g ( x ) = g ( x - ) ~ ~ \\right \\} \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\Big ( 1 + ( t _ { i + 1 } - t _ { i } ) ^ { \\frac { \\beta } { \\alpha } } \\Big ) ^ { - 1 } & = \\bigg [ \\prod _ { i = 1 } ^ n ( t _ { i + 1 } - t _ { i } ) ^ { 1 - \\frac { \\beta } { \\alpha } } \\bigg ] \\int _ { [ 0 , \\infty ) ^ n } \\prod _ { i = 1 } ^ n e ^ { - ( t _ { i + 1 } - t _ { i } ) l _ i } E _ { \\beta / \\alpha } \\Big ( - l _ i ^ \\frac { \\beta } { \\alpha } \\Big ) d l _ i . \\\\ \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\mathrm { r } _ { X - Y } \\left ( d \\right ) \\coloneqq \\# \\left \\{ \\left ( x , y \\right ) \\in X \\times Y : \\ , d = x - y \\right \\} . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { - \\log b _ c ^ { ( n ) } } \\log P ( B _ 2 ^ { ( n ) } ) = - \\lceil \\ell _ 4 \\varepsilon \\rceil . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\lim _ { J \\downarrow x } \\frac { f ( J ) } { \\mathcal L ( J ) } = s ( x ) , ; \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} & J _ k ( x , y ) = \\int _ { ( \\R ^ n ) ^ { k - 1 } } J ( x , y _ 1 ) J ( y _ 1 , y _ 2 ) \\cdots J ( y _ { k - 1 } , y ) d y _ { k - 1 } \\cdots d y _ 1 . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ s R _ { 2 n + 1 } ^ \\lambda ( x ) = b _ n ^ \\lambda \\ , \\sum _ { k = 0 } ^ { n } \\frac { ( - n ) _ { k } ( n + \\lambda + 1 ) _ { k } } { ( \\lambda + \\frac { 1 } { 2 } ) _ { k } k ! } ( - \\Delta ) ^ s \\bigg \\{ \\frac { x } { \\left ( 1 + x ^ 2 \\right ) ^ { k + \\frac { \\lambda } { 2 } + 1 } } \\bigg \\} , \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} h ( s ( y ) ) + y s ( y ) & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & + ( 1 + O ( y ^ { - \\kappa } ) ) ( K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } ( u + 1 ) y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) \\tfrac { 1 } { u } ( K u \\Gamma ( u + 2 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } ( u + 1 ) ^ { u / ( u + 1 ) } y ^ { u / ( u + 1 ) } , \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} \\left ( \\frac { t } { \\log ( 1 + t ) } \\right ) ^ k ( 1 + t ) ^ { x - 1 } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( n - k + 1 ) } ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 3 ( x ) + \\frac { 2 } { 6 ^ 4 } ( 3 ^ 4 x - 3 3 ) ( 7 - 2 ^ 4 x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 1 - 3 ^ 5 x ) \\\\ & + \\frac { 2 } { 6 ^ 6 } ( 3 ^ 6 x - 3 0 2 ) ( 2 7 - 2 ^ 6 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 7 6 } { 6 6 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 9 6 2 3 3 9 2 6 9 3 6 0 9 9 7 3 9 9 1 8 7 7 8 4 9 2 2 4 5 2 1 3 0 9 } { 4 2 5 6 2 3 6 9 9 6 1 5 6 9 2 2 6 1 4 6 6 8 0 2 4 2 3 6 2 2 1 1 0 2 5 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} g _ { N , \\mathrm { e v e n } } ( y _ 1 , y ^ \\prime ) : = \\begin{cases} g _ N ( y _ 1 , y ^ \\prime ) & \\\\ g _ N ( - y _ 1 , y ^ \\prime ) & , \\end{cases} \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\lambda _ { k } \\big ( - \\Delta _ { ( 1 - \\varepsilon ) \\Omega } ^ { D } + m ( h ) \\big ) = ( 1 - \\varepsilon ) ^ { - 2 } \\lambda _ { k } ( - \\Delta _ { \\Omega } ^ { D } ) + o _ { h \\to 0 } ^ { \\varepsilon } ( 1 ) , \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} 2 \\pi - 2 = ( K _ X + H ) \\cdot H , \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} p \\in [ 2 , \\infty ] , q \\in [ 2 , \\infty ) , ( p , q ) \\ne \\left ( 2 , \\frac { 4 d - 2 } { 2 d - 3 } \\right ) , \\frac { 2 s } { p } + \\frac { d } { q } = \\frac { d } { 2 } . \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} J ( t , A _ t ; U , V ) = y _ t ^ { t , A _ t ; U , V } , ~ t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} & ( T _ 0 - q _ 0 ^ { - 1 } ) ( T _ 0 + q _ 1 ) = 0 , ( T _ i - q ^ { - 1 } ) ( T _ i + q ) = 0 ( i \\neq 0 ) , \\\\ & T _ k T _ { k - 1 } T _ k = T _ { k - 1 } T _ k T _ { k - 1 } \\quad ( k \\neq 0 , 1 ) , \\\\ & ( T _ 0 T _ 1 ) ^ 2 = ( T _ 1 T _ 0 ) ^ 2 , T _ i T _ j = T _ j T _ i \\quad ( | i - j | > 1 ) . \\end{align*}"}
-{"id": "10033.png", "formula": "\\begin{align*} \\mathbb { P } _ t ^ { \\alpha , i } ( g ) ( x ) = x _ i P _ t ^ { \\alpha + e _ i } \\left ( \\frac { g } { \\cdot _ i } \\right ) ( x ) . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\sum _ { g \\in G } g ( M [ \\alpha ] ) g ^ { - 1 } = M ( \\sum _ { g \\in G } g ( \\alpha ) g ^ { - 1 } ) , \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} X _ t = g ( t ) + \\int _ 0 ^ t K ( t - s ) b ( X _ s ) d s + \\int _ 0 ^ t K ( t - s ) \\sigma ( X _ s ) d W _ s , \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\int _ 0 ^ T M ( d s , \\theta _ s ) = \\int _ 0 ^ T \\left \\{ M ( s , \\theta _ s ) + 1 \\right \\} \\theta _ s d W _ s \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} L _ { \\nu } ( p ) = l _ { \\nu } x _ { m } , \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} a \\psi ( S _ 1 ) \\mathtt { C } \\mathtt { S } ( \\phi ) = ( \\rho ^ a ) \\mathtt { C } \\mathtt { S } ( \\phi ) = \\rho ^ { a \\phi } . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} f ( 0 ) & = 1 , \\\\ f ( \\alpha + 1 ) & = f ( \\alpha ) + 1 + \\alpha , \\\\ f ( \\lambda ) & = \\textstyle \\sup _ { \\alpha < \\lambda } f ( \\alpha ) \\qquad . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\sup \\Big \\{ \\left | \\left | T ( t , x ) \\right | \\right | ~ \\big | ~ F _ i ( t , x ) \\leq 0 ~ ~ \\forall i = \\overline { 1 , l } \\Big \\} \\leq C . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\Phi ( n _ k ) - \\Phi ( n _ { k - 1 } ) = e ^ { k ^ { 1 - \\epsilon } } - e ^ { ( k - 1 ) ^ { 1 - \\epsilon } } . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\norm { H _ 0 ( w ^ { ( k ) } ) } - \\Psi _ { \\mu _ k } ( w ^ { ( k ) } ) & \\leq \\norm { \\phi _ 0 ( w ^ { ( k ) } ) } - \\norm { \\phi _ { \\mu _ k } ( w ^ { ( k ) } ) } \\\\ & \\leq \\norm { \\phi _ 0 ( w ^ { ( k ) } ) - \\phi _ { \\mu _ k } ( w ^ { ( k ) } ) } \\\\ & = \\norm { p _ 0 ( x ^ { ( k ) } - y ^ { ( k ) } ) - p _ { \\mu _ k } ( x ^ { ( k ) } - y ^ { ( k ) } ) } \\leq \\sqrt { \\vartheta } \\mu _ k , \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\chi _ k ( r ) : = \\chi \\left ( \\frac { 4 } { \\rho _ { k + 1 } - \\rho _ k } ( r - \\rho _ { k 1 } ) \\right ) , \\tilde \\chi _ k ( r ) : = \\chi \\left ( \\frac { 4 } { \\rho _ { k + 1 } - \\rho _ k } ( \\rho _ { k 3 } - r ) \\right ) . \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} = \\sum _ { n \\geq 0 } \\frac { - 1 } { z ^ { n + 1 } } \\left \\langle \\psi _ 1 ^ { n - 1 } e ^ { \\tau _ 2 / z } T _ j , \\tau '^ { ( l ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 1 , d } = \\frac { - 1 } { z } \\left \\langle \\frac { e ^ { \\tau _ 2 / z } T _ j } { z + \\psi } \\tau '^ { ( l ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , l + 1 , d } \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} { \\cal D } _ { \\nu } w | _ { \\partial X } = { \\cal T } ^ { ( k ) } f \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} E f ( \\xi ) = \\frac { \\partial } { \\partial r } \\left ( \\int _ \\Omega \\frac { f ^ { p - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\delta _ r ( \\xi ) | ^ { Q - \\alpha } } d \\eta + \\lambda \\int _ \\Omega \\frac { f ^ { p - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\delta _ r ( \\xi ) | ^ { Q - \\alpha - 1 } } d \\eta \\right ) _ { r = 1 } . \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} a _ { m + 1 } \\triangleq \\sum _ { k = 1 } ^ { m } \\| u _ m \\| _ \\textsc { C C } , \\ \\ I _ { m } \\triangleq [ a _ { m } , a _ { m + 1 } ] , \\ \\ I \\triangleq \\overline { \\bigcup _ { m = 1 } ^ { \\infty } I _ { m } } . \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} L ( 1 , \\chi ) = L ( 1 , \\chi , Y ) \\left ( 1 + \\mathcal { O } \\left ( \\frac { 1 } { ( \\log q ) ^ 2 } \\right ) \\right ) , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} C _ n T = C _ n [ ( I - P ) T + P T ] = C _ n ( I - P ) T = C _ n T _ { \\rm r e g } , \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} D _ M = D _ { C ( I ) } \\otimes D _ E \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} { \\mathcal { S } } \\ , { } ^ t \\ ! ( \\sigma ( M ) ) & = \\{ ( y _ 1 , \\dots , y _ n ) \\in \\mathbb { R } ^ n \\mid y _ j = k _ j \\ , ( j = 1 , \\dots , t ) \\} , \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { h } \\oplus \\mathfrak { m } , \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\overline { U } _ { 2 k + 1 , 2 } ( x ; q ) = ( x q ) ^ 2 U _ { 2 k + 1 , 2 k - 1 } ( x q ; q ) + U _ { 2 k + 1 , 2 k + 1 } ( x q ; q ) . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} D _ I = \\biggl \\{ \\theta \\in \\{ \\pm 1 \\} ^ { \\mathcal { D } } : \\Bigl | \\langle T b _ { I } ^ { ( \\theta ) } , b _ { I } ^ { ( \\theta ) } \\rangle - \\sum _ { K \\in \\mathcal { B } _ I } \\langle T h _ { K } , h _ { K } \\rangle \\Bigr | > \\eta _ 0 \\biggr \\} , I \\in \\mathcal { D } _ { \\leq n } . \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} | \\mathcal { C } | = \\sum _ { \\ell = 0 } ^ { \\log \\log n + 1 } | \\mathcal { C } _ { \\ell } | \\leq 2 ^ { 1 6 \\sqrt { n } + 1 } + 2 ^ { 1 6 7 \\sqrt { n } } ( \\log \\log n + 1 ) \\log n \\leq 2 ^ { 1 6 8 \\sqrt { n } } . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} \\sum _ { U ^ a V ^ b W ^ c \\in \\mathcal { B } _ n } U ^ a V ^ b W ^ c X ^ a Y ^ b Z ^ c & = ( U X + V Y + W Z ) ^ n \\\\ & = \\prod _ { i \\in S _ n } \\left ( U ^ { 2 ^ i } X ^ { 2 ^ i } + V ^ { 2 ^ i } Y ^ { 2 ^ i } + W ^ { 2 ^ i } Z ^ { 2 ^ i } \\right ) . \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} | I _ { n _ k } ( x ) | ^ { 2 s ( M ) - 1 } \\geq ( K _ 3 ^ { 2 k - 1 } K _ 1 ^ k ) ^ { 2 s ( M ) - 1 } C _ M ^ { - k } \\mu ( I _ { n _ k } ( x ) ) \\left ( \\prod _ { j = 1 } ^ k a _ { n _ j } ^ { - d } \\right ) ^ { { 2 } s ( M ) - 1 } . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} w ( p , n ) = O ( x ^ { 1 / 3 + o ( 1 ) } ) \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} \\mathfrak { H } : = \\left ( \\begin{array} { c c c c c c c c c c } H _ 3 & H _ { 2 } & H _ 1 & H _ 0 & & & & & & \\\\ & H _ 3 & H _ 2 & H _ 1 & H _ 0 & & & & & \\\\ & & H _ 3 & H _ 2 & H _ 1 & H _ 0 & & & & \\\\ & & & H _ 3 & H _ 2 & H _ 1 & H _ 0 & & & \\\\ & & & & H _ 3 & H _ 2 & H _ 1 & H _ 0 & & \\\\ & & & & & H _ 3 & H _ 2 & H _ 1 & H _ 0 & \\\\ & & & & & & H _ 3 & H _ 2 & H _ 1 & H _ 0 \\end{array} \\right ) \\in \\mathbb F ^ { 7 \\times 2 0 } . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } F ( n , k - 1 ) - \\sum _ { n = 0 } ^ { m - 1 } F ( n , k ) = G ( m , k ) . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} & \\chi _ i = \\sum _ { j = 1 } ^ { m } \\left ( \\left ( 1 - \\psi _ { i - j + 1 } \\right ) ^ 2 \\left ( 1 - \\phi _ { i - j + 1 } \\right ) ^ 2 + { \\psi _ { i - j + 1 } } ^ 2 \\left ( 1 - \\phi _ { i - j + 1 } \\right ) ^ 2 + \\vphantom { U } \\right . \\\\ & \\qquad \\qquad \\left . \\vphantom { U } \\left ( 1 - \\psi _ { i - j + 1 } \\right ) ^ 2 { \\phi _ { i - j + 1 } } ^ 2 + { \\psi _ { i - j + 1 } } ^ 2 { \\phi _ { i - j + 1 } } ^ 2 \\right ) . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} & \\log P ( n - a _ n - S _ n ( n - h ( n ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) = \\log P ( \\mathrm { B i n } ( n - a _ n , 1 - \\pi _ n ( n - h ( n ) ) ) \\geq \\lceil \\varepsilon f _ 4 ( n ) \\rceil ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\sim _ e \\lceil \\varepsilon f _ 4 ( n ) \\rceil \\log b _ c ^ { ( n ) ' } . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} u _ { t t } ( 1 + u _ x ^ 2 ) - 2 u _ t u _ x u _ { t x } + u _ { x x } ( 1 + u _ t ^ 2 ) = 0 . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 4 ( x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 2 - 3 ^ 5 x ) \\\\ & \\qquad + \\frac { 2 } { 6 ^ 6 } ( 3 ^ 6 x - 3 0 4 ) ( 2 7 - 2 ^ 6 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 8 8 } { 2 1 1 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 5 6 4 1 1 0 5 4 1 4 7 3 3 8 6 5 5 8 5 2 2 2 2 2 7 9 1 5 3 7 1 4 1 2 5 7 0 3 } { 6 3 1 6 3 9 7 5 7 6 8 6 3 4 1 1 7 2 4 2 5 4 5 0 3 5 6 7 3 5 9 9 9 0 2 3 5 7 9 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} c _ k ( E ) = \\frac { 1 } { k ! } ( \\frac { \\sqrt { - 1 } } { 2 \\pi } ) ^ k t r ( \\alpha _ E ^ k ) \\in H ^ k ( A , \\wedge ^ k A ^ { \\perp } ) . \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} J = \\left \\lfloor \\frac { \\log ( L / L _ 0 ) } { \\log ( 1 + \\tilde B ) } \\right \\rfloor B = ( L / L _ 0 ) ^ { 1 / J } - 1 . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\sigma ( h _ 0 ) = T \\big ( \\tilde { \\sigma } ( t _ 0 , \\langle \\zeta , h _ 0 - \\phi ( t _ 0 ) \\rangle ) \\big ) \\in V ^ m , \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 + x ^ 2 ) ^ \\gamma } \\ ; \\ ; \\ ; \\ ; \\gamma = k + \\frac { \\lambda + 1 } 2 { \\rm o r } \\frac { x } { ( 1 + x ^ 2 ) ^ \\gamma } \\ ; \\ ; \\ ; \\ ; \\gamma = k + \\frac { \\lambda } 2 + 1 . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\phi _ { t _ - } & = \\sum _ j ( \\sigma _ { 2 ^ s - 1 + j 2 ^ s } + \\sigma _ { 2 ^ s + j 2 ^ s } ) \\\\ \\phi _ { t _ + } & = \\sum _ j \\sum _ { i = t } ^ { 2 ^ s } \\sigma _ { i + j 2 ^ s } . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{gather*} \\Phi _ a ^ { \\prime } = \\rho _ a ^ i ( x ) p _ i + \\alpha ^ { \\prime } _ a ( x ) , \\\\ H = \\frac { 1 } { 2 } g ^ { i j } ( x ) p _ i p _ j + V ^ { \\prime } ( x ) . \\end{gather*}"}
-{"id": "3878.png", "formula": "\\begin{align*} ( \\N ^ 2 f ) ^ { 1 , 1 } ( \\cdot , J \\cdot ) _ { i j } = & \\ \\tfrac { 1 } { 2 } \\left [ ( \\N ^ 2 f ) _ { i k } + ( \\N ^ 2 f ) _ { p q } J _ i ^ p J _ k ^ q \\right ] J _ j ^ k \\\\ = & \\ \\tfrac { 1 } { 2 } \\left [ J _ j ^ k ( \\N ^ 2 f ) _ { i k } - ( \\N ^ 2 f ) _ { p j } J _ i ^ p \\right ] \\\\ = & \\ \\tfrac { 1 } { 2 } \\left [ \\N _ i ( J _ j ^ k \\N _ k f ) - \\N _ j ( J _ i ^ k \\N _ k f ) \\right ] \\\\ = & \\ - \\tfrac { 1 } { 2 } ( d d ^ c f ) _ { i j } . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\mathcal { V } ^ { B D } [ v ; \\eta ] ( t ) = \\lim _ { \\lambda \\downarrow 0 } \\frac { \\mathcal { V } [ v + \\lambda \\eta ] ( t ) - \\mathcal { V } [ v ] ( t ) } { \\lambda } \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} F ( \\psi _ 1 , \\psi _ 2 ) = f _ 2 + \\frac 1 2 \\left ( \\psi _ 1 ^ 2 - \\psi _ 1 \\right ) + ( \\rho \\nu \\psi _ 1 - \\lambda ) \\psi _ 2 + \\frac { \\nu ^ 2 } { 2 } \\psi ^ 2 _ 2 . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} 1 = | \\xi _ { 0 } | \\leq \\| G _ { [ - j _ 0 , j _ 0 ] } ( x _ 0 , E ) \\| \\| R _ { [ - j _ 0 , j _ 0 ] } H ( x _ 0 ) R _ { \\mathbb { Z } \\setminus [ - j _ 0 , j _ 0 ] } \\xi \\| . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} U ^ { ( \\theta ) } T g - g = \\sum _ { \\substack { I , I ' \\in \\mathcal { D } _ { \\leq n } \\\\ I \\neq I ' } } a _ { I ' } \\frac { \\langle T b _ { I ' } ^ { ( \\theta ) } , b _ I ^ { ( \\theta ) } \\rangle } { \\langle T b _ I ^ { ( \\theta ) } , b _ I ^ { ( \\theta ) } \\rangle } b _ I ^ { ( \\theta ) } . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} \\left \\langle \\psi _ 1 ^ n T _ a , T _ j , \\tau ^ { ( k ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , k + 2 , 0 } = \\sum _ { u + v = k } \\sum _ { x + y = u - 1 } \\frac { k ! } { v ! x ! y ! } \\tau _ 2 ( 0 ) ^ x \\left \\langle \\psi _ 1 ^ { n - y + 1 } ( T _ a \\cup \\tau _ 2 ^ { y - 1 } ) , T _ j , \\tau _ 2 , \\tau '^ { ( v ) } \\right \\rangle ^ \\textnormal { c o h } _ { 0 , v + 3 , 0 } \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\mathrm { w d } ( \\mathcal { T } ) = \\max \\lbrace \\# \\mathrm { L y r } _ n { \\mathcal { T } } \\mid n _ \\ast < n < n _ \\ast + \\ell _ \\ast + \\ell + d \\rbrace . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { ( Q _ 1 ( q ( t ) ; q ( t ) ) _ \\infty } { ( Q _ 2 ( q ( t ) ) ; q ( t ) ) _ \\infty } = ( 1 - Q _ 0 ) ^ { \\alpha _ 2 - \\alpha _ 1 } \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} X ( s ) = \\sqrt { 2 } ( \\cosh s e ^ { i \\theta _ 1 } + \\sinh s e ^ { - i \\theta _ 2 } , \\sinh s e ^ { - i \\theta _ 1 } + \\cosh s e ^ { i \\theta _ 2 } ) . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} _ { \\mathcal { H } } \\langle f , h \\rangle _ { \\bar { \\mathcal { H } } } = \\mathbb { E } [ Z \\cdot I ( f ) ] = \\mathbb { E } \\left [ Z \\cdot \\int _ { 0 } ^ { 1 } f _ { s } d B _ { s } \\right ] = \\int _ { 0 } ^ { 1 } f _ { s } \\ , \\mathbb { E } [ Z d B _ { s } ] = \\int _ { 0 } ^ { 1 } f _ { s } d h _ { s } . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} I d _ n = I d _ 1 ^ { \\leftrightarrow n } = \\overbrace { I d _ { 1 } \\leftrightarrow \\cdots \\leftrightarrow I d _ { 1 } } ^ { n \\mbox { \\footnotesize t i m e s } } \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} [ M _ { S ' } , \\mu _ { S ' } ] ( I ^ G _ { M _ S } ( \\rho ) ) = ( \\mathrm { I n d } ^ G _ { P _ { S ' } } \\circ [ \\mu _ { S ' } ] ) ( \\bigoplus _ { w \\in W _ { \\rho } } \\delta ^ { \\frac { 1 } { 2 } } _ { P _ { S ' } } \\otimes I ^ { M _ { S ' } } _ { w ( M _ S ) } ( w ( \\rho ) ) ) \\otimes [ 1 ] [ | \\cdot | ^ { \\langle \\rho _ G , \\mu _ { S ' } - \\mu \\rangle } ] . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\delta _ j = \\| T _ j ( \\hat \\Sigma - \\Sigma ) T _ j \\Vert _ \\infty , T _ j = | R _ j | ^ { 1 / 2 } + g _ j ^ { - 1 / 2 } P _ j , \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} F _ \\epsilon ( t ) = \\frac { j _ \\epsilon ^ 2 } { 2 t ^ 2 } - \\ln t . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\sup _ { m \\geq 1 } \\| \\dot { \\tilde { h } } ^ { ( m ) } \\| _ { \\infty ; [ 0 , a _ { m + 1 } ] } = \\sup _ { m \\geq 1 } \\left \\{ \\frac { \\| u _ { m } \\| _ { } } { | I _ { m } | } \\cdot \\| \\dot { \\bar { h } } _ { m } \\| _ { \\infty ; [ 0 , 1 ] } \\right \\} . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\det \\left ( \\varphi | _ { u _ i ^ \\bot } \\right ) = \\frac { h _ { K } ( u _ i ) } { h _ { \\varphi K } ( v _ i ) } . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\tilde f ( z ) : = \\prod _ { k = 0 } ^ { m - 1 } f ( e ^ { 2 \\pi i k / m } \\cdot z ) . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\mathcal { P } _ n ( A ) = \\biggl \\{ \\sum _ { j = 1 } ^ m S _ n ( a _ { 1 , j } , \\dotsc , a _ { n , j } ) : a _ { 1 , j } , \\dotsc , a _ { n , j } \\in A , \\ m \\in \\mathbb { N } \\biggr \\} \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\pi _ { \\beta , \\rho } ( g ) = & \\pi ^ { \\beta , \\rho } \\left ( l ( \\overline T ) ( 1 + \\varpi ^ l S ) _ r \\right ) \\\\ = & \\tau \\left ( \\varpi ^ { - ( l - 1 ) } B ( S , \\beta ) + \\varpi ^ { - l } B ( \\lambda ( \\overline T ) , \\beta ) \\right ) \\cdot \\rho ( Y ) \\cdot \\pi ^ { \\beta } ( v , 1 ) \\\\ = & \\tau \\left ( \\varpi ^ { - l } B ( T , \\beta ) - 2 ^ { - 1 } \\varpi ^ { - 1 } B ( T ^ 2 , \\beta ) \\right ) \\cdot \\rho ( Y ) \\cdot \\pi ^ { \\beta } ( v , 1 ) . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{gather*} \\nabla _ { X _ 1 } ( J X _ 1 ) = J \\nabla _ { X _ 1 } X _ 1 = - X _ 1 , \\nabla _ { X _ 2 } ( J X _ 2 ) = J \\nabla _ { X _ 2 } X _ 2 = - X _ 2 , \\\\ \\nabla _ { X _ 1 } ( J X _ 2 ) = J \\nabla _ { X _ 1 } X _ 2 = 0 , \\nabla _ { X _ 2 } ( J X _ 1 ) = J \\nabla _ { X _ 2 } X _ 1 = 0 . \\end{gather*}"}
-{"id": "2124.png", "formula": "\\begin{align*} t _ 0 : = \\sqrt { \\dfrac { 6 \\ , | \\theta _ L | } { \\lambda _ n ( C ) } } , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} C = 2 \\delta + 3 C ' = 2 \\delta + 4 . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} E _ \\varphi ( \\Phi ) : = \\left \\{ x \\in ( 0 , 1 ) : \\lim _ { n \\to \\infty } { S _ n \\varphi ( x ) \\over \\Phi ( n ) } = 1 \\right \\} . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} & P _ { \\textrm { i s o u t } } ^ { \\textrm { I S A N C } } = \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace + \\textrm { P r } ^ { \\textrm { N O M A } } \\left \\lbrace \\textrm { F D N = 0 } \\right \\rbrace . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\widetilde I ( f ) = \\int _ G f ( g ) i ( g ) d g . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ 0 ^ t e ^ { - \\mathrm { i } ( t - \\tau ) \\mathcal { H } } \\chi F ( \\tau , x ) \\ , d \\tau \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\int _ { B } | \\nabla v | ^ { p - 2 } \\Big ( \\frac { \\partial v ^ i } { \\partial x _ \\alpha } \\frac { \\partial \\Psi ^ j } { \\partial x _ \\beta } \\delta _ { i j } - \\Gamma _ { i j } ^ l ( v ) \\frac { \\partial v ^ i } { \\partial x _ \\alpha } \\frac { \\partial v ^ j } { \\partial x _ \\beta } \\Psi ^ l \\Big ) \\delta ^ { \\alpha \\beta } d x = 0 , \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} p _ e ( \\rho ) = a \\rho ^ \\gamma \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\overline { r } ( n ; \\theta ) + 1 } \\mathfrak { J } _ k ^ { ( 1 ) } ( n ; \\theta ) \\leq \\frac { S _ 1 ( \\gamma ) } { [ n \\theta ( 1 - \\theta ) ] ^ { ( 1 + \\varepsilon ( \\gamma ) ) } } + \\frac { S _ 2 ( \\gamma ) + S _ 3 ( \\gamma ) } { [ n \\theta ( 1 - \\theta ) ] ^ { ( 1 + \\varepsilon ( \\gamma ) ) } } \\log \\left ( \\frac { n } { \\theta ( 1 - \\theta ) } \\right ) \\ . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\sum _ { \\omega _ 0 \\in A } \\gamma ( \\omega _ { - \\infty } ^ { - 1 } , \\omega _ 0 , \\omega _ { 1 } ^ { \\infty } ) = 1 , \\textrm { f o r a l l } \\omega \\in \\Omega \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} Q ( \\theta ) _ t + \\theta p _ \\theta ( \\rho ) u _ y = ( \\frac { \\kappa \\theta _ y } { v } ) _ y + \\frac { \\varepsilon u _ y ^ 2 } { v } + \\frac { \\mu | \\mathbf { w } _ y | ^ 2 } { v } + \\frac { \\nu | \\mathbf { h } _ y | ^ 2 } { v } . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\min \\{ \\| x - z \\| _ 1 : z \\in \\Delta _ { n - 1 } , \\bar { A } z = \\bar { u } \\} \\le \\frac { 2 \\| \\bar { A } x - \\bar { u } \\| _ v } { \\Phi _ v ( \\bar { A } ) } . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} \\hat { \\Delta } g ( y ) : = \\Delta _ { y ^ \\prime } g ( y ) + \\alpha ( y ^ \\prime ) \\partial _ { y _ 1 } ^ 2 g ( y ) + \\sum _ { j = 2 } ^ n \\beta _ j ( y ^ \\prime ) \\partial _ { y _ j } \\partial _ { y _ 1 } g ( y ) + \\gamma ( y ^ \\prime ) \\partial _ { y _ 1 } g ( y ) \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\| X \\| _ { \\Psi } = \\inf \\{ t > 0 \\ , : \\ , \\mathbb { E } \\Psi ( X / t ) \\leq 1 \\} . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} & \\mathcal A ( t ) : = \\left ( \\begin{array} { c c c } 0 & - 1 \\\\ - \\div ( A ^ { ( 4 ) } ( t ) \\nabla ( \\cdot ) ) + p ( t ) \\cdot \\nabla ( \\cdot ) & - 2 q ( t ) \\cdot \\nabla ( \\cdot ) \\end{array} \\right ) \\ , , \\\\ & X : = H ^ 1 _ D ( \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) ) \\times L ^ 2 ( \\Omega ^ { ( 4 ) } ) \\ , , \\\\ & Y : = \\mathcal H \\times H ^ 1 _ D ( \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) ) \\ , , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} d \\sigma _ \\Sigma : = ( \\overline { p } \\omega _ 2 - \\overline { q } \\omega _ 1 ) \\wedge \\omega , ~ ~ ~ ~ d \\overline { \\sigma _ \\Sigma } : = \\frac { X _ 3 u } { l } \\omega _ 1 \\wedge \\omega _ 2 - \\frac { ( X _ 3 u ) ^ 2 } { 2 l ^ 2 } ( \\overline { p } \\omega _ 2 - \\overline { q } \\omega _ 1 ) \\wedge \\omega . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} d _ a = \\sum _ { i = 1 } ^ d a _ i \\nabla _ { X _ i } , \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} i \\partial _ t - ( - \\Delta ) ^ s u = - | u | ^ \\alpha u , u ( 0 ) = u _ 0 , [ 0 , \\infty ) \\times \\R ^ d , \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\langle L h , h \\rangle _ X & = \\sum _ { | \\alpha | \\leq k } \\langle \\partial _ x ^ \\alpha L h , \\partial _ x ^ \\alpha h \\rangle _ 0 \\\\ & = \\sum _ { | \\alpha | \\leq k } \\langle L \\partial _ x ^ \\alpha h , \\partial _ x ^ \\alpha h \\rangle _ 0 \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} F ( x ) = \\inf \\left \\{ \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i g ( x _ i ) \\ , : \\ , \\lambda _ i \\geq 0 , \\ , \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i = 1 , \\ , x = \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i x _ i \\right \\} . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\widehat { f } _ a = F _ b \\circ \\widehat { \\varphi } _ a , \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\begin{cases} w _ { t t } - \\mathrm { e } ^ { 2 t } \\Delta w + M ^ 2 w = g ( t , x ) , & x \\in \\mathbb { R } ^ n , t > 0 , \\\\ w ( 0 , x ) = w _ 0 ( x ) , & x \\in \\mathbb { R } ^ n , \\\\ w _ t ( 0 , x ) = w _ 1 ( x ) , & x \\in \\mathbb { R } ^ n , \\end{cases} \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} h : S = S ^ { N - 1 } \\to \\R ^ j , h ( e ) = [ ( g _ e , \\phi _ 1 ) _ { L ^ 2 ( \\Omega ) } , \\dots , ( g _ e , \\phi _ j ) _ { L ^ 2 ( \\Omega ) } ] . \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} \\begin{aligned} \\max \\limits _ { B \\leq 2 f _ { \\rm c } \\frac { 1 - \\cos \\theta _ { \\rm c } } { 1 + \\cos \\theta _ { \\rm c } } } R = B \\log _ 2 \\left ( 1 + \\frac { P _ { \\rm R } } { B N _ 0 | \\cos \\theta _ { \\rm c } | \\log \\sqrt { \\frac { f _ { \\rm c } + B / 2 } { f _ { \\rm c } - B / 2 } } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\dfrac { \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } \\ln R ( n ) } { \\sum _ { n \\geq 1 } \\dfrac { R ( n ) ^ { t } } { n ^ { s } } } & = \\sum _ { p } \\dfrac { p ^ t \\ln { p } ( p ^ s - 1 ) } { ( p ^ s + p ^ t - 1 ) ( p ^ s - 1 ) } \\\\ & = \\sum _ { p } \\dfrac { p ^ { t } } { p ^ s - 1 + p ^ t } \\ln { p } \\ , \\ , \\ , : = \\ , \\mathcal { T } ( s , \\ , t ) . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} ( \\omega \\pi ) ( x ' ) = - \\int _ { \\mathbb R ^ 2 } ( \\nabla _ { y ' } \\Psi ) ( x ' , y ' ) \\cdot \\big [ \\omega F + 2 ( \\nabla _ { y ' } \\omega ) \\pi \\big ] ( y ' ) \\ , d y ' - \\int _ { \\mathbb R ^ 2 } \\Psi ( x ' , y ' ) \\big [ ( \\nabla _ H \\omega ) \\cdot F + ( \\Delta _ H \\omega ) \\pi \\big ] ( y ' ) \\ , d y ' . \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\overline { Q } _ { k , - \\frac { 1 } { 2 } } ( x ; q ) = - ( x q ) ^ { - \\frac { 1 } { 2 } } \\overline { Q } _ { k , \\frac { 1 } { 2 } } ( x ; q ) \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} & | \\tilde { \\mathcal { L } } ^ d _ { \\leq x } | - O _ { d , r } ( x ^ { d - 1 } \\log x ) + O _ { d } ( x ^ { d - 1 + O ( 1 / ( \\log \\log x ) } ) \\\\ = & ( 1 + O _ { d , r } ( x ^ { - 1 + O ( 1 / \\log \\log x ) } ) ) c _ d x ^ d , \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} A ^ { \\sup } = \\bigoplus _ { X \\in \\mathcal { X } } X . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} | \\Lambda ( f _ 1 , \\ldots , f _ { n + 1 } ) | \\lesssim \\sum _ { Q \\in \\mathcal { S } } | Q | \\prod _ { j = 1 } ^ { n + 1 } \\langle | f _ j | _ { X _ j } \\rangle _ Q . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 1 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = i ( i R _ { 1 2 } - i R _ { 1 4 } ) = - R _ { 1 2 } + R _ { 1 4 } = - 1 = u _ 1 ; \\\\ \\frac { R _ { 3 2 } } { - i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 3 4 } } { - i - \\omega ( 4 , 4 ) } u _ 4 = - R _ { 3 2 } + R _ { 3 4 } = 1 = u _ 3 . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\bar { Z } = \\begin{pmatrix} a & c & 0 & d \\cr c & b & c & 0 \\cr 0 & c & b & c \\cr d & 0 & c & a \\end{pmatrix} . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} \\Gamma _ p ( M ) : = \\{ h \\in C ^ 0 ( D _ 1 ^ 2 , W ^ { 1 , p } ( M , \\mathbb { R } ^ 2 ) ) \\mid h _ y \\equiv y y \\in S ^ 1 \\} . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} ( \\rho _ f ( G _ F ) \\cap \\Gamma ( k - 1 ) ) / ( \\rho _ f ( G _ F ) \\cap \\Gamma ( k ) ) = \\Gamma ( k - 1 ) / \\Gamma ( k ) . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} P ( f ) = \\Phi _ 0 \\left ( f ^ { * n } \\right ) \\left ( f \\in \\mathcal { T } ( G ) \\right ) . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} v = ( \\lambda _ 1 + y - \\Sigma ) ^ { - 1 / 2 } F \\frac { z } { \\sqrt { n } } \\| v \\| ^ 2 = \\frac { z } { \\sqrt { n } } \\sum _ { k = 1 } ^ r \\frac { \\lambda _ k } { \\lambda _ 1 + y - \\lambda _ k } > \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} & H _ { T ^ { N + 1 } } ^ * ( \\textnormal { p t } ; \\mathbb { Q } ) = \\mathbb { Q } [ \\lambda _ 0 , \\dots , \\lambda _ N ] & & H _ { T } ^ * ( \\textnormal { p t } ; \\mathbb { Q } ) = \\mathbb { Q } [ z ] \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 1 ) } = & \\rho + \\kappa ^ 2 \\\\ ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 2 ) } = & \\frac { 2 } { 3 } D \\rho \\\\ ( \\sigma ^ { a b } n _ { a b } ) ^ { ( 3 ) } = & \\frac { 3 } { 8 } D ^ 2 \\rho + \\frac { 1 } { 3 0 } | \\alpha | ^ 2 - \\frac { 1 1 } { 4 5 } | \\beta | ^ 2 . \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} B [ v , \\varpi ] = \\langle A v , \\varpi \\rangle _ \\mathcal { H } . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\left ( X _ j ^ s \\right ) ^ * = Y _ j ^ s , 1 \\leq j \\leq m \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} Q = \\log ( 1 + e ^ { R } ) . \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} w \\ = \\ \\mu _ 1 b _ 1 + ( \\lambda _ 2 + \\mu _ 2 ) b _ 2 + \\cdots + ( \\lambda _ t + \\mu _ t ) b _ t \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\lambda _ j = c _ 0 + 2 \\sum _ { k = 1 } ^ { \\frac { n - 1 } { 2 } } c _ { k } \\cos \\left ( \\frac { 2 \\pi j k } { n } \\right ) , \\textrm { f o r } j = 0 , \\ldots , \\ n - 1 \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { M \\to \\infty } \\mathop { \\max } \\limits _ { 1 \\le k \\le K } \\left \\{ { \\eta _ { K , M } ^ { ( 3 ) } + \\eta _ { K , M } ^ { ( 4 ) } } \\right \\} = \\mathop { \\max } \\limits _ { 1 \\le k \\le K } \\left \\{ { \\mathop { \\lim } \\limits _ { M \\to \\infty } \\eta _ { K , M } ^ { ( 3 ) } + \\mathop { \\lim } \\limits _ { M \\to \\infty } \\eta _ { K , M } ^ { ( 4 ) } } \\right \\} \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} d \\widetilde { C } _ { q } ( K , Q , \\cdot ) = h _ K ^ { p } d \\mu . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} R ^ * ( = R ^ * ( \\omega ) ) = 2 c \\rho . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\Omega ^ * _ j \\ = \\ h ^ * _ { d - j } \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla f + ( E + v \\wedge B ) \\cdot \\nabla _ p f = 0 \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } ( h _ 0 ^ { ( 2 ) } - h ^ { ( 2 ) } ) \\tilde X ^ i d S ^ 2 = 0 . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\sum _ { j } \\binom { n } { 3 j } - \\sum _ { j } \\binom { n } { 3 j + 1 } = \\begin{cases} ( - 1 ) ^ n & \\\\ 0 & \\\\ ( - 1 ) ^ { n - 1 } & \\end{cases} \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} R _ 1 ( y _ 0 , y _ 2 , x _ 0 , x _ 1 ) = x _ 0 ^ 2 S _ 1 ( y _ 0 , y _ 2 , x _ 0 , x _ 1 ) , \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} m _ 1 ( x , \\xi ) : = 1 + | x | ^ 2 + | \\xi | ^ 2 . \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b _ 1 , \\mu } ( \\mathrm { R e d } _ { b _ 1 } ( \\pi ) ) = \\mathrm { M a n t } _ { G , b _ 1 , \\mu } ( L J ( \\delta ^ { \\frac { 1 } { 2 } } _ { P _ { b _ 1 } } \\otimes J ^ G _ { P ^ { o p } _ { b _ 1 } } I ^ G _ { M _ { b _ 1 } } ( \\rho _ 1 \\boxtimes \\rho _ 2 ) ) ) . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} { \\cal E } ( t ) = \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\sqrt { 1 + p ^ 2 } \\ , f ( t , x , p ) \\ , d x \\ , d p + \\frac { 1 } { 2 } \\int _ { \\R ^ 3 } ( | E ( t , x ) | ^ 2 + | B ( t , x ) | ^ 2 ) \\ , d x \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} a : = \\lim _ { x \\to \\infty } \\frac { f ( x ) } { x ^ { n / p ^ { m } } } \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ S \\mathcal { L } _ 2 ( x , ( B - \\frac { \\sqrt { \\kappa } } { 2 } A ^ 2 ) _ { \\tan } ) : \\tilde { B } _ { \\tan } d x d t = 0 , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\mathbb { R } _ { U _ 1 ^ n } ( d ) & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { 1 } { 2 } \\log \\max \\left ( \\mu _ { n , i } , ~ \\frac { \\sigma ^ 2 } { \\theta _ n } \\right ) , \\\\ d & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\min \\left ( \\theta _ n , \\frac { \\sigma ^ 2 } { \\mu _ { n , i } } \\right ) , \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} & ( q + 1 ) \\widetilde { m } ^ q ( \\nabla _ y \\widetilde { m } ) ^ T \\nabla _ y ^ 2 \\widetilde { w } ( \\Lambda + \\nabla _ y \\widetilde { w } ) + ( q + 1 ) \\widetilde { m } ^ q ( \\nabla _ y \\widetilde { m } ) ^ T \\nabla _ y V = ( q + 1 ) \\widetilde { m } ^ { q - 1 } | \\nabla _ y \\widetilde { m } | ^ 2 . \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } e ^ { 2 \\pi i n \\theta } = 0 . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} 2 \\langle \\nabla ^ L _ { X _ i } X _ j , X _ k \\rangle _ L = \\langle [ X _ i , X _ j ] , X _ k \\rangle _ L - \\langle [ X _ j , X _ k ] , X _ i \\rangle _ L + \\langle [ X _ k , X _ i ] , X _ j \\rangle _ L , \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} { \\cal L } f ( x ) = \\int _ { \\R } \\left ( f ( x + z ) - f ( x ) - { \\bf 1 } _ { | z | < 1 } ( z \\cdot \\nabla f ( x ) ) \\right ) \\nu ( d z ) \\ , , f \\in C ^ 2 _ b ( \\R ) \\ , , \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\widehat { \\pi } _ j = \\frac { \\sum _ { \\boldsymbol { y } \\in \\boldsymbol { \\mathcal { N } } } { y _ j } } { \\sum _ { \\boldsymbol { y } \\in \\boldsymbol { \\mathcal { N } } } \\vert \\boldsymbol { y } \\vert } . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} \\left \\Vert a \\right \\Vert _ { \\mathcal { P } _ n } = \\left \\Vert \\alpha ^ { - 1 } ( \\alpha a ) \\right \\Vert _ { \\mathcal { P } _ n } \\le \\left \\vert \\alpha \\right \\vert ^ { - 1 } \\left \\Vert \\alpha a \\right \\Vert _ { \\mathcal { P } _ n } , \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\rho _ 1 = ( - 1 , - 2 , 3 ) , \\rho _ 2 = ( 1 , - 3 , 2 ) , \\rho _ 3 = ( 2 , - 3 , 1 ) . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} P _ 0 ( h , \\mu ) \\widetilde u = { \\rm O p } _ h \\left ( e ^ { i \\varphi / h } \\Phi _ { \\epsilon , \\delta } A _ M \\right ) f + { \\rm O p } _ h \\left ( e ^ { i \\varphi / h } A ^ \\sharp _ M \\right ) f + { \\cal E } _ M f \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\dot { y } & = A _ { y y } \\Delta y + A _ { y z } \\Delta z + B _ { y h } \\Delta u _ h + \\Delta u _ b + d _ y \\\\ \\Delta \\dot { z } & = A _ { z y } \\Delta y + A _ { z z } \\Delta z + B _ { z h } \\Delta u _ h + d _ z \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\min _ { x \\in S } \\ ( \\max _ { i = 1 } ^ { n } x _ { i } ) ( \\max _ { j = 1 } ^ { n } ( 1 / x _ { j } ) ) . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\hat h _ n = \\left \\{ \\begin{array} { c c } h _ n , & n < N _ 3 , \\\\ h _ { N } , & n \\ge N _ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} \\begin{aligned} - 1 6 L \\mathcal Q ( t ) = & ~ { } 2 ( 9 + \\tilde \\sigma ) \\int g ^ 2 + ( - 3 + 5 \\tilde \\sigma ) \\int g _ x ^ 2 \\\\ & ~ { } + 4 ( - 1 + \\tilde \\sigma ) \\int g _ { x x } ^ 2 + ( 1 + \\tilde \\sigma ) \\int g _ { x x x } ^ 2 + O \\left ( \\frac 1 L \\int ( g ^ 2 + g _ x ^ 2 + g _ { x x } ^ 2 ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\ln \\lambda = \\lim _ { k \\rightarrow \\infty } \\sup _ { \\theta \\in \\mathbb { R } / \\mathbb { Z } } \\frac { 1 } { k } \\ln \\| A _ k ( \\theta ) \\| , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\varphi _ 0 ( x ) : = \\inf _ { t \\in ( 0 , 1 ] } \\varphi _ t ( x ) = \\lim _ { t \\downarrow 0 } \\varphi _ t ( x ) , \\\\ \\psi _ 1 ( x ) : = \\inf _ { t \\in [ 0 , 1 ) } \\psi _ t ( x ) = \\lim _ { t \\uparrow 1 } \\psi _ t ( x ) . \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} | | H f | | _ { L ^ 2 ( \\mathbb { R } ) } = \\pi | | f | | _ { L ^ 2 ( \\mathbb { R } ) } \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} 0 < & ~ \\big ( p ( \\hat { x } , \\hat { t } ) - 2 \\big ) \\big \\langle X \\xi _ x , \\xi _ x \\big \\rangle - \\big ( p ( \\hat { y } , \\hat { t } ) - 2 \\big ) \\big \\langle Y \\xi _ y , \\xi _ y \\big \\rangle \\\\ & + \\sum _ { i = 1 } ^ n \\lambda _ i \\big ( X - Y \\big ) + 2 \\langle \\mu , \\delta \\hat { x } + \\delta \\hat { y } \\rangle - r \\alpha / 2 , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\eta \\int _ { E ( k ) } | \\nabla _ y \\psi | ^ 2 d y \\leq & \\int _ { E ( k ) } ( \\nabla _ y \\psi ) ^ T A ^ T \\nabla _ y \\psi d y = \\int _ { E ( k ) } - \\phi ^ T \\nabla _ y \\psi \\leq c _ \\Lambda \\int _ { E ( k ) } | \\nabla _ y \\psi | d y . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\gamma > \\sigma \\quad \\mbox { i f $ L $ i s e v e n t u a l l y i n c r e a s i n g } , \\quad \\gamma = \\sigma \\quad \\mbox { i f $ L $ i s e v e n t u a l l y d e c r e a s i n g } . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} f ^ N = i ( \\pm \\gamma ) \\varepsilon g \\ ; . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\mathrm { d i a m } ( \\Pi [ w ] ) = \\mathrm { d i a m } ( \\varphi _ { w } ( K ) ) \\le | r _ { w } | < 2 ^ { n m ( \\chi + \\delta ) } , \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} H ^ \\delta ( x ) = \\begin{cases} 0 ~ & \\mbox { f o r } ~ x \\le - \\delta , \\\\ 1 ~ & \\mbox { f o r } ~ x \\ge \\delta , \\end{cases} ( H ^ \\delta ) ' ( x ) \\ge 0 , ~ \\mbox { f o r } ~ ~ | x | \\le \\delta , \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\nabla _ { x x } \\mu _ f ( x , t ) = 1 2 \\ , { \\rm d i a g } ( a _ 1 x _ 1 ^ 2 , \\ldots , a _ n x _ n ^ 2 ) + 2 B + 4 \\ , t ^ 2 { \\rm d i a g } ( a ) \\ , . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\left \\{ \\mathfrak q : u ( \\mathfrak q ) = \\begin{array} { c } \\mathfrak p ' \\\\ \\updownarrow \\\\ \\mathfrak p '' \\end{array} , \\mathtt { O u t } ( \\mathfrak q ) = I , \\mbox { a n d } \\mathtt { I n } ( \\mathfrak q ) = J \\right \\} \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} f ( z , t ) = 1 + g _ { ( a ) } ( z , t ) + g _ { ( b ) } ( z , t ) + g _ { ( c ) } ( z , t ) , \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} | | g | | _ { L ^ s ( \\Omega ) } = | | I _ { \\alpha , \\Omega } ( g ^ { q ' - 1 } ) | | _ { L ^ s ( \\Omega ) } \\le C | | g ^ { q ' - 1 } | | _ { L ^ t ( \\Omega ) } \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\beta _ n ( 1 , q ) = \\frac { 1 } { ( q ^ 2 ; q ^ 2 ) _ n } , \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\{ \\mathfrak q _ 1 \\leftrightarrow \\mathfrak q _ 2 : u ( \\mathfrak q _ 1 ) = \\mathfrak p _ 1 , u ( \\mathfrak q _ 2 ) = \\mathfrak p _ 2 , \\mathtt { I n } ( \\mathfrak q _ 1 ) = I _ 1 , \\mathtt { I n } ( \\mathfrak q _ 2 ) = I _ 2 , \\mathtt { O u t } ( \\mathfrak q _ 1 ) = J _ 1 , \\mbox { a n d } \\mathtt { O u t } ( \\mathfrak q _ 2 ) = J _ 2 \\} \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} v _ { r ' } ( X ) = v _ { r ' } ( \\widetilde { Q } - Q + ( \\widetilde { P } - 1 ) ( \\widetilde { Q } - Q ) ) = v _ { r ' } ( \\widetilde { Q } - Q ) \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} H ^ { - 1 } ( A ^ \\bullet ( q = 0 ) ) \\rightarrow H ^ { - 1 } ( Q ^ \\bullet ) \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} \\Phi _ m : = \\begin{cases} \\exp \\left ( - \\gamma \\sum _ { k = 1 } ^ { m - 1 } \\phi _ \\delta ( \\norm { A _ k } _ { B ( \\mathcal { K } ) } ) \\right ) I \\ , , & m \\le N \\ , , \\\\ \\exp \\left ( - \\gamma \\sum _ { k = 1 } ^ { N - 1 } \\phi _ \\delta ( \\norm { A _ k } _ { B ( \\mathcal { K } ) } ) \\right ) I \\ , , & m > N \\ , , \\end{cases} \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\displaystyle f ( y ) = \\left \\{ \\begin{array} { r c } 0 , & \\mbox { i f } y \\leq 0 , \\\\ \\frac { C _ { \\alpha } } { y ^ { \\alpha } ( 1 + y ) } , & \\mbox { i f } y > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\mathcal { F } _ k ( f ) ( x ) = b _ k \\int _ 0 ^ \\infty \\widetilde { f } ( s ) \\mathcal { J } _ { \\gamma _ k + d / 2 - 1 } ( s | x | ) s ^ { 2 \\gamma _ k + d } d s . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\mathcal { E } = \\left \\{ ( a , b ) \\in [ 0 , + \\infty ) \\times \\mathbb { R } ~ ~ \\big | ~ b > 0 a = 0 \\right \\} \\ , . \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\mathrm { s u p p } \\ , ( \\omega ( 1 - \\theta ) ) = \\mathrm { s u p p } \\ , ( \\omega - \\theta ) \\subset \\mathrm { s u p p } ( \\omega ) \\setminus B ( x ' _ 0 ; 2 r ) \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} X = \\bigl ( \\{ n - 1 , n \\} X \\bigr ) \\epsilon _ { k + 2 } + \\{ n - 1 \\} ( X \\gamma _ { k } ) + \\{ n \\} ( X \\gamma _ k ) + \\{ n - 1 , n \\} ( X \\epsilon _ k ) . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} W ^ { M _ S , N _ S } = \\{ w \\in W ^ { \\mathrm { r e l } } : w ( M _ S \\cap B ) \\subset B , w ^ { - 1 } ( N _ S \\cap B ) \\subset B \\} \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} P _ { \\textrm { o u t , N } } ^ { \\textrm { I S A O C } } = & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F H N = 0 , N D N = 0 , F D F = 1 , F D N = 1 } \\right \\rbrace \\\\ & + \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace \\times \\left ( 1 - \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 1 , F D N = 1 } \\right \\rbrace \\right ) . \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} f ( x , y ) = f _ X ( x ) f _ Y ( y ) . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} w _ b = w _ { ( 1 2 3 ) } w _ a w _ { ( 1 2 3 ) } ^ { - 1 } w _ c = w _ { ( 1 2 3 ) } ^ { - 1 } w _ a w _ { ( 1 2 3 ) } . \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} A ( t ) : = \\sup _ { k > 0 } \\bigg \\{ \\sup _ { ( \\varphi , W ) \\in \\mathcal { A } _ t ( k ) } \\| \\varphi \\| _ { L ^ \\infty } \\bigg \\} < + \\infty . \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} | \\partial _ t u _ \\lambda - J _ \\lambda \\partial _ t u _ \\lambda | = \\lambda | \\alpha _ \\lambda ( \\partial _ t u _ \\lambda ) | \\ , , | \\partial _ t v _ \\lambda - J _ { \\Gamma \\lambda } \\partial _ t v _ \\lambda | = \\lambda | \\alpha _ { \\Gamma \\lambda } ( \\partial _ t v _ \\lambda ) | \\ , , \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} \\gamma & = \\{ \\{ i _ { 2 s p + 1 } , \\overline { i _ { 2 s p + 1 } } \\} , \\ldots , \\{ i _ { ( 2 s + t ) p } , \\overline { i _ { ( 2 s + t ) p } } \\} \\} \\\\ \\delta & = \\{ [ i _ { ( 2 s + t ) p + 1 } , \\overline { i _ { ( 2 s + t ) p + 1 } } ] , \\ldots , [ i _ { r p } , \\overline { i _ { r p } } ] \\} . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} M _ i ^ { ( n ) } ( t ) = \\sum _ { s = 1 } ^ { t } I _ i ^ { ( n ) } ( s ) . \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} z ^ { \\pm } _ { l _ { \\pm } } ( x , t ) = 0 \\ \\ { \\rm f o r \\ a l l } \\ \\ ( x , t ) \\in S _ { \\pm } \\cap ( \\R ^ N \\times ( - \\infty , 0 ] ) \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s & R _ { 2 n + 1 } ^ \\lambda ( x ) = ( 2 s + 1 ) \\ , b _ n ^ \\lambda \\ , x \\\\ & \\times \\sum _ { k = 0 } ^ { n } \\frac { ( - n ) _ { k } ( n + \\lambda + 1 ) _ { k } } { ( \\lambda + \\frac { 1 } { 2 } ) _ { k } k ! } \\ , A _ s ^ { k + \\frac { \\lambda } { 2 } + 1 } { } _ { 2 } F _ { 1 } \\Big ( s + k + \\frac { \\lambda } { 2 } + 1 , s + \\frac 3 2 ; \\frac { 3 } { 2 } ; - x ^ 2 \\Big ) , \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { 4 } ( \\overline { p } \\frac { l } { l _ L } ) ^ 2 - \\frac { 3 } { 4 } ( \\overline { q } \\frac { l } { l _ L } ) ^ 2 - \\frac { 1 } { 4 } \\right ] L = - \\overline { q } ^ 2 L - ( \\frac { 1 } { 4 } \\overline { p } ^ 2 - \\frac { 3 } { 4 } \\overline { q } ^ 2 ) \\frac { ( X _ 3 u ) ^ 2 } { l ^ 2 } + O ( L ^ { - \\frac { 1 } { 2 } } ) ~ ~ { \\rm a s } ~ ~ L \\rightarrow + \\infty , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} W _ j L = ( L - 2 i Z ) W _ j . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} M _ s ^ \\eta = \\mathcal { M } _ { s \\wedge \\tau } \\theta _ { s \\wedge \\tau } Q \\eta ( Y _ { s \\wedge \\tau } , B _ { s \\wedge \\tau } ) . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} ( 1 + z ) L _ n ^ { ( \\delta ) } ( z q ; q ) = a _ { n + 1 } L _ { n + 1 } ^ { ( \\delta ) } ( z ; q ) + a _ n L _ n ^ { ( \\delta ) } ( z ; q ) , \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\log \\kappa _ M ^ { N , h } ( y , x ) = \\sum \\limits _ { n = 1 } ^ { M } \\sqrt { \\lambda _ n ^ { N , h } } \\phi _ n ^ { N , h } ( x ) \\psi _ n ^ { N , h } ( y ) . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} \\lambda _ c = \\sup \\big \\{ \\lambda : \\lim _ { t \\rightarrow + \\infty } P _ \\lambda ( \\eta _ t ( O ) = 1 ) = 0 \\big \\} , \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} S + S ' = \\sum _ { w } \\mu \\hat \\oplus \\mu ' ( \\mathrm { c i r c u i t } ( w ) ) w , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} q ^ { m } \\sum _ { n = 1 } ^ { \\frac { m - 1 } { 2 } } \\frac { [ 3 n ] ( q ; q ^ 2 ) _ { n } ( q ^ { m } ; q ^ 2 ) _ { n } ^ 2 q ^ { - { n + 1 \\choose 2 } - \\frac { ( 2 n + 1 ) ( m - 1 ) } { 2 } } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } \\equiv 0 \\pmod { [ m ] \\Phi _ m ( q ) ^ 2 } . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\rho ( X ^ { ( 1 , 1 ) } ) \\ = \\ 1 \\ , \\rho ( X ^ { ( 2 , 2 ) } ) \\ \\le \\ 1 \\ , . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} - \\frac { \\Delta u } { 2 } + \\langle x , \\nabla u \\rangle = \\lambda u \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { n - 1 } { 2 } } [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 q ^ { - { k + 1 \\choose 2 } } } { ( q ; q ) _ k ^ 2 ( q ^ 2 ; q ^ 2 ) _ k } & \\equiv [ n ] q ^ { \\frac { 1 - n } { 2 } } \\pmod { [ n ] \\Phi _ n ( q ) ^ 2 } , \\\\ [ 5 p t ] \\sum _ { k = 0 } ^ { n - 1 } [ 3 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 q ^ { - { k + 1 \\choose 2 } } } { ( q ; q ) _ k ^ 2 ( q ^ 2 ; q ^ 2 ) _ k } & \\equiv [ n ] q ^ { \\frac { 1 - n } { 2 } } \\pmod { [ n ] \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} F _ n ( 0 ) = - { q ^ { n - 1 } ( 1 - a q ^ n ) ( 1 - a b q ^ { n + 1 } ) \\over ( 1 - a b q ^ { 2 n } ) ( 1 - a b q ^ { 2 n + 1 } ) } \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\Delta v _ 1 = \\Delta v _ 2 = 0 \\R ^ n . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} r _ { i j } \\propto 1 - \\left ( 1 - \\frac { w _ { i j } } { \\sum _ { j = 1 } ^ n w _ { i j } } \\right ) ^ { \\lambda _ i } , \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} ( a ; q ) _ k & = \\begin{cases} 1 ; ~ & k = 0 , \\\\ ( 1 - a ) ( 1 - a q ) ( 1 - a q ^ 2 ) \\cdots ( 1 - a q ^ { k - 1 } ) ; & k \\in \\mathbb { N } : = \\{ 1 , 2 , 3 , \\cdots \\} \\end{cases} \\\\ \\ ( a ; q ) _ { \\infty } & = \\prod _ { k = 0 } ^ { \\infty } ( 1 - a q ^ k ) . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} \\dfrac { 1 } { ( 1 - x ) ^ { \\nu + 1 } } = \\sum _ { k = \\nu } ^ { + \\infty } { k \\choose \\nu } x ^ { k - \\nu } = \\sum _ { j = 0 } ^ { + \\infty } { j + \\nu \\choose \\nu } x ^ { j } ( \\nu = 0 , 1 , 2 , . . . ) . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\| f ( t ) \\| _ 2 ^ 2 & = e ^ { - 2 \\Re \\lambda _ 1 t } \\| v _ 1 + \\alpha e ^ { - ( \\lambda _ 2 - \\lambda _ 1 ) t } v _ 2 \\| _ 2 ^ 2 \\\\ & = \\tfrac 1 { 1 - \\alpha ^ 2 } e ^ { - 2 \\Re \\lambda _ 1 t } \\Big ( 1 - 2 \\alpha ^ 2 e ^ { - \\Re ( \\lambda _ 2 - \\lambda _ 1 ) t } \\cos \\big ( \\Im ( \\lambda _ 2 - \\lambda _ 1 ) t \\big ) + \\alpha ^ 2 e ^ { - 2 \\Re ( \\lambda _ 2 - \\lambda _ 1 ) t } \\Big ) \\ , . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} Q _ { m _ { 1 } , \\ldots , m _ { d } } ^ { ( j ) } / / \\xi _ { j } / / \\left ( 1 , \\binom { j } { d } , \\ldots , j \\right ) ^ { T } , \\| \\xi _ { j } \\| = 1 . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\rho \\overline { \\mathcal { K } } _ \\zeta = \\rho \\mathcal { K } _ \\zeta ( \\overline { \\theta } ) , \\Omega \\times ( 0 , T ) . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} & ( \\frac { Q } { p } + \\frac { \\alpha - Q } 2 ) \\int _ { \\Omega } f ^ { p } ( \\xi ) d \\xi \\\\ = & - \\frac { \\lambda } 2 \\int _ { \\Omega } \\int _ { \\Omega } \\frac { f ^ { p - 1 } ( \\xi ) f ^ { p - 1 } ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta d \\xi + \\frac 1 { p } \\int _ { \\partial \\Omega } ( E \\cdot \\nu ) f ^ { p } ( \\xi ) d \\sigma , \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} w '' ( t ) & = - A U _ { m + 1 } ' ( t ) + ( V _ m ( t ) - w ( t ) ) ' \\\\ & = - A ( w ( t ) - V _ m ( t ) ) + V _ m ' ( t ) - w ' ( t ) \\\\ & = - A w ( t ) - w ' ( t ) + ( A V _ m ( t ) + V _ m ' ( t ) ) \\\\ & = - A w ( t ) - w ' ( t ) - V _ { m - 1 } ' ( t ) , \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{gather*} \\boldsymbol { \\Upsilon } _ { \\frac { n } { 2 } } : = \\mathrm { d i a g } ( \\lambda _ { 3 } , \\lambda _ { 5 } , \\ldots , \\lambda _ { n + 1 } ) \\\\ \\boldsymbol { \\Delta } _ { \\frac { n } { 2 } } : = \\mathrm { d i a g } ( \\lambda _ { 2 } , \\lambda _ { 4 } , \\ldots , \\lambda _ { n } ) \\end{gather*}"}
-{"id": "3705.png", "formula": "\\begin{align*} S : = \\{ ( \\xi , \\eta ) \\in \\R _ + \\times \\R _ + \\mid \\lim _ { \\tau \\rightarrow \\infty } ( u ( \\tau ; \\xi , \\eta ) , v ( \\tau ; \\xi , \\eta ) ) = ( u ^ * , v ^ * ) \\} , \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} { p } ( t , y ) & : = - [ A ^ { ( 4 ) } ( t , y ) \\nabla ( \\mathrm { d e t } D \\Phi ^ { - 1 } ( t , y ) ) + \\partial _ t ( { q } ( t , y ) \\mathrm { d e t } D \\Phi ^ { - 1 } ( t , y ) ) ] \\mathrm { d e t } D \\Phi ( t , \\Phi ^ { - 1 } ( t , y ) ) \\ , , \\\\ { q } ( t , y ) & : = - \\dot { \\Phi } ( t , \\Phi ^ { - 1 } ( t , y ) ) \\ , , \\\\ { g } ( t , y ) & : = f ( t , \\Phi ^ { - 1 } ( t , y ) ) \\ , . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\frac { ( \\alpha t ) ^ k } { k ! } k ^ \\beta ( \\ln k ) ^ { \\mu _ 1 } \\cdots ( \\ln _ m k ) ^ { \\mu _ m } \\asymp ( \\alpha t ) ^ \\beta ( \\ln ( \\alpha t ) ) ^ { \\mu _ 1 } \\cdots ( \\ln _ m ( \\alpha t ) ) ^ { \\mu _ m } \\quad \\mbox { a s $ t \\to \\infty $ } , \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} R ( s , N ) = \\frac { 1 } { N } \\left \\{ 1 \\leq i \\neq j \\leq N : ~ \\| x _ i - x _ j \\| \\leq s / N \\right \\} , s \\geq 0 , \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} T x = x + \\omega . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} & \\lim _ { h \\to 0 } \\mathbb { E } \\left [ \\left ( \\frac { M ( x + h ) - M ( x ) } { h } - \\frac { d } { d x } M ( x ) \\right ) ^ 2 \\right ] \\\\ & = \\lim _ { h \\to 0 } \\mathbb { E } \\left [ \\int \\left ( \\frac { \\mu ( s , x + h ) - \\mu ( s , x ) } { h } - \\nu ( s , x ) \\right ) ^ 2 d [ B ] _ s \\right ] = 0 . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} L ( { \\cal T } ( i ) ) \\leq L ( { \\cal T } ( i - 1 ) ) + 2 ( s _ n + 8 r _ n ) \\leq \\sum _ { l = 1 } ^ { i } T _ l + 2 ( i - 1 ) ( s _ n + 8 r _ n ) . \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} A \\vec { D } = & - \\vec { B } \\times \\vec { C } \\\\ A > & \\max \\{ | \\vec { B } | , | \\vec { C } | , | \\vec { D } | \\} \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} K = \\Lambda + \\nabla _ y \\widetilde { w } \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} \\frac { \\partial ^ k f } { \\partial y ^ { j _ 1 } \\dots \\partial y ^ { j _ k } } ( g . z ) = 0 \\mbox { f o r } k \\le n - 1 . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\displaystyle P \\left ( \\zeta _ t \\neq \\emptyset , \\forall t > 0 ~ \\bigg \\vert ~ \\left \\{ \\lim _ { n \\rightarrow \\infty } X _ n = \\infty \\right \\} ^ { c } \\right ) = 0 . \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\mathfrak { g } ^ { ( l ) } \\triangleq \\bigoplus _ { k = 1 } ^ { l } { \\cal L } _ { k } \\subseteq T ^ { ( l ) } . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} Y \\equiv \\max \\{ Y _ i ~ ; ~ i = 1 , \\ldots , M \\} . \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\omega _ { 2 } ( \\sigma ) = : \\max \\Bigg \\{ C _ { I I } \\sigma \\Big ( \\frac { 1 } { | \\Omega \\cap B ( \\sigma ) | } \\int _ { \\Omega \\cap B ( \\sigma ) } | g | ^ { q } \\Big ) ^ { \\frac { 1 } { q } } , ~ ~ ~ ~ \\tilde { \\omega } _ { 2 } ( \\sigma ) \\Bigg \\} . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} \\lambda \\int _ \\Omega | y _ \\lambda | ^ 2 + \\int _ \\Omega | \\nabla y _ \\lambda | ^ 2 = \\int _ \\Omega y y _ \\lambda \\leq \\frac { 1 } { 4 \\delta } \\norm { y } _ { V ^ * } ^ 2 + \\delta \\norm { y _ \\lambda } _ V ^ 2 \\ , , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} z ( x , t ) & = e ^ { A t } z _ 0 ( x ) + \\int _ 0 ^ { t } e ^ { A ( t - s ) } g ( x , s ) \\ , d s . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\sigma ^ { j } \\nu [ \\omega | _ { n m } ] = \\nu ( \\sigma ^ { - j } [ \\omega | _ { m - j } ] ) \\cdot \\left ( \\prod _ { k = 1 } ^ { n - 1 } \\nu [ ( \\sigma ^ { k m - j } \\omega ) | _ { m } ] \\right ) \\cdot \\nu [ ( \\sigma ^ { n m - j } \\omega ) | _ { j } ] \\ : . \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} u _ t + f ( x , u ) _ x = \\varepsilon u _ { x x } . \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c c } \\frac { \\partial u ( x , t ) } { \\partial t } = & f \\left ( u , v \\right ) + \\varepsilon \\Delta u ( x , t ) \\\\ & \\\\ \\frac { \\partial v ( x , t ) } { \\partial t } = & g \\left ( u , v \\right ) + \\varepsilon d \\Delta v ( x , t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\sum \\limits _ { \\lambda _ 1 \\vdash n } m ( \\lambda _ 1 ) \\left [ \\prod \\limits _ { k = 1 } ^ { \\ell ( \\lambda _ 1 ) } \\chi _ { b _ k } ( t ) \\right ] P _ { \\ell ( \\lambda _ 1 ) } ( t ) . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} p ( 0 ) = y { \\rm \\ a n d \\ } p ( 1 ) \\in \\{ a _ 1 , \\ldots , a _ { m _ 0 } \\} . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\varphi _ 0 + R \\varphi _ 0 = - \\frac { n - 1 } { n } \\tau ^ { 2 a _ 0 } \\varphi ^ { N - 1 } _ 0 + \\left ( | \\sigma + L W _ 0 | ^ { 2 } + k _ 0 ^ { 2 } \\right ) \\varphi ^ { - N - 1 } _ 0 . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\overline { \\{ v \\in C _ { } ^ \\infty ( \\overline { \\Omega } ) ^ 2 : _ H \\overline { v } = 0 \\} } ^ { \\norm { \\cdot } _ { L ^ p ( \\Omega ) } } . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} e ^ { s B _ i ( t ) } f ( \\xi ) = f ( z _ i ( s , t , \\xi ) ) , \\ s \\in \\mathbb { R } , \\ t \\geq 0 , \\ \\xi \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\exp \\left \\{ - \\sum _ { i = j } ^ { n } \\frac 1 { \\left ( i + e _ { [ k ] } ^ 1 \\right ) \\ln \\left ( i + e _ { [ k ] } ^ 1 \\right ) \\dots \\ln _ k \\left ( i + e _ { [ k ] } ^ 1 \\right ) } \\right \\} \\le \\frac { \\ln _ { k } ( j + e _ { [ k ] } ^ 1 ) } { \\ln _ { k } ( n + 1 + e _ { [ k ] } ^ 1 ) } , \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} { \\rm d i v } ( | x | ^ { \\sigma } | \\nabla u | ^ { p - 2 } \\nabla u ) = | x | ^ { - \\tau } u ^ q | \\nabla u | ^ m \\mathrm { i n } \\ \\Omega ^ * : = \\Omega \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} \\sup _ n \\ | | n ^ { - 1 / 2 } \\sum _ { i = 1 } ^ n X _ i | | _ { s , \\Omega } \\le \\inf _ { v > s } \\left \\{ \\ Z [ \\alpha ] ( s , v ) \\cdot | | X | | _ { v , \\Omega } \\ \\right \\} , \\ v > s . \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} \\langle k _ 1 , b _ 1 , f _ 1 \\rangle \\in ^ * \\langle k _ 2 , b _ 2 , f _ 2 \\rangle & \\iff b _ 1 b _ 2 \\in h \\big ( \\big \\{ a _ 1 a _ 2 : f _ 1 ( a _ 1 ) \\in f _ 2 ( a _ 2 ) \\big \\} \\big ) , \\\\ \\langle k _ 1 , b _ 1 , f _ 1 \\rangle = ^ * \\langle k _ 2 , b _ 2 , f _ 2 \\rangle & \\iff b _ 1 b _ 2 \\in h \\big ( \\big \\{ a _ 1 a _ 2 : f _ 1 ( a _ 1 ) = f _ 2 ( a _ 2 ) \\big \\} \\big ) , \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\phi ( z , t ) , \\ \\widetilde { \\phi } ( z , t ) \\ \\ ( z = n \\cdot x - c t ) \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\sum _ { k \\ne j } ( C _ { k k } - C _ { j j } ) X _ { k k } = - 1 . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} m ( u ) = x \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} Q _ { 2 } \\leq \\sum _ { k = 1 } ^ { m } k ^ { 2 H + 1 } | I _ { k } | ^ { 2 ( 1 - H ) } . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\Big | \\mu _ { \\beta , 0 } ^ + \\big ( \\eta ( 0 , 0 ) = + 1 \\big ) - \\mu _ { \\beta , 0 } ^ { [ - n , n ] ^ 2 , + } \\big ( \\eta ( 0 , 0 ) = + 1 \\big ) \\Big | \\leq C e ^ { - \\hat { \\theta } n } . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\begin{bmatrix} - d _ { n - 1 } ^ { \\mathbb { H } } & 0 \\\\ - f _ { n - 1 } & d _ n ^ { \\mathbb { G } } \\end{bmatrix} : { \\begin{array} { l c l } \\mathbb { H } _ { n - 1 } & \\longrightarrow & \\mathbb { H } _ { n - 2 } \\\\ \\bigoplus & \\searrow & \\bigoplus \\\\ \\mathbb { G } _ n & \\longrightarrow & \\mathbb { G } _ { n - 1 } \\end{array} } \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { y } _ { \\mathcal { Q } _ { k _ { i } } } & = \\mathbf { y } _ { \\mathcal { Q } } - \\mathbf { A } \\left ( \\sum _ { j = 1 } ^ { i - 1 } ( \\mathbf { H } ) _ { k _ { j } } \\tilde { x } _ { k _ { j } } \\right ) \\\\ & = \\mathbf { A } \\left ( \\mathbf { H x } - \\sum _ { j = 1 } ^ { i - 1 } ( \\mathbf { H } ) _ { k _ { j } } \\tilde { x } _ { k _ { j } } \\right ) + \\mathbf { A n } + \\mathbf { n } _ q \\end{aligned} \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} f _ { 1 } = \\frac { f _ { 1 } ^ { \\prime } } { c _ { 1 } + c _ { 2 } \\left ( f _ { 1 } ^ { \\prime } \\right ) ^ { 2 } } \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\sigma ( 1 ) = 4 & & \\sigma ( 2 ) = 2 & & \\delta \\geq 6 , \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\bar A ( t ) = \\sup _ { s > 0 } \\{ s t - A ( s ) \\} , t > 0 . \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} \\mathcal { L } ( \\phi ) : = \\varepsilon ^ 2 \\phi _ { x x } + f ( \\phi ) = 0 , \\phi \\bigl ( - \\tfrac 1 2 \\ell \\bigr ) = \\phi \\bigl ( \\tfrac 1 2 \\ell \\bigr ) = 0 , \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} w ( \\gamma , T ) = ( \\prod _ { j \\in [ k ] } t _ { i _ j i _ { j + 1 } } ) ^ { \\frac { 1 } { k } } , i _ { k + 1 } \\equiv i _ 1 . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\frac { \\partial ( q - t q ' ( u ) ) } { \\partial y _ 0 } = \\dots = \\frac { \\partial ( q - t q ' ( u ) ) } { \\partial y _ 5 } = 0 , \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} a d _ x ^ * \\kappa = 0 \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} | J _ 1 - J ^ { ( P ) } _ 1 | = \\sum _ { \\frac { \\eta _ 1 n } { 2 N } \\leq k _ 1 , k _ 2 \\leq \\frac { 2 \\eta _ 2 n } { N } } | B _ { l _ 1 , l _ 2 } - P o i ( k _ 1 ; n p _ { l _ 1 } ) P o i ( k _ 2 ; n p _ { l _ 2 } ) | \\Delta ( k _ 1 , q _ { l _ 1 } ) \\Delta ( k _ 2 , q _ { l _ 2 } ) \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{gather*} v _ r ^ { \\alpha } ( X _ l ) \\ge v _ r ^ { \\alpha } ( R - P _ l Q _ l ) - r i - v ^ { \\alpha } ( R _ i ) = c _ l ^ { \\alpha } , \\\\ v _ r ^ { \\alpha } ( Y _ l ) \\ge v _ r ^ { \\alpha } ( R - P _ l Q _ l ) = c _ l ^ { \\alpha } + r i + v ^ { \\alpha } ( R _ i ) . \\end{gather*}"}
-{"id": "5273.png", "formula": "\\begin{align*} Z = \\hat { Z } ^ \\ast . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\dim _ H B ( \\{ s _ n \\} , \\{ t _ n \\} , N ) = \\dim _ H B ( \\{ s _ { n } \\} , \\{ t _ { n } \\} , 1 ) . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} V _ t = \\int _ 0 ^ \\infty U _ t ( x ) \\mu ( d x ) . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\| \\hat P _ j - P _ j \\| _ 2 = \\sqrt { 2 } \\| ( I - P _ j ) \\hat P _ j \\| _ 2 \\leq \\sqrt { 2 } g _ j ^ { - 1 / 2 } \\Vert | R _ j | ^ { - 1 / 2 } \\hat P _ j \\Vert _ 2 . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} A = \\frac { a ^ 2 + b ^ 2 } { 2 a ^ 2 b ^ 2 } \\ , \\ \\ B = \\frac { a ^ 2 - b ^ 2 } { 2 a ^ 2 b ^ 2 } = \\frac { c ^ 2 } { 2 a ^ 2 b ^ 2 } \\ . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} H ( \\rho ) : = & \\sum _ { \\{ \\alpha , \\beta \\} \\in \\mathcal { S } ( N ) } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha } ( \\rho ) + \\ell _ { \\beta } ( \\rho ) ) } + ( - 1 ) ^ { \\alpha \\cdot \\beta } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} V _ 0 ^ { ( 1 ) } ( t ) & = e ^ { - t A } v _ 0 , \\\\ V _ m ^ { ( 1 ) } ( t ) & = ( - 1 ) ^ { m } \\sum _ { j = 1 } ^ m \\begin{pmatrix} m - 1 \\\\ j - 1 \\end{pmatrix} \\frac { ( - t A ) ^ j } { j ! } e ^ { - t A } v _ 0 , m \\in \\N \\\\ V _ 0 ^ { ( 2 ) } ( t ) & = 0 , \\\\ V _ m ^ { ( 2 ) } ( t ) & = ( - 1 ) ^ { m } \\sum _ { k = 0 } ^ { m - 1 } \\begin{pmatrix} m - 1 \\\\ k \\end{pmatrix} \\frac { ( - t A ) ^ k } { k ! } e ^ { - t A } u _ 1 , m \\in \\N \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} s = \\# \\mathcal { F } _ 0 ( r ) = \\begin{cases} 5 1 5 & r = 4 , \\\\ 3 5 7 & r \\ge 6 , , \\end{cases} \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} P \\left ( d _ \\pi f \\ast \\chi _ \\pi \\right ) & = \\varphi \\bigl ( ( f \\ast d _ \\pi \\chi _ \\pi ) ^ { * n } , d _ \\pi \\chi _ \\pi , \\dotsc , d _ \\pi \\chi _ \\pi \\bigr ) \\\\ & = \\varphi \\bigl ( f ^ { * n } \\ast d _ \\pi \\chi _ \\pi , d _ \\pi \\chi _ \\pi , \\dotsc , d _ \\pi \\chi _ \\pi \\bigr ) . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} J _ i ( f ; v ^ 1 , v ^ 2 ) : = \\dfrac { \\alpha _ i } { 2 } \\iint _ { \\mathcal { O } _ { i , d } \\times ( 0 , T ) } | y - y _ { i , d } | ^ 2 d x d t + \\dfrac { \\mu _ i } { 2 } \\iint _ { \\mathcal { O } _ i \\times ( 0 , T ) } | v ^ i | ^ 2 d x d t , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} n ! { \\| h _ n ( . , t , x ) \\| } _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 \\leq \\frac { 1 } { n ! } C _ 1 e ^ { - ( 2 - \\delta ) \\mu _ 1 t } \\int _ { [ 0 , t ] ^ { 2 n } } \\prod _ { i = 1 } ^ n \\gamma ( t _ i - s _ i ) \\prod _ { i = 1 } ^ n \\Big ( t _ { \\tau ( i + 1 ) } - t _ { \\tau ( i ) } \\Big ) ^ { - \\beta / \\alpha } d { \\bf t } d { \\bf s } . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} L ( { \\cal C } _ n ) \\geq \\sum _ { l = 1 } ^ { N } T _ l , \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} \\gamma \\left ( \\textnormal { c o n f l u e n c e } \\left ( \\widetilde { J ^ { K \\textnormal { t h } } } \\right ) ( z , Q ) \\right ) = \\widetilde { J ^ \\textnormal { c o h } } ( z , Q ) \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\frac { d } { d x } \\sigma ^ 2 ( x ) & \\le \\frac { d } { d x } g _ { 7 + } ( x ) + \\frac 1 { 2 ^ 6 } = \\frac { 1 7 4 5 9 4 7 } { 1 3 9 9 6 8 } - 3 0 x + \\frac 1 { 2 ^ 6 } \\\\ & \\le \\frac { 1 7 4 5 9 4 7 } { 1 3 9 9 6 8 } - 3 0 c + \\frac 1 { 2 ^ 6 } = - \\frac { 6 2 4 9 7 } { 9 3 0 7 8 7 2 } < 0 \\quad \\hbox { a . e . } \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} Q _ { 1 , k } \\leq C _ { H } | I _ k | \\left ( \\sum _ { j = 1 } ^ k | I _ { k } | \\right ) ^ { 1 - 2 H } \\| \\dot { \\bar { h } } _ { k } \\| _ { \\infty ; [ 0 , 1 ] } ^ { 2 } \\leq C _ { H , l _ 0 } \\frac { | I _ k | } { \\left ( \\sum _ { j = 1 } ^ m | I _ { k } | \\right ) ^ { 2 H - 1 } } , \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} h ' ( s ( y ) ) = - y . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} g v _ { i , j } & = \\sum _ { 1 \\le k , l \\le \\chi ( 1 ) } \\delta _ { i , k } A ( g ) _ { l , j } v _ { k , l } \\\\ & = \\sum _ { 1 \\le l \\le \\chi ( 1 ) } A ( g ) _ { l , j } v _ { i , l } , \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} | \\Lambda _ { P _ 1 , \\dots , P _ { m _ 1 } } ^ { Q _ 1 , \\dots , Q _ { m _ 2 } } ( F _ c ; \\Psi ) | & = q ^ { \\delta _ 3 } | \\Lambda _ { P _ 1 , \\dots , P _ { m _ 1 } } ^ { Q _ 1 , \\dots , Q _ { m _ 2 } } ( q ^ { - \\delta _ 3 } f _ c , f _ 1 , \\dots , f _ { m _ 1 } ; \\Psi ) | \\\\ & \\leq q ^ { \\delta _ 3 } ( b _ 1 \\| q ^ { - \\delta _ 3 } f _ c \\| _ { U ^ s } ^ { b _ 2 } + b _ 3 ) \\\\ & \\leq q ^ { ( 1 - b _ 2 ) \\delta _ 3 - b _ 2 \\delta _ 4 } b _ 1 + q ^ { \\delta _ 3 } b _ 3 , \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 + 2 } } \\ge c _ 0 \\big ( \\| { \\mathbb { T } } \\| ^ \\ell _ { \\dot { B } _ { 2 , 1 } ^ { \\frac n 2 } } \\big ) ^ { \\frac { 1 } { 1 - \\theta _ { 1 } } } , \\theta _ 1 = \\frac { 4 } { n - 2 \\sigma + 2 } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} f ( x ) = x - e ^ { - 3 } x ^ 3 . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} D _ b ^ { k ' } ( I _ b ^ k f ) ( x ) = f ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } W ^ n _ g ( \\sqrt { p } \\ , b , p T ) = W _ \\star \\left ( \\frac { b } { \\gamma _ g ^ n } , T \\right ) \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{gather*} \\{ H , \\Phi _ a \\} = \\lambda ^ b _ { a } \\Phi _ b , \\end{gather*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { W } ^ { \\gamma , q } } : = \\| | \\nabla | ^ \\gamma u \\| _ { L ^ q } < \\infty . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\rho _ K ( \\xi ) = \\max \\{ a \\ge 0 : a \\xi \\in K \\} , \\xi \\in S ^ { n - 1 } . \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } z _ j = z _ * = ( 0 , \\cdots , 0 , z _ { * , N } ) , \\ \\ \\lim _ { j \\rightarrow \\infty } \\tilde { z } _ j = \\tilde { z } _ * = ( \\tilde { z } _ { * , 1 } , \\cdots , \\tilde { z } _ { * , N } ) . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n } ^ { \\infty } \\sum _ { m } ^ { \\infty } b ^ 1 _ \\nu ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n q ^ n , \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} U ' = U ' _ { ( r + 1 ) } = U ' _ { ( r ) } U ' _ { r + 1 } , \\ \\ \\ \\ \\chi ' = \\chi ' _ { ( r + 1 ) } = \\chi ' _ { ( r ) } \\chi ' _ { r + 1 } . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} H _ 3 = ( 1 , \\dots , 1 \\ , | \\ , 1 , \\dots , 1 , - 1 ) . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\mathcal { F } ( t _ 1 , t _ 2 , t _ 3 ) = \\frac { 1 } { 2 } ( t _ 0 t _ 1 ^ 2 + t _ 0 ^ 2 t _ 2 ) + \\sum _ { d = 1 } ^ \\infty e ^ { d t _ 1 } N _ d \\frac { t _ 2 ^ { 3 d - 1 } } { ( 3 d - 1 ) ! } \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} \\omega _ { 3 } = \\frac { - c _ { 4 } \\omega _ { 4 } } { \\left ( \\frac { c _ { 5 } } { c _ { 6 } } \\right ) ^ { \\frac { 1 } { 4 } } \\omega _ { 4 } ^ { \\frac { 3 } { 2 c _ { 1 } } } - a } . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\langle [ u , v ] + [ v , u ] , w \\rangle = \\pi ( w ) \\langle u , v \\rangle \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\sigma ( x _ 0 : \\ldots : x _ 5 ) = ( x _ 0 : \\ldots : x _ 4 : \\xi x _ 5 ) \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} U _ 0 ( x ' , 0 , y ) = 0 . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} B \\leftrightarrow \\left \\{ \\sum _ { i = 1 } ^ { j } b _ { i , j } e _ i : \\ j \\in [ d ] \\right \\} . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\begin{aligned} Y & = S G _ y G _ d D _ { y z } + S G _ y ( 1 - C _ b ) \\hat d _ x D _ x - S G _ y C _ b \\hat d _ b D _ b \\\\ S & = \\frac { 1 } { 1 + G _ y C _ b \\sigma _ y \\bar { y } + G _ y G _ h C _ h } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} p _ { \\geq s } ( n ) = \\Theta _ s ( n ^ { - ( s - 1 ) / 2 } ) p ( n ) , \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} T ^ N = \\{ ( z _ 1 , \\ldots , z _ N ) \\in \\C ^ N \\mid | z _ i | = 1 \\ ; \\ ; i = 1 , \\ldots , N \\} \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\epsilon _ { 1 , n } & : = C ( 1 + \\| ( y ^ n , p ^ { 1 , n } , p ^ { 2 , n } , f ^ n ) \\| _ Y ^ 2 + \\| ( y , p ^ 1 , p ^ 2 , f ) \\| _ Y ^ 2 ) \\| ( y ^ n , p ^ { 1 , n } , p ^ { 2 , n } , f ^ n ) - ( y , p ^ 1 , p ^ 2 , f ) \\| _ Y ^ 2 \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} H ( s ) : = F ( s ) / ( s F ' ( s ) ) \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\lambda _ j = 0 . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} V = f _ 1 J X _ 1 + f _ 2 J X _ 2 . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\{ z \\mid ( w _ 1 ( z , 0 ) , w _ 2 ( z , 0 ) ) \\in S \\} = \\{ z \\mid z _ N = \\theta _ 0 \\} \\ , ( = : H ) . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\delta _ i ^ \\varepsilon \\delta _ { j } ^ \\eta = \\delta _ { j - 1 } ^ \\eta \\delta _ i ^ \\varepsilon \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} u _ { p q } ( r , \\varphi ) = \\cos ( p \\varphi ) J _ p ( \\lambda _ { p q } r ) v _ { p q } ( r , \\varphi ) = \\sin ( p \\varphi ) J _ p ( \\lambda _ { p q } r ) . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\hat { \\eta } _ k ^ ( t ) = \\frac { 1 } { | \\mathcal { N } _ k \\cup \\{ k \\} | } \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } \\eta _ l ( t ) , \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\iint _ Q \\rho _ 3 ^ 2 | y _ { x x } | ^ 2 d x d t & \\leq C \\left ( \\| y _ 0 \\| _ { H _ 0 ^ 1 ( I ) } ^ 2 + \\iint _ Q \\rho ^ 2 | G | ^ 2 d x d t + \\iint _ { \\mathcal { O } \\times ( 0 , T ) } \\rho _ 1 ^ 2 | f | ^ 2 d x d t \\right . \\\\ & + \\left . \\iint _ Q \\rho _ 0 ^ 2 | y | ^ 2 d x d t + \\sum _ { i = 1 } ^ 2 \\iint _ Q \\rho _ 0 ^ 2 | p ^ i | ^ 2 d x d t \\right ) \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathbf E ( N , p , q ; I , J ) \\\\ \\updownarrow \\\\ \\mathbf E ( N , p , r ; J ' , K ) \\end{array} = \\delta _ { J , J ' } \\mathbf E ( N , p , r ; I , K ) , \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = 0 } E _ p ( u _ t ) = \\int _ { \\Omega } \\langle | \\nabla u | ^ { p - 2 } \\nabla u , \\nabla \\psi \\rangle d \\mu = 0 . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} R ( n , d , \\epsilon ) = \\mathbb { R } _ { X } ( d ) + Q ^ { - 1 } ( \\epsilon ) \\sqrt { \\frac { \\mathbb { V } ( d ) } { n } } + O \\left ( \\frac { \\log n } { n } \\right ) , \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} M = h _ { m + 1 } ^ q f _ m ^ { r _ - } f _ { m + 1 } ^ { r _ + } . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} d ( x , y ) = d ( x , x + F _ { l } ( w , x ) ) \\leq C \\| w \\| _ { \\textsc { C C } } . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\binom { a } { t } \\ , \\geq \\ , \\left ( \\frac { a } { t } \\right ) ^ t \\ , \\geq \\ , \\left ( \\frac { \\sqrt { d } + \\alpha } { 2 \\left ( \\tau + 1 / \\sqrt { d } \\right ) } \\right ) ^ { \\tau \\sqrt { d } } \\ , = : \\ , \\gamma _ { \\alpha \\tau } ( d ) \\ , \\geq \\ , 1 \\ , . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} & p ^ + \\succeq W ^ { + , \\sigma ^ * - \\eta } \\succeq p ^ + - \\frac { \\delta } { 2 } \\varphi ^ + \\ \\ { \\rm o n } \\ \\ S _ + , \\\\ & p ^ - \\preceq U ^ { - } \\preceq p ^ - + \\frac { \\delta } { 2 } \\varphi ^ - \\ \\ { \\rm o n } \\ \\ S _ - \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} | f | _ \\infty = \\sup \\{ | f _ 1 | _ \\infty , \\ | f _ 2 | _ \\infty , \\cdots , | f _ n | _ \\infty \\} . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} C : = \\sum _ { i = 1 } ^ n \\nabla ^ 2 f _ { i i } \\ , . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} h ( x ) & \\le g _ { 6 - } ( x ) \\Bigr | _ { x = \\frac { 8 5 7 } { 2 0 5 9 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 5 1 6 9 5 9 6 0 1 1 4 7 6 7 3 0 5 3 2 1 7 7 3 8 6 3 2 4 9 1 5 8 9 3 1 9 0 9 4 9 } { 1 8 0 4 4 2 3 5 8 7 6 7 0 4 3 4 6 4 4 3 3 5 5 4 1 0 0 5 6 9 9 8 8 7 6 1 0 2 1 7 1 3 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} \\bigoplus \\limits _ { s \\in S } V _ s \\cong \\bigoplus \\limits ^ k _ { i = 1 } \\mathrm { I n d } ^ { \\Lambda } _ { \\mathrm { s t a b } ( s _ i ) } V _ { s _ i } , \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} C = \\# ( e _ { \\geq m + 2 } ) + \\# ( h _ { \\geq m + 2 } ) + \\# ( f _ { \\geq m + 2 } ) . \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h ^ { 1 ^ * } _ { 1 1 k } + \\bar h ^ { 1 ^ * } _ { 2 2 k } = 0 , \\\\ & \\bar h ^ { 2 ^ * } _ { 1 1 k } + \\bar h ^ { 2 ^ * } _ { 2 2 k } = 0 , \\\\ & ( \\bar \\lambda _ 1 - 3 \\bar \\lambda _ 2 ) \\bar h ^ { 1 ^ * } _ { 1 1 k } = 0 \\end{cases} \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} \\mathbf f _ { \\mathbf D _ u } ^ i ( x ) = \\underbrace { ( f _ { k } * \\cdots * f _ { l } ) } _ { | \\mathbf D _ u | } ( x ) , \\ , \\mathbf g _ { \\mathbf D _ u } ^ i ( x ) = \\underbrace { ( g _ { k } * \\cdots * g _ { l } ) } _ { | \\mathbf D _ u | } ( x ) , \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} \\alpha ( 1 , x ) = \\alpha ( x , 1 ) = \\epsilon ( x ) . \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} \\underset { \\bar { { \\bf w } } \\in \\mathcal { W } } { \\min } ~ \\underset { \\bar { \\bf x } \\in \\mathcal { X } } { \\max } ~ \\phi ^ { ( n ) } ( \\bar { \\bf w } , \\bar { \\bf x } ) = \\underset { \\bar { { \\bf x } } \\in \\mathcal { X } } { \\min } ~ \\underset { \\bar { \\bf w } \\in \\mathcal { W } } { \\max } ~ \\phi ^ { ( n ) } ( \\bar { \\bf w } , \\bar { \\bf x } ) \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\textstyle ( 1 - \\delta _ { a i } ) \\sum _ j k _ { j i } P _ { a k j i } - ( 1 - \\delta _ { a k } ) \\sum _ j k _ { j k } P _ { a i j k } = 0 \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\int _ { B _ 0 } \\vert u \\vert ^ { p ^ * } \\ , d \\mu \\leq \\sum _ { k = - \\infty } ^ { \\infty } a _ k ^ { p ^ * } \\mu ( B _ 0 \\cap ( E _ k \\setminus E _ { k - 1 } ) ) . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} T \\left ( \\mathcal { F } ^ { \\sharp } _ { [ N _ 1 , h _ 2 ] } \\right ) ~ = ~ \\left \\{ \\varphi _ { f ^ { \\sharp } } ~ \\Big | ~ f ^ { \\sharp } \\in \\mathcal { F } ^ { \\sharp } _ { [ N _ 1 , h _ 2 ] } \\right \\} ~ \\subseteq ~ \\mathcal { I } _ { [ N _ 1 , h _ 2 ] } ~ . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } ( - 1 ) ^ k \\frac { [ n + 3 k ] ( - q ^ { n + 1 } ; q ) _ { k - 1 } } { [ 2 k ] } { 2 k \\brack k } \\equiv ( 1 + q ^ n ) \\left ( q ^ { n \\choose 2 } - ( - q ; q ) _ { n - 1 } \\right ) \\pmod { \\Phi _ n ( q ) ^ 2 } , \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} \\eta _ { m \\rho } ^ \\zeta = ( \\rho , \\rho u ) = \\vartheta \\rho ^ { \\vartheta - 1 } \\int _ { - 1 } ^ 1 \\zeta '' ( u + \\rho ^ \\vartheta s ) s [ 1 - s ^ 2 ] _ + ^ \\Lambda d s . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{gather*} \\mathbf { q } _ { 1 } = \\alpha ( \\mu _ { k } \\mathbf { I } _ { \\frac { n } { 2 } } - \\boldsymbol { \\Upsilon } _ { \\frac { n } { 2 } } ) ^ { - 1 } \\mathbf { u } , \\\\ \\mathbf { q } _ { 2 } = \\mathbf { 0 } \\end{gather*}"}
-{"id": "5786.png", "formula": "\\begin{align*} \\sum \\limits _ { b \\in \\mathbf { B } ( G , \\mu ) } \\mathrm { M a n t } _ { G , b , \\mu } ( \\mathrm { R e d } _ b ( \\pi ) ) = [ \\pi ] [ r _ { - \\mu } \\circ \\mathrm { L L } ( \\pi ) | _ { W _ { E _ { \\{ \\mu \\} _ G } } } ] , \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} U _ { 2 k , 2 a - 1 } ( x ; q ) = U _ { 2 k , 2 a } ( x ; q ) - U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) , \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in \\Gamma } \\sharp \\{ \\gamma ^ { \\prime } \\in \\Gamma : \\gamma ^ { \\prime } V \\cap \\gamma V \\not = \\emptyset \\} \\le \\sup _ { x \\in G } \\sharp \\{ \\gamma ^ { \\prime } \\in \\Gamma : \\gamma ^ { \\prime } \\in x V ^ { 2 } \\} = m < \\infty ~ . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} x _ { N + k } < x _ N e ^ { - 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } x _ { N + i } ^ 2 } \\le x _ N e ^ { - 2 x _ N ^ 2 \\sum _ { i = 0 } ^ { k - 1 } h _ { N + i } } < x _ N e ^ { - 2 x _ N ^ 2 S } < 0 . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} c ^ 2 _ { \\mu } ( z ) : = \\int \\int \\frac { 1 } { R ( z , w , \\zeta ) ^ 2 } d \\mu ( w ) d \\mu ( \\zeta ) , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { n - 1 } { 2 } } [ 4 k + 1 ] \\frac { ( q ; q ^ 2 ) _ k ^ 3 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 3 } q ^ { \\frac { k ( n ^ 2 - 2 n k - n - 2 ) } { 4 } } \\equiv 0 \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } e _ { q ( t ) , \\lambda _ { q ( t ) } } ( - Q ) = Q ^ \\mu \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} \\mathcal { I } u ( x ) : = \\sup _ { a \\in \\mathcal { A } } \\inf _ { b \\in \\mathcal { B } } \\{ - { \\rm t r } a _ { a b } ( x ) D ^ 2 u ( x ) - I _ { a b } [ x , u ] + b _ { a b } ( x ) \\cdot D u ( x ) + c _ { a b } ( x ) u ( x ) + f _ { a b } ( x ) \\} = 0 \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} y = S [ u ] , y _ h = S [ u + h ] \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 = 0 , \\ \\ \\ \\ \\bar \\lambda _ 1 \\bar \\lambda _ 2 \\neq 0 , \\\\ & S < \\sup H ^ 2 = \\bar H ^ 2 \\leq 3 S - 2 \\ \\ \\ \\ S < \\sup H ^ 2 < \\dfrac 4 3 S , \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\widetilde { I } [ x , \\Lambda ; { w } ] = \\int _ { \\mathcal { Y } ^ d } e ^ { \\frac { | \\Lambda + \\nabla { w } ( y ) | ^ 2 } { 2 } + V ( x , y ) } d y . \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\nu _ { i } [ w _ { 1 } . . . w _ { l } ] = p _ { w _ { 1 } , i } \\cdot . . . \\cdot p _ { w _ { l } , i } w _ { 1 } , . . . , w _ { l } \\in \\Lambda _ { i } ^ { m } \\ : . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\Delta = & E [ Y ] + \\frac { E [ Y ^ 2 ] } { 2 E [ Y ] } \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} & ( n - 1 ) ! V ( k ) \\\\ = & \\sum _ { r , s \\in \\mathbb { Z } } ( - 1 ) ^ { n + k + s } \\sum _ { m = 1 } ^ { n - 1 } { k \\choose n - s - m + r } { n - k \\choose m - r } M ( r - s a ) ^ { n - 1 } \\\\ = & \\sum _ { 0 \\le s \\le n \\atop 1 \\le r \\le n - 1 } ( - 1 ) ^ { n + k + s } \\sum _ { m = 1 } ^ { n - 1 } { k \\choose n - s - m + r } { n - k \\choose m - r } M ( r - s a ) ^ { n - 1 } \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} h _ I = \\chi _ { I ^ + } - \\chi _ { I ^ - } , I \\in \\mathcal { D } , \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\limsup _ { X \\to \\infty } \\frac { \\# \\{ p \\in S p l _ X ( f ) \\mid f ( \\lfloor p \\alpha \\rfloor ) \\equiv 0 \\bmod p \\} } { 2 \\# S p l _ X ( f ) } \\le \\alpha - ( \\alpha - \\kappa ) = \\kappa \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} - 2 a = \\left ( 1 + a ^ { 2 } \\right ) \\left ( \\frac { f _ { 1 } } { f _ { 1 } ^ { \\prime } } \\right ) \\left ( \\frac { f _ { 2 } ^ { \\prime \\prime } } { f _ { 2 } ^ { \\prime } } \\right ) + \\left ( \\frac { f _ { 1 } ^ { \\prime \\prime } } { f _ { 1 } ^ { \\prime } } \\right ) \\left ( \\frac { f _ { 2 } } { f _ { 2 } ^ { \\prime } } \\right ) , \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} f _ 1 ( q , z _ 2 ^ s ) & = - \\frac { q \\xi ( q ) } { \\xi ' ( q ) } - q ^ 2 \\log \\frac { q \\xi ' ( q ) - \\xi ( q ) } { \\xi ( q ) } + q ^ 2 - \\frac { 2 \\xi ( q ) q } { \\xi ' ( q ) } + \\frac { q [ q \\xi ' ( q ) - \\xi ( q ) ] } { \\xi ' ( q ) } \\\\ & = \\frac { - 4 q \\xi ( q ) + 2 q ^ 2 \\xi ' ( q ) } { \\xi ' ( q ) } - q ^ 2 \\log \\Big ( 1 + \\frac { q \\xi ' ( q ) - 2 \\xi ( q ) } { \\xi ( q ) } \\Big ) \\\\ & < \\frac { - 4 q \\xi ( q ) + 2 q ^ 2 \\xi ' ( q ) } { \\xi ' ( q ) } - q ^ 2 \\frac { \\frac { 2 [ q \\xi ' ( q ) - 2 \\xi ( q ) ] } { \\xi ( q ) } } { 2 + \\frac { q \\xi ' ( q ) - 2 \\xi ( q ) } { \\xi ( q ) } } = 0 . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon } ^ 2 = \\exp ( - ( \\varepsilon _ 0 + o ( 1 ) ) \\gamma _ \\varepsilon ^ 2 ) \\ , . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left ( \\frac { 1 - q ^ t } { z } \\right ) ^ b J _ b \\left ( q ^ t , \\left ( \\frac { 1 - q ^ t } { z } \\right ) ^ { N + 1 } Q \\right ) = g _ b ( z , Q ) \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\log \\rho _ { g } ^ { \\left ( p , q \\right ) } \\left ( f \\right ) = \\underset { \\sigma \\rightarrow + \\infty } { \\overline { \\lim } } \\left [ \\log ^ { \\left [ p + 1 \\right ] } M _ { g } ^ { - 1 } \\left ( \\sigma \\right ) - \\log ^ { \\left [ q + 1 \\right ] } M _ { f } ^ { - 1 } ( \\sigma ) \\right ] , \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} S ^ \\tau _ F \\otimes S ^ { ( 1 ^ m ) } _ F = \\left ( S ^ \\tau _ F \\otimes S ^ { ( 1 ^ m ) } _ F \\right ) ( \\overline { H ^ \\tau } \\otimes 1 ) \\subseteq S ^ { \\tilde { \\tau } } _ F ( \\overline { H ^ \\tau } \\otimes 1 ) . \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\sum _ { \\{ \\alpha _ 2 , \\beta _ 2 \\} \\in \\mathcal { S } _ \\varphi } \\left ( e ^ { \\frac { 1 } { 2 } ( \\ell _ { \\alpha _ 2 } ( \\phi ) + \\ell _ { \\beta _ 2 } ( \\phi ) ) } + 1 \\right ) ^ { - 1 } = 0 . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} S _ X ( f ) & = \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\sum _ { \\gamma _ { \\alpha } } f ( \\gamma _ { \\alpha } ) \\\\ & = \\sum _ { \\alpha } ''' w \\left ( \\frac { N ( \\alpha ) } { X } \\right ) \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { ( c ) } - \\int _ { ( 1 - c ) } \\right ) \\frac { L ' ( \\chi _ { \\alpha } , s ) } { L ( \\chi _ { \\alpha } , s ) } f \\left ( - i \\left ( s - \\frac { 1 } { 2 } \\right ) \\right ) d s . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} h ( s ( y ) ) & = ( 1 + O ( ( s ( y ) ) ^ { \\kappa } ) ) \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ( y ) ^ { - u } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ( K u \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } \\\\ & = ( 1 + O ( y ^ { - \\kappa } ) ) ( K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) ) ^ { 1 / ( u + 1 ) } u ^ { - u / ( u + 1 ) } y ^ { u / ( u + 1 ) } . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} K _ 1 \\leq 2 & & K _ 2 = \\delta - 1 & & C = \\delta + 7 & & C ' = 3 \\delta + 2 \\\\ K _ 1 = 1 & & K _ 2 = \\delta - 1 & & C = 2 \\delta + 3 & & C ' = C + 1 \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\hat \\varepsilon ^ 2 _ { s } = \\frac { \\hat \\sigma ^ 2 } { s } \\Big [ 1 + 2 \\ , \\sum \\limits _ { k = 1 } ^ { s - 1 } ( 1 - \\frac { k } { s } ) \\ , \\hat \\rho ( k ; \\hat \\theta ) \\Big ] , \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\begin{cases} \\frac { | \\Lambda _ i + ( \\widetilde { w } _ i ( x , y _ i ) ) _ { y _ i } | ^ 2 } { 2 } + V _ i ( x , y _ i ) = \\ln \\widetilde { m } _ i ( x , y _ i ) + \\widetilde { H } _ i ( x , \\Lambda _ i ) , \\\\ \\big ( \\widetilde { m } _ i ( x , y _ i ) ( \\Lambda _ i + ( \\widetilde { w } _ i ( x , y _ i ) ) _ { y _ i } ) \\big ) _ { y _ i } = 0 , \\\\ \\int _ 0 ^ 1 \\widetilde { m } _ i ( x , y _ i ) d y = 1 . \\end{cases} \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} 1 / \\omega _ i \\le 1 / \\omega _ 0 = \\sqrt { 1 + 2 \\tau _ 0 \\tilde { \\gamma } _ { G } } , \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\mathrm { M a n t } _ { G , b , \\mu } ( L J ( I ^ { M _ b } _ M ( \\rho ) ) ) = [ I ^ G _ M ( \\rho ) ] \\left [ \\bigoplus _ { ( M , \\mu ' ) \\in \\mathrm { R e l } ^ { G , \\mu } _ { M , b } } r _ { - \\mu ' } \\circ L L ( \\rho ) \\right ] \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { n - 1 } \\lambda _ j = \\prod _ { j = 0 } ^ { n - 1 } f ( \\omega ^ j ) , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} \\check { y } _ { \\mathrm { d } , i } [ n ] & = \\sum _ { k \\in \\mathcal { K } } \\sum _ { \\ell \\in \\mathcal { L } _ { m _ i , k } } h _ { i , k , \\ell } x _ { \\mathrm { d } , k } [ n - d _ { m _ i , k , \\ell } ] + z _ { \\mathrm { d } , i } [ n ] + e _ { \\mathrm { d } , i } [ n ] . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} L = - \\sum _ { i = 1 } ^ d X _ i ^ * X _ i + V , \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { k > r } \\frac { \\tilde \\lambda _ k } { \\tilde \\lambda _ 1 - \\tilde \\lambda _ k } = \\frac { 1 } { \\sqrt { n } } \\sum _ { k > r } \\frac { \\lambda _ k } { \\tilde \\lambda _ 1 - \\lambda _ k } \\leq \\frac { 1 } { \\sqrt { n } } \\sum _ { k > r } \\frac { \\lambda _ k } { \\lambda _ 1 - \\lambda _ k } \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} P = \\frac { ( T - M ) } { ( T - b ' ) } \\times P _ { \\rm T x } . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} g = \\sum _ { \\alpha } h ^ \\lambda _ \\alpha Q ^ \\lambda _ \\alpha \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\aligned \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta \\psi + R \\psi = & - \\frac { n - 1 } { n } \\tau ^ { 2 a } t ^ { 2 N } t _ 0 ^ 2 \\psi ^ { N - 1 } \\\\ & + \\bigg [ | \\sigma + L W | ^ 2 + \\bigg ( 2 \\max \\big \\{ \\| \\varphi _ 0 \\| _ { L ^ \\infty } , \\ , 2 \\big \\} - \\| \\varphi \\| _ { L ^ \\infty } \\bigg ) _ + \\bigg ( \\| \\sigma + L W _ 0 \\| ^ 2 _ { L ^ \\infty } + k _ 0 ^ 2 \\bigg ) \\bigg ] \\psi ^ { - N - 1 } , \\\\ - \\frac { 1 } { 2 } L ^ * L W = & \\frac { n - 1 } { n } t ^ N t _ 0 \\psi ^ N d \\tau ^ { a } . \\endaligned \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } ^ d } \\nabla _ { \\widetilde { \\Lambda } } \\widetilde { m } _ 1 d y = 0 \\ \\ \\int _ { \\mathcal { Y } ^ d } \\widetilde { m } _ 1 \\nabla _ { \\widetilde { \\Lambda } } \\widehat { H } d y = \\nabla _ { \\widetilde { \\Lambda } } \\widehat { H } . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} - \\nu ( \\ell _ 1 ) - \\nu ( \\ell _ 2 ) = 2 + 2 > 2 + 1 = \\alpha _ 1 ^ + + \\alpha _ 2 ^ + . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} L ^ H _ \\varepsilon = o ( 1 ) L ^ g _ \\varepsilon = o ( 1 ) \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} \\left ( \\Delta \\tilde { u } _ \\varepsilon \\right ) ( z ) = \\frac { ( \\Delta u _ \\varepsilon ) ( z _ \\varepsilon ) } { ( \\Delta u _ \\varepsilon ) ( y _ \\varepsilon ) } = \\frac { \\Psi ' _ { N _ \\varepsilon } ( z _ \\varepsilon ) } { \\Psi ' _ { N _ \\varepsilon } ( y _ \\varepsilon ) } \\ , . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( t ) + F ( t ) = R ( t ) , t \\geq 0 \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} \\mathcal D ' : = & \\Bigl \\{ \\{ ( x , y ) | x < 0 , y > 0 \\} , \\{ ( 0 , y ) | y > 0 \\} , \\{ ( x , y ) | x > 0 , y > 0 \\} \\\\ & \\ , \\ , \\ , \\ , \\{ ( x , 0 ) | x < 0 \\} , \\ , \\ , \\ , \\ , \\ , \\{ ( 0 , 0 ) \\} , \\ , \\ , \\ , \\ , \\{ ( x , 0 ) | x > 0 \\} , \\\\ & \\ , \\ , \\ , \\ , \\{ ( x , y ) | x < 0 , y < 0 \\} , \\{ ( 0 , y ) | y < 0 \\} , \\{ ( x , y ) | x > 0 , y < 0 \\} \\Bigr \\} \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { a _ i - 1 } ( 1 - \\frac { i \\pi _ q ( i ) - i j } { q ^ i } ) \\le 2 a _ i q ^ { - \\lceil \\frac { i } { 2 } \\rceil } + \\frac { i } { q ^ i } \\frac { a _ i ( a _ i - 1 ) } { 2 } \\le 4 a _ i ^ 2 i q ^ { - \\lceil \\frac { i } { 2 } \\rceil } . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} J ^ 2 = \\left ( \\sum _ { j = 1 } ^ { n - 1 } e _ { j + 1 } e _ j ^ T \\right ) \\left ( \\sum _ { j = 1 } ^ { n - 1 } e _ { j + 1 } e _ j ^ T \\right ) = \\sum _ { j = 2 } ^ { n - 1 } ( e _ { j + 1 } e _ j ^ T ) ( e _ j e _ { j - 1 } ^ T ) = \\sum _ { j = 1 } ^ { n - 2 } e _ { j + 2 } e _ j ^ T . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { U } _ i ^ F } { \\partial d _ i } = \\frac { B w _ i \\sum _ { j \\in \\mathcal { N } / \\{ i \\} } w _ j d _ j } { \\left ( \\sum _ { j \\in \\mathcal { N } } w _ j d _ j \\right ) ^ 2 } - \\left ( c _ i + \\lambda _ i \\right ) , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} | g ( \\xi \\gamma ) - g ( \\xi ) | = & \\left | \\int _ \\Omega ( g ( \\xi \\eta ^ { - 1 } ) ) ^ { q ' - 1 } ( | \\eta \\gamma | ^ { \\alpha - Q } - | \\eta | ^ { \\alpha - Q } ) d \\eta \\right | \\\\ \\leq & \\| g \\| _ { L ^ \\infty ( \\Omega ) } ^ { q ' - 1 } \\int _ \\Omega \\left | | \\eta \\gamma | ^ { \\alpha - Q } - | \\eta | ^ { \\alpha - Q } \\right | d \\eta . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} h _ { j } \\check { m } ^ { - j } = ( 2 e _ { 2 } , \\ , e _ { 1 } + 2 e _ { 2 } ) . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} b _ 1 b _ 0 & = ( L ( b _ 0 ) + x _ 1 b _ 0 + d x _ 0 ) a _ 0 = ( x _ 0 L ( a _ 0 ) + d x _ 0 ) a _ 0 = d ( x _ 0 a _ 0 ) . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\mu _ { \\beta , 0 } ^ + \\big ( f \\mid \\eta = - 1 ( - \\infty , 0 ) \\times \\{ 0 \\} \\big ) > 0 . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\langle D \\Phi _ { t } ( x ; h ) , l \\rangle _ { \\bar { \\mathcal { H } } } = J _ { t } ( x ; h ) \\cdot \\int _ { 0 } ^ { t } J _ { s } ^ { - 1 } ( x ; h ) \\cdot V ( \\Phi _ { s } ( x ; h ) ) d l _ { s } , \\ \\ \\ \\mathrm { f o r \\ a l l } \\ l \\in \\bar { \\mathcal { H } } , \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\chi _ k ( z ) = z ^ k ( z \\in \\mathbb { T } ) . \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\varphi ( x + x ^ { \\prime } ) = \\varphi ( x ) x ^ { \\prime } \\in B _ { l ( x ) } . \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in G _ q } \\vert R ( \\chi ) \\vert ^ 2 = \\sum _ { \\chi \\in G _ q } q _ m q _ n \\chi ( m ) \\overline { \\chi ( n ) } = \\phi ( q ) \\left ( \\sum _ { m = n \\bmod q } q _ m q _ n \\right ) \\geq q ^ { 1 - o ( 1 ) } \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\widehat { J } ( \\xi ) = 1 - A | \\xi | ^ \\S + o ( | \\xi | ^ \\S ) \\quad \\mbox { a s $ \\xi \\to 0 $ } , \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\varphi _ { \\cal A } ( \\alpha , x ) : = \\displaystyle \\sum _ { i = 1 } ^ p \\alpha _ i ^ 2 w _ i ^ T \\nabla ^ 2 f ( x ) w _ i + 2 \\sum _ { 1 \\le i < j \\le p } ^ p \\alpha _ i \\ , \\alpha _ j w _ i ^ T \\nabla ^ 2 f ( x ) w _ j . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\mathcal { C } _ { n } ( z ) = - 4 \\int _ { 0 } ^ { z } \\int _ { 0 } ^ { \\zeta } \\mathcal { C } _ { n - 1 } ( \\zeta ^ { \\prime } ) d \\zeta ^ { \\prime } d \\zeta - \\frac { ( 2 ^ { 2 n + 1 } - 1 ) } { ( 2 n + 2 ) ! } | B _ { 2 n + 2 } | , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\textbf { y } _ k = \\lvert \\textbf { s } \\rvert \\beta \\left ( \\textbf { x } _ k \\right ) \\textbf { W } _ { k } ^ \\textbf { a } ( \\textbf { x } _ k ) + \\textbf { z } _ { k } . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} v _ 1 ^ { ( \\ell ) } = v _ 2 ^ { ( \\ell ) } G , \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} m : = \\sup _ { \\phi \\in \\mathcal { G } _ 0 } \\| \\phi \\| < \\infty . \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} \\omega _ i \\subset \\mathcal { O } _ { j , d } \\ \\ \\ \\ \\omega _ j \\cap \\mathcal { O } _ { i , d } = \\emptyset , \\ \\ \\ \\ ( i , j ) = ( 1 , 2 ) \\ \\ ( i , j ) = ( 2 , 1 ) . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\textup { s p a c e - t i m e } & = B \\times S ^ 1 \\\\ \\textup { g a u g e g r o u p } & = U ( n ) \\\\ \\textup { m a t t e r } & = T ^ * M \\ , . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\int _ 0 ^ x \\mathrm { e } ^ { - \\gamma t } t ^ { \\nu } \\mathbf { L } _ { \\nu + n } ( t ) \\ , \\mathrm { d } t & = \\int _ 0 ^ x \\mathrm { e } ^ { - \\gamma t } \\frac { 1 } { t ^ { n + 1 } } t ^ { \\nu + n + 1 } \\mathbf { L } _ { \\nu + n } ( t ) \\ , \\mathrm { d } t \\\\ & > \\frac { \\mathrm { e } ^ { - \\gamma x } } { x ^ { n + 1 } } \\int _ 0 ^ x t ^ { \\nu + n + 1 } \\mathbf { L } _ { \\nu + n } ( t ) \\ , \\mathrm { d } t = \\mathrm { e } ^ { - \\gamma x } x ^ { \\nu } \\mathbf { L } _ { \\nu + n + 1 } ( x ) , \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\rho ( H ) & \\geq x ^ { \\mathrm { T } } ( \\mathcal { A } x ) = \\frac { 1 } { m } \\sum _ { \\{ i _ 1 , i _ 2 , \\ldots , i _ r \\} \\in E ( H ) } \\sqrt [ r ] { d _ { i _ 1 } d _ { i _ 2 } \\cdots d _ { i _ r } } \\\\ & \\geq \\Bigg ( \\prod _ { \\{ i _ 1 , i _ 2 , \\ldots , i _ r \\} \\in E ( H ) } \\sqrt [ r ] { d _ { i _ 1 } d _ { i _ 2 } \\cdots d _ { i _ r } } \\Bigg ) ^ { \\frac { 1 } { m } } . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\lvert \\lambda \\rvert ^ { 1 / 2 } \\nabla _ H ( \\lambda - \\Delta ) ^ { - 1 } \\left ( Q \\overline { f } \\otimes 1 \\right ) = \\lvert \\lambda \\rvert ^ { 1 / 2 } \\int _ 0 ^ \\infty e ^ { - \\lambda t } \\left ( \\nabla _ H e ^ { t \\Delta _ H } Q \\overline { f } \\otimes e ^ { t \\Delta _ z } 1 \\right ) \\ , d t , \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} & { \\rm d e t } \\left [ \\begin{array} { l l } 1 - p _ + ^ 2 ( k ) - p _ - ^ 2 ( k ) & - 2 p _ - ( k ) p _ + ( k ) \\\\ & \\\\ - 2 p _ - ( k ) p _ + ( k ) & 1 - p _ + ^ 2 ( k ) - p _ - ^ 2 ( k ) \\end{array} \\right ] \\\\ & = \\left [ 1 - ( p _ + ( k ) + p _ - ( k ) ) ^ 2 \\right ] \\left [ 1 - ( p _ + ( k ) - p _ - ( k ) ) ^ 2 \\right ] \\ge 0 . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} \\partial _ t \\sigma - \\Delta \\sigma + c \\sigma h ( \\varphi _ 1 ) + c \\sigma _ 2 \\left ( h ( \\varphi _ 1 ) - h ( \\varphi _ 2 ) \\right ) + b ( \\sigma - w ) = 0 \\ , , \\sigma ( 0 ) = \\sigma _ 0 \\ , , \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} F = F _ a = \\textrm { c o n v } \\left ( x \\mapsto \\inf _ { y \\in E } \\left \\{ f ( y ) + \\langle G ( y ) , x - y \\rangle + \\varphi _ { y } ( x ) + a | x - y | ^ 2 \\right \\} \\right ) \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} ( T ^ { * * } ) _ { \\rm r e g } = ( I - P ) T ^ { * * } , ( T ^ { * * } ) _ { \\rm s i n g } = P T ^ { * * } , \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} L ( { \\cal T } ( 1 ) ) = L ( { \\cal C } _ 1 ) = T _ 1 . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} F _ { l } ( \\Gamma _ { t } , x ) \\triangleq \\sum _ { k = 1 } ^ { l } \\sum _ { i _ { 1 } , \\ldots , i _ { k } = 1 } ^ { d } V _ { ( i _ { 1 } , \\ldots , i _ { k } ) } ( x ) \\int _ { 0 < t _ 1 < \\cdots < t _ k < t } d B _ { t _ 1 } ^ { i _ { 1 } } \\cdots d B _ { t _ k } ^ { i _ { k } } , \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} b _ { I } ^ { ( \\theta ) } = \\sum _ { K \\in \\mathcal { B } _ I } \\theta _ K h _ K , I \\in \\mathcal { D } _ { \\leq n } , \\ \\theta \\in \\{ \\pm 1 \\} ^ { \\mathcal { D } } . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\tau ( W _ m ) = \\begin{cases} 1 & m = ( 0 , 0 ) \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\Vert t _ m f ( n ) \\Vert _ { { \\ell } ^ { p , \\infty } } \\leq C \\Vert f \\Vert _ { \\ell ^ 1 } , C = \\Vert k \\Vert _ { \\ell ^ { p , \\infty } } , \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} \\dim \\Xi _ { \\chi } = \\sharp X _ w ( 1 ) _ { P _ { 1 / 2 } } = ( q - 1 ) q ^ { n - 1 } . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} a _ j \\int _ { 0 } ^ { \\infty } w ( z ; \\ , q ) \\left ( L _ j ^ { ( \\delta ) } ( z ; \\ , q ) \\right ) ^ 2 d z = \\int _ { 0 } ^ { \\infty } w ( z ; \\ , q ) ( 1 + z ) L _ { n } ^ { ( \\delta ) } ( z q ; \\ , q ) L _ j ^ { ( \\delta ) } ( z ; \\ , q ) d z , \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\mathcal { I } ^ { G , \\mu } _ { M _ { S _ 2 } , b _ { S _ 2 } } = \\{ ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\in \\mathcal { C } _ { M _ { S _ 2 } } : ( M _ { S _ 2 } , \\mu _ { S _ 2 } ) \\in \\mathcal { I } ^ { M _ { S _ 1 } , \\mu _ { S _ 1 } } _ { M _ { S _ 2 } , b _ { S _ 2 } } \\ , \\ \\ , \\ ( M _ { S _ 1 } , \\mu _ { S _ 1 } ) \\in \\mathcal { I } ^ { G , \\mu } _ { M _ { S _ 1 } , b _ { S _ 1 } } \\} . \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} ( \\Upsilon u ) _ k & : = A _ { k - 1 } ^ * u _ { k - 1 } + B _ k u _ k + A _ k u _ { k + 1 } k \\in \\mathbb { N } \\setminus \\{ 1 \\} , \\\\ ( \\Upsilon u ) _ 1 & : = B _ 1 u _ 1 + A _ 1 u _ 2 \\ , , \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} y ( t ) = G ( y ( t ) ) : = e ^ { t \\Delta } U _ 0 + F ( y ) ( t ) , \\ t \\geq 0 , \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} y _ { \\textrm { F N } } = \\sqrt { \\beta _ { \\textrm { F } } \\eta P _ \\textrm { B } d _ { \\textrm { B F } } ^ { - \\alpha } d _ { \\textrm { N F } } ^ { - \\alpha } | h _ { \\textrm { B F } } | ^ 2 } h _ { \\textrm { N F } } x _ { \\textrm { N } } + n _ { \\textrm { N } } , \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{gather*} D \\gamma = 0 , \\end{gather*}"}
-{"id": "6145.png", "formula": "\\begin{align*} B _ 3 ^ { ( n ) } & \\subseteq \\bigcup _ { t = \\lfloor p _ n ^ { - 1 } \\rfloor } ^ { \\lfloor c n \\rfloor } \\{ S _ n ' ( t ) \\leq t \\} \\cup \\bigcup _ { t = \\lfloor c n \\rfloor } ^ { \\lfloor n - p _ n ^ { - 1 } \\rfloor \\wedge \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } \\{ S _ n ' ( t ) \\leq t \\} \\cup \\bigcup _ { t = \\lfloor n - p _ n ^ { - 1 } \\rfloor \\wedge \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } ^ { \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } \\{ S _ n ' ( t ) \\leq t \\} , \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} \\delta _ { a b } \\int e ^ { - H } z _ a g ( a , z - 1 ) = \\delta _ { a b } g ( a , 0 ) . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\frac { \\theta _ { q ( t ) } ( Q _ 1 ( q ( t ) ) } { \\theta _ { q ( t ) } ( Q _ 2 ( q ( t ) ) } = Q _ 0 ^ { \\alpha _ 2 - \\alpha _ 1 } \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} | x _ { \\bar N } | < | x _ 0 | ^ { { 3 } ^ { \\bar N } } \\prod _ { j = 0 } ^ { \\bar N - 1 } h _ { j } ^ { { 3 } ^ { \\bar N - 1 - j } } . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} \\left | e ^ { \\frac { 1 } { 2 } \\int _ 0 ^ t B _ i ^ 2 ( s ) d s } f \\right | _ q \\leq | f | _ q , \\ \\forall f \\in L ^ q , \\ t \\geq 0 , \\ i = 1 , 2 , . . . , N . \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} P ( 0 ) = 0 , \\ P ' ( q ) > 0 \\ \\mbox { f o r a l l } \\ q \\geq 0 , \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} I _ { n , m } \\risingdotseq \\cup _ { k = 0 } ^ { d - 1 } \\left \\{ \\left . \\frac { k + \\epsilon } { d } \\right | \\epsilon \\in I _ { N , M } \\right \\} , \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} \\begin{cases} a _ { 0 , j } = h a _ { 0 , j } + k a _ { 1 , j } + t c _ { 0 , j } j = 0 , 1 , 2 \\\\ a _ { 1 , j } = h a _ { 1 , j } + k a _ { 2 , j } + t c _ { 1 , j } j = 0 , 1 , 2 \\\\ a _ { 2 , j } = h a _ { 2 , j } + k a _ { 3 , j } + t c _ { 2 , j } j = 0 , 1 , 2 \\\\ \\xi ^ 2 b _ 0 = h b _ 0 + k b _ 1 + t d _ 0 \\\\ \\xi ^ 2 b _ 1 = h b _ 1 + k b _ 2 + t d _ 1 \\\\ \\xi ^ 2 b _ 2 = h b _ 2 + k b _ 3 + t d _ 2 \\end{cases} \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\textnormal { S o l } \\left ( E _ q \\right ) = \\left \\{ X _ q \\in \\left ( \\mathbb { C } \\{ Q , Q ^ { - 1 } \\} \\right ) ^ n \\ , \\middle | \\ , q ^ { Q \\partial _ Q } X _ q ( Q ) = A _ q ( Q ) X _ q ( Q ) \\right \\} \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\vert u ( x _ k ) \\vert \\leq \\bigg ( \\sum _ { i = 0 } ^ { k - k _ 0 - 1 } \\vert u ( x _ { k - i } ) - u ( x _ { k - ( i + 1 ) } ) \\vert \\bigg ) + \\vert u ( x _ { k _ 0 } ) \\vert \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\widetilde { f } ( Q ) = ( Q - 1 ) ^ { \\frac { 1 } { 4 } ( 1 + i ) } ( Q - i ) ^ { - \\frac { 1 } { 2 } } ( Q + 1 ) ^ { \\frac { 1 } { 4 } ( 1 - i ) } \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\frac { G ( w ) } { F ( w ) } \\int \\frac { F ( z ) \\overline { F _ 2 ( z ) } } { z - w } d \\nu ( z ) = \\int \\frac { G ( z ) \\overline { F _ 2 ( z ) } } { z - w } d \\nu ( z ) \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} P [ C _ 0 ( \\alpha \\times L ) ] = & P [ A _ \\rho ( \\alpha ) \\oplus B _ \\rho ( \\alpha ) ] \\\\ = & R _ L [ B _ \\rho ( \\alpha ) ] + P [ A _ \\rho ( \\alpha ) ] = \\overline { R _ L [ B _ \\rho ( \\alpha ) ] } + \\overline { P [ A _ \\rho ( \\alpha ) ] } \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} I I ^ L _ 1 = \\left ( \\begin{array} { c c } h _ { 1 1 } , & h _ { 1 2 } \\\\ h _ { 2 1 } , & h _ { 2 2 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} Z _ m ( x _ { i } ) - & Z _ n ( x _ { i } ) \\ = \\ Z _ m ( x _ { i - 1 } ) - Z _ n ( x _ { i - 1 } ) \\\\ & + \\left \\{ \\begin{array} { l l } S ( x _ { i } , y _ { i } ) \\ , y _ { i } ^ { - 1 / \\alpha ( Z _ m ( x _ { i - 1 } ) ) } & \\mbox { i f } i \\neq i _ k \\mbox { f o r a l l } k \\\\ S ( x _ { i } , y _ { i } ) \\big ( y _ { i } ^ { - 1 / \\alpha ( Z _ m ( x _ { i - 1 } ) ) } - y _ { i } ^ { - 1 / \\alpha ( Z _ n ( x _ { i - 1 } ) ) } \\big ) & \\mbox { i f } i = i _ k \\mbox { f o r s o m e } k \\end{array} . \\right . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\iota _ \\partial \\omega ^ 2 = \\omega + \\sum a _ i \\cdot d b _ i \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\lim _ { \\ell \\rightarrow \\infty } \\beta _ { \\ell } = t _ 1 . \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\int _ \\Omega \\rho ( t , x ) d x = \\int _ \\Omega \\rho _ 0 ( x ) d x , \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\displaystyle \\mathcal K ( z _ 1 , z _ 2 , a , b ) : = \\displaystyle \\int _ { a } ^ { b } ( b - \\tau ) ^ { z _ 1 - 1 } ( \\tau - a ) ^ { z _ 2 - 1 } d \\tau = ( b - a ) ^ { z _ 1 + z _ 2 - 1 } { \\bf B } ( z _ 1 , z _ 2 ) , \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} - h ''' ( s ) & = K u \\Gamma ( u + 3 ) s ^ { - u - 3 } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 - 3 } ) + O ( s ^ { - 3 } ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K u \\Gamma ( u + 3 ) \\zeta ( u + 1 ) s ^ { - u - 3 } . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\mathcal { X } ( t ) + \\int ^ t _ { 0 } \\mathcal { X } ( \\tau ) d \\tau \\leqslant C _ 0 \\sum _ { j = 1 } ^ { m } \\int ^ t _ { 0 } \\mathcal { X } ( \\tau ) ^ { \\alpha _ j } d \\tau + C _ 0 \\sum _ { k = 1 } ^ { n } \\mathcal { X } ( t ) ^ { \\beta _ k } + C _ 0 \\sum _ { k = 1 } ^ { n } \\mathcal { X } ( 0 ) ^ { \\beta _ k } + C _ 0 \\mathcal { X } ( 0 ) , \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} [ L : K ] & \\geq \\sum _ { i \\leq s } e ( u ^ { 0 } _ { i } / v ^ { 0 } ) \\ , f ( u ^ { 0 } _ { i } / v ^ { 0 } ) \\\\ & \\geq \\sum _ { i \\leq s } e ( u ^ { 0 } _ { i } / v ^ { 0 } ) \\ , \\sum _ { j \\leq r _ { i } } e ( \\bar { u } _ { i , j } / \\bar { v } ) \\ , f ( \\bar { u } _ { i , j } / \\bar { v } ) \\\\ & = \\sum _ { i \\leq s } \\sum _ { j \\leq r _ { i } } e ( u _ { i , j } / v ) \\ , f ( u _ { i , j } / v ) , \\tag * { } \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} \\hat { B } ^ * = \\frac { \\lambda } { 1 + p \\frac { | \\mathcal { N } | - 1 } { a | \\mathcal { N } | } } - \\frac { 1 } { \\frac { | \\mathcal { N } | - 1 } { a | \\mathcal { N } | } } . \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} W ' ( h x ) & = W ' ( h a ) + h W '' ( h a ) ( x - a ) + \\frac 1 2 h ^ 2 W ''' ( h a ) ( x - a ) ^ 2 + \\frac 1 6 h ^ 3 ( x - a ) ^ 3 \\nu _ { h , a } ^ 1 ( x ) \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} H _ 0 ( K _ \\bullet , \\iota _ \\tau ) = A / A ^ + \\cong A ^ 0 . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } X _ 1 + \\dot { \\gamma } _ 3 X _ 2 + \\omega ( \\dot { \\gamma } ( t ) ) X _ 3 . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} \\zeta ( 2 \\mu c _ \\alpha ) & = 4 \\mu ^ 2 c _ \\alpha ^ 2 + c _ \\alpha \\\\ - \\zeta & = - 4 \\mu c _ \\alpha | \\alpha | a \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ 5 a _ i A _ i ( \\varphi ( M ) ( y ) ) = \\sum _ { i = 0 } ^ 5 \\psi _ i ( M ) ( a ) A _ i ( y ) \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} ( - g ) \\cdot \\eta ( \\tilde \\sigma ) = g \\cdot \\sigma , \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} W _ { h } ( x ) = \\frac { ( \\delta _ { + } - \\delta _ { - } ) ^ { 2 } } { h ^ { 2 } } V \\big ( \\delta _ { - } + x ( \\delta _ { + } - \\delta _ { - } ) \\big ) . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\log \\kappa ( y , x ) : = e ^ { - | x - y | } x \\in [ 0 , 1 ] y \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\varphi _ N ( \\Gamma ) = \\exp ( \\Gamma ) \\int _ 0 ^ { \\Gamma } \\exp ( - s ) \\frac { s ^ N } { N ! } d s \\ , . \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} & - q _ * ^ { - 1 } t ^ { \\frac { N + 2 A } { 2 } + 1 } ( \\partial _ r ^ 2 u _ 0 ) ( x , t ) \\\\ & = - [ M _ { 0 , 1 } + o ( 1 ) ] t ( \\partial _ r ^ 2 U ) ( | x | ) + \\left [ \\frac { N + 2 A } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] \\partial _ r ^ 2 [ U F ] ( | x | ) + O ( t ^ { - 1 } ) \\\\ & \\ge \\left [ \\frac { N + 2 A } { 2 } M _ { 0 , 1 } + o ( 1 ) \\right ] \\partial _ r ^ 2 [ U F ] ( | x | ) + O ( t ^ { - 1 } ) \\ge \\frac { N + 2 A } { 4 N } M _ { 0 , 1 } U ( r _ * ) > 0 \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} E _ M : = \\{ x \\in ( 0 , 1 ) : a _ { n _ k } ( x ) = u _ { k } , \\ 1 \\leq a _ j ( x ) \\leq M j \\neq n _ k \\} . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} h b _ 1 g _ p ^ { \\pm 1 } b _ 2 = c _ 1 g _ q ^ { \\pm 1 } c _ 2 . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left | \\frac { t ^ { 2 H l _ { 0 } } } { \\lambda _ { t } ^ { ( l ) } } \\right | ^ { 2 q } : \\| U _ { t } ^ { ( l ) } \\| _ { \\textsc { H S } } < r \\right ] & = \\frac { 1 } { t ^ { 4 q H ( l - l _ { 0 } ) } } \\mathbb { E } \\left [ \\left | \\frac { t ^ { 2 H l } } { \\lambda _ { t } ^ { ( l ) } } \\right | ^ { 2 q } : \\| U _ { t } ^ { ( l ) } \\| _ { \\textsc { H S } } < r \\right ] \\\\ & \\leq \\frac { C _ { 1 , q , l } } { t ^ { 4 q H ( l - l _ { 0 } ) } } . \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} f _ { \\sigma ( 1 ) } \\ast \\dotsb \\ast f _ { \\sigma ( n ) } \\ast \\chi _ \\pi = f \\ast h _ \\lambda \\ast \\dotsb \\ast h _ \\lambda \\ast \\chi _ \\pi = \\alpha _ \\lambda ( \\pi ) ^ { n - 1 } f \\ast \\chi _ \\pi \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\log \\kappa _ M ^ { N , h } ( y , x ) : = \\sum \\limits _ { n = 1 } ^ { M } \\sqrt { \\lambda _ n ^ { N , h } } \\phi _ n ^ { N , h } ( x ) \\psi _ n ( y ) . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} & c _ { l i + 1 } \\dots c _ { l ( i + 1 ) } \\\\ = \\ , & c _ 1 \\dots c _ { l ( i + 1 ) } - c _ 1 \\dots c _ { l i } \\times 1 0 ^ l \\\\ = \\ , & \\lfloor 1 0 ^ { l ( i + 1 ) } a / b \\rfloor - \\lfloor 1 0 ^ { l i } a / b \\rfloor \\times 1 0 ^ l \\\\ = \\ , & 1 0 ^ { l ( i + 1 ) } a / b - s _ { i + 1 } / b - ( 1 0 ^ { l i } a / b - s _ { i } / b ) 1 0 ^ l \\\\ = \\ , & ( 1 0 ^ l s _ i - s _ { i + 1 } ) / b , \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} V ( u ( \\ell _ 0 ) ) = \\frac { 1 } { 2 \\pi r } \\int _ { \\S ^ 1 ( r ) } V ( u ( \\ell ) ) { \\rm d } \\ell . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ k [ 3 k + 1 ] { 2 k \\brack k } ^ 3 \\frac { ( - q ; q ) _ n ^ 3 } { ( - q ; q ) _ k ^ 3 } \\equiv 0 \\pmod { ( 1 + q ^ n ) ^ 2 [ 2 n + 1 ] { 2 n \\brack n } } , \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\overline { f ^ { \\not \\ominus } ( z , t ) } - ( 1 - z ) ( 1 - z t ) \\overline { f ( z , t ) } - z + 1 = 0 . \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\int _ { D } \\kappa ( y , x ) \\nabla u ( y , x ) \\cdot \\nabla v ( x ) \\mathrm { d } x = \\int _ { D } f ( x ) v ( x ) \\mathrm { d } x \\forall v \\in V . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} i ^ { \\tau _ { \\epsilon } } + k ^ { \\tau _ { \\epsilon } } = i + \\delta + \\epsilon - k < i + \\delta + \\epsilon - ( i + j ) = \\delta + \\epsilon - j < j = j ^ { \\tau _ { \\epsilon } } \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\| F ^ { * } d y \\| & = \\sqrt { \\langle d y ^ { 1 } \\wedge \\cdots \\wedge d y ^ { n } , d y ^ { 1 } \\wedge \\cdots \\wedge d y ^ { n } \\rangle } \\\\ & = \\sqrt { \\mathrm { d e t } \\left ( \\left ( \\langle d y ^ { i } , d y ^ { j } \\rangle \\right ) _ { 1 \\leqslant i , j \\leqslant n } \\right ) } \\\\ & = \\sqrt { \\mathrm { d e t } \\left ( \\frac { d y } { d x } \\left ( \\frac { d y } { d x } \\right ) ^ { * } \\right ) } . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} m \\log _ 2 \\Bigl ( \\frac { n } { 2 m } \\Bigr ) & = f ( m ) \\ge f \\Bigl ( \\frac { k } { \\log _ 2 ( n / k + 1 ) } \\Bigr ) \\\\ & = \\frac { k } { \\log _ 2 ( n / k + 1 ) } \\log _ 2 \\Bigl ( \\frac { n \\log _ 2 ( n / k + 1 ) } { 2 k } \\Bigr ) \\ge k , \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\varphi _ t : = \\sup _ { \\mathcal { S } _ { ( 0 , \\ell ) } ( \\varphi , \\psi ) } u _ t . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} \\Sigma _ { p / q } = \\sigma _ { p / q } \\Bigl ( \\frac { ( p - q - 1 ) / 2 } { p - q } \\Bigr ) . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} e ^ { - t \\mathcal { A } } ( 0 , - u _ { 1 } ) + \\int _ 0 ^ t e ^ { - ( t - s ) \\mathcal { A } } ( 0 , A e ^ { s A } ( u _ { 0 } + u _ { 1 } ) ) \\ , d s = ( U _ 1 ( t ) , U _ 1 ' ( t ) ) . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} Z ( \\lambda , y ) : = \\begin{pmatrix} \\lambda & s ( y ) ^ T \\\\ s ( y ) & S ( y ) \\end{pmatrix} \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} f ^ { \\not \\ominus } ( z , t ) = z f ( z , 1 ) ( f ( z , t ) - 1 ) + \\left ( \\frac { z } { 1 - z } \\right ) \\left ( f ( z , t ) - 1 \\right ) \\left ( \\frac { f ( z , 1 ) f ^ { \\not \\ominus } ( z , f ( z , 1 ) ) - t f ^ { \\not \\ominus } ( z , t ) } { f ( z , 1 ) - t } \\right ) . \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\begin{aligned} S ( 1 - \\dfrac 3 2 S ) + \\dfrac 3 2 S \\bar H ^ 2 - \\dfrac 3 4 \\bar H ^ 4 - \\dfrac { \\bar H ^ 2 \\sqrt { ( 4 S - 3 \\bar H ^ 2 ) \\bar H ^ 2 } } 4 = 0 . \\end{aligned} \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n x _ j z _ j \\leq \\sum _ { j = 1 } ^ n x _ j y _ j \\max _ { 1 \\leq k \\leq n } \\Big ( \\sum _ { i = 1 } ^ k z _ i \\Big ) \\Big ( \\sum _ { i = 1 } ^ k y _ i \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\ , & \\frac { N - 4 \\sigma } { 4 \\mu - 2 } = \\frac { N + 2 - 4 \\nu } { 4 \\mu - 2 } + 1 > \\frac { N + 2 - 4 \\left ( \\frac { N } { 4 } \\right ) } { 4 \\mu - 2 } + 1 > 1 ; \\\\ \\ , & \\frac { N - 4 \\sigma } { N + 2 - 4 \\nu } = \\frac { 4 \\mu - 2 } { N + 2 - 4 \\nu } + 1 > 1 ; \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} y _ i = F \\left ( y _ { i - 1 } , \\ldots , y _ { i - p } , x _ { i - d } , \\ldots , x _ { i - d - m + 1 } , e _ { i - 1 } , \\ldots , e _ { i - g } \\right ) + e _ i , \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} \\Delta ( e _ x e _ y e _ z e _ t ) = \\Delta ( e _ x e _ y ) \\Delta ( e _ z e _ t ) = \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\mu \\bigl ( X _ { 0 } = \\omega _ { 0 } \\bigm | X _ { - \\infty } ^ { - 1 } = \\omega _ { - \\infty } ^ { - 1 } \\bigr ) \\ ; = \\ ; P \\bigl ( \\omega _ { 0 } \\bigm | \\omega _ { - \\infty } ^ { - 1 } \\bigr ) \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ m a ^ 0 _ { i j } X _ i X _ j U _ 0 = 0 \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 [ w _ 1 ] ( x ^ * ) = \\mathcal { A } _ 1 [ w _ 1 ] ( x ^ * ) = \\left [ - \\Delta w _ 1 - ( p - 2 ) \\frac { \\langle ( D ^ 2 w _ 1 ) ( \\nabla u _ 1 ) , \\nabla u _ 1 \\rangle } { | \\nabla u _ 1 | ^ 2 } \\right ] ( x ^ * ) \\geq 0 . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\widetilde { \\phi } ( \\cdot ) = \\phi ( \\cdot - \\theta _ 0 ) . \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} L : = \\{ j \\in \\mathbb N : \\ ; 1 \\leq j \\leq ( n - 1 ) , \\ ; x _ j < \\widehat x _ j \\} \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\sup _ { 0 < u \\le \\lambda } ( u x - \\log L _ { X } ( u ) ) = Q ( u ^ { \\ast } ) \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| A ^ { k / 2 } w ( t ) \\| ^ 2 & = - 2 ( A ^ { k / 2 } w ( t ) , A ^ { k / 2 } w ' ( t ) ) \\\\ & = - 2 ( A ^ { k / 2 } w ( t ) , A ^ { k / 2 + 1 } w ( t ) ) \\\\ & = - 2 \\| A ^ { \\frac { k + 1 } { 2 } } w ( t ) \\| ^ 2 . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} D F ( p ^ { \\pm } ) = \\left ( \\begin{array} { c c } f _ { 1 , u } ( p ^ { \\pm } ) & f _ { 1 , v } ( p ^ { \\pm } ) \\\\ f _ { 2 , u } ( p ^ { \\pm } ) & f _ { 2 , v } ( p ^ { \\pm } ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\Delta \\lhd \\Gamma , \\Gamma \\cap T ( \\mathbb { Z } ^ d ) = \\Delta , [ \\Gamma : \\Delta ] \\leq 2 ^ d d ! . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| \\mathbf { f } _ { \\boldsymbol { \\psi } } \\left ( \\tilde { \\boldsymbol { \\psi } } ^ k ( t ) \\right ) \\right \\| _ 2 \\le C _ { \\mathbf { f } } . \\end{aligned} \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } { \\frac { a } { a + b n } \\binom { a + b n } { n } z ^ { n } } & = x ^ { a } = x ^ { a _ { 1 } } x ^ { a _ { 2 } } \\cdots x ^ { a _ { r } } \\\\ & = \\prod _ { i = 1 } ^ { r } { \\left ( \\sum _ { n = 0 } ^ { \\infty } { \\frac { a _ { i } } { a _ { i } + b n } \\binom { a _ { i } + b n } { n } z ^ { n } } \\right ) } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } { \\left ( \\sum _ { n _ { 1 } + n _ { 2 } + \\cdots + n _ { r } = n } \\prod _ { i = 1 } ^ { r } { \\frac { a _ { i } } { a _ { i } + b n _ { i } } \\binom { a _ { i } + b n _ { i } } { n _ { i } } } \\right ) } z ^ { n } . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} I _ { s , N } \\left ( \\alpha \\right ) \\sim \\sum _ { n \\in \\mathbb { Z } } c _ { n } e \\left ( n \\alpha \\right ) \\qquad \\mathrm { w h e r e } c _ { n } \\coloneqq \\begin{cases} \\sin \\left ( 2 \\pi n s / N \\right ) / \\left ( \\pi n \\right ) & \\mathrm { i f \\ , } n \\neq 0 , \\\\ 2 s / N & \\mathrm { i f \\ , } n = 0 , \\end{cases} \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} R _ { p ^ { \\ast } q ^ { \\ast } k l } = \\sum _ i \\left ( h ^ { p ^ { \\ast } } _ { i k } h ^ { q ^ { \\ast } } _ { i l } - h ^ { p ^ { \\ast } } _ { i l } h ^ { q ^ { \\ast } } _ { i k } \\right ) . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\mathcal { G } _ { x } ( \\eta , y ) : = \\big \\{ \\lambda \\in \\mathbb { Y } \\ | \\ \\eta + \\nabla \\ ! g ( x ) \\lambda = 0 , \\ , y \\ ! - \\ ! \\Pi _ { \\mathcal { K } } ( y \\ ! + \\ ! \\lambda ) = 0 \\big \\} . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\begin{cases} \\dd { x } _ s ^ { t , \\bar { x } ; u , v } = \\bigl [ f _ 1 ( x _ s ^ { t , \\bar { x } ; u , v } ) + f _ 2 ( u _ s , u _ { s - r } , v _ s , v _ { s - r } ) \\bigr ] \\dd s + \\sigma ( x _ s ^ { t , \\bar { x } ; u , v } ) \\dd B _ s , ~ s \\in ( t , T ] \\\\ x _ s ^ { t , \\bar { x } ; u , v } = \\bar { x } , ~ u _ s = \\bar { u } _ s , ~ v _ s = \\bar { v } _ s , ~ s \\in [ t - r , t ] , \\end{cases} \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} d _ k = { \\sqrt { k ^ 2 + 1 } \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } c _ k \\over \\hat \\sigma \\sinh \\sqrt { k ^ 2 + 1 } ( \\rho _ s - \\eta _ s ) } . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} [ \\mu _ { S ' } ] ( I ^ { M _ { S ' } } _ { w ( M _ S ) } ( w ( \\rho ) ) ) = \\oplus ^ k _ { i = 1 } [ \\pi _ i ] [ r _ { - \\mu _ { S ' } } \\circ L L ( \\pi _ i ) \\otimes | \\cdot | ^ { - \\langle \\rho _ { M _ { S ' } } , \\mu _ { S ' } \\rangle } ] \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} \\bar A : = \\sum _ { i = 1 } ^ m \\bar y _ i A _ i , \\ \\ \\bar a : = \\sum _ { i = 1 } ^ m \\bar y _ i a _ i . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} U ^ { ( P ) } _ { t o t } = U ^ { ( P ) } _ { t o t } ( n ) : = \\bigcap _ { l = 1 } ^ { N } U ^ { ( P ) } _ l \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\Theta ( g h ) ( x ) & = \\mathcal { I } _ { g h } ( x ) = ( g h ) x ( g h ) ^ { - 1 } = ( g h ) x ( h ^ { - 1 } g ^ { - 1 } ) \\\\ & = g ( h x h ^ { - 1 } ) g ^ { - 1 } = \\mathcal { I } _ g ( \\mathcal { I } _ h ( x ) ) = ( \\mathcal { I } _ g \\circ \\mathcal { I } _ h ) ( x ) \\\\ & = ( \\Theta ( g ) \\circ \\Theta ( h ) ) ( x ) . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} X = b ^ M \\cdot S ( X ) . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} & O _ { r } ( 1 ) \\sum _ { c _ 2 \\ldots c _ d \\leq x } c _ 2 ^ { d - 2 } c _ 3 ^ { d - 3 } \\ldots c _ { d - 2 } ^ 2 c _ { d - 1 } ( x / ( c _ 2 c _ 3 \\ldots c _ d ) ) ^ { d - 1 } \\\\ = \\ & O _ r ( 1 ) x ^ { d - 1 } \\sum _ { c _ 2 c _ 3 \\ldots c _ d \\leq x } c _ 2 ^ { - 1 } c _ 3 ^ { - 2 } \\ldots c _ { d - 1 } ^ { - ( d - 2 ) } c _ { d } ^ { - ( d - 1 ) } \\\\ \\leq \\ & O _ { d , r } ( 1 ) x ^ { d - 1 } \\log x . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\gamma _ f ^ { \\pm } ( s ) = \\Gamma _ \\R \\ ! \\left ( s + \\frac { 1 \\mp ( - 1 ) ^ k \\epsilon } 2 + \\nu \\right ) \\Gamma _ \\R \\ ! \\left ( s + \\frac { 1 \\mp \\epsilon } 2 - \\nu \\right ) . \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} V _ { \\varepsilon , z _ \\varepsilon } ^ 2 = ( 1 - \\eta _ \\varepsilon ) U _ { \\varepsilon , z _ \\varepsilon } ^ 2 \\ , , \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} ( m + 2 ) & \\left ( \\# ( e _ { \\geq m + 2 } ) + \\# ( h _ { \\geq m + 2 } ) + \\# ( f _ { \\geq m + 2 } ) \\right ) \\\\ & + ( m + 1 ) \\left ( \\# ( e _ { m + 1 } ) + \\# ( h _ { m + 1 } ) + \\# ( f _ { m + 1 } ) \\right ) + m \\# ( f _ m ) \\\\ = { } & C + ( m + 1 ) \\left ( \\# ( e _ { \\geq m + 1 } ) + \\# ( h _ { \\geq m + 1 } ) + \\# ( f _ { \\geq m } ) \\right ) - \\# ( f _ m ) \\\\ = { } & C + ( m + 1 ) ( k + 1 ) - \\# ( f _ m ) . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\varrho ( x , v , z _ 1 , \\dots , z _ N ) \\propto \\exp \\Big \\{ - \\Phi ( x ) - \\frac { m } { 2 } v ^ 2 - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ N z _ k ^ 2 \\Big \\} . \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\Delta _ h K & = ( \\Delta _ h K * L ) ( 0 ) K + d ( \\Delta _ h K * L ) * K , \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} D \\phi ( y _ 1 + y _ 2 ) = D \\phi _ 1 ( y _ 1 ) + D \\phi _ 2 ( y _ 2 ) \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} \\N ( R , K _ m ) \\leq { \\binom { r } { m } } \\left ( \\frac { n _ 1 } r \\right ) ^ m + \\delta \\cdot k ^ m . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} n ( x ) - f ( x , y , v ) v & = 0 , \\\\ f ( x , y , v ) v - \\frac { a } { G _ { 1 } } \\varphi _ { 1 } ( y ) - \\frac { p } { G _ { 1 } } \\varphi _ { 1 } ( y ) \\varphi _ { 2 } ( z ) & = 0 , \\\\ k \\varphi _ { 1 } ( y ) - \\frac { u } { G _ { 2 } } v & = 0 , \\\\ c \\varphi _ { 1 } ( y ) \\varphi _ { 2 } ( z ) - \\frac { b } { G _ { 3 } } \\varphi _ { 2 } ( z ) & = 0 . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{gather*} ( m _ 1 + m _ 2 ) ( \\alpha _ 1 + \\alpha _ 2 ) + ( m _ 3 + m _ 4 ) ( \\alpha _ 3 + \\alpha _ 4 ) = 2 m , \\\\ ( m _ 3 + m _ 4 ) ( \\alpha _ 1 + \\alpha _ 2 ) + ( m _ 1 + m _ 2 ) ( \\alpha _ 3 + \\alpha _ 4 ) = 2 m , \\end{gather*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\Lambda = \\big \\{ a \\in \\mathbb { R } ^ { n - m } \\big | ( a , 0 ) \\in G ( T _ { p i j } ^ { 0 , \\rho } ) \\big \\} . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} 2 f _ { i j } ^ { 2 } + 2 \\alpha R i c ' _ { i j } f _ { i j } & = \\frac { 2 \\alpha } { p } f _ { i j } ^ { 2 } + 2 \\alpha \\left ( \\frac { 1 } { q } f _ { i j } ^ { 2 } + R i c ' _ { i j } f _ { i j } \\right ) \\geq \\frac { 2 \\alpha } { p } f _ { i j } ^ { 2 } - \\frac { \\alpha q } { 2 } { R i c ' } _ { i j } ^ { 2 } , \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\mathbb { P } _ { x _ 0 , k } ^ { \\epsilon , \\hat { v } } \\bigl \\{ \\zeta ^ { \\epsilon } ( \\tau _ { D } ^ { \\epsilon , \\hat { v } } ) = k _ 0 \\ , \\vert \\ , \\tau _ { D } ^ { \\epsilon , \\hat { v } } < \\infty \\bigr \\} = 1 , \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\nu _ 2 ( f ) = \\min \\{ \\nu _ 1 ( f _ i ) + i \\gamma _ 1 \\} . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} e _ 2 ( y ) = e _ 3 ( y ) + \\frac 1 2 h ^ 2 W '' ( h a ) y ^ 2 \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } R _ { n } ^ { \\lambda } ( x ) R _ { m } ^ { \\lambda } ( x ) \\ , d x = \\gamma _ { n } ^ { \\lambda } \\delta _ { n m } . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} ( G \\to \\pi { G } ) : = ( G , \\pi { G } ) \\in V \\times V . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\Phi \\varPsi ( \\mathcal { C } _ 2 ) ( c ) & = \\varPsi ( \\mathcal { C } _ 2 ) ( \\Phi ( c ) ) \\\\ & = \\widetilde { \\mathcal { C } _ 2 } \\mu c _ \\mu = \\Phi ( c ) . ) \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} t _ i & i = 1 , \\dots , n \\\\ e ^ \\alpha & \\alpha \\in C ^ * : = C \\setminus \\{ \\ 0 , \\ 1 \\} \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} ( 1 + p a ) ^ { - 1 } : = \\sum ^ { \\infty } _ { \\nu = 0 } ( - p a ) ^ { \\nu } \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} T _ n ( x ) : = { 1 \\over x + n } \\ \\ \\ x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} d Y _ t = 2 \\sqrt { f ( t ) } d t + n ^ { - 1 / 2 } d W _ t , t \\in [ 0 , 1 ] , \\ \\ f \\in \\Theta , \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{gather*} D \\eta ^ { ( n - 1 ) } ( e ) = \\iota _ { \\rho ( e ) } \\widetilde { h } . \\end{gather*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\left < \\sigma \\right > = \\eta ( \\sigma ) [ \\sigma ] . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} \\Phi _ n ^ { ( 1 ) } ( z ) = \\frac { q ^ n ( q ^ { - n } - q ^ \\gamma ) } { ( 1 - q ^ \\gamma ) } p _ n ( z ; a , b | q ) + \\frac { q ^ { n + \\gamma } ( 1 - q ^ { - n } ) } { ( 1 - q ^ \\gamma ) } p _ { n - 1 } ( z ; a , b q | q ) , \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Psi _ 0 ( Y _ j ) = X _ j , \\\\ \\Psi _ 1 ( Y _ j ) = Y _ j , \\end{array} \\right . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} M _ { X } \\cong \\prod _ { j = 1 } ^ { l } G ( V ^ { \\gamma , t _ { j } } _ { t _ { j } - 1 } ) . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} w _ m : = W ^ { \\frac { p _ m } { r _ m } } \\widehat { w } ^ { \\frac { p _ m } { \\delta _ m } } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{gather*} [ e _ 1 , f e _ 2 ] = f [ e _ 1 , e _ 2 ] + \\rho ( e _ 1 ) f \\cdot e _ 2 , \\end{gather*}"}
-{"id": "552.png", "formula": "\\begin{align*} i ^ \\ast \\circ \\nu ( t _ 2 ) = \\sigma ^ \\ast ( t _ 2 ) = s _ 1 , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { S } } _ { n } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { \\sin ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n + 1 } } , \\\\ \\widetilde { \\mathcal { C } } _ { n } ( z ) & = \\sum _ { k = 1 } ^ { \\infty } \\frac { \\cos ( 2 \\pi k z ) } { ( \\pi k ) ^ { 2 n } } , \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\to 0 } \\sup _ { t \\in [ a , b ] } | F _ \\epsilon ( t ) - F ( t ) | & = \\limsup _ { \\epsilon \\to 0 } \\sup _ { t \\in [ a , b ] } \\bigg | \\frac { j _ \\epsilon ^ 2 } { 2 t ^ 2 } - \\frac { j ^ 2 } { 2 t ^ 2 } \\bigg | \\leq \\lim _ { \\epsilon \\to 0 } \\frac { | j _ \\epsilon ^ 2 - j ^ 2 | } { 2 a ^ 2 } = 0 . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} = - \\sum \\limits _ { ( M _ { S ' } , \\mu _ { S ' } ) \\in X \\setminus \\{ ( M _ S , \\mu _ S ) \\} } \\ , \\ \\sum \\limits _ { b \\in Y _ { ( M _ { S ' } , \\mu _ { S ' } ) } } ( - 1 ) ^ { L _ { M _ { S ' } , M _ b } } . \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ 3 ( x ) + \\frac { 2 } { 6 ^ 4 } ( 3 ^ 4 x - 3 3 ) ( 7 - 2 ^ 4 x ) + \\frac { 2 } { 6 ^ 5 } ( 2 ^ 5 x - 1 3 ) ( 1 0 1 - 3 ^ 5 x ) \\\\ & + \\frac { 2 } { 6 ^ 6 } ( 2 ^ 6 x - 2 6 ) ( 3 0 3 - 3 ^ 6 x ) \\Bigr ) \\Bigr | _ { x = \\frac { 2 7 } { 6 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 6 } - \\eta \\\\ & = - \\frac { 1 9 0 6 3 4 9 5 2 1 5 4 4 4 9 3 8 7 3 2 6 1 5 4 9 9 7 1 9 2 0 3 4 9 } { 4 2 5 6 2 3 6 9 9 6 1 5 6 9 2 2 6 1 4 6 6 8 0 2 4 2 3 6 2 2 1 1 0 2 5 6 0 0 0 } < 0 , \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\mathbf { s } _ \\mathcal { Q } } = \\frac { 2 } { \\pi } \\left ( ^ { - 1 } \\left ( \\mathbf { K } \\mathfrak { R } \\{ \\mathbf { C } _ { \\mathbf { s } } \\} \\mathbf { K } \\right ) + j ^ { - 1 } \\left ( \\mathbf { K } \\mathfrak { I } \\{ \\mathbf { C } _ { \\mathbf { s } } \\} \\mathbf { K } \\right ) \\right ) . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} q ^ n - 1 = \\prod _ { d | n } \\Phi _ d ( q ) . \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} X _ i u = 0 ( Y _ i + p _ i ) u = 0 ( Z + q ) u = 0 \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} & - 1 - \\frac { v } { v _ { 1 } } + \\frac { f ( x , y _ { 1 } , v _ { 1 } ) } { f ( x , y , v ) } + \\frac { v } { v _ { 1 } } \\frac { f ( x , y , v ) } { f ( x , y _ { 1 } , v _ { 1 } ) } \\\\ & = \\left ( 1 - \\frac { f ( x , y , v ) } { f ( x , y _ { 1 } , v _ { 1 } ) } \\right ) \\left ( \\frac { f ( x , y _ { 1 } , v _ { 1 } ) } { f ( x , y , v ) } - \\frac { v } { v _ { 1 } } \\right ) \\leq 0 . \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\beta _ { j , j ' } = \\beta \\hat { \\otimes } 1 : S { \\frak C } ( D _ { r _ i } ^ j ) & \\to S C _ 0 ( V _ { j , j ' } , C l ( V _ { j , j ' } ) ) \\hat { \\otimes } { \\frak C } ( D _ { r _ i } ^ j ) \\\\ & \\cong S { \\frak C } ( E _ { r _ i } ^ { j . j ' } ) \\\\ & \\hookrightarrow S { \\frak C } ( V _ { j ' } ) \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} R _ X ( \\mathbf { x } ) = \\prod _ { i = 1 } ^ { p - 1 } \\bigl ( \\langle \\mathbf { 1 } _ X , \\mathbf { x } \\rangle - ( i p - 1 ) \\bigr ) . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\| a \\| _ { \\mathrm { H S } } = \\| u - \\mathbf { 1 } \\| _ { \\mathrm { H S } } \\leq C _ { 1 , l , M } . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\mathcal { K } ^ { \\Sigma , L } \\frac { 1 } { \\sqrt { L } } d \\sigma _ { \\Sigma , L } + \\sum _ { i = 1 } ^ n \\int _ { \\gamma _ i } k ^ { L , s } _ { \\gamma _ i , \\Sigma } \\frac { 1 } { \\sqrt { L } } d { s } _ L = 2 \\pi \\frac { \\chi ( \\Sigma ) } { \\sqrt { L } } . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { F H N } } ^ { \\textrm { O F D M A } } = \\frac { P _ \\textrm { N } { | h _ { \\textrm { B N } } | ^ 2 } } { { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } \\left ( 1 - \\theta \\right ) } + \\frac { \\beta _ { \\textrm { F } } \\eta P _ \\textrm { B } | h _ { \\textrm { B F } } | ^ 2 | h _ { \\textrm { N F } } | ^ 2 } { d _ { \\textrm { B F } } ^ { \\alpha } d _ { \\textrm { N F } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 \\left ( 1 - \\theta \\right ) } . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\omega _ 1 { } ^ 2 = 4 \\bar z _ 1 \\ , \\theta ^ 1 . \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} & ( \\sum _ g g ^ { - 1 } ( \\alpha ) g ) ( v _ i , \\sigma v _ i ) \\\\ = \\ , & ( v _ i , \\sigma v _ i ) \\left ( \\begin{array} { c c } \\alpha _ 1 - \\alpha _ 2 - \\alpha _ 5 + \\alpha _ 6 & \\alpha _ 2 - \\alpha _ 3 + \\alpha _ 4 - \\alpha _ 6 \\\\ - \\alpha _ 2 + \\alpha _ 3 + \\alpha _ 4 - \\alpha _ 5 & \\alpha _ 1 - \\alpha _ 3 + \\alpha _ 5 - \\alpha _ 6 \\end{array} \\right ) \\\\ = & \\ , ( v _ i , \\sigma v _ i ) { \\bf { A } } , , \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} H = \\int h _ t d B _ t \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left | \\frac { t ^ { 2 H l _ { 0 } } } { \\lambda _ { t } ^ { ( l ) } } \\right | ^ { q } ; \\| U _ { t } ^ { ( l ) } \\| _ { \\textsc { H S } } < r \\right ] = I _ t + J _ t , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} | \\partial _ j e ^ { t \\Delta } g | _ \\beta \\leq c t ^ { \\frac { 3 } { 2 } \\left ( \\frac { 1 } { \\beta } - \\frac { 1 } { \\alpha } \\right ) - \\frac { 1 } { 2 } } | g | _ \\alpha , \\ u \\in L ^ \\alpha ( \\mathbb { R } ^ 3 ) , \\ j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} Q _ { m _ { 1 } , \\ldots , m _ { d } } ^ { ( j ) } = Q _ { m _ { 1 } , \\ldots , m _ { d - 1 } } \\cap \\left [ x _ { 2 } = \\frac { d } { j - d + 1 } x _ { 1 } - \\frac { d } { j - d + 1 } \\left ( x _ { 1 } ^ { 0 } + \\cdots + \\frac { 1 } { d } \\binom { j + 1 } { d - 1 } x _ { d } ^ { 0 } + m _ { d } \\right ) \\right ] , \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty m ^ s \\int | \\nabla f _ m | ^ 2 d x d m = \\int \\left ( \\frac { \\sin \\pi s } { \\pi } \\int _ 0 ^ \\infty \\frac { m ^ s } { ( | \\xi | ^ 2 + m ) ^ 2 } d m \\right ) | \\xi | ^ 2 | \\hat { f } ( \\xi ) | ^ 2 d \\xi = s \\| f \\| ^ 2 _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{gather*} y _ s = \\frac { 1 } { s } \\sum _ { i = 1 } ^ s x _ i , \\end{gather*}"}
-{"id": "3781.png", "formula": "\\begin{align*} & \\sigma ^ * > 0 , \\ \\phi ( - \\infty ) = p ^ + , \\ \\phi ( + \\infty ) = p ^ - \\ \\ { \\rm a n d } \\\\ & \\phi ( x _ 1 - c t - a ) \\preceq U ( x , t ) \\preceq \\phi ( x _ 1 - c t - b ) , \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\norm { \\pi } ( A ) = \\sup \\Big \\{ \\sum _ { i = 1 } ^ { \\infty } \\norm { \\pi ( A _ i ) } \\big | A = \\bigcup _ { i = 1 } ^ { \\infty } A _ i , A _ i \\in \\mathcal { B } ( \\Omega ) , A _ i \\cap A _ j = \\emptyset , i , j \\in \\mathbb { N } \\Big \\} \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} h ( x ) & \\le g _ { 7 - } ( x ) \\Bigr | _ { x = \\frac { 2 0 2 } { 4 8 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 7 } - \\eta \\\\ & = - \\frac { 1 0 3 6 9 8 7 9 4 7 2 3 9 2 1 2 0 1 6 5 9 1 6 7 2 0 5 7 6 7 3 1 3 } { 3 0 5 8 1 8 5 1 0 0 9 4 2 3 8 1 4 3 4 2 4 2 9 5 0 7 4 7 5 0 7 0 1 4 4 3 2 0 0 0 } < 0 , \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\hat u ^ { R } _ 1 ( t ) - \\hat u ^ { R } _ 2 ( t ) = ( k _ 2 ( t ) - k _ 1 ( t ) ) \\hat S ( t ) \\ , \\quad \\Omega \\setminus \\Gamma ( t ) \\ , . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} r _ { \\textrm { N } } = { R } / { \\log \\left ( 1 + \\frac { P _ { \\textrm { B } } \\lambda _ { \\textrm { B N } } } { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ 2 } \\right ) } . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} f _ q ( Q ) = e _ { q , \\lambda _ q } ( Q ) \\frac { \\prod _ i ( \\beta _ i Q ; q ) _ \\infty } { \\prod _ i ( \\alpha _ i Q ; q ) _ \\infty } \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{gather*} A _ \\lambda \\partial _ t ( u _ \\lambda , v _ \\lambda ) + B _ \\lambda ( u _ \\lambda , v _ \\lambda ) = ( g _ \\lambda , g _ { \\Gamma \\lambda } ) - ( T _ \\lambda \\pi ( f ) , T _ \\lambda \\pi _ \\Gamma ( f _ \\Gamma ) ) \\quad ( 0 , T ) \\ , , \\\\ ( u _ \\lambda , v _ \\lambda ) ( 0 ) = ( u _ 0 , u _ { 0 | \\Gamma } ) \\ , . \\end{gather*}"}
-{"id": "7619.png", "formula": "\\begin{align*} c _ 0 ( \\Gamma ) = \\widehat { \\Z \\Gamma } \\otimes _ { \\Z _ p } \\Q _ p = \\Big \\{ \\sum _ { \\gamma } x _ { \\gamma } \\gamma \\mid x _ { \\gamma } \\in \\Q _ p \\ ; \\ ; | x _ { \\gamma } | \\to 0 \\ ; \\ ; \\gamma \\to \\infty \\Big \\} \\ ; . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\Omega : t ^ k \\longmapsto \\ , & \\left ( \\frac { z } { 1 - z } \\right ) ( f ( z , t ) - 1 ) \\left ( f ( z , 1 ) ^ { k } + t f ( z , 1 ) ^ { k - 1 } + \\cdots + t ^ { k } \\right ) \\\\ = \\ , & \\left ( \\frac { z } { 1 - z } \\right ) ( f ( z , t ) - 1 ) \\left ( \\frac { f ( z , 1 ) ^ { k + 1 } - t ^ { k + 1 } } { f ( z , 1 ) - t } \\right ) . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\langle | f | \\rangle _ { A , Q } = \\inf \\{ \\lambda > 0 : \\frac 1 { | Q | } \\int _ Q A ( | f | / \\lambda ) \\le 1 \\} . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\frac { 2 } { p } + \\frac { n } { q } = \\frac { n } { 2 } - s \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} E : Y ^ 2 = 4 X ^ 3 - g _ 2 X - g _ 3 \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} H _ 3 = ( 1 , \\dots , 1 \\ , | \\ , 1 , \\dots , 1 , - 1 ) . \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} U _ k ( x ) = \\frac { 1 } { k ! } \\sum _ { i = 0 } ^ k ( - 1 ) ^ i { k \\choose i } M ( x - i ) ^ k . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\bigcap _ { \\delta > 0 } \\overline { g ^ 0 _ t ( F _ { t + \\delta } \\setminus F _ t ) } & = \\bigcap _ { \\delta > 0 } \\overline { \\iota _ t \\circ g _ t ( F _ { t + \\delta } \\setminus F _ t ) } = \\iota _ t \\left ( \\bigcap _ { \\delta > 0 } \\overline { g _ t ( F _ { t + \\delta } \\setminus F _ t ) } \\right ) \\\\ & = \\iota _ t ( \\{ \\xi ( t ) \\} ) = \\{ U ( t ) \\} . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\hat { \\mathbf { x } } = \\mathbf { W } ^ H \\mathbf { y } _ { \\mathcal { Q } } , \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\sum _ { m \\equiv b \\mod { q } } \\lambda ( m ) g ( m ) = \\frac { 1 } { q } \\sum _ { d \\mid q } \\sum _ { m = 1 } ^ { \\infty } \\lambda ( m ) S ( b , m ; d ) \\check { g } _ d ( m ) , \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { S } x \\theta ( x , y , t ) d z = - \\int _ { S } x ( - \\Psi _ y \\theta _ x + \\Psi _ x \\theta _ y ) d z \\ , , \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} e _ { M } + e _ { N } = e _ { A } + e _ { B } = 2 e _ { A \\cap B } + e _ { A \\setminus B } + e _ { B \\setminus A } . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} P ( t ) = \\sum _ { k \\geq 0 } x _ k ( t _ 0 , \\cdots , t _ s ) t ^ k = \\det A ( t ) \\ ( A ( t ) = A ( t _ 0 , \\cdots , t _ s , t ) ) \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} & \\underset { n _ 0 , n _ d \\ge 0 , d \\not \\in C _ J , \\atop n _ 0 + \\sum _ { d \\not \\in C _ J } n _ d = \\# C _ J - 1 } { \\sum } \\frac { ( 2 \\pi i ) ^ { n _ 0 } } { \\underset { d \\not \\in C _ J } { \\prod } { ( \\alpha _ d - a _ J ) ^ { n _ d } } } \\epsilon ^ { \\# C _ J - 1 } + O ( \\epsilon ^ { \\# C _ J } ) . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\begin{aligned} { } \\| \\tilde { u } - L \\| _ { L ^ { \\infty } ( \\tilde { \\Omega } \\cap B ( \\sigma ) ) } & = \\sup _ { p \\in \\tilde { \\Omega } \\cap B ( \\sigma ) } \\Big | \\frac { ( u - L _ { \\nu } ) ( \\delta _ { \\sigma ^ { \\nu } } ( p ) ) } { \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) } - L ( p ) \\Big | \\\\ & = \\frac { 1 } { \\sigma ^ { \\nu } \\omega ( \\sigma ^ { \\nu } ) } \\| u - L _ { \\nu + 1 } \\| _ { L ^ { \\infty } ( \\Omega \\cap B ( \\sigma ^ { \\nu + 1 } ) ) } , \\end{aligned} \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} & D ^ { i j } _ l ( x ) \\ \\ ( i , j = 1 , 2 , \\cdots , N ) , \\ \\ q _ l ( x ) \\in \\R ^ N \\ \\ ( l = 1 , 2 , \\cdots , m ) , \\\\ & F ( x , u _ 1 , \\cdots , u _ m ) = ( f _ 1 , f _ 2 , \\cdots , f _ m ) \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\delta \\widehat H + \\frac { 1 } { 2 } \\hat h ^ { a b } ( \\delta \\hat \\sigma ) _ { a b } = - \\widehat \\Delta f + 2 f + \\hat \\nabla ^ b ( P ^ c \\hat h _ { c b } ) . \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } \\phi \\big [ \\big ( \\exp ( \\frac { t } { 2 } B ) \\exp ( t A ) \\exp ( \\frac { t } { 2 } B ) \\big ) ^ \\frac { 1 } { t } \\big ] = & \\ \\phi \\big [ \\lim _ { t \\searrow 0 } \\big ( \\exp ( \\frac { t } { 2 } B ) \\exp ( t A ) \\exp ( \\frac { t } { 2 } B ) \\big ) ^ \\frac { 1 } { t } \\big ] \\\\ = & \\ \\phi [ \\exp ( A + B ) ] , \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} E _ i \\{ \\check { S } _ k ( t ) \\} & = \\mu _ { \\eta ^ , i } t \\\\ _ i \\{ \\check { S } _ k ( t ) \\} & \\leq \\sigma ^ 2 _ { \\eta ^ , i } \\xi t . \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} \\beta _ { j , j ' } = \\beta \\hat { \\otimes } 1 : S _ { r _ i } { \\frak C } ( D _ { r _ i } ^ j ) \\to S _ { r _ { i + 1 } } { \\frak C } ( D _ { r _ { i + 1 } } ^ { j ' } ) . \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} \\langle \\theta _ { i } , \\theta _ { j } \\rangle _ { t } = \\int _ { 0 } ^ { t } \\omega _ { i } ( W _ { s } ) \\cdot \\omega _ { j } ( W _ { s } ) \\ , d s \\ , . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} b ( \\gamma ^ { - n } z _ 0 ) = b ( z _ 0 ) = 0 \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} { \\mathcal U } _ { [ L , M ] } ~ : = ~ \\Big \\{ u _ { 0 } \\in { \\bf L } ^ { \\infty } ( \\mathbb { R } ) \\ \\big | \\ \\mbox { S u p p \\ , } ( u _ { 0 } ) \\subset [ - L , L ] \\ , \\ \\| u _ { 0 } \\| _ { { \\bf L } ^ { \\infty } \\left ( \\mathbb { R } \\right ) } \\leq M \\Big \\} , \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} \\det ( M ' ) & = \\mu \\vert \\alpha \\vert ( - 1 ) ^ { \\vert \\alpha \\vert - 1 } + \\tfrac { 1 } { 2 \\mu c _ \\alpha } \\lambda \\vert \\alpha \\vert ( - 1 ) ^ { \\vert \\alpha \\vert - 1 } \\\\ & = | \\alpha | ( - 1 ) ^ { \\vert \\alpha \\vert - 1 } \\left ( \\mu + \\tfrac { \\lambda } { 2 \\mu c _ \\alpha } \\right ) \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} f ( a ) = \\sum _ i f _ i \\norm { i } { a } \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} u _ k ^ { \\epsilon } ( t ) = v _ k ^ { \\epsilon } \\bigl ( X _ { k } ^ { \\epsilon , v _ k } ( t ) \\bigr ) , \\forall t , k = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 0 } ~ ~ \\Big | ~ ~ y _ 0 + \\int _ { 0 } ^ { \\cdot } b _ { j ^ { \\delta } ( s ) } ( s , y ^ { \\delta } ( s ) ) d s + \\int _ { 0 } ^ { \\cdot } \\sigma _ { j ^ { \\delta } ( s ) } ( s , y ^ { \\delta } ( s ) ) d B ^ { \\delta } ( s ) ~ ~ \\\\ & - ~ ~ \\Big ( y _ 0 + \\int _ { 0 } ^ { \\cdot } b _ { j ( s ) } ( s , y ( s ) ) d s + \\int _ { 0 } ^ { \\cdot } \\sigma _ { i ( s ) } ( s , y ( s ) ) d B ( s ) ~ ~ \\Big ) ~ ~ \\Big | _ { ( 0 , T ) } ~ ~ = ~ ~ 0 . \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\widehat { w } = \\Big ( \\prod _ { i = 1 } ^ { m - 1 } w _ i ^ { \\frac 1 { p _ i } } \\Big ) ^ { \\varrho } \\in A _ { \\frac { 1 - r } { r } \\varrho } \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} F _ { } ( t ) = \\frac { 1 } { 2 } \\log \\max \\left ( t , ~ \\frac { \\sigma ^ 2 } { \\theta } \\right ) , \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\dim P _ { z } \\Pi \\mu = \\min \\{ 1 , \\frac { h } { - \\log r } \\} \\ : . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} ( \\delta f ) ( x _ 0 , x _ 1 , \\ldots , x _ n ) = \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { i } \\rho ( x _ i ) f ( x _ 0 , \\ldots , \\hat { x _ i } , \\ldots , x _ k ) \\\\ + \\sum _ { 0 \\leq i < j \\leq n } ( - 1 ) ^ { i + j } f ( [ x _ i , x _ j ] , x _ 0 , \\ldots , \\hat { x _ i } , \\ldots , \\hat { x _ j } , \\ldots , x _ k ) , \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\small \\textbf { R } _ k = \\mathbb { E } \\left \\lbrace \\textit { \\textbf { h } } _ k ( f ) \\textit { \\textbf { h } } _ k ^ { \\rm H } ( f ) \\right \\rbrace = \\sum \\limits _ { p = 1 } ^ { N _ { \\rm S } + 1 } | \\alpha _ { k , p } | ^ 2 \\textbf { a } \\left ( \\theta _ { k , p } \\right ) \\textbf { a } ^ { \\rm H } \\left ( \\theta _ { k , p } \\right ) . \\normalsize \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\| U { \\mathcal T } \\{ c _ k \\} _ { k = 1 } ^ \\infty \\| \\leq K \\| U \\{ c _ k \\} _ { k = 1 } ^ \\infty \\| \\ \\mbox { f o r a l l f i n i t e s e q u e n c e s $ \\{ c _ k \\} _ { k = 1 } ^ \\infty $ } . \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\mathcal { Q } = \\bigcup _ { ( { r _ 0 } , { \\beta } ) \\in \\mathcal { Q } \\times \\mathcal { B } \\colon { r _ 0 \\in { U _ \\beta } } } { \\bigg ( { U _ { \\beta } \\cap { B \\left ( { r _ 0 } , { \\widetilde { \\delta } _ { { \\epsilon } , { r _ 0 } , { s ( { r _ 0 } ) } } } \\right ) } } \\bigg ) } \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} f _ T ( z ) \\ : = \\ \\sum _ { j = 0 } ^ { d + 1 } f _ { j - 1 } \\ , z ^ j \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} 0 \\leq p _ D ( t , x , y ) = p _ D ( t , y , x ) \\leq p ( t , x , y ) \\ , . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\frac { B _ \\varepsilon ^ k } { \\varphi _ { { N } _ \\varepsilon } ( B _ \\varepsilon ^ 2 ) } = o ( 1 ) \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\nu _ { i + 1 } = [ \\nu _ i ; \\nu _ { i + 1 } ( \\phi _ { i } ) = \\gamma _ { i + 1 } ] . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} P _ { z } \\Pi \\sigma ^ { j } \\nu ( B ( x , \\eta ) ) = \\sum _ { u \\in \\Lambda ^ { j } } P _ { z } \\Pi \\sigma ^ { j } ( \\nu | _ { [ u ] } ) ( B ( x , \\eta ) ) \\le | \\Lambda | ^ { m } \\eta ^ { ( 1 - \\delta ) ( \\beta - c _ { 1 } \\delta ) } < \\eta ^ { ( 1 - \\delta ) \\alpha } , \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} Q ^ N : = \\sum _ { k = 0 } ^ d \\frac { 1 } { k ! } ( N \\log ( Q ) ) ^ d \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\left [ \\phi _ i \\star _ \\tau , \\phi _ j \\star _ \\tau \\right ] \\phi _ k = 0 \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} G ( m , N , a , \\varepsilon , s ) \\leq 2 \\sum _ { k = 1 } ^ { ( { m ( 1 + \\epsilon ) \\over 2 } ) ^ { { 1 \\over a } } } k ^ { - d s } \\sum _ { i _ 2 \\cdots i _ n \\in \\widetilde { A } ( m , n , k , a , \\epsilon ) } \\prod _ { k = 2 } ^ n i _ k ^ { - d s } , \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} Q ( T _ 1 + T _ 2 ) & = \\{ \\ , \\{ f , Q ( h + k ) \\} : \\ , \\{ f , h \\} \\in T _ 1 , \\ , \\ , \\{ f , k \\} \\in T _ 2 \\ , \\} \\\\ & = \\{ \\ , \\{ f , Q h + Q k ) \\} : \\ , \\{ f , h \\} \\in T _ 1 , \\ , \\ , \\{ f , k \\} \\in T _ 2 \\ , \\} = Q T _ { 1 } + Q T _ { 2 } . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\mathrm { r e s } ( \\overline { C } ^ * _ { \\mathbb { Q } } ) = \\ , \\ _ k \\overline { C } ^ * _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\int _ { \\R } { \\mathcal H } ^ { N - 1 } \\left ( \\varphi ^ { - 1 } ( s ) \\right ) { \\rm d } s = \\int _ \\Omega \\vert \\nabla \\varphi ( x ) \\vert { \\rm d } x . \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} ( 1 - t ) E [ e ^ { X t } ] = \\frac { - t } { \\log ( 1 - t ) } = \\sum _ { n = 0 } ^ \\infty b _ n ( - 1 ) ^ n \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} & \\liminf _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P ( S _ n ( \\kappa _ n ( x ) ) \\leq \\lfloor \\kappa _ n ( x ) \\rfloor - a _ n ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\geq \\liminf _ { n \\to \\infty } \\frac { 1 } { b _ c ^ { ( n ) } } \\log P \\left ( \\frac { n - a _ n - S _ n ( \\kappa _ n ( x ) ) } { f _ 2 ( n ) } > x ( 1 + \\delta ) \\right ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\geq - \\inf _ { y \\in ( x ( 1 + \\delta ) , + \\infty ) } H ( \\ell _ 2 y ) = - H ( \\ell _ 2 x ( 1 + \\delta ) ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\chi _ { - 2 k - 1 } ( A , z _ 1 , \\ldots , z _ { - 2 k - 1 } ) = { \\rm d i a g } ( A , z _ 1 , \\ldots , z _ { - 2 k - 1 } ) \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\Gamma ( 1 + t ) = t \\ , \\Gamma ( t ) t > 0 \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} \\lim _ { x \\to - \\infty } U ( x ) = 1 , \\lim _ { x \\to + \\infty } U ( x ) = 0 . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\lambda _ { \\varepsilon } ( x ) = \\frac { 1 } { \\varepsilon } \\lambda \\left ( \\frac { x } { \\varepsilon } \\right ) , \\lambda \\in \\C \\infty \\left ( \\mathbb { R } , [ 0 , 1 ] \\right ) , \\int _ { \\mathbb { R } } \\lambda ( x ) \\ ; d x = 1 , \\lambda ( x ) = 0 \\ ; \\forall x \\not \\in \\left [ - 1 , 1 \\right ] . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} P \\{ X \\in B \\} = \\int _ B f ( x ) d x , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} C _ { H } ( X ) & = \\{ f \\in C ^ { \\ast } ( X ) \\mid f ( \\ast ) _ { s } ^ f s \\in \\mathcal { H } \\} , \\\\ C _ { P } ( X ) & = \\{ f \\in C ^ { \\ast } ( X ) \\mid f ( \\ast ) _ { s } ^ f s \\in \\mathcal { P } \\} , \\\\ C _ { L } ( X ) & = \\{ f \\in C ^ { \\ast } ( X ) \\mid f ( \\ast ) _ { s } ^ f s \\in \\mathcal { L } \\} . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} \\begin{gathered} \\\\ \\iff \\sum c _ i \\nu _ i = 0 \\\\ \\iff \\sum c _ i \\eta ( \\sigma _ i ) = 0 . \\end{gathered} \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} u _ m ( x , t ) & = U _ m ^ - ( x , t ) \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\| \\hat Q _ r ( \\hat \\Sigma - \\mu _ r I ) R _ r \\| _ 2 & = \\sqrt { \\sum _ { j \\in \\mathcal { I } _ r } ( \\hat \\lambda _ j - \\mu _ r ) ^ 2 \\| ( \\hat u _ j \\otimes \\hat u _ j ) R _ r \\| _ 2 ^ 2 } \\\\ & \\leq C x m _ r \\mu _ r \\sqrt { \\sum _ { j \\in \\mathcal { I } _ r } \\| ( \\hat u _ j \\otimes \\hat u _ j ) R _ r \\| _ 2 ^ 2 } = C x m _ r \\mu _ r \\| \\hat Q _ r R _ r \\| _ 2 . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\int _ { I _ 2 } d \\lambda ^ { ( 2 ) } ( \\lambda _ 2 - \\lambda _ 1 ) & = \\frac { 1 } { 3 ! } , \\\\ \\int _ { I _ 2 } d \\lambda ^ { ( 2 ) } ( \\lambda _ 2 - \\lambda _ 1 ) \\ln ( \\lambda _ 2 - \\lambda _ 1 ) & = - \\frac { 5 } { 3 6 } . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} [ b , T ] _ { l } ( f _ { 1 } , \\cdots , f _ { m } ) = T ( f _ { 1 } , \\cdots , b f _ { l } , \\cdots , f _ { m } ) - b T ( f _ { 1 } , \\cdots , f _ { m } ) . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\langle \\alpha _ 1 , \\dots , \\alpha _ n \\rangle ^ \\textnormal { c o h } _ { 0 , n , 0 } = \\left \\{ \\begin{aligned} \\int _ X \\alpha _ 1 \\cup & \\alpha _ 2 \\cup \\alpha _ 3 & & \\textnormal { i f } n = 3 \\\\ & 0 & & \\textnormal { o t h e r w i s e . } \\end{aligned} \\right . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\theta ( G ) = \\theta ( J _ { n } ^ p ) \\cdot \\theta ( C _ 5 ) ^ 3 = 5 ^ { 3 / 2 } \\biggl ( { p \\over ( p + 1 ) ^ 2 } + o ( 1 ) \\biggr ) n ^ 2 . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} y - x = \\int _ { 0 } ^ { 1 } V ( \\Phi _ { t } ( x ; h ) ) d h _ { t } . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\frac { R _ { 1 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 1 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = - i ( - R _ { 1 2 } + R _ { 1 4 } ) = i R _ { 1 2 } - i R _ { 1 4 } = i = u _ 1 ; \\\\ \\frac { R _ { 3 2 } } { i - \\omega ( 2 , 2 ) } u _ 2 + \\frac { R _ { 3 4 } } { i - \\omega ( 4 , 4 ) } u _ 4 = i R _ { 3 2 } - i R _ { 3 4 } = - i = u _ 3 . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\dot v & = k \\int ^ { \\infty } _ { 0 } f _ { 2 } ( \\tau ) e ^ { - \\alpha _ { 2 } \\tau } \\varphi _ { 1 } ( y ( t - \\tau ) ) d \\tau - u v ( t ) \\\\ & \\leq k M _ { 2 } G _ { 2 } - u v ( t ) , \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\frac { \\partial \\pi _ { t + 1 } } { \\partial m } ( z ) = s _ t ( z ) \\cdot \\frac { \\partial \\pi _ t } { \\partial m } ( z ) + c [ C _ t ( z ) ] \\cdot \\left [ 1 - \\frac { \\partial \\pi _ t } { \\partial m } ( z ) \\right ] , \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} I _ { w , \\gamma , N } = \\left \\{ \\xi \\in \\R \\ : \\ \\widehat { w } ( \\xi ) \\leq 1 - \\sqrt [ N ] { 1 - \\gamma } \\right \\} . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} x _ t = x _ 0 + \\int _ 0 ^ t \\vartheta _ s \\ , \\mathrm { d } { s } + 2 c _ { H } ^ { B , R } \\int _ 0 ^ t \\varphi _ s \\delta B _ s ^ { \\frac { H } { 2 } + \\frac { 1 } { 2 } } + \\int _ 0 ^ t \\psi _ s \\delta R _ s ^ H \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} x ^ * = [ - 0 . 6 7 7 8 4 7 2 5 8 7 7 9 \\ 0 . 9 1 5 7 5 7 2 1 3 5 0 6 \\ { - 1 . 6 7 6 5 6 7 4 7 1 0 9 2 } \\ { - 1 . 1 2 9 3 9 0 4 2 9 4 0 2 } \\ 0 . 7 6 9 4 7 8 5 7 4 8 1 5 \\ 0 . 7 4 0 9 3 3 6 1 7 8 5 9 ] \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\Delta u = Q ( x , u , \\nabla u ) , \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} \\sum _ j H ^ { p ^ { \\ast } } _ { , i j } \\omega _ j = d H ^ { p ^ { \\ast } } _ { , i } + \\sum _ j H ^ { p ^ { \\ast } } _ { , j } \\omega _ { j i } + \\sum _ { q } H ^ { q ^ { \\ast } } _ { , i } \\omega _ { q ^ { \\ast } p ^ { \\ast } } , \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} \\delta \\big ( K _ { G / G _ { i - 1 } } ( X ^ { G _ { i - 1 } } ) \\big ) = \\delta \\big ( K _ { G / G _ { i - 1 } } ( X ^ { G _ i } ) \\big ) = \\delta \\big ( K _ { G / G _ i } ( X ^ { G _ i } ) \\big ) . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} ( 1 + g ( r ) ) \\exp ( r ^ 2 ) - ( 1 + g ( 0 ) ) = 2 \\int _ 0 ^ r s H ( s ) \\exp ( s ^ 2 ) d s \\ , , \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} L _ { 1 ( n ) } ( q , b , q ) = \\frac { 1 } { 2 ( 1 - b ) } \\alpha ^ { * } _ n ( q , b , q ) + \\frac { 1 } { 2 ( 1 + b ) } \\alpha ^ { * } _ n ( q , - b , q ) , \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} f \\in D ( \\Delta _ { l o c } ) \\ \\ \\nabla f \\in D ( { \\rm d i v } _ { l o c } ) \\Delta f = { \\rm d i v } ( \\nabla f ) , \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} d ( y e _ { x ^ y } - e _ x y ) = y d ( e _ { x ^ y } ) - d ( e _ x ) y = y ( 1 - x ^ y ) - ( 1 - x ) y = y - y x ^ y - y + x y = x y - y x ^ y . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\log \\rho ( A ) = \\max _ { \\mu = [ \\mu _ { i , j } ] \\in \\Omega ( n ) } \\sum _ { i , j \\in [ n ] } \\mu _ { i j } \\log \\frac { a _ { i j } \\sum _ { k = 1 } ^ n \\mu _ { i k } } { \\mu _ { i j } } . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\Big \\{ \\frac { \\langle w , ( \\hat \\Sigma - \\tilde \\Sigma ) w \\rangle } { \\langle v , v \\rangle } \\geq 0 \\Big \\} \\cap \\Big \\{ \\tilde x \\geq \\frac { 2 z } { \\sqrt { n } } \\Big \\} = \\Big \\{ \\frac { \\langle w , ( \\hat \\Sigma - \\tilde \\Sigma ) w \\rangle } { \\langle v , v \\rangle ^ 2 } \\geq 0 \\Big \\} \\cap \\tilde { \\mathcal { E } } _ { 2 z } , \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} & y _ i = \\sum _ { j = 1 } ^ { m } U \\left [ j , k _ { i - j + 1 } \\right ] , \\\\ & k _ i = 1 + \\operatorname { r o u n d } \\left ( \\frac { \\left ( n - 1 \\right ) \\left ( x _ i - x _ \\mathrm { m i n } \\right ) } { x _ \\mathrm { m a x } - x _ \\mathrm { m i n } } \\right ) , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\mathrm { S I N R } _ { A _ i , E } = \\frac { \\mathrm { S N R } _ { A _ i , E } } { \\sum _ { D _ k \\in \\mathcal D _ u ^ i } \\mathrm { S N R } _ { D _ { k } ^ { \\ , t } , E } + 1 } . \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} C ( n , s ) = \\left ( \\int _ { \\mathbb R ^ n } \\frac { 1 - \\cos ( \\zeta _ 1 ) } { | \\zeta | ^ { n + s } } \\ , d \\zeta \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\frac { v o l ( I _ { n , m } ^ { \\tilde { n } } \\cap \\hat { \\mathfrak { D } } _ { \\tilde { n } } ) } { v o l ( \\hat { \\mathfrak { D } } _ { \\tilde { n } } ) } = \\frac { 1 } { \\sqrt { { \\tilde { n } } } } v o l ( \\{ ( x _ 1 , \\dots , x _ { \\tilde { n } } ) \\in ( 0 , 1 ) ^ { \\tilde { n } } \\mid x _ i \\in I _ { n , m } , \\sum x _ i \\in \\mathbb { Z } \\} ) . \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\varphi _ { n , k } ( r , l ) = \\int _ r ^ l \\int _ t ^ l \\left ( \\frac { s _ k ( \\tau ) } { s _ k ( t ) } \\right ) ^ { n - 1 } d \\tau \\ , d t . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} z ( 1 - z ) \\mathsf { F } _ { z z } \\left ( \\tfrac { 1 - \\sqrt { \\delta } } { 2 } , \\tfrac { 1 - \\sqrt { \\delta } } { 2 } ; 1 ; z \\right ) + \\left [ 1 - \\left ( 2 - \\sqrt { \\delta } \\right ) z \\right ] \\mathsf { F } _ z \\left ( \\tfrac { 1 - \\sqrt { \\delta } } { 2 } , \\tfrac { 1 - \\sqrt { \\delta } } { 2 } ; 1 ; z \\right ) - \\left ( \\tfrac { 1 - \\sqrt { \\delta } } { 2 } \\right ) ^ 2 \\mathsf { F } \\left ( \\tfrac { 1 - \\sqrt { \\delta } } { 2 } , \\tfrac { 1 - \\sqrt { \\delta } } { 2 } ; 1 ; z \\right ) = 0 . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} \\sqrt { \\lambda _ k } \\langle \\hat u _ j , u _ k \\rangle & = \\frac { \\sqrt { \\lambda _ k } } { \\hat \\lambda _ j - \\lambda _ k } \\langle \\hat u _ j , E u _ k \\rangle \\\\ & = \\frac { \\lambda _ k } { \\hat \\lambda _ j - \\lambda _ k } \\big ( \\bar \\eta _ { k j } \\sqrt { \\lambda _ j } \\langle \\hat u _ j , u _ j \\rangle + \\sum _ { l \\neq j } \\bar \\eta _ { k l } \\sqrt { \\lambda _ l } \\langle \\hat u _ j , u _ l \\rangle \\big ) \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d t ^ \\ell } V _ m ( t ) = ( - A ) ^ \\ell V _ { m , \\ell } ( t ) , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} h = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log | X ^ { \\Gamma _ n } | \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} \\left [ \\mathbf { R } _ { n } \\right ] _ { k , \\ell } = \\sqrt { \\frac { 2 } { n + 3 } } \\sin \\left [ \\frac { ( k + 1 ) ( \\ell + 1 ) \\pi } { n + 3 } \\right ] \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\mathcal { R } ^ L ( p , B , \\boldsymbol { d } ) = \\lambda f ( \\boldsymbol { d } ) , \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\bar { w } _ \\varepsilon ( 0 ) = 0 - r _ \\varepsilon \\bar { w } ' _ { \\varepsilon } ( r _ \\varepsilon ) = \\int _ 0 ^ { r _ \\varepsilon } ( \\Delta \\bar { w } _ \\varepsilon ) ~ r d r \\ , , \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} & d _ Q ( 1 4 , 6 ) = d _ Q ( 1 5 , 7 ) = 7 , d _ Q ( 1 7 , 6 ) = d _ Q ( 1 9 , 7 ) = 9 , \\\\ & d _ Q ( 1 7 , 7 ) = 8 d _ Q ( 2 0 , 7 ) = 1 0 . \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} m _ 1 ( z , t ) = 1 - z - t + t ^ 2 z - t ^ 2 z ^ 2 \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\chi ( j ) = \\left ( \\frac { j } { n } \\right ) : = \\begin{cases} + 1 ; \\\\ - 1 ; \\\\ \\phantom { + } 0 . \\end{cases} \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} \\mathrm { M a n t _ { G , b , \\mu } } ( \\mathrm { R e d } _ b ( \\pi ) ) = [ \\pi ] [ ( r _ { ( - 1 , 0 ^ { n _ 1 - 1 } ) } \\boxtimes r _ { ( - 1 , 0 ^ { n _ 2 - 1 } ) } ) \\circ ( L L ( \\rho _ 1 ) + L L ( \\rho _ 2 ) ) \\otimes | \\cdot | ^ { 2 - n _ 1 - n _ 2 } ] . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} g _ 2 ' ( u ) = \\frac { [ \\xi ' ( u ) - \\xi ' ( q ) ] ( 1 + z _ 2 - q - u z _ 2 ) - ( u - q ) ( \\xi ' ( 1 ) - \\xi ' ( q ) ) } { ( 1 + z _ 2 ) ( 1 + z _ 2 - q - u z _ 2 ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} & \\xi ^ { x , 0 } ( y ) = \\begin{cases} \\xi ( y ) & y \\neq x , \\\\ 0 & y = x , \\end{cases} \\\\ & \\xi ^ { x , y } _ { a , b } ( z ) = \\begin{cases} \\xi ( z ) & z \\neq x , \\\\ b \\xi ( x ) + a \\xi ( y ) & z = x \\end{cases} \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathbb { E } \\left [ \\int \\left ( h _ s - \\mu ( s , \\theta _ s ) \\right ) \\alpha _ s \\frac { \\partial } { \\partial x } \\mu ( s , \\theta ^ H _ s ) d [ B ] _ s \\right ] \\\\ = \\mathbb { E } \\left [ \\int \\left ( h _ s - \\mu ( s , \\theta ^ H _ s ) \\right ) \\frac { \\partial } { \\partial x } \\mu ( s , \\theta ^ H _ s ) d B _ s \\int \\alpha _ s d B _ s \\right ] = 0 \\end{array} \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} n = 0 , h + f = 0 . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} Y ^ \\prime _ N : = P _ N T _ N ^ \\prime P _ N ^ * \\ , . \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\lim _ { R \\to 0 ^ + } \\Phi ( A , M , \\nu , R ) = 0 ; \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { B } ) : = \\max \\lbrace \\mathrm { d p } ( v ) \\mid v \\in \\mathcal { B } \\rbrace . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} F _ 0 & : = B ( x , r _ 0 ) ^ c , \\\\ F _ { n + 1 } & : = B [ x , r _ n ] \\smallsetminus B ( x , r _ { n + 1 } ) \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\tilde { N } _ \\varepsilon = N _ \\varepsilon - 1 \\ , . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} L \\left ( { \\cal C } _ { t o t } \\right ) = I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} \\sigma ( \\theta _ 1 , \\theta _ 2 , \\phi ) = ( r ( \\theta _ 1 , \\theta _ 2 ) \\cos ( \\phi ) , r ( \\theta _ 1 , \\theta _ 2 ) \\sin ( \\phi ) , z ( \\theta _ 1 , \\theta _ 2 ) ) \\ ; , \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\Sigma ( W ) = \\left \\{ g \\in G ( \\mathbb { R } ) \\mathop { | } \\dim ( g . V \\cap \\mathcal { F } \\cap \\pi ^ { - 1 } ( W ) ) = \\dim ( V ) \\right \\} . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} R ( x , y , z ) = x - \\ln \\frac { \\cosh \\tfrac { y } { 2 } + \\cosh \\tfrac { x + z } { 2 } } { \\cosh \\tfrac { y } { 2 } + \\cosh \\tfrac { x - z } { 2 } } \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} R ( \\gamma ) = \\frac { n } { u ^ 2 } \\ ( ( n - 1 ) - ( n - 1 ) ( u ' ) ^ 2 - 2 u u '' \\ ) . \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\{ i _ 1 , i ' _ 1 , \\ldots , i _ { \\lambda _ 1 } , i ' _ { \\lambda _ 1 } \\} & = \\{ 1 , \\ldots , \\lambda _ 1 , s + 1 , \\ldots , \\lambda _ 1 + s \\} \\\\ \\{ i _ { \\lambda _ 1 + 1 } , i ' _ { \\lambda _ 1 + 1 } , \\ldots , i _ { s } , i ' _ { s } \\} & = \\{ \\lambda _ 1 + 1 , \\ldots , s , \\lambda _ 1 + 1 + s , \\ldots , 2 s \\} \\\\ \\{ j _ 1 , \\ldots , j _ k \\} & = \\{ 2 s + 1 , \\ldots , r \\} , \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} g ( z ) : = \\frac { ( 1 - \\alpha ^ 2 ) \\ , ( \\frac { 1 } { b } + b z ^ 2 ) } { 1 - 2 \\alpha z + z ^ 2 } \\ , . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\delta & \\leq | T ( e _ \\xi ) ( x _ \\xi , y _ \\xi ) | = | T ( e _ \\xi ) ( x _ \\xi , y _ \\xi ) - T ( e _ \\eta ) ( x _ \\xi , y _ \\eta ) | \\\\ & = | R ( e _ \\xi \\restriction _ { \\alpha \\times \\{ y _ \\xi \\} } ) ( x _ \\xi ) - R ( e _ \\eta \\restriction _ { \\alpha \\times \\{ y _ \\eta \\} } ) ( x _ \\xi ) | \\\\ & \\leq \\| R ( e _ \\xi \\restriction _ { \\alpha \\times \\{ y _ \\xi \\} } ) - R ( e _ \\eta \\restriction _ { \\alpha \\times \\{ y _ \\eta \\} } ) \\| < \\delta , \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\frac { 1 } { a _ n } \\log _ p \\chi _ n = h _ p + \\varepsilon _ n \\ ; \\varepsilon _ n \\to 0 \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} \\wp ( x ) \\cdot \\left ( a ( - 2 ) b - b ( - 2 ) a \\right ) \\otimes m = \\wp ( x ) \\cdot \\left ( 2 a ( - 2 ) b - T ( a ( - 1 ) b ) \\right ) \\otimes m , \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\hat { \\mathcal { U } } ^ L = \\ln \\left ( \\frac { \\lambda } { p + \\frac { a | \\mathcal { N } | } { | \\mathcal { N } | - 1 } } \\right ) + \\frac { a | \\mathcal { N } | } { | \\mathcal { N } | - 1 } + p - \\lambda . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} c _ 2 : = \\max _ { t \\in [ t _ 0 , t _ 1 ] } | \\ddot { s } ^ { ( 1 ) } ( t ) | \\ , . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} ( a , u ) ( g , m ) = ( a g , f _ 3 ( a , g ) u + f _ 4 ( a , g ) m ) = ( a , u ) , \\\\ ( g , m ) ( a , u ) = ( g a , f _ 3 ( g , a ) m + f _ 4 ( g , a ) u ) = ( a , u ) . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} v ( x ) = e ^ { \\rho \\cdot x } ( 1 + r ) \\Omega , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { - \\log ( \\beta ) \\beta ^ { 1 / x } x ^ { q - 1 } } { q ( 1 - \\beta ^ { 1 / x } ) } = 0 . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} x ^ 2 + 2 \\xi _ \\beta x - 1 = 0 . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\tau ( K _ 1 ) + \\tau ( K _ 2 ) - \\epsilon _ 1 - \\epsilon _ 2 = 1 . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\frac { d } { d t } & \\left ( \\int _ I a ( 0 , t , x ) \\rho _ 3 ^ 2 | y _ x | ^ 2 d x \\right ) + \\int _ I \\rho _ 3 ^ 2 | y _ t | ^ 2 d x \\\\ & \\leq C \\left ( \\int _ I \\rho ^ 2 | G | ^ 2 d x + \\int _ { \\mathcal { O } } \\rho _ 1 ^ 2 | f | ^ 2 d x + \\int _ I \\rho _ 0 ^ 2 | y | ^ 2 d x + \\sum _ { i = 1 } ^ 2 \\int _ I \\rho _ 0 ^ 2 | p ^ i | ^ 2 d x + \\int _ I \\rho _ 2 ^ 2 | y _ x | ^ 2 d x \\right ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ \\Omega ( 1 + g ( u _ \\varepsilon ) ) \\exp ( u _ \\varepsilon ^ 2 ) ~ d x = \\int _ \\Omega ( 1 + g ( u _ 0 ) ) \\exp ( u _ 0 ^ 2 ) ~ d x \\ , . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} \\frac { d } { d t } S ( t ) f = S ( t ) \\Omega f \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\tilde A _ 0 : = \\begin{pmatrix} 1 & c ^ T \\\\ c & C \\end{pmatrix} , \\ \\tilde A _ 1 : = \\begin{pmatrix} 1 & a _ 1 ^ T \\\\ a _ 1 & A _ m \\end{pmatrix} , \\ldots , \\ \\tilde A _ m : = \\begin{pmatrix} 1 & a _ m ^ T \\\\ a _ m & A _ m \\end{pmatrix} . \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\gamma _ { A , M , \\nu } = \\lim _ { R \\to + \\infty } \\Phi \\left ( A , M , \\nu , R \\right ) . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} r _ - ( \\Re E _ j ) - r _ + ( \\Re E _ j ) = f ( \\Re E _ j ) , j = 0 , 1 , 2 , \\ldots , n ; \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} ( A _ { n , 4 } V _ { n , 4 } + \\epsilon V _ { n , 4 } ) ( \\zeta , \\eta , y , x ) \\leq K _ 4 - K _ { 4 , \\zeta } \\ ; \\zeta ^ 2 - K _ { 4 , \\eta } \\ ; \\eta ^ 2 - K _ { 4 , y } \\ ; y ^ 2 - K _ { 4 , x } \\ ; x ^ 2 + \\delta | y | + | x | . \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} \\alpha \\left ( u _ 1 '' + \\frac { \\mu _ 1 } { \\alpha ^ 2 } \\left ( \\alpha - v ^ + \\right ) v ^ + - k \\omega u _ 1 u _ 2 \\right ) & = v '' + d u _ 2 '' + \\frac { \\mu _ 1 } { \\alpha } \\left ( \\alpha - v \\right ) v ^ + - \\alpha k \\omega u _ 1 u _ 2 \\\\ & = \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d + v \\right ) v ^ - + d u _ 2 '' - \\alpha k \\omega u _ 1 u _ 2 \\\\ & = d u _ 2 '' + \\frac { \\mu _ 2 } { d ^ 2 } \\left ( d - v ^ - \\right ) v ^ - - \\alpha k \\omega u _ 1 u _ 2 \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\hat H ( t _ l ) \\triangleq \\hat I _ t ^ { 1 - \\alpha } h ( t _ l ) = \\Gamma ( 1 - \\alpha ) ^ { - 1 } \\sum _ { k = 1 } ^ K w _ k h ( t _ l - s _ k ) , l = 0 , \\cdots , N . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} B _ 3 ^ { ( n ) } : = \\bigcup _ { t = \\lfloor p _ n ^ { - 1 } \\rfloor } ^ { \\lfloor n - \\varepsilon f _ 1 ( n ) \\rfloor } \\{ S _ n ( t ) + a _ n - t \\leq 0 \\} , \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\{ t _ 0 , t _ 1 , t _ { n - 1 } \\} = \\{ 1 , \\cos \\theta , \\cos ( ( n - 1 ) \\theta ) \\} , \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} - r f \\varphi h '' - 2 r f \\varphi ' h ' + [ ( n - 2 ) f \\varphi '' - m \\varphi f '' - 2 m \\varphi ' f ' ] h = 0 . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} ( 1 - c b _ 0 ) y _ 3 ^ 2 + c b _ 4 y _ 3 y _ 5 - c b _ 2 y _ 5 ^ 2 = 0 . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\i [ L , a ^ ( \\phi ) ] \\Psi = a ^ ( \\i h _ 0 \\phi ) \\Psi \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} v o l ( D _ { 1 , a } \\cap \\mathfrak { D } ( f , \\sigma ) ) = v o l ( D _ { 2 , a } \\cap \\mathfrak { D } ( f , \\sigma ) ) & = \\begin{cases} 2 a & a \\le 1 / 2 , \\\\ 1 & 1 / 2 < a < 1 , \\end{cases} \\\\ v o l ( D _ { 3 , a } \\cap \\mathfrak { D } ( f , \\sigma ) ) = v o l ( D _ { 4 , a } \\cap \\mathfrak { D } ( f , \\sigma ) ) & = \\begin{cases} 0 & a \\le 1 / 2 , \\\\ 2 a - 1 & 1 / 2 < a < 1 . \\end{cases} \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} h ( x ) & \\le g _ { 9 - } ( x ) \\Bigr | _ { x = \\frac { 2 4 1 6 9 } { 5 8 0 2 5 } } + \\frac 1 { 1 0 \\cdot 6 ^ 9 } - \\eta \\\\ & = - \\frac { 5 9 0 2 4 7 0 4 5 2 8 5 2 5 7 7 0 0 0 4 3 1 7 8 6 2 0 0 6 4 5 8 9 4 6 8 3 4 5 7 } { 2 4 7 6 2 8 0 5 0 6 0 3 3 5 3 1 9 3 2 3 7 6 7 3 5 5 7 8 5 4 7 8 1 0 5 7 2 0 3 9 8 9 5 6 8 0 0 0 0 } < 0 , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\min f _ 0 ( x ) h ( x ) = 0 , \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} I _ b ^ k \\left ( D _ b ^ { k ' } f \\right ) ( x ) = f ( x ) , \\mbox { a . e . } x \\in [ a , b ] . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} ( 1 , 1 ) _ K = 1 \\mbox { a n d } ( f _ i x , y ) _ K = ( x , e '' _ i ( y ) ) _ K \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } u _ { p } ( y _ { i , p } ) = m _ i , \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\partial ^ 2 _ { \\xi } u ^ h - \\div D W ( \\nabla u ^ h ) = f ^ h \\quad \\mbox { i n } ( 0 , \\xi _ h ) \\times S ^ h , \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} N = \\frac { a ^ 2 - b } { 2 ( k + a ) } , \\ \\ \\ \\ \\ k + a \\neq 0 . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\int _ { S ^ 2 } W _ 0 ( \\frac { 2 } { 3 } \\tilde { \\nabla } \\tilde { X } ^ i \\tilde { \\nabla } W _ 0 + 2 \\tilde { X } ^ i W _ 0 ) d S ^ 2 = 0 . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\left ( ( n + 1 ) ^ \\xi - \\phi _ G ( n + 1 ) \\right ) z ^ n \\ = \\ \\frac { h _ \\xi \\ , z ^ \\xi + h _ { \\xi - 1 } \\ , z ^ { \\xi - 1 } + \\dots + h _ 0 } { ( 1 - z ) ^ \\xi } \\ , , \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\d Z _ t ^ n = F _ n ( Z _ t ^ n ) \\d t + \\sigma \\d W _ t \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} a _ d + \\frac { 1 } { \\left ( 1 - q ^ d \\right ) ^ 3 } \\sum _ { k = 0 } ^ 3 \\binom { 3 } { k } ( - q ^ d ) ^ k k = a _ { d - 1 } \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} d A ^ n _ t = | \\nabla ^ 2 u _ n ( t , \\Phi _ n ^ { - 1 } ( t , Y ^ n _ t ) ) | ^ 2 d t , \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\frac 1 { \\eta _ i } & = \\frac { r } { ( m - 1 ) ( r - 1 ) } \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { \\theta _ i } + \\frac 1 { ( m - 1 ) ( \\frac { 1 - r } { r } \\varrho ) ' } \\\\ & = 1 - \\frac { r } { ( m - 1 ) ( r - 1 ) } \\sum _ { i = 1 } ^ { m - 1 } \\frac 1 { \\delta _ i } + \\frac 1 { m - 1 } - \\frac 1 { m - 1 } \\frac { r } { ( 1 - r ) \\varrho } \\\\ & = 1 - \\frac { r } { ( m - 1 ) ( r - 1 ) } \\Big ( \\frac { 1 - r } { r } - \\frac 1 { \\varrho } \\Big ) + \\frac 1 { m - 1 } - \\frac 1 { m - 1 } \\frac { r } { ( 1 - r ) \\varrho } \\\\ & = 1 . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} \\mathbb P \\left \\{ \\limsup _ { n \\to \\infty } \\frac { \\xi _ n } { \\sqrt { 2 \\ln n } } = 1 \\right \\} = 1 . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\dim V \\dim W = 0 . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} \\Omega ( t _ 0 , x _ 0 ) = \\Big \\{ ( t , x ) \\in [ 0 , \\infty ) \\times \\mathbb { R } ^ n : t \\in [ 0 , t _ 0 ] , \\ , | x - x _ 0 | \\leqslant ( t _ 0 - t ) \\Big \\} . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} f ( x ^ { k _ { l } } + \\lambda _ { k _ { l } } d ^ { k _ { l } } ) - f ( x ^ { k _ { l } } ) = f ( x ^ { k _ { l } + 1 } ) - f ( x ^ { k _ { l } } ) > - \\beta \\lambda _ { k _ { l } } \\| d ^ { k _ { l } } \\| ^ { 2 } . \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { q } ( v ) } \\le C ( [ \\vec v ] _ { A _ { \\vec q , \\vec r } } ) \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { q _ i } ( v _ i ) } \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\textrm { I M F } _ 1 = \\lim _ { m \\rightarrow \\infty } ( I - W ) ^ { m } s \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} A ( f ) = \\{ 1 , \\rho ( F _ { 1 } ) ^ { 2 } , \\dots , \\rho ( F _ { r } ) ^ { 2 } \\} \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} E _ 0 ( Q ) = - 2 \\pi \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\check { g } _ d ( m ) = \\frac { 2 \\pi i ^ { k } } { d } \\int _ { 0 } ^ { \\infty } g ( x ) J _ { k - 1 } \\left ( \\frac { 4 \\pi } { d } \\sqrt { x m } \\right ) \\ , d x , \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\int _ { B _ { x _ \\varepsilon } ( \\rho _ \\varepsilon ) } \\Psi _ { N _ \\varepsilon } ( u _ \\varepsilon ) d y = \\frac { 4 \\pi ( 1 + o ( 1 ) ) } { \\gamma _ \\varepsilon ^ 2 \\lambda _ \\varepsilon } \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\sup _ I \\langle \\sigma \\rangle _ I \\langle w \\rangle _ I \\le \\sup _ I \\inf _ { x \\in I } M ( \\sigma ) ( x ) \\langle w \\rangle _ I \\le \\sup _ I \\langle M ( \\sigma ) w \\rangle _ I = 1 . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} H ( x _ 0 ) ( m , n ) = \\phi _ { m - n } ( T ^ { m } x _ 0 ) + \\overline { \\phi _ { n - m } ( T ^ { n } x _ 0 ) } , m \\neq n \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} R _ { n , s } ( f ) = \\sum _ { k = 0 } ^ { + \\infty } \\alpha _ { ( 2 s + 1 ) n + k } \\ \\epsilon _ { n , k } ^ { ( s ) } , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} A ^ { * n } ( A ^ * A ) ^ m A ^ { n } = ( A ^ * A ) ^ { m + n } \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\sum \\limits _ { b \\in B ( G , \\mu ) } \\mathcal { M } _ { G , b , \\mu } = ( G , \\mu ) . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\nu _ 1 : = [ \\nu _ 0 ; \\nu _ 1 ( x ) = \\gamma _ 1 ] . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} 1 < p ^ s _ { 1 } , \\ldots p ^ s _ { n } , q ^ s _ { n + 1 } < \\infty , { \\frac { 1 } { q _ { n + 1 } ^ s } } = \\sum _ { k = 1 } ^ n { \\frac { 1 } { p _ k ^ s } } \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} [ e ^ { - t L _ V ^ k } \\phi ^ { k , i } ] ( | x | ) = o \\left ( [ e ^ { - t L _ V } \\phi ^ { 0 , 1 } ] ( | x | ) \\right ) \\quad \\mbox { a s } t \\to \\infty \\quad \\mbox { i f } M _ { 0 , 1 } > 0 \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\mathcal { H } _ 0 & : P = P _ 0 \\\\ \\mathcal { H } _ 1 & : P = P _ 1 . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} d & = \\dim ( X _ 0 ) \\geq \\dim \\bigl ( ( X _ 0 \\cap X _ 1 ) + ( X _ 0 \\cap X _ { 2 k } ) \\bigr ) \\\\ & = \\dim ( X _ 0 \\cap X _ 1 ) + \\dim ( X _ 0 \\cap X _ { 2 k } ) \\geq 2 \\bigl ( ( k + 1 ) d - n \\bigr ) , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} 1 - F \\left ( { \\tau - \\hat \\theta _ { { \\rm { M L E } } } ^ { ( u ) } } \\right ) = S \\ge 1 - q \\left ( \\theta \\right ) - { \\epsilon _ 1 } = \\frac { 1 } { 2 } \\left [ { 1 - q \\left ( \\theta \\right ) } \\right ] \\ge { \\epsilon ^ * } , \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} l ( { \\rm L } _ { 2 } ( q ^ { 1 / r } ) ) = \\max \\{ \\O ( q ^ { 1 / r } - 1 ) + f / r , \\O ( q ^ { 1 / r } + 1 ) + 1 \\} . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\phi _ { \\omega + n + 1 } = \\phi _ \\omega - 1 - \\sum _ { i = 1 } ^ n y ^ { \\frac { 1 + \\ldots + p ^ i } { p ^ i } } \\in K [ x ] . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\| J _ k \\| _ { L ^ 1 } & = \\int _ { \\R ^ n } \\left | \\int _ { ( \\R ^ n ) ^ { k - 1 } } J ( x - y _ 1 ) J ( y _ 1 - y _ 2 ) \\cdots J ( y _ { k - 1 } ) d y _ { k - 1 } \\cdots d y _ 1 \\right | d x \\\\ & \\leq \\int _ { ( \\R ^ n ) ^ k } | J ( x - y _ 1 ) | | J ( y _ 1 - y _ 2 ) | \\cdots | J ( y _ { k - 1 } ) | d x d y _ { 1 } \\cdots d y _ { k - 1 } = \\| J \\| _ { L ^ 1 } ^ k . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} & \\alpha { } = a + _ { V ( \\delta { } ) } & \\gamma { } & = c + _ { V ( \\delta { } ) } \\\\ & \\beta { } = b + _ { V ( \\delta { } ) } & \\widetilde { \\gamma } & = \\widetilde { c } + _ { V } \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} Q _ r = \\sum _ { j \\in \\mathcal { I } _ r } P _ j . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} y ^ p = \\underbrace { x _ { n - 1 } x _ n + f _ 2 ( x _ 1 , \\dots , x _ { n - 2 } ) } _ { } \\ ; + \\ ; . \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} \\Delta ^ 2 u _ 1 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } - \\frac { 2 0 u _ 1 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 , \\\\ \\Delta ^ 2 u _ 2 & = \\left ( R _ 1 ^ 2 + R _ 2 ^ 2 \\right ) \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } - \\frac { 2 0 u _ 2 } { 1 + | u | ^ 2 } | \\nabla u | ^ 4 , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} w _ i ( x ) = \\chi _ i ( u ( x ) ) , \\forall x \\in \\Omega . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} & C \\bullet X + c _ j x _ j = - 1 \\\\ & A _ i \\bullet X + a _ { i j } x _ j \\le 0 \\ \\ \\forall \\ i = 1 , \\ldots , m \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\begin{aligned} h _ { i j k l n } ^ { p ^ { \\ast } } - h _ { i j k n l } ^ { p ^ { \\ast } } & = \\sum _ { m } h _ { m j k } ^ { p ^ { \\ast } } R _ { m i l n } + \\sum _ { m } h _ { i m k } ^ { p ^ { \\ast } } R _ { m j l n } + \\sum _ { m } h _ { i j m } ^ { p ^ { \\ast } } R _ { m k l n } \\\\ & \\ \\ + \\sum _ { m } h _ { i j k } ^ { m ^ { \\ast } } R _ { m ^ { \\ast } p ^ { \\ast } l n } . \\end{aligned} \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} a _ { i , j } = \\chi _ E ( i , j ) = \\begin{cases} 1 & ( i , j ) \\in E , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} & \\partial _ t ^ \\alpha \\psi ( t _ 1 ) \\approx b _ { 1 , 0 } ( \\psi ( t _ 1 ) - \\psi ( t _ 0 ) ) , \\\\ & \\partial _ t ^ \\alpha \\psi ( t _ n ) \\approx \\sum _ { k = 1 } ^ { n - 1 } ( b _ { n , k - 1 } - b _ { n , k } ) \\psi ( t _ k ) + b _ { n , n - 1 } \\psi ( t _ n ) - b _ { n , 0 } \\psi ( t _ 0 ) , n = 2 , \\cdots , N , \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} h , _ { y _ j } = f e ^ { l ( \\hat { x _ i } ) } . \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} \\left ( \\mu _ 1 ^ \\star \\right ) _ { | \\left [ 0 , 1 \\right ] } = M _ 1 \\mathbf { 1 } _ { \\left [ 0 , r _ 1 \\right ] } + M _ 1 \\mathbf { 1 } _ { \\left [ r _ 1 + 2 r _ 0 + 2 r _ 2 , 1 \\right ] } \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} B - \\hat \\nu ( s ) = \\begin{cases} A _ 1 \\xi ' ( s ) + \\frac 1 { A _ 1 q + A _ 2 ( 1 - q ) + \\Delta } , & s \\le q , \\\\ A _ 1 \\xi ' ( q ) + A _ 2 ( \\xi ' ( s ) - \\xi ' ( q ) ) + \\frac 1 { A _ 1 q + A _ 2 ( 1 - q ) + \\Delta } , & q \\le s \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\boxtimes ^ k _ { i = 1 } [ M _ { S _ i } , \\mu _ { S _ i } ] = [ M _ S , \\mu _ S ] \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} \\sum _ { \\widetilde { Q } \\in \\mathcal { D } } \\mu ( \\widetilde { Q } ) = \\mu ( Q ) = \\Theta ( Q ) l ( Q ) . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} u ( t ) = 0 \\hbox { o n } \\partial _ D \\Omega \\ , , ( A \\nabla u ( t ) ) \\cdot n = 0 \\hbox { o n } \\partial _ N \\Omega \\cup \\Gamma ( t ) \\ , , \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} R _ { u , w } : = ( B _ - u B _ + ) / B _ + \\cap ( B _ + w B _ + ) / B _ + \\subset G / B _ + . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\max _ { i = 1 , 2 , \\ldots , n } h _ { i } \\leq x \\right ) = ( 1 - \\mathrm { e } ^ { - x } ) ^ n . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to + \\infty } \\overline { W } ' _ { \\rho , \\nu } ( - \\rho ) = \\sqrt { \\frac 1 3 + \\nu ^ 2 \\left ( \\frac 2 3 \\nu - 1 \\right ) } . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} { \\rm ( A S ) } \\left \\{ \\begin{array} { l l } { \\rm a . } \\ ; X \\ ; \\mbox { i s a t t r a c t e d t o a s t a b l e l a w o f e x p o n e n t } \\ ; 0 < \\alpha \\leq 2 . \\\\ { \\rm b . } \\ ; E X = 0 \\ ; \\mbox { i f } \\ ; E | X | < \\infty . \\\\ { \\rm c . } \\mbox { \\ ; \\ ; t h e r e e x i s t s } \\rho : = \\lim P [ S _ n > 0 ] , \\end{array} \\right . \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i j } = w _ { i } ^ { \\ast } + v _ { j } ^ { \\ast } \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} [ ( 1 + g ( t ) ) \\exp ( t ^ 2 ) ] ' = 2 t H ( t ) \\exp ( t ^ 2 ) \\ , . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\sum _ { l \\in \\overline S } \\frac { R _ { i l } ( \\lambda _ 0 ) } { \\lambda _ 0 - \\omega ( l , l ) } u _ l = c u _ i , \\forall i \\in S , \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\Gamma ( n - l - \\varepsilon ) \\sin ( \\pi \\varepsilon ) = \\left \\{ \\begin{array} { c c } - \\pi & l = n \\\\ 0 & l < n \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} \\rho ' ( t ) = A \\rho ( t ) + G ( \\rho ( t ) ) \\qquad \\mbox { f o r } \\ ; \\ ; t > 0 , \\qquad \\rho ( 0 ) = \\rho _ 0 . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\rho ( x ) : = \\sup \\bigg \\{ r > 0 : \\frac { 1 } { r ^ { d - 2 } } \\int _ { B ( x , r ) } V ( y ) \\ , d y \\leq 1 \\bigg \\} , x \\in \\mathbb R ^ d , \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} J E ' ( Q _ { a , c } ) & = \\partial _ x ( - H \\partial _ x Q _ { a , c } - \\frac 1 2 Q _ { a , c } ^ 2 + W ( h x ) Q _ { a , c } ) \\\\ & = \\partial _ x ( - c Q _ { a , c } + W ( h x ) Q _ { a , c } ) \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} & a _ k \\sim b _ k \\ , \\ , \\ , \\ , \\mbox { a s $ k \\to \\infty $ } \\quad \\Leftrightarrow \\quad \\lim _ { k \\to \\infty } \\frac { a _ k } { b _ k } = 1 \\\\ & f ( t ) \\sim g ( t ) \\ , \\ , \\ , \\ , \\mbox { a s $ t \\to \\infty $ } \\quad \\Leftrightarrow \\quad \\lim _ { t \\to \\infty } \\frac { f ( t ) } { g ( t ) } = 1 . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d Y _ t & = & ( A Y _ t + \\Pi _ V \\alpha ( \\psi ( t _ 0 + t ) + Y _ t ) ) d t + \\sigma ( \\psi ( t _ 0 + t ) + Y _ { t - } ) d X _ t \\medskip \\\\ Y _ 0 & = & v _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\epsilon _ { n , 0 } ^ { ( s ) } = \\dfrac { 1 } { 4 } \\omega _ { n , 0 } ^ { } ( s ) , \\ \\epsilon _ { n , 1 } ^ { ( s ) } = \\dfrac { 1 } { 4 } \\omega _ { n , 1 } ^ { ( s ) } , \\ \\epsilon _ { n , k } ^ { ( s ) } = \\dfrac { 1 } { 4 } ( \\omega _ { n , k } ^ { ( s ) } - \\omega _ { n , k - 2 } ^ { ( s ) } ) , \\ k = 2 , 3 , . . . \\ . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} | e ^ { s B _ i } f | _ q = \\left ( \\int _ { \\mathbb { R } ^ d } | f ( z _ i ( s , t , \\xi ) | ^ q d \\xi \\right ) ^ \\frac { 1 } { q } = | f | _ q , \\ i = 1 , 2 , . . . , N , \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\vert \\nabla ( w _ i ) ^ 2 \\vert \\leq 4 \\sqrt { \\lambda _ 0 } ^ { - 1 } J ( u ) ( x ) , \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} { u _ k ^ { - d + 1 } \\over u _ k ^ { - d } } | I _ { n _ k } ( x ) | = u _ k | I _ { n _ k } ( x ) | , \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\alpha \\ , B ^ i + \\alpha ' y ^ i = 0 , \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} Y & = \\{ ( x , y ) \\in X \\mid y \\geq 0 \\} , \\\\ Z & = \\{ ( x , y ) \\in X \\mid y \\leq 0 \\} , \\\\ A & = \\{ ( x , y ) \\in \\mathbb R ^ 2 \\mid x \\geq 0 , \\ | y | \\leq x \\} , \\\\ B & = \\{ ( x , y ) \\in A \\mid y \\geq 0 \\} , \\\\ C & = \\{ ( x , y ) \\in A \\mid y \\leq 0 \\} . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\check { S } _ k ( t ) & = \\sum _ { l \\in \\mathcal { N } _ k \\cup \\{ k \\} } w _ { k l } \\left ( \\check { S } _ l ( t - 1 ) + \\eta _ l ^ ( t ) \\right ) . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} C ( \\Theta , X , Y ) : = \\Big \\langle \\sum _ { l = 1 } ^ m \\Theta _ l F _ l ( X ) , Y - X \\Big \\rangle \\geq 0 . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { L } \\chi ( T ^ n x ) \\leq C \\epsilon ^ { d } \\sum _ { n = 1 } ^ { L } f ( T ^ n x ) \\leq C \\epsilon ^ { d } \\sum _ { n = 1 } ^ { L } \\sum _ { 0 \\leq | l _ j | < R } \\hat { f } ( l _ 1 , \\ldots , l _ d ) e ^ { 2 \\pi i \\langle T ^ n x , l \\rangle } \\\\ \\leq C \\epsilon ^ { d } \\left ( L + \\sum _ { 0 < \\lvert k \\mid < \\frac { 1 } { \\epsilon } } \\Big \\lvert \\sum _ { n = 1 } ^ { L } e ^ { 2 \\pi i \\langle T ^ n x , k \\rangle } \\Big \\lvert \\right ) . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} u ( x ) = e ^ { i \\vartheta } \\mu ^ { \\frac { 2 s } { \\alpha } } h ( \\mu x + y _ 0 ) , \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\gamma _ { 1 } & = \\{ q ( m + 1 ) + ( q - \\frac { e - 4 } { 2 } + k ) d + \\frac { e } { 2 } \\mid 0 \\leq k \\leq ( e - 3 ) \\} \\\\ \\gamma _ { 2 } & = \\{ k ( m + d ) \\mid 0 \\leq k \\leq 2 q + 1 \\} . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\mathcal { W } _ { i } = \\{ w \\in \\Lambda _ { i } ^ { m } \\ : : \\ : 2 ^ { - m ( h _ { i } + \\delta ) } \\le \\mu _ { i } [ w ] \\le 2 ^ { - m ( h _ { i } - \\delta ) } \\} \\ : . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} V _ U ( d ) = \\mathbb { V } _ U ( d ) . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { m } s ( X _ t ) = \\sum _ { u = 1 } ^ { l } r ( Y _ u ) , \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\beta ( L + \\pi ( V ) - \\pi ( \\tau _ V ^ t ( V ) ) ) = - \\log \\Delta _ { \\omega _ { - t } | \\omega } . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} G ( m , N , a , \\varepsilon , s ) & \\leq 6 \\cdot C _ 1 C _ 2 ^ { N - 2 } \\varepsilon ( { m \\over 3 } ) ^ { 1 - d s \\over a } \\sum _ { k = 1 } ^ { ( { m ( 1 + \\epsilon ) \\over 2 } ) ^ { { 1 \\over a } } } k ^ { - d s } \\\\ & \\leq 6 \\cdot 3 ^ { d s - 1 \\over a } C _ 1 C _ 2 ^ { N - 2 } \\varepsilon m ^ { 1 - d s \\over a } \\zeta ( d s ) \\\\ & = C _ 1 C _ 2 ^ { N - 1 } \\varepsilon m ^ { 1 - d s \\over a } , \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\tau ( x y ) = \\tau ( y x ) . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} w _ j : = \\phi ^ { 2 \\alpha _ j } z _ j \\ \\ \\alpha _ 1 = 1 / ( 2 k ) \\ \\ \\ \\alpha _ 2 = 1 / 2 . \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} \\mathcal { I } ^ { \\alpha } [ f ] ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ 0 ^ t ( t - s ) ^ { \\alpha - 1 } f ( s ) \\ , d s , \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} \\alpha _ j \\alpha _ k + \\alpha _ k \\alpha _ j = 2 \\delta _ { j , k } \\mathbb { I } _ N , 1 \\leq j , k \\leq n + 1 , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\mathcal { Z } _ p : = \\left \\{ f = f ( t , \\xi ) : \\ t ^ { 1 - \\frac { 1 } { p } - \\gamma } f \\in C _ b ( [ 0 , \\infty ) ; L ^ p ( \\mathbb { R } ^ 2 ) ) , \\ t ^ { \\frac { 3 } { 2 } - \\frac { 1 } { p } - \\gamma } \\partial _ j f \\in C _ b ( [ 0 , \\infty ) ; L ^ p ( \\mathbb { R } ^ 2 ) ) , \\ j = 1 , 2 \\right \\} . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} ( \\lambda - G _ 1 ) Q ^ { \\lambda } _ { \\epsilon } ( x ) = ( \\lambda - G _ 1 ) Q ^ { \\lambda } * q _ { \\epsilon } ^ { 1 } ( x ) = q _ { \\epsilon } ^ { 1 } ( x ) . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} R _ L ( F _ n ) ( x , y ) & = R _ L ( \\sum _ { i = 1 } ^ { m } f _ i \\otimes g _ i ) ( x , y ) = R ( \\sum _ { i = 1 } ^ { m } ( f _ i \\otimes g _ i ) \\restriction _ { [ 0 , \\alpha ] \\times { y } } ) ( x ) \\\\ & = \\sum _ { i = 1 } ^ { m } R ( f _ i ) ( x ) \\cdot g _ i ( y ) = \\sum _ { i = 1 } ^ { m } ( R ( f _ i ) \\otimes g _ i ) ( x , y ) \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } N _ \\varepsilon = + \\infty \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} \\Phi _ { 1 } ( x ; ( h _ { 0 } + \\varepsilon \\cdot \\gamma ) \\sqcup \\alpha ) = \\Phi _ { 1 } \\left ( \\Phi _ { 1 } ( x ; h _ { 0 } + \\varepsilon \\cdot \\gamma ) ; \\alpha \\right ) , \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\Phi ( a ) = \\sum _ { j = 1 } ^ m \\Phi ( a _ j ^ n ) = \\sum _ { j = 1 } ^ m P ( a _ j ) , \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} u = \\tau ( J ( \\tau ) - \\lambda I ) ^ { - 1 } P _ 1 L ^ * L P _ 1 u \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} & q ^ { - k + 1 } + q ^ { - 2 \\lambda _ { r + 1 } + k + 1 } ( q _ 0 q _ 1 ^ { - 1 } - 1 ) + \\sum _ { m = k + 1 } ^ { \\lambda _ { r + 1 } } q ^ { - 2 \\lambda _ { r + 1 } - k + 2 m } ( q ^ { - 1 } - q ) \\\\ & + \\sum _ { m = 1 } ^ { k - 1 } q ^ { - 2 \\lambda _ { r + 1 } - k + 2 + 2 m } ( q ^ { - 1 } - q ) + \\sum _ { m = 1 } ^ { k - 1 } q ^ { - 2 \\lambda _ { r + 1 } - k + 2 + 2 m } ( q ^ { - 1 } - q ) ( q _ 0 q _ 1 ^ { - 1 } - 1 ) . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} y \\cong ( \\xi _ 2 , . . . , \\xi _ k ) = ( x _ { 2 , 1 } , . . . , x _ { 2 , m _ 2 } , . . . . , x _ { k , 1 } , . . . , x _ { k , m _ k } ) . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} H ( t ) = 1 + g ( t ) + \\frac { g ' ( t ) } { 2 t } \\ , , \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } v ( x ) & = \\frac { 2 ^ { 2 s } \\Gamma ( s + \\gamma ) \\Gamma ( s + \\frac { 1 } { 2 } ) } { \\sqrt { \\pi } \\Gamma ( \\gamma ) } \\ , { } _ { 2 } F _ { 1 } \\Big ( s + \\gamma , s + \\frac { 1 } { 2 } ; \\frac { 1 } { 2 } ; - x ^ 2 \\Big ) . \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { E C } ( x , y , z ) = ( h , c ) , \\\\ ( h , c ) \\in \\Sigma ^ u , \\\\ ( h , c ) = \\mathcal { E C } ( 0 , 0 , \\pm | M | ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\| ( I - Q _ r ) \\hat Q _ r ( I - Q _ r ) \\| _ 2 = \\| ( I - Q _ r ) \\hat Q _ r \\hat Q _ r ( I - Q _ r ) \\| _ 2 \\leq \\| \\hat Q _ r ( I - Q _ r ) \\| _ 2 ^ 2 = \\| \\hat Q _ r - Q _ r \\| _ 2 ^ 2 / 2 \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} 2 b ^ 2 l ^ 2 \\ , \\lambda ^ { 2 } - 2 \\left [ b ^ 2 l ^ 2 + a c d ( l + 2 ) \\right ] \\lambda + c ( l + 2 ) = 0 . \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\mu ( x - a ) = \\delta \\geq \\mu ( x - b ) \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\dot { p } _ { 1 } = \\frac { \\left ( c _ { 1 } + c _ { 2 } p _ { 1 } ^ { 2 } \\right ) ^ { 2 } } { c _ { 1 } - c _ { 2 } \\left ( p _ { 1 } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - p t } E _ \\nu ( - t ^ \\nu ) d t = p ^ { \\nu - 1 } ( 1 + p ^ \\nu ) ^ { - 1 } . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{gather*} \\pi ( a ( \\phi ) ) = a ( \\sqrt { 1 - T } \\phi \\oplus 0 ) + a ^ * ( 0 \\oplus \\sqrt { T } \\ , \\overline { \\phi } ) , \\end{gather*}"}
-{"id": "6514.png", "formula": "\\begin{align*} U _ { \\varepsilon _ { n } } ^ { j , k } ( x _ { j } + r e ^ { i \\vartheta _ { j } } ) = z ( r ) e ^ { i \\vartheta _ { j } } \\ , \\ , \\ , \\ , \\ , B _ { \\frac { T _ { \\varepsilon _ { n } } } { 2 0 d _ { k } } , \\frac { T _ { \\varepsilon _ { n } } } { 1 0 d _ { k } } } ( x _ { j } ) \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} a ^ { k l } = 0 \\ \\ { \\rm f o r } \\ \\ k \\in \\Lambda , \\ l \\not \\in \\Lambda . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} & | G _ { [ 0 , M ] } ( E , T ^ { n _ { 0 } - \\frac { M } { 2 } } x ) ( m , n ) | = \\\\ & | G _ { [ n _ { 0 } - \\frac { M } { 2 } , n _ { 0 } + \\frac { M } { 2 } ] } ( E , x ) ( m + n _ { 0 } - \\frac { M } { 2 } , n + n _ { 0 } - \\frac { M } { 2 } ) | < e ^ { M ^ { 1 - } - c _ 3 | m - n | \\chi _ { | m - n | > \\frac { M } { 1 0 } } } . \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} f _ i = \\frac { \\partial f } { \\partial x ^ i } = - \\frac { x ^ i } { | x | ^ 2 } , f _ { i j } = \\frac { \\partial ^ 2 f } { \\partial x ^ i \\partial x ^ j } = - \\frac { \\delta _ { i j } } { | x | ^ 2 } + \\frac { 2 x ^ i x ^ j } { | x | ^ 4 } . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} | B | = \\frac { q ^ { r ( r + 1 ) / 2 } ( q - 1 ) ^ { r } } { ( r + 1 , q - 1 ) } . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\tau ^ { \\ast } = \\frac { \\sigma _ w ^ 2 } { \\rho } ( \\sigma _ w ^ 2 + \\rho ) \\ln ( 1 + \\frac { \\rho } { \\sigma _ w ^ 2 } ) . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { F D F = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B F } } ^ { \\alpha } \\sigma _ { \\textrm { F } } ^ { 2 } \\theta } { \\lambda _ { \\textrm { B F } } \\rho } \\left ( 2 ^ { \\frac { R } { \\theta } } - 1 \\right ) \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\int _ 0 ^ T g _ s ^ { ( 1 ) } \\ , \\mathrm { d } ^ { - } H _ s = \\int _ 0 ^ T g _ s ^ { ( 1 ) } g _ s ^ { ( 2 ) } \\ , \\mathrm { d } ^ { - } h _ s \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\dfrac { \\mathrm { d } u } { \\mathrm { d } t } = \\nu ( u ) \\left ( \\nabla H ( u ) \\times \\nabla C ( u ) \\right ) - \\alpha ( u ) ( H ( u ) - h ) \\left [ \\nabla C ( u ) \\times \\left ( \\nabla H ( u ) \\times \\nabla C ( u ) \\right ) \\right ] , \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} ( \\log f _ r ) _ r = \\frac { f _ { r r } } { f _ r } = \\frac { \\phi _ r } { \\phi } = ( \\log \\phi ) _ r \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} d ( H _ 0 - V ( B _ 1 ) ) = d ( \\{ B _ 2 , B _ 3 , B _ 4 \\} ) . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} H = \\left \\{ \\pm \\left [ \\begin{array} [ c ] { c c } a & b \\\\ 0 & a ^ { 1 / 2 } \\end{array} \\right ] : a > 0 , b \\in \\mathbb { R } \\right \\} ~ . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} S = \\lambda _ { 1 } ^ { 2 } + 3 \\lambda _ { 2 } ^ { 2 } + 4 \\lambda ^ 2 , \\ \\ H \\lambda _ { 1 } = K + \\lambda _ { 1 } ^ { 2 } + \\lambda _ { 2 } ^ { 2 } + 2 \\lambda ^ 2 , \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\mu _ S | _ { Z ( \\widehat { M _ S } ) ^ { \\Gamma } } = \\gamma ( \\mu _ S ) | _ { Z ( \\widehat { M _ S } ) ^ { \\Gamma } } . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { 1 \\leq j \\leq n } \\frac { f ( I _ j ^ n ) } { \\mathcal L ( I _ j ^ n ) } { \\mathcal L ( I _ j ^ n \\cap J ) } = \\int _ J s ( x ) d x , \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\frac { f _ { 1 } ^ { \\prime \\prime } } { f _ { 1 } ^ { \\prime } } = f _ { 1 } \\left ( \\frac { c _ { 3 } } { f _ { 1 } ^ { \\prime } } + c _ { 4 } f _ { 1 } ^ { \\prime } \\right ) , \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} h ( a b ) f _ 3 ( a , b ) = f _ 3 ( a b , b ^ { - 1 } a ^ { - 1 } ) f _ 3 ( a , b ) = f _ 3 ( a , a ^ { - 1 } ) = g _ 3 ( a , b ) h ( a ) , \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} T _ N : = P _ N ^ * Y _ N P _ N \\ , , \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} D u ( x ) = \\frac { 1 } { 2 } x x = \\frac { 1 } { 2 } x ^ 2 , \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} | D ^ 2 u _ j | ^ 2 - 2 \\theta a _ { 0 , j } a _ { 1 , j } \\geq \\theta \\left [ \\left ( \\sum _ { i = 0 } ^ 3 a _ { i , j } \\right ) ^ 2 - 2 a _ { 0 , j } a _ { 1 , j } \\right ] . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } p _ { \\omega } ( n ) q ^ n & = q \\ , \\omega ( q ) , \\sum _ { n = 0 } ^ { \\infty } p _ { \\nu } ( n ) q ^ n = \\nu ( - q ) . \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} a \\stackrel { } { \\mapsto } c _ a : = ( f _ a ( \\alpha _ i ) , i = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow + \\infty } p \\| \\nabla u _ { p } \\| _ 2 ^ 2 = 8 \\pi \\sum _ { i = 1 } ^ k m _ i ^ 2 , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\textit { w h e r e } \\bold { B } _ { x , x ; i } = \\left [ \\begin{array} { c | c | c } \\bold { B } _ { x , x ; i , 1 } & \\dots & \\bold { B } _ { x , x ; i , p _ x } \\end{array} \\right ] \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\mathbb { T } ^ d } m _ \\epsilon ( x ) \\phi _ \\epsilon ( x ) d x = \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\mathbb { T } ^ d } F ^ { - 1 } _ \\epsilon ( w _ \\epsilon ( x ) ) \\phi _ \\epsilon ( x ) d x = \\int _ { \\mathbb { T } ^ d } \\int _ { \\mathcal { Y } ^ d } F ^ { - 1 } ( w ( x , y ) ) \\phi ( x , y ) d y d x \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} P _ \\sharp { \\rm O p } _ h ( \\chi ) U ^ { - 1 } = { \\rm O p } _ h ( \\chi ) U ^ { - 1 } P _ 0 + { \\cal O } ( h ^ \\infty ) \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\nu _ \\varepsilon = \\min \\left ( | x _ \\varepsilon - y _ \\varepsilon | , d ( y _ \\varepsilon , \\partial \\Omega ) \\right ) \\ , . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\inf _ { \\tau \\leq s } \\tau ^ { - \\varepsilon } L ( \\tau ) = \\frac { 1 } { \\sup _ { \\tau \\leq s } \\tau ^ \\varepsilon K ( \\tau ) } \\sim \\frac { 1 } { s ^ \\varepsilon K ( s ) } = s ^ { - \\varepsilon } L ( s ) \\quad ( s \\to \\infty ) , \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\aligned & f ( t ) = t ^ 4 - 2 S t ^ 3 - 6 S ( S - 1 ) t ^ 2 + 2 S ( 2 - 3 S ) ^ 2 t - ( 2 - 3 S ) ^ 2 S ^ 2 , \\endaligned \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} U ^ - ( x , t ) & = \\inf _ { S \\in \\mathcal S } \\sup _ { A \\in \\mathcal { A C } } \\mathbb E \\Big [ e ^ { - r ( T - t ) } g \\big ( X ( T ) \\big ) \\Big ] , \\\\ U ^ + ( x , t ) & = \\sup _ { S \\in \\mathcal S } \\inf _ { A \\in \\mathcal { A C } } \\mathbb E \\Big [ e ^ { - r ( T - t ) } g \\big ( X ( T ) \\big ) \\Big ] \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} A : = - \\langle e _ 1 , \\nabla _ H ( \\frac { X _ 3 u } { | \\nabla _ H u | } ) \\rangle - \\overline { p } [ X _ 1 ( \\overline { p } ) + X _ 2 ( \\overline { q } ) ] - \\overline { p } ^ 2 \\frac { ( X _ 3 u ) ^ 2 } { l ^ 2 } + 2 \\overline { q } \\frac { X _ 3 u } { l } . \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} v _ \\Psi ( t ) = \\int _ 0 ^ t S ( t - s ) B ( s , v _ \\Psi ( s ) + \\Psi ( s ) ) d s . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} & \\lim \\limits _ { h \\rightarrow 0 } \\frac { \\int _ B G ( x + h , z ) P _ 2 ( y , z ) d z - \\int _ B G ( x , z ) P _ 2 ( y , z ) d z } { h } \\\\ & = \\lim \\limits _ { h \\rightarrow 0 } \\int _ B \\frac { G ( x + h , z ) - G ( x , z ) } { h } P _ 2 ( y , z ) d z = \\int _ B \\partial _ x G ( x , z ) P _ 2 ( y , z ) d z . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } e _ { q ^ t , \\lambda _ { q ^ t } } ( Q ) = Q ^ \\mu \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} { 1 \\over ( { m \\over 3 ^ i } ) ^ { 1 / a } C _ 4 ( a ) } < \\epsilon \\leq 1 / 3 , \\ \\ i = 0 , \\dots , N - 2 . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\Sigma ( W ) \\cap \\Gamma = \\Sigma ' ( W ) \\cap \\Gamma \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} & f ( \\rho ) ^ { \\frac { 1 } { 4 } } g ( \\rho ) ^ { \\frac { 1 } { 4 } } A ^ * g ( \\rho ) ^ { \\frac { 1 } { 2 } } f ( \\rho ) ^ { \\frac { 1 } { 2 } } A g ( \\rho ) ^ { \\frac { 1 } { 4 } } f ( \\rho ) ^ { \\frac { 1 } { 4 } } \\\\ & = f ( \\rho ) ^ { \\frac { 1 } { 4 } } g ( \\rho ) ^ { \\frac { 1 } { 4 } } A ^ * g ( \\rho ) ^ { \\frac { 3 } { 4 } } f ( \\rho ) ^ { \\frac { 3 } { 4 } } ( f ( \\rho ) g ( \\rho ) ) ^ { - 1 } g ( \\rho ) ^ { \\frac { 3 } { 4 } } f ( \\rho ) ^ { \\frac { 3 } { 4 } } A f ( \\rho ) ^ { \\frac { 1 } { 4 } } g ( \\rho ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} b _ \\Omega ( x ; v ) = \\sqrt { g _ B ( v , v ) } \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} C & \\textup { i s a n a t w o r s t n o d a l c u r v e o f g e n u s $ g $ } \\ , , \\\\ p _ 1 , \\dots , p _ n \\in C & \\textup { a r e s m o o t h d i s t i n c t m a r k e d p o i n t s o f $ C $ } \\ , , \\\\ f & \\textup { i s a m a p o f d e g r e e $ \\deg f : = f _ * [ C ] = d $ \\ , . } \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\P ( F ) = \\P ( \\{ \\omega ' _ { j _ 1 , k _ 1 , 1 } \\} ) = p _ 2 , \\P ( \\{ \\omega ' _ { j _ 1 , k _ 1 , 3 } \\} ) = p _ 1 - p _ 2 . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} B _ P = \\hat \\nu _ P ( 0 ) + \\frac 1 { \\nu _ P ( [ 0 , 1 ] ) } . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} [ y ^ { p ^ r } _ { 1 2 } , y ^ { p ^ s } _ { 1 1 2 2 } ] = [ x ^ { p ^ r } _ { 1 2 } , x ^ { p ^ s } _ { 1 1 2 2 } ] = x ^ { p ^ r } _ { 1 2 } ( 1 - x ^ { - p ^ r } _ { 1 2 } x ^ { p ^ s } _ { 1 1 2 2 } x ^ { p ^ r } _ { 1 2 } x ^ { - p ^ s } _ { 1 1 2 2 } ) x ^ { p ^ s } _ { 1 1 2 2 } . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi _ t ( x ) = \\Big ( 1 - 2 d \\lambda \\big [ ( b - 1 ) + a \\big ] \\Big ) \\xi _ t ( x ) . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\partial _ t u = \\kappa _ 1 \\Delta u + \\gamma \\nabla F ( u ) \\times \\Delta u - \\kappa _ 2 ( 1 + \\mu \\cdot F ( u ) ) \\nabla F ( u ) \\\\ \\frac { \\partial u } { \\partial \\nu } = 0 \\\\ u ( \\cdot , 0 ) = u _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\Delta u _ 1 = a _ { 1 , 1 } ( x ) - \\sigma \\frac { \\langle x , \\nabla u _ 1 \\rangle } { | x | ^ 2 } - \\left ( p - 2 \\right ) \\frac { \\langle ( D ^ 2 u _ 1 ) ( \\nabla u _ 1 ) , \\nabla u _ 1 \\rangle } { | \\nabla u _ 1 | ^ 2 } \\ \\mathcal G \\cap B _ { \\rho _ 0 / 2 } ( x _ 0 ) . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\widetilde { m } ( x , y ) = \\prod _ { i = 1 } ^ { d } \\widetilde { m } _ i ( x , y _ i ) , \\ \\widetilde { w } ( x , y ) = \\sum _ { i = 1 } ^ { d } \\widetilde { w } _ i ( x , y _ i ) , \\ \\widetilde { H } ( x , \\Lambda ) = \\sum _ { i = 1 } ^ { d } \\widetilde { H } _ i ( x , \\Lambda _ { i } ) , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} E _ { P } \\left \\{ \\psi _ { P , \\mathbf { a } } \\left [ \\mathbf { G , B } ; \\mathcal { G } \\right ] g ( \\mathbf { A } , \\mathbf { G } , \\mathbf { B } ) \\right \\} = 0 g E _ { P } \\left [ g ( \\mathbf { A } , \\mathbf { G } , \\mathbf { B } ) | \\mathbf { G } , \\mathbf { B } \\right ] = 0 \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { z ^ n q ^ { n ^ 2 + n } } { ( z q ; q ^ 2 ) _ { n + 1 } } \\right ) ^ 2 = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } b _ \\omega ^ 3 ( m , n ) z ^ m q ^ n = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } , \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b ( n ) z ^ n = \\prod _ { j = 1 } ^ { \\infty } ( 1 - z ^ j ) ^ { - \\gamma ( j ) } , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\begin{aligned} \\L \\Theta & \\leq - \\frac { \\gamma } { 2 m } v ^ 2 - \\frac { 1 } { m } \\sum _ { k = 1 } ^ N \\lambda _ k z _ k ^ 2 - a _ 1 \\sum _ { k > N } k ^ { - 2 s } \\lambda _ k z _ k ^ 2 + a , \\end{aligned} \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} B _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) & + B _ \\Omega ( z _ 0 , y _ n ) - B _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , y _ n ) \\\\ & = B _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) + B _ \\Omega ( z _ 0 , y _ n ) - B _ \\Omega ( z _ 0 , \\gamma _ n y _ n ) \\\\ & \\geq B _ \\Omega ( \\gamma _ n ^ { - 1 } z _ 0 , z _ 0 ) - B _ \\Omega ( \\gamma _ n y _ n , y _ n ) \\rightarrow \\infty . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} A Q _ m = Q _ { m + 1 } R _ { m + 1 } \\hat { H } ^ { ( h ) } _ m R ^ { - 1 } _ m . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } \\mathcal I ( t ) = & ~ \\frac { \\sigma } { 2 L } \\int \\varphi ' u _ x ^ 2 + \\frac { 1 } { 2 L } ( \\sigma - 1 ) \\int \\varphi ' u ^ 2 + \\frac { 1 } { L } \\int \\varphi ' u ( 1 - \\partial _ x ^ 2 ) ^ { - 1 } u \\\\ & ~ { } - \\frac 1 { L ( p + 1 ) } \\int \\varphi ' u ^ { p + 1 } + \\frac { 1 } { L } \\int \\varphi ' u ( 1 - \\partial _ x ^ 2 ) ^ { - 1 } \\left ( u ^ p \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} & 0 = T ^ { D } ( e _ { - } , v _ { + } , w _ { + } ) = \\langle D _ { e _ { - } } v _ { + } - D _ { v _ { + } } e _ { - } - [ e _ { - } , v _ { + } ] , w _ { + } \\rangle + \\langle v _ { + } , D _ { w _ { + } } e _ { - } \\rangle \\\\ & = \\langle D _ { e _ { - } } v _ { + } - [ e _ { - } , v _ { + } ] , w _ { + } \\rangle , \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\mathcal { U } _ { j } = \\{ w \\in \\mathcal { U } \\ : : \\ : \\sigma ^ { j } \\nu [ w ] \\ge \\exp ( - n m ( h + c _ { 1 } \\delta / \\epsilon ) ) \\} \\ : . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\Omega \\left [ f ^ { \\not \\ominus } ( z , t ) \\right ] = \\left ( \\frac { z } { 1 - z } \\right ) \\left ( f ( z , t ) - 1 \\right ) \\left ( \\frac { f ( z , 1 ) f ^ { \\not \\ominus } ( z , f ( z , 1 ) ) - t f ^ { \\not \\ominus } ( z , t ) } { f ( z , 1 ) - t } \\right ) . \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\mathbf E ( N , p , q ; I , J ) \\leftrightarrow \\mathbf E ( N , p ' , q ' ; I ' , J ' ) = \\mathbf E ( N , p + p ' , q + q ' ; I I ' , J J ' ) , \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\left \\| w _ 1 ^ 2 - w _ 2 ^ 2 \\right \\| & = \\left \\| \\left ( w _ 1 + w _ 2 \\right ) * \\left ( w _ 1 - w _ 2 \\right ) \\right \\| \\le \\frac { 2 p r } { \\sqrt { \\Delta x } } \\| w _ 1 - w _ 2 \\| \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} K = f _ { 1 } f _ { 2 } f _ { 1 } ^ { \\prime \\prime } f _ { 2 } ^ { \\prime \\prime } - \\left ( f _ { 1 } ^ { \\prime } f _ { 2 } ^ { \\prime } \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} \\sup \\bigg \\{ \\int _ X \\langle u , d \\mu \\rangle | u \\colon X \\to \\mathbb { R } ^ m 1 \\bigg \\} = \\int _ X \\langle u _ 0 , d \\mu \\rangle . \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\beta _ { p } ( n ) \\geq \\begin{cases} 1 - \\frac { 1 } { p } & p > 2 \\\\ \\frac { 1 } { 4 } & p = 2 . \\end{cases} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\tilde { \\omega } _ { 1 } ( \\sigma ) = \\max \\{ \\omega _ { \\nabla \\psi } ( \\sigma ) , \\sigma ^ { \\alpha } \\} . \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} f _ { i } ( \\omega ) = - \\frac { 1 } { m } 1 _ { \\mathcal { W } _ { i } } ( \\omega | _ { m } ) \\log \\mu _ { i } [ \\omega | _ { m } ] , \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} \\mathbb { R } ^ { N } \\backslash E = \\bigcup _ { n = 1 } ^ { \\infty } F _ { n } ( ( \\mathbb { R } ^ { d } ) ^ { n } ) \\backslash E \\subseteq \\mathcal { P } , \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} \\hat { l } _ { a b } = & l _ { a b } - \\frac { 1 } { 2 } ( \\sigma ^ { c d } l _ { c d } ) \\sigma _ { a b } \\\\ \\hat { n } _ { a b } = & l _ { a b } - \\frac { 1 } { 2 } ( \\sigma ^ { c d } l _ { c d } ) \\sigma _ { a b } \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} f _ 1 ( x ) ^ q = f _ 2 ( x ) ^ q . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\tilde { V } _ m : = V _ { m + 1 } - V _ m , m \\ge 0 . \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} ( f _ i , e _ j ) _ { R T } = \\delta _ { i j } , ( q ^ h , q ^ { h ' } ) _ { R T } = q ^ { - ( h , h ' ) } , ( f _ i , q ^ { h ' } ) _ { R T } = ( q ^ h , e _ i ) _ { R T } = 0 \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} A _ { m m } ( x ) = v ( T ^ { m } x ) , \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\C _ { \\epsilon , \\mu } f ( z ) : = \\int _ { | z - w | > \\epsilon } \\frac { f ( w ) } { z - w } d \\mu ( w ) . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} \\Delta ( \\widetilde { \\chi _ k u } ) & = \\Delta ( \\widetilde { \\chi _ k \\eta _ k u } ) = \\chi _ k \\Delta ( \\widetilde { \\eta _ k u } ) + 2 \\nabla \\chi _ k \\nabla ( \\widetilde { \\eta _ k u } ) + ( \\Delta \\chi _ k ) ( \\widetilde { \\eta _ k u } ) \\\\ & = ( \\widetilde { \\chi _ k \\mathcal { H } u } ) + 2 \\nabla \\chi _ k \\nabla ( \\widetilde { \\eta _ k u } ) + ( \\Delta \\chi _ k ) ( \\widetilde { \\eta _ k u } ) , \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} | X ^ { \\Gamma _ n } | = \\chi ( \\Gamma _ n , X ) : = \\prod _ i | H ^ i ( \\Gamma _ n , X ) | ^ { ( - 1 ) ^ i } \\ ; . \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\begin{aligned} f ' ( X ) & = a X ^ { a - 1 } ( X - A ) + ( n - a ) X ^ a ( X - A ) ^ { n - a - 1 } \\\\ & = X ^ { a - 1 } X ^ { n - a - 1 } ( n X - a A ) , \\end{aligned} \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} { \\rm l i m } _ { L \\rightarrow + \\infty } \\frac { k _ { \\gamma } ^ { L } } { \\sqrt { L } } = \\frac { | \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) | } { \\frac { 1 } { 2 } \\left ( - e ^ { - \\gamma _ 3 } \\dot { \\gamma _ 1 } + e ^ { \\gamma _ 3 } \\dot { \\gamma _ 2 } \\right ) ^ 2 + \\dot { \\gamma } _ 3 ^ 2 } , ~ ~ i f ~ ~ \\omega ( \\dot { \\gamma } ( t ) ) = 0 ~ ~ a n d ~ ~ \\frac { d } { d t } ( \\omega ( \\dot { \\gamma } ( t ) ) ) \\neq 0 . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ n } ( A P ) ( x ) \\overline { Q ( x ) } d x & = \\int _ { \\mathbb { T } ^ n } \\left ( \\sum _ \\xi e ^ { i 2 \\pi ( x , \\xi ) } { a ( x , \\xi ) \\delta _ { m , \\xi } } \\right ) \\overline { Q ( x ) } d x \\\\ & = \\int _ { \\mathbb { T } ^ n } e ^ { i 2 \\pi ( x , m ) } a ( x , m ) \\overline { Q ( x ) } d x = \\int _ { \\mathbb { T } ^ n } e ^ { i 2 \\pi ( x , m ) - i 2 \\pi k x } a ( x , m ) d x . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\mathcal E _ \\lambda ( \\varphi _ \\lambda ) : = \\frac { A } 2 \\int _ D | \\nabla \\varphi _ \\lambda | ^ 2 + B \\int _ D \\psi _ \\lambda ( \\varphi _ \\lambda ) \\ , . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\phi ( F ( C ) ) = & \\ \\phi [ \\lambda ( F ( C ) ) ] \\\\ \\geq & \\ \\phi ( - v ) \\\\ = & \\ \\phi \\big [ D \\big ( \\tau \\lambda ( F ( A ) ) + ( 1 - \\tau ) \\lambda ( F ( B ) ) \\big ) \\big ] \\\\ \\geq & \\ \\tau \\phi [ \\lambda ( F ( A ) ) ] + ( 1 - \\tau ) \\phi [ \\lambda ( F ( B ) ) ] \\\\ = & \\ \\tau \\phi [ F ( A ) ] + ( 1 - \\tau ) \\phi [ F ( B ) ] . \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\phi \\big ( \\exp ( H + \\sum _ { j = 1 } ^ m p _ j \\log X _ j ) \\big ) = & \\ \\phi \\Big ( \\exp \\big ( H + r \\sum _ { j = 1 } ^ m \\frac { p _ j } { r } ( \\log X _ j - \\log C _ j ) + \\sum _ { j = 1 } ^ m p _ j \\log C _ j \\big ) \\Big ) \\\\ = & \\ \\phi \\Big ( \\exp \\big ( L + r \\sum _ { j = 1 } ^ m \\frac { p _ j } { r } ( \\log X _ j - \\log C _ j ) \\big ) \\Big ) \\\\ \\leq & \\ \\sum _ { j = 1 } ^ m \\frac { p _ j } { r } \\phi \\big ( \\exp ( L + r \\log X _ j - r \\log C _ j ) \\big ) , X _ j = A _ j , B _ j . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\Lambda _ { \\lambda ' } = \\epsilon \\Lambda _ \\lambda , ~ ~ \\Lambda _ { \\lambda ' } = \\epsilon ( 1 - \\lambda ) ^ { \\frac 1 2 } \\Lambda _ { \\lambda } , ~ ~ \\Lambda _ { \\lambda ' } = \\epsilon \\lambda ^ { \\frac 1 2 } \\Lambda _ { \\lambda } , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { j = 1 } ^ { J } A _ { j } \\mathbf { 1 } _ { E ^ { j } } ( t ) , t = ( t _ 1 , \\ldots , t _ s ) \\in \\Omega _ S , \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\nabla \\Phi ^ { - 1 } ( t , x ) = [ \\nabla \\Phi ( t , \\Phi ^ { - 1 } ( t , x ) ) ] ^ { - 1 } = [ I + \\nabla u ( t , \\Phi ^ { - 1 } ( t , x ) ) ] ^ { - 1 } , \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} ( \\forall \\ , k \\ge n _ \\ast + \\ell _ \\ast ) \\ ( \\forall \\ , k \\le n \\le k + d ) \\ \\# \\mathrm { L y r } _ { n + \\ell } { \\mathcal { B } ( k + \\ell ) } = \\# \\mathrm { L y r } _ n { \\mathcal { B } ( k ) } . \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} E ( G , r ) : = \\{ x : d ( x , G ) < r \\} \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} ( a ) = 2 \\pi \\int _ { \\lambda \\varepsilon _ { n } } ^ { \\frac { T _ { \\varepsilon _ { n } } } { 2 0 d _ { k } } } p \\frac { f _ { 0 } ^ { 2 } } { r } d r = \\underbrace { 2 \\pi \\int _ { \\lambda \\varepsilon _ { n } } ^ { \\frac { T _ { \\varepsilon _ { n } } } { 2 0 d _ { k } } } ( p - p _ { 0 } ) \\frac { f _ { 0 } ^ { 2 } } { r } d r } _ { ( 1 ) } + \\underbrace { 2 \\pi p _ { 0 } \\int _ { \\lambda \\varepsilon _ { n } } ^ { \\frac { T _ { \\varepsilon _ { n } } } { 2 0 d _ { k } } } \\frac { f _ { 0 } ^ { 2 } } { r } d r } _ { ( 2 ) } . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\omega _ { 3 } ^ { 2 } = \\frac { 1 } { c _ { 7 } \\omega _ { 4 } ^ { \\frac { 6 } { c _ { 1 } } } + c _ { 8 } } , c _ { 7 } , c _ { 8 } \\in \\mathbb { R } , c _ { 7 } \\neq 0 . \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} f = ( x _ 1 + P ( x _ 2 , x _ 3 ) , x _ 2 + Q ( x _ 3 ) , x _ 3 + d ) \\circ ( a _ 1 x _ 1 , a _ 2 x _ 2 , a _ 3 x _ 3 ) \\in M _ \\alpha \\rtimes L _ \\alpha , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} l _ { a b } = & - r \\tilde \\sigma _ { a b } - \\frac { 2 } { 3 } r ^ 3 \\alpha _ { a b } + O ( r ^ 4 ) \\\\ = & ( - r \\tilde \\sigma _ { a b } - \\frac { 1 } { 3 } r ^ 3 \\alpha _ { a b } ) - \\frac { 1 } { 3 } r ^ 3 \\alpha _ { a b } + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} M _ k ( E / F , n ) = \\prod _ { \\ell \\mid n } M _ k ( E / F , \\ell ) . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\chi _ { F ^ { \\ast } A } = f ^ { \\ast } ( \\chi _ { A } ) , \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} P \\{ X \\leq a , \\ , \\ , Y \\leq b \\} = P \\{ X \\leq a \\} \\cdot P \\{ Y \\leq b \\} . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} T _ { \\rm o p } = ( I - P _ m ) T , T _ { \\rm m u l } = P _ m T , \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} U _ 1 = \\{ z = b _ 1 \\omega _ 1 + b _ 2 \\omega _ 2 ; ~ ~ b _ 2 \\in [ 1 / 3 0 , 2 9 / 3 0 ] , b _ 1 \\in [ 0 , 1 ] \\} \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} | a | : = \\sum _ { j = 1 } ^ { n } | a _ j | e _ j , \\ , \\ , | a | ^ \\gamma : = \\sum _ { j = 1 } ^ { n } | a _ j | ^ \\gamma e _ j \\quad \\mbox { a n d } | | a | | _ 1 = \\sum _ { j = 1 } ^ { n } | a _ j | = t r ( | a | ) . \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} 0 = A ^ t \\tilde { W } + \\tilde { W } ' + \\tilde { W } A . \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\nabla _ { t x } \\mu _ f ( x , t ) = \\sum _ { j = 1 } ^ n \\dfrac { \\partial ^ 2 \\mu _ f ( x , t ) } { \\partial t \\partial x _ j } e ^ j , \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\alpha ^ { J _ { 0 } } = - \\frac { | B ( y _ { 2 } , r ) | } { | B ( \\frac { y _ { 1 } + y _ { 2 } } { 2 } , 2 ^ { J _ { 0 } } r ) | } \\langle f _ { 2 } \\rangle _ { B ( y _ { 2 } , r ) } . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} \\mathcal N _ x : = \\{ L \\in H ^ { 1 , 0 } _ x ( M ) : \\mathcal L ( L , L ) = 0 \\} . \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\mu ( f ) ( x _ 1 , x _ 2 ) & = ( 1 + x _ 1 ) ( 1 + x _ 2 ) \\left [ \\frac { x _ 1 } { 1 + x _ 1 } + \\left ( \\frac { x _ 1 } { 1 + x _ 1 } \\right ) \\left ( \\frac { x _ 2 } { 1 + x _ 2 } \\right ) \\right ] = \\\\ & = ( 1 + x _ 2 ) x _ 1 + x _ 1 x _ 2 , \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} _ { \\mathcal { H } } \\langle f , h \\rangle _ { \\bar { \\mathcal { H } } } = \\int _ { 0 } ^ { 1 } f _ { s } \\ , d h _ { s } . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} v ( t , y ( t , x ) ) = \\frac { 1 } { \\rho ( t , x ) } , \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} \\phi _ { i } ( x , \\lambda ) = \\sum _ { k = 0 } ^ { q _ { i } - 1 } ( \\lambda - \\lambda _ { 0 } ) ^ { k } \\xi _ { k , i } ( x ) + ( \\lambda - \\lambda _ { 0 } ) ^ { m _ { 0 } } ( R _ { \\lambda } ( \\widetilde { L } _ { 0 } ) \\eta _ { i } ) ( x ) , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} [ L _ { \\alpha + \\hat e _ i } , A _ j ] = [ [ L _ \\alpha , A _ i ] , A _ j ] = - [ [ A _ j , L _ \\alpha ] , A _ i ] - [ [ A _ i , A _ j ] , L _ \\alpha ] = [ [ L _ \\alpha , A _ j ] , A _ i ] = [ L _ { \\alpha + \\hat e _ j } , A _ i ] \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} d ( x , y ) \\leq \\| h \\| _ { \\bar { \\mathcal { H } } } & = \\left \\| \\left ( \\int _ { 0 } ^ { \\cdot } ( V ^ * ( V V ^ * ) ^ { - 1 } ) ( z _ { s } ) d s \\right ) \\cdot ( y - x ) \\right \\| _ { \\bar { \\mathcal { H } } } \\\\ & = \\| \\gamma \\| _ { \\bar { \\mathcal { H } } } | y - x | \\leq C _ { H , V } | y - x | . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} | X ( s ) | ^ 2 & = 2 \\big ( ( \\cosh s \\cos \\theta _ 1 + \\sinh s \\cos \\theta _ 2 ) ^ 2 + ( \\cosh s \\sin \\theta _ 1 - \\sinh s \\sin \\theta _ 2 ) ^ 2 \\\\ & \\qquad + ( \\sinh s \\cos \\theta _ 1 + \\cosh s \\cos \\theta _ 2 ) ^ 2 + ( - \\sinh s \\sin \\theta _ 1 + \\cosh s \\sin \\theta _ 2 ) ^ 2 \\big ) \\\\ & = 2 \\big ( 2 ( \\cosh ^ 2 s + \\sinh ^ 2 s ) + 4 \\sinh s \\cosh s \\cos ( \\theta _ 1 + \\theta _ 2 ) \\big ) \\\\ & = 4 \\cosh 2 s + 4 \\sinh 2 s \\cos ( \\theta _ 1 + \\theta _ 2 ) . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\widetilde { m } ( x , y ) = e ^ { \\frac { | P + \\nabla u _ 0 ( x ) + \\nabla _ y u _ 1 ( x , y ) | } { 2 } + V ( x , y ) - \\overline { H } ( P ) } . \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} A ^ { \\otimes k } = \\underbrace { A \\otimes A \\otimes \\ldots \\otimes A } _ { ( ) } . \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} p _ 0 ( t , x , y ) & = p ( t , x , y ) \\ , , \\\\ p _ n ( t , x , y ) & = \\int _ 0 ^ t \\int _ { \\R } p _ { n - 1 } ( t - s , x , z ) b ( z ) \\partial _ z p ( s , z , y ) \\ , d z \\ , d s \\ , , n \\ge 1 \\ , , \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\psi _ n ( \\xi ) & = \\exp \\left \\{ n \\mathrm { L o g } \\left [ \\psi \\left ( \\frac { \\xi } { \\sqrt { n \\sigma ^ 2 } } \\right ) \\right ] \\right \\} \\\\ & = \\exp \\left \\{ n \\mathrm { L o g } \\left [ 1 - \\frac { 1 } { 2 n } \\xi ^ 2 + \\frac { \\alpha _ 3 } { 6 \\sigma ^ 3 n ^ { 3 / 2 } } ( i \\xi ) ^ 3 + \\rho _ { n , \\delta } ( \\xi ) \\right ] \\right \\} \\\\ & = e ^ { - \\xi ^ 2 / 2 } \\exp \\left \\{ \\frac { \\alpha _ 3 } { 6 \\sqrt { n } \\sigma ^ 3 } ( i \\xi ) ^ 3 \\right \\} e ^ { \\tau _ { n , \\delta } ( \\xi ) } \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} T = u + P ( u ) = u + P _ { q _ 1 } \\ast S _ { r _ 1 } \\oplus \\cdots \\oplus P _ { q _ L } \\ast S _ { r _ L } , \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} J _ 2 \\leq C _ 5 r _ n ^ 2 \\left ( \\frac { n } { N } \\right ) ^ 2 \\exp \\left ( - 2 C \\frac { n } { N } \\right ) = C _ 5 \\left ( \\frac { r _ n ^ 2 n ^ 2 } { N ^ 3 } \\right ) N \\exp \\left ( - 2 C \\frac { n } { N } \\right ) . \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\textrm { S I N R } _ k = \\frac { \\frac { P } { N _ { \\rm M S } } \\eta ^ 2 | \\hat { \\textbf { h } } _ k ^ { \\rm H } \\textbf { K } \\hat { \\textbf { h } } _ k | ^ 2 } { \\sigma ^ 2 + \\frac { P } { N _ { \\rm M S } } \\eta ^ 2 | \\hat { \\textbf { e } } _ k ^ { \\rm H } \\textbf { K } \\hat { \\textbf { h } } _ k | ^ 2 + \\frac { P } { N _ { \\rm M S } } \\eta ^ 2 \\sum \\limits _ { k ' \\neq k } | \\textit { \\textbf { h } } _ k ^ { \\rm H } \\textbf { F } \\textbf { K } \\hat { \\textbf { h } } _ { k ' } | ^ 2 } . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\ln \\left [ F ( \\bar N ) \\right ] & \\le 2 \\cdot 3 ^ { \\bar N } \\ln | x _ 0 | + \\frac { 2 } { 3 } \\cdot 3 ^ { \\bar N } \\sum _ { j = 0 } ^ { \\bar N } 3 ^ { - j } \\ln h _ { j } \\\\ & = \\frac 2 3 \\cdot 3 ^ { \\bar N } \\left [ \\ln | x _ 0 | ^ 3 + \\sum _ { j = 0 } ^ { \\bar N } 3 ^ { - j } \\ln h _ { j } \\right ] . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} D _ n : = \\begin{bmatrix} B _ n & O _ n \\\\ C _ n & B _ n \\end{bmatrix} \\mbox { f o r e v e r y } n \\ge 1 , \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\mathbb { K } _ 1 ( t - s ) \\mathbb { K } _ 2 ( s ) \\ , d s = 1 , t > 0 . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} H _ i ( \\omega _ { i , m } ( K ) ) & = \\sum _ { t = 1 } ^ m \\big ( H _ i ( \\omega _ { i , t } ( K ) ) - H _ i ( \\omega _ { i , t } ( 0 ) ) \\big ) \\\\ & = \\sum _ { t = 1 } ^ m \\tilde { x } _ { i , t } \\nabla _ t H _ i ( u _ t ) \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} M + \\sum _ { I \\subset [ n + 1 ] } y _ I ^ - \\cdot \\triangle _ I = \\sum _ { I \\subset [ n + 1 ] } y _ I ^ + \\cdot \\triangle _ I \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} \\delta _ j : = \\min _ { z \\in \\partial K } \\abs { p _ j ( z ) - a _ j } > 0 , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } c ^ n _ { \\alpha \\ , \\beta } ( x ) s _ { n \\ , \\beta } ( x ) = s _ { \\alpha } ( x ) , x \\in U _ { \\alpha \\ , \\beta } \\ , , \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} w _ { \\nu _ \\Lambda } = ( \\mathbf { a } : \\mathbf { a } ^ { \\mathbf { T } } : \\nabla v ) \\nu - q ( \\mathbf { a } \\nu ) \\quad \\mbox { o n } \\quad \\Gamma _ c \\times ( 0 , T ) , \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} ( \\mu ^ 0 \\hat \\oplus \\mu ^ 1 ) ( c ) = \\begin{array} { c } \\mu ^ 0 ( \\mathfrak p _ 1 ) \\hat \\oplus \\mu ^ 1 ( \\mathfrak p _ 1 ) \\leftrightarrow \\cdots \\leftrightarrow \\mu ^ 0 ( \\mathfrak p _ k ) \\hat \\oplus \\mu ^ 1 ( \\mathfrak p _ k ) \\\\ \\updownarrow \\\\ \\mu ^ 0 ( \\mathfrak q _ 1 ) \\hat \\oplus \\mu ^ 1 ( \\mathfrak q _ 1 ) \\leftrightarrow \\cdots \\leftrightarrow \\mathfrak \\mu ^ 0 ( \\mathfrak q _ \\ell ) \\hat \\oplus \\mu ^ 1 ( \\mathfrak q _ \\ell ) \\end{array} . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\mu ( \\varepsilon _ i ( x ) ) = a _ i ( - x ^ { - 2 } ) F _ i ( - x ^ { - 2 } ) = 1 - b _ i ( - x ^ { - 2 } ) f _ i ( - x ^ { - 2 } ) \\ { \\rm i n } \\ { \\cal A } \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\nabla u = 0 D ^ 2 u \\ge 0 \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\| X \\| _ { - s } ^ 2 = x ^ 2 + v ^ 2 + \\sum _ { k \\geq 1 } k ^ { - 2 s } z _ k ^ 2 . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & - p ^ i _ t - ( ( D _ 1 a ( y _ x , t , x ) y _ x + a ( y _ x , t , x ) ) p ^ i _ x ) _ { x } + D _ 1 F ( y , y _ x ) p ^ i - ( D _ 2 F ( y , y _ x ) p ^ i ) _ x = \\alpha _ i ( y - y _ { i , d } ) { 1 } _ { \\mathcal { O } _ { i , d } } \\ \\ Q , \\\\ & p ^ i ( 0 , t ) = p ^ i ( L , t ) = 0 \\ \\ \\ ( 0 , T ) , \\\\ & p ^ i ( T ) = 0 \\ \\ \\ I . \\end{array} \\right . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\aligned \\int _ M \\partial _ t u _ n \\cdot e _ i \\ , d M = & \\kappa _ 1 \\int _ M \\Delta u _ n \\cdot e _ i \\ , d M + \\gamma \\int _ M \\nabla F ( u _ n ) \\times \\Delta u _ n \\cdot e _ i \\ , d M \\\\ & - \\kappa _ 2 \\int _ M ( 1 + \\mu \\cdot F ( u _ n ) ) \\nabla F ( u _ n ) \\cdot e _ i \\ , d M \\\\ u _ n ( 0 , \\cdot ) = u _ { 0 n } \\endaligned \\end{array} \\right . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\mathcal { O } _ \\delta : = \\{ \\rho \\in h ^ { 4 + \\alpha } ( \\Bbb S ) : \\ , \\| \\rho \\| _ { h ^ { 4 + \\alpha } ( \\Bbb S ) } < \\delta \\} . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} u ^ * \\begin{pmatrix} * & & \\\\ & * & \\\\ & & \\ddots \\end{pmatrix} u , \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} N ( 1 , 1 ) = \\begin{cases} \\emptyset & \\mbox { i f } n = m = 3 , 4 \\\\ \\nu ^ { 0 } _ + , \\nu ^ { 0 } _ - & n \\geq 5 \\end{cases} \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} Q ^ { ( n ) } ( t ) & = P ^ { ( n ) } ( t ) + 4 \\bigl [ \\cdot \\arctan \\cdot \\bigr ] ^ { ( n - 1 ) } ( t ) \\\\ & = P ^ { ( n ) } ( t ) + 4 \\Biggl [ t \\frac { ( - 1 ) ^ { n - 1 } ( n - 1 ) ! } { ( 1 + t ^ { 2 } ) ^ { n / 2 } } \\sin ( n \\pi / 2 - n \\arctan t ) \\\\ & + \\frac { ( - 1 ) ^ { n } ( n - 1 ) ! } { ( 1 + t ^ { 2 } ) ^ { ( n - 1 ) / 2 } } \\sin ( ( n - 1 ) \\pi / 2 - ( n - 1 ) \\arctan t ) \\Biggr ] . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 \\frac { 1 } { 4 } \\frac { v o l ( D _ i ) } { v o l ( \\mathfrak { D } ( f , i d ) ) } = A \\quad ( 0 \\le A < 1 ) , \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} U _ n ^ * ( A ^ * A ) ^ m U _ n = ( A ^ * A ) ^ m , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} M ( x ) = \\int \\mu ( t , x ) d B _ t . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} g '' ( q ) = - \\xi '' ( q ) + \\frac 1 { ( A _ 2 ( 1 - q ) + \\Delta ) ^ 2 } = - \\xi '' ( q ) + \\frac 1 { ( 1 + z _ 2 ) ^ 2 \\Delta ^ 2 } \\ge 0 . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\Pr { B _ k } { M _ k } \\leq \\frac { c } { M _ k ^ { 5 / 4 } } \\cdot \\frac { 1 } { 4 c M _ k ^ { - 1 / 4 } } = \\frac { 1 } { 4 M _ k } \\cdot \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\begin{aligned} \\operatorname { C o v } \\left ( \\hat { \\textbf { x } } \\right ) - \\textbf { I } ^ { - 1 } \\left ( \\textbf { x } \\right ) \\succeq \\textbf { 0 } , \\end{aligned} \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} D = \\{ v \\in \\R ^ 2 \\mid v = t _ 1 e _ 1 + t _ 2 v _ 2 | t _ 1 | \\le x | t _ 2 | \\le y \\} . \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\in O \\right ) & \\geq P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\in ( \\varepsilon _ 1 , \\varepsilon _ 2 ] \\right ) \\\\ & = P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq \\varepsilon _ 2 \\right ) - P \\left ( \\frac { n - A _ n ^ * } { f _ 2 ( n ) } \\leq \\varepsilon _ 1 \\right ) . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} \\MoveEqLeft \\lim _ { \\theta \\to - \\infty } \\mathbb { P } ( \\mathit { S I N R } > \\theta ) = \\lim _ { \\theta \\to - \\infty } \\sum _ { n = 1 } ^ \\infty f _ { N _ i } ( n ) \\mathbb { P } ( \\mathit { S I N R } > \\theta \\mid n ) \\\\ & = \\sum _ { n = 1 } ^ \\infty f _ { N _ i } ( n ) = 1 - f _ { N _ i } ( 0 ) . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} [ a ( z ) , b ( w ) ] & = \\sum _ { j \\in \\Z _ { \\geq 0 } } [ a ( j ) b ] ( w ) \\partial _ w ^ { ( j ) } \\delta ( z , w ) , \\\\ [ a ( m ) , b ( n ) ] & = \\sum _ { j \\in \\Z _ { \\geq 0 } } \\binom { m } { j } \\left [ a ( j ) b \\right ] ( m + n - j ) . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\tau ^ { \\Gamma } _ p ( M ) = \\lim _ { n \\to \\infty } ( \\Gamma : \\Gamma _ n ) ^ { - 1 } \\log _ p \\chi ( \\Gamma _ n , M ) \\ ; \\Q _ p \\ ; . \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} X \\times D _ { \\ell } \\ni ( x , z ) \\mapsto U ( x , z ) : = u _ { \\log | z | } ( x ) , \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} ( Q T ) ^ { * } = T ^ { * } Q , \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\{ ( M _ S , \\mu _ S ) \\in \\mathcal { C } _ G : \\exists ( M _ b , \\mu _ b ) \\in \\mathcal { T } _ { G , b , \\mu } \\ , \\ \\ , \\ \\theta _ { M _ b } ( \\mu _ b ) = \\theta _ { M _ S } ( \\mu _ S ) , \\ , \\ \\mu _ b \\sim _ { M _ b } \\mu _ S \\} . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} m _ { j } ( x , y ) : = \\sup _ { ( u , v ) \\in E } \\{ f ( u , v ) + \\langle G ( u , v ) , ( x , y ) - ( u , v ) \\rangle \\} = \\max \\left \\{ | x | , 2 ( y - j - 1 ) \\right \\} , \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\alpha _ { H _ i } & : = \\sup _ { u : \\hat { c } _ i ^ T u \\leq 1 } \\{ \\beta ~ | ~ H _ i ^ * \\big ( \\nabla H _ i ( u ) \\big ) \\geq \\beta H _ i ( u ) \\} \\\\ & = \\sup _ { u : \\hat { c } _ i ^ T u \\leq 1 } \\{ \\beta ~ | ~ \\langle \\nabla H _ i ( u ) , u \\rangle \\geq ( 1 + \\beta ) H _ i ( u ) \\} \\\\ & = \\inf _ { u : \\hat { c } _ i ^ T u \\leq 1 } \\frac { \\langle \\nabla H _ i ( u ) , u \\rangle } { H _ i ( u ) } - 1 \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} Q ^ D : = P ^ { - 1 } \\textnormal { d i a g } ( Q ^ { \\lambda _ i } ) P \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\begin{cases} V _ { n , 2 } ( y , x ) & \\triangleq \\delta \\left ( \\frac { y ^ 2 } { 2 } + U ( x ) \\right ) + x y + C _ V , \\\\ V _ { n , 3 } ( \\eta , y , x ) & \\triangleq \\Gamma _ 1 ( \\eta ) + V _ { n , 2 } ( y , x ) , \\\\ V _ { n , 4 } ( \\zeta , \\eta , y , x ) & \\triangleq \\Gamma _ 2 ( \\zeta , \\eta ) + V _ { n , 2 } ( y , x ) , \\end{cases} \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\omega = M ^ W d , M ^ W \\in \\mathcal { W } ^ { N D } [ y _ h ] , y _ h = S [ u + h ] \\Omega _ T . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ G ( x ) - G ( y ) : x , y \\in E \\} = \\R ^ n . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} \\sigma _ k ^ \\ast ( s _ 1 ) = k . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\big ( a ^ k ( l ) , c _ k ( l ) \\big ) = \\begin{cases} ( a ^ { k - 1 } ( l ) , c _ { k - 1 } ( l ) ) , & ~ l \\in \\cup _ { i = 1 } ^ { k - 1 } E _ i \\\\ z ^ k \\big ( X ( t _ { k - 1 } ) \\big ) , & ~ l \\in E _ k . \\end{cases} \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} \\phi _ i \\star _ \\tau \\phi _ j = \\sum _ { \\alpha , \\beta = 0 } ^ N \\left ( \\partial _ { t _ i } \\partial _ { t _ j } \\partial _ { t _ \\alpha } \\mathcal { F } \\right ) G ^ { \\alpha \\beta } \\phi _ \\beta \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} F ( \\psi _ 1 , \\psi _ 2 ) ( t ) & = - \\lambda \\langle K ( t + \\cdot ) , h \\rangle + \\frac { \\nu ^ 2 } 2 \\langle K ( t + \\cdot ) , h \\rangle ^ 2 \\\\ & + ( \\nu ^ 2 \\langle K ( t + \\cdot ) , h \\rangle - \\lambda ) \\psi _ 2 ( t ) + \\frac { \\nu ^ 2 } 2 \\psi _ 2 ( t ) ^ 2 . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} Q _ { 3 , k } & \\leq C _ { H , l _ { 0 } } | I _ { k } | ^ { 2 ( 1 - H ) } \\int _ { 0 } ^ { 1 } \\left ( \\sum _ { j = 1 } ^ k | I _ { j } | \\right ) ^ { 2 H - 1 } \\cdot \\left | \\int _ { 0 } ^ { v _ 1 } \\frac { d v _ 2 } { ( \\sum _ { j = 1 } ^ k | I _ { j } | ) ^ { H - \\frac { 1 } { 2 } } ( v _ 1 - v _ 2 ) ^ { H - \\frac { 1 } { 2 } } } \\right | ^ { 2 } d v _ 1 \\\\ & \\leq C _ { H , l _ { 0 } } | I _ { k } | ^ { 2 ( 1 - H ) } . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} 0 & = P ^ d \\subset \\cdots \\subset P ^ 1 \\subset P ^ 0 = H . \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\tau = \\ ? \\ ? & \\{ U \\colon U \\cap X _ { n } \\boldsymbol { X } _ { n } n \\in \\omega , \\\\ & \\top \\in U n \\in \\omega \\bigcup _ { n \\leq m } X _ { m } \\subseteq U \\} . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\frac { \\log ( ( n p _ n b _ c ^ { ( n ) ' } ) / a _ c ^ { ( n ) } ) } { n p _ n } & = \\frac { \\log ( n p _ n ) } { n p _ n } + \\frac { \\log b _ c ^ { ( n ) ' } } { n p _ n } - \\frac { \\log a _ c ^ { ( n ) } } { n p _ n } \\\\ & \\sim _ e - 1 - \\frac { \\log a _ c ^ { ( n ) } } { n p _ n } \\sim _ e - 1 , \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\gamma _ { v _ { + } + v _ { - } } ( s _ { + } { \\otimes } s _ { - } ) = \\gamma _ { v _ { + } } ( s _ { + } ) { \\otimes } s _ { - } + ( - 1 ) ^ { | s _ { + } | } s _ { + } { \\otimes } \\gamma _ { v _ { - } } ( s _ { - } ) . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\nu _ \\beta = [ \\nu _ \\theta ; \\nu _ \\beta ( \\phi _ \\beta ) = \\gamma _ \\beta ] \\mbox { a n d } \\nu _ \\alpha = [ \\nu _ \\beta ; \\nu _ \\alpha ( \\phi _ \\alpha ) = \\gamma _ \\alpha ] , \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} \\mathcal { T } _ R ( x ) \\doteq \\begin{cases} x & | x | _ E \\leq R \\\\ \\frac { R x } { | x | _ E } & | x | _ E > R . \\end{cases} \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\frac { \\# D _ { a b } } { \\# \\{ 0 , 1 \\} ^ d } & \\ , = \\ , \\frac { 1 } { 2 ^ d } \\sum _ { k = a } ^ { b } \\binom { d } { k } \\ , = \\ , \\P \\biggl \\{ \\alpha \\leq \\frac { 1 } { \\sqrt { d } } \\ , \\sum _ { j = 1 } ^ d Z _ j \\leq \\beta \\biggr \\} \\\\ & \\ , \\geq \\ , \\underbrace { \\Phi ( \\beta ) - \\Phi ( \\alpha ) } _ { = : C _ { \\alpha \\beta } } - \\frac { 2 \\ , C _ 0 } { \\sqrt { d } } \\ , = : \\ , r _ 0 ( \\alpha , \\beta , d ) \\ , . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} p ^ { - 1 } ( b ) & \\rightarrow \\{ ( e , \\gamma ) \\in E \\times _ B B ^ I \\ | \\ \\gamma ( 0 ) = b , \\gamma ( 1 ) = e \\} \\\\ f & \\mapsto ( f , t \\mapsto p ( f ) ) \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} { z } \\ , p _ n ^ { ( \\alpha ) } ( z ) \\ = \\ a _ { n + 1 } p ^ { ( \\alpha ) } _ { n + 1 } ( z ) + b _ { n } p ^ { ( \\alpha ) } _ { n - 1 } ( z ) \\ , \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} \\Gamma _ { k m } ( t ) = \\left \\{ \\begin{array} { l } \\gamma _ { k m } ( t ) , \\quad m \\neq k \\\\ - \\sum \\nolimits _ { j : j \\neq k } \\gamma _ { k j } ( t ) , m = k \\end{array} \\right . \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\delta \\tau = \\frac { d } { d s } | _ { s = 0 } \\tau \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} u ( t , X _ t ) & = u ( 0 , X _ 0 ) + \\int _ 0 ^ t ( u _ t + b \\cdot \\nabla u + \\frac { 1 } { 2 } \\Delta u ) ( s , X _ s ) d s + \\int _ 0 ^ t \\nabla u ( s , X _ s ) d B _ s \\\\ & = u ( 0 , X _ 0 ) - \\int _ 0 ^ t b ( s , X _ s ) d s + \\int _ 0 ^ t \\nabla u ( s , X _ s ) d B _ s \\\\ & = u ( 0 , X _ 0 ) - X _ t + X _ 0 + B _ t + \\int _ 0 ^ t \\nabla u ( s , X _ s ) d B _ s . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} p = \\sum _ { i = 0 } ^ s \\left ( \\sum _ { j = 1 } ^ { r _ i } a _ { i j } \\textbf { Q } ^ { \\lambda _ { i j } } \\right ) q ^ i = \\sum _ { 0 \\leq i \\leq s , 1 \\leq j \\leq r _ i } a _ { i j } \\textbf Q ^ { \\lambda _ { i j } ' } , \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} S ( N ) = \\sum _ { n = 1 } ^ { \\infty } \\lambda _ g ( n ) \\chi ( n ) W \\left ( \\frac { n } { N } \\right ) , \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} \\det ( \\nabla ^ 2 h ( u ) + h ( u ) \\ , { \\rm I d } ) = \\mbox { $ \\frac 1 n $ } \\ , h ( u ) ^ { p - 1 } \\left \\| \\nabla h _ K ( u ) + h _ K ( u ) \\ , u \\right \\| _ Q ^ { n - q } \\cdot f ( u ) . \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} M c _ { \\sigma ( \\chi ) } ( \\sum _ g g ( \\alpha ) g ^ { - 1 } ) & = \\sigma ( m ) | G | ^ { - 1 } \\sum _ { h , g } \\sigma ( \\overline { \\chi ( h ) } ) g ( \\alpha ) h g ^ { - 1 } \\\\ & = | G | ^ { - 1 } \\sigma ( m ) \\sum _ { h , g } \\sigma ( \\overline { \\chi ( h g ) } ) g ( \\alpha ) h \\\\ & = | G | ^ { - 1 } \\sigma ( m ) \\sum _ h \\sigma ( \\overline { \\chi ( h ) } ) \\{ \\sum _ { g } \\sigma ( { \\chi ( g ) } ) g ^ { - 1 } ( \\alpha ) \\} h . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { ( - 1 ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( - q ) _ n ( 1 - q ^ { 2 n + 1 } ) } = \\frac { 1 } { 2 } \\sum _ { n \\ge 0 } q ^ { n ^ 2 + n / 2 } ( 1 + q ^ { n + 1 / 2 } ) \\sum _ { | j | \\le n } q ^ { j ^ 2 / 2 } \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\bigvee _ { i = 1 } ^ { 3 } ( x \\to y _ { i } ) \\lor ( y _ { i } \\to x ) \\thickapprox 1 \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} \\frac { \\left | { \\sum } ^ * ~ L ( 1 , \\chi , Y ) | R ( \\chi ) | ^ 2 \\right | } { { \\sum } ^ * ~ | R ( \\chi ) | ^ 2 } & \\geq e ^ \\gamma \\log X \\left ( 1 - \\frac { 1 } { \\log X } + \\mathcal { O } \\left ( \\frac { 1 } { ( \\log X ) ^ 2 } \\right ) \\right ) \\\\ & = e ^ \\gamma \\left ( \\log _ 2 q + \\log _ 3 q - \\log B - 1 + \\mathcal { O } \\left ( \\frac { 1 } { \\log _ 2 q } \\right ) \\right ) , \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} u ( x , t ) = \\phi ( n \\cdot x - c t - \\theta _ 0 ) \\ \\ { \\rm f o r \\ a l l } \\ \\ ( x , t ) \\in \\R ^ N \\times \\R . \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} f ( n ) \\coloneqq \\sum _ { M = 1 } ^ { \\infty } 1 _ { \\Xi _ M } ( n ) f _ M ( n ) = \\begin{cases} 0 , & \\ n \\notin \\bigcup \\limits _ { K = 1 } ^ \\infty \\Psi _ K \\\\ f _ M ( n ) , & \\ M = \\min \\{ K \\in \\N : n \\in \\Psi _ K \\} \\end{cases} \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\gamma _ \\textnormal { e q } \\left ( \\textnormal { c o n f l u e n c e } \\left ( J ^ { K \\textnormal { t h } , \\textnormal { e q } } \\right ) ( z , Q ) \\right ) = J ^ { \\textnormal { c o h } , \\textnormal { e q } } ( z , Q ) \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} i \\partial _ t u + \\Delta u + V ( x , t ) u = 0 , ( x , t ) \\in \\R ^ n \\times ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{gather*} P r ( f , \\sigma ) = P r ( f , \\sigma \\nu ) , \\\\ P r _ D ( f , \\sigma ) = P r _ D ( f , \\sigma \\nu ) , \\\\ P r ( f , \\sigma , \\{ k _ j \\} ) = P r ( f , \\sigma \\nu , \\{ k _ j ' \\} ) , \\\\ P r ( f , \\sigma , \\{ k _ j \\} , L , \\{ R _ i \\} ) = P r ( f , \\sigma \\nu , \\{ k _ j ' \\} , L , \\{ R _ i \\} ) . \\end{gather*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\widehat { M } _ s : = \\min \\{ m \\in \\mathbb { Z } _ { > 0 } \\colon { m } > I _ s \\} \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} M ' _ { | x | ^ 2 } ( t ) = 8 s \\| u ( t ) \\| ^ 2 _ { \\dot { H } ^ s } - \\frac { 4 d \\alpha } { \\alpha + 2 } \\| u ( t ) \\| ^ { \\alpha + 2 } _ { L ^ { \\alpha + 2 } } = 4 d \\alpha E ( u ( t ) ) - 2 ( d \\alpha - 4 s ) \\| u ( t ) \\| ^ 2 _ { \\dot { H } ^ s } . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\beta _ k ( x ) = \\sup \\Big \\{ \\epsilon \\geq 0 \\big | \\exists _ { \\mathcal { C } \\in C _ { n , k } ( \\epsilon ) } \\exists _ { W \\in G _ { m , k } } \\exists _ { T \\in O ( V _ { \\mathcal { C } } , W ) } & \\forall _ { y \\in ( x + \\mathcal { C } ) \\cap B ( x , \\epsilon ) } \\\\ & u ( x ) - u ( y ) = T ( x - y ) \\Big \\} , \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\ddot { v } ( t ) - \\div ( A ^ { ( 4 ) } \\nabla v ( t ) ) + { p } ( t ) \\cdot \\nabla v ( t ) - 2 { q } ( t ) \\cdot \\nabla \\dot { v } ( t ) = { g } ( t ) \\hbox { i n } \\Omega ^ { ( 4 ) } \\setminus \\Gamma ^ { ( 4 ) } ( t _ 0 ) \\ , , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\kappa + v = ( \\kappa + v _ 1 ) + v _ 2 , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} I ( \\gamma _ \\varepsilon ) : = \\gamma _ \\varepsilon ^ { - 4 } + A ( \\gamma _ \\varepsilon ) / 2 + 4 \\gamma _ \\varepsilon ^ { - 3 } \\exp ( - 1 - M ) B ( \\gamma _ \\varepsilon ) S \\ , , \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} M ( x ) = \\int \\mu ( t , x ) d B _ t \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\hat { \\eta } _ k ^ ( t ) = \\begin{cases} \\boldsymbol { \\eta } _ k ( \\frac { | \\mathcal { N } _ k | + 1 } { 2 } ) & \\ ! , | \\mathcal { N } _ k | \\ \\\\ \\frac { 1 } { 2 } \\left ( \\boldsymbol { \\eta } _ k ( \\frac { | \\mathcal { N } _ k | + 1 } { 2 } ) + \\boldsymbol { \\eta } _ k ( \\frac { | \\mathcal { N } _ k | + 1 } { 2 } + 1 ) \\right ) & \\ ! , | \\mathcal { N } _ k | \\ \\end{cases} , \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\nabla F = \\nabla c \\circ P + v . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} S _ { l } ( w ) _ { \\cdot , \\cdot } : \\{ ( s , t ) \\in [ 0 , 1 ] ^ { 2 } ; \\ , s \\le t \\} \\rightarrow T ^ { ( l ) } , ( s , t ) \\mapsto S _ { l } ( w ) _ { s , t } : = 1 + \\sum _ { n = 1 } ^ { l } \\mathbf { w } _ { s , t } ^ { n } . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} ( \\theta _ \\epsilon ^ \\beta e ^ \\beta + e ^ { \\alpha + \\beta } ) ( \\theta _ \\iota ^ \\beta e ^ \\beta + e ^ { \\alpha + \\beta } ) = \\theta _ \\epsilon ^ \\beta \\theta _ \\iota ^ \\beta c _ \\beta t _ \\beta + c _ { \\alpha + \\beta } t _ { \\alpha + \\beta } + b _ { \\beta , \\alpha + \\beta } ( \\theta _ \\epsilon ^ \\beta + \\theta _ \\iota ^ \\beta ) e ^ \\alpha \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { \\sum _ { \\substack { q \\\\ q \\le x } } | \\lambda _ f ( q ) | ^ 2 } { \\# \\{ q : q \\le x \\} } = 1 . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial r } | \\eta ^ { - 1 } \\delta _ r ( \\xi ) | ^ 4 \\right | _ { r = 1 } = & 4 | z - w | ^ 2 [ x ( x - u ) + y ( y - v ) ] \\\\ & + 4 ( t - s + 2 ( y u - x v ) ) ( t + ( y u - x v ) ) . \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} M _ { \\tau } ^ g = e ^ { \\int _ 0 ^ { \\tau } V ( Y _ v ) d v } g ( Y _ { \\tau } ) . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} X _ m = V ( w ^ m + x y ^ { m - 1 } + y z ^ { m - 1 } + z x ^ { m - 1 } ) . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} E ( u ( t , \\cdot ) ) = \\frac 1 2 \\int _ { M } ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 ) - \\rho _ 1 \\log \\int _ { M } e ^ { u - \\overline { u } } - \\rho _ 2 \\log \\int _ { M } e ^ { - u + \\overline { u } } , \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} & r ( z , t ) > 0 t < \\mathcal { T } ( z ) , & & \\lim _ { t \\to \\mathcal { T } ( z ) } r ( z , t ) = 0 . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} { \\rm E n d s } ( S ) = { \\rm E n d s } _ { \\infty } ( S ) = \\mathcal { B } \\sqcup \\mathcal { U } , \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} ( \\left \\| \\overline { \\nabla } f \\right \\| ^ { 2 } - \\partial _ { t } f ) ^ { 2 } & = \\frac { 1 } { \\alpha ^ { 2 } } \\left ( \\frac { G } { t _ { 0 } } \\right ) ^ { 2 } + \\frac { 2 ( \\alpha - 1 ) } { \\alpha ^ { 2 } } \\left \\| \\overline { \\nabla } f \\right \\| ^ { 2 } \\left ( \\frac { G } { t _ { 0 } } \\right ) + \\frac { ( \\alpha - 1 ) ^ { 2 } } { \\alpha } \\left \\| \\overline { \\nabla } f \\right \\| ^ { 4 } . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} K ( f ) ( \\xi ) = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } ^ 2 } \\frac { ( \\xi - \\bar { \\xi } ) ^ \\perp } { | \\xi - \\bar { \\xi } | ^ 2 } f ( \\bar { \\xi } ) d \\bar { \\xi } , \\ \\xi \\in \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} u ( t , x ) = 0 \\partial _ n u ( t , x ) = 0 , ( t , x ) \\in \\mathbb R \\times \\partial \\Omega , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\xi ( T ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma } \\xi ( z ) ( T - z I ) ^ { - 1 } d z \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\P ( | X - Y | \\geq 1 - \\delta ) = \\P ( A ) = 2 \\delta / ( 1 + \\delta ) . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} Z _ { n + 1 , x } = \\begin{cases} \\displaystyle \\frac { X _ { n + 1 , x } } { W _ { n , x } } \\geq \\frac { X _ { n + 1 , x } } { X _ n } , & \\mbox { i f } W _ { n , x } \\geq t ^ { \\ast } , \\\\ \\displaystyle \\frac { X _ { n + 1 , x } - t ^ { \\ast } } { W _ { n , x } + t ^ { \\ast } } + \\frac { t ^ { \\ast } } { \\Tilde { t } } \\geq \\frac { X _ { n + 1 , x } - t ^ { \\ast } } { \\Tilde { t } } + \\frac { t ^ { \\ast } } { \\Tilde { t } } \\geq \\frac { X _ { n + 1 , x } } { X _ n } , & \\mbox { i f } W _ { n , x } < t ^ { \\ast } . \\end{cases} \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} \\Big | \\sum _ { k = 1 } ^ K e _ { 1 , k } \\prod _ { j = 2 } ^ n e _ { j } \\Big | _ { Y _ { n + 1 } } \\le \\Big | \\sum _ { k = 1 } ^ K e _ { 1 , k } \\Big | _ { X _ 1 } \\prod _ { j = 2 } ^ n | e _ { j } | _ { X _ { j } } . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} p _ e ( \\rho ) = a v ^ { - \\gamma } , \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\frac { d z _ 2 } { d q } = \\frac { \\xi '' ( q ) ( 1 - q ) - [ \\xi ' ( 1 ) - \\xi ' ( q ) + ( 1 - q ) \\xi '' ( q ) ] \\frac { 1 - \\xi ( q ) - \\xi ' ( q ) ( 1 - q ) } { ( 1 - q ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] } } { ( 1 - q ) [ \\xi ' ( 1 ) - \\xi ' ( q ) ] ( \\frac { 2 + z _ 2 } { z _ 2 ^ 3 } \\log ( 1 + z _ 2 ) - \\frac 2 { z _ 2 ^ 2 } ) } . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\frac { \\delta ^ 2 \\lvert \\textbf { s } \\rvert ^ 2 } { 3 2 C _ { \\mathbf { I } } C _ e ^ 2 c ( 0 ) \\sigma _ z ^ 2 } \\ ! \\cdot \\ ! \\frac { \\epsilon _ S ^ 2 \\sigma _ z ^ 2 } { \\lvert \\textbf { s } \\rvert ^ 2 } \\ ! \\ge \\ ! \\frac { \\delta ^ 2 } { \\sum \\limits _ { i \\ge 1 } \\ ! \\frac { 3 2 C _ { \\mathbf { I } } e ^ { 2 L ( T + \\frac { \\epsilon _ { b } } { K _ { S , 0 } + 1 } ) } } { ( i + K _ { S , 0 } ) ^ 2 } } \\overset { \\Delta } { = } R . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\begin{aligned} & w _ m = \\sum _ { i _ 1 = 0 } ^ { \\alpha _ 1 } \\cdots \\sum _ { i _ 8 = 0 } ^ { \\alpha _ 8 } a _ { i _ 1 \\cdots i _ 8 } ( y _ 1 ^ { p ^ s } ) ^ { i _ 1 } ( y _ 2 ^ { p ^ s } ) ^ { i _ 2 } \\cdots ( y _ 8 ^ { p ^ s } ) ^ { i _ 8 } \\\\ & \\in \\mathbb { F } _ p [ y _ 1 ^ { p ^ s } , y _ 2 ^ { p ^ s } , \\cdots , y _ 8 ^ { p ^ s } ] \\backslash \\mathbb { F } _ p [ y _ 1 ^ { p ^ { s + 1 } } , y _ 2 ^ { p ^ { s + 1 } } , \\cdots , y _ 8 ^ { p ^ { s + 1 } } ] , \\end{aligned} \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} f ( S ) = 2 ( S - 1 ) ( S - 2 ) S ^ 2 = 0 , \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\hat { Q } _ r ( I - Q _ r ) & = \\hat { Q } _ r ( \\hat \\Sigma - \\mu _ r I + \\Sigma - \\hat \\Sigma ) R _ r \\\\ & = \\hat Q _ r ( \\hat \\Sigma - \\mu _ r I ) R _ r - Q _ r E R _ r - ( \\hat { Q } _ r - Q _ r ) E R _ r \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} V \\cap G L R = \\oplus _ { \\chi ( \\ne 1 ) \\in \\Psi } L R ( \\chi ) , \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} ( X ^ 2 + 1 ) P _ r + X ^ { 2 ^ r } + X = 0 . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { \\ell ^ p } : = ( \\sum _ { n ' \\in \\mathbb { Z } ^ n } | f ( n ' ) | ^ p ) ^ { \\frac { 1 } { p } } < \\infty , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\theta _ { \\max } \\triangleq \\max _ { w \\in [ - \\pi , \\pi ] } ~ \\frac { \\sigma ^ 2 } { g ( w ) } = \\frac { \\sigma ^ 2 } { ( a - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\alpha _ { f } ( x ) = \\lim _ { n \\to \\infty } \\max \\{ 1 , h _ { H } ( f ^ { n } ( x ) ) \\} ^ { 1 / n } \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} \\sum _ { i , j , k , p } ( \\bar h ^ { p ^ { \\ast } } _ { i j k } ) ^ 2 = 4 - 4 \\bar H ^ 2 + \\frac { 3 } { 2 } \\bar H ^ 4 = ( 2 - H ^ 2 ) ^ 2 + \\dfrac 1 2 \\bar H ^ 4 > 0 . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} \\big ( K \\circ v _ j \\big ) ^ { ( q ) } & = \\sum \\frac { q ! } { m _ 1 ! \\dots m _ q ! } ( K ^ { ( M _ q ) } \\circ v _ j ) \\prod _ { s = 1 } ^ q \\left ( \\frac { v _ j ^ { ( s ) } } { s ! } \\right ) ^ { m _ s } \\\\ & = \\sum c _ { m _ 1 , \\ldots , m _ q } \\frac { K ^ { ( M _ q ) } \\circ v _ j } { F ^ { M _ q } } \\prod _ { s = 1 } ^ q \\big ( f _ 0 ^ { ( s - 1 ) } \\big ) ^ { m _ s } , \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } x _ { N _ 1 + 2 + m } = L \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\textrm { s p a n } \\{ \\xi _ y - \\xi _ z : y , z \\in E ^ { * } \\} = X ; \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} s _ i \\varpi _ i & = - \\varpi _ i + \\sum _ { h } \\varpi _ h . \\\\ s _ i \\varpi _ h & = \\varpi _ h , \\textup { f o r $ h \\ne i $ . } \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} w = ( M ) = ( m + 1 ) ( k + 1 ) - r _ - . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\mathbf E ( M , p , q ; I , J ) \\hat \\oplus \\mathbf E ( N , p , q ; I ' , J ' ) = \\mathbf E ( M + N , p , q ; I , J ) + \\mathbf E ( M + N , p , q ; [ i ' _ { 1 } , \\dots , i ' _ { p } ] + [ M ^ { p } ] , [ j ' _ { 1 } , \\dots , j ' _ { q } ] + [ M ^ { q } ] ) , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} T = \\{ r _ { 1 } ^ { j _ { 1 } } r _ { 2 } ^ { - j _ { 2 } } \\ : : \\ : 0 \\le j _ { 1 } , j _ { 2 } < m \\} \\ : . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} y = S u \\ , , S : L ^ q ( \\Omega _ T ) \\to L ^ q ( \\Omega ; H ^ 1 ( 0 , T ) ) \\cap L ^ \\infty ( 0 , T ; V ) \\ , , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} i \\psi _ t + \\psi _ { y y } = | \\psi | ^ 2 \\psi + G , \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} K ( t ) = \\sum _ { k = 1 } ^ N c _ k e ^ { - \\lambda _ k t } , \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} x - e ^ { - 3 } x ^ 3 = e ^ { - 1 } , \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\phi ( \\langle e , h \\rangle ) = h \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} | | \\dot { \\gamma } | | _ { \\Sigma , L } = | \\overline { q } \\frac { \\dot { \\gamma } _ 1 } { \\gamma _ 1 } - \\overline { p } \\dot { \\gamma } _ 3 | , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\imath _ V ( d _ { a , V } ( w ) ) & = \\imath _ V \\left ( d w + a L ( w ) - b w \\right ) = L ( w ) + \\imath _ V ( a ) L ( w ) - a \\imath _ V ( L ( w ) ) - \\imath _ V ( b ) w = 0 . \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} T V ^ { { 1 \\over p _ f } } ( S _ T u _ 0 , \\mathbb { R } ) ~ \\leq ~ { \\tilde { \\gamma } _ { [ L , M ] } } \\left ( 1 + { 1 \\over T } \\right ) ~ = ~ { \\tilde { \\gamma } _ { [ L , M , T ] } } \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\partial _ i w ( x ' ) = \\left ( \\int _ { - h } ^ 0 | v ( x ' , z ) | ^ q \\ , d z \\right ) ^ { 1 / q - 1 } \\int _ { - h } ^ 0 | v ( x ' , z ) | ^ { q - 2 } ( \\partial _ i v ( x ' , z ) \\cdot v ( x ' , z ) ) \\ , d z . \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} \\begin{array} { r c l } T _ 3 = \\displaystyle \\dfrac { t ^ 4 } { 1 2 0 } \\sum _ { i = 1 } ^ { k - 1 } f _ { i i i i } , & \\hbox { a n d } & T _ 4 = \\displaystyle \\dfrac { t ^ 4 } { 3 6 } \\ , \\sum _ { \\substack { i , j = 1 \\\\ j > i } } ^ { k - 1 } { f } _ { i i j j } . \\end{array} \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} \\Theta _ { i } ( \\mathbf D _ u , \\beta ) = \\frac { e ^ { - \\frac { 2 ^ { \\frac { n r _ s } { \\beta } } - 1 } { \\gamma _ { A _ i , B } } } } { \\frac { \\gamma _ { A _ i , E } } { \\gamma _ { A _ i , B } } 2 ^ { \\frac { n r _ s } { \\beta } } + 1 } , \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} & \\overline { U } _ { 2 k , 2 a } ( x ; q ) - \\overline { U } _ { 2 k , 2 a - 2 } ( x ; q ) \\\\ & = ( x q ) ^ { 2 a } U _ { 2 k , 2 k - 2 a } ( x q ; q ) + ( x q ) ^ { 2 a - 2 } U _ { 2 k , 2 k - 2 a + 2 } ( x q ; q ) . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} ( 1 - \\beta ) \\delta _ 3 ^ { ( s ) } - \\beta \\delta _ 4 ^ { ( s ) } + \\sum _ { i = 1 } ^ { s - 2 } \\delta _ 3 ^ { ( s - i ) } & \\leq - \\gamma \\beta ^ 2 2 ^ { - 2 s ^ 2 } \\left ( 1 - 2 ^ { 1 + 2 s - 2 s ^ 2 } - \\sum _ { i = 1 } ^ { s - 2 } 2 ^ { 1 + 2 ( s - i ) - 2 s ^ 2 } \\right ) \\\\ & < - \\gamma \\beta ^ 2 2 ^ { - 2 s ^ 2 } \\left ( 1 - 2 ^ { - 3 } \\right ) , \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} \\mathcal { I } ( \\mu ) = \\inf \\bigg \\{ { \\int _ { X \\times X } } d ( x , y ) d \\norm { \\pi } ( x , y ) \\Big | \\pi \\in \\Gamma ( \\mu ) \\bigg \\} . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} \\Psi _ { 0 } \\left ( T _ { 0 } \\right ) & = \\frac { \\left ( \\lambda T _ { 0 } + 1 \\right ) ^ { p } - 1 } { \\lambda ^ { p } } \\\\ \\Psi _ { 1 } \\left ( T _ { 0 } , T _ { 1 } \\right ) & = \\frac { 1 } { \\lambda ^ { p } } \\left [ \\frac { \\left ( \\lambda T _ { 1 } + F \\left ( T _ { 0 } \\right ) \\right ) ^ { p } } { \\lambda T _ { 0 } + 1 } - G \\left ( \\frac { \\left ( \\lambda T _ { 0 } + 1 \\right ) ^ { p } - 1 } { \\lambda ^ { p } } \\right ) \\right ] \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} \\nabla ^ 2 Y + R ( Y , \\gamma ' ) \\gamma ' = 0 . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } e _ { q , \\lambda _ q } ( Q ) = Q ^ \\mu \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\min f _ 0 ( x ) g _ i ( x ) = 0 , i \\in I , g _ j ( x ) \\le 0 , j \\in J , x \\in A . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} u _ m ( t ) : = c _ s \\frac { 1 } { - \\Delta + m } u ( t ) = c _ s \\mathcal { F } ^ { - 1 } \\left ( \\frac { \\hat { u } ( t ) } { | \\xi | ^ 2 + m } \\right ) , m > 0 , \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} U _ { \\varepsilon , z _ \\varepsilon } ( y ) ^ 2 = \\gamma _ \\varepsilon ^ 2 - 2 \\tilde { t } _ \\varepsilon + \\frac { \\tilde { t } _ \\varepsilon ^ 2 } { \\gamma _ \\varepsilon ^ 2 } + \\frac { 2 \\tilde { S } _ { 0 , \\varepsilon } } { \\gamma _ \\varepsilon ^ 2 } + O \\left ( ( | A ( \\gamma _ \\varepsilon ) | + \\gamma _ \\varepsilon ^ { - 4 } ) ( 1 + \\tilde { t } _ \\varepsilon ( y ) ^ 2 ) + | y - z _ \\varepsilon | \\right ) \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} B ^ { \\prime } _ { n } ( f _ { 1 } , f _ { 2 } ) : & = \\frac { f _ { 1 } ( x ) f _ { 2 } ( y ) - f _ { 1 } ( y ) f _ { 2 } ( x ) } { x - y } = \\sum _ { i , j = 1 } ^ { n } \\alpha _ { i j } ^ { \\prime } x ^ { i - 1 } y ^ { j - 1 } \\in F [ x , y ] \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} \\Theta ^ * _ \\mu ( z ) = 0 \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\| \\hat { Q } _ r ( \\Sigma - \\lambda _ r I ) \\hat { Q } _ r \\| _ 1 & \\leq \\sum _ { s \\geq 1 } \\| \\hat { Q } _ r Q _ s ( \\Sigma - \\mu _ r I ) \\hat { Q } _ r \\| _ 1 \\\\ & = \\sum _ { s \\neq r } \\| ( \\mu _ s - \\mu _ r ) \\hat { Q } _ r Q _ s \\hat { Q } _ r \\| _ 1 \\\\ & = \\sum _ { s \\neq r } | \\mu _ r - \\mu _ s | \\| \\hat { Q } _ r Q _ s \\| _ 2 ^ 2 \\\\ & \\leq \\sum _ { \\substack { s < r _ 0 : \\\\ s \\neq r } } | \\mu _ r - \\mu _ s | \\| \\hat { Q } _ r Q _ s \\| _ 2 ^ 2 + 2 | \\mu _ r - \\mu _ { r _ 0 } | \\| \\hat { Q } _ r Q _ { \\geq r _ 0 } \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} u ^ \\delta ( t , x ) = \\begin{cases} 1 , & x \\le - \\delta , \\\\ u _ m ( x ) = 1 - H ^ \\delta ( x ) , & | x | \\le \\delta , \\\\ 0 , & x \\ge \\delta , \\end{cases} \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} S ^ { ' } _ { g } : = S _ { g } \\setminus \\{ g h _ { j } \\check { m } ^ { - j } , g m ^ { - j } : j \\leq J \\} \\subset S _ { g } , \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} r _ \\mathrm { l e n s } = \\sum _ { k \\in \\mathcal { K } } \\log _ 2 ( 1 + \\gamma _ k ) \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\Delta u _ j = \\sum _ { i = 0 } ^ 3 a _ { i , j } ( x ) x \\in \\mathcal G \\cap B _ { \\rho _ 0 / 2 } ( x _ 0 ) . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} S _ { t } \\bar u ~ = ~ \\lim _ { \\varepsilon \\to 0 } S ^ { \\varepsilon } _ { t } \\bar u , \\bar u \\in D . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} x = \\sinh t - \\frac 1 2 | \\tilde s | ^ { 2 } e ^ t , \\ ; \\ ; y ^ i = e ^ { t } s ^ i \\ ; \\ ; ( i = 1 , \\dots , n - 1 ) , \\ ; \\ ; z = \\cosh t + \\frac 1 2 | \\tilde s | ^ 2 e ^ t . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} \\mathrm { R e } ( \\ell _ { \\gamma } ( \\rho ) ) : = \\inf _ { x \\in \\mathbb { H } ^ 3 } d _ { \\mathbb { H } ^ 3 } ( x , \\rho ( \\gamma ) \\cdot x ) \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} f = g \\ast h \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} w \\left ( y , z \\right ) = \\frac { c _ { 1 } } { \\sqrt { \\left \\vert H _ { 0 } \\right \\vert } } \\sqrt { y + a z } c _ { 1 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\eta _ { } \\left ( \\theta , \\phi = \\frac { \\pi } { 2 } \\right ) & = - \\min { \\left \\{ 1 2 \\left ( \\frac { \\theta } { \\theta _ { } } \\right ) ^ 2 , \\eta _ { \\max } \\right \\} } \\\\ \\eta _ { } \\left ( \\theta = 0 , \\phi \\right ) & = - \\min { \\left \\{ 1 2 \\left ( \\frac { \\phi - \\frac { \\pi } { 2 } } { \\phi _ { } } \\right ) ^ 2 , \\eta _ { \\max } \\right \\} } , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} h _ { a l g } ( \\widetilde { \\phi \\upharpoonright _ { H } } ) = h _ { a l g } ( \\xi ) + h _ { a l g } ( \\widetilde { \\phi } \\upharpoonright _ { H G ' / G ' } ) . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} v a r _ { P _ { \\alpha } } \\left [ \\chi _ { P _ { \\alpha } , a , e f f } ^ { 1 } ( \\mathbf { V } ; \\mathcal { G } ) \\right ] = v a r _ { P _ { \\alpha } } \\left [ \\psi _ { P _ { \\alpha } , a } ( \\mathbf { O } ; \\mathcal { G } ) \\right ] - v a r _ { P _ { \\alpha } } \\left [ \\Delta _ { P _ { \\alpha } } \\left ( \\mathbf { O } \\right ) \\right ] \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\mathrm { d p } ( \\mathcal { T } ^ 4 { R _ 2 ^ 4 } ) = 2 , \\mathrm { w d } ( \\mathcal { T } ^ 4 { R _ 2 ^ 4 } ) = 5 0 , \\mathrm { I C } ( \\mathcal { T } ^ 4 { R _ 2 ^ 4 } ) = 8 9 . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\omega _ 1 \\cap \\mathcal { O } _ { 2 , d } = \\emptyset \\ \\ \\ \\ \\ \\omega _ 2 \\cap \\mathcal { O } _ { 1 , d } = \\emptyset \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\beta = \\min \\{ 1 , \\frac { h _ { 1 } } { - \\log r _ { 1 } } + \\frac { h _ { 2 } } { - \\log r _ { 2 } } \\} . \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\frac { d } { d t } | v ( t ) | _ { H ^ { - 1 } } ^ 2 + 2 | v ( t ) | _ H ^ 2 = & - 2 b ( v ( t ) + z ( t ) , \\mathcal { M } ( \\Phi ) ( t ) , ( - A ) ^ { - 1 } v ( t ) ) \\\\ & - 2 b ( \\mathcal { M } ( \\Psi ) ( t ) , v ( t ) + z ( t ) , ( - A ) ^ { - 1 } v ( t ) ) . \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\Gamma ( k ) _ { i , j } & = \\Gamma ( k ) _ { i , 1 } \\otimes \\Gamma ( k ) _ { 1 , j } \\\\ & = \\Gamma ( k ) _ { i , 1 } + \\Gamma ( k ) _ { 1 , j } . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} 0 & = \\int _ { B _ t } | \\nabla v _ 0 | ^ { p - 2 } \\Big ( | \\nabla v _ 0 | ^ 2 - \\Gamma _ { i j } ^ l \\nabla v _ 0 ^ i \\cdot \\nabla v _ 0 ^ j v _ 0 ^ l \\Big ) \\eta d x \\\\ & \\geq a _ { \\kappa } ( 2 R ) \\int _ { B _ t } | \\nabla v _ 0 | ^ { p - 2 } h _ { i j } \\nabla v _ 0 ^ i \\cdot \\nabla v _ 0 ^ j d x \\\\ & \\geq c a _ { \\kappa } ( 2 R ) \\int _ { B _ t } | \\nabla v _ 0 | ^ { p } d x , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} T ^ { D ^ { ( 1 ) } } ( u , v , w ) = \\frac { 1 } { 2 } \\sum _ { ( u , v , w ) \\ ; \\mathrm { c y c l i c } } \\eta ( u , v , \\mathcal J w ) \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\hat \\lambda _ { { \\cal A } , j } ^ { ( i ) } = \\frac { { 1 - \\alpha } } { { 1 - { \\alpha ^ i } } } \\left ( { \\alpha \\sum \\limits _ { l = 1 } ^ { i - 1 } { { \\alpha ^ { K - l - 1 } } u _ j ^ { ( l ) } } + u _ j ^ { ( i ) } } \\right ) = \\frac { { \\alpha - { \\alpha ^ i } } } { { 1 - { \\alpha ^ i } } } \\hat \\lambda _ { { \\cal A } , j } ^ { ( i - 1 ) } + \\frac { { 1 - \\alpha } } { { 1 - { \\alpha ^ i } } } u _ j ^ { ( i ) } , \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} H _ 4 = ( 1 , \\dots , 1 , - 1 \\ , | \\ , 1 , \\dots , 1 , - 1 ) . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} ( T x ) _ { d } = x _ { d } + \\omega , x = ( x _ 1 , \\ldots , x _ d ) . \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & \\alpha _ 2 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ \\alpha _ 2 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} d _ 2 ^ { M a y } ( b _ { 3 , 0 } ) = b _ { 2 , 1 } h _ 1 + b _ { 2 , 0 } h _ 3 . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} 2 \\langle [ u , u ] , v \\rangle = \\pi ( v ) \\langle u , u \\rangle . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} \\sigma = \\sigma _ s + \\epsilon \\phi ( x , y , t ) , p = p _ s + \\epsilon \\psi ( x , y , t ) , \\eta = \\eta _ s + \\epsilon \\xi ( x , t ) , \\rho = \\rho _ s + \\epsilon \\zeta ( x , t ) , \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { k = 1 : k \\neq j ^ * } ^ { r _ n } \\frac { \\lambda _ k ^ { ( n ) } } { \\lambda _ { j ^ * } ^ { ( n ) } - \\lambda _ k ^ { ( n ) } } \\geq \\epsilon \\qquad \\forall n \\geq 1 . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} u ( z ) = \\sum _ { i = 1 } ^ k t _ i u ( z _ i ) . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} & \\{ ( x _ 1 , \\dots , x _ { \\tilde { n } } ) \\mid 0 \\le x _ i \\le 1 , x _ i \\in I _ { n , m } \\ , ( ^ \\forall i ) , \\sum x _ i \\in \\mathbb { Z } \\} \\\\ = & \\mathop { \\cup } _ { \\sigma \\in S _ { \\tilde { n } } } \\{ ( x _ 1 , \\dots , x _ { \\tilde { n } } ) \\mid 0 \\le x _ { \\sigma ( 1 ) } \\le \\dots \\le x _ { \\sigma ( { \\tilde { n } } ) } \\le 1 , x _ i \\in I _ { n , m } \\ , ( ^ \\forall i ) , \\sum x _ i \\in \\mathbb { Z } \\} , \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} ( m \\otimes \\nu ) \\cdot \\tau = - ( \\tau \\cdot m ) \\otimes \\nu + m \\otimes ( \\nu \\cdot \\tau ) . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\nabla _ { z \\partial z } S ^ \\textnormal { c o h } ( \\tau , z ) z ^ { - \\mu } z ^ \\rho = 0 \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\mathcal { F } ( \\tau , Q ) = \\sum _ { \\substack { n \\geq 0 \\\\ d \\in H _ 2 ( X ; \\mathbb { Z } ) } } \\frac { 1 } { n ! } \\langle \\tau , \\dots , \\tau \\rangle ^ \\textnormal { c o h } _ { 0 , n , d } Q ^ d \\in \\mathbb { Q } [ \\ ! [ \\tau ] \\ ! ] \\otimes \\mathbb { C } [ \\ ! [ Q ] \\ ! ] \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} | \\Lambda ( f _ 1 , \\ldots , f _ { n + 1 } ) | \\lesssim \\sum _ { Q \\in \\mathcal { S } } | Q | \\prod _ { j = 1 } ^ { n + 1 } \\langle | f _ j | \\rangle _ Q , \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} D u ( v ( a ) ) D v ( a ) = 0 a \\in \\Lambda . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} f _ { i _ 1 , \\ldots , i _ d } u _ { i _ 2 } \\cdots u _ { i _ d } = a _ { i _ 1 } g _ { i _ 1 , \\ldots , i _ d } v _ { i _ 2 } \\cdots v _ { i _ d } \\textrm { f o r a l l } i _ 1 , \\ldots , i _ d \\in [ n ] . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\mathbf { y } _ { \\mathcal { Q } } \\mathbf { x } } = \\sigma _ x ^ 2 \\mathbf { A } \\mathbf { H } . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\boldsymbol { L } \\mathbb { D } \\boldsymbol { w } _ { \\kappa } ( x ) = \\kappa \\boldsymbol { w } _ { \\kappa } ( x ) \\\\ \\\\ \\boldsymbol { w } _ { \\kappa } \\in L ^ { 2 } \\left ( \\mathcal { K } _ { N } , \\mathbb { C } \\right ) \\oplus L ^ { 2 } \\left ( \\mathcal { K } _ { N } , \\mathbb { C } \\right ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} f ^ { * * } ( t ) = \\frac { 1 } { t } \\int _ 0 ^ t f ^ * ( s ) d s . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} \\int _ { D } \\kappa _ M ^ { N , h } ( y , x ) \\nabla u _ M ^ { N , h } ( y , x ) \\cdot \\nabla v ( x ) \\mathrm { d } x = \\int _ { D } f ( x ) v ( x ) \\mathrm { d } x \\forall v \\in V , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\partial _ Q \\mathcal { X } ( Q ) = A ( Q ) \\mathcal { X } ( Q ) \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} C _ { 2 l + 1 } ^ { ( 1 + \\alpha ) } ( x ) = \\frac { \\Gamma ( 2 l + 2 + \\alpha ) 2 ^ { 2 l + 1 } } { \\Gamma ( 1 + \\alpha ) ( 2 l + 1 ) ! } x ^ { 2 l + 1 } + O ( x ^ { 2 l - 1 } ) \\ . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} \\Sigma _ { j \\in \\mathbb { Z } } \\eta _ j ( t ) = 1 , \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\Q ^ { \\times } _ p = \\mu _ { p - 1 } \\times ( 1 + p \\Z _ p ) \\times p ^ { \\Z } \\ ; p \\neq 2 \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} D \\log \\abs { \\det D F \\circ G } ( x ) ( v ) & = \\frac { d } { d t } \\log \\abs { \\det \\big ( D F ( G ( F ( a , b ) + t D u ( v ( a ) ) ^ * ( b ' ) \\big ) } = \\\\ & = \\frac { d } { d t } \\log \\abs { \\det ( D F ( a , b + t b ' ) ) } = \\frac { d } { d t } \\log \\det H ( b + t b ' ) . \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} { } | u | \\le 1 , \\partial _ n u > 0 , \\lim _ { x _ n \\to \\pm \\infty } u ( x ' , x _ n ) = \\pm 1 . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\Big \\lceil \\frac { k + t } { r } \\Big \\rceil - \\Big \\lceil \\frac k r \\Big \\rceil = \\Big \\lceil u + 1 + \\frac { v + 1 - s } { r } \\Big \\rceil - ( u + 1 ) , \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } | e _ N ( x ) | ^ 2 d x = \\int _ { - 1 } ^ 1 | S ( t ) \\breve e _ N ( t ) | ^ 2 \\frac { d x } { d t } d t = \\int _ { - 1 } ^ 1 | \\breve e _ N ( t ) | ^ 2 \\omega _ \\lambda ( t ) d t . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} R _ 1 & = \\Delta u _ 1 + \\frac { 2 ( u _ 1 + u _ 2 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 , \\\\ R _ 2 & = \\Delta u _ 2 + \\frac { 2 ( u _ 2 - u _ 1 ) } { 1 + | u | ^ 2 } | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} \\mathrm { m o d } _ q ( d \\varphi ) & = \\phi ( l ) \\cdot \\mathrm { F i x } ^ + ( \\phi ( \\gamma ) ) - \\mathrm { F i x } ^ + ( \\phi ( \\gamma ) ) \\\\ & = \\mathrm { F i x } ^ + ( \\phi ( l \\gamma l ^ { - 1 } ) ) - \\mathrm { F i x } ^ + ( \\phi ( \\gamma ) ) \\\\ & = \\mathrm { F i x } ^ + ( \\phi ( d \\varphi _ * \\gamma ) ) - \\mathrm { F i x } ^ + ( \\phi ( \\gamma ) ) . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\gamma _ t \\equiv \\int _ 0 ^ t ( V ^ * ( V V ^ * ) ^ { - 1 } ) ( z _ s ) d s = \\int _ 0 ^ t g ( ( 1 - s ) x + s y ) d s , \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\sigma - \\sigma _ h \\| _ 0 + \\| p - p _ h \\| _ 0 + \\| u _ h - P _ { V _ h } u _ h \\| _ 0 & \\leq C h ^ { r + 2 } \\| u \\| _ { r + 3 } , \\\\ \\| u - u _ h \\| _ 0 & \\leq C h ^ { r + 1 } \\| u \\| _ { r + 2 } , \\\\ | \\sigma - \\sigma _ h | _ { \\div } & \\leq C h ^ { r + 1 } \\| \\div \\sigma \\| _ { r + 1 } . \\end{aligned} \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\rho _ { 2 ^ s } & = \\sum _ j \\sum _ { i = 2 ^ s } ^ { 2 ^ { s + 1 } - 1 } \\sigma _ { i + j 2 ^ { s + 1 } } \\\\ \\rho _ { t _ - } & = \\sum _ j \\sigma _ { 2 ^ s - 1 + j 2 ^ s } . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} { t r ( R _ X ^ i ) t r ( R _ Y ^ j ) } = c _ { i , j } ( \\mathbb { A } ) { t r ( R _ X ^ i \\circ R _ Y ^ j ) } , \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} B _ c ( x ) ^ \\dagger K _ { c , \\lambda } ( x ) B _ c ( x ) = K _ { c , \\lambda } ( b ) \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\begin{aligned} \\| P _ h u - u _ h \\| _ 0 ^ 2 & \\leq C h ^ 2 ( \\| \\div \\sigma \\| _ 1 \\| v \\| _ 1 + ( \\| A \\| _ { 1 , \\infty } + \\| A \\| _ { 2 , \\infty } ) ( \\| \\sigma \\| _ 1 + \\| p \\| _ 1 ) ( \\| \\tau \\| _ 1 + \\| q \\| _ 1 ) ) \\\\ & \\leq C h ^ 2 \\| A \\| _ { 2 , \\infty } ( \\| \\div \\sigma \\| _ 1 + \\| \\sigma \\| _ 1 + \\| p \\| _ 1 ) ( \\| v \\| _ 1 + \\| \\tau \\| _ 1 + \\| q \\| _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\gamma = \\{ ( 1 , \\ldots , n ) \\} \\in S _ n \\ , . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} h _ { w , x } ( \\gamma ) = r \\gamma + \\alpha ( w , x ) \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} - \\Delta _ g u _ i = \\sum _ { j = 1 } ^ n a _ { i j } \\ , \\rho _ j \\left ( \\frac { e ^ { u _ j } } { \\int _ { M } e ^ { u _ j } } - \\frac { 1 } { | M | } \\right ) , i = 1 , \\dots , n , \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} - Z '' - \\left [ \\mu _ { 1 } \\left ( 1 - 2 \\frac { v ^ + } { \\alpha } \\right ) \\mathbf { 1 } _ { v > 0 } + \\frac { 1 } { d } \\mu _ { 2 } \\left ( 1 + 2 \\frac { v ^ - } { d } \\right ) \\mathbf { 1 } _ { v < 0 } \\right ] Z = \\lambda \\sigma ( v ) Z \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\partial _ { \\bf n } \\mu = 0 \\qquad ( 0 , T ) \\times \\Gamma \\ , , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} h _ K ^ { p - 1 } \\ , d \\mu = d \\widetilde { C } _ q ( K , \\cdot ) . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} K _ \\lambda ( x ) = \\frac { 1 } { 4 \\pi } \\frac { e ^ { - \\lambda ^ { 1 / 2 } \\lvert x \\rvert } } { \\lvert x \\rvert } , x \\in \\R ^ 3 \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} d _ n = n \\cdot d _ { n - 1 } + ( - 1 ) ^ n , \\ , \\ , ( n \\geq 0 ) , ( \\textnormal { s e e } \\ , \\ , [ 3 , 4 , 7 ] ) . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} e ^ { - \\gamma s q _ n } = \\min _ { i , j \\in I _ 1 \\cup I _ 2 } | \\sin \\pi ( 2 \\theta + ( i + j ) \\alpha ) | . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} \\mathrm { r e s } ( \\overline { C } ^ * _ { \\mathbb { Q } } ) \\ , \\ = \\ , \\ _ k \\overline { C } ^ * _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\partial _ { y _ j } \\partial _ { y _ 1 } g _ { D , \\mathrm { o d d } } ( y ) = ( \\partial _ { y _ j } \\partial _ { y _ 1 } g _ D ) _ { \\mathrm { e v e n } } ( y ) , \\partial _ { y _ j } \\partial _ { y _ 1 } g _ { N , \\mathrm { e v e n } } ( y ) = ( \\partial _ { y _ j } \\partial _ { y _ 1 } g _ N ) _ { \\mathrm { o d d } } ( y ) \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} Q _ T = ( 0 , T ) \\times \\Omega , \\ \\Omega \\subset R ^ N \\ \\mbox { a d o m a i n } , \\ N = 1 , 2 , 3 . \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\frac { 1 } { d + 1 } \\theta ^ { - i + 1 } + \\frac { ( 1 + \\mu ) d } { d + 1 } \\theta ^ { - i - 1 } = \\theta ^ { - i } , \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} f \\left ( s \\right ) = \\overset { \\infty } { \\underset { n = 1 } { \\sum } } a _ { n } e ^ { s \\lambda _ { n } } \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} \\dot z ^ j _ * ( t ) & = g ^ { k j } ( t , z _ * ( t ) ) \\\\ & \\cdot \\Big ( \\lambda _ k + \\dot \\gamma ^ i ( t ) \\frac { \\partial f _ i } { \\partial z ^ k } ( t , z _ * ( t ) ) + \\int _ t ^ \\omega \\dot \\gamma ^ i ( s ) \\frac { d } { d s } \\Big ( \\frac { \\partial f _ i } { \\partial z ^ k } ( s , z _ * ( s ) ) \\Big ) d s \\\\ & - \\int _ t ^ { \\omega } \\frac { \\partial L } { \\partial z ^ k } ( s , z _ * ( s ) , \\dot z _ * ( s ) ) d s - a _ k ( t , z _ * ( t ) ) \\Big ) . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{gather*} \\xi : = \\mathcal { V } [ v + \\eta ] - \\mathcal { V } [ v ] - \\mathcal { V } ^ { B D } [ v ; \\eta ] \\\\ \\intertext { a n d o b s e r v i n g t h a t $ | \\xi | \\le | \\mathcal { V } [ v + \\eta ] - \\mathcal { V } [ v ] | + | \\mathcal { V } ^ { B D } [ v ; \\eta ] | $ o r } \\xi : = \\mathcal { V } [ v + \\eta ] - \\mathcal { V } [ v ] - M \\eta \\qquad M \\in \\mathcal { V } ^ { N D } [ v + \\eta ] , \\end{gather*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\theta _ p ( v , 0 , r ) = \\theta _ p ( u , x ) r > 0 , \\frac { \\partial v } { \\partial r } = 0 . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} F ( a + \\varpi ^ l b ) & = F ( a ) + \\varpi ^ l \\frac { \\partial F } { \\partial x } ( a ) b + \\varpi ^ { 2 l } V ( a , b ) \\\\ & = \\varpi ^ { 2 l } V ( a . b ) \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} L _ k = ( L _ { k - 2 } + ( 2 ^ k - 2 ^ { k - 2 } ) ) \\cup ( L _ { k - 1 } + 2 ^ k ) , k \\geq 2 . \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\underbrace { \\left [ \\begin{matrix} | b _ 1 ^ 1 | ^ 2 & \\cdots & | b _ 1 ^ n | ^ 2 \\\\ \\vdots & & \\vdots \\\\ | b _ q ^ 1 | ^ 2 & \\cdots & | b _ q ^ n | ^ 2 \\end{matrix} \\right ] } _ B \\left [ \\begin{matrix} d _ 1 \\\\ \\vdots \\\\ d _ n \\end{matrix} \\right ] = \\left [ \\begin{matrix} 0 \\\\ \\vdots \\\\ 0 \\end{matrix} \\right ] \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} P ( x , y ) : = \\begin{pmatrix} y & ( n - 1 ) x \\\\ x & y \\end{pmatrix} . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\| D _ t \\| _ { \\infty } = \\max _ { 1 \\leq j \\leq n } \\sum _ { i = 1 } ^ n \\| \\partial _ j P _ { i , t } \\| _ { \\infty } \\leq 1 + \\alpha _ t , \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\widetilde { H } ( x , \\Lambda ) = \\ln \\widetilde { I } [ x , \\Lambda ; \\widetilde { w } ] , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\langle S ( \\lambda ) ( x ) , w \\rangle = \\lambda ( ( \\delta _ x - \\delta _ { x _ 0 } ) w ) , \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} z _ i ( s , t , \\xi ) = \\sigma _ { i } ( t ) s \\mathbf { 1 } _ d + \\xi , \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\mbox { t h e t o t a l m a s s } m _ 0 : = \\int _ S \\theta _ 0 ( z ) d z , \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} \\frac { 2 } { q } + \\frac { d } { p } = 1 . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} & S ^ { K \\textnormal { t h } } \\left . \\right . _ \\circ \\left . \\right . P _ j ^ { - 1 } q ^ { Q _ j \\partial _ { Q _ j } } = \\mathcal { A } _ j \\left . \\right . _ \\circ \\left . \\right . S ^ { K \\textnormal { t h } } & & P _ j ^ { - 1 } q ^ { Q _ j \\partial _ { Q _ j } } \\left . \\right . _ \\circ \\left . \\right . T ^ { K \\textnormal { t h } } = T ^ { K \\textnormal { t h } } \\left . \\right . _ \\circ \\left . \\right . \\mathcal { A } _ j \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\frac { 1 } { 8 \\pi } \\int _ { \\Sigma _ r } ( k ^ { ( 1 ) } - h ^ { ( 1 ) } ) d S ^ 2 = \\frac { 4 \\pi } { 3 } ( \\bar R i c ( e _ 0 , e _ 0 ) + \\frac { 1 } { 2 } \\bar R ) \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} d _ { \\max } = \\frac { \\sigma ^ 2 } { a ^ 2 - 1 } . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\sum \\limits _ { b \\in B ( G , \\mu ) } \\sum \\limits _ { ( M _ S , \\mu _ S ) \\in \\mathcal { R } _ { G , b , \\mu } } ( - 1 ) ^ { L _ { M _ S , M _ b } } ( M _ S , \\mu _ S ) = ( G , \\mu ) . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\Delta v - \\lambda v = f \\R ^ 3 , f \\in C ^ \\infty _ c ( \\R ^ 3 ) , \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} b ( \\gamma ^ { - 1 } z ) = \\lim _ { n \\rightarrow \\infty } B _ \\Omega ( \\gamma ^ { - 1 } z , y _ n ) - B _ \\Omega ( y _ n , z _ 0 ) = \\lim _ { n \\rightarrow \\infty } B _ \\Omega ( z , \\gamma y _ n ) - B _ \\Omega ( y _ n , z _ 0 ) = b ( z ) \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} & \\textrm { P r } ^ { \\textrm { O F D M A } } \\left \\lbrace \\textrm { N D N = 0 } \\right \\rbrace \\to \\frac { d _ { \\textrm { B N } } ^ { \\alpha } \\sigma _ { \\textrm { N } } ^ { 2 } ( 1 - \\theta ) } { \\lambda _ { \\textrm { B N } } ( 1 - \\rho ) } \\left ( 2 ^ { \\frac { R } { 1 - \\theta } } - 1 \\right ) \\frac { 1 } { P _ { \\textrm { B } } } . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\tilde \\lambda _ j = \\lambda _ { j - 1 } - A ( \\lambda _ { j - 1 } - \\lambda _ j ) \\quad 0 < A < 1 . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\frac { 1 } { t } = \\frac { \\| \\psi \\| _ { L ^ \\infty } } { \\| \\varphi \\| _ { L ^ \\infty } } = \\kappa . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\tilde \\Lambda ^ x _ { t } = \\int _ 0 ^ t \\tilde f _ s ( x ) d s \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} \\lvert n + 1 - i - ( n + 1 - j ) \\rvert = \\lvert i - j \\rvert . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} G _ i ( u ) & = \\frac { L _ i } { ( e - 1 ) \\ln ( 1 + \\frac { U _ i ( e - 1 ) } { L _ i } ) } \\big ( 1 - ( 1 + \\frac { U _ i ( e - 1 ) } { L _ i } ) ^ u \\big ) + \\frac { L _ i } { e - 1 } u \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} u _ 0 ( x ) = a \\sin ( 2 \\pi x ) - ( \\pi a ) ^ 2 , \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} \\inf _ { y \\in E \\cap B ( 0 , \\eta ) } \\{ f ( y ) + \\langle G ( y ) , x - y \\rangle + \\widetilde { \\varphi } _ { y } ( x ) \\} = \\inf _ { y \\in E } \\{ f ( z ) + \\langle G ( y ) , x - y \\rangle + \\widetilde { \\varphi } _ { y } ( x ) \\} . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} y _ i \\left ( X ^ * \\right ) + y _ i \\left ( X ^ { * p } + X ^ { * q } - x _ \\mathrm { m i n } \\right ) = y \\left ( X ^ { * p } \\right ) + y \\left ( X ^ { * q } \\right ) , \\forall p , q , p \\neq q . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{gather*} \\boldsymbol { \\Sigma } _ { \\frac { n } { 2 } } : = \\mathrm { d i a g } \\left ( \\lambda _ { 3 } , \\lambda _ { 5 } , \\ldots , \\lambda _ { n + 1 } \\right ) + \\lambda _ { 1 } \\mathbf { u } \\mathbf { u } ^ { \\top } \\\\ \\boldsymbol { \\Gamma } _ { \\frac { n } { 2 } } : = \\mathrm { d i a g } \\left ( \\lambda _ { 2 } , \\lambda _ { 4 } , \\ldots , \\lambda _ { n } \\right ) + \\lambda _ { n + 2 } \\mathbf { v } \\mathbf { v } ^ { \\top } \\end{gather*}"}
-{"id": "4301.png", "formula": "\\begin{align*} ( x _ 2 + d x _ 3 ^ p ) ^ { a ' } x _ 2 ^ { b ' } R _ 1 ( x _ 2 + d x _ 3 ^ p , x _ 3 ) = - x _ 2 ^ a ( x _ 2 + d x _ 3 ^ p ) ^ b R _ 3 ( x _ 2 , x _ 3 ) . \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\partial _ \\tau v _ g ^ n ( y , \\tau ) - \\frac { 1 } { 2 } \\partial _ { y y } v _ g ^ n ( y , \\tau ) + ( y + a _ n ' ( y ) ) \\partial _ y v _ g ^ n ( y , \\tau ) = \\left | \\partial _ y w _ g ^ n ( y , \\tau ) \\right | ^ 2 , \\ : \\forall ( y , \\tau ) \\in \\mathbb { R } \\times [ 0 , \\infty ) , \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} f _ \\epsilon ^ \\frac { Q - \\alpha } { Q + \\alpha } ( \\xi ) = B \\int _ { \\mathbb { H } ^ n } \\frac { f _ \\epsilon ( \\eta ) } { | \\eta ^ { - 1 } \\xi | ^ { Q - \\alpha } } d \\eta , \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} K = - 1 + \\frac { 1 } { 4 } | H _ 0 | ^ 2 + O ( r ^ 2 ) \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\Theta _ j : = \\begin{pmatrix} 1 & \\frac { m } { n } j \\\\ 0 & 1 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 & 0 \\\\ n - 1 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 + \\frac { m } { n } ( n - 1 ) j & \\frac { m } { n } j \\\\ n - 1 & 1 \\end{pmatrix} , j \\in \\mathbb { Z } \\ , . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\Lambda _ { n } ( \\theta ) : = \\log E \\left [ \\mathrm { e } ^ { \\theta \\mathrm { B i n } ( n - a _ n , \\pi ( n ) ) } \\right ] & = ( n - a _ n ) \\log E \\left [ \\mathrm { e } ^ { \\theta \\mathrm { B e } ( \\pi ( n ) ) } \\right ] \\\\ & = ( n - a _ n ) \\log ( 1 + \\pi ( n ) ( \\mathrm { e } ^ \\theta - 1 ) ) . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} ( \\nu - \\mu \\tilde \\sigma ) \\eta _ s + \\mu \\tilde \\sigma \\rho _ s = \\mu \\sqrt { \\bar \\sigma ^ 2 - \\hat \\sigma ^ 2 } . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} \\tilde { B } ^ * = \\frac { a | \\mathcal { N } | ^ 2 T } { | \\mathcal { N } | - 1 } , \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} ( y ( g ) , \\gamma ^ \\star , \\delta ^ \\star ) { z } & = \\sum _ { E \\subseteq \\{ 1 , \\ldots , s \\} } ( \\prod _ { e \\in E } y _ e ) ( g ^ { ( \\tau _ E ) z } , \\gamma ^ \\star , \\delta ^ \\star ) \\\\ & = \\sum _ { E ' \\subseteq \\{ 1 , \\ldots , s \\} } ( \\prod _ { e \\in E } y _ e ) ( ( g ^ { z } ) ^ { \\tau _ { E ' } } , \\gamma ^ \\star , \\delta ^ \\star ) , \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} \\{ x \\in \\mathbb { Z } ^ N : \\ \\| x - u _ i \\| _ 2 \\le R _ i \\} = \\{ x \\in \\mathbb { Z } ^ n : \\ \\| x - u _ i \\| _ 2 < \\sqrt { R _ i ^ 2 + 1 } \\} . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} ( - 1 ) ^ { m - s } \\pi \\sum _ { i = 0 } ^ { s } ( - 1 ) ^ { i } { m + s - 1 + i \\choose 2 s - 1 } { 2 s + 1 \\choose i } . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} ( \\Upsilon E ( \\{ \\lambda \\} u ) _ 1 & = \\lambda P _ 1 E ( \\{ \\lambda \\} ) u \\\\ & = 0 \\ , , \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} ( \\cup _ { i = \\frac { 3 q - 6 } { 1 0 } } ^ { \\frac { 3 q - 1 6 } { 1 0 } + t } \\mathbb { C } _ { a } ) \\cap - q ( \\cup _ { i = 0 } ^ { \\frac { 3 q - 1 6 } { 1 0 } } \\mathbb { C } _ { a } ) & = \\mathbb { C } _ \\frac { q ^ 2 - q + 3 } { 5 } , \\\\ ( \\cup _ { i = 0 } ^ { \\frac { 3 q - 1 6 } { 1 0 } } \\mathbb { C } _ { a } ) \\cap - q ( \\cup _ { i = \\frac { 3 q - 6 } { 1 0 } } ^ { \\frac { 3 q - 1 6 } { 1 0 } + t } \\mathbb { C } _ { a } ) & = \\mathbb { C } _ \\frac { 2 q ^ 2 - 2 q + 1 } { 5 } , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} & \\nabla ^ { F } _ { X } ( \\omega \\otimes l ) = ( \\nabla ^ { F } _ { X } \\omega ) \\otimes l + \\omega \\otimes \\nabla ^ { F } _ { X } l \\\\ & \\nabla ^ { F } _ { X } l = { \\mathcal L } _ { X } l + \\frac { 1 } { 2 } \\mathrm { t r a c e } ( \\nabla ^ { F } X ) l . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} e _ i = ( 0 , \\ldots , 0 , \\mathop 1 \\limits _ { i } , 0 , \\ldots , 0 ) . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} { \\rm C o v } _ { \\rho , \\Phi } ^ { f , g } ( A , B ) & : = \\Phi \\left ( f ( \\rho ) A ^ * B \\right ) - \\Phi \\left ( f ( \\rho ) g ( \\rho ) A ^ * \\right ) \\Phi \\left ( f ( \\rho ) g ( \\rho ) ^ 2 \\right ) ^ { - 1 } \\Phi \\left ( f ( \\rho ) g ( \\rho ) B \\right ) \\\\ { \\rm V a r } _ { \\rho , \\Phi } ^ { f , g } ( A ) & : = { \\rm C o v } _ { \\rho , \\Phi } ^ { f , g } ( A , A ) , \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} ( x _ n ^ { \\mu } , t _ n ^ { \\mu } ) = ( x _ { 1 , n } ^ { \\mu } , x _ { n } ^ { \\mu \\prime } , t _ n ^ { \\mu } ) \\in S _ { \\mu } \\ \\ ( n = 1 , 2 , 3 , \\cdots ) \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\delta ( \\alpha , \\theta ) = \\limsup _ { k \\to \\infty } - \\frac { \\ln | | 2 \\theta + k \\alpha | | _ { \\R / \\Z } } { | k | } \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} & ( u _ 1 , u _ 2 , \\cdots , u _ k ) \\preceq ( v _ 1 , v _ 2 , \\cdots , u _ k ) \\ \\ { \\rm i f } \\ \\ u _ i \\leq v _ i , \\ \\ ( i = 1 , 2 , \\cdots , k ) , \\\\ & u : = ( u _ 1 , u _ 2 , \\cdots , u _ k ) \\prec v : = ( v _ 1 , v _ 2 , \\cdots , v _ k ) \\ \\ { \\rm i f } \\ \\ u \\preceq v \\ \\ { \\rm a n d } \\ \\ u \\not = v , \\\\ & ( u _ 1 , u _ 2 , \\cdots , u _ k ) \\ll ( v _ 1 , v _ 2 , \\cdots , v _ k ) \\ \\ { \\rm i f } \\ \\ u _ i < v _ i , \\ \\ ( i = 1 , 2 , \\cdots , k ) , \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} & U _ { 2 k , 2 a - 1 } ( x ; q ) = U _ { 2 k , 2 a } ( x ; q ) - U ^ { 2 a } _ { 2 k , 2 a } ( x ; q ) \\\\ & = ( - x q ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) - x q ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) + x q ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a - 1 } ( x ^ 2 ; q ^ 2 ) \\\\ & = ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a } ( x ^ 2 ; q ^ 2 ) + x q ( - x q ^ 3 ; q ^ 2 ) _ \\infty \\overline { Q } _ { k , a - 1 } ( x ^ 2 ; q ^ 2 ) . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} I = I _ { 0 } + I _ { 1 } p + \\ldots + I _ { N - 1 } p ^ { N - 1 } , \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\limsup _ { | n ' | \\rightarrow \\infty } \\sup _ { \\xi \\in \\mathbb { T } ^ n } | m ( n ' , \\xi ) | = 0 . \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} g = g _ D + g _ N \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} D ^ \\mu _ X ( x ) = \\langle \\{ x \\} , \\mu _ 1 ( 0 ) \\rangle . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\O ^ { 1 , v } _ 2 ( s ) = ( \\eta _ v + 1 ) s ^ v , ~ \\Gamma ^ { 1 v } _ { 2 k } ( t ) = \\eta _ v \\delta ^ v _ k . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } W _ { k , a } ( n ) q ^ n & = \\sum _ { N _ 1 \\geq N _ 2 \\geq \\cdots \\geq N _ { k - 1 } \\geq 0 } \\frac { q ^ { N _ 1 ^ 2 + N _ 2 ^ 2 + \\cdots + N _ { k - 1 } ^ 2 + 2 N _ { a } + 2 N _ { a + 2 } + \\cdots + 2 N _ { k - 2 } } } { ( q ^ 2 ; q ^ 2 ) _ { N _ 1 - N _ 2 } \\cdots ( q ^ 2 ; q ^ 2 ) _ { N _ { k - 2 } - N _ { k - 1 } } ( q ^ 2 ; q ^ 2 ) _ { N _ { k - 1 } } } \\\\ [ 6 p t ] & \\qquad = \\frac { ( - q ; q ^ 2 ) _ \\infty ( q ^ a , q ^ { 2 k + 2 - a } , q ^ { 2 k + 2 } ; q ^ { 2 k + 2 } ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\nu ' } \\left [ \\zeta \\left ( \\frac { 3 } { 2 } + 3 \\nu ' \\right ) \\zeta \\left ( \\frac { 3 } { 2 } + 2 \\nu ' + \\nu \\right ) ^ { - 1 } A ( \\nu ' , \\nu ) \\right ] _ { \\nu ' = \\nu = r } = : C _ 1 ( r ) . \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} u = S _ l ( \\alpha ) \\otimes w , \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\lambda _ q q ^ \\mathbb { Z } : = \\left \\{ \\lambda _ q q ^ k \\ , \\middle | \\ , k \\in \\mathbb { Z } \\right \\} \\subset \\mathbb { C } \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} x - \\sigma _ { \\alpha _ i } ( x ) = a _ i \\alpha _ i \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c c c c c c } & & & 1 & 1 & 0 & \\cdots & 0 \\\\ & I _ { n - i } & & \\vdots & \\vdots & \\vdots & & \\vdots \\\\ & & & 1 & 1 & 0 & \\cdots & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} { L } _ { n } ^ { ( \\delta - 1 ) } ( z , q ) = q ^ { - n } { L } _ { n } ^ { ( \\delta ) } ( z , q ) - q ^ { - n } { L } _ { n - 1 } ^ { ( \\delta ) } ( z , q ) . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ T _ i \\right ] = \\frac { n } { \\epsilon _ { k , n } ( \\delta ) } + \\frac { \\mathbb { E } \\left [ Z \\right ] } { \\epsilon _ { k , n } ( \\delta ) } \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\varphi ^ { \\pm } = { } ^ t ( \\varphi ^ { \\pm } _ 1 , \\varphi ^ { \\pm } _ 2 , \\cdots , \\varphi ^ { \\pm } _ m ) \\gg { } ^ t ( 0 , 0 , \\cdots , 0 ) , \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} N _ { \\Sigma ( W ) } ( T ) = \\left \\{ \\gamma \\in \\Gamma \\cap \\Sigma ( W ) \\mathop { | } H ( \\gamma ) \\leq T \\right \\} \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} I ( s ) = \\sqrt { \\cosh ^ 2 2 s - \\sinh ^ 2 2 s \\cos ^ 2 ( \\theta _ 1 + \\theta _ 2 ) } e ^ { 1 - \\cosh 2 s - \\sinh 2 s \\cos ( \\theta _ 1 + \\theta _ 2 ) } . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} \\Theta _ j - M \\frac { \\langle \\nabla z _ j , \\nabla w _ j \\rangle } { z _ j \\phi ^ { 2 \\alpha _ j } } = \\Psi _ j - M \\frac { | \\nabla z _ j | ^ 2 } { z _ j } - 2 \\alpha _ j ( M + 2 ) \\frac { \\langle \\nabla z _ j , \\nabla \\phi \\rangle } { \\phi } \\omega . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} a = \\sum _ { j = 1 } ^ m S _ n \\left ( a _ { 1 , j } , \\dotsc , a _ { n , j } \\right ) = \\sum _ { j = 1 } ^ m \\left \\Vert a _ { 1 , j } \\right \\Vert \\dotsb \\left \\Vert a _ { n , j } \\right \\Vert S _ n \\left ( b _ { 1 , j } , \\dotsc , b _ { n , j } \\right ) , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\sum _ { n \\ge 1 } \\Omega _ \\Pi ^ \\circ ( n ) \\ , z ^ n \\ = \\ \\frac { \\Omega ^ * _ \\Pi ( z ) } { ( 1 - z ) ^ { d + 1 } } \\ , . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\Phi \\left ( \\sum _ i g [ \\sigma _ i ] \\right ) & = | g | \\sum \\int _ { \\sigma _ i } \\Omega \\\\ & = | g | \\int _ { \\partial \\tau } \\Omega \\\\ & = 0 \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} F ( x ) & = \\sum _ t \\big ( F ( \\hat { x } _ t ) - F ( \\hat { x } _ { t - 1 } ) \\big ) \\\\ & = \\sum _ t x _ t \\nabla _ t F ( \\hat { x } _ { t - 1 } ) \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} y \\in Y ( X ) \\mapsto x ^ * [ y ] \\in X ^ * , x ^ * [ y ] ( x ) = \\tau ( y x ) , x \\in X , \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\sigma ^ 2 & = \\frac { \\langle w , ( \\Sigma - 1 / ( 2 r ) ( F \\otimes F ) ) w \\rangle } { \\langle v , v \\rangle ^ 2 } \\\\ & = \\bigg ( \\sum _ { j = 1 } ^ r \\frac { \\lambda _ j ^ 2 } { ( \\lambda _ 1 + y - \\lambda _ j ) ^ 2 } - \\frac { 1 } { 2 r } \\bigg ( \\sum _ { j = 1 } ^ r \\frac { \\lambda _ j } { \\lambda _ 1 + y - \\lambda _ j } \\bigg ) ^ 2 \\bigg ) \\bigg ( \\sum _ { j = 1 } ^ r \\frac { \\lambda _ j } { \\lambda _ 1 + y - \\lambda _ j } \\bigg ) ^ { - 2 } \\geq \\frac { 1 } { 2 r } , \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\P ( C ' _ { k _ 2 , j _ 1 } \\cap B ' ) + \\P ( C ' _ { k _ 2 , j _ 2 } \\cap B ' ) & = 0 = \\P ( C _ { k _ 2 , j _ 1 } \\cap B ) + \\P ( C _ { k _ 2 , j _ 2 } \\cap B ) . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\int _ { \\mathfrak { g } ^ { ( l ) } } \\varphi ( u ) d u = \\int _ { \\mathbb { R } ^ { N } } d y \\int _ { M _ { x , y } } K ( u , x ) \\varphi ( u ) m _ { x , y } ( d u ) , \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} M ^ { - 1 } \\sum _ { k = 1 } ^ { m ( x ) } ( g \\circ \\phi ) ( x , k ) \\leq { } & \\sum _ { k = 1 } ^ { m ( x ) } ( f \\circ \\phi ) ( x , k ) \\\\ & \\leq M \\sum _ { k = 1 } ^ { m ( x ) } ( g \\circ \\phi ) ( x , k ) , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} h ( s ) & = K \\Gamma ( u + 1 ) s ^ { - u } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 } ) + O ( 1 ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u } , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} R _ i ^ j = { d - 1 \\over 2 } \\delta _ i ^ j + { \\nabla _ 1 } _ i E ^ j = { \\eta _ i \\over { \\eta _ r } + 1 } \\delta _ i ^ j . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\langle C ^ { ( 1 + \\alpha ) } _ { k + 1 } ( z / c ) , z ^ j { \\bar z } ^ l \\rangle _ { \\alpha } = 0 \\ , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\bigl ( \\tfrac { p + 1 - u } { p + 1 } + o ( 1 ) \\bigr ) a _ 1 + \\bigl ( \\tfrac { ( p + 1 ) ! } { ( p + 1 - u ) ! } + o ( 1 ) \\bigr ) \\tfrac { ( - 1 ) ^ u } { n ^ u } a _ { p + 1 } & \\geq - 1 \\qquad u \\in \\{ 0 , \\dotsc , p \\} \\\\ - \\bigl ( 1 + o ( 1 ) \\bigr ) \\tfrac { 1 } { n } a _ 1 + \\bigl ( ( p + 1 ) ! + o ( 1 ) \\bigr ) \\tfrac { ( - 1 ) ^ { p + 1 } } { n ^ { p + 1 } } a _ { p + 1 } & \\geq - 1 \\qquad u = p + 1 , \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} R _ { i , j } ( G , S , \\lambda ) : = \\sum _ { \\beta \\in \\mathcal { B } _ { i , j } } w ( \\beta , \\lambda ) . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ m ( - 1 ) ^ j \\binom { 2 m } { 2 j } = \\begin{cases} ( - 1 ) ^ { m / 2 } 2 ^ { m } & \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} \\ss ( G ) = \\prod _ { i = 1 } ^ m T _ i \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\xi _ 1 ^ k = \\frac { k } { k + 1 } , k = 1 , . . . , s . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} ( H ( x , y , z ) - h ) \\cdot & [ \\nabla C ( x , y , z ) \\times ( \\nabla C ( x , y , z ) \\times \\nabla H ( x , y , z ) ) ] \\\\ & = ( H ( x , y , z ) - h ) \\cdot y ^ { \\lambda - 1 } \\cdot \\left [ \\begin{array} { c } y ( \\lambda x ^ 2 - y ^ 2 - z ^ 2 ) \\\\ x ( - \\lambda x ^ 2 + y ^ 2 - \\lambda z ^ 2 ) \\\\ ( \\lambda + 1 ) x y z \\end{array} \\right ] . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x } & & \\bigoplus _ { k = 1 } ^ { m } w _ { k } x ^ { - } A _ { k } x . \\end{aligned} \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} B _ { R _ { 1 } , R _ { 2 } } ( b ) = \\left \\lbrace R _ { 1 } < | x - b | < R _ { 2 } \\right \\rbrace \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} f _ j ^ { - 1 } ( 0 ) \\cap O _ j \\setminus V = \\varnothing . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\frac 1 { \\tau } - \\frac 1 { \\delta _ { m + 1 } } = \\frac 1 { \\widetilde { \\delta } _ { m + 1 } } - \\frac 1 { \\delta _ { m + 1 } } = \\frac 1 { p _ { m + 1 } } - \\frac 1 { q _ { m + 1 } } = \\frac 1 { q } - \\frac 1 { p } = \\frac 1 { s } - \\frac 1 { p } , \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} g ^ T ( X , Y ) = \\tfrac { 1 } { 2 } d \\eta ( X , I Y ) . \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\begin{cases} \\frac { ( P + ( u _ \\epsilon ( x ) ) _ x ) ^ 2 } { 2 } + V ( x , \\frac { x } { \\epsilon } ) = \\ln m _ \\epsilon ( x ) + \\overline { H } _ \\epsilon ( P ) , \\ & \\ \\mathbb { T } , \\\\ - ( m _ \\epsilon ( x ) ( P + ( u _ \\epsilon ( x ) ) _ x ) ) _ x = 0 , \\ & \\ \\mathbb { T } , \\\\ \\int _ 0 ^ 1 u _ \\epsilon ( x ) d x = 0 , \\int _ 0 ^ 1 m _ \\epsilon ( x ) d x = 1 . \\end{cases} \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\mathcal { U } : = \\left \\{ [ U ( g ) _ { n } ] _ { n \\in \\mathbb { N } } \\in { \\rm E n d s } ( S ) : g \\in G \\right \\} \\subset { \\rm E n d s } ( S ) . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} \\mathcal { H } = \\frac 1 2 \\sum _ { \\{ j , k \\} \\in E ( G ) } w _ { j k } ( X _ j X _ k + Y _ j Y _ k ) \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} h \\left ( R _ { n } \\right ) = f \\left ( R _ { n } \\right ) - g \\left ( R _ { n } \\right ) , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\nu ( z ) = g ( z ) \\mu ( z ) . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} { { \\bf { 1 } } ^ T } { \\bf { W } } = { { \\bf { 1 } } ^ T } , \\ ; { \\bf { W 1 } } = { \\bf { 1 } } \\ ; \\ ; 0 < { \\sigma _ 2 } \\left ( { \\bf { W } } \\right ) < 1 , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\chi ( r ) = \\left \\{ \\begin{array} { l l } r ^ 2 & r \\leq 1 , \\\\ 0 & r \\geq 2 , \\end{array} \\right . \\chi '' ( r ) \\leq 2 r \\geq 0 . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} P ( V ) : = \\langle h \\cdot g : h \\in V g \\in T V \\rangle , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} T M = S ( T M ) \\oplus \\mathrm { R a d } \\ ; T M . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\widetilde { \\gamma } : = \\beta { } ^ { - 1 } \\circ \\gamma \\circ \\alpha \\colon { } V \\to { } \\mathbb { C } ^ n { } \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} A | z | ^ 2 - B \\Re ( z ^ 2 ) = \\frac { ( a ^ 2 + b ^ 2 ) ( x ^ 2 + y ^ 2 ) } { 2 a ^ 2 b ^ 2 } - \\frac { ( a ^ 2 - b ^ 2 ) ( x ^ 2 - y ^ 2 ) } { 2 a ^ 2 b ^ 2 } = \\frac { x ^ 2 } { a ^ 2 } + \\frac { y ^ 2 } { b ^ 2 } \\ . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} T _ j m _ { x _ 2 , y , h _ 2 } & = T _ { j } T _ { x _ 1 } T _ { x _ 2 } T _ y T _ { d _ i } T _ { h _ 1 } T _ { h _ 2 } \\\\ & = T _ { x _ 1 } T _ { x _ 2 } T _ { j } T _ y T _ { d _ i } T _ { h _ 1 } T _ { h _ 2 } \\\\ & = \\begin{cases} T _ { x _ 1 } T _ { x _ 2 } T _ { s _ j y } T _ { d _ i } T _ { h _ 1 } T _ { h _ 2 } & \\ell ( s _ j y ) > \\ell ( y ) , \\\\ ( q - 1 ) T _ { x _ 1 } T _ { x _ 2 } T _ { y } T _ { d _ i } T _ { h _ 1 } T _ { h _ 2 } + q T _ { x _ 1 } T _ { x _ 2 } T _ { s _ j y } T _ { d _ i } T _ { h _ 1 } T _ { h _ 2 } & \\ell ( s _ j y ) < \\ell ( y ) , \\end{cases} \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\partial _ x ^ i \\mathbb { L } ( A _ t ; Z _ t , W _ t ) & = \\lim _ { h \\downarrow 0 } \\frac { \\mathbb { L } ( A _ { t } ^ { ( h e _ i ) } ; Z _ { t } , W _ { t } ) - \\mathbb { L } ( A _ t ; Z _ t , W _ t ) } { h } . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\min ~ \\varphi ( x ) : = g ( x ) + r ( x ) \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\Pi _ * ( \\hat { l } _ q ) = \\tfrac { t } { \\mathrm { g c d } ( s , t ) } \\cdot l + k _ l \\cdot m _ q \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\Psi ( t _ 0 ) = i d \\ , , \\Psi ( t ) _ { \\lfloor _ { \\partial \\Omega } } = i d \\ , , \\Psi ( t ) ( \\Gamma ^ { ( 2 ) } ( t ) ) = \\Gamma ^ { ( 2 ) } ( t _ 0 ) \\ , , \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} { } \\langle \\rho _ { G \\setminus M _ { S _ 1 } } , \\mu _ { S _ 1 } \\rangle = \\langle \\rho _ { G \\setminus M _ { S _ 1 } } , \\mu _ { S _ 2 } \\rangle , \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\frac { m - 1 } { 2 } } [ 3 n ] \\frac { ( 1 - q ) ^ 2 ( q ; q ^ 2 ) _ { n } ( q ^ { m + 2 } ; q ^ 2 ) _ { n - 1 } ^ 2 q ^ { m - { n + 1 \\choose 2 } - \\frac { ( 2 n + 1 ) ( m - 1 ) } { 2 } } } { ( q ; q ) _ { n } ^ 2 ( q ^ 2 ; q ^ 2 ) _ { n } } \\equiv 0 \\pmod { \\Phi _ m ( q ) } . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} h _ { a l g } ( \\bar \\phi ) \\geq \\log [ \\pi O : { \\bar \\phi } ^ { \\ - 1 } ( \\pi O ) ] . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } C _ { n } ^ { \\lambda } ( t ) C _ { m } ^ { \\lambda } ( t ) \\omega _ \\lambda ( t ) \\ , d t = \\gamma _ { n } ^ { \\lambda } \\delta _ { n m } , \\gamma _ { n } ^ { \\lambda } = \\frac { \\pi 2 ^ { 1 - 2 \\lambda } \\Gamma ( n + 2 \\lambda ) } { n ! \\left ( n + \\lambda \\right ) \\Gamma ^ 2 ( \\lambda ) } . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p ^ r - 1 } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 3 } { k ! ^ 3 } ( 3 k + 1 ) 2 ^ { 2 k } & \\equiv p ^ r \\pmod { p ^ { r + 3 } } \\quad . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\dim _ H B ( \\{ s _ n \\} , \\{ t _ n \\} , 1 ) \\leq \\liminf _ { \\ell \\to \\infty } \\frac { \\sum _ { i = 1 } ^ \\ell \\log t _ i } { d \\sum _ { i = 1 } ^ { \\ell + 1 } \\log s _ i - \\log t _ { \\ell + 1 } } . \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} z ( P _ \\lambda ) & = \\sum _ { i = 0 } ^ r a _ i z ( P _ { e p ^ i } ) \\\\ & = a _ 0 + \\sum _ { i = 1 } ^ r \\Big ( a _ i ( i + 1 ) p ^ i - a _ i i p ^ { i - 1 } \\Big ) \\\\ & = z ( P _ n ) . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} w ^ { \\sigma } ( x , t ) : = u ( x + \\rho + \\sigma n , t + \\tau ) \\preceq u ( x , t ) \\ \\ { \\rm f o r \\ a l l } \\ \\ x \\in \\R ^ N , \\ t \\in \\R . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\mathcal { B } _ { I ^ + } = \\{ K ^ + : K \\in \\mathcal { B } _ I \\} \\quad \\mathcal { B } _ { I ^ - } = \\{ K ^ - : K \\in \\mathcal { B } _ I \\} , I \\in \\mathcal { D } _ k . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} g w _ j & = ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) \\times ( A ( g ) ) \\\\ & = \\sum _ { h \\in G } c _ { j , h } ( w _ 1 , \\dots , w _ { \\chi ( 1 ) } ) \\times ( A ( g h ) ) ) , \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} \\eta _ { m } ^ \\zeta ( \\rho , \\rho u ) = \\int _ { - 1 } ^ 1 \\zeta ' ( u + \\rho ^ \\vartheta s ) [ 1 - s ^ 2 ] _ + ^ \\Lambda d s . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} Y = Y [ \\{ X _ i \\} ] ( v ) : = \\sup _ i | | X _ i | | _ { v , \\Omega } , \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} [ U ' _ { n } ] = \\sum _ { m \\in I } [ U ' _ { n , m } ] = \\sum _ { m = 1 } ^ n \\L ^ { m d _ 1 } \\cdot ( \\L ^ { d _ 2 } - 1 ) \\L ^ { ( n - m ) d _ 2 } + \\L ^ { ( n + 1 ) d _ 1 } \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\mathcal { X } _ q = \\widetilde { X } \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} d _ { 1 } ( x , y ) = T ^ { H } d _ { T } ( x , y ) , \\ \\ \\ \\forall T > 0 , \\ x , y \\in \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} 1 / | x | : = l ( x ) + r ( x ) \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} M _ A f ( x ) = \\sup _ { Q \\ni x } \\langle | f | \\rangle _ { A , Q } , \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\frac { d t _ 1 } { d s } & = - \\frac { \\eta - a } { 2 } - \\frac { 1 } { \\sqrt { \\Gamma } } \\Bigg \\{ \\sigma ^ 2 [ ( - a + \\eta ) ^ 2 - 1 ] s \\\\ & \\quad + \\frac { 1 } { 2 } [ ( - a + \\eta ) ( a ^ 2 - 1 ) + 2 \\eta ] \\Bigg \\} , \\\\ \\frac { d ^ 2 t _ 1 } { d s ^ 2 } & = \\frac { \\sigma ^ 2 ( 2 a \\eta - a ^ 2 + 1 ) ^ 2 } { \\Gamma ^ { \\frac { 3 } { 2 } } } \\geq 0 . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} f ( y ) = x _ { m } \\leq ^ { \\boldsymbol { X } ^ { \\infty } _ { n } } x _ { ( t + 1 ) n } . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f _ + & = - \\partial _ x f _ + + \\frac 1 2 ( f _ - - f _ + ) \\ , , \\\\ \\partial _ t f _ - & = \\partial _ x f _ - - \\frac 1 2 ( f _ - - f _ + ) \\ , . \\end{cases} \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\lambda _ n ( \\nabla _ { x x } \\mu _ f ( x , t ) ) & \\ge & 1 2 \\displaystyle \\min _ { i = 1 , \\ldots , n } a _ i x _ i ^ 2 + 2 \\lambda _ n ( B ) + 4 t ^ 2 a _ k \\\\ & & \\\\ & \\ge & 2 \\lambda _ n ( B ) + 4 t ^ 2 a _ k \\ , , \\end{array} \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} | D _ j | = \\left | \\int ( T ( \\omega _ j ) - T ( \\omega _ { j - 1 } ) ) f ( y _ j ) \\ldots f ( y _ n ) d y _ { j } \\ldots d y _ { n + 1 } \\right | \\leq H _ j \\\\ \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} b _ n : = r _ n \\sqrt { n N } , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\ , \\Big ( \\frac { \\partial f _ j } { \\partial z ^ k } ( t , z ) \\Big ) = l \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} W _ m = \\left [ \\frac { w \\left ( \\frac { x _ i - x _ j } { l _ m } \\right ) } { l _ m } \\cdot \\frac { 1 } { n } \\right ] _ { i , \\ j = 0 } ^ { n - 1 } = \\left [ \\frac { w \\left ( \\frac { i - j } { ( n - 1 ) l _ m } \\right ) } { l _ m } \\cdot \\frac { 1 } { n } \\right ] _ { i , \\ j = 0 } ^ { n - 1 } \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\theta _ q ( Q ) = ( q ; q ) _ \\infty ( - Q ; q ) _ \\infty \\left ( \\frac { - q } { Q } ; q \\right ) _ \\infty \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\tau ( A ) = \\sum _ { j = 1 } ^ \\infty \\langle A e _ i , e _ i \\rangle \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } m ( x ) = \\infty , \\ , \\ , \\ , \\textrm { a n d } m ( x ) \\geq 0 \\textrm { f o r a l l } x \\in \\R ^ n \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} ( m _ { i , j } ) \\times ( \\sum _ { g \\in G } g ^ { - 1 } ( \\alpha ) A ( g ) ) = 0 ^ { ( \\chi ( 1 ) , 1 ) } . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} V ( y ) = \\inf \\Bigl \\{ I _ { 0 T } ( \\varphi ) \\ , \\bigl \\vert \\ , \\varphi ( 0 ) = x _ 0 , \\ , \\ , T > 0 , \\ , \\ , \\varphi ( T ) = y \\ , \\ , \\ , \\ , \\ , \\ , y \\in \\partial D \\Bigr \\} \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} | H _ 0 | ^ 2 = - ( \\Delta Y _ 0 ) ^ 2 - ( \\Delta Y _ 4 ) ^ 2 + \\sum _ { i = 1 } ^ 3 ( \\Delta Y _ i ) ^ 2 + 4 . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} 0 & = ( - d ^ * f ) ( a e _ x e _ y ) = f ( d ( a e _ x e _ y ) ) = f ( d ( a ) e _ x e _ y ) + f ( a d ( e _ x ) e _ y ) - f ( a e _ x d ( e _ y ) ) \\\\ & = f ( a ( 1 - x ) e _ y ) - f ( a e _ x ( 1 - y ) ) = f ( a e _ y ) - f ( a x e _ y ) - f ( a e _ x ) + f ( a e _ x y ) , \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} J ' _ 2 ( f ; v ^ 1 , v ^ 2 ) ( 0 , \\hat { v } ^ 2 ) = 0 , \\ \\ \\ \\forall \\hat { v } ^ 2 \\in L ^ 2 ( \\mathcal { O } _ 2 \\times ( 0 , T ) ) . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} H _ { a l g } ( \\phi , U ) = \\limsup _ { n \\to \\infty } \\frac { \\log \\mu ( T _ n ( \\phi , U ) ) } { n } , \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} & P r ( f , \\sigma , \\{ k _ j \\} ) = \\frac { v o l ( \\mathfrak { D } ( f , \\sigma , \\{ k _ j \\} ) ) } { v o l ( \\mathfrak { D } ( f , \\sigma ) ) } , \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} a : = \\frac { \\gamma } { m ^ 2 } + \\frac { 1 } { m } \\sum _ { k = 1 } ^ N \\lambda _ k + \\sum _ { k > N } \\lambda _ k k ^ { - 2 s } , a _ 1 : = 1 - \\frac { 1 } { \\gamma m } \\sum _ { k > N } \\frac { c _ k } { k ^ { - 2 s } \\lambda _ k } - \\frac { m } { \\gamma } \\sum _ { k > N } \\frac { k ^ { - 2 s } c _ k } { \\lambda _ k } . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\| \\psi _ n - \\psi _ n ^ N \\| _ { L ^ 2 ( \\Omega ) } \\lesssim \\frac { 1 } { \\sqrt { \\lambda _ n ^ { N } } } N ^ { - 1 / 2 } \\quad n = 1 , \\cdots , M . \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\theta : = \\frac { \\alpha } { m } - \\frac { \\alpha } { p } = 1 - \\frac { ( d - 2 s ) \\alpha } { 4 s } > 0 , q \\leq n = \\frac { d q } { d - s q } . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} z ( 1 - z ) & = \\frac { \\big [ ( t - b ) ^ 2 - ( y - x ) ^ 2 \\big ] \\big [ ( t + b + 2 ) ^ 2 - ( t - b ) ^ 2 \\big ] } { \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] ^ 2 } = \\frac { 4 ( 1 + t ) ( 1 + b ) \\big [ ( t - b ) ^ 2 - ( y - x ) ^ 2 \\big ] } { \\big [ ( t + b + 2 ) ^ 2 - ( y - x ) ^ 2 \\big ] ^ 2 } , \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} s p t ( \\nu ) = \\Sigma = \\lim _ { i \\to \\infty } S i n g ( u _ i ) , \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\begin{array} { r c l } T _ 1 & = & \\displaystyle \\mu _ h ( x _ k , t ) = h ( x _ k ) + \\frac { t ^ 2 } { 6 } \\ , h '' ( x _ k ) + \\frac { t ^ 4 } { 1 2 0 } h ^ { ( 4 ) } ( x _ k ) \\\\ & & \\\\ & = & \\displaystyle f ( x ) + \\frac { t ^ 2 } { 6 } \\ , f _ { k k } ( x ) + \\frac { t ^ 4 } { 1 2 0 } f _ { k k k k } . \\\\ \\end{array} \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\frac { d t _ 1 } { d s } \\Big | _ { s = s ^ \\star } = 0 , \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} a ( w , x ) : = \\lim _ { y \\to x } \\frac { f _ w ( x ) - f _ w ( y ) } { ( x - y ) ^ r } , \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} e ^ { \\beta ^ n } ( 1 - \\varepsilon _ n ) ^ a < S _ n \\varphi ( x ) < e ^ { \\beta ^ n } ( 1 + \\varepsilon _ n ) ^ a + \\sum _ { i = 1 } ^ { n - 1 } ( 1 + \\varepsilon _ i ) ^ a \\cdot e ^ { \\beta ^ i } < e ^ { \\beta ^ n } ( 1 + a \\varepsilon _ n + o ( 1 ) ) . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\P \\Big ( \\| \\hat P _ j - P _ j \\| _ 2 > 2 - 2 \\delta \\Big ) & \\geq \\P \\Big ( \\Big \\{ \\| \\tilde P _ j - P _ j \\| _ 2 > 2 - 2 \\delta + \\| \\tilde P _ j - \\hat P _ j \\| _ 2 \\Big \\} \\cap \\tilde { \\mathcal { E } } _ { z _ 1 , z _ 2 } \\Big ) \\\\ & \\geq \\P \\Big ( \\Big \\{ \\| \\tilde P _ 1 - P _ 1 \\| _ 2 > 2 - \\delta \\Big \\} \\cap \\tilde { \\mathcal { E } } _ { z _ 1 , z _ 2 } \\Big ) - e ^ { - t } = 1 - e ^ { - t } - \\P ( \\tilde { \\mathcal { E } } _ { z _ 1 , z _ 2 } ^ c ) . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} - \\phi ' ( s ) = K u \\Gamma ( u + 1 ) s ^ { - u - 1 } ( 1 + O ( s ^ { \\epsilon / 2 } ) ) \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} E [ e ^ { ( X _ 1 + X _ 2 + \\cdots + X _ k ) t } ] & = \\sum _ { n = 0 } ^ \\infty E [ ( X _ 1 + \\cdots + X _ k ) ^ n ] \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} \\frac { \\partial m } { \\partial t } = \\lambda _ 1 m \\times H _ { e f f } - \\lambda _ 2 m \\times ( m \\times H _ { e f f } ) \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} & \\begin{cases} \\partial _ t \\mathbb { L } ( A _ t ) + \\sup _ { v \\in \\mathrm { V } } \\inf _ { u \\in \\mathrm { U } } \\mathcal { H } ( A _ t , u , v , ( \\mathbb { L } , \\partial _ x \\mathbb { L } , \\partial _ { x x } \\mathbb { L } ) ( A _ t ) ) = 0 , \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ t \\in [ 0 , T ) , ~ A _ t \\in \\Lambda \\\\ \\mathbb { L } ( A _ T ) = m ( A _ T ) , ~ A _ T \\in \\Lambda _ T , \\end{cases} \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} P r _ D ( f , i d ) = \\frac { v o l ( D \\cap \\mathfrak { D } ( f , i d ) ) } { v o l ( \\mathfrak D ( f , i d ) ) } . \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} d S ( K , \\cdot ) = n h _ K ^ { p - 1 } d \\mu , \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\tau _ \\epsilon ( i ) & = \\begin{cases} ( \\delta + \\epsilon ) - i & \\mbox { $ i $ o d d } \\\\ i & \\mbox { $ i $ e v e n } \\end{cases} \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\Psi ' _ { N _ \\varepsilon } ( u _ \\varepsilon ( \\check { z } _ \\varepsilon ) ) = ( 1 + o ( 1 ) ) 2 u _ \\varepsilon ( \\check { z } _ \\varepsilon ) \\varphi _ { N _ \\varepsilon - 1 } ( u _ \\varepsilon ( \\check { z } _ \\varepsilon ) ^ 2 ) \\ , , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} d _ { T V } ( \\zeta _ { 1 } , \\zeta _ { 2 } ) = \\sup \\{ | \\zeta _ { 1 } ( E ) - \\zeta _ { 2 } ( E ) | \\ : : \\ : E \\subset \\mathbb { R } \\} \\ : . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\varphi _ { N } ( \\Gamma ) = \\delta \\exp ( \\Gamma ) \\implies \\forall T \\in [ 0 , \\Gamma ] \\ , , \\varphi _ { N } ( T ) \\le \\delta \\exp ( T ) \\ , , \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} ( A ^ * + B ^ * ) W ( ( A + B ) X _ 0 - I ) = 0 . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} K \\varphi \\triangleq \\begin{cases} C _ { H } \\cdot I _ { 0 ^ { + } } ^ { 1 } \\left ( t ^ { H - \\frac { 1 } { 2 } } \\cdot I _ { 0 ^ { + } } ^ { H - \\frac { 1 } { 2 } } \\left ( s ^ { \\frac { 1 } { 2 } - H } \\varphi ( s ) \\right ) ( t ) \\right ) , & H > \\frac { 1 } { 2 } ; \\\\ C _ { H } \\cdot I _ { 0 ^ { + } } ^ { 2 H } \\left ( t ^ { \\frac { 1 } { 2 } - H } \\cdot I _ { 0 ^ { + } } ^ { \\frac { 1 } { 2 } - H } \\left ( s ^ { H - \\frac { 1 } { 2 } } \\varphi ( s ) \\right ) ( t ) \\right ) , & H \\leq \\frac { 1 } { 2 } , \\end{cases} \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} \\lambda = \\lambda _ r + \\lambda _ s , \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} r _ i ( P ^ { - ( r - 1 ) } \\mathcal { A } ( H ) P ) & = \\sum _ { \\{ i , i _ 2 , \\ldots , i _ r \\} \\in E ( H ) } \\frac { \\sqrt [ r ] { d _ i d _ { i _ 2 } \\cdots d _ { i _ r } } } { d _ i } \\\\ & \\leq \\max _ { \\{ i _ 1 , i _ 2 , \\ldots , i _ r \\} \\in E ( H ) } \\left \\{ \\sqrt [ r ] { d _ { i _ 1 } d _ { i _ 2 } \\cdots d _ { i _ r } } \\right \\} . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} b _ i = 1 + \\left ( n - 1 \\right ) \\frac { x _ i - x _ \\mathrm { m i n } } { x _ \\mathrm { m a x } - x _ \\mathrm { m i n } } . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\Lambda _ 0 ( \\mathbb { D } ^ 2 ) = \\frac { 4 \\pi } { \\lambda _ 1 ( \\mathbb { D } ^ 2 ) } < \\pi e \\ , , \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} \\bar a ( t ) = h ^ { - 1 } \\bar A ( h t ) \\ , , \\bar c ( t ) = \\bar C ( h t ) \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} A _ { \\inf } \\{ 1 \\} = \\varprojlim _ r A _ { \\inf } \\{ 1 \\} \\otimes _ { A _ { \\inf } , \\tilde \\theta _ r } W _ r ( R ) \\ . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} ( f \\nu ) \\cdot g \\partial _ { x _ i } = - \\partial _ { x _ i } ( f g ) \\nu , \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} h ( x ) & \\le \\Bigl ( g _ { 1 0 - } ( x ) + \\frac { 2 } { 6 ^ { 1 1 } } ( 2 ^ { 1 1 } x - 8 5 3 ) ( 7 3 7 8 9 - 3 ^ { 1 1 } x ) \\Bigr ) \\Bigr | _ { x = \\frac { 7 3 7 8 9 } { 3 ^ { 1 1 } } } \\\\ & + \\frac 1 { 1 0 \\cdot 6 ^ { 1 1 } } - \\eta \\\\ & = - \\frac { 2 9 3 4 7 5 4 6 7 1 0 9 0 8 4 1 6 4 5 4 7 6 1 3 5 0 7 8 0 0 3 9 7 3 2 6 0 7 } { 6 5 1 4 3 8 3 8 7 4 9 6 6 9 3 7 6 0 0 4 3 3 9 5 3 6 6 8 1 9 0 1 4 7 4 2 0 8 9 6 2 0 4 8 0 0 0 } < 0 , \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\epsilon } \\left ( \\epsilon > t \\right ) = & \\int _ { t } ^ { \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left ( \\frac { - x ^ 2 } { 2 } \\right ) d x \\\\ < & \\int _ { t } ^ { \\infty } \\frac { x } { t } \\exp \\left ( \\frac { - x ^ 2 } { 2 } \\right ) d x \\\\ = & \\frac { 1 } { t } \\exp \\left ( \\frac { - t ^ 2 } { 2 } \\right ) \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} d v ( t ) = [ A v ( t ) - B ( v ( t ) + \\Psi ( t ) ) ] d t , \\ \\ v ( 0 ) = 0 . \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} ( \\theta ' \\otimes { \\rm i d } \\otimes { \\rm i d } ) ( \\varphi ( w ^ { \\theta } ) ) & = ( \\theta ' \\otimes { \\rm i d } \\otimes { \\rm i d } ) ( { \\rm i d } \\otimes \\theta ^ { 2 } \\otimes \\theta ) ( \\varphi ( w ) ) \\\\ & = \\lambda ^ { - 1 } ( \\theta ^ { 3 } \\otimes \\theta ^ { 2 } \\otimes \\theta ) ( w ) = \\lambda ^ { - 1 } ( \\theta ^ { 2 } \\otimes \\theta \\otimes { \\rm i d } ) ( \\lambda w ) = w ^ { \\theta } , \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} p _ i = \\sum _ { j = 1 } ^ { r _ i } a _ { i j } \\textbf { Q } ^ { \\lambda _ { i j } } \\mbox { w i t h } \\nu \\left ( a _ { i j } \\textbf { Q } ^ { \\lambda _ { i j } } \\right ) \\geq \\nu ( p _ i ) \\mbox { f o r e v e r y } j , 1 \\leq j \\leq r _ i , \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\| \\pi _ { N } ^ { \\lambda } u - u \\| _ { H ^ 1 ( \\mathbb R ) } & \\le \\big \\| \\sqrt { 1 + x ^ 2 } \\ , ( \\pi _ { N } ^ { \\lambda } u - u ) ' \\big \\| _ { L ^ 2 ( \\mathbb R ) } + \\| \\pi _ { N } ^ { \\lambda } u - u \\| _ { L ^ 2 ( \\mathbb R ) } \\\\ & \\le c N ^ { 1 - m } | u | _ { { \\mathbb B } ^ m _ { \\lambda } ( \\mathbb R ) } , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} ( H _ { \\lambda , \\alpha , \\theta } u ) ( n ) = u ( { n + 1 } ) + u ( { n - 1 } ) + 2 \\lambda v ( \\theta + n \\alpha ) u ( n ) , v ( \\theta ) = 2 \\cos 2 \\pi \\theta , \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} q _ { l } \\left ( \\omega \\right ) = \\int _ { 0 } ^ { \\omega } \\eta ' \\left ( \\bar \\omega \\right ) f _ { l } ' \\left ( \\bar \\omega \\right ) \\ ; d \\bar \\omega , q _ { r } \\left ( \\omega \\right ) = \\int _ { 0 } ^ { \\omega } \\eta ' \\left ( \\bar \\omega \\right ) f _ { r } ' \\left ( \\bar \\omega \\right ) \\ ; d \\bar \\omega . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} \\mathcal A ( \\rho ) v _ 0 = 0 \\hbox { i n } \\ ; \\ ; \\Omega _ s , v _ 0 = \\gamma \\mathcal K ( \\rho ) \\hbox { o n } \\ ; \\ ; \\Gamma _ s , \\partial _ y v _ 0 = 0 \\hbox { o n } \\ ; \\ ; \\Gamma _ 0 . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { M - 1 } & ( - 1 ) ^ k c _ j \\left ( \\frac { q _ j ^ 2 } { N } \\right ) ^ { s - m _ 0 - \\frac 1 2 } P _ f ( s - m _ 0 ; \\delta + m _ 0 , 0 ) \\Delta _ f \\ ! \\left ( s - m _ 0 , \\frac 1 { q _ j } , \\cos ^ { ( \\delta + m _ 0 + k ) } \\right ) \\\\ & - \\eta \\frac { ( - 2 \\pi N ) ^ { m _ 0 } } { m _ 0 ! } P _ f ( s - m _ 0 ; \\delta + m _ 0 , m _ 0 ) \\Delta _ { \\bar { f } } \\ ! \\left ( s , - \\frac { q } { N } , \\cos ^ { ( \\delta ) } \\right ) \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\liminf _ { x \\to \\alpha ^ { + } } v ( x ) = 0 , \\liminf _ { x \\to \\beta ^ { - } } v ( x ) = 0 , v ( x ) > 0 x \\in \\left ] \\alpha , \\beta \\right [ . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} 2 ( F _ i \\cap [ 0 , 1 ] ^ d ) \\subseteq 2 ( F _ i ) \\cap [ 0 , 2 ] ^ d = ( 2 F ) _ i \\cap [ 0 , 2 ] ^ d , \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} h ( s ) & = K \\Gamma ( u + 1 ) s ^ { - u } \\zeta ( u + 1 ) ( 1 + O ( s ^ { \\epsilon / 2 } ) ) + O ( s ^ { - u / 2 } ) + O ( 1 ) \\\\ & = ( 1 + O ( s ^ { \\epsilon / 2 } ) ) K \\Gamma ( u + 1 ) \\zeta ( u + 1 ) s ^ { - u } , \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} x t _ q ( X ) = \\frac { 1 } { \\binom { q } { 2 } } \\sup _ { x _ 1 , . . . , x _ q \\in X } \\sum _ { 1 \\leq i < j \\leq q } d ( x _ i , x _ j ) . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} \\frac { p _ 1 ( V ) - ( 2 k + 1 ) c ^ 2 } { 2 } = 0 , \\end{align*}"}