{"id": "2324.png", "formula": "\\begin{align*} | W ( g , f ) ( x , \\omega ) | & = 2 ^ d | V _ { g ^ \\vee } f ( 2 x , 2 \\omega ) | = 2 ^ d | \\langle f , M _ { 2 \\omega } T _ { 2 x } g ^ \\vee \\rangle | \\\\ & \\leq 2 ^ d \\norm { f } _ 2 \\norm { M _ { 2 \\omega } T _ { 2 x } g ^ \\vee } _ 2 = 2 ^ d \\norm { f } _ 2 \\norm { g } _ 2 . \\end{align*}"}
{"id": "3034.png", "formula": "\\begin{align*} \\frac { \\partial g ^ { i } } { \\partial y _ { j k } ^ { \\sigma } } + \\frac { \\partial g ^ { j } } { \\partial y _ { k i } ^ { \\sigma } } + \\frac { \\partial g ^ { k } } { \\partial y _ { i j } ^ { \\sigma } } = 0 . \\end{align*}"}
{"id": "5547.png", "formula": "\\begin{align*} A - \\lambda = 0 \\end{align*}"}
{"id": "2219.png", "formula": "\\begin{align*} \\| G \\| _ { C _ W ( [ 0 , T ] ; H ) } = \\Big ( \\sup _ { s \\in [ 0 , T ] } \\mathbb { E } \\big [ \\| G ( s ) \\| ^ 2 \\big ] \\Big ) ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "1335.png", "formula": "\\begin{align*} A ( \\delta _ { 1 } ) \\approx \\frac { 2 } { 3 } p _ { i } R , \\ , \\ , A ( \\delta _ { 2 } ) \\approx \\frac { 1 } { 3 } p _ { i } R , \\ , \\ , p _ { i + 1 } = \\frac { 4 } { 3 } p _ { i } , \\ , \\ , A ( \\eta ) \\approx \\frac { 1 } { 2 } p _ { i } R . \\end{align*}"}
{"id": "6274.png", "formula": "\\begin{align*} y [ n ] & = \\sum _ { i \\in \\mathcal { K } } \\nolimits \\sqrt { p _ i } s [ n ] h _ i + w [ n ] = \\sqrt { K \\bar { p } } h ' s [ n ] + w [ n ] , \\ n = 1 , 2 , \\cdots , N , \\end{align*}"}
{"id": "8348.png", "formula": "\\begin{align*} \\Lambda ( \\rho ) = \\int _ 0 ^ \\infty \\exp \\left ( - \\rho y ^ { - 1 } - y \\right ) d y \\sim \\rho ^ { 1 / 4 } \\exp ( - 2 \\sqrt { \\rho } ) \\sqrt { \\pi } , \\textrm { a s } \\rho \\to \\infty . \\end{align*}"}
{"id": "3427.png", "formula": "\\begin{align*} | D _ k f ( x ) - D _ k f ( y ) | & = \\bigg | \\int _ { \\Bbb R ^ N } ( D _ k ( x , z ) - D _ k ( y , z ) ) f ( z ) d \\omega ( z ) \\bigg | \\\\ & \\leqslant \\| f \\| _ { L ^ \\infty } \\bigg ( \\int _ { \\| x - z \\| \\leqslant r ^ { 4 - k } } + \\int _ { \\| y - z \\| \\leqslant r ^ { 4 - k } } \\bigg ) \\frac { ( r ^ k \\| x - y \\| ) ^ s } { V _ k ( x ) + V _ k ( z ) } d \\omega ( z ) , \\end{align*}"}
{"id": "5522.png", "formula": "\\begin{align*} \\sigma _ r ( h ) : = S _ r \\sigma ( h ) , h \\in H . \\end{align*}"}
{"id": "4653.png", "formula": "\\begin{align*} \\partial _ t u + \\partial _ x ( - | D | u + u ^ 2 ) = 0 , \\end{align*}"}
{"id": "3640.png", "formula": "\\begin{align*} \\left | \\sum _ { \\substack { | \\Im ( \\rho ) | \\leq T \\\\ \\ : \\Re ( \\rho ) > \\sigma } } \\frac { x ^ { \\rho - 1 } } { \\rho } \\right | & \\leq \\frac { 2 N ( \\sigma , T ) } { T } \\\\ & \\le 2 \\bigg [ C _ 1 ( \\sigma ) \\exp \\left ( \\frac { B _ 2 ( 5 - 8 \\sigma ) } { 3 } r \\right ) \\left ( B _ 2 r \\right ) ^ { 5 - 2 \\sigma } \\\\ & \\qquad \\qquad \\qquad + C _ 2 ( \\sigma ) \\exp \\left ( - B _ 2 r \\right ) ( B _ 2 ) ^ 2 r ^ 2 \\bigg ] \\\\ & = s _ 2 '' ( x , \\sigma ) , \\ . \\end{align*}"}
{"id": "3686.png", "formula": "\\begin{align*} B _ t + a B J _ x + b J B _ x + \\mu \\Lambda ^ \\alpha B = & \\ 0 , \\ \\ \\ \\ a , b \\in \\mathbb R \\\\ B _ x = & \\ \\mathcal H J \\end{align*}"}
{"id": "3493.png", "formula": "\\begin{align*} \\frac { 1 } { y ^ { \\sigma _ 1 } } \\sum _ { n \\leq x } \\frac { 1 } { n ^ { \\sigma _ 2 } ( y + n ) ^ { \\sigma _ 3 } } & \\ll \\frac { 1 } { y ^ { \\sigma _ 1 + \\sigma _ 3 } } \\sum _ { n \\leq x } \\frac { 1 } { n ^ { \\sigma _ 2 } } \\\\ & \\ll \\begin{cases} x ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ x ^ { - \\sigma _ 1 - \\sigma _ 3 } ( \\log x ) & ( \\sigma _ 2 = 1 ) \\\\ x ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "3453.png", "formula": "\\begin{align*} f ( x ) & = - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q } w ( Q ) \\psi _ { j } ( x , x _ { Q } ) q _ { j } f ( x _ { Q } ) + R _ 1 ( f ) ( x ) + R _ M ( x ) \\\\ & = T _ M ( f ) ( x ) + R _ 1 ( f ) ( x ) + R _ M ( x ) . \\end{align*}"}
{"id": "242.png", "formula": "\\begin{align*} \\nabla ( f _ h ) ( x ) = - \\int _ 0 ^ { + \\infty } e ^ { - t } P _ t ^ { \\Sigma } ( \\nabla ( h ) ) ( x ) d t . \\end{align*}"}
{"id": "8029.png", "formula": "\\begin{align*} \\left \\{ \\mathfrak { P } _ { \\ell } \\chi F , \\mathfrak { P } _ { \\ell } \\chi G \\right \\} _ { \\ell } ^ { \\widetilde { \\Sigma } } [ \\widetilde { \\psi } ] = \\left \\langle E _ { \\widetilde { \\Sigma } } , \\left ( \\mathfrak { P } _ { \\ell } \\chi F \\right ) ^ { ( 1 ) } [ \\widetilde { \\psi } ] \\otimes \\left ( \\mathfrak { P } _ { \\ell } \\chi G \\right ) ^ { ( 1 ) } [ \\widetilde { \\psi } ] \\right \\rangle . \\end{align*}"}
{"id": "1739.png", "formula": "\\begin{align*} \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ { [ \\hat t ( n ) ] } ) } \\stackrel { ( \\ref { n u 2 } ) , ( \\ref { e m b _ n u } ) } { \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } } 2 ^ { ( ( 1 \u2010 \\lambda ) \\mu _ * \u2010 \\lambda \\alpha _ * ) k _ * \\hat t ( n ) } \\stackrel { ( \\ref { m t p r 0 e q } ) } { = } n ^ { \u2010 \\hat \\nu } . \\end{align*}"}
{"id": "5247.png", "formula": "\\begin{align*} ( A \\otimes B ) ( A \\ , { } ^ I \\ ! \\otimes ^ I B ) = ( A \\otimes B ) E , ( A \\ , { } ^ I \\ ! \\otimes ^ I B ) ( A \\otimes B ) = E ( A \\otimes B ) . \\end{align*}"}
{"id": "8550.png", "formula": "\\begin{align*} H \\psi _ { \\pm } ( x , k ) = \\left ( - \\partial _ { x x } + V \\right ) \\psi _ { \\pm } ( x , k ) = k ^ { 2 } \\psi _ { \\pm } ( x , k ) \\end{align*}"}
{"id": "1312.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } \\frac { C \\sqrt { k _ { n } } } { \\epsilon k _ { n } } \\geq \\lim _ { n \\to \\infty } \\frac { | \\hat { I } _ { \\geq \\epsilon } ( k _ { n } ) | } { k _ { n } } . \\end{align*}"}
{"id": "1655.png", "formula": "\\begin{align*} \\mathrm { C h } _ \\lambda ( u , v ) : = \\mathrm { C h } ( u , v ) + \\lambda \\left \\langle u , v \\right \\rangle _ { L ^ 2 ( M ) } , u , v \\in W ^ { 1 } ( M ) , \\end{align*}"}
{"id": "3942.png", "formula": "\\begin{align*} s = \\frac { u + 2 + 2 \\sqrt { u + 1 } } { u } , \\end{align*}"}
{"id": "6311.png", "formula": "\\begin{align*} \\mathrm { v a r } [ Z _ i ] & \\stackrel { ( a ) } { = } \\mathbb { E } [ X ^ 2 ] \\mathbb { E } [ Y ^ { - 2 } ] = \\int _ { \\sqrt { \\mu _ i / \\vartheta _ i } } ^ { \\infty } y ^ { - 2 } f _ Y ( y ) \\mathrm { d } y \\stackrel { ( b ) } { = } 2 \\int _ { \\sqrt { \\mu _ i / \\vartheta _ i } } ^ { \\infty } \\frac { 1 } { y } e ^ { \\frac { \\mu _ i } { \\vartheta _ i } - y ^ 2 } \\mathrm { d } y \\stackrel { ( c ) } { = } e ^ { \\frac { \\mu _ i } { \\vartheta _ i } } \\mathrm { E i } \\Big ( \\frac { \\mu _ i } { \\vartheta _ i } \\Big ) , \\end{align*}"}
{"id": "4760.png", "formula": "\\begin{align*} \\frac { n - d } { n - 1 } R _ 1 + \\frac { 1 } { n - 1 } \\left ( R _ 2 + \\cdots + R _ n \\right ) = ( 1 , \\cdots , 1 ) . \\end{align*}"}
{"id": "2760.png", "formula": "\\begin{align*} ( q _ 1 , r _ 1 ) = \\left ( \\frac { 2 p } { 1 + s _ c ( p - 1 ) } , \\frac { 2 N p } { N + \\gamma } \\right ) ( q _ 2 , r _ 2 ) = \\left ( \\frac { 2 p } { 1 - s _ c } , \\frac { 2 N p } { N + \\gamma + 2 s _ c p } \\right ) ; \\end{align*}"}
{"id": "639.png", "formula": "\\begin{align*} h ( x , m ) \\ = \\ x m + x + m , \\end{align*}"}
{"id": "5442.png", "formula": "\\begin{align*} \\tilde M _ 1 ( p , \\tau , s , u _ 0 ) = p b _ { \\sup } | 1 - p | \\Big ( m ^ * ( \\tau , s , u _ 0 ) - \\frac { a _ { \\sup } } { b _ { \\inf } } | \\Omega | \\Big ) . \\end{align*}"}
{"id": "316.png", "formula": "\\begin{align*} & \\frac { d \\rho } { d t } = \\frac { \\partial } { \\partial S } \\mathcal H _ 0 ( \\rho , S ) + \\frac { \\partial } { \\partial S } \\mathcal H _ 1 ( \\rho , S ) \\circ d W ( t ) , \\\\ & \\frac { d S } { d t } = - \\frac { \\partial } { \\partial \\rho } \\mathcal H _ 0 ( \\rho , S ) - \\frac { \\partial } { \\partial \\rho } \\mathcal H _ 1 ( \\rho , S ) \\circ d W ( t ) . \\end{align*}"}
{"id": "98.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 3 / 2 } \\cdot ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) . \\end{align*}"}
{"id": "2087.png", "formula": "\\begin{align*} \\widehat { \\mathbf { 1 } _ { P _ { \\textbf { c } } } } ( \\pmb { \\xi } ) = \\frac { \\exp { ( 2 \\pi i \\langle \\pmb { \\xi } , \\mathbf { c } \\rangle ) } } { p ^ { 2 k n } } \\sum _ { \\mathbf { v } \\in P _ { \\mathbf { c } } } \\exp { \\Big ( 2 \\pi i \\frac { \\langle \\pmb { \\xi } , \\mathbf { v } - \\mathbf { c } \\rangle } { p ^ { 2 k } } \\Big ) } . \\end{align*}"}
{"id": "5460.png", "formula": "\\begin{align*} p = \\frac { 4 \\beta } { ( \\chi - \\beta ) ^ 2 } . \\end{align*}"}
{"id": "3918.png", "formula": "\\begin{align*} F ( k + 3 ) = - \\frac { \\sum _ { l = 0 } ^ { k } ( l + 1 ) ( l + 2 ) F ( l ) F ( k - l ) } { 2 ( k + 1 ) ( k + 2 ) ( k + 3 ) } \\end{align*}"}
{"id": "4859.png", "formula": "\\begin{align*} m : = \\frac { y _ 2 + y _ 1 } 2 \\ell : = y _ 2 - y _ 1 \\end{align*}"}
{"id": "3967.png", "formula": "\\begin{align*} & e ^ { i ( 1 - x ) ( k \\pi + c _ k + i d _ k ) + O ( k ^ { - 1 } ) } - e ^ { - i ( 1 - x ) ( k \\pi + c _ k + i d _ k ) + O ( k ^ { - 1 } ) } \\\\ & = 2 i \\sin ( ( 1 - x ) ( k \\pi + c _ k + i d _ k ) ) + O ( k ^ { - 1 } ) \\\\ & \\sim _ { + \\infty } 2 i \\sin ( k \\pi ( 1 - x ) ) + O ( k ^ { - 1 } ) , \\end{align*}"}
{"id": "2174.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { v ( x ) - v ( y ) } { \\vert x - y \\vert ^ { \\alpha + d } } d x d y + \\int _ { \\mathbb { R } ^ { d } } g ( u ) v \\ d x = \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , u ) v \\ d x , \\ \\ W ^ { \\alpha , G } ( \\mathbb { R } ^ { d } ) . \\end{align*}"}
{"id": "1150.png", "formula": "\\begin{align*} \\textnormal { R e s } _ { k = \\eta } T _ { \\eta , 1 } = \\frac { 3 } { 4 \\left ( D _ { 0 } ^ { - 2 } r ^ { - 1 } ( \\eta ) - 1 \\right ) } \\begin{pmatrix} \\tilde { T } _ { \\eta , 1 } ^ { ( 1 1 ) } & \\tilde { T } _ { \\eta , 1 } ^ { ( 1 2 ) } \\\\ \\tilde { T } _ { \\eta , 1 } ^ { ( 2 1 ) } & \\tilde { T } _ { \\eta , 1 } ^ { ( 2 2 ) } \\end{pmatrix} \\end{align*}"}
{"id": "4659.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { x _ 1 ( t ) - x _ 2 ( t ) } { \\sqrt { t } } = \\alpha _ 1 - \\alpha _ 2 , \\end{align*}"}
{"id": "8665.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\Phi ^ { i j } ( p , q ) \\left ( \\frac { q _ i } { q ^ 0 } - \\frac { p _ i } { p ^ 0 } \\right ) = \\sum _ { j = 1 } ^ 3 \\Phi ^ { i j } ( p , q ) \\left ( \\frac { q _ j } { q ^ 0 } - \\frac { p _ j } { p ^ 0 } \\right ) = 0 , \\end{align*}"}
{"id": "1216.png", "formula": "\\begin{align*} ( 1 + \\varepsilon ) \\vert B \\vert ^ d & \\geq ( 1 + \\varepsilon ) \\mathcal { H } ^ { d } _ { \\infty } ( B ) \\geq ( 1 + \\varepsilon ) \\mathcal { H } ^ { d } _ { \\infty } ( E ) \\geq \\sum _ { n \\geq 0 } \\vert L _ n \\vert ^ d \\\\ & \\geq \\sum _ { n \\geq 0 } \\mathcal { L } ^ d ( L _ n ) \\geq \\mathcal { L } ^ d ( E ) = \\mathcal { L } ^ d ( B ) = \\vert B \\vert ^ d . \\end{align*}"}
{"id": "961.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x _ 1 } ( 0 ) & = 2 n \\gamma _ { n , s } \\int _ { \\R ^ n _ + \\setminus B _ r ^ + } \\frac { r ^ { 2 s } y _ 1 u ( y ) } { ( \\vert y \\vert ^ 2 - r ^ 2 ) ^ s \\vert y \\vert ^ { n + 2 } } \\dd y , \\end{align*}"}
{"id": "2288.png", "formula": "\\begin{align*} V _ g f ( x , \\omega ) = e ^ { - 2 \\pi i x \\cdot \\omega } V _ { \\widehat { g } } \\widehat { f } ( \\omega , - x ) . \\end{align*}"}
{"id": "2711.png", "formula": "\\begin{align*} \\mathbf { B } = ( [ - 1 , 1 ] ^ n ) _ { n > 0 } . \\end{align*}"}
{"id": "2612.png", "formula": "\\begin{align*} \\norm { f } _ { S _ 0 } = \\norm { V _ g f } _ { L ^ { 1 , 1 } } = \\iint _ { \\R ^ { 2 d } } | V _ g f ( x , \\omega ) | \\ , d ( x , \\omega ) < \\infty . \\end{align*}"}
{"id": "2509.png", "formula": "\\begin{align*} \\langle f , \\rho ( x , \\omega , \\tau ) g \\rangle = e ^ { - 2 \\pi i \\tau } \\langle f , \\rho ( x , \\omega ) g \\rangle = e ^ { - 2 \\pi i \\tau } A ( f , g ) ( x , \\omega ) = e ^ { - 2 \\pi i \\tau } e ^ { \\pi i x \\cdot \\omega } V _ g f ( x , \\omega ) . \\end{align*}"}
{"id": "5983.png", "formula": "\\begin{align*} A _ { n } ^ { \\lambda , \\beta } ( x ) = \\sum _ { m = 0 } ^ { n } \\frac { B _ { n } ^ { m } } { m ! } C _ { m } ^ { \\lambda , \\beta } ( x ) . \\end{align*}"}
{"id": "2013.png", "formula": "\\begin{align*} \\lambda ^ { \\Q _ { \\alpha } } ( \\nu ) : = \\inf \\left \\{ { \\Q } _ { \\alpha } ( f , f ) \\ ; \\Biggl | \\ ; f \\in \\C , ~ \\int _ { E } f ^ { 2 } { \\rm d } \\nu = 1 \\right \\} . \\end{align*}"}
{"id": "4610.png", "formula": "\\begin{align*} G \\cap \\partial \\pi _ v ( Q ) = \\pi _ v ( G ^ + ) \\cap \\pi _ v ( G ^ - ) \\cap \\partial \\pi _ v ( Q ) = \\pi _ v ( G ^ + \\cap Q _ v ^ - ) \\cap \\pi _ v ( G ^ - \\cap Q _ v ^ + ) = \\pi _ v ( G ^ + \\cap G ^ - ) . \\end{align*}"}
{"id": "3197.png", "formula": "\\begin{align*} \\nabla _ { \\xi } ^ 2 \\ , \\varphi ( r , t , \\xi ) \\ = \\ \\frac { 1 } { t } \\frac { 1 } { ( r - \\xi ) ^ 2 } \\ > \\ 0 \\end{align*}"}
{"id": "825.png", "formula": "\\begin{align*} 2 R < \\int _ { y _ 0 } ^ \\infty \\rho d y \\leq \\int _ { y _ 0 } ^ \\infty y ^ { - \\beta } d y = \\frac { 1 } { \\beta - 1 } y _ 0 ^ { 1 - \\beta } . \\end{align*}"}
{"id": "2500.png", "formula": "\\begin{align*} ( x , \\omega , e ^ { 2 \\pi i \\tau } ) \\circledcirc ( x ' , \\omega ' , e ^ { 2 \\pi i \\tau ' } ) = ( x + x ' , \\omega + \\omega ' , e ^ { 2 \\pi i ( \\tau + \\tau ' + x ' \\cdot \\omega ) } ) . \\end{align*}"}
{"id": "8377.png", "formula": "\\begin{align*} \\partial _ t \\zeta ( x , t , y , s ; v ) = v _ 1 \\frac { \\sqrt { s } } { 2 \\sqrt { t } } \\ , \\Psi - v _ 2 \\frac { \\sqrt { 1 - s } } { 2 \\sqrt { 1 - t } } , \\end{align*}"}
{"id": "3222.png", "formula": "\\begin{align*} \\tilde \\mu _ { | S + T | } = \\tilde \\mu _ { | S | } \\boxplus \\tilde \\mu _ { | T | } . \\end{align*}"}
{"id": "2799.png", "formula": "\\begin{align*} L _ + Q = - ( 2 p - 2 ) \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ { p } \\right ) Q ^ { p - 1 } , \\end{align*}"}
{"id": "3511.png", "formula": "\\begin{align*} D _ { 1 1 } & = \\frac { ( a t _ 3 ) ^ { 1 - s _ 1 } } { s _ 1 + s _ 3 - 1 } \\int _ 1 ^ { a t _ 3 } \\frac { 1 } { v ^ { s _ 2 } ( a t _ 3 + v ) ^ { s _ 3 } } d v \\\\ & + \\frac { s _ 3 } { s _ 1 + s _ 3 - 1 } \\int _ 1 ^ { a t _ 3 } \\frac { 1 } { v ^ { s _ 2 - 1 } } \\int _ { a t _ 3 } ^ \\infty \\frac { 1 } { u ^ { s _ 1 } ( u + v ) ^ { s _ 3 + 1 } } d u d v \\\\ & = D _ { 1 1 1 } + D _ { 1 1 2 } , \\end{align*}"}
{"id": "5779.png", "formula": "\\begin{align*} p _ 1 : = ( - \\Delta ) ^ { - 1 } \\partial _ i \\partial _ j ( \\rho u _ i u _ j ) \\mbox { a n d } p _ 2 : = ( \\lambda + 2 \\nu ) \\operatorname { d i v } \\mathbf { u } . \\end{align*}"}
{"id": "2968.png", "formula": "\\begin{align*} \\sum _ { B \\subset S _ 8 } E ^ { ( S _ 4 , B ) } _ { n , \\ell } = \\frac { \\ell _ 1 \\left ( n - \\ell _ 4 \\right ) \\left ( \\sum _ { j = 0 } ^ 6 n ^ j p _ j \\left ( \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 \\right ) \\right ) } { n ^ 8 ( n - 1 ) ( n - 2 ) ( n - 3 ) ( n - 4 ) ( n - 5 ) ( n - 6 ) } , \\end{align*}"}
{"id": "6506.png", "formula": "\\begin{align*} M _ { n + 1 } ^ { ( 2 m ) } = f _ n ^ { ( 2 m ) } + h _ n ^ { ( 2 m ) } + g _ n ^ { ( 2 m ) } M _ n ^ { ( 2 m ) } . \\end{align*}"}
{"id": "7066.png", "formula": "\\begin{align*} A ^ K _ i ( t ) = \\widetilde A ^ K _ i ( t ) + 2 C ( \\bar b + \\bar d ) t + 2 \\bar p t \\leq & \\max _ { 0 \\leq j \\leq 1 / \\delta _ K } \\widetilde A ^ K _ j ( 0 ) + 2 C ( \\bar b + \\bar d ) t + 2 \\bar p t \\\\ = & \\max _ { 0 \\leq j \\leq 1 / \\delta _ K } \\beta ^ K _ j ( 0 ) + 2 C ( \\bar b + \\bar d ) t + 2 \\bar p t . \\end{align*}"}
{"id": "7914.png", "formula": "\\begin{align*} \\partial ^ 2 _ { x _ l x _ k } \\Phi \\in T _ { y _ 0 } ^ { \\perp } X , \\forall \\ l , k = 1 , \\dots n , \\end{align*}"}
{"id": "5698.png", "formula": "\\begin{align*} I ( \\alpha \\cup { \\gamma _ { 0 } } , \\alpha ) = 2 + 2 w ( \\alpha ) . \\end{align*}"}
{"id": "6514.png", "formula": "\\begin{align*} H ^ { ( 2 m ) } _ n = \\begin{cases} O ( n ^ { - m ( 1 - 2 \\alpha ) } ) & \\mbox { i f $ - 2 + m ( 1 - 2 \\alpha ) < - 1 $ } , \\\\ O ( n ^ { - m ( 1 - 2 \\alpha ) } \\log n ) & \\mbox { i f $ - 2 + m ( 1 - 2 \\alpha ) = - 1 $ } , \\\\ O ( n ^ { - 1 } ) & \\mbox { i f $ - 2 + m ( 1 - 2 \\alpha ) > - 1 $ } , \\end{cases} \\end{align*}"}
{"id": "7923.png", "formula": "\\begin{align*} F = \\int _ { l } ^ { u } f ^ { 2 } ( x ) \\pi _ { l , u } ( x ) \\mathrm { d } x . \\end{align*}"}
{"id": "1732.png", "formula": "\\begin{align*} 2 ^ { \u2010 m _ 0 ' ( ( 1 \u2010 \\lambda ) ( s _ * + 1 / q \u2010 1 / p _ 1 ) + \\lambda ( 1 / q \u2010 1 / p _ 0 ) ) } = 2 ^ { \u2010 m _ 0 ' ( 1 \u2010 \\lambda ) s _ * } \\stackrel { ( \\ref { h a t _ s i g m a _ d e f } ) } { = } n ^ { \u2010 \\hat \\sigma } . \\end{align*}"}
{"id": "2611.png", "formula": "\\begin{align*} \\langle f , h \\rangle = \\iint _ { \\R ^ { 2 d } } V _ { g _ 0 } f ( z ) \\overline { V _ { g _ 0 } h ( z ) } \\ , d z , \\end{align*}"}
{"id": "8808.png", "formula": "\\begin{align*} T ^ { \\epsilon , L } : = \\lim _ { t \\uparrow \\infty } T _ t , \\end{align*}"}
{"id": "7579.png", "formula": "\\begin{align*} c _ { i , j } = \\dfrac { 4 } { \\pi ^ 2 } \\int _ D f ( x , y ) T _ i ( x ) T _ j ( y ) \\omega ( x , y ) d x d y = \\dfrac { 4 } { \\pi ^ 2 } \\int _ { 0 } ^ { \\pi } \\int _ { 0 } ^ { \\pi } f ( \\cos \\theta _ x , \\cos \\theta _ y ) \\cos i \\theta _ x \\cos j \\theta _ y d \\theta _ x d \\theta _ y , \\end{align*}"}
{"id": "7336.png", "formula": "\\begin{align*} & u ( \\mu x _ 1 + ( 1 - \\mu ) x _ 2 , t ) \\le \\max \\{ u ( \\lambda _ 1 y _ 1 + ( 1 - \\lambda _ 1 ) z _ 1 , t ) , u ( \\lambda _ 2 y _ 2 + ( 1 - \\lambda _ 2 ) z _ 2 , t ) \\} \\\\ \\le & \\ , \\max _ { i = 1 , 2 } \\max \\{ u ( y _ i , t ) , u ( z _ i , t ) \\} < h + \\varepsilon . \\end{align*}"}
{"id": "7865.png", "formula": "\\begin{align*} ( D \\setminus \\{ \\bar { b } _ n \\} ) \\cup ( E \\setminus \\{ \\bar { b } _ n \\} ) \\cup ( F \\setminus \\{ \\bar { b } _ n \\} ) & = ( D \\cup E \\cup F ) \\setminus \\{ \\bar { b } _ n \\} \\\\ & = \\mathbb { N } \\setminus \\{ \\bar { b } _ n \\} \\end{align*}"}
{"id": "7506.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R } ( \\rho ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\partial { R } } \\log F ( s ) \\ d s \\end{align*}"}
{"id": "2891.png", "formula": "\\begin{align*} \\widetilde { K } = \\{ Q ^ - ( \\cdot + x ( t ) , t ) , \\ ; t \\in \\mathbb { R } \\} \\end{align*}"}
{"id": "2770.png", "formula": "\\begin{align*} \\frac { \\| U ^ 1 \\| _ 2 ^ { N + \\gamma - ( N - 2 ) p } \\| \\nabla U ^ 1 \\| _ 2 ^ { N p - ( N + \\gamma ) } } { \\iint \\frac { | U ^ 1 ( x ) | ^ p | U ^ 1 ( y ) | ^ p } { | x - y | ^ { N - \\gamma } } d x d y } = C _ { G N } ^ { - 1 } = \\frac { \\| Q \\| _ 2 ^ { N + \\gamma - ( N - 2 ) p } \\| \\nabla Q \\| _ 2 ^ { N p - ( N + \\gamma ) } } { \\iint \\frac { | Q ( x ) | ^ p | Q ( y ) | ^ p } { | x - y | ^ { N - \\gamma } } d x d y } . \\end{align*}"}
{"id": "4889.png", "formula": "\\begin{align*} \\frac { f '' ( z ) } { f ( z ) } = L ' ( z ) + L ( z ) ^ 2 = \\frac { m ( m - 1 ) } { z ^ 2 } + O ( | z | ^ { - 3 } ) \\hbox { a s $ z \\to \\infty $ . } \\end{align*}"}
{"id": "2406.png", "formula": "\\begin{align*} D ^ * f = ( \\langle f , e _ \\gamma \\rangle ) _ { \\gamma \\in \\Gamma } . \\end{align*}"}
{"id": "598.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ n ) \\ = \\ g ( h _ 1 ( x _ 1 , \\ldots , x _ n ) , \\ldots , h _ m ( x _ 1 , \\ldots , x _ n ) ) . \\end{align*}"}
{"id": "7038.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ L \\| p _ j \\| ^ 2 \\le C _ 1 ^ 2 \\sum _ { j = 1 } ^ L \\| p _ j \\| _ { H ^ 2 } ^ 2 \\le C _ 1 ^ 2 \\sum _ { j = 1 } ^ \\infty \\| p _ j \\| _ { H ^ 2 } ^ 2 \\le C _ 1 ^ 2 . \\end{align*}"}
{"id": "8738.png", "formula": "\\begin{align*} p ( x ) = p _ { 1 0 } ( x ) \\cdot p _ { 2 0 } ( x ) + \\bigl ( p _ { \\pi _ 1 1 } ( x ) - p _ { \\pi _ 1 0 } ( x ) \\bigr ) \\cdot \\prod _ { i \\neq \\pi _ 1 } p _ { i 0 } ( x ) + \\cdots + \\bigl ( p _ { \\pi _ { 2 n } n } ( x ) - p _ { \\pi _ { 2 n } ( n - 1 ) } ( x ) \\bigr ) \\cdot \\prod _ { i \\neq \\pi _ { d n } } p _ { i n } ( x ) , \\end{align*}"}
{"id": "3980.png", "formula": "\\begin{align*} - \\lambda ^ 3 - 2 7 \\lambda ^ 2 + ( - 1 2 l ^ 2 \\pi ^ 2 + 1 2 l m \\pi ^ 2 - 1 2 m ^ 2 \\pi ^ 2 ) \\lambda - 1 6 i l ^ 3 \\pi ^ 3 + 2 4 i l ^ 2 m \\pi ^ 3 + 2 4 i l m ^ 2 \\pi ^ 3 - 1 6 i m ^ 3 \\pi ^ 3 = 0 , \\end{align*}"}
{"id": "8519.png", "formula": "\\begin{align*} t ^ c _ j = Q ^ c \\left ( \\prod _ { i \\in \\mathcal { N } _ c \\setminus \\{ j \\} } \\operatorname { s g n } ( \\phi ^ v _ i ( t ^ v _ i ) ) \\sum _ { i \\in \\mathcal { N } _ c \\setminus \\{ j \\} } | \\phi ^ v _ i ( t ^ v _ i ) | \\right ) . \\end{align*}"}
{"id": "5355.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { \\| \\tilde a _ i \\| ^ 2 } \\in \\mathbb { R } _ { > 0 } . \\end{align*}"}
{"id": "6033.png", "formula": "\\begin{align*} \\widetilde { L } _ n ' - \\widetilde { L } _ { n - 1 } ' - \\widetilde { L } _ { n - 1 } = 0 . \\end{align*}"}
{"id": "500.png", "formula": "\\begin{align*} \\varepsilon _ { \\delta _ { 1 } \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 1 } a _ { 2 } a _ { 3 } } = c _ { 1 } ^ { ( 1 , 2 ) } c _ { 1 } ^ { ( 1 , 3 ) } c _ { 1 } ^ { ( 2 , 3 ) } \\end{align*}"}
{"id": "3719.png", "formula": "\\begin{align*} & \\left \\| \\lambda _ q ^ { \\frac { \\alpha } { 2 } } e ^ { - \\mu \\lambda _ q ^ \\alpha t } \\| B _ q ( 0 ) \\| _ { L ^ { 2 } } \\right \\| _ { L ^ { 2 } ( 0 , T ; l ^ 2 ) } \\\\ \\leq & \\ \\left \\| e ^ { - \\mu \\lambda _ q ^ \\alpha t } B _ q ( 0 ) \\right \\| _ { L ^ { 2 } ( 0 , T ; H ^ { \\frac { \\alpha } { 2 } } ) } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { L ^ { 2 } } \\end{align*}"}
{"id": "9315.png", "formula": "\\begin{align*} - \\begin{bmatrix} H & - A ^ T \\\\ A & 0 \\end{bmatrix} \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} + \\begin{bmatrix} - \\mathbf { g } \\\\ \\mathbf { b } \\end{bmatrix} \\in N _ D ( \\mathbf { x } , \\mathbf { y } ) , \\end{align*}"}
{"id": "4737.png", "formula": "\\begin{align*} | \\partial _ t \\widetilde { N } _ 3 | \\leq \\frac { | \\dot { \\mathfrak { q } } _ 1 | + | \\dot { \\mathfrak { q } } _ 2 | } { ( \\mathfrak { q } _ 1 - \\mathfrak { q } _ 2 ) ^ 4 } \\ll \\frac { 1 } { ( \\mathfrak { q } _ 1 - \\mathfrak { q } _ 2 ) ^ 4 } = N _ 3 . \\end{align*}"}
{"id": "6561.png", "formula": "\\begin{align*} \\frac { d } { d t } G ( t ) & = \\zeta _ { 2 } ( T ) \\| \\omega ( t ) \\| _ { L ^ { \\infty } } \\\\ & \\leq \\zeta _ { 2 } ( T ) \\big ( \\| \\omega ( 0 ) \\| _ { L ^ { \\infty } } + C ( T ) \\int _ { 0 } ^ { t } \\| \\nabla j ( \\tau ) \\| _ { L ^ { \\infty } } \\ , d \\tau \\big ) \\\\ & \\leq \\zeta _ { 2 } ( T ) \\big ( \\| \\omega ( 0 ) \\| _ { L ^ { \\infty } } + C ( T ) G ( t ) \\big ) . \\end{align*}"}
{"id": "5237.png", "formula": "\\begin{align*} \\frac { \\| f ( C _ r ) \\| } { | C _ r | } \\ge \\frac { w ( 2 d k - d + 2 e ) + ( 1 - 2 w ) e + ( 1 - d w ) ( 2 d k - d ) / d } { 2 d k - d + 2 e } = \\frac { 2 k + e - 1 } { 2 d k - d + 2 e } . \\end{align*}"}
{"id": "9006.png", "formula": "\\begin{align*} p ' _ 0 & = \\min \\left ( \\frac { 1 } { N } , \\frac { 2 ^ \\rho - 1 } { K - 1 } \\right ) , \\\\ p _ w & = \\frac { 1 - N p ' _ 0 } { N ^ K } , w \\in 0 : K - 1 , \\end{align*}"}
{"id": "6384.png", "formula": "\\begin{align*} f ( \\underline T ) = V ^ * f ( \\underline N ) V , \\textup { f o r e v e r y } f \\in \\mathcal R ( X ) , \\end{align*}"}
{"id": "4191.png", "formula": "\\begin{align*} H ( v ) : = \\beta ( v ) \\omega _ 0 ( v , u ) , \\end{align*}"}
{"id": "561.png", "formula": "\\begin{align*} \\int _ { \\Omega _ R } | b ( \\rho ) | ^ \\gamma = \\int _ { \\{ \\rho > \\overline \\rho - \\alpha _ 1 \\} } | b ( \\rho ) | ^ \\gamma \\leq c \\int _ { \\{ \\rho > \\overline \\rho - \\alpha _ 1 \\} } P ( \\rho ) \\leq c \\int _ { \\{ \\rho > \\overline \\rho - \\alpha _ 1 \\} } \\left ( P ( \\rho ) - P ( r ) - P ' ( r ) ( \\rho - r ) \\right ) \\end{align*}"}
{"id": "8321.png", "formula": "\\begin{align*} B ( y , u ) : = B _ + ( y , u ) \\cup B _ - ( y , u ) , \\end{align*}"}
{"id": "8609.png", "formula": "\\begin{align*} \\mu ^ { \\# } ( k , \\ell , m , n ) : = \\int \\overline { \\mathcal { K } ^ { \\# } ( x , k ) } \\mathcal { K } ^ { \\# } ( x , \\ell ) \\overline { \\mathcal { K } ^ { \\# } ( x , m ) } \\mathcal { K } ^ { \\# } ( x , n ) \\ , d x \\end{align*}"}
{"id": "8991.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Ddot { \\xi } ( s ) & = \\dot { \\xi } ( s ) + f ^ \\prime ( \\xi ( s ) ) , s \\in ( 0 , b ) , \\\\ \\xi ( 0 ) & = x _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "862.png", "formula": "\\begin{align*} \\bar { \\Delta } ^ { \\rm P } _ { \\rm P r o a c t i v e } = - 1 - \\tau _ { \\rm f } + \\mathbb { E } T ^ { \\rm P r o a c } _ j + \\mathbb { E } \\tau ^ { \\rm P r o a c } _ { V _ { j - 1 } } . \\end{align*}"}
{"id": "6421.png", "formula": "\\begin{align*} \\mathcal { T } ( * ^ V , L ) & = - \\tfrac 1 4 \\sum _ { i < j } \\left ( \\lambda _ i \\lambda _ j \\ , w _ i w _ j \\otimes 1 + 1 \\otimes v _ i v _ j \\right ) \\circ \\left ( \\mu _ { i j } \\ , w _ i w _ j \\otimes 1 + 1 \\otimes v _ i v _ j \\right ) \\\\ & = \\tfrac 1 4 \\sum _ { i < j } \\left ( - \\lambda _ i \\lambda _ j + w _ i w _ j \\otimes v _ i v _ j \\right ) \\circ \\left ( - \\mu _ { i j } + w _ i w _ j \\otimes v _ i v _ j \\right ) , \\end{align*}"}
{"id": "7060.png", "formula": "\\begin{align*} \\tau ' _ { K } = \\inf \\Big \\{ t \\ge 0 , \\exists i \\in \\{ 0 , 1 , \\cdots , { 1 \\over \\delta _ K } - 1 \\} ; N ^ K _ i ( t \\log K ) < K ^ a \\Big \\} . \\end{align*}"}
{"id": "3809.png", "formula": "\\begin{align*} ( f _ { g _ 1 } ( g _ i x ) , f _ { g _ 2 } ( g _ i x ) , \\ldots , f _ { g _ n } ( g _ i x ) ) = F ( g _ i x ) = \\tau _ 2 ( g _ i , F ( x ) ) = ( f _ { g _ i . g _ 1 } ( x ) , f _ { g _ i . g _ 2 } ( x ) , \\ldots , f _ { g _ i . g _ n } ( x ) ) \\end{align*}"}
{"id": "1356.png", "formula": "\\begin{align*} \\mathcal { C } _ R = \\{ u \\in C _ c ^ \\infty ( \\overline { W _ R } ) \\mid u = 0 \\} . \\end{align*}"}
{"id": "7658.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta P _ { \\tilde { \\Omega } _ { \\varepsilon } } w + \\tilde { \\lambda } _ 0 \\underline { m } P _ { \\tilde { \\Omega } _ { \\varepsilon } } w = f _ 0 ( w ) & \\tilde { \\Omega } _ { \\varepsilon } \\ ; , \\\\ P _ { \\tilde { \\Omega } _ { \\varepsilon } } w = 0 & \\partial \\tilde { \\Omega } _ { \\varepsilon } \\ ; . \\end{cases} \\end{align*}"}
{"id": "1999.png", "formula": "\\begin{align*} \\sigma ( h _ { n , p } ^ { - 1 } ) = ( 1 , 3 , \\dots , 2 n - 1 ) ^ p ( 2 , 4 , \\dots , 2 n ) ^ { - p } \\in \\mathfrak { S } _ { 2 n } . \\end{align*}"}
{"id": "7289.png", "formula": "\\begin{align*} \\alpha _ 1 = \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ { 5 k - 1 } X ^ { \\frac { 3 ( 5 k - 1 ) ^ 2 + 5 k - 1 } { 2 } } = - X \\sum ( - 1 ) ^ k X ^ { \\frac { 7 5 k ^ 2 - 2 5 k } { 2 } } = - X \\alpha ( X ^ { 2 5 } ) . \\end{align*}"}
{"id": "1663.png", "formula": "\\begin{align*} \\mathcal A = \\big \\{ { \\bf y } \\in \\R ^ { 4 d } : 4 \\ , C _ 0 \\ge c _ 0 \\mathcal W ( x , v ) + c _ 0 \\mathcal W ( x ' , v ' ) \\big \\} , \\Gamma = \\big \\{ { \\bf y } \\in \\R ^ { 4 d } : r ( { \\bf y } ) \\ge R _ 0 \\big \\} , \\end{align*}"}
{"id": "3753.png", "formula": "\\begin{align*} R ( X , Y ) A _ { i } ( Z ) + R ( Y , Z ) A _ { i } ( X ) + R ( Z , X ) A _ { i } ( Y ) = 0 , \\end{align*}"}
{"id": "1103.png", "formula": "\\begin{align*} & r _ { \\eta } = r ( \\eta ) e ^ { 2 i t g ( \\eta ) } D ^ { - 2 } _ { 0 } ( \\eta , \\xi ) \\exp \\left [ i \\nu ( \\eta ) \\log ( 2 t g '' \\left ( \\eta \\right ) ) \\right ] , \\\\ & r _ { - \\eta } = r ( - \\eta ) e ^ { 2 i t g ( - \\eta ) } D ^ { - 2 } _ { 0 } ( - \\eta , \\xi ) \\exp \\left [ i \\nu ( - \\eta ) \\log ( - 2 t g '' \\left ( - \\eta \\right ) ) \\right ] . \\end{align*}"}
{"id": "8371.png", "formula": "\\begin{align*} L = \\omega ^ 2 - F ^ 2 - d x ^ 2 - d y ^ 2 \\end{align*}"}
{"id": "1249.png", "formula": "\\begin{align*} \\mu \\left ( \\bigcup _ { i \\in \\mathbb { N } } \\bigcup _ { L \\in \\mathcal { F } _ { 2 , i } } L \\right ) = 1 . \\end{align*}"}
{"id": "937.png", "formula": "\\begin{align*} E ( i , j ) = D ^ { - 1 } ( i , j ) - \\frac { D ^ { - 1 } ( i , \\ell ) D ^ { - 1 } ( \\ell , j ) } { D ^ { - 1 } ( \\ell , \\ell ) } , i , j \\in { \\bf m } , \\end{align*}"}
{"id": "3205.png", "formula": "\\begin{align*} P _ { i n } = P _ t | h _ 1 | ^ 2 . \\end{align*}"}
{"id": "3201.png", "formula": "\\begin{align*} \\tau ( y ) & = \\sup \\{ 0 < \\delta < c \\ ; | \\ ; ( \\mathcal B _ d ( x , \\delta ) \\times \\mathcal B _ { n - d } ( y , \\varepsilon ) ) \\cap X \\\\ & \\} . \\end{align*}"}
{"id": "1826.png", "formula": "\\begin{align*} E _ n = { A _ n ( i ) \\over ( 1 + i ) ^ { n - 1 } } , \\end{align*}"}
{"id": "8130.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 9 } ( P G ) } c ( g ( \\gamma ) ) = 3 5 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "5922.png", "formula": "\\begin{align*} \\omega _ Q ^ a = d q ^ i \\wedge d p _ i ^ a = - d \\theta ^ a _ Q , \\end{align*}"}
{"id": "1882.png", "formula": "\\begin{align*} \\Gamma : = \\mathrm { S L } _ n ( { \\Z [ 1 / p ] } ) < \\mathrm { S L } _ n ( \\R ) \\times \\mathrm { S L } _ n ( \\Q _ p ) , \\end{align*}"}
{"id": "5960.png", "formula": "\\begin{align*} \\mu _ { j k } = \\frac { a _ { j k } } { 2 \\rho a d } , \\nu _ { j k } = \\frac { b _ { j k } } { 2 \\rho \\omega a d } , ( j , k ) = 1 , 3 . \\end{align*}"}
{"id": "853.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\left ( t \\right ) & = t - U \\left ( t \\right ) \\\\ & = t - t ^ S _ { j - 1 } - \\tau _ { \\rm f } + \\tau ^ { \\rm R e a c } _ { V _ { j - 1 } } , t \\in \\mathcal { I } _ j . \\end{aligned} \\end{align*}"}
{"id": "5671.png", "formula": "\\begin{align*} J ^ m ( x \\cdot y ) \\cdot J ^ { m + 1 } x = J ^ m y \\end{align*}"}
{"id": "752.png", "formula": "\\begin{align*} X = \\begin{pmatrix} ( g ^ 1 ) ^ T \\\\ ( g ^ 2 ) ^ T \\end{pmatrix} \\begin{pmatrix} g ^ 1 & g ^ 2 \\end{pmatrix} = \\begin{pmatrix} \\| g ^ 1 \\| ^ 1 & \\langle g ^ 1 , g ^ 2 \\rangle \\\\ \\langle g ^ 1 , g ^ 2 \\rangle & \\| g ^ 2 \\| ^ 2 \\end{pmatrix} , \\end{align*}"}
{"id": "1453.png", "formula": "\\begin{align*} \\alpha U ( z ) = \\left [ \\begin{array} { c } a z + b u _ 0 \\\\ c z + d u _ 0 \\end{array} \\right ] = \\left [ \\begin{array} { c } ( a z + b u _ 0 ) ( c z + d u _ 0 ) ^ { - 1 } u _ 0 \\\\ u _ 0 \\end{array} \\right ] u _ 0 ^ { - 1 } ( c z + d u _ 0 ) . \\end{align*}"}
{"id": "7878.png", "formula": "\\begin{align*} \\nu _ F = \\sum _ { H \\in H ( n - m ) } \\nu _ { F , H } . \\end{align*}"}
{"id": "8625.png", "formula": "\\begin{align*} & \\mathcal { K } _ + ( x , k ) = b _ + ^ + ( k ) e ^ { i k x } + b _ + ^ - ( k ) e ^ { - i k x } , \\\\ & \\mbox { w i t h } b ^ + _ + ( k ) : = \\mathbf { 1 } _ { + } ( k ) T ( k ) + \\mathbf { 1 } _ { - } ( k ) , b _ + ^ - ( k ) : = \\mathbf { 1 } _ { - } ( k ) R _ + ( - k ) , \\end{align*}"}
{"id": "940.png", "formula": "\\begin{align*} N _ i ^ { { \\bf m } * } = \\sum _ { j \\in \\bf m } a _ { i j } ^ { \\bf m } N _ j , \\end{align*}"}
{"id": "5738.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( h ) \\backslash V ( T ' \\cup B ) | & \\geq d _ { G } ( h ) - | N _ { G } ( h ) \\cap V ( B ) | \\\\ & - | N _ { G } ( h ) \\cap ( V ( T ' ) \\backslash \\{ u _ { l + 1 } , \\cdots , u _ { j - 1 } \\} | \\\\ & \\geq ( t + 3 ) - 2 - ( t - \\lfloor \\frac { j - l - 1 } { 2 } \\rfloor ) \\\\ & \\geq 1 + \\lfloor \\frac { j - l - 1 } { 2 } \\rfloor . \\end{aligned} \\end{align*}"}
{"id": "7095.png", "formula": "\\begin{align*} i _ a : ( U , \\tau _ 0 \\restriction U ) & \\to ( G , \\tau ) \\\\ i _ a ( x ) & = a \\cdot x \\end{align*}"}
{"id": "6590.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( d ) } \\sum _ { \\psi \\bmod d } \\psi ( m h ) \\overline { \\psi } ( - n k ) = \\left \\{ \\begin{array} { c l } 1 & d | m h + n k \\\\ \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
{"id": "4496.png", "formula": "\\begin{align*} \\Gamma ( 3 , x ) = \\Big ( x ^ 2 + 2 ( x + 1 ) \\Big ) e ^ { - x } , \\ \\ \\Gamma ( s , x ) \\sim x ^ { s - 1 } e ^ { - x } \\ \\ x \\to \\infty , \\ \\ s > 1 . \\end{align*}"}
{"id": "5103.png", "formula": "\\begin{align*} h _ A ( \\tfrac { 1 } { x } ) = h _ A ( x ) = h _ A ( y ) . \\end{align*}"}
{"id": "2526.png", "formula": "\\begin{align*} F ( x , \\omega ) \\langle \\pi ( x , \\omega ) f , g \\rangle = 0 , ( x , \\omega ) \\end{align*}"}
{"id": "2589.png", "formula": "\\begin{align*} \\langle \\widetilde { f } , \\varphi \\rangle = \\frac { 1 } { \\overline { \\langle g , \\widetilde { g } \\rangle } } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) \\langle M _ \\omega T _ x \\widetilde { g } , \\varphi \\rangle \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "8965.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { T _ { x , \\xi } } p \\left ( f \\left ( \\xi ( s ) \\right ) - u \\left ( \\xi ( s ) \\right ) \\right ) d s = u \\left ( \\xi ( t _ 1 ) \\right ) , \\end{align*}"}
{"id": "5795.png", "formula": "\\begin{align*} \\mathcal { Z } _ K = \\bigcup _ { \\sigma \\in K } ( D ^ 2 , S ^ 1 ) ^ \\sigma \\subseteq ( D ^ 2 ) ^ m \\end{align*}"}
{"id": "2478.png", "formula": "\\begin{align*} \\L ^ \\circ = r ^ { - 2 } \\L . \\end{align*}"}
{"id": "1112.png", "formula": "\\begin{align*} E _ { 1 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\left ( I + \\mu ( s ) \\right ) ( J ^ E ( s ) - I ) d s : = I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
{"id": "9504.png", "formula": "\\begin{align*} M _ { p , \\alpha } = \\{ a \\in S ( B ) : p ( a ) = \\alpha \\} \\end{align*}"}
{"id": "2594.png", "formula": "\\begin{align*} \\langle V ^ * _ g F , f \\rangle & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\langle M _ \\omega T _ x g , f \\rangle \\ , d ( x , \\omega ) \\\\ & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\overline { V _ g f ( x , \\omega ) } \\ , d ( x , \\omega ) \\\\ & = \\langle F , V _ g f \\rangle . \\end{align*}"}
{"id": "1023.png", "formula": "\\begin{align*} \\sup _ { B _ R ( y ) } \\frac { u ( x ) } { x _ 1 } \\leqslant C \\sup _ { B _ R ( y ) } u = C \\sup _ { B _ { R / y _ 1 } ( e _ 1 ) } \\tilde u & \\leqslant C \\inf _ { B _ { R / y _ 1 } ( e _ 1 ) } \\tilde u = C \\inf _ { B _ R ( y ) } u \\leqslant C \\inf _ { B _ R ( y ) } \\frac { u ( x ) } { x _ 1 } \\end{align*}"}
{"id": "6273.png", "formula": "\\begin{align*} \\hat { K } = \\arg \\max _ { k } f ( \\mathbf { y } ; \\mathcal { H } _ k ) . \\end{align*}"}
{"id": "5118.png", "formula": "\\begin{align*} A = n \\ , \\frac { 2 \\gamma a } { 1 - e ^ { - 2 \\gamma a } } \\ , e ^ { - 2 \\gamma a } B = n \\ , \\frac { 2 \\gamma a } { 1 - e ^ { - 2 \\gamma a } } . \\end{align*}"}
{"id": "3158.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\alpha _ k \\frac { \\alpha _ k ^ \\intercal x _ k - \\beta _ k } { \\norm { \\alpha _ k } _ 2 ^ 2 } . \\end{align*}"}
{"id": "4615.png", "formula": "\\begin{align*} \\sum _ { \\substack { F \\in \\mathcal { F } \\\\ { G \\in \\mathcal { G } } } } \\varphi ( F , G ) \\geq \\rho ( d , d - k - 1 ) | \\mathcal { F } | = \\rho ( d , d - k - 1 ) f _ { d - 1 } ( P ) . \\end{align*}"}
{"id": "1540.png", "formula": "\\begin{align*} = \\int _ { G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\phi ( \\tilde { \\tau } _ m ( g \\times h ) , s ) \\mathbf { f } ( h ) \\mathbf { d } h . \\end{align*}"}
{"id": "3306.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( ( F _ 2 / T _ 2 ) | _ { D _ 1 \\cap D _ 2 } ) = \\sum _ { l = 1 } ^ { j - 1 } \\frac { ( - 1 ) ^ { l - 1 } a _ 1 ^ l } { l ! } \\alpha _ { n - j + l } ( F _ 2 / T _ 2 ) . \\end{align*}"}
{"id": "4204.png", "formula": "\\begin{align*} 0 & = - \\int _ M | \\nabla \\dot u | ^ 2 \\ , d V _ g + 2 \\int _ M V \\dot u \\ , r \\ , d V _ g + \\int _ M V \\dot u ^ 2 \\ , d V _ g \\\\ & \\ , \\ , + \\int _ M V r ^ 2 \\ , d V _ g + \\int _ M | \\nabla r | ^ 2 \\ , d V _ g . \\end{align*}"}
{"id": "527.png", "formula": "\\begin{align*} \\begin{aligned} & \\begin{bmatrix} \\mathcal { B G } & \\mathcal { A B G } & \\dots & \\mathcal { A } ^ { n - 1 } \\mathcal { B G } \\end{bmatrix} \\\\ & = \\begin{bmatrix} B _ 1 g _ 1 ^ \\top & A _ 1 B _ 1 g _ 1 ^ \\top & \\cdots & A _ 1 ^ { n - 1 } B _ 1 g _ 1 ^ \\top \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ B _ m g _ m ^ \\top & A _ m B _ m g _ m ^ \\top & \\cdots & A _ m ^ { n - 1 } B _ m g _ m ^ \\top \\end{bmatrix} . \\end{aligned} \\end{align*}"}
{"id": "6375.png", "formula": "\\begin{align*} L \\sigma _ { \\widetilde { X } } = \\left ( \\mathcal { O } _ { \\widetilde { X } } , \\alpha ^ \\ast \\mathcal { T } ( - 1 ) , M , \\alpha ^ \\ast \\mathcal { O } ( 1 ) , \\mathcal { O } _ { L _ 1 } , \\dots , \\mathcal { O } _ { L _ 7 } , \\mathcal { B } _ 1 , \\mathcal { B } _ 2 , \\mathcal { B } _ 3 , \\mathcal { B } _ 4 \\right ) \\end{align*}"}
{"id": "577.png", "formula": "\\begin{align*} f ^ { - 1 } ( Z ) \\ = \\ \\Pi ( \\Gamma _ f \\ \\cap \\ ( \\R ^ n \\times Z ) ) \\end{align*}"}
{"id": "3209.png", "formula": "\\begin{align*} { \\rm S N R } _ { D _ 2 , D _ 1 } ^ { x _ 2 } = \\frac { P _ { H } | h _ { 1 2 } | ^ 2 } { \\sigma ^ 2 } . \\end{align*}"}
{"id": "3372.png", "formula": "\\begin{align*} \\theta \\Big ( \\sum _ { i = 0 } ^ { + \\infty } x _ i t ^ { i } , \\sum _ { j = 0 } ^ { + \\infty } y _ j t ^ { j } \\Big ) \\Big ( \\sum _ { k = 0 } ^ { + \\infty } V _ k t ^ { k } \\Big ) = \\sum _ { s = 0 } ^ { + \\infty } \\sum _ { i + j + k = s } \\theta ( x _ i , y _ j ) v _ k t ^ { s } , \\ ; \\forall x _ i , y _ j \\in L , \\ ; v _ k \\in V . \\end{align*}"}
{"id": "2268.png", "formula": "\\begin{align*} \\norm { f } _ 2 = \\norm { \\widehat { f } } _ 2 . \\end{align*}"}
{"id": "5840.png", "formula": "\\begin{align*} X _ { t + s } - X _ t = \\begin{cases} 1 & G ^ + _ { t , s } ( x ) \\cap \\{ X _ t = x \\} , \\\\ - 1 & G ^ - _ { t , s } ( x ) \\cap \\{ X _ t = x \\} , \\end{cases} \\end{align*}"}
{"id": "890.png", "formula": "\\begin{align*} S _ q = \\begin{cases} U _ 0 \\bot p U _ 0 & q = p , \\\\ H _ 2 & . \\end{cases} \\end{align*}"}
{"id": "3701.png", "formula": "\\begin{align*} & \\left \\| \\lambda _ q ^ s e ^ { - \\mu \\lambda _ q ^ \\alpha t } \\| B _ q ( 0 ) \\| _ { L ^ 2 } \\right \\| _ { L ^ { \\frac { \\alpha } { s + \\alpha - \\frac 5 2 } } ( 0 , T ; l ^ 2 ) } \\\\ \\leq & \\ \\left \\| e ^ { - \\mu \\lambda _ q ^ \\alpha t } B _ q ( 0 ) \\right \\| _ { L ^ { \\frac { \\alpha } { s + \\alpha - \\frac 5 2 } } ( 0 , T ; H ^ s ) } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } \\end{align*}"}
{"id": "6244.png", "formula": "\\begin{align*} t _ i = t _ j \\textrm { f o r } 0 \\leq i < j \\leq n - 1 . \\end{align*}"}
{"id": "185.png", "formula": "\\begin{align*} \\mathcal { A } _ \\delta ( f ) ( x ) = - f ^ \\prime ( x ) + \\delta | x | ^ { \\delta - 1 } \\operatorname { s i g n } ( x ) f ( x ) . \\end{align*}"}
{"id": "5398.png", "formula": "\\begin{align*} P ^ { K B } _ \\mu ( T , f ) & = P ^ { B } _ \\mu ( T , f ) , ~ \\underline { C P } ^ K _ \\mu ( T , f ) = \\underline { C P } _ \\mu ( T , f ) , \\\\ \\overline { C P } ^ K _ \\mu ( T , f ) & \\leq \\overline { C P } _ \\mu ( T , f ) , ~ P ^ { K P } _ \\mu ( T , f ) = P ^ { P } _ \\mu ( T , f ) . \\end{align*}"}
{"id": "1881.png", "formula": "\\begin{align*} D ^ n ( x ) = x \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } L ( n , k ) x ^ { 2 k } y ^ { n - 2 k } . \\end{align*}"}
{"id": "5268.png", "formula": "\\begin{align*} \\varphi \\circ S ^ 2 = \\varphi ( - \\nu ) , \\psi \\circ S ^ 2 = \\psi ( - \\nu ) . \\end{align*}"}
{"id": "8388.png", "formula": "\\begin{align*} \\P ( B \\mbox { i s n o t u n i q u e } ) \\leq \\sum _ { B ' \\in \\mathfrak B _ { r - 1 } , B ' \\neq B } \\P ( \\sigma | _ B = \\sigma | _ { B ' } ) \\leq n ^ d q ^ { - ( r - 1 ) ^ { d } } \\ , , \\end{align*}"}
{"id": "4757.png", "formula": "\\begin{align*} ( \\underbrace { 1 , 1 , \\cdots , 1 } _ { n } ) = \\frac { 1 } { ( n - 1 ) } \\sum \\limits _ { i = 1 } ^ n R _ { i } . \\end{align*}"}
{"id": "9150.png", "formula": "\\begin{align*} H ( d ( x _ n , y ) ) & \\leq G ( d ( x _ N , y ) ) + \\sum _ { i = N } ^ \\infty \\varepsilon _ i \\\\ & \\leq \\beta _ H ( \\delta / 2 ) / 2 + \\beta _ H ( \\delta / 2 ) / 2 \\\\ & \\leq \\beta _ H ( \\delta / 2 ) \\end{align*}"}
{"id": "135.png", "formula": "\\begin{align*} \\left ( x ^ 3 - \\omega ( x ) x ^ 2 \\right ) \\left ( x - \\frac 1 2 \\omega ( x ) \\right ) ^ k = L _ v ^ k ( x ^ 3 - \\omega ( x ) x ^ 2 ) , \\mbox { f o r a l l } k \\geq 1 . \\end{align*}"}
{"id": "3986.png", "formula": "\\begin{align*} \\{ \\lambda ^ h _ k \\} _ { | k | \\geq k _ 0 } \\cap \\{ \\lambda ^ p _ k \\} _ { k \\geq k _ 0 } = \\emptyset , \\ \\ \\{ \\lambda ^ h _ k \\} _ { | k | \\geq k _ 0 } \\cap \\big ( \\{ \\lambda _ 0 \\} \\cup \\{ \\widehat \\lambda _ n \\} _ { n = 1 } ^ { n _ 0 } \\big ) = \\emptyset , \\ \\ \\{ \\lambda ^ p _ k \\} _ { k \\geq k _ 0 } \\cap \\big ( \\{ \\lambda _ 0 \\} \\cup \\{ \\widehat \\lambda _ n \\} _ { n = 1 } ^ { n _ 0 } \\big ) = \\emptyset . \\end{align*}"}
{"id": "2352.png", "formula": "\\begin{align*} \\dfrac { d } { d \\omega } M _ \\omega f \\Big | _ { \\omega = 0 } = 2 \\pi i X f \\dfrac { d } { d x } T _ x f \\Big | _ { x = 0 } = - 2 \\pi i P f . \\end{align*}"}
{"id": "3901.png", "formula": "\\begin{align*} E _ 1 ( f , p ) = \\sum _ { z \\leq p } \\frac { 1 } { p } \\sum _ { \\substack { 0 < b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { m } - v ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 \\leq b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } = 0 . \\end{align*}"}
{"id": "5631.png", "formula": "\\begin{align*} [ H , E ^ \\pm ] = \\pm 2 E ^ \\pm , ~ ~ ~ ~ ~ [ E ^ + , E ^ - ] = - H . \\end{align*}"}
{"id": "5705.png", "formula": "\\begin{align*} w ( \\alpha ) - ( w ( \\beta ) + w i n d ( \\eta _ { \\kappa _ { + } } ) - 1 ) = \\# ( \\gamma _ { 0 } \\cap { \\pi ( v \\cap { [ - s _ { * } , s _ { * } ] \\times S ^ { 3 } } ) } ) = \\# ( \\mathbb { R } \\times \\gamma _ { 0 } \\cap v ) . \\end{align*}"}
{"id": "6131.png", "formula": "\\begin{align*} \\varphi ^ { 1 } _ { s } & = 2 j ( 2 j - 1 ) \\cdot s ^ { 2 j - 2 } w ^ { 1 } , \\\\ \\psi ^ { 1 } _ { t } & = 2 k ( 2 k - 1 ) \\cdot t ^ { 2 k - 2 } w ^ { 1 } . \\end{align*}"}
{"id": "6963.png", "formula": "\\begin{align*} \\widehat { D } ^ * \\Phi ( x , y ) ( y ^ * ) \\subset \\begin{cases} \\widehat { D } ^ * g ( x ) ( y ^ * ) & y ^ * \\in \\widehat { \\mathcal N } _ D ( g ( x ) - y ) , \\\\ \\varnothing & \\end{cases} \\end{align*}"}
{"id": "5713.png", "formula": "\\begin{align*} 1 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 0 } \\cup { \\gamma _ { 2 } } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) - 1 0 . \\end{align*}"}
{"id": "6983.png", "formula": "\\begin{align*} \\left . \\begin{aligned} & \\nabla g ( \\bar x ) ^ * y ^ * = 0 , \\ , \\nabla g ( \\bar x ) ^ * \\hat z ^ * = 0 , \\\\ & y ^ * \\in \\mathcal N _ D ( g ( \\bar x ) ; \\nabla g ( \\bar x ) u ) \\end{aligned} \\right \\} \\quad \\Longrightarrow \\hat z ^ * \\notin D _ \\textup { s u b } \\mathcal N _ D ( g ( \\bar x ) , y ^ * ) \\left ( \\frac { \\nabla g ( \\bar x ) u } { \\norm { \\nabla g ( \\bar x ) u } } \\right ) \\end{align*}"}
{"id": "3864.png", "formula": "\\begin{align*} \\tilde \\sigma _ 0 ( x ) = \\tilde h - h - \\int _ 0 ^ x \\sigma _ 0 ( t ) \\ , d t . \\end{align*}"}
{"id": "7586.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k - n , l - m ) } | \\leq \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 } \\begin{cases} \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 0 , 0 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } n = 2 s , m = 2 r , \\\\ \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 1 , - 1 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } n = 2 s , m = 2 r + 1 , \\\\ \\Gamma _ { 1 , - 1 } [ s ] ( i ) \\Gamma _ { 0 , 0 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } n = 2 s + 1 , m = 2 r , \\\\ \\Gamma _ { 1 , - 1 } [ s ] ( i ) \\Gamma _ { 1 , - 1 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } n = 2 s + 1 , m = 2 r + 1 , \\end{cases} \\end{align*}"}
{"id": "4963.png", "formula": "\\begin{align*} \\ < P s i > _ n ( t , x ) = - i \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ d } a _ { n } ( t - t ' , x - x ' ) \\dot { W } ( t ' , x ' ) d t ' d x ' = - i \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ d } a _ { n } ( t - t ' , x - x ' ) W ( d t ' , d x ' ) , \\end{align*}"}
{"id": "6753.png", "formula": "\\begin{align*} D ^ 2 f _ { - 1 } & = \\frac { 1 } { \\bar { h } _ { - 1 } } \\left [ \\frac { 1 } { h _ { - 1 } } u _ { - 2 } - \\left ( \\frac { 1 } { h _ { - 1 } } + \\frac { 1 } { h _ { 0 } } \\right ) u _ { - 1 } + \\frac { 1 } { h _ { 0 } } v _ 0 \\right ] , \\end{align*}"}
{"id": "9312.png", "formula": "\\begin{align*} \\sum _ { n = K ( N ) } 2 ^ N | f _ n | ^ 2 \\leq \\frac { 1 } { 2 ^ { N - 1 } } \\| f \\| ^ 2 _ \\omega . \\end{align*}"}
{"id": "1505.png", "formula": "\\begin{align*} P _ n : = P ^ 0 _ n = \\left \\{ \\left [ \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ] \\in G _ { n } : c = 0 \\right \\} , \\end{align*}"}
{"id": "7584.png", "formula": "\\begin{align*} c _ { i , j } ^ { ( r , s ) } & = \\dfrac { 1 } { 2 i } \\left ( c _ { i - 1 , j } ^ { ( r + 1 , s ) } - c _ { i + 1 , j } ^ { ( r + 1 , s ) } \\right ) \\end{align*}"}
{"id": "3842.png", "formula": "\\begin{align*} y ^ { ( m ) } = y ^ { [ m ] } + \\sum _ { j = 1 } ^ m f _ { m , j } y ^ { ( j - 1 ) } \\in L _ { 2 , l o c } ( \\mathbb R _ + ) , \\end{align*}"}
{"id": "3494.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) = \\sum _ { m \\leq a t _ 3 } \\sum _ { n < m } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + \\begin{cases} O ( t _ 3 ^ { \\frac { 1 } { 2 } - \\sigma _ 1 - \\sigma _ 3 } ) & ( \\sigma _ 2 > \\frac { 3 } { 2 } ) \\\\ O ( t _ 3 ^ { \\frac { 1 } { 2 } - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 ) & ( \\sigma _ 2 = \\frac { 3 } { 2 } ) \\\\ O ( t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) & ( \\sigma _ 2 < \\frac { 3 } { 2 } ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "2855.png", "formula": "\\begin{align*} \\int _ \\sigma ^ \\tau \\delta ( t ) d t \\lesssim R \\left ( \\delta ( \\sigma ) + \\delta ( \\tau ) \\right ) = R _ 2 \\left ( 1 + \\sup _ { \\sigma \\le t \\le \\tau } | x ( t ) | \\right ) \\left ( \\delta ( \\sigma ) + \\delta ( \\tau ) \\right ) . \\end{align*}"}
{"id": "3063.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 3 + x _ 2 ^ 3 y _ 2 ^ 3 + a ( x _ 1 ^ 3 y _ 2 ^ 3 + x _ 2 ^ 3 y _ 1 ^ 3 ) + b x _ 1 ^ 2 y _ 1 ^ 2 x _ 2 y _ 2 + c x _ 1 y _ 1 x _ 2 ^ 2 y _ 2 ^ 2 = 0 \\ , , \\end{align*}"}
{"id": "7782.png", "formula": "\\begin{align*} \\int _ { \\Omega } g _ 1 ( w ) d \\mu = \\sum \\limits _ { k \\in T } \\int _ { [ k ] } g _ 1 ( w ) d \\mu = \\sum \\limits _ { k \\in T } \\mu ( [ k ] ) \\log p _ { k } = \\sum \\limits _ { k \\in T } p _ k \\log p _ k . \\end{align*}"}
{"id": "498.png", "formula": "\\begin{align*} \\forall j _ { 1 } , j _ { 2 } \\in \\mathcal { L } ^ { ( r - 1 ) } , \\beta _ { 1 } , \\beta _ { 2 } \\in ( \\mathcal { L } ^ { ( r - 1 ) } ) ^ { \\mathtt { C } } : \\ , h ( ( \\mathcal { L } ^ { ( r - 1 ) } ) _ { \\gamma _ { r } \\omega } ^ { j _ { 1 } j _ { 2 } } ) = h ( ( \\mathcal { L } ^ { ( r - 1 ) } ) _ { \\beta _ { 1 } \\beta _ { 2 } } ^ { l _ { r } m } ) = 0 \\end{align*}"}
{"id": "5257.png", "formula": "\\begin{align*} a _ { ( 1 ) } S ( a _ { ( 2 ) } ) a _ { ( 3 ) } = a , a _ { ( 3 ) } S ^ { - 1 } ( a _ { ( 2 ) } ) a _ { ( 1 ) } = a , \\forall a \\in A . \\end{align*}"}
{"id": "2575.png", "formula": "\\begin{align*} \\langle \\ell _ { } ' , f \\rangle = - \\langle \\ell _ { } , f ' \\rangle = \\int _ { - a } ^ 0 \\overline { f ' ( x ) } \\ , d x - \\int _ 0 ^ a \\overline { f ' ( x ) } \\ , d x = 2 \\overline { f ( 0 ) } . \\end{align*}"}
{"id": "3726.png", "formula": "\\begin{align*} \\begin{cases} B _ x ( x , t ) \\geq 0 \\ \\ \\mbox { f o r } \\ x \\in ( - X ( x _ 0 , t ) , X ( x _ 0 , t ) ) ; \\\\ B _ x ( x , t ) \\leq 0 \\ \\ \\mbox { f o r } \\ x \\in ( - \\infty , - X ( x _ 0 , t ) ) \\cap ( X ( x _ 0 , t ) , \\infty ) . \\end{cases} \\end{align*}"}
{"id": "6899.png", "formula": "\\begin{align*} \\bar { \\Phi } ( \\bar { X } _ W ) = \\frac { q ^ { d - i } - q ^ { k - 1 } } { q ^ d - q ^ { k - 1 } } = q ^ { - i } \\pm q ^ { - \\Omega ( d ) } \\end{align*}"}
{"id": "2724.png", "formula": "\\begin{align*} U T = V S \\in R [ X _ 1 , \\dots , X _ { n - 2 } ] . \\end{align*}"}
{"id": "1596.png", "formula": "\\begin{align*} R ^ { 1 , k } e _ { k } & = g _ { 2 k - 1 } z _ { 2 k - 1 } + g _ { 2 k } z _ { 2 k } , \\\\ R ^ { 2 , k } e _ k & = ( - 1 ) ^ { 2 k - 1 + 1 } g _ { 2 k - 1 } z _ { 2 k - 1 } + ( - 1 ) ^ { 2 k + 1 } g _ { 2 k } z _ { 2 k } , \\end{align*}"}
{"id": "1279.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { M ^ { 2 } } { | \\Lambda ( M , \\Gamma ) | } = 2 \\mathrm { V o l } ( Y , \\lambda ) . \\end{align*}"}
{"id": "8648.png", "formula": "\\begin{align*} n a + ( n - 1 ) b = c \\end{align*}"}
{"id": "4928.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ d } f ( x ) : = \\mathbb E _ { z \\sim \\mathcal D } \\left [ f ( x ; z ) \\right ] , \\end{align*}"}
{"id": "5855.png", "formula": "\\begin{align*} \\theta = \\sum _ { i \\in J } \\left ( \\frac { \\bar \\alpha _ { i } } { \\rho _ { i } } - 1 \\right ) > 0 . \\end{align*}"}
{"id": "211.png", "formula": "\\begin{align*} A _ \\alpha ( p _ \\alpha f ) ( x ) & = \\sum _ { k = 1 } ^ d \\left ( R _ k ^ { \\alpha } ( \\partial _ k ( p _ \\alpha ) , f ) ( x ) + R _ k ^ { \\alpha } ( \\partial _ k ( f ) , p _ \\alpha ) ( x ) \\right ) \\\\ & \\quad + \\sum _ { k = 1 } ^ d \\left ( \\partial _ k ( p _ \\alpha ) ( x ) D _ k ^ { \\alpha - 1 } ( f ) ( x ) + p _ \\alpha ( x ) \\partial _ k D ^ { \\alpha - 1 } _ k ( f ) ( x ) \\right ) . \\end{align*}"}
{"id": "5410.png", "formula": "\\begin{align*} f ( n , k , \\ell ) = \\left ( \\sum _ { t \\leq j \\leq \\ell } \\binom { \\ell } { j } \\right ) \\binom { \\ell } { t } \\ell k ^ { \\ell - t - 1 } \\sum _ { 0 \\leq i \\leq k - \\ell } \\binom { n - t } { i } \\end{align*}"}
{"id": "5622.png", "formula": "\\begin{align*} \\mathcal { Q } ^ \\bullet = \\bigoplus _ { p = 0 } ^ \\infty \\biguplus _ { q = 0 } ^ \\infty \\biguplus _ { r = 0 } ^ \\infty \\mathcal { Q } ^ { p q r } . \\end{align*}"}
{"id": "2974.png", "formula": "\\begin{align*} \\Psi _ { \\mathbf i } = \\varphi _ 2 \\left ( \\mathbf { i } _ { 1 : 4 } \\right ) \\cdot \\varphi _ 2 \\left ( \\mathbf { i } _ { 5 : 8 } \\right ) . \\end{align*}"}
{"id": "7285.png", "formula": "\\begin{align*} \\begin{gathered} ( 1 + X ^ 1 + X ^ { 1 + 1 } + \\ldots ) ( 1 + X ^ 2 + X ^ { 2 + 2 } + \\ldots ) \\ldots ( 1 + X ^ k + X ^ { k + k } + \\ldots ) \\\\ = \\frac { 1 } { 1 - X } \\frac { 1 } { 1 - X ^ 2 } \\ldots \\frac { 1 } { 1 - X ^ k } = \\frac { 1 } { X ^ k ! } . \\end{gathered} \\end{align*}"}
{"id": "1209.png", "formula": "\\begin{align*} \\mbox { f o r e v e r y $ U \\in \\mathcal { W } _ 1 $ , } \\ \\ \\ \\ \\eta ( U ) = \\frac { \\mu ( B ^ { [ U ] } ) } { \\sum _ { \\widetilde { U } \\in \\mathcal { W } _ 1 } \\mu ( B ^ { [ \\widetilde { U } ] } ) } . \\end{align*}"}
{"id": "3813.png", "formula": "\\begin{align*} \\psi \\colon \\thinspace \\begin{cases} \\widehat { \\delta } \\longmapsto \\alpha \\\\ a _ i \\longmapsto g ^ { b _ i } , \\ ; \\end{cases} \\end{align*}"}
{"id": "8690.png", "formula": "\\begin{align*} \\begin{aligned} & \\phi ( a _ { 1 p ^ t _ 1 } , \\ldots , a _ { \\omega ^ t - 1 p ^ t _ { \\omega ^ t - 1 } } , u _ { \\omega ^ t p ^ t _ { \\omega ^ t } } , a _ { \\omega ^ t + 1 p ^ t _ { \\omega ^ t + 1 } } , \\ldots , a _ { d p ^ t _ d } ) \\\\ & - \\phi ( a _ { 1 p ^ t _ 1 } , \\ldots , a _ { \\omega ^ t - 1 p ^ t _ { \\omega ^ t - 1 } } , u _ { \\omega ^ t p ^ t _ { \\omega ^ { t } } - 1 } , a _ { \\omega ^ t + 1 p ^ t _ { \\omega ^ t + 1 } } , \\ldots , a _ { d p ^ t _ d } ) , \\end{aligned} \\end{align*}"}
{"id": "2195.png", "formula": "\\begin{align*} m _ 1 = \\inf \\limits _ { \\mathcal { M } } J = J ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ + + s _ { \\widehat { w } ^ - } \\widehat { w } ^ - ) . \\end{align*}"}
{"id": "8916.png", "formula": "\\begin{align*} C _ h ( X ) | _ A = C _ h = C _ h ( A ) \\end{align*}"}
{"id": "4038.png", "formula": "\\begin{align*} F ( \\mu _ k ) & = 2 \\sin ( k \\pi + c _ k + i d _ k ) + O ( k ^ { - 1 } ) \\\\ & = 2 ( - 1 ) ^ k \\sin ( c _ k + i d _ k ) + O ( k ^ { - 1 } ) . \\end{align*}"}
{"id": "7366.png", "formula": "\\begin{align*} \\gamma ^ \\ast ( t ) = \\frac { R ^ 2 } { \\pi R ^ 2 t - r + 2 R } . \\end{align*}"}
{"id": "85.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 \\cdot \\frac { ( 2 ^ { ( n _ { 2 , \\nu _ 2 } + 1 ) / 2 } + 1 ) ( 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 ) } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "100.png", "formula": "\\begin{align*} \\mathcal { R } [ L ' ] ( F , \\diamond ) _ { \\ast } = \\begin{cases} \\mathcal { R } _ { L ' } ( F , 2 ) _ { I } & ( F \\neq \\Q ( \\sqrt { - 1 } ) , \\Q ( \\sqrt { - 3 } ) ) , \\\\ \\mathcal { R } _ { L ' } ( \\Q ( \\sqrt { - 1 } ) , 4 ) & ( F = \\Q ( \\sqrt { - 1 } ) ) , \\\\ \\mathcal { R } _ { L ' } ( \\Q ( \\sqrt { - 3 } ) , 6 ) & ( F = \\Q ( \\sqrt { - 3 } ) ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "3613.png", "formula": "\\begin{align*} d [ c a _ { 1 , i } b _ { 1 , j } ] = \\min \\{ d ( c a _ { 1 , i } b _ { 1 , j } ) , \\alpha _ { i } , \\beta _ { j } \\} \\quad d ^ { \\prime } [ c a _ { 1 , i } b _ { 1 , j } ] = \\min \\{ d ( c a _ { 1 , i } b _ { 1 , j } ) , \\alpha _ { i } , \\beta _ { j } ^ { \\prime } \\} \\ , , \\end{align*}"}
{"id": "3392.png", "formula": "\\begin{align*} f = \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\phi _ { k } * \\psi _ { k } * f , \\end{align*}"}
{"id": "9203.png", "formula": "\\begin{align*} ( r ) _ \\circ ( n ) : = j ( 2 k _ 0 , 2 ^ { n + 1 } - 1 ) , \\end{align*}"}
{"id": "1073.png", "formula": "\\begin{align*} X _ { R , L } ( k ) = \\sqrt { k ^ 2 - C _ { R , L } ^ 2 } , \\Omega ( k ) = 2 ( 2 k ^ 2 + C _ { R , L } ^ 2 ) X _ { R , L } ( k ) . \\end{align*}"}
{"id": "1168.png", "formula": "\\begin{align*} \\mathcal { \\psi } = \\zeta ^ { i \\nu \\sigma _ 3 } e ^ { - \\frac { i } { 4 } \\zeta ^ 2 \\sigma _ 3 } \\left ( I + m ^ { P C } _ { \\eta , 1 } \\zeta ^ { - 1 } + \\mathcal { O } ( \\zeta ^ { - 2 } ) \\right ) , { \\rm a s } \\zeta \\rightarrow \\infty . \\end{align*}"}
{"id": "1428.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { s + m + 1 } a _ i e _ i \\mapsto h _ 1 | _ { \\bar { \\sigma } } \\left ( \\sum _ { i = 0 } ^ m a _ i e _ i + a g \\left ( \\frac { 1 } { a } \\sum _ { i = 0 } ^ { s } a _ { i + m + 1 } e _ i \\right ) \\right ) . \\end{align*}"}
{"id": "3027.png", "formula": "\\begin{align*} d _ { i } ^ { \\prime } = \\frac { \\partial } { \\partial x ^ { i } } + \\sum _ { k = 0 } ^ { r - 1 } \\sum _ { j _ { 1 } \\leq \\ldots \\leq j _ { k } } \\frac { \\partial } { \\partial y _ { j _ { 1 } \\ldots j _ { k } } ^ { \\sigma } } y _ { j _ { 1 } \\ldots j _ { k } i } ^ { \\sigma } . \\end{align*}"}
{"id": "1416.png", "formula": "\\begin{align*} { { d \\hat z _ i } \\over d t } = \\hat u _ i , \\end{align*}"}
{"id": "5370.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , n - 1 : \\left \\langle c , c ^ { ( i ) } \\right \\rangle = 0 . \\end{align*}"}
{"id": "1892.png", "formula": "\\begin{align*} { } ^ { w _ { s _ o } g _ { s _ o } } C _ { s _ o } = { } ^ { q _ { s _ o } g _ { s _ o } } C _ { s _ o } , { } ^ { w g } C = { } ^ { ( e , \\dots , e , q _ { s _ o } , e , \\dots ) } ( { } ^ g C ) . \\end{align*}"}
{"id": "5439.png", "formula": "\\begin{align*} a _ { \\inf } > \\begin{cases} \\frac { \\mu \\chi ^ 2 } { 4 } , & \\\\ \\mu ( \\chi - 1 ) , & \\\\ \\end{cases} \\end{align*}"}
{"id": "8687.png", "formula": "\\begin{align*} C ^ { o } ( \\kappa , { \\bf P } _ { Y _ 1 } ) = C ^ { \\infty } ( \\kappa , { \\bf P } _ { Y _ 1 } ) = C ^ { \\infty } ( \\kappa ) = \\mbox { ( \\ref { l l _ 3 } ) } , \\ : \\forall { \\bf P } _ { Y _ 1 } \\end{align*}"}
{"id": "2738.png", "formula": "\\begin{align*} m _ { 0 , \\lambda } ( \\Z ( \\{ P _ 1 , P _ 2 , P _ 3 \\} , B ) ) = 1 > 0 , \\end{align*}"}
{"id": "6177.png", "formula": "\\begin{align*} U = [ U _ 1 , u _ { n + 1 } ] , \\ \\ V = \\left [ \\begin{array} { c c } V _ { 1 1 } & v _ { 1 2 } \\\\ v _ { 2 1 } & v _ { 2 2 } \\end{array} \\right ] , \\ \\ \\Sigma = \\left [ \\begin{array} { c c } \\Sigma _ { 1 } & 0 \\\\ 0 & \\sigma _ { n + 1 } \\end{array} \\right ] , \\end{align*}"}
{"id": "8011.png", "formula": "\\begin{align*} \\chi _ { \\ell / r } = \\widetilde { \\pi } _ { \\ell / r } \\circ \\chi \\circ \\pi _ { \\ell / r } | _ { \\Sigma } ^ { - 1 } . \\end{align*}"}
{"id": "2130.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\binom { c _ n k n } l } { \\prod _ { i = 1 } ^ l ( k n - c _ n k n + i ) } = \\frac 1 { l ! } ( \\frac c { 1 - c } ) ^ l . \\end{align*}"}
{"id": "6252.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } ( e ^ { \\lambda ( q , t ) } - 1 ) - 1 = \\mathcal { E } ( q , \\lambda ( q , t ) ) . \\end{align*}"}
{"id": "1344.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\omega \\chi ^ k ( x , \\omega ) = \\mathcal { L } ^ \\omega \\Pi ^ k ( x ) , \\ ; ( k = 1 , \\dots , d ) \\end{align*}"}
{"id": "7515.png", "formula": "\\begin{align*} \\arg \\zeta ( \\sigma ) = \\pi \\end{align*}"}
{"id": "8223.png", "formula": "\\begin{align*} [ z ^ n ] ( P _ 1 + P _ 2 + \\cdots + P _ k ) & = [ z ^ n ] \\Big ( 1 - \\frac { z F _ { k , 1 } ( A ) } { A } \\Big ) = - [ z ^ { n - 1 } ] A ^ { - 1 } \\\\ & = \\frac { 1 } { n - 1 } [ t ^ { n - 2 } ] t ^ { - 2 } ( 1 - t ) ^ { - k ( n - 1 ) } = \\frac { 1 } { n - 1 } [ t ^ { n } ] ( 1 - t ) ^ { - k ( n - 1 ) } \\\\ & = \\frac { 1 } { n - 1 } \\binom { ( k + 1 ) ( n - 1 ) } { n } = \\frac { k } { n } \\binom { ( k + 1 ) ( n - 1 ) } { n - 1 } . \\end{align*}"}
{"id": "7797.png", "formula": "\\begin{align*} X _ 1 ( f ) = X _ 2 ( f ) = X _ 3 ( f ) \\quad . \\end{align*}"}
{"id": "1157.png", "formula": "\\begin{align*} G : = \\left \\{ \\begin{aligned} & \\begin{pmatrix} 1 & 0 \\\\ - r e ^ { 2 i t g } & 1 \\end{pmatrix} , & k \\in U _ 1 \\cup U _ 2 , \\\\ & \\begin{pmatrix} 1 & - r ^ { * } e ^ { - 2 i t g } \\\\ 0 & 1 \\end{pmatrix} , & k \\in U _ 1 ^ { * } \\cup U _ 2 ^ { * } , \\\\ & I , & \\textnormal { e l s e w h e r e } \\end{aligned} \\right . \\end{align*}"}
{"id": "2058.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow + \\infty } \\| f \\| ^ 2 _ { p , \\lambda , t } = \\int _ { \\{ p \\Psi _ 1 < - t \\} } | f | ^ 2 , \\ \\forall t \\in [ 0 , + \\infty ) . \\end{align*}"}
{"id": "6910.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { \\boldsymbol { \\Theta } , \\ : \\mathbf { w } } & \\biggl | \\Bigl ( \\mathbf { g } ^ T \\boldsymbol { \\Theta } \\mathbf { H } + \\mathbf { h } _ d ^ T \\Bigr ) \\mathbf { w } \\biggr | ^ 2 \\\\ \\textrm { s . t . } & \\| \\mathbf { w } \\| ^ 2 \\leqslant 1 \\\\ & \\phi _ n \\in [ 0 , 2 \\pi ) , \\ : \\forall n = 1 , 2 , \\ldots , N . \\end{aligned} \\end{align*}"}
{"id": "5241.png", "formula": "\\begin{align*} A \\otimes ^ I B = E ( A \\otimes B ) E \\end{align*}"}
{"id": "3555.png", "formula": "\\begin{align*} \\Phi _ m [ C ] = \\Theta ( C ) . \\end{align*}"}
{"id": "7080.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 } C ( X _ i , X _ i ) X = \\sum _ { i = 1 } ^ { 2 } ( \\nabla _ { X } ( ( \\nabla _ { X _ i } A ) X _ i ) + [ R ( X _ i , X ) , A ] X _ i - T _ i X ) . \\end{align*}"}
{"id": "6994.png", "formula": "\\begin{align*} f _ { R | C } = \\int f _ { R | C B } f _ { B | C } d b , \\end{align*}"}
{"id": "4623.png", "formula": "\\begin{gather*} \\gamma _ X \\alpha _ X = \\alpha _ X ( S ^ 2 \\otimes \\gamma _ X ) , \\\\ \\beta _ X \\gamma _ X = ( S ^ 2 \\otimes \\gamma _ X ) \\beta _ X , \\\\ c _ X ^ R = c _ X ^ L , \\end{gather*}"}
{"id": "8000.png", "formula": "\\begin{align*} t ( \\mathbf { P } _ { \\ ! \\sigma } ) = t ( \\mathbf { p } _ { \\sigma _ d } ; \\ldots ; \\mathbf { p } _ { \\sigma _ 1 } ) = \\sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\bigg ( H ( \\mathbf { p } _ { \\sigma _ { n } } ) + \\int \\ ! \\varphi _ n ^ { \\sigma } \\ , \\mathrm { d } \\mathbf { p } _ { \\sigma _ n } \\bigg ) , \\end{align*}"}
{"id": "1449.png", "formula": "\\begin{align*} \\Gamma _ 1 ^ 0 = \\Gamma _ 1 ( \\mathcal { N } ) = \\left \\{ \\gamma = \\left [ \\begin{array} { c c c } a & b & c \\\\ g & e & f \\\\ h & l & d \\end{array} \\right ] \\in G ( \\mathcal { O } ) : \\gamma \\equiv \\left [ \\begin{array} { c c c } 1 _ m & \\ast & \\ast \\\\ 0 & 1 _ r & \\ast \\\\ 0 & 0 & 1 _ m \\end{array} \\right ] \\mathcal { N } \\right \\} . \\end{align*}"}
{"id": "9098.png", "formula": "\\begin{align*} 1 + u ^ m \\alpha _ m t + \\cdots + u ^ { n - 1 } \\alpha _ { n - 1 } t ^ { n - 1 } = 1 + u ^ { m - 1 } \\beta _ m t ^ m + \\cdots + u ^ { n - 2 } \\beta _ { n - 1 } t ^ { n - 1 } \\end{align*}"}
{"id": "5788.png", "formula": "\\begin{align*} \\| Y \\| ^ p _ { W ^ { k , p } \\delta } = \\sum _ { \\ell = 0 } ^ { k } \\int _ { \\Sigma _ \\pm } \\vert \\nabla ^ \\ell Y \\vert ^ p _ { \\Sigma _ i } . \\end{align*}"}
{"id": "5819.png", "formula": "\\begin{align*} \\gamma = \\min \\{ \\gamma _ \\circ , \\gamma _ \\star \\} , \\kappa = \\tfrac 1 9 \\min \\{ \\kappa _ \\circ , \\kappa _ \\star \\} . \\end{align*}"}
{"id": "5199.png", "formula": "\\begin{align*} \\Gamma ^ { k } = \\{ \\chi \\in \\Gamma : \\forall x \\in \\mathfrak { P } ^ { k } \\implies \\chi ( x ) = 1 \\} . \\end{align*}"}
{"id": "6167.png", "formula": "\\begin{align*} b _ 1 & = - 1 - z _ 1 - z _ 2 - z _ 3 , \\\\ b _ 2 & = \\frac { 3 } { 5 } + ( z _ 1 + z _ 2 + z _ 3 ) + ( z _ 1 z _ 2 + z _ 2 z _ 3 + z _ 3 z _ 1 ) , \\\\ \\left ( z _ 1 + z _ 2 + \\frac { 2 } { 3 } + z _ 1 z _ 2 \\right ) z _ 3 & = - \\frac { 1 } { 3 } - \\frac { 2 } { 3 } ( z _ 1 + z _ 2 ) - z _ 1 z _ 2 . \\end{align*}"}
{"id": "6516.png", "formula": "\\begin{align*} - 2 ( 1 - \\alpha ) + m ( 1 - 2 \\alpha ) \\geq - 2 ( 1 - \\alpha ) + 2 ( 1 - 2 \\alpha ) = - 2 \\alpha > - 1 . \\end{align*}"}
{"id": "2486.png", "formula": "\\begin{align*} \\mathbf { 1 } _ r = ( 0 , 0 , 1 ) \\mathbf { h } _ r ^ { - 1 } = ( - x , - \\omega , e ^ { - 2 \\pi i \\tau } ) , \\end{align*}"}
{"id": "9202.png", "formula": "\\begin{align*} j ( n ^ 0 , m ^ 0 ) : = \\begin{cases} \\min u \\leq _ 0 ( n + m ) ^ 2 + 3 n + m [ 2 u = _ 0 ( n + m ) ^ 2 + 3 n + m ] & , \\\\ 0 ^ 0 & . \\end{cases} \\end{align*}"}
{"id": "4170.png", "formula": "\\begin{align*} \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} \\right ) \\geq \\frac { 1 } { 2 ^ { 1 - \\frac { 1 } { p } } } \\omega _ { p } ( A + B ) , \\end{align*}"}
{"id": "5524.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\| \\sigma _ r ( h ) - \\sigma ( h ) \\| _ { L _ 2 ^ 0 ( H ) } = 0 \\end{align*}"}
{"id": "2200.png", "formula": "\\begin{align*} m _ 0 = \\inf \\limits _ { \\mathcal { N } } J \\geq \\inf \\limits _ { \\mathcal { M } } J = m _ 1 . \\end{align*}"}
{"id": "152.png", "formula": "\\begin{align*} \\left ( \\lambda E - \\mathcal { A } \\right ) ^ { - \\frac { r } { 2 } } f = \\frac { 1 } { \\Gamma ( \\frac { r } { 2 } ) } \\int _ 0 ^ { + \\infty } \\dfrac { e ^ { - \\lambda t } } { t ^ { 1 - \\frac { r } { 2 } } } P _ t ( f ) d t , \\end{align*}"}
{"id": "7227.png", "formula": "\\begin{align*} F ' _ { s , t , x } : = \\{ v \\in \\R ^ 3 : s < { \\mathcal T } _ { t , x _ 1 , v _ 1 } - 2 , | v | < \\delta ^ { - \\beta } / 2 , | v ^ \\perp - v ^ \\perp _ \\ast | \\check \\tau _ { t , x } > 2 \\sqrt { \\mathcal T _ { t , x _ 1 , v _ 1 } - s } \\} , \\\\ F '' _ { s , t , x } : = \\{ v \\in \\R ^ 3 : s < { \\mathcal T } _ { t , x _ 1 , v _ 1 } - 3 , | v | < \\delta ^ { - \\beta } / 3 , | v ^ \\perp - v ^ \\perp _ \\ast | \\check \\tau _ { t , x } > 3 \\sqrt { \\mathcal T _ { t , x _ 1 , v _ 1 } - s } \\} . \\end{align*}"}
{"id": "9106.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 , \\lambda \\in \\mathcal { P } ( n ) } t ^ n \\ = \\prod _ { n \\geq 1 } \\frac { 1 } { ( 1 - t ^ n ) } \\end{align*}"}
{"id": "5283.png", "formula": "\\begin{align*} \\check { S } ( \\varphi ( - a ) ) = \\varphi ( - \\sigma ^ { \\varphi } ( \\delta _ { \\varphi } S ( a ) ) ) . \\end{align*}"}
{"id": "5574.png", "formula": "\\begin{align*} \\| 1 _ { \\varepsilon A } + x \\| \\ = \\ \\| y - P _ { B \\cup D } ( y ) \\| & \\ \\leqslant \\ \\mathbf C _ \\ell \\| y - P _ { \\Lambda } ( y ) \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , p } \\widehat { \\sigma } _ m ( y ) \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , p } \\| x + 1 _ { \\delta B } \\| . \\end{align*}"}
{"id": "6748.png", "formula": "\\begin{align*} \\mathbf A \\tilde { \\mathbf V } ^ { n + 4 } = 4 \\tilde { \\mathbf V } ^ { n + 3 } - 3 \\tilde { \\mathbf V } ^ { n + 2 } + \\frac { 4 } { 3 } \\tilde { \\mathbf V } ^ { n + 1 } - \\frac { 1 } { 4 } \\tilde { \\mathbf V } ^ { n } + k \\mathbf b ^ { n + 4 } + k \\mathbf q ( \\tilde { \\mathbf V } ^ { n + 4 } ) , \\end{align*}"}
{"id": "5817.png", "formula": "\\begin{align*} \\gamma _ { k } ^ { \\langle n \\rangle } = \\frac { \\left ( \\sum _ { m \\in \\mathbb { M } _ k } \\sqrt { \\eta _ { m k } } \\left | \\hat { g } _ { m k , n } \\right | ^ 2 \\right ) ^ 2 } { \\sum _ { m \\in \\mathbb { M } _ k } \\eta _ { m k } ( \\beta _ { m k } - \\alpha _ { m k } ) \\alpha _ { m k } + \\frac { 1 } { \\gamma _ t } } . \\end{align*}"}
{"id": "7802.png", "formula": "\\begin{align*} t : = \\log ( 1 + \\pi ) = \\sum _ { m = 1 } ^ { \\infty } ( - 1 ) ^ { m + 1 } \\frac { \\pi ^ m } { m } . \\end{align*}"}
{"id": "1576.png", "formula": "\\begin{align*} \\frac { c _ k ( \\mu ) L ( \\mu , \\mathbf { f } , \\chi ) } { \\pi ^ { \\beta } } \\mathbf { f } ( g _ { \\mathbf { h } } \\cdot g _ { \\infty } ) = \\mathfrak { P } _ k ( w ) \\langle \\mathbf { h } '' , \\mathbf { f } \\rangle . \\end{align*}"}
{"id": "4908.png", "formula": "\\begin{align*} & \\leq f ( Z _ T ) + G _ 0 G ^ 2 \\eta _ 0 ^ 2 \\\\ & \\leq f ( G _ 0 ^ { - 2 } \\sigma ^ 2 \\log ( 1 / \\delta ) ) + G _ 0 G ^ 2 \\eta _ 0 ^ 2 \\\\ & = G _ 0 ^ { - 1 } \\sigma ^ 2 \\log ( 1 / \\delta ) + G _ 0 G ^ 2 \\eta _ 0 ^ 2 \\\\ \\end{align*}"}
{"id": "7439.png", "formula": "\\begin{align*} 0 = \\int _ { \\mathbb { R } } \\rho _ 3 ( t , u ) G ( t , u ) d u - \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\rho _ { 3 } ( s , u ) \\partial _ s G ( s , u ) d u d s - \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\rho _ { 5 } ( s , u ) [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( s , u ) d u d s . \\end{align*}"}
{"id": "6025.png", "formula": "\\begin{align*} 0 & = \\widetilde { H } ' _ { n + 1 } + 2 \\widetilde { H } _ n + 2 x \\widetilde { H } ' _ n + \\widetilde { H } '' _ n \\\\ & = - 2 ( n + 1 ) \\widetilde { H } _ n + 2 \\widetilde { H } _ n + 2 x \\widetilde { H } _ n ' + \\widetilde { H } '' _ n \\\\ & = \\widetilde { H } '' _ n + 2 x \\widetilde { H } ' _ n - 2 n \\widetilde { H } _ n . \\end{align*}"}
{"id": "6758.png", "formula": "\\begin{align*} \\begin{aligned} u _ { - 1 } - v _ { - 1 } = \\frac { ( h _ 0 - \\delta ) ^ 2 } { 2 } ( u '' _ { \\delta } - v '' _ { \\delta } ) & - \\frac { ( h _ 0 - \\delta ) ^ 3 } { 6 } ( u ''' _ { \\delta } - v ''' _ { \\delta } ) \\\\ & + \\frac { ( h _ 0 - \\delta ) ^ 4 } { 2 4 } ( u '''' _ { \\delta } - v '''' _ { \\delta } ) + \\mathcal O ( ( h _ 0 - \\delta ) ^ 5 ) , \\end{aligned} \\end{align*}"}
{"id": "5147.png", "formula": "\\begin{align*} \\log ( \\kappa ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) ) = - \\log ( A ( \\tfrac { \\eta } { n } ) ) + \\log ( B ( \\tfrac { \\eta } { n } ) ) = - \\log ( A ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) ) - 2 \\log ( \\tanh ( \\tfrac { \\eta } { 2 } ) ) + \\log ( h ( \\eta ) ) , \\end{align*}"}
{"id": "3136.png", "formula": "\\begin{align*} F _ 1 & = x _ 1 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 = 0 \\ , , \\\\ F _ 2 & = x _ 2 ^ 2 + x _ 4 ^ 2 - x _ 5 ^ 2 = 0 \\ , , \\\\ F _ 3 & = x _ 3 ^ 2 + x _ 4 x _ 5 = 0 \\ , . \\end{align*}"}
{"id": "3559.png", "formula": "\\begin{align*} \\Phi _ m [ C ] = \\Theta ( C ) . \\end{align*}"}
{"id": "9336.png", "formula": "\\begin{align*} e _ { \\lambda } ^ { x } ( t ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( x ) _ { k , \\lambda } } { k ! } t ^ { k } , ( \\mathrm { s e e } \\ [ 9 , 1 0 , 1 3 ] ) , \\end{align*}"}
{"id": "995.png", "formula": "\\begin{align*} f _ \\varepsilon ( x ) & = \\int _ { \\R ^ n } \\bar f ( y ) \\eta _ \\varepsilon ( x - y ) \\dd y = \\int _ { \\R ^ n _ + } \\bar f ( y ) \\big ( \\eta _ \\varepsilon ( x - y ) - \\eta _ \\varepsilon ( x _ \\ast - y ) \\big ) \\dd y . \\end{align*}"}
{"id": "4459.png", "formula": "\\begin{align*} a | z \\rangle = z | z \\rangle = z \\sum _ { n = 0 } ^ { \\infty } A _ { n } | n \\rangle = \\sum _ { n = 1 } ^ { \\infty } z A _ { n - 1 } | n - 1 \\rangle . \\end{align*}"}
{"id": "3037.png", "formula": "\\begin{align*} 2 ( p - 1 ) = 2 n ( p ^ \\prime - 1 ) + \\sum \\frac { n } { n _ i } ( n _ i - 1 ) \\ , . \\end{align*}"}
{"id": "4305.png", "formula": "\\begin{align*} \\partial _ \\tau w = \\partial _ y ^ 2 w + \\frac { d + 1 } { y } \\partial _ y w - \\beta ( \\tau ) \\Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 , \\end{align*}"}
{"id": "5373.png", "formula": "\\begin{align*} K _ { \\mathfrak { G } } = ( 2 \\eta + 1 ) ^ { n - 1 } \\end{align*}"}
{"id": "3623.png", "formula": "\\begin{align*} \\nu _ 3 ( t ) = \\frac { 1 } { c \\log ^ { 2 / 3 } | t | ( \\log \\log | t | ) ^ { 1 / 3 } } \\end{align*}"}
{"id": "4217.png", "formula": "\\begin{align*} \\Vert \\eta \\Vert _ \\gamma = \\max _ { t \\in [ 0 , T ] } e ^ { - \\gamma t } \\ , \\vert \\eta ( t ) \\vert , \\eta \\in C ^ 0 ( [ 0 , T ] ; \\R ^ d ) , \\end{align*}"}
{"id": "9425.png", "formula": "\\begin{align*} { \\mathcal N } _ q u = M ^ { ( q ) } _ 1 ( d _ q \\oplus d _ { q - 1 } ^ * u , u ) + d _ { q - 1 } M ^ { ( q ) } _ 2 ( u , u ) \\end{align*}"}
{"id": "874.png", "formula": "\\begin{align*} \\mathbb { E } G ^ { \\rm R e a c } _ j = \\frac { \\mathbb { E } \\left ( T ^ { \\rm R e a c } _ j \\right ) ^ 2 } { 2 } + \\mathbb { E } T ^ { \\rm R e a c } _ j \\left ( \\mathbb { E } \\tau ^ { \\rm R e a c } _ { V _ { j - 1 } } - \\tau _ { \\rm f } - \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
{"id": "7511.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { T } \\left [ \\log \\left | - \\frac { 1 } { 2 } - \\epsilon + i t \\right | - \\log \\left | - \\frac { 1 } { 2 } + \\epsilon + i t \\right | \\right ] \\ d t + \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\left [ \\arg \\left ( \\sigma - 1 + i T \\right ) - \\arg ( \\sigma - 1 ) \\right ] \\ d \\sigma = 0 \\end{align*}"}
{"id": "7847.png", "formula": "\\begin{align*} \\langle y \\rangle _ { T ^ { * 3 } } & = ~ \\{ y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( j - r ) } y \\} ~ , ~ ~ ~ \\langle y \\rangle _ { T ^ { * 3 } } = j - r + 1 , \\\\ \\langle z \\rangle _ { T ^ { * 3 } } & = ~ \\{ z , T ^ { * 3 } z , \\ldots , T ^ { * 3 ( n - 2 j - 3 + r ) } z \\} ~ , ~ ~ ~ \\langle z \\rangle _ { T ^ { * 3 } } = n - 2 j - 2 + r , \\end{align*}"}
{"id": "9047.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty d ^ 2 ( \\rho ^ n , \\rho ^ { n + 1 } ) \\leq 2 \\tau ( E ( \\rho ^ 0 , \\phi ^ { 0 } ) - \\inf _ { ( \\rho , \\phi ) \\in \\mathcal { A } } E ( \\rho , \\phi ) ) . \\end{align*}"}
{"id": "2096.png", "formula": "\\begin{align*} \\mathcal { W } _ m : = \\{ { \\bf c } \\in \\Z ^ n : | c _ i | \\leq H ^ i , m ^ 2 \\mid \\mathrm { d i s c } ( f _ { { \\bf c } } ) \\} . \\end{align*}"}
{"id": "5623.png", "formula": "\\begin{align*} \\mathcal { Q } ^ \\bullet _ \\star = \\bigoplus _ { p = 0 } ^ \\infty \\biguplus _ { q = 0 } ^ \\infty \\biguplus _ { r = 0 } ^ \\infty \\mathcal { Q } ^ { p q r } _ \\star , \\end{align*}"}
{"id": "5867.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } v _ \\eta ( \\xi _ 0 ) \\stackrel { a . s . } { = } 0 . \\end{align*}"}
{"id": "6381.png", "formula": "\\begin{align*} D _ { \\underline T } = \\sum _ { i = 1 } ^ n T _ i \\otimes E _ i . \\end{align*}"}
{"id": "8226.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ 1 } [ z ^ n ] P _ 1 \\Big | _ { x _ 1 = x _ 2 = x _ 3 = 1 } & = \\frac { 1 } { n } [ t ^ { n - 1 } ] ( 1 + ( 2 n - 1 ) t ) ( 1 + t ) ^ { 4 n - 3 } \\\\ & = \\frac { 1 } { n } \\Big ( \\binom { 4 n - 3 } { n - 1 } + ( 2 n - 1 ) \\binom { 4 n - 3 } { n - 2 } \\Big ) . \\end{align*}"}
{"id": "7318.png", "formula": "\\begin{align*} \\begin{aligned} & \\Phi \\ \\ ( x _ \\varepsilon , y _ \\varepsilon , t _ \\varepsilon ) \\in \\R ^ n \\times \\R ^ n \\times ( 0 , \\infty ) \\ \\\\ & ( x _ \\varepsilon , y _ \\varepsilon , t _ \\varepsilon ) \\to ( x _ 0 , x _ 0 , t _ 0 ) , u ( x _ \\varepsilon , t _ \\varepsilon ) \\to u ( x _ 0 , t _ 0 ) \\quad \\ \\varepsilon \\to 0 . \\end{aligned} \\end{align*}"}
{"id": "6218.png", "formula": "\\begin{align*} | \\tau _ { i } ( \\mathbf { V } ^ { T } _ { 1 1 } ) - \\hat { \\sigma } _ { i } ( \\hat { \\mathbf { V } } ^ { T } _ { 1 1 } ) | \\leq \\| \\mathbf { V } ^ { T } _ { 1 1 } - \\hat { \\mathbf { V } } ^ { T } _ { 1 1 } \\| _ { 2 } = \\| \\mathbf { V } _ { 1 1 } - \\hat { \\mathbf { V } } _ { 1 1 } \\| _ { 2 } \\leq \\| \\mathbf { V } _ { 1 1 } - \\hat { \\mathbf { V } } _ { 1 1 } \\| _ F \\leq \\epsilon _ v , \\end{align*}"}
{"id": "6376.png", "formula": "\\begin{align*} L \\sigma ^ \\prime _ \\mathcal { X } = \\left ( \\mathcal { E } ^ \\prime _ p , \\mathcal { E } _ { p _ 1 } , \\mathcal { E } _ { p _ 2 } , \\mathcal { E } _ { p _ 3 } , \\mathcal { E } _ { p _ 4 } , \\Phi ( L \\sigma _ { \\widetilde { X } } ) \\right ) \\end{align*}"}
{"id": "6292.png", "formula": "\\begin{align*} p _ { \\widehat { K } } ( \\hat { k } ) & = \\mathbb { P } \\big ( \\mathrm { r o u n d } ( \\widehat { K } ^ \\mathbb { R } ) = \\hat { k } \\big ) , \\end{align*}"}
{"id": "4043.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ { t } + \\rho _ { x } = 0 \\quad \\mbox { i n } ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) , t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ { 0 } ( x ) , x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "6478.png", "formula": "\\begin{align*} c ( \\alpha ) = - \\frac { 2 ( 2 \\alpha ^ 2 + 1 ) } { 3 ( 1 - 4 \\alpha ) } . \\end{align*}"}
{"id": "8590.png", "formula": "\\begin{align*} \\left \\langle h , \\int _ 0 ^ t \\left [ \\int _ \\R e ^ { - i k ^ { 2 } s } \\overline { \\phi } ( k ) \\overline { \\mathcal { Q } } ( y , k ) F ( s , y ) \\ , d y \\right ] d s \\right \\rangle = & \\int _ 0 ^ t \\int _ \\R \\Big ( \\int _ \\R e ^ { i k ^ { 2 } s } \\phi ( k ) \\mathcal { Q } ( y , k ) h ( k ) \\ , d k \\Big ) \\overline { F } ( s , y ) \\ , d y d s . \\end{align*}"}
{"id": "6813.png", "formula": "\\begin{align*} \\boldsymbol { A } \\Psi _ { r n j } = \\mathfrak { a } ( p ^ { 1 - r } ) \\Psi _ { r n j } r n j \\end{align*}"}
{"id": "3157.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { Q _ s \\in \\mathfrak { Q } _ s , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) \\geq \\sup _ { Q _ s \\in \\mathfrak { Q } _ s ' , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) , \\end{aligned} \\end{align*}"}
{"id": "2447.png", "formula": "\\begin{align*} - S J S ^ T = - J \\ ; \\Leftrightarrow S J S ^ T = J \\end{align*}"}
{"id": "3736.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb S ^ 1 } B _ x J _ { x } B _ x \\ , d x \\right | \\leq & \\ \\| J _ x \\| _ { L ^ \\infty } \\| B _ x \\| ^ 2 _ { L ^ 2 } , \\\\ \\left | \\int _ { \\mathbb S ^ 1 } B J _ x B _ { x x } \\ , d x \\right | \\lesssim & \\ \\| J _ x \\| _ { L ^ \\infty } \\| B _ { x x } \\| _ { L ^ \\infty } \\| B _ x \\| _ { L ^ 2 } . \\end{align*}"}
{"id": "1877.png", "formula": "\\begin{align*} G = \\{ a \\rightarrow a x , \\ ; \\ ; x \\rightarrow 1 + x ^ 2 \\} . \\end{align*}"}
{"id": "5463.png", "formula": "\\begin{align*} m ^ * ( \\tau , s , u _ 0 ) = \\frac { a _ { \\sup } } { b _ { \\inf } } { \\rm a n d } \\tilde M _ 1 ( p , \\tau , s , u _ 0 ) = 0 \\forall \\ , \\ , \\tau \\ge s . \\end{align*}"}
{"id": "2781.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d \\lambda ^ 2 } \\Big | _ { \\lambda = 0 } I [ Q + \\lambda h ] \\ge 0 . \\end{align*}"}
{"id": "7905.png", "formula": "\\begin{align*} h _ t ( x ) = \\sum _ { p = 1 } ^ { q } \\sum _ { s = 1 } ^ m \\bigg ( e ^ { - i \\langle x , \\xi _ p \\rangle } g _ s ( \\xi _ p ) \\rho _ { s t } ( V _ p ) + e ^ { i \\langle x , \\xi _ p \\rangle } g _ s ( - \\xi _ p ) \\rho _ { s t } ( - V _ p ) \\bigg ) , \\end{align*}"}
{"id": "7097.png", "formula": "\\begin{align*} & = \\sum _ { j = 1 } ^ { r _ i } j ^ 2 - \\sum _ { j = 1 } ^ { r _ i } j - r _ i \\sum _ { j = 1 } ^ { r _ i } j + \\frac { r _ i ( r _ i + 1 ) } { 2 } \\sum _ { j = 1 } ^ { r _ i } 1 \\\\ & = \\frac { r _ i ( r _ i + 1 ) ( 2 r _ i + 1 ) } { 6 } - \\frac { r _ i ( r _ i + 1 ) } { 2 } - \\frac { r ^ 2 _ i ( r _ i + 1 ) } { 2 } + \\frac { r ^ 2 _ i ( r _ i + 1 ) } { 2 } \\\\ & = r _ i ( r _ i + 1 ) \\left [ \\frac { 1 } { 6 } ( 2 r _ i + 1 ) - \\frac { 1 } { 2 } \\right ] \\\\ & = r _ i ( r _ i + 1 ) \\left [ \\frac { r _ i } { 3 } - \\frac { 1 } { 3 } \\right ] \\\\ & = \\frac { 1 } { 3 } r _ i ( r ^ 2 _ i - 1 ) \\end{align*}"}
{"id": "7123.png", "formula": "\\begin{align*} \\tilde { f } ^ { \\prime } ( s , y ) = f _ 1 ( s , y ) + f _ 2 ( s , y ) , \\end{align*}"}
{"id": "4949.png", "formula": "\\begin{align*} { \\rm S h t } _ \\infty ( G , b , \\mu ) = \\varprojlim _ K { \\rm S h t } _ K ( G , b , \\mu ) , \\end{align*}"}
{"id": "3734.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ 1 } B _ { x x } J B _ x \\ , d x = - \\int _ { \\mathbb S ^ 1 } B _ x J _ { x } B _ x \\ , d x - \\int _ { \\mathbb S ^ 1 } B _ x J B _ { x x } \\ , d x \\end{align*}"}
{"id": "3472.png", "formula": "\\begin{gather*} e _ { P _ j } ( K / k ) = l ^ { \\beta _ j } , 1 \\leq \\beta _ j \\leq n , 1 \\leq j \\leq r , \\quad e _ { \\infty } ( K / k ) = l ^ { t } , 0 \\leq t \\leq n \\end{gather*}"}
{"id": "8893.png", "formula": "\\begin{align*} \\tilde \\psi ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) ) = \\sum _ { n = 0 } ^ { \\lfloor \\max ( t _ j ) \\rfloor } ( h _ n ^ * \\psi _ 1 ) ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) ) . \\end{align*}"}
{"id": "7009.png", "formula": "\\begin{align*} \\| p \\| ' : = \\max _ { | z | = 1 } | p ( z ) | \\mbox { a n d } \\| p \\| '' : = \\max _ { | z | = 3 } | p ( z ) | . \\end{align*}"}
{"id": "7219.png", "formula": "\\begin{align*} | \\check x ^ \\perp | = | x ^ \\perp - \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } v ^ \\perp | = \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } | v _ * ^ \\perp ( t , x , v _ 1 ) - v ^ \\perp | \\geq \\frac 1 2 \\check \\tau _ { t , x } | v _ * ^ \\perp ( t , x , v _ 1 ) - v ^ \\perp | . \\end{align*}"}
{"id": "342.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H ^ 1 _ R } \\mathcal L ( \\rho , m , S ) = \\sup _ { S \\in H ^ 1 _ R } \\inf _ { ( \\rho , m ) } \\mathcal L ( \\rho , m , S ) \\end{align*}"}
{"id": "3452.png", "formula": "\\begin{align*} & S \\Big ( \\sum \\limits _ { | j | > n } \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { Q } ( \\cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) \\Big ) ( x ) \\\\ & \\qquad = \\Big ( \\sum \\limits _ { Q ' } \\Big | q _ { Q ' } \\Big ( \\sum \\limits _ { | j | > n } \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { Q } ( \\cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) \\Big ) ( x _ { Q ' } ) \\Big | ^ 2 \\chi _ { Q ' } ( x ) \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "5680.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) - I ( \\alpha , \\beta , Z ' ) = < c _ { 1 } ( \\xi ) + 2 \\mathrm { P D } ( \\Gamma ) , Z - Z ' > . \\end{align*}"}
{"id": "7357.png", "formula": "\\begin{align*} \\begin{aligned} & { 1 \\over q } \\varphi ( x _ q , t _ q ) ^ { 1 - q } ( \\lambda h _ q + ( 1 - \\lambda ) k _ q ) + \\mu ( w _ q ( x _ q , t _ q ) ) \\\\ & \\geq - w _ q ( x _ q , t _ q ) ^ { 1 - q } \\left ( \\lambda u ( y _ q , t _ q ) ^ { q - 1 } + ( 1 - \\lambda ) u ( z _ q , t _ q ) ^ { q - 1 } \\right ) \\omega ( | X _ q | ) . \\end{aligned} \\end{align*}"}
{"id": "3119.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 & = x ^ 6 + a x ^ 4 y ^ 2 + b x ^ 2 y ^ 4 + y ^ 6 \\ , ; \\\\ z ^ 3 y ^ 2 & = x ( x ^ 4 + a x ^ 2 y ^ 2 + y ^ 4 ) \\ , ; \\\\ z ^ 3 y ^ 3 & = x ^ 6 + a x ^ 3 y ^ 3 + y ^ 6 \\ , ; \\\\ z ^ 3 y ^ 2 & = x ^ 5 + y ^ 5 \\ , ; \\\\ z ^ 3 y ^ 3 & = x ^ 6 + y ^ 6 \\ , ; \\\\ z ^ 3 y ^ 2 & = x ( x ^ 4 + y ^ 4 ) \\ , . \\end{align*}"}
{"id": "7014.png", "formula": "\\begin{align*} R ( z ) = \\frac { z - \\beta _ { k } } { \\beta _ 1 - \\beta _ { k } } r ( z ) - \\frac { z - \\beta _ { 1 } } { \\beta _ 1 - \\beta _ { k } } s ( z ) , \\end{align*}"}
{"id": "6434.png", "formula": "\\begin{align*} \\Psi _ N ( q z , q z , 1 / z , 1 / z ) & = \\sum _ { k = 0 } ^ N q ^ k z ^ k { N \\brack k } _ q , \\\\ \\Phi _ N ( q z , q z , 1 / z , 1 / z ) & = \\frac { \\Psi _ N ( q z , q z , 1 / z , 1 / z ) } { ( q ; q ) _ N } = \\sum _ { k = 0 } ^ N \\frac { q ^ k z ^ k } { ( q ; q ) _ k ( q ; q ) _ { N - k } } . \\end{align*}"}
{"id": "1907.png", "formula": "\\begin{align*} s '' ( t + k \\tau '' ) & = s ( t + k \\tau '' ) + s ' ( t + k \\tau '' ) = s ( t + k \\alpha \\tau ) + s ' ( t + k \\alpha ' \\tau ' ) \\\\ & = k \\alpha \\nu + s ( t ) + k \\alpha ' \\nu ' + s ' ( t ) = k ( \\alpha \\nu + \\alpha ' \\nu ' ) + s '' ( t ) . \\end{align*}"}
{"id": "6744.png", "formula": "\\begin{align*} \\hat { V } ( t , S _ f ( t ) ) = V ^ * ( S _ f ( t ) ) , \\frac { \\partial \\hat { V } } { \\partial S } ( t , S _ f ( t ) ) = \\frac { \\partial V ^ * } { \\partial S } ( S _ f ( t ) ) , \\end{align*}"}
{"id": "81.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "5697.png", "formula": "\\begin{align*} 2 - J _ { 0 } ( u ) = I ( \\delta _ { k + 1 } , \\delta _ { k } ) - J _ { 0 } ( \\delta _ { k + 1 } , \\delta _ { k } ) = \\mu _ { \\mathrm { d i s c } } ( \\delta _ { k + 1 } ) - \\mu _ { \\mathrm { d i s c } } ( \\delta _ { k } ) . \\end{align*}"}
{"id": "5369.png", "formula": "\\begin{align*} \\gamma _ M ( g ) = g - \\sigma _ M ( g ) c , \\end{align*}"}
{"id": "3338.png", "formula": "\\begin{align*} D ( x _ 1 , x _ 2 ) f ( x _ 3 ) - \\theta ( x _ 1 , x _ 3 ) f ( x _ 2 ) + \\theta ( x _ 2 , x _ 3 ) f ( x _ 1 ) - f ( [ x _ 1 , x _ 2 , x _ 3 ] ) = 0 , \\end{align*}"}
{"id": "5047.png", "formula": "\\begin{align*} \\varphi ( \\theta ) = \\sum _ { n \\not = 0 } \\varphi _ n e ^ { i n \\theta } , \\varphi _ n = \\frac { x _ n + i y _ n } { 2 \\sqrt { n } } \\textrm { f o r } n > 0 \\end{align*}"}
{"id": "9104.png", "formula": "\\begin{align*} \\alpha _ n ( x y ) = x \\alpha _ n ( y ) + \\alpha _ 1 ( x ) \\alpha _ { n - 1 } ( y ) + \\dots + \\alpha _ n ( x ) y . \\end{align*}"}
{"id": "6882.png", "formula": "\\begin{align*} \\lambda _ \\ell ( N ^ j _ k ) & = \\prod \\limits _ { s = 1 } ^ j \\left ( 1 - \\prod ^ { k - s + j } _ { i = \\ell } \\delta _ i \\right ) \\\\ & = \\prod \\limits _ { s = 1 } ^ j \\left ( 1 - \\prod ^ { k - s + j } _ { i = \\ell } \\frac { ( q ^ i - 1 ) ( q ^ { n - i + 1 } - 1 ) } { ( q ^ { i + 1 } - 1 ) ( q ^ { n - i } - 1 ) } \\right ) . \\end{align*}"}
{"id": "4511.png", "formula": "\\begin{align*} c _ 2 ( \\log t _ { k + 1 } ) ^ 2 - c _ 2 ( \\log t _ { k } ) ^ 2 = c _ 2 ( \\log t _ 0 ) ^ 2 \\big ( w _ { k + 1 } ^ 2 - w _ { k } ^ 2 \\big ) . \\end{align*}"}
{"id": "1913.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v \\big ( ( u - v ) f \\big ) - \\Delta _ v f + { \\rm { d i v } } _ v ( f L [ f ] ) = 0 , \\\\ & \\partial _ t \\rho + { \\rm { d i v } } _ x ( \\rho u ) = 0 , \\\\ & \\partial _ t ( \\rho u ) + { \\rm { d i v } } _ x ( \\rho u \\otimes u ) + \\nabla _ x \\rho ^ \\gamma - { \\rm { d i v } } _ x \\mathbb { S } ( \\nabla u ) = - \\int _ { \\mathbb { R } ^ 3 } ( u - v ) f \\ , d v , \\end{aligned} \\right . \\end{align*}"}
{"id": "3562.png", "formula": "\\begin{align*} u \\leqq v & \\ , : \\Longleftrightarrow \\ , u _ i \\leq v _ i , \\ ; i = 1 , \\ldots , K , \\\\ u < v & \\ , : \\Longleftrightarrow \\ , u _ i < v _ i , \\ ; i = 1 , \\ldots , K . \\end{align*}"}
{"id": "6534.png", "formula": "\\begin{align*} S _ n : = \\sum _ { j = 1 } ^ n f ( j ) \\mbox { a n d } I _ n : = \\int _ 1 ^ n f ( t ) \\ , d t . \\end{align*}"}
{"id": "3769.png", "formula": "\\begin{align*} \\mathbb { K } _ \\psi ^ \\pi ( w ) \\xi \\{ \\chi , k \\} = \\epsilon ( \\chi ^ { - 1 } \\pi , \\psi ) \\xi \\{ \\chi ^ { - 1 } \\varpi _ \\pi , - n ( \\chi ^ { - 1 } \\pi , \\psi ) - k \\} , \\end{align*}"}
{"id": "1134.png", "formula": "\\begin{align*} P ( \\xi , k ) : = \\left \\{ \\begin{aligned} & \\Delta _ { \\eta } Q _ 1 ^ { - 1 } \\left ( \\Psi ^ { A i } _ 0 \\right ) ^ { - 1 } \\zeta ^ \\frac { \\sigma _ 3 } { 4 } , & \\mathbb { C } _ { + } \\cap U _ \\delta ( \\eta ) , \\\\ & \\Delta _ { \\eta } Q _ 2 ^ { - 1 } \\left ( \\Psi ^ { A i } _ 0 \\right ) ^ { - 1 } \\zeta ^ \\frac { \\sigma _ 3 } { 4 } , & \\mathbb { C } _ { - } \\cap U _ \\delta ( \\eta ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "2982.png", "formula": "\\begin{align*} \\sum _ { r \\ne r ' } ^ d \\sum _ { q _ 1 = 1 } ^ { r - 1 } \\sum _ { p _ 1 = 1 } ^ { q _ 1 - 1 } \\sum _ { \\substack { q _ 3 = 1 \\\\ q _ 3 \\ne p _ 1 , q _ 1 } } ^ { r ' - 1 } 1 = \\frac { d ^ 5 } { 1 2 } ( 1 + o ( 1 ) ) . \\end{align*}"}
{"id": "2966.png", "formula": "\\begin{align*} \\mathbf { R } = \\big ( 1 _ { \\{ q _ \\ell \\in \\mathbf { p } _ { k , j } \\} } \\big ) _ { \\ell = 1 , \\dots , 2 k , j = 1 , \\dots , 4 } . \\end{align*}"}
{"id": "4537.png", "formula": "\\begin{align*} G ( z , \\chi ) : & = \\prod _ { p } \\left ( 1 - \\frac { \\chi ( p ) } { p ^ { z } } \\right ) \\left ( 1 - \\frac { \\chi ( p ) ^ { 2 } } { p ^ { 2 z } } \\right ) \\left ( \\sum _ { \\alpha = 0 } ^ { \\infty } \\frac { a ( p ^ { \\alpha } ) \\chi ( p ^ { \\alpha } ) } { p ^ { \\alpha z } } \\right ) , \\end{align*}"}
{"id": "8645.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sigma _ j ( e _ i v ) = 0 \\sum _ { j = 1 } ^ n \\sigma _ j ( e _ i ) \\sigma _ j ( v ) = 0 \\end{align*}"}
{"id": "8332.png", "formula": "\\begin{align*} \\varphi \\in C ^ \\infty ( \\R ) , \\varphi ( s ) = 0 ~ \\forall s \\in ( - \\infty , - 1 ] , \\varphi ( s ) = 1 ~ \\forall s \\in [ 0 , \\infty ) , \\varphi ' ( s ) \\geq 0 ~ \\forall s \\in \\R . \\end{align*}"}
{"id": "5411.png", "formula": "\\begin{align*} 2 \\pi ( 2 - 2 g ) = \\sum _ { p \\in \\sigma } 2 \\pi - k _ p \\pi \\end{align*}"}
{"id": "1638.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { ( c _ 2 - \\lambda ) t } t ^ { - 1 / 2 } P _ { s + \\alpha t } f ( x ) d t = \\int _ 0 ^ \\delta e ^ { ( c _ 2 - \\lambda ) t } t ^ { - 1 / 2 } P _ { s + \\alpha t } f ( x ) d t + \\int _ \\delta ^ \\infty e ^ { ( c _ 2 - \\lambda ) t } t ^ { - 1 / 2 } P _ { s + \\alpha t } f ( x ) d t \\end{align*}"}
{"id": "4072.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\rho ^ 2 ( t , 1 ) d t & = - \\int _ { 0 } ^ { 1 } x ( \\rho ^ 2 ( T , x ) - \\rho _ 0 ^ 2 ( x ) ) d x + \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } \\rho ^ 2 d x d t - 2 \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } x \\rho u _ x d x d t \\\\ & \\leq 2 \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } \\rho ^ 2 d x d t + \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } u _ x ^ 2 d x d t + \\int _ { 0 } ^ { 1 } \\rho _ 0 ^ 2 ( x ) d x . \\end{align*}"}
{"id": "2277.png", "formula": "\\begin{align*} T _ x f ( t ) = f ( t - x ) \\textnormal { a n d } M _ \\omega f ( t ) = f ( t ) \\ , e ^ { 2 \\pi i \\omega \\cdot t } . \\end{align*}"}
{"id": "504.png", "formula": "\\begin{align*} Y ( \\mathcal { I } ) _ { \\beta \\omega } ^ { m j } = - Y ( \\mathcal { I } ) _ { \\beta \\alpha } ^ { m i } \\cdot Y ( \\mathcal { I } ) _ { \\alpha \\omega } ^ { m i } \\cdot Y ( \\mathcal { I } ) _ { \\beta \\alpha } ^ { i j } \\cdot Y ( \\mathcal { I } ) _ { \\alpha \\omega } ^ { i j } \\in \\mathbb { C } , ( j , \\beta ) \\in \\mathfrak { c } . \\end{align*}"}
{"id": "5282.png", "formula": "\\begin{align*} \\varphi ( S ^ { - 1 } ( b ) a ) = \\varphi ( S ^ { - 1 } ( a ' ) b ) , \\forall b \\in A . \\end{align*}"}
{"id": "1777.png", "formula": "\\begin{align*} C ^ k _ { \\rm d i f f } ( G ) : = \\left \\{ \\begin{array} { l } c : G ^ { \\times ( k + 1 ) } \\to \\mathbb { C } ~ { \\rm s m o o t h } , \\\\ ~ c ( g g _ 0 , \\ldots , g g _ k ) = c ( g _ 0 , \\ldots , g _ k ) , \\ \\forall g , g _ 0 , \\ldots , g _ k \\in G \\end{array} \\right \\} . \\end{align*}"}
{"id": "8663.png", "formula": "\\begin{align*} \\Phi ( p , q ) : = \\frac { \\Lambda ( p , q ) } { p ^ 0 q ^ 0 } \\mathcal { S } ( p , q ) \\end{align*}"}
{"id": "6793.png", "formula": "\\begin{align*} \\dfrac { d l } { d \\theta } = & \\dfrac { \\partial l } { \\partial \\theta } + \\dfrac { \\partial l } { \\partial p ( \\theta ) } \\dfrac { \\partial p ( \\theta ) } { \\partial \\theta } . \\end{align*}"}
{"id": "9199.png", "formula": "\\begin{align*} \\exists y ^ \\sigma A _ 0 ( y ) \\rightarrow s = _ \\rho t \\equiv \\forall y ^ \\sigma ( A _ 0 ( y ) \\rightarrow s = _ \\rho t ) \\end{align*}"}
{"id": "103.png", "formula": "\\begin{align*} \\lambda _ i \\ : x _ i \\ : y _ { i + 1 } = \\mu _ i \\ : x _ { i + 1 } \\ : y _ i \\end{align*}"}
{"id": "5963.png", "formula": "\\begin{align*} \\mathcal { V } _ { i j } = \\psi _ j ( r _ i , s _ i ) , ( i , j ) = 1 , . . . , N _ { e p } . \\end{align*}"}
{"id": "758.png", "formula": "\\begin{align*} \\frac { f ( x ^ 2 ) } { f ( x ^ 1 ) } \\leq \\frac { 2 L ( f ^ 1 ) ^ { - 1 } - 2 \\mu _ p ( f ^ 1 ) ^ { 2 \\theta - 2 } } { 2 L ( f ^ 1 ) ^ { - 1 } + \\mu _ p ( f ^ 1 ) ^ { 2 \\theta - 2 } } = \\frac { 2 L - 2 \\mu _ p ( f ^ 1 ) ^ { 2 \\theta - 1 } } { 2 L + \\mu _ p ( f ^ 1 ) ^ { 2 \\theta - 1 } } . \\end{align*}"}
{"id": "232.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\Sigma } ( f ) ( x ) = - \\langle x ; \\nabla ( f ) ( x ) \\rangle + \\Delta ^ { \\Sigma } ( f ) ( x ) , \\end{align*}"}
{"id": "2872.png", "formula": "\\begin{align*} \\begin{cases} | \\alpha _ + ( t ) | \\lesssim e ^ { - c _ 1 ^ - t } , & c _ 0 < c _ 1 \\le e _ 0 e _ 0 < c _ 0 < c _ 1 , \\\\ | \\alpha _ + ( t ) - A e ^ { - e _ 0 t } | \\lesssim e ^ { - c _ 1 t } , & c _ 0 \\le e _ 0 < c _ 1 . \\end{cases} \\end{align*}"}
{"id": "539.png", "formula": "\\begin{align*} \\begin{aligned} \\Xi _ k & = \\begin{bmatrix} y _ { 1 , [ k , k + d _ 1 - 1 ] } ^ \\top & \\dots & y _ { m , [ k , k + d _ m - 1 ] } ^ \\top \\end{bmatrix} ^ \\top . \\end{aligned} \\end{align*}"}
{"id": "886.png", "formula": "\\begin{align*} F = \\sum _ { 0 \\leq T \\in \\Lambda _ n } a ( F , T ) \\ , q ^ T & = \\sum _ { t _ i } \\left ( \\sum _ { t _ { i j } } a ( F , T ) \\prod _ { i < j } q _ { i j } ^ { 2 t _ { i j } } \\right ) \\prod _ { i = 1 } ^ n q _ i ^ { t _ { i j } } \\\\ & \\in \\mathbb { C } [ q _ { i j } ^ { - 1 } , q _ { i j } ] [ \\ ! [ q _ 1 , \\ldots , q _ n ] \\ ! ] . \\end{align*}"}
{"id": "1188.png", "formula": "\\begin{align*} \\mu \\Big ( \\bigcup _ { i = 1 } ^ { N _ \\Omega } B _ { n _ i } \\Big ) \\geq C \\mu ( \\Omega ) . \\end{align*}"}
{"id": "5538.png", "formula": "\\begin{align*} \\| b ( t , h ) - b ( t , g ) \\| & \\leq \\| a ( h ) - a ( g ) \\| + \\sum _ { j = 1 } ^ r \\| \\sigma ^ j ( h ) - \\sigma ^ j ( g ) \\| \\ , | \\dot { B } _ m ^ j ( t ) | \\\\ & \\leq L \\| h - g \\| + L \\| h - g \\| \\sum _ { j = 1 } ^ r Y _ m ^ j = ( L + Z _ { r , m } ) \\| h - g \\| . \\end{align*}"}
{"id": "4563.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } ( S _ n / ( \\sigma \\sqrt { n } ) > x ) } { 1 - \\Phi ( x ) } = 1 + o \\big ( 1 \\big ) , \\ \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "7491.png", "formula": "\\begin{align*} & ( \\Delta _ h \\phi ) _ { j k } = - \\sum _ { p = - M / 2 } ^ { M / 2 - 1 } \\sum _ { q = - M / 2 } ^ { M / 2 - 1 } \\left ( ( \\varrho ^ { x } _ { p } ) ^ { 2 } + ( \\varrho ^ { y } _ { q } ) ^ { 2 } \\right ) \\widehat { \\phi } _ { p q } \\ , e ^ { i \\frac { 2 j p \\pi } { M } } e ^ { i \\frac { 2 k q \\pi } { M } } , \\\\ & ( L _ { z } ^ h \\phi ) _ { j k } = - i \\Big ( x _ { j } ( \\partial _ { y } ^ { h } \\phi ) _ { j k } - y _ { k } ( \\partial _ { x } ^ { h } \\phi ) _ { j k } \\Big ) , j , k = 0 , 1 , \\cdots , M - 1 , \\end{align*}"}
{"id": "158.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\left ( \\psi \\left ( \\frac { t } { n } \\right ) \\right ) ^ n = \\exp \\left ( \\psi ' ( 0 ) t \\right ) . \\end{align*}"}
{"id": "8080.png", "formula": "\\begin{align*} \\left \\langle a , f \\right \\rangle : = \\sum _ { n \\in \\mathbb { Z } } a _ n f _ { - n } . \\end{align*}"}
{"id": "787.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla \\phi \\ , d \\mu = \\int _ { \\partial \\Omega } \\phi f \\ , d \\nu \\end{align*}"}
{"id": "3904.png", "formula": "\\begin{align*} T ( u , p ) = \\sum _ { \\substack { 0 < a < p \\\\ g c d ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } . \\end{align*}"}
{"id": "6462.png", "formula": "\\begin{align*} E [ X _ { n + 1 } \\mid \\mathcal { F } _ n ] & = \\alpha \\cdot \\dfrac { S _ n } { n } , \\\\ \\intertext { a n d } E [ S _ { n + 1 } \\mid \\mathcal { F } _ n ] & = \\left ( 1 + \\dfrac { \\alpha } { n } \\right ) S _ n . \\end{align*}"}
{"id": "4727.png", "formula": "\\begin{align*} \\partial _ t \\mathcal { N } ( t ) & \\leq t ^ 3 ( 3 t ^ { - 1 } - 2 e _ 0 ) \\sum _ { i = 1 } ^ n | a _ i ^ - ( t ) | ^ 2 + O \\Bigg ( t \\bigg ( \\sum _ { i = 1 } ^ n | a _ i ^ - ( t ) | ^ 2 \\bigg ) ^ { 1 / 2 } \\Bigg ) . \\end{align*}"}
{"id": "4638.png", "formula": "\\begin{align*} \\lambda _ { ( j - 1 ) k + i } - \\lambda _ { ( j - 1 ) k + ( i + 1 ) } & = \\phi ( a _ { i , j } ) - \\phi ( a _ { i + 1 , j } ) \\\\ & = \\sum \\limits _ { s = i } ^ k a _ { s , j } + \\sum \\limits _ { s = 1 } ^ { i - 1 } a _ { s , j + 1 } - \\left ( \\sum \\limits _ { s = i + 1 } ^ k a _ { s , j } + \\sum \\limits _ { s = 1 } ^ { i } a _ { s , j + 1 } \\right ) \\\\ & = a _ { i , j } - a _ { i , j + 1 } . \\end{align*}"}
{"id": "2838.png", "formula": "\\begin{align*} y _ R ( 0 ) = \\int R ^ 2 \\varphi \\left ( \\frac { x } { R } \\right ) | u _ 0 | ^ 2 d x \\le y _ R ( t ) , \\forall t \\ge 0 , \\ ; \\forall R \\ge R _ 0 , \\end{align*}"}
{"id": "1675.png", "formula": "\\begin{align*} \\overline { H } _ n ^ { \\star } ( \\mathbf { s } ) = \\overline { H } _ n ^ { \\star } ( s _ 1 , s _ 2 , \\ldots , s _ r ) : = \\sum \\limits _ { 0 \\leq k _ 1 \\leq k _ 2 \\leq \\cdots \\leq k _ r \\leq n - 1 } \\prod _ { j = 1 } ^ { r } \\frac { 1 } { ( 2 k _ j + 1 ) ^ { s _ j } } . \\end{align*}"}
{"id": "5382.png", "formula": "\\begin{align*} c _ a = 1 . \\end{align*}"}
{"id": "64.png", "formula": "\\begin{align*} L ' \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } L ' _ { v , j } ( \\pi ^ j ) & ( v = 2 , p _ 1 , \\cdots , p _ k ) , \\\\ L ' _ { v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\\\ \\end{cases} \\\\ K _ { \\ell } \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } K _ { \\ell , v , j } ( \\pi ^ j ) & ( v = 2 , p _ 1 , \\cdots , p _ k ) , \\\\ K _ { \\ell , v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
{"id": "5860.png", "formula": "\\begin{align*} \\theta _ k = \\sum _ { j : \\ j \\le k , \\ j \\in J } \\left ( \\frac { \\bar \\alpha _ { j } } { \\rho _ { j } } - 1 \\right ) . \\end{align*}"}
{"id": "4262.png", "formula": "\\begin{align*} { \\bf H } [ g _ b ] ( x ) ~ = ~ \\O ( 1 ) \\cdot | x | ^ { - 1 } , { d ^ k \\over d x ^ k } \\left ( { \\bf H } [ g _ b ] \\right ) ( x ) ~ = ~ \\O ( 1 ) \\cdot x ^ { - ( k + 1 ) } , k = 1 , 2 , 3 . \\end{align*}"}
{"id": "7807.png", "formula": "\\begin{align*} v ^ m = \\frac { \\pi ^ m } { \\pi _ 1 ^ m } = \\frac { ( \\pi _ 1 ^ 2 + 2 \\pi _ 1 ) ^ m } { \\pi _ 1 ^ m } = ( \\pi _ 1 + 2 ) ^ m . \\end{align*}"}
{"id": "835.png", "formula": "\\begin{align*} B _ { \\rho } = B _ { \\rho } ( ( x _ { 0 } , y _ { 0 } ) , r _ { 0 } ) \\subset \\widetilde B \\subset C _ { \\omega } ^ 2 B _ { \\rho } = B _ { \\rho } ( ( x _ { 0 } , y _ { 0 } ) , C _ \\omega ^ 2 r _ { 0 } ) . \\end{align*}"}
{"id": "7190.png", "formula": "\\begin{align*} \\limsup _ { t \\to T _ \\ast } \\| \\rho ( t , \\cdot ) \\| _ { W ^ { 1 , \\infty } } = \\infty . \\end{align*}"}
{"id": "9134.png", "formula": "\\begin{align*} \\forall x ^ * \\in \\Gamma _ k \\forall l \\leq m \\left ( \\norm { x _ { n + l } - x ^ * } < e ^ A \\norm { x _ n - x ^ * } + ( 2 M + 1 ) e ^ A \\sum _ { i = n } ^ { n + l - 1 } \\mu _ i + \\frac { 1 } { r + 1 } \\right ) \\end{align*}"}
{"id": "8427.png", "formula": "\\begin{align*} D _ { U } Y _ { t } ^ { x ; l ^ { \\epsilon } } ( W ) = \\lim _ { \\delta \\to 0 } \\frac { Y _ { t } ^ { x ; l ^ { \\epsilon } } ( W + \\delta U ) - Y _ { t } ^ { x ; l ^ { \\epsilon } } ( W ) } { \\delta } . \\end{align*}"}
{"id": "2798.png", "formula": "\\begin{align*} L _ + f _ * = \\lambda _ 0 \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 1 } . \\end{align*}"}
{"id": "2453.png", "formula": "\\begin{align*} \\det \\left ( \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} I & I \\\\ i I & - i I \\end{pmatrix} \\right ) = \\left ( \\tfrac { 1 } { \\sqrt { 2 } } \\right ) ^ { 2 d } \\det ( - 2 i I ) = 2 ^ { - d } ( - 2 i ) ^ { d } = ( - i ) ^ d \\end{align*}"}
{"id": "4456.png", "formula": "\\begin{align*} a | m \\rangle = \\sqrt { m } | m - 1 \\rangle , a ^ { \\dagger } | m \\rangle = \\sqrt { m + 1 } | m + 1 \\rangle . \\end{align*}"}
{"id": "7687.png", "formula": "\\begin{align*} \\mathcal { C } \\coloneqq \\{ \\mathbf { x } \\in \\mathcal { A } : \\nabla f = 0 \\} \\ ; . \\end{align*}"}
{"id": "5311.png", "formula": "\\begin{align*} g \\omega = \\omega ( S ^ { - 1 } ( g _ { ( 3 ) } ) - g _ { ( 1 ) } ) g _ { ( 2 ) } . \\end{align*}"}
{"id": "633.png", "formula": "\\begin{align*} \\abs { \\frac { f _ 0 ( x , n ) - g _ 0 ( x , n ) } { h _ 0 ( x , n ) + 1 } - \\alpha ( n ) } \\ & = \\ \\abs { \\frac { u ( 2 x + 1 , n ) - v ( 2 x + 1 , n ) } { w ( 2 x + 1 , n ) + 1 } - \\alpha ( n ) } \\\\ [ 1 5 p t ] & \\leq \\ \\frac { 1 } { 2 x + 1 + 1 } \\\\ [ 1 5 p t ] & = \\ \\frac { 1 } { 2 ( x + 1 ) } . \\end{align*}"}
{"id": "218.png", "formula": "\\begin{align*} C = - x _ k p _ \\alpha ( x ) \\partial _ k ( f ) ( x ) + f ( x ) \\left ( - p _ \\alpha ( x ) - x _ k \\partial _ k ( p _ \\alpha ) ( x ) \\right ) , \\end{align*}"}
{"id": "7555.png", "formula": "\\begin{align*} \\liminf _ { T \\to \\infty } \\frac { N _ 0 ( T ) } { N ( T ) } = 1 \\end{align*}"}
{"id": "9463.png", "formula": "\\begin{align*} f _ 1 = Y ^ { p ^ s } + \\tfrac { c _ 1 } { c _ 0 } \\Delta ^ { p ^ { s - 1 } } X ^ { p ^ s - 2 p ^ { s - 1 } } + \\tfrac { c _ 2 } { c _ 0 } Y ^ { p ^ { s - 1 } } X ^ { p ^ s - p ^ { s - 1 } } + \\tfrac { c _ 3 } { c _ 0 } \\Delta ^ { p ^ { s - 2 } } X ^ { p ^ { s } - 2 p ^ { s - 2 } } + \\ldots + \\tfrac { c _ r } { c _ 0 } Y X ^ { p ^ { s } - 1 } \\end{align*}"}
{"id": "5249.png", "formula": "\\begin{align*} x \\widetilde { m } : = \\sum _ i z _ i ( y _ i m ) , x \\in A \\otimes B , x E = \\sum _ i z _ i y _ i \\textrm { f o r } y _ i \\in A \\ , { } ^ I \\ ! \\otimes ^ I B , z _ i \\in A \\otimes B . \\end{align*}"}
{"id": "5861.png", "formula": "\\begin{align*} \\underline { k } = \\max \\{ j \\in J : j \\leq k \\} . \\end{align*}"}
{"id": "4100.png", "formula": "\\begin{align*} H ( Z _ t ) & = H ( Z _ 0 ) + \\int _ 0 ^ t ( H ' ( Z _ s ) \\cdot \\widetilde \\mu ( Z _ s ) + \\tfrac { 1 } { 2 } H '' ( Z _ s ) \\cdot \\widetilde \\sigma ^ 2 ( Z _ s ) ) \\ , d s + \\int _ 0 ^ t H ' ( Z _ s ) \\cdot \\widetilde \\sigma ( Z _ s ) \\ , d W _ s \\\\ & = x _ 0 + \\int _ 0 ^ t \\mu ( H ( Z _ s ) ) \\ , d s + \\int _ 0 ^ t \\sigma ( H ( Z _ s ) ) \\ , d W _ s . \\end{align*}"}
{"id": "8986.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { e ^ { - t } u \\left ( \\xi ( t ) \\right ) - e ^ { - ( t + h ) } u ( \\xi ( t + h ) ) } { h } \\leq \\frac { e ^ { - t } \\phi \\left ( \\xi ( t ) \\right ) - e ^ { - ( t + h ) } \\phi ( \\xi ( t + h ) ) } { h } \\\\ \\Longrightarrow & \\frac { 1 } { h } \\int _ t ^ { t + h } e ^ { - s } C _ p \\left | \\dot { \\xi } ( s ) \\right | ^ { q } d s \\leq \\frac { e ^ { - t } \\phi \\left ( \\xi ( t ) \\right ) - e ^ { - ( t + h ) } \\phi ( \\xi ( t + h ) ) } { h } \\end{aligned} \\end{align*}"}
{"id": "4210.png", "formula": "\\begin{align*} H '' _ f ( t ) & = - 2 \\int _ M ( | \\nabla v | ^ 2 + V _ 1 v ^ 2 ) \\ , d V _ g + 6 t \\int _ M q v ^ 2 \\ , d V _ g \\\\ & + 1 2 t ^ 2 \\int _ M q v r _ t \\ , d V _ g + 8 t ^ 3 \\int _ M q v \\dot { r } _ t \\ , d V _ g + t ^ 4 \\int _ M q v \\ddot { r } _ t \\ , d V _ g . \\end{align*}"}
{"id": "3381.png", "formula": "\\begin{align*} & \\partial _ { \\rho } ( f ) ( x _ 1 , \\cdots , x _ { p + 1 } ) \\\\ & = \\sum _ { 1 \\leq i < j \\leq p + 1 } ( - 1 ) ^ { i + j } f ( [ x _ i , x _ j ] , x _ 1 , \\cdots , \\widehat { x _ i } , \\cdots , \\widehat { x _ j } , \\cdots , x _ { p + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ { p + 1 } ( - 1 ) ^ { i + 1 } \\rho ( x _ i ) f ( x _ 1 , \\cdots , \\widehat { x _ i } , \\cdots , x _ { p + 1 } ) , \\end{align*}"}
{"id": "572.png", "formula": "\\begin{align*} a _ 0 + a _ 1 \\phi + \\cdots + a _ d \\phi ^ d \\ = \\ 0 , \\end{align*}"}
{"id": "7533.png", "formula": "\\begin{align*} \\frac { T } { 4 } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) \\ d \\sigma & = \\epsilon T \\log \\left ( \\frac { T } { 2 } \\right ) + \\epsilon \\ \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "1884.png", "formula": "\\begin{align*} \\pi _ G ( G \\times I K ^ { - 1 } \\cap \\Gamma ) & \\subseteq \\pi _ G ( G \\times ( W \\tau ( F ) ) \\cap \\Gamma ) \\\\ & = \\pi _ G ( G \\times W \\cap \\Gamma ) F \\\\ & = \\Lambda F \\end{align*}"}
{"id": "9371.png", "formula": "\\begin{align*} - \\int _ { 0 } ^ { t } \\frac { 1 } { x } \\log _ { \\lambda } ( 1 - x ) d x & = \\int _ { 0 } ^ { t } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } } { n ! } ( 1 ) _ { n , \\frac { 1 } { \\lambda } } ( - 1 ) ^ { n - 1 } x ^ { n - 1 } d x \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } } { ( n - 1 ) ! n ^ { 2 } } ( 1 ) _ { n , \\frac { 1 } { \\lambda } } ( - 1 ) ^ { n - 1 } t ^ { n } = \\mathrm { L i } _ { 2 , \\lambda } ( t ) . \\end{align*}"}
{"id": "1880.png", "formula": "\\begin{align*} G = \\{ x \\rightarrow x y , \\ ; \\ ; y \\rightarrow z , \\ ; \\ ; z \\rightarrow 2 y z \\} . \\end{align*}"}
{"id": "2366.png", "formula": "\\begin{align*} \\widehat { F _ { ( \\xi , \\eta ) } } ( x , \\omega ) = F _ { ( \\xi , \\eta ) } ( - \\omega , x ) , \\forall ( \\xi , \\eta ) \\in \\R ^ { 2 d } . \\end{align*}"}
{"id": "2452.png", "formula": "\\begin{align*} S + J S J ^ { - 1 } = \\begin{pmatrix} E & F \\\\ - F & E \\end{pmatrix} = \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} I & I \\\\ i I & - i I \\end{pmatrix} \\begin{pmatrix} E + i F & 0 \\\\ 0 & E - i F \\end{pmatrix} \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} I & - i I \\\\ I & i I \\end{pmatrix} \\end{align*}"}
{"id": "587.png", "formula": "\\begin{align*} \\begin{cases} \\ ( W _ 1 \\times I _ 1 ) \\cup ( Z _ 1 \\times J _ 1 ) \\\\ [ 8 p t ] \\ ( W _ 2 \\times I _ 2 ) \\cup ( Z _ 2 \\times J _ 2 ) \\end{cases} \\end{align*}"}
{"id": "822.png", "formula": "\\begin{align*} \\mu _ \\omega ( B _ \\rho ( ( x _ 0 , 0 ) , r ) ) = ( 1 + a ) ^ { - 1 } \\ r ^ { 1 + a } \\ , \\mu _ Z ( B ( x _ 0 , r ) ) . \\end{align*}"}
{"id": "7560.png", "formula": "\\begin{align*} [ s _ 1 , x _ { a _ 1 } ( t ) ] & = x _ { - a _ 1 } \\left ( ( v _ 1 ^ { - 2 } ) t \\right ) x _ { a _ 1 } ( t ) , \\quad \\\\ [ s _ 2 , x _ { 2 a _ 2 } ( t ) ] & = x _ { - 2 a _ 2 } \\left ( ( v _ 2 ^ { - 2 } ) t \\right ) x _ { 2 a _ 2 } ( t ) . \\end{align*}"}
{"id": "8601.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ 2 \\ , \\mu ^ \\# ( { \\bf k } ) = \\sqrt { 2 \\pi } \\ , \\delta _ 0 ( k _ 1 - k _ 2 + k _ 3 - k _ 4 ) + \\mu ^ \\# _ L ( { \\bf k } ) + \\mu ^ \\# _ R ( { \\bf k } ) , \\end{align*}"}
{"id": "6134.png", "formula": "\\begin{align*} U ^ { 1 } : \\R ^ { 2 } \\to \\R ^ { 3 } ; \\ U ^ { 1 } ( s , t ) = \\left ( ( u ^ { 1 } ( s , t ) , v ^ { 1 } ( s , t ) , w ^ { 1 } ( s , t ) \\right ) \\end{align*}"}
{"id": "812.png", "formula": "\\begin{align*} \\left ( \\int _ { \\overline \\Omega } g _ k ^ p \\ , d \\mu \\right ) ^ { 1 / p } \\le \\sum _ { j = k } ^ { N ( k ) } \\lambda _ { j , k } \\left ( \\int _ { \\overline \\Omega } g _ { u _ j } ^ p \\ , d \\mu \\right ) ^ { 1 / p } = \\sum _ { j = k } ^ { N ( k ) } \\lambda _ { j , k } \\left ( I _ { f _ j } ( u _ j ) + p \\int _ { \\partial \\Omega } u _ j f _ j \\ , d \\nu \\right ) ^ { 1 / p } . \\end{align*}"}
{"id": "1139.png", "formula": "\\begin{align*} E ( k ) = I + \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\frac { ( I + \\mu ( s ) ) ( J ^ E ( s ) - I ) } { s - k } d s , \\end{align*}"}
{"id": "8047.png", "formula": "\\begin{align*} \\omega _ \\ell ( u , v ) = \\omega _ { \\ell } ( u ) = \\rho ' ( - u ) , \\omega _ { r } ( u , v ) = \\omega _ r ( v ) = \\frac { \\widetilde { \\gamma } ' ( \\rho \\gamma ^ { - 1 } ( v ) ) } { \\gamma ' ( \\gamma ^ { - 1 } ( v ) ) } \\rho ' ( \\gamma ^ { - 1 } ( v ) ) . \\end{align*}"}
{"id": "6367.png", "formula": "\\begin{align*} v _ i = \\frac { 1 } { k - ( i - 1 ) } ( \\rho _ 1 + ( k - i ) v _ { i - 1 } ) \\end{align*}"}
{"id": "7755.png", "formula": "\\begin{align*} \\varepsilon _ 0 = \\varepsilon _ 0 ( \\delta ) = C _ p \\delta ^ { p } . \\end{align*}"}
{"id": "6892.png", "formula": "\\begin{align*} { k + j \\choose k } _ q \\left ( N _ k ^ j ( V , V ' ) - \\frac { 1 } { { k + j \\choose k } _ q } I ( V , V ' ) \\right ) = \\sum \\limits _ { i = 1 } ^ j ( - 1 ) ^ { i } { j \\choose i } _ q a _ i . \\end{align*}"}
{"id": "1723.png", "formula": "\\begin{align*} \\tilde \\nu = \\frac { \\mu _ * ( 1 / p _ 0 \u2010 1 / 2 ) + \\alpha _ * ( 1 / p _ 1 \u2010 1 / 2 ) } { \\gamma _ * ( 1 / p _ 1 \u2010 1 / p _ 0 ) } + \\frac 1 2 \u2010 \\frac 1 q . \\end{align*}"}
{"id": "4041.png", "formula": "\\begin{align*} e ^ { - \\alpha _ { 1 , k } - i \\alpha _ { 2 , k } } = 1 + O ( | k | ^ { - 1 } ) , . \\end{align*}"}
{"id": "1440.png", "formula": "\\begin{align*} \\mathfrak { i } ( x ) ^ { \\ast } = I ^ { - 1 } \\mathfrak { i } ( x ^ { \\ast } ) I , \\ , \\ , \\ , \\ , I : = \\left [ \\begin{array} { c c } - \\alpha & 0 \\\\ 0 & 1 \\end{array} \\right ] , \\end{align*}"}
{"id": "2278.png", "formula": "\\begin{align*} \\langle T _ x f , T _ x g \\rangle = \\langle f , g \\rangle \\textnormal { a n d } \\langle M _ \\omega f , M _ \\omega g \\rangle = \\langle f , g \\rangle . \\end{align*}"}
{"id": "8272.png", "formula": "\\begin{align*} \\mathbf { M } _ { u } \\mathbf { M } _ { v } = \\sum _ { w \\in \\mathfrak { B } _ { p + q } } c _ { u , v } ^ w \\mathbf { M } _ { w } . \\end{align*}"}
{"id": "7044.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { k = 1 } ^ L q _ k \\widetilde { \\phi _ k } \\Big \\| _ { H ^ 2 } ^ 2 & \\le \\Big ( \\sum _ { k = 1 } ^ L \\| q _ k \\widetilde { \\phi _ k } \\| _ { H ^ 2 } \\Big ) ^ 2 \\le \\Big ( \\sum _ { k = 1 } ^ L \\| q _ k \\| _ \\infty \\| \\phi _ k \\| _ b \\Big ) ^ 2 \\\\ & \\le C \\Big ( \\sum _ { k = 1 } ^ L \\| q _ k \\| _ { H ^ 2 } \\| \\phi _ k \\| _ b \\Big ) ^ 2 \\\\ & \\le C \\Big ( \\sum _ { k = 1 } ^ L \\| q _ k \\| _ { H ^ 2 } ^ 2 \\Big ) \\Big ( \\sum _ { k = 1 } ^ L \\| \\phi _ k \\| _ b ^ 2 \\Big ) . \\end{align*}"}
{"id": "506.png", "formula": "\\begin{align*} \\Delta _ { u } = ( - c _ { u , 1 } + c _ { u , 2 } + \\tau - c _ { u , 3 } \\cdot \\tau ) ^ { 2 } - 4 c _ { u , 2 } ( c _ { u , 1 } - 1 ) \\cdot ( c _ { u , 3 } \\cdot c _ { u , 2 } ^ { - 1 } - 1 ) \\cdot \\tau \\end{align*}"}
{"id": "8540.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u + H _ 1 u + u = a ( x ) u ^ 2 + u ^ 3 , \\end{align*}"}
{"id": "463.png", "formula": "\\begin{align*} \\prod _ { \\iota \\in 2 ^ { V ' } } h _ { x y } ^ \\iota ( c ) = \\sum _ { d \\in D ( \\Gamma _ { x y } ) ( c ) } \\prod _ { \\iota \\in 2 ^ { V ' } } g ^ { \\iota } ( d _ \\iota ) . \\end{align*}"}
{"id": "2257.png", "formula": "\\begin{align*} \\widetilde { e } _ m ^ { M , N } - \\widetilde { e } _ { m - 1 } ^ { M , N } + k A ^ 2 \\widetilde { e } _ m ^ { M , N } = - k P _ N A P F ( X _ m ^ { M , N } ) + k P _ N A P F ( X ( t _ m ) ) , \\widetilde { e } _ 0 ^ { M , N } = 0 , \\end{align*}"}
{"id": "1372.png", "formula": "\\begin{align*} n ( 0 ) = n ( 1 ) = 1 . \\end{align*}"}
{"id": "4502.png", "formula": "\\begin{align*} \\varepsilon _ 2 ( x , \\sigma _ 1 , T _ 0 , T ) = 2 x ^ { - 1 / 2 } ( S _ 0 + B _ 1 ( T _ 0 , T ) ) + \\Big ( x ^ { \\sigma _ 1 - 1 } - x ^ { - 1 / 2 } \\Big ) B _ 1 ( H _ 0 , T ) , \\end{align*}"}
{"id": "5366.png", "formula": "\\begin{align*} q = L \\cap H _ i . \\end{align*}"}
{"id": "3781.png", "formula": "\\begin{align*} \\Big ( r _ l \\big ( \\epsilon ( X , \\Pi _ { \\chi _ 0 } \\otimes \\Pi _ { \\pi _ F } , \\psi _ F ) \\big ) \\Big ) ^ l = \\epsilon \\big ( X , \\chi _ 0 r _ l ( \\pi _ F ) , \\overline { \\psi } _ F \\big ) ^ l . \\end{align*}"}
{"id": "5874.png", "formula": "\\begin{align*} S _ t ( ( Y , S , \\widetilde W ) _ 0 , d W _ \\cdot ) = e ^ { - \\mu t } S _ 0 + e ^ { - \\mu t } \\int _ 0 ^ t e ^ { \\mu s } F ^ 2 \\left ( Y _ s \\left ( ( Y , S , \\widetilde W ) _ 0 , d W _ \\cdot \\right ) \\right ) d W _ s , t \\ge 0 . \\end{align*}"}
{"id": "6069.png", "formula": "\\begin{align*} \\begin{cases} x _ { s } u ^ { j } _ { s } + y _ { s } v ^ { j } _ { s } + z _ { s } w ^ { j } _ { s } = F ^ { j } , \\\\ x _ { s } u ^ { j } _ { t } + y _ { s } v ^ { j } _ { t } + z _ { s } w ^ { j } _ { t } + x _ { t } u ^ { j } _ { s } + y _ { t } v ^ { j } _ { s } + z _ { t } w ^ { j } _ { s } = G ^ { j } , \\\\ x _ { t } u ^ { 1 } _ { t } + y _ { t } v ^ { 1 } _ { t } + z _ { t } w ^ { 1 } _ { t } = H ^ { j } , \\end{cases} \\end{align*}"}
{"id": "9536.png", "formula": "\\begin{align*} ( y S _ 0 ) ^ * ( p _ { - 1 } + y \\bar c ) & = ( y S _ 0 ) ^ * ( \\frac { E [ y \\bar c ] } { E [ y ] } y ) \\\\ & = ( \\frac { y } { E [ y ] } E [ y ] S _ 0 ) ^ * ( \\frac { y } { E [ y ] } E [ y \\bar c ] ) \\\\ & = \\frac { y } { E [ y ] } ( E [ y ] S _ 0 ) ^ * ( E [ y \\bar c ] ) \\end{align*}"}
{"id": "4244.png", "formula": "\\begin{align*} 0 \\ge H _ y = { } & u _ { x x y } + \\tau ^ 2 u _ { y y y } + 2 \\tau u _ { x y y } \\\\ = { } & ( \\tau ^ 2 - 1 ) u _ { y y y } - 2 \\tau u _ { x x x } - f ' ( u ) u _ y \\\\ = { } & - \\frac { ( 1 - \\tau ^ 2 ) ^ 2 + 4 \\tau ^ 2 } { 2 \\tau } u _ { x x x } - ( \\tau ^ 2 - 1 ) f ' ( u ) u _ y - f ' ( u ) u _ y \\\\ = { } & - \\frac { ( 1 + \\tau ^ 2 ) ^ 2 } { 2 \\tau } u _ { x x x } - \\tau ^ 2 f ' ( u ) u _ y . \\end{align*}"}
{"id": "1201.png", "formula": "\\begin{align*} { \\widetilde { E } _ { m } ^ { [ \\alpha , \\beta ] , \\rho , \\varepsilon } = \\left \\{ x \\in \\mathbb { R } ^ d : \\ \\underline \\dim _ { { \\rm l o c } } ( m , x ) \\in [ \\alpha , \\beta ] \\ a n d \\ \\ \\forall r \\leq \\rho , \\ m ( B ( x , r ) ) \\leq r ^ { \\underline { \\dim } _ { \\mathrm { l o c } } ( m , x ) - \\varepsilon } \\right \\} } \\end{align*}"}
{"id": "128.png", "formula": "\\begin{align*} U ^ i V ^ j = 0 \\mbox { f o r a l l } i \\geq 1 \\mbox { a n d } j \\geq 2 \\end{align*}"}
{"id": "9288.png", "formula": "\\begin{align*} \\Phi ( z ^ * , n , m , l , k ) : = \\max ( t ^ * _ u ( n , m , l , k ) ( z ^ * ) , t ^ * _ v ( n , m , l , k ) ( z ^ * ) ) . \\end{align*}"}
{"id": "1269.png", "formula": "\\begin{align*} \\mathrm { E C C } ( Y , \\lambda , \\Gamma ) = \\bigoplus _ { * : \\ , \\ , \\mathbb { Z } / d \\ , \\ , \\mathrm { g r a d i n g } } \\mathrm { E C C } _ { * } ( Y , \\lambda , \\Gamma ) . \\end{align*}"}
{"id": "3922.png", "formula": "\\begin{align*} K _ { 1 } = \\frac { 2 - \\sigma } { \\sigma } K n _ { x } { R e _ { x } } ^ { 1 / 2 } \\end{align*}"}
{"id": "9291.png", "formula": "\\begin{align*} C ( x , t ) = \\frac { C _ { 0 } } { 2 ( \\pi D t ) ^ { 1 / 2 } } \\exp \\left ( - \\frac { x ^ { 2 } } { 4 D t } \\right ) \\end{align*}"}
{"id": "7216.png", "formula": "\\begin{align*} K _ { s , t , x } : = \\{ v : s < { \\mathcal T } _ { t , x , v } - 1 , | v | < \\delta ^ { - \\beta } , \\check \\tau _ { t , x } \\langle v ^ \\perp \\rangle < \\langle x ^ \\perp \\rangle / 4 \\} , \\end{align*}"}
{"id": "863.png", "formula": "\\begin{align*} { \\bar { \\Delta } ^ { \\rm P } _ { \\rm P r o a c t i v e } } = - 1 - { \\tau _ { \\rm f } } - \\frac { { \\left ( { { \\tau _ { \\rm c } } + { n _ m } + \\mathcal { T } } \\right ) { \\epsilon _ m } } } { { 1 - { \\epsilon _ m } } } + \\frac { 2 } { { 1 - { \\epsilon _ m } } } \\left ( { { \\tau _ { \\rm c } } + { n _ 1 } + \\mathcal { T } + \\sum \\limits _ { i = 1 } ^ { m - 1 } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) { \\epsilon _ i } } } \\right ) . \\end{align*}"}
{"id": "8385.png", "formula": "\\begin{align*} \\chi _ { \\mathbf L } = \\P ( \\mathcal E _ { \\mathsf k } ( U , \\mathbf L ) \\mid \\mathcal L ( U ) = \\mathbf L ) \\ , , \\end{align*}"}
{"id": "4306.png", "formula": "\\begin{align*} \\beta ( \\tau ) = - \\frac { 1 } { 2 } \\frac { \\mu _ \\tau } { \\mu ( \\tau ) } , \\end{align*}"}
{"id": "9310.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ 1 \\Lambda ( x ) d x = \\sum _ { n = 1 } ^ \\infty | I _ n | c _ n = \\alpha ^ { - 1 } \\sum _ { n = 1 } ^ \\infty \\lambda _ n - \\lambda _ { n + 1 } \\\\ = \\lim _ { M \\to \\infty } \\alpha ^ { - 1 } \\sum _ { n = 1 } ^ M \\lambda _ n - \\lambda _ { n + 1 } = \\lim _ { M \\to \\infty } \\alpha ^ { - 1 } ( \\lambda _ 1 - \\lambda _ { M + 1 } ) \\\\ = \\alpha ^ { - 1 } \\lambda _ 1 \\end{gather*}"}
{"id": "222.png", "formula": "\\begin{align*} F _ { \\alpha , x , t } ( u ) = \\frac { 1 } { \\left ( 1 - e ^ { - \\alpha t } \\right ) ^ { \\frac { d } { \\alpha } } } \\frac { p _ \\alpha ( u ) } { p _ \\alpha ( x ) } p _ \\alpha \\left ( \\dfrac { x - u e ^ { - t } } { ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { 1 } { \\alpha } } } \\right ) . \\end{align*}"}
{"id": "4387.png", "formula": "\\begin{align*} b ( \\tau ) \\partial _ \\tau z = \\partial ^ 2 _ \\xi z + \\frac { d + 1 } { \\xi } \\partial _ \\xi z - 3 ( d - 2 ) ( 2 Q + \\xi ^ 2 Q ^ 2 ) z + \\bar { B } ( z ) + \\mu ( \\tau ) \\Lambda _ \\xi Q + \\mu ( \\tau ) \\Lambda _ \\xi z , \\end{align*}"}
{"id": "7363.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { \\gamma } ^ \\ast ( s ) = - \\pi \\gamma ^ \\ast ( s ) ^ 2 & \\ s > 0 , \\\\ \\gamma ^ \\ast ( 0 ) = r , & \\end{array} \\right . \\end{align*}"}
{"id": "6114.png", "formula": "\\begin{align*} C = \\min \\left [ \\left ( \\frac { m + 1 } { 4 ( m + 2 ) } \\right ) ^ { 1 / ( m + 2 ) } , \\left ( \\frac { ( n + 2 ) ( n + 1 ) } { 4 ( m + 2 ) ^ 2 } \\right ) ^ { 1 / ( n + 2 ) } \\right ] \\ , . \\end{align*}"}
{"id": "335.png", "formula": "\\begin{align*} & \\theta _ { i j } ( \\rho ) [ v _ { i j } - ( \\lambda _ i - \\lambda _ j ) ] = 0 , \\ ; \\forall ( i , j ) \\in E , \\\\ & \\ < \\dot \\lambda , \\rho \\ > - \\frac 1 4 \\sum _ { i j } v _ { i j } ^ 2 \\theta _ { i j } ( \\rho ) + \\frac 1 2 \\sum _ { i j } ( \\Sigma _ i - \\Sigma _ j ) ( \\lambda _ { i } - \\lambda _ { j } ) \\theta _ { i j } ( \\rho ) d W ^ { \\delta } ( t ) = 0 , \\ ; \\mathcal L ^ 1 \\ ; \\end{align*}"}
{"id": "7271.png", "formula": "\\begin{align*} \\boxed { \\alpha = \\sum _ { n = 0 } ^ \\infty \\frac { \\alpha ^ { ( n ) } ( 0 ) } { n ! } X ^ n . } \\end{align*}"}
{"id": "3139.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 & = 0 \\ , , \\\\ x _ 2 ^ 2 + j x _ 4 ^ 2 + j ^ 2 x _ 5 ^ 2 & = 0 \\ , , ( j ^ 3 = 1 . ) \\\\ x _ 3 ^ 2 + j ^ 2 x _ 4 ^ 2 + j x _ 5 ^ 2 & = 0 \\ , . \\end{align*}"}
{"id": "3117.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + z y ^ 4 + x ^ 5 = 0 \\ , . \\end{align*}"}
{"id": "8974.png", "formula": "\\begin{align*} u + \\left | u ' \\right | ^ 2 - \\left ( 1 - | x | \\right ) = 0 \\end{align*}"}
{"id": "8456.png", "formula": "\\begin{align*} \\delta ^ * ( \\alpha \\delta ) ^ * = ( \\alpha \\delta ) ^ * . \\end{align*}"}
{"id": "8679.png", "formula": "\\begin{align*} C _ n ( \\kappa , { \\bf P } _ { Y _ 1 } & ) = \\sup _ { \\frac { 1 } { n } { \\bf E } \\big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\big \\} \\leq \\kappa } \\sum _ { t = 1 } ^ n I ( X _ t , V ^ { t - 1 } ; Y _ t | Y ^ { t - 1 } ) \\\\ = & \\sup _ { \\frac { 1 } { n } { \\bf E } \\big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\big \\} \\leq \\kappa } H ( Y ^ n ) - H ( V ^ n ) \\in [ 0 , \\infty ] \\end{align*}"}
{"id": "1876.png", "formula": "\\begin{align*} 2 ^ n { { \\rm d } ^ n \\over { \\rm d } x ^ n } E ( x ) = P _ n ( E ( x ) ) . \\end{align*}"}
{"id": "2722.png", "formula": "\\begin{align*} \\frac { - V ( w ) } { U ( w ) } S ( w ) + T ( w ) = 0 , \\end{align*}"}
{"id": "7319.png", "formula": "\\begin{align*} \\nabla \\varphi ( y _ \\varepsilon , t _ \\varepsilon ) = { 4 \\over \\varepsilon } | x _ \\varepsilon - y _ \\varepsilon | ^ 2 ( y _ \\varepsilon - x _ \\varepsilon ) , \\nabla ^ 2 \\varphi ( y _ \\varepsilon , t _ \\varepsilon ) \\leq { 4 \\over \\varepsilon } | x _ \\varepsilon - y _ \\varepsilon | ^ 2 I + { 8 \\over \\varepsilon } ( x _ \\varepsilon - y _ \\varepsilon ) \\otimes ( x _ \\varepsilon - y _ \\varepsilon ) . \\end{align*}"}
{"id": "8113.png", "formula": "\\begin{align*} T B ( f ) = \\sum _ { \\gamma \\in \\Gamma ( G ) } t b ( f ( \\gamma ) ) . \\end{align*}"}
{"id": "482.png", "formula": "\\begin{align*} Y ( \\mathcal { I } ) _ { \\alpha \\beta } ^ { i j } : = c _ { 1 } c _ { 2 } \\cdot \\frac { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\alpha } ^ { i } ) } { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\beta } ^ { i } ) } \\cdot \\frac { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\beta } ^ { j } ) } { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\alpha } ^ { j } ) } , i , j \\in \\mathcal { I } , \\ , \\alpha , \\beta \\in \\mathcal { I } ^ { \\mathtt { C } } \\end{align*}"}
{"id": "958.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + c u & = 0 B _ \\rho ^ + . \\end{align*}"}
{"id": "377.png", "formula": "\\begin{align*} F _ * J = \\tilde J F _ * . \\end{align*}"}
{"id": "6110.png", "formula": "\\begin{align*} x ^ m w _ { y y } \\ , + \\varepsilon y ^ n w _ { x x } = 0 , \\varepsilon = 1 \\ \\textrm { o r } \\ - 1 \\ , . \\end{align*}"}
{"id": "9477.png", "formula": "\\begin{align*} | \\mathcal { B C } _ { ( s , s + d , s + 2 d ) } | & = | \\mathcal { C S } _ { ( s , s + d , s + 2 d ) } | = | \\mathcal { D D } _ { ( s , s + d , s + 2 d ) } | \\\\ & = \\sum _ { i = 0 } ^ { ( s - 1 ) / 2 } \\binom { ( s + d - 3 ) / 2 } { \\lfloor i / 2 \\rfloor } \\binom { ( s + d - 1 ) / 2 - \\lfloor i / 2 \\rfloor } { ( s - 1 ) / 2 - i } . \\end{align*}"}
{"id": "2932.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { x } { 2 } 1 _ { \\{ x \\} } + \\frac { x + 1 } { 2 } 1 _ { \\{ x \\} } . \\end{align*}"}
{"id": "9414.png", "formula": "\\begin{align*} \\varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) & = \\frac { 1 } { \\omega ( p ) } \\omega ( p b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) \\\\ & = \\frac { 1 } { \\omega ( p ) } \\omega ( p a _ 1 \\cdots a _ n ) \\tau ( b _ 0 ) \\cdots \\tau ( b _ n ) \\\\ & = \\varphi ( a _ 1 \\cdots a _ n ) \\varphi ( b _ 0 ) \\cdots \\varphi ( b _ n ) . \\end{align*}"}
{"id": "1932.png", "formula": "\\begin{align*} & f ( 0 , x , v ) = f _ { 0 } ( x , v ) , \\rho ( 0 , x ) = \\rho _ 0 ( x ) , u ( 0 , x ) = u _ 0 ( x ) , \\\\ & \\gamma ^ - f ( t , x , v ) \\big | _ { ( 0 , T ) \\times \\Sigma ^ - } = g ( t , x , v ) , \\\\ & \\rho ( t , x ) \\big | _ { ( 0 , T ) \\times \\Gamma _ { \\rm { i n } } } = \\rho _ B ( x ) , u ( t , x ) \\big | _ { ( 0 , T ) \\times \\partial \\Omega } = u _ B ( x ) , \\end{align*}"}
{"id": "3331.png", "formula": "\\begin{align*} ( 1 + x ) ^ { h _ 1 } w ^ { h _ 2 } \\left ( \\eta _ { A , 1 } ^ { h _ 1 } \\eta _ { A , 2 } ^ { h _ 2 } \\right ) \\left ( \\eta _ { B , 1 } ^ { h _ 1 } \\eta _ { B , 2 } ^ { h _ 2 } \\right ) = ( 1 + q _ 1 x _ { f _ { \\lambda } , d } ) ^ { h _ 1 } ( 1 + q _ 2 x _ { f _ { \\lambda } , d } ) ^ { h _ 2 } = 1 + ( h _ 1 q _ 1 + h _ 2 q _ 2 ) x _ { f _ { \\lambda } , d } \\end{align*}"}
{"id": "4594.png", "formula": "\\begin{align*} \\Big \\| \\langle M \\rangle _ n - ( n a _ n ^ 2 \\sigma ^ 2 + v _ n ) \\Big \\| _ t \\leq c _ 1 \\Big \\| \\sum _ { k = 1 } ^ { n - 1 } ( \\frac { a _ k T _ k } { k } ) ^ 2 \\Big \\| _ t \\leq c _ 2 \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { k ^ 2 } \\| a _ k T _ k \\| _ { 2 t } ^ 2 . \\end{align*}"}
{"id": "3912.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } a _ k x ^ k = \\sum _ { m = 0 } ^ n \\binom { n } { m } \\nabla ^ m b _ n ( x - 1 ) ^ { m } . \\end{align*}"}
{"id": "7375.png", "formula": "\\begin{align*} V ' ( r ^ { 1 - \\beta } ) - { 1 \\over r ^ { 1 - \\beta } } V ( r ^ { 1 - \\beta } ) = { 1 \\over r ^ { 1 - \\beta } } \\left ( \\int _ 0 ^ { r ^ { 1 - \\beta } } s V '' ( s ) \\ , d s - V ( 0 ) \\right ) \\leq 0 \\end{align*}"}
{"id": "2107.png", "formula": "\\begin{align*} \\nu ^ + : k & \\longrightarrow [ 0 , \\infty ] , \\\\ \\alpha & \\mapsto \\max \\{ \\nu ( \\alpha ) , 0 \\} , \\end{align*}"}
{"id": "8196.png", "formula": "\\begin{align*} \\hbox { i f } f = a ^ 2 + a b + b ^ 2 , \\hbox { t h e n } \\vert a \\vert + \\vert b \\vert \\leq \\sqrt { 4 f } \\hbox { a n d } \\vert a b \\vert \\geq \\sqrt { f / 3 } . \\end{align*}"}
{"id": "7479.png", "formula": "\\begin{align*} & e _ { r } ^ { n } : = \\left \\| G ( \\phi ^ { n } ) - \\mu ^ n \\phi ^ { n } \\right \\| _ { \\infty } < \\varepsilon _ { r } \\end{align*}"}
{"id": "1145.png", "formula": "\\begin{align*} & I _ { 1 } = - \\frac { 1 } { 2 \\pi i } \\oint _ { \\partial U _ { \\delta } ( \\pm \\eta ) } \\left ( J ^ E ( s ) - I \\right ) d s , \\\\ & I _ { 2 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma ^ { ( 3 ) } \\backslash U _ { \\delta ( \\pm \\eta ) } } \\left ( J ^ E ( s ) - I \\right ) d s , \\\\ & I _ { 3 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\mu ( s ) ( J ^ E ( s ) - I ) d s . \\end{align*}"}
{"id": "4972.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ t u - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = 0 , \\\\ & \\mbox { d i v } u = 0 , \\\\ & u ( 0 , x ) = f ^ { \\omega } ( x ) . \\endaligned \\right . \\end{align*}"}
{"id": "8224.png", "formula": "\\begin{align*} \\frac { [ z ^ n ] \\frac { \\partial ^ 2 } { \\partial x _ h ^ 2 } ( P _ 1 + P _ 2 + P _ 3 ) | _ { x _ 1 = x _ 2 = x _ 3 = 1 } + [ z ^ n ] \\frac { \\partial } { \\partial x _ h } ( P _ 1 + P _ 2 + P _ 3 ) | _ { x _ 1 = x _ 2 = x _ 3 = 1 } } { [ z ^ n ] ( P _ 1 + P _ 2 + P _ 3 ) | _ { x _ 1 = x _ 2 = x _ 3 = 1 } } , \\end{align*}"}
{"id": "8263.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { F } _ { 1 \\bar { 4 } \\ , \\bar { 2 } 3 } ) = \\iota \\otimes \\mathbf { F } _ { 1 \\bar { 4 } \\ , \\bar { 2 } 3 } + \\mathbf { F } _ { 1 } \\otimes \\mathbf { F } _ { \\bar { 3 } \\ , \\bar { 1 } 2 } + \\mathbf { F } _ { 1 \\bar { 2 } } \\otimes \\mathbf { F } _ { \\bar { 1 } 2 } + \\mathbf { F } _ { 1 \\bar { 3 } \\ , \\bar { 2 } } \\otimes \\mathbf { F } _ { 1 } + \\mathbf { F } _ { 1 \\bar { 4 } \\ , \\bar { 2 } 3 } \\otimes \\iota . \\end{align*}"}
{"id": "8314.png", "formula": "\\begin{align*} \\left \\{ ~ ~ \\begin{aligned} & \\int _ 0 ^ T ( v - y ) \\dd ( y - u ) \\geq 0 & & \\forall v \\in C ( [ 0 , T ] ; Z ) , \\\\ & y ( t ) \\in Z \\forall t \\in [ 0 , T ] , & & y ( 0 ) = y _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "1234.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ m p _ i \\mu \\circ f _ i ^ { - 1 } . \\end{align*}"}
{"id": "4562.png", "formula": "\\begin{align*} \\bigg | \\ln \\frac { \\mathbf { P } ( S _ n / ( \\sigma \\sqrt { n } ) > x ) } { 1 - \\Phi ( x ) } \\bigg | = O \\bigg ( \\frac { 1 + x ^ 3 } { \\sqrt { n } } \\bigg ) , \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "7302.png", "formula": "\\begin{align*} x + y + z & = 3 , \\\\ x ^ 2 + y ^ 2 + z ^ 2 & = 1 5 , \\\\ x ^ 3 + y ^ 3 + z ^ 3 & = 4 5 . \\end{align*}"}
{"id": "7620.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\Delta P & = u ^ { \\frac { \\alpha + 1 } { \\alpha } } ( \\frac { 1 } { n } | h | ^ { 2 } - H ^ { 2 } ) - \\frac { 1 } { n } u ^ { \\frac { 1 } { \\alpha } } \\overline { \\mathrm { R i c } } ( \\nu , \\lambda \\partial _ { r } ^ { T } ) \\\\ & + \\frac { 1 } { \\alpha } \\bar { g } ( \\lambda \\partial _ { r } , \\nabla \\log u ) - \\frac { 1 } { n \\alpha } u ^ { \\frac { \\alpha + 1 } { \\alpha } } | \\nabla \\log u | ^ { 2 } . \\end{align*}"}
{"id": "247.png", "formula": "\\begin{align*} \\mathcal { L } ( g _ j ) ( x ) = - \\langle x ; \\nabla ( g _ j ) ( x ) \\rangle + \\Delta ( g _ j ) ( x ) . \\end{align*}"}
{"id": "9032.png", "formula": "\\begin{align*} \\begin{aligned} K : & = \\{ \\rho = ( \\rho _ 1 , \\cdots , \\rho _ s ) , u = ( u _ 1 , \\cdots , u _ s ) : \\\\ & \\partial _ t \\rho _ i + \\nabla \\cdot ( \\rho _ i u _ i ) = 0 , ( \\rho _ i u _ i ) \\cdot \\mathbf { n } = 0 \\partial \\Omega \\times [ 0 , 1 ] , \\\\ & \\rho _ i \\in \\mathcal { P } ( \\Omega ) , \\rho _ i ( x , 0 ) = \\rho _ i ^ 0 ( x ) , \\rho _ i ( x , 1 ) = \\rho _ i ^ 1 ( x ) \\} . \\end{aligned} \\end{align*}"}
{"id": "4902.png", "formula": "\\begin{align*} \\frac { f '' ( z ) } { f ( z ) } = L ' ( z ) + L ( z ) ^ 2 \\sim c s ( s + 1 ) z ^ { - 2 - s } . \\end{align*}"}
{"id": "2604.png", "formula": "\\begin{align*} f _ n = V ^ * _ g F _ n = \\iint _ { \\R ^ { 2 d } } F _ n ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) = \\iint _ { K _ n } V _ g f ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "8892.png", "formula": "\\begin{align*} \\psi _ n ( z _ 0 , \\ldots , z _ { q - 1 } ) = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( z _ 0 , \\ldots , z _ i , h _ n ( z _ i ) , \\ldots , h _ n ( z _ { q - 1 } ) ) . \\end{align*}"}
{"id": "2973.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Lambda _ { n , 1 } = 0 , \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } = 4 . \\end{align*}"}
{"id": "1874.png", "formula": "\\begin{align*} P _ n ( x ) = 2 ^ n x ^ { n + 1 } E _ n \\left ( { 1 + x ^ 2 \\over 2 x ^ 2 } \\right ) . \\end{align*}"}
{"id": "751.png", "formula": "\\begin{align*} f ( x ^ i ) = f ^ i , \\nabla f ( x ^ i ) = g ^ i \\ \\ i \\in I , \\end{align*}"}
{"id": "5982.png", "formula": "\\begin{align*} \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; x ) = \\frac { \\mathrm { e } ^ { z x } } { l _ { \\pi _ { \\lambda , \\beta } } ( z ) } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { n ! } A _ { n } ^ { \\lambda , \\beta } ( x ) , \\end{align*}"}
{"id": "7520.png", "formula": "\\begin{align*} \\log \\Gamma ( s ) = \\left ( s - \\frac { 1 } { 2 } \\right ) \\log s - s - \\frac { 1 } { 2 } \\log ( 2 \\pi ) + \\mathcal { O } \\left ( \\frac { 1 } { s } \\right ) \\end{align*}"}
{"id": "6630.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq m < n < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( \\frac { 1 } { 2 } + s _ 1 - z ) } p ^ { n ( \\frac { 1 } { 2 } + s _ 2 + z ) } } = \\frac { \\tau _ B ( p ) } { p ^ { \\frac { 1 } { 2 } + s _ 2 + z } } + O \\bigg ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\bigg ) . \\end{align*}"}
{"id": "4979.png", "formula": "\\begin{align*} j ( ( \\eta , \\xi ) ) = \\left ( \\left ( { \\eta ( x ) \\atop \\eta ( x ) } \\right ) , \\left ( { \\xi ( x ) \\atop \\xi ( x ) } \\right ) \\right ) . \\end{align*}"}
{"id": "6747.png", "formula": "\\begin{align*} \\mathbf b = & p ( t , S _ 0 ) V ( t , S _ 0 ) \\bar { \\mathbf L } _ 2 [ : , 1 ] + w ( t , S _ 0 ) V ( t , S _ 0 ) \\bar { \\mathbf L } _ 1 [ : , 1 ] \\\\ & + p ( t , S _ { M + 1 } ) V ( t , S _ { M + 1 } ) \\bar { \\mathbf L } _ 2 [ : , M + 2 ] + w ( t , S _ { M + 1 } ) V ( t , S _ { M + 1 } ) \\bar { \\mathbf L } _ 1 [ : , M + 2 ] + g ( t , \\mathbf S ) , \\end{align*}"}
{"id": "6524.png", "formula": "\\begin{align*} \\binom { j + m } { j } & \\sim \\dfrac { j ^ m } { m ! } ( j \\to \\infty ) , \\\\ t _ n ^ { ( 2 m ) } & \\sim m ! \\cdot ( 2 m - 1 ) ! ! \\cdot ( \\log n ) ^ { m } ( n \\to \\infty ) . \\end{align*}"}
{"id": "1143.png", "formula": "\\begin{align*} & E _ 1 ^ { ( 1 2 ) } = \\frac { 3 } { 4 t \\left ( D _ { 0 } ^ { - 2 } r ^ { - 1 } ( \\eta ) - 1 \\right ) } \\tilde { T } _ { \\eta , 1 } ^ { ( 1 2 ) } - \\frac { 3 } { 4 t \\left ( D _ { 0 } ^ { - 2 } r ^ { - 1 } ( - \\eta ) - 1 \\right ) } \\tilde { T } _ { - \\eta , 1 } ^ { ( 1 2 ) } + \\mathcal { O } ( t ^ { - 2 } ) , \\end{align*}"}
{"id": "6965.png", "formula": "\\begin{align*} D ^ * \\Phi ( ( x , y ) ; ( u , v ) ) ( y ^ * ) \\subset \\begin{cases} \\bigcup \\limits _ { w \\in D g ( x ) ( u ) } D ^ * g ( x ; ( u , w ) ) ( y ^ * ) & y ^ * \\in \\mathcal N _ D ( g ( x ) - y ; w - v ) , \\\\ \\varnothing & \\end{cases} \\end{align*}"}
{"id": "2778.png", "formula": "\\begin{align*} R ( h ) & = \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | Q + h | ^ p \\right ) | Q + h | ^ { p - 2 } ( Q + h ) \\\\ & - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) \\left ( Q ^ { p - 1 } + \\frac { p - 2 } { 2 } Q ^ { p - 2 } \\bar { h } + \\frac { p } { 2 } Q ^ { p - 2 } h \\right ) \\\\ & - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * \\left ( \\frac { p } { 2 } Q ^ { p - 1 } \\bar { h } + \\frac { p } { 2 } Q ^ { p - 1 } h \\right ) \\right ) Q ^ { p - 1 } . \\end{align*}"}
{"id": "7040.png", "formula": "\\begin{align*} b _ j : = q _ j + \\Big ( 1 - \\sum _ { k = 1 } ^ { L } \\phi _ k q _ k \\Big ) e _ j . \\end{align*}"}
{"id": "7378.png", "formula": "\\begin{align*} \\begin{aligned} V _ { r r } ( r , p ) & = 2 \\beta r ^ { - 3 } | p | ^ 2 + a ( 1 - \\beta ) ^ { \\alpha - 1 } \\beta ( 1 - \\alpha ) ( \\beta - \\alpha \\beta - 1 ) r ^ { \\beta ( 1 - \\alpha ) - 2 } | p | ^ \\alpha \\\\ & = r ^ { - 3 } | p | ^ \\alpha \\left ( 2 \\beta | p | ^ { 2 - \\alpha } + a ( 1 - \\beta ) ^ { \\alpha - 1 } ( \\beta - \\alpha \\beta ) ( \\beta - \\alpha \\beta - 1 ) r ^ { \\beta ( 1 - \\alpha ) + 1 } \\right ) . \\end{aligned} \\end{align*}"}
{"id": "370.png", "formula": "\\begin{align*} \\mathcal L ( \\rho , m , S ) & : = \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > + \\mathcal A ( \\rho , m ) \\\\ & \\quad - \\int _ 0 ^ 1 ( \\ < \\dot S , \\rho \\ > + \\ < m , \\nabla _ G S \\ > ( 1 + \\dot W ^ { \\delta } ( t ) ) ) d t , \\end{align*}"}
{"id": "38.png", "formula": "\\begin{align*} ( d \\theta _ { t } ^ { c } ) ^ { n } \\wedge \\theta \\wedge \\theta _ { t } ^ { c } = \\sum ^ n _ { k = 1 } \\binom n k ( d \\theta _ { t } ^ { c } ) ^ { n - k } \\wedge d \\zeta ^ k _ t \\wedge \\theta \\wedge \\theta ^ c + \\sum ^ n _ { k = 1 } \\binom n k ( d \\theta _ { t } ^ { c } ) ^ { n - k } \\wedge d \\zeta ^ k _ t \\wedge \\theta \\wedge \\zeta _ t \\end{align*}"}
{"id": "4990.png", "formula": "\\begin{align*} \\partial _ y ( B \\circ A ^ { r _ 0 } \\circ H ) ( x , y ) = \\partial _ y B ( x , \\tilde { y } ) = \\partial _ 2 B ( x , \\tilde { y } ) \\cdot \\partial _ y \\tilde { y } = O ( \\varepsilon ) \\cdot O ( \\varepsilon ) = O ( \\varepsilon ^ { 2 } ) \\end{align*}"}
{"id": "834.png", "formula": "\\begin{align*} \\int _ A \\langle \\nabla u ( x , y ) , \\nabla v ( x , y ) \\rangle _ { \\rho , ( x , y ) } \\mu _ \\omega ( x , y ) = \\int _ A \\langle \\nabla u ( x , y ) , \\nabla v ( x , y ) \\rangle _ { ( x , y ) } \\ , d ( \\mu \\times \\mathcal { L } ^ 1 ) ( x , y ) . \\end{align*}"}
{"id": "7349.png", "formula": "\\begin{align*} \\begin{aligned} & \\varphi _ t ( x _ q , t _ q ) + F ( w _ q ( x _ q , t _ q ) , \\nabla \\varphi ( x _ q , t _ q ) , \\nabla ^ 2 \\varphi ( x _ q , t _ q ) - C _ q \\varepsilon I , W _ \\star [ x _ 0 , t _ 0 ] ) \\\\ & \\geq \\lambda \\frac { u ( y _ q , t _ q ) ^ { q - 1 } } { \\varphi ( x _ q , t _ q ) ^ { q - 1 } } D _ { 1 , q } + ( 1 - \\lambda ) \\frac { u ( z _ q , t _ q ) ^ { q - 1 } } { \\varphi ( x _ q , t _ q ) ^ { q - 1 } } D _ { 2 , q } , \\end{aligned} \\end{align*}"}
{"id": "2544.png", "formula": "\\begin{align*} \\mathbf { a } _ M ( z , \\tau ) \\mathbf { a } _ M ( z ' , \\tau ' ) = ( M z , \\tau ) ( M z ' , \\tau ' ) = ( M ( z + z ' ) , \\tau + \\tau ' + \\tfrac { 1 } { 2 } \\sigma ( M z , M z ' ) ) \\end{align*}"}
{"id": "1615.png", "formula": "\\begin{align*} R ^ { 1 , \\operatorname { l i n } } & = \\nu u _ 0 + \\rho _ 0 w , \\\\ R ^ { 1 , \\operatorname { c o r r } } & = \\rho _ 0 w _ c + \\nu w _ c . \\end{align*}"}
{"id": "5629.png", "formula": "\\begin{align*} L _ { i j } \\rhd x ^ h = ( a _ { i k } x ^ k + a _ { 0 i } ) \\delta ^ h _ j - ( a _ { j k } x ^ k + a _ { 0 j } ) \\delta ^ h _ i , \\end{align*}"}
{"id": "1652.png", "formula": "\\begin{align*} G _ \\lambda ^ { B , T } f ( x ) : = \\int _ B g _ \\lambda ^ { B , T } ( x , y ) f ( y ) \\ : \\mu ( d y ) \\end{align*}"}
{"id": "4468.png", "formula": "\\begin{align*} ( a ^ { \\dagger } a ) _ { k , \\lambda } | m \\rangle = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( a ^ { \\dagger } ) ^ { l } a ^ { l } | m \\rangle = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( m ) _ { l } | m \\rangle . \\end{align*}"}
{"id": "3310.png", "formula": "\\begin{align*} c l ( G _ i ) = \\sum _ p n _ { i , p } c l ( G _ { i , p } ) \\end{align*}"}
{"id": "3706.png", "formula": "\\begin{align*} I _ { 2 1 } \\leq & \\sum _ { q \\geq 0 } \\int _ 0 ^ { \\frac { t } 2 } c _ q \\lambda _ q ^ { \\alpha + \\beta - 2 \\gamma } e ^ { - \\mu \\lambda _ q ^ \\alpha ( t - \\tau ) } \\| B ( \\tau ) \\| ^ 2 _ { H ^ { \\frac 5 2 - \\alpha + \\gamma } } \\ , d \\tau \\\\ \\lesssim & \\int _ 0 ^ { \\frac { t } 2 } ( t - \\tau ) ^ { - \\frac { \\alpha + \\beta - 2 \\gamma } { \\alpha } } \\| B ( \\tau ) \\| ^ 2 _ { H ^ { \\frac 5 2 - \\alpha + \\gamma } } \\ , d \\tau . \\end{align*}"}
{"id": "6718.png", "formula": "\\begin{align*} g _ 1 ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + g _ r ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ r ) ^ { q ^ d } = 0 \\end{align*}"}
{"id": "479.png", "formula": "\\begin{align*} \\mathbf { L } _ { \\mathrm { e x } } : = \\begin{array} { c } \\begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 1 & 3 \\\\ 0 & 1 & 0 & 0 & 2 & 2 & 4 \\\\ 0 & 0 & 1 & 1 & 4 & 5 & 0 \\end{pmatrix} \\\\ \\begin{array} { c c c c c c c } { \\scriptstyle i } & { \\scriptstyle j } & { \\scriptstyle m } & { \\scriptstyle \\alpha } & { \\scriptstyle \\beta } & { \\scriptstyle \\gamma } & { \\scriptstyle \\delta } \\end{array} \\end{array} \\end{align*}"}
{"id": "3371.png", "formula": "\\begin{align*} \\Big [ \\sum _ { i = 0 } ^ { + \\infty } x _ i t ^ { i } , \\sum _ { j = 0 } ^ { + \\infty } y _ j t ^ { j } , \\sum _ { k = 0 } ^ { + \\infty } z _ k t ^ { k } \\Big ] = \\sum _ { s = 0 } ^ { + \\infty } \\sum _ { i + j + k = s } [ x _ i , y _ j , z _ k ] t ^ { s } , \\ ; \\forall x _ i , y _ j , z _ k \\in L . \\end{align*}"}
{"id": "8900.png", "formula": "\\begin{align*} \\check H _ { c t } ^ q ( X ; A _ X ) = H Y ^ q _ b ( X , A ) . \\end{align*}"}
{"id": "4415.png", "formula": "\\begin{align*} R e ( ( z ^ 0 ) ^ * A w ) = c _ 1 \\ ; \\ ; \\forall w \\in S ^ { n } _ \\beta \\Rightarrow R e ( ( z ^ 0 ) ^ * A w ^ 0 ) = c _ 1 \\end{align*}"}
{"id": "4424.png", "formula": "\\begin{align*} \\begin{array} { l l } z _ 1 2 + z _ 2 ( 3 + i ) = \\eta \\\\ z _ 1 ( 1 + i ) + z _ 2 3 = \\eta \\\\ z _ 1 + z _ 2 = 1 \\end{array} \\end{align*}"}
{"id": "4256.png", "formula": "\\begin{align*} T ( t ) f ( x ) = \\begin{cases} f ( x - t ) & \\mbox { f o r } x > t \\\\ 0 & \\mbox { f o r } x \\leq t . \\end{cases} \\end{align*}"}
{"id": "4426.png", "formula": "\\begin{align*} & u _ { t t } + u _ { x x x x } + F ( u _ t ) + G ( u ) = f ( x , t ) , ( c , d ) \\times ( 0 , T ] \\\\ & u ( c , t ) = 0 = u ( d , t ) , t ( 0 , T ] \\\\ & u _ { x x } ( c , t ) = 0 = u _ { x x } ( d , t ) , t ( 0 , T ] \\\\ & u ( x , 0 ) = u _ 0 ( x ) , u _ t ( x , 0 ) = u _ 1 ( x ) x ( c , d ) . \\\\ \\end{align*}"}
{"id": "6833.png", "formula": "\\begin{align*} | x _ 3 ' + x _ 4 ' i | ^ 2 + | ( x _ 3 ' + x _ 4 ' i ) ( x + y i ) - ( x _ 1 ' + x _ 2 ' i ) w | ^ 2 = w \\end{align*}"}
{"id": "4580.png", "formula": "\\begin{align*} \\hat { \\theta } _ n = \\frac { \\sum _ { k = 1 } ^ n X _ { k - 1 } X _ k } { \\sum _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } . \\end{align*}"}
{"id": "6497.png", "formula": "\\begin{align*} M ^ { ( 2 \\ell - 1 ) } _ n & \\sim C ' _ \\ell \\cdot n ^ { \\ell - 1 / 2 } ( \\log n ) ^ { \\ell - 1 } ( n \\to \\infty ) \\end{align*}"}
{"id": "8750.png", "formula": "\\begin{align*} t _ I = \\prod _ { i \\in I } \\Biggl ( \\sum _ { j = 0 } ^ n a _ { i j } \\lambda _ { i j } \\Biggr ) \\cdot \\prod _ { i ' \\notin I } \\Biggl ( \\sum _ { j = 0 } ^ n \\lambda _ { i ' j } \\Biggr ) = \\sum _ { e \\in E } \\Biggl ( \\prod _ { i \\in I } a _ { i e ( i ) } \\Biggr ) y _ e = : L _ I ( y ) I \\subseteq \\{ 1 , \\ldots , d \\} . \\end{align*}"}
{"id": "4023.png", "formula": "\\begin{align*} \\lambda ^ p _ k : = - \\mu _ k : = - k ^ 2 \\pi ^ 2 - 2 c _ k k \\pi - 2 i d _ k k \\pi - ( c _ k ^ 2 - d _ k ^ 2 ) - 2 i c _ k d _ k , \\forall k \\geq k _ 0 . \\end{align*}"}
{"id": "3972.png", "formula": "\\begin{align*} & \\eta ^ { \\prime \\prime \\prime } ( x ) - \\lambda \\eta ^ { \\prime \\prime } ( x ) - 2 \\lambda \\eta ^ { \\prime } ( x ) + \\lambda ^ 2 \\eta ( x ) = 0 , \\ \\ \\forall x \\in ( 0 , 1 ) , \\\\ & \\eta ( 0 ) = 0 , \\ \\ \\eta ( 1 ) = 0 , \\ \\ \\eta ^ { \\prime \\prime } ( 0 ) = \\eta ^ { \\prime \\prime } ( 1 ) . \\end{align*}"}
{"id": "4106.png", "formula": "\\begin{align*} \\begin{aligned} | \\sigma _ n ( x ) | ^ 2 & \\le c \\cdot \\min \\bigl ( \\sqrt { n } \\ , ( 1 + x ^ 2 ) , \\sigma ^ 2 ( x ) \\bigr ) , \\\\ | \\mu _ n ( x ) | & \\le c \\cdot \\min \\bigl ( \\sqrt { n } \\ , ( 1 + | x | ) , | \\mu ( x ) | \\bigr ) . \\end{aligned} \\end{align*}"}
{"id": "314.png", "formula": "\\begin{align*} \\mathcal S ( \\rho _ t , \\Phi _ t ) & = \\ < \\rho ( 0 ) , \\Phi ( 0 ) \\ > - \\ < \\rho ( T ) , \\Phi ( T ) \\ > + \\int _ 0 ^ T \\ < \\partial _ t \\Phi ( t ) , \\rho _ t \\ > + \\mathcal H _ 0 ( \\rho _ t , \\Phi _ t ) d t \\\\ & + \\int _ 0 ^ T \\mathcal H _ 1 ( \\rho _ t , \\Phi _ t ) \\dot W _ { \\delta } d t . \\end{align*}"}
{"id": "8186.png", "formula": "\\begin{align*} S ( H _ 5 , 5 f ) = \\frac { 7 f - 1 0 d + 2 + \\varepsilon _ d } { 5 } \\hbox { a n d } N _ 5 ' ( f , H ) = - \\frac { 4 } { 3 } d + \\frac { 1 + \\varepsilon _ d } { 6 } , \\end{align*}"}
{"id": "5261.png", "formula": "\\begin{align*} \\sum _ i \\varphi ( A b _ i ) a _ i = 0 , \\end{align*}"}
{"id": "3817.png", "formula": "\\begin{align*} \\theta _ 1 ( a _ 1 ) & = \\\\ \\theta _ 2 ( c ) & = . \\end{align*}"}
{"id": "8451.png", "formula": "\\begin{align*} ( [ a , b ] [ c , d ] ) [ e , f ] = [ u a , v d ] [ e , f ] = [ i ( u a ) , ( j q ) f ] , \\end{align*}"}
{"id": "8796.png", "formula": "\\begin{align*} \\Bigl \\{ ( f , \\phi , \\lambda , \\delta ) \\Bigm | \\check { b } \\bigl ( V ( \\lambda ) \\bigr ) \\leq \\phi \\leq \\hat { b } \\bigl ( V ( \\lambda ) \\bigr ) , \\ f = A ( \\lambda ) , \\ ( \\lambda _ i , \\delta _ i ) \\in ( \\ref { e q : S O S 2 - l o g } ) , \\ ; i = 1 , 2 \\Bigr \\} . \\end{align*}"}
{"id": "5614.png", "formula": "\\begin{align*} \\Xi _ { \\perp \\star } : = \\{ X \\in \\Xi _ \\star ~ | ~ \\mathbf { g } _ \\star ( X , \\Xi _ { t \\star } ) = 0 \\} \\Omega _ { t , \\star } : = \\{ \\omega \\in \\Omega _ \\star ~ | ~ \\mathbf { g } _ \\star ^ { - 1 } ( \\omega , \\Omega _ { \\perp \\star } ) = 0 \\} . \\end{align*}"}
{"id": "2667.png", "formula": "\\begin{align*} \\int _ { P _ R } F ( z ) \\overline { z ^ \\beta } e ^ { - \\pi | z | ^ 2 } \\ , d z = \\sum _ { \\alpha \\geq 0 } c _ \\alpha \\int _ { P _ R } z ^ \\alpha \\overline { z ^ \\beta } e ^ { - \\pi | z | ^ 2 } \\ , d z = c _ \\beta \\ , \\mu _ { \\beta , R } . \\end{align*}"}
{"id": "6225.png", "formula": "\\begin{align*} M _ A = \\sum _ { j = 1 } ^ d \\Big ( P _ { 2 j - 1 } A \\tau _ { j } ^ { - 1 } + P _ { 2 j } A \\tau _ { j } \\Big ) . \\end{align*}"}
{"id": "2491.png", "formula": "\\begin{align*} \\int _ { \\mathbf { H } _ r } F ( \\mathbf { h } _ r ) \\ , d \\mathbf { h } _ r = \\int _ { \\R ^ { 2 d } } \\int _ 0 ^ 1 F ( x , \\omega , e ^ { 2 \\pi i \\tau } ) \\ , d \\tau \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "9042.png", "formula": "\\begin{align*} \\frac { ( Y ( \\theta ) ) ^ 2 } { X ( \\theta ) } - \\theta \\frac { ( Y ^ 0 ) ^ 2 } { X ^ 0 } - ( 1 - \\theta ) \\frac { ( Y ^ 1 ) ^ 2 } { X ^ 1 } = - \\theta ( 1 - \\theta ) \\frac { ( X ^ 1 Y ^ 0 - X ^ 0 Y ^ 1 ) ^ 2 } { X ^ 0 X ^ 1 X ( \\theta ) } . \\end{align*}"}
{"id": "4587.png", "formula": "\\begin{align*} e ^ { - \\lambda ^ 2 / 2 } \\mathbf { E } \\left ( e ^ { - \\lambda \\mathcal { N } } \\mathbf { 1 } _ { \\{ \\mathcal { N } > 0 \\} } \\right ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ 0 ^ { \\infty } e ^ { - ( y + \\lambda ) ^ 2 / 2 } d y = 1 - \\Phi \\left ( \\lambda \\right ) \\end{align*}"}
{"id": "1608.png", "formula": "\\begin{align*} M : = M _ 0 \\max \\lbrace ( 3 d ) ^ \\frac { 1 } { p } , ( 3 d ) ^ \\frac { 1 } { p ' } \\rbrace , \\end{align*}"}
{"id": "9067.png", "formula": "\\begin{align*} I _ 3 & \\leq | \\partial _ { \\phi } \\mathcal { F } _ h ( u ^ * ) | \\cdot | \\tilde \\phi | \\\\ & \\leq \\frac { 1 } { h } ( | \\epsilon | + \\beta _ a ^ { - 1 } + \\beta _ b ^ { - 1 } ) C _ \\phi ^ * \\cdot \\gamma | z | ( | c | + k | d | ) h ^ 2 = C _ 0 C ^ * _ \\phi h . \\end{align*}"}
{"id": "1590.png", "formula": "\\begin{align*} \\div ( \\beta ( \\rho ) u ) = \\beta ' ( \\rho ) \\div ( \\rho u ) + \\left [ \\beta ( \\rho ) - \\rho \\beta ' ( \\rho ) \\right ] \\div u , \\end{align*}"}
{"id": "5416.png", "formula": "\\begin{align*} a ' + a '' & = 1 , \\\\ b ' + b '' & = \\tfrac { \\sin \\alpha } { \\sin \\gamma } , \\\\ c ' + c '' & = \\tfrac { \\sin \\beta } { \\sin \\gamma } . \\end{align*}"}
{"id": "7819.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\lambda _ j ^ { 2 \\beta \\left ( \\frac { \\sigma } { 2 } + n + \\tau + \\frac { 1 } { 2 } - \\gamma \\right ) } \\widetilde { \\lambda } _ j ^ { - \\alpha } \\eqsim _ { \\text ( \\alpha , \\beta , \\gamma , \\sigma , n , \\tau ) } \\sum _ { j = 1 } ^ \\infty j ^ { \\frac { 4 } { d } \\left [ \\beta \\left ( n + \\tau + \\frac { 1 + \\sigma } { 2 } \\right ) - \\beta \\gamma - \\frac { \\alpha } { 2 } \\right ] } , \\end{align*}"}
{"id": "5452.png", "formula": "\\begin{align*} { ( \\mathcal { M } ( \\tilde u _ 0 , a , b ) u ) ( t , \\cdot ) } & = e ^ { - A ( t - s ) } \\tilde u _ 0 - \\chi \\int _ s ^ t e ^ { - A ( t - \\tau ) } \\nabla \\cdot \\left ( \\frac { u ( \\tau , \\cdot ) } { v ( \\tau , \\cdot ) } \\nabla v ( \\tau , \\cdot ) \\right ) d \\tau \\\\ & + \\int _ s ^ t e ^ { - A ( t - \\tau ) } u ( \\tau , \\cdot ) \\big [ { \\mu } + a ( \\tau , \\cdot ) - b ( \\tau , \\cdot ) u ( \\tau , \\cdot ) \\big ] d \\tau , \\end{align*}"}
{"id": "1806.png", "formula": "\\begin{align*} \\sigma ( X , [ \\alpha ] ) : = \\int _ X c _ X L ( X ) \\wedge \\psi _ X ^ * ( \\alpha ) , \\end{align*}"}
{"id": "6135.png", "formula": "\\begin{align*} S _ { \\varepsilon } = \\left \\{ R _ { \\varepsilon } ( s , t ) = R ( s , t ) + 2 \\varepsilon U ^ { 1 } ( s , t ) \\right \\} \\end{align*}"}
{"id": "6070.png", "formula": "\\begin{align*} S ^ { A , B } _ { \\varepsilon } = \\left \\{ R ^ { A , B } _ { \\varepsilon } ( s , t ) = R ( s , t ) + \\varepsilon \\left ( A \\times R ( s , t ) + B \\right ) \\right \\} \\end{align*}"}
{"id": "6426.png", "formula": "\\begin{align*} \\Gamma ^ + : = \\{ \\xi \\in \\Gamma ( S ^ + ( T N ) \\otimes E ) : P _ { > 0 } \\left ( \\xi | _ { \\partial N } \\right ) = 0 \\} . \\end{align*}"}
{"id": "1300.png", "formula": "\\begin{align*} | \\Lambda ( M , \\Gamma ) | = \\sum _ { n = 0 } ^ { \\infty } | \\Lambda _ { ( n , \\infty ) } ( M , \\Gamma ) | = \\sum _ { n = 0 } ^ { \\infty } | \\Lambda _ { ( 0 , \\infty ) } ( M - n R , \\Gamma - n [ \\gamma ] ) | . \\end{align*}"}
{"id": "2441.png", "formula": "\\begin{align*} [ \\pi ( \\l ' ) , \\pi ( \\l ) ] = \\left ( 1 - e ^ { 2 \\pi i \\sigma ( \\l ' , \\l ) } \\right ) \\pi ( \\l ' ) \\pi ( \\l ) . \\end{align*}"}
{"id": "3195.png", "formula": "\\begin{align*} \\mu ( x _ 0 ) = u _ 0 ^ \\star ( x _ 0 ) , \\end{align*}"}
{"id": "5011.png", "formula": "\\begin{align*} D _ { k n } \\le { \\log 2 \\over \\pi K } + { 1 \\over \\pi | \\bar { u } _ { k n } | } \\sum _ { k = 1 } ^ K { 1 \\over c } = { \\log 2 \\over \\pi K } + { K \\over \\pi | \\bar { u } _ { k n } | c } . \\end{align*}"}
{"id": "9254.png", "formula": "\\begin{align*} \\forall x ^ X \\left ( \\exists y ^ X , z ^ X ( z \\in A y \\land x = _ X y + _ X z ) \\rightarrow ( x - _ X J ^ A _ { 1 } x ) \\in A ( J ^ A _ { 1 } x ) \\right ) , \\end{align*}"}
{"id": "2600.png", "formula": "\\begin{align*} \\widetilde { f } = \\frac { 1 } { \\langle g , \\widetilde { g } \\rangle } V _ g ^ * V _ { \\widetilde { g } } f \\in M ^ { p , q } \\end{align*}"}
{"id": "6144.png", "formula": "\\begin{align*} Q ( x ) = \\sum _ { k = 0 } ^ m q _ k x ^ k \\end{align*}"}
{"id": "664.png", "formula": "\\begin{align*} \\begin{cases} \\ E ^ \\prime ( n ) \\ = \\ \\min \\{ E ( i ) : 0 \\leq i \\leq n \\} \\\\ [ 8 p t ] \\ k ( n ) \\ = \\ \\min \\{ i : 0 \\leq i \\leq n , \\ E ( i ) = E ^ \\prime ( n ) \\} \\\\ [ 8 p t ] \\ A ^ \\prime ( n ) \\ = \\ A ( k ( n ) ) \\end{cases} . \\end{align*}"}
{"id": "7727.png", "formula": "\\begin{align*} u = x + t , \\ ; v = t - x , \\end{align*}"}
{"id": "6442.png", "formula": "\\begin{align*} ( x , b ) + ( x ' , b ' ) = ( \\rho ( x + \\varphi ( b , x ' ) + \\gamma ( b , b ' ) , b + b ' ) , b + b ' ) \\end{align*}"}
{"id": "60.png", "formula": "\\begin{align*} \\mathrm { r k } ( L ' _ { 2 , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( L ' _ { 2 , 1 } ) = 2 , \\\\ \\mathrm { r k } ( K _ { \\ell , 2 , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( K _ { \\ell , 2 , 1 } ) = 1 . \\end{align*}"}
{"id": "5729.png", "formula": "\\begin{align*} 0 \\leq | \\xi | \\leq \\epsilon t + L _ \\epsilon t ^ { 1 / 2 } , \\ \\forall t \\geq 0 , \\ \\xi \\in \\partial F ^ + ( t ) = \\partial F ( t ) . \\end{align*}"}
{"id": "5069.png", "formula": "\\begin{align*} G = & \\{ ( x , r ) : x \\in \\mathbb { R } ^ k , 0 < r < d ( x , K ) , z _ 1 , z _ 2 \\in K \\\\ & d ( x , z _ 1 ) , d ( x , z _ 2 ) \\leq d ( x , K ) + \\epsilon r \\\\ & | \\theta | > \\cos ^ { - 1 } \\left ( 2 \\left ( \\frac { 1 - ( 2 \\delta + \\epsilon ) } { 1 + 2 \\delta } \\right ) ^ 2 - 1 \\right ) \\} \\end{align*}"}
{"id": "620.png", "formula": "\\begin{align*} C ( k ) \\ = \\ \\frac { 1 } { A ( k + c ) } ( k + c \\geq c \\implies A ( k + c ) \\neq 0 ) . \\end{align*}"}
{"id": "6856.png", "formula": "\\begin{align*} X = X ( 0 ) \\cup \\ldots \\cup X ( d ) , \\end{align*}"}
{"id": "6447.png", "formula": "\\begin{align*} T _ h ( \\lambda ) ( u ) ( { x } ) & = \\frac { \\eta _ 0 ^ 1 } { 4 \\pi } \\int _ { B _ 1 } \\frac { \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \\\\ & + \\frac { \\eta _ 0 ^ 2 } { 4 \\pi } \\int _ { B _ 2 } \\frac { \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \\end{align*}"}
{"id": "1340.png", "formula": "\\begin{align*} R ( Q ) = R _ 0 ( Q ) \\cap \\{ \\C ^ \\vee ( Q ) Q \\} \\cap \\{ \\C ^ \\vee ( Q ^ + ) Q ^ + \\} . \\end{align*}"}
{"id": "4293.png", "formula": "\\begin{align*} \\partial _ s v = \\partial _ { \\xi } ^ 2 v + \\frac { d + 1 } { \\xi } \\partial _ \\xi v + \\frac { 1 } { 2 } \\frac { \\lambda _ s } { \\lambda } \\Lambda _ \\xi v - 3 ( d - 2 ) v ^ 2 - ( d - 2 ) \\xi ^ 2 v ^ 3 , \\end{align*}"}
{"id": "861.png", "formula": "\\begin{align*} \\bar { \\Delta } _ { \\rm P r o a c t i v e } = \\frac { \\mathbb { E } \\sum _ { t \\in \\mathcal { I } _ j } \\Delta { \\left ( t \\right ) } } { \\mathbb { E } T ^ { \\rm P r o a c } _ j } = \\frac { \\mathbb { E } T ^ { \\rm P r o a c } _ j } { 2 \\mathbb { E } \\left ( T ^ { \\rm P r o a c } _ j \\right ) ^ 2 } + \\mathbb { E } \\tau ^ { \\rm P r o a c } _ { V _ j } - \\tau _ { \\rm f } - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "6752.png", "formula": "\\begin{align*} D ^ 2 v _ 0 = \\frac { 1 } { \\bar { h } _ { 0 } } \\left [ \\frac { 1 } { h _ { 0 } } v _ { - 1 } - \\left ( \\frac { 1 } { h _ { 0 } } + \\frac { 1 } { h _ { 1 } } \\right ) v _ 0 + \\frac { 1 } { h _ { 1 } } v _ 1 \\right ] , \\end{align*}"}
{"id": "9201.png", "formula": "\\begin{align*} ( \\lambda x ^ \\rho . t ) ( s ^ \\rho ) = _ \\tau t [ s / x ] . \\end{align*}"}
{"id": "689.png", "formula": "\\begin{align*} \\widehat { \\kappa } _ { 2 k ; \\alpha } ^ { ( L ) } : = \\frac { \\kappa _ { 2 k ; \\alpha } ^ { ( \\ell ) } } { ( K _ { 2 ; \\alpha } ^ { ( \\ell ) } ) ^ k } . \\end{align*}"}
{"id": "3303.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F ) = \\alpha _ { n - j } ( F _ 1 ) + \\alpha _ { n - j } ( F _ 2 ) - \\alpha _ { n - j } ( F | _ { D _ 1 \\cap D _ 2 } ) . \\end{align*}"}
{"id": "7067.png", "formula": "\\begin{align*} g ^ K _ { i } ( t ) = \\Delta _ K A ^ K _ { i } ( t \\wedge \\theta _ K ) + \\frac { \\| p \\| _ { } } { \\underline p } A ^ K _ { i + 1 } ( t \\wedge \\theta _ K ) , \\end{align*}"}
{"id": "5975.png", "formula": "\\begin{align*} E _ { \\beta } ( z ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { \\Gamma ( \\beta n + 1 ) } , z \\in \\mathbb { C } . \\end{align*}"}
{"id": "8911.png", "formula": "\\begin{align*} H o m _ { S h } ( \\Z _ { U , X } ^ \\# , \\mathcal I ) = H o m _ { P S h } ( \\Z _ { U , X } , \\mathcal I ) = \\mathcal I ( U ) \\end{align*}"}
{"id": "7461.png", "formula": "\\begin{align*} E ( \\phi ( \\cdot , t ) ) + \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 + 2 \\int _ 0 ^ t \\eta ( s ) \\| \\dot { \\phi } ( \\cdot , s ) \\| ^ 2 \\mathrm { d } s = \\mathcal { F } ( 0 ) = E ( \\phi _ 0 ) + \\| v _ 0 \\| ^ 2 , \\forall t > 0 . \\end{align*}"}
{"id": "6001.png", "formula": "\\begin{align*} \\frac { 1 } { g ( t ) } \\exp ( x t ) = \\sum _ { n = 0 } ^ { \\infty } A _ { n } ( x ) \\frac { t ^ { n } } { n ! } , \\end{align*}"}
{"id": "4730.png", "formula": "\\begin{align*} \\mathcal { P } _ i ( t , y ) = \\frac { \\mathcal { R } _ i ( y - \\mathfrak { q } _ i ( t ) ) } { [ \\mathfrak { q } _ 1 ( t ) - \\mathfrak { q } _ 2 ( t ) ] ^ 2 } \\end{align*}"}
{"id": "7444.png", "formula": "\\begin{align*} E ( \\phi ) = \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi ( \\mathbf { x } ) | ^ 2 + V ( \\mathbf { x } ) | \\phi ( \\mathbf { x } ) | ^ 2 + \\frac { \\beta } { 2 } | \\phi ( \\mathbf { x } ) | ^ 4 - \\Omega \\overline { \\phi } ( \\mathbf { x } ) L _ z \\phi ( \\mathbf { x } ) \\right ) \\mathrm { d } \\mathbf { x } \\end{align*}"}
{"id": "6127.png", "formula": "\\begin{align*} S = \\left \\{ R ( s , t ) = ( s , t , s ^ { 2 j } + t ^ { 2 k } ) ; \\ \\ ( s , t ) \\in \\R ^ { 2 } \\right \\} , \\end{align*}"}
{"id": "917.png", "formula": "\\begin{align*} a | m \\rangle = \\sqrt { m } | m - 1 \\rangle , a ^ { + } | m \\rangle = \\sqrt { m + 1 } | m + 1 \\rangle . \\end{align*}"}
{"id": "2427.png", "formula": "\\begin{align*} S ^ { - 1 } f = S ^ { - 1 } S ( S ^ { - 1 } f ) = \\sum _ { \\gamma \\in \\Gamma } \\langle f , S ^ { - 1 } e _ \\gamma \\rangle S ^ { - 1 } e _ \\gamma . \\end{align*}"}
{"id": "898.png", "formula": "\\begin{align*} a ( \\widetilde E _ 2 ^ { ( 3 ) } , T ) = 0 . \\end{align*}"}
{"id": "9328.png", "formula": "\\begin{align*} \\begin{cases} ( G _ { 2 2 } - G _ { 2 1 } G _ { 1 1 } ^ { - 1 } G _ { 1 2 } ) \\mathbf { y } = \\mathbf { b } _ { \\mathbf { y } } - G _ { 2 1 } G _ { 1 1 } ^ { - 1 } \\mathbf { b } _ { \\mathbf { x } } \\\\ \\mathbf { x } = G _ { 1 1 } ^ { - 1 } ( \\mathbf { b } _ { \\mathbf { x } } - G _ { 1 2 } \\mathbf { y } ) . \\end{cases} \\end{align*}"}
{"id": "4165.png", "formula": "\\begin{align*} \\omega _ { p } ^ { p } ( A ) \\leq \\frac { 1 } { 2 ^ { p - 2 } } \\underset { i , j = 1 } { \\overset { 2 } { \\sum } } \\omega _ { p } ^ { p } ( a _ { i j } ) \\end{align*}"}
{"id": "6819.png", "formula": "\\begin{align*} K _ { g _ { \\textnormal { p o i n } } } = \\frac { 2 \\pi \\ , \\chi ( \\Sigma ) } { \\mu _ { g _ { \\textnormal { p o i n } } } ( \\Sigma ) } = \\frac { 4 \\pi \\ , ( 1 - p ) } { \\mu _ { g _ { \\textnormal { p o i n } } } ( \\Sigma ) } , \\end{align*}"}
{"id": "771.png", "formula": "\\begin{align*} W ' & = w _ 2 , w _ 3 , \\ldots , w _ n \\mbox { o r } W ' = \\emptyset \\mbox { i f } | W | = 1 . \\\\ W '' & = w _ 1 , w _ 2 , \\ldots , w _ { n - 1 } \\mbox { o r } W '' = \\emptyset \\mbox { i f } | W | = 1 . \\\\ W ^ * & = w _ n , w _ { n - 1 } , \\ldots , w _ 1 . \\end{align*}"}
{"id": "6510.png", "formula": "\\begin{align*} \\bar { g } _ n ^ { ( 2 m ) } \\sim ( j _ 0 + 1 ) ^ m A _ { 2 m \\alpha } n ^ { - m ( 1 - 2 \\alpha ) } \\end{align*}"}
{"id": "7.png", "formula": "\\begin{align*} \\kappa _ 2 ( y ) & = \\lim _ { N \\to \\infty } \\left ( 2 H _ { 2 N + 1 } - 2 \\ln \\left ( 2 N + 1 \\right ) - H _ { N } + \\ln { \\frac { N } { \\epsilon _ 2 } } + 2 \\ln \\left ( \\frac { 2 N + 1 } { N } \\right ) \\right ) + \\ln { \\lvert y \\rvert } \\\\ & = \\gamma + \\ln { \\left ( \\frac { 4 \\lvert y \\rvert } { \\epsilon _ 2 } \\right ) } . \\end{align*}"}
{"id": "2835.png", "formula": "\\begin{align*} \\dot { y } _ R ( t ) = 2 R \\Im \\int e ^ { - i t } Q \\nabla \\varphi \\left ( \\frac { x } { R } \\right ) \\cdot e ^ { i t } \\nabla Q d x = 0 \\delta ( u ( t ) ) = 0 u ( t ) = e ^ { i t } Q . \\end{align*}"}
{"id": "8434.png", "formula": "\\begin{align*} \\big | x + \\eta \\big ( b ( x ) - b ( 0 ) \\big ) \\big | ^ { 2 } & = | x | ^ { 2 } + 2 \\eta \\left \\langle b ( x ) - b ( 0 ) , x \\right \\rangle + \\eta ^ { 2 } | b ( x ) - b ( 0 ) | ^ { 2 } \\\\ & \\leq \\left ( 1 - 2 \\theta _ { 1 } \\eta + \\theta _ { 2 } ^ { 2 } \\eta ^ { 2 } \\right ) | x | ^ { 2 } + 2 K \\eta , \\end{align*}"}
{"id": "5554.png", "formula": "\\begin{align*} d _ K ( h ) = d _ K ( h ^ + - h ^ - ) \\leq d _ K ( h ^ + ) + \\| h ^ - \\| = \\| h ^ - \\| , \\end{align*}"}
{"id": "327.png", "formula": "\\begin{align*} \\mathcal S ( \\rho _ t , \\Phi _ t ) & = \\ < \\rho ( 0 ) , \\Phi ( 0 ) \\ > - \\ < \\rho ( 1 ) , \\Phi ( 1 ) \\ > + \\int _ 0 ^ 1 \\ < \\partial _ t \\Phi ( t ) , \\rho _ t \\ > + \\mathcal H _ 0 ( \\rho _ t , \\Phi _ t ) d t \\\\ & + \\int _ 0 ^ 1 \\mathcal H _ 1 ( \\rho _ t , \\Phi _ t ) \\circ d W ( t ) . \\end{align*}"}
{"id": "7127.png", "formula": "\\begin{align*} \\tilde { f } ' ( s , y ) \\le f _ 2 ( s , y ) \\le s ^ { 2 H - 1 } K = K _ s \\end{align*}"}
{"id": "2715.png", "formula": "\\begin{align*} P _ { \\mathcal { M } } = \\sum _ { I } { X ^ I } \\in \\R [ X _ 1 , \\ldots , X _ n ] , \\end{align*}"}
{"id": "3244.png", "formula": "\\begin{align*} s ( r , 0 ) = \\begin{cases} 0 , 0 < r \\leq \\lambda _ 1 ( \\mu ) ; \\\\ s _ 0 , \\quad r \\in ( \\lambda _ 1 ( \\mu ) , \\lambda _ 2 ( \\mu ) ) ; \\\\ + \\infty , , r \\geq \\lambda _ 2 ( \\mu ) . \\end{cases} \\end{align*}"}
{"id": "5768.png", "formula": "\\begin{align*} \\Delta _ { \\hat { g } _ { i , \\delta } } \\hat { f } _ { i } ^ { j } = 0 , \\ \\ \\pi _ { i } ^ { - 1 } ( B _ { 2 R } ( p _ { i } , g _ { i , \\delta } ) ) . \\end{align*}"}
{"id": "2741.png", "formula": "\\begin{align*} m _ { 0 , \\lambda } ( V ) = 0 , \\end{align*}"}
{"id": "9439.png", "formula": "\\begin{align*} u \\ ! + \\ ! \\varPhi _ q \\ , d _ q \\ , \\varPsi _ { \\mu , q } W _ q u = v ^ { ( 0 ) } . \\end{align*}"}
{"id": "6762.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf e & \\approx \\mathcal O ( h ) \\begin{bmatrix} \\mathbf 0 \\\\ G ( \\mathbf S _ 2 , S _ { m + 1 } ) \\end{bmatrix} + \\mathcal O ( h ) \\begin{bmatrix} \\mathbf 0 \\\\ G ( \\mathbf S _ 2 , S _ { m + 2 } ) \\end{bmatrix} + \\sum _ { j = m + 3 } ^ M \\mathcal O ( h ^ 5 ) \\begin{bmatrix} \\mathbf 0 \\\\ G ( \\mathbf S _ 2 , S _ { j } ) \\end{bmatrix} . \\end{aligned} \\end{align*}"}
{"id": "2382.png", "formula": "\\begin{align*} C _ { \\mathbf { F } _ N } ^ T C _ { \\mathbf { F } _ N } = \\begin{pmatrix} \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) ^ 2 & \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) \\sin ( 2 \\pi k / N ) \\\\ \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) \\sin ( 2 \\pi k / N ) & \\sum _ { k = 0 } ^ { N - 1 } \\sin ( 2 \\pi k / N ) ^ 2 \\end{pmatrix} . \\end{align*}"}
{"id": "6816.png", "formula": "\\begin{align*} A _ f ( X , Y ) : = D _ X ( D _ Y ( f ) ) - P ^ { \\textnormal { T a n } ( f ) } ( D _ X ( D _ Y ( f ) ) ) \\equiv ( D _ X ( D _ Y ( f ) ) ) ^ { \\perp _ { f } } , \\end{align*}"}
{"id": "7068.png", "formula": "\\begin{align*} \\left | p ( ( \\ell + i + 1 ) \\delta _ K ) - p ( ( \\ell + i ) \\delta _ K ) \\right | = & p ( ( \\ell + i ) \\delta _ K ) \\frac { \\left | p ( ( \\ell + i + 1 ) \\delta _ K ) - p ( ( \\ell + i ) \\delta _ K ) \\right | } { p ( ( \\ell + i ) \\delta _ K ) } \\\\ \\leq & \\frac { \\| p \\| _ { } \\delta _ K } { \\underline { p } } p ( ( \\ell + i ) \\delta _ K ) \\end{align*}"}
{"id": "8587.png", "formula": "\\begin{align*} J _ { \\ast } : = \\chi _ { \\ast } ( x ) \\int _ { k \\geq 0 } \\mathcal { K } _ { \\ast } ^ { \\# } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k , \\end{align*}"}
{"id": "3933.png", "formula": "\\begin{align*} s ( \\nu ) = s ( \\nu \\circ \\alpha _ { \\Phi } ^ { \\sigma \\to \\tau } ) . \\end{align*}"}
{"id": "2823.png", "formula": "\\begin{align*} \\dot { y } = \\Re \\int _ { \\mathbb { R } ^ N } | x | ^ 2 \\dot { u } \\bar { u } d x \\\\ = - 2 \\Im \\int _ { \\mathbb { R } ^ N } | x | ^ 2 ( \\Delta u ) \\bar { u } d x \\\\ = 4 \\Im \\int _ { \\mathbb { R } ^ N } x \\cdot \\nabla u \\bar { u } d x , \\end{align*}"}
{"id": "8448.png", "formula": "\\begin{align*} L ' _ { b } \\wedge L ' _ { c } = ( L ' _ { b } \\wedge L ' _ { b } ) \\wedge L ' _ { c } = L ' _ { s x b } . \\end{align*}"}
{"id": "343.png", "formula": "\\begin{align*} \\sup _ { S \\in H _ R ^ 1 } \\mathcal L ( \\rho , m , S ) = \\mathcal A ( \\rho , m ) + R \\mathcal E ( \\rho , m ) , \\end{align*}"}
{"id": "8262.png", "formula": "\\begin{align*} \\mathbf { F } _ { 1 \\bar { 2 } } \\mathbf { F } _ { \\bar { 2 } 1 } = \\mathbf { F } _ { 1 \\bar { 2 } \\ , \\bar { 4 } 3 } + \\mathbf { F } _ { 1 \\bar { 4 } \\ , \\bar { 2 } 3 } + \\mathbf { F } _ { 1 \\bar { 4 } 3 \\bar { 2 } } + \\mathbf { F } _ { \\bar { 4 } 1 \\bar { 2 } 3 } + \\mathbf { F } _ { \\bar { 4 } 1 3 \\bar { 2 } } + \\mathbf { F } _ { \\bar { 4 } 3 1 \\bar { 2 } } \\end{align*}"}
{"id": "6324.png", "formula": "\\begin{align*} \\langle \\Delta _ { \\emph { C K } } z , x \\otimes y \\rangle = \\langle z , x \\star y \\rangle , \\langle x \\otimes y , \\Delta _ { \\emph { G L } } z \\rangle = \\langle x y , z \\rangle \\end{align*}"}
{"id": "203.png", "formula": "\\begin{align*} M _ \\alpha ( g ) ( x ) = g ( x ) p _ \\alpha ( x ) . \\end{align*}"}
{"id": "5753.png", "formula": "\\begin{align*} h _ { s t } = g _ { s t } , s , t = 1 , \\dots , m . \\end{align*}"}
{"id": "5977.png", "formula": "\\begin{align*} \\pi _ { \\lambda , \\beta } = \\int _ { 0 } ^ { \\infty } \\pi _ { \\lambda \\tau } \\ , \\mathrm { d \\nu _ { \\beta } ( \\tau ) , } \\quad \\forall \\lambda > 0 . \\end{align*}"}
{"id": "5333.png", "formula": "\\begin{align*} D ( u ^ { \\epsilon } , \\eta ^ { \\epsilon } , \\tau ^ { \\epsilon } ) = \\mu _ 1 \\| \\nabla u ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 + \\mu _ 2 \\| \\mathrm { d i v } u ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 + \\nu [ \\beta ( L - 1 ) + 2 \\bar { \\mathfrak { z } } ] \\| \\nabla \\eta ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 + \\frac { \\beta A _ 0 } { 4 k ^ 2 } \\| \\tau ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 + \\frac { \\beta \\nu } { 2 k ^ 2 } \\| \\nabla \\tau ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 . \\end{align*}"}
{"id": "3501.png", "formula": "\\begin{align*} & \\frac { 1 } { \\abs { \\Gamma ( s _ 3 + 1 ) } } \\int _ { ( \\frac { 1 } { 2 } ) } \\abs * { \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { s _ 1 + s _ 3 + z } } \\abs { d z } \\\\ & = \\frac { 1 } { \\abs { \\Gamma ( s _ 3 + 1 ) } } \\int _ { - \\infty } ^ \\infty \\frac { \\abs { \\Gamma ( \\sigma _ 3 + \\frac { 3 } { 2 } + i ( t _ 3 + w ) ) \\Gamma ( - \\frac { 1 } { 2 } - i w ) } } { \\abs { \\sigma _ 1 + \\sigma _ 3 + \\frac { 1 } { 2 } + i ( t _ 1 + t _ 3 + w ) } } d w \\\\ & \\ll t _ 3 ^ { - \\sigma _ 3 - \\frac { 1 } { 2 } } + t _ 3 ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "492.png", "formula": "\\begin{align*} h ( \\mathcal { I } ) \\cdot h ( \\mathcal { I } _ { \\alpha \\beta } ^ { i j } ) = h ( \\mathcal { I } _ { \\alpha } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\beta } ^ { j } ) + Y _ { \\alpha \\beta } ^ { i j } \\cdot h ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\alpha } ^ { j } ) + \\frac { h ( \\mathcal { I } _ { \\alpha } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\beta } ^ { j } ) } { Y _ { \\alpha \\beta } ^ { i j } } + h ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\alpha } ^ { j } ) . \\end{align*}"}
{"id": "7725.png", "formula": "\\begin{gather*} \\phi [ T ] = u [ T ] . \\end{gather*}"}
{"id": "5342.png", "formula": "\\begin{align*} & \\epsilon \\phi ^ { \\epsilon } \\partial _ t u ^ { \\epsilon } \\rightarrow 0 \\ ; \\ ; \\ ; \\ ; L ^ \\infty ( 0 , T ; H ^ 1 ) , \\\\ & \\epsilon \\phi ^ { \\epsilon } u ^ { \\epsilon } \\cdot \\nabla u ^ { \\epsilon } \\rightarrow 0 \\ ; \\ ; \\ ; \\ ; L ^ \\infty ( \\mathbb { R } ^ + ; H ^ 2 ) , \\ ; \\ ; \\ ; \\ ; T > 0 , \\end{align*}"}
{"id": "1666.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\psi _ z ( u ) \\ , \\d u \\ge & \\int _ { \\R ^ d } \\big ( \\varphi ( | u | ) \\wedge \\varphi ( | u | + | z | ) \\big ) \\ , \\d u \\ge \\int _ { \\R ^ d } \\varphi ( | u | + | z | ) \\ , \\d u = \\int _ { \\{ | u | \\ge | z | \\} } \\varphi ( u ) \\ , \\d u , \\end{align*}"}
{"id": "8351.png", "formula": "\\begin{align*} g ( p ) = r = \\log \\left ( \\frac { | \\log p | ^ 2 } { 4 } \\right ) + o ( 1 ) \\sim 2 \\log | \\log p | , \\qquad \\textrm { a s } p \\to 0 . \\end{align*}"}
{"id": "4812.png", "formula": "\\begin{align*} ( \\phi ^ 2 + \\phi * \\phi ) ( a ) & = c ^ 2 ( 1 + J ' ( \\alpha ^ 2 , \\alpha ^ 2 ) ) \\alpha ^ 4 ( a ) + \\bar c ^ 2 ( 1 + J ' ( \\bar \\alpha ^ 2 , \\bar \\alpha ^ 2 ) ) \\bar \\alpha ^ 4 ( a ) + 2 ( 1 + J ( \\alpha ^ 2 , \\bar \\alpha ^ 2 ) ) \\\\ & = c ^ 2 ( 1 + J ' ( \\alpha ^ 2 , \\alpha ^ 2 ) ) \\bar \\alpha ^ 4 ( a ) + \\bar c ^ 2 ( 1 + J ' ( \\bar \\alpha ^ 2 , \\bar \\alpha ^ 2 ) ) \\alpha ^ 4 ( a ) \\\\ & = 2 \\mathrm { R e } ( c ^ 2 ( 1 + J ' ( \\alpha ^ 2 , \\alpha ^ 2 ) ) ) \\alpha ^ 4 ( a ) . \\end{align*}"}
{"id": "1199.png", "formula": "\\begin{align*} K _ { 1 / 3 } ^ { ( 0 ) } = \\pi \\left ( \\left \\{ x \\in \\Lambda ^ \\mathbb { N } : \\liminf _ { k \\to + \\infty } \\frac { S _ k \\phi ( x ) } { k } = 0 \\right \\} \\right ) , \\end{align*}"}
{"id": "7268.png", "formula": "\\begin{align*} \\boxed { \\exp \\Bigl ( \\sum \\alpha _ k \\Bigr ) = \\prod \\exp ( \\alpha _ k ) . } \\end{align*}"}
{"id": "1981.png", "formula": "\\begin{align*} \\widehat { \\Phi ^ { - 1 } } \\big ( x { \\widehat { \\Phi } } ( x ) \\big ) = \\frac { 1 } { \\widehat { \\Phi } ( x ) } \\end{align*}"}
{"id": "8079.png", "formula": "\\begin{align*} F _ { H } [ \\partial _ { \\Sigma , \\epsilon } \\chi ^ { * } \\phi ] = F _ { H } [ \\rho ^ { * } _ { ( 1 ) } \\partial _ { \\widetilde { \\Sigma } , \\widetilde { \\epsilon } } \\phi ] , \\end{align*}"}
{"id": "1483.png", "formula": "\\begin{align*} = | j ( g _ 1 , z _ 0 ) j ( g _ 2 , z _ 0 ) \\det ( B ( z _ 0 , z _ 0 ) ^ { - 1 } B ( g _ 1 z _ 0 , g _ 2 z _ 0 ) ) | ^ { - 2 } \\delta ( z _ 0 ) \\delta ( z _ 0 ) \\end{align*}"}
{"id": "5343.png", "formula": "\\begin{align*} \\partial _ t u + u \\cdot \\nabla u + \\nabla \\big ( \\beta ( L - 1 ) \\eta + \\bar { \\mathfrak { z } } ( \\eta ) ^ { 2 } \\big ) - \\mu _ 1 \\Delta u - \\mu _ 2 \\nabla \\mathrm { d i v } u - \\frac { \\beta } { k } \\mathrm { d i v } \\tau = - \\nabla \\pi _ 1 , \\end{align*}"}
{"id": "6707.png", "formula": "\\begin{align*} \\bigl ( ~ _ r \\mathcal { F } _ s ( \\alpha ) \\bigr ) ^ { q ^ d } = \\sum _ { n = 0 } ^ { \\infty } \\biggl ( \\prod _ { m = 1 } ^ { n + d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \\biggr ) \\alpha ^ { q ^ { n + d } } \\end{align*}"}
{"id": "36.png", "formula": "\\begin{align*} J _ { t } = J - U \\otimes \\zeta _ { t } - V \\otimes ( \\zeta _ { t } \\circ J ) . \\end{align*}"}
{"id": "6598.png", "formula": "\\begin{align*} \\mathcal { U } ( h , k ) = \\frac { 1 } { 2 } \\sum _ { \\substack { 1 \\leq c \\leq C \\\\ ( c , h k ) = 1 } } \\mu ( c ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , c ) = 1 \\\\ m h \\neq n k } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { 1 \\leq d < \\infty \\\\ ( d , m h n k ) = 1 \\\\ d | m h \\pm n k } } \\phi ( d ) W \\left ( \\frac { c d } { Q } \\right ) . \\end{align*}"}
{"id": "7740.png", "formula": "\\begin{gather*} f ( x ) = \\sum _ { n \\in \\mathbb { Z } } f _ n e ^ { i n x } \\ ; \\textrm { a n d } \\ ; \\tilde { \\phi } ( x ) = \\sum _ { n \\in \\mathbb { Z } } a _ n e ^ { i n x } , \\end{gather*}"}
{"id": "99.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ 2 \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "7993.png", "formula": "\\begin{align*} \\boldsymbol { \\mu } _ n ^ { \\sigma } \\coloneqq \\big ( \\mu _ n ^ { \\sigma } ( i ) \\big ) _ { i \\in \\mathcal { I } _ n ^ { \\sigma } } , \\ ; \\mu _ n ^ { \\sigma } ( i ) \\coloneqq \\sum _ { j \\in \\mathcal { I } : \\ , \\Pi _ n ^ { \\sigma } j = i } \\mu ( j ) . \\end{align*}"}
{"id": "4171.png", "formula": "\\begin{align*} \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} \\right ) & = \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ - B & 0 \\end{bmatrix} \\right ) \\\\ & \\geq \\frac { 1 } { 2 ^ { 1 - \\frac { 1 } { p } } } \\omega _ { p } ( A - B ) , \\end{align*}"}
{"id": "189.png", "formula": "\\begin{align*} F _ \\delta ( 0 ) = \\int _ { - \\infty } ^ 0 \\left ( \\int _ x ^ 0 f _ \\delta ( y ) d y \\right ) p _ \\delta ( x ) d x - \\int _ 0 ^ { + \\infty } \\left ( \\int _ 0 ^ { x } f _ \\delta ( y ) d y \\right ) p _ \\delta ( x ) d x . \\end{align*}"}
{"id": "350.png", "formula": "\\begin{align*} \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho , m , S ) = \\mathcal A ( \\rho , m ) + \\mathbb I _ { C _ F ( \\rho ^ a , \\rho ^ b ) } ( \\rho , m ) , \\end{align*}"}
{"id": "5142.png", "formula": "\\begin{align*} I _ n = \\left ( \\max \\{ 3 . 0 8 , \\tfrac { \\eta _ n } { 2 } \\} , \\min \\{ \\tfrac { 4 n } { 3 } , 2 \\eta _ n \\} \\right ) , 4 \\leq n \\in \\N I _ n = ( 3 . 3 , 4 . 8 ) , n = 3 . \\end{align*}"}
{"id": "753.png", "formula": "\\begin{align*} & b _ 1 = \\frac { \\left ( L - \\mu \\right ) \\left ( \\alpha + \\mu _ p \\left ( 1 - L t _ 1 \\right ) \\right ) } { \\alpha \\left ( L - \\mu + \\mu _ p \\right ) } \\\\ & b _ 2 = b _ 1 - \\left ( \\frac { \\alpha } { L - \\mu } b _ 1 \\right ) ^ 2 , \\end{align*}"}
{"id": "7125.png", "formula": "\\begin{align*} f _ 2 ( s , y ) : = s ^ { 2 H - 1 } \\Big ( \\alpha _ 2 \\frac { 1 } { \\tilde { \\theta } ^ { \\rho } } ( - \\tilde { \\theta } \\rho + \\tilde { \\theta } + 1 ) y ^ { - \\tilde { \\theta } \\rho + \\tilde { \\theta } } + H \\tilde { \\sigma } ( \\tilde { \\theta } + 1 ) y ^ { - 2 } \\Big ) . \\end{align*}"}
{"id": "680.png", "formula": "\\begin{align*} K _ { \\alpha \\alpha } ^ { ( 1 ) } = C _ b + C _ W K _ { \\alpha \\beta } ^ { ( 0 ) } , K _ { \\alpha \\beta } ^ { ( 0 ) } : = \\frac { 1 } { n _ 0 } \\sum _ { j = 1 } ^ { n _ 0 } x _ { j ; \\alpha } x _ { j ; \\beta } , x _ \\alpha , x _ \\beta \\in \\R ^ { n _ 0 } . \\end{align*}"}
{"id": "7836.png", "formula": "\\begin{align*} T ^ { * 3 ( j - l - 2 ) } x = \\sum _ { i = 0 } ^ { 3 l + 6 } \\alpha _ { i } e _ { i } , T ^ { * 3 ( j - r - l - 2 ) } y = \\sum _ { i = 0 } ^ { 3 l + 6 } \\beta _ { i } e _ { i } \\quad T ^ { * 3 ( n - 2 j + r - l - 3 ) } z = \\sum _ { i = 0 } ^ { 3 l + 6 } \\gamma _ { i } e _ { i } . \\end{align*}"}
{"id": "5853.png", "formula": "\\begin{align*} \\nu ( ( - \\infty , z ] ) = \\frac { R } { L ^ { 1 / \\rho } } ( z \\vee 0 ) ^ { 1 / \\rho } , z \\in \\R , \\end{align*}"}
{"id": "6287.png", "formula": "\\begin{align*} B _ { \\xi } \\big ( 1 + K _ l , 1 + \\widehat { K ' } \\big ) ( 1 + K _ l + \\widehat { K ' } ) ! = \\xi K _ l ! \\widehat { K ' } ! . \\end{align*}"}
{"id": "2662.png", "formula": "\\begin{align*} e _ \\alpha ( z ) = \\left ( \\frac { \\pi ^ { | \\alpha | } } { \\alpha ! } \\right ) ^ { 1 / 2 } z ^ \\alpha = \\prod _ { k = 1 } ^ d \\left ( \\frac { \\pi ^ { \\alpha _ k } } { \\alpha _ k ! } \\right ) ^ { 1 / 2 } z _ k ^ { \\alpha _ k } , \\end{align*}"}
{"id": "4446.png", "formula": "\\begin{align*} ( a ^ { \\dagger } a ) _ { k , \\lambda } = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( a ^ { \\dagger } ) ^ { l } a ^ { l } , \\end{align*}"}
{"id": "3658.png", "formula": "\\begin{align*} W _ s : = \\{ \\psi < s \\} . \\end{align*}"}
{"id": "1131.png", "formula": "\\begin{align*} \\Gamma ^ { ( 3 ) } \\cap U _ { \\delta } ( \\eta ) = \\Gamma ^ { \\eta , \\delta } _ { j } , j = 1 , 2 , 3 , 4 . \\end{align*}"}
{"id": "6312.png", "formula": "\\begin{align*} \\omega _ 1 \\triangleq \\mathbb { E } [ T ^ 2 ] = K ' \\mathbb { E } [ Z ^ 2 ] , \\end{align*}"}
{"id": "7304.png", "formula": "\\begin{align*} \\partial _ k \\bigl ( \\alpha ( \\beta _ 1 , \\ldots , \\beta _ n ) \\bigr ) = \\sum _ { i = 1 } ^ n ( \\partial _ k \\beta _ i ) a _ i \\beta _ 1 ^ { a _ 1 } \\ldots \\beta _ i ^ { a _ i - 1 } \\ldots \\beta _ n ^ { a _ n } = \\sum _ { i = 1 } ^ n ( \\partial _ k \\beta _ i ) ( \\partial _ i \\alpha ) ( \\beta _ 1 , \\ldots , \\beta _ n ) . \\end{align*}"}
{"id": "6485.png", "formula": "\\begin{align*} E [ ( S _ { n + 1 } ) ^ { 2 m - 1 } ] & = \\sum _ { \\ell = 1 } ^ m \\left \\{ \\binom { 2 m - 1 } { 2 \\ell - 1 } + \\frac { \\alpha } { n } \\binom { 2 m - 1 } { 2 \\ell - 2 } \\right \\} E [ ( S _ { n } ) ^ { 2 \\ell - 1 } ] . \\end{align*}"}
{"id": "8737.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\leq p _ i ^ L = u _ { i 0 } ( x ) \\leq \\cdots \\leq u _ { i n } ( x ) = p _ i ( x ) , \\\\ u _ { i j } ( x ) \\leq a _ { i j } j = 0 , \\ldots , n , \\\\ u _ { i 0 } ( x ) = a _ { i 0 } \\leq \\cdots \\leq a _ { i n } , \\\\ \\deg ( u _ { i j } ) \\leq \\tau . \\end{aligned} \\end{align*}"}
{"id": "4314.png", "formula": "\\begin{align*} \\varepsilon _ j = \\| \\phi _ { j , b , \\beta } \\| ^ { - 2 } _ { L ^ 2 _ { \\rho _ { \\beta } } } \\langle \\varepsilon , \\phi _ { j , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } . \\end{align*}"}
{"id": "7159.png", "formula": "\\begin{align*} \\theta = \\frac 1 2 \\min \\{ 2 ^ { - d } \\beta ^ { - d } , 1 \\} \\qquad R = \\max \\{ 4 ( \\theta / 2 ) ^ { - 1 / d } , 8 \\} \\ , . \\end{align*}"}
{"id": "4425.png", "formula": "\\begin{align*} \\begin{array} { l l } w _ 1 2 + w _ 2 ( 1 + i ) = \\theta \\\\ w _ 1 ( 3 + i ) + w _ 2 3 = \\theta \\\\ w _ 1 + w _ 2 = 1 \\end{array} \\end{align*}"}
{"id": "5162.png", "formula": "\\begin{align*} \\vartheta _ 2 ( z , \\tau ) = \\sum _ { k \\in \\Z } e ^ { \\pi i ( k + 1 / 2 ) ^ 2 \\tau } e ^ { ( 2 k + 1 ) \\pi i z } \\vartheta _ 4 ( z , \\tau ) = \\sum _ { k \\in \\Z } ( - 1 ) ^ k e ^ { \\pi i k ^ 2 \\tau } e ^ { 2 k \\pi i z } . \\end{align*}"}
{"id": "8877.png", "formula": "\\begin{align*} H X _ b ^ 1 ( X , A ) = \\mathcal C _ f ( X , A ) / A = \\begin{cases} A ^ { e ( X ) - 1 } & e ( X ) < \\infty \\\\ \\bigoplus _ \\N A & e ( X ) = \\infty \\end{cases} \\end{align*}"}
{"id": "8109.png", "formula": "\\begin{align*} { \\mathcal L } ( \\varphi ) = \\sum _ { ( d , e ) \\in D _ { 1 } ( P G ) } \\varepsilon ( d , e ) \\ell _ { D } ( d , e ) . \\end{align*}"}
{"id": "3004.png", "formula": "\\begin{align*} \\omega _ n ( T _ { \\mathfrak u } ( x _ { i , k } ) & , T _ { \\mathfrak u } ( x _ { j , l } ) \\\\ & = \\omega _ n \\Big ( \\big ( \\frac { 2 ^ { k - 1 } } { ( k - 1 ) ! } u _ i \\log ^ { k - 1 } | u _ i | , \\ldots , u _ i , \\ldots , 0 \\big ) , \\big ( \\frac { 2 ^ { l - 1 } } { ( l - 1 ) ! } u _ j \\log ^ { l - 1 } | u _ j | , \\ldots , u _ j , \\ldots , 0 \\big ) \\Big ) \\\\ & = ( - 1 ) ^ k \\langle u _ i , u _ j \\rangle = \\omega _ n ( x _ { i , k } , x _ { j , l } ) . \\end{align*}"}
{"id": "4687.png", "formula": "\\begin{align*} & ( - \\mathcal { L } B _ { i j , 0 } ) ' = - a _ { i j } \\sigma _ i \\Lambda Q + 2 p \\kappa _ 0 \\sigma _ j ( y Q ^ { p - 1 } ) ' , \\\\ & \\int B _ { i j , 0 } Q ' = 0 , \\quad \\lim _ { y \\rightarrow + \\infty } B _ { i j , 0 } ( y ) = 0 , \\\\ & \\lim _ { y \\rightarrow - \\infty } B _ { i j , 0 } ( y ) = - \\frac { a _ { i j } \\sigma _ i ( p - 2 ) } { p - 1 } \\int Q , \\end{align*}"}
{"id": "5165.png", "formula": "\\begin{align*} P _ 3 ( t ) = \\vartheta _ 3 ( 0 , i t ) \\vartheta _ 3 ( 0 , i / t ) . \\end{align*}"}
{"id": "4437.png", "formula": "\\begin{align*} H ' ( t ) & \\leq - a _ 1 \\int _ U | u _ t ( x , t ) | ^ 2 \\ , d x - a _ 1 \\int _ U | u _ t ( x , t ) | ^ m \\ , d x \\\\ & + ( \\frac { 3 } { 2 } + a ^ 2 _ 2 B ) \\epsilon \\int _ U ( u _ t ^ 2 ( x , t ) ) \\ , d x - \\frac { 1 } { 4 } \\epsilon \\int _ U ( u _ { x x } ( x , t ) ) ^ 2 \\ , d x \\\\ & + a _ 2 \\epsilon ( \\gamma \\delta | | u | | ^ 2 _ 2 + c ( \\delta ) | | u _ t | | ^ m _ m ) - \\epsilon E ( t ) , \\end{align*}"}
{"id": "2334.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z ^ d } f ( t + k ) = \\sum _ { l \\in \\Z ^ d } \\widehat { f } ( l ) e ^ { 2 \\pi i l \\cdot t } . \\end{align*}"}
{"id": "3840.png", "formula": "\\begin{align*} ( \\sigma ^ { ( i ) } y ^ { ( k ) } ) ^ { ( k + 1 ) } + ( \\sigma ^ { ( i ) } y ^ { ( k + 1 ) } ) ^ { ( k ) } & = \\sum _ { s = 0 } ^ { i + 1 } ( - 1 ) ^ s ( C _ { i + 1 } ^ s - 2 C _ i ^ { s - 1 } ) ( \\sigma y ^ { ( s + k ) } ) ^ { ( i + 1 - s + k ) } , \\\\ ( ( \\sigma ^ { ( i ) } y ^ { ( k ) } ) ^ { ( k + 1 ) } + ( \\sigma ^ { ( i ) } y ^ { ( k + 1 ) } ) ^ { ( k ) } , z ) & = \\sum _ { s = 0 } ^ { i + 1 } ( - 1 ) ^ { i + k + 1 } ( C _ { i + 1 } ^ s - 2 C _ i ^ s ) ( \\sigma y ^ { ( s + k ) } , z ^ { ( i + 1 - s + k ) } ) , \\end{align*}"}
{"id": "6337.png", "formula": "\\begin{align*} \\frac { \\Gamma ' ( z ) } { \\Gamma ( z ) } = \\log z - \\frac { 1 } { 2 z } - \\int _ 0 ^ \\infty \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { t } + \\frac { 1 } { e ^ t - 1 } \\right ) e ^ { - t z } \\ , d t , \\qquad \\Re z > 0 , \\end{align*}"}
{"id": "2161.png", "formula": "\\begin{align*} - ( a ( \\vert \\nabla u \\vert ^ { p } ) \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u ) + V ( x ) b ( \\vert u \\vert ^ { p } ) \\vert u \\vert ^ { p - 2 } = K ( x ) f ( u ) , \\ \\ \\ \\mathbb { R } ^ { d } , \\end{align*}"}
{"id": "7254.png", "formula": "\\begin{align*} 3 b d - c ^ 2 = & c ^ 2 ( \\dfrac { 3 b d } { c ^ 2 } - 1 ) = c ^ 2 \\left ( \\dfrac { 3 ( k + 1 ) ( n - k - 1 ) } { ( n - k ) ( k + 2 ) } - 1 \\right ) \\\\ = & c ^ 2 \\cdot \\dfrac { - 2 k ^ 2 + 2 ( n - 2 ) k + ( n - 3 ) } { ( n - k ) ( k + 2 ) } . \\end{align*}"}
{"id": "3758.png", "formula": "\\begin{align*} H ^ { 2 n - k } _ { - \\alpha } ( M , \\R ) & \\cong H ^ { \\alpha } _ k ( M , \\R ) , \\\\ H ^ { n - p , n - q } _ { B C , - \\alpha } ( X , \\C ) & \\cong H ^ { B C , \\alpha } _ { p , q } ( X , \\C ) , \\\\ H ^ { n - p , n - q } _ { A , - \\alpha } ( X , \\C ) & \\cong H ^ { A , \\alpha } _ { p , q } ( X , \\C ) . \\end{align*}"}
{"id": "5390.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { t _ { n } } \\int _ { 0 } ^ { t _ { n } } \\left \\{ \\sum _ { k = 0 } ^ { n - 1 } \\varphi \\left ( X _ { t _ { k } } \\right ) \\mathbf { 1 } _ { \\left [ t _ { k } , t _ { k + 1 } \\right ) } ( s ) \\right \\} \\mathrm { d } s = \\mathbb { E } ( \\varphi ( \\bar { X } ) ) , \\end{align*}"}
{"id": "2563.png", "formula": "\\begin{align*} S _ { \\widehat { S } g , S \\L } = \\widehat { S } S _ { g , \\L } \\widehat { S } ^ { - 1 } \\end{align*}"}
{"id": "3956.png", "formula": "\\begin{align*} \\begin{cases} - \\sigma _ t - \\sigma _ x - v _ { x } = f & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ - v _ { t } - v _ { x x } - v _ x - \\sigma _ x = g & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\sigma ( t , 0 ) = \\sigma ( t , 1 ) & t \\in ( 0 , T ) , \\\\ v ( t , 0 ) = v ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\sigma ( T , x ) = \\sigma _ T ( x ) , \\ \\ v ( T , x ) = v _ T ( x ) & x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
{"id": "4145.png", "formula": "\\begin{align*} \\hat { \\sigma } ( \\xi ) = \\int _ 0 ^ \\xi ( \\xi - \\zeta ) \\partial _ \\xi ^ 2 \\hat { \\sigma } ( \\zeta ) d \\zeta . \\end{align*}"}
{"id": "2246.png", "formula": "\\begin{align*} \\Theta _ { \\alpha } ( s ) : = \\int _ 0 ^ s \\tilde { E } _ { k , N } ( \\lceil s \\rceil - \\sigma ) ( s - \\sigma ) ^ { - \\alpha } \\ , \\dd W ( \\sigma ) . \\end{align*}"}
{"id": "520.png", "formula": "\\begin{align*} { \\scriptstyle \\begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 0 & 1 & - c & c \\cdot \\xi & c \\cdot ( \\xi - 1 ) \\\\ 0 & 0 & 1 & 1 & - c \\cdot ( \\xi + 1 ) & - c \\cdot ( 1 + \\xi ^ { - 1 } ) & - c \\end{pmatrix} } ^ { \\mathtt { T } } , c \\in \\mathbb { C } . \\end{align*}"}
{"id": "7959.png", "formula": "\\begin{align*} - \\Delta u _ { \\varepsilon } = P _ { \\epsilon } ( \\mu _ T - T ) . \\end{align*}"}
{"id": "9531.png", "formula": "\\begin{align*} E [ \\sum _ { t = 0 } ^ T K _ t ^ * ( p _ { t } + \\Delta y _ { t + 1 } , y _ t ) ] \\ge & E [ \\sum _ { t = 0 } ^ T K _ t ^ * ( E _ { t } \\Delta y _ { t + 1 } , E _ { t } y _ t ) ] . \\end{align*}"}
{"id": "6302.png", "formula": "\\begin{align*} f \\big ( \\Re \\{ \\mathbf { y } \\} ; \\mathcal { H } _ k \\big ) = \\ ! \\int \\ ! f \\big ( \\Re \\{ \\mathbf { y } \\} | h ' ; \\mathcal { H } _ k \\big ) f _ { H ' } ( h ' ) \\mathrm { d } h ' , \\ \\ f \\big ( \\Im \\{ \\mathbf { y } \\} ; \\mathcal { H } _ k \\big ) = \\ ! \\int \\ ! f \\big ( \\Im \\{ \\mathbf { y } \\} | h ' ; \\mathcal { H } _ k \\big ) f _ { H ' } ( h ' ) \\mathrm { d } h ' \\end{align*}"}
{"id": "1770.png", "formula": "\\begin{align*} \\operatorname { t r } _ \\gamma ( f ) = \\sum _ { \\alpha \\in O _ \\gamma } f ( \\alpha ) = \\sum _ { \\beta \\in G / C _ \\gamma } f ( \\beta \\gamma \\beta ^ { - 1 } ) , \\end{align*}"}
{"id": "5136.png", "formula": "\\begin{align*} \\psi ( x ) = 1 + x - x \\coth ( x ) . \\end{align*}"}
{"id": "9344.png", "formula": "\\begin{align*} F _ { n } ( x ) = \\sum _ { k = 0 } ^ { n } S _ { 2 } ( n , k ) k ! x ^ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 1 2 ] ) , \\end{align*}"}
{"id": "5816.png", "formula": "\\begin{align*} \\gamma _ { k } ^ { \\langle n \\rangle } = \\frac { \\left ( \\sum _ { m \\in \\mathbb { M } _ k } \\sqrt { \\eta _ { m k } } \\alpha _ { m k } \\right ) ^ 2 } { \\sum _ { m \\in \\mathbb { M } _ k } \\eta _ { m k } \\beta _ { m k } \\alpha _ { m k } + \\frac { 1 } { \\gamma _ t } } . \\end{align*}"}
{"id": "6245.png", "formula": "\\begin{align*} t _ 0 = 0 , \\dotsc , \\ t _ { n - 1 } = 0 , \\ t _ i - t _ j = 0 \\textrm { f o r } 0 \\leq i < j \\leq n - 1 . \\end{align*}"}
{"id": "7666.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + \\tilde { \\lambda } _ 0 \\underline { m } \\ , u = 0 \\R ^ n \\\\ u ( \\mathbf { 0 } ) = 1 \\ ; u > 0 \\ ; \\R ^ n \\end{cases} \\end{align*}"}
{"id": "7766.png", "formula": "\\begin{align*} \\| ( a , b ) - ( p , 0 ) \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } = \\varepsilon \\leq \\tilde \\varepsilon , \\end{align*}"}
{"id": "5864.png", "formula": "\\begin{align*} \\chi _ i = \\begin{cases} \\frac { \\bar \\alpha _ { i - 1 } } { \\lambda _ i } , & i \\in \\{ 1 , \\dots , n \\} \\setminus J , \\\\ \\frac { \\bar \\alpha _ i } { \\mu _ i } , & i \\in J . \\end{cases} \\end{align*}"}
{"id": "2648.png", "formula": "\\begin{align*} \\norm { X g } _ 2 \\norm { P g } _ 2 = \\norm { X g } _ 2 \\norm { X \\widehat { g } } _ 2 \\geq \\frac { 1 } { 4 \\pi } \\norm { g } _ 2 ^ 2 , \\end{align*}"}
{"id": "6027.png", "formula": "\\begin{align*} H _ { - \\alpha } ( x ) = \\mathfrak { D } ^ { - \\alpha } _ { { \\rm r i g h t } , 2 x } 1 ( x ) = \\frac 1 { \\Gamma ( \\alpha ) } \\int _ 0 ^ \\infty e ^ { - 2 x t - t ^ 2 } \\ , \\frac { d t } { t ^ { 1 - \\alpha } } \\end{align*}"}
{"id": "1116.png", "formula": "\\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \\xi ) - \\theta ( k , \\xi ) ] \\sigma _ 3 } . \\end{align*}"}
{"id": "5479.png", "formula": "\\begin{align*} \\Phi ( d , e , t ) = e ^ { \\gamma t } d + \\varphi _ { \\gamma } ( t ) e \\end{align*}"}
{"id": "3749.png", "formula": "\\begin{align*} \\delta ( \\mathfrak { r i c } ) = - \\frac { 1 } { 2 } d ( \\mathfrak { s } ) , \\end{align*}"}
{"id": "3210.png", "formula": "\\begin{align*} \\mu _ { s s } = \\frac { \\Gamma ( m _ { s s } + 1 / 2 ) } { \\Gamma ( m _ { s s } ) } \\sqrt { \\frac { \\Omega _ { s s } } { m _ { s s } } } , \\mu _ { s 1 } = \\frac { \\Gamma ( m _ { s 1 } + 1 / 2 ) } { \\Gamma ( m _ { s 1 } ) } \\sqrt { \\frac { \\Omega _ { s 1 } } { m _ { s 1 } } } . \\end{align*}"}
{"id": "7977.png", "formula": "\\begin{align*} \\Delta ( K ; \\alpha , \\beta ) = \\Big [ \\frac { ( K ^ { \\alpha + \\beta } - 1 ) ^ 3 } { \\alpha + \\beta } \\Big ] ^ 2 - \\frac { ( K ^ { 2 \\alpha } - 1 ) ^ 3 } { 2 \\alpha } \\cdot \\frac { ( K ^ { 2 \\beta } - 1 ) ^ 3 } { 2 \\beta } \\leq \\frac { K ^ { 6 \\alpha + 6 \\beta } } { ( \\alpha + \\beta ) ^ 2 } - \\frac { K ^ { 6 \\alpha } } { 2 5 6 | \\alpha \\beta | } . \\end{align*}"}
{"id": "2937.png", "formula": "\\begin{align*} \\tilde N _ { n } ^ 2 ( k ) = \\sum \\limits _ { \\substack { \\mathbf { p } _ { k , 1 } , \\mathbf { p } _ { k , 2 } \\in \\mathcal { P } ( d , k ) } } \\tilde N _ { n , \\mathbf { p } _ { k , 1 } } ( k ) \\tilde N _ { n , \\mathbf { p } _ { k , 2 } } ( k ) = \\sum _ { \\mu = 0 } ^ k \\tilde N _ { n , \\mu } ( k ) , \\end{align*}"}
{"id": "5881.png", "formula": "\\begin{align*} b ( z ) & = b ( y ) + D b ( y ) ( z - y ) + Q _ 1 ( y , z - y ) , z , y \\in \\R ^ 2 , \\\\ \\sigma ( z ) & = \\sigma ( y ) + Q _ 2 ( y , z - y ) , z , y \\in \\R ^ 2 , \\end{align*}"}
{"id": "3699.png", "formula": "\\begin{align*} \\partial _ t B _ q + a \\Delta _ q ( B J _ x ) + b \\Delta _ q ( J B _ x ) + \\mu \\Lambda ^ \\alpha B _ q = 0 . \\end{align*}"}
{"id": "3517.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) & = \\sum _ { k = 3 } ^ \\infty \\left ( \\sum _ { k / 2 < m \\leq k - 1 } \\frac { 1 } { m ^ { s _ 1 } ( k - m ) ^ { s _ 2 } k ^ { \\sigma _ 3 } } \\right ) \\frac { 1 } { k ^ { i t _ 3 } } \\\\ & = \\overline { \\sum _ { k = 3 } ^ \\infty \\overline { a _ k } k ^ { i t _ 3 } } \\\\ \\end{align*}"}
{"id": "993.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s \\bar u + k \\bar u & = \\bar f B _ 1 . \\end{align*}"}
{"id": "5173.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) & = \\lim _ { t \\to \\infty } \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ { N } \\frac { \\sum _ { i = 1 } ^ { R _ \\ell ^ \\pi ( t ) } ( \\rho _ \\ell ( \\frac { ( T _ { \\ell i } ^ \\pi ) ^ 2 } { 2 } + T _ { \\ell i } ^ \\pi Z _ { \\ell i } ^ \\pi ) + c _ \\ell ) } { t } , \\end{align*}"}
{"id": "3599.png", "formula": "\\begin{align*} A _ { j } & \\le \\dfrac { R _ { j + 1 } - S _ { j } } { 2 } + e \\le \\dfrac { 0 - ( - 2 e ) } { 2 } + e = 2 e \\le d [ ( - 1 ) ^ { j / 2 } a _ { 1 , j } ] \\ , , \\end{align*}"}
{"id": "555.png", "formula": "\\begin{align*} w _ R = - \\chi _ R ( v + \\nabla \\Psi _ 0 ) \\end{align*}"}
{"id": "7936.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ t a _ i ^ \\ell = \\sum _ { a \\in S } a ^ \\ell = \\frac { 1 } { \\ell + 1 } t n ^ { \\ell } + O _ \\ell \\left ( t n ^ { \\ell - 1 } \\right ) - \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } E _ k . \\end{align*}"}
{"id": "6298.png", "formula": "\\begin{align*} \\widehat { K ' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } & = K ' + \\sum _ { i \\in \\mathcal { K } ' } \\frac { X _ i / Y _ i } { \\sqrt { 2 N _ 1 \\varrho \\vartheta _ i } } + V = K ' + \\sum _ { i \\in \\mathcal { K } ' } \\frac { Z _ i } { \\sqrt { 2 N _ 1 \\varrho \\vartheta _ i } } + V , \\end{align*}"}
{"id": "4213.png", "formula": "\\begin{align*} \\int _ M \\langle \\nabla v _ m , \\nabla v _ m \\rangle \\ , d V _ g = 0 . \\end{align*}"}
{"id": "2105.png", "formula": "\\begin{align*} ( \\xi _ i , ( x _ 0 , \\ldots , x _ { i - 1 } , x _ { i + 1 } , \\ldots , x _ n ) ) & \\to ( \\xi _ i ^ { q _ 0 } x _ 0 , \\ldots , \\xi _ i ^ { q _ { i - 1 } } x _ { i - 1 } , \\xi ^ { q _ { i + 1 } } x _ { i + 1 } , \\ldots , \\xi _ i ^ { q _ { n } } x _ n ) , \\end{align*}"}
{"id": "5455.png", "formula": "\\begin{align*} \\| ( \\mathcal { M } ( \\tilde u _ 0 , a , b _ n ) u ) ( t , \\cdot ) - ( \\mathcal { M } ( \\tilde u _ 0 , a , b _ 0 ) u ) ( t , \\cdot ) \\| _ { C ^ 0 ( \\bar \\Omega ) } & = \\| \\int _ s ^ t e ^ { - A ( t - \\tau ) } u ^ 2 ( \\tau , \\cdot ) \\big [ b _ n ( \\tau , \\cdot ) - b _ 0 ( \\tau , \\cdot ) \\big ] d \\tau \\| _ { C ^ 0 ( \\bar \\Omega ) } \\\\ & \\le R ^ 2 \\| b _ n - b _ 0 \\| _ { \\mathcal { X } _ T } T . \\end{align*}"}
{"id": "6400.png", "formula": "\\begin{align*} U = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\ , : \\ , \\mathbb C ^ m \\oplus \\mathbb C ^ n \\rightarrow \\mathbb C ^ m \\oplus \\mathbb C ^ n . \\end{align*}"}
{"id": "3565.png", "formula": "\\begin{align*} \\gamma \\circ ( \\mu ^ T \\Gamma ) = ( \\Gamma \\mu ^ S ) \\circ ( \\gamma S ) \\circ ( T \\gamma ) \\end{align*}"}
{"id": "4233.png", "formula": "\\begin{align*} D = \\frac { 4 \\pi ^ 2 } { A ^ 2 } . \\end{align*}"}
{"id": "2804.png", "formula": "\\begin{align*} \\sum _ { j _ = 1 } ^ N \\lambda _ j \\partial _ { x _ j } Q + \\lambda _ { N + 1 } i Q + \\lambda _ { N + 2 } \\mathcal { Y } _ + + \\lambda _ { N + 3 } f = 0 , \\end{align*}"}
{"id": "9215.png", "formula": "\\begin{align*} \\norm { J ^ A _ \\gamma x - J ^ A _ \\gamma y } & \\leq \\norm { ( J ^ A _ \\gamma x - J ^ A _ \\gamma y ) + s ( \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma ^ { - 1 } ( y - J ^ A _ \\gamma y ) ) } \\\\ & = \\norm { ( J ^ A _ \\gamma x - J ^ A _ \\gamma y ) + s \\gamma ^ { - 1 } ( ( x - y ) - ( J ^ A _ \\gamma x - J ^ A _ \\gamma y ) ) } \\\\ & = \\norm { s \\gamma ^ { - 1 } ( x - y ) + ( 1 - s \\gamma ^ { - 1 } ) ( J ^ A _ \\gamma x - J ^ A _ \\gamma y ) } \\end{align*}"}
{"id": "576.png", "formula": "\\begin{align*} \\big \\{ ( x , x ^ \\prime , y , y ^ \\prime ) \\ \\ y ^ \\prime = f ^ \\prime ( x ^ \\prime ) \\big \\} \\approx \\Gamma _ { f ^ \\prime } \\times \\R ^ { n } \\times \\R ^ { m } . \\end{align*}"}
{"id": "6593.png", "formula": "\\begin{align*} \\sum _ { \\substack { c > C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) = \\sum _ { c | q } \\mu ( c ) - \\sum _ { \\substack { c \\le C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) = \\left \\lfloor \\frac { 1 } { q } \\right \\rfloor - \\sum _ { \\substack { c \\le C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) . \\end{align*}"}
{"id": "2032.png", "formula": "\\begin{align*} L _ t ^ h = \\exp \\left ( M _ t - \\frac { \\ , 1 \\ , } { 2 } \\ < M ^ c \\ > _ t \\right ) \\prod _ { 0 < s \\le t } \\frac { h ( X _ s ) } { h ( X _ { s - } ) } \\exp \\left ( 1 - \\frac { h ( X _ s ) } { h ( X _ { s - } ) } \\right ) , \\end{align*}"}
{"id": "6981.png", "formula": "\\begin{align*} u _ k : = \\frac { x _ k - \\bar x } { \\norm { x _ k - \\bar x } } , v _ k : = \\frac { y _ k - \\bar y } { \\norm { y _ k - \\bar y } } , y _ k ^ * : = \\frac { \\norm { y _ k - \\bar y } } { \\norm { x _ k - \\bar x } } \\lambda _ k \\end{align*}"}
{"id": "408.png", "formula": "\\begin{align*} B ^ { i j } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & B _ { 0 } ^ { i j } & 0 \\\\ 0 & 0 & 0 \\\\ \\end{array} \\right ) \\in \\mathbb { M } _ { N \\times N } , \\end{align*}"}
{"id": "8013.png", "formula": "\\begin{align*} \\bar { \\rho } \\circ d _ { \\Sigma } = d _ { \\Sigma _ 0 } \\circ \\bar { \\rho } , \\end{align*}"}
{"id": "224.png", "formula": "\\begin{align*} \\dfrac { p _ \\alpha ( v ) } { p _ { \\alpha } ( x ) ^ 2 } \\dfrac { 1 } { ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { d } { \\alpha } } } D ^ { \\alpha - 1 } ( p _ \\alpha ) ( x ) p _ \\alpha \\left ( \\dfrac { x - v e ^ { - t } } { ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { 1 } { \\alpha } } } \\right ) = \\dfrac { p _ \\alpha ( v ) } { p _ { \\alpha } ( x ) } \\dfrac { ( - x ) } { ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { d } { \\alpha } } } p _ \\alpha \\left ( \\dfrac { x - v e ^ { - t } } { ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { 1 } { \\alpha } } } \\right ) , \\end{align*}"}
{"id": "1846.png", "formula": "\\begin{align*} W ( n , k + 1 ) = M ( n , k ) . \\end{align*}"}
{"id": "2001.png", "formula": "\\begin{align*} ( u _ 1 , u _ 2 , u _ 3 ) & = ( | x _ 1 | ^ 2 - | x _ 2 | ^ 2 , 2 \\mathrm { R e } ( x _ 1 \\bar { x _ 2 } ) , 2 \\mathrm { I m } ( x _ 1 \\bar { x _ 2 } ) ) \\\\ & = ( r _ 1 ^ 2 - r _ 2 ^ 2 , 2 r _ 1 r _ 2 \\cos ( \\theta _ 1 - \\theta _ 2 ) , 2 r _ 1 r _ 2 \\sin ( \\theta _ 1 - \\theta _ 2 ) ) . \\end{align*}"}
{"id": "1831.png", "formula": "\\begin{align*} W _ n ( x , y ) = \\sum _ { k = 0 } ^ { \\lfloor ( n - 1 ) / 2 \\rfloor } W ( n , k + 1 ) x ^ { 2 k + 2 } y ^ { n - 2 k - 1 } . \\end{align*}"}
{"id": "9017.png", "formula": "\\begin{align*} & \\partial _ t \\rho _ i = \\nabla \\cdot \\left [ D _ i ( x ) \\left ( \\nabla \\rho _ i + z _ i \\rho _ i \\nabla \\phi \\right ) \\right ] , i = 1 , \\cdots , s , \\\\ & - \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ) = f ( x ) + \\sum _ { i = 1 } ^ { s } z _ i \\rho _ i . \\end{align*}"}
{"id": "1994.png", "formula": "\\begin{align*} R H _ \\infty \\subsetneq \\bigcap _ { s > 1 } R H _ s \\subset \\bigcup _ { s > 1 } R H _ s = A _ \\infty . \\end{align*}"}
{"id": "4349.png", "formula": "\\begin{align*} g = 2 \\alpha , \\end{align*}"}
{"id": "6976.png", "formula": "\\begin{align*} \\frac { y _ k } { \\norm { x _ k - \\bar x } } = \\frac { g ( x _ k ) - z _ k } { \\norm { x _ k - \\bar x } } = \\frac { g ( x _ k ) - g ( \\bar x ) } { \\norm { x _ k - \\bar x } } - \\frac { z _ k - g ( \\bar x ) } { \\norm { x _ k - \\bar x } } \\to v - v = 0 , \\end{align*}"}
{"id": "2410.png", "formula": "\\begin{align*} S f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , e _ \\gamma \\rangle e _ \\gamma . \\end{align*}"}
{"id": "9118.png", "formula": "\\begin{align*} \\lambda _ { I J K } = \\sum _ { i \\in I } \\sum _ { j \\in J } \\lambda _ { i j k } \\end{align*}"}
{"id": "6644.png", "formula": "\\begin{align*} \\Sigma _ 2 : = p ^ { ( w - 1 ) h _ p } \\sum _ { m = \\max \\{ 0 , k _ p - h _ p \\} } ^ { \\infty } \\sum _ { n = m + h _ p - k _ p + 1 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( 2 - \\alpha - w ) } p ^ { n ( 1 - \\beta ) } } . \\end{align*}"}
{"id": "9481.png", "formula": "\\begin{align*} \\mathcal { B C } _ { ( s _ 1 , s _ 2 , \\dots , s _ p ) } = \\mathcal { C S } _ { ( s _ 1 , s _ 2 , \\dots , s _ p ) } = \\mathcal { D D } _ { ( s _ 1 , s _ 2 , \\dots , s _ p ) } . \\end{align*}"}
{"id": "7261.png", "formula": "\\begin{align*} A _ 3 = & 1 8 t \\theta _ 2 - 9 a d = \\dfrac { 9 b c ( 3 a c - b ^ 2 ) ( 3 b d - c ^ 2 ) } { 3 a c ^ 3 + 3 b ^ 3 d - 2 b ^ 2 c ^ 2 } - 9 a d \\\\ = & \\dfrac { 9 ( b ^ 3 c ^ 3 - 3 a b c ^ 4 - 3 b ^ 4 c d + 1 1 a b ^ 2 c ^ 2 d - 3 a ^ 2 c ^ 3 d - 3 a b ^ 3 d ^ 2 ) } { 3 a c ^ 3 + 3 b ^ 3 d - 2 b ^ 2 c ^ 2 } . \\end{align*}"}
{"id": "1767.png", "formula": "\\begin{align*} B \\Phi ( a _ 0 \\otimes \\ldots \\otimes a _ { k - 1 } ) : = & \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ { i ( k - 1 ) } \\Phi ( 1 , a _ i , \\cdots , a _ { k - 1 } , a _ 0 , \\ldots , a _ { i - 1 } ) \\\\ & - ( - 1 ) ^ { i ( k - 1 ) } \\Phi ( a _ i , 1 , a _ { i + 1 } , \\cdots , a _ { k - 1 } , a _ 0 , \\cdots , a _ { i - 1 } ) . \\end{align*}"}
{"id": "5360.png", "formula": "\\begin{align*} \\rho _ c ( x ) = \\frac { \\left \\langle c , z - x \\right \\rangle } { \\| c \\| } . \\end{align*}"}
{"id": "497.png", "formula": "\\begin{align*} \\Upsilon _ { \\omega } ^ { \\sigma ( + ) } = \\left \\{ \\tau ^ { 2 \\cdot \\varepsilon _ { 2 } } - 1 , C ^ { - 1 } \\right \\} , \\Upsilon _ { \\omega } ^ { \\sigma ( - ) } = \\left \\{ \\tau - 1 , - \\tau ^ { \\varepsilon _ { 1 } } - 1 \\right \\} . \\end{align*}"}
{"id": "8546.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi + \\phi + V ( x ) \\phi \\pm \\phi ^ 3 = 0 , ( u , u _ t ) ( 0 ) = ( u _ 0 , u _ 1 ) , \\end{align*}"}
{"id": "6871.png", "formula": "\\begin{align*} \\left ( \\sum \\limits _ { j = 1 } ^ k \\norm { f _ j } \\right ) ^ 2 & \\leq k \\sum \\limits _ { j = 1 } ^ k \\norm { f _ j } ^ 2 \\\\ & \\leq k \\langle f , f \\rangle - k \\sum \\limits _ { i \\neq j \\neq 0 } \\langle f _ i , f _ j \\rangle \\\\ & \\leq k \\langle f , f \\rangle + c \\gamma \\sum \\limits _ { i \\neq j \\neq 0 } \\norm { f _ i } \\norm { f _ j } \\\\ & \\leq k \\langle f , f \\rangle + c \\gamma \\left ( \\sum \\limits _ { j = 1 } ^ k \\norm { f _ j } \\right ) ^ 2 \\end{align*}"}
{"id": "9051.png", "formula": "\\begin{align*} & \\frac { \\rho ^ { n + 1 } _ i - \\rho ^ n _ i } { \\tau } = \\nabla \\cdot \\left ( D _ i ( \\nabla \\rho ^ { n + 1 } _ i + z _ i \\nabla \\phi ^ { n + 1 } ) \\right ) + O ( \\tau ) , \\\\ & \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ^ { n + 1 } ) = f + \\sum _ { i = 1 } ^ s z _ i \\rho _ i ^ { n + 1 } . \\end{align*}"}
{"id": "2196.png", "formula": "\\begin{align*} \\varphi _ \\pm ( w ) & = \\langle J ^ { ' } ( w ) , w ^ \\pm \\rangle = \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { w ( x ) - w ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { w ^ \\pm ( x ) - w ^ \\pm ( y ) } { \\vert x - y \\vert ^ { \\alpha + d } } d x d y \\\\ & + \\int _ { \\mathbb { R } ^ { d } } g ( w ) w ^ \\pm d x - \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , w ^ \\pm ) w ^ \\pm d x . \\end{align*}"}
{"id": "8499.png", "formula": "\\begin{align*} & \\int _ { ( \\ell _ 1 - 1 ) 2 ^ { - k _ 1 } } ^ { \\ell _ 1 2 ^ { - k _ 1 } } \\int _ { ( \\ell _ 2 - 1 ) 2 ^ { - k _ 2 } } ^ { \\ell _ 2 2 ^ { - k _ 2 } } f _ { \\rho , L } ( x , y ) \\ , d y \\ , d x \\\\ & = 2 ^ { - k _ 1 - k _ 2 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\bigl \\{ 1 + \\rho \\sin ( 2 ^ { L - k _ 1 + 1 } \\pi u ) \\sin ( 2 ^ { L - k _ 2 + 1 } \\pi v ) \\bigr \\} \\ , d v \\ , d u = 2 ^ { - k _ 1 - k _ 2 } , \\end{align*}"}
{"id": "2.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 n } \\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ r r ^ { 4 n - 2 k } P _ { 2 k } \\left ( 4 n + 1 , 2 n + 1 \\right ) = 0 . \\end{align*}"}
{"id": "606.png", "formula": "\\begin{align*} \\ = \\ . \\end{align*}"}
{"id": "3553.png", "formula": "\\begin{align*} \\Pi ( S ) = C ( \\Sigma ) . \\end{align*}"}
{"id": "3796.png", "formula": "\\begin{align*} L _ 1 & \\cong 2 ^ { - 1 } A ( 0 , \\ , 0 ) \\bot \\cdots \\bot 2 ^ { - 1 } A ( 0 , \\ , 0 ) \\bot 2 ^ { - 1 } A ( 0 , \\ , 0 ) \\ , , \\\\ L _ 1 & \\cong 2 ^ { - 1 } A ( 0 , \\ , 0 ) \\bot \\cdots \\bot 2 ^ { - 1 } A ( 0 , \\ , 0 ) \\bot 2 ^ { - 1 } A ( 2 , \\ , 2 \\rho ) \\ , . \\end{align*}"}
{"id": "3526.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs { \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) + E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 + O \\left ( \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) E ( s _ 1 , s _ 2 , s _ 3 ) } d t _ 3 \\right ) \\\\ & \\quad + O \\left ( \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\right ) . \\end{align*}"}
{"id": "2869.png", "formula": "\\begin{align*} e ^ { e _ 0 t } \\alpha _ + ( t ) = \\alpha _ + ( 0 ) + \\int _ 0 ^ t e ^ { e _ 0 s } B ( R , \\mathcal { Y } _ - ) d s . \\end{align*}"}
{"id": "2568.png", "formula": "\\begin{align*} \\langle T _ x \\ell , f \\rangle = \\langle \\ell , T _ { - x } f \\rangle \\langle M _ \\omega \\ell , f \\rangle = \\langle \\ell , M _ { - \\omega } f \\rangle . \\end{align*}"}
{"id": "2776.png", "formula": "\\begin{align*} L _ + = - \\Delta + 1 - p \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * \\left ( Q ^ { p - 1 } \\cdot \\right ) \\right ) Q ^ { p - 1 } - ( p - 1 ) \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 2 } , \\end{align*}"}
{"id": "1646.png", "formula": "\\begin{align*} V ^ \\nu _ { 2 , p } ( x ) = \\int _ M g _ \\lambda ( x , y ) \\left ( \\int _ M g _ \\lambda ( z , y ) \\nu ( d z ) \\right ) ^ { q - 1 } \\mu ( d y ) , x \\in M , \\end{align*}"}
{"id": "1680.png", "formula": "\\begin{align*} q ' ( x ) = 1 - \\frac { e } { \\frac { e } { 2 } x + e + \\frac { 1 } { 2 } } > 0 \\end{align*}"}
{"id": "3978.png", "formula": "\\begin{align*} r _ 2 = \\frac { 1 } { 3 } ( - \\lambda + 4 i l \\pi - 2 i m \\pi ) , \\ \\ \\ r _ 3 = \\frac { 1 } { 3 } ( - \\lambda - 2 i l \\pi + 4 i m \\pi ) . \\end{align*}"}
{"id": "5810.png", "formula": "\\begin{align*} \\tilde { y } _ { k , n } [ t ] = \\sum _ { m = 1 } ^ M \\tilde { g } _ { m k , n } [ t ] \\tilde { x } _ { m , n } [ t ] + \\tilde { z } _ { k , n } [ t ] , \\ : \\ : k \\in \\{ 1 , \\ldots , K \\} . \\end{align*}"}
{"id": "8431.png", "formula": "\\begin{gather*} q _ { 1 } = \\theta _ { 1 } \\beta + \\beta K + \\theta _ { 1 } ^ { 1 - \\beta } | b ( 0 ) | ^ { \\beta } + \\frac { C _ { d , \\alpha } \\beta ( 3 - \\beta ) \\sqrt { d } \\sigma _ { d - 1 } } { 2 ( 2 - \\alpha ) } + \\frac { C _ { d , \\alpha } \\beta \\sigma _ { d - 1 } } { \\alpha - \\beta } + \\left ( \\frac { \\theta _ { 1 } } { 4 } \\right ) ^ { 1 - \\beta } \\left ( \\frac { C _ { d , \\alpha } \\sigma _ { d - 1 } } { \\alpha - 1 } \\right ) ^ { \\beta } , \\end{gather*}"}
{"id": "138.png", "formula": "\\begin{align*} x ^ 3 = \\gamma _ 1 ( x ) x ^ 2 + \\gamma _ 2 ( x ) x \\end{align*}"}
{"id": "1477.png", "formula": "\\begin{align*} \\iota ( z _ 1 , z _ 2 ) = \\end{align*}"}
{"id": "5145.png", "formula": "\\begin{align*} \\quad \\psi '' ( \\eta ) = \\tfrac { 1 } { n ^ 2 } f '' ( \\tfrac { \\eta } { n } ) - f '' ( \\eta ) , \\eta \\in ( \\eta _ n / 2 , 2 \\eta _ n ) . \\end{align*}"}
{"id": "5683.png", "formula": "\\begin{align*} \\mathcal { M } _ { k } ^ { J } ( \\alpha , \\beta ) : = \\{ \\ , u \\in \\mathcal { M } ^ { J } ( \\alpha , \\beta ) \\ , | \\ , I ( u ) = k \\ , \\ , \\} \\end{align*}"}
{"id": "2274.png", "formula": "\\begin{align*} I ( \\omega ) = e ^ { - \\frac { \\pi \\omega ^ 2 } { M } + C } = C _ 1 \\ , e ^ { - \\frac { \\pi \\omega ^ 2 } { M } } . \\end{align*}"}
{"id": "2665.png", "formula": "\\begin{align*} \\mu _ { \\alpha , \\infty } = \\prod _ { k = 1 } ^ d \\left ( \\int _ 0 ^ \\infty \\left ( \\frac { s } { \\pi } \\right ) ^ { \\alpha _ k } e ^ { - s } \\ , d s \\right ) = \\prod _ { k = 1 } ^ d \\frac { \\Gamma ( \\alpha _ k ) } { \\pi ^ { \\alpha _ k } } = \\prod _ { k = 1 } ^ d \\frac { \\alpha _ k ! } { \\pi ^ { \\alpha _ k } } = \\frac { \\alpha ! } { \\pi ^ { | \\alpha | } } . \\end{align*}"}
{"id": "7920.png", "formula": "\\begin{align*} \\sum _ { l , k } ( \\partial ^ 2 _ { x _ l x _ k } f _ \\alpha ) ( \\partial _ { t _ j } \\varphi ^ l ) ( \\partial _ { t _ i } \\varphi ^ k ) + \\sum _ { l } ( \\partial _ { x _ l } f _ \\alpha ) ( \\partial ^ 2 _ { t _ i t _ j } \\varphi ^ l ) = 0 . \\end{align*}"}
{"id": "4230.png", "formula": "\\begin{align*} - \\Delta u = { } & f ( u ) , \\textrm { i n } \\Omega , \\\\ u = { } & 0 , \\textrm { o n } \\partial \\Omega . \\end{align*}"}
{"id": "3590.png", "formula": "\\begin{align*} S ( e , z ^ * ) = [ ( Q \\widetilde { T } ) ^ * ( z ^ * ) ] ( e \\otimes \\cdot ) \\ , \\ , ( e , z ^ * ) \\in E \\times Q ( X ) ^ * . \\end{align*}"}
{"id": "5014.png", "formula": "\\begin{align*} X _ { k } & = | \\pi _ 1 \\tau - \\pi _ 1 c _ { k } | , \\\\ Y _ { k } & = | \\pi _ 2 \\tau - \\pi _ 2 c _ { k } | = | \\pi _ 2 c _ { k } | . \\end{align*}"}
{"id": "4390.png", "formula": "\\begin{align*} H _ 0 '' + \\frac { d + 1 } { \\xi } H _ 0 ' - 3 ( d - 2 ) ( 2 Q _ \\sigma + \\xi ^ 2 Q _ \\sigma ^ 2 ) H _ 0 = 0 , \\end{align*}"}
{"id": "9456.png", "formula": "\\begin{align*} N _ { 1 0 } + \\tfrac { B ( 0 , 1 ) ^ p } { B ( 0 ) ^ p } - \\tfrac { B ( - 1 , 1 ) } { B ( - 1 ) } = 0 . \\end{align*}"}
{"id": "4882.png", "formula": "\\begin{align*} w = \\frac { K + 1 } { K - 1 } - \\frac { 2 K } { ( K - 1 ) z } , z = \\phi ( w ) = \\frac { 2 K } { K + 1 - ( K - 1 ) w } , \\end{align*}"}
{"id": "3333.png", "formula": "\\begin{align*} \\frac { \\tau ( n ) } { \\tau ( n ) \\log n } = \\frac { 1 } { \\log n } . \\end{align*}"}
{"id": "24.png", "formula": "\\begin{gather*} \\omega ( x _ i ^ \\pm ( u ) ) = x _ { i } ^ \\mp ( u ) , \\omega ( h _ i ( u ) ) = h _ i ( u ) \\\\ \\varsigma ( x _ i ^ \\pm ( u ) ) = x _ i ^ \\pm ( - u ) , \\varsigma ( h _ i ( u ) ) = h _ i ( - u ) \\end{gather*}"}
{"id": "5596.png", "formula": "\\begin{align*} h \\rhd ( a \\star b ) = ( h _ { \\widehat { ( 1 ) } } \\rhd a ) \\star ( h _ { \\widehat { ( 2 ) } } \\rhd b ) , \\end{align*}"}
{"id": "5143.png", "formula": "\\begin{align*} \\tfrac { 1 } { n } A ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) = \\tanh ( \\tfrac { \\eta } { 2 } ) \\ , \\psi ( \\eta ) . \\end{align*}"}
{"id": "1794.png", "formula": "\\begin{align*} \\langle \\Phi ^ P _ { t } , \\operatorname { I n d } ( D ) \\rangle = \\frac { \\sum _ { w \\in W _ K } ( - 1 ) ^ w e ^ { w \\cdot \\mu } ( t ) } { \\Delta ^ { M } _ T ( t ) } . \\end{align*}"}
{"id": "2061.png", "formula": "\\begin{align*} \\int _ { \\{ q \\Psi < - t \\} } | f | ^ 2 e ^ { - \\varphi _ 0 } \\geq e ^ { - t } C ( q \\Psi , \\varphi _ 0 , J _ q , f ) = e ^ { - t } C ( \\Psi , \\varphi _ 0 , J _ q , f ) . \\end{align*}"}
{"id": "3418.png", "formula": "\\begin{align*} | \\langle T \\chi _ 0 , \\psi \\rangle | & = | - \\langle T \\chi _ 1 , \\psi \\rangle | \\\\ & = \\Big | \\int _ { \\R ^ N } \\int _ { \\R ^ N } [ K ( x , y ) - K ( x _ 0 , y ) ] \\chi _ 1 ( y ) \\psi ( x ) d \\omega ( y ) d \\omega ( x ) \\Big | . \\end{align*}"}
{"id": "1029.png", "formula": "\\begin{align*} w _ M ( 2 e _ 1 ) & = - C M \\int _ { \\R ^ n \\setminus B _ 1 ( 2 e _ 1 ) } \\frac { \\zeta _ 2 ( y ) } { ( \\vert y \\vert ^ 2 - 1 ) ^ s \\vert y \\vert ^ n } \\dd y . \\end{align*}"}
{"id": "3240.png", "formula": "\\begin{align*} \\omega ( z ) = z - \\frac { | \\lambda | ^ 2 } { z + H _ 1 ( \\omega ( z ) ) } \\end{align*}"}
{"id": "788.png", "formula": "\\begin{align*} I ( v ) : = \\int _ { \\overline { \\Omega } } | \\nabla v | ^ { p } \\ , d \\mu - p \\int _ { \\partial { \\Omega } } v f \\ , d \\nu \\end{align*}"}
{"id": "1824.png", "formula": "\\begin{align*} D _ n ( u , v ) = \\sum _ { L } 2 ^ { f _ 2 ( T ) } u ^ { f _ { 0 } ( T ) } ( 2 v ) ^ { f _ 1 ( T ) } , \\end{align*}"}
{"id": "7774.png", "formula": "\\begin{align*} \\Box \\tilde \\phi _ { k + 1 } = \\left ( | \\tilde \\phi _ { k + 1 , t } | ^ 2 - | \\tilde \\phi _ { k + 1 , x } | ^ 2 \\right ) \\tilde \\phi _ { k + 1 } + \\mathbf { 1 } _ { \\omega } f _ { k } ^ { \\tilde \\phi ^ { \\perp } _ { k + 1 } } + e _ k , \\ ; \\tilde \\phi _ { k + 1 } [ 0 ] = ( a , b ) \\end{align*}"}
{"id": "2337.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z ^ d } f ( k - x ) e ^ { 2 \\pi i \\omega ( k - x ) } = \\sum _ { l \\in \\Z ^ d } \\widehat { f } ( l - \\omega ) e ^ { - 2 \\pi i x \\cdot l } . \\end{align*}"}
{"id": "7834.png", "formula": "\\begin{align*} X _ { 1 } & = ~ \\{ x , T ^ { * 3 } x , \\ldots , T ^ { * 3 ( 3 j - n - r ) } x , y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( 3 j - n - 2 r ) } y \\} \\\\ X _ { 2 } & = ~ \\{ T ^ { * 3 ( 3 j - n - r + 1 ) } x , \\ldots , T ^ { * 3 j } x , T ^ { * 3 ( 3 j - n - 2 r + 1 ) } y , \\ldots , T ^ { * 3 ( j - 1 - r ) } y , z , \\ldots , T ^ { * 3 ( n - 2 j - 2 + r ) } z \\} . \\end{align*}"}
{"id": "8969.png", "formula": "\\begin{align*} \\begin{aligned} \\phi ( x + h ) - 2 \\phi ( x ) + \\phi ( x - h ) & \\leq u ( x + h ) - 2 u ( x ) + u ( x - h ) \\\\ & \\leq C \\left ( 1 + \\frac { 2 } { C _ 0 } \\right ) | h | ^ 2 \\end{aligned} \\end{align*}"}
{"id": "4773.png", "formula": "\\begin{align*} R ^ 2 + R ^ \\# & = ( R _ 0 ) ^ 2 + ( R _ 2 ) ^ 2 + ( R _ 0 ) ^ \\# + ( R _ 2 ) ^ \\# \\\\ & = ( n - 1 ) ( R _ 0 ) ^ 2 + ( R _ 2 ) ^ 2 + ( R _ 2 ) ^ \\# . \\end{align*}"}
{"id": "8670.png", "formula": "\\begin{align*} 1 - F _ N ( x s _ N ) = \\exp \\left ( - \\frac { 1 } { 2 } x ^ 2 Q _ N ( x ) \\right ) \\left ( 1 - \\Phi ( x ) + \\vartheta \\lambda _ N e ^ { - x ^ 2 / 2 } \\right ) . \\end{align*}"}
{"id": "7377.png", "formula": "\\begin{align*} V ( r , p ) = \\beta r ^ { - 1 } | p | ^ 2 + a ( 1 - \\beta ) ^ { \\alpha - 1 } r ^ { \\beta ( 1 - \\alpha ) } | p | ^ \\alpha \\end{align*}"}
{"id": "2031.png", "formula": "\\begin{align*} | M _ { \\zeta } ^ { h _ m } - M _ { \\zeta } ^ { h _ n } | & = | N _ { \\zeta } ^ { h _ m } - N _ { \\zeta } ^ { h _ n } | \\\\ & \\leq \\int _ 0 ^ { \\zeta } \\left | h - \\ 1 _ { K _ n } h _ n \\right | ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } _ 1 ^ * + { \\mu } _ 2 + e ^ { \\| F _ 1 \\| _ { \\infty } } N ( F _ 2 ) \\mu _ H } & \\to 0 \\quad m , n \\to \\infty \\end{align*}"}
{"id": "8725.png", "formula": "\\begin{align*} x _ 1 ^ { \\alpha _ 1 } ( 1 - x _ 1 ) ^ { \\beta _ 1 } \\cdots x _ n ^ { \\alpha _ n } ( 1 - x _ n ) ^ { \\beta _ n } , \\sum _ { i = 1 } ^ n ( \\alpha _ i + \\beta _ i ) = \\delta , \\end{align*}"}
{"id": "8018.png", "formula": "\\begin{align*} \\phi ( u , v ) = \\int _ 0 ^ u \\psi ( - u ' ) \\ , \\mathrm { d } u ' . \\end{align*}"}
{"id": "129.png", "formula": "\\begin{align*} U ' = \\{ u + 2 u _ 0 u \\ ; | \\ ; u \\in U \\} , \\ ; \\ ; V ' = \\{ v - 2 ( u _ 0 + u _ 0 ^ 2 ) v \\ ; | \\ ; v \\in V \\} . \\end{align*}"}
{"id": "4134.png", "formula": "\\begin{align*} I _ 2 ( u ) = \\int _ { \\R } u ^ 2 ( x , t ) d x \\end{align*}"}
{"id": "118.png", "formula": "\\begin{align*} ( v _ 1 * \\ldots * v _ k ) . u = v _ 1 ( \\ldots ( v _ k u ) \\ldots ) , \\mbox { f o r a l l } v _ 1 , \\ldots , v _ k \\in V \\mbox { a n d } u \\in L ( A ) . \\end{align*}"}
{"id": "9224.png", "formula": "\\begin{align*} \\forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \\left ( x = _ X x ' \\land y = _ X y ' \\rightarrow \\chi _ A x y = _ 0 \\chi _ A x ' y ' \\right ) . \\end{align*}"}
{"id": "6683.png", "formula": "\\begin{align*} L i _ { K , \\mathfrak { s } } ( { \\bf z } ) : = \\sum _ { i _ 1 > \\cdots > i _ r > 0 } \\frac { z _ 1 ^ { i _ 1 } \\cdots z _ r ^ { i _ r } } { ( \\theta ^ { q ^ { i _ 1 } } - \\theta ) ^ { s _ 1 } \\cdots ( \\theta ^ { q ^ { i _ r } } - \\theta ) ^ { s _ r } } \\in \\mathbb { C } _ { \\infty } . \\end{align*}"}
{"id": "353.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) \\in C _ F ( \\rho ^ 0 , \\rho ^ 1 ) } \\{ \\mathcal A ( \\rho , m ) \\} = \\sup _ { S \\in H ^ 1 } \\inf _ { ( \\rho , m ) } \\mathcal L ( \\rho , m , S ) \\end{align*}"}
{"id": "4926.png", "formula": "\\begin{align*} \\begin{aligned} \\bar x _ t & = \\alpha _ t x _ t + ( 1 - \\alpha _ t ) \\bar x _ { t - 1 } \\\\ x _ { t + 1 } & = x _ t - \\alpha _ t \\eta _ t g _ t , \\end{aligned} \\end{align*}"}
{"id": "2209.png", "formula": "\\begin{align*} \\dot { H } ^ \\alpha = \\big \\{ v \\in \\dot { H } : | v | _ \\alpha < \\infty \\big \\} , H ^ \\alpha = \\big \\{ v \\in H : \\| v \\| _ \\alpha < \\infty \\big \\} , \\end{align*}"}
{"id": "7608.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { \\partial f } { \\partial \\kappa _ { i } } \\kappa _ { i } ^ { 2 } \\geq H _ { 1 } H _ { k } ^ { \\frac { 1 } { k } } \\geq H _ { k } ^ { \\frac { 2 } { k } } = f ^ { 2 } . \\end{align*}"}
{"id": "2625.png", "formula": "\\begin{align*} Z \\varphi _ s ( x , - \\omega ) = \\sum _ { k \\in \\Z } e ^ { - \\pi s ( x + k ) ^ 2 } e ^ { 2 \\pi i k \\omega } . \\end{align*}"}
{"id": "4454.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( n + 1 , k ) = S _ { 2 , \\lambda } ( n , k - 1 ) + ( k - n \\lambda ) S _ { 2 , \\lambda } ( n , k ) , ( \\mathrm { S e e } \\ [ 4 ] ) , \\end{align*}"}
{"id": "5170.png", "formula": "\\begin{align*} A ( b ) = \\sum _ { k = 0 } ^ { 1 } \\left | Z _ { b ^ { - 1 } } \\phi \\left ( \\tfrac { 1 } { 4 } + \\tfrac { k } { 2 } , \\tfrac { 1 } { 2 } \\right ) \\right | ^ 2 = 2 \\left | Z _ { b ^ { - 1 } } \\phi \\left ( \\tfrac { 1 } { 4 } , \\tfrac { 1 } { 2 } \\right ) \\right | ^ 2 . \\end{align*}"}
{"id": "6957.png", "formula": "\\begin{align*} \\mathcal N _ Q ( \\bar x ; u ) : = \\left \\{ \\eta \\in \\mathbb X \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ u _ k \\} _ { k \\in \\N } \\subset \\mathbb X , \\ , \\exists \\{ t _ k \\} _ { k \\in \\N } \\subset \\R _ + , \\ , \\exists \\{ \\eta _ k \\} _ { k \\in \\N } \\subset \\mathbb X \\colon \\\\ & u _ k \\to u , \\ , t _ k \\searrow 0 , \\ , \\eta _ k \\to \\eta , \\ , \\eta _ k \\in \\widehat { \\mathcal N } _ Q ( \\bar x + t _ k u _ k ) \\ , \\forall k \\in \\N \\end{aligned} \\right \\} \\end{align*}"}
{"id": "7356.png", "formula": "\\begin{align*} \\begin{aligned} & { 1 \\over q } ( \\lambda h _ q + ( 1 - \\lambda ) k _ q ) + w _ q ( x _ q , t _ q ) ^ { q - 1 } \\mu ( w _ q ( x _ q , t _ q ) ) \\\\ & \\geq - \\lambda v ( y _ q , t _ q ) ^ { 1 - { 1 \\over q } } \\omega ( | X _ q | ) - ( 1 - \\lambda ) v ( z _ q , t _ q ) ^ { 1 - { 1 \\over q } } \\omega ( | X _ q | ) , \\end{aligned} \\end{align*}"}
{"id": "2924.png", "formula": "\\begin{align*} \\tilde M _ n ( k ) = \\sum \\limits _ { \\substack { A \\subset \\{ 1 , \\dots , d \\} \\\\ | A | = k } } \\tilde M _ { n , A } , \\tilde N _ n ( k ) = \\sum \\limits _ { \\substack { A \\subset \\{ 1 , \\dots , d \\} \\\\ | A | = k } } \\tilde N _ { n , A } . \\end{align*}"}
{"id": "9223.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , x ^ X , p ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land p = _ X J ^ A _ \\gamma x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "6873.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { \\mathbb { E } [ f ] } ( D ^ k _ i f ) \\right | = \\left | \\underset { P _ i } { \\mathbb { E } } [ D ^ k _ i f ] - \\mathbb { E } [ f ] \\right | \\leq \\| D ^ k _ i f - \\mathbb { E } [ f ] \\| _ { \\infty } . \\end{align*}"}
{"id": "7535.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) = - \\frac { \\epsilon T } { 2 \\pi } \\log \\pi + \\frac { \\epsilon T } { 2 \\pi } \\log \\left ( \\frac { T } { 2 } \\right ) - \\frac { \\epsilon T } { 2 \\pi } + \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\arg \\zeta ( \\sigma + i T ) \\ d \\sigma + \\left ( \\frac { \\epsilon } { \\pi } \\right ) \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "2793.png", "formula": "\\begin{align*} \\Phi _ 1 ( h _ 1 ) \\ge c \\| h _ 1 \\| _ 2 ^ 2 , \\forall \\ ; h _ 1 \\int \\left ( \\partial _ { x _ j } Q \\right ) h _ 1 = \\int ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p ) Q ^ { p - 1 } h _ 1 = 0 , \\forall 1 \\le j \\le N . \\end{align*}"}
{"id": "854.png", "formula": "\\begin{align*} \\bar { \\Delta } _ { \\rm R e a c t i v e } = \\frac { \\mathbb { E } \\sum _ { t \\in \\mathcal { I } _ j } \\Delta { \\left ( t \\right ) } } { \\mathbb { E } T ^ { \\rm R e a c } _ j } = \\frac { \\mathbb { E } T ^ { \\rm P r o a c } _ j } { 2 \\mathbb { E } \\left ( T ^ { \\rm R e a c } _ j \\right ) ^ 2 } + \\mathbb { E } \\tau ^ { \\rm R e a c } _ { V _ j } - \\tau _ { \\rm f } - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "9555.png", "formula": "\\begin{align*} R \\leq \\min \\begin{cases} 2 C , \\\\ C + C _ { 0 } + \\frac { 1 } { 2 } \\log \\left ( \\frac { 1 - \\rho _ c ^ 2 + ( 1 - \\rho ) ( 1 + \\rho - 2 \\rho _ c ^ 2 ) P } { 1 - \\rho _ c ^ 2 } \\right ) , \\\\ \\frac { 1 } { 2 } \\log \\left ( 1 + 2 ( 1 + \\rho ) P \\right ) , \\\\ 2 C _ { 0 } + \\frac { 1 } { 2 } \\log \\left ( 1 + 2 ( 1 + \\rho - 2 \\rho _ c ^ 2 ) P \\right ) , \\end{cases} \\end{align*}"}
{"id": "3766.png", "formula": "\\begin{align*} F ( z _ 1 , z _ 2 ) = ( \\alpha z _ 1 + t z _ 2 ^ m , \\beta z _ 2 ) , 0 < | \\alpha | < | \\beta | < 1 , t ( \\alpha - \\beta ^ m ) = 0 , \\ m \\ge 1 . \\end{align*}"}
{"id": "5429.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ 4 } = \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 = \\sqrt { \\frac { s _ { + } ( a ) s _ { + } ( c ) } { s _ { + } ( b ) } } \\sqrt { \\frac { s _ { + } ( a ) s _ { + } ( b ) } { s _ { + } ( c ) } } \\sqrt { \\frac { s _ { + } ( b ) s _ { + } ( c ) } { s _ { + } ( a ) } } = \\sqrt { s _ + ( a ) s _ + ( b ) s _ + ( c ) } . \\end{align*}"}
{"id": "5531.png", "formula": "\\begin{align*} \\eta _ m ^ { j , k } : = \\frac { 1 } { \\delta _ m } \\big ( B ^ j ( k \\delta _ m ) - B ^ j ( ( k - 1 ) \\delta _ m ) \\big ) , j = 1 , \\ldots , r k = 1 , \\ldots , m \\end{align*}"}
{"id": "3259.png", "formula": "\\begin{align*} s - t = \\frac { 1 - \\sqrt { 1 - 4 | \\lambda | ^ 2 h ( s ) ^ 2 } } { 2 h ( s ) } . \\end{align*}"}
{"id": "8237.png", "formula": "\\begin{align*} \\frac { [ z ^ n ] \\frac { \\partial ^ 2 } { \\partial x _ h ^ 2 } ( N _ 1 + N _ 2 ) | _ { x _ 1 = x _ 2 = 1 } + [ z ^ n ] \\frac { \\partial } { \\partial x _ h } ( N _ 1 + N _ 2 ) | _ { x _ 1 = x _ 2 = 1 } } { [ z ^ n ] ( N _ 1 + N _ 2 ) | _ { x _ 1 = x _ 2 = 1 } } , \\end{align*}"}
{"id": "3844.png", "formula": "\\begin{align*} ( y ^ { [ n ] } , z ) = ( - 1 ) ^ m ( y ^ { [ m ] } , z ^ { ( m ) } ) + \\sum _ { j = 0 } ^ m \\sum _ { s = 0 } ^ { m - 1 } ( q _ { j , s } y ^ { ( j ) } , z ^ { ( s ) } ) . \\end{align*}"}
{"id": "2198.png", "formula": "\\begin{align*} 0 = \\langle J ^ { ' } ( \\widehat { w } ) , \\widehat { w } \\rangle + \\lambda _ + \\langle \\varphi ^ { * } _ { \\widehat { w } ^ + } , \\widehat { w } \\rangle + \\lambda _ - \\langle \\varphi ^ { * } _ { \\widehat { w } ^ - } , \\widehat { w } \\rangle = \\lambda _ + \\langle \\varphi ^ { * } _ { \\widehat { w } ^ + } , \\widehat { w } \\rangle + \\lambda _ - \\langle \\varphi ^ { * } _ { \\widehat { w } ^ - } , \\widehat { w } \\rangle , \\end{align*}"}
{"id": "645.png", "formula": "\\begin{align*} \\Xi \\ = \\ e \\end{align*}"}
{"id": "1510.png", "formula": "\\begin{align*} \\alpha _ { \\mathfrak { n } } ( q ^ { \\ast } h q , s , \\chi ) = \\frac { \\prod _ { i = 1 } ^ { m - r } L _ { \\mathfrak { n } } ( 2 s - 4 m + 2 r + 2 i + 1 , \\chi ^ 2 ) } { \\prod _ { i = 0 } ^ { m - 1 } L _ { \\mathfrak { n } } ( 2 s - 2 i , \\chi ^ 2 ) } \\prod _ { p } P _ { h , q , p } ( \\chi ^ { \\ast } ( p ) p ^ { - s } ) , \\end{align*}"}
{"id": "4998.png", "formula": "\\begin{align*} f _ { \\zeta } ( x ) = \\begin{cases} \\eta \\circ \\xi ( x ) & x \\in [ \\eta ( 0 ) , 0 ] , \\\\ \\eta ( x ) & x \\in [ 0 , \\xi \\circ \\eta ( 0 ) ] \\end{cases} \\end{align*}"}
{"id": "5074.png", "formula": "\\begin{align*} f ( x ) - f ( y _ 1 ) & \\geq d ( x , y _ 1 ) - \\epsilon r \\\\ & = r - \\epsilon r \\\\ & = r ( 1 - \\epsilon ) \\\\ & > 0 . \\end{align*}"}
{"id": "6835.png", "formula": "\\begin{align*} S p e c ( U D ) \\in \\{ 1 \\} \\cup \\bigcup \\limits _ { i = 1 } ^ k [ \\lambda _ i ( U D ) + O _ k ( \\gamma ) , \\lambda _ i ( U D ) - O _ k ( \\gamma ) ] , \\end{align*}"}
{"id": "814.png", "formula": "\\begin{align*} \\rho ( x , y ) : = \\min \\{ 1 , \\ , y ^ { - \\beta } \\} , \\omega ( x , y ) = \\min \\{ 1 , \\ , y ^ { - 2 \\beta } \\} . \\end{align*}"}
{"id": "1114.png", "formula": "\\begin{align*} q ( x , t ) = 2 i D ^ { - 2 } _ { \\infty } ( \\xi ) \\left [ E _ { 1 } ^ { ( 1 2 ) } + \\lim _ { k \\rightarrow \\infty } k \\Delta _ { L } ^ { ( 1 2 ) } ( k ) \\right ] , \\end{align*}"}
{"id": "8971.png", "formula": "\\begin{align*} u ( x ) + | u ' ( x ) | ^ p - f ( x ) = 0 \\end{align*}"}
{"id": "6457.png", "formula": "\\begin{align*} T _ 0 ( u ) ( { x } ) = \\sum _ { k = 1 } ^ { k = n } \\frac { \\eta _ 0 ^ k } { 4 \\pi } \\int _ { B _ k } \\frac { 1 } { | x - y | } u ( y ) d y \\end{align*}"}
{"id": "3275.png", "formula": "\\begin{align*} \\mu _ { T } \\{ \\lambda \\in \\mathbb { C } : | \\lambda | \\leq r \\} = \\begin{cases} 0 , & r < \\lambda _ 1 ( \\mu _ { | T | } ) ; \\\\ 1 + S ^ { \\langle - 1 \\rangle } _ { \\mu _ { T ^ * T } } ( r ^ { - 2 } ) , & \\lambda _ 1 ( \\mu _ { | T | } ) < r < \\lambda _ 2 ( \\mu _ { | T | } ) ; \\\\ 1 , & r \\geq \\lambda _ 2 ( \\mu _ { | T | } ) . \\end{cases} \\end{align*}"}
{"id": "4654.png", "formula": "\\begin{align*} u _ { \\lambda _ 0 , t _ 0 , x _ 0 } ( t , x ) = \\frac { 1 } { \\lambda _ 0 ^ { 1 / ( p - 1 ) } } u \\bigg ( \\frac { t - t _ 0 } { \\lambda _ 0 ^ 2 } , \\frac { x - x _ 0 } { \\lambda _ 0 } \\bigg ) . \\end{align*}"}
{"id": "8193.png", "formula": "\\begin{align*} S ( a , b , 2 f ) = s ( a , b , 2 f ) \\end{align*}"}
{"id": "8508.png", "formula": "\\begin{align*} [ a , q * w ] _ + \\in [ \\Gamma _ { [ c - k ] } ( B ) , \\Gamma _ { k } ( B ) ] _ + = 0 . \\end{align*}"}
{"id": "8668.png", "formula": "\\begin{align*} P _ 0 = \\frac { n _ 0 } { \\gamma } , e _ 0 = n _ 0 \\left ( \\frac { K _ 3 ( \\gamma ) } { K _ 2 ( \\gamma ) } - \\frac { 1 } { \\gamma } \\right ) = n _ 0 \\left ( \\frac { K _ 1 ( \\gamma ) } { K _ 2 ( \\gamma ) } + \\frac { 3 } { \\gamma } \\right ) , \\end{align*}"}
{"id": "1628.png", "formula": "\\begin{align*} \\Gamma ( F ( f _ 1 , . . . , f _ n ) , g ) = \\sum _ { i = 1 } ^ n \\frac { \\partial F } { \\partial x _ i } ( f _ 1 , . . . , f _ n ) \\Gamma ( f _ i , g ) \\end{align*}"}
{"id": "2333.png", "formula": "\\begin{align*} \\mathcal { P } f ( t ) = \\mathcal { P } _ 1 f ( t ) . \\end{align*}"}
{"id": "2926.png", "formula": "\\begin{align*} \\mathcal { P } ( d , k ) = \\left \\{ \\mathbf p _ k = ( p _ 1 , \\dots , p _ k ) \\in \\{ 1 , \\dots , d \\} ^ k : p _ 1 < \\cdots < p _ k \\right \\} . \\end{align*}"}
{"id": "3459.png", "formula": "\\begin{align*} | T ( a ) ( x ) | & = \\bigg | \\int _ Q K ( x , y ) a ( y ) d \\omega ( y ) \\bigg | = \\bigg | \\int _ Q [ K ( x , y ) - k ( x , x _ Q ) ] a ( y ) d \\omega ( y ) \\bigg | \\\\ & \\leqslant C \\Big ( \\frac { \\l ( Q ) } { \\| x - x _ Q \\| } \\Big ) ^ \\varepsilon \\frac { 1 } { \\omega ( x , d ( x , x _ Q ) ) } \\| a \\| _ 1 . \\end{align*}"}
{"id": "325.png", "formula": "\\begin{align*} & \\inf _ { \\rho , v } \\ ; [ \\int _ 0 ^ 1 \\frac 1 2 \\ < v _ t , v _ t \\ > _ { \\theta ( \\rho _ t ) } d t ] \\\\ & \\ ; \\ ; d \\rho ( t ) + d i v _ G ^ { \\theta } ( \\rho ( t ) v ( t ) ) + d i v _ G ^ { \\theta } ( \\rho ( t ) \\nabla _ G \\Sigma ) \\circ d W _ t = 0 \\\\ & \\ ; \\ ; \\rho ( 0 , \\omega ) = \\rho _ a , \\ ; \\rho ( 1 , \\omega ) = \\rho _ b , \\end{align*}"}
{"id": "7061.png", "formula": "\\begin{align*} \\tau _ K = \\tau _ K ( L ) = \\inf \\left \\{ t \\geq 0 : \\exists i \\in \\{ 0 , 1 , \\cdots , { 1 \\over \\delta _ K } - 1 \\} , \\ , | \\beta ^ K _ { i + 1 } ( t ) - \\beta ^ K _ i ( t ) | > L \\delta _ K \\right \\} , \\end{align*}"}
{"id": "8835.png", "formula": "\\begin{align*} | N _ { H _ { 1 } } ( s ) - N _ { H _ { 1 } } ( v _ { t } ) | = | N _ { H _ { 1 } } ( s ) \\cap N _ { F } ( v _ { t } ) | + | N _ { H _ { 1 } } ( s ) \\cap N _ { H _ { 2 } } ( v _ { t } ) | , \\end{align*}"}
{"id": "7860.png", "formula": "\\begin{align*} [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( I ) & = [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G \\cup H \\cup I ) \\\\ & = [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( \\mathbb { N } ) = \\{ \\bar { a } _ n \\} \\end{align*}"}
{"id": "7178.png", "formula": "\\begin{align*} \\alpha = C _ k = c _ k = C _ \\Phi = c _ \\Phi = 1 . \\end{align*}"}
{"id": "4402.png", "formula": "\\begin{align*} \\| f \\| _ { X ^ a _ { \\xi _ 0 } } = \\sup _ { \\xi \\in [ 0 , \\xi _ 0 ] } \\sum _ { i = 0 } ^ 2 \\frac { \\left | ( \\xi \\partial _ \\xi ) ^ i f ( \\xi ) \\right | } { \\langle \\xi \\rangle ^ { a } } , \\end{align*}"}
{"id": "4251.png", "formula": "\\begin{align*} ( T _ 2 ( t ) - T _ 1 ( t ) ) f ) ( x _ n ) = \\langle T _ 2 ( t ) f - T _ 1 ( t ) f , \\delta _ { x _ n } \\rangle \\geq 0 \\end{align*}"}
{"id": "2029.png", "formula": "\\begin{align*} N _ t ^ { h _ n } : = - \\int _ 0 ^ t h _ n ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } ^ n } , M _ t ^ { h _ n } : = h _ n ( X _ t ) - h _ n ( X _ 0 ) - N _ t ^ { h _ n } . \\end{align*}"}
{"id": "6823.png", "formula": "\\begin{align*} - \\triangle _ { g _ { \\textnormal { p o i n } , j } } ( u _ { j } ) = e ^ { 2 u _ { j } } \\ , K _ { \\tilde F _ { t _ { j } } ^ * ( g _ { \\textnormal { e u c } } ) } \\textnormal { o n } \\ , \\ , \\ , \\Sigma . \\end{align*}"}
{"id": "3050.png", "formula": "\\begin{align*} x ^ \\prime = \\alpha _ 2 \\beta _ 1 z \\ , , y ^ \\prime = \\alpha _ 1 \\beta _ 1 y \\ , , z ^ \\prime = \\alpha _ 1 \\beta _ 2 x \\ , , w ^ \\prime = \\alpha _ 2 \\beta _ 2 w \\ , . \\end{align*}"}
{"id": "5348.png", "formula": "\\begin{align*} f ( x ) = c _ 1 x _ 1 + \\ldots + c _ n x _ n . \\end{align*}"}
{"id": "3410.png", "formula": "\\begin{align*} S & \\lesssim \\sum \\limits _ { j = 1 } ^ \\infty \\int _ { t 2 ^ { j - 1 } \\leqslant d ( x , y ) < t 2 ^ j } \\frac 1 { \\omega ( x , t + d ( x , y ) ) } \\bigg ( \\frac { t } { t + d ( x , y ) } \\bigg ) ^ { \\varepsilon } d \\omega ( x ) \\\\ & + \\int _ { d ( x , y ) < t } \\frac 1 { \\omega ( x , t + d ( x , y ) ) } d \\omega ( x ) . \\end{align*}"}
{"id": "6804.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } u \\in C \\left ( \\left [ 0 , T \\right ] , \\mathcal { H } _ { s } \\right ) \\cap C ^ { 1 } \\left ( \\left [ 0 , T \\right ] , \\mathcal { H } _ { s } \\right ) ; \\\\ \\\\ u _ { t } = - \\gamma \\boldsymbol { D } ^ { \\alpha } u - \\beta u , t \\in \\left [ 0 , T \\right ] ; \\\\ \\\\ u ( x , 0 ) = f _ { 0 } ( x ) \\in \\mathcal { H } _ { s } , \\end{array} \\right . \\end{align*}"}
{"id": "2420.png", "formula": "\\begin{align*} \\langle f , S ^ { - 1 } f \\rangle = \\sum _ { \\gamma \\in \\Gamma } a _ \\gamma \\langle e _ \\gamma , S ^ { - 1 } f \\rangle = \\sum _ { \\gamma \\in \\Gamma } | a _ \\gamma | ^ 2 . \\end{align*}"}
{"id": "2603.png", "formula": "\\begin{align*} \\norm { f } _ { M ^ { p , q } } = \\norm { V _ { g _ 0 } f } _ { L ^ { p , q } } \\leq \\norm { ( 1 + | z | ) ^ n \\ , V _ { g _ 0 } f } _ \\infty \\norm { ( 1 + | z | ) ^ { - n } } _ { L ^ { p , q } } . \\end{align*}"}
{"id": "6755.png", "formula": "\\begin{align*} \\begin{aligned} D ^ 2 v _ 0 = D ^ 2 f _ 0 - \\frac { ( h _ 0 - \\delta ) ^ 2 } { h _ 0 ( h _ 0 + h _ 1 ) } ( u '' _ { \\delta } - v '' _ { \\delta } ) & + \\frac { ( h _ 0 - \\delta ) ^ 3 } { 3 h _ 0 ( h _ 0 + h _ 1 ) } ( u ''' _ { \\delta } - v ''' _ { \\delta } ) \\\\ & - \\frac { ( h _ 0 - \\delta ) ^ 4 } { 1 2 h _ 0 ( h _ 0 + h _ 1 ) } ( u '''' _ { \\delta } - v '''' _ { \\delta } ) + \\mathcal O ( h ^ 3 ) , \\\\ \\end{aligned} \\end{align*}"}
{"id": "5307.png", "formula": "\\begin{align*} X ( v \\otimes a b ) = ( X ( v \\otimes a ) ) ( 1 \\otimes b ) , v \\in V , a , b \\in A . \\end{align*}"}
{"id": "9391.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mu ^ { v _ N } _ { B _ N + \\theta A _ N } = \\left ( \\lim _ { N \\to \\infty } \\mu ^ { v _ N } _ { \\theta A _ N } \\right ) \\rhd \\left ( \\lim _ { N \\to \\infty } \\mu ^ { v _ N } _ { B _ N } \\right ) = \\delta _ \\theta \\rhd \\mu _ { s c } . \\end{align*}"}
{"id": "50.png", "formula": "\\begin{align*} \\det M _ { t } = \\frac { \\left [ ( 1 - t ) + ( 2 + 4 t ) \\left \\vert w \\right \\vert ^ { 2 } + ( 1 - t ) \\left \\vert w \\right \\vert ^ { 4 } \\right ] ^ 2 } { \\left ( 1 + \\left \\vert w \\right \\vert ^ { 2 } \\right ) ^ { 8 } } . \\end{align*}"}
{"id": "4775.png", "formula": "\\begin{align*} ( n - 1 ) r ^ 2 & = \\theta r \\\\ \\sum _ k S _ { i , k } & = 0 \\\\ S _ { i , j } ^ 2 + \\sum _ k S _ { i , k } S _ { k , j } & = \\theta S _ { i , j } i < j . \\end{align*}"}
{"id": "7589.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k - 2 s - 2 , l ) } | & \\le \\dfrac { 1 } { 2 i } \\left ( | c _ { i - 1 , j } ^ { ( k - 2 s - 1 , l ) } | + | c _ { i + 1 , j } ^ { ( k - 2 s - 1 , l ) } | \\right ) \\\\ & \\leq \\dfrac { 1 } { 2 i } \\frac { 4 V _ { k , l } } { \\pi ^ 2 j } \\left ( \\Gamma _ { 1 , - 2 } [ s ] ( i ) + \\Gamma _ { 1 , 0 } [ s ] ( i ) \\right ) = \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 j } \\Gamma _ { 0 , - 2 } [ s ] ( i ) . \\end{align*}"}
{"id": "4758.png", "formula": "\\begin{align*} R _ { 2 } + R _ { { \\ell + 1 } } = ( \\underbrace { 1 , 0 , 1 , 0 , \\cdots , 1 , 0 } _ { \\ell + 1 } , \\underbrace { * , \\cdots , * } _ { n - ( \\ell + 1 ) } ) . \\end{align*}"}
{"id": "8631.png", "formula": "\\begin{align*} \\mu _ { R , 2 } ( k , \\ell , m , n ) : = \\int \\overline { \\mathcal { K } _ S ( x , k ) } \\mathcal { K } _ S ( x , \\ell ) \\overline { \\mathcal { K } _ S ( x , m ) } \\mathcal { K } _ S ( x , n ) \\ , d x - \\mu _ { S } ( k , \\ell , m , n ) . \\end{align*}"}
{"id": "415.png", "formula": "\\begin{align*} \\Phi _ { 0 } ^ { 2 } ( t ) : = C _ { 1 } e ^ { C _ { 1 } \\int _ { 0 } ^ { t } ( \\mu _ { 0 } ( \\tau ) + \\mu _ { 1 } ( \\tau ) ) d t } . \\end{align*}"}
{"id": "5650.png", "formula": "\\begin{align*} x \\circ y = \\alpha _ 3 ^ { - 1 } ( \\alpha _ 1 x \\ast \\alpha _ 2 y ) \\end{align*}"}
{"id": "3385.png", "formula": "\\begin{align*} \\theta _ { \\rho } ( x , y ) = \\rho ( y ) \\rho ( x ) . \\end{align*}"}
{"id": "8675.png", "formula": "\\begin{align*} \\max \\{ n _ { \\max } , m _ { \\max } \\} = O \\left ( \\sqrt { R _ n ^ { - 1 } ( \\mathcal { N } _ n ^ 2 + \\mathcal { M } _ n ^ 2 ) } \\right ) . \\end{align*}"}
{"id": "9190.png", "formula": "\\begin{align*} T = ( 1 - \\alpha ) I d + \\alpha N \\end{align*}"}
{"id": "7427.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\int _ { \\Omega _ 2 ^ j ( x ) } \\sum _ { z = x - \\ell } ^ { x - 1 } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta ) - f ( \\eta _ { 1 , x , j , z } ) ] d \\nu _ { b } \\Big | , \\end{align*}"}
{"id": "8139.png", "formula": "\\begin{align*} M ( f , H ) : = { 1 \\over \\# X _ f ^ - ( H ) } \\sum _ { \\chi \\in X _ f ^ - ( H ) } \\vert L ( 1 , \\chi ) \\vert ^ 2 . \\end{align*}"}
{"id": "5244.png", "formula": "\\begin{align*} \\sum _ i c _ i d _ i = a , \\sum _ j e _ j f _ j = b . \\end{align*}"}
{"id": "9186.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u - P \\| ^ * _ { L ^ 2 ( B _ { \\eta } ) } \\leq \\eta ^ 2 , \\\\ & \\Delta P = f _ { B _ { \\eta } } , \\\\ & | P ( 0 ) | + | D P ( 0 ) | + | D ^ 2 P ( 0 ) | \\leq \\bar { C } , \\end{aligned} \\end{align*}"}
{"id": "7965.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\mathrm { A r e a } ( \\Gamma _ i ) = + \\infty \\ , , \\lim _ { i \\rightarrow \\infty } \\mathrm { i n d e x } ( \\Gamma _ i ) = + \\infty \\ , . \\end{align*}"}
{"id": "7693.png", "formula": "\\begin{align*} 1 = \\int _ { \\R ^ N } m ' u ^ 2 = \\int _ { \\R ^ N } m ' _ { \\ast } u _ { \\ast } ^ 2 \\ ; , \\end{align*}"}
{"id": "7475.png", "formula": "\\begin{align*} \\tilde { \\phi } ^ { n + 1 } = \\frac { ( \\tau \\eta ^ n - 2 ) \\phi ^ { n - 1 } + 4 \\phi ^ { n } + 2 \\tau ^ 2 ( - G ( \\phi ^ n ) + \\lambda ^ n \\phi ^ n ) } { 2 + \\tau \\eta ^ n } \\end{align*}"}
{"id": "6183.png", "formula": "\\begin{align*} \\| S \\| _ F ^ 2 = \\sum _ { t = 1 } ^ { p } \\| S _ { t , : } \\| ^ 2 _ { 2 } = \\sum _ { t = 1 } ^ { p } \\left \\| \\frac { C _ { i _ { t } , : } } { \\sqrt { p P _ { i _ { t } } } } \\right \\| ^ 2 _ { 2 } = \\sum _ { t = 1 } ^ { p } \\frac { \\| C _ { i _ { t } , : } \\| ^ 2 _ { 2 } } { p \\frac { \\| C _ { i _ { t } , : } \\| ^ 2 _ { 2 } } { | | C | | _ F ^ 2 } } = \\sum _ { t = 1 } ^ { p } \\frac { \\| C \\| _ F ^ 2 } { p } = \\| C \\| _ F ^ 2 . \\end{align*}"}
{"id": "2654.png", "formula": "\\begin{align*} \\varphi ( k , l ) = \\varphi ( k , 0 ) + \\kappa ( 0 , l ) = \\left ( \\varphi ( 0 , 0 ) + \\kappa ( k , 0 ) \\right ) + \\kappa ( 0 , l ) . \\end{align*}"}
{"id": "412.png", "formula": "\\begin{align*} \\Lambda = \\left ( \\begin{array} { c c c } \\mathbb { I } _ { n \\times n } & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & \\mathbb { I } _ { p \\times p } \\end{array} \\right ) \\end{align*}"}
{"id": "206.png", "formula": "\\begin{align*} D ^ { \\alpha - 1 } ( p _ \\alpha ) ( x ) = - x p _ \\alpha ( x ) , ( D ^ { \\alpha - 1 } ) ^ * ( p _ \\alpha ) ( x ) = x p _ \\alpha ( x ) . \\end{align*}"}
{"id": "7613.png", "formula": "\\begin{align*} \\nabla _ { i } h _ { j l } = \\nabla _ { l } h _ { i j } + \\bar { R } _ { \\nu j l i } , \\end{align*}"}
{"id": "2285.png", "formula": "\\begin{align*} g _ 0 ( t ) = 2 ^ { d / 4 } e ^ { - \\pi t ^ 2 } , \\end{align*}"}
{"id": "4941.png", "formula": "\\begin{align*} h = i _ r ( h ) ^ { - 1 } \\cdot b \\cdot \\phi ( i _ r ( h ) ) . \\end{align*}"}
{"id": "9313.png", "formula": "\\begin{align*} { \\begin{bmatrix} H + D & - A ^ T \\\\ A & 0 \\end{bmatrix} } \\begin{bmatrix} \\Delta \\mathbf { x } \\\\ \\Delta \\mathbf { y } \\end{bmatrix} = \\begin{bmatrix} f _ { \\mathbf { x } } \\\\ f _ { \\mathbf { y } } \\end{bmatrix} , \\end{align*}"}
{"id": "8156.png", "formula": "\\begin{align*} s ( b , c , d ) + s ( d , b , c ) + s ( c , d , b ) = { b ^ 2 + c ^ 2 + d ^ 2 - 3 \\vert b c d \\vert \\over 1 2 b c d } . \\end{align*}"}
{"id": "3877.png", "formula": "\\begin{align*} \\left \\{ \\begin{pmatrix} B & A \\\\ - A ^ * & C \\end{pmatrix} \\mid B ^ * = - B , C ^ * = - C \\right \\} . \\end{align*}"}
{"id": "6645.png", "formula": "\\begin{align*} \\Sigma _ 1 \\ll p ^ { ( \\frac { 3 } { 2 } - \\epsilon ) h _ p } \\sum _ { m = 0 } ^ { k _ p - h _ p - 1 } \\frac { p ^ { \\frac { 1 } { 2 } - \\epsilon } } { p ^ { m ( - \\frac { 1 } { 2 } - \\varepsilon + \\epsilon ) } } \\end{align*}"}
{"id": "5301.png", "formula": "\\begin{align*} \\lambda \\varphi ( b b ^ * ) = \\varphi ( b b ^ * \\delta _ { \\varphi } ) = \\varphi ( \\nu \\delta _ { \\varphi } b b ^ * ) = \\nu _ r \\lambda \\varphi ( b b ^ * ) . \\end{align*}"}
{"id": "3488.png", "formula": "\\begin{align*} \\sum _ { y < m \\leq M } g ( m ) e ^ { 2 \\pi i f ( m ) } = \\int _ y ^ M \\frac { 1 } { u ^ { \\overline { s _ 1 } } ( u + n ) ^ { \\overline { s _ 3 } } } d u + O \\left ( \\frac { 1 } { y ^ { \\sigma _ 1 } ( y + n ) ^ { \\sigma _ 3 } } \\right ) . \\end{align*}"}
{"id": "3341.png", "formula": "\\begin{align*} & \\ ; [ T u + u , T v + v , T w + w ] _ { L \\oplus V } = [ T u , T v , T w ] + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v + D ( T u , T v ) w . \\end{align*}"}
{"id": "4341.png", "formula": "\\begin{align*} \\tilde K _ 0 = - \\frac { \\| \\phi _ { 1 , b , \\beta } \\| ^ { 2 } \\| \\phi _ { 0 , b , \\beta } \\| ^ { - 2 } \\beta ' } { 4 \\beta } \\frac { \\varepsilon _ 1 } { c _ { 1 , 0 } } + O ( L ) , \\end{align*}"}
{"id": "9154.png", "formula": "\\begin{align*} F _ 2 ( x ) : = D ( T ^ \\circ x , S x ) . \\end{align*}"}
{"id": "7926.png", "formula": "\\begin{align*} S _ n = n ^ { 1 / 2 } + O ( n ^ { 1 1 / 4 0 + \\varepsilon } ) . \\end{align*}"}
{"id": "5203.png", "formula": "\\begin{align*} \\| f \\| _ { { F _ { r t } ^ s } ( K ) } & = \\| q ^ { s j } \\Delta _ j f \\| _ { L ^ r ( \\ell ^ t ( K ) ) } \\\\ & = \\Bigg \\| \\Bigg \\{ \\sum _ { j = 0 } ^ { \\infty } | q ^ { s j } \\Delta _ j f | ^ t \\Bigg \\} ^ { \\frac { 1 } { t } } \\Bigg \\| _ { L ^ r ( K ) } \\\\ & = \\Bigg ( \\int _ { K } \\Bigg \\{ \\sum _ { j = 0 } ^ { \\infty } q ^ { s j t } | \\Delta _ j f | ^ t \\Bigg \\} ^ { \\frac { r } { t } } d x \\Bigg ) ^ { \\frac { 1 } { r } } . \\end{align*}"}
{"id": "697.png", "formula": "\\begin{align*} J _ { z _ { \\alpha } ^ { ( \\ell _ 0 ) } } z _ { \\alpha } ^ { ( L + 1 ) } = \\prod _ { \\ell = \\ell _ 0 } ^ { L } J _ { z _ { \\alpha } ^ { ( \\ell ) } } z _ { \\alpha } ^ { ( \\ell + 1 ) } , \\end{align*}"}
{"id": "6499.png", "formula": "\\begin{align*} F ^ { ( 2 m - 1 ) } _ n & \\sim A _ { m - 1 / 2 } n ^ { m - 1 / 2 } \\sum _ { j = 1 } ^ { n - 1 } \\dfrac { ( 2 m - 1 ) ( m - 1 ) C ' _ { m - 1 } j ^ { m - 3 / 2 } ( \\log j ) ^ { m - 2 } } { A _ { m - 1 / 2 } ( j + 1 ) ^ { m - 1 / 2 } } \\\\ & \\sim ( 2 m - 1 ) ( m - 1 ) C ' _ { m - 1 } n ^ { m - 1 / 2 } \\sum _ { j = 1 } ^ { n - 1 } \\dfrac { ( \\log j ) ^ { m - 2 } } { j } \\\\ & \\sim ( 2 m - 1 ) C ' _ { m - 1 } n ^ { m - 1 / 2 } ( \\log n ) ^ { m - 1 } \\\\ & = C ' _ m n ^ { m - 1 / 2 } ( \\log n ) ^ { m - 1 } . \\end{align*}"}
{"id": "1851.png", "formula": "\\begin{align*} L _ { n + 1 } ( x , y ) = \\sum _ { k = 0 } ^ n { n \\choose k } L _ { k } ( x , y ) M _ { n - k } ( x , y ) , \\end{align*}"}
{"id": "198.png", "formula": "\\begin{align*} q _ \\alpha ( t ) = \\left ( e ^ { - \\alpha t } \\dfrac { ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } + e ^ { - \\alpha t } } { ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } } \\right ) ^ { \\frac { 1 } { \\alpha } } & = \\dfrac { e ^ { - t } } { ( 1 - e ^ { - \\alpha t } ) ^ { 1 - \\frac { 1 } { \\alpha } } } \\left ( ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } + e ^ { - \\alpha t } \\right ) ^ { \\frac { 1 } { \\alpha } } . \\end{align*}"}
{"id": "6692.png", "formula": "\\begin{align*} \\Psi ^ { ( - 1 ) } = \\Phi \\Psi \\end{align*}"}
{"id": "1500.png", "formula": "\\begin{align*} L ( s , \\mathbf { f } , \\chi ) = \\Lambda _ { \\mathfrak { n } } ( s , \\chi ) D ( s , \\mathbf { f } , \\chi ) , \\Lambda _ { \\mathfrak { n } } ( s , \\chi ) = \\prod _ { i = 0 } ^ { n - 1 } L _ { \\mathfrak { n } } ( 2 s - 2 i , \\chi ^ 2 ) . \\end{align*}"}
{"id": "1682.png", "formula": "\\begin{align*} \\overline { H } _ n ( s _ 1 , \\ldots , s _ r ) = \\sum \\limits _ { k _ 1 = 0 \\atop p = 2 k _ 1 + 1 } ^ { n - r } \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } + \\sum \\limits _ { k _ 1 = 0 \\atop p \\neq 2 k _ 1 + 1 } ^ { n - r } \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } & = \\frac { c _ { \\frac { p - 1 } { 2 } } } { p ^ { s _ 1 } } + \\sum \\limits _ { k _ 1 = 0 \\atop p \\neq 2 k _ 1 + 1 } ^ { n - r } \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } . \\end{align*}"}
{"id": "5117.png", "formula": "\\begin{align*} A = C _ { b , \\gamma } ^ 2 \\ , \\frac { 1 } { b } \\ , \\frac { 1 - e ^ { - \\frac { 2 \\gamma } { b } } } { 1 - e ^ { - 2 \\gamma a } } \\ , e ^ { - 2 \\gamma a } B = C _ { b \\gamma } ^ 2 \\ , \\frac { 1 } { b } \\ , \\frac { 1 - e ^ { - \\frac { 2 \\gamma } { b } } } { 1 - e ^ { - 2 \\gamma a } } . \\end{align*}"}
{"id": "8854.png", "formula": "\\begin{align*} \\bar \\pi = \\{ \\mathcal F \\in \\hat S : \\{ \\mathcal F \\} \\curlywedge \\pi \\} . \\end{align*}"}
{"id": "9259.png", "formula": "\\begin{align*} & \\forall x ^ X , { x ' } ^ X , \\gamma ^ 1 , { \\gamma ' } ^ 1 \\big ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land x \\in \\mathrm { d o m } ( J ^ A _ \\gamma ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\land \\gamma = _ \\mathbb { R } \\gamma ' \\land x = _ X x ' \\rightarrow J ^ A _ \\gamma x = _ X J ^ A _ { \\gamma ' } x ' \\big ) . \\end{align*}"}
{"id": "6518.png", "formula": "\\begin{align*} f ^ { ( 4 ) } _ n = F ^ { ( 4 ) } _ n = 0 \\mbox { f o r $ n \\geq 2 $ , i f $ \\alpha = 0 $ o r $ - 1 / 2 $ . } \\end{align*}"}
{"id": "5357.png", "formula": "\\begin{align*} H _ c = \\left \\{ x \\in \\mathbb { R } ^ n \\middle | \\left \\langle c , x - z \\right \\rangle = 0 \\right \\} , \\end{align*}"}
{"id": "9255.png", "formula": "\\begin{align*} \\forall x ^ X , p ^ X \\left ( x \\in \\mathrm { d o m } ( J ^ A _ 1 ) \\land p = _ X J ^ A _ { 1 } x \\rightarrow ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "1352.png", "formula": "\\begin{align*} \\int _ { W ' } \\langle a \\nabla y ^ k , \\nabla \\phi \\rangle \\ ; d x = 0 \\end{align*}"}
{"id": "3320.png", "formula": "\\begin{align*} a _ i & = ^ \\# \\{ ( \\alpha , \\beta ) \\in S \\cap { \\Bbb Z } ^ 2 \\mid \\alpha = i \\} \\\\ b _ i & = ^ \\# \\{ ( \\alpha , \\beta ) \\in T \\cap { \\Bbb Z } ^ 2 \\mid \\alpha = i \\} . \\end{align*}"}
{"id": "9035.png", "formula": "\\begin{align*} \\rho ^ 0 = \\rho ^ { i n } , \\rho ^ { n + 1 } = \\arg \\min _ { \\rho \\in [ \\mathcal { P } ( \\Omega ) ] ^ 2 } \\left \\{ \\frac { 1 } { 2 \\tau } d ^ 2 ( \\rho ^ n , \\rho ) + \\mathcal { E } ( \\rho ) \\right \\} . \\end{align*}"}
{"id": "6242.png", "formula": "\\begin{align*} x _ 1 = 0 , \\dotsc , \\ x _ n = 0 , \\ x _ i - x _ j = 0 \\textrm { f o r } 1 \\leq i < j \\leq n . \\end{align*}"}
{"id": "6503.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ n = 0 . \\end{align*}"}
{"id": "9114.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left \\| \\overline { \\mathcal { M } } \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } G _ { i } ^ { \\prime } \\mathcal { X } _ { i } \\right ) \\right \\| _ { } \\right ] \\leq C _ { 9 } K \\left ( \\sqrt { \\frac { p _ { 2 } \\lceil \\sqrt { p _ 3 } \\rceil } { n } } + \\sqrt { \\frac { p _ { 1 } \\lceil \\sqrt { p _ 3 } \\rceil } { n } } + \\sqrt [ 4 ] { \\frac { p _ { 1 } \\lceil \\sqrt { p _ 3 } \\rceil ^ 2 p _ { 3 } } { n ^ 2 } } \\right ) . \\end{align*}"}
{"id": "3891.png", "formula": "\\begin{align*} \\sum _ { z \\leq p } \\left ( \\frac { 1 } { p } \\sum _ { \\gcd ( k , p - 1 ) = 1 } 1 \\right ) \\left ( \\frac { 1 } { p } \\sum _ { \\gcd ( n , p - 1 ) = 1 } 1 \\right ) = \\left ( \\frac { \\varphi ( p - 1 ) } { p - 1 } \\right ) ^ 2 p + O \\left ( 1 \\right ) . \\end{align*}"}
{"id": "7197.png", "formula": "\\begin{align*} | \\nabla ^ j _ \\xi \\frac { d ^ \\ell } { d ^ \\ell p } \\hat { \\psi } _ \\xi ( p ) | \\lesssim _ { j , l , M } \\begin{cases} \\frac { 1 } { 1 + | \\xi | ^ { 2 + j } } \\frac { 1 } { 1 + | p | ^ M } , & , \\\\ \\frac { 1 } { 1 + | p | ^ M } & . \\end{cases} \\end{align*}"}
{"id": "5505.png", "formula": "\\begin{align*} d _ K ( \\eta _ r ( t ; x ) ) = d _ K ( \\xi ( r + t ; r , x ) ) \\leq \\Phi _ { \\beta + L , 0 } ( d _ K ( x ) , \\epsilon , t ) , ( t , x ) \\in [ 0 , u - r ] \\times X . \\end{align*}"}
{"id": "7917.png", "formula": "\\begin{align*} \\partial _ { t _ j } ( \\Phi \\circ \\varphi ) = \\sum _ { l } ( \\partial _ { x _ l } \\Phi ) ( \\partial _ { t _ j } \\varphi ^ { l } ) , \\end{align*}"}
{"id": "4000.png", "formula": "\\begin{align*} \\begin{aligned} \\xi ^ { \\prime } ( x ) + \\eta ^ { \\prime } ( x ) & = \\lambda \\xi ( x ) , x \\in ( 0 , 1 ) , \\\\ \\eta ^ { \\prime \\prime } ( x ) + \\eta ^ { \\prime } ( x ) + \\xi ^ { \\prime } ( x ) & = \\lambda \\eta ( x ) , x \\in ( 0 , 1 ) , \\\\ \\xi ( 0 ) & = \\xi ( 1 ) , \\\\ \\eta ( 0 ) = 0 , \\ \\ \\eta ( 1 ) & = 0 . \\end{aligned} \\end{align*}"}
{"id": "5566.png", "formula": "\\begin{align*} A ^ * = \\frac { \\kappa } { 2 } \\frac { d ^ 2 } { d x ^ 2 } - \\frac { d } { d x } . \\end{align*}"}
{"id": "3097.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + z y f _ 4 ( x , y ) + f _ 6 ( x , y ) = 0 \\ , . \\end{align*}"}
{"id": "8735.png", "formula": "\\begin{align*} \\phi ( s ) = \\prod _ { i \\in 1 } ^ d \\Bigl ( \\sum _ { j \\in 1 } ^ m \\alpha _ { i j } s _ { i j } + \\beta _ i \\Bigr ) , \\end{align*}"}
{"id": "2172.png", "formula": "\\begin{align*} F ( x , t ) & = \\int _ 0 ^ t f ( x , s ) d s = \\int _ 0 ^ t \\frac { f ( x , s ) } { \\vert s \\vert ^ { g ^ + - 1 } } \\vert s \\vert ^ { g ^ + - 1 } d s \\\\ & \\geq \\frac { f ( x , t ) } { \\vert t \\vert ^ { g ^ + - 1 } } \\int _ 0 ^ t \\vert s \\vert ^ { g ^ + - 1 } d s = \\frac { 1 } { g ^ + } f ( x , t ) t \\\\ & \\geq 0 \\ \\ \\ \\ ( \\ t \\leq s < 0 \\ \\ f ( t ) < 0 ) . \\end{align*}"}
{"id": "5054.png", "formula": "\\begin{align*} \\Big \\{ \\alpha = Q + i p \\in \\Sigma \\ , | \\ , \\forall j \\leq \\ell , { \\rm I m } \\sqrt { p ^ 2 - 2 j } \\in ( \\beta / 2 - \\gamma , \\beta / 2 ) \\Big \\} \\end{align*}"}
{"id": "3649.png", "formula": "\\begin{align*} T & = t _ 0 x ^ { \\nu _ 2 ( t _ 0 ) } \\\\ & = \\exp ( \\theta \\sqrt { \\log x } ) \\cdot \\exp \\left ( \\frac { \\log x } { 3 . 3 5 9 \\theta \\sqrt { \\log x } } \\left ( 1 - \\frac { 8 . 0 2 \\log ( \\theta \\sqrt { \\log x } ) } { \\theta \\sqrt { \\log x } } \\right ) \\right ) \\\\ & = \\exp \\left ( \\sqrt { \\log x } \\left ( \\theta + \\frac { 1 } { 3 . 3 5 9 \\theta } \\left ( 1 - \\frac { 8 . 0 2 \\log ( \\theta \\sqrt { \\log x } ) } { \\theta \\sqrt { \\log x } } \\right ) \\right ) \\right ) . \\end{align*}"}
{"id": "5777.png", "formula": "\\begin{align*} \\liminf _ { R \\rightarrow \\infty } M ^ 3 _ { \\ , p , \\ , q } ( R ) \\leq \\delta D ( \\mathbf { u } ) \\end{align*}"}
{"id": "4539.png", "formula": "\\begin{align*} M ( X ; \\chi , u ) = R e s _ { z = 1 } L ( z , \\chi ) L ( 2 z , \\chi ^ { 2 } ) G ( z , \\chi ) F _ { u } ( z , \\chi ) \\frac { X ^ { z } } { z } + \\sum _ { j = 1 } ^ 4 \\int _ j - \\int _ 5 + O \\left ( \\frac { X ^ { 1 + \\varepsilon } \\log X } { T } \\right ) , \\end{align*}"}
{"id": "2207.png", "formula": "\\begin{align*} \\| X ( t _ m ) - X _ m ^ { M , N } \\| _ { L ^ p ( \\Omega ; H ) } = O ( \\lambda _ N ^ { - \\frac \\gamma 2 } + k ^ { \\frac \\gamma 4 } ) , \\ ; \\gamma \\in \\big ( \\tfrac d 2 , 4 \\big ] , \\end{align*}"}
{"id": "5097.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } \\frac { \\sinh ( x ) } { x ^ 2 } = \\frac { \\cosh ( x ) x ^ 2 - 2 x \\sinh ( x ) } { x ^ 4 } = \\frac { \\sinh ( x ) } { x ^ 3 } ( x \\coth ( x ) - 2 ) . \\end{align*}"}
{"id": "3670.png", "formula": "\\begin{align*} E _ n = \\{ \\{ u , v \\} : u , v \\in [ n ] , u \\ne v \\} , \\end{align*}"}
{"id": "6024.png", "formula": "\\begin{align*} \\widetilde { H } _ n ' ( x ) = - 2 n \\widetilde { H } _ { n - 1 } ( x ) \\end{align*}"}
{"id": "4178.png", "formula": "\\begin{align*} s ( A , B ) : = \\sup _ { x \\in A } d _ M ( x , B ) : = \\sup _ { x \\in A } \\inf _ { y \\in B } d _ M ( x , y ) , \\end{align*}"}
{"id": "1444.png", "formula": "\\begin{align*} \\mathbb { H } = \\R \\oplus \\R \\mathbf { i } \\oplus \\R \\mathbf { j } \\oplus \\R \\mathbf { i j } , \\ , \\ , \\ , \\ , \\ , \\mathbf { i } ^ 2 = \\mathbf { j } ^ 2 = - 1 , \\mathbf { i j } = - \\mathbf { j i } , \\end{align*}"}
{"id": "8149.png", "formula": "\\begin{align*} M _ { d _ 0 } ( f , H ) = M ( d _ 0 f , H _ { d _ 0 } ) . \\end{align*}"}
{"id": "9488.png", "formula": "\\begin{align*} | \\mathcal { C S } _ { ( s , t ) } | = \\binom { ( s - 2 ) / 2 + ( t - 1 ) / 2 } { ( s - 2 ) / 2 } + \\binom { ( s - 2 ) / 2 + ( t - 3 ) / 2 } { ( s - 2 ) / 2 } . \\end{align*}"}
{"id": "9501.png", "formula": "\\begin{align*} f ( x ) = \\tau _ x ( f ) = \\int f d m = \\int f d \\lambda D _ y = \\lambda f ( y ) = \\lambda \\tau _ y ( f ) \\end{align*}"}
{"id": "3633.png", "formula": "\\begin{align*} \\frac { 2 N ( \\sigma , T ) } { T } & \\le 2 \\left [ C _ 1 ( \\sigma ) T ^ { ( 5 - 8 \\sigma ) / 3 } \\log ^ { 5 - 2 \\sigma } T + \\frac { C _ 2 ( \\sigma ) \\log ^ 2 T } { T } \\right ] , \\\\ & \\le 2 \\bigg [ C _ 1 ( \\sigma ) \\exp \\left ( \\frac { B _ 2 ( 5 - 8 \\sigma ) } { 3 } \\sqrt { \\log x } \\right ) \\left ( B _ 2 \\sqrt { \\log x } \\right ) ^ { 5 - 2 \\sigma } \\\\ & \\qquad \\qquad \\qquad + C _ 2 ( \\sigma ) \\exp \\left ( - B _ 2 \\sqrt { \\log x } \\right ) ( B _ 2 ) ^ 2 \\log x \\bigg ] \\\\ & = s _ 2 ' ( x , \\sigma ) , \\ . \\end{align*}"}
{"id": "1179.png", "formula": "\\begin{align*} \\Psi ^ { A i } ( \\zeta ) \\sim \\frac { e ^ { \\frac { i \\pi } { 1 2 } } } { 2 \\sqrt { \\pi } } \\zeta ^ { - \\frac { \\sigma _ 3 } { 4 } } \\sum _ { j = 0 } ^ { \\infty } \\begin{pmatrix} ( - 1 ) ^ { j } s _ { j } & s _ { j } \\\\ - ( - 1 ) ^ { j } \\nu _ j & \\nu _ j \\end{pmatrix} e ^ { - \\frac { i \\pi } { 4 } \\sigma _ 3 } \\left ( \\frac { 2 } { 3 } \\zeta ^ { \\frac { 3 } { 2 } } \\right ) ^ { - j } , \\zeta \\rightarrow \\infty . \\end{align*}"}
{"id": "6751.png", "formula": "\\begin{align*} D ^ 2 f _ 0 = \\frac { 1 } { \\bar { h } _ { 0 } } \\left [ \\frac { 1 } { h _ { 0 } } u _ { - 1 } - \\left ( \\frac { 1 } { h _ { 0 } } + \\frac { 1 } { h _ { 1 } } \\right ) v _ 0 + \\frac { 1 } { h _ { 1 } } v _ 1 \\right ] , \\end{align*}"}
{"id": "5794.png", "formula": "\\begin{align*} \\operatorname { T o r } ^ S _ \\ast ( k [ K ] , k ) _ { \\subseteq J } = \\bigoplus _ { I \\subseteq J } \\operatorname { T o r } ^ S _ \\ast ( k [ K ] , k ) _ I \\end{align*}"}
{"id": "8901.png", "formula": "\\begin{align*} \\check H ^ q _ { c t } ( X ; A _ X ) = \\check H ( \\nu ( X ) ; A _ { \\nu ( X ) } ) \\end{align*}"}
{"id": "3428.png", "formula": "\\begin{align*} | G _ k ( f ) ( x ) | & = | \\int _ { \\R ^ N } D _ k ( x , y ) D ^ M _ k ( f ) ( y ) d \\omega ( y ) | \\leqslant C \\frac { 1 } { V _ k ( x ) } \\| f \\| _ \\infty \\\\ & \\lesssim r ^ k \\frac { 1 } { V _ 0 ( x ) } \\| f \\| _ \\infty \\lesssim r ^ k \\| f \\| _ \\infty \\end{align*}"}
{"id": "2138.png", "formula": "\\begin{align*} \\begin{aligned} & k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\sum _ { s = 1 } ^ \\infty ( 1 - c ) ^ { k ( s + 1 ) } \\frac 1 { ( s + 1 ) k - l } = \\\\ & - k ( 1 - c ) ^ k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l ( \\frac c { 1 - c } ) ^ l \\frac 1 { k - l } + k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l c ^ l \\int _ 0 ^ { 1 - c } \\frac { y ^ { k - l - 1 } } { 1 - y ^ k } d y - \\\\ & c ^ k \\log ( 1 - ( 1 - c ) ^ k ) . \\end{aligned} \\end{align*}"}
{"id": "6606.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq e < \\infty \\\\ ( e , g ) = 1 } } \\frac { \\mu ( e ) } { e } \\sum _ { \\substack { a | g \\\\ ( e a , \\frac { m h } { g } \\cdot \\frac { n k } { g } ) = 1 } } \\frac { \\mu ( a ) } { \\phi ( e a ) } R ( 0 ; a e , m n h k / g ^ 2 ) = \\frac { \\phi ( m n h k ) } { m n h k } . \\end{align*}"}
{"id": "2680.png", "formula": "\\begin{align*} \\widehat { g } ( \\omega ) = C \\ , e ^ { - \\gamma \\omega ^ 2 } \\prod _ { k = 1 } ^ \\infty \\frac { e ^ { 2 \\pi i \\delta _ k \\omega } } { 1 + 2 \\pi i \\delta _ k \\omega } . \\end{align*}"}
{"id": "3272.png", "formula": "\\begin{align*} S _ { \\mu _ { T ^ * T } } ^ { \\langle - 1 \\rangle } \\left ( \\frac { 1 } { | \\lambda | ^ { 2 } } \\right ) = \\psi _ { \\mu _ { T ^ * T } } \\left ( - \\frac { 1 } { s ( | \\lambda | , 0 ) ^ 2 } \\right ) . \\end{align*}"}
{"id": "7425.png", "formula": "\\begin{align*} + \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\sum _ { z = x - \\ell } ^ { x - 1 } \\int _ { \\Omega _ 2 ^ j ( x ) } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta ) - f ( \\eta ^ { z , z - j \\ell } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "4766.png", "formula": "\\begin{align*} \\langle R ( x \\wedge y ) , z \\wedge w \\rangle + \\langle R ( z \\wedge x ) , y \\wedge w \\rangle + \\langle R ( y \\wedge z ) , x \\wedge w \\rangle = 0 . \\end{align*}"}
{"id": "9251.png", "formula": "\\begin{align*} \\forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \\left ( x = _ X x ' \\land y = _ X y ' \\rightarrow \\chi _ A x y = _ 0 \\chi _ A x ' y ' \\right ) \\end{align*}"}
{"id": "1997.png", "formula": "\\begin{align*} \\rho ( g _ n ^ 2 ) & = \\left ( \\begin{array} { c c } \\cos ( \\frac { 2 \\pi } { n } ) & - \\sin ( \\frac { 2 \\pi } { n } ) \\\\ \\sin ( \\frac { 2 \\pi } { n } ) & \\cos ( \\frac { 2 \\pi } { n } ) \\end{array} \\right ) , \\\\ \\sigma ( g _ n ^ 2 ) & = ( 1 , 3 , \\dots , 2 n - 1 ) ( 2 , 4 , \\dots , 2 n ) \\\\ \\tau ( g _ n ^ 2 ) & = 1 , \\end{align*}"}
{"id": "6213.png", "formula": "\\begin{align*} 1 - \\left ( \\frac { 1 } { \\eta _ j } + \\frac { 1 } { \\eta _ { j - 1 } } \\right ) 1 0 \\epsilon \\| C \\| _ F ^ 2 \\leq \\sum _ { a , b = 1 } ^ { j - 1 } t _ { a , b } + \\sum _ { a , b = 1 } ^ { j } t _ { a , b } - \\sum _ { b = 1 } ^ { j - 1 } \\sum _ { a = 1 } ^ { j } t _ { a , b } - \\sum _ { a = 1 } ^ { j - 1 } \\sum _ { b = 1 } ^ { j } t _ { a , b } = t _ { j , j } , \\end{align*}"}
{"id": "3155.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal e _ { i _ k } \\frac { e _ { i _ k } ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal e _ { i _ k } } _ 2 ^ 2 } . \\end{align*}"}
{"id": "313.png", "formula": "\\begin{align*} \\mathcal I ( \\rho ^ 0 , \\rho ^ T ) = \\inf \\{ \\mathcal S ( \\rho _ t , \\Phi _ t ) | ( - \\Delta _ { \\rho _ t } ) ^ { \\dagger } \\Phi _ t \\in \\mathcal T _ { \\rho _ t } \\mathcal P _ { o } ( G ) , \\rho ( 0 ) = \\rho ^ 0 , \\rho ( T ) = \\rho ^ T \\} \\end{align*}"}
{"id": "1562.png", "formula": "\\begin{align*} \\mathfrak { A } _ { \\omega } ( \\Gamma ) = \\{ h \\in \\mathfrak { A } _ { \\omega } : h | _ { \\omega } \\gamma = h \\gamma \\in \\Gamma \\} , \\end{align*}"}
{"id": "7284.png", "formula": "\\begin{align*} \\boxed { \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ k ) ^ 3 = \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k ( 2 k + 1 ) X ^ { \\frac { k ^ 2 + k } { 2 } } . } \\end{align*}"}
{"id": "9484.png", "formula": "\\begin{align*} b c _ s ( n ) = d d _ s ( 2 n ) + d d _ s ( 2 n - s ) + d d _ s ( 2 n - 4 s ) + \\cdots = \\sum _ { i \\geq 0 } d d _ s ( 2 n - i ^ 2 s ) , \\end{align*}"}
{"id": "7848.png", "formula": "\\begin{align*} \\delta = \\underset { m \\geq 2 , n } { \\sup } ~ \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\frac { w _ { k + m } \\cdots w _ { k + n } } { w _ { m } \\cdots w _ { n } } \\bigg ) ^ { 2 } < \\infty . \\end{align*}"}
{"id": "387.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; f ( x ) : = \\varphi ( x ) + g ( x ) \\mbox { s u b j e c t t o } \\ ; \\ ; x \\in \\R ^ n . \\end{align*}"}
{"id": "8894.png", "formula": "\\begin{align*} \\psi _ 2 ( \\bar z ) + h _ 2 ^ * \\psi _ 1 ( \\bar z ) & = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 2 ( z _ i ) , \\ldots , h _ 2 ( z _ { q - 1 } ) ) \\\\ & + \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 2 ( z _ 0 ) , \\ldots , h _ 2 ( z _ i ) , h _ 3 ( z _ i ) , \\ldots , h _ 3 ( z _ { q - 1 } ) ) \\\\ & = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 3 ( z _ i ) , \\ldots , h _ 3 ( z _ { q - 1 } ) ) + d _ { q - 2 } \\chi _ 2 ( \\bar z ) \\\\ & = \\psi _ 3 ( \\bar z ) + d _ { q - 2 } \\chi _ 2 ( \\bar z ) . \\end{align*}"}
{"id": "997.png", "formula": "\\begin{align*} \\int _ { B _ \\varepsilon ( x ) } y _ 1 \\eta _ \\varepsilon ( x - y ) \\dd y & = \\int _ { B _ \\varepsilon ( x ) } ( y _ 1 - x _ 1 ) \\eta _ \\varepsilon ( x - y ) \\dd y + x _ 1 \\int _ { B _ \\varepsilon ( x ) } \\eta _ \\varepsilon ( x - y ) \\dd y = x _ 1 \\end{align*}"}
{"id": "1225.png", "formula": "\\begin{align*} d X _ t = - \\beta _ t \\nabla U ( X _ t ) \\ , d t + \\sqrt { 2 } d W _ t , \\end{align*}"}
{"id": "2941.png", "formula": "\\begin{align*} X _ { n , r } : = X _ { n , r } ( k ) : = \\begin{cases} \\frac 1 { \\delta _ n ( k ) } \\sum \\limits _ { \\substack { \\mathbf p _ k \\in \\mathcal P ( d , k ) \\\\ p _ k = r } } \\tilde M _ { n , \\mathbf { p } _ k } & , r \\ge k , \\\\ 0 & , r < k . \\end{cases} \\end{align*}"}
{"id": "8052.png", "formula": "\\begin{align*} \\Psi ^ n _ { ( \\Sigma , U ) } ( f ) [ \\rho ^ { * } _ { ( 1 ) } \\psi ] = \\Psi ^ n _ { ( \\widetilde { \\Sigma } , \\widetilde { U } ) } ( \\rho _ { * } ( \\omega _ { \\ell } | _ { \\Sigma } ^ { \\mu - 1 } f ) ) [ \\psi ] . \\end{align*}"}
{"id": "8563.png", "formula": "\\begin{align*} T ( 0 ) = \\frac { 2 a } { 1 + a ^ 2 } , R _ + ( 0 ) = \\frac { 1 - a ^ 2 } { 1 + a ^ 2 } , \\mbox { a n d } R _ - ( 0 ) = \\frac { a ^ 2 - 1 } { 1 + a ^ 2 } . \\end{align*}"}
{"id": "8319.png", "formula": "\\begin{align*} B _ + ( y , u ) : = \\{ t \\in [ 0 , T ] \\colon y ( t ) = r \\exists \\varepsilon > 0 y - u = \\mathrm { c o n s t } [ t , t + \\varepsilon ) \\} , \\end{align*}"}
{"id": "4559.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { 1 } { 2 } \\delta ^ 2 \\right ) d ' \\binom { | G - B | } { 2 } & > \\left ( \\max \\left \\{ \\frac { r - 2 } { r - 1 } k , ( h - 1 ) \\right \\} - 2 \\eta + \\frac { 1 } { 2 } \\delta ^ 2 \\right ) \\binom { | G - B | } { 2 } \\\\ & > \\max \\left \\{ k , \\frac { r - 1 } { r - 2 } ( h - 1 ) \\right \\} t _ { r - 1 } ( | G - B | ) . \\end{align*}"}
{"id": "5633.png", "formula": "\\begin{align*} H \\rhd u ^ i = \\lambda u ^ i , ~ ~ ~ ~ ~ ~ ~ ~ ~ E ^ \\pm \\rhd u ^ i = \\delta _ 0 ^ i \\frac { 1 } { \\sqrt { a } } u ^ \\pm - \\delta ^ i _ \\mp \\sqrt { a } u ^ 0 . \\end{align*}"}
{"id": "4891.png", "formula": "\\begin{align*} \\frac { f '' ( z ) } { f ( z ) } = L ' ( z ) + L ( z ) ^ 2 \\sim c s ( s + 1 ) z ^ { - 2 - s } . \\end{align*}"}
{"id": "4865.png", "formula": "\\begin{align*} \\alpha = P ^ { - 1 } \\Big ( \\frac n \\beta \\Big ) \\in ( c , n ] , c : = \\max \\{ q _ 2 , P ^ { - 1 } ( 1 / \\Gamma ) \\} . \\end{align*}"}
{"id": "1118.png", "formula": "\\begin{align*} D ( k , \\xi ) : = \\exp \\left \\{ \\frac { X _ { \\eta } ( k ) } { 2 \\pi i } \\left ( \\int _ { - \\eta } ^ { - C _ R } + \\int _ { C _ R } ^ { \\eta } \\right ) \\frac { \\log r _ { + } ( s ) } { X _ { \\eta + } ( s ) ( s - k ) } d s \\right \\} , \\end{align*}"}
{"id": "532.png", "formula": "\\begin{align*} \\Xi _ k = \\begin{bmatrix} x _ { 1 , k } & x _ { 1 , k } + T _ s x _ { 2 , k } & x _ { 3 , k } & x _ { 3 , k } + T _ s x _ { 4 , k } \\end{bmatrix} ^ \\top . \\end{align*}"}
{"id": "320.png", "formula": "\\begin{align*} \\tau _ n : = \\inf \\{ t \\in [ 0 , T ] : \\| S ^ n \\| _ { \\mathcal C ( [ 0 , t ] ; \\mathbb R ^ N \\times \\mathbb R ^ N ) } \\ge n \\} \\wedge \\inf \\{ t \\in [ 0 , T ] : \\min _ { i = 1 } ^ N \\min _ { s \\in [ 0 , t ] } \\rho _ i ^ n ( s ) \\le \\frac 1 { n } \\} , \\end{align*}"}
{"id": "7941.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } L _ n ^ { 1 / 2 } n ^ { 5 / 4 } & \\leqslant \\sum _ { m = 0 } ^ { \\ell - 1 } \\sum _ { \\substack { p _ { m } ^ 2 - 1 < n \\leqslant p _ { m + 1 } ^ 2 - 1 } } L _ n ^ { 1 / 2 } n ^ { 5 / 4 } . \\end{align*}"}
{"id": "6745.png", "formula": "\\begin{align*} \\partial _ t V = \\mathcal L V + g + \\rho \\max \\{ V ^ * - V , 0 \\} , \\end{align*}"}
{"id": "3118.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 = f _ 6 ( x , y ) \\ , . \\end{align*}"}
{"id": "2822.png", "formula": "\\begin{align*} \\begin{cases} u _ 0 | x | u _ 0 \\in L ^ 2 , \\\\ u _ 0 \\in L _ { r a d } ^ 2 ( \\mathbb { R } ^ N ) ( N , p , \\gamma ) ( \\ref { p a r a m e t e r : r a n g e f o r r a d i a l c a s e } ) , \\end{cases} \\end{align*}"}
{"id": "6858.png", "formula": "\\begin{align*} \\delta _ i = \\frac { ( q ^ i - 1 ) ( q ^ { n - i + 1 } - 1 ) } { ( q ^ { i + 1 } - 1 ) ( q ^ { n - i } - 1 ) } , \\end{align*}"}
{"id": "8173.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) = \\frac { 2 \\pi ^ 2 \\mu ( d _ 0 ) \\phi ( d _ 0 ) } { d _ 0 ^ 2 p } \\sum _ { \\delta \\mid d _ 0 } \\frac { \\delta \\mu ( \\delta ) } { \\phi ( \\delta ) } S ( H _ \\delta , \\delta p ) , \\hbox { w h e r e } S ( H _ \\delta , \\delta f ) = \\sum _ { h \\in H _ \\delta } s ( h , \\delta f ) , \\end{align*}"}
{"id": "5654.png", "formula": "\\begin{align*} \\left ( \\prescript { ( 1 \\ , 2 ) } { } N _ r ^ A \\right ) ^ { - 1 } = \\prescript { ( 1 \\ , 2 ) } { } N _ l ^ A . \\end{align*}"}
{"id": "5176.png", "formula": "\\begin{align*} t = \\sum _ { i = 1 } ^ { R _ \\ell ^ { \\pi } ( t ) } T _ { \\ell i } ^ \\pi + \\eta _ \\ell ^ \\pi ( t ) , \\end{align*}"}
{"id": "7408.png", "formula": "\\begin{align*} & \\Omega _ { j , x } ^ 2 : = \\{ \\eta \\in \\Omega _ x : \\ ; \\eta ( x ) = 0 , \\eta ( z _ { 1 j } ) = 1 , \\eta ( x + r ) = 1 \\} ; \\\\ & \\Omega _ { j , x } ^ 3 : = \\{ \\eta \\in \\Omega _ x : \\ ; \\eta ( x ) = 0 , \\eta ( z _ { 1 j } ) = 0 , \\eta ( x + r ) = 1 \\} ; \\\\ & \\Omega _ { j , x } ^ 4 : = \\{ \\eta \\in \\Omega _ x : \\ ; \\eta ( x ) = 1 , \\eta ( z _ { 1 j } ) = 1 , \\eta ( x + r ) = 0 \\} . \\end{align*}"}
{"id": "6551.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } W + \\mathcal { L } W = f , \\\\ & W ( x , 0 ) = W _ { 0 } ( x ) . \\endaligned \\right . \\end{align*}"}
{"id": "6184.png", "formula": "\\begin{align*} P ' _ { j } = \\sum \\limits _ { t = 1 } ^ { p } \\frac { \\mathcal { D } _ { C _ { i _ { t } , : } } ( j ) } { p } = \\frac { 1 } { \\| C \\| _ F ^ 2 } \\sum \\limits _ { t = 1 } ^ { p } \\frac { | C _ { i _ t , j } | ^ 2 } { p \\frac { \\| C _ { i _ t , : } \\| ^ 2 _ { 2 } } { \\| C \\| _ F ^ 2 } } = \\frac { 1 } { \\| C \\| _ F ^ 2 } \\sum \\limits _ { t = 1 } ^ { p } \\frac { | C _ { i _ t , j } | ^ 2 } { p P _ { i _ t } } = \\frac { \\| S _ { : , j } \\| ^ 2 _ { 2 } } { \\| S \\| _ F ^ 2 } . \\end{align*}"}
{"id": "3725.png", "formula": "\\begin{align*} \\frac { d } { d t } X ( x _ 0 , t ) = J ( X ( x _ 0 , t ) , t ) = - H B _ x ( X ( x _ 0 , t ) , t ) = - \\Lambda B ( X ( x _ 0 , t ) , t ) . \\end{align*}"}
{"id": "1261.png", "formula": "\\begin{align*} \\mathcal { M } _ { k } ^ { J } ( \\alpha , \\beta ) : = \\{ \\ , u \\in \\mathcal { M } ^ { J } ( \\alpha , \\beta ) \\ , | \\ , I ( u ) = k \\ , \\ , \\} \\end{align*}"}
{"id": "8596.png", "formula": "\\begin{align*} A : = \\big \\{ [ T ( \\cdot ) - T ( 0 ) ] \\mathbf { 1 } _ + ( \\cdot ) , \\ , \\ , - R _ + ( - \\cdot ) \\mathbf { 1 } _ - ( \\cdot ) , \\ , \\ , R _ - ( \\cdot ) \\mathbf { 1 } _ + ( \\cdot ) , - [ T ( - \\cdot ) - T ( 0 ) ] \\mathbf { 1 } _ - ( \\cdot ) , \\ , \\ , 1 , \\ , \\ , - 1 \\big \\} . \\end{align*}"}
{"id": "4257.png", "formula": "\\begin{align*} T _ n ( t ) f ( x ) = \\begin{cases} \\Big ( \\frac { x - t } { x } \\Big ) ^ { \\frac { 1 } { n } } f ( x - t ) & \\mbox { f o r } x > t \\\\ 0 & \\mbox { f o r } x \\leq t . \\end{cases} \\end{align*}"}
{"id": "7979.png", "formula": "\\begin{align*} [ f ] _ { x , \\lambda ^ 2 g , C ^ k _ * } = [ f ] _ { x , g , C ^ k _ * } ; & \\ & [ \\lambda ^ 2 \\beta ] _ { x , \\lambda ^ 2 g , C ^ k _ * } = [ \\beta ] _ { x , g , C ^ k _ * } \\ , . \\end{align*}"}
{"id": "6128.png", "formula": "\\begin{align*} h ^ { 1 } ( s , t ) = \\sqrt { j ( 2 j - 1 ) } | s | ^ { j - 1 } \\psi ^ { 1 } ( s , t ) - i \\sqrt { k ( 2 k - 1 ) } | t | ^ { k - 1 } \\varphi ^ { 1 } ( s , t ) , \\end{align*}"}
{"id": "5649.png", "formula": "\\begin{align*} \\Delta v - \\varepsilon ^ { - 2 } \\left ( e ^ { v } - \\left ( ( { x ^ { \\lambda } } ) ^ { 2 } + y ^ { 2 } \\right ) e ^ { - v } \\right ) = \\Delta v - \\varepsilon ^ { - 2 } \\left ( e ^ { v } - \\left ( { x } ^ { 2 } + y ^ { 2 } \\right ) e ^ { - v } \\right ) = 0 . \\end{align*}"}
{"id": "4381.png", "formula": "\\begin{align*} \\langle \\phi _ 0 , \\partial _ \\tau \\phi _ 0 \\rangle _ { L ^ 2 _ \\rho } = - \\frac { \\gamma b _ \\tau } { b } \\| \\phi _ 0 \\| ^ 2 _ { L ^ 2 _ \\rho } ( 1 + O ( b ^ { 1 - \\frac { \\epsilon } { 2 } } ) ) . \\end{align*}"}
{"id": "9407.png", "formula": "\\begin{align*} \\varphi ( a _ 1 \\cdots a _ n ) & = \\frac { 1 } { \\tau ' ( p ) } \\tau ' ( p a _ 1 \\cdots a _ n ) = \\frac { 1 } { \\tau ' ( p ) } \\tau ' ( p ) \\tau ( a _ 1 \\cdots a _ n ) = 0 . \\end{align*}"}
{"id": "4485.png", "formula": "\\begin{align*} \\theta ( x ) = \\sum _ { p \\le x } \\log p , \\ E _ { \\theta } ( x ) = \\Big | \\frac { \\theta ( x ) - x } { x } \\Big | , \\ \\pi ( x ) = \\sum _ { p \\le x } 1 , \\ \\ \\ E _ { \\pi } ( x ) = \\Big | \\frac { \\pi ( x ) - \\textrm { L i } ( x ) } { \\frac { x } { \\log x } } \\Big | , \\end{align*}"}
{"id": "1584.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } b ( \\cos \\theta ) | v - v _ * | ^ \\gamma f ( v ' ) d \\sigma d v = \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } b ( \\cos \\theta ) \\frac 1 { \\cos ^ { 3 + \\gamma } ( \\theta / 2 ) } | v - v _ * | ^ \\gamma f ( v ) d \\sigma d v . \\end{align*}"}
{"id": "4670.png", "formula": "\\begin{align*} H ( Q ( \\cdot - x _ 1 ) + \\sigma Q ( \\cdot - x _ 2 ) ) = 2 H ( Q ) - \\frac { \\sigma \\kappa _ 0 } { ( x _ 1 - x _ 2 ) ^ 2 } + O \\bigg ( \\frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \\bigg ) , \\end{align*}"}
{"id": "4147.png", "formula": "\\begin{align*} \\nu : = 2 + \\theta < 5 / 2 + a . \\end{align*}"}
{"id": "8164.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) & = \\frac { 1 } { \\# X _ p ^ - ( H ) } \\sum _ { \\chi \\in X _ p ^ - ( H ) } | L ( 1 , \\chi ' ) | ^ 2 = ( 1 + o ( 1 ) ) \\frac { \\pi ^ 2 } { 6 } \\prod _ { q \\mid d _ 0 } \\left ( 1 - \\frac { 1 } { q ^ 2 } \\right ) . \\end{align*}"}
{"id": "5835.png", "formula": "\\begin{align*} X ^ { y _ { i } } _ H - \\pi _ 1 ( y _ { i } ) = X ^ { y _ { i } } _ { r \\frac { H } { r } } - \\pi _ 1 ( y _ { i } ) \\leq \\Big ( v _ + - \\frac { \\theta } { 2 r } \\Big ) r \\frac { H } { r } = \\Big ( v _ + - \\frac { \\theta } { 2 r } \\Big ) H . \\end{align*}"}
{"id": "5056.png", "formula": "\\begin{align*} \\Psi _ { \\alpha , \\nu , \\tilde { \\nu } } = \\lim _ { t \\to + \\infty } e ^ { t ( \\Delta _ { \\alpha } + | \\nu | + | \\tilde { \\nu } | ) } e ^ { - t { \\bf H } } \\Psi _ { \\alpha , \\nu , \\tilde { \\nu } } ^ 0 \\end{align*}"}
{"id": "1096.png", "formula": "\\begin{align*} m ^ { ( 3 ) } _ { + } ( x , t , k ) = m ^ { ( 3 ) } _ { - } ( x , t , k ) J ^ { ( 3 ) } ( x , t , k ) , k \\in \\Gamma ^ { ( 3 ) } , \\end{align*}"}
{"id": "7328.png", "formula": "\\begin{align*} & . \\ \\liminf _ { \\alpha \\to 0 } | \\tilde { x } - \\tilde { y } | > 0 , \\\\ & . \\ \\liminf _ { \\alpha \\to 0 } | \\tilde { x } - \\tilde { y } | = 0 , \\ \\ \\exists \\alpha _ i \\to 0 \\ \\ \\lim _ { \\alpha _ i \\to 0 } | \\tilde { x } - \\tilde { y } | = 0 . \\end{align*}"}
{"id": "9084.png", "formula": "\\begin{align*} & p _ { | h _ { 1 } | , \\cdots , | h _ { N } | } ( x _ { 1 } , \\dots , x _ { N } ) \\\\ & = \\prod _ { \\substack { k = 1 \\\\ ( \\mu _ { 1 } \\triangleq 0 ) } } ^ { N } \\ ! \\ ! \\frac { 2 x _ { k } } { \\sigma ^ { 2 } ( 1 - \\mu _ { k } ^ { 2 } ) } e ^ { - \\frac { x _ { k } ^ { 2 } + \\mu _ { k } ^ { 2 } x _ { 1 } ^ { 2 } } { \\sigma ^ { 2 } ( 1 - \\mu _ { k } ^ { 2 } ) } } I _ { 0 } \\left ( { \\frac { 2 ~ \\mu _ { k } x _ { 1 } x _ { k } } { \\sigma ^ { 2 } ( 1 - \\mu _ { k } ^ { 2 } ) } } \\right ) \\ ! , \\end{align*}"}
{"id": "7716.png", "formula": "\\begin{align*} A ( z ) = \\abs { \\{ n \\in [ 1 , N ] : a _ { n } = z \\} } \\end{align*}"}
{"id": "6105.png", "formula": "\\begin{align*} \\left ( b _ 2 \\mathbb { N } + b _ 1 \\mathbb { Z } \\right ) \\ , \\cap \\ , \\left ( m b _ 1 \\mathbb { N } + \\pi \\mathbb { Z } \\right ) \\ , = \\ , \\emptyset . \\end{align*}"}
{"id": "9271.png", "formula": "\\begin{align*} A : x \\mapsto \\begin{cases} \\partial \\varphi ( x ) & x \\in ( 0 , \\pi / 2 ) , \\\\ \\emptyset & , \\end{cases} \\end{align*}"}
{"id": "7229.png", "formula": "\\begin{align*} w = \\delta ^ { \\beta } \\langle \\check \\tau _ { t , x } \\rangle ( v ^ \\perp - v ^ \\perp _ \\ast ( t , x , v _ 1 ) ) , \\end{align*}"}
{"id": "6148.png", "formula": "\\begin{align*} ( m + 1 ) P ( x ) + x P ' ( x ) = \\mu _ { n , m } ( 1 - x ) ^ n . \\end{align*}"}
{"id": "4968.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } d x \\ , \\mathbb { E } \\bigg [ \\Big | \\mathcal { F } ^ { - 1 } \\Big ( \\big ( 1 + | . | ^ 2 \\big ) ^ { - s + \\frac { \\kappa } { 2 } } \\mathcal { F } \\Big ( \\rho \\Big [ \\ < I P s i 2 > _ { n } ( t , . ) - \\ < I P s i 2 > _ { n } ( s , . ) \\Big ] \\Big ) \\Big ) ( x ) \\Big | ^ { 2 p } \\bigg ] \\lesssim \\Bigg ( \\sum _ { k = 1 } ^ { 8 } | \\tilde { I _ k } | \\Bigg ) ^ p , \\end{align*}"}
{"id": "5153.png", "formula": "\\begin{align*} - \\frac { \\partial ^ 2 } { \\partial t ^ 2 } \\left [ \\log ( \\cosh ( t ) - 1 ) \\right ] & = - \\frac { \\partial } { \\partial t } \\frac { \\sinh ( t ) } { \\cosh ( t ) - 1 } \\\\ & = - \\frac { \\cosh ( t ) \\ , ( \\cosh ( t ) - 1 ) - \\sinh ( t ) \\sinh ( t ) } { ( \\cosh ( t ) - 1 ) ^ 2 } \\\\ & = \\frac { \\cosh ( t ) } { ( \\cosh ( t ) + 1 ) ^ 2 } > 0 . \\end{align*}"}
{"id": "517.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { R } _ { 0 } ( \\tau ) } ( \\mathcal { I } ) = \\left ( 1 + \\tau \\cdot \\mathbf { 1 } ^ { \\mathtt { T } } \\cdot \\left ( \\mathbf { r } _ { 0 } ( 0 ) _ { \\kappa ( \\mathcal { I } ) } ^ { \\kappa ( \\mathcal { I } ) } \\right ) ^ { - 1 } \\cdot \\mathbf { 1 } \\right ) \\cdot \\Delta _ { \\mathbf { R } _ { 0 } ( 0 ) } ( \\mathcal { I } ) = \\Delta _ { \\mathbf { R } _ { 0 } ( 0 ) } ( \\mathcal { I } ) \\in \\mathbb { C } \\end{align*}"}
{"id": "8117.png", "formula": "\\begin{align*} T B _ { 9 } ( f ) = 3 T B _ { 5 } ( f ) . \\end{align*}"}
{"id": "7340.png", "formula": "\\begin{align*} x _ j = \\lambda y _ j + ( 1 - \\lambda ) z _ j , w _ \\star ( x _ j , t _ j ) = \\min \\{ u ( y _ j , t _ j ) , u ( z _ j , t _ j ) \\} . \\end{align*}"}
{"id": "5600.png", "formula": "\\begin{align*} \\mathcal { T } _ \\star ^ { p , r } : = \\underbrace { \\Omega _ \\star \\otimes _ { \\mathcal { X } _ \\star } \\ldots \\otimes _ { \\mathcal { X } _ \\star } \\Omega _ \\star } _ { p } \\otimes _ { \\mathcal { X } _ \\star } \\underbrace { \\Xi _ \\star \\otimes _ { \\mathcal { X } _ \\star } \\ldots \\otimes _ { \\mathcal { X } _ \\star } \\Xi _ \\star } _ { r } \\end{align*}"}
{"id": "7045.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\Big \\| \\Big ( q - \\sum _ { k = 1 } ^ { L } q _ k \\widetilde { \\phi _ k } \\Big ) e _ j \\Big \\| _ { H ^ 2 } ^ 2 & \\le \\frac { C } { \\delta ^ { 2 \\max m _ j } } \\Big ( \\frac { 1 } { \\delta ^ 2 } \\log \\frac { 1 } { \\delta } \\Big ) ^ 2 . \\end{align*}"}
{"id": "4969.png", "formula": "\\begin{align*} \\alpha _ d = \\left \\{ \\begin{array} { l } 1 / 4 \\ d = 1 \\\\ 5 / 6 \\ d = 2 \\\\ 1 7 / 1 2 \\ d = 3 \\end{array} \\right . \\ , . \\end{align*}"}
{"id": "3249.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 ^ + } \\frac { h ( s ) } { s } = \\lim _ { s \\rightarrow 0 ^ + } \\int _ \\mathbb { R } \\frac { 1 } { s ^ 2 + u ^ 2 } d \\mu ( u ) = \\frac { 1 } { \\lambda _ 1 ( \\mu ) ^ 2 } , \\end{align*}"}
{"id": "4755.png", "formula": "\\begin{align*} A _ { i j } = \\begin{cases} O _ { m _ i \\times m _ j } , & a _ { i , j } = 0 \\\\ E _ { m _ i \\times m _ j } , & a _ { i , j } = 1 \\end{cases} \\end{align*}"}
{"id": "3319.png", "formula": "\\begin{align*} F _ \\ell & = F / x ^ \\ell F , \\\\ G _ \\ell & = G / y ^ \\ell G , \\\\ A _ \\ell & = A / ( x ^ \\ell F \\cap A ) , \\\\ B _ \\ell & = B / ( y ^ \\ell G \\cap B ) . \\end{align*}"}
{"id": "4242.png", "formula": "\\begin{align*} u _ { x x x } + u _ { y y x } = { } & 0 , \\\\ u _ { x x y } + u _ { y y y } = { } & - f ' ( u ) u _ y . \\end{align*}"}
{"id": "2114.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac 1 { \\alpha _ k } \\frac k { k + 1 } - \\frac { 1 - \\alpha _ k } { \\alpha _ k } \\le \\liminf _ { n \\to \\infty } E _ { n , k } ( X _ { n , k } ^ { \\mathcal { T } } | W _ { \\mathcal { T } ( n , k ; R _ n ) } ) \\le \\\\ & \\limsup _ { n \\to \\infty } E _ { n , k } ( X _ { n , k } ^ { \\mathcal { T } } | W _ { \\mathcal { T } ( n , k ; R _ n ) } ) \\le \\frac 1 { \\alpha _ k } \\frac k { k + 1 } . \\end{aligned} \\end{align*}"}
{"id": "7636.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ 0 = \\lim _ { n \\to + \\infty } \\min _ { B ^ 1 ( \\mathbf { x } ) \\subset B ^ { A _ n } ( \\mathbf { 0 } ) } \\lambda ^ 1 ( B ^ 1 ( \\mathbf { x } ) , B ^ { A _ n } ( \\mathbf { 0 } ) ) = \\lim _ { n \\to + \\infty } \\lambda ^ 1 ( B ^ 1 ( \\mathbf { 0 } ) , B ^ { A _ n } ( \\mathbf { 0 } ) ) , \\end{align*}"}
{"id": "1863.png", "formula": "\\begin{align*} { \\rm G e n } ( a ^ { - 1 } x , t ) = a ^ { - 1 } x \\cos ( t ) + a ^ { - 1 } \\sin ( t ) . \\end{align*}"}
{"id": "8495.png", "formula": "\\begin{gather*} F ( u ) = \\frac { u z ^ 2 ( 1 + z ) H ( 0 ) + 2 u z ^ { 3 } - u ^ { 2 } z + u ^ { 2 } + z ^ { 2 } - u } { u ^ { 2 } z ^ { 2 } + 2 u z ^ { 3 } + u ^ { 2 } + z ^ { 2 } - u } , \\\\ G ( u ) = - \\frac { z \\left ( H ( 0 ) ( u z ^ { 2 } + u - 1 ) + 2 u z ^ { 2 } + z \\right ) } { u ^ { 2 } z ^ { 2 } + 2 u z ^ { 3 } + u ^ { 2 } + z ^ { 2 } - u } , \\\\ H ( u ) = \\frac { z \\left ( z H ( 0 ) ( u z - u + 1 ) - u ^ { 2 } z + u ^ { 2 } - u \\right ) } { u ^ { 2 } z ^ { 2 } + 2 u z ^ { 3 } + u ^ { 2 } + z ^ { 2 } - u } . \\end{gather*}"}
{"id": "1262.png", "formula": "\\begin{align*} \\partial _ { J } \\langle \\alpha \\rangle = \\sum _ { \\beta : \\mathrm { a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } [ \\beta ] = \\Gamma } \\# ( \\mathcal { M } _ { 1 } ^ { J } ( \\alpha , \\beta ) / \\mathbb { R } ) \\cdot \\langle \\beta \\rangle . \\end{align*}"}
{"id": "4655.png", "formula": "\\begin{align*} - | D | Q - Q + Q ^ p = 0 . \\end{align*}"}
{"id": "8363.png", "formula": "\\begin{align*} u \\in \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } . \\smallskip \\end{align*}"}
{"id": "6330.png", "formula": "\\begin{align*} \\nabla _ { \\beta _ 1 } \\cdots \\nabla _ { \\beta _ { n - 1 } } \\partial _ { \\beta _ n } = \\Gamma ^ \\alpha _ { \\beta _ 1 , \\ldots , \\beta _ n } \\partial _ \\alpha \\end{align*}"}
{"id": "8994.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & x ( s ) = \\frac { 1 } { 2 } + A e ^ { 2 t } + B e ^ { - t } , \\\\ & y ( s ) = 2 A e ^ { 2 t } - B e ^ { - t } . \\end{aligned} \\right . \\end{align*}"}
{"id": "6236.png", "formula": "\\begin{align*} & \\lambda { } ^ 3 + ( 2 \\beta - \\beta \\mu + 5 \\delta ) \\lambda { } ^ 2 + ( 2 \\delta ( \\beta + 4 \\delta + 2 ) + d \\beta { } ^ 2 - \\beta ( \\beta + 4 \\delta ) \\mu ) \\lambda \\\\ & + 2 \\delta ( 2 \\delta ( \\delta + 1 ) + d \\beta ^ 2 - \\beta ( \\beta + 2 \\delta ) \\mu ) = 0 , \\ \\ \\mu \\in { \\rm S p e c } ( { \\bf A } ( G ) ) , \\end{align*}"}
{"id": "27.png", "formula": "\\begin{align*} \\left ( \\frac { \\sin ( t \\sqrt { \\Delta _ { E _ 1 } } ) } { \\sqrt { \\Delta _ { E _ 1 } } } f \\right ) ( x ) = \\left ( \\frac { \\sin ( t \\sqrt { \\Delta _ { E _ 2 } } ) } { \\sqrt { \\Delta _ { E _ 2 } } } f \\right ) ( x ) , \\end{align*}"}
{"id": "9505.png", "formula": "\\begin{align*} \\phi ( p ) ( T ( a ) ) = \\tau ( p , 1 ) p ( a ) \\end{align*}"}
{"id": "6796.png", "formula": "\\begin{align*} \\rho \\Bigg ( \\begin{bmatrix} \\theta \\\\ p ( \\theta ) \\\\ \\end{bmatrix} \\Bigg ) = \\begin{bmatrix} \\theta - \\alpha \\Delta \\theta \\\\ p ( \\theta ) - \\alpha \\Delta p ( \\theta ) \\\\ \\end{bmatrix} , \\end{align*}"}
{"id": "781.png", "formula": "\\begin{align*} X _ n + 1 = \\varepsilon _ { n - 1 } u _ n ^ 2 ( a + b ) . \\end{align*}"}
{"id": "1142.png", "formula": "\\begin{align*} E = I + k ^ { - 1 } E _ 1 + \\mathcal { O } ( k ^ { - 2 } ) , k \\rightarrow \\infty . \\end{align*}"}
{"id": "2512.png", "formula": "\\begin{align*} \\lim _ { | x _ n | + | \\omega _ n | + | \\tau _ n | \\to 0 } \\norm { e ^ { 2 \\pi i \\tau _ n } M _ { \\omega _ n / 2 } T _ { x _ n } M _ { \\omega _ n / 2 } f - f } _ 2 = 0 . \\end{align*}"}
{"id": "6500.png", "formula": "\\begin{align*} E [ ( S _ { n + 1 } ) ^ { 2 m } ] & = 1 + \\sum _ { \\ell = 1 } ^ m \\left \\{ \\binom { 2 m } { 2 \\ell } + \\frac { \\alpha } { n } \\binom { 2 m } { 2 \\ell - 1 } \\right \\} E [ ( S _ { n } ) ^ { 2 \\ell } ] . \\end{align*}"}
{"id": "3149.png", "formula": "\\begin{align*} \\Phi ( t , x ) & = : \\Phi ( z , R _ 1 , R _ 2 ) \\\\ & = \\eta ^ L ( t ( z ) , x _ 0 ) + Z ( z , r _ 1 ( z , R _ 1 , R _ 2 ) , r _ 2 ( z , R _ 1 , R _ 2 ) ) \\tau ( z ) + R _ 1 n ( z ) + R _ 2 b ( z ) \\end{align*}"}
{"id": "1565.png", "formula": "\\begin{align*} \\mathfrak { D } _ { \\omega } ^ d f = ( \\omega \\otimes \\tau ^ d ) ( \\eta ( z ) ) ^ { - 1 } \\mathfrak { C } ^ d [ \\omega ( \\eta ( z ) ) f ] . \\end{align*}"}
{"id": "5588.png", "formula": "\\begin{align*} \\overset { \\bullet } { v } _ 1 ( t ) & = - \\mu v _ 1 ( t ) \\\\ \\overset { \\bullet } { v } _ 2 ( t ) & = C v _ 2 ( t - \\Delta ) - \\mu v _ 2 ( t ) \\\\ & \\vdots \\\\ \\overset { \\bullet } { v } _ N ( t ) & = C v _ N ( t - \\Delta ) - \\mu v _ N ( t ) \\end{align*}"}
{"id": "603.png", "formula": "\\begin{align*} & 0 ! = 1 , \\\\ [ 1 5 p t ] & ( n + 1 ) ! = \\circ ( S \\circ P _ 1 ^ 2 , P _ 2 ^ 2 ) ( n , n ! ) . \\end{align*}"}
{"id": "614.png", "formula": "\\begin{align*} \\abs { A ( x ) - \\alpha } \\ = \\ \\abs { A ( x _ 0 ) - \\alpha } \\ \\leq \\ \\frac { 1 } { x _ 0 + 1 } \\ \\leq \\ \\frac { 1 } { x + 1 } . ] \\end{align*}"}
{"id": "4280.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { t \\in [ \\tau _ 0 , 0 ] } \\left \\| w ^ { ( k ) } ( t , \\cdot ) - w ( t , \\cdot ) \\right \\| _ { H ^ 2 ( \\R \\backslash \\{ t , 0 \\} ) } ~ = ~ 0 . \\end{align*}"}
{"id": "3705.png", "formula": "\\begin{align*} & \\sup _ { t \\in ( 0 , T ) } t ^ { \\frac { \\beta } { \\alpha } } \\| B ( t ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha + \\beta } } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } + C C ( T ) \\left ( \\sup _ { t \\in ( 0 , T ) } t ^ { \\frac { \\beta } { \\alpha } } \\| B ( t ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha + \\beta } } \\right ) ^ 2 \\end{align*}"}
{"id": "1788.png", "formula": "\\begin{align*} \\operatorname { A S } ( D ) : = \\widehat { A } \\Big ( \\frac { R _ X } { 2 \\pi i } \\Big ) \\operatorname { t r } \\Big ( \\exp \\big ( \\frac { R _ F } { 2 \\pi i } \\big ) \\Big ) . \\end{align*}"}
{"id": "8470.png", "formula": "\\begin{align*} \\lambda \\mathbf { \\mathrm { v } } _ u = \\sum _ { w \\sim u } \\mathbf { \\mathrm { v } } _ w . \\end{align*}"}
{"id": "8812.png", "formula": "\\begin{align*} \\nu ^ { \\epsilon , L } ( d \\varphi ) = e ^ { - V _ 0 ^ { \\epsilon , L } ( \\varphi ) } \\gamma ^ { \\epsilon , L } ( d \\varphi ) . \\end{align*}"}
{"id": "4489.png", "formula": "\\begin{align*} \\beta = \\frac 1 2 \\ \\ | \\gamma | \\le H _ 0 \\end{align*}"}
{"id": "6463.png", "formula": "\\begin{align*} a _ n : = \\prod _ { j = 1 } ^ { n - 1 } \\left ( 1 + \\dfrac { \\alpha } { j } \\right ) = \\prod _ { j = 1 } ^ { n - 1 } \\dfrac { j + \\alpha } { j } = \\dfrac { \\Gamma ( n + \\alpha ) } { \\Gamma ( n ) \\Gamma ( 1 + \\alpha ) } . \\end{align*}"}
{"id": "4146.png", "formula": "\\begin{align*} \\hat z ( 0 , \\tau ) = 0 , \\tau \\in [ - T , T ] . \\end{align*}"}
{"id": "4445.png", "formula": "\\begin{align*} \\chi _ { k } ( n ) : = \\begin{cases} 1 & n \\equiv k , \\ ; k + 1 \\ ! \\ ! \\pmod { 4 k + 2 } , \\\\ - 1 & n \\equiv - k , \\ ; - k - 1 \\ ! \\ ! \\pmod { 4 k + 2 } , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "4770.png", "formula": "\\begin{align*} \\sum _ { \\xi , \\eta } \\langle X _ \\alpha , [ X _ \\xi , X _ \\eta ] \\rangle \\langle X _ \\beta , [ \\tilde R X _ \\xi , \\tilde T X \\eta ] \\rangle = \\sum _ { \\xi , \\eta } r _ \\xi t _ \\eta \\langle X _ \\alpha , [ X _ \\xi , X _ \\eta ] \\rangle \\langle X _ \\beta , [ X _ \\xi , X \\eta ] \\rangle = 0 , \\end{align*}"}
{"id": "3916.png", "formula": "\\begin{align*} D T \\left [ f ''' ( \\eta ) \\right ] = ( k + 1 ) ( k + 2 ) ( k + 3 ) F ( k ) \\end{align*}"}
{"id": "8138.png", "formula": "\\begin{align*} T B _ { k } ( f ) = q \\cdot T B _ { j } ( f ) . \\end{align*}"}
{"id": "3229.png", "formula": "\\begin{align*} h ( s ) = \\int _ \\mathbb { R } \\frac { s } { s ^ 2 + u ^ 2 } d \\mu ( u ) , s > 0 . \\end{align*}"}
{"id": "5584.png", "formula": "\\begin{align*} \\| 1 _ A \\| \\ = \\ \\| z - 1 _ { B \\backslash E } - P _ E ( z ) \\| & \\ \\leqslant \\ \\mathbf C _ \\ell \\| z - P _ E ( z ) \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , r p } \\sigma ^ { \\mathcal { U } _ X , R } _ m ( z ) \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , r p } \\| 1 _ B \\| . \\end{align*}"}
{"id": "8041.png", "formula": "\\begin{align*} \\rho ^ * _ + \\Psi _ \\Sigma ( f ) = \\frac { 1 } { 2 } \\left ( \\Pi ( f ) + \\Phi ( * d f ) \\right ) . \\end{align*}"}
{"id": "6317.png", "formula": "\\begin{align*} \\zeta ^ \\wedge _ { \\mathcal { L } _ \\Delta , p } ( s ) = \\int _ { G _ \\Delta ^ + ( \\Q _ p ) } | \\det g | _ p ^ s d \\mu , \\end{align*}"}
{"id": "1612.png", "formula": "\\begin{align*} \\chi _ Q ( x ) = 1 \\operatorname { d i s t } ( x , \\partial Q ) > \\alpha \\varepsilon \\chi _ Q ( x ) = 0 \\operatorname { d i s t } ( x , \\partial Q ) < \\frac { \\alpha } { 2 } \\varepsilon , \\\\ \\psi _ Q ( x ) = 1 \\operatorname { d i s t } ( x , \\partial Q ) > \\frac { \\alpha } { 2 } \\varepsilon \\psi _ Q ( x ) = 0 \\operatorname { d i s t } ( x , \\partial Q ) < \\frac { \\alpha } { 4 } \\varepsilon , \\end{align*}"}
{"id": "9503.png", "formula": "\\begin{align*} \\phi ( p , \\lambda ) ( T ( a ) ) = \\tau ( p , \\lambda ) , a \\in F _ { p , \\lambda } . \\end{align*}"}
{"id": "6935.png", "formula": "\\begin{align*} R _ h ^ 2 ( w ) = \\sum _ { i \\in I _ h } r _ { h , i } ^ 2 ( w ) \\ , . \\end{align*}"}
{"id": "3237.png", "formula": "\\begin{align*} \\omega ( z ) = z + H _ 2 ( z + H _ 1 ( \\omega ( z ) ) ) , z \\in \\mathbb { C } ^ + \\end{align*}"}
{"id": "6249.png", "formula": "\\begin{align*} \\lambda ( q , t ) = t + q \\frac { t ^ 3 } { 3 ! } + ( q + q ^ 2 ) \\frac { t ^ 4 } { 4 ! } + \\dotsb \\end{align*}"}
{"id": "9193.png", "formula": "\\begin{align*} J ^ A _ \\gamma : = ( I d + \\gamma A ) ^ { - 1 } . \\end{align*}"}
{"id": "3657.png", "formula": "\\begin{align*} G _ m ( r ) : = \\begin{cases} \\log r & m = k \\\\ - r ^ { - 2 \\left ( \\frac { k } { m } - 1 \\right ) } & m < k , \\end{cases} \\end{align*}"}
{"id": "2915.png", "formula": "\\begin{align*} \\hat U _ { i p } = \\frac { R _ { i p } } { n + 1 } , R _ { i p } = \\sum _ { j = 1 } ^ { n } 1 _ { \\{ X _ { j p } \\le X _ { i p } \\} } , p = 1 , \\ldots , d . \\end{align*}"}
{"id": "4460.png", "formula": "\\begin{align*} A _ { n } & = \\frac { z } { \\sqrt { n } } A _ { n - 1 } = \\frac { z } { \\sqrt { n } } \\frac { z } { \\sqrt { n - 1 } } A _ { n - 2 } = \\cdots = \\frac { z ^ { n } } { \\sqrt { n ! } } A _ { 0 } . \\end{align*}"}
{"id": "8879.png", "formula": "\\begin{align*} p : = \\iota \\circ \\pi : I _ 0 & \\to I _ 0 \\\\ ( x , y ) & \\mapsto ( x + y , 0 ) \\end{align*}"}
{"id": "59.png", "formula": "\\begin{align*} L ' \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } L ' _ { 2 , j } ( \\pi ^ j ) & ( v = 2 ) , \\\\ L ' _ { v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\\\ \\end{cases} \\\\ K _ { \\ell } \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } K _ { \\ell , 2 , j } ( \\pi ^ j ) & ( v = 2 ) , \\\\ K _ { \\ell , v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
{"id": "431.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } ( h , U ^ { t } ) _ { \\mathbb { R } ^ { N } } d x & = \\lambda _ { h } ( U ^ { t } ) = \\langle \\psi ( h ) , U ^ { t } \\rangle _ { s } = \\lim _ { j \\rightarrow \\infty } \\langle \\psi ( h ) , U ^ { j } ( t ) \\rangle _ { s } \\\\ & = \\lim _ { j \\rightarrow \\infty } \\lambda _ { h } ( U ^ { j } ( t ) ) = \\lim _ { j \\rightarrow \\infty } \\int _ { \\mathbb { R } ^ { d } } ( h , U ^ { j } ( t ) ) _ { \\mathbb { R } ^ { N } } d x = \\int _ { \\mathbb { R } ^ { d } } ( h , U ( t ) ) _ { \\mathbb { R } ^ { N } } d x , \\end{align*}"}
{"id": "4843.png", "formula": "\\begin{align*} \\frac { n ^ 2 } { \\gamma } \\int _ \\R \\int _ \\R K ( \\alpha [ x - y ] ) \\ , d \\rho ( y ) d \\rho ( x ) = \\frac { n ^ 2 } { \\gamma \\alpha } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\alpha K ( \\alpha [ x - y ] ) \\ , d y d x . \\end{align*}"}
{"id": "8558.png", "formula": "\\begin{align*} & T \\left ( - k \\right ) = T ( k ) , \\ \\ \\ R _ { \\pm } \\left ( - k \\right ) = \\overline { R _ { \\pm } ( k ) , } \\\\ & \\left | R _ { \\pm } ( k ) \\right | ^ { 2 } + \\left | T ( k ) \\right | ^ { 2 } = 1 , \\ \\ T ( k ) \\overline { R _ { - } ( k ) } + \\overline { T ( k ) } R _ { + } ( k ) = 0 . \\end{align*}"}
{"id": "8878.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i ( & \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 1 ( z _ i ) , \\ldots , h _ 1 ( z _ { q - 1 } ) ) + \\varphi ( h _ 1 ( z _ 0 ) , \\ldots , h _ 1 ( z _ i ) , h _ 2 ( z _ i ) , \\ldots , h _ 2 ( z _ { q - 1 } ) ) \\\\ & - \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 2 ( z _ i ) , \\ldots , h _ 2 ( z _ { q - 1 } ) ) ) \\end{align*}"}
{"id": "5497.png", "formula": "\\begin{align*} & e ^ { \\gamma ( t - s ) } \\varphi _ { \\gamma } ( s - r ) + \\varphi _ { \\gamma } ( t - s ) = e ^ { \\gamma ( t - s ) } \\int _ 0 ^ { s - r } e ^ { \\gamma ( s - r - u ) } d u + \\int _ 0 ^ { t - s } e ^ { \\gamma ( t - s - u ) } d u \\\\ & = \\int _ 0 ^ { s - r } e ^ { \\gamma ( t - r - u ) } d u + \\int _ { s - r } ^ { t - r } e ^ { \\gamma ( t - r - u ) } d u = \\int _ 0 ^ { t - r } e ^ { \\gamma ( t - r - u ) } d u = \\varphi _ { \\gamma } ( t - r ) \\end{align*}"}
{"id": "4457.png", "formula": "\\begin{align*} | z \\rangle = \\sum _ { n = 0 } ^ { \\infty } A _ { n } | n \\rangle . \\end{align*}"}
{"id": "2256.png", "formula": "\\begin{align*} J _ 2 & \\leq C ( \\lambda _ N ^ { - \\frac { 2 + \\ell ( \\gamma ) } 2 } + k ^ { \\frac { 2 + \\ell ( \\gamma ) } 4 } ) \\| F ( X ( t _ m ) ) \\| _ { L ^ p ( \\Omega ; H ^ { \\ell ( \\gamma ) } ) } \\\\ & \\leq C ( \\lambda _ N ^ { - \\frac { 2 + \\ell ( \\gamma ) } 2 } + k ^ { \\frac { 2 + \\ell ( \\gamma ) } 4 } ) \\sup _ { s \\in [ 0 , T ] } \\| F ( X ( s ) ) \\| _ { L ^ p ( \\Omega ; H ^ { \\ell ( \\gamma ) } ) } \\leq C ( \\lambda _ N ^ { - \\frac \\gamma 2 } + k ^ { \\frac \\gamma 4 } ) , \\end{align*}"}
{"id": "7864.png", "formula": "\\begin{align*} [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( K ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( L ) & = [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J \\cup K \\cup L ) \\\\ & = [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( \\mathbb { N } ) = \\{ \\bar { b } _ n \\} \\end{align*}"}
{"id": "5381.png", "formula": "\\begin{align*} c _ { M a p } = 4 n ^ 2 m + 2 m ^ 2 n + 1 6 n m - 1 2 m . \\end{align*}"}
{"id": "4108.png", "formula": "\\begin{align*} | \\sigma _ n ( x ) - \\sigma _ n ( y ) | = \\Bigl | \\frac { \\sigma ( x ) - \\sigma ( y ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } } + f _ n ( x , y ) \\Bigr | \\leq c \\cdot ( 1 + | x | ^ { \\ell _ { \\sigma } } + | y | ^ { \\ell _ { \\sigma } } ) \\cdot | x - y | + | f _ n ( x , y ) | . \\end{align*}"}
{"id": "7562.png", "formula": "\\begin{align*} \\psi : E _ d & \\to K \\\\ g = \\sum _ { F \\subset [ n ] , | F | = d } \\beta _ F x ^ F & \\mapsto \\sum _ { F \\subset [ n ] , | F | = d } \\beta _ F \\alpha _ F = g ( \\partial _ { 1 } , \\cdots , \\partial _ n ) f \\end{align*}"}
{"id": "2129.png", "formula": "\\begin{align*} \\prod _ { t = 0 } ^ { c _ n k n - l - 1 } \\frac { k ( n - s - 1 ) - t } { k n - t } = ( 1 + o ( 1 ) ) ( 1 - c ) ^ { k ( s + 1 ) } , \\ \\ n \\to \\infty . \\end{align*}"}
{"id": "5361.png", "formula": "\\begin{align*} \\bar x = \\arg \\min \\left \\{ \\rho _ c ( x ) \\middle | x \\in M \\right \\} . \\end{align*}"}
{"id": "8990.png", "formula": "\\begin{align*} \\min \\left \\{ I _ b [ \\gamma ] ; \\gamma \\in \\mathrm { A C } \\left ( [ 0 , b ] ; [ - 1 , 1 ] \\right ) , \\gamma ( 0 ) = x _ 0 , \\gamma ( b ) = \\gamma ( b ) \\right \\} . \\end{align*}"}
{"id": "1580.png", "formula": "\\begin{align*} M ( t , x ) = \\int _ { \\R ^ 3 } f ( t , x , v ) d v , E ( t , x ) = \\int _ { \\R ^ 3 } f ( t , x , v ) | v | ^ 2 d v , H ( t , x ) = \\int _ { \\R ^ 3 } f ( t , x , v ) \\ln f ( t , x , v ) d v . \\end{align*}"}
{"id": "8475.png", "formula": "\\begin{align*} E : = \\{ v \\in V ( H _ k ) \\setminus L ' : | N _ 1 ( v ) \\cap L ' | \\leq k - 1 \\} . \\end{align*}"}
{"id": "1497.png", "formula": "\\begin{align*} K _ 1 ( \\mathfrak { n } ) \\xi K _ 1 ( \\mathfrak { n } ) = \\bigcup _ { y \\in Y } K _ 1 ( \\mathfrak { n } ) y . \\end{align*}"}
{"id": "9530.png", "formula": "\\begin{align*} E _ t \\Delta y _ { t + 1 } & \\in \\partial _ { x } H _ t ( x _ { t } , y _ t ) , \\\\ \\Delta x _ t & \\in \\partial _ { y } [ - H _ t ] ( x _ { t } , y _ t ) , \\end{align*}"}
{"id": "8177.png", "formula": "\\begin{align*} \\sum _ { h \\in H _ { d _ 0 } } s ( h , \\delta f ) = \\frac { \\phi ( d _ 0 ) } { \\phi ( \\delta ) } \\sum _ { h \\in H _ { \\delta } } s ( h , \\delta f ) \\end{align*}"}
{"id": "7224.png", "formula": "\\begin{align*} | \\Psi _ { s , t } ( x , v ) - v | = | \\Psi _ { s , t } ( x , v ) - \\zeta ( \\Psi _ { s , t } ( x , v ) ) | \\leq \\frac { | \\tilde Y _ { s , t } ( x , \\Psi _ { s , t } ( x , v ) ) | } { t - s } \\leq \\frac { | \\tilde Y _ { s , t } ( x , v ) | } { t - s } + \\frac 1 2 | \\Psi _ { s , t } ( x , v ) - v | , \\end{align*}"}
{"id": "3818.png", "formula": "\\begin{align*} 1 = \\psi \\left ( c ^ 2 [ a _ 1 , b _ 1 ] \\right ) = x \\sigma _ 1 x \\sigma _ 1 y g z y ^ { - 1 } g ^ { - 1 } z ^ { - 1 } = x \\ldotp \\sigma _ 1 x \\sigma _ 1 ^ { - 1 } \\ldotp A _ { 1 , 2 } \\ldotp y \\ldotp g z g ^ { - 1 } \\ldotp g y ^ { - 1 } g ^ { - 1 } \\ldotp z ^ { - 1 } . \\end{align*}"}
{"id": "3583.png", "formula": "\\begin{align*} ( F ( f ) ) ( t ) : = f _ 0 ( t ) F _ Z ( f ( t ) ) f \\in X t \\in \\partial A . \\end{align*}"}
{"id": "2320.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\int _ { - \\infty } ^ \\infty W \\psi ( x , p ) \\ , d p \\ , d x = 1 , \\end{align*}"}
{"id": "9416.png", "formula": "\\begin{align*} \\tau ' ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots c _ { n - 1 } a _ n b _ n ) = \\tau ( b _ 0 \\tau ' ( a _ 1 ) c _ 1 a _ 2 \\cdots c _ { n - 1 } a _ n b _ n ) = 0 . \\end{align*}"}
{"id": "6676.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq \\tilde { \\jmath } < \\infty \\\\ p | \\tilde { \\jmath } \\Rightarrow p | j } } \\frac { ( j , \\tilde { \\jmath } ) } { \\tilde { \\jmath } } & = \\prod _ { p | j } \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { p ^ { \\min \\{ j _ p , \\nu \\} } } { p ^ { \\nu } } = \\prod _ { p | j } \\bigg ( j _ p + \\frac { 1 } { 1 - \\frac { 1 } { p } } \\bigg ) \\ll j ^ { \\varepsilon } \\end{align*}"}
{"id": "5309.png", "formula": "\\begin{align*} \\sum _ { r s } v _ { ( 0 ; r s ) } \\otimes b S ( v _ { ( 1 ; r s ) } ) a = ( 1 \\otimes b ) ( X ^ - ( v \\otimes a ) ) , v \\in V , a , b \\in A . \\end{align*}"}
{"id": "2465.png", "formula": "\\begin{align*} M = \\begin{pmatrix} \\alpha & 0 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\alpha } & 0 & 0 \\\\ 0 & 0 & \\alpha & 0 \\\\ 0 & 0 & 0 & \\frac { 1 } { \\alpha } \\end{pmatrix} . \\end{align*}"}
{"id": "9076.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ 1 ^ { i n } ( x ) = 2 + 1 2 ( x - 0 . 5 ) ^ 2 , \\rho _ 2 ^ { i n } ( x ) = 1 + 2 x , \\\\ & \\partial _ x \\phi ( 0 , t ) = 0 , \\ \\partial _ x \\phi ( 1 , t ) = 0 . \\end{aligned} \\end{align*}"}
{"id": "4422.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\sum _ { i = 1 } ^ { m } z _ i a _ { i j } = \\eta , \\ ; \\ ; \\ ; j = 1 , 2 , . . . , n \\\\ \\displaystyle \\sum _ { i = 1 } ^ { m } z _ i = 1 \\\\ \\end{array} \\end{align*}"}
{"id": "436.png", "formula": "\\begin{align*} \\widetilde { A } ^ { 0 } ( U ) = \\left ( \\begin{array} { c c c c } 1 & & & \\\\ & \\rho \\mathbb { I } _ { 3 } & & \\\\ & & \\rho e _ { \\theta } & \\\\ & & & \\tau \\mathbb { I } _ { 3 } \\end{array} \\right ) , \\widetilde { D } ( U ) = \\left ( \\begin{array} { c c } \\mathbb { O } _ { 5 \\times 5 } & \\\\ & \\mathbb { I } _ { 3 } \\end{array} \\right ) , \\end{align*}"}
{"id": "3168.png", "formula": "\\begin{align*} \\mathcal { U } _ k = \\left \\lbrace j : \\frac { \\norm { E _ j ^ \\intercal ( A x _ k - b ) } _ 2 ^ 2 } { \\norm { A x _ k - b } _ 2 ^ 2 \\norm { A ^ \\intercal E _ j } _ F ^ 2 } \\geq \\epsilon _ k \\right \\rbrace . \\end{align*}"}
{"id": "567.png", "formula": "\\begin{align*} | b ^ { ( j ) } ( S _ \\varepsilon ( \\rho ) ) | \\leq h ^ j = \\begin{cases} \\max _ { [ 0 , \\bar \\rho - \\alpha _ 0 ] } | b ^ { ( j ) } | & \\rho \\leq \\bar \\rho - \\alpha _ 0 \\\\ | b ^ { ( j ) } ( \\rho ) | & \\rho > \\bar \\rho - \\alpha _ 0 . \\end{cases} \\end{align*}"}
{"id": "6966.png", "formula": "\\begin{align*} \\begin{aligned} M ( x , y ) & : = \\{ ( ( \\tilde x , g ( \\tilde x ) ) , ( 0 , \\tilde y ) ) \\ , | \\ , \\tilde x = x , \\ , \\tilde y \\in - D , y = g ( \\tilde x ) + \\tilde y \\} \\\\ & = \\begin{cases} \\{ ( ( x , g ( x ) ) , ( 0 , y - g ( x ) ) ) \\} & g ( x ) - y \\in D , \\\\ \\varnothing & \\end{cases} \\end{aligned} \\end{align*}"}
{"id": "2220.png", "formula": "\\begin{align*} J ( u ) = \\frac 1 2 \\| \\nabla u \\| ^ 2 + \\int _ D \\Phi ( u ) \\ , \\dd x , \\end{align*}"}
{"id": "4748.png", "formula": "\\begin{align*} S _ 1 + T ^ 2 S _ 2 \\leq \\sum _ { \\gamma \\in \\mathcal { I } } \\| g _ { \\gamma } \\| _ { L ^ 2 } ^ 2 ( 2 \\N ) ^ { - p _ 3 \\gamma } + T ^ 2 \\sum _ { \\gamma \\in \\mathcal { I } } \\| f _ { \\gamma } \\| ^ 2 ( 2 \\N ) ^ { - p _ 2 \\gamma } : = A < \\infty . \\end{align*}"}
{"id": "4377.png", "formula": "\\begin{align*} B ( \\varepsilon ^ + ) = - 3 ( n - 4 ) ( 1 + | y | ^ 2 Q _ b ) e ^ { 2 \\int _ { \\tau _ 0 } ^ \\tau \\lambda _ { M , b } d \\tau ' } \\phi _ { M , b } ^ 2 f ^ 2 ( y ) - ( n - 4 ) | y | ^ 2 e ^ { 3 \\int _ { \\tau _ 0 } ^ \\tau \\lambda _ { M , b } d \\tau ' } \\phi _ { M , b } ^ 3 f ^ 3 ( y ) \\end{align*}"}
{"id": "7969.png", "formula": "\\begin{align*} & \\theta _ g ( x ^ i , 1 ; \\| \\Sigma ^ i \\| _ g ) - \\theta ( x ^ i ; \\| \\Sigma ^ i \\| _ g ) \\leq \\delta ; \\\\ & \\theta ( x ^ 0 ; \\| \\Sigma ^ 0 \\| _ g ) = \\theta ( x ^ 1 ; \\| \\Sigma ^ 1 \\| _ g ) = \\theta ( x ^ 2 ; \\| \\Sigma ^ 2 \\| _ g ) . \\end{align*}"}
{"id": "1331.png", "formula": "\\begin{align*} J _ { 0 } ( \\hat { \\alpha } \\cup { ( \\gamma , N ) } , \\hat { \\alpha } \\cup { ( \\gamma , N - p _ { i } ) } \\cup { ( \\delta _ { 1 } , 1 ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 1 . \\end{align*}"}
{"id": "6996.png", "formula": "\\begin{align*} \\rho = \\sum _ i p _ i P _ i , \\end{align*}"}
{"id": "3965.png", "formula": "\\begin{align*} \\begin{cases} \\| \\xi _ { \\lambda ^ h _ k } \\| _ { L ^ 2 ( 0 , 1 ) } \\leq C , \\\\ \\| \\xi _ { \\lambda ^ h _ k } \\| _ { H ^ { - s } _ { p e r } ( 0 , 1 ) } \\leq C | k | ^ { - s } , \\ \\ 0 < s < 1 , \\\\ \\| \\xi _ { \\lambda ^ h _ k } \\| _ { H ^ { - s } _ { p e r } ( 0 , 1 ) } \\leq C | k | ^ { - 1 } , \\ \\ s \\geq 1 , \\\\ \\| \\eta _ { \\lambda ^ h _ k } \\| _ { L ^ 2 ( 0 , 1 ) } \\leq C | k | ^ { - 1 } . \\end{cases} \\end{align*}"}
{"id": "5555.png", "formula": "\\begin{align*} h ( x ) < 0 x < x _ 0 , h ( x _ 0 ) = 0 h ( x ) > 0 x > x _ 0 . \\end{align*}"}
{"id": "6290.png", "formula": "\\begin{align*} \\mathrm { v a r } [ \\widehat { K } _ { \\mathrm { a - c p t - d } } ^ \\mathbb { R } ] \\ ! = \\ ! \\frac { \\psi } { 2 } \\Big ( \\frac { 1 } { N _ 2 \\bar { \\gamma } _ c ' } \\ ! - \\ ! \\frac { \\mathrm { l i } ( 1 \\ ! - \\ ! \\xi ) } { 1 - \\xi } \\sum _ { i \\in \\mathcal { K } ' } \\frac { 1 } { 1 \\ ! + \\ ! N _ 1 \\bar { \\gamma } _ i } \\Big ) \\end{align*}"}
{"id": "8691.png", "formula": "\\begin{align*} x _ 1 ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 \\geq \\max \\left \\{ \\begin{aligned} & \\max \\{ x _ 1 ^ 2 + x _ 2 ^ 2 - 1 , 4 x _ 1 ^ 2 + 4 x _ 2 ^ 2 - 1 6 \\} + x _ 3 ^ 2 - 1 \\\\ & 4 \\max \\bigl \\{ x _ 1 ^ 2 + x _ 2 ^ 2 - 1 , 4 x _ 1 ^ 2 + 4 x _ 2 ^ 2 - 1 6 \\bigr \\} + 1 6 x _ 3 ^ 2 - 6 4 \\end{aligned} \\right \\} . \\end{align*}"}
{"id": "4750.png", "formula": "\\begin{align*} \\mathcal { A } ( \\phi ) = \\mathcal { A } ( \\phi ^ { - 1 } ) . \\end{align*}"}
{"id": "55.png", "formula": "\\begin{align*} 0 \\to H ^ 0 ( k a \\L - 2 ( j _ 2 + 1 ) B _ 2 ) \\to H ^ 0 ( k a \\L - 2 j _ 2 B _ 2 ) \\to \\bigoplus _ { i = 1 } ^ { s _ 2 } M _ { k a + 4 j _ 2 } ( \\Gamma _ i ) , \\\\ 0 \\to H ^ 0 ( k a \\L - \\frac { k } { 2 } B _ 2 - ( j _ 4 + 1 ) B _ 4 ) \\to H ^ 0 ( k a \\L - \\frac { k } { 2 } B _ 2 - j _ 4 B _ 4 ) \\to \\bigoplus _ { i = 1 } ^ { s _ 4 } M _ { k a + 4 j _ 4 } ( \\Gamma _ i ) . \\end{align*}"}
{"id": "2466.png", "formula": "\\begin{align*} A D ^ T - B C ^ T = \\begin{pmatrix} \\alpha ^ 2 & 0 \\\\ 0 & \\frac { 1 } { \\alpha ^ 2 } \\end{pmatrix} \\neq I , \\alpha \\in \\R , \\ , \\alpha \\neq \\pm 1 . \\end{align*}"}
{"id": "9508.png", "formula": "\\begin{align*} \\| H _ { g , s } + s g \\| & = 1 , \\\\ p ( H _ { g , s } + s g & ) = - \\frac { p ( g ) } { | p ( g ) | } , \\\\ \\| H _ { g , s } \\| & \\le 1 + | p ( g ) | . \\end{align*}"}
{"id": "7545.png", "formula": "\\begin{align*} \\Im \\left \\{ \\log \\zeta ( s ) \\right \\} = \\sum _ { | \\gamma - t | \\leq 1 } \\Im \\left \\{ \\log ( s - \\rho ) \\right \\} + \\mathcal { O } ( \\log t ) \\end{align*}"}
{"id": "7162.png", "formula": "\\begin{align*} \\lambda _ \\mu : = v ^ \\mu ( p ) - u ^ \\mu ( p ) \\qquad w ^ \\mu : = \\lambda _ \\mu ^ { - 1 } ( v ^ \\mu - u ^ \\mu ) \\in W ^ { 1 , 2 } _ { l o c } ( B _ 1 ) \\ , . \\end{align*}"}
{"id": "3180.png", "formula": "\\begin{align*} \\begin{bmatrix} 2 & 1 & 0 \\\\ - 1 & 2 & 3 \\end{bmatrix} \\quad \\mathrm { a n d } \\begin{bmatrix} 1 & - 3 & 6 \\\\ 0 & 1 & - 5 \\end{bmatrix} , \\end{align*}"}
{"id": "43.png", "formula": "\\begin{align*} g _ t ( v , v ) = d \\theta ^ c ( J v , v ) + d \\zeta ( J v , v ) = 1 + d \\zeta ( J v , v ) > 0 \\end{align*}"}
{"id": "4228.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\left | \\mathcal { F } _ N ( \\mathbf { h } _ N , \\mathbf { g } _ N ) - \\mathcal { F } ( \\mathbf { h } _ N , \\mathbf { g } _ N ) \\right | = 0 . \\end{align*}"}
{"id": "417.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { 0 \\leq \\tau \\leq t } \\| ( u , v , w ) ( \\tau ) \\| _ { s } ^ { 2 } & + \\int _ { 0 } ^ { t } \\| v ( \\tau ) \\| _ { s + 1 } ^ { 2 } d \\tau + \\\\ & + \\int _ { 0 } ^ { t } \\| \\left ( u _ { t } ( \\tau ) , v _ { t } ( \\tau ) , w _ { t } ( \\tau ) \\right ) \\| _ { s - 1 } ^ { 2 } d \\tau \\leq M ^ { 2 } \\end{aligned} \\end{align*}"}
{"id": "5948.png", "formula": "\\begin{align*} \\norm { | f | ^ 2 - | g | ^ 2 } _ 2 \\le \\big \\| \\ , | f | + | g | \\ , \\big \\| _ 4 \\cdot \\big \\| \\ , | f | - | g | \\ , \\big \\| _ 4 & \\le C ( \\norm { f } _ 2 + \\norm { g } _ 2 ) \\ , \\big \\| \\ , | f | - | g | \\ , \\big \\| _ 4 \\\\ & \\le 2 C \\ , \\big \\| \\ , | f | - | g | \\ , \\big \\| _ 4 . \\end{align*}"}
{"id": "7027.png", "formula": "\\begin{align*} a _ 1 ( z ) : = \\prod _ { j = 1 } ^ n ( z - \\xi _ j ) ^ { m _ j } . \\end{align*}"}
{"id": "9338.png", "formula": "\\begin{align*} \\log _ { \\lambda } ( 1 + t ) = \\sum _ { k = 1 } ^ { \\infty } \\lambda ^ { k - 1 } ( 1 ) _ { k , \\frac { 1 } { \\lambda } } \\frac { t ^ { k } } { k ! } , ( \\mathrm { s e e } \\ [ 8 , 1 4 ] ) . \\end{align*}"}
{"id": "8071.png", "formula": "\\begin{align*} \\mathrm { W F } ( W _ { \\Sigma } ) = \\left \\{ ( r , r ; \\xi , - \\xi ) \\in \\dot { T } ^ { * } \\Sigma ^ 2 \\ , | \\ , \\xi > 0 \\right \\} , \\end{align*}"}
{"id": "6454.png", "formula": "\\begin{align*} \\lambda _ h = \\lambda _ 0 - i \\frac { { \\lambda _ 0 } ^ \\frac { 5 } { 2 } } { 4 \\pi } \\left [ { ( \\eta _ 0 ^ 1 U _ 1 + \\eta _ 0 ^ 2 U _ 2 ) U _ 0 } \\right ] h + \\mathcal { O } ( h ^ 2 ) . \\end{align*}"}
{"id": "5128.png", "formula": "\\begin{align*} \\frac { \\widetilde { f } ' ( \\gamma ) } { \\widetilde { f } ( \\gamma ) } = a + \\frac { 1 } { \\gamma } - a \\coth ( \\gamma a ) = a + a \\ , \\underbrace { \\left ( \\frac { 1 } { \\gamma a } - \\coth ( \\gamma a ) \\right ) } _ { 0 > , \\ , \\searrow , \\ , > - 1 } , \\end{align*}"}
{"id": "1712.png", "formula": "\\begin{align*} ( 1 \u2010 \\lambda ) \\left ( s _ * + \\frac 1 q \u2010 \\frac { 1 } { p _ 1 } \\right ) + \\lambda \\left ( \\frac 1 q \u2010 \\frac { 1 } { p _ 0 } \\right ) = ( 1 \u2010 \\lambda ) s _ * . \\end{align*}"}
{"id": "4025.png", "formula": "\\begin{align*} \\begin{cases} C _ 1 = e ^ { m _ 2 } - e ^ { m _ 3 } , \\\\ C _ 2 = e ^ { m _ 3 } - e ^ { m _ 1 } , \\\\ C _ 3 = e ^ { m _ 1 } - e ^ { m _ 2 } . \\end{cases} \\end{align*}"}
{"id": "6048.png", "formula": "\\begin{align*} R _ i f ( x ) & = \\partial _ { x _ i } \\widetilde { \\mathcal { O } } ^ { - 1 / 2 } f ( x ) = - e ^ { - | x | ^ 2 } ( \\partial _ { x _ i } - 2 x _ i ) ( \\mathcal { O } + 2 d \\ , I _ d ) ^ { - 1 / 2 } ( e ^ { | \\cdot | ^ 2 } f ( \\cdot ) ) ( x ) \\\\ & = - e ^ { - | x | ^ 2 } ( \\partial _ { x _ i } - 2 x _ i ) \\mathcal { O } ^ { - 1 / 2 } \\Lambda ( e ^ { | \\cdot | ^ 2 } f ( \\cdot ) ) ( x ) . \\end{align*}"}
{"id": "6030.png", "formula": "\\begin{align*} \\widetilde { \\mathfrak { L } } _ \\alpha f & = \\widetilde { Q } _ { \\alpha } ^ { - 1 } \\delta _ x ^ { \\alpha - 1 } ( \\delta _ x ^ { \\alpha - 1 } ) ^ * \\widetilde { Q } _ { \\alpha } f \\\\ & = \\widetilde { Q } _ { \\alpha } ^ { - 1 } ( \\delta _ x ^ { \\alpha } ) ^ * \\delta _ x ^ { \\alpha } \\widetilde { Q } _ { \\alpha } f + \\frac 1 2 f = \\widetilde { Q } _ { \\alpha } ^ { - 1 } Q { \\mathfrak { L } } _ \\alpha Q ^ { - 1 } \\widetilde { Q } _ { \\alpha } f + \\frac 1 2 f \\end{align*}"}
{"id": "9064.png", "formula": "\\begin{align*} \\tilde m _ { l , j + 1 / 2 } = \\frac { 1 } { h } \\sum _ { p = 1 } ^ j \\tilde \\rho _ { l , p } = \\left \\{ \\begin{array} { l l } \\frac { 1 } { h } b _ j \\gamma , & l = i , \\ ; j _ 1 \\leq j \\leq j _ { k + 1 } - 1 , \\\\ 0 , & , \\end{array} \\right . \\end{align*}"}
{"id": "4081.png", "formula": "\\begin{align*} 1 - c _ 2 ( | x _ 1 | ^ { \\alpha } + . . . + | x _ { d - 1 } | ^ { \\alpha } ) \\leq f ( x ) \\leq 1 - c _ 1 ( | x _ 1 | ^ { \\alpha } + . . . + | x _ { d - 1 } | ^ { \\alpha } ) , \\ ; \\ ; \\nabla f ( 0 ) = 0 . \\end{align*}"}
{"id": "6358.png", "formula": "\\begin{align*} \\begin{cases} x _ { t - 1 } ^ { i _ { t - 1 } } = u _ t \\\\ y _ { t - 1 } ^ { i _ { t - 1 } } = u _ t ^ { i _ t } v _ t ^ { i _ { t - 1 } } \\end{cases} \\begin{cases} x _ { t - 1 } ^ { i _ { t - 1 } } = x _ t ^ { i _ { t - 1 } } y _ t \\\\ y _ { t - 1 } ^ { i _ { t - 1 } } = y _ t ^ { i _ t } \\end{cases} \\end{align*}"}
{"id": "3598.png", "formula": "\\begin{align*} d [ c a _ { 1 , i } b _ { 1 , j } ] = \\min \\{ d ( c a _ { 1 , i } b _ { 1 , j } ) , \\alpha _ { i } , \\beta _ { j } \\} \\ , , c \\in F ^ { \\times } \\ , . \\end{align*}"}
{"id": "433.png", "formula": "\\begin{align*} A ^ { 0 } ( U ^ { k } ) V ^ { k } _ { t } + A ^ { i } ( U ^ { k } ) \\partial _ { i } V ^ { k } - & B ^ { i j } ( U ^ { k } ) \\partial _ { i } \\partial _ { j } V ^ { k } + D ( U ^ { k } ) V ^ { k } = F ( U ^ { k } ; D _ { x } U ^ { k } ) , \\\\ \\left . U ^ { k } \\right \\rvert _ { t = 0 } = & U _ { 0 } \\end{align*}"}
{"id": "3614.png", "formula": "\\begin{align*} \\beta _ { n - 1 } = \\beta _ { n - 1 } + 0 \\ge d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] + d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] > 2 e + S _ { n } - R _ { n + 2 } = 2 e - 1 \\ , . \\end{align*}"}
{"id": "5107.png", "formula": "\\begin{align*} \\psi ( x ) > 1 - 2 x ^ 2 + 6 x ^ 2 \\cdot 1 ^ 2 - 6 x \\cdot 1 . 1 = 4 x ^ 2 - 6 . 6 x + 1 = 4 \\cdot ( 1 . 5 - 0 . 8 2 5 ) ^ 2 - 1 . 7 2 2 5 = 0 . 1 > 0 . \\end{align*}"}
{"id": "3957.png", "formula": "\\begin{align*} - V ^ \\prime ( t ) = A ^ * V ( t ) + F ( t ) , \\ \\ \\forall t \\in ( 0 , T ) , V ( T ) = V _ T , \\end{align*}"}
{"id": "546.png", "formula": "\\begin{align*} b ( \\rho _ 0 ) , b ' ( \\rho _ 0 ) & \\in L ^ \\frac { 3 } { 2 } ( \\Omega ) \\\\ b ( \\rho ) , b ' ( \\rho ) & \\in L ^ \\infty ( 0 , T ; L ^ \\frac { 3 } { 2 } ( \\Omega ) ) \\cap L ^ \\frac { 5 } { 2 } ( Q _ T ) . \\end{align*}"}
{"id": "2343.png", "formula": "\\begin{align*} f _ { a , b } ( x ) = e ^ { - 2 \\pi i b \\cdot x } f ( x + a ) = M _ { - b } T _ { - a } f ( x ) , \\end{align*}"}
{"id": "2570.png", "formula": "\\begin{align*} \\langle \\tfrac { 1 } { x } , f \\rangle = \\lim _ { \\varepsilon \\to 0 } \\int _ { | x | > \\varepsilon } \\frac { \\overline { f ( x ) } } { x } \\ , d x . \\end{align*}"}
{"id": "4803.png", "formula": "\\begin{align*} \\phi ^ 2 + \\phi * \\phi & = ( s ^ 2 ( \\ell - 1 ) + t ^ 2 ( m - 1 ) + ( \\ell - 1 ) ( m - 1 ) ) \\delta _ { ( 0 , 0 ) } \\\\ & + ( s ^ 2 ( \\ell - 1 ) + ( \\ell - 2 ) ( m - 1 ) + 2 t ( m - 1 ) ) \\chi _ A \\\\ & + ( t ^ 2 ( m - 1 ) + ( \\ell - 1 ) ( m - 2 ) + 2 s ( \\ell - 1 ) ) \\chi _ B \\\\ & + ( 1 + ( \\ell - 2 ) ( m - 2 ) + 2 s t + 2 s ( \\ell - 2 ) + 2 t ( m - 2 ) ) \\chi _ C . \\end{align*}"}
{"id": "3806.png", "formula": "\\begin{align*} \\xi _ 1 ( 0 ) = f ( \\tau ( g , \\xi ( 0 ) ) ) = f ( \\tau ( g , m _ 1 ) ) \\stackrel { ( \\ref { e q : e q u i v _ m 1 } ) } { = } \\tau _ 1 ( g , f ( m _ 1 ) ) = \\tau _ 1 ( g , f \\circ \\xi ( 0 ) ) = \\xi _ 2 ( 0 ) . \\end{align*}"}
{"id": "6043.png", "formula": "\\begin{align*} \\widetilde { P } _ n ^ { \\alpha , \\beta } ( x ) = \\frac { ( - 1 ) ^ { n } } { 2 ^ n n ! } ( 1 - x ) ^ { \\alpha } ( 1 + x ) ^ { \\beta } \\frac { d ^ n } { d x ^ n } \\big ( ( 1 - x ) ^ { - \\alpha + n } ( 1 + x ) ^ { - \\beta + n } \\big ) \\end{align*}"}
{"id": "1558.png", "formula": "\\begin{align*} \\Gamma \\backslash \\mathcal { B } \\stackrel { \\sim } \\longrightarrow \\mathcal { F } ( \\Omega ) , \\Gamma = \\{ \\gamma \\in U ( \\mathcal { T } ) : L \\gamma = L , u _ i \\gamma - u _ i \\in L \\} . \\end{align*}"}
{"id": "6805.png", "formula": "\\begin{gather*} F ( u ) - F ( w ) = ( \\beta + 1 ) ( u ^ { 2 } - w ^ { 2 } ) - ( u ^ { 3 } - w ^ { 3 } ) + P \\left ( \\boldsymbol { D } \\right ) ( u ^ { m } - w ^ { m } ) \\\\ = ( \\beta + 1 ) ( u - w ) \\left ( u + w \\right ) - ( u - w ) ( u ^ { 2 } + u w + w ^ { 2 } ) + P \\left ( \\boldsymbol { D } \\right ) ( ( u - w ) q ( u , w ) ) , \\end{gather*}"}
{"id": "2576.png", "formula": "\\begin{align*} \\partial ^ \\alpha t ^ \\beta ( M _ \\omega T _ x g ( t ) ) = \\sum _ { \\gamma _ 1 \\leq \\alpha } \\sum _ { \\gamma _ 2 \\leq \\beta } \\begin{pmatrix} \\alpha \\\\ \\gamma _ 1 \\end{pmatrix} \\begin{pmatrix} \\beta \\\\ \\gamma _ 2 \\end{pmatrix} x ^ { \\gamma _ 2 } ( 2 \\pi i \\omega ) ^ { \\gamma _ 1 } M _ \\omega T _ x ( \\partial ^ { \\alpha - \\gamma _ 1 } t ^ { \\beta - \\gamma _ 2 } g ( t ) ) , \\end{align*}"}
{"id": "359.png", "formula": "\\begin{align*} \\mathcal A ( \\widetilde \\rho , \\widetilde m ) = \\frac 1 2 \\int _ 0 ^ 1 \\frac { ( ( \\Sigma _ 1 - \\Sigma _ 2 ) \\theta _ { 1 2 } ( \\widetilde \\rho ) \\dot W ^ { \\delta } ) ^ 2 } { \\theta _ { 1 2 } ( \\widetilde \\rho ) } + ( m _ { 2 3 } + \\frac 1 2 \\rho _ 1 ( \\Sigma _ 2 - \\Sigma _ 3 ) \\dot W ^ { \\delta } ) ^ 2 d t . \\end{align*}"}
{"id": "699.png", "formula": "\\begin{align*} - \\norm { \\xi } ^ 2 = z _ { i ; \\alpha } ^ { ( \\ell + 1 ) } . \\end{align*}"}
{"id": "5707.png", "formula": "\\begin{align*} \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } ) = 5 , \\ , \\ , \\ , \\ , \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) = 7 \\end{align*}"}
{"id": "5124.png", "formula": "\\begin{align*} A = \\frac { 2 \\gamma a ( 1 - e ^ { - 2 \\gamma n a } ) ( 1 - e ^ { - \\gamma n a } ) ^ 2 } { ( 1 - e ^ { - 2 \\gamma a } ) ( 1 - e ^ { - 4 \\gamma n a } ) } \\ , e ^ { - 2 a \\gamma } B = \\frac { 2 \\gamma a ( 1 - e ^ { - 2 \\gamma n a } ) ( 1 + e ^ { - \\gamma n a } ) ^ 2 } { ( 1 - e ^ { - 2 \\gamma a } ) ( 1 - e ^ { - 4 \\gamma n a } ) } . \\end{align*}"}
{"id": "194.png", "formula": "\\begin{align*} \\tau _ { \\delta , k } ( x ) = \\frac { 1 } { 2 \\varphi _ { \\delta } ( \\| x \\| ^ 2 ) } \\int _ { \\| x \\| ^ 2 } ^ { + \\infty } \\varphi _ { \\delta } ( t ) d t . \\end{align*}"}
{"id": "8819.png", "formula": "\\begin{align*} \\nabla u _ { \\frac { t } { 2 } } ( \\varphi ) = - Q _ t \\nabla V _ t ( Q _ t \\varphi ) \\quad \\nabla ^ 2 u _ { \\frac { t } { 2 } } ( \\varphi ) = - Q _ t \\nabla ^ 2 V _ t ( Q _ t \\varphi ) Q _ t . \\end{align*}"}
{"id": "847.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } R _ j & = \\frac { \\epsilon _ m } { 1 - \\epsilon _ m } , \\\\ \\mathbb { E } R _ j ^ 2 & = \\frac { \\epsilon _ m ^ 2 + \\epsilon _ m } { \\left ( 1 - \\epsilon _ m \\right ) ^ 2 } . \\end{aligned} \\end{align*}"}
{"id": "801.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla \\phi \\ , d \\mu = \\int _ { \\partial \\Omega } \\phi f \\ , d \\nu . \\end{align*}"}
{"id": "136.png", "formula": "\\begin{align*} x ^ i x ^ j = \\frac 1 2 \\left [ y ^ i x ^ j + y ^ j x ^ i \\right ] , \\end{align*}"}
{"id": "154.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\gamma ( f ) ( x ) = - \\langle x ; \\nabla ( f ) ( x ) \\rangle + \\Delta ( f ) ( x ) , \\end{align*}"}
{"id": "7165.png", "formula": "\\begin{align*} ( \\xi ^ \\mu ( y ' ) - \\eta ^ \\mu ( y ' ) ) ( 1 + R ^ \\mu ( x ' ) ) = \\frac { H _ u ^ \\mu ( F ^ \\mu ( x ' ) ) - H _ v ^ \\mu ( F ^ \\mu ( x ' ) ) } { \\sqrt { 1 + | D ' H _ 0 ( x ' , 0 ) | ^ 2 } } , \\end{align*}"}
{"id": "731.png", "formula": "\\begin{align*} K _ { \\alpha \\alpha } ^ { ( \\ell ) } = : \\frac { 1 } { a \\ell } + \\epsilon ^ { ( \\ell ) } , a : = - 6 \\frac { \\sigma _ 3 } { \\sigma _ 1 } > 0 , \\end{align*}"}
{"id": "8794.png", "formula": "\\begin{align*} \\max _ { \\pi \\in \\Omega ' } \\hat { B } \\bigl ( \\psi ( T ) \\bigr ) \\Bigl ( Z \\bigl ( s ( T ) \\bigr ) \\Bigr ) \\leq \\phi \\leq \\min _ { \\pi \\in \\Omega ' } \\hat { B } ( \\psi ) \\bigl ( Z ( s ) \\bigr ) , \\ V _ 1 ^ { - 1 } ( s _ 1 ) \\in ( \\ref { e q : S O S 1 } ) , \\ s _ { 2 1 } \\in [ a _ { 2 0 } , a _ { 2 1 } ] , \\ s _ { i 0 } = \\end{align*}"}
{"id": "1026.png", "formula": "\\begin{align*} \\zeta _ 1 ( x ) = \\begin{cases} 0 & \\R ^ n _ + \\setminus B _ 1 ( 2 e _ 1 ) , \\\\ 1 & \\R ^ n _ - \\setminus \\{ x _ 1 > - 1 \\} \\end{cases} \\qquad { \\mbox { a n d } } \\zeta _ 2 ( x ) = \\begin{cases} 0 & \\R ^ n _ + \\setminus B _ 1 ( 2 e _ 1 ) , \\\\ 1 & B _ { 1 / 2 } ( - 2 e _ 1 ) . \\end{cases} \\end{align*}"}
{"id": "5421.png", "formula": "\\begin{align*} \\phi _ { i , \\Delta } ( \\Delta ) = \\Phi _ i ( \\Delta ) . \\end{align*}"}
{"id": "2581.png", "formula": "\\begin{align*} \\partial ^ \\alpha t ^ \\beta f ( t ) = \\sum _ { \\gamma _ 1 \\leq \\alpha } \\sum _ { \\gamma _ 2 \\leq \\beta } \\begin{pmatrix} \\alpha \\\\ \\gamma _ 1 \\end{pmatrix} \\begin{pmatrix} \\beta \\\\ \\gamma _ 2 \\end{pmatrix} \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\ , x ^ { \\gamma _ 2 } ( 2 \\pi i \\omega ) ^ { \\gamma _ 1 } M _ \\omega T _ x ( \\partial ^ { \\alpha - \\gamma _ 1 } t ^ { \\beta - \\gamma _ 2 } g ( t ) ) \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "7875.png", "formula": "\\begin{align*} ( \\nabla F ( u ) ) ^ \\# = ( v _ 1 ( u ) , v _ 2 ( u ) , \\dots , v _ m ( u ) ) , \\end{align*}"}
{"id": "5197.png", "formula": "\\begin{align*} T _ k f ( x ) = \\int _ { | y | > q ^ { k } } f ( x - y ) \\dfrac { \\Omega ( y ) } { | y | } d y , ~ ~ ~ ~ ~ k \\in \\mathbb { Z } , \\end{align*}"}
{"id": "4245.png", "formula": "\\begin{align*} - \\frac { 2 \\tau ^ 4 } { ( 1 + \\tau ^ 2 ) ^ 2 } f ' ( u ) u _ y \\le \\tau u _ { x x x } = { } & - 3 \\tau u _ { x y } \\rho '' - \\tau u _ y \\rho ''' \\\\ = { } & \\frac { 3 \\rho '' } { 2 } ( - H + u _ { x x } + \\tau ^ 2 u _ { y y } ) - \\tau u _ y \\rho ''' . \\end{align*}"}
{"id": "8303.png", "formula": "\\begin{align*} E _ { \\text i } ^ \\pm ( \\kappa _ x ) = \\frac { \\kappa _ 1 \\eta _ 1 } { 2 } \\frac { J _ \\pm ( \\kappa _ x ) } { \\kappa _ { 1 z } } \\end{align*}"}
{"id": "985.png", "formula": "\\begin{align*} U : = \\bigg \\{ y \\in B _ \\rho ( e _ 1 ) u ( y ) > \\frac { u ( a ) } 2 \\bigg \\} . \\end{align*}"}
{"id": "2450.png", "formula": "\\begin{align*} \\min _ { | x | = 1 } x \\cdot ( S ^ T S + I ) x = \\min _ { | x | = 1 } ( x \\cdot S ^ T S x + | x | ^ 2 ) = \\min _ { | x | = 1 } ( | S x | ^ 2 + | x | ^ 2 ) > 1 . \\end{align*}"}
{"id": "5461.png", "formula": "\\begin{align*} \\frac { 1 } { \\inf _ { t > \\tau , x \\in \\Omega } v ^ { q + 1 - \\varepsilon _ 0 / 2 } ( t , x ) } = \\sup _ { t > \\tau , x \\in \\Omega } \\frac { 1 } { v ^ { q + 1 - \\varepsilon _ 0 / 2 } ( t , x ) } \\le \\sup _ { t > \\tau } \\frac { \\big ( \\int _ \\Omega u ^ { - p } \\big ) ^ { \\frac { q + 1 - \\varepsilon _ 0 / 2 } { p } } } { \\big ( \\delta _ 0 | \\Omega | \\big ) ^ { q + 1 - \\varepsilon _ 0 / 2 } } . \\end{align*}"}
{"id": "1993.png", "formula": "\\begin{align*} \\lVert m ^ \\natural f \\rVert _ { B U O } & = \\lVert - M ^ \\natural ( - f ) \\rVert _ { B U O } = \\lVert M ^ \\natural ( - f ) \\rVert _ { B L O } \\\\ & \\leq C ( C _ d ) \\norm { - f } _ { B M O } = C \\norm { f } _ { B M O } . \\end{align*}"}
{"id": "604.png", "formula": "\\begin{align*} ( x , y ) \\ = \\ \\bigg [ \\frac { x } { y + 1 } \\bigg ] \\end{align*}"}
{"id": "277.png", "formula": "\\begin{align*} \\chi ( x , t ) : = \\frac { 1 } { \\sqrt { 1 + t } } \\chi _ { * } \\left ( \\frac { x } { \\sqrt { 1 + t } } \\right ) , x \\in \\R , \\ \\ t \\ge 0 , \\end{align*}"}
{"id": "1952.png", "formula": "\\begin{align*} ( f g ) _ { i _ 1 \\dotsm i _ m } : = f _ { i _ 1 \\dotsm i _ m } g _ 0 + f _ 0 g _ { i _ 1 \\dotsm i _ m } + \\sum _ { j = 1 } ^ { m - 1 } f _ { i _ 1 \\dotsm i _ j } g _ { i _ { j + 1 } \\dotsm i _ m } . \\end{align*}"}
{"id": "4948.png", "formula": "\\begin{align*} \\Omega _ G = \\pi _ 1 ( G ) _ I \\to \\Omega _ { G ' } = \\pi _ 1 ( G ' ) _ I \\to X _ * ( T ) _ I \\to 0 \\end{align*}"}
{"id": "3476.png", "formula": "\\begin{align*} \\zeta _ { M T , 2 } ( s _ 1 , s _ 2 , s _ 3 ) = 2 ^ { - s _ 3 } \\zeta ( s _ 1 + s _ 2 + s _ 3 ) + \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) + \\zeta _ { A V , 2 } ( s _ 2 , s _ 1 , s _ 3 ) , \\end{align*}"}
{"id": "696.png", "formula": "\\begin{align*} J _ { \\theta _ \\mu } z _ { \\alpha } ^ { ( L + 1 ) } = J _ { \\theta _ \\mu } z _ { \\alpha } ^ { ( \\ell _ 0 ) } J _ { z _ { \\alpha } ^ { ( \\ell _ 0 ) } } z _ { \\alpha } ^ { ( L + 1 ) } . \\end{align*}"}
{"id": "3546.png", "formula": "\\begin{align*} \\partial M ( t ) = \\Gamma ( t ) \\end{align*}"}
{"id": "7795.png", "formula": "\\begin{align*} T ( X ) = \\big ( T _ 1 ( X _ 1 ) , T _ 2 ( X _ { 1 : 2 } ) , \\ldots , T _ N ( X ) \\big ) . \\end{align*}"}
{"id": "767.png", "formula": "\\begin{align*} \\mathfrak { W } _ \\alpha & : \\ell ^ \\infty ( \\N ) \\to C ^ \\alpha ( S ^ 1 ) , \\\\ [ \\mathfrak { W } _ \\alpha & ( c ) ] ( z ) : = \\sum _ { k = 0 } ^ \\infty c _ k 2 ^ { - \\alpha k } z ^ { 2 ^ k } . \\end{align*}"}
{"id": "5665.png", "formula": "\\begin{align*} \\theta ^ { - 1 } \\prescript { \\tau \\sigma \\tau ^ { - 1 } } { } N _ v ^ \\ast \\theta \\subseteq \\prescript { \\sigma } { } N _ l ^ \\circ \\iff \\sigma = \\varepsilon \\ ; \\ ; ( 1 \\ , 3 ) . \\end{align*}"}
{"id": "4166.png", "formula": "\\begin{align*} \\omega _ { p } ^ { p } ( A ) \\leq \\underset { i , j = 1 } { \\overset { 2 } { \\sum } } \\omega _ { p } ^ { p } ( a _ { i j } ) \\end{align*}"}
{"id": "6297.png", "formula": "\\begin{align*} F _ { \\widehat { K } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } | K } ( \\hat { k } ) & = \\sum _ { k ' = 0 } ^ { K } F _ { \\widehat { K } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } | K ' } ( \\hat { k } ) \\ \\ ! p _ { K ' | K } ( k ' ) , \\end{align*}"}
{"id": "9268.png", "formula": "\\begin{align*} \\tilde { x } ^ L : = \\frac { L x } { \\max _ \\mathbb { R } \\{ \\norm { x } _ X , L \\} } . \\end{align*}"}
{"id": "4677.png", "formula": "\\begin{align*} \\varphi _ { i j } ( t , y ) : = \\psi \\bigg ( \\frac { y - x _ i ( t ) } { x _ { i j } ^ 3 ( t ) } \\bigg ) , \\end{align*}"}
{"id": "309.png", "formula": "\\begin{align*} \\frac { d \\rho _ i } { d t } + d i v _ G ^ { \\theta } ( \\rho v ) = 0 , \\end{align*}"}
{"id": "7820.png", "formula": "\\begin{align*} \\mathbf 1 _ { ( 0 , T ) } \\otimes x _ n \\to \\mathbf 1 _ { ( 0 , T ) } \\otimes x , \\\\ \\mathcal A ( \\mathbf 1 _ { ( 0 , T ) } \\otimes x _ n ) = \\mathbf 1 _ { ( 0 , T ) } \\otimes A x _ n \\to \\mathbf 1 _ { ( 0 , T ) } \\otimes y . \\end{align*}"}
{"id": "416.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\langle A ^ { 0 } U ^ { \\delta } _ { t } + A ^ { i } \\partial _ { i } U ^ { \\delta } + D U ^ { \\delta } - B ^ { i j } \\partial _ { i } \\partial _ { j } U ^ { \\delta } - \\delta \\Lambda \\Delta U ^ { \\delta } , \\phi \\rangle d t = \\int _ { 0 } ^ { T } \\langle f , \\phi \\rangle \\end{align*}"}
{"id": "1485.png", "formula": "\\begin{align*} f ( \\gamma z ) = j ( \\gamma , z ) ^ k f ( z ) . \\end{align*}"}
{"id": "6986.png", "formula": "\\begin{align*} u \\in \\mathbb S _ { \\mathbb X } , \\ , \\nabla g ( \\bar x ) u \\in \\mathcal T _ D ( g ( \\bar x ) ) , \\ , \\nabla g ( \\bar x ) ^ * y ^ * = 0 , \\ , y ^ * \\in \\mathcal N _ D ( g ( \\bar x ) ; \\nabla g ( \\bar x ) u ) \\Longrightarrow y ^ * = 0 \\end{align*}"}
{"id": "7471.png", "formula": "\\begin{align*} \\lambda ^ n : = \\begin{cases} \\mu ^ 0 - \\| v _ 0 \\| ^ 2 = \\mu ^ 0 , & n = 0 , \\\\ \\mu ^ n - \\| ( \\phi ^ { n } - \\phi ^ { n - 1 } ) / \\tau \\| ^ 2 , & n = 1 , 2 , \\ldots , \\end{cases} \\end{align*}"}
{"id": "888.png", "formula": "\\begin{align*} a ( \\widetilde E _ 2 ^ { ( 4 ) } , T ) = a ( \\mathrm { g e n u s } \\ \\Theta ^ { ( 4 ) } ( S ^ { ( p ) } ) , T ) . \\end{align*}"}
{"id": "4448.png", "formula": "\\begin{align*} ( x ) _ { 0 , \\lambda } = 1 , ( x ) _ { n , \\lambda } = x ( x - \\lambda ) \\cdots \\big ( x - ( n - 1 ) \\lambda \\big ) , ( n \\ge 1 ) . \\end{align*}"}
{"id": "8175.png", "formula": "\\begin{align*} M _ { d _ 0 } ( f , H ) = M ( d _ 0 f , H _ { d _ 0 } ) = { 2 \\pi ^ 2 \\over d _ 0 ^ 2 f ^ 2 } \\sum _ { \\delta \\mid d _ 0 f } \\delta \\mu ( d _ 0 f / \\delta ) \\sum _ { h \\in H _ { d _ 0 } } s ( h , \\delta ) . \\end{align*}"}
{"id": "4334.png", "formula": "\\begin{align*} \\tilde \\tau = 2 \\beta ( \\tau ) \\tau , \\end{align*}"}
{"id": "2769.png", "formula": "\\begin{align*} u _ n ( x ) = U ^ 1 ( x - x _ n ^ 1 ) + r _ n ^ 1 ( x ) \\end{align*}"}
{"id": "9440.png", "formula": "\\begin{align*} \\varPhi _ q \\ , d _ q \\ , \\varPsi _ { \\mu , q } W _ q u = \\varPhi _ q \\ , d _ q \\ , \\varPsi _ \\mu B _ q ( w , u ) , \\end{align*}"}
{"id": "893.png", "formula": "\\begin{align*} & \\lim _ { m \\to \\infty } F _ q ( T , q ^ { k _ m - n - 1 } ) = F _ q ( T , q ^ { 2 - n - 1 } ) . \\\\ \\end{align*}"}
{"id": "2201.png", "formula": "\\begin{align*} m _ 0 = \\inf \\limits _ { \\mathcal { N } } J \\leq \\inf \\limits _ { \\mathcal { M } } J = m _ 1 . \\end{align*}"}
{"id": "5359.png", "formula": "\\begin{align*} \\rho _ c ( x ) = \\| \\pi _ c ( x ) - x \\| . \\end{align*}"}
{"id": "3556.png", "formula": "\\begin{align*} r = \\frac { y \\pm \\sqrt { y ^ 2 - 4 a v ^ 2 } } { 2 a v } . \\end{align*}"}
{"id": "7445.png", "formula": "\\begin{align*} \\| \\phi \\| ^ 2 : = \\int _ { \\mathbb { R } ^ d } | \\phi ( \\mathbf { x } ) | ^ 2 \\mathrm { d } \\mathbf { x } = 1 . \\end{align*}"}
{"id": "7803.png", "formula": "\\begin{align*} \\log a : = \\sum _ { m = 1 } ^ { \\infty } ( - 1 ) ^ { m + 1 } \\frac { ( a - 1 ) ^ m } { m } \\end{align*}"}
{"id": "8764.png", "formula": "\\begin{align*} ~ \\begin{aligned} & u _ { i ' j ' } ( x ) \\leq u ^ t _ { i ' j ' } ( x ) \\leq \\min \\bigl \\{ m ^ t _ { i ' j ' } , f _ { i ' } ( x ) \\bigr \\} , \\\\ & m ^ { t } _ { i ' \\tau ( i ' , t ) - 1 } \\leq m ^ t _ { i ' j ' } \\leq m ^ { t } _ { i ' \\tau ( i ' , t ) } t < t ' a _ { i ' j ' - 1 } \\leq m ^ { t ' } _ { i ' j ' } \\leq a _ { i ' j ' } . \\end{aligned} \\end{align*}"}
{"id": "6773.png", "formula": "\\begin{align*} & C _ 1 = 1 / \\left [ ( S _ { \\max } - S _ { \\min } ) - \\frac { \\sqrt { \\pi } } { 2 } \\frac { 1 - \\beta } { \\beta } \\alpha \\left ( \\left ( \\frac { S _ { \\max } - K } { \\alpha } \\right ) - \\left ( \\frac { S _ { \\min } - K } { \\alpha } \\right ) \\right ) \\right ] , \\\\ & C _ 2 = \\left [ \\frac { \\sqrt { \\pi } } { 2 } \\frac { 1 - \\beta } { \\beta } \\alpha \\left ( \\frac { S _ { \\min } - K } { \\alpha } \\right ) - S _ { \\min } \\right ] C _ 1 . \\end{align*}"}
{"id": "8752.png", "formula": "\\begin{align*} K : = \\Bigl \\{ \\lambda \\begin{pmatrix} 1 \\\\ y \\end{pmatrix} \\Bigm | y \\in F , \\ \\lambda \\geq 0 \\Bigr \\} , \\end{align*}"}
{"id": "9142.png", "formula": "\\begin{align*} \\Phi ( k , n ) = \\phi \\left ( \\left \\lceil 2 C ( k + 1 ) \\right \\rceil - 1 , \\max \\{ \\theta ( M \\varpi ( k ) + M - 1 ) , n \\} \\right ) \\end{align*}"}
{"id": "3635.png", "formula": "\\begin{align*} A ( x , \\sigma ) & : = \\frac { s _ 1 ( x , \\sigma ) + s _ 2 ( x , \\sigma ) + s _ 3 ( x , \\sigma ) } { \\log ^ { \\frac { 5 - 2 \\sigma } { 2 } } ( x ) \\exp \\left ( \\frac { B _ 2 ( x _ 0 ) ( 5 - 8 \\sigma ) } { 3 } \\sqrt { \\log x } \\right ) } \\end{align*}"}
{"id": "2384.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) ^ 2 + \\sum _ { k = 0 } ^ { N - 1 } \\sin ( 2 \\pi k / N ) ^ 2 = N . \\end{align*}"}
{"id": "922.png", "formula": "\\begin{align*} ( a ^ { + } ) ^ { k } a ^ { k } & = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k , m ) ( a ^ { + } a ) _ { m , \\lambda } \\\\ & = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k , m ) ( \\hat { n } ) _ { m , \\lambda } = ( \\hat { n } ) _ { k } , \\end{align*}"}
{"id": "3981.png", "formula": "\\begin{align*} & \\mathcal B ^ * _ \\rho \\Phi _ { \\lambda ^ p _ k } = \\xi _ { \\lambda ^ p _ k } ( 1 ) , \\forall k \\geq k _ 0 , \\\\ & \\mathcal B ^ * _ \\rho \\Phi _ { \\lambda ^ h _ k } = \\xi _ { \\lambda ^ h _ k } ( 1 ) , \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "9459.png", "formula": "\\begin{align*} N _ { 1 j } + \\tfrac { B ( j , 1 ) ^ p } { B ( j ) ^ p } - \\tfrac { B ( j - 1 , 1 ) } { B ( j - 1 ) } = 0 . \\end{align*}"}
{"id": "1196.png", "formula": "\\begin{align*} \\dim _ { H } ( \\limsup _ { \\underline { i } \\in \\Lambda ^ * } B ( f _ { \\underline { i } } ( x ) , c _ { \\underline { i } } ^ { \\delta } ) ) = \\frac { \\dim _ H ( K ) } { \\delta } . \\end{align*}"}
{"id": "3175.png", "formula": "\\begin{align*} \\varphi _ R ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } , \\zeta _ { k - 1 } + 1 ) . \\end{align*}"}
{"id": "3923.png", "formula": "\\begin{align*} A _ { r _ j } ( x - a _ j ) \\circ A _ { r _ j } ( x - a _ j ) = - A _ { r _ j } ' ( x - a _ j ) + F _ { r _ j } ( x - a _ j ) \\end{align*}"}
{"id": "8813.png", "formula": "\\begin{align*} \\alpha _ p : = \\begin{cases} ( p - 1 ) ^ { 1 / 2 } \\quad p \\in [ 2 , \\infty ) \\\\ \\frac { \\pi } { 2 } \\quad p \\in [ 1 , 2 ) . \\end{cases} \\end{align*}"}
{"id": "5742.png", "formula": "\\begin{align*} h ( x , t ) = e ^ { \\sqrt { \\lambda } t } \\psi _ \\lambda ( x ) , \\end{align*}"}
{"id": "2744.png", "formula": "\\begin{align*} Q ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , z _ n ) = 0 : \\end{align*}"}
{"id": "1518.png", "formula": "\\begin{align*} x \\phi _ 1 x ^ { \\ast } = y \\phi _ 2 y ^ { \\ast } = \\left [ \\begin{array} { c c c } 0 & 0 & e _ t \\\\ 0 & \\zeta I _ r & 0 \\\\ e _ t ^ { \\ast } & 0 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c c } 0 & 0 & e _ t \\\\ 0 & 0 _ r & 0 \\\\ e _ t ^ { \\ast } & 0 & 0 \\end{array} \\right ] , 0 \\leq t \\leq m . \\end{align*}"}
{"id": "4815.png", "formula": "\\begin{align*} ( \\phi ^ 2 + \\phi * \\phi ) ( a ) & = c ^ 2 ( 1 + J ' ( \\alpha , \\alpha ) ) \\alpha ^ 2 ( a ) + \\bar c ^ 2 ( 1 + J ' ( \\bar \\alpha , \\bar \\alpha ) ) \\bar \\alpha ^ 2 ( a ) + 2 ( 1 + J ( \\alpha , \\bar \\alpha ) ) \\\\ & = c ^ 2 ( 1 + J ' ( \\alpha , \\alpha ) ) \\bar \\alpha ( a ) + \\bar c ^ 2 ( 1 + J ' ( \\bar \\alpha , \\bar \\alpha ) ) \\alpha ( a ) . \\end{align*}"}
{"id": "7720.png", "formula": "\\begin{gather*} \\Box y = 2 \\left ( | \\phi _ t | ^ 2 - | \\phi _ x | ^ 2 \\right ) ( y - 1 ) + a ( x ) y _ t , \\\\ y ( 0 , x ) = 1 \\textrm { a n d } y _ t ( 0 , x ) = 0 , \\ ; \\forall x \\in \\S , \\end{gather*}"}
{"id": "212.png", "formula": "\\begin{align*} A _ \\alpha ( p _ \\alpha f ) ( x ) = d p _ \\alpha ( x ) f ( x ) + \\langle x ; p _ \\alpha ( x ) \\nabla ( f ) ( x ) \\rangle + \\langle x ; f ( x ) \\nabla ( p _ \\alpha ) ( x ) \\rangle + \\sum _ { k = 1 } ^ d \\partial _ k D _ k ^ { \\alpha - 1 } ( p _ \\alpha f ) ( x ) . \\end{align*}"}
{"id": "9368.png", "formula": "\\begin{align*} \\beta _ { n , \\lambda } ( x ) & = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( x ) _ { n - k , \\lambda } \\beta _ { k , \\lambda } \\\\ & = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( x ) _ { n - k , \\lambda } \\sum _ { j = 0 } ^ { k } \\frac { \\lambda ^ { j } ( 1 ) _ { j + 1 , \\frac { 1 } { \\lambda } } } { j + 1 } S _ { 2 , \\lambda } ( k , j ) \\\\ & = \\sum _ { j = 0 } ^ { n } \\frac { \\lambda ^ { j } ( 1 ) _ { j + 1 , \\frac { 1 } { \\lambda } } } { j + 1 } \\sum _ { k = j } ^ { n } \\binom { n } { k } S _ { 2 , \\lambda } ( k , j ) ( x ) _ { n - k , \\lambda } . \\end{align*}"}
{"id": "3211.png", "formula": "\\begin{align*} k _ 1 = \\frac { \\left ( \\mu ^ { ( 2 ) } _ { X _ 1 } \\right ) ^ 2 } { \\mu _ { X _ 1 } ^ { ( 4 ) } - \\left ( \\mu ^ { ( 2 ) } _ { X _ 1 } \\right ) ^ 2 } , \\theta _ 1 = \\frac { \\mu _ { X _ 1 } ^ { ( 4 ) } - \\left ( \\mu ^ { ( 2 ) } _ { X _ 1 } \\right ) ^ 2 } { \\mu ^ { ( 2 ) } _ { X _ 1 } } , \\end{align*}"}
{"id": "423.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { T } : X _ { T _ { 0 } } ^ { s } ( g _ { 2 } , M ) & \\rightarrow X _ { T _ { 0 } } ^ { s } ( g _ { 2 } , M ) \\\\ U & \\mapsto V \\end{aligned} \\end{align*}"}
{"id": "2545.png", "formula": "\\begin{align*} \\mathbf { a } _ M ( ( z , \\tau ) ( z ' , \\tau ' ) ) = \\mathbf { a } _ M ( z + z ' , \\tau + \\tau ' + \\tfrac { 1 } { 2 } \\sigma ( z , z ' ) ) = ( M ( z + z ' ) , \\tau + \\tau ' + \\tfrac { 1 } { 2 } \\sigma ( z , z ' ) ) . \\end{align*}"}
{"id": "7365.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { \\gamma } ^ \\ast ( s ) = - \\pi R ^ 2 & \\ 0 < s < \\tilde { t } \\\\ \\gamma ^ \\ast ( 0 ) = r , & \\end{array} \\right . \\quad \\left \\{ \\begin{array} { l l } \\dot { \\gamma } ^ \\ast ( s ) = - \\pi \\gamma ^ \\ast ( s ) ^ 2 & \\ \\tilde { t } < s < t , \\\\ \\gamma ^ \\ast ( \\tilde { t } ) = R . & \\end{array} \\right . \\end{align*}"}
{"id": "3285.png", "formula": "\\begin{align*} \\mu _ T ( \\{ \\lambda : | \\lambda | \\leq r \\} ) = \\frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + r ^ 2 } , \\end{align*}"}
{"id": "9300.png", "formula": "\\begin{align*} \\Sigma = \\bigsqcup _ { 1 \\leq i \\leq r } \\tilde { p } _ i , \\end{align*}"}
{"id": "4602.png", "formula": "\\begin{align*} \\varphi ( P , G ) = \\frac { m ( B \\cap P ) } { m ( B ) } , \\end{align*}"}
{"id": "4621.png", "formula": "\\begin{gather*} \\gamma _ { X ^ * } = ( \\gamma _ X ^ { - 1 } ) ^ * . \\end{gather*}"}
{"id": "2142.png", "formula": "\\begin{align*} E _ { n , k } ( X _ { n , k } ^ { \\mathcal { T } } | W _ { \\mathcal { T } ( n , k ; R _ n ) } ) = E _ { n , k } ( Y ^ { ( n , k ) } _ k | W _ { \\mathcal { T } ( n , k ; R _ n ) } ) \\le \\frac { E _ { n , k } Y ^ { ( n , k ) } _ k } { P _ { n , k } ( W _ { \\mathcal { T } ( n , k ; R _ n ) } ) } , \\end{align*}"}
{"id": "5440.png", "formula": "\\begin{align*} T _ { \\max } ( s , u _ 0 ) = \\infty \\end{align*}"}
{"id": "2969.png", "formula": "\\begin{align*} \\sum _ { B \\subset S _ 8 } E ^ { ( S _ 4 , B ) } _ { n , \\ell } & = \\frac { \\ell _ 1 \\left ( n - \\ell _ 4 \\right ) \\left ( \\sum _ { j = 0 } ^ 6 n ^ j p _ j \\left ( \\ell _ 1 , \\ldots , \\ell _ 4 \\right ) \\right ) } { n ^ 8 ( n - 1 ) ( n - 2 ) ( n - 3 ) ( n - 4 ) ( n - 5 ) ( n - 6 ) ( n - 7 ) } , \\end{align*}"}
{"id": "5708.png", "formula": "\\begin{align*} 2 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) - n . \\end{align*}"}
{"id": "148.png", "formula": "\\begin{align*} P ^ { \\nu _ \\alpha } _ t ( f ) ( x ) = \\int _ { \\mathbb { R } ^ d } f ( x e ^ { - t } + ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { 1 } { \\alpha } } y ) \\mu _ \\alpha ( d y ) . \\end{align*}"}
{"id": "4267.png", "formula": "\\begin{align*} v _ t + a _ n ( t , x ) \\cdot v _ x ~ ~ = 0 , v ( 0 , \\cdot ) ~ = ~ \\bar { v } ~ \\in ~ H ^ 2 \\bigl ( \\R \\backslash [ - \\delta _ 0 \\tau , \\delta _ 0 \\tau ] \\bigr ) , \\end{align*}"}
{"id": "6656.png", "formula": "\\begin{align*} D _ { 2 , m } = \\tau _ { A \\smallsetminus \\{ \\alpha \\} } ( p ^ { m } ) \\tau _ { B \\smallsetminus \\{ \\beta \\} \\cup \\{ - \\alpha \\} } ( p ^ m ) . \\end{align*}"}
{"id": "3693.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } t ^ { \\frac { \\beta } { \\alpha } } \\| B ( t ) \\| _ { H ^ { \\frac 5 2 - \\alpha + \\beta } ( \\mathbb S ^ 1 ) } = 0 \\ \\ \\forall \\ \\ \\beta > 0 . \\end{align*}"}
{"id": "2905.png", "formula": "\\begin{align*} - \\partial _ r ^ { k + 2 } Q - \\partial _ r ^ k \\left ( \\frac { N - 1 } { r } \\partial _ r Q \\right ) + \\partial _ r ^ k Q - \\partial _ r ^ k \\left [ \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 1 } \\right ] = 0 , \\end{align*}"}
{"id": "8697.png", "formula": "\\begin{align*} v _ { i j } = ( a _ { i 0 } , \\ldots , a _ { i j - 1 } , a _ { i j } , \\ldots , a _ { i j } ) j = 0 , \\ldots , n , \\end{align*}"}
{"id": "889.png", "formula": "\\begin{align*} \\widetilde E _ 2 ^ { ( n ) } = \\mathrm { g e n u s } \\ \\Theta ^ { ( n ) } ( S ^ { ( p ) } ) . \\end{align*}"}
{"id": "1293.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda _ { ( n , \\infty ) } ( M , \\Gamma ) | } { | \\Lambda ( M , \\Gamma ) | } = 0 \\end{align*}"}
{"id": "8404.png", "formula": "\\begin{align*} \\partial _ t u + ( \\mathbf { v } _ \\varepsilon \\nabla c _ { \\varepsilon } - \\mathbf { \\tilde { v } } _ A \\nabla c _ A ) + \\frac { \\mu _ { \\varepsilon } - \\mu _ A } { \\varepsilon } = \\mathbf { w } | _ \\Gamma \\ , \\nabla c _ { A , 0 } - \\mathbf { w } | _ \\Gamma \\ , \\mathbf { Q } - \\mathcal { C } . \\end{align*}"}
{"id": "9009.png", "formula": "\\begin{align*} X ^ { m } [ T ] = \\lim _ { \\ell \\to \\infty } X ^ { B _ { \\ell } } _ { B _ m } [ T ] = X ^ { B _ { M _ m } } _ { B _ m } [ T ] . \\end{align*}"}
{"id": "8802.png", "formula": "\\begin{align*} \\bar { s } _ { i j } = a _ { i 0 } z _ { i 0 } + \\sum _ { k = 1 } ^ j ( a _ { i k } - a _ { i k - 1 } ) z _ { i k } j = 0 , \\ldots , n . \\end{align*}"}
{"id": "987.png", "formula": "\\begin{align*} v ( x ) & : = \\left ( 1 - \\frac \\theta 2 \\right ) ^ { - n - 2 } u ( a ) \\zeta ( x ) - u ( x ) x \\in \\R ^ n . \\end{align*}"}
{"id": "193.png", "formula": "\\begin{align*} \\varphi _ \\delta ( t ) = \\exp \\left ( - t ^ { \\frac { \\delta } { 2 } } \\right ) , t \\in ( 0 , + \\infty ) , \\end{align*}"}
{"id": "7915.png", "formula": "\\begin{align*} \\sum _ { l } ( \\partial _ { t _ i } \\varphi ^ l ) ( \\partial _ { t _ j } \\varphi ^ l ) = \\delta _ { i j } , \\end{align*}"}
{"id": "5131.png", "formula": "\\begin{align*} \\frac { 2 a \\ , e ^ { 2 \\gamma a } \\left ( e ^ { 2 n \\gamma a } - n \\ , e ^ { n \\gamma a } - 1 \\right ) } { \\left ( e ^ { a \\gamma n } - 1 \\right ) ^ 3 } = 0 \\Longleftrightarrow e ^ { 2 n \\gamma a } - n \\ , e ^ { n \\gamma a } - 1 = 0 . \\end{align*}"}
{"id": "2947.png", "formula": "\\begin{align*} \\forall ~ \\ell \\in \\{ 1 , \\dots , k - 1 \\} , i \\in \\{ 1 , \\ldots , 4 \\} ~ \\exists ~ ( \\ell ' , j ) \\neq ( \\ell , i ) ~ p _ { \\ell , i } = p _ { \\ell ' , j } . \\end{align*}"}
{"id": "204.png", "formula": "\\begin{align*} \\mathcal { F } ( T ^ \\alpha _ t ( g ) ) ( \\xi ) = \\mathcal { F } ( g ) ( e ^ { - t } \\xi ) \\frac { \\varphi _ \\alpha ( \\xi ) } { \\varphi _ \\alpha ( e ^ { - t } \\xi ) } . \\end{align*}"}
{"id": "8865.png", "formula": "\\begin{align*} \\check H ^ q _ { c t } ( \\Z ^ n ; A ) = \\begin{cases} A \\oplus A & n = 1 , q = 0 \\\\ A & n \\not = 1 , q = 0 \\vee q = n - 1 \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "3174.png", "formula": "\\begin{align*} W _ k , \\zeta _ k = \\varphi _ R ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) \\in \\mathbb { R } ^ { n \\times n _ k } \\times \\mathfrak { Z } . \\end{align*}"}
{"id": "8343.png", "formula": "\\begin{align*} \\P ( w \\ge r ) = \\P ( \\log ( 1 + \\gamma h ) \\ge r ) = \\P ( \\gamma h \\ge e ^ r - 1 ) = \\Lambda ( e ^ r - 1 ) . \\end{align*}"}
{"id": "9356.png", "formula": "\\begin{align*} - \\frac { \\log _ { \\lambda } ( 1 - t ) } { ( 1 - t ) ^ { r } } = \\sum _ { n = 1 } ^ { \\infty } H _ { n , \\lambda } ^ { ( r ) } t ^ { n } , ( r \\ge 1 ) , H _ { 0 , \\lambda } ^ { ( r ) } = 0 . \\end{align*}"}
{"id": "2135.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( \\mathcal { S } ( n , k ; M _ n ) ) = k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\sum _ { s = 1 } ^ \\infty ( 1 - c ) ^ { k ( s + 1 ) } \\frac 1 { k ( s + 1 ) - l } + \\\\ & ( 1 - c ) ^ k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l ( \\frac c { 1 - c } ) ^ l , \\ \\ M _ n \\sim c n , \\ c \\in ( 0 , 1 ) . \\end{aligned} \\end{align*}"}
{"id": "9403.png", "formula": "\\begin{align*} \\tau _ t ( a _ n \\cdots a _ 1 v b _ 1 \\cdots b _ m ) = t \\cdot \\tau ' ( a _ n \\cdots a _ 1 v b _ 1 \\cdots b _ m ) + o ( t ) . \\end{align*}"}
{"id": "9187.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u - P _ m \\| ^ * _ { L ^ 2 ( B _ { \\eta ^ m } ) } \\leq \\eta ^ { 2 m } , \\\\ & \\Delta P _ m = f _ { B _ { \\eta ^ m } } , \\\\ & | ( P _ m - P _ { m - 1 } ) ( 0 ) | + \\eta ^ { m - 1 } | D ( P _ m - P _ { m - 1 } ) ( 0 ) | + \\eta ^ { 2 ( m - 1 ) } | D ^ 2 ( P _ m - P _ { m - 1 } ) ( 0 ) | \\leq \\bar { C } \\eta ^ { 2 ( m - 1 ) } , \\end{aligned} \\end{align*}"}
{"id": "9277.png", "formula": "\\begin{align*} \\forall x , y \\in \\mathrm { d o m } A ( x = y \\rightarrow H ( A x , A y ) = 0 ) . \\end{align*}"}
{"id": "3358.png", "formula": "\\begin{align*} C _ { T } ^ { 2 n - 1 } ( V , L ) = \\begin{cases} C ^ { 2 n - 1 } ( V , L ) , n \\geq 1 , \\\\ L \\wedge L , \\quad \\ ; \\ ; \\ ; n = 0 . \\end{cases} \\end{align*}"}
{"id": "471.png", "formula": "\\begin{align*} \\vartheta _ { \\mathbf { d } } : \\mathbb { C } [ x _ { \\alpha } : \\ , \\alpha \\in [ n ] ] \\longrightarrow \\mathbb { C } ( \\mathbf { t } ) , \\quad \\vartheta _ { \\mathbf { d } } ( x _ { \\alpha } ) : = \\mathbf { d } ( \\mathbf { t } ) _ { \\alpha } , \\ , \\alpha \\in [ n ] \\end{align*}"}
{"id": "2582.png", "formula": "\\begin{align*} P ( x , \\omega ) & = \\sum _ { \\gamma _ 1 \\leq \\alpha } \\sum _ { \\gamma _ 2 \\leq \\beta } \\begin{pmatrix} \\alpha \\\\ \\gamma _ 1 \\end{pmatrix} \\begin{pmatrix} \\beta \\\\ \\gamma _ 2 \\end{pmatrix} | x ^ { \\gamma _ 2 } | \\ , | ( 2 \\pi i \\omega ) ^ { \\gamma _ 1 } | \\\\ & = \\prod _ { k = 1 } ^ d ( 1 + | x _ k | ) ^ { \\beta _ k } ( 1 + 2 \\pi | \\omega _ k | ) ^ { \\alpha _ k } . \\end{align*}"}
{"id": "6072.png", "formula": "\\begin{align*} K ( p ) > 0 \\quad \\forall p \\in \\overline { \\Omega } \\setminus \\{ p _ 1 , \\ , \\cdots \\ , , p _ n \\} \\ \\ \\mathrm { a n d } \\qquad \\\\ \\mathrm { o r d e r } _ { p _ j } ( K ) = 2 \\ , \\mathrm { o r d e r } _ { p _ j } ( e ) = 2 \\ , \\mathrm { o r d e r } _ { p _ j } ( g ) \\ \\ \\forall j \\in \\{ 1 , \\cdots , n \\} \\ , , \\end{align*}"}
{"id": "3145.png", "formula": "\\begin{align*} \\partial _ z ( \\kappa | \\partial _ t \\eta ^ L | ^ 2 ) = 0 \\quad x _ 0 \\in \\ell , \\end{align*}"}
{"id": "2147.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , j , l } ) = \\frac k { k ( n - j + 1 ) - l } , \\ j \\in [ n - 1 ] . \\end{align*}"}
{"id": "8240.png", "formula": "\\begin{align*} \\partial _ { h _ 1 , \\dots , h _ { 2 s } , k } \\phi ( x ) = \\partial _ { h _ 1 , \\dots , h _ { 2 s } , k ' } \\phi ( x ) = \\partial _ { h _ 1 , \\dots , h _ { 2 s } , k } \\phi ( x + k ' ) & = \\partial _ { h _ 1 , \\dots , h _ { 2 s } , k ' } \\phi ( x + k ) \\\\ & = v _ { 1 , 0 , \\mathbf { h } } - v _ { 1 , 1 , \\mathbf { h } } - ( v _ { 1 , 0 , \\mathbf { h } } - v _ { 1 , 1 , \\mathbf { h } } ) \\\\ & = 0 \\end{align*}"}
{"id": "1206.png", "formula": "\\begin{align*} { m \\left ( \\bigcup _ { r > 0 } A ( r ) \\right ) = m ( A ) . } \\end{align*}"}
{"id": "7383.png", "formula": "\\begin{align*} D _ { 2 , q } : = | \\nabla \\varphi ( x _ q , t _ q ) | \\left ( m ( \\{ u ( \\cdot , t _ q ) < u ( z _ q , t _ q ) \\} ) - m ( \\{ u _ { \\star , \\lambda } ( \\cdot , t _ 0 ) < u _ { \\star , \\lambda } ( x _ 0 , t _ 0 ) \\} ) \\right ) . \\end{align*}"}
{"id": "4530.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } d _ k ( n + a ( n ) ) = x Q _ k ( \\log x ) + O ( x ^ { 3 / 4 + \\varepsilon } ) , \\end{align*}"}
{"id": "7817.png", "formula": "\\begin{align*} \\Psi ' = ( \\gamma - 1 - ( k - j ) ) \\Phi _ { \\gamma - 1 - ( k - j ) - 1 , j } - \\Phi _ { \\gamma - 1 - ( k - j ) , j + 1 } . \\end{align*}"}
{"id": "201.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } \\dfrac { e ^ { - t } } { \\sqrt { 1 - e ^ { - 2 t } } } d t = \\frac { \\pi } { 2 } . \\end{align*}"}
{"id": "6627.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 ^ * ( h , k ) = J _ { 1 1 } + J _ { 2 1 } + J _ { 2 3 } + J _ { 3 1 } + J _ { 3 3 } + O \\big ( X ^ { - \\frac { 1 } { 2 } + \\varepsilon } Q ^ { \\frac { 5 } { 2 } } ( h k ) ^ { \\varepsilon } \\big ) + O \\big ( X ^ { \\varepsilon } Q ^ { \\frac { 3 } { 2 } + \\varepsilon } ( h k ) ^ { \\varepsilon } \\big ) , \\end{align*}"}
{"id": "589.png", "formula": "\\begin{align*} I \\bigg ( X , \\frac { P } { Q } \\bigg ) \\ = \\ I \\bigg ( X _ + , \\frac { P } { Q } \\bigg ) - I \\bigg ( X _ - , - \\frac { P } { Q } \\bigg ) . \\end{align*}"}
{"id": "4416.png", "formula": "\\begin{align*} R e ( z ^ * A w ^ 0 ) = c _ 2 \\ ; \\ ; \\forall z \\in S ^ { m } _ \\alpha \\Rightarrow R e ( ( z ^ 0 ) ^ * A w ^ 0 ) = c _ 2 \\end{align*}"}
{"id": "9355.png", "formula": "\\begin{align*} - \\frac { \\log _ { \\lambda } ( 1 - t ) } { 1 - t } = \\sum _ { n = 1 } ^ { \\infty } H _ { n , \\lambda } t ^ { n } , ( \\mathrm { s e e } \\ [ 9 ] ) . \\end{align*}"}
{"id": "8454.png", "formula": "\\begin{align*} L ' _ { b } \\wedge L ' _ { p c } = L ' _ { i u b } . \\end{align*}"}
{"id": "969.png", "formula": "\\begin{align*} \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } & = \\int _ 0 ^ \\infty \\frac 1 { ( z _ 1 + 1 ) ^ { 1 + 2 s } } \\left ( \\int _ { \\R ^ { n - 1 } } \\frac { \\dd \\tilde z ' } { \\big ( 1 + \\vert \\tilde z ' \\vert ^ 2 \\big ) ^ { \\frac { n + 2 s } 2 } } \\right ) \\dd z _ 1 . \\end{align*}"}
{"id": "5474.png", "formula": "\\begin{align*} \\mathcal { T } ( T ) u _ 0 = u ( T , \\cdot ; 0 , u _ 0 ) \\forall \\ , u _ 0 \\in \\mathcal { E } . \\end{align*}"}
{"id": "4983.png", "formula": "\\begin{align*} \\iota ( \\eta , \\xi ) ( x , y ) = \\Lambda \\left ( \\begin{pmatrix} ( \\eta ^ { r _ 0 } \\circ \\xi ) ^ { r _ 1 } \\circ \\eta ( x ) \\\\ ( \\eta ^ { r _ 0 } \\circ \\xi ) ^ { r _ 1 - 1 } \\circ \\eta ( x ) \\end{pmatrix} , \\begin{pmatrix} \\eta ^ { r _ 0 } \\circ \\xi ( x ) \\\\ x \\end{pmatrix} \\right ) . \\end{align*}"}
{"id": "6921.png", "formula": "\\begin{align*} \\mathbf { w } _ 2 = \\sqrt { \\frac { 1 } { N _ s } } \\textbf { a } _ 2 ^ * ( \\theta _ d ) . \\end{align*}"}
{"id": "4698.png", "formula": "\\begin{align*} & \\partial _ y \\Bigg [ \\bigg ( \\bigg | \\sum _ { k = 1 } ^ n \\sigma _ k \\R _ k \\bigg | ^ { p - 1 } - \\sum _ { k = 1 } ^ n \\R _ k ^ { p - 1 } \\bigg ) \\eta _ i \\frac { \\tau _ i ( A _ { i j , 0 } ) } { x _ { i j } ^ 2 } \\Bigg ] \\\\ & = \\partial _ y \\Bigg [ \\bigg ( \\sum ^ n _ { \\substack { k = 1 , \\\\ k \\not = i } } \\frac { ( p - 1 ) \\kappa _ 0 \\sigma _ k \\tau _ i ( Q ^ { p - 2 } A _ { i j , 0 } ) } { x ^ 2 _ { i k } x ^ 2 _ { i j } } \\bigg ) \\eta _ i \\Bigg ] + O ( \\Gamma ) = \\frac { \\partial _ y ( \\tau _ i f _ { i j , 4 } ) } { x ^ 2 _ { i j } } + O ( \\Gamma ) , \\end{align*}"}
{"id": "1742.png", "formula": "\\begin{align*} \\begin{array} { c } X _ { p _ 1 } ( \\Omega ) = \\left \\{ f : \\Omega \\rightarrow \\R : \\ ; \\left \\| \\frac { \\nabla ^ r f } { g } \\right \\| _ { L _ { p _ 1 } ( \\Omega ) } < \\infty \\right \\} , \\\\ \\| f \\| _ { X _ { p _ 1 } ( \\Omega ) } = \\left \\| \\frac { \\nabla ^ r f } { g } \\right \\| _ { L _ { p _ 1 } ( \\Omega ) } . \\end{array} \\end{align*}"}
{"id": "5313.png", "formula": "\\begin{align*} \\pi ( a ) ( b \\otimes \\chi ) = a b \\otimes \\chi , \\pi ( \\omega ) ( b \\otimes \\chi ) = b _ { ( 2 ) } \\otimes \\omega ( b _ { ( 3 ) } - S ^ { - 1 } ( b _ { ( 1 ) } ) ) \\chi , \\end{align*}"}
{"id": "5689.png", "formula": "\\begin{align*} A ( \\alpha ) = \\sum m _ { i } A ( \\alpha _ { i } ) = \\sum m _ { i } \\int _ { \\alpha _ { i } } \\lambda . \\end{align*}"}
{"id": "7851.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { n } x _ { 2 i + 1 } e _ { 2 i + 1 } + \\sum _ { i = 0 } ^ { \\infty } x _ { 2 i } e _ { 2 i } , \\end{align*}"}
{"id": "2821.png", "formula": "\\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \\| \\nabla u _ 0 \\| _ 2 > \\| \\nabla Q \\| _ 2 , \\end{align*}"}
{"id": "6102.png", "formula": "\\begin{align*} J X ' ( \\theta ) = \\lambda H ( \\theta ) X ( \\theta ) , \\end{align*}"}
{"id": "127.png", "formula": "\\begin{align*} 2 e ( x y ) + 4 ( e x ) ( e y ) = x y . \\end{align*}"}
{"id": "241.png", "formula": "\\begin{align*} \\varphi _ \\mu ( r \\theta ) = \\exp \\left ( - \\frac { r ^ 2 } { 2 } \\langle \\theta ; \\Sigma ( \\theta ) \\rangle \\right ) . \\end{align*}"}
{"id": "4070.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) + p ( t ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "5092.png", "formula": "\\begin{align*} c _ { ( k , l ) , ( k ' , l ' ) } = \\langle M _ { l / a } T _ { k / b } \\ , g , \\ , M _ { l ' / a } T _ { k ' / b } \\ , g \\rangle , k , k ' , l , l ' \\in \\Z . \\end{align*}"}
{"id": "8608.png", "formula": "\\begin{align*} & f ^ { \\# } ( t , k ) = f ^ { \\# } ( 0 , k ) \\\\ & \\pm i \\int _ { 0 } ^ { t } \\iiint e ^ { i s ( - k ^ 2 + \\ell ^ 2 - m ^ 2 + n ^ 2 ) } f ^ { \\# } ( s , \\ell ) \\overline { f ^ { \\# } ( s , m ) } f ^ { \\# } ( s , n ) \\mu ^ { \\# } ( k , \\ell , m , n ) \\ , d n d m d \\ell d s \\end{align*}"}
{"id": "9093.png", "formula": "\\begin{align*} \\alpha = 1 + \\alpha _ 1 t + \\alpha _ { 2 } t ^ { 2 } + \\cdots + \\alpha _ { n - 1 } t ^ { n - 1 } \\end{align*}"}
{"id": "3204.png", "formula": "\\begin{align*} { P _ { H } ( P _ { i n } ) = \\frac { P _ { t h } \\left ( \\frac { 1 } { 1 + e ^ { - a ( \\rho P _ { i n } - b ) } } - \\frac { 1 } { 1 + e ^ { a b } } \\right ) } { 1 - \\frac { 1 } { 1 + e ^ { a b } } } } , \\end{align*}"}
{"id": "6861.png", "formula": "\\begin{align*} \\lambda _ { n - k + 1 } ( M _ i ^ 2 ) \\leq \\max _ { f \\in V ^ i } \\left \\{ \\frac { \\langle f , M _ i ^ 2 f \\rangle } { \\langle f , f \\rangle } \\right \\} = \\max _ { f \\in V ^ i } \\left \\{ \\frac { \\norm { ( M - \\lambda _ i I ) f } _ 2 ^ 2 } { \\langle f , f \\rangle } \\right \\} \\leq c _ i ^ 2 \\end{align*}"}
{"id": "6088.png", "formula": "\\begin{align*} \\widetilde { \\left ( \\overline { L } Z \\right ) } \\ , \\frac { \\partial \\widetilde { h } ^ { j } } { \\partial \\overline { \\zeta } } = \\frac { \\widetilde { A } } { \\widetilde { C } } \\ , \\widetilde { h } ^ { j } - \\frac { \\widetilde { B } } { \\widetilde { C } } \\ , \\overline { \\widetilde { h } ^ { j } } + \\widetilde { M } , \\end{align*}"}
{"id": "1070.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Phi ^ { \\pm \\infty } _ x + i k \\sigma _ 3 \\Phi ^ { \\pm \\infty } = Q ^ { \\pm \\infty } \\Phi ^ { \\pm \\infty } , \\\\ & \\Phi ^ { \\pm \\infty } _ t + 4 i k ^ 3 \\sigma _ 3 \\Phi ^ { \\pm \\infty } = V ^ { \\pm \\infty } \\Phi ^ { \\pm \\infty } , \\end{aligned} \\right . \\end{align*}"}
{"id": "6452.png", "formula": "\\begin{align*} \\lambda _ h = \\lambda _ 0 + { \\lambda _ 0 } ^ 2 \\large \\left \\langle ( T _ 0 - T _ h ( \\lambda _ 0 ) ) u _ 0 , u _ 0 \\large \\right \\rangle + \\mathcal { O } ( h ^ 2 ) \\end{align*}"}
{"id": "6007.png", "formula": "\\begin{align*} f ( z ) & : = \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; 0 ) = \\frac { 1 } { l _ { \\pi _ { \\lambda , \\beta } } ( \\log ( 1 + z ) ) } = \\frac { 1 } { E _ { \\beta } ( \\lambda z ) } , \\\\ g ( z ) & : = \\mathrm { \\mathrm { e } } ^ { x \\log ( 1 + z ) } . \\end{align*}"}
{"id": "7842.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { 3 j - 1 } \\beta _ { i } e _ { i } , \\quad ~ \\beta _ { 3 j - 3 } , \\beta _ { 3 j - 2 } \\beta _ { 3 j - 1 } \\neq 0 . \\end{align*}"}
{"id": "2567.png", "formula": "\\begin{align*} \\norm { f } _ m = \\sup \\{ | \\partial ^ \\alpha f ( x ) | \\mid x \\in \\Omega , | \\alpha | \\leq m \\} . \\end{align*}"}
{"id": "4981.png", "formula": "\\begin{align*} | | F | | _ y = \\underset { ( x , y ) \\in W } { \\operatorname { s u p } } | | \\partial _ y F ( x , y ) | | . \\end{align*}"}
{"id": "7438.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { | x | < M n } \\sup _ { s \\in [ 0 , T ] } | [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x - 1 } { n } ) - [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x } { n } ) | \\leq \\limsup _ { M \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\frac { 2 M K } { n } = 0 . \\end{align*}"}
{"id": "7201.png", "formula": "\\begin{align*} R ^ { ( b ) } ( \\xi ) = \\eta ( | \\xi | ^ 2 ) | \\xi | e ^ { - | \\xi | } \\hat { \\psi } _ 0 ( 0 ) + \\eta ( | \\xi | ^ 2 ) \\left ( \\hat { \\phi } ( \\xi ) | \\xi | \\hat { \\psi } _ \\xi ( 0 ) - | \\xi | e ^ { - | \\xi | } \\hat { \\psi } _ 0 ( 0 ) \\right ) . \\end{align*}"}
{"id": "5694.png", "formula": "\\begin{align*} I ( \\alpha , \\beta ) - J _ { 0 } ( \\alpha , \\beta ) = 2 c _ { 1 } ( \\xi | _ { * } , \\tau ) + \\sum _ { i } \\mu _ { \\tau } ( \\alpha _ { i } ^ { m _ { i } } ) - \\sum _ { j } \\mu _ { \\tau } ( \\beta _ { j } ^ { n _ { j } } ) \\end{align*}"}
{"id": "3042.png", "formula": "\\begin{align*} \\gamma ^ \\prime = 3 2 4 - 1 2 \\Theta - 4 \\Delta \\ , ; t = 4 8 0 - 1 2 \\Theta \\ , ; t ^ \\prime = 1 2 0 \\ , . \\end{align*}"}
{"id": "3510.png", "formula": "\\begin{align*} D _ { 1 3 2 } \\ll t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\left ( t _ 3 ^ { - \\sigma _ 3 } + 1 \\right ) \\begin{cases} 1 & ( \\sigma _ 2 > \\frac { 3 } { 2 } ) \\\\ \\log t _ 3 & ( \\sigma _ 2 = \\frac { 3 } { 2 } ) \\\\ t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 2 } & ( \\sigma _ 2 < \\frac { 3 } { 2 } ) \\\\ \\end{cases} \\\\ \\end{align*}"}
{"id": "7299.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ k \\frac { \\rho _ k } { k } Y ^ k & = - \\sum _ { i = 1 } ^ n \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k - 1 } \\frac { ( X _ i Y ) ^ k } { k } = - \\sum _ { i = 1 } ^ n \\log ( 1 + X _ i Y ) \\overset { \\eqref { f u n c l o g } } { = } - \\log \\Bigl ( \\prod _ { i = 1 } ^ n ( 1 + X _ i Y ) \\Bigr ) \\\\ & \\overset { \\eqref { v i e t a 1 } } { = } - \\log \\Bigl ( 1 + \\sum _ { i = 1 } ^ n \\sigma _ i Y ^ i \\Bigr ) = \\sum _ { l = 1 } ^ \\infty \\frac { ( - 1 ) ^ l } { l } \\Bigl ( \\sum _ { i = 1 } ^ n \\sigma _ i Y ^ i \\Bigr ) ^ l . \\end{align*}"}
{"id": "3057.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\pm x _ 2 \\ , , x _ 2 ^ \\prime = x _ 1 \\ , , y _ 1 ^ \\prime = \\pm y _ 2 \\ , , y _ 2 ^ \\prime = y _ 1 \\ , . \\end{align*}"}
{"id": "3092.png", "formula": "\\begin{align*} z _ 1 ^ \\prime = z _ 2 \\ , , z _ 2 ^ \\prime = z _ 3 \\ , , z _ 3 ^ \\prime = z _ 1 \\ , , z _ 4 ^ \\prime = z _ 4 \\ , , z _ 5 ^ \\prime = z _ 5 \\ , ; \\end{align*}"}
{"id": "8275.png", "formula": "\\begin{align*} \\sum _ { p \\leq x , f ( x ) = q } \\mu _ { P } ( p , x ) = \\sum _ { y \\leq q , \\ , g ( y ) = p } \\mu _ { Q } ( y , q ) \\end{align*}"}
{"id": "7052.png", "formula": "\\begin{align*} \\beta _ \\varepsilon ( t , x ) = \\varepsilon \\log u _ \\varepsilon ( t , x ) , \\quad u _ \\varepsilon ( t , x ) = \\exp \\left ( \\frac { \\beta _ \\varepsilon ( t , x ) } { \\varepsilon } \\right ) , \\end{align*}"}
{"id": "3391.png", "formula": "\\begin{align*} [ u , v , w ] _ T = D _ { \\rho } ( T u , T v ) w + \\theta _ { \\rho } ( T v , T w ) u - \\theta _ { \\rho } ( T u , T w ) v . \\end{align*}"}
{"id": "2764.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\limsup _ { n \\to \\infty } \\| r _ n ^ l \\| _ { L ^ s ( \\mathbb { R } ^ N ) } = 0 , \\forall s \\in \\left ( 2 , 2 ^ * \\right ) , 2 ^ * = \\begin{cases} \\frac { 2 N } { N - 2 } , & N \\ge 3 , \\\\ + \\infty , & N = 1 , 2 . \\end{cases} \\end{align*}"}
{"id": "45.png", "formula": "\\begin{align*} J = J _ 0 + \\frac \\partial { \\partial x } \\otimes d ^ c \\left ( \\frac 1 2 \\log ( 1 + \\left | w \\right | ^ 2 ) \\right ) + \\frac \\partial { \\partial y } \\otimes d ^ c \\left ( \\frac 1 2 \\log ( 1 + \\left | w \\right | ^ 2 ) \\right ) \\circ J _ 0 . \\end{align*}"}
{"id": "63.png", "formula": "\\begin{align*} K _ { \\ell } \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } K _ { \\ell , p _ i , j } ( \\pi ^ j ) & ( v = p _ i \\ \\mathrm { f o r } \\ i = k ' + 1 , \\cdots , k ) , \\\\ K _ { \\ell , v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
{"id": "991.png", "formula": "\\begin{align*} C = C ' \\big ( 1 + \\rho ^ { 2 s } \\| c \\| _ { L ^ \\infty ( B _ \\rho ^ + ) } \\big ) \\end{align*}"}
{"id": "2811.png", "formula": "\\begin{align*} | \\alpha ( t ) | = \\frac { 1 } { \\| Q \\| _ 2 ^ 2 } \\Big | \\int Q h _ 1 d x \\Big | + O ( \\widetilde { \\delta } ^ 2 ) . \\end{align*}"}
{"id": "5682.png", "formula": "\\begin{align*} \\mathrm { E C C } ( Y , \\lambda , \\Gamma ) : = \\bigoplus _ { \\alpha : \\mathrm { a d m i s s i b e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } { [ \\alpha ] = \\Gamma } } \\mathbb { Z } _ { 2 } \\langle \\alpha \\rangle . \\end{align*}"}
{"id": "4932.png", "formula": "\\begin{align*} \\eta _ n ( B ) = \\frac { 1 } { \\mu ( A ' _ n ) } \\mu ( B \\cap A ' _ n ) \\le \\frac { 1 } { \\mu ( A ' _ n ) } \\mu \\Big ( \\bigcup _ { m \\in \\mathcal { I } _ n \\atop B \\cap B _ m \\ne \\varnothing } B _ m \\Big ) \\le \\frac { \\mu ( ( 1 + \\theta ) B ) } { \\mu ( A ' _ n ) } < \\frac { C ^ 2 ( 1 + \\theta ) ^ s \\mu ( B ) } { C ^ { - 2 } 5 ^ { - s } ( 1 - 1 / 2 n ) } , \\end{align*}"}
{"id": "6914.png", "formula": "\\begin{align*} \\textbf { w } _ m = \\left [ e ^ { j \\psi _ { 1 m } } , e ^ { j \\psi _ { 2 m } } , \\ldots , e ^ { j \\psi _ { N _ s m } } \\right ] ^ T . \\end{align*}"}
{"id": "2847.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { T } \\int _ { t _ 0 ( \\varepsilon ) } ^ T ( 8 s _ c ( p - 1 ) \\delta ( t ) + A _ R ( u ( t ) ) d t \\Big | = \\Big | \\frac { 1 } { T } \\int _ { t _ 0 ( \\varepsilon ) } ^ T \\ddot { y } _ R ( t ) d t \\Big | \\le \\frac { C R } { T } , \\end{align*}"}
{"id": "8880.png", "formula": "\\begin{align*} h _ t : I _ 0 & \\to I _ 0 \\\\ ( x , y ) & \\mapsto \\begin{cases} ( x + t , y - t ) & t < y \\\\ ( x + y , 0 ) & t \\ge y . \\end{cases} \\end{align*}"}
{"id": "783.png", "formula": "\\begin{align*} \\alpha ' = a _ 0 + a _ 1 \\alpha + \\cdots + a _ { d - 1 } \\alpha ^ { d - 1 } \\end{align*}"}
{"id": "4229.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathbf { h } _ * , \\mathbf { g } _ * ) \\leq \\liminf _ { N \\to \\infty } \\mathcal { F } _ N ( \\mathbf { h } _ N , \\mathbf { g } _ N ) \\leq \\limsup _ { N \\to \\infty } \\mathcal { F } _ N ( \\mathbf { h } _ N , \\mathbf { g } _ N ) \\leq \\lim _ { N \\to \\infty } \\mathcal { F } _ N ( \\mathbf { h } , \\mathbf { g } ) = \\mathcal { F } ( \\mathbf { h } , \\mathbf { g } ) . \\end{align*}"}
{"id": "3787.png", "formula": "\\begin{align*} \\int _ { X _ E } r _ l ( V ) \\begin{pmatrix} g & 0 \\\\ 0 & 1 \\end{pmatrix} r _ l ( \\mathcal { S } ) ( g ) \\ , d g = \\int _ { X _ F } r _ l ( V ) \\begin{pmatrix} g & 0 \\\\ 0 & 1 \\end{pmatrix} r _ l ( \\mathcal { S } ) ( g ) \\ , d g \\end{align*}"}
{"id": "1461.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\rho ( \\alpha z ) \\\\ u _ { 0 2 } \\end{array} \\right ] = R \\left [ \\begin{array} { c } \\alpha z \\\\ u _ { 0 1 } \\end{array} \\right ] \\mu ( \\alpha z ) ^ { - 1 } = R \\alpha \\left [ \\begin{array} { c } z \\\\ u _ { 0 1 } \\end{array} \\right ] \\lambda ( \\alpha , z ) ^ { - 1 } \\mu ( \\alpha z ) ^ { - 1 } = \\end{align*}"}
{"id": "1349.png", "formula": "\\begin{align*} \\frac { y _ j } { t ^ { d + 1 } } \\partial _ i \\eta \\left ( \\frac { y } { t } \\right ) = \\partial _ i \\left ( \\frac { y _ j } { t ^ d } \\eta \\left ( \\frac { y } { t } \\right ) \\right ) , \\end{align*}"}
{"id": "8192.png", "formula": "\\begin{align*} s ( a , b , f ) = \\frac { f - 1 } { 1 2 f } , \\ S ( H , f ) = \\frac { f - 1 } { 1 2 } \\hbox { a n d } N ( f , H ) = - 1 + 1 2 S ( H , f ) = - 1 . \\end{align*}"}
{"id": "6021.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { U } ( \\widetilde { \\mathcal { O } } f ) & = \\widetilde { U } \\Big [ \\Big ( \\frac { d ^ 2 } { d x ^ 2 } + 2 x \\frac { d } { d x } \\Big ) f \\Big ] \\\\ & = \\Big ( \\frac { d } { d x } + x \\Big ) \\Big ( \\frac { d } { d x } - x \\Big ) \\widetilde { U } f \\\\ & = - \\mathcal { H } ( \\widetilde { U } f ) - \\widetilde { U } f . \\end{aligned} \\end{align*}"}
{"id": "3761.png", "formula": "\\begin{align*} d _ { \\beta } d ^ c _ { \\beta } F & = - \\frac { 1 } { 2 } \\Big ( \\Delta _ { g _ 0 } u + \\big \\langle 2 t _ 0 \\alpha _ 0 - \\theta _ { g _ 0 } , d u \\big \\rangle _ { g _ 0 } + t _ 0 ( 1 - t _ 0 ) | \\alpha _ 0 | _ { g _ 0 } ^ 2 u \\Big ) e ^ { t _ 0 f } \\ , F _ 0 \\wedge F _ 0 . \\end{align*}"}
{"id": "7275.png", "formula": "\\begin{align*} \\boxed { ( 1 + \\alpha ) ^ c = \\sum _ { k = 0 } ^ \\infty \\binom { c } { k } \\alpha ^ k . } \\end{align*}"}
{"id": "5838.png", "formula": "\\begin{align*} \\mathcal { U } _ L = \\left \\{ \\exists \\ , n \\in \\N , n \\geq L \\colon | X _ n - n v | > \\tfrac 1 2 \\varepsilon n \\right \\} \\end{align*}"}
{"id": "743.png", "formula": "\\begin{align*} \\sum _ { A , B } \\xi _ { A } \\xi _ { B } \\Delta _ { A B } ^ { ( \\ell + 1 ) } = \\sum _ { Q } \\Delta _ { Q } ^ { ( \\ell + 1 ) } \\sum _ { Q ^ { ( 1 ) } + \\cdots + Q ^ { ( n _ { \\ell + 1 } ) } = Q } \\prod _ { i = 1 } ^ { n _ { \\ell + 1 } } \\binom { q _ i } { Q ^ { ( i ) } } \\xi _ { i ; Q ^ { ( i ) } } , \\end{align*}"}
{"id": "8440.png", "formula": "\\begin{align*} D _ { s } X _ { t } ^ { x ; l ^ \\epsilon } = \\nabla _ { x } X _ { t } ^ { x ; l ^ \\epsilon } ( \\nabla _ { x } X _ { \\gamma ^ { \\epsilon } _ { s } } ^ { x ; l ^ \\epsilon } ) ^ { - 1 } . \\end{align*}"}
{"id": "651.png", "formula": "\\begin{align*} \\begin{cases} \\ f _ 0 ( x ) \\ = \\ u ( 2 ( x + 1 ) + 1 ) \\\\ [ 8 p t ] \\ g _ 0 ( x ) \\ = \\ v ( 2 ( x + 1 ) + 1 ) \\\\ [ 8 p t ] \\ h _ 0 ( x ) \\ = \\ w ( 2 ( x + 1 ) + 1 ) \\end{cases} . \\end{align*}"}
{"id": "473.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { L } ( \\mathbf { t } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } ) = \\mathbf { t } ^ { \\mathbf { m } _ { 0 } } \\cdot \\Delta _ { \\mathbf { L } ( \\mathbf { 1 } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { 1 } ) } ( \\mathcal { I } ) \\cdot \\prod _ { \\alpha \\in \\mathcal { I } } \\mathbf { t } ^ { \\psi ( \\alpha ) } , \\quad \\mathcal { I } \\in \\wp _ { k } [ n ] . \\end{align*}"}
{"id": "1321.png", "formula": "\\begin{align*} J _ { 0 } ( \\hat { \\alpha } \\cup { ( \\gamma , N ) } , \\hat { \\alpha } \\cup { ( \\delta , 1 ) \\cup { ( \\gamma , N - p _ { i } ) } } ) = 1 , \\end{align*}"}
{"id": "5252.png", "formula": "\\begin{align*} A \\otimes ^ I ( A \\otimes ^ I A ) = ( A \\otimes ^ I A ) \\otimes ^ I A = A \\otimes ^ I A \\otimes ^ I A , \\end{align*}"}
{"id": "1503.png", "formula": "\\begin{align*} x = \\left [ \\begin{array} { c c c c c } a _ 1 & a _ 2 & b _ 1 & c _ 1 & c _ 2 \\\\ a _ 3 & a _ 4 & b _ 2 & c _ 3 & c _ 4 \\\\ g _ 1 & g _ 2 & e & f _ 1 & f _ 2 \\\\ h _ 1 & h _ 2 & l _ 1 & d _ 1 & d _ 2 \\\\ h _ 3 & h _ 4 & l _ 2 & d _ 3 & d _ 4 \\end{array} \\right ] \\end{align*}"}
{"id": "351.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho , m , S ) & = \\inf _ { ( \\rho , m ) } \\{ \\mathcal A ( \\rho , m ) + \\mathbb I _ { C _ F ( \\rho ^ a , \\rho ^ b ) } ( \\rho , m ) \\} \\\\ & = \\inf _ { ( \\rho , m ) \\in C _ F ( \\rho ^ 0 , \\rho ^ 1 ) } \\{ \\mathcal A ( \\rho , m ) \\} . \\end{align*}"}
{"id": "8420.png", "formula": "\\begin{gather*} D _ { U } \\big ( \\langle F _ { t } ( W ) , G _ { t } ( W ) \\rangle \\big ) = \\langle D _ { U } F _ { t } ( W ) , G _ { t } ( W ) \\rangle + \\langle F _ { t } ( W ) , D _ { U } G _ { t } ( W ) \\rangle , \\intertext { a n d } D _ { U } \\nabla \\phi ( F _ { t } ( W ) ) = \\nabla _ { D _ { U } F _ { t } ( W ) } \\nabla \\phi ( F _ { t } ( W ) ) . \\end{gather*}"}
{"id": "29.png", "formula": "\\begin{align*} \\theta ( U ) = 1 , \\theta ^ c ( V ) = 1 , \\theta ( V ) = 0 , \\theta ^ c ( U ) = 0 . \\end{align*}"}
{"id": "6636.png", "formula": "\\begin{align*} I _ 1 = I _ { 1 1 } + I _ { 1 2 } , \\end{align*}"}
{"id": "6941.png", "formula": "\\begin{align*} \\eta _ { { \\rm r h s } , 1 } ( E ) = h _ E \\Vert f - \\Pi _ { E , q - 1 } f \\Vert _ { 0 , E } \\ , . \\end{align*}"}
{"id": "2718.png", "formula": "\\begin{align*} \\dim \\mathbb { S } ^ \\lambda = \\frac { n ! } { \\prod _ { i , j } h _ { i , j } ( \\lambda ) } , \\end{align*}"}
{"id": "8957.png", "formula": "\\begin{align*} C _ p q \\left | \\dot { \\xi } ( s ) \\right | ^ { q - 1 } = \\left | D u ( \\xi ( s ) ) \\right | = ( f ( \\xi ( s ) ) - u ( \\xi ( s ) ) ) ^ \\frac { 1 } { p } . \\end{align*}"}
{"id": "8406.png", "formula": "\\begin{align*} \\partial _ t c _ A = \\partial _ t ( \\zeta \\circ d _ \\Gamma ) ( c ^ { i n } - \\chi _ + + \\chi _ - ) + ( \\zeta \\circ d _ \\Gamma ) \\theta _ { 0 } ^ { \\prime } ( \\rho ) ( - \\frac { V } { \\varepsilon } - \\partial _ { t } ^ { \\Gamma } h _ { \\varepsilon } ) + ( \\zeta \\circ d _ \\Gamma ) ( \\varepsilon ^ 2 \\partial _ t c ^ { i n } _ 2 + \\varepsilon ^ 3 \\partial _ t c ^ { i n } _ 3 ) \\end{align*}"}
{"id": "2828.png", "formula": "\\begin{align*} \\sqrt { y ( t ) } - \\sqrt { y ( 0 ) } = \\int _ 0 ^ t \\frac { \\dot { y } ( s ) } { \\sqrt { y ( s ) } } d s \\le - C \\left ( \\dot { y } ( t ) - \\dot { y } ( 0 ) \\right ) \\le C \\dot { y } ( 0 ) \\end{align*}"}
{"id": "2681.png", "formula": "\\begin{align*} \\widehat { g } ( \\omega ) = C \\ , \\prod _ { k = 1 } ^ M \\frac { e ^ { 2 \\pi i \\delta _ k \\omega } } { 1 + 2 \\pi i \\delta _ k \\omega } , M \\in \\N . \\end{align*}"}
{"id": "7745.png", "formula": "\\begin{align*} - \\phi _ { t , u v } = ( \\phi _ u \\cdot \\phi _ v ) \\phi _ t + ( \\phi _ u \\cdot \\phi _ { t , v } + \\phi _ v \\cdot \\phi _ { t , u } ) \\phi + \\frac { 1 } { 4 } a \\phi _ { t t } = : G ( t , x ) , \\end{align*}"}
{"id": "2490.png", "formula": "\\begin{align*} \\int _ { \\mathbf { H } } F ( \\mathbf { h } ) \\ , d \\mathbf { h } = \\int _ { \\mathbf { H } } F ( \\mathbf { h } _ 0 \\bullet \\mathbf { h } ) \\ , d \\mathbf { h } \\end{align*}"}
{"id": "8532.png", "formula": "\\begin{align*} f ( t , x ) : = e ^ { - i t H } u ( t , x ) , H : = - \\partial _ { x x } + V , \\end{align*}"}
{"id": "9011.png", "formula": "\\begin{align*} X _ v [ T ] : = \\lim _ { m \\to \\infty } X ^ { m } _ v [ T ] = X ^ { m _ v } _ v [ T ] , v \\in V _ G , \\end{align*}"}
{"id": "4408.png", "formula": "\\begin{align*} W ( y ) = y ^ { - ( d + 1 ) } e ^ { 2 \\beta \\frac { y ^ 2 } { 4 } } . \\end{align*}"}
{"id": "8111.png", "formula": "\\begin{align*} \\tau _ { \\varphi } ( x , y ) = \\frac { \\varphi ( x ) - \\varphi ( y ) } { \\| \\varphi ( x ) - \\varphi ( y ) \\| } . \\end{align*}"}
{"id": "3692.png", "formula": "\\begin{align*} B _ t + J B _ x + \\mu \\Lambda ^ \\alpha B = 0 , \\ \\ \\ B _ x = \\mathcal H J . \\end{align*}"}
{"id": "526.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j i { r \\choose { i - 1 } } { { r - i } \\choose { j - i } } ( - 1 ) ^ { j - i } = r + 1 . \\end{align*}"}
{"id": "3105.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + z ( a x ^ 4 + b x ^ 2 y ^ 2 + c y ^ 4 ) + x ( d x ^ 4 + e x ^ 2 y ^ 2 + f y ^ 4 ) = 0 \\ , . \\end{align*}"}
{"id": "7994.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { I } _ { n } ^ { \\sigma } } ( \\mu _ n ^ { \\sigma } ( i ) ) ^ q \\prod _ { \\ell = 1 } ^ n \\big ( \\lambda _ i ^ { ( \\sigma _ { \\ell } ) } \\big ) ^ { T _ { \\ell } ^ { \\boldsymbol { \\mu } , \\sigma } ( q ) - T _ { \\ell - 1 } ^ { \\boldsymbol { \\mu } , \\sigma } ( q ) } = 1 . \\end{align*}"}
{"id": "8170.png", "formula": "\\begin{align*} S ' ( H , f ) = \\sum _ { 1 \\neq h \\in H } s ( h , f ) \\hbox { a n d } N ( f , H ) : = - 3 + \\frac { 2 } { f } + 1 2 S ' ( H , f ) . \\end{align*}"}
{"id": "5265.png", "formula": "\\begin{align*} \\varphi _ S ( a ) = \\varphi ( a \\delta _ { \\varphi } ) , \\varphi _ { S ^ { - 1 } } ( a ) = \\varphi ( \\delta _ { \\varphi } a ) , \\forall a \\in A . \\end{align*}"}
{"id": "3113.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + a z y ^ 2 ( x ^ 3 + y ^ 3 ) + x ^ 6 + 2 0 x ^ 3 y ^ 3 - 8 y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "8977.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = - 2 u ' \\left ( \\gamma ( t ) \\right ) = 2 \\left ( f \\left ( \\gamma ( t ) \\right ) - u \\left ( \\gamma ( t ) \\right ) \\right ) ^ \\frac { 1 } { 2 } \\end{align*}"}
{"id": "7323.png", "formula": "\\begin{align*} \\Phi ( x , y , t ) : = u ( x , t ) - v ( y , t ) - \\frac { | x - y | ^ 4 } { \\varepsilon ^ 4 } - \\alpha ( | x | ^ 2 + | y | ^ 2 ) - \\frac { \\lambda } { T - t } \\end{align*}"}
{"id": "8250.png", "formula": "\\begin{align*} v \\xi ^ { - 1 } = s _ 1 \\cdots s _ k s _ { k + 1 } \\cdots s _ m u \\xi ^ { - 1 } \\ \\ z \\eta ^ { - 1 } = s _ { k + 1 } \\cdots s _ m u \\xi ^ { - 1 } , \\end{align*}"}
{"id": "7386.png", "formula": "\\begin{align*} ( u ( x ) - u ( y ) ) u _ + ( x ) = ( u _ + ( x ) - u _ + ( y ) ) u _ + ( x ) + u _ - ( y ) u _ + ( x ) . \\end{align*}"}
{"id": "2289.png", "formula": "\\begin{align*} V _ g ( M _ \\eta T _ \\xi f ) ( x , \\omega ) & = \\langle M _ \\eta T _ \\xi f , M _ \\omega T _ x g \\rangle \\\\ & = \\langle f , T _ { - \\xi } M _ { - \\eta } M _ \\omega T _ x g \\rangle \\\\ & = \\langle f , e ^ { 2 \\pi i \\xi \\cdot ( - \\eta + \\omega ) } M _ { \\omega - \\eta } T _ { x - \\xi } g \\rangle \\\\ & = e ^ { - 2 \\pi i \\xi \\cdot ( \\omega - \\eta ) } V _ g f ( x - \\xi , \\omega - \\eta ) \\end{align*}"}
{"id": "1491.png", "formula": "\\begin{align*} \\mathbf { f } ( g ) = j ( k _ { \\infty } , z _ 0 ) ^ { - k } \\sum _ { \\tau \\in S } \\det ( q _ { \\infty } ) ^ { - k } c ( \\tau , q ; \\mathbf { f } ) e _ { \\infty } ( \\lambda ( q ^ { \\ast } \\tau q ) z _ 0 ) e _ { \\mathbb { A } } ( \\lambda ( \\tau \\sigma ) ) . \\end{align*}"}
{"id": "9102.png", "formula": "\\begin{align*} \\alpha = 1 + D t ^ m + \\alpha _ { m + 1 } t ^ { m + 1 } + \\cdots + \\alpha _ { m n } t ^ { m n } \\end{align*}"}
{"id": "5862.png", "formula": "\\begin{align*} \\underline { k } = j + 1 . \\end{align*}"}
{"id": "483.png", "formula": "\\begin{align*} \\mathfrak { c } _ { \\mathcal { J } } : = ( \\mathfrak { c } _ { r } ) _ { \\gamma } ^ { l } \\times ( \\mathfrak { c } _ { c } ) _ { l } ^ { \\gamma } . \\end{align*}"}
{"id": "7615.png", "formula": "\\begin{align*} \\nabla _ { j } u & = h _ { j l } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) , \\\\ \\nabla _ { i } \\nabla _ { j } u & = \\nabla _ { i } h _ { j l } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) + \\lambda ' h _ { i j } - h _ { i l } h _ { j l } \\bar { g } ( \\lambda \\partial _ { r } , \\nu ) \\\\ & = \\nabla _ { l } h _ { i j } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) + \\bar { R } _ { \\nu j l i } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) + \\lambda ' h _ { i j } - u h _ { i l } h _ { j l } . \\end{align*}"}
{"id": "7029.png", "formula": "\\begin{align*} \\| a _ 1 \\widetilde f + p \\| ^ { 2 } _ { b } : = \\| \\widetilde { f } \\| ^ 2 _ { H ^ 2 } + \\| p \\| ^ 2 _ { H ^ 2 } . \\end{align*}"}
{"id": "1154.png", "formula": "\\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \\xi ) - \\theta ( k , \\xi ) ] \\sigma _ 3 } . \\end{align*}"}
{"id": "6140.png", "formula": "\\begin{align*} Q ( x ) = \\sum _ { k = 0 } ^ m { n + m + 1 \\choose k } x ^ k ( 1 - x ) ^ { m - k } = \\sum _ { k = 0 } ^ m { n + k \\choose k } x ^ k . \\end{align*}"}
{"id": "5210.png", "formula": "\\begin{align*} \\| T _ k f \\| _ { L ^ r ( K ) } & \\leq \\sum _ { i = 1 } ^ { \\infty } | \\lambda _ i | \\| B f \\| _ { L ^ r ( K ) } , \\\\ & \\leq C ^ \\prime \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\| B f \\| _ { L ^ r ( K ) } . \\end{align*}"}
{"id": "2258.png", "formula": "\\begin{align*} \\widetilde { e } _ m ^ { M , N } = k \\sum _ { j = 1 } ^ m E _ { k , N } ^ { m - j + 1 } P _ N A P ( F ( X ( t _ j ) ) - F ( X _ j ^ { M , N } ) ) . \\end{align*}"}
{"id": "5236.png", "formula": "\\begin{align*} \\| f ( C _ r ) \\| \\ge w ( d k + e ) + 1 / d + ( 1 - d w ) ( d k - 1 ) / d = w ( e + 1 ) + k . \\end{align*}"}
{"id": "2962.png", "formula": "\\begin{align*} | \\mathcal { V } _ { k } ( \\ell ) | = \\binom { d } { 2 k - \\ell } \\cdot \\binom { 2 k - \\ell - 1 } { k - 1 } \\cdot \\binom { k - 1 } { \\ell } . \\end{align*}"}
{"id": "9103.png", "formula": "\\begin{align*} \\alpha ' = 1 + D t ^ m + \\alpha _ { 2 m } ' t ^ { 2 m } + \\cdots + \\alpha _ { m n } ' t ^ { m n } \\end{align*}"}
{"id": "1658.png", "formula": "\\begin{align*} \\psi _ \\xi ( u ) : = \\varphi ( u ) \\wedge \\varphi ( u + \\xi ) , \\end{align*}"}
{"id": "3239.png", "formula": "\\begin{align*} ( s - t ) ^ 2 - \\frac { s - t } { h ( s ) } + r ^ 2 = 0 . \\end{align*}"}
{"id": "8095.png", "formula": "\\begin{align*} \\left \\langle \\mathrm { P V } \\left ( \\tfrac { 1 } { x } \\right ) , f \\right \\rangle = \\lim _ { \\epsilon \\searrow 0 } \\int _ { \\mathbb { R } \\setminus ( - \\epsilon , \\epsilon ) } \\frac { f ( x ) } { x } \\ , \\mathrm { d } x . \\end{align*}"}
{"id": "3215.png", "formula": "\\begin{align*} k _ { P _ H } = \\frac { \\left ( \\mathbb { E } [ Y ] - \\frac { 1 } { 1 + e ^ { a b } } \\right ) ^ 2 } { \\mathbb { E } [ Y ^ 2 ] - \\mathbb { E } [ Y ] ^ 2 } , \\end{align*}"}
{"id": "3659.png", "formula": "\\begin{align*} G _ m ( t ) : = \\begin{cases} \\log t & m = k \\\\ - t ^ { 2 - \\frac { 2 k } { m } } & m < k . \\end{cases} \\end{align*}"}
{"id": "8346.png", "formula": "\\begin{align*} p = \\exp \\left ( - ( e ^ r - 1 ) / c \\right ) \\end{align*}"}
{"id": "1363.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\frac { 1 } { | W ' _ R | } \\int _ { W ' _ R } \\chi _ \\epsilon ( x , \\omega ) \\eta ( x ) d x = 0 . \\end{align*}"}
{"id": "686.png", "formula": "\\begin{align*} a : = - 6 \\frac { \\sigma _ 3 } { \\sigma _ 1 } . \\end{align*}"}
{"id": "6109.png", "formula": "\\begin{align*} \\left ( b _ 2 \\mathbb { N } + b _ 1 \\Z \\right ) \\cap \\pi \\Z = \\emptyset \\ , . \\end{align*}"}
{"id": "6530.png", "formula": "\\begin{align*} \\frac { ( n + m - 1 ) ! } { n ^ m ( n - 1 ) ! } \\cdot \\frac { m } { ( \\log n ) ^ m } \\sum ^ { n - 1 } _ { j = 1 } \\frac { j ^ m j ! } { ( j + m ) ! } \\cdot \\frac { ( \\log j ) ^ { m - 1 } } { j } - 1 . \\\\ \\end{align*}"}
{"id": "5345.png", "formula": "\\begin{align*} \\partial _ t u + u \\cdot \\nabla u - \\mu _ 1 \\Delta u + \\nabla \\pi & = \\frac { \\beta } { k } \\mathrm { d i v } \\tau , \\\\ \\mathrm { d i v } u & = 0 , \\end{align*}"}
{"id": "3321.png", "formula": "\\begin{align*} \\ \\ & \\mbox { $ c _ i = q ' _ i $ f o r $ i = 1 , 2 , \\ldots , m u $ , w h e r e $ q ' _ i $ i s d e f i n e d i n ( \\ref { q } ) . } \\\\ \\ \\ & \\mbox { T h e c o n d i t i o n E M U f o r $ m \\Delta _ { \\overline { t } , \\overline { u } , \\overline { s } } $ i s n o t s a t i s f i e d i f a n d o n l y i f t h e r e e x i s t s $ i $ } \\\\ & \\mbox { s a t i s f y i n g $ 0 < i < m u $ a n d $ e _ i > i $ . } \\end{align*}"}
{"id": "3597.png", "formula": "\\begin{align*} \\alpha _ { i } = \\min \\{ ( R _ { i + 1 } - R _ { i } ) / 2 + e , R _ { i + 1 } - R _ { i } + d [ - a _ { i , i + 1 } ] \\} \\ , . \\end{align*}"}
{"id": "5837.png", "formula": "\\begin{align*} \\bar { v } = v _ { \\min } + \\frac { v _ { \\max } - v _ { \\min } } { \\sqrt { \\ell _ k } } \\leq v _ k + \\frac { \\varepsilon _ \\bullet } { \\sqrt { \\ell _ k } } \\leq v _ { k + 1 } \\end{align*}"}
{"id": "446.png", "formula": "\\begin{align*} A ^ { 0 } u _ { t } ^ { \\epsilon } + A ^ { i } \\partial _ { i } u ^ { \\epsilon } + D u ^ { \\epsilon } - B ^ { i j } \\partial _ { i } \\partial _ { j } u ^ { \\epsilon } = f ^ { \\epsilon } + F ^ { \\epsilon } , \\end{align*}"}
{"id": "9541.png", "formula": "\\begin{align*} \\C \\cap L ^ 0 _ + = \\{ 0 \\} , \\end{align*}"}
{"id": "6705.png", "formula": "\\begin{align*} \\psi ^ { ( - 1 ) } = \\Phi \\psi . \\end{align*}"}
{"id": "8234.png", "formula": "\\begin{align*} \\frac { 1 } { n - 1 } & \\sum _ { r = 0 } ^ { n - \\ell } \\binom { 4 ( h - 1 ) ( n - 1 ) + r - 1 } { r } \\binom { 3 n - 3 - r } { \\ell } \\binom { 2 ( k + 1 - 2 h ) ( n - 1 ) } { n - r - \\ell } \\\\ & - \\frac { 1 } { 2 n - 1 } \\sum _ { r = 0 } ^ { n - \\ell } \\binom { 2 ( h - 1 ) ( 2 n - 1 ) + r - 1 } { r } \\binom { 3 n - 2 - r } { \\ell } \\binom { ( k + 1 - 2 h ) ( 2 n - 1 ) } { n - r - \\ell } \\end{align*}"}
{"id": "5821.png", "formula": "\\begin{align*} \\hat { \\mu } _ \\phi ( k ) = \\frac { 1 } { Z ( \\phi ) } \\frac { \\phi ^ k } { g ( k ) ! } , k \\in \\mathbb { N } _ { 0 } , \\end{align*}"}
{"id": "5979.png", "formula": "\\begin{align*} \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; \\cdot ) : \\mathbb { R } \\longrightarrow \\mathbb { C } , \\ ; x \\mapsto \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; x ) = { \\displaystyle \\frac { \\mathrm { e } ^ { x z } } { l _ { \\pi _ { \\lambda , \\beta } } ( z ) } } = { \\displaystyle \\frac { \\mathrm { e } ^ { x z } } { E _ { \\beta } ( \\lambda ( \\mathrm { e } ^ { z } - 1 ) ) } } . \\end{align*}"}
{"id": "8542.png", "formula": "\\begin{align*} { \\| \\mathcal { F } f ( t , \\cdot ) \\| } _ { L ^ \\infty } = { \\| w ( t , \\cdot ) \\| } _ { L ^ \\infty } \\leq ( C _ 0 / 2 ) \\varepsilon + C \\varepsilon ^ 3 \\leq C _ 0 \\varepsilon . \\end{align*}"}
{"id": "2416.png", "formula": "\\begin{align*} B ^ { - 1 } \\norm { f } _ \\mathcal { H } ^ 2 \\leq \\langle S ^ { - 1 } f , f \\rangle = \\sum _ { \\gamma \\in \\Gamma } | \\langle f , S ^ { - 1 } e _ \\gamma \\rangle | ^ 2 \\leq A ^ { - 1 } \\norm { f } _ \\mathcal { H } ^ 2 . \\end{align*}"}
{"id": "1583.png", "formula": "\\begin{align*} ( ( L f , f ) ) _ k = ( L f , f ) _ { L ^ 2 _ k } + \\eta \\int _ 0 ^ \\infty ( \\mathcal { S } _ L ( \\tau ) L f , \\mathcal { S } _ L ( \\tau ) f ) _ { L ^ 2 _ v } d \\tau \\sim \\| f \\| _ { H ^ s _ { k + \\gamma / 2 } } ^ 2 + k ^ s \\| f \\| _ { L ^ 2 _ { k + \\gamma / 2 } } ^ 2 , \\end{align*}"}
{"id": "33.png", "formula": "\\begin{align*} & J ( U ) = V , \\quad \\ , J ( V ) = - U , \\\\ & J ( e _ j ) = e _ { n + j } , J ( e _ { n + j } ) = - e _ j . \\end{align*}"}
{"id": "2518.png", "formula": "\\begin{align*} \\rho ( x , \\omega , e ^ { 2 \\pi i \\tau } ) = e ^ { 2 \\pi i \\tau } \\rho ( x , \\omega ) . \\end{align*}"}
{"id": "69.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - n + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 2 } ^ { n } ( q _ v ^ i - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - n ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 2 } ^ { n + 1 } ( q _ v ^ i - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & = \\frac { q _ v ^ { n _ { v , \\nu _ v } } - 1 } { q _ v ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "7404.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { x } \\sup _ { s \\in [ 0 , T ] } | \\Phi _ { n } ( s , \\tfrac { x } { n } ) | \\leq M _ 1 \\quad \\textrm { a n d } \\quad \\| \\Phi _ { n } \\| _ { \\infty } : = \\sup _ { ( s , u ) \\in [ 0 , T ] \\times \\mathbb { R } } | \\Phi _ n ( s , u ) | \\leq M _ 2 , \\end{align*}"}
{"id": "8929.png", "formula": "\\begin{align*} H Y ^ 0 _ b ( U , A ) = \\ker d _ 0 = A ( U ) \\end{align*}"}
{"id": "1086.png", "formula": "\\begin{align*} q ( x , t ) = 2 i \\lim _ { k \\rightarrow \\infty } [ k m ( x , t , k ) ] _ { 1 2 } , ( x , t ) \\in \\mathbb { R } \\times [ 0 , + \\infty ) . \\end{align*}"}
{"id": "3910.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } a _ k c _ k = \\sum _ { m = 0 } ^ n \\binom { n } { m } d _ m \\nabla ^ m b _ n . \\end{align*}"}
{"id": "2851.png", "formula": "\\begin{align*} \\forall R \\ge R _ 1 + | x ( t ) | \\Rightarrow | A _ R ( u ( t ) ) | \\le C \\varepsilon = 4 s _ c ( p - 1 ) \\delta _ 0 \\le 4 s _ c ( p - 1 ) \\delta ( t ) , \\forall t \\notin D _ { \\delta _ 0 } . \\end{align*}"}
{"id": "3491.png", "formula": "\\begin{align*} E ( s _ 1 , s _ 3 ; y , n , M ) = O \\left ( \\frac { 1 } { y ^ { \\sigma _ 1 } ( y + n ) ^ { \\sigma _ 3 } } \\right ) \\end{align*}"}
{"id": "293.png", "formula": "\\begin{align*} z _ { t } + ( \\beta \\chi z ) _ { x } - z _ { x x } & = \\lambda _ x , \\ \\ x \\in \\R , \\ t > 0 , \\\\ z ( x , 0 ) & = z _ { 0 } ( x ) , \\ \\ x \\in \\R , \\end{align*}"}
{"id": "6808.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } u \\in C \\left ( \\left [ 0 , T \\right ] , D o m ( \\boldsymbol { A } ) \\right ) \\cap C ^ { 1 } \\left ( \\left [ 0 , T \\right ] , \\mathcal { X } \\right ) ; \\\\ \\\\ u _ { t } = \\boldsymbol { A } u + f ( t ) , \\ \\ t \\in \\left [ 0 , T \\right ] ; \\\\ \\\\ u ( 0 ) = u _ { 0 } \\in \\mathcal { X } . \\end{array} \\right . \\end{align*}"}
{"id": "7600.png", "formula": "\\begin{align*} \\frac { \\lambda ( r ) '' } { \\lambda ( r ) } + \\frac { c - \\lambda ( r ) '^ { 2 } } { \\lambda ( r ) ^ { 2 } } \\geq 0 . \\end{align*}"}
{"id": "7032.png", "formula": "\\begin{align*} f _ k ( z ) = a _ { k } ^ { \\# } ( z ) \\widetilde { f } ( z ) + \\frac { p ( z ) - p ( \\xi ) } { z - \\xi _ k } \\in H ^ 2 . \\end{align*}"}
{"id": "2199.png", "formula": "\\begin{align*} \\left ( \\widehat { w } ( x ) - \\widehat { w } ( y ) \\right ) = \\left ( \\widehat { w } ^ \\pm ( x ) - \\widehat { w } ^ \\pm ( y ) \\right ) , \\ \\ \\ x , y \\in \\mathbb { R } ^ d . \\end{align*}"}
{"id": "1733.png", "formula": "\\begin{align*} m _ t ^ * = \\max \\{ \\hat m _ t \u2010 \\varepsilon | t \u2010 t _ * ( n ) | , \\ , 0 \\} . \\end{align*}"}
{"id": "1332.png", "formula": "\\begin{align*} A ( \\delta _ { 1 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , A ( \\delta _ { 2 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , p _ { i + 1 } = \\frac { 3 } { 2 } p _ { i } , \\ , \\ , A ( \\eta ) \\approx \\frac { 1 } { 4 } p _ { i } . \\end{align*}"}
{"id": "1212.png", "formula": "\\begin{align*} \\mbox { f o r e v e r y $ B \\in \\mathcal { G } ^ { U } $ , } \\ \\ \\ \\eta ( B ) = \\eta ( U ) \\times \\frac { m _ { E _ U } ^ { s _ 2 } ( B ) } { \\sum _ { B ' \\in \\mathcal { G } ^ { U } } m _ { E _ U } ^ { s _ 2 } ( B ' ) } . \\end{align*}"}
{"id": "411.png", "formula": "\\begin{align*} A ^ { 0 } U _ { t } ^ { \\delta } + A ^ { i } \\partial _ { i } U ^ { \\delta } + D U ^ { \\delta } - B ^ { i j } \\partial _ { i } \\partial _ { j } U ^ { \\delta } = f + \\delta \\Lambda \\Delta U ^ { \\delta } , \\end{align*}"}
{"id": "950.png", "formula": "\\begin{align*} \\Gamma _ 1 = \\{ n > n ( E ) : J _ n \\subset L ^ c \\} , \\Gamma _ 2 = \\{ n > n ( E ) : J _ n \\subset L \\} , \\end{align*}"}
{"id": "5949.png", "formula": "\\begin{align*} \\norm { f } _ 2 ^ 2 \\norm { g } _ 2 ^ 2 - | \\langle f , g \\rangle | ^ 2 = ( r ^ 2 + \\norm { h } _ 2 ^ 2 ) - r ^ 2 = \\norm { h } _ 2 ^ 2 . \\end{align*}"}
{"id": "2876.png", "formula": "\\begin{align*} \\int _ { n } ^ { n + 1 } B ( h , R ) d s & = \\frac { 1 } { 2 } \\int _ n ^ { n + 1 } \\int _ { \\mathbb { R } ^ N } ( L _ + h _ 1 ) R _ 1 d x d s + \\frac { 1 } { 2 } \\int _ n ^ { n + 1 } \\int _ { \\mathbb { R } ^ N } ( L _ - h _ 2 ) R _ 2 d x d s \\\\ & \\lesssim \\| \\left \\langle \\nabla \\right \\rangle h \\| _ { S ( ( n , + \\infty ) , L ^ 2 ) } \\| \\left \\langle \\nabla \\right \\rangle R \\| _ { S ' ( ( n , + \\infty ) , L ^ 2 ) } \\\\ & \\lesssim e ^ { - ( c _ 0 + c _ 1 ) n } , \\end{align*}"}
{"id": "3033.png", "formula": "\\begin{align*} \\lambda = h d \\mu . \\end{align*}"}
{"id": "8859.png", "formula": "\\begin{align*} \\tilde \\varphi : S & \\to A \\\\ s & \\mapsto \\sum _ { i = 0 } ^ { n - 1 } \\varphi ( a _ i , a _ { i + 1 } ) \\end{align*}"}
{"id": "2956.png", "formula": "\\begin{align*} \\Delta \\left ( \\mathbf { P } _ k \\right ) = ( \\mathbf p _ { k - 1 , 1 } \\cup \\mathbf p _ { k - 1 , 2 } ) \\cap ( \\mathbf p _ { k - 1 , 3 } \\cup \\mathbf p _ { k - 1 , 4 } ) \\end{align*}"}
{"id": "5688.png", "formula": "\\begin{align*} U _ { J , z } \\langle \\alpha \\rangle = \\sum _ { \\beta : \\mathrm { a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } [ \\beta ] = \\Gamma } \\# \\{ \\ , u \\in \\mathcal { M } _ { 2 } ^ { J } ( \\alpha , \\beta ) / \\mathbb { R } ) \\ , | \\ , ( 0 , z ) \\in u \\ , \\} \\cdot \\langle \\beta \\rangle . \\end{align*}"}
{"id": "3305.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F | _ { D _ 1 \\cap D _ 2 } ) = \\alpha _ { n - j } ( T _ i | _ { D _ 1 \\cap D _ 2 } ) + \\alpha _ { n - j } ( ( F _ i / T _ i ) | _ { D _ 1 \\cap D _ 2 } ) . \\end{align*}"}
{"id": "97.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ 2 \\cdot \\frac { ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) ( 2 ^ { n _ { 2 , \\nu _ 2 } - 2 } - 1 ) } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "4752.png", "formula": "\\begin{align*} A ( \\Gamma ) = \\left ( \\begin{array} { c c c c } 0 & a _ { 1 , 2 } & \\cdots & a _ { 1 , n } \\\\ a _ { 2 , 1 } & 0 & \\cdots & a _ { 2 , n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ { n , 1 } & a _ { n , 2 } & \\cdots & 0 \\end{array} \\right ) \\end{align*}"}
{"id": "3323.png", "formula": "\\begin{align*} { \\rm d e t } \\left ( \\begin{array} { c c } f _ i & f _ { i - 1 } \\\\ g _ i & g _ { i - 1 } \\end{array} \\right ) = 1 . \\end{align*}"}
{"id": "6182.png", "formula": "\\begin{align*} \\| C \\| _ F = \\| S \\| _ F = \\| W \\| _ F . \\end{align*}"}
{"id": "7968.png", "formula": "\\begin{align*} l _ 0 : = \\sup \\big \\{ l \\in \\mathbb { N } ^ + : & \\| u \\| _ { L ^ 2 ( A ^ g ( \\mathbf { 0 } , \\tau / K , \\tau ) ; \\| \\Sigma ^ 0 \\| _ g ) } \\leq \\delta _ 1 ' \\cdot \\tau ^ { 1 + n / 2 } , \\\\ & \\ \\forall \\tau = K ^ { - m } \\in [ K ^ { - l + 1 } , 1 ] , \\ m \\big \\} . \\end{align*}"}
{"id": "1123.png", "formula": "\\begin{align*} m ^ { ( 3 ) } ( x , t , k ) = m ^ { ( 2 ) } ( x , t , k ) D ^ { \\sigma _ 3 } ( k , \\xi ) G ( x , t , k ) D ^ { - \\sigma _ 3 } ( k , \\xi ) , \\end{align*}"}
{"id": "1370.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { \\tau } , ~ ~ \\mbox { t h e r e c i p r o c a l o f t h e m o m e n t u m r e l a x a t i o n t i m e } . \\end{align*}"}
{"id": "646.png", "formula": "\\begin{align*} f ( 0 ) = 1 , f ( k ) = k ( k > 0 ) . \\end{align*}"}
{"id": "8799.png", "formula": "\\begin{align*} ~ \\begin{aligned} \\phi \\Bigl ( \\Pi \\bigl ( ( a _ { - i } , s _ i ) ; p ^ { \\tau _ { i j } } \\bigr ) \\Bigr ) & - \\phi \\Bigl ( \\Pi \\bigl ( ( a _ { - i } , u _ i ) ; p ^ { \\tau _ { i j } } \\bigr ) \\Bigr ) \\leq \\phi \\Bigl ( \\Pi \\bigl ( ( a _ { - i } , s _ i ) ; p ^ { \\tau _ { i j + 1 } - 1 } \\bigr ) \\Bigr ) \\\\ & - \\phi \\Bigl ( \\Pi \\bigl ( ( a _ { - i } , u _ i ) ; p ^ { \\tau _ { i j + 1 } - 1 } \\bigr ) \\Bigr ) , \\end{aligned} \\end{align*}"}
{"id": "2548.png", "formula": "\\begin{align*} \\mathcal { D } _ L M _ \\omega T _ x f ( t ) & = | \\det ( L ) | ^ { - 1 / 2 } e ^ { 2 \\pi i \\omega \\cdot ( L ^ { - 1 } t ) } f ( L ^ { - 1 } t - x ) \\\\ & = | \\det ( L ) | ^ { - 1 / 2 } e ^ { 2 \\pi i ( L ^ { - T } \\omega ) \\cdot t } f ( L ^ { - 1 } ( t - L x ) ) \\\\ & = M _ { L ^ { - T } \\omega } T _ { L x } \\mathcal { D } _ L f ( t ) . \\end{align*}"}
{"id": "2543.png", "formula": "\\begin{align*} \\left ( \\rho ( x , \\omega , \\tau ) f \\right ) ^ \\vee ( t ) = e ^ { 2 \\pi i \\tau } e ^ { \\pi i x \\cdot \\omega } T _ { - x } M _ { - \\omega } f ( - t ) = \\rho ( - x , - \\omega , \\tau ) f ^ \\vee ( t ) . \\end{align*}"}
{"id": "3440.png", "formula": "\\begin{align*} I \\ ! I _ 2 = \\int _ { \\| u - y \\| > t } \\frac 1 { V ( x , y , t + d ( x , y ) ) } \\Big ( \\frac { t } { t + d ( x , y ) } \\Big ) ^ { \\varepsilon _ 0 } \\frac 1 { V ( u , y , s + d ( u , y ) ) } \\Big ( \\frac { s } { s + \\| u - y \\| } \\Big ) ^ { \\varepsilon _ 0 } d \\omega ( u ) \\end{align*}"}
{"id": "6192.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } \\geq \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } - 2 \\sqrt { k } \\theta \\| S \\| ^ 2 _ F . \\end{align*}"}
{"id": "4832.png", "formula": "\\begin{align*} I _ { \\chi , h } ( \\xi , \\eta ) = \\int \\chi ( x ) e ^ { i x \\cdot ( \\xi + \\eta ) / h } d x . \\end{align*}"}
{"id": "1736.png", "formula": "\\begin{align*} S \\underset { \\mathfrak { Z } _ 0 } { \\asymp } n ^ { \u2010 \\beta _ { j _ * } } = \\varphi ( t _ n , \\ , m _ n , \\ , n ) ; \\end{align*}"}
{"id": "2239.png", "formula": "\\begin{align*} \\Big ( k \\sum _ { j = 1 } ^ m \\| A E _ { k , N } ^ j v \\| ^ 2 \\Big ) ^ { \\frac 1 2 } \\leq C \\| v \\| , \\forall v \\in H . \\end{align*}"}
{"id": "9478.png", "formula": "\\begin{align*} & | \\mathcal { B C } _ { ( s , s + d , s + 2 d ) } | = | \\mathcal { C S } _ { ( s , s + d , s + 2 d ) } | \\\\ & ~ ~ = \\sum _ { i = 0 } ^ { ( s - 1 ) / 2 } \\binom { ( d - 1 ) / 2 + i } { \\lfloor i / 2 \\rfloor } \\left ( \\binom { ( s + d - 2 ) / 2 } { ( d - 1 ) / 2 + i } + \\binom { ( s + d - 4 ) / 2 } { ( d - 1 ) / 2 + i } \\right ) . \\end{align*}"}
{"id": "3603.png", "formula": "\\begin{align*} u _ { 1 } = \\min \\{ v _ { 1 } , v _ { 2 } \\} = 0 \\quad u _ { 2 } = \\min \\{ v _ { 2 } , 2 ( r _ { 2 } - r _ { 1 } ) + v _ { 1 } \\} = 0 \\ , . \\end{align*}"}
{"id": "1230.png", "formula": "\\begin{align*} & \\underline \\dim ( \\mu , x ) = \\liminf _ { r \\rightarrow 0 ^ { + } } \\frac { \\log \\mu ( B ( x , r ) ) } { \\log r } \\\\ \\mbox { a n d } \\ \\ \\ \\ & \\overline \\dim ( \\mu , x ) = \\limsup _ { r \\rightarrow 0 ^ { + } } \\frac { \\log \\mu ( B ( x , r ) ) } { \\log r } . \\end{align*}"}
{"id": "438.png", "formula": "\\begin{align*} A ^ { 0 } _ { 1 } ( U ) = \\rho , A ^ { 0 } _ { 3 } ( U ) = \\left ( \\begin{array} { c c } \\frac { \\rho e _ { \\theta } } { \\theta } & \\\\ & \\frac { \\tau } { \\kappa \\theta } \\mathbb { I } _ { 3 \\times 3 } \\end{array} \\right ) \\quad \\mbox { a n d } A ^ { 0 } _ { 2 } ( U ) = \\rho \\mathbb { I } _ { 3 \\times 3 } . \\end{align*}"}
{"id": "857.png", "formula": "\\begin{align*} { \\bar { \\Delta } _ { \\rm C l a s s i c a l - A R Q } } = - \\frac { 1 } { 2 } + { n _ 1 } \\left ( { \\frac { 2 } { { 1 - { \\epsilon _ 1 } } } - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
{"id": "2298.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "4729.png", "formula": "\\begin{align*} \\Psi _ { V _ 0 } = \\mathcal { E } _ { V _ 0 } - \\sum _ { i = 1 } ^ 2 \\dot { \\mathfrak { q } } _ i \\sigma _ i \\partial _ y R _ i + \\sum _ { \\substack { i , j = 1 , \\\\ j \\not = i } } ^ 2 \\frac { a _ { i j } \\sigma _ i } { ( \\mathfrak { q } _ i - \\mathfrak { q } _ j ) ^ 3 } \\Lambda R _ i + \\sum _ { i = 1 } ^ 2 \\dot { \\mathfrak { q } } _ i ( \\mathcal { P } _ i + \\mathcal { Q } _ i ) \\end{align*}"}
{"id": "9069.png", "formula": "\\begin{align*} \\mathcal { I } ( \\phi ) = & \\frac { 1 } { 8 h } \\epsilon _ 1 ( \\phi _ 2 - c ^ a - c ^ a _ 1 \\phi _ 1 ) ^ 2 + \\frac { 1 } { 8 h } \\sum _ { j = 2 } ^ { N - 1 } \\epsilon _ j ( \\phi _ { j + 1 } - \\phi _ { j - 1 } ) ^ 2 + \\frac { 1 } { 8 h } \\epsilon _ N ( c ^ b + c ^ b _ N \\phi _ N - \\phi _ { N - 1 } ) ^ 2 \\\\ & + \\frac { 1 } { 8 \\beta _ a } ( c ^ a + ( 1 + c ^ a _ 1 ) \\phi _ 1 ) ^ 2 + \\frac { 1 } { 8 \\beta _ b } ( c ^ b + ( 1 + c ^ b _ 1 ) \\phi _ N ) ^ 2 . \\end{align*}"}
{"id": "7369.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t + \\big \\{ m ( K \\cap \\{ u ( \\cdot , t ) < u ( x , t ) \\} ) - 1 \\big \\} | \\nabla u | = 0 & \\\\ u ( x , 0 ) = \\varphi _ 0 ( | x | ) & \\ \\R ^ n , \\end{array} \\right . \\end{align*}"}
{"id": "7370.png", "formula": "\\begin{align*} \\omega _ n \\min \\{ R , r ^ \\ast \\} ^ n = 1 , \\end{align*}"}
{"id": "9523.png", "formula": "\\begin{gather*} x _ t \\ge 0 , \\ R _ t \\le y _ t , \\ x _ t ( R _ t - y _ t ) = 0 t = 0 , \\ldots , T , \\\\ y _ T \\ge 0 , \\ \\sum _ { t = 0 } ^ T x _ t \\le 1 , \\ y _ T ( \\sum _ { t = 0 } ^ T x _ t - 1 ) = 0 . \\end{gather*}"}
{"id": "4142.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int x u ^ 2 ( x , t ) d x = 2 I _ 3 ( u ) \\end{align*}"}
{"id": "881.png", "formula": "\\begin{align*} i _ X \\phi = i _ X d \\phi = 0 , \\end{align*}"}
{"id": "2845.png", "formula": "\\begin{align*} \\lim \\limits _ { T \\to \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\delta ( t ) d t = 0 . \\end{align*}"}
{"id": "8241.png", "formula": "\\begin{align*} H ( x , y , h _ 1 , h _ 2 , h _ 3 , z ) : = \\big ( & \\Delta _ { - h _ 1 , - 2 h _ 2 } B ( y + 2 z ) \\Delta _ { - h _ 2 , - h _ 3 } B ( y + z ) \\Delta _ { - h _ 1 , - h _ 2 } C ( x + y + 2 z ) \\\\ & \\Delta _ { - h _ 3 } C ( x + y + z ) \\Delta _ { - h _ 1 } D ( 2 x + y + 2 z ) \\Delta _ { h _ 2 , - h _ 3 } D ( 2 x + y + z ) \\big ) \\end{align*}"}
{"id": "8180.png", "formula": "\\begin{align*} M ( p , H ) = \\frac { \\pi ^ 2 } { 6 } \\times \\frac { a + 1 } { a - 1 } \\times \\left ( 1 - \\frac { ( d - 1 ) a + 1 } { p } \\right ) . \\end{align*}"}
{"id": "2807.png", "formula": "\\begin{align*} \\Im \\int Q v = 0 , \\Re \\int ( \\partial _ { x _ k } Q ) v = 0 , \\ ; k = 1 , . . . , N , \\end{align*}"}
{"id": "2867.png", "formula": "\\begin{align*} | \\alpha _ \\pm | = | B ( h , \\mathcal { Y } _ \\mp ) | \\lesssim \\Big | \\frac { 1 } { 2 } \\int ( L _ + h _ 1 ) \\mathcal { Y } _ 1 d x \\Big | + \\Big | \\frac { 1 } { 2 } \\int ( L _ - h _ 2 ) \\mathcal { Y } _ 2 d x \\Big | \\lesssim \\| h ( t ) \\| _ { L _ x ^ 2 } \\lesssim e ^ { - c _ 0 t } . \\end{align*}"}
{"id": "3393.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\sum \\limits _ { Q } | Q | \\phi _ { Q } ( x - x _ Q ) \\psi _ { Q } \\ast f ( x _ Q ) , \\end{align*}"}
{"id": "6461.png", "formula": "\\begin{align*} & P ( X _ { n + 1 } = \\pm 1 \\mid \\mathcal { F } _ n ) \\\\ & = \\dfrac { \\# \\{ i = 1 , \\ldots , n : X _ i = \\pm 1 \\} } { n } \\cdot p + \\dfrac { \\# \\{ i = 1 , \\ldots , n : X _ i = \\mp 1 \\} } { n } \\cdot ( 1 - p ) \\\\ & = \\dfrac { n \\pm S _ n } { 2 n } \\cdot p + \\dfrac { n \\mp S _ n } { 2 n } \\cdot ( 1 - p ) = \\dfrac { 1 } { 2 } \\left ( 1 \\pm \\alpha \\cdot \\dfrac { S _ n } { n } \\right ) . \\end{align*}"}
{"id": "5314.png", "formula": "\\begin{align*} \\sum _ i \\omega _ i ( b _ { ( 3 ) } c S ^ { - 1 } ( b _ { ( 1 ) } ) ) a _ i b _ { ( 2 ) } d = 0 , \\forall b , c , d \\in A . \\end{align*}"}
{"id": "8652.png", "formula": "\\begin{align*} f ( x ) = \\frac { ( x ^ 3 - 9 x ) ( a x + b ) - 9 ( 1 - x ^ 2 ) ( c x + d ) } { - ( x ^ 3 - 9 x ) ( c x + d ) - 3 ( 1 - x ^ 2 ) ( a x + b ) } \\end{align*}"}
{"id": "5408.png", "formula": "\\begin{align*} \\frac { 1 } { n h } \\sum _ { k = 0 } ^ { n - 1 } \\partial _ { \\theta ^ { 2 } } f ( X _ { t _ { k } } , \\theta _ { 0 } ) \\int _ { t _ { k } } ^ { t _ { k + 1 } } \\big ( f ( X _ { t } , \\theta _ { 0 } ) - f ( X _ { t _ { k } } , \\theta _ { 0 } ) \\big ) \\mathrm { d } t \\rightarrow 0 , \\end{align*}"}
{"id": "8078.png", "formula": "\\begin{align*} \\mathfrak { A } \\chi \\theta _ { \\Sigma , \\epsilon } ( F _ H ) = \\theta _ { \\widetilde { \\Sigma } , \\widetilde { \\epsilon } } \\mathfrak { A } _ { \\ell } ( \\rho , \\chi ) ( F _ H ) + \\mathfrak { I } _ S ( \\widetilde { \\mathcal { M } } ) , \\end{align*}"}
{"id": "1745.png", "formula": "\\begin{align*} \\left [ \\nabla ^ { E } _ V , \\mathbf { c } ( V ' ) \\right ] = \\mathbf { c } ( \\nabla ^ { T X } _ V V ' ) , V , V ' \\in C ^ \\infty ( X , T X ) \\end{align*}"}
{"id": "7958.png", "formula": "\\begin{align*} \\mu _ { \\tau , T } : = \\sum _ { t = 1 } ^ { \\lfloor T / \\tau \\rfloor } \\delta _ { B _ { t \\tau } ^ H } \\tau , \\end{align*}"}
{"id": "4279.png", "formula": "\\begin{align*} w _ t + a _ n ( t , x ) \\cdot w _ x ~ = ~ F ^ { ( k ) } ( t , x ) , w ( \\tau _ 0 , \\cdot ) ~ = ~ \\overline { w } ( \\cdot ) \\end{align*}"}
{"id": "1941.png", "formula": "\\begin{align*} B ( x ) : = \\max \\left \\lbrace \\beta _ 1 ( x ) , \\beta _ 2 ( x ) , \\beta _ 3 ( x ) \\right \\rbrace , \\end{align*}"}
{"id": "257.png", "formula": "\\begin{align*} x _ { \\kappa , c } ( \\sum _ { i \\in I } w _ i ) = \\sum _ { i \\in I } u _ i w _ i , w _ i \\in F _ i . \\end{align*}"}
{"id": "784.png", "formula": "\\begin{align*} ( a b ( a + b ) x ) ' = ( a b ) ' ( 1 + x ^ 2 ) . \\end{align*}"}
{"id": "2222.png", "formula": "\\begin{align*} \\| P \\big ( & F ( X ^ { n } ( t ) ) - F ( X ^ { n } ( s ) ) \\big ) \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & \\leq C \\| X ^ { n } ( t ) - X ^ { n } ( s ) \\| _ { L ^ { 2 p } ( \\Omega ; H ) } \\Big ( 1 + \\big ( \\sup _ { s \\in [ 0 , T ] } \\| X ^ { n } ( s ) \\| _ { L ^ { 2 p } ( \\Omega ; H ^ { \\delta _ 0 } ) } \\big ) \\Big ) ^ 2 \\\\ & \\leq C | t - s | ^ { \\frac { \\delta _ 0 } 4 } , \\end{align*}"}
{"id": "3684.png", "formula": "\\begin{align*} B = \\nabla \\times ( - \\Delta ) ^ { - 1 } J . \\end{align*}"}
{"id": "6119.png", "formula": "\\begin{align*} a _ { n + 2 } ( x ) = \\frac { - \\varepsilon } { A ^ n _ 1 } \\sum _ { k = m + 2 } ^ \\infty \\left [ k + ( m + 2 ) \\right ] \\left [ k + ( m + 1 ) \\right ] a _ { k + ( m + 2 ) } x ^ { k } \\ , . \\end{align*}"}
{"id": "3514.png", "formula": "\\begin{align*} D _ 1 \\ll ( t _ 3 ^ { - \\sigma _ 3 } + 1 ) \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > \\frac { 3 } { 2 } ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 & ( \\sigma _ 2 = \\frac { 3 } { 2 } ) \\\\ t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < \\frac { 3 } { 2 } ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "8269.png", "formula": "\\begin{align*} \\Delta ' ( \\mathbf { F } _ { u } ) = & \\sum _ { p = 0 } ^ n \\sum _ { u \\leq v _ { 1 } \\times v _ { 2 } } \\mathbf { M } _ { v _ { 1 } } \\otimes \\mathbf { M } _ { v _ { 2 } } = \\sum _ { p = 0 } ^ n \\sum _ { u _ { ( 1 ) } ^ p \\leq v _ { 1 } } \\mathbf { M } _ { v _ { 1 } } \\otimes \\sum _ { u _ { ( 2 ) } ^ p \\leq v _ { 2 } } \\mathbf { M } _ { v _ { 2 } } \\\\ = & \\sum _ { p = 0 } ^ n \\mathbf { F } _ { u _ { ( 1 ) } ^ p } \\otimes \\mathbf { F } _ { u _ { ( 2 ) } ^ p } = \\Delta ( \\mathbf { F } _ { u } ) , \\end{align*}"}
{"id": "8967.png", "formula": "\\begin{align*} \\int _ { t } ^ { T } C _ p q \\left | \\dot { \\xi } ( s ) \\right | ^ q d s = C _ p ( q - 1 ) p ^ q \\left [ f ( \\xi ( T ) ) - u ( \\xi ( T ) ) - \\left ( f ( \\xi ( t ) ) - u ( \\xi ( t ) ) \\right ) \\right ] + f ( \\xi ( t ) ) - f ( \\xi ( T ) ) . \\end{align*}"}
{"id": "7007.png", "formula": "\\begin{align*} R ( \\beta _ j ) A ( \\beta _ j ) = 1 , ( R A ) ^ { ( k ) } ( \\beta _ j ) = 0 \\quad 1 \\le k \\le \\nu _ j - 1 ; \\end{align*}"}
{"id": "3160.png", "formula": "\\begin{align*} \\epsilon _ k = \\frac { 1 } { 2 } \\max _ { j \\in \\lbrace 1 , \\ldots , n \\rbrace } \\frac { | e _ j ^ \\intercal ( A x _ k - b ) | ^ 2 } { \\norm { A x _ k - b } _ 2 ^ 2 \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 } + \\frac { 1 } { 2 \\norm { A } _ F ^ 2 } ; \\end{align*}"}
{"id": "1062.png", "formula": "\\begin{align*} q | _ { \\mathcal { R } _ { \\xi , I I } } ( x , t ) = D ^ { - 2 } _ { \\infty } ( \\xi ) \\sqrt { - \\frac { x } { 6 t } } + t ^ { - 1 } f _ { \\mathcal { R } _ { \\xi , I I } } ( x , t ) + \\mathcal { O } ( t ^ { - 2 } ) . \\end{align*}"}
{"id": "7228.png", "formula": "\\begin{align*} s - \\mathcal T _ { t , x _ 1 , v _ 1 } = \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } - ( t - s ) \\geq \\frac 1 2 \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } \\geq \\frac 1 4 \\check \\tau _ { t , x } s \\geq t - \\frac { \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } } 2 = : s ^ \\ast _ { t , x , v _ 1 } . \\end{align*}"}
{"id": "2708.png", "formula": "\\begin{align*} \\langle f _ 1 \\otimes f _ 2 , g _ 1 \\otimes g _ 2 \\rangle = \\langle f _ 1 , g _ 1 \\rangle _ 1 \\langle f _ 2 , g _ 2 \\rangle _ 2 , \\forall f _ 1 , g _ 1 \\in \\mathcal { H } _ 1 , \\ ; \\forall f _ 2 , g _ 2 \\in \\mathcal { H } _ 2 . \\end{align*}"}
{"id": "2514.png", "formula": "\\begin{align*} \\rho ( x , \\omega , e ^ { 2 \\pi i \\tau } ) = e ^ { 2 \\pi i \\tau } e ^ { \\pi i x \\cdot \\omega } T _ x M _ \\omega . \\end{align*}"}
{"id": "8658.png", "formula": "\\begin{align*} a p ( x , y ) ^ 2 + b q ( x , y ) ^ 2 + c r ( x , y ) ^ 2 = 0 \\end{align*}"}
{"id": "9405.png", "formula": "\\begin{align*} \\tau ( a _ 1 \\cdots a _ n ) & = 0 , \\\\ \\varphi ( a _ 1 \\cdots a _ n ) & = \\varphi ( a _ 1 ) \\cdots \\varphi ( a _ n ) . \\end{align*}"}
{"id": "7671.png", "formula": "\\begin{align*} \\Delta v _ { \\varepsilon } ( \\mathbf { x } ) = \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\frac { ( N - 1 ) } { | \\mathbf { x } - \\mathbf { y } _ { \\varepsilon } | } , \\end{align*}"}
{"id": "1703.png", "formula": "\\begin{align*} \\| f \\| _ { p _ i , q , T } = \\left ( \\sum \\limits _ { j = 1 } ^ n \\| f | _ { E _ j } \\| _ { Y _ q ( E _ j ) } ^ { p _ i } \\right ) ^ { \\frac { 1 } { p _ i } } , i = 0 , \\ , 1 . \\end{align*}"}
{"id": "8107.png", "formula": "\\begin{align*} \\left \\langle ( \\rho _ 0 ^ { * } \\partial _ { \\Sigma _ 0 } ) ^ { \\otimes 2 } E _ { \\mathcal { M } _ 0 } , f \\otimes g \\right \\rangle = \\int _ { \\mathcal { I } } f \\wedge d g \\end{align*}"}
{"id": "1563.png", "formula": "\\begin{align*} g ( w ) = \\det \\left ( U ( w ) ^ { - 1 } \\mathfrak { p } ( w ) \\right ) ^ k \\det ( \\mathfrak { p } ( w ) ) ^ { - k } f ( w ) . \\end{align*}"}
{"id": "3030.png", "formula": "\\begin{align*} \\frac { \\partial f _ { 0 } } { \\partial y _ { j _ { 1 } \\ldots j _ { k } } ^ { \\sigma } } - d _ { i } f _ { \\sigma } ^ { i , j _ { 1 } \\ldots j _ { k } } - f _ { \\sigma } ^ { j _ { k } , j _ { 1 } \\ldots j _ { k - 1 } } & = 0 \\qquad \\mathrm { S y m } ( j _ { 1 } \\ldots j _ { k } ) , \\ , \\ , k \\leq s , \\\\ \\frac { \\partial f _ { 0 } } { \\partial y _ { j _ { 1 } \\ldots j _ { s + 1 } } ^ { \\sigma } } - f _ { \\sigma } ^ { j _ { s + 1 } , j _ { 1 } \\ldots j _ { s } } & = 0 \\qquad \\mathrm { S y m } ( j _ { 1 } \\ldots j _ { s + 1 } ) . \\end{align*}"}
{"id": "5339.png", "formula": "\\begin{align*} \\sum \\limits _ { \\ell = 0 } ^ { 2 } \\frac { d } { d t } \\langle \\nabla ^ { \\ell } u ^ { \\epsilon } , \\epsilon \\nabla ^ { \\ell + 1 } \\phi ^ { \\epsilon } \\rangle + \\frac { 3 } { 4 } \\| \\nabla \\phi ^ { \\epsilon } \\| _ { H ^ 2 _ { \\frac { P ' ( \\rho ^ { \\epsilon } ) } { \\rho ^ { \\epsilon } } } } ^ 2 \\lesssim D ( u ^ { \\epsilon } , \\eta ^ { \\epsilon } , \\tau ^ { \\epsilon } ) + \\| \\mathrm { d i v } u ^ { \\epsilon } \\| _ { H ^ 2 } ^ 2 , \\end{align*}"}
{"id": "6612.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { c | m h \\pm e ^ { \\xi } n k | } { g x Q } W \\left ( \\frac { c | m h \\pm e ^ { \\xi } n k | } { g x Q } \\right ) x ^ { w - 1 } \\ , d x = \\left ( \\frac { c | m h \\pm e ^ { \\xi } n k | } { g Q } \\right ) ^ { w } \\widetilde { W } ( 1 - w ) . \\end{align*}"}
{"id": "5797.png", "formula": "\\begin{align*} H ^ n ( \\mathcal { Z } _ K ; k ) \\cong \\bigoplus _ { n = 2 | J | - i } \\operatorname { T o r } ^ S _ i ( k [ K ] , k ) _ J . \\end{align*}"}
{"id": "1915.png", "formula": "\\begin{align*} \\mathbb { S } ( \\nabla u ) = \\mu _ 1 ( \\nabla _ x u + \\nabla _ x ^ \\top u ) + \\mu _ 2 { \\rm { d i v } } u \\mathbb { I } , \\end{align*}"}
{"id": "6916.png", "formula": "\\begin{align*} r = h _ 1 B _ 1 ( \\theta _ 1 ) s _ 1 + h _ 2 B _ 2 ( \\theta _ 2 ) s _ 2 + n , \\end{align*}"}
{"id": "799.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ \\Omega | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla \\phi _ k \\ , d \\mu = \\int _ \\Omega | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla v \\ , d \\mu . \\end{align*}"}
{"id": "2325.png", "formula": "\\begin{align*} W ( \\widehat { f } , \\widehat { g } ) ( x , \\omega ) & = 2 ^ d e ^ { 4 \\pi i x \\cdot \\omega } \\langle \\widehat { f } , M _ { 2 \\omega } T _ { 2 x } \\widehat { g } ^ \\vee \\rangle \\\\ & = 2 ^ d e ^ { 4 \\pi i x \\cdot \\omega } \\langle f , T _ { - 2 \\omega } M _ { 2 x } g ^ \\vee \\rangle \\\\ & = 2 ^ d e ^ { - 4 \\pi i x \\cdot \\omega } \\langle f , M _ { 2 x } T _ { - 2 \\omega } g ^ \\vee \\rangle \\\\ & = W ( f , g ) ( - \\omega , x ) . \\end{align*}"}
{"id": "7065.png", "formula": "\\begin{align*} { d \\widetilde A ^ K _ { i } ( t ) \\over d t } < \\bar p e ^ { \\varepsilon _ K \\log K } \\sum _ { \\ell = - \\lfloor 1 / 2 \\delta _ K \\rfloor } ^ { 1 / \\delta _ K - 1 - \\lfloor 1 / 2 \\delta _ K \\rfloor } h _ { K } G ( h _ { K } \\ell ) \\exp ( \\log K ( \\widetilde A ^ K _ { i + \\ell } ( t ) - \\widetilde A ^ K _ { i } ( t ) ) - 2 \\bar p . \\end{align*}"}
{"id": "4455.png", "formula": "\\begin{align*} ( a ^ { \\dagger } a ) ^ { k } = \\sum _ { l = 0 } ^ { k } S _ { 2 } ( k , l ) ( a ^ { \\dagger } ) ^ { l } a ^ { l } , ( \\mathrm { s e e } \\ [ 3 , 7 ] ) . \\end{align*}"}
{"id": "5109.png", "formula": "\\begin{align*} \\psi ( x ) > q _ 1 ( \\sqrt { 0 . 5 } ) + 0 = 0 . \\end{align*}"}
{"id": "406.png", "formula": "\\begin{align*} A ^ { 0 } = \\left ( \\begin{array} { c c c } A _ { 1 } ^ { 0 } & 0 & 0 \\\\ 0 & A _ { 2 } ^ { 0 } & 0 \\\\ 0 & 0 & A _ { 3 } ^ { 0 } \\\\ \\end{array} \\right ) \\in \\mathbb { M } _ { N \\times N } , \\end{align*}"}
{"id": "4400.png", "formula": "\\begin{align*} { H } ^ { - 1 } f ( \\xi ) = \\Lambda Q ( \\xi ) \\int _ 0 ^ \\xi \\frac { { L } f ( \\xi ' ) } { \\Lambda Q ( \\xi ' ) } d \\xi ' , \\end{align*}"}
{"id": "5911.png", "formula": "\\begin{align*} \\left ( \\frac { \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , | \\lambda _ i | } { 2 k } \\right ) ^ 2 \\geq \\frac { 1 } { \\displaystyle { \\frac { 2 k ( 2 k - 1 ) } { 2 } } } \\ , \\sum \\limits _ { i < j } \\ , | \\lambda _ i | \\ , | \\lambda _ j | , \\end{align*}"}
{"id": "7942.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } L _ n ^ { 1 / 2 } n ^ { 5 / 4 } & \\ll \\sum _ { m = 0 } ^ { \\ell - 1 } p _ { m + 1 } ^ { 9 / 4 } ( p _ { m + 1 } - p _ m ) ^ { 3 / 2 } \\\\ & \\leqslant \\sum _ { m = 0 } ^ { \\ell - 1 } p _ { m + 1 } ^ { 1 1 / 4 } ( p _ { m + 1 } - p _ m ) \\\\ & \\leqslant p _ { \\ell } ^ { 1 1 / 4 } \\sum _ { m = 0 } ^ { \\ell - 1 } ( p _ { m + 1 } - p _ m ) \\\\ & \\leqslant p _ { \\ell } ^ { 1 5 / 4 } . \\end{align*}"}
{"id": "3397.png", "formula": "\\begin{align*} f = \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\widetilde { D } _ { k } D _ { k } ( f ) = \\sum \\limits _ { k = - \\infty } ^ { \\infty } D _ { k } \\bar { D } _ { k } ( f ) , \\end{align*}"}
{"id": "9519.png", "formula": "\\begin{align*} L ( x , p , y ) & = \\begin{cases} + \\infty & , \\\\ E l ( x , y ) - \\langle x , p \\rangle & , \\\\ - \\infty & . \\end{cases} \\end{align*}"}
{"id": "4648.png", "formula": "\\begin{align*} H ^ { \\theta \\sim \\alpha } = \\{ h \\in H \\mid ( \\theta - \\alpha ) ^ n ( h ) = 0 \\ ; n \\ge 1 \\} \\end{align*}"}
{"id": "5885.png", "formula": "\\begin{align*} \\alpha ' = \\alpha \\rho > \\alpha . \\end{align*}"}
{"id": "1053.png", "formula": "\\begin{align*} P _ M ( \\bar f ^ { \\lambda + M } ) = b ( \\lambda ) \\dots b ( \\lambda + M - 1 ) \\bar f ^ { \\lambda } . \\end{align*}"}
{"id": "4303.png", "formula": "\\begin{align*} \\varepsilon ( y , \\tau ) = w ( y , \\tau ) - Q _ { b ( \\tau ) } ( y ) . \\end{align*}"}
{"id": "5564.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X ( t ) & = & \\big ( \\frac { \\kappa } { 2 } \\frac { d ^ 2 } { d x ^ 2 } X ( t ) + \\frac { d } { d x } X ( t ) + \\alpha \\big ) d t + \\sigma d W ( t ) \\medskip \\\\ X ( 0 ) & = & h \\end{array} \\right . \\end{align*}"}
{"id": "3785.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ E _ r \\big ( \\widetilde { V } , \\widetilde { \\mathcal { S } } \\big ) q _ F ^ { - \\frac { r } { 2 } f } X ^ { - f r } = \\gamma ( X , \\pi _ E , \\widehat { \\sigma _ E } , \\psi _ E ) \\sum _ { r \\in \\mathbb { Z } } c ^ E _ r ( V , \\mathcal { S } ) q _ F ^ { \\frac { r } { 2 } f } X ^ { f r } \\end{align*}"}
{"id": "5593.png", "formula": "\\begin{align*} \\mathcal { F } = 1 \\otimes 1 + \\mathcal { O } ( \\nu ) \\in ( U \\Xi \\otimes U \\Xi ) [ [ \\nu ] ] \\end{align*}"}
{"id": "2737.png", "formula": "\\begin{align*} M ^ { ( n - k , k ) } \\cong _ { \\mathfrak { S } _ { n } } \\bigoplus _ { j = 0 } ^ { k } \\mathbb { S } ^ { n - j , j } \\end{align*}"}
{"id": "9453.png", "formula": "\\begin{align*} f _ 2 = Y ^ { p ^ s } - \\tfrac { b _ 1 } { a _ 0 } Y ^ { p ^ { s - 1 } } X ^ { p ^ s - p ^ { s - 1 } } - \\tfrac { b _ 2 } { a _ 0 } \\Delta ^ { p ^ { s - 2 } } X ^ { p ^ s - 2 p ^ { s - 2 } } \\ldots - \\tfrac { b _ { r - 1 } } { a _ 0 } \\Delta X ^ { p ^ s - 2 } - \\tfrac { b _ r } { a _ 0 } Y X ^ { p ^ s - 1 } . \\end{align*}"}
{"id": "4332.png", "formula": "\\begin{align*} \\Psi ( \\tau ) = b ( \\tau ) \\exp \\left ( \\left ( \\frac { 2 } { \\alpha } - 1 \\right ) \\left ( \\int _ { \\tau _ 0 } ^ \\tau 2 \\beta ( \\tau ' ) d \\tau ' + \\tau _ 0 \\right ) \\right ) \\Psi ( \\tau _ 0 ) = 1 , \\end{align*}"}
{"id": "3025.png", "formula": "\\begin{align*} d Z _ { \\lambda } = 0 \\quad \\mathrm { i f \\ , a n d \\ , o n l y \\ , i f } E _ { \\lambda } = 0 , \\end{align*}"}
{"id": "4434.png", "formula": "\\begin{align*} H ( t ) = E ( t ) + \\epsilon \\int _ U u u _ t ( x , t ) \\ , d x \\end{align*}"}
{"id": "3572.png", "formula": "\\begin{align*} S _ 2 & = { \\bigcup } _ { u = 0 } ^ { n - 1 } { \\bigcup } _ { z = 1 } ^ { e } U ( \\nu _ { u , z } ) = { \\bigcup } _ { b = 1 } ^ c { \\bigcup } _ { u = n _ { b - 1 } } ^ { n _ b - 1 } { \\bigcup } _ { z = q _ b } ^ { e } U ( \\nu _ { u , z } ) \\\\ & = { \\bigcup } _ { z = q _ c } ^ { e } { \\bigcup } _ { b = b _ z } ^ { c } { \\bigcup } _ { u = n _ { b - 1 } } ^ { n _ b - 1 } U ( \\nu _ { u , z } ) = { \\bigcup } _ { z = q _ c } ^ { e } { \\bigcup } _ { u = n _ { b _ z - 1 } } ^ { n - 1 } U ( \\nu _ { u , z } ) , \\end{align*}"}
{"id": "7852.png", "formula": "\\begin{align*} \\bigg \\| \\frac { T ^ { * 2 K } x } { x _ { 2 K } w _ { 2 K - 1 } \\cdots w _ { 0 } } - e _ { 0 } \\bigg \\| ^ { 2 } & \\leq \\frac { \\mu ^ { 2 } \\delta } { w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } } \\sum _ { i = 0 } ^ { \\infty } w _ { i + 2 K } ^ { 2 } ~ \\bigg | \\frac { x _ { i + 2 K + 1 } } { x _ { 2 K } } \\bigg | ^ { 2 } \\\\ & \\leq \\frac { \\mu ^ { 2 } \\delta } { w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } } \\sum _ { i = 0 } ^ { \\infty } w _ { i + 2 K } ^ { 2 } < C \\epsilon , \\end{align*}"}
{"id": "6057.png", "formula": "\\begin{align*} X = \\left [ X _ 1 , \\left [ X _ 2 , \\ , \\cdots \\ , \\left [ X _ { n - 1 } , X _ { n } \\right ] \\right ] \\right ] \\end{align*}"}
{"id": "7106.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & ( \\lambda ^ 2 - 4 ) \\left [ ( \\lambda ^ 2 - 4 ) P _ { n - 2 } - 2 ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 2 } \\right ] - 2 ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\\\ P _ { n } ( \\lambda ) = & ( \\lambda ^ 2 - 4 ) ^ 2 P _ { n - 2 } - 2 \\times 2 ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\end{align*}"}
{"id": "3206.png", "formula": "\\begin{align*} { \\rm S I N R } _ { D _ 1 } ^ { x _ 2 } = \\frac { | h _ 1 | ^ 2 ( 1 - \\rho ) P _ t \\alpha _ 2 } { | h _ 1 | ^ 2 ( 1 - \\rho ) P _ t \\alpha _ 1 + | h _ { S I } | ^ 2 P _ H + \\sigma ^ 2 } , \\end{align*}"}
{"id": "440.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } U _ { t } + A ^ { i } \\partial _ { i } U - D U & = F , \\\\ \\left . V \\right \\rvert _ { t = 0 } & = U _ { 0 } , \\end{aligned} \\end{align*}"}
{"id": "6652.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq m < n < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) } { p ^ { m \\beta } p ^ { n ( 1 - \\beta ) } } = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\tau _ { A \\cup \\{ - \\beta \\} } ( p ^ m ) \\tau _ B ( p ^ m ) } { p ^ { m } } - \\sum _ { m = 0 } ^ { \\infty } \\frac { \\tau _ { A } ( p ^ m ) \\tau _ B ( p ^ m ) } { p ^ { m } } . \\end{align*}"}
{"id": "9221.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land p \\in A ( x - _ X \\gamma p ) \\rightarrow p = _ X A _ \\gamma x \\right ) . \\end{align*}"}
{"id": "6546.png", "formula": "\\begin{align*} \\epsilon ' = b _ 1 ^ * - \\frac { k - r } { r } \\epsilon = \\frac { k } { r } - b _ 2 ^ * \\ , . \\end{align*}"}
{"id": "7889.png", "formula": "\\begin{align*} \\int _ { Q _ 1 } \\varphi _ { - } ( x ) \\nu _ { R , L } ( x , w ) \\ d x & \\leq \\int _ { Q _ 1 } \\varphi _ { - } ( x ) \\frac { n _ L ( C _ { R / L } ( x ) ) } { | C _ R | } \\ d x \\\\ & \\leq \\int _ { Q _ 1 } \\int _ { C _ { R / L } ( x ) } \\frac { \\varphi ( y ) } { | C _ R | } \\ d n _ L ( y ) \\ d x = \\frac { 1 } { L ^ n } \\int _ U \\varphi ( y ) \\ d n _ L ( y ) \\\\ & \\leq \\int _ { Q _ 1 } \\varphi _ { + } ( x ) \\frac { n _ L ( C _ { R / L } ( x ) ) } { | C _ R | } \\ d x . \\end{align*}"}
{"id": "557.png", "formula": "\\begin{align*} | J _ { 4 , a } | + | J _ { 4 , c } | & \\leq \\frac { p ' ( \\varrho ) } { \\varrho } \\varepsilon ^ { - 1 } \\| \\rho - \\varrho \\| _ { L ^ \\infty ( 0 , T ; L ^ 2 ( \\Omega _ R ) ) } \\left ( \\| v \\| _ { L ^ \\infty ( 0 , T ; L ^ { q _ 1 } ( \\Omega _ R ) ) } + \\| \\nabla \\Psi \\| _ { L ^ \\infty ( 0 , T ; L ^ { q _ 1 } ( \\Omega _ R ) ) } \\right ) \\| \\nabla s \\| _ { L ^ 1 ( 0 , T ; L ^ { { q _ 2 } } ( \\Omega _ R ) ) } \\\\ & \\leq c \\varepsilon ^ { 1 - \\frac { 2 } { q _ 2 } } \\end{align*}"}
{"id": "2085.png", "formula": "\\begin{align*} \\min \\{ v _ p ( u _ i ) , k \\} = \\min \\{ v _ p ( u _ 1 ) + ( i - 1 ) b , k \\} \\end{align*}"}
{"id": "6960.png", "formula": "\\begin{align*} \\ker D ^ * \\Phi ( \\bar x , \\bar y ) = \\{ 0 \\} \\end{align*}"}
{"id": "7495.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { j k } ^ { n + 1 } = \\sum _ { p = - M / 2 } ^ { M / 2 - 1 } \\sum _ { q = - M / 2 } ^ { M / 2 - 1 } \\frac { \\widehat { ( \\mathcal { H } ^ n ) } _ { p q } } { 1 + \\frac { \\tau } { 2 } \\eta ^ n + \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n + \\frac { \\tau ^ 2 } { 4 } \\left ( ( \\varrho ^ { x } _ { p } ) ^ 2 + ( \\varrho ^ { y } _ { q } ) ^ 2 \\right ) } \\ , e ^ { i \\frac { 2 j p \\pi } { M } } e ^ { i \\frac { 2 k q \\pi } { M } } . \\end{align*}"}
{"id": "3061.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = j ^ 2 x _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 1 ^ \\prime = j y _ 1 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , , \\left ( j = e ^ { \\frac { 2 \\pi i } { 3 } } \\right ) \\end{align*}"}
{"id": "2069.png", "formula": "\\begin{align*} f _ { \\alpha } ( z ) = \\sum _ { \\beta \\leq \\alpha } a _ { \\beta } g _ { \\beta } ( z ) . \\end{align*}"}
{"id": "3741.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb S ^ 1 } B _ x D ^ m J D ^ m B \\ , d x \\right | \\leq & \\ \\| B _ x \\| _ { L ^ \\infty } \\| D ^ m B \\| _ { L ^ 2 } \\| D ^ m J \\| _ { L ^ 2 } \\\\ \\left | \\int _ { \\mathbb S ^ 1 } B D ^ m J D ^ m B _ x \\ , d x \\right | \\leq & \\ \\| B \\| _ { L ^ \\infty } \\| D ^ m B _ x \\| _ { L ^ 2 } \\| D ^ m J \\| _ { L ^ 2 } \\\\ \\left | \\int _ { \\mathbb S ^ 1 } J D ^ m B _ x D ^ m B \\ , d x \\right | \\leq & \\ \\| J _ x \\| _ { L ^ \\infty } \\| D ^ m B \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
{"id": "8696.png", "formula": "\\begin{align*} g ( x ) : = \\Bigl ( x _ 1 - \\frac { 7 } { 4 } \\Bigr ) ^ 2 - \\frac { 7 } { 4 } x _ 1 + h ( x _ 2 ) + 3 ( - x _ 2 + 2 ) - \\left ( \\frac { 3 } { 4 } \\right ) ^ 2 - \\frac { 5 } { 4 } , \\end{align*}"}
{"id": "4095.png", "formula": "\\begin{align*} \\widetilde \\mu ( x ) = \\mu ( x ) , \\ , \\widetilde \\sigma ( x ) = \\sigma ( x ) . \\end{align*}"}
{"id": "5195.png", "formula": "\\begin{align*} \\Omega ( \\mathfrak { p } ^ j x ) & = \\Omega ( x ) ~ ~ j \\in \\mathbb { Z } , \\\\ \\int \\limits _ { \\mathfrak { D ^ { * } } } \\Omega ( x ) d x & = 0 . \\end{align*}"}
{"id": "8694.png", "formula": "\\begin{align*} \\begin{aligned} L _ 1 ( f ) & : = 4 f _ 1 ^ { 0 . 8 } + 2 ^ { 2 . 8 } f _ 2 ^ { 0 . 8 } - 2 ^ { 2 . 8 } , \\\\ L _ 2 ( f ) & : = 2 ^ { 2 . 8 } f _ 1 ^ { 0 . 8 } + 4 f _ 2 ^ { 0 . 8 } - 2 ^ { 2 . 8 } , \\\\ R _ 1 ( f ) & : = - \\bigl ( ( 2 ^ { 0 . 6 } - 1 ) f _ 1 - 2 ^ { 0 . 6 } + 2 \\bigr ) - \\bigl ( ( 2 ^ { 0 . 6 } - 1 ) f _ 2 - 2 ^ { 0 . 6 } + 2 \\bigr ) + 1 , \\\\ R _ 2 ( f ) & : = - 2 ^ { 0 . 6 } \\bigl ( ( 2 ^ { 0 . 6 } - 1 ) f _ 1 - 2 ^ { 0 . 6 } + 2 \\bigr ) - 2 ^ { 0 . 6 } \\bigl ( ( 2 ^ { 0 . 6 } - 1 ) f _ 2 - 2 ^ { 0 . 6 } + 2 \\bigr ) + 2 ^ { 1 . 2 } , \\end{aligned} \\end{align*}"}
{"id": "3879.png", "formula": "\\begin{align*} \\Omega \\coloneqq \\{ w \\in W _ H \\mid w ( S _ G ) = S _ G \\} . \\end{align*}"}
{"id": "1521.png", "formula": "\\begin{align*} w ( g _ 1 \\times g _ 2 ) = \\left [ \\begin{array} { c c c c c c c c c c } 0 & 0 & 0 & - 1 _ t & 0 & - 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & u & 0 & 0 & u ' \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \\zeta & 0 & 0 \\\\ 1 _ t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & v & 0 & 0 & v ' \\end{array} \\right ] . \\end{align*}"}
{"id": "8160.png", "formula": "\\begin{align*} _ { q _ 1 , q _ 2 , p } : = \\sum _ { x \\bmod p } \\max ( q _ 1 x , q _ 2 x ) \\end{align*}"}
{"id": "8247.png", "formula": "\\begin{align*} t = ( i , \\bar { j } ) ( \\bar { i } , j ) = \\Big ( \\bar { u ( i _ p ) } , \\bar { u ( i _ p + 1 ) } \\Big ) \\Big ( u ( i _ p ) , u ( i _ p + 1 ) \\Big ) = u s _ { i _ p } u ^ { - 1 } , \\end{align*}"}
{"id": "6571.png", "formula": "\\begin{align*} \\widetilde { f } ( s ) : = \\int _ 0 ^ { \\infty } f ( x ) x ^ { s - 1 } \\ , d x . \\end{align*}"}
{"id": "8360.png", "formula": "\\begin{align*} y _ n = \\sum _ { i = 1 } ^ n h _ i \\prod _ { j = i + 1 } ^ n g _ j + y _ 0 \\prod _ { j = 1 } ^ n g _ j . \\end{align*}"}
{"id": "4527.png", "formula": "\\begin{align*} \\tau ( a _ i ) = - 4 a _ i c _ 1 + a _ i + 4 ( a _ i , c _ 1 ) c _ 1 . \\end{align*}"}
{"id": "1075.png", "formula": "\\begin{align*} \\chi _ { R , L } ( k ) : \\mathbb { C } \\backslash [ - C _ { R , L } , C _ { R , L } ] \\rightarrow \\mathbb { C } , \\chi _ { R , L } ( k ) = \\left ( \\frac { k - C _ { R , L } } { k + C _ { R , L } } \\right ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
{"id": "5526.png", "formula": "\\begin{align*} B _ m ^ j ( t ) : = B ^ j ( [ t ] _ m ^ - ) + \\frac { t - [ t ] _ m ^ - } { \\delta _ m } ( B ^ j ( [ t ] _ m ^ + ) - B ^ j ( [ t ] _ m ^ - ) ) , t \\in [ 0 , T ] . \\end{align*}"}
{"id": "7754.png", "formula": "\\begin{gather*} \\alpha _ k = \\alpha + 0 + k S _ 0 + k \\tau _ 0 \\in [ - 1 5 \\pi , 0 ) , \\\\ \\beta _ k = \\beta _ 0 - k S _ 0 - k \\tau _ 0 \\in [ 3 \\pi , 1 5 \\pi ] , \\\\ x _ k \\in [ x _ { k - 1 } + S _ 0 / 2 , x _ { k - 1 } + S _ 0 ] , \\end{gather*}"}
{"id": "8255.png", "formula": "\\begin{align*} v ' _ { i _ 1 } = \\bar { r } , \\ v ' _ { i _ 2 } = \\bar { r - 1 } , \\ \\cdots , \\ v ' _ { i _ r } = \\bar { 1 } , \\ v ' _ { i _ { r + 1 } } = r + 1 , \\ v ' _ { i _ { r + 2 } } = r + 2 , \\ \\cdots , \\ v ' _ { i _ { p _ 2 } } = p _ 2 , \\end{align*}"}
{"id": "921.png", "formula": "\\begin{align*} x ^ { k } \\bigg ( \\frac { d } { d x } \\bigg ) ^ { k } f ( x ) = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k , m ) \\bigg ( x \\frac { d } { d x } \\bigg ) _ { m , \\lambda } f ( x ) . \\end{align*}"}
{"id": "815.png", "formula": "\\begin{align*} d _ \\rho ( ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) ) = \\inf _ \\gamma \\int _ \\gamma \\rho \\ , d s , \\end{align*}"}
{"id": "8039.png", "formula": "\\begin{align*} \\Phi ( f ) [ \\psi , \\pi ] = \\int _ \\Sigma f \\psi \\ , \\mathrm { d } V _ \\Sigma , \\Pi ( f ) [ \\psi , \\pi ] = \\int _ \\Sigma f \\pi \\ , \\mathrm { d } V _ \\Sigma . \\end{align*}"}
{"id": "4156.png", "formula": "\\begin{align*} \\int w ( x , t ) d x = \\int \\sigma ( x ) d x \\end{align*}"}
{"id": "6634.png", "formula": "\\begin{align*} \\mathcal { U } ^ 2 ( h , k ) = I _ 1 + I _ 2 + O \\left ( X ^ { \\varepsilon } Q ^ { 1 + \\varepsilon } \\frac { ( h k ) ^ \\varepsilon ( h , k ) } { ( h k ) ^ { 1 / 2 } } + ( X C h k ) ^ { \\varepsilon } k X ^ 2 Q ^ { - 9 7 } \\right ) , \\end{align*}"}
{"id": "6368.png", "formula": "\\begin{align*} v _ i = v _ { i - 1 } + v i = 1 , \\dots , k - 1 \\rho _ 1 = v _ { k - 1 } + v \\end{align*}"}
{"id": "6377.png", "formula": "\\begin{align*} F \\cdot K _ { \\widetilde { X } } = 4 k - 1 C _ { i , 0 } \\cdot K _ { \\widetilde { X } } = 1 C _ { i , j } \\cdot K _ { \\widetilde { X } } = 0 C _ { i , k } \\cdot K _ { \\widetilde { X } } = - 1 \\end{align*}"}
{"id": "1633.png", "formula": "\\begin{align*} G _ \\lambda f : = \\int _ 0 ^ \\infty e ^ { - \\lambda t } P _ t f d t \\end{align*}"}
{"id": "2070.png", "formula": "\\begin{align*} f _ { \\alpha } ^ { ( \\beta ) } ( o ) = f ^ { ( \\beta ) } ( o ) . \\end{align*}"}
{"id": "4052.png", "formula": "\\begin{align*} \\epsilon \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } _ x \\bar { \\sigma } _ x + \\int _ { 0 } ^ { 1 } u ^ { \\epsilon } _ x \\bar { \\sigma } + \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } _ x \\bar { \\sigma } + \\lambda \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } \\bar { \\sigma } = \\int _ { 0 } ^ { 1 } f \\bar { \\sigma } . \\end{align*}"}
{"id": "2987.png", "formula": "\\begin{align*} \\mathcal H = \\mathcal D _ - \\oplus K \\oplus \\mathcal D _ + . \\end{align*}"}
{"id": "6041.png", "formula": "\\begin{align*} \\delta _ { \\alpha , \\beta } ^ * ( \\delta _ { \\alpha , \\beta } g ) & = \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } ) ' \\big ) ' + x g ' \\\\ & - \\Big ( \\frac { \\alpha } { 2 } + \\frac { \\beta } { 2 } + 1 \\Big ) g - \\frac { [ ( \\frac { \\beta } { 2 } - \\frac { \\alpha } { 2 } ) - ( \\frac { \\alpha } { 2 } + \\frac { \\beta } { 2 } ) x ] ^ 2 } { 1 - x ^ 2 } . \\end{align*}"}
{"id": "4144.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\int x ^ 2 w ( x , t ) d x = 2 t ( I _ 3 ( \\phi ) - I _ 3 ( \\varphi ) ) . \\end{align*}"}
{"id": "452.png", "formula": "\\begin{align*} | \\frac { | \\phi ( x , y ) \\cap A \\times B | } { | \\phi ( x , y ) \\cap X _ i \\times Y _ j | } - \\frac { | A \\times B | } { | X _ i \\times Y _ j | } | = O ( q ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "3311.png", "formula": "\\begin{align*} p _ \\ell ( G _ 1 / T ( G _ 1 ) ) = p _ \\ell ( E | _ { D _ 1 } ) , \\end{align*}"}
{"id": "1546.png", "formula": "\\begin{align*} \\widetilde { c } _ k ( s ) = \\alpha ( s ) \\int _ { \\mathfrak { B } } \\det ( I + z \\bar { z } ) ^ { s + k } \\textbf { d } z , \\end{align*}"}
{"id": "4033.png", "formula": "\\begin{align*} ( p _ k + i q _ k ) ^ 2 = ( p ^ 2 _ k - q ^ 2 _ k ) + i 2 p _ k q _ k = a _ k + i b _ k , \\end{align*}"}
{"id": "2044.png", "formula": "\\begin{align*} \\mathcal { P } \\left ( \\lambda \\right ) = \\lambda I + T , \\lambda \\in \\mathbf { C } , \\end{align*}"}
{"id": "768.png", "formula": "\\begin{align*} \\mathrm { T r } _ \\omega ( P \\mathfrak { W } _ { 1 / 2 } ( c _ 1 ) ( 1 - P ) [ \\mathfrak { W } _ { 1 / 2 } ( c _ 2 ) ] ^ * P ) = \\omega \\circ \\log _ 2 ( c _ 1 c _ 2 ^ * ) . \\end{align*}"}
{"id": "7561.png", "formula": "\\begin{align*} k ' = & ( p , p + 1 , k _ 2 - 1 , p + 1 , p + 1 , p , p + 1 , p + 1 ) , \\\\ k ^ { \\mu _ 1 } = & ( p , p + 1 , k _ 2 + 1 , p + 1 , p + 1 , p , p + 1 , p + 1 ) , \\\\ k ^ { \\mu _ 2 } = & ( p , p + 1 , k _ 2 - 1 , p + 1 , p + 3 , p , p + 1 , p + 1 ) , \\\\ k ^ { \\mu _ 3 } = & ( p , p + 1 , k _ 2 - 1 , p + 1 , p + 1 , p , p + 1 , p + 3 ) . \\end{align*}"}
{"id": "6874.png", "formula": "\\begin{align*} f ' = f _ 0 ' + f _ i + \\ldots + f _ k , \\end{align*}"}
{"id": "5786.png", "formula": "\\begin{align*} \\sum _ { k : \\hat { u } _ { \\nu k } ( l ) = 0 } u _ { \\nu k } + \\sum _ { k : \\hat { u } _ { \\nu k } ( l ) = 1 } ( 1 - u _ { \\nu k } ) \\ge w _ { \\nu , \\nu , 0 } , \\ , \\forall l \\in S _ \\nu . \\end{align*}"}
{"id": "843.png", "formula": "\\begin{align*} Y = \\sqrt { P } X + Z , \\end{align*}"}
{"id": "9172.png", "formula": "\\begin{align*} \\mathcal { J } \\leq \\log \\log \\log X , \\alpha _ { 1 } = \\frac { 1 } { ( \\log \\log X ) ^ { 2 } } , \\mbox { a n d } \\sum _ { p \\leq X ^ { 1 / ( \\log \\log X ) ^ { 2 } } } \\frac { 1 } { p } \\leq \\log \\log X . \\end{align*}"}
{"id": "2204.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dd u - \\Delta w \\dd t = \\dd W ( t ) , & \\ ; \\ ; D \\times ( 0 , T ] , \\\\ w = - \\Delta u + f ( u ) , & \\ ; \\ ; D \\times ( 0 , T ] , \\\\ \\frac { \\partial u } { \\partial n } = \\frac { \\partial w } { \\partial n } = 0 , & \\ ; \\ ; \\partial D \\times ( 0 , T ] , \\\\ u ( 0 , x ) = u _ 0 , & \\ ; \\ ; D , \\end{array} \\right . \\end{align*}"}
{"id": "1148.png", "formula": "\\begin{align*} \\tilde { P } ( \\xi , k ) : = P ( \\xi , k , 1 ) = \\Delta _ { \\eta } { \\tilde { Q } } ^ { - 1 } ( \\xi , k ) ( \\Psi _ 0 ^ { A i } ) ^ { - 1 } ( f ( \\xi , k ) ) ^ { \\frac { \\sigma _ 3 } { 4 } } , \\end{align*}"}
{"id": "8952.png", "formula": "\\begin{align*} e ^ { - t } u ( y ) = e ^ { - t } u \\left ( \\xi ( t ) \\right ) = \\int _ t ^ { t + b } e ^ { - s } \\left ( C _ p \\left | \\dot { \\xi } ( s ) \\right | ^ q + f \\left ( \\xi ( s ) \\right ) \\right ) d s + e ^ { - ( b + t ) } u ( \\xi ( b + t ) ) . \\end{align*}"}
{"id": "3704.png", "formula": "\\begin{align*} & \\int _ 0 ^ t t ^ { \\frac { \\beta } { \\alpha } } ( t - \\tau ) ^ { - 1 + \\frac { \\beta } { \\alpha } } \\tau ^ { - \\frac { 2 \\beta } { \\alpha } } \\ , d \\tau \\\\ = & \\int _ 0 ^ 1 ( 1 - \\tau ' ) ^ { - 1 + \\frac { \\beta } { \\alpha } } ( \\tau ' ) ^ { - 1 + ( 1 - \\frac { 2 \\beta } { \\alpha } ) } \\ , d \\tau ' \\\\ \\lesssim & \\ 1 \\end{align*}"}
{"id": "8904.png", "formula": "\\begin{align*} A _ \\ast ( U ) = \\begin{cases} A & U \\mbox { i n f i n i t e } \\\\ 0 & U \\mbox { f i n i t e } \\end{cases} \\end{align*}"}
{"id": "4152.png", "formula": "\\begin{align*} N _ 2 : = \\sup _ { [ - T , T ] } \\| u ( t ) \\| _ { Z _ { s , 4 + \\theta } } \\end{align*}"}
{"id": "8981.png", "formula": "\\begin{align*} v ( x ) : = \\left \\{ \\begin{aligned} & \\max \\{ u ( x ) , \\phi ( x ) + \\epsilon ^ 2 \\} , x \\in \\mathrm { B ( x _ 0 , 2 \\epsilon ) } \\cap \\overline { \\Omega } , \\\\ & u ( x ) , x \\in \\overline { \\Omega } \\setminus \\mathrm { B ( x _ 0 , 2 \\epsilon ) } . \\end{aligned} \\right . \\end{align*}"}
{"id": "2022.png", "formula": "\\begin{align*} R ^ A g _ x ( z ) \\leq \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ n } = 1 . \\end{align*}"}
{"id": "1790.png", "formula": "\\begin{align*} { \\rm A S } _ g ( D ) : = \\frac { \\widehat { A } \\big ( \\frac { R _ { X ^ g } } { 2 \\pi i } \\big ) \\operatorname { t r } \\big ( g \\exp ( \\frac { R ^ F } { 2 \\pi i } ) \\big ) \\exp ( \\operatorname { t r } ( \\frac { R ^ L } { 2 \\pi i } ) ) } { \\operatorname { d e t } \\big ( 1 - g \\exp ( - \\frac { R ^ { \\mathcal { N } } } { 2 \\pi i } ) \\big ) ^ { \\frac { 1 } { 2 } } } . \\end{align*}"}
{"id": "1012.png", "formula": "\\begin{align*} \\partial _ 1 u ( a ) = \\tau \\partial _ 1 \\big \\vert _ { x = a } \\big ( \\zeta ( x ) ( \\rho - \\vert x \\vert ) ^ { - n - 2 } \\big ) = \\tau ( \\partial _ 1 \\zeta ( a ) ) ( \\rho - \\vert a \\vert ) ^ { - n - 2 } . \\end{align*}"}
{"id": "4616.png", "formula": "\\begin{gather*} \\psi _ { X \\otimes Y , Z } = ( \\psi _ { X , Z } \\otimes Y ) ( X \\otimes \\psi _ { Y , Z } ) , \\psi _ { X , Y \\otimes Z } = ( Y \\otimes \\psi _ { X , Z } ) ( \\psi _ { X , Y } \\otimes Z ) \\end{gather*}"}
{"id": "7871.png", "formula": "\\begin{align*} G \\cap I & = ( ( F \\setminus \\{ \\bar { b } _ n \\} ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( L ) ) \\cap ( D _ 1 \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) ) \\\\ & \\subseteq ( ( F \\setminus \\{ \\bar { b } _ n \\} ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( L ) ) \\cap ( ( D \\setminus \\{ \\bar { b } _ n \\} ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) ) \\end{align*}"}
{"id": "8260.png", "formula": "\\begin{align*} \\mathfrak { B } _ 1 \\times \\mathfrak { B } _ { 2 , \\emptyset } = [ 1 2 3 , \\bar 1 \\ , 3 2 ] , \\mathfrak { B } _ 1 \\times \\mathfrak { B } _ { 2 , \\{ 1 \\} } = [ 1 \\bar { 2 } 3 , \\bar 1 \\ , \\bar 3 2 ] , \\mathfrak { B } _ 1 \\times \\mathfrak { B } _ { 2 , \\{ 2 \\} } = [ 1 3 \\bar { 2 } , \\bar 1 2 \\bar { 3 } ] \\end{align*}"}
{"id": "3481.png", "formula": "\\begin{align*} \\sum _ { a < n \\leq b } g ( n ) e ^ { 2 \\pi i f ( n ) } & = \\sum _ { \\substack { \\nu \\in \\mathbb { Z } \\\\ \\alpha - \\eta < \\nu < \\beta + \\eta } } \\int _ a ^ b g ( x ) e ^ { 2 \\pi i ( f ( x ) - \\nu x ) } d x \\\\ & \\quad + O \\left ( g ( a ) \\log ( \\beta - \\alpha + 2 ) \\right ) + O \\left ( \\abs { g ^ \\prime ( a ) } \\right ) \\end{align*}"}
{"id": "307.png", "formula": "\\begin{align*} I ( \\rho ) = \\frac 1 2 \\sum _ { i = 1 } ^ N \\sum _ { j \\in N ( i ) } \\widetilde \\omega _ { i j } | \\log ( \\rho _ i ) - \\log ( \\rho _ j ) | ^ 2 \\widetilde \\theta _ { i j } ( \\rho ) , \\end{align*}"}
{"id": "5662.png", "formula": "\\begin{align*} \\theta \\varphi \\theta ^ { - 1 } & = ( \\tau \\sigma \\tau ^ { - 1 } , ( \\alpha _ { ( \\tau \\sigma ^ { - 1 } ) 1 } \\varphi _ { \\tau 1 } \\alpha ^ { - 1 } _ { \\tau 1 } , \\alpha _ { ( \\tau \\sigma ^ { - 1 } ) 2 } \\varphi _ { \\tau 2 } \\alpha ^ { - 1 } _ { \\tau 2 } , \\alpha _ { ( \\tau \\sigma ^ { - 1 } ) 3 } \\varphi _ { \\tau 3 } \\alpha ^ { - 1 } _ { \\tau 3 } ) ) \\\\ & = ( \\tau \\sigma \\tau ^ { - 1 } , ( \\beta _ 1 , \\beta _ 2 , \\beta _ 3 ) ) . \\end{align*}"}
{"id": "3534.png", "formula": "\\begin{align*} \\zeta _ { M T , 2 } ( s _ 1 , s _ 2 , s _ 3 ) = \\sum _ { m \\leq b t _ 3 } \\sum _ { n \\leq b t _ 3 } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + O ( t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) . \\end{align*}"}
{"id": "901.png", "formula": "\\begin{align*} \\widetilde \\alpha _ q ( 2 T , 2 S ) = 2 ^ { m \\delta _ { 2 , p } } \\alpha _ q ( S , T ) \\end{align*}"}
{"id": "4359.png", "formula": "\\begin{align*} H \\left ( T _ { i + 1 } \\right ) = T _ i . \\end{align*}"}
{"id": "8784.png", "formula": "\\begin{align*} s ( T ) _ { i j } = a _ { i 0 } + \\sum _ { k = 1 } ^ j ( a _ { i k } - a _ { i k - 1 } ) ( 1 - z _ { i n _ i + 1 - k } ) j = 0 , \\ldots , n _ i , \\end{align*}"}
{"id": "4660.png", "formula": "\\begin{align*} \\Lambda f ( y ) = \\frac { \\partial f _ c ( y ) } { \\partial c } \\bigg | _ { c = 1 } = \\frac { 1 } { p - 1 } f ( y ) + y f ' ( y ) , \\end{align*}"}
{"id": "4237.png", "formula": "\\begin{align*} K _ E = g '' ( 0 ) = \\frac { 2 a ^ 3 } { \\sqrt { D } } , \\ \\partial _ s K _ E = g ''' ( 0 ) = \\frac { 6 a ^ 4 b } { D } . \\end{align*}"}
{"id": "1413.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k \\hat q _ { j _ i j _ { i + 1 } } = 1 . \\end{align*}"}
{"id": "2487.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( | x - x _ n | , | \\omega - \\omega _ n | , | \\tau - \\tau _ n | ) = ( 0 , 0 , \\Z ) \\Leftrightarrow \\lim _ { n \\to \\infty } | \\mathbf { h } _ r - { \\mathbf { h } _ r } _ n | = ( 0 , 0 , 1 ) = \\mathbf { 1 } _ r . \\end{align*}"}
{"id": "8463.png", "formula": "\\begin{align*} \\begin{aligned} x _ { k + 1 } - x _ \\ast & = \\frac { 1 } { q } \\left ( \\sum _ { i = 1 } ^ { q } y _ k ^ { ( i ) } \\right ) - x _ \\ast \\\\ & = \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } ( y _ k ^ { ( i ) } - x _ \\ast ) \\\\ & = \\frac { 1 } { q } ( Q _ 1 + Q _ 2 + \\cdots + Q _ q ) ( x _ { k } - x _ \\ast ) . \\end{aligned} \\end{align*}"}
{"id": "4167.png", "formula": "\\begin{align*} \\norm { R e ( e ^ { i \\theta } A ) } _ { p } & = \\norm { \\begin{bmatrix} R e ( e ^ { i \\theta } A _ { 1 1 } ) & \\frac { 1 } { 2 } ( e ^ { i \\theta } A _ { 1 2 } + e ^ { - i \\theta } A _ { 2 1 } ^ { \\ast } ) \\\\ \\frac { 1 } { 2 } ( e ^ { i \\theta } A _ { 2 1 } + e ^ { - i \\theta } A _ { 1 2 } ^ { \\ast } ) & R e ( e ^ { i \\theta } A _ { 2 2 } ) \\end{bmatrix} } _ { p } , \\end{align*}"}
{"id": "4499.png", "formula": "\\begin{align*} \\Sigma _ { a } ^ { b } = 2 \\sum _ { \\substack { 0 < \\gamma < T \\\\ a \\le \\beta < b } } \\frac { x ^ { \\beta - 1 } } { \\gamma } , \\end{align*}"}
{"id": "3655.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { 2 \\log \\log t } \\leq 1 . 0 4 4 2 5 = \\gamma , . \\end{align*}"}
{"id": "5884.png", "formula": "\\begin{align*} f ^ 1 ( \\phi ( x _ 0 ) ) = 0 , \\quad g ( 0 ) = 0 \\end{align*}"}
{"id": "2270.png", "formula": "\\begin{align*} M ^ T = M M > 0 , \\end{align*}"}
{"id": "9399.png", "formula": "\\begin{align*} \\tau _ t ( a _ 1 \\cdots a _ n ) = o ( 1 ) \\end{align*}"}
{"id": "3443.png", "formula": "\\begin{align*} \\begin{aligned} | q _ { t } ( z , y ) - q _ { j } ( z , y ) | & \\leqslant C \\frac { ( r ^ { - j + 1 } - r ^ { - j } ) } { { r ^ { - j } } } \\frac 1 { V ( z , y , r ^ { - j } + d ( z , y ) ) } \\frac { r ^ { - j } } { r ^ { - j } + \\| z - y \\| } \\\\ & \\leqslant C ( r - 1 ) \\frac 1 { V ( z , y , r ^ { - j } + d ( z , y ) ) } \\frac { r ^ { - j } } { r ^ { - j } + \\| z - y \\| } . \\end{aligned} \\end{align*}"}
{"id": "8243.png", "formula": "\\begin{align*} \\mathbf { M } _ { u } = \\sum _ { u \\leq v } \\mu _ { \\mathfrak { B } _ n } ( u , v ) \\mathbf { F } _ { v } , \\mathbf { F } _ { u } = \\sum _ { u \\leq v } \\mathbf { M } _ { v } . \\end{align*}"}
{"id": "8682.png", "formula": "\\begin{align*} & \\frac { 1 } { n } { \\bf E } \\Big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\Big \\} = \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } t r \\Big ( \\Gamma _ t P _ t \\Gamma _ t ^ T + D _ t K _ { Z _ t } D _ t ^ T \\Big ) . \\end{align*}"}
{"id": "9123.png", "formula": "\\begin{align*} A F _ k \\supseteq A F _ { k + 1 } F = \\bigcap _ { k \\in \\mathbb { N } } A F _ k . \\end{align*}"}
{"id": "7033.png", "formula": "\\begin{align*} \\| f _ k \\| ^ { 2 } _ { H ^ 2 } & \\leq 2 ( \\| a _ { k } ^ { \\# } \\| _ { \\infty } ^ { 2 } \\| \\widetilde { f } \\| ^ { 2 } _ { H ^ 2 } + \\widetilde { c } _ { k } \\| p \\| ^ { 2 } _ { H ^ 2 } ) \\\\ & \\leq c _ { k } ( \\| \\widetilde { f } \\| ^ { 2 } _ { H ^ 2 } + \\| p \\| _ { H ^ 2 } ^ { 2 } ) \\\\ & = c _ { k } \\| f \\| _ { b } ^ { 2 } , \\end{align*}"}
{"id": "7924.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } X _ { t } = \\theta f ( X _ { t } ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ { t } + \\mathrm { d } L _ { t } , \\\\ & X _ { 0 } = x \\in [ l , \\infty ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "4927.png", "formula": "\\begin{align*} \\frac { L } { 2 \\eta _ T } \\sum _ { t = 1 } ^ { T } \\frac { \\eta _ t ^ 2 } { \\alpha _ t ^ 4 } \\norm { g _ t } ^ 2 , \\end{align*}"}
{"id": "9515.png", "formula": "\\begin{align*} p ( [ - M _ x ] , [ H ] ) & = p ( [ - M _ x ] , 0 ) + p ( 0 , [ H ] ) - p ( 0 , 0 ) \\\\ & = f _ - ( [ M _ x ] ) + f _ + ( [ h ] ) - f ( 0 ) \\\\ & = [ \\gamma ( f ) ] \\\\ & = \\Gamma ( f ) . \\end{align*}"}
{"id": "5228.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { l = 0 } ^ { \\infty } \\| T _ k f _ j \\| ^ t _ { L ^ r ( K ) } \\bigg ) ^ { 1 / t } \\leq C q ^ { - k } \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\bigg ( \\sum _ { l = 0 } ^ { \\infty } \\| f _ j \\| ^ t _ { L ^ r ( K ) } \\bigg ) ^ { 1 / t } . \\\\ \\end{align*}"}
{"id": "5031.png", "formula": "\\begin{align*} \\Delta _ { G } & = \\{ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , \\ ; 1 \\leq i \\leq p + q - 1 \\ , , \\widehat { x } _ { p + q } \\} \\\\ \\intertext { a n d } \\Delta _ { K } & = \\left \\{ \\begin{cases} \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , \\ ; 1 \\leq i \\leq p - 1 \\ , , \\ ; \\widehat { x } _ { p - 1 } + \\widehat { x } _ { p } \\ , , \\\\ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , \\ ; p + 1 \\leq i \\leq p + q - 1 \\ , , \\ ; \\widehat { x } _ { p + q } \\ , , \\end{cases} \\right \\} \\ , , \\end{align*}"}
{"id": "9337.png", "formula": "\\begin{align*} ( x ) _ { 0 , \\lambda } = 1 , ( x ) _ { k , \\lambda } = x ( x - \\lambda ) ( x - 2 \\lambda ) \\cdots ( x - ( k - 1 ) \\lambda ) , ( k \\ge 1 ) . \\end{align*}"}
{"id": "1296.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda ( M ) | } { | \\Lambda ( M , \\Gamma ) | } = | H _ { 1 } ( Y ) | \\end{align*}"}
{"id": "402.png", "formula": "\\begin{align*} \\Psi _ { 0 } ^ { 2 } ( t ) : = C _ { 1 } e ^ { C _ { 1 } \\int _ { 0 } ^ { t } ( \\mu _ { 0 } ( \\tau ) + \\mu _ { 1 } ( \\tau ) ) d \\tau , } \\end{align*}"}
{"id": "6301.png", "formula": "\\begin{align*} \\hat { K } & = \\arg \\max _ { k } \\Big ( f \\big ( \\Re \\{ \\mathbf { y } \\} ; \\mathcal { H } _ k \\big ) ^ 2 + f \\big ( \\Im \\{ \\mathbf { y } \\} ; \\mathcal { H } _ k \\big ) ^ 2 \\Big ) , \\end{align*}"}
{"id": "2436.png", "formula": "\\begin{align*} S _ { g , \\L } ^ { - 1 } \\pi ( \\l ) g = \\pi ( \\l ) S _ { g , \\L } ^ { - 1 } g . \\end{align*}"}
{"id": "9363.png", "formula": "\\begin{align*} \\mathcal { E } _ { n , \\lambda } ( x ) = \\sum _ { k = 0 } ^ { n } k ! S _ { 2 , \\lambda } ( n , k ) \\sum _ { j = 0 } ^ { k } \\binom { x } { j } \\bigg ( - \\frac { 1 } { 2 } \\bigg ) ^ { k - j } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "6987.png", "formula": "\\begin{align*} \\mathcal N _ { \\mathbf T ( u ) } ( w _ s ( u , 0 ) ) & = \\mathcal N _ { \\mathcal T _ D ( g ( \\bar x ) ) } ( \\nabla g ( \\bar x ) u ; w _ s ( u , 0 ) ) \\\\ & \\subset \\mathcal N _ { \\mathcal T _ D ( g ( \\bar x ) ) } ( \\nabla g ( \\bar x ) u ) \\cap [ w _ s ( u , 0 ) ] ^ { \\perp } = \\mathcal N _ D ( g ( \\bar x ) ; \\nabla g ( \\bar x ) u ) \\cap [ w _ s ( u , 0 ) ] ^ \\perp \\end{align*}"}
{"id": "8703.png", "formula": "\\begin{align*} \\hat { \\phi } ^ \\omega ( f _ 1 , \\ldots , f _ d ) : = \\phi ( 0 ) + \\sum _ { i = 1 } ^ d \\Biggl ( \\phi \\biggl ( \\sum _ { j = 1 } ^ i e _ { \\omega _ j } \\biggr ) - \\phi \\biggl ( \\sum _ { j = 1 } ^ { i - 1 } e _ { \\omega _ j } \\biggr ) \\Biggr ) f _ { \\omega _ i } . \\end{align*}"}
{"id": "3326.png", "formula": "\\begin{align*} e _ { - ( f ' + i ) } \\ge e _ { - f ' } - 1 + e _ { - i } = f + f ' + e _ { - i } \\end{align*}"}
{"id": "3894.png", "formula": "\\begin{align*} E _ 1 ( x ) = \\sum _ { x \\leq p \\leq 2 x } \\frac { 1 } { p } \\sum _ { \\substack { 0 < b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { m } - v ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 \\leq b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } = 0 . \\end{align*}"}
{"id": "808.png", "formula": "\\begin{align*} N ^ { 1 , p } _ * ( \\Omega ) = \\left \\{ u \\in N ^ { 1 , p } ( \\Omega ) : \\int _ \\Omega u \\ , d \\mu = 0 \\right \\} , \\end{align*}"}
{"id": "9432.png", "formula": "\\begin{align*} F \\ ! = \\ ! ( f , u _ 0 ) \\in \\mathcal { F } ^ { k , \\mathbf { s } ( s , \\lambda , \\lambda ' , \\delta ) } _ { T , \\varLambda ^ q } \\times C ^ { 2 s + k + 1 , \\lambda , \\delta } _ { \\varLambda ^ q } \\end{align*}"}
{"id": "432.png", "formula": "\\begin{align*} \\begin{aligned} U ^ { j } \\overset { \\ast } { \\rightharpoonup } U ^ { 0 } ~ \\mbox { i n } ~ L ^ { \\infty } ( 0 , T _ { 0 } ; H ^ { s } ) , & v ^ { j } \\rightharpoonup v ^ { 0 } ~ \\mbox { i n } ~ L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s + 1 } ) , \\\\ U _ { t } ^ { j } \\rightharpoonup U _ { 1 } ~ \\mbox { i n } ~ & L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s - 1 } ) . \\end{aligned} \\end{align*}"}
{"id": "5135.png", "formula": "\\begin{align*} \\kappa ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) = \\big ( \\coth ( \\tfrac { \\eta } { 2 } ) \\ , e ^ { \\frac { \\eta } { n } } \\big ) ^ 2 , \\end{align*}"}
{"id": "973.png", "formula": "\\begin{align*} I _ 1 & : = c _ { n , s } \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { B \\setminus B _ \\varepsilon ( x ) } \\left ( \\frac 1 { \\vert x - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\right ) \\big ( u ( x ) - u ( y ) \\big ) \\dd y \\\\ I _ 2 & : = c _ { n , s } \\int _ { \\R ^ n _ + \\setminus B } \\left ( \\frac 1 { \\vert x - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\right ) \\left ( u ( x ) - u ( y ) \\right ) \\dd y . \\end{align*}"}
{"id": "2922.png", "formula": "\\begin{align*} \\frac { T _ n ( 2 ) - \\nu _ n ( 2 ) } { \\delta _ n ( 2 ) } = \\frac { \\tilde T _ n ( 2 ) } { \\delta _ n ( 2 ) } . \\end{align*}"}
{"id": "8615.png", "formula": "\\begin{align*} B _ 1 ( k ) & : = \\int _ { 0 } ^ { t } i s k e ^ { - i s k ^ 2 } \\iiint u ^ { \\# } ( \\ell ) \\overline { u ^ { \\# } } ( n ) u ^ { \\# } ( m ) \\ , \\mu _ { R , 1 } ^ { \\# , ( 2 ) } \\left ( k , \\ell , n , m \\right ) \\ , d \\ell d m d n \\ , d s , \\\\ B _ 2 ( k ) & : = \\int _ { 0 } ^ { t } e ^ { - i s k ^ 2 } \\iiint u ^ { \\# } ( \\ell ) \\overline { u ^ { \\# } } ( n ) u ^ { \\# } ( m ) \\ , \\partial _ { k } \\mu _ { R , 1 } ^ { \\# , ( 2 ) } \\left ( k , \\ell , n , m \\right ) \\ , d \\ell d m d n \\ , d s . \\end{align*}"}
{"id": "2387.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N _ 1 } \\cos ( 2 \\pi k / N ) ^ 2 = \\sum _ { k = 0 } ^ { N - 1 } \\sin ( 2 \\pi k / N ) ^ 2 = \\frac { N } { 2 } . \\end{align*}"}
{"id": "5129.png", "formula": "\\begin{align*} n \\tanh ( \\tfrac { 1 } { 2 } ) = n - \\frac { 2 n } { e + 1 } > \\frac { e + e ^ { - 1 } } { 2 } = \\cosh ( 1 ) . \\end{align*}"}
{"id": "348.png", "formula": "\\begin{align*} & \\mathcal A ( \\rho ^ * , m ^ * ) = \\mathcal A ( \\rho ^ { * , \\infty } , m ^ { * , \\infty } ) , \\limsup _ { R \\to + \\infty } R \\mathcal E ( \\rho ^ { * , R } , m ^ { * , R } ) = 0 . \\end{align*}"}
{"id": "4089.png", "formula": "\\begin{align*} G = G _ { \\xi , \\alpha , \\nu } . \\end{align*}"}
{"id": "6371.png", "formula": "\\begin{align*} 2 v _ 0 + \\rho _ 2 + \\rho _ 3 + \\rho _ 4 = 0 \\end{align*}"}
{"id": "2445.png", "formula": "\\begin{align*} S ^ T J S = J \\ ; \\Leftrightarrow \\ ; S ^ T J = J S ^ { - 1 } \\ ; \\Leftrightarrow \\ ; J = S ^ { - T } J S ^ { - 1 } . \\end{align*}"}
{"id": "8993.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { x } ( s ) & = y ( s ) , \\\\ \\dot { y } ( s ) & = y ( s ) + f ^ \\prime ( x ( s ) ) , \\\\ x ( 0 ) & = x _ 0 , \\\\ y ( 0 ) & = y _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "4713.png", "formula": "\\begin{align*} & | \\dot { x } _ i ( t ) - \\mu _ i ( t ) | \\lesssim \\frac { 1 } { t ^ { 5 / 4 - \\delta _ 0 } } , \\\\ & \\bigg | \\dot { \\mu } _ i ( t ) + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { x ^ 3 _ { i j } ( t ) } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\mu _ k ( t ) } { x ^ 3 _ { i j } ( t ) } \\bigg | \\lesssim \\frac { 1 } { t ^ { 9 / 4 - \\delta _ 0 } } . \\end{align*}"}
{"id": "7763.png", "formula": "\\begin{gather*} \\phi [ T ] = u [ T ] . \\end{gather*}"}
{"id": "1832.png", "formula": "\\begin{align*} L ( x , t ) = \\frac { \\sqrt { 1 - x } } { \\sqrt { 1 - x } \\cosh { ( t \\sqrt { 1 - x } ) } - \\sinh { t \\sqrt { 1 - x } } } . \\end{align*}"}
{"id": "4030.png", "formula": "\\begin{align*} \\Phi _ { \\lambda ^ p _ k } ( x ) = \\begin{pmatrix} \\xi _ { \\lambda ^ p _ k } ( x ) \\\\ \\eta _ { \\lambda ^ p _ k } ( x ) \\end{pmatrix} , \\forall k \\geq k _ 0 . \\end{align*}"}
{"id": "1837.png", "formula": "\\begin{align*} A ' ( \\bar { x } , \\bar { y } , t ) = L ^ 2 ( x , y , t ) , \\end{align*}"}
{"id": "5625.png", "formula": "\\begin{align*} & e _ { B + a } x ^ h - x ^ h e _ { B + a } - \\frac { \\partial f ^ a } { \\partial x ^ h } = 0 , a = 1 , \\ldots , k \\\\ & e _ \\alpha x ^ h - x ^ h e _ \\alpha - x ^ \\mu \\tau ^ { \\mu h } ( e _ \\alpha ) = 0 , \\alpha = 1 , \\ldots , B \\\\ & t ^ \\alpha _ \\ell e _ \\alpha = 0 , \\ell = 1 , \\ldots , B + k - n \\\\ & e _ \\alpha e _ \\beta - e _ \\beta e _ \\alpha - C ^ \\gamma _ { \\alpha \\beta } e _ \\gamma = 0 , \\\\ & e _ \\alpha \\xi ^ i - \\xi ^ i e _ \\alpha = 0 \\end{align*}"}
{"id": "4994.png", "formula": "\\begin{align*} \\norm { \\mathcal { R } ^ { n } Z } _ { y } = O ( \\epsilon ^ { 2 ^ { n } } ) \\end{align*}"}
{"id": "1494.png", "formula": "\\begin{align*} \\langle \\mathbf { f , h } \\rangle = \\int _ { G ( \\Q ) \\backslash ( G ( \\mathbb { A } _ { \\mathbf { h } } ) / K _ 1 ( \\mathfrak { n } ) \\times \\mathfrak { Z } ) } \\mathbf { f } ( g , z ) \\overline { \\mathbf { h } ( g , z ) } \\delta ( z ) ^ k \\mathbf { d } g _ { \\mathbf { h } } \\mathbf { d } z . \\end{align*}"}
{"id": "661.png", "formula": "\\begin{align*} A ( x ) \\ = \\ A ^ \\prime ( f ( x ) ) \\end{align*}"}
{"id": "7787.png", "formula": "\\begin{align*} N _ { \\delta } ( G ( h _ i ) ) & \\leq 2 m + \\delta ^ { - 1 } \\sum \\limits _ { j = 0 } ^ { m - 1 } R _ { h _ i } [ j \\delta , ( j + 1 ) \\delta ] \\\\ & \\leq 2 ( \\delta ^ { - 1 } + 1 ) + \\sum \\limits _ { j = 0 } ^ { m - 1 } h _ { q } \\delta ^ { \\sigma - 1 } \\\\ & \\leq \\delta ^ { \\sigma - 2 } ( 4 + 2 m h _ { q } ) . \\end{align*}"}
{"id": "7399.png", "formula": "\\begin{align*} \\overrightarrow { \\eta } ^ { \\ell } ( x ) : = \\frac { 1 } { \\ell } \\sum _ { y = 1 } ^ { \\ell } \\eta ( x + y ) \\ ; \\ ; \\ ; \\ ; \\overleftarrow { \\eta } ^ { \\ell } ( x ) : = \\frac { 1 } { \\ell } \\sum _ { y = - \\ell } ^ { - 1 } \\eta ( x + y ) . \\end{align*}"}
{"id": "6925.png", "formula": "\\begin{align*} L ( d ) = \\frac { L _ 0 } { d _ 0 ^ { - \\alpha } } , \\end{align*}"}
{"id": "3377.png", "formula": "\\begin{align*} & [ T u , T v , T _ 1 ( w ) ] + [ T u , T _ 1 ( v ) , T w ] + [ T _ 1 ( u ) , T v , T w ] \\\\ & = T \\Big ( D ( T u , T _ 1 ( v ) ) w + D ( T _ 1 ( u ) , T v ) w + \\theta ( T v , T _ 1 ( w ) ) u + \\theta ( T _ 1 ( v ) , T w ) u \\\\ & - \\theta ( T u , T _ 1 ( w ) ) v - \\theta ( T _ 1 ( u ) , T w ) v \\Big ) + T _ 1 \\Big ( D ( T u , T v ) w + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v \\Big ) , \\end{align*}"}
{"id": "6197.png", "formula": "\\begin{align*} \\sum _ { t = l + 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } < ( k - l ) \\alpha \\| W \\| _ F ^ 2 . \\end{align*}"}
{"id": "6870.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k \\| f _ j \\| \\leq O ( \\sqrt { k } \\| f \\| ) . \\end{align*}"}
{"id": "9263.png", "formula": "\\begin{align*} \\forall n ^ 0 , x ^ X , y ^ X \\exists z ^ X , w ^ X \\left ( y \\in A x \\rightarrow \\left ( w \\in A z \\land x = _ X z + _ X \\gamma _ n w \\right ) \\right ) . \\end{align*}"}
{"id": "6591.png", "formula": "\\begin{align*} \\mathcal { L } ( h , k ) = \\mathcal { L } ^ 0 ( h , k ) + \\mathcal { L } ^ r ( h , k ) , \\end{align*}"}
{"id": "9357.png", "formula": "\\begin{align*} - t e _ { \\lambda } ^ { - r } ( t ) & = \\sum _ { k = 1 } ^ { \\infty } H _ { k , \\lambda } ^ { ( r ) } ( - 1 ) ^ { k } k ! \\frac { 1 } { k ! } \\big ( e _ { \\lambda } ( t ) - 1 \\big ) ^ { k } \\\\ & = \\sum _ { k = 1 } ^ { \\infty } H _ { k , \\lambda } ^ { ( r ) } ( - 1 ) ^ { k } k ! \\sum _ { n = k } ^ { \\infty } S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\bigg ( \\sum _ { k = 1 } ^ { n } H _ { k , \\lambda } ^ { ( r ) } ( - 1 ) ^ { k } k ! S _ { 2 , \\lambda } ( n , k ) \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "8171.png", "formula": "\\begin{align*} M ( p , H ) = \\frac { \\pi ^ 2 } { 6 } \\left ( 1 + \\frac { N ( p , H ) } { p } \\right ) = \\frac { \\pi ^ 2 } { 6 } \\left ( \\left ( 1 - \\frac { 1 } { p } \\right ) \\left ( 1 - \\frac { 2 } { p } \\right ) + \\frac { 1 2 S ' ( H , p ) } { p } \\right ) . \\end{align*}"}
{"id": "255.png", "formula": "\\begin{align*} \\underset { R \\rightarrow + \\infty } { \\lim } G _ R ( y ) = y . \\end{align*}"}
{"id": "2079.png", "formula": "\\begin{align*} \\widehat { \\psi _ { p ^ { 2 k } } } ( \\mathbf { u } ) : = \\frac { 1 } { p ^ { 2 k n } } \\sum _ { \\mathbf { c } \\in ( \\mathbb { Z } / p ^ { 2 k } \\mathbb { Z } ) ^ n } \\psi _ { p ^ { 2 k } } ( f _ \\mathbf { c } ) \\exp \\left ( \\frac { 2 \\pi i \\langle \\mathbf { c } , \\mathbf { u } \\rangle } { p ^ { 2 k } } \\right ) , \\end{align*}"}
{"id": "8092.png", "formula": "\\begin{align*} F _ { m _ { \\Lambda } ^ { * } H _ { \\mathbb { M } ^ 2 } } [ 0 ] = F _ { H _ { \\mathbb { M } ^ 2 } } [ 0 ] . \\end{align*}"}
{"id": "5741.png", "formula": "\\begin{align*} g _ \\gamma ( x ) = \\begin{cases} e ^ { - \\frac { 1 } { ( x ( 1 - x ) ) ^ \\gamma } } & x \\in ( 0 , 1 ) , \\\\ 0 & x \\notin ( 0 , 1 ) . \\end{cases} \\end{align*}"}
{"id": "3179.png", "formula": "\\begin{align*} \\chi _ { \\ell } = \\begin{cases} 1 & x _ { \\ell + 1 } \\neq x _ { \\ell } , \\\\ 0 & \\mathrm { o t h e r w i s e } , \\end{cases} \\end{align*}"}
{"id": "3578.png", "formula": "\\begin{align*} h - x ' & = ( s _ 1 - c _ 0 + p ) + { \\sum } _ { i = 1 } ^ { t - 1 } ( p - 1 ) p ^ i + ( s _ { t + 1 } - c _ t - 1 ) p ^ t \\\\ & \\ \\ \\ \\ + { \\sum } _ { i = t + 1 } ^ { e - 1 } ( s _ { i + 1 } - c _ i ) p ^ i , \\end{align*}"}
{"id": "3361.png", "formula": "\\begin{align*} & [ T u , T v , T _ 1 ( w ) ] + [ T u , T _ 1 ( v ) , T w ] + [ T _ 1 ( u ) , T v , T w ] \\\\ = & T \\Big ( D ( T u , T _ 1 ( v ) ) w + D ( T _ 1 ( u ) , T v ) w + \\theta ( T v , T _ 1 ( w ) ) u + \\theta ( T _ 1 ( v ) , T w ) u \\\\ & - \\theta ( T u , T _ 1 ( w ) ) v - \\theta ( T _ 1 ( u ) , T w ) v \\Big ) + T _ 1 \\Big ( D ( T u , T v ) w + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v \\Big ) , \\end{align*}"}
{"id": "5414.png", "formula": "\\begin{align*} \\operatorname { h r m } ( \\Delta ) = \\sum _ { T _ i \\in \\Delta } \\operatorname { h r m } ( T _ i ) \\end{align*}"}
{"id": "5451.png", "formula": "\\begin{align*} \\mathcal { Y } ( T ) = \\{ a ( \\cdot , \\cdot ) \\in \\mathcal { X } _ T \\ , : \\ , | a ( t , x ) - a ( t ^ { ' } , x ) | \\le B _ 1 | t - t ^ { ' } | ^ \\gamma , \\ , \\ , \\alpha \\le a ( t , x ) \\le B _ 2 \\ , \\ , \\forall \\ , t , t ^ { ' } \\in [ s , s + T ] , \\ , \\ , x \\in \\bar \\Omega \\} , \\end{align*}"}
{"id": "726.png", "formula": "\\begin{align*} \\sigma _ j : = \\frac { 1 } { j ! } \\frac { d ^ j } { d x ^ j } \\bigg | _ { x = 0 } \\sigma ( x ) \\end{align*}"}
{"id": "1409.png", "formula": "\\begin{align*} { \\rm A d } _ q v = q v q ^ * . \\end{align*}"}
{"id": "4733.png", "formula": "\\begin{align*} & \\partial _ t \\epsilon - \\partial _ y ( | D | \\epsilon + \\epsilon - p | V _ 0 | ^ { p - 1 } \\epsilon ) \\\\ & \\qquad \\qquad + \\partial _ y \\big [ | V _ 0 + \\epsilon | ^ { p - 1 } ( V + \\epsilon ) - | V _ 0 | ^ { p - 1 } V _ 0 - p | V _ 0 | ^ { p - 1 } \\epsilon \\big ] + \\Psi _ { V _ 0 } = 0 , \\end{align*}"}
{"id": "3639.png", "formula": "\\begin{align*} \\int _ { \\exp ( 5 8 ) } ^ x \\frac { | \\theta ( t ) - t | } { t \\log ^ 2 t } t & \\le \\int _ { \\exp ( 5 8 ) } ^ { x } \\frac { A _ 1 \\log ^ B t \\exp ( - C u ( t ) ) } { \\log ^ 2 t } t \\\\ & \\le \\int _ { \\exp ( 5 8 ) } ^ { x } h ' ( t ) t \\\\ & \\leq A _ 1 x \\log ^ { - \\alpha } x \\exp ( - C u ( x ) ) \\\\ & = \\log ^ { 1 - B - \\alpha } x \\cdot A _ 1 x \\log ^ { B - 1 } x \\exp ( - C u ( x ) ) \\\\ & \\le \\log ^ { 1 - B - \\alpha } x _ 0 \\cdot A _ 1 x \\log ^ { B - 1 } x \\exp ( - C u ( x ) ) , \\end{align*}"}
{"id": "6018.png", "formula": "\\begin{align*} U \\Big [ \\Big ( - \\frac { d } { d x } + 2 x \\Big ) f \\Big ] & = \\Big ( - \\frac { d } { d x } + x \\Big ) U f \\\\ U \\Big [ \\frac { d } { d x } f \\Big ] & = \\Big ( \\frac { d } { d x } + x \\Big ) U f . \\end{align*}"}
{"id": "4901.png", "formula": "\\begin{align*} f ( z ) - 1 \\sim c z ^ { - s } , L ( z ) = \\frac { f ' ( z ) } { f ( z ) } \\sim f ' ( z ) \\sim - c s z ^ { - 1 - s } , F ( z ) \\sim \\frac { z ^ { 1 + s } } { c s } , \\end{align*}"}
{"id": "5443.png", "formula": "\\begin{align*} | \\Omega | & = \\int _ \\Omega u ^ { \\frac { 1 } { r } } ( t , x ; s , u _ 0 ) u ^ { - \\frac { 1 } { r } } ( t , x ; s , u _ 0 ) d x \\\\ & \\le \\left ( \\int _ \\Omega u ( t , x ; s , u _ 0 ) d x \\right ) ^ { \\frac { p } { p + 1 } } \\left ( \\int _ \\Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \\right ) ^ { \\frac { 1 } { p + 1 } } \\forall \\ , t > s . \\end{align*}"}
{"id": "1930.png", "formula": "\\begin{align*} \\int ^ { T } _ { 0 } & \\int _ { \\Omega } \\big ( \\rho u \\cdot \\partial _ t \\phi + ( \\rho u \\otimes u ) : \\nabla \\phi + \\rho ^ \\gamma { \\rm { d i v } } \\phi + \\delta \\rho ^ \\beta { \\rm { d i v } } \\phi - \\varepsilon \\nabla \\rho \\cdot \\nabla u \\cdot \\phi \\\\ & - Z _ \\varepsilon : \\nabla \\phi - \\mathbb { S } ( \\nabla u ) : \\nabla \\phi + ( j - n u ) \\cdot \\phi \\big ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 u _ 0 \\cdot \\phi ( 0 , x ) \\ , d x = 0 , \\end{align*}"}
{"id": "3368.png", "formula": "\\begin{align*} \\begin{cases} T _ 1 ( u ) + [ \\mathfrak { X } , T u ] = T D ( \\mathfrak { X } ) u + T ^ { ' } _ 1 ( u ) , \\\\ [ \\mathfrak { X } , T _ 1 ( u ) ] = T ^ { ' } _ 1 D ( \\mathfrak { X } ) ( u ) . \\end{cases} \\end{align*}"}
{"id": "1304.png", "formula": "\\begin{align*} \\mathrm { E C H } ( Y , \\lambda , 0 ) \\cong \\bigoplus _ { k = 0 } ^ { \\infty } \\mathbb { F } \\langle \\alpha _ { k } \\rangle \\bigoplus \\bigoplus _ { j = 1 } ^ { m } \\mathbb { F } \\langle \\beta _ { j } \\rangle \\end{align*}"}
{"id": "1910.png", "formula": "\\begin{align*} \\bigwedge _ { i = 1 } ^ m \\left ( \\nu \\delta ^ { \\tau } \\odot ^ \\flat n _ i \\delta ^ { t _ i } \\right ) = \\bigwedge _ { i = j } ^ m \\left ( \\nu \\delta ^ { \\tau } \\odot ^ \\flat n _ i \\delta ^ { t _ i } \\right ) = \\bigwedge _ { i = j } ^ m \\left ( \\nu - n _ i \\right ) \\delta ^ \\tau = ( \\nu - n _ j ) \\delta ^ \\tau . \\end{align*}"}
{"id": "9212.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) . \\end{align*}"}
{"id": "8762.png", "formula": "\\begin{align*} s _ { i k } = m ^ 1 _ { i 0 } + \\sum _ { t = 1 } ^ { l _ i } \\Bigl ( ( m ^ { t + 1 } _ { i 0 } - m ^ t _ { i k } ) \\delta _ { i t } + \\sum _ { j ' = 1 } ^ k ( m ^ t _ { i j ' } - m ^ t _ { i j ' - 1 } ) z _ { i j ' } \\Bigr ) , \\end{align*}"}
{"id": "2881.png", "formula": "\\begin{align*} \\partial _ t \\mathcal { V } _ k ^ A + \\mathcal { L } \\mathcal { V } _ k ^ A = i R ( \\mathcal { V } _ k ^ A ) + O ( e ^ { - ( k + 1 ) e _ 0 t } ) \\mathcal { S } ( \\mathbb { R } ^ N ) . \\end{align*}"}
{"id": "4297.png", "formula": "\\begin{align*} \\partial _ \\tau w = \\partial _ y ^ 2 w + \\frac { d + 1 } { y } \\partial _ y w - \\frac { 1 } { 2 } \\Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 . \\end{align*}"}
{"id": "2280.png", "formula": "\\begin{align*} M _ \\omega T _ x f ( t ) = f ( t - x ) \\ , e ^ { 2 \\pi i \\omega \\cdot t } = f ( t - x ) \\ , e ^ { 2 \\pi i \\omega \\cdot ( t - x ) } e ^ { 2 \\pi i \\omega \\cdot x } = e ^ { 2 \\pi i \\omega \\cdot x } \\ T _ x M _ \\omega f ( t ) . \\end{align*}"}
{"id": "2182.png", "formula": "\\begin{align*} m _ 0 = \\lim _ { n \\rightarrow + \\infty } J ( u _ n ) & \\geq \\liminf _ { n \\rightarrow + \\infty } J ( t _ { \\widehat { u } } u _ n ) \\geq J ( t _ { \\widehat { u } } \\widehat { u } ) \\\\ & \\geq m _ 0 . \\end{align*}"}
{"id": "6803.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\partial _ { t } u ( t ) = F \\left ( t , u \\left ( t \\right ) \\right ) \\\\ \\\\ u ( 0 ) = \\phi \\in Y \\end{array} \\right . \\end{align*}"}
{"id": "4716.png", "formula": "\\begin{align*} & m _ { i j } = - \\frac { 3 a _ { i j } } { ( \\alpha _ i - \\alpha _ j ) ^ 4 } , i \\not = j , \\\\ & m _ { i i } = \\sum ^ n _ { \\substack { j = 1 \\\\ j \\not = i } } \\frac { 3 a _ { i j } } { ( \\alpha _ i - \\alpha _ j ) ^ 4 } , \\forall i = 1 , \\ldots , n . \\end{align*}"}
{"id": "3301.png", "formula": "\\begin{align*} Q ( x ) ^ { - 1 } = \\frac { 1 - e ^ { - x } } { x } = \\sum _ { l \\geq 1 } \\frac { ( - 1 ) ^ { l - 1 } x ^ { l - 1 } } { l ! } , \\end{align*}"}
{"id": "8982.png", "formula": "\\begin{align*} u ( x ) = \\inf \\left \\lbrace \\int _ 0 ^ \\infty e ^ { - s } \\left ( C _ p | \\dot { \\gamma } ( s ) | ^ { q } + f ( \\gamma ( s ) ) \\right ) d s : \\gamma \\in \\mathrm { A C } ( [ 0 , \\infty ) ; \\overline { \\Omega } ) , \\gamma ( 0 ) = x \\right \\rbrace , \\end{align*}"}
{"id": "5180.png", "formula": "\\begin{align*} T _ { \\ell , \\pi ^ \\star _ { O F } } ^ { a v } = \\lim _ { t \\to \\infty } \\frac { t } { R _ \\ell ^ \\star ( t ) } = \\lim _ { t \\to \\infty } \\frac { t } { h _ \\ell ( t ) } \\frac { h _ \\ell ( t ) } { R _ { \\ell } ^ \\star ( t ) } \\stackrel { ( a ) } { = } \\frac { \\mu _ { \\ell } } { f _ \\ell ^ \\star } , \\end{align*}"}
{"id": "8271.png", "formula": "\\begin{align*} \\mathbf { M } _ { u } \\mathbf { M } _ { v } = \\sum _ { w \\in \\mathfrak { B } _ { p + q } } \\sum _ { v \\leq v ' } \\mu ( v , v ' ) b _ { u , v ' } ^ w \\mathbf { M } _ { w } . \\end{align*}"}
{"id": "2344.png", "formula": "\\begin{align*} \\int _ \\R \\left ( x f ( x ) \\right ) \\overline { f ' ( x ) } \\ , d x = x | f ( x ) | ^ 2 \\Bigg | _ { x = - \\infty } ^ \\infty - \\int _ \\R \\left ( | f ( x ) | ^ 2 + x f ' ( x ) \\overline { f ( x ) } \\right ) \\ , d x . \\end{align*}"}
{"id": "3028.png", "formula": "\\begin{align*} ( \\pi ^ { r + 1 , r } ) ^ { * } \\rho = h \\rho + \\sum _ { k = 1 } ^ { q } p _ { k } \\rho , \\end{align*}"}
{"id": "6582.png", "formula": "\\begin{align*} \\mathcal { D } ( h , k ) : = \\frac { 1 } { 2 } \\sum _ { \\substack { 1 \\leq q < \\infty \\\\ ( q , h k ) = 1 } } W \\left ( \\frac { q } { Q } \\right ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , q ) = 1 \\\\ m h = n k } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { 1 \\leq c \\leq C , d \\geq 1 \\\\ c d = q \\\\ d | m h \\pm n k } } \\phi ( d ) \\mu ( c ) , \\end{align*}"}
{"id": "6830.png", "formula": "\\begin{align*} x _ { t _ { m ( k ) } } \\big ( s ( t _ { m ( k ) } ) \\big ) = \\varepsilon _ { t _ { m ( k ) } } = 1 . \\end{align*}"}
{"id": "965.png", "formula": "\\begin{align*} \\int _ { \\R ^ n _ + } \\frac { \\dd y } { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } & = x _ 1 ^ { - 2 s } \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } . \\end{align*}"}
{"id": "4619.png", "formula": "\\begin{gather*} c ^ L _ X = ( c ^ R _ X ) ^ { - 1 } \\end{gather*}"}
{"id": "3675.png", "formula": "\\begin{align*} C = \\left ( a _ 1 , a _ 2 , \\ldots , a _ k , \\overline { a _ 1 } , \\overline { a _ 2 } , \\ldots , \\overline { a _ k } \\right ) , \\end{align*}"}
{"id": "6594.png", "formula": "\\begin{align*} \\sum _ { \\substack { c > C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) = \\begin{cases} \\displaystyle - \\sum _ { \\substack { c \\leq C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) & q > 1 \\\\ \\\\ \\hphantom { - - - } 0 & q = 1 . \\end{cases} \\end{align*}"}
{"id": "6642.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq n < m < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( 1 - \\alpha ) } p ^ { n ( 2 - \\beta - w ) } } = \\frac { \\tau _ A ( p ) } { p ^ { 1 - \\alpha } } - \\frac { \\tau _ A ( p ) } { p ^ { 3 - \\alpha - w } } + O \\left ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\right ) . \\end{align*}"}
{"id": "2805.png", "formula": "\\begin{align*} \\Phi ( h _ n ) \\to 0 , \\| h _ n \\| _ 2 ^ 2 = 1 , \\end{align*}"}
{"id": "190.png", "formula": "\\begin{align*} ( - \\mathcal { L } _ \\delta ) ( F _ \\delta ) = g , \\end{align*}"}
{"id": "5154.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial \\eta } A ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) \\Bigr | _ { \\eta = \\eta _ n } > 0 \\end{align*}"}
{"id": "3656.png", "formula": "\\begin{align*} T : = \\frac { 1 } { \\max _ { t \\ge H } \\left ( \\frac { x ^ { - \\nu _ 3 ( t ) } } { t } \\right ) } = \\min _ { t \\ge H } \\left ( x ^ { \\nu _ 3 ( t ) } t \\right ) . \\end{align*}"}
{"id": "7647.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ 0 = \\frac { \\int _ { T } | \\nabla \\bar { u } | ^ 2 } { \\int _ { T } \\tilde { m } _ 0 \\bar { u } ^ 2 } = \\frac { \\int _ { \\R ^ N } | \\nabla \\bar { u } | ^ 2 } { \\int _ { \\R ^ N } \\tilde { m } _ 0 \\bar { u } ^ 2 } \\ ; . \\end{align*}"}
{"id": "4742.png", "formula": "\\begin{align*} ( \\partial _ t - \\mathcal { L } ) U + Q \\lozenge U = F , U | _ { t = 0 } = G , \\end{align*}"}
{"id": "1043.png", "formula": "\\begin{align*} & q = r \\wedge s & & & & \\ , \\\\ & \\lambda _ 5 = \\xi _ 5 s - \\eta _ 5 r , & & \\lambda _ 6 = \\xi _ 6 s - \\eta _ 6 r , & & \\\\ & a _ { 5 5 } = \\xi _ 5 \\eta _ 5 , & & a _ { 6 6 } = \\xi _ 6 \\eta _ 6 , & & a _ { 5 6 } = \\xi _ 5 \\eta _ 6 + \\xi _ 6 \\eta _ 5 \\ , , \\end{align*}"}
{"id": "4490.png", "formula": "\\begin{align*} \\zeta ( s ) = \\sum _ { n \\ge 1 } n ^ { - s } \\ \\ \\Re s > 1 . \\end{align*}"}
{"id": "2997.png", "formula": "\\begin{align*} ( a ( x ) - a ( y ) ) \\langle x , x - y \\rangle + a ( y ) | x - y | ^ { 2 } & = a ( y ) ( | x - y | ^ { 2 } + \\langle x , x - y \\rangle ) \\\\ & = a ( y ) ( \\langle x - y , x - y \\rangle - \\langle x , x - y \\rangle \\\\ & a ( y ) \\langle x - y , - y \\rangle \\\\ & = a ( y ) ( | y | ^ 2 - \\langle x , y \\rangle ) \\\\ & \\geq a ( y ) ( | y | ^ 2 - | x | | y | ) \\geq 0 \\end{align*}"}
{"id": "4751.png", "formula": "\\begin{align*} \\ell _ { \\mathfrak { B } } ( \\Psi ^ k ( t ) ) \\leq 1 + \\sum _ { i = 0 } ^ { k - 1 } \\ell _ { \\mathfrak { B } } ( \\Psi ^ i ( a _ { \\Psi } ) ) \\leq 1 + s ( k + 1 ) ^ { n + 1 } . \\end{align*}"}
{"id": "5299.png", "formula": "\\begin{align*} \\psi = \\varphi _ S = \\varphi _ { S ^ { - 1 } } . \\end{align*}"}
{"id": "7468.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\delta \\mathcal { L } _ { \\sigma } } { \\delta \\overline { \\phi } } ( \\phi _ { \\star } , \\chi _ { \\star } ) = \\frac { \\delta E } { \\delta \\overline { \\phi } } ( \\phi _ { \\star } ) - \\chi _ { \\star } \\phi _ { \\star } + \\sigma R ^ \\prime ( \\| \\phi _ { \\star } \\| ^ 2 - 1 ) \\phi _ { \\star } = 0 , \\\\ & \\frac { \\partial \\mathcal { L } _ { \\sigma } } { \\partial \\chi } ( \\phi _ { \\star } , \\chi _ { \\star } ) = \\| \\phi _ { \\star } \\| ^ { 2 } - 1 = 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "5918.png", "formula": "\\begin{align*} \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 3 & = n - 3 , \\\\ \\lambda _ 1 \\ , \\lambda _ 2 + \\lambda _ 2 \\ , \\lambda _ 3 + \\lambda _ 3 \\ , \\lambda _ 1 & = - ( 2 n - 3 ) , \\\\ \\lambda _ 1 \\ , \\lambda _ 2 \\ , \\lambda _ 3 & = n - 1 - 4 s _ 2 t _ 1 ( s _ 1 + t _ 2 ) . \\end{align*}"}
{"id": "2417.png", "formula": "\\begin{align*} f = S ( S ^ { - 1 } f ) = \\sum _ { \\gamma \\in \\Gamma } \\langle S ^ { - 1 } f , e _ \\gamma \\rangle e _ \\gamma = \\sum _ { \\gamma \\in \\Gamma } \\langle f , S ^ { - 1 } e _ \\gamma \\rangle e _ \\gamma \\end{align*}"}
{"id": "2014.png", "formula": "\\begin{align*} & \\overline { \\mu } _ 1 : = N \\left ( e ^ { F + [ u ] } \\ ! \\ ! - ( F + [ u ] ) \\ ! - 1 + F _ 1 \\right ) \\mu _ H + \\mu _ { 1 } + \\frac { \\ , 1 \\ , } { 2 } \\mu _ { \\ < u \\ > } ^ c , \\overline { \\mu } _ 2 : = N ( F _ { 2 } ) \\mu _ H + \\mu _ { 2 } , \\\\ & \\overline { \\mu } _ 1 ^ * : = N ( e ^ { F _ 1 + [ u ] } - ( F _ 1 + [ u ] ) - 1 + F _ 1 ) \\mu _ H + \\mu _ { 1 } + \\frac { \\ , 1 \\ , } { 2 } \\mu _ { \\ < u \\ > } ^ c , \\overline { \\mu } _ 2 ^ * : = \\mu _ { 2 } , \\end{align*}"}
{"id": "1135.png", "formula": "\\begin{align*} \\Delta _ \\eta m ^ { - 1 } _ \\eta = I + \\mathcal { O } ( t ^ { - 1 } ) , k \\in \\partial U _ { \\delta } ( \\eta ) , t \\rightarrow \\infty , \\end{align*}"}
{"id": "1893.png", "formula": "\\begin{align*} \\pi _ j = \\prod _ { i = 1 } ^ r \\nu _ i ^ { n _ { j i } } \\end{align*}"}
{"id": "8837.png", "formula": "\\begin{align*} r & \\geq | N _ { F - E ( F _ { 0 } ) } ( z ^ { - } _ { 1 } ) \\cap V ( D _ { 2 } ) | \\\\ & \\geq | N _ { F _ { 0 } } [ z _ { 2 } ] | - | N _ { F - E ( F _ { 0 } ) } ( z _ { 1 } ) \\cap N _ { F _ { 0 } } [ z _ { 2 } ] | \\\\ & \\geq k ' + 1 - \\bigg ( \\bigg \\lceil \\frac { k ' } { 2 } \\bigg \\rceil - 1 \\bigg ) \\\\ & = \\bigg \\lfloor \\frac { k ' } { 2 } \\bigg \\rfloor + 2 = \\bigg \\lceil \\frac { k ' } { 2 } \\bigg \\rceil + 1 . \\end{align*}"}
{"id": "4704.png", "formula": "\\begin{align*} \\| D \\| _ { H ^ 1 } \\lesssim & \\sup _ { \\substack { i \\not = j , k \\geq 1 \\\\ 2 \\leq k + \\ell \\leq 3 } } \\frac { 1 } { d ^ 3 } \\Big ( \\big \\| ( \\partial _ y ^ k \\varphi _ { i j } ) ( \\partial _ y ^ \\ell B _ { i j , 0 } ) \\big \\| _ { L ^ 2 } \\Big ) + \\frac { 1 } { d ^ 5 } \\lesssim \\frac { 1 } { d ^ 5 } . \\end{align*}"}
{"id": "3582.png", "formula": "\\begin{align*} & V : = \\{ \\delta _ t \\otimes z ^ * : t \\in B _ 2 , z ^ * \\in V _ Z \\} \\subseteq W \\subseteq U , \\\\ & x _ 1 ^ * : = \\delta _ { s _ 0 } \\otimes z _ 1 ^ * , e ( t ) : = e _ A ( t ) e _ Z \\ , \\ , t \\in \\partial A \\end{align*}"}
{"id": "4431.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\int _ U ( u _ t ( t ) ) ^ 2 \\ , d x + \\frac { 1 } { 2 } \\int _ U ( u _ { x x } ( t ) ) ^ 2 \\ , d x , \\end{align*}"}
{"id": "3846.png", "formula": "\\begin{align*} P ' ( x ) + P ( x ) \\tilde F ( x ) = F ( x ) P ( x ) , x \\in \\mathbb R _ + . \\end{align*}"}
{"id": "7006.png", "formula": "\\begin{align*} p ( x ) = y _ 1 \\delta _ { 1 } ( x ) + y _ 2 \\delta _ 2 ( x ) + \\cdots + y _ t \\delta _ { t } ( x ) , \\end{align*}"}
{"id": "9152.png", "formula": "\\begin{align*} F _ 1 ( x ) : = \\norm { x - J ^ S _ { \\mu _ 0 } ( x + \\mu _ 0 T ^ \\circ x ) } \\end{align*}"}
{"id": "4917.png", "formula": "\\begin{align*} X ^ 2 - 3 \\sigma { \\log ( 1 / \\delta ) } X - \\left ( ( \\kappa + L ) \\sqrt { \\sum _ { t = 1 } ^ T \\| g _ t \\| ^ 2 } + 3 ( G ^ 2 + G \\tilde G ) { \\log ( 1 / \\delta ) } \\right ) \\leq 0 , \\end{align*}"}
{"id": "8714.png", "formula": "\\begin{align*} \\biggl \\{ ( s , m ) \\biggm | s \\in Q , \\ m _ { ( I , j ) } = \\prod _ { i \\in I } s _ { i j _ i } , \\ ; I \\subseteq \\{ 1 , \\ldots , d \\} | I | \\geq 2 , j \\in \\{ 1 , \\ldots , n \\} ^ d \\biggr \\} . \\end{align*}"}
{"id": "4874.png", "formula": "\\begin{align*} F _ 2 ( z ) = \\frac { d ^ { n - 2 } } { d z ^ { n - 2 } } \\left ( z ^ { n - 1 } ( z - 1 ) ^ { n - 1 } \\right ) \\end{align*}"}
{"id": "9004.png", "formula": "\\begin{align*} N p ' _ 0 + N \\sum _ { w = 0 } ^ { K - 1 } \\binom { K - 1 } { w } ( N - 1 ) ^ w p _ w = 1 . \\end{align*}"}
{"id": "3774.png", "formula": "\\begin{align*} H ^ 1 \\big ( \\Gamma , C _ c ^ \\infty ( X _ E \\setminus X _ F , \\mathbb { \\overline { F } } _ l ) \\big ) = 0 . \\end{align*}"}
{"id": "2979.png", "formula": "\\begin{align*} \\tilde \\Lambda _ { n , 2 } ^ { ( 1 , 3 ) } = \\frac { 1 6 } { n ^ 4 \\delta ^ 2 _ n ( 2 ) \\delta ^ 2 _ n ( 3 ) } \\sum _ { r \\ne r ' } ^ d \\sum _ { q _ 1 = 1 } ^ { r - 1 } \\sum _ { p _ 1 = 1 } ^ { q _ 1 - 1 } \\sum _ { \\substack { q _ 3 = 1 \\\\ q _ 3 \\ne p _ 1 , q _ 1 } } ^ { r ' - 1 } \\sum _ { \\mathbf { i } \\in \\mathcal { J } ^ 2 } \\varphi _ 2 ( \\mathbf i _ { 1 : 4 } ) ^ 3 \\varphi _ 2 ( \\mathbf i _ { 5 : 8 } ) ^ 2 . \\end{align*}"}
{"id": "8713.png", "formula": "\\begin{align*} \\begin{aligned} & w _ { ( D , j ) } \\geq 0 w _ { ( I , j ) } = 0 & & I \\subset D , \\ ; j \\in E , \\\\ & y _ { ( I , j ) } = \\sum _ { I ' \\supseteq I , j ' \\in E ( I , j ) } w _ { ( I ' , j ' ) } & & I \\subseteq D , \\ ; j \\in E . \\end{aligned} \\end{align*}"}
{"id": "809.png", "formula": "\\begin{align*} I _ { \\min } ( f ) = \\inf _ { v \\in N ^ { 1 , p } _ * ( \\Omega ) } I _ f ( v ) . \\end{align*}"}
{"id": "4270.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\sup _ { \\tau \\in ( 0 , T ] } \\left \\| z ^ { ( k ) } ( \\tau , \\cdot ) \\right \\| _ { H ^ 2 ( \\R \\backslash \\{ 0 \\} ) } ~ < ~ \\infty . \\end{align*}"}
{"id": "1506.png", "formula": "\\begin{align*} \\mathbf { E } _ l ( x , s ) = \\mathbf { E } ^ m _ l ( x , s ; \\chi ) = \\sum _ { \\gamma \\in P _ m \\backslash G _ { 2 m } } \\varphi ( \\gamma x , s ) , \\end{align*}"}
{"id": "1939.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\dfrac { \\mathrm { d } } { \\mathrm { d } t } x ( t ) & = - \\nabla \\varphi ( x ( t ) ) \\\\ x ( 0 ) & = x _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "1395.png", "formula": "\\begin{align*} E _ i ( x ) = \\alpha + \\int _ 0 ^ x ( n _ i - b _ i ) ( y ) d y , \\quad \\forall x \\in [ 0 , 1 ] , i = 1 , 2 . \\end{align*}"}
{"id": "8960.png", "formula": "\\begin{align*} \\tilde { \\xi } ( s ) = \\left \\{ \\begin{aligned} & \\xi ( s ) 0 \\leq s < t _ 0 \\\\ & \\xi ( t _ 0 ) s \\geq t _ 0 \\end{aligned} \\right . \\end{align*}"}
{"id": "8741.png", "formula": "\\begin{align*} \\mathcal { M } _ K : = \\biggl \\{ ( \\lambda , y ) \\in \\R _ + ^ { | V | + | E | } \\biggm | \\lambda \\in \\Lambda , \\ y _ e = \\prod _ { v \\in e } \\lambda _ v , \\ e \\in E \\biggr \\} . \\end{align*}"}
{"id": "4494.png", "formula": "\\begin{align*} J _ { a , b } ( T ) = \\int _ { T } ^ { \\infty } \\frac { \\log ^ a y } { y ^ b } d y . \\end{align*}"}
{"id": "7564.png", "formula": "\\begin{align*} f = \\sum _ { ( i _ 1 , i _ 2 , i _ 3 , i _ 4 ) \\in { \\{ 1 , 2 , \\ldots , k \\} } ^ 4 } a _ { i _ 1 , i _ 2 , i _ 3 , i _ 4 } x _ { i _ 1 } y _ { i _ 2 } z _ { i _ 3 } w _ { i _ 4 } , a _ { i _ 1 , i _ 2 , i _ 3 , i _ 4 } \\in K . \\end{align*}"}
{"id": "4844.png", "formula": "\\begin{align*} K _ \\alpha ( x ) : = \\alpha K ( \\alpha x ) \\end{align*}"}
{"id": "8006.png", "formula": "\\begin{align*} \\partial _ u \\partial _ v \\phi = : P \\phi = 0 . \\end{align*}"}
{"id": "240.png", "formula": "\\begin{align*} \\partial _ r \\left ( \\varphi _ \\mu \\right ) ( r \\theta ) = - r \\langle \\theta ; \\Sigma ( \\theta ) \\rangle \\varphi _ \\mu ( r \\theta ) . \\end{align*}"}
{"id": "1579.png", "formula": "\\begin{align*} \\frac { c _ k ( \\mu ) L ( \\mu , \\mathbf { f } , \\chi ) } { \\pi ^ { \\beta } \\langle \\mathbf { f , f } \\rangle } = \\frac { \\langle \\mathbf { h } _ w , \\mathbf { f } \\rangle } { \\langle \\mathbf { f , f } \\rangle } \\left ( \\frac { \\mathbf { f } ( g _ { \\mathbf { h } } \\cdot g _ { \\infty } ) } { \\mathfrak { P } _ k ( w ) } \\right ) ^ { - 1 } \\in \\overline { \\Q } , \\end{align*}"}
{"id": "6246.png", "formula": "\\begin{align*} \\sigma = [ b _ 1 , \\ldots , b _ { m - s } , a _ { i + 1 } , a _ 1 , \\ldots , a _ { i } , a _ { i + 2 } , \\ldots , a _ s ] ; \\end{align*}"}
{"id": "5291.png", "formula": "\\begin{align*} b ^ * \\delta _ { \\varphi } ^ { n } ( ( ( \\sigma ^ { \\varphi } ) ^ n ) ( S ^ { 2 n } ( b ) ) ) = 0 . \\end{align*}"}
{"id": "5482.png", "formula": "\\begin{align*} d _ K ( x + y ) & = \\inf _ { z \\in K } \\| ( x + y ) - z \\| = \\inf _ { z \\in K } \\| ( x - z ) + y \\| \\leq \\inf _ { z \\in K } \\big ( \\| x - z \\| + \\| y \\| \\big ) \\\\ & = \\inf _ { z \\in K } \\| x - z \\| + \\| y \\| = d _ K ( x ) + \\| y \\| , \\end{align*}"}
{"id": "2181.png", "formula": "\\begin{align*} J ( u _ n ) & = J ( \\Vert u _ n \\Vert v _ n ) \\\\ & \\leq g ^ + \\max \\left \\lbrace \\Vert u _ n \\Vert ^ { g ^ - } \\Vert v _ n \\Vert ^ { g ^ - } , \\Vert u _ n \\Vert ^ { g ^ + } \\Vert v _ n \\Vert ^ { g ^ + } \\right \\rbrace - \\int _ { \\mathbb { R } ^ { d } } K ( x ) F ( x , \\Vert u _ n \\Vert v _ n ) d x \\\\ & \\leq g ^ + \\max \\left \\lbrace \\Vert u _ n \\Vert ^ { g ^ - } , \\Vert u _ n \\Vert ^ { g ^ + } \\right \\rbrace - \\int _ { \\mathbb { R } ^ { d } } K ( x ) F ( x , \\Vert u _ n \\Vert v _ n ) d x , \\end{align*}"}
{"id": "444.png", "formula": "\\begin{align*} 2 \\langle A ^ { 0 } \\partial _ { x } ^ { \\alpha } u _ { t } , \\partial _ { x } ^ { \\alpha } u \\rangle = \\frac { d } { d t } \\langle A ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle - \\langle \\partial _ { x } ^ { \\alpha } u , ( \\partial _ { t } A ^ { 0 } ) \\partial _ { x } ^ { \\alpha } u \\rangle , \\end{align*}"}
{"id": "7081.png", "formula": "\\begin{align*} \\nabla _ Y E _ 3 ^ { \\top } = \\langle \\xi , E _ 3 ^ { \\perp } \\rangle A Y + \\langle Y , E _ 1 ^ { \\top } \\rangle E _ 1 ^ { \\top } - \\langle Y , E _ 2 ^ { \\top } \\rangle E _ 2 ^ { \\top } . \\end{align*}"}
{"id": "4681.png", "formula": "\\begin{align*} a _ { i j } = \\frac { 4 \\sigma _ i \\sigma _ j \\kappa _ 0 ( p - 1 ) \\int Q ^ p } { ( p - 3 ) \\int Q ^ 2 } . \\end{align*}"}
{"id": "4164.png", "formula": "\\begin{align*} \\norm { T } _ { p } ^ { p } \\leq \\overset { n } { \\underset { i , j = 1 } { \\sum } } \\norm { T _ { i j } } _ { p } ^ { p } \\leq n ^ { 2 - p } \\norm { T } _ { p } ^ { p } , \\end{align*}"}
{"id": "5803.png", "formula": "\\begin{align*} \\mathbf { g } _ { m k } [ t ] & = \\Bigl [ g _ { m k , 0 } [ t ] , \\ldots , g _ { m k , L _ { m k } - 1 } [ t ] \\Bigr ] ^ T \\\\ & = \\sqrt { \\beta _ { m k } } \\mathbf { h } _ { m k } [ t ] , \\end{align*}"}
{"id": "3253.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } \\frac { t } { s ( r , t ) } = \\frac { \\lambda _ 1 ( \\mu ) ^ 2 - r ^ 2 } { \\lambda _ 1 ( \\mu ) ^ 2 } . \\end{align*}"}
{"id": "7665.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + \\tilde { \\lambda } _ 0 \\underline { m } \\ , u = 0 & \\R ^ n \\ ; , \\\\ u ( \\mathbf { 0 } ) = 1 \\ ; u > 0 & \\R ^ n \\ ; . \\end{cases} \\end{align*}"}
{"id": "76.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq q _ v ^ { ( n _ { v , \\nu _ v } - 1 ) / 2 } + 1 . \\end{align*}"}
{"id": "7966.png", "formula": "\\begin{align*} G ^ { \\Sigma ^ 1 } _ { \\Sigma ^ 0 , g } ( x ) : = \\begin{cases} \\sup \\{ t \\geq 0 : \\exp ^ g _ x ( s \\nu ^ 0 ( x ) ) \\in U ^ 1 s \\in [ 0 , t ] \\} , & \\ x \\in U ^ 1 ; \\\\ \\inf \\{ t \\leq 0 : \\exp ^ g _ x ( s \\nu ^ 0 ( x ) ) \\in N \\setminus U ^ 1 s \\in [ t , 0 ] \\} , & \\ x \\notin U ^ 1 . \\end{cases} \\end{align*}"}
{"id": "1318.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , M - 1 ) } , \\hat { \\alpha } \\cup { ( \\delta , 1 ) } \\cup { ( \\gamma , M - p _ { i } - 1 ) } ) = 2 \\end{align*}"}
{"id": "9038.png", "formula": "\\begin{align*} \\mathcal { A } : = \\left \\{ ( \\rho , \\phi ) : - \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ) = f ( x ) + \\sum _ { i = 1 } ^ { s } z _ i \\rho _ i , \\alpha \\phi + \\beta \\frac { \\partial \\phi } { \\partial \\mathbf { n } } = \\phi ^ b , x \\in \\partial \\Omega , \\rho \\in [ \\mathcal { P } ( \\Omega ) ] ^ s \\right \\} . \\end{align*}"}
{"id": "2631.png", "formula": "\\begin{align*} \\iint _ { \\mathcal { Q } \\times \\mathcal { Q } } | Z f ( x , \\omega ) | ^ 2 \\ , d ( x , \\omega ) = \\norm { f } _ 2 ^ 2 . \\end{align*}"}
{"id": "4860.png", "formula": "\\begin{align*} \\int _ { - \\ell / 2 } ^ { \\ell / 2 } z ^ 2 K _ \\alpha '' ( z ) \\ , d z = \\int _ { - \\alpha \\ell / 2 } ^ { \\alpha \\ell / 2 } y ^ 2 K '' ( y ) \\ , d y \\geq \\int _ { - c } ^ c y ^ 2 K '' ( y ) \\ , d y = c ' > 0 . \\end{align*}"}
{"id": "9260.png", "formula": "\\begin{align*} x ' = _ X x = _ X y + \\gamma \\gamma ^ { - 1 } ( x - y ) = _ X y + \\gamma ' \\gamma ^ { - 1 } ( x - y ) . \\end{align*}"}
{"id": "8742.png", "formula": "\\begin{align*} \\chi ( e ' ) _ v = \\begin{cases} 1 & v \\in e ' \\\\ 0 & v \\notin e ' , \\end{cases} \\chi ( e ' ) _ e = \\begin{cases} 1 & e = e ' \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "6578.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\ell } ( h , k ) = \\mathcal { I } _ { \\ell } ^ * ( h , k ) + O ( X ^ { \\varepsilon } Q ^ { 1 + \\varepsilon } ( h k ) ^ { \\varepsilon } ) , \\end{align*}"}
{"id": "167.png", "formula": "\\begin{align*} \\mu ( d x ) = \\frac { 1 } { Z ( \\mu ) } \\exp \\left ( - V ( x ) \\right ) d x , \\end{align*}"}
{"id": "5029.png", "formula": "\\begin{align*} \\mathrm { S p e c } ( \\Delta _ 0 ) & = \\{ c _ { \\gamma } \\ ; ; \\ ; \\gamma \\in \\widehat { G } \\ ; \\mathrm { s . t . } \\ ; \\mathrm { m u l t } ( \\mathrm { t r i v . r e p r . } , \\mathrm { R e s } ^ { G } _ { K } ( \\rho _ { \\gamma } ) ) \\neq 0 \\} \\ , , \\\\ \\intertext { a n d } \\mathrm { S p e c } ( \\Delta _ 1 ) & = \\{ c _ { \\gamma } \\ ; ; \\ ; \\gamma \\in \\widehat { G } \\ ; \\mathrm { s . t . } \\ ; \\mathrm { m u l t } ( \\rho , \\mathrm { R e s } ^ { G } _ { K } ( \\rho _ { \\gamma } ) ) \\neq 0 \\} \\ , . \\end{align*}"}
{"id": "4319.png", "formula": "\\begin{align*} \\| \\varepsilon ( \\cdot , \\tau ) \\| _ { L ^ \\infty } = o ( b ^ { - 1 } ) , \\tau \\to + \\infty . \\end{align*}"}
{"id": "1969.png", "formula": "\\begin{align*} \\phi \\prec \\varepsilon _ A : = \\phi , \\varepsilon _ A \\succ \\phi : = \\phi , \\phi \\succ \\varepsilon _ A : = 0 , \\varepsilon _ A \\prec \\phi : = 0 . \\end{align*}"}
{"id": "5943.png", "formula": "\\begin{align*} ( J _ L ) _ b ^ \\xi = \\mathcal { G } ( v _ b , \\xi _ Q ) = g _ { i j } v ^ i _ b \\Lambda ^ j . \\end{align*}"}
{"id": "1422.png", "formula": "\\begin{align*} S _ n ( a ) = \\{ x \\in [ 0 , 1 ) : 1 - 2 ^ n ( 1 - | a | ) \\leq x < 1 \\} , \\ \\ n = 1 , 2 , \\cdots . \\end{align*}"}
{"id": "5892.png", "formula": "\\begin{align*} \\mathcal { D } ^ { ( k ) } \\mathcal { X } _ t = \\left ( \\mathcal { D } ^ { j _ 1 , j _ 2 , \\dots , j _ k } _ { r _ 1 , r _ 2 , \\dots , r _ k } \\mathcal { X } _ t \\right ) _ { \\substack { j _ 1 , j _ 2 , \\dots , j _ k \\in \\{ 1 , 2 , \\dots , d \\} \\\\ r _ 1 , r _ 2 , \\dots , r _ k \\in [ 0 , T ] } } . \\end{align*}"}
{"id": "6740.png", "formula": "\\begin{align*} g _ 1 ( t ) \\mathcal { L } _ { K , \\mathfrak { s } } ( \\boldsymbol { \\alpha } ) + g _ 2 ( t ) \\mathcal { L } _ { K , w } ( \\beta ) = 0 . \\end{align*}"}
{"id": "7344.png", "formula": "\\begin{align*} v _ t + G _ \\beta ( v , \\nabla v , \\nabla ^ 2 v , K \\cap \\{ v ( \\cdot , t ) \\leq v ( x , t ) \\} ) = 0 \\end{align*}"}
{"id": "1184.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\zeta } _ t ( E ) = \\inf \\left \\{ \\sum _ { n \\in \\mathbb { N } } \\zeta ( \\vert B _ n \\vert ) : \\ , \\vert B _ n \\vert \\leq t , \\ B _ n E \\subset \\bigcup _ { n \\in \\mathbb { N } } B _ n \\right \\} . \\end{align*}"}
{"id": "9332.png", "formula": "\\begin{align*} \\| { S } ( \\widehat { \\Theta } ) - S ( \\Theta ) \\| _ 2 = \\| D _ A \\| _ 2 < \\| D _ C \\| _ 2 = \\| { \\mathcal { N } } _ { C } ( \\widehat { \\Theta } ) - { \\mathcal { N } } _ { C } ( \\Theta ) \\| _ 2 . \\end{align*}"}
{"id": "4876.png", "formula": "\\begin{align*} F _ 3 ( z ) = ( z - K ) H _ n \\left ( \\frac { K + 1 } { K - 1 } - \\frac { 2 K } { ( K - 1 ) z } \\right ) \\end{align*}"}
{"id": "1841.png", "formula": "\\begin{align*} L _ n \\left ( { 4 x \\over ( 1 + x ) ^ 2 } \\right ) = { 1 \\over ( 1 + x ) ^ n } \\sum _ { k = 0 } ^ n { n \\choose k } ( 1 - x ) ^ { n - k } 2 ^ k A _ k ( x ) . \\end{align*}"}
{"id": "3934.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } s ( \\nu _ 0 \\circ \\alpha _ { T \\Psi } ^ { 0 \\to 1 } | \\nu _ 1 ) = 0 , \\end{align*}"}
{"id": "6005.png", "formula": "\\begin{align*} f ^ { ( n - k ) } ( z ) & = \\sum _ { j = 0 } ^ { n - k } h ^ { ( j ) } \\big ( l _ { \\pi _ { \\lambda , \\beta } } ( z ) \\big ) B _ { n - k , j } \\big ( l ' _ { \\pi _ { \\lambda , \\beta } } ( z ) , l '' _ { \\pi _ { \\lambda , \\beta } } ( z ) , \\dots , l _ { \\pi _ { \\lambda , \\beta } } ^ { ( n - k - j + 1 ) } ( z ) \\big ) . \\\\ & = \\sum _ { j = 0 } ^ { n - k } \\frac { ( - 1 ) ^ { j } j ! } { ( l _ { \\pi _ { \\lambda , \\beta } } ( z ) ) ^ { j + 1 } } B _ { n - k , j } \\big ( ( l _ { \\pi _ { \\lambda , \\beta } } ^ { ( i ) } ( z ) ) _ { i = 1 } ^ { n - k - j + 1 } \\big ) . \\end{align*}"}
{"id": "8862.png", "formula": "\\begin{align*} d _ 0 \\psi = \\begin{array} { c | c c c c } & 0 & 1 & 2 & 3 \\\\ \\hline 0 & 0 & 1 & 2 & 1 \\\\ 1 & 2 & 0 & 1 & 0 \\\\ 2 & 1 & 2 & 0 & 2 \\\\ 3 & 2 & 0 & 1 & 0 \\end{array} \\end{align*}"}
{"id": "2636.png", "formula": "\\begin{align*} Z f \\overline { Z g } = \\sum _ { k , l } \\langle Z f \\overline { Z g } , e _ { k , l } \\rangle \\ , e _ { k , l } \\end{align*}"}
{"id": "2183.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { u _ n ( x ) - u _ n ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { u _ n ( x ) - u _ n ( y ) } { \\vert x - y \\vert ^ { \\alpha + d } } d x d y + \\int _ { \\mathbb { R } ^ { d } } g ( u _ n ) u _ n d x = \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , u _ n ) u _ n d x , \\ \\ n \\in \\mathbb { N } . \\end{align*}"}
{"id": "5449.png", "formula": "\\begin{align*} \\mathcal { B } ( u _ 0 , r ) = \\{ u \\in C ^ 0 ( \\bar \\Omega ) \\ , | \\ , \\| u - u _ 0 \\| _ \\infty \\le r \\} . \\end{align*}"}
{"id": "1868.png", "formula": "\\begin{align*} a ^ 2 D ^ { n + 1 } ( x ) = ( 1 + x ^ 2 ) D ^ n ( a ^ 2 ) . \\end{align*}"}
{"id": "1741.png", "formula": "\\begin{align*} \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ { [ \\hat t ( n ) ] } ) } \\stackrel { ( \\ref { n u 2 } ) , ( \\ref { e m b _ n u } ) } { \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } } 2 ^ { ( ( 1 \u2010 \\lambda ) \\mu _ * \u2010 \\lambda \\alpha _ * ) k _ * \\hat t ( n ) } \\stackrel { ( \\ref { h a t _ m t } ) } { = } n ^ { ( ( 1 \u2010 \\lambda ) \\mu _ * \u2010 \\lambda \\alpha _ * ) / \\gamma _ * } = n ^ { \u2010 \\theta _ 2 } \\end{align*}"}
{"id": "5432.png", "formula": "\\begin{align*} \\dot x _ 1 ( t ) = & x _ 2 ( t ) \\\\ \\dot x _ 2 ( t ) = & - \\theta _ 1 x _ 1 ( t ) - \\theta _ 2 + \\theta _ 3 u ( t ) \\\\ y ( t ) = & x _ 1 ( t ) , \\end{align*}"}
{"id": "6496.png", "formula": "\\begin{align*} C _ { \\ell } = C _ { \\ell } ( \\alpha , \\beta ) : = \\dfrac { ( 2 \\ell - 1 ) ! ! } { ( 1 - 2 \\alpha ) ^ { \\ell - 1 } } \\cdot { \\dfrac { \\beta } { \\Gamma ( 1 + \\alpha ) } } ( \\ell \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "8867.png", "formula": "\\begin{align*} h _ 1 : U _ 2 \\ast I & \\to U _ 2 \\\\ ( ( x , t ) , ( i , j ) ) & \\mapsto \\begin{cases} ( x , \\hat i t ) & t < 0 \\\\ ( x , t ) & t \\ge 0 . \\end{cases} \\end{align*}"}
{"id": "2806.png", "formula": "\\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \\end{align*}"}
{"id": "331.png", "formula": "\\begin{align*} \\mathcal A ( \\rho , m ) : = \\int _ 0 ^ 1 \\frac 1 4 \\sum _ { ( i , j ) \\in E } L ( \\theta _ { i j } ( \\rho ) , m _ { i j } ) d t , \\end{align*}"}
{"id": "8167.png", "formula": "\\begin{align*} M ( f , \\{ 1 \\} ) = { \\pi ^ 2 \\over 6 } \\times \\left \\{ \\prod _ { q \\mid f } \\left ( 1 - { 1 \\over q ^ 2 } \\right ) - { 3 \\over f } \\prod _ { q \\mid f } \\left ( 1 - { 1 \\over q } \\right ) \\right \\} \\ \\ \\ \\ \\ \\hbox { ( $ f > 2 $ ) . } \\end{align*}"}
{"id": "6431.png", "formula": "\\begin{align*} ( L _ i ) _ K = \\begin{pmatrix} K & e ^ { ( i ) } _ 1 \\times A \\\\ e _ 2 ^ { ( i ) } \\times A & K \\end{pmatrix} , \\end{align*}"}
{"id": "3053.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = - x _ 1 \\ , , y _ 1 ^ \\prime = - y _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , . \\end{align*}"}
{"id": "4190.png", "formula": "\\begin{align*} \\bigsqcup _ { i = 1 } ^ { \\lfloor T \\rfloor } B ^ M _ \\epsilon ( x _ i ) & \\subseteq B ^ M _ \\epsilon ( K ) , \\intertext { a n d t h u s } \\sum _ { i = 1 } ^ { \\lfloor T \\rfloor } \\mathrm { V o l } \\left ( B ^ M _ \\epsilon ( x _ i ) \\right ) & \\leq \\mathrm { V o l } \\left ( B ^ M _ \\epsilon ( K ) \\right ) . \\end{align*}"}
{"id": "1004.png", "formula": "\\begin{align*} \\frac { ( - \\Delta ) ^ s v ( h e _ 1 ) } h & = \\frac { c _ { n , s } } 2 \\int _ { \\R ^ n } \\bigg ( 2 \\partial ^ h _ 1 v ( 0 ) - \\partial ^ h _ 1 v ( y ) - \\partial ^ h _ 1 v ( - y ) - \\frac { v ( y ) + v ( - y ) } h \\bigg ) \\frac { \\dd y } { \\vert y \\vert ^ { n + 2 s } } \\\\ & = ( - \\Delta ) ^ s \\partial ^ h _ 1 v ( 0 ) . \\end{align*}"}
{"id": "9197.png", "formula": "\\begin{align*} \\mathrm { Q F } \\mathrm { E R } : \\quad \\frac { A _ 0 \\rightarrow s = _ \\rho t } { A _ 0 \\rightarrow r [ s / x ^ \\rho ] = _ \\tau r [ t / x ^ \\rho ] } \\end{align*}"}
{"id": "6020.png", "formula": "\\begin{align*} \\widetilde { U } \\Big [ \\Big ( \\frac { d } { d x } + 2 x \\Big ) f \\Big ] & = \\Big ( \\frac { d } { d x } + x \\Big ) \\widetilde { U } f \\\\ \\widetilde { U } \\Big [ \\frac { d } { d x } f \\Big ] & = \\Big ( \\frac { d } { d x } - x \\Big ) \\widetilde { U } f . \\end{align*}"}
{"id": "1447.png", "formula": "\\begin{align*} K _ v = \\left \\{ \\gamma = \\left [ \\begin{array} { c c c } a & b & c \\\\ g & e & f \\\\ h & l & d \\end{array} \\right ] \\in G ( \\mathcal { O } _ v ) : \\gamma \\equiv \\left [ \\begin{array} { c c c } 1 _ m & \\ast & \\ast \\\\ 0 & 1 _ r & \\ast \\\\ 0 & 0 & 1 _ m \\end{array} \\right ] p _ v ^ { n _ v } \\right \\} . \\end{align*}"}
{"id": "9473.png", "formula": "\\begin{align*} H _ 0 = \\mathbb { N } , H _ i = \\langle p ^ { 2 i - 1 } + 2 , p H _ { i - 1 } \\rangle , i = 1 , \\ldots , s . \\end{align*}"}
{"id": "8825.png", "formula": "\\begin{align*} \\mathbb { E } _ { U , \\xi } \\left [ \\left \\| \\nabla \\phi _ { \\mu } ^ { 1 / \\bar { \\rho } } ( x _ { t ^ * } ) \\right \\| _ 2 ^ 2 \\right ] \\leq \\frac { \\bar { \\rho } } { \\bar { \\rho } - \\rho } \\frac { \\left ( \\phi _ { \\mu } ^ { 1 / \\bar { \\rho } } ( x _ 0 ) - \\underset { x } { \\min } \\ \\phi _ { \\mu } ( x ) \\right ) + 2 ( n ^ 2 + 2 n ) \\bar { \\rho } L _ { f , 0 } ^ 2 \\sum _ { t = 0 } ^ { T } \\alpha _ t ^ 2 } { \\sum _ { t = 0 } ^ T \\alpha _ t } . \\end{align*}"}
{"id": "593.png", "formula": "\\begin{align*} \\deg ( \\pi ^ n ) \\ = \\ n + 1 . \\end{align*}"}
{"id": "4058.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } u _ x \\bar { \\sigma } - \\int _ { 0 } ^ { 1 } \\rho \\bar { \\sigma } _ x + \\lambda \\int _ { 0 } ^ { 1 } \\rho \\bar { \\sigma } = \\int _ { 0 } ^ { 1 } f \\bar { \\sigma } . \\end{align*}"}
{"id": "3468.png", "formula": "\\begin{align*} | R _ j ( x , y ) - R _ j ( x , y ' ) | & \\leqslant c | y _ j - y _ j ' | \\int _ 0 ^ \\infty | h _ t ( x , y ) - h _ t ( x , y ' ) | \\frac { d t } { t \\sqrt t } \\\\ & \\leqslant c | y _ j - y _ j ' | \\bigg ( \\int _ 0 ^ { \\| x - y \\| ^ 2 } + \\int _ { \\| x - y \\| ^ 2 } ^ \\infty \\bigg ) | h _ t ( x , y ) - h _ t ( x , y ' ) | \\frac { d t } { t \\sqrt t } \\\\ & = : I \\ ! I _ 1 + I \\ ! I _ 2 . \\end{align*}"}
{"id": "5385.png", "formula": "\\begin{align*} \\hat { b } _ { t _ { n } . h } ( x ) = \\frac { \\sum _ { k = 0 } ^ { n - 1 } \\left ( t _ { n } - t _ { k } \\right ) ^ { 1 - 2 H } K \\left ( \\left ( X _ { t _ { k } } - x \\right ) / h \\right ) \\left ( X _ { t _ { k + 1 } } - X _ { t _ { k } } \\right ) } { \\sum _ { k = 0 } ^ { n - 1 } \\left ( t _ { n } - t _ { k } \\right ) ^ { 1 - 2 H } K \\left ( \\left ( X _ { t _ { k } } - x \\right ) / h \\right ) \\left ( t _ { k + 1 } - t _ { k } \\right ) } . \\end{align*}"}
{"id": "4085.png", "formula": "\\begin{align*} M _ 1 : = \\begin{pmatrix} 1 & 2 & 1 \\\\ 1 & 2 & 2 \\\\ 1 & 1 & 1 \\end{pmatrix} , \\ M _ 2 : = \\begin{pmatrix} 0 & 2 & 1 \\\\ 2 & 1 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix} , \\ M _ 3 : = \\begin{pmatrix} 2 & 2 & 0 \\\\ 1 & 1 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix} , \\ M _ 4 : = \\begin{pmatrix} 0 & 2 & 2 \\\\ 0 & 0 & 0 \\\\ 2 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "4124.png", "formula": "\\begin{align*} \\pi ( p ) = \\sum _ { i = 1 } ^ k ( \\beta ^ { - 1 } \\alpha _ i ) \\pi ( p _ i ) + \\sum _ { i = k + 1 } ^ d ( - \\beta ^ { - 1 } \\beta _ i ) \\pi ( p _ i ) . \\end{align*}"}
{"id": "721.png", "formula": "\\begin{align*} K _ { 1 1 } ^ { ( 1 ) } = \\frac { C _ W } { n _ 0 } \\end{align*}"}
{"id": "4219.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\max _ { t \\in [ 0 , T ] } R _ j ( t ) = 0 . \\end{align*}"}
{"id": "3089.png", "formula": "\\begin{align*} \\sum z _ i = 0 \\ , , \\end{align*}"}
{"id": "2726.png", "formula": "\\begin{align*} m _ { i , \\lambda } ( \\Z ( P , B ) ) = 0 \\end{align*}"}
{"id": "6589.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( d ) } \\sum _ { \\psi \\bmod d } \\psi ( m h ) \\overline { \\psi } ( n k ) = \\left \\{ \\begin{array} { c l } 1 & d | m h - n k \\\\ \\\\ 0 & \\end{array} \\right . \\end{align*}"}
{"id": "3502.png", "formula": "\\begin{align*} D _ { 2 1 } & = \\frac { 1 } { 2 \\pi i \\Gamma ( s _ 3 ) } \\int _ { a t _ 3 } ^ \\infty \\int _ v ^ \\infty \\int _ { ( c ) } \\frac { \\Gamma ( s _ 3 + z ) \\Gamma ( - z ) } { v ^ { s _ 2 - z } u ^ { s _ 1 + s _ 3 + z } } d z d u d v \\\\ & = \\frac { ( a t _ 3 ) ^ { 2 - s _ 1 - s _ 2 - s _ 3 } } { 2 \\pi i ( s _ 1 + s _ 2 + s _ 3 - 2 ) \\Gamma ( s _ 3 ) } \\int _ { ( c ) } \\frac { \\Gamma ( s _ 3 + z ) \\Gamma ( - z ) } { s _ 1 + s _ 3 - 1 + z } d z . \\end{align*}"}
{"id": "9066.png", "formula": "\\begin{align*} \\mathcal { F } _ h ( u ) = & \\frac { h } { 2 \\tau } \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { s } \\frac { \\hat { m } ^ 2 _ { i , j } } { \\rho _ { i , j } } D ^ { - 1 } _ { i , j } + h \\sum _ { j = 1 } ^ { N } \\left ( \\sum _ { i = 1 } ^ s \\rho _ { i , j } \\log \\rho _ { i , j } + \\frac { \\epsilon _ j } { 8 h ^ 2 } ( \\phi _ { j + 1 } - \\phi _ { j - 1 } ) ^ 2 \\right ) \\\\ & + \\frac { 1 } { 8 \\beta _ a } ( \\phi _ 0 + \\phi _ 1 ) ^ 2 + \\frac { 1 } { 8 \\beta _ b } ( \\phi _ N + \\phi _ { N + 1 } ) ^ 2 , \\end{align*}"}
{"id": "5945.png", "formula": "\\begin{align*} \\min _ { | z | = 1 } \\norm { f - z g } _ { L ^ p } \\le C \\big \\| \\ , | f | - | g | \\ , \\big \\| _ { L ^ p } ^ \\gamma \\cdot ( \\norm { f } _ { L ^ p } + \\norm { g } _ { L ^ p } ) ^ { 1 - \\gamma } \\ \\ \\forall \\ , f , g \\in V . \\end{align*}"}
{"id": "4922.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { \\bar x _ t - x _ t } ^ 2 & = \\norm { ( 1 - \\alpha _ t ) \\Gamma _ { t - 1 } \\sum _ { k = 1 } ^ { t - 1 } \\frac { \\alpha _ k } { \\Gamma _ k } \\frac { ( \\eta _ k - \\gamma _ k ) } { \\alpha _ k } \\nabla f ( \\bar x _ k ) } ^ 2 \\\\ & \\leq ( 1 - \\alpha _ t ) ^ 2 \\Gamma _ { t - 1 } \\sum _ { k = 1 } ^ { t - 1 } \\frac { \\alpha _ k } { \\Gamma _ k } \\frac { ( \\eta _ k - \\gamma _ k ) ^ 2 } { \\alpha _ k ^ 2 } \\norm { \\nabla f ( \\bar x _ k ) } ^ 2 \\end{aligned} \\end{align*}"}
{"id": "6785.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p _ 0 } i _ { k _ j } = - ( v _ 1 + 1 ) + ( v _ 1 + 1 ) + \\sum _ { j = 1 } ^ { y - 2 } ( - ( v _ { q _ j } + 1 ) + ( v _ { q _ j } + 1 ) ) - ( v _ { q _ { y - 1 } } + 1 ) = - ( v _ { q _ { y - 1 } } + 1 ) . \\end{align*}"}
{"id": "5127.png", "formula": "\\begin{align*} \\frac { \\widetilde { f } ' ( \\gamma ) } { \\widetilde { f } ( \\gamma ) } = - \\frac { \\widetilde { h } ' ( \\gamma ) } { \\widetilde { h } ( \\gamma ) } . \\end{align*}"}
{"id": "8031.png", "formula": "\\begin{align*} \\left \\langle ( \\partial _ { \\Sigma _ 0 , \\epsilon } ^ { * } F ) ^ { ( n ) } [ \\phi ] , h \\right \\rangle & = \\left \\langle K ^ { \\otimes n } , h \\otimes F ^ { ( n ) } [ \\partial _ { \\Sigma _ 0 , \\epsilon } \\phi ] \\right \\rangle \\end{align*}"}
{"id": "2773.png", "formula": "\\begin{align*} \\inf _ { ( \\theta _ 0 , x _ 0 ) \\in \\mathbb { R } \\times \\mathbb { R } ^ N } \\| u _ n - e ^ { i \\theta _ 0 } Q ( \\cdot - x _ 0 ) \\| _ { H ^ 1 } \\le \\| r _ n ^ 1 \\| _ { H ^ 1 } = o _ n ( 1 ) , \\end{align*}"}
{"id": "8468.png", "formula": "\\begin{align*} 2 e ( N _ 1 ( u ) ) + e ( N _ 1 ( u ) , N _ 2 ( u ) ) & \\leq ( 2 k - 1 ) d ( u ) + k d _ 2 ( u ) \\\\ & \\leq ( 2 k - 1 ) d ( u ) + k ( n - d ( u ) - 1 ) \\\\ & = ( k - 1 ) d ( u ) + k ( n - 1 ) . \\end{align*}"}
{"id": "6435.png", "formula": "\\begin{align*} \\omega ( \\pi ) : = \\lambda _ { \\# ( \\pi ) } \\cdot \\prod _ { i = 1 } ^ { \\# ( \\pi ) - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } - 1 ) . \\end{align*}"}
{"id": "3108.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + a z x ^ 2 y ^ 2 + x ( x ^ 4 + y ^ 4 ) = 0 \\ , . \\end{align*}"}
{"id": "5800.png", "formula": "\\begin{align*} y = A x _ 0 + z , \\end{align*}"}
{"id": "1427.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\bar { \\sigma } _ j & = - \\sigma _ j \\\\ \\frac { \\partial } { \\partial a _ i } \\bar { \\sigma } _ j & = ( 1 - t ) \\frac { \\partial } { \\partial a _ i } \\sigma _ j \\end{align*}"}
{"id": "7906.png", "formula": "\\begin{align*} R ( \\varphi _ t ) ( x ) = \\frac { 1 } { 2 } \\sum _ { p = 1 } ^ { q _ t } \\bigg ( e ^ { - i \\langle x , \\xi _ p ^ t \\rangle } \\phi _ t ( \\xi _ p ^ t ) | V _ p ^ t | + e ^ { i \\langle x , \\xi _ p ^ t \\rangle } \\overline { \\phi _ t ( \\xi _ p ^ t ) } | V _ p ^ t | \\bigg ) , \\end{align*}"}
{"id": "6067.png", "formula": "\\begin{align*} d R _ { \\varepsilon } ^ { 2 } = d R ^ { 2 } + o ( \\varepsilon ^ m ) , \\ \\ \\varepsilon \\to 0 . \\end{align*}"}
{"id": "7708.png", "formula": "\\begin{align*} n A = F \\cup ( r , a n - q ) \\cup a n - G ( n + 1 ) A = F \\cup ( r , a ( n + 1 ) - q ) \\cup a ( n + 1 ) - G \\ , . \\end{align*}"}
{"id": "5001.png", "formula": "\\begin{align*} \\mathcal { P } _ { k } = \\{ \\zeta ^ { \\bar { w } } ( I _ k ) \\bar { w } \\prec \\bar { s } _ { k } \\zeta ^ { \\bar { w } } ( J _ { k } ) \\bar { w } \\prec \\bar { t } _ { k } \\} . \\end{align*}"}
{"id": "7317.png", "formula": "\\begin{align*} \\Phi ( x , y , t ) : = u ( x , t ) - \\varphi ( y , t ) - { | x - y | ^ 4 \\over \\varepsilon } . \\end{align*}"}
{"id": "434.png", "formula": "\\begin{align*} \\widetilde { A } ^ { 0 } ( U ) U _ { t } + \\widetilde { A } ^ { i } ( U ) \\partial _ { i } U + \\widetilde { D } ( U ) U - \\widetilde { B } ^ { i j } ( U ) \\partial _ { i } \\partial _ { j } U = \\widetilde { F } ( U , D _ { x } U ) , \\end{align*}"}
{"id": "9148.png", "formula": "\\begin{align*} \\sum _ { i = \\xi ( \\delta ) } ^ \\infty \\varepsilon _ i < \\delta \\end{align*}"}
{"id": "7962.png", "formula": "\\begin{align*} \\norm { G _ 2 } _ { \\ell ^ 1 ( \\Z ^ { d \\times 2 } ) } & = \\norm { ( f _ 1 * f _ 2 ) F _ 2 } _ { \\ell ^ 1 ( \\Z ^ { d \\times 2 } ) } \\le \\norm { f _ 1 * f _ 2 } _ { \\ell ^ { \\alpha } ( \\Z ^ { d } ) } \\norm { F _ 2 } _ { \\ell ^ { \\Lambda _ 2 } ( \\Z ^ { d } ) } . \\end{align*}"}
{"id": "324.png", "formula": "\\begin{align*} & d \\rho _ i = \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ i - S _ j ) \\theta _ { i j } ( \\rho ) \\circ d W _ t ; \\\\ & d S _ i + ( \\frac 1 2 \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ i - S _ j ) ^ 2 \\frac { \\partial \\theta _ { i j } } { \\partial \\rho _ i } + \\frac 1 8 \\frac { \\partial } { \\partial \\rho _ i } I ( \\rho ) ) \\circ d W _ t + ( \\mathbb V _ i + \\sum _ { j \\in N ( i ) } \\mathbb W _ { i j } \\rho _ j ) d t = 0 , \\end{align*}"}
{"id": "3351.png", "formula": "\\begin{align*} G r ( ( \\phi , \\psi ) ) = \\Big \\{ \\Big ( ( x , u ) , ( \\phi ( x ) , \\psi ( u ) \\Big ) \\ ; | \\ ; x \\in L , u \\in V \\Big \\} \\subset ( L \\oplus V ) \\oplus ( L ^ { ' } \\oplus V ^ { ' } ) , \\end{align*}"}
{"id": "4071.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { 0 } ^ { 1 } x ( \\rho ^ 2 ( T , x ) - \\rho _ 0 ^ 2 ( x ) ) d x + \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\rho ^ 2 ( t , 1 ) d t - \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } \\rho ^ 2 d x d t + \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { 1 } x \\rho u _ x d x d t = 0 . \\end{align*}"}
{"id": "1192.png", "formula": "\\begin{align*} s ( \\mu , \\mathcal { B } , \\mathcal { U } ) & = \\sup \\left \\{ s \\geq 0 : \\ ( B _ n ) _ { n \\in \\mathcal { N } _ \\mu ( \\mathcal { B } , \\mathcal { U } , s ) } \\mbox { i s } \\mu \\mbox { - a . c . } \\right \\} . \\end{align*}"}
{"id": "3983.png", "formula": "\\begin{align*} & \\mathcal B ^ * _ u \\Phi _ { \\lambda ^ p _ k } = \\xi _ { \\lambda ^ p _ k } ( 1 ) + \\eta ^ \\prime _ { \\lambda ^ p _ k } ( 1 ) , \\forall k \\geq k _ 0 , \\\\ & \\mathcal B ^ * _ u \\Phi _ { \\lambda ^ h _ k } = \\xi _ { \\lambda ^ h _ k } ( 1 ) + \\eta ^ \\prime _ { \\lambda ^ h _ k } ( 1 ) , \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "9136.png", "formula": "\\begin{align*} J ^ A _ \\gamma x = J ^ A _ { \\lambda \\gamma } ( \\lambda x + ( 1 - \\lambda ) J ^ A _ \\gamma x ) . \\end{align*}"}
{"id": "5157.png", "formula": "\\begin{align*} Z _ \\alpha f ( x , \\omega ) = \\alpha ^ { - 1 / 2 } \\sum _ { k \\in \\Z } f \\left ( \\frac { x + k } { \\alpha } \\right ) e ^ { 2 \\pi i k \\omega } , \\alpha > 0 . \\end{align*}"}
{"id": "8804.png", "formula": "\\begin{align*} \\gamma ^ { \\epsilon , L } ( d \\varphi ) \\propto \\exp \\left [ - \\frac { \\epsilon ^ d } { 2 } \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\left ( \\varphi _ x ( - \\Delta ^ { \\epsilon } + m ) \\varphi _ x \\right ) \\right ] d \\varphi = \\exp \\left [ - \\frac { 1 } { 2 } ( \\varphi , A _ { \\epsilon } \\varphi ) \\right ] d \\varphi , \\end{align*}"}
{"id": "2039.png", "formula": "\\begin{align*} \\Phi ( s , t ) = \\Phi ( \\phi , s , t ) : = \\sup _ { r > 0 } \\left \\{ \\frac { s } { r } - \\frac { t } { \\phi ( r ) } \\right \\} . \\end{align*}"}
{"id": "9227.png", "formula": "\\begin{align*} p = _ X J ^ A _ 1 x \\leftrightarrow ( x - p ) \\in A p \\end{align*}"}
{"id": "1283.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { c _ { \\langle \\alpha _ { k } \\rangle } ( Y , \\lambda ) ^ { 2 } } { I ( \\alpha _ { k } ^ { \\Gamma } ) } = \\lim _ { k \\to \\infty } \\frac { A ( \\alpha _ { k } ^ { \\Gamma } ) ^ { 2 } } { 2 | \\Lambda ( A ( \\alpha _ { k } ^ { \\Gamma } ) , \\Gamma ) | } = \\mathrm { V o l } ( Y , \\lambda ) . \\end{align*}"}
{"id": "5275.png", "formula": "\\begin{align*} \\Delta : M ( A ) \\rightarrow M ( A \\otimes ^ I A ) = \\Delta ( 1 ) M ( A \\otimes A ) \\Delta ( 1 ) \\subseteq M ( A \\otimes A ) \\end{align*}"}
{"id": "1908.png", "formula": "\\begin{align*} \\bigoplus _ { i = 1 } ^ m \\left ( n _ i \\delta ^ { t _ i } \\odot ^ \\sharp \\nu \\delta ^ \\tau \\right ) = \\left ( \\bigoplus _ { i = 1 } ^ j ( n _ i - \\nu ) \\delta ^ { t _ i } \\right ) \\oplus ( n _ { j + 1 } - \\nu ) \\delta ^ { + \\infty } . \\end{align*}"}
{"id": "1730.png", "formula": "\\begin{align*} 2 ^ { ( \\mu _ * ( 1 \u2010 \\lambda ) \u2010 \\alpha _ * \\lambda ) k _ * t } \\stackrel { \\eqref { h a t _ n u _ d e f } } { = } n ^ { \u2010 \\hat \\nu } ; \\end{align*}"}
{"id": "4338.png", "formula": "\\begin{align*} \\left \\langle \\varepsilon , \\phi _ { 0 , b , \\beta } \\left ( \\frac { d + 2 } { 2 \\beta } - \\frac { y ^ 2 } { 2 } \\right ) \\right \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } = \\frac { \\gamma \\varepsilon _ 1 } { \\beta } \\| \\phi _ { 0 , b , \\beta } \\| ^ 2 _ { L ^ 2 _ { \\rho _ \\beta } } + O ( b ^ { \\frac { \\alpha } { 2 } + \\delta } ) \\end{align*}"}
{"id": "4158.png", "formula": "\\begin{align*} \\int w ( x , t ) d x = 0 \\end{align*}"}
{"id": "5191.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ N \\left ( \\frac { \\rho _ \\ell T _ { \\ell , \\pi } ^ { a v } } { 2 } + \\frac { c _ \\ell } { T _ { \\ell , \\pi } ^ { a v } } \\right ) \\implies \\Gamma ( \\tilde { \\pi } _ { O F } ^ \\star ) \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ N \\left ( \\frac { \\rho _ \\ell T _ { \\ell , \\tilde { \\pi } _ { O F } ^ \\star } ^ { a v } } { 2 } + \\frac { c _ \\ell } { T _ { \\ell , \\tilde { \\pi } _ { O F } ^ \\star } ^ { a v } } \\right ) , \\end{align*}"}
{"id": "6195.png", "formula": "\\begin{align*} \\| C \\hat { V } \\| _ F ^ 2 = \\| C X \\Sigma _ { \\hat { V } } \\| _ F ^ 2 \\leq \\| C X \\| _ F ^ 2 \\| \\Sigma _ { \\hat { V } } \\| ^ 2 _ { 2 } = \\| C X \\| _ F ^ 2 \\| \\hat { V } ^ { T } \\hat { V } \\| _ { 2 } . \\end{align*}"}
{"id": "1325.png", "formula": "\\begin{align*} 2 Q _ { \\tau } ( Z , Z _ { \\gamma } ) + 2 \\lfloor p _ { i } \\theta \\rfloor = 0 \\end{align*}"}
{"id": "7932.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } S ( k ) & = \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } | S \\cap [ 1 , k ] | = \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } \\left ( \\frac { k } { n } t + E _ k \\right ) \\\\ & = \\frac { t } { n ( \\ell + 1 ) } \\left ( B _ { \\ell + 1 } ( n + 1 ) - B _ { \\ell + 1 } \\right ) + \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } E _ k , \\end{align*}"}
{"id": "7967.png", "formula": "\\begin{align*} \\theta _ { | \\Sigma ^ i | } ( x ^ i , 1 ) - \\theta _ { | \\Sigma ^ i | } ( x ^ i ) \\leq \\delta , & \\ & \\theta _ { | \\Sigma ^ 1 | } ( x ^ 1 ) = \\theta _ { | \\Sigma ^ 0 | } ( x ^ 0 ) . \\end{align*}"}
{"id": "6098.png", "formula": "\\begin{align*} \\begin{array} { l l } \\beta _ { 1 } ( r , \\theta ) = { ( - 2 z _ { s t } \\cos \\theta - z _ { t t } \\sin \\theta ) } , & \\gamma _ { 1 } ( r , \\theta ) = { - z _ { t t } \\cos \\theta } , \\\\ \\beta _ { 2 } ( r , \\theta ) = { z _ { s s } \\cos \\theta } , & \\gamma _ { 2 } ( r , \\theta ) = { - z _ { t t } \\sin \\theta } , \\\\ \\beta _ { 3 } ( r \\theta ) = { z _ { s s } \\sin \\theta } , & \\gamma _ { 3 } ( r , \\theta ) = { z _ { s s } \\cos \\theta + 2 z _ { s t } \\sin \\theta } . \\end{array} \\end{align*}"}
{"id": "5806.png", "formula": "\\begin{align*} \\mathbf { x } _ m [ t ] = \\mathbf { D } ^ { - 1 } \\tilde { \\mathbf { x } } _ m [ t ] = \\frac { 1 } { N } \\mathbf { D } ^ { * } \\tilde { \\mathbf { x } } _ m [ t ] . \\end{align*}"}
{"id": "2007.png", "formula": "\\begin{align*} \\lambda ( \\langle b \\rangle ) = \\lambda ( b ) = \\lambda ( \\mathfrak { c } ( \\Gamma ( b ) ) ) , \\end{align*}"}
{"id": "329.png", "formula": "\\begin{align*} & \\inf _ { \\rho , v } \\ ; [ \\int _ 0 ^ 1 \\frac 1 2 \\ < v _ t , v _ t \\ > _ { \\theta ( \\rho _ t ) } d t ] \\\\ & \\ ; \\ ; d \\rho ( t ) + d i v _ G ^ { \\theta } ( \\rho ( t ) v ( t ) ) + d i v _ G ^ { \\theta } ( \\rho ( t ) \\nabla _ G \\Sigma ) d W _ t ^ { \\delta } = 0 \\\\ & \\ ; \\ ; \\rho ( 0 , \\omega ) = \\rho _ a , \\ ; \\rho ( 1 , \\omega ) = \\rho _ b . \\end{align*}"}
{"id": "1294.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda _ { ( \\infty , m ) } ( M , \\Gamma ) | } { | \\Lambda ( M , \\Gamma ) | } = 0 \\end{align*}"}
{"id": "7090.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( 1 \\otimes X ) & = 1 \\otimes X \\\\ \\tilde { \\Phi } ( X \\otimes 1 - 1 \\otimes X ) & = \\Bar { X } \\otimes 1 . \\end{align*}"}
{"id": "4808.png", "formula": "\\begin{align*} | J ' ( \\alpha , \\alpha ) | ^ 2 = q . \\end{align*}"}
{"id": "8961.png", "formula": "\\begin{align*} \\begin{aligned} & - \\left ( e ^ { - s } C _ p q \\left | \\dot { \\xi } ( s ) \\right | ^ { q - 2 } \\dot { \\xi } ( s ) \\right ) ^ \\prime + e ^ { - s } D f \\left ( \\xi ( s ) \\right ) = 0 \\\\ \\Longrightarrow & - \\left ( e ^ { - s } C _ p q \\left | \\dot { \\xi } ( s ) \\right | ^ { q - 2 } \\dot { \\xi } ( s ) \\right ) ^ \\prime \\cdot \\dot { \\xi } ( s ) e ^ s + D f \\left ( \\xi ( s ) \\right ) \\cdot \\dot { \\xi } ( s ) = 0 . \\end{aligned} \\end{align*}"}
{"id": "7129.png", "formula": "\\begin{align*} x _ t = x + \\int _ 0 ^ t \\tilde { f } ( s , x _ s ) d s - \\tilde { \\sigma } B _ t ^ H , 0 \\le t \\le 1 , x > 0 , \\end{align*}"}
{"id": "8199.png", "formula": "\\begin{align*} M _ 6 ( p , H ) = { \\pi ^ 2 \\over 9 } \\left ( 1 + \\frac { c _ a } { p } \\right ) , \\hbox { w h e r e } c _ a = \\begin{cases} - 2 a - 1 & \\hbox { i f $ a \\equiv 0 \\pmod 6 $ } , \\\\ - 3 & \\hbox { i f $ a \\equiv 2 , 3 \\pmod 6 $ } , \\\\ 2 a + 1 & \\hbox { i f $ a \\equiv 5 \\pmod 6 $ } . \\end{cases} \\end{align*}"}
{"id": "3665.png", "formula": "\\begin{align*} h h ' & = s + ( 2 + \\delta ) A s ^ { 1 + \\delta } + ( 1 + \\delta ) A ^ 2 s ^ { 1 + 2 \\delta } \\\\ h h '' & = ( 1 + \\delta ) \\delta A s ^ \\delta + ( 1 + \\delta ) \\delta A ^ 2 s ^ { 2 \\delta } \\\\ ( h ' ) ^ 2 & = 1 + 2 ( 1 + \\delta ) A s ^ \\delta + ( 1 + \\delta ) ^ 2 A ^ 2 s ^ { 2 \\delta } , \\end{align*}"}
{"id": "2323.png", "formula": "\\begin{align*} W ( M _ \\eta T _ \\xi f ) ( x , \\omega ) = W f ( x - \\xi , \\omega - \\eta ) . \\end{align*}"}
{"id": "6547.png", "formula": "\\begin{align*} \\lim \\limits _ { H ^ 2 \\rightarrow + \\infty } \\frac { r } { k - r } w _ 2 = \\ & H ^ 2 \\left ( \\frac { k } { r } - \\frac { 1 } { 2 } \\right ) + \\frac { H ^ 2 r } { 2 ( k - r ) } \\left ( \\frac { k } { r } - \\epsilon \\right ) \\left ( - \\frac { k } { r } - \\epsilon + 1 \\right ) \\end{align*}"}
{"id": "8580.png", "formula": "\\begin{align*} \\big ( \\mathcal { F } ^ \\# \\big ) ^ { - 1 } [ \\phi ] ( x ) = \\int \\mathcal { K } ^ \\# ( x , k ) \\phi ( k ) \\ , d k . \\end{align*}"}
{"id": "6724.png", "formula": "\\begin{align*} h _ { 1 , r - 1 } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + R _ { r - 1 } = 0 . \\end{align*}"}
{"id": "2468.png", "formula": "\\begin{align*} D _ L = \\left \\{ \\begin{pmatrix} L & 0 \\\\ 0 & L ^ { - T } \\end{pmatrix} \\mid \\det ( L ) \\neq 0 \\right \\} . \\end{align*}"}
{"id": "142.png", "formula": "\\begin{align*} c _ { \\alpha , d } = \\dfrac { - \\alpha ( \\alpha - 1 ) \\Gamma \\left ( \\frac { \\alpha + d } { 2 } \\right ) } { 4 \\cos \\left ( \\frac { \\alpha \\pi } { 2 } \\right ) \\Gamma \\left ( \\frac { \\alpha + 1 } { 2 } \\right ) \\pi ^ { \\frac { d - 1 } { 2 } } \\Gamma ( 2 - \\alpha ) } . \\end{align*}"}
{"id": "7182.png", "formula": "\\begin{align*} g _ { R , X _ \\ast , V _ \\ast } ( t , x , v ) = h _ { V _ \\ast } ( t , x - X _ \\ast + ( R - t ) V _ \\ast , v ) . \\end{align*}"}
{"id": "7912.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } | | K _ { x , L } ( u , v ) - K _ x ( u - v ) | | _ { C ^ k ( C _ { R + 1 } \\times C _ { R + 1 } ) } = 0 . \\end{align*}"}
{"id": "3378.png", "formula": "\\begin{align*} \\Big ( \\phi _ t = I d _ L + t [ \\mathfrak { X } , - ] + \\sum _ { i = 2 } ^ { + \\infty } \\phi _ i t ^ { i } , \\ ; \\psi _ t = I d _ V + t D ( \\mathfrak { X } ) ( - ) + \\sum _ { i = 2 } ^ { + \\infty } \\psi _ i t ^ { i } \\Big ) , \\end{align*}"}
{"id": "9151.png", "formula": "\\begin{align*} D ( 0 , A ( x ) ) = 0 x \\in \\mathrm { z e r } A \\end{align*}"}
{"id": "7042.png", "formula": "\\begin{align*} b _ j = a _ 1 \\Big ( q - \\sum _ { k = 1 } ^ { L } q _ k \\widetilde { \\phi _ k } \\Big ) e _ j + q _ j . \\end{align*}"}
{"id": "5727.png", "formula": "\\begin{align*} ( \\sigma u ) ( x ) = u ( \\sigma ^ { - 1 } ( x ) ) , \\ \\forall \\sigma \\in G , \\ u \\in H ^ 1 ( M ) , \\ x \\in M . \\end{align*}"}
{"id": "3445.png", "formula": "\\begin{align*} \\| T _ j T ^ * _ k f \\| _ { L ^ 2 ( \\R ^ N , \\omega ) } ^ 2 & \\lesssim \\sum \\limits _ { \\sigma \\in G } ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \\varepsilon } \\int _ { \\mathbb R ^ N } \\big ( M f ( \\sigma ( x ) ) \\big ) ^ 2 d \\omega ( x ) \\\\ & = \\sum \\limits _ { \\sigma \\in G } ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \\varepsilon } \\int _ { \\mathbb R ^ N } \\big ( M f ( x ) \\big ) ^ 2 d \\omega ( x ) \\\\ & \\lesssim ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \\varepsilon } \\| f \\| _ { L ^ 2 ( \\R ^ N , \\omega ) } ^ 2 , \\end{align*}"}
{"id": "6826.png", "formula": "\\begin{align*} X ( s , \\varphi ) : = e ^ { i \\varphi } \\cdot \\eta ( s ) , \\forall \\ , ( s , \\varphi ) \\in [ a , b ] \\times [ 0 , 2 \\pi ] , \\end{align*}"}
{"id": "8702.png", "formula": "\\begin{align*} \\hat { b } ( s ) : = \\min _ { \\omega \\in \\Omega } \\biggl \\{ a _ { 1 0 } a _ { 2 0 } + \\sum _ { t : \\omega _ t = 1 } a _ { 2 p ^ t _ 2 } \\bigl ( s _ { 1 p ^ t _ 1 } - s _ { 1 p ^ { t - 1 } _ 1 } \\bigr ) + \\sum _ { t : \\omega _ t = 2 } a _ { 1 p ^ t _ 1 } \\bigl ( s _ { 2 p ^ t _ 2 } - s _ { 2 p ^ { t - 1 } _ 2 } \\bigr ) \\biggr \\} , \\end{align*}"}
{"id": "6685.png", "formula": "\\begin{align*} \\mathbb { C } _ { \\infty } ( ( t ) ) & \\rightarrow \\mathbb { C } _ { \\infty } ( ( t ) ) \\\\ f : = \\sum _ { i } a _ i t ^ i & \\mapsto \\sum _ { i } a _ i ^ { q ^ { n } } t ^ i = : f ^ { ( n ) } . \\end{align*}"}
{"id": "1807.png", "formula": "\\begin{align*} \\widehat { A } ( X , [ \\alpha ] ) : = \\int _ X c _ X \\widehat { A } ( X ) \\wedge \\psi _ X ^ * ( \\alpha ) \\end{align*}"}
{"id": "1943.png", "formula": "\\begin{align*} g ( x ) = f ( x ) - \\langle h ( x ) , \\lambda ( x ) \\rangle + \\beta \\norm { h ( x ) } ^ 2 \\end{align*}"}
{"id": "1415.png", "formula": "\\begin{align*} \\hat q _ { j i } = ( \\hat q _ { i j } ) ^ * . \\end{align*}"}
{"id": "6607.png", "formula": "\\begin{align*} \\mathcal { U } ^ 1 ( h , k ) = Q \\sum _ { \\substack { 1 \\leq c \\leq C \\\\ ( c , h k ) = 1 } } \\frac { \\mu ( c ) } { c } \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , c ) = 1 \\\\ m h \\neq n k } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\frac { \\phi ( m n h k ) } { m n h k } \\int _ 0 ^ { \\infty } W ( u ) \\ , d u . \\end{align*}"}
{"id": "978.png", "formula": "\\begin{align*} \\vert ( - \\Delta ) ^ s ( \\zeta \\circ Q _ \\rho ) ( x ) \\vert & = C \\int _ { B _ { 3 / 4 } ( - 2 e _ 1 / \\rho ) } \\frac { ( \\zeta \\circ Q _ \\rho ) ( y ) } { \\vert x - y \\vert ^ { n + 2 s } } \\dd y \\leqslant C x \\in B _ { 3 / 4 } \\end{align*}"}
{"id": "1160.png", "formula": "\\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) G ( x , t , k ) , \\end{align*}"}
{"id": "2933.png", "formula": "\\begin{align*} & \\phantom { { } = { } } \\frac 1 { n ^ 2 ( n - 3 ) } \\Big \\{ ( 4 n \\ell - n ^ 2 ) \\dfrac { \\ell - Z _ { \\{ 3 , 4 \\} } ( \\ell ) } { n - 2 } - 3 \\ell ^ 2 - n ^ 2 \\dfrac { \\ell - Z _ { \\{ 3 , 4 \\} } ( \\ell ) } { n - 2 } Z _ { \\{ 3 , 4 \\} } ( \\ell ) + n \\ell Z _ { \\{ 3 , 4 \\} } ( \\ell ) \\Big \\} \\\\ & = \\frac 1 { n ^ 2 ( n - 2 ) ( n - 3 ) } \\Big \\{ F _ { n 1 } + F _ { n 2 } \\Big \\} , \\end{align*}"}
{"id": "5043.png", "formula": "\\begin{align*} \\hat { K } ^ { \\lambda , \\nu } ( z ; w ) : = \\sum _ { w = 1 } ^ k \\sum _ { i = 0 } ^ { \\infty } K ^ j _ { ( i ) } ( \\lambda ; \\nu ) z ^ i w ^ j . \\end{align*}"}
{"id": "5003.png", "formula": "\\begin{align*} [ \\eta , \\xi ] ( x ) \\equiv \\eta \\circ \\xi ( x ) - \\xi \\circ \\eta ( x ) = o ( x ^ 3 ) x = 0 . \\end{align*}"}
{"id": "3889.png", "formula": "\\begin{align*} ( a - b ) + 2 ( ( n - b + 1 ) + ( n - a + 1 ) ) < \\# M ^ { [ 1 ] } = 4 n - a - b + 1 . \\end{align*}"}
{"id": "1623.png", "formula": "\\begin{align*} H ^ { Q , j } ( x ) = \\pm \\mu ^ { \\frac { d - 1 } { p ' } - 1 } \\tilde { \\Phi } ( \\mu \\bar { x } ) = \\pm \\mu ^ { \\frac { d - 1 } { p } } \\alpha _ i x _ { \\bar { j } } \\end{align*}"}
{"id": "2438.png", "formula": "\\begin{align*} S _ { g , \\L } ^ { - 1 } f = \\sum _ { \\l \\in \\L } \\langle f , \\pi ( \\l ) \\widetilde { g } \\rangle \\ \\pi ( \\l ) \\widetilde { g } . \\end{align*}"}
{"id": "3827.png", "formula": "\\begin{align*} ( \\beta _ 1 , \\ldots , \\beta _ n ) : = ( b _ 1 , \\ldots , b _ n ) E ^ { - 1 } , \\end{align*}"}
{"id": "4024.png", "formula": "\\begin{align*} \\lambda ^ h _ k : = - \\tilde \\mu _ k : = - 1 - \\alpha _ { 1 , k } - i ( 2 k \\pi + \\alpha _ { 2 , k } ) , \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "7627.png", "formula": "\\begin{align*} u ^ { - \\frac { \\alpha + 1 } { \\alpha } } \\bar { g } ( \\lambda \\partial _ { r } , \\nabla P ) = u ^ { - \\frac { \\alpha + 1 } { \\alpha } } \\lambda ^ { 2 } \\sum _ { i } r _ { i } ^ { 2 } - \\bar { g } ( \\lambda \\partial _ { r } , \\nabla \\log u ) \\end{align*}"}
{"id": "1008.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s ( \\tilde u - \\tau \\varphi ^ { ( 2 ) } ) ( a ) = - C \\int _ { \\R ^ n _ + } \\bigg ( \\frac 1 { \\vert a - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert a _ \\ast - y \\vert ^ { n + 2 s } } \\bigg ) ( \\tilde u - \\tau \\varphi ^ { ( 2 ) } ) ( y ) \\dd y . \\end{align*}"}
{"id": "9111.png", "formula": "\\begin{align*} f \\circ g = \\sum _ { i = 0 } ^ { m - 1 } ( - 1 ) ^ { i ( n - 1 ) } f \\circ _ i g , [ f , g ] = f \\circ g - ( - 1 ) ^ { ( m - 1 ) ( n - 1 ) } g \\circ f \\end{align*}"}
{"id": "7291.png", "formula": "\\begin{align*} \\sum & p ( n ) X ^ n = \\frac { 1 } { \\alpha } = \\frac { \\alpha ( X ^ { 2 5 } ) } { \\alpha ( X ^ 5 ) ^ 6 } \\alpha ( \\zeta X ) \\alpha ( \\zeta ^ 2 X ) \\alpha ( \\zeta ^ 3 X ) \\alpha ( \\zeta ^ 4 X ) \\\\ & = \\frac { \\alpha ( X ^ { 2 5 } ) } { \\alpha ( X ^ 5 ) ^ 6 } ( \\alpha _ 0 + \\zeta \\alpha _ 1 + \\zeta ^ 2 \\alpha _ 2 ) ( \\alpha _ 0 + \\zeta ^ 2 \\alpha _ 1 + \\zeta ^ 4 \\alpha _ 2 ) ( \\alpha _ 0 + \\zeta ^ 3 \\alpha _ 1 + \\zeta \\alpha _ 2 ) ( \\alpha _ 0 + \\zeta ^ 4 \\alpha _ 1 + \\zeta ^ 3 \\alpha _ 2 ) . \\end{align*}"}
{"id": "4909.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { t } \\frac { \\alpha _ k } { \\Gamma _ k } = \\frac { 1 } { \\Gamma _ t } ~ ~ ~ ~ ~ \\implies ~ ~ ~ ~ ~ \\Gamma _ t \\sum _ { k = 1 } ^ t \\frac { \\alpha _ k } { \\Gamma _ k } = 1 . \\end{align*}"}
{"id": "1914.png", "formula": "\\begin{align*} L [ f ] = \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } K ( x , y ) f ( t , y , w ) ( w - v ) \\ , d w d y , \\end{align*}"}
{"id": "5367.png", "formula": "\\begin{align*} q = g + \\tau ' c \\end{align*}"}
{"id": "453.png", "formula": "\\begin{align*} \\mu _ n ( L / A ) = \\frac { [ L ' : \\sigma ( L ' ) ] } { [ L : \\sigma ( L ) ] _ { i n s e p } } . \\end{align*}"}
{"id": "9437.png", "formula": "\\begin{align*} ( \\varPhi _ q \\ , f ) ( x , t ) = \\int _ { { \\mathbb R } ^ n } f ( y , t ) \\wedge \\phi _ q ( x , y ) \\end{align*}"}
{"id": "5858.png", "formula": "\\begin{align*} \\bar \\alpha _ { i + 1 } / \\rho _ { i + 1 } = \\rho _ { i + 1 , k ( i + 1 ) } ^ { - 1 } / \\rho _ { i + 1 } = \\rho _ { i , k ( i + 1 ) } ^ { - 1 } . \\end{align*}"}
{"id": "1707.png", "formula": "\\begin{align*} T _ { t , m } = \\{ E \\in T _ { t , j , m } : \\ ; j \\in \\hat J _ t \\} , \\hat T _ { t , m } = \\{ E \\cap E ' : \\ ; E \\in T _ { t , m } , \\ ; E ' \\in T _ { t , m + 1 } \\} , \\end{align*}"}
{"id": "4480.png", "formula": "\\begin{align*} \\langle z | ( a ^ { \\dagger } a ) _ { k , \\lambda } | z \\rangle & = e ^ { - \\frac { | z | ^ { 2 } } { 2 } } \\cdot e ^ { - \\frac { | z | ^ { 2 } } { 2 } } \\sum _ { m , n = 0 } ^ { \\infty } \\frac { \\bar { z } ^ { m } z ^ { n } } { \\sqrt { m ! } \\sqrt { n ! } } ( n ) _ { k , \\lambda } \\langle m | n \\rangle \\\\ & = e ^ { - | z | ^ { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\frac { | z | ^ { 2 n } } { n ! } ( n ) _ { k , \\lambda } . \\end{align*}"}
{"id": "427.png", "formula": "\\begin{align*} & \\| \\mathcal { T } ^ { 2 } ( U ^ { 2 } ) - \\mathcal { T } ^ { 2 } ( U ^ { 1 } ) \\| _ { s - 1 } ^ { 2 } = \\| \\mathcal { T } ( V ^ { 2 } ) - \\mathcal { T } ( V ^ { 1 } ) \\| _ { s - 1 } ^ { 2 } \\\\ & \\leq C \\left ( T _ { 0 } \\sup _ { 0 \\leq t \\leq T _ { 0 } } \\| \\mathcal { T } ( U ^ { 2 } ) - \\mathcal { T } ( U ^ { 1 } ) \\| _ { s - 1 } ^ { 2 } + \\int _ { 0 } ^ { T _ { 0 } } \\| ( \\hat { v } ^ { 2 } - \\hat { v } ^ { 1 } ) ( \\tau ) \\| _ { s } ^ { 2 } d \\tau \\right ) . \\end{align*}"}
{"id": "1653.png", "formula": "\\begin{align*} \\frac { | \\nabla _ y p _ t ^ B ( x , \\cdot ) | ( y ) } { p _ t ^ B ( x , y ) } = | \\nabla _ y \\log p _ t ^ B ( x , \\cdot ) | ( y ) \\leq \\frac { c _ { B , T } } { \\sqrt { t } } \\Big ( 1 + \\frac { \\varrho ( x , y ) } { \\sqrt { t } } \\Big ) \\end{align*}"}
{"id": "3029.png", "formula": "\\begin{align*} ( \\pi ^ { s + 1 , s } ) ^ { * } \\rho = f _ { 0 } \\omega _ { 0 } + \\sum _ { k = 0 } ^ { s } f _ { \\sigma } ^ { i , j _ { 1 } \\ldots j _ { k } } \\omega _ { j _ { 1 } \\ldots j _ { k } } ^ { \\sigma } \\wedge \\omega _ { i } + \\eta , \\end{align*}"}
{"id": "1423.png", "formula": "\\begin{align*} 1 - | a | x \\geq | a | - x \\geq | a | - ( 1 - 2 ^ { n - 1 } ( 1 - | a | ) ) = ( 2 ^ { n - 1 } - 1 ) ( 1 - | a | ) . \\end{align*}"}
{"id": "7631.png", "formula": "\\begin{align*} \\lambda ^ 1 ( m ) = \\inf _ { \\substack { u \\in H ^ 1 _ 0 ( \\Omega ) \\\\ \\int _ { \\Omega } m u ^ 2 > 0 } } \\frac { \\int _ { \\Omega } | \\nabla u | ^ 2 } { \\int _ { \\Omega } m u ^ 2 } . \\end{align*}"}
{"id": "1659.png", "formula": "\\begin{align*} \\lim _ { | x | + | v | \\to \\infty } \\mathcal W ( x , v ) = \\infty \\end{align*}"}
{"id": "3849.png", "formula": "\\begin{align*} p ' _ { k , j } + p _ { k , j - 1 } + p _ { k , m } \\tilde f _ { m , j } + \\tilde f _ { k , j } = f _ { k , j } + f _ { k , m + 1 } p _ { m + 1 , j } + p _ { k + 1 , j } . \\end{align*}"}
{"id": "4364.png", "formula": "\\begin{align*} b ^ { - \\frac { \\gamma } { 2 } - \\frac { 1 } { 2 } } \\partial _ \\xi \\phi _ { i , i n t , \\beta } \\left ( \\frac { y _ 0 } { \\sqrt { b } } \\right ) = b ^ { - \\frac { \\gamma } { 2 } } \\frac { \\phi _ { i , i n t , \\beta } \\left ( \\frac { y _ 0 } { \\sqrt { b } } \\right ) } { \\phi _ { i , o u t , \\beta } ( y _ 0 ) } \\partial _ y \\phi _ { i , o u t , \\beta } ( y _ 0 ) , \\end{align*}"}
{"id": "4036.png", "formula": "\\begin{align*} \\sqrt { a _ k ^ 2 + b _ k ^ 2 } = \\left [ ( 1 + \\alpha _ { 1 , k } ) ^ 2 + ( 2 k \\pi + \\alpha _ { 2 , k } ) ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } = \\left [ 4 k ^ 2 \\pi ^ 2 + O ( k ) \\right ] ^ { \\frac { 1 } { 2 } } = 2 | k \\pi | + O ( 1 ) , \\ \\ \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "6714.png", "formula": "\\begin{align*} _ { s + 1 } \\mathcal { F } _ s ( \\alpha ) ^ { q ^ d } = \\sum _ { n \\geq 0 } \\sum _ { H \\geq h \\geq 1 } f _ h ( t ) \\theta ^ { h q ^ { n + d } } \\alpha ^ { q ^ { n + d } } = \\sum _ { H \\geq h \\geq 1 } f _ h ( t ) \\sum _ { n \\geq 0 } \\theta ^ { h q ^ { n + d } } \\alpha ^ { q ^ { n + d } } . \\end{align*}"}
{"id": "6028.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 ( y ) = \\exp \\bigg [ - \\int _ 1 ^ x \\frac { 1 } { 2 } ( 1 - \\frac { \\alpha + 1 } { y } ) \\ , d y \\bigg ] = e ^ { 1 / 2 } x ^ { ( \\alpha + 1 ) / 2 } e ^ { - x / 2 } . \\end{align*}"}
{"id": "8886.png", "formula": "\\begin{align*} \\psi ( z _ 0 , \\ldots , z _ { q - 1 } ) = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( z _ 0 , \\ldots , z _ i , p ( z _ i ) , \\ldots , p ( z _ { q - 1 } ) ) . \\end{align*}"}
{"id": "6359.png", "formula": "\\begin{align*} l = d _ 1 i _ 1 + \\dots + d _ n i _ n 0 \\leq \\sum _ { t > t _ 0 } d _ t i _ t < i _ { t _ 0 } \\ . \\end{align*}"}
{"id": "8958.png", "formula": "\\begin{align*} C _ p q \\left | \\dot { \\xi } ( 0 ) \\right | ^ { q - 1 } = ( f ( \\xi ( 0 ) ) - u ( \\xi ( 0 ) ) ) ^ \\frac { 1 } { p } \\end{align*}"}
{"id": "9447.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } H _ \\mu u ' \\ ! + \\ ! \\varPhi _ q d _ q { \\mathcal N } _ q u ' & = & \\varPhi _ q d _ q f & ( x , t ) \\in { \\mathbb R } ^ n \\times ( 0 , T ) , \\\\ u & = & u _ 0 , & ( x , t ) \\in \\mathbb { R } ^ n \\times \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
{"id": "2522.png", "formula": "\\begin{align*} \\pi ( F ) = \\pi ( F _ r ) = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\pi ( x , \\omega , 0 ) \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "852.png", "formula": "\\begin{align*} U \\left ( t \\right ) = U \\left ( t ^ S _ { j - 1 } \\right ) = t ^ S _ { j - 1 } + \\tau _ { \\rm f } - \\tau ^ { \\rm R e a c } _ { V _ { j - 1 } } , t \\in \\mathcal { I } _ j . \\end{align*}"}
{"id": "5773.png", "formula": "\\begin{align*} \\partial _ x ^ 2 F = \\sum _ j \\nabla ^ 2 \\phi _ j ( x ) \\cdot f _ j ( x ) + 2 d \\phi _ j ( x ) \\otimes d f _ j ( x ) + \\phi _ j ( x ) \\cdot \\nabla ^ 2 f _ j ( x ) . \\end{align*}"}
{"id": "4541.png", "formula": "\\begin{align*} c ( r , k ) : = L ( 2 , \\chi _ 0 ^ { 2 } ) G ( 1 , \\chi _ 0 ) F _ { u } ( 1 , \\chi _ 0 ) . \\end{align*}"}
{"id": "9301.png", "formula": "\\begin{align*} K _ Y + \\pi _ { * } ^ { - 1 } \\varepsilon D _ 1 + \\pi _ { * } ^ { - 1 } \\varepsilon D _ 2 + \\sum _ { i = 1 } ^ n a _ i E _ i = \\pi ^ { * } ( K _ X + \\varepsilon D ) \\end{align*}"}
{"id": "3163.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal e _ { i _ k } \\frac { e _ { i _ k } ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 } . \\end{align*}"}
{"id": "4047.png", "formula": "\\begin{align*} \\lambda ^ p _ k & = - k ^ 2 \\pi ^ 2 + O ( 1 ) , k \\geq k _ 0 , \\\\ \\lambda ^ h _ k & = - 1 - i 2 k \\pi + O ( | k | ^ { - 1 } ) , | k | \\geq k _ 0 . \\end{align*}"}
{"id": "3325.png", "formula": "\\begin{align*} f _ i = ( n - 2 ) f _ { i - 1 } + g _ { i - 1 } = ( n - 2 ) f _ { i - 1 } + f _ { i - 1 } - f _ { i - 2 } = ( n - 1 ) f _ { i - 1 } - f _ { i - 2 } . \\end{align*}"}
{"id": "1250.png", "formula": "\\begin{align*} \\mu \\left ( \\bigcup _ { i \\in \\mathbb { N } } \\bigcup _ { L \\in \\mathcal { F } _ { k , i } } L \\right ) = 1 . \\end{align*}"}
{"id": "4396.png", "formula": "\\begin{align*} M f ( y ) = \\sup _ { y \\in \\mathcal { I } } \\frac { \\int _ { \\mathcal { I } } | f ( y ' ) ( y ' ) ^ \\gamma | ( y ' ) ^ { 1 + \\omega } e ^ { - \\frac { ( y ' ) ^ 2 } { 4 } } d y ' } { \\int _ { \\mathcal { I } } ( y ' ) ^ { 1 + \\omega } e ^ { - \\frac { ( y ' ) ^ 2 } { 4 } } d y ' } , \\omega = \\sqrt { d ^ 2 - 1 2 d + 2 4 } , \\end{align*}"}
{"id": "625.png", "formula": "\\begin{align*} A ( x ) \\ = \\ a + ( b - a ) \\frac { 2 \\phi ( x ) + 1 } { 2 x + 2 } , \\end{align*}"}
{"id": "3513.png", "formula": "\\begin{align*} D _ { 1 1 2 } & \\ll \\left ( t _ 3 ^ { - \\sigma _ 3 } + 1 \\right ) \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 - 1 } & ( \\sigma _ 2 > \\frac { 5 } { 2 } ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 - 1 } \\log t _ 3 & ( \\sigma _ 2 = \\frac { 5 } { 2 } ) \\\\ t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < \\frac { 5 } { 2 } ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "2192.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , w _ { n } ^ { \\pm } ) w _ { n } ^ { \\pm } \\ d x = \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , w ^ { \\pm } ) w ^ { \\pm } \\ d x . \\end{align*}"}
{"id": "225.png", "formula": "\\begin{align*} D ^ { \\alpha - 1 } \\left ( ( P _ t ^ { \\nu _ \\alpha } ) ^ * ( f ) \\right ) ( x ) + \\frac { 1 } { p _ \\alpha ( x ) } R ^ \\alpha \\left ( p _ \\alpha , ( P _ t ^ { \\nu _ \\alpha } ) ^ * ( f ) \\right ) ( x ) & = \\dfrac { - x e ^ { - \\alpha t } } { ( 1 - e ^ { - \\alpha t } ) } ( P ^ { \\nu _ \\alpha } _ t ) ^ { * } ( f ) ( x ) \\\\ & + \\dfrac { e ^ { - t } } { \\left ( 1 - e ^ { - \\alpha t } \\right ) } ( P ^ { \\nu _ \\alpha } _ t ) ^ { * } ( h f ) ( x ) , \\end{align*}"}
{"id": "3207.png", "formula": "\\begin{align*} { \\rm S I N R } _ { D _ 1 } ^ { x _ 1 } = \\frac { | h _ 1 | ^ 2 ( 1 - \\rho ) P _ t \\alpha _ 1 } { | h _ { S I } | ^ 2 P _ H + \\sigma ^ 2 } . \\end{align*}"}
{"id": "5573.png", "formula": "\\begin{align*} | A \\backslash B | \\ \\leqslant \\ \\frac { 1 } { \\inf _ n w ( n ) } \\sum _ { n \\in A \\backslash B } w ( n ) \\cdot 1 \\ = \\ \\frac { 1 } { \\inf _ n w ( n ) } \\| 1 _ { A \\backslash B } \\| . \\end{align*}"}
{"id": "5660.png", "formula": "\\begin{align*} \\phi ^ { - 1 } \\rho & = ( \\tau , ( \\phi _ { \\tau ^ { - 1 } 1 } ^ { - 1 } \\alpha _ 1 , \\phi _ { \\tau ^ { - 1 } 2 } ^ { - 1 } , \\phi _ { \\tau ^ { - 1 } 3 } ^ { - 1 } \\alpha _ 3 ) ) \\\\ & = ( \\tau , ( ( \\phi _ { \\tau ^ { - 1 } 1 } ^ { - 1 } \\alpha _ 1 , \\varepsilon , \\phi _ { \\tau ^ { - 1 } 3 } ^ { - 1 } \\alpha _ 3 ) ) = \\psi \\in \\prescript { \\tau } { } N _ l ^ A \\end{align*}"}
{"id": "341.png", "formula": "\\begin{align*} \\mathcal L ( \\rho , m , S ) & : = \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > + \\mathcal A ( \\rho , m ) \\\\ & \\quad - \\int _ 0 ^ 1 ( \\ < \\dot S , \\rho \\ > + \\ < m , \\nabla _ G S \\ > + \\ < \\nabla _ G \\Sigma , \\nabla _ G S \\ > \\dot W ^ { \\delta } ( t ) ) d t . \\end{align*}"}
{"id": "1606.png", "formula": "\\begin{align*} \\| D ^ m \\Psi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } & = \\mu ^ { \\frac { d - 1 } { p } + m - \\frac { d - 1 } { r } } \\| D ^ m \\Phi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } = \\mu ^ { \\frac { d - 1 } { p } + m - \\frac { d - 1 } { r } } \\| D ^ m ( \\eta _ \\ell ) \\ast \\varphi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } \\\\ & = \\frac { \\mu ^ { \\frac { d - 1 } { p } + m - \\frac { d - 1 } { r } } } { \\ell ^ m } \\| ( D ^ m \\eta ) _ \\ell \\ast \\varphi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } = C ( a , \\zeta , m ) \\mu ^ { \\frac { d - 1 } { p } + m - \\frac { d - 1 } { r } } \\end{align*}"}
{"id": "5770.png", "formula": "\\begin{align*} ( \\hat y _ i ^ 1 , \\dots , \\hat y _ i ^ n ) : = ( - \\hat f _ i ^ 1 , \\dots , - \\hat f _ i ^ m , \\varphi _ { i , \\delta } ^ { m + 1 } , \\dots , \\varphi _ { i , \\delta } ^ n ) : B _ { R } ^ n ( 0 , \\hat g _ { i , \\delta } ^ * ) \\to \\mathbb R ^ n \\end{align*}"}
{"id": "2180.png", "formula": "\\begin{align*} v _ n : = \\frac { u _ n } { \\Vert u _ n \\Vert } \\ \\ n \\in \\mathbb { N } . \\end{align*}"}
{"id": "3779.png", "formula": "\\begin{align*} \\overline { \\mathbb { K } } _ { \\overline { \\psi } _ F ^ l } ^ { r _ l ( \\pi _ F ) ^ { ( l ) } } ( w ) \\xi \\{ \\chi , 0 \\} = \\epsilon ( \\chi ^ { - 1 } r _ l ( \\pi _ F ) ^ { ( l ) } , \\overline { \\psi } _ F ^ l ) \\xi \\{ \\chi , - n ( \\chi ^ { - 1 } r _ l ( \\pi _ F ) ^ { ( l ) } , \\overline { \\psi } _ F ^ l ) \\} , \\end{align*}"}
{"id": "5811.png", "formula": "\\begin{align*} \\ddot { m } _ 1 ^ k = \\arg \\max _ { m = 1 , \\ldots , M } \\bigl ( \\mathbf { b } _ k \\bigr ) , \\end{align*}"}
{"id": "6970.png", "formula": "\\begin{align*} \\forall k \\in \\N \\colon x _ k : = \\frac 1 k , y _ k : = \\frac 1 { k ^ 2 } , x _ k ^ * : = 1 , \\lambda _ k : = \\frac k 2 , \\end{align*}"}
{"id": "6009.png", "formula": "\\begin{align*} C _ { n } ^ { \\lambda , \\beta } ( x ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\Bigg [ \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ { i } i ! B _ { n - k , i } \\left ( \\left ( \\tilde { M } _ { \\lambda , \\beta } ( j ) \\right ) _ { j = 1 } ^ { n - k - i + 1 } \\right ) \\Bigg ] ( x ) _ { k } , \\end{align*}"}
{"id": "4378.png", "formula": "\\begin{align*} \\Phi ( y , \\tau ) = \\frac { 1 } { 2 } \\Lambda _ y Q _ b \\left ( y \\right ) \\left [ \\frac { b _ \\tau } { b } - 1 \\right ] \\end{align*}"}
{"id": "4432.png", "formula": "\\begin{align*} E ' ( t ) = - \\int _ U F ( u _ t ( t ) ) u _ t ( t ) \\ , d x \\leq 0 . \\end{align*}"}
{"id": "5903.png", "formula": "\\begin{align*} \\psi _ { Q _ { J - 2 A } } ( x ) = \\frac { x } { 2 } \\Big [ 2 x ^ { 2 k - 1 } + ( - 1 ) ^ { k - 1 } x ^ 2 \\det U _ x + ( - 1 ) ^ k \\det V _ x \\Big ] . \\end{align*}"}
{"id": "5317.png", "formula": "\\begin{align*} \\check { \\Delta } ( \\omega ) \\Delta ( a ) = \\sum \\Delta ( a ' ) \\check { \\Delta } ( \\omega ' ) , \\end{align*}"}
{"id": "1604.png", "formula": "\\begin{align*} \\Psi _ { a , \\zeta , \\mu , \\sigma } ( x ) & : = \\mu ^ { \\frac { d - 1 } { p } } \\Phi ( \\mu x ) \\\\ \\tilde { \\Psi } _ { a , \\zeta , \\mu , \\sigma } ( x ) & : = \\mu ^ { \\frac { d - 1 } { p ' } } \\tilde { \\Phi } ( \\mu x ) . \\end{align*}"}
{"id": "8190.png", "formula": "\\begin{align*} N _ { 1 0 5 } ' ( f , H ) = - \\frac { 1 } { 5 7 6 } \\times \\begin{cases} 4 3 7 d + 1 3 9 & \\hbox { i f $ d \\equiv 1 \\pmod { 1 2 } $ } , \\\\ 5 3 5 d - 6 4 4 & \\hbox { i f $ d \\equiv 5 \\pmod { 1 2 } $ } , \\\\ 9 7 d - 3 2 4 & \\hbox { i f $ d \\equiv 7 \\pmod { 1 2 } $ } , \\\\ 1 9 5 d + 1 3 & \\hbox { i f $ d \\equiv 1 1 \\pmod { 1 2 } $ . } \\end{cases} \\end{align*}"}
{"id": "8425.png", "formula": "\\begin{align*} Y _ { t } ^ { x ; l ^ { \\epsilon } } : = X _ { \\gamma _ { t } ^ { \\varepsilon } } ^ { x ; l ^ { \\epsilon } } , t \\geq l _ { 0 } ^ { \\epsilon } . \\end{align*}"}
{"id": "3181.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - W _ k ( W _ k ^ \\intercal A ^ \\intercal A W _ k ) ^ \\dagger W _ k ^ \\intercal A ^ \\intercal ( A x _ k - b ) , \\end{align*}"}
{"id": "8525.png", "formula": "\\begin{align*} L ( b _ { d _ c } ) & = \\log \\left ( \\frac { \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\prod _ { i = 1 } ^ { d _ c - 1 } \\psi _ i } { \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\prod _ { i = 1 } ^ { d _ c - 1 } \\psi _ i } \\right ) = 2 \\tanh ^ { - 1 } \\prod _ { i = 1 } ^ { d _ c - 1 } \\psi _ i , \\end{align*}"}
{"id": "6569.png", "formula": "\\begin{align*} \\delta = Q ^ { - 9 9 } . \\end{align*}"}
{"id": "8985.png", "formula": "\\begin{align*} \\frac { e ^ { - t } u \\left ( \\xi ( t ) \\right ) - e ^ { - ( t + h ) } u ( \\xi ( t + h ) ) } { h } = \\frac { 1 } { h } \\int _ t ^ { t + h } e ^ { - s } C _ p \\left | \\dot { \\xi } ( s ) \\right | ^ { q } d s . \\end{align*}"}
{"id": "9274.png", "formula": "\\begin{align*} \\forall x ^ X , z ^ X \\left ( z \\in A x \\land \\norm { z } _ X = \\norm { A ^ \\circ _ X x } _ X \\rightarrow A ^ \\circ _ X x = _ X z \\right ) . \\end{align*}"}
{"id": "1034.png", "formula": "\\begin{align*} u ( x ) & = \\gamma _ { n , s } \\int _ { \\R ^ n \\setminus B _ r } \\bigg ( \\frac { r ^ 2 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - r ^ 2 } \\bigg ) ^ s \\frac { u ( y ) } { \\vert x - y \\vert ^ n } \\dd y \\end{align*}"}
{"id": "5473.png", "formula": "\\begin{align*} \\| u ( t , \\cdot ; - n , u _ n ) \\| _ { C ^ { \\theta } } = \\sup _ { x \\in \\Omega } { | u ( t , x ; - n , u _ 0 ) | } + \\sup { \\frac { | u ( t , x ; - n , u _ 0 ) - u ( t , y ; - n , u _ 0 ) | } { | x - y | ^ { \\theta } } } \\le \\tilde M _ 3 ^ * \\end{align*}"}
{"id": "712.png", "formula": "\\begin{align*} K _ { ( i j ) } ^ { ( \\ell ) } : = \\partial _ { x _ { i ; \\alpha _ 1 } } \\partial _ { x _ { j ; \\alpha _ 2 } } K _ { \\alpha _ 1 \\alpha _ 2 } ^ { ( \\ell ) } \\bigg | _ { x _ { \\alpha _ 1 } = x _ { \\alpha _ 2 } = x _ \\alpha } \\end{align*}"}
{"id": "7376.png", "formula": "\\begin{align*} u _ t - \\Delta u + a | \\nabla u | ^ \\alpha = 0 \\end{align*}"}
{"id": "7011.png", "formula": "\\begin{align*} B ( z ) = B _ 0 + \\dots + B _ k z ^ k = B _ k ( z - \\beta _ 1 ) \\cdots ( z - \\beta _ k ) , \\beta _ i \\neq \\beta _ j . \\end{align*}"}
{"id": "5169.png", "formula": "\\begin{align*} B ( b ) = \\frac { \\pi } { 2 } \\left | \\sqrt { b } \\ , \\vartheta _ 3 ( 0 , i b ) \\vartheta _ 3 ( 0 , i b ) \\right | ^ 2 = \\frac { \\pi } { 2 } \\left | \\vartheta _ 3 ( 0 , i b ^ { - 1 } ) \\vartheta _ 3 ( 0 , i b ) \\right | ^ 2 = \\frac { \\pi } { 2 } \\ , \\vartheta _ 3 ( 0 , i b ^ { - 1 } ) ^ 2 \\vartheta _ 3 ( 0 , i b ) ^ 2 . \\end{align*}"}
{"id": "6929.png", "formula": "\\begin{align*} F : V \\to \\mathbb { R } \\ , , F ( v ) = \\int _ \\Omega f \\ , v \\ , ; \\end{align*}"}
{"id": "1446.png", "formula": "\\begin{align*} G = G _ n ( \\Q ) = \\{ g \\in \\mathrm { S L } _ n ( \\mathbb { B } ) : g ^ { \\ast } \\phi g = \\phi \\} , \\phi = \\left [ \\begin{array} { c c c } 0 & 0 & - 1 _ m \\\\ 0 & \\zeta \\cdot 1 _ r & 0 \\\\ 1 _ m & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "7630.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = \\lambda m u & \\Omega \\ , , \\\\ u = 0 & \\partial \\Omega \\end{cases} \\end{align*}"}
{"id": "1509.png", "formula": "\\begin{align*} c ( h , q , s ) = A ( n ) \\chi ( \\det ( q _ { \\mathbf { h } } ) ) ^ { - 1 } \\det ( q _ { \\infty } ) ^ s | \\det ( q ) | _ { \\mathbf { h } } ^ { 2 m - 1 - s } \\alpha _ { \\mathfrak { n } } ( q _ { \\mathbf { h } } ^ { \\ast } h q _ { \\mathbf { h } } , s , \\chi ) \\xi ( q _ { \\infty } q _ { \\infty } ^ { \\ast } , h , s + l , s - l ) . \\end{align*}"}
{"id": "3014.png", "formula": "\\begin{align*} [ D _ { y ^ 4 z } \\cap D _ { y ^ 3 z ^ 2 } ] = [ \\ell _ y ] + [ \\gamma _ { v y ^ 4 , y ^ 4 z } ] + [ \\gamma _ { w y ^ 4 , y ^ 4 z } ] + [ \\gamma _ { x y ^ 4 , y ^ 4 z } ] . \\end{align*}"}
{"id": "170.png", "formula": "\\begin{align*} \\Gamma _ 2 ( f ) ( x ) & = \\sum _ { j , k } \\left ( \\partial ^ 2 _ { j , k } ( f ) ( x ) \\right ) ^ 2 + \\langle \\nabla ( f ) ( x ) ; \\operatorname { H e s s } ( V ) ( x ) \\nabla ( f ) ( x ) \\rangle \\geq \\kappa \\Gamma ( f ) ( x ) . \\end{align*}"}
{"id": "7484.png", "formula": "\\begin{align*} \\phi ^ 1 = \\phi ^ 0 + \\tau \\dot { \\phi } \\big | _ { t = 0 } + \\frac { \\tau ^ 2 } { 2 } \\ddot { \\phi } \\big | _ { t = 0 } = \\phi ^ 0 + \\frac { \\tau ^ 2 } { 2 } \\left ( - G ( \\phi ^ 0 ) + \\chi ^ 0 \\phi ^ 0 \\right ) , \\end{align*}"}
{"id": "1933.png", "formula": "\\begin{align*} \\underset { x } { \\min } f ( x ) \\textrm { s u b j e c t t o } h ( x ) = 0 , \\end{align*}"}
{"id": "6988.png", "formula": "\\begin{align*} P _ \\theta [ \\eta \\le \\eta _ \\alpha ] = P _ \\theta [ C ( \\eta ) \\le C ( \\eta _ \\alpha ) ] = P _ \\theta [ C ( \\eta ) \\le \\alpha ] = \\alpha . \\end{align*}"}
{"id": "5785.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ { \\nu k } \\ge w _ { \\nu , \\nu , l } , & \\hat { u } _ { \\nu k } ( l ) = 1 \\\\ u _ { \\nu k } \\le 1 - w _ { \\nu , \\nu , l } , & \\end{array} \\right . ~ ~ \\forall l \\in S _ { \\nu } , k = 1 , \\cdots , n _ \\nu ^ 1 , \\end{align*}"}
{"id": "8490.png", "formula": "\\begin{align*} f ^ { t } _ k = \\frac { N ^ t _ { 3 k + 1 } } { D _ t } , g ^ { t } _ k = \\frac { N ^ t _ { 3 k + 2 } } { D _ t } , h ^ { t } _ k = \\frac { N ^ t _ { 3 k + 3 } } { D _ t } , \\end{align*}"}
{"id": "855.png", "formula": "\\begin{align*} { \\bar { \\Delta } _ { \\rm N o n - A R Q } } = - \\frac { 1 } { 2 } + \\frac { { { n _ 1 } } } { { 1 - { \\epsilon _ 1 } } } + \\frac { { { n _ 1 } } } { 2 } , \\end{align*}"}
{"id": "7306.png", "formula": "\\begin{align*} J ( \\alpha \\circ \\beta ) ( 0 ) = J ( \\alpha ) ( 0 ) J ( \\beta ) ( 0 ) \\ne 0 \\end{align*}"}
{"id": "8044.png", "formula": "\\begin{align*} w _ { * } ^ { ( \\mu - 1 ) } f : = \\left [ \\left ( \\frac { d w } { d z } \\right ) ^ { \\mu - 1 } \\cdot f \\right ] \\circ w ^ { - 1 } \\end{align*}"}
{"id": "8756.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { \\sqrt { 2 } } = \\lambda _ 1 \\phi ^ 1 + \\sum _ { k > 1 } \\lambda _ k \\phi ^ k & \\leq \\lambda _ 1 \\phi ^ 1 + ( \\lambda _ 2 + \\cdots + \\lambda _ k ) \\sqrt { \\frac { \\lambda _ 2 y ^ 2 + \\cdots + \\lambda _ k y ^ k } { \\lambda _ 2 + \\cdots + \\lambda _ k } } \\\\ & < \\sqrt { \\lambda _ 1 y ^ 1 + \\cdots + \\lambda _ k y ^ k } = \\frac { 1 } { \\sqrt { 2 } } , \\end{aligned} \\end{align*}"}
{"id": "306.png", "formula": "\\begin{align*} \\mathcal L ( \\rho , v ) = \\int _ 0 ^ 1 \\bigl [ \\frac 1 2 \\ < v , v \\ > _ { \\theta ( \\rho ) } - \\mathcal V ( \\rho ) - \\mathcal W ( \\rho ) + \\alpha L ( \\rho ) - \\beta I ( \\rho ) \\bigr ] d t , \\end{align*}"}
{"id": "1223.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } ( U _ { p , q , \\delta } ) \\geq \\widetilde { C } q ^ { - 2 } = \\mathcal { L } \\Big ( B \\Big ( \\frac { p } { q } , q ^ { - 2 } \\Big ) \\Big ) . \\end{align*}"}
{"id": "4462.png", "formula": "\\begin{align*} 1 & = \\langle z | z \\rangle = \\overline { A } _ { 0 } \\sum _ { m = 0 } ^ { \\infty } \\frac { \\overline { z } ^ { m } } { \\sqrt { m ! } } A _ { 0 } \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { \\sqrt { n ! } } \\langle m | n \\rangle \\\\ & = | A _ { 0 } | ^ { 2 } \\sum _ { n = 0 } ^ { \\infty } \\frac { | z | ^ { 2 n } } { n ! } = | A _ { 0 } | ^ { 2 } e ^ { | z | ^ { 2 } } . \\end{align*}"}
{"id": "535.png", "formula": "\\begin{align*} \\begin{aligned} y _ { i , k + d _ i } & = h _ i ( \\boldsymbol { f } _ O ^ { d _ i - 1 } \\left ( \\boldsymbol { f } ( \\mathbf { x } _ k , \\mathbf { u } _ k ) \\right ) ) . \\end{aligned} \\end{align*}"}
{"id": "3970.png", "formula": "\\begin{align*} \\eta ^ { \\prime \\prime } ( x ) + \\lambda \\xi ( x ) - \\lambda \\eta ( x ) = 0 , \\ \\ \\forall x \\in ( 0 , 1 ) . \\end{align*}"}
{"id": "464.png", "formula": "\\begin{align*} | \\frac { | \\rho _ u W ( u ) \\cap W ' ( u ) | } { | \\rho _ u W ( u ) | } - \\frac { | W ' ( u ) | } { | W ( u ) ^ - | } | = O ( q ^ { - 2 ^ { - | u | - 1 } } ) \\end{align*}"}
{"id": "4412.png", "formula": "\\begin{align*} R e ( x ^ * y ) = 0 \\\\ \\end{align*}"}
{"id": "7278.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty ( 1 + \\alpha ^ { - 1 } X ^ { 2 k - 1 } ) & = \\prod _ { k = 0 } ^ \\infty ( 1 + \\alpha ^ { - 1 } X ^ { 2 k + 1 } ) = \\sum _ { k = 0 } ^ \\infty \\frac { \\alpha ^ { - k } X ^ { k ^ 2 } } { ( 1 - X ^ 2 ) \\ldots ( 1 - X ^ { 2 k } ) } \\\\ & = \\prod _ { k = 1 } ^ \\infty \\frac { 1 } { 1 - X ^ { 2 k } } \\sum _ { k = 0 } ^ \\infty \\alpha ^ { - k } X ^ { k ^ 2 } \\prod _ { l = 0 } ^ \\infty ( 1 - X ^ { 2 l + 2 k + 2 } ) . \\end{align*}"}
{"id": "4578.png", "formula": "\\begin{align*} 1 - \\Phi \\left ( x + \\varepsilon _ n \\right ) = \\Big ( 1 - \\Phi ( x ) \\Big ) \\exp \\Big \\{ \\theta c \\ , ( 1 + x ) \\varepsilon _ n \\Big \\} , \\end{align*}"}
{"id": "4199.png", "formula": "\\begin{align*} M \\subset \\R \\times M _ 0 , g ( t , x ) = c ( t , x ) ( d t ^ 2 + g _ 0 ( x ) ) , \\end{align*}"}
{"id": "8571.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow 0 ^ - } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = \\chi _ - ( x ) T ( 0 ) m _ - ( x , 0 ) + \\chi _ + ( x ) [ m _ + ( x , 0 ) + R _ + ( 0 ) m _ + ( x , 0 ) ] . \\end{align*}"}
{"id": "9420.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } } x ^ k \\ d \\mu ^ { v _ N } _ { A _ N + U _ N B _ N U _ N ^ * } ( x ) = \\tau ^ { v _ N } _ { A _ N , U _ N B _ N U _ N ^ * } ( ( X + Y ) ^ k ) = \\varphi ( ( X + Y ) ^ k ) + o ( 1 ) \\\\ & = \\int _ { \\mathbb { R } } x ^ k \\ d \\nu _ 1 \\prescript { } { \\mu _ 1 } { \\boxplus } ^ { } _ { \\mu _ 2 } \\mu _ 2 ( x ) . \\end{align*}"}
{"id": "7633.png", "formula": "\\begin{align*} - \\Delta w = \\tilde { \\lambda } _ 0 \\tilde { m } _ 0 w \\R ^ N \\ , , \\end{align*}"}
{"id": "8286.png", "formula": "\\begin{align*} \\prod _ { i \\in V _ 1 } \\left ( \\zeta _ { 2 k } ^ { \\mu _ i } - \\zeta _ { 2 k } ^ { - \\mu _ i } \\right ) & = \\prod _ { i \\in V _ 1 } \\sum _ { l _ i \\in \\{ \\pm 1 \\} } l _ i \\zeta _ { 2 k } ^ { l _ i \\mu _ i } \\end{align*}"}
{"id": "3103.png", "formula": "\\begin{align*} x ^ \\prime : y ^ \\prime : z = \\alpha x : \\beta y : \\gamma _ 1 z \\ , ; \\gamma _ 1 = \\frac { k } { \\alpha l } = \\frac { \\beta } { \\alpha } \\gamma \\ , . \\end{align*}"}
{"id": "2579.png", "formula": "\\begin{align*} f ( t ) = e ^ t \\cos ( e ^ t ) . \\end{align*}"}
{"id": "3395.png", "formula": "\\begin{align*} \\begin{aligned} I & = \\sum \\limits _ { k = - \\infty } ^ { \\infty } D _ { k } = \\left ( \\sum \\limits _ { k = - \\infty } ^ { \\infty } D _ { k } \\right ) \\left ( \\sum \\limits _ { j = - \\infty } ^ { \\infty } D _ { j } \\right ) \\\\ & = \\sum \\limits _ { | k - j | \\leqslant N } D _ { k } D _ { j } + \\sum \\limits _ { | k - j | > N } D _ { k } D _ { j } = T _ { N } + R _ { N } , \\end{aligned} \\end{align*}"}
{"id": "7322.png", "formula": "\\begin{align*} & g _ R ( x ) = 0 { g _ R ( x ) \\over | x | } \\to 1 \\\\ & \\sup \\{ | \\nabla g _ R ( x ) | + | \\nabla ^ 2 g _ R ( x ) | : x \\in \\R ^ n , \\ R > 0 \\} < \\infty . \\end{align*}"}
{"id": "7279.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty ( 1 + \\alpha ^ { - 1 } X ^ { 2 k - 1 } ) ( 1 - X ^ { 2 k } ) & = \\sum _ { k = - \\infty } ^ \\infty \\alpha ^ { - k } X ^ { k ^ 2 } \\sum _ { l = 0 } ^ \\infty \\frac { ( - 1 ) ^ l X ^ { l ^ 2 + l + 2 k l } } { ( 1 - X ^ 2 ) \\ldots ( 1 - X ^ { 2 l } ) } \\\\ & = \\sum _ { l = 0 } ^ \\infty \\frac { ( - \\alpha X ) ^ l } { ( 1 - X ^ 2 ) \\ldots ( 1 - X ^ { 2 l } ) } \\sum _ { k = - \\infty } ^ \\infty X ^ { ( k + l ) ^ 2 } \\alpha ^ { - k - l } . \\end{align*}"}
{"id": "7712.png", "formula": "\\begin{align*} \\begin{aligned} A _ { n , n } & = a \\cdot \\Big ( [ 0 , 2 n + 5 ] \\cup [ 2 n + 7 , 4 n + 8 ] \\Big ) \\\\ & = A + a \\cdot \\Big ( [ 0 , 2 n + 4 ] \\cup [ 2 n + 7 , 4 n + 7 ] \\Big ) \\ , , \\end{aligned} \\end{align*}"}
{"id": "8944.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( e ^ { - t } C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } \\dot { \\xi } ( t ) \\right ) = e ^ { - t } D f \\left ( \\xi ( t ) \\right ) , \\end{align*}"}
{"id": "4500.png", "formula": "\\begin{align*} 2 \\sum _ { 0 < \\gamma < T } \\frac { x ^ { \\beta - 1 } } { \\gamma } = \\Sigma _ { 0 } ^ { \\sigma _ 1 } + \\Sigma _ { \\sigma _ 1 } ^ { \\sigma _ 2 } + \\Sigma _ { \\sigma _ 2 } ^ 1 . \\end{align*}"}
{"id": "609.png", "formula": "\\begin{align*} A ( x ) \\ = \\ \\frac { f ( x ) - g ( x ) } { h ( x ) + 1 } ( x = 0 , 1 , 2 , \\ldots ) , \\end{align*}"}
{"id": "6702.png", "formula": "\\begin{align*} \\mathcal { L i } _ { K , \\mathfrak { s } } ( { \\bf z } ) ^ { ( - 1 ) } = \\frac { z _ r } { ( \\theta - t ) ^ { s _ r } } \\mathcal { L i } _ { K , ( s _ 1 , \\ldots , s _ { r - 1 } ) } ( z _ 1 , \\ldots , z _ { r - 1 } ) + \\mathcal { L i } _ { K , \\mathfrak { s } } ( { \\bf z } ) \\end{align*}"}
{"id": "1259.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) = \\epsilon ( \\alpha ) - \\epsilon ( \\beta ) \\ , \\ , \\ , \\mathrm { m o d } \\ , \\ , 2 . \\end{align*}"}
{"id": "4397.png", "formula": "\\begin{align*} e ^ { \\frac { - r \\frac { y ^ 2 } { 4 } + \\frac { 1 } { 2 } ( r ) ^ { \\frac { 1 } { 2 } } ( y x ) - r \\frac { x ^ 2 } { 4 } } { 1 - r } } e ^ { - \\frac { x ^ 2 } { 4 } } \\leq e ^ { \\frac { 1 } { 2 } ( r ) ^ { \\frac { 1 } { 2 } } ( y x ) } e ^ { - \\frac { x ^ 2 } { 4 } } = e ^ { - \\frac { x } { 2 } ( \\frac { x } { 2 } - \\sqrt { r } y ) } \\end{align*}"}
{"id": "6995.png", "formula": "\\begin{align*} f _ { R | | C } = \\int f _ { R | C B } f _ B d b . \\end{align*}"}
{"id": "5356.png", "formula": "\\begin{align*} H _ c ^ + = \\left \\{ x \\in \\mathbb { R } ^ n \\middle | \\left \\langle c , x - z \\right \\rangle \\leqslant 0 \\right \\} \\end{align*}"}
{"id": "8405.png", "formula": "\\begin{align*} | J _ 5 | = & \\Big | - \\varepsilon \\ ! \\int _ 0 ^ t \\ ! \\int _ \\Omega \\mathcal { C } ( \\mu _ { \\varepsilon } - \\mu _ A ) \\ , \\mathrm { d } x \\ , \\mathrm { d } \\varsigma \\Big | \\leq C \\varepsilon ^ { \\frac { 7 } { 2 } } \\| ( \\mu _ { \\varepsilon } - \\mu _ A ) \\| _ { L ^ 2 ( \\Omega \\times ( 0 , t ) ) } \\\\ \\le & C \\varepsilon ^ 7 + \\eta \\| ( \\mu _ { \\varepsilon } - \\mu _ A ) \\| _ { L ^ 2 ( \\Omega \\times ( 0 , t ) ) } ^ 2 . \\end{align*}"}
{"id": "4764.png", "formula": "\\begin{align*} \\theta R = R ^ 2 + R ^ \\# . \\end{align*}"}
{"id": "685.png", "formula": "\\begin{align*} \\sigma _ 1 \\sigma _ 3 < 0 , \\sigma _ j : = \\frac { 1 } { j ! } \\frac { d ^ j } { d t ^ j } \\bigg | _ { t = 0 } \\sigma ( t ) . \\end{align*}"}
{"id": "7015.png", "formula": "\\begin{align*} \\left | \\frac { \\beta _ k } { z - \\beta _ k } \\right | \\le \\frac { 1 } { 2 M - 1 } + 1 = \\frac { 2 M } { 2 M - 1 } \\le 2 . \\end{align*}"}
{"id": "596.png", "formula": "\\begin{align*} & ( 0 , y ) = 0 , \\\\ [ 8 p t ] & ( x + 1 , y ) = ( ( x , y ) , y ) . \\end{align*}"}
{"id": "1031.png", "formula": "\\begin{align*} \\tilde u _ { \\bar M + \\varepsilon } = \\tilde u + h ^ { ( \\varepsilon ) } \\geqslant a - \\varepsilon > 0 B _ 1 ( 2 e _ 1 ) \\end{align*}"}
{"id": "6103.png", "formula": "\\begin{align*} J X _ \\varepsilon ' = \\lambda H _ { \\varepsilon } ( \\theta ) X _ \\varepsilon \\end{align*}"}
{"id": "2216.png", "formula": "\\begin{align*} \\| g ( u ) v \\| _ { 1 } \\leq C \\| v \\| _ { 1 } ( 1 + \\| u \\| _ { \\iota } ^ 2 ) , f o r \\ ; u \\in H ^ \\iota , v \\in H ^ 1 . \\end{align*}"}
{"id": "344.png", "formula": "\\begin{align*} \\mathcal E ( \\rho , m ) & : = \\sup _ { S \\in H _ 1 ^ 1 } \\{ \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > \\\\ & - \\int _ 0 ^ 1 ( \\ < \\dot S , \\rho \\ > + \\ < m , \\nabla _ G S \\ > + \\ < \\nabla _ G \\Sigma , \\nabla _ G S \\ > _ { \\theta ( \\rho ) } \\dot W ^ { \\delta } ( t ) ) d t \\} . \\end{align*}"}
{"id": "3610.png", "formula": "\\begin{align*} 2 e + R _ { n + 1 } - 1 = 2 e > 2 e - 1 = 2 \\left ( e - \\left \\lfloor \\dfrac { 1 - 1 } { 2 } \\right \\rfloor \\right ) - 1 = G _ { n } \\ , . \\end{align*}"}
{"id": "261.png", "formula": "\\begin{align*} i _ { V } \\omega = \\sqrt { - 1 } \\overline { \\partial } \\theta _ { V } ^ { ( \\omega ) } \\quad \\int _ { X } \\theta _ { V } ^ { ( \\omega ) } \\omega ^ { n } = 0 , \\end{align*}"}
{"id": "4647.png", "formula": "\\begin{align*} S & = { \\sqrt { \\Delta ( { \\mathbb K } ) } } \\sum _ { d \\vert \\frac { m } { p } } \\left ( \\mu ( d ) \\lambda ^ { ( n / d ) ^ s } + \\mu ( p d ) \\lambda ^ { n / ( d p ) } \\right ) \\\\ & = { \\sqrt { \\Delta ( { \\mathbb K } ) } } \\sum _ { d \\vert \\frac { m } { p } } \\pm \\lambda ^ { ( n / ( p d ) ) ^ s } \\left ( \\lambda ^ { ( n / p d ) ^ s ( p ^ s - 1 ) } - 1 \\right ) . \\end{align*}"}
{"id": "7985.png", "formula": "\\begin{align*} 0 \\leq u ( x ) \\leq 1 , E ( u ) : = \\int _ M | \\nabla u | ^ 2 d x \\leq C _ 1 \\ , . \\end{align*}"}
{"id": "7064.png", "formula": "\\begin{align*} \\mathbb { P } \\Big ( \\sup _ { t \\le T \\wedge \\theta _ { K } } \\sup _ { i } | M ^ K _ i ( t ) | \\geq A \\Big ) & \\leq \\sum _ { i = 0 } ^ { 1 / \\delta _ K - 1 } \\mathbb { P } \\Big ( \\sup _ { t \\le T \\wedge \\theta _ { K } } | M ^ K _ i ( t ) | \\geq A \\Big ) \\\\ \\leq & \\frac { 1 } { A ^ 2 } \\sum _ { i = 0 } ^ { 1 / \\delta _ K - 1 } \\mathbb { E } \\Big ( \\sup _ { t \\le T \\wedge \\theta _ { K } } | M ^ K _ i ( t ) | ^ 2 \\Big ) , \\end{align*}"}
{"id": "7741.png", "formula": "\\begin{gather*} | a _ n | = \\frac { f _ n } { c _ E - n ^ 2 } \\lesssim \\frac { 1 } { n ^ 2 } \\sqrt { \\delta E ( 0 ) } , \\ ; \\forall n \\in \\mathbb { Z } \\setminus \\{ 0 , \\pm 1 \\} , \\\\ | a _ 0 | = \\left | \\frac { f _ 0 } { c _ E } \\right | \\lesssim \\sqrt { \\frac { \\delta } { E ( 0 ) } } \\ ; \\textrm { a n d } \\ ; | a _ { \\pm 1 } | = \\left | \\frac { 2 \\pi f _ { \\pm 1 } } { E ( 0 ) - 2 \\pi } \\right | \\lesssim \\frac { \\sqrt { \\delta E ( 0 ) } } { 2 \\pi - E ( 0 ) } . \\end{gather*}"}
{"id": "1708.png", "formula": "\\begin{align*} d _ { C _ 1 C n } ( M , \\ , Y _ q ( \\Omega ) ) \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } \\sum \\limits _ { t = 0 } ^ { \\hat t ( n ) } \\sum \\limits _ { m = 0 } ^ \\infty d _ { k _ { t , m } } ( W _ { t , m } , \\ , l _ q ^ { \\nu _ { t , m } } ) + \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ { \\hat t ( n ) + 1 } ) } . \\end{align*}"}
{"id": "3814.png", "formula": "\\begin{align*} \\varepsilon \\circ \\psi ( [ a _ { 2 i - 1 } , a _ { 2 i } ] ) = \\varepsilon \\bigl ( w _ { 2 i - 1 } \\ldotp g ^ { b _ { 2 i - 1 } } w _ { 2 i } g ^ { - b _ { 2 i - 1 } } \\ldotp g ^ { b _ { 2 i } } w _ { 2 i - 1 } ^ { - 1 } g ^ { - b _ { 2 i } } \\ldotp w _ { 2 i } ^ { - 1 } \\bigr ) = \\varepsilon ( [ w _ { 2 i - 1 } , w _ { 2 i } ] ) = 0 . \\end{align*}"}
{"id": "7525.png", "formula": "\\begin{align*} \\Im \\left \\{ \\log \\Gamma \\left ( \\frac { \\sigma + i T } { 2 } \\right ) \\right \\} & = \\frac { T } { 4 } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) + \\left ( \\frac { \\sigma - 1 } { 2 } \\right ) \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) - \\frac { T } { 2 } + o ( 1 ) \\end{align*}"}
{"id": "513.png", "formula": "\\begin{align*} h ( \\mathcal { I } _ { \\beta } ^ { j } ) \\cdot h ( \\mathcal { I } _ { \\alpha \\beta } ^ { i j } ) = h ( \\mathcal { J } _ { j } ^ { \\beta } ) \\cdot h ( \\mathcal { J } _ { \\alpha j } ^ { i \\beta } ) = 0 , \\quad ( j , \\beta ) \\in ( \\mathcal { I } \\setminus \\mathcal { J } ) \\times ( \\mathcal { J } \\setminus \\mathcal { I } ) . \\end{align*}"}
{"id": "9022.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ 0 ( \\rho , \\phi ) ( t ) & = - \\int _ { \\Omega } \\sum _ { i = 1 } ^ s D _ i ( x ) \\rho _ i | \\nabla ( \\log \\rho _ i + z _ i \\phi ) | ^ 2 d x + \\frac { 1 } { 2 } \\int _ { \\partial \\Omega } \\epsilon ( x ) \\left [ \\phi ( \\partial _ n \\phi ) _ t - ( \\partial _ n \\phi ) \\phi _ t \\right ] d s . \\end{align*}"}
{"id": "5470.png", "formula": "\\begin{align*} u _ 1 ^ * ( x ) = u ( 1 , x ; 0 , u _ 0 ^ * , a ^ * , b ^ * ) . \\end{align*}"}
{"id": "2050.png", "formula": "\\begin{align*} w _ r \\left ( \\rho , s \\right ) = e ^ { \\rho \\omega _ r s } \\left [ 1 + f _ r \\left ( s \\right ) + \\frac { f _ { r 1 } \\left ( s \\right ) } { \\rho } + \\frac { H _ r \\left ( \\rho , s \\right ) } { \\rho ^ 2 } \\right ] \\end{align*}"}
{"id": "3671.png", "formula": "\\begin{align*} E ( X ) ^ g = \\{ \\{ u ^ g , v ^ g \\} \\mid \\{ u , v \\} \\in E ( X ) \\} . \\end{align*}"}
{"id": "7111.png", "formula": "\\begin{align*} d x ( t ) = ( \\alpha _ { - 1 } x ( t ) ^ { - 1 } - \\alpha _ { 0 } + \\alpha _ { 1 } x ( t ) - \\alpha _ { 2 } t ^ { 2 H - 1 } x ( t ) ^ { \\rho } ) d t + \\sigma x ( t ) ^ { \\theta } d B _ t ^ H \\end{align*}"}
{"id": "2248.png", "formula": "\\begin{align*} \\Big ( \\mathbb { E } \\big [ \\sup _ { 1 \\leq m \\leq M } \\| Y _ m ^ { M , N } \\| ^ { 2 p } \\big ] \\Big ) ^ { \\frac 1 p } + \\Big \\| k \\sum _ { j = 1 } ^ M \\| A Y _ j ^ { M , N } \\| ^ 2 \\Big \\| _ { L ^ p ( \\Omega ; \\mathbb { R } ) } + \\Big \\| k \\sum _ { j = 1 } ^ M \\| \\nabla ( Y _ j ^ { M , N } ) ^ 2 \\| ^ 2 \\Big \\| _ { L ^ p ( \\Omega ; \\mathbb { R } ) } < \\infty . \\end{align*}"}
{"id": "7592.png", "formula": "\\begin{align*} \\| f - { \\mathsf { C } } _ { d _ x , d _ y } [ f ] \\| _ 1 \\le \\dfrac { 4 V _ { k , l } } { k l \\pi ^ 2 } \\begin{cases} \\Pi _ { 1 } [ s ] ( d _ x ) \\Pi _ { 1 } [ r ] ( d _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , l = 2 r , \\\\ \\Pi _ { 1 } [ s ] ( d _ x ) \\Pi _ { 0 } [ r ] ( d _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , l = 2 r + 1 , \\\\ \\Pi _ { 0 } [ s ] ( d _ x ) \\Pi _ { 1 } [ r ] ( d _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 , l = 2 r , \\\\ \\Pi _ { 0 } [ s ] ( d _ x ) \\Pi _ { 0 } [ r ] ( d _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 , l = 2 r + 1 , \\end{cases} \\end{align*}"}
{"id": "8137.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 1 0 } ( H G ) } c ( g ( \\gamma ) ) = 1 8 \\cdot 7 + 1 7 \\cdot 7 = 2 4 5 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "4253.png", "formula": "\\begin{align*} T ( t ) f : = \\sigma \\lim _ { \\mathclap { n \\to \\infty } } T _ n ( t ) f \\end{align*}"}
{"id": "5634.png", "formula": "\\begin{align*} u ^ i \\star w ^ j = u ^ i w ^ j + \\mathrm { i } \\nu ( \\delta ^ i _ - - \\delta ^ i _ + ) u ^ i \\bigg ( \\frac { 1 } { \\sqrt { a } } \\delta ^ j _ 0 w ^ + - 2 \\sqrt { a } \\delta ^ j _ - w ^ 0 \\bigg ) + \\delta ^ i _ + \\delta ^ j _ - 2 \\nu ^ 2 u ^ + w ^ + . \\end{align*}"}
{"id": "3991.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) + \\widetilde p ( t ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "6951.png", "formula": "\\begin{align*} \\mathcal { A } _ { s ' } = \\mathcal { A } ^ 1 _ m + \\mathcal { A } ^ { 2 , f i n } _ m = \\mathcal { A } ^ { 1 , i n i t } _ m + \\mathcal { C } _ m + \\mathcal { D } _ m + \\mathcal { A } ^ { 2 , f i n } _ m , \\end{align*}"}
{"id": "3248.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } ( s ( r , t ) - t ) = \\lim _ { t \\rightarrow } F _ \\sigma ( i t ) - i t = 0 . \\end{align*}"}
{"id": "6633.png", "formula": "\\begin{align*} \\Sigma _ 2 : = p ^ { ( w - 1 ) h _ p } \\sum _ { m = \\max \\{ 0 , k _ p - h _ p \\} } ^ { \\infty } \\sum _ { n = m + h _ p - k _ p + 1 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( \\frac { 1 } { 2 } + s _ 1 - z ) } p ^ { n ( \\frac { 1 } { 2 } + s _ 2 + z ) } } . \\end{align*}"}
{"id": "7142.png", "formula": "\\begin{align*} \\tau _ n = \\inf \\Big \\{ 0 \\le t \\le 1 ; x _ t \\notin [ \\frac { 1 } { n } , n ] \\Big \\} ( \\inf \\emptyset = \\infty ) \\end{align*}"}
{"id": "8579.png", "formula": "\\begin{align*} { \\big \\| f ^ \\# \\big \\| } _ { L ^ 2 } = { \\| f \\| } _ { L ^ 2 } , \\forall f \\in L ^ { 2 } \\end{align*}"}
{"id": "1986.png", "formula": "\\begin{align*} M _ t = 1 + t h + ( h \\triangleleft h ) \\frac { t ^ 2 } { 2 } + ( ( h \\triangleleft h ) \\triangleleft h ) \\frac { t ^ 3 } { 6 } + \\dotsb = 1 + \\sum _ { n = 1 } ^ \\infty R ^ { ( n - 1 ) } _ { \\triangleleft h } ( h ) \\frac { t ^ n } { n ! } . \\end{align*}"}
{"id": "6404.png", "formula": "\\begin{align*} \\| ( P _ i ^ { \\perp } + z P _ i ) x \\| = \\| ( P _ i ^ { \\perp } + z P _ i ) ( x _ 1 \\oplus x _ 2 ) \\| = \\| P _ i ^ { \\perp } x _ 2 \\oplus z P _ i x _ 1 \\| = \\sqrt { \\| P _ i ^ { \\perp } x _ 2 \\| ^ 2 + | z | \\| P _ i x _ 1 \\| ^ 2 } < 1 . \\end{align*}"}
{"id": "5471.png", "formula": "\\begin{align*} 0 \\le u _ n ( x ) = u _ n ( x ) - u ( x ) + u ( x ) \\leq \\varepsilon + u ( x ) \\forall \\ , n \\ge N _ \\varepsilon , \\ , \\ , x \\in \\bar \\Omega . \\end{align*}"}
{"id": "6263.png", "formula": "\\begin{align*} | B | & \\geq | N ( u ) \\cap N ( v ) | - h \\\\ & \\geq | N ( u ) | + | N ( v ) | - | V | - h \\\\ & \\geq 2 \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 1 0 h } \\right ) n - \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 1 0 h } \\right ) n - h \\\\ & = \\left ( \\frac { 1 } { 2 } - \\frac { 3 } { 1 0 h } \\right ) n - h . \\end{align*}"}
{"id": "6387.png", "formula": "\\begin{align*} \\Omega _ { t _ n } = \\{ ( z _ 1 , \\dots , z _ n ) \\in \\Omega \\ , : \\ , z _ n = t _ n \\} \\end{align*}"}
{"id": "337.png", "formula": "\\begin{align*} & m _ { i j } = \\theta _ { i j } ( \\rho ) ( \\nabla _ G S ) _ { i j } , \\ ; \\forall ( i , j ) \\in E , \\ ; \\\\ & \\ < \\dot S , \\rho \\ > + \\frac 1 4 \\sum _ { i j } ( S _ i - S _ j ) \\theta _ { i j } ( \\rho ) + \\sum _ { i j } ( \\Sigma _ i - \\Sigma _ j ) ( S _ { i } - S _ { j } ) \\theta _ { i j } ( \\rho ) d W ^ { \\delta } ( t ) = 0 . \\end{align*}"}
{"id": "3399.png", "formula": "\\begin{align*} \\begin{aligned} \\Big ( { \\| x - x ' \\| \\over \\| x - y \\| } \\Big ) ^ \\varepsilon \\frac { 1 } { \\omega ( B ( x , \\| x - y \\| ) ) } & \\leqslant C \\Big ( \\frac { \\| x - x ' \\| } { \\| x - y \\| } \\Big ) ^ \\varepsilon \\frac 1 { \\omega ( B ( x , d ( x , y ) ) ) } \\Big ( \\frac { d ( x , y ) } { \\| x - y \\| } \\Big ) ^ { N } \\\\ & \\leqslant C \\Big ( \\frac { \\| x - x ' \\| } { \\| x - y \\| } \\Big ) ^ \\varepsilon \\frac 1 { \\omega ( B ( x , d ( x , y ) ) ) } . \\end{aligned} \\end{align*}"}
{"id": "496.png", "formula": "\\begin{align*} \\Upsilon _ { \\omega } ^ { + } = \\left \\{ \\tau ^ { - 1 } - 1 , \\tau \\theta ^ { - 1 } \\cdot \\Omega \\right \\} , \\quad \\Upsilon _ { \\omega } ^ { - } = \\left \\{ \\tau - 1 , \\Omega \\right \\} , \\Omega \\in \\{ - \\vartheta \\tau ^ { - 1 } - 1 \\} \\cup \\mathbb { C } . \\end{align*}"}
{"id": "7822.png", "formula": "\\begin{align*} [ D _ h \\Psi ] ( t ) : = \\frac { \\Psi ( t + h ) - \\Psi ( t ) } { h } . \\end{align*}"}
{"id": "8953.png", "formula": "\\begin{align*} \\begin{aligned} u \\left ( \\xi ( 0 ) \\right ) & = \\int _ 0 ^ { t } e ^ { - s } \\left ( C _ p \\left | \\dot { \\xi } ( s ) \\right | ^ q + f \\left ( \\xi ( s ) \\right ) \\right ) d s + e ^ { - t } u ( \\xi ( t ) ) \\\\ & = \\int _ 0 ^ { t } e ^ { - s } \\left ( C _ p \\left | \\dot { \\xi } ( s ) \\right | ^ q + f \\left ( \\xi ( s ) \\right ) \\right ) d s + e ^ { - t } u ( y ) \\end{aligned} \\end{align*}"}
{"id": "2642.png", "formula": "\\begin{align*} \\langle X g , M _ l T _ k g \\rangle & = \\langle g , X M _ l T _ k g \\rangle \\\\ & = k \\langle g , M _ l T _ k g \\rangle + \\langle g , M _ l T _ k X g \\rangle \\\\ & = 0 + \\langle T _ { - k } M _ { - l } g , X g \\rangle , \\end{align*}"}
{"id": "2730.png", "formula": "\\begin{align*} \\pi _ 2 ( S ( \\underline { P } , B ) ) = \\{ x \\in B \\ ; \\vert \\ ; \\bigvee _ { j = 1 } ^ { q } ( P _ j ( x ) \\leq 0 ) \\} . \\end{align*}"}
{"id": "9286.png", "formula": "\\begin{align*} [ J ^ { \\chi _ A } ] _ \\mathcal { M } : = \\lambda \\alpha \\in \\mathbb { N } ^ \\mathbb { N } , x \\in X . \\begin{cases} J ^ A _ { r _ \\alpha } x & r _ \\alpha > 0 x \\in \\mathrm { d o m } ( J ^ A _ { r _ \\alpha } ) , \\\\ 0 & , \\end{cases} \\end{align*}"}
{"id": "5167.png", "formula": "\\begin{align*} \\sqrt { - i \\tau } \\ , \\vartheta _ 2 ( 0 , \\tau ) = \\vartheta _ 4 ( 0 , - 1 / \\tau ) , \\tau \\in \\mathbb { H } . \\end{align*}"}
{"id": "1645.png", "formula": "\\begin{align*} h _ p ( r ) : = \\Big ( 1 + \\log _ + \\frac { 1 } { r } \\Big ) ^ { 1 - p } , r > 0 , \\end{align*}"}
{"id": "9486.png", "formula": "\\begin{align*} | \\mathcal { B C } _ { ( s , t ) } | = | \\mathcal { S C } _ { ( s , t ) } | = \\binom { \\lfloor s / 2 \\rfloor + \\lfloor t / 2 \\rfloor } { \\lfloor s / 2 \\rfloor } . \\end{align*}"}
{"id": "884.png", "formula": "\\begin{align*} \\Delta _ { g _ M } f = \\Delta _ B f . \\end{align*}"}
{"id": "1586.png", "formula": "\\begin{align*} S ( z ) = | \\mathbb { S } ^ 1 | \\int _ { 0 } ^ { \\frac \\pi 2 } \\sin \\theta \\left [ \\frac 1 { \\cos ^ 3 ( \\theta / 2 ) } B \\left ( \\frac { | z | } { \\cos ( \\theta / 2 ) } , \\cos \\theta \\right ) - B ( | z | , c o s \\theta ) \\right ] . \\end{align*}"}
{"id": "7607.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { \\partial f } { \\partial \\kappa _ { i } } \\kappa _ { i } ^ { 2 } & = \\frac { 1 } { k } H _ { k } ^ { \\frac { 1 - k } { k } } \\sum _ { i = 1 } ^ { n } \\frac { \\partial H _ { k } } { \\partial \\kappa _ { i } } \\kappa _ { i } ^ { 2 } \\\\ & = \\frac { n } { k } H _ { 1 } H _ { k } ^ { \\frac { 1 } { k } } - \\frac { n - k } { k } H _ { k } ^ { \\frac { 1 - k } { k } } H _ { k + 1 } . \\end{align*}"}
{"id": "4925.png", "formula": "\\begin{align*} X = \\frac { 3 \\sigma \\log ( 1 / \\delta ) \\pm \\sqrt { ( 3 \\sigma \\log ( 1 / \\delta ) ) ^ 2 + 4 A } } { 2 } \\end{align*}"}
{"id": "830.png", "formula": "\\begin{align*} \\mu _ \\omega ( 2 B _ \\rho ) \\le \\mu _ \\omega ( B _ \\rho ( \\infty , 3 r ) ) = C _ { \\beta , a } \\ , \\mu ( Z ) \\ , r ^ { \\tfrac { 2 \\beta - a - 1 } { \\beta - 1 } } & = C _ { \\beta , a } \\ , \\mu _ \\omega ( B _ \\rho ( \\infty , r / 2 ) ) \\\\ & \\le C _ { \\beta , a } \\ , \\mu _ \\omega ( B _ \\rho ) , \\end{align*}"}
{"id": "8500.png", "formula": "\\begin{align*} r ( x , y ) = ( \\sigma _ x ( y ) , \\tau _ y ( x ) ) \\end{align*}"}
{"id": "9040.png", "formula": "\\begin{align*} | X ( \\theta ) | ^ 2 - \\theta | X ^ 0 | ^ 2 - ( 1 - \\theta ) | X ^ 1 | ^ 2 = - \\theta ( 1 - \\theta ) | X ^ 1 - X ^ 0 | ^ 2 . \\end{align*}"}
{"id": "1337.png", "formula": "\\begin{align*} q _ { i ' } \\leq M - q _ { i } \\leq q _ { i + 1 } - q _ { i } = \\frac { 1 } { 2 } q _ { i } < q _ { i } . \\end{align*}"}
{"id": "2850.png", "formula": "\\begin{align*} \\forall R \\ge R _ 0 + | X ( t ) | = R _ 0 + | x ( t ) | \\Rightarrow | A _ R ( u ( t ) ) | \\le 4 s _ c ( p - 1 ) \\delta ( t ) , \\ ; \\forall t \\in D _ { \\delta _ 0 } . \\end{align*}"}
{"id": "2380.png", "formula": "\\begin{align*} \\mathbf { F } _ N = \\{ \\left ( \\cos ( 2 \\pi k / N ) , \\sin ( 2 \\pi k / N ) \\right ) \\mid k = 0 , \\ldots N - 1 \\} \\end{align*}"}
{"id": "3256.png", "formula": "\\begin{align*} h _ \\lambda ( t ) = h ( s ( | \\lambda | , t ) ) , t > 0 . \\end{align*}"}
{"id": "5560.png", "formula": "\\begin{align*} d _ K ( \\alpha _ { \\rm H J M } ) & = d _ K \\bigg ( \\frac { 1 } { z _ 6 } e ^ { - 2 z _ 6 \\cdot } \\bigg ) \\leq \\bigg \\| \\frac { 1 } { z _ 6 } e ^ { - 2 z _ 6 \\cdot } - \\frac { 1 } { z _ 6 } e ^ { - z _ 7 \\cdot } \\bigg \\| \\\\ & = \\frac { 1 } { z _ 6 } \\| e ^ { - 2 z _ 6 \\cdot } - e ^ { - z _ 7 \\cdot } \\| , \\end{align*}"}
{"id": "6333.png", "formula": "\\begin{align*} \\Gamma ^ k _ { i j } = \\partial ^ k _ \\gamma \\Gamma ^ \\gamma _ { \\alpha \\beta } \\partial ^ \\alpha _ i \\partial ^ \\beta _ j + \\partial ^ k _ \\gamma \\partial ^ \\gamma _ { i j } \\end{align*}"}
{"id": "6327.png", "formula": "\\begin{align*} \\langle \\nabla _ V T , S \\rangle = \\langle \\nabla T , V \\otimes S \\rangle \\end{align*}"}
{"id": "2846.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\varphi _ R } { \\partial x _ i \\partial x _ j } ( x ) = ( \\Delta ^ 2 \\varphi _ R ) ( x ) = \\frac { \\partial ^ 2 \\varphi _ R } { \\partial x _ i ^ 2 } ( x ) - 2 = 2 N - \\Delta \\varphi _ R ( x ) = 0 , 2 x - \\nabla \\varphi _ R ( x ) = 0 , \\end{align*}"}
{"id": "8074.png", "formula": "\\begin{align*} ( F _ H ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } \\star _ { \\ell } ( G _ H ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } = ( F _ H \\star _ { H , \\ell } G _ H ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } . \\end{align*}"}
{"id": "1774.png", "formula": "\\begin{align*} H P ^ i ( \\mathcal { C } ( \\mathbb { R } ^ n ) ) = \\left \\{ \\begin{array} { l l } 0 , & i \\not \\equiv n \\ ( m o d \\ 2 ) , \\\\ \\mathbb { C } , & i \\equiv n \\ ( m o d \\ 2 ) . \\end{array} \\right . \\end{align*}"}
{"id": "3373.png", "formula": "\\begin{align*} T _ t = \\sum _ { i = 0 } ^ { + \\infty } T _ i t ^ { i } , \\ ; T _ i \\in H o m _ { \\mathbb { K } } ( V ; L ) , \\end{align*}"}
{"id": "2139.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { E _ { n , k } Y ^ { ( n , k ) } _ i } { n k } = \\frac i { k + 1 } . \\end{align*}"}
{"id": "5647.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta \\Phi & = \\sigma \\varepsilon ^ { - 2 } ( 1 - ( x ^ { 2 } + y ^ { 2 } ) ) & \\Omega , \\\\ \\Phi & = \\sigma g & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
{"id": "8236.png", "formula": "\\begin{align*} [ z ^ n x _ h ^ { \\ell } ] ( N _ 1 + N _ 2 + \\cdots + N _ k ) = [ z ^ n x _ h ^ { \\ell } ] \\left ( \\sqrt { \\frac { z F _ { k , 1 } ( B ) } { B } } - \\frac { z F _ { k , 1 } ( B ) } { B } \\right ) . \\end{align*}"}
{"id": "8820.png", "formula": "\\begin{align*} \\partial _ t [ \\nabla S _ t ( \\varphi ) w ] = - \\nabla ^ 2 u _ t ( \\varphi ) [ \\nabla S _ t ( \\varphi ) w ] . \\end{align*}"}
{"id": "6664.png", "formula": "\\begin{align*} R _ { 2 1 } = J _ { 1 1 } + O \\big ( ( h k ) ^ { \\varepsilon } ( h , k ) ^ { 1 / 2 } Q ^ { - 9 6 } \\big ) . \\end{align*}"}
{"id": "7089.png", "formula": "\\begin{align*} \\alpha _ { s _ m } ( f ) = \\alpha _ { s _ { m , \\beta ^ i _ m , W ^ 0 _ { V , i } } } ( f _ { { W ^ 0 _ { V , i } } } ) . \\end{align*}"}
{"id": "1636.png", "formula": "\\begin{align*} p _ { t + s } ( x , y ) = \\int _ M p _ t ( x , z ) p _ s ( z , y ) \\mu ( d z ) , s , t > 0 , x , y \\in M . \\end{align*}"}
{"id": "7102.png", "formula": "\\begin{align*} = & - \\frac { 1 } { 4 } \\lambda ^ { n - 1 } \\left ( \\lambda ^ 2 - 4 \\right ) ( 3 ) + \\lambda ^ 3 P _ { n - 3 } ( \\lambda ) \\\\ & \\vdots \\\\ = & - \\frac { 1 } { 4 } \\lambda ^ { n - 1 } \\left ( \\lambda ^ 2 - 4 \\right ) ( n ) + \\lambda ^ n P _ { 0 } ( \\lambda ) \\end{align*}"}
{"id": "2264.png", "formula": "\\begin{align*} M x ^ 2 = x \\cdot M x M ^ { - T } = ( M ^ { - 1 } ) ^ T = ( M ^ T ) ^ { - 1 } . \\end{align*}"}
{"id": "2481.png", "formula": "\\begin{align*} \\pi ( \\l ) \\pi ( \\l ' ) = e ^ { - 2 \\pi i \\omega ' \\cdot x } \\pi ( \\l + \\l ' ) . \\end{align*}"}
{"id": "9543.png", "formula": "\\begin{align*} \\tilde S _ 0 ( \\bar x ) - \\bar x \\cdot \\bar c - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } \\le 0 . \\end{align*}"}
{"id": "7361.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\varphi _ t + | \\varphi _ r | \\pi \\min \\{ R ^ 2 , r ^ 2 \\} = 0 & \\ r > 0 , t > 0 , \\\\ \\varphi ( 0 , t ) = 1 & \\ t > 0 , \\\\ \\varphi ( r , 0 ) = r + 1 & \\ r \\ge 0 . \\end{array} \\right . \\end{align*}"}
{"id": "3576.png", "formula": "\\begin{align*} \\Lambda _ 0 & = \\{ - p ^ e \\} , \\\\ \\Lambda _ z & = \\Big \\{ { \\sum } _ { i = e - z } ^ { e - 1 } a _ i ( p ^ e - p ^ { i } ) + p ^ e - 2 p ^ { e - z } \\colon 0 \\le a _ i < p \\Big \\} , 0 < z < e ; \\\\ \\Lambda _ e & = \\Big \\{ { \\sum } _ { i = 1 } ^ { e - 1 } a _ i ( p ^ e - p ^ i ) - p ^ e - 2 \\colon 0 \\le a _ i < p \\Big \\} . \\end{align*}"}
{"id": "5981.png", "formula": "\\begin{align*} C _ { 0 } ^ { \\lambda , 1 } ( x ) & = 1 \\\\ C _ { 1 } ^ { \\lambda , 1 } ( x ) & = x - \\lambda \\\\ C _ { 2 } ^ { \\lambda , 1 } ( x ) & = x ^ { 2 } - ( 1 + 2 \\lambda ) x + \\lambda ^ { 2 } \\\\ C _ { 3 } ^ { \\lambda , 1 } ( x ) & = x ^ { 3 } - ( 3 + 3 \\lambda ) x ^ { 2 } + ( 2 + 3 \\lambda + 3 \\lambda ^ { 2 } ) x - \\lambda ^ { 3 } . \\end{align*}"}
{"id": "5586.png", "formula": "\\begin{align*} 0 & = \\lambda \\frac { \\bar { G } ( q ^ * ) } { \\sum _ { j = 1 } ^ { N } \\bar { G } ( q ^ * ) } - \\mu q ^ * \\\\ & = \\lambda \\frac { \\bar { G } ( q ^ * ) } { N \\bar { G } ( q ^ * ) } - \\mu q ^ * \\\\ & = \\frac { \\lambda } { N } - \\mu q ^ * \\end{align*}"}
{"id": "9014.png", "formula": "\\begin{align*} E = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } ( f + \\sum _ { i = 1 } ^ s z _ i \\rho _ i ) \\phi \\bigg ) d x + B , \\end{align*}"}
{"id": "3984.png", "formula": "\\begin{align*} L _ k : = - \\big [ 2 i k \\pi + O ( 1 ) \\big ] e ^ { - 1 + O ( k ^ { - 1 } ) } , \\end{align*}"}
{"id": "6420.png", "formula": "\\begin{align*} L ^ * \\big ( * ^ V ( v _ i \\wedge v _ j ) \\big ) \\otimes 1 + 1 \\otimes * ^ V ( v _ i \\wedge v _ j ) & = \\tfrac 1 2 \\left ( \\mu _ { i j } \\ , \\omega ^ W _ \\C w _ i w _ j \\otimes 1 + 1 \\otimes \\omega ^ V _ \\C v _ i v _ j \\right ) \\\\ & = \\tfrac 1 2 \\left ( \\mu _ { i j } \\ , w _ i w _ j \\omega ^ W _ \\C \\otimes 1 + 1 \\otimes v _ i v _ j \\omega ^ V _ \\C \\right ) \\\\ & = \\tfrac 1 2 \\left ( \\mu _ { i j } \\ , w _ i w _ j \\otimes 1 + 1 \\otimes v _ i v _ j \\right ) , \\end{align*}"}
{"id": "4567.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\left ( W _ n > x \\right ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) , \\ \\ \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "9124.png", "formula": "\\begin{align*} \\exists k \\in \\mathbb { N } \\forall p \\in A F _ k \\forall l \\leq m \\left ( H ( d ( x _ { n + l } , p ) ) < G ( d ( x _ n , p ) ) + \\sum _ { i = n } ^ { n + l - 1 } \\varepsilon _ i + \\frac { 1 } { r + 1 } \\right ) . \\end{align*}"}
{"id": "3760.png", "formula": "\\begin{align*} 0 < T ( F ) = \\frac { t ( t - 1 ) } { 2 } \\tau \\Big ( | \\alpha | ^ 2 \\ F \\wedge F \\Big ) . \\end{align*}"}
{"id": "1241.png", "formula": "\\begin{align*} { m \\left ( \\bigcup _ { r > 0 } A ( r ) \\right ) = m ( A ) . } \\end{align*}"}
{"id": "2507.png", "formula": "\\begin{align*} \\rho _ \\kappa ( \\mathbf { h } ) = \\rho \\circ \\delta _ \\kappa ( \\mathbf { h } ) = \\rho ( \\kappa x , \\omega , \\kappa \\tau ) = e ^ { 2 \\pi i \\kappa \\tau } e ^ { \\pi i \\kappa x \\cdot \\omega } T _ { \\kappa x } M _ { \\omega } . \\end{align*}"}
{"id": "2458.png", "formula": "\\begin{align*} A B ^ T = B A ^ T , \\ C D ^ T = D C ^ T A D ^ T - B C ^ T = I . \\end{align*}"}
{"id": "821.png", "formula": "\\begin{align*} d _ \\rho ( ( x , y ) , \\infty ) = \\begin{cases} \\frac { 1 } { ( \\beta - 1 ) \\ , y ^ { \\beta - 1 } } & y \\ge 1 , \\\\ \\frac { \\beta } { \\beta - 1 } - y & 0 \\le y \\le 1 . \\end{cases} \\end{align*}"}
{"id": "2892.png", "formula": "\\begin{align*} \\partial _ t h + \\mathcal { L } h = R ( h ) \\end{align*}"}
{"id": "7792.png", "formula": "\\begin{align*} \\overline { \\dim } _ B \\big ( G ( { \\textbf { f } + \\textbf { g } } ) \\big ) \\le \\max \\{ \\overline { \\dim } _ B \\big ( G ( { \\textbf { f } } ) \\big ) , \\overline { \\dim } _ B \\big ( G ( { \\textbf { g } } ) \\big ) \\} = 1 . 5 , \\end{align*}"}
{"id": "1574.png", "formula": "\\begin{align*} \\pi ^ { - \\beta } \\mathbf { T } ( g , h ) = \\sum _ { j = 1 } ^ e \\overline { \\mathbf { g } _ j ( g ) } \\mathbf { h } _ j ( h ) , \\end{align*}"}
{"id": "7989.png", "formula": "\\begin{align*} P ( \\boldsymbol { \\varphi } ) = \\max _ { \\sigma \\in \\mathcal { A } } \\ , \\sup _ { \\mathbf { P } _ { \\ ! \\sigma } \\in \\mathcal { Q } ^ { \\sigma } } \\ , \\sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\bigg ( H ( \\mathbf { p } _ { \\sigma _ { n } } ) + \\int \\ ! \\varphi _ n ^ { \\sigma } \\ , \\mathrm { d } \\mathbf { p } _ { \\sigma _ n } \\bigg ) . \\end{align*}"}
{"id": "8477.png", "formula": "\\begin{align*} M \\mathbf { \\mathrm { y } } _ u = \\sum _ { \\substack { w \\sim u \\\\ w \\in E } } \\mathbf { \\mathrm { v } } _ w \\geq \\frac { 3 } { 4 k } \\lambda \\mathbf { \\mathrm { v } } _ u = \\frac { 3 } { 4 k } \\lambda \\mathbf { \\mathrm { y } } _ u . \\end{align*}"}
{"id": "6407.png", "formula": "\\begin{align*} V _ { \\widetilde S } = \\{ ( z _ 1 , \\dots , z _ n , \\Pi _ { i = 1 } ^ n z _ i ) \\ , : \\ , ( z _ 1 , \\dots , z _ n ) \\in V _ S \\} . \\end{align*}"}
{"id": "4607.png", "formula": "\\begin{align*} \\mathcal { Q } _ v = \\{ \\pi _ v ( G ^ + ) \\cap \\pi _ v ( G ^ - ) \\mid G ^ + \\in Q _ v ^ + , G ^ - \\in Q _ v ^ - \\} . \\end{align*}"}
{"id": "2875.png", "formula": "\\begin{align*} \\partial _ t \\Phi ( h ) = 2 B ( \\partial _ t h , h ) = 2 B ( - \\mathcal { L } h + R , h ) = 2 B ( R , h ) . \\end{align*}"}
{"id": "2632.png", "formula": "\\begin{align*} \\int _ \\mathcal { Q } | Z f ( x , \\omega ) | ^ 2 \\ , d \\omega = \\sum _ { k \\in \\Z ^ d } | f ( x + k ) | ^ 2 . \\end{align*}"}
{"id": "2206.png", "formula": "\\begin{align*} X _ m ^ { M , N } - X ^ { M , N } _ { m - 1 } + k A ( A X _ m ^ { M , N } + P _ N F ( X _ m ^ { M , N } ) ) = P _ N \\Delta W _ m , \\ ; X _ 0 ^ { M , N } = P _ N X _ 0 . \\end{align*}"}
{"id": "9361.png", "formula": "\\begin{align*} \\frac { 2 } { 2 - t } ( 1 - t ) ^ { x } & = \\frac { 1 } { 1 - \\frac { t } { 2 } } ( 1 - t ) ^ { x } = \\sum _ { l = 0 } ^ { \\infty } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { l } t ^ { l } \\sum _ { k = 0 } ^ { \\infty } \\binom { x } { k } ( - 1 ) ^ { k } t ^ { k } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = 0 } ^ { n } \\binom { x } { k } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { n - k } ( - 1 ) ^ { k } \\bigg ) t ^ { n } . \\end{align*}"}
{"id": "6209.png", "formula": "\\begin{align*} \\begin{aligned} \\| Z _ { j } ^ { T } V _ j \\| _ F ^ 2 \\geq j - \\frac { 1 0 \\epsilon \\| C \\| _ F ^ 2 } { \\eta _ j } . \\end{aligned} \\end{align*}"}
{"id": "2202.png", "formula": "\\begin{align*} m _ 0 = J ( \\widehat { u } ) = \\inf \\limits _ { \\mathcal { N } } J = \\inf \\limits _ { \\mathcal { M } } J = J ( \\widehat { w } ) = m _ 1 . \\end{align*}"}
{"id": "4104.png", "formula": "\\begin{align*} \\begin{aligned} ( X _ u - X _ s ) \\cdot \\mu ( X _ u ) + \\frac { 1 } { 2 } ( p - 1 ) \\cdot \\sigma ^ 2 ( X _ u ) & = X _ u \\mu ( X _ u ) + \\frac { 1 } { 2 } ( p - 1 ) \\cdot \\sigma ^ 2 ( X _ u ) - X _ s \\mu ( X _ u ) \\\\ & \\le X _ u \\mu ( X _ u ) + \\frac { 1 } { 2 } ( p _ 0 - 1 ) \\cdot \\sigma ^ 2 ( X _ u ) + | X _ s | \\cdot | \\mu ( X _ u ) | \\\\ & \\le c \\cdot \\bigl ( 1 + X _ u ^ 2 + | X _ s | \\cdot ( 1 + | X _ u | ^ { \\ell _ \\mu + 1 } ) \\bigr ) . \\end{aligned} \\end{align*}"}
{"id": "393.png", "formula": "\\begin{align*} \\begin{aligned} f _ { 1 } ( U , D _ { x } v ) & = g _ { 1 } ( U , D _ { x } v ) - A ^ { i } _ { 1 2 } ( u , v ) \\partial _ { i } v \\\\ f _ { 2 } ( U , D _ { x } U ) & = g _ { 2 } ( U , D _ { x } U ) - A ^ { i } _ { 2 1 } ( u , v ) \\partial _ { i } u - A ^ { i } _ { 2 2 } ( u , v ) \\partial _ { i } v . \\end{aligned} \\end{align*}"}
{"id": "2401.png", "formula": "\\begin{align*} D : \\ell ^ 2 ( \\Gamma ) \\to \\mathcal { H } , D ( c _ \\gamma ) _ { \\gamma \\in \\Gamma } = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma e _ \\gamma \\end{align*}"}
{"id": "8712.png", "formula": "\\begin{align*} \\begin{aligned} x _ i - x ^ L _ i & \\geq 0 & x ^ U _ i - x _ i & \\geq 0 , i = 1 , \\ldots , d , \\\\ f _ { i } ( x ) - a _ { i 0 } ( x ) & \\geq 0 & a _ { i n } ( x ) - f _ { i } ( x ) & \\geq 0 , i = 1 , \\ldots , d , \\\\ f _ { i } ( x ) - u _ { i j } ( x ) & \\geq 0 & a _ { i j } ( x ) - u _ { i j } ( x ) & \\geq 0 , i = 1 , \\ldots , d , \\ ; j = 0 , \\ldots , n . \\end{aligned} \\end{align*}"}
{"id": "3615.png", "formula": "\\begin{align*} \\alpha _ { n + 2 } = 2 e - 1 \\quad R _ { n + 3 } \\in \\{ 0 , 1 \\} \\ , . \\end{align*}"}
{"id": "5363.png", "formula": "\\begin{align*} g \\notin H _ i ^ + . \\end{align*}"}
{"id": "9031.png", "formula": "\\begin{align*} E ( \\rho , \\phi ) = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } \\epsilon ( x ) | \\nabla \\phi | ^ 2 \\bigg ) d x + \\frac { \\alpha } { 2 \\beta } \\int _ { \\partial \\Omega } \\epsilon ( x ) | \\phi | ^ 2 d s , \\end{align*}"}
{"id": "7280.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty ( 1 + \\alpha ^ { - 1 } X ^ { 2 k - 1 } ) ( 1 - X ^ { 2 k } ) = \\prod _ { k = 0 } ^ \\infty \\frac { 1 } { 1 + \\alpha X ^ { 2 k + 1 } } \\sum _ { k = - \\infty } ^ \\infty X ^ { k ^ 2 } \\alpha ^ { k } = \\prod _ { k = 1 } ^ \\infty \\frac { 1 } { 1 + \\alpha X ^ { 2 k - 1 } } \\sum _ { k = - \\infty } ^ \\infty X ^ { k ^ 2 } \\alpha ^ { k } . \\end{align*}"}
{"id": "8905.png", "formula": "\\begin{align*} \\alpha ^ * A _ Y = \\alpha ^ * \\circ \\rho _ Y ^ * A _ \\ast = ( \\rho _ Y \\circ \\alpha ) ^ * A _ \\ast = \\rho _ X ^ * A _ \\ast = A _ X . \\end{align*}"}
{"id": "5617.png", "formula": "\\begin{align*} \\Pi _ \\star ^ \\mathcal { F } : = \\mathrm { p r } _ { \\perp \\star } \\circ \\nabla ^ \\mathcal { F } | _ { \\Xi _ { t \\star } \\otimes _ { \\mathcal { X } _ \\star } \\Xi _ { t \\star } } \\colon \\Xi _ { t \\star } \\otimes _ { \\mathcal { X } _ \\star } \\Xi _ { t \\star } \\rightarrow \\Xi _ { \\perp \\star } \\end{align*}"}
{"id": "1905.png", "formula": "\\begin{align*} R ( x ) & = \\left \\{ \\theta \\in \\Theta : \\P ( \\theta | x ) \\geq \\frac { \\lambda _ { B N } } { ( \\lambda _ { B N } - \\lambda _ { B P } ) + \\lambda _ { N P } } \\right \\} . \\end{align*}"}
{"id": "1641.png", "formula": "\\begin{align*} F ( u ) \\in \\mathcal { D } ( \\mathcal { L } ^ { ( 1 ) } ) \\mathcal { L } ^ { ( 1 ) } F ( u ) = F ' ( u ) \\mathcal { L } u + F '' ( u ) \\Gamma ( u ) ; \\end{align*}"}
{"id": "2057.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { p , \\lambda , t } = \\int _ { \\{ p \\Psi _ 1 < - t \\} } | f | ^ 2 + \\int _ { \\{ 0 > p \\Psi _ 1 \\geq - t \\} } | f | ^ 2 e ^ { - \\lambda ( p \\Psi _ 1 + t ) } . \\end{align*}"}
{"id": "228.png", "formula": "\\begin{align*} \\Phi _ t ( x , u ) = ( x + t u , u ) . \\end{align*}"}
{"id": "4651.png", "formula": "\\begin{align*} \\dim _ { \\C } V _ k = \\sum _ { \\pi \\ , \\atop W ( \\pi ) = - 1 } \\dim V _ { \\pi } \\ ; . \\end{align*}"}
{"id": "8419.png", "formula": "\\begin{align*} D _ { U } F _ { t } ( W ) = \\lim _ { \\epsilon \\to 0 } \\frac { F _ { t } ( W + \\epsilon U ) - F _ { t } ( W ) } { \\epsilon } \\end{align*}"}
{"id": "458.png", "formula": "\\begin{align*} \\prod _ { u \\in J } [ \\S ( u ) : \\S _ 0 ( u ) ] = [ \\prod _ { u \\in J } \\S ( u ) B : B ] \\leq [ \\S ( V ) : \\S _ 0 ( V ) ] \\leq \\prod _ { u \\in J } [ \\S ( u ) : \\S _ 0 ( u ) ] \\end{align*}"}
{"id": "3522.png", "formula": "\\begin{align*} & \\frac { s _ 3 } { 2 \\pi i ( s _ 1 + s _ 3 - 1 ) \\Gamma ( s _ 3 + 1 ) } \\sum _ { n > a t _ 3 } \\int _ n ^ \\infty \\int _ { ( c ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { u ^ { s _ 1 + s _ 3 + 1 + z } n ^ { s _ 2 - 1 - z } } d z d u . \\\\ & = \\frac { s _ 3 } { 2 \\pi i ( s _ 1 + s _ 3 - 1 ) \\Gamma ( s _ 3 + 1 ) } \\sum _ { n > a t _ 3 } \\frac { 1 } { n ^ { s _ 1 + s _ 2 + s _ 3 - 1 } } \\int _ { ( c ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { s _ 1 + s _ 3 + z } d z . \\end{align*}"}
{"id": "9517.png", "formula": "\\begin{align*} \\N ^ \\perp : = \\{ v \\in L ^ 1 \\mid \\langle x , v \\rangle = 0 \\forall x \\in \\N \\cap L ^ \\infty \\} . \\end{align*}"}
{"id": "643.png", "formula": "\\begin{align*} \\abs { \\frac { f ( x + 1 , n ) } { x + 1 } - \\alpha ( n ) } \\ & = \\ \\abs { \\frac { f ( x + 1 , n ) } { x + 1 } - \\frac { 1 } { n ! } } \\\\ [ 1 5 p t ] & = \\ \\frac { 1 } { x + 1 } \\abs { f ( x + 1 , n ) - \\frac { x + 1 } { n ! } } \\\\ [ 1 5 p t ] & = \\ \\frac { 1 } { x + 1 } \\abs { \\bigg [ \\frac { x + 1 } { n ! } \\bigg ] - { \\frac { x + 1 } { n ! } } } \\\\ [ 1 5 p t ] & \\leq \\ \\frac { 1 } { x + 1 } . \\end{align*}"}
{"id": "7057.png", "formula": "\\begin{align*} \\sup _ { K \\geq 1 } \\ , \\sum _ { j = 0 } ^ { { 1 \\over \\delta _ K } - 1 } h _ K G \\left ( \\overline { ( i _ K - j ) \\delta _ K } \\ , \\log K \\right ) e ^ { L \\ , \\log K \\ , \\rho ( j \\delta _ K , i _ K \\delta _ K ) } = \\ : \\ \\overline G ( \\alpha ) < + \\infty . \\end{align*}"}
{"id": "1295.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\bigcup _ { p _ { i } \\in S _ { \\theta } } \\Lambda _ { ( p _ { i } , \\infty ) } ( M , \\Gamma ) | } { | \\Lambda ( M , \\Gamma ) | } = 0 \\end{align*}"}
{"id": "5080.png", "formula": "\\begin{align*} \\left ( \\frac { B } { A } \\right ) ' = 0 \\Longleftrightarrow \\frac { A ' } { A } = \\frac { B ' } { B } . \\end{align*}"}
{"id": "8989.png", "formula": "\\begin{align*} u ( x ) = \\inf \\left \\{ \\left . I [ \\gamma ] : = \\int _ 0 ^ \\infty e ^ { - s } \\left ( \\frac { \\left ( \\dot { \\gamma } ( s ) \\right ) ^ 2 } { 2 } + f ( \\gamma ( s ) ) \\right ) d s \\right | \\gamma \\in \\mathrm { A C } ( [ 0 , \\infty ) ; [ - 1 , 1 ] ) , \\gamma ( 0 ) = x \\right \\} . \\end{align*}"}
{"id": "6952.png", "formula": "\\begin{align*} W _ { i } \\upharpoonright t '' = \\Phi _ i ^ { \\Gamma _ A } \\upharpoonright t '' , 1 = \\Gamma _ A ( \\langle x _ m , y _ m \\rangle ) = \\Psi _ i ^ { W _ i \\upharpoonright t '' } ( \\langle x _ m , y _ m \\rangle ) . \\end{align*}"}
{"id": "8575.png", "formula": "\\begin{align*} & \\sqrt { 2 \\pi } \\mathcal { K } ^ { \\# } ( x , k ) = \\mathcal { K } ^ { \\# } _ S ( x , k ) + \\mathcal { K } ^ { \\# } _ R ( x , k ) , \\end{align*}"}
{"id": "4700.png", "formula": "\\begin{align*} & \\int _ { | y | < \\frac { 1 } { 2 | x _ i - x _ k | } } Q ' ( \\tau _ i ^ { - 1 } \\tau _ k Q ) ^ { p - 1 } A _ { i j , 0 } = \\int _ { | y | < \\frac { 1 } { 2 | x _ i - x _ k | } } \\bigg ( \\frac { \\kappa _ 0 } { ( x _ k - x _ i ) ^ 2 } \\bigg ) ^ { p - 1 } + O \\bigg ( \\frac { 1 } { d ^ 3 } \\bigg ) \\\\ & = \\int Q ' A _ { i j } \\bigg ( \\frac { c _ 0 } { ( x _ k - x _ i ) ^ 2 } \\bigg ) ^ { p - 1 } + O \\bigg ( \\frac { 1 } { d ^ 3 } \\bigg ) = O \\bigg ( \\frac { 1 } { d ^ 3 } \\bigg ) , \\end{align*}"}
{"id": "3593.png", "formula": "\\begin{align*} & V : = P _ { i _ 0 } ^ * ( V _ { i _ 0 } ) , \\ , \\ , z _ 1 ^ * : = P _ { i _ 0 } ^ * ( x _ { i _ 0 } ^ * ) , \\ , \\ , e : = J _ { i _ 0 } ( e _ { i _ 0 } ) \\ , \\ , \\ , \\ , F : = J _ { i _ 0 } F _ { i _ 0 } P _ { i _ 0 } \\end{align*}"}
{"id": "8551.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\left | e ^ { - i k x } \\psi _ + ( x , k ) - 1 \\right | = 0 , \\lim _ { x \\rightarrow - \\infty } \\left | e ^ { i k x } \\psi _ - ( x , k ) - 1 \\right | = 0 . \\end{align*}"}
{"id": "5273.png", "formula": "\\begin{align*} \\check { A } = \\{ \\varphi ( a - ) \\mid a \\in A \\} . \\end{align*}"}
{"id": "5427.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ 1 & = \\sqrt { \\frac { s _ { \\pm } ( a ) s _ { \\pm } ( c ) } { s _ { \\pm } ( b ) } } \\\\ \\lambda _ 2 & = \\sqrt { \\frac { s _ { \\pm } ( a ) s _ { \\pm } ( b ) } { s _ { \\pm } ( c ) } } \\\\ \\lambda _ 3 & = \\sqrt { \\frac { s _ { \\pm } ( b ) s _ { \\pm } ( c ) } { s _ { \\pm } ( a ) } } \\end{cases} \\end{align*}"}
{"id": "5994.png", "formula": "\\begin{align*} \\rho _ { \\pi _ { \\lambda , \\beta } } ( - z , \\cdot ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( z ) _ { k } } { k ! } Q _ { k } ^ { \\pi _ { \\lambda , \\beta } } ( \\cdot ) . \\end{align*}"}
{"id": "5825.png", "formula": "\\begin{align*} v _ + : = \\inf \\big \\{ v \\in \\mathbb { R } \\colon \\liminf _ { H \\to \\infty } p _ H ( v ) = 0 \\big \\} , \\end{align*}"}
{"id": "9157.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 1 } ^ { \\infty } \\lambda _ f ( n ) n ^ { ( \\kappa - 1 ) / 2 } e ( n z ) . \\end{align*}"}
{"id": "4931.png", "formula": "\\begin{align*} \\phi _ t ( \\mu _ n , Q , y ) = \\int _ { Q _ { \\alpha } ( y ) } \\rho ( x , y ) ^ { - t } d \\mu _ n ( x ) + \\int _ { Q \\setminus Q _ { \\alpha } ( y ) } \\rho ( x , y ) ^ { - t } d \\mu _ n ( x ) . \\end{align*}"}
{"id": "709.png", "formula": "\\begin{align*} \\kappa _ { 2 k ; \\alpha } ^ { ( \\ell + 1 ) } = \\kappa _ { k } \\big ( \\underbrace { \\Delta _ { \\alpha \\alpha } ^ { ( \\ell ) } , \\ldots , \\Delta _ { \\alpha \\alpha } ^ { ( \\ell ) } } _ { k } \\big ) , \\end{align*}"}
{"id": "369.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho , m , S ) = \\sup _ { S \\in H ^ 1 } \\inf _ { ( \\rho , m ) } \\mathcal L ( \\rho , m , S ) \\end{align*}"}
{"id": "1058.png", "formula": "\\begin{align*} & \\mathcal { R } _ { \\xi , I } : = \\left \\{ ( x , t ) | \\xi < - \\frac { C _ { L } ^ 2 } { 2 } \\right \\} , \\mathcal { R } _ { \\xi , I I } : = \\left \\{ ( x , t ) | - \\frac { C ^ 2 _ L } { 2 } < \\xi < - \\frac { C _ { R } ^ 2 } { 2 } \\right \\} , \\\\ & \\mathcal { R } _ { \\xi , I I I } : = \\left \\{ ( x , t ) | \\frac { - C _ { R } ^ 2 } { 2 } < \\xi < \\frac { C _ { R } ^ 2 } { 2 } \\right \\} , \\mathcal { R } _ { \\xi , I V } : = \\left \\{ ( x , t ) | \\xi > \\frac { C _ { R } ^ 2 } { 2 } \\right \\} , \\ \\xi = \\frac { x } { 1 2 t } , \\end{align*}"}
{"id": "6517.png", "formula": "\\begin{align*} f ^ { ( 4 ) } _ n = \\frac { s ^ { ( 2 ) } _ n } { s ^ { ( 4 ) } _ { n + 1 } } \\left \\{ \\binom { 4 } { 2 } + \\frac { \\alpha } { n } \\binom { 4 } { 1 } \\right \\} M ^ { ( 2 ) } _ n . \\end{align*}"}
{"id": "1388.png", "formula": "\\begin{align*} | R _ 1 | = \\left | \\alpha \\left ( \\frac { n _ 2 ^ 2 } { n _ 2 + 1 } - \\frac { n _ 1 ^ 2 } { n _ 1 + 1 } \\right ) \\right | \\le C | n _ 1 - n _ 2 | \\le C \\frac { | n _ 1 - n _ 2 | ( x ) } { ( 1 - x ) ^ \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "3812.png", "formula": "\\begin{align*} ( g \\circ f ) ( \\tau ( a , b ) ) & = ( g \\circ f ) \\left ( - a , a + b + \\frac { 1 } { 4 k } \\right ) = g \\left ( - a , - k a + 2 k \\left ( a + b + \\frac { 1 } { 4 k } \\right ) \\right ) \\\\ & = g \\left ( - a , k a + 2 k b + \\frac { 1 } { 2 } \\right ) = ( g \\circ \\tau _ 1 ) \\left ( a , k a + 2 k b \\right ) = g ( a , k a + 2 k b ) = ( g \\circ f ) ( a , b ) . \\end{align*}"}
{"id": "5889.png", "formula": "\\begin{align*} Y _ 0 = y . \\end{align*}"}
{"id": "7075.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\bar R ( X , Y ) Z & = & \\langle Y , Z \\rangle X - \\langle X , Z \\rangle Y + 2 \\langle Z , E _ 3 \\rangle ( \\langle X , E _ 3 \\rangle Y - \\langle Y , E _ 3 \\rangle X ) \\\\ \\\\ & & + 2 ( \\langle X , Z \\rangle \\langle Y , E _ 3 \\rangle - \\langle Y , Z \\rangle \\langle X , E _ 3 \\rangle ) E _ 3 , \\end{array} \\end{align*}"}
{"id": "9469.png", "formula": "\\begin{align*} H = \\{ b _ 1 a _ 1 + \\cdots + b _ m a _ m : \\ ; b _ i \\in \\mathbb { N } i = 1 , \\ldots , m \\} \\end{align*}"}
{"id": "5830.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 : = \\left \\{ \\exists \\ , y \\in I _ { H } ( w ) \\cap \\mathbb { L } : X ^ { y } _ { H ^ \\prime } - \\pi _ 1 ( y ) \\geq v ^ { \\prime } H \\right \\} , \\end{align*}"}
{"id": "287.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) : = \\mathcal { F } [ f ] ( \\xi ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { \\R } e ^ { - i x \\xi } f ( x ) d x , \\check { g } ( x ) : = \\mathcal { F } ^ { - 1 } [ g ] ( x ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { \\R } e ^ { i x \\xi } g ( \\xi ) d \\xi . \\end{align*}"}
{"id": "762.png", "formula": "\\begin{align*} b & : C _ k ( \\mathfrak { A } ) \\to C _ { k - 1 } ( \\mathfrak { A } ) , \\\\ b & ( a _ 0 \\otimes a _ 1 \\otimes \\cdots \\otimes a _ k ) : = \\sum _ { j = 0 } ^ { k - 1 } ( - 1 ) ^ { j } a _ 0 \\otimes \\cdots \\otimes a _ j a _ { j + 1 } \\otimes \\cdots \\otimes \\cdots \\otimes a _ k + \\\\ & \\qquad \\qquad \\qquad \\qquad + ( - 1 ) ^ k a _ k a _ 0 \\otimes a _ 1 \\otimes \\cdots \\otimes a _ { k - 1 } . \\end{align*}"}
{"id": "5648.png", "formula": "\\begin{align*} u ( s ) - u ( t ) \\rightarrow 0 | s | - | t | = c , \\quad | s | , | t | \\rightarrow \\infty \\end{align*}"}
{"id": "2326.png", "formula": "\\begin{align*} & \\ , \\iint _ { \\R ^ { 2 d } } W ( f _ 1 , g _ 1 ) ( x , \\omega ) \\overline { W ( f _ 2 , g _ 2 ) ( x , \\omega ) } \\ , d x d \\omega \\\\ = & \\ , 2 ^ { 2 d } \\iint _ { \\R ^ { 2 d } } V _ { g _ 1 ^ \\vee } f _ 1 ( 2 x , 2 \\omega ) \\overline { V _ { g _ 2 ^ \\vee } f _ 2 ( 2 x , 2 \\omega ) } \\ , d x d \\omega \\\\ = & \\ , \\langle f _ 1 , f _ 2 \\rangle \\overline { \\langle g ^ \\vee _ 1 , g ^ \\vee _ 2 \\rangle } \\\\ = & \\ , \\langle f _ 1 , f _ 2 \\rangle \\overline { \\langle g _ 1 , g _ 2 \\rangle } \\end{align*}"}
{"id": "7402.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon _ 1 \\rightarrow 0 ^ { + } } \\limsup _ { n \\rightarrow \\infty } \\mathbb { E } _ { \\mu _ n } \\Big [ \\int _ 0 ^ T n ^ { \\gamma - 1 } \\sum _ { x , y : | x - y | \\geq \\varepsilon n } & [ \\overleftarrow { \\eta } _ t ^ { \\varepsilon _ 1 n } ( y ) \\overrightarrow { \\eta } _ t ^ { \\varepsilon _ 1 n } ( y + 1 ) \\\\ - & \\eta _ t ^ n ( y ) \\eta _ t ^ n ( y + 1 ) ] F ( t , \\tfrac { x } { n } , \\tfrac { y } { n } ) ( c _ { \\gamma } ) ^ { - 1 } p ( x - y ) d t \\Big ] = 0 . \\end{align*}"}
{"id": "2588.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\overline { \\langle g , \\widetilde { g } \\rangle } } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "6599.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq e < \\infty \\\\ ( e , g ) = 1 } } \\mu ( e ) \\sum _ { a | g } a \\mu ( a ) \\sum _ { \\substack { 1 \\leq b < \\infty \\\\ e a b | \\frac { m h } { g } \\pm \\frac { n k } { g } } } b W \\left ( \\frac { c e a b } { Q } \\right ) . \\end{align*}"}
{"id": "1522.png", "formula": "\\begin{align*} V _ t = \\{ \\beta \\times \\gamma \\in P _ { n } ^ t \\times P _ { n } ^ t : \\kappa _ t \\pi _ t ( \\beta ) = \\pi _ t ( \\gamma ) \\kappa _ t \\} , \\kappa _ t = \\left [ \\begin{array} { c c c } 0 & 0 & - 1 _ t \\\\ 0 & 1 _ r & 0 \\\\ 1 _ t & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "3506.png", "formula": "\\begin{align*} D _ { 1 2 1 } \\ll t _ 3 ^ { - \\sigma _ 1 - \\sigma } \\begin{cases} 1 & ( \\sigma _ 2 > 0 ) \\\\ \\log t _ 3 & ( \\sigma _ 2 = 0 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 2 } & ( \\sigma _ 2 < 0 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "1888.png", "formula": "\\begin{align*} X ^ 1 = \\lim _ S \\left ( X _ S ^ 1 \\times \\prod _ { s \\in S \\setminus V } \\{ o _ s \\} \\right ) . \\end{align*}"}
{"id": "3456.png", "formula": "\\begin{align*} \\| S _ { c w } ( T _ M f ) \\| _ { L ^ p ( \\omega ) } & = \\bigg \\| \\bigg \\{ \\sum \\limits _ { k \\in \\Bbb Z } \\sum \\limits _ { Q ' \\in Q ^ k } | D _ k T _ { M } f ( x _ { Q ' } ) | ^ 2 \\chi _ { Q ' } ( x ) \\bigg \\} ^ { \\frac 1 2 } \\bigg \\| _ { L ^ p } \\\\ & \\lesssim \\bigg \\| \\bigg \\{ \\sum \\limits _ { j \\in \\Bbb Z } \\sum \\limits _ { Q \\in Q ^ j } | q _ Q f ( x _ { Q } ) | ^ 2 \\chi _ { Q } ( x ) \\bigg \\} ^ { \\frac 1 2 } \\bigg \\| _ { L ^ p } \\\\ & \\lesssim \\| S ( f ) \\| _ p . \\end{align*}"}
{"id": "2127.png", "formula": "\\begin{align*} \\prod _ { t = 0 } ^ { c _ n k n - l - 1 } \\frac { k ( n - s - 1 ) - t } { k n - t } = \\prod _ { i = ( 1 - c _ n ) k n + l + 1 } ^ { k n } ( 1 - \\frac { k ( s + 1 ) } i ) , \\end{align*}"}
{"id": "4896.png", "formula": "\\begin{align*} S f ' = f ^ 2 + S ' f - c = P _ 2 ( f ) = ( f - A _ 1 ) ( f - A _ 2 ) , A _ 1 , A _ 2 \\in \\C , - c = A _ 1 A _ 2 . \\end{align*}"}
{"id": "9305.png", "formula": "\\begin{align*} \\lambda ( L ) : = \\prod _ { i = 1 } ^ n \\prod _ { j \\in J _ i } \\left ( \\frac { a _ { i j } } { - b _ { i j } } \\right ) ^ { d _ { i j } } . \\end{align*}"}
{"id": "4564.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { i = 1 } ^ n \\mathbf { E } ( \\eta _ i ^ 2 | \\mathcal { F } _ { i - 1 } ) - n \\sigma ^ 2 \\bigg \\| _ { \\infty } \\leq N ^ 2 , \\end{align*}"}
{"id": "268.png", "formula": "\\begin{align*} & \\sigma _ { 0 } ( t ) : = - t \\qquad ( - \\infty < t < + \\infty ) , \\\\ & \\sigma _ { 1 } ( t ) : = - \\log ( t + 1 ) \\qquad ( - 1 < t < + \\infty ) , \\end{align*}"}
{"id": "3364.png", "formula": "\\begin{align*} & [ T u , T v , T _ 1 ( w ) ] = - [ T v , T _ 1 ( w ) , T u ] - [ T _ 1 ( w ) , T u , T v ] = [ T _ 1 ( w ) , T v , T u ] + [ T u , T _ 1 ( w ) , T v ] . \\end{align*}"}
{"id": "8577.png", "formula": "\\begin{align*} \\mathcal { K } _ { 0 } ^ { \\# } ( x , k ) & = T ( 0 ) m _ { + } ( x , 0 ) e ^ { i k x } , \\end{align*}"}
{"id": "5813.png", "formula": "\\begin{align*} \\tilde { x } _ { m , n } = \\sqrt { P _ d } \\sum _ { k = 1 } ^ { K } \\sqrt { \\eta _ { m k } } \\hat { g } _ { m k , n } ^ * s _ { k , n } , \\end{align*}"}
{"id": "4707.png", "formula": "\\begin{align*} \\frac { \\alpha _ i } { 4 } = \\sum _ { \\substack { j = 1 \\\\ j \\not = i } } ^ n \\frac { a _ { i j } } { ( \\alpha _ i - \\alpha _ j ) ^ 3 } , \\end{align*}"}
{"id": "3091.png", "formula": "\\begin{align*} z _ 1 ^ \\prime = z _ 2 \\ , , z _ 2 ^ \\prime = z _ 3 \\ , , z _ 3 ^ \\prime = z _ 1 \\ , , z _ 4 ^ \\prime = z _ 5 \\ , , z _ 5 ^ \\prime = z _ 4 \\ , , \\end{align*}"}
{"id": "8537.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi + m ^ 2 \\phi + V ( x ) \\phi = \\phi ^ 3 , ( \\phi , \\phi _ t ) ( 0 ) = ( \\phi _ 0 , \\phi _ 1 ) , \\end{align*}"}
{"id": "4209.png", "formula": "\\begin{align*} \\int _ M ( 2 q u _ 0 v + | \\nabla v | ^ 2 + V _ 1 v ^ 2 ) \\ , d V _ g & = - \\int _ M | \\nabla v | ^ 2 \\ , d V _ g - \\int _ M V _ 1 v ^ 2 \\ , d V _ g , \\\\ \\int _ M ( q v r _ t + V _ 1 r _ t ^ 2 + | \\nabla r _ t | ^ 2 + t q r _ t ^ 2 ) \\ , d V _ g & = 0 . \\end{align*}"}
{"id": "9164.png", "formula": "\\begin{align*} \\sum _ { ( p , 2 ) = 1 } ( \\log p ) \\chi _ { 8 p } ( m \\ell ) V \\left ( \\frac { m } { p } \\right ) \\Phi \\left ( \\frac { p } { X } \\right ) = & \\sum ^ { \\infty } _ { n = 1 } \\chi _ { 8 n } ( m \\ell ) \\Lambda ( n ) V \\left ( \\frac { m } { n } \\right ) \\Phi \\left ( \\frac { n } X \\right ) + O \\left ( \\sum _ { \\substack { p ^ j \\leq X ^ { 1 + \\varepsilon } , j \\geq 2 } } ( \\log p ) \\Phi \\left ( \\frac { p ^ j } X \\right ) \\right ) . \\end{align*}"}
{"id": "7550.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) = \\left ( \\frac { 1 } { 2 } - \\left ( \\frac { 1 } { 2 } - \\epsilon \\right ) \\right ) N _ 0 ( T ) \\end{align*}"}
{"id": "8511.png", "formula": "\\begin{align*} ( x + y ) * z = x * ( \\lambda _ { x } ^ { - 1 } ( y ) * z ) + \\lambda _ { x } ^ { - 1 } ( y ) * z + x * z \\end{align*}"}
{"id": "2175.png", "formula": "\\begin{align*} J ( w ) & = J ( w ^ + ) + J ( w ^ - ) \\\\ & + \\int _ { ( w ^ - ) } \\int _ { ( w ^ + ) } G \\left ( \\frac { w ^ + ( x ) - w ^ - ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { d x d y } { \\vert x - y \\vert ^ { d } } \\\\ & + \\int _ { ( w ^ + ) } \\int _ { ( w ^ - ) } G \\left ( \\frac { w ^ - ( x ) - w ^ + ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { d x d y } { \\vert x - y \\vert ^ { d } } . \\end{align*}"}
{"id": "1988.png", "formula": "\\begin{align*} R ^ { ( n - 1 ) } _ { \\triangleleft h } ( h ) & = \\sum _ { k = n } ^ \\infty \\bigg ( \\sum _ { i _ 1 + \\dotsm + i _ n = k } ( i _ 1 + 1 ) ( i _ 1 + i _ 2 + 1 ) \\dotsm ( i _ 1 + \\dotsb + i _ { n - 1 } + 1 ) h _ { i _ 1 } \\dotsm h _ { i _ n } \\bigg ) x ^ k . \\end{align*}"}
{"id": "1027.png", "formula": "\\begin{align*} \\bar M : = \\sup \\{ M \\geqslant 0 \\tilde u _ M > 0 B _ 1 ( 2 e _ 1 ) \\} \\end{align*}"}
{"id": "6859.png", "formula": "\\begin{align*} ( M ) \\subseteq \\bigcup _ { i = 1 } ^ k \\left [ \\lambda _ i - c _ i , \\lambda _ i + c _ i \\right ] \\end{align*}"}
{"id": "58.png", "formula": "\\begin{align*} L \\otimes \\Z _ v & = L _ { v , 0 } , \\\\ K _ { \\ell } \\otimes \\Z _ v & = K _ { \\ell , v , 0 } . \\end{align*}"}
{"id": "2444.png", "formula": "\\begin{align*} ( S S ' ) ^ T J S S ' = S '^ T ( S ^ T J S ) S ' = S '^ T J S ' = J . \\end{align*}"}
{"id": "7249.png", "formula": "\\begin{align*} a = C _ n ^ { k - 1 } , b = C _ n ^ { k } , c = C _ n ^ { k + 1 } , d = C _ n ^ { k + 2 } . \\end{align*}"}
{"id": "7578.png", "formula": "\\begin{align*} f ( x , y ) = \\sum _ { i = 0 } ^ { \\infty } { \\vphantom { \\sum } } ' \\sum _ { j = 0 } ^ { \\infty } { \\vphantom { \\sum } } ' c _ { i , j } T _ i ( x ) T _ j ( y ) , \\end{align*}"}
{"id": "4532.png", "formula": "\\begin{align*} T ( x ; k , r ) = B ( r , k ) x + O ( ( r x ) ^ { 1 / 2 + \\varepsilon } ) , \\ B ( r , k ) = O ( 1 / r ) . \\end{align*}"}
{"id": "1764.png", "formula": "\\begin{align*} \\Phi ( a _ k , a _ 0 , \\dots , a _ { k - 1 } ) = ( - 1 ) ^ k \\Phi ( a _ 0 , a _ 1 , \\dots , a _ k ) , \\ \\ \\ \\forall a _ 0 , \\dots , a _ k \\in A . \\end{align*}"}
{"id": "9317.png", "formula": "\\begin{align*} 0 \\in \\begin{bmatrix} H \\mathbf { x } + \\mathbf { g } - A ^ T \\mathbf { y } + \\partial _ \\mathbf { x } I _ D ( \\mathbf { x } , \\mathbf { y } ) + \\rho ( \\mathbf { x } - \\mathbf { x } _ k ) \\\\ A \\mathbf { x } - \\mathbf { b } + \\partial _ \\mathbf { y } I _ D ( \\mathbf { x } , \\mathbf { y } ) + \\delta ( \\mathbf { y } - \\mathbf { y } _ k ) \\end{bmatrix} \\end{align*}"}
{"id": "8798.png", "formula": "\\begin{align*} & \\theta _ k = \\sum _ { I \\in \\mathcal { I } _ k } c ^ k _ I \\biggl ( \\sum _ { e \\in E } \\Bigl ( \\prod _ { i \\in I } a _ { i e _ i } \\Bigr ) w _ e \\biggr ) , & & k = 1 , \\ldots , \\kappa , \\\\ & w \\geq 0 , \\ \\sum _ { e \\in E } w _ e = 1 , \\ \\lambda _ { i j } = \\sum _ { e \\in E : e _ i = j } w _ e , & & i = 1 , \\ldots , d , \\ ; j = 0 , \\ldots , n , \\\\ & f = A ( \\lambda ) , \\ ( \\lambda _ i , \\delta _ i ) \\in ( \\ref { e q : S O S 2 - l o g } ) , & & i = 1 , \\ldots , d . \\end{align*}"}
{"id": "3276.png", "formula": "\\begin{align*} \\mu _ { T } \\{ \\lambda \\in \\mathbb { C } : | \\lambda | \\leq r \\} = \\frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + r ^ 2 } , \\end{align*}"}
{"id": "5132.png", "formula": "\\begin{align*} \\gamma = \\frac { \\log \\left ( n + \\sqrt { n ^ 2 + 4 } \\right ) - \\log ( 2 ) } { a \\ , n } . \\end{align*}"}
{"id": "9491.png", "formula": "\\begin{align*} | \\mathcal { F } ( a + b , - b \\ , ; \\ , \\{ U \\} , \\{ D \\} ) | = \\sum _ { i = 0 } ^ { a - 1 } \\binom { a + b - 2 } { \\lfloor i / 2 \\rfloor } \\binom { a + b - 1 - \\lfloor i / 2 \\rfloor } { a - i - 1 } . \\end{align*}"}
{"id": "8097.png", "formula": "\\begin{align*} \\overline { u } _ n ( x ) = \\frac { ( - 1 ) ^ n } { n ! } \\frac { d ^ n } { d x ^ n } \\lim _ { \\epsilon \\searrow 0 } \\frac { 1 } { x \\mp i \\epsilon } . \\end{align*}"}
{"id": "5123.png", "formula": "\\begin{align*} A = C _ { b / 2 , \\gamma } ^ 2 \\ , \\frac { a ( 1 - e ^ { - 2 \\gamma / b } ) ( 1 - e ^ { - \\gamma / b } ) ^ 2 } { 1 - e ^ { - 2 \\gamma a } } \\ , e ^ { - 2 a \\gamma } B = C _ { b / 2 , \\gamma } ^ 2 \\ , \\frac { a ( 1 - e ^ { - 2 \\gamma / b } ) ( 1 + e ^ { - \\gamma / b } ) ^ 2 } { 1 - e ^ { - 2 \\gamma a } } . \\end{align*}"}
{"id": "4863.png", "formula": "\\begin{align*} D ( \\varphi ) \\leq \\frac C { n ^ 2 } \\sum _ { i = 1 } ^ n \\alpha \\log \\Big ( \\max \\Big \\{ \\frac 1 { \\alpha \\ell _ i } , e \\Big \\} \\Big ) . \\end{align*}"}
{"id": "295.png", "formula": "\\begin{align*} | r _ { 0 } ( x ) | & \\le C \\int _ { - \\infty } ^ { x } | z _ { 0 } ( y ) | d y \\le C \\int _ { - \\infty } ^ { x } ( 1 + | y | ) ^ { - \\alpha } d y \\\\ & \\le C \\int _ { - \\infty } ^ { x } ( 1 - y ) ^ { - \\alpha } d y \\le C ( 1 - x ) ^ { - ( \\alpha - 1 ) } = C ( 1 + | x | ) ^ { - ( \\alpha - 1 ) } . \\end{align*}"}
{"id": "3600.png", "formula": "\\begin{align*} d [ - b _ { j - 1 } b _ { j } ] = \\min \\limits _ { k \\in [ 1 , i ] ^ { E } } \\{ d [ - b _ { k - 1 } b _ { k } ] \\} \\le d [ ( - 1 ) ^ { i / 2 } b _ { 1 , i } ] \\le 1 - R _ { i + 2 } \\end{align*}"}
{"id": "6464.png", "formula": "\\begin{align*} E [ S _ n ] = \\beta a _ n \\mbox { f o r a n y $ n \\in \\mathbb { N } $ } . \\end{align*}"}
{"id": "9161.png", "formula": "\\begin{align*} \\sum _ { p \\le x } \\frac { ( \\log p ) ^ j } { p } = \\frac { ( \\log x ) ^ j } { j } + O ( ( \\log x ) ^ { j - 1 } ) . \\end{align*}"}
{"id": "3344.png", "formula": "\\begin{gather*} T = \\begin{pmatrix} 0 & a & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ b & c & d & e \\\\ f & g & h & k \\end{pmatrix} \\end{gather*}"}
{"id": "5570.png", "formula": "\\begin{align*} A - \\lambda = 0 \\end{align*}"}
{"id": "6862.png", "formula": "\\begin{align*} \\delta ^ k _ { - 1 } = 1 , ~ \\delta _ j ^ k = \\prod \\limits _ { i = k - j } ^ k \\delta _ i , ~ \\gamma ^ k _ j = \\gamma \\sum \\limits ^ { j - 1 } _ { i = - 1 } \\delta ^ k _ i . \\end{align*}"}
{"id": "9494.png", "formula": "\\begin{align*} | \\mathcal { F } ( a + b - 1 , - b \\ , ; \\ , \\{ U \\} , \\emptyset ) | & = \\sum _ { i = 0 } ^ { a - 1 } \\binom { a + b - 2 } { \\lfloor i / 2 \\rfloor , b + \\lfloor ( i - 1 ) / 2 \\rfloor , a - i - 1 } , \\\\ | \\mathcal { F } ( a + b - 1 , - b - 1 \\ , ; \\ , \\{ U \\} , \\emptyset ) | & = \\sum _ { i = 0 } ^ { a - 2 } \\binom { a + b - 2 } { \\lfloor i / 2 \\rfloor , b + \\lfloor ( i + 1 ) / 2 \\rfloor , a - i - 2 } . \\end{align*}"}
{"id": "259.png", "formula": "\\begin{align*} \\prod _ { i \\in I _ \\kappa ^ * } c _ i = ( - 1 ) ^ { \\frac { | I _ \\kappa ^ * | - \\sum _ { i \\in I ^ * } c _ i } { 2 } } . \\end{align*}"}
{"id": "7598.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n _ y - l _ y } { \\vphantom { \\sum } } ' \\sum _ { i = n _ x - l _ x + 1 } ^ { \\infty } \\left | c _ { i , j } \\right | \\le ( n _ y - l _ y + 1 ) \\dfrac { 4 V _ k } { 2 k \\pi ^ 2 } \\begin{cases} \\Pi _ { 1 } [ s ] ( n _ x - l _ x ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , \\\\ \\Pi _ { 0 } [ s ] ( n _ x - l _ x ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 , \\end{cases} \\end{align*}"}
{"id": "8445.png", "formula": "\\begin{align*} b b ^ { - 1 } u ^ { - 1 } u b c ^ { - 1 } = b ( u b ) ^ { - 1 } ( u b ) c ^ { - 1 } = b b ^ { - 1 } b c ^ { - 1 } c c ^ { - 1 } = b c ^ { - 1 } . \\end{align*}"}
{"id": "4461.png", "formula": "\\begin{align*} | z \\rangle = A _ { 0 } \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { \\sqrt { n ! } } | n \\rangle . \\end{align*}"}
{"id": "4710.png", "formula": "\\begin{align*} \\alpha _ { i } = \\sum _ { j = i } ^ k \\theta _ { 0 , j } , \\alpha _ { n + 1 - i } = - \\alpha _ { i } , \\forall i = 1 , \\ldots , k . \\end{align*}"}
{"id": "946.png", "formula": "\\begin{align*} | B | \\leq | B _ { \\ell + k _ \\delta } | \\leq \\gamma | D _ \\ell \\setminus D _ { \\ell + k _ \\delta } | = \\gamma | D _ \\ell | - \\gamma | D _ { \\ell + k _ \\delta } | \\leq \\gamma | D _ \\ell | , \\end{align*}"}
{"id": "8436.png", "formula": "\\begin{align*} \\tilde { Y } _ { k + 1 } = \\tilde { Y } _ { k } - \\eta \\tilde { Y } _ { k } + \\frac { \\eta ^ { 1 / \\alpha } } { \\sigma } \\widetilde { Z } _ { k + 1 } , k = 0 , 1 , 2 , \\dots , \\end{align*}"}
{"id": "749.png", "formula": "\\begin{align*} z _ k ^ { ( \\ell + 1 ) } = \\sum _ { j = 1 } ^ { n _ { \\ell } } W _ { k j } ^ { ( \\ell + 1 ) } \\sigma ( z _ { j } ^ { ( \\ell ) } ) . \\end{align*}"}
{"id": "5318.png", "formula": "\\begin{align*} \\check { \\Delta } ( \\omega ) \\Delta ( a ) ( \\chi \\otimes b ) = \\sum \\Delta ( a ' ) \\check { \\Delta } ( \\omega ' ) ( \\chi \\otimes b ) . \\end{align*}"}
{"id": "9540.png", "formula": "\\begin{align*} f ^ \\infty ( x , u , \\omega ) = V ^ \\infty \\left ( u - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\cdot \\Delta s _ { t + 1 } ( \\omega ) - \\bar c ( \\omega ) \\cdot \\bar x + S _ 0 ^ \\infty ( \\bar x ) , \\omega \\right ) + \\sum _ { t = 0 } ^ { T - 1 } \\delta _ { D _ t ^ \\infty ( \\omega ) } ( x _ t , \\omega ) . \\end{align*}"}
{"id": "339.png", "formula": "\\begin{align*} 0 & = \\ < \\rho , \\dot S + \\nabla _ { \\rho } \\mathcal H ( \\rho , S ) + \\nabla _ { \\rho } \\mathcal H _ 1 ( \\rho , S ) \\dot W ^ { \\delta } \\ > \\\\ & = \\ < \\rho , \\dot S \\ > + \\mathcal H ( \\rho , S ) + \\mathcal H _ 1 ( \\rho , S ) \\dot W ^ { \\delta } \\le 0 , \\end{align*}"}
{"id": "8480.png", "formula": "\\begin{align*} \\frac { q + 1 } { | G | } \\sum _ { \\eta \\in S _ { \\ell } ^ \\wedge } R ( \\alpha \\eta ) ( \\mathbf { d } ( a ) ) & = \\frac { q + 1 } { | G | } \\left ( \\alpha ( a ) \\sum _ { \\eta \\in S _ { \\ell } ^ \\wedge } \\eta ( a ) + \\alpha ( a ^ { - 1 } ) \\sum _ { \\eta \\in S _ { \\ell } ^ \\wedge } \\eta ( a ^ { - 1 } ) \\right ) . \\end{align*}"}
{"id": "3357.png", "formula": "\\begin{align*} & f ( v _ 1 , v _ 1 , v _ 2 ) = 0 , \\\\ & f ( v _ 1 , v _ 2 , v _ 3 ) + f ( v _ 2 , v _ 3 , v _ 1 ) + f ( v _ 3 , v _ 1 , v _ 2 ) = 0 , \\\\ & f ( v _ 1 , v _ 2 , [ u _ 1 , u _ 2 , u _ 3 ] _ T ) + D _ T ( v _ 1 , v _ 2 ) f ( u _ 1 , u _ 2 , u _ 3 ) \\\\ = & f ( [ v _ 1 , v _ 2 , u _ 1 ] _ T , u _ 2 , u _ 3 ) + f ( u _ 1 , [ v _ 1 , v _ 2 , u _ 2 ] _ T , u _ 3 ) + f ( u _ 1 , u _ 2 , [ v _ 1 , v _ 2 , u _ 3 ] _ T ) \\\\ & + \\theta _ T ( u _ 2 , u _ 3 ) f ( v _ 1 , v _ 2 , u _ 1 ) - \\theta _ T ( u _ 1 , u _ 3 ) f ( v _ 1 , v _ 2 , u _ 2 ) + D _ T ( u _ 1 , u _ 2 ) f ( v _ 1 , v _ 2 , u _ 3 ) . \\end{align*}"}
{"id": "6616.png", "formula": "\\begin{align*} \\int _ { ( - \\epsilon ) } & \\int _ { ( \\epsilon / 2 ) } \\left | \\zeta ( 1 + w ) \\widetilde { W } ( 1 - w ) \\mathcal { H } ( z , - w ) \\frac { e ^ { \\delta z } - e ^ { - \\delta z } } { 2 \\delta z } \\right | \\ , | d z | \\ , | d w | \\\\ & \\ll \\int _ { ( - \\epsilon ) } \\int _ { ( \\epsilon / 2 ) } | w | ^ { - 9 8 } | z | ^ { \\varepsilon - 1 } \\min \\left \\{ 1 , \\frac { 1 } { \\delta | z | } \\right \\} \\ , | d z | \\ , | d w | \\ll \\left ( \\frac { 1 } { \\delta } \\right ) ^ { \\varepsilon } . \\end{align*}"}
{"id": "1407.png", "formula": "\\begin{align*} \\hat \\xi ^ b = \\omega ^ b + \\epsilon ( \\dot p ^ b + \\omega ^ b \\times p ^ b ) \\end{align*}"}
{"id": "613.png", "formula": "\\begin{align*} A ( 0 ) \\ = \\ A ( x _ 0 ) , \\ A ( 1 ) = A ( x _ 0 ) , \\ldots , A ( x _ 0 - 1 ) = A ( x _ 0 ) . \\end{align*}"}
{"id": "6663.png", "formula": "\\begin{align*} R _ 1 = J _ { 2 3 } + O \\big ( ( X h k ) ^ { \\varepsilon } k ^ { 1 / 2 } Q ^ { - 9 6 } \\big ) . \\end{align*}"}
{"id": "595.png", "formula": "\\begin{align*} ( x , y ) \\ = \\ x y . \\end{align*}"}
{"id": "2587.png", "formula": "\\begin{align*} \\langle f , \\varphi \\rangle = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\langle M _ \\omega T _ x g , \\varphi \\rangle \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "219.png", "formula": "\\begin{align*} \\mathcal { L } ( f ) ( x ) & = \\frac { 1 } { \\alpha } \\bigg ( - \\langle x ; \\nabla ( f ) ( x ) \\rangle + \\sum _ { k = 1 } ^ d \\partial _ k D ^ { \\alpha - 1 } _ { k } ( f ) ( x ) + \\frac { 1 } { p _ \\alpha ( x ) } \\sum _ { k = 1 } ^ d \\partial _ k R _ k ^ \\alpha ( p _ \\alpha , f ) ( x ) \\\\ & \\quad + \\sum _ { k = 1 } ^ d \\left ( \\frac { \\partial _ k ( p _ \\alpha ) ( x ) } { p _ \\alpha ( x ) } D _ k ^ { \\alpha - 1 } ( f ) ( x ) + \\partial _ k D _ k ^ { \\alpha - 1 } ( f ) ( x ) \\right ) \\bigg ) , \\end{align*}"}
{"id": "8462.png", "formula": "\\begin{align*} Q _ { j } = H _ { i _ j } H _ { i _ { j - 1 } } \\cdots H _ { i _ 1 } , \\ , Q _ 0 = I , \\ , j = 1 , \\ldots , q . \\end{align*}"}
{"id": "5415.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 f ( v _ i ) = 1 2 , \\end{align*}"}
{"id": "8279.png", "formula": "\\begin{align*} \\pi ( M _ p ) = \\sum _ { q \\in Q } \\Big ( \\sum _ { y \\leq q , \\ , g ( y ) = p } \\mu _ { Q } ( y , q ) \\Big ) F _ q . \\end{align*}"}
{"id": "321.png", "formula": "\\begin{align*} & \\min _ { t \\in [ 0 , T ] } \\min _ { i = 1 } ^ N \\rho _ i ( t ) > 0 , a . s . \\\\ & \\max _ { i j \\in E } | S _ i ( t ) - S _ j ( t ) | \\le \\sqrt { \\frac { C _ { K i n } } { \\min \\limits _ { t \\in [ 0 , T ] } \\min \\limits _ { i = 1 } ^ N \\rho _ i ( t ) } } < \\infty , a . s . \\end{align*}"}
{"id": "5447.png", "formula": "\\begin{align*} \\lim _ { t \\to s + } \\int _ \\Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x = \\int _ \\Omega u ^ { - p } _ 0 ( x ) d x . \\end{align*}"}
{"id": "2936.png", "formula": "\\begin{align*} M _ n ( k ) & = \\frac { 1 } { n } \\sum _ { \\mathbf { p } _ k \\in \\mathcal { P } ( d , k ) } \\sum _ { i \\neq j } ^ n \\left \\{ \\prod _ { \\ell = 1 } ^ k I ^ { ( p _ \\ell ) } _ { i , j } - \\left ( \\frac { - 1 } { 6 n } \\right ) ^ { k } \\right \\} . \\end{align*}"}
{"id": "2938.png", "formula": "\\begin{align*} \\tilde N _ { n , \\mu } ( k ) = \\sum \\limits _ { ( \\mathbf { p } _ { k , 1 } , \\mathbf { p } _ { k , 2 } ) \\in \\mathcal { Z } _ k ( \\mu ) } \\tilde N _ { n , \\mathbf { p } _ { k , 1 } } ( k ) \\tilde N _ { n , \\mathbf { p } _ { k , 2 } } ( k ) . \\end{align*}"}
{"id": "3780.png", "formula": "\\begin{align*} r _ l \\big ( \\epsilon ( X , { \\rm i n d } _ { \\mathcal { W } _ E } ^ { \\mathcal { W } _ F } ( \\Pi _ { \\widetilde { \\chi _ 0 } } \\otimes \\Pi _ { \\pi _ E } ) , \\psi _ F ) \\big ) = r _ l \\Big ( \\epsilon \\big ( X , ( \\Pi _ { \\chi _ 0 } \\otimes \\Pi _ { \\pi _ F } ) \\otimes { \\rm i n d } _ { \\mathcal { W } _ E } ^ { \\mathcal { W } _ F } ( 1 _ E ) , \\psi _ F \\big ) \\Big ) = \\Big ( r _ l \\big ( \\epsilon ( X , \\Pi _ { \\chi _ 0 } \\otimes \\Pi _ { \\pi _ F } , \\psi _ F ) \\big ) \\Big ) ^ l . \\end{align*}"}
{"id": "1575.png", "formula": "\\begin{align*} \\frac { c _ k ( \\mu ) L ( \\mu , \\mathbf { f } , \\chi ) } { \\pi ^ { \\beta } } \\mathbf { f } ( g ) = \\sum _ { j = 1 } ^ e \\langle \\mathbf { h } _ j , \\mathbf { f } \\rangle \\cdot \\mathbf { g } _ j ( g ) . \\end{align*}"}
{"id": "1959.png", "formula": "\\begin{align*} \\Lambda ( \\phi ) : = \\phi ( \\mathbf { 1 } ) + \\sum _ { w \\in \\N ^ * } \\phi ( w ) x _ w , \\end{align*}"}
{"id": "6684.png", "formula": "\\begin{align*} L i _ { K , \\mathfrak { s } _ 1 } ( { \\bf z _ 1 } ) L i _ { K , \\mathfrak { s } _ 2 } ( { { \\bf z } _ 2 } ) = \\sum _ { ( { { \\bf v } _ 1 , { \\bf v } _ 2 } ) } L i _ { K , { { \\bf v } _ 1 + { \\bf v } _ 2 } } ( { { \\bf z } _ 3 } ) . \\end{align*}"}
{"id": "6472.png", "formula": "\\begin{align*} \\mu _ k : = \\begin{cases} 0 & ( k = 2 m - 1 ) , \\\\ ( 2 m - 1 ) ! ! : = \\prod _ { j = 1 } ^ m ( 2 j - 1 ) & ( k = 2 m ) , \\end{cases} ( m \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "1756.png", "formula": "\\begin{align*} \\mathcal { A } ^ \\infty _ G ( X , E ) : = \\big ( \\mathcal { C } ( G ) \\hat { \\otimes } \\Psi ^ { - \\infty } ( S , E | _ S ) \\big ) ^ { K \\times K } . \\end{align*}"}
{"id": "8647.png", "formula": "\\begin{align*} a + b = c + d \\end{align*}"}
{"id": "1904.png", "formula": "\\begin{align*} R _ k ( x ) = \\left \\{ 0 \\leq i \\leq N : \\bigg | \\frac { ( N - n ) } { 2 } + x - i \\bigg | < k \\right \\} & \\end{align*}"}
{"id": "2131.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\binom { c _ n k n } l \\frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } = \\frac 1 { l ! } ( \\frac c { 1 - c } ) ^ l ( 1 - c ) ^ { k ( s + 1 ) } . \\end{align*}"}
{"id": "1634.png", "formula": "\\begin{align*} P _ t ( x , A ) : = P _ t \\mathbf { 1 } _ A ( x ) , t > 0 , x \\in M , A \\subset M \\ , \\end{align*}"}
{"id": "4952.png", "formula": "\\begin{align*} z ^ m = ( k ^ + ) ^ m \\cdot ( g _ 0 ^ { - 1 } \\cdot \\phi ( g ^ + ) ^ m \\cdot g _ 0 ) . \\end{align*}"}
{"id": "9169.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x ^ { 1 / 2 } \\\\ p \\nmid d } } \\frac { ( \\lambda ^ 2 _ f ( p ) - 2 ) } { p ^ { 1 + 2 / \\log x } } \\frac { \\log ( x / p ^ 2 ) } { \\log x } = \\sum _ { \\substack { p \\leq x ^ { 1 / 2 } \\\\ p \\nmid d } } \\frac { \\lambda ^ 2 _ f ( p ) - 2 } { p ^ { 1 + 2 / \\log x } } - 2 \\sum _ { \\substack { p \\leq x ^ { 1 / 2 } \\\\ p \\nmid d } } \\frac { ( \\lambda ^ 2 _ f ( p ) - 2 ) } { p ^ { 1 + 2 / \\log x } } \\frac { \\log p } { \\log x } . \\end{align*}"}
{"id": "6956.png", "formula": "\\begin{align*} \\mathcal T _ Q ( \\bar x ) : = \\left \\{ d \\in \\mathbb X \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ d _ k \\} _ { k \\in \\N } \\subset \\mathbb X , \\ , \\exists \\{ t _ k \\} _ { k \\in \\N } \\subset \\R _ + \\colon \\\\ & d _ k \\to d , \\ , t _ k \\searrow 0 , \\ , \\bar x + t _ k d _ k \\in Q \\ , \\forall k \\in \\N \\end{aligned} \\right \\} \\end{align*}"}
{"id": "716.png", "formula": "\\begin{align*} \\partial ^ i f = \\partial ^ i f ( z ) = \\frac { \\partial ^ i } { \\partial z ^ i } f ( z ) . \\end{align*}"}
{"id": "1965.png", "formula": "\\begin{align*} \\Delta _ \\prec ( a _ 1 \\dotsm a _ n ) : = a _ 1 \\dotsm a _ n \\otimes \\ 1 + \\sum _ { 1 \\in S \\subsetneq [ n ] } w _ S \\otimes w _ { J ^ S _ 1 } \\vert \\dotsm \\vert w _ { J ^ S _ m } \\end{align*}"}
{"id": "3131.png", "formula": "\\begin{align*} F _ i = a _ i x _ 1 ^ 2 + b _ i x _ 2 ^ 2 + c _ i x _ 3 ^ 3 + d _ i x _ 4 ^ 4 + e _ i x _ 5 ^ 2 = 0 \\ , . \\end{align*}"}
{"id": "9089.png", "formula": "\\begin{align*} p _ { | \\dot { h } | , | h | } ( \\dot { x } , x ) & = \\sum _ { i = 1 } ^ N p _ { | \\dot { h _ i } | } ( \\dot { x } ) \\\\ [ - 0 . 2 e m ] & \\ ! \\ ! \\times \\underbrace { \\int _ 0 ^ x \\ ! \\ ! \\cdots \\ ! \\ ! \\int _ 0 ^ x } _ { ( N - 1 ) - { \\rm f o l d } } \\ ! \\ ! \\ ! \\ ! p _ { | h _ { 1 } | , \\cdots , | h _ { N } | } ( x _ 1 , \\dots , x _ i = x , \\cdots , x _ N ) \\\\ [ - 0 . 2 e m ] & \\times \\underbrace { d x _ 1 \\cdots d x _ k \\cdots d x _ N } _ { \\substack { ( N - 1 ) - { \\rm f o l d } \\\\ k \\neq i } } , \\end{align*}"}
{"id": "3008.png", "formula": "\\begin{align*} \\begin{aligned} u & = t _ 1 t _ 2 t _ 3 t _ 4 t _ 5 & v & = t _ 1 ^ 5 \\\\ w & = t _ 2 ^ 5 & x & = t _ 3 ^ 5 \\\\ y & = t _ 4 ^ 5 & z & = t _ 5 ^ 5 . \\end{aligned} \\end{align*}"}
{"id": "4032.png", "formula": "\\begin{align*} \\Phi _ { \\lambda ^ h _ k } ( x ) = \\begin{pmatrix} \\xi _ { \\lambda ^ h _ k } ( x ) \\\\ \\eta _ { \\lambda ^ h _ k } ( x ) \\end{pmatrix} , \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "6408.png", "formula": "\\begin{align*} & g _ 1 = ( z _ 1 - z _ { 1 1 } ( p ) ) ( z _ 1 - z _ { 1 2 } ( p ) ) \\dots ( z _ 1 - z _ { 1 \\ , k } ( p ) ) \\ , , \\\\ & g _ 2 = ( z _ 2 - z _ { 2 1 } ( p ) ) ( z _ 2 - z _ { 2 2 } ( p ) ) \\dots ( z _ 2 - z _ { 2 \\ , k } ( p ) ) \\ , , \\\\ & \\vdots \\\\ & g _ { n } = ( z _ { n } - z _ { n \\ , 1 } ( p ) ) ( z _ { n } - z _ { n \\ , 2 } ( p ) ) \\dots ( z _ { n } - z _ { n \\ , k } ( p ) ) \\ , . \\end{align*}"}
{"id": "5966.png", "formula": "\\begin{align*} \\boldsymbol { b } ^ n = \\boldsymbol { b } ^ n + \\mathcal { M } ^ n _ { 1 D } \\boldsymbol { q } , \\boldsymbol { q } = [ q _ 1 , . . . , q _ { N _ { e p } ^ { 1 D } } ] ^ T , \\end{align*}"}
{"id": "3193.png", "formula": "\\begin{align*} \\tau _ { j + 1 } = \\tau _ j + \\nu ( \\tau _ j ) + 1 , \\end{align*}"}
{"id": "8552.png", "formula": "\\begin{align*} m _ { \\pm } ( x , k ) = e ^ { \\mp i k x } \\psi _ { \\pm } ( x , k ) . \\end{align*}"}
{"id": "2571.png", "formula": "\\begin{align*} \\langle \\tfrac { 1 } { x } , f \\rangle = - \\lim _ { \\varepsilon \\to 0 } \\int _ { | x | > \\varepsilon } \\log | x | \\ , \\overline { f ' ( x ) } \\ , d x + ( \\overline { f ( - \\varepsilon ) } - \\overline { f ( \\varepsilon ) } ) \\log ( \\varepsilon ) . \\end{align*}"}
{"id": "2655.png", "formula": "\\begin{align*} \\kappa ( k , l ) = \\kappa ( 0 , l ) + \\kappa ( k , 0 ) + k \\cdot l = \\kappa ( 0 , l ) + \\kappa ( k , 0 ) . \\end{align*}"}
{"id": "5998.png", "formula": "\\begin{align*} | S ( n , k ) | & \\leq \\frac { 1 } { k ! } \\sum _ { l _ { 1 } + \\dots + l _ { k } = n } \\frac { n ! } { l _ { 1 } ! \\dots l _ { k } ! } | g ^ { ( l _ { 1 } ) } ( 0 ) | \\dots | g ^ { ( l _ { k } ) } ( 0 ) | \\\\ & \\leq \\frac { 1 } { k ! } \\sum _ { l _ { 1 } + \\dots + l _ { k } = n } \\frac { n ! l _ { 1 } ! \\dots l _ { k } ! } { l _ { 1 } ! \\dots l _ { k } ! } G _ { \\varepsilon } ^ { k } \\varepsilon ^ { - n } \\\\ & \\leq \\frac { n ! } { k ! } G _ { \\varepsilon } ^ { k } 2 ^ { n } \\varepsilon ^ { - n } . \\end{align*}"}
{"id": "462.png", "formula": "\\begin{align*} \\prod _ { \\iota \\in 2 ^ { V ' } } h _ { x y } ^ \\iota ( c ) = \\sum _ { \\xi \\in \\Xi } \\prod _ { \\iota \\in 2 ^ { V ' } } g ^ { \\iota } ( \\xi ( \\iota ) ) . \\end{align*}"}
{"id": "3775.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( \\widetilde { W _ v } , \\widetilde { U _ v } ) ^ l q _ F ^ { - l r / 2 } X ^ { - l r } = \\omega _ \\sigma ( - 1 ) ^ { n - 2 } \\gamma ( X , \\pi , \\sigma , \\psi ) ^ l \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( W _ v , U _ v ) ^ l q _ F ^ { l r / 2 } X ^ { l r } \\end{align*}"}
{"id": "8461.png", "formula": "\\begin{align*} H _ i = I - 2 \\frac { a _ { i } a _ { i } ^ T } { a _ { i } ^ T a _ { i } } , i = 1 , 2 , \\ldots , m . \\end{align*}"}
{"id": "2865.png", "formula": "\\begin{align*} \\alpha _ + ( t ) = B ( h ( t ) , \\mathcal { Y } _ - ) , \\alpha _ - ( t ) = B ( h ( t ) , \\mathcal { Y } _ + ) , \\end{align*}"}
{"id": "3165.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal w _ k \\frac { w _ k ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal w _ k } _ 2 ^ 2 } . \\end{align*}"}
{"id": "9297.png", "formula": "\\begin{align*} \\dim _ K H ^ 2 _ { \\mathrm { c r i s } } ( Z / W ) _ 0 = 1 . \\end{align*}"}
{"id": "2677.png", "formula": "\\begin{align*} \\mathfrak { F } _ { ( \\alpha , \\beta ) } ( e ^ { - | t | } ) = \\{ \\alpha \\Z \\times \\beta \\Z \\mid \\alpha \\beta < 1 \\} \\end{align*}"}
{"id": "4187.png", "formula": "\\begin{align*} k \\rho \\mu ( u ) = \\rho \\mu ( u ^ k ) = \\omega ( u ^ k ) = k \\omega ( u ) , \\end{align*}"}
{"id": "7599.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n _ x - l _ x } { \\vphantom { \\sum } } ' \\sum _ { j = n _ y - l _ y + 1 } ^ { \\infty } \\left | c _ { i , j } \\right | \\le ( n _ x - l _ x + 1 ) \\dfrac { 4 V _ l } { 2 l \\pi ^ 2 } \\begin{cases} \\Pi _ { 1 } [ r ] ( n _ y - l _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } l = 2 r , \\\\ \\Pi _ { 0 } [ r ] ( n _ y - l _ y ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } l = 2 r + 1 , \\end{cases} \\end{align*}"}
{"id": "8334.png", "formula": "\\begin{align*} \\int _ 0 ^ T ( z - \\hat { \\eta } _ i ) \\dd ( \\hat { \\eta } _ i - h _ i ) & = \\int _ 0 ^ t ( z - \\hat { \\eta } _ i ) \\dd ( \\hat { \\eta } _ i - h _ i ) + \\int _ t ^ T ( z - \\hat { \\eta } _ i ) \\dd ( \\hat { \\eta } _ i - h _ i ) \\\\ & = \\int _ 0 ^ t ( z - \\hat { \\eta } _ i ) \\dd ( \\hat { \\eta } _ i - h _ i ) \\\\ & = \\int _ 0 ^ t ( z - \\eta _ i ) \\dd ( \\eta _ i - h _ i ) \\ge 0 , \\end{align*}"}
{"id": "7326.png", "formula": "\\begin{align*} & h - k = \\frac { \\lambda } { ( T - \\tilde { t } ) ^ 2 } , \\ p = \\frac { 4 | \\tilde { x } - \\tilde { y } | ^ 2 ( \\tilde { x } - \\tilde { y } ) } { \\varepsilon ^ 4 } , \\\\ & \\begin{pmatrix} X & 0 \\\\ 0 & - Y \\end{pmatrix} \\le \\begin{pmatrix} Z & - Z \\\\ - Z & Z \\end{pmatrix} + \\rho \\begin{pmatrix} Z & - Z \\\\ - Z & Z \\end{pmatrix} ^ 2 , \\end{align*}"}
{"id": "4971.png", "formula": "\\begin{align*} f ^ { \\omega } = \\sum _ { j \\in \\N } g _ { j } ( \\omega ) \\Box _ j f . \\end{align*}"}
{"id": "3058.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 3 + & x _ 2 ^ 3 y _ 2 ^ 3 + a x _ 1 y _ 1 x _ 2 y _ 2 ( x _ 1 y _ 1 + x _ 2 y _ 2 ) + b y _ 1 y _ 2 ( x _ 1 ^ 3 y _ 2 + x _ 2 ^ 3 y _ 1 ) + \\\\ + & c x _ 1 x _ 2 ( x _ 1 y _ 2 ^ 3 + x _ 2 y _ 1 ^ 3 ) = 0 \\ , . \\end{align*}"}
{"id": "4946.png", "formula": "\\begin{align*} ( g \\dot { \\gamma } ^ { - 1 } ) ^ { - 1 } b \\phi ( g \\dot { \\gamma } ^ { - 1 } ) = \\dot { \\gamma } g ^ { - 1 } b \\phi ( g ) \\phi ( \\dot { \\gamma } ^ { - 1 } ) = \\dot { \\gamma } k _ 1 ^ o \\omega k \\phi ( \\dot { \\gamma } ^ { - 1 } ) = \\end{align*}"}
{"id": "2817.png", "formula": "\\begin{align*} | \\dot { X } _ j | = O ( \\delta + \\delta \\delta ^ * ) | \\dot { \\alpha } | = O ( \\delta + \\delta \\delta ^ * ) . \\end{align*}"}
{"id": "935.png", "formula": "\\begin{align*} \\sum _ { l = m } ^ { k } S _ { 2 , \\lambda } ( k + 1 , l + 1 ) S _ { 1 , \\lambda } ( l , m ) & = \\sum _ { l = m } ^ { k } \\sum _ { p = l } ^ { k } \\binom { k } { p } ( 1 - \\lambda ) _ { k - p , \\lambda } S _ { 2 , \\lambda } ( p , l ) S _ { 1 , \\lambda } ( l , m ) . \\\\ & = \\sum _ { p = m } ^ { k } \\binom { k } { p } ( 1 - \\lambda ) _ { k - p , \\lambda } \\sum _ { l = m } ^ { p } S _ { 2 , \\lambda } ( p , l ) S _ { 1 , \\lambda } ( l , m ) \\\\ & = \\binom { k } { m } ( 1 - \\lambda ) _ { k - m , \\lambda } . \\end{align*}"}
{"id": "704.png", "formula": "\\begin{align*} \\mathbb P ( S _ n ^ c ) = O ( n ^ { - \\infty } ) . \\end{align*}"}
{"id": "6198.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { l } \\bar { \\sigma } ^ 2 _ { t } = \\sum _ { t = 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } - \\sum _ { t = l + 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } \\\\ > \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } - 2 \\sqrt { k } \\theta \\| S \\| ^ 2 _ F - ( k - l ) \\alpha \\| W \\| _ F ^ 2 . \\end{align*}"}
{"id": "5490.png", "formula": "\\begin{align*} \\frac { 1 } { t } d _ K ( a + t \\beta a ) = \\frac { 1 } { t } | ( a + t \\beta a ) - a | = | \\beta | a . \\end{align*}"}
{"id": "8568.png", "formula": "\\begin{align*} \\chi _ + ( x ) = \\int _ { - \\infty } ^ { x } \\Phi ( y ) \\ , d y , \\chi _ + ( x ) + \\chi _ - ( x ) = 1 . \\end{align*}"}
{"id": "7510.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) & = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { T } \\left [ \\log \\left | - \\frac { 1 } { 2 } - \\epsilon + i t \\right | - \\log \\left | - \\frac { 1 } { 2 } + \\epsilon + i t \\right | \\right ] \\ d t + \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\left [ \\arg \\left ( \\sigma - 1 + i T \\right ) - \\arg ( \\sigma - 1 ) \\right ] \\ d \\sigma \\end{align*}"}
{"id": "6827.png", "formula": "\\begin{align*} - \\triangle _ { g _ { \\textnormal { p o i n } , j } } ( u _ { j } ) = e ^ { 2 u _ { j } } \\ , K _ { \\tilde F _ { t _ { j } } ^ * ( g _ { \\textnormal { e u c } } ) } \\textnormal { o n } \\ , \\ , \\ , \\Sigma \\end{align*}"}
{"id": "4899.png", "formula": "\\begin{align*} L ( z ) = \\frac { f ' ( z ) } { f ( z ) } \\sim c z ^ { - n } , \\frac { f '' ( z ) } { f ( z ) } = L ' ( z ) + L ( z ) ^ 2 \\sim - n c z ^ { - n - 1 } , F ( z ) \\sim - \\frac { z ^ n } { c } . \\end{align*}"}
{"id": "3605.png", "formula": "\\begin{align*} d ( a _ { 1 , 2 } b _ { 1 , 2 } ) \\ge d [ a _ { 1 , 2 } b _ { 1 , 2 } ] \\ge 2 e - 1 = \\alpha _ { 2 } \\end{align*}"}
{"id": "5504.png", "formula": "\\begin{align*} d _ K ( \\xi ( t ; s , x ) ) = d _ K ( \\eta _ s ( t - s ; x ) ) \\leq \\Phi _ { \\beta + L , 0 } ( d _ K ( x ) , \\epsilon , t - s ) . \\end{align*}"}
{"id": "1982.png", "formula": "\\begin{align*} \\widehat { \\Phi ^ { - 1 } } ( x ) = \\frac { 1 } { 1 + \\widehat { \\kappa } ( x ) } . \\end{align*}"}
{"id": "7093.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } s _ k & = \\sum _ { k = 1 } ^ { p - 1 } \\binom { p - 1 } { k } f ( x , v ) ^ k f ( y , v ) ^ { p - 1 - k } y - \\sum _ { k = 1 } ^ { p - 1 } \\binom { p - 1 } { k } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k } x \\\\ & = \\sum _ { k = 1 } ^ { p - 1 } \\binom { p - 1 } { k } f ( x , v ) ^ k f ( y , v ) ^ { p - 1 - k } y + \\sum _ { k = 1 } ^ { p - 1 } \\binom { p - 1 } { k - 1 } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k } x . \\end{align*}"}
{"id": "8135.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 6 } ( H G ) } c ( g ( \\gamma ) ) = 2 \\cdot 7 + 7 = 2 1 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "338.png", "formula": "\\begin{align*} & \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > \\\\ = & \\int _ 0 ^ 1 ( \\ < m , \\nabla _ G S \\ > + \\ < \\rho , \\dot S \\ > ) d t + \\int _ 0 ^ 1 \\ < \\nabla _ G \\Sigma , \\nabla _ G S \\ > _ { \\theta ( \\rho ) } d W ^ { \\delta } ( t ) \\\\ \\le & \\mathcal A ( \\rho , m ) + \\int _ 0 ^ 1 \\Big ( \\ < \\rho , \\dot S \\ > + \\frac 1 2 \\| \\nabla _ G S \\| _ { \\theta ( \\rho ) } ^ 2 + \\ < \\nabla _ G \\Sigma , \\nabla _ G S \\ > _ { \\theta ( \\rho ) } \\dot { W ^ { \\delta } ( t ) } \\Big ) d t \\le \\mathcal A ( \\rho , m ) . \\end{align*}"}
{"id": "6626.png", "formula": "\\begin{align*} J _ { 3 } = J _ { 3 1 } + J _ { 3 2 } + J _ { 3 3 } . \\end{align*}"}
{"id": "1276.png", "formula": "\\begin{align*} c _ { \\sigma } ( Y , \\lambda ) = \\inf \\{ L > 0 | \\ , \\sigma \\in \\mathrm { I m } ( i _ { L } : \\mathrm { E C H } ^ { L } ( Y , \\lambda , \\Gamma ) \\to \\mathrm { E C H } ( Y , \\lambda , \\Gamma ) ) \\ , \\} \\end{align*}"}
{"id": "4792.png", "formula": "\\begin{align*} ( S ^ 2 ) _ { a , b } = \\sum _ c \\phi ( a ^ { - 1 } c ) \\phi ( c ^ { - 1 } b ) = \\sum _ g \\phi ( g ) \\phi ( g ^ { - 1 } a ^ { - 1 } b ) = ( \\phi * \\phi ) ( a ^ { - 1 } b ) = \\Delta ( \\phi * \\phi ) ( a ^ { - 1 } , b ) . \\end{align*}"}
{"id": "8034.png", "formula": "\\begin{align*} \\left \\langle E , ( \\partial _ { \\Sigma _ 0 , \\epsilon } ^ { * } F ) ^ { ( 1 ) } [ \\phi ] \\otimes ( \\partial _ { \\Sigma _ 0 , \\epsilon } ^ { * } G ) ^ { ( 1 ) } [ \\phi ] \\right \\rangle = \\left \\langle E _ { \\Sigma _ 0 } , F ^ { ( 1 ) } [ \\partial _ { \\Sigma _ 0 , \\epsilon } \\phi ] \\otimes G ^ { ( 1 ) } [ \\partial _ { \\Sigma _ 0 , \\epsilon } \\phi ] \\right \\rangle . \\end{align*}"}
{"id": "6851.png", "formula": "\\begin{align*} Y = Z _ { 2 h ( Y ) } \\circ \\cdots \\circ Z _ { 1 } , \\end{align*}"}
{"id": "6003.png", "formula": "\\begin{align*} ( f g ) ^ { ( n ) } = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } f ^ { ( n - k ) } g ^ { ( k ) } , \\end{align*}"}
{"id": "9489.png", "formula": "\\begin{align*} \\displaystyle | \\mathcal { S C } _ { ( s , s + d , s + 2 d ) } | = \\begin{cases} & \\displaystyle \\sum _ { i = 0 } ^ { \\lfloor s / 4 \\rfloor } \\binom { ( s + d - 1 ) / 2 } { i , d / 2 + i , ( s - 1 ) / 2 - 2 i } \\\\ & \\\\ & \\displaystyle \\sum _ { i = 0 } ^ { \\lfloor s / 2 \\rfloor } \\binom { \\lfloor ( s + d - 1 ) / 2 \\rfloor } { \\lfloor i / 2 \\rfloor , \\lfloor ( d + i ) / 2 \\rfloor , \\lfloor s / 2 \\rfloor - i } \\end{cases} \\end{align*}"}
{"id": "5659.png", "formula": "\\begin{align*} \\phi \\psi & = ( \\sigma \\tau , ( \\phi _ { \\tau ^ { - 1 } 1 } \\psi _ 1 , \\phi _ { \\tau ^ { - 1 } 2 } , \\phi _ { \\tau ^ { - 1 } 3 } \\psi _ 3 ) ) \\\\ & = ( \\sigma \\tau , ( \\phi _ { \\tau ^ { - 1 } 1 } \\psi _ 1 , \\varepsilon , \\phi _ { \\tau ^ { - 1 } 3 } \\psi _ 3 ) ) \\in \\prescript { \\sigma \\tau } { } N _ l ^ A \\end{align*}"}
{"id": "3950.png", "formula": "\\begin{align*} \\norm { \\phi } _ { H ^ s _ { { p e r } } ( I ) } = \\left ( \\sum _ { m \\in \\mathbb { Z } } \\left ( 1 + \\frac { 4 \\pi ^ 2 m ^ { 2 } } { \\mod { I } ^ 2 } \\right ) ^ s \\mod { c _ m } ^ 2 \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "8287.png", "formula": "\\begin{align*} & 2 ^ { \\abs { V } } q ^ { \\sum _ { v \\in V } ( w _ { v } + 3 ) / 4 } \\widehat { Z } _ { \\Gamma } ( q ) \\\\ = \\ , & \\sum _ { l \\in 2 \\Z ^ V + \\delta } q ^ { - { } ^ t \\ ! l W ^ { - 1 } l / 4 } \\prod _ { v \\in V } \\mathrm { v . p . } \\int _ { \\abs { z _ v } = 1 } \\frac { z _ v ^ { l _ v - 1 } } { ( z _ v - 1 / z _ v ) ^ { \\deg ( v ) - 2 } } \\frac { d z _ v } { 2 \\pi \\sqrt { - 1 } } . \\end{align*}"}
{"id": "4061.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( \\tilde { \\rho } , \\tilde { u } ) \\| _ { L ^ 2 ( L ^ 2 ) \\times L ^ 2 ( L ^ 2 ) } & = \\| \\Lambda _ 1 ^ * ( p ) \\| _ { L ^ 2 ( L ^ 2 ) \\times L ^ 2 ( L ^ 2 ) } \\\\ & \\leq \\| \\Lambda _ 1 ^ * \\| \\ \\| p \\| _ { L ^ 2 ( 0 , T ) } . \\end{aligned} \\end{align*}"}
{"id": "8238.png", "formula": "\\begin{align*} [ z ^ n ] & \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } ( N _ 1 + N _ 2 ) | _ { x _ 1 = x _ 2 = 1 } \\\\ & = \\frac { 1 } { n - 1 } [ t ^ n ] 3 ( n - 1 ) ( 3 n - 4 ) t ^ 2 ( 1 - t ) ^ { - 4 ( n - 1 ) } - \\frac { 1 } { 2 n - 1 } [ t ^ n ] ( 3 n - 2 ) ( 3 n - 3 ) t ^ 2 ( 1 - t ) ^ { - ( 4 n - 2 ) } \\\\ & = 3 ( 3 n - 4 ) [ t ^ { n - 2 } ] ( 1 - t ) ^ { - 4 ( n - 1 ) } - \\frac { ( 3 n - 2 ) ( 3 n - 3 ) } { 2 n - 1 } [ t ^ { n - 2 } ] ( 1 - t ) ^ { - ( 4 n - 2 ) } \\\\ & = 3 ( 3 n - 4 ) \\binom { 5 n - 7 } { n - 2 } - \\frac { ( 3 n - 2 ) ( 3 n - 3 ) } { 2 n - 1 } \\binom { 5 n - 5 } { n - 2 } . \\end{align*}"}
{"id": "9059.png", "formula": "\\begin{align*} \\mathcal { H } ( \\rho , m ) = & \\frac { h } { 2 \\tau } \\sum _ { i = 1 } ^ s \\sum _ { j = 1 } ^ N \\frac { \\hat m ^ 2 _ { i , j } } { \\rho _ { i , j } } D ^ { - 1 } _ { i , j } + h \\sum _ { i = 1 } ^ s \\sum _ { j = 1 } ^ { N } \\rho _ { i , j } \\log \\rho _ { i , j } , \\\\ \\mathcal { I } ( \\phi ) = & \\frac { 1 } { 8 h } \\sum _ { j = 1 } ^ { N } \\epsilon _ j ( \\phi _ { j + 1 } - \\phi _ { j - 1 } ) ^ 2 + \\frac { 1 } { 8 \\beta _ a } ( \\phi _ 0 + \\phi _ 1 ) ^ 2 + \\frac { 1 } { 8 \\beta _ b } ( \\phi _ N + \\phi _ { N + 1 } ) ^ 2 . \\end{align*}"}
{"id": "1183.png", "formula": "\\begin{align*} \\Psi ^ { A i } ( \\zeta ) \\sim \\zeta ^ { - \\frac { \\sigma _ 3 } { 4 } } \\Psi ^ { A i } _ { 0 } \\left ( I + \\sum _ { j = 1 } ^ { \\infty } \\frac { \\mathcal { K } _ j } { \\zeta ^ { 3 j / 2 } } \\right ) , \\end{align*}"}
{"id": "267.png", "formula": "\\begin{align*} & \\widetilde { f } ( \\widetilde { t } ) \\geqq \\widetilde { f } ( 0 ) = f ( \\alpha ) = 0 , \\\\ & \\widetilde { f } ( \\widetilde { t } ) \\geqq { A _ { 0 } \\widetilde { t } } . \\end{align*}"}
{"id": "9183.png", "formula": "\\begin{align*} \\sum _ { p \\in \\mathcal { S } ( j ) } ( \\log p ) \\left | L ( \\tfrac { 1 } { 2 } , f \\otimes \\chi _ { 8 p } ) \\right | ^ 2 \\left | \\mathcal { M } ( p ) \\right | ^ 2 \\ll X \\exp \\Big ( \\frac { 4 } { \\alpha _ j } \\Big ) ( \\mathcal { I } - j ) e ^ { - 1 2 2 / \\alpha _ { j + 1 } } \\ll ( \\mathcal { I } - j ) e ^ { - 4 2 / \\alpha _ { j + 1 } } X \\ll e ^ { - 1 / ( 1 0 \\alpha _ { j } ) } X . \\end{align*}"}
{"id": "5323.png", "formula": "\\begin{align*} \\mathrm { d i v } _ x \\mathbf { S } ( u ) = \\mu ^ S \\Delta _ x u + \\Big ( \\mu ^ B + \\frac { \\mu ^ S } { 3 } \\Big ) \\nabla _ x \\mathrm { d i v } _ x u = \\mu _ 1 \\Delta _ x u + \\mu _ 2 \\nabla _ x \\mathrm { d i v } _ x u , \\end{align*}"}
{"id": "1955.png", "formula": "\\begin{align*} ( f \\bullet g ) \\bullet h ( x ) = h ( x ) ( f \\bullet g ) ( x h ( x ) ) = h ( x ) g ( x h ( x ) ) f ( x h ( x ) g ( x h ( x ) ) ) . \\end{align*}"}
{"id": "7186.png", "formula": "\\begin{align*} E ( t , x ) = - ( \\nabla \\phi * _ x \\rho [ f ( t , \\cdot ) ] ) ( x ) . \\end{align*}"}
{"id": "2233.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { \\mathrm { d } } { \\mathrm { d } t } ( X _ 1 - X _ 2 ) + A \\big ( A ( X _ 1 - X _ 2 ) + F ( X _ 1 ) - F ( X _ 2 ) \\big ) = 0 , & \\\\ ( X _ 1 - X _ 2 ) ( 0 ) = 0 . & \\end{array} \\right . \\end{align*}"}
{"id": "7458.png", "formula": "\\begin{align*} \\mathcal { F } ( t ) = E ( \\phi ( \\cdot , t ) ) + \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 . \\end{align*}"}
{"id": "6631.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq n < m < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( - \\frac { 1 } { 2 } + s _ 1 + w - z ) } p ^ { n ( \\frac { 3 } { 2 } + s _ 2 - w + z ) } } = \\frac { \\tau _ A ( p ) } { p ^ { - \\frac { 1 } { 2 } + s _ 1 + w - z } } + O \\bigg ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\bigg ) . \\end{align*}"}
{"id": "3611.png", "formula": "\\begin{align*} d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] = \\min \\{ d ( - a _ { 1 , n + 2 } b _ { 1 , n } ) , \\alpha _ { n + 2 } \\} > G _ { n } \\ , . \\end{align*}"}
{"id": "454.png", "formula": "\\begin{align*} \\mu _ d ( \\phi ( x , b ) ) = \\nu ( \\phi ) \\mu _ d ( c l _ B ( \\phi ( x , b ) ) ) . \\end{align*}"}
{"id": "8624.png", "formula": "\\begin{align*} \\mathcal { K } _ + ( x , k ) & = \\begin{cases} T ( k ) e ^ { i k x } & k \\geq 0 \\\\ e ^ { i k x } + R _ + ( - k ) e ^ { - i k x } & k < 0 , \\end{cases} \\\\ \\mathcal { K } _ - ( x , k ) & = \\begin{cases} e ^ { i k x } + R _ - ( k ) e ^ { - i k x } & k \\geq 0 \\\\ T ( - k ) e ^ { i k x } & k < 0 , \\end{cases} \\end{align*}"}
{"id": "9496.png", "formula": "\\begin{align*} \\mathfrak { b c } = \\mathfrak { c s } = \\sum _ { i = 0 } ^ { ( s - 1 ) / 2 } \\binom { ( d - 1 ) / 2 + i } { \\lfloor i / 2 \\rfloor } \\left ( \\binom { ( s + d - 2 ) / 2 } { ( d - 1 ) / 2 + i } + \\binom { ( s + d - 4 ) / 2 } { ( d - 1 ) / 2 + i } \\right ) . \\end{align*}"}
{"id": "9071.png", "formula": "\\begin{align*} \\min _ u \\mathcal { F } _ h ( u ) , s . t . A u = b , S u \\geq \\delta , \\end{align*}"}
{"id": "4467.png", "formula": "\\begin{align*} ( a ^ { \\dagger } a ) _ { k , \\lambda } | m \\rangle = ( a ^ { \\dagger } a ) ( a ^ { \\dagger } a - \\lambda ) \\cdots ( a ^ { \\dagger } a - ( k - 1 ) \\lambda | m \\rangle = ( m ) _ { k , \\lambda } | m \\rangle , \\end{align*}"}
{"id": "2214.png", "formula": "\\begin{align*} \\ , \\dd Z ( t ) + A ^ 2 Z ( t ) \\ , \\dd t = \\dd W ( t ) , & \\ ; Z ( 0 ) = ( I - P ) X _ 0 , \\end{align*}"}
{"id": "1014.png", "formula": "\\begin{align*} U : = \\bigg \\{ x \\in B _ \\rho ^ + \\frac { u ( x ) } { \\zeta ( x ) } > \\frac { \\partial _ 1 u ( a ) } { 2 \\partial _ 1 \\zeta ( a ) } \\bigg \\} . \\end{align*}"}
{"id": "1801.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { c _ Y } ( A ) = { } ^ b { \\rm T r } _ { Y } \\left ( \\Phi _ A ( e ) \\right ) . \\end{align*}"}
{"id": "6672.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) = \\mathcal { I } _ 0 ( h , k ) + \\mathcal { I } _ 1 ( h , k ) + \\mathcal { E } ( h , k ) , \\end{align*}"}
{"id": "1825.png", "formula": "\\begin{align*} 2 ^ n E _ n ( u , v ) = A _ n ( x , y ) , \\end{align*}"}
{"id": "2510.png", "formula": "\\begin{align*} \\rho ( \\mathbf { h } ^ { - 1 } ) = \\rho ( - x , - \\omega - \\tau ) = e ^ { - 2 \\pi i \\tau } M _ { - \\omega / 2 } T _ { - x } M _ { - \\omega / 2 } = \\rho ( \\mathbf { h } ) ^ * . \\end{align*}"}
{"id": "4365.png", "formula": "\\begin{align*} b ^ { - \\frac { 1 } { 2 } } \\partial _ \\xi \\phi _ { i , i n t , \\beta } \\left ( \\frac { y _ 0 } { \\sqrt { b } } \\right ) \\phi _ { i , o u t , \\beta } ( y _ 0 ) - \\phi _ { i , i n t , \\beta } \\left ( \\frac { y _ 0 } { \\sqrt { b } } \\right ) \\partial _ y \\phi _ { i , o u t , \\beta } ( y _ 0 ) = 0 \\end{align*}"}
{"id": "1175.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\psi _ { 1 1 } } { d \\eta ^ 2 } + \\left ( \\frac { 1 } { 2 } - \\frac { \\eta ^ 2 } { 4 } + i \\nu \\right ) \\psi _ { 1 1 } = 0 . \\end{align*}"}
{"id": "7737.png", "formula": "\\begin{align*} \\int _ { \\R } | \\phi _ x | ^ 2 ( t , x _ 1 ) \\psi ( t ) \\ , d t - \\int _ { \\R } | \\phi _ x | ^ 2 ( t , \\bar x ) \\psi ( t ) \\ , d t & = 2 \\int _ { \\bar x } ^ { x _ 1 } \\int _ { \\R } ( - \\phi _ { t } \\cdot \\phi _ x \\psi ' ( t ) + a ( x ) \\phi _ t \\cdot \\phi _ x \\psi ( t ) ) \\ , d t d x \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; + \\int _ { \\R } | \\phi _ t | ^ 2 ( t , \\bar x ) \\psi ( t ) \\ , d t - \\int _ { \\R } | \\phi _ t | ^ 2 ( t , x _ 1 ) \\psi ( t ) \\ , d t . \\end{align*}"}
{"id": "5055.png", "formula": "\\begin{align*} \\Psi ^ 0 _ { \\alpha , \\nu , \\tilde \\nu } : = \\mathbf { L } _ { - \\nu } ^ 0 \\tilde { \\mathbf { L } } _ { - \\tilde \\nu } ^ 0 \\ : \\Psi ^ 0 _ \\alpha , \\end{align*}"}
{"id": "923.png", "formula": "\\begin{align*} ( \\hat { n } ) _ { k } = ( a ^ { + } ) ^ { k } a ^ { k } & = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k , m ) ( a ^ { + } a ) _ { m , \\lambda } = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k , m ) ( \\hat { n } ) _ { m , \\lambda } . \\end{align*}"}
{"id": "2369.png", "formula": "\\begin{align*} | V _ g \\left ( M _ \\eta T _ \\xi f \\right ) ( x , \\omega ) | = | V _ g f ( x - \\xi , \\omega - \\eta ) | . \\end{align*}"}
{"id": "5400.png", "formula": "\\begin{align*} e ^ { - m _ i ( x ) s + f _ { m _ i ( x ) } ( x ) } = \\mu _ i ( \\overline { B } ( x , \\gamma _ i ) ) \\leq \\hat { \\mu } ( \\overline { B } ( x , \\gamma _ i ) ) \\leq C \\mu _ i ( \\overline { B } ( x , \\gamma _ i ) ) = C e ^ { - m _ i ( x ) s + f _ { m _ i ( x ) } ( x ) } , ~ ~ \\forall x \\in K _ i . \\end{align*}"}
{"id": "9528.png", "formula": "\\begin{align*} f ^ * ( v , y , \\omega ) & = \\sup \\{ x \\cdot v - l ( x , y , \\omega ) \\} \\\\ & = \\sum _ { t = 0 } ^ T K _ t ^ * ( v _ { t } + \\Delta y _ { t + 1 } , y _ t , \\omega ) . \\end{align*}"}
{"id": "6351.png", "formula": "\\begin{align*} v = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { r } a _ j \\ ; \\rho _ { i _ j } \\end{align*}"}
{"id": "7750.png", "formula": "\\begin{align*} S _ 0 : = \\left ( \\frac { 1 } { 2 \\widetilde C } \\right ) ^ 2 . \\end{align*}"}
{"id": "8618.png", "formula": "\\begin{align*} A _ { 1 , 1 } ( k ) & = \\int _ 0 ^ t s \\ , \\phi _ { 2 } ( k ) e ^ { - i s k ^ 2 } \\int \\overline { \\mathcal { K } _ { \\# , R } ' ( x , k ) } \\partial _ x \\big [ u _ { M _ { 1 } ( s ) } \\overline { u _ { M _ { 2 } } ( s ) } u _ { M _ { 3 } } ( s ) \\big ] \\ , d x \\ , d s . \\end{align*}"}
{"id": "8310.png", "formula": "\\begin{align*} d ( D , { \\sf S N R } ) = \\sqrt { \\eta \\lambda D / N } , \\end{align*}"}
{"id": "6864.png", "formula": "\\begin{align*} \\norm { Y f _ \\ell - \\prod \\limits _ { s = 1 } ^ j \\left ( 1 - \\delta _ { k - 2 s + i _ s - \\ell } ^ { k - 2 s + i _ s } \\right ) f _ \\ell } \\leq \\norm { g _ \\ell } \\sum \\limits _ { s = 1 } ^ j \\gamma ^ { k - 2 s + i _ s } _ { k - 2 s + i _ s - \\ell } \\prod \\limits _ { t = 1 } ^ { s - 1 } \\left ( 1 - \\delta _ { k - 2 t + i _ t - \\ell } ^ { k - 2 t + i _ t } \\right ) \\leq ( j + k ) j \\gamma \\norm { g _ \\ell } . \\end{align*}"}
{"id": "5955.png", "formula": "\\begin{align*} n _ { k } = \\boldsymbol { n } , k = \\{ 1 , 3 \\} , n _ { k } = \\boldsymbol { r } \\times \\boldsymbol { n } , k = 5 , \\Gamma ^ { b o d y } , \\end{align*}"}
{"id": "4107.png", "formula": "\\begin{align*} & 2 x \\cdot \\mu _ n ( x ) + ( p _ 0 - 1 ) \\cdot ( \\sigma _ n ( x ) ) ^ 2 \\\\ & \\qquad = \\frac { 2 x \\cdot \\mu ( x ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } } + \\frac { ( p _ 0 - 1 ) \\cdot \\sigma ^ 2 ( x ) } { ( 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } ) ^ 2 } \\le \\frac { 2 x \\cdot \\mu ( x ) + ( p _ 0 - 1 ) \\cdot \\sigma ^ 2 ( x ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } } \\end{align*}"}
{"id": "3176.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } \\frac { e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } } _ 2 ^ 2 } , \\end{align*}"}
{"id": "1068.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} 0 & q \\\\ q & 0 \\end{pmatrix} , V = 4 k Q + 2 i k \\sigma _ 3 ( Q _ x - Q ^ 2 ) + 2 Q ^ 3 - Q _ { x x } . \\end{align*}"}
{"id": "1719.png", "formula": "\\begin{align*} 2 ^ { \u2010 \\alpha _ * k _ * l } \\cdot 2 ^ { \u2010 m _ l ( 1 / q \u2010 1 / p _ 0 ) } = 2 ^ { \\mu _ * k _ * l } \\cdot 2 ^ { \u2010 m _ l ( s _ * + 1 / q \u2010 1 / p _ 1 ) } . \\end{align*}"}
{"id": "262.png", "formula": "\\begin{align*} x _ { i } ( q _ { i } ) : = - \\log { h _ { i } } ( q _ { i } , q _ { i } ) . \\end{align*}"}
{"id": "3947.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = h ( t ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "4699.png", "formula": "\\begin{align*} & \\int _ { | y | < \\frac { 1 } { 2 | x _ i - x _ k | } } Q ' ( \\tau _ i ^ { - 1 } \\tau _ k Q ) ^ { p - 1 } A _ { i j , 0 } = \\int _ { | y | < \\frac { 1 } { 2 | x _ i - x _ k | } } Q ' A _ { i j } \\bigg ( \\frac { \\kappa _ 0 } { [ y - ( x _ k - x _ i ) ] ^ 2 } \\bigg ) ^ { p - 1 } + O \\bigg ( \\frac { 1 } { d ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "9374.png", "formula": "\\begin{align*} \\phi _ { n , \\lambda } ( x ) = e ^ { - x } \\bigg ( x \\frac { d } { d x } \\bigg ) _ { n , \\lambda } e ^ { x } = \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) x ^ { k } , ( n \\ge 1 ) . \\end{align*}"}
{"id": "4241.png", "formula": "\\begin{align*} H = u _ { e e } = u _ { x x } + \\tau ^ 2 u _ { y y } + 2 \\tau u _ { x y } . \\end{align*}"}
{"id": "5469.png", "formula": "\\begin{align*} u ( t _ n , x ; s , u _ n , a , b ) & = u ( t _ n , x ; t _ n - 1 , u ( t _ n - 1 , \\cdot ; s _ n , u _ n , a , b ) , a , b ) \\\\ & = u ( 1 , x ; 0 , u ( t _ n - 1 , \\cdot ; s _ n , u _ n , a , b ) , a _ n , b _ n ) . \\end{align*}"}
{"id": "180.png", "formula": "\\begin{align*} \\mathcal { A } ( G ( u ) ) = u . \\end{align*}"}
{"id": "8365.png", "formula": "\\begin{align*} u ^ { \\frac { l } { s + n h + 1 } } u = u ^ { \\frac { l + 1 } { s + n h + 1 } } u ^ { \\frac { s + n h } { s + n h + 1 } } \\in u ^ { \\frac { l + 1 } { s + n h + 1 } } \\cdot \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } B . \\end{align*}"}
{"id": "8669.png", "formula": "\\begin{align*} W _ s ( x ) = V _ s ( x ) + \\sum _ { j = 1 } ^ s V _ { j - 1 } ( x ) W _ s ^ { ( j ) } ( x ^ { p ^ { e _ { 1 } + \\dots + e _ j } } ) , \\end{align*}"}
{"id": "4846.png", "formula": "\\begin{align*} E ^ \\alpha ( \\rho ) & = \\frac 1 2 \\int _ \\R \\int _ \\R K _ \\alpha ( x - y ) \\ , d \\rho ( y ) d \\rho ( x ) + \\int _ \\R Q _ \\alpha ( x ) \\ , d \\rho ( x ) , \\end{align*}"}
{"id": "3267.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } - i t \\cdot \\omega _ 2 ( i t ) & = \\lim _ { t \\rightarrow 0 ^ + } \\frac { t | \\lambda | ^ 2 } { s ( | \\lambda | , t ) - t } \\\\ & = \\lim _ { t \\rightarrow 0 ^ + } \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , t ) / t - 1 } \\\\ & = \\lambda _ 1 ( \\mu ) ^ 2 - | \\lambda | ^ 2 . \\end{align*}"}
{"id": "4064.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( \\check { \\rho } , \\check { u } ) \\| _ { L ^ 2 ( L ^ 2 ) \\times L ^ 2 ( L ^ 2 ) } & = \\| \\Lambda _ 2 ^ * ( \\rho _ 0 , u _ 0 ) \\| _ { L ^ 2 ( L ^ 2 ) \\times L ^ 2 ( L ^ 2 ) } \\\\ & \\leq \\| \\Lambda _ 2 ^ * \\| \\ \\| ( \\rho _ 0 , u _ 0 ) \\| _ { L ^ 2 ( 0 , 1 ) \\times L ^ 2 ( 0 , 1 ) } . \\end{aligned} \\end{align*}"}
{"id": "7465.png", "formula": "\\begin{align*} ( a ) \\ ; R ( \\zeta ) = \\frac { 1 } { 2 } | \\zeta | ^ 2 ( b ) \\ ; R ( \\zeta ) = \\frac { 1 } { 2 } ( \\max ( \\zeta , 0 ) ) ^ 2 . \\end{align*}"}
{"id": "1343.png", "formula": "\\begin{align*} \\frac { \\mathcal { H } _ { d - 1 } ( W ' ( \\omega ) \\cap O ) } { | W ' ( \\omega ) \\cap O | ^ \\frac { d - 1 } { d } } \\geq \\frac { \\sum _ { l = 1 } ^ N \\mathcal { H } _ { d - 1 } ( B ( x _ { j _ l } , \\rho ' ) \\cap O ) } { \\sum _ { l = 1 } ^ N | B ( x _ { j _ l } , \\rho ' ) \\cap O | ^ \\frac { d - 1 } { d } } \\geq C _ . \\end{align*}"}
{"id": "1768.png", "formula": "\\begin{align*} \\langle [ \\Phi ] , [ P ] \\rangle \\colon = \\frac { 1 } { k ! } \\sum _ { i _ 0 , \\cdots , i _ { 2 k } = 1 } ^ m \\Phi ( p _ { i _ 0 i _ 1 } , p _ { i _ 1 i _ 2 } , . . . , p _ { i _ { 2 k } i _ 0 } ) \\end{align*}"}
{"id": "9479.png", "formula": "\\begin{align*} | \\mathcal { D D } _ { ( s , s + d , s + 2 d ) } | = \\sum _ { i = 0 } ^ { \\lfloor ( s - 1 ) / 2 \\rfloor } \\binom { \\lfloor ( s + d - 2 ) / 2 \\rfloor } { \\lfloor i / 2 \\rfloor , \\lfloor ( d + i ) / 2 \\rfloor , \\lfloor ( s - 1 ) / 2 \\rfloor - i } . \\end{align*}"}
{"id": "6224.png", "formula": "\\begin{align*} A = \\sum _ { j = 1 } ^ { 2 d } P _ { j } A , \\end{align*}"}
{"id": "7764.png", "formula": "\\begin{align*} \\Box \\phi _ k = \\left ( | \\phi _ { k , t } | ^ 2 - | \\phi _ { k , x } | ^ 2 \\right ) \\phi _ { k } + \\mathbf { 1 } _ { \\omega } f _ { k - 1 } ^ { \\phi ^ { \\perp } _ k } , \\ ; \\phi _ k [ 0 ] = ( a , b ) , \\end{align*}"}
{"id": "2155.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M ) } | A ^ { ( n ) } _ { M , n , l } ) = 0 . \\end{align*}"}
{"id": "2751.png", "formula": "\\begin{align*} \\inf _ { 0 \\not = u \\in H ^ 1 ( \\mathbb { R } ^ N ) } \\frac { \\| u \\| _ 2 ^ { ( N + \\gamma ) - ( N - 2 ) p } \\| \\nabla u \\| _ 2 ^ { N p - ( N + \\gamma ) } } { \\int _ { \\mathbb { R } ^ N } \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | u | ^ p \\right ) | u | ^ p d x } \\end{align*}"}
{"id": "4342.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d \\tau } \\| \\varepsilon _ - \\| _ { L ^ 2 _ { \\rho } } ^ 2 = \\left \\langle \\partial _ \\tau \\varepsilon _ - , \\varepsilon _ - \\right \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } + \\beta ' \\left \\langle \\varepsilon _ - , \\varepsilon _ - \\left ( \\frac { d + 2 } { 2 \\beta } - \\frac { y ^ 2 } { 2 } \\right ) \\right \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } . \\end{align*}"}
{"id": "8476.png", "formula": "\\begin{align*} \\sum _ { u \\sim w } \\mathbf { \\mathrm { v } } _ w = \\lambda \\mathbf { \\mathrm { v } } _ u = \\sum _ { \\substack { w \\sim u \\\\ w \\in L ' \\cup R } } \\mathbf { \\mathrm { v } } _ w + \\sum _ { \\substack { w \\sim u \\\\ w \\in E } } \\mathbf { \\mathrm { v } } _ w . \\end{align*}"}
{"id": "4806.png", "formula": "\\begin{align*} ( \\phi | \\psi ) = \\frac { 1 } { | A | } \\sum _ { a \\in A } \\bar \\phi ( a ) \\psi ( a ) . \\end{align*}"}
{"id": "8979.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - x ^ 2 & \\leq 0 \\ , ( - 1 , 1 ) , \\\\ u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - x ^ 2 & \\geq 0 [ - 1 , 1 ] . \\end{aligned} \\right . \\end{align*}"}
{"id": "4050.png", "formula": "\\begin{align*} \\lambda \\rho + \\rho _ x + u _ x = f , \\\\ \\lambda u + \\rho _ x - u _ { x x } + u _ x = g . \\end{align*}"}
{"id": "4172.png", "formula": "\\begin{align*} \\frac { \\omega _ { p } ( T ) } { 2 ^ { 1 - \\frac { 1 } { p } } } \\leq \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} \\right ) \\leq \\frac { \\omega _ { p } ( T ) } { 2 ^ { - \\frac { 1 } { p } } } . \\end{align*}"}
{"id": "2024.png", "formula": "\\begin{align*} e ^ { - \\| F _ 2 \\| _ { \\infty } } R ^ Y ( h _ n \\overline { \\mu } _ 2 ) ( x ) & \\leq R ^ Y h _ n ( { \\mu } _ 2 + N ( 1 - e ^ { - F _ 2 } ) \\mu _ H ) ( x ) \\\\ & \\leq R ^ Y h _ n ( { \\mu } _ 2 + N ( e ^ { F _ 1 } ( 1 - e ^ { - F _ 2 } ) \\mu _ H ) ( x ) \\\\ & = R ^ Y ( h _ n \\ 1 _ { K _ n } \\overline { \\mu } _ 1 ^ * ) ( x ) - h _ n ( x ) + 1 . \\end{align*}"}
{"id": "8770.png", "formula": "\\begin{align*} \\tilde { S } : = \\Bigl \\{ ( x , s , \\delta ) \\Bigm | \\bigl ( Z ( s ) , \\delta \\bigr ) \\in ( \\ref { e q : I n c - 1 } ) , \\ ( x , s _ { \\cdot n } ) \\in W , \\ s _ { i j } = \\min \\bigl \\{ b _ { i j } ( \\delta ) , s _ { i n } \\bigr \\} \\ ; \\forall i \\ ; \\forall j \\Bigr \\} . \\end{align*}"}
{"id": "3095.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + z ^ 2 y ^ 2 f _ 2 ( x , y ) + z y f _ 4 ( x , y ) + f _ 6 ( x , y ) = 0 \\ , , \\end{align*}"}
{"id": "7024.png", "formula": "\\begin{align*} R A + \\sum _ { j = 1 } ^ L \\bar y _ j S B _ j \\equiv 1 . \\end{align*}"}
{"id": "9417.png", "formula": "\\begin{align*} & \\tau ' ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) \\\\ = & \\tau ' ( b _ 0 a _ 1 ( c _ 1 + \\tau ( b _ 1 ) ) a _ 2 \\cdots ( c _ { n - 1 } + \\tau ( b _ { n - 1 } ) ) a _ n b _ n ) \\\\ = & \\tau ' ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots c _ { n - 1 } a _ n b _ n ) \\\\ & + \\sum _ { r = 1 } ^ n \\sum _ { 1 \\leq k _ 1 < \\cdots < k _ r \\leq n - 1 } \\tau ' ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots \\hat { c } _ { k _ 1 } \\cdots \\hat { c } _ { k _ r } \\cdots c _ { n - 1 } a _ n b _ n ) \\tau ( b _ { k _ 1 } ) \\cdots \\tau ( b _ { k _ r } ) \\end{align*}"}
{"id": "6880.png", "formula": "\\begin{align*} N _ k ^ j f _ \\ell = \\lambda _ \\ell f _ \\ell , \\end{align*}"}
{"id": "7645.png", "formula": "\\begin{align*} \\underline { m } \\int _ { B ^ 1 ( \\mathbf { 0 } ) } \\tilde { u } _ n ^ 2 + \\underline { m } \\int _ { B ^ 1 ( \\mathbf { 0 } ) ^ c } \\tilde { u } _ n ^ 2 = \\underline { m } \\ ; . \\end{align*}"}
{"id": "947.png", "formula": "\\begin{align*} | s _ \\delta | = | d _ { n } ^ \\delta ( C _ { n } ^ \\delta , J _ { n } ^ \\delta ) | \\geq k _ \\delta + 1 , n \\in \\Lambda , \\end{align*}"}
{"id": "5667.png", "formula": "\\begin{align*} \\alpha ( x \\cdot y ) \\cdot \\beta x = \\gamma y \\end{align*}"}
{"id": "5559.png", "formula": "\\begin{align*} \\alpha _ { \\rm H J M } = \\frac { 1 } { z _ 6 } e ^ { - z _ 6 \\cdot } ( 1 - e ^ { - z _ 6 \\cdot } ) = \\frac { 1 } { z _ 6 } ( e ^ { - z _ 6 \\cdot } - e ^ { - 2 z _ 6 \\cdot } ) . \\end{align*}"}
{"id": "2226.png", "formula": "\\begin{align*} \\sup _ { m \\geq n } \\| \\widetilde { v } ^ { n , m } ( t ) \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } \\leq & \\| E ( t ) P _ n P X _ 0 \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } + \\sup _ { m \\geq n } \\Big \\| \\int _ 0 ^ t E ( t - s ) P _ n A P F ( X ^ m ( s ) ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } \\\\ \\leq & C \\| X _ 0 \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } + C \\leq C ( X _ 0 , \\gamma , T ) . \\end{align*}"}
{"id": "7771.png", "formula": "\\begin{align*} \\Box \\phi _ 1 = \\left ( | \\phi _ { 1 , t } | ^ 2 - | \\phi _ { 1 , x } | ^ 2 \\right ) \\phi _ { 1 } + \\mathbf { 1 } _ { \\omega } f _ { 0 } ^ { \\ \\phi ^ { \\perp } _ 1 } , \\ ; \\phi _ 1 [ 0 ] = ( a , b ) . \\end{align*}"}
{"id": "5379.png", "formula": "\\begin{align*} c _ { M a p } = \\left ( c _ G + c _ { F _ k } \\right ) m , \\end{align*}"}
{"id": "4944.png", "formula": "\\begin{align*} ( \\phi - 1 ) \\kappa ( g ) + \\kappa ( b ) = \\kappa ( k ) + \\kappa ( \\mu ) . \\end{align*}"}
{"id": "1872.png", "formula": "\\begin{align*} P _ n ( x ) = ( x - i ) ^ { n + 1 } A _ n \\left ( { x + i \\over x - i } \\right ) . \\end{align*}"}
{"id": "1899.png", "formula": "\\begin{align*} \\max _ { \\substack { x , y \\in X \\\\ x \\ne y } } \\frac { \\beta _ { v _ i } ( x ) - \\beta _ { v _ i } ( y ) } { d ( x , y ) } = \\max _ { \\substack { x ' , y ' \\in X \\\\ x ' \\ne y ' } } \\frac { \\beta _ { v _ i ' } ( x ' ) - \\beta _ { v _ i ' } ( y ' ) } { d ( x ' , y ' ) } \\end{align*}"}
{"id": "629.png", "formula": "\\begin{align*} A ( x , n ) \\ = \\ A _ n ( x ) \\end{align*}"}
{"id": "8731.png", "formula": "\\begin{align*} X = \\bigl \\{ x \\in \\R ^ k \\bigm | g _ i ( x ) \\geq 0 , \\ i = 1 , \\ldots , m \\bigr \\} , \\end{align*}"}
{"id": "1263.png", "formula": "\\begin{align*} \\mathrm { i n d } ( u ) : = - \\chi ( u ) + 2 c _ { 1 } ( \\xi | _ { [ u ] } , \\tau ) + \\sum _ { k } \\mu _ { \\tau } ( \\gamma _ { k } ^ { + } ) - \\sum _ { l } \\mu _ { \\tau } ( \\gamma _ { l } ^ { - } ) . \\end{align*}"}
{"id": "2115.png", "formula": "\\begin{align*} \\begin{aligned} P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M ) } ) = \\sum _ { j = 1 } ^ { n - 1 } \\sum _ { l = 1 } ^ k \\frac { \\binom M l ( k ) _ l ( k ( j - 1 ) ) _ { M - l } } { ( k n ) _ M } \\frac 1 { n - j } . \\end{aligned} \\end{align*}"}
{"id": "4203.png", "formula": "\\begin{align*} u = 1 + \\dot u + r , \\end{align*}"}
{"id": "480.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { M } } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { M } } ( \\mathcal { I } _ { \\alpha \\beta } ^ { i j } ) = c _ { 1 } \\Delta _ { \\mathbf { M } } ( \\mathcal { I } _ { \\alpha } ^ { i } ) \\cdot \\Delta _ { \\mathbf { M } } ( \\mathcal { I } _ { \\beta } ^ { j } ) + c _ { 2 } \\Delta _ { \\mathbf { M } } ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot \\Delta _ { \\mathbf { M } } ( \\mathcal { I } _ { \\alpha } ^ { j } ) \\end{align*}"}
{"id": "1019.png", "formula": "\\begin{align*} ( w - v ) ( x ) ( x ) & = \\begin{cases} 0 & x \\in \\R ^ n _ + \\cap \\{ v \\geqslant 0 \\} , \\\\ u ( x ) - \\tau d ^ { - n - 2 } \\big ( 1 - \\frac \\theta 2 \\big ) ^ { - n - 2 } \\zeta ( x ) , & x \\in \\R ^ n _ + \\cap \\{ v < 0 \\} . \\end{cases} \\end{align*}"}
{"id": "4854.png", "formula": "\\begin{align*} S _ { N _ \\alpha } : = \\Big [ z _ 1 + \\frac { N _ \\alpha - 1 } \\alpha , z _ 2 \\Big ) ^ 2 . \\end{align*}"}
{"id": "5475.png", "formula": "\\begin{align*} u ( \\cdot , t ; 0 , u ^ * ) = u ^ * ( \\cdot ) \\ , \\ , t \\ge 0 . \\end{align*}"}
{"id": "7738.png", "formula": "\\begin{align*} \\tilde { \\phi } ( x ) : = \\int _ { \\R } \\phi ( t , x ) \\psi ( t ) \\ , d t . \\end{align*}"}
{"id": "6202.png", "formula": "\\begin{align*} \\| X X ^ { T } - \\hat { V } \\hat { V } ^ { T } \\| _ F = \\| X ( I - \\Sigma _ { \\hat { V } } ^ { 2 } ) X ^ { T } \\| _ F \\\\ = \\| I - \\Sigma _ { \\hat { V } } ^ { 2 } \\| _ F \\\\ = \\| Y ( I - \\Sigma _ { \\hat { V } } ^ { 2 } ) Y ^ { T } \\| _ F \\\\ = \\| I - \\hat { V } ^ { T } \\hat { V } \\| _ F \\\\ \\leq \\xi . \\end{align*}"}
{"id": "7634.png", "formula": "\\begin{align*} \\begin{cases} - k _ { \\varepsilon } \\Delta \\tilde { \\Psi } _ { \\varepsilon } + | \\nabla \\tilde { \\Psi } _ { \\varepsilon } | ^ 2 - \\tilde { \\lambda } _ 0 \\underline { m } = 0 & \\Omega \\ ; , \\\\ \\tilde { \\Psi } _ { \\varepsilon } = - k _ { \\varepsilon } \\log ( w ) ( \\frac { \\mathbf { x } - \\mathbf { x } _ { \\varepsilon } } { k _ { \\varepsilon } } ) & \\partial \\Omega \\ ; , \\end{cases} \\end{align*}"}
{"id": "1953.png", "formula": "\\begin{align*} ( x g ( x ) ) _ i : = x _ i g ( x ) = \\sum _ { w \\in \\N ^ * \\cup \\{ \\mathbf { 1 } \\} } g _ { w } x _ { i w } = x _ i + \\sum _ { w \\in \\N ^ * } g _ { w } x _ i x _ w . \\end{align*}"}
{"id": "5611.png", "formula": "\\begin{align*} \\Xi _ t : = \\{ X \\in \\Xi ~ | ~ X ( f ^ a ) = 0 a = 1 , \\ldots , k \\} \\end{align*}"}
{"id": "249.png", "formula": "\\begin{align*} \\hat { \\mu } _ { \\varepsilon } \\left ( \\xi \\right ) : = \\hat { \\mu } ( \\xi ) \\exp \\left ( - \\dfrac { \\varepsilon ^ 2 \\langle \\xi ; \\Sigma ( \\xi ) \\rangle } { 2 } \\right ) , \\end{align*}"}
{"id": "6019.png", "formula": "\\begin{align*} \\begin{aligned} U ( \\mathcal { O } f ) & = U \\bigg [ \\Big ( - \\frac { d } { d x } + 2 x \\Big ) \\frac { d } { d x } f \\bigg ] \\\\ & = \\Big ( - \\frac { d } { d x } + x \\Big ) \\Big ( \\frac { d } { d x } + x \\Big ) U f \\\\ & = \\frac 1 2 \\Big ( \\Big ( \\frac { d } { d x } + x \\Big ) \\Big ( - \\frac { d } { d x } + x \\Big ) + \\Big ( - \\frac { d } { d x } + x \\Big ) \\Big ( \\frac { d } { d x } + x \\Big ) \\Big ) U f - U f \\\\ & = \\mathcal { H } ( U f ) - U f . \\end{aligned} \\end{align*}"}
{"id": "5832.png", "formula": "\\begin{align*} \\theta = \\frac { v _ { + } - v _ { - } } { 6 } \\in \\left ( 0 , \\frac { \\lambda } { 6 } \\right ) \\end{align*}"}
{"id": "4652.png", "formula": "\\begin{align*} \\langle v , w \\rangle = v \\cup \\ast w \\ ; v , w \\in V _ k \\ ; . \\end{align*}"}
{"id": "7308.png", "formula": "\\begin{align*} \\det ( 1 _ n + D A ) = \\sum _ { I \\subseteq N } \\det ( A _ I ) X _ I , \\end{align*}"}
{"id": "4818.png", "formula": "\\begin{align*} \\begin{aligned} & H _ L ( w ^ d ) = [ \\ , U _ { 1 } \\ , U _ { 2 } \\ , ] \\begin{bmatrix} S & 0 \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} V _ 1 \\\\ V _ 2 \\end{bmatrix} , \\\\ & H _ L ( \\tilde { w } ^ d ) = [ \\ , \\tilde { U } _ { 1 } \\ , \\tilde { U } _ { 2 } \\ , ] \\begin{bmatrix} \\tilde { S } & 0 \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} \\tilde { V } _ 1 \\\\ \\tilde { V } _ 2 \\end{bmatrix} \\end{aligned} \\end{align*}"}
{"id": "9557.png", "formula": "\\begin{align*} | \\mathcal { A } | _ { | \\mathcal { B } ( 1 ) } = \\sum _ { k _ { \\Omega } \\in \\mathcal { B } ( 1 ) } \\phi ( k _ { \\Omega } ) . \\end{align*}"}
{"id": "80.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 1 / 2 } ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) . \\end{align*}"}
{"id": "3784.png", "formula": "\\begin{align*} r _ l \\big ( \\epsilon ( \\widetilde { \\chi _ 0 } \\pi _ E , \\psi _ F ) \\big ) = \\epsilon \\big ( \\chi r _ l ( \\pi _ F ) ^ { ( l ) } , \\overline { \\psi } _ F ^ l \\big ) . \\end{align*}"}
{"id": "3826.png", "formula": "\\begin{align*} r _ I ( \\underline { \\lambda } , \\sigma ) = \\frac { \\chi ( ( V _ f ^ I ) ^ { \\langle ( \\underline { \\lambda } , \\sigma ) \\rangle } ) } { m _ I | { \\rm K e r } \\ , A _ \\sigma \\cap G | } \\ , . \\end{align*}"}
{"id": "10.png", "formula": "\\begin{align*} u _ n = \\sum _ { j = 1 } ^ { J } [ T _ n ^ j ] \\phi ^ j + \\omega _ n ^ { J } , \\end{align*}"}
{"id": "6727.png", "formula": "\\begin{align*} f _ 1 L i _ { K , \\mathfrak { s } _ 1 } ( \\alpha ) + \\cdots + f _ r L i _ { K , \\mathfrak { s } _ r } ( \\alpha ) = 0 \\end{align*}"}
{"id": "7200.png", "formula": "\\begin{align*} R ^ { ( a ) } ( \\xi , t | \\xi | ) = \\eta ( | \\xi | ^ 2 ) \\hat { \\phi } ( \\xi ) t | \\xi | ^ 2 \\frac { \\hat { \\psi } _ \\xi ( t | \\xi | ) - \\hat { \\psi } _ \\xi ( 0 ) } { t | \\xi | } = \\eta ( | \\xi | ^ 2 ) \\hat { \\phi } ( \\xi ) t | \\xi | ^ 2 h _ \\xi ( t | \\xi | ) , \\end{align*}"}
{"id": "8189.png", "formula": "\\begin{align*} s ( 3 \\cdot 2 ^ k , 2 ^ k + ( - 1 ) ^ k f ) = & \\frac { 3 \\cdot 2 ^ k } { 1 2 ( 2 ^ k + ( - 1 ) ^ k f ) } + \\frac { 2 ^ k + ( - 1 ) ^ k f } { 3 6 \\cdot 2 ^ k } - \\frac { ( - 1 ) ^ k } { 4 } + \\frac { 1 } { 3 6 \\cdot 2 ^ k \\cdot ( 2 ^ k + ( - 1 ) ^ k f ) } \\\\ & + ( - 1 ) ^ k s ( 1 , 3 \\cdot 2 ^ k ) . \\end{align*}"}
{"id": "2140.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { E \\tau ^ { ( N , A ) } } N = \\frac 1 { A + 1 } . \\end{align*}"}
{"id": "6315.png", "formula": "\\begin{align*} \\Big \\{ \\omega _ 1 = \\frac { s ^ 2 \\nu } { \\nu - 2 } , \\omega _ 2 ( t ) = \\frac { K _ { \\nu / 2 } ( \\sqrt { \\nu } t s ) ( \\sqrt { \\nu } t s ) ^ { \\nu / 2 } } { 2 ^ { \\nu / 2 - 1 } \\Gamma ( \\nu / 2 ) } \\Big \\} . \\end{align*}"}
{"id": "918.png", "formula": "\\begin{align*} \\bigg ( x \\frac { d } { d x } \\bigg ) _ { n , \\lambda } f ( x ) = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) x ^ { k } \\bigg ( \\frac { d } { d x } \\bigg ) ^ { k } f ( x ) , ( \\mathrm { s e e } \\ [ 7 ] ) , \\end{align*}"}
{"id": "6588.png", "formula": "\\begin{align*} \\mathcal { D } ( h , k ) = \\mathcal { I } ^ * _ 0 ( h , k ) + O \\left ( \\left ( Q + \\frac { Q ^ 2 } { C } \\right ) \\frac { ( X H K ) ^ \\varepsilon } { \\sqrt { H K } } \\right ) . \\end{align*}"}
{"id": "6386.png", "formula": "\\begin{align*} \\Omega = V _ S \\cap \\mathbb D ^ n = \\{ ( z _ 1 \\dots , z _ n ) \\in \\mathbb D ^ n \\ , : \\ , f _ i ( z _ 1 , \\dots , z _ n ) = 0 \\ , , \\ , 1 \\leq i \\leq n \\} . \\end{align*}"}
{"id": "6788.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\ & f ( x ) \\\\ \\enspace & h ( x ) = 0 \\\\ & g ( x ) \\le 0 , \\end{aligned} \\end{align*}"}
{"id": "2103.png", "formula": "\\begin{align*} p _ 0 q _ 0 + \\cdots + p _ n q _ n = m d _ i , \\end{align*}"}
{"id": "871.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\{ \\varsigma _ { Q _ j , i } = 1 \\right \\} \\subseteq \\left \\{ \\varsigma _ { Q _ j , m } = 1 \\right \\} , i \\le m , \\\\ & \\left \\{ \\varsigma _ { Q _ j , i } = 0 \\right \\} \\subseteq \\left \\{ \\varsigma _ { Q _ j , r } = 0 \\right \\} , r \\le i . \\end{aligned} \\end{align*}"}
{"id": "6564.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) : = \\sum _ { q = 1 } ^ { \\infty } W \\left ( \\frac { q } { Q } \\right ) \\sideset { } { ^ \\flat } \\sum _ { \\chi \\bmod q } \\chi ( h ) \\overline { \\chi } ( k ) \\sum _ { m = 1 } ^ { \\infty } \\frac { \\tau _ A ( m ) \\chi ( m ) } { \\sqrt { m } } V \\left ( \\frac { m } { X } \\right ) \\sum _ { n = 1 } ^ { \\infty } \\frac { \\tau _ B ( n ) \\overline { \\chi } ( n ) } { \\sqrt { n } } V \\left ( \\frac { n } { X } \\right ) , \\end{align*}"}
{"id": "388.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; f ( x ) : = \\varphi ( x ) - ( g \\circ F ) ( x ) \\mbox { s u b j e c t t o } \\ ; \\ ; x \\in \\R ^ n . \\end{align*}"}
{"id": "1549.png", "formula": "\\begin{align*} \\widetilde { c } _ k ( ( s - k ) / 2 ) j ( g _ { \\infty } , z _ 0 ) ^ { - k } f ( z ) = c _ k ( s ) \\mathbf { f } ( h _ { \\mathbf { h } } \\cdot g _ { \\infty } ) , \\end{align*}"}
{"id": "3466.png", "formula": "\\begin{align*} \\| f ( x ) \\| _ p ^ p \\leqslant C \\sum \\limits _ { l = - \\infty } ^ \\infty 2 ^ { l p } \\mu ( { \\Omega } _ l ) ^ { p } \\leq C \\| S _ { c w } ( f ) \\| ^ p _ p \\leqslant C \\| f \\| ^ p _ { H _ { c w } ^ p } \\leqslant C \\| f \\| _ { H _ d ^ p } ^ p \\end{align*}"}
{"id": "6885.png", "formula": "\\begin{align*} \\Pr _ { V ' \\subset W } [ \\mathcal { E } _ i ( W ) ~ | ~ W ] = q ^ { i ^ 2 } \\frac { { j \\choose i } _ q { k \\choose k - i } _ q } { { k + j \\choose k } _ q } , \\end{align*}"}
{"id": "5177.png", "formula": "\\begin{align*} \\hat { p } _ \\ell = \\frac { ( p _ \\ell / \\mu _ \\ell ) } { \\sum _ { i = 1 } ^ { N } ( p _ i / \\mu _ i ) } , \\end{align*}"}
{"id": "2368.png", "formula": "\\begin{align*} F _ { ( \\xi , \\eta ) } ( x , \\omega ) = e ^ { 2 \\pi i x \\cdot \\omega } \\ , V _ g ( M _ \\eta T _ \\xi f ) ( x , \\omega ) \\ , V _ g ( M _ \\eta T _ \\xi f ) ( - x , - \\omega ) . \\end{align*}"}
{"id": "4885.png", "formula": "\\begin{align*} \\frac { b _ { k + 1 } } { b _ k } = \\frac { ( n - k - 1 ) ( n + 2 + k ) } { ( k + 1 ) ( k + 2 ) ( - 2 ) } = \\frac { ( k + 1 - n ) ( k + 2 + n ) } { 2 ( k + 1 ) ( k + 2 ) } \\hbox { f o r $ k = 0 , \\ldots , n - 2 $ . } \\end{align*}"}
{"id": "6962.png", "formula": "\\begin{align*} \\forall u \\in \\mathbb S _ { \\mathbb X } \\colon \\ker D ^ * _ \\gamma \\Phi ( ( \\bar x , \\bar y ) ; ( u , 0 ) ) = \\{ 0 \\} . \\end{align*}"}
{"id": "2068.png", "formula": "\\begin{align*} \\overline { \\{ g _ { \\alpha } : \\alpha \\in S _ 2 \\} } = A ^ 2 ( D , e ^ { - \\varphi } ) . \\end{align*}"}
{"id": "5624.png", "formula": "\\begin{align*} ( x ^ 0 ) ^ { * _ \\star } : = x ^ 0 , ~ ~ ~ ~ ~ ( x ^ i ) ^ { * _ \\star } : = x ^ \\mu \\tau ^ { \\mu i } ( S ( \\beta ) ) , ~ ~ ~ ~ ~ ( \\xi ^ i ) ^ { * _ \\star } : = \\xi ^ j \\tau ^ { j i } ( S ( \\beta ) ) , ~ ~ ~ ~ ~ ( \\partial ' _ i ) ^ { * _ \\star } : = - \\tau ^ { i j } ( \\beta ^ { - 1 } ) \\partial ' _ j , \\end{align*}"}
{"id": "1306.png", "formula": "\\begin{align*} \\hat { I } _ { ( n , \\infty ) } ( k ) : = \\bigcup _ { m = 0 } ^ { \\infty } \\hat { I } _ { ( n , m ) } ( k ) \\end{align*}"}
{"id": "1728.png", "formula": "\\begin{align*} 2 ^ { \\mu _ * k _ * t } \\cdot 2 ^ { \u2010 m _ t ( s _ * + 1 / q \u2010 1 / p _ 1 ) } = 2 ^ { \u2010 \\alpha _ * k _ * t } \\cdot 2 ^ { \u2010 m _ t ( 1 / q \u2010 1 / p _ 0 ) } \\stackrel { \\eqref { h a t _ t h e t a _ d e f } } { = } n ^ { \u2010 q \\hat \\theta / 2 } ; \\end{align*}"}
{"id": "273.png", "formula": "\\begin{align*} \\sigma _ \\varepsilon : = \\Big ( \\frac { \\varepsilon ^ d } { a _ \\varepsilon ^ { d - 2 } } \\Big ) ^ { \\frac { 1 } { 2 } } , \\ d \\geqslant 3 ; \\quad { \\sigma _ \\varepsilon } : = \\varepsilon \\left | \\log \\frac { { a _ \\varepsilon } } { \\varepsilon } \\right | ^ { \\frac { 1 } { 2 } } , \\ d = 2 , \\end{align*}"}
{"id": "6913.png", "formula": "\\begin{align*} \\textbf { a } _ m ( \\theta ) = \\left [ e ^ { - j 2 \\pi f _ 0 \\tau _ { 1 m } ( \\theta ) } , \\ldots , e ^ { - j 2 \\pi f _ 0 \\tau _ { N _ s m } ( \\theta ) } \\right ] ^ T , \\end{align*}"}
{"id": "3098.png", "formula": "\\begin{align*} x ^ \\prime : y ^ \\prime : z ^ \\prime = ( a _ 1 x + b _ 1 y ) ( a _ 2 x + b _ 2 y ) : ( a _ 2 x + b _ 2 y ) ^ 2 : z y \\ , . \\end{align*}"}
{"id": "5441.png", "formula": "\\begin{align*} m ^ * ( \\tau , s , u _ 0 ) = { \\rm m a x } \\Big \\{ \\int _ { \\Omega } u ( \\tau , x ; s , u _ 0 ) d x , \\frac { a _ { \\sup } } { b _ { \\inf } } | \\Omega | \\Big \\} \\end{align*}"}
{"id": "4046.png", "formula": "\\begin{align*} \\mathcal D ( A ) = \\Big \\{ \\Phi = ( \\xi , \\eta ) \\in \\dot { H } ^ 1 ( 0 , 1 ) \\times H ^ 2 ( 0 , 1 ) : \\xi ( 0 ) = \\xi ( 1 ) , \\ \\eta ( 0 ) = \\eta ( 1 ) = 0 \\Big \\} , \\end{align*}"}
{"id": "5608.png", "formula": "\\begin{align*} \\mathcal { X } ^ { M _ c } : = \\mathcal { X } / \\mathcal { C } ^ c = \\{ [ g ] : = g + \\mathcal { C } ^ c ~ | ~ g \\in \\mathcal { X } \\} , \\end{align*}"}
{"id": "4093.png", "formula": "\\begin{align*} ( G ' \\cdot \\mu + \\tfrac { 1 } { 2 } G '' \\cdot \\sigma ^ 2 ) ( \\xi _ i - ) & = \\mu ( \\xi _ i - ) - \\alpha _ i \\cdot \\sigma ^ 2 ( \\xi _ i ) = ( \\mu ( \\xi _ i - ) + \\mu ( \\xi _ i + ) ) / 2 = ( G ' \\cdot \\mu + \\tfrac { 1 } { 2 } G '' \\cdot \\sigma ^ 2 ) ( \\xi _ i ) \\\\ & = \\mu ( \\xi _ i + ) + \\alpha _ i \\cdot \\sigma ^ 2 ( \\xi _ i ) = ( G ' \\cdot \\mu + \\tfrac { 1 } { 2 } G '' \\cdot \\sigma ^ 2 ) ( \\xi _ i + ) . \\end{align*}"}
{"id": "1439.png", "formula": "\\begin{align*} \\overline { \\cdot } : \\mathbb { B } \\to \\mathbb { B } : a + b \\zeta + c \\xi + d \\zeta \\xi \\mapsto \\overline { a + b \\zeta + c \\xi + d \\zeta \\xi } = a - b \\zeta - c \\xi - d \\zeta \\xi . \\end{align*}"}
{"id": "2829.png", "formula": "\\begin{align*} \\int _ t ^ \\infty \\delta ( s ) d s = - \\frac { 1 } { 8 s _ c ( p - 1 ) } \\int _ t ^ \\infty \\ddot { y } ( s ) d s \\lesssim e ^ { - c t } , \\ ; \\forall t \\ge 0 . \\end{align*}"}
{"id": "5222.png", "formula": "\\begin{align*} T _ k ( f ) ( x ) & = \\sum \\limits _ { i = 1 } ^ { \\infty } \\lambda _ i B f , \\end{align*}"}
{"id": "9490.png", "formula": "\\begin{align*} | \\mathcal { F } ( a + b , - b \\ , ; \\ , \\{ U \\} , \\emptyset ) | = \\sum _ { i = 0 } ^ { a } \\binom { a + b - 1 } { \\lfloor i / 2 \\rfloor , b + \\lfloor ( i - 1 ) / 2 \\rfloor , a - i } . \\end{align*}"}
{"id": "4873.png", "formula": "\\begin{align*} z y '' ( z ) = y ( z ) ; \\end{align*}"}
{"id": "2634.png", "formula": "\\begin{align*} Z ( S _ g f ) = | Z g | ^ 2 Z f . \\end{align*}"}
{"id": "5717.png", "formula": "\\begin{align*} w ( \\gamma _ { 1 } \\cup { \\gamma _ { 3 } } ) = w ( \\gamma _ { 2 } \\cup { \\gamma _ { 3 } } ) = w ( \\gamma _ { 1 } \\cup { \\gamma _ { 4 } } ) = w ( \\gamma _ { 2 } \\cup { \\gamma _ { 4 } } ) = 3 . \\end{align*}"}
{"id": "7629.png", "formula": "\\begin{align*} \\lambda ^ { 2 } F \\sum _ { i } r _ { i } ^ { 2 } - \\lambda F ^ { i j } r _ { j } \\nabla _ { i } u = \\lambda ^ { 2 } \\sum _ { i } ( f - f ^ { i } \\kappa _ { i } ) r _ { i } ^ { 2 } . \\end{align*}"}
{"id": "8780.png", "formula": "\\begin{align*} s _ { i } ( x ) = \\bigl ( \\min \\{ a _ { i j ' } , a _ { i 0 } \\} , \\ldots , \\min \\{ a _ { i j ' } , a _ { i n } \\} \\bigr ) = ( a _ { i 0 } , \\ldots , a _ { i j ' } , a _ { i j ' } , \\ldots , a _ { i j ' } ) = : v _ { i j ' } . \\end{align*}"}
{"id": "5620.png", "formula": "\\begin{align*} g \\rhd x ^ 0 = \\epsilon ( g ) x ^ 0 , ~ g \\rhd x ^ i = x ^ \\mu \\tau ^ { \\mu i } ( g ) , ~ g \\rhd \\xi ^ i = \\xi ^ j \\tau ^ { j i } ( g ) , ~ g \\rhd \\partial _ i = \\tau ^ { i j } ( S ( g ) ) \\partial _ j . \\end{align*}"}
{"id": "6704.png", "formula": "\\begin{align*} \\Psi ^ { ( - 1 ) } _ { \\mathfrak { s } , \\boldsymbol { \\alpha } } = \\Phi _ { \\mathfrak { s } , \\boldsymbol { \\alpha } } \\Psi _ { \\mathfrak { s } , \\boldsymbol { \\alpha } } \\end{align*}"}
{"id": "2791.png", "formula": "\\begin{align*} B ( \\mathcal { Y } _ - , h ) = B ( \\mathcal { Y } _ + , h ) = 0 \\Leftrightarrow \\int \\mathcal { Y } _ 1 h _ 2 = \\int \\mathcal { Y } _ 2 h _ 1 = 0 , \\end{align*}"}
{"id": "3921.png", "formula": "\\begin{align*} \\kappa ^ { * } _ { n e w } = \\kappa ^ { * } - \\frac { ( f ' ( \\tilde { \\infty } ) - 1 . ) } { g } \\end{align*}"}
{"id": "4482.png", "formula": "\\begin{align*} \\phi _ { k , \\lambda } = \\frac { 1 } { e } \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } ( n ) _ { k , \\lambda } = \\frac { 1 } { e } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { ( n - 1 ) ! } ( n - \\lambda ) _ { k - 1 , \\lambda } , ( k \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "7121.png", "formula": "\\begin{align*} \\tau _ { \\infty } : = \\lim _ { n \\rightarrow \\infty } \\tau _ n \\le 1 . \\end{align*}"}
{"id": "7877.png", "formula": "\\begin{align*} A ' ( e _ 1 ) = \\begin{bmatrix} 1 & e _ 1 \\cdot w _ 2 & \\cdots & e _ 1 \\cdot w _ n \\\\ 0 & & & \\\\ \\vdots & & \\bigg [ w _ { t _ 1 } \\cdot w _ { t _ 2 } - ( e _ 1 \\cdot w _ { t _ 1 } ) ( e _ 1 \\cdot w _ { t _ 2 } ) \\bigg ] _ { ( m - 1 ) \\times ( m - 1 ) } & \\\\ 0 & & & \\end{bmatrix} . \\end{align*}"}
{"id": "2100.png", "formula": "\\begin{align*} \\lambda \\star ( x _ 0 , \\dots , x _ n ) = \\left ( \\lambda ^ { q _ 0 } x _ 0 , \\dots , \\lambda ^ { q _ n } x _ n \\right ) , \\ \\ \\lambda \\in k ^ \\ast . \\end{align*}"}
{"id": "7602.png", "formula": "\\begin{align*} 0 = \\int _ { M ^ { n } } \\mathrm { d i v } ( H X ^ { T } - \\nabla \\langle X , \\nu \\rangle ) \\geq - ( n - 1 ) \\int _ { M ^ { n } } \\langle X , \\nabla H \\rangle . \\end{align*}"}
{"id": "1292.png", "formula": "\\begin{align*} \\Lambda ( M ) : = \\bigcup _ { \\Gamma \\in H _ { 1 } ( M ) } \\Lambda ( M , \\Gamma ) \\end{align*}"}
{"id": "6194.png", "formula": "\\begin{align*} \\hat { V } = X \\Sigma _ { \\hat { V } } Y ^ { T } , \\end{align*}"}
{"id": "6924.png", "formula": "\\begin{align*} \\gamma _ { m a x } = \\frac { N _ s P _ t \\Bigl | \\sum _ { n = 1 } ^ N | g _ n | | h _ n | + | h _ d | \\Bigr | ^ 2 } { \\sigma _ n ^ 2 } , \\end{align*}"}
{"id": "2439.png", "formula": "\\begin{align*} [ \\pi ( \\l ' ) , \\pi ( \\l ) ] = \\left ( 1 - e ^ { 2 \\pi i ( \\omega \\cdot x ' - x \\cdot \\omega ' ) } \\right ) \\pi ( \\l ' ) \\pi ( \\l ) , \\l = ( x , \\omega ) , \\ \\l ' = ( x ' , \\omega ' ) . \\end{align*}"}
{"id": "3574.png", "formula": "\\begin{align*} \\lambda ( x ) & = \\min \\Big \\{ h \\in { \\bigcup } _ { z = 0 } ^ { e } \\Lambda _ z \\colon h \\equiv - x \\ ( \\bmod { \\ p ^ e } ) \\Big \\} , \\\\ \\overline { \\lambda } ( x ) & = \\frac { 1 } { 2 } ( n p ^ e + \\lambda ( x ) ) . \\end{align*}"}
{"id": "6528.png", "formula": "\\begin{align*} \\frac { 1 } { t _ n ^ { ( 2 m ) } } \\sum ^ { n - 1 } _ { j = 1 } \\frac { 1 } { 2 j } \\cdot d _ j ^ { ( 2 \\ell ) } = O \\left ( \\dfrac { 1 } { t _ n ^ { ( 2 m ) } } \\right ) = O ( ( \\log n ) ^ { - m } ) . \\end{align*}"}
{"id": "1754.png", "formula": "\\begin{align*} \\nu _ { V , W , m } ( f ) : = \\sup _ { g \\in G } \\lVert | 1 + L ( g ) | ^ m \\Theta ( g ) ^ { - 1 } L _ V R _ W \\big ( f \\big ) ( g ) \\rVert , \\ f \\in C _ c ^ \\infty ( G ) . \\end{align*}"}
{"id": "1435.png", "formula": "\\begin{align*} G : = G _ n : = \\{ g \\in \\mathrm { G L } ( V , \\mathbb { D } ) : \\langle g x , g y \\rangle = \\langle x , y \\rangle \\} . \\end{align*}"}
{"id": "116.png", "formula": "\\begin{align*} \\left ( x ^ 3 - \\omega ( x ) x ^ 2 \\right ) \\left ( x - \\frac 1 2 \\omega ( x ) \\right ) ^ k = L _ v ^ k ( x ^ 3 - \\omega ( x ) x ^ 2 ) , \\mbox { f o r a l l } k \\geq 1 . \\end{align*}"}
{"id": "2910.png", "formula": "\\begin{align*} \\partial \\varphi ( x , y ) & = \\varphi ( \\overline { [ x , y ] } ) - \\overline { [ \\varphi ( x ) , y ] } - \\overline { [ x , \\varphi ( y ) ] } \\\\ \\theta _ \\varphi ( x , y ) & = \\theta ( \\varphi ( x ) , y ) + \\theta ( x , \\varphi ( y ) ) . \\end{align*}"}
{"id": "4007.png", "formula": "\\begin{align*} & \\eta ^ { \\prime \\prime \\prime } ( x ) - \\lambda \\eta ^ { \\prime \\prime } ( x ) - 2 \\lambda \\eta ^ { \\prime } ( x ) + \\lambda ^ 2 \\eta ( x ) = 0 , \\ \\ \\forall x \\in ( 0 , 1 ) , \\\\ & \\eta ( 0 ) = 0 , \\ \\ \\eta ( 1 ) = 0 , \\ \\ \\eta ^ { \\prime \\prime } ( 0 ) = \\eta ^ { \\prime \\prime } ( 1 ) . \\end{align*}"}
{"id": "2812.png", "formula": "\\begin{align*} \\delta ( u ( t ) ) & = \\Big | \\| \\nabla u ( t ) \\| _ 2 ^ 2 - \\| \\nabla Q \\| _ 2 ^ 2 \\Big | \\\\ & = \\Big | 2 \\alpha \\| \\nabla Q \\| _ 2 ^ 2 + 2 \\int Q h _ 1 d x \\Big | + O ( \\widetilde { \\delta } ^ 2 ) \\\\ & = 2 \\left ( \\| \\nabla Q \\| _ 2 ^ 2 + \\| Q \\| _ 2 ^ 2 \\right ) | \\alpha ( t ) | + O ( \\widetilde { \\delta } ^ 2 ) . \\end{align*}"}
{"id": "9048.png", "formula": "\\begin{align*} \\begin{aligned} ( \\rho ^ { n + 1 } , \\phi ^ { n + 1 } ) = & \\arg \\min _ { ( \\rho , \\phi ) \\in \\mathcal { A } , m } \\left \\{ \\frac { 1 } { 2 \\tau } \\sum _ { i = 1 } ^ s \\int _ 0 ^ 1 \\int _ { \\Omega } F ( \\rho _ i , m _ i ) D ^ { - 1 } _ i d x d t + E ( \\rho ( \\cdot , 1 ) , \\phi ( \\cdot , 1 ) ) \\right \\} , \\\\ & \\partial _ t \\rho _ i + \\nabla \\cdot ( m _ i ) = 0 , m _ i \\cdot \\mathbf { n } = 0 , x \\in \\partial \\Omega , \\ ; \\rho ( x , 0 ) = \\rho ^ n . \\end{aligned} \\end{align*}"}
{"id": "2691.png", "formula": "\\begin{align*} \\norm { K } _ { o p } ^ 2 = \\sup _ { \\stackrel { f \\in \\mathcal { H } } { \\norm { f } _ \\mathcal { H } = 1 } } \\norm { K f } _ \\mathcal { H } ^ 2 \\leq \\sum _ { n \\in \\N } | c _ n | ^ 2 \\norm { K e _ n } _ \\mathcal { H } ^ 2 \\leq \\sum _ { n \\in \\N } | c _ n | ^ 2 \\sum _ { n \\in \\N } \\norm { K e _ n } _ \\mathcal { H } ^ 2 = \\norm { K } _ { H . S . } ^ 2 . \\end{align*}"}
{"id": "7405.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\rightarrow 0 ^ { + } } \\limsup _ { n \\rightarrow \\infty } \\mathbb { E } _ { \\mu _ n } \\Big [ \\Big | \\int _ { 0 } ^ { t } \\frac { 1 } { n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) [ \\eta _ { s } ( x ) - \\overleftarrow { \\eta } _ { s } ^ { \\ell } ( x ) ] \\eta _ { s } ( x + 1 ) d s \\Big | \\Big ] = 0 . \\end{align*}"}
{"id": "0.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 n } \\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ r r ^ { 4 n - 2 k } P _ { 2 k } \\left ( 4 n + 1 , 2 n + 1 \\right ) = \\frac { ( ( 2 n ) ! ) ^ 2 } { 2 } - \\frac { P _ { 4 n } \\left ( 4 n + 1 , 2 n + 1 \\right ) } { 2 } . \\end{align*}"}
{"id": "2827.png", "formula": "\\begin{align*} \\inf _ { 0 \\not = f \\in H ^ 1 ( \\mathbb { R } ^ N ) } \\frac { \\| \\nabla f \\| _ 2 ^ { N p - ( N + \\gamma ) } \\| f \\| _ 2 ^ { N + \\gamma - ( N - 2 ) p } } { \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | f | ^ p \\right ) | f | ^ p d x } , \\end{align*}"}
{"id": "8497.png", "formula": "\\begin{align*} f _ k & = [ u ^ k ] F ( u ) = \\frac { z ( z + 1 ) } { ( 1 + z ^ 2 ) s _ 1 ^ k } , \\\\ g _ k & = [ u ^ k ] G ( u ) = \\frac { z ^ 2 ( 2 z s _ 1 + 1 ) } { ( 1 + z ^ 2 ) s _ 1 ^ { k + 2 } } , \\mbox { a n d } \\\\ h _ k & = [ u ^ k ] H ( u ) = \\frac { z ( ( z - 1 ) s _ 1 + 1 ) } { ( 1 + z ^ 2 ) s _ 1 ^ { k + 1 } } . \\end{align*}"}
{"id": "986.png", "formula": "\\begin{align*} u ( x ) \\leqslant \\tau \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } d ^ { - n - 2 } = \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } u ( a ) x \\in B _ { \\theta d / 2 } ( a ) . \\end{align*}"}
{"id": "8572.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow 0 ^ + } \\mathcal { K } ( x , k ) = - \\lim _ { k \\rightarrow 0 ^ - } \\mathcal { K } ( x , k ) . \\end{align*}"}
{"id": "7537.png", "formula": "\\begin{align*} f ( s ) = e ^ { A + B s } \\prod _ { n = 1 } ^ \\infty ( 1 - s / \\rho _ n ) e ^ { s / \\rho _ n } \\end{align*}"}
{"id": "967.png", "formula": "\\begin{align*} \\tilde c _ { n , s } : = c _ { n , s } \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } . \\end{align*}"}
{"id": "6318.png", "formula": "\\begin{align*} - \\Phi _ r ( \\{ X _ I \\} ) = \\frac { \\sum _ { I \\in \\mathcal { P } [ r ] } ( - 1 ) ^ { | I | + 1 } X _ I \\prod _ { J \\in \\mathcal { P } [ r ] \\atop J \\neq I } ( 1 - X _ J ) } { \\prod _ { J \\in \\mathcal { P } [ r ] } ( 1 - X _ J ) } . \\end{align*}"}
{"id": "5818.png", "formula": "\\begin{align*} P L _ { m k } = \\begin{cases} - L - 3 5 \\log _ { 1 0 } ( d _ { m k } ) , & d _ { m k } > d _ 1 \\\\ - L - 1 0 \\log _ { 1 0 } ( d _ 1 ^ { 1 . 5 } d _ { m k } ^ 2 ) , & d _ 0 < d _ { m k } \\leq d _ 1 \\\\ - L - 1 0 \\log _ { 1 0 } ( d _ 1 ^ { 1 . 5 } d _ 0 ^ 2 ) , & d _ { m k } \\leq d _ 0 \\end{cases} , \\end{align*}"}
{"id": "1695.png", "formula": "\\begin{align*} 1 _ { S _ a } S 1 _ { S _ b } = S _ { G ( a , b ) } . \\end{align*}"}
{"id": "999.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { ( - \\Delta ) ^ s v ( h e _ 1 ) } h = - 2 c _ { n , s } ( n + 2 s ) \\int _ { \\R ^ n _ + } \\frac { y _ 1 v ( y ) } { \\vert y \\vert ^ { n + 2 s + 2 } } \\dd y . \\end{align*}"}
{"id": "8617.png", "formula": "\\begin{align*} i k \\mathcal { K } ^ { \\# } _ { R } ( x , k ) = \\partial _ x \\mathcal { K } _ { \\# , R } ' ( x , k ) - \\mathcal { K } _ { \\# , R } '' ( x , k ) , \\end{align*}"}
{"id": "9287.png", "formula": "\\begin{align*} [ J ^ { \\chi _ A } ] _ \\mathcal { M } : = \\lambda \\alpha \\in \\mathbb { N } ^ \\mathbb { N } , x \\in X . \\begin{cases} J ^ A _ { r _ \\alpha } x & r _ \\alpha > 0 , \\rho > - r _ \\alpha / 2 x \\in \\mathrm { d o m } ( J ^ A _ { r _ \\alpha } ) , \\\\ 0 & , \\end{cases} \\end{align*}"}
{"id": "6849.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { X ( k ) } = \\sum \\limits _ { \\tau \\in X ( k ) } \\pi _ k ( \\tau ) f ( \\tau ) g ( \\tau ) , \\end{align*}"}
{"id": "3072.png", "formula": "\\begin{align*} x _ 1 ^ 3 f _ 3 ( y _ 1 , y _ 2 ) + x _ 2 ^ 3 \\varphi _ 3 ( y _ 1 , y _ 2 ) = 0 \\ , . \\end{align*}"}
{"id": "9119.png", "formula": "\\begin{align*} x ^ * \\in \\Gamma : = \\{ x ^ * \\in X \\mid T ( x ^ * ) \\cap S ( x ^ * ) \\neq \\emptyset \\} . \\end{align*}"}
{"id": "8941.png", "formula": "\\begin{align*} \\begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \\leq \\int _ 0 ^ T e ^ { - s } L \\left ( \\xi ( s ) + \\left ( 1 - \\frac { s } { T } \\right ) h , - \\left ( \\dot { \\xi } ( s ) - \\frac { h } { T } \\right ) \\right ) \\\\ & + e ^ { - s } L \\left ( \\xi ( s ) - \\left ( 1 - \\frac { s } { T } \\right ) h , - \\left ( \\dot { \\xi } ( s ) + \\frac { h } { T } \\right ) \\right ) - 2 e ^ { - s } L \\left ( \\xi ( s ) , - \\dot { \\xi } ( s ) \\right ) d s . \\end{aligned} \\end{align*}"}
{"id": "6186.png", "formula": "\\begin{align*} N _ { t , : } = \\frac { M _ { i _ t , : } } { \\sqrt { p P _ { i _ t } } } , \\ \\ t \\in [ p ] . \\end{align*}"}
{"id": "1061.png", "formula": "\\begin{align*} f _ { \\mathcal { R } _ { \\xi , I I } } ( x , t ) : = \\frac { 3 } { 4 } D _ { \\infty } ^ { - 2 } \\left ( \\frac { \\tilde { T } _ { \\eta , 1 } ^ { ( 1 2 ) } } { D _ 0 ^ { - 2 } r ^ { - 1 } ( \\eta ) - 1 } - \\frac { \\tilde { T } _ { - \\eta , 1 } ^ { ( 1 2 ) } } { D _ 0 ^ { - 2 } r ^ { - 1 } ( - \\eta ) - 1 } \\right ) . \\end{align*}"}
{"id": "1215.png", "formula": "\\begin{align*} \\eta ( A ) & = \\eta ( B ) \\cdot \\frac { \\sum _ { V \\in \\mathcal { W } _ { p + 1 } : V \\cap A \\neq \\emptyset } \\mu ( B ^ { [ V ] } ) } { \\sum _ { V ' \\in \\mathcal { F } ^ { B } } \\mu ( B ^ { [ V ^ { \\prime } ] } ) } \\leq 2 Q _ { d , 4 } \\frac { \\eta ( B ) } { \\mu ( B ) } \\mu ( 5 A ) . \\end{align*}"}
{"id": "9023.png", "formula": "\\begin{align*} \\begin{aligned} E = E _ 0 - \\frac { 1 } { 2 } \\left [ \\int _ { \\Gamma _ D } \\epsilon ( x ) \\phi ^ b _ D \\partial _ n \\phi d s - \\int _ { \\Gamma _ N } \\phi ^ b _ N \\phi d s - \\frac { 1 } { \\beta _ R } \\int _ { \\Gamma _ R } \\phi ^ b _ R \\phi d s \\right ] . \\end{aligned} \\end{align*}"}
{"id": "3841.png", "formula": "\\begin{align*} y ^ { [ j ] } = y ^ { ( j ) } , j = \\overline { 0 , m - 1 } . \\end{align*}"}
{"id": "2002.png", "formula": "\\begin{align*} M ( k ) = \\left ( \\begin{array} { c c c } - 1 & 0 & 0 \\\\ 0 & \\cos ( \\frac { 2 \\pi k } { n } ) & \\sin ( \\frac { 2 \\pi k } { n } ) \\\\ 0 & \\sin ( \\frac { 2 \\pi k } { n } ) & - \\cos ( \\frac { 2 \\pi k } { n } ) \\end{array} \\right ) . \\end{align*}"}
{"id": "5958.png", "formula": "\\begin{align*} \\omega ^ 2 a _ { j k } - i \\omega b _ { j k } = \\frac { \\mathcal { F } \\{ F _ { j k } \\} } { \\mathcal { F } \\{ x _ k \\} } , \\end{align*}"}
{"id": "490.png", "formula": "\\begin{align*} Y _ { \\alpha \\beta } ^ { i j } = - \\frac { h ( \\mathcal { I } _ { \\alpha } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\beta } ^ { j } ) } { h ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\alpha } ^ { j } ) } \\end{align*}"}
{"id": "8666.png", "formula": "\\begin{align*} & \\sum _ { i , j = 1 } ^ 3 \\sigma ^ { i j } \\big ( u _ 0 \\hat { p } _ i - u _ i \\big ) \\big ( u _ 0 \\hat { p } _ j - u _ j \\big ) \\geq \\sum _ { i , j = 1 } ^ 3 \\frac 1 2 u _ 0 ^ 2 \\sigma ^ { i j } \\hat { p } _ i \\hat { p } _ j - \\sum _ { i , j = 1 } ^ 3 \\sigma ^ { i j } u _ i u _ j . \\end{align*}"}
{"id": "7002.png", "formula": "\\begin{align*} | \\det { \\mathfrak { S } } | = | B _ K | ^ N | A ( \\beta _ 1 ) \\cdots A ( \\beta _ K ) | = | A _ N | ^ K | B ( \\alpha _ 1 ) \\cdots B ( \\alpha _ N ) | . \\end{align*}"}
{"id": "7699.png", "formula": "\\begin{align*} a = \\prod _ { p \\in P } p ^ { \\mathsf v _ p ( a ) } \\ , , \\quad \\end{align*}"}
{"id": "2664.png", "formula": "\\begin{align*} \\int _ { P _ R } | z ^ \\alpha | ^ 2 e ^ { - \\pi | z | ^ 2 } \\ , d z = \\prod _ { k = 1 } ^ d \\left ( 2 \\pi \\int _ 0 ^ R r ^ { 2 \\alpha _ k + 1 } e ^ { - \\pi r _ k ^ 2 } \\ , d r _ k \\right ) = \\mu _ { \\alpha , R } \\end{align*}"}
{"id": "6670.png", "formula": "\\begin{align*} \\Sigma _ { c , a , e , d , g _ 1 , g _ 2 , g _ 3 , g _ 4 } \\ll ( X Q L a e d g _ 1 g _ 3 g _ 4 ) ^ { \\varepsilon } \\left ( \\frac { L Q } { c } \\right ) \\mathop { \\sum _ { S _ 5 > 0 } \\sum _ { S _ 6 > 0 } } _ { S _ 6 \\leq S _ 5 } \\frac { S _ 6 ^ { \\varepsilon } } { S _ 5 ^ { j _ 1 - \\varepsilon } } \\bigg ( 1 + \\frac { X c Q ^ { \\vartheta - 1 } } { L } \\bigg ) ^ { j _ 1 - 1 } \\\\ \\times \\Big ( Q ^ { \\vartheta } L S _ 5 + a e L ^ 2 S _ 5 ^ 2 \\Big ) . \\end{align*}"}
{"id": "1624.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) = - \\left \\langle \\mathcal { L } f , g \\right \\rangle _ { L ^ 2 ( M ) } , f \\in \\mathcal { D } ( \\mathcal { L } ) , \\ g \\in \\mathcal { D } ( \\mathcal { E } ) . \\end{align*}"}
{"id": "2066.png", "formula": "\\begin{align*} \\sum _ { \\gamma < \\beta } b _ { \\gamma } ^ { \\beta } g _ { \\alpha } ^ { ( \\gamma ) } ( o ) + g _ { \\alpha } ^ { ( \\beta ) } ( o ) = \\int _ D g _ { \\alpha } \\overline { { g } _ { \\beta } } e ^ { - \\varphi } , \\ \\forall \\beta < \\alpha . \\end{align*}"}
{"id": "7974.png", "formula": "\\begin{align*} \\begin{cases} u _ j / d _ j \\to \\hat { u } _ \\infty & \\ C ^ 2 _ { l o c } ( \\bar { \\Sigma } ' ) ; \\\\ ( f _ j - f ) / ( ( 1 + f ) d _ j ) \\to \\hat { f } _ \\infty & \\ C ^ { k , \\alpha } ( M ) . \\end{cases} \\end{align*}"}
{"id": "8395.png", "formula": "\\begin{align*} \\mu _ { \\varepsilon } = - \\varepsilon \\Delta c _ { \\varepsilon } + \\frac { 1 } { \\varepsilon } f ^ { \\prime } ( c _ { \\varepsilon } ) , \\end{align*}"}
{"id": "2803.png", "formula": "\\begin{align*} \\Phi ( h ) = B ( h , h ) = \\lambda _ { N + 3 } ^ 2 B ( f , f ) = \\lambda _ { N + 3 } ^ 2 \\Phi ( f ) \\le 0 . \\end{align*}"}
{"id": "450.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 3 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle \\leq \\frac { C \\epsilon } { 2 } \\| \\nabla v \\| _ { m } ^ { 2 } + C \\left \\lbrace \\| f _ { 3 } \\| _ { m } ^ { 2 } + ( \\mu _ { 0 } ( t ) + \\mu _ { 1 } ( t ) ) \\left ( \\| w \\| _ { m } ^ { 2 } + \\| v \\| _ { m } ^ { 2 } \\right ) \\right \\rbrace . \\end{align*}"}
{"id": "4169.png", "formula": "\\begin{align*} \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} \\right ) & = \\omega _ { p } \\left ( U \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} U ^ { \\ast } \\right ) \\\\ & = \\omega _ { p } \\left ( \\begin{bmatrix} 0 & B \\\\ A & 0 \\end{bmatrix} \\right ) \\end{align*}"}
{"id": "5859.png", "formula": "\\begin{align*} m = \\begin{cases} 3 , & \\rho < 1 , \\\\ 4 , & \\rho \\ge 1 . \\end{cases} \\end{align*}"}
{"id": "7654.png", "formula": "\\begin{align*} \\mathbf { y } _ { n _ k } \\cap B _ 1 ( \\mathbf { 0 } ) = \\emptyset , \\varepsilon < \\frac { \\underline { m } } { \\underline { m } + \\overline { m } } \\ ; . \\end{align*}"}
{"id": "345.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho , m , S ) = \\sup _ { S \\in H ^ 1 } \\inf _ { ( \\rho , m ) } \\mathcal L ( \\rho , m , S ) . \\end{align*}"}
{"id": "200.png", "formula": "\\begin{align*} I _ { 2 , \\lambda } ( f ) ( x , y ) = \\int _ { 0 } ^ { + \\infty } \\dfrac { e ^ { - ( 1 + \\lambda ) t } } { \\sqrt { 1 - e ^ { - 2 t } } } f \\left ( x e ^ { - t } + y \\sqrt { 1 - e ^ { - 2 t } } \\right ) d t . \\end{align*}"}
{"id": "8671.png", "formula": "\\begin{align*} \\alpha _ n & = \\sqrt { 2 \\log n } , & \\beta _ n & = \\alpha _ n - ( 2 \\alpha _ n ) ^ { - 1 } \\bigl ( \\log \\log n + \\log ( 4 \\pi ) \\bigr ) . \\end{align*}"}
{"id": "5207.png", "formula": "\\begin{align*} B _ i f = \\sum \\limits _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ \\int \\limits _ { | y | = q ^ { j + 1 } } f ( x - y ) a _ i ( y ) d y . \\end{align*}"}
{"id": "2338.png", "formula": "\\begin{align*} f ( t ) = \\int _ { - 1 / 2 } ^ { 1 / 2 } \\widehat { f } ( \\omega ) e ^ { 2 \\pi i t \\cdot \\omega } \\ , d \\omega , \\end{align*}"}
{"id": "7030.png", "formula": "\\begin{align*} f ( \\xi _ k ) = \\lim _ { r \\to 1 ^ { - } } f ( r \\xi _ k ) = p ( \\xi _ k ) , \\end{align*}"}
{"id": "1398.png", "formula": "\\begin{align*} | E _ 1 ( 1 ) - E _ 2 ( 1 ) | \\le & \\left | \\int _ 0 ^ 1 \\Big [ ( n _ 1 - b _ 1 ) - ( n _ 2 - b _ 2 ) \\Big ] d y \\right | \\\\ \\le & \\int _ 0 ^ 1 | n _ 1 - n _ 2 | ( y ) d y + \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } \\\\ = & | n _ 1 - n _ 2 | ( \\xi ) + \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } , \\quad \\exists \\xi \\in [ 0 , 1 ] , \\end{align*}"}
{"id": "6243.png", "formula": "\\begin{align*} y _ 1 = 0 , \\dotsc , \\ y _ n = 0 , \\ y _ i - y _ j = 0 \\textrm { f o r } 1 \\leq i < j \\leq n \\end{align*}"}
{"id": "1099.png", "formula": "\\begin{align*} & U _ { \\xi } = U _ { \\eta } \\cup U _ { - \\eta } , \\\\ & U _ { \\eta } = \\left \\{ k : | k - \\eta | < \\varrho \\right \\} , U _ { - \\eta } = \\left \\{ k : | k + \\eta | < \\varrho \\right \\} , \\end{align*}"}
{"id": "8005.png", "formula": "\\begin{align*} \\frac { A _ { \\sigma _ n } ( i ) } { B _ { \\sigma _ n } } \\leq p _ { \\sigma _ n } ( i ) < \\frac { A _ { \\sigma _ n } ( i ) + 1 } { B _ { \\sigma _ n } } \\ ; \\ ; A _ { \\sigma _ n } ( 1 ) = B _ { \\sigma _ n } - \\sum _ { i \\in \\mathcal { I } _ n ^ { \\sigma } \\setminus \\{ 1 \\} } A _ { \\sigma _ n } ( i ) , \\end{align*}"}
{"id": "428.png", "formula": "\\begin{align*} A _ { 2 } ^ { 0 } ( U ^ { 2 } ) \\hat { v } _ { t } + A _ { 2 2 } ^ { i } ( U ^ { 2 } ) \\partial _ { i } \\hat { v } - B _ { 0 } ^ { i j } ( U ^ { 2 } ) \\partial _ { i } \\partial _ { j } \\hat { v } = \\hat { g } , \\end{align*}"}
{"id": "2748.png", "formula": "\\begin{align*} a _ t = \\frac { - R ( z ' _ 1 , \\dots , z ' _ { i - 1 } , t , z _ { i + 1 } , \\dots , z _ { n - 1 } ) } { Q ( z ' _ 1 , \\dots , z ' _ { i - 1 } , t , z _ { i + 1 } , \\dots , z _ { n - 1 } ) } \\end{align*}"}
{"id": "2373.png", "formula": "\\begin{align*} | V _ g f ( x , \\omega ) | = | \\langle f , M _ \\omega T _ x g \\rangle | \\leq \\norm { f } _ 2 \\norm { g } _ 2 = 1 . \\end{align*}"}
{"id": "8254.png", "formula": "\\begin{align*} w _ i = w ( \\xi _ r ) = ( u \\times v ) ( r ) = v _ { r - p } [ p ] \\quad w _ j = w ( \\xi _ s ) = ( u \\times v ) ( s ) = v _ { s - p } [ p ] . \\end{align*}"}
{"id": "171.png", "formula": "\\begin{align*} \\mu _ m ( d x ) = c _ { m , d } \\left ( 1 + \\| x \\| ^ 2 \\right ) ^ { - m - d / 2 } d x , \\end{align*}"}
{"id": "4157.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int x w ( x , t ) d x = \\frac 1 2 ( \\| \\phi \\| ^ 2 - \\| \\varphi \\| ^ 2 ) . \\end{align*}"}
{"id": "503.png", "formula": "\\begin{align*} g _ { \\delta _ { 1 } \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 1 } a _ { 2 } a _ { 3 } } : = \\frac { Y ( \\mathcal { I } _ { \\delta _ { 1 } } ^ { a _ { 1 } } ) _ { \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 3 } a _ { 2 } } + 1 } { Y ( \\mathcal { I } ) _ { \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 3 } a _ { 2 } } + 1 } . \\end{align*}"}
{"id": "1090.png", "formula": "\\begin{align*} D ( k , \\xi ) : = \\exp \\left \\{ \\frac { X _ { L } ( k ) } { 2 \\pi i } \\left [ \\left ( \\int _ { - \\eta } ^ { - C _ L } + \\int _ { C _ L } ^ { \\eta } \\right ) \\frac { \\log ( 1 - r ( s ) r ^ { * } ( s ) ) } { X _ { L } ( s ) ( s - k ) } d s + \\left ( \\int _ { - C _ L } ^ { - C _ R } + \\int _ { C _ R } ^ { C _ L } \\right ) \\frac { \\log r _ { + } ( s ) } { X _ { L + } ( s ) ( s - k ) } d s \\right ] \\right \\} , \\end{align*}"}
{"id": "7752.png", "formula": "\\begin{align*} b _ z ( x ) : = b ( x - z ) . \\end{align*}"}
{"id": "6798.png", "formula": "\\begin{align*} \\Omega _ m = \\{ \\theta \\ : | \\ : \\theta = x _ { 1 : m } , x \\in \\Omega _ n \\} . \\end{align*}"}
{"id": "6967.png", "formula": "\\begin{align*} \\forall k \\in \\N \\colon \\eta _ k \\in \\widehat { \\partial } \\varphi ( x _ k ' ) + \\widehat { D } ^ * \\Phi ( x _ k , y _ k ) \\left ( \\frac { \\norm { x _ k - \\bar x } } { \\norm { y _ k - \\bar y } } y _ k ^ * \\right ) . \\end{align*}"}
{"id": "4629.png", "formula": "\\begin{align*} A ( y + z ) = A ( y ) y \\in \\R ^ d z \\in \\Gamma ; \\end{align*}"}
{"id": "6869.png", "formula": "\\begin{align*} \\rho ^ k _ \\ell = \\prod \\limits ^ { k - \\ell } _ { i = 1 } \\left ( 1 - \\delta _ { k - \\ell - i } ^ { k - i } \\right ) , \\ \\ \\rho _ { } = \\min _ { 0 \\leq \\ell \\leq k } \\{ \\rho ^ k _ \\ell \\} . \\end{align*}"}
{"id": "9107.png", "formula": "\\begin{align*} \\sum _ { i } \\left ( t ^ { 2 i } \\prod _ { n \\geq 1 } \\frac { 1 } { ( 1 - t ^ n ) } \\right ) = \\frac { t ^ { 2 } } { 1 - t ^ { 2 } } \\prod _ { n \\geq 1 } \\frac { 1 } { ( 1 - t ^ n ) } . \\end{align*}"}
{"id": "4849.png", "formula": "\\begin{align*} C \\sqrt { \\frac { n ^ 3 \\log n } { P ^ { - 1 } ( n ) } } \\bigg / \\bigg ( c \\frac { n ^ 2 } { P ^ { - 1 } ( n ) } \\bigg ) = \\frac C c \\sqrt { P ^ { - 1 } ( n ) \\frac { \\log n } n } . \\end{align*}"}
{"id": "4151.png", "formula": "\\begin{align*} F ( t ^ { \\ast } ) = 0 . \\end{align*}"}
{"id": "9007.png", "formula": "\\begin{align*} ~ ~ & I ( Q _ n ; M ) , \\\\ ~ ~ & ( p _ 1 , p _ 2 , \\dots , p _ { K - 1 } ) , \\\\ ~ ~ & - p _ w \\leq 0 , \\forall w \\in 1 : K - 1 , \\\\ & \\begin{aligned} N \\sum _ { w = 0 } ^ { K - 1 } & \\binom { K - 1 } { w } ( N - 1 ) ^ w p _ w - 1 = 0 . \\end{aligned} \\end{align*}"}
{"id": "4795.png", "formula": "\\begin{align*} a = a ^ \\sigma , \\epsilon ( a ) = 0 , a \\odot a = a , a ^ 2 = ( \\lambda - \\mu ) a + \\mu w + ( r - \\mu ) e , \\end{align*}"}
{"id": "6453.png", "formula": "\\begin{align*} ( T _ 0 - T _ h ( \\lambda ) ) ( u _ 0 ) ( x ) & = \\frac { \\eta _ 0 ^ 1 } { 4 \\pi } \\int _ { B _ 1 } \\big ( 1 - \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } \\big ) \\frac { u _ 0 ( y ) } { | x - y | } d y \\\\ & + \\frac { \\eta _ 0 ^ 2 } { 4 \\pi } \\int _ { B _ 2 } \\big ( 1 - \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } \\big ) \\frac { u _ 0 ( y ) } { | x - y | } d y \\end{align*}"}
{"id": "95.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ 2 \\cdot 3 \\cdot \\frac { ( 2 ^ { ( n _ { 2 , \\nu _ 2 } + 1 ) / 2 } + 1 ) ( 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 ) } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "5725.png", "formula": "\\begin{align*} \\int _ M | \\nabla _ g u ( x ) | ^ 2 { \\rm d } v _ g - \\mu \\int _ M \\frac { u ^ 2 ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \\rm d } v _ g + \\int _ M u ^ 2 ( x ) { \\rm d } v _ g = \\lambda \\int _ M \\alpha ( x ) \\xi _ x u ( x ) { \\rm d } v _ g , \\end{align*}"}
{"id": "1018.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s v ( x ) + c ( x ) v ( x ) & = - ( - \\Delta ) ^ s u ( x ) - c ( x ) u ( x ) + c ( x ) \\tau d ^ { - n - 2 } \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } \\zeta ( x ) \\\\ & \\geqslant - x _ 1 - C \\tau d ^ { - n - 2 } \\| c ^ - \\| _ { L ^ \\infty ( B _ \\rho ^ + ) } \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } \\zeta ( x ) . \\end{align*}"}
{"id": "3379.png", "formula": "\\begin{align*} & T _ 1 ( u ) = T ^ { ' } _ 1 ( u ) + T D ( \\mathfrak { X } ) u - [ \\mathfrak { X } , T u ] _ \\mathfrak { g } \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; = T ^ { ' } _ 1 ( u ) + ( d _ T \\mathfrak { X } ) ( u ) , \\ ; \\forall u \\in V , \\end{align*}"}
{"id": "3594.png", "formula": "\\begin{align*} | z ^ * ( F ( e ) ) | \\leq \\rho _ { i _ 0 } ( x _ { i _ 0 } ^ * ) = \\rho ( z _ 1 ^ * ) . \\end{align*}"}
{"id": "3343.png", "formula": "\\begin{align*} \\ ; [ T u + u , T v + v , T w + w ] _ { L \\oplus V } & = T \\Big ( \\theta ( T v , T w ) u - \\theta ( T u , T w ) v + D ( T u , T v ) w \\Big ) \\\\ & + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v + D ( T u , T v ) w \\in G r ( T ) . \\end{align*}"}
{"id": "5890.png", "formula": "\\begin{align*} \\mathcal { X } = f ( W ( h _ 1 ) , \\dots , W ( h _ m ) ) \\end{align*}"}
{"id": "3018.png", "formula": "\\begin{align*} D ^ { x } = \\sum _ { \\substack { \\mathbf { m } \\\\ } } \\nu _ x ( \\mathbf { m } ) \\ , D _ { \\mathbf { m } } , \\end{align*}"}
{"id": "547.png", "formula": "\\begin{align*} b _ \\alpha ( s ) = \\begin{cases} b ( s ) & s \\in [ 0 , \\bar \\rho - \\alpha ] \\\\ b ( \\bar \\rho - \\alpha ) & s \\in ( \\bar \\rho - \\alpha , \\bar \\rho ) . \\end{cases} \\end{align*}"}
{"id": "7309.png", "formula": "\\begin{align*} \\boxed { \\frac { 1 } { \\det ( 1 _ n - D A ) } = \\sum _ { k _ 1 , \\ldots , k _ n \\ge 0 } c _ { k _ 1 , \\ldots , k _ n } X _ 1 ^ { k _ 1 } \\ldots X _ n ^ { k _ n } , } \\end{align*}"}
{"id": "6334.png", "formula": "\\begin{align*} \\widetilde \\Gamma ^ l _ { i j k } \\stackrel { ( i j k ) } { = } \\partial _ \\lambda ^ l \\widetilde \\Gamma ^ \\lambda _ { \\alpha \\beta \\gamma } \\partial ^ \\alpha _ i \\partial ^ \\beta _ j \\partial ^ \\gamma _ k + 2 \\partial ^ l _ \\lambda \\Gamma ^ \\lambda _ { ( \\alpha \\beta ) } \\partial ^ \\alpha _ i \\partial ^ \\beta _ { j k } + \\partial ^ l _ \\lambda \\partial ^ \\lambda _ { i j k } \\end{align*}"}
{"id": "1527.png", "formula": "\\begin{align*} K _ { v } ^ N ( \\mathfrak { n } ) = \\left \\{ \\gamma = \\left [ \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ] \\in G _ N ( \\mathbb { A } _ { v } ) \\cap M _ N ( \\mathcal { O } _ v ) : \\gamma \\equiv 1 _ N \\mathfrak { n } _ v \\right \\} . \\end{align*}"}
{"id": "5791.png", "formula": "\\begin{align*} k [ K ] = S / \\big ( v _ { i _ 1 } \\ ! \\cdots v _ { i _ t } \\ , : \\ , \\{ i _ 1 , \\ldots , i _ t \\} \\in { \\rm M F } ( K ) \\big ) . \\end{align*}"}
{"id": "3681.png", "formula": "\\begin{align*} \\underbrace { \\frac { 1 } { n ! } \\sum _ { g \\in \\mathrm { S y m } ( n ) } 2 ^ { c ( g _ E ) } } _ { } = \\underbrace { \\frac { 1 } { n ! } \\sum _ { \\substack { g \\in \\mathrm { S y m } ( n ) \\\\ | g | \\ \\mathrm { o d d } } } 2 ^ { c ( g _ E ) } } _ { } + \\underbrace { \\frac { 1 } { n ! } \\sum _ { \\substack { g \\in \\mathrm { S y m } ( n ) \\\\ | g | \\ \\mathrm { e v e n } } } 2 ^ { c ( g _ E ) } } _ { } . \\end{align*}"}
{"id": "2266.png", "formula": "\\begin{align*} z ^ 2 = z \\cdot z = z ^ T z , z \\in \\C ^ d . \\end{align*}"}
{"id": "8887.png", "formula": "\\begin{align*} d ( ( s _ 1 , t _ 1 ) , ( s _ 2 , t _ 2 ) ) = | s _ 2 - s _ 1 | + | t _ 2 - t _ 1 | . \\end{align*}"}
{"id": "586.png", "formula": "\\begin{align*} ( ( W _ 1 \\times I _ 1 ) \\cup ( Z _ 1 \\times J _ 1 ) ) \\cap ( ( W _ 2 \\times I _ 2 ) \\cup ( Z _ 2 \\times J _ 2 ) ) \\ = \\ \\emptyset . \\end{align*}"}
{"id": "5038.png", "formula": "\\begin{align*} \\rho _ { \\beta * } ( E _ { \\theta } ) \\cdot ( \\rho _ { \\beta } ( g ) \\cdot v _ { \\beta } ) & = { \\frac { d } { d t } } _ { | t = 0 } \\left ( \\rho _ { \\beta } ( g g ^ { - 1 } \\exp ( t \\ , E _ { \\theta } ) g ) \\cdot v _ { \\beta } \\right ) \\\\ & = \\rho _ { \\beta } ( g ) \\cdot \\rho _ { \\beta * } ( \\mathrm { A d } ( g ^ { - 1 } ) \\cdot E _ { \\theta } ) \\cdot v _ { \\beta } \\\\ & = \\rho _ { \\beta } ( g ) \\cdot 0 = 0 \\ , . \\end{align*}"}
{"id": "2150.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { E _ { n , k } ( \\tau ( A ^ { ( n ) } _ { c _ n k n , n , l } ) | \\mathcal { W A } ^ { ( n , k ) } _ { M _ n , n , l } ) } { n k } = \\frac { 1 - c } { k - l + 1 } . \\end{align*}"}
{"id": "1551.png", "formula": "\\begin{align*} \\sum _ { G ( \\mathcal { O } _ { \\mathbf { h } } ) \\backslash \\mathfrak { X } / G ( \\mathcal { O } _ { \\mathbf { h } } ) } \\lambda _ { \\mathbf { f } } ( \\xi ) \\cdot \\chi _ { \\mathbf { h } } ( \\det ( r ) ) | \\det ( r ) | _ { \\mathbf { h } } ^ { - s } \\mathbf { f } ( g ) = D ( s , \\mathbf { f } , \\chi ) \\mathbf { f } ( g ) . \\end{align*}"}
{"id": "8003.png", "formula": "\\begin{align*} \\overline { S } _ n ^ { \\boldsymbol { \\mu } , \\sigma } - \\sum _ { k = n + 1 } ^ { d } C _ { k } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\bigg ( \\int \\ ! \\log \\boldsymbol { \\mu } _ k ^ { \\sigma } \\ , \\mathrm { d } \\mathbf { p } _ { \\sigma _ k } + \\sum _ { \\ell = 1 } ^ { n } \\chi _ { \\ell } ^ { \\sigma } ( \\mathbf { p } _ { \\sigma _ k } ) \\big ( \\overline { S } _ { \\ell } ^ { \\boldsymbol { \\mu } , \\sigma } - \\overline { S } _ { \\ell - 1 } ^ { \\boldsymbol { \\mu } , \\sigma } \\big ) \\bigg ) , \\end{align*}"}
{"id": "3863.png", "formula": "\\begin{align*} h = - \\tilde \\sigma _ 0 ( 0 ) + \\tilde h . \\end{align*}"}
{"id": "9129.png", "formula": "\\begin{align*} \\norm { p - J ^ S _ { \\mu _ i } ( p + \\mu _ i a ) } & \\leq \\norm { p - q } + \\norm { q - J ^ S _ { \\mu _ i } ( q + \\mu _ i a ) } + \\norm { J ^ S _ { \\mu _ i } ( q + \\mu _ i a ) - J ^ S _ { \\mu _ i } ( p + \\mu _ i a ) } \\\\ & \\leq 2 \\norm { p - q } + \\norm { q - J ^ S _ { \\mu _ i } ( q + \\mu _ i a ) } \\\\ & \\leq \\frac { 2 } { \\omega ( k ) + 1 } + \\frac { 1 } { \\delta ( k ) + 1 } \\\\ & \\leq \\frac { 2 } { 4 ( k + 1 ) } + \\frac { 1 } { 2 ( k + 1 ) } \\\\ & = \\frac { 1 } { k + 1 } \\end{align*}"}
{"id": "7466.png", "formula": "\\begin{align*} g ( \\chi ) = \\inf \\limits _ { \\phi } \\mathcal { L } _ { \\sigma } ( \\phi , \\chi ) , \\end{align*}"}
{"id": "7339.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { \\star , \\lambda } ( x , t ) - u _ { q , \\lambda } ( x , t ) \\\\ & \\leq { 1 \\over q } \\max \\left \\{ u ( y _ j , t ) , u ( z _ j , t ) \\right \\} \\max \\{ { \\lambda } ^ { { 1 \\over q } - 1 } ( 1 - \\lambda ) , ( 1 - \\lambda ) ^ { { 1 \\over q } - 1 } \\lambda \\} ( 1 - r ^ q ) + { 1 \\over j } , \\end{aligned} \\end{align*}"}
{"id": "2267.png", "formula": "\\begin{align*} | z | ^ 2 = \\overline { z } \\cdot z = z ^ * z , z ^ * = \\overline { z } ^ T . \\end{align*}"}
{"id": "5134.png", "formula": "\\begin{align*} a \\ , \\gamma = \\frac { \\log \\left ( n + \\sqrt { n ^ 2 + 4 } \\right ) - \\log ( 2 ) } { n } . \\end{align*}"}
{"id": "3218.png", "formula": "\\begin{align*} \\lambda _ 1 ( T ) = \\lambda _ 2 ( \\mu _ { | T | } ) , \\qquad \\lambda _ 2 ( T ) = \\lambda _ 1 ( \\mu _ { | T | } ) . \\end{align*}"}
{"id": "6336.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { \\ell = 1 } ^ n d _ \\ell - \\sum _ { d _ \\ell \\geq \\delta } d _ \\ell = \\frac { 1 } { 2 } \\left ( \\sum _ { d _ \\ell < \\delta } d _ \\ell - \\sum _ { d _ \\ell \\geq \\delta } d _ \\ell \\right ) \\geq \\frac { \\delta d } { 2 } \\end{align*}"}
{"id": "8469.png", "formula": "\\begin{align*} \\lambda ^ 2 & \\leq \\max _ { u \\in V ( H _ k ) } \\big \\{ \\sum _ { w \\in V ( H _ k ) } \\ A ^ 2 _ { u , w } \\big \\} = \\max _ { u \\in V ( H _ k ) } \\big \\{ \\sum _ { v \\in N ( u ) } d ( v ) \\big \\} \\\\ & = \\max _ { u \\in V ( H _ k ) } \\big \\{ d ( u ) + 2 e ( N _ 1 ( u ) ) + e ( N _ 1 ( u ) , N _ 2 ( u ) ) \\} \\\\ & \\leq k d ( u ) + k ( n - 1 ) \\leq 2 k ( n - 1 ) . \\end{align*}"}
{"id": "1115.png", "formula": "\\begin{align*} g ( k , \\xi ) : = 4 X ^ 3 _ \\eta ( k ) , X _ { \\eta } ( k ) = \\sqrt { k ^ 2 - \\eta ^ 2 } , \\eta = \\sqrt { - 2 \\xi } \\in ( C _ R , C _ L ) . \\end{align*}"}
{"id": "6437.png", "formula": "\\begin{align*} M = M _ 1 \\oplus \\big ( \\bigoplus _ { \\alpha \\in \\Phi } M _ { \\overline { \\chi } ^ { \\langle \\alpha , \\lambda \\rangle } } \\big ) \\end{align*}"}
{"id": "6036.png", "formula": "\\begin{align*} P _ n ^ { ( \\alpha , \\beta ) } ( x ) & = \\frac { ( - 1 ) ^ { n } } { 2 ^ { n } n ! } ( 1 - x ) ^ { - \\alpha } ( 1 + x ) ^ { - \\beta } \\frac { d ^ n } { d x ^ n } \\Big ( ( 1 - x ) ^ { \\alpha + n } ( 1 + x ) ^ { \\beta + n } \\Big ) \\\\ & = \\frac { ( - 1 ) ^ { n } } { 2 ^ n } ( 1 - x ) ^ { - \\alpha } ( 1 + x ) ^ { - \\beta } \\frac { 1 } { 2 \\pi i } \\int _ C \\frac { ( 1 - r ) ^ { \\alpha + n } ( 1 + r ) ^ { \\beta + n } } { ( r - x ) ^ { n + 1 } } \\ , d r \\end{align*}"}
{"id": "3062.png", "formula": "\\begin{align*} x ^ \\prime = j x \\ , , y ^ \\prime = y \\ , , z ^ \\prime = j ^ 2 z \\ , , w ^ \\prime = w \\ , . \\end{align*}"}
{"id": "7711.png", "formula": "\\begin{align*} \\begin{aligned} A _ { m , n } & : = m \\{ 0 , 2 a \\} + \\{ 0 , 2 a , 3 a \\} + \\{ 0 , a , a ( 2 n + 5 ) \\} \\\\ & = \\big ( \\{ 0 \\} \\cup a \\cdot [ 2 , 2 m + 3 ] \\big ) + a \\cdot \\{ 0 , 1 , 2 n + 5 \\} \\\\ & = a \\cdot \\big ( [ 0 , 2 m + 4 ] \\cup \\{ 2 n + 5 \\} \\cup [ 2 n + 7 , 2 m + 2 n + 8 ] \\big ) \\ , . \\end{aligned} \\end{align*}"}
{"id": "4954.png", "formula": "\\begin{align*} ( D , D ^ + ) = ( ( \\prod _ { i \\in I } C ^ + _ i ) [ 1 / ( \\varpi _ i ) ] , \\prod _ { i \\in I } C ^ + _ i ) . \\end{align*}"}
{"id": "5150.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to 0 ^ + } \\sinh ( x ) - x ^ 2 \\coth ( x ) = 0 - \\lim \\limits _ { x \\to 0 ^ + } x \\cdot x \\ , \\coth ( x ) = - 0 \\cdot 1 = 0 . \\end{align*}"}
{"id": "4548.png", "formula": "\\begin{align*} f _ 1 \\geq 1 = I , f _ 4 , f _ 5 \\geq h - 1 \\geq h - e _ 6 = e _ 1 + \\dots + e _ 5 , f _ 6 \\geq h . \\end{align*}"}
{"id": "2370.png", "formula": "\\begin{align*} | F _ { ( - \\xi , - \\eta ) } ( 0 , 0 ) | = | V _ g \\left ( M _ { - \\eta } T _ { - \\xi } f \\right ) ( 0 , 0 ) | ^ 2 = | V _ g f ( \\xi , \\eta ) | ^ 2 , \\forall ( \\xi , \\eta ) \\in \\R ^ { 2 d } . \\end{align*}"}
{"id": "8934.png", "formula": "\\begin{align*} d _ { q - 1 } \\tilde \\varphi ( x _ 0 , \\ldots , x _ q ) & = \\sum _ { i = 0 } ^ q ( - 1 ) ^ i \\varphi ( 0 , x _ 0 , \\ldots , \\hat x _ i , \\ldots , x _ q ) \\\\ & = \\sum _ { i = 1 } ^ { q + 1 } ( - 1 ) ^ { i + 1 } \\varphi ( 0 , x _ 0 , \\ldots , \\hat x _ { i - 1 } , \\ldots , x _ q ) + d _ q \\varphi ( 0 , x _ 0 , \\ldots , x _ q ) \\\\ & = \\varphi ( x _ 0 , \\ldots , x _ q ) \\end{align*}"}
{"id": "5224.png", "formula": "\\begin{align*} B f ( x ) = \\sum \\limits _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } g _ j \\ast f ( x ) . \\end{align*}"}
{"id": "5300.png", "formula": "\\begin{align*} ( \\varphi ( - a ) \\check { \\delta } _ { \\check { \\varphi } } ) ( b ) = \\varphi ( \\kappa ^ { - 1 } ( b ) a ) = ( \\varphi \\circ S ^ 2 ) ( b \\kappa ( a ) ) ) = \\varphi ( b \\kappa ( a ) \\nu ^ { - 1 } ) , a , b \\in A . \\end{align*}"}
{"id": "6382.png", "formula": "\\begin{align*} \\sigma _ T ( { \\underline T } , \\mathcal X ) = \\{ \\lambda = ( \\lambda _ 1 , . . . , \\lambda _ n ) \\in \\mathbb { C } ^ n : { \\underline T } - \\lambda \\} . \\end{align*}"}
{"id": "2918.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 ( 2 ) & = \\frac { ( n - 2 ) ^ 2 ( n - 1 ) ( 8 n + 1 ) } { 3 2 4 0 0 n ^ 2 ( n + 1 ) ^ 2 } = \\frac 2 { 9 0 ^ 2 } - \\frac { 1 1 } { 6 4 8 0 n } + \\frac { 1 6 1 } { 3 2 4 0 0 n ^ 2 } + O ( n ^ { - 3 } ) , \\\\ \\sigma _ n ^ 2 ( 3 ) & = \\frac { ( n - 2 ) ( n - 1 ) ( 1 6 n ^ 5 - 9 6 n ^ 4 + 3 5 9 n ^ 3 - 2 6 9 n ^ 2 - 9 6 3 n - 3 7 0 ) } { 5 8 3 2 0 0 0 \\cdot n ^ 4 ( n + 1 ) ^ 3 } \\\\ & = \\frac 2 { 9 0 ^ 3 } - \\frac { 1 } { 3 0 3 7 5 n } + \\frac { 1 2 0 7 } { 5 8 3 2 0 0 0 n ^ 2 } + O ( n ^ { - 3 } ) . \\end{align*}"}
{"id": "8921.png", "formula": "\\begin{align*} B = W _ 1 ^ c \\cap \\cdots \\cap W _ n ^ c = \\left ( \\bigcup _ i W _ i \\right ) ^ c = \\emptyset . \\end{align*}"}
{"id": "3416.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\langle K ( x , y ) , \\lambda _ \\delta ( x , y ) \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle = 0 . \\end{align*}"}
{"id": "4216.png", "formula": "\\begin{align*} M ^ p _ p ( \\mu , x _ 0 ) = \\int _ { B } ( d ( x , x _ 0 ) ) ^ p \\ , d \\mu < \\infty , \\end{align*}"}
{"id": "65.png", "formula": "\\begin{align*} \\mathrm { r k } ( L ' _ { 2 , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( L ' _ { 2 , 1 } ) = 2 , \\\\ \\mathrm { r k } ( L ' _ { v , 0 } ) & = n , \\ \\mathrm { r k } ( L ' _ { v , 1 } ) = 1 ( v = p _ 1 , \\dots , p _ k ) , \\\\ \\mathrm { r k } ( K _ { \\ell , v , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( K _ { \\ell , v , 1 } ) = 1 ( v = 2 , p _ 1 , \\dots , p _ k ) . \\end{align*}"}
{"id": "8198.png", "formula": "\\begin{align*} M _ 2 ( p , H ) = { \\pi ^ 2 \\over 8 } \\left ( 1 - ( - 1 ) ^ { a } \\frac { 2 a + 1 } { p } \\right ) , \\end{align*}"}
{"id": "5205.png", "formula": "\\begin{align*} \\Omega = \\sum _ { i = 1 } ^ { \\infty } \\lambda _ i a _ i , \\end{align*}"}
{"id": "4222.png", "formula": "\\begin{align*} ( \\partial _ t - \\sigma \\Delta ) \\ \\widehat { \\mu } _ t = - \\div \\left ( \\widehat { v } ( t , \\cdot ) \\ , \\widehat { \\mu } _ t \\right ) . \\end{align*}"}
{"id": "6779.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m i _ { k _ j } = - ( v _ 1 + 1 ) + ( v _ 1 + 1 ) + \\sum _ { j = 1 } ^ { z ' } ( - ( v _ { q _ j } + 1 ) + ( v _ { q _ j } + 1 ) ) = 0 . \\end{align*}"}
{"id": "8715.png", "formula": "\\begin{align*} \\begin{aligned} 1 & - 3 x _ 2 + ( - 3 x _ 1 ) ( - 3 x _ 2 + 1 ) + ( - 3 x _ 1 + 1 ) ( - x _ 2 ^ 3 + 3 x _ 2 ^ 2 ) + ( - x _ 2 ^ 3 + 3 x _ 2 ^ 2 ) \\geq \\\\ & ( - 0 . 7 5 x _ 1 + 0 . 5 ) \\bigl ( ( 1 - x _ 2 ) ^ 3 + 0 . 7 5 x _ 2 \\bigr ) + \\bigl ( ( 1 - x _ 1 ) ^ 3 + 0 . 7 5 x _ 1 - 0 . 5 \\bigr ) 0 . 5 \\\\ & + ( 1 - x _ 1 ) ^ 3 ( - 0 . 7 5 x _ 2 ) . \\end{aligned} \\end{align*}"}
{"id": "2519.png", "formula": "\\begin{align*} \\Phi ( x , \\omega , \\tau ) = \\sum _ { k \\in \\Z } \\Phi _ k ( x , \\omega ) e ^ { 2 \\pi i k \\tau } , \\end{align*}"}
{"id": "6806.png", "formula": "\\begin{align*} u ( t ) = \\boldsymbol { V } ( t ) f _ { 0 } + \\int \\nolimits _ { 0 } ^ { t } \\boldsymbol { V } ( t - \\tau ) F ( u ( \\tau ) ) d \\tau , \\end{align*}"}
{"id": "2644.png", "formula": "\\begin{align*} \\langle X g , P g \\rangle = \\sum _ { k , l \\in \\Z } \\langle P g , T _ { - k } M _ { - l } g \\rangle \\langle T _ { - k } M _ { - l } g , X g \\rangle = \\langle P g , X g \\rangle . \\end{align*}"}
{"id": "7595.png", "formula": "\\begin{align*} | c _ { i , j } | \\le \\dfrac { 4 V _ l } { \\pi ^ 2 } \\begin{cases} \\Gamma _ { 0 , 0 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } l = 2 r , \\\\ \\Gamma _ { 1 , - 1 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } l = 2 r + 1 . \\end{cases} \\end{align*}"}
{"id": "8176.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) = { 2 \\pi ^ 2 \\mu ( d _ 0 ) \\over d _ 0 ^ 2 p } \\sum _ { \\delta \\mid d _ 0 } \\delta \\mu ( \\delta ) \\sum _ { h \\in H _ { d _ 0 } } s ( h , \\delta p ) \\end{align*}"}
{"id": "4466.png", "formula": "\\begin{align*} ( a ^ { \\dagger } a ) _ { k , \\lambda } = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( a ^ { \\dagger } ) ^ { l } a ^ { l } , ( k \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "7957.png", "formula": "\\begin{align*} \\mu _ T : = \\int _ 0 ^ T \\delta _ { B _ t ^ H } \\ , \\d t , \\end{align*}"}
{"id": "5514.png", "formula": "\\begin{align*} \\Phi _ { \\delta } ^ { x _ 0 } ( d , e , t ) = e ^ { \\beta t } d + \\varphi _ { \\beta } ( t ) ( e + 2 B ) \\end{align*}"}
{"id": "1817.png", "formula": "\\begin{align*} x = v + \\sqrt { v ^ 2 - u } , \\ ; \\ ; y = v - \\sqrt { v ^ 2 - u } . \\end{align*}"}
{"id": "4054.png", "formula": "\\begin{align*} u _ x + \\rho _ x + \\lambda \\rho = f , \\end{align*}"}
{"id": "9499.png", "formula": "\\begin{align*} 1 = | h ( z ) | = | ( f ( z ) + g ( z ) | / 2 \\le ( | f ( z ) | + | g ( z ) | ) / 2 \\le 1 \\end{align*}"}
{"id": "6466.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\Gamma ( s ) } = 0 \\mbox { f o r $ s = 0 , - 1 , - 2 , \\ldots $ } . \\end{align*}"}
{"id": "251.png", "formula": "\\begin{align*} S _ n = \\frac { 1 } { \\sqrt { n } } \\sum _ { k = 1 } ^ n X _ k , \\end{align*}"}
{"id": "7437.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { | x | \\geq M n } \\sup _ { s \\in [ 0 , T ] } | [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x - 1 } { n } ) - [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x } { n } ) | = 0 . \\end{align*}"}
{"id": "511.png", "formula": "\\begin{align*} Y ( \\mathcal { A } ) _ { \\omega \\beta } ^ { i v } = \\left ( Y ( \\mathcal { A } ) _ { \\alpha \\kappa _ { 2 } } ^ { g v } \\cdot Y ( \\mathcal { A } ) _ { \\kappa _ { 2 } \\beta } ^ { g v } \\right ) \\cdot \\left ( Y ( \\mathcal { A } ) _ { \\beta \\kappa _ { 2 } } ^ { g i } \\cdot Y ( \\mathcal { A } ) _ { \\kappa _ { 2 } \\alpha } ^ { g i } \\right ) \\cdot Y ( \\mathcal { A } ) _ { \\omega \\alpha } ^ { i v } \\end{align*}"}
{"id": "7360.png", "formula": "\\begin{align*} u _ 0 ( x ) = | x | + 1 , \\end{align*}"}
{"id": "2106.png", "formula": "\\begin{align*} D = \\sum c _ i D _ i , \\end{align*}"}
{"id": "3890.png", "formula": "\\begin{align*} N _ { f } ( p ) = c ( f , p ) \\left ( \\frac { \\varphi ( p - 1 ) } { p - 1 } \\right ) ^ 2 p + O \\left ( p ^ { 1 - 2 \\varepsilon } \\right ) , \\end{align*}"}
{"id": "5062.png", "formula": "\\begin{align*} \\varphi ( a ^ 2 \\cdot v _ 0 ) = [ \\varphi ( a ) , a \\cdot v _ 0 ] + [ a , \\varphi ( a \\cdot v _ 0 ) ] = 2 \\lambda ( a \\cdot v _ 0 ) a \\cdot v _ 0 + [ a , [ \\zeta , a ] ] , \\end{align*}"}
{"id": "7172.png", "formula": "\\begin{align*} \\int _ { \\Omega \\cap B _ { \\gamma / 2 } } e ^ { \\delta | w | } d x \\leq \\sum _ { j = 0 } ^ \\infty \\int _ { \\Omega \\cap B _ { \\gamma / 2 } } \\delta ^ j | w | ^ j / j ! \\leq c \\sum _ { j = 0 } ^ \\infty ( \\delta j ) ^ j / j ! \\leq c ( d , \\chi ) . \\end{align*}"}
{"id": "1946.png", "formula": "\\begin{align*} g ( x _ k ) \\leq g ( x ) = f ( x ) . \\end{align*}"}
{"id": "7729.png", "formula": "\\begin{gather*} \\phi _ u ( u , v ) = \\phi _ u ( t , x ) = \\frac { 1 } { 2 } \\left ( \\phi _ t + \\phi _ x \\right ) ( t , x ) , \\ ; \\ ; \\phi _ v ( t , x ) = \\frac { 1 } { 2 } \\left ( - \\phi _ x + \\phi _ t \\right ) ( t , x ) , \\\\ \\phi _ u \\cdot \\phi _ v = \\frac { 1 } { 4 } ( - | \\phi _ x | ^ 2 + | \\phi _ t | ^ 2 ) , \\ ; \\ ; \\phi _ { u v } = \\frac { 1 } { 4 } ( \\phi _ { t t } - \\phi _ { x x } ) . \\end{gather*}"}
{"id": "3102.png", "formula": "\\begin{align*} z ^ 3 x ^ 3 + z x f _ 4 ( x , y ) + f _ 6 ( x , y ) = 0 \\ , , \\end{align*}"}
{"id": "4200.png", "formula": "\\begin{align*} V = c ^ { - \\frac { n - 2 } { 4 } } \\Delta _ g ( c ^ { \\frac { n - 2 } { 4 } } ) . \\end{align*}"}
{"id": "7946.png", "formula": "\\begin{align*} | E _ a | & \\leqslant 5 2 n ^ { 1 / 4 } ( 1 + ( a / n ) ^ { 1 / 2 } n ^ { 1 / 8 } ) ( 1 + L _ { + } ^ { 1 / 2 } n ^ { - 1 / 8 } ) ~ ~ ( ~ ~ L _ { + } = \\max \\{ 0 , n ^ { 1 / 2 } - t \\} ) \\\\ & \\ll \\begin{cases} n ^ { 1 / 4 } + a ^ { 1 / 2 } n ^ { - 1 / 8 } , ~ & t \\geqslant n ^ { 1 / 2 } - n ^ { 1 / 4 } \\\\ \\left ( n ^ { 1 / 8 } + a ^ { 1 / 2 } n ^ { - 1 / 4 } \\right ) \\sqrt { n ^ { 1 / 2 } - t } , ~ & t < n ^ { 1 / 2 } - n ^ { 1 / 4 } . \\end{cases} \\end{align*}"}
{"id": "8630.png", "formula": "\\begin{align*} \\mu _ { R , 1 } ( k , \\ell , m , n ) : = \\sum _ { ( A , B , C , D ) \\in \\mathcal { X } _ { R } } \\int \\overline { \\mathcal { K } _ { A } ( x , k ) } \\mathcal { K } _ { B } ( x , \\ell ) \\overline { \\mathcal { K } _ { C } ( x , m ) } \\mathcal { K } _ { D } ( x , n ) \\ , d x , \\\\ \\mathcal { X } _ { R } : = \\left \\{ ( A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 } ) \\in \\{ S , R \\} ^ 4 \\ , : \\ , \\exists \\ , j = 0 , 1 , 2 , 3 \\ , : \\ , A _ { j } = R \\right \\} , \\end{align*}"}
{"id": "2768.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\iint \\frac { | u _ n ( x ) | ^ p | u _ n ( y ) | ^ p } { | x - y | ^ { N - \\gamma } } d x d y = \\lim _ { l \\to \\infty } \\sum _ { j = 1 } ^ l \\iint \\frac { | U ^ j ( x ) | ^ p | U ^ j ( y ) | ^ p } { | x - y | ^ { N - \\gamma } } d x d y . \\end{align*}"}
{"id": "476.png", "formula": "\\begin{align*} \\Psi ( P ) : = \\left \\{ \\mathbf { e } \\in \\mathbb { Z } ^ { d } : \\ , \\exists c _ { \\mathbf { e } } \\in \\mathbb { C } ^ { \\times } : c _ { \\mathbf { e } } \\mathbf { t } ^ { \\mathbf { e } } \\in \\mathrm { S u p p } ( P ) \\right \\} . \\end{align*}"}
{"id": "7616.png", "formula": "\\begin{align*} \\nabla _ { j } \\Phi & = \\bar { g } ( \\lambda \\partial _ { r } , e _ { j } ) , \\\\ \\nabla _ { i } \\nabla _ { j } \\Phi & = \\lambda ' \\delta _ { i j } - h _ { i j } \\bar { g } ( \\lambda \\partial _ { r } , \\nu ) = \\lambda ' \\delta _ { i j } - u h _ { i j } , \\end{align*}"}
{"id": "2471.png", "formula": "\\begin{align*} P = P ^ T , Q = Q ^ T , D = Q L P + L ^ { - T } = C A ^ { - 1 } B + A ^ { - T } . \\end{align*}"}
{"id": "6354.png", "formula": "\\begin{align*} \\frac r a = \\alpha _ 1 - \\frac { 1 } { \\alpha _ 2 - \\frac { 1 } { \\alpha _ 3 - \\frac { 1 } { ( . . . ) } } } \\eqqcolon [ \\alpha _ 1 , . . . , \\alpha _ n ] \\end{align*}"}
{"id": "4679.png", "formula": "\\begin{align*} \\R _ i ( t , y ) = Q _ { 1 + \\mu _ i ( t ) } ( y - x _ i ( t ) ) . \\end{align*}"}
{"id": "7010.png", "formula": "\\begin{align*} 0 < \\eta \\le \\inf \\Big \\{ | A ( z ) | : z \\notin \\bigcup _ { j = 1 } ^ { n } \\mathfrak B _ j \\Big \\} . \\end{align*}"}
{"id": "735.png", "formula": "\\begin{align*} K _ { \\alpha \\alpha } ^ { ( 0 ) } = \\frac { 1 } { n _ 0 } \\norm { x _ \\alpha } ^ 2 = K = \\frac { 1 } { n _ 0 } \\norm { x _ \\beta } ^ 2 = K _ { \\beta \\beta } ^ { ( 0 ) } , K > 0 . \\end{align*}"}
{"id": "6774.png", "formula": "\\begin{align*} \\pi = \\tau _ { r _ { 1 } } ^ { - ( v _ { 1 } + 1 ) } \\cdot \\tau _ { r _ { 1 } + 1 } ^ { v _ { 1 } + 1 } \\cdot \\tau _ { r _ { 2 } } ^ { - ( v _ { 2 } + 1 ) } \\cdot \\tau _ { r _ { 2 } + 1 } ^ { v _ { 2 } + 1 } \\cdots \\tau _ { r _ { z } } ^ { - ( v _ { z } + 1 ) } \\cdot \\tau _ { r _ { z } + 1 } ^ { v _ { z } + 1 } . \\end{align*}"}
{"id": "8283.png", "formula": "\\begin{align*} I ( A ( x , y ) , z ) = I ( x , I ( y , z ) ) . \\end{align*}"}
{"id": "7701.png", "formula": "\\begin{align*} \\max \\big \\{ \\max \\mathsf L ( a ) / \\min \\mathsf L ( a ) \\colon a \\in S \\big \\} = n _ t / n _ 1 \\min \\Delta ( S ) = \\gcd ( n _ 2 - n _ 1 , \\ldots , n _ t - n _ { t - 1 } ) \\ , . \\end{align*}"}
{"id": "818.png", "formula": "\\begin{align*} \\int _ \\gamma g _ 0 \\ , d s _ \\rho = \\int _ \\gamma g _ 0 \\ , \\rho \\ , d s \\end{align*}"}
{"id": "6389.png", "formula": "\\begin{align*} V _ { S ^ { \\prime } } = \\{ ( z _ 1 , \\dots , z _ { n - 1 } ) \\in \\mathbb C ^ { n - 1 } \\ , : \\ , f _ i ( z _ 1 , \\dots , z _ { n - 1 } , t _ n ) = 0 \\ , , \\ , 1 \\leq i \\leq n \\} . \\end{align*}"}
{"id": "4761.png", "formula": "\\begin{align*} \\mathcal { X } = & \\mathcal { M } ( D _ 2 , \\cdots , D _ { 5 } , D _ 7 , D _ 8 , D _ { 9 } , D _ { 1 1 } , D _ { 1 2 } , D _ { 1 3 } , D _ { 1 6 } , D _ { 1 9 } , D _ { 2 0 } , D _ { 2 3 } ) \\\\ \\mathcal { Y } = & \\mathcal { M } ( D _ 6 , D _ { 1 0 } , D _ { 1 4 } , D _ { 1 5 } , D _ { 1 7 } , D _ { 1 8 } , D _ { 2 1 } , D _ { 2 2 } , D _ { 2 4 } ) . \\end{align*}"}
{"id": "7707.png", "formula": "\\begin{align*} \\max ( A ) - \\max ( B ) & = \\max ( C ) - \\max ( D ) \\ , \\\\ \\min ( A ) - \\min ( B ) & = \\min ( C ) - \\min ( D ) \\ , . \\end{align*}"}
{"id": "9021.png", "formula": "\\begin{align*} E _ 0 = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } ( f + \\sum _ { i = 1 } ^ s z _ i \\rho _ i ) \\phi \\bigg ) d x , \\end{align*}"}
{"id": "2521.png", "formula": "\\begin{align*} \\rho ( F _ r ) & = \\iint _ { \\R ^ { 2 d } } \\int _ 0 ^ 1 F ( x , \\omega ) e ^ { - 2 \\pi i \\tau } \\rho ( x , \\omega ) e ^ { 2 \\pi i \\tau } \\ , d \\tau \\ , d ( x , \\omega ) \\\\ & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\rho ( x , \\omega ) \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "8494.png", "formula": "\\begin{align*} F ( u ) & = 1 + z \\sum \\limits _ { k \\geq 1 } u ^ k \\biggl ( 1 + \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } g _ \\ell + \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } h _ \\ell \\biggr ) \\\\ & = 1 + \\frac { z u } { 1 - u } + z \\sum \\limits _ { k \\geq 0 } \\frac { u ^ { k + 1 } } { 1 - u } g _ k + z \\sum \\limits _ { k \\geq 0 } \\frac { u ^ { k + 1 } } { 1 - u } h _ k \\\\ & = 1 + \\frac { z u } { 1 - u } ( 1 + G ( u ) + H ( u ) ) , \\\\ G ( u ) & = \\frac { z } { u } ( F ( u ) + H ( u ) - 1 - H ( 0 ) ) , \\\\ H ( u ) & = z ( F ( u ) + G ( u ) ) . \\end{align*}"}
{"id": "5642.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = \\varepsilon ^ { - 2 } \\left ( e ^ { u } - \\left ( x ^ { 2 } + y ^ { 2 } \\right ) e ^ { - u } \\right ) & \\Omega , \\\\ u & = g & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
{"id": "4182.png", "formula": "\\begin{align*} V ^ \\omega : = \\{ w \\in T _ y M \\ | \\ \\omega _ y ( w , v ) = 0 , \\ \\forall v \\in V \\} \\end{align*}"}
{"id": "1935.png", "formula": "\\begin{align*} g ( x - \\alpha \\nabla g ( x ) ) & \\leq g ( x ) + \\langle - \\alpha \\nabla g ( x ) , \\nabla g ( x ) \\rangle + \\dfrac { L _ g } { 2 } \\norm { \\alpha \\nabla g ( x ) } ^ 2 \\\\ & = g ( x ) + \\ ! \\left ( \\dfrac { \\alpha L _ g } { 2 } - 1 \\right ) \\ ! \\alpha \\norm { \\nabla g ( x ) } ^ 2 . \\end{align*}"}
{"id": "3086.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = a ^ 2 \\varepsilon ^ 4 x _ 2 \\ , , x _ 2 ^ \\prime = \\frac { \\varepsilon } { a } x _ 1 \\ , , y _ 1 ^ \\prime = \\varepsilon ^ 3 y _ 2 \\ , , y _ 2 ^ \\prime = \\frac { \\varepsilon ^ 2 } { a } y _ 1 \\ , , \\end{align*}"}
{"id": "3680.png", "formula": "\\begin{align*} | S | = K \\frac { n ! } { 2 } = \\sum _ { \\substack { g \\in \\mathrm { S y m } ( n ) \\\\ | g | \\ \\mathrm { e v e n } } } 2 ^ { c ( g _ E ) - 1 } \\end{align*}"}
{"id": "8766.png", "formula": "\\begin{align*} \\begin{aligned} s ^ t _ { i ' j ' } & = G ^ { t } _ { i ' j ' } ( z , \\delta , w ) : = a _ { i ' \\tau ( i ' , t - 1 ) } ( \\delta _ { i t - 1 } - \\delta _ { i t } ) \\\\ & + \\sum _ { j = \\tau ( i ' , t - 1 ) + 1 } ^ { \\tau ( i ' , t ) - 1 } ( m ^ { t } _ { i ' j } - m ^ { t } _ { i ' j - 1 } ) z _ { i j } + ( m ^ t _ { i ' j ' } - m ^ t _ { i ' \\tau ( i ' t ) - 1 } ) w ^ t _ { i ' j ' } . \\end{aligned} \\end{align*}"}
{"id": "6875.png", "formula": "\\begin{align*} \\Phi ( S ) = \\underset { v \\sim \\pi _ k | _ S } { \\mathbb { E } } \\left [ M ( v , X ( k ) \\setminus S ) \\right ] , \\end{align*}"}
{"id": "5828.png", "formula": "\\begin{align*} D _ { m } = \\hat { D } _ { m } \\cap \\bar { D } _ { m } \\end{align*}"}
{"id": "9352.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - x ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } F _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 1 2 ] ) . \\end{align*}"}
{"id": "5246.png", "formula": "\\begin{align*} \\left ( \\forall y \\in A \\otimes ^ I B : x y = 0 \\right ) \\Rightarrow x E = 0 , \\left ( \\forall y \\in A \\otimes ^ I B : y x = 0 \\right ) \\Rightarrow E x = 0 . \\end{align*}"}
{"id": "6583.png", "formula": "\\begin{align*} \\mathcal { U } ( h , k ) : = \\frac { 1 } { 2 } \\sum _ { \\substack { 1 \\leq q < \\infty \\\\ ( q , h k ) = 1 } } W \\left ( \\frac { q } { Q } \\right ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , q ) = 1 \\\\ m h \\neq n k } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { 1 \\leq c \\leq C , d \\geq 1 \\\\ c d = q \\\\ d | m h \\pm n k } } \\phi ( d ) \\mu ( c ) , \\end{align*}"}
{"id": "8732.png", "formula": "\\begin{align*} p ( x ) = \\phi ( g _ { j ( 1 ) } ( x ) , \\ldots , g _ { j ( d ) } ( x ) ) , \\deg ( p ) \\leq \\delta , \\end{align*}"}
{"id": "3335.png", "formula": "\\begin{align*} [ x , y , z ] = [ [ x , y ] , z ] , \\forall x , y , z \\in L . \\end{align*}"}
{"id": "5185.png", "formula": "\\begin{align*} \\Delta _ { \\ell } ^ { a v } ( t ) = \\frac { 1 } { t } \\int _ { 0 } ^ { t } \\Delta _ { \\ell } ( i ) d i . \\end{align*}"}
{"id": "2042.png", "formula": "\\begin{align*} \\mathcal { M } \\left ( \\lambda \\right ) = \\lambda ^ 2 G + \\lambda D + C , \\lambda \\in \\mathbf { C } , \\end{align*}"}
{"id": "2040.png", "formula": "\\begin{align*} F ( r , t ) : = F ( f , r , t ) = \\sup _ { s > 0 } \\left [ \\frac { r } { s } - \\frac { t } { f ( s ) } \\right ] . \\end{align*}"}
{"id": "5990.png", "formula": "\\begin{align*} \\| \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( f ( z ) , \\cdot ) \\| _ { q , \\kappa , \\pi _ { \\lambda , \\beta } } ^ { 2 } & = \\sum _ { n = 0 } ^ { \\infty } ( n ! ) ^ { 1 + \\kappa } 2 ^ { n q } \\frac { | z | ^ { 2 n } } { ( n ! ) ^ { 2 } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { 2 ^ { n \\kappa } } \\frac { \\left ( 2 ^ { \\kappa } 2 ^ { q } | z | ^ { 2 } \\right ) ^ { n } } { ( n ! ) ^ { 1 - \\kappa } } . \\end{align*}"}
{"id": "17.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\mu ( \\mathbb { E } ^ M ) = \\lim _ { n \\to \\infty } \\mu ( \\mathbb { P } ^ M ) = 0 , \\end{align*}"}
{"id": "1698.png", "formula": "\\begin{align*} M = \\left \\{ f : \\Omega \\rightarrow \\R , \\ ; \\ ; \\left \\| \\frac { \\nabla ^ r f } { g } \\right \\| _ { L _ { p _ 1 } ( \\Omega ) } \\le 1 , \\ ; \\ ; \\| w f \\| _ { L _ { p _ 0 } ( \\Omega ) } \\le 1 \\right \\} , \\end{align*}"}
{"id": "4020.png", "formula": "\\begin{align*} \\mod { \\frac { I _ 2 ( w ) } { G ( w ) } } & = \\mod { \\frac { \\cos ( w ) } { \\sin ( w ) } } \\frac { \\mod { O ( w ^ { - 1 } ) e ^ { - w ^ 2 + 1 + O ( w ^ { - 1 } ) } + O ( w ^ { - 1 } ) e ^ { O ( w ^ { - 1 } ) } } } { \\mod { e ^ { - w ^ 2 + 1 } - 1 } } \\leq \\frac { C } { \\mod { w } } \\mod { \\frac { \\cos ( w ) } { \\sin ( w ) } } \\frac { \\mod { e ^ { - w ^ 2 + 1 } } + 1 } { \\mod { e ^ { - w ^ 2 + 1 } - 1 } } , \\end{align*}"}
{"id": "5970.png", "formula": "\\begin{align*} x _ { k } ( t ) = e ^ { - 2 \\pi ^ { 2 } s ^ { 2 } \\left ( t - t _ { 0 } \\right ) ^ { 2 } } , t _ 0 = \\sqrt { \\frac { \\log ( \\epsilon ) } { - 2 \\pi ^ 2 s ^ 2 } } , , s = \\sqrt { \\frac { - f _ r ^ { 2 } } { 2 \\ln ( r ) } } , \\end{align*}"}
{"id": "5110.png", "formula": "\\begin{align*} \\psi ( x ) > q _ 1 ( 1 . 0 1 ) + 6 \\ , q _ 2 ( 1 . 1 5 8 ) = 0 . 0 5 7 5 8 4 > 0 . \\end{align*}"}
{"id": "9322.png", "formula": "\\begin{align*} \\nabla _ { \\mathbf { x } } \\mathcal { L } _ k ( \\mathbf { x } , \\mathbf { y } ) = ( H + \\rho I ) \\mathbf { x } - A ^ T \\mathbf { y } + \\mathbf { g } - \\rho { \\mathbf { x } _ k } - \\begin{bmatrix} 0 \\\\ \\frac { \\mu } { x _ { \\bar { d } + 1 } } \\\\ \\vdots \\\\ \\frac { \\mu } { x _ { d } } \\end{bmatrix} = 0 ; \\\\ - \\nabla _ { \\mathbf { y } } \\mathcal { L } _ k ( \\mathbf { x } , \\mathbf { y } ) = ( A \\mathbf { x } + \\delta ( \\mathbf { y } - \\mathbf { y } _ k ) - \\mathbf { b } ) = 0 . \\end{align*}"}
{"id": "3471.png", "formula": "\\begin{gather*} e _ { \\infty } ( K / k ) = \\frac { l ^ n } { \\gcd \\big ( \\frac { d _ P } w \\frac { { \\eta } ^ w - 1 } { { \\eta } - 1 } , l ^ n \\big ) } . \\end{gather*}"}
{"id": "4756.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c c } A _ { 1 1 } & A _ { 1 2 } & \\cdots & A _ { 1 n } \\\\ A _ { 2 1 } & A _ { 2 2 } & \\cdots & A _ { 2 n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ { n 1 } & A _ { n 2 } & \\cdots & A _ { n n } \\end{array} \\right ) \\left ( \\begin{array} { c c c c } C _ 1 \\\\ C _ 2 \\\\ \\vdots \\\\ C _ n \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } X _ 1 \\\\ X _ 2 \\\\ \\vdots \\\\ X _ n \\end{array} \\right ) . \\end{align*}"}
{"id": "8383.png", "formula": "\\begin{align*} \\mathfrak L ( U , \\iota ) = \\mathfrak L ( | U | , \\iota ) = \\{ \\mathbf L \\in \\mathfrak L ( U ) : \\mu _ U ( \\mathbf L ) \\geq \\iota \\} \\ , . \\end{align*}"}
{"id": "3720.png", "formula": "\\begin{align*} \\widehat B _ t ^ N ( k , t ) = i \\sum _ { m + n = k , | m | , | n | , | k | \\leq N } m | n | \\widehat B ( m ) \\widehat B ( n ) - \\mu | k | ^ \\alpha \\widehat B ( k ) . \\end{align*}"}
{"id": "3128.png", "formula": "\\begin{align*} a _ i x _ 1 ^ 2 + b _ i x _ 2 ^ 2 + \\psi _ i ( x _ 3 , x _ 4 , x _ 5 ) = 0 \\ , . \\end{align*}"}
{"id": "9168.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\geq 1 \\\\ ( m , 2 ) = 1 } } \\frac { \\lambda _ f ( l _ 1 m ^ 2 ) } { m ^ { 1 + 2 s } } = \\lambda _ f ( l _ 1 ) \\zeta ( 2 + 4 s ) ^ { - 1 } L ( 1 + 2 s , \\operatorname { s y m } ^ 2 f ) \\prod _ { p | 2 l } \\left ( 1 + O \\left ( \\frac { 1 } { p ^ { 1 + 2 s } } \\right ) \\right ) ^ { - 1 } \\end{align*}"}
{"id": "3438.png", "formula": "\\begin{align*} | S | & \\leqslant \\int _ { \\| u - y \\| \\leqslant t } | f _ t ( x , u ) - f _ t ( x , y ) | \\cdot | g _ s ( u , y ) | d \\omega ( u ) \\\\ & \\quad + \\int _ { \\| u - y \\| > t } \\big ( | f _ t ( x , u ) | + | f _ t ( x , y ) | \\big ) \\cdot | g _ s ( u , y ) | d \\omega ( u ) \\\\ & = : I + I \\ ! I , \\end{align*}"}
{"id": "6064.png", "formula": "\\begin{align*} c _ j = \\inf _ { z \\in Z _ j ( V _ j ) } \\abs { H _ j ' ( z ) } \\quad M _ j = \\sup _ { ( s , t ) \\in V _ j } \\abs { D \\Phi _ j ( s , t ) } \\end{align*}"}
{"id": "2752.png", "formula": "\\begin{align*} - \\Delta Q + Q - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 1 } = 0 . \\end{align*}"}
{"id": "6548.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } u + ( u \\cdot \\nabla ) u + \\nabla p = ( b \\cdot \\nabla ) b , x \\in \\mathbb { R } ^ { 2 } , \\ , t > 0 , \\\\ & \\partial _ { t } b + ( u \\cdot \\nabla ) b + \\mathcal { L } b = ( b \\cdot \\nabla ) u , \\\\ & \\nabla \\cdot u = 0 , \\ \\ \\ \\nabla \\cdot b = 0 , \\\\ & u ( x , 0 ) = u _ { 0 } ( x ) , \\ , \\ , b ( x , 0 ) = b _ { 0 } ( x ) , \\endaligned \\right . \\end{align*}"}
{"id": "1471.png", "formula": "\\begin{align*} \\alpha \\left [ \\begin{array} { c c } z & 1 \\\\ 1 & - \\overline { z } \\end{array} \\right ] = \\left [ \\begin{array} { c c } \\alpha z & 1 \\\\ 1 & - \\overline { \\alpha z } \\end{array} \\right ] \\left [ \\begin{array} { c c } \\lambda ( \\alpha , z ) & 0 \\\\ 0 & \\overline { \\lambda ( \\alpha , z ) } \\end{array} \\right ] , \\end{align*}"}
{"id": "8146.png", "formula": "\\begin{align*} h _ K ^ - = \\frac { w _ K } { \\Pi _ { d _ 0 } ( p , H ) } \\left ( \\frac { p } { 4 \\pi ^ 2 } \\right ) ^ { m / 4 } \\prod _ { \\chi \\in X _ K ^ - } L ( 1 , \\chi ' ) \\leq w _ K \\left ( \\frac { p M _ { d _ 0 } ( p , H ) } { 4 \\pi ^ 2 D _ { d _ 0 } ( p , H ) } \\right ) ^ { m / 4 } . \\end{align*}"}
{"id": "3486.png", "formula": "\\begin{align*} \\frac { d w } { d u } & = 1 - \\frac { 1 } { 2 \\pi } \\left ( \\frac { t _ 1 } { u } + \\frac { t _ 3 } { u + n } \\right ) , \\\\ \\frac { d ^ 2 w } { d u ^ 2 } & = \\frac { 1 } { 2 \\pi } \\left ( \\frac { t _ 1 } { u ^ 2 } + \\frac { t _ 3 } { ( u + n ) ^ 2 } \\right ) > 0 . \\end{align*}"}
{"id": "4613.png", "formula": "\\begin{align*} f _ k ( Q ) - f _ k ( \\pi _ v ( Q ) ) \\geq { \\lceil \\frac { d } { 2 } \\rceil \\choose d - k - 1 } + { \\lfloor \\frac { d } { 2 } \\rfloor \\choose d - k - 1 } = 2 \\rho ( d , d - k - 1 ) . \\end{align*}"}
{"id": "3169.png", "formula": "\\begin{align*} \\max _ { j \\in \\lbrace 1 , \\ldots , \\epsilon \\rbrace } \\frac { \\norm { E _ j ^ \\intercal ( A x _ 0 - b ) } _ 2 ^ 2 } { \\norm { A x _ 0 - b } _ 2 ^ 2 \\norm { A ^ \\intercal E _ j } _ F ^ 2 } \\geq \\sum _ { j = 1 } ^ \\epsilon \\frac { \\norm { A ^ \\intercal E _ j } _ F ^ 2 } { \\norm { A } _ F ^ 2 } \\frac { \\norm { E _ j ^ \\intercal ( A x _ k - b ) } _ 2 ^ 2 } { \\norm { A x _ 0 - b } _ 2 ^ 2 \\norm { A ^ \\intercal E _ j } _ F ^ 2 } = \\frac { 1 } { \\norm { A } _ F ^ 2 } . \\end{align*}"}
{"id": "2794.png", "formula": "\\begin{align*} \\Phi _ 1 ( f _ n ) & = \\frac { 1 } { 2 } \\int | \\nabla f _ n | ^ 2 + \\frac { 1 } { 2 } \\int | f _ n | ^ 2 - \\frac { p } { 2 } \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * ( Q ^ { p - 1 } f _ n ) \\right ) ( Q ^ { p - 1 } f _ n ) \\\\ & - \\frac { p - 1 } { 2 } \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ { p } \\right ) Q ^ { p - 2 } | f _ n | ^ 2 = o _ n ( 1 ) , \\end{align*}"}
{"id": "2391.png", "formula": "\\begin{align*} \\norm { f - \\sum _ { k = 1 } ^ N e _ { \\pi ( k ) } } _ \\mathcal { B } < \\varepsilon , N \\geq N _ 0 . \\end{align*}"}
{"id": "5645.png", "formula": "\\begin{align*} \\Delta | \\nabla u | ^ 2 & = < \\nabla \\Delta u , \\nabla u > + | \\nabla ^ 2 u | ^ 2 \\\\ & = \\varepsilon ^ { - 2 } ( e ^ u + r ^ 2 e ^ { - u } ) | \\nabla u | ^ 2 - \\varepsilon ^ { - 2 } e ^ { - u } < \\nabla r ^ 2 , \\nabla u > + | \\nabla ^ 2 u | ^ 2 . \\end{align*}"}
{"id": "7082.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\sum _ { i , j = 1 } ^ { 2 } \\langle [ R ( X _ i , X _ j ) , A ] X _ i , A X _ j \\rangle & = & \\sum _ { i , j = 1 } ^ { 2 } \\langle [ R ( \\widetilde X _ i , \\widetilde X _ j ) , A ] \\widetilde X _ i , A \\widetilde X _ j \\rangle \\\\ \\\\ & = & - \\langle R ( \\widetilde X _ 1 , \\widetilde X _ 2 ) \\widetilde X _ 1 , \\widetilde X _ 2 \\rangle ( \\lambda _ 1 - \\lambda _ 2 ) ^ 2 \\\\ \\\\ & = & 2 K ( | A | ^ 2 - 2 f ^ 2 ) , \\end{array} \\end{align*}"}
{"id": "1618.png", "formula": "\\begin{align*} \\tilde { \\varphi } = \\alpha _ 1 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 1 ) } + \\alpha _ 2 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 2 ) } + \\alpha _ 3 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 3 ) } \\end{align*}"}
{"id": "5612.png", "formula": "\\begin{align*} \\Omega _ { \\perp \\star } : = \\{ \\omega \\in \\Omega _ \\star ~ | ~ \\langle \\Xi _ { t \\star } , \\omega \\rangle _ \\star = 0 \\} , \\end{align*}"}
{"id": "7359.png", "formula": "\\begin{align*} u _ t + | \\nabla u | m ( K \\cap \\{ u ( \\cdot , t ) < u ( x , t ) \\} ) = 0 , \\end{align*}"}
{"id": "6700.png", "formula": "\\begin{align*} \\Psi ^ { - 1 } | _ { t = \\theta } = \\begin{pmatrix} \\tilde { \\pi } ^ s & 0 \\\\ - \\tilde { \\pi } ^ s L i _ { K , s } ( \\alpha ) & \\tilde { \\pi } ^ s \\end{pmatrix} . \\end{align*}"}
{"id": "358.png", "formula": "\\begin{align*} \\mathcal A ( \\rho , m ) = \\frac 1 2 \\int _ 0 ^ 1 \\Big ( \\frac { m _ { 1 2 } ^ 2 } { \\theta _ { 1 2 } ( \\rho ) } + \\frac { m _ { 2 3 } ^ 2 } { \\theta _ { 2 3 } ( \\rho ) } \\Big ) d t , \\end{align*}"}
{"id": "1313.png", "formula": "\\begin{align*} I ( \\alpha _ { k _ { n } } ) - J _ { 0 } ( \\alpha _ { k _ { n } } ) = 2 n c _ { s \\gamma } + 2 \\lfloor n \\theta \\rfloor + 1 \\end{align*}"}
{"id": "4084.png", "formula": "\\begin{align*} 0 \\ne g \\cdot v = g \\cdot ( u + a e _ t ) = g \\cdot u + g \\cdot a e _ t = a g _ t . \\end{align*}"}
{"id": "5928.png", "formula": "\\begin{align*} ( \\lambda + 2 \\nu ) \\partial _ { 1 1 } \\phi ^ 1 + ( \\lambda + \\nu ) \\partial _ { 1 2 } \\phi ^ 2 + \\nu \\partial _ { 2 2 } \\phi ^ 1 & = 0 , \\\\ \\nu \\partial _ { 1 1 } \\phi ^ 2 + ( \\lambda + \\nu ) \\partial _ { 1 2 } \\phi ^ 1 + ( \\lambda + 2 \\nu ) \\partial _ { 2 2 } \\phi ^ 2 & = 0 . \\end{align*}"}
{"id": "4585.png", "formula": "\\begin{align*} \\Big ( \\lambda + c _ \\alpha \\ , \\lambda ^ 2 \\epsilon _ n \\Big ) \\Big ( 1 + \\delta _ n ( \\lambda ) \\Big ) = x \\sqrt { 1 - \\delta _ n ( \\lambda ) } . \\end{align*}"}
{"id": "4022.png", "formula": "\\begin{align*} \\mod { \\frac { I _ 1 ( \\mu ) } { G ( \\mu ) } } & = \\frac { 1 } { \\mod { \\sin ( \\mu ^ { \\frac { 1 } { 2 } } ) } } \\mod { \\frac { O ( \\mu ^ { - \\frac { 1 } { 2 } } ) e ^ { - \\mu + 1 + O ( \\mu ^ { - \\frac { 1 } { 2 } } ) } + O ( \\mu ^ { - 1 } ) e ^ { O ( \\mu ^ { - \\frac { 1 } { 2 } } ) } } { e ^ { - \\mu + 1 } - 1 } } \\leq \\frac { C } { \\mod { \\mu } ^ { \\frac { 1 } { 2 } } } \\frac { 1 } { \\mod { \\sin ( \\mu ^ { \\frac { 1 } { 2 } } ) } } \\frac { \\mod { e ^ { - \\mu + 1 } } + 1 } { \\mod { e ^ { - \\mu + 1 } - 1 } } , \\end{align*}"}
{"id": "174.png", "formula": "\\begin{align*} \\mathcal { E } _ m ( g , \\tilde { f } _ m ) = \\langle f ; g \\rangle _ { L ^ 2 ( \\mu _ m ) } , \\tilde { f } _ m = \\int _ 0 ^ { + \\infty } \\mathcal { P } ^ m _ t ( f ) d t . \\end{align*}"}
{"id": "9081.png", "formula": "\\begin{align*} \\begin{cases} \\ ! h _ { 1 } = \\sigma x _ { 0 } + j \\sigma y _ { 0 } \\\\ \\ ! h _ { k } = \\sigma \\left ( { \\sqrt { 1 - \\mu _ { k } ^ { 2 } } x _ { k } + \\mu _ { k } x _ { 0 } } \\right ) \\\\ \\quad \\ : \\ : + j \\sigma \\left ( { \\sqrt { 1 - \\mu _ { k } ^ { 2 } } y _ { k } + \\mu _ { k } y _ { 0 } } \\right ) \\ : \\ : k = 2 , { \\cdots } , N , \\end{cases} \\end{align*}"}
{"id": "8064.png", "formula": "\\begin{align*} \\left \\langle E ^ { i j } _ 0 , ( { m _ { \\Lambda } } _ * f ) \\otimes ( { m _ { \\Lambda } } _ * g ) \\right \\rangle = \\Lambda ^ { 2 - \\mu _ i + \\mu _ j } \\left ( \\mathfrak { P } _ { \\ell } m _ { \\Lambda } \\left \\{ \\Psi ^ i ( f ) , \\Psi ^ j ( g ) \\right \\} _ { \\ell } ^ { \\Sigma _ 0 } \\right ) [ 0 ] . \\end{align*}"}
{"id": "6227.png", "formula": "\\begin{align*} \\Psi _ { n + 1 } ( { \\bf x } ) \\equiv ( M _ A \\Psi _ { n } ) ( { \\bf x } ) = \\sum _ { j = 1 } ^ { d } \\Big ( P _ { 2 j - 1 } A \\Psi _ { n } ( { \\bf x } + { \\bf e } _ j ) + P _ { 2 j } A \\Psi _ { n } ( { \\bf x } - { \\bf e } _ j ) \\Big ) . \\end{align*}"}
{"id": "770.png", "formula": "\\begin{align*} P ( x ) = A x ^ 4 + B x ^ 3 + C x ^ 2 + 1 \\end{align*}"}
{"id": "5126.png", "formula": "\\begin{align*} \\frac { f ' ( \\gamma ) } { f ( \\gamma ) } = - a + \\frac { 1 } { \\gamma } - a \\coth ( \\gamma a ) = - a + \\frac { 1 } { \\gamma } \\underbrace { ( 1 - \\gamma a \\coth ( \\gamma a ) ) } _ { < 0 , \\ , \\searrow } , \\end{align*}"}
{"id": "6624.png", "formula": "\\begin{align*} \\mathcal { K } ( s _ 1 , - s _ 1 - \\alpha - \\beta , 2 ) = \\left ( \\frac { h } { k } \\right ) ^ { s _ 1 } \\mathcal { K } ( 0 , - \\alpha - \\beta , 2 ) . \\end{align*}"}
{"id": "7105.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & \\lambda ^ 2 P _ { n - 1 } - \\lambda ( \\lambda ^ 2 - 4 ) ^ { n - 1 } - 4 P _ { n - 1 } - 2 ( \\lambda ^ 2 - 4 ) ^ { n - 1 } - 2 ( \\lambda ^ 2 - 4 ) ^ { n - 1 } - \\lambda ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\\\ = & ( \\lambda ^ 2 - 4 ) P _ { n - 1 } - 2 ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\end{align*}"}
{"id": "1197.png", "formula": "\\begin{align*} & \\bullet \\limsup _ { B \\in \\mathcal { Q } } B = [ 0 , 1 ] . \\\\ & \\bullet \\delta \\geq 1 , \\dim _ H ( \\limsup _ { B \\in \\mathcal { Q } } B ^ { \\delta } ) = \\frac { 1 } { \\delta } . \\end{align*}"}
{"id": "8474.png", "formula": "\\begin{align*} k n & < e ( S _ 1 ( x ) , L _ 1 ( x ) \\setminus \\{ z \\} ) + | N _ 1 ( x ) \\cap N _ 1 ( z ) | \\mathbf { \\mathrm { v } } _ z + \\frac { \\epsilon ^ 2 n } { 2 } \\\\ & \\leq ( k + \\epsilon ) n - | N _ 1 ( x ) \\cap N _ 1 ( z ) | + \\left ( 1 - \\frac { 1 } { 1 6 k ^ 3 } \\right ) | N _ 1 ( x ) \\cap N _ 1 ( z ) | + \\frac { \\epsilon ^ 2 n } { 2 } \\\\ & = k n + \\epsilon n + \\frac { \\epsilon ^ 2 n } { 2 } - \\frac { | N _ 1 ( x ) \\cap N _ 1 ( z ) | } { 1 6 k ^ 3 } , \\end{align*}"}
{"id": "8947.png", "formula": "\\begin{align*} u ( x ) = \\inf \\left \\lbrace \\int _ 0 ^ \\infty e ^ { - s } \\left ( C _ p | \\dot { \\gamma } ( s ) | ^ { q } + f ( \\gamma ( s ) ) \\right ) d s : \\gamma \\in \\mathrm { A C } ( [ 0 , \\infty ) ; \\overline { \\Omega } ) , \\gamma ( 0 ) = x \\right \\rbrace , \\end{align*}"}
{"id": "8917.png", "formula": "\\begin{align*} \\bar \\varphi ( z ) = \\frac { \\varphi ( z _ - ) ( z _ + - z ) + \\varphi ( z _ + ) ( z - z _ - ) } { z _ + - z _ - } . \\end{align*}"}
{"id": "5174.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sum _ { \\ell = 1 } ^ { N } \\frac { \\gamma _ \\ell R _ \\ell ^ \\pi ( t ) } { t } \\le 1 , \\ \\ \\end{align*}"}
{"id": "6493.png", "formula": "\\begin{align*} M ^ { ( 2 m - 1 ) } _ n & = M ^ { ( 2 m - 1 ) } _ { j _ 0 + 1 } { \\bar g } ^ { ( 2 m - 1 ) } _ n + \\bar { g } _ n ^ { ( 2 m - 1 ) } \\sum _ { j = j _ 0 + 1 } ^ { n - 1 } \\frac { f _ j ^ { ( 2 m - 1 ) } } { \\bar { g } _ { j + 1 } ^ { ( 2 m - 1 ) } } , \\end{align*}"}
{"id": "1020.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s w + c w & = ( - \\Delta ) ^ s v + c v + ( - \\Delta ) ^ s ( w - v ) \\\\ & \\geqslant - C \\bigg ( 1 + \\tau d ^ { - n - 2 } + ( \\theta d ) ^ { - n - 2 s - 2 } \\bigg ) x _ 1 \\\\ & \\geqslant - C \\bigg ( ( \\theta d ) ^ { - n - 2 s - 2 } + \\tau d ^ { - n - 2 } \\bigg ) x _ 1 \\end{align*}"}
{"id": "982.png", "formula": "\\begin{align*} \\int _ { H _ \\rho } \\frac { ( y _ 1 + 1 / \\rho ) \\tilde u ( y ) } { \\vert Q _ \\rho ( a ) - y \\vert ^ { n + 2 s + 2 } } \\dd y & = \\rho ^ { - n - 1 } \\int _ { \\R ^ n _ + } \\frac { z _ 1 u ( z ) } { \\vert Q _ \\rho ( a ) - z / \\rho + e _ 1 / \\rho \\vert ^ { n + 2 s + 2 } } \\dd z . \\end{align*}"}
{"id": "8318.png", "formula": "\\begin{align*} I ( y ) : = \\{ t \\in [ 0 , T ] \\colon | y ( t ) | < r \\} , \\end{align*}"}
{"id": "5424.png", "formula": "\\begin{align*} u _ 1 & = v _ 1 & w _ 1 & = v _ 1 \\\\ u _ 2 & = - \\lambda _ 3 v _ 2 & w _ 2 & = ( \\lambda _ 2 + 1 ) u _ 3 + u _ 2 = ( \\lambda _ 3 \\lambda _ 2 + \\lambda _ 2 + 1 ) v _ 2 + ( \\lambda _ 2 + 1 ) v _ 3 - \\lambda _ 3 v _ 2 \\\\ u _ 3 & = ( \\lambda _ 3 + 1 ) v _ 2 + v _ 3 \\end{align*}"}
{"id": "4790.png", "formula": "\\begin{align*} \\phi = \\phi ^ \\sigma , \\epsilon ( \\phi ) = 0 , \\langle \\phi , 1 \\rangle = 0 , \\phi ^ 2 + \\phi * \\phi = \\theta \\phi + | | \\phi | | _ 2 ^ 2 \\delta _ e , \\end{align*}"}
{"id": "6282.png", "formula": "\\begin{align*} \\mathrm { v a r } \\big [ \\widehat { K } _ { \\mathrm { a - c p t - f } } ^ \\mathbb { R } \\big ] = \\upsilon ^ 2 \\bigg ( \\ ! \\Big ( 1 \\ ! - \\ ! \\frac { \\pi } { 4 } \\Big ) \\ ! \\sum _ { i \\in \\mathcal { K } } \\ ! \\Big ( 1 \\ ! + \\ ! \\frac { 1 } { N _ 1 \\bar { \\gamma } _ i } \\Big ) \\ ! + \\ ! \\sum _ { i \\in \\mathcal { K } } \\ ! \\frac { 1 } { 2 N _ 1 \\bar { \\gamma } _ i } \\ ! + \\ ! \\frac { 1 } { 2 N _ 2 \\bar { \\gamma } _ c } \\ ! \\bigg ) , \\end{align*}"}
{"id": "9145.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\norm { x _ n - x _ { n + 1 } } / \\mu _ n = 0 . \\end{align*}"}
{"id": "2194.png", "formula": "\\begin{align*} m _ 1 = \\lim _ { n \\rightarrow + \\infty } J ( w _ n ) & \\geq \\liminf _ { n \\rightarrow + \\infty } J ( t _ { \\widehat { w } ^ + } w ^ + _ n + s _ { \\widehat { w } ^ - } w ^ - _ n ) \\\\ & \\geq J ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ + + s _ { \\widehat { w } ^ - } \\widehat { w } ^ - ) \\\\ & \\geq m _ 1 . \\end{align*}"}
{"id": "2753.png", "formula": "\\begin{align*} \\ker L _ + = \\{ \\partial _ { x _ 1 } Q , . . . , \\partial _ { x _ N } Q \\} L _ + ( \\ref { l i n e a r i z e d o p e r a t o r L + } ) , \\end{align*}"}
{"id": "2225.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } \\| Z ^ m ( t ) - Z ^ { n } ( t ) \\| _ { L ^ p ( \\Omega ; H ^ \\beta ) } = & \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } \\| P ^ m ( I - P _ n ) Z ( t ) \\| _ { L ^ p ( \\Omega ; H ^ \\beta ) } \\\\ \\leq & C \\lim _ { n \\rightarrow \\infty } \\lambda _ { n + 1 } ^ { - \\frac { \\gamma - \\beta } 2 } \\sup _ { t \\in [ 0 , T ] } \\| Z ( t ) \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } . \\end{align*}"}
{"id": "7246.png", "formula": "\\begin{align*} E _ k ^ 2 ( x ) \\geq E _ { k - 1 } ( x ) E _ { k + 1 } ( x ) , k = 1 , 2 \\cdots , n - 1 , \\end{align*}"}
{"id": "3227.png", "formula": "\\begin{align*} h _ \\lambda ( t ) = h ( - i \\omega _ 1 ( i t ) ) = h ( W ( t ) ) . \\end{align*}"}
{"id": "7373.png", "formula": "\\begin{align*} \\phi ( x , t ) : = \\sigma \\left ( \\max \\{ \\gamma ( x ) - C t , 0 \\} \\right ) , \\end{align*}"}
{"id": "8242.png", "formula": "\\begin{align*} I ( x , z , h _ 1 , h _ 2 , h _ 3 , k , y ) : = \\big ( & \\Delta _ { - h _ 1 , - 2 h _ 2 , 2 k } B ( y + 2 z ) \\Delta _ { - h _ 2 , - h _ 3 , k } B ( y + z ) \\\\ & \\Delta _ { - h _ 1 , - h _ 2 , 2 k } C ( x + y + 2 z ) \\Delta _ { - h _ 3 , k } C ( x + y + z ) \\\\ & \\Delta _ { - h _ 1 , 2 k } D ( 2 x + y + 2 z ) \\Delta _ { h _ 2 , - h _ 3 , k } D ( 2 x + y + z ) \\big ) , \\end{align*}"}
{"id": "2310.png", "formula": "\\begin{align*} | A f ( x , \\omega ) | < A f ( 0 , 0 ) = \\norm { f } _ 2 ^ 2 , \\end{align*}"}
{"id": "4103.png", "formula": "\\begin{align*} \\begin{aligned} | X _ t - X _ s | ^ { p } & = \\int _ s ^ t p \\cdot | X _ u - X _ s | ^ { p - 2 } \\cdot \\bigl ( ( X _ u - X _ s ) \\cdot \\mu ( X _ u ) + \\frac { 1 } { 2 } ( p - 1 ) \\cdot \\sigma ^ 2 ( X _ u ) \\bigr ) \\ , d u \\\\ & + \\int _ s ^ t p \\cdot ( X _ u - X _ s ) | X _ u - X _ s | ^ { p - 2 } \\sigma ( X _ u ) \\ , d W _ u . \\end{aligned} \\end{align*}"}
{"id": "6113.png", "formula": "\\begin{align*} D ^ j \\left ( X ^ { q ( m + 2 ) } \\right ) = \\frac { A ^ m _ q } { A ^ m _ { q - j } } X ^ { ( q - j ) ( m + 2 ) } \\ , . \\end{align*}"}
{"id": "3739.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb S ^ 1 } [ D ^ m , B ] J _ x D ^ m B \\ , d x \\right | \\lesssim & \\| J _ x \\| _ { L ^ \\infty } \\| D ^ m B \\| ^ 2 _ { L ^ 2 } + \\| B _ x \\| _ { L ^ \\infty } \\| D ^ { m - 1 } J _ x \\| _ { L ^ 2 } \\| D ^ m B \\| _ { L ^ 2 } , \\\\ \\left | \\int _ { \\mathbb S ^ 1 } [ D ^ m , J ] B _ x D ^ m B \\ , d x \\right | \\lesssim & \\| B _ x \\| _ { L ^ \\infty } \\| D ^ m J \\| _ { L ^ 2 } \\| D ^ m B \\| _ { L ^ 2 } + \\| J _ x \\| _ { L ^ \\infty } \\| D ^ { m - 1 } B _ x \\| _ { L ^ 2 } \\| D ^ m B \\| _ { L ^ 2 } . \\end{align*}"}
{"id": "3703.png", "formula": "\\begin{align*} I _ 2 \\leq & \\sum _ { q \\geq 0 } \\int _ 0 ^ t c _ q \\lambda _ q ^ { \\alpha - \\beta } e ^ { - \\mu \\lambda _ q ^ \\alpha ( t - \\tau ) } \\| B ( \\tau ) \\| ^ 2 _ { H ^ { \\frac 5 2 - \\alpha + \\beta } } \\ , d \\tau \\\\ \\lesssim & \\int _ 0 ^ t ( t - \\tau ) ^ { - 1 + \\frac { \\beta } { \\alpha } } \\| B ( \\tau ) \\| ^ 2 _ { H ^ { \\frac 5 2 - \\alpha + \\beta } } \\ , d \\tau . \\end{align*}"}
{"id": "7288.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p ( n ) X ^ n = \\alpha ^ { - 1 } = \\frac { ( \\alpha ^ 3 ) ^ 3 } { ( \\alpha ^ 5 ) ^ 2 } \\equiv \\frac { ( \\alpha _ 0 + \\alpha _ 1 ) ^ 3 } { \\alpha ( X ^ 5 ) ^ 2 } \\pmod { 5 } . \\end{align*}"}
{"id": "6116.png", "formula": "\\begin{align*} w ( x , y ) = \\sum _ { j = 0 } ^ \\infty \\alpha _ j ( x ) y ^ j \\ , , \\end{align*}"}
{"id": "7563.png", "formula": "\\begin{align*} \\dim _ K ( E \\cap I ) _ d & = \\dim _ K I _ d - \\dim _ K ( x _ 1 ^ 2 , \\ldots , x _ n ^ 2 ) _ d \\\\ & = ( \\dim _ K R _ d - 1 ) - \\dim _ K ( x _ 1 ^ 2 , \\ldots , x _ n ^ 2 ) _ d \\\\ & = \\dim _ K E _ d - 1 . \\end{align*}"}
{"id": "6288.png", "formula": "\\begin{align*} \\widehat { K '' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } \\stackrel { K _ l \\rightarrow \\infty } { = } \\xi \\big ( 1 + \\widehat { K ' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } \\big ) / ( 1 - \\xi ) ^ 2 \\end{align*}"}
{"id": "6403.png", "formula": "\\begin{align*} \\langle ( P _ i ^ { \\perp } + z P _ i ) \\xi , U _ i ^ * \\xi \\rangle = z _ i . \\end{align*}"}
{"id": "6401.png", "formula": "\\begin{align*} V = \\{ ( z , w ) \\in \\mathbb D ^ 2 \\ , : \\ , \\det ( A ^ * + A z w - ( z + w ) I ) = 0 \\} , \\end{align*}"}
{"id": "5264.png", "formula": "\\begin{align*} \\Delta \\circ \\sigma ^ { \\varphi } = ( S ^ 2 \\otimes \\sigma ^ { \\varphi } ) \\circ \\Delta , \\Delta \\circ \\sigma ^ { \\psi } = ( \\sigma ^ { \\psi } \\otimes S ^ { - 2 } ) \\circ \\Delta \\end{align*}"}
{"id": "7556.png", "formula": "\\begin{align*} \\kappa = 1 \\end{align*}"}
{"id": "6743.png", "formula": "\\begin{align*} \\begin{cases} & - \\mathcal L \\hat { V } - g \\geq 0 , \\\\ & \\hat { V } - V ^ * \\geq 0 , \\\\ & ( - \\mathcal L \\hat { V } - g ) \\cdot ( \\hat { V } - V ^ * ) = 0 , \\end{cases} \\end{align*}"}
{"id": "4369.png", "formula": "\\begin{align*} \\left | \\phi _ { i , o u t , \\beta } ( y ) - \\sum _ { j = 0 } ^ i c _ { i , j } ( \\sqrt { b } ) ^ { 2 j - \\gamma } T _ { j } \\left ( \\frac { y } { \\sqrt { b } } \\right ) \\right | \\le C y ^ { 2 i + 2 - \\gamma } | \\ln y | b ^ { 1 - \\frac { \\epsilon } { 2 } } . \\end{align*}"}
{"id": "8619.png", "formula": "\\begin{align*} B _ { 1 , \\phi _ { 2 } } ( k ) : = \\int _ { 0 } ^ { t } \\phi _ { 2 } ( k ) i s k e ^ { - i s k ^ { 2 } } \\iiint u ^ { \\# } ( \\ell ) \\overline { u ^ { \\# } } ( n ) u ^ { \\# } ( m ) \\ , \\mu _ { R , 1 } ^ { \\# , ( 2 ) } \\left ( k , \\ell , n , m \\right ) \\ , d \\ell d m d n \\ , d s . \\end{align*}"}
{"id": "2392.png", "formula": "\\begin{align*} \\norm { f - \\sum _ { k = 1 } ^ N e _ { \\pi ( k ) } } _ \\mathcal { B } < \\frac { \\varepsilon } { 2 } , \\forall N \\geq N _ 0 . \\end{align*}"}
{"id": "5985.png", "formula": "\\begin{align*} z ^ { n } = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } A _ { k } ^ { \\lambda , \\beta } ( x ) M _ { \\lambda , \\beta } ( n - k ) . \\end{align*}"}
{"id": "7622.png", "formula": "\\begin{align*} \\bar { g } ( \\nabla P , \\nabla \\log u ) = \\bar { g } ( \\lambda \\partial _ { r } , \\nabla \\log u ) - u ^ { \\frac { \\alpha + 1 } { \\alpha } } | \\nabla \\log u | ^ { 2 } . \\end{align*}"}
{"id": "8635.png", "formula": "\\begin{align*} I [ g _ 1 , g _ 2 , g _ 3 ] ( t , k ) = \\frac { \\pi } { | t | } e ^ { - i t k ^ 2 } \\int e ^ { i t ( - p + \\epsilon _ 0 k ) ^ 2 } g _ 1 ( \\epsilon _ 1 ( - p + \\epsilon _ 0 k ) ) \\overline { g _ 2 ( \\epsilon _ 2 ( - p + \\epsilon _ 0 k ) ) } \\\\ \\times g _ 3 ( \\epsilon _ 3 ( - p + \\epsilon _ 0 k ) ) \\mathrm { p . v . } \\frac { \\widehat { \\phi } ( p ) } { p } \\ , d p + \\mathcal { O } ( | t | ^ { - 1 - \\rho } ) \\end{align*}"}
{"id": "2699.png", "formula": "\\begin{align*} f ( t ) = e ^ { - \\pi t \\cdot A t + 2 \\pi b \\cdot t + c } , \\end{align*}"}
{"id": "2816.png", "formula": "\\begin{align*} - \\dot { \\theta } \\| Q \\| _ 2 ^ 2 = O ( \\delta + \\delta \\delta ^ * ) \\ ; \\Rightarrow \\ ; | \\dot { \\theta } | = O ( \\delta + \\delta \\delta ^ * ) . \\end{align*}"}
{"id": "4394.png", "formula": "\\begin{align*} \\gamma ^ 2 - d \\gamma + 3 ( d - 2 ) = 0 . \\end{align*}"}
{"id": "8392.png", "formula": "\\begin{align*} B _ { I _ j } ( v ) \\cap B ' _ { I _ { j ' } } = \\emptyset \\mbox { f o r a l l } 1 \\leq j , j ' \\leq \\ell \\mbox { a n d } B ' _ { I _ j } \\cap B ' _ { I _ { j ' } } = \\emptyset \\mbox { f o r } 1 \\leq j \\neq j ' \\leq \\ell \\ , . \\end{align*}"}
{"id": "4470.png", "formula": "\\begin{align*} \\langle z | ( a ^ { \\dagger } a ) _ { k , \\lambda } | z \\rangle & = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) \\langle z | ( a ^ { \\dagger } ) ^ { l } a ^ { l } | z \\rangle = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( \\overline { z } ) ^ { l } z ^ { l } \\langle z | z \\rangle \\\\ & = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) | z | ^ { 2 l } = \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) . \\end{align*}"}
{"id": "7212.png", "formula": "\\begin{align*} x - ( t - s ) v - X ( s ) = \\check x _ { t , x , v } - ( \\mathcal T _ { t , x , v } - s ) v + X ( \\mathcal T _ { t , x , v } ) - X ( s ) . \\end{align*}"}
{"id": "4501.png", "formula": "\\begin{align*} \\Sigma _ 0 ^ { \\sigma _ 1 } = 2 \\sum _ { \\substack { 0 < \\gamma < T \\\\ 0 \\le \\beta < \\sigma _ 1 } } \\frac { x ^ { \\beta - 1 } } { \\gamma } \\le \\varepsilon _ 2 ( x , \\sigma _ 1 , T _ 0 , T ) , \\end{align*}"}
{"id": "7661.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta h _ { \\varepsilon } + \\tilde { \\lambda } _ 0 \\underline { m } \\ , h _ { \\varepsilon } = 0 & \\Omega _ { \\varepsilon } \\ ; , \\\\ h _ { \\varepsilon } = w & \\partial \\Omega _ { \\varepsilon } \\ ; , \\end{cases} \\end{align*}"}
{"id": "3271.png", "formula": "\\begin{align*} s ( | \\lambda | , 0 ) ^ 2 + | \\lambda | ^ 2 = \\frac { s ( | \\lambda | , 0 ) } { h ( s ( | \\lambda | , 0 ) ) } , \\end{align*}"}
{"id": "5501.png", "formula": "\\begin{align*} \\eta _ s ( t ; x ) : = \\xi ( s + t ; s , x ) , t \\in [ 0 , u - s ] . \\end{align*}"}
{"id": "3607.png", "formula": "\\begin{align*} 1 - R _ { n + 2 } = 0 + ( 1 - R _ { n + 2 } ) = d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] + d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] > 2 e + S _ { n } - R _ { n + 2 } \\ , , \\end{align*}"}
{"id": "3985.png", "formula": "\\begin{align*} J _ k : = - \\frac { 1 } { k \\pi } ( \\sqrt { \\mod { k \\pi } } - i \\sqrt { \\mod { k \\pi } } + O ( 1 ) ) e ^ { - \\alpha _ { 1 , k } - i ( 2 k \\pi + \\alpha _ { 2 , k } ) + O ( \\mod { k } ^ { - 1 } ) } e ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { \\sqrt { | k | } } + i \\sqrt { \\mod { k \\pi } } + O ( \\mod { k } ^ { - \\frac { 1 } { 2 } } ) } , \\end{align*}"}
{"id": "2467.png", "formula": "\\begin{align*} U _ P = \\left \\{ \\begin{pmatrix} I & P \\\\ 0 & I \\end{pmatrix} \\mid P = P ^ T \\right \\} , V _ Q = \\left \\{ \\begin{pmatrix} I & 0 \\\\ Q & I \\end{pmatrix} \\mid Q = Q ^ T \\right \\} \\end{align*}"}
{"id": "3883.png", "formula": "\\begin{align*} w _ 1 ( M ) & = ( 2 n - 2 - a ) t _ 1 - t _ 2 - \\dots - t _ b \\\\ & + ( 2 n - 2 - b ) s _ 1 - s _ 2 - \\dots - s _ a . \\end{align*}"}
{"id": "2890.png", "formula": "\\begin{align*} d _ k ( u , v ) = \\| u - v \\| _ { X ^ k } . \\end{align*}"}
{"id": "3575.png", "formula": "\\begin{align*} { \\rm s p } _ 0 ( \\tau ) = p ^ e \\big ( \\mathbb { N } + \\frac { n } { 2 } \\big ) + \\frac { 1 } { 2 } { \\bigcup } _ { z = 0 } ^ { e } \\Lambda _ z = p ^ e \\mathbb { N } + \\big \\{ \\overline { \\lambda } ( x ) \\colon 0 \\le x < p ^ e \\big \\} . \\end{align*}"}
{"id": "4004.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\overline { \\eta ( x ) } \\eta ^ \\prime ( x ) d x = i \\int _ 0 ^ 1 \\Im ( \\overline { \\eta ( x ) } \\eta ^ \\prime ( x ) ) d x . \\end{align*}"}
{"id": "1217.png", "formula": "\\begin{align*} \\begin{cases} c ( \\mu , d , s ) \\mathcal { H } ^ s _ { \\infty } ( \\Omega ) \\leq \\mathcal { H } ^ { \\tilde { \\mu } , s } _ { \\infty } ( \\Omega ) \\leq \\mathcal { H } ^ { s } _ { \\infty } ( \\Omega ) s < \\alpha \\\\ \\mathcal { H } ^ { \\tilde { \\mu } , s } _ { \\infty } ( \\Omega ) = 0 s > \\alpha . \\end{cases} \\end{align*}"}
{"id": "8749.png", "formula": "\\begin{align*} s _ { \\cdot 0 } = a _ { i 0 } , s _ { \\cdot j } = \\sum _ { k = 0 } ^ n \\min \\{ a _ { \\cdot k } , a _ { \\cdot j } \\} \\lambda _ { \\cdot k } j = 1 , \\ldots , n . \\end{align*}"}
{"id": "6887.png", "formula": "\\begin{align*} a _ j = \\sum \\limits _ { i = 1 } ^ { j } ( - 1 ) ^ i { j \\choose i } _ q b _ i , \\end{align*}"}
{"id": "8289.png", "formula": "\\begin{align*} & M \\left ( a ( \\mu + \\alpha ) ^ 2 + b ( \\mu + \\alpha ) \\right ) - M \\left ( a ( \\mu + M M ^ * \\alpha ) ^ 2 + b ( \\mu + M M ^ * \\alpha ) \\right ) & & \\\\ = \\ , & M \\left ( ( 1 - ( M M ^ * ) ^ 2 ) a \\alpha ^ 2 + ( 1 - M M ^ * ) ( b + 2 a \\mu ) \\alpha \\right ) & & \\\\ \\equiv \\ , & ( 1 - M M ^ * ) \\left ( ( 1 + M M ^ * ) a \\alpha ^ 2 + b \\alpha \\right ) & & \\bmod k \\Z . \\end{align*}"}
{"id": "7451.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ { k } - s \\nabla f \\left ( u _ { k } \\right ) + \\gamma \\left ( u _ { k } - u _ { k - 1 } \\right ) , k = 1 , 2 , \\ldots , \\end{align*}"}
{"id": "3225.png", "formula": "\\begin{align*} \\omega _ 1 ( z ) = z + H _ { \\mu _ 2 } ( z + H _ { \\mu _ 1 } ( \\omega _ 1 ( z ) ) ) , \\omega _ 2 ( z ) = z + H _ { \\mu _ 1 } ( z + H _ { \\mu _ 2 } ( \\omega _ 2 ( z ) ) ) . \\end{align*}"}
{"id": "2532.png", "formula": "\\begin{align*} \\Phi \\natural \\Phi ^ { \\xi , \\eta } = e ^ { - \\tfrac { \\pi } { 2 } ( \\xi ^ 2 + \\eta ^ 2 ) } \\Phi . \\end{align*}"}
{"id": "5691.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) + J _ { 0 } ( \\beta , \\gamma , Z ' ) = J _ { 0 } ( \\alpha , \\gamma , Z + Z ' ) . \\end{align*}"}
{"id": "373.png", "formula": "\\begin{align*} & \\dot { \\widetilde \\rho _ 2 } + \\widetilde m _ { 2 3 } ( 1 + \\dot W ^ { \\delta } ) = 0 , \\\\ & \\dot { \\widetilde \\rho _ 3 } + \\widetilde m _ { 3 2 } ( 1 + \\dot W ^ { \\delta } ) = 0 . \\end{align*}"}
{"id": "2488.png", "formula": "\\begin{align*} ( x , 0 , 1 ) \\circ ( 0 , \\omega , 1 ) = ( x , \\omega , e ^ { - \\pi i x \\cdot \\omega } ) ( 0 , \\omega , 1 ) \\circ ( x , 0 , 1 ) = ( x , \\omega , e ^ { \\pi i x \\cdot \\omega } ) . \\end{align*}"}
{"id": "8964.png", "formula": "\\begin{align*} \\begin{aligned} c _ 0 f \\left ( \\xi ( t _ 1 ) \\right ) & \\leq \\left ( T _ { x , \\xi } - t _ 1 \\right ) p ( 1 - c _ 0 ) f ( \\xi ( t _ 1 ) ) \\\\ \\frac { c _ 0 } { p ( 1 - c _ 0 ) } & \\leq T _ { x , \\xi } - t _ 1 \\\\ \\frac { c _ 0 } { p ( 1 - c _ 0 ) } & \\leq T _ { x , \\xi } . \\end{aligned} \\end{align*}"}
{"id": "551.png", "formula": "\\begin{align*} \\Omega _ 1 & = \\{ x \\in \\Omega : \\rho ( t , x ) \\in [ 0 , \\alpha _ 1 ] \\} , \\\\ \\Omega _ 2 & = \\{ x \\in \\Omega : \\rho ( t , x ) \\in ( \\alpha _ 1 , \\bar \\rho - \\alpha _ 1 ] \\} , \\\\ \\Omega _ 3 & = \\{ x \\in \\Omega : \\rho ( t , x ) \\in ( \\bar \\rho - \\alpha _ 1 , \\bar \\rho ) \\} , \\\\ \\end{align*}"}
{"id": "9441.png", "formula": "\\begin{align*} d _ { q - 1 } \\tilde p = ( I - \\varPhi _ q \\ , d _ q ) \\Big ( \\varPsi _ { \\mu , q } f + \\varPsi _ { \\mu , q , 0 } u _ 0 - \\varPsi _ \\mu B _ q ( w , u ) \\Big ) . \\end{align*}"}
{"id": "256.png", "formula": "\\begin{align*} \\nabla ( \\tilde { h } _ R ) ( x ) = \\frac { 1 } { R } \\nabla ( \\psi ) ( \\frac { x } { R } ) h ( x ) + \\psi \\left ( \\frac { x } { R } \\right ) \\nabla ( h ) ( x ) . \\end{align*}"}
{"id": "5959.png", "formula": "\\begin{align*} \\mu _ { j k } = \\frac { a _ { j k } } { \\frac { 1 } { 2 } \\pi \\rho R ^ 2 } , \\nu _ { j k } = \\frac { b _ { j k } } { \\frac { 1 } { 2 } \\pi \\rho \\omega R ^ 2 } , ( j , k ) = 1 , 3 . \\end{align*}"}
{"id": "5444.png", "formula": "\\begin{align*} \\mathcal { E } = \\left \\{ u \\in C ^ 0 ( \\bar \\Omega ) \\ , | \\ , u \\ge 0 , \\ , \\ , { \\int _ \\Omega u ( x ) d x \\le M _ 0 ^ * } , \\int _ \\Omega u ^ { - p } ( x ) d x \\le M _ 1 ^ * , \\ , \\ , \\int _ \\Omega u ^ { q } ( x ) d x \\le M _ 2 ^ * \\right \\} \\end{align*}"}
{"id": "5907.png", "formula": "\\begin{align*} \\det S = ( - 1 ) ^ { n - 2 k } \\det Q _ S . \\end{align*}"}
{"id": "5996.png", "formula": "\\begin{align*} | s ( n , k ) | & \\le \\frac { 1 } { k ! } \\sum _ { l _ { 1 } + \\dots + l _ { k } = n } \\frac { n ! } { l _ { 1 } ! \\dots l _ { k } ! } | f ^ { ( l _ { 1 } ) } ( 0 ) | \\dots | f ^ { ( l _ { k } ) } ( 0 ) | \\\\ & \\leq \\frac { 1 } { k ! } \\sum _ { l _ { 1 } + \\dots + l _ { k } = n } \\frac { n ! l _ { 1 } ! \\dots l _ { k } ! } { l _ { 1 } ! \\dots l _ { k } ! } F _ { \\varepsilon } ^ { k } \\varepsilon ^ { - n } \\\\ & \\leq \\frac { n ! } { k ! } F _ { \\varepsilon } ^ { k } 2 ^ { n } \\varepsilon ^ { - n } . \\end{align*}"}
{"id": "4597.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > \\Phi ^ { - 1 } ( 1 - \\kappa _ n / 2 ) \\Big ) \\sim \\kappa _ n / 2 \\end{align*}"}
{"id": "3199.png", "formula": "\\begin{align*} \\dim ( X \\cup Y ) = \\max \\{ \\dim ( X ) , \\dim ( Y ) \\} \\end{align*}"}
{"id": "2649.png", "formula": "\\begin{align*} \\norm { X g } _ 2 \\norm { P g } _ 2 = \\infty . \\end{align*}"}
{"id": "8935.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u + H ( x , D u ) & \\leq 0 \\Omega , \\\\ u + H ( x , D u ) & \\geq 0 \\overline { \\Omega } , \\end{aligned} \\right . \\end{align*}"}
{"id": "1369.png", "formula": "\\begin{align*} u < c , u = c \\quad u > c , \\quad . \\end{align*}"}
{"id": "4852.png", "formula": "\\begin{align*} \\int _ \\R M \\leq \\int _ { - \\sigma } ^ \\sigma K = 2 \\int _ 0 ^ \\sigma K \\leq C \\int _ 0 ^ \\sigma - \\log x \\ , d x = C \\sigma ( | \\log \\sigma | + 1 ) . \\end{align*}"}
{"id": "4028.png", "formula": "\\begin{align*} \\eta ^ { \\prime \\prime } ( x ) = m ^ 2 _ 1 ( e ^ { m _ 2 } - e ^ { m _ 3 } ) e ^ { m _ 1 x } + m ^ 2 _ 2 ( e ^ { m _ 3 } - e ^ { m _ 1 } ) e ^ { m _ 2 x } + m _ 3 ^ 2 ( e ^ { m _ 1 } - e ^ { m _ 2 } ) e ^ { m _ 3 x } . \\end{align*}"}
{"id": "5496.png", "formula": "\\begin{align*} e ^ { \\gamma ( t - s ) } \\varphi _ { \\gamma } ( s - r ) + \\varphi _ { \\gamma } ( t - s ) = \\varphi _ { \\gamma } ( t - r ) . \\end{align*}"}
{"id": "6424.png", "formula": "\\begin{align*} R _ 1 = \\begin{bmatrix} \\frac { 4 b ^ 2 - 3 \\varphi ^ 2 } { 4 b ^ 4 } & - \\frac { \\varphi ' } { b ^ 2 } \\\\ - \\frac { \\varphi ' } { b ^ 2 } & - \\frac { \\varphi '' } { \\varphi } \\end{bmatrix} , R _ 2 = R _ 3 = \\begin{bmatrix} \\frac { \\varphi ^ 2 } { 4 b ^ 4 } & \\frac { \\varphi ' } { 2 b ^ 2 } \\\\ \\frac { \\varphi ' } { 2 b ^ 2 } & 0 \\end{bmatrix} , \\end{align*}"}
{"id": "8866.png", "formula": "\\begin{align*} p _ 1 : U _ 2 & \\to X \\times \\R _ { \\ge 0 } \\\\ ( x , t ) & \\mapsto \\begin{cases} ( x , t ) & t \\ge 0 \\\\ ( x , 0 ) & t < 0 \\end{cases} \\end{align*}"}
{"id": "7652.png", "formula": "\\begin{align*} \\eta _ k \\coloneqq \\begin{cases} 1 & \\overline { B } _ { R _ k } ( \\mathbf { 0 } ) \\ ; , \\\\ 1 - \\bigl ( | \\mathbf { x } | - R _ k ) ^ 2 & B _ { R _ k + 1 } ( \\mathbf { 0 } ) \\setminus B _ { R _ k } ( \\mathbf { 0 } ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "4529.png", "formula": "\\begin{align*} Q ( x ) = C x + O ( x ^ { 3 / 4 + \\varepsilon } ) , \\end{align*}"}
{"id": "8757.png", "formula": "\\begin{align*} \\phi \\leq \\sqrt { ( 1 - \\delta ) ( z _ 1 - \\delta ) } + z _ 2 , \\ 1 = z _ 0 \\geq z _ 1 \\geq \\delta \\geq z _ 2 \\geq 0 , \\ \\delta \\in \\{ 0 , 1 \\} , \\ f = z _ 1 + z _ 2 . \\end{align*}"}
{"id": "4343.png", "formula": "\\begin{align*} \\langle \\varepsilon _ - , \\phi _ { j , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } = 0 , j = 0 j = 1 , \\end{align*}"}
{"id": "4593.png", "formula": "\\begin{align*} M _ { n , k } = \\sum _ { i = 1 } ^ { k } \\Big ( a _ n \\alpha _ { i } X _ { \\beta _ { i } } ( Z _ { i } - 1 ) + a _ i \\varepsilon _ i \\Big ) , \\end{align*}"}
{"id": "7348.png", "formula": "\\begin{align*} \\lambda a _ q + ( 1 - \\lambda ) b _ q = w _ q ( x _ q , t _ q ) ^ q , \\end{align*}"}
{"id": "4357.png", "formula": "\\begin{align*} \\partial _ x ^ k \\epsilon ( x ) = \\partial _ x ^ k ( h _ + e ^ { \\lambda _ 1 x } ) + O ( e ^ { 3 \\lambda _ 1 x } ) , x \\to + \\infty . \\end{align*}"}
{"id": "6123.png", "formula": "\\begin{align*} \\alpha _ 1 ( x ) = h ^ 3 ( x ^ { m + 2 } ) + x h ^ 4 ( x ^ { m + 2 } ) \\ , . \\end{align*}"}
{"id": "5477.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X ( t ) & = & \\big ( A X ( t ) + \\alpha ( X ( t ) ) \\big ) d t + \\sigma ( X ( t ) ) d W ( t ) \\medskip \\\\ X ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "6022.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { O } } f & = - \\widetilde { U } ^ { - 1 } \\mathcal { H } ( \\widetilde { U } f ) - f \\\\ & = - \\widetilde { U } ^ { - 1 } U \\mathcal { O } U ^ { - 1 } \\widetilde { U } f - 2 f \\\\ & = - e ^ { - x ^ 2 } \\mathcal { O } ( e ^ { ( \\cdot ) ^ 2 } f ( \\cdot ) ) - 2 f \\end{align*}"}
{"id": "3595.png", "formula": "\\begin{align*} T _ { 0 } ^ { ( i ) } = \\dfrac { R _ { i + 1 } - R _ { i } } { 2 } + e , T _ { j } ^ { ( i ) } = \\begin{cases} R _ { i + 1 } - R _ { j } + d ( - a _ { j } a _ { j + 1 } ) & \\ , , \\\\ R _ { j + 1 } - R _ { i } + d ( - a _ { j } a _ { j + 1 } ) & \\ , , \\end{cases} \\end{align*}"}
{"id": "4988.png", "formula": "\\begin{align*} \\tilde { p } ( A , B ) & = \\left ( H ^ { - 1 } \\circ ( B \\circ A ^ { r _ 0 } ) ^ { r _ 1 } \\circ A \\circ H , H ^ { - 1 } \\circ B \\circ A ^ { r _ 0 } \\circ H \\right ) , \\end{align*}"}
{"id": "8829.png", "formula": "\\begin{align*} \\min _ { \\sigma \\in \\mathbb { R } } f ( \\sigma ; k ) \\coloneqq \\frac { 1 } { m } \\sum _ { j = 1 } ^ m F ( \\sigma , \\xi _ j ; k ) + \\delta _ { \\left [ \\sigma _ { \\min } , \\sigma _ { \\max } \\right ] } \\left ( \\sigma \\right ) . \\end{align*}"}
{"id": "3643.png", "formula": "\\begin{align*} t u ' ( t ) & = \\frac { 3 \\log \\log t - 1 } { 5 \\log ^ { 5 / 2 } t ( \\log \\log t ) ^ { 6 / 5 } } \\le 1 . 6 3 \\cdot 1 0 ^ { - 5 } t \\ge \\exp ( 5 8 ) \\end{align*}"}
{"id": "2010.png", "formula": "\\begin{align*} \\Psi _ t ( x , y ) : = f ( t , x ) g ( t , d ( x , y ) ) . \\end{align*}"}
{"id": "8983.png", "formula": "\\begin{align*} \\min \\left \\{ I _ b [ \\gamma ] ; \\gamma \\in \\mathrm { A C } \\left ( [ 0 , b ] ; \\Omega \\right ) , \\gamma ( 0 ) = x , \\gamma ( b ) = \\xi ( b ) \\right \\} . \\end{align*}"}
{"id": "2888.png", "formula": "\\begin{align*} d _ k ( u , v ) = \\sup _ { t \\ge t _ k } \\| u - v \\| _ { S ( [ t , + \\infty ) , \\dot { H } ^ { s _ c } ) } e ^ { ( k + \\frac { 1 } { 2 } ) \\frac { 1 } { p - 2 } e _ 0 t } . \\end{align*}"}
{"id": "5910.png", "formula": "\\begin{align*} \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , \\lambda ^ 2 _ i = t r a c e \\ Q _ S ^ 2 = n ^ 2 - 2 n + 2 k . \\end{align*}"}
{"id": "9308.png", "formula": "\\begin{align*} u ^ { T } ( \\sum _ { i } p _ i y _ i ) ( \\sum _ { i } p _ i y _ i ) ^ { T } u & = \\left ( \\sum _ { i } p _ i ( u ^ { T } y _ i ) \\right ) ^ { 2 } \\\\ & \\leq \\left ( \\sum _ { i } p _ { i } \\right ) \\cdot \\left ( \\sum _ { i } p _ { i } ( u ^ { T } y _ i ) ^ { 2 } \\right ) & \\textrm { ( C a u c h y - - S c h w a r z ) } \\\\ & = u ^ { T } \\left ( \\sum _ { i } p _ i y _ { i } y _ { i } ^ { T } \\right ) u . \\end{align*}"}
{"id": "4301.png", "formula": "\\begin{align*} \\beta ( \\tau ) = - \\frac { 1 } { 2 } \\frac { \\mu _ \\tau } { \\mu ( \\tau ) } , \\end{align*}"}
{"id": "5999.png", "formula": "\\begin{align*} { \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { x ^ { n } } { n ! } \\right ) ^ { \\gamma } } & \\leq \\left ( \\sum _ { n = 0 } ^ { \\infty } \\frac { x ^ { n } } { n ! } \\right ) ^ { 2 - \\gamma } \\left ( \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { x ^ { n } } { n ! } \\right ) ^ { 2 } \\right ) ^ { \\gamma - 1 } \\\\ & \\leq ( \\mathrm { e } ^ { x } ) ^ { 2 - \\gamma } ( \\mathrm { e } ^ { 2 x } ) ^ { \\gamma - 1 } = \\mathrm { e } ^ { \\gamma x } \\end{align*}"}
{"id": "5290.png", "formula": "\\begin{align*} \\check { \\varphi } ( \\omega ^ * \\omega ) = \\omega ^ * ( S ^ { - 1 } ( a ) ) = \\varphi ( a ^ * a ) , \\omega = \\varphi ( - a ) , a \\in A , \\end{align*}"}
{"id": "2539.png", "formula": "\\begin{align*} f = \\sum _ { k = 1 } ^ n c _ k \\pi ( x _ k , \\omega _ k , 0 ) g \\in \\mathcal { H } _ g . \\end{align*}"}
{"id": "1022.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s \\tilde u ( x ) + \\tilde c \\tilde u = 0 , B _ { 2 R / y _ 1 } ( e _ 1 ) . \\end{align*}"}
{"id": "2884.png", "formula": "\\begin{align*} \\varepsilon _ { k + 1 } & = - i \\left ( R ( \\mathcal { V } _ { k + 1 } ^ A ) - R ( \\mathcal { V } _ k ^ A ) \\right ) + \\varepsilon _ k + e ^ { - ( k + 1 ) e _ 0 t } \\left ( \\mathcal { L } - ( k + 1 ) e _ 0 \\right ) \\mathcal { Z } _ { k + 1 } ^ A \\\\ & = - i \\left ( R ( \\mathcal { V } _ { k + 1 } ^ A ) - R ( \\mathcal { V } _ k ^ A ) \\right ) + O \\left ( e ^ { - ( k + 2 ) e _ 0 t } \\right ) . \\end{align*}"}
{"id": "8312.png", "formula": "\\begin{align*} ( u + \\Delta y ) ^ \\perp : = \\left \\{ z \\in H _ 0 ^ 1 ( \\Omega ) \\colon \\int _ \\Omega u z - \\nabla y \\cdot \\nabla z \\dd x = 0 \\right \\} \\end{align*}"}
{"id": "2314.png", "formula": "\\begin{align*} A b _ 0 ( x , 0 ) = \\lim _ { \\omega \\to 0 } \\frac { \\sin ( \\pi \\omega ) ( 1 - | x | ) } { \\pi \\omega } = 1 - | x | . \\end{align*}"}
{"id": "6886.png", "formula": "\\begin{align*} \\Pr [ T ' _ i = T ~ | ~ W ] = \\Pr [ V ' = T ~ | ~ W ~ ~ \\mathcal { E } _ i ( W ) ] \\end{align*}"}
{"id": "7316.png", "formula": "\\begin{align*} m \\left ( \\bigcap _ { \\mu > 0 } \\bigcup _ { \\delta \\leq \\mu } K \\cap \\left ( \\{ f < a \\} \\setminus \\{ f _ \\delta < a _ \\delta \\} \\right ) \\right ) = 0 . \\end{align*}"}
{"id": "6054.png", "formula": "\\begin{align*} \\begin{array} { l l @ { } l l } & \\displaystyle x _ k - x _ \\ell & \\\\ & \\displaystyle v _ { i j } - x _ j \\leq t _ i \\ , , & i \\in [ m ] j \\in [ n ] \\\\ & \\displaystyle x _ { 1 } + \\dots + x _ { n } = 0 & \\\\ & \\displaystyle t _ { 1 } + \\dots + t _ { m } = p ^ * / n & \\end{array} \\end{align*}"}
{"id": "3964.png", "formula": "\\begin{align*} \\begin{dcases} \\| \\xi _ { \\lambda ^ p _ k } \\| _ { L ^ 2 ( 0 , 1 ) } \\leq C k ^ { - 1 } , \\\\ \\| \\xi _ { \\lambda ^ p _ k } \\| _ { H ^ { - s } _ { p e r } ( 0 , 1 ) } \\leq C k ^ { - s - 1 } , \\ \\ 0 < s < 1 , \\\\ \\| \\xi _ { \\lambda ^ p _ k } \\| _ { H ^ { - s } _ { p e r } ( 0 , 1 ) } \\leq C k ^ { - 2 } , \\ \\ s \\geq 1 , \\\\ \\| \\eta _ { \\lambda ^ p _ k } \\| _ { L ^ 2 ( 0 , 1 ) } \\leq C . \\end{dcases} \\end{align*}"}
{"id": "4366.png", "formula": "\\begin{align*} \\phi _ { i , b , \\beta } ( y ) = \\sum _ { j = 0 } ^ i c _ { i , j } ( 2 \\beta ) ^ j ( \\sqrt { b } ) ^ { 2 j - \\gamma } T _ { j } \\left ( \\frac { y } { \\sqrt { b } } \\right ) + \\tilde \\phi _ { i , b } ( y ) , \\end{align*}"}
{"id": "8276.png", "formula": "\\begin{align*} M _ { p } : = \\sum _ { p \\leq p ' } \\mu _ { P } ( p , p ' ) F _ { p ' } , F _ p : = \\sum _ { p \\leq p ' } M _ { p ' } . \\end{align*}"}
{"id": "7195.png", "formula": "\\begin{align*} \\| \\Psi \\| _ { Y _ T } = \\sup _ { s \\in [ 0 , T ] , x \\in \\R ^ 3 } | \\Psi ( s , x ) | \\langle \\check \\tau _ { s , x } ^ 2 + d _ { s , x } ^ 2 + | x ^ \\perp | ^ 2 \\rangle + | \\nabla \\Psi ( s , x ) | \\langle \\check \\tau _ { s , x } ^ 3 + d _ { s , x } ^ 3 + | x ^ \\perp | ^ 3 \\rangle . \\end{align*}"}
{"id": "5077.png", "formula": "\\begin{align*} \\left | L \\left ( \\frac { w _ 1 + w _ 2 } { 2 } \\right ) \\right | \\geq L \\left ( \\frac { w _ 1 + w _ 2 } { 2 } \\right ) = \\frac { L ( w _ 1 ) + L ( w _ 2 ) } { 2 } \\geq r ( 1 - ( 2 \\delta + \\epsilon ) ) . \\end{align*}"}
{"id": "8140.png", "formula": "\\begin{align*} h _ K ^ - = w _ K \\left ( \\frac { p } { 4 \\pi ^ 2 } \\right ) ^ { m / 4 } \\prod _ { \\chi \\in X _ K ^ - } L ( 1 , \\chi ) \\leq w _ K \\left ( { p M ( p , H ) \\over 4 \\pi ^ 2 } \\right ) ^ { m / 4 } , \\end{align*}"}
{"id": "3667.png", "formula": "\\begin{align*} F _ \\nu ( t ) : = \\begin{cases} t ^ { 2 - 2 \\nu } & \\nu < 1 \\\\ \\log t & \\nu = 1 \\\\ - t ^ { 2 - 2 \\nu } & \\nu > 1 . \\end{cases} \\end{align*}"}
{"id": "3078.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 3 + x _ 1 ^ 3 y _ 2 ^ 3 + x _ 2 ^ 3 y _ 1 ^ 3 + a ^ 3 x _ 2 ^ 3 y _ 2 ^ 3 = 0 \\ , . \\end{align*}"}
{"id": "6300.png", "formula": "\\begin{align*} 2 ^ { \\frac { \\nu } { 2 } - 1 } \\omega _ 2 \\Gamma \\Big ( \\frac { \\nu } { 2 } \\Big ) & = K _ { \\frac { \\nu } { 2 } } ( 2 6 \\sqrt { \\omega _ 1 ( \\nu - 2 ) } ) ( 2 6 \\sqrt { \\omega _ 1 ( \\nu - 2 ) } ) ^ { \\frac { \\nu } { 2 } } , \\end{align*}"}
{"id": "422.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } ( U ) V _ { t } + A ^ { i } ( U ) \\partial _ { i } V - B ^ { i j } ( U ) \\partial _ { i } \\partial _ { j } V + D ( U ) V & = F ( U ; D _ { x } U ) , \\\\ \\left . V \\right \\rvert _ { t = 0 } & = U _ { 0 } \\end{aligned} \\end{align*}"}
{"id": "6879.png", "formula": "\\begin{align*} \\delta _ i = \\frac { ( q ^ i - 1 ) ( q ^ { n - i + 1 } - 1 ) } { ( q ^ { i + 1 } - 1 ) ( q ^ { n - i } - 1 ) } . \\end{align*}"}
{"id": "7385.png", "formula": "\\begin{align*} \\partial _ t u - L ^ { ( n ) } u = f , - L ^ { ( n ) } u = f \\end{align*}"}
{"id": "4328.png", "formula": "\\begin{align*} \\partial _ \\beta \\| \\phi _ { 1 , b , \\beta } \\| ^ { 2 } _ { L ^ 2 _ { \\rho _ \\beta } } = \\left ( 2 \\frac { 1 } { \\beta } + \\frac { d + 2 } { 2 \\beta } - \\left ( \\frac { 2 } { \\beta } \\left ( \\frac { d } { 2 } - \\gamma + 2 \\right ) - \\frac { 1 } { \\beta } \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) \\right ) \\right ) \\| \\phi _ { 1 , b , \\beta } \\| _ { \\rho _ \\beta } ^ 2 + O ( b ^ { 1 - \\frac { \\epsilon } { 2 } } ) \\end{align*}"}
{"id": "8896.png", "formula": "\\begin{align*} \\psi ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) ) : = \\psi _ { \\lfloor \\max ( t _ j ) \\rfloor + 1 } ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) ) \\end{align*}"}
{"id": "6087.png", "formula": "\\begin{align*} Z ( r , \\theta ) = Z ( 0 , 0 ) + r ^ { \\mu _ { j } } \\cdot e ^ { i \\theta } + O ( r ^ { 2 \\mu _ { j } } ) \\end{align*}"}
{"id": "34.png", "formula": "\\begin{align*} J _ { t } ( U ) = V , \\ , J _ { t } ( V ) = - U . \\end{align*}"}
{"id": "6161.png", "formula": "\\begin{align*} x _ 1 + x _ 2 & = 1 , \\\\ x _ 1 z _ 1 + x _ 2 z _ 2 & = - 1 , \\\\ x _ 1 z _ 1 ^ 2 + x _ 2 z _ 2 ^ 2 & = \\frac { 2 } { 3 } . \\end{align*}"}
{"id": "2687.png", "formula": "\\begin{align*} | U | \\geq \\sup _ { p > 2 } \\left ( \\tfrac { p } { 2 } \\right ) ^ { \\frac { 2 d } { p - 2 } } ( 1 - \\varepsilon ) ^ { \\frac { p } { p - 2 } } \\geq 2 ^ d ( 1 - \\varepsilon ) ^ 2 , ( p = 4 ) . \\end{align*}"}
{"id": "2457.png", "formula": "\\begin{align*} \\sigma = d x \\wedge d \\omega = \\sum _ { k = 1 } ^ d d x _ k \\wedge d \\omega _ k . \\end{align*}"}
{"id": "6055.png", "formula": "\\begin{align*} T _ 1 = \\mathtt { ( D : 1 0 , ( C : 4 , ( B : 2 , A : 2 ) : 2 ) : 6 ) } \\quad T _ 2 = \\mathtt { ( A : 1 0 , ( B : 4 , ( C : 2 , D : 2 ) : 2 ) : 6 ) } \\end{align*}"}
{"id": "4430.png", "formula": "\\begin{align*} & u _ { x x x x } + G ( u ) = f ( x ) U , \\\\ & u ( c ) = 0 = u ( d ) , \\\\ & u _ { x x } ( c ) = 0 = u _ { x x } ( d ) . \\\\ \\end{align*}"}
{"id": "7825.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { 2 j + t } \\alpha _ { i } e _ { i } , \\quad \\alpha _ { 2 j + t } \\neq 0 . \\end{align*}"}
{"id": "122.png", "formula": "\\begin{align*} 2 ( x y ) ( x z ) + x ^ 2 ( y z ) = 0 . \\end{align*}"}
{"id": "9324.png", "formula": "\\begin{align*} \\underbrace { \\begin{bmatrix} H + \\rho I + \\Theta ^ \\dagger & - A ^ T \\\\ A & \\delta I \\end{bmatrix} } _ { \\mathcal { N } _ { \\rho , \\delta , \\Theta } } \\begin{bmatrix} \\Delta \\mathbf { x } \\\\ \\Delta \\mathbf { y } \\end{bmatrix} = \\begin{bmatrix} \\xi _ d ^ 1 \\\\ \\xi ^ 2 _ d + X ^ { - 1 } _ { \\mathcal { C } } \\xi _ { \\mu , \\sigma } \\\\ \\xi _ p \\\\ \\end{bmatrix} , \\end{align*}"}
{"id": "654.png", "formula": "\\begin{align*} ( x , y ) \\ = \\ x ^ y \\end{align*}"}
{"id": "5376.png", "formula": "\\begin{align*} \\mathcal { L } _ { m a p } = \\left [ 1 , \\ldots , m \\right ] . \\end{align*}"}
{"id": "8100.png", "formula": "\\begin{align*} \\widehat { ( \\partial _ { \\Sigma _ 0 , \\epsilon } ^ { * } ) ^ { \\otimes 2 } \\overline { u } _ n } ( \\xi , \\eta ) & = - 4 \\xi _ u \\eta _ u \\delta ( \\xi _ u + \\eta _ u ) \\widehat { \\overline { u } } ( \\xi _ u ) \\\\ & = \\frac { ( - i ) ^ { n } \\xi _ u ^ { n + 2 } } { n ! } [ \\alpha - i \\pi \\mathrm { s g n } ( \\xi _ u ) ] . \\end{align*}"}
{"id": "4945.png", "formula": "\\begin{align*} ( \\phi - 1 ) \\kappa ( g ) = \\kappa ( k ) - ( \\phi - 1 ) c _ { b , \\mu } . \\end{align*}"}
{"id": "2378.png", "formula": "\\begin{align*} C _ { \\mathbf { F } } = \\begin{pmatrix} v _ { 1 , 1 } & \\ldots & v _ { 1 , d } \\\\ & \\ddots & \\\\ v _ { K , 1 } & \\ldots & v _ { K , d } \\end{pmatrix} D _ \\mathbf { F } = C _ \\mathbf { F } ^ T . \\end{align*}"}
{"id": "2409.png", "formula": "\\begin{align*} \\widetilde { s } _ { N } = \\sum _ { \\substack { k , l \\in \\Z ^ d \\\\ | k | ^ 2 + | l | ^ 2 \\leq N } } c _ { k , l } M _ { \\beta l } T _ { \\alpha k } g \\end{align*}"}
{"id": "5567.png", "formula": "\\begin{align*} \\frac { \\kappa } { 2 } u '' + u ' - \\lambda u = 0 , u ( 0 ) = u ( 1 ) = 0 \\end{align*}"}
{"id": "3426.png", "formula": "\\begin{align*} T _ k \\bigg ( \\frac 1 { T _ k ( 1 ) } \\bigg ) ( x ) & = \\int _ { \\Bbb R ^ N } \\theta ( r ^ k \\| x - y \\| ) \\frac 1 { T _ k ( 1 ) ( y ) } d \\omega ( y ) \\\\ & \\sim \\int _ { \\Bbb R ^ N } \\theta ( r ^ k \\| x - y \\| ) \\frac 1 { V _ k ( y ) } d \\omega ( y ) \\\\ & \\sim \\frac 1 { V _ k ( x ) } \\int _ { \\Bbb R ^ N } \\theta ( r ^ k \\| x - y \\| ) d \\omega ( y ) \\\\ & = \\frac 1 { V _ k ( x ) } T _ k ( 1 ) ( x ) \\sim 1 . \\end{align*}"}
{"id": "9217.png", "formula": "\\begin{align*} \\mathcal { V } ^ \\omega \\vdash \\forall x ^ X , \\gamma ^ 1 , { \\gamma ' } ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ' > _ \\mathbb { R } 0 \\land \\gamma = _ \\mathbb { R } \\gamma ' \\rightarrow J ^ A _ \\gamma x = _ X J ^ A _ { \\gamma ' } x \\right ) . \\end{align*}"}
{"id": "7667.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { \\tilde { \\Psi } _ { \\varepsilon } ( \\mathbf { x } ) } { | \\mathbf { x } - \\mathbf { x } _ { \\varepsilon } | } = \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } , \\qquad \\mathbf { x } \\in \\partial \\Omega . \\end{align*}"}
{"id": "3696.png", "formula": "\\begin{align*} \\chi ( \\xi ) = \\begin{cases} 1 , \\ \\ \\ \\mbox { f o r } \\ \\ | \\xi | \\leq \\frac 3 4 \\\\ 0 , \\ \\ \\ \\mbox { f o r } \\ \\ | \\xi | \\geq 1 . \\end{cases} \\end{align*}"}
{"id": "7379.png", "formula": "\\begin{align*} \\begin{aligned} & 2 \\beta | p | ^ { 2 - \\alpha } + a ( 1 - \\beta ) ^ { \\alpha - 1 } ( \\beta - \\alpha \\beta ) ( \\beta - \\alpha \\beta - 1 ) r ^ { \\beta ( 1 - \\alpha ) + 1 } \\\\ & \\leq 2 L ^ { 2 - \\alpha } + a ( 1 - \\beta ) ^ { \\alpha - 1 } ( \\alpha - \\alpha ^ 2 ) ( \\alpha - \\alpha ^ 2 - 1 ) \\min \\{ c _ 0 ^ { \\alpha ( 1 - \\alpha ) + 1 } , c _ 0 ^ { 2 - \\alpha } \\} \\end{aligned} \\end{align*}"}
{"id": "5814.png", "formula": "\\begin{align*} \\gamma _ { k } ^ { \\langle n \\rangle } = \\frac { \\left ( \\sum _ { m = 1 } ^ M \\sqrt { \\eta _ { m k } } \\alpha _ { m k } \\right ) ^ 2 } { \\sum _ { m = 1 } ^ M \\beta _ { m k } \\sum _ { k ' = 1 } ^ { K } \\eta _ { m k ' } \\alpha _ { m k ' } + \\frac { 1 } { \\gamma _ t } } , \\end{align*}"}
{"id": "4339.png", "formula": "\\begin{align*} \\langle \\phi _ 0 , \\partial _ \\tau \\phi _ 0 \\rangle _ { L ^ 2 _ \\rho } = - \\frac { \\gamma b _ \\tau } { b } \\| \\phi _ 0 \\| ^ 2 _ { L ^ 2 _ \\rho } ( 1 + O ( b ^ { 1 - \\frac { \\epsilon } { 2 } } ) ) . \\end{align*}"}
{"id": "9426.png", "formula": "\\begin{align*} { \\mathcal N } _ 1 u = ( u \\cdot \\nabla ) u = \\star ( \\star d _ 1 u \\wedge u ) + d _ 0 | u | ^ 2 / 2 , \\end{align*}"}
{"id": "6233.png", "formula": "\\begin{align*} & \\det ( I + u ( J B ^ { \\prime } + E ) ) \\\\ & = \\det ( I + u F _ 0 ) \\det ( ( I + u F _ 1 ) ( I + u F _ 2 ) - u ^ 2 ( I + u F _ 1 ) B _ 1 ( I + u F _ 1 ) ^ { - 1 } B _ 2 ) . \\end{align*}"}
{"id": "849.png", "formula": "\\begin{align*} \\begin{aligned} T ^ { \\rm R e a c } _ j & = t _ j ^ S - t _ { j - 1 } ^ S = \\tau ^ { \\rm R e a c } _ { m } R _ j + \\tau ^ { \\rm R e a c } _ { V _ j } + \\tau _ { \\rm f } . \\end{aligned} \\end{align*}"}
{"id": "1916.png", "formula": "\\begin{align*} & f ( 0 , x , v ) = f _ 0 ( x , v ) , ( x , v ) \\in \\Omega \\times \\mathbb { R } ^ 3 , \\\\ & \\rho ( 0 , x ) = \\rho _ 0 ( x ) , u ( 0 , x ) = u _ 0 ( x ) , x \\in \\Omega , \\end{align*}"}
{"id": "6942.png", "formula": "\\begin{align*} \\Vert v - I _ h ^ C v \\Vert _ { k , E } \\lesssim h _ E ^ k \\vert v \\vert _ { 1 , D _ E } , k = 0 , 1 \\ , , \\end{align*}"}
{"id": "9237.png", "formula": "\\begin{align*} \\begin{cases} \\forall x ^ X , y ^ X \\left ( \\langle x - _ X y , J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y \\rangle _ X \\geq _ \\mathbb { R } \\norm { J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y } ^ 2 _ X \\right ) , \\\\ \\forall p ^ X , x ^ X \\left ( ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { 1 } x \\right ) . \\end{cases} \\end{align*}"}
{"id": "1242.png", "formula": "\\begin{align*} \\mu ( E \\cap B \\cap A ) & = \\mu ( E \\cap B ) + \\mu ( A \\cap B ) - \\mu \\left ( ( E \\cap B ) \\cup ( A \\cap B ) \\right ) \\\\ & \\geq ( c + 1 - \\frac { c } { 2 } - 1 ) \\mu ( B ) = \\frac { c } { 2 } \\mu ( B ) > 0 , \\end{align*}"}
{"id": "3519.png", "formula": "\\begin{align*} \\frac { s _ 3 } { s _ 1 + s _ 3 - 1 } \\sum _ { n \\leq a t _ 3 } \\frac { 1 } { n ^ { s _ 2 - 1 } } \\int _ { a t _ 3 } ^ \\infty \\frac { d u } { u ^ { s _ 1 } ( u + n ) ^ { s _ 3 + 1 } } & \\ll \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 2 ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 & ( \\sigma _ 2 = 2 ) \\\\ t _ 3 ^ { 2 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 2 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "8489.png", "formula": "\\begin{align*} f _ k & = [ u ^ k ] F ( u ) = \\frac { N _ 3 ^ { t - k + 1 } } { D _ t } ( - 1 ) ^ { k - 1 } , \\\\ g _ k & = [ u ^ k ] G ( u ) = \\frac { N _ 2 ^ { t - k } } { D _ t } ( - 1 ) ^ { k } , \\mbox { a n d } \\\\ h _ k & = [ u ^ k ] H ( u ) = \\frac { N _ 2 ^ { t - k + 1 } } { D _ t } ( - 1 ) ^ { k - 1 } . \\end{align*}"}
{"id": "2191.png", "formula": "\\begin{align*} \\lim \\limits _ { \\vert ( t , s ) \\vert \\rightarrow + \\infty } \\frac { F ( x , t w ^ + + s w ^ - ) } { \\max \\lbrace \\vert t \\vert ^ { g ^ + } , \\vert s \\vert ^ { g ^ + } \\rbrace } = + \\infty , \\ \\ \\ x \\in \\mathbb { R } ^ d . \\end{align*}"}
{"id": "4130.png", "formula": "\\begin{align*} \\prod _ { i = 2 } ^ { d - 1 } ( \\ell _ i + 1 ) \\le A _ P ^ { \\frac { d - 2 } { d - 1 } } . \\end{align*}"}
{"id": "3307.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F ) \\geq \\alpha _ { n - j } ( F _ 1 / T _ 1 ) + \\alpha _ { n - j } ( F _ 2 / T _ 2 ) - \\sum _ { l = 1 } ^ { j - 1 } \\frac { ( - 1 ) ^ { l - 1 } a _ 1 ^ l } { l ! } \\alpha _ { n - j + l } ( F _ 2 / T _ 2 ) . \\end{align*}"}
{"id": "5851.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 1 } ^ n \\frac { \\alpha _ { i - 1 } } { \\lambda _ i } . \\end{align*}"}
{"id": "6277.png", "formula": "\\begin{align*} z _ i [ n ] = \\sqrt { p } h _ i v [ n ] + w _ i ' [ n ] , \\ n = 1 , 2 , \\cdots , N _ 1 , \\end{align*}"}
{"id": "2121.png", "formula": "\\begin{align*} \\binom { c _ n k n } l \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\le C \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n - l } } \\le C \\frac { ( k j ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n - l } } . \\end{align*}"}
{"id": "3138.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 & = 0 \\ , , \\\\ x _ 1 ^ 2 + i x _ 2 ^ 2 - x _ 3 ^ 2 - i x _ 4 ^ 2 & = 0 \\ , , ( i ^ 2 = - 1 . ) \\\\ x _ 1 ^ 2 - x _ 2 ^ 2 + x _ 3 ^ 3 - x _ 4 ^ 2 & = 0 \\ , . \\end{align*}"}
{"id": "6096.png", "formula": "\\begin{align*} \\varphi ^ { j } = u ^ { j } + z _ { s } w ^ { j } , \\psi ^ { j } = v ^ { j } + z _ { t } w ^ { j } \\ , . \\end{align*}"}
{"id": "4407.png", "formula": "\\begin{align*} \\tilde \\gamma = \\frac { 1 } { 2 } ( d + \\sqrt { d ^ 2 - 1 2 d + 2 4 } ) . \\end{align*}"}
{"id": "4840.png", "formula": "\\begin{align*} P ( x ) = \\begin{cases} - ( x + 1 ) ^ 2 & x < - 1 \\\\ 0 & - 1 \\leq x \\leq 1 \\\\ ( x - 1 ) ^ 2 & x > 1 , \\end{cases} P ^ { - 1 } ( y ) = \\begin{cases} - \\sqrt { | y | } - 1 & y < 0 \\\\ \\sqrt y + 1 & y > 0 . \\end{cases} \\end{align*}"}
{"id": "3673.png", "formula": "\\begin{align*} A _ n = \\{ ( u , v ) : u , v \\in [ n ] , u \\ne v \\} , \\end{align*}"}
{"id": "9509.png", "formula": "\\begin{align*} [ f ( - M _ x ) ] [ g ( H ) ] & = \\big ( [ f ( 0 ) ] + [ - M _ x ] [ f _ 1 ( - M _ x ) ] \\big ) \\big ( [ g ( 0 ) ] + [ H ] [ g _ 1 ( H ) ] \\big ) \\\\ \\\\ & = f ( 0 ) g ( 0 ) + g ( 0 ) [ - M _ x ] [ f _ 1 ( - M _ x ) ] + f ( 0 ) [ H ] [ g _ 1 ( H ) ] \\\\ \\\\ & = f ( 0 ) g ( 0 ) + g ( 0 ) [ f ( - M _ x ) - f ( 0 ) ] + f ( 0 ) [ g ( H ) - g ( 0 ) ] \\\\ \\\\ & = g ( 0 ) [ f ( - M _ x ) ] + f ( 0 ) [ g ( H ) ] - f ( 0 ) g ( 0 ) . \\end{align*}"}
{"id": "6283.png", "formula": "\\begin{align*} \\mathrm { v a r } \\big [ \\widehat { K } _ { \\ ! \\mathrm { a - c p t - f } } ^ \\mathbb { R } \\big ] \\ ! = \\ ! \\upsilon ^ 2 \\bigg ( \\ ! \\Big ( 1 \\ ! - \\ ! \\frac { \\pi } { 4 } \\Big ) \\Big ( 1 \\ ! + \\ ! \\frac { 1 } { N _ 1 \\bar { \\gamma } ' } \\Big ) \\ ! K \\ ! + \\ ! \\frac { 1 } { 2 N _ 1 \\bar { \\gamma } ' } \\ ! + \\ ! \\frac { 1 } { 2 N _ 2 \\bar { \\gamma } _ c } \\bigg ) = \\frac { 4 - \\pi } { \\pi } K + \\frac { 1 + \\frac { N _ 1 \\bar { \\gamma } ' } { N _ 2 \\bar { \\gamma } _ c } } { \\pi ( 1 + N _ 1 \\bar { \\gamma } ' ) } \\end{align*}"}
{"id": "3411.png", "formula": "\\begin{align*} S \\lesssim \\sum \\limits _ { j = 1 } ^ \\infty 2 ^ { - j ( \\varepsilon ) } + 1 \\lesssim 1 . \\end{align*}"}
{"id": "4136.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int u ^ 2 ( x , t ) d x = 0 . \\end{align*}"}
{"id": "6889.png", "formula": "\\begin{align*} S _ k ^ j = \\frac { 1 } { q ^ { j ^ 2 } { k \\choose k - j } _ q } \\sum \\limits _ { i = 0 } ^ { j } ( - 1 ) ^ { j - i } q ^ { { j - i \\choose 2 } } { j \\choose i } _ q { k + i \\choose i } _ q N _ k ^ { i } , \\end{align*}"}
{"id": "7838.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { 3 j + 1 } \\alpha _ { i } e _ { i } , ~ \\alpha _ { 3 j + 1 } \\neq 0 \\quad ~ \\langle x \\rangle _ { T ^ { * 3 } } = j + 1 . \\end{align*}"}
{"id": "2554.png", "formula": "\\begin{align*} \\widehat { D } _ { L , m } f ( x ) = \\mu ( D _ L ) f ( x ) = i ^ m \\sqrt { | \\det ( L ^ { - 1 } ) | } \\ , f ( L ^ { - 1 } x ) = i ^ m \\mathcal { D } _ L f ( x ) , \\det ( L ) \\neq 0 . \\end{align*}"}
{"id": "4555.png", "formula": "\\begin{align*} 2 ( 1 - \\delta ) k ' \\cdot \\frac { 2 } { 3 } ( n - 1 ) & \\le d ( v _ 1 ) + d ( v _ 2 ) \\\\ & = e ( \\{ v _ 1 , v _ 2 \\} , S ) + e \\bigl ( \\{ v _ 1 , v _ 2 \\} , V - ( S \\cup \\{ v _ 1 , v _ 2 \\} ) \\bigr ) + 2 w ( v _ 1 v _ 2 ) \\\\ & \\le \\left ( k ' + h _ 5 + \\frac { 1 } { 2 } \\right ) | S | + \\biggl ( k ' + \\frac { h - h _ 5 } { 2 } \\biggr ) ( n - | S | - 2 ) + 2 k . \\end{align*}"}
{"id": "1420.png", "formula": "\\begin{align*} r _ { N / \\{ w _ 1 , w _ 2 \\} } ( z ) & = r ( z \\cup \\{ w _ 1 , w _ 2 \\} ) - r ( \\{ w _ 1 , w _ 2 \\} ) \\\\ & = \\rho ( \\langle z , w _ 1 , w _ 2 \\rangle ) - \\rho ( \\langle w _ 1 , w _ 2 \\rangle ) \\\\ & = \\rho ( W ) - \\rho ( W ) = 0 . \\end{align*}"}
{"id": "2661.png", "formula": "\\begin{align*} \\norm { V _ g f } _ 2 = \\norm { f } _ 2 \\norm { g } _ 2 \\end{align*}"}
{"id": "810.png", "formula": "\\begin{align*} v _ k : = \\sum _ { j = k } ^ { N ( k ) } \\lambda _ { j , k } u _ j , g _ k : = \\sum _ { j = k } ^ { N ( k ) } \\lambda _ { j , k } g _ { u _ j } \\end{align*}"}
{"id": "3024.png", "formula": "\\begin{align*} \\rho : H _ 2 ( X _ \\psi , Q ) \\to \\mathbf { N } : = W / \\left ( n \\left ( \\ker r \\right ) \\right ) . \\end{align*}"}
{"id": "5599.png", "formula": "\\begin{align*} \\mathcal { T } ^ { p , r } : = \\underbrace { \\Omega \\otimes _ \\mathcal { X } \\ldots \\otimes _ \\mathcal { X } \\Omega } _ { p } \\otimes _ \\mathcal { X } \\underbrace { \\Xi \\otimes _ \\mathcal { X } \\ldots \\otimes _ \\mathcal { X } \\Xi } _ { r } \\end{align*}"}
{"id": "7503.png", "formula": "\\begin{align*} \\| \\phi ( \\cdot , t ) \\| ^ 2 \\equiv \\| \\phi _ 0 \\| ^ 2 = 1 , \\frac { \\mathrm { d } } { \\mathrm { d } t } E ( \\phi ( \\cdot , t ) ) = - 2 \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 , \\forall t \\geq 0 . \\end{align*}"}
{"id": "2506.png", "formula": "\\begin{align*} \\rho ( \\mathbf { h } ) = \\rho ( x , \\omega , \\tau ) = e ^ { 2 \\pi i \\tau } M _ { \\omega / 2 } T _ x M _ { \\omega / 2 } = e ^ { 2 \\pi i \\tau } e ^ { \\pi i x \\cdot \\omega } T _ x M _ \\omega . \\end{align*}"}
{"id": "1842.png", "formula": "\\begin{align*} D ^ k ( \\bar { x } \\bar { y } ^ { - 1 } ) = \\bar { x } \\bar { y } ^ { - 1 } ( \\bar { y } - \\bar { x } ) ^ k , \\end{align*}"}
{"id": "3763.png", "formula": "\\begin{align*} \\hat u \\circ \\gamma = C \\hat u \\end{align*}"}
{"id": "3829.png", "formula": "\\begin{align*} E ^ { [ 1 ] } ( f , G \\rtimes S ) ( t , \\overline { t } ) = \\sum \\widehat { m } _ { \\underline { \\lambda } , \\widetilde { \\underline { \\lambda } } } ( - 1 ) ^ { n _ { \\underline { \\lambda } } } ( t \\overline { t } ) ^ { { \\rm a g e } ( \\underline { \\lambda } ) - \\frac { n - n _ { \\underline { \\lambda } } } { 2 } } \\left ( \\frac { \\overline { t } } { t } \\right ) ^ { { \\rm a g e } ( \\widetilde { \\underline { \\lambda } } ) - \\frac { n - n _ { \\widetilde { \\underline { \\lambda } } } } { 2 } } , \\end{align*}"}
{"id": "5035.png", "formula": "\\begin{align*} \\Delta _ G & = \\left \\{ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , 1 \\leq i \\leq n - 1 \\ , , \\ ; 2 \\ , \\widehat { x } _ { n } \\right \\} \\ , , \\\\ \\intertext { a n d } \\Delta _ K & = \\left \\{ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , 1 \\leq i \\leq n - 1 \\right \\} \\ , , \\end{align*}"}
{"id": "2743.png", "formula": "\\begin{align*} m _ { 0 , \\lambda } ( V ) = 0 \\ \\mbox { i f $ \\lambda = 1 ^ n $ } . \\end{align*}"}
{"id": "9529.png", "formula": "\\begin{align*} p _ { t } + \\Delta y _ { t + 1 } & \\in \\partial _ { x } H _ t ( x _ { t } , y _ t ) , \\\\ \\Delta x _ t & \\in \\partial _ { y } [ - H _ t ] ( x _ { t } , y _ t ) , \\end{align*}"}
{"id": "1773.png", "formula": "\\begin{align*} f \\cdot g ( \\xi ) = f ( \\xi ) g ( \\xi ) , \\ \\mathcal { F } ( f \\ast g ) = \\mathcal { F } ( f ) \\cdot \\mathcal { F } ( g ) . \\end{align*}"}
{"id": "2907.png", "formula": "\\begin{align*} J ( z ) = | 1 + z | ^ p - \\left ( 1 + \\frac { p } { 2 } z + \\frac { p } { 2 } \\bar { z } \\right ) , z \\in \\mathbb { C } \\end{align*}"}
{"id": "5744.png", "formula": "\\begin{align*} \\mu ( A _ m ) \\leq \\sum _ { \\ell = 1 } ^ \\infty \\mu ( C _ { m , \\ell } ) & \\leq \\sum _ { \\ell = 1 } ^ \\infty 4 ^ { \\ell } \\mu ( B _ { m , \\ell } ) \\leq C _ 5 \\sum _ { \\ell = 1 } ^ \\infty 4 ^ { \\ell } e ^ { - c _ 1 m \\ell } \\leq C _ 5 \\sum _ { \\ell = 1 } ^ \\infty e ^ { ( 2 - c _ 1 m ) \\ell } . \\end{align*}"}
{"id": "3263.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } \\frac { i t } { \\omega _ 2 ( i t ) } = \\frac { | \\lambda | ^ 2 - \\lambda _ 2 ( \\mu ) } { | \\lambda | ^ 2 } . \\end{align*}"}
{"id": "9140.png", "formula": "\\begin{align*} \\frac { 1 } { k + 1 } & \\geq \\norm { T _ { \\lambda _ N } x _ N - z } \\\\ & \\geq \\vert \\norm { T _ { \\lambda _ N } x _ N } - \\norm { z } \\vert \\\\ & = \\norm { z } - \\norm { T _ { \\lambda _ N } x _ N } \\\\ & \\geq \\norm { T ^ \\circ x _ N } - \\norm { T _ { \\lambda _ N } x _ N } \\\\ & = \\vert \\norm { T ^ \\circ x _ N } - \\norm { T _ { \\lambda _ N } x _ N } \\vert \\end{align*}"}
{"id": "8593.png", "formula": "\\begin{align*} & \\partial _ { x } \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } e ^ { i k x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k \\\\ & = \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } i k e ^ { i k x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k + \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } e ^ { i k x } \\partial _ { x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k . \\end{align*}"}
{"id": "7909.png", "formula": "\\begin{align*} ( k _ w ( t ) ) _ { i _ 1 i _ 2 } = \\frac { t ^ 2 } { 2 } \\cdot ( k _ w '' ( \\xi _ { i _ 1 i _ 2 } ) ) _ { i _ 1 i _ 2 } . \\end{align*}"}
{"id": "5388.png", "formula": "\\begin{align*} \\hat { b } _ { n , h } ( x ) = \\sum _ { k = 0 } ^ { n - 1 } W _ { n , k } ( X _ { t _ { k } } , x ) \\diamond ( \\frac { X _ { t _ { k + 1 } } - X _ { t _ { k } } } { \\alpha _ { n } } ) , \\end{align*}"}
{"id": "7876.png", "formula": "\\begin{align*} A ( e _ 1 ) = \\begin{bmatrix} 1 & e _ 1 \\cdot w _ 2 & \\cdots & e _ 1 \\cdot w _ n \\\\ w _ 2 \\cdot e _ 1 & 1 & \\cdots & w _ 2 \\cdot w _ n \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ w _ n \\cdot e _ 1 & w _ n \\cdot w _ 2 & \\cdots & 1 \\end{bmatrix} . \\end{align*}"}
{"id": "6320.png", "formula": "\\begin{align*} \\zeta ^ \\wedge _ { \\mathcal { L } _ f , p } ( s ) = \\int _ { \\Z _ p \\setminus \\{ 0 \\} } | a | _ p ^ { ( n + 2 ) s - 4 n } \\mu _ 2 ( G _ 2 ^ + ( a ) ) d \\mu _ 1 ( a ) . \\end{align*}"}
{"id": "3498.png", "formula": "\\begin{align*} D _ { 2 2 } & = \\frac { s _ 2 2 ^ { - s _ 3 } } { s _ 1 + s _ 3 - 1 } \\int _ { a t _ 3 } ^ \\infty \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 1 + s _ 2 + s _ 3 } } d v \\\\ & + \\frac { s _ 2 s _ 3 } { s _ 1 + s _ 3 - 1 } \\int _ { a t _ 3 } ^ \\infty \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 } } \\int _ v ^ \\infty \\frac { 1 } { u ^ { s _ 1 } ( u + v ) ^ { s _ 3 + 1 } } d u d v \\\\ & = D _ { 2 2 1 } + D _ { 2 2 2 } , \\end{align*}"}
{"id": "5562.png", "formula": "\\begin{align*} \\| g _ n \\| ^ 2 = \\int _ 0 ^ 1 g _ n ' ( x ) ^ 2 e ^ { \\gamma x } d x = \\int _ 0 ^ 1 f _ n ( x ) ^ 2 e ^ { \\gamma x } d x \\leq e ^ { \\gamma } \\int _ 0 ^ 1 f _ n ( x ) ^ 2 d x = \\frac { e ^ { \\gamma } } { 2 } . \\end{align*}"}
{"id": "3318.png", "formula": "\\begin{align*} \\sum _ { \\alpha , n } c _ { \\alpha , n } x _ { \\alpha , n } = 0 \\ \\ ( c _ { \\alpha , n } \\in K ) . \\end{align*}"}
{"id": "1308.png", "formula": "\\begin{align*} \\hat { I } _ { \\geq \\epsilon } ( k ) : = \\{ \\ , \\ , \\alpha _ { k ' } \\ , \\ , | \\ , \\ , k ' \\leq k , \\ , \\ , A ( \\alpha _ { k ' + 1 } ) - A ( \\alpha _ { k ' } ) \\geq \\epsilon \\ , \\ , \\} \\end{align*}"}
{"id": "207.png", "formula": "\\begin{align*} T _ { s + t } ^ { \\alpha } ( f ) = ( T _ t ^ { \\alpha } \\circ T _ s ^ \\alpha ) ( f ) = ( T _ s ^ \\alpha \\circ T _ t ^ { \\alpha } ) ( f ) . \\end{align*}"}
{"id": "6002.png", "formula": "\\begin{align*} g ( t ) = \\sum _ { n = 0 } ^ { \\infty } g _ { n } \\frac { t ^ { n } } { n ! } \\end{align*}"}
{"id": "5206.png", "formula": "\\begin{align*} T _ k ( f ) ( x ) & = \\sum \\limits _ { i = 1 } ^ { \\infty } \\sum \\limits _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } \\lambda _ i \\int \\limits _ { | y | = q ^ { j + 1 } } f ( x - y ) a _ i ( y ) d y , \\\\ & = \\sum \\limits _ { i = 1 } ^ { \\infty } \\lambda _ i B _ i f , \\end{align*}"}
{"id": "8797.png", "formula": "\\begin{align*} \\begin{aligned} & ( f _ 1 , f _ 2 , \\phi ) = \\sum _ { j = 0 } ^ 2 \\sum _ { k = 0 } ^ 2 w _ { j k } ( a _ { 1 j } , a _ { 2 k } , a _ { 1 j } a _ { 2 k } ) , \\ \\sum _ { j = 0 } ^ 2 \\sum _ { k = 0 } ^ 2 w _ { j k } = 1 , \\ w \\geq 0 , \\\\ & \\sum _ { j = 0 } ^ 2 w _ { j 0 } \\leq \\delta _ 1 , \\ \\sum _ { j = 0 } ^ 2 w _ { j 2 } \\leq 1 - \\delta _ 1 , \\ \\delta _ 1 \\in \\{ 0 , 1 \\} , \\\\ & \\sum _ { k = 0 } ^ 2 w _ { 0 k } \\leq \\delta _ 2 , \\ \\sum _ { k = 0 } ^ 2 w _ { 2 k } \\leq 1 - \\delta _ 2 , \\ \\delta _ 2 \\in \\{ 0 , 1 \\} . \\end{aligned} \\end{align*}"}
{"id": "4970.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ t u - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = 0 , \\\\ & \\mbox { d i v } u = 0 , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\endaligned \\right . \\end{align*}"}
{"id": "823.png", "formula": "\\begin{align*} 2 R \\leq \\int _ { y _ 0 } ^ \\infty \\rho d y = 1 - y _ 0 + \\int _ 1 ^ \\infty y ^ { - \\beta } d y = 1 - y _ 0 + \\frac { 1 } { \\beta - 1 } . \\end{align*}"}
{"id": "1473.png", "formula": "\\begin{align*} R = \\left [ \\begin{array} { c c c c c c } 1 _ { m _ 1 } & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 / 2 & 0 & 0 & - \\zeta ^ { - 1 } & 0 \\\\ 0 & 0 & 0 & 1 _ { m _ 2 } & 0 & 0 \\\\ 0 & 0 & - 1 _ { m _ 1 } & 0 & 0 & 0 \\\\ 0 & - 1 / 2 & 0 & 0 & - \\zeta ^ { - 1 } & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 _ { m _ 2 } \\end{array} \\right ] . \\end{align*}"}
{"id": "1752.png", "formula": "\\begin{align*} \\Theta ( g ) : = \\langle v _ 0 , \\pi ( g ) v _ 0 \\rangle . \\end{align*}"}
{"id": "3589.png", "formula": "\\begin{align*} | [ ( Q \\widetilde { T } ) ^ * ( z ^ * ) ] ( e _ k \\otimes y ^ * ) - \\langle u ^ * ( y ^ * ) , Q ^ * ( z ^ * ) \\rangle | & = | \\langle ( T ( e _ k ) ) ^ * ( y ^ * ) - u ^ * ( y ^ * ) , Q ^ * ( z ^ * ) \\rangle | \\\\ & \\leq \\| ( T ( e _ k ) ) ^ * - u ^ * \\| \\| y ^ * \\| \\| Q ^ * ( z ^ * ) \\| \\longrightarrow 0 \\end{align*}"}
{"id": "2780.png", "formula": "\\begin{align*} \\left \\langle \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | Q | ^ p \\right ) Q ^ { p - 1 } , h _ 1 \\right \\rangle = 0 , \\end{align*}"}
{"id": "1889.png", "formula": "\\begin{align*} \\Gamma = \\{ ( \\gamma , \\gamma ^ * ) \\mid \\gamma \\in \\Gamma _ G \\} \\end{align*}"}
{"id": "2920.png", "formula": "\\begin{align*} \\bar \\delta _ n ( k ) = \\sqrt { \\sigma _ n ^ 2 ( k ) \\cdot \\binom { d } k } , \\delta _ n ( k ) = \\sqrt { \\frac { 2 } { 9 0 ^ k } \\cdot \\binom { d } k } . \\end{align*}"}
{"id": "7696.png", "formula": "\\begin{align*} - \\Delta u _ { \\ast } = \\tilde { \\lambda } _ 0 \\tilde { m } _ 0 u _ { \\ast } \\R ^ N \\ , . \\end{align*}"}
{"id": "1511.png", "formula": "\\begin{align*} \\xi ( y , h , s + l , s - l ) = \\frac { \\Gamma _ t ( 2 s - 2 m + 1 ) } { \\Gamma _ { m - q } ( s + l ) \\Gamma _ { m - p } ( s - l ) } \\omega ( y , h , s + l , s - l ) , \\end{align*}"}
{"id": "1464.png", "formula": "\\begin{align*} \\phi _ { \\infty } = \\left [ \\begin{array} { c c c c } 0 & 0 & 0 & - 1 _ { 2 m } \\\\ 0 & 0 & - 1 _ r & 0 \\\\ 0 & 1 _ r & 0 & 0 \\\\ 1 _ { 2 m } & 0 & 0 & 0 \\end{array} \\right ] , \\psi _ { \\infty } = \\left [ \\begin{array} { c c c c } 0 & 0 & 0 & J _ m ' \\\\ 0 & 1 _ r & 0 & 0 \\\\ 0 & 0 & 1 _ r & 0 \\\\ - J _ m ' & 0 & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "4009.png", "formula": "\\begin{align*} m ^ 3 + \\mu m ^ 2 + 2 \\mu m + \\mu ^ 2 = 0 . \\end{align*}"}
{"id": "9010.png", "formula": "\\begin{align*} X ^ { m ' } _ { B _ m } [ T ] = \\lim _ { \\ell \\to \\infty } ( X ^ { B _ { \\ell } } _ { B _ { m ' } } ) _ { B _ m } [ T ] = \\lim _ { \\ell \\to \\infty } X ^ { B _ { \\ell } } _ { B _ m } [ T ] = X ^ m [ T ] = X ^ m _ { B _ m } [ T ] , \\end{align*}"}
{"id": "4091.png", "formula": "\\begin{align*} \\widetilde \\mu = ( G ' \\cdot \\mu + \\tfrac { 1 } { 2 } G '' \\cdot \\sigma ^ 2 ) \\circ G ^ { - 1 } \\ , \\ , \\widetilde \\sigma = ( G ' \\cdot \\sigma ) \\circ G ^ { - 1 } \\end{align*}"}
{"id": "1109.png", "formula": "\\begin{align*} E ( k ) = I + \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\frac { ( I + \\mu ( s ) ) ( J ^ E ( s ) - I ) } { s - k } d s , \\end{align*}"}
{"id": "3182.png", "formula": "\\begin{align*} W _ k , \\zeta _ k = \\varphi _ C ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) \\in \\mathbb { R } ^ { d \\times n _ k } \\times \\mathfrak { Z } . \\end{align*}"}
{"id": "1748.png", "formula": "\\begin{align*} \\mathcal { A } _ G ( X ) : = \\big ( C _ c ( G ) \\hat { \\otimes } C ( S \\times S ) \\big ) ^ { K \\times K } . \\end{align*}"}
{"id": "8954.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 1 } & = \\frac { e ^ t } { C _ p q } \\left | \\int _ 0 ^ t e ^ { - s } D f \\left ( \\xi ( s ) \\right ) d s + \\tilde { c } \\right | \\\\ \\left | \\dot { \\xi } ( t ) \\right | & = \\left ( \\frac { e ^ t } { C _ p q } \\right ) ^ \\frac { 1 } { q - 1 } \\left | \\int _ 0 ^ t e ^ { - s } D f \\left ( \\xi ( s ) \\right ) d s + \\tilde { c } \\right | ^ \\frac { 1 } { q - 1 } , \\end{aligned} \\end{align*}"}
{"id": "3274.png", "formula": "\\begin{align*} S _ { \\mu _ { T ^ * T } } \\left ( \\psi _ { \\mu _ { T ^ * T } } \\left ( - \\frac { 1 } { s ( | \\lambda | , 0 ) ^ 2 } \\right ) \\right ) = - \\frac { 1 } { s ( | \\lambda | , t ) ^ 2 } \\cdot - \\frac { s ( | \\lambda | , t ) ^ 2 } { | \\lambda | ^ 2 } = \\frac { 1 } { | \\lambda | ^ 2 } \\end{align*}"}
{"id": "4226.png", "formula": "\\begin{align*} \\mathcal { F } _ N ( \\mathbf { h } _ * , \\mathbf { g } _ * ) = \\min _ { ( \\mathbf { h } , \\mathbf { g } ) \\in E ^ m } \\mathcal { F } _ N ( \\mathbf { h } , \\mathbf { g } ) . \\end{align*}"}
{"id": "2765.png", "formula": "\\begin{align*} \\| u _ n \\| _ 2 ^ 2 = \\sum _ { j = 1 } ^ l \\| U ^ j \\| _ 2 ^ 2 + \\| r _ n ^ l \\| _ 2 ^ 2 + o _ n ( 1 ) , \\end{align*}"}
{"id": "9110.png", "formula": "\\begin{align*} ( f \\smile g ) ( x _ 1 \\otimes \\cdots x _ { n + m } ) = f ( x _ 1 \\otimes \\cdots x _ n ) g ( x _ { n + 1 } \\otimes \\cdots x _ { n + m } ) . \\end{align*}"}
{"id": "8162.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in X _ p } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) \\sum _ { k = 1 } ^ { p - 1 } \\vert S ( k , \\chi ) \\vert ^ 2 = \\sum _ { \\chi \\in X _ p } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) \\sum _ { k = 1 } ^ { p - 1 } \\left \\vert \\sum _ { l = 1 } ^ k \\chi ( l ) \\right \\vert ^ 2 \\\\ = \\sum _ { \\chi \\in X _ p } \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { 1 \\leq l _ 1 , l _ 2 \\leq k } \\chi ( q _ 1 l _ 1 ) \\overline { \\chi ( q _ 2 l _ 2 ) } = ( p - 1 ) ^ 2 { \\mathcal { A } } ( q _ 1 , q _ 2 , p ) , \\end{align*}"}
{"id": "3536.png", "formula": "\\begin{align*} S _ 2 & \\ll \\sum _ { \\substack { m _ 1 , n _ 1 , m _ 2 , n _ 2 \\leq b T \\\\ m _ 1 + n _ 1 = m _ 2 + n _ 2 } } \\frac { 1 } { m _ 1 ^ { \\sigma _ 1 } m _ 2 ^ { \\sigma _ 1 } n _ 1 ^ { \\sigma _ 2 } n _ 2 ^ { \\sigma _ 2 } ( m _ 1 + n _ 1 ) ^ { 2 \\sigma _ 3 - 1 } } \\\\ & \\ll T ^ { 4 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } . \\end{align*}"}
{"id": "3315.png", "formula": "\\begin{align*} \\frac { v ^ { - 1 } w - 1 } { v - 1 } = - v ^ { - 1 } w + x \\in F , \\end{align*}"}
{"id": "1397.png", "formula": "\\begin{align*} E _ i ( 1 ) = \\alpha + \\int _ 0 ^ 1 ( n _ i - b _ i ) ( y ) d y , i = 1 , 2 . \\end{align*}"}
{"id": "4581.png", "formula": "\\begin{align*} \\Bigg | \\ln \\frac { \\mathbf { P } \\big ( ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \\sigma \\big ) } { 1 - \\Phi ( x ) } \\Bigg | \\leq c \\ , \\Bigg ( \\frac { x ^ 3 } { \\sqrt { n } } + ( 1 + x ) \\frac { \\ln n } { \\sqrt { n } } \\Bigg ) . \\end{align*}"}
{"id": "9242.png", "formula": "\\begin{align*} - \\gamma ^ { - 1 } \\norm { p - J ^ A _ \\gamma x } ^ 2 & = \\gamma ^ { - 1 } \\langle J ^ A _ \\gamma x - p , p - J ^ A _ \\gamma x \\rangle \\\\ & = \\langle J ^ A _ \\gamma x - p , \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma ^ { - 1 } ( x - p ) \\rangle \\\\ & \\geq \\tilde { \\rho } \\norm { \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma ^ { - 1 } ( x - p ) } ^ 2 \\\\ & = \\tilde { \\rho } / \\gamma ^ 2 \\norm { p - J ^ A _ \\gamma x } ^ 2 . \\end{align*}"}
{"id": "7534.png", "formula": "\\begin{align*} \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) = \\frac { \\pi } { 2 } + \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "6746.png", "formula": "\\begin{align*} \\mathcal L V + g + \\rho \\max \\{ V ^ * - V , 0 \\} = 0 , \\end{align*}"}
{"id": "6763.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial S } ( t , S _ f ( t ) ) = \\frac { \\partial V ^ * } { \\partial S } ( S _ f ( t ) ) . \\end{align*}"}
{"id": "1865.png", "formula": "\\begin{align*} P _ { n + 1 } ( x ) = ( 1 + x ^ 2 ) \\sum _ { k = 0 } ^ n { n \\choose k } Q _ k ( x ) Q _ { n - k } ( x ) . \\end{align*}"}
{"id": "6176.png", "formula": "\\begin{align*} \\begin{aligned} U ^ { T } C V & = \\Sigma = { \\rm d i a g } \\{ \\sigma _ 1 , \\cdots , \\sigma _ t \\} , \\\\ { U } ^ { T } _ { A } A V _ { A } & = \\Sigma _ { A } = { \\rm d i a g } \\{ \\sigma ^ { A } _ 1 , \\cdots , \\sigma ^ { A } _ n \\} , \\end{aligned} \\end{align*}"}
{"id": "3489.png", "formula": "\\begin{align*} \\sum _ { y < m \\leq M } \\frac { 1 } { m ^ { s _ 1 } ( m + n ) ^ { s _ 3 } } & = \\int _ y ^ M \\frac { d u } { u ^ { s _ 1 } ( u + n ) ^ { s _ 3 } } + O \\left ( \\frac { 1 } { y ^ { \\sigma _ 1 } ( y + n ) ^ { \\sigma _ 3 } } \\right ) . \\end{align*}"}
{"id": "4915.png", "formula": "\\begin{align*} f ( x _ { t + 1 } ) - f ( x _ t ) \\leq - \\eta _ t \\| g _ t \\| ^ 2 + \\frac { L \\eta _ t ^ 2 } { 2 } \\| g _ t \\| ^ 2 = ( - \\eta _ t + \\frac { L \\eta _ t ^ 2 } { 2 } ) \\| g _ t \\| ^ 2 \\end{align*}"}
{"id": "298.png", "formula": "\\begin{align*} \\psi ( x , t ) = U [ \\psi _ { 0 } ] ( x , t , 0 ) + D [ u , \\psi ] ( x , t ) , \\end{align*}"}
{"id": "607.png", "formula": "\\begin{align*} f ( n ) \\ = \\ 2 ^ n . ] \\end{align*}"}
{"id": "802.png", "formula": "\\begin{align*} ( | z | ^ { p - 2 } z - | w | ^ { p - 2 } w ) \\cdot ( z - w ) \\ge \\begin{cases} C ' | z - w | ^ p , & p \\ge 2 \\\\ C '' ( | z | + | w | ) ^ { p - 2 } | z - w | ^ 2 , & p \\le 2 . \\end{cases} \\end{align*}"}
{"id": "5409.png", "formula": "\\begin{align*} \\frac { 1 } { n h } \\sum _ { k = 0 } ^ { n - 1 } \\partial _ { \\theta ^ { 2 } } f ( X _ { t _ { k } } , \\theta _ { 0 } ) \\delta B _ { t _ { k } } \\rightarrow 0 , \\end{align*}"}
{"id": "1931.png", "formula": "\\begin{align*} & \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v ( ( u - v ) f ) + { \\rm { d i v } } _ v ( f L [ f ] ) - \\Delta _ v f = 0 , \\\\ & \\partial _ t \\rho + { \\rm { d i v } } _ x ( \\rho u ) = 0 , \\\\ & \\partial _ t ( \\rho u ) + { \\rm { d i v } } ( \\rho u \\otimes u ) + \\nabla \\rho ^ { \\gamma } + \\delta \\nabla \\rho ^ { \\beta } - { \\rm { d i v } } \\mathbb { S } ( \\nabla u ) = - \\int _ { \\mathbb { R } ^ 3 } ( u - v ) f \\ , d v , \\end{align*}"}
{"id": "1347.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } D _ i v ( \\tau _ x \\omega ) \\partial _ j \\eta ( x ) d x & = - \\int _ { \\mathbb { R } ^ d } v ( \\tau _ x \\omega ) \\partial _ i \\partial _ j \\eta ( x ) d x \\\\ & = - \\int _ { \\mathbb { R } ^ d } v ( \\tau _ x \\omega ) \\partial _ j \\partial _ i \\eta ( x ) d x \\\\ & = \\int _ { \\mathbb { R } ^ d } D _ j v ( \\tau _ x \\omega ) \\partial _ i \\eta ( x ) d x . \\end{align*}"}
{"id": "3298.png", "formula": "\\begin{align*} \\alpha _ { d - j } ( E | _ D ) = { } & \\alpha _ { d - j } ( E ) - \\alpha _ { d - j } ( E ( - a ) ) \\\\ = { } & \\sum _ { l = 1 } ^ { j } \\frac { ( - 1 ) ^ { l - 1 } a ^ l } { l ! } \\alpha _ { d - j + l } ( E ) \\end{align*}"}
{"id": "2578.png", "formula": "\\begin{align*} \\lim _ { | x | , | \\omega | \\to 0 } \\langle M _ \\omega T _ x f , g \\rangle = \\lim _ { | x | , | \\omega | \\to 0 } \\langle f , T _ { - x } M _ { - \\omega } g \\rangle = \\langle f , g \\rangle , \\end{align*}"}
{"id": "5772.png", "formula": "\\begin{align*} d ^ 2 f = - ( \\partial _ y F ) ^ { - 1 } \\cdot \\left ( \\partial _ x ^ 2 F + \\partial _ y ( \\partial _ x F ) \\cdot d f \\right ) - ( \\partial _ y F ) ^ { - 1 } \\cdot \\left ( \\partial _ x ( \\partial _ y F ) \\cdot d f + \\partial _ y ^ 2 F \\cdot ( d f ) ^ 2 \\right ) , \\end{align*}"}
{"id": "8432.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| P _ { t } ( x , \\cdot ) - \\mu \\| _ { \\mathrm { T V } } = 0 , \\end{align*}"}
{"id": "419.png", "formula": "\\begin{align*} \\| h ( U ) \\| _ { \\bar { s } } = \\| h ( U ) \\| _ { L ^ { \\infty } } + \\| D _ { x } h ( U ) \\| _ { s - 1 } \\leq C _ { h } ( g _ { 2 } ) \\left ( 1 + \\| D _ { x } U \\| _ { s - 1 } \\right ) . \\end{align*}"}
{"id": "6947.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f & \\Omega \\ , , \\\\ \\ \\ \\ , u = g & \\Gamma \\ , , \\end{cases} \\end{align*}"}
{"id": "7879.png", "formula": "\\begin{align*} \\Phi ( R , r ) \\equiv \\frac { 1 } { | C _ r | } \\int _ { C _ { R - r } } N ( x , r ; F ) \\ d x = \\frac { 1 } { | C _ r | } \\int _ { C _ { R - r } } N ( r ; \\tau _ x ( F ) ) \\ d x , \\end{align*}"}
{"id": "5006.png", "formula": "\\begin{align*} | I _ { k n } | = | p _ { k n - 1 } - \\rho _ * q _ { k n - 1 } | = { 1 \\over R _ { k n } q _ { k n - 1 } + q _ { k n - 2 } } , \\end{align*}"}
{"id": "9258.png", "formula": "\\begin{align*} \\forall x ^ X , p ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land p = _ X J ^ A _ { \\gamma } x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\right ) , \\end{align*}"}
{"id": "6526.png", "formula": "\\begin{align*} d _ j ^ { ( 2 \\ell ) } \\sim m ! \\cdot ( 2 \\ell - 1 ) ! ! \\cdot j ^ { - ( m - \\ell ) } ( \\log j ) ^ { \\ell } ( j \\to \\infty ) . \\end{align*}"}
{"id": "7286.png", "formula": "\\begin{align*} \\prod _ { k \\ , \\nmid \\ , m } \\frac { 1 } { 1 - X ^ m } = \\prod _ { m = 1 } ^ \\infty \\frac { 1 - X ^ { k m } } { 1 - X ^ m } = ( 1 + X + \\ldots + X ^ { k - 1 } ) ( 1 + X ^ 2 + \\ldots + X ^ { 2 ( k - 1 ) } ) \\ldots . \\end{align*}"}
{"id": "4310.png", "formula": "\\begin{align*} c _ { \\ell , 0 } \\| \\phi _ { \\ell , \\phi , \\beta } \\| ^ { - 2 } _ { L ^ 2 _ { \\rho _ \\beta } } \\langle \\varepsilon , \\phi _ { \\ell , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } = - \\| \\phi _ { 0 , \\phi , \\beta } \\| ^ { - 2 } _ { L ^ 2 _ { \\rho _ \\beta } } \\langle \\varepsilon , \\phi _ { 0 , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } \\end{align*}"}
{"id": "4001.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\overline { \\xi ( x ) } \\xi ^ { \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\xi ( x ) } \\eta ^ { \\prime } ( x ) d x & = \\lambda \\int _ { 0 } ^ { 1 } | \\xi ( x ) | ^ 2 d x \\\\ \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\eta ^ { \\prime \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\eta ^ { \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\xi ^ { \\prime } ( x ) d x & = \\lambda \\int _ { 0 } ^ { 1 } | \\eta ( x ) | ^ 2 d x . \\end{align*}"}
{"id": "6313.png", "formula": "\\begin{align*} \\omega _ 2 ( t ) & \\triangleq \\mathbb { E } [ e ^ { \\imath t T } ] = \\mathbb { E } \\big [ e ^ { \\imath t \\sum _ { i \\in \\mathcal { K } ' } Z _ i } \\big ] = \\mathbb { E } [ e ^ { \\imath t Z } ] ^ { K ' } \\stackrel { ( a ) } { = } \\bigg [ \\int _ { - \\infty } ^ { \\infty } \\big ( \\cos ( t z ) + \\imath \\sin ( t z ) \\big ) f _ Z ( z ) \\mathrm { d } z \\bigg ] ^ { K ' } \\\\ & \\stackrel { ( b ) } { = } \\bigg [ 2 \\int _ { 0 } ^ { \\infty } \\cos ( t z ) f _ Z ( z ) \\mathrm { d } z \\bigg ] ^ { K ' } , \\ \\forall t \\ge 0 , \\end{align*}"}
{"id": "7983.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n | \\tilde { \\nu } ^ j | + | \\tilde { \\nu } ^ z - 1 | \\leq C _ n ( [ \\tilde { g } - g ] _ { C ^ 3 , M _ 0 } + | u | + | d u | ) \\ , ; \\end{align*}"}
{"id": "2844.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to + \\infty } \\frac { x ( t ) } { t } = 0 . \\end{align*}"}
{"id": "5289.png", "formula": "\\begin{align*} \\omega ^ * ( a ) = \\overline { \\omega ( S ( a ) ^ * ) } , \\omega \\in \\check { A } , a \\in A . \\end{align*}"}
{"id": "9132.png", "formula": "\\begin{align*} T ^ \\circ x ^ * = \\mu _ 0 ^ { - 1 } ( x ^ * + \\mu _ 0 T ^ \\circ x ^ * - x ^ * ) \\in S x ^ * . \\end{align*}"}
{"id": "1992.png", "formula": "\\begin{align*} A _ 1 ( X ) = A _ \\infty \\cap e ^ { B L O ( X ) } , R H _ \\infty ( X ) = e ^ { B U O ( X ) } . \\end{align*}"}
{"id": "5980.png", "formula": "\\begin{align*} C _ { n } ^ { \\lambda , \\beta } ( x ) = \\frac { \\mathrm { d } ^ { n } } { \\mathrm { d } z ^ { n } } \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( f ( z ) ; x ) \\Big | _ { z = 0 } . \\end{align*}"}
{"id": "4420.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\sum _ { j = 1 } ^ { n } a _ { i j } w _ j = \\theta , \\ ; \\ ; \\ ; i = 1 , 2 , . . . , m \\\\ \\displaystyle \\sum _ { j = 1 } ^ { n } w _ j = 1 \\\\ \\lvert a r g w _ j \\rvert \\leq \\beta , \\ ; \\ ; \\ ; j = 1 , 2 , . . . , n \\end{array} \\end{align*}"}
{"id": "8640.png", "formula": "\\begin{align*} \\mathcal { N } _ S = \\mathcal { N } _ + ^ \\delta + \\mathcal { N } _ - ^ \\delta + \\mathcal { N } _ + ^ { \\mathrm { p . v . } } + \\mathcal { N } _ - ^ { \\mathrm { p . v . } } \\end{align*}"}
{"id": "841.png", "formula": "\\begin{align*} r _ n = \\frac { R } { 2 } ( 1 + 2 ^ { - n } ) , k _ n = k _ 0 + d ( 1 - 2 ^ { - n } ) . \\end{align*}"}
{"id": "4786.png", "formula": "\\begin{align*} n = m ^ 2 , \\theta = m - 2 \\ell + 1 , \\hat \\theta = \\frac { | m - 2 \\ell + 1 | } { m } \\sqrt { \\frac { m + 1 } { \\ell ( m - 1 ) ( m - \\ell + 1 ) } } . \\end{align*}"}
{"id": "8188.png", "formula": "\\begin{align*} s ( 2 ^ k + ( - 1 ) ^ k f , 3 f ) = & \\frac { 2 ^ k + ( - 1 ) ^ k f } { 3 6 f } + \\frac { f } { 4 ( 2 ^ k + ( - 1 ) ^ k f ) } - \\frac { ( - 1 ) ^ k } { 4 } + \\frac { 1 } { 3 6 ( 2 ^ k + ( - 1 ) ^ k f ) f } \\\\ & + ( - 1 ) ^ k s ( 3 \\cdot 2 ^ k , 2 ^ k + ( - 1 ) ^ k f ) \\end{align*}"}
{"id": "2951.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 } & = \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } + \\sum _ { \\lambda = 1 } ^ { k - 1 } \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , \\lambda } , \\end{align*}"}
{"id": "2045.png", "formula": "\\begin{align*} { { x } } \\left ( t \\right ) = \\mathcal { U } \\left ( t \\right ) x _ 0 , t \\ge 0 , \\end{align*}"}
{"id": "1814.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty A _ n ( x , y ) { t ^ n \\over n ! } = x y \\ , { e ^ { x t } - e ^ { y t } \\over x e ^ { y t } - y e ^ { x t } } . \\end{align*}"}
{"id": "106.png", "formula": "\\begin{align*} U ^ 2 \\subseteq V , \\ ; U V \\subseteq U , \\ ; V ^ 2 \\subseteq U , \\ ; U V ^ 2 = 0 . \\end{align*}"}
{"id": "7544.png", "formula": "\\begin{align*} \\log \\zeta ( s ) = \\sum _ { | \\gamma - t | \\leq 1 } \\log ( s - \\rho ) + \\mathcal { O } ( \\log t ) \\ \\ \\ \\ \\ \\ \\ \\ \\ ( - 1 \\leq \\sigma \\leq 2 ) \\end{align*}"}
{"id": "6657.png", "formula": "\\begin{align*} \\mathfrak { G } _ p = \\Sigma _ 1 \\times \\left ( 1 - \\frac { 1 } { p } \\right ) \\prod _ { \\substack { \\hat { \\alpha } \\neq \\alpha \\\\ \\hat { \\beta } \\neq \\beta } } \\left ( 1 - \\frac { 1 } { p ^ { 1 + \\hat { \\alpha } + \\hat { \\beta } } } \\right ) \\prod _ { \\hat { \\alpha } \\neq \\alpha } \\left ( 1 - \\frac { 1 } { p ^ { 1 + \\hat { \\alpha } - \\alpha } } \\right ) \\prod _ { \\hat { \\beta } \\neq \\beta } \\left ( 1 - \\frac { 1 } { p ^ { 1 + \\hat { \\beta } - \\beta } } \\right ) , \\end{align*}"}
{"id": "2896.png", "formula": "\\begin{align*} C _ { G N } ^ { - 1 } \\triangleq \\inf _ { 0 \\not = u \\in H ^ 1 ( \\mathbb { R } ^ N ) } Z [ u ] \\end{align*}"}
{"id": "2517.png", "formula": "\\begin{align*} \\rho ( x , \\omega ) = \\rho ( x , \\omega , e ^ { 2 \\pi i 0 } ) \\end{align*}"}
{"id": "5569.png", "formula": "\\begin{align*} Z ( t ) = e ^ { B t } \\bigg [ z + e ^ { - B t } c W ( t ) + \\int _ 0 ^ t e ^ { - B s } ( b + B c W ( s ) ) d s \\bigg ] , t \\in [ 0 , T ] . \\end{align*}"}
{"id": "3186.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k + W _ k ( W _ k ^ \\intercal A ^ \\intercal A W _ k ) ^ \\dagger W _ k ^ \\intercal A ^ \\intercal ( b - A x _ k ) . \\end{align*}"}
{"id": "6108.png", "formula": "\\begin{align*} w ^ l = \\frac { \\varphi ^ l _ s + \\psi ^ l _ t - F ^ l - H ^ l } { z _ { s s } + z _ { t t } } \\ , , \\end{align*}"}
{"id": "229.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left ( \\int _ 0 ^ { \\frac { w } { t } } z e ^ { - z } e ^ { z t } d z \\right ) d t & = \\int _ 0 ^ 1 \\left ( \\int _ 0 ^ w \\frac { y } { t } e ^ { - \\frac { y } { t } } e ^ { y } \\frac { d y } { t } \\right ) d t , \\\\ & = \\int _ 0 ^ w y e ^ { y } \\left ( \\int _ 0 ^ 1 e ^ { - \\frac { y } { t } } \\frac { d t } { t ^ 2 } \\right ) d y = \\int _ 0 ^ w y e ^ { y } \\frac { e ^ { - y } } { y } d y = w . \\end{align*}"}
{"id": "5420.png", "formula": "\\begin{align*} & \\alpha + \\alpha ' = \\pi & & & & \\alpha + \\alpha ' < \\pi & & & & \\alpha + \\alpha ' < \\pi \\\\ & \\beta + \\beta ' < \\pi & & o r & & \\beta + \\beta ' = \\pi & & o r & & \\beta + \\beta ' < \\pi \\\\ & \\gamma + \\gamma ' < \\pi & & & & \\gamma + \\gamma ' < \\pi & & & & \\gamma + \\gamma ' = \\pi \\end{align*}"}
{"id": "5189.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) & \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ { N } \\Bigg ( \\lim _ { t \\to \\infty } \\frac { \\sum _ { i = 1 } ^ { R _ \\ell ^ \\pi ( t ) } \\rho _ \\ell ( { T _ { \\ell i } ^ \\pi } ) ^ 2 } { 2 t } + \\lim _ { t \\to \\infty } \\frac { c _ \\ell R _ \\ell ^ \\pi ( t ) } { t } \\Bigg ) . \\end{align*}"}
{"id": "2833.png", "formula": "\\begin{align*} \\ddot { y } _ R ( t ) = - 8 s _ c ( p - 1 ) \\delta ( t ) + A _ R ( u ( t ) ) , \\end{align*}"}
{"id": "1886.png", "formula": "\\begin{align*} \\mu _ x ^ s = \\abs { x } _ s \\cdot \\mu ^ s \\end{align*}"}
{"id": "9518.png", "formula": "\\begin{align*} E f ( x , \\bar u ) + E f ^ * ( p , y ) = E [ x \\cdot p ] + E [ \\bar u \\cdot y ] . \\end{align*}"}
{"id": "8833.png", "formula": "\\begin{align*} | Q - \\{ s \\} | & \\leq d e g _ { G } ( s ) - | N _ { H _ { 1 } } ( s ) - Q | - | N _ { F } ( s ) - Q | \\\\ & \\leq d _ { 1 } - | N _ { H _ { 1 } } ( s ) \\cap ( D ' - \\{ v _ { t } \\} ) | - | N _ { H _ { 1 } } ( s ) \\cap \\{ v _ { t } \\} | - | N _ { F } ( s ) - Q | \\\\ & \\leq d _ { 1 } - ( d _ { n } - r + 1 ) - | N _ { H _ { 1 } } ( s ) \\cap \\{ v _ { t } \\} | - | N _ { F } ( s ) - Q | \\\\ & = d _ { 1 } - d _ { n } + r - 1 - | N _ { H _ { 1 } } ( s ) \\cap \\{ v _ { t } \\} | - | N _ { F } ( s ) - Q | . \\end{align*}"}
{"id": "407.png", "formula": "\\begin{align*} A ^ { i } = \\left ( \\begin{array} { c c c } A _ { 1 1 } ^ { i } & A _ { 1 2 } ^ { i } & 0 \\\\ A _ { 2 1 } ^ { i } & A _ { 2 2 } ^ { i } & A _ { 2 3 } ^ { i } \\\\ 0 & A _ { 3 2 } ^ { i } & A _ { 3 3 } ^ { i } \\\\ \\end{array} \\right ) \\in \\mathbb { M } _ { N \\times N } , \\end{align*}"}
{"id": "281.png", "formula": "\\begin{align*} \\left \\| u ( \\cdot , t ) - t ^ { - \\frac { 1 } { 2 } } f _ { M } \\left ( ( \\cdot ) t ^ { - \\frac { 1 } { 2 } } \\right ) \\right \\| _ { L ^ { \\infty } } = \\left ( \\left \\| \\widetilde { f } _ { M } ( \\cdot ) \\right \\| _ { L ^ { \\infty } } + o ( 1 ) \\right ) t ^ { - 1 } \\log t a s \\ \\ t \\to \\infty . \\end{align*}"}
{"id": "4961.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i \\partial _ t u - \\Delta u = \\rho ^ 2 | u | ^ 2 + \\langle \\nabla \\rangle ^ { - \\alpha } \\dot { W } \\ , , t \\in [ 0 , T ] \\ , , \\ , x \\in \\R ^ d \\ , , \\\\ u _ 0 = \\phi \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "7789.png", "formula": "\\begin{align*} \\underline { \\dim } _ B \\big ( G ( h _ i ) ) \\big ) = \\varliminf _ { \\delta \\rightarrow 0 } \\frac { \\log \\Big ( N _ { \\delta } ( G ( h _ i ) ) \\Big ) } { - \\log ( \\delta ) } \\geq 2 - \\sigma ~ ~ ~ ~ ~ ~ \\forall ~ ~ i \\in \\{ 1 , 2 , \\cdots , M \\} . \\end{align*}"}
{"id": "6759.png", "formula": "\\begin{align*} & D _ 4 ^ 2 f _ { - 2 } = D _ 4 ^ 2 u _ { - 2 } + c _ 2 ( v _ 0 - u _ 0 ) , \\\\ & D _ 4 ^ 2 f _ { - 1 } = D _ 4 ^ 2 u _ { - 1 } + c _ 1 ( v _ 0 - u _ 0 ) + c _ 2 ( v _ 1 - u _ 1 ) , \\\\ & D _ 4 ^ 2 f _ { 0 } = D _ 4 ^ 2 v _ { 0 } + c _ { - 2 } ( u _ { - 2 } - v _ { - 2 } ) + c _ { - 1 } ( u _ { - 1 } - v _ { - 1 } ) , \\\\ & D _ 4 ^ 2 f _ { 1 } = D _ 4 ^ 2 v _ { 1 } + c _ { - 2 } ( u _ { - 1 } - v _ { - 1 } ) . \\end{align*}"}
{"id": "3730.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ 1 } B _ x J B \\ , d x = - \\frac 1 2 \\int _ { \\mathbb S ^ 1 } B J _ x B \\ , d x . \\end{align*}"}
{"id": "8973.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u + \\left | u ' \\right | ^ 2 - \\left ( 1 - | x | \\right ) & \\leq 0 ( - 1 , 1 ) , \\\\ u + \\left | u ' \\right | ^ 2 - \\left ( 1 - | x | \\right ) & \\geq 0 [ - 1 , 1 ] . \\end{aligned} \\right . \\end{align*}"}
{"id": "2033.png", "formula": "\\begin{align*} L _ t ^ h = \\frac { h ( X _ t ) } { h ( X _ 0 ) } \\exp \\left ( A _ t ^ { \\overline { \\mu } } \\right ) . \\end{align*}"}
{"id": "429.png", "formula": "\\begin{align*} 0 < \\alpha _ { 0 } : = C T _ { 0 } < \\tfrac { 1 } { 2 } , \\end{align*}"}
{"id": "7130.png", "formula": "\\begin{align*} B ^ H _ t = \\int ^ t _ 0 K _ H ( t , s ) d W _ s , t \\ge 0 . \\end{align*}"}
{"id": "5009.png", "formula": "\\begin{align*} \\left | { \\mathfrak { u } ^ I _ { k n } \\over | \\bar { u } _ { k n } | } - { 1 \\over 1 + \\rho _ * } \\right | = \\left | { 1 \\over | \\bar { u } _ { k n } | } \\sum _ { i = 0 } ^ { | \\bar { u } _ { k n } | - 1 } \\chi _ { I } ( T _ * ^ i ( p ) ) - { 1 \\over 1 + \\rho _ * } \\int _ { - 1 } ^ { \\rho _ * } \\chi _ { I } ( x ) d x \\right | \\le D _ { { k n } } , \\end{align*}"}
{"id": "4639.png", "formula": "\\begin{align*} \\lambda _ { ( t - 1 ) k + i } - \\lambda _ { ( t - 1 ) k + ( i + 1 ) } = a _ { i , t } . \\end{align*}"}
{"id": "8225.png", "formula": "\\begin{align*} [ z ^ n ] \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } ( P _ 1 + P _ 2 + P _ 3 ) | _ { x _ 1 = x _ 2 = x _ 3 = 1 } & = \\frac { 1 } { n - 1 } [ t ^ n ] 2 ( n - 1 ) ( 2 n - 3 ) t ^ 2 ( 1 + t ) ^ { 4 n - 6 } \\\\ & = 2 ( 2 n - 3 ) [ t ^ { n - 2 } ] ( 1 + t ) ^ { 4 n - 6 } = 2 ( 2 n - 3 ) \\binom { 4 n - 6 } { n - 2 } . \\end{align*}"}
{"id": "7345.png", "formula": "\\begin{align*} \\begin{aligned} & h _ q + G _ \\beta ( a _ q , \\eta _ q , Y _ q , K \\cap \\{ v ( \\cdot , t _ q ) \\leq v ( y _ q , t _ q ) \\} ) \\geq 0 , \\\\ & k _ q + G _ \\beta ( b _ q , \\zeta _ q , Z _ q , K \\cap \\{ v ( \\cdot , t _ q ) \\leq v ( z _ q , t _ q ) \\} ) \\geq 0 , \\end{aligned} \\end{align*}"}
{"id": "5.png", "formula": "\\begin{align*} \\lim _ { y \\to \\infty } \\kappa _ 1 ( y ) & = \\lim _ { y \\to \\infty } \\left ( \\psi \\left ( \\frac { 1 + \\lvert y \\rvert } { 2 } \\right ) - \\ln { \\lvert y \\rvert } + \\ln { \\epsilon _ 1 } \\right ) , \\end{align*}"}
{"id": "3743.png", "formula": "\\begin{align*} \\left [ R _ m ( x , y ) , R _ m ( z , t ) \\right ] = R _ m ( R _ m ( x , y ) z , t ) + R _ m ( z , R _ m ( x , y ) t ) , \\ ; \\forall ( x , y , z , t ) \\in ( T _ m M ) ^ 4 , \\end{align*}"}
{"id": "2308.png", "formula": "\\begin{align*} \\overline { A f ( - x , - \\omega ) } = A f ( x , \\omega ) . \\end{align*}"}
{"id": "5745.png", "formula": "\\begin{align*} S _ 1 = \\{ j \\in S _ 0 ; | I _ j \\cap Z _ { \\psi _ \\lambda } | \\geq \\kappa p _ 0 \\} , \\end{align*}"}
{"id": "2415.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma } | \\langle f , S ^ { - 1 } e _ \\gamma \\rangle | ^ 2 = \\sum _ { \\gamma \\in \\Gamma } | \\langle S ^ { - 1 } f , e _ \\gamma \\rangle | ^ 2 = \\langle S ( S ^ { - 1 } f ) , S ^ { - 1 } f \\rangle = \\langle S ^ { - 1 } f , f \\rangle . \\end{align*}"}
{"id": "7184.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } \\nabla \\phi * _ x \\rho [ h _ { V _ \\ast } ( s , \\cdot ) ] ( 0 ) = \\lim _ { R \\rightarrow \\infty } \\nabla \\phi * _ x \\rho [ g _ { R , X _ \\ast , V _ \\ast } ( R , \\cdot ) ] ( X _ \\ast ) . \\end{align*}"}
{"id": "2668.png", "formula": "\\begin{align*} V _ { g _ 0 } ( M _ \\eta T _ \\xi g _ 0 ) ( x , \\omega ) = e ^ { - \\pi i ( x + \\xi ) \\cdot ( \\omega - \\eta ) } e ^ { - \\frac { \\pi } { 2 } ( ( x - \\xi ) ^ 2 + ( \\omega - \\eta ) ^ 2 ) } . \\end{align*}"}
{"id": "6008.png", "formula": "\\begin{align*} f ^ { ( n - k ) } ( z ) & = \\sum _ { j = 0 } ^ { n - k } \\frac { ( - 1 ) ^ { j } j ! } { ( l _ { \\pi _ { \\lambda , \\beta } } ( z ) ) ^ { j + 1 } } B _ { n - k , j } \\left ( \\left ( \\sum _ { p = i } ^ { \\infty } \\frac { p ! } { ( p - i ) ! } \\frac { \\lambda ^ { p } z ^ { p - i } } { \\Gamma ( p \\beta + 1 ) } \\right ) _ { i = 1 } ^ { n - k - j + 1 } \\right ) \\end{align*}"}
{"id": "6480.png", "formula": "\\begin{align*} E [ R ] \\begin{cases} < + \\infty & ( - 1 < \\alpha < - 1 / 2 ) , \\\\ = + \\infty & ( - 1 / 2 \\leq \\alpha \\leq 1 / 2 ) . \\end{cases} \\end{align*}"}
{"id": "8382.png", "formula": "\\begin{align*} \\sigma | _ { B _ 1 } = \\sigma | _ { B _ 4 } \\mbox { a n d } \\sigma | _ { B _ 3 } = \\sigma | _ { B _ 6 } \\end{align*}"}
{"id": "7542.png", "formula": "\\begin{align*} \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } = \\sum _ { \\rho , | \\gamma - t | < 1 } \\frac { 1 } { s - \\rho } + \\mathcal { O } ( \\log t ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( - 1 \\leq \\sigma \\leq 2 ) . \\end{align*}"}
{"id": "8333.png", "formula": "\\begin{align*} h _ i ( s ) : = c \\varphi \\left ( \\frac { s - t } { 1 / i } \\right ) \\forall s \\in [ 0 , T ] \\forall i \\in \\mathbb { N } . \\end{align*}"}
{"id": "9331.png", "formula": "\\begin{align*} & \\lim _ { \\mu \\to 0 } \\lambda _ { m i n } ( \\delta I + ( \\Theta ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } = \\frac { \\rho } { \\delta \\rho + 1 } \\\\ & \\lim _ { \\mu \\to 0 } \\lambda _ { m a x } ( \\delta I + ( \\Theta ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } = \\frac { 1 } { \\delta } . \\end{align*}"}
{"id": "6143.png", "formula": "\\begin{align*} p _ k = ( - 1 ) ^ k \\frac { n + m + 1 } { k + m + 1 } { n + m \\choose m } { n \\choose k } \\quad ( 0 \\leq k \\leq n ) \\end{align*}"}
{"id": "2704.png", "formula": "\\begin{align*} | B f ( z ) | e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } \\leq \\norm { V _ { g _ 0 } f } _ \\infty \\leq \\norm { f } _ 2 \\norm { g _ 0 } _ 2 = \\norm { f } _ 2 , \\end{align*}"}
{"id": "7119.png", "formula": "\\begin{align*} x _ t - y _ t = 0 , 0 \\le t \\le 1 . \\end{align*}"}
{"id": "6294.png", "formula": "\\begin{align*} \\widehat { K } = \\arg \\max _ { k \\in \\mathbb { N } } f _ { \\widehat { K } ^ \\mathbb { R } | K = k } ( \\hat { k } ) . \\end{align*}"}
{"id": "8584.png", "formula": "\\begin{align*} \\partial _ k f ^ \\# ( k ) & = \\int _ \\R \\partial _ k \\overline { \\mathcal { K } ^ \\# _ 0 ( x , k ) } f ( x ) \\ , d x + \\int _ \\R \\chi _ + ( x ) \\partial _ k \\overline { \\mathcal { K } ^ \\# _ + ( x , k ) } f ( x ) \\ , d x \\\\ & + \\int _ \\R \\chi _ + ( x ) \\partial _ k \\overline { \\mathcal { K } ^ \\# _ - ( x , k ) } f ( x ) \\ , d x + \\int _ \\R \\partial _ k \\overline { \\mathcal { K } ^ \\# _ R ( x , k ) } f ( x ) \\ , d x . \\end{align*}"}
{"id": "8106.png", "formula": "\\begin{align*} N \\rho ( [ \\mathfrak { A } _ c i _ 1 ( F ) , \\mathfrak { A } _ c i _ 2 ( G ) ] _ { \\mathfrak { A } _ c ( \\mathcal { I } ) } ) & = [ \\mathfrak { A } \\chi _ 1 \\circ N \\rho _ 1 ( F ) , \\mathfrak { A } \\chi _ 2 \\circ N \\rho _ 2 ( G ) ] _ { \\mathfrak { A } ( \\mathcal { M } ) } \\\\ & = 0 . \\end{align*}"}
{"id": "4273.png", "formula": "\\begin{align*} g _ b ( x ) ~ = ~ \\chi _ { \\strut [ 0 , \\infty [ } ( x ) \\cdot \\phi ( x , b ) , x \\in \\R , b \\geq 0 , \\end{align*}"}
{"id": "3252.png", "formula": "\\begin{align*} r ^ 2 & = \\lim _ { t \\rightarrow 0 ^ + } \\left ( - ( s ( r , t ) - t ) ^ 2 + \\frac { s ( r , t ) - t } { h ( s ( r , t ) ) } \\right ) \\\\ & = \\lim _ { s \\rightarrow 0 ^ + } \\frac { s } { h ( s ) } \\lim _ { t \\rightarrow 0 ^ + } \\left ( 1 - \\frac { t } { s ( r , t ) } \\right ) . \\end{align*}"}
{"id": "3415.png", "formula": "\\begin{align*} \\begin{aligned} p & = \\langle K ( x , y ) , ( 1 - \\lambda _ \\delta ( x , y ) ) \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle \\\\ & + \\langle K ( x , y ) , \\lambda _ \\delta ( x , y ) \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle \\\\ & = : p _ { 1 , \\delta } + p _ { 2 , \\delta } . \\end{aligned} \\end{align*}"}
{"id": "6152.png", "formula": "\\begin{align*} a _ k = ( - 1 ) ^ k \\sum _ { \\nu = k } ^ n { m + \\nu \\choose \\nu } { \\nu \\choose k } . \\end{align*}"}
{"id": "8152.png", "formula": "\\begin{align*} s ( c ^ * , d ) = s ( c , d ) c c ^ * \\equiv 1 \\pmod d \\end{align*}"}
{"id": "7892.png", "formula": "\\begin{align*} W _ L = ( \\nabla ^ X F _ L , ( \\nabla ^ X ) ^ 2 F _ L ) \\end{align*}"}
{"id": "3455.png", "formula": "\\begin{align*} & | D _ k T _ { M } f ( x ) | ^ 2 \\chi _ { Q ' } ( x ) \\\\ & \\lesssim \\sum \\limits _ { \\sigma \\in G } \\sum \\limits _ { j = - \\infty } ^ \\infty r ^ { - | j - k | \\varepsilon } r ^ { [ - k - ( - k \\vee - j ) ] { \\bf N } ( 1 - \\frac 1 \\theta ) } \\bigg \\{ M \\Big ( \\sum \\limits _ { Q _ j } | q _ { j } f ( x _ { Q _ j } ) | ^ \\theta \\chi _ { Q } \\Big ) ( x ) \\bigg \\} ^ { 2 / \\theta } \\chi _ { Q ' } ( x ) , \\end{align*}"}
{"id": "2538.png", "formula": "\\begin{align*} U \\Big ( \\sum _ { k = 1 } ^ n c _ k \\pi ( x _ k , \\omega _ k , 0 ) g \\Big ) = \\sum _ { k = 1 } ^ n c _ k \\rho ( x _ k , \\omega _ k ) \\varphi . \\end{align*}"}
{"id": "6356.png", "formula": "\\begin{align*} A _ l \\coloneqq \\left [ 0 \\to M _ { l - a - 1 } \\to M _ { l - 1 } \\oplus M _ { l - a } \\to M _ l \\to 0 \\right ] , \\end{align*}"}
{"id": "3865.png", "formula": "\\begin{align*} H = - \\tilde \\sigma _ 0 ( 1 ) + \\tilde H \\end{align*}"}
{"id": "4747.png", "formula": "\\begin{align*} \\norm { u _ \\beta ( t , \\cdot ) } { L ^ 2 } \\leq M ( t ) \\Bigg \\{ a _ { \\beta } ( t ) + \\sum \\limits _ { l = 1 } ^ { | \\beta | } \\tilde { M } ( t ) ^ l \\bigg ( \\sum _ { \\substack { 0 \\leq | \\alpha | \\leq | \\beta | - l \\\\ \\alpha < \\beta } } a _ \\alpha ( t ) \\sum _ { \\substack { \\theta _ 1 + \\dots + \\theta _ l = \\beta - \\alpha \\\\ \\theta _ i \\not = \\mathbf { 0 } , \\ , i = 1 , \\dots , l } } \\prod _ { i = 1 } ^ { l } \\| q _ { \\theta _ i } \\| _ { L ^ \\infty } \\bigg ) \\Bigg \\} , \\end{align*}"}
{"id": "5731.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( v ) \\cap ( V ( T ' ) \\backslash \\{ u _ { 1 } , \\cdots , u _ { s } \\} ) | & = | N _ { G } ( v ) \\cap ( X \\backslash \\{ u _ { 1 } , \\cdots , u _ { s } \\} ) | \\\\ & \\leq | X | - \\lfloor \\frac { s } { 2 } \\rfloor \\\\ & \\leq t - \\lfloor \\frac { s } { 2 } \\rfloor \\end{aligned} \\end{align*}"}
{"id": "7047.png", "formula": "\\begin{align*} | b _ j ( z ) | ^ 2 \\le 2 \\Big ( | q _ j ( z ) | ^ 2 + \\big | 1 - \\sum _ { k = 1 } ^ L \\phi _ k ( z ) q _ k ( z ) \\big | ^ 2 | e _ j ( z ) | ^ 2 \\Big ) , \\end{align*}"}
{"id": "7816.png", "formula": "\\begin{align*} \\bigl [ A ^ { b + \\frac { \\sigma } { 2 } } S ( \\ , \\cdot \\ , ) \\bigr ] ^ { ( j ) } ( t ) & = \\bigl [ S ( \\ , \\cdot \\ , - \\varepsilon ) A ^ { b + \\frac { \\sigma } { 2 } } S ( \\varepsilon ) \\bigr ] ^ { ( j ) } ( t ) \\\\ & = ( - A ) ^ j S ( t - \\varepsilon ) A ^ { b + \\frac { \\sigma } { 2 } } S ( \\varepsilon ) = ( - 1 ) ^ j A ^ { j + b + \\frac { \\sigma } { 2 } } S ( t ) . \\end{align*}"}
{"id": "4718.png", "formula": "\\begin{align*} \\partial _ t z _ k = \\frac { ( 1 + \\omega _ k ^ 1 ) z _ k } { t } + O \\bigg ( \\frac { 1 } { t ^ { 5 / 4 - \\delta _ 0 } } \\bigg ) . \\end{align*}"}
{"id": "2362.png", "formula": "\\begin{align*} \\langle \\widehat { g _ 1 } , T _ x M _ \\omega \\widehat { g _ 2 } \\rangle = \\langle g _ 1 , M _ x T _ { - \\omega } g _ 2 \\rangle = V _ { g _ 2 } f _ 2 ( - \\omega , x ) . \\end{align*}"}
{"id": "7660.png", "formula": "\\begin{align*} w = P _ { \\tilde { \\Omega } _ { \\varepsilon } } w + e ^ { - \\beta _ { \\varepsilon } \\tilde { \\Psi } _ { \\varepsilon } ( \\mathbf { x } _ { \\varepsilon } ) } \\tilde { V } _ { \\varepsilon } \\tilde { \\Omega } _ { \\varepsilon } \\ ; . \\end{align*}"}
{"id": "981.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ s ( \\tilde u - \\tau \\varphi ^ { ( 1 ) } ) ( a ) + c _ \\rho ( a ) ( \\tilde u - \\tau \\varphi ^ { ( 1 ) } ) ( a ) \\\\ & \\qquad = - C \\int _ { H _ \\rho } \\left ( \\frac 1 { \\vert a - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert Q _ \\rho ( a ) - y \\vert ^ { n + 2 s } } \\right ) \\big ( \\tilde u ( y ) - \\tau \\varphi ^ { ( 1 ) } ( y ) \\big ) \\dd y . \\end{align*}"}
{"id": "8776.png", "formula": "\\begin{align*} U _ { i J ( i ) } : = \\bigl \\{ ( x , u _ i ) \\bigm | u _ { i j } \\geq a _ { i j } j \\leq \\min ( J _ i ) , \\ u _ { i j } = u _ { i n } j \\geq \\max ( J _ i ) , \\ u _ { i j } ( x ; J ) = u _ { i j } ( x ) , \\ ( x , u _ { \\cdot n } ) \\in W _ J \\bigr \\} \\end{align*}"}
{"id": "3830.png", "formula": "\\begin{align*} \\ell _ { 2 m + \\tau } ( y ) : = & y ^ { ( 2 m + \\tau ) } + \\sum _ { k = 0 } ^ { m - 1 } ( - 1 ) ^ { i _ { 2 k } + k } ( \\sigma _ { 2 k } ^ { ( i _ { 2 k } ) } ( x ) y ^ { ( k ) } ) ^ { ( k ) } \\\\ + & \\sum _ { k = 0 } ^ { m + \\tau - 2 } ( - 1 ) ^ { i _ { 2 k + 1 } + k + 1 } \\bigl [ ( \\sigma _ { 2 k + 1 } ^ { ( i _ { 2 k + 1 } ) } ( x ) y ^ { ( k ) } ) ^ { ( k + 1 ) } + ( \\sigma _ { 2 k + 1 } ^ { ( i _ { 2 k + 1 } ) } ( x ) y ^ { ( k + 1 ) } ) ^ { ( k ) } \\bigr ] , \\ : x \\in \\mathbb R _ + , \\end{align*}"}
{"id": "9466.png", "formula": "\\begin{align*} N _ { 1 j } + \\tfrac { C ( j , 2 ) ^ p } { C ( j ) ^ p } - \\tfrac { C ( j - 1 , 2 ) } { C ( j - 1 ) } = 0 . \\end{align*}"}
{"id": "35.png", "formula": "\\begin{align*} J _ { t } | _ { Q _ t } ( w ) & = \\pi _ { t } \\circ J \\circ \\pi _ { t } ^ { - 1 } ( w ) \\\\ & = \\pi _ { t } \\circ J ( w + \\zeta _ { t } ( w ) V ) \\\\ & = J ( w + \\zeta _ { t } ( w ) V ) - \\zeta _ { t } ( J ( w + \\zeta _ { t } ( w ) V ) ) \\\\ & = J ( w ) - \\zeta _ { t } ( w ) U - \\zeta _ { t } ( J ( w ) ) V , \\end{align*}"}
{"id": "6323.png", "formula": "\\begin{align*} W _ { \\mathbf { 1 } , \\mathbf { f } } ( X , Y ) = \\frac { 1 } { X ^ { 4 n } Y ^ { n + 2 } \\prod _ { i = 1 } ^ r ( 1 - X ^ { f _ i } ) } \\Phi _ r ( \\{ X _ I \\} _ { I \\subseteq [ r ] } ) , \\end{align*}"}
{"id": "1463.png", "formula": "\\begin{align*} \\Phi = \\left [ \\begin{array} { c c c c } 0 & 0 & 0 & - 1 _ { 2 m } \\\\ 0 & 0 & - 1 _ r & 0 \\\\ 0 & 1 _ r & 0 & 0 \\\\ 1 _ { 2 m } & 0 & 0 & 0 \\end{array} \\right ] , \\Psi = \\left [ \\begin{array} { c c c c } 0 & 0 & 0 & J _ m ' I _ m ' \\\\ 0 & - \\alpha ^ { - 1 } & 0 & 0 \\\\ 0 & 0 & 1 _ r & 0 \\\\ - J _ m ' I _ m ' & 0 & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "6733.png", "formula": "\\begin{align*} \\mathcal { L } _ { K , ( i s _ 1 + j s _ 2 , i s _ 2 ) } ( \\alpha _ 1 \\alpha _ 3 , \\alpha _ 2 ) = \\sum _ { \\substack { i _ 1 > i _ 2 > 0 \\\\ i _ 1 \\neq N } } \\frac { \\alpha _ 1 ^ { q ^ { i _ 1 } } \\alpha _ 2 ^ { q ^ { i _ 2 } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { i s _ 1 + j s _ 2 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { i s _ 2 } } + \\sum _ { N = i _ 1 > i _ 2 > 0 } \\frac { \\alpha _ 1 ^ { q ^ { i _ 1 } } \\alpha _ 2 ^ { q ^ { i _ 2 } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { i s _ 1 + j s _ 2 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { i s _ 2 } } \\end{align*}"}
{"id": "7919.png", "formula": "\\begin{align*} \\sum _ l ( \\partial _ { x _ l } f _ \\alpha ) ( \\partial _ { t _ j } \\varphi ^ l ) = 0 , \\end{align*}"}
{"id": "5661.png", "formula": "\\begin{align*} \\theta \\varphi \\theta ^ { - 1 } & = ( \\tau , T ) ( \\sigma , R ) ( \\tau , T ) ^ { - 1 } \\\\ & = ( \\tau , T ) ( \\sigma \\tau ^ { - 1 } , R ^ { \\tau ^ { - 1 } } ( T ^ { - 1 } ) ^ { \\tau ^ { - 1 } } ) \\\\ & = ( \\tau \\sigma \\tau ^ { - 1 } , T ^ { \\sigma \\tau ^ { - 1 } } R ^ { \\tau ^ { - 1 } } ( T ^ { - 1 } ) ^ { \\tau ^ { - 1 } } ) . \\end{align*}"}
{"id": "8642.png", "formula": "\\begin{align*} P _ T ( X ) = P ( X ) - T Q ( X ) \\end{align*}"}
{"id": "6592.png", "formula": "\\begin{align*} \\mathcal { L } ^ 0 ( h , k ) : = \\sum _ { \\substack { 1 \\leq q < \\infty \\\\ ( q , h k ) = 1 } } W \\left ( \\frac { q } { Q } \\right ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , q ) = 1 } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { c > C , d \\geq 1 \\\\ c d = q } } \\mu ( c ) \\end{align*}"}
{"id": "2249.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\sup _ { 1 \\leq m \\leq M } \\| Y _ m ^ { M , N } \\| ^ p _ { L ^ 6 } \\Big ] \\leq C ( p , T , \\gamma , X _ 0 ) . \\end{align*}"}
{"id": "7170.png", "formula": "\\begin{align*} \\Psi ( \\nu ) \\leq \\sum _ { \\mu = 1 } ^ \\nu c \\chi ^ \\mu \\leq c ( d , \\chi ) \\chi ^ \\nu . \\end{align*}"}
{"id": "2763.png", "formula": "\\begin{align*} \\delta ( t ) = \\Big | \\| \\nabla u ( t ) \\| _ 2 ^ 2 - \\| \\nabla Q \\| _ 2 ^ 2 \\Big | , \\end{align*}"}
{"id": "4421.png", "formula": "\\begin{align*} I m ( \\displaystyle \\sum _ { i = 1 } ^ { m } \\overline { z _ i } a _ { i k } ) = I m ( \\displaystyle \\sum _ { i = 1 } ^ { m } \\overline { z _ i } a _ { i l } ) : = p _ 1 \\ ; \\ ; \\forall k \\not = l \\end{align*}"}
{"id": "4787.png", "formula": "\\begin{align*} \\phi ^ \\sigma ( g ) = \\phi ( g ^ { - 1 } ) \\quad \\quad \\epsilon ( \\phi ) = \\phi ( e ) . \\end{align*}"}
{"id": "5964.png", "formula": "\\begin{align*} \\int \\int \\nabla \\phi _ k \\cdot \\nabla v ~ d x d z = \\int _ { \\Gamma ^ { N B C } } q ~ v ~ d \\Gamma , \\end{align*}"}
{"id": "6509.png", "formula": "\\begin{align*} F ^ { ( 2 m ) } _ n & : = \\bar { g } _ n ^ { ( 2 m ) } \\sum _ { j = j _ 0 + 1 } ^ { n - 1 } \\dfrac { f _ j ^ { ( 2 m ) } } { \\bar { g } _ { j + 1 } ^ { ( 2 m ) } } , H _ n ^ { ( 2 m ) } : = \\bar { g } _ n ^ { ( 2 m ) } \\sum _ { j = j _ 0 + 1 } ^ { n - 1 } \\dfrac { h _ j ^ { ( 2 m ) } } { \\bar { g } _ { j + 1 } ^ { ( 2 m ) } } . \\end{align*}"}
{"id": "8654.png", "formula": "\\begin{align*} 3 a ^ 2 - 1 0 a d + 3 d ^ 2 + b ^ 2 + 1 0 b c + 9 c ^ 2 = 0 \\end{align*}"}
{"id": "1211.png", "formula": "\\begin{align*} S _ U : = \\bigcup _ { 0 \\leq k \\leq \\lfloor \\frac { \\beta - \\alpha } { \\varepsilon _ 2 } \\rfloor + 1 } E ^ { [ \\theta _ k , \\theta _ { k + 1 } ] , \\rho _ { U } , \\varepsilon _ 2 } _ { \\mu } \\cap E _ U \\cap \\left \\{ x \\in \\R ^ d : \\ \\overline { \\dim } _ { \\mathrm { l o c } } ( m _ { E _ U } ^ { s _ 2 } , x ) \\leq d \\right \\} . \\end{align*}"}
{"id": "951.png", "formula": "\\begin{align*} \\Gamma _ { 1 , y , s } = \\bigcup _ { \\delta = 1 } ^ d \\Gamma _ { 1 , y , s } ^ { \\delta } , \\Gamma _ { 1 , y , s } ^ { \\delta } = \\{ n \\in \\Gamma _ { 1 , y , s } : J _ { n } ^ { \\delta } \\subset ( L ^ { \\delta } ) ^ c \\} . \\end{align*}"}
{"id": "7590.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k - 2 s , l - 2 r - 1 ) } | & \\leq \\dfrac { 1 } { 2 j } \\left ( | c _ { i , j - 1 } ^ { ( k - 2 s , l - 2 r ) } | + | c _ { i , j + 1 } ^ { ( k - 2 s , l - 2 r ) } | \\right ) \\\\ & \\leq \\dfrac { 1 } { 2 j } \\frac { 4 V _ { k , l } } { \\pi ^ 2 } \\Gamma _ { 0 , 0 } [ s ] ( i ) \\left ( \\Gamma _ { 0 , - 1 } [ r ] ( j ) + \\Gamma _ { 0 , 1 } [ r ] ( j ) \\right ) = \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 } \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 1 , - 1 } [ r ] ( j ) . \\end{align*}"}
{"id": "1125.png", "formula": "\\begin{align*} m ^ { ( 3 ) } _ { + } ( x , t , k ) = m ^ { ( 3 ) } _ { - } ( x , t , k ) J ^ { ( 3 ) } ( x , t , k ) , k \\in \\Gamma ^ { ( 3 ) } , \\end{align*}"}
{"id": "5234.png", "formula": "\\begin{align*} & ( w ( b _ r - d + e ) + k + 1 / ( d - 1 ) ) ( 2 d k - d + 2 ) - 2 k ( b _ r + d k - d + e ) \\\\ = & \\frac { ( b _ r - d ) ( d ( d - 4 ) ( 2 k - 1 ) + 4 ( k - 1 ) ) + d ( 2 k + 2 - d ) } { 2 d - 2 } \\ge 0 . \\end{align*}"}
{"id": "8200.png", "formula": "\\begin{align*} S ( H , f ) = \\frac { f - 1 } { 1 2 } \\hbox { a n d } S ( H _ 2 , 2 f ) = \\frac { 2 f + c _ a ' } { 1 2 } \\hbox { w h e r e } c _ a ' : = \\begin{cases} - 3 a - 2 & \\hbox { i f $ a \\equiv 0 \\pmod 2 $ } , \\\\ 3 a + 1 & \\hbox { i f $ a \\equiv 1 \\pmod 2 $ } \\end{cases} \\end{align*}"}
{"id": "9101.png", "formula": "\\begin{align*} \\gamma _ p = 1 + v ^ m u ^ { m - 1 } \\beta _ m t ^ m + \\cdots + v ^ { n - 1 } u ^ { n - 2 } \\beta _ { n - 1 } t ^ { n - 1 } \\quad w _ p = v ^ m u ^ m \\end{align*}"}
{"id": "4074.png", "formula": "\\begin{align*} g ( z + t ) - Q _ z ( t ) = \\int _ z ^ { z + t } [ A + g '' ( u ) ] ( z + t - u ) \\ , d u , \\ \\ ( z , t ) \\in E . \\end{align*}"}
{"id": "5078.png", "formula": "\\begin{align*} \\left | \\left | \\frac { w _ 1 + w _ 2 } { 2 } \\right | \\right | = r \\sqrt { \\frac { 1 + \\cos ( \\theta ) } { 2 } } . \\end{align*}"}
{"id": "5008.png", "formula": "\\begin{align*} \\mathfrak { u } _ { k n } ^ I : = \\sum _ { i = 0 } ^ { | \\bar u _ { k n } | - 1 } \\chi _ { I } ( T _ * ^ i ( p ) ) \\end{align*}"}
{"id": "5681.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) = \\epsilon ( \\alpha ) - \\epsilon ( \\beta ) \\ , \\ , \\ , \\mathrm { m o d } \\ , \\ , 2 . \\end{align*}"}
{"id": "2903.png", "formula": "\\begin{align*} \\left ( - \\Delta + 1 + \\frac { N - 1 } { r ^ 2 } - \\left ( p - 1 \\right ) \\left ( | \\cdot | ^ { N - \\gamma } * Q ^ p \\right ) Q ^ { p - 2 } \\right ) \\left ( - \\partial _ r Q \\right ) = - \\partial _ r \\left [ | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ] Q ^ { p - 1 } . \\end{align*}"}
{"id": "618.png", "formula": "\\begin{align*} x \\abs { ( A ( x ) + B ( x ) ) - ( \\alpha + \\beta ) } \\ & \\leq \\ \\frac { 2 x } { x + 1 } \\\\ [ 1 1 p t ] & = \\ \\frac { 2 } { 1 + \\frac { 1 } { x } } \\\\ [ 1 1 p t ] & \\leq \\ 2 \\end{align*}"}
{"id": "4276.png", "formula": "\\begin{align*} \\left | a ^ { ( n ) } _ 1 ( t ) - a ^ { ( n ) } _ 2 ( t ) - { 1 \\over 2 } \\cdot \\left ( \\sigma ^ { ( n ) } _ 1 ( t ) + \\sigma ^ { ( n ) } _ 2 ( t ) \\right ) \\right | ~ = ~ \\left | w ^ { ( n ) } ( t , t + ) - w ^ { ( n ) } ( t , 0 - ) \\right | ~ \\leq ~ M _ 0 \\sqrt { | t | } , \\end{align*}"}
{"id": "6732.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\geq i \\geq 0 \\\\ m \\geq j \\geq 0 } } g _ { i j } ( t ) \\bigl ( \\sum _ { r _ { i j } \\geq h \\geq 1 } \\mathcal { L } _ { K , \\mathfrak { s } _ { i j , h } } ( { \\boldsymbol \\alpha _ { i j , h } } ) \\bigr ) = 0 \\end{align*}"}
{"id": "6814.png", "formula": "\\begin{align*} \\left \\{ \\Psi _ { r n 1 } \\left ( x \\right ) \\right \\} ^ { 2 } & = p ^ { - r } \\chi _ { p } \\left ( 2 p ^ { - 1 } \\left ( p ^ { r } x - n \\right ) \\right ) \\Omega \\left ( \\left \\vert p ^ { r } x - n \\right \\vert _ { p } \\right ) \\\\ & = p ^ { r } \\left \\{ \\Psi _ { r n 1 } \\left ( x \\right ) \\right \\} ^ { 2 } = p ^ { \\frac { r } { 2 } } \\Psi _ { r n 2 } \\left ( x \\right ) , \\end{align*}"}
{"id": "6215.png", "formula": "\\begin{align*} \\begin{aligned} \\| V _ l - \\hat { V } \\| _ F \\leq \\| V _ l - Z \\| _ F + \\| Z - \\hat { V } \\| _ F \\leq \\sqrt { \\frac { 4 0 k \\epsilon } { \\eta } } \\| C \\| _ F + \\xi . \\end{aligned} \\end{align*}"}
{"id": "4083.png", "formula": "\\begin{align*} \\dim ( ( X + V ) / V ) + \\dim ( ( Y + W ) / W ) = \\\\ \\dim ( X + Y + V + W ) + \\dim ( ( X + V ) \\cap ( Y + W ) ) - \\dim ( V + W ) - \\dim ( V \\cap W ) = \\\\ \\dim ( ( X + Y + V + W ) / ( V + W ) ) + \\dim ( ( X + V ) \\cap ( Y + W ) ) - \\dim ( V \\cap W ) \\geq \\\\ \\dim ( ( X + Y + V + W ) / ( V + W ) ) + \\dim ( ( ( X \\cap Y ) + ( V \\cap W ) ) / ( V \\cap W ) ) , \\end{align*}"}
{"id": "6756.png", "formula": "\\begin{align*} \\begin{aligned} D ^ 2 u _ { - 1 } = D ^ 2 f _ { - 1 } + \\frac { \\delta ^ 2 } { h _ 0 ( h _ { - 1 } + h _ 0 ) } & ( u '' _ { \\delta } - v '' _ { \\delta } ) + \\frac { \\delta ^ 3 } { 3 h _ 0 ( h _ { - 1 } + h _ 0 ) } ( u ''' _ { \\delta } - v ''' _ { \\delta } ) \\\\ & + \\frac { \\delta ^ 4 } { 1 2 h _ 0 ( h _ { - 1 } + h _ 0 ) } ( u '''' _ { \\delta } - v '''' _ { \\delta } ) + \\mathcal O ( h ^ 3 ) , \\end{aligned} \\end{align*}"}
{"id": "4577.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( a _ n S _ n \\leq - x \\sqrt { v _ n + a _ n ^ 2 \\sum _ { i = 1 } ^ n ( Z _ i - 1 ) ^ 2 } \\ , \\Big ) } { \\Phi \\left ( - x \\right ) } = 1 + o ( 1 ) \\end{align*}"}
{"id": "9348.png", "formula": "\\begin{align*} e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 1 0 , 1 4 ] ) . \\end{align*}"}
{"id": "1961.png", "formula": "\\begin{align*} \\Delta ( a _ 1 \\dotsm a _ n ) : = \\sum _ { S \\subseteq [ n ] } w _ S \\otimes w _ { J ^ S _ 1 } \\vert \\dotsm \\vert w _ { J ^ S _ m } , \\end{align*}"}
{"id": "1166.png", "formula": "\\begin{align*} \\mathcal { \\psi } _ { + } ( \\zeta ) = \\mathcal { \\psi } _ { - } ( \\zeta ) J ^ { \\mathcal { \\psi } } , \\zeta \\in \\mathbb { R } , \\end{align*}"}
{"id": "7103.png", "formula": "\\begin{align*} P _ 0 ( \\lambda ) = & \\begin{vmatrix} \\lambda & - 1 & - 1 \\\\ - 1 & \\lambda & - 2 \\\\ - 1 & - 2 & \\lambda \\end{vmatrix} \\\\ P _ 0 ( \\lambda ) = & \\lambda ^ 3 - 6 \\lambda - 4 \\end{align*}"}
{"id": "214.png", "formula": "\\begin{align*} \\partial _ k D _ k ^ { \\alpha - 1 } ( p _ \\alpha f ) ( x ) = A + B + C , \\end{align*}"}
{"id": "878.png", "formula": "\\begin{align*} n _ i = a _ i + n _ { \\min } - 1 , i = 1 , 2 , \\cdots , \\left | \\mathcal { A } \\right | . \\end{align*}"}
{"id": "6460.png", "formula": "\\begin{align*} X _ { n + 1 } & = \\begin{cases} X _ { U _ n } & \\mbox { w i t h p r o b a b i l i t y $ p $ } , \\\\ - X _ { U _ n } & \\mbox { w i t h p r o b a b i l i t y $ 1 - p $ } . \\\\ \\end{cases} \\end{align*}"}
{"id": "235.png", "formula": "\\begin{align*} \\operatorname { V a r } _ { \\mu } ( f _ j ) = \\operatorname { C o v } _ { \\mu } ( g _ j + \\varepsilon f , g _ j + \\varepsilon f ) = \\operatorname { V a r } _ { \\mu } ( g _ j ) + 2 \\varepsilon \\operatorname { C o v } _ { \\mu } ( g _ j , f ) + \\varepsilon ^ 2 \\operatorname { V a r } _ { \\mu } ( f ) , \\end{align*}"}
{"id": "6776.png", "formula": "\\begin{align*} \\pi = \\tau _ { r _ { 1 } } ^ { - ( v _ { 1 } + 1 ) } \\cdots \\tau _ { r _ { j } } ^ { - ( v _ { j } + 1 ) } \\cdot \\tau _ { r _ { j + 1 } + 1 } ^ { v _ { j + 1 } + 1 } \\cdots \\tau _ { r _ { z } + 1 } ^ { v _ { z } + 1 } . \\end{align*}"}
{"id": "6911.png", "formula": "\\begin{align*} & \\textbf { s } _ m ( t ) = \\\\ & \\biggl [ \\Re \\left [ s _ m [ t ] e ^ { j 2 \\pi f _ 0 t } e ^ { j \\psi _ { 1 m } } \\right ] , \\ldots , \\Re \\left [ s _ m [ t ] e ^ { j 2 \\pi f _ 0 t } e ^ { j \\psi _ { N _ s m } } \\right ] \\biggr ] ^ T . \\end{align*}"}
{"id": "5952.png", "formula": "\\begin{align*} \\phi = \\phi ^ R + \\phi _ 0 + \\phi _ 7 , \\end{align*}"}
{"id": "8751.png", "formula": "\\begin{align*} \\Bigl \\{ ( \\phi , s , t , \\lambda , y ) \\in \\R ^ { m + d ( n + 1 ) + 2 ^ d + | V | + | E | } \\Bigm | \\phi = A ( t ) , \\ t = L ( y ) , \\ s \\in S ( \\lambda ) , \\ ( \\lambda , y ) \\in ( \\Lambda ) \\Bigr \\} \\end{align*}"}
{"id": "2729.png", "formula": "\\begin{align*} S ( \\underline { P } , B ) = \\{ ( \\omega , x ) \\in \\Omega \\times B \\ ; \\vert \\ ; \\omega \\underline { P } ( x ) \\leq 0 \\} . \\end{align*}"}
{"id": "6867.png", "formula": "\\begin{align*} \\norm { \\Gamma ' g _ { \\ell } } \\leq \\norm { g _ \\ell } \\sum \\limits _ { s = 2 } ^ { j } \\gamma ^ { i _ s + \\ell - 2 s } _ { i _ s - 2 s } \\prod _ { t = 2 } ^ { s - 1 } \\left ( 1 - \\delta ^ { i _ t + \\ell - 2 t } _ { i _ t - 2 t } \\right ) . \\end{align*}"}
{"id": "3512.png", "formula": "\\begin{align*} D _ { 1 1 1 } \\ll \\frac { t _ 3 ^ { 1 - \\sigma _ 1 } } { t _ 3 } \\int _ 1 ^ { a t _ 3 } \\frac { 1 } { v ^ { \\sigma _ 2 } ( a t _ 3 + v ) ^ { \\sigma _ 3 } } d v \\ll \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 = 1 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) \\\\ \\end{cases} \\end{align*}"}
{"id": "7638.png", "formula": "\\begin{align*} w ( r ) = \\begin{cases} A \\ , r ^ { 1 - N / 2 } J _ { N / 2 - 1 } \\left ( \\sqrt { \\tilde { \\lambda } _ 0 \\overline { m } } r \\right ) & 0 \\le r < 1 \\ ; , \\\\ B \\ , r ^ { 1 - N / 2 } K _ { N / 2 - 1 } \\left ( \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } r \\right ) & r \\ge 1 \\ ; , \\\\ \\end{cases} \\end{align*}"}
{"id": "1609.png", "formula": "\\begin{align*} g _ j ^ Q = \\frac { 1 } { | Q | } \\int _ { Q } g _ j ( x ) \\ , d x . \\end{align*}"}
{"id": "7486.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { \\tau } { 2 } \\eta ^ n + \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n \\right ) \\phi ^ { n + 1 } - \\frac { \\tau ^ 2 } { 4 } \\Delta \\phi ^ { n + 1 } = \\mathcal { H } ^ n . \\end{align*}"}
{"id": "5973.png", "formula": "\\begin{align*} \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; x ) : = \\frac { \\mathrm { e } ^ { z x } } { l _ { \\pi _ { \\lambda , \\beta } } ( z ) } = \\frac { \\mathrm { e } ^ { z x } } { E _ { \\beta } ( \\lambda ( e ^ { z } - 1 ) ) } , z \\in \\mathbb { C } , \\ ; x \\in \\mathbb { R } , \\end{align*}"}
{"id": "2930.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ q ( a _ i + b _ i ) = \\sum _ { A \\subset S _ q } \\prod _ { j \\in A } a _ j \\prod _ { j \\notin A } b _ j , \\end{align*}"}
{"id": "6521.png", "formula": "\\begin{align*} M _ n ^ { ( 2 m ) } = I ^ { ( 2 m ) } _ n + J ^ { ( 2 m ) } _ n + K ^ { ( 2 m ) } _ n , \\end{align*}"}
{"id": "2293.png", "formula": "\\begin{align*} X = \\{ M _ \\omega T _ x g _ 0 \\mid ( x , \\omega ) \\in \\R ^ { 2 d } \\} \\end{align*}"}
{"id": "5782.png", "formula": "\\begin{align*} = \\int _ { \\R ^ 3 } | \\nabla \\mathbf { v } | ^ 2 \\varphi _ R \\ , d x + \\sum _ { i , j = 1 } ^ { 3 } \\int _ { \\R ^ 3 } \\partial _ { j } v _ i \\cdot v _ i \\partial _ { j } \\varphi _ R \\ , d x . \\end{align*}"}
{"id": "1678.png", "formula": "\\begin{align*} x \\geq 3 2 7 5 , \\end{align*}"}
{"id": "4053.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } u ^ { \\epsilon } _ x \\bar { v } _ x + \\int _ { 0 } ^ { 1 } u ^ { \\epsilon } _ x \\bar { v } + \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } _ x \\bar { v } + \\lambda \\int _ { 0 } ^ { 1 } u ^ { \\epsilon } \\bar { v } = \\int _ { 0 } ^ { 1 } g \\bar { v } . \\end{align*}"}
{"id": "3920.png", "formula": "\\begin{align*} g = \\frac { f ' _ { s ' _ { n } } ( \\tilde { \\infty } ) - f ' _ { s _ { n } } ( \\tilde { \\infty } ) } { \\delta \\kappa } \\end{align*}"}
{"id": "8355.png", "formula": "\\begin{align*} e ^ r - 1 = | \\log p | ( 1 + o ( 1 ) ) , \\end{align*}"}
{"id": "1921.png", "formula": "\\begin{align*} \\big ( f ( 0 , x , v ) , \\rho ( 0 , x ) , u ( 0 , x ) \\big ) = ( f _ { 0 } ( x , v ) , \\rho _ 0 ( x ) , u _ 0 ( x ) \\big ) , \\end{align*}"}
{"id": "8381.png", "formula": "\\begin{align*} \\xi ( x , t , y , s ; 1 , v ) = \\zeta ( X , t , Y , s ; 1 , v ) , \\end{align*}"}
{"id": "4458.png", "formula": "\\begin{align*} a | z \\rangle = \\sum _ { n = 0 } ^ { \\infty } A _ { n } a | n \\rangle = \\sum _ { n = 1 } ^ { \\infty } A _ { n } \\sqrt { n } | n - 1 \\rangle , \\end{align*}"}
{"id": "5871.png", "formula": "\\begin{align*} \\Theta ^ t W _ s = W _ { t + s } - W _ t , s \\ge 0 , \\end{align*}"}
{"id": "1922.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ 0 \\in L ^ 2 ( \\Omega ) , \\rho _ 0 \\in W ^ { 1 , 2 } ( \\Omega ) , \\\\ & 0 < \\underline { \\rho } \\leq \\rho _ 0 ( x ) \\leq \\bar { \\rho } < \\infty , x \\in \\Omega , \\\\ & 0 < \\underline { \\rho } \\leq \\rho _ B ( x ) \\leq \\bar { \\rho } < \\infty , x \\in \\Gamma _ { \\rm { i n } } . \\end{aligned} \\right . \\end{align*}"}
{"id": "687.png", "formula": "\\begin{align*} K _ * = 0 , C _ b = 0 , C _ W = \\sigma _ 1 ^ { - 2 } . \\end{align*}"}
{"id": "5113.png", "formula": "\\begin{align*} h _ B ( \\tfrac { \\eta } { n } ) = h _ B ( \\tfrac { 1 } { \\eta } ) \\Longleftrightarrow \\frac { \\eta } { n } = \\frac { 1 } { \\eta } \\Longleftrightarrow \\eta = \\sqrt { n } . \\end{align*}"}
{"id": "3548.png", "formula": "\\begin{align*} \\rho _ m ( x , t ) = | t | ^ { - m / 2 } \\Phi _ m ( x / | t | ^ { 1 / 2 } ) . \\end{align*}"}
{"id": "2332.png", "formula": "\\begin{align*} A f ( x , \\omega ) = \\frac { e ^ { - \\pi i ( \\omega - i ) | x | } } { 2 \\pi i ( \\omega - i ) } . \\end{align*}"}
{"id": "2212.png", "formula": "\\begin{align*} ( F ' ( v ) ( \\psi ) \\big ) ( x ) & = f ' ( v ( x ) ) \\psi ( x ) = \\big ( 3 v ^ 2 ( x ) - 1 \\big ) \\psi ( x ) , x \\in D , \\\\ \\big ( F '' ( v ) ( \\psi _ 1 , \\psi _ 2 ) \\big ) ( x ) & = f '' ( v ( x ) ) \\psi _ 1 ( x ) \\psi _ 2 ( x ) = 6 v ( x ) \\psi _ 1 ( x ) \\psi _ 2 ( x ) , x \\in D . \\end{align*}"}
{"id": "7079.png", "formula": "\\begin{align*} C ( X , Y ) = C ( Y , X ) + [ R ( X , Y ) , A ] . \\end{align*}"}
{"id": "637.png", "formula": "\\begin{align*} g ( x , n ) \\ & = \\ C ( 0 , h _ 0 ( x , n + 1 ) ) \\\\ [ 1 2 p t ] & = \\ \\bigg [ \\frac { 0 } { h _ 0 ( x , n ) + 1 } + \\frac { 1 } { 2 } \\bigg ] \\\\ [ 1 2 p t ] & = \\ \\bigg [ \\frac { 1 } { 2 } \\bigg ] \\\\ [ 1 2 p t ] & = \\ 0 . \\end{align*}"}
{"id": "1862.png", "formula": "\\begin{align*} { \\rm G e n } ( x , t ) = { { \\rm G e n } ( a ^ { - 1 } x , t ) \\over { \\rm G e n } ( a ^ { - 1 } , t ) } , \\end{align*}"}
{"id": "750.png", "formula": "\\begin{align*} f ^ \\star : = \\inf _ { x \\in \\mathbb { R } ^ n } f ( x ) , \\end{align*}"}
{"id": "8036.png", "formula": "\\begin{align*} \\rho ( x + t _ + ( x ) ) - \\rho ( x - t _ + ( x ) ) = 2 \\int _ { x - t _ + ( x ) } ^ { x + t _ + ( x ) } \\frac { \\mathrm { d } x ' } { t _ + ( x ' ) } = \\frac { 4 t _ + ( x ) } { t _ + ( x + c ) } , \\end{align*}"}
{"id": "1816.png", "formula": "\\begin{align*} G = \\{ u \\rightarrow 2 u v , \\ ; \\ ; v \\rightarrow u \\} , \\end{align*}"}
{"id": "9369.png", "formula": "\\begin{align*} \\frac { \\log _ { \\lambda } ( 1 + t ) } { t } ( 1 + t ) ^ { x } & = \\sum _ { k = 0 } ^ { \\infty } \\beta _ { k , \\lambda } ( x ) \\frac { 1 } { k ! } \\big ( \\log _ { \\lambda } ( 1 + t ) \\big ) ^ { k } \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\beta _ { k , \\lambda } ( x ) \\sum _ { n = k } ^ { \\infty } S _ { 1 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = 0 } ^ { n } \\beta _ { k , \\lambda } ( x ) S _ { 1 , \\lambda } ( n , k ) \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "5666.png", "formula": "\\begin{align*} \\theta \\varphi \\theta ^ { - 1 } & = ( \\tau , \\boldsymbol \\varepsilon ) ( \\sigma , R ) ( \\tau , \\boldsymbol \\varepsilon ) ^ { - 1 } \\\\ & = ( \\tau \\sigma \\tau ^ { - 1 } , R ^ { \\tau ^ { - 1 } } ) \\\\ & = ( \\tau \\sigma \\tau ^ { - 1 } , ( \\varphi _ { \\tau 1 } , \\varphi _ { \\tau 2 } , \\varphi _ { \\tau 3 } ) ) \\\\ & = ( \\tau \\sigma \\tau ^ { - 1 } , ( \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ) ) . \\end{align*}"}
{"id": "1165.png", "formula": "\\begin{align*} m ^ { P C } _ { \\eta } = \\mathcal { \\psi } ( \\zeta ) \\mathcal { P } \\zeta ^ { - i \\nu \\sigma _ 3 } e ^ { \\frac { i } { 4 } \\zeta ^ 2 \\sigma _ 3 } , \\end{align*}"}
{"id": "4867.png", "formula": "\\begin{align*} ( K * \\tilde f ) '' ( x ) = \\int _ { \\R } \\big [ \\tilde f ( x + z ) - \\tilde f ( x ) - z \\tilde f ' ( x ) \\big ] \\ , K '' ( z ) d z , \\end{align*}"}
{"id": "9214.png", "formula": "\\begin{align*} 0 = \\norm { p - J ^ A _ \\gamma x + \\vert \\gamma \\vert ( \\gamma ^ { - 1 } ( x - p ) - \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) ) } \\geq \\norm { p - J ^ A _ \\gamma x } . \\end{align*}"}
{"id": "698.png", "formula": "\\begin{align*} n _ 1 , \\ldots , n _ L = n \\gg 1 , \\end{align*}"}
{"id": "6204.png", "formula": "\\begin{align*} \\| \\hat { V } - Z \\| _ F = \\| R - I \\| _ F < \\xi . \\end{align*}"}
{"id": "4663.png", "formula": "\\begin{align*} ( f , g ) = \\int _ { \\mathbb { R } } f ( x ) g ( x ) d x . \\end{align*}"}
{"id": "807.png", "formula": "\\begin{align*} I _ f ( u ) : = \\int _ { \\overline { \\Omega } } g _ u ^ p \\ , d \\mu - p \\int _ { \\partial \\Omega } u f \\ , d \\nu . \\end{align*}"}
{"id": "6545.png", "formula": "\\begin{align*} \\frac { 1 } { r - k } \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s - r \\right ) \\overset { \\eqref { b o u n d f o r s } } { > } \\ & \\frac { 1 } { r - k } \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + \\frac { k ^ 2 } { 2 r } H ^ 2 - r - 1 + \\frac { 1 } { r } \\right ) \\\\ = \\ & \\frac { r - k } { 2 r } H ^ 2 + \\frac { 1 } { r - k } \\left ( - r - 1 + \\frac { 1 } { r } \\right ) \\\\ > \\ & 1 \\ , . \\end{align*}"}
{"id": "4184.png", "formula": "\\begin{align*} 0 = \\omega ( v _ i \\# u ) = \\omega ( v _ i ) + \\omega ( u ) . \\end{align*}"}
{"id": "8689.png", "formula": "\\begin{align*} \\begin{aligned} & f ^ L _ i \\leq a _ { i 0 } ( x ) \\leq \\ldots \\leq a _ { i n _ i } ( x ) \\leq f ^ U _ i , \\\\ & u _ { i j } ( x ) \\leq \\min \\bigl \\{ f _ i ( x ) , a _ { i j } ( x ) \\bigr \\} j \\in \\{ 0 , \\ldots , n _ i \\} , \\\\ & u _ { i 0 } ( x ) = a _ { i 0 } ( x ) , u _ { i n _ i } ( x ) = f _ { i } ( x ) . \\end{aligned} \\end{align*}"}
{"id": "54.png", "formula": "\\begin{align*} \\mathcal { R } _ L ( \\Q ( \\sqrt { - 1 } ) , 4 ) & = \\mathcal { R } _ L ( F , 4 ) _ I \\coprod \\mathcal { R } _ L ( F , 4 ) _ { I I } , \\\\ \\mathcal { R } _ L ( \\Q ( \\sqrt { - 3 } ) , 2 ) & = \\mathcal { R } _ L ( \\Q ( \\sqrt { - 3 } ) , 2 ) _ { I I } , \\\\ \\mathcal { R } _ L ( F , 2 ) & = \\mathcal { R } _ L ( F , 2 ) _ I \\coprod \\mathcal { R } _ L ( F , 2 ) _ { I I } \\coprod \\mathcal { R } _ L ( F , 2 ) _ { I I I } \\coprod \\mathcal { R } _ L ( F , 2 ) _ { I V } \\coprod \\mathcal { R } _ L ( F , 2 ) _ { V } , \\end{align*}"}
{"id": "1793.png", "formula": "\\begin{align*} \\langle \\Phi ^ P _ { e } , \\operatorname { I n d } _ { C ^ * _ r ( G ) } ( D ) \\rangle = \\frac { 1 } { | W _ { M \\cap K } | } \\cdot \\sum _ { w \\in W _ K } m \\left ( \\sigma ^ { M } ( w \\cdot \\mu ) \\right ) , \\end{align*}"}
{"id": "4076.png", "formula": "\\begin{align*} g ( x ) - y = g ( z + t ) - Q _ z ( t ) = \\int _ z ^ { z + t } ( g '' ( u ) + A ) ( z + t - u ) \\ , d u , \\end{align*}"}
{"id": "5997.png", "formula": "\\begin{align*} u ( z ) & = \\sum _ { n = 0 } ^ { \\infty } u _ { n } z ^ { n } \\end{align*}"}
{"id": "8716.png", "formula": "\\begin{align*} \\begin{aligned} & u _ n ( x ) = p ( x ) u _ j ( x ) \\leq p ( x ) u ( x ) \\leq a ( x ) & & j \\in \\{ 0 , \\ldots , n \\} , \\\\ & p ^ L = u _ { 0 } ( x ) = a _ 0 ( x ) \\leq \\cdots \\leq a _ n ( x ) = p ^ U , \\\\ & \\deg ( u _ j ) \\leq c _ 1 \\deg ( a _ j ) \\leq c _ 2 & & j \\in \\{ 0 , \\ldots , n \\} . \\end{aligned} \\end{align*}"}
{"id": "5724.png", "formula": "\\begin{align*} { \\sf F i x } _ X ( G ) = \\{ u \\in X : \\sigma u = u , \\ \\forall \\sigma \\in G \\} \\end{align*}"}
{"id": "1364.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( T _ { \\gamma ( t ) } G ) \\ 1 { \\gamma ( t ) \\in W ' ( \\omega ) } \\gamma ' ( t ) d t = 0 , \\end{align*}"}
{"id": "6868.png", "formula": "\\begin{align*} \\norm { \\left ( 1 - \\delta ^ { i _ 1 + \\ell - 2 } _ { i _ 1 - 2 } \\right ) \\Gamma ' g _ { \\ell } + \\Gamma g _ \\ell } & \\leq \\left ( 1 - \\delta ^ { i _ 1 + \\ell - 2 } _ { i _ 1 - 2 } \\right ) \\norm { \\Gamma ' } \\norm { g _ { \\ell } } + \\norm { \\Gamma } \\norm { g _ \\ell } \\\\ & \\leq \\sum \\limits _ { s = 1 } ^ j \\gamma _ { i _ s - 2 s } ^ { i _ s + \\ell - 2 s } \\prod \\limits _ { t = 1 } ^ { s - 1 } \\left ( 1 - \\delta ^ { i _ t + \\ell - 2 t } _ { i _ t - 2 t } \\right ) \\norm { g _ \\ell } , \\end{align*}"}
{"id": "7440.png", "formula": "\\begin{align*} \\forall k \\geq 1 , 0 = - \\int _ 0 ^ T \\int _ { \\mathbb { R } } \\rho _ { 3 } ( s , u ) \\partial _ s G _ { k } ( s , u ) d u d s - & \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\rho _ { 5 } ( s , u ) [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ { k } ] ( s , u ) d u d s . \\end{align*}"}
{"id": "7681.png", "formula": "\\begin{align*} - \\Delta g + \\tilde { \\lambda } _ 0 \\underline { m } \\ , g = f _ 0 ( g ) \\R ^ n . \\end{align*}"}
{"id": "8891.png", "formula": "\\begin{align*} h _ n : X \\ast I & \\to X \\ast I \\\\ ( x , ( s , t ) ) & \\mapsto \\begin{cases} ( x , ( s + n , t - n ) ) & n < t \\\\ ( x , ( s + t , 0 ) ) & n \\ge t . \\end{cases} \\end{align*}"}
{"id": "7574.png", "formula": "\\begin{align*} \\psi _ { \\eta } ' ( 0 ) = 1 . \\end{align*}"}
{"id": "5926.png", "formula": "\\begin{align*} \\sum _ a \\Gamma _ a \\lrcorner \\omega _ { Q , L } ^ a = d E _ L , \\end{align*}"}
{"id": "3122.png", "formula": "\\begin{align*} \\Delta _ 5 ( \\lambda _ 1 , \\lambda _ 2 , \\lambda _ 3 ) = 0 \\ , , \\end{align*}"}
{"id": "8634.png", "formula": "\\begin{align*} \\mu _ { R , 2 } ^ { ( 0 ) } : = \\int \\overline { \\chi _ + ( x ) \\mathcal { K } _ + ( x , k ) } \\chi _ - ( x ) \\mathcal { K } _ - ( x , \\ell ) \\overline { \\mathcal { K } _ S ( x , m ) } \\mathcal { K } _ S ( x , n ) \\ , d x ; \\end{align*}"}
{"id": "1451.png", "formula": "\\begin{align*} \\alpha U ( z ) = U ( z ' ) \\lambda ( \\alpha , z ) . \\end{align*}"}
{"id": "6258.png", "formula": "\\begin{align*} { \\bf T } _ { s } ( { \\bf A } ) = \\left \\{ W = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) : \\bigcap \\limits _ { i = 1 } ^ { s } \\operatorname { A n n } ( \\theta _ { i } ) \\cap \\operatorname { A n n } ( { \\bf A } ) = 0 \\right \\} , \\end{align*}"}
{"id": "5843.png", "formula": "\\begin{align*} g _ c ( x ) = \\frac { 1 } { \\sqrt { 2 \\pi c } } e ^ { - \\frac { x ^ 2 } { 2 c } } , x \\in \\R . \\end{align*}"}
{"id": "4315.png", "formula": "\\begin{align*} \\varepsilon ( \\tau ) = \\sum _ { j = 0 } ^ { \\ell } \\hat { \\varepsilon } _ j ( \\tau ) \\phi _ { j , \\infty } + \\hat { \\varepsilon } _ - ( \\tau ) : = \\hat { \\varepsilon } _ + + \\hat { \\varepsilon } _ - , \\end{align*}"}
{"id": "6122.png", "formula": "\\begin{align*} \\alpha _ 0 ( x ) = h ^ 1 ( x ^ { m + 2 } ) + x h ^ 2 ( x ^ { m + 2 } ) \\ , . \\end{align*}"}
{"id": "6825.png", "formula": "\\begin{align*} ( \\tilde f _ k ) ^ * ( g _ { \\textnormal { e u c } } ) = e ^ { 2 \\tilde u _ k } \\ , g _ { \\textnormal { p o i n } , j _ k } \\textnormal { o n } \\ , \\ , \\Sigma \\end{align*}"}
{"id": "642.png", "formula": "\\begin{align*} & f ( x , 0 ) \\ = \\ x , \\\\ [ 5 p t ] & f ( x , n + 1 ) \\ = \\ \\bigg [ \\frac { f ( x , n ) } { n + 1 } \\bigg ] , \\\\ [ 5 p t ] & f ( x , n ) \\ \\leq x \\end{align*}"}
{"id": "7715.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { \\# \\{ A \\in \\mathcal A ( H ) \\colon \\max ( A ) \\le x \\} } { \\# \\{ A \\in H \\colon \\max ( A ) \\le x \\} } = 1 \\ , . \\end{align*}"}
{"id": "8502.png", "formula": "\\begin{align*} a * b = \\lambda _ a ( b ) - b = - a + a \\circ b - b . \\end{align*}"}
{"id": "9034.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t \\rho _ 1 = \\nabla \\cdot \\left ( \\nabla \\rho _ 1 + \\rho _ 1 \\nabla ( U _ 1 + \\phi ) \\right ) , \\\\ & \\partial _ t \\rho _ 2 = \\nabla \\cdot \\left ( \\nabla \\rho _ 2 + \\rho _ 2 \\nabla ( U _ 2 - \\phi ) \\right ) , \\\\ & - \\Delta \\phi = \\rho _ 1 - \\rho _ 2 , \\end{aligned} \\end{align*}"}
{"id": "6863.png", "formula": "\\begin{align*} \\delta _ j ^ k = \\prod \\limits _ { i = k - j } ^ k \\delta _ i , ~ \\gamma ^ k _ j = \\gamma \\sum \\limits ^ { j - 1 } _ { i = - 1 } \\delta ^ k _ i , \\end{align*}"}
{"id": "9454.png", "formula": "\\begin{align*} F _ 3 ( { \\bf \\underline { t } } ) = t _ 2 ^ 2 - t _ 1 ^ p + ( 2 b _ 1 / a _ 0 ) \\cdot t _ 1 ^ { ( p + 1 ) / 2 } X ^ { p ^ s - p ^ { s - 1 } } . \\end{align*}"}
{"id": "3538.png", "formula": "\\begin{align*} \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & \\ll \\begin{cases} T ^ { 4 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } & ( \\frac { 3 } { 2 } < \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 < 2 ) \\\\ \\log T & ( \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 = 2 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "1172.png", "formula": "\\begin{align*} \\left ( \\frac { d \\psi } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 \\psi \\right ) = \\beta ^ { m a t } \\psi . \\end{align*}"}
{"id": "3627.png", "formula": "\\begin{align*} \\sum _ { \\substack { | \\Im ( \\rho ) | \\leq T \\\\ \\Re ( \\rho ) > \\sigma } } \\frac { x ^ { \\Re ( \\rho ) - 1 } } { | \\Im ( \\rho ) | } & \\leq 2 \\sum _ { k = 0 } ^ { K - 1 } \\frac { x ^ { - \\nu _ 1 \\left ( t _ k \\right ) } } { t _ k } N _ 0 ( \\sigma , t _ { k + 1 } ) \\\\ & = s _ 2 ( x , \\sigma , K ) , \\ . \\end{align*}"}
{"id": "892.png", "formula": "\\begin{align*} a ( E _ k ^ { ( n ) } , T ) & = ( - 1 ) ^ { [ ( n + 1 ) / 2 ] } 2 ^ { n - [ n / 2 ] } \\frac { k } { B _ k } \\prod _ { i = 1 } ^ { [ n / 2 ] } \\frac { 2 k - 2 i } { B _ { 2 k - 2 i } } \\prod _ q F _ q ( T , q ^ { k - n - 1 } ) \\\\ & \\times \\begin{cases} \\frac { B _ { k - n / 2 , \\chi _ T } } { k - n / 2 } & n \\\\ 1 & n \\end{cases} \\end{align*}"}
{"id": "9343.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - x ( e ^ { t } - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } F _ { n } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 4 , 6 , 1 2 , 1 6 ] ) . \\end{align*}"}
{"id": "28.png", "formula": "\\begin{align*} \\omega = d \\theta ^ c - \\theta \\wedge \\theta ^ c . \\end{align*}"}
{"id": "7961.png", "formula": "\\begin{align*} & \\frac 1 { \\lambda _ 1 } + \\frac 1 { \\lambda _ 2 } + \\frac 1 { \\Lambda _ 2 } = 2 , \\\\ & \\frac 1 { \\lambda _ k } + \\frac 1 { \\Lambda _ k } = 1 \\mbox { i f } \\ ; 3 \\le k \\le p . \\end{align*}"}
{"id": "9176.png", "formula": "\\begin{align*} \\sum ^ { \\mathcal { J } } _ { j = 1 } 2 \\alpha _ { j } \\lceil e ^ 2 \\alpha ^ { - 3 / 4 } _ j \\rceil \\leq \\lceil 4 e ^ 2 k 1 0 ^ { - M / 4 } \\rceil + 1 . \\end{align*}"}
{"id": "83.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } \\Bigr \\} ^ { - 1 } \\\\ & \\leq 3 \\cdot 2 ^ { 3 / 2 } \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "3900.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq a < p \\\\ \\gcd ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } = 0 \\end{align*}"}
{"id": "7706.png", "formula": "\\begin{align*} A + D + E = [ \\mathsf F ( S ) + 1 + m , \\mathsf F ( S ) + 1 + 2 M - m ] = C + B + E \\ , , \\end{align*}"}
{"id": "2442.png", "formula": "\\begin{align*} \\sigma ( S \\l ' , S \\l ) = \\sigma ( \\l ' , \\l ) . \\end{align*}"}
{"id": "1298.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda ( M ) | } { M ^ { 2 } } = \\lim _ { M \\to \\infty } \\sum _ { \\Gamma \\in H _ { 1 } ( Y ) } \\frac { | \\Lambda ( M , \\Gamma ) | } { M ^ 2 } = \\frac { | H _ { 1 } ( Y ) | } { 2 \\mathrm { V o l } ( Y , \\lambda ) } . \\end{align*}"}
{"id": "2020.png", "formula": "\\begin{align*} e _ A ( t ) : = \\exp ( A _ t ) , t \\ge 0 . \\end{align*}"}
{"id": "7513.png", "formula": "\\begin{align*} \\xi ( s ) = \\frac { 1 } { 2 } s ( s - 1 ) \\pi ^ { - s / 2 } \\Gamma \\left ( \\frac { s } { 2 } \\right ) \\zeta ( s ) \\end{align*}"}
{"id": "2223.png", "formula": "\\begin{align*} \\| v ^ { n } ( t ) \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } & \\leq \\| E ( t ) P _ n P X _ 0 \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } + \\Big \\| \\int _ 0 ^ t E ( t - s ) P _ n A P F ( X ^ { n } ( s ) ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } \\\\ & \\leq \\| X _ 0 \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } + C \\leq C ( T , \\gamma , X _ 0 , p ) \\end{align*}"}
{"id": "6559.png", "formula": "\\begin{align*} H _ { 2 } ( t ) = C ( \\| \\nabla K ( t ) \\| _ { L _ { t } ^ { 1 } L ^ { 2 } } \\| \\omega \\| _ { L _ { t } ^ { \\infty } L ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\| j \\| _ { L _ { t } ^ { \\infty } L ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\| j \\| _ { L _ { t } ^ { 1 } L ^ { \\infty } } ^ { \\frac { 1 } { 2 } } + \\| \\nabla ^ { 2 } K ( t ) \\| _ { L _ { t } ^ { 1 } L ^ { 1 } } \\| u \\| _ { L _ { t } ^ { \\infty } L ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\| j \\| _ { L _ { t } ^ { 2 } L ^ { \\infty } } ) . \\end{align*}"}
{"id": "7387.png", "formula": "\\begin{align*} Q ( x _ 0 , r _ n ) = \\lbrace y \\in \\R ^ d : 0 \\le \\min _ i ( y _ i - ( x _ 0 ) _ i ) \\le \\max _ i ( y _ i - ( x _ 0 ) _ i ) < r _ n \\rbrace . \\end{align*}"}
{"id": "426.png", "formula": "\\begin{align*} \\| \\hat { f } _ { l } \\| _ { s - 1 } & \\leq C \\left ( \\| \\hat { u } ^ { 2 } - \\hat { u } ^ { 1 } \\| _ { s - 1 } + \\| \\hat { v } ^ { 2 } - \\hat { v } ^ { 1 } \\| _ { s } + \\| \\hat { w } ^ { 2 } - \\hat { w } ^ { 1 } \\| _ { s - 1 } \\right ) , l = 1 , 3 , \\\\ \\| \\hat { f } _ { 2 } \\| _ { s - 2 } & \\leq C \\| ( \\hat { u } ^ { 2 } - \\hat { u } ^ { 1 } , \\hat { v } ^ { 2 } - \\hat { v } ^ { 1 } , \\hat { w } ^ { 2 } - \\hat { w } ^ { 1 } ) \\| _ { s - 1 } , \\end{align*}"}
{"id": "5656.png", "formula": "\\begin{align*} \\varphi = ( \\varphi ^ { - 1 } ) ^ { - 1 } = ( \\sigma ^ { - 1 } , ( \\alpha ^ { - 1 } _ { \\sigma 1 } , \\alpha ^ { - 1 } _ { \\sigma 2 } , \\alpha ^ { - 1 } _ { \\sigma 3 } ) ) . \\end{align*}"}
{"id": "9175.png", "formula": "\\begin{align*} { \\mathcal P } _ j ( p ) = & \\sum _ { \\substack { X ^ { \\alpha _ { j - 1 } } < q \\leq X ^ { \\alpha _ { j } } \\\\ q } } \\frac { \\chi _ { 8 p } ( q ) \\lambda _ f ( q ) } { \\sqrt { q } } , { \\mathcal M } _ j ( p , \\alpha ) = E _ { e ^ 2 \\alpha ^ { - 3 / 4 } _ j } \\Big ( \\alpha { \\mathcal P } _ j ( p ) \\Big ) , { \\mathcal M } ( p , \\alpha ) = ( \\log X ) ^ { 1 / 2 } \\prod ^ { \\mathcal { J } } _ { j = 1 } { \\mathcal M } _ j ( p , \\alpha ) . \\end{align*}"}
{"id": "1222.png", "formula": "\\begin{align*} U _ { p , q , \\delta } = F _ { T } ( \\Omega _ { N _ { p , q , \\delta } } ) . \\end{align*}"}
{"id": "4181.png", "formula": "\\begin{align*} \\delta _ H ( f _ i ( L _ i ) , f _ 0 ( L _ 0 ) ) = \\delta _ H ( f _ i ( \\phi _ i ( ( L _ 0 ) ) , f _ 0 ( L _ 0 ) ) \\leq d _ { C ^ 0 } ( f _ i \\circ \\phi _ i , f _ 0 ) , \\end{align*}"}
{"id": "7688.png", "formula": "\\begin{align*} \\partial _ { x _ i x _ j } f ( \\mathbf { x } ) = o ( 1 ) \\Omega \\to 0 ^ { + } , \\ ; n \\to + \\infty \\ ; , \\end{align*}"}
{"id": "8067.png", "formula": "\\begin{align*} \\mathfrak { P } _ { \\ell } \\rho ^ { ( t ) } \\Psi _ { \\Sigma } ( f ) = \\Psi _ { \\Sigma } ( \\rho ^ { ( t ) } _ { * } f ) = \\Psi _ { \\Sigma } ( f ) - t \\Psi _ { \\Sigma } ( \\mathcal { L } _ X f ) + \\mathcal { O } ( t ^ 2 ) . \\end{align*}"}
{"id": "8581.png", "formula": "\\begin{align*} m ( D ) = \\big ( \\mathcal { F } ^ \\# \\big ) ^ { - 1 } m ( k ) \\mathcal { F } ^ \\# , \\end{align*}"}
{"id": "1190.png", "formula": "\\begin{align*} s _ \\mu ( B , U ) = \\sup \\left \\{ s \\geq 0 \\ : \\ \\mathcal { H } ^ { \\mu , s } _ { \\infty } \\left ( U \\right ) \\geq \\mu ( B ) \\right \\} . \\end{align*}"}
{"id": "3987.png", "formula": "\\begin{align*} \\{ 1 \\} \\cup \\{ e ^ { \\lambda ^ h _ k t } \\} _ { | k | \\geq k _ 0 } \\cup \\{ e ^ { \\widehat \\lambda _ { n _ l } t } \\} _ { l = 1 } ^ { l _ 0 } \\end{align*}"}
{"id": "9080.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\begin{dcases} 1 , & x = 0 , 0 \\leq y \\leq 1 , \\\\ - 1 , & x = 1 , 0 \\leq y \\leq 1 , \\\\ - 2 x + 1 , & 0 \\leq x \\leq 1 , y = 0 , \\\\ - 2 x + 1 , & 0 \\leq x \\leq 1 , y = 1 . \\end{dcases} \\end{align*}"}
{"id": "4246.png", "formula": "\\begin{align*} 0 \\ge { } & \\Delta Q _ A \\\\ = { } & \\Delta u _ { e e } - A \\Delta u \\\\ = { } & - f ' ( u ) u _ { e e } - f '' ( u ) u _ e ^ 2 + A f ( u ) \\\\ \\ge { } & - A f ' ( u ) u - \\sigma A f ' ( u ) x _ 1 + A f ( u ) \\\\ > { } & A ( - f ' ( u ) u + f ( u ) ) - A f ( 0 ) \\ge 0 , \\end{align*}"}
{"id": "6360.png", "formula": "\\begin{align*} D _ s \\equiv \\sum _ { t = s + 1 } ^ n h ^ { ( s ) } _ t E _ t + h ^ { ( s ) } _ { n + 1 } A \\equiv h ^ { ( s ) } _ 0 B + \\sum _ { t = 1 } ^ { s - 1 } h ^ { ( s ) } _ t E _ t s = 1 , . . . , n . \\end{align*}"}
{"id": "4685.png", "formula": "\\begin{align*} \\mathcal { L } \\mathfrak { A } _ { i j } = \\sum ^ n _ { k = 1 } \\mu _ k E _ { i j k } + F _ { i j } , \\end{align*}"}
{"id": "1255.png", "formula": "\\begin{align*} Q _ { \\tau } ( Z _ { 1 } + Z _ { 2 } , Z _ { 1 } + Z _ { 2 } ) = Q _ { \\tau } ( Z _ { 1 } , Z _ { 1 } ) + 2 Q _ { \\tau } ( Z _ { 1 } , Z _ { 2 } ) + Q _ { \\tau } ( Z _ { 2 } , Z _ { 2 } ) . \\end{align*}"}
{"id": "6802.png", "formula": "\\begin{align*} ( \\mathcal { F } ( \\mathcal { F } \\varphi ) ) ( \\xi ) = \\varphi ( - \\xi ) , \\end{align*}"}
{"id": "1267.png", "formula": "\\begin{align*} P _ { \\gamma } ^ { \\mathrm { i n } } ( M ) = P _ { \\gamma } ^ { \\mathrm { o u t } } ( M ) : = \\begin{cases} ( 2 , . . . , \\ , 2 ) & \\mathrm { i f } \\ , \\ , M \\ , \\ , \\mathrm { i s \\ , \\ , e v e n } , \\\\ ( 2 , . . . , \\ , 2 , \\ , 1 ) & \\mathrm { i f } \\ , \\ , M \\ , \\ , \\mathrm { i s \\ , \\ , o d d } , \\end{cases} \\end{align*}"}
{"id": "5133.png", "formula": "\\begin{align*} a = \\frac { \\log \\left ( n + \\sqrt { n ^ 2 + 4 } \\right ) - \\log ( 2 ) } { \\gamma \\ , n } . \\end{align*}"}
{"id": "9397.png", "formula": "\\begin{align*} \\tau ' ( a _ n \\cdots a _ 1 v b _ 1 \\cdots b _ m ) = \\tau ( a _ n \\cdots a _ 1 \\tau ' ( v ) b _ 1 \\cdots b _ m ) . \\end{align*}"}
{"id": "7272.png", "formula": "\\begin{align*} \\boxed { \\log \\Bigl ( \\prod ( 1 + \\alpha _ k ) \\Bigr ) = \\sum \\log ( 1 + \\alpha _ k ) . } \\end{align*}"}
{"id": "9252.png", "formula": "\\begin{align*} & \\forall \\gamma ^ 1 , x ^ X \\big ( \\gamma > _ \\mathbb { R } 0 \\land \\exists y ^ X , z ^ X ( z \\in A y \\land x = _ X y + _ X \\gamma z ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\rightarrow \\gamma ^ { - 1 } ( x - _ X J ^ A _ { \\gamma } x ) \\in A ( J ^ A _ { \\gamma } x ) \\big ) , \\end{align*}"}
{"id": "6629.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { m } ) \\left ( 1 + \\frac { p ^ w } { p ^ { 2 } ( p - 1 ) } - \\frac { 1 } { p - 1 } \\right ) } { p ^ { m ( 1 + s _ 1 + s _ 2 ) } } \\ll \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { p ^ { m ( 1 + \\varepsilon ) } } \\ll \\frac { 1 } { p ^ { 1 + \\varepsilon } } . \\end{align*}"}
{"id": "188.png", "formula": "\\begin{align*} C _ { 1 , p , \\delta , r } : = \\int _ 0 ^ { + \\infty } p _ \\delta ( x ) ( G _ \\delta ( x ) ) ^ r d x < + \\infty , \\end{align*}"}
{"id": "6647.png", "formula": "\\begin{align*} R _ 0 = R _ 1 + R _ 2 + R _ 3 + R _ 4 , \\end{align*}"}
{"id": "6780.png", "formula": "\\begin{align*} - k _ { r } \\leqslant \\sum _ { j = 1 } ^ { r } i _ { k _ { j } } \\leqslant 0 f o r 1 \\leqslant r \\leqslant m . \\end{align*}"}
{"id": "7940.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\sum _ { k = 1 } ^ { n } | E _ { n , k } | \\ll \\sum _ { n = 1 } ^ { N } ( n ^ { 1 1 / 8 } + L _ n ^ { 1 / 2 } n ^ { 5 / 4 } ) \\ll N ^ { 1 9 / 8 } + \\sum _ { n = 1 } ^ { N } L _ n ^ { 1 / 2 } n ^ { 5 / 4 } . \\end{align*}"}
{"id": "8290.png", "formula": "\\begin{align*} \\partial _ t X _ t \\cdot \\nu ( X _ t ) : = - H _ { E ( t ) } ^ \\alpha ( X _ t ) , \\ ; \\ ; \\ ; \\mbox { f o r a l l $ X _ t \\in \\partial E ( t ) $ a n d } t \\in [ 0 , T ] , \\end{align*}"}
{"id": "4198.png", "formula": "\\begin{align*} \\ell = \\bigcup \\limits _ { i = 1 } ^ { 5 } ( W _ { i } - \\{ 0 \\} ) = \\bigcup \\limits _ { j = 1 } ^ { 5 } ( W _ { j } ^ { \\prime } - \\{ 0 \\} ) , \\end{align*}"}
{"id": "6471.png", "formula": "\\begin{align*} D \\left ( \\dfrac { S _ n } { \\sqrt { n / ( 1 - 2 \\alpha ) } } \\right ) & \\leq \\begin{cases} \\dfrac { C _ { \\alpha } \\log n } { \\sqrt { n } } & ( - 1 < \\alpha \\leq 0 ) , \\\\ [ 4 m m ] \\dfrac { C _ { \\alpha } \\log n } { n ^ { ( 1 - 2 \\alpha ) / 2 } } & ( 0 \\leq \\alpha < 1 / 2 ) , \\\\ [ 4 m m ] \\end{cases} \\\\ D \\left ( \\dfrac { S _ n } { \\sqrt { n \\log n } } \\right ) & \\leq \\dfrac { C _ { 1 / 2 } \\log \\log n } { \\sqrt { \\log n } } ( \\alpha = 1 / 2 ) . \\end{align*}"}
{"id": "7619.png", "formula": "\\begin{align*} \\mathcal { L } \\Phi = F ^ { i j } ( \\lambda ' \\delta _ { i j } - u h _ { i j } ) = \\lambda ' \\sum _ { i } F ^ { i i } - u F = \\lambda ' \\sum _ { i } F ^ { i i } - u ^ { \\frac { \\alpha + 1 } { \\alpha } } F ^ { 2 } . \\end{align*}"}
{"id": "101.png", "formula": "\\begin{align*} X _ 0 ( T ) & = \\big \\{ \\ , \\lambda \\in X ^ \\star ( T ) \\langle \\lambda , \\alpha ^ \\vee \\rangle = 0 \\alpha \\in R ^ + \\ , \\big \\} \\\\ X _ 1 ( T ) & = \\big \\{ \\ , \\lambda \\in X ^ \\star ( T ) 0 \\leq \\langle \\lambda , \\alpha ^ \\vee \\rangle \\leq p - 1 \\alpha \\in R ^ + \\ , \\big \\} \\end{align*}"}
{"id": "746.png", "formula": "\\begin{align*} & Y ^ { ( \\ell ) } [ ( \\partial _ \\mu z ) ^ { \\gamma _ 1 } , \\ldots , ( \\partial _ \\mu z ) ^ { \\gamma _ m } ; f _ 1 , \\ldots , f _ m ] : = \\sum _ { \\mu \\leq \\ell } \\lambda _ \\mu \\prod _ { t = 1 } ^ m \\left \\{ \\frac { C _ W } { n _ { \\ell } } \\sum _ { k = 1 } ^ { n _ \\ell } f _ { t ; k } ^ { ( \\ell ) } ( \\partial _ \\mu z _ k ^ { ( \\ell ) } ) ^ { \\gamma _ t } \\right \\} \\end{align*}"}
{"id": "1860.png", "formula": "\\begin{align*} G = \\{ a \\rightarrow a x , \\ ; \\ ; x \\rightarrow 1 + x ^ 2 \\} , \\end{align*}"}
{"id": "74.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq q _ v ^ { - 1 / 2 } ( q _ v ^ { n _ { v , \\nu _ v / 2 } } + 1 ) . \\end{align*}"}
{"id": "2942.png", "formula": "\\begin{align*} \\tilde M _ { n , \\mathbf { p } _ { k , 1 } } \\cdot { \\tilde M } _ { n , \\mathbf { p } _ { k , 2 } } & = \\frac { 4 } { n ^ 2 } \\sum _ { \\mathbf { i } \\in \\mathcal { J } } \\prod _ { \\ell = 1 } ^ { k } \\tilde I ^ { ( p _ { \\ell , 1 } ) } _ { i _ 1 , i _ 2 } \\tilde I ^ { ( p _ { \\ell , 2 } ) } _ { i _ 3 , i _ 4 } , \\end{align*}"}
{"id": "8416.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { N - 1 } \\sup _ { h \\in \\mathrm { L i p } ( 1 ) } \\left | \\tilde { Q } _ { i - 1 } \\big ( P _ { \\eta } - \\tilde { Q } _ { 1 } \\big ) P _ { ( N - i ) \\eta } h ( x ) \\right | \\leq C ( 1 + | x | ) \\ , \\eta ^ { 2 / \\alpha - 1 } . \\end{gather*}"}
{"id": "2259.png", "formula": "\\begin{align*} | \\widetilde { e } _ m ^ { M , N } | _ { - 1 } ^ 2 + k \\sum _ { j = 1 } ^ m | \\widetilde { e } ^ { M , N } _ j | _ 1 ^ 2 \\leq C k \\sum _ { j = 1 } ^ m \\| \\widetilde { X } _ j ^ { M , N } - X ( t _ j ) \\| ^ 2 \\big ( 1 + \\| \\widetilde { X } _ j ^ { M , N } \\| _ V ^ 4 + \\| X ( t _ j ) \\| _ V ^ 4 \\big ) . \\end{align*}"}
{"id": "658.png", "formula": "\\begin{align*} \\abs { f ( x ) - g ( x ) } \\ = \\ h ( X , x ) , \\end{align*}"}
{"id": "2735.png", "formula": "\\begin{align*} R ^ t \\pi _ { 1 * } ( \\Q _ S ) ) _ \\lambda = 0 . \\end{align*}"}
{"id": "6660.png", "formula": "\\begin{align*} D _ { 2 , m , n } = \\tau _ { A \\smallsetminus \\{ \\alpha \\} } ( p ^ m ) \\tau _ { B \\smallsetminus \\{ \\beta \\} \\cup \\{ - \\alpha \\} } ( p ^ n ) . \\end{align*}"}
{"id": "5204.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\int _ Y | F _ k ( y ) | d \\nu ( y ) \\bigg ) ^ r \\bigg ) ^ { 1 / r } ~ \\leq ~ \\int _ Y \\bigg ( \\sum _ { k = 0 } ^ { \\infty } | F _ k ( y ) | ^ r \\bigg ) ^ { 1 / r } d \\nu ( y ) . \\end{align*}"}
{"id": "5722.png", "formula": "\\begin{align*} { \\sf F i x } _ M ( G ) = \\{ x \\in M : \\sigma ( x ) = x , \\forall \\sigma \\in G \\} \\end{align*}"}
{"id": "4819.png", "formula": "\\begin{align*} \\sigma _ { } ( ( I + \\tilde { F } \\tilde { F } ^ \\top ) ^ { - \\frac { 1 } { 2 } } ) & = \\frac { 1 } { \\sigma _ { } ( ( I + \\tilde { F } \\tilde { F } ^ \\top ) ^ { \\frac { 1 } { 2 } } ) } \\\\ & = \\frac { 1 } { \\sqrt { \\sigma _ { } ( I + \\tilde { F } \\tilde { F } ^ \\top ) } } \\\\ & = \\frac { 1 } { \\sqrt { \\lVert ( I + \\tilde { F } \\tilde { F } ^ \\top ) \\rVert _ 2 } } , \\end{align*}"}
{"id": "6250.png", "formula": "\\begin{align*} \\mathcal { L } ( q , \\lambda ( q , t ) ) = \\mathcal { E } ( q , \\lambda ( q , t ) ) \\end{align*}"}
{"id": "996.png", "formula": "\\begin{align*} f _ \\varepsilon ( x ) = \\int _ { B _ \\varepsilon ^ + ( x ) } \\bar f ( y ) \\big ( \\eta _ \\varepsilon ( x - y ) - \\eta _ \\varepsilon ( x _ \\ast - y ) \\big ) \\dd y \\geqslant - ( M + \\varepsilon ) \\int _ { B _ \\varepsilon ^ + ( x ) } y _ 1 \\big ( \\eta _ \\varepsilon ( x - y ) - \\eta _ \\varepsilon ( x _ \\ast - y ) \\big ) \\dd y . \\end{align*}"}
{"id": "4774.png", "formula": "\\begin{align*} R ^ 2 + R ^ \\# = \\theta R \\end{align*}"}
{"id": "754.png", "formula": "\\begin{align*} \\alpha = \\sqrt { \\left ( L - \\mu \\right ) \\left ( \\mu _ p t _ 1 \\left ( \\mu _ p - \\mu \\right ) \\left ( L t _ 1 - 2 \\right ) + \\left ( L - \\mu \\right ) \\right ) } . \\end{align*}"}
{"id": "7353.png", "formula": "\\begin{align*} \\begin{aligned} & h _ q + G _ \\beta ^ \\ast ( v ( y _ q , t _ q ) , 0 , Y _ q , K \\cap \\{ v ( \\cdot , t _ q ) \\leq v ( y _ q , t _ q ) \\} ) \\geq 0 , \\\\ & k _ q + G _ \\beta ^ \\ast ( v ( z _ q , t _ q ) , 0 , Z _ q , K \\cap \\{ v ( \\cdot , t _ q ) \\leq v ( z _ q , t _ q ) \\} ) \\geq 0 , \\end{aligned} \\end{align*}"}
{"id": "1401.png", "formula": "\\begin{align*} \\hat q ^ * = q ^ * + q _ d ^ * \\epsilon . \\end{align*}"}
{"id": "6834.png", "formula": "\\begin{align*} S _ n Q _ { n + 1 } - S _ { n + 1 } Q _ n = ( - 1 ) ^ n w \\end{align*}"}
{"id": "8293.png", "formula": "\\begin{align*} \\| u \\| _ { C _ b ^ m } = \\sum _ { k = 0 } ^ m \\| D ^ k u \\| _ { L ^ \\infty } : = \\sum _ { k = 0 } ^ m \\sup _ { x \\in \\R ^ { N - 1 } } | D ^ k u ( x ) | . \\end{align*}"}
{"id": "2859.png", "formula": "\\begin{align*} \\delta ( t _ n ' ) = \\big | \\| \\nabla u ( t _ n ' ) \\| _ 2 ^ 2 - \\| \\nabla Q \\| _ 2 ^ 2 \\big | \\to 0 , \\end{align*}"}
{"id": "7092.png", "formula": "\\begin{align*} s _ k & = \\binom { p - 1 } { k } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k - 1 } ( f ( x , v ) y - f ( y , v ) x ) \\\\ & = \\binom { p - 1 } { k } f ( x , v ) ^ { k } f ( y , v ) ^ { p - k - 1 } y - \\binom { p - 1 } { k } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k } x . \\end{align*}"}
{"id": "3401.png", "formula": "\\begin{align*} \\begin{aligned} | f ( x ) - f ( x ' ) | & \\leqslant C \\Big ( \\frac { \\| x - x ' \\| } { r } \\Big ) ^ \\beta \\Big \\{ \\frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \\Big ( \\frac { r } { r + \\boldsymbol { \\| x - x _ 0 \\| } } \\Big ) ^ \\gamma \\\\ & \\qquad + \\frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \\Big ( \\frac { r } { r + \\boldsymbol { \\| x ' - x _ 0 \\| } } \\Big ) ^ \\gamma \\Big \\} ; \\end{aligned} \\end{align*}"}
{"id": "1130.png", "formula": "\\begin{align*} f ( \\xi , k ) = - 2 \\cdot 6 ^ { \\frac { 2 } { 3 } } \\eta \\left ( k - \\eta \\right ) \\left ( 1 + \\frac { k - \\eta } { 2 \\eta } + \\mathcal { O } \\left ( \\left ( k - \\eta \\right ) ^ 2 \\right ) \\right ) . \\end{align*}"}
{"id": "6674.png", "formula": "\\begin{align*} I _ { A \\cup \\{ - \\beta \\} , \\{ \\alpha \\} } ( n ) = { \\tau } _ { A \\smallsetminus \\{ \\alpha \\} \\cup \\{ - \\beta \\} } ( n ) \\end{align*}"}
{"id": "8082.png", "formula": "\\begin{align*} [ a ( z ) , b ( w ) ] = \\sum _ { j = 0 } ^ { N - 1 } \\partial _ w ^ { ( j ) } \\delta ( z - w ) c ^ j ( w ) , \\end{align*}"}
{"id": "7682.png", "formula": "\\begin{align*} g = a \\ , w a \\in \\R . \\end{align*}"}
{"id": "9122.png", "formula": "\\begin{align*} \\forall n , m \\in \\mathbb { N } \\forall p \\in F \\left ( H ( d ( x _ { n + m } , p ) ) \\leq G ( d ( x _ n , p ) ) + \\sum _ { i = n } ^ { n + m - 1 } \\varepsilon _ i \\right ) , \\end{align*}"}
{"id": "3219.png", "formula": "\\begin{align*} \\mu _ { T } \\{ \\lambda \\in \\mathbb { C } : | \\lambda | \\leq r \\} = \\begin{cases} 0 , & r < \\lambda _ 1 ( \\mu _ { | T | } ) ; \\\\ 1 + S ^ { \\langle - 1 \\rangle } _ { \\mu _ { T ^ * T } } ( r ^ { - 2 } ) , & \\lambda _ 1 ( \\mu _ { | T | } ) < r < \\lambda _ 2 ( \\mu _ { | T | } ) ; \\\\ 1 , & r \\geq \\lambda _ 2 ( \\mu _ { | T | } ) . \\end{cases} \\end{align*}"}
{"id": "491.png", "formula": "\\begin{align*} Y _ { \\alpha \\beta } ^ { i j } = \\frac { h ( \\mathcal { I } ) \\cdot h ( \\mathcal { I } _ { \\alpha \\beta } ^ { i j } ) } { h ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\alpha } ^ { j } ) } - 1 . \\end{align*}"}
{"id": "4519.png", "formula": "\\begin{align*} a _ 0 + a _ { 1 / 2 } + \\alpha a = b = b ^ 2 = ( a _ 0 + a _ { 1 / 2 } + \\alpha a ) ^ 2 = a _ 0 ^ 2 + a _ { 1 / 2 } ^ 2 + \\alpha ^ 2 a + 2 a _ 0 a _ { 1 / 2 } + \\alpha a _ { 1 / 2 } . \\end{align*}"}
{"id": "1897.png", "formula": "\\begin{align*} Z _ { t _ 1 } \\subseteq \\mathring { Z } _ { t _ 1 ' } = T ( Z _ { t _ 1 ' } ) + C \\subseteq 0 + C = \\mathring { N } _ t \\end{align*}"}
{"id": "7949.png", "formula": "\\begin{align*} n o r m ( \\pi ^ { \\prime } ) = \\prod _ { u = \\rho _ 1 } ^ { k _ 1 - 1 } \\prod _ { r = 0 } ^ { \\rho _ 1 - 1 } s _ { u - r } \\cdot \\prod _ { u = k _ 1 } ^ { k _ 2 - 1 } \\prod _ { r = 0 } ^ { \\rho _ 2 - 1 } s _ { u - r } \\cdot \\prod _ { u = k _ 2 } ^ { k _ 3 - 1 } \\prod _ { r = 0 } ^ { \\rho _ 3 - 1 } s _ { u - r } \\cdots \\prod _ { u = k _ { m - 1 } } ^ { k _ m - 1 } \\prod _ { r = 0 } ^ { \\rho _ m - 1 } s _ { u - r } , \\end{align*}"}
{"id": "7870.png", "formula": "\\begin{align*} G \\cup H \\cup I & = ( \\mathbb { N } \\setminus \\{ \\bar { b } _ n \\} ) \\cup \\{ \\bar { b } _ n \\} = \\mathbb { N } , \\end{align*}"}
{"id": "4875.png", "formula": "\\begin{align*} z ( z - 1 ) y '' ( z ) = n ( n - 1 ) y ( z ) ; \\end{align*}"}
{"id": "6888.png", "formula": "\\begin{align*} b _ j = \\sum \\limits _ { i = 1 } ^ { j } ( - 1 ) ^ i q ^ { { j - i \\choose 2 } } { j \\choose i } _ q a _ i \\end{align*}"}
{"id": "2306.png", "formula": "\\begin{align*} R f ( x , \\omega ) = \\widehat { V _ f f } ( \\omega , - x ) . \\end{align*}"}
{"id": "3002.png", "formula": "\\begin{align*} \\lhd \\ , T ( x ) , T ( y ) \\ , \\rhd = \\lhd \\ , x , y \\ , \\rhd , \\quad x , y \\in Z _ 2 . \\end{align*}"}
{"id": "7655.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | \\tilde { u } _ n - w | ^ 2 = & \\int _ { B _ R ( \\mathbf { 0 } ) } | \\tilde { u } _ n - w | ^ 2 + \\int _ { B _ R ^ c ( \\mathbf { 0 } ) } | \\tilde { u } _ n - w | ^ 2 \\le \\\\ & \\int _ { B _ R ( \\mathbf { 0 } ) } | \\tilde { u } _ n - w | ^ 2 + 2 \\varepsilon \\le 3 \\varepsilon \\ ; , \\end{align*}"}
{"id": "6438.png", "formula": "\\begin{align*} N _ { k + 1 } = c _ 1 ( N _ k ^ * + 2 ) + c _ 2 ( N _ k ^ * + 4 ) + \\cdots + c _ r ( N _ k ^ * + 2 r ) \\end{align*}"}
{"id": "8903.png", "formula": "\\begin{align*} H ' = \\{ ( \\rho ( x ) , \\varphi ( x ) ) \\mid x \\in A \\} \\end{align*}"}
{"id": "2972.png", "formula": "\\begin{align*} \\Lambda _ n = \\Lambda _ { n , 1 } + \\Lambda _ { n , 2 } , \\end{align*}"}
{"id": "5628.png", "formula": "\\begin{align*} [ L _ { i j } , L _ { h k } ] = a _ { j h } L _ { i k } - a _ { i h } L _ { j k } - a _ { j k } L _ { i h } + a _ { i k } L _ { j h } , \\end{align*}"}
{"id": "4327.png", "formula": "\\begin{align*} \\int ( w ( \\tau _ 0 ) - Q _ b ) \\phi _ { 1 , b , \\beta } \\frac { y ^ 2 } { 2 } \\rho _ \\beta d y = \\left ( \\frac { 2 } { \\beta } \\left ( \\frac { d } { 2 } - \\gamma + 2 \\right ) - \\frac { 1 } { \\beta } \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) \\right ) \\| \\phi _ { 1 , b , \\beta } \\| _ { L ^ 2 _ { \\rho _ \\beta } } ^ { 2 } + O ( b ^ { 1 - \\frac { \\epsilon } { 2 } } ) . \\end{align*}"}
{"id": "5507.png", "formula": "\\begin{align*} \\alpha ( t , x ) = a _ { \\pi ( j ) } ( x ) , ( t , x ) \\in [ s _ { j - 1 } , s _ j ) \\times X . \\end{align*}"}
{"id": "3570.png", "formula": "\\begin{align*} V ( \\alpha ) & = \\Big \\{ { \\sum } _ { i = 1 } ^ m ( 1 - p ^ { - d _ i } ) \\colon \\alpha \\le ( d _ 1 , \\ldots , d _ m ) , \\ d _ 1 = z \\Big \\} , \\\\ \\Gamma _ z & = \\Big \\{ { \\sum } _ { i = 0 } ^ { z - 1 } a _ i ( p ^ z - p ^ i ) \\colon 0 \\le a _ i < p \\Big \\} . \\end{align*}"}
{"id": "5026.png", "formula": "\\begin{align*} B _ { k n } ( x , y ) = \\begin{bmatrix} \\xi _ { k n } ( x ) + E _ { k n } ( x , y ) \\\\ x \\end{bmatrix} \\end{align*}"}
{"id": "6115.png", "formula": "\\begin{align*} w ( x , y ) = \\sum _ { p = 0 } ^ \\infty \\left [ H ^ 1 _ { p } ( x ^ { m + 2 } ) + x H ^ 2 _ { p } ( x ^ { m + 2 } ) + y H ^ 3 _ { p } ( x ^ { m + 2 } ) + x y H ^ 4 _ { p } ( x ^ { m + 2 } ) \\right ] y ^ { p ( n + 2 ) } \\ , , \\end{align*}"}
{"id": "7036.png", "formula": "\\begin{align*} \\delta ^ 2 & \\leq ( 1 + \\eta ) \\sum _ { j = 1 } ^ { \\infty } | \\phi _ j ( \\xi _ k ) | ^ 2 + \\Big ( 1 + \\frac { 1 } { \\eta } \\Big ) c _ k \\frac { | z - \\xi _ k | ^ 2 } { 1 - | z | ^ 2 } \\sum _ { j = 1 } ^ { \\infty } \\| \\phi _ j \\| _ { b } ^ { 2 } \\\\ & \\leq ( 1 + \\eta ) \\sum _ { j = 1 } ^ { \\infty } | \\phi _ j ( \\xi _ k ) | ^ 2 + \\Big ( 1 + \\frac { 1 } { \\eta } \\Big ) c _ k \\frac { | z - \\xi _ k | ^ 2 } { 1 - | z | ^ 2 } . \\end{align*}"}
{"id": "1480.png", "formula": "\\begin{align*} = \\left [ \\begin{array} { c } \\rho ( g _ 1 , g _ 2 ) \\iota ( z _ 1 , z _ 2 ) \\\\ 1 \\end{array} \\right ] \\lambda ( \\rho ( g _ 1 , g _ 2 ) , \\iota ( z _ 1 , z _ 2 ) ) \\times \\end{align*}"}
{"id": "2282.png", "formula": "\\begin{align*} \\pi ( \\l ) = M _ \\omega T _ x , \\l = ( x , \\omega ) \\in \\R ^ { 2 d } \\ , . \\end{align*}"}
{"id": "850.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } T ^ { \\rm R e a c } _ j = \\tau ^ { \\rm R e a c } _ { m } \\mathbb { E } R _ j + \\mathbb { E } \\tau ^ { \\rm R e a c } _ { V _ j } + \\tau _ { \\rm f } , \\\\ & \\mathbb { E } \\left ( T ^ { \\rm R e a c } _ j \\right ) ^ 2 = \\mathbb { E } \\left ( \\tau ^ { \\rm R e a c } _ { m } R _ j + \\tau ^ { \\rm R e a c } _ { V _ j } + \\tau _ { \\rm f } \\right ) ^ 2 . \\\\ \\end{aligned} \\end{align*}"}
{"id": "4765.png", "formula": "\\begin{align*} R ^ 2 + R ^ \\# = \\theta R , \\end{align*}"}
{"id": "4101.png", "formula": "\\begin{align*} G ( X _ t ) & = G ( x _ 0 ) + \\int _ 0 ^ t ( G ' ( X _ s ) \\cdot \\mu ( X _ s ) + \\tfrac { 1 } { 2 } G '' ( X _ s ) \\cdot \\sigma ^ 2 ( X _ s ) ) \\ , d s + \\int _ 0 ^ t G ' ( X _ s ) \\cdot \\sigma ( X _ s ) \\ , d W _ s \\\\ & = G ( x _ 0 ) + \\int _ 0 ^ t \\widetilde \\mu ( G ( X _ s ) ) \\ , d s + \\int _ 0 ^ t \\widetilde \\sigma ( G ( X _ s ) ) \\ , d W _ s . \\end{align*}"}
{"id": "9323.png", "formula": "\\begin{align*} F _ k ^ { \\mu , \\sigma } ( \\mathbf { x } , \\mathbf { y } , \\mathbf { s } ) : = \\begin{bmatrix} ( H + \\rho I ) \\mathbf { x } - A ^ T \\mathbf { y } + \\mathbf { g } - \\rho { \\mathbf { x } _ k } - \\begin{bmatrix} 0 \\\\ \\mathbf { s } \\end{bmatrix} \\\\ A \\mathbf { x } + \\delta ( \\mathbf { y } - \\mathbf { y } _ k ) - \\mathbf { b } \\\\ S X _ { \\mathcal { C } } \\mathbf { e } - \\sigma \\mu \\mathbf { e } \\end{bmatrix} . \\end{align*}"}
{"id": "6443.png", "formula": "\\begin{align*} D _ h ( r u + z , r v + z ) = D _ { h ( r \\cdot + z ) + Q \\log r } ( u , v ) , \\forall u , v \\in \\mathbb C . \\end{align*}"}
{"id": "4281.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\sup _ { \\tau \\in [ \\tau _ 0 , 0 ] } \\left \\| z ^ { ( k ) } ( \\tau , \\cdot ) \\right \\| _ { H ^ 2 ( \\R \\backslash \\{ t , 0 \\} ) } ~ < ~ \\sum ^ { \\infty } _ { k = 1 } \\beta _ k ( \\tau ) ~ < ~ \\infty . \\end{align*}"}
{"id": "1819.png", "formula": "\\begin{align*} G = \\{ u \\rightarrow 2 u v , \\ ; \\ ; v \\rightarrow u \\} . \\end{align*}"}
{"id": "7536.png", "formula": "\\begin{align*} \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } = B - \\frac { 1 } { s - 1 } + \\frac { 1 } { 2 } \\log \\pi - \\frac { \\Gamma ' ( s / 2 + 1 ) } { 2 \\ \\Gamma ( s / 2 + 1 ) } + \\sum _ { \\rho } \\left ( \\frac { 1 } { s - \\rho } + \\frac { 1 } { \\rho } \\right ) \\end{align*}"}
{"id": "624.png", "formula": "\\begin{align*} P _ x ( X ) = \\ A _ 0 ( x ) X ^ k + A _ 1 ( x ) X ^ { k - 1 } + \\cdots + A _ { k - 1 } ( x ) X + A _ k ( x ) ( x = 0 , 1 , 2 , \\ldots ) . \\end{align*}"}
{"id": "5604.png", "formula": "\\begin{align*} \\mathbf { g } _ \\star ( X , Y ) = \\langle X \\star \\langle Y , \\mathbf { g } ^ A \\rangle _ \\star , \\mathbf { g } _ A \\rangle _ \\star . \\end{align*}"}
{"id": "5921.png", "formula": "\\begin{align*} h = \\frac { 4 r ^ 2 } { 8 m - 4 r } . \\end{align*}"}
{"id": "5582.png", "formula": "\\begin{align*} \\| 1 _ A \\| & \\ = \\ \\| z - 1 _ { B \\backslash E } - P _ E ( z ) \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ \\ell \\| z - P _ E ( z ) \\| \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , d k p } \\sigma ^ { \\mathcal { U } _ X , L } _ m ( z ) \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { \\lambda , d k p } \\| 1 _ B \\| . \\end{align*}"}
{"id": "3324.png", "formula": "\\begin{align*} f _ i = g _ i + f _ { i - 1 } \\end{align*}"}
{"id": "998.png", "formula": "\\begin{align*} u _ \\varepsilon ( x ) = \\int _ { \\R ^ n _ + } \\bar u ( y ) \\big ( \\eta _ \\varepsilon ( x - y ) - \\eta _ \\varepsilon ( x _ \\ast - y ) \\big ) \\dd y \\geqslant 0 x \\in \\R ^ n _ + \\end{align*}"}
{"id": "4599.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \\Big ) } { 1 - \\Phi \\left ( x \\right ) } = \\exp \\bigg \\{ \\theta _ 1 C \\frac { \\ln n + x ^ { 3 } } { \\sqrt { n } } \\bigg \\} \\end{align*}"}
{"id": "3216.png", "formula": "\\begin{align*} \\theta _ { P _ H } = \\frac { \\mathbb { E } [ Y ^ 2 ] - \\mathbb { E } [ Y ] ^ 2 } { \\mathbb { E } [ Y ] - \\frac { 1 } { 1 + e ^ { a b } } } . \\end{align*}"}
{"id": "6787.png", "formula": "\\begin{align*} \\tau _ { k _ { 1 } } ^ { - j } \\cdot \\tau _ { k _ { 2 } } ^ { j } = t _ { k _ { 1 } } ^ { k _ { 1 } - j } \\cdot t _ { k _ { 2 } } ^ { j } \\end{align*}"}
{"id": "8530.png", "formula": "\\begin{align*} H \\left ( u \\right ) : = \\int \\frac { 1 } { 2 } \\left | \\partial _ x u \\right | ^ { 2 } + \\frac { 1 } { 2 } V \\left | u \\right | ^ { 2 } \\pm \\frac { 1 } { 4 } \\left | u \\right | ^ { 4 } \\ , d x . \\end{align*}"}
{"id": "502.png", "formula": "\\begin{align*} \\mathfrak { m } _ { \\delta _ { 1 } \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 1 } a _ { 2 } a _ { 3 } } = \\left ( Y ( \\mathcal { I } _ { \\delta _ { 1 } } ^ { a _ { 1 } } ) _ { \\delta _ { 3 } \\delta _ { 2 } } ^ { a _ { 2 } a _ { 3 } } + 1 \\right ) \\cdot \\left ( Y ( \\mathcal { I } ) _ { \\delta _ { 2 } \\delta _ { 1 } } ^ { a _ { 1 } a _ { 2 } } + 1 \\right ) \\cdot \\left ( Y ( \\mathcal { I } ) _ { \\delta _ { 3 } \\delta _ { 1 } } ^ { a _ { 1 } a _ { 3 } } + 1 \\right ) \\end{align*}"}
{"id": "6488.png", "formula": "\\begin{align*} j _ 0 ( \\delta ) & : = \\begin{cases} k & \\mbox { i f $ \\delta = - k $ f o r s o m e $ k \\in \\mathbb { N } $ , } \\\\ 0 & \\mbox { o t h e r w i s e } , \\\\ \\end{cases} \\end{align*}"}
{"id": "8181.png", "formula": "\\begin{align*} M _ 2 ( p , H ) = \\frac { \\pi ^ 2 } { 8 } \\times \\begin{cases} \\frac { a + 1 } { a - 1 } \\times \\left ( 1 - \\frac { d } { p } \\right ) & \\hbox { i f $ a $ i s o d d , } \\\\ 1 - \\frac { ( d - 1 ) a + 1 } { p } & \\hbox { i f $ a $ i s e v e n . } \\\\ \\end{cases} \\end{align*}"}
{"id": "1923.png", "formula": "\\begin{align*} n = \\int _ { \\{ | v | > G \\} } \\frac { G ^ { \\kappa _ 0 } } { G ^ { \\kappa _ 0 } } f \\ , d v + \\int _ { \\{ | v | \\leq G \\} } f \\ , d v \\leq \\frac { 1 } { G ^ { \\kappa _ 0 } } \\int _ { \\mathbb { R } ^ 3 } | v | ^ { \\kappa _ 0 } f \\ , d v + \\| f \\| _ { L ^ \\infty ( \\mathbb { R } ^ 3 ) } ( 2 G ) ^ 3 . \\end{align*}"}
{"id": "210.png", "formula": "\\begin{align*} A _ \\alpha ( p _ \\alpha ) ( x ) = 0 . \\end{align*}"}
{"id": "8717.png", "formula": "\\begin{align*} p ( x ) = \\phi \\bigl ( f _ { 1 } ( x ) , \\ldots , f _ d ( x ) \\bigr ) , \\end{align*}"}
{"id": "1359.png", "formula": "\\begin{align*} & \\left ( \\int _ E | u | ^ { \\frac { d q ^ \\# } { d - q ^ \\# \\zeta } } d x \\right ) ^ { \\frac { d - 1 } { d } } = \\norm { u ^ { \\bar { d } } } _ { L ^ \\frac { d } { d - \\zeta } ( W _ R ) } \\\\ & \\leq C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { \\nabla u ^ { \\bar { d } } } _ { L ^ 1 ( W _ R ) } = \\bar { d } C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { u ^ { \\bar { d } - 1 } \\nabla u } _ { L ^ 1 ( W _ R ) } . \\end{align*}"}
{"id": "4035.png", "formula": "\\begin{align*} p _ k = \\left ( \\frac { \\sqrt { a ^ 2 _ k + b ^ 2 _ k } + a _ k } { 2 } \\right ) ^ { \\frac { 1 } { 2 } } , q _ k = \\left ( \\frac { \\sqrt { a ^ 2 _ k + b ^ 2 _ k } - a _ k } { 2 } \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "2537.png", "formula": "\\begin{align*} \\pi ( \\Phi ) \\pi ( \\xi , \\eta , 0 ) \\pi ( \\Phi ) = \\pi ( \\Phi ) \\pi ( \\Phi ^ { \\xi , \\eta } ) = \\pi ( \\Phi \\natural \\Phi ^ { \\xi , \\eta } ) = e ^ { - \\tfrac { \\pi } { 2 } ( \\xi ^ 2 + \\eta ^ 2 ) } \\pi ( \\Phi ) . \\end{align*}"}
{"id": "8201.png", "formula": "\\begin{align*} N _ 2 ( f , H ) = - f - 4 S ( H , f ) + 8 S ( H _ 2 , 2 f ) = ( - 1 ) ^ { a - 1 } ( 2 a + 1 ) . \\end{align*}"}
{"id": "4830.png", "formula": "\\begin{align*} e ^ { - i t _ 0 P } - e ^ { - i t _ 0 h ^ { - 1 } ( h P - i Q ) } = h ^ { - 1 } \\int _ 0 ^ { t _ 0 } e ^ { - i ( t _ 0 - t ) P } Q e ^ { - i t h ^ { - 1 } ( h P - i Q ) } d t , \\end{align*}"}
{"id": "3803.png", "formula": "\\begin{align*} \\norm { A - A \\mathbb { P } } & = \\norm { A ( I - \\mathbb { P } ) } = \\norm { A ( I - Q _ { W _ k } \\ ! Q _ { W _ k } ^ T ) ( I - \\mathbb { P } ) } \\\\ & \\leq \\norm { A ( I - Q _ { W _ k } \\ ! Q _ { W _ k } ^ T ) } ~ \\norm { I - \\mathbb { P } } . \\end{align*}"}
{"id": "7761.png", "formula": "\\begin{gather*} \\phi [ T ] = ( p , 0 ) . \\end{gather*}"}
{"id": "9044.png", "formula": "\\begin{align*} X ( \\theta ) \\log X ( \\theta ) = & \\theta X ^ 0 \\log X ^ 0 - \\theta ( 1 - \\theta ) ^ 2 \\frac { X ^ 0 } { ( \\tilde X ^ 0 ) ^ 2 } ( X ^ 1 - X ^ 0 ) ^ 2 \\\\ & + ( 1 - \\theta ) X ^ 1 \\log X ^ 1 - \\theta ^ 2 ( 1 - \\theta ) \\frac { X ^ 1 } { ( \\tilde X ^ 1 ) ^ 2 } ( X ^ 1 - X ^ 0 ) ^ 2 , \\end{align*}"}
{"id": "1133.png", "formula": "\\begin{align*} Q ( \\xi , k ) : = \\left \\{ \\begin{aligned} & Q _ 1 = \\sigma _ 1 r ^ { - \\frac { \\sigma _ 3 } { 2 } } D ^ { \\sigma _ 3 } \\sigma _ 1 \\sigma _ 3 , & \\mathbb { C } _ { + } \\cap U _ \\delta ( \\eta ) , \\\\ & Q _ 2 = \\sigma _ 1 ( r ^ * ) ^ { - \\frac { \\sigma _ 3 } { 2 } } D ^ { - \\sigma _ 3 } , & \\mathbb { C } _ { - } \\cap U _ \\delta ( \\eta ) , \\\\ \\end{aligned} \\right . \\end{align*}"}
{"id": "7409.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\rightarrow 0 ^ { + } } \\limsup _ { n \\rightarrow \\infty } \\mathbb { E } _ { \\mu _ n } \\Big [ \\Big | \\int _ { 0 } ^ { t } \\frac { 1 } { n } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\overleftarrow { \\eta } _ { s } ^ { \\ell } ( x ) [ \\eta _ { s } ( x + 1 ) - \\overrightarrow { \\eta } _ { s } ^ { \\varepsilon n } ( x + 1 ) ] d s \\Big | \\Big ] = 0 . \\end{align*}"}
{"id": "6348.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\log x < | x - n | \\leq x \\\\ \\Lambda ( n ) \\neq 0 } } \\frac { 1 } { | x - n | } = \\int _ { \\log x } ^ { x } \\frac { d p ( u ) } { u } = \\left [ \\frac { p ( u ) } { u } \\right ] _ { \\log x } ^ { x } + \\int _ { \\log x } ^ { x } \\frac { p ( u ) } { u ^ 2 } \\ , d u . \\end{align*}"}
{"id": "2388.png", "formula": "\\begin{align*} C _ { \\mathbf { F } _ N } ^ T C _ { \\mathbf { F } _ N } = \\begin{pmatrix} \\frac { N } { 2 } & 0 \\\\ 0 & \\frac { N } { 2 } \\end{pmatrix} , \\end{align*}"}
{"id": "6719.png", "formula": "\\begin{align*} g _ 1 ' ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + g _ r ' ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ r ) ^ { q ^ d } = 0 \\end{align*}"}
{"id": "5483.png", "formula": "\\begin{align*} \\frac { 1 } { t } d _ K \\big ( x + t ( A x + v ) \\big ) & = \\frac { 1 } { t } d _ K \\bigg ( S _ t x + t v + t \\bigg ( A x - \\frac { S _ t x - x } { t } \\bigg ) \\bigg ) \\\\ & \\leq \\frac { 1 } { t } d _ K ( S _ t x + t v ) + \\bigg \\| A x - \\frac { S _ t x - x } { t } \\bigg \\| , \\end{align*}"}
{"id": "3539.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs { \\Sigma _ 2 ( s _ 1 , s _ 2 , s _ 3 ) E ( s _ 1 , s _ 2 , s _ 3 ) } d t _ 3 \\\\ & \\ll \\begin{cases} T ^ { \\frac { 5 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\frac { 3 } { 2 } < \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 < 2 ) \\\\ ( T \\log T ) ^ \\frac { 1 } { 2 } & ( \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 = 2 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "817.png", "formula": "\\begin{align*} \\mu _ \\omega ( A ) = \\int _ A \\omega \\ , d \\mu _ X \\end{align*}"}
{"id": "1844.png", "formula": "\\begin{align*} D ^ n ( y ) = M _ n ( x , y ) . \\end{align*}"}
{"id": "4406.png", "formula": "\\begin{align*} \\tilde \\phi _ { i , \\beta } ( y ) = \\left \\{ \\begin{array} { r c l } & & K _ 0 y ^ { - \\tilde \\gamma } ( 1 + O ( y ^ 2 ) ) y \\to 0 , \\\\ & & K _ { \\infty } y ^ { 2 i - \\gamma } ( \\ln y + O ( 1 ) ) y \\to + \\infty , \\end{array} \\right . \\end{align*}"}
{"id": "9296.png", "formula": "\\begin{align*} \\phi _ { i j k } ^ { - 1 } \\tilde { F } _ i \\phi _ { i j k } ( y ) & = ( \\mathrm { i d } - p \\psi _ { i j k } ) \\tilde { F } _ i ( \\mathrm { i d } + p \\psi _ { i j k } ) ( y ) \\\\ & = ( \\mathrm { i d } - p \\psi _ { i j k } ) ( \\tilde { F } _ i ( y ) + p \\psi _ { i j k } ( y ) ^ p ) \\\\ & = \\tilde { F } _ i ( y ) - p \\psi _ { i j k } ( y ^ p ) + p \\psi _ { i j k } ( y ) ^ p \\\\ & = \\tilde { F } _ i ( y ) + p \\psi _ { i j k } ( y ) ^ p , \\end{align*}"}
{"id": "4684.png", "formula": "\\begin{align*} \\mathcal { L } A _ { i j , 0 } = p \\kappa _ 0 \\sigma _ j Q ^ { p - 1 } , \\end{align*}"}
{"id": "3940.png", "formula": "\\begin{align*} \\partial _ t \\gamma _ T ^ { s \\to t } \\circ \\alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) & = \\gamma _ T ^ { s \\to t } ( - \\i [ \\widetilde { H } _ t , \\alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) ] ) + \\gamma _ T ^ { s \\to t } ( T \\alpha _ { V ( t ) } ^ { ( t - s ) T } \\circ \\delta _ { V ( t ) } ( A ) + \\i [ \\widetilde { H } _ t , \\alpha _ { V ( t ) } ^ { ( t - s ) T } ( A ) ] ) \\\\ & = T \\gamma _ T ^ { s \\to t } \\circ \\alpha _ { V ( t ) } ^ { ( t - s ) T } \\circ \\delta _ { V ( t ) } ( A ) . \\end{align*}"}
{"id": "1547.png", "formula": "\\begin{align*} \\widetilde { c } _ k ( s ) = \\alpha ( s ) \\pi ^ { \\frac { n ( n - 1 ) } { 2 } } \\frac { \\Gamma ( 2 \\lambda + 1 ) \\Gamma ( 2 \\lambda + 3 ) \\ldots \\Gamma ( 2 \\lambda + 2 n - 3 ) } { \\Gamma ( 2 \\lambda + n ) \\Gamma ( 2 \\lambda + n + 1 ) \\ldots \\Gamma ( 2 \\lambda + 2 n - 2 ) } , \\end{align*}"}
{"id": "1442.png", "formula": "\\begin{align*} \\mathfrak { i } : M _ n ( \\mathbb { B } ) \\stackrel { \\sim } \\longrightarrow \\{ x \\in M _ { 2 n } ( \\mathbb { K } ) : \\overline { x } I _ n ' J _ n ' = I _ n ' J _ n ' x \\} . \\end{align*}"}
{"id": "1632.png", "formula": "\\begin{align*} \\left \\langle f , ( \\lambda - \\mathcal { L } ^ { ( p ) } ) \\varphi \\right \\rangle = 0 , \\varphi \\in \\mathcal { A } ( M \\setminus K ) , \\end{align*}"}
{"id": "1290.png", "formula": "\\begin{align*} \\Lambda _ { ( n , \\infty ) } ( M , \\Gamma ) : = \\bigcup _ { m = 0 } ^ { \\infty } \\Lambda _ { ( n , m ) } ( M , \\Gamma ) \\end{align*}"}
{"id": "8598.png", "formula": "\\begin{align*} \\prod _ \\ast b _ j ( x , k _ j ) : = \\overline { b _ 1 ( x , k _ 1 ) } b _ 2 ( x , k _ 2 ) \\overline { b _ 3 ( x , k _ 3 ) } b _ 4 ( x , k _ 4 ) . \\end{align*}"}
{"id": "6502.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ n = \\frac { M ^ { ( 2 ) } _ 1 } { n } \\prod ^ { n - 1 } _ { j = 1 } \\left ( 1 + \\frac { 2 \\alpha } { j } \\right ) = - \\frac { \\Gamma ( n + 2 \\alpha ) } { \\Gamma ( n + 1 ) \\Gamma ( 2 \\alpha ) } \\sim - \\frac { n ^ { - ( 1 - 2 \\alpha ) } } { \\Gamma ( 2 \\alpha ) } , \\end{align*}"}
{"id": "6898.png", "formula": "\\begin{align*} \\alpha _ j = \\frac { ( q ^ { d - i - j } - 1 ) \\ldots ( q ^ { d - k + 1 - i } - 1 ) } { ( q ^ { d - j } - 1 ) \\ldots ( q ^ { d - k + 1 } - 1 ) } = q ^ { - i ( k - j ) } + o _ { q , d } ( 1 ) . \\end{align*}"}
{"id": "7371.png", "formula": "\\begin{align*} G _ \\beta ( r , p , X , A ) = F ( p , X , A ) \\end{align*}"}
{"id": "9360.png", "formula": "\\begin{align*} \\frac { 2 } { 2 - t } ( 1 - t ) ^ { x } & = \\sum _ { k = 0 } ^ { \\infty } \\mathcal { E } _ { k , \\lambda } ( x ) \\frac { 1 } { k ! } \\big ( \\log _ { \\lambda } ( 1 - t ) \\big ) ^ { k } \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\mathcal { E } _ { k , \\lambda } ( x ) \\sum _ { n = k } ^ { \\infty } S _ { 1 , \\lambda } ( n , k ) \\frac { ( - 1 ) ^ { n } t ^ { n } } { n ! } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { n } S _ { 1 , \\lambda } ( n , k ) \\mathcal { E } _ { k , \\lambda } ( x ) \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "2109.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) = - ( 1 - c ) ^ k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l ( \\frac c { 1 - c } ) ^ l \\frac l { k - l } + \\\\ & k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l c ^ l \\int _ 0 ^ { 1 - c } \\frac { y ^ { k - l - 1 } } { 1 - y ^ k } d y - c ^ k \\log ( 1 - ( 1 - c ) ^ k ) . \\end{aligned} \\end{align*}"}
{"id": "597.png", "formula": "\\begin{align*} ( x , y ) \\ = \\ \\abs { x - y } \\end{align*}"}
{"id": "9211.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , x ^ X , p ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land p = _ X J ^ A _ { \\gamma } x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\right ) \\end{align*}"}
{"id": "6794.png", "formula": "\\begin{align*} \\nabla l = \\nabla f _ { 1 : m } + P ^ T \\nabla f _ { m + 1 : n } . \\end{align*}"}
{"id": "3906.png", "formula": "\\begin{align*} \\delta _ f ( z ) = \\lim _ { x \\to \\infty } \\frac { \\pi _ { f } ( x , z ) } { \\pi ( x ) } = c _ f ( z ) a _ 2 , \\end{align*}"}
{"id": "7709.png", "formula": "\\begin{align*} z = u _ 1 \\cdot \\ldots \\cdot u _ k v _ 1 \\cdot \\ldots \\cdot v _ { \\ell } z ' = u _ 1 \\cdot \\ldots \\cdot u _ k w _ 1 \\cdot \\ldots \\cdot w _ { m } \\ , , \\end{align*}"}
{"id": "5417.png", "formula": "\\begin{align*} ( a + 1 ) a ' & = a ' + a a ' = 1 , \\\\ ( b + 1 ) b ' & = b ' + b b ' = \\tfrac { \\sin \\alpha } { \\sin \\gamma } , \\\\ ( c + 1 ) c ' & = c ' + c c ' = \\tfrac { \\sin \\beta } { \\sin \\gamma } . \\end{align*}"}
{"id": "2841.png", "formula": "\\begin{align*} M [ u ] = M [ Q ] , E [ u ] = E [ Q ] , \\| \\nabla u \\| _ 2 < \\| \\nabla Q \\| _ 2 , \\end{align*}"}
{"id": "2879.png", "formula": "\\begin{align*} \\widetilde { h } ( t ) = h ( t ) - A e ^ { - e _ 0 t } \\mathcal { Y } _ + \\end{align*}"}
{"id": "3094.png", "formula": "\\begin{align*} z _ 1 ^ \\prime = z _ 2 \\ , , z _ 2 ^ \\prime = z _ 1 \\ , , z _ 3 ^ \\prime = z _ 4 \\ , , z _ 4 ^ \\prime = z _ 3 \\ , , z _ 5 ^ \\prime = z _ 5 \\ , . \\end{align*}"}
{"id": "3958.png", "formula": "\\begin{align*} \\begin{aligned} \\xi ^ { \\prime } ( x ) + \\eta ^ { \\prime } ( x ) & = \\lambda \\xi ( x ) , x \\in ( 0 , 1 ) , \\\\ \\eta ^ { \\prime \\prime } ( x ) + \\eta ^ { \\prime } ( x ) + \\xi ^ { \\prime } ( x ) & = \\lambda \\eta ( x ) , x \\in ( 0 , 1 ) , \\\\ \\xi ( 0 ) & = \\xi ( 1 ) , \\\\ \\eta ( 0 ) = 0 , \\ \\ \\eta ( 1 ) & = 0 . \\end{aligned} \\end{align*}"}
{"id": "8065.png", "formula": "\\begin{align*} T _ \\Sigma ( f ) [ \\psi ] : = \\frac { 1 } { 2 } \\int _ \\mathcal { I } f ( s ) \\psi ^ 2 ( s ) \\sqrt { \\gamma ' ( s ) } \\mathrm { d } V _ \\Sigma ( s ) . \\end{align*}"}
{"id": "6178.png", "formula": "\\begin{align*} x _ { \\rm T L S } = - \\frac { v _ { 1 2 } } { v _ { 2 2 } } , \\ \\ \\left [ \\begin{array} { c } x _ { \\rm T L S } \\\\ - 1 \\\\ \\end{array} \\right ] = - \\frac { v _ { n + 1 } } { v _ { 2 2 } } . \\end{align*}"}
{"id": "213.png", "formula": "\\begin{align*} D _ k ^ { \\alpha - 1 } ( p _ \\alpha f ) ( x ) = p _ \\alpha ( x ) D _ k ^ { \\alpha - 1 } ( f ) ( x ) + f ( x ) D _ k ^ { \\alpha - 1 } ( p _ \\alpha ) ( x ) + R _ k ^ \\alpha ( p _ \\alpha , f ) ( x ) . \\end{align*}"}
{"id": "4399.png", "formula": "\\begin{align*} T _ { i + 1 } = H ^ { - 1 } T _ i , T _ 0 = c \\Lambda _ \\xi Q \\end{align*}"}
{"id": "9327.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { \\mathbf { x } \\in \\mathbb { R } ^ { d _ 1 } } \\ ; & \\frac { 1 } { 2 } \\mathbf { x } ^ T H \\mathbf { x } + \\mathbf { g } ^ T \\mathbf { x } \\\\ \\hbox { s . t . } \\ ; & A \\mathbf { x } = \\mathbf { b } , \\ ; \\mathbf { x } - \\mathbf { z } = 0 \\\\ & \\mathbf { z } \\geq 0 \\\\ \\end{aligned} \\end{align*}"}
{"id": "4622.png", "formula": "\\begin{gather*} ( X , \\alpha , \\beta ) \\otimes ( X ' , \\alpha ' , \\beta ' ) = ( X \\otimes X ' , \\alpha '' , \\beta '' ) , \\end{gather*}"}
{"id": "6034.png", "formula": "\\begin{align*} ( n + 1 ) \\widetilde { L } ' _ { n + 1 } - \\widetilde { L } _ n + ( \\alpha - 1 - x - 2 n ) \\widetilde { L } ' _ n + ( n - \\alpha ) \\widetilde { L } ' _ { n - 1 } = 0 . \\end{align*}"}
{"id": "5070.png", "formula": "\\begin{align*} f ( y ) & = d ( y , K ) \\\\ & \\leq d ( y , z ) \\\\ & = d ( x , z ) - d ( x , y ) \\\\ & \\leq d ( x , K ) + \\epsilon r - d ( x , y ) \\\\ & = f ( x ) + \\epsilon r - d ( x , y ) . \\end{align*}"}
{"id": "6017.png", "formula": "\\begin{align*} \\begin{aligned} \\Big ( \\frac { d } { d x } + x \\Big ) ^ \\alpha u ( x ) & = e ^ { - x ^ 2 / 2 } \\Big ( \\frac { d } { d x } \\Big ) ^ \\alpha \\big [ e ^ { ( \\cdot ) ^ 2 / 2 } u ( \\cdot ) \\big ] ( x ) \\\\ \\Big ( - \\frac { d } { d x } + x \\Big ) ^ \\alpha v ( x ) & = e ^ { x ^ 2 / 2 } \\Big ( - \\frac { d } { d x } \\Big ) ^ \\alpha \\big [ e ^ { - ( \\cdot ) ^ 2 / 2 } v ( \\cdot ) \\big ] ( x ) . \\end{aligned} \\end{align*}"}
{"id": "3712.png", "formula": "\\begin{align*} & \\| B ^ k \\| _ { L ^ { 2 } ( 0 , T ; H ^ { \\frac 5 2 - \\frac { \\alpha } { 2 } } ) } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } + C \\| B ^ { k - 1 } \\| _ { L ^ { \\frac { \\alpha } { ( s + \\alpha - \\frac 5 2 ) } } ( 0 , T ; H ^ s ) } \\| B ^ { k } \\| _ { L ^ { \\frac { \\alpha } { ( s + \\alpha - \\frac 5 2 ) } } ( 0 , T ; H ^ s ) } \\end{align*}"}
{"id": "4693.png", "formula": "\\begin{align*} L _ A ( t , y ) = & - \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\bigg ( \\frac { \\dot { x } _ i ( t ) \\partial _ y A _ { i j } ( t , y - x _ i ( t ) ) } { x ^ 2 _ { i j } ( t ) } + \\frac { 2 \\dot { x } _ { i j } ( t ) A _ { i j } ( t , y - x _ { i } ( t ) ) } { x ^ 3 _ { i j } ( t ) } \\bigg ) , \\\\ & + \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\frac { ( \\partial _ t \\vec { x } ( t ) , \\partial _ t \\vec { \\mu } ( t ) ) \\cdot \\nabla _ { \\vec { x } , \\vec { \\mu } } \\mathfrak { A } _ { i j } ( \\vec { x } ( t ) , \\vec { \\mu } ( t ) , y - x _ i ( t ) ) } { x ^ 2 _ { i j } ( t ) } , \\end{align*}"}
{"id": "6855.png", "formula": "\\begin{align*} C _ k = V ^ 0 _ k \\oplus \\ldots \\oplus V ^ k _ k . \\end{align*}"}
{"id": "4183.png", "formula": "\\begin{align*} | h ' _ i ( y ) | & = | h ' _ i ( y ) - h ' _ i ( x ) | \\\\ & = \\left | \\int _ 0 ^ { d _ { L _ 0 } ( x , y ) } ( d h ' _ i ) _ { \\gamma ( t ) } \\left ( \\dot { \\gamma } ( t ) \\right ) d t \\right | \\\\ & \\leq | | d h ' _ i | | \\int _ 0 ^ { d _ { L _ 0 } ( x , y ) } | \\dot { \\gamma } ( t ) | d t \\\\ & \\leq \\mathrm { D i a m } ( L _ 0 ) | | d h ' _ i | | , \\end{align*}"}
{"id": "2101.png", "formula": "\\begin{align*} ( \\xi _ 0 , \\cdots , \\xi _ n ) \\bullet [ x _ 0 : \\dots : x _ n ] = \\left [ \\xi _ 0 x _ 0 : \\dots : \\xi _ n x _ n \\right ] . \\end{align*}"}
{"id": "2616.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { M } } = \\inf \\sum _ { \\gamma \\in \\Gamma } | c _ \\gamma | , \\end{align*}"}
{"id": "4299.png", "formula": "\\begin{align*} u ( r , t ) = \\frac { 1 } { \\mu ( t ) } w ( y , \\tau ) , y = \\frac { r } { \\sqrt { \\mu ( t ) } } \\frac { d \\tau } { d t } = \\frac { 1 } { \\mu ( t ) } . \\end{align*}"}
{"id": "4993.png", "formula": "\\begin{align*} \\mathcal { R } ( Z ) = \\Pi \\tilde { \\mathcal { R } } ( Z ) . \\end{align*}"}
{"id": "7853.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { i = n + 1 } ^ { \\infty } \\frac { x _ { 2 i } w _ { 2 i - 1 } \\cdots w _ { 2 i - 2 n } } { x _ { 2 n } w _ { 2 n - 1 } \\cdots w _ { 0 } } e _ { 2 i - 2 n } \\bigg \\| ^ { 2 } & = \\sum _ { i = n + 1 } ^ { \\infty } \\bigg | \\frac { x _ { 2 i } w _ { 2 i - 1 } \\cdots w _ { 2 i - 2 n } } { x _ { 2 n } w _ { 2 n - 1 } \\cdots w _ { 0 } } \\bigg | ^ { 2 } \\\\ & \\leq \\sum _ { i = n + 1 } ^ { \\infty } \\bigg | \\frac { x _ { 2 i } } { x _ { 2 n } } \\bigg | ^ { 2 } \\bigg ( \\frac { w _ { 2 i - 1 } } { w _ { 0 } } \\bigg ) ^ { 2 } . \\end{align*}"}
{"id": "5503.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\eta } _ s ( t ) & = & A \\eta _ s ( t ) + a ( \\eta _ s ( t ) ) \\\\ \\eta _ s ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "2976.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 1 , 1 ) } = \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 1 , 3 ) } = \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 3 , 3 ) } = 1 , \\\\ & \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 1 , 2 ) } = \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 2 , 2 ) } = \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } ^ { ( 2 , 3 ) } = 0 . \\end{align*}"}
{"id": "3170.png", "formula": "\\begin{align*} \\varphi _ R ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } , \\zeta _ { k - 1 } + 1 ) . \\end{align*}"}
{"id": "3425.png", "formula": "\\begin{align*} & q ( x ) - q ( x ' ) \\\\ & = \\int _ { \\R ^ N } [ K ( x , y ) - K ( x ' , y ) ] \\eta ( y ) [ f ( y ) - f ( x ) ] d \\omega ( y ) + [ f ( x ) - f ( x ' ) ] \\int _ { \\R ^ N } K ( x ' , y ) \\xi ( y ) d \\omega ( y ) \\\\ & : = I + I \\ ! I . \\end{align*}"}
{"id": "5670.png", "formula": "\\begin{align*} J ^ r ( x \\cdot y ) \\cdot J ^ s x = J ^ t y \\end{align*}"}
{"id": "1792.png", "formula": "\\begin{align*} E _ { X / A N } = M \\times _ { M \\cap K } \\big ( S _ { \\mathfrak { p } \\cap \\mathfrak { m } } \\otimes S _ { \\mathfrak { k } / ( \\mathfrak { k } \\cap \\mathfrak { m } ) } \\otimes W | _ S \\big ) \\cong M \\times _ { M \\cap K } \\big ( S _ { \\mathfrak { p } \\cap \\mathfrak { m } } \\otimes \\widetilde { W } \\big ) , \\end{align*}"}
{"id": "7476.png", "formula": "\\begin{align*} \\tilde { \\phi } ^ { n + 1 } = \\phi ^ n - \\frac { 2 \\tau ^ 2 } { 2 + \\tau \\eta ^ n } \\Big ( G ( \\phi ^ n ) - \\lambda ^ n \\phi ^ n \\Big ) + \\frac { 2 - \\tau \\eta ^ n } { 2 + \\tau \\eta ^ n } \\Big ( \\phi ^ n - \\phi ^ { n - 1 } \\Big ) . \\end{align*}"}
{"id": "7933.png", "formula": "\\begin{align*} \\sum _ { a \\in S } B _ \\ell ( a ) & = \\left ( B _ \\ell ( n + 1 ) - \\frac { \\ell } { \\ell + 1 } \\frac { B _ { \\ell + 1 } ( n + 1 ) } { n } \\right ) t + \\frac { \\ell t } { ( \\ell + 1 ) n } B _ { \\ell + 1 } - \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } E _ k . \\end{align*}"}
{"id": "8473.png", "formula": "\\begin{align*} k n ( 1 - \\epsilon ) < d ( z ) + e ( S _ 1 , L _ 1 \\cup L _ 2 ) + \\frac { \\epsilon ^ 2 n } { 2 } = e ( S _ 1 , L ) + \\frac { \\epsilon ^ 2 n } { 2 } . \\end{align*}"}
{"id": "46.png", "formula": "\\begin{align*} \\left \\| \\frac { \\ 6 } { \\ 6 x } \\right \\| = \\left \\| \\frac { \\ 6 } { \\ 6 y } \\right \\| = 1 , \\end{align*}"}
{"id": "8438.png", "formula": "\\begin{gather*} \\lim _ { x \\downarrow 0 } \\frac { \\log \\left ( 1 - x + \\frac { c \\alpha } { ( 2 - \\alpha ) \\sigma ^ 2 } \\ , x ^ { 2 / \\alpha } \\right ) + x } { x ^ { 2 / \\alpha } } = \\frac { c \\alpha } { ( 2 - \\alpha ) \\sigma ^ 2 } . \\end{gather*}"}
{"id": "4044.png", "formula": "\\begin{align*} \\begin{cases} u _ { t } - u _ { x x } = 0 \\quad \\mbox { i n } ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u ( t , 0 ) = u ( t , 1 ) = 0 , \\quad \\ , t \\in ( 0 , T ) , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) , x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
{"id": "1204.png", "formula": "\\begin{align*} { m ( E _ { m } ^ { [ \\alpha , \\beta ] , \\varepsilon } ) = m ( \\left \\{ x : \\underline { \\dim } _ { \\mathrm { l o c } } ( m , x ) \\in [ \\alpha , \\beta ] \\right \\} ) } . \\end{align*}"}
{"id": "2919.png", "formula": "\\begin{align*} T _ n ( k ) = \\sum \\limits _ { \\substack { A \\subset \\{ 1 , \\dots , d \\} \\\\ | A | = k } } S _ { n , A } ^ { \\rm M } . \\end{align*}"}
{"id": "4912.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } Z _ t \\leq \\frac { 3 } { 4 } \\lambda \\sum _ { t = 1 } ^ { T } Y _ t ^ 2 + \\frac { 1 } { \\lambda } \\log ( 1 / \\delta ) \\end{align*}"}
{"id": "3297.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ m = E \\end{align*}"}
{"id": "1998.png", "formula": "\\begin{align*} x _ { 2 k - 1 } ( t ) = \\omega ^ { ( k - 1 ) } x _ 1 ( t ) , \\ x _ { 2 k } ( t ) = \\omega ^ { ( k - 1 ) } x _ 2 ( t ) , \\end{align*}"}
{"id": "3189.png", "formula": "\\begin{align*} \\Gamma = \\left \\lbrace 1 - \\sup _ { \\lbrace \\cup _ { j = 1 } ^ s U _ j : U _ j \\in \\mathfrak { U } _ { i _ j } \\rbrace } \\min _ { H \\in \\mathcal { H } } \\det ( H ^ \\intercal H ) : \\lbrace \\mathfrak { U } _ { i _ 1 } , \\ldots , \\mathfrak { U } _ { i _ s } \\rbrace \\in \\mathfrak { P } , ~ s = 1 , \\ldots , r \\right \\rbrace . \\end{align*}"}
{"id": "2779.png", "formula": "\\begin{align*} \\Phi ( h ) = \\frac { 1 } { 2 } \\int ( L _ + h _ 1 ) h _ 1 d x + \\frac { 1 } { 2 } \\int ( L _ - h _ 2 ) h _ 2 d x . \\end{align*}"}
{"id": "4796.png", "formula": "\\begin{align*} ( \\chi _ A * \\chi _ B ) ( g ) = | A \\cap ( g - B ) | . \\end{align*}"}
{"id": "3850.png", "formula": "\\begin{align*} p ' _ { k , j } + p _ { k , j - 1 } + ( \\tilde f _ { k , j } - \\tilde f _ { k , m + 1 } \\tilde f _ { m , j } ) = p _ { k + 1 , j } + ( f _ { k , j } - f _ { k , m + 1 } f _ { m , j } ) , k = \\overline { m + 1 , 2 m } , \\ : j = \\overline { 1 , m } . \\end{align*}"}
{"id": "3540.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs { \\zeta _ { M T , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\zeta _ { M T , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + \\begin{cases} O ( T ^ { \\frac { 5 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) & ( \\frac { 3 } { 2 } < \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 < 2 ) \\\\ O ( ( T \\log T ) ^ \\frac { 1 } { 2 } ) & ( \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 = 2 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "161.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\alpha , 1 } ( f ) ( x ) = x f '' ( x ) + ( \\alpha - x ) f ' ( x ) . \\end{align*}"}
{"id": "959.png", "formula": "\\begin{align*} u ( 0 ) = \\gamma _ { n , s } \\int _ { \\R ^ n \\setminus B _ r } \\frac { r ^ { 2 s } u ( y ) } { ( \\vert y \\vert ^ 2 - r ^ 2 ) ^ s \\vert y \\vert ^ n } \\dd y r \\in ( 0 , 1 ] \\end{align*}"}
{"id": "3689.png", "formula": "\\begin{align*} \\int B ( t , x ) \\ , d x = 0 . \\end{align*}"}
{"id": "3926.png", "formula": "\\begin{align*} \\dot { a } _ j ( 0 ) = \\dot { a } _ { j , 0 } ( j = 1 , \\ldots , N ) , \\dot { b } _ j ( 0 ) = \\dot { b } _ { j , 0 } ( j = 1 , \\ldots , M ) \\end{align*}"}
{"id": "5951.png", "formula": "\\begin{align*} r _ \\nu ( x , y ) = 2 ^ { 1 / 2 } \\sin ( 2 \\pi \\nu y ) \\ , e ^ { 2 \\pi i n _ \\nu x } , \\end{align*}"}
{"id": "5457.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\cdot \\frac { d } { d t } \\int _ { \\Omega } u ^ { - p } ( t , x ) = & - ( p + 1 ) \\int _ { \\Omega } u ^ { - p - 2 } ( t , x ) | \\nabla u ( t , x ) | ^ 2 + ( p + 1 ) \\chi \\int _ { \\Omega } \\frac { u ^ { - p - 1 } ( t , x ) } { v ( t , x ) } \\nabla u ( t , x ) \\cdot \\nabla v ( t , x ) \\\\ & - \\int _ { \\Omega } a ( t , x ) u ^ { - p } ( t , x ) + \\int _ { \\Omega } b ( t , x ) u ^ { - p + 1 } ( t , x ) \\forall \\ , t > s . \\end{align*}"}
{"id": "1869.png", "formula": "\\begin{align*} G = \\{ a \\rightarrow a v , \\ ; \\ ; v \\rightarrow u , \\ ; \\ ; u \\rightarrow 2 u v \\} . \\end{align*}"}
{"id": "8239.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ 1 } [ z ^ n ] N _ 1 \\Big | _ { x _ 1 = x _ 2 = 1 } & = \\frac { 1 } { n } [ t ^ { n - 1 } ] \\left ( 1 + ( 3 n - 7 ) t - ( 9 n - 8 ) t ^ 2 \\right ) ( 1 - t ) ^ { 1 - 4 n } \\\\ & = \\frac { 1 } { n } \\left [ \\binom { 5 n - 3 } { n - 1 } + ( 3 n - 7 ) \\binom { 5 n - 4 } { n - 2 } - ( 9 n - 8 ) \\binom { 5 n - 5 } { n - 3 } \\right ] \\\\ & = \\frac { 2 ( 3 n ^ 2 - n - 1 ) } { ( n - 1 ) ( n - 2 ) } \\binom { 5 n - 5 } { n - 3 } . \\end{align*}"}
{"id": "6162.png", "formula": "\\begin{align*} z _ 1 & = \\frac { - 5 + \\sqrt { - 1 1 } } { 6 } , \\\\ z _ 2 & = \\frac { - 5 - \\sqrt { - 1 1 } } { 6 } \\end{align*}"}
{"id": "2986.png", "formula": "\\begin{align*} \\int _ { K _ 1 } ( g \\circ \\varphi ) d \\mu = \\int _ { K _ 2 } g d ( \\Phi ^ * \\mu ) . \\end{align*}"}
{"id": "2550.png", "formula": "\\begin{align*} \\mu ( S _ 1 S _ 2 ) ^ { - 1 } \\mu ( S _ 1 ) \\mu ( S _ 2 ) \\ , \\rho ( x , \\omega ) \\ , \\mu ( S _ 2 ) ^ { - 1 } \\mu ( S _ 1 ) ^ { - 1 } \\mu ( S _ 1 S _ 2 ) = \\rho ( ( S _ 1 S _ 2 ) ^ { - 1 } S _ 1 S _ 2 ( x , \\omega ) ) = \\rho ( x , \\omega ) . \\end{align*}"}
{"id": "8267.png", "formula": "\\begin{align*} \\mathbf { M } _ { u } : = \\sum _ { u \\leq v } \\mu _ { \\mathfrak { B } _ n } ( u , v ) \\mathbf { F } _ v , \\mathbf { F } _ { u } = \\sum _ { u \\leq v } \\mathbf { M } _ { v } . \\end{align*}"}
{"id": "4120.png", "formula": "\\begin{align*} \\pi ( p ) = \\sum _ { i = 1 } ^ d c _ i \\pi ( p _ i ) h ( p ) > \\sum _ { i = 1 } ^ d c _ i h ( p _ i ) \\end{align*}"}
{"id": "8788.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { t \\notin T _ { i k } } \\lambda _ { i t } \\leq \\delta _ { i k } , \\ \\sum _ { t \\notin \\bar { T } _ { i k } } \\lambda _ { i t } \\leq 1 - \\delta _ { i k } i \\in \\{ 1 , \\ldots , d \\} \\ , k \\in \\{ 1 , \\ldots , \\lceil \\log _ 2 l _ i \\rceil \\} , \\\\ & \\lambda \\in \\Lambda , \\ \\delta \\in \\{ 0 , 1 \\} ^ { \\sum _ { i = 1 } ^ d \\lceil \\log _ 2 l _ i \\rceil } , \\end{aligned} \\end{align*}"}
{"id": "1208.png", "formula": "\\begin{align*} s _ k = \\min \\left \\{ s ( \\mu , \\mathcal { B } , \\mathcal { U } ) , \\alpha \\right \\} - \\varepsilon _ { k } . \\end{align*}"}
{"id": "6439.png", "formula": "\\begin{align*} \\sigma _ \\gamma ( p \\otimes q ) = v ^ * ( \\varphi _ 1 ( p ) \\otimes \\varphi _ 2 ( q ) ) v \\end{align*}"}
{"id": "6580.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) = \\mathcal { L } ( h , k ) + \\mathcal { D } ( h , k ) + \\mathcal { U } ( h , k ) , \\end{align*}"}
{"id": "2012.png", "formula": "\\begin{align*} e _ { A } ( t ) : = \\exp ( A _ t ) , t \\ge 0 . \\end{align*}"}
{"id": "6739.png", "formula": "\\begin{align*} f _ 1 L i _ { K , \\mathfrak { s } } ( \\boldsymbol { \\alpha } ) + f _ 2 L i _ { K , w } ( \\beta ) = 0 . \\end{align*}"}
{"id": "3975.png", "formula": "\\begin{align*} - z ^ 3 - \\lambda z ^ 2 + 2 \\lambda z + \\lambda ^ 2 = 0 . \\end{align*}"}
{"id": "1277.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { c _ { \\sigma _ { k } } ( Y , \\lambda ) ^ { 2 } } { I ( \\sigma _ { k } ) } = \\mathrm { V o l } ( Y , \\lambda ) . \\end{align*}"}
{"id": "3586.png", "formula": "\\begin{align*} & B : = \\{ t \\in \\rho A : | z _ 0 ^ * ( f _ k ( t ) ) - z _ 0 ^ * ( f _ k ( t _ 0 ) ) | < 1 , \\ , k = 1 , \\ldots , n \\} , \\\\ & D : = \\{ z ^ * \\in \\Gamma _ { Z } : | z ^ * ( f _ k ( t _ 0 ) ) - z _ 0 ^ * ( f _ k ( t _ 0 ) ) | < 1 , \\ , k = 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "5655.png", "formula": "\\begin{align*} \\varphi = ( ( 1 \\ , 2 ) , ( \\alpha _ 2 ^ { - 1 } , \\varepsilon , \\alpha _ 3 ^ { - 1 } ) ) \\in \\prescript { ( 1 \\ , 2 ) } { } N _ l ^ A . \\end{align*}"}
{"id": "6335.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k d _ \\ell \\leq k ( k - 1 ) + \\sum _ { \\ell = k + 1 } ^ n \\min ( k , d _ \\ell ) \\ ; . \\end{align*}"}
{"id": "8526.png", "formula": "\\begin{align*} L ( b _ { d _ c } ) & = \\left ( \\prod _ { i = 1 } ^ { d _ c - 1 } \\operatorname { s g n } \\psi _ i \\right ) 2 \\tanh ^ { - 1 } \\exp \\left ( \\sum _ { i = 1 } ^ { d _ c - 1 } \\log | \\psi _ i | \\right ) . \\end{align*}"}
{"id": "8513.png", "formula": "\\begin{align*} [ k a , b ] _ + & = [ ( k - 1 ) a + a , b ] _ + \\\\ & = ( k - 1 ) a + [ a , b ] _ + - ( k - 1 ) a + [ ( k - 1 ) a , b ] _ + \\end{align*}"}
{"id": "8594.png", "formula": "\\begin{align*} & \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } e ^ { i k x } \\partial _ { x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k = I _ { 1 } ( t , x ) + I _ { 2 } ( t , x ) , \\\\ & I _ 1 ( t , x ) : = \\int \\mathbf { 1 } _ { \\{ 0 \\leq k \\leq \\frac { 1 } { \\sqrt { t } } \\} } e ^ { i k x } \\partial _ { x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k , \\\\ & I _ 2 ( t , x ) : = \\int \\mathbf { 1 } _ { \\{ k \\geq \\frac { 1 } { \\sqrt { t } } \\} } e ^ { i k x } \\partial _ { x } \\mathfrak { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k . \\end{align*}"}
{"id": "3159.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { Q _ s \\in \\mathfrak { Q } _ s , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) \\geq \\sup _ { Q _ s \\in \\mathfrak { Q } _ s ' , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) , \\end{aligned} \\end{align*}"}
{"id": "5175.png", "formula": "\\begin{align*} Z _ { \\ell i } ^ \\pi = w _ { \\ell i } ^ \\pi + d _ { \\ell i } , \\end{align*}"}
{"id": "2939.png", "formula": "\\begin{align*} \\tilde N _ { n , \\mathbf { p } _ { k , 1 } } ( k ) \\tilde N _ { n , \\mathbf { p } _ { k , 2 } } ( k ) = \\frac { 1 } { n ^ 2 } \\sum _ { i , j = 1 } ^ n \\prod _ { \\ell = 1 } ^ { k } \\tilde I ^ { ( p _ { \\ell , 1 } ) } _ { i , i } \\tilde I ^ { ( p _ { \\ell , 2 } ) } _ { j , j } . \\end{align*}"}
{"id": "3650.png", "formula": "\\begin{align*} B _ 0 = \\left ( \\frac { 2 } { 3 c } \\right ) ^ { 3 / 5 } \\left ( \\frac { 5 } { 3 } \\right ) ^ { 1 / 5 } = 0 . 0 7 6 3 3 \\ldots \\quad B _ 1 = 0 . 0 8 2 2 8 . \\end{align*}"}
{"id": "6998.png", "formula": "\\begin{align*} \\| p \\| : = \\max _ { 0 \\leq j \\leq n } | p _ j | \\end{align*}"}
{"id": "6786.png", "formula": "\\begin{align*} - k _ { p } \\leqslant \\sum _ { j = 1 } ^ { p } i _ { k _ { j } } \\leqslant 0 f o r 1 \\leqslant p \\leqslant m - 1 . \\end{align*}"}
{"id": "6489.png", "formula": "\\begin{align*} A _ { \\delta } & : = \\begin{cases} k ! & \\mbox { i f $ \\delta = - k $ f o r s o m e $ k \\in \\mathbb { N } $ , } \\\\ [ 2 m m ] \\dfrac { 1 } { \\Gamma ( 1 + \\delta ) } & \\mbox { o t h e r w i s e . } \\\\ \\end{cases} \\end{align*}"}
{"id": "3903.png", "formula": "\\begin{align*} E _ 2 ( f , p ) = \\sum _ { z \\leq p } \\frac { 1 } { p } \\sum _ { \\substack { 0 < a < p \\\\ g c d ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 < b < p \\\\ \\gcd ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } \\ll p ^ { 1 - 2 \\varepsilon } . \\end{align*}"}
{"id": "4283.png", "formula": "\\begin{align*} A _ { \\lambda } ( x , t ) = \\lambda A \\left ( \\lambda x , { \\lambda } ^ 2 t \\right ) , \\lambda > 0 . \\end{align*}"}
{"id": "4486.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } E _ { \\theta } ( x ) = 0 , \\ \\ \\lim _ { x \\to \\infty } E _ { \\pi } ( x ) = 0 . \\end{align*}"}
{"id": "6726.png", "formula": "\\begin{align*} h _ { 1 , j ' - 1 } ( \\theta ^ { q ^ N } ) \\biggl ( \\prod _ { \\substack { l = 1 \\\\ l \\neq m _ a } } ^ { d - 1 } ( \\theta ^ { q ^ { l + N - m _ u } } - & \\theta ^ { q ^ N } ) ^ { c ( l ) q ^ { d - l } } \\alpha _ 1 ^ { q ^ { n + d } } \\biggr ) + \\cdots \\\\ & \\cdots + h _ { r - j ' + 1 , j ' - 1 } ( \\theta ^ { q ^ N } ) \\biggl ( \\prod _ { \\substack { l = 1 \\\\ l \\neq m _ u } } ^ { d - 1 } ( \\theta ^ { q ^ { l + N - m _ u } } - \\theta ^ { q ^ N } ) ^ { c ( l ) q ^ { d - l } } \\alpha _ { r - j ' + 1 } ^ { q ^ { n + d } } \\biggr ) = 0 . \\end{align*}"}
{"id": "1621.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( x ) = \\alpha _ 1 x _ 1 B _ { \\frac { 1 } { 8 } } ( P _ 1 ) , \\tilde { \\Phi } ( x ) = \\alpha _ 2 x _ 1 B _ { \\frac { 1 } { 8 } } ( P _ 2 ) \\tilde { \\Phi } ( x ) = \\alpha _ 3 x _ 1 B _ { \\frac { 1 } { 8 } } ( P _ 3 ) . \\end{align*}"}
{"id": "6016.png", "formula": "\\begin{align*} - c _ \\alpha \\lim _ { y \\to 0 ^ + } y ^ { 1 - 2 \\alpha } U _ y ( x , y ) = ( \\mathfrak { D } _ { { \\rm l e f t } , a } ) ^ \\alpha u ( x ) \\end{align*}"}
{"id": "2234.png", "formula": "\\begin{align*} l ( \\gamma ) = \\left \\{ \\begin{array} { l l } 2 , & f o r \\ ; \\gamma \\in ( \\frac d 2 , 2 ] , \\\\ 3 , & f o r \\ ; \\gamma \\in ( 2 , 3 ] , \\\\ 4 , & f o r \\ ; \\gamma \\in ( 3 , 4 ] , \\end{array} \\right . \\ ; a n d \\ell ( \\gamma ) = \\left \\{ \\begin{array} { l l } 0 , & f o r \\ ; \\gamma \\in ( \\frac d 2 , 2 ] , \\\\ 2 , & f o r \\ ; \\gamma \\in ( 2 , 4 ] . \\end{array} \\right . \\end{align*}"}
{"id": "6973.png", "formula": "\\begin{align*} \\forall x \\in \\R \\colon \\varphi ( x ) : = - x , \\Phi ( x ) : = \\begin{cases} \\R & x \\leq 0 , \\\\ [ x ^ 2 , \\infty ) & x = \\frac 1 k k \\in \\N , \\\\ \\varnothing & \\end{cases} \\end{align*}"}
{"id": "6723.png", "formula": "\\begin{align*} h _ { 1 , j } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + h _ { r - j , j } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ { r - j } ) ^ { q ^ d } + R _ j = 0 \\end{align*}"}
{"id": "6577.png", "formula": "\\begin{align*} \\sum _ { m = 2 } ^ { \\infty } \\frac { \\tau _ { A _ { s _ 1 } \\smallsetminus U _ { s _ 1 } \\cup ( V _ { s _ 2 } ) ^ { - } } ( p ^ { m } ) \\tau _ { B _ { s _ 2 } \\smallsetminus V _ { s _ 2 } \\cup ( U _ { s _ 1 } ) ^ { - } } ( p ^ { m } ) } { p ^ { m } } \\ll \\frac { 1 } { p ^ { 2 - \\varepsilon } } \\end{align*}"}
{"id": "4783.png", "formula": "\\begin{align*} n = ( s + 1 ) ( s t + 1 ) , \\theta = s - t - 1 , \\hat \\theta = \\frac { | s - t - 1 | } { s } \\sqrt { \\frac { s t + t + 1 } { ( s + 1 ) ( t s + 1 ) t ( t + 1 ) } } . \\end{align*}"}
{"id": "7796.png", "formula": "\\begin{align*} \\mathcal V _ { p } ^ p ( \\P , \\Q ) : = \\max \\big \\{ V _ p ( \\P , \\Q ) , V _ p ( \\Q , \\P ) \\big \\} , \\end{align*}"}
{"id": "9320.png", "formula": "\\begin{align*} r _ k ( \\mathbf { x } , \\mathbf { y } ) : = \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} - \\Pi _ { D } ( \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} - \\begin{bmatrix} H \\mathbf { x } + \\mathbf { g } - A ^ T \\mathbf { y } + \\rho ( \\mathbf { x } - \\mathbf { x } _ k ) \\\\ A \\mathbf { x } - \\mathbf { b } + \\delta ( \\mathbf { y } - \\mathbf { y } _ k ) \\end{bmatrix} ) . \\end{align*}"}
{"id": "4683.png", "formula": "\\begin{align*} A _ { i j } ( t , y ) = A _ { i j , 0 } ( y ) + \\mathfrak { A } _ { i j } ( \\vec { x } ( t ) , \\vec { \\mu } ( t ) , y ) . \\end{align*}"}
{"id": "3045.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = a _ 1 x _ 1 \\ , , x _ 2 ^ \\prime = a _ 2 x _ 2 \\ , , y _ 1 ^ \\prime = b _ 1 y _ 1 \\ , , y _ 2 ^ \\prime = b _ 2 y _ 2 \\ , . \\end{align*}"}
{"id": "8923.png", "formula": "\\begin{align*} \\check H _ { c t } ^ q ( \\Z _ { \\ge 0 } ; A ) = \\begin{cases} A & q = 0 \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "1403.png", "formula": "\\begin{align*} q q _ d ^ * + q _ d q ^ * = q ^ * q _ d + q _ d ^ * q = 0 . \\end{align*}"}
{"id": "934.png", "formula": "\\begin{align*} \\sum _ { n = k } ^ { \\infty } S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } & = \\frac { 1 } { k ! } \\big ( e _ { \\lambda } ( t ) - 1 \\big ) ^ { k } = \\frac { 1 } { k ! } \\sum _ { m = 0 } ^ { k } \\binom { k } { m } ( - 1 ) ^ { k - m } e _ { \\lambda } ^ { m } ( t ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\frac { 1 } { k ! } \\sum _ { m = 0 } ^ { k } \\binom { k } { m } ( - 1 ) ^ { k - m } ( m ) _ { n , \\lambda } \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "7801.png", "formula": "\\begin{align*} h ^ i _ { \\mathbb { Q } _ p } : H ^ i ( K _ n , V ) \\xrightarrow { \\sim } H ^ i _ { \\Phi \\Gamma } ( \\mathbf { D } ( V ) ) : = H ^ i ( C ^ { \\bullet } ( \\mathbf { D } ( V ) ) ) . \\end{align*}"}
{"id": "5456.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\int _ \\Omega u _ t ^ q ( t , x ) d x & = - ( q - 1 ) \\int _ \\Omega u ^ { q - 2 } ( t , x ) | \\nabla u ( t , x ) | ^ 2 d x - \\chi ( q - 1 ) \\int _ \\Omega \\frac { u ^ { q - 1 } ( t , x ) } { v ( t , x ) } \\nabla u ( t , x ) \\cdot \\nabla v ( t , x ) d x \\\\ & \\ , \\ , \\ , \\ , + \\int _ \\Omega a ( t , x ) u ^ q ( t , x ) - \\int _ \\Omega b ( t , x ) u ^ { q + 1 } ( t , x ) d x \\forall \\ , t > s , \\end{align*}"}
{"id": "5405.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } X _ { t } = f ( X _ { t } , \\theta ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ { t } + \\mathrm { d } L _ { t } , \\\\ & X _ { 0 } = x \\in [ a , \\infty ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "9078.png", "formula": "\\begin{align*} f ( x , y ) = \\begin{dcases} 3 2 , & \\frac { 5 } { 8 } \\leq x \\leq \\frac { 7 } { 8 } , \\frac { 5 } { 8 } \\leq y \\leq \\frac { 7 } { 8 } , \\\\ 0 , & e l s e . \\\\ \\end{dcases} \\end{align*}"}
{"id": "7380.png", "formula": "\\begin{align*} \\phi _ t ( x , t ) - \\Delta \\phi ( x , t ) + a | \\nabla \\phi ( x , t ) | ^ \\alpha = - a k ^ \\alpha - { n - 1 \\over | x | } + a k ^ \\alpha \\leq 0 . \\end{align*}"}
{"id": "296.png", "formula": "\\begin{align*} | r _ { 0 } ( x ) | & \\le C \\left | \\int _ { - \\infty } ^ { x } z _ { 0 } ( y ) d y \\right | = C \\left | \\int _ { - \\infty } ^ { x } z _ { 0 } ( y ) d y - \\int _ { \\R } z _ { 0 } ( y ) d y \\right | = C \\left | \\int _ { x } ^ { \\infty } z _ { 0 } ( y ) d y \\right | \\\\ & \\le C \\int _ { x } ^ { \\infty } ( 1 + | y | ) ^ { - \\alpha } d y \\le C \\int _ { x } ^ { \\infty } ( 1 + y ) ^ { - \\alpha } d y \\le C ( 1 + x ) ^ { - ( \\alpha - 1 ) } = C ( 1 + | x | ) ^ { - ( \\alpha - 1 ) } . \\end{align*}"}
{"id": "7122.png", "formula": "\\begin{align*} \\tilde { g } _ 2 ( s , y ) & = s ^ { 2 H - 1 } y ^ { - \\tilde { \\theta } \\rho + \\tilde { \\theta } + 1 } \\Big ( \\alpha _ 2 \\frac { 1 } { \\tilde { \\theta } ^ { \\rho } } - ( H + \\epsilon ) \\tilde { \\sigma } ( \\tilde { \\theta } + 1 ) y ^ { \\tilde { \\theta } \\rho - \\tilde { \\theta } - 2 } \\Big ) \\\\ & \\ge h _ 1 s ^ { 2 H - 1 } y ^ { - \\alpha } \\end{align*}"}
{"id": "4520.png", "formula": "\\begin{align*} ( \\alpha 1 + x ) ( \\beta 1 + y ) = ( \\alpha \\beta + f ( x , y ) ) 1 + \\alpha y + \\beta x , \\end{align*}"}
{"id": "8322.png", "formula": "\\begin{align*} A ( y , u ) : = \\{ t \\in [ 0 , T ) \\colon | y ( t ) | = r \\nexists \\varepsilon > 0 y - u = \\mathrm { c o n s t } [ t , t + \\varepsilon ) \\} . \\end{align*}"}
{"id": "4029.png", "formula": "\\begin{align*} & \\xi ( x ) \\\\ = & \\frac { \\eta ^ { \\prime \\prime } ( x ) + \\mu \\eta ( x ) } { \\mu } \\\\ = & \\Big ( \\frac { m _ 1 ^ 2 + \\mu } { \\mu } \\Big ) ( e ^ { m _ 2 } - e ^ { m _ 3 } ) e ^ { m _ 1 x } + \\Big ( \\frac { m ^ 2 _ 2 + \\mu } { \\mu } \\Big ) ( e ^ { m _ 3 } - e ^ { m _ 1 } ) e ^ { m _ 2 x } + \\Big ( \\frac { m _ 3 ^ 2 + \\mu } { \\mu } \\Big ) ( e ^ { m _ 1 } - e ^ { m _ 2 } ) e ^ { m _ 3 x } . \\end{align*}"}
{"id": "2017.png", "formula": "\\begin{align*} M ^ { F ^ u } _ { t } = M _ { t } ^ { F } + M _ { t } ^ { - u , j } + M _ { t } ^ { - u , \\kappa } , t < \\zeta \\end{align*}"}
{"id": "9383.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n + 1 } S _ { 2 , \\lambda } ( n + 1 , k ) ( k - 1 ) ! x ^ { k } = \\sum _ { k = 1 } ^ { n + 1 } \\big \\{ S _ { 2 , \\lambda } ( n , k - 1 ) + ( k - n \\lambda ) S _ { 2 , \\lambda } ( n , k ) \\big \\} ( k - 1 ) ! x ^ { k } \\\\ & = \\sum _ { k = 1 } ^ { n } k ! S _ { 2 , \\lambda } ( n , k ) x ^ { k } + x \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) k ! x ^ { k } - n \\lambda \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( k - 1 ) ! x ^ { k } \\\\ & = ( 1 + x ) F _ { n , \\lambda } ( x ) - n \\lambda \\int _ { 0 } ^ { x } \\frac { F _ { n , \\lambda } ( t ) } { t } d t , ( n \\ge 1 ) . \\end{align*}"}
{"id": "9261.png", "formula": "\\begin{align*} 0 = \\norm { ( J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x ' ) + \\gamma ( \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma '^ { - 1 } ( x ' - J ^ A _ { \\gamma ' } x ' ) ) } \\geq \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x ' } . \\end{align*}"}
{"id": "9233.png", "formula": "\\begin{align*} \\lambda ^ { - 1 } ( x - J ^ A _ \\lambda x ) = _ X \\frac { \\gamma } { \\lambda \\gamma } ( x - J ^ A _ \\gamma x ) = _ X \\frac { 1 } { \\gamma } \\left ( \\frac { \\gamma } { \\lambda } x + \\left ( 1 - \\frac { \\gamma } { \\lambda } \\right ) J ^ A _ \\lambda x - J ^ A _ \\lambda x \\right ) . \\end{align*}"}
{"id": "7565.png", "formula": "\\begin{align*} f & = a _ 1 x _ 1 y _ 1 z _ 1 w _ 1 + a _ 2 x _ 1 y _ 1 z _ 1 w _ 2 + a _ 3 x _ 1 y _ 1 z _ 2 w _ 1 + a _ 4 x _ 1 y _ 1 z _ 2 w _ 2 + a _ 5 x _ 1 y _ 2 z _ 1 w _ 1 + a _ 6 x _ 1 y _ 2 z _ 1 w _ 2 \\\\ & + a _ 7 x _ 1 y _ 2 z _ 2 w _ 1 + a _ 8 x _ 1 y _ 2 z _ 2 w _ 2 + a _ 9 x _ 2 y _ 1 z _ 1 w _ 1 + a _ { 1 0 } x _ 2 y _ 1 z _ 1 w _ 2 + a _ { 1 1 } x _ 2 y _ 1 z _ 2 w _ 1 + a _ { 1 2 } x _ 2 y _ 1 z _ 2 w _ 2 \\\\ & + a _ { 1 3 } x _ 2 y _ 2 z _ 1 w _ 1 + a _ { 1 4 } x _ 2 y _ 2 z _ 1 w _ 2 + a _ { 1 5 } x _ 2 y _ 2 z _ 2 w _ 1 + a _ { 1 6 } x _ 2 y _ 2 z _ 2 w _ 2 , \\end{align*}"}
{"id": "3278.png", "formula": "\\begin{align*} s ^ 2 + | \\lambda | ^ 2 - \\frac { s } { h ( s ) } = 0 \\end{align*}"}
{"id": "5799.png", "formula": "\\begin{align*} \\widetilde { A } _ 1 = \\mathrm { p r o j } _ { \\subseteq U } ( \\varphi ( A _ 1 ) ) \\quad \\widetilde { A } _ 2 = \\mathrm { p r o j } _ { \\subseteq V } ( \\varphi ( A _ 2 ) ) \\end{align*}"}
{"id": "2221.png", "formula": "\\begin{align*} - \\big < F ( v ^ { n } ( t ) + Z ^ { n } ) , \\ ; \\dot { v } ^ { n } ( t ) \\big > \\leq & - \\frac { \\dd } { \\dd t } \\int _ D \\Phi ( v ^ { n } ( t ) ) \\ , \\dd x + \\frac 1 2 \\| \\dot { v } ^ { n } ( t ) \\| _ { - 1 } ^ 2 \\\\ & + C ( \\gamma ) \\big ( 1 + \\| ( v ^ n ( t ) ) ^ 2 \\| _ 1 + \\| A v ^ { n } ( t ) \\| \\big ) ^ 2 \\big ( 1 + \\| Z ^ { n } ( t ) \\| _ \\gamma ^ 3 \\big ) ^ 2 . \\end{align*}"}
{"id": "7785.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to 0 } \\frac { \\log r } { q } = \\lim \\limits _ { q \\to \\infty } \\frac { 1 } { q } \\log \\prod \\limits _ { i = 1 } ^ { q } C _ { w _ i } = \\sum \\limits _ { k \\in T } p _ k \\log C _ k . \\end{align*}"}
{"id": "2746.png", "formula": "\\begin{align*} a _ t = \\frac { - R ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , z _ { j - 1 } , t , z _ { j + 1 } , \\dots z _ n ) } { Q ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , z _ { j - 1 } , t , z _ { j + 1 } , \\dots z _ n ) } , \\end{align*}"}
{"id": "7276.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ \\infty ( 1 + \\alpha X ^ k ) & = \\sum _ { k = 0 } ^ \\infty \\frac { \\alpha ^ k X ^ { \\binom { k } { 2 } } } { X ^ k ! } , \\\\ \\prod _ { k = 1 } ^ \\infty \\frac { 1 } { 1 - \\alpha X ^ k } & = \\sum _ { k = 0 } ^ \\infty \\frac { \\alpha ^ k X ^ k } { X ^ k ! } . \\end{align*}"}
{"id": "7099.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { m } r ^ 2 _ i ( r ^ 2 _ i - 1 ) - \\frac { 1 5 } { 8 } m _ 2 - \\frac { 3 } { 4 } m _ 3 - \\frac { 1 } { 2 } m _ 4 & = \\sum _ { i = 1 } ^ { n } \\lambda ^ 2 _ i \\\\ \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { m } r ^ 2 _ i ( r ^ 2 _ i - 1 ) - \\frac { 1 5 } { 8 } m _ 2 - \\frac { 3 } { 4 } m _ 3 - \\frac { 1 } { 2 } m _ 4 - \\lambda ^ 2 _ 1 & = \\sum _ { i = 2 } ^ { n } \\lambda ^ 2 _ i \\end{align*}"}
{"id": "2989.png", "formula": "\\begin{align*} L = A + i V \\end{align*}"}
{"id": "2390.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\norm { f - \\sum _ { k = 1 } ^ N e _ { \\pi ( k ) } } _ { \\mathcal { B } } = 0 . \\end{align*}"}
{"id": "108.png", "formula": "\\begin{align*} 2 e ( x y ) + 4 ( e x ) ( e y ) = x y . \\end{align*}"}
{"id": "9367.png", "formula": "\\begin{align*} \\beta _ { n , \\lambda } ( x ) & = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\beta _ { k , \\lambda } ( x ) _ { n - k , \\lambda } \\\\ & = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( x ) _ { k , \\lambda } \\beta _ { n - k , \\lambda } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "6346.png", "formula": "\\begin{align*} \\left ( \\frac { x } { n } \\right ) ^ c \\Lambda ( n ) \\leq \\left ( \\frac { x } { n } \\right ) ^ c \\log n = \\frac { x } { n } \\cdot \\frac { e \\log n } { e ^ { \\log n / \\log x } } \\leq \\frac { x } { n } \\log x , \\end{align*}"}
{"id": "1827.png", "formula": "\\begin{align*} L _ n ( x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } L ( n , k ) x ^ { k } , \\end{align*}"}
{"id": "4479.png", "formula": "\\begin{align*} \\frac { \\partial f ( t ) } { \\partial t } & = e _ { \\lambda } ^ { 1 - \\lambda } ( t ) | z | ^ { 2 } f ( t ) = e _ { \\lambda } ^ { 1 - \\lambda } ( t ) | z | ^ { 2 } \\sum _ { k = 0 } ^ { \\infty } \\frac { t ^ { k } } { k ! } \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) \\\\ & = | z | ^ { 2 } \\sum _ { k = 0 } ^ { \\infty } \\bigg ( \\sum _ { l = 0 } ^ { k } \\binom { k } { l } ( 1 - \\lambda ) _ { k - l , \\lambda } \\phi _ { l , \\lambda } ( | z | ^ { 2 } ) \\bigg ) \\frac { t ^ { k } } { k ! } . \\end{align*}"}
{"id": "4603.png", "formula": "\\begin{align*} \\varphi _ k ( P ) = \\sum _ { G \\in \\mathcal { G } _ k } \\varphi ( P , G ) . \\end{align*}"}
{"id": "6279.png", "formula": "\\begin{align*} \\tilde { h } _ i = \\mathbf { v } ^ H \\mathbf { w } _ i ' / ( N _ 1 \\sqrt { p } ) \\sim \\mathcal { C N } \\Big ( 0 , \\frac { 1 } { N _ 1 \\varrho } \\Big ) . \\end{align*}"}
{"id": "8721.png", "formula": "\\begin{align*} M _ { ( k , \\tau ) } = \\Bigl \\{ z \\in \\R ^ { \\binom { k + \\tau } { \\tau } } \\Bigm | z = ( 1 , x _ 1 , x _ 2 , \\ldots , x ^ \\tau _ n ) , \\ x \\in [ 0 , 1 ] ^ k \\Bigr \\} , \\end{align*}"}
{"id": "8169.png", "formula": "\\begin{align*} M _ 2 ( f , \\{ 1 \\} ) = { \\pi ^ 2 \\over 8 } \\times \\left \\{ \\prod _ { q \\mid f } \\left ( 1 - { 1 \\over q ^ 2 } \\right ) - { 1 \\over f } \\prod _ { q \\mid f } \\left ( 1 - { 1 \\over q } \\right ) \\right \\} \\ \\ \\ \\ \\ \\hbox { ( $ f > 2 $ o d d ) } . \\end{align*}"}
{"id": "948.png", "formula": "\\begin{align*} V _ j = E _ j \\setminus \\bigcup _ { i < j } E _ i . \\end{align*}"}
{"id": "6310.png", "formula": "\\begin{align*} \\mathrm { v a r } [ \\widehat { K } _ { \\mathrm { a - c p t - d } } ^ \\mathbb { R } ] = \\frac { \\psi } { 2 } \\Big ( \\sum _ { i \\in \\mathcal { K } ' } \\frac { \\mathrm { v a r } [ Z _ i ] } { N _ 1 \\varrho \\vartheta _ i } + \\frac { 1 } { N _ 2 \\bar { \\gamma } _ c ' } \\Big ) . \\end{align*}"}
{"id": "3874.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i < j \\leq 2 n } ( t _ i - t _ j ) ^ 2 = \\prod _ { \\alpha \\in \\Phi _ H } \\alpha ( t ) ^ 2 \\prod _ { i = 1 } ^ n t _ i ^ 2 . \\end{align*}"}
{"id": "6364.png", "formula": "\\begin{align*} d = - \\frac { 4 k - 1 } { 4 k + 1 } d _ j = - \\frac { k - ( j - 1 ) } { 2 k + 1 } \\end{align*}"}
{"id": "4678.png", "formula": "\\begin{align*} V ( t , y ) & = \\Theta ( \\vec { x } ( t ) , \\vec { \\mu } ( t ) , y ) = \\sum _ { i = 1 } ^ n \\sigma _ i Q _ { 1 + \\mu _ { i } ( t ) } ( y - x _ { i } ( t ) ) + \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\frac { A _ { i j } ( t , y - x _ { i } ( t ) ) } { x ^ 2 _ { i j } ( t ) } \\\\ & + \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\frac { B _ { i j } ( t , y - x _ { i } ( t ) ) } { x _ { i j } ^ 3 ( t ) } \\varphi _ { i j } ( t , y ) , \\end{align*}"}
{"id": "1605.png", "formula": "\\begin{align*} \\| \\Psi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } = \\mu ^ { \\frac { d - 1 } { p } - \\frac { d - 1 } { r } } \\| \\Phi \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } = \\mu ^ { \\frac { d - 1 } { p } - \\frac { d - 1 } { r } } \\| \\Phi \\| _ { L ^ r ( [ 0 , 1 ] ^ { d - 1 } ) } \\leq M _ 1 \\mu ^ { \\frac { d - 1 } { p } - \\frac { d - 1 } { r } } . \\end{align*}"}
{"id": "4851.png", "formula": "\\begin{align*} L ( x ) : = \\left \\{ \\begin{aligned} & K ( \\sigma ) + ( x - \\sigma ) K ' ( \\sigma ) & & 0 \\leq x \\leq \\sigma \\\\ & K ( x ) & & x > \\sigma , \\end{aligned} \\right . M : = K - L \\end{align*}"}
{"id": "6154.png", "formula": "\\begin{align*} d _ k = ( - 1 ) ^ k \\sum _ { \\nu = 0 } ^ m ( - 1 ) ^ { \\nu } { n + \\nu \\choose \\nu } { n + 1 \\choose k - \\nu } . \\end{align*}"}
{"id": "7873.png", "formula": "\\begin{align*} \\max _ { 0 \\leq | \\alpha | \\leq 3 } \\sup _ { x \\in C _ { R + 1 } } | D _ x ^ { \\alpha } D _ y ^ { \\alpha } K _ { i _ 1 i _ 2 } ( x , y ) | _ { y = x } | \\leq M , \\ i _ 1 , i _ 2 = 1 , 2 , \\dots , m . \\end{align*}"}
{"id": "2086.png", "formula": "\\begin{align*} P _ { \\mathbf { c } } = \\Big \\{ \\mathbf { v } \\in ( \\mathbb { Z } / p ^ { 2 k } \\mathbb { Z } ) ^ n : \\mathbf { D } _ { \\mathbf { c } } \\cdot \\frac { \\mathbf { v } - \\mathbf { c } } { p ^ k } \\equiv 0 \\mod p ^ k \\Big \\} . \\end{align*}"}
{"id": "7691.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ 0 \\le \\frac { 1 } { \\int _ { \\R ^ N } m ' u ^ 2 } \\le \\frac { 1 } { \\int _ { \\R ^ N } m u ^ 2 } = \\tilde { \\lambda } _ 0 \\ ; , \\end{align*}"}
{"id": "4063.png", "formula": "\\begin{align*} \\Lambda _ 2 ( f , g ) = ( \\sigma ( 0 , \\cdot ) , v ( 0 , \\cdot ) ) , \\end{align*}"}
{"id": "5512.png", "formula": "\\begin{align*} X ( t ; x ) = S _ t x + \\int _ 0 ^ t S _ { t - s } \\alpha ( X ( s ; x ) ) d s + \\int _ 0 ^ t S _ { t - s } \\sigma ( X ( s ; x ) ) d W ( s ) , t \\in [ 0 , T ] . \\end{align*}"}
{"id": "6903.png", "formula": "\\begin{align*} U ^ k _ i f ( x _ k ) = \\frac { 1 } { R ( k , i ) } \\sum \\limits _ { x _ i < x _ k } f ( x _ i ) \\end{align*}"}
{"id": "8452.png", "formula": "\\begin{align*} [ a , b ] ( [ c , d ] [ e , f ] ) = [ a , b ] [ p c , q f ] = [ ( i u ) a , j ( q f ) ] . \\end{align*}"}
{"id": "8816.png", "formula": "\\begin{align*} \\partial _ t p _ t = \\nabla \\left ( p _ t \\nabla \\log \\frac { d p _ t } { d \\gamma ^ { \\epsilon , L } } \\right ) , p _ 0 = \\nu ^ { \\epsilon , L } , \\end{align*}"}
{"id": "1588.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } F ( v , v _ * , v ' , v _ * ' ) B ( | v - v _ * | , \\cos \\theta ) d v d v _ * d \\sigma = \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } F ( v ' , v _ * ' , v , v _ * ) B ( | v - v _ * | , \\cos \\theta ) d v d v _ * d \\sigma \\end{align*}"}
{"id": "724.png", "formula": "\\begin{align*} S _ { ( 0 0 ) } ^ { ( \\ell ) } = O ( n ^ { - 1 } ) , \\end{align*}"}
{"id": "6363.png", "formula": "\\begin{align*} K _ { \\widetilde { X } } = \\tau ^ \\ast ( K _ X ) + d \\cdot F + \\sum _ { i = 1 } ^ 4 \\sum _ { j = 1 } ^ k d _ j \\cdot C _ { i j } \\end{align*}"}
{"id": "9185.png", "formula": "\\begin{align*} \\Delta u = f ~ ~ ~ ~ \\mbox { i n } ~ B _ 1 . \\end{align*}"}
{"id": "8364.png", "formula": "\\begin{align*} u ^ { \\frac { s + n h } { s + n h + 1 } } \\in \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } B . \\end{align*}"}
{"id": "1740.png", "formula": "\\begin{align*} 2 ^ { \\hat m _ 0 } = n < 2 ^ { m ' _ 0 } \\le n ^ { q / 2 } = 2 ^ { \\overline { m } _ 0 } . \\end{align*}"}
{"id": "5446.png", "formula": "\\begin{align*} u ( t , \\cdot ; s , u _ 0 ) & = e ^ { - A ( t - s ) } u _ 0 - \\chi \\int _ s ^ t e ^ { - A ( t - \\tau ) } \\nabla \\cdot \\left ( \\frac { u ( \\tau , \\cdot ; s , u _ 0 ) } { v ( \\tau , \\cdot ; s , u _ 0 ) } \\nabla v ( \\tau , \\cdot ; s , u _ 0 ) \\right ) d \\tau \\\\ & + \\int _ s ^ t e ^ { - A ( t - \\tau ) } u ( \\tau , \\cdot ; s , u _ 0 ) \\big [ { \\mu } + a ( \\tau , \\cdot ) - b ( \\tau , \\cdot ) u ( \\tau , \\cdot ; s , u _ 0 ) \\big ] d \\tau \\end{align*}"}
{"id": "9547.png", "formula": "\\begin{align*} ( 1 - \\mu ^ k ) [ w ^ k - z _ 1 ^ k + ( m - 1 ) ( \\hat { \\mathcal { A } } ( x ^ k ) ^ { m - 2 } ) ^ T ( x ^ k - z _ 2 ^ k ) ] + \\mu ^ k ( x ^ k - x ^ { ( 0 ) } ) = 0 \\end{align*}"}
{"id": "8763.png", "formula": "\\begin{align*} ~ \\begin{aligned} u _ { i j } ( x ) & \\leq \\min \\{ a _ { i j } , f _ i ( x ) \\} & & f _ { i } ( x ) \\in [ a _ { i \\tau ( i , t - 1 ) } , a _ { i \\tau ( i , t ) } ] , \\\\ u _ { i j } ( x ) & = - \\infty & & . \\\\ \\end{aligned} \\end{align*}"}
{"id": "1074.png", "formula": "\\begin{align*} \\Delta ( k ) = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c } \\chi _ { R , L } ( k ) + \\chi _ { R , L } ^ { - 1 } ( k ) & i \\left ( \\chi _ { R , L } ( k ) - \\chi _ { R , L } ^ { - 1 } ( k ) \\right ) \\\\ - i \\left ( \\chi _ { R , L } ( k ) - \\chi _ { R , L } ^ { - 1 } ( k ) \\right ) & \\chi _ { R , L } ( k ) + \\chi _ { R , L } ^ { - 1 } ( k ) \\end{array} \\right ) , \\end{align*}"}
{"id": "7426.png", "formula": "\\begin{align*} f ( \\eta ) - f ( \\eta ^ { z , z - j \\ell } ) = \\sum _ { k = 0 } ^ 2 [ f ( \\eta _ { k , x , j , z } ) - f ( \\eta _ { k + 1 , x , j , z } ) ] . \\end{align*}"}
{"id": "4435.png", "formula": "\\begin{align*} | \\epsilon \\int _ U u u _ t ( x , t ) \\ , d x | & \\leq \\frac { \\epsilon } { 2 } ( \\int _ U u ( x , t ) ^ 2 \\ , d x ) + \\frac { \\epsilon } { 2 } \\int _ U ( u _ t ( x , t ) ) ^ 2 \\ , d x \\\\ & \\leq \\frac { \\epsilon B ^ 2 } { 2 } \\int _ U ( u _ { x x } ( x , t ) ) ^ 2 \\ , d x + \\frac { \\epsilon B ^ 2 } { 2 } \\int _ U ( u _ t ( x , t ) ) ^ 2 \\ , d x \\\\ & \\leq \\epsilon B ^ 2 E ( t ) . \\end{align*}"}
{"id": "8510.png", "formula": "\\begin{align*} b * ( n a ) & = b * ( ( n - 1 ) a + a ) = b * ( n - 1 ) a + ( n - 1 ) a + b * a - ( n - 1 ) a . \\end{align*}"}
{"id": "3254.png", "formula": "\\begin{align*} r ^ 2 & = - ( s - t ) ^ 2 + \\frac { s - t } { h ( s ) } \\\\ & = \\frac { s ^ 2 \\Big ( 1 - s h ( s ) \\Big ) } { s h ( s ) } + s t \\Big ( 2 s h ( s ) - 1 \\Big ) - t ^ 2 s h ( s ) . \\end{align*}"}
{"id": "6649.png", "formula": "\\begin{align*} R _ 2 = R _ { 2 1 } + R _ { 2 2 } + R _ { 2 3 } + R _ { 2 4 } , \\end{align*}"}
{"id": "4984.png", "formula": "\\begin{align*} \\Lambda ( Z ) = ( \\Lambda ^ { - 1 } \\circ A \\circ \\Lambda , \\Lambda ^ { - 1 } \\circ B \\circ \\Lambda ) \\end{align*}"}
{"id": "9028.png", "formula": "\\begin{align*} & - \\nabla \\cdot ( \\epsilon ( x ) \\phi ) = f ( x ) + \\sum _ { i = 1 } ^ s z _ i \\rho _ i , \\quad , x \\in \\Omega , \\\\ & \\alpha \\phi = \\phi _ b , x \\in \\partial \\Omega , \\end{align*}"}
{"id": "954.png", "formula": "\\begin{align*} u ( x _ \\ast ) = - u ( x ) x \\in \\R ^ n . \\end{align*}"}
{"id": "4404.png", "formula": "\\begin{align*} S _ j = L ( S _ j ) = L ( 0 ) + D L ( S _ j ) , \\end{align*}"}
{"id": "6061.png", "formula": "\\begin{align*} \\phi _ 1 ( s , t ) = M ( s , t ) \\left ( t ^ { 2 l } + \\sum _ { j = 1 } ^ { 2 l } A _ j ( s ) t ^ { 2 l - j } \\right ) \\end{align*}"}
{"id": "4232.png", "formula": "\\begin{align*} a ^ 2 ( x - h ) ^ 2 + b ( x - h ) ( y - k ) + c ^ 2 ( y - k ) ^ 2 = 1 , \\end{align*}"}
{"id": "6610.png", "formula": "\\begin{align*} \\mathcal { U } ( h , k ) = - \\mathcal { L } ^ 0 ( h , k ) + \\mathcal { U } ^ 2 ( h , k ) + \\mathcal { U } ^ r ( h , k ) + O \\big ( ( X h k ) ^ \\varepsilon X C \\big ) + O \\bigg ( Q \\frac { ( X C H K ) ^ { \\varepsilon } } { \\sqrt { H K } } \\bigg ) . \\end{align*}"}
{"id": "1438.png", "formula": "\\begin{align*} \\zeta ^ 2 = \\alpha , \\xi ^ 2 = \\beta , \\zeta \\xi = - \\xi \\zeta . \\end{align*}"}
{"id": "1492.png", "formula": "\\begin{align*} \\langle f , h \\rangle = \\int _ { \\Gamma \\backslash \\mathfrak { Z } } f ( z ) \\overline { h ( z ) } \\delta ( z ) ^ k \\mathbf { d } z , \\end{align*}"}
{"id": "3040.png", "formula": "\\begin{align*} x = 1 2 6 - \\Theta \\ , , \\alpha = 6 0 - 2 \\Theta \\ , , g = 5 3 1 - \\frac { 6 5 } { 2 } \\Theta + \\frac { \\Theta ^ 2 } { 2 } - \\Delta \\ , . \\end{align*}"}
{"id": "7626.png", "formula": "\\begin{align*} \\nabla P = \\lambda \\partial _ { r } ^ { T } - u ^ { \\frac { \\alpha + 1 } { \\alpha } } \\nabla \\log u , \\end{align*}"}
{"id": "3735.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ 1 } B _ { x x } J B _ x \\ , d x = - \\frac 1 2 \\int _ { \\mathbb S ^ 1 } B _ x J _ { x } B _ x \\ , d x . \\end{align*}"}
{"id": "2721.png", "formula": "\\begin{align*} V _ n ( I ) = \\mathrm { Z e r } ( \\phi _ n ( f _ 1 ) , \\ldots , \\phi _ n ( f _ k ) ) \\subset \\mathrm { R } ^ n , n > 0 . \\end{align*}"}
{"id": "9141.png", "formula": "\\begin{align*} \\begin{cases} \\Psi _ 0 ( 0 , k , g , \\Phi , \\chi , \\tilde \\xi ) = 0 \\\\ \\Psi _ 0 ( n + 1 , k , g , \\Phi , \\chi , \\tilde \\xi ) = \\Phi ( \\chi ^ M _ g ( \\Psi _ 0 ( n , k , g , \\Phi , \\chi , \\tilde \\xi ) , 8 k + 7 , \\tilde \\xi ( 8 k + 7 ) ) \\end{cases} \\end{align*}"}
{"id": "6139.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 0 } ^ n { n + m + 1 \\choose k } x ^ { n - k } ( 1 - x ) ^ k = \\sum _ { k = 0 } ^ n ( - 1 ) ^ k \\frac { n + m + 1 } { k + m + 1 } { n + m \\choose m } { n \\choose k } x ^ k , \\end{align*}"}
{"id": "5685.png", "formula": "\\begin{align*} \\mathrm { i n d } ( u ) : = - \\chi ( u ) + 2 c _ { 1 } ( \\xi | _ { [ u ] } , \\tau ) + \\sum _ { k } \\mu _ { \\tau } ( \\gamma _ { k } ^ { + } ) - \\sum _ { l } \\mu _ { \\tau } ( \\gamma _ { l } ^ { - } ) . \\end{align*}"}
{"id": "8353.png", "formula": "\\begin{align*} p = \\Lambda ( e ^ r - 1 ) \\sim b \\ , ( e ^ r - 1 ) ^ { 1 - \\alpha } \\end{align*}"}
{"id": "3584.png", "formula": "\\begin{align*} \\widetilde { T } ( x ) ( t ) : = [ ( T ( x ) ) ( t ) ] z _ 0 ( x \\in X , \\ , t \\in K ) . \\end{align*}"}
{"id": "5079.png", "formula": "\\begin{align*} G _ t = & \\{ ( x , r ) : x \\in \\mathbb { R } ^ k , 0 < r < d ( x , K ) , z _ 1 , z _ 2 \\in K \\\\ & d ( x , z _ 1 ) , d ( x , z _ 2 ) < d ( x , K ) + \\epsilon r + t \\\\ & | \\theta | > \\cos ^ { - 1 } \\left ( 2 \\left ( \\frac { 1 - ( 2 \\delta + \\epsilon ) } { 1 + 2 \\delta } \\right ) ^ 2 - 1 \\right ) \\} , \\end{align*}"}
{"id": "9192.png", "formula": "\\begin{align*} J ( x ) : = \\left \\{ j \\in X ^ * \\mid \\langle x , j \\rangle = \\norm { x } ^ 2 = \\norm { j } ^ 2 \\right \\} \\end{align*}"}
{"id": "4309.png", "formula": "\\begin{align*} \\rho _ \\beta ( y ) = \\frac { ( 2 \\beta ) ^ \\frac { d + 2 } { 2 } } { ( 4 \\pi ) ^ { \\frac { d + 2 } { 2 } } } y ^ { d + 1 } e ^ { - ( 2 \\beta ) \\frac { y ^ 2 } { 4 } } . \\end{align*}"}
{"id": "8058.png", "formula": "\\begin{align*} \\Psi ^ i ( { t _ c } _ { * } f ) = \\mathfrak { P } _ { \\ell } t _ c \\Psi ^ i ( f ) . \\end{align*}"}
{"id": "9246.png", "formula": "\\begin{align*} \\mathrm { d o m } J ^ A _ \\gamma = \\mathrm { r a n } ( I d + \\gamma A ) \\end{align*}"}
{"id": "2560.png", "formula": "\\begin{align*} \\ker ( \\pi ^ { M p } ) = \\{ \\pm I \\} . \\end{align*}"}
{"id": "2860.png", "formula": "\\begin{align*} | x ( \\tau ) - x ( \\sigma ) | = | X ( \\tau ) - X ( \\sigma ) | \\le \\int _ \\sigma ^ \\tau | \\dot { X } ( t ) | d t \\lesssim \\int _ { \\sigma } ^ \\tau \\delta ( t ) d t . \\end{align*}"}
{"id": "9350.png", "formula": "\\begin{align*} \\frac { t } { e _ { \\lambda } ( t ) - 1 } e _ { \\lambda } ^ { x } ( t ) = \\sum _ { n = 0 } ^ { \\infty } \\beta _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 2 , 3 ] ) . \\end{align*}"}
{"id": "1651.png", "formula": "\\begin{align*} W _ 0 ^ 2 ( M ) = \\left \\lbrace u \\in W _ 0 ^ 1 ( M ) : \\Delta _ \\mu u \\in L ^ 2 ( M ) \\right \\rbrace , \\end{align*}"}
{"id": "2434.png", "formula": "\\begin{align*} f & = \\sum _ { \\l \\in \\L } \\langle f , \\pi ( \\l ) g \\rangle \\ \\pi ( \\l ) \\widetilde { g } \\\\ & = \\sum _ { \\l \\in \\L } \\langle f , \\pi ( \\l ) \\widetilde { g } \\rangle \\ \\pi ( \\l ) g , \\end{align*}"}
{"id": "291.png", "formula": "\\begin{align*} J ( \\xi , t ) : = \\left | \\exp \\left ( \\frac { i \\gamma t \\xi ^ { 3 } } { 1 + \\xi ^ { 2 } } \\right ) - \\exp \\left ( \\frac { - t \\xi ^ { 4 } } { 1 + \\xi ^ { 2 } } \\right ) \\right | ^ { 2 } . \\end{align*}"}
{"id": "3431.png", "formula": "\\begin{align*} | ( I - S _ { k + M } ) f ( x ) | & = | f ( x ) - S _ { k + M } f ( x ) | \\\\ & = \\bigg | \\int _ { \\Bbb R ^ N } S _ { k + M } ( x , y ) [ f ( x ) - f ( y ) ] d \\omega ( y ) \\bigg | \\\\ & \\leqslant C r ^ { - ( M + k ) s } \\| f \\| _ s \\end{align*}"}
{"id": "2999.png", "formula": "\\begin{align*} \\| u _ { \\varphi ( n ) } ( \\delta ) - u _ { \\varphi ( n ) } ( 0 ) \\| ^ { 2 } _ { L ^ { 2 } } & = \\| u ( \\delta ) \\| ^ { 2 } _ { L ^ { 2 } } + \\| u ( 0 ) \\| ^ { 2 } _ { L ^ { 2 } } - 2 R e \\langle u _ { \\varphi ( n ) } ( \\xi ) - u _ { \\varphi ( n ) } ( 0 ) \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "9427.png", "formula": "\\begin{align*} { \\mathcal N } _ 1 u = ( u \\cdot \\nabla ) u \\ , b + \\frac { ( 1 - b ) \\ , \\nabla | u | ^ 2 + ( \\mathrm { d i v } u ) u } { 2 } = \\star ( \\star d _ 1 u \\wedge u ) \\ , b + \\frac { d _ 0 | u | ^ 2 - ( d _ 0 ^ * u ) u } { 2 } \\end{align*}"}
{"id": "1328.png", "formula": "\\begin{align*} J _ { 0 } ( \\hat { \\alpha } \\cup { ( \\gamma , N ) } \\cup { ( \\delta _ { 1 } , 1 ) } , \\hat { \\alpha } \\cup { ( \\gamma , N - p _ { i } ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 1 . \\end{align*}"}
{"id": "425.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } ( V ^ { 2 } ) W _ { t } + A ^ { i } ( V ^ { 2 } ) \\partial _ { i } W - & B ^ { i j } ( V ^ { 2 } ) \\partial _ { i } \\partial _ { j } W + D ( V ^ { 2 } ) W = \\widehat { R } , \\\\ \\left . W \\right \\rvert _ { t = 0 } & = 0 , \\end{aligned} \\end{align*}"}
{"id": "1001.png", "formula": "\\begin{align*} \\partial ^ h _ 1 v ( x ) : = \\frac { v ( x + h e _ 1 ) - v ( x ) } h \\end{align*}"}
{"id": "328.png", "formula": "\\begin{align*} & d \\rho _ i ( t ) + \\sum _ { j \\in N ( i ) } \\theta ( \\rho _ i , \\rho _ j ) ( S _ j - S _ i ) d t + \\sum _ { j \\in N ( i ) } \\theta ( \\rho _ i , \\rho _ j ) ( \\Sigma _ j - \\Sigma _ i ) \\circ d W _ t = 0 , \\\\ & d S _ i ( t ) + \\sum _ { j \\in N ( i ) } \\frac 1 2 ( S _ j - S _ i ) ^ 2 \\frac { \\partial \\theta } { \\partial \\rho _ i } ( \\rho _ i , \\rho _ j ) d t + \\sum _ { j \\in N ( i ) } ( S _ i - S _ j ) ( \\Sigma _ i - \\Sigma _ j ) \\frac { \\partial \\theta } { \\partial \\rho _ i } ( \\rho _ i , \\rho _ j ) \\circ d W _ t = 0 . \\end{align*}"}
{"id": "7516.png", "formula": "\\begin{align*} \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) + \\arg \\left ( \\sigma - 1 + i T \\right ) = \\arctan \\left ( \\frac { T } { \\sigma } \\right ) + \\arctan \\left ( \\frac { T } { \\sigma - 1 } \\right ) + \\pi \\end{align*}"}
{"id": "2350.png", "formula": "\\begin{align*} X f ( x ) = x f ( x ) P f ( x ) = \\tfrac { 1 } { 2 \\pi i } f ' ( x ) . \\end{align*}"}
{"id": "3709.png", "formula": "\\begin{align*} & \\sup _ { t \\in [ 0 , T ) } \\| B ( t ) \\| _ { L ^ 2 } \\\\ \\geq & \\ \\left ( \\sum _ { q \\geq 0 } e ^ { - 2 \\mu \\lambda _ q ^ \\alpha t } \\| B _ q ( 0 ) \\| ^ 2 _ { L ^ 2 } \\right ) ^ { \\frac 1 2 } \\\\ & - C \\left ( \\sup _ { t \\in [ 0 , T ) } \\| B ( t ) \\| _ { L ^ 2 } \\right ) \\left ( \\sup _ { t \\in ( 0 , T ) } t ^ { \\frac { m - ( \\frac 5 2 - \\alpha ) } { \\alpha } } \\| B ( t ) \\| _ { H ^ m } \\right ) \\end{align*}"}
{"id": "3612.png", "formula": "\\begin{align*} N _ { 1 } ^ { n } ( c ) , \\ ; \\ ; N _ { 1 } ^ { n } ( c \\tilde { c } ^ { \\# } ) , \\quad \\ ; c = ( - 1 ) ^ { ( n + 1 ) / 2 } a _ { 1 , n + 2 } \\quad \\tilde { c } = ( - 1 ) ^ { ( n + 1 ) / 2 } a _ { 1 , n + 1 } \\ , \\end{align*}"}
{"id": "1595.png", "formula": "\\begin{align*} \\| D ^ k ( \\mu ^ a g _ \\mu ) \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } = \\mu ^ { a + k - \\frac { d - 1 } { r } } \\| D ^ k g \\| _ { L ^ r ( \\mathbb { R } ^ { d - 1 } ) } . \\end{align*}"}
{"id": "9431.png", "formula": "\\begin{align*} w ( 0 ) = \\gamma \\geq 0 , \\ , w ' ( 0 ) = 0 , \\ , w ( y ) = y ^ { - 2 } \\mbox { a s } y \\to + \\infty , \\end{align*}"}
{"id": "4626.png", "formula": "\\begin{gather*} \\kappa : = \\left ( \\sum S ( R '' ) R ' \\right ) \\mathbf { r } ^ { - 1 } = K ^ { - 1 } . \\end{gather*}"}
{"id": "8890.png", "formula": "\\begin{align*} p : = \\iota _ 0 \\circ \\pi : X \\ast I & \\to X \\ast I \\\\ ( x , ( s , t ) ) & \\mapsto ( x , ( s + t , 0 ) ) \\end{align*}"}
{"id": "2156.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M _ n ) } ) = \\Big ( k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\Big ) \\Big ( \\sum _ { s = 1 } ^ \\infty ( 1 - c ) ^ { k ( s + 1 ) } \\frac 1 { k s } \\Big ) , \\\\ & \\ M _ n \\sim c n , \\ c \\in ( 0 , 1 ) , \\end{aligned} \\end{align*}"}
{"id": "8839.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { t + 1 } e _ { F _ { 0 } } ( X _ { i } , S ) \\geq t k ' - 2 t l . \\end{align*}"}
{"id": "9108.png", "formula": "\\begin{align*} ( f \\circ _ i g ) \\circ _ j h = \\begin{cases} ( f \\circ _ j h ) \\circ _ { i + l - 1 } h & 0 \\leq j \\leq i - 1 \\\\ f \\circ _ i ( g \\circ _ { j - i } h ) & i \\leq j \\leq n - 1 , \\end{cases} \\end{align*}"}
{"id": "7798.png", "formula": "\\begin{align*} \\mathbb E [ Y _ 1 ^ 2 ] = \\mathbb E [ Y _ 2 ^ 2 ] = \\mathbb E [ Y _ 3 ^ 2 ] . \\end{align*}"}
{"id": "1380.png", "formula": "\\begin{align*} n _ { i x } = \\frac { \\tilde { E } _ i n _ i ^ 3 } { n _ i ^ 2 - 1 } , i = 1 , 2 . \\end{align*}"}
{"id": "8457.png", "formula": "\\begin{align*} ( \\gamma \\delta ) ^ * = ( \\gamma ^ * \\delta ) ^ * = ( \\alpha ^ * \\beta ^ * \\delta ) ^ * . \\end{align*}"}
{"id": "1979.png", "formula": "\\begin{align*} \\Phi = \\varepsilon _ { \\mathbb C } + \\Phi \\succ \\beta , \\end{align*}"}
{"id": "7196.png", "formula": "\\begin{align*} E ( T , X ( T ) ) & = - ( \\nabla \\phi \\ast ( G \\ast \\mathcal S + \\mathcal S ) ) ( t , X ( t ) ) , \\\\ \\lim _ { s \\rightarrow \\infty } \\nabla \\phi * \\rho [ h _ { V ( T ) } ] ) ( s , 0 ) & = \\lim _ { R \\rightarrow \\infty } ( \\nabla \\phi \\ast ( G \\ast S _ { R , X ( T ) , V ( T ) } + S _ { R , X ( T ) , V ( T ) } ) ( R , X ( T ) ) . \\end{align*}"}
{"id": "9314.png", "formula": "\\begin{align*} \\mathcal { L } ( \\mathbf { x } , \\mathbf { y } ) = \\frac { 1 } { 2 } \\mathbf { x } ^ T H \\mathbf { x } + \\mathbf { g } ^ T \\mathbf { x } - \\mathbf { y } ^ T ( A \\mathbf { x } - \\mathbf { b } ) + I _ D ( \\mathbf { x } , \\mathbf { y } ) , \\end{align*}"}
{"id": "5879.png", "formula": "\\begin{align*} D = ( - R , R ) \\times ( - L ' , L ' ) , \\end{align*}"}
{"id": "6314.png", "formula": "\\begin{align*} E \\big [ ( s \\mathcal { T } ( \\nu ) ) ^ 2 \\big ] & = \\frac { s ^ 2 \\nu } { \\nu - 2 } , \\\\ \\big ( s \\mathcal { T } ( \\nu ) \\big ) & = \\frac { K _ { \\nu / 2 } ( \\sqrt { \\nu } t s ) ( \\sqrt { \\nu } t s ) ^ { \\nu / 2 } } { 2 ^ { \\nu / 2 - 1 } \\Gamma ( \\nu / 2 ) } , \\ \\forall t \\ge 0 . \\end{align*}"}
{"id": "2988.png", "formula": "\\begin{align*} S ( z ) = I _ E + i \\alpha ( L ^ * - z I ) ^ { - 1 } \\alpha : E \\to E , z \\in \\mathbb C _ + . \\end{align*}"}
{"id": "6136.png", "formula": "\\begin{align*} ( 1 - x ) ^ { n + 1 } \\sum _ { k = 0 } ^ m { n + k \\choose k } x ^ k + x ^ { m + 1 } \\sum _ { k = 0 } ^ n { m + k \\choose k } ( 1 - x ) ^ k = 1 , \\end{align*}"}
{"id": "6632.png", "formula": "\\begin{align*} \\Sigma _ 1 : = p ^ { ( w - 1 ) h _ p } \\sum _ { m = 0 } ^ { k _ p - h _ p - 1 } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( \\frac { 1 } { 2 } + s _ 1 - z ) } p ^ { n ( \\frac { 1 } { 2 } + s _ 2 + z ) } } \\end{align*}"}
{"id": "1782.png", "formula": "\\begin{align*} \\operatorname { t r } _ g ( f ) : = \\int _ { G / Z _ g } f ( h g h ^ { - 1 } ) d ( h Z _ g ) . \\end{align*}"}
{"id": "1543.png", "formula": "\\begin{align*} j ( \\tilde { \\tau } _ m ( g _ { \\infty } \\times h _ { \\infty } ) , z _ 0 \\times z _ 0 ) = j ( \\tilde { \\tau } _ m , g _ { \\infty } z _ 0 \\times h _ { \\infty } z _ 0 ) j ( g _ { \\infty } , z _ 0 ) \\overline { j ( h _ { \\infty } , z _ 0 ) } . \\end{align*}"}
{"id": "2900.png", "formula": "\\begin{align*} \\| \\nabla Q \\| _ 2 ^ 2 + \\| Q \\| _ 2 ^ 2 - \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ p d x = 0 . \\end{align*}"}
{"id": "4087.png", "formula": "\\begin{align*} \\mu _ n ( x ) = \\frac { \\mu ( x ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } } \\mbox { a n d } \\sigma _ n ( x ) = \\frac { \\sigma ( x ) } { 1 + n ^ { - 1 / 2 } | x | ^ { \\ell _ \\mu } } \\end{align*}"}
{"id": "5209.png", "formula": "\\begin{align*} B f ( x ) = \\sum \\limits _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } g _ j \\ast f ( x ) . \\end{align*}"}
{"id": "7277.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ \\infty \\frac { 1 } { 1 - \\alpha X ^ k } = \\sum _ { k = 0 } ^ \\infty \\frac { \\alpha ^ k } { X ^ k ! } . \\end{align*}"}
{"id": "3866.png", "formula": "\\begin{align*} l _ { 4 - s , 1 } + l _ { 4 , s + 1 } , s = \\overline { 0 , i _ 0 - 1 } , \\end{align*}"}
{"id": "5866.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } v _ \\eta ( x ) = 0 , \\end{align*}"}
{"id": "6923.png", "formula": "\\begin{align*} \\phi _ n ^ \\star = \\mod \\Bigl [ \\phi _ { d } - ( \\phi _ { h , n } + \\phi _ { g , n } ) , 2 \\pi \\Bigr ] , \\end{align*}"}
{"id": "7055.png", "formula": "\\begin{align*} \\lim _ { K \\rightarrow + \\infty } p ( x _ K ) \\sum _ { j = 0 } ^ { { 1 \\over \\delta _ K } - 1 } h _ K G \\left ( \\overline { ( i _ K - j ) \\delta _ K } \\ , \\log K \\right ) & = \\lim _ { K \\rightarrow + \\infty } p ( x _ K ) \\sum _ { \\ell = - \\lfloor 1 / 2 \\delta _ K \\rfloor } ^ { 1 / \\delta _ K - 1 - \\lfloor 1 / 2 \\delta _ K \\rfloor } h _ K G \\left ( h _ K \\ell \\right ) \\\\ & = p ( x ) \\int _ { \\mathbb { R } } G ( y ) \\ , d y = p ( x ) , \\end{align*}"}
{"id": "6927.png", "formula": "\\begin{align*} \\begin{cases} L u : = - \\nabla \\cdot ( \\mu \\nabla u ) + \\boldsymbol { \\beta } \\cdot \\nabla u + \\sigma u = f & \\Omega \\ , , \\\\ u = 0 & \\Gamma \\ , , \\end{cases} \\end{align*}"}
{"id": "6281.png", "formula": "\\begin{align*} \\widehat { K } _ { \\mathrm { a - c p t - f } } ^ \\mathbb { R } = 2 \\Re \\{ \\zeta _ \\mathcal { K } \\} / \\sqrt { \\pi \\bar { p } \\big ( 1 + \\frac { 1 } { N _ 1 \\bar { \\gamma } ' } \\big ) } , \\end{align*}"}
{"id": "6307.png", "formula": "\\begin{align*} \\sum _ { k '' = 0 } ^ { K _ l } p _ { K '' | K ' } ( k '' ) & = \\ ! \\sum _ { k '' = 0 } ^ { \\infty } \\ ! \\ ! p _ { K '' | K ' } ( k '' ) \\ ! - \\ ! \\ ! \\sum _ { k '' = 0 } ^ { \\infty } \\ ! \\ ! p _ { K '' | K ' } ( k '' \\ ! + \\ ! K _ l \\ ! + \\ ! 1 ) \\\\ & = \\frac { 1 } { 1 - \\xi } \\bigg ( 1 - B _ { \\xi } \\big ( 1 + K _ l , 1 + K ' \\big ) \\frac { ( 1 + K _ l + K ' ) ! } { K _ l ! K ' ! } \\bigg ) , \\end{align*}"}
{"id": "5856.png", "formula": "\\begin{align*} \\bar \\chi = \\sum _ { i \\notin J } \\frac { \\bar \\alpha _ { i - 1 } } { \\lambda _ i } + \\sum _ { i \\in J } \\frac { \\bar \\alpha _ i } { \\mu _ i } . \\end{align*}"}
{"id": "737.png", "formula": "\\begin{align*} K _ { \\alpha \\alpha } ^ { ( \\ell ) } = K _ { \\beta \\beta } ^ { ( \\ell ) } = K , \\end{align*}"}
{"id": "9116.png", "formula": "\\begin{align*} \\bar x _ 1 ^ 2 & = \\sum _ { j = 2 } ^ { 2 m } ( - 1 ) ^ j ( \\bar x _ 1 + \\bar x _ j ) ( \\bar x _ 1 + \\bar x _ { j + 1 } ) \\\\ \\bar x _ i ^ 2 & = ( \\bar x _ 1 + \\bar x _ i ) ^ 2 + \\sum _ { j = 2 } ^ { i - 1 } ( - 1 ) ^ { i + j } ( \\bar x _ 1 + \\bar x _ j ) ( \\bar x _ 1 + \\bar x _ { j + 1 } ) + \\sum _ { j = i } ^ { 2 m } ( - 1 ) ^ { i + j + 1 } ( \\bar x _ 1 + \\bar x _ j ) ( \\bar x _ 1 + \\bar x _ { j + 1 } ) . \\end{align*}"}
{"id": "4575.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( a _ n S _ n \\geq x \\sqrt { v _ n + n a _ n ^ 2 \\sigma ^ 2 } \\ , \\Big ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) \\ \\ \\ a n d \\ \\ \\ \\frac { \\mathbf { P } \\Big ( a _ n S _ n \\leq - x \\sqrt { v _ n + n a _ n ^ 2 \\sigma ^ 2 } \\ , \\Big ) } { \\Phi \\left ( - x \\right ) } = 1 + o ( 1 ) \\end{align*}"}
{"id": "4643.png", "formula": "\\begin{align*} F ( X ) = X ^ k - a _ 1 X ^ { k - 1 } - \\cdots - a _ k , \\end{align*}"}
{"id": "1697.png", "formula": "\\begin{align*} D _ i = \\pm \\prod _ { j \\neq i } ( c _ j - c _ i ) ^ { - 1 } . \\end{align*}"}
{"id": "7072.png", "formula": "\\begin{align*} \\lefteqn { \\beta _ j ^ K ( t _ K ) - \\beta _ { i _ K } ^ K ( t _ K ) } \\\\ & = \\widetilde { \\beta } ^ K ( t _ K , j \\delta _ K ) - \\widetilde { \\beta } ^ K ( t _ K , x _ K ) - \\left ( \\beta _ { i _ K } ^ K ( t _ K ) - \\widetilde { \\beta } ^ K ( t _ K , x _ K ) \\right ) \\\\ & \\leq \\varphi ( t _ K , j \\delta _ K ) - \\varphi ( t _ K , x _ K ) + \\widetilde { M } ^ K ( t _ K , j \\delta _ K ) - \\widetilde { M } ^ K ( t _ K , x _ K ) - \\left ( \\beta _ { i _ K } ^ K ( t _ K ) - \\widetilde { \\beta } ^ K ( t _ K , x _ K ) \\right ) \\end{align*}"}
{"id": "9109.png", "formula": "\\begin{align*} ( f \\smile g ) \\circ _ j h = \\begin{cases} ( f \\circ _ i h ) \\smile g & 0 \\leq i \\leq m - 1 \\\\ f \\smile ( g \\circ _ { i - m } h ) & m \\leq i \\leq m + n - 1 . \\end{cases} \\end{align*}"}
{"id": "7364.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { \\gamma } ^ \\ast ( s ) = - \\pi R ^ 2 & \\ s > 0 , \\\\ \\gamma ^ \\ast ( 0 ) = r , & \\end{array} \\right . \\end{align*}"}
{"id": "8015.png", "formula": "\\begin{align*} ( - 1 ) \\cdot * _ \\Sigma i _ \\Sigma ^ * \\Pi _ \\ell d , = : \\partial _ \\Sigma : \\mathfrak { E } ( \\mathcal { M } ) \\longrightarrow \\mathfrak { E } ( \\Sigma ) \\end{align*}"}
{"id": "2533.png", "formula": "\\begin{align*} \\pi ( \\xi , \\eta , 0 ) \\pi ( \\Phi ) = \\pi ( \\Phi ^ { \\xi , \\eta } ) . \\end{align*}"}
{"id": "728.png", "formula": "\\begin{align*} \\frac { d } { d t } K ( t ) = - a K ( t ) ^ 2 , \\end{align*}"}
{"id": "6615.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\delta } \\int _ { - \\delta } ^ { \\delta } | 1 + e ^ { \\xi } r | ^ { - \\omega } + | 1 - e ^ { \\xi } r | ^ { - \\omega } \\ , d \\xi = \\frac { 1 } { 2 \\pi i } \\int _ { ( c ) } \\mathcal { H } ( z , \\omega ) r ^ { - z } \\frac { e ^ { \\delta z } - e ^ { - \\delta z } } { 2 \\delta z } \\ , d z , \\end{align*}"}
{"id": "3509.png", "formula": "\\begin{align*} D _ { 1 3 1 } \\ll t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\begin{cases} 1 & ( \\sigma _ 2 > 1 ) \\\\ \\log t _ 3 & ( \\sigma _ 2 = 1 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 2 } & ( \\sigma _ 2 < 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "4658.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } x _ 1 ( t ) - x _ { 2 } ( t ) = + \\infty , \\end{align*}"}
{"id": "1975.png", "formula": "\\begin{align*} \\Lambda ( \\phi \\succ \\gamma ) = \\sum _ { u , v \\in \\N ^ * } f _ v g _ u x _ u x _ v = g ( x ) f ( x ) . \\end{align*}"}
{"id": "2698.png", "formula": "\\begin{align*} \\norm { L ( m ) } _ { o p } = \\norm { m } _ \\infty \\norm { L ( m ) ^ { - 1 } } _ { o p } \\leq \\norm { L ( m ^ { - 1 } ) } _ { o p } = \\norm { m ^ { - 1 } } _ \\infty \\end{align*}"}
{"id": "5758.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| w ^ j \\right \\| _ { C ^ { 2 , \\alpha } \\left ( B _ { 3 r / 5 } ^ { m } ( 0 ) \\right ) } \\leq C ( r ) \\left \\| \\Delta _ { h } z ^ j \\right \\| _ { C ^ { 0 , \\alpha } \\left ( B _ { 3 r / 5 } ^ { m } ( 0 ) \\right ) } \\leq C ( r ) \\Psi ( \\tau \\ , | \\ , n , \\alpha , Q ) . \\end{aligned} \\end{align*}"}
{"id": "5513.png", "formula": "\\begin{align*} \\Phi _ { \\delta } ^ { x _ 0 } ( d , e , t ) = e ^ { ( \\beta + L ) t } d + \\varphi _ { \\beta + L } ( t ) e \\end{align*}"}
{"id": "6754.png", "formula": "\\begin{align*} D ^ 2 u _ { - 1 } & = \\frac { 1 } { \\bar { h } _ { - 1 } } \\left [ \\frac { 1 } { h _ { - 1 } } u _ { - 2 } - \\left ( \\frac { 1 } { h _ { - 1 } } + \\frac { 1 } { h _ { 0 } } \\right ) u _ { - 1 } + \\frac { 1 } { h _ { 0 } } u _ 0 \\right ] . \\end{align*}"}
{"id": "4724.png", "formula": "\\begin{align*} W _ 2 = W _ { 2 , 1 } + W _ { 2 , 2 } + W _ { 2 , 3 } + W _ { 2 , 4 } + W _ { 2 , 5 } , \\end{align*}"}
{"id": "693.png", "formula": "\\begin{align*} \\partial _ { x _ { 0 ; \\alpha } } z _ { i ; \\alpha } ^ { ( \\ell ) } : = z _ { i ; \\alpha } ^ { ( \\ell ) } . \\end{align*}"}
{"id": "382.png", "formula": "\\begin{align*} W - V = a ^ j Y _ j - b ^ k \\overline { Y } _ k = 0 , \\end{align*}"}
{"id": "3101.png", "formula": "\\begin{align*} \\beta ^ 3 \\gamma ^ 3 = \\beta \\gamma l = k \\ , ; \\gamma = \\frac { k } { \\beta l } \\ , ; k ^ 2 = l ^ 3 \\ , . \\end{align*}"}
{"id": "4005.png", "formula": "\\begin{align*} i \\int _ 0 ^ 1 \\left ( \\Im ( \\overline { \\xi ( x ) } \\xi ^ \\prime ( x ) ) + \\Im ( \\overline { \\eta ( x ) } \\eta ^ \\prime ( x ) ) \\right ) d x + \\int _ { 0 } ^ { 1 } { \\xi ^ \\prime ( x ) } \\overline { \\eta ( x ) } d x - \\int _ { 0 } ^ { 1 } \\overline { \\xi ^ \\prime ( x ) } { \\eta ( x ) } d x - \\int _ { 0 } ^ { 1 } | \\eta ^ { \\prime } ( x ) | ^ 2 d x \\\\ = \\lambda \\int _ { 0 } ^ { 1 } | \\xi ( x ) | ^ 2 d x + \\lambda \\int _ { 0 } ^ { 1 } | \\eta ( x ) | ^ 2 d x , \\end{align*}"}
{"id": "6413.png", "formula": "\\begin{align*} \\Omega _ { \\Sigma } ^ * = \\{ ( z _ 1 , \\dots , z _ { n } ) \\in \\mathbb D ^ n \\ , : ( z _ 1 , \\dots , z _ { n } ) \\in \\sigma _ T ( P _ 1 ^ { \\perp } U _ 1 ^ * + z P _ 1 U _ 1 ^ * , \\dots , P _ n ^ { \\perp } U _ n ^ * + z P _ n U _ n ^ * ) \\} . \\end{align*}"}
{"id": "3691.png", "formula": "\\begin{align*} B _ t + B J _ x + \\mu \\Lambda ^ \\alpha B = 0 , \\ \\ \\ B _ x = \\mathcal H J \\end{align*}"}
{"id": "86.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 ^ { \\frac { 1 } { 2 } } ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) \\end{align*}"}
{"id": "932.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( k , p ) = & \\sum _ { m = p } ^ { k } \\bigg \\{ \\binom { k } { m } ( - 1 ) ^ { k - m } \\langle 1 \\rangle _ { k - m , \\lambda } S _ { 2 , \\lambda } ( m + 1 , p + 1 ) \\\\ & + \\lambda k \\binom { k - 1 } { m } ( - 1 ) ^ { k - m - 1 } \\langle 1 \\rangle _ { k - m - 1 , \\lambda } S _ { 2 , \\lambda } ( m + 1 , p + 1 ) \\bigg \\} . \\end{align*}"}
{"id": "1338.png", "formula": "\\begin{align*} q _ { i ' } < E ( \\alpha _ { k } ) = M - q _ { i } < \\frac { 1 } { 3 } q _ { i } < q _ { i } \\end{align*}"}
{"id": "8412.png", "formula": "\\begin{align*} P _ { N \\eta } h ( x ) - \\tilde { Q } _ N h ( x ) = \\sum _ { i = 1 } ^ { N } \\tilde { Q } _ { i - 1 } \\big ( P _ { \\eta } - \\tilde { Q } _ { 1 } \\big ) P _ { ( N - i ) \\eta } h ( x ) . \\end{align*}"}
{"id": "3948.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) + p ( t ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) . \\end{dcases} \\end{align*}"}
{"id": "5028.png", "formula": "\\begin{align*} \\beta _ p \\colon N Y ^ { o p } _ p \\times _ { Y _ 0 } B ( Y _ 0 \\sl f ) \\to N Y ^ { o p } _ p = N Y ^ { o p } _ p \\times _ { Y _ 0 } Y _ 0 . \\end{align*}"}
{"id": "5839.png", "formula": "\\begin{align*} G ^ + _ { t , s } ( x ) & = \\left \\{ Z ^ \\alpha _ { t , s } ( x ) = 1 , Z ^ { \\beta } _ { t , s } ( x ) = Z ^ { \\Lambda } _ { t , s } ( x - 1 ) = Z ^ { \\Lambda } _ { t , s } ( x + 1 ) = 0 \\right \\} , \\\\ G ^ - _ { t , s } ( x ) & = \\left \\{ Z ^ \\beta _ { t , s } ( x ) = 1 , Z ^ { \\alpha } _ { t , s } ( x ) = Z ^ { \\Lambda } _ { t , s } ( x - 1 ) = Z ^ { \\Lambda } _ { t , s } ( x + 1 ) = 0 \\right \\} , \\\\ G ^ 0 _ { t , s } ( x ) & = \\left \\{ Z ^ \\alpha _ { t , s } ( x ) = Z ^ { \\beta } _ { t , s } ( x ) = 0 \\right \\} , \\end{align*}"}
{"id": "3966.png", "formula": "\\begin{align*} \\norm { \\xi _ { \\lambda ^ p _ k } - \\phi _ { k } } ^ 2 _ { L ^ 2 } = \\norm { \\xi _ { \\lambda ^ p _ k } } ^ 2 _ { L ^ 2 } \\leq C k ^ { - 2 } , \\ \\ \\forall k \\geq k _ 0 . \\end{align*}"}
{"id": "519.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { r } _ { 0 } ( \\tau ) } ( [ k ] _ { \\alpha } ^ { i } ) ^ { - 1 } \\cdot \\Delta _ { \\mathbf { r } _ { 0 } ( \\tau ) } ( \\mathcal { I } ) = \\det ( \\mathbf { r } _ { ( \\alpha ) } ( \\tau ) ) \\end{align*}"}
{"id": "8347.png", "formula": "\\begin{align*} g ( p ) = r = \\log \\left ( 1 + c \\ , | \\log p | \\right ) \\sim \\log | \\log p | , \\textrm { a s } p \\to 0 . \\end{align*}"}
{"id": "3927.png", "formula": "\\begin{align*} \\zeta _ 2 ( z \\pm 2 \\ell ) = \\zeta _ 2 ( z ) \\pm \\frac { \\pi } { \\delta } , \\wp _ 2 ( z \\pm 2 \\ell ) = \\wp ( z ) , f _ 2 ( z \\pm 2 \\ell ) = f _ 2 ( z ) \\pm \\frac { 2 \\pi } { \\delta } \\zeta _ 2 ( z ) + \\bigg ( \\frac { \\pi } { \\delta } \\bigg ) ^ 2 \\end{align*}"}
{"id": "7087.png", "formula": "\\begin{align*} \\begin{gathered} ( f _ { { W ^ 0 _ V } , i } ^ { t ^ 2 _ m } ) ^ * ( f _ { { W ^ 0 _ V } , i } ^ { t ^ 2 _ m } ) _ * \\omega ^ m _ { X ^ { t ^ 2 _ m } _ { W ^ 0 _ V , i } / W _ { V , i } } \\longrightarrow \\omega ^ m _ { X ^ { t ^ 2 _ m } _ { W ^ 0 _ { V , i } } / W ^ 0 _ { V , i } } , \\\\ ( f ^ { t ^ 2 _ m } ) ^ * f _ * ^ { t ^ 2 _ m } \\omega ^ m _ { U ^ { t ^ 2 _ m } / V } \\longrightarrow \\omega ^ m _ { U ^ { t ^ 2 _ m } / V } \\end{gathered} \\end{align*}"}
{"id": "5787.png", "formula": "\\begin{align*} \\sum _ { l \\in S _ { \\nu } \\cup \\{ 0 \\} } w _ { \\nu , \\nu ' , l } = 1 , \\ , \\forall \\nu ' \\in { \\cal P } . \\end{align*}"}
{"id": "5609.png", "formula": "\\begin{align*} \\Xi _ { \\mathcal { C } ^ c } : = \\{ X \\in \\Xi ~ | ~ X ( \\mathcal { C } ^ c ) \\subseteq \\mathcal { C } ^ c \\} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\Xi _ { \\mathcal { C } \\mathcal { C } ^ c } : = \\{ X \\in \\Xi ~ | ~ X ( \\mathcal { X } ) \\subseteq \\mathcal { C } ^ c \\} , \\end{align*}"}
{"id": "8020.png", "formula": "\\begin{align*} \\partial _ { \\Sigma } \\chi ^ { * } \\phi _ 0 = \\rho ^ { * } _ { ( 1 ) } \\partial _ { \\Sigma _ 0 } \\phi _ 0 = \\psi . \\end{align*}"}
{"id": "3366.png", "formula": "\\begin{align*} \\begin{cases} [ z _ 1 , [ \\mathfrak { X } , z _ 2 ] , [ \\mathfrak { X } , z _ 3 ] ] + [ [ \\mathfrak { X } , z _ 1 ] , z _ 2 , [ \\mathfrak { X } , z _ 3 ] ] + [ [ \\mathfrak { X } , z _ 1 ] , [ \\mathfrak { X } , z _ 2 ] , z _ 3 ] = 0 , \\\\ [ [ \\mathfrak { X } , z _ 1 ] , [ \\mathfrak { X } , z _ 2 ] , [ \\mathfrak { X } , z _ 3 ] ] = 0 , \\ ; \\ ; z _ 1 , z _ 2 , z _ 3 \\in L . \\end{cases} \\end{align*}"}
{"id": "7220.png", "formula": "\\begin{align*} F _ { s , t , x } : = \\{ v \\in \\R ^ 3 : s < { \\mathcal T } _ { t , x _ 1 , v _ 1 } - 1 , | v | < \\delta ^ { - \\beta } , | v _ * ^ \\perp ( t , x , v _ 1 ) - v ^ \\perp | \\check \\tau _ { t , x } > \\sqrt { \\mathcal T _ { t , x _ 1 , v _ 1 } - s } \\} . \\end{align*}"}
{"id": "4269.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\left \\| w ^ { ( k ) } ( t , \\cdot ) - w ( t , \\cdot ) \\right \\| _ { H ^ 2 ( \\R \\backslash \\{ 0 \\} ) } ~ = ~ 0 . \\end{align*}"}
{"id": "4174.png", "formula": "\\begin{align*} \\omega _ { p } ( A ) & = \\underset { \\alpha \\in \\mathbb { R } } { s u p } \\norm { R e ( e ^ { i \\alpha } A ) } _ { p } \\\\ & = \\underset { \\theta \\in \\mathbb { R } } { s u p } \\norm { R e ( e ^ { i ( \\theta - \\frac { \\pi } { 2 } ) } A ) } _ { p } \\\\ & = \\underset { \\theta \\in \\mathbb { R } } { s u p } \\norm { I m ( e ^ { i \\theta } A ) } _ { p } . \\end{align*}"}
{"id": "3537.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs { \\zeta _ { M T , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\int _ 2 ^ T \\abs { \\Sigma _ 2 ( s _ 1 , s _ 2 , s _ 3 ) + E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\int _ 2 ^ T \\abs { \\Sigma _ 2 ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 + O \\left ( \\int _ 2 ^ T \\abs { \\Sigma _ 2 ( s _ 1 , s _ 2 , s _ 3 ) E ( s _ 1 , s _ 2 , s _ 3 ) } d t _ 3 \\right ) \\\\ & + O \\left ( \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\right ) . \\end{align*}"}
{"id": "3009.png", "formula": "\\begin{align*} v + w + x + y + z & = 5 \\psi u \\\\ v w x y z & = u ^ 5 , \\end{align*}"}
{"id": "1833.png", "formula": "\\begin{align*} G = \\{ x \\rightarrow x y , y \\rightarrow x ^ 2 \\} , \\end{align*}"}
{"id": "2283.png", "formula": "\\begin{align*} f ( t ) = \\begin{cases} e ^ { 4 \\pi i t } + e ^ { 8 \\pi i t } , & - 3 \\leq t \\leq - 1 \\\\ e ^ { 2 \\pi i t } , & - 1 \\leq t \\leq 1 \\\\ e ^ { 2 \\pi i t ^ 2 } , & 1 \\leq t \\leq 4 \\\\ 0 , & \\end{cases} . \\end{align*}"}
{"id": "5850.png", "formula": "\\begin{align*} \\alpha _ k = ( \\alpha _ { k - 1 } \\rho _ k ) \\wedge 1 . \\end{align*}"}
{"id": "8846.png", "formula": "\\begin{align*} & ( B ^ { - 1 } ) _ 0 ( { \\bf b } ) = e ^ { - b _ 0 } , ( B ^ { - 1 } ) _ 1 ( { \\bf b } ) = 2 e ^ { - \\frac { 3 } { 2 } b _ 0 } b _ 1 , ( B ^ { - 1 } ) _ 2 ( { \\bf b } ) = e ^ { - 2 b _ 0 } ( - 4 b _ 2 + 8 b _ 1 ^ 2 ) , \\\\ & ( B ^ { - 1 } ) _ 3 ( { \\bf b } ) = e ^ { - \\frac { 5 } { 2 } b _ 0 } \\bigl ( 8 b _ 3 - 6 0 b _ 2 b _ 1 + 5 0 b _ 1 ^ 3 \\bigr ) . \\end{align*}"}
{"id": "5774.png", "formula": "\\begin{align*} \\nabla _ { d f _ j ( X ) } \\left ( \\nabla ^ 2 \\beta ( Y , Y ) \\right ) & = d f _ j ( X ) \\left ( Y Y \\beta \\right ) - d f _ j ( X ) \\left ( ( \\nabla _ Y Y ) \\beta \\right ) \\\\ & = Y Y \\left < d f _ j ( X ) , \\nabla \\beta \\right > - ( \\nabla _ Y Y ) d f _ j ( X ) \\beta . \\end{align*}"}
{"id": "1911.png", "formula": "\\begin{align*} p \\odot ^ \\flat p ' & = ( \\underline { - 2 \\delta ^ { - 1 } } \\wedge - 3 \\delta ^ { - 1 } \\wedge - 5 \\delta ^ { - 1 } ) \\oplus ( \\underline { - 1 \\delta ^ { 2 } } \\wedge - 3 \\delta ^ { 2 } ) \\oplus ( \\underline { 1 \\delta ^ { 4 } } ) \\\\ & = - 2 \\delta ^ { - 1 } \\oplus - 1 \\delta ^ { 2 } \\oplus 1 \\delta ^ { 4 } . \\end{align*}"}
{"id": "632.png", "formula": "\\begin{align*} \\begin{cases} \\ f _ 0 ( x , n ) \\ = \\ u ( 2 x + 1 , n ) \\\\ [ 8 p t ] \\ g _ 0 ( x , n ) \\ = \\ v ( 2 x + 1 , n ) \\\\ [ 8 p t ] \\ h _ 0 ( x , n ) \\ = \\ w ( 2 x + 1 , n ) \\end{cases} . \\end{align*}"}
{"id": "6125.png", "formula": "\\begin{align*} u _ s + ( m + 2 ) s ^ { m + 1 } w _ s & = 0 \\\\ u _ t + v _ s + ( m + 2 ) s ^ { m + 1 } w _ t + \\varepsilon ( n + 2 ) t ^ { n + 1 } w _ s & = 0 \\\\ v _ t + \\varepsilon ( n + 2 ) t ^ { n + 1 } w _ t & = 0 \\end{align*}"}
{"id": "5571.png", "formula": "\\begin{align*} \\| x \\| \\ = \\ \\left \\| x - P _ A ( x ) + \\sum _ { n \\in A } e _ n ^ * ( x ) e _ n \\right \\| & \\ \\leqslant \\ \\sup _ { \\delta } \\left \\| x - P _ A ( x ) + 1 _ { \\delta A } \\right \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ { \\lambda , i b } \\left \\| x - P _ A ( x ) + 1 _ { \\varepsilon B } \\right \\| . \\end{align*}"}
{"id": "4363.png", "formula": "\\begin{align*} \\phi _ { i , , \\beta } ( \\xi ) = \\sum _ { j = 0 } ^ i c _ { i , j } ( 2 \\beta ) ^ j b ^ j T _ j ( \\xi ) + \\tilde \\lambda \\sum _ { j = 0 } ^ i b ^ { j + 1 } \\left ( c _ { i , j } ( 2 \\beta ) ^ { j + 1 } T _ { j + 1 } ( \\xi ) + S _ j ( \\xi ) \\right ) + b R _ i ( \\xi ) , \\end{align*}"}
{"id": "2299.png", "formula": "\\begin{align*} \\widetilde { f } = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "4633.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 0 } ^ \\infty d _ k ( n ) q ^ n = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty ^ k } { ( q ; q ) _ \\infty ^ { 3 k + 1 } } , \\end{align*}"}
{"id": "5545.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d Z ( t ) & = & \\ell ( Z ( t ) ) d t + a ( Z ( t ) ) d W ( t ) \\medskip \\\\ Z ( 0 ) & = & z , \\end{array} \\right . \\end{align*}"}
{"id": "5016.png", "formula": "\\begin{align*} Z _ * ( x , y ) = \\Lambda \\left ( \\begin{pmatrix} ( \\eta _ * ^ { r _ 0 } \\circ \\xi _ * ) ^ { r _ 1 } \\circ \\eta _ * ( x ) \\\\ ( \\eta _ * ^ { r _ 0 } \\circ \\xi _ * ) ^ { r _ 1 - 1 } \\circ \\eta _ * ( x ) \\end{pmatrix} , \\begin{pmatrix} \\eta _ * ^ { r _ 0 } \\circ \\xi _ * ( x ) \\\\ x \\end{pmatrix} \\right ) . \\end{align*}"}
{"id": "913.png", "formula": "\\begin{align*} S _ { 1 , \\lambda } ( k + 1 , l ) = S _ { 1 , \\lambda } ( k , l - 1 ) - ( k - l \\lambda ) S _ { 1 , \\lambda } ( k , l ) , \\end{align*}"}
{"id": "3728.png", "formula": "\\begin{align*} K _ 2 \\leq & - \\frac { 1 } { 2 \\pi X ( x _ 0 , t ) } \\int _ { - X ( x _ 0 , t ) } ^ { X ( x _ 0 , t ) } B _ y ( y , t ) \\ , d y \\\\ = & - \\frac { 1 } { 2 \\pi X ( x _ 0 , t ) } \\left ( B ( X ( x _ 0 , t ) , t ) - B ( - X ( x _ 0 , t ) , t ) \\right ) \\\\ = & - \\frac { 1 } { \\pi X ( x _ 0 , t ) } . \\end{align*}"}
{"id": "4714.png", "formula": "\\begin{align*} \\frac { b _ { i j k } \\mu _ k ( t ) } { x ^ 3 _ { i j } ( t ) } & = \\frac { b _ { i j k } \\alpha _ k } { ( \\alpha _ i - \\alpha _ j ) ^ 3 t ^ { 2 } } \\frac { 1 + [ \\mu _ k ( t ) - \\alpha _ k / \\sqrt { t } ] / ( \\alpha _ { k } \\sqrt { t } ) } { ( 1 + [ x _ { i j } ( t ) - ( \\alpha _ i - \\alpha _ j ) \\sqrt { t } ] / [ ( \\alpha _ i - \\alpha _ j ) \\sqrt { t } ] ) ^ 3 } \\\\ & = \\frac { b _ { i j k } \\alpha _ k } { ( \\alpha _ i - \\alpha _ j ) ^ 3 t ^ { 2 } } + O \\bigg ( \\frac { 1 } { t ^ { 9 / 4 } } \\bigg ) . \\end{align*}"}
{"id": "9079.png", "formula": "\\begin{align*} \\partial _ t \\rho = & \\nabla \\cdot \\left ( \\nabla \\rho + \\rho \\nabla \\phi \\right ) , \\\\ - \\Delta \\phi = & \\rho , \\end{align*}"}
{"id": "7160.png", "formula": "\\begin{align*} \\int _ Q { \\rm d i v } ( \\phi ) \\ , d x = 0 \\quad \\phi \\in C ^ 1 _ c ( \\Omega \\cap B _ 1 ; \\R ^ d ) \\ , . \\end{align*}"}
{"id": "2271.png", "formula": "\\begin{align*} I ( \\omega ) = \\int _ { \\R } e ^ { - \\pi M x ^ 2 } e ^ { - 2 \\pi i \\omega x } \\ , d x . \\end{align*}"}
{"id": "8847.png", "formula": "\\begin{align*} & v ( x , { \\bf s } _ { \\rm e v e n } ) = 0 , \\\\ & e ^ { u ( x , { \\bf s } _ { \\rm e v e n } ) } = \\sum _ { k = 1 } \\frac 1 k \\sum _ { j _ 1 , \\dots , j _ k \\in \\mathbb { Z } ^ { \\rm e v e n } _ { \\geq 0 } , \\atop j _ 1 + \\cdots + j _ k = 2 k - 2 } { \\rm w t } ( j _ 1 ) \\cdots { \\rm w t } ( j _ k ) \\binom { j _ 1 } { j _ 1 / 2 } \\cdots \\binom { j _ k } { j _ k / 2 } s _ { j _ 1 } \\cdots s _ { j _ k } , \\end{align*}"}
{"id": "9474.png", "formula": "\\begin{align*} I _ { H _ s } = ( p _ 1 ( { \\bf \\underline { t } } ) , \\ldots , p _ s ( { \\bf \\underline { t } } ) ) , \\end{align*}"}
{"id": "2516.png", "formula": "\\begin{align*} \\langle \\pi ( F ) f , g \\rangle = \\int _ { \\mathbf { H } } F ( \\mathbf { h } ) \\langle \\pi ( \\mathbf { h } ) f , g \\rangle \\ , d \\mathbf { h } , \\end{align*}"}
{"id": "2980.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { i } \\in \\mathcal { J } ^ 2 } \\varphi _ 2 ( \\mathbf i _ { 1 : 4 } ) ^ 3 \\varphi _ 2 ( \\mathbf i _ { 5 : 8 } ) ^ 2 = \\sum _ { \\mathbf { i } \\in \\mathcal { I } _ 2 ^ 2 } \\varphi _ 2 ( \\mathbf i _ { 1 : 4 } ) ^ 3 \\varphi _ 2 ( \\mathbf i _ { 5 : 8 } ) ^ 2 + o ( n ^ 4 ) = \\frac { n ^ 4 } { 4 } \\cdot \\frac { 1 } { 9 0 ^ 5 } ( 1 + o ( 1 ) ) , \\end{align*}"}
{"id": "4227.png", "formula": "\\begin{align*} \\inf _ { ( \\mathbf { h } , \\mathbf { g } ) \\in E ^ m } \\mathcal { F } _ N ( \\mathbf { h } , \\mathbf { g } ) = \\liminf _ { j \\to \\infty } \\mathcal { F } _ N ( \\mathbf { h } _ j , \\mathbf { g } _ j ) \\ge \\mathcal { F } _ N ( \\mathbf { h } _ * , \\mathbf { g } _ * ) . \\end{align*}"}
{"id": "8531.png", "formula": "\\begin{align*} \\left \\Vert u _ { 0 } \\right \\Vert _ { H ^ { 1 } } + \\left \\Vert x u _ { 0 } \\right \\Vert _ { L ^ { 2 } } \\leq \\varepsilon \\end{align*}"}
{"id": "5714.png", "formula": "\\begin{align*} 1 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } \\cup { \\gamma _ { 2 } } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 4 } ) = 1 2 - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 4 } ) \\end{align*}"}
{"id": "8008.png", "formula": "\\begin{align*} \\mathbb { M } ^ 2 _ { \\ell } \\ni \\gamma = \\left \\{ ( u _ 0 , v ) \\in \\mathbb { M } ^ 2 \\ , | \\ , v \\in \\mathbb { R } \\right \\} \\end{align*}"}
{"id": "5389.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } f _ { s } \\mathrm { d } B ^ { H } _ { s } = \\lim _ { \\alpha _ { n } \\rightarrow 0 } \\sum _ { k = 0 } ^ { n - 1 } f _ { t _ { k } } \\diamond ( B ^ { H } _ { t _ { k + 1 } } - B ^ { H } _ { t _ { k } } ) . \\end{align*}"}
{"id": "7026.png", "formula": "\\begin{align*} \\| f \\| ^ { 2 } _ { H ^ 2 } = \\sum _ { k = 0 } ^ { \\infty } | a _ k | ^ 2 . \\end{align*}"}
{"id": "6829.png", "formula": "\\begin{align*} s ( t _ { m ( k ) } ) = s ' ( t _ { k ' } ) . \\end{align*}"}
{"id": "322.png", "formula": "\\begin{align*} & d \\rho _ i = \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ i - S _ j ) \\theta _ { i j } ( \\rho ) d t ; \\\\ & d S _ i + ( \\sum _ { j \\in N ( i ) } \\frac 1 2 \\omega _ { i j } ( S _ i - S _ j ) ^ 2 \\frac { \\partial \\theta _ { i j } } { \\partial \\rho _ i } + \\frac 1 8 \\frac { \\partial } { \\partial \\rho _ i } I ( \\rho ) + \\mathbb V _ i + \\sum _ { j \\in N ( i ) } \\mathbb W _ { i j } \\rho _ j ) d t + \\sigma _ i d W _ t = 0 . \\end{align*}"}
{"id": "1013.png", "formula": "\\begin{align*} U : = \\bigg \\{ x \\in B _ \\rho ^ + \\frac { u ( x ) } { \\zeta ( x ) } > \\frac { u ( a ) } { 2 \\zeta ( a ) } \\bigg \\} ; \\end{align*}"}
{"id": "2853.png", "formula": "\\begin{align*} \\int _ \\sigma ^ \\tau \\delta ( t ) d t \\lesssim \\int _ \\sigma ^ \\tau \\ddot { y } _ R ( t ) d t = \\dot { y } _ R ( \\tau ) - \\dot { y } _ R ( \\sigma ) . \\end{align*}"}
{"id": "5095.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\rightarrow 0 ^ + } x \\ , \\coth ( x ) & = \\lim \\limits _ { x \\rightarrow 0 ^ + } x + \\lim \\limits _ { x \\rightarrow 0 ^ + } \\Big ( \\frac { e ^ { 2 x } - e ^ 0 } { 2 x - 0 } \\Big ) ^ { - 1 } = 0 + \\frac { 1 } { \\exp ' ( 0 ) } = 1 . \\end{align*}"}
{"id": "1734.png", "formula": "\\begin{align*} d _ n ( W _ { t , m } , \\ , l _ q ^ { \\nu _ { t , m } } ) \\underset { \\mathfrak { Z } _ 0 } { \\asymp } \\varphi _ i ( t , \\ , m , \\ , n ) : = 2 ^ { \\kappa _ { 1 , i } t + \\kappa _ { 2 , i } m } n ^ { \\sigma _ i } \\end{align*}"}
{"id": "7859.png", "formula": "\\begin{align*} J \\cap K & = ( ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cap ( B \\setminus \\{ \\bar { a } _ n \\} ) ) \\cup \\\\ & \\qquad \\qquad \\cup ( ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cap [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) ) \\\\ & \\qquad \\qquad \\cup ( ( B \\setminus \\{ \\bar { a } _ n \\} ) \\cap [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) ) \\\\ & \\qquad \\qquad \\cup ( [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) \\cap [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) ) \\end{align*}"}
{"id": "904.png", "formula": "\\begin{align*} _ p \\left ( \\frac { B _ { k _ m - n / 2 , \\chi _ T } } { k _ m - n / 2 } \\right ) & = _ p \\left ( B _ { k _ m - n / 2 , \\chi _ T } \\right ) - _ p \\left ( k _ m - n / 2 \\right ) \\\\ & \\geq - 1 - _ p \\left ( 2 - n / 2 \\right ) , \\end{align*}"}
{"id": "7548.png", "formula": "\\begin{align*} \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\Im \\left \\{ \\log \\zeta ( \\sigma + i T ) \\right \\} \\ d \\sigma = 2 \\epsilon \\ \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "3573.png", "formula": "\\begin{align*} \\theta _ z & = \\big ( ( z - q _ c ) ^ { s _ c } , \\ldots , ( z - q _ { b _ z } ) ^ { s _ { b _ z } } \\big ) , \\\\ \\Omega _ z & = \\Big \\{ 1 + p ^ z \\delta _ { n - r } + { \\sum } _ { i = 1 } ^ r p ^ { d ' _ i } \\colon ( d ' _ 1 , \\ldots , d ' _ r ) \\le \\theta _ z , \\ d ' _ r = 0 \\Big \\} . \\end{align*}"}
{"id": "2898.png", "formula": "\\begin{align*} - \\Delta Q + Q - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 1 } = 0 , \\end{align*}"}
{"id": "9232.png", "formula": "\\begin{align*} 0 \\geq \\langle x - y , - ( u - v ) \\rangle = - \\langle x - y , u - v \\rangle \\end{align*}"}
{"id": "9030.png", "formula": "\\begin{align*} E ( \\rho , \\psi ) = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } \\epsilon ( x ) | \\nabla \\psi | ^ 2 \\bigg ) d x , \\end{align*}"}
{"id": "2970.png", "formula": "\\begin{align*} \\frac { \\tilde M _ n ( 2 ) } { \\delta _ n ( 2 ) } + \\frac { \\tilde M _ n ( 3 ) } { \\delta _ n ( 3 ) } = \\delta ^ { - 1 } _ n ( 3 ) \\cdot \\sum _ { \\ell = 1 } ^ { d } \\sum _ { q = 1 } ^ { \\ell - 1 } \\left [ \\frac { \\delta _ n ( 3 ) } { \\delta _ n ( 2 ) } \\cdot \\tilde M _ { n , \\{ q , \\ell \\} } + \\sum _ { p = 1 } ^ { q - 1 } \\tilde M _ { n , \\{ p , q , \\ell \\} } \\right ] . \\end{align*}"}
{"id": "8736.png", "formula": "\\begin{align*} u _ { i 0 } ( x ) = p _ i ^ L , u _ { i j } ( x ) = \\sum _ { k \\in C _ j } g _ { k } ( x ) j = 1 , \\ldots , n . \\end{align*}"}
{"id": "12.png", "formula": "\\begin{align*} m _ 0 : = \\max \\{ m : \\lim _ { n \\to \\infty } a _ n ^ { m , j , k } = \\infty \\} . \\end{align*}"}
{"id": "4561.png", "formula": "\\begin{align*} e ( G ' ) > e ( G ) - \\delta ( A ( n ) ) = e ( G ) - ( e ( A ( n ) ) - e ( A ( n - 1 ) ) ) , \\end{align*}"}
{"id": "4888.png", "formula": "\\begin{align*} L ( z ) = \\frac { f ' ( z ) } { f ( z ) } = \\frac { m } z + O ( | z | ^ { - 2 } ) \\hbox { a s $ z \\to \\infty $ , } \\end{align*}"}
{"id": "9065.png", "formula": "\\begin{align*} & ( h + 2 \\beta _ a \\epsilon ( a ) ) \\tilde \\phi _ 0 + ( h - 2 \\beta _ a \\epsilon ( a ) ) \\tilde \\phi _ 1 = 0 , \\\\ & - \\epsilon _ { j - 1 / 2 } \\tilde \\phi _ { j - 1 } + 2 \\hat \\epsilon _ j \\tilde \\phi _ j - \\epsilon _ { j + 1 / 2 } \\tilde \\phi _ { j + 1 } = h ^ 2 z _ i \\tilde \\rho _ { i , j } j = 1 , \\cdots , N , \\\\ & ( h - 2 \\beta _ b \\epsilon ( b ) ) \\tilde \\phi _ N + ( h + 2 \\beta _ b \\epsilon ( b ) ) \\tilde \\phi _ { N + 1 } = 0 . \\end{align*}"}
{"id": "5491.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } ( t ) & = & A \\xi ( t ) + \\alpha ( t , \\xi ( t ) ) \\\\ \\xi ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "7333.png", "formula": "\\begin{align*} u _ { \\star , \\lambda } ( x , t ) = u ( x , t ) ( x , t ) \\in \\R ^ n \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "7145.png", "formula": "\\begin{align*} \\mathbb { E } [ ( x _ 1 ^ n - \\mathbb { E } [ x ^ n _ 1 ] ) ^ 2 ] & = \\mathbb { E } \\Big [ \\int _ 0 ^ 1 \\Big ( \\mathbb { E } [ D _ s x ^ n _ 1 \\vert \\mathcal { F } _ s ] \\Big ) ^ 2 d s \\Big ] \\\\ & \\le \\mathbb { E } \\Big [ \\int _ 0 ^ 1 \\mathbb { E } [ ( D _ s x ^ n _ 1 ) ^ 2 \\vert \\mathcal { F } _ s ] d s \\Big ] = \\int _ 0 ^ 1 \\mathbb { E } [ ( D _ s x ^ n _ 1 ) ^ 2 ] d s . \\end{align*}"}
{"id": "7054.png", "formula": "\\begin{align*} h _ K : = \\delta _ K \\log K \\ll 1 . \\end{align*}"}
{"id": "1005.png", "formula": "\\begin{align*} \\bigg \\vert \\frac { ( - \\Delta ) ^ s v ( h e _ 1 ) } h - ( - \\Delta ) ^ s \\partial _ 1 v ( 0 ) \\bigg \\vert & = \\big \\vert ( - \\Delta ) ^ s ( \\partial ^ h _ 1 v - \\partial _ 1 v ) ( 0 ) \\big \\vert \\\\ & \\leqslant C \\Big ( \\| \\partial ^ h _ 1 v - \\partial _ 1 v \\| _ { L ^ \\infty ( \\R ^ n ) } + \\| D ^ 2 \\partial ^ h _ 1 v - D ^ 2 \\partial _ 1 v \\| _ { L ^ \\infty ( \\R ^ n ) } \\Big ) . \\end{align*}"}
{"id": "4637.png", "formula": "\\begin{align*} \\lambda _ { ( j - 1 ) k + 1 } - \\lambda _ { ( j - 1 ) k + 2 } & = \\phi ( a _ { 1 , j } ) - \\phi ( a _ { 2 , j } ) \\\\ & = \\sum \\limits _ { s = 1 } ^ k a _ { s , j } - \\left ( \\sum \\limits _ { s = 2 } ^ k a _ { s , j } + a _ { 1 , j + 1 } \\right ) \\\\ & = a _ { 1 , j } - a _ { 1 , j + 1 } . \\end{align*}"}
{"id": "6720.png", "formula": "\\begin{align*} & g _ 1 ' ( t ) ^ { ( - 1 ) } ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + g _ r ' ( t ) ^ { ( - 1 ) } ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ r ) ^ { q ^ d } \\\\ & + g _ 1 ' ( t ) ^ { ( - 1 ) } \\prod _ { j = 1 } ^ { d - 1 } ( \\theta ^ { q ^ j } - t ) ^ { c ( j ) q ^ { d - j } } \\alpha _ 1 ^ { q ^ { d } } + \\cdots + g _ { r } ' ( t ) ^ { ( - 1 ) } \\prod _ { j = 1 } ^ { d - 1 } ( \\theta ^ { q ^ j } - t ) ^ { c ( j ) q ^ { d - j } } \\alpha _ r ^ { q ^ { d } } = 0 \\end{align*}"}
{"id": "8073.png", "formula": "\\begin{align*} \\mathfrak { A } _ { \\ell } ( \\Sigma , \\mathcal { M } ) = \\left \\{ ( F _ H ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } \\subset \\mathfrak { F } _ c ( \\Sigma ) [ [ \\hbar ] ] \\ , | \\ , \\beta _ { H ' - H } F _ H = F _ { H ' } \\right \\} \\end{align*}"}
{"id": "8996.png", "formula": "\\begin{align*} Q ^ { [ k ] } _ n : = \\phi _ n ( k , F ) , \\forall n \\in 1 : N , \\end{align*}"}
{"id": "8920.png", "formula": "\\begin{align*} \\mathcal B : = \\{ A _ 1 ^ { \\varepsilon _ 1 } \\cap \\cdots \\cap A _ n ^ { \\varepsilon _ n } : \\epsilon _ i = 1 , 2 \\} . \\end{align*}"}
{"id": "1060.png", "formula": "\\begin{align*} & g ( k , \\xi ) : = 4 X ^ 3 _ \\eta ( k ) . \\end{align*}"}
{"id": "5663.png", "formula": "\\begin{align*} \\theta \\prescript { \\sigma } { } N _ l ^ \\circ \\theta ^ { - 1 } \\subseteq \\prescript { \\tau \\sigma \\tau ^ { - 1 } } { } N _ v ^ \\ast \\iff \\sigma = \\varepsilon \\ ; \\ ; ( 1 \\ , 3 ) . \\end{align*}"}
{"id": "8411.png", "formula": "\\begin{gather*} W _ { 1 } ( \\mu _ { 1 } , \\mu _ { 2 } ) = \\sup _ { h \\in \\mathrm { L i p } ( 1 ) } | \\mu _ { 1 } ( h ) - \\mu _ { 2 } ( h ) | , \\end{gather*}"}
{"id": "7557.png", "formula": "\\begin{align*} x _ { w \\cdot a } ( t ) : = { } ^ { s _ w } x _ a ( t ) \\end{align*}"}
{"id": "1481.png", "formula": "\\begin{align*} \\lambda ( \\rho ( g _ 1 , g _ 2 ) , \\iota ( z _ 1 , z _ 2 ) ) S ^ { - 1 } B ( z _ 1 , z _ 2 ) \\left [ \\begin{array} { c c } \\lambda ( g _ 1 , z _ 1 ) & 0 \\\\ 0 & \\overline { \\lambda ( g _ 2 , z _ 2 ) } \\end{array} \\right ] ^ { - 1 } B ( g _ 1 z _ 1 , g _ 2 z _ 2 ) ^ { - 1 } S = 1 , \\end{align*}"}
{"id": "5391.png", "formula": "\\begin{align*} M ( n , \\alpha , \\varepsilon , Z , T , f ) = \\inf \\{ \\sum _ { i } e ^ { - \\alpha n _ i + f _ { n _ i } ( x _ i ) } : Z \\subset \\cup _ { i } B _ { n _ i } ( x _ i , \\varepsilon ) \\} , \\end{align*}"}
{"id": "4991.png", "formula": "\\begin{align*} \\tilde { \\mathcal { R } } Z = \\Lambda ( \\tilde { p } Z ) . \\end{align*}"}
{"id": "3012.png", "formula": "\\begin{align*} \\begin{aligned} R ^ { x y , v } _ { x ^ 4 y } & = [ \\varphi _ { x ^ 4 y } ] - [ \\gamma _ { x ^ 4 y , x ^ 4 v } ] - [ \\gamma _ { x ^ 4 y , x ^ 3 y v } ] \\\\ R ^ { x y , w } _ { x ^ 4 y } & = [ \\varphi _ { x ^ 4 y } ] - [ \\gamma _ { x ^ 4 y , x ^ 4 w } ] - [ \\gamma _ { x ^ 4 y , x ^ 3 y w } ] \\\\ R ^ { x y , z } _ { x ^ 4 y } & = [ \\varphi _ { x ^ 4 y } ] - [ \\gamma _ { x ^ 4 y , x ^ 4 z } ] - [ \\gamma _ { x ^ 4 y , x ^ 3 y z } ] \\\\ \\end{aligned} \\end{align*}"}
{"id": "2873.png", "formula": "\\begin{align*} \\partial _ t \\left ( e ^ { - e _ 0 t } \\alpha _ - \\right ) = e ^ { - e _ 0 t } B ( R , \\mathcal { Y } _ + ) \\Rightarrow e ^ { - e _ 0 t } \\alpha _ - ( t ) = \\int _ t ^ \\infty e ^ { - e _ 0 s } B ( R , \\mathcal { Y } _ + ) d s . \\end{align*}"}
{"id": "176.png", "formula": "\\begin{align*} \\mathcal { E } _ m ^ \\sigma ( \\tilde { f } _ m ^ \\sigma , g ) = \\langle f ; g \\rangle _ { L ^ 2 ( \\mu _ m ) } , \\tilde { f } _ m ^ \\sigma = \\int _ 0 ^ { + \\infty } \\mathcal { P } ^ { m , \\sigma } _ t ( f ) d t . \\end{align*}"}
{"id": "538.png", "formula": "\\begin{align*} \\begin{matrix} \\Xi _ { k + 1 } = \\mathcal { A } \\Xi _ k + \\mathcal { B } \\mathbf { v } _ k , & \\mathbf { y } _ k = \\mathcal { C } \\Xi _ k , \\end{matrix} \\end{align*}"}
{"id": "556.png", "formula": "\\begin{align*} \\| \\sqrt \\rho u \\| _ { L ^ \\infty ( 0 , T ; L ^ 2 ( \\Omega _ R ) ) } & \\leq c , \\\\ \\left \\| \\frac { \\rho - \\varrho } { \\varepsilon } \\right \\| _ { L ^ \\infty ( 0 , T ; L ^ 2 ( \\Omega _ R ) ) } & \\leq c , \\\\ \\nu ^ \\frac { 1 } { 2 } \\| \\nabla u \\| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \\Omega _ R ) ) } & \\leq c \\end{align*}"}
{"id": "8150.png", "formula": "\\begin{align*} D _ { d _ 0 } ( p , H ) \\ = 1 + o ( 1 ) . \\end{align*}"}
{"id": "4631.png", "formula": "\\begin{align*} \\mathcal { L } _ 0 = - ( \\widehat { A } \\nabla ) , \\end{align*}"}
{"id": "2435.png", "formula": "\\begin{align*} \\pi ( \\l ' ) ^ { - 1 } S _ { g , \\L } \\ , \\pi ( \\l ' ) f & = \\sum _ { \\l \\in \\L } \\langle \\pi ( \\l ' ) f , \\pi ( \\l ) g \\rangle \\ , \\pi ( \\l ' ) ^ { - 1 } \\pi ( \\l ) g \\\\ & = \\sum _ { \\l \\in \\L } \\langle f , \\pi ( \\l ' ) ^ { - 1 } \\pi ( \\l ) g \\rangle \\ , \\pi ( \\l ' ) ^ { - 1 } \\pi ( \\l ) g \\\\ & = \\sum _ { \\l \\in \\L } \\langle f , \\pi ( \\l - \\l ' ) g \\rangle \\ , \\pi ( \\l - \\l ' ) g \\\\ & = S _ { g , \\L } f . \\end{align*}"}
{"id": "1245.png", "formula": "\\begin{align*} E _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\varepsilon } = \\bigcup _ { n \\geq 1 } E _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\frac { 1 } { n } , \\varepsilon } \\ \\ \\ \\mbox { a n d } \\ \\ \\ \\ F _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\varepsilon } = \\bigcup _ { n \\geq 1 } F _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\frac { 1 } { n } , \\varepsilon } . \\end{align*}"}
{"id": "579.png", "formula": "\\begin{align*} \\begin{cases} \\ 0 \\ \\leq x \\ \\leq \\ 1 \\\\ \\ 0 \\ \\leq y \\ \\leq \\ 1 \\end{cases} . \\end{align*}"}
{"id": "9381.png", "formula": "\\begin{align*} & \\sum _ { p = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { n - 1 } ( k - \\lambda + 1 ) _ { p , \\lambda } \\frac { t ^ { p } } { p ! } = \\sum _ { k = 0 } ^ { n - 1 } e _ { \\lambda } ^ { k + 1 - \\lambda } ( t ) \\\\ & = e _ { \\lambda } ^ { 1 - \\lambda } ( t ) \\frac { e _ { \\lambda } ^ { n } ( t ) - 1 } { e _ { \\lambda } ( t ) - 1 } = \\sum _ { p = 0 } ^ { \\infty } \\bigg ( \\frac { \\beta _ { p + 1 } ( n + 1 - \\lambda ) - \\beta _ { p + 1 , \\lambda } ( 1 - \\lambda ) } { p + 1 } \\bigg ) \\frac { t ^ { p } } { p ! } . \\end{align*}"}
{"id": "7151.png", "formula": "\\begin{align*} R _ { H } ( t , s ) = \\int _ { 0 } ^ { t \\wedge s } K _ { H } ( t , u ) K _ { H } ( s , u ) d u , \\end{align*}"}
{"id": "5394.png", "formula": "\\begin{align*} M ( \\alpha , \\varepsilon , Z , T , f ) & = \\lim _ { n \\to \\infty } M ( n , \\alpha , \\varepsilon , Z , T , f ) , \\\\ \\underline { R } ( \\alpha , \\varepsilon , Z , T , f ) & = \\liminf _ { n \\to \\infty } R ( n , \\alpha , \\varepsilon , Z , T , f ) , \\\\ \\overline { R } ( \\alpha , \\varepsilon , Z , T , f ) & = \\limsup _ { n \\to \\infty } R ( n , \\alpha , \\varepsilon , Z , T , f ) , \\\\ M ^ P ( \\alpha , \\varepsilon , Z , T , f ) & = \\lim _ { n \\to \\infty } M ^ P ( n , \\alpha , \\varepsilon , Z , T , f ) . \\end{align*}"}
{"id": "8728.png", "formula": "\\begin{align*} p = 1 + 1 \\cdot ( - 3 x _ 2 ) + ( - 3 x _ 1 ) \\cdot ( - 3 x _ 2 + 1 ) + ( - 3 x _ 1 + 1 ) \\cdot ( - x _ 2 ^ 3 + 3 x _ 2 ^ 2 ) + ( - x _ 1 ^ 3 + 3 x _ 1 ^ 2 ) \\cdot ( 1 - x _ 2 ) ^ 3 . \\end{align*}"}
{"id": "191.png", "formula": "\\begin{align*} \\mathcal { L } _ \\delta ( f ) ( x ) = f '' ( x ) - \\delta | x | ^ { \\delta - 1 } \\operatorname { s g n } ( x ) f ' ( x ) . \\end{align*}"}
{"id": "8656.png", "formula": "\\begin{align*} R ( T ) & = 9 p r ^ 3 - 9 p q ^ 2 r + 2 7 q r ^ 2 s - 3 q ^ 3 s + ( 9 p q r ^ 2 - p q ^ 3 - 9 r ^ 3 s + 9 q ^ 2 r s ) T \\end{align*}"}
{"id": "4609.png", "formula": "\\begin{align*} \\pi _ v ( G ^ + \\cap G ^ - ) & = \\pi _ v ( ( G ^ + \\cap Q _ v ^ - ) \\cap ( G ^ - \\cap Q _ v ^ + ) ) = \\pi _ v ( G ^ + \\cap Q _ v ^ - ) \\cap \\pi _ v ( G ^ - \\cap Q _ v ^ + ) , \\\\ \\pi _ v ( G ^ + \\cap Q _ v ^ - ) & = \\pi _ v ( G ^ + \\cap ( Q _ v ^ + \\cap Q _ v ^ - ) ) = \\pi _ v ( G ^ + ) \\cap \\partial \\pi _ v ( Q ) , \\\\ \\pi _ v ( G ^ - \\cap Q _ v ^ + ) & = \\pi _ v ( G ^ - \\cap ( Q _ v ^ + \\cap Q _ v ^ - ) ) = \\pi _ v ( G ^ - ) \\cap \\partial \\pi _ v ( Q ) . \\end{align*}"}
{"id": "2790.png", "formula": "\\begin{align*} \\int \\left ( \\partial _ { x _ 1 } Q \\right ) h _ 1 = . . . = \\int \\left ( \\partial _ { x _ N } Q \\right ) h _ 1 = \\int Q h _ 2 , \\end{align*}"}
{"id": "6800.png", "formula": "\\begin{align*} u _ { t } = D u _ { x x } - u \\left ( u - \\kappa \\right ) \\left ( u - 1 \\right ) - \\varepsilon u _ { x } ^ { m } , \\end{align*}"}
{"id": "8932.png", "formula": "\\begin{align*} \\tilde \\varphi : V ^ { q + 1 } & \\to A \\\\ ( x _ 0 , \\ldots , x _ q ) & \\mapsto \\begin{cases} 0 & \\exists i \\in \\{ 0 , \\ldots , q \\} : x _ i \\not \\in U \\\\ \\varphi ( x _ 0 , \\ldots , x _ q ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "1566.png", "formula": "\\begin{align*} \\Delta _ { \\mu } ^ p \\mathbf { E } _ { \\mu } ( g , \\mu ) = C \\cdot \\mathbf { E } _ l ( g , \\mu ) , \\ , \\ , \\ , \\ , \\ , C \\in \\overline { \\Q } ^ { \\times } . \\end{align*}"}
{"id": "912.png", "formula": "\\begin{align*} ( x ) _ { n } = \\sum _ { k = 0 } ^ { n } S _ { 1 , \\lambda } ( n , k ) ( x ) _ { k , \\lambda } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "9278.png", "formula": "\\begin{align*} H ^ * [ P , Q , \\varepsilon ] : = \\forall p \\in P \\exists q \\in Q \\left ( \\norm { p - q } \\leq \\varepsilon \\right ) \\end{align*}"}
{"id": "6538.png", "formula": "\\begin{align*} s _ n ^ 2 & : = \\sum _ { i = 1 } ^ n \\dfrac { \\Gamma ( i ) ^ 2 } { \\Gamma ( i + \\alpha ) ^ 2 } , \\intertext { a n d } \\sigma _ n ^ 2 & : = \\begin{cases} \\dfrac { \\Gamma ( n ) ^ 2 } { \\Gamma ( n + \\alpha ) ^ 2 } \\cdot \\dfrac { n } { 1 - 2 \\alpha } & ( - 1 < \\alpha < 1 / 2 ) , \\\\ [ 4 m m ] \\dfrac { \\Gamma ( n ) ^ 2 } { \\Gamma ( n + \\alpha ) ^ 2 } \\cdot n \\log n & ( \\alpha = 1 / 2 ) . \\end{cases} \\end{align*}"}
{"id": "7411.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] f ( \\eta ) d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "7606.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { \\partial f } { \\partial \\kappa _ { i } } = \\frac { 1 } { k } H _ { k } ^ { \\frac { 1 - k } { k } } \\sum _ { i = 1 } ^ { n } \\frac { \\partial H _ { k } } { \\partial \\kappa _ { i } } = H _ { k } ^ { \\frac { 1 - k } { k } } H _ { k - 1 } \\geq 1 , \\end{align*}"}
{"id": "2395.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma } A e _ \\gamma = A \\sum _ { \\gamma \\in \\Gamma } e _ \\gamma = A f . \\end{align*}"}
{"id": "3126.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = - x _ 1 \\ , , x _ i ^ \\prime = x _ i ( i = 2 , 3 , 4 , 5 ) \\end{align*}"}
{"id": "779.png", "formula": "\\begin{align*} u _ { n + 1 } & = \\varepsilon _ n u _ n ^ 2 , \\\\ v _ { n + 1 } & = \\varepsilon _ n u _ n v _ n + 1 . \\\\ \\end{align*}"}
{"id": "115.png", "formula": "\\begin{align*} x ^ k = a _ k ^ { ( k ) } v ^ k + b _ { k - 1 } ^ { ( k ) } L _ v ^ { k - 1 } ( u ) + c _ { k - 2 } ^ { ( k ) } L _ v ^ { k - 2 } ( u ^ 2 ) , \\end{align*}"}
{"id": "880.png", "formula": "\\begin{align*} \\Delta _ { g _ M } f + h e ^ f = c , \\end{align*}"}
{"id": "6922.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { \\boldsymbol { \\Theta } } & N _ s \\biggl | \\sum _ { n = 1 } ^ N g _ n c _ n h _ n + h _ d \\biggr | ^ 2 \\\\ \\textrm { s . t . } & \\phi _ n \\in [ 0 , 2 \\pi ) , \\ : \\forall n = 1 , 2 , \\ldots , N , \\end{aligned} \\end{align*}"}
{"id": "6795.png", "formula": "\\begin{align*} \\| \\theta _ 1 - \\theta _ 2 \\| ^ 2 + \\| P \\theta _ 1 - P \\theta _ 2 \\| ^ 2 = \\| x _ 1 - x _ 2 \\| ^ 2 . \\end{align*}"}
{"id": "2345.png", "formula": "\\begin{align*} \\int _ \\R | f ( x ) | ^ 2 \\ , d x = - \\int _ \\R x f ( x ) \\overline { f ' ( x ) } \\ , d x - \\int _ \\R x \\overline { f ( x ) } f ' ( x ) \\ , d x = - 2 \\Re \\left ( \\int _ \\R x f ( x ) \\overline { f ' ( x ) } \\ , d x \\right ) , \\end{align*}"}
{"id": "8268.png", "formula": "\\begin{align*} \\mathbf { M } _ { 1 2 \\overline { 3 } } = \\mathbf { F } _ { 1 2 \\overline { 3 } } - \\mathbf { F } _ { 2 1 \\overline { 3 } } - \\mathbf { F } _ { \\overline { 1 } 2 \\overline { 3 } } + \\mathbf { F } _ { \\overline { 1 } \\ , \\overline { 2 } \\ , \\overline { 3 } } \\quad \\mathbf { M } _ { 1 \\bar { 3 } 2 } = \\mathbf { F } _ { 1 \\bar { 3 } 2 } - \\mathbf { F } _ { 2 \\overline { 3 } 1 } - \\mathbf { F } _ { \\overline { 1 } \\ , \\overline { 3 } 2 } + \\mathbf { F } _ { \\overline { 1 } \\ , \\overline { 3 } \\ , \\overline { 2 } } . \\end{align*}"}
{"id": "3437.png", "formula": "\\begin{align*} & S = \\int _ { \\Bbb R ^ N } f _ t ( x , u ) g _ s ( u , y ) d \\omega ( u ) = \\int _ { \\Bbb R ^ N } \\Big ( f _ t ( x , u ) - f _ t ( x , y ) \\Big ) g _ s ( u , y ) d \\omega ( u ) . \\end{align*}"}
{"id": "4006.png", "formula": "\\begin{align*} \\Re ( \\lambda ) = - \\frac { \\| \\eta ^ \\prime \\| ^ 2 _ { L ^ 2 } } { \\| \\xi \\| ^ 2 _ { L ^ 2 } + \\| \\eta \\| ^ 2 _ { L ^ 2 } } < 0 , \\end{align*}"}
{"id": "5302.png", "formula": "\\begin{align*} \\varphi _ S ( a ^ * a ) = \\varphi ( a ^ * a \\delta _ { \\varphi } ) = \\varphi ( \\delta _ { \\varphi } ^ { 1 / 2 } a ^ * a \\delta _ { \\varphi } ^ { 1 / 2 } ) \\geq 0 , \\end{align*}"}
{"id": "9379.png", "formula": "\\begin{align*} \\frac { d } { d x } \\sum _ { k = 1 } ^ { p + 1 } S _ { 2 , \\lambda } ( p + 1 , k ) \\frac { x ^ { k } } { k } = \\frac { d } { d x } \\bigg ( e ^ { - x } \\sum _ { n = 0 } ^ { \\infty } \\big ( ( 1 - \\lambda ) _ { p , \\lambda } + ( 2 - \\lambda ) _ { p , \\lambda } + \\cdots + ( n - \\lambda ) _ { p , \\lambda } \\big ) \\frac { x ^ { n } } { n ! } \\bigg ) . \\end{align*}"}
{"id": "3273.png", "formula": "\\begin{align*} \\psi _ { \\mu _ { T ^ * T } } \\left ( - \\frac { 1 } { s ( | \\lambda | , 0 ) ^ 2 } \\right ) = \\frac { s ( | \\lambda | , 0 ) ^ 2 } { s ( | \\lambda | , 0 ) ^ 2 + | \\lambda | ^ 2 } - 1 = - \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , 0 ) ^ 2 + | \\lambda | ^ 2 } . \\end{align*}"}
{"id": "4667.png", "formula": "\\begin{align*} & Q ( y ) = \\frac { \\kappa _ 0 } { y ^ 2 } + \\frac { g ( y ) } { y ^ 4 } + O \\bigg ( \\frac { 1 } { y ^ 6 } \\bigg ) , \\ ; \\ ; | y | \\rightarrow + \\infty , \\\\ & Q ' ( y ) = - \\frac { 2 \\kappa _ 0 } { y ^ 3 } + O \\bigg ( \\frac { 1 } { | y | ^ 5 } \\bigg ) , \\ ; \\ ; | y | \\rightarrow + \\infty . \\end{align*}"}
{"id": "2863.png", "formula": "\\begin{align*} \\partial _ t h + \\mathcal { L } h = R , ( x , t ) \\in \\mathbb { R } ^ N \\times ( t _ 0 , + \\infty ) , \\end{align*}"}
{"id": "8591.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb { R } , \\ , k \\in \\mathbb { R } } \\left | \\left \\langle x \\right \\rangle ^ { \\beta } \\partial _ { k } ^ { j } \\mathcal { Q } ( x , k ) \\right | < \\infty , j = 0 , 1 , \\end{align*}"}
{"id": "4128.png", "formula": "\\begin{align*} | C \\cap F | = O _ d \\left ( A _ { F } ^ { \\frac { d - 2 } { d - 1 } } \\log A _ F \\right ) . \\end{align*}"}
{"id": "9257.png", "formula": "\\begin{align*} & \\forall \\gamma ^ 1 , x ^ X \\big ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\\\ & \\qquad \\qquad \\qquad \\land \\exists y ^ X , z ^ X ( z \\in A y \\land x = _ X y + _ X \\gamma z ) \\rightarrow \\gamma ^ { - 1 } ( x - _ X J ^ A _ { \\gamma } x ) \\in A ( J ^ A _ { \\gamma } x ) \\big ) , \\end{align*}"}
{"id": "5549.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X ( t ) & = & \\big ( \\frac { d } { d x } X ( t ) + \\alpha _ { \\rm H J M } ( X ( t ) ) \\big ) d t + \\sigma ( X ( t ) ) d W _ t \\medskip \\\\ X ( 0 ) & = & h , \\end{array} \\right . \\end{align*}"}
{"id": "1226.png", "formula": "\\begin{align*} \\theta ( r ) & \\ge \\sqrt { \\psi ^ \\prime ( \\varphi ' ( \\mu _ { \\min } ) ) } \\int _ 0 ^ r \\sqrt { s } \\ , d s \\\\ & = \\frac { 2 } { 3 } \\sqrt { \\psi ^ \\prime ( \\varphi ' ( \\mu _ { \\min } ) ) } r ^ { 3 / 2 } . \\end{align*}"}
{"id": "5037.png", "formula": "\\begin{align*} [ X , \\mathrm { A d } ( g ^ { - 1 } ) \\cdot E _ { \\theta } ] & = \\mathrm { A d } ( g ^ { - 1 } ) \\cdot [ \\mathrm { A d } ( g ) \\cdot X , E _ { \\theta } ] \\\\ & = \\mathrm { A d } ( g ^ { - 1 } ) \\cdot [ w ^ { - 1 } \\cdot X , E _ { \\theta } ] \\\\ & = \\theta ( w ^ { - 1 } \\cdot X ) \\ , \\mathrm { A d } ( g ^ { - 1 } ) \\cdot E _ { \\theta } \\\\ & = ( w \\cdot \\theta ) ( X ) \\ , \\mathrm { A d } ( g ^ { - 1 } ) \\cdot E _ { \\theta } \\ , . \\end{align*}"}
{"id": "9092.png", "formula": "\\begin{align*} p _ { | h _ 1 | , | h _ 2 | } ( x _ 1 , x _ 2 ) & = \\frac { 4 x _ 1 x _ 2 } { \\sigma ^ { 4 } ( 1 - \\mu _ 2 ^ { 2 } ) } e ^ { - \\frac { x _ 1 ^ 2 + x _ 2 ^ 2 } { \\sigma ^ { 2 } ( 1 - \\mu _ 2 ^ 2 ) } } \\\\ [ - 0 . 2 e m ] & \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\times I _ { 0 } \\left ( { \\frac { 2 \\mu _ 2 x _ 1 x _ 2 } { \\sigma ^ { 2 } ( 1 - \\mu _ 2 ^ 2 ) } } \\right ) \\ ! , x _ 1 , x _ 2 \\geq 0 . \\end{align*}"}
{"id": "5820.png", "formula": "\\begin{align*} L f ( \\eta ) = \\displaystyle \\sum _ { x \\in \\mathbb { Z } } g ( \\eta ( x ) ) \\sum _ { y \\in \\mathbb { Z } } p ( x , y ) [ f ( \\eta ^ { x y } ) - f ( \\eta ) ] , \\end{align*}"}
{"id": "942.png", "formula": "\\begin{align*} N _ { n , i } = N _ { n , i _ 1 } ^ 1 \\otimes \\cdots \\otimes N _ { n , i _ d } ^ d , \\end{align*}"}
{"id": "4821.png", "formula": "\\begin{align*} \\begin{aligned} & y _ t = 0 . 2 y _ { t - 1 } + 0 . 2 4 y _ { t - 2 } + 2 u _ { t - 1 } + n _ t , \\\\ & y _ t = 0 . 7 y _ { t - 1 } - 0 . 1 2 y _ { t - 2 } + 1 u _ { t - 1 } + n _ t , \\end{aligned} \\end{align*}"}
{"id": "6654.png", "formula": "\\begin{align*} \\Sigma _ 0 = \\sum _ { m = 0 } ^ { \\infty } \\Big ( D _ { 1 , m } + D _ { 2 , m } + D _ { 3 , m } \\Big ) \\frac { 1 } { p ^ m } , \\end{align*}"}
{"id": "5216.png", "formula": "\\begin{align*} g _ j \\ast f _ 1 ( x ) & = \\int \\limits _ { K } a ( y ) \\Phi _ { A _ j } ( y ) f _ 1 ( x - y ) d y \\\\ & = \\int \\limits _ { \\mathfrak { D } ^ * } a ( y ) f _ 1 ( x - y ) d y , \\end{align*}"}
{"id": "2124.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\limsup _ { n \\to \\infty } k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { j = 1 } ^ { n - N } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } = 0 . \\end{align*}"}
{"id": "5581.png", "formula": "\\begin{align*} \\left \\| x - \\sum _ { n = 1 } ^ { \\lceil \\lambda m \\rceil } e ^ * _ { \\rho _ x ( n ) } ( x ) e _ { \\rho _ x ( n ) } \\right \\| & \\ \\leqslant \\ \\mathbf C _ \\ell \\left \\| x - \\sum _ { n = 1 } ^ { m } e ^ * _ { \\rho _ x ( n ) } ( x ) e _ { \\rho _ x ( n ) } \\right \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ \\ell C \\sigma ^ { \\mathcal { U } _ X , L } _ m ( x ) . \\end{align*}"}
{"id": "9498.png", "formula": "\\begin{align*} | \\mathcal { F } _ 2 | & = \\sum _ { i = 0 } ^ { ( s - 2 ) / 2 } \\binom { ( s + d - 3 ) / 2 } { \\left \\lfloor i / 2 \\right \\rfloor } \\binom { ( s + d - 3 ) / 2 - \\left \\lfloor i / 2 \\right \\rfloor } { ( s - 2 ) / 2 - i } , \\\\ | \\mathcal { F } _ 1 | + | \\mathcal { F } _ 3 | & = \\sum _ { i = 0 } ^ { ( s - 2 ) / 2 } \\binom { ( s + d - 5 ) / 2 } { \\left \\lfloor i / 2 \\right \\rfloor } \\binom { ( s + d - 1 ) / 2 - \\left \\lfloor i / 2 \\right \\rfloor } { ( s - 2 ) / 2 - i } , \\end{align*}"}
{"id": "8459.png", "formula": "\\begin{align*} A x = b , A \\in \\mathbb { R } ^ { m \\times n } , b \\in \\mathbb { R } ^ { m } , \\end{align*}"}
{"id": "6157.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m { n + m + 1 \\choose k } x ^ k ( 1 - x ) ^ { m - k } = \\sum _ { k = 0 } ^ m { n + k \\choose k } x ^ k = ( n + m + 1 ) { n + m \\choose m } \\sum _ { k = 0 } ^ m \\frac { ( - 1 ) ^ k } { k + n + 1 } { m \\choose k } ( 1 - x ) ^ k . \\end{align*}"}
{"id": "4696.png", "formula": "\\begin{align*} f _ { 1 + \\mu _ j } & = f + \\int _ 0 ^ 1 \\frac { \\mu _ j } { 1 + s \\mu _ j } ( \\Lambda f ) _ { 1 + s \\mu _ j } = f + \\mu _ j ( \\Lambda f ) + \\frac { \\mu _ j ^ 2 } { 2 } \\int _ 0 ^ 1 \\frac { \\big [ ( \\Lambda ^ 2 f ) _ { 1 + s \\mu _ j } - ( \\Lambda f ) _ { 1 + s \\mu _ j } \\big ] } { ( 1 + s \\mu _ j ) ^ 2 } \\ , \\dd s . \\end{align*}"}
{"id": "5418.png", "formula": "\\begin{align*} & a ' = \\tfrac { 1 } { a + 1 } , & a '' & = \\tfrac { a } { a + 1 } , \\\\ & b ' = \\tfrac { \\sin \\alpha } { \\sin \\gamma } \\tfrac { 1 } { b + 1 } , & b '' & = \\tfrac { \\sin \\alpha } { \\sin \\gamma } \\tfrac { b } { b + 1 } , \\\\ & c ' = \\tfrac { \\sin \\beta } { \\sin \\gamma } \\tfrac { 1 } { c + 1 } , & c '' & = \\tfrac { \\sin \\beta } { \\sin \\gamma } \\tfrac { c } { c + 1 } . \\\\ \\end{align*}"}
{"id": "56.png", "formula": "\\begin{align*} K _ { \\ell , v , j } & = L _ { v , j } ( j \\neq \\nu _ v ) , \\\\ \\mathrm { r k } ( K _ { \\ell , v , \\nu _ v } ) & = n _ { v , \\nu _ v } - 1 . \\end{align*}"}
{"id": "7389.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho ( t , u ) = [ - ( - \\Delta ) ^ { \\gamma / 2 } \\rho ^ m ] ( t , u ) , \\ ; \\ ; u \\in \\mathbb { R } , \\ ; t \\in [ 0 , T ] , \\\\ & \\rho ( 0 , u ) = g ( u ) , \\ ; \\ ; u \\in \\mathbb { R } , \\end{cases} \\end{align*}"}
{"id": "7393.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho ( t , u ) = [ - ( - \\Delta ) ^ { \\gamma / 2 } \\rho ^ 2 ] ( t , u ) , \\ ; \\ ; u \\in \\mathbb { R } , \\ ; t \\in [ 0 , T ] , \\\\ & \\rho ( 0 , u ) = g ( u ) , \\ ; \\ ; u \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
{"id": "5833.png", "formula": "\\begin{align*} X _ { ( j + 1 ) K } ^ y - X _ { j K } ^ y \\geq \\Big ( v _ + + \\frac { \\theta } { 2 r } \\Big ) K \\end{align*}"}
{"id": "8259.png", "formula": "\\begin{align*} \\mathfrak { B } _ 2 \\times \\mathfrak { B } _ { 1 , \\emptyset } = [ 1 2 3 , \\bar 1 \\ , \\bar 2 3 ] \\mathfrak { B } _ 2 \\times \\mathfrak { B } _ { 1 , \\{ 1 \\} } = [ 1 2 \\bar { 3 } , \\bar 1 \\ , \\bar 2 \\ , \\bar 3 ] , \\end{align*}"}
{"id": "1345.png", "formula": "\\begin{align*} D _ i v ( \\omega ) = - \\int _ { \\mathbb { R } ^ d } f ( \\tau _ x \\omega ) \\partial _ i \\phi ( x ) d x . \\end{align*}"}
{"id": "5515.png", "formula": "\\begin{align*} \\liminf _ { t \\downarrow 0 } \\frac { 1 } { t } d _ K \\big ( S _ t h + t ( \\alpha ( h ) - \\rho ( h ) + \\sigma ( h ) u ) \\big ) = 0 , h \\in K u \\in U _ 0 . \\end{align*}"}
{"id": "199.png", "formula": "\\begin{align*} & F _ 1 ( x , y , t ) = x e ^ { - t } - \\frac { e ^ { - \\alpha t } y } { ( 1 - e ^ { - \\alpha t } ) ^ { 1 - \\frac { 1 } { \\alpha } } } , \\\\ & F _ 2 ( x , y , t ) = \\nabla ( f ) \\left ( x e ^ { - t } + ( 1 - e ^ { - \\alpha t } ) ^ { \\frac { 1 } { \\alpha } } y \\right ) . \\end{align*}"}
{"id": "5059.png", "formula": "\\begin{align*} [ { \\bf L } _ n , \\chi _ T ] = [ { \\bf L } ^ 0 _ n , \\chi _ T ] = i \\chi _ T ' { \\bf A } _ n , \\end{align*}"}
{"id": "5974.png", "formula": "\\begin{align*} l _ { \\pi _ { \\lambda } } ( z ) = \\int _ { \\mathbb { R } } \\mathrm { e } ^ { z x } \\ , \\mathrm { d } \\pi _ { \\lambda } ( x ) = \\mathrm { e } ^ { - \\lambda } \\sum _ { k = 0 } ^ { \\infty } \\mathrm { e } ^ { z k } \\frac { \\lambda ^ { k } } { k ! } = \\exp \\left ( \\lambda ( \\mathrm { e } ^ { z } - 1 ) \\right ) . \\end{align*}"}
{"id": "6613.png", "formula": "\\begin{align*} \\frac { c | m h \\pm n k | } { g x Q } W \\left ( \\frac { c | m h \\pm n k | } { g x Q } \\right ) = \\frac { 1 } { 2 \\delta } \\int _ { - \\delta } ^ { \\delta } \\frac { c | m h \\pm e ^ { \\xi } n k | } { g x Q } W \\left ( \\frac { c | m h \\pm e ^ { \\xi } n k | } { g x Q } \\right ) \\ , d \\xi + O \\left ( \\frac { c n k \\delta } { g x Q } \\right ) . \\end{align*}"}
{"id": "5595.png", "formula": "\\begin{align*} ( \\mathcal { F } \\otimes 1 ) ( \\Delta \\otimes \\mathrm { i d } ) ( \\mathcal { F } ) = & ( 1 \\otimes \\mathcal { F } ) ( \\mathrm { i d } \\otimes \\Delta ) ( \\mathcal { F } ) , \\\\ ( \\epsilon \\otimes \\mathrm { i d } ) ( \\mathcal { F } ) = & 1 = ( \\mathrm { i d } \\otimes \\epsilon ) ( \\mathcal { F } ) . \\end{align*}"}
{"id": "4389.png", "formula": "\\begin{align*} H _ { 0 } ( \\xi ) = \\Lambda _ \\xi Q _ \\sigma ( \\xi ) , \\end{align*}"}
{"id": "5598.png", "formula": "\\begin{align*} ( s \\otimes _ \\star t ) ^ * = ( \\overline { \\mathcal { R } } _ 1 \\rhd s ^ * ) \\otimes _ \\star ( \\overline { \\mathcal { R } } _ 2 \\rhd t ^ * ) , ( s \\otimes _ \\star t ) ^ { * _ \\star } = ( \\overline { \\mathcal { R } } _ 1 \\rhd s ^ { * _ \\star } ) \\otimes _ \\star ( \\overline { \\mathcal { R } } _ 2 \\rhd t ^ { * _ \\star } ) . \\end{align*}"}
{"id": "7262.png", "formula": "\\begin{align*} & ( \\dfrac { 1 2 b c } { a d } + 2 0 - \\dfrac { c ^ 3 } { a d ^ 2 } - \\dfrac { b ^ 3 } { a ^ 2 d } - \\dfrac { 1 2 c ^ 2 } { b d } - \\dfrac { 1 2 b ^ 2 } { a c } - \\dfrac { 3 a c } { b ^ 2 } - \\dfrac { 3 b d } { c ^ 2 } ) \\\\ = & \\dfrac { 4 ( n + 1 ) ^ 2 [ - ( n + 5 ) k ^ 2 + ( n ^ 2 + 4 n - 5 ) k - n ^ 2 + 1 ] } { k ^ 2 ( k + 1 ) ( k + 2 ) ( n - k + 1 ) ( n - k ) ( n - k - 1 ) ^ 2 } . \\end{align*}"}
{"id": "1600.png", "formula": "\\begin{align*} \\psi ( x ) : = \\mu ^ { \\frac { d - 1 } { p } } \\varphi ( \\mu x ) \\\\ \\tilde { \\psi } ( x ) : = \\mu ^ { \\frac { d - 1 } { p ' } } \\tilde { \\varphi } ( \\mu x ) \\end{align*}"}
{"id": "4196.png", "formula": "\\begin{align*} | | \\psi _ i | | _ H & \\leq \\int _ 0 ^ 1 \\left ( \\max _ { x \\in M } H _ i ( x ) - \\min _ { x \\in M } H _ i ( x ) \\right ) d t \\\\ & = | \\phi ( x _ i ) | \\left ( \\max _ { | v | \\leq \\delta } H ( v ) - \\min _ { | v | \\leq \\delta } H ( v ) \\right ) \\\\ & \\leq 2 \\delta | \\phi ( x _ i ) | \\end{align*}"}
{"id": "8951.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - t } u ( x ) \\leq & \\int _ t ^ { t + b } e ^ { - s } \\left ( C _ p \\left | \\dot { \\xi } ( s ) - \\frac { x - y } { b } \\right | ^ q + f \\left ( \\xi ( s ) + \\frac { b + t - s } { b } ( x - y ) \\right ) \\right ) d s \\\\ & + e ^ { - ( b + t ) } u ( \\xi ( b + t ) ) . \\end{aligned} \\end{align*}"}
{"id": "1476.png", "formula": "\\begin{align*} \\Omega _ 1 \\times \\Omega _ 2 \\to \\Omega _ N , U _ 1 \\times U _ 2 \\mapsto R ^ { - 1 } \\left [ \\begin{array} { c c } U _ 1 & 0 \\\\ 0 & \\mathfrak { J } \\overline { U _ 2 } \\end{array} \\right ] , \\mathfrak { J } = \\left [ \\begin{array} { c c c c } J _ { m } ' & 0 & 0 & 0 \\\\ 0 & 0 & - 1 _ r & 0 \\\\ 0 & 1 _ r & 0 & 0 \\\\ 0 & 0 & 0 & J _ { m } ' \\end{array} \\right ] . \\end{align*}"}
{"id": "4703.png", "formula": "\\begin{align*} D = - \\sum _ { \\substack { i , j = 1 , \\\\ i \\not = j } } ^ n \\partial _ y \\Bigg [ \\bigg ( \\frac { \\tau _ i ( \\mathcal { L } B _ { i j , 0 } ) } { x _ { i j } ^ 3 } \\bigg ) ( \\varphi _ { i j } - 1 ) \\Bigg ] - \\sum _ { \\substack { i , j = 1 , \\\\ i \\not = j } } ^ n \\partial _ y \\bigg ( \\frac { \\tau _ i ( [ | D | , \\varphi _ { i j } ] B _ { i j , 0 } ) } { x _ { i j } ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "8743.png", "formula": "\\begin{align*} \\mathcal { M } ' _ K : = \\Bigl \\{ ( \\lambda , y ) \\in \\{ 0 , 1 \\} ^ { | V | + | E | } \\Bigm | \\lambda \\in \\Lambda , \\ y _ e = \\prod _ { v \\in e } \\lambda _ v , \\ e \\in E \\Bigr \\} . \\end{align*}"}
{"id": "3930.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\ , \\sup _ { \\tau \\in [ 0 , 1 ] } S \\left ( \\omega _ { V ( 0 ) } \\circ \\alpha _ T ^ { 0 \\to \\tau } \\bigg | \\omega _ { V ( \\tau ) } \\right ) = 0 . \\end{align*}"}
{"id": "6565.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) \\sim \\sum _ { \\ell = 0 } ^ { \\min \\{ | A | , | B | \\} } \\mathcal { I } _ \\ell ( h , k ) , Q \\to \\infty . \\end{align*}"}
{"id": "1699.png", "formula": "\\begin{align*} g ( x ) = { \\rm d i s t } ^ { \u2010 \\beta } ( x , \\ , \\Gamma ) , \\ ; \\ ; w ( x ) = { \\rm d i s t } ^ { \u2010 \\sigma } ( x , \\ , \\Gamma ) , \\ ; \\ ; v ( x ) = { \\rm d i s t } ^ { \u2010 \\lambda } ( x , \\ , \\Gamma ) , \\end{align*}"}
{"id": "4878.png", "formula": "\\begin{align*} z ^ 2 ( z - 1 ) ( z - K ) y '' ( z ) = K n ( n + 1 ) y ( z ) . \\end{align*}"}
{"id": "6815.png", "formula": "\\begin{gather*} G ^ { \\prime } ( t ) = \\int \\limits _ { \\mathbb { Q } _ { p } } u _ { t } ( x , t ) w ( x ) d x = - \\gamma \\int \\limits _ { \\mathbb { Q } _ { p } } ( \\boldsymbol { D } _ { x } ^ { \\alpha } u ) ( x , t ) w ( x ) d x \\\\ + \\int \\limits _ { \\mathbb { Q } _ { p } } F ( u ( x , t ) ) w ( x ) d x + \\int \\limits _ { \\mathbb { Q } _ { p } } ( \\boldsymbol { D } _ { x } ^ { \\alpha _ { 1 } } u ^ { 3 } ) ( x , t ) w ( x ) d x . \\end{gather*}"}
{"id": "677.png", "formula": "\\begin{align*} D _ \\alpha ^ { J } : = d _ 1 ^ { j _ 1 } \\cdots d _ m ^ { j _ { p } } \\bigg | _ { x = x _ \\alpha } \\end{align*}"}
{"id": "2341.png", "formula": "\\begin{align*} | I _ m \\cap I _ n | = 0 , \\end{align*}"}
{"id": "9171.png", "formula": "\\begin{align*} \\sum _ { \\substack { p > x ^ { 1 / 2 } \\\\ p | d } } \\frac { \\lambda ^ 2 _ f ( p ) - 2 } { p } \\ll \\frac 1 { x ^ { 1 / 2 } } \\sum _ { \\substack { p > x ^ { 1 / 2 } \\\\ p | d } } 1 \\ll \\frac { \\log X } { x ^ { 1 / 2 } \\log x } . \\end{align*}"}
{"id": "1759.png", "formula": "\\begin{align*} \\operatorname { I n d _ c } ( D ) : = [ P _ { Q } ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { A } _ G ^ c ( X , E ) ) \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "3648.png", "formula": "\\begin{align*} \\log ^ 3 t & \\ge B _ 1 ^ 2 \\log x \\log t = \\frac { \\log x \\log t } { R _ 1 } \\ge \\frac { \\log x } { R _ 1 } \\left ( \\log t + D ( 1 - 2 \\log \\log t ) \\right ) . \\end{align*}"}
{"id": "3789.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r \\big ( r _ l ( \\widetilde { V } ) , r _ l ( \\widetilde { \\mathcal { S } } ) \\big ) q _ F ^ { - \\frac { r } { 2 } } X ^ { - e f r } = r _ l \\big ( \\gamma ( X , \\pi _ E , \\widehat { \\sigma _ E } , \\psi _ E ) \\big ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r \\big ( r _ l ( V ) , r _ l ( \\mathcal { S } ) \\big ) q _ F ^ { \\frac { r } { 2 } } X ^ { e f r } \\end{align*}"}
{"id": "8864.png", "formula": "\\begin{align*} \\hat t d ( x , x _ 0 ) = \\hat t d ( \\gamma _ x ( 1 ) , \\gamma _ x ( 0 ) ) = d ( \\gamma _ x ( \\hat t ) , \\gamma _ x ( 0 ) ) \\le S \\end{align*}"}
{"id": "2091.png", "formula": "\\begin{align*} \\widehat { \\psi _ { p ^ { 2 k } } } ( \\mathbf { u } ) = \\frac { 1 } { p ^ { 2 n k } } \\sum _ { \\mathbf { c } \\in ( \\mathbb { Z } / p ^ { 2 k } \\mathbb { Z } ) ^ n } \\psi _ { p ^ { 2 k } } ( f _ \\mathbf { c } ) \\exp \\left ( \\frac { 2 \\pi i ( c _ 1 + n ) u _ 1 } { p ^ { 2 k } } \\right ) . \\end{align*}"}
{"id": "2880.png", "formula": "\\begin{align*} \\partial _ t \\widetilde { h } + \\mathcal { L } \\widetilde { h } = R \\end{align*}"}
{"id": "4834.png", "formula": "\\begin{align*} K ( x ) : = - \\log | \\tanh x | \\end{align*}"}
{"id": "8155.png", "formula": "\\begin{align*} \\hbox { $ s ( b , c , d ) = s ( a b , a c , d ) $ f o r a n y $ a \\in { \\mathbb Z } $ w i t h $ \\gcd ( a , d ) = 1 $ . } \\end{align*}"}
{"id": "2591.png", "formula": "\\begin{align*} \\langle f , \\varphi \\rangle = \\frac { 1 } { \\overline { \\langle g , \\widetilde { g } \\rangle } } \\iint _ { \\R ^ { 2 d } } \\langle f , M _ \\omega T _ x g \\rangle \\overline { V _ { \\widetilde { g } } \\varphi ( x , \\omega ) } \\ , d ( x , \\omega ) = \\langle \\widetilde { f } , \\varphi \\rangle , \\end{align*}"}
{"id": "5895.png", "formula": "\\begin{align*} x _ j = x _ 1 + \\frac { 2 t _ 1 } { 1 + \\lambda } x _ 2 + \\frac { 2 t _ 2 } { 1 + \\lambda } x _ 4 + \\cdots + \\frac { 2 t _ { \\frac { j - 1 } { 2 } } } { 1 + \\lambda } x _ { j - 1 } , \\end{align*}"}
{"id": "1371.png", "formula": "\\begin{align*} T = 1 \\mbox { a n d } J = 1 , \\end{align*}"}
{"id": "1750.png", "formula": "\\begin{align*} \\mathcal { A } ^ c _ G ( X , E ) : = \\big ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\Psi ^ { - \\infty } ( S , E | _ S ) \\big ) ^ { K \\times K } , \\end{align*}"}
{"id": "2645.png", "formula": "\\begin{align*} 0 = \\langle ( P X - X P ) g , g \\rangle = \\frac { 1 } { 2 \\pi i } \\norm { g } _ 2 ^ 2 . \\end{align*}"}
{"id": "2832.png", "formula": "\\begin{align*} \\dot { y } _ R ( t ) = 2 \\Im \\int _ { \\mathbb { R } ^ N } \\nabla \\varphi _ R ( x ) \\cdot \\nabla u \\bar { u } d x = 2 R \\Im \\int _ { \\mathbb { R } ^ N } \\nabla \\varphi \\left ( \\frac { x } { R } \\right ) \\cdot \\nabla u \\bar { u } d x , \\end{align*}"}
{"id": "7930.png", "formula": "\\begin{align*} \\sum _ { a \\leqslant k \\leqslant n } k ^ { \\ell - 1 } = \\frac { 1 } { \\ell } B _ \\ell ( n + 1 ) - \\frac { 1 } { \\ell } B _ \\ell ( a ) . \\end{align*}"}
{"id": "6223.png", "formula": "\\begin{align*} ( \\tau _ j f ) ( { \\bf x } ) = f ( { \\bf x } - { \\bf e } _ { j } ) , \\end{align*}"}
{"id": "3294.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ l = E \\end{align*}"}
{"id": "2028.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } h _ n ( X _ t ) - h _ n ( X _ 0 ) = M _ t ^ { h _ n } + N _ t ^ { h _ n } , \\\\ N _ t ^ { h _ n } = - \\int _ 0 ^ t h _ n ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } ^ n } \\end{array} \\right . \\end{align*}"}
{"id": "1758.png", "formula": "\\begin{align*} P _ { Q } : = \\left ( \\begin{array} { c c } S _ { + } ^ 2 & S _ { + } ( I + S _ { + } ) Q \\\\ S _ { - } D ^ + & I - S _ { - } ^ 2 \\end{array} \\right ) . \\end{align*}"}
{"id": "7703.png", "formula": "\\begin{align*} H _ 0 = \\big \\{ \\{ k \\} \\colon k \\in S \\big \\} H _ 0 ' = \\big \\{ \\{ k \\} \\colon k \\in S ' \\big \\} \\end{align*}"}
{"id": "8114.png", "formula": "\\begin{align*} T B _ { k } ( f ) = \\sum _ { \\gamma \\in \\Gamma _ { k } ( G ) } t b ( f ( \\gamma ) ) . \\end{align*}"}
{"id": "7031.png", "formula": "\\begin{align*} f ( z ) & = ( z - \\xi _ k ) a _ { k } ^ { \\# } ( z ) \\widetilde { f } ( z ) + ( p ( z ) - p ( \\xi _ k ) ) + p ( \\xi _ k ) \\\\ & = ( z - \\xi _ k ) \\Big ( a _ { k } ^ { \\# } ( z ) \\widetilde { f } ( z ) + \\frac { p ( z ) - p ( \\xi _ k ) } { z - \\xi _ k } \\Big ) + p ( \\xi _ k ) \\\\ & = ( z - \\xi _ k ) f _ { k } ( z ) + f ( \\xi _ k ) , \\end{align*}"}
{"id": "7849.png", "formula": "\\begin{align*} \\frac { w _ { k + r } } { w _ { r } } = \\begin{cases} \\frac { 1 } { 2 ^ { k } } , & ~ k , r ~ \\\\ \\frac { 1 } { 2 ^ { k + 2 } } , & ~ k , r ~ \\\\ \\frac { 1 } { 2 ^ { k } } , & ~ k ~ ~ r ~ \\\\ \\frac { 1 } { 2 ^ { k - 2 } } , & ~ k ~ ~ r ~ \\end{cases} \\end{align*}"}
{"id": "2262.png", "formula": "\\begin{align*} x ^ 2 = x \\cdot x = x ^ T x . \\end{align*}"}
{"id": "1351.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\sum _ { j \\neq i } \\partial _ j \\left ( \\frac { y _ j } { t ^ d } \\eta \\left ( \\frac { y } { t } \\right ) \\right ) + \\frac { y _ i } { t ^ { d + 1 } } \\partial _ i \\eta \\left ( \\frac { y } { t } \\right ) d t = - \\int _ 0 ^ 1 \\frac { 1 } { t ^ { d - 1 } } \\frac { d } { d t } \\left ( \\eta \\left ( \\frac { y } { t } \\right ) \\right ) d t = - \\eta ( y ) \\end{align*}"}
{"id": "7903.png", "formula": "\\begin{align*} \\frac { 1 } { | B _ R | } \\int _ { B _ R } | | \\Theta ( v ) | | ^ 2 \\ d v & \\leq \\frac { C ( m , k ) } { | B _ R | } \\sum _ { i _ 1 , i _ 2 , j _ 1 , j _ 2 } \\int _ { B _ R } ( k _ { i _ 1 i _ 2 } ( x _ { j _ 1 } - x _ { j _ 2 } + v ) ) ^ 2 \\ d v \\\\ & \\leq \\frac { C ( m , k ) } { | B _ R | } \\sum _ { 1 \\leq i _ 1 , i _ 2 \\leq m } \\int _ { B _ { R + T } } ( k _ { i _ 1 i _ 2 } ) ^ 2 ( v ) \\ d v , \\end{align*}"}
{"id": "2792.png", "formula": "\\begin{align*} \\Phi _ 1 ( h _ 1 ) = \\frac { 1 } { 2 } \\int \\left ( L _ + h _ 1 \\right ) h _ 1 , \\Phi _ 2 ( h _ 2 ) = \\frac { 1 } { 2 } \\int \\left ( L _ - h _ 2 \\right ) h _ 2 . \\end{align*}"}
{"id": "705.png", "formula": "\\begin{align*} Z _ i = Z _ { i ; | | } + Z _ { i ; \\perp } , Z _ { i ; | | } \\in \\ker ( \\kappa ^ { ( \\ell ) } ) , \\ , Z _ { i ; \\perp } \\in \\ker ( \\kappa ^ { ( \\ell ) } ) ^ \\perp \\end{align*}"}
{"id": "1570.png", "formula": "\\begin{align*} \\mathfrak { P } _ k ( z _ 0 \\times z _ 0 ) ^ { - 1 } \\overline { \\mathbf { f } ( g _ { \\mathbf { h } } \\times h _ { \\mathbf { h } } ) } = \\sum _ { j = 1 } ^ e \\mathfrak { P } _ k ( z _ 0 ) ^ { - 1 } \\overline { \\mathbf { g } _ j ( g _ { \\mathbf { h } } ) } \\mathfrak { P } _ k ( z _ 0 ) ^ { - 1 } h _ j ( \\mathbf { h } ) . \\end{align*}"}
{"id": "68.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - n + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 2 } ^ { n } ( q _ v ^ i - ( - 1 ) ^ i ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - n ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 2 } ^ { n + 1 } ( q _ v ^ i - ( - 1 ) ^ i ) \\Bigr \\} ^ { - 1 } \\\\ & = \\frac { q _ v ^ { n _ { v , \\nu _ v } } - ( - 1 ) ^ { n _ { v , \\nu _ v } } } { q _ v ^ { n + 1 } - ( - 1 ) ^ { n + 1 } } . \\end{align*}"}
{"id": "3076.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = - j x _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 1 ^ \\prime = - y _ 1 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , . \\end{align*}"}
{"id": "4598.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - \\Phi ^ { - 1 } ( 1 - \\kappa _ n / 2 ) \\Big ) \\sim \\kappa _ n / 2 , \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "9244.png", "formula": "\\begin{align*} \\forall x ^ X , y ^ X , { x ' } ^ X , { y ' } ^ { X } \\left ( x = _ X x ' \\land y = _ X y ' \\rightarrow \\chi _ A x y = _ 0 \\chi _ A x ' y ' \\right ) \\end{align*}"}
{"id": "5295.png", "formula": "\\begin{align*} a \\otimes b = \\sum _ { i = 1 } ^ n \\Delta ( p _ i ) ( 1 \\otimes q _ i ) , \\end{align*}"}
{"id": "7648.png", "formula": "\\begin{align*} \\bar { u } = A w \\qquad . \\end{align*}"}
{"id": "3114.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + a z y ^ 5 + x ^ 6 + b y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "7480.png", "formula": "\\begin{align*} & d _ { v } ^ { n } : = \\frac { \\left \\| \\phi ^ { n } - \\phi ^ { n - 1 } \\right \\| _ { \\infty } } { \\tau } < \\varepsilon _ { v } \\end{align*}"}
{"id": "4347.png", "formula": "\\begin{align*} \\hat { V } [ A , \\eta ] ( \\tau ) = \\left [ - A b ^ { \\frac { \\alpha } { 2 } + \\eta } ( \\tau ) , A b ^ { \\frac { \\alpha } { 2 } + \\eta } ( \\tau ) \\right ] ^ { \\ell } . \\end{align*}"}
{"id": "9304.png", "formula": "\\begin{align*} D \\cdot C _ n = E _ Z \\cdot C _ n - C _ n ^ 2 = - ( K _ Z + C _ n ) \\cdot C _ n = - \\deg ( K _ { C _ n } ) . \\end{align*}"}
{"id": "5976.png", "formula": "\\begin{align*} l _ { \\pi _ { \\lambda , \\beta } } ( z ) = \\int _ { \\mathbb { R } } \\mathrm { e } ^ { z x } \\ , \\mathrm { d } \\pi _ { \\lambda , \\beta } ( x ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\big ( \\mathrm { e } ^ { z } \\lambda \\big ) ^ { k } } { k ! } E _ { \\beta } ^ { ( k ) } \\big ( - \\lambda \\big ) = E _ { \\beta } \\big ( \\lambda ( \\mathrm { e } ^ { z } - 1 ) \\big ) . \\end{align*}"}
{"id": "4753.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c c } 0 & a _ { 1 , 2 } & \\cdots & a _ { 1 , n } \\\\ a _ { 2 , 1 } & 0 & \\cdots & a _ { 2 , n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ { n , 1 } & a _ { n , 2 } & \\cdots & 0 \\end{array} \\right ) \\left ( \\begin{array} { c c c c } c _ 1 \\\\ c _ 2 \\\\ \\vdots \\\\ c _ n \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } x _ 1 \\\\ x _ 2 \\\\ \\vdots \\\\ x _ n \\end{array} \\right ) . \\end{align*}"}
{"id": "671.png", "formula": "\\begin{align*} \\alpha \\ = \\ \\sum \\limits _ { n = 0 } ^ \\infty \\ \\frac { f ( n ) } { 2 ^ n } ( f ( n ) \\in \\{ 0 , 1 \\} ) . \\end{align*}"}
{"id": "7071.png", "formula": "\\begin{align*} L _ 0 = C ( A + \\| \\beta _ 0 \\| _ \\infty + 3 + T ) \\end{align*}"}
{"id": "2322.png", "formula": "\\begin{align*} & \\ , W ( M _ { \\eta _ 1 } T _ { \\xi _ 1 } f , \\ , M _ { \\eta _ 2 } T _ { \\xi _ 2 } g ) \\\\ = & \\ , e ^ { \\pi i ( \\xi _ 1 - \\xi _ 2 ) \\cdot ( \\eta _ 1 + \\eta _ 2 ) } e ^ { 2 \\pi i x \\cdot ( \\eta _ 1 - \\eta _ 2 ) } e ^ { - 2 \\pi i \\omega \\cdot ( \\xi _ 1 - \\xi _ 2 ) } \\ , W ( f , g ) ( x - \\tfrac { \\xi _ 1 + \\xi _ 2 } { 2 } , \\omega - \\tfrac { \\eta _ 1 + \\eta _ 2 } { 2 } ) . \\end{align*}"}
{"id": "4711.png", "formula": "\\begin{align*} - \\sum _ { \\substack { i , j = 1 , \\\\ \\i \\not = j } } ^ n \\frac { a _ { i j } } { ( \\alpha _ i - \\alpha _ j ) ^ 2 } = - C _ 0 \\sum _ { i = 1 } ^ { 2 k } \\sum _ { \\ell = 1 } ^ { 2 k - i + 1 } \\frac { ( - 1 ) ^ { i - 1 } } { ( \\theta _ \\ell + \\cdots + \\theta _ { \\ell + i - 1 } ) ^ 2 } , \\end{align*}"}
{"id": "155.png", "formula": "\\begin{align*} \\varphi ( \\xi _ 1 , \\xi _ 2 ) = ( \\varphi _ X ( \\xi _ 1 ) \\varphi _ X ( \\xi _ 2 ) ) ^ { 1 - z } \\varphi _ X ( \\xi _ 1 + \\xi _ 2 ) ^ { z } . \\end{align*}"}
{"id": "2628.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Q } } Z f ( x , \\omega ) \\ , d \\omega & = \\int _ \\mathcal { Q } \\left ( \\sum _ { k \\in \\Z ^ d } f ( x + k ) e ^ { - 2 \\pi i k \\cdot \\omega } \\right ) \\ , d \\omega \\\\ & = \\sum _ { k \\in \\Z ^ d } \\int _ \\mathcal { Q } f ( x + k ) e ^ { - 2 \\pi i k \\cdot \\omega } \\ , d \\omega = \\sum _ { k \\in \\Z ^ d } f ( x + k ) \\int _ \\mathcal { Q } e ^ { - 2 \\pi i k \\cdot \\omega } \\ , d \\omega \\\\ & = f ( x ) , \\end{align*}"}
{"id": "227.png", "formula": "\\begin{align*} \\langle ( P ^ { \\nu _ \\alpha } _ t ) ^ * ( f ) - f ; g \\rangle _ { L ^ 2 ( \\mu _ \\alpha ) } & = \\int _ { 0 } ^ t \\left \\langle ( D ^ { \\alpha - 1 } ) ^ { * } ( f ) + \\frac { 1 } { p _ \\alpha } ( R ^ { \\alpha } ) ^ * ( p _ \\alpha , f ) ; \\nabla P _ s ^ { \\nu _ \\alpha } ( g ) \\right \\rangle _ { L ^ 2 ( \\mu _ \\alpha ) } d s . \\end{align*}"}
{"id": "6648.png", "formula": "\\begin{align*} \\frac { e ^ { \\delta z } - e ^ { - \\delta z } } { 2 \\delta z } = 1 + O ( \\delta | z | ) . \\end{align*}"}
{"id": "694.png", "formula": "\\begin{align*} \\frac { \\kappa _ { ( 1 1 ) ( 0 0 ) } ^ { ( \\ell + 1 ) } } { K _ { ( 1 1 ) } ^ { ( \\ell ) } K _ { ( 0 0 ) } ^ { ( \\ell ) } } & = - \\frac { 1 } { 3 } \\xi ( 1 + O ( \\ell ^ { - 1 } ) ) + O ( n ^ { - 2 } ) \\\\ \\frac { \\kappa _ { ( 1 1 ) ( 1 1 ) } ^ { ( \\ell + 1 ) } } { ( K _ { ( 1 1 ) } ^ { ( \\ell ) } ) ^ 2 } & = \\frac { 8 } { 3 } \\xi ( 1 + O ( \\ell ^ { - 1 } ) ) + O ( n ^ { - 2 } ) \\\\ \\frac { \\kappa _ { ( 1 1 ) ( 2 2 ) } ^ { ( \\ell + 1 ) } } { K _ { ( 1 1 ) } ^ { ( \\ell ) } K _ { ( 2 2 ) } ^ { ( \\ell ) } } & = \\frac { 2 } { 3 } \\xi ( 1 + O ( \\ell ^ { - 1 } ) ) + O ( n ^ { - 2 } ) , \\end{align*}"}
{"id": "8644.png", "formula": "\\begin{align*} P ' Q & = - \\left ( \\sum _ { i = 1 } ^ n P _ i \\right ) \\left ( \\sum _ { j = 1 } ^ n - \\lambda _ j ^ 2 P _ j \\right ) \\\\ & = - \\sum _ { j = 1 } ^ n \\lambda _ j ^ 2 P _ j ^ 2 - \\sum _ { \\substack { 1 \\leq i , j \\leq n \\\\ i \\neq j } } \\lambda _ j ^ 2 P _ i P _ j \\end{align*}"}
{"id": "907.png", "formula": "\\begin{align*} ( a ^ { + } ) ^ { k } a ^ { k } = \\sum _ { l = 0 } ^ { k } S _ { 1 , \\lambda } ( k , l ) ( a ^ { + } a ) _ { l , \\lambda } . \\end{align*}"}
{"id": "4252.png", "formula": "\\begin{align*} \\big ( \\sup _ { n \\in \\N } S _ n T _ n \\big ) f \\geq \\big ( \\sup _ { n \\in \\N } S _ n T _ m \\big ) f = \\sigma \\lim _ n ( S _ n T _ m ) f = \\sigma \\lim _ n S _ n ( T _ m f ) = S T _ m f , \\end{align*}"}
{"id": "7367.png", "formula": "\\begin{align*} u ( x , t ) = \\begin{cases} \\frac { | x | } { 1 + \\pi | x | t } + 1 & \\\\ \\frac { R ^ 2 } { \\pi R ^ 2 t - { | x | } + 2 R } + 1 & \\\\ | x | - \\pi R ^ 2 t + 1 & \\end{cases} \\end{align*}"}
{"id": "2690.png", "formula": "\\begin{align*} \\norm { K } _ { H . S . } = \\left ( \\sum _ { n \\in \\N } \\norm { K e _ n } _ \\mathcal { H } ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "1531.png", "formula": "\\begin{align*} \\mathbf { E } ( \\mathfrak { g } , s ) : = E _ k ^ { \\tilde { \\tau } _ m } ( \\mathfrak { g } , s ; \\chi , \\mathfrak { n } ) : = \\sum _ { \\gamma \\in P _ N \\backslash G _ N } \\phi ( \\gamma \\mathfrak { g } , s ) , \\mathfrak { g } \\in G _ N \\end{align*}"}
{"id": "5433.png", "formula": "\\begin{align*} \\ddot x _ 1 ( t ) = - \\theta _ 1 x _ 1 ( t ) - \\theta _ 2 \\dot x _ 1 ( t ) + \\theta _ 3 u ( t ) \\end{align*}"}
{"id": "4240.png", "formula": "\\begin{align*} u _ x = { } & 0 \\\\ u _ { x x } + u _ y \\rho '' = { } & 0 \\\\ u _ { x x x } + 3 u _ { x y } \\rho '' + u _ y \\rho ''' = { } & 0 . \\end{align*}"}
{"id": "2413.png", "formula": "\\begin{align*} f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , S ^ { - 1 } e _ \\gamma \\rangle e _ \\gamma \\end{align*}"}
{"id": "3596.png", "formula": "\\begin{align*} d [ c a _ { i , j } ] : = \\min \\{ d ( c a _ { i , j } ) , \\alpha _ { i - 1 } , \\alpha _ { j } \\} \\ , , c \\in F ^ { \\times } \\ , . \\end{align*}"}
{"id": "9347.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( n , k ) = S _ { 2 , \\lambda } ( n - 1 , k - 1 ) + ( k - ( n - 1 ) \\lambda ) S _ { 2 , \\lambda } ( n - 1 , k ) , ( \\mathrm { s e e } \\ [ 8 ] ) , \\end{align*}"}
{"id": "5494.png", "formula": "\\begin{align*} \\xi ( s ; r , x ) & = S _ { s - r } x + \\int _ r ^ s S _ { s - u } \\alpha ( u , \\xi ( u ; r , x ) ) d u , s \\in [ r , \\infty ) . \\end{align*}"}
{"id": "8949.png", "formula": "\\begin{align*} e ^ { - t } C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } \\dot { \\xi } ( t ) = \\int _ 0 ^ t e ^ { - s } D f \\left ( \\xi ( s ) \\right ) d s + \\tilde { c } \\end{align*}"}
{"id": "4536.png", "formula": "\\begin{align*} H ( z , \\chi ) = L ( z , \\chi ) L ( 2 z , \\chi ^ { 2 } ) G ( z , \\chi ) , \\ ( \\Re z > 1 ) \\end{align*}"}
{"id": "7835.png", "formula": "\\begin{align*} x _ { 1 } & = T ^ { * 3 ( j - 1 ) } x - \\alpha _ { 0 } e _ { 0 } = \\sum _ { i = 1 } ^ { 3 } \\alpha _ { i } e _ { i } , \\\\ y _ { 1 } & = T ^ { * 3 ( j - 1 - r ) } y - \\beta _ { 0 } e _ { 0 } = \\sum _ { i = 1 } ^ { 2 } \\beta _ { i } e _ { i } , \\\\ z _ { 1 } & = T ^ { * 3 ( n - 2 j - 2 + r ) } z - \\gamma _ { 0 } e _ { 0 } = \\sum _ { i = 1 } ^ { 2 } \\gamma _ { i } e _ { i } . \\end{align*}"}
{"id": "3220.png", "formula": "\\begin{align*} a = \\left ( \\int _ 0 ^ \\infty \\frac { 1 } { u } d \\mu ( u ) \\right ) ^ { - 1 } , b = \\int _ 0 ^ \\infty u \\ , d \\mu ( u ) . \\end{align*}"}
{"id": "2462.png", "formula": "\\begin{align*} S ^ T J S = \\begin{pmatrix} A ^ T C - C ^ T A & A ^ T D - C ^ T B \\\\ B ^ T C - D ^ T A & B ^ T D - D ^ T B \\end{pmatrix} = \\begin{pmatrix} 0 & I \\\\ - I & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "9372.png", "formula": "\\begin{align*} \\mathrm { L i } _ { 2 } ( 1 - e _ { \\lambda } ( - t ) ) & = - \\int _ { 0 } ^ { 1 - e _ { \\lambda } ( - t ) } \\frac { 1 } { x } \\log _ { \\lambda } ( 1 - x ) d x \\\\ & = \\int _ { 0 } ^ { t } \\frac { - t } { e _ { \\lambda } ( - t ) - 1 } e _ { \\lambda } ^ { 1 - \\lambda } ( - t ) d t \\\\ & = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\beta _ { n - 1 , \\lambda } ( 1 - \\lambda ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "3192.png", "formula": "\\begin{align*} A x = b , \\end{align*}"}
{"id": "7857.png", "formula": "\\begin{align*} ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cup ( B \\setminus \\{ \\bar { a } _ n \\} ) \\cup ( C \\setminus \\{ \\bar { a } _ n \\} ) & = ( A \\cup B \\cup C ) \\setminus \\{ \\bar { a } _ n \\} \\\\ & = \\mathbb { N } \\setminus \\{ \\bar { a } _ n \\} \\end{align*}"}
{"id": "1834.png", "formula": "\\begin{align*} L _ n ( x , y ) = D ^ n ( x ) . \\end{align*}"}
{"id": "7922.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } X _ { t } = \\theta f ( X _ { t } ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ { t } + \\mathrm { d } L _ { t } - \\mathrm { d } R _ { t } , \\\\ & X _ { 0 } = x \\in [ l , u ] , \\end{aligned} \\right . \\end{align*}"}
{"id": "435.png", "formula": "\\begin{align*} \\widetilde { A } ( \\xi ; U ) : = \\sum _ { i = 1 } ^ { 3 } \\widetilde { A } ^ { i } ( U ) \\xi _ { i } \\quad \\mbox { a n d } \\widetilde { B } ( \\xi ; U ) : = \\sum _ { i , j = 1 } ^ { 3 } \\widetilde { B } ^ { i j } ( U ) \\xi _ { i } \\xi _ { j } , \\end{align*}"}
{"id": "1002.png", "formula": "\\begin{align*} \\frac { 2 v ( h e _ 1 ) - v ( h e _ 1 + y ) - v ( h e _ 1 - y ) } h & = 2 \\partial ^ h _ 1 v ( 0 ) - \\partial ^ h _ 1 v ( y ) - \\partial ^ h _ 1 v ( - y ) - \\frac { v ( y ) + v ( - y ) } h \\end{align*}"}
{"id": "3804.png", "formula": "\\begin{align*} \\norm { A - \\mathbb { S } A } & = \\norm { ( I - \\mathbb { S } ) A } = \\norm { ( I - \\mathbb { S } ) ( I - Q _ { Z _ k } Q _ { Z _ k } ^ T ) A } \\\\ & \\le \\norm { ( I - \\mathbb { S } ) } \\ , \\norm { ( I - Q _ { Z _ k } Q _ { Z _ k } ^ T ) A } . \\end{align*}"}
{"id": "7478.png", "formula": "\\begin{align*} \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ n } { \\tau ^ 2 } + \\eta ^ n \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ n } { 2 \\tau } = 0 . \\end{align*}"}
{"id": "9120.png", "formula": "\\begin{align*} x _ \\lambda = J ^ S _ \\mu ( x _ \\lambda + \\mu T _ \\lambda x _ \\lambda ) \\end{align*}"}
{"id": "1512.png", "formula": "\\begin{align*} \\Gamma _ m ( s ) = \\pi ^ { m ( m - 1 ) } \\prod _ { i = 0 } ^ { m - 1 } \\Gamma ( s - 2 i ) , m \\in \\Z , \\end{align*}"}
{"id": "5516.png", "formula": "\\begin{align*} \\eta : = \\frac { \\delta } { 2 \\phi _ { \\delta } ^ { x _ 0 } ( T ) } , \\end{align*}"}
{"id": "5836.png", "formula": "\\begin{align*} v _ { k _ \\bullet } = v _ + - 2 \\varepsilon _ \\bullet , v _ { k + 1 } = v _ k + \\frac { \\varepsilon _ \\bullet } { k ^ 2 } , \\ ; \\ ; k \\geq k _ \\bullet . \\end{align*}"}
{"id": "9321.png", "formula": "\\begin{align*} r ( \\mathbf { x } , \\mathbf { y } ) : = \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} - \\Pi _ { D } ( \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} - \\begin{bmatrix} H \\mathbf { x } + \\mathbf { g } - A ^ T \\mathbf { y } \\\\ A \\mathbf { x } - \\mathbf { b } \\end{bmatrix} ) , \\end{align*}"}
{"id": "5019.png", "formula": "\\begin{align*} \\partial _ x \\pi _ 2 \\tilde A _ { k n } & = O ( 1 ) , \\\\ \\partial _ y \\pi _ 2 \\tilde A _ { k n } & = \\epsilon ^ { 2 ^ { k n } } \\tilde b ^ { ( 1 + O ( \\alpha ^ n ) ) | \\bar \\upsilon _ { k n } | } , \\end{align*}"}
{"id": "9341.png", "formula": "\\begin{align*} H _ { n } ^ { ( 1 ) } = H _ { n } , H _ { n } ^ { ( r ) } = \\sum _ { k = 1 } ^ { n } H _ { k } ^ { ( r - 1 ) } , ( r \\ge 2 ) . \\end{align*}"}
{"id": "2243.png", "formula": "\\begin{align*} \\mathcal { O } _ { k , N } ( t ) : = \\int _ 0 ^ t \\tilde { E } _ { k , N } ( t - s ) \\textmd { d } W ( s ) . \\end{align*}"}
{"id": "1.png", "formula": "\\begin{align*} P _ n ( n + 1 , x ) = \\frac { \\Gamma ( x ) } { \\Gamma ( x - n - 1 ) } \\left ( \\psi ( x ) - \\psi ( x - n - 1 ) \\right ) . \\end{align*}"}
{"id": "7189.png", "formula": "\\begin{align*} \\rho ( t , x ) & = G * _ { t , x } S + S \\end{align*}"}
{"id": "2272.png", "formula": "\\begin{align*} I ' ( \\omega ) & = \\int _ { \\R } ( - 2 \\pi i x ) e ^ { - \\pi M x ^ 2 } e ^ { - 2 \\pi i \\omega x } \\ , d x = \\frac { i } { M } \\int _ { \\R } \\dfrac { d } { d x } ( e ^ { - \\pi M x ^ 2 } ) e ^ { - 2 \\pi i \\omega x } \\ , d x \\\\ & \\stackrel { } { = } - \\frac { i } { M } \\int _ { \\R } e ^ { - \\pi M x ^ 2 } \\dfrac { d } { d x } ( e ^ { - 2 \\pi i \\omega x } ) \\ , d x = - \\frac { 2 \\pi \\omega } { M } I ( \\omega ) . \\end{align*}"}
{"id": "1488.png", "formula": "\\begin{align*} \\mathbf { f } ( \\alpha g k _ { \\infty } k ) = j ( k _ { \\infty } , z _ 0 ) ^ { - k } \\mathbf { f } ( g ) , \\end{align*}"}
{"id": "1753.png", "formula": "\\begin{align*} L ( g ) : = | | X | | \\end{align*}"}
{"id": "2035.png", "formula": "\\begin{align*} L ^ { h _ \\alpha } _ t & = \\exp \\left ( M ^ { ( \\alpha ) } _ t - \\frac { \\ , 1 \\ , } { 2 } \\ < M ^ { ( \\alpha ) , c } \\ > _ t \\right ) \\prod _ { 0 < s \\leq t } \\frac { h _ { \\alpha } ( X _ s ) } { h _ { \\alpha } ( X _ { s - } ) } \\exp \\left ( 1 - \\frac { h _ { \\alpha } ( X _ s ) } { h _ { \\alpha } ( X _ { s - } ) } \\right ) \\\\ & = \\frac { h _ { \\alpha } ( X _ t ) } { h _ { \\alpha } ( X _ 0 ) } \\exp \\left ( A _ t ^ { \\overline { \\mu } } + \\alpha \\int _ 0 ^ t \\left ( h _ { \\alpha } ^ { - 1 } - 1 \\right ) ( X _ s ) { \\rm d } s \\right ) . \\end{align*}"}
{"id": "2419.png", "formula": "\\begin{align*} f = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma e _ \\gamma \\end{align*}"}
{"id": "6639.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { 1 } { p ^ w } \\right ) \\left ( 1 + \\frac { p ^ { w - 1 } - 1 } { p ( p - 1 ) } \\right ) = 1 + \\frac { 1 } { p ^ { 3 - w } } + O \\left ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\right ) . \\end{align*}"}
{"id": "505.png", "formula": "\\begin{align*} \\Psi ( \\mathfrak { m } _ { \\alpha \\omega _ { 1 } \\omega _ { 2 } } ^ { i m _ { 1 } m _ { 2 } } ) = \\Psi ( Y _ { \\omega _ { 2 } \\omega _ { 1 } } ^ { m _ { 1 } m _ { 2 } } + 1 ) . \\end{align*}"}
{"id": "8948.png", "formula": "\\begin{align*} \\min \\left \\{ I _ b [ \\gamma ] ; \\gamma \\in \\mathrm { A C } \\left ( [ 0 , b ] ; \\Omega \\right ) , \\gamma ( 0 ) = x , \\gamma ( b ) = \\xi ( b ) \\right \\} . \\end{align*}"}
{"id": "1805.png", "formula": "\\begin{align*} \\begin{aligned} & \\Phi ^ P _ { Y , g } ( A _ 0 , A _ 1 , \\dots , A _ m ) \\\\ : = & \\int _ { h \\in M / Z _ M ( g ) } \\int _ { K N } \\int _ { G ^ { \\times m } } C \\big ( H ( g _ 1 . . . g _ m k ) , H ( g _ 2 . . . g _ m k ) , \\dots , H ( g _ m k ) \\big ) \\\\ & \\operatorname { T r } \\Big ( A _ 0 \\big ( k h g h ^ { - 1 } n k ^ { - 1 } ( g _ 1 \\dots g _ m ) ^ { - 1 } \\big ) \\circ A _ 1 ( g _ 1 ) \\dots \\circ A _ m ( g _ m ) \\Big ) d g _ 1 \\cdots d g _ m d k d n d h . \\end{aligned} \\end{align*}"}
{"id": "1406.png", "formula": "\\begin{align*} \\dot { \\hat q } \\equiv { d { \\hat q } \\over d t } = { 1 \\over 2 } \\hat q \\hat \\xi ^ b , \\end{align*}"}
{"id": "1460.png", "formula": "\\begin{align*} R \\left [ \\begin{array} { c } z _ 1 \\\\ u _ { 0 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c } z _ 2 \\\\ u _ { 0 2 } \\end{array} \\right ] \\mu ( z _ 1 ) , \\end{align*}"}
{"id": "243.png", "formula": "\\begin{align*} \\langle \\nabla ( f _ h ) ( x ) ; u \\rangle = - \\int _ 0 ^ { + \\infty } e ^ { - t } P _ t ^ { \\Sigma } ( \\langle u ; \\nabla ( h ) \\rangle ) ( x ) d t . \\end{align*}"}
{"id": "6076.png", "formula": "\\begin{align*} \\varphi ^ { j } & = R _ { s } \\cdot U ^ { j } = x _ { s } u ^ { j } + y _ { s } v ^ { j } + z _ { s } w ^ { j } , \\\\ \\psi ^ { j } & = R _ { t } \\cdot U ^ { j } = x _ { t } u ^ { j } + y _ { t } v ^ { j } + z _ { t } w ^ { j } . \\end{align*}"}
{"id": "1858.png", "formula": "\\begin{align*} { { \\rm d } ^ n \\over { \\rm d } x ^ n } \\tan ( x ) = P _ n ( \\tan ( x ) ) . \\end{align*}"}
{"id": "92.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ 2 \\cdot 3 ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) . \\end{align*}"}
{"id": "3997.png", "formula": "\\begin{align*} \\int _ { T - \\tau } ^ { T } \\mod { \\widetilde { U } ^ p ( t ) } ^ 2 d t = \\int _ { 0 } ^ { \\tau } \\mod { \\widetilde { U } ^ p ( T - t ) } ^ 2 d t & \\geq C \\sum _ { k \\geq N } \\mod { a _ k } ^ 2 | 1 - e ^ { \\tilde { \\lambda } _ k } | ^ 2 e ^ { 2 \\Re ( \\tilde { \\lambda } _ k ) \\tau } \\\\ & \\geq C \\sum _ { k \\geq N } \\mod { a _ k } ^ 2 e ^ { 2 \\Re ( \\tilde { \\lambda } _ k ) \\tau } , \\end{align*}"}
{"id": "3282.png", "formula": "\\begin{align*} L ( U ^ \\mu ; z _ 0 , r ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi U ^ \\mu ( z _ 0 + r e ^ { i \\theta } ) d \\theta . \\end{align*}"}
{"id": "2003.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ 1 ( - t ) \\\\ u _ 2 ( - t ) \\\\ u _ 3 ( - t ) \\end{array} \\right ) = M ( - 1 ) \\left ( \\begin{array} { c } u _ 1 ( t ) \\\\ u _ 2 ( t ) \\\\ u _ 3 ( t ) \\end{array} \\right ) \\end{align*}"}
{"id": "6536.png", "formula": "\\begin{align*} S ' _ n : = \\sum _ { j = n _ 0 } ^ n f ( j ) \\mbox { a n d } I ' _ n : = \\int _ { n _ 0 } ^ n f ( t ) \\ , d t . \\end{align*}"}
{"id": "2913.png", "formula": "\\begin{align*} H _ 2 : C _ A = \\Pi _ 2 A \\in I _ d ( 2 ) , \\end{align*}"}
{"id": "4114.png", "formula": "\\begin{align*} \\begin{aligned} & | \\sigma _ n ^ 2 ( z ) - \\sigma _ n ^ 2 ( y ) | \\\\ & \\le ( | \\sigma _ n ( z ) | + | \\sigma _ n ( y ) | ) \\cdot | \\sigma _ n ( z ) - \\sigma _ n ( y ) | \\le ( | \\sigma ( z ) | + | \\sigma ( y ) | ) \\cdot | \\sigma _ n ( z ) - \\sigma _ n ( y ) | \\\\ & \\le c \\cdot \\bigl ( ( 1 + | z | ^ { 2 \\ell _ \\sigma + 1 } + | y | ^ { 2 \\ell _ \\sigma + 1 } ) \\cdot \\bigl ( | z - y | + n ^ { - 1 / 2 } \\bigr ) . \\end{aligned} \\end{align*}"}
{"id": "7211.png", "formula": "\\begin{align*} \\tau _ x & = { \\mathcal T } _ { t , x , 0 } , \\check \\tau _ { t , x } = \\check { \\mathcal T } _ { t , x , 0 } , \\\\ { \\mathcal T } _ { t , x , v } & = \\tau _ { x - \\check { \\mathcal T } _ { t , x , v } v } \\check { \\mathcal T } _ { t , x , v } > 0 . \\end{align*}"}
{"id": "5384.png", "formula": "\\begin{align*} { } \\left \\{ \\begin{aligned} \\mathrm { d } X _ { t } & = b ( X _ { t } ) \\mathrm { d } t + \\sigma \\mathrm { d } B ^ { H } _ { t } , \\\\ X _ { 0 } & = x , \\end{aligned} \\right . \\end{align*}"}
{"id": "3845.png", "formula": "\\begin{align*} y ^ { [ m ] } = y ^ { ( m ) } + ( - 1 ) ^ m \\sum _ { j = 0 } ^ { m - 1 } q _ { j , m } y ^ { ( j ) } . \\end{align*}"}
{"id": "4015.png", "formula": "\\begin{align*} \\eta ( x ) = C _ 1 e ^ { m _ 1 x } + C _ 2 e ^ { m _ 2 x } + C _ 3 e ^ { m _ 3 x } , \\ \\ \\ x \\in ( 0 , 1 ) , \\end{align*}"}
{"id": "2163.png", "formula": "\\begin{align*} - ( \\triangle ) ^ \\alpha u + V ( x ) u = K ( x ) f ( u ) , \\ \\ \\ \\mathbb { R } ^ { d } , \\end{align*}"}
{"id": "3196.png", "formula": "\\begin{align*} \\varphi ( r , t , \\xi ) \\ , = \\ , - \\frac { 1 } { t } \\cdot \\log \\left ( r - \\xi \\right ) \\end{align*}"}
{"id": "8367.png", "formula": "\\begin{align*} g _ { v } ( u , w ) = \\frac { 1 } { 2 } \\left . \\frac { \\partial ^ 2 } { \\partial s \\partial t } L ( v + t u + s w ) \\right | _ { t = s = 0 } \\end{align*}"}
{"id": "2658.png", "formula": "\\begin{align*} \\pi ( z ' ) ^ { - 1 } \\pi ( z ) \\pi ( z ' ) = e ^ { 2 \\pi i \\sigma ( z ' , z ) } \\pi ( z ) V _ { \\pi ( z ' ) g } ( \\pi ( z ' ) f ) ( z ) & = \\langle f , \\pi ( z ' ) ^ { - 1 } \\pi ( z ) \\pi ( z ' ) g \\rangle \\\\ & = e ^ { - 2 \\pi i \\sigma ( z ' , z ) } V _ g f ( z ) . \\end{align*}"}
{"id": "1426.png", "formula": "\\begin{align*} q \\colon [ 0 , 1 ] \\times \\Delta _ d & \\to \\Delta _ { d + 1 } \\\\ ( t , b _ 1 , \\dots , b _ d ) & \\mapsto ( ( 1 - t ) ( 1 - \\sum _ { i = 1 } ^ d b _ i ) , ( 1 - t ) b _ 1 , \\dots , ( 1 - t ) b _ d ) . \\end{align*}"}
{"id": "8783.png", "formula": "\\begin{align*} s _ { i j } = a _ { i 0 } z _ { i 0 } + \\sum _ { k = 1 } ^ j ( a _ { i k } - a _ { i k - 1 } ) z _ { i k } j = 0 , \\ldots , n _ i . \\end{align*}"}
{"id": "4358.png", "formula": "\\begin{align*} H = \\partial _ \\xi ^ 2 + \\frac { d + 1 } { \\xi } \\partial _ \\xi - 3 ( d - 2 ) \\left ( 2 Q ( \\xi ) + \\xi ^ 2 Q ^ 2 ( \\xi ) \\right ) . \\end{align*}"}
{"id": "5544.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d Y ( t ) & = & L ( Y ( t ) ) d t + \\sigma ( Y ( t ) ) d W ( t ) \\medskip \\\\ Y ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "5308.png", "formula": "\\begin{align*} X ( v \\otimes a ) = \\sum _ s \\delta _ s ( v ) ( 1 \\otimes a ) , v \\in V , a \\in A . \\end{align*}"}
{"id": "4600.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - x \\Big ) } { 1 - \\Phi \\left ( x \\right ) } = \\exp \\bigg \\{ \\theta _ 2 C \\frac { \\ln n + x ^ { 3 } } { \\sqrt { n } } \\bigg \\} , \\end{align*}"}
{"id": "1302.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\bigcup _ { p _ { i } \\in S _ { \\theta } , \\ , \\ , p _ { i } < p _ { l } } \\Lambda _ { ( p _ { i } , \\infty ) } ( M , \\Gamma ) | } { | \\Lambda ( M , \\Gamma ) | } = 0 . \\end{align*}"}
{"id": "3345.png", "formula": "\\begin{align*} [ N x , N y , N z ] = & N \\Big ( [ N x , N y , z ] + [ x , N y , N z ] + [ N x , y , N z ] \\\\ - & N \\Big ( [ N x , y , z ] + [ x , N y , z ] + [ x , y , N z ] - N [ x , y , z ] \\Big ) \\Big ) . \\end{align*}"}
{"id": "4595.png", "formula": "\\begin{align*} \\hat { \\theta } _ n - \\theta = \\frac { \\sum _ { k = 1 } ^ n X _ { k - 1 } \\varepsilon _ k } { \\sum _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } . \\end{align*}"}
{"id": "5114.png", "formula": "\\begin{align*} c _ { ( k , l ) , ( k ' , l ' ) } = \\langle M _ { l / a } T _ { k / b } \\ , g , \\ , M _ { l ' / a } T _ { k ' / b } \\ , g \\rangle , k , k ' , l , l ' \\in \\Z . \\end{align*}"}
{"id": "7910.png", "formula": "\\begin{align*} k _ w ( t ) = \\frac { t ^ 2 } { 2 } \\big ( k _ w '' ( 0 ) + \\sqrt { t } R \\big ) . \\end{align*}"}
{"id": "4221.png", "formula": "\\begin{align*} \\widehat { v } ( t , x ) = H _ 1 \\ast \\widehat { \\mu } _ t ( x ) + \\frac 1 m \\sum _ { i = 1 } ^ m K _ 1 ( \\widehat { Y } ^ i ( t ) - x ) . \\end{align*}"}
{"id": "7155.png", "formula": "\\begin{align*} B _ { t } ^ { H } = \\int _ { 0 } ^ { t } K _ { H } ( t , s ) d W _ { s } . \\end{align*}"}
{"id": "5392.png", "formula": "\\begin{align*} R ( n , \\alpha , \\varepsilon , Z , T , f ) = \\inf \\{ \\sum _ { i } e ^ { - \\alpha n + f _ { n } ( x _ i ) } : Z \\subset \\cup _ { i } B _ { n } ( x _ i , \\varepsilon ) \\} , \\end{align*}"}
{"id": "2315.png", "formula": "\\begin{align*} A g _ 0 ( 0 , \\omega ) = e ^ { - \\frac { \\pi } { 2 } \\omega ^ 2 } , \\end{align*}"}
{"id": "7292.png", "formula": "\\begin{align*} \\sum p ( 5 n + 4 ) X ^ { 5 n + 4 } = \\frac { \\alpha ( X ^ { 2 5 } ) } { \\alpha ( X ^ 5 ) ^ 6 } ( \\alpha _ 0 ^ 2 \\alpha _ 2 ^ 2 - 3 \\alpha _ 0 \\alpha _ 1 ^ 2 \\alpha _ 2 + \\alpha _ 1 ^ 4 ) \\overset { \\eqref { s h o r t } } { = } 5 \\frac { \\alpha ( X ^ { 2 5 } ) } { \\alpha ( X ^ 5 ) ^ 6 } \\alpha _ 1 ^ 4 \\overset { \\eqref { a l p h a 1 } } { = } 5 X ^ 4 \\frac { \\alpha ( X ^ { 2 5 } ) ^ 5 } { \\alpha ( X ^ 5 ) ^ 6 } . \\end{align*}"}
{"id": "5734.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( h ) \\backslash V ( B \\cup T ' ) | & = d _ { G } ( h ) - | N _ { G } ( h ) \\cap V ( B ) | \\\\ & - | N _ { G } ( h ) \\cap ( V ( T ' ) \\backslash \\{ u _ { l + 1 } , \\cdots , u _ { r } \\} ) | \\\\ & \\geq t + 1 - ( t - \\lfloor \\frac { r - l } { 2 } \\rfloor ) \\\\ & \\geq 1 + \\lfloor \\frac { r - l } { 2 } \\rfloor , \\end{aligned} \\end{align*}"}
{"id": "4862.png", "formula": "\\begin{align*} D ( \\varphi ) : = \\frac 1 2 \\sum _ { i = 1 } ^ n \\int _ \\R ( K _ \\alpha * \\varphi _ i ) \\varphi _ i = \\frac 1 { 2 n ^ 2 } \\sum _ { i = 1 } ^ n \\frac 1 { \\ell _ i ^ 2 } \\iint _ { ( 0 , \\ell _ i ) ^ 2 } K _ \\alpha ( x - y ) \\ , d y d x . \\end{align*}"}
{"id": "9016.png", "formula": "\\begin{align*} & \\partial _ t \\rho _ i = \\nabla \\cdot \\left [ D _ i ( x ) \\left ( \\nabla \\rho _ i + \\frac { z _ i e } { k _ B T } \\rho _ i \\nabla \\phi \\right ) \\right ] , i = 1 , \\cdots , s , \\\\ & - \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ) = 4 \\pi \\left ( f ( x ) + \\sum _ { i = 1 } ^ { s } z _ i e \\rho _ i \\right ) . \\end{align*}"}
{"id": "2097.png", "formula": "\\begin{align*} H _ k ( x ) = \\prod _ { \\nu \\in M _ k } \\max \\{ 1 , | x | _ \\nu \\} h _ k ( x ) = \\log H _ k ( x ) = \\sum _ { \\nu \\in M _ k } \\log | x | _ \\nu . \\end{align*}"}
{"id": "4857.png", "formula": "\\begin{align*} \\| K _ \\alpha * \\rho \\| _ \\infty \\leq \\| \\rho \\| _ \\infty \\int _ \\R K _ \\alpha = \\int _ \\R K . \\end{align*}"}
{"id": "2464.png", "formula": "\\begin{align*} S ^ T J S = J \\end{align*}"}
{"id": "2606.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { V _ { g _ 0 } f _ n - F } _ { L ^ { p , q } } = 0 . \\end{align*}"}
{"id": "7022.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\le | \\langle \\mathbf { v } _ i , \\mathbf { y } \\rangle _ { \\C ^ { L } } | = \\frac { 1 } { \\delta } \\Big | \\sum _ { j = 1 } ^ L B _ j ( \\alpha _ i ) \\overline y _ j \\Big | \\mbox { f o r a l l $ 1 \\leq i \\leq n $ . } \\end{align*}"}
{"id": "7205.png", "formula": "\\begin{align*} V _ { s , t } ( x , v ) & = v + \\tilde W _ { s , t } ( x , v ) , \\\\ X _ { s , t } ( x , v ) & = x - ( t - s ) v + \\tilde { Y } _ { s , t } ( x , v ) . \\end{align*}"}
{"id": "960.png", "formula": "\\begin{align*} \\gamma _ { n , s } : = \\frac { \\sin ( \\pi s ) \\Gamma ( n / 2 ) } { \\pi ^ { \\frac n 2 + 1 } } . \\end{align*}"}
{"id": "3525.png", "formula": "\\begin{align*} \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T \\\\ & + \\begin{cases} O \\left ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } \\log T \\right ) & ( \\frac { 1 } { 2 } < \\sigma _ 1 + \\sigma _ 3 < 1 ) \\\\ O ( \\log T ) ^ 2 & ( \\sigma _ 1 + \\sigma _ 3 = 1 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "8576.png", "formula": "\\begin{align*} & \\mathcal { K } ^ { \\# } _ S ( x , k ) : = \\chi _ 0 ( x ) \\mathcal { K } ^ { \\# } _ 0 ( x , k ) + \\chi _ + ( x ) \\mathcal { K } ^ { \\# } _ + ( x , k ) + \\chi _ - ( x ) \\mathcal { K } ^ { \\# } _ - ( x , k ) \\end{align*}"}
{"id": "4477.png", "formula": "\\begin{align*} \\langle z | ( a ^ { \\dagger } a ) _ { k , \\lambda } | z \\rangle = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) | z | ^ { 2 l } = \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) . \\end{align*}"}
{"id": "4886.png", "formula": "\\begin{align*} L = \\frac { f ' } f , F ( z ) = z - \\frac { f ( z ) } { f ' ( z ) } , F ' = \\frac { f f '' } { ( f ' ) ^ 2 } . \\end{align*}"}
{"id": "4974.png", "formula": "\\begin{align*} \\begin{pmatrix} - a _ { 0 } ( 0 ) & - a _ { 0 } ( 0 ) ^ { 2 } & - a _ { 0 } ( 0 ) ^ { 3 } & - a _ { 0 } ( 0 ) ^ { 4 } \\\\ a _ { 0 } ^ { ( 1 ) } ( b _ { 0 } ( 0 ) ) - a _ { 0 } ^ { ( 1 ) } ( 0 ) & - 2 a _ { 0 } ( 0 ) a _ { 0 } ^ { ( 1 ) } ( 0 ) & - 3 a _ { 0 } ( 0 ) ^ { 2 } a _ { 0 } ^ { ( 1 ) } ( 0 ) & - 4 a _ { 0 } ( 0 ) ^ { 3 } a _ { 0 } ^ { 1 } ( 0 ) \\\\ A & B & C & D \\\\ E & F & G & H \\end{pmatrix} \\end{align*}"}
{"id": "3544.png", "formula": "\\begin{align*} C ( \\Sigma ) : = \\{ r x : x \\in \\Sigma , \\ , r > 0 \\} \\end{align*}"}
{"id": "4231.png", "formula": "\\begin{align*} - \\Delta u = { } & 1 , \\textrm { i n } \\Omega , \\\\ u = { } & 0 , \\textrm { o n } \\partial \\Omega . \\end{align*}"}
{"id": "9520.png", "formula": "\\begin{align*} l ( x , y , \\omega ) & = \\inf \\{ f ( x , u , \\omega ) - u \\cdot y \\} \\\\ & = \\begin{cases} + \\infty & , \\\\ f _ 0 ( x , \\omega ) + y \\cdot H ( x , \\omega ) & , \\\\ - \\infty & \\end{cases} \\end{align*}"}
{"id": "6766.png", "formula": "\\begin{align*} & \\mathbf a _ 2 = [ 0 , \\ ; \\ldots , \\ ; 0 , \\ ; p ( S _ { m - 1 } ) C '' _ 1 , \\ ; p ( S _ { m } ) C '' _ 2 , \\ ; p ( S _ { m + 1 } ) C '' _ 3 , \\ ; p ( S _ { m + 2 } ) C '' _ 4 , \\ ; 0 , \\ ; \\ldots , \\ ; 0 ] ^ T , \\\\ & \\mathbf a _ 1 = [ 0 , \\ ; \\ldots , \\ ; 0 , \\ ; w ( S _ { m - 1 } ) C ' _ 1 , \\ ; w ( S _ { m } ) C ' _ 2 , \\ ; w ( S _ { m + 1 } ) C ' _ 3 , \\ ; w ( S _ { m + 2 } ) C ' _ 4 , \\ ; 0 , \\ ; \\ldots , \\ ; 0 ] ^ T . \\end{align*}"}
{"id": "5618.png", "formula": "\\begin{align*} \\nabla _ t ^ \\mathcal { F } : = \\mathrm { p r } _ { t \\star } \\circ \\nabla ^ \\mathcal { F } | _ { \\Xi _ { t \\star } \\otimes _ { \\mathbb { K } [ [ \\nu ] ] } \\Xi _ { t \\star } } \\colon \\Xi _ { t \\star } \\otimes _ { \\mathbb { K } [ [ \\nu ] ] } \\Xi _ { t \\star } \\rightarrow \\Xi _ { t \\star } \\end{align*}"}
{"id": "828.png", "formula": "\\begin{align*} H _ R = \\left ( \\frac { 1 } { ( \\beta - 1 ) R } \\right ) ^ { 1 / ( \\beta - 1 ) } . \\end{align*}"}
{"id": "5519.png", "formula": "\\begin{align*} S _ r \\Sigma : = \\Sigma \\circ \\pi _ r , \\Sigma \\in L _ 2 ^ 0 ( H ) . \\end{align*}"}
{"id": "4588.png", "formula": "\\begin{align*} 1 - \\Phi \\left ( \\overline { \\lambda } \\right ) = \\Big ( 1 - \\Phi ( x ) \\Big ) \\exp \\bigg \\{ \\theta _ { 1 } c _ { 6 } \\ , ( 1 + x ) \\Big ( x ^ 2 ( \\epsilon _ n + \\delta _ n ) + x \\delta _ n \\sqrt { | \\ln \\delta _ n | } \\Big ) \\bigg \\} , \\end{align*}"}
{"id": "3723.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int | B | \\ , d x = \\frac { d } { d t } \\int B \\ , d x = \\int B _ x J \\ , d x - \\mu \\int \\Lambda ^ \\alpha B \\ , d x \\end{align*}"}
{"id": "939.png", "formula": "\\begin{align*} 0 \\leq ( - 1 ) ^ { i + j } E ( i , j ) = ( - 1 ) ^ { i + j } C ^ { - 1 } ( i , j ) \\leq ( - 1 ) ^ { i + j } D ^ { - 1 } ( i , j ) , i , j \\in { \\bf m } . \\end{align*}"}
{"id": "6525.png", "formula": "\\begin{align*} I _ n ^ { ( 2 m ) } \\sim \\frac { 1 } { t ^ { ( 2 m ) } _ n } \\left ( 1 + \\sum ^ { \\infty } _ { j = 1 } \\binom { j + m } { j } ^ { - 1 } \\right ) = O ( ( \\log n ) ^ { - m } ) . \\end{align*}"}
{"id": "622.png", "formula": "\\begin{align*} \\mu f ( x _ 1 , \\ldots , x _ k , x _ { k + 1 } ) \\ = \\ x _ { k + 1 } \\end{align*}"}
{"id": "6032.png", "formula": "\\begin{align*} ( n + 1 ) \\widetilde { L } _ { n + 1 } + ( \\alpha - 1 - x - 2 n ) \\widetilde { L } _ n + ( n - \\alpha ) \\widetilde { L } _ { n - 1 } = 0 , \\end{align*}"}
{"id": "9050.png", "formula": "\\begin{align*} L ( \\rho , \\phi , m , v , \\xi ) : = & \\frac { 1 } { 2 \\tau } \\sum _ { i = 1 } ^ s \\int _ { \\Omega } F ( \\rho _ i , m _ i ) D ^ { - 1 } _ i d x + E ( \\rho , \\phi ) + \\int _ { \\partial \\Omega } \\xi ( \\alpha \\phi + \\beta \\partial _ n \\phi - \\phi ^ b ) d s \\\\ & + \\sum _ { i = 1 } ^ s \\int _ { \\Omega } v _ i ( \\rho _ i - \\rho _ i ^ n + \\nabla \\cdot m _ i ) d x + \\int _ { \\Omega } v _ { s + 1 } ( f + \\sum _ { i = 1 } ^ s z _ i \\rho _ i + \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ) ) d x . \\end{align*}"}
{"id": "4039.png", "formula": "\\begin{align*} \\sin ( c _ k + i d _ k ) = O ( k ^ { - 1 } ) , . \\end{align*}"}
{"id": "8778.png", "formula": "\\begin{align*} P _ { i J ( i ) } : = \\bigl \\{ u _ i \\in P _ i \\bigm | u _ { i j } = a _ { i j } j \\leq \\min ( J _ i ) , \\ u _ { i j } = u _ { i n } j \\geq \\max ( J _ i ) \\bigr \\} . \\end{align*}"}
{"id": "8026.png", "formula": "\\begin{align*} \\Xi _ \\pm ^ { n } = \\left \\{ ( s _ 1 , \\ldots , s _ n ; \\xi _ 1 , \\ldots , \\xi _ n ) \\in T ^ { * } \\Sigma ^ n \\ , | \\ , \\pm \\xi _ i \\ge 0 , 0 \\leq i \\leq 1 \\right \\} , \\end{align*}"}
{"id": "8317.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 ^ + } \\frac { \\S ( u + \\alpha h ) ( t ) - \\S ( u ) ( t ) } { \\alpha } = \\S ' ( u ; h ) ( t ) \\forall t \\in [ 0 , T ] . \\end{align*}"}
{"id": "5266.png", "formula": "\\begin{align*} \\Delta ( \\delta _ { \\varphi } ) = \\Delta ( 1 ) ( \\delta _ { \\varphi } \\otimes \\delta _ { \\varphi } ) \\Delta ( 1 ) , S ( \\delta _ { \\varphi } ) = \\delta _ { \\varphi } ^ { - 1 } , \\varepsilon ( \\delta _ { \\varphi } a ) = \\varepsilon ( a \\delta _ { \\varphi } ) = \\varepsilon ( a ) , \\forall a \\in A , \\end{align*}"}
{"id": "3682.png", "formula": "\\begin{align*} u _ t + ( u \\cdot \\nabla ) u - ( B \\cdot \\nabla ) B + \\nabla P = & \\ \\nu \\Delta u , \\\\ B _ t + ( u \\cdot \\nabla ) B - ( B \\cdot \\nabla ) u + \\nabla \\times ( ( \\nabla \\times B ) \\times B ) = & \\ \\mu \\Delta B , \\\\ \\nabla \\cdot u = 0 , \\ \\ \\nabla \\cdot B = & \\ 0 , \\end{align*}"}
{"id": "3716.png", "formula": "\\begin{align*} \\tilde B ^ { k + 1 } _ t + B ^ k \\tilde J ^ { k + 1 } _ x + \\tilde B ^ k J ^ k _ x - J ^ k \\tilde B ^ { k + 1 } _ x - \\tilde J ^ k B _ x ^ k + \\mu \\Lambda ^ \\alpha \\tilde B ^ { k + 1 } = 0 \\end{align*}"}
{"id": "5049.png", "formula": "\\begin{align*} v ( z ) = - \\sum _ { n = - 1 } ^ \\infty v _ n z ^ { n + 1 } . \\end{align*}"}
{"id": "3581.png", "formula": "\\begin{align*} & B : = \\{ t \\in \\chi A : | z _ 1 ^ * ( f _ k ( t ) ) - z _ 1 ^ * ( f _ k ( t _ 1 ) ) | < 1 , \\ , k = 1 , \\ldots , n \\} , \\\\ & D : = \\{ z ^ * \\in \\Gamma _ { z _ 0 ^ * } : | z ^ * ( f _ k ( t _ 1 ) ) - z _ 1 ^ * ( f _ k ( t _ 1 ) ) | < 1 , \\ , k = 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "2302.png", "formula": "\\begin{align*} \\widetilde { f } = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "9390.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { n } ( - 1 ) ^ { k - 2 } ( k - 2 ) ! S _ { 2 } ( n , k ) = n - 1 . \\end{align*}"}
{"id": "9464.png", "formula": "\\begin{align*} f _ 2 = \\left ( \\tfrac { c _ 0 } { 2 c _ 1 } \\right ) \\cdot \\frac { { \\bar f _ 2 } + f _ 1 ^ 2 } { X ^ { p ^ s - 2 p ^ { s - 1 } } } . \\end{align*}"}
{"id": "569.png", "formula": "\\begin{align*} \\big \\{ ( x , y ) \\in \\R ^ 2 \\ y = e ^ x \\big \\} \\end{align*}"}
{"id": "4848.png", "formula": "\\begin{align*} Q ( 0 ) = Q ' ( 0 ) = 0 , Q '' ( x ) = \\sum _ { k = 1 } ^ \\infty \\big ( \\varphi _ { e ^ { - k } } ( x + k ) + \\varphi _ { e ^ { - k } } ( x - k ) \\big ) , \\end{align*}"}
{"id": "1128.png", "formula": "\\begin{align*} \\chi _ \\eta ( k ) = \\left ( \\frac { k - \\eta } { k + \\eta } \\right ) ^ { \\frac { 1 } { 4 } } = \\left ( \\frac { k - \\sqrt { - 2 \\xi } } { k + \\sqrt { - 2 \\xi } } \\right ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
{"id": "366.png", "formula": "\\begin{align*} & \\theta _ { i j } ( \\rho ) [ v _ { i j } - ( \\lambda _ i - \\lambda _ j ) ( 1 + \\dot W ^ { \\delta } ) ] = 0 , \\ ; \\forall ( i , j ) \\in E , \\\\ & \\ < \\dot \\lambda , \\rho \\ > + \\frac 1 4 \\sum _ { i j } v _ { i j } ^ 2 \\theta _ { i j } ( \\rho ) + \\frac 1 2 \\sum _ { i j } v _ { i j } ( \\lambda _ j - \\lambda _ i ) \\theta _ { i j } ( \\rho ) \\\\ & + \\frac 1 2 \\sum _ { i j } v _ { i j } ( \\lambda _ j - \\lambda _ i ) \\theta _ { i j } ( \\rho ) d W ^ { \\delta } ( t ) = 0 , \\ ; \\mathcal L ^ 1 \\ ; \\end{align*}"}
{"id": "2862.png", "formula": "\\begin{align*} V ( h ) & = p \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * ( Q ^ { p - 1 } \\Re h ) \\right ) Q ^ { p - 1 } \\\\ & + \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) \\left ( \\left ( \\frac { p } { 2 } - 1 \\right ) Q ^ { p - 2 } \\bar { h } + \\frac { p } { 2 } Q ^ { p - 2 } h \\right ) \\end{align*}"}
{"id": "157.png", "formula": "\\begin{align*} \\mathcal { E } \\left ( G _ { 0 ^ + } ( u ) , v \\right ) = \\langle u ; v \\rangle _ H . \\end{align*}"}
{"id": "675.png", "formula": "\\begin{align*} \\begin{cases} \\ f ( x ) = s _ x \\\\ [ 3 p t ] \\ g ( x ) = s _ x + 2 ^ { - x } \\end{cases} . \\end{align*}"}
{"id": "7269.png", "formula": "\\begin{align*} \\exp ( \\alpha + \\beta ) & = \\sum \\frac { ( \\alpha + \\beta ) ^ n } { n ! } = \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n \\binom { n } { k } \\frac { \\alpha ^ k \\beta ^ { n - k } } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n \\frac { \\alpha ^ k \\beta ^ { n - k } } { k ! ( n - k ) ! } = \\sum \\frac { \\alpha ^ n } { n ! } \\cdot \\sum \\frac { \\beta ^ n } { n ! } = \\exp ( \\alpha ) \\exp ( \\beta ) . \\end{align*}"}
{"id": "2412.png", "formula": "\\begin{align*} \\norm { ( I - B ^ { - 1 } S ) } _ { o p } = \\sup _ { \\norm { f } _ \\mathcal { H } = 1 } | \\langle ( I - B ^ { - 1 } S ) f , f \\rangle | \\leq \\frac { B - A } { B } = 1 - \\frac { A } { B } < 1 . \\end{align*}"}
{"id": "1336.png", "formula": "\\begin{align*} A ( \\delta _ { 1 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , A ( \\delta _ { 2 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , p _ { i + 1 } = \\frac { 3 } { 2 } p _ { i } , \\ , \\ , A ( \\eta ) \\approx \\frac { 1 } { 2 } p _ { i } . \\end{align*}"}
{"id": "449.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 2 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle & + G _ { 0 } \\left ( \\| v \\| _ { m } ^ { 2 } + \\| \\nabla v \\| _ { m } ^ { 2 } \\right ) \\leq \\frac { 4 C \\epsilon } { 2 } \\| \\nabla v \\| _ { m } ^ { 2 } + C \\left \\lbrace \\| f _ { 2 } \\| _ { m - 1 } ^ { 2 } \\right . \\\\ & + C \\left . ( \\mu _ { 0 } ( t ) + \\mu _ { 1 } ( t ) ) \\left ( \\| u \\| _ { m } ^ { 2 } + \\| v \\| _ { m } ^ { 2 } + \\| w \\| _ { m } ^ { 2 } \\right ) \\right \\rbrace , \\end{align*}"}
{"id": "7388.png", "formula": "\\begin{align*} V ( \\R ^ d | \\R ^ d ) = \\left \\lbrace v \\in L ^ 2 ( \\R ^ d ) : ( v ( x ) - v ( y ) ) K _ s ^ { 1 / 2 } ( x , y ) \\in L ^ 2 ( \\R ^ d \\times \\R ^ d ) \\right \\rbrace \\end{align*}"}
{"id": "3880.png", "formula": "\\begin{align*} \\omega _ 1 \\colon & t _ i \\leftrightarrow s _ i , \\\\ \\omega _ 2 \\colon & \\begin{cases} s _ i \\mapsto s _ i , t _ i \\mapsto t _ i , & ( 1 \\leq i \\leq n - 1 ) \\\\ t _ n \\mapsto - t _ n , s _ n \\mapsto - s _ n . & \\end{cases} \\end{align*}"}
{"id": "8706.png", "formula": "\\begin{align*} F ( x ) = \\begin{pmatrix} f _ { 1 1 } ( x ) & \\cdots & f _ { 1 r _ 2 } ( x ) \\\\ \\vdots & \\ddots & \\vdots \\\\ f _ { r _ 1 1 } ( x ) & \\cdots & f _ { r _ 1 r _ 2 } ( x ) \\end{pmatrix} G ( x ) = \\begin{pmatrix} g _ { 1 1 } ( x ) & \\cdots & g _ { 1 q _ 2 } ( x ) \\\\ \\vdots & \\ddots & \\vdots \\\\ g _ { q _ 1 1 } ( x ) & \\cdots & g _ { q _ 1 q _ 2 } ( x ) \\end{pmatrix} , \\end{align*}"}
{"id": "1373.png", "formula": "\\begin{align*} \\underline { b } : = \\inf _ { x \\in [ 0 , 1 ] } b ( x ) \\mbox { a n d } \\bar { b } : = \\sup _ { x \\in [ 0 , 1 ] } b ( x ) , \\end{align*}"}
{"id": "7018.png", "formula": "\\begin{align*} \\left | \\frac { z - \\beta _ { k } } { \\beta _ 1 - \\beta _ { k } } \\right | \\le \\frac { 1 } { M } + 2 = \\frac { 2 M + 1 } { M } \\mbox { f o r a l l $ | z | = 1 $ } . \\end{align*}"}
{"id": "8638.png", "formula": "\\begin{align*} \\big | I _ 0 ( t , K ) \\big | = \\Big | \\ , \\mathrm { p . v . } \\int _ \\R \\big [ H ( q ; t ) - H ( K ; t ) \\big ] \\frac { \\psi ( K - q ) } { K - q } \\chi _ 1 \\big ( ( K - q ) t ^ 3 \\big ) \\ , d q \\Big | \\\\ \\lesssim t \\ , \\int _ \\R \\frac { 1 } { | K - q | ^ { 1 / 2 } } \\chi _ 1 \\big ( ( K - q ) t ^ 3 \\big ) \\ , d q \\lesssim | t | ^ { - 1 / 2 } . \\end{align*}"}
{"id": "3810.png", "formula": "\\begin{align*} F ( \\tau ( g , x ) ) & = F ( g x ) = ( f _ { g _ 1 } ( g x ) , \\ldots , f _ { g _ n } ( g x ) ) = ( f ( g _ 1 ( g x ) ) , \\ldots , f ( g _ n ( g x ) ) ) \\\\ & = ( f ( ( g \\ldotp g _ 1 ) x ) , \\ldots , f ( ( g \\ldotp g _ n ) x ) ) = ( f _ { g \\ldotp g _ 1 } ( x ) , \\ldots , f _ { g \\ldotp g _ n } ( x ) ) \\\\ & = \\tau _ 2 ( g , ( f _ { g _ 1 } ( x ) , \\ldots , f _ { g _ n } ( x ) ) ) = \\tau _ 2 ( g , F ( x ) ) . \\end{align*}"}
{"id": "4535.png", "formula": "\\begin{align*} F _ { u } ( z , \\chi ) \\ll \\prod _ { p \\mid u } ( 1 + a ( p ^ l ) p ^ { - 1 / 2 } ) \\ll d ( u ) , \\ \\Re z \\geq 1 / 2 , \\ \\ ( u = \\prod _ { p } p ^ l ) . \\end{align*}"}
{"id": "6219.png", "formula": "\\begin{align*} \\begin{aligned} & \\ \\ O ( p { \\rm l o g } \\ , m + p { \\rm l o g } \\ , n + p ^ 3 + n p l + n d ^ 2 + d ^ 3 ) \\\\ & = O \\left ( \\frac { k ^ { 8 } } { \\epsilon ^ { 4 } \\delta } { \\rm l o g } \\ , m + \\frac { k ^ { 1 6 } } { \\epsilon ^ { 8 } \\delta ^ 2 } { \\rm l o g } \\ , n + \\frac { k ^ { 2 4 } } { \\epsilon ^ { 1 2 } \\delta ^ 3 } + \\frac { k ^ { 8 } } { \\epsilon ^ { 4 } \\delta } n l + n d ^ 2 + d ^ 3 \\right ) . \\end{aligned} \\end{align*}"}
{"id": "1789.png", "formula": "\\begin{align*} f _ { \\varphi } ( x _ 0 , \\cdots , x _ k ) & : = \\int _ { G ^ { \\times ( k + 1 ) } } c _ X ( g _ 0 ^ { - 1 } x _ 0 ) \\cdots c _ X ( g _ k ^ { - 1 } x _ k ) \\varphi ( g _ 0 , g _ 1 , \\cdots , g _ k ) { \\rm d } g _ 0 \\cdots { \\rm d } g _ k , \\\\ \\Phi ( \\varphi ) & : = ( d _ { x _ 1 } d _ { x _ 2 } \\cdots d _ { x _ k } f _ { \\varphi } ) ( x , \\cdots , x ) . \\end{align*}"}
{"id": "7053.png", "formula": "\\begin{align*} \\rho ( x , y ) & = \\min \\left \\{ | x ' - y ' | , \\ x ' = x 1 , \\ y ' = y 1 \\right \\} \\\\ & = \\min \\big ( | x - y | , 1 - | x - y | \\big ) . \\end{align*}"}
{"id": "1840.png", "formula": "\\begin{align*} L ( x , y , t ) = \\frac { x y \\sqrt { y ^ 2 - x ^ 2 } } { \\sqrt { y ^ 2 - x ^ 2 } \\cosh { ( t \\sqrt { y ^ 2 - x } ) } - y \\sinh { t \\sqrt { y ^ 2 - x ^ 2 } } } . \\end{align*}"}
{"id": "5941.png", "formula": "\\begin{align*} \\tilde { L } ( q , v ) = \\frac { 1 } { 2 } \\mathcal { G } ( v , v ) = \\frac { 1 } { 2 } g _ { i j } ( q ) v ^ i v ^ j . \\end{align*}"}
{"id": "456.png", "formula": "\\begin{align*} \\mu ( \\phi ( x , b ) ) = \\frac { \\mu ( Y ( b ) ) } { | H | } . \\end{align*}"}
{"id": "5484.png", "formula": "\\begin{align*} \\frac { 1 } { t } d _ K ( S _ t x + t v ) & = \\frac { 1 } { t } d _ K \\bigg ( x + t ( A x + v ) - t \\bigg ( A x - \\frac { S _ t x - x } { t } \\bigg ) \\bigg ) \\\\ & \\leq \\frac { 1 } { t } d _ K \\big ( x + t ( A x + v ) \\big ) + \\bigg \\| A x - \\frac { S _ t x - x } { t } \\bigg \\| , \\end{align*}"}
{"id": "6567.png", "formula": "\\begin{align*} \\tau _ E ( m ) : = \\sum _ { m _ 1 \\cdots m _ j = m } m _ 1 ^ { - \\xi _ 1 } \\cdots m _ j ^ { - \\xi _ j } , \\end{align*}"}
{"id": "7469.png", "formula": "\\begin{align*} \\xi ( t ) = \\left \\{ \\begin{aligned} & b , & t \\leq t _ s , \\\\ & b / ( t - t _ s + 1 ) , & t > t _ s . \\end{aligned} \\right . \\end{align*}"}
{"id": "4395.png", "formula": "\\begin{align*} \\mathbf { I } _ \\zeta ( z ) = \\frac { \\left ( \\frac { 1 } { 2 } z \\right ) ^ \\zeta } { \\Gamma ( \\zeta + \\frac { 1 } { 2 } ) \\Gamma ( \\frac { 1 } { 2 } ) } \\int _ 0 ^ \\pi \\cosh \\left ( z \\cos \\theta \\right ) \\sin ^ { 2 \\zeta } ( \\theta ) d \\theta , \\end{align*}"}
{"id": "2421.png", "formula": "\\begin{align*} \\langle f , S ^ { - 1 } f \\rangle = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma \\langle e _ \\gamma , S ^ { - 1 } f \\rangle = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma \\overline { a _ \\gamma } = \\langle c , a \\rangle _ { \\ell ^ 2 } \\end{align*}"}
{"id": "1597.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb { R } ^ { d - 1 } } \\Psi _ { a , \\zeta , \\mu , \\sigma } \\tilde { \\Psi } _ { a , \\zeta , \\mu , \\sigma } \\ , d x - 1 \\right | & < \\zeta , \\\\ \\left | \\int _ { \\mathbb { R } ^ { d - 1 } } \\beta ( | a | ^ \\frac { 1 } { p } \\Psi _ { a , \\zeta , \\mu , \\sigma } ) \\tilde { \\Psi } _ { a , \\zeta , \\mu , \\sigma } \\ , d x - \\sigma | a | ^ \\frac { 1 } { p } \\right | & < \\zeta \\\\ \\int _ { \\mathbb { R } ^ { d - 1 } } \\tilde { \\Psi } _ { a , \\zeta , \\mu , \\sigma } \\ , d x & = 0 , \\end{align*}"}
{"id": "8915.png", "formula": "\\begin{align*} \\overline { V _ 1 ^ c } \\cap \\overline { V _ 2 ^ c } = ( [ 0 , 0 . 2 5 ] + \\Z ) \\cap ( [ 0 . 5 , 0 . 7 5 ] + \\Z ) = \\emptyset \\end{align*}"}
{"id": "3003.png", "formula": "\\begin{align*} \\| \\mathcal { T } _ { u , \\lambda } ( x ) \\| _ { X } & \\leq \\| x \\| _ { X } + | \\lambda | \\ , | \\omega ( x , u ) | \\| u \\| _ { X } \\\\ & \\leq \\big ( 1 + | \\lambda | \\ , \\| \\omega \\| \\ , \\| u \\| _ { X } ^ 2 \\big ) \\| x \\| _ { X } . \\end{align*}"}
{"id": "7953.png", "formula": "\\begin{align*} \\dot { \\varrho } _ { m a j _ { \\alpha } ( \\pi ) } ( \\pi ^ { \\prime } ) = \\begin{cases} m a j _ { \\alpha } ( \\pi ) & \\pi _ { \\mu } ^ { \\circ } \\neq 1 1 \\leq \\alpha \\leq \\mu - 2 \\\\ 0 & \\alpha = \\mu - 1 \\pi _ { \\mu } ^ { \\circ } = 1 . \\end{cases} \\end{align*}"}
{"id": "2342.png", "formula": "\\begin{align*} f ( t ) = \\int _ \\R \\widehat { f } ( \\omega ) e ^ { 2 \\pi i t \\omega } \\ , d \\omega , \\end{align*}"}
{"id": "2132.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { j = n - N + 1 } ^ { n - 1 } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } = \\\\ & k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\sum _ { s = 1 } ^ { N - 1 } ( 1 - c ) ^ { k ( s + 1 ) } \\frac 1 { k ( s + 1 ) - l } . \\end{aligned} \\end{align*}"}
{"id": "2736.png", "formula": "\\begin{align*} ( E _ r ^ { s , t } ) _ \\lambda = 0 . \\end{align*}"}
{"id": "3370.png", "formula": "\\begin{align*} [ \\mathfrak { X } , T D ( \\mathfrak { X } ) ( u ) - [ \\mathfrak { X } , T ( u ) ] ] = 0 . \\end{align*}"}
{"id": "7776.png", "formula": "\\begin{align*} \\Box \\phi _ { k + 1 } = \\left ( | \\phi _ { k + 1 , t } | ^ 2 - | \\phi _ { k + 1 , x } | ^ 2 \\right ) \\phi _ { k + 1 } + \\mathbf { 1 } _ { \\omega } f _ { k } ^ { \\phi ^ { \\perp } _ { k + 1 } } , \\ ; \\phi _ { k + 1 } [ 0 ] = ( a , b ) . \\end{align*}"}
{"id": "7839.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { 3 j + 1 } \\beta _ { i } e _ { i } , \\quad ~ \\beta _ { 3 j } , \\beta _ { 3 j + 1 } \\neq 0 . \\end{align*}"}
{"id": "4168.png", "formula": "\\begin{align*} \\omega _ { p } ^ { p } ( A ) & = \\underset { \\theta \\in \\mathbb { R } } { \\sup } \\norm { R e ( e ^ { i \\theta } A ) } _ { p } ^ { p } \\\\ & \\leq \\frac { 1 } { 2 ^ { p - 2 } } \\underset { i = j } { \\sum } \\omega _ { p } ^ { p } ( a _ { i j } ) \\end{align*}"}
{"id": "8605.png", "formula": "\\begin{align*} \\int _ { \\R } \\prod _ \\ast ( - m _ + ( x , 0 ) ) e ^ { i x k _ j } \\ , d x & = \\int _ { \\R } m _ + ^ 4 ( x , 0 ) e ^ { i x ( - k _ 1 + k _ 2 - k _ 3 + k _ 4 ) } \\ , d x \\\\ & = \\sqrt { 2 \\pi } \\ , \\delta ( k _ 1 - k _ 2 + k _ 3 - k _ 4 ) \\\\ & + \\int _ { \\R } \\big ( m _ + ^ 4 ( x , 0 ) - 1 \\big ) e ^ { i x ( - k _ 1 + k _ 2 - k _ 3 + k _ 4 ) } \\ , d x . \\end{align*}"}
{"id": "4735.png", "formula": "\\begin{align*} \\dot { \\mathfrak { p } } _ 1 = ( \\partial _ t \\epsilon , R _ 1 ) - \\dot { \\mathfrak { q } _ 1 } ( \\epsilon , \\partial _ y R _ 1 ) + \\frac { 1 } { 2 } \\int \\partial _ t \\psi _ 1 \\epsilon ^ 2 + \\int \\psi _ 1 \\partial _ t \\epsilon \\epsilon . \\end{align*}"}
{"id": "5039.png", "formula": "\\begin{align*} \\mathcal K = \\left ( \\begin{array} { c c } H & \\sigma \\\\ \\tau & K \\end{array} \\right ) , \\end{align*}"}
{"id": "1326.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , N ) \\cup { ( \\delta _ { 1 } , 1 ) } } , \\hat { \\alpha } \\cup { ( \\gamma , N - p _ { i } ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 2 . \\end{align*}"}
{"id": "7307.png", "formula": "\\begin{align*} \\alpha = \\sum _ { k _ 1 , \\ldots , k _ n \\ge 0 } c _ { k _ 1 , \\ldots , k _ n } \\beta _ 1 ^ { k _ 1 } \\ldots \\beta _ n ^ { k _ n } \\end{align*}"}
{"id": "3242.png", "formula": "\\begin{align*} ( s - t ) ^ 2 - \\frac { s - t } { h ( s ) } + | \\lambda | ^ 2 = 0 . \\end{align*}"}
{"id": "6581.png", "formula": "\\begin{align*} \\mathcal { L } ( h , k ) : = \\frac { 1 } { 2 } \\sum _ { \\substack { 1 \\leq q < \\infty \\\\ ( q , h k ) = 1 } } W \\left ( \\frac { q } { Q } \\right ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , q ) = 1 } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { c > C , d \\geq 1 \\\\ c d = q \\\\ d | m h \\pm n k } } \\phi ( d ) \\mu ( c ) , \\end{align*}"}
{"id": "5476.png", "formula": "\\begin{align*} | u ( t , x ; 0 , u ^ * ) - u ^ * ( x ) | & \\le | u ( t , x ; 0 , u ^ * ) - u ( t , x ; 0 , u _ { n _ k } ) | + | u ( t , x ; 0 , u _ { n _ k } ) - u ^ * ( x ) | ( \\ , \\ , k \\gg 1 ) \\\\ & = | u ( t , x ; 0 , u ^ * ) - u ( t , x ; 0 , u _ { n _ k } ) | + | u ( \\tau _ { n _ k } , x ; 0 , u _ { n _ k } ) - u ^ * ( x ) | \\\\ & \\le | u ( t , x ; 0 , u ^ * ) - u ( t , x ; 0 , u _ { n _ k } ) | + | u ( \\tau _ { n _ k } , x ; 0 , u _ { n _ k } ) - u ( \\tau _ { n _ k } , x ; 0 , u ^ * ) | \\\\ & + | u ( \\tau _ { n _ k } , x ; 0 , u ^ * ) - u ^ * ( x ) | \\\\ & \\le 3 \\epsilon \\forall \\ , x \\in \\bar \\Omega . \\end{align*}"}
{"id": "2895.png", "formula": "\\begin{align*} Z [ u ] = \\frac { \\| u \\| _ 2 ^ { ( N + \\gamma ) - ( N - 2 ) p } \\| \\nabla u \\| _ 2 ^ { N p - ( N + \\gamma ) } } { \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | u | ^ p \\right ) | u | ^ p d x } . \\end{align*}"}
{"id": "9422.png", "formula": "\\begin{align*} G _ { 0 } C _ { 0 } = \\mathrm { I d } _ { S _ n } , \\sum _ { i = 0 } ^ { g } G _ { 2 i } C _ { 2 ( g - i ) } = 0 \\end{align*}"}
{"id": "3996.png", "formula": "\\begin{align*} \\widetilde { U } ( t ) = \\widetilde { U } ^ p ( t ) + \\widetilde { U } ^ h ( t ) = U ( t ) - U ( t - 1 ) . \\end{align*}"}
{"id": "7678.png", "formula": "\\begin{align*} \\phi _ { \\varepsilon } = C _ { \\varepsilon } \\ , \\tilde u _ { \\varepsilon } + \\hat { \\phi } _ { \\varepsilon } \\end{align*}"}
{"id": "7673.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { \\varepsilon _ { n _ k } } } | 2 w - e ^ { - \\beta _ { \\varepsilon _ { n _ k } } \\Psi _ { \\varepsilon _ { n _ k } } ( \\mathbf { q } ) } V _ { \\varepsilon _ { n _ k } } | | V _ { \\varepsilon _ { n _ k } } | = o \\left ( e ^ { D \\sigma _ 1 \\beta _ { \\varepsilon _ { n _ k } } \\Psi _ { \\varepsilon _ { n _ k } } ( \\mathbf { q } ) } \\right ) \\end{align*}"}
{"id": "4398.png", "formula": "\\begin{align*} \\int e ^ { - \\frac { z ^ 2 } { 4 } } ( \\sqrt { r } y - z \\sqrt { 1 - r } ) ^ { \\omega + 3 } d z = - \\frac { 1 } { ( \\omega + 4 ) \\sqrt { 1 - r } } \\int \\frac { z } { 2 } e ^ { - \\frac { z ^ 2 } { 4 } } ( \\sqrt { r } y - z \\sqrt { 1 - r } ) ^ { \\omega + 4 } d z \\end{align*}"}
{"id": "6821.png", "formula": "\\begin{align*} w _ F = \\frac { 1 } { 2 } \\Big { ( } \\nabla ^ F ( \\vec H _ F ) + \\ , \\langle ( A _ F ) , \\vec H _ { F } \\rangle _ { g _ { \\textnormal { e u c } } } ^ { \\sharp _ { g _ F } } + \\ , \\langle ( A _ F ^ 0 ) , \\vec H _ { F } \\rangle _ { g _ { \\textnormal { e u c } } } ^ { \\sharp _ { g _ F } } \\Big { ) } , \\end{align*}"}
{"id": "6015.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } [ ( D _ { \\rm l e f t } ) ^ { \\alpha } u ( x ) ] v ( x ) \\ , d x = \\int _ \\mathbb { R } u ( x ) [ ( D _ { \\rm r i g h t } ) ^ { \\alpha } v ( x ) ] \\ , d x . \\end{align*}"}
{"id": "5711.png", "formula": "\\begin{align*} 1 = \\mathrm { i n d } ( v _ { + } ^ { \\infty } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) - 6 = n - 4 \\end{align*}"}
{"id": "9380.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p + 1 } S _ { 2 , \\lambda } ( p + 1 , k ) \\frac { x ^ { k } } { k } = e ^ { - x } \\sum _ { n = 0 } ^ { \\infty } \\big ( ( 1 - \\lambda ) _ { p , \\lambda } + ( 2 - \\lambda ) _ { p , \\lambda } + \\cdots + ( n - \\lambda ) _ { p , \\lambda } \\big ) \\frac { x ^ { n } } { n ! } . \\end{align*}"}
{"id": "5229.png", "formula": "\\begin{align*} \\delta ( D ) : = \\liminf _ { n \\to \\infty } \\frac { | D \\cap [ - n , n ] | } { 2 n + 1 } . \\end{align*}"}
{"id": "5616.png", "formula": "\\begin{align*} \\Omega _ { M _ c \\star } = \\Omega _ { \\mathcal { C } ^ c \\star } / \\Omega _ { \\mathcal { C } \\mathcal { C } ^ c \\star } = \\{ [ \\omega ] = \\omega + \\Omega _ { \\mathcal { C } \\mathcal { C } ^ c \\star } ~ | ~ \\omega \\in \\Omega _ { \\mathcal { C } ^ c \\star } \\} . \\end{align*}"}
{"id": "692.png", "formula": "\\begin{align*} \\partial _ { x _ { j ; \\alpha } } z _ { i ; \\alpha } ^ { ( \\ell ) } , j = 0 , \\ldots , n _ 0 , \\end{align*}"}
{"id": "8256.png", "formula": "\\begin{align*} v '' _ { i _ 1 } = \\bar { p _ 2 - r + 1 } , \\ v '' _ { i _ 2 } = \\bar { p _ 2 - r + 2 } , \\ \\cdots , \\ v '' _ { i _ r } = \\bar { p _ 2 } , \\ v '' _ { i _ { r + 1 } } = p _ 2 - r , \\ v '' _ { i _ { r + 2 } } = p _ 2 - r - 1 , \\ \\cdots , \\ v '' _ { i _ { p _ 2 } } = 1 . \\end{align*}"}
{"id": "7907.png", "formula": "\\begin{align*} G = ( g _ 1 , \\dots , g _ m ) \\equiv ( x _ 1 , \\dots , x _ { m - 1 } , 1 - | x _ m | ^ 2 - | x _ { m + 1 } | ^ 2 - \\dots | x _ n | ^ 2 ) , \\end{align*}"}
{"id": "8815.png", "formula": "\\begin{align*} \\partial _ t U _ t F ( \\varphi ) = \\Delta U _ t F ( \\varphi ) - \\left ( A _ { \\epsilon } \\varphi , \\nabla U _ t F ( \\varphi ) \\right ) . \\end{align*}"}
{"id": "8919.png", "formula": "\\begin{align*} \\varphi : \\Z & \\to \\R \\\\ z & \\mapsto \\sum _ { n = 1 } ^ { | z | } \\frac 1 n . \\end{align*}"}
{"id": "2985.png", "formula": "\\begin{align*} c ( w ) = \\min \\left ( \\left \\{ k \\in \\omega : { h } ^ { k + 1 } ( { t } _ { w } ) = \\ ; \\uparrow \\right \\} \\right ) . \\end{align*}"}
{"id": "9218.png", "formula": "\\begin{align*} \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x } & \\leq \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x + \\vert \\gamma \\vert ( \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma '^ { - 1 } ( x - J ^ A _ { \\gamma ' } x ) ) } \\\\ & = \\norm { \\left ( 1 - \\frac { \\gamma } { \\gamma ' } \\right ) x + \\left ( \\frac { \\gamma } { \\gamma ' } - 1 \\right ) J ^ A _ { \\gamma ' } x } \\\\ & = \\left \\vert 1 - \\frac { \\gamma } { \\gamma ' } \\right \\vert \\norm { x - J ^ A _ { \\gamma ' } x } \\end{align*}"}
{"id": "9303.png", "formula": "\\begin{align*} H ^ i ( \\mathcal { Y } _ K , \\mathcal { A } _ { \\mathcal { Y } _ K } ^ { \\otimes m } ( - \\mathcal { E } _ K ) ) = 0 \\end{align*}"}
{"id": "5260.png", "formula": "\\begin{align*} c \\otimes a = \\sum _ i ( p _ i \\otimes 1 ) \\Delta ( q _ i ) , d \\otimes b = \\sum _ j \\Delta ( z _ j ) ( w _ j \\otimes 1 ) . \\end{align*}"}
{"id": "4340.png", "formula": "\\begin{align*} \\langle \\varepsilon , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } = \\sum _ { j = 0 } ^ { M } \\varepsilon _ j \\langle \\phi _ j , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } + \\langle \\varepsilon _ { - } , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } \\end{align*}"}
{"id": "7739.png", "formula": "\\begin{align*} \\tilde { \\phi } _ { x x } ( x ) & = \\int _ { \\R } \\phi _ { x x } ( t , x ) \\psi ( t ) \\ , d t \\\\ & = \\int _ { \\R } \\left ( - | \\phi _ x | ^ 2 \\phi + \\phi _ { t t } + | \\phi _ t | ^ 2 \\phi + a ( x ) \\phi _ t \\right ) ( t , x ) \\psi ( t ) \\ , d t \\\\ & = \\int _ { \\R } - | \\phi _ x | ^ 2 \\phi \\psi ( t ) - \\phi _ { t } \\psi _ t ( t ) + ( | \\phi _ t | ^ 2 \\phi + a ( x ) \\phi _ t ) \\psi ( t ) \\ , d t . \\end{align*}"}
{"id": "7730.png", "formula": "\\begin{align*} - \\phi _ { u v } = ( \\phi _ u \\cdot \\phi _ v ) \\phi + \\frac { 1 } { 4 } a \\phi _ t = : F ( t , x ) = F ( u , v ) . \\end{align*}"}
{"id": "769.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathfrak { c } _ \\omega ( a _ 0 \\wedge a _ 1 \\wedge a _ 2 \\wedge a _ 3 ) = & \\mathrm { T r } _ \\omega ( P a _ 0 ( 1 - P ) a _ 1 P a _ 2 ( 1 - P ) a _ 3 P ) - \\\\ & - \\mathrm { T r } _ \\omega ( P a _ 0 ( 1 - P ) a _ 3 P a _ 2 ( 1 - P ) a _ 1 P ) = \\\\ = & \\lim _ { N \\to \\omega } \\frac { 1 } { \\log ( N + 2 ) } \\sum _ { k = 0 } ^ N \\sum _ { m = k } ^ \\infty k a _ { 0 , k } a _ { 2 , m } \\left ( a _ { 1 , - m } a _ { 3 , - k } - a _ { 1 , - k } a _ { 3 , - m } \\right ) , \\end{align*}"}
{"id": "7296.png", "formula": "\\begin{gather*} ( a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 + a _ 4 ^ 2 ) ( b _ 1 ^ 2 + b _ 2 ^ 2 + b _ 3 ^ 2 + b _ 4 ^ 2 ) = ( a _ 1 b _ 1 + a _ 2 b _ 2 + a _ 3 b _ 3 + a _ 4 b _ 4 ) ^ 2 \\\\ + ( a _ 1 b _ 2 - a _ 2 b _ 1 + a _ 3 b _ 4 + a _ 4 b _ 3 ) ^ 2 + ( a _ 1 b _ 3 - a _ 3 b _ 1 + a _ 4 b _ 2 - a _ 2 b _ 4 ) ^ 2 + ( a _ 1 b _ 4 - a _ 4 b _ 1 + a _ 2 b _ 3 - a _ 3 b _ 2 ) ^ 2 \\end{gather*}"}
{"id": "3843.png", "formula": "\\begin{align*} y ^ { [ k ] } = ( y ^ { [ k - 1 ] } ) ' - f _ { k , m + 1 } y ^ { ( m ) } - \\sum _ { j = 1 } ^ m ( f _ { k , j } - f _ { k , m + 1 } f _ { m , j } ) y ^ { ( j - 1 ) } , k = \\overline { m + 1 , 2 m } . \\end{align*}"}
{"id": "5378.png", "formula": "\\begin{align*} c _ c = 2 . \\end{align*}"}
{"id": "7921.png", "formula": "\\begin{align*} T ^ T T = A ^ { - 1 } . \\end{align*}"}
{"id": "3035.png", "formula": "\\begin{align*} & Z _ { \\lambda } = \\Theta _ { \\lambda } + \\frac { 1 } { 2 } P _ { \\sigma \\nu } \\omega ^ { \\sigma } \\wedge \\omega ^ { \\nu } + Q _ { \\sigma , \\nu } ^ { j } \\omega ^ { \\sigma } \\wedge \\omega _ { j } ^ { \\nu } + \\frac { 1 } { 2 } R _ { \\sigma , \\nu } ^ { i , j } \\omega _ { i } ^ { \\sigma } \\wedge \\omega _ { j } ^ { \\nu } , \\end{align*}"}
{"id": "481.png", "formula": "\\begin{align*} c _ { 1 } : = \\mathrm { s i g n } \\left [ ( i - j ) ( \\alpha - \\beta ) ( i - \\beta ) ( \\alpha - j ) \\right ] , \\ , c _ { 2 } : = \\mathrm { s i g n } \\left [ ( i - j ) ( \\alpha - \\beta ) ( i - \\alpha ) ( j - \\beta ) \\right ] . \\end{align*}"}
{"id": "8370.png", "formula": "\\begin{align*} d f _ p = \\lambda g _ { v } ( v , \\cdot ) = g _ { \\lambda v } ( \\lambda v , \\cdot ) \\end{align*}"}
{"id": "3065.png", "formula": "\\begin{align*} x - z = 0 \\ , , x - j ^ 2 z = 0 \\ , , x - j z = 0 \\ , , \\end{align*}"}
{"id": "9524.png", "formula": "\\begin{align*} p _ t - ( \\Delta y _ { t + 1 } + A ^ * _ { t + 1 } y _ { t + 1 } , B ^ * _ { t + 1 } y _ { t + 1 } ) \\in \\partial L _ t ( X _ t , U _ t ) , \\\\ \\Delta X _ { t } = A _ t X _ { t - 1 } + B _ t U _ { t - 1 } + W _ t \\end{align*}"}
{"id": "3178.png", "formula": "\\begin{align*} \\mathcal { F } _ { k } ^ j = \\sigma ( \\zeta _ { j - k } , x _ { j - k + 1 } , W _ { j - k + 1 } , \\ldots , W _ { j - 1 } , \\zeta _ { j - 1 } , x _ j ) , \\end{align*}"}
{"id": "2861.png", "formula": "\\begin{align*} i \\partial _ t h + \\Delta h - h + V ( h ) + R ( h ) = 0 , \\end{align*}"}
{"id": "1557.png", "formula": "\\begin{align*} h ( \\beta \\otimes \\alpha ) p ( x , w ) = p ( \\beta x \\alpha , w ) = p ( x \\beta \\alpha , w ) = p ( x \\alpha \\beta , w ) . \\end{align*}"}
{"id": "8288.png", "formula": "\\begin{align*} \\lim _ { q \\to \\zeta _ k } \\widehat { Z } _ { \\Gamma } ( q ) = \\ , & ( - 1 ) ^ { \\abs { V _ 1 } } \\zeta _ { 4 k } ^ { - \\sum _ { v \\in V } ( w _ { v } + 3 ) - \\sum _ { i \\in V _ 1 } 1 / w _ i } \\lim _ { t \\to + 0 } F ( f _ 2 ; t ) \\end{align*}"}
{"id": "31.png", "formula": "\\begin{align*} T M = Q \\oplus \\Sigma = Q \\oplus \\langle U \\rangle \\oplus \\langle V \\rangle , \\end{align*}"}
{"id": "2215.png", "formula": "\\begin{align*} \\| A ^ { - \\frac \\delta 2 } P x \\| \\leq C \\| x \\| _ { L ^ 1 } , \\ ; \\| A ^ { - \\frac 1 2 } P x \\| \\leq C \\| x \\| _ { L ^ { \\frac 6 5 } } , \\ ; \\forall x \\in L ^ 2 ( D ) . \\end{align*}"}
{"id": "7614.png", "formula": "\\begin{align*} \\bar { R } _ { \\nu j l i } = - \\left ( \\frac { \\lambda '' } { \\lambda } + \\frac { c - \\lambda '^ { 2 } } { \\lambda ^ { 2 } } \\right ) ( \\delta _ { i j } r _ { l } - \\delta _ { j l } r _ { i } ) r _ { \\nu } , \\end{align*}"}
{"id": "5652.png", "formula": "\\begin{align*} ( \\sigma , R ) ( \\tau , S ) = ( \\sigma \\tau , R ^ { \\tau } S ) \\end{align*}"}
{"id": "7208.png", "formula": "\\begin{align*} \\check x _ { t , x , v } : = x - \\check { \\mathcal T } _ { t , x , v } v . \\end{align*}"}
{"id": "868.png", "formula": "\\begin{align*} \\left \\{ R _ j = a \\right \\} = \\left \\{ \\varsigma _ { Q _ { j } , m } = 1 \\right \\} \\bigcap _ { i \\in [ a ] } \\left \\{ \\varsigma _ { i + Q _ { j - 1 } , m } = 0 \\right \\} . \\end{align*}"}
{"id": "4833.png", "formula": "\\begin{align*} e ^ { - i t _ 0 P } - e ^ { - i t _ 0 h ^ { - 1 } ( h P - i Q ) } = h ^ { - 1 } \\int _ 0 ^ { t _ 0 } e ^ { - i t h ^ { - 1 } ( h P - i Q ) } Q e ^ { - i ( t _ 0 - t ) P } d t , \\end{align*}"}
{"id": "4153.png", "formula": "\\begin{align*} r _ k = \\begin{cases} \\frac { k \\theta + 1 } { 2 } , & \\mbox { i f } \\ \\ k = 1 , . . . , 4 \\\\ 5 / 2 + \\frac { a \\theta } { 2 } , & \\mbox { i f } \\ \\ k = 5 \\\\ 5 / 2 + a \\theta , & \\mbox { i f } \\ \\ k = 6 , \\end{cases} \\end{align*}"}
{"id": "3791.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ { - r } \\big ( r _ l ( \\pi _ F ) ^ { ( l ) } ( w _ n ) W , \\sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \\big ) q _ F ^ { - \\frac { r } { 2 } } X ^ { - r } = \\gamma \\big ( X , r _ l ( \\pi _ F ) ^ { ( l ) } , \\sigma _ F ^ { ( l ) } , \\overline { \\psi } _ F ^ l \\big ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( W , W ' ) q _ F ^ { \\frac { r } { 2 } } X ^ { r } . \\end{align*}"}
{"id": "1731.png", "formula": "\\begin{align*} 2 ^ { ( \\mu _ * ( 1 \u2010 \\tilde \\lambda ) \u2010 \\alpha _ * \\tilde \\lambda ) k _ * t } \\cdot n ^ { \u2010 \\frac 1 2 + \\frac 1 q } \\stackrel { \\eqref { t i l d e _ n u _ d e f } } { = } n ^ { \u2010 \\tilde \\nu } . \\end{align*}"}
{"id": "23.png", "formula": "\\begin{align*} \\langle f ( t ) , g ( s ) \\rangle : = - \\mathrm { R e s } _ t ( f ( t ) , g ( t ) ) , \\end{align*}"}
{"id": "1581.png", "formula": "\\begin{align*} ( ( f , g ) ) _ k : = ( f , g ) _ { L ^ 2 _ k } + \\eta \\int _ 0 ^ { + \\infty } ( \\mathcal { S } _ L ( \\tau ) f , \\mathcal { S } _ L ( \\tau ) g ) _ { L ^ 2 _ v } d \\tau . \\end{align*}"}
{"id": "8959.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - s } \\left ( C _ p \\left | \\dot { \\tilde { \\xi } } ( s ) \\right | ^ q + f \\left ( \\tilde { \\xi } ( s ) \\right ) \\right ) d s = f ( \\xi ( 0 ) ) = u ( \\xi ( 0 ) ) = u ( x ) . \\end{align*}"}
{"id": "6118.png", "formula": "\\begin{align*} \\alpha _ { n + 2 } ( x ) = \\frac { - \\varepsilon } { A ^ n _ 1 } D \\alpha _ 0 ( x ) = \\frac { - \\varepsilon } { A ^ n _ 1 } \\sum _ { j = 2 } ^ \\infty j ( j - 1 ) a _ j x ^ { j - ( m + 2 ) } , \\end{align*}"}
{"id": "5537.png", "formula": "\\begin{align*} b ( t , h ) : = a ( h ) + \\sum _ { j = 1 } ^ r \\sigma ^ j ( h ) \\dot { B } _ m ^ j ( t ) . \\end{align*}"}
{"id": "2899.png", "formula": "\\begin{align*} - \\frac { N - 2 } { 2 } \\| \\nabla Q \\| _ 2 ^ 2 - \\frac { N } { 2 } \\| Q \\| _ 2 ^ 2 + \\frac { N + \\gamma } { 2 p } \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p } d x = 0 . \\end{align*}"}
{"id": "8093.png", "formula": "\\begin{align*} \\mathrm { s d } ( u ) = \\inf \\left \\{ \\delta \\in \\mathbb { R } \\ , | \\ , \\lim _ { \\lambda \\rightarrow 0 } \\lambda ^ { \\delta } u ( \\lambda x ) = 0 \\right \\} . \\end{align*}"}
{"id": "3504.png", "formula": "\\begin{align*} D _ { 1 4 } & = \\frac { ( a t _ 3 ) ^ { 1 - s _ 1 } ( a t _ 3 + 1 ) ^ { - s _ 3 } } { s _ 1 + s _ 3 - 1 } + \\frac { s _ 3 } { s _ 1 + s _ 3 - 1 } \\int _ { a t _ 3 } ^ \\infty \\frac { 1 } { u ^ { s _ 1 } ( u + 1 ) ^ { s _ 3 + 1 } } d u . \\end{align*}"}
{"id": "6620.png", "formula": "\\begin{align*} J _ 1 = J _ { 1 1 } + J _ { 1 2 } + J _ { 1 3 } + O \\big ( X ^ { - \\frac { 1 } { 2 } + \\varepsilon } Q ^ { \\frac { 5 } { 2 } } ( h k ) ^ { \\varepsilon } \\big ) . \\end{align*}"}
{"id": "5327.png", "formula": "\\begin{align*} \\mathbf { C } _ i ( \\psi ) ( x , t ) : = \\int _ D \\psi ( x , q , t ) U ' _ i ( \\frac { 1 } { 2 } | q _ i | ^ 2 ) q _ i q _ i ^ T \\ ; \\mathrm { d } q , \\ ; \\ ; i = 1 , \\ldots , K . \\end{align*}"}
{"id": "7187.png", "formula": "\\begin{align*} f ( t , x , v ) = - \\int _ 0 ^ t E ( s , X _ { s , t } ( x , v ) ) \\cdot \\nabla _ v \\mu ( V _ { s , t } ( x , v ) ) - \\int _ 0 ^ t e _ 0 \\nabla \\Phi ( X _ { s , t } ( x , v ) - X ( s ) ) \\cdot \\nabla _ v \\mu ( V _ { s , t } ( x , v ) ) . \\end{align*}"}
{"id": "6172.png", "formula": "\\begin{align*} ( z - z _ 1 ) \\cdots ( z - z _ { r + 1 } ) & = 1 \\cdot z ^ { r + 1 } + Z _ 1 ( z _ 1 , \\ldots , z _ { r + 1 } ) z ^ r + \\cdots + Z _ { r + 1 } ( z _ 1 , \\ldots , z _ { r + 1 } ) , \\\\ \\phi _ { r + 1 } ( z ) + b _ 1 \\phi _ r ( z ) + \\cdots + b _ r \\phi _ 1 ( z ) & = 1 \\cdot z ^ { r + 1 } + B _ 1 ( b _ 1 , \\ldots , b _ r ) z ^ r + \\cdots + B _ { r + 1 } ( b _ 1 , \\ldots , b _ r ) . \\end{align*}"}
{"id": "5431.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { k _ c } \\phi _ { t _ { r , i } + \\ell } \\phi ^ \\top _ { t _ { r , i } + \\ell } \\ge C _ d I _ p . \\end{align*}"}
{"id": "4584.png", "formula": "\\begin{align*} \\Psi _ k ( \\lambda ) = \\sum _ { i = 1 } ^ k \\ln \\mathbf { E } \\big ( e ^ { \\lambda \\xi _ i } \\big | \\mathcal { F } _ { i - 1 } \\big ) . \\end{align*}"}
{"id": "8453.png", "formula": "\\begin{align*} L ' _ { v d } \\wedge L ' _ { e } = L ' _ { i v d } . \\end{align*}"}
{"id": "7971.png", "formula": "\\begin{align*} ( \\gamma \\cdot f ) ( x ) & : = \\langle \\frac { d } { d t } \\Big | _ { t = 0 } \\gamma ( \\exp ^ g _ y ( t \\cdot f ( y ) \\nu ( y ) ) ) , \\nu ( \\gamma ( y ) ) \\rangle _ g \\Big | _ { y = \\gamma ^ { - 1 } ( x ) } \\\\ & = f ( \\gamma ^ { - 1 } ( x ) ) \\cdot \\langle \\gamma _ * \\nu ( \\gamma ^ { - 1 } ( x ) ) , \\nu ( x ) \\rangle _ g , \\end{align*}"}
{"id": "9487.png", "formula": "\\begin{align*} | \\mathcal { D D } _ { ( s , t ) } | = \\binom { ( s - 2 ) / 2 + ( t - 1 ) / 2 } { ( s - 2 ) / 2 } . \\end{align*}"}
{"id": "5208.png", "formula": "\\begin{align*} g _ j ( y ) = a ( y ) \\Phi _ { A _ j } ( y ) , \\end{align*}"}
{"id": "7416.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega _ 1 ( x ) } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta ) - f ( \\eta _ { 1 , x , r } ) ] d \\nu _ { b } \\Big | , \\end{align*}"}
{"id": "7530.png", "formula": "\\begin{align*} T \\log \\left ( \\frac { a ^ 2 + T ^ 2 } { 4 } \\right ) = 2 T \\log \\left ( \\frac { T } { 2 } \\right ) + \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "988.png", "formula": "\\begin{align*} ( w - v ) ( x ) & = \\begin{cases} 0 , & x \\in \\{ v > 0 \\} \\cap \\R ^ n _ + \\\\ u ( x ) - ( 1 - \\theta / 2 ) ^ { - n - 2 } u ( a ) \\zeta ( x ) , & x \\in \\{ v \\leqslant 0 \\} \\cap \\R ^ n _ + . \\end{cases} \\end{align*}"}
{"id": "2577.png", "formula": "\\begin{align*} \\lim _ { | x | , | \\omega | \\to 0 } \\norm { D ^ \\alpha X ^ \\beta ( M _ \\omega T _ x g - g ) } _ \\infty = 0 , \\end{align*}"}
{"id": "485.png", "formula": "\\begin{align*} Y _ { \\alpha \\beta } ^ { i j } = - Y _ { \\alpha \\omega } ^ { i j } \\cdot Y _ { \\omega \\beta } ^ { i j } = - Y _ { \\alpha \\omega } ^ { i m } \\cdot Y _ { \\omega \\delta } ^ { m j } \\cdot Y _ { \\omega \\beta } ^ { i m } \\cdot Y _ { \\omega \\beta } ^ { i m } . \\end{align*}"}
{"id": "6147.png", "formula": "\\begin{align*} ( n + 1 ) Q ( x ) - ( 1 - x ) Q ' ( x ) = \\mu _ { n , m } x ^ m . \\end{align*}"}
{"id": "8277.png", "formula": "\\begin{align*} \\pi ( M _ { p } ) = \\begin{cases} M _ { y } , & y \\in Q \\ \\ p = g ( y ) , \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
{"id": "1813.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty A _ n ( x , y ) { t ^ n \\over n ! } = { y - x \\over 1 - x y ^ { - 1 } e ^ { ( y - x ) t } } . \\end{align*}"}
{"id": "7415.png", "formula": "\\begin{align*} f ( \\eta ) - f ( \\eta ^ { x + 1 , x + 1 + r } ) = \\sum _ { k = 0 } ^ 2 [ f ( \\eta _ { k , x , r } ) - f ( \\eta _ { k + 1 , x , r } ) ] . \\end{align*}"}
{"id": "6238.png", "formula": "\\begin{align*} K _ 1 \\cap \\dotsb \\cap K _ j \\cap \\dotsb \\cap K _ t = G _ m \\cap G _ { s ( K _ 1 ) } \\cap \\dotsb \\cap G _ { s ( K _ t ) } = G _ { s ( K _ 1 ) } \\cap \\dotsb \\cap G _ { s ( K _ t ) } \\end{align*}"}
{"id": "2006.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ 1 ( \\bar { T } ) \\\\ u _ 2 ( \\bar { T } ) \\\\ u _ 3 ( \\bar { T } ) \\end{array} \\right ) = M ( 2 p - 1 ) \\left ( \\begin{array} { c } u _ 1 ( \\bar { T } ) \\\\ u _ 2 ( \\bar { T } ) \\\\ u _ 3 ( \\bar { T } ) \\end{array} \\right ) , \\end{align*}"}
{"id": "2914.png", "formula": "\\begin{align*} H _ k : C _ A = \\Pi _ { k } A \\in I _ d ( k ) , \\end{align*}"}
{"id": "6207.png", "formula": "\\begin{align*} \\begin{aligned} & ~ ~ ~ ~ \\| C - C Z _ { j } Z _ { j } ^ { T } \\| ^ 2 _ F \\\\ & < \\| C - C _ { ( j ) } \\| ^ 2 _ F + \\epsilon \\| C \\| _ F ^ 2 + 2 ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) ( 2 + \\xi ) \\| C \\| ^ 2 _ F + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) ^ 2 \\| C \\| ^ 2 _ F \\\\ & = \\| C - C _ { ( j ) } \\| ^ 2 _ F + \\left [ \\epsilon + 2 ( 2 + \\xi ) ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) ^ 2 \\right ] \\| C \\| ^ 2 _ F . \\end{aligned} \\end{align*}"}
{"id": "2179.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } K ( x ) \\frac { f ( x , u ) } { \\vert u \\vert ^ { g ^ + - 1 } } \\vert u \\vert ^ { g ^ + - 1 } u d x & = \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha + d } } d x d y \\\\ & + \\int _ { \\mathbb { R } ^ { d } } g ( u ) u d x . \\end{align*}"}
{"id": "4055.png", "formula": "\\begin{align*} - u _ { x x } + u _ x + \\rho _ x + \\lambda u = g , \\end{align*}"}
{"id": "7063.png", "formula": "\\begin{align*} \\frac { N ^ K _ j ( t \\log K ) } { N ^ K _ { i } ( t \\log K ) } = \\exp ( \\log K ( \\beta _ { j } ( t ) - \\beta _ { i } ( t ) ) \\le e ^ { L \\ , \\rho ( j \\delta _ K , i \\delta _ K ) \\ , \\log K } . \\end{align*}"}
{"id": "6906.png", "formula": "\\begin{align*} D _ { k + 1 } U ^ { k + 1 } _ { i } = ( 1 - \\delta ^ k _ { k - i } ) U ^ { k } _ { i } + \\delta _ { k - i } ^ k U ^ { k } _ { i - 1 } D _ { i } + \\sum \\limits _ { j = - 1 } ^ { k - i - 1 } U ^ k _ { k - j - 1 } \\Gamma _ j U ^ { k - j - 1 } _ { i } \\end{align*}"}
{"id": "1639.png", "formula": "\\begin{align*} \\delta = \\frac { \\int _ 0 ^ \\infty e ^ { - \\lambda \\tau } P _ { s + \\tau } f ( x ) d \\tau } { \\sup _ { t > 0 } P _ t f ( x ) } \\end{align*}"}
{"id": "7935.png", "formula": "\\begin{align*} B _ \\ell ( a ) = a ^ \\ell + O _ \\ell \\left ( a ^ { \\ell - 1 } \\right ) . \\end{align*}"}
{"id": "1220.png", "formula": "\\begin{align*} C \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } ( \\Omega _ { k } ) \\leq \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } ( K _ { 1 / 3 } ) \\leq \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } \\Big ( \\bigcup _ { I \\in \\mathcal { K } _ k } I \\Big ) = \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } ( \\Omega _ { k } ) \\le 1 , \\end{align*}"}
{"id": "6513.png", "formula": "\\begin{align*} \\frac { s ^ { ( 2 \\ell ) } _ n } { s ^ { ( 2 m ) } _ { n + 1 } } & = \\dfrac { ( 2 \\ell - 1 ) ! ! } { ( 2 m - 1 ) ! ! } \\cdot ( 1 - 2 \\alpha ) ^ { m - \\ell } \\cdot \\dfrac { n ^ { \\ell } } { ( n + 1 ) ^ m } \\\\ & \\sim \\dfrac { ( 2 \\ell - 1 ) ! ! } { ( 2 m - 1 ) ! ! } \\cdot ( 1 - 2 \\alpha ) ^ { m - \\ell } \\cdot n ^ { \\ell - m } ( n \\to \\infty ) , \\end{align*}"}
{"id": "5822.png", "formula": "\\begin{align*} L f ( \\eta ) = \\sum _ { x , y \\in \\Z } p ( x , y ) \\left [ f ( \\eta ^ { x y } ) - f ( \\eta ) \\right ] , \\end{align*}"}
{"id": "9247.png", "formula": "\\begin{align*} x \\in \\mathrm { r a n } ( I d + \\gamma A ) & \\exists y \\left ( x \\in y + \\gamma A y \\right ) \\\\ & \\exists y , z \\left ( z \\in A y \\land x = y + \\gamma z \\right ) \\\\ & \\exists y , z \\left ( z \\in A y \\land z = \\frac { 1 } { \\gamma } ( x - y ) \\right ) . \\end{align*}"}
{"id": "1891.png", "formula": "\\begin{align*} \\log _ { q _ s } \\abs { \\psi ( t ) } _ s = \\langle \\psi , \\omega ( t ) \\rangle \\end{align*}"}
{"id": "2286.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\norm { T _ x f - f } _ 2 = 0 \\end{align*}"}
{"id": "4811.png", "formula": "\\begin{align*} n = q , \\quad \\theta = \\hat \\theta = 0 \\end{align*}"}
{"id": "4825.png", "formula": "\\begin{align*} - \\widetilde { P } _ h ( z ) ^ * : = - h P - i Q - ( - \\bar { z } ) , \\end{align*}"}
{"id": "8004.png", "formula": "\\begin{align*} ( n + 1 ) ^ { - \\# \\mathcal { I } } e ^ { n H ( \\mathbf { p } ) } \\leq \\# \\{ ( i _ 1 , \\ldots , i _ n ) \\in \\mathcal { I } ^ n : \\tau ( i _ 1 , \\ldots , i _ n ) = \\mathbf { p } \\} \\leq e ^ { n H ( \\mathbf { p } ) } . \\end{align*}"}
{"id": "5577.png", "formula": "\\begin{align*} | B | + N \\ = \\ k - s + N \\ \\leqslant \\ ( N + 1 ) m - s \\ = \\ | A | - s \\ = \\ | F | . \\end{align*}"}
{"id": "9165.png", "formula": "\\begin{align*} L ( s , \\operatorname { s y m } ^ 2 f ) = & \\zeta ( 2 s ) \\sum _ { n \\geq 1 } \\frac { \\lambda ( n ^ 2 ) } { n ^ s } = \\prod _ { p } \\left ( 1 - \\frac { \\lambda ( p ^ 2 ) } { p ^ s } + \\frac { \\lambda ( p ^ 2 ) } { p ^ { 2 s } } - \\frac { 1 } { p ^ { 3 s } } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "1737.png", "formula": "\\begin{align*} A ^ \\varepsilon = A ^ \\varepsilon ( n ) = \\{ ( t , \\ , m ) : \\ ; 0 \\le t \\le \\hat t ( n ) , \\ ; m \\ge m ^ * _ t \\} \\end{align*}"}
{"id": "5984.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { x ^ { n } } { n ! } z ^ { n } & = \\sum _ { m = 0 } ^ { \\infty } \\frac { z ^ { m } } { m ! } A _ { m } ^ { \\lambda , \\beta } ( x ) \\cdot \\sum _ { k = 0 } ^ { \\infty } \\frac { z ^ { k } } { k ! } M _ { \\lambda , \\beta } ( n ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { n ! } \\left ( \\sum _ { k = 0 } ^ { n } \\binom { n } { k } A _ { k } ^ { \\lambda , \\beta } ( x ) M _ { \\lambda , \\beta } ( n - k ) \\right ) . \\end{align*}"}
{"id": "2784.png", "formula": "\\begin{align*} \\ker \\mathcal { L } = \\{ i Q , \\partial _ { x _ 1 } Q , . . . , \\partial _ { x _ N } Q \\} . \\end{align*}"}
{"id": "1341.png", "formula": "\\begin{align*} \\# \\{ \\{ z , z ' \\} \\mid z \\in A , \\ ; z ' \\in \\mathcal { C } ^ \\vee ( Q ) - A , \\ ; | z - z ' | = \\delta \\} \\geq c _ 1 n ^ { - 1 } \\# A \\end{align*}"}
{"id": "6353.png", "formula": "\\begin{align*} D ( [ \\C ^ 2 / G ] ) = \\left \\langle \\phi ( D ( Y ) ) ^ \\perp , \\phi ( D ( Y ) ) \\right \\rangle . \\end{align*}"}
{"id": "6469.png", "formula": "\\begin{align*} M _ n = \\dfrac { S _ n } { a _ n } \\to M _ { \\infty } \\mbox { a . s . a n d i n $ L ^ 2 $ a s $ n \\to \\infty $ . } \\end{align*}"}
{"id": "392.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } _ { 1 } ( u , v ) \\partial _ { t } u + A ^ { i } _ { 1 1 } ( u , v ) \\partial _ { i } u + A ^ { i } _ { 1 2 } ( u , v ) \\partial _ { i } v & = g _ { 1 } ( U , D _ { x } v ) , \\\\ A ^ { 0 } _ { 2 } ( u , v ) \\partial _ { t } v + A ^ { i } _ { 2 1 } ( u , v ) \\partial _ { i } u + A ^ { i } _ { 2 2 } ( u , v ) \\partial _ { i } v - B ^ { i j } ( u , v ) \\partial _ { i } \\partial _ { j } v & = g _ { 2 } ( U , D _ { x } U ) , \\end{aligned} \\end{align*}"}
{"id": "8055.png", "formula": "\\begin{align*} \\left \\{ \\Psi ^ i ( z _ 1 ) , \\Psi ^ j ( z _ 2 ) \\right \\} [ \\psi ] : = \\int _ { \\Sigma ^ 2 } E ( y _ 1 , y _ 2 ) K ^ i _ { \\psi } ( z _ 1 , y _ 1 ) K ^ j _ { \\psi } ( z _ 2 , y _ 2 ) \\ , \\mathrm { d } y _ 1 \\mathrm { d } y _ 2 \\end{align*}"}
{"id": "910.png", "formula": "\\begin{align*} ( x ) _ { n } = \\sum _ { k = 0 } ^ { n } S _ { 1 } ( n , k ) x ^ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 1 , 2 , 3 , 9 , 1 0 ] ) , \\end{align*}"}
{"id": "2885.png", "formula": "\\begin{align*} i \\partial _ t w + \\Delta w - w + S ( \\mathcal { V } _ { l ( k ) } ^ A + w ) - S ( \\mathcal { V } _ { l ( k ) } ^ A ) = - \\varepsilon _ { l ( k ) } = O ( e ^ { - ( l ( k ) + 1 ) e _ 0 t } ) , \\end{align*}"}
{"id": "420.png", "formula": "\\begin{align*} \\Phi _ { 0 } ^ { 2 } ( t ) \\leq C _ { 1 } e ^ { C _ { 2 } \\left ( t + M t ^ { 1 / 2 } \\right ) } = : \\Phi _ { 1 } ^ { 2 } ( t ) , \\end{align*}"}
{"id": "5938.png", "formula": "\\begin{align*} J _ L ( g , g \\cdot \\xi _ 1 , \\dots , g \\cdot \\xi _ k ) = \\big ( { \\rm A d } ^ * _ { g ^ { - 1 } } \\big ) ^ k \\pmb { F } \\ell ( \\xi _ 1 , \\dots , \\xi _ k ) . \\end{align*}"}
{"id": "5341.png", "formula": "\\begin{align*} & \\partial _ t u ^ { \\epsilon } \\rightarrow \\partial _ t u \\ ; \\ ; \\ ; \\ ; L ^ \\infty ( 0 , T ; H ^ 1 ) , \\\\ & \\frac { 1 } { \\epsilon } P ' ( \\rho ^ \\epsilon ) \\nabla \\phi ^ \\epsilon \\rightarrow v \\ ; \\ ; \\ ; \\ ; L ^ { \\infty } ( 0 , T ; H ^ 1 ) , \\ ; \\ ; \\ ; \\ ; T > 0 , \\end{align*}"}
{"id": "4675.png", "formula": "\\begin{align*} w _ t - \\partial _ y ( | D | w + w - | w | ^ { p - 1 } w ) = 0 . \\end{align*}"}
{"id": "3123.png", "formula": "\\begin{align*} \\begin{vmatrix} F _ 1 ^ 1 & F _ 1 ^ 2 & F _ 1 ^ 3 & F _ 1 ^ 4 & F _ 1 ^ 5 \\\\ F _ 2 ^ 1 & F _ 2 ^ 2 & F _ 2 ^ 3 & F _ 2 ^ 4 & F _ 2 ^ 5 \\\\ F _ 3 ^ 1 & F _ 3 ^ 2 & F _ 3 ^ 3 & F _ 3 ^ 4 & F _ 3 ^ 5 \\end{vmatrix} = 0 \\ , , \\end{align*}"}
{"id": "2719.png", "formula": "\\begin{align*} h _ { i , j } ( \\lambda ) = \\lambda _ i + \\lambda ' _ j - i - j + 1 , \\end{align*}"}
{"id": "460.png", "formula": "\\begin{align*} h _ { x y } ( a ) = \\sum _ { b \\in \\Lambda _ { x y } ( V ' ) ( a ) } g ( b ) . \\end{align*}"}
{"id": "9316.png", "formula": "\\begin{align*} ( \\mathbf { x } _ { k + 1 } , \\mathbf { y } _ { k + 1 } ) = \\mathcal { P } ( \\mathbf { x } _ { k } , \\mathbf { y } _ { k } ) , \\hbox { w h e r e } \\mathcal { P } = ( I + \\Sigma ^ { - 1 } T _ { \\mathcal { L } } ) ^ { - 1 } \\hbox { a n d } \\Sigma : = b l o c k d i a g ( \\rho I _ d , \\delta I _ m ) . \\end{align*}"}
{"id": "2705.png", "formula": "\\begin{align*} q ( z ) = \\pi A ' z ^ 2 + 2 \\pi b ' \\cdot z + c ' . \\end{align*}"}
{"id": "7171.png", "formula": "\\begin{align*} \\bigg ( \\int _ { \\Omega \\cap B _ { \\gamma / 2 } } | w | ^ j d x \\bigg ) ^ { 1 / j } \\leq c ( d , \\chi ) j \\forall j = 1 , 2 , \\ldots \\end{align*}"}
{"id": "1009.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s ( \\tilde u - \\tau \\varphi ^ { ( 2 ) } ) ( a ) & \\leqslant - C a _ 1 \\int _ { B _ { 1 / 2 } ^ + } \\frac { y _ 1 ( \\tilde u - \\tau \\varphi ^ { ( 2 ) } ) ( y ) } { \\vert a _ \\ast - y \\vert ^ { n + 2 s + 2 } } \\dd y \\\\ & \\leqslant - C a _ 1 \\bigg ( \\int _ { B _ { 1 / 2 } ^ + } y _ 1 \\tilde u ( y ) \\dd y - \\tau \\bigg ) \\\\ & = - C a _ 1 \\bigg ( \\frac 1 { \\rho ^ { n + 1 } } \\int _ { B _ { \\rho / 2 } ^ + } y _ 1 u ( y ) \\dd y - \\tau \\bigg ) . \\end{align*}"}
{"id": "3054.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 3 + x _ 1 y _ 1 x _ 2 y _ 2 ( a x _ 1 y _ 1 + b x _ 2 y _ 2 ) + & c x _ 2 ^ 3 y _ 2 ^ 3 + x _ 1 ^ 3 y _ 1 y _ 2 ^ 2 + x _ 1 x _ 2 ^ 2 y _ 1 ^ 3 + \\\\ + & d x _ 2 ^ 3 y _ 1 ^ 2 y _ 2 + e x _ 1 ^ 2 x _ 2 y _ 2 ^ 3 = 0 \\ , . \\end{align*}"}
{"id": "237.png", "formula": "\\begin{align*} \\operatorname { C o v } _ { \\mu } ( g _ j , f ) = \\mathcal { E } _ { \\Sigma } ( g _ j , f ) . \\end{align*}"}
{"id": "869.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { P } \\left ( R _ j = a \\right ) & = \\mathbb { P } \\left ( \\varsigma _ { Q _ { j } , m } = 1 \\right ) \\cdot \\prod _ { i \\in [ a ] } \\mathbb { P } \\left ( \\varsigma _ { i + Q _ { j - 1 } , m } = 0 \\right ) \\\\ & = \\left ( 1 - \\epsilon _ m \\right ) \\epsilon ^ a _ m . \\end{aligned} \\end{align*}"}
{"id": "8851.png", "formula": "\\begin{align*} \\Delta ^ q _ R : = \\{ ( x _ 0 , \\ldots , x _ q ) \\mid d ( x _ i , x _ j ) \\le R \\forall i , j \\} . \\end{align*}"}
{"id": "8701.png", "formula": "\\begin{align*} \\begin{aligned} \\check { b } ( s ) : = \\max _ { \\omega \\in \\Omega } \\biggl \\{ a _ { 1 0 } a _ { 2 n } & + \\sum _ { t : \\omega _ t = 1 } a _ { 2 ( n - p ^ t _ 2 ) } \\bigl ( s _ { 1 p ^ { t - 1 } _ 1 } - s _ { 1 p ^ { t } _ 1 } \\bigr ) + \\\\ & \\sum _ { t : \\omega _ t = 2 } a _ { 1 p ^ t _ 1 } \\bigl ( a _ { 2 ( n - p _ 2 ^ t ) } - a _ { 2 ( n - p _ 2 ^ { t - 1 } ) } - s _ { 2 p ^ { t } _ 2 } + s _ { 2 p ^ { t - 1 } _ 2 } \\bigr ) \\biggr \\} , \\end{aligned} \\end{align*}"}
{"id": "9099.png", "formula": "\\begin{align*} u ^ { i - 1 } \\beta _ i ( x y ) - u ^ { i - 1 } x \\beta _ i ( y ) - u ^ { i - 2 } \\beta _ m ( x ) \\beta _ { i - m } ( y ) - \\dots - u ^ { i - 1 } \\beta _ i ( x ) y = 0 \\end{align*}"}
{"id": "6597.png", "formula": "\\begin{align*} \\frac { c } { Q } \\int _ { 1 } ^ { \\infty } W \\left ( \\frac { c x } { Q } \\right ) \\ , d x = \\int _ { 0 } ^ { \\infty } W ( x ) \\ , d x - \\int _ { 0 } ^ { c / Q } W ( x ) \\ , d x = \\int _ { 0 } ^ { \\infty } W ( x ) \\ , d x + O \\left ( \\frac { c } { Q } \\right ) . \\end{align*}"}
{"id": "3869.png", "formula": "\\begin{align*} \\lim _ { X \\rightarrow + \\infty } \\frac { N ( \\cup _ { p > M } \\mathcal { W } _ p ( V ) , X ) } { X ^ { \\dim V } } = O \\left ( \\frac { 1 } { \\log M } \\right ) , \\end{align*}"}
{"id": "5762.png", "formula": "\\begin{align*} \\| \\operatorname { H e s s } \\hat { f } ^ j \\| _ { C ^ { 0 , \\alpha } ( B _ { R } ( \\hat x , \\hat h _ \\delta ) ) } \\leq \\Psi ( \\delta \\mid n , r , v , \\rho , \\alpha , Q ) , \\ \\ j = 1 , \\dots , n , \\end{align*}"}
{"id": "3477.png", "formula": "\\begin{align*} \\mathcal { D } ^ \\prime & = \\left \\{ ( s _ 1 , s _ 2 , s _ 3 ) \\mid \\sigma _ 1 \\geq 0 , \\sigma _ 2 \\geq 0 , \\sigma _ 3 > 0 , t _ 1 \\geq 0 , t _ 2 \\geq 0 , 2 \\leq t _ 3 \\leq T , \\right . \\\\ & \\left . \\sigma _ 1 + \\sigma _ 3 \\leq 1 , \\sigma _ 2 + \\sigma _ 3 \\leq 1 , \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 > 3 / 2 , \\right . \\\\ & \\left . s _ 1 + s _ 3 \\neq 1 , s _ 2 + s _ 3 \\neq 1 , s _ 1 + s _ 2 + s _ 3 \\neq 2 \\right \\} . \\end{align*}"}
{"id": "2856.png", "formula": "\\begin{align*} M [ u ] = M [ v _ 1 ] = M [ Q ] , \\ ; E [ u ] = E [ v _ 1 ] = E [ Q ] , \\ ; \\| \\nabla v _ 1 \\| _ { L ^ 2 } = \\| \\nabla Q \\| _ 2 , \\end{align*}"}
{"id": "1436.png", "formula": "\\begin{align*} \\int _ { G _ { n } ( \\Q ) \\backslash G _ { n } ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { E } ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h = c _ k ( s ) D ( s , \\mathbf { f } , \\chi ) \\mathbf { f } ( g ) , \\end{align*}"}
{"id": "690.png", "formula": "\\begin{align*} C _ 4 = \\frac { 2 } { 3 } , C _ 6 = \\frac { 2 8 } { 1 5 } , C _ 8 = \\frac { 8 7 5 6 } { 3 1 5 } \\end{align*}"}
{"id": "7722.png", "formula": "\\begin{gather*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } , \\ ; \\phi [ 0 ] = u [ 0 ] , \\end{gather*}"}
{"id": "8533.png", "formula": "\\begin{align*} i \\partial _ t u - \\partial _ { x x } u + V ( x ) u \\pm b _ 1 ( x ) \\left | u \\right | ^ { 2 } u + b _ 2 ( x ) u ^ 3 + b _ 3 ( x ) | u | u ^ 2 + b _ 4 ( x ) | u | ^ 3 = 0 \\end{align*}"}
{"id": "8557.png", "formula": "\\begin{align*} \\frac { R _ { \\pm } ( k ) } { T ( k ) } = \\frac { 1 } { 2 i k } \\int e ^ { \\mp 2 i k x } V ( x ) m _ { \\mp } ( x , k ) \\ , d x , \\end{align*}"}
{"id": "568.png", "formula": "\\begin{align*} \\big \\{ ( x , y ) \\in \\R ^ 2 \\ \\exists \\ n \\in \\N , \\ y = n x \\big \\} \\end{align*}"}
{"id": "4695.png", "formula": "\\begin{align*} C = & \\partial _ y \\Bigg [ \\eta _ i \\int _ 0 ^ 1 \\frac { p ( p - 1 ) } { 2 } \\bigg | 1 + s \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n \\frac { \\sigma _ i \\sigma _ j \\R _ j } { \\R _ i } \\bigg | ^ { p - 3 } \\bigg ( 1 + s \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n \\frac { \\sigma _ i \\sigma _ j \\R _ j } { \\R _ i } \\bigg ) \\times \\sigma _ i \\R _ i ^ p \\bigg ( \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n \\frac { \\sigma _ i \\sigma _ j \\R _ j } { \\R _ i } \\bigg ) ^ 2 \\ , \\dd s \\Bigg ] . \\end{align*}"}
{"id": "3557.png", "formula": "\\begin{align*} \\int n ( y ) \\Phi _ 1 ( y ) \\ , d y = \\int \\Phi _ 1 ( y ) \\ , d y = 1 . \\end{align*}"}
{"id": "6679.png", "formula": "\\begin{align*} d _ F \\circ \\Delta _ { a _ 1 } \\circ \\cdots \\circ \\Delta _ { a _ r } \\bigl ( ~ _ r F _ s ( z ) \\bigr ) = \\Delta _ { b _ 1 - 1 } \\circ \\cdots \\circ \\Delta _ { b _ s - 1 } \\bigl ( ~ _ r F _ s ( z ) \\bigr ) . \\end{align*}"}
{"id": "6388.png", "formula": "\\begin{align*} V _ { S _ { t _ n } } = \\{ ( z _ 1 , \\dots , z _ { n - 1 } , t _ n ) \\in \\mathbb C ^ n \\ , : \\ , f _ i ( z _ 1 , \\dots , z _ { n - 1 } , t _ n ) = 0 \\ , , \\ , i = 1 , \\dots , n \\} \\subseteq V _ S . \\end{align*}"}
{"id": "9295.png", "formula": "\\begin{align*} P = W _ { - 1 } \\overset { W ( - 1 , 0 ] } { \\leadsto } W _ { 0 } \\overset { W ( 0 , 1 ] } { \\leadsto } \\cdots \\overset { W ( d - 2 , d - 1 ] } { \\leadsto } W _ { d - 1 } \\overset { W ( d - 1 , d ] } { \\leadsto } W _ { d } = Q \\end{align*}"}
{"id": "5916.png", "formula": "\\begin{align*} \\psi _ S ( x ) = ( x + 1 ) ^ { n - 3 } \\ , \\Big [ ( x + 1 ) ^ 3 - n ( x + 1 ) ^ 2 + 4 s _ 2 t _ 1 ( s _ 1 + t _ 2 ) \\Big ] . \\end{align*}"}
{"id": "4214.png", "formula": "\\begin{align*} \\Gamma _ { g _ 1 , k } ( f ) = \\Gamma _ { g _ 2 , k } ( f ) \\end{align*}"}
{"id": "3773.png", "formula": "\\begin{align*} \\epsilon ( X , \\pi , \\pi ' , \\psi _ K ) = ( q _ K ^ { \\frac { 1 } { 2 } } X ) ^ { n ( \\pi , \\pi ' , \\psi _ K ) } \\epsilon ( \\pi , \\pi ' , \\psi _ K ) . \\end{align*}"}
{"id": "5250.png", "formula": "\\begin{align*} \\widetilde { M } ( A \\otimes ^ I B ) = \\widetilde { M } ( A \\otimes B ) \\cap M ( A \\otimes ^ I B ) , \\end{align*}"}
{"id": "6587.png", "formula": "\\begin{align*} 1 + \\sum _ { q = 1 } ^ { \\infty } p ^ { - q w } \\phi ^ { \\star } ( p ^ q ) & = 1 + \\sum _ { q = 1 } ^ { \\infty } p ^ { - q w } \\sum _ { c d = p ^ q } \\phi ( d ) \\mu ( c ) = \\left ( \\frac { 1 } { 1 - p ^ { 1 - w } } \\right ) \\left ( 1 - p ^ { - w } \\right ) ^ 2 . \\end{align*}"}
{"id": "7051.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty | b _ j ( z ) | ^ 2 \\le \\frac { C } { \\delta ^ { 2 \\max m _ j } } \\Big ( 1 + \\frac { 1 } { \\delta ^ 2 } \\log \\frac { 1 } { \\delta } \\Big ) ^ 2 . \\end{align*}"}
{"id": "5042.png", "formula": "\\begin{align*} \\begin{array} { l c l } K ^ { \\frac { \\theta } { 2 } , \\nu } = 0 \\ m = 2 & & K ^ { \\theta , \\mu } = \\delta _ { 0 , \\mu } ( 1 + z ^ { m - 2 } ) . \\end{array} \\end{align*}"}
{"id": "5480.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } ( t ) & = & A \\xi ( t ) + \\alpha ( \\xi ( t ) ) \\\\ \\xi ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "8233.png", "formula": "\\begin{align*} [ z ^ n ] N _ h & = [ z ^ { n - 1 / 2 } ] \\sqrt { B } ( 1 - B ) ^ { h - 1 } \\\\ & = \\frac { 1 } { n - \\frac 1 2 } [ t ^ { n - 3 / 2 } ] \\frac { 1 } { 2 \\sqrt { t } } ( 1 - ( 2 h - 1 ) t ) ( 1 - t ) ^ { h - 2 } ( 1 - t ) ^ { - 2 k ( n - 1 / 2 ) } \\\\ & = \\frac { 1 } { 2 n - 1 } [ t ^ { n - 1 } ] ( 1 - ( 2 h - 1 ) t ) ( 1 - t ) ^ { - 2 k n + k + h - 2 } \\\\ & = \\frac { 1 } { 2 n - 1 } \\binom { ( 2 k + 1 ) n - k - h } { n - 1 } - \\frac { 2 h - 1 } { 2 n - 1 } \\binom { ( 2 k + 1 ) n - k - h - 1 } { n - 2 } \\\\ & = \\frac { k + 1 - h } { 2 k n - k - h + 1 } \\binom { ( 2 k + 1 ) n - k - h - 1 } { n - 1 } \\end{align*}"}
{"id": "5692.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) - J _ { 0 } ( \\alpha , \\beta , Z ' ) = < - c _ { 1 } ( \\xi ) + 2 \\mathrm { P D } ( \\Gamma ) , Z - Z ' > . \\end{align*}"}
{"id": "6338.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ n ( 2 + p _ \\ell - p _ { \\ell - 1 } ) ^ 2 \\leq n ^ { 3 / 2 } , \\end{align*}"}
{"id": "8661.png", "formula": "\\begin{align*} & P = R ^ 2 - 2 m S U \\\\ & Q = - 2 R S + 2 k S U + m U ^ 2 \\\\ & S ^ 2 + 2 R U - k U ^ 2 = 0 \\end{align*}"}
{"id": "6768.png", "formula": "\\begin{align*} \\begin{aligned} - f '' + f + 1 & \\geq 0 , \\\\ f - f ^ * & \\geq 0 , \\\\ ( - f '' + f + 1 = 0 ) \\vee ( f - f ^ * & = 0 ) , \\end{aligned} \\end{align*}"}
{"id": "2852.png", "formula": "\\begin{align*} \\ddot { y } _ R ( t ) = 8 s _ c ( p - 1 ) \\delta ( t ) + A _ R ( u ( t ) ) \\ge 4 s _ c ( p - 1 ) \\delta ( t ) , \\forall t \\in [ \\delta , \\tau ] , \\end{align*}"}
{"id": "759.png", "formula": "\\begin{align*} f ( x ) - f ( x ^ \\star ) & = \\int _ 0 ^ 1 \\tfrac { 1 } { \\lambda } \\langle \\nabla f ( \\lambda x + ( 1 - \\lambda ) x ^ \\star ) , \\lambda x + ( 1 - \\lambda ) x ^ \\star - x ^ \\star \\rangle d \\lambda \\\\ & \\geq \\int _ 0 ^ 1 \\tfrac { \\gamma } { \\lambda } \\left ( f ( \\lambda x + ( 1 - \\lambda ) x ^ \\star ) - f ( x ^ \\star ) + \\tfrac { \\mu _ s \\lambda ^ 2 } { 2 } \\| x - x ^ \\star \\| ^ 2 \\right ) d \\lambda \\\\ & \\geq \\tfrac { \\gamma \\mu _ s } { 4 } \\| x - x ^ \\star \\| ^ 2 , \\end{align*}"}
{"id": "1458.png", "formula": "\\begin{align*} \\overline { j ( \\alpha , z _ 1 ) } \\delta ( \\alpha z _ 1 , \\alpha z _ 2 ) j ( \\alpha , z _ 2 ) = \\delta ( z _ 1 , z _ 2 ) . \\end{align*}"}
{"id": "2053.png", "formula": "\\begin{align*} { \\rho } _ n = \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi + \\xi _ n , n \\in \\mathbf { N } . \\end{align*}"}
{"id": "9200.png", "formula": "\\begin{align*} \\forall y ^ \\sigma ( A _ 0 ( y ) \\rightarrow r [ s / x ^ \\rho ] = _ \\tau r [ t / x ^ \\rho ] ) \\equiv \\exists y ^ \\sigma A _ 0 ( y ) \\rightarrow r [ s / x ^ \\rho ] = _ \\tau r [ t / x ^ \\rho ] , \\end{align*}"}
{"id": "3384.png", "formula": "\\begin{align*} & \\widetilde { d } _ T ( f ) ( u _ 1 , \\cdots , u _ { p + 1 } ) \\\\ & = \\sum _ { 1 \\leq i < j \\leq p + 1 } ( - 1 ) ^ { i + j } f ( \\rho ( T u _ i ) u _ j - \\rho ( T u _ j ) u _ i , u _ 1 , \\cdots , \\widehat { u _ i } , \\cdots , \\widehat { u _ j } , \\cdots , u _ { p + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ { p + 1 } ( - 1 ) ^ { i + 1 } \\Big ( [ T u _ i , f ( u _ 1 , \\cdots , \\widehat { u _ i } , \\cdots , u _ { p + 1 } ) ] + T \\rho ( f ( u _ 1 , \\cdots , \\widehat { u _ i } , \\cdots , u _ { p + 1 } ) ) ( u _ i ) \\Big ) . \\end{align*}"}
{"id": "1472.png", "formula": "\\begin{align*} \\mathbf { d } z = \\delta ( z ) ^ { - n + 1 } \\prod _ { h \\leq k } [ ( i / 2 ) d z _ { h k } \\wedge d \\overline { z } _ { h k } ] , \\end{align*}"}
{"id": "6638.png", "formula": "\\begin{align*} I _ 1 = R _ 0 + O \\bigg ( \\frac { ( X C h k ) ^ { \\varepsilon } Q ^ { 2 } ( h , k ) } { C \\sqrt { h k } } \\bigg ) + O \\Big ( X ^ { \\varepsilon } Q ^ { \\frac { 3 } { 2 } } h ^ { \\varepsilon } k ^ { \\varepsilon } \\Big ) , \\end{align*}"}
{"id": "5932.png", "formula": "\\begin{align*} \\sum _ a X _ a ( J ^ a _ \\xi ) = 0 \\end{align*}"}
{"id": "4484.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } E _ { \\psi } ( x ) = 0 . \\end{align*}"}
{"id": "2168.png", "formula": "\\begin{align*} [ u ] _ { ( \\alpha , G ) } = \\inf \\bigg { \\{ } \\lambda > 0 : \\ \\overline { \\rho } ( \\alpha ; \\frac { u } { \\lambda } ) \\leq 1 \\bigg { \\} } . \\end{align*}"}
{"id": "858.png", "formula": "\\begin{align*} \\bar { \\Delta } ^ { \\rm P } _ { \\rm R e a c t i v e } = - 1 - \\tau _ { \\rm f } + \\mathbb { E } T ^ { \\rm R e a c } _ j + \\mathbb { E } \\tau ^ { \\rm R e a c } _ { V _ { j - 1 } } . \\end{align*}"}
{"id": "2235.png", "formula": "\\begin{align*} X _ m ^ { M , N } - X _ { m - 1 } ^ { M , N } + k A ^ 2 X _ m ^ { M , N } + k P _ N A F ( X _ m ^ { M , N } ) = P _ N \\Delta W _ m , \\ ; X _ 0 ^ { M , N } = P _ N X _ 0 , \\ ; \\end{align*}"}
{"id": "5686.png", "formula": "\\begin{align*} P ( m ) : = \\begin{cases} ( 2 , . . . , \\ , 2 ) & \\mathrm { i f } \\ , \\ , m \\ , \\ , \\mathrm { i s \\ , \\ , e v e n } , \\\\ ( 2 , . . . , \\ , 2 , \\ , 1 ) & \\mathrm { i f } \\ , \\ , m \\ , \\ , \\mathrm { i s \\ , \\ , o d d } , \\end{cases} \\end{align*}"}
{"id": "7596.png", "formula": "\\begin{align*} f ( x , y ) \\sim \\sum _ { j = 0 } ^ { \\infty } { \\vphantom { \\sum } } ' g _ j ( x ) T _ j ( y ) , \\mbox { w h e r e } ~ ~ ~ g _ j ( x ) = \\sum _ { i = 0 } ^ { \\infty } { \\vphantom { \\sum } } ' c _ { i , j } T _ i ( x ) . \\end{align*}"}
{"id": "2287.png", "formula": "\\begin{align*} \\lim _ { \\omega \\to 0 } \\norm { M _ \\omega f - f } _ 2 = \\lim _ { \\omega \\to 0 } \\norm { T _ \\omega \\widehat { f } - \\widehat { f } } _ 2 = 0 . \\end{align*}"}
{"id": "8246.png", "formula": "\\begin{align*} \\mathfrak { B } ^ J = \\{ w \\in \\mathfrak { B } _ n \\ , | \\ , 0 < w _ 1 < \\cdots < w _ { p _ 1 } , w _ { p _ 1 + 1 } < \\cdots < w _ { p _ 2 } , \\ldots , w _ { p _ k + 1 } < \\cdots < w _ { n } \\} . \\end{align*}"}
{"id": "2849.png", "formula": "\\begin{align*} | A _ R ( u ( t ) ) | & = \\Big | A _ R ( u ) - A _ R \\left ( e ^ { i ( t + \\theta ( t ) ) } Q ( x - X ( t ) ) \\right ) \\Big | \\\\ & \\lesssim e ^ { - C R _ 0 } \\delta ( t ) + \\delta ( t ) ^ 2 + \\delta ( t ) ^ { 2 p } , R \\ge R _ 0 + | X ( t ) | . \\end{align*}"}
{"id": "3887.png", "formula": "\\begin{align*} w _ 1 ( M ) & = ( a - 3 ) t _ 1 - t _ 2 - \\dots - t _ { b - 1 } \\\\ & + ( b - 3 ) s _ 1 - s _ 2 - \\dots - s _ { a - 1 } \\\\ & = ( a - 3 ) \\beta _ 1 + \\dots + ( a - b - 1 ) \\beta _ { b - 1 } + \\dots + ( a - b - 1 ) \\beta _ { n - 2 } + \\frac { 1 } { 2 } \\left ( a - b - 1 \\right ) \\left ( \\beta _ { n - 1 } + \\beta _ n \\right ) \\\\ & + ( b - 3 ) \\gamma _ 1 + \\dots + ( b - a - 1 ) \\gamma _ { a - 1 } + \\dots + ( b - a - 1 ) \\gamma _ { n - 2 } + \\frac { 1 } { 2 } ( b - a - 1 ) ( \\gamma _ { n - 1 } + \\gamma _ n ) . \\end{align*}"}
{"id": "9384.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n + 1 } S _ { 2 , \\lambda } ( n + 1 , k ) k ! x ^ { k - 1 } & = \\frac { d } { d x } \\sum _ { k = 1 } ^ { n + 1 } S _ { 2 , \\lambda } ( n + 1 , k ) ( k - 1 ) ! x ^ { k } \\\\ & = - \\frac { n \\lambda } { x } \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) k ! x ^ { k } + F _ { n , \\lambda } ( x ) + ( 1 + x ) F _ { n , \\lambda } ^ { \\prime } ( x ) , \\end{align*}"}
{"id": "5664.png", "formula": "\\begin{align*} \\theta ^ { - 1 } \\varphi \\theta & = ( \\tau , T ) ^ { - 1 } ( \\tau \\sigma \\tau ^ { - 1 } , R ) ( \\tau , T ) \\\\ & = ( \\tau ^ { - 1 } , ( T ^ { - 1 } ) ^ { \\tau ^ { - 1 } } ) ( \\tau \\sigma , R ^ { \\tau } T ) \\\\ & = ( \\sigma , ( T ^ { - 1 } ) ^ { \\sigma } R ^ { \\tau } T ) . \\end{align*}"}
{"id": "4202.png", "formula": "\\begin{align*} \\int _ M V \\ , d V _ g = 0 . \\end{align*}"}
{"id": "8956.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - s } \\left ( C _ p \\left | \\dot { \\tilde { \\xi } } ( s ) \\right | ^ q + f \\left ( \\tilde { \\xi } ( s ) \\right ) \\right ) d s = \\int _ 0 ^ { \\infty } e ^ { - s } f ( \\xi ( 0 ) ) d s = f ( \\xi ( 0 ) ) . \\end{align*}"}
{"id": "2185.png", "formula": "\\begin{align*} 0 = J ^ { ' } ( \\widehat { u } ) + \\lambda _ 0 \\varphi ^ * _ { \\widehat { u } } \\ \\ \\ ( W ^ { \\alpha , G } ( \\mathbb { R } ^ d ) ) ^ * , \\ \\ \\ \\varphi ^ * _ { \\widehat { u } } \\in \\partial \\varphi ( \\widehat { u } ) . \\end{align*}"}
{"id": "6145.png", "formula": "\\begin{align*} q _ k = { n + k \\choose k } ( 0 \\leq k \\leq m ) . \\end{align*}"}
{"id": "1144.png", "formula": "\\begin{align*} E _ { 1 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\left ( I + \\mu ( s ) \\right ) ( J ^ E ( s ) - I ) d s : = I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
{"id": "6609.png", "formula": "\\begin{align*} \\mathcal { U } ^ 1 ( h , k ) = - \\mathcal { L } ^ 0 ( h , k ) + O \\big ( ( X h k ) ^ \\varepsilon X C \\big ) + O \\bigg ( Q \\frac { ( X C H K ) ^ { \\varepsilon } } { \\sqrt { H K } } \\bigg ) . \\end{align*}"}
{"id": "4817.png", "formula": "\\begin{align*} n = q , \\quad \\hat \\theta = \\frac { | 1 + J ' ( \\alpha , \\alpha ) | } { \\sqrt { 2 q ( q - 1 ) } } , \\end{align*}"}
{"id": "1875.png", "formula": "\\begin{align*} P _ n ( 1 ) = 2 ^ n E _ { n } , \\end{align*}"}
{"id": "1365.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\langle T _ { \\gamma ( t ) } G , \\gamma ' ( t ) \\rangle \\ 1 { \\gamma ( t ) \\in W ' } d t = 0 , \\end{align*}"}
{"id": "7312.png", "formula": "\\begin{align*} \\frac { 1 } { \\det ( 1 _ 3 - D A ) } & = \\frac { 1 } { 1 + X Z + Y Z + X Y } = \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k ( X Y + Y Z + Z X ) ^ k \\\\ & = \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k \\sum _ { a + b + c = k } \\frac { k ! } { a ! b ! c ! } X ^ { a + c } Y ^ { a + b } Z ^ { b + c } . \\end{align*}"}
{"id": "4018.png", "formula": "\\begin{align*} F ( \\mu ) = 0 , \\end{align*}"}
{"id": "6427.png", "formula": "\\begin{align*} \\big ( D _ E ^ + \\big ) = \\tfrac 1 4 \\big ( - \\sigma ( N ) + 2 \\chi ( M ) + 3 \\sigma ( M ) + 2 b _ 0 ( \\partial M ) + 2 b _ 2 ( \\partial M ) \\big ) , \\end{align*}"}
{"id": "3505.png", "formula": "\\begin{align*} D _ { 1 2 } & = \\frac { s _ 2 ( a t _ 3 ) ^ { 1 - s _ 1 } } { s _ 1 + s _ 3 - 1 } \\int _ 1 ^ { a t _ 3 } \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 + 1 } ( a t _ 3 + v ) ^ { s _ 3 } } d v \\\\ & + \\frac { s _ 2 s _ 3 } { s _ 1 + s _ 3 - 1 } \\int _ 1 ^ { a t _ 3 } \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 } } \\int _ { a t _ 3 } ^ \\infty \\frac { 1 } { u ^ { s _ 1 } ( u + v ) ^ { s _ 3 + 1 } } d u d v \\\\ & = D _ { 1 2 1 } + D _ { 1 2 2 } , \\end{align*}"}
{"id": "702.png", "formula": "\\begin{align*} \\forall b \\in [ B ] , \\abs { \\pi _ b } = 2 \\quad i _ { \\pi _ b ( 1 ) } = i _ { \\pi _ b ( 2 ) } \\end{align*}"}
{"id": "4449.png", "formula": "\\begin{align*} ( x ) _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 4 ] ) , \\end{align*}"}
{"id": "3799.png", "formula": "\\begin{align*} D _ A = \\begin{bmatrix} D _ 1 \\\\ 0 _ { m - n , n } \\end{bmatrix} D _ B = \\begin{bmatrix} D _ 2 & 0 _ { n , m - n } & 0 _ { n , \\ell - m } \\\\ 0 _ { m - n , n } & I _ { m - n } & 0 _ { m - n , \\ell - m } \\end{bmatrix} . \\end{align*}"}
{"id": "2671.png", "formula": "\\begin{align*} B f ( w ) & = e ^ { - \\pi i \\xi \\cdot \\eta } e ^ { \\frac { \\pi } { 2 } | w | ^ 2 } V _ { g _ 0 } f ( \\xi , - \\eta ) \\\\ & = e ^ { - \\pi i \\xi \\cdot \\eta } e ^ { \\frac { \\pi } { 2 } | w | ^ 2 } e ^ { - \\pi i \\xi \\cdot ( - \\eta ) } \\langle f , \\rho ( \\xi , - \\eta , 0 ) g _ 0 \\rangle \\\\ & = e ^ { \\frac { \\pi } { 2 } | w | ^ 2 } \\langle f , \\rho ( \\overline { w } , 0 ) g _ 0 \\rangle . \\end{align*}"}
{"id": "4492.png", "formula": "\\begin{align*} b _ 1 = 0 . 1 0 3 8 , \\ b _ 2 = 0 . 2 5 7 3 , \\ \\ b _ 3 = 9 . 3 6 7 5 . \\end{align*}"}
{"id": "1289.png", "formula": "\\begin{align*} \\Lambda _ { ( n , m ) } ( M , \\Gamma ) : = \\{ \\ , \\ , \\alpha \\in \\Lambda ( M , \\Gamma ) \\ , | \\ , ( E ( \\alpha ) , H ( \\alpha ) ) = ( n , m ) \\ , \\} \\end{align*}"}
{"id": "2396.png", "formula": "\\begin{align*} \\norm { A f - \\sum _ { \\gamma \\in F } A e _ \\gamma } _ { \\mathcal { B } _ 2 } = \\norm { A ( f - \\sum _ { \\gamma \\in F } e _ \\gamma ) } _ { \\mathcal { B } _ 2 } \\leq \\norm { A } _ { o p } \\norm { f - \\sum _ { \\gamma \\in F } e _ \\gamma } _ { \\mathcal { B } _ 1 } < \\varepsilon , \\end{align*}"}
{"id": "8083.png", "formula": "\\begin{align*} a ( z ) b ( w ) \\sim \\sum _ { j = 0 } ^ { N - 1 } \\frac { c ^ j ( w ) } { ( z - w ) ^ { j + 1 } } . \\end{align*}"}
{"id": "1200.png", "formula": "\\begin{align*} \\dim _ H \\Big ( \\limsup _ { B \\in \\mathcal { Q } } B ^ { \\delta } \\cap K _ { 1 / 3 } ^ { ( 0 ) } \\Big ) = \\min \\left \\{ \\frac { \\log 2 } { \\log 3 } , \\frac { 1 } { \\delta } \\right \\} . \\end{align*}"}
{"id": "5192.png", "formula": "\\begin{align*} X = \\begin{cases} \\alpha , & \\ 0 . 5 , \\\\ \\epsilon & \\ 0 . 5 , \\end{cases} \\end{align*}"}
{"id": "8105.png", "formula": "\\begin{align*} \\left [ \\mathfrak { A } _ c i _ 1 ( \\mathfrak { A } _ c ( \\mathcal { I } _ 1 ) ) , \\mathfrak { A } _ c i _ 2 ( \\mathfrak { A } _ c ( \\mathcal { I } _ 2 ) ) \\right ] = 0 . \\end{align*}"}
{"id": "6046.png", "formula": "\\begin{align*} ( 1 - x ^ 2 ) Y '' + [ - \\alpha + \\beta - ( \\alpha + \\beta + 2 ) x ] \\ , Y ' + ( n + 1 ) ( n - \\alpha - \\beta ) Y = 0 , \\end{align*}"}
{"id": "7394.png", "formula": "\\begin{align*} \\forall ( t , u ) \\in [ 0 , T ] \\times \\mathbb { R } , G ( t , u ) = \\sum _ { j = 0 } ^ { k } t ^ { j } G _ j ( u ) . \\end{align*}"}
{"id": "7805.png", "formula": "\\begin{align*} ( - 1 ) ^ { m + 1 } \\frac { ( x v ) ^ m } { m } = ( - 1 ) ^ { m + 1 } ( m - 1 ) ! \\cdot x ^ m \\cdot \\frac { v ^ m } { m ! } . \\end{align*}"}
{"id": "5737.png", "formula": "\\begin{align*} \\begin{aligned} ( \\lfloor \\frac { a - 1 } { 2 } \\rfloor + \\lfloor \\frac { j - l } { 2 } \\rfloor + 1 & - \\lceil \\frac { p - 1 } { 2 } \\rceil ) + \\frac { a } { 2 } + l - a + p \\\\ & \\geq \\lfloor \\frac { j - l } { 2 } \\rfloor + l + \\lceil \\frac { p } { 2 } \\rceil \\\\ & \\geq \\lfloor \\frac { j - l } { 2 } \\rfloor + l + \\lceil \\frac { j - l + 1 } { 2 } \\rceil \\\\ & \\geq j , \\\\ \\end{aligned} \\end{align*}"}
{"id": "2818.png", "formula": "\\begin{align*} \\delta ^ * = O ( \\delta + \\delta \\delta ^ * ) , \\forall t \\in D _ { \\delta _ 0 } , \\end{align*}"}
{"id": "8493.png", "formula": "\\begin{align*} \\begin{array} { l l } f _ 0 & = 1 , \\mbox { a n d } f _ k = z f _ 0 + z \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } g _ \\ell + z \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } h _ \\ell , k \\geq 1 , \\\\ g _ k & = z f _ { k + 1 } + z h _ { k + 1 } , k \\geq 0 , \\\\ h _ k & = z f _ k + z g _ k , k \\geq 0 . \\end{array} \\end{align*}"}
{"id": "4924.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\eta _ t \\norm { \\nabla f ( \\bar x _ t ) } ^ 2 & \\leq f ( x _ 1 ) - f ( x _ { T + 1 } ) + 2 L \\sum _ { t = 1 } ^ T { \\eta _ t ^ 2 } \\norm { \\nabla f ( \\bar x _ t ) } ^ 2 \\\\ \\end{align*}"}
{"id": "1709.png", "formula": "\\begin{align*} \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ t ) } \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } \\sum \\limits _ { l = t } ^ \\infty \\sum \\limits _ { m = 0 } ^ \\infty d _ 0 ( W _ { l , m } , \\ , l _ q ^ { \\nu _ { l , m } } ) . \\end{align*}"}
{"id": "6946.png", "formula": "\\begin{align*} \\eta ^ 2 ( E ) = \\eta _ { \\rm r e s } ^ 2 ( E ) + \\eta _ { \\rm l o s s } ^ 2 ( E ) + \\sum _ { k = 1 } ^ 6 \\eta _ { { \\rm c o e f } , k } ^ 2 ( E ) + \\sum _ { k = 1 } ^ 2 \\eta _ { { \\rm r h s } , k } ^ 2 ( E ) \\ , , \\end{align*}"}
{"id": "2679.png", "formula": "\\begin{align*} \\widehat { g } ( \\omega ) = C \\ , e ^ { - \\gamma \\omega ^ 2 } e ^ { - 2 \\pi i \\delta \\omega } \\prod _ { k = 1 } ^ \\infty \\frac { e ^ { 2 \\pi i \\delta _ k \\omega } } { 1 + 2 \\pi i \\delta _ k \\omega } , \\end{align*}"}
{"id": "7913.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\sup _ { u , v \\in C _ { R + 1 } } | K _ { x , L } ( u , v ) - K _ x ( u - v ) | = 0 , \\end{align*}"}
{"id": "7886.png", "formula": "\\begin{align*} P _ d ( x ) = \\sum _ { | J | = d } \\sqrt { \\binom { d } { J } } a _ { J } x ^ J , \\end{align*}"}
{"id": "5396.png", "formula": "\\begin{align*} R _ \\mu ( n , \\alpha , \\varepsilon , \\delta , T , f ) = \\inf \\{ \\sum _ { i } e ^ { - \\alpha n + f _ { n } ( x _ i ) } : \\mu \\big ( \\cup _ { i } B _ { n } ( x _ i , \\varepsilon ) \\big ) \\geq 1 - \\delta \\} , \\end{align*}"}
{"id": "8534.png", "formula": "\\begin{align*} u \\left ( t , x \\right ) = \\frac { e ^ { i \\frac { x ^ { 2 } } { 4 t } } } { \\sqrt { - 2 i t } } ( \\mathcal { F } f ) \\left ( t , - \\frac { x } { 2 t } \\right ) + \\mathcal { O } \\big ( { \\big \\| \\partial _ { k } ( \\mathcal { F } f ) ( t ) \\big \\| } _ { L _ k ^ 2 } \\ , t ^ { - \\frac { 3 } { 4 } } \\big ) , t \\geq 1 , \\end{align*}"}
{"id": "4384.png", "formula": "\\begin{align*} b ( \\tau ) \\partial _ \\tau v = \\partial _ { \\xi } ^ 2 v + \\frac { d + 1 } { \\xi } \\partial _ \\xi v - 3 ( d - 2 ) ( 2 Q + \\xi ^ 2 Q ^ 2 ) v + B ( v ) + \\theta ( \\tau ) \\Lambda _ \\xi Q + \\theta ( \\tau ) \\Lambda _ \\xi v , \\end{align*}"}
{"id": "5987.png", "formula": "\\begin{align*} \\langle \\ ! \\langle C _ { n } ^ { \\lambda , \\beta } , Q _ { m } ^ { \\pi _ { \\lambda , \\beta } } \\rangle \\ ! \\rangle _ { \\pi _ { \\lambda , \\beta } } & = \\int _ { \\mathbb { R } } m ! \\sum _ { k = m } ^ { \\infty } \\frac { S ( k , m ) } { k ! } \\left ( \\nabla ^ { k } C _ { n } ^ { \\lambda , \\beta } ( x ) \\right ) \\ , \\mathrm { d } \\pi _ { \\lambda , \\beta } ( x ) \\\\ & = m ! \\sum _ { k = m } ^ { \\infty } \\frac { S ( k , m ) } { k ! } k ! s ( n , k ) \\\\ & = m ! \\sum _ { k = m } ^ { n } s ( n , k ) S ( k , m ) \\\\ & = n ! \\delta _ { n , m } . \\end{align*}"}
{"id": "1555.png", "formula": "\\begin{align*} \\mathcal { B } = \\{ z \\in M _ n ( \\C ) : 1 - z ^ { \\ast } z > 0 \\} , \\end{align*}"}
{"id": "6309.png", "formula": "\\begin{align*} f _ { Y _ i } ( y ) & = \\frac { f _ { | \\hat { h } _ i | } ( y ) } { 1 - F _ { | \\hat { h } _ i | } ( \\sqrt { \\mu _ i / \\vartheta } ) } = 2 y e ^ { \\mu _ i / \\vartheta _ i - y ^ 2 } \\end{align*}"}
{"id": "3496.png", "formula": "\\begin{align*} & \\int _ c ^ x ( f ( u , u ) - f ( u , c ) ) d u + \\int _ c ^ b ( f ( u , x ) - f ( u , c ) ) d u \\\\ & = \\int _ c ^ x f ( u , u ) d u + \\int _ c ^ b f ( u , x ) d u - \\int _ c ^ b f ( u , c ) d u \\\\ & = \\int _ c ^ x f ( u , u ) d u + F _ 2 ( x ) - F _ 2 ( c ) . \\end{align*}"}
{"id": "11.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( h _ n ^ j h _ n ^ k ) ^ { - \\frac { 1 } { 2 } } \\Big | B _ n ^ j ( R ) \\cap B _ n ^ k ( R ) \\Big | = 0 , \\end{align*}"}
{"id": "8926.png", "formula": "\\begin{align*} C ^ q ( U , A ) = \\varinjlim _ { A _ 1 \\sqcup \\cdots \\sqcup A _ n = U } C ^ q _ { A _ 1 , \\ldots , A _ n } ( U , A ) \\end{align*}"}
{"id": "9265.png", "formula": "\\begin{align*} \\norm { w } = \\frac { \\norm { x - z } } { \\vert \\gamma _ n \\vert } \\leq \\frac { \\norm { x } + \\norm { z } } { \\vert \\gamma _ n \\vert } \\leq \\frac { 2 L } { \\vert \\gamma _ n \\vert } . \\end{align*}"}
{"id": "8913.png", "formula": "\\begin{align*} \\varphi : U & \\to A \\\\ x & \\mapsto \\begin{cases} \\varphi _ 1 ( x ) & x \\in U _ 1 \\\\ \\varphi _ 2 ( x ) & x \\in U _ 2 \\setminus U _ 1 \\\\ \\vdots & \\vdots \\\\ \\varphi _ n ( x ) & x \\in U _ n \\cap U _ 1 ^ c \\cap \\cdots \\cap U _ { n - 1 } ^ c \\\\ 0 & x \\in U _ 1 ^ c \\cap \\cdots \\cap U _ n ^ c \\end{cases} \\end{align*}"}
{"id": "8062.png", "formula": "\\begin{align*} \\left \\langle m _ { \\Lambda } ^ { * } E ^ { i j } _ 0 , f \\otimes g \\right \\rangle \\equiv \\Lambda ^ { - 2 } \\left \\langle E ^ { i j } _ 0 , ( { m _ { \\Lambda } } _ * f ) \\otimes ( { m _ { \\Lambda } } _ * g ) \\right \\rangle = \\left \\langle E ^ { i j } _ 0 , f \\otimes g \\right \\rangle \\end{align*}"}
{"id": "2377.png", "formula": "\\begin{align*} \\mathbf { F } = \\{ v _ k = ( v _ { k , 1 } , \\ldots , v _ { k , d } ) \\mid k = 1 , \\ldots , K \\} , \\end{align*}"}
{"id": "2059.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { q , \\lambda , t } = \\int _ { \\{ q \\Psi < - t \\} } | f | ^ 2 e ^ { - \\varphi _ 0 } + \\int _ { \\{ 0 > q \\Psi \\geq - t \\} } | f | ^ 2 e ^ { - \\varphi _ 0 - \\lambda ( q \\Psi + t ) } . \\end{align*}"}
{"id": "1170.png", "formula": "\\begin{align*} \\left ( \\frac { d \\psi } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 \\psi \\right ) \\psi ^ { - 1 } = \\left [ \\frac { d m ^ { P C } _ { \\eta } } { d \\zeta } + m ^ { P C } _ { \\eta } \\frac { i \\nu } { \\zeta } \\sigma _ 3 \\right ] \\left ( m ^ { P C } _ { \\eta } \\right ) ^ { - 1 } + \\frac { i \\zeta } { 2 } \\left [ \\sigma _ 3 , m ^ { P C } _ { \\eta } \\right ] \\left ( m ^ { P C } _ { \\eta } \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "380.png", "formula": "\\begin{align*} g = \\exp ( t ^ { s } \\xi _ { s } ) \\dots \\exp ( t ^ { 1 } \\xi _ 1 ) . \\end{align*}"}
{"id": "5437.png", "formula": "\\begin{align*} a _ { \\inf } > \\begin{cases} \\frac { \\mu \\chi ^ 2 } { 4 } { \\rm i f } 0 < \\chi \\leq 2 \\cr \\mu ( \\chi - 1 ) { \\rm i f } \\chi > 2 . \\end{cases} \\end{align*}"}
{"id": "2459.png", "formula": "\\begin{align*} A ^ T C = C ^ T A , \\ B ^ T D = D ^ T B A ^ T D - C ^ T B = I , \\end{align*}"}
{"id": "209.png", "formula": "\\begin{align*} ( T _ t ^ \\alpha ) ^ * ( f ) = P ^ { \\nu _ \\alpha } _ t ( f ) . \\end{align*}"}
{"id": "4243.png", "formula": "\\begin{align*} 0 = H _ x = { } & u _ { e e x } \\\\ = { } & u _ { x x x } + \\tau ^ 2 u _ { y y x } + 2 \\tau u _ { x y x } \\\\ = { } & ( 1 - \\tau ^ 2 ) u _ { x x x } - 2 \\tau u _ { y y y } - 2 \\tau f ' ( u ) u _ y . \\end{align*}"}
{"id": "9428.png", "formula": "\\begin{align*} u = - v ( r , t ) x , \\end{align*}"}
{"id": "955.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + c u & = 0 \\Omega ^ + . \\end{align*}"}
{"id": "8399.png", "formula": "\\begin{align*} \\mathbf { v } _ { i } ( \\rho , x , t ) & = \\tilde { \\mathbf { v } } _ { i } ( \\rho , x , t ) + \\eta ( \\rho ) d _ { \\Gamma } ( x , t ) \\hat { \\mathbf { v } } _ { i } ( x , t ) , i = 0 , 1 , 2 , \\\\ p _ { j } ( \\rho , x , t ) & = \\tilde { p } _ { j } ( \\rho , x , t ) + \\eta ( \\rho ) d _ { \\Gamma } ( x , t ) \\hat { p } _ { j } ( x , t ) , j = - 1 , 0 , 1 , \\end{align*}"}
{"id": "657.png", "formula": "\\begin{align*} \\abs { \\frac { \\abs { f ( x ) - g ( x ) } } { \\abs { x + 1 } } - \\abs { \\alpha } } \\ & = \\ \\abs { \\frac { \\abs { f ( x ) - g ( x ) } } { x + 1 } - \\alpha } \\\\ [ 1 1 p t ] & \\leq \\ \\abs { \\frac { f ( x ) - g ( x ) } { x + 1 } - \\alpha } \\\\ [ 1 1 p t ] & < \\ \\frac { 1 } { x + 1 } . \\end{align*}"}
{"id": "691.png", "formula": "\\begin{align*} n _ 1 , \\ldots , n _ L = n \\gg 1 , \\end{align*}"}
{"id": "8307.png", "formula": "\\begin{align*} d ( D ) = \\sqrt { \\lambda D / N } , \\end{align*}"}
{"id": "4838.png", "formula": "\\begin{align*} [ q _ 1 , q _ 2 ] : = \\{ x \\in \\R \\mid Q ( x ) = 0 \\} \\end{align*}"}
{"id": "5809.png", "formula": "\\begin{align*} \\tilde { \\mathbf { y } } _ k [ t ] = \\mathbf { D } \\mathbf { y } _ k [ t ] . \\end{align*}"}
{"id": "2692.png", "formula": "\\begin{align*} \\norm { K } _ { H . S . } = \\norm { k } _ 2 . \\end{align*}"}
{"id": "4379.png", "formula": "\\begin{align*} \\partial _ \\tau \\epsilon _ 0 \\| \\phi _ 0 \\| ^ 2 _ \\rho + \\epsilon _ 0 ( \\tau ) \\partial _ \\tau \\| \\phi _ 0 \\| ^ 2 _ { \\rho } = \\langle \\partial _ \\tau \\epsilon , \\phi _ 0 \\rangle _ { L ^ 2 _ \\rho } + \\langle \\epsilon , \\partial _ \\tau \\phi _ 0 \\rangle _ { L ^ 2 _ \\rho } . \\end{align*}"}
{"id": "7043.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\| b _ j \\| _ b ^ 2 = \\sum _ { j = 1 } ^ \\infty \\| q _ j \\| _ { H ^ 2 } ^ 2 + \\sum _ { j = 1 } ^ \\infty \\Big \\| \\Big ( q - \\sum _ { k = 1 } ^ { L } q _ k \\widetilde { \\phi _ k } \\Big ) e _ j \\Big \\| _ { H ^ 2 } ^ 2 . \\end{align*}"}
{"id": "571.png", "formula": "\\begin{align*} \\big \\{ ( x , y , z ) \\ x = y = z \\big \\} . ] \\end{align*}"}
{"id": "1751.png", "formula": "\\begin{align*} & \\mathcal { A } ^ c _ G ( X , E ) \\\\ \\cong & \\big \\{ \\Phi : G \\to \\Psi ^ { - \\infty } ( S , E | _ S ) , K \\times K \\big \\} . \\end{align*}"}
{"id": "6445.png", "formula": "\\begin{align*} \\Delta u + k ^ 2 ( 1 + \\eta ) u = 0 \\end{align*}"}
{"id": "7896.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sum _ { 1 \\leq i _ 1 , i _ 2 \\leq m } \\frac { 1 } { | B _ R | } \\int _ { B _ R } ( k _ { i _ 1 i _ 2 } ( x ) ) ^ 2 d x = 0 , \\end{align*}"}
{"id": "8278.png", "formula": "\\begin{align*} \\pi ( M _ p ) = \\sum _ { p \\leq x } \\mu _ { P } ( p , x ) F _ { f ( x ) } = \\sum _ { q \\in Q } \\Big ( \\sum _ { p \\leq x , \\ f ( x ) = q } \\mu _ { P } ( p , x ) \\Big ) F _ q . \\end{align*}"}
{"id": "3663.png", "formula": "\\begin{align*} B _ 2 : = 2 + \\delta , B _ 3 : = 2 \\left [ \\frac { k } { m } + \\delta \\left ( \\frac { k } { m } - \\frac { 1 } { 2 } \\right ) - \\frac { \\delta ^ 2 } { 4 } \\right ] . \\end{align*}"}
{"id": "4531.png", "formula": "\\begin{align*} T ( x ; k , r ) : = \\sum _ { m \\leq x , \\ , m \\equiv k \\ , ( m o d \\ , r ) } a ( m ) \\end{align*}"}
{"id": "8054.png", "formula": "\\begin{align*} \\Psi _ { ( \\Sigma , U ) } ( f ) & = \\Psi _ { ( \\Sigma , U ) } \\left ( \\mathfrak { D } ^ { ( \\mu ) } ( i , i _ U ) ( f | _ { \\Sigma } ) \\right ) \\\\ & = \\mathfrak { P } _ { \\ell } ( i , i _ U ) \\left ( \\Psi _ { ( \\Sigma _ f , U _ f ) } ( f | _ { \\Sigma } ) \\right ) . \\end{align*}"}
{"id": "9395.png", "formula": "\\begin{align*} \\tau _ t ( a _ 1 \\cdots a _ k ) = \\tau ( a _ 1 \\ldots a _ k ) + t \\cdot \\tau ' ( a _ 1 \\ldots a _ k ) + o ( t ) . \\end{align*}"}
{"id": "6415.png", "formula": "\\begin{align*} \\phantom { \\quad x , y , z , w \\in V . } \\langle R _ { x , y } ( z ) , w \\rangle = \\langle R ( x \\wedge y ) , w \\wedge z \\rangle , \\quad x , y , z , w \\in V . \\end{align*}"}
{"id": "2479.png", "formula": "\\begin{align*} \\L ^ \\circ = r ^ { - 1 } J S ^ { - T } \\Z ^ { 2 d } = r ^ { - 1 } \\underbrace { J S ^ { - T } J ^ { - 1 } } _ { = S } \\Z ^ { 2 d } = r ^ { - 2 } r S \\Z ^ { 2 d } = r ^ { - 2 } \\L . \\end{align*}"}
{"id": "7414.png", "formula": "\\begin{align*} + \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega _ 1 ( x ) } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta ) - f ( \\eta ^ { x + 1 , x + 1 + r } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "2480.png", "formula": "\\begin{align*} \\sum _ { \\l ^ \\circ \\in \\L ^ \\circ } A g ( \\l ^ \\circ ) = - \\sum _ { \\l ^ \\circ \\in \\L ^ \\circ } A g ( \\l ^ \\circ ) = 0 . \\end{align*}"}
{"id": "7800.png", "formula": "\\begin{align*} ( x , y ) _ { F , p ^ n } : = \\frac { \\rho _ { F } ( x ) ( \\sqrt [ p ^ n ] { y } ) } { \\sqrt [ p ^ n ] { y } } \\in \\mu _ { p ^ n } \\ \\ ( x , y \\in F ^ { \\times } ) , \\end{align*}"}
{"id": "3328.png", "formula": "\\begin{align*} e '' _ { - f _ { \\lambda + 2 } } = 2 f _ { \\lambda + 2 } - g _ { \\lambda + 2 } + 1 = 2 f _ { \\lambda + 2 } - ( f _ { \\lambda + 2 } - f _ { \\lambda + 1 } ) + 1 = f _ { \\lambda + 1 } + f _ { \\lambda + 2 } + 1 . \\end{align*}"}
{"id": "8122.png", "formula": "\\begin{align*} T B _ { 1 4 } ( f ) = 2 T B _ { 6 } ( f ) . \\end{align*}"}
{"id": "57.png", "formula": "\\begin{align*} M = \\bigoplus _ { j = 0 } ^ 1 M _ j ( \\pi ^ j ) , \\end{align*}"}
{"id": "8505.png", "formula": "\\begin{align*} B ^ { \\left [ n + 1 \\right ] } = \\left \\langle \\bigcup _ { i = 1 } ^ n B ^ { [ i ] } * B ^ { [ n - i ] } \\right \\rangle _ + \\end{align*}"}
{"id": "3765.png", "formula": "\\begin{align*} F ( z _ 1 , z _ 2 ) = ( \\l _ 1 z _ 1 z _ 2 ^ s + P ( z _ 2 ) + \\lambda _ 2 z _ 2 ^ { \\frac { s k } { k - 1 } } , z _ 2 ^ k ) , \\end{align*}"}
{"id": "2152.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { E _ { n , k } ( \\tau ( A ^ { ( n ) } _ { c _ n k n , n - s , l } ) | \\mathcal { W A } ^ { ( n , k ) } _ { M _ n , n - s , l } ) } { n k } = \\frac { 1 - c } { k + 1 } . \\end{align*}"}
{"id": "3068.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = j y _ 2 \\ , , y _ 2 ^ \\prime = x _ 1 \\ , , x _ 2 ^ \\prime = y _ 1 \\ , , y _ 1 ^ \\prime = j ^ 2 x _ 2 \\ , . \\end{align*}"}
{"id": "6345.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } \\left ( - \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } \\right ) \\frac { x ^ s } { s } \\ , d s + O ^ * \\left ( \\sum _ { n = 1 } ^ \\infty \\left ( \\frac { x } { n } \\right ) ^ c \\Lambda ( n ) \\min \\left ( 0 . 5 0 1 , \\frac { 1 } { \\pi T \\left | \\log \\frac { x } { n } \\right | } \\right ) \\right ) . \\end{align*}"}
{"id": "5759.png", "formula": "\\begin{align*} \\frac { \\partial z ^ { k } } { \\partial x ^ j } = \\left ( \\left \\{ \\frac { \\partial x ^ s } { \\partial z ^ t } \\right \\} \\right ) / \\operatorname { d e t } ( B ) , \\end{align*}"}
{"id": "6891.png", "formula": "\\begin{align*} a _ i = ( - 1 ) ^ i q ^ { i ^ 2 } { k \\choose k - i } _ q S _ k ^ i ( V , V ' ) . \\end{align*}"}
{"id": "9029.png", "formula": "\\begin{align*} & - \\nabla \\cdot ( \\epsilon ( x ) \\psi ) = \\tilde { f } ( x ) + \\sum _ { i = 1 } ^ s z _ i \\rho _ i , \\quad , x \\in \\Omega , \\\\ & \\psi = 0 , x \\in \\partial \\Omega , \\end{align*}"}
{"id": "722.png", "formula": "\\begin{align*} \\kappa _ { ( 0 0 ) ( 0 0 ) } ^ { ( \\ell ) } = \\frac { 2 } { 3 n \\ell a ^ 2 } ( 1 + O ( \\ell ^ { - 1 } ) ) . \\end{align*}"}
{"id": "4503.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < \\gamma < T \\\\ \\beta \\le 1 / 2 } } \\frac { 1 } { \\gamma } = \\sum _ { \\substack { 0 < \\gamma < T } } \\frac { 1 } { \\gamma } - \\sum _ { \\substack { H _ 0 \\le \\gamma < T \\\\ \\beta > 1 / 2 } } \\frac { 1 } { \\gamma } \\le \\sum _ { \\substack { 0 < \\gamma < T } } \\frac { 1 } { \\gamma } - \\sum _ { \\substack { H _ 0 \\le \\gamma < T \\\\ 1 / 2 < \\beta < \\sigma } } \\frac { 1 } { \\gamma } . \\end{align*}"}
{"id": "5917.png", "formula": "\\begin{align*} f ( x ) = x ^ 3 - ( n - 3 ) \\ , x ^ 2 - ( 2 n - 3 ) \\ , x - n + 1 + 4 s _ 2 t _ 1 ( s _ 1 + t _ 2 ) . \\end{align*}"}
{"id": "6908.png", "formula": "\\begin{align*} r = \\sqrt { P _ t } \\Bigl ( \\mathbf { g } ^ T \\boldsymbol { \\Theta } \\mathbf { H } + \\mathbf { h } _ d ^ T \\Bigr ) \\mathbf { w } s + n . \\end{align*}"}
{"id": "2772.png", "formula": "\\begin{align*} U ^ 1 ( x ) = e ^ { i \\theta _ 2 } Q ( x + x _ 2 ) \\ ; \\Rightarrow \\ ; u _ n ( x + x _ n ^ 1 ) = e ^ { i \\theta _ 2 } Q ( x + x _ 2 ) + r _ n ^ 1 ( x + x _ n ^ 1 ) . \\end{align*}"}
{"id": "4930.png", "formula": "\\begin{align*} u ( r , \\theta ) = ( r ^ { - \\alpha } - r ^ { \\alpha } ) \\mathrm { s i n } ( \\alpha \\theta ) \\end{align*}"}
{"id": "9194.png", "formula": "\\begin{align*} 0 & = \\norm { x - J ^ A _ { \\gamma } ( x + \\gamma u ) + \\gamma ( u - \\gamma ^ { - 1 } ( ( x + \\gamma u ) - J ^ A _ { \\gamma } ( x + \\gamma u ) ) ) } \\\\ & \\geq \\norm { x - J ^ A _ { \\gamma } ( x + \\gamma u ) } . \\end{align*}"}
{"id": "2162.png", "formula": "\\begin{align*} - M \\left ( \\int _ { \\Omega } g ( \\vert \\nabla u \\vert ) d x \\right ) \\Delta _ g u = f ( u ) , \\ \\ \\Omega , \\end{align*}"}
{"id": "4097.png", "formula": "\\begin{align*} \\begin{aligned} ( \\widetilde \\sigma ( x ) - \\widetilde \\sigma ( y ) ) ^ 2 & = ( \\widetilde \\sigma ( x ) - \\widetilde \\sigma ( z ) + \\widetilde \\sigma ( z ) - \\widetilde \\sigma ( y ) ) ^ 2 \\\\ & \\leq ( 1 + \\delta _ { x , y } ) \\cdot ( \\widetilde \\sigma ( x ) - \\widetilde \\sigma ( z ) ) ^ 2 + \\Bigl ( 1 + \\frac { 1 } { \\delta _ { x , y } } \\Bigr ) \\cdot ( \\widetilde \\sigma ( z ) - \\widetilde \\sigma ( y ) ) ^ 2 , \\end{aligned} \\end{align*}"}
{"id": "478.png", "formula": "\\begin{align*} \\chi ( \\mathcal { I } \\mid _ { \\alpha \\beta } ^ { i j } ) : = \\left \\{ h ( \\mathcal { I } ) \\cdot h ( \\mathcal { I } _ { \\alpha \\beta } ^ { i j } ) , h ( \\mathcal { I } _ { \\alpha } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\beta } ^ { j } ) , h ( \\mathcal { I } _ { \\beta } ^ { i } ) \\cdot h ( \\mathcal { I } _ { \\alpha } ^ { j } ) \\right \\} \\end{align*}"}
{"id": "6568.png", "formula": "\\begin{align*} \\tau _ { E _ s } ( m ) = m ^ { - s } \\tau _ E ( m ) . \\end{align*}"}
{"id": "2102.png", "formula": "\\begin{align*} \\xi ^ \\alpha \\cdot [ x _ 0 : \\dots : x _ n ] = \\left [ \\xi ^ { \\alpha m / q _ 0 } x _ 0 : \\dots : \\xi ^ { \\alpha m / q _ n } x _ n \\right ] , \\end{align*}"}
{"id": "4188.png", "formula": "\\begin{align*} \\max _ { x \\in L _ 0 } h _ i ( x ) - \\min _ { y \\in L _ 0 } h _ i ( y ) & = \\left | \\int _ 0 ^ { d _ { L _ 0 } ( x _ i , y _ i ) } ( d h _ i ) _ { \\gamma _ i ( t ) } ( \\dot { \\gamma } _ i ( t ) ) d t \\right | \\\\ & \\leq | | d h _ i | | \\int _ 0 ^ { d _ { L _ 0 } ( x _ i , y _ i ) } | \\dot { \\gamma } _ i ( t ) | d t \\\\ & \\leq \\mathrm { D i a m } ( L _ 0 ) | | d h _ i | | . \\end{align*}"}
{"id": "6107.png", "formula": "\\begin{align*} ( l - 1 ) b _ 2 + ( t _ \\sigma + 1 ) b _ 1 = m b _ 1 s _ \\sigma + ( q - l p ) \\pi \\ + r _ { p , q } , \\end{align*}"}
{"id": "2688.png", "formula": "\\begin{align*} f ( x ) = \\mathcal { O } ( e ^ { - a \\pi x ^ 2 } ) \\widehat { f } ( \\omega ) = \\mathcal { O } ( e ^ { - b \\pi \\omega ^ 2 } ) , \\end{align*}"}
{"id": "3861.png", "formula": "\\begin{align*} P _ U = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , L _ U = \\begin{bmatrix} 1 & 0 \\\\ h & 1 \\end{bmatrix} , h \\in \\mathbb C . \\end{align*}"}
{"id": "7751.png", "formula": "\\begin{align*} b ( x ) = 1 \\textrm { o n } \\left [ \\frac { S _ 0 } { 8 } , \\frac { 3 S _ 0 } { 8 } \\right ] , \\ ; \\ ; b ( x ) = 0 \\textrm { o n } \\S \\setminus \\left [ 0 , \\frac { S _ 0 } { 2 } \\right ] , \\end{align*}"}
{"id": "5610.png", "formula": "\\begin{align*} \\Xi ^ { M _ c } : = \\Xi _ { \\mathcal { C } ^ c } / \\Xi _ { \\mathcal { C } \\mathcal { C } ^ c } : = \\{ [ X ] : = X + \\Xi _ { \\mathcal { C } \\mathcal { C } ^ c } ~ | ~ X \\in \\Xi _ { \\mathcal { C } ^ c } \\} \\end{align*}"}
{"id": "8787.png", "formula": "\\begin{align*} M : = \\Bigl \\{ ( \\gamma _ i , \\delta _ i ) \\Bigm | \\gamma _ { i t } = \\prod _ { k \\in \\eta _ i ( t ) } \\delta _ { i k } \\prod _ { k \\notin \\eta _ i ( t ) } ( 1 - \\delta _ { i k } ) \\ ; t \\in \\{ 1 , \\ldots , 2 ^ { \\lceil \\log _ 2 l _ i \\rceil } \\} , \\ \\delta _ i \\in [ 0 , 1 ] ^ { \\lceil \\log _ 2 l _ i \\rceil } \\Bigr \\} . \\end{align*}"}
{"id": "2218.png", "formula": "\\begin{align*} \\| v \\nabla g ( u ) \\| & \\leq C \\| v \\nabla u \\| ( 1 + \\| u \\| _ V ) \\leq C \\| v \\| _ 1 ( 1 + \\| u \\| _ { \\iota } ^ 2 ) . \\end{align*}"}
{"id": "9225.png", "formula": "\\begin{align*} \\forall x ^ X , p ^ X , \\gamma ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land p = _ X J ^ A _ \\gamma x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "3404.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } f ( x ) d \\omega ( x ) = 0 . \\end{align*}"}
{"id": "2707.png", "formula": "\\begin{align*} V _ { g _ 0 } f ( 0 , - \\omega ) = B f ( i \\omega ) e ^ { - \\frac { \\pi } { 2 } \\omega ^ 2 } . \\end{align*}"}
{"id": "5190.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ N \\left ( \\frac { \\rho _ \\ell \\beta _ \\ell ^ \\pi } { 2 T _ { \\ell , \\pi } ^ { a v } } + \\frac { \\rho _ \\ell T _ { \\ell , \\pi } ^ { a v } } { 2 } + \\frac { c _ \\ell } { T _ { \\ell , \\pi } ^ { a v } } \\right ) , \\end{align*}"}
{"id": "1101.png", "formula": "\\begin{align*} \\zeta ( k ) = t ^ { 1 / 2 } \\sqrt { 2 g '' ( \\eta ) } ( k - \\eta ) , \\end{align*}"}
{"id": "6412.png", "formula": "\\begin{align*} \\left \\| f _ * \\left ( \\Phi _ 1 ( e ^ { i \\theta } ) , \\dots , \\Phi _ { n } ( e ^ { i \\theta } ) \\right ) \\right \\| = \\max _ { \\theta } \\{ | f _ * ( \\lambda _ 1 , \\dots , \\lambda _ { n } ) | : ( \\lambda _ 1 , \\dots , \\lambda _ { n } ) \\in \\sigma _ T ( \\Phi ( e ^ { i \\theta } ) , \\dots , \\Phi _ { n } ( e ^ { i \\theta } ) ) \\} . \\end{align*}"}
{"id": "6767.png", "formula": "\\begin{align*} \\mathbf A = \\left ( \\frac { 2 5 } { 1 2 } \\mathbf I - k \\mathbf L \\right ) , \\tilde { \\mathbf y ^ { n + 4 , ( \\ell ) } } = k \\tilde { \\mathbf b } ^ { n + 4 } + 4 \\tilde { \\mathbf V } ^ { n + 3 , ( \\ell ) } - 3 \\tilde { \\mathbf V } ^ { n + 2 , ( \\ell ) } + \\frac { 4 } { 3 } \\tilde { \\mathbf V } ^ { n + 1 , ( \\ell ) } - \\frac { 1 } { 4 } \\tilde { \\mathbf V } ^ { n , ( \\ell ) } . \\end{align*}"}
{"id": "3212.png", "formula": "\\begin{align*} k _ { P _ H } = \\frac { \\left ( \\frac { ( 1 + e ^ { a b } ) ^ { k _ 1 - 1 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { k _ 1 } } - \\frac { 1 } { 1 + e ^ { a b } } \\right ) ^ 2 } { \\frac { ( 1 + e ^ { a b } ) ^ { k _ 1 - 2 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { k _ 1 } } - \\frac { ( 1 + e ^ { a b } ) ^ { 2 k _ 1 - 2 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { 2 k _ 1 } } } , \\end{align*}"}
{"id": "8179.png", "formula": "\\begin{align*} N _ { d _ 0 } ( f , \\{ 1 \\} ) = - \\frac { 3 \\phi ( d _ 0 ) ^ 2 } { B } + \\frac { 3 } { B f } S , \\end{align*}"}
{"id": "8014.png", "formula": "\\begin{align*} \\chi \\circ d ( x ) & = ( - \\bar { \\rho } \\circ \\pi _ { \\ell } ^ { \\overline { \\Sigma } } \\circ d ( x ) , \\bar { \\rho } \\circ \\pi _ r ^ { \\overline { \\Sigma } } \\circ d ( x ) ) \\\\ & = ( - d _ { \\Sigma _ 0 } \\circ \\bar { \\rho } \\circ \\pi _ { \\ell } ^ { \\overline { \\Sigma } } ( x ) , d _ { \\Sigma _ 0 } \\circ \\bar { \\rho } \\circ \\pi _ { r } ^ { \\overline { \\Sigma } } ( x ) ) \\\\ & = d _ 0 \\circ \\chi ( x ) , \\end{align*}"}
{"id": "623.png", "formula": "\\begin{align*} P ( X ) \\ = \\ \\alpha _ 0 X ^ k + \\alpha _ 1 X ^ { k - 1 } + \\cdots + \\alpha _ { k - 1 } X + \\alpha _ k \\end{align*}"}
{"id": "6275.png", "formula": "\\begin{align*} \\widehat { K } _ { \\mathrm { u - c p t } } ^ \\mathbb { R } & = \\big ( | \\mathbf { s } ^ H \\mathbf { y } | ^ 2 / N ^ 2 - \\sigma ^ 2 / N \\big ) / \\bar { p } . \\end{align*}"}
{"id": "2916.png", "formula": "\\begin{align*} I _ { i , j } ^ { ( p ) } = \\frac { 2 n + 1 } { 6 n } + \\frac { R _ { i p } ( R _ { i p } - 1 ) } { 2 n ( n + 1 ) } + \\frac { R _ { j p } ( R _ { j p } - 1 ) } { 2 n ( n + 1 ) } - \\frac { \\max ( R _ { i p } , R _ { j p } ) } { n + 1 } . \\end{align*}"}
{"id": "6276.png", "formula": "\\begin{align*} \\mathrm { v a r } \\big [ \\widehat { K } ^ \\mathbb { R } _ { \\mathrm { u - c p t } } \\big ] & = ( K \\bar { p } + \\sigma ^ 2 / N ) ^ 2 / \\bar { p } ^ 2 = K ^ 2 + \\frac { 2 K } { N \\bar { \\gamma } } + \\frac { 1 } { N ^ 2 \\bar { \\gamma } ^ 2 } \\end{align*}"}
{"id": "6329.png", "formula": "\\begin{align*} \\nabla _ { \\gamma _ 1 , \\ldots , \\gamma _ 2 } f = \\partial _ { \\gamma _ 1 } \\nabla _ { \\gamma _ 2 , \\ldots , \\gamma _ n } f - \\sum _ { k = 1 } ^ n \\nabla _ { \\gamma _ 2 , \\ldots , \\gamma _ { k - 1 } , \\alpha , \\gamma _ { k + 1 } , \\ldots , \\gamma _ n } f \\Gamma ^ \\alpha _ { \\gamma _ 1 \\gamma _ k } \\end{align*}"}
{"id": "3545.png", "formula": "\\begin{align*} \\Gamma ( t ) = F ( \\Sigma , t ) . \\end{align*}"}
{"id": "2184.png", "formula": "\\begin{align*} \\varphi ( u ) & = \\langle J ^ { ' } ( u ) , u \\rangle = \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\frac { d x d y } { \\vert x - y \\vert ^ { d } } \\\\ & + \\int _ { \\mathbb { R } ^ { d } } g ( u ) u d x - \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , u ) u d x . \\end{align*}"}
{"id": "2940.png", "formula": "\\begin{align*} \\prod _ { \\ell = 1 } ^ { k } \\tilde I ^ { ( p _ { \\ell , 1 } ) } _ { i , i } \\tilde I ^ { ( p _ { \\ell , 2 } ) } _ { j , j } = \\Big ( \\prod _ { p \\in \\mathbf { p } _ { k , 1 } \\cap \\mathbf { p } _ { k , 2 } } \\tilde I ^ { ( p ) } _ { i , i } \\tilde I ^ { ( p ) } _ { j , j } \\Big ) \\cdot \\Big ( \\prod _ { p \\in \\mathbf { p } _ { k , 1 } \\setminus \\mathbf { p } _ { k , 2 } } \\tilde I ^ { ( p ) } _ { i , i } \\Big ) \\cdot \\Big ( \\prod _ { p \\in \\mathbf { p } _ { k , 2 } \\setminus \\mathbf { p } _ { k , 1 } } \\tilde I ^ { ( p ) } _ { j , j } \\Big ) , \\end{align*}"}
{"id": "2381.png", "formula": "\\begin{align*} C _ { \\mathbf { F } _ N } = \\begin{pmatrix} \\cos ( 2 \\pi 0 / N ) & \\sin ( 2 \\pi 0 / N ) \\\\ \\vdots & \\vdots \\\\ \\cos ( 2 \\pi ( N - 1 ) / N ) & \\sin ( 2 \\pi ( N - 1 ) / N ) \\end{pmatrix} . \\end{align*}"}
{"id": "7225.png", "formula": "\\begin{align*} X _ { s , t } ( x , \\Psi _ { s , t } ( x , v ) ) = x - ( t - s ) v . \\end{align*}"}
{"id": "7139.png", "formula": "\\begin{align*} 0 \\le R _ H ( s _ 1 , s _ 2 ) = \\frac { 1 } { 2 } ( s _ 1 ^ { 2 H } + s _ 2 ^ { 2 H } - \\vert s _ 1 - s _ 2 \\vert ^ { 2 H } ) \\le 1 , H < \\frac { 1 } { 2 } \\end{align*}"}
{"id": "2834.png", "formula": "\\begin{align*} e ^ { - i \\theta } u = ( 1 + \\alpha ) e ^ { i t } Q + e ^ { i t } h = e ^ { i t } Q + e ^ { i t } \\left ( \\alpha Q + h \\right ) . \\end{align*}"}
{"id": "230.png", "formula": "\\begin{align*} \\hat { \\gamma } _ { \\Sigma } ( \\xi ) = \\exp \\left ( - \\dfrac { \\langle \\xi ; \\Sigma ( \\xi ) \\rangle } { 2 } \\right ) . \\end{align*}"}
{"id": "8686.png", "formula": "\\begin{align*} \\lim _ { n \\longrightarrow \\infty } & \\frac { 1 } { n } { \\bf E } \\Big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\Big \\} = \\lim _ { n \\longrightarrow \\infty } \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } t r \\Big ( \\Gamma P _ t \\Gamma ^ T + D K _ { Z } D ^ T \\Big ) \\\\ = & t r \\Big ( \\Gamma P \\Gamma ^ T + D K _ { Z } D ^ T \\Big ) , \\ ; \\forall P _ 1 \\succeq 0 . \\end{align*}"}
{"id": "4173.png", "formula": "\\begin{align*} R e ( e ^ { i ( \\theta - \\frac { \\pi } { 2 } ) } A ) & = \\frac { e ^ { i ( \\theta - \\frac { \\pi } { 2 } ) } A + e ^ { - i ( \\theta - \\frac { \\pi } { 2 } ) } A ^ { \\ast } } { 2 } \\\\ & = \\frac { - i e ^ { i \\theta } A + i e ^ { - i \\theta } A ^ { \\ast } } { 2 } \\\\ & = \\frac { e ^ { i \\theta } A - e ^ { - i \\theta } A ^ { \\ast } } { 2 i } \\\\ & = I m ( e ^ { i \\theta } A ) . \\end{align*}"}
{"id": "4809.png", "formula": "\\begin{align*} J ' ( \\bar \\alpha , \\alpha ) = \\sum _ { t \\neq 0 , 1 } \\alpha ( ( 1 - t ) / t ) = \\sum _ { y \\neq 0 , - 1 } \\alpha ( y ) = - \\alpha ( - 1 ) . \\end{align*}"}
{"id": "7521.png", "formula": "\\begin{align*} \\log \\Gamma ( s ) = \\left ( s - \\frac { 1 } { 2 } \\right ) \\log s - s - \\frac { 1 } { 2 } \\log ( 2 \\pi ) + o ( 1 ) \\end{align*}"}
{"id": "2056.png", "formula": "\\begin{align*} K ^ { \\varphi _ 0 } _ { \\xi , \\Psi , \\lambda } ( t ) = \\frac { 1 } { \\| f _ t \\| _ { \\lambda , t } ^ 2 } . \\end{align*}"}
{"id": "7362.png", "formula": "\\begin{align*} \\varphi ( r , t ) = \\inf \\big \\{ \\varphi ( \\gamma ( t ) , 0 ) \\mid \\gamma ( 0 ) = r , | \\dot { \\gamma } ( s ) | \\le \\pi \\min \\{ R ^ 2 , \\gamma ( s ) ^ 2 \\} \\ \\ s \\in [ 0 , t ] \\big \\} . \\end{align*}"}
{"id": "1243.png", "formula": "\\begin{align*} { { E } _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\rho , \\varepsilon } = \\left \\{ x \\in \\mathbb { R } ^ d : \\ \\underline \\dim ( \\mu , x ) \\in [ \\alpha , \\gamma ] \\forall r \\leq \\rho , \\ \\mu ( B ( x , r ) ) \\leq r ^ { \\underline { \\dim } ( \\mu , x ) - \\varepsilon } \\right \\} } , \\end{align*}"}
{"id": "8211.png", "formula": "\\begin{align*} F _ { k , k / 2 + 1 } = 1 ( k ) G _ { k , ( k + 1 ) / 2 } = 1 ( k ) . \\end{align*}"}
{"id": "2978.png", "formula": "\\begin{align*} A ( p _ 1 , q _ 1 , p _ 2 , q _ 2 , q _ 3 , q _ 4 ) = \\sum _ { \\mathbf i \\in \\mathcal J } \\Psi _ { \\mathbf i } \\cdot \\varphi _ 2 ( \\mathbf i _ { 1 : 4 } ) \\varphi _ 2 ( \\mathbf i _ { \\{ 1 , 2 , 5 , 6 \\} } ) \\varphi _ 2 ( \\mathbf i _ { \\{ 1 , 2 , 7 , 8 \\} } ) , \\end{align*}"}
{"id": "2857.png", "formula": "\\begin{align*} u ( \\cdot + x ( t _ n ) , t _ n ' ) = u ( \\cdot + x ( t _ n ) , t _ n + t _ n ' - t _ n ) \\to e ^ { i t _ 0 } e ^ { i \\theta _ 0 } Q ( \\cdot - x _ 0 ) . \\end{align*}"}
{"id": "4040.png", "formula": "\\begin{align*} \\left ( 1 - e ^ { - \\alpha _ { 1 , k } - i 2 k \\pi - i \\alpha _ { 2 , k } + O ( \\mod { k } ^ { - 1 } ) } \\right ) + O ( \\mod { k } ^ { - 1 } ) = 0 , \\end{align*}"}
{"id": "1383.png", "formula": "\\begin{align*} | ( E _ 1 - E _ 2 ) _ x | ( x ) = & | n _ 1 - n _ 2 - ( b _ 1 - b _ 2 ) | ( x ) \\\\ \\le & C \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } , x \\in [ 0 , \\delta _ 0 ) . \\end{align*}"}
{"id": "4441.png", "formula": "\\begin{align*} ( - 1 ) ^ { n + 1 } \\sum _ { m = 1 } ^ M f ( m ) B _ { 2 n + \\nu + 1 } \\left ( \\dfrac { m } { M } \\right ) \\end{align*}"}
{"id": "3908.png", "formula": "\\begin{align*} c ( f , p ) = \\lim _ { p \\to \\infty } \\frac { \\mathcal { S } _ p } { p } . \\end{align*}"}
{"id": "4117.png", "formula": "\\begin{align*} R & = \\frac { 2 ( 1 + t ^ 2 ) } { 1 - t ^ 2 } , \\\\ Q ^ \\prime & = \\frac { 4 ( 1 - 1 4 t ^ 2 + t ^ 4 ) } { ( 1 - t ^ 2 ) ^ 2 } , \\end{align*}"}
{"id": "3080.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = a x _ 2 \\ , , x _ 2 ^ \\prime = x _ 1 \\ , , y _ 1 ^ \\prime = y _ 2 \\ , , y _ 2 ^ \\prime = \\frac { 1 } { a } y _ 1 \\ , , \\end{align*}"}
{"id": "7514.png", "formula": "\\begin{align*} \\left | \\xi \\left ( \\frac { 1 } { 2 } - \\epsilon + i t \\right ) \\right | = \\left | \\xi \\left ( \\frac { 1 } { 2 } + \\epsilon - i t \\right ) \\right | = \\left | \\overline { \\xi \\left ( \\frac { 1 } { 2 } + \\epsilon + i t \\right ) } \\right | = \\left | \\xi \\left ( \\frac { 1 } { 2 } + \\epsilon + i t \\right ) \\right | \\end{align*}"}
{"id": "7894.png", "formula": "\\begin{align*} | | F ( x ) | | \\leq \\tau + \\tau \\cdot | | F | | _ { C ^ { 1 + \\beta } ( C _ { R + 1 / 2 } ) } = W \\tau , \\end{align*}"}
{"id": "221.png", "formula": "\\begin{align*} D ^ { \\alpha - 1 } \\left ( ( P _ t ^ { \\nu _ \\alpha } ) ^ * ( f ) \\right ) ( x ) + \\frac { 1 } { p _ \\alpha ( x ) } R ^ \\alpha \\left ( p _ \\alpha , ( P _ t ^ { \\nu _ \\alpha } ) ^ * ( f ) \\right ) ( x ) & = \\dfrac { - x e ^ { - \\alpha t } } { ( 1 - e ^ { - \\alpha t } ) } ( P ^ { \\nu _ \\alpha } _ t ) ^ { * } ( f ) ( x ) \\\\ & \\quad + \\dfrac { e ^ { - t } } { \\left ( 1 - e ^ { - \\alpha t } \\right ) } ( P ^ { \\nu _ \\alpha } _ t ) ^ { * } ( x f ) ( x ) , \\end{align*}"}
{"id": "8094.png", "formula": "\\begin{align*} \\overline { u } _ n ( x ) = \\frac { ( - 1 ) ^ n } { n ! } \\frac { d ^ n } { d x ^ n } \\left [ \\alpha \\delta ( x ) + \\mathrm { P V } \\left ( \\frac { 1 } { x } \\right ) \\right ] \\end{align*}"}
{"id": "1537.png", "formula": "\\begin{align*} = \\int _ { U _ t ( \\mathbb { A } ) P ^ t _ n ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\int _ { U _ t ( \\Q ) \\backslash U _ t ( \\mathbb { A } ) } \\sum _ { \\xi , \\gamma } \\phi ( \\tilde { \\tau } _ t ( \\xi n g \\times \\gamma h ) , s ) \\mathbf { f } ( n h ) \\mathbf { d } n \\mathbf { d } h . \\end{align*}"}
{"id": "2015.png", "formula": "\\begin{align*} N _ t ^ { R _ { \\alpha } \\nu } = \\alpha \\int _ 0 ^ t R _ { \\alpha } \\nu ( X _ s ) { \\rm d } s - A _ t ^ { \\nu } , t \\in [ 0 , \\zeta [ \\end{align*}"}
{"id": "8841.png", "formula": "\\begin{align*} & \\theta _ 1 ( { \\bf v } ; z ) : = \\sum _ { p \\geq 0 } \\theta _ { 1 , p } ( { \\bf v } ) z ^ p = - 2 e ^ { z v } \\sum _ { m \\geq 0 } \\Bigl ( \\gamma - \\frac 1 2 u + \\psi ( m + 1 ) \\Bigr ) e ^ { m u } \\frac { z ^ { 2 m } } { m ! ^ 2 } , \\\\ & \\theta _ 2 ( { \\bf v } ; z ) : = \\sum _ { p \\geq 0 } \\theta _ { 2 , p } ( { \\bf v } ) z ^ p = z ^ { - 1 } \\biggl ( \\sum _ { m \\geq 0 } e ^ { m u + z v } \\frac { z ^ { 2 m } } { m ! ^ 2 } - 1 \\biggr ) , \\end{align*}"}
{"id": "1327.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , p _ { i + 1 } ) } \\cup { ( \\delta _ { 1 } , 1 ) } , \\hat { \\alpha } \\cup { ( \\gamma , p _ { i + 1 } - p _ { i } ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 4 . \\end{align*}"}
{"id": "1838.png", "formula": "\\begin{align*} A ' ( \\bar { x } , \\bar { y } , t ) = \\bar { x } \\bar { y } e ^ { ( \\bar { x } + \\bar { y } ) t } { ( \\bar { x } - \\bar { y } ) ^ 2 \\over ( \\bar { x } e ^ { \\bar { y } t } - \\bar { y } e ^ { \\bar { x } t } ) ^ 2 } . \\end{align*}"}
{"id": "2871.png", "formula": "\\begin{align*} | e ^ { e _ 0 t } \\alpha _ + ( t ) | \\lesssim \\sum _ { n = [ t ] } ^ \\infty \\Bigg | \\int _ n ^ { n + 1 } e ^ { e _ 0 s } B ( R , \\mathcal { Y } _ - ) d s \\Bigg | \\le \\sum _ { n = [ t ] } ^ \\infty e ^ { ( e _ 0 - c _ 1 ) n } \\lesssim e ^ { ( e _ 0 - c _ 1 ) t } , \\end{align*}"}
{"id": "7474.png", "formula": "\\begin{align*} & \\tilde { \\phi } ^ 1 = \\phi ^ 0 + \\tau \\dot { \\phi } \\big | _ { t = 0 } + \\frac { \\tau ^ 2 } { 2 } \\ddot { \\phi } \\big | _ { t = 0 } = \\phi ^ 0 + \\frac { \\tau ^ 2 } { 2 } \\left ( - G ( \\phi ^ 0 ) + \\lambda ^ 0 \\phi ^ 0 \\right ) , \\\\ & \\phi ^ 1 = \\tilde { \\phi } ^ 1 / \\| \\tilde { \\phi } ^ 1 \\| . \\end{align*}"}
{"id": "529.png", "formula": "\\begin{align*} \\Xi _ { k + 1 } & = \\mathcal { A } \\Xi _ k + \\mathcal { B } \\mathcal { G } ( \\hat { \\Psi } _ k ( \\mathbf { u } , { \\tilde { \\Xi } } ) + \\hat { E } _ k ( \\mathbf { u } , { \\tilde { \\Xi } } ) + \\hat { D } _ k ( \\omega ) ) , \\\\ { \\mathbf { \\tilde { y } } _ k } & { = \\mathcal { C } \\Xi _ k + \\mathbf { w } _ k } , \\end{align*}"}
{"id": "9219.png", "formula": "\\begin{align*} \\left \\vert 1 - \\frac { \\gamma } { \\gamma ' } \\right \\vert = \\left \\vert \\frac { \\gamma ' - \\gamma } { \\gamma ' } \\right \\vert \\leq \\frac { \\vert \\gamma ' - \\gamma \\vert } { \\gamma ' } \\leq \\frac { 2 ^ { - j } } { 2 ^ { - l ' } } . \\end{align*}"}
{"id": "7767.png", "formula": "\\begin{gather*} \\varphi _ 0 [ T ] = - \\phi _ 0 [ T ] + ( p , 0 ) + \\left ( \\phi _ 0 [ T ] - ( p , 0 ) \\right ) ^ p . \\end{gather*}"}
{"id": "2446.png", "formula": "\\begin{align*} S ^ { - T } J S ^ { - 1 } = J & \\ ; \\Leftrightarrow \\ ; ( S ^ { - T } J S ^ { - 1 } ) ^ { - 1 } = J ^ { - 1 } \\ ; \\Leftrightarrow \\ ; S J ^ { - 1 } S ^ { T } = J ^ { - 1 } . \\end{align*}"}
{"id": "7893.png", "formula": "\\begin{align*} h ( x , W _ L ( x ) ) = C ( n , m , X ) + \\big | \\det A _ L ( x ) \\big | ^ { - \\frac { ( n - m ) } { 2 } } | | \\nabla ^ X F _ L ( x ) | | ^ { ( n - m ) ( m - 1 ) } | | ( \\nabla ^ X ) ^ 2 F _ L ( x ) | | ^ { ( n - m ) } . \\end{align*}"}
{"id": "9480.png", "formula": "\\begin{align*} \\mu _ 1 + \\mu _ { j - \\ell + 1 } ' - ( j - \\ell + 1 ) = ( \\mu _ 1 + \\mu _ 1 ' - 1 ) - ( \\mu _ k + \\mu _ 1 ' - k ) , \\end{align*}"}
{"id": "9126.png", "formula": "\\begin{align*} & \\Gamma _ k : = \\bigg \\{ x ^ * \\in X _ 0 \\mid \\exists y ^ * \\bigg ( \\vert \\norm { y ^ * } - \\norm { T ^ \\circ x ^ * } \\vert \\leq \\frac { 1 } { k + 1 } \\land H ^ * \\left [ y ^ * , T x ^ * , \\frac { 1 } { k + 1 } \\right ] \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\land \\forall i \\leq k \\left ( \\norm { x ^ * - J ^ S _ { \\mu _ i } ( x ^ * + \\mu _ i y ^ * ) } \\leq \\frac { 1 } { k + 1 } \\right ) \\bigg ) \\bigg \\} \\end{align*}"}
{"id": "5430.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ 1 ' & = \\sqrt { \\frac { s _ { - } ( a ) s _ { + } ( c ) } { s _ { + } ( b ) } } = \\sqrt { \\frac { s _ + ( c ) } { s _ + ( a ) s _ + ( b ) } } = \\frac { 1 } { \\lambda _ 2 } \\\\ \\lambda _ 2 ' & = \\sqrt { \\frac { s _ { - } ( a ) s _ { + } ( b ) } { s _ { + } ( c ) } } = \\sqrt { \\frac { s _ + ( b ) } { s _ + ( a ) s _ + ( c ) } } = \\frac { 1 } { \\lambda _ 1 } \\\\ \\lambda _ 3 ' & = \\sqrt { \\frac { s _ { + } ( b ) s _ { + } ( c ) } { s _ { - } ( a ) } } = \\sqrt { s _ + ( a ) s _ + ( b ) s _ + ( c ) } = \\frac { 1 } { \\lambda _ 4 } . \\end{cases} \\end{align*}"}
{"id": "8781.png", "formula": "\\begin{align*} \\Phi ^ { Q ( J ) } : = \\bigl \\{ ( s , \\phi ) \\bigm | \\phi = \\phi ( s _ { 1 n _ 1 } , \\ldots , s _ { d n _ d } ) , \\ s \\in Q ^ J \\bigr \\} . \\end{align*}"}
{"id": "3564.png", "formula": "\\begin{align*} c ^ w _ i ( e ) \\coloneqq \\begin{cases} w ( e ) & o ( e ) = \\eta _ i \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "5305.png", "formula": "\\begin{align*} \\varphi = \\varphi \\circ S . \\end{align*}"}
{"id": "486.png", "formula": "\\begin{align*} \\mathcal { N } ( \\mathcal { A } ; \\mathcal { H } ) : = \\left \\{ ( m , \\omega ) : \\ , \\omega \\in \\mathcal { N } _ { \\mathcal { I } ; \\mathcal { A } } m \\in \\mathcal { N } ^ { \\mathcal { I } ; \\mathcal { H } } \\right \\} . \\end{align*}"}
{"id": "4576.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( a _ n S _ n \\geq x \\sqrt { v _ n + a _ n ^ 2 \\sum _ { i = 1 } ^ n ( Z _ i - 1 ) ^ 2 } \\ , \\Big ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) \\end{align*}"}
{"id": "3232.png", "formula": "\\begin{align*} k ( s , 0 ) = s \\bigg ( \\frac { 1 } { h ( s ) } - s \\bigg ) = \\frac { n ( s ) } { d ( s ) } \\end{align*}"}
{"id": "1787.png", "formula": "\\begin{align*} \\langle \\operatorname { t r } _ e , \\operatorname { I n d } ( D _ \\mu ) \\rangle = \\langle \\widehat { A } ( \\mathfrak { g } , K ) \\wedge \\operatorname { c h } ( V _ \\mu ) _ { \\mathfrak { p } ^ * } , [ V ] \\rangle , \\end{align*}"}
{"id": "7455.png", "formula": "\\begin{align*} \\mathrm { R e } \\langle \\phi , \\ddot { \\phi } \\rangle + \\eta ( t ) \\mathrm { R e } \\langle \\phi , \\dot { \\phi } \\rangle & = - \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi | ^ 2 + V | \\phi | ^ 2 + \\beta | \\phi | ^ 4 - \\Omega \\overline { \\phi } L _ z \\phi \\right ) \\mathrm { d } \\mathbf { x } + \\lambda _ { \\phi } ( t ) \\| \\phi \\| ^ 2 . \\end{align*}"}
{"id": "5375.png", "formula": "\\begin{align*} \\mathfrak { I } ( z , \\eta , \\delta ) = \\left \\lbrace \\rho _ c ( \\gamma _ M ( g ) ) \\middle | g \\in \\mathfrak { G } ( z , \\eta , \\delta ) \\right \\rbrace . \\end{align*}"}
{"id": "8876.png", "formula": "\\begin{align*} H X _ b ^ { q + 1 } ( X , A ) = H Y _ b ^ q ( X , A ) \\end{align*}"}
{"id": "990.png", "formula": "\\begin{align*} C = C ' \\big ( 1 + ( \\theta d ) ^ { 2 s } \\| c \\| _ { L ^ \\infty ( B _ { \\rho } ( e _ 1 ) ) } \\big ) , \\end{align*}"}
{"id": "2120.png", "formula": "\\begin{align*} \\begin{aligned} & k \\sum _ { j = 1 } ^ { n - 1 } \\sum _ { l = 1 } ^ k \\frac { \\binom { M _ n } l ( k ) _ l ( k ( j - 1 ) ) _ { M _ n - l } } { ( k n ) _ { M _ n } } \\frac 1 { k ( n - j + 1 ) - l } = \\\\ & k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { j = 1 } ^ { n - N } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } + \\\\ & k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { j = n - N + 1 } ^ { n - 1 } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } . \\end{aligned} \\end{align*}"}
{"id": "5527.png", "formula": "\\begin{align*} \\dot { B } _ m ^ j ( t ) = \\frac { B ^ j ( [ t ] _ m ^ + ) - B ^ j ( [ t ] _ m ^ - ) } { \\delta _ m } , t \\in [ 0 , T ] . \\end{align*}"}
{"id": "391.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } _ { 1 } ( u , v ) \\partial _ { t } \\hat { u } & + A ^ { i } _ { 1 1 } ( u , v ) \\partial _ { i } \\hat { u } = f _ { 1 } ( U , D _ { x } v ) , \\\\ A ^ { 0 } _ { 2 } ( u , v ) \\partial _ { t } \\hat { v } & - B ^ { i j } _ { 2 2 } ( u , v ) \\partial _ { i } \\partial _ { j } \\hat { v } = f _ { 2 } ( U , D _ { x } U ) , \\end{aligned} \\end{align*}"}
{"id": "1533.png", "formula": "\\begin{align*} \\phi ( \\mathfrak { g } , s ) = \\chi _ v ( \\det ( d _ p ) ) ^ { - 1 } | \\det ( d _ p ) | _ { v } ^ { - s } j ( \\mathfrak { g } , z _ 0 ) ^ { - k } | j ( \\mathfrak { g } , z _ 0 ) | ^ { k - s } . \\end{align*}"}
{"id": "6555.png", "formula": "\\begin{align*} \\| K ( t ) \\| _ { L ^ { \\infty } } \\leq & C \\| K ( t ) \\| _ { L ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\| K ( t ) \\| _ { \\dot { H } ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\\\ \\leq & C t ^ { - \\frac { 1 } { 4 } } g ( A _ { t } ) ^ { - \\frac { 1 } { 4 } } t ^ { - \\frac { 3 } { 4 } } g ( A _ { t } ) ^ { - \\frac { 3 } { 4 } } \\\\ = & C t ^ { - 1 } g ( A _ { t } ) ^ { - 1 } . \\end{align*}"}
{"id": "3625.png", "formula": "\\begin{align*} \\sum _ { | \\Im ( \\rho ) | \\leq T } \\frac { x ^ { \\Re ( \\rho ) - 1 } } { | \\Im ( \\rho ) | } = \\sum _ { \\substack { | \\Im ( \\rho ) | \\leq T \\\\ \\Re ( \\rho ) \\leq \\sigma } } \\frac { x ^ { \\Re ( \\rho ) - 1 } } { | \\Im ( \\rho ) | } + \\sum _ { \\substack { | \\Im ( \\rho ) | \\leq T \\\\ \\Re ( \\rho ) > \\sigma } } \\frac { x ^ { \\Re ( \\rho ) - 1 } } { | \\Im ( \\rho ) | } . \\end{align*}"}
{"id": "5991.png", "formula": "\\begin{align*} | \\varphi ( z ) | \\leq \\sum _ { n = 0 } ^ { \\infty } | \\varphi _ { n } C _ { n } ^ { \\lambda , \\beta } ( z ) | & \\leq \\sum _ { n = 0 } ^ { \\infty } | \\varphi _ { n } | | C _ { n } ^ { \\lambda , \\beta } ( z ) | \\leq \\mathrm { e } ^ { \\varepsilon | z | } C _ { \\varepsilon } \\sum _ { n = 0 } ^ { \\infty } n ! | \\varphi _ { n } | \\sigma _ { \\varepsilon } ^ { - n } . \\end{align*}"}
{"id": "3314.png", "formula": "\\begin{align*} \\ell C = ( \\pi ^ { - 1 } ( U _ 1 ) \\cap \\ell C ) \\cup ( \\pi ^ { - 1 } ( U _ 2 ) \\cap \\ell C ) \\end{align*}"}
{"id": "6266.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ l \\frac { | S _ i | ^ 2 } { 1 6 h ^ 2 } \\leq 2 5 h ^ 2 \\varepsilon n ^ 2 . \\end{align*}"}
{"id": "928.png", "formula": "\\begin{align*} \\langle x \\rangle _ { 0 , \\lambda } = 1 , \\langle x \\rangle _ { n , \\lambda } = x ( x + \\lambda ) \\cdots ( x + ( n - 1 ) \\lambda ) , ( n \\ge 1 ) . \\end{align*}"}
{"id": "6881.png", "formula": "\\begin{align*} \\lambda _ \\ell = q ^ { \\ell j } \\frac { { k + j - \\ell \\choose j } _ q } { { k + j \\choose j } _ q } \\frac { { n - k - \\ell \\choose j } _ q } { { n - k \\choose j } _ q } \\approx q ^ { - \\ell j } . \\end{align*}"}
{"id": "3622.png", "formula": "\\begin{align*} \\frac { 1 } { R ( t ) \\log t } & = \\frac { 6 \\cdot 0 . 0 4 9 6 2 } { \\log t } \\left ( \\frac { 1 - a _ 1 ( t ) } { 1 + a _ 2 ( t ) } \\right ) \\\\ & = \\frac { 6 \\cdot 0 . 0 4 9 6 2 } { \\log t } \\left ( 1 - \\frac { a _ 1 ( t ) + a _ 2 ( t ) } { 1 + a _ 2 ( t ) } \\right ) \\\\ & \\geq \\frac { 1 } { 3 . 3 5 9 \\log t } \\left ( 1 - \\frac { 8 . 0 2 \\log \\log t } { \\log t } \\right ) \\end{align*}"}
{"id": "7048.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty | b _ j ( z ) | ^ 2 \\le 2 \\sum _ { j = 1 } ^ \\infty | q _ j ( z ) | ^ 2 + 2 \\big | 1 - \\sum _ { k = 1 } ^ L \\phi _ k ( z ) q _ k ( z ) \\big | ^ 2 \\Big ( \\sum _ { j = 1 } ^ \\infty | e _ j ( z ) | ^ 2 \\Big ) . \\end{align*}"}
{"id": "5752.png", "formula": "\\begin{align*} \\pi _ { * } ( \\nabla y ^ { j } ) = \\bar { \\nabla } z ^ { j } , j = 1 , \\dots , m , \\end{align*}"}
{"id": "4665.png", "formula": "\\begin{align*} \\dot { x } _ i \\sim \\mu _ i , \\quad \\dot { \\mu } _ i + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { ( x _ i - x _ j ) ^ 3 } \\sim 0 , \\end{align*}"}
{"id": "9229.png", "formula": "\\begin{align*} ( ( y + x ) - x ) = _ X y ( ( y ' + x ' ) - x ' ) = _ X y ' . \\end{align*}"}
{"id": "3807.png", "formula": "\\begin{align*} p _ { \\tau _ { 1 } } \\circ \\xi _ { 1 } ( t ) & = p _ { \\tau _ { 1 } } \\circ f ( \\tau ( g , \\xi ( t ) ) ) = \\overline { f } \\circ p _ { \\tau } ( \\tau ( g , \\xi ( t ) ) ) = \\overline { f } \\circ p _ { \\tau } ( \\xi ( t ) ) = p _ { \\tau _ { 1 } } ( f \\circ \\xi ( t ) ) = p _ { \\tau _ { 1 } } ( \\tau _ 1 ( g , f \\circ \\xi ( t ) ) ) \\\\ & = p _ { \\tau _ { 1 } } \\circ \\xi _ { 2 } ( t ) . \\end{align*}"}
{"id": "6671.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\leq Q ^ { \\vartheta } \\\\ ( d , g _ 3 g _ 4 c a e ) = 1 } } \\frac { 1 } { d ^ { 1 - \\varepsilon } } \\cdot d \\ll Q ^ { \\varepsilon } . \\end{align*}"}
{"id": "7194.png", "formula": "\\begin{align*} d _ { s , x } = d _ { s , x _ 1 } : = [ x _ 1 - X _ 1 ^ T ( s ) ] _ + . \\end{align*}"}
{"id": "8758.png", "formula": "\\begin{align*} \\begin{aligned} z _ { i 0 } = 1 z _ { i j } = \\frac { s _ { i j } - s _ { i j - 1 } } { a _ { i j } - a _ { i j - 1 } } j = 1 , \\ldots , n . \\end{aligned} \\end{align*}"}
{"id": "5700.png", "formula": "\\begin{align*} L _ { A _ { S } } : = - i \\partial _ { t } - A _ { S } : C ^ { \\infty } ( S ^ { 1 } ; \\mathbb { C } ) \\to C ^ { \\infty } ( S ^ { 1 } ; \\mathbb { C } ) . \\end{align*}"}
{"id": "5743.png", "formula": "\\begin{align*} A _ m = \\{ ( x , \\xi ) \\in S ^ * B ^ n _ 1 ; \\ ; ( m \\sqrt { \\lambda } ) ^ { s } \\leq n ( x , \\xi ) < ( ( m + 1 ) \\sqrt { \\lambda } ) ^ { s } \\} . \\end{align*}"}
{"id": "6977.png", "formula": "\\begin{align*} \\forall x \\in \\mathbb X \\colon \\Phi ( x ) : = \\bigl ( \\Omega - x ^ 1 , S ( x ^ 1 ) - x ^ 2 \\bigr ) , \\end{align*}"}
{"id": "215.png", "formula": "\\begin{align*} A = \\partial _ k R _ k ^ \\alpha ( p _ \\alpha , f ) ( x ) , B = \\partial _ k \\left ( p _ \\alpha ( x ) D _ k ^ { \\alpha - 1 } ( f ) \\right ) ( x ) , C = \\partial _ k \\left ( f ( x ) D _ k ^ { \\alpha - 1 } ( p _ \\alpha ) \\right ) ( x ) . \\end{align*}"}
{"id": "8191.png", "formula": "\\begin{align*} H = \\left \\{ 1 , a / b , b / a \\right \\} , \\end{align*}"}
{"id": "7343.png", "formula": "\\begin{align*} \\begin{aligned} & \\begin{pmatrix} \\lambda Y _ q & 0 \\\\ 0 & ( 1 - \\lambda ) Z _ q \\end{pmatrix} \\\\ & \\geq \\begin{pmatrix} \\lambda ^ 2 X _ q & \\lambda ( 1 - \\lambda ) X _ q \\\\ \\lambda ( 1 - \\lambda ) X _ q & ( 1 - \\lambda ) ^ 2 X _ q \\end{pmatrix} - \\varepsilon \\begin{pmatrix} \\lambda ^ 2 X _ q & \\lambda ( 1 - \\lambda ) X _ q \\\\ \\lambda ( 1 - \\lambda ) X _ q & ( 1 - \\lambda ) ^ 2 X _ q \\end{pmatrix} ^ 2 , \\end{aligned} \\end{align*}"}
{"id": "7448.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mu \\phi ( \\mathbf { x } ) = - \\frac 1 2 \\Delta \\phi ( \\mathbf { x } ) + V ( \\mathbf { x } ) \\phi ( \\mathbf { x } ) + \\beta | \\phi ( \\mathbf { x } ) | ^ 2 \\phi ( \\mathbf { x } ) - \\Omega L _ z \\phi ( \\mathbf { x } ) , \\mathbf { x } \\in \\mathbb { R } ^ d , \\\\ & \\| \\phi \\| ^ 2 = 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "8501.png", "formula": "\\begin{align*} a \\circ ( b + c ) = a \\circ b - a + a \\circ c \\end{align*}"}
{"id": "6374.png", "formula": "\\begin{align*} \\sigma ^ \\prime _ { \\mathcal { X } } = \\left ( \\mathcal { E } ^ \\prime _ p , \\mathcal { E } _ { p _ 1 } , \\mathcal { E } _ { p _ 2 } , \\mathcal { E } _ { p _ 3 } , \\mathcal { E } _ { p _ 4 } , \\Phi ( \\sigma _ { \\widetilde { X } } ) \\right ) \\end{align*}"}
{"id": "3475.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs * { \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + \\begin{cases} O ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } \\log T ) & ( \\frac { 1 } { 2 } < \\sigma _ 1 + \\sigma _ 3 \\leq \\frac { 3 } { 4 } ) \\\\ O ( T ^ \\frac { 1 } { 2 } ) & ( \\frac { 3 } { 4 } < \\sigma _ 1 + \\sigma _ 3 \\leq 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "4955.png", "formula": "\\begin{align*} \\Phi _ i = g _ i ^ { - 1 } \\cdot b \\cdot \\phi ( g _ i ) . \\end{align*}"}
{"id": "6984.png", "formula": "\\begin{align*} \\begin{aligned} y ^ * & \\in \\mathcal N _ { \\mathcal T _ D ( g ( \\bar x ) ) } ( \\nabla g ( \\bar x ) u ) \\cap \\ker \\nabla g ( \\bar x ) ^ * , \\\\ z ^ * & \\in \\mathcal T _ { \\mathcal N _ { \\mathcal T _ D ( g ( \\bar x ) ) } ( \\nabla g ( \\bar x ) u ) } ( y ^ * ) , \\end{aligned} \\end{align*}"}
{"id": "2908.png", "formula": "\\begin{align*} K ( z ) = | 1 + z | ^ { p - 2 } ( 1 + z ) - \\left ( 1 + \\frac { p } { 2 } z + \\frac { p - 2 } { 2 } \\bar { z } \\right ) , z \\in \\mathbb { C } \\end{align*}"}
{"id": "6158.png", "formula": "\\begin{align*} \\frac { ( 1 - \\alpha ) ^ { ( n + 1 ) } } { ( n + 1 ) ! } \\sum _ { k = 0 } ^ m \\frac { n + 1 } { n + k + 1 } \\frac { { \\alpha } ^ { ( k ) } } { k ! } + \\frac { { \\alpha } ^ { ( m + 1 ) } } { ( m + 1 ) ! } \\sum _ { k = 0 } ^ n \\frac { m + 1 } { m + k + 1 } \\frac { ( 1 - \\alpha ) ^ { ( k ) } } { k ! } = 1 . \\end{align*}"}
{"id": "3435.png", "formula": "\\begin{align*} \\langle U _ { L ' _ 1 , L ' _ 2 } g _ 1 , T U _ { L _ 1 , L _ 2 } f _ 1 \\rangle & \\quad = \\sum \\limits _ { k = L _ 1 } ^ { L _ 2 } \\sum \\limits _ { k ' = L ' _ 1 } ^ { L ' _ 2 } \\Big \\langle D ^ { M } _ { k ' } g _ 1 , D ^ * _ { k ' } T D _ { k } D ^ { M } _ { k } f _ 1 \\Big \\rangle . \\end{align*}"}
{"id": "5087.png", "formula": "\\begin{align*} M _ \\omega T _ x = e ^ { 2 \\pi i \\omega x } T _ x M _ \\omega . \\end{align*}"}
{"id": "3420.png", "formula": "\\begin{align*} \\textup { ( v ) } \\ \\Big | \\int _ { \\R ^ N } f _ 2 ( y ) d \\omega ( y ) \\Big | = \\Big | - \\int _ { \\R ^ N } f _ 1 ( y ) d \\omega ( y ) - \\int _ { \\R ^ N } f _ 3 ( y ) d \\omega ( y ) \\Big | \\lesssim \\Big ( \\frac { r } { R } \\Big ) ^ \\gamma . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
{"id": "900.png", "formula": "\\begin{align*} ( \\star ) \\qquad \\lim _ { m \\to \\infty } \\frac { B _ { k _ m } } { k _ m } = \\lim _ { m \\to \\infty } \\frac { B _ { 2 k _ m - 2 } } { 2 k _ m - 2 } = \\frac { ( 1 - p ) B _ 2 } { 2 } = \\frac { 1 - p } { 1 2 } . \\end{align*}"}
{"id": "8016.png", "formula": "\\begin{align*} ( - 1 ) \\cdot * _ \\Sigma i ^ * _ \\Sigma \\Pi _ \\ell d \\phi ( s ) = ( \\partial _ \\Sigma \\phi ) ( s ) = \\frac { 1 } { \\sqrt { \\gamma ' ( s ) } } ( \\partial _ u \\phi ) ( - s , \\gamma ( s ) ) . \\end{align*}"}
{"id": "6818.png", "formula": "\\begin{align*} - \\triangle _ { g } ( u ) + K _ { g _ { \\textnormal { p o i n } } } \\ , e ^ { - 2 u } = K _ g \\textnormal { o n } \\ , \\ , \\ , \\Sigma \\end{align*}"}
{"id": "389.png", "formula": "\\begin{align*} A ^ { 0 } ( U ) U _ { t } + A ^ { i } ( U ) \\partial _ { i } U - B ^ { i j } ( U ) \\partial _ { i } \\partial _ { j } U + D ( U ) U = F ( U ; D _ { x } U ) , \\end{align*}"}
{"id": "4512.png", "formula": "\\begin{align*} C _ { 1 , k } = c _ 1 \\Big ( e ^ { \\frac { 1 6 w _ { k + 1 } } { 3 R } } w _ { k + 1 } ^ { 3 + \\frac { 4 } { R \\log t _ 0 } } - e ^ { \\frac { 1 6 w _ k } { 3 R } } w _ k ^ { 3 + \\frac { 4 } { R \\log t _ 0 } } \\Big ) , \\ C _ { 2 , k } = c _ 2 \\big ( w _ { k + 1 } ^ 2 - w _ { k } ^ 2 \\big ) . \\end{align*}"}
{"id": "7467.png", "formula": "\\begin{align*} \\frac { \\delta E ( \\phi ) } { \\delta \\overline { \\phi } } = - \\frac 1 2 \\Delta \\phi + V \\phi + \\beta | \\phi | ^ 2 \\phi - \\Omega L _ z \\phi \\end{align*}"}
{"id": "2639.png", "formula": "\\begin{align*} ( P X - X P ) g = \\frac { 1 } { 2 \\pi i } g , \\end{align*}"}
{"id": "8308.png", "formula": "\\begin{align*} C ( \\vect { H } , { \\sf S N R } ) = \\sum _ { n = 1 } ^ { N } \\log _ 2 \\ ! \\left ( 1 + \\left ( \\nu - \\frac { 1 } { \\lambda _ n ( \\vect { H } ) } \\right ) ^ { \\ ! \\ ! + } \\ ! \\ ! \\ ! \\lambda _ n ( \\vect { H } ) \\right ) \\end{align*}"}
{"id": "5073.png", "formula": "\\begin{align*} | A ( x ) - A ( x + r v ) | & = | L ( x ) + C - ( L ( x + r v ) + C ) | \\\\ & = | L ( x ) - L ( x + r v ) | \\\\ & = | L ( r v ) | \\\\ & = r | L ( v ) | . \\end{align*}"}
{"id": "6919.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { \\mathbf { w } _ 1 } & \\Bigl | \\textbf { a } _ 1 ^ T ( \\theta _ I ) \\mathbf { w } _ 1 \\Bigl | ^ 2 \\\\ \\textrm { s . t . } & \\| \\mathbf { w } _ 1 \\| ^ 2 \\leqslant 1 . \\end{aligned} \\end{align*}"}
{"id": "6206.png", "formula": "\\begin{align*} \\begin{aligned} & ~ ~ ~ ~ \\| C - C Z _ { j } Z _ { j } ^ { T } \\| ^ 2 _ F \\\\ & < \\left [ \\| C - C \\hat { V } _ j \\hat { V } _ j ^ { T } \\| _ F + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) \\| C \\| _ F \\right ] ^ 2 \\\\ & = \\| C - C \\hat { V } _ j \\hat { V } _ j ^ { T } \\| ^ 2 _ F + 2 ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) \\| C - C \\hat { V } _ j \\hat { V } _ j ^ { T } \\| _ F \\| C \\| _ F + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) ^ 2 \\| C \\| ^ 2 _ F . \\end{aligned} \\end{align*}"}
{"id": "3690.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int B ( t , x ) \\ , d x = & \\int \\left ( - a B J _ x - b J B _ x - \\mu \\Lambda ^ \\alpha B \\right ) \\ , d x \\\\ = & \\int ( a - b ) B _ x J \\ , d x \\\\ = & \\ ( a - b ) \\int J \\mathcal H J \\ , d x \\\\ = & \\ 0 \\end{align*}"}
{"id": "7841.png", "formula": "\\begin{align*} X _ { 1 } & = ~ \\{ x , T ^ { * 3 } x , \\ldots , T ^ { * 3 ( 3 j - n + 2 ) } x , y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( 3 j - n + 2 ) } y \\} \\\\ X _ { 2 } & = ~ \\{ T ^ { * 3 ( 3 j - n + 3 ) } x , \\ldots , T ^ { * 3 j } x , T ^ { * 3 ( 3 j - n + 3 ) } y , \\ldots , T ^ { * 3 j } y , z , \\ldots , T ^ { * 3 ( n - 2 j - 3 ) } z \\} . \\end{align*}"}
{"id": "8245.png", "formula": "\\begin{align*} \\mathfrak { B } _ J = \\begin{cases} \\mathfrak { S } _ { p _ 2 } \\times \\cdots \\times \\mathfrak { S } _ { n - p _ k } , & p _ 1 = 0 , \\\\ \\mathfrak { B } _ { p _ 1 } \\times \\mathfrak { S } _ { p _ 2 - p _ 1 } \\cdots \\times \\mathfrak { S } _ { n - p _ k } , & p _ 1 > 0 . \\end{cases} \\end{align*}"}
{"id": "1030.png", "formula": "\\begin{align*} h ^ { ( \\varepsilon ) } ( x ) = - \\varepsilon \\zeta _ 2 ( x ) \\geqslant - \\varepsilon \\R ^ n \\setminus B _ 1 ( 2 e _ 1 ) . \\end{align*}"}
{"id": "401.png", "formula": "\\begin{align*} \\mu _ { 1 } ( t ) : = \\| \\partial _ { t } A ^ { 0 } \\| _ { s - 1 } + \\sum _ { i , j = 1 } ^ { d } \\| \\partial _ { t } B ^ { i j } \\| _ { s - 1 } . \\end{align*}"}
{"id": "264.png", "formula": "\\begin{align*} u _ { t } = \\left ( u _ { 0 } ^ { * } + t ( \\phi - R ) \\right ) ^ { * } . \\end{align*}"}
{"id": "413.png", "formula": "\\begin{align*} \\begin{aligned} u , w & \\in L ^ { \\infty } ( 0 , T ; H ^ { m } ) , \\\\ v & \\in L ^ { \\infty } ( 0 , T ; H ^ { m } ) \\cap L ^ { 2 } ( 0 , T ; H ^ { m + 1 } ) , \\\\ u _ { t } , v _ { t } , w _ { t } & \\in L ^ { 2 } ( 0 , T ; H ^ { m - 1 } ) . \\end{aligned} \\end{align*}"}
{"id": "8366.png", "formula": "\\begin{align*} \\varphi \\Bigl ( u ^ { \\frac { l } { s + n h + 1 } } \\Bigr ) \\cdot u = \\varphi \\Bigl ( u ^ { \\frac { l + 1 } { s + n h + 1 } } u ^ { \\frac { s + n h } { s + n h + 1 } } \\Bigr ) \\in \\varphi \\Bigl ( u ^ { \\frac { l + 1 } { s + n h + 1 } } \\Bigr ) \\cdot \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } . \\end{align*}"}
{"id": "9195.png", "formula": "\\begin{align*} \\mathrm { d e g } ( 0 ) : = 0 , \\quad \\mathrm { d e g } ( \\tau ( \\rho ) ) : = \\max \\{ \\mathrm { d e g } ( \\tau ) , \\mathrm { d e g } ( \\rho ) + 1 \\} . \\end{align*}"}
{"id": "8487.png", "formula": "\\begin{gather*} [ z ^ n ] [ u ^ k ] F ( u ) = \\frac { k } { n } { 2 n - k - 1 \\choose n - 1 } , \\\\ [ z ^ n ] [ u ^ k ] G ( u ) = \\frac { k + 3 } { n + 1 } { 2 n - k - 2 \\choose n } , \\\\ [ z ^ n ] [ u ^ k ] H ( u ) = \\frac { k + 1 } { n } { 2 n - k - 2 \\choose n - 1 } . \\end{gather*}"}
{"id": "1683.png", "formula": "\\begin{align*} v _ p \\left ( \\sum \\limits _ { k _ 1 = 0 \\atop p \\neq 2 k _ 1 + 1 } ^ { n - r } \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } \\right ) \\geq \\min \\limits _ { 0 \\leq k _ 1 \\leq n - r \\atop p \\neq 2 k _ 1 + 1 } v _ p \\left ( \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } \\right ) = \\min \\limits _ { 0 \\leq k _ 1 \\leq n - r \\atop p \\neq 2 k _ 1 + 1 } v _ p ( c _ { k _ 1 } ) > v _ p \\left ( \\frac { c _ { \\frac { p - 1 } { 2 } } } { p ^ { s _ 1 } } \\right ) . \\end{align*}"}
{"id": "4497.png", "formula": "\\begin{align*} \\psi ( x ) = x - \\sum _ { | \\gamma | \\le T ^ { \\star } } \\frac { x ^ { \\varrho } } { \\varrho } + O ^ { \\star } \\left ( M \\frac { x } { T } ( \\log x ) ^ { 1 - \\omega } \\right ) \\ \\ x \\ge x _ M . \\end{align*}"}
{"id": "1810.png", "formula": "\\begin{align*} D ( f g ) = f D ( g ) + D ( f ) g . \\end{align*}"}
{"id": "7050.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty | q _ j ( z ) | ^ 2 = \\sum _ { j = 1 } ^ L | q _ j ( z ) | ^ 2 \\le \\sum _ { j = 1 } ^ L \\| q _ j \\| _ \\infty ^ 2 \\le C \\sum _ { j = 1 } ^ L \\| q _ j \\| _ { H ^ 2 } ^ 2 \\le \\frac { C } { \\delta ^ { 2 \\max m _ j } } \\end{align*}"}
{"id": "6138.png", "formula": "\\begin{align*} x ^ { m + 1 } P ( x ) + ( 1 - x ) ^ { n + 1 } Q ( x ) = 1 . \\end{align*}"}
{"id": "3444.png", "formula": "\\begin{align*} \\| T _ j T ^ * _ k f \\| _ { L ^ 2 ( \\R ^ N , \\omega ) } ^ 2 & = \\int _ { \\mathbb R ^ N } \\bigg | \\int _ { \\mathbb R ^ N } T _ j T ^ * _ k ( x , y ) f ( y ) d \\omega ( y ) \\bigg | ^ 2 d \\omega ( x ) \\end{align*}"}
{"id": "7113.png", "formula": "\\begin{align*} x _ t = x _ 0 + \\int _ 0 ^ t \\big ( \\alpha _ { - 1 } x _ s ^ { - 1 } - \\alpha _ 0 + \\alpha _ 1 x _ s - \\alpha _ 2 s ^ { 2 H - 1 } x _ s ^ { \\rho } \\big ) d s + \\int _ 0 ^ t \\sigma x ^ { \\theta } _ s d B _ s ^ H , \\end{align*}"}
{"id": "9181.png", "formula": "\\begin{align*} ( \\mathcal { S } ( 0 ) ) \\ll & \\sum _ { 2 < p \\leq X } \\sum ^ { \\mathcal { J } } _ { l = 1 } \\Big ( \\alpha ^ { 3 / 4 } _ { 1 } { | \\mathcal M } _ { 1 , l } ( p ) | \\Big ) ^ { 2 \\lceil 1 / ( 1 0 \\alpha _ { 1 } ) \\rceil } \\Phi \\left ( \\frac { p } X \\right ) . \\end{align*}"}
{"id": "8374.png", "formula": "\\begin{align*} \\tilde { x } ^ 0 & = u \\ , , \\\\ \\tilde { x } ^ 1 & = v - \\frac { 1 } { 2 } h _ { i j } \\dot { M } ^ i _ k M ^ j _ l x ^ k x ^ l \\ , , \\\\ \\tilde { x } ^ i & = M ^ i _ j x ^ j \\ , , \\end{align*}"}
{"id": "5115.png", "formula": "\\begin{align*} c _ { ( k , l ) , ( k ' , l ' ) } = e ^ { - 2 \\pi i k ( l ' - l ) / ( a b ) } \\langle g , \\ , M _ { ( l ' - l ) / a } T _ { ( k ' - k ) / b } \\ , g \\rangle \\end{align*}"}
{"id": "3848.png", "formula": "\\begin{align*} p _ { k , m } = f _ { k , m + 1 } - \\tilde f _ { k , m + 1 } , k = \\overline { m + 1 , 2 m } . \\end{align*}"}
{"id": "5690.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) : = - c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z ) + \\sum _ { i } \\sum _ { k = 1 } ^ { m _ { i } - 1 } \\mu _ { \\tau } ( \\alpha _ { i } ^ { k } ) - \\sum _ { j } \\sum _ { k = 1 } ^ { n _ { j } - 1 } \\mu _ { \\tau } ( \\beta _ { j } ^ { k } ) . \\end{align*}"}
{"id": "804.png", "formula": "\\begin{align*} & \\ \\int _ { B ( x , r ) } ( | \\nabla u | ^ { p - 2 } \\nabla u - | \\nabla v | ^ { p - 2 } \\nabla v ) \\cdot ( \\nabla u - \\nabla v ) \\ , d \\mu \\\\ & = \\int _ { B ( x , r ) } | \\nabla u | ^ { p - 2 } \\nabla u \\cdot ( \\nabla u - \\nabla v ) - | \\nabla v | ^ { p - 2 } \\nabla v \\cdot ( \\nabla u - \\nabla v ) \\ , d \\mu \\\\ & = \\int _ { B ( x , r ) } | \\nabla u | ^ { p - 2 } \\nabla u \\cdot ( \\nabla u - \\nabla v ) \\ , d \\mu . \\end{align*}"}
{"id": "720.png", "formula": "\\begin{align*} K _ { ( 1 0 ) } ^ { ( \\ell + 1 ) } = K _ { ( 1 0 ) } ^ { ( 1 ) } \\prod _ { \\ell ' = 1 } ^ \\ell \\chi _ { | | } ^ { ( \\ell ' ) } . \\end{align*}"}
{"id": "3982.png", "formula": "\\begin{align*} J : = \\frac { 1 } { k \\pi e ^ { \\frac { 1 } { \\sqrt { k } } } } \\big ( - \\alpha _ { 1 , k } + 2 i { k \\pi } + O ( 1 ) \\big ) e ^ { - \\alpha _ { 1 , k } - i ( 2 k \\pi + \\alpha _ { 2 , k } ) + O ( k ^ { - \\frac { 1 } { 2 } } ) } \\times e ^ { - \\frac { 1 } { 2 } - i \\sqrt { k \\pi } + O ( k ^ { - \\frac { 1 } { 2 } } ) } , \\end{align*}"}
{"id": "5321.png", "formula": "\\begin{align*} Y = \\sum ( \\omega _ { ( 1 ) } ' \\theta ) ( S ^ { - 1 } ( a _ { ( 3 ) } ' ) - a _ { ( 1 ) } ' ) \\otimes a _ { ( 2 ) } ' \\otimes \\omega _ { ( 2 ) } ' ( S ^ { - 1 } ( a _ { ( 6 ) } ' ) - a _ { ( 4 ) } ' ) \\otimes a _ { ( 5 ) } ' b . \\end{align*}"}
{"id": "3059.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = y _ 2 \\ , , y _ 2 ^ \\prime = - i x _ 1 \\ , , x _ 2 ^ \\prime = i y _ 1 \\ , , y _ 1 ^ \\prime = x _ 2 \\ , . \\end{align*}"}
{"id": "565.png", "formula": "\\begin{align*} \\varepsilon ^ { - 2 } \\int _ 0 ^ \\tau \\int _ { \\Omega _ R } p ( \\rho ) \\left ( \\rho - \\frac { 1 } { | \\Omega _ R | } \\int _ { \\Omega _ R } \\rho \\right ) = \\sum _ { i = 1 } ^ 4 I _ i , \\end{align*}"}
{"id": "8362.png", "formula": "\\begin{align*} u \\in \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } B . \\end{align*}"}
{"id": "6200.png", "formula": "\\begin{align*} \\| C X \\| _ F ^ 2 > \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } - 2 \\xi \\| C \\| ^ 2 _ F . \\end{align*}"}
{"id": "6409.png", "formula": "\\begin{align*} z _ i ^ { \\widetilde k } \\in { } \\ , \\{ 1 , z _ i , \\dots , z _ i ^ { { \\widetilde k } - 1 } \\} + { R a n } \\ , M _ { z _ { n + 1 } } \\ , , \\ ; i = 1 , \\dots , n . \\end{align*}"}
{"id": "8695.png", "formula": "\\begin{align*} \\begin{aligned} & \\phi ( f _ 1 , 1 ) + \\phi ( 2 , f _ 2 ) - \\phi ( 2 , 1 ) - L _ 1 ( f ) = ( 2 ^ { 0 . 6 } - f _ 1 ^ { 0 . 6 } ) - 2 ^ { 0 . 6 } f _ 2 ^ { 0 . 6 } \\\\ & \\leq \\min \\bigl \\{ - f _ 1 ^ { 0 . 6 } - f _ 2 ^ { 0 . 6 } + 1 , 2 ^ { 0 . 6 } ( f _ 1 ^ { 0 . 6 } - 2 ^ { 0 . 6 } ) - 2 ^ { 0 . 6 } f _ 2 ^ { 0 . 6 } \\bigr \\} \\leq \\min \\bigl \\{ R _ 1 ( f ) , R _ 2 ( f ) \\bigr \\} . \\end{aligned} \\end{align*}"}
{"id": "611.png", "formula": "\\begin{align*} \\frac { f ( x ) - g ( x ) } { h ( x ) + 1 } \\ = \\ \\frac { r } { s } . ] \\end{align*}"}
{"id": "4436.png", "formula": "\\begin{align*} H ' ( t ) & \\leq - a _ 1 \\int _ U | u _ t ( x , t ) | ^ 2 \\ , d x - a _ 1 \\int _ U | u _ t ( x , t ) | ^ m \\ , d x \\\\ & + ( \\frac { 3 } { 2 } + ( a _ 2 ^ 2 B ) ) \\epsilon \\int _ U ( u _ t ( x , t ) ) ^ 2 \\ , d x - \\frac { 1 } { 4 } \\epsilon \\int _ U ( u _ { x x } ( x , t ) ) ^ 2 \\ , d x \\\\ & - a _ 2 \\epsilon \\int _ U | u _ t | ^ { m - 1 } | u | ( x , t ) \\ , d x - \\epsilon E ( t ) . \\end{align*}"}
{"id": "4125.png", "formula": "\\begin{align*} h ( p ) = \\sum _ { i = 1 } ^ d c _ i h ( p _ i ) + \\beta ^ { - 1 } \\lambda > \\sum _ { i = 1 } ^ d c _ i h ( p _ i ) , \\end{align*}"}
{"id": "2935.png", "formula": "\\begin{align*} M _ { n , \\mathbf p _ k } = \\frac 1 n \\sum _ { i \\ne j } ^ n \\prod _ { \\ell = 1 } ^ k I _ { i , j } ^ { ( p _ \\ell ) } , N _ { n , \\mathbf p _ k } = \\frac 1 n \\sum _ { i = 1 } ^ n \\prod _ { \\ell = 1 } ^ k I _ { i , i } ^ { ( p _ \\ell ) } . \\end{align*}"}
{"id": "6228.png", "formula": "\\begin{align*} \\overline { \\zeta } \\left ( A , T ^ d _ N , u \\right ) = \\det \\Big ( I _ { 2 d N ^ d } - u M _ A \\Big ) ^ { - 1 / N ^ d } . \\end{align*}"}
{"id": "7148.png", "formula": "\\begin{align*} \\int _ 0 ^ t g ( x _ s ) d B ^ H _ s = \\int _ 0 ^ t - H s ^ { 2 H - 1 } g ^ { \\prime } ( x _ s ) d s + \\int _ 0 ^ t g ( x _ s ) d ^ { \\circ } B _ s ^ H \\end{align*}"}
{"id": "7173.png", "formula": "\\begin{align*} \\theta ( 1 - \\delta ) \\chi / ( \\chi - \\delta ) = \\min \\{ 1 / 2 e , \\theta / 2 \\} . \\end{align*}"}
{"id": "6735.png", "formula": "\\begin{align*} g _ 1 ( t ) \\Omega ^ n \\mathcal { L i } _ { K , n } ( \\alpha ) + g _ 2 ( t ) \\Omega ^ n \\mathcal { L i } _ { C , n } ( \\beta ) = 0 . \\end{align*}"}
{"id": "1490.png", "formula": "\\begin{align*} \\int _ { U ( \\Q ) \\backslash U ( \\mathbb { A } ) } \\mathbf { f } ( u g ) \\mathbf { d } u = 0 , \\end{align*}"}
{"id": "4726.png", "formula": "\\begin{align*} & P _ { i , 1 } = ( \\dot { x } _ i - \\mu _ i ) \\sigma _ i \\partial _ y \\widetilde { R } _ i , \\\\ & P _ { i , 2 } = - \\bigg ( \\dot { \\mu } _ i + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { x ^ 3 _ { i j } } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\mu _ k } { x ^ 3 _ { i j } } \\bigg ) \\frac { \\sigma _ i \\Lambda \\widetilde { R } _ i } { 1 + \\mu _ i } . \\end{align*}"}
{"id": "3148.png", "formula": "\\begin{align*} \\partial _ { r _ 1 } Z | _ { r _ 1 = r _ 2 = 0 } = \\partial _ { r _ 2 } Z | _ { r _ 1 = r _ 2 = 0 } = 0 \\quad z > 0 . \\end{align*}"}
{"id": "3115.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + z y ^ 5 + x ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "7783.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log \\prod \\limits _ { i = 1 } ^ { n } p _ { w _ i } = \\sum \\limits _ { k \\in T } p _ k \\log p _ k . \\end{align*}"}
{"id": "1334.png", "formula": "\\begin{align*} A ( \\delta _ { 1 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , A ( \\delta _ { 2 } ) \\approx \\frac { 1 } { 2 } p _ { i } R , \\ , \\ , p _ { i + 1 } = \\frac { 3 } { 2 } p _ { i } , \\ , \\ , A ( \\eta ) \\approx \\frac { 1 } { 4 } p _ { i } . \\end{align*}"}
{"id": "739.png", "formula": "\\begin{align*} J _ { x _ \\alpha } z _ { \\alpha } ^ { ( L + 1 ) } = W ^ { ( L + 1 ) } D ^ { ( L ) } W ^ { ( L ) } \\cdots D ^ { ( 1 ) } W ^ { ( 1 ) } , \\end{align*}"}
{"id": "4296.png", "formula": "\\begin{align*} u ( r , t ) = \\frac { 1 } { T - t } w \\left ( \\frac { r } { \\sqrt { T - t } } , \\tau \\right ) , \\tau = - \\log ( T - t ) . \\end{align*}"}
{"id": "5181.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { N } \\bigg ( \\frac { 2 \\rho _ { \\ell } \\mu _ { \\ell } } { p _ \\ell } + \\frac { c _ \\ell p _ \\ell } { \\mu _ \\ell } \\bigg ) \\le \\sum _ { \\ell = 1 } ^ { N } \\bigg ( \\frac { 2 \\rho _ { \\ell } \\mu _ { \\ell } } { f _ \\ell ^ \\star } + \\frac { c _ \\ell f _ \\ell ^ \\star } { \\mu _ \\ell } \\bigg ) . \\end{align*}"}
{"id": "5149.png", "formula": "\\begin{align*} \\sinh ( 2 t ) - 4 t ^ 2 \\coth ( 2 t ) < \\max \\ , \\{ \\ , \\sinh ( 2 \\cdot \\tfrac { 4 } { 3 } ) - 4 \\cdot \\tfrac { 1 6 } { 9 } \\coth ( 2 \\cdot \\tfrac { 4 } { 3 } ) , \\ , 0 \\ , \\} = 0 , \\end{align*}"}
{"id": "7242.png", "formula": "\\begin{align*} a ( z ^ * ) = ( \\hat { \\phi } ( k ) ) ^ { - 1 } > 1 . \\end{align*}"}
{"id": "6059.png", "formula": "\\begin{align*} Z ( x , y ) = x + i Q ( x , y ) = x + i \\int _ 0 ^ y P ( x , \\tau ) d \\tau \\ , . \\end{align*}"}
{"id": "3838.png", "formula": "\\begin{align*} ( \\ell _ n ( y ) , z ) = ( - 1 ) ^ m ( y ^ { ( m + \\tau ) } , z ^ { ( m ) } ) + \\sum _ { r , j = 0 } ^ m ( q _ { r , j } y ^ { ( r ) } , z ^ { ( j ) } ) , z \\in \\mathfrak D , \\end{align*}"}
{"id": "7998.png", "formula": "\\begin{align*} \\widehat { P } ( s ) = N ^ { 1 - s } \\cdot \\max \\big \\{ N a ^ t b ^ { 1 - t } + c ^ t d ^ { 1 - t } , N b + d \\big \\} , \\end{align*}"}
{"id": "8069.png", "formula": "\\begin{align*} \\partial _ \\Sigma ^ * \\Psi ( f ) \\star _ H \\partial _ \\Sigma ^ * \\Psi ( g ) = \\partial _ \\Sigma ^ * \\Psi ( f ) \\cdot \\partial _ \\Sigma ^ * \\Psi ( g ) + \\hbar \\left \\langle ( \\partial _ \\Sigma \\otimes \\partial _ \\Sigma ) [ \\tfrac { i } { 2 } E + H ] , f \\otimes g \\right \\rangle _ { \\Sigma ^ 2 } . \\end{align*}"}
{"id": "9542.png", "formula": "\\begin{align*} S _ 0 ^ \\infty ( - \\bar x ) + \\bar x \\cdot \\bar c + \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } > 0 \\end{align*}"}
{"id": "4433.png", "formula": "\\begin{align*} E ' ( t ) & = - \\int _ U ( F ( u _ t ) u _ t ) ( x , t ) \\ , d x \\\\ & \\leq - a _ 1 ( \\int _ U ( u _ t ( x , t ) ) ^ 2 \\ , d x + \\int _ U | u _ t ( x , t ) | ^ m \\ , d x ) . \\end{align*}"}
{"id": "1278.png", "formula": "\\begin{align*} \\Lambda ( M , \\Gamma ) : = \\{ \\ , \\ , \\alpha \\ , \\ , | \\ , \\ , \\alpha \\ , \\ , \\mathrm { i s \\ , \\ , a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , s u c h \\ , \\ , t h a t \\ , \\ , } [ \\alpha ] = \\Gamma \\mathrm { \\ , \\ , a n d } \\ , \\ , A ( \\alpha ) < M \\ , \\ , \\} \\end{align*}"}
{"id": "3944.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + V _ 0 \\rho _ x + Q _ 0 u _ x = 0 \\ & ( 0 , + \\infty ) \\times ( 0 , L ) , \\\\ u _ t - \\frac { \\nu } { Q _ 0 } u _ { x x } + V _ 0 u _ x + a \\gamma Q _ 0 ^ { \\gamma - 2 } \\rho _ x = 0 \\ & ( 0 , + \\infty ) \\times ( 0 , L ) . \\end{dcases} \\end{align*}"}
{"id": "5930.png", "formula": "\\begin{align*} \\Phi ^ { T Q } _ g ( v _ q ) & = T _ q \\Phi _ g ( v _ q ) , \\\\ \\Phi ^ { T ^ 1 _ k Q } _ g ( v _ { 1 q } , \\dots , v _ { k q } ) & = ( T _ q \\Phi _ g ( v _ { 1 q } ) , \\dots , T _ q \\Phi _ g ( v _ { k q } ) ) . \\end{align*}"}
{"id": "1405.png", "formula": "\\begin{align*} r ^ n = q ^ * r ^ o q , \\end{align*}"}
{"id": "6836.png", "formula": "\\begin{align*} \\lambda _ i ( U D ) = \\frac { R ( k - 1 , i ) } { R ( k , i ) } . \\end{align*}"}
{"id": "6093.png", "formula": "\\begin{align*} S = \\left \\{ R ( s , t ) = \\left ( s , t , z ( s , t ) \\right ) ; \\ ( s , t ) \\in \\R ^ { 2 } \\right \\} , \\end{align*}"}
{"id": "1894.png", "formula": "\\begin{align*} d ( ( a , e _ H ) , ( \\gamma _ a , \\gamma _ a ^ * ) ) = \\max \\{ d _ G ( a , \\gamma _ a ) , d _ H ( \\gamma _ a ^ * , e _ H ) \\} \\leq D , \\end{align*}"}
{"id": "7158.png", "formula": "\\begin{align*} u = \\alpha x _ d ^ + + \\beta x _ d ^ - \\end{align*}"}
{"id": "8132.png", "formula": "\\begin{align*} \\kappa ( g ) = \\sum _ { ( d , e ) \\in D _ { 2 } ( H G ) } | g ( d ) \\cap g ( e ) | = 7 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "3825.png", "formula": "\\begin{align*} \\zeta _ { h _ { \\vert ( V _ f ^ I ) ^ { \\langle ( \\underline { \\lambda } , \\sigma ) \\rangle } / C _ { G \\rtimes S } ( \\underline { \\lambda } , \\sigma ) } } ( t ) = ( 1 - t ^ { m _ I } ) ^ { r _ I ( \\underline { \\lambda } , \\sigma ) } \\ , , \\end{align*}"}
{"id": "714.png", "formula": "\\begin{align*} d _ 0 : = \\end{align*}"}
{"id": "8428.png", "formula": "\\begin{align*} D _ { U _ { i } } Y _ { s } ^ { x ; l ^ { \\epsilon } } = \\frac { s - l _ { 0 } ^ { \\epsilon } } { t - l _ { 0 } ^ { \\epsilon } } \\ , \\nabla _ { v _ i } Y _ { s } ^ { x ; l ^ { \\epsilon } } , l _ { 0 } ^ { \\epsilon } \\leq s \\leq t . \\end{align*}"}
{"id": "2695.png", "formula": "\\begin{align*} M = \\begin{pmatrix} m _ 0 & m _ { - 1 } & m _ { - 2 } & \\ldots & \\ldots & m _ { - k } \\\\ m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & \\ldots & m _ { - ( k - 1 ) } \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\ m _ l & \\ldots & \\ldots & \\ldots & \\ldots & \\cdot \\end{pmatrix} \\end{align*}"}
{"id": "4515.png", "formula": "\\begin{align*} E _ { \\psi } ( x ) \\le \\varepsilon _ s ( x ) : = \\varepsilon _ 1 ( x , T ) + 2 ( S _ 0 + B _ 1 ( T _ 0 , T ) ) x ^ { - 1 / 2 } \\end{align*}"}
{"id": "7250.png", "formula": "\\begin{align*} \\sigma _ 1 ^ 2 = z _ 1 ^ 2 + z _ 2 ^ 2 + z _ 3 ^ 2 + 2 \\sigma _ 2 , \\sigma _ 2 ^ 2 = z _ 1 ^ 2 z _ 2 ^ 2 + z _ 1 ^ 2 z _ 3 ^ 2 + z _ 2 ^ 2 z _ 3 ^ 2 + 2 \\sigma _ 1 \\sigma _ 3 , \\end{align*}"}
{"id": "6440.png", "formula": "\\begin{align*} \\sigma _ \\gamma ( a \\otimes b ) = \\widetilde { \\phi } _ \\gamma ( a ) ( b ) \\ ; , \\end{align*}"}
{"id": "3191.png", "formula": "\\begin{align*} \\nu ( j ) = \\min \\left \\lbrace k \\geq 0 : y _ { j } \\in \\mathcal C _ k ^ j \\right \\rbrace . \\end{align*}"}
{"id": "7612.png", "formula": "\\begin{align*} 1 \\leq \\sum _ { i } \\frac { \\partial f ^ { * } } { \\partial \\kappa _ { i } } ( \\kappa ^ { * } ) = \\frac { 1 } { ( f ( \\kappa ) ) ^ { 2 } } \\frac { \\partial f } { \\partial \\kappa _ { i } } ( \\kappa ) \\kappa _ { i } ^ { 2 } . \\end{align*}"}
{"id": "6174.png", "formula": "\\begin{align*} \\frac { 1 7 2 } { 4 4 1 } f \\left ( \\frac { 1 } { 2 } \\right ) - \\frac { 1 6 2 5 } { 1 6 1 1 } f \\left ( \\frac { 1 } { 5 } \\right ) + \\frac { 4 2 5 9 2 } { 2 6 3 1 3 } f \\left ( - \\frac { 2 7 } { 4 4 } \\right ) = - \\frac { 1 } { 4 \\pi \\sqrt { - 1 } } \\int _ { S ^ 1 } f ( z ) e ^ { - \\frac { 2 } { z } } d z \\end{align*}"}
{"id": "3998.png", "formula": "\\begin{align*} ( \\sigma _ T , v _ T ) & = \\sum _ { k \\geq K _ 0 } a _ k \\Phi _ { \\lambda ^ p _ { k } } + \\sum _ { | k | \\geq K _ 0 } b _ k \\Phi _ { \\lambda ^ h _ k } \\\\ & = \\sum _ { k \\geq K _ 0 } a _ k ( \\xi _ { \\lambda ^ p _ k } , \\eta _ { \\lambda ^ p _ k } ) + \\sum _ { | k | \\geq K _ 0 } b _ k ( \\xi _ { \\lambda ^ h _ k } , \\eta _ { \\lambda ^ h _ k } ) , \\end{align*}"}
{"id": "7217.png", "formula": "\\begin{align*} \\begin{aligned} | \\tilde Y _ { s , t } ( x , v ) | & \\lesssim \\frac { \\delta ( t - s ) } { 1 + \\frac { \\langle x ^ \\perp \\rangle ^ 2 } { \\langle v ^ \\perp \\rangle ^ 2 } } , \\\\ | \\nabla _ v \\tilde Y _ { s , t } ( x , v ) | & \\lesssim \\log ( 2 + t ) \\frac { \\delta ( t - s ) } { 1 + \\frac { \\langle x ^ \\perp \\rangle } { \\langle v ^ \\perp \\rangle } } . \\end{aligned} \\end{align*}"}
{"id": "9497.png", "formula": "\\begin{align*} \\mathfrak { b c } = \\sum _ { i = 0 } ^ { s / 2 } \\binom { ( s + d - 1 ) / 2 } { \\lfloor i / 2 \\rfloor , ( d + 1 ) / 2 + \\lfloor ( i - 1 ) / 2 \\rfloor , s / 2 - i } . \\end{align*}"}
{"id": "1478.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\iota ( g _ 1 z _ 2 , g _ 2 z _ 2 ) \\\\ 1 \\end{array} \\right ] = R ^ { - 1 } \\left [ \\begin{array} { c c } U ( g _ 1 z _ 1 ) & 0 \\\\ 0 & \\mathfrak { J } \\overline { U ( g _ 2 z _ 2 ) } \\end{array} \\right ] B ( g _ 1 z _ 1 , g _ 2 z _ 2 ) ^ { - 1 } S \\end{align*}"}
{"id": "5754.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\nabla } z ^ { j } ( h _ { s t } ) ( \\bar { q } ) & = \\pi _ { * } ( \\nabla y ^ { j } ) ( h ( \\bar { \\nabla } z ^ { s } , \\bar { \\nabla } z ^ { t } ) ) ( \\pi ( q ) ) = \\nabla y ^ { j } ( g ( \\nabla y ^ { s } , \\nabla y ^ { t } ) ) ( q ) \\\\ & = y ^ { s } ( \\nabla y ^ { j } , \\nabla y ^ { t } ) ( q ) + y ^ { t } ( \\nabla y ^ { s } , \\nabla y ^ { j } ) ( q ) \\end{aligned} \\end{align*}"}
{"id": "1466.png", "formula": "\\begin{align*} g z = ( a z + b u _ 0 ) ( c z + d u _ 0 ) ^ { - 1 } u _ 0 , \\lambda ( g , z ) = u _ 0 ^ { - 1 } ( c z + d u _ 0 ) , g = \\left [ \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ] \\in G _ { \\infty } . \\end{align*}"}
{"id": "7001.png", "formula": "\\begin{align*} \\det { \\mathfrak { S } } = A _ N ^ K B _ K ^ N \\prod _ { i = 1 } ^ { N } \\prod _ { j = 1 } ^ { K } ( \\alpha _ i - \\beta _ j ) . \\end{align*}"}
{"id": "5912.png", "formula": "\\begin{align*} ( 2 k - 1 ) \\ , \\left ( \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , | \\lambda _ i | \\right ) ^ 2 & \\geq 4 k \\ , \\sum \\limits _ { i < j } \\ , | \\lambda _ i | \\ , | \\lambda _ j | , \\\\ & = 2 k \\ , \\left [ \\left ( \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , | \\lambda _ i | \\right ) ^ 2 - \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , \\lambda ^ 2 _ i \\right ] , \\end{align*}"}
{"id": "156.png", "formula": "\\begin{align*} G _ { 0 ^ + } ( u ) : = \\int _ { 0 } ^ { + \\infty } P _ t ( u ) d t , u \\in H _ 0 , \\end{align*}"}
{"id": "5058.png", "formula": "\\begin{align*} R ( \\alpha ) = - \\Big ( \\pi \\mu \\frac { \\Gamma ( \\frac { \\gamma ^ 2 } { 4 } ) } { \\Gamma ( 1 - \\frac { \\gamma ^ 2 } { 4 } ) } \\Big ) ^ { 2 \\frac { ( Q - \\alpha ) } { \\gamma } } \\frac { \\Gamma ( - \\frac { \\gamma ( Q - \\alpha ) } { 2 } ) \\Gamma ( - \\frac { 2 ( Q - \\alpha ) } { \\gamma } ) } { \\Gamma ( \\frac { \\gamma ( Q - \\alpha ) } { 2 } ) \\Gamma ( \\frac { 2 ( Q - \\alpha ) } { \\gamma } ) } . \\end{align*}"}
{"id": "7794.png", "formula": "\\begin{align*} \\Delta _ 1 ( \\Gamma ) = \\sum _ { b \\in B ( \\Gamma ) } \\sum _ { c _ 1 , c _ 2 \\in C ( b ) } e _ { c _ 1 c _ 2 } ( \\Gamma ) . \\end{align*}"}
{"id": "2505.png", "formula": "\\begin{align*} U \\ , \\pi _ 1 ( \\mathbf { h } ) \\ , U ^ { - 1 } = \\pi _ 2 ( \\mathbf { h } ) \\forall \\mathbf { h } \\in \\mathbf { H } . \\end{align*}"}
{"id": "2810.png", "formula": "\\begin{align*} e ^ { - i t - i \\theta ( t ) } u ( x + X ( t ) , t ) = ( 1 + \\alpha ( t ) ) Q ( x ) + h ( t , x ) , \\ ; \\forall t \\in D _ { \\delta _ 0 } , \\end{align*}"}
{"id": "6132.png", "formula": "\\begin{align*} j ( 2 j - 1 ) \\cdot s ^ { 2 j - 2 } \\psi ^ { 1 } _ { t } - k ( 2 k - 1 ) \\cdot t ^ { 2 k - 2 } \\varphi ^ { 1 } _ { s } = 0 , \\\\ \\varphi ^ { 1 } _ { t } + \\psi ^ { 1 } _ { s } = 0 , \\end{align*}"}
{"id": "5075.png", "formula": "\\begin{align*} f ( x ) - f ( y _ 1 ) & = | f ( x ) - f ( y _ 1 ) | \\\\ & \\leq d ( x , y _ 1 ) \\\\ & = r . \\end{align*}"}
{"id": "1610.png", "formula": "\\begin{align*} \\Theta ^ { Q , j } = 0 , \\\\ W ^ { Q , j } = 0 . \\end{align*}"}
{"id": "165.png", "formula": "\\begin{align*} \\partial _ { \\alpha , \\beta } ( f ) ( x ) : = \\sqrt { 1 - x ^ 2 } f ' ( x ) . \\end{align*}"}
{"id": "5695.png", "formula": "\\begin{align*} I ( \\alpha , \\beta ) - J _ { 0 } ( \\alpha , \\beta ) = \\sum _ { i } \\mu _ { \\mathrm { d i s c } } ( \\alpha _ { i } ^ { m _ { i } } ) - \\sum _ { j } \\mu _ { \\mathrm { d i s c } } ( \\beta _ { j } ^ { n _ { j } } ) . \\end{align*}"}
{"id": "6203.png", "formula": "\\begin{align*} \\| \\hat { V } ^ { T } \\hat { V } - I \\| _ F = \\| R ^ { T } R - I \\| _ F \\leq \\xi . \\end{align*}"}
{"id": "2551.png", "formula": "\\begin{align*} S = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\in S p ( \\R , 2 d ) \\det ( A ) \\neq 0 . \\end{align*}"}
{"id": "1567.png", "formula": "\\begin{align*} L ( s , \\mathbf { f } , \\chi ) \\mathbf { f } ( g ) = c _ k ( s ) \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathfrak { E } ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h . \\end{align*}"}
{"id": "1906.png", "formula": "\\begin{align*} \\P ( \\theta \\in A | x ) & = \\max ( 0 , \\alpha ^ A - 0 . 4 \\delta ) , \\\\ \\P ( \\theta \\in B | x ) & = \\max ( 0 , \\alpha ^ B - 0 . 4 \\delta ) , \\\\ \\P ( \\theta \\in A \\cup B | x ) & = \\min ( 1 , \\alpha ^ A + \\alpha ^ B - 0 . 8 \\delta ) , \\end{align*}"}
{"id": "9351.png", "formula": "\\begin{align*} \\frac { 2 } { e _ { \\lambda } ( t ) + 1 } e _ { \\lambda } ^ { x } ( t ) = \\sum _ { n = 0 } ^ { \\infty } \\mathcal { E } _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 2 ] ) . \\end{align*}"}
{"id": "3141.png", "formula": "\\begin{align*} 2 \\sum ( x ^ 4 y z + y ^ 3 z ^ 3 ) - 2 \\sum x ^ 4 y ^ 2 + \\sum x ^ 3 y ^ 2 z - 6 x ^ 2 y ^ 2 z ^ 2 = 0 \\ , . \\end{align*}"}
{"id": "2928.png", "formula": "\\begin{align*} \\mathcal J = \\mathcal J _ n = \\left \\{ \\mathbf { i } = ( i _ 1 , i _ 2 , i _ 3 , i _ 4 ) \\in \\{ 1 , \\dots , n \\} ^ 4 : i _ 1 < i _ 2 , i _ 3 < i _ 4 \\right \\} . \\end{align*}"}
{"id": "9230.png", "formula": "\\begin{align*} x + u = _ X J ^ A _ 1 ( x + u ) + ( ( x + u ) - J ^ A _ 1 ( x + u ) ) \\end{align*}"}
{"id": "8789.png", "formula": "\\begin{align*} \\left \\{ ( \\lambda , \\delta ) \\in \\Lambda \\times \\{ 0 , 1 \\} ^ { \\lceil \\log _ 2 l _ i \\rceil } \\left | ( \\gamma _ i , \\delta _ i ) \\in M , \\ \\sum _ { j \\in \\varpi _ i ( t ) } \\lambda _ { i j } \\geq \\gamma _ { i t } \\ ; t \\in \\{ 1 , \\ldots , l _ i \\} , \\ \\gamma _ { i t } = 0 \\{ l _ i + 1 , \\ldots 2 ^ { \\lceil \\log _ 2 l _ i \\rceil } \\right . \\right \\} \\end{align*}"}
{"id": "3075.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ( y _ 1 ^ 2 + a y _ 2 ^ 2 ) + x _ 2 ^ 3 y _ 2 ( a y _ 1 ^ 2 + y _ 2 ^ 2 ) = 0 \\ , . \\end{align*}"}
{"id": "2717.png", "formula": "\\begin{align*} \\Sigma _ { d , n } = \\left \\{ P \\in \\R [ X _ 1 , \\ldots , X _ n ] _ { \\leq d } \\ ; \\mid \\ ; P = \\sum _ { 0 \\le i \\le d } a _ i \\sigma _ { i , n } , a _ i \\in \\R , 0 \\leq i \\leq d \\right \\} . \\end{align*}"}
{"id": "7874.png", "formula": "\\begin{align*} W = ( \\nabla F , \\nabla ^ 2 F ) , \\end{align*}"}
{"id": "4777.png", "formula": "\\begin{align*} A _ { u , v } = \\begin{cases} 1 & u , v \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "9096.png", "formula": "\\begin{align*} \\alpha \\alpha ' \\alpha ^ { - 1 } \\alpha '^ { - 1 } = 1 + ( \\alpha _ m \\alpha ' _ { m ' } - \\alpha ' _ { m ' } \\alpha _ { m } ) t ^ { m + m ' } + \\cdots \\end{align*}"}
{"id": "2252.png", "formula": "\\begin{align*} \\| X ( t _ m ) - & \\widetilde { X } _ m ^ { M , N } \\| _ { L ^ p ( \\Omega ; H ) } = \\| ( E ( t _ m ) - E _ { k , N } ^ m ) X _ 0 \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\Big \\| \\int _ 0 ^ { t _ m } E ( t _ m - s ) A P F ( X ( s ) ) \\ , \\dd s - k \\sum _ { j = 1 } ^ m E _ { k , N } ^ { m - j + 1 } A P F ( X ( t _ j ) ) \\Big \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\Big \\| \\sum _ { j = 1 } ^ m \\int _ { t _ { j - 1 } } ^ { t _ j } \\big ( E ( t _ m - s ) - E _ { k , N } ^ { m - j + 1 } \\big ) \\ , \\dd W ( s ) \\Big \\| _ { L ^ p ( \\Omega ; H ) } \\\\ = : & I + J + K . \\end{align*}"}
{"id": "8818.png", "formula": "\\begin{align*} \\partial _ t [ U _ { \\frac { t } { 2 } } F ( \\varphi ) ] = \\frac { 1 } { 2 } \\Delta U _ { \\frac { t } { 2 } } F ( \\varphi ) - \\frac { 1 } { 2 } \\left ( A _ { \\epsilon } \\varphi , \\nabla U _ { \\frac { t } { 2 } } F ( \\varphi ) \\right ) , \\end{align*}"}
{"id": "8315.png", "formula": "\\begin{align*} 0 \\le \\int _ s ^ \\tau ( v - y ) \\dd ( y - u ) = - \\varepsilon ( ( y - u ) ( \\tau ) - ( y - u ) ( s ) ) . \\end{align*}"}
{"id": "1673.png", "formula": "\\begin{align*} H _ n ^ { \\star } ( \\mathbf { s } ) = H _ n ^ { \\star } ( s _ 1 , s _ 2 , \\ldots , s _ r ) : = \\sum \\limits _ { 1 \\leq k _ 1 \\leq k _ 2 \\leq \\cdots \\leq k _ r \\leq n } \\prod _ { j = 1 } ^ { r } \\frac { 1 } { k _ j ^ { s _ j } } . \\end{align*}"}
{"id": "4939.png", "formula": "\\begin{align*} h = \\lambda ^ { - 1 } \\phi ( \\lambda ) . \\end{align*}"}
{"id": "7163.png", "formula": "\\begin{align*} H _ u ( x ' , u ( x ' , x _ d ) ) = x _ d , \\quad u ( x ' , H _ u ( x ' , y _ d ) ) = y _ { d } \\ , , \\end{align*}"}
{"id": "8622.png", "formula": "\\begin{align*} \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) = \\mathcal { K } _ { S } ( x , k ) + \\mathcal { K } _ { R } ( x , k ) , \\end{align*}"}
{"id": "6100.png", "formula": "\\begin{align*} X ' { \\theta ) } = \\lambda \\Lambda ( \\theta ) X ( \\theta ) . \\end{align*}"}
{"id": "1622.png", "formula": "\\begin{align*} \\tilde { \\Theta } ^ { Q , j } ( x ) & = \\Psi ( x _ { \\bar { j } } , x _ 1 , \\dots , x _ { k - 1 } , x _ { k + 1 } , \\dots , x _ { \\bar { j } - 1 } , x _ { \\bar { j } + 1 } , \\dots , x _ d ) , \\\\ \\tilde { H } ^ { Q , j } ( x ) & = \\pm \\tilde { \\Psi } ( x _ { \\bar { j } } , x _ 1 , \\dots , x _ { k - 1 } , x _ { k + 1 } , \\dots , x _ { \\bar { j } - 1 } , x _ { \\bar { j } + 1 } , \\dots , x _ d ) . \\end{align*}"}
{"id": "1404.png", "formula": "\\begin{align*} \\hat q = q + { \\epsilon \\over 2 } q p ^ b , \\end{align*}"}
{"id": "8313.png", "formula": "\\begin{align*} v ( t - ) & : = \\lim _ { [ 0 , T ] \\ni s \\to t ^ - } v ( s ) \\forall t \\in ( 0 , T ] , v ( 0 - ) : = v ( 0 ) , \\\\ v ( t + ) & : = \\lim _ { [ 0 , T ] \\ni s \\to t ^ + } v ( s ) \\forall t \\in [ 0 , T ) , v ( T + ) : = v ( T ) . \\end{align*}"}
{"id": "2312.png", "formula": "\\begin{align*} A b _ 0 ( x , \\omega ) = \\begin{cases} \\frac { \\sin ( \\pi \\omega ( 1 - | x | ) ) } { \\pi \\omega } , & | x | \\leq 1 \\\\ 0 , & \\end{cases} \\end{align*}"}
{"id": "7003.png", "formula": "\\begin{align*} \\min \\{ | A ( \\beta _ j ) | , | B ( \\alpha _ i ) | , 1 \\leq i \\leq N , 1 \\leq j \\leq K \\} = \\delta > 0 . \\end{align*}"}
{"id": "5643.png", "formula": "\\begin{align*} \\Delta u = \\varepsilon ^ { - 2 } \\left ( e ^ { u } - \\left ( x ^ { 2 } + y ^ { 2 } \\right ) e ^ { - u } \\right ) \\mathbb { R } ^ 2 \\end{align*}"}
{"id": "6982.png", "formula": "\\begin{align*} \\left . \\begin{aligned} & \\nabla g ( \\bar x ) ^ * y ^ * = 0 , \\ , \\nabla g ( \\bar x ) ^ * \\hat z ^ * = 0 , \\\\ & y ^ * \\in \\mathcal N _ D ( g ( \\bar x ) ; \\nabla g ( \\bar x ) u ) , \\ , \\hat z ^ * \\in D \\mathcal N _ D ( g ( \\bar x ) , y ^ * ) ( 0 ) \\end{aligned} \\right \\} \\quad \\Longrightarrow \\hat z ^ * = 0 \\end{align*}"}
{"id": "6722.png", "formula": "\\begin{align*} & h _ i ( t ) : = g _ i ' ( t ) - g _ i ' ( t ) ^ { ( - 1 ) } \\ \\\\ & R : = - \\prod _ { m = 1 } ^ { d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m ) q ^ { d - m } } \\bigl ( g _ 1 ' ( t ) ^ { ( - 1 ) } \\alpha _ 1 ^ { q ^ { d } } + \\cdots + g _ { r - 1 } ' ( t ) ^ { ( - 1 ) } \\alpha _ { r - 1 } ^ { q ^ { d } } + \\alpha _ r ^ { q ^ { d } } \\bigr ) \\end{align*}"}
{"id": "9162.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\log p \\cdot \\chi ( p ) = \\delta _ { \\chi = \\chi _ 0 } x + O ( \\sqrt { x } \\left ( \\log 2 q x ) ^ 2 \\right ) , \\end{align*}"}
{"id": "4271.png", "formula": "\\begin{align*} { u ^ { - } _ { 1 } ( t ) + u ^ { + } _ 1 ( t ) \\over 2 } ~ = ~ \\dot { y } _ 1 ( t ) ~ > ~ \\dot { y } _ 2 ( t ) ~ = ~ { u ^ { - } _ { 2 } ( t ) + u ^ { + } _ 2 ( t ) \\over 2 } \\ , . \\end{align*}"}
{"id": "175.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\sigma _ m ( f ) ( x ) = ( 1 + \\| x \\| ^ 2 ) \\Delta ( f ) ( x ) + 2 \\left ( 1 - m - \\frac { d } { 2 } \\right ) \\langle x ; \\nabla ( f ) ( x ) \\rangle , \\end{align*}"}
{"id": "5597.png", "formula": "\\begin{align*} s \\star a = ( \\overline { \\mathcal { R } } _ 1 \\rhd a ) \\star ( \\overline { \\mathcal { R } } _ 2 \\rhd s ) \\end{align*}"}
{"id": "1322.png", "formula": "\\begin{align*} 2 = I ( \\alpha _ { k + 1 } , \\alpha _ { k } ) = & I ( \\hat { \\alpha } \\cup { ( \\gamma , M ) } , \\hat { \\alpha } \\cup { ( \\delta , 1 ) } \\cup { ( \\gamma , M - p _ { i } ) } ) \\\\ = & c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z , Z ) + 2 ( M - p _ { i } ) Q _ { \\tau } ( Z , Z _ { \\gamma } ) \\\\ & + \\sum _ { k = M - p _ { i } + 1 } ^ { M } ( 2 \\lfloor k \\theta \\rfloor + 1 ) - \\mu _ { \\tau } ( \\delta ) . \\end{align*}"}
{"id": "1448.png", "formula": "\\begin{align*} G ( \\mathbb { A } ) = \\bigcup _ { j } G ( \\Q ) t _ j K _ 1 ( \\mathfrak { n } ) G ( \\R ) . \\end{align*}"}
{"id": "6341.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k d _ \\ell \\leq k ( k - 1 ) + \\sum _ { \\ell = k + 1 } ^ n \\min ( k , d _ \\ell ) . \\end{align*}"}
{"id": "4409.png", "formula": "\\begin{align*} f '' ( \\xi ) = a ( \\xi ) f ' ( \\xi ) + + b ( \\xi ) f ( \\xi ) \\end{align*}"}
{"id": "1303.png", "formula": "\\begin{align*} \\sum _ { n = p _ { i } + 1 } ^ { p _ { i + 1 } } | \\Lambda _ { ( n , \\infty ) } ( M , \\Gamma ) | \\geq N | \\Lambda _ { ( p _ { i + 1 } , \\infty ) } ( M , \\Gamma ) | \\end{align*}"}
{"id": "8059.png", "formula": "\\begin{align*} \\left \\langle E ^ { i j } _ 0 , { t _ { c } } _ { * } f \\otimes { t _ { c } } _ { * } g \\right \\rangle = \\left ( \\mathfrak { P } _ { \\ell } t _ c \\left \\{ \\Psi ^ i ( f ) , \\Psi ^ j ( g ) \\right \\} _ { \\ell } ^ { \\Sigma _ 0 } \\right ) [ 0 ] . \\end{align*}"}
{"id": "523.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq w \\leq h \\\\ 1 \\leq z \\leq s } } ( - 1 ) ^ { w + z } \\sum _ { \\substack { y _ 1 + . . . + y _ w = h \\\\ x _ 1 + . . . + x _ z = s } } \\sum _ { \\substack { 0 \\leq p _ w \\leq . . . \\leq p _ 1 \\leq i - h d \\\\ i - h d + 1 \\leq v _ 1 \\leq . . . \\leq v _ z \\leq r + 1 - j d } } \\prod _ { a = 1 } ^ w i _ a \\alpha ^ { i _ a } _ { r - ( h - y _ 1 - . . . - y _ a ) d - p _ a } \\prod _ { b = 1 } ^ z x _ b \\alpha ^ { x _ b } _ { r - ( j - x _ b - . . . - x _ z ) d - v _ b } . \\end{align*}"}
{"id": "9521.png", "formula": "\\begin{align*} \\begin{gathered} A ^ * y + c = p , \\\\ A x - b \\in K , y \\in K ^ * , ( A x - b ) \\cdot y = 0 , \\end{gathered} \\end{align*}"}
{"id": "461.png", "formula": "\\begin{align*} \\sum _ { c \\in D ( \\Gamma _ { x y } ) } \\prod _ { \\iota \\in 2 ^ { V ' } } h _ { x y } ^ \\iota ( c ) = \\sum _ { c \\in D ( \\Gamma _ { x y } ) } \\prod _ { \\iota \\in 2 ^ { V ' } } \\sum _ { b \\in \\Lambda _ { x y } ( V ' , \\iota ) ( c _ \\iota ) } g ^ \\iota ( b ) \\end{align*}"}
{"id": "4767.png", "formula": "\\begin{align*} \\textstyle S _ 0 ( \\bigwedge ^ 2 V ) = U _ 1 \\oplus U _ 2 \\oplus U _ 3 , \\end{align*}"}
{"id": "7818.png", "formula": "\\begin{align*} f ( s , r ; x ) z : = [ \\Gamma ( \\gamma ) ] ^ { - 1 } ( z , ( s - r ) ^ { \\gamma - 1 } [ S ( s - r ) ] ^ * x ) _ H , z \\in H . \\end{align*}"}
{"id": "5063.png", "formula": "\\begin{align*} \\varphi ( a ^ 3 \\cdot v _ 0 ) = [ \\varphi ( a ) , a ^ 2 \\cdot v _ 0 ] + [ a , \\varphi ( a ^ 2 \\cdot v _ 0 ) ] = 3 \\lambda ( a \\cdot v _ 0 ) a ^ 2 \\cdot v _ 0 , \\end{align*}"}
{"id": "7784.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log \\prod \\limits _ { i = 1 } ^ { n } C _ { w _ i } = \\sum \\limits _ { k \\in T } p _ k \\log C _ k . \\end{align*}"}
{"id": "349.png", "formula": "\\begin{align*} \\mathcal A ( \\rho ^ { * , \\infty } , S ^ { * , \\infty } ) = \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho ^ { * , \\infty } , m ^ { * , \\infty } , S ) \\ge \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H ^ 1 } \\mathcal L ( \\rho , m , S ) , \\end{align*}"}
{"id": "1239.png", "formula": "\\begin{align*} \\sum _ { s , t = 1 } ^ { Q } \\mu ( L _ { B , s } \\cap L _ { B , t } ) \\leq \\frac { C } { \\mu ( B ) } \\left ( \\sum _ { n = 1 } ^ { Q } \\mu ( L _ { B , n } ) \\right ) ^ 2 . \\end{align*}"}
{"id": "8158.png", "formula": "\\begin{align*} L ( 1 , \\chi ) = \\frac { \\pi } { 2 f } \\sum _ { a = 1 } ^ { f - 1 } \\chi ( a ) \\cot \\left ( \\frac { \\pi a } { f } \\right ) \\ \\ \\ \\ \\ ( \\chi \\in X _ f ^ - ) . \\end{align*}"}
{"id": "3299.png", "formula": "\\begin{align*} \\beta _ { d - j } ( E | _ D ) \\coloneqq \\alpha _ { d - j } ( E | _ D ) + \\frac { a } { 2 } \\alpha _ { d - j + 1 } ( E | _ D ) + \\sum _ { l = 1 } ^ { j - 1 } \\frac { ( - 1 ) ^ { l - 1 } B _ { 2 l } } { ( 2 l ) ! } a ^ { 2 l } \\alpha _ { d - j + 1 + l } ( E | _ D ) , \\end{align*}"}
{"id": "7890.png", "formula": "\\begin{align*} \\int _ U \\varphi ( x ) \\bar { \\nu } ( x ) d x & \\geq \\bigg ( \\frac { R + T } { R } \\bigg ) ^ n \\cdot \\int _ { Q _ 1 } \\varphi _ { + } ( x ) \\bar { \\nu } ( x ) \\ d x - C ( n ) \\cdot ( T / R ) \\cdot ( \\sup _ U \\varphi ) \\cdot | | \\bar { \\nu } | | _ { L ^ \\infty ( Q _ 1 ) } \\cdot | Q _ 1 | \\\\ & - \\omega _ \\varphi ( \\delta ) \\cdot | | \\bar { \\nu } | | _ { L ^ \\infty ( Q _ 1 ) } \\cdot | Q _ 1 | . \\end{align*}"}
{"id": "3700.png", "formula": "\\begin{align*} \\partial _ t B _ q + a B J _ { x , q } + b J B _ { x , q } + \\mu \\Lambda ^ \\alpha B _ q = a [ B , \\Delta _ q ] J _ x - b [ J , \\Delta _ q ] B _ x . \\end{align*}"}
{"id": "4126.png", "formula": "\\begin{align*} ( g _ 1 ' , g _ 2 ' , \\dotsc , g _ d ' ) = S _ 1 ( p _ { g _ 1 } ) + S _ 2 ( p _ { g _ 2 } ) + \\dotsb + S _ d ( p _ { g _ d } ) . \\end{align*}"}
{"id": "7916.png", "formula": "\\begin{align*} \\sum _ l ( \\partial ^ 2 _ { t _ i t _ j } \\varphi ^ l ) ( \\partial _ { t _ s } \\varphi ^ l ) = 0 , \\forall \\ i , j , s = 1 , \\dots , n - m . \\end{align*}"}
{"id": "6214.png", "formula": "\\begin{align*} \\| V _ l - Z \\| _ F ^ 2 = { \\rm T r } [ ( V _ l - Z ) ^ { T } ( V _ l - Z ) ] = 2 l - 2 { \\rm T r } ( Z ^ { T } V _ l ) \\leq 2 l - 2 \\left [ l - \\frac { 2 0 l \\epsilon \\| C \\| _ F ^ 2 } { \\eta } \\right ] = \\frac { 4 0 l \\epsilon \\| C \\| _ F ^ 2 } { \\eta } . \\end{align*}"}
{"id": "7900.png", "formula": "\\begin{align*} \\tilde { \\Lambda } = \\begin{bmatrix} \\Lambda & \\Theta ( v ) \\\\ \\Theta ^ T ( v ) & \\Lambda \\end{bmatrix} \\end{align*}"}
{"id": "9502.png", "formula": "\\begin{align*} \\phi ( p , - \\lambda ) = \\phi ( p , \\lambda ) , \\ , \\ , \\tau ( p , - \\lambda ) = - \\tau ( p , \\lambda ) \\end{align*}"}
{"id": "1791.png", "formula": "\\begin{align*} \\mathfrak { p } = ( \\mathfrak { p } \\cap \\mathfrak { m } ) \\oplus \\mathfrak { a } \\oplus ( \\mathfrak { k } / ( \\mathfrak { k } \\cap \\mathfrak { m } ) ) , \\end{align*}"}
{"id": "2254.png", "formula": "\\begin{align*} X ( t _ j ) = E ( t _ j - s ) X ( s ) - \\int _ { s } ^ { t _ j } E ( t _ j - \\sigma ) A F ( X ( \\sigma ) ) \\ , \\dd \\sigma + \\int _ { s } ^ { t _ j } E ( t _ j - \\sigma ) \\ , \\dd W ( \\sigma ) , \\end{align*}"}
{"id": "5030.png", "formula": "\\begin{align*} \\mathrm { S p i n } ( 2 p ) \\cdot \\mathrm { S p i n } ( 2 q + 1 ) = & \\big \\{ \\psi \\in \\mathrm { S p i n } ( 2 p + 2 q + 1 ) \\ , ; \\ , \\psi = \\varphi \\ , \\phi \\ , , \\\\ & \\qquad \\varphi \\in \\mathrm { S p i n } ( 2 p ) \\ , , \\phi \\in \\mathrm { S p i n } ( 2 q + 1 ) \\big \\} \\ , , \\end{align*}"}
{"id": "3652.png", "formula": "\\begin{align*} \\log x ( 1 + 2 \\log \\log t ) & = \\frac { \\log ^ { 5 / 3 } t ( \\log \\log x ) ^ { 1 / 3 } ( 1 + 2 \\log \\log t ) } { B _ 0 ^ { 5 / 3 } } \\\\ & = 3 c \\log ^ { 5 / 3 } t ( \\log \\log t ) ^ { 4 / 3 } \\cdot \\left ( 1 + \\frac { 1 } { 2 \\log \\log t } \\right ) \\cdot \\left ( \\frac { \\frac { 3 } { 5 } \\log \\log x } { \\log \\log t } \\right ) ^ { 1 / 3 } \\\\ & \\geq 3 c \\log ^ { 5 / 3 } t ( \\log \\log t ) ^ { 4 / 3 } \\cdot \\left ( \\frac { \\frac { 3 } { 5 } \\log \\log x } { \\log \\log t } \\right ) ^ { 1 / 3 } , \\end{align*}"}
{"id": "3284.png", "formula": "\\begin{align*} s ^ 2 + r ^ 2 - \\frac { s } { h ( s ) } = 0 , \\end{align*}"}
{"id": "8090.png", "formula": "\\begin{align*} \\frac { d ^ { n } } { d \\hbar ^ n } \\omega _ { H , \\psi } ( \\Psi ^ i ( z _ 1 ) \\star \\Psi ^ j ( z _ 2 ) ) | _ { \\hbar = 0 } = \\int _ { \\Sigma ^ { 2 n } } \\big [ W _ { \\Sigma } ( y _ 1 , y _ 2 ) \\cdots W _ { \\Sigma } ( y _ { 2 n - 1 } , y _ { 2 n } ) \\\\ K ^ i _ { n , \\psi } ( z , y _ 1 , \\ldots , y _ { 2 n - 1 } ) K ^ j _ { n , \\psi } ( z , y _ 2 , \\ldots , y _ { 2 n } ) \\big ] \\ , \\mathrm { d } y _ 1 \\cdots \\mathrm { d } y _ { 2 n } . \\end{align*}"}
{"id": "3070.png", "formula": "\\begin{align*} ( x _ 1 y _ 1 + x _ 2 y _ 2 ) ^ 3 + b ( x _ 1 y _ 1 - x _ 2 y _ 2 ) ( x _ 1 y _ 2 - x _ 2 y _ 1 ) ( x _ 1 y _ 2 + x _ 2 y _ 2 ) = 0 \\ , . \\end{align*}"}
{"id": "8573.png", "formula": "\\begin{align*} m _ + ( x , 0 ) = - m _ - ( x , 0 ) . \\end{align*}"}
{"id": "3221.png", "formula": "\\begin{align*} \\tilde \\sigma ( B ) = \\textstyle { \\frac 1 2 } ( \\sigma ( B ) + \\sigma ( - B ) ) , \\end{align*}"}
{"id": "7657.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta P _ { \\Omega _ { \\varepsilon } } w + \\tilde { \\lambda } _ 0 \\underline { m } P _ { \\Omega _ { \\varepsilon } } w = f _ 0 ( w ) & \\Omega _ { \\varepsilon } \\ ; , \\\\ P _ { \\Omega _ { \\varepsilon } } w = 0 & \\partial \\Omega _ { \\varepsilon } \\ ; , \\end{cases} \\end{align*}"}
{"id": "1386.png", "formula": "\\begin{align*} | I _ 1 | = \\left | h ( n _ 1 , E _ 1 ) \\frac { ( 1 - x ) ^ \\frac { 1 } { 2 } } { n _ 1 - 1 } \\frac { ( 1 - x ) ^ \\frac { 1 } { 2 } } { n _ 2 - 1 } \\frac { n _ 2 - n _ 1 } { ( 1 - x ) ^ \\frac { 1 } { 2 } } \\right | \\le C \\frac { | n _ 1 - n _ 2 | } { ( 1 - x ) ^ \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "317.png", "formula": "\\begin{align*} \\tau ^ * ( \\rho ( 0 ) , S ( 0 ) ) = + \\infty , \\ ; \\ ; \\lim _ { t \\to \\tau ^ * } \\min _ { i = 1 } ^ N \\rho _ i ( t ) = 0 \\ ; \\ ; \\lim _ { t \\to \\tau ^ * } S ( t ) = \\infty , \\ ; \\ ; \\end{align*}"}
{"id": "3433.png", "formula": "\\begin{align*} | H _ k f ( x ) | & = \\bigg | \\int _ { \\Bbb R ^ N } H _ k ( x , y ) [ f ( y ) - f ( x ) ] d \\omega ( y ) \\bigg | \\\\ & \\leqslant C \\int _ { | x - y | \\leqslant r ^ { 6 - ( k - M ) } } r ^ { - M } ( V _ { k - M - 1 } ( x ) ) ^ { - 1 } \\| x - y \\| ^ s \\| f \\| _ s d \\omega ( y ) \\\\ & \\leqslant C r ^ { - M } r ^ { - ( k - M ) s } \\| f \\| _ s . \\end{align*}"}
{"id": "3279.png", "formula": "\\begin{align*} s h ( s ) = \\frac { s ^ 2 } { s ^ 2 + r ^ 2 } . \\end{align*}"}
{"id": "3707.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\frac { t } { 2 } } t ^ { \\frac { \\beta } { \\alpha } } ( t - \\tau ) ^ { - 1 + \\frac { 2 \\gamma - \\beta } { \\alpha } } \\tau ^ { - \\frac { 2 \\gamma } { \\alpha } } \\ , d \\tau \\\\ = & \\int _ 0 ^ { \\frac { 1 } { 2 } } ( 1 - \\tau ' ) ^ { - 1 + \\frac { 2 \\gamma - \\beta } { \\alpha } } ( \\tau ' ) ^ { - \\frac { 2 \\gamma } { \\alpha } } \\ , d \\tau ' \\\\ \\lesssim & \\int _ 0 ^ { \\frac { 1 } { 2 } } ( \\tau ' ) ^ { - \\frac { 2 \\gamma } { \\alpha } } \\ , d \\tau ' \\lesssim 1 \\end{align*}"}
{"id": "4923.png", "formula": "\\begin{align*} \\eta _ t \\norm { \\nabla f ( \\bar x _ t ) } ^ 2 & \\leq f ( x _ t ) - f ( x _ { t + 1 } ) + { L \\eta _ t ^ 2 } \\norm { \\nabla f ( \\bar x _ t ) } ^ 2 + \\frac { L } { 2 } ( 1 - \\alpha _ t ) ^ 2 \\Gamma _ { t - 1 } \\sum _ { k = 1 } ^ { t - 1 } \\frac { \\alpha _ k } { \\Gamma _ k } \\frac { ( \\eta _ k - \\gamma _ k ) ^ 2 } { \\alpha _ k ^ 2 } \\norm { \\nabla f ( \\bar x _ k ) } ^ 2 . \\end{align*}"}
{"id": "2912.png", "formula": "\\begin{align*} H : C = \\Pi _ d , \\end{align*}"}
{"id": "5808.png", "formula": "\\begin{align*} \\mathbf { y } _ k [ t ] = \\sum _ { m = 1 } ^ M \\mathbf { g } _ { m k } ^ N [ t ] \\otimes \\mathbf { x } _ m [ t ] + \\mathbf { z } _ k [ t ] , \\end{align*}"}
{"id": "3265.png", "formula": "\\begin{align*} ( s - t ) ^ 2 - \\frac { s - t } { h ( s ) } + | \\lambda | ^ 2 = 0 , \\end{align*}"}
{"id": "5221.png", "formula": "\\begin{align*} \\| T _ k f \\| _ { L ^ r ( K ) } & = \\lim _ { j \\to \\infty } \\| T _ k f _ j \\| _ { L ^ r ( K ) } \\\\ & \\leq C q ^ { - k } \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\lim _ { j \\to \\infty } \\| f _ j \\| _ { L ^ r ( K ) } \\\\ & \\leq C q ^ { - k } \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\| f \\| _ { L ^ r ( K ) } , \\end{align*}"}
{"id": "6467.png", "formula": "\\begin{align*} E [ ( S _ n ) ^ 2 ] \\sim \\begin{cases} \\dfrac { n } { 1 - 2 \\alpha } & ( - 1 < \\alpha < 1 / 2 ) , \\\\ [ 3 m m ] n \\log n & ( \\alpha = 1 / 2 ) , \\\\ [ 1 m m ] \\dfrac { n ^ { 2 \\alpha } } { ( 2 \\alpha - 1 ) \\Gamma ( 2 \\alpha ) } & ( 1 / 2 < \\alpha < 1 ) . \\end{cases} \\end{align*}"}
{"id": "1213.png", "formula": "\\begin{align*} \\eta ( V ) = \\eta ( B ) \\times \\frac { \\mu ( B ^ { [ V ] } ) } { \\sum _ { V ^ { \\prime } \\in \\mathcal { F } ^ { B } } \\mu ( B ^ { [ V ^ { \\prime } ] } ) } . \\end{align*}"}
{"id": "5909.png", "formula": "\\begin{align*} S E ( G ) = n - 2 k + \\sum \\limits ^ { 2 k } _ { i = 1 } \\ , | \\lambda _ i | , \\end{align*}"}
{"id": "1470.png", "formula": "\\begin{align*} g z = ( a z + b ) ( c z + d ) ^ { - 1 } , \\lambda ( g , z ) = c z + d , g = \\left [ \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ] . \\end{align*}"}
{"id": "8782.png", "formula": "\\begin{align*} \\begin{aligned} z _ { i 0 } = 1 z _ { i j } = \\frac { s _ { i j } - s _ { i j - 1 } } { a _ { i j } - a _ { i j - 1 } } j = 1 , \\ldots , n _ i . \\end{aligned} \\end{align*}"}
{"id": "4697.png", "formula": "\\begin{align*} J _ 1 = & - \\sum _ { \\substack { i , j = 1 , \\\\ i \\not = j } } ^ n \\Bigg [ \\partial _ y \\bigg ( \\frac { p \\sigma _ j ( \\tau _ i Q ) ^ { p - 1 } } { x _ { i j } ^ 2 } - \\frac { 2 p \\kappa _ 0 \\sigma _ j \\partial _ y [ \\tau _ i ( y Q ^ { p - 1 } ) ] } { x _ { i j } ^ 3 } \\bigg ) + \\frac { a _ { i j } \\sigma _ i \\tau _ i ( \\Lambda Q ) } { x _ { i j } ^ 3 } \\Bigg ] . \\end{align*}"}
{"id": "4234.png", "formula": "\\begin{align*} g '' ( 0 ) = \\frac { 2 a ^ 3 } { \\sqrt { D } } , \\ g ''' ( 0 ) = \\frac { 6 a ^ 4 b } { D } . \\end{align*}"}
{"id": "3387.png", "formula": "\\begin{align*} & \\omega ( x , y , z ) + \\omega ( y , z , x ) + \\omega ( z , x , y ) \\\\ = & \\varphi ( [ x , y ] , z ) - \\rho ( z ) \\varphi ( x , y ) + \\varphi ( [ y , z ] , x ) - \\rho ( x ) \\varphi ( y , z ) + \\varphi ( [ z , x ] , y ) - \\rho ( y ) \\varphi ( z , x ) \\\\ = & \\partial _ { \\rho } ( \\varphi ) ( x , y , z ) = 0 . \\end{align*}"}
{"id": "8306.png", "formula": "\\begin{align*} J _ { \\text r } ^ - ( \\kappa _ x , \\kappa _ y ) = J _ + ( \\kappa _ x , \\kappa _ y ) R _ - ( \\kappa _ x , \\kappa _ y ) . \\end{align*}"}
{"id": "4031.png", "formula": "\\begin{align*} \\begin{cases} m _ { 1 } = - k ^ 2 \\pi ^ 2 - 2 c _ k k \\pi - 2 i d _ k k \\pi + O ( 1 ) , \\forall k \\geq k _ 0 , \\\\ m _ { 2 } = - \\frac { 1 } { 2 } + d _ k - i ( k \\pi + c _ k ) + O ( k ^ { - 1 } ) , \\forall k \\geq k _ 0 , \\\\ m _ { 3 } = - \\frac { 1 } { 2 } - d _ k + i ( k \\pi + c _ k ) + O ( k ^ { - 1 } ) , \\forall k \\geq k _ 0 . \\end{cases} \\end{align*}"}
{"id": "836.png", "formula": "\\begin{align*} 2 ^ { - 2 \\beta / ( \\beta - 1 ) } H _ R ^ { - 2 \\beta } = H _ { R / 2 } ^ { - 2 \\beta } \\le \\omega ( x , y ) \\le H _ R ^ { - 2 \\beta } . \\end{align*}"}
{"id": "3406.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { j } ( x , x _ { Q } ) q _ { j } h ( x _ { Q } ) , \\end{align*}"}
{"id": "3988.png", "formula": "\\begin{align*} \\mathcal W _ { [ a , a + T ] } = \\big ( \\{ 1 \\} \\cup \\{ e ^ { \\lambda ^ h _ k t } \\} _ { | k | \\geq k _ 0 } \\cup \\{ e ^ { \\widehat \\lambda _ { n _ l } t } \\} _ { l = 1 } ^ { l _ 0 } \\big ) \\ L ^ 2 ( a , a + T ) , \\ a \\in \\mathbb R , \\end{align*}"}
{"id": "9461.png", "formula": "\\begin{align*} N _ { 2 j } + L _ { 2 j } \\cdot \\tfrac { B ( j , 1 ) } { B ( j ) } + \\tfrac { B ( j , 3 ) ^ p } { B ( j ) ^ p } - \\tfrac { B ( j - 1 , 3 ) } { B ( j - 1 ) } = 0 . \\end{align*}"}
{"id": "1203.png", "formula": "\\begin{align*} { E _ { m } ^ { [ \\alpha , \\beta ] , \\varepsilon } = \\bigcup _ { n \\geq 1 } E _ { m } ^ { [ \\alpha , \\beta ] , \\frac { 1 } { n } , \\varepsilon } } . \\end{align*}"}
{"id": "1181.png", "formula": "\\begin{align*} \\Psi ^ { A i } ( \\zeta ) \\sim \\zeta ^ { - \\frac { \\sigma _ 3 } { 4 } } \\Psi ^ { A i } _ { 0 } \\left ( I + \\sum _ { j = 1 } ^ { \\infty } \\frac { \\left ( \\Psi ^ { A i } _ { 0 } \\right ) ^ { - 1 } \\Psi ^ { A i } _ { j } \\left ( \\frac { 2 } { 3 } \\right ) ^ { - j } } { \\zeta ^ { 3 j / 2 } } \\right ) , \\zeta \\rightarrow \\infty . \\end{align*}"}
{"id": "5529.png", "formula": "\\begin{align*} Y _ m ^ j : = \\sup _ { t \\in [ 0 , T ] } | \\dot { B } _ m ^ j ( t ) | , j = 1 , \\ldots , r . \\end{align*}"}
{"id": "7209.png", "formula": "\\begin{align*} W _ { s , t } ( x - \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } v , v ) & = V _ { s , t } ( x , v ) - v , \\\\ Y _ { s , t } ( x - \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } v , v ) & = X _ { s , t } ( x , v ) - ( x - ( t - s ) v ) . \\end{align*}"}
{"id": "4794.png", "formula": "\\begin{align*} \\phi = \\alpha + \\frac { r } { 1 - n } ( 1 - \\delta _ e ) \\end{align*}"}
{"id": "6047.png", "formula": "\\begin{align*} \\widetilde { P } _ { n + 1 } ^ { \\alpha , \\beta } ( x ) = ( A _ 1 x + A _ 0 ) \\widetilde { P } _ n ^ { \\alpha , \\beta } ( x ) + B \\widetilde { P } _ { n - 1 } ^ { \\alpha , \\beta } ( x ) . \\end{align*}"}
{"id": "4272.png", "formula": "\\begin{align*} v _ x ( \\tau , x ) ~ = ~ u _ x \\bigl ( t , x + y _ 2 ( t ) \\bigr ) , v _ { \\tau } ( \\tau , x ) ~ = ~ { u _ t \\bigl ( t , x + y _ 2 ( t ) \\bigr ) + \\dot { y } _ 2 ( t ) \\cdot u _ x \\bigl ( t , x + y _ 2 ( t ) \\bigr ) \\over \\dot { y } _ 1 ( t ) - \\dot { y } _ 2 ( t ) } . \\end{align*}"}
{"id": "4119.png", "formula": "\\begin{align*} \\pi ( p ) = \\sum _ { i = 1 } ^ d c _ i \\pi ( p _ i ) h ( p ) < \\sum _ { i = 1 } ^ d c _ i h ( p _ i ) \\end{align*}"}
{"id": "5438.png", "formula": "\\begin{align*} \\lim _ { t \\to s + } \\| u ( t , \\cdot ; s , u _ 0 ) - u _ 0 ( \\cdot ) \\| _ { C ^ 0 ( \\bar \\Omega ) } = 0 , \\end{align*}"}
{"id": "775.png", "formula": "\\begin{align*} \\langle A , B \\rangle = \\langle A \\rangle \\langle B \\rangle + \\langle A '' \\rangle \\langle B ' \\rangle . \\end{align*}"}
{"id": "6372.png", "formula": "\\begin{align*} \\left ( x _ 4 ^ 2 + x _ 3 ^ 4 + g _ 2 ( x _ 1 , x _ 2 ) x _ 3 ^ 2 + g _ 4 ( x _ 1 , x _ 2 ) = 0 \\right ) \\subset \\P ( 1 , 1 , 1 , 2 ) \\end{align*}"}
{"id": "5778.png", "formula": "\\begin{align*} \\begin{cases} - \\nu \\Delta \\mathbf { u } + \\operatorname { d i v } ( \\mathbf { u } \\otimes \\mathbf { u } ) + \\nabla \\pi = 0 , \\\\ - \\Delta \\mathbf { b } + \\operatorname { d i v } ( \\mathbf { u } \\otimes \\mathbf { b } ) - \\operatorname { d i v } ( \\mathbf { b } \\otimes \\mathbf { u } ) = 0 , \\\\ \\operatorname { d i v } \\mathbf { u } = 0 = \\operatorname { d i v } \\mathbf { b } , \\end{cases} \\end{align*}"}
{"id": "6392.png", "formula": "\\begin{align*} s _ i ( z ) = \\sum _ { 1 \\leq k _ 1 \\leq k _ 2 \\dots \\leq k _ i \\leq n } z _ { k _ 1 } \\dots z _ { k _ i } \\ ; , \\ ; i = 1 , \\dots , n - 1 p ( z ) = \\prod _ { i = 1 } ^ { n } z _ i \\ , . \\end{align*}"}
{"id": "2693.png", "formula": "\\begin{align*} U ^ { - 1 } = \\sum _ { k = 0 } ^ \\infty ( I - U ) ^ k . \\end{align*}"}
{"id": "655.png", "formula": "\\begin{align*} g ( k + 1 ) \\ = \\ \\begin{cases} \\ 3 g ( k ) 6 g ( k ) + 3 \\leq h ( k , 2 \\cdot 3 ^ { k + 1 } - 1 ) \\\\ [ 8 p t ] \\ 3 g ( k ) + 2 \\end{cases} , \\end{align*}"}
{"id": "368.png", "formula": "\\begin{align*} \\min _ { ( \\rho , m ) \\in C _ F ( \\rho ^ a , \\rho ^ b ) } \\mathcal A ( \\rho , m ) & = \\sup _ { S } \\Big \\{ \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > : \\sup _ { \\rho } \\{ \\ < \\dot S , \\rho \\ > \\\\ & + \\frac 1 4 \\sum _ { i j } ( S _ { i } - S _ { j } ) ^ 2 \\theta _ { i j } ( \\rho ) ( 1 + \\dot { W ^ { \\delta } } ( t ) ) ^ 2 \\} = 0 \\Big \\} . \\end{align*}"}
{"id": "4746.png", "formula": "\\begin{align*} \\| u _ { \\mathbf { 0 } } ( t , \\cdot ) \\| _ { L ^ 2 } \\leq M ( t ) \\left ( \\| g _ { \\mathbf { 0 } } \\| _ { L ^ 2 } + \\int _ 0 ^ t \\| f _ { \\mathbf { 0 } } ( s , \\cdot ) \\| _ { L ^ 2 } \\ , d s \\right ) \\leq M ( t ) \\left ( \\| g _ { \\mathbf { 0 } } \\| _ { L ^ 2 } + t \\| f _ { \\mathbf { 0 } } \\| \\right ) = M ( t ) a _ { \\mathbf { 0 } } ( t ) . \\end{align*}"}
{"id": "3631.png", "formula": "\\begin{align*} F ( x , \\sigma ) = \\left [ \\left ( \\frac { \\log x } { R } \\right ) ^ { B } \\exp \\left ( - C ' \\sqrt { \\frac { \\log x } { R } } \\right ) \\right ] ^ { - 1 } \\end{align*}"}
{"id": "266.png", "formula": "\\begin{align*} f ( t ) : = \\exp \\left ( - \\sigma ( t ) \\right ) \\geqq A _ { 0 } ( t - \\alpha ) ( \\alpha \\leqq { t } < \\beta ) , \\end{align*}"}
{"id": "5908.png", "formula": "\\begin{align*} p _ 1 & = \\frac { 1 } { n } ( a _ 1 + a _ 2 + a _ 3 + \\cdots + a _ n ) , \\\\ p _ 2 & = \\frac { 1 } { \\frac { n ( n - 1 ) } { 2 } } ( a _ 1 a _ 2 + a _ 1 a _ 3 + \\cdots + a _ 1 a _ n + a _ 2 a _ 3 + \\cdots + a _ { n - 1 } a _ n ) , \\\\ \\vdots \\\\ p _ n & = a _ 1 a _ 2 \\cdots a _ n . \\end{align*}"}
{"id": "3759.png", "formula": "\\begin{align*} d _ { \\alpha } d ^ { c } _ { \\alpha } F = 0 , \\end{align*}"}
{"id": "3046.png", "formula": "\\begin{align*} x ^ \\prime = a _ 2 b _ 1 x _ 1 \\ , , y ^ \\prime = a _ 1 b _ 1 y \\ , , z ^ \\prime = a _ 1 b _ 2 z \\ , , w ^ \\prime = a _ 2 b _ 2 w \\ , . \\end{align*}"}
{"id": "9046.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ^ 0 = \\rho ^ { \\rm i n } , ( \\rho ^ { n + 1 } , \\phi ^ { n + 1 } ) = & \\arg \\min _ { ( \\rho , \\phi ) \\in \\mathcal { A } } \\left \\{ \\frac { 1 } { 2 \\tau } d ^ 2 ( \\rho ^ n , \\rho ) + E ( \\rho , \\phi ) \\right \\} . \\end{aligned} \\end{align*}"}
{"id": "734.png", "formula": "\\begin{align*} C _ b = 0 , C _ W = \\frac { 2 } { a _ + ^ 2 + a _ - ^ 2 } . \\end{align*}"}
{"id": "1721.png", "formula": "\\begin{align*} \\hat \\theta = \\frac { \\mu _ * ( 1 / q \u2010 1 / p _ 0 ) + \\alpha _ * ( s _ * + 1 / q \u2010 1 / p _ 1 ) } { \\mu _ * + \\alpha _ * + \\gamma _ * ( s _ * + 1 / p _ 0 \u2010 1 / p _ 1 ) } , \\end{align*}"}
{"id": "4657.png", "formula": "\\begin{align*} x _ { i } ( t ) = \\alpha _ i \\sqrt { t } + \\beta _ i \\log t + \\gamma _ i \\end{align*}"}
{"id": "2255.png", "formula": "\\begin{align*} R _ F ( X ( s ) , X ( t _ j ) ) = \\int _ 0 ^ 1 F '' \\big ( X ( s ) + \\lambda ( X ( t _ j ) - X ( s ) ) \\big ) ( X ( t _ j ) - X ( s ) , X ( t _ j ) - X ( s ) ) ( 1 - \\lambda ) \\ , \\dd \\lambda . \\end{align*}"}
{"id": "8700.png", "formula": "\\begin{align*} m ( f ) \\leq \\prod _ { i ' = 1 } ^ d f _ { i ' } ^ L + \\sum _ { i = 1 } ^ d \\Biggl ( \\prod _ { i ' \\in \\{ \\omega _ 1 , \\ldots , \\omega _ { i - 1 } \\} } f ^ U _ { i ' } \\Biggr ) \\cdot \\Biggl ( \\prod _ { i ' \\in \\{ \\omega _ { i + 1 } , \\ldots , \\omega _ { d } \\} } f ^ L _ { i ' } \\Biggr ) \\cdot ( f _ { \\omega _ i } - f _ { \\omega _ i } ^ L ) , \\end{align*}"}
{"id": "5332.png", "formula": "\\begin{align*} E ( \\phi ^ { \\epsilon } , u ^ { \\epsilon } , \\eta ^ { \\epsilon } , \\tau ^ { \\epsilon } ) = \\| \\phi ^ { \\epsilon } \\| _ { H ^ 3 _ { P ' ( \\rho ^ { \\epsilon } ) } } ^ 2 + \\| u ^ { \\epsilon } \\| _ { H ^ 3 _ { \\rho ^ { \\epsilon } } } ^ 2 + [ \\beta ( L - 1 ) + 2 \\bar { \\mathfrak { z } } ] \\| ( \\eta ^ { \\epsilon } - 1 ) \\| _ { H ^ 3 } ^ 2 + \\frac { \\beta } { 2 k ^ 2 } \\| \\tau ^ { \\epsilon } \\| _ { H ^ 3 } ^ 2 , \\end{align*}"}
{"id": "8087.png", "formula": "\\begin{align*} ( \\Psi _ { \\Sigma } ( s ) \\star _ { H , \\ell } \\Psi _ { \\Sigma } ( s ' ) ) [ \\psi ] = \\psi ( s ) \\psi ( s ' ) - \\frac { \\hbar } { 4 \\pi } \\frac { 1 } { ( s - s ' ) ^ 2 } . \\end{align*}"}
{"id": "3976.png", "formula": "\\begin{align*} r _ 1 + r _ 2 + r _ 3 & = - \\lambda , \\\\ r _ 1 r _ 2 + r _ 2 r _ 3 + r _ 1 r _ 3 & = - 2 \\lambda , \\\\ r _ 1 r _ 2 r _ 3 & = \\lambda ^ 2 . \\end{align*}"}
{"id": "2569.png", "formula": "\\begin{align*} \\langle \\partial ^ \\alpha \\ell , f \\rangle = ( - 1 ) ^ \\alpha \\langle \\ell , \\partial ^ \\alpha f \\rangle . \\end{align*}"}
{"id": "9068.png", "formula": "\\begin{align*} I _ 1 + I _ 2 + I _ 3 & \\leq \\gamma h k { \\rm l o g } \\delta ( b - a ) + \\frac { \\gamma k N ^ 4 } { 2 \\tau h } D _ { i , j _ { k + 1 } } ^ { - 1 } + C _ 1 + C _ 0 C ^ * _ \\phi h \\\\ & = \\frac { 1 } { 2 N } { \\rm l o g } \\delta ( b - a ) + \\frac { N ^ 3 } { 4 \\tau h ^ 2 } D _ { i , j _ { k + 1 } } ^ { - 1 } + C _ 1 + C _ 0 C ^ * _ \\phi h < 0 \\end{align*}"}
{"id": "5676.png", "formula": "\\begin{align*} J ( x \\cdot y ) \\cdot y = J x \\end{align*}"}
{"id": "4642.png", "formula": "\\begin{align*} { } ( T ^ n ) = \\# \\{ { \\bf x } \\in X \\mid T ^ n ( { \\bf x } ) = { \\bf x } \\} = { } \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 0 \\end{matrix} \\right ) ^ n = L _ n \\end{align*}"}
{"id": "929.png", "formula": "\\begin{align*} a ^ { + } ( \\hat { n } ) _ { k , \\lambda } a = ( \\hat { n } - 1 ) _ { k , \\lambda } \\hat { n } = \\hat { n } ( \\hat { n } - 1 ) _ { k , \\lambda } , \\quad ( k \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "3032.png", "formula": "\\begin{align*} E _ { \\lambda } = p _ { 1 } d \\rho . \\end{align*}"}
{"id": "2963.png", "formula": "\\begin{align*} p _ { k , 2 } = q _ { 2 k - \\ell } , p _ { j , 2 } = q _ { r _ j } ( j = 1 , \\dots , k - 1 ) \\end{align*}"}
{"id": "2455.png", "formula": "\\begin{align*} 0 < 1 < \\det ( S ^ T S + I ) & = \\det ( S ^ T ( S + J S J ^ { - 1 } ) ) \\\\ & = \\det ( S ) \\det ( E + i F ) \\det ( E - i F ) \\\\ & = \\det ( S ) \\det ( E + i F ) \\ , \\overline { \\det ( E + i F ) } \\\\ & = \\det ( S ) | \\det ( E + i F ) | ^ 2 \\end{align*}"}
{"id": "9166.png", "formula": "\\begin{align*} \\Lambda ( s , \\operatorname { s y m } ^ 2 f ) = & \\pi ^ { - 3 s / 2 } \\Gamma \\left ( \\frac { s + 1 } { 2 } \\right ) \\Gamma \\left ( \\frac { s + \\kappa - 1 } { 2 } \\right ) \\Gamma \\left ( \\frac { s + \\kappa } { 2 } \\right ) L ( s , \\operatorname { s y m } ^ 2 f ) . \\end{align*}"}
{"id": "5679.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) + I ( \\beta , \\gamma , Z ' ) = I ( \\alpha , \\gamma , Z + Z ' ) . \\end{align*}"}
{"id": "9443.png", "formula": "\\begin{align*} \\mathcal { N } _ q ( u ' ) - \\mathcal { N } _ q ( u '' ) = \\mathcal { B } _ q ( u '' , u ' - u '' ) + ( 1 / 2 ) \\ , \\mathcal { B } _ q ( u ' - u '' , u ' - u '' ) . \\end{align*}"}
{"id": "7244.png", "formula": "\\begin{align*} \\lim _ { z \\downarrow 0 } z \\tilde f ( - i z ) & = \\lim _ { z \\downarrow 0 } \\int _ 0 ^ \\infty f ( t ) z e ^ { - z t } \\dd t = \\lim _ { z \\downarrow 0 } \\int _ 0 ^ \\infty f ' ( t ) e ^ { - z t } \\dd t - \\left [ f e ^ { - z t } \\right ] _ 0 ^ \\infty \\\\ & = \\lim _ { z \\downarrow 0 } \\int _ 0 ^ \\infty f ' ( t ) e ^ { - z t } \\dd t - \\left [ f e ^ { - z t } \\right ] _ 0 ^ \\infty = \\int _ 0 ^ \\infty f ' ( t ) \\dd t - f ( 0 ) = \\lim _ { t \\to \\infty } f ( t ) . \\end{align*}"}
{"id": "8483.png", "formula": "\\begin{align*} \\pi ^ { '' } _ { q , i } : = \\frac { ( q + 1 ) _ { \\ell } - 1 } { 2 } - \\pi ' _ { q , i } = \\frac { ( q + 1 ) _ \\ell \\cdot \\ell ^ { - i } - 1 } { 2 } \\ , , \\end{align*}"}
{"id": "7297.png", "formula": "\\begin{align*} \\beta _ { k + 1 } : = \\beta _ k + X ^ k \\nu , & & \\gamma _ { k + 1 } : = \\gamma _ k + X ^ k \\mu \\end{align*}"}
{"id": "4197.png", "formula": "\\begin{align*} g x ( \\ell _ { \\varphi ^ n ( j ) } s ) & = x ( g ^ { - 1 } \\ell _ { \\varphi ^ n ( j ) } s ) \\\\ & = x ( \\ell _ { \\varphi ^ { n - 1 } ( j ) } \\ell _ { \\varphi ^ { n - 1 } ( j ) } ^ { - 1 } g ^ { - 1 } \\ell _ { \\varphi ^ n ( j ) } s ) \\\\ & = y _ j ( h _ { n - 1 , j } ^ { - 1 } \\ell _ { \\varphi ^ { n - 1 } ( j ) } ^ { - 1 } g ^ { - 1 } \\ell _ { \\varphi ^ n ( j ) } s ) \\\\ & = y _ j ( h _ { n , j } ^ { - 1 } s ) \\\\ & = x ( \\ell _ { \\varphi ^ n ( j ) } s ) . \\end{align*}"}
{"id": "5004.png", "formula": "\\begin{align*} | \\lambda _ { k n } | = ( \\theta _ 0 \\theta _ 2 \\cdots \\theta _ { n - 1 } ) ^ k , \\end{align*}"}
{"id": "7997.png", "formula": "\\begin{align*} A _ i & = \\mathrm { d i a g } ( a , b , 1 / N ) , t _ i = ( 0 , 0 , ( i - 1 ) / N ) \\ ; i = 1 , \\ldots , N ; \\\\ A _ { N + 1 } & = \\mathrm { d i a g } ( c , d , 1 / N ) , t _ { N + 1 } = ( 1 - c , b , 0 ) . \\end{align*}"}
{"id": "8922.png", "formula": "\\begin{align*} V _ i : = \\bigcup _ { ( \\varepsilon _ 1 , \\ldots , \\varepsilon _ n ) \\in \\{ 1 , 2 \\} ^ n , \\varepsilon _ i = 2 } A _ 1 ^ { \\varepsilon _ 1 } \\cap \\cdots \\cap A _ n ^ { \\varepsilon _ n } . \\end{align*}"}
{"id": "1455.png", "formula": "\\begin{align*} U ( z _ 1 ) ^ { \\ast } H U ( z _ 2 ) = \\lambda ( \\alpha , z _ 1 ) ^ { \\ast } U ( \\alpha z _ 1 ) ^ { \\ast } H U ( \\alpha z _ 2 ) \\lambda ( \\alpha , z _ 2 ) , \\end{align*}"}
{"id": "2878.png", "formula": "\\begin{align*} \\| h _ \\perp ( t ) \\| _ { H _ x ^ 1 } ^ 2 \\simeq B ( h _ \\perp , h _ \\perp ) = \\Phi ( h ) - 2 \\alpha _ + \\alpha _ - \\lesssim e ^ { - ( c _ 0 + c _ 1 ) t } + e ^ { - 2 c _ 1 ^ - t } \\lesssim e ^ { - ( c _ 0 + c _ 1 ) t } \\end{align*}"}
{"id": "5940.png", "formula": "\\begin{align*} L ( g , g \\cdot \\xi _ 1 , \\dots , g \\cdot \\xi _ k ) = \\frac { 1 } { 2 } \\mathcal { G } ( \\xi _ 1 , \\xi _ 1 ) + \\dots + \\frac { 1 } { 2 } \\mathcal { G } ( \\xi _ k , \\xi _ k ) = \\ell ( \\xi _ 1 , \\dots , \\xi _ k ) . \\end{align*}"}
{"id": "4439.png", "formula": "\\begin{align*} F ( q ) : = \\sum _ { n = 0 } ^ { \\infty } ( q ) _ n \\end{align*}"}
{"id": "3140.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 & = 0 \\ , , \\\\ x _ 1 ^ 2 + \\varepsilon x _ 2 ^ 2 + \\varepsilon ^ 2 x _ 3 ^ 2 + \\varepsilon ^ 3 x _ 4 ^ 2 + \\varepsilon ^ 4 x _ 5 ^ 2 & = 0 \\ , , ( \\varepsilon ^ 5 = 1 . ) \\\\ \\varepsilon ^ 4 x _ 1 ^ 2 + \\varepsilon ^ 3 x _ 2 ^ 2 + \\varepsilon ^ 2 x _ 3 ^ 2 + \\varepsilon x _ 4 ^ 2 + x _ 5 ^ 2 & = 0 \\ , . \\end{align*}"}
{"id": "5718.png", "formula": "\\begin{align*} I ( \\delta _ { 1 3 } , \\delta _ { 1 2 } ) - J ( \\delta _ { 1 3 } , \\delta _ { 1 2 } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } \\cup { \\gamma _ { 4 } } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } \\cup { \\gamma _ { 3 } } ) = 2 \\end{align*}"}
{"id": "6698.png", "formula": "\\begin{align*} \\Bigl ( ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d - 1 } } \\Bigr ) ^ { ( - 1 ) } = \\frac { \\prod _ { i = 1 } ^ { s + 1 } \\Bigl ( ( \\theta - t ) ^ { q ^ { d - 2 } } \\mathbb { D } _ { a _ i - 2 } ^ { q ^ { d - a _ i } } \\Bigr ) } { \\prod _ { j = 1 } ^ s \\Bigl ( ( \\theta - t ) ^ { q ^ { d - 2 } } \\mathbb { D } _ { b _ j - 2 } ^ { q ^ { d - b _ j } } \\Bigr ) } \\alpha ^ { q ^ { d - 2 } } + ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d - 1 } } . \\end{align*}"}
{"id": "2501.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & \\omega _ 1 & \\ldots & \\omega _ d & \\tau \\\\ 0 & 1 & \\ldots & 0 & x _ 1 \\\\ \\vdots & & \\ddots & & \\vdots \\\\ 0 & 0 & \\ldots & 1 & x _ d \\\\ 0 & 0 & \\ldots & 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "8898.png", "formula": "\\begin{align*} \\check H _ { c t } ^ q ( M , A ) = H _ { s i n g } ^ q ( S ^ { n - 1 } ; A ) \\end{align*}"}
{"id": "1017.png", "formula": "\\begin{align*} v ( x ) : = \\tau d ^ { - n - 2 } \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } \\zeta ( x ) - u ( x ) x \\in \\R ^ n . \\end{align*}"}
{"id": "9008.png", "formula": "\\begin{align*} \\lambda _ w & = 0 , ~ w \\in 1 : K - 1 , \\\\ \\nu & = y _ { K - 1 } / N , \\end{align*}"}
{"id": "4348.png", "formula": "\\begin{align*} Q '' ( \\xi ) + \\frac { d + 1 } { \\xi } Q ' _ \\xi - 3 ( d - 2 ) Q ^ 2 - ( d - 2 ) \\xi ^ 2 Q ^ 3 = 0 , Q ( 0 ) = - 1 Q ' ( 0 ) = 0 . \\end{align*}"}
{"id": "1983.png", "formula": "\\begin{align*} \\widehat \\beta ( x ) = 1 - \\widehat { \\Phi ^ { - 1 } } \\big ( x { \\widehat { \\Phi } } ( x ) \\big ) . \\end{align*}"}
{"id": "5151.png", "formula": "\\begin{align*} \\cosh ( \\eta ) - \\tfrac { n } { \\eta ^ 2 } > n - \\tfrac { n } { 1 } = 0 . \\end{align*}"}
{"id": "6220.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } K ( s , t ) f ( t ) d t = g ( s ) , \\ \\ c \\leq s \\leq d , \\end{align*}"}
{"id": "4649.png", "formula": "\\begin{align*} \\langle h , h ' \\rangle = \\langle \\theta h , h ' \\rangle + \\langle h , \\theta h ' \\rangle \\ ; . \\end{align*}"}
{"id": "7421.png", "formula": "\\begin{align*} \\forall x \\in \\mathbb { Z } , \\overleftarrow { \\eta } ^ { \\ell } ( x ) - \\overleftarrow { \\eta } ^ { \\varepsilon n } ( x ) = \\frac { 1 } { m \\ell } \\sum _ { j = 1 } ^ { m - 1 } \\sum _ { z = x - \\ell } ^ { x - 1 } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] . \\end{align*}"}
{"id": "7980.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z g _ { i j } ^ z = - 2 A ^ z _ { i j } ; \\\\ \\partial _ z A _ { i j } ^ z = - A ^ z _ { i k } A ^ z _ { j l } g _ z ^ { k l } + R ^ z _ { i j } , \\end{cases} \\end{align*}"}
{"id": "3161.png", "formula": "\\begin{align*} \\mathcal { U } _ k = \\left \\lbrace j : \\frac { | e _ j ^ \\intercal ( A x _ k - b ) | ^ 2 } { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 \\norm { A x _ k - b } _ 2 ^ 2 } \\geq \\epsilon _ k \\right \\rbrace ; \\end{align*}"}
{"id": "594.png", "formula": "\\begin{align*} ( x , y ) \\ = \\ x + y . \\end{align*}"}
{"id": "7618.png", "formula": "\\begin{align*} \\mathcal { L } u = \\bar { g } ( \\lambda \\partial _ { r } , \\nabla F ) + F ^ { i j } \\bar { R } _ { \\nu j l i } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) + \\lambda ' F - u F ^ { i j } h _ { i l } h _ { j l } . \\end{align*}"}
{"id": "1835.png", "formula": "\\begin{align*} G = \\{ u \\rightarrow 2 u v , \\ ; \\ ; v \\rightarrow u \\} , \\end{align*}"}
{"id": "1700.png", "formula": "\\begin{align*} h ( t ) = t ^ \\theta , 0 \\le \\theta < d . \\end{align*}"}
{"id": "1310.png", "formula": "\\begin{align*} \\hat { I } _ { > 2 , < \\epsilon } ( k ) : = \\{ \\ , \\ , \\alpha _ { k ' } \\ , \\ , | \\ , \\ , k ' \\leq k , \\ , \\ , J ( \\alpha _ { k ' + 1 } , \\alpha _ { k ' } ) > 2 , \\ , \\ , \\ , \\alpha _ { k ' + 1 } , \\ , \\ , \\alpha _ { k ' } \\ , \\ , \\mathrm { s a t i s f y \\ , \\ , 2 , 3 , 4 , 5 , 6 \\ , \\ , i n \\ , \\ , L e m m a \\ , \\ , \\ref { m a i n l e m m a } } \\} \\end{align*}"}
{"id": "1434.png", "formula": "\\begin{align*} \\sum _ { r , q } s _ r s _ q = & \\sum _ { r , q \\notin \\{ i , j \\} } s _ r s _ q + s _ i \\sum _ { r \\notin \\{ i , j \\} } s _ r + s _ j \\sum _ { r \\notin \\{ i , j \\} } s _ r + s _ i s _ j \\\\ < & \\underbrace { \\sum _ { r , q \\notin \\{ i , j \\} } s _ r s _ q + ( s _ i + 1 ) \\sum _ { r \\notin \\{ i , j \\} } s _ r + ( s _ j - 1 ) \\sum _ { r \\notin \\{ i , j \\} } s _ r + ( s _ i + 1 ) ( s _ j - 1 ) } _ { = A } - 1 . \\end{align*}"}
{"id": "3295.png", "formula": "\\begin{align*} P ( E , m ) = \\sum _ { j = 0 } ^ d \\alpha _ { d - j } ( E ) \\frac { m ^ { d - j } } { ( d - j ) ! } , \\end{align*}"}
{"id": "8709.png", "formula": "\\begin{align*} \\begin{aligned} F ( x ) \\otimes G ( x ) & \\preceq A ^ 0 ( x ) \\otimes B ^ 0 ( x ) + \\sum _ { t : \\omega _ t = 1 } \\Bigl ( U ^ { p ^ t _ 1 } ( x ) - U ^ { p ^ { t - 1 } _ 1 } ( x ) \\Bigr ) \\otimes B ^ { p ^ t _ 2 } ( x ) \\\\ & + \\sum _ { t : \\omega _ t = 2 } A ^ { p ^ t _ 1 } ( x ) \\otimes \\Bigl ( V ^ { p ^ t _ 2 } ( x ) - V ^ { p ^ { t - 1 } _ 2 } ( x ) \\Bigr ) x \\in X . \\end{aligned} \\end{align*}"}
{"id": "5374.png", "formula": "\\begin{align*} c _ G = 4 n ^ 2 + 5 n - 9 , \\end{align*}"}
{"id": "2281.png", "formula": "\\begin{align*} \\norm { M _ \\omega T _ x f } _ p = \\norm { f } _ p \\ . \\end{align*}"}
{"id": "3902.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq b < p \\\\ \\gcd ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } = 0 \\end{align*}"}
{"id": "244.png", "formula": "\\begin{align*} \\partial _ k \\partial _ \\ell \\left ( f _ h \\right ) ( x ) = - \\int _ 0 ^ { + \\infty } e ^ { - 2 t } P ^ { \\Sigma } _ t ( \\partial _ k \\partial _ \\ell ( h ) ) ( x ) d t . \\end{align*}"}
{"id": "2830.png", "formula": "\\begin{align*} y _ R ( t ) = \\int _ { \\mathbb { R } ^ N } \\varphi _ R ( x ) | u ( x , t ) | ^ 2 d x = \\int _ { \\mathbb { R } ^ N } R ^ 2 \\varphi \\left ( \\frac { x } { R } \\right ) | u ( x , t ) | ^ 2 d x , \\end{align*}"}
{"id": "6737.png", "formula": "\\begin{align*} g _ 1 ( t ) \\Omega ^ w \\mathcal { L i } _ { K , \\mathfrak { s } } ( { \\boldsymbol \\alpha } ) + g _ 2 ( t ) \\cdot 1 = 0 \\end{align*}"}
{"id": "1155.png", "formula": "\\begin{align*} m ^ { ( 1 ) } _ { + } ( x , t , k ) = m ^ { ( 1 ) } _ { - } ( x , t , k ) J ^ { ( 1 ) } ( x , t , k ) , k \\in \\mathbb { R } , \\end{align*}"}
{"id": "3837.png", "formula": "\\begin{align*} K ( I ) \\subseteq \\{ 0 , 1 , \\ldots , n - 2 \\} , K ( I ) = \\varnothing & , \\\\ \\left . \\begin{array} { r c l } \\nu = 2 k \\in K ( I ) & \\Leftrightarrow & i _ { 2 k } = m - k , \\\\ \\nu = 2 k + 1 \\in K ( I ) & \\Leftrightarrow & i _ { 2 k + 1 } = m - k - 1 \\end{array} \\right \\} & . \\end{align*}"}
{"id": "8722.png", "formula": "\\begin{align*} p _ { 1 n ( 1 ) } \\cdot m _ { 2 j } = m _ { 1 0 } \\cdot m _ { 2 j } + \\ldots + m _ { 1 n ( 1 ) } \\cdot m _ { 2 j } . \\end{align*}"}
{"id": "8824.png", "formula": "\\begin{align*} \\mathbb { E } _ { U , \\xi } \\left [ \\phi _ { \\mu } ^ { 1 / \\bar { \\rho } } ( x _ { t + 1 } ) \\right ] \\leq & \\ \\mathbb { E } _ { U , \\xi } \\left [ \\phi _ { \\mu } ^ { 1 / \\bar { \\rho } } ( x _ { t } ) \\right ] - \\frac { \\alpha _ t ( \\bar { \\rho } - \\rho ) } { \\bar { \\rho } } \\mathbb { E } _ { U , \\xi } \\left [ \\left \\| \\nabla \\phi _ { \\mu } ^ { 1 / \\bar { \\rho } } ( x _ t ) \\right \\| _ 2 ^ 2 \\right ] \\\\ & + 2 ( n ^ 2 + 2 n ) \\bar { \\rho } \\alpha _ t ^ 2 L _ { f , 0 } ^ 2 , \\end{align*}"}
{"id": "405.png", "formula": "\\begin{align*} A ^ { 0 } U _ { t } + A ^ { i } \\partial _ { i } U - B ^ { i j } \\partial _ { i } \\partial _ { j } U + D u = F \\end{align*}"}
{"id": "2493.png", "formula": "\\begin{align*} ( F _ 1 * F _ 2 ) ( x _ 0 , \\omega _ 0 , \\tau _ 0 ) = \\int _ { \\R ^ { 2 d } } \\int _ { \\R } F _ 1 ( x , \\omega , \\tau ) F _ 2 ( x _ 0 - x , \\omega _ 0 - \\omega , \\tau _ 0 - \\tau + \\tfrac { 1 } { 2 } ( x \\cdot \\omega _ 0 - x _ 0 \\cdot \\omega ) ) \\ , d \\tau \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "3776.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( \\widetilde { W _ v ^ l } , \\widetilde { U _ v ^ l } ) q _ F ^ { - r / 2 } X ^ { - r } = \\omega _ { \\sigma ^ { ( l ) } } ( - 1 ) ^ { n - 2 } \\gamma ( X , \\pi ^ { ( l ) } , \\sigma ^ { ( l ) } , \\psi ^ l ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( W _ v ^ l , U _ v ^ l ) q _ F ^ { r / 2 } X ^ r . \\end{align*}"}
{"id": "9276.png", "formula": "\\begin{align*} H ( P , Q ) : = \\max \\left \\{ \\sup _ { p \\in P } \\inf _ { q \\in Q } \\norm { p - q } , \\sup _ { q \\in Q } \\inf _ { p \\in P } \\norm { p - q } \\right \\} \\end{align*}"}
{"id": "1727.png", "formula": "\\begin{align*} 2 ^ { \\mu _ * k _ * t } \\cdot 2 ^ { \u2010 m _ t ( s _ * + 1 / q \u2010 1 / p _ 1 ) } = 2 ^ { \u2010 \\alpha _ * k _ * t } \\cdot 2 ^ { \u2010 m _ t ( 1 / q \u2010 1 / p _ 0 ) } \\stackrel { \\eqref { h a t _ t h e t a _ d e f } } { = } n ^ { \u2010 \\hat \\theta } ; \\end{align*}"}
{"id": "1761.png", "formula": "\\begin{align*} V _ { { \\rm C M } } ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
{"id": "9460.png", "formula": "\\begin{align*} L _ { 2 j } + \\tfrac { B ( j , 2 ) ^ p } { B ( j ) ^ p } - \\tfrac { B ( j - 1 , 2 ) } { B ( j - 1 ) } = 0 \\end{align*}"}
{"id": "5896.png", "formula": "\\begin{align*} x _ i = x _ 2 - \\frac { 2 s _ 2 } { 1 + \\lambda } x _ 3 - \\frac { 2 s _ 3 } { 1 + \\lambda } x _ 5 - \\cdots - \\frac { 2 s _ { \\frac { i } { 2 } } } { 1 + \\lambda } x _ { i - 1 } . \\end{align*}"}
{"id": "5565.png", "formula": "\\begin{align*} A = \\frac { \\kappa } { 2 } \\frac { d ^ 2 } { d x ^ 2 } + \\frac { d } { d x } \\end{align*}"}
{"id": "1644.png", "formula": "\\begin{align*} G _ \\lambda \\nu ( x ) : = \\int _ M g _ \\lambda ( x , y ) \\nu ( d y ) , x \\in M , \\end{align*}"}
{"id": "5699.png", "formula": "\\begin{align*} I ( \\alpha \\cup { \\gamma _ { 0 } } , \\alpha ) = I ( \\gamma _ { 0 } ) + 2 \\# ( \\mathbb { R } \\times \\alpha \\cap u ) = 2 + 2 w ( \\alpha ) . \\end{align*}"}
{"id": "5534.png", "formula": "\\begin{align*} Z _ { r , m } : = L \\sum _ { j = 1 } ^ r Y _ m ^ j . \\end{align*}"}
{"id": "7702.png", "formula": "\\begin{align*} A + [ 0 , \\max ( A ) ] = [ 0 , 2 \\max ( A ) ] = [ 0 , \\max ( A ) ] + [ 0 , \\max ( A ) ] \\ , , \\end{align*}"}
{"id": "9041.png", "formula": "\\begin{align*} X ( \\theta ) \\log X ( \\theta ) - \\theta X ^ 0 \\log X ^ 0 - ( 1 - \\theta ) X ^ 1 \\log X ^ 1 = - \\theta ( 1 - \\theta ) ( X ^ 1 - X ^ 0 ) ^ 2 g ( X ^ 0 , X ^ 1 ; \\theta ) , \\end{align*}"}
{"id": "9335.png", "formula": "\\begin{align*} L _ 1 ( \\Theta ) : = - ( \\delta I + A ( \\rho I + ( \\delta I + ( \\Theta ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } ) ^ { - 1 } A ^ T ) , \\end{align*}"}
{"id": "521.png", "formula": "\\begin{align*} \\int _ { \\beta } c _ 1 ( T _ X ) = r ( n + g - 1 ) . \\end{align*}"}
{"id": "1482.png", "formula": "\\begin{align*} \\delta ( \\iota ( z _ 1 , z _ 2 ) ) = \\delta ( \\rho ( g _ 1 , g _ 2 ) \\iota ( z _ 0 , z _ 0 ) ) = | j ( \\rho ( g _ 1 , g _ 2 ) , \\iota ( z _ 0 , z _ 0 ) ) | ^ { - 2 } \\delta ( \\iota ( z _ 0 , z _ 0 ) ) \\end{align*}"}
{"id": "3738.png", "formula": "\\begin{align*} & \\frac 1 2 \\frac { d } { d t } \\| D ^ m B \\| _ { L ^ 2 } ^ 2 + \\int _ { \\mathbb S ^ 1 } [ D ^ m , B ] J _ x D ^ m B \\ , d x + \\int _ { \\mathbb S ^ 1 } B D ^ m J _ x D ^ m B \\ , d x \\\\ & - \\int _ { \\mathbb S ^ 1 } [ D ^ m , J ] B _ x D ^ m B \\ , d x - \\int _ { \\mathbb S ^ 1 } J D ^ m B _ x D ^ m B \\ , d x + \\mu \\| \\Lambda ^ { \\frac { \\alpha } { 2 } } D ^ m B \\| _ { L ^ 2 } ^ 2 = 0 . \\end{align*}"}
{"id": "4176.png", "formula": "\\begin{align*} \\omega _ { p } ( T ) & \\geq \\norm { \\cos ( 2 \\pi ) R e ( T ) - \\sin ( 2 \\pi ) I m ( T ) } _ { p } \\\\ & = \\frac { 1 } { 2 } \\norm { T + T ^ { \\ast } } _ { p } \\end{align*}"}
{"id": "6937.png", "formula": "\\begin{align*} \\tilde { R } _ h ^ 2 ( w ) = \\sum _ { i \\in I _ h } \\tilde { r } _ { h , i } ^ 2 ( w ) \\ , , \\end{align*}"}
{"id": "6781.png", "formula": "\\begin{align*} i _ { k _ { 1 } } + i _ { k _ { 2 } } - ( v _ 1 + 1 ) + ( v _ 1 + 1 ) = 0 . \\end{align*}"}
{"id": "7573.png", "formula": "\\begin{align*} L _ { \\eta } \\psi _ { \\eta } = 0 , \\end{align*}"}
{"id": "2064.png", "formula": "\\begin{align*} \\sum _ { \\gamma < \\alpha } b _ { \\gamma } ^ { \\alpha } f ^ { ( \\gamma ) } ( o ) + f ^ { ( \\alpha ) } ( o ) = \\int _ D f \\overline { { g } _ { \\alpha } } e ^ { - \\varphi } . \\end{align*}"}
{"id": "4223.png", "formula": "\\begin{align*} \\mathrm { L a w } ( \\widehat { X } ( t ) ) = \\widehat { \\mu } _ t . \\end{align*}"}
{"id": "7334.png", "formula": "\\begin{align*} u _ { q , \\lambda } ( x , t ) = \\inf \\left \\{ \\left ( \\lambda u ( y , t ) ^ q + ( 1 - \\lambda ) u ( z , t ) ^ q \\right ) ^ { { 1 \\over q } } : \\lambda y + ( 1 - \\lambda ) z = x \\right \\} \\end{align*}"}
{"id": "5201.png", "formula": "\\begin{align*} \\Delta _ j f = \\mathcal { F } ^ { - 1 } \\Big ( \\phi _ j \\mathcal { F } f \\Big ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ j \\in \\mathbb { N } _ 0 , ~ ~ f \\in S ^ { \\prime } ( K ) . \\end{align*}"}
{"id": "9013.png", "formula": "\\begin{align*} E = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } ( f + \\sum _ { i = 1 } ^ s z _ i \\rho _ i ) \\phi \\bigg ) d x + B , \\end{align*}"}
{"id": "7698.png", "formula": "\\begin{align*} q = \\max \\mathsf L ( A ) / \\min \\mathsf L ( A ) \\ , . \\end{align*}"}
{"id": "2094.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} 0 & 1 & 1 & \\dots & 1 \\\\ 1 & 0 & 1 & \\dots & 1 \\\\ \\vdots & & \\ddots & & \\vdots \\\\ 1 & 1 & 1 & \\dots & 0 \\end{pmatrix} , \\end{align*}"}
{"id": "6993.png", "formula": "\\begin{align*} f _ { R C B } = f _ { R | C B } f _ { B | C } f _ B . \\end{align*}"}
{"id": "3873.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Phi _ H ^ { + } } \\alpha & = 2 ( n - 1 ) t _ 1 + 2 ( n - 2 ) t _ 2 + \\dots + 2 t _ { n - 1 } \\\\ & = \\sum _ { k = 1 } ^ { n - 2 } k ( 2 n - k - 1 ) \\alpha _ k + \\frac { n ( n - 1 ) } { 2 } ( \\alpha _ { n - 1 } + \\alpha _ n ) . \\end{align*}"}
{"id": "1421.png", "formula": "\\begin{align*} \\C _ H ( A ) & : = \\{ v \\in \\C \\ , : \\ , S _ H ( v ) = A \\} \\\\ \\C _ R ( V ) & : = \\{ v \\in \\C \\ , : \\ , S _ R ( v ) = V \\} \\end{align*}"}
{"id": "5288.png", "formula": "\\begin{align*} \\varphi ( \\delta _ { \\varphi } ^ * a ) = \\overline { \\varphi ( a ^ * \\delta _ { \\varphi } ) } = \\overline { \\varphi ( S ( a ) ) } = \\varphi ( S ^ { - 1 } ( a ^ * ) ) = \\varphi ( \\delta _ { \\varphi } a ^ * ) . \\end{align*}"}
{"id": "6623.png", "formula": "\\begin{align*} & \\tau _ { A _ { s _ 1 } \\smallsetminus \\{ \\alpha + s _ 1 \\} \\cup \\{ - \\beta - s _ 2 \\} } ( p ^ m ) \\tau _ { B _ { s _ 2 } \\smallsetminus \\{ \\beta + s _ 2 \\} \\cup \\{ - \\alpha - s _ 1 \\} } ( p ^ m ) \\\\ & = \\tau _ { A \\smallsetminus \\{ \\alpha \\} \\cup \\{ - \\beta - s _ 1 - s _ 2 \\} } ( p ^ m ) \\tau _ { B _ { s _ 1 + s _ 2 } \\smallsetminus \\{ \\beta + s _ 1 + s _ 2 \\} \\cup \\{ - \\alpha \\} } ( p ^ m ) . \\end{align*}"}
{"id": "4122.png", "formula": "\\begin{align*} q _ 1 = \\sum _ { i = 1 } ^ k \\alpha _ i p _ i , q _ 2 = \\beta p + \\sum _ { i = k + 1 } ^ d \\beta _ i p _ i . \\end{align*}"}
{"id": "8323.png", "formula": "\\begin{align*} \\int _ s ^ \\tau z \\dd ( y - u ) = 0 \\forall \\ ; 0 \\le s < \\tau \\le T . \\end{align*}"}
{"id": "851.png", "formula": "\\begin{align*} U \\left ( t ^ S _ { j - 1 } \\right ) = t ^ G _ { j } = t ^ S _ { j - 1 } + \\tau _ { \\rm f } - \\tau ^ { \\rm R e a c } _ { V _ { j - 1 } } . \\end{align*}"}
{"id": "6799.png", "formula": "\\begin{align*} u _ { t } = - \\gamma \\boldsymbol { D } _ { x } ^ { \\alpha } u - u ^ { 3 } + \\left ( \\beta + 1 \\right ) u ^ { 2 } - \\beta u , \\end{align*}"}
{"id": "4175.png", "formula": "\\begin{align*} \\omega _ { p } \\left ( \\begin{bmatrix} 0 & A \\\\ B & 0 \\end{bmatrix} \\right ) \\leq 2 ^ { \\frac { 1 } { p } } \\omega _ { p } ( B ) + \\min ( \\omega _ { p } ( A + B ) , \\omega _ { p } ( A - B ) ) . \\end{align*}"}
{"id": "6703.png", "formula": "\\begin{align*} \\mathcal { L i } ^ { \\ast } _ { K , \\mathfrak { s } } ( { \\bf z } ) : = \\sum _ { i _ 1 \\geq i _ 2 \\geq \\cdots \\geq i _ r > 0 } \\frac { z _ 1 ^ { q ^ { i _ 1 } } z _ 2 ^ { q ^ { i _ 2 } } \\cdots z _ r ^ { q ^ { i _ r } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { s _ 1 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { s _ 2 } \\cdots ( \\theta ^ { q ^ { i _ r } } - t ) ^ { s _ r } } \\in \\mathbb { T } \\end{align*}"}
{"id": "3750.png", "formula": "\\begin{align*} \\nabla _ { X } e _ i = A _ { i } ( X ) + \\sum _ { j = 1 } ^ { r } B _ { i } ^ j ( X ) e _ j , \\end{align*}"}
{"id": "2113.png", "formula": "\\begin{align*} \\begin{aligned} & \\gamma ^ { ( k ) } ( c ) \\approx c + ( 1 - c ) \\sum _ { i = 1 } ^ k ( 1 - c ) ^ { k - i + 1 } c ^ { i - 1 } \\binom k { i - 1 } \\frac 1 { k + 2 - i } + \\frac { ( 1 - c ) c ^ k } { k + 1 } = \\\\ & c + ( 1 - c ) E \\frac 1 { k + 1 - X _ { k , c } 1 _ { X _ { k , c } \\neq k } } . \\end{aligned} \\end{align*}"}
{"id": "403.png", "formula": "\\begin{align*} J _ { 0 } ^ { 2 } ( t ) : = \\| u _ { 0 } \\| _ { m } ^ { 2 } + \\int _ { 0 } ^ { t } \\| f ( \\tau ) \\| _ { m - 1 } ^ { 2 } d \\tau , \\end{align*}"}
{"id": "1523.png", "formula": "\\begin{align*} \\tilde { V } _ t = \\{ \\beta \\times \\gamma \\in P _ { n } ^ t \\times P _ { n } ^ t : \\pi _ t ( \\beta ) = \\pi _ t ( \\gamma ) \\} . \\end{align*}"}
{"id": "2400.png", "formula": "\\begin{align*} S : \\mathcal { H } \\to \\mathcal { H } , S f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , e _ \\gamma \\rangle e _ \\gamma . \\end{align*}"}
{"id": "2530.png", "formula": "\\begin{align*} \\Phi ^ { \\xi , \\eta } ( x , \\omega ) = \\langle \\rho ( \\xi , \\eta ) \\varphi , \\rho ( x , \\omega ) \\varphi \\rangle \\end{align*}"}
{"id": "5051.png", "formula": "\\begin{align*} h _ { p , q } = \\frac { 1 } { 2 } ( \\delta _ { q , p - n } + \\delta _ { q , p + n } ) \\end{align*}"}
{"id": "4763.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( - R _ 2 + R _ 3 + 2 R _ 4 + R _ 6 \\right ) = & ( 0 , 1 , 1 , 1 , 1 , 1 , 1 ) \\in R ( A ( D _ 6 ) ) \\\\ \\frac { 1 } { 2 } \\left ( R _ 1 - R _ 2 + R _ 5 + 2 R _ 6 \\right ) = & ( 1 , 1 , 1 , 1 , 1 , 1 ) \\in R ( A ( D _ { 1 4 } ) ) \\\\ \\frac { 1 } { 2 } \\left ( R _ 4 + R _ 5 + R _ 6 \\right ) = & ( 1 , 1 , 1 , 1 , 1 , 1 ) \\in R ( A ( D _ { 1 5 } ) ) \\\\ \\frac { 1 } { 2 } \\left ( - R _ 1 + R _ 2 + 2 R _ 3 + R _ 5 \\right ) = & ( 1 , 1 , 1 , 1 , 1 , 1 , 1 ) \\in R ( A ( D _ { 1 7 } ) ) . \\end{align*}"}
{"id": "7174.png", "formula": "\\begin{align*} \\int _ { \\Omega _ r } D ( u - f ) \\cdot A ( x ) D ( u - f ) \\ , d x & = \\int _ { \\Omega _ r } D u \\cdot A ( x ) D u \\ , d x - \\int _ { \\Omega _ r } D f \\cdot A ( x ) D f \\ , d x \\\\ & \\le ( 1 + C _ g r ^ \\alpha ) \\left ( \\int _ { \\Omega _ r } | D u | ^ 2 \\ , d x - \\frac { 1 - C _ g r ^ \\alpha } { 1 + C _ g r ^ \\alpha } \\int _ { \\Omega _ r } | D f | ^ 2 \\ , d x \\right ) \\\\ & \\le ( 1 + C _ g r ^ \\alpha ) \\left ( | \\Omega _ r | + C _ g r ^ \\alpha \\int _ { \\Omega _ r } | D f | ^ 2 \\ , d x \\right ) . \\end{align*}"}
{"id": "9061.png", "formula": "\\begin{align*} \\xi _ 0 - \\xi _ 1 = 0 , \\xi _ 1 - \\xi _ 3 = 0 , \\cdots , \\xi _ { N - 1 } - \\xi _ { N + 1 } = 0 , \\xi _ { N } + \\xi _ { N + 1 } = 0 , \\end{align*}"}
{"id": "4206.png", "formula": "\\begin{align*} 0 & = - \\int _ M | \\nabla \\dot u | ^ 2 \\ , d V _ g + \\int _ M V \\dot u \\ , r \\ , d V _ g + \\int _ M V \\dot u ^ 2 \\ , d V _ g . \\end{align*}"}
{"id": "153.png", "formula": "\\begin{align*} \\left ( - \\mathcal { A } \\right ) ^ { - \\frac { r } { 2 } } f = \\frac { 1 } { \\Gamma ( \\frac { r } { 2 } ) } \\int _ 0 ^ { + \\infty } t ^ { \\frac { r } { 2 } - 1 } P _ t ( f ) d t . \\end{align*}"}
{"id": "5253.png", "formula": "\\begin{align*} \\varepsilon ( a b ) = \\varepsilon ( a ) \\varepsilon ( b ) \\textrm { w h e n e v e r } a \\in A _ t ^ s , b \\in { } ^ s _ t A , \\end{align*}"}
{"id": "2149.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , n , l } ) = \\begin{cases} 1 , \\ \\ l \\in [ k - 1 ] ; \\\\ 0 , \\ \\ l = k . \\end{cases} \\end{align*}"}
{"id": "5445.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\limsup _ { t - s \\to \\infty } \\int _ \\Omega u ( t , x ; s , u _ 0 ) \\le M _ 0 ^ * \\cr \\displaystyle \\limsup _ { t - s \\to \\infty } \\int _ \\Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \\le M _ 1 ^ * \\cr \\displaystyle \\limsup _ { t - s \\to \\infty } \\int _ \\Omega u ^ { q } ( t , x ; s , u _ 0 ) d x \\le M _ 2 ^ * . \\end{cases} \\end{align*}"}
{"id": "7314.png", "formula": "\\begin{align*} \\begin{aligned} & G _ \\beta ( r , p , X , A ) \\\\ & = { 1 \\over 1 - \\beta } r ^ \\beta F \\left ( r ^ { 1 - \\beta } , ( 1 - \\beta ) r ^ { - \\beta } p , ( 1 - \\beta ) r ^ { - \\beta } X + ( \\beta ^ 2 - \\beta ) r ^ { - \\beta - 1 } p \\otimes p , A \\right ) \\end{aligned} \\end{align*}"}
{"id": "8220.png", "formula": "\\begin{align*} [ z ^ n x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] P _ h & = [ z ^ { n - 1 } x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] f ( A ) \\\\ & = \\frac { 1 } { ( n - 1 ) } [ t ^ { n - 2 } x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] f ' ( t ) \\Phi ( t ) ^ n . \\end{align*}"}
{"id": "9329.png", "formula": "\\begin{align*} S ( \\Theta ) : = \\begin{bmatrix} H + \\rho I + ( \\delta I + ( \\Theta ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } & A ^ T \\\\ A & - \\delta I \\end{bmatrix} . \\end{align*}"}
{"id": "4892.png", "formula": "\\begin{align*} | p - q | \\leq 1 , p + q = 1 + s \\geq 2 , 2 p + r = 2 + s = p + q + 1 , r = q - p + 1 , 0 \\leq r \\leq 2 . \\end{align*}"}
{"id": "6448.png", "formula": "\\begin{align*} \\lambda _ 0 T _ 0 ( u _ 0 ) = u _ 0 \\end{align*}"}
{"id": "7118.png", "formula": "\\begin{align*} ( x _ t - y _ t ) ^ 2 & = 2 \\int _ 0 ^ t \\big ( b ( s , x _ s ) - b ( s , y _ s ) \\big ) ( x _ s - y _ s ) d s \\\\ & \\le 2 \\int _ 0 ^ t K _ s ( x _ s - y _ s ) ^ 2 d s . \\end{align*}"}
{"id": "3083.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = x _ 2 \\ , , x _ 2 ^ \\prime = x _ 1 \\ , , y _ 1 ^ \\prime = y _ 1 \\ , , y _ 2 ^ \\prime = - y _ 2 \\ , . \\end{align*}"}
{"id": "9364.png", "formula": "\\begin{align*} \\mathcal { E } _ { n , \\lambda } \\bigg ( \\frac { 1 } { 2 } \\bigg ) & = \\sum _ { k = 0 } ^ { n } k ! S _ { 2 , \\lambda } ( n , k ) \\sum _ { j = 0 } ^ { k } \\binom { \\frac { 1 } { 2 } } { j } \\bigg ( - \\frac { 1 } { 2 } \\bigg ) ^ { k - j } \\\\ & = \\sum _ { k = 0 } ^ { n } k ! S _ { 2 , \\lambda } ( n , k ) ( - 1 ) ^ { k } \\sum _ { j = 0 } ^ { k } \\binom { 2 j } { j } \\frac { 1 } { ( 1 - 2 j ) 2 ^ { k + j } } . \\end{align*}"}
{"id": "4212.png", "formula": "\\begin{align*} \\int _ M \\langle \\nabla v _ m , \\nabla v _ m \\rangle \\ , d V _ g + \\int _ M V \\ , v _ m \\ , d V _ g = 0 , \\end{align*}"}
{"id": "238.png", "formula": "\\begin{align*} \\partial _ { \\xi _ j } \\left ( \\varphi _ \\mu \\right ) ( \\xi ) = - \\langle \\Sigma ( e _ j ) ; \\xi \\rangle \\varphi _ \\mu ( \\xi ) , \\end{align*}"}
{"id": "5968.png", "formula": "\\begin{align*} t _ i = i \\Delta t , i = 0 , 1 , . . . N . \\end{align*}"}
{"id": "786.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ \\theta v = f Z , \\end{align*}"}
{"id": "5328.png", "formula": "\\begin{align*} \\tau _ 1 : = \\frac { \\beta } { } \\bigg ( \\frac { 1 } { k } \\mathbb { T } ( x , t ) - L \\eta ( x , t ) \\mathbb { I } \\bigg ) , \\end{align*}"}
{"id": "3834.png", "formula": "\\begin{align*} \\mathcal D _ F = \\{ y \\colon y ^ { [ k ] } \\in A C _ { l o c } ( \\mathbb R _ + ) , \\ , k = \\overline { 0 , n - 1 } \\} . \\end{align*}"}
{"id": "5703.png", "formula": "\\begin{align*} \\eta _ { \\pm } ( s , t ) = e ^ { \\kappa _ { \\mp } s } \\eta _ { \\kappa _ { \\mp } } ( t ) + O ( e ^ { ( \\kappa _ { \\mp } + \\epsilon _ { \\mp } ) s } ) \\end{align*}"}
{"id": "468.png", "formula": "\\begin{align*} \\mathfrak { G } ( \\mathbf { L } ) : = \\left \\{ \\mathcal { I } \\in \\wp _ { k } [ n ] : \\ , \\Delta _ { \\mathbf { L } } ( \\mathcal { I } ) \\neq 0 \\right \\} \\end{align*}"}
{"id": "4827.png", "formula": "\\begin{align*} T ( z ) = { \\rm t r } ^ \\flat ( e ^ { - i t _ 0 h ^ { - 1 } \\widetilde { P } _ h ( z ) } \\widetilde { R } _ h ( z ) ) \\end{align*}"}
{"id": "6410.png", "formula": "\\begin{align*} H ^ 2 ( \\mu ) = { } \\ , \\{ z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\dots z _ { n } ^ { i _ { n } } \\ , : \\ , 0 \\leq i _ 1 , \\dots , i _ { n } \\leq { \\widetilde k } - 1 \\} + { R a n } \\ , M _ { z } \\ , , \\end{align*}"}
{"id": "6414.png", "formula": "\\begin{align*} \\phantom { , \\quad w _ 1 , w _ 2 \\in W } L ( w _ 1 \\wedge w _ 2 ) = l ( w _ 1 ) \\wedge l ( w _ 2 ) , \\quad w _ 1 , w _ 2 \\in W , \\end{align*}"}
{"id": "4869.png", "formula": "\\begin{align*} \\int _ \\R g ( x + z ) K '' ( z ) \\ , d z = \\int _ \\R g ( x - y ) \\tilde K '' ( y ) \\ , d z = ( \\tilde K '' * g ) ( x ) = ( \\tilde K * g ) '' ( x ) . \\end{align*}"}
{"id": "363.png", "formula": "\\begin{align*} \\rho _ 1 ( t ) - \\rho _ 1 ( 0 ) = \\int _ 0 ^ t m _ { 2 1 } ( s ) ( 1 + \\dot { W ^ { \\delta } } ( s ) ) d s . \\end{align*}"}
{"id": "3260.png", "formula": "\\begin{align*} \\omega _ 2 ( i t ) = i \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , t ) - t } = \\frac { | \\lambda | ^ 2 } { i t - i \\omega _ 1 ( i t ) } . \\end{align*}"}
{"id": "3394.png", "formula": "\\begin{align*} S _ d ( f ) ( x ) = \\left \\{ \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\sum \\limits _ { Q } | \\psi _ { Q } \\ast f ( x _ Q ) | ^ 2 \\chi _ Q ( x ) \\right \\} ^ { 1 / 2 } . \\end{align*}"}
{"id": "297.png", "formula": "\\begin{align*} J [ \\lambda ] ( x , t , \\tau ) : = \\int _ { \\R } G ( x - y , t - \\tau ) \\eta ( y , \\tau ) ^ { - 1 } \\lambda ( y , \\tau ) d y = \\left ( G ( t - \\tau ) * ( \\eta ^ { - 1 } \\lambda \\right ) ( \\tau ) ) ( x ) . \\end{align*}"}
{"id": "8206.png", "formula": "\\begin{align*} s \\left ( a , \\frac { a ^ d - 1 } { a - 1 } \\right ) = \\frac { ( f - 1 ) ( f - a ^ 2 - 1 ) } { 1 2 a f } = O \\left ( f ^ { 1 - \\frac { 1 } { d - 1 } } \\right ) \\hbox { f o r $ d \\geq 3 $ o d d a n d $ a \\neq - 1 , 0 , 1 $ , } \\end{align*}"}
{"id": "6332.png", "formula": "\\begin{align*} \\widetilde \\Gamma ^ \\lambda _ { \\alpha \\beta \\gamma } \\stackrel { ( \\alpha \\beta \\gamma ) } { = } \\partial _ \\gamma \\Gamma ^ \\lambda _ { \\alpha \\beta } + \\Gamma ^ \\lambda _ { ( \\gamma \\sigma ) } \\Gamma ^ \\sigma _ { \\alpha \\beta } \\end{align*}"}
{"id": "1324.png", "formula": "\\begin{align*} 2 Q _ { \\tau } ( Z , Z _ { \\gamma } ) + 2 ( \\lfloor M \\theta \\rfloor - \\lfloor ( M - p _ { i } ) \\theta \\rfloor ) = 0 \\end{align*}"}
{"id": "8324.png", "formula": "\\begin{align*} \\int _ s ^ \\tau z \\dd ( y - u ) = \\int _ s ^ \\tau \\min \\{ 0 , z \\} \\dd ( y - u ) = \\int _ s ^ \\tau \\max \\{ 0 , - z \\} \\dd ( u - y ) \\geq 0 . \\end{align*}"}
{"id": "5112.png", "formula": "\\begin{align*} \\psi ( x ) > q _ 1 ( 1 . 6 ) + 6 \\ , q _ 2 ( 1 . 5 ) = 0 . 3 8 > 0 . \\end{align*}"}
{"id": "723.png", "formula": "\\begin{align*} S _ { ( i j ) } ^ { ( \\ell ) } : = \\kappa _ { ( i j ) } ^ { ( \\ell ) } - K _ { ( i j ) } ^ { ( \\ell ) } \\end{align*}"}
{"id": "9037.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { \\tau } d _ W ( \\rho _ i ^ 0 , \\ \\rho _ i ^ 1 ) ^ 2 = & \\inf \\left \\{ \\int _ 0 ^ { \\tau } \\int _ { \\Omega } \\rho _ i D ^ { - 1 } _ i | u _ i | ^ 2 d x d t \\right \\} , \\\\ & \\partial _ t \\rho _ i + \\nabla \\cdot ( \\rho _ i u _ i ) = 0 , \\\\ & ( \\rho _ i u _ i ) \\cdot \\mathbf { n } = 0 , x \\in \\partial \\Omega , \\\\ & \\rho _ i ( x , 0 ) = \\rho ^ 0 _ i , \\ \\rho _ i ( x , \\tau ) = \\rho ^ 1 _ i . \\end{aligned} \\end{align*}"}
{"id": "5397.png", "formula": "\\begin{align*} M _ \\mu ( \\alpha , \\varepsilon , \\delta , T , f ) & = \\lim _ { n \\to \\infty } M _ \\mu ( n , \\alpha , \\varepsilon , \\delta , T , f ) , \\\\ \\underline { M } _ \\mu ( \\alpha , \\varepsilon , \\delta , T , f ) & = \\liminf _ { n \\to \\infty } R _ \\mu ( n , \\alpha , \\varepsilon , \\delta , T , f ) , \\\\ \\overline { M } _ \\mu ( \\alpha , \\varepsilon , \\delta , T , f ) & = \\limsup _ { n \\to \\infty } R _ \\mu ( n , \\alpha , \\varepsilon , \\delta , T , f ) . \\end{align*}"}
{"id": "4160.png", "formula": "\\begin{align*} \\tilde P = \\sup _ { [ - T , T ] } \\{ \\| u _ 1 ( t ) \\| _ { H ^ s } + \\| u _ 2 ( t ) \\| _ { H ^ s } + \\| w ( t ) \\| _ { Z _ { s , 5 } } \\} < \\infty . \\end{align*}"}
{"id": "9137.png", "formula": "\\begin{align*} \\Phi ( k , n ) = \\phi \\left ( \\left \\lceil 2 C ( k + 1 ) \\right \\rceil - 1 , \\max \\{ \\theta ( M \\varpi ( k ) + M - 1 ) , n \\} \\right ) \\end{align*}"}
{"id": "2296.png", "formula": "\\begin{align*} f = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) ( M _ \\omega T _ x g ) \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "4353.png", "formula": "\\begin{align*} \\epsilon '' + ( d - 4 ) \\epsilon ' - ( d - 2 ) \\epsilon ( \\epsilon + 1 ) ( \\epsilon - 1 ) = 0 . \\end{align*}"}
{"id": "508.png", "formula": "\\begin{align*} Y ( \\mathcal { B } ) _ { \\alpha \\beta } ^ { i j } = - Y ( \\mathcal { B } ) _ { \\alpha v } ^ { i \\omega } Y ( \\mathcal { B } ) _ { v \\beta } ^ { i \\omega } Y ( \\mathcal { B } ) _ { \\alpha v } ^ { \\omega j } Y ( \\mathcal { B } ) _ { v \\beta } ^ { \\omega j } \\end{align*}"}
{"id": "4016.png", "formula": "\\begin{align*} \\begin{aligned} & C _ 1 + C _ 2 + C _ 3 = 0 , \\\\ & C _ 1 e ^ { m _ 1 } + C _ 2 e ^ { m _ 2 } + C _ 3 e ^ { m _ 3 } = 0 , \\\\ & C _ 1 m _ 1 ^ 2 \\left ( 1 - e ^ { m _ 1 } \\right ) + C _ 2 m _ 2 ^ 2 \\left ( 1 - e ^ { m _ 2 } \\right ) + C _ 3 m _ 3 ^ 2 \\left ( 1 - e ^ { m _ 3 } \\right ) = 0 . \\end{aligned} \\end{align*}"}
{"id": "6269.png", "formula": "\\begin{align*} R _ 3 ^ * = \\left \\{ u \\in V ( F ) \\colon \\phi _ { x y } ( u ) \\in T \\right \\} \\mbox { a n d } R _ 4 ^ * = \\left ( \\cup _ { i = 1 } ^ s \\left \\{ c _ 2 ^ i , \\ldots , c _ { 2 k - 2 } ^ i \\right \\} \\right ) \\setminus R _ 3 ^ * . \\end{align*}"}
{"id": "7895.png", "formula": "\\begin{align*} \\nabla ( | | G ( x ) | | ^ 2 ) = 2 ( g _ 1 ( x ) \\nabla g _ 1 ( x ) + \\dots g _ m ( x ) \\nabla g _ m ( x ) ) . \\end{align*}"}
{"id": "2685.png", "formula": "\\begin{align*} g _ 0 ( t _ 1 , t _ 2 ) = 2 ^ { 1 / 2 } e ^ { - \\pi ( t _ 1 ^ 2 + t _ 2 ^ 2 ) } = g _ 0 ( t _ 1 ) \\otimes g _ 0 ( t _ 2 ) . \\end{align*}"}
{"id": "2273.png", "formula": "\\begin{align*} \\log ( I ( \\omega ) ) = - \\frac { \\pi \\omega ^ 2 } { M } + C , \\end{align*}"}
{"id": "3626.png", "formula": "\\begin{align*} \\int _ { H } ^ { t _ 0 } \\frac { x ^ { - \\nu _ 1 ( t ) } } { t } N ( \\sigma , t ) \\le \\frac { x ^ { - \\nu _ 1 ( t _ 0 ) } } { t _ 0 } \\int _ H ^ { t _ 0 } N ( \\sigma , t ) = \\frac { x ^ { - \\nu _ 1 ( t _ 0 ) } } { t _ 0 } N ( \\sigma , t _ 0 ) \\end{align*}"}
{"id": "8821.png", "formula": "\\begin{align*} \\nabla S _ t ( \\varphi ) w = | \\alpha _ w ( t ) | \\ge \\exp \\left ( \\int _ 0 ^ t \\dot { \\lambda } _ { 2 s } d s \\right ) = e ^ { \\frac { \\lambda _ { 2 t } } { 2 } } , \\end{align*}"}
{"id": "5929.png", "formula": "\\begin{align*} L ( \\phi _ i , v ^ i _ a ) = \\frac { 1 } { 2 } \\left ( \\sum _ { a , b = 1 } ^ 4 \\eta _ { a b } v ^ 1 _ a v ^ 1 _ b - m ^ 2 \\phi _ 1 ^ 2 \\right ) + \\frac { 1 } { 2 } \\left ( \\sum _ { a , b = 1 } ^ 4 \\eta _ { a b } v ^ 2 _ a v ^ 2 _ b - m ^ 2 \\phi _ 2 ^ 2 \\right ) - \\frac { 1 } { 4 } g ( \\phi _ 1 ^ 2 + \\phi _ 2 ^ 2 ) ^ 2 . \\end{align*}"}
{"id": "8849.png", "formula": "\\begin{align*} \\Delta _ R : = \\{ ( x , y ) \\in X \\times X | d ( x , y ) \\le R \\} \\end{align*}"}
{"id": "8437.png", "formula": "\\begin{align*} \\tilde { Y } _ { k + 1 } = ( 1 - \\eta ) ^ { k + 1 } x + \\frac { \\eta ^ { 1 / \\alpha } } { \\sigma } \\sum _ { i = 0 } ^ { k } ( 1 - \\eta ) ^ { i } \\widetilde { Z } _ { k + 1 - i } . \\end{align*}"}
{"id": "3977.png", "formula": "\\begin{align*} 3 r _ 1 + 2 i l \\pi + 2 i m \\pi = - \\lambda , \\ \\ \\ r _ 1 = \\frac { 1 } { 3 } ( - \\lambda - 2 i l \\pi - 2 i m \\pi ) , \\end{align*}"}
{"id": "773.png", "formula": "\\begin{align*} \\langle W \\rangle = w _ 1 \\langle W ' \\rangle + \\langle ( W ' ) ' \\rangle \\mbox { f o r } | W | \\geq 2 . \\end{align*}"}
{"id": "5603.png", "formula": "\\begin{align*} \\mathcal { T } ^ \\mathcal { F } _ \\star ( X , Y ) = \\langle X \\otimes _ { \\mathcal { X } _ \\star } Y , \\mathrm { T } ^ \\mathcal { F } \\rangle _ \\star , ~ \\mathrm { R } ^ \\mathcal { F } _ \\star ( X , Y , Z ) = \\langle X \\otimes _ { \\mathcal { X } _ \\star } Y \\otimes _ { \\mathcal { X } _ \\star } Z , \\mathrm { R } ^ \\mathcal { F } \\rangle _ \\star \\end{align*}"}
{"id": "7604.png", "formula": "\\begin{align*} | M ^ { n } | = \\int _ { M ^ { n } } H ^ { 1 - \\alpha } . \\end{align*}"}
{"id": "5158.png", "formula": "\\begin{align*} ( a b ) ^ { - 1 } \\sum _ { k , l \\in \\Z } V _ g g \\left ( \\frac { k } { a } , \\frac { l } { b } \\right ) e ^ { 2 \\pi i ( k x + l \\omega ) } = \\sum _ { k = 0 } ^ { n - 1 } \\left | Z _ b g \\left ( \\frac { x + k } { n } , \\omega \\right ) \\right | ^ 2 . \\end{align*}"}
{"id": "8297.png", "formula": "\\begin{align*} e ( \\vect { r } ) = \\begin{cases} e _ { \\text i } ( \\vect { r } ) + e _ { \\text r } ( \\vect { r } ) & r _ z < d _ 1 \\\\ e _ { \\text t } ( \\vect { r } ) & r _ z > d _ 1 . \\end{cases} \\end{align*}"}
{"id": "820.png", "formula": "\\begin{align*} \\int _ X g _ { u , \\rho } ^ 2 \\ , d \\mu _ \\omega = \\int _ X g _ u ^ 2 \\ , d \\mu _ X . \\end{align*}"}
{"id": "2716.png", "formula": "\\begin{align*} N _ { \\ell , n } = X _ 1 ^ \\ell + \\cdots + X _ n ^ \\ell . \\end{align*}"}
{"id": "2959.png", "formula": "\\begin{align*} \\Lambda ^ { ( h : h ' ) } _ { n , 2 } = \\Lambda ^ { ( h : h ' ) } _ { n , 2 , 0 } + o ( 1 ) = \\Lambda ^ { ( h : h ' ) } _ { n , 2 , 0 } ( \\mathcal S _ k ) + \\Lambda ^ { ( h : h ' ) } _ { n , 2 , 0 } ( \\mathcal { W } _ k ( 0 ) \\setminus \\mathcal { S } _ k ) + o ( 1 ) \\end{align*}"}
{"id": "1517.png", "formula": "\\begin{align*} x = \\left [ \\begin{array} { c c c } a _ 1 & \\frac { b _ 1 } { 2 } + b _ 2 \\zeta ^ { - 1 } & a _ 2 \\end{array} \\right ] , y = \\left [ \\begin{array} { c c c } - c _ 1 & - \\frac { b _ 1 } { 2 } + b _ 2 \\zeta ^ { - 1 } & c _ 2 \\end{array} \\right ] , x \\phi _ 1 x ^ { \\ast } = y \\phi _ 2 y ^ { \\ast } . \\end{align*}"}
{"id": "6537.png", "formula": "\\begin{align*} \\frac { m } { ( \\log n ) ^ m } \\displaystyle \\sum ^ n _ { j = 1 } \\frac { ( \\log j ) ^ { m - 1 } } { j } & = 1 + O ( ( \\log n ) ^ { - m } ) ( n \\to \\infty ) . \\end{align*}"}
{"id": "2065.png", "formula": "\\begin{align*} b _ { \\beta } ^ { \\alpha } g _ { \\beta } ^ { ( \\beta ) } ( o ) + \\sum _ { \\beta < \\gamma < \\alpha } b _ { \\gamma } ^ { \\alpha } g _ { \\beta } ^ { ( \\gamma ) } ( o ) = - g _ { \\beta } ^ { ( \\alpha ) } ( o ) , \\forall \\beta < \\alpha , \\end{align*}"}
{"id": "4308.png", "formula": "\\begin{align*} Q _ { b ( \\tau ) } ( y ) = \\frac { 1 } { b ( \\tau ) } Q \\left ( \\frac { | y | } { \\sqrt { b ( \\tau ) } } \\right ) , \\end{align*}"}
{"id": "1047.png", "formula": "\\begin{align*} \\Delta ( D _ { i j } ) & = \\sum _ { P \\cap ( k < l ) } D _ { i j } ^ { k l } \\otimes D _ { k l } ^ { 1 2 } - ( a _ { i 5 } a _ { j 6 } + q ^ { - 1 } a _ { i 6 } a _ { j 5 } ) \\otimes D _ { 5 6 } \\\\ & - ( 1 + q ^ { - 2 } ) \\sum _ { 5 \\leq k \\leq 6 } a _ { i k } a _ { j k } \\otimes D _ { k k } \\ , . \\end{align*}"}
{"id": "8927.png", "formula": "\\begin{align*} \\varphi | _ { A _ { i _ 0 } \\cap B _ { j _ 0 } \\times \\cdots \\times A _ { i _ q } \\cap B _ { j _ q } } = \\psi | _ { A _ { i _ 0 } \\cap B _ { j _ 0 } \\times \\cdots \\times A _ { i _ q } \\cap B _ { j _ q } } \\end{align*}"}
{"id": "6370.png", "formula": "\\begin{align*} v = \\rho _ 2 + \\rho _ 3 + 4 v _ 0 \\end{align*}"}
{"id": "2473.png", "formula": "\\begin{align*} P _ { 2 , 3 } = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\det ( P _ { 2 , 3 } ) = 1 . \\end{align*}"}
{"id": "1457.png", "formula": "\\begin{align*} \\lambda ( \\alpha , z _ 1 ) ^ { \\ast } \\eta ( \\alpha z _ 1 , \\alpha z _ 2 ) \\lambda ( \\alpha , z _ 2 ) = \\eta ( z _ 1 , z _ 2 ) , \\end{align*}"}
{"id": "1529.png", "formula": "\\begin{align*} p _ { i 1 } ( - h ) + p _ { i 2 } ( - \\zeta ^ { - 1 } g ) + p _ { i 3 } ( a - 1 ) \\equiv 0 \\mathfrak { n } , i = 1 , 2 , 3 . \\end{align*}"}
{"id": "6487.png", "formula": "\\begin{align*} M _ { n + 1 } ^ { ( 2 m - 1 ) } = f _ n ^ { ( 2 m - 1 ) } + g ^ { ( 2 m - 1 ) } _ n M _ n ^ { ( 2 m - 1 ) } . \\end{align*}"}
{"id": "4051.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda \\rho + \\rho _ x + u _ x - \\epsilon \\rho _ { x x } = f , \\\\ \\lambda u + \\rho _ x + u _ x - u _ { x x } = g , \\end{aligned} \\end{align*}"}
{"id": "5920.png", "formula": "\\begin{align*} n = h + 2 m \\end{align*}"}
{"id": "5235.png", "formula": "\\begin{align*} & ( w ( b _ r - d + e ) + k + 1 / ( d - 1 ) ) ( d - 1 ) - ( b _ r + d k - d + e ) \\\\ = & ( ( b _ r - d ) ( d - 4 ) + d - 2 k - 2 ) / 2 \\ge 0 . \\end{align*}"}
{"id": "8871.png", "formula": "\\begin{align*} d _ { q - 1 } ( h \\varphi ) + h ( d _ q \\varphi ) = \\beta ^ * \\varphi - \\alpha ^ * \\varphi . \\end{align*}"}
{"id": "5130.png", "formula": "\\begin{align*} \\frac { B } { A } = e ^ { 2 \\gamma a } \\coth \\left ( \\frac { \\gamma n a } { 2 } \\right ) ^ 2 . \\end{align*}"}
{"id": "5211.png", "formula": "\\begin{align*} \\| B f \\| _ { L ^ 2 ( K ) } & = \\bigg \\| \\sum _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ g _ j \\ast f \\bigg \\| _ { L ^ 2 ( K ) } \\\\ & \\leq ~ \\sum _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ \\| g _ j \\ast f \\| _ { L ^ 2 ( K ) } \\\\ & \\leq ~ \\sum _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ \\| f \\| _ { L ^ 2 ( K ) } \\\\ & = ~ q ^ { - k } ( q - 1 ) ^ { - 1 } \\| f \\| _ { L ^ 2 ( K ) } . \\end{align*}"}
{"id": "6549.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } u + ( u \\cdot \\nabla ) u + ( - \\Delta ) ^ { \\alpha } u + \\nabla p = ( b \\cdot \\nabla ) b , x \\in \\mathbb { R } ^ { 2 } , \\ , t > 0 , \\\\ & \\partial _ { t } b + ( u \\cdot \\nabla ) b + ( - \\Delta ) ^ { \\beta } b = ( b \\cdot \\nabla ) u , \\\\ & \\nabla \\cdot u = 0 , \\ \\ \\ \\nabla \\cdot b = 0 , \\\\ & u ( x , 0 ) = u _ { 0 } ( x ) , \\ \\ b ( x , 0 ) = b _ { 0 } ( x ) , \\endaligned \\right . \\end{align*}"}
{"id": "3342.png", "formula": "\\begin{align*} & [ T u , T v , T w ] = T \\Big ( \\theta ( T v , T w ) u - \\theta ( T u , T w ) v + D ( T u , T v ) w \\Big ) . \\end{align*}"}
{"id": "9290.png", "formula": "\\begin{align*} \\frac { \\partial C ( x , t ) } { \\partial t } = D \\frac { \\partial ^ { 2 } C ( x , t ) } { \\partial x ^ { 2 } } \\end{align*}"}
{"id": "2813.png", "formula": "\\begin{align*} O ( \\widetilde { \\delta } ^ s ) = \\Phi ( \\alpha Q + h ) = \\alpha ^ 2 \\Phi ( Q ) + \\Phi ( h ) + 2 \\alpha B ( Q , h ) = \\alpha ^ 2 \\Phi ( Q ) + \\Phi ( h ) , \\end{align*}"}
{"id": "7430.png", "formula": "\\begin{align*} & n ^ { \\gamma - 2 } \\sum _ { | x | \\leq 2 b _ G n } \\sum _ { | y | \\leq 2 b _ G n } \\sup _ { s \\in [ 0 , T ] } C _ { G , \\delta } | x - y | ^ { \\delta } n ^ { - \\delta } p ( y - x ) \\\\ \\lesssim & n ^ { \\gamma - 2 - \\delta } \\sum _ { | x | \\leq 2 b _ G n } \\sum _ { z \\neq 0 } | z | ^ { \\delta - \\gamma - 1 } = 2 ( 2 b _ G + 1 ) n ^ { \\gamma - 1 - \\delta } \\sum _ { z = 1 } ^ { \\infty } z ^ { \\delta - \\gamma - 1 } . \\end{align*}"}
{"id": "9280.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { C } ^ \\omega \\vdash F ( \\underline { a } ) \\Rightarrow \\\\ \\mathcal { C } ^ { \\omega - } + ( \\mathrm { B R } ) \\vdash \\forall \\underline { a } , \\underline { y } ( F ' ) _ D ( \\underline { t } \\underline { a } , \\underline { y } , \\underline { a } ) \\end{cases} \\end{align*}"}
{"id": "7551.png", "formula": "\\begin{align*} \\epsilon N _ 0 ( T ) = - \\frac { \\epsilon T } { 2 \\pi } \\log \\pi + \\frac { \\epsilon T } { 2 \\pi } \\log \\left ( \\frac { T } { 2 } \\right ) - \\frac { \\epsilon T } { 2 \\pi } + \\frac { \\epsilon } { \\pi } \\ \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "3257.png", "formula": "\\begin{align*} ( s - t ) ^ 2 - \\frac { s - t } { h ( s ) } + | \\lambda | ^ 2 = 0 . \\end{align*}"}
{"id": "4814.png", "formula": "\\begin{align*} \\phi = c \\alpha + \\bar c \\bar \\alpha . \\end{align*}"}
{"id": "1156.png", "formula": "\\begin{align*} m ^ { ( 2 ) } ( x , t , k ) = m ^ { ( 1 ) } ( x , t , k ) G ( x , t , k ) , \\end{align*}"}
{"id": "3771.png", "formula": "\\begin{align*} \\epsilon \\big ( X , \\rho , \\psi _ K \\big ) \\epsilon \\big ( q _ K ^ { - 1 } X ^ { - 1 } , \\rho ^ \\vee , \\psi _ K \\big ) = \\rm d e t \\big ( \\rho ( - 1 ) \\big ) , \\end{align*}"}
{"id": "681.png", "formula": "\\begin{align*} K _ { } ^ 2 : = n _ 0 C _ W ^ { - 1 } ( K _ * - C _ b ) \\end{align*}"}
{"id": "1486.png", "formula": "\\begin{align*} ( f | _ k \\gamma ) ( z ) = \\sum _ { \\tau \\in S _ + } c ( \\tau , \\gamma , f ; v , w ) e ( \\lambda ( \\mathfrak { i } ( \\tau ) u ) ) , z = ( u , v , w ) \\in \\mathfrak { Z } , \\end{align*}"}
{"id": "4057.png", "formula": "\\begin{align*} \\epsilon \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } _ x \\bar { \\sigma } _ x + \\int _ { 0 } ^ { 1 } u ^ { \\epsilon } _ x \\bar { \\sigma } - \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } \\bar { \\sigma } _ x + \\lambda \\int _ { 0 } ^ { 1 } \\rho ^ { \\epsilon } \\bar { \\sigma } = \\int _ { 0 } ^ { 1 } f \\bar { \\sigma } . \\end{align*}"}
{"id": "6280.png", "formula": "\\begin{align*} \\mathbb { E } [ \\zeta _ \\mathcal { K } ] & \\stackrel { ( a ) } { = } \\mathbb { E } \\big [ \\Re \\{ \\zeta _ \\mathcal { K } \\} \\big ] = \\sum _ { i \\in \\mathcal { K } } \\sqrt { \\bar { p } / \\beta _ i } \\mathbb { E } \\big [ | \\hat { h } _ i | ] \\stackrel { ( b ) } { = } \\frac { \\sqrt { \\pi \\bar { p } } } { 2 } \\sum _ { i \\in \\mathcal { K } } \\sqrt { 1 + \\frac { 1 } { N _ 1 \\bar { \\gamma } _ i } } \\stackrel { ( c ) } { \\approx } \\frac { K \\sqrt { \\pi \\bar { p } } } { 2 } \\sqrt { 1 + \\frac { 1 } { N _ 1 \\bar { \\gamma } ' } } , \\end{align*}"}
{"id": "8038.png", "formula": "\\begin{align*} \\left \\{ \\Phi ( \\mathbf { x } ) , \\Pi ( \\mathbf { y } ) \\right \\} ^ \\Sigma _ { \\mathrm { c a n } } = \\delta _ \\Sigma ( \\mathbf { x } , \\mathbf { y } ) , \\end{align*}"}
{"id": "4351.png", "formula": "\\begin{align*} Z '' + \\frac { d - 3 } { \\xi } Z ' - \\frac { ( d - 2 ) } { \\xi ^ 2 } Z ( Z - 1 ) ( Z - 2 ) = 0 . \\end{align*}"}
{"id": "4011.png", "formula": "\\begin{align*} m _ 1 & = - \\frac { 1 } { 3 } \\bigg [ \\mu + \\left ( \\mu + \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } + \\frac { 3 \\sqrt { 3 } } { 8 } \\mu ^ { - 1 / 2 } + O ( \\mu ^ { - 3 / 2 } ) \\right ) \\\\ & + \\left ( \\mu - \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } - \\frac { 3 \\sqrt { 3 } } { 8 } \\mu ^ { - 1 / 2 } + O ( \\mu ^ { - 1 } ) \\right ) \\bigg ] \\\\ & = - \\frac { 1 } { 3 } \\left ( 3 \\mu - 3 + O ( \\mu ^ { - 1 } ) \\right ) \\\\ & = - \\mu + 1 + O ( \\mu ^ { - 1 } ) , \\end{align*}"}
{"id": "4608.png", "formula": "\\begin{align*} G _ 1 \\cap G _ 2 = \\pi _ v ( G _ 1 ^ + ) \\cap \\pi _ v ( G _ 1 ^ - ) \\cap \\pi _ v ( G _ 2 ^ + ) \\cap \\pi _ v ( G _ 2 ^ - ) = \\pi _ v ( G _ 1 ^ + \\cap G _ 2 ^ + ) \\cap \\pi _ v ( G _ 1 ^ - \\cap G _ 2 ^ - ) . \\end{align*}"}
{"id": "2797.png", "formula": "\\begin{align*} L _ + f _ * = \\lambda _ 0 \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ { p } \\right ) Q ^ { p - 1 } + \\sum _ { j = 1 } ^ { N } \\lambda _ j \\partial _ { x _ j } Q + \\lambda _ { N + 1 } f _ * . \\end{align*}"}
{"id": "1094.png", "formula": "\\begin{align*} m ^ { ( 2 ) } _ { + } ( x , t , k ) = m ^ { ( 2 ) } _ { - } ( x , t , k ) J ^ { ( 2 ) } ( x , t , k ) , k \\in \\mathbb { R } , \\end{align*}"}
{"id": "8133.png", "formula": "\\begin{align*} \\ell _ { D } ( d , e ) = c _ { D } ^ { + } ( d , e ) - c _ { D } ^ { - } ( d , e ) \\equiv c _ { D } ^ { + } ( d , e ) + c _ { D } ^ { - } ( d , e ) \\equiv | d \\cap e | \\pmod 2 . \\end{align*}"}
{"id": "1268.png", "formula": "\\begin{align*} P _ { \\gamma } ^ { \\mathrm { i n } } ( M ) : = P _ { \\theta } ^ { \\mathrm { i n } } ( M ) , \\ , \\ , \\ , \\ , P _ { \\gamma } ^ { \\mathrm { o u t } } ( M ) : = P _ { \\theta } ^ { \\mathrm { o u t } } ( M ) . \\end{align*}"}
{"id": "8565.png", "formula": "\\begin{align*} & \\mbox { i f $ k > 0 $ } , \\widetilde { f } ( k ) = - \\frac { \\alpha k } { \\sqrt { 2 \\pi } } \\int f ( x ) \\psi _ + ( x , 0 ) \\ , d x + O ( k ^ 2 ) \\\\ & \\mbox { i f $ k < 0 $ } , \\widetilde { f } ( k ) = \\frac { \\alpha k } { \\sqrt { 2 \\pi } } \\int f ( x ) \\psi _ - ( x , 0 ) \\ , d x + O ( k ^ 2 ) , \\end{align*}"}
{"id": "4493.png", "formula": "\\begin{align*} \\Gamma ( s , x ) = \\int _ x ^ { \\infty } t ^ { s - 1 } e ^ { - t } d t , \\ \\ \\Re s > 0 . \\end{align*}"}
{"id": "2151.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { N \\to \\infty } \\lim _ { n \\to \\infty } \\\\ & \\sum _ { j = 1 } ^ { n - N } \\sum _ { l = 1 } ^ k \\frac { \\binom { c _ n k n } l ( k ) _ l ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac k { k ( n - j + 1 ) - l } \\frac { E _ { n , k } ( \\tau ( A ^ { ( n ) } _ { c _ n k n , j , l } ) | \\mathcal { W A } ^ { ( n , k ) } _ { M _ n , j , l } ) } { n k } = 0 . \\end{aligned} \\end{align*}"}
{"id": "2128.png", "formula": "\\begin{align*} \\begin{aligned} & \\log \\prod _ { i = ( 1 - c _ n ) k n + l + 1 } ^ { k n } ( 1 - \\frac { k ( s + 1 ) } i ) = \\sum _ { i = ( 1 - c _ n ) k n + l + 1 } ^ { k n } \\log ( 1 - \\frac { k ( s + 1 ) } i ) = \\\\ & - k ( s + 1 ) \\sum _ { i = ( 1 - c _ n ) k n + l + 1 } ^ { k n } \\frac 1 i + o ( 1 ) = k ( s + 1 ) \\log ( 1 - c _ n ) + o ( 1 ) , \\ \\ n \\to \\infty . \\end{aligned} \\end{align*}"}
{"id": "6080.png", "formula": "\\begin{align*} g ( g \\varphi ^ { j } _ { s } + \\lambda \\psi ^ { j } _ { s } ) + \\lambda ( g \\varphi ^ { j } _ { t } + \\lambda \\psi ^ { j } _ { t } ) = - \\eta ^ { t } \\Lambda \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} + g ^ { 2 } F ^ { j } + g \\lambda G ^ { j } + \\lambda ^ { 2 } H ^ { j } , \\end{align*}"}
{"id": "8301.png", "formula": "\\begin{align*} R _ - ( \\kappa _ x ) & = \\frac { 1 - \\alpha } { 1 + \\alpha } = \\frac { \\mu _ 2 \\kappa _ { 1 z } - \\mu _ 1 \\kappa _ { 2 z } } { \\mu _ 2 \\kappa _ { 1 z } + \\mu _ 1 \\kappa _ { 2 z } } \\\\ T _ + ( \\kappa _ x ) & = \\frac { 2 } { 1 + \\alpha } = \\frac { 2 \\mu _ 2 \\kappa _ { 1 z } } { \\mu _ 2 \\kappa _ { 1 z } + \\mu _ 1 \\kappa _ { 2 z } } . \\end{align*}"}
{"id": "8429.png", "formula": "\\begin{align*} Y _ { l _ { t } ^ { \\epsilon } } ^ { x ; l ^ { \\epsilon } } = X _ { t } ^ { x ; l ^ { \\epsilon } } \\quad \\nabla _ { v } Y _ { l _ { t } ^ { \\epsilon } } ^ { x ; l ^ { \\epsilon } } = \\nabla _ { v } X _ { t } ^ { x ; l ^ { \\epsilon } } . \\end{align*}"}
{"id": "3363.png", "formula": "\\begin{align*} & [ T _ 1 ( u ) , T _ 1 ( v ) , T _ 1 ( w ) ] = T _ 1 \\Big ( D ( T _ 1 ( u ) , T _ 1 ( v ) ) w + \\theta ( T _ 1 ( v ) , T _ 1 ( w ) ) u - \\theta ( T _ 1 ( u ) , T _ 1 ( w ) ) v \\Big ) . \\end{align*}"}
{"id": "9281.png", "formula": "\\begin{align*} \\widehat { 0 } : = 0 , \\ ; \\widehat { X } : = 0 , \\ ; \\widehat { \\tau ( \\xi ) } : = \\widehat { \\tau } ( \\widehat { \\xi } ) . \\end{align*}"}
{"id": "3881.png", "formula": "\\begin{align*} \\beta _ i = t _ i - t _ { i + 1 } & = ( t _ i - s _ i ) + \\boxed { ( s _ i - t _ { i + 1 } ) } \\\\ & = \\boxed { ( t _ i - s _ { i + 1 } ) } + ( s _ { i + 1 } - t _ { i + 1 } ) . \\end{align*}"}
{"id": "5259.png", "formula": "\\begin{align*} \\Delta ( A ) ( 1 \\otimes A ) = \\Delta ( A ) ( A \\otimes 1 ) = \\Delta ( 1 ) ( A \\otimes A ) , ( A \\otimes 1 ) \\Delta ( A ) = ( 1 \\otimes A ) \\Delta ( A ) = ( A \\otimes A ) \\Delta ( 1 ) . \\end{align*}"}
{"id": "5220.png", "formula": "\\begin{align*} \\| T _ k f _ j - T _ k f _ { j ^ \\prime } \\| _ { L ^ r ( K ) } & = \\| T _ k ( f _ j - f _ { j ^ \\prime } ) \\| _ { L ^ r ( K ) } \\\\ & \\leq C q ^ { - k } \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\| f _ j - f _ { j ^ \\prime } \\| _ { L ^ r ( K ) } , \\end{align*}"}
{"id": "2767.png", "formula": "\\begin{align*} \\| Q \\| _ 2 ^ 2 = \\lim _ { n \\to \\infty } \\| u _ n \\| _ 2 ^ 2 \\ge \\sum _ { j = 1 } ^ l \\| U ^ j \\| _ 2 ^ 2 , \\| \\nabla Q \\| _ 2 ^ 2 = \\lim _ { n \\to \\infty } \\| \\nabla u _ n \\| _ 2 ^ 2 \\ge \\sum _ { j = 1 } ^ l \\| \\nabla U ^ j \\| _ 2 ^ 2 . \\end{align*}"}
{"id": "7869.png", "formula": "\\begin{align*} ( F \\setminus \\{ \\bar { b } _ n \\} ) \\cup D _ 1 \\cup D _ 2 & = ( F \\setminus \\{ \\bar { b } _ n \\} ) \\cup ( D \\setminus \\{ \\bar { b } _ n \\} ) \\\\ & = ( F \\cup D ) \\setminus \\{ \\bar { a } _ n \\} = \\mathbb { N } \\setminus \\{ \\bar { a } _ n \\} \\end{align*}"}
{"id": "4048.png", "formula": "\\begin{align*} \\omega _ 0 = \\sup \\big \\{ \\Re ( \\lambda ) : \\lambda \\in \\sigma ( A ) \\big \\} < 0 . \\end{align*}"}
{"id": "7091.png", "formula": "\\begin{align*} s _ k & = - \\frac { f ( x , y ) } { k } \\binom { p - 2 } { k - 1 } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k - 1 } v & = - \\frac { 1 } { k } \\binom { p - 2 } { k - 1 } f ( x , v ) ^ { k - 1 } f ( y , v ) ^ { p - k - 1 } f ( x , y ) v . \\end{align*}"}
{"id": "8230.png", "formula": "\\begin{align*} 1 - \\frac { N _ 1 } { x _ 1 z } ( N _ 1 + N _ 2 + \\cdots + N _ { k + 1 - h } ) = \\frac { x _ h z } { N _ h } , \\\\ 1 - \\frac { N _ 1 } { x _ 1 z } ( N _ 1 + N _ 2 + \\cdots + N _ { k - h } ) = \\frac { x _ { h + 1 } z } { N _ { h + 1 } } . \\end{align*}"}
{"id": "885.png", "formula": "\\begin{align*} & v _ j \\coloneqq u _ { j + 1 } - u _ j , \\ , \\\\ & v _ r \\coloneqq u _ 1 - u _ r , \\end{align*}"}
{"id": "1056.png", "formula": "\\begin{align*} & A _ { p , q } = \\partial _ p \\partial _ q - \\partial _ { p + 1 } \\partial _ { q - 1 } { \\rm w h e r e } ( p , q ) \\in [ 1 , k - 1 ] \\times [ 2 , k ] \\\\ & \\mathcal { T } ^ m : = \\partial _ 1 \\partial _ { m - 1 } + \\partial _ m E { \\rm w h e r e } E : = \\sum _ { h = 1 } ^ k \\sigma _ h \\partial _ h { \\rm a n d } m \\in [ 2 , k ] \\end{align*}"}
{"id": "9534.png", "formula": "\\begin{align*} y & \\in \\partial V ( u - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } - \\bar c \\cdot \\bar x + S _ 0 ( \\bar x ) ) , \\\\ p _ t + y \\Delta s _ { t + 1 } & \\in N _ { D _ t } ( x _ t ) t = 0 , \\dots , T , \\\\ p _ { - 1 } + y \\bar c & \\in \\partial ( y S _ 0 ) ( \\bar x ) \\end{align*}"}
{"id": "1489.png", "formula": "\\begin{align*} \\mathbf { f } ( \\alpha g _ { \\mathbf { h } } k , \\alpha z ) = j ( \\alpha , z ) ^ k \\mathbf { f } ( g _ { \\mathbf { h } } , z ) . \\end{align*}"}
{"id": "5508.png", "formula": "\\begin{align*} d _ K ( \\xi ( s _ j ; s _ 0 , x ) ) & = d _ K ( \\xi ( s _ j ; s _ { j - 1 } , \\xi ( s _ { j - 1 } ; s _ 0 , x ) ) ) \\\\ & \\leq \\Phi _ { \\beta + L , 0 } ( d _ K ( \\xi ( s _ { j - 1 } ; s _ 0 , x ) ) , \\epsilon , s _ j - s _ { j - 1 } ) \\\\ & \\leq \\Phi _ { \\beta + L , 0 } ( \\Phi _ { \\beta + L , 0 } ( d _ K ( x ) , \\epsilon , s _ { j - 1 } - s _ 0 ) , \\epsilon , s _ j - s _ { j - 1 } ) \\\\ & = \\Phi _ { \\beta + L , 0 } ( d _ K ( x ) , \\epsilon , s _ j - s _ 0 ) , \\end{align*}"}
{"id": "1113.png", "formula": "\\begin{align*} & I _ { 1 } = - \\frac { 1 } { 2 \\pi i } \\oint _ { \\partial U _ \\xi } \\left ( J ^ E ( s ) - I \\right ) d s , \\\\ & I _ { 2 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma ^ { ( 3 ) } \\backslash U _ \\xi } \\left ( J ^ E ( s ) - I \\right ) d s , \\\\ & I _ { 3 } = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\mu ( s ) ( J ^ E ( s ) - I ) d s . \\end{align*}"}
{"id": "8611.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\ast } ( s , k ) = \\frac { 1 } { ( 2 \\pi ) ^ { 2 } } \\iiint e ^ { i s ( - k ^ 2 + \\ell ^ 2 - m ^ 2 + n ^ 2 ) } f ^ { \\# } ( s , \\ell ) \\overline { f ^ { \\# } ( s , m ) } f ^ { \\# } ( s , n ) & \\\\ \\times \\mu ^ { \\# } _ { \\ast } ( k , \\ell , m , n ) \\ , d n d m d \\ell , & \\ast \\in \\{ 0 ; L ; R , \\ ! 1 ; R , \\ ! 2 \\} \\end{align*}"}
{"id": "1704.png", "formula": "\\begin{align*} T _ { t , j , 0 } = \\{ \\Omega _ { t , j } \\} , { \\rm c a r d } \\ , T _ { t , j , m } \\le c \\cdot 2 ^ m , \\end{align*}"}
{"id": "1202.png", "formula": "\\begin{align*} { E _ { m } ^ { [ \\alpha , \\beta ] , \\rho , \\varepsilon } = \\left \\{ x \\in \\widetilde { E } _ { m } ^ { [ \\alpha , \\beta ] , \\rho , \\varepsilon } : \\forall r \\leq \\rho , \\ \\frac { 3 } { 4 } m ( B ( x , r ) ) \\leq m ( B ( x , r ) \\cap \\widetilde { E } _ { m } ^ { [ \\alpha , \\beta ] , \\rho , \\varepsilon } ) \\right \\} } \\end{align*}"}
{"id": "5193.png", "formula": "\\begin{align*} \\frac { \\Gamma ( \\pi _ { s r } ) } { \\Gamma ( \\pi _ { t p } ) } = \\frac { 1 } { 2 } \\left ( \\frac { \\sigma ^ 2 } { \\mu ^ 2 } + 1 \\right ) . \\end{align*}"}
{"id": "1710.png", "formula": "\\begin{align*} \\begin{array} { c } d _ { C _ 1 C n } ( M , \\ , Y _ q ( \\Omega ) ) \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } \\sum \\limits _ { t = 0 } ^ { \\hat t ( n ) } \\sum \\limits _ { m = 0 } ^ \\infty d _ { k _ { t , m } } ( W _ { t , m } , \\ , l _ q ^ { \\nu _ { t , m } } ) + \\\\ + \\sum \\limits _ { t = 0 } ^ { \\hat t ( n ) } d _ { s _ t } ( 2 ^ { \u2010 \\alpha _ * k _ * t } B _ { p _ 0 } ^ { \\nu _ { t , 0 } } , \\ , l _ q ^ { \\nu _ { t , 0 } } ) + \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ { \\hat t ( n ) + 1 } ) } . \\end{array} \\end{align*}"}
{"id": "3892.png", "formula": "\\begin{align*} E _ 0 ( x ) = \\sum _ { x \\leq p \\leq 2 x } \\frac { 1 } { p } \\sum _ { \\substack { 0 \\leq a < p \\\\ \\gcd ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 < \\leq b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { m } - v ) } { p } } = 0 . \\end{align*}"}
{"id": "8357.png", "formula": "\\begin{align*} \\tau _ n = \\tau _ { n - 1 } + k _ { n - 1 } ( \\tau _ 0 - 1 ) . \\end{align*}"}
{"id": "442.png", "formula": "\\begin{align*} \\begin{aligned} A _ { 1 } ^ { 0 } ( u , v ) \\partial _ { t } u & + A ^ { i } _ { 1 1 } ( u , v ) \\partial _ { i } u = f _ { 1 } ( U , D _ { x } v ) , \\\\ A _ { 2 } ^ { 0 } ( u , v ) \\partial _ { t } v & - B _ { 2 2 } ^ { i j } ( u , v ) \\partial _ { i } \\partial _ { j } v = f _ { 2 } ( U , D _ { x } U ) . \\end{aligned} \\end{align*}"}
{"id": "5730.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( v ) \\setminus V ( S ' ) | & = d _ { G } ( v ) - | N _ { G } ( v ) \\cap V ( S ' ) | \\\\ & \\geq t - | Y _ { S ' } | \\\\ & \\geq | Y | - | Y _ { S } | \\\\ & \\geq | N _ { T } ( \\phi ^ { - 1 } ( v ) ) \\cap ( Y \\backslash Y _ { S } ) | \\\\ & \\geq d _ { T } ( \\phi ^ { - 1 } ( v ) ) - d _ { S } ( \\phi ^ { - 1 } ( v ) ) . \\\\ \\end{aligned} \\end{align*}"}
{"id": "7137.png", "formula": "\\begin{align*} R _ H ( s _ 1 , s _ 2 ) = \\mathbb { E } [ B _ { s _ 1 } ^ H \\cdot B _ { s _ 2 } ^ H ] \\end{align*}"}
{"id": "8422.png", "formula": "\\begin{align*} S _ { t } ( l ) : = l _ { t } \\end{align*}"}
{"id": "3470.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 0 } ^ { l - 1 } \\zeta _ l ^ i \\varphi ^ i ( \\xi ) \\neq 0 , \\end{align*}"}
{"id": "2617.png", "formula": "\\begin{align*} f ( t ) = \\int _ \\R \\widehat { f } ( \\omega ) e ^ { 2 \\pi i \\omega t } \\ , d t , \\end{align*}"}
{"id": "6717.png", "formula": "\\begin{align*} P _ { { \\bf b } _ s , d _ s } \\biggl \\{ g _ 0 ( t ) + g _ 1 ( t ) \\biggl ( \\sum _ { n = 0 } ^ { \\infty } \\Bigl ( \\prod _ { l = 1 } ^ { d _ 1 - 1 } ( \\theta ^ { q ^ { l + n } } - t ) ^ { c _ 1 ( l ) q ^ { d - l } } & \\alpha _ 1 ^ { q ^ { n + d } } \\Bigr ) \\biggr ) + \\\\ & \\cdots + g _ s ( t ) \\biggl ( \\sum _ { n = 0 } ^ { \\infty } \\Bigl ( \\prod _ { l = 1 } ^ { d _ s - 1 } ( \\theta ^ { q ^ { l + n } } - t ) ^ { c _ s ( l ) q ^ { d - l } } \\alpha _ s ^ { q ^ { n + d } } \\Bigr ) \\biggr ) \\biggr \\} = 0 . \\end{align*}"}
{"id": "5924.png", "formula": "\\begin{align*} \\flat _ { \\omega } ( \\pmb { X } ) = d H . \\end{align*}"}
{"id": "6974.png", "formula": "\\begin{align*} \\begin{aligned} x _ k & \\to \\bar x , & y _ k & \\to \\bar y , & \\lambda _ k & \\to \\lambda , & \\\\ \\eta _ k & \\to 0 , & \\frac { x _ k - \\bar x } { \\norm { x _ k - \\bar x } } & \\to u , & \\frac { y _ k - \\bar y } { \\norm { x _ k - \\bar x } } & \\to 0 , & \\end{aligned} \\end{align*}"}
{"id": "6104.png", "formula": "\\begin{align*} u ^ { 2 } & = r ^ { 2 \\lambda _ { p } - 1 } \\alpha _ { 1 } ( \\theta ) + r ^ { 2 \\lambda _ { p } - 2 m + 1 } \\alpha _ { 2 } ( \\theta ) , \\\\ v ^ { 2 } & = r ^ { 2 \\lambda _ { p } - 1 } \\beta _ { 1 } ( \\theta ) + r ^ { 2 \\lambda _ { p } - 2 m + 1 } \\beta _ { 2 } ( \\theta ) , \\\\ \\end{align*}"}
{"id": "8401.png", "formula": "\\begin{align*} \\operatorname { d i v } \\mathbf { v } _ A = & \\ ; \\operatorname { d i v } ( \\zeta \\circ d _ \\Gamma ) ( \\mathbf { v } _ A ^ { i n } - \\mathbf { v } _ A ^ + \\chi _ + - \\mathbf { v } _ A ^ - \\chi _ - ) + ( \\zeta \\circ d _ \\Gamma ) \\operatorname { d i v } \\ , \\mathbf { v } _ A ^ { i n } \\\\ & + ( 1 - \\zeta \\circ d _ \\Gamma ) ( \\operatorname { d i v } \\ , \\mathbf { v } _ A ^ + \\chi _ + + \\operatorname { d i v } \\ , \\mathbf { v } _ A ^ - \\chi _ - ) \\triangleq I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
{"id": "3793.png", "formula": "\\begin{align*} \\gamma ( X , \\pi _ E , \\widehat { \\sigma _ E } , \\psi _ E ) = \\prod _ { j = 1 } ^ t \\gamma ( X , \\pi _ E , Q ( \\widehat { D _ i } ) , \\psi _ E ) \\end{align*}"}
{"id": "1694.png", "formula": "\\begin{align*} & { _ { s + 1 } } F _ s \\left ( \\left \\{ 1 \\right \\} ^ { s } , 1 - n ; \\left \\{ 2 \\right \\} ^ { s } ; 1 \\right ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } H _ k ( s ) , \\\\ & { _ { s + 1 } } F _ s \\left ( \\left \\{ 1 \\right \\} ^ { s } , 1 - n ; \\left \\{ 2 \\right \\} ^ { s } ; - 1 \\right ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } H _ k ( \\overline { s } ) . \\end{align*}"}
{"id": "4967.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\ < P s i 2 > _ n ( t _ 1 , y ) \\overline { \\ < P s i 2 > _ n ( t _ 2 , \\tilde { y } ) } \\Big ] = \\Big | \\mathbb { E } \\Big [ \\ < P s i > _ n ( t _ 1 , y ) \\overline { \\ < P s i > _ n ( t _ 2 , \\tilde { y } ) } \\Big ] \\Big | ^ 2 + \\Big | \\mathbb { E } \\Big [ \\ < P s i > _ n ( t _ 1 , y ) \\ < P s i > _ n ( t _ 2 , \\tilde { y } ) \\Big ] \\Big | ^ 2 . \\end{align*}"}
{"id": "8842.png", "formula": "\\begin{align*} & \\sum _ { i , j \\geq 1 } \\frac { \\omega _ { i , j } } { \\lambda ^ { i + 1 } \\mu ^ { j + 1 } } = \\frac { { \\rm t r } \\ , R ( \\lambda ) R ( \\mu ) - 1 } { ( \\lambda - \\mu ) ^ 2 } , \\end{align*}"}
{"id": "668.png", "formula": "\\begin{align*} A ( x ) \\ = \\ \\frac { g ( x ) + f ( x ) } { 2 } E ( x ) \\ = \\ \\frac { g ( x ) - f ( x ) } { 2 } . \\end{align*}"}
{"id": "2825.png", "formula": "\\begin{align*} E [ u ] = E [ u _ 0 ] = E [ Q ] \\end{align*}"}
{"id": "6777.png", "formula": "\\begin{align*} \\pi = \\tau _ { r _ { 1 } } ^ { - ( v _ { 1 } + 1 ) } \\cdot \\tau _ { r _ { q _ 1 } + 1 } ^ { v _ { 1 } + 1 } \\cdot \\tau _ { r _ { q _ 1 + 1 } } ^ { - ( v _ { q _ 1 } + 1 ) } \\cdot \\tau _ { r _ { q _ 2 } + 1 } ^ { v _ { q _ 1 } + 1 } \\cdots \\tau _ { r _ { q _ { z ' } + 1 } } ^ { - ( v _ { q _ { z ' } } + 1 ) } \\cdot \\tau _ { r _ { z } + 1 } ^ { v _ { q _ { z ' } } + 1 } . \\end{align*}"}
{"id": "4870.png", "formula": "\\begin{align*} ( \\tilde K * g ) ( \\xi ) = \\int _ { B ( 0 , \\delta ) ^ c } g ( \\xi - y ) \\tilde K ( y ) \\ , d z = \\int _ { B ( 0 , \\delta ) ^ c } g ( \\xi - y ) K ( y ) \\ , d z = ( K * g ) ( \\xi ) . \\end{align*}"}
{"id": "6520.png", "formula": "\\begin{align*} E [ ( S _ { n + 1 } ) ^ { 2 m } ] & = 1 + \\sum ^ m _ { \\ell = 1 } \\left \\{ \\binom { 2 m } { 2 \\ell } + \\frac { 1 } { 2 n } \\binom { 2 m } { 2 \\ell - 1 } \\right \\} E [ ( S _ n ) ^ { 2 \\ell } ] . \\end{align*}"}
{"id": "1864.png", "formula": "\\begin{align*} P _ { n + 2 } ( x ) = 2 \\sum _ { k = 0 } ^ n { n \\choose k } P _ k ( x ) P _ { n + 1 - k } ( x ) . \\end{align*}"}
{"id": "7253.png", "formula": "\\begin{align*} \\dfrac { a } { b } = \\dfrac { C _ n ^ { k - 1 } } { C _ n ^ { k } } = \\dfrac { k } { n - k + 1 } , \\dfrac { b } { c } = \\dfrac { C _ n ^ { k } } { C _ n ^ { k + 1 } } = \\dfrac { k + 1 } { n - k } , \\dfrac { c } { d } = \\dfrac { C _ n ^ { k + 1 } } { C _ n ^ { k + 2 } } = \\dfrac { k + 2 } { n - k - 1 } . \\end{align*}"}
{"id": "4831.png", "formula": "\\begin{align*} { \\rm t r } ^ \\flat ( P ( h ) ) = \\mathcal { O } ( h ^ { - 2 n - m } ) . \\end{align*}"}
{"id": "7144.png", "formula": "\\begin{align*} x ^ n _ 1 = \\mathbb { E } [ x ^ n _ 1 ] + \\int _ 0 ^ 1 \\mathbb { E } [ D _ s x ^ n _ 1 \\vert \\mathcal { F } _ s ] d W _ s , \\end{align*}"}
{"id": "4123.png", "formula": "\\begin{align*} \\beta p = \\sum _ { i = 1 } ^ k \\alpha _ i p _ i - \\sum _ { i = k + 1 } ^ d \\beta _ i p _ i + \\lambda e _ d . \\end{align*}"}
{"id": "7571.png", "formula": "\\begin{align*} \\phi ( - d ( L \\phi ) ) = \\phi ( - d ( \\eta _ 0 ) ) = \\phi ( 0 ) = L \\phi = \\eta _ 0 . \\end{align*}"}
{"id": "5592.png", "formula": "\\begin{align*} f ^ a ( x ) = 0 , a = 1 , 2 , . . . , k < n . \\end{align*}"}
{"id": "8219.png", "formula": "\\begin{align*} [ x _ k ^ { \\ell _ k } ] G _ { k , 1 } ^ { - \\ell _ 1 } & = [ x _ k ^ { \\ell _ k } ] ( F _ { k , 2 } - x _ k ) ^ { - \\ell _ 1 } = [ x _ k ^ { \\ell _ k } ] F _ { k , 2 } ^ { - \\ell _ 1 } \\Big ( 1 - \\frac { x _ k } { F _ { k , 2 } } \\Big ) ^ { - \\ell _ 1 } \\\\ & = F _ { k , 2 } ^ { - \\ell _ 1 } \\binom { - \\ell _ 1 } { \\ell _ k } \\Big ( - \\frac { 1 } { F _ { k , 2 } } \\Big ) ^ { \\ell _ k } = \\binom { \\ell _ 1 + \\ell _ k - 1 } { \\ell _ k } F _ { k , 2 } ^ { - \\ell _ 1 - \\ell _ k } . \\end{align*}"}
{"id": "6053.png", "formula": "\\begin{align*} t _ i + x _ j \\ = \\ v _ { i j } \\end{align*}"}
{"id": "3039.png", "formula": "\\begin{align*} r = 1 8 \\ , , h = 6 \\ , , n = 3 6 - \\Theta \\ , , y = 9 6 \\ , . \\end{align*}"}
{"id": "8585.png", "formula": "\\begin{align*} \\int _ \\R \\partial _ k \\overline { \\mathcal { K } ^ \\# _ 0 ( x , k ) } f ( x ) \\ , d x = T ( 0 ) \\int _ \\R m _ + ( x , 0 ) ( - i x ) e ^ { i x k } f ( x ) \\ , d x \\end{align*}"}
{"id": "9198.png", "formula": "\\begin{align*} \\Sigma _ 1 \\mathrm { E R } : \\quad \\frac { \\exists y ^ \\sigma A _ 0 ( y ) \\rightarrow s = _ \\rho t } { \\exists y ^ \\sigma A _ 0 ( y ) \\rightarrow r [ s / x ^ \\rho ] = _ \\tau r [ t / x ^ \\rho ] } \\end{align*}"}
{"id": "2414.png", "formula": "\\begin{align*} f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , e _ \\gamma \\rangle S ^ { - 1 } e _ \\gamma , \\end{align*}"}
{"id": "4590.png", "formula": "\\begin{align*} \\underline { \\lambda } = x - c _ { 2 } \\theta ( x ) \\big ( x ^ 2 ( \\epsilon _ n + \\delta _ n ) + x \\delta _ n \\sqrt { | \\ln \\delta _ n | } \\big ) \\in [ 0 , \\ , o ( \\min \\{ \\epsilon _ n ^ { - 1 } , \\delta _ n ^ { - 1 } \\} \\ , ] , \\end{align*}"}
{"id": "5900.png", "formula": "\\begin{align*} \\psi _ { Q _ S } ( x ) = \\det ( Q _ S - x I ) = \\det \\Big ( Q _ { J - 2 A } - ( x + 1 ) I \\Big ) = \\psi _ { Q _ { J - 2 A } } ( x + 1 ) \\end{align*}"}
{"id": "4772.png", "formula": "\\begin{align*} ( R \\# T ) ( e _ i \\wedge e _ j ) = \\frac { 1 } { 2 } \\sum _ k ( r _ { i , k } t _ { j , k } + t _ { i , k } r _ { j , k } ) e _ i \\wedge e _ j . \\end{align*}"}
{"id": "1231.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\mu ( L _ { B , n } ) = + \\infty \\end{align*}"}
{"id": "6852.png", "formula": "\\begin{align*} M = \\sum \\limits _ { Y \\in \\mathcal Y } \\alpha _ Y Y \\end{align*}"}
{"id": "2868.png", "formula": "\\begin{align*} \\partial _ t \\left ( e ^ { e _ 0 t } \\alpha _ + \\right ) = e ^ { e _ 0 t } B ( R , \\mathcal { Y } _ - ) , \\end{align*}"}
{"id": "8145.png", "formula": "\\begin{align*} \\Pi _ { d _ 0 } ( f , H ) : = \\prod _ { q \\mid d _ 0 } \\prod _ { \\chi \\in X _ f ^ - ( H ) } \\left ( 1 - \\frac { \\chi ( q ) } { q } \\right ) D _ { d _ 0 } ( f , H ) : = \\Pi _ { d _ 0 } ( f , H ) ^ { 4 / m } . \\end{align*}"}
{"id": "6086.png", "formula": "\\begin{align*} \\begin{array} { l l } A & = ( L ^ { 2 } R \\times \\overline { L } R ) \\cdot ( L R \\times \\overline { L } R ) = O ( r ^ { 5 m _ { j } - 1 1 } ) , \\\\ B & = ( L ^ { 2 } R \\times L R ) \\cdot ( L R \\times \\overline { L } R ) = O ( r ^ { 5 m _ { j } - 1 1 } ) , \\\\ C & = ( L R \\times \\overline { L } R ) \\cdot ( L R \\times \\overline { L } R ) = O ( r ^ { 4 m _ { j } - 8 } ) . \\end{array} \\end{align*}"}
{"id": "3067.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = x _ 2 ( j x _ 2 , j ^ 2 x _ 2 ) \\ , , x _ 2 ^ \\prime = x _ 1 \\ , , y _ 1 ^ \\prime = y _ 2 ( j ^ 2 y _ 2 , j y _ 2 ) \\ , , y _ 2 ^ \\prime = y _ 1 \\ , , \\end{align*}"}
{"id": "8120.png", "formula": "\\begin{align*} T B _ { 1 0 } ( f ) = 5 T B _ { 6 } ( f ) , \\end{align*}"}
{"id": "6990.png", "formula": "\\begin{align*} \\widehat { \\beta } = ( X ' X ) ^ { - 1 } X ' y . \\end{align*}"}
{"id": "2660.png", "formula": "\\begin{align*} V _ { g _ 0 } f ( x , - \\omega ) = e ^ { \\pi i x \\cdot \\omega } e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } B f ( z ) . \\end{align*}"}
{"id": "5606.png", "formula": "\\begin{align*} \\nabla ^ \\mathcal { F } _ X Y : = \\nabla _ { \\overline { \\mathcal { F } } _ 1 \\rhd X } ( \\overline { \\mathcal { F } } _ 2 \\rhd Y ) , ~ X , Y \\in \\Xi _ \\star \\end{align*}"}
{"id": "7692.png", "formula": "\\begin{align*} \\frac { 1 } { \\int _ { \\R ^ N } m ' u ^ 2 } = \\tilde { \\lambda } _ 0 \\ ; . \\end{align*}"}
{"id": "7134.png", "formula": "\\begin{align*} D _ u x _ t ^ n & = \\int ^ t _ u \\exp \\Big ( \\int _ s ^ t \\tilde { f } ' ( r , x _ r ^ n ) d r \\Big ) \\tilde { f } ' _ n ( s , x _ s ^ n ) K _ H ( s , u ) d s + K _ H ( t , u ) \\\\ & = J _ 1 ^ n ( t , u ) + J _ 2 ^ n ( t , u ) + K _ H ( t , u ) , u < t , \\lambda \\times , \\end{align*}"}
{"id": "2991.png", "formula": "\\begin{align*} { L ^ { \\phantom { s } } \\ ! \\ ! } ^ { \\scriptscriptstyle { | | } } : = A + i | V | = A + i \\frac { \\alpha ^ 2 } { 2 } . \\end{align*}"}
{"id": "2759.png", "formula": "\\begin{align*} \\| u \\| _ { S ' ( \\dot { H } ^ { - s } ) } = \\inf \\left \\lbrace \\| u \\| _ { L _ t ^ { q ' } L _ x ^ { r ' } } : ( q , r ) ( \\ref { S t r i c h a r t z p a i r } ) ( \\ref { r e s t r i c t i o n o n d u a l S t r i c h a r t z p a i r } ) \\right \\rbrace . \\end{align*}"}
{"id": "4478.png", "formula": "\\begin{align*} \\frac { \\partial f ( t ) } { \\partial t } = \\sum _ { k = 1 } ^ { \\infty } \\frac { t ^ { k - 1 } } { ( k - 1 ) ! } \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) = \\sum _ { k = 0 } ^ { \\infty } \\phi _ { k + 1 , \\lambda } ( | z | ^ { 2 } ) \\frac { t ^ { k } } { k ! } . \\end{align*}"}
{"id": "956.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = c _ { n , s } \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { \\R ^ n _ + \\setminus B _ \\varepsilon ( x ) } \\left ( \\frac 1 { \\vert x - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\right ) \\big ( u ( x ) - u ( y ) \\big ) \\dd y + \\frac { c _ { 1 , s } } s u ( x ) x _ 1 ^ { - 2 s } \\end{align*}"}
{"id": "7683.png", "formula": "\\begin{align*} \\| f _ 0 ( g _ k ) \\| _ { L ^ { 2 } ( \\tilde { \\Omega } _ { \\varepsilon _ k } ) } = \\| f _ 0 ( g _ k ) \\| _ { L ^ { 2 } ( B ^ 1 ( \\mathbf { 0 } ) ) } = o ( 1 ) \\ ; \\varepsilon _ k \\to 0 \\end{align*}"}
{"id": "6133.png", "formula": "\\begin{align*} \\left ( \\sqrt { j ( 2 j - 1 ) } s ^ { j - 1 } \\right ) L \\psi ^ { 1 } - i \\sqrt { k ( 2 k - 1 ) } t ^ { k - 1 } L \\varphi ^ { 1 } = 0 . \\end{align*}"}
{"id": "5776.png", "formula": "\\begin{align*} D ( \\mathbf { u } , \\mathbf { b } ) : = \\int _ { \\mathbb { R } ^ 3 } | \\nabla \\mathbf { u } | ^ 2 + | \\nabla \\mathbf { b } | ^ 2 \\ , d x \\leq \\mathcal { C } \\liminf _ { R \\rightarrow \\infty } M ^ 3 _ { \\ , p , \\ , q } ( R ) . \\end{align*}"}
{"id": "131.png", "formula": "\\begin{align*} x ^ r + \\gamma _ 1 \\omega ( x ) x ^ { r - 1 } + \\cdots + \\gamma _ { r - 1 } \\omega ( x ) ^ { r - 1 } x = 0 , \\end{align*}"}
{"id": "3238.png", "formula": "\\begin{align*} H _ 1 ( z ) = F _ { { \\mu } } ( z ) - z , H _ 2 ( z ) = F _ { \\nu } ( z ) - z . \\end{align*}"}
{"id": "6278.png", "formula": "\\begin{align*} \\hat { h } _ i = \\mathbf { v } ^ H \\mathbf { z } _ i / ( N _ 1 \\sqrt { p } ) \\sim \\mathcal { C N } \\Big ( 0 , \\beta _ i + \\frac { 1 } { N _ 1 \\varrho } \\Big ) , \\end{align*}"}
{"id": "8231.png", "formula": "\\begin{align*} \\frac { N _ 1 N _ { k + 1 - h } } { x _ 1 z } = \\frac { x _ { h + 1 } z } { N _ { h + 1 } } - \\frac { x _ h z } { N _ h } . \\end{align*}"}
{"id": "9400.png", "formula": "\\begin{align*} \\tau _ t ( a _ 1 \\cdots a _ k ) = \\tau ( a _ 1 \\ldots a _ k ) + o ( 1 ) . \\end{align*}"}
{"id": "2542.png", "formula": "\\begin{align*} U ^ { - 1 } \\rho ( x , \\omega , \\tau ) U = \\pi ( x , \\omega , \\tau ) . \\end{align*}"}
{"id": "6014.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { e ^ { - t \\mathfrak { D } _ { { \\rm l e f t } , a } } u ( x ) - u ( x ) } { t } & = - \\mathfrak { D } _ { { \\rm l e f t } , a } u ( x ) \\\\ \\lim _ { t \\to 0 ^ + } \\frac { e ^ { - t \\mathfrak { D } _ { { \\rm r i g h t } , a } } v ( x ) - v ( x ) } { t } & = - \\mathfrak { D } _ { { \\rm r i g h t } , a } v ( x ) . \\end{align*}"}
{"id": "5748.png", "formula": "\\begin{align*} \\| \\operatorname { H e s s } y ^ j \\| _ { C ^ { 0 , \\alpha } ( B _ { r } ( p ) ) } \\leq \\tau ( n , r , \\alpha , Q ) , \\ \\ j = 1 , \\dots , m , \\end{align*}"}
{"id": "6146.png", "formula": "\\begin{align*} x ^ m \\bigl ( ( m + 1 ) P ( x ) + x P ' ( x ) \\bigr ) = ( 1 - x ) ^ n \\bigl ( ( n + 1 ) Q ( x ) - ( 1 - x ) Q ' ( x ) \\bigr ) . \\end{align*}"}
{"id": "2165.png", "formula": "\\begin{align*} g ^ - _ { * } \\leq \\frac { g _ * ( t ) t } { G _ * ( t ) } \\leq g ^ + _ { * } , \\ \\ t > 0 , \\ \\ g ^ + _ { * } = \\frac { d g ^ + } { d - g ^ + } \\ \\ g ^ - _ { * } = \\frac { d g ^ - } { d - g ^ - } . \\end{align*}"}
{"id": "4075.png", "formula": "\\begin{align*} \\Phi ( z , t ) = ( x , y ) : = \\big ( z + t , Q _ z ( t ) \\big ) , \\ \\ ( z , t ) \\in E . \\end{align*}"}
{"id": "1084.png", "formula": "\\begin{align*} & r _ { + } ( k ) = - r ^ * _ { - } ( k ) , & k \\in I _ { R } , \\\\ & r _ { + } ( k ) = \\frac { 1 } { r _ { - } ^ { * } ( k ) } , \\vert r _ { \\pm } ( k ) \\vert = 1 , & k \\in I _ { L } \\backslash I _ { R } , \\\\ & r _ { + } ( k ) = r _ { - } ( k ) , & k \\in \\mathbb { R } \\backslash ( I _ L \\cup I _ R ) . \\end{align*}"}
{"id": "4068.png", "formula": "\\begin{align*} \\begin{dcases} \\lambda \\sigma _ n - ( \\sigma _ n ) _ x - ( u _ n ) _ x = f _ n & ( 0 , 1 ) , \\\\ \\lambda u _ n - ( \\sigma _ n ) _ x - ( u _ n ) _ x - ( u _ n ) _ { x x } = g _ n & ( 0 , 1 ) , \\\\ \\sigma _ n ( 0 ) = \\sigma _ n ( 1 ) , \\ \\ u _ n ( 0 ) = u _ n ( 1 ) = 0 . \\end{dcases} \\end{align*}"}
{"id": "5993.png", "formula": "\\begin{align*} \\langle \\ ! \\langle \\varphi , \\rho _ { \\pi _ { \\lambda , \\beta } } ( z , \\cdot ) \\rangle \\ ! \\rangle _ { \\pi _ { \\lambda , \\beta } } = \\int _ { \\mathbb { R } } \\varphi ( x - z ) \\ , \\mathrm { d } \\pi _ { \\lambda , \\beta } ( x ) , \\quad \\varphi \\in ( N ) _ { \\pi _ { \\lambda , \\beta } } ^ { 1 } . \\end{align*}"}
{"id": "8027.png", "formula": "\\begin{align*} \\mathrm { W F } \\left ( \\left \\langle E _ { \\Sigma } , F ^ { ( k + 1 ) } [ \\psi ] \\otimes G ^ { ( m + 1 ) } [ \\psi ] \\right \\rangle \\right ) \\cap ( \\Xi ^ { ( k + m ) } _ + \\cup \\Xi ^ { ( k + m ) } _ - ) = \\emptyset , \\end{align*}"}
{"id": "1163.png", "formula": "\\begin{align*} m ^ { ( 1 ) } \\rightarrow \\Delta _ { R } ( k ) , \\vert J ^ { ( 1 ) } ( x , t , k ) - I \\vert = \\mathcal { O } \\left ( e ^ { - 8 t k _ 2 ( 3 k _ 1 ^ 2 - k _ 2 ^ 2 + 3 \\xi ) } \\right ) . \\end{align*}"}
{"id": "6669.png", "formula": "\\begin{align*} V \\left ( \\frac { g _ 2 g _ 4 v } { K X } \\right ) = \\frac { 1 } { 2 \\pi i } \\Psi \\Big ( \\frac { v } { X Q ^ { \\vartheta } } \\Big ) \\int _ { ( \\varepsilon ) } \\left ( \\frac { X K } { g _ 2 g _ 4 v } \\right ) ^ { s _ 4 } \\widetilde { V } ( s _ 4 ) \\ , d s _ 4 . \\end{align*}"}
{"id": "8810.png", "formula": "\\begin{align*} ( \\varphi , \\phi ) _ M : = \\sum _ { i j } M _ { i j } \\varphi _ i \\phi _ j , ( \\varphi ) _ M ^ 2 : = ( \\varphi , \\varphi ) _ M , \\Delta _ M F : = ( \\nabla , \\nabla ) _ M F ; \\end{align*}"}
{"id": "1229.png", "formula": "\\begin{align*} \\mathcal { H } ^ { s } _ { t } ( E ) = \\inf \\left \\{ \\sum _ { n \\in \\mathbb { N } } \\vert B _ n \\vert ^ s : \\ ( B _ n ) _ { n \\in \\N } \\mbox { c l o s e d b a l l s , } \\vert B _ n \\vert \\leq t E \\subset \\bigcup _ { n \\in \\mathbb { N } } B _ n \\right \\} . \\end{align*}"}
{"id": "3487.png", "formula": "\\begin{align*} \\Re { \\left ( \\int _ y ^ M g ( u ) e ^ { 2 \\pi i \\left ( f ( u ) - u \\right ) } d u \\right ) } & = \\Re { \\left ( \\int _ y ^ M \\frac { e ^ { - 2 \\pi i w ( u ) } } { u ^ { \\sigma _ 1 } ( u + n ) ^ { \\sigma _ 3 } } d u \\right ) } \\\\ & = \\int _ { w ( y ) } ^ { w ( M ) } \\frac { 1 } { u ^ { \\sigma _ 1 } ( u + n ) ^ { \\sigma _ 3 } } \\frac { \\cos ( 2 \\pi w ) d w } { 1 - \\frac { 1 } { 2 \\pi } \\left ( \\frac { t _ 1 } { u } + \\frac { t _ 3 } { u + n } \\right ) } . \\end{align*}"}
{"id": "9353.png", "formula": "\\begin{align*} F _ { n , \\lambda } ( x ) = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) k ! x ^ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 1 2 ] ) . \\end{align*}"}
{"id": "7412.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta ) - f ( \\eta ^ { x + 1 , x + 1 + r } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "7117.png", "formula": "\\begin{align*} x _ t - y _ t = \\int _ 0 ^ t \\big ( b ( s , x _ s ) - b ( s , y _ s ) \\big ) d s . \\end{align*}"}
{"id": "9125.png", "formula": "\\begin{align*} H ^ * [ P , Q , \\varepsilon ] : = \\forall p \\in P \\exists q \\in Q \\left ( \\norm { p - q } \\leq \\varepsilon \\right ) . \\end{align*}"}
{"id": "1042.png", "formula": "\\begin{align*} D i f f ^ { i + j - 1 } & \\ni [ m _ a , D ^ i ] \\circ D ^ j + D ^ i \\circ [ m _ a , D ^ j ] = m _ a \\circ D ^ i \\circ D ^ j - D ^ i \\circ m _ a \\circ D ^ j \\\\ & + D ^ i \\circ m _ a \\circ D ^ j - D ^ i \\circ D ^ j \\circ m _ a = m _ a \\circ D ^ i \\circ D ^ j - D ^ i \\circ D ^ j \\circ m _ a = [ m _ a , D ^ i \\circ D ^ j ] \\ \\end{align*}"}
{"id": "5935.png", "formula": "\\begin{align*} \\tau = \\mathcal { T } \\circ \\mathcal { S } \\colon J ^ { - 1 } ( \\mu ) & \\to \\pi ^ * \\left ( ( T ^ 1 _ k ) ^ * ( Q / G ) \\right ) \\end{align*}"}
{"id": "566.png", "formula": "\\begin{align*} b _ \\alpha ( s ) = \\begin{cases} b ( s ) & s \\leq \\bar \\rho - \\alpha , \\\\ b ( \\bar \\rho - \\alpha ) & s > \\bar \\rho - \\alpha \\end{cases} \\end{align*}"}
{"id": "7502.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { \\phi } = \\frac { 1 } { 2 } \\Delta \\phi - V \\phi - \\beta | \\phi | ^ { 2 } \\phi + \\Omega L _ z \\phi + \\mu _ { \\phi } ( t ) \\phi , \\mathbf { x } \\in \\mathbb { R } ^ d , \\ ; t \\geq 0 , \\\\ & \\phi ( \\mathbf { x } , 0 ) = \\phi _ { 0 } ( \\mathbf { x } ) , \\mathbf { x } \\in \\mathbb { R } ^ d , \\end{aligned} \\right . \\end{align*}"}
{"id": "3602.png", "formula": "\\begin{align*} \\delta = s ^ { 2 } ( 1 + r \\pi ^ { d ( \\delta ) } ) = s ^ { 2 } ( 1 + r \\pi ^ { d ( c ) } ) , \\quad \\ ; r , s \\in \\mathcal { O } _ { F } ^ { \\times } \\ , \\end{align*}"}
{"id": "8144.png", "formula": "\\begin{align*} M _ { d _ 0 } ( f , H ) : = { 1 \\over \\# X _ f ^ - ( H ) } \\sum _ { \\chi \\in X _ f ^ - ( H ) } \\vert L ( 1 , \\chi ' ) \\vert ^ 2 \\end{align*}"}
{"id": "3979.png", "formula": "\\begin{align*} \\lambda = - 3 \\pm i \\sqrt { 4 \\pi ^ 2 ( l ^ 2 - l m + m ^ 2 ) - 9 } . \\end{align*}"}
{"id": "2353.png", "formula": "\\begin{align*} \\langle X f , f \\rangle = \\int _ \\R x f ( x ) \\overline { f ( x ) } \\ , d x = \\int _ \\R f ( x ) \\overline { ( x f ( x ) ) } \\ , d x = \\langle f , X f \\rangle , \\end{align*}"}
{"id": "7350.png", "formula": "\\begin{align*} C _ q = { C \\over q } \\varphi ( x _ q , t _ q ) ^ { 1 - q } , \\end{align*}"}
{"id": "1159.png", "formula": "\\begin{align*} m ^ { ( 2 ) } \\rightarrow \\Delta _ { R } ( k ) , \\vert J ^ { ( 2 ) } ( x , t , k ) - I \\vert = \\mathcal { O } \\left ( e ^ { - \\gamma t } \\right ) , \\end{align*}"}
{"id": "8566.png", "formula": "\\begin{align*} \\widetilde { f } ( 0 + ) = \\frac { 2 a } { 1 + a ^ 2 } \\frac { 1 } { \\sqrt { 2 \\pi } } \\int f ( x ) \\psi _ + ( x , 0 ) \\ , d x \\mbox { a n d } \\widetilde { f } ( 0 - ) = \\frac { 1 } { a } \\widetilde { f } ( 0 + ) . \\end{align*}"}
{"id": "7210.png", "formula": "\\begin{align*} Y _ { s , t } ( x , v ) = X _ { s , t } ( x + \\check \\tau _ { t , x } v , v ) - ( x + ( s - \\tau _ { x } ) v ) , \\end{align*}"}
{"id": "2073.png", "formula": "\\begin{align*} \\overline { \\{ g _ { \\alpha } : \\alpha \\in S _ 2 \\} } = A ^ 2 ( D , e ^ { - \\varphi } ) , \\end{align*}"}
{"id": "8759.png", "formula": "\\begin{align*} s _ { i j } = a _ { i 0 } z _ { i 0 } + \\sum _ { k = 1 } ^ j ( a _ { i k } - a _ { i k - 1 } ) z _ { i k } j = 0 , \\ldots , n . \\end{align*}"}
{"id": "635.png", "formula": "\\begin{align*} f ( x , n ) \\ & = \\ C ( 0 , h _ 0 ( x , n ) ) \\\\ [ 1 2 p t ] & = \\ \\bigg [ \\frac { 0 } { h _ 0 ( x , n ) + 1 } + \\frac { 1 } { 2 } \\bigg ] \\\\ [ 1 2 p t ] & = \\ \\bigg [ \\frac { 1 } { 2 } \\bigg ] \\\\ [ 1 2 p t ] & = \\ 0 . \\end{align*}"}
{"id": "9057.png", "formula": "\\begin{align*} \\tiny { \\frac { ( X ( \\theta ) + Y ( \\theta ) ) ^ 2 } { Z ( \\theta ) } - \\theta \\frac { ( X ^ 0 + Y ^ 0 ) ^ 2 } { Z ^ 0 } - ( 1 - \\theta ) \\frac { ( X ^ 1 + Y ^ 1 ) ^ 2 } { Z ^ 1 } = - \\theta ( 1 - \\theta ) \\frac { [ Z ^ 1 ( X ^ 0 + Y ^ 0 ) - Z ^ 0 ( X ^ 1 + Y ^ 1 ) ] ^ 2 } { Z ^ 0 Z ^ 1 Z ( \\theta ) } . } \\end{align*}"}
{"id": "2358.png", "formula": "\\begin{align*} M _ b T _ a e ^ { - \\pi c t ^ 2 } = e ^ { - \\pi c ( t - a ) ^ 2 } e ^ { 2 \\pi i b t } . \\end{align*}"}
{"id": "1161.png", "formula": "\\begin{align*} G : = \\left \\{ \\begin{aligned} & \\begin{pmatrix} 1 & 0 \\\\ - r e ^ { 2 i t \\theta } & 1 \\end{pmatrix} , & k \\in U _ 1 , \\\\ & \\begin{pmatrix} 1 & - r ^ * e ^ { - 2 i t \\theta } \\\\ & 1 \\end{pmatrix} , & k \\in U _ 1 ^ * , \\\\ & I , \\textnormal { e l s e w h e r e } \\end{aligned} \\right . \\end{align*}"}
{"id": "2515.png", "formula": "\\begin{align*} \\pi ( F ) = \\int _ { \\mathbf { H } } F ( \\mathbf { h } ) \\pi ( \\mathbf { h } ) \\ , d \\mathbf { h } . \\end{align*}"}
{"id": "9213.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 > _ \\mathbb { R } 0 , { \\gamma ' } ^ 1 > _ \\mathbb { R } 0 , x ^ X , { x ' } ^ X \\left ( x = _ X x ' \\land \\gamma = _ \\mathbb { R } \\gamma ' \\rightarrow J ^ A _ \\gamma x = _ X J ^ A _ { \\gamma ' } x ' \\right ) . \\end{align*}"}
{"id": "563.png", "formula": "\\begin{align*} I _ 6 \\leq & c \\int _ 0 ^ \\tau \\frac { 1 } { | \\Omega _ R | } \\int _ { \\Omega _ R } p ( \\rho ) \\varepsilon ^ { - 2 } \\int _ { \\Omega _ R } \\left ( P ( \\rho ) - P ( r ) - P ' ( r ) ( \\rho - r ) \\right ) \\\\ \\leq & c \\int _ 0 ^ \\tau \\frac { 1 } { | \\Omega _ R | } \\int _ { \\Omega _ R } p ( \\rho ) \\mathcal E ( \\rho , u | r , U ) ( t ) \\end{align*}"}
{"id": "848.png", "formula": "\\begin{align*} t _ j ^ S = t _ { j - 1 } ^ S + \\tau ^ { \\rm R e a c } _ { m } R _ j + \\tau ^ { \\rm R e a c } _ { V _ j } + \\tau _ { \\rm f } . \\end{align*}"}
{"id": "1429.png", "formula": "\\begin{align*} \\frac { ( \\delta ( v _ 1 ) + \\cdots + \\delta ( v _ n ) ) ^ 2 - \\delta ( v _ 1 ) ^ 2 - \\cdots - \\delta ( v _ n ) ^ 2 } { 2 } = \\frac { 1 - ( \\delta ( v _ 1 ) ^ 2 + \\cdots + \\delta ( v _ n ) ^ 2 ) } { 2 } . \\end{align*}"}
{"id": "2449.png", "formula": "\\begin{align*} S ^ T J S = J \\end{align*}"}
{"id": "1669.png", "formula": "\\begin{align*} \\big ( 4 \\theta _ 0 - ( \\theta ^ * ) ^ 2 \\big ) \\bigg ( \\frac { 1 } { 8 } | x | ^ 2 + \\frac { 1 } { ( \\theta ^ * ) ^ 2 + 4 \\theta _ 0 } | v | ^ 2 \\bigg ) & \\le \\theta _ 0 | x | ^ 2 + | v | ^ 2 + \\theta ^ * \\ < x , v \\ > \\\\ & \\le \\Big ( 1 \\vee \\theta _ 0 + \\frac { \\theta ^ * } { 2 } \\Big ) \\big ( | x | ^ 2 + | v | ^ 2 \\big ) \\end{align*}"}
{"id": "1625.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ { \\mathcal { D } ( \\mathcal { L } ) } : = \\left \\| ( \\lambda - \\mathcal { L } ) f \\right \\| _ { L ^ 2 ( M ) } . \\end{align*}"}
{"id": "6451.png", "formula": "\\begin{align*} | | T _ h ( \\lambda ) - T _ 0 | | & \\leq | | ( T _ h ( \\lambda ) - T _ 0 ) | _ { B _ 1 } + ( T _ h ( \\lambda ) - T _ 0 ) | _ { B _ 2 } | | \\\\ & \\leq | | T _ h ( \\lambda ) - T _ 0 | | _ { L ^ 2 ( B _ 1 ) } + | | T _ h ( \\lambda ) - T _ 0 | | _ { L ^ 2 ( B _ 2 ) } \\\\ & \\leq C _ 1 h + C _ 2 h \\\\ & = ( C _ 1 + C _ 2 ) h \\end{align*}"}
{"id": "9433.png", "formula": "\\begin{align*} \\partial _ t \\| u ( \\cdot , t ) \\| ^ 2 _ { L ^ 2 _ { \\varLambda ^ q } ( { \\mathbb R } ^ n ) } + \\mu \\ , \\sum _ { j = 1 } ^ n \\| \\partial _ j u ( \\cdot , t ) \\| ^ 2 _ { L ^ 2 _ { \\varLambda ^ { q } } ( { \\mathbb R } ^ n ) } = ( B _ q ( w , u ) ( \\cdot , t ) , u ( \\cdot , t ) ) ^ 2 _ { L ^ 2 _ { \\varLambda ^ { q } } ( { \\mathbb R } ^ n ) } . \\end{align*}"}
{"id": "6151.png", "formula": "\\begin{align*} U _ { n , m } ( x ) = x ^ { m + 1 } \\sum _ { k = 0 } ^ n { m + k \\choose k } \\sum _ { \\nu = 0 } ^ k ( - 1 ) ^ { \\nu } { k \\choose \\nu } x ^ { \\nu } = \\sum _ { k = m + 1 } ^ { n + m + 1 } a _ { k - m - 1 } x ^ k \\end{align*}"}
{"id": "9180.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\mathcal { J } } \\sum _ { p \\in \\mathcal { S } ( j ) } ( \\log p ) \\left | L ( \\tfrac { 1 } { 2 } , f \\otimes \\chi _ { 8 p } ) \\right | ^ 2 | \\mathcal { M } ( p ) | ^ 2 \\ll X . \\end{align*}"}
{"id": "3961.png", "formula": "\\begin{align*} \\sigma ( A ^ * ) : = \\Big \\{ \\lambda ^ p _ { k } , \\ , k \\geq k _ 0 ; \\ \\ \\lambda ^ h _ k , \\ , | k | \\geq k _ 0 \\Big \\} \\cup \\Big \\{ \\lambda _ 0 \\Big \\} \\cup \\Big \\{ \\widehat \\lambda _ n , \\ , 1 \\leq n \\leq n _ 0 \\Big \\} . \\end{align*}"}
{"id": "7240.png", "formula": "\\begin{align*} \\nabla _ v \\mu ( v ) = - v \\psi ( v ) , \\end{align*}"}
{"id": "3287.png", "formula": "\\begin{align*} \\begin{bmatrix} i \\delta _ 1 & 0 \\\\ 0 & i \\delta _ 2 \\end{bmatrix} = \\Omega _ 2 \\left ( \\begin{bmatrix} i \\varepsilon & 0 \\\\ 0 & i \\varepsilon \\end{bmatrix} \\right ) , \\end{align*}"}
{"id": "3532.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs { \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 + O \\left ( \\int _ 2 ^ T \\abs { \\Sigma _ 1 ( s _ 1 , s _ 2 , s _ 3 ) E ( s _ 1 , s _ 2 , s _ 3 ) } d t _ 3 \\right ) \\\\ & \\quad + O \\left ( \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\right ) . \\end{align*}"}
{"id": "4324.png", "formula": "\\begin{align*} \\hat { V } _ A ( \\tau ) = \\left [ - A e ^ { \\left [ \\frac { \\alpha } { 2 } - l + \\frac { 2 \\eta } { \\alpha } \\left ( \\frac { \\alpha } { 2 } - l \\right ) \\right ] \\tau } , A e ^ { \\left [ \\frac { \\alpha } { 2 } - l + \\frac { 2 \\eta } { \\alpha } \\left ( \\frac { \\alpha } { 2 } - l \\right ) \\right ] \\tau } \\right ] ^ { l - 1 } . \\end{align*}"}
{"id": "4362.png", "formula": "\\begin{align*} \\left ( H - b \\beta \\Lambda \\right ) \\phi _ { i , , \\beta } = 2 \\beta b \\left ( \\frac { \\alpha } { 2 } - i + \\tilde \\lambda \\right ) \\phi _ { i , , \\beta } , \\end{align*}"}
{"id": "8724.png", "formula": "\\begin{align*} \\begin{aligned} & p _ 1 ( x ) p _ 2 ( x ) = 1 \\times 1 + ( - 2 x _ 1 ) \\times 1 + x _ 1 ^ 2 \\times 1 + ( 1 - 2 x _ 1 + x _ 1 ^ 2 ) \\times ( - 2 x _ 2 ) + ( 1 - 2 x _ 1 + x _ 1 ^ 2 ) \\times x _ 2 ^ 2 , \\\\ & ( 1 - 2 x _ 1 + x _ 1 ^ 2 ) ( - 2 x _ 2 ) = 1 \\times ( - 2 x _ 2 ) + ( - 2 x _ 1 ) \\times ( - 2 x _ 2 ) + x ^ 2 ( - 2 x _ 2 ) , \\\\ & ( 1 - 2 x _ 1 + x _ 1 ^ 2 ) x _ 2 ^ 2 = 1 \\times x _ 2 ^ 2 + ( - 2 x _ 1 ) \\times x _ 2 ^ 2 + x _ 1 ^ 2 \\times x _ 2 ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "236.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\Sigma } ( f _ j , f _ j ) = \\mathcal { E } _ { \\Sigma } ( g _ j , g _ j ) + 2 \\varepsilon \\mathcal { E } _ { \\Sigma } ( g _ j , f ) + \\varepsilon ^ 2 \\mathcal { E } _ { \\Sigma } ( f , f ) . \\end{align*}"}
{"id": "2620.png", "formula": "\\begin{align*} \\int _ \\mathcal { Q } | Z f ( x , \\omega ) | \\ , d x \\leq \\int _ \\mathcal { Q } \\sum _ { k \\in \\Z ^ d } | f ( x + k ) | \\ , d x = \\norm { f } _ 1 , \\end{align*}"}
{"id": "394.png", "formula": "\\begin{align*} \\tau \\left [ q _ { t } + v \\cdot \\nabla q - q \\cdot \\nabla v + ( \\nabla \\cdot v ) q \\right ] + q = - \\kappa \\nabla \\theta , \\end{align*}"}
{"id": "5230.png", "formula": "\\begin{align*} | S _ p \\cup \\{ p \\} | = | S \\cup \\{ 0 \\} | = d + 1 . \\end{align*}"}
{"id": "5518.png", "formula": "\\begin{align*} \\phi _ { \\delta } ^ { x _ 0 } ( t ) = e ^ { \\mu t } , t \\in [ 0 , T ] , \\end{align*}"}
{"id": "8328.png", "formula": "\\begin{align*} \\int _ 0 ^ T f \\dd ( \\eta - \\S ' ( u ; h ) ) = 0 \\forall f \\in G [ 0 , T ] . \\end{align*}"}
{"id": "7000.png", "formula": "\\begin{align*} \\mathfrak { S } \\bf x = \\bf e , \\end{align*}"}
{"id": "5100.png", "formula": "\\begin{align*} \\pi f _ A ( t ) + \\pi f _ A ( t ^ { - 1 } ) = 1 , \\end{align*}"}
{"id": "5528.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } _ { r , m } ( t ) & = & A \\xi _ { r , m } ( t ) + \\alpha ( \\xi _ { r , m } ( t ) ) - \\rho ( \\xi _ { r , m } ( t ) ) + \\sum _ { j = 1 } ^ r \\sigma ^ j ( \\xi _ { r , m } ( t ) ) \\dot { B } _ m ^ j ( t ) \\medskip \\\\ \\xi _ { r , m } ( 0 ) & = & x . \\end{array} \\right . \\end{align*}"}
{"id": "8503.png", "formula": "\\begin{align*} 0 = I _ 0 \\subseteq I _ 1 \\subseteq . . . \\subseteq I _ n = B \\end{align*}"}
{"id": "443.png", "formula": "\\begin{align*} A ^ { 0 } ( \\partial _ { x } ^ { \\alpha } u ) _ { t } - B ^ { i j } \\partial _ { i } \\partial _ { j } ( \\partial _ { x } ^ { \\alpha } u ) & = A ^ { 0 } \\partial _ { x } ^ { \\alpha } \\left ( ( A ^ { 0 } ) ^ { - 1 } [ f - A ^ { i } \\partial _ { i } u - D u ] \\right ) \\\\ & + A ^ { 0 } G _ { \\alpha } ( ( A ^ { 0 } ) ^ { - 1 } B ^ { i j } , \\partial _ { i } \\partial _ { j } u ) . \\end{align*}"}
{"id": "7146.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\varliminf } \\vert x _ 1 ^ n - \\mathbb { E } [ x ^ n _ 1 ] \\vert = \\underset { n \\rightarrow \\infty } { \\varliminf } \\vert x _ 1 - \\mathbb { E } [ x ^ n _ 1 ] \\vert . \\end{align*}"}
{"id": "8918.png", "formula": "\\begin{align*} \\check H _ { c t } ^ 1 ( X ; A ) = \\frac { \\mathcal C _ h ( X , S ^ 1 ) } { \\mathcal C _ h ( X , \\R ) } . \\end{align*}"}
{"id": "9231.png", "formula": "\\begin{align*} 0 & = \\norm { x - J ^ A _ 1 ( x + u ) + ( u - 1 ^ { - 1 } ( ( x + u ) - J ^ A _ 1 ( x + u ) ) ) } \\\\ & \\geq \\norm { x - J ^ A _ 1 ( x + u ) } \\end{align*}"}
{"id": "5119.png", "formula": "\\begin{align*} A = n \\ , \\frac { 2 \\gamma a } { e ^ { - 2 \\gamma a } - 1 } B = n \\ , \\frac { 2 \\gamma a } { e ^ { - 2 \\gamma a } - 1 } \\ , e ^ { - 2 \\gamma a } , \\gamma < 0 . \\end{align*}"}
{"id": "6038.png", "formula": "\\begin{align*} \\delta _ { \\alpha , \\beta } g & = \\mathfrak { D } _ { { \\rm l e f t } , \\frac { ( \\alpha + \\beta + 2 ) x - ( \\beta - \\alpha ) } { 2 ( 1 - x ^ 2 ) } } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ) \\\\ - \\delta ^ * _ { \\alpha , \\beta } g & = \\mathfrak { D } _ { { \\rm r i g h t } , - \\frac { ( \\alpha + \\beta ) x - ( \\beta - \\alpha ) } { 2 ( 1 - x ^ 2 ) } } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ) . \\end{align*}"}
{"id": "3382.png", "formula": "\\begin{align*} [ T u , T v ] = T \\Big ( \\rho ( T u ) v - \\rho ( T v ) u \\Big ) , \\ ; \\forall u , v \\in V . \\end{align*}"}
{"id": "1538.png", "formula": "\\begin{align*} = \\int _ { U _ t ( \\mathbb { A } ) P ^ t _ n ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\int _ { U _ t ( \\Q ) \\backslash U _ t ( \\mathbb { A } ) } \\sum _ { \\xi , \\gamma } \\phi ( \\tilde { \\tau } _ t ( \\xi g \\times \\gamma h ) , s ) \\mathbf { f } ( n h ) \\mathbf { d } n \\mathbf { d } h , \\end{align*}"}
{"id": "7569.png", "formula": "\\begin{align*} u \\rhd _ s v = v s v ^ { - 1 } u \\in G . \\end{align*}"}
{"id": "1187.png", "formula": "\\begin{align*} \\underline \\dim _ { { \\rm l o c } } ( \\mu , x ) = \\liminf _ { r \\rightarrow 0 ^ { + } } \\frac { \\log ( \\mu ( B ( x , r ) ) ) } { \\log ( r ) } \\ \\mbox { a n d } \\ \\overline \\dim _ { { \\rm l o c } } ( \\mu , x ) = \\limsup _ { r \\rightarrow 0 ^ { + } } \\frac { \\log ( \\mu ( B ( x , r ) ) ) } { \\log ( r ) } . \\end{align*}"}
{"id": "4289.png", "formula": "\\begin{align*} \\lambda _ { \\ell } ( t ) = C ( u _ 0 ) ( 1 + o ( 1 ) ) ( T - t ) ^ { \\frac { 2 \\ell } { \\alpha } } t \\to T . \\end{align*}"}
{"id": "1879.png", "formula": "\\begin{align*} G = \\{ x \\rightarrow x y , \\ ; \\ ; y \\rightarrow x ^ 2 \\} \\end{align*}"}
{"id": "447.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = 0 } ^ { m } \\langle ( A _ { 1 } ^ { 0 } \\partial _ { x } ^ { \\alpha } \\left ( ( A _ { 1 } ^ { 0 } ) ^ { - 1 } f _ { 1 } \\right ) , 2 \\partial _ { x } ^ { \\alpha } u \\rangle \\leq C \\| f _ { 1 } \\| _ { m } \\| u \\| _ { m } , \\end{align*}"}
{"id": "2425.png", "formula": "\\begin{align*} f = S ^ { - 1 / 2 } S ( S ^ { - 1 / 2 } f ) = \\sum _ { \\gamma \\in \\Gamma } \\langle f , S ^ { - 1 / 2 } e _ \\gamma \\rangle S ^ { - 1 / 2 } e _ \\gamma , \\end{align*}"}
{"id": "5263.png", "formula": "\\begin{align*} \\varphi ( a b ) = \\varphi ( b \\sigma ^ { \\varphi } ( a ) ) , \\psi ( a b ) = \\psi ( b \\sigma ^ { \\psi } ( a ) ) , \\forall a , b \\in A . \\end{align*}"}
{"id": "5426.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ 3 \\lambda _ 2 & = \\frac { ( a - 2 ) \\pm \\sqrt { a ^ 2 - 4 a } } { 2 } \\\\ \\lambda _ 2 \\lambda _ 1 & = \\frac { ( b - 2 ) \\pm \\sqrt { b ^ 2 - 4 b } } { 2 } \\\\ \\lambda _ 1 \\lambda _ 3 & = \\frac { ( c - 2 ) \\pm \\sqrt { c ^ 2 - 4 c } } { 2 } \\end{cases} \\end{align*}"}
{"id": "3816.png", "formula": "\\begin{align*} 2 \\varepsilon ( w _ { \\widehat { \\delta } } ) + \\varepsilon ( \\Delta _ { 4 k + 2 } ^ 2 ) = 0 . \\end{align*}"}
{"id": "976.png", "formula": "\\begin{align*} C = C ' \\left ( 1 + \\rho ^ { 2 s } \\| c \\| _ { L ^ \\infty ( B _ \\rho ( e _ 1 ) ) } \\right ) \\end{align*}"}
{"id": "5027.png", "formula": "\\begin{align*} D _ { p q } = N Y ^ { o p } _ p \\times _ { Y _ 0 } Y _ 1 \\times _ { Y _ 0 } N X _ q \\end{align*}"}
{"id": "649.png", "formula": "\\begin{align*} \\begin{cases} \\ \\\\ [ 5 p t ] \\ \\end{cases} \\in \\ \\R _ { \\ell } . \\end{align*}"}
{"id": "7088.png", "formula": "\\begin{align*} \\alpha _ { s _ { m , W ^ 0 _ { V , i } } } ( f _ { { W ^ 0 _ { V , i } } } ) = \\alpha _ { s ^ 0 _ m } ( f _ U ) . \\end{align*}"}
{"id": "2669.png", "formula": "\\begin{align*} B ( M _ { - \\eta } T _ \\xi g _ 0 ) ( z ) = e ^ { - \\pi i \\xi \\cdot \\eta } e ^ { - \\frac { \\pi } { 2 } | w | ^ 2 } K _ w ( z ) . \\end{align*}"}
{"id": "6260.png", "formula": "\\begin{align*} | X | > | Y _ { t ^ s } | & = \\frac { n - | Z | - | X \\setminus X _ { t ^ s } | - | Y \\setminus Y _ { t ^ s } | } { 2 } \\\\ & = \\frac { n - s t ( 2 k - 1 ) - 2 t ^ s n ^ { \\frac { 1 } { s + 1 } } } { 2 } \\\\ & = \\frac { n - s \\alpha n ^ { \\frac { 1 } { s + 1 } } ( 2 k - 1 ) - 2 ( \\alpha n ^ { \\frac { 1 } { s + 1 } } ) ^ s n ^ { \\frac { 1 } { s + 1 } } } { 2 } \\\\ & > \\frac { n } { 2 } - k s \\alpha n ^ { \\frac { 1 } { s + 1 } } - \\alpha ^ s n . \\end{align*}"}
{"id": "5719.png", "formula": "\\begin{align*} \\mu _ { \\mathrm { d i s c } } ( \\delta _ { 1 4 } ) - \\mu _ { \\mathrm { d i s c } } ( \\delta _ { 1 3 } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } \\cup { \\gamma _ { 3 } } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } \\cup { \\gamma _ { 4 } } ) = 2 \\end{align*}"}
{"id": "3457.png", "formula": "\\begin{align*} \\frac 1 { \\omega ( P ) ^ { \\frac 2 p - 1 } } \\sum \\limits _ { Q \\subseteq P } \\mu ( Q ) | \\psi _ Q ( q _ { j } ( \\cdot , y ) ) ( x _ Q ) | ^ 2 & \\lesssim \\sup _ { x \\in P } \\frac 1 { \\omega ( B ( x , r ^ { - j } ) ) ^ { \\frac 2 p } } \\sum \\limits _ { k = k _ 0 } ^ { \\infty } r ^ { - 2 | k - j | \\varepsilon } \\\\ & \\leqslant C _ j \\sup _ { x \\in P } \\frac 1 { \\omega ( B ( x , r ^ { - j } ) ) ^ { \\frac 2 p } } . \\end{align*}"}
{"id": "2670.png", "formula": "\\begin{align*} V _ { g _ 0 } h _ \\alpha ( x , - \\omega ) = e ^ { \\pi i x \\cdot \\omega } B h _ \\alpha ( z ) e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } = e ^ { \\pi i x \\cdot \\omega } e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } e _ \\alpha . \\end{align*}"}
{"id": "7668.png", "formula": "\\begin{align*} v _ { \\varepsilon } ( \\mathbf { x } _ { \\varepsilon } ) = \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\ , | \\mathbf { x } _ { \\varepsilon } - \\mathbf { y } _ { \\varepsilon } | = \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\ , ( | \\mathbf { x } _ { \\varepsilon } - \\bar { \\mathbf { x } } _ { \\varepsilon } | + | \\bar { \\mathbf { x } } _ { \\varepsilon } - \\mathbf { y } _ { \\varepsilon } | ) \\ge \\end{align*}"}
{"id": "355.png", "formula": "\\begin{align*} \\dot \\rho _ 1 + m _ { 1 2 } & = ( \\Sigma _ 1 - \\Sigma _ 2 ) \\theta _ { 1 2 } ( \\rho ) \\dot W ^ { \\delta } , \\\\ \\dot \\rho _ 2 + m _ { 2 1 } + m _ { 2 3 } & = ( \\Sigma _ 2 - \\Sigma _ 1 ) \\theta _ { 2 1 } ( \\rho ) \\dot W ^ { \\delta } + ( \\Sigma _ 2 - \\Sigma _ 3 ) \\theta _ { 2 3 } ( \\rho ) \\dot W ^ { \\delta } , \\\\ \\dot \\rho _ 3 + m _ { 3 2 } & = ( \\Sigma _ 3 - \\Sigma _ 2 ) \\theta _ { 3 2 } ( \\rho ) \\dot W ^ { \\delta } . \\end{align*}"}
{"id": "352.png", "formula": "\\begin{align*} \\inf _ { ( \\rho , m ) } \\mathcal L ( \\rho , m , S ) = \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > - \\int _ 0 ^ 1 \\max ( H ( \\dot S , \\nabla _ G S ) , 0 ) d t , \\end{align*}"}
{"id": "7413.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega - \\Omega _ 1 ( x ) } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta ) - f ( \\eta ^ { x + 1 , x + 1 + r } ) ] d \\nu _ { b } \\Big | \\end{align*}"}
{"id": "2319.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty W \\psi ( x , p ) \\ , d p = | \\psi ( x ) | ^ 2 , \\\\ \\int _ { - \\infty } ^ \\infty W \\psi ( x , p ) \\ , d x = | \\phi ( p ) | ^ 2 ; \\end{align*}"}
{"id": "8369.png", "formula": "\\begin{align*} C _ { v } ( v , u , w ) = C _ { v } ( u , v , w ) = C _ { v } ( u , w , v ) = 0 , \\end{align*}"}
{"id": "7862.png", "formula": "\\begin{align*} J \\cup K \\cup L & = ( \\mathbb { N } \\setminus \\{ \\bar { a } _ n \\} ) \\cup \\{ \\bar { a } _ n \\} = \\mathbb { N } , \\end{align*}"}
{"id": "6712.png", "formula": "\\begin{align*} \\bigl ( ~ _ { s + 1 } \\mathcal { F } _ s ( \\alpha ) \\bigr ) ^ { q ^ d } & = \\sum _ { n = 0 } ^ { \\infty } \\biggl ( \\prod _ { m = 1 } ^ { n + d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \\alpha ^ { q ^ { n + d } } \\biggr ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\biggl ( \\prod _ { l = 1 } ^ { d - 1 } ( \\theta ^ { q ^ { l + n } } - t ) ^ { c ( l ) q ^ { d - l } } \\alpha ^ { q ^ { n + d } } \\biggr ) . \\end{align*}"}
{"id": "727.png", "formula": "\\begin{align*} a : = - \\frac { 6 \\sigma _ 3 } { \\sigma _ 1 } > 0 . \\end{align*}"}
{"id": "842.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } u ( k _ n , r _ n ) = u ( k _ 0 + d , R / 2 ) , \\end{align*}"}
{"id": "845.png", "formula": "\\begin{align*} \\bar { \\Delta } ^ { \\rm P } \\triangleq \\lim \\limits _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\Delta \\left ( t ^ S _ j - 1 \\right ) . \\end{align*}"}
{"id": "5231.png", "formula": "\\begin{align*} m _ 1 + 2 m _ 2 + \\cdots + ( d k + e ) m _ { d k + e } = b _ r + d k - d + e . \\end{align*}"}
{"id": "5422.png", "formula": "\\begin{align*} \\cal { M } ( \\phi _ { 1 , \\Delta } ) & = P \\circ \\begin{bmatrix} \\lambda _ 1 & 0 \\\\ \\lambda _ 1 + 1 & - 1 \\end{bmatrix} \\circ P ^ { - 1 } , \\\\ \\cal { M } ( \\phi _ { 2 , \\Delta } ) & = P \\circ \\begin{bmatrix} \\lambda _ 2 & 0 \\\\ - ( \\lambda _ 2 + 1 ) & - 1 \\end{bmatrix} \\circ P ^ { - 1 } , \\\\ \\cal { M } ( \\phi _ { 3 , \\Delta } ) & = P \\circ \\begin{bmatrix} - 1 & 0 \\\\ 0 & \\lambda _ 3 \\end{bmatrix} \\circ P ^ { - 1 } \\end{align*}"}
{"id": "6544.png", "formula": "\\begin{align*} Q _ { } - Q _ { } = \\ & \\sqrt { \\left ( \\frac { 1 } { k } ( s - r ) \\right ) ^ 2 + ( 2 H ^ 2 + 4 ) } - \\sqrt { \\left ( \\frac { 1 } { k } ( s - r ) - 1 \\right ) ^ 2 + ( 2 H ^ 2 + 4 ) } \\ \\ + \\\\ & \\sqrt { \\left ( \\frac { 1 } { r - k } \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s - r \\right ) \\right ) ^ 2 + ( 2 H ^ 2 + 4 ) } \\ \\ - \\\\ & \\sqrt { \\left ( \\frac { 1 } { r - k } \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s - r \\right ) - 1 \\right ) ^ 2 + ( 2 H ^ 2 + 4 ) } \\end{align*}"}
{"id": "8936.png", "formula": "\\begin{align*} u + H ( x , D u ) = 0 \\end{align*}"}
{"id": "1717.png", "formula": "\\begin{align*} \\nu _ * = \\alpha _ * + \\frac { \\gamma _ * } { p _ 0 } \u2010 \\frac { \\gamma _ * } { q } . \\end{align*}"}
{"id": "7104.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & \\lambda A _ { ( 2 n + 1 ) ( 2 n + 1 ) } + 2 A _ { ( 2 n ) ( 2 n + 1 ) } - 1 A _ { 1 ( 2 n + 1 ) } \\end{align*}"}
{"id": "9158.png", "formula": "\\begin{align*} L ( s , f \\otimes \\chi _ d ) & = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda _ f ( n ) \\chi _ d ( n ) } { n ^ s } = \\prod _ { p \\nmid d } \\left ( 1 - \\frac { \\lambda _ f ( p ) \\chi _ d ( p ) } { p ^ s } + \\frac { 1 } { p ^ { 2 s } } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "6428.png", "formula": "\\begin{align*} \\begin{aligned} \\big ( D _ E ^ + \\big ) = - \\big ( D _ E ^ - \\big ) & = - \\int _ { N } a ( D _ E ^ - ) + \\frac { h ( \\nu B \\nu ) + \\eta ( \\nu B \\nu ) } { 2 } \\\\ & = \\int _ { N } a ( D _ E ^ + ) + \\frac { h ( B ) - \\eta ( B ) } { 2 } , \\end{aligned} \\end{align*}"}
{"id": "1195.png", "formula": "\\begin{align*} R _ { \\tau } ( x , r ) = \\prod _ { i = 1 } ^ d [ x _ i - \\frac { 1 } { 2 } r ^ { \\tau _ i } , x _ i + \\frac { 1 } { 2 } r ^ { \\tau _ i } ] . \\end{align*}"}
{"id": "7828.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { 3 j + t } \\alpha _ { i } e _ { i } , \\quad \\alpha _ { 3 j + t } \\neq 0 . \\end{align*}"}
{"id": "4096.png", "formula": "\\begin{align*} \\begin{aligned} | \\widetilde \\mu ( x ) - \\widetilde \\mu ( y ) | = | \\mu ( x ) - \\mu ( y ) | \\leq c \\cdot ( 1 + | x | ^ { \\ell _ \\mu } + | y | ^ { \\ell _ \\mu } ) \\cdot | x - y | . \\end{aligned} \\end{align*}"}
{"id": "8718.png", "formula": "\\begin{align*} p ( x ) = \\phi ( g _ { j ( 1 ) } ( x ) , \\ldots , g _ { j ( T ) ( T ) } ( x ) ) , \\deg ( p ) \\leq \\delta , \\end{align*}"}
{"id": "6980.png", "formula": "\\begin{align*} \\forall t \\in \\R \\colon S ( t ) : = \\begin{cases} \\{ - t ^ 2 \\} & t \\leq 0 , \\\\ \\{ \\sqrt t \\} & t > 0 . \\end{cases} \\end{align*}"}
{"id": "805.png", "formula": "\\begin{align*} \\int _ { B ( x , r ) } | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla ( u - v ) \\ , d \\mu = \\int _ { B ( x , r ) } ( u - v ) f \\ , d \\nu . \\end{align*}"}
{"id": "1185.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\zeta } ( E ) = \\lim _ { t \\to 0 ^ + } \\mathcal { H } ^ { \\zeta } _ t ( E ) . \\end{align*}"}
{"id": "8688.png", "formula": "\\begin{align*} g | _ S = \\begin{cases} g ( x ) & x \\in S \\\\ - \\infty & . \\end{cases} \\end{align*}"}
{"id": "8336.png", "formula": "\\begin{align*} \\int _ a ^ b c \\dd g = c ( g ( b ) - g ( a ) ) \\int _ a ^ b f \\dd c = 0 \\end{align*}"}
{"id": "6575.png", "formula": "\\begin{align*} \\ell : = \\frac { | m h \\pm n k | } { d } . \\end{align*}"}
{"id": "8827.png", "formula": "\\begin{align*} \\underset { x \\in \\mathbb { R } ^ n } { } \\ c ^ \\top x + \\frac { 1 } { 2 } x ^ \\top Q x + \\| D x \\| _ 1 + \\delta _ { \\mathcal { K } } ( x ) , \\ A x = b , \\end{align*}"}
{"id": "2126.png", "formula": "\\begin{align*} \\binom { c _ n k n } l \\frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } = \\frac { \\binom { c _ n k n } l } { \\prod _ { i = 1 } ^ l ( k n - c _ n k n + i ) } \\prod _ { t = 0 } ^ { c _ n k n - l - 1 } \\frac { k ( n - s - 1 ) - t } { k n - t } . \\end{align*}"}
{"id": "3529.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) = \\sum _ { m \\leq a t _ 3 } \\sum _ { n < m } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + O ( t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) . \\end{align*}"}
{"id": "6854.png", "formula": "\\begin{align*} H ^ 0 = C _ 0 , H ^ i = ( D _ i ) , V _ k ^ i = U ^ { k } _ i H ^ i . \\end{align*}"}
{"id": "1079.png", "formula": "\\begin{align*} & \\textnormal { d e t } \\Phi _ { R , L } ( x , t , k ) = 1 , k \\in \\mathbb { R } \\backslash [ - C , C ] . \\end{align*}"}
{"id": "2656.png", "formula": "\\begin{align*} A \\norm { c } _ { \\ell ^ 2 } ^ 2 \\leq \\norm { \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma f _ \\gamma } _ \\mathcal { H } ^ 2 \\leq B \\norm { c } _ { \\ell ^ 2 } ^ 2 , \\forall c = ( c _ \\gamma ) _ { \\gamma \\in \\Gamma } \\in \\ell ^ 2 ( \\Gamma ) . \\end{align*}"}
{"id": "8529.png", "formula": "\\begin{align*} i \\partial _ t u - \\partial _ { x x } u + V ( x ) u \\pm \\left | u \\right | ^ { 2 } u = 0 , u ( 0 , x ) = u _ { 0 } ( x ) , \\end{align*}"}
{"id": "6284.png", "formula": "\\begin{align*} \\rho = - \\bar { p } ( 1 - \\xi ) / \\mathrm { l i } ( 1 - \\xi ) , \\ \\ \\mu _ i = - \\vartheta _ i \\ln ( 1 - \\xi ) , \\ \\forall i , \\end{align*}"}
{"id": "48.png", "formula": "\\begin{align*} \\left [ z ^ 1 , z ^ 2 \\right ] = \\left [ \\cos s \\ , e ^ { i \\Phi _ 1 } , \\sin s \\ , e ^ { i \\Phi _ 2 } \\right ] . \\end{align*}"}
{"id": "916.png", "formula": "\\begin{align*} ( a ^ { + } a ) ^ { k } = \\sum _ { l = 0 } ^ { k } S _ { 2 } ( k , l ) ( a ^ { + } ) ^ { l } a ^ { l } , ( \\mathrm { s e e } \\ [ 4 , 5 , 8 ] ) . \\end{align*}"}
{"id": "2011.png", "formula": "\\begin{align*} M _ { t } ^ { u } = M _ { t } ^ { u , c } + M _ { t } ^ { u , j } \\end{align*}"}
{"id": "8464.png", "formula": "\\begin{align*} \\| x _ { k + 1 } - x _ \\ast \\| \\leq \\frac { 1 } { q } \\sum _ { j = 1 } ^ { q } \\| Q _ { j } ( x _ k - x _ \\ast ) \\| = \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } \\| ( x _ k - x _ \\ast ) \\| = \\| ( x _ k - x _ \\ast ) \\| . \\end{align*}"}
{"id": "3641.png", "formula": "\\begin{align*} \\left | \\frac { \\psi ( x ) - x } { x } \\right | & \\leq A ( x _ 0 , \\sigma ) \\cdot \\left ( B _ 2 r \\right ) ^ { 5 - 2 \\sigma } \\exp \\left ( \\frac { B _ 2 ( 5 - 8 \\sigma ) } { 3 } r \\right ) \\\\ & = A ( x _ 0 , \\sigma ) \\cdot \\left ( B _ 2 \\frac { \\log ^ { 3 / 5 } x } { ( \\log \\log x ) ^ { 1 / 5 } } \\right ) ^ { 5 - 2 \\sigma } \\exp \\left ( \\frac { B _ 2 ( 5 - 8 \\sigma ) } { 3 } \\frac { \\log ^ { 3 / 5 } x } { ( \\log \\log x ) ^ { 1 / 5 } } \\right ) . \\end{align*}"}
{"id": "8583.png", "formula": "\\begin{align*} { \\| u \\| } _ { H ^ 1 } ^ 2 & = - \\int u u _ { x x } \\ , d x + \\int u ^ 2 = \\int u H u \\ , d x - \\int u V u \\ , d x + \\int u ^ 2 \\ , d x \\\\ & = { \\big \\| | k | u ^ \\sharp \\big \\| } _ { L ^ 2 } ^ 2 - \\int u V u \\ , d x + { \\| u ^ \\sharp \\| } _ { L ^ 2 } ^ 2 . \\end{align*}"}
{"id": "7341.png", "formula": "\\begin{align*} x _ q = \\lambda y _ q + ( 1 - \\lambda ) z _ q , w _ q ( x _ q , t _ q ) = \\left ( \\lambda u ( y _ q , t _ q ) ^ q + ( 1 - \\lambda ) u ( z _ q , t _ q ) ^ q \\right ) ^ { \\frac { 1 } { q } } . \\end{align*}"}
{"id": "7008.png", "formula": "\\begin{align*} S ( \\alpha _ i ) B ( \\alpha _ i ) = 1 , ( S B ) ^ { ( \\ell ) } ( \\alpha _ i ) = 0 \\quad 1 \\le \\ell \\le \\mu _ i - 1 . \\end{align*}"}
{"id": "3571.png", "formula": "\\begin{align*} { \\rm s p } _ 0 ( \\tau ) = \\frac { 1 } { 2 } p ^ e S , S = { \\bigcup } _ { u = 0 } ^ n { \\bigcup } _ { z = 1 } ^ { e } { \\bigcup } _ { \\alpha \\in \\Xi _ { u , z } } U ( \\alpha ) . \\end{align*}"}
{"id": "7908.png", "formula": "\\begin{align*} \\Delta f _ i + f _ i = 0 . \\end{align*}"}
{"id": "1771.png", "formula": "\\begin{align*} f \\ast g ( x ) = \\int _ { \\mathbb { R } ^ n } f ( y ) g ( x - y ) d y . \\end{align*}"}
{"id": "1855.png", "formula": "\\begin{align*} M _ { n + 1 } ( x ) = x \\sum _ { k = 0 } ^ n { n \\choose k } M _ { k } ( x ) M _ { n - k } ( x ) . \\end{align*}"}
{"id": "2291.png", "formula": "\\begin{align*} \\norm { V _ g f } _ 2 = \\norm { f } _ 2 \\norm { g } _ 2 \\ , . \\end{align*}"}
{"id": "1718.png", "formula": "\\begin{align*} \\nu _ * = \\frac { \\alpha _ * ( s _ * + 1 / q \u2010 1 / p _ 1 ) + \\mu _ * ( 1 / q \u2010 1 / p _ 0 ) } { s _ * + 1 / p _ 0 \u2010 1 / p _ 1 } . \\end{align*}"}
{"id": "9526.png", "formula": "\\begin{align*} p _ t - ( \\Delta y _ { t + 1 } + A ^ * _ { t + 1 } y _ { t + 1 } , B ^ * _ { t + 1 } y _ { t + 1 } ) \\in \\partial L _ t ( X _ t , U _ t ) , \\\\ \\Delta X _ { t } = A _ t X _ { t - 1 } + B _ t U _ { t - 1 } + W _ t \\end{align*}"}
{"id": "6782.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p _ 0 + 1 } i _ { k _ j } = - ( v _ 1 + 1 ) + ( v _ 1 + 1 ) + \\sum _ { j = 1 } ^ { y - 1 } ( - ( v _ { q _ j } + 1 ) + ( v _ { q _ j } + 1 ) ) = 0 . \\end{align*}"}
{"id": "7273.png", "formula": "\\begin{align*} \\log \\Bigl ( \\frac { 1 } { 1 - X } \\Bigr ) = - \\log ( 1 - X ) = \\sum _ { n = 1 } ^ \\infty \\frac { X ^ n } { n } . \\end{align*}"}
{"id": "4835.png", "formula": "\\begin{align*} E _ { n } ^ { \\mathrm { m i n } } : = \\inf \\left [ \\sup _ { \\| f \\| \\leq 1 } \\sup _ { x \\in \\R } \\left | f ( x ) - \\sum _ { j = 1 } ^ { l } \\sum _ { k = 0 } ^ { m _ { j } - 1 } f ^ { ( k ) } ( a _ { j } ) \\ , \\phi _ { j k } ( x ) \\right | \\right ] , \\end{align*}"}
{"id": "1620.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d - 1 } } \\psi \\tilde \\psi \\ , d x & = 1 , \\\\ \\int _ { \\mathbb { R } ^ { d - 1 } } \\beta ( | g _ j ^ Q | ^ \\frac { 1 } { p } \\psi ) \\tilde \\psi \\ , d x & = ( - 1 ) ^ { j + 1 } | g _ j ^ Q | ^ \\frac { 1 } { p } \\\\ \\int _ { \\mathbb { R } ^ { d - 1 } } \\tilde \\psi \\ , d x & = 0 . \\end{align*}"}
{"id": "493.png", "formula": "\\begin{align*} \\Upsilon _ { \\omega } ^ { + } : = \\left \\{ Y _ { \\alpha \\omega } ^ { m i } , Y _ { \\beta \\omega } ^ { m j } \\right \\} , \\quad \\Upsilon _ { \\omega } ^ { - } : = \\left \\{ Y _ { \\alpha \\omega } ^ { m j } , Y _ { \\beta \\omega } ^ { m i } \\right \\} , \\quad \\Upsilon _ { \\omega } : = \\Upsilon _ { \\omega } ^ { + } \\cup \\Upsilon _ { \\omega } ^ { - } \\end{align*}"}
{"id": "2808.png", "formula": "\\begin{align*} J ( \\sigma , X , u ) = \\left ( \\begin{matrix} J _ 1 ( \\sigma , X , u ) \\\\ J _ 2 ( \\sigma , X , u ) \\end{matrix} \\right ) = \\left ( \\begin{matrix} \\Im \\int e ^ { - i \\sigma } u ( x + X ) Q ( x ) d x \\\\ \\Re \\int e ^ { - i \\sigma } u ( x + X ) \\nabla Q ( x ) d x \\end{matrix} \\right ) , \\end{align*}"}
{"id": "8143.png", "formula": "\\begin{align*} L ( 1 , \\chi ) = L ( 1 , \\chi ' ) \\prod _ { q \\mid d _ 0 } \\left ( 1 - \\frac { \\chi ( q ) } { q } \\right ) ^ { - 1 } \\end{align*}"}
{"id": "1348.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\left ( \\sum _ { j = 1 } ^ d \\int _ 0 ^ 1 x _ j U _ j ^ k ( \\tau _ { t x } \\omega ) d t \\right ) \\partial _ i \\eta ( x ) d x = \\int _ 0 ^ 1 \\sum _ { j = 1 } ^ d \\int _ { \\mathbb { R } ^ d } \\frac { y _ j } { t ^ { d + 1 } } U _ j ^ k ( \\tau _ y \\omega ) \\partial _ i \\eta \\left ( \\frac { y } { t } \\right ) d y d t \\end{align*}"}
{"id": "278.png", "formula": "\\begin{align*} \\chi _ { * } ( x ) : = \\frac { 1 } { \\beta } \\frac { ( e ^ { \\beta M / 2 } - 1 ) e ^ { - x ^ { 2 } / 4 } } { \\sqrt { \\pi } + ( e ^ { \\beta M / 2 } - 1 ) \\int _ { x / 2 } ^ { \\infty } e ^ { - y ^ { 2 } } d y } , M : = \\int _ { \\R } u _ { 0 } ( x ) d x \\neq 0 , \\end{align*}"}
{"id": "2870.png", "formula": "\\begin{align*} e ^ { e _ 0 t } \\alpha _ + ( t ) = \\int _ t ^ \\infty e ^ { e _ 0 s } B ( R , \\mathcal { Y } _ - ) d s . \\end{align*}"}
{"id": "4627.png", "formula": "\\begin{gather*} \\beta ( v ) = \\sum R '' \\otimes R ' v \\end{gather*}"}
{"id": "6691.png", "formula": "\\begin{align*} \\sigma f = f ^ { ( - 1 ) } \\sigma , f \\in \\overline { k } ( t ) . \\end{align*}"}
{"id": "2943.png", "formula": "\\begin{align*} X _ { n , p _ k } ^ 2 = \\frac 4 { n ^ 2 \\delta _ n ^ 2 ( k ) } \\sum _ { \\substack { \\mathbf { p } _ { k , 1 } \\in \\mathcal P ( d , k ) : p _ { k , 1 } = p _ k \\\\ \\mathbf { p } _ { k , 2 } \\in \\mathcal P ( d , k ) : p _ { k , 2 } = p _ k } } \\sum _ { \\mathbf { i } \\in \\mathcal { J } } \\Big ( \\prod _ { \\ell = 1 } ^ { k - 1 } \\tilde I ^ { ( p _ { \\ell , 1 } ) } _ { i _ 1 , i _ 2 } \\tilde I ^ { ( p _ { \\ell , 2 } ) } _ { i _ 3 , i _ 4 } \\Big ) \\cdot \\tilde I ^ { ( p _ { k } ) } _ { i _ 1 , i _ 2 } \\tilde I ^ { ( p _ { k } ) } _ { i _ 3 , i _ 4 } . \\end{align*}"}
{"id": "7507.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\partial { R _ \\epsilon } } \\log \\left ( \\frac { s } { 2 } \\right ) \\ d s \\end{align*}"}
{"id": "7996.png", "formula": "\\begin{align*} T _ u ^ { \\sigma } ( q ) = \\frac { - 1 } { \\log 2 } \\left ( q \\cdot \\log \\Big ( \\frac { u } { 1 - u } \\Big ) + \\log \\left ( \\frac { 1 } { 2 } \\sqrt { 1 + 4 \\Big ( \\frac { 1 - u } { u ^ 2 } \\Big ) ^ q } - \\frac { 1 } { 2 } \\right ) \\right ) . \\end{align*}"}
{"id": "5575.png", "formula": "\\begin{align*} | B | + ( \\lambda - 1 ) \\max B & \\ \\leqslant \\ k - | \\{ 1 , 2 , \\ldots , k \\} \\cap A | + ( \\lambda - 1 ) k \\\\ & \\ \\leqslant \\ \\lambda m - | \\{ 1 , 2 , \\ldots , k \\} \\cap A | \\\\ & \\ \\leqslant \\ | A | - | \\{ 1 , 2 , \\ldots , k \\} \\cap A | \\ = \\ | F | . \\end{align*}"}
{"id": "1462.png", "formula": "\\begin{align*} \\rho ( \\alpha ) \\left [ \\begin{array} { c } \\rho ( z ) \\\\ u _ { 0 2 } \\end{array} \\right ] \\mu ( z ) \\lambda ( \\alpha , z ) ^ { - 1 } \\mu ( \\alpha z ) ^ { - 1 } = \\left [ \\begin{array} { c } \\rho ( \\alpha ) \\rho ( z ) \\\\ u _ { 0 2 } \\end{array} \\right ] \\lambda ( \\rho ( \\alpha ) , \\rho ( z ) ) \\mu ( z ) \\lambda ( \\alpha , z ) ^ { - 1 } \\mu ( \\alpha z ) ^ { - 1 } . \\end{align*}"}
{"id": "3500.png", "formula": "\\begin{align*} D _ { 2 3 } & = \\frac { s _ 3 } { s _ 1 + s _ 3 } \\int _ { a t _ 3 } ^ \\infty \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 1 + s _ 2 + s _ 3 } } d v \\\\ & + \\frac { s _ 3 } { 2 \\pi i \\Gamma ( s _ 3 + 1 ) } \\left ( \\int _ { a t _ 3 } ^ \\infty \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 1 + s _ 2 + s _ 3 } } d v \\right ) \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { s _ 1 + s _ 3 + z } d z . \\end{align*}"}
{"id": "4472.png", "formula": "\\begin{align*} a ^ { \\dagger } a \\ , e _ { \\lambda } ^ { a ^ { \\dagger } a - \\lambda } ( t ) = e _ { \\lambda } ^ { a ^ { \\dagger } a - \\lambda } ( t ) a ^ { \\dagger } a = a ^ { \\dagger } e _ { \\lambda } ^ { a a ^ { \\dagger } - \\lambda } ( t ) a = a ^ { \\dagger } e _ { \\lambda } ^ { a ^ { \\dagger } a + 1 - \\lambda } ( t ) a . \\end{align*}"}
{"id": "7669.png", "formula": "\\begin{align*} \\ge \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\ , | \\mathbf { x } _ { \\varepsilon } - \\bar { \\mathbf { x } } _ { \\varepsilon } | = \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\ , d ( \\mathbf { x } _ { \\varepsilon } , \\partial \\Omega ) ; . \\end{align*}"}
{"id": "9163.png", "formula": "\\begin{align*} L ( \\tfrac { 1 } { 2 } , f \\otimes \\chi _ { 8 p } ) = & 2 \\sum ^ { \\infty } _ { \\substack { n = 1 } } \\frac { \\lambda _ f ( n ) \\chi _ { 8 p } ( n ) } { \\sqrt { n } } V \\left ( \\frac { n } { p } \\right ) , \\end{align*}"}
{"id": "4383.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u - 3 ( n - 4 ) u ^ 2 - ( n - 4 ) | x | ^ 2 u ^ 3 . \\end{align*}"}
{"id": "2749.png", "formula": "\\begin{align*} \\| \\lambda ^ \\frac { \\gamma + 2 } { 2 ( p - 1 ) } u _ 0 ( \\lambda x ) \\| _ { \\dot { H } ^ { s _ c } ( \\mathbb { R } ^ N ) } = \\| u _ 0 \\| _ { \\dot { H } ^ { s _ c } ( \\mathbb { R } ^ N ) } , \\end{align*}"}
{"id": "1920.png", "formula": "\\begin{align*} \\int ^ { T } _ { 0 } \\int _ { \\Omega } & \\big ( \\rho u \\cdot \\partial _ t \\phi + \\rho u \\otimes u : \\nabla _ x \\phi + \\rho ^ { \\gamma } { \\rm { d i v } } _ x \\phi - \\mathbb { S } ( \\nabla _ x u ) : \\nabla _ x \\phi \\\\ & + ( j - n u ) \\cdot \\phi \\big ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 u _ 0 \\cdot \\phi ( 0 , x ) \\ , d x = 0 , \\end{align*}"}
{"id": "6688.png", "formula": "\\begin{align*} & \\Bigl ( ~ _ { s + 1 } F _ s ( 1 , m , \\ldots , m ; 1 + m , \\ldots , 1 + m ) ( z ^ { q ^ { - m + 1 } } ) \\Bigr ) ^ { q ^ m } \\\\ & = \\biggl ( \\sum _ { i \\geq 0 } \\frac { D _ i ( m ) _ i \\cdots ( m ) _ i } { ( 1 + m ) _ i \\cdots ( 1 + m ) _ i D _ i } z ^ { q ^ { i - m + 1 } } \\biggr ) ^ { q ^ m } = \\biggl ( \\sum _ { i \\geq 0 } \\frac { 1 } { [ i + m ] ^ { n q ^ { - m } } } z ^ { q ^ { i - m + 1 } } \\biggr ) ^ { q ^ m } = \\sum _ { i \\geq 0 } \\frac { z ^ { q ^ { i + 1 } } } { [ i + m ] ^ { s } } . \\end{align*}"}
{"id": "3044.png", "formula": "\\begin{align*} x _ 1 ^ \\prime & = a _ { 1 1 } x _ 1 + a _ { 1 2 } x _ 2 \\ , , x _ 2 ^ \\prime = a _ { 2 1 } x _ 1 + a _ { 2 2 } x _ 2 \\ , ; \\\\ y _ 1 ^ \\prime & = b _ { 1 1 } y _ 1 + b _ { 1 2 } y _ 2 \\ , , y _ 2 ^ \\prime = b _ { 2 1 } y _ 1 + b _ { 2 2 } y _ 2 \\ , . \\end{align*}"}
{"id": "7804.png", "formula": "\\begin{align*} \\log a = \\log ( 1 + x v ) = \\sum _ { m = 1 } ^ { \\infty } ( - 1 ) ^ { m + 1 } \\frac { ( x v ) ^ m } { m } . \\end{align*}"}
{"id": "3601.png", "formula": "\\begin{align*} R _ { n + 1 } - S _ { n } + d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] = 1 - S _ { n } \\le d [ a _ { 1 , n } b _ { 1 , n } ] \\ , , \\end{align*}"}
{"id": "4099.png", "formula": "\\begin{align*} | H ' ( x ) - H ' ( y ) | & = \\Bigl | \\frac { 1 } { G ' ( G ^ { - 1 } ( x ) ) } - \\frac { 1 } { G ' ( G ^ { - 1 } ( y ) ) } \\Bigr | \\leq c _ 1 \\cdot | G ' ( G ^ { - 1 } ( y ) ) - G ' ( G ^ { - 1 } ( x ) ) | \\\\ & \\leq c _ 2 \\cdot | G ^ { - 1 } ( y ) - G ^ { - 1 } ( x ) | \\leq c _ 3 \\cdot | x - y | . \\end{align*}"}
{"id": "5651.png", "formula": "\\begin{align*} B ( x _ 1 , x _ 2 ) = A ( x _ 1 , x _ 2 ) ( \\sigma , R ) = \\alpha _ 3 ^ { - 1 } A ( \\alpha _ 1 x _ { \\sigma ^ { - 1 } 1 } , \\alpha _ 2 x _ { \\sigma ^ { - 1 } 2 } ) \\end{align*}"}
{"id": "4571.png", "formula": "\\begin{align*} \\left \\langle X \\right \\rangle _ 0 = 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\left \\langle X \\right \\rangle _ k = \\sum _ { i = 1 } ^ k \\mathbf { E } ( \\xi _ i ^ 2 | \\mathcal { F } _ { i - 1 } ) , \\ \\ \\ \\ k = 1 , . . . , n . \\end{align*}"}
{"id": "6841.png", "formula": "\\begin{align*} M ( v , X ( k ) \\setminus S ) = \\sum \\limits _ { y \\in X ( k ) \\setminus S } M ( v , y ) \\end{align*}"}
{"id": "4546.png", "formula": "\\begin{align*} \\delta ( G ) | K | - k \\cdot 2 \\binom { r - 1 } { 2 } a ^ 2 \\le e ( K , V - V ( K ) ) \\le ( ( r - 2 ) a k - 1 ) | A | + ( r - 2 ) a k ( n - | K | - | A | ) , \\end{align*}"}
{"id": "7374.png", "formula": "\\begin{align*} ( r ^ \\beta V ( r ^ { 1 - \\beta } ) ) '' = \\beta ( 1 - \\beta ) r ^ { - 1 } \\left ( V ' ( r ^ { 1 - \\beta } ) - r ^ { \\beta - 1 } V ( r ^ { 1 - \\beta } ) \\right ) + ( 1 - \\beta ) ^ 2 r ^ { - \\beta } V '' ( r ^ { 1 - \\beta } ) . \\end{align*}"}
{"id": "5747.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( Z _ { \\psi _ \\lambda } \\cap B ^ n _ 1 ) & \\leq C \\sum _ { m = 0 } ^ \\infty e ^ { - c m } J _ { ( m + 1 ) \\lfloor \\sqrt { \\lambda } \\rfloor } . \\end{align*}"}
{"id": "3634.png", "formula": "\\begin{align*} \\frac { 4 . 3 1 2 8 } { T } \\log ^ { 0 . 6 } x & \\le \\frac { 4 . 3 1 2 8 } { \\exp ( B _ 2 \\sqrt { \\log x } ) } \\log ^ { 0 . 6 } x = 4 . 3 1 2 8 \\log ^ { 0 . 6 } x \\exp ( - B _ 2 \\sqrt { \\log x } ) \\\\ & = s _ 3 ' ( x , \\sigma ) , \\ . \\end{align*}"}
{"id": "3109.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + z ( x ^ 4 + a y ^ 4 ) + y ( b x ^ 4 + c y ^ 4 ) = 0 \\ , . \\end{align*}"}
{"id": "9146.png", "formula": "\\begin{align*} \\forall x , y \\in X \\left ( x = y \\rightarrow T x = T y \\right ) \\end{align*}"}
{"id": "1443.png", "formula": "\\begin{align*} \\mathrm { G L } _ n ( \\mathbb { B } ) = \\{ g \\in M _ n ( \\mathbb { B } ) : \\det ( g ) \\neq 0 \\} , \\mathrm { S L } _ n ( \\mathbb { B } ) = \\{ g \\in M _ n ( \\mathbb { B } ) : \\det ( g ) = 1 \\} . \\end{align*}"}
{"id": "615.png", "formula": "\\begin{align*} A ( \\phi ( x ) ) \\ & = \\ \\frac { f ( \\phi ( x ) ) - g ( \\phi ( x ) ) } { h ( \\phi ( x ) ) + 1 } \\\\ [ 1 1 p t ] & = \\ \\frac { ( f \\circ \\phi ) ( x ) - ( g \\circ \\phi ) ( x ) } { ( h \\circ \\phi ) ( x ) + 1 } \\end{align*}"}
{"id": "1297.png", "formula": "\\begin{align*} | \\Lambda ( M ) | = \\sum _ { \\Gamma \\in H _ { 1 } ( Y ) } | \\Lambda ( M , \\Gamma ) | \\end{align*}"}
{"id": "9393.png", "formula": "\\begin{align*} \\tau ( a _ 1 \\cdots a _ n ) & = 0 , \\\\ \\ \\ \\ \\tau ' ( a _ 1 \\cdots a _ n ) & = \\sum _ { j = 1 } ^ n \\tau ( a _ 1 \\cdots a _ { j - 1 } \\tau ' ( a _ j ) a _ { j + 1 } \\cdots a _ n ) . \\end{align*}"}
{"id": "9392.png", "formula": "\\begin{align*} \\tau ( a _ 1 \\cdots a _ n ) = 0 . \\end{align*}"}
{"id": "6809.png", "formula": "\\begin{align*} u ( t ) = \\boldsymbol { S } ( t ) u _ { 0 } + \\int \\nolimits _ { 0 } ^ { t } \\boldsymbol { S } ( t - \\tau ) f ( \\tau ) ) d \\tau , \\end{align*}"}
{"id": "6303.png", "formula": "\\begin{align*} \\mathbb { E } \\bigg [ \\frac { \\rho } { | \\hat { h } _ i | ^ 2 } \\Big | | \\hat { h } _ i | ^ 2 \\ge \\mu _ i \\bigg ] & = \\frac { \\rho } { \\mathbb { P } ( | \\hat { h } _ i | ^ 2 \\ge \\mu _ i ) } \\int \\limits _ { \\mu _ i } ^ { \\infty } \\frac { e ^ { - x / \\vartheta _ i } } { \\vartheta _ i x } \\mathrm { d } x = - \\frac { \\rho } { ( 1 - \\xi ) \\vartheta _ i } \\mathrm { l i } ( 1 - \\xi ) , \\end{align*}"}
{"id": "5755.png", "formula": "\\begin{align*} \\nabla _ { \\nabla y ^ { k } } \\nabla y ^ { l } = \\widetilde { \\bar { \\nabla } _ { \\bar { \\nabla } z ^ { k } } \\bar { \\nabla } z ^ { l } } + A ( \\nabla y ^ { k } , \\nabla y ^ { l } ) , \\end{align*}"}
{"id": "3447.png", "formula": "\\begin{align*} & E _ { j } ( x , z ) - E _ { j } ( x , z ' ) \\\\ & = \\sum \\limits _ { Q \\in Q ^ j } \\int _ { Q } \\psi _ { Q } ( x , x _ { Q } ) \\Big [ \\big ( q _ { Q } ( y , z ) - q _ { Q } ( x _ { Q } , z ) \\big ) - \\big ( q _ { Q } ( y , z ' ) - q _ { Q } ( x _ { Q } , z ' ) \\Big ] d \\omega ( y ) . \\end{align*}"}
{"id": "3132.png", "formula": "\\begin{align*} \\Delta _ 5 ( \\lambda ) = \\sum \\pm u _ { 1 1 } u _ { 2 2 } u _ { 3 3 } u _ { 4 4 } u _ { 5 5 } \\end{align*}"}
{"id": "4632.png", "formula": "\\begin{align*} \\widehat { A } + ( \\widehat { A } ) ^ T = 2 I . \\end{align*}"}
{"id": "7637.png", "formula": "\\begin{align*} \\begin{cases} w '' ( r ) + \\frac { N - 1 } { r } w ' ( r ) + \\tilde { \\lambda } _ 0 \\overline { m } w ( r ) = 0 & 0 < r < 1 \\ ; , \\\\ w '' ( r ) + \\frac { N - 1 } { r } w ' ( r ) - \\tilde { \\lambda } _ 0 \\underline { m } w ( r ) = 0 & r > 1 \\ ; , \\\\ w ' ( 0 ) = 0 \\ ; , \\\\ \\lim _ { r \\to + \\infty } w ( r ) = 0 \\ ; , \\\\ w ( r ) \\in C ^ 1 ( \\R ^ { + } ) \\ ; . \\end{cases} \\end{align*}"}
{"id": "8544.png", "formula": "\\begin{align*} \\mu ^ { \\# } ( k , \\ell , m , n ) : = \\int \\overline { \\mathcal { K } ^ { \\# } ( x , k ) } \\mathcal { K } ^ { \\# } ( x , \\ell ) \\overline { \\mathcal { K } ^ { \\# } ( x , m ) } \\mathcal { K } ( x , n ) \\ , d x . \\end{align*}"}
{"id": "6926.png", "formula": "\\begin{align*} \\mathbf { h } = \\sqrt { \\frac { K } { K + 1 } } \\mathbf { h } _ { L O S } + \\sqrt { \\frac { 1 } { K + 1 } } \\mathbf { h } _ { N L O S } , \\end{align*}"}
{"id": "400.png", "formula": "\\begin{align*} \\mu _ { 0 } ( t ) : = \\sum _ { i = 1 } ^ { d } \\| A ^ { i } \\| _ { \\bar { s } } ^ { 2 } + \\sum _ { i , j = 1 } ^ { d } \\| B ^ { i j } \\| _ { \\bar { s } } ^ { 2 } + \\sum _ { i = 1 } ^ { d } \\| A ^ { i } \\| _ { \\bar { s } } + \\| D \\| _ { \\bar { s } } ^ { 2 } + \\| D \\| _ { \\bar { s } } + 1 \\end{align*}"}
{"id": "4907.png", "formula": "\\begin{align*} \\frac { d } { d x } f ( x ) = \\frac { \\sigma \\sqrt { \\log ( 1 / \\delta ) } } { \\sqrt { x } } - G _ 0 , \\end{align*}"}
{"id": "4110.png", "formula": "\\begin{align*} \\begin{aligned} | f _ n ( x , y ) | & \\leq n ^ { - 1 / 2 } | | y | ^ { \\ell _ \\mu } - | x | ^ { \\ell _ \\mu } | \\cdot | \\sigma ( y ) | \\le 4 | x - y | \\cdot \\frac { 1 + | y | ^ { \\ell _ \\sigma + 1 } } { 1 + | y | } \\leq c \\cdot ( 1 + | y | ^ { \\ell _ \\sigma } ) \\cdot | x - y | . \\end{aligned} \\end{align*}"}
{"id": "7597.png", "formula": "\\begin{align*} \\tilde { g } _ j ( x ) = g _ j ( x ) + \\sum _ { k _ y = 1 } ^ { \\infty } ( - 1 ) ^ { k _ y } \\left ( g _ { 2 k _ y n _ y - j } ( x ) + g _ { 2 k _ y n _ y + j } ( x ) \\right ) . \\end{align*}"}
{"id": "5870.png", "formula": "\\begin{align*} S _ t = S _ t ( Y _ 0 , d W _ \\cdot ) , \\end{align*}"}
{"id": "102.png", "formula": "\\begin{align*} \\tau ( s , \\mu + \\eta ) = \\bigoplus _ { 1 \\leq i \\leq n } \\tilde { \\omega } _ { r f } ^ { \\sum _ { j ' = 0 } ^ { r f - 1 } \\alpha ' _ { ( s , \\mu ) , j ' , i } p ^ { j ' } } . \\end{align*}"}
{"id": "5219.png", "formula": "\\begin{align*} | \\{ x \\in K : | B f ( x ) | > \\lambda \\} | & \\leq \\lambda ^ { - 1 } ~ \\| f \\| _ { L ^ 1 ( K ) } + 4 q \\lambda ^ { - 1 } ~ \\| f \\| _ { L ^ 1 ( K ) } \\\\ & = ( 1 + 4 q ) \\lambda ^ { - 1 } ~ \\| f \\| _ { L ^ 1 ( K ) } . \\end{align*}"}
{"id": "3909.png", "formula": "\\begin{align*} 0 = \\sum _ { z < p } \\Psi ( z ) \\Psi ( f ( z ) ) . \\end{align*}"}
{"id": "7952.png", "formula": "\\begin{align*} \\dot { \\varrho } _ { L _ { \\alpha _ j } } ( \\pi ^ { \\prime } ) = \\varrho _ { \\alpha } ( \\pi ) . \\end{align*}"}
{"id": "5706.png", "formula": "\\begin{align*} I ( \\delta _ { k + 1 } \\cup { \\gamma } , \\delta _ { k } \\cup { \\gamma } ) = I ( \\delta _ { k + 1 } , \\delta _ { k } ) + 2 \\# ( \\mathbb { R } \\times \\gamma \\cap { v ^ { z } } ) = 2 + 2 \\# ( \\mathbb { R } \\times \\gamma \\cap { v ^ { z } } ) . \\end{align*}"}
{"id": "7501.png", "formula": "\\begin{align*} & d _ { a } ^ { n } : = \\frac { \\big \\| \\tilde { \\phi } ^ { n + 1 } - 2 \\phi ^ { n } + \\phi ^ { n - 1 } \\big \\| _ { \\infty } } { \\tau ^ 2 } , n \\geq 1 . \\end{align*}"}
{"id": "5336.png", "formula": "\\begin{align*} ( u , \\eta , \\tau ) \\big | _ { t = 0 } = ( u _ 0 , \\eta _ 0 , \\tau _ 0 ) . \\end{align*}"}
{"id": "5579.png", "formula": "\\begin{align*} \\sum _ { n \\in D } w ( n ) | y _ n | & \\ = \\ \\sum _ { n \\in D \\cap ( A \\cup B ) ^ c } w ( n ) | x _ n | + t \\sum _ { n \\in D \\cap B } w ( n ) \\\\ & \\ \\leqslant \\ \\sum _ { n \\in D \\cap ( A \\cup B ) ^ c } w ( n ) | x _ n | + t \\sum _ { n \\in E } 1 \\\\ & \\ = \\ \\sum _ { n \\in G } w ( n ) | z _ n | \\ \\leqslant \\ \\| z \\| _ \\omega . \\end{align*}"}
{"id": "9470.png", "formula": "\\begin{align*} \\phi : T \\rightarrow k [ H ] , t _ i \\mapsto Y ^ { a _ i } i = 1 , \\ldots , m . \\end{align*}"}
{"id": "8096.png", "formula": "\\begin{align*} \\mathrm { W F } ( \\overline { u } _ n ) = \\left \\{ \\begin{array} { l @ { \\quad : \\quad } l } \\{ 0 \\} \\times \\mathbb { R } _ { < 0 } & \\alpha = i \\pi \\\\ \\{ 0 \\} \\times \\mathbb { R } _ { > 0 } & \\alpha = - i \\pi \\\\ \\{ 0 \\} \\times \\mathbb { R } \\setminus \\{ 0 \\} & \\end{array} \\right . \\end{align*}"}
{"id": "6706.png", "formula": "\\begin{align*} \\det \\Phi ( \\xi ^ { ( - i ) } ) \\neq 0 \\end{align*}"}
{"id": "7098.png", "formula": "\\begin{align*} & ( j - 1 ) ^ 2 + \\cdots + 2 ^ 2 + 1 ^ 2 + 1 ^ 2 + 2 ^ 2 + \\cdots + ( r _ i - j ) ^ 2 \\\\ = & \\frac { j ( j - 1 ) ( 2 j - 1 ) } { 6 } + \\frac { ( r _ i - j ) ( r _ i - j + 1 ) ( 2 r _ i - 2 j + 1 ) } { 6 } \\\\ = & \\frac { 1 } { 6 } [ 2 j ^ 3 - 3 j ^ 2 + j + 2 r ^ 3 _ i + 3 r ^ 2 _ i - 6 r ^ 2 _ i j + 6 r _ i j ^ 2 - 6 r _ i j + r _ i - 2 j ^ 3 + 3 j ^ 2 - j ] \\\\ = & \\frac { 1 } { 6 } \\left [ 2 r ^ 3 _ i + 3 r ^ 2 _ i + r _ i - 6 r ^ 2 _ i j + 6 r _ i j ^ 2 - 6 r _ i j \\right ] \\ \\end{align*}"}
{"id": "1087.png", "formula": "\\begin{align*} g ( k , \\xi ) : = ( 4 k ^ 2 + 1 2 \\xi + 2 C _ { L } ^ 2 ) X _ { L } ( k ) , X _ { L } ( k ) = \\sqrt { k ^ 2 - C _ { L } ^ 2 } . \\end{align*}"}
{"id": "1088.png", "formula": "\\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) e ^ { i t [ g ( k , \\xi ) - \\theta ( k , \\xi ) ] \\sigma _ 3 } . \\end{align*}"}
{"id": "8753.png", "formula": "\\begin{align*} \\mathcal { H } : = \\Biggl \\{ \\prod _ { i = 1 } ^ d \\bigl [ a _ { i \\tau ( i , t _ i - 1 ) } , a _ { i \\tau ( i , t _ i ) } \\bigr ] \\Biggm | t = ( t _ 1 , \\ldots , t _ d ) \\in \\prod _ { i = 1 } ^ d \\{ 1 , \\ldots , l _ i \\} \\Biggr \\} . \\end{align*}"}
{"id": "1968.png", "formula": "\\begin{align*} \\phi * \\psi = \\phi \\succ \\psi + \\phi \\prec \\psi . \\end{align*}"}
{"id": "6969.png", "formula": "\\begin{align*} \\forall x \\in \\R \\colon \\Phi ( x ) : = \\begin{cases} \\R & x \\leq 0 , \\\\ [ x ^ 2 , \\infty ) & x > 0 \\end{cases} \\end{align*}"}
{"id": "7934.png", "formula": "\\begin{align*} B _ \\ell ( n + 1 ) - \\frac { \\ell } { \\ell + 1 } \\frac { B _ { \\ell + 1 } ( n + 1 ) } { n } = \\frac { 1 } { \\ell + 1 } n ^ { \\ell } + O _ \\ell \\left ( n ^ { \\ell - 1 } \\right ) \\end{align*}"}
{"id": "5869.png", "formula": "\\begin{align*} Z = \\frac { 1 } { \\lambda _ 1 } + \\frac { \\rho _ 1 } { \\lambda _ 2 } + \\frac { \\rho _ 1 \\rho _ 2 } { \\lambda _ 3 } + \\frac { \\rho _ 1 \\rho _ 2 \\rho _ 3 } { \\lambda _ 0 } . \\end{align*}"}
{"id": "9299.png", "formula": "\\begin{align*} \\Sigma = \\bigsqcup _ { g \\in ( \\mathbb { Z } / l \\mathbb { Z } ) } g ( \\tilde { p } ) . \\end{align*}"}
{"id": "2684.png", "formula": "\\begin{align*} \\sigma ( z ) = z \\prod _ { \\gamma \\in \\Gamma \\backslash 0 } \\left ( 1 - \\frac { z } { \\gamma } \\right ) e ^ { \\frac { z } { \\gamma } + \\frac { z ^ 2 } { 2 \\gamma ^ 2 } } . \\end{align*}"}
{"id": "3702.png", "formula": "\\begin{align*} & \\| B \\| _ { L ^ { \\infty } ( 0 , T ; H ^ { \\frac 5 2 - \\alpha } ) } \\\\ \\geq & \\left ( \\sum _ { q \\geq 0 } \\lambda _ q ^ { 2 ( \\frac 5 2 - \\alpha ) } e ^ { - 2 \\mu \\lambda _ q ^ \\alpha t } \\| B _ q ( 0 ) \\| ^ 2 _ { L ^ 2 } \\right ) ^ { \\frac 1 2 } - C \\| B \\| ^ 2 _ { L ^ { 2 } ( 0 , T ; H ^ { \\frac 5 2 - \\frac { \\alpha } { 2 } } ) } . \\end{align*}"}
{"id": "6397.png", "formula": "\\begin{align*} \\Lambda = \\{ ( s _ 1 , \\dots , s _ { n - 1 } , p ) \\in \\mathbb G _ n \\ , : \\ , f _ i ( s _ 1 , \\dots , s _ { n - 1 } , p ) = 0 \\ , , \\ , 1 \\leq i \\leq 1 \\} . \\end{align*}"}
{"id": "4504.png", "formula": "\\begin{align*} \\Sigma _ { a } ^ { b } = 2 \\sum _ { \\substack { H _ a \\le \\gamma < T \\\\ a \\le \\beta < b } } \\frac { x ^ { \\beta - 1 } } { \\gamma } . \\end{align*}"}
{"id": "3150.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { z } \\Phi & = \\tau + R _ 1 ( T b - \\kappa \\tau ) - R _ 2 T n + \\mathcal O ( R _ 1 ^ 2 , R _ 2 ^ 2 ) , \\\\ \\partial _ { R _ 1 } \\Phi & = n + \\mathcal O ( R _ 1 , R _ 2 ) , \\\\ \\partial _ { R _ 2 } \\Phi & = b + \\mathcal O ( R _ 1 , R _ 2 ) , \\end{cases} \\end{align*}"}
{"id": "4672.png", "formula": "\\begin{align*} & \\int _ { y < ( x _ 1 + x _ 2 ) / 2 } R _ 1 R _ 2 ^ p = \\kappa _ 0 \\int _ { y < ( x _ 1 + x _ 2 ) / 2 } \\frac { Q ^ p ( y - x _ 2 ) } { ( y - x _ 1 ) ^ 2 } \\ , \\dd y + O \\bigg ( \\int _ { y < ( x _ 1 + x _ 2 ) / 2 } \\frac { Q ^ p ( y - x _ 2 ) } { ( y - x _ 1 ) ^ 4 } \\ , \\dd y \\bigg ) \\\\ & = \\kappa _ 0 \\int _ { y < ( x _ 1 + x _ 2 ) / 2 } \\frac { Q ^ p ( y - x _ 2 ) } { ( y - x _ 1 ) ^ 2 } \\ , \\dd y + O \\bigg ( \\frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "7175.png", "formula": "\\begin{align*} \\partial _ s h _ { V _ \\ast } + ( v - V _ \\ast ) \\cdot \\nabla _ x h _ { V _ \\ast } - \\nabla ( \\phi * _ x \\rho [ h _ { V _ \\ast } ] ) \\cdot \\nabla _ v \\mu & = - e _ 0 \\nabla \\Phi ( x ) \\cdot \\nabla _ v \\mu , h _ { V _ \\ast } ( 0 , \\cdot ) = 0 . \\end{align*}"}
{"id": "6997.png", "formula": "\\begin{align*} A ( z ) R ( z ) + B ( z ) S ( z ) = 1 \\mbox { f o r a l l $ z \\in \\C $ . } \\end{align*}"}
{"id": "3283.png", "formula": "\\begin{align*} \\mu ( \\{ \\lambda : | \\lambda - z _ 0 | \\leq r \\} ) = - r \\frac { d } { d r } L ( U ^ \\mu ; z _ 0 , r ) . \\end{align*}"}
{"id": "4266.png", "formula": "\\begin{align*} - 4 \\delta _ 0 ~ \\leq ~ a _ n ( t , 0 + ) ~ = ~ - { \\sigma _ n ( t ) \\over 2 } ~ \\leq ~ - 2 \\delta _ 0 , 2 \\delta _ 0 ~ \\leq ~ a _ n ( t , 0 - ) ~ = ~ { \\sigma _ n ( t ) \\over 2 } ~ \\leq ~ 4 \\delta _ 0 . \\end{align*}"}
{"id": "9105.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } t ^ n \\sum _ { \\lambda \\in \\mathcal { P } ( n ) } L ( \\lambda ) = \\left ( \\sum _ { n \\geq 1 } \\frac { a _ n t ^ { n } } { 1 - t ^ { n } } \\right ) \\prod _ { n \\geq 1 } \\frac { 1 } { ( 1 - t ^ n ) } . \\end{align*}"}
{"id": "3202.png", "formula": "\\begin{align*} \\boldsymbol { \\Phi } = { \\rm { d i a g } } ( e ^ { j \\phi _ 1 } , e ^ { j \\phi _ 2 } , \\dots , e ^ { j \\phi _ N } ) \\end{align*}"}
{"id": "4325.png", "formula": "\\begin{align*} \\int ( w ( \\tau _ 0 ) - Q _ b ) \\phi _ { 1 , b , \\beta } \\rho _ \\beta d y = \\varepsilon _ 1 ( \\tau _ 0 ) \\| \\phi _ { 1 , b , \\beta } \\| ^ 2 . \\end{align*}"}
{"id": "3107.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + z [ a ( x ^ 4 + y ^ 4 ) + b x ^ 2 y ^ 2 ] + x ( x ^ 4 - y ^ 4 ) = 0 \\ , . \\end{align*}"}
{"id": "8536.png", "formula": "\\begin{align*} H _ n : = - \\partial _ { x x } - n ( n + 1 ) ^ 2 ( x ) , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
{"id": "5436.png", "formula": "\\begin{align*} u _ k = c _ { 1 , k } y _ k + c _ { 2 , k } y _ { k - 1 } + c _ { 3 , k } u _ { k - 1 } + c _ { 4 , k } r _ k , \\end{align*}"}
{"id": "839.png", "formula": "\\begin{align*} \\ell _ \\rho ( \\gamma \\vert _ { [ 0 , t ] } ) = \\int _ 0 ^ t \\rho ( y + s ) \\ , d s = \\begin{cases} \\frac { 1 } { \\beta - 1 } \\left [ \\frac { 1 } { y ^ { \\beta - 1 } } - \\frac { 1 } { ( y + t ) ^ { \\beta - 1 } } \\right ] & y \\ge 1 , \\\\ 1 - y + \\frac { 1 } { \\beta - 1 } \\left [ 1 - \\frac { 1 } { ( t + y ) ^ { \\beta - 1 } } \\right ] & y < 1 \\le y + t , \\\\ t & y + t < 1 . \\end{cases} \\end{align*}"}
{"id": "1936.png", "formula": "\\begin{align*} g ( x _ 0 ) - \\underline { g } & \\geq \\sum _ { k = 0 } ^ K g ( x _ k ) - g ( x _ { k + 1 } ) \\\\ & \\geq K \\min \\left ( c _ 1 \\tau _ 1 \\min ( \\underline { \\alpha _ 1 } , \\underline { t _ 1 } ) \\varepsilon _ 1 ^ 2 , c _ 2 \\tau _ 2 ^ 2 \\min \\left ( \\min \\left ( \\alpha _ { 0 2 } , \\dfrac { 3 | 2 c _ 2 - 1 | \\varepsilon _ 2 } { M _ g } \\right ) , \\underline { t _ 2 } \\right ) ^ 2 \\varepsilon _ 2 \\right ) . \\end{align*}"}
{"id": "5353.png", "formula": "\\begin{align*} H _ { i } ^ + = \\left \\lbrace x \\in \\mathbb { R } ^ n \\middle | \\left \\langle \\tilde a _ i , x \\right \\rangle \\leqslant b _ i \\right \\rbrace . \\end{align*}"}
{"id": "6166.png", "formula": "\\begin{align*} f _ 2 ( x , y ) = 3 ( ( 1 5 y ^ 2 + 1 5 y + 5 ) x ^ 2 + ( 1 5 y ^ 2 + 1 4 y + 4 ) x + ( 5 y ^ 2 + 4 y + 1 ) ) . \\end{align*}"}
{"id": "8614.png", "formula": "\\begin{align*} A _ 1 ( k ) : = & \\int _ { 0 } ^ { t } i s k e ^ { - i s k ^ 2 } \\iiint u ^ { \\# } ( \\ell ) \\overline { u ^ { \\# } } ( n ) u ^ { \\# } ( m ) \\ , \\mu ^ { \\# , ( 1 ) } _ { R , 1 } \\left ( k , \\ell , n , m \\right ) \\ , d \\ell d m d n \\ , d s , \\\\ A _ 2 ( k ) : = & \\int _ { 0 } ^ { t } e ^ { - i s k ^ 2 } \\iiint u ^ { \\# } ( \\ell ) \\overline { u ^ { \\# } } ( n ) u ^ { \\# } ( m ) \\ , \\partial _ { k } \\mu ^ { \\# , ( 1 ) } _ { R , 1 } \\left ( k , \\ell , n , m \\right ) \\ , d \\ell d m d n \\ , d s , \\end{align*}"}
{"id": "5824.png", "formula": "\\begin{align*} I _ H ( w ) = w + [ 0 , \\lambda H ) \\times \\{ 0 \\} . \\end{align*}"}
{"id": "2339.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k \\in \\Z } f ( k ) \\frac { \\sin ( \\pi ( t - k ) ) } { \\pi ( t - k ) } , t \\in \\R . \\end{align*}"}
{"id": "3733.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ 1 } B J _ { x x } B _ x \\ , d x = - \\int _ { \\mathbb S ^ 1 } B _ x J _ { x } B _ x \\ , d x - \\int _ { \\mathbb S ^ 1 } B J _ { x } B _ { x x } \\ , d x , \\end{align*}"}
{"id": "7651.png", "formula": "\\begin{align*} \\forall \\ , R > 0 \\ ; : \\ ; \\lim _ { k \\to + \\infty } \\sup _ { \\mathbf { y } \\in \\R ^ N } \\int _ { \\mathbf { y } + B _ R ( \\mathbf { y } ) } \\rho _ { n _ k } = 0 \\ ; , \\end{align*}"}
{"id": "9075.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\rho _ 1 = & \\partial _ x \\left ( \\partial _ x \\rho _ 1 + \\rho _ 1 \\partial _ x \\phi \\right ) , \\\\ \\partial _ t \\rho _ 2 = & \\partial _ x \\left ( \\partial _ x \\rho _ 2 - 2 \\rho _ 2 \\partial _ x \\phi \\right ) , \\\\ - \\partial _ x ^ 2 \\phi = & 1 2 ( x - 0 . 5 ) ^ 2 + \\rho _ 1 - 2 \\rho _ 2 . \\end{aligned} \\end{align*}"}
{"id": "3246.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } \\frac { t } { s ( r , t ) } = \\frac { \\lambda _ 1 ( \\mu ) ^ 2 - r ^ 2 } { \\lambda _ 1 ( \\mu ) ^ 2 } . \\end{align*}"}
{"id": "5639.png", "formula": "\\begin{align*} \\Pi ^ \\mathcal { F } _ \\star ( X , Y ) = - \\frac { 1 } { 2 c } \\ , \\mathbf { g } _ { t \\star } ( X , Y ) \\ , V _ \\perp = - \\frac { 1 } { 2 c } \\ , \\mathbf { g } _ { t \\star } ( X , Y ) \\star V _ \\perp \\end{align*}"}
{"id": "742.png", "formula": "\\begin{align*} D _ { z , A } : = \\prod _ { i = 1 } ^ m D _ { z ; A ^ { ( i ) } } , \\xi _ { B } : = \\prod _ { i = 1 } ^ m \\prod _ { j = 1 } ^ q \\xi _ { i ; \\beta _ j ^ { ( i ) } } , \\Delta _ { A B } ^ { ( \\ell + 1 ) } : = \\prod _ { j = 1 } ^ q \\Delta _ { \\alpha _ j ^ { ( i ) } \\beta _ j ^ { ( i ) } } ^ { ( \\ell + 1 ) } . \\end{align*}"}
{"id": "2328.png", "formula": "\\begin{align*} \\mathcal { T } _ s ^ { - 1 } F ( x , t ) = F ( \\tfrac { x + t } { 2 } , x - t ) . \\end{align*}"}
{"id": "8588.png", "formula": "\\begin{align*} { \\big \\| e ^ { i t H } h \\big \\| } _ { L ^ \\infty _ x } = \\left \\Vert \\int \\mathcal { K } ^ { \\# } ( x , k ) e ^ { i k ^ 2 t } h ^ { \\# } ( k ) \\ , d k \\right \\Vert _ { L ^ \\infty _ x } \\lesssim \\frac { 1 } { \\sqrt { t } } { \\big \\| h ^ \\# \\big \\| } _ { L ^ \\infty _ k } + \\frac { 1 } { t ^ { \\frac { 3 } { 4 } } } { \\big \\| \\partial _ k h ^ \\# \\big \\| } _ { L ^ 2 _ k } . \\end{align*}"}
{"id": "648.png", "formula": "\\begin{align*} f _ \\R ( k ) \\ = \\ \\zeta ( k + 2 ) ( k = 0 , 1 , \\ldots ) . \\end{align*}"}
{"id": "4956.png", "formula": "\\begin{align*} \\Phi ' _ i = h ' _ i \\cdot \\Psi ' _ i \\cdot \\phi ( h ' _ i ) ^ { - 1 } \\end{align*}"}
{"id": "132.png", "formula": "\\begin{align*} a ^ { n + 1 } = \\gamma _ 1 a + \\gamma _ 2 a ^ 2 + \\dots + \\gamma _ { n } a ^ { n } \\end{align*}"}
{"id": "4976.png", "formula": "\\begin{align*} E & = O ( \\epsilon ) \\\\ F & = O ( \\epsilon ) \\\\ G & = 6 a _ { 0 } ^ { ( 1 ) } ( b _ { 0 } ( 0 ) ) + O ( \\epsilon ) \\\\ H & = O ( \\epsilon ) \\end{align*}"}
{"id": "47.png", "formula": "\\begin{align*} f ( w , x , y ) = \\frac 1 2 \\left ( \\frac { 1 - \\left | w \\right | ^ 2 } { 1 + \\left | w \\right | ^ 2 } \\right ) ^ 2 , \\end{align*}"}
{"id": "374.png", "formula": "\\begin{align*} \\mathcal A ( \\rho , m ) = \\frac 1 2 \\int _ 0 ^ 1 \\Big ( \\frac { m _ { 1 2 } ^ 2 } { \\theta _ { 1 2 } ( \\rho ) } + \\frac { m _ { 2 3 } ^ 2 } { \\theta _ { 2 3 } ( \\rho ) } \\Big ) d t , \\end{align*}"}
{"id": "5530.png", "formula": "\\begin{align*} Y _ m ^ j \\leq \\sum _ { k = 1 } ^ m | \\eta _ m ^ { j , k } | \\end{align*}"}
{"id": "3617.png", "formula": "\\begin{align*} 1 = T _ { j } = R _ { j + 1 } - R _ { n + 1 } + d ( - a _ { j } a _ { j + 1 } ) = R _ { j + 1 } + d ( - a _ { j } a _ { j + 1 } ) \\ , , \\end{align*}"}
{"id": "3969.png", "formula": "\\begin{align*} \\Phi _ { \\widehat \\lambda _ n } : = ( \\xi _ { \\widehat \\lambda _ n } , \\eta _ { \\widehat \\lambda _ n } ) , \\ 1 \\leq n \\leq n _ 0 . \\end{align*}"}
{"id": "7587.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k , l ) } | & \\le \\dfrac { 1 } { 4 } \\left ( \\dfrac { 1 } { i } ( | c _ { i - 1 , j } ^ { ( k + 1 , l ) } | + | c _ { i + 1 , j } ^ { ( k + 1 , l ) } | ) + \\dfrac { 1 } { j } ( | c _ { i , j - 1 } ^ { ( k , l + 1 ) } | + | c _ { i , j + 1 } ^ { ( k , l + 1 ) } | ) \\right ) \\\\ & \\le \\dfrac { 1 } { 4 } \\left ( \\dfrac { 8 V _ { k , l } } { \\pi ^ 2 i j } + \\dfrac { 8 V _ { k , l } } { \\pi ^ 2 i j } \\right ) = \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 i j } , ~ ~ ~ ~ i , j \\ge 1 . \\end{align*}"}
{"id": "872.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\{ \\varsigma _ { Q _ j , m } = 1 , \\varsigma _ { Q _ j , i } = 1 \\right \\} \\bigcap _ { r \\in [ i - 1 ] } \\left \\{ \\varsigma _ { Q _ j , r } = 0 \\right \\} \\\\ & = \\left \\{ \\varsigma _ { Q _ j , i } = 1 , \\varsigma _ { Q _ j , i - 1 } = 0 \\right \\} \\\\ & = \\left \\{ \\varsigma _ { Q _ j , i } = 1 \\right \\} / \\left \\{ \\varsigma _ { Q _ j , i - 1 } = 1 \\right \\} . \\end{aligned} \\end{align*}"}
{"id": "1766.png", "formula": "\\begin{align*} b \\Phi ( a _ 0 , a _ 1 ) = \\Phi ( a _ 0 a _ 1 ) - \\Phi ( a _ 1 a _ 0 ) . \\end{align*}"}
{"id": "4566.png", "formula": "\\begin{align*} \\Bigg | \\ln \\frac { \\mathbf { P } \\left ( S _ n / ( \\sqrt { n } \\sigma ) > x \\right ) } { 1 - \\Phi \\left ( x \\right ) } \\Bigg | = O \\bigg ( \\frac { x ^ 3 } { \\sqrt { n } } \\bigg ) , \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "5464.png", "formula": "\\begin{align*} | \\Omega | & = \\int _ \\Omega u ^ { \\frac { p } { p + 1 } } ( t , x ; s , u _ 0 ) u ^ { - \\frac { p } { p + 1 } } ( t , x ; s , u _ 0 ) d x \\\\ & \\le \\big ( \\int _ \\Omega u ( t , x ; s , u _ 0 ) d x \\big ) ^ { \\frac { p } { p + 1 } } \\big ( \\int _ \\Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x \\big ) ^ { \\frac { 1 } { p + 1 } } \\quad \\forall \\ , t > s . \\end{align*}"}
{"id": "4238.png", "formula": "\\begin{align*} \\frac { \\sqrt { D } } { 2 a ( a ^ 2 + c ^ 2 ) } = { } & \\frac { \\sqrt { D } } { 2 a ^ 3 + ( 2 a ) ^ { - 1 } ( D + b ^ 2 ) } \\\\ = { } & K _ E ^ { - 1 } \\left ( \\frac { \\pi ^ { 2 / 3 } } { K _ E ^ { 4 / 3 } A ^ { 2 / 3 } } + 1 + \\frac { ( \\partial _ s K _ E ) ^ 2 } { 9 K _ E ^ 4 } \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "2422.png", "formula": "\\begin{align*} \\norm { c } _ { \\ell ^ 2 } ^ 2 & = \\norm { c - a + a } _ { \\ell ^ 2 } ^ 2 \\\\ & = \\norm { c - a } _ { \\ell ^ 2 } ^ 2 + \\norm { a } _ { \\ell ^ 2 } ^ 2 + \\langle c - a , a \\rangle _ { \\ell ^ 2 } + \\langle a , c - a \\rangle _ { \\ell ^ 2 } \\\\ & = \\norm { c - a } _ { \\ell ^ 2 } ^ 2 + \\norm { a } _ { \\ell ^ 2 } ^ 2 \\geq \\norm { a } _ { \\ell ^ 2 } ^ 2 , \\end{align*}"}
{"id": "5448.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| u ( t , \\cdot ; s , u _ n , a _ n , b _ n ) - u ( t , \\cdot ; s , u _ 0 , a _ 0 , b _ 0 ) \\| _ \\infty = 0 \\ , \\ , { \\rm u n i f o r m l y \\ , \\ , i n } \\ , \\ , t \\in [ s , s + T ] \\end{align*}"}
{"id": "4611.png", "formula": "\\begin{gather*} \\dim Q / X ^ + = d - \\ell ^ + - 2 = \\ell ^ - , \\\\ \\dim Q / X ^ - = d - \\ell ^ - - 2 = \\ell ^ + . \\end{gather*}"}
{"id": "5500.png", "formula": "\\begin{align*} \\alpha ( t , x ) = a ( x ) , ( t , x ) \\in [ r , u ) \\times X . \\end{align*}"}
{"id": "7603.png", "formula": "\\begin{align*} \\langle X , \\nabla H \\rangle = - \\frac { 1 } { \\alpha } \\langle X , \\nu \\rangle ^ { - \\frac { \\alpha + 1 } { \\alpha } } h _ { i j } \\langle X , e _ { i } \\rangle \\langle X , e _ { j } \\rangle \\leq 0 , \\end{align*}"}
{"id": "6189.png", "formula": "\\begin{align*} \\begin{aligned} \\| C \\hat { V } \\| _ F ^ 2 \\geq \\sum ^ { l } _ { t = 1 } \\bar { \\sigma } ^ 2 _ { t } - \\frac { 2 \\theta \\| S \\| ^ 2 _ F } { \\sqrt { \\alpha } } - \\theta ( k + \\sqrt { k } \\xi ) \\| C \\| ^ 2 _ F . \\end{aligned} \\end{align*}"}
{"id": "8338.png", "formula": "\\begin{align*} \\int _ a ^ b f \\dd g = 0 \\forall f \\in G [ a , b ] . \\end{align*}"}
{"id": "3396.png", "formula": "\\begin{align*} f = \\sum \\limits _ { k = - \\infty } ^ { \\infty } T _ { N } ^ { - 1 } D _ { k } ^ { N } D _ { k } ( f ) = \\sum \\limits _ { k = - \\infty } ^ { \\infty } D _ { k } D _ { k } ^ { N } T _ { N } ^ { - 1 } ( f ) \\end{align*}"}
{"id": "5553.png", "formula": "\\begin{align*} d _ K ( h ) = \\inf _ { g \\in K } \\| h - g \\| = \\eta . \\end{align*}"}
{"id": "1674.png", "formula": "\\begin{align*} \\overline { H } _ n ( \\mathbf { s } ) = \\overline { H } _ n ( s _ 1 , s _ 2 , \\ldots , s _ r ) : = \\sum \\limits _ { 0 \\leq k _ 1 < k _ 2 < \\cdots < k _ r \\leq n - 1 } \\prod _ { j = 1 } ^ { r } \\frac { 1 } { ( 2 k _ j + 1 ) ^ { s _ j } } , \\end{align*}"}
{"id": "2227.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\widetilde { e } ^ { n , m } ( t ) + A ^ 2 \\widetilde { e } ^ { n , m } ( t ) = - P _ n A P ( F ( X ^ m ( t ) ) - F ( X ^ { n } ( t ) ) ) . \\end{align*}"}
{"id": "7489.png", "formula": "\\begin{align*} & \\chi ^ 0 = \\chi _ 0 : = \\frac { \\int _ { \\mathbb { R } ^ d } \\left ( \\frac { 1 } { 2 } | \\nabla \\phi _ 0 | ^ 2 + V | \\phi _ 0 | ^ 2 + \\beta | \\phi _ 0 | ^ 4 - \\Omega \\overline { \\phi } _ 0 L _ z \\phi _ 0 \\right ) \\mathrm { d } \\mathbf { x } } { \\| \\phi _ 0 \\| ^ 2 } + \\sigma R ^ \\prime ( \\| \\phi _ 0 \\| ^ 2 - 1 ) . \\end{align*}"}
{"id": "3023.png", "formula": "\\begin{align*} n ^ 0 _ \\beta = \\left ( - 1 \\right ) ^ { \\dim \\left ( M _ \\beta \\right ) } e \\left ( M _ \\beta \\right ) , \\end{align*}"}
{"id": "2251.png", "formula": "\\begin{align*} \\Psi _ k ^ { M , N } ( t ) : = E ( t ) - E _ { k , N } ^ m , \\ ; \\ ; \\ ; t \\in [ t _ { m - 1 } , t _ m ) , \\ ; m \\in \\{ 1 , 2 , \\cdots , M \\} . \\end{align*}"}
{"id": "7473.png", "formula": "\\begin{align*} & \\frac { \\tilde { \\phi } ^ { n + 1 } - 2 \\phi ^ n + \\phi ^ { n - 1 } } { \\tau ^ 2 } + \\eta ^ n \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ { n - 1 } } { 2 \\tau } = - G ( \\phi ^ n ) + \\lambda ^ n \\phi ^ n , \\mathbf { x } \\in \\mathbb { R } ^ d , \\\\ & \\phi ^ { n + 1 } = \\tilde { \\phi } ^ { n + 1 } / \\| \\tilde { \\phi } ^ { n + 1 } \\| , n \\geq 1 . \\end{align*}"}
{"id": "4300.png", "formula": "\\begin{align*} \\partial _ \\tau w = \\partial _ y ^ 2 w + \\frac { d + 1 } { y } \\partial _ y w - \\beta ( \\tau ) \\Lambda _ y w - 3 ( d - 2 ) w ^ 2 - ( d - 2 ) y ^ 2 w ^ 3 , \\end{align*}"}
{"id": "2363.png", "formula": "\\begin{align*} F ( x , \\omega ) = e ^ { 2 \\pi i x \\cdot \\omega } V _ g f ( x , \\omega ) \\ , V _ g f ( - x , - \\omega ) \\end{align*}"}
{"id": "226.png", "formula": "\\begin{align*} \\dfrac { ( D ^ { \\alpha - 1 } ) ^ * ( p _ \\alpha f ) ( x ) } { p _ \\alpha ( x ) } = ( D ^ { \\alpha - 1 } ) ^ { * } ( f ) ( x ) + x f ( x ) + \\frac { 1 } { p _ \\alpha ( x ) } ( R ^ { \\alpha } ) ^ * ( p _ \\alpha , f ) ( x ) , \\end{align*}"}
{"id": "4236.png", "formula": "\\begin{align*} u _ y ( 0 , 0 ) \\le w _ y ( 0 , 0 ) = \\frac { \\sqrt { D } } { 2 a ( a ^ 2 + c ^ 2 ) } . \\end{align*}"}
{"id": "9468.png", "formula": "\\begin{align*} N _ { 2 j } + L _ { 2 j } \\cdot \\tfrac { C ( j , 2 ) } { C ( j ) } + \\tfrac { C ( j , 4 ) ^ p } { C ( j ) ^ p } - \\tfrac { C ( j - 1 , 4 ) } { C ( j - 1 ) } = 0 . \\end{align*}"}
{"id": "2372.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 d } } | k ( x , t ) | ^ 2 \\ , d ( x , t ) = | \\Omega | | T | . \\end{align*}"}
{"id": "5045.png", "formula": "\\begin{align*} \\Psi _ { Q + i P , \\nu , \\tilde { \\nu } } = { \\bf L } _ { - \\nu _ k } \\dots { \\bf L } _ { - \\nu _ 1 } \\tilde { \\bf L } _ { - \\tilde { \\nu } _ { k ' } } \\dots \\tilde { \\bf L } _ { - \\tilde { \\nu } _ 1 } \\Psi _ { Q + i P } \\end{align*}"}
{"id": "3541.png", "formula": "\\begin{align*} [ V _ 2 ] [ V _ 4 ] = \\left ( \\begin{matrix} 1 & 0 & t _ 1 & a + c t _ 1 + d t _ 3 & t _ 3 \\\\ 0 & 1 & t _ 2 & b + c t _ 2 + d t _ 4 & t _ 4 \\end{matrix} \\right ) . \\end{align*}"}
{"id": "3647.png", "formula": "\\begin{align*} \\frac { } { t } \\left ( \\frac { x ^ { - \\nu _ 2 ( t ) } } { t } \\right ) = \\frac { x ^ { - \\nu _ 2 ( t ) } \\log x } { R _ 1 t ^ 2 } \\left ( \\frac { D ( 1 - 2 \\log \\log t ) + \\log t } { \\log ^ 3 t } - \\frac { R _ 1 } { \\log x } \\right ) . \\end{align*}"}
{"id": "3819.png", "formula": "\\begin{align*} f ( \\lambda ^ { w _ 1 } x _ 1 , \\ldots , \\lambda ^ { w _ n } x _ n ) = \\lambda ^ d f ( x _ 1 , \\ldots , x _ n ) \\end{align*}"}
{"id": "8001.png", "formula": "\\begin{align*} S ( \\mathbf { P } _ { \\ ! \\sigma } ) = S ( \\mathbf { p } _ { \\sigma _ d } ; \\ldots ; \\mathbf { p } _ { \\sigma _ 1 } ) = - \\sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\int \\ ! \\log \\boldsymbol { \\mu } _ n ^ { \\sigma } \\ , \\mathrm { d } \\mathbf { p } _ { \\sigma _ n } . \\end{align*}"}
{"id": "2580.png", "formula": "\\begin{align*} f ( t ) = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x g ( t ) \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "8261.png", "formula": "\\begin{align*} \\mathfrak { B } _ 1 \\times \\mathfrak { B } _ { 2 , \\{ 1 , 2 \\} } = [ 1 \\bar { 3 } \\ , \\bar { 2 } , \\bar 1 \\ , \\bar 2 \\ , \\bar 3 ] . \\end{align*}"}
{"id": "1662.png", "formula": "\\begin{align*} z : = x - x ' , w : = v - v ' , q : = z + \\alpha ^ { - 1 } w , r ( { \\bf y } ) : = \\alpha _ 0 | z | + | q | . \\end{align*}"}
{"id": "1614.png", "formula": "\\begin{align*} f ^ j ( x ) & : = \\sum _ { Q \\in \\mathcal { Q } } \\psi _ Q ( x ) | g _ j ^ Q | ^ \\frac { 1 } { p } , \\\\ h ^ j ( x ) & : = \\sum _ { Q \\in \\mathcal { Q } } \\mathbb { 1 } _ Q ( x ) \\Theta ^ { Q , j } ( \\lambda x ) . \\end{align*}"}
{"id": "6750.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} v ( x ) , & x + \\delta > 0 , \\\\ u ( x ) , & x + \\delta \\leq 0 , \\end{cases} \\end{align*}"}
{"id": "6770.png", "formula": "\\begin{align*} - f '' + f + 1 - \\rho \\max ( f ^ * - f , 0 ) = 0 , \\end{align*}"}
{"id": "7718.png", "formula": "\\begin{align*} \\Box y = 2 \\langle \\Box \\phi , \\phi \\rangle - 2 \\left ( | \\phi _ t | ^ 2 - | \\phi _ x | ^ 2 \\right ) = 2 \\left ( | \\phi _ t | ^ 2 - | \\phi _ x | ^ 2 \\right ) ( y - 1 ) . \\end{align*}"}
{"id": "688.png", "formula": "\\begin{align*} n _ 1 , \\ldots , n _ L = n \\gg 1 , \\end{align*}"}
{"id": "385.png", "formula": "\\begin{align*} f ( W , b \\ ; ; \\ ; x ) = W ^ N \\sigma ^ { N - 1 } \\big ( . \\ ; . \\ ; . \\big ( W ^ 2 \\sigma ^ 1 ( W ^ 1 x - b ^ 1 ) - b ^ 2 \\big ) . \\ ; . \\ ; . \\big ) - b ^ N \\end{align*}"}
{"id": "4111.png", "formula": "\\begin{align*} \\begin{aligned} d X ^ x _ t & = \\mu ( X ^ x _ t ) \\ , d t + \\sigma ( X ^ x _ t ) \\ , d W _ t , t \\in [ 0 , \\infty ) , \\\\ X ^ x _ 0 & = x , \\end{aligned} \\end{align*}"}
{"id": "1572.png", "formula": "\\begin{align*} \\psi _ Z ( s ) = \\prod _ { h = 1 } ^ m \\prod _ { i = 1 } ^ { r _ { 2 h } } \\left ( s - i + 2 h - 1 \\right ) \\end{align*}"}
{"id": "2927.png", "formula": "\\begin{align*} \\sum \\limits _ { \\mathbf { p } _ k \\in \\mathcal { P } ( d , k ) } f ( \\mathbf { p } _ k ) & = \\sum \\limits _ { 1 \\le p _ 1 < \\cdots < p _ k \\le d } f ( \\mathbf { p } _ k ) = \\sum _ { p _ { k } = 1 } ^ { d } \\sum _ { p _ { k - 1 } = 1 } ^ { p _ k - 1 } \\cdots \\sum _ { p _ 1 = 1 } ^ { p _ 2 - 1 } f ( \\mathbf { p } _ k ) . \\end{align*}"}
{"id": "6423.png", "formula": "\\begin{align*} \\mathcal { T } ( R , L ) = \\mathcal { T } ( R + \\tau \\ , * , L ) - \\tau \\ ; \\mathcal { T } ( * , L ) \\succeq 0 , \\end{align*}"}
{"id": "5256.png", "formula": "\\begin{align*} \\varepsilon \\circ S = \\varepsilon . \\end{align*}"}
{"id": "8126.png", "formula": "\\begin{align*} \\kappa ( f ) = \\sum _ { ( d , e ) \\in D _ { 1 } ( P G ) } | f ( d ) \\cap f ( e ) | \\end{align*}"}
{"id": "1467.png", "formula": "\\begin{align*} \\Phi ' = J _ n , \\Psi ' = \\mathrm { d i a g } [ 1 , 1 , 1 , - \\alpha , - \\alpha , - \\alpha ] , \\end{align*}"}
{"id": "4951.png", "formula": "\\begin{align*} z = k ^ + \\cdot ( g _ 0 ^ { - 1 } \\cdot \\phi ( g ^ + ) \\cdot g _ 0 ) . \\end{align*}"}
{"id": "336.png", "formula": "\\begin{align*} \\dot \\rho = \\nabla _ { S } \\mathcal H ( \\rho , S ) + \\nabla _ { S } \\mathcal H _ 1 ( \\rho , S ) \\dot W ^ { \\delta } , \\ ; \\\\ \\dot S = - \\nabla _ { \\rho } \\mathcal H ( \\rho , S ) - \\nabla _ { \\rho } \\mathcal H _ 1 ( \\rho , S ) \\dot W ^ { \\delta } , \\end{align*}"}
{"id": "181.png", "formula": "\\begin{align*} \\langle \\mathcal { A } ( \\tilde { f } ) ; v \\rangle _ H = \\langle f ; v \\rangle _ H , \\end{align*}"}
{"id": "2800.png", "formula": "\\begin{align*} f _ * = - \\frac { \\lambda _ 0 } { 2 ( p - 1 ) } Q + \\sum _ { j = 1 } ^ N \\mu _ j \\left ( \\partial _ { x _ j } Q \\right ) \\end{align*}"}
{"id": "3637.png", "formula": "\\begin{align*} h ( t ) = A _ 1 t \\log ^ { - \\alpha } t \\exp \\left ( - C u ( t ) \\right ) , \\end{align*}"}
{"id": "3698.png", "formula": "\\begin{align*} \\sup _ { t \\in ( 0 , \\infty ) } t ^ { \\frac { s } { \\alpha } } \\left \\| e ^ { - \\mu t \\Lambda ^ { \\alpha } } * f \\right \\| _ { H ^ s } \\leq & \\ C ( \\alpha , s ) \\| f \\| _ { L ^ 2 } , \\\\ \\lim _ { t \\to 0 } t ^ { \\frac { s } { \\alpha } } \\left \\| e ^ { - \\mu t \\Lambda ^ { \\alpha } } * f \\right \\| _ { H ^ s } = & \\ 0 . \\end{align*}"}
{"id": "5186.png", "formula": "\\begin{align*} C _ { \\ell } ^ { a v } ( t ) = \\frac { c _ { \\ell } R _ \\ell ( t ) } { t } , \\end{align*}"}
{"id": "5428.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ 1 & = \\sqrt { \\frac { s _ { + } ( a ) s _ { + } ( c ) } { s _ { + } ( b ) } } \\\\ \\lambda _ 2 & = \\sqrt { \\frac { s _ { + } ( a ) s _ { + } ( b ) } { s _ { + } ( c ) } } \\\\ \\lambda _ 3 & = \\sqrt { \\frac { s _ { + } ( b ) s _ { + } ( c ) } { s _ { + } ( a ) } } \\end{cases} \\end{align*}"}
{"id": "7995.png", "formula": "\\begin{align*} \\dim _ { \\mathrm H } F = \\sup _ { \\mathbf { p } _ { \\sigma _ 2 } \\in \\mathcal { P } _ { 2 } ^ { \\sigma } } \\ ; t ( \\mathbf { p } _ { \\sigma _ 2 } ; \\ , ( \\mathbf { p } _ { \\sigma _ 2 } ) _ 1 ^ { \\sigma } ) , \\end{align*}"}
{"id": "2351.png", "formula": "\\begin{align*} \\langle X f , f \\rangle = \\int _ { \\R } X f ( x ) \\overline { f ( x ) } \\ , d x = \\int _ \\R x | f ( x ) | ^ 2 \\ , d x . \\end{align*}"}
{"id": "6792.png", "formula": "\\begin{align*} \\overline { M } = \\sqrt { 1 + \\| P \\| } \\overline { L } . \\end{align*}"}
{"id": "6949.png", "formula": "\\begin{align*} \\mathcal { A } ^ 1 _ m = \\mathcal { A } ^ { 1 , i n i t } _ m + \\mathcal { C } _ m + \\mathcal { D } _ m , \\end{align*}"}
{"id": "9398.png", "formula": "\\begin{align*} \\varphi ( a _ n \\cdots a _ 1 v b _ 1 \\cdots b _ m ) = \\tau ( a _ n b _ m ) \\cdots \\tau ( a _ 1 b _ 1 ) \\varphi ( v ) \\end{align*}"}
{"id": "5793.png", "formula": "\\begin{align*} \\operatorname { T o r } ^ S _ * ( k [ K ] , k ) = { H } _ * \\big ( \\operatorname { K o s } ^ { { k [ K ] } } ( v _ 1 , \\ldots , v _ m ) \\big ) \\end{align*}"}
{"id": "3360.png", "formula": "\\begin{align*} \\phi \\circ \\theta _ T ( u , v ) = \\theta _ { T ^ { ' } } ( \\psi ( u ) , \\psi ( v ) ) \\circ \\phi , \\forall u , v \\in V . \\end{align*}"}
{"id": "5878.png", "formula": "\\begin{align*} F _ j ^ i ( y ) & = \\partial _ k f ^ i ( f ^ { - 1 } ( y ) ) \\sigma ^ k _ j ( f ^ { - 1 } ( y ) ) , y \\in f ( U ) , \\\\ G ^ i ( y ) & = \\frac { 1 } { 2 } \\partial _ { j k } ^ 2 f ^ i ( f ^ { - 1 } ( y ) ) \\left ( \\ , \\sigma ^ j ( f ^ { - 1 } ( y ) ) \\cdot \\sigma ^ k ( f ^ { - 1 } ( y ) ) \\ , \\right ) , y \\in f ( U ) , \\end{align*}"}
{"id": "5965.png", "formula": "\\begin{align*} \\mathcal { A } ^ n = \\mathcal { D } _ x ^ T \\mathcal { M } ^ n \\mathcal { D } _ x + \\mathcal { D } _ z ^ T \\mathcal { M } ^ n \\mathcal { D } _ z , \\end{align*}"}
{"id": "8730.png", "formula": "\\begin{align*} \\begin{aligned} 1 + 1 \\cdot ( - 3 x _ 2 ) + ( - 3 x _ 1 ) \\cdot & ( - 3 x _ 2 + 1 ) + ( - 3 x _ 1 + 1 ) \\cdot ( - x _ 2 ^ 3 + 3 x _ 2 ^ 2 ) + ( - x _ 2 ^ 3 + 3 x _ 2 ^ 2 ) \\geq \\\\ & 0 . 5 ( - 0 . 7 5 x _ 1 + 0 . 5 ) + 0 . 5 ( 1 - x _ 1 ) ^ 3 + 0 . 5 ( - 0 . 7 5 x _ 2 + 0 . 5 ) + 0 . 5 ( 1 - x _ 2 ) ^ 3 - 0 . 7 5 . \\end{aligned} \\end{align*}"}
{"id": "1757.png", "formula": "\\begin{align*} K _ i ( \\mathcal { A } ^ \\infty _ G ( X , E ) ) \\cong K _ i ( C ^ * _ G ( X , E ) ) , \\ i = 0 , 1 . \\end{align*}"}
{"id": "5712.png", "formula": "\\begin{align*} 1 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } \\cup { \\gamma _ { 2 } } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) = 1 2 - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) \\end{align*}"}
{"id": "466.png", "formula": "\\begin{align*} | \\frac { | \\rho _ u W ( u ) \\cap W ' ( u ) | } { | \\rho _ u W ( u ) | } - \\frac { | W ' ( u ) | } { | W ( u ) ^ - | } | = O ( q ^ { - 2 ^ { - | u | - 1 } } ) \\end{align*}"}
{"id": "4992.png", "formula": "\\begin{align*} \\Pi ( A , B ) ( x , y ) = \\left ( \\begin{pmatrix} a ( x , y ) \\\\ h ( x , y ) \\end{pmatrix} , \\begin{pmatrix} b ( x , y ) + c x + d x ^ { 2 } + e x ^ { 3 } + f x ^ { 4 } \\\\ x \\end{pmatrix} \\right ) \\end{align*}"}
{"id": "1532.png", "formula": "\\begin{align*} \\phi ( \\mathfrak { g } , s ) = \\chi _ v ( \\det ( d _ p ) ) ^ { - 1 } | \\det ( d _ p ) | _ { v } ^ { - s } j ( \\mathfrak { g } , z _ 0 ) ^ { - k } | j ( \\mathfrak { g } , z _ 0 | ^ { k - s } . \\end{align*}"}
{"id": "455.png", "formula": "\\begin{align*} \\mu ( \\phi ( x , b ) ) = \\nu ' ( \\phi ( x , b ) ) \\mu ( c l _ B ( \\phi ( x , b ) ) ) \\end{align*}"}
{"id": "8895.png", "formula": "\\begin{align*} \\tilde \\psi ( \\bar z ) & = \\psi _ { \\lfloor \\max ( t _ j ) \\rfloor + 1 } ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) ) \\\\ & + \\sum _ { n = 1 } ^ { \\lfloor \\max ( t _ j ) \\rfloor } d _ { q - 2 } \\chi _ n ( ( x _ 0 , ( s _ 0 , t _ 0 ) ) , \\ldots , ( x _ { q - 1 } , ( s _ { q - 1 } , t _ { q - 1 } ) ) \\end{align*}"}
{"id": "4943.png", "formula": "\\begin{align*} g ^ { - 1 } b \\phi ( g ) = k ^ o _ 1 \\omega k . \\end{align*}"}
{"id": "4661.png", "formula": "\\begin{align*} f _ c ( y ) = c ^ { \\frac { 1 } { p - 1 } } f ( c y ) . \\end{align*}"}
{"id": "398.png", "formula": "\\begin{align*} \\| \\mathcal { T } ( V ^ { m } ) - \\mathcal { T } ( V ^ { n } ) \\| _ { y } & \\leq \\frac { \\beta _ { 0 } } { 2 ^ { [ \\tfrac { n } { 2 } ] } } \\left ( \\frac { 1 } { 2 ^ { [ \\tfrac { m - 1 } { 2 } ] - [ \\tfrac { n } { 2 } ] } } + \\frac { 1 } { 2 ^ { [ \\tfrac { m - 2 } { 2 } ] - [ \\tfrac { n } { 2 } ] } } + \\cdots + \\frac { 1 } { 2 ^ { [ \\tfrac { n + 1 } { 2 } ] - [ \\tfrac { n } { 2 } ] } } + 1 \\right ) \\\\ & \\leq \\frac { \\beta _ { 0 } } { 2 ^ { [ \\tfrac { n } { 2 } ] } } \\left ( 1 + 2 \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { 2 ^ { k } } \\right ) . \\end{align*}"}
{"id": "3495.png", "formula": "\\begin{align*} \\int _ c ^ x H ( v ) d v & = \\int _ c ^ x \\int _ v ^ b \\frac { \\partial f } { \\partial v } ( u , v ) d u d v \\\\ & = \\int _ c ^ x \\int _ c ^ u \\frac { \\partial f } { \\partial v } ( u , v ) d v d u + \\int _ x ^ b \\int _ c ^ x \\frac { \\partial f } { \\partial v } ( u , v ) d v d u . \\end{align*}"}
{"id": "1513.png", "formula": "\\begin{align*} \\xi ( y , h , 2 l , 0 ) = 2 ^ { 2 - 2 m } ( 2 \\pi i ) ^ { 2 m l } \\Gamma _ m ^ { - 1 } ( 2 l ) \\det ( h ) ^ { l - \\frac { 2 m - 1 } { 2 } } e ( i \\lambda ( h y ) ) . \\end{align*}"}
{"id": "2383.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) \\sin ( 2 \\pi k / N ) = 0 , \\end{align*}"}
{"id": "179.png", "formula": "\\begin{align*} \\langle \\mathcal { A } ( G ( u ) ) ; v \\rangle _ H = \\langle u ; v \\rangle _ H . \\end{align*}"}
{"id": "477.png", "formula": "\\begin{align*} h ( \\mathcal { I } ) : = \\Delta _ { \\mathbf { L } ( \\mathbf { t } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } ) , \\mathcal { I } \\in \\mathfrak { G } ( \\mathbf { L } ) \\end{align*}"}
{"id": "1724.png", "formula": "\\begin{align*} \\hat \\sigma = s _ * \\cdot \\frac { \\frac 1 q \u2010 \\frac { 1 } { p _ 0 } } { \u2010 s _ * \u2010 \\frac { 1 } { p _ 0 } + \\frac { 1 } { p _ 1 } + \\frac { 2 s _ * } { q } } . \\end{align*}"}
{"id": "5815.png", "formula": "\\begin{align*} \\gamma _ { k } ^ { \\langle n \\rangle } = \\frac { \\left ( \\sum _ { m = 1 } ^ M \\sqrt { \\eta _ { m k } } \\alpha _ { m k } \\right ) ^ 2 } { \\sum _ { m = 1 } ^ M \\beta _ { m k } \\eta _ { m k } \\alpha _ { m k } + \\frac { 1 } { \\gamma _ t } } . \\end{align*}"}
{"id": "516.png", "formula": "\\begin{align*} \\mathcal { T } _ { 0 } : = \\mathcal { J } , \\quad \\mathcal { T } _ { u } : = ( \\mathcal { T } _ { u - 1 } ) _ { i _ { u } } ^ { \\alpha _ { u } } \\end{align*}"}
{"id": "6832.png", "formula": "\\begin{align*} | x _ 3 + x _ 4 i | ^ 2 + | ( x _ 1 + x _ 2 i ) w - ( x _ 3 + x _ 4 i ) ( x + y i ) | ^ 2 = w \\end{align*}"}
{"id": "5995.png", "formula": "\\begin{align*} \\| \\delta _ { z } \\| _ { - q , - 1 , \\pi _ { \\lambda , \\beta } } ^ { 2 } & = \\sum _ { n = 0 } ^ { \\infty } ( n ! ) ^ { - 2 } 2 ^ { - n q } | C _ { n } ^ { \\lambda , \\beta } ( z ) | ^ { 2 } \\\\ & \\leq C _ { \\varepsilon } ^ { 2 } \\mathrm { e } ^ { 2 \\varepsilon | z | } \\sum _ { n = 0 } ^ { \\infty } 2 ^ { - n q } \\sigma _ { \\varepsilon } ^ { - 2 n } , \\end{align*}"}
{"id": "6357.png", "formula": "\\begin{align*} \\rho _ { t } = \\frac { 1 } { { i } _ { t - 1 } } \\left ( \\rho _ { n + 1 } + i _ { t } \\cdot \\rho _ { t - 1 } \\right ) t = 1 , \\dots , n \\end{align*}"}
{"id": "1121.png", "formula": "\\begin{align*} m ^ { ( 2 ) } ( x , t , k ) = D _ { \\infty } ^ { \\sigma _ 3 } ( \\xi ) m ^ { ( 1 ) } ( x , t , k ) D ^ { - \\sigma _ 3 } ( k , \\xi ) . \\end{align*}"}
{"id": "2608.png", "formula": "\\begin{align*} \\norm { f - f _ n } _ { M ^ { p , q } } = \\norm { g _ 0 } _ 2 ^ { - 2 } \\norm { V _ { g _ 0 } ^ * ( F - V _ { g _ 0 } f ) } _ { M ^ { p , q } } \\leq C \\norm { F - V _ { g _ 0 } f _ n } _ { L ^ { p , q } } . \\end{align*}"}
{"id": "3795.png", "formula": "\\begin{align*} \\ell _ { \\le i } = \\begin{cases} 0 & i < - 1 \\ , , \\\\ I _ { - 1 } & i = - 1 \\ , , \\\\ I _ { - 1 } \\bot I _ 0 \\bot P _ 0 & i = 0 \\ , , \\\\ \\ell & i \\ge 1 \\ , . \\\\ \\end{cases} \\end{align*}"}
{"id": "9512.png", "formula": "\\begin{align*} A & = \\Big ( [ f _ - ( - M _ x ) ] [ g _ - ( - M _ x ) ] + [ f _ + ( H ) ] [ g _ + ( H ) ] \\Big ) \\\\ & = \\Big ( [ f _ - g _ - ( - M _ x ) ] + [ f _ + g _ + ( H ) ] - f ( 0 ) g ( 0 ) \\Big ) + f ( 0 ) g ( 0 ) \\\\ & = [ \\gamma ( f g ) ] + f ( 0 ) g ( 0 ) \\\\ & = \\Gamma ( f g ) + f ( 0 ) g ( 0 ) , \\end{align*}"}
{"id": "5351.png", "formula": "\\begin{align*} \\left \\langle \\tilde a _ i , x \\right \\rangle = b _ i \\ ; ( 1 \\leqslant i \\leqslant m ) . \\end{align*}"}
{"id": "8504.png", "formula": "\\begin{align*} ( a + b ) \\circ c = a \\circ c - c + b \\circ c \\end{align*}"}
{"id": "3945.png", "formula": "\\begin{align*} \\alpha : = \\left ( a \\gamma Q ^ { \\gamma - 3 } _ 0 \\right ) ^ { - 1 / 2 } , \\ \\ \\ \\beta : = \\frac { Q _ 0 V ^ 2 _ 0 } { \\nu } , \\ \\ \\ \\delta : = \\frac { Q _ 0 V _ 0 } { \\nu } , \\end{align*}"}
{"id": "6511.png", "formula": "\\begin{align*} H ^ { ( 2 m ) } _ n \\sim \\frac { ( 1 - 4 \\alpha ) c _ { 2 m } } { m ( 1 - 2 \\alpha ) - 1 } \\cdot n ^ { - 1 } ( n \\to \\infty ) . \\end{align*}"}
{"id": "7490.png", "formula": "\\begin{align*} \\frac { \\phi ^ { n + 1 } - \\phi ^ { n } } { \\tau } & = \\left ( \\frac 1 2 \\Delta - \\vartheta ^ n \\right ) \\phi ^ { n + 1 } + \\bigg ( \\vartheta ^ n - V - \\beta | \\phi ^ n | ^ 2 + \\Omega L _ z + \\chi ^ n - \\sigma R ^ \\prime ( \\| \\phi ^ n \\| ^ 2 - 1 ) \\bigg ) \\phi ^ n , \\\\ \\frac { \\chi ^ { n + 1 } - \\chi ^ { n } } { \\xi ^ n \\tau } & = 1 - \\| \\phi ^ { n + 1 } \\| ^ 2 , n = 0 , 1 , \\ldots \\end{align*}"}
{"id": "1670.png", "formula": "\\begin{align*} c ^ * > c _ 0 ^ * : = \\frac { ( \\lambda _ 1 + \\gamma ) ^ 2 ( \\lambda _ 2 - \\lambda _ 1 ) ^ 2 } { 4 ( 2 \\lambda _ 1 \\lambda _ 2 - \\lambda _ 1 ^ 2 + 4 \\lambda _ 2 \\gamma + 3 \\gamma ^ 2 ) } \\end{align*}"}
{"id": "5223.png", "formula": "\\begin{align*} \\| T _ k f \\| _ { F _ { r t } ^ s ( K ) } & \\leq \\sum \\limits _ { i = 1 } ^ { \\infty } | \\lambda _ i | \\| B f \\| _ { F _ { r t } ^ s ( K ) } \\\\ & \\leq C \\| \\Omega \\| _ { H ^ 1 ( \\mathfrak { D } ^ * ) } \\| B f \\| _ { F _ { r t } ^ s ( K ) } , \\\\ \\end{align*}"}
{"id": "2535.png", "formula": "\\begin{align*} \\Phi \\natural \\Phi ^ { \\xi , \\eta } ( x , \\omega ) = \\iint _ { \\R ^ { 2 d } } \\langle \\varphi , \\rho ( x ' , \\omega ' ) \\varphi \\rangle \\langle \\rho ( \\xi , \\eta ) \\varphi , \\rho ( x - x ' , \\omega - \\omega ' ) \\varphi \\rangle e ^ { \\pi i ( x \\cdot \\omega ' - x ' \\cdot \\omega ) } \\ , d ( x ' , \\omega ' ) . \\end{align*}"}
{"id": "1425.png", "formula": "\\begin{align*} \\hat { \\sigma } : \\Delta _ { d + 1 } & \\to \\R ^ N ; \\\\ ( a _ 0 , a _ 1 , \\dots , a _ d ) & \\mapsto A \\sigma ( a _ 1 / A , \\dots , a _ d / A ) , A = \\sum _ { i = 0 } ^ d a _ i . \\end{align*}"}
{"id": "6686.png", "formula": "\\begin{align*} \\mathcal { L i } _ { K , s } ( z ) ^ { ( - 1 ) } = \\frac { z } { \\theta - t } + \\mathcal { L i } _ { K , s } ( z ) . \\end{align*}"}
{"id": "1465.png", "formula": "\\begin{align*} \\mathfrak { Z } = \\mathfrak { Z } _ n = \\mathfrak { Z } _ { m , r } = \\{ z \\in \\C _ n ^ n : U ( z ) \\in \\Omega \\} , U ( z ) = \\left [ \\begin{array} { c } z \\\\ u _ 0 \\end{array} \\right ] , u _ 0 = \\left [ \\begin{array} { c c } 0 & 1 _ r \\\\ 1 _ { 2 m } & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "5972.png", "formula": "\\begin{align*} \\dot { x } _ k ( t ) = - 4 \\pi ^ 2 s ^ 2 ( t - t _ 0 ) e ^ { - 2 \\pi ^ 2 s ^ 2 ( t - t _ 0 ) ^ 2 } , \\mathcal { F } \\{ x _ k ( t ) \\} = \\hat { x } _ k ( f ) = \\frac { 1 } { s \\sqrt { 2 \\pi } } e ^ { \\frac { - f ^ 2 } { 2 s ^ 2 } } . \\end{align*}"}
{"id": "3860.png", "formula": "\\begin{align*} F = \\begin{bmatrix} a & 1 \\\\ b & - a \\end{bmatrix} , \\tilde F = \\begin{bmatrix} \\tilde a & 1 \\\\ \\tilde b & - \\tilde a \\end{bmatrix} , \\end{align*}"}
{"id": "2666.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { \\alpha \\in \\N _ 0 ^ d } c _ \\alpha z ^ \\alpha . \\end{align*}"}
{"id": "7221.png", "formula": "\\begin{align*} X _ { s , t } ( x , \\Psi _ { s , t } ( x , v ) ) = x - ( t - s ) v . \\end{align*}"}
{"id": "3281.png", "formula": "\\begin{align*} U ^ \\mu ( \\lambda ) = \\int _ \\mathbb { C } \\log \\frac { 1 } { | \\lambda - z | } d \\mu ( z ) , \\lambda \\in \\mathbb { C } , \\end{align*}"}
{"id": "457.png", "formula": "\\begin{align*} \\mu ( \\phi ( x , b ) ) = \\frac { 1 } { | H | } \\mu ( X ( b ) ) = \\frac { | \\Sigma | } { | G | } \\mu ( X ( b ) ) = \\nu ' ( \\phi ( x , b ) ) \\mu ( X ( b ) ) \\end{align*}"}
{"id": "9402.png", "formula": "\\begin{align*} \\tau _ t ( a _ n \\cdots a _ 1 v b _ 1 \\cdots b _ m ) = t \\cdot \\tau ( a _ n \\cdots a _ 1 \\tau ' ( v ) b _ 1 \\cdots b _ m ) + o ( t ) \\end{align*}"}
{"id": "2148.png", "formula": "\\begin{align*} X _ { n , k } ^ { \\mathcal { S } } = M + \\tau ( A ^ { ( n ) } _ { M , n , l } ) \\ \\ W _ { \\mathcal { S } ( n , k ; M ) } \\cap A ^ { ( n ) } _ { M , n , l } . \\end{align*}"}
{"id": "3093.png", "formula": "\\begin{align*} z _ 1 ^ \\prime = z _ 2 \\ , , z _ 2 ^ \\prime = z _ 2 \\ , , z _ 3 ^ \\prime = z _ 3 \\ , , z _ 4 ^ \\prime = z _ 5 \\ , , z _ 5 ^ \\prime = z _ 4 \\ , . \\end{align*}"}
{"id": "7748.png", "formula": "\\begin{align*} - 2 f _ 3 ( t , x ) & = \\int _ { t - d } ^ { t + d } \\int _ { \\lambda \\in \\R } e ^ { i s \\lambda } \\hat { g } ( \\lambda ) d \\lambda d s \\\\ & = \\int _ { t - d } ^ { t + d } \\int _ { | \\lambda | \\leq 1 } e ^ { i s \\lambda } \\hat { g } ( \\lambda ) d \\lambda d s + \\int _ { t - d } ^ { t + d } \\int _ { | \\lambda | > 1 } e ^ { i s \\lambda } \\hat { g } ( \\lambda ) d \\lambda d s . \\end{align*}"}
{"id": "9389.png", "formula": "\\begin{align*} f ( t ) = ( 1 + t ) \\log _ { \\lambda } ( 1 + t ) - t . \\end{align*}"}
{"id": "3467.png", "formula": "\\begin{align*} | R _ j ( x , y ) | & \\lesssim | y _ j - x _ j | \\int _ 0 ^ \\infty \\frac 1 { V ( x , y , \\sqrt t ) } \\frac t { \\| x - y \\| ^ 2 } e ^ { - c d ( x , y ) ^ 2 / t } \\frac { d t } { t \\sqrt t } \\\\ & \\leqslant \\frac 1 { \\| x - y \\| } \\bigg ( \\int _ 0 ^ { d ( x , y ) ^ 2 } + \\int _ { d ( x , y ) ^ 2 } ^ \\infty \\bigg ) \\frac 1 { V ( x , y , \\sqrt t ) } e ^ { - c d ( x , y ) ^ 2 / t } \\frac { d t } { \\sqrt t } \\\\ & = : I _ 1 + I _ 2 . \\end{align*}"}
{"id": "3020.png", "formula": "\\begin{align*} \\beta = \\sum m _ { \\mathbf { t } } \\ell ^ { \\mathbf { t } } + \\sum n _ { i , j } \\gamma _ { \\mathbf { m _ 1 } , \\mathbf { m _ 2 } } + \\sum p _ { \\mathbf { s } , \\mathbf { t } } \\sigma _ { \\mathbf { s } , \\mathbf { t } } \\end{align*}"}
{"id": "2476.png", "formula": "\\begin{align*} \\pi ( ( x _ 1 , x _ 2 ) ; ( \\omega _ 1 , \\omega _ 2 ) ) = M _ { ( \\omega _ 1 , \\omega _ 2 ) } T _ { ( x _ 1 , x _ 2 ) } \\end{align*}"}
{"id": "1441.png", "formula": "\\begin{align*} \\mathfrak { i } : \\mathbb { B } \\stackrel { \\sim } \\longrightarrow \\{ x \\in M _ 2 ( \\mathbb { K } ) : \\overline { x } I J = I J x \\} . \\end{align*}"}
{"id": "7078.png", "formula": "\\begin{align*} ( \\nabla _ X A ) Y = ( \\nabla _ Y A ) X + 2 \\langle \\xi , E ^ { \\perp } _ 3 \\rangle ( \\langle Y , E ^ { \\top } _ 3 \\rangle X - \\langle X , E ^ { \\top } _ 3 \\rangle Y ) . \\end{align*}"}
{"id": "6694.png", "formula": "\\begin{align*} \\langle a \\rangle _ n : = \\begin{cases} \\mathbb { D } _ { n + a - 1 } ^ { q ^ { - ( a - 1 ) } } & , \\\\ 1 / \\mathbb { L } _ { - a - n } ^ { q ^ n } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "5159.png", "formula": "\\begin{align*} Z _ { b ^ { - 1 / 2 } } \\varphi ( x , \\omega ) = ( 2 b ) ^ { 1 / 4 } \\sum _ { k \\in \\Z } e ^ { - \\pi b ( x - k ) ^ 2 } e ^ { 2 \\pi i k \\omega } = ( 2 b ) ^ { 1 / 4 } e ^ { - \\pi b x ^ 2 } \\vartheta _ 3 ( \\omega - i b x , i b ) , \\end{align*}"}
{"id": "1271.png", "formula": "\\begin{align*} U _ { J , z } \\langle \\alpha \\rangle = \\sum _ { \\beta : \\mathrm { a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } [ \\beta ] = \\Gamma } \\# \\{ \\ , u \\in \\mathcal { M } _ { 2 } ^ { J } ( \\alpha , \\beta ) / \\mathbb { R } ) \\ , | \\ , ( 0 , z ) \\in u \\ , \\} \\cdot \\langle \\beta \\rangle . \\end{align*}"}
{"id": "5280.png", "formula": "\\begin{align*} \\check { \\sigma } ^ { \\check { \\varphi } } ( \\varphi ( - a ) ) = \\varphi ( - S ^ 2 ( a ) \\delta _ { \\varphi } ^ { - 1 } ) , \\check { \\delta } _ { \\check { \\varphi } } ( a ) = \\varepsilon ( \\sigma ^ { \\varphi } ( a ) ) , \\check { \\nu } = \\nu ^ { - 1 } . \\end{align*}"}
{"id": "3962.png", "formula": "\\begin{align*} \\Phi _ { \\lambda ^ p _ k } : = ( \\xi _ { \\lambda ^ p _ k } , \\eta _ { \\lambda ^ p _ k } ) \\ \\ \\lambda ^ p _ k , \\ \\ \\forall k \\geq k _ 0 , \\end{align*}"}
{"id": "4331.png", "formula": "\\begin{align*} \\tau ^ * = \\sup \\{ \\tau _ 1 \\ge \\tau _ 0 ( \\varepsilon , b , \\beta ) ( \\tau ) \\in V _ 1 [ A , \\eta , \\tilde \\eta ] ( \\tau ) , \\forall \\tau \\in [ \\tau _ 0 , \\tau _ 1 ] \\} . \\end{align*}"}
{"id": "1550.png", "formula": "\\begin{align*} \\phi _ v ( \\tilde { \\tau } _ m ( h _ v \\times 1 ) ) = \\chi _ { v } ( \\det ( r ) ) | \\det ( r ) | _ v ^ { - s } . \\end{align*}"}
{"id": "8554.png", "formula": "\\begin{align*} m _ \\pm ( x , k ) = 1 \\pm \\int _ x ^ { \\pm \\infty } D _ k ( \\pm ( y - x ) ) V ( y ) m _ \\pm ( y , k ) \\ , d y , D _ k ( x ) = \\frac { e ^ { 2 i k x } - 1 } { 2 i k } ; \\end{align*}"}
{"id": "8975.png", "formula": "\\begin{align*} \\begin{aligned} & u ' + 2 u ' u '' + 1 = 0 \\\\ \\Longrightarrow & u '' = \\frac { - 1 - u ' } { 2 u ' } = \\frac { 1 } { - 2 u ' } - \\frac { 1 } { 2 } \\to + \\infty \\end{aligned} \\end{align*}"}
{"id": "8884.png", "formula": "\\begin{align*} \\tilde \\psi ( \\bar z ) = \\psi _ { \\max ( y _ j ) } \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) + \\sum _ { t = 1 } ^ { \\max ( y _ j ) - 1 } d _ { q - 2 } \\chi _ t \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) \\end{align*}"}
{"id": "3485.png", "formula": "\\begin{align*} A & = \\left ( \\sum _ { n \\leq x } \\sum _ { n < m \\leq y } + \\sum _ { n \\leq x } \\sum _ { y < m \\leq M } + \\sum _ { x < n \\leq N } \\sum _ { n < m \\leq M } \\right ) \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } \\\\ & = \\sum _ { n \\leq x } \\sum _ { n < m \\leq y } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + A _ 1 + A _ 2 , \\end{align*}"}
{"id": "6339.png", "formula": "\\begin{align*} 1 6 ( p _ { n + 1 } - p _ n ) \\sum _ { \\ell = 1 } ^ n ( p _ \\ell - p _ { \\ell - 1 } ) ^ 2 \\leq ( p _ n - 1 ) ^ 2 . \\end{align*}"}
{"id": "9358.png", "formula": "\\begin{align*} - t e _ { \\lambda } ^ { - r } ( t ) & = - t \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - r ) _ { n , \\lambda } } { n ! } t ^ { n } = t \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\langle r \\rangle _ { n , \\lambda } } { n ! } t ^ { n } \\\\ & = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\langle r \\rangle _ { n - 1 , \\lambda } \\frac { t ^ { n } } { ( n - 1 ) ! } = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\langle r \\rangle _ { n - 1 , \\lambda } n \\frac { t ^ { n } } { n ! } , \\end{align*}"}
{"id": "2075.png", "formula": "\\begin{align*} \\mathrm { i n d e x } ( f ) \\gg _ n H ^ { \\frac { n ^ 2 - n - 2 } { 2 } - ( n - 1 ) } = H ^ { \\frac { n ( n - 3 ) } { 2 } } . \\end{align*}"}
{"id": "2026.png", "formula": "\\begin{align*} \\overline { \\mu } _ 1 ^ * & = \\overline { \\mu } _ 1 + N ( e ^ { F _ 1 } - e ^ { F _ 1 - F _ 2 } - F _ 2 ) \\mu _ H \\\\ & \\leq \\overline { \\mu } _ 1 + N \\left ( e ^ { F _ 1 } ( 1 - e ^ { - F _ 2 } ) \\right ) \\mu _ H \\leq \\overline { \\mu } _ 1 + e ^ { \\| F _ 1 \\| _ { \\infty } } N ( F _ 2 ) \\mu _ H \\in S _ { D _ 0 } ^ 1 ( { \\bf Y } ) . \\end{align*}"}
{"id": "5172.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } \\Delta _ { \\ell , \\pi } ^ { a v } ( t ) = \\lim _ { t \\to \\infty } \\sum _ { i = 1 } ^ { R _ \\ell ^ \\pi ( t ) } \\frac { \\frac { ( T _ { \\ell i } ^ \\pi ) ^ 2 } { 2 } + T _ { \\ell i } ^ \\pi Z _ { \\ell i } ^ \\pi } { t } . \\end{align*}"}
{"id": "9292.png", "formula": "\\begin{align*} u _ { n + 1 } = - \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ { n } u _ { j } u _ { n - j } , \\ , n = 0 , 1 , \\ldots , \\ , u _ { 0 } = 1 \\end{align*}"}
{"id": "8561.png", "formula": "\\begin{align*} \\mathcal { K } ( x , k ) : = \\frac { 1 } { \\sqrt { 2 \\pi } } \\begin{cases} T ( k ) \\psi _ + ( x , k ) & k \\geq 0 \\\\ T ( - k ) \\psi _ - ( x , - k ) & k < 0 , \\end{cases} \\end{align*}"}
{"id": "903.png", "formula": "\\begin{align*} \\alpha _ p ( U _ 0 \\bot p U _ 0 , T ) = 2 ( 1 + p ^ { - 1 } ) ^ 2 p . \\end{align*}"}
{"id": "3162.png", "formula": "\\begin{align*} \\max _ { j \\in \\lbrace 1 , \\ldots , n \\rbrace } \\frac { | e _ j ^ \\intercal ( A x _ 0 - b ) | ^ 2 } { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 \\norm { A x _ 0 - b } _ 2 ^ 2 } \\geq \\sum _ { j = 1 } ^ n \\frac { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 } { \\norm { A } _ F ^ 2 } \\frac { | e _ j ^ \\intercal ( A x _ 0 - b ) | ^ 2 } { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 \\norm { A x _ 0 - b } _ 2 ^ 2 } = \\frac { 1 } { \\norm { A } _ F ^ 2 } . \\end{align*}"}
{"id": "3106.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + z [ a ( x ^ 4 + y ^ 4 ) + b x ^ 2 y ^ 2 ] + x [ c ( x ^ 4 + y ^ 4 ) + d x ^ 2 y ^ 2 ] = 0 \\ , . \\end{align*}"}
{"id": "9401.png", "formula": "\\begin{align*} o ( t ) = \\tau _ t ( a _ 1 \\cdots a _ n ) + \\sum _ { r = 1 } ^ n ( - 1 ) ^ r \\sum _ { 1 \\leq k _ 1 < \\ldots < k _ r \\leq n } \\left ( \\prod _ { i = 1 } ^ { r } \\tau _ t ( a _ { k _ i } ) \\right ) \\tau _ t ( a _ 1 \\cdots \\hat { a } _ { k _ 1 } \\cdots \\hat { a } _ { k _ r } \\cdots a _ n ) \\end{align*}"}
{"id": "876.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum \\limits _ { i = 1 } ^ { N } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) } , \\\\ & \\sum \\limits _ { i = 1 } ^ { N } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) \\left ( { 2 { \\tau _ { \\rm c } } + 2 \\mathcal { T } + { n _ { i + 1 } } + { n _ i } } \\right ) } . \\end{aligned} \\end{align*}"}
{"id": "8291.png", "formula": "\\begin{align*} \\partial _ t X _ t = \\Big ( \\dot { x } ( t ) , \\dot { x } ( t ) \\cdot \\nabla u ( t , x ( t ) ) + \\partial _ t u ( t , x ( t ) ) \\Big ) . \\end{align*}"}
{"id": "4906.png", "formula": "\\begin{align*} \\frac { a _ 1 } { a _ 1 } = 1 \\leq 1 + \\log ( 1 + a _ 1 ) \\end{align*}"}
{"id": "5277.png", "formula": "\\begin{align*} \\check { \\Delta } ( \\omega ) ( \\chi \\otimes 1 ) = \\varphi ( - b _ { ( 2 ) } ) \\otimes \\varphi ( - S ^ { - 1 } ( b _ { ( 1 ) } ) a ) . \\end{align*}"}
{"id": "9174.png", "formula": "\\begin{align*} E _ { y } ( x ) = \\sum _ { j = 0 } ^ { 2 \\lceil y \\rceil } \\frac { x ^ { j } } { j ! } . \\end{align*}"}
{"id": "1376.png", "formula": "\\begin{align*} E ( 0 ) = \\alpha , E ( 1 ) < \\alpha , \\end{align*}"}
{"id": "1967.png", "formula": "\\begin{align*} \\phi \\prec \\psi : = m _ A ( \\phi \\otimes \\psi ) \\Delta _ \\prec , \\phi \\succ \\psi : = m _ A ( \\phi \\otimes \\psi ) \\Delta _ \\succ \\end{align*}"}
{"id": "4405.png", "formula": "\\begin{align*} \\phi _ { i , o u t , \\beta } ( y ) = \\phi _ { i , \\infty , \\beta } ( y ) + \\tilde { \\lambda } ( \\tilde { \\phi } _ { i , \\beta } ( y ) + R _ { i , 1 } ( y ) ) + R _ { i , 2 } ( y ) , \\end{align*}"}
{"id": "1803.png", "formula": "\\begin{align*} { \\rm t r } _ g ^ Y ( \\kappa ) : = \\int _ { G / Z _ g } \\int _ Y c _ Y ( h g h ^ { - 1 } y ) { \\rm t r } ( h g h ^ { - 1 } \\kappa ( h g ^ { - 1 } h ^ { - 1 } y , y ) ) d x \\ , d ( h Z ) . \\end{align*}"}
{"id": "8710.png", "formula": "\\begin{align*} ~ \\alpha _ { i 0 } \\preceq _ i \\cdots \\preceq _ i \\alpha _ { i n } \\ ; \\ ; \\ ; \\upsilon _ { i 0 } = \\alpha _ { i 0 } \\upsilon _ { i j } \\preceq _ i \\upsilon _ { i n } \\upsilon _ { i j } \\preceq _ i \\alpha _ { i j } \\ ; \\ ; j \\in \\{ 1 , \\ldots , n \\} , \\end{align*}"}
{"id": "4014.png", "formula": "\\begin{align*} \\begin{cases} m _ 1 = - \\mu + 1 + O ( \\mu ^ { - 1 } ) , \\\\ m _ 2 = - \\frac { 1 } { 2 } - i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) , \\\\ m _ 3 = - \\frac { 1 } { 2 } + i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) , \\end{cases} \\end{align*}"}
{"id": "8386.png", "formula": "\\begin{align*} \\P ( \\cup _ { \\emptyset \\neq U ' \\subset U , W ' \\subset U _ { \\mathsf k ' } } \\{ \\mathcal L ( W ' ) = \\mathcal L ( U ' ) \\} \\mid \\mathcal L ( U ) = \\mathbf L ) \\geq \\chi _ { \\mathbf L } \\ , . \\end{align*}"}
{"id": "3356.png", "formula": "\\begin{align*} [ x + u , y + v , z + v ] _ { \\overline { T } } = [ u , v , w ] _ T + \\theta _ T ( v , w ) x - \\theta _ T ( u , w ) y + D _ T ( u , v ) z . \\end{align*}"}
{"id": "8999.png", "formula": "\\begin{align*} ( M \\rightarrow Q _ n ^ { [ M ] } ) : = I ( M ; Q _ n ) , \\end{align*}"}
{"id": "8586.png", "formula": "\\begin{align*} & \\phi = \\phi _ S + \\phi _ R , \\phi _ S = \\phi _ 0 + \\phi _ + + \\phi _ - , \\\\ & \\phi _ \\ast ( x ) : = \\frac { 1 } { \\sqrt { 2 \\pi } } \\chi _ \\ast ( x ) \\int \\mathcal { K } _ \\ast ^ \\# ( x , k ) \\ , ( \\mathcal { F } ^ \\# \\phi ) ( k ) \\ , d k , \\ast \\in \\{ 0 , + , - \\} \\\\ & \\phi _ R ( x ) : = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int \\mathcal { K } _ R ^ \\# ( x , k ) \\ , ( \\mathcal { F } ^ \\# \\phi ) ( k ) \\ , d k . \\end{align*}"}
{"id": "3268.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } \\frac { i t } { \\omega _ 2 ( i t ) } & = \\lim _ { t \\rightarrow 0 ^ + } \\frac { s ( | \\lambda | , t ) t - t ^ 2 } { | \\lambda | ^ 2 } = \\frac { | \\lambda | ^ 2 - \\lambda _ 2 ( \\mu ) } { | \\lambda | ^ 2 } . \\end{align*}"}
{"id": "8539.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi + \\sin \\phi = 0 , \\end{align*}"}
{"id": "7790.png", "formula": "\\begin{align*} \\| ( S f - S g ) ( t _ j ) - ( S f - S g ) ( t _ { j + 1 } ) \\| & = \\| \\alpha _ k ( f - g ) ( P _ k ^ { - 1 } ( t _ j ) ) - \\alpha _ k ( f - g ) ( P _ k ^ { - 1 } ( t _ { j + 1 } ) ) \\| \\\\ & = | \\alpha _ k | \\| ( f - g ) ( P _ k ^ { - 1 } ( t _ j ) ) - ( f - g ) ( P _ k ^ { - 1 } ( t _ { j + 1 } ) ) \\| \\\\ & \\leq \\alpha _ { \\max } \\| ( f - g ) ( P _ k ^ { - 1 } ( t _ j ) ) - ( f - g ) ( P _ k ^ { - 1 } ( t _ { j + 1 } ) ) \\| . \\end{align*}"}
{"id": "3616.png", "formula": "\\begin{align*} \\min \\{ d ( - a _ { n + 1 } a _ { n + 2 } ) , 2 e , \\alpha _ { n + 2 } \\} & = \\min \\{ d ( - a _ { n + 1 } a _ { n + 2 } ) , \\alpha _ { n } , \\alpha _ { n + 2 } \\} \\\\ & = d [ - a _ { n + 1 } a _ { n + 2 } ] \\ge 1 - R _ { n + 2 } = 2 e - 1 \\ , . \\end{align*}"}
{"id": "3554.png", "formula": "\\begin{align*} ( r a - r ^ { - 1 } ) v = y , \\end{align*}"}
{"id": "9318.png", "formula": "\\begin{align*} - \\begin{bmatrix} H + \\rho I & - A ^ T \\\\ A & \\delta I \\end{bmatrix} \\begin{bmatrix} \\mathbf { x } \\\\ \\mathbf { y } \\end{bmatrix} + \\begin{bmatrix} - \\mathbf { g } + \\rho \\mathbf { x } _ k \\\\ \\mathbf { b } + \\delta \\mathbf { y } _ k \\end{bmatrix} \\in N _ D ( \\mathbf { x } , \\mathbf { y } ) . \\end{align*}"}
{"id": "6915.png", "formula": "\\begin{align*} B _ m ( \\theta ) = \\textbf { a } _ m ^ T ( \\theta ) \\textbf { w } _ m = \\sum _ { n = 1 } ^ { N _ s } e ^ { - j 2 \\pi f _ 0 \\tau _ { n m } ( \\theta ) } e ^ { j \\psi _ { n m } } . \\end{align*}"}
{"id": "2750.png", "formula": "\\begin{align*} s _ c = \\frac { N } { 2 } - \\frac { \\gamma + 2 } { 2 ( p - 1 ) } , \\end{align*}"}
{"id": "2347.png", "formula": "\\begin{align*} \\left ( \\int _ \\R | f ' ( x ) | ^ 2 \\ , d x \\right ) ^ { 1 / 2 } = \\left ( \\int _ \\R | ( 2 \\pi i \\omega ) \\widehat { f } ( \\omega ) | ^ 2 \\ , d \\omega \\right ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "9117.png", "formula": "\\begin{align*} x _ i [ y _ j - ( \\alpha _ 1 g _ 1 + \\cdots + \\alpha _ r g _ r ) ] & = ( \\beta _ 1 g _ 1 + \\cdots + \\beta _ r g _ r ) y _ j \\\\ y _ j [ x _ i - ( \\beta _ 1 g _ 1 + \\cdots + \\beta _ r g _ r ) ] & = ( \\alpha _ 1 g _ 1 + \\cdots + \\alpha _ r g _ r ) x _ i \\end{align*}"}
{"id": "4284.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ r ^ 2 u + \\frac { d + 1 } { r } \\partial _ r u - 3 ( d - 2 ) u ^ 2 - ( d - 2 ) r ^ 2 u ^ 3 , ( r , t ) \\in \\R _ + \\times \\R _ + . \\end{align*}"}
{"id": "4386.png", "formula": "\\begin{align*} \\mathcal { P } ( v ) : = \\partial _ \\xi ^ 2 v + \\frac { d + 1 } { \\xi } \\partial _ \\xi v - 3 ( d - 2 ) ( 2 Q + \\xi ^ 2 Q ^ 2 ) v + \\bar B ( v ) + \\theta ( \\tau ) \\Lambda _ \\xi Q + \\theta ( \\tau ) \\Lambda _ \\xi v - b ( \\tau ) \\partial _ \\tau v , \\end{align*}"}
{"id": "2171.png", "formula": "\\begin{align*} \\langle ( - \\triangle _ { g } ) ^ { \\alpha } u , v \\rangle = \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\bigg { ( } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { \\alpha } } \\bigg { ) } \\frac { v ( x ) - v ( y ) } { | x - y | ^ { \\alpha } } \\frac { d x d y } { | x - y | ^ { d } } , \\end{align*}"}
{"id": "89.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 ^ { 1 / 2 } \\cdot \\frac { 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 } { 2 ^ { n + 1 } - 1 } \\end{align*}"}
{"id": "3924.png", "formula": "\\begin{align*} \\zeta _ 2 ( z ) \\wp _ 2 ( z ) = - \\frac 1 2 \\big ( \\wp _ 2 ' ( z ) + f _ 2 ' ( z ) \\big ) \\end{align*}"}
{"id": "4282.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } & & \\partial _ t A _ \\mu ( x , t ) + \\partial ^ \\nu F _ { \\mu \\nu } ( x , t ) + \\left [ A ^ \\nu , F _ { \\mu , \\nu } \\right ] ( x , t ) = 0 , ~ ~ t > 0 , \\\\ & & A _ \\mu ( x , 0 ) = A _ { \\mu , 0 } ( x ) . \\end{array} \\right . \\end{align*}"}
{"id": "7358.png", "formula": "\\begin{align*} \\varphi _ t ( x _ q , t _ q ) = { 1 \\over q } \\varphi ( x _ q , t _ q ) ^ { 1 - q } \\psi _ t ( x _ q , t _ q ) = { 1 \\over q } \\varphi ( x _ q , t _ q ) ^ { 1 - q } ( \\lambda h _ q + ( 1 - \\lambda ) k _ q ) , \\end{align*}"}
{"id": "5218.png", "formula": "\\begin{align*} B f _ 1 ( x ) & = \\sum \\limits _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } g _ j \\ast f _ 1 ( x ) \\\\ & = 0 . \\end{align*}"}
{"id": "6931.png", "formula": "\\begin{align*} a _ h ( w , v ) = \\sum _ { E \\in { \\cal T } _ h } \\sum _ { \\iota \\in I ^ E } [ \\mu \\nabla w \\cdot \\nabla v + \\boldsymbol { \\beta } \\cdot \\nabla w \\ , v + \\sigma w v ] ( \\xi ^ E _ \\iota ) \\ , \\omega ^ E _ \\iota \\ , , \\end{align*}"}
{"id": "3968.png", "formula": "\\begin{align*} \\norm { \\eta _ { \\lambda ^ h _ k } - \\tilde { \\psi } _ k } _ { L ^ 2 } = \\norm { \\eta _ { \\lambda ^ h _ k } } _ { L ^ 2 } \\leq C | k | ^ { - 1 } , \\end{align*}"}
{"id": "367.png", "formula": "\\begin{align*} \\dot \\rho = \\nabla _ { S } \\mathcal H ( \\rho , S ) ( 1 + \\dot W ^ { \\delta } ) ^ 2 , \\ ; \\\\ \\dot S = - \\nabla _ { \\rho } \\mathcal H ( \\rho , S ) ( 1 + \\dot W ^ { \\delta } ) ^ 2 , \\end{align*}"}
{"id": "8104.png", "formula": "\\begin{align*} \\left [ \\mathfrak { A } \\chi _ 1 \\left ( \\mathfrak { A } ( \\mathcal { M } _ 1 ) \\right ) , \\mathfrak { A } \\chi _ 2 \\left ( \\mathfrak { A } ( \\mathcal { M } _ 2 ) \\right ) \\right ] = 0 . \\end{align*}"}
{"id": "2574.png", "formula": "\\begin{align*} ( \\log ( | x | ) ) ' = \\tfrac { 1 } { x } . \\end{align*}"}
{"id": "5535.png", "formula": "\\begin{align*} \\varphi _ { \\gamma + Z _ { r , m } } ( t ) \\leq \\varphi _ { \\gamma + Z _ { r , m } } ( T ) = \\frac { 1 } { \\gamma + Z _ { r , m } } \\big ( e ^ { ( \\gamma + Z _ { r , m } ) T } - 1 \\big ) \\leq \\frac { e ^ { \\gamma T } } { \\gamma } e ^ { Z _ { r , m } T } , t \\in [ 0 , T ] . \\end{align*}"}
{"id": "6972.png", "formula": "\\begin{align*} \\forall k \\in \\N \\colon x _ k : = \\frac 1 k , y _ k : = \\frac { 1 } { k ^ 2 } , x _ k ^ * : = 1 , \\lambda _ k : = \\frac { k } { 2 } , \\end{align*}"}
{"id": "5226.png", "formula": "\\begin{align*} \\Delta _ l ( g _ j \\ast f ) & = \\Delta _ l ( f \\ast g _ j ) \\\\ & = \\mathcal { F } ^ { - 1 } \\phi _ l \\mathcal { F } ( f \\ast g _ j ) \\\\ & = ( \\mathcal { F } ^ { - 1 } \\phi _ l \\mathcal { F } f ) \\ast g _ j , \\end{align*}"}
{"id": "1092.png", "formula": "\\begin{align*} D _ { \\infty } ( \\xi ) : = \\exp \\left \\{ - \\frac { 1 } { 2 \\pi i } \\left [ \\left ( \\int _ { - \\eta } ^ { - C _ L } + \\int _ { C _ L } ^ { \\eta } \\right ) \\frac { \\log ( 1 - r ( s ) r ^ { * } ( s ) ) } { X _ { L } ( s ) } d s + \\left ( \\int _ { - C _ L } ^ { - C _ R } + \\int _ { C _ R } ^ { C _ L } \\right ) \\frac { \\log r _ { + } ( s ) } { X _ { L + } ( s ) } d s \\right ] \\right \\} . \\end{align*}"}
{"id": "6687.png", "formula": "\\begin{align*} \\Bigl ( ~ _ { s + 1 } F _ s ( 1 , m , \\ldots , m ; 1 + m , \\ldots , 1 + m ) ( z ^ { q ^ { - m + 1 } } ) \\Bigr ) ^ { q ^ m } = \\sum _ { i \\geq 0 } \\frac { z ^ { q ^ { i + 1 } } } { [ i + m ] ^ { s } } . \\end{align*}"}
{"id": "1164.png", "formula": "\\begin{align*} m ^ { P C } _ { \\eta , + } ( \\zeta ) = m ^ { P C } _ { \\eta , - } ( \\zeta ) J ^ { P C } ( \\zeta ) , \\zeta \\in \\Sigma ^ { p c } , \\end{align*}"}
{"id": "8248.png", "formula": "\\begin{align*} t = ( i , \\bar { j } ) ( \\bar { i } , j ) = \\Big ( u ( i _ p + 1 ) , u ( i _ p ) \\Big ) \\Big ( \\bar { u ( i _ p + 1 ) } , \\bar { u ( i _ p ) } \\Big ) = u s _ { i _ p } u ^ { - 1 } . \\end{align*}"}
{"id": "1554.png", "formula": "\\begin{align*} U ( \\mathcal { T } ) : = \\{ g \\in \\mathrm { G L } _ { 2 n } ( K ) : g ^ { \\ast } \\mathcal { T } g = \\mathcal { T } \\} . \\end{align*}"}
{"id": "1366.png", "formula": "\\begin{align*} y ^ i ( x , \\omega ) = \\int _ 0 ^ 1 \\langle \\nabla y ^ i ( \\gamma _ \\omega ( t ) , \\omega ) , \\gamma _ \\omega ' ( t ) \\rangle d t = \\sum _ { k = 1 } ^ d \\int _ 0 ^ 1 D _ k y ^ i ( 0 , \\tau _ { \\gamma _ \\omega ( t ) } \\omega ) \\gamma _ { \\omega , k } ' ( t ) \\ 1 { \\gamma _ \\omega ( t ) \\in W ' ( \\omega ) } d t , \\end{align*}"}
{"id": "8636.png", "formula": "\\begin{align*} I ( t , K ) : = e ^ { - i t K ^ 2 } a ( K ) \\ , \\mathrm { p . v . } \\int _ \\R e ^ { i q ^ 2 t } G ( q ) \\frac { \\psi ( K - q ) } { K - q } \\ , d q \\end{align*}"}
{"id": "4813.png", "formula": "\\begin{align*} n = q , \\theta = \\hat \\theta = 0 . \\end{align*}"}
{"id": "1367.png", "formula": "\\begin{align*} \\langle v , y ( x , \\omega ) \\rangle & = \\sum _ { i = 1 } ^ d \\sum _ { k = 1 } ^ d \\int _ 0 ^ 1 v _ i D _ k y ^ i ( 0 , \\tau _ { \\gamma ( t ) } \\omega ) \\gamma _ { \\omega , k } ' ( t ) \\ 1 { \\gamma _ \\omega ( t ) \\in W ' ( \\omega ) } d t \\\\ & = \\sum _ { k = 1 } ^ d \\int _ 0 ^ 1 \\langle v , D _ k y ( 0 , \\tau _ { \\gamma _ \\omega ( t ) } \\omega ) \\rangle \\ 1 { \\gamma _ \\omega ( t ) \\in W ' } \\gamma _ \\omega ' ( t ) d t . \\end{align*}"}
{"id": "8659.png", "formula": "\\begin{align*} P - \\alpha Q = u ( R + \\alpha S + \\alpha ^ 2 U ) ^ 2 \\end{align*}"}
{"id": "763.png", "formula": "\\begin{align*} | \\Delta _ j ( 1 \\otimes f - f \\otimes 1 ) & | _ { C ^ \\alpha ( X \\times X ) } = | \\Delta _ j ( 1 \\otimes f - f \\otimes 1 ) | _ { C ^ \\alpha ( U _ j ) } \\leq \\\\ \\leq & | \\Delta _ j | _ { C ^ \\alpha ( U _ j ) } \\| 1 \\otimes f - f \\otimes 1 \\| _ { C ^ 0 ( U _ j ) } + \\| \\Delta _ j \\| _ { C ^ 0 ( U _ j ) } | 1 \\otimes f - f \\otimes 1 ) | _ { C ^ \\alpha ( U _ j ) } \\lesssim \\\\ \\lesssim & \\| f \\| _ { C ^ \\beta ( X ) } j ^ { - \\gamma } , \\end{align*}"}
{"id": "3422.png", "formula": "\\begin{align*} I \\ ! I _ 1 & = \\bigg ( \\int _ { \\| y - x _ 0 \\| \\leqslant \\frac { R } { 4 } } + \\int _ { d ( y , x _ 0 ) \\leqslant \\frac { R } { 4 } \\leqslant \\| y - x _ 0 \\| } \\bigg ) [ K ( x , y ) - K ( x , x _ 0 ) ] f _ 2 ( y ) d \\omega ( y ) \\\\ & = : I \\ ! I _ { 1 1 } + I \\ ! I _ { 1 2 } . \\end{align*}"}
{"id": "1078.png", "formula": "\\begin{align*} \\sigma _ 1 \\overline { \\Phi _ { R , L } ( x , t , \\overline { k } ) } \\sigma _ 1 = \\Phi _ { R , L } ( x , t , k ) , \\overline { \\Phi _ { R , L } ( x , t , - \\overline { k } ) } = \\Phi _ { R , L } ( x , t , k ) . \\end{align*}"}
{"id": "617.png", "formula": "\\begin{align*} \\abs { ( A ( x ) + B ( x ) ) - ( \\alpha + \\beta ) } \\ & \\leq \\ \\abs { A ( x ) - \\alpha } + \\abs { B ( x ) - \\beta } \\\\ [ 1 1 p t ] & \\leq \\ \\frac { 1 } { x + 1 } + \\frac { 1 } { x + 1 } \\\\ [ 1 1 p t ] & = \\ \\frac { 2 } { x + 1 } \\end{align*}"}
{"id": "6379.png", "formula": "\\begin{align*} Z ( S ) = \\{ P \\in \\mathbb A _ { \\mathbb C } ^ n \\ , : \\ , f ( P ) = 0 f \\in S \\} . \\end{align*}"}
{"id": "7676.png", "formula": "\\begin{align*} \\frac { \\tilde { \\lambda } _ 0 + B } { 1 - A } = \\tilde { \\lambda } _ 0 + ( \\tilde { \\lambda } _ 0 A + B ) \\frac { 1 } { 1 - A } = \\tilde { \\lambda } _ 0 + ( \\tilde { \\lambda } _ 0 A + B ) \\sum _ { k = 0 } ^ { + \\infty } A ^ k \\end{align*}"}
{"id": "9245.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , x ^ X , p ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land p = _ X J ^ A _ { \\gamma } x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "3090.png", "formula": "\\begin{align*} \\sum z _ i ^ 2 = 0 \\ , , \\sum z _ i ^ 3 = 0 \\footnote { O n e c a n c o n s i d e r t h e c o o r d i n a t e s o f a p o i n t o n t h e c u r v e a s t h e r o o t s o f a n e q u a t i o n o f t h e 5 t h d e g r e e , w h i c h h a s b e e n b r o u g h t i n t o B R I N G ' s n o r m a l f o r m b y T s c h i r n h a u s t r a n s f o r m a t i o n . } \\ , . \\end{align*}"}
{"id": "4045.png", "formula": "\\begin{align*} \\dot L ^ 2 ( 0 , 1 ) : = \\Big \\{ \\phi \\in L ^ 2 ( 0 , 1 ) : \\int _ 0 ^ 1 \\phi = 0 \\Big \\} . \\end{align*}"}
{"id": "585.png", "formula": "\\begin{align*} \\begin{cases} \\ \\ W _ 1 \\ = \\ X _ 1 \\cup Y _ 1 , W _ 2 \\ = \\ X _ 2 \\cup Y _ 2 \\\\ [ 8 p t ] \\ Z _ 1 \\ \\ \\ = \\ X _ 1 \\cap Y _ 1 , Z _ 2 \\ \\ = \\ X _ 2 \\cap Y _ 2 \\end{cases} \\end{align*}"}
{"id": "8775.png", "formula": "\\begin{align*} \\Phi ^ P : = \\bigl \\{ ( u , \\phi ) \\in \\R ^ { d \\times ( n + 1 ) } \\times \\R \\bigm | \\phi = \\phi ( u _ { 1 n } , \\ldots , u _ { d n } ) , \\ u \\in P \\bigr \\} . \\end{align*}"}
{"id": "1937.png", "formula": "\\begin{align*} g ( x _ 0 ) - \\underline { g } & \\geq \\sum _ { k = 0 } ^ K g ( x _ k ) - g ( x _ { k + 1 } ) \\\\ & \\geq K c _ 1 \\tau _ 1 \\min ( \\underline { \\alpha _ 1 } , t _ 1 ( x _ k ) ) \\norm { \\nabla g ( x _ k ) } ^ 2 \\\\ & \\geq K c _ 1 \\tau _ 1 \\min ( \\underline { \\alpha _ 1 } , \\underline { t _ 1 } ) \\varepsilon _ 1 ^ 2 . \\end{align*}"}
{"id": "6954.png", "formula": "\\begin{align*} P _ f ( x ) = \\bigcup _ { j \\in \\omega } ( f ^ { - 1 } ) ^ { ( j ) } ( x ) . \\end{align*}"}
{"id": "3905.png", "formula": "\\begin{align*} \\pi ( x ) = \\# \\{ p \\leq x : p \\} , \\end{align*}"}
{"id": "2946.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Lambda _ { n , 1 } = 0 , \\lim _ { n \\to \\infty } \\Lambda _ { n , 2 } = \\frac 1 2 . \\end{align*}"}
{"id": "4278.png", "formula": "\\begin{align*} w ^ { ( 1 ) } ( t , x ) ~ = ~ \\begin{cases} \\overline { w } ( x ) & \\qquad \\mathrm { i f } ~ ~ x \\in ( t , 0 ) \\cup ( 0 , \\infty ) , \\\\ [ 2 m m ] \\overline { w } ( x + \\tau _ 0 - t ) & \\qquad \\mathrm { i f } ~ ~ x \\in ( - \\infty , t ) , \\end{cases} \\end{align*}"}
{"id": "7593.png", "formula": "\\begin{align*} \\| f - { \\mathsf { C } } _ { d _ x , d _ y } [ f ] \\| _ 1 = \\int _ D \\left | f ( x , y ) - { \\mathsf { C } } _ { d _ x , d _ y } [ f ] ( x , y ) \\right | d x d y \\le 4 \\sum _ { i = d _ x + 1 } ^ { \\infty } \\sum _ { j = d _ y + 1 } ^ { \\infty } \\left | c _ { i , j } \\right | , \\end{align*}"}
{"id": "8486.png", "formula": "\\begin{align*} f _ k & = [ u ^ k ] F ( u ) = \\frac { 2 ^ k z ^ k } { ( 1 + \\sqrt { 1 - 4 z } ) ^ k } , \\\\ g _ k & = [ u ^ k ] G ( u ) = \\frac { 2 ^ k z ^ k ( 1 - 2 z - \\sqrt { 1 - 4 z } ) } { ( 1 + \\sqrt { 1 - 4 z } ) ^ { k + 1 } } , \\mbox { a n d } \\\\ h _ k & = [ u ^ k ] H ( u ) = \\frac { 2 ^ { k + 1 } z ^ { k + 1 } } { ( 1 + \\sqrt { 1 - 4 z } ) ^ { k + 1 } } . \\end{align*}"}
{"id": "9053.png", "formula": "\\begin{align*} \\frac { \\phi _ 0 + \\phi _ 1 } { 2 } - \\beta _ a \\epsilon ( a ) \\frac { \\phi _ 1 - \\phi _ 0 } { h } = \\phi ^ b ( a ) , \\frac { \\phi _ { N + 1 } + \\phi _ N } { 2 } + \\beta _ b \\epsilon ( b ) \\frac { \\phi _ { N + 1 } - \\phi _ N } { h } = \\phi ^ b ( b ) . \\end{align*}"}
{"id": "9275.png", "formula": "\\begin{align*} x _ { n + 1 } = J ^ S _ { \\mu _ n } ( x _ n + \\mu _ n T _ { \\lambda _ n } x _ n ) \\end{align*}"}
{"id": "3171.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } \\frac { e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , n ) + 1 } } _ 2 ^ 2 } , \\end{align*}"}
{"id": "6943.png", "formula": "\\begin{align*} \\Vert \\varphi \\Vert _ { 0 , E , \\omega } = \\left ( \\sum _ { \\iota \\in I ^ E } \\varphi ^ 2 ( \\xi ^ E _ \\iota ) \\ , \\omega ^ E _ \\iota \\right ) ^ { 1 / 2 } \\ , . \\end{align*}"}
{"id": "1410.png", "formula": "\\begin{align*} { \\rm A d } _ { \\hat q } \\hat v = \\hat q \\hat v \\hat q ^ * . \\end{align*}"}
{"id": "6492.png", "formula": "\\begin{align*} \\prod _ { j = j _ 0 ( \\delta ) + 1 } ^ { n - 1 } \\left ( 1 + \\dfrac { \\delta } { j } \\right ) = \\prod _ { j = k + 1 } ^ { n - 1 } \\dfrac { j - k } { j } = \\dfrac { \\Gamma ( n - k ) \\times k ! } { \\Gamma ( n ) } . \\end{align*}"}
{"id": "3048.png", "formula": "\\begin{align*} x _ 1 ^ \\prime : x _ 2 ^ \\prime = x _ 1 : x _ 2 \\ , , y _ 1 ^ \\prime : y _ 2 ^ \\prime = y _ 1 : y _ 2 \\ , , \\end{align*}"}
{"id": "5831.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 : = \\left \\{ \\exists \\ , y ^ { \\prime } \\in \\left ( w + [ - 4 \\lambda H , 5 \\lambda H ) \\times \\{ H ^ \\prime \\} \\right ) \\cap \\mathbb { L } : X ^ { y ^ { \\prime } } _ { H - H ^ \\prime } - \\pi _ 1 ( y ^ \\prime ) \\geq \\left ( v - v ^ { \\prime } \\right ) H \\right \\} . \\end{align*}"}
{"id": "283.png", "formula": "\\begin{align*} V ( x , t ) & : = - \\kappa d V _ { * } \\left ( \\frac { x } { \\sqrt { 1 + t } } \\right ) ( 1 + t ) ^ { - 1 } \\log ( 1 + t ) , \\\\ V _ { * } ( x ) & : = \\frac { 1 } { \\sqrt { 4 \\pi } } \\frac { d } { d x } \\left ( \\eta _ { * } ( x ) e ^ { - \\frac { x ^ { 2 } } { 4 } } \\right ) = \\frac { 1 } { 4 \\sqrt { \\pi } } \\left ( \\beta \\chi _ { * } ( x ) - x \\right ) \\eta _ { * } ( x ) e ^ { - \\frac { x ^ { 2 } } { 4 } } , \\\\ d & : = \\int _ { \\R } ( \\eta _ { * } ( y ) ) ^ { - 1 } ( \\chi _ { * } ( y ) ) ^ { 3 } d y , \\kappa : = \\frac { \\beta ^ { 2 } \\gamma } { 8 } . \\end{align*}"}
{"id": "4999.png", "formula": "\\begin{align*} \\rho ( f ) = \\cfrac { 1 } { r _ 0 + \\cfrac { 1 } { r _ 1 + \\cfrac { 1 } { r _ 2 + \\dotsb } } } . \\end{align*}"}
{"id": "7719.png", "formula": "\\begin{align*} \\Box \\phi = \\left ( | \\phi _ t | ^ 2 - | \\phi _ x | ^ 2 \\right ) \\phi + a ( x ) \\phi _ t , \\ ; \\phi [ 0 ] \\in \\mathbb { S } ^ k \\times T \\mathbb { S } ^ k . \\end{align*}"}
{"id": "2558.png", "formula": "\\begin{align*} \\widehat { V } _ Q ^ { - 1 } = \\widehat { V } _ { - Q } \\widehat { D } _ { L , m } ^ { - 1 } = \\widehat { D } _ { L ^ { - 1 } , - m } . \\end{align*}"}
{"id": "8771.png", "formula": "\\begin{align*} s _ { i j } = \\min \\bigl \\{ m _ { i j t _ i ( H ) } , s _ { i n } \\bigr \\} i \\in \\{ 1 , \\ldots , d \\} \\ ; j \\in \\{ 0 , \\ldots , n \\} . \\end{align*}"}
{"id": "276.png", "formula": "\\begin{align*} u _ { t } + u u _ { x } - u _ { x x } = 0 , \\ \\ x \\in \\R , \\ t > 0 . \\end{align*}"}
{"id": "6304.png", "formula": "\\begin{align*} p _ { K '' | K } ( k '' ) & = \\frac { \\binom { K } { k '' } } { \\binom { Q } { K } } \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { \\forall \\mathcal { I } , \\mathcal { J } \\subset \\mathcal { Q } : \\\\ \\mathcal { I } \\cap \\mathcal { J } = \\emptyset , \\\\ | \\mathcal { I } | = k '' , | \\mathcal { J } | = K - k '' } } \\ ! \\ ! \\ ! \\ ! \\prod _ { i \\in \\mathcal { I } } \\xi _ i \\prod _ { j \\in \\mathcal { J } } ( 1 - \\xi _ j ) \\end{align*}"}
{"id": "3243.png", "formula": "\\begin{align*} K ( \\omega , z ) = ( \\omega - z ) \\left ( \\omega - z - \\frac { 1 } { G _ { { \\mu } } ( \\omega ) } \\right ) , z , \\omega \\in \\mathbb { C } ^ + . \\end{align*}"}
{"id": "6918.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { \\boldsymbol { \\Theta } , \\mathbf { w } _ m } & \\biggl | \\sum _ { n = 1 } ^ N g _ n c _ n h _ n B _ r ( \\theta _ I ) + h _ d B _ d ( \\theta _ d ) \\biggr | ^ 2 \\\\ \\textrm { s . t . } & \\| \\mathbf { w } _ m \\| ^ 2 \\leqslant 1 , \\ : m = 1 , 2 , \\\\ & \\phi _ n \\in [ 0 , 2 \\pi ) , \\ : \\forall n = 1 , 2 , \\ldots , N . \\end{aligned} \\end{align*}"}
{"id": "7485.png", "formula": "\\begin{align*} \\mathcal { H } ^ n & = \\left ( - 1 + \\frac { \\tau } { 2 } \\eta ^ n - \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n \\right ) \\phi ^ { n - 1 } + \\frac { \\tau ^ 2 } { 4 } \\Delta \\phi ^ { n - 1 } \\\\ & \\quad \\ ; + 2 \\phi ^ n + \\tau ^ 2 \\left ( \\Big ( \\vartheta ^ n + \\chi ^ n - \\sigma R ^ \\prime ( \\| \\phi ^ n \\| ^ 2 - 1 ) \\Big ) \\phi ^ n - \\frac { 1 } { 2 } \\Delta \\phi ^ n - G ( \\phi ^ n ) \\right ) \\end{align*}"}
{"id": "7133.png", "formula": "\\begin{align*} D _ u x ^ n _ t = \\int _ u ^ t \\tilde { f } ' _ n ( s , x _ s ^ n ) D _ u x _ s ^ n d s + K _ H ( t , u ) \\end{align*}"}
{"id": "5844.png", "formula": "\\begin{align*} \\psi _ c ( x ) = \\int _ { - \\infty } ^ x g _ c ( x ' ) d x ' , x \\in \\R . \\end{align*}"}
{"id": "1809.png", "formula": "\\begin{align*} G = \\{ x \\rightarrow x y , \\ ; \\ ; y \\rightarrow x y \\} . \\end{align*}"}
{"id": "5578.png", "formula": "\\begin{align*} y & \\ : = \\ P _ { ( A \\cup B ) ^ c } ( x ) + t 1 _ { B } \\ = : \\ ( y _ 1 , y _ 2 , \\ldots ) \\\\ z & \\ : = \\ P _ { ( A \\cup B ) ^ c } ( x ) + t 1 _ { F } \\ = : \\ ( z _ 1 , z _ 2 , \\ldots ) . \\end{align*}"}
{"id": "3721.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi ^ N ( k , t ) = & \\ i \\sum _ { \\substack { m + n = k , \\\\ | m | , | n | , | k | \\leq N } } m | n | e ^ { \\frac 1 2 \\mu ( | k | ^ \\alpha - | m | ^ \\alpha - | n | ^ \\alpha ) t } \\xi ^ N ( m ) \\xi ^ N B ( n ) \\\\ & - \\frac 1 2 \\mu | k | ^ \\alpha \\xi ^ N ( k ) . \\end{align*}"}
{"id": "1218.png", "formula": "\\begin{align*} 1 = \\sum _ { I \\in \\mathcal { K } _ k } \\vert I \\vert ^ { \\frac { \\log 2 } { \\log 3 } } . \\end{align*}"}
{"id": "9262.png", "formula": "\\begin{align*} x _ 0 \\in \\mathrm { d o m } A , \\ ; x _ { n + 1 } = J ^ A _ { \\gamma _ n } x _ n \\end{align*}"}
{"id": "6373.png", "formula": "\\begin{align*} \\sigma _ { \\mathcal { X } } = \\left ( \\mathcal { E } _ p , \\mathcal { E } _ { p _ 1 } , \\mathcal { E } _ { p _ 2 } , \\mathcal { E } _ { p _ 3 } , \\mathcal { E } _ { p _ 4 } , \\Phi ( \\sigma _ { \\widetilde { X } } ) \\right ) , \\end{align*}"}
{"id": "5735.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( v ) \\cap ( V ( H ) \\cup \\{ u _ { 1 } , \\cdots , u _ { l - 1 } \\} ) | & \\geq d _ { G } ( v ) - | N _ { G } ( v ) \\cap V ( B ) | \\\\ & - | N _ { G } ( v ) \\cap ( V ( T ' ) \\setminus \\{ u _ { 1 } , \\cdots , u _ { l - 1 } \\} ) | \\\\ & \\geq ( t + 3 ) - 2 - ( t - \\lfloor \\frac { l - 1 } { 2 } \\rfloor ) \\\\ & \\geq \\lfloor \\frac { l - 1 } { 2 } \\rfloor + 1 . \\end{aligned} \\end{align*}"}
{"id": "3875.png", "formula": "\\begin{align*} \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\end{align*}"}
{"id": "8790.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { j \\notin L _ { i k } } \\lambda _ { i j } \\leq \\delta _ { i k } , \\ \\sum _ { j \\notin R _ { i k } } \\lambda _ { i j } \\leq 1 - \\delta _ { i k } k \\in K _ i , \\\\ & \\lambda _ i \\in \\Lambda _ i , \\ \\delta _ i \\in \\{ 0 , 1 \\} ^ { \\lceil \\log _ 2 l _ i \\rceil } , \\end{aligned} \\end{align*}"}
{"id": "250.png", "formula": "\\begin{align*} X _ \\varepsilon = _ { \\cal L } X _ \\mu + Z _ \\varepsilon , \\end{align*}"}
{"id": "3531.png", "formula": "\\begin{align*} W _ 1 T = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + \\begin{cases} O ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } ) & ( \\frac { 3 } { 2 } > \\sigma _ 2 > 1 ) \\\\ O ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } ( \\log T ) ^ 2 ) & ( \\sigma _ 2 = 1 ) \\\\ O ( T ^ { 4 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } ) & ( \\sigma _ 2 < 1 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "600.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) \\ & = \\ g ( P _ 2 ^ 2 ( x _ 1 , x _ 2 ) , P _ 1 ^ 2 ( x _ 1 , x _ 2 ) ) \\\\ [ 1 1 p t ] & = \\ g ( x _ 2 , x _ 1 ) . \\end{align*}"}
{"id": "32.png", "formula": "\\begin{align*} g = g ^ T \\oplus g ^ \\Sigma , \\end{align*}"}
{"id": "8110.png", "formula": "\\begin{align*} S _ { k } ( G ) = \\bigcup _ { ( d , e ) \\in D _ { k } ( G ) } d \\times e \\cup e \\times d . \\end{align*}"}
{"id": "2389.png", "formula": "\\begin{align*} f = \\lim _ { N \\to \\infty } \\sum _ { k = 1 } ^ N e _ { \\pi ( k ) } . \\end{align*}"}
{"id": "4.png", "formula": "\\begin{align*} \\tanh { x } = 8 x \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { 4 x ^ 2 + \\pi ^ 2 \\left ( 2 k + 1 \\right ) ^ 2 } , x \\in \\mathbb { C } . \\end{align*}"}
{"id": "7013.png", "formula": "\\begin{align*} | \\det { \\mathfrak { S } } | & = | A ( \\beta _ 1 ) \\cdots A ( \\beta _ { k } ) | \\geqslant c _ j ^ k \\prod _ { i = 1 } ^ { k } | \\beta _ i - \\alpha _ j | ^ { m _ j } \\\\ & = c _ j ^ k | B ( \\alpha _ j ) | ^ { m _ j } \\geqslant c _ j ^ k \\delta ^ { m _ j } \\geqslant c _ { j } ^ { K } \\delta ^ { m _ j } . \\end{align*}"}
{"id": "6801.png", "formula": "\\begin{align*} \\varphi ( x + x ^ { \\prime } ) = \\varphi ( x ) x ^ { \\prime } \\in B _ { l ( x ) } ^ { n } . \\end{align*}"}
{"id": "5709.png", "formula": "\\begin{align*} 1 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 0 } \\cup { \\gamma _ { 1 } } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) = n + 3 - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) . \\end{align*}"}
{"id": "3367.png", "formula": "\\begin{align*} \\begin{cases} \\theta ( z _ 1 , [ \\mathfrak { X } , z _ 2 ] ) D ( \\mathfrak { X } ) + \\theta ( [ \\mathfrak { X } , z _ 1 ] , z _ 2 ) D ( \\mathfrak { X } ) + \\theta ( [ \\mathfrak { X } , z _ 1 ] , [ \\mathfrak { X } , z _ 2 ] ) = 0 , \\\\ \\theta ( [ \\mathfrak { X } , z _ 1 ] , [ \\mathfrak { X } , z _ 2 ] ) D ( \\mathfrak { X } ) = 0 . \\end{cases} \\end{align*}"}
{"id": "7625.png", "formula": "\\begin{align*} ( \\sum _ { l } r _ { l } ^ { 2 } ) ( \\sum _ { i } F ^ { i i } ) - F ^ { i j } r _ { i } r _ { j } = ( \\sum _ { l } r _ { l } ^ { 2 } ) ( \\sum _ { i } f ^ { i } ) - f ^ { i } r _ { i } ^ { 2 } \\geq 0 , \\end{align*}"}
{"id": "8484.png", "formula": "\\begin{align*} \\begin{array} { l } f _ 0 = 1 , \\mbox { a n d } f _ k = z \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } f _ \\ell + z \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } g _ \\ell + z \\sum \\limits _ { \\ell = 0 } ^ { k - 1 } h _ \\ell , k \\geq 1 , \\\\ g _ k = z f _ { k + 1 } + z g _ { k + 1 } + z h _ { k + 1 } , k \\geq 0 , \\\\ h _ k = z f _ k + z g _ k + z h _ k , k \\geq 0 . \\\\ \\end{array} \\end{align*}"}
{"id": "7021.png", "formula": "\\begin{align*} \\mathbf { v } _ i = \\frac { 1 } { \\delta } ( B _ 1 ( \\alpha _ i ) , B _ 2 ( \\alpha _ i ) , \\dots , B _ L ( \\alpha _ i ) ) \\end{align*}"}
{"id": "1661.png", "formula": "\\begin{align*} \\Phi \\big ( ( x , v ) , ( x ' , v ' ) \\big ) : = \\big ( ( | x - x ' | + | v - v ' | ) \\wedge 1 \\big ) \\big ( \\mathcal W ( x , v ) + \\mathcal W ( x ' , v ' ) \\big ) \\end{align*}"}
{"id": "4804.png", "formula": "\\begin{align*} B ^ \\perp = \\{ \\alpha \\in X ( A ) \\mid \\alpha ( b ) = 1 b \\in B \\} . \\end{align*}"}
{"id": "9554.png", "formula": "\\begin{align*} H ' ( v ( s ) , \\mu ( s ) ) \\left [ \\begin{array} { c } \\frac { d v } { d s } \\\\ \\frac { d \\mu } { d s } \\\\ \\end{array} \\right ] = 0 , \\ \\| ( \\frac { d v } { d s } , \\frac { d \\mu } { d s } ) \\| = 1 , \\ v ( 0 ) = v ^ { ( 0 ) } , \\ \\mu ( 0 ) = 1 , \\ \\frac { d \\mu } { d s } ( 0 ) < 0 , \\end{align*}"}
{"id": "6909.png", "formula": "\\begin{align*} R = \\log \\left ( 1 + \\frac { P _ t \\Bigl | \\bigl ( \\mathbf { g } ^ T \\boldsymbol { \\Theta } \\mathbf { H } + \\mathbf { h } _ d ^ T \\bigr ) \\mathbf { w } \\Bigr | ^ 2 } { \\sigma _ n ^ 2 } \\right ) \\end{align*}"}
{"id": "2317.png", "formula": "\\begin{align*} A ( f , g ) ( x , \\omega ) & = e ^ { \\pi i x \\cdot \\omega } V _ g f ( x , \\omega ) = e ^ { \\pi i x \\cdot \\omega } e ^ { - 2 \\pi i x \\cdot \\omega } V _ { \\widehat { g } } \\widehat { f } ( \\omega , - x ) \\\\ & = e ^ { - \\pi i x \\cdot \\omega } V _ { \\widehat { g } } \\widehat { f } ( \\omega , - x ) = A ( \\widehat { f } , \\widehat { g } ) ( \\omega , - x ) . \\end{align*}"}
{"id": "4373.png", "formula": "\\begin{align*} b ( \\tau ) \\sim e ^ { \\frac { 2 } { \\alpha } ( \\frac { \\alpha } { 2 } - l ) \\tau } . \\end{align*}"}
{"id": "710.png", "formula": "\\begin{align*} \\Delta _ { \\alpha \\alpha } ^ { ( \\ell ) } = \\frac { C _ W } { n _ { \\ell } } \\sum _ { j = 1 } ^ { n _ \\ell } X _ j . \\end{align*}"}
{"id": "528.png", "formula": "\\begin{align*} \\Xi _ { k + 1 } & = \\mathcal { A } \\Xi _ k + \\mathcal { B } \\mathcal { G } ( \\Psi ( \\mathbf { u } _ k , { \\tilde { \\Xi } _ k } ) + E ( \\mathbf { u } _ k , { \\tilde { \\Xi } _ k } ) + D ( \\omega _ k ) ) , \\\\ { \\mathbf { \\tilde { y } } _ k } & { = \\mathcal { C } \\Xi _ k + \\mathbf { w } _ k } , \\end{align*}"}
{"id": "7884.png", "formula": "\\begin{align*} P _ d ( x ) = \\sum _ { | J | = d } p _ J x ^ J , \\ Q _ d ( x ) = \\sum _ { | J | = d } q _ J x ^ J , \\ x ^ J = x _ 0 ^ { j _ 0 } \\cdot \\ldots \\cdot x _ n ^ { j _ n } . \\end{align*}"}
{"id": "1198.png", "formula": "\\begin{align*} \\pi : x = ( x _ n ) _ { n \\in \\mathbb { N } } \\mapsto \\lim _ { n \\to + \\infty } f _ { ( x _ 1 , . . . , x _ n ) } ( 0 ) . \\end{align*}"}
{"id": "8662.png", "formula": "\\begin{align*} p ^ { \\mu } q _ { \\mu } : = - p ^ 0 q ^ 0 + \\sum _ { i = 1 } ^ 3 p _ i q _ i . \\end{align*}"}
{"id": "6271.png", "formula": "\\begin{align*} n ' = | T | + | ( U \\cup V ) \\setminus ( U _ 4 \\cup V _ 4 ) | & \\leq 3 0 h ^ 2 \\varepsilon n + 1 6 0 h ^ 3 \\varepsilon n ^ { \\frac { 3 } { 2 } } + 3 \\cdot ( 6 h ) ^ { s + 3 } \\varepsilon ^ { \\frac { s + 1 } { 2 } } n ^ { \\frac { s + 2 } { 2 } } . \\end{align*}"}
{"id": "6554.png", "formula": "\\begin{align*} \\| K ( t ) \\| _ { \\dot { H } ^ { s } } ^ { 2 } = & \\int _ { \\mathbb { R } ^ { 2 } } { | \\xi | ^ { 2 s } | \\widehat { K } ( \\xi , t ) | ^ { 2 } } \\ , d \\xi \\\\ = & \\int _ { \\mathbb { R } ^ { 2 } } { | \\xi | ^ { 2 s } e ^ { - 2 t | \\xi | ^ { 2 } g ( \\xi ) } } \\ , d \\xi \\\\ \\leq & C t ^ { - ( s + 1 ) } g ( A _ { t } ) ^ { - ( s + 1 ) } . \\end{align*}"}
{"id": "9049.png", "formula": "\\begin{align*} \\begin{aligned} ( \\rho ^ { n + 1 } , \\phi ^ { n + 1 } ) = & \\arg \\min _ { ( \\rho , \\phi ) \\in \\mathcal { A } , m } \\left \\{ \\frac { 1 } { 2 \\tau } \\sum _ { i = 1 } ^ s \\int _ { \\Omega } F ( \\rho _ i , m _ i ) D ^ { - 1 } _ i d x + E ( \\rho , \\phi ) \\right \\} , \\\\ & \\rho _ i - \\rho _ i ^ n + \\nabla \\cdot ( m _ i ) = 0 , m _ i \\cdot \\mathbf { n } = 0 , x \\in \\partial \\Omega . \\end{aligned} \\end{align*}"}
{"id": "40.png", "formula": "\\begin{align*} d \\omega _ { t } = \\theta \\wedge d \\theta _ { t } ^ { c } = \\theta \\wedge \\omega _ { t } . \\end{align*}"}
{"id": "7728.png", "formula": "\\begin{align*} D = \\{ ( u , v ) : | u | \\leq 1 8 \\pi , | v | \\leq 1 8 \\pi \\} . \\end{align*}"}
{"id": "6540.png", "formula": "\\begin{align*} \\left | \\dfrac { \\sqrt { s _ n ^ 2 } } { \\sqrt { \\sigma _ n ^ 2 } } - 1 \\right | = \\dfrac { | s _ n ^ 2 - \\sigma _ n ^ 2 | } { \\sqrt { \\sigma _ n ^ 2 } \\cdot ( \\sqrt { s _ n ^ 2 } + \\sqrt { \\sigma _ n ^ 2 } ) } \\leq \\dfrac { | s _ n ^ 2 - \\sigma _ n ^ 2 | } { \\sigma _ n ^ 2 } , \\end{align*}"}
{"id": "1317.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { | \\hat { I } _ { = 2 , < \\epsilon } ( k _ { n } ) | } { k _ { n } } = 1 . \\end{align*}"}
{"id": "4920.png", "formula": "\\begin{align*} \\Gamma _ t = \\prod _ { k = 1 } ^ t ( 1 - \\alpha _ t ) , ~ ~ ~ \\forall t \\geq 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\Gamma _ 1 = 1 . \\end{align*}"}
{"id": "9396.png", "formula": "\\begin{align*} o ( t ) = & \\tau _ t \\Big ( ( a _ 1 - \\tau _ t ( a _ 1 ) ) \\cdots ( a _ n - \\tau _ t ( a _ n ) ) \\Big ) \\\\ = & \\tau _ t ( a _ 1 \\cdots a _ n ) + \\sum _ { r = 1 } ^ n ( - 1 ) ^ r \\sum _ { 1 \\leq k _ 1 < \\ldots < k _ r \\leq n } \\left ( \\prod _ { i = 1 } ^ { r } \\tau _ t ( a _ { k _ i } ) \\right ) \\tau _ t ( a _ 1 \\cdots \\hat { a } _ { k _ 1 } \\cdots \\hat { a } _ { k _ r } \\cdots a _ n ) \\end{align*}"}
{"id": "1598.png", "formula": "\\begin{align*} \\varphi = \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 1 ) } - \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 2 ) } \\end{align*}"}
{"id": "4247.png", "formula": "\\begin{align*} \\rho _ c ( x ) = \\rho ( x ) - a x ^ 4 < P ( x ) , \\textrm { f o r } | x | \\le c / 2 . \\end{align*}"}
{"id": "3329.png", "formula": "\\begin{align*} e _ { - i } & = \\lfloor \\frac { 2 n - 1 } { n - 1 } i \\rfloor - i + 1 = \\lfloor \\frac { n } { n - 1 } i \\rfloor + 1 \\\\ a _ i & \\ge ( n - 1 ) i + 1 . \\end{align*}"}
{"id": "7483.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { \\tau } { 2 } \\eta ^ n + \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n \\right ) \\tilde { \\phi } ^ { n + 1 } - \\frac { \\tau ^ 2 } { 4 } \\Delta \\tilde { \\phi } ^ { n + 1 } = \\mathcal { H } ^ n . \\end{align*}"}
{"id": "9482.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b c _ { s } ( n ) q ^ n = \\sum _ { n = 0 } ^ { \\infty } c s _ { s } ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ { s } ; q ^ { s } ) ^ { ( s - 1 ) / 2 } _ \\infty } { ( - q ^ s ; q ^ s ) _ \\infty } . \\end{align*}"}
{"id": "7850.png", "formula": "\\begin{align*} \\bigg \\| \\frac { T ^ { * 2 K } x } { x _ { 2 K } w _ { 2 K - 1 } \\cdots w _ { 0 } } - e _ { 0 } \\bigg \\| ^ { 2 } & \\leq \\frac { \\mu ^ { 2 } \\delta } { w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } } \\sum _ { i = 0 } ^ { \\infty } w _ { 2 i + 2 K + 1 } ^ { 2 } ~ \\bigg | \\frac { x _ { 2 i + 2 K + 2 } } { x _ { 2 K } } \\bigg | ^ { 2 } \\\\ & \\leq \\frac { \\mu ^ { 2 } \\delta } { w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } } \\sum _ { i = 0 } ^ { \\infty } w _ { 2 i + 2 K + 1 } ^ { 2 } < C \\epsilon , \\end{align*}"}
{"id": "7303.png", "formula": "\\begin{align*} ( \\alpha + \\beta ) \\circ \\gamma = ( \\alpha \\circ \\gamma ) + ( \\beta \\circ \\gamma ) , \\\\ ( \\alpha \\cdot \\beta ) \\circ \\gamma = ( \\alpha \\circ \\gamma ) \\cdot ( \\beta \\circ \\gamma ) \\end{align*}"}
{"id": "4692.png", "formula": "\\begin{align*} \\Psi _ V = & \\sum _ { i = 1 } ^ n \\big [ \\dot { \\mu } _ i \\sigma _ i \\Lambda \\R _ i - ( \\dot { x } _ i - \\mu _ i ) \\sigma _ i \\partial _ y \\R _ i \\big ] + I + J + K + L , \\end{align*}"}
{"id": "4680.png", "formula": "\\begin{align*} \\Psi _ V = & E _ V - \\sum _ { i = 1 } ^ n ( \\dot { x } _ i - \\mu _ i ) \\sigma _ i \\partial _ y \\widetilde { R } _ i + \\sum _ { i = 1 } ^ n \\bigg ( \\dot { \\mu } _ i + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { x ^ 3 _ { i j } } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\mu _ k } { x ^ 3 _ { i j } } \\bigg ) \\frac { \\sigma _ i \\Lambda \\widetilde { R } _ i } { 1 + \\mu _ i } , \\end{align*}"}
{"id": "1083.png", "formula": "\\begin{align*} & a _ { + } ( k ) = a ^ * _ { - } ( k ) , b _ { + } ( k ) = - b ^ * _ { - } ( k ) , & k \\in I _ { R } , \\\\ & a _ { + } ( k ) = b _ { - } ^ { * } ( k ) , b _ { + } ( k ) = a ^ * _ { - } ( k ) , & k \\in I _ { L } \\backslash I _ { R } , \\\\ & a _ { + } ( k ) = a _ { - } ( k ) , b _ { + } ( k ) = b _ { - } ( k ) , & k \\in \\mathbb { R } \\backslash ( I _ L \\cup I _ R ) . \\end{align*}"}
{"id": "3955.png", "formula": "\\begin{align*} A ^ * = \\begin{pmatrix} \\partial _ x & \\partial _ x \\\\ [ 4 p t ] \\partial _ x & \\partial _ { x x } + \\partial _ x \\end{pmatrix} , \\end{align*}"}
{"id": "8514.png", "formula": "\\begin{align*} b \\circ a \\circ \\overline { b } = b \\circ a \\circ \\overline { b } \\circ \\overline { a } + \\lambda _ { b \\circ a \\circ \\overline { b } \\circ \\overline { a } } ( a ) \\in T _ + ( A ) . \\end{align*}"}
{"id": "2742.png", "formula": "\\begin{align*} m _ { 0 , \\lambda } ( V ) = 0 , \\mbox { i f $ \\lambda \\neq ( n ) , 1 ^ n $ } . \\end{align*}"}
{"id": "8650.png", "formula": "\\begin{align*} \\Delta ( P - T Q ) = ( - 1 ) ^ { \\frac { n } { 2 } + 1 } ( n - 1 ) ^ { n - 1 } n ^ n T ^ { n - 2 } ( T - ( n - 2 ) ^ { n - 2 } ) ^ 2 \\end{align*}"}
{"id": "6455.png", "formula": "\\begin{align*} \\lambda _ h = \\lambda _ 0 - i \\frac { \\eta _ 0 } { 4 \\pi } { \\lambda _ 0 } ^ \\frac { 5 } { 2 } { U _ 0 } ^ 2 h + \\mathcal { O } ( h ^ 2 ) . \\end{align*}"}
{"id": "5638.png", "formula": "\\begin{align*} \\mathrm { R } _ t { } ^ \\delta _ { \\alpha \\beta \\gamma } = \\frac { \\mathbf { g } _ { \\alpha \\gamma } \\delta ^ \\delta _ \\beta - \\mathbf { g } _ { \\beta \\gamma } \\delta ^ \\delta _ \\alpha } { 2 c } , ~ ~ ~ ~ ~ \\mathrm { R i c } _ t { } _ { \\beta \\gamma } = \\mathrm { R } _ t { } ^ \\alpha _ { \\alpha \\beta \\gamma } = - \\frac { \\mathbf { g } _ { \\beta \\gamma } } { 2 c } , ~ ~ ~ ~ ~ \\mathfrak { R } _ t = \\mathrm { R i c } _ t { } _ { \\beta \\beta } = - \\frac { 1 } { c } . \\end{align*}"}
{"id": "4965.png", "formula": "\\begin{align*} & \\overline { \\ < P s i > _ { m , n } ( t , \\tilde { y } ) - \\ < P s i > _ { m , n } ( s , \\tilde { y } ) } \\\\ & = i \\int _ s ^ t \\int _ { \\mathbb { R } ^ d } \\overline { a _ { m , n } ( t - t ' , \\tilde { y } - x ' ) } W ( d t ' , d x ' ) + i \\int _ 0 ^ s \\int _ { \\mathbb { R } ^ d } \\Big [ \\overline { a _ { m , n } ( t - t ' , \\tilde { y } - x ' ) - a _ { m , n } ( s - t ' , \\tilde { y } - x ' ) } \\Big ] W ( d t ' , d x ' ) . \\end{align*}"}
{"id": "4098.png", "formula": "\\begin{align*} & 2 ( x - y ) \\cdot ( \\widetilde \\mu ( x ) - \\widetilde \\mu ( y ) ) + ( p _ 1 - 1 ) \\cdot ( \\widetilde \\sigma ( x ) - \\widetilde \\sigma ( y ) ) ^ 2 \\\\ & \\qquad = 2 ( y - x ) \\cdot ( \\widetilde \\mu ( y ) - \\widetilde \\mu ( x ) ) + ( p _ 1 - 1 ) \\cdot ( \\widetilde \\sigma ( y ) - \\widetilde \\sigma ( x ) ) ^ 2 , \\end{align*}"}
{"id": "2340.png", "formula": "\\begin{align*} \\sum _ { l \\in \\Z } \\widehat { f } ( \\omega + l ) = \\sum _ { k \\in \\Z } f ( k ) e ^ { - 2 \\pi i k \\omega } . \\end{align*}"}
{"id": "3469.png", "formula": "\\begin{gather*} e _ { P _ j } ( K / k ) = l ^ { \\beta _ j } , 1 \\leq \\beta _ j \\leq n , 1 \\leq j \\leq r , \\quad e _ { \\infty } ( K / k ) = l ^ { t ' } , 0 \\leq t ' \\leq n \\end{gather*}"}
{"id": "7112.png", "formula": "\\begin{align*} d x ( t ) = ( \\alpha _ { - 1 } x ( t ) ^ { - 1 } - \\alpha _ { 0 } + \\alpha _ { 1 } x ( t ) - \\alpha _ { 2 } x ( t ) ^ { \\rho } ) d t + \\sigma x ( t ) ^ { \\theta } d ^ { \\circ } B _ t ^ H \\end{align*}"}
{"id": "2489.png", "formula": "\\begin{align*} ( x , \\omega , \\tau ) \\bullet ( x ' , \\omega ' , \\tau ' ) = ( x + x ' , \\omega + \\omega ' , \\tau + \\tau ' + \\tfrac { 1 } { 2 } ( x ' \\cdot \\omega - x \\cdot \\omega ' ) ) \\end{align*}"}
{"id": "2540.png", "formula": "\\begin{align*} U \\pi ( x , \\omega , \\tau ) f & = U \\Big ( \\sum _ { k = 1 } ^ n c _ k e ^ { 2 \\pi i \\tau } \\pi ( x , \\omega , 0 ) \\pi ( x _ k , \\omega _ k , 0 ) g \\Big ) \\\\ & = \\sum _ { k = 1 } ^ n c _ k e ^ { 2 \\pi i \\tau } \\rho ( x , \\omega ) \\rho ( x _ k , \\omega _ k ) \\varphi \\\\ & = \\rho ( x , \\omega , \\tau ) U f . \\end{align*}"}
{"id": "6347.png", "formula": "\\begin{align*} \\left | \\log \\frac { x } { y } \\right | = \\int _ { \\min ( x , y ) } ^ { \\max ( x , y ) } \\frac { d u } { u } \\geq \\frac { | x - y | } { \\max ( x , y ) } , y > 0 . \\end{align*}"}
{"id": "5065.png", "formula": "\\begin{align*} [ a \\cdot v _ 0 , \\varphi ( a ^ 2 \\cdot v _ 0 ) ] = [ [ \\zeta '' , a ] , a ] \\cdot ( a \\cdot v _ 0 ) = { \\rm I I } ( [ [ \\zeta '' , a ] , a ] , a ) \\cdot v _ 0 . \\end{align*}"}
{"id": "8746.png", "formula": "\\begin{align*} w _ { 1 2 } = \\lambda _ 1 \\lambda _ 2 , w _ { 1 3 } = \\lambda _ 1 \\lambda _ 3 , w _ { 2 3 } = \\lambda _ 2 \\lambda _ 3 , w _ { 1 2 3 } = \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 . \\end{align*}"}
{"id": "3569.png", "formula": "\\begin{align*} \\mathbb { Z } _ p ^ { v } \\cong \\frac { A ( \\tau ) / N } { p ( A ( \\tau ) / N ) } = \\frac { A ( \\tau ) / N } { ( p A ( \\tau ) + N ) / N } \\cong \\frac { A ( \\tau ) } { p A ( \\tau ) + N } . \\end{align*}"}
{"id": "4148.png", "formula": "\\begin{align*} \\hat { \\sigma } ( \\xi ) = \\int _ 0 ^ \\xi ( \\xi - \\zeta ) \\partial _ \\xi ^ 2 \\hat { \\sigma } ( \\zeta ) d \\zeta . \\end{align*}"}
{"id": "1859.png", "formula": "\\begin{align*} { { \\rm d } ^ n \\over { \\rm d } x ^ n } \\sec ( x ) = Q _ n ( \\tan ( x ) ) \\sec ( x ) . \\end{align*}"}
{"id": "7223.png", "formula": "\\begin{align*} X _ { s , t } ( x , v ) = x - ( t - s ) \\zeta _ { s , t , x } ( v ) . \\end{align*}"}
{"id": "3654.png", "formula": "\\begin{align*} B _ 1 ' = \\left ( \\frac { 2 } { 3 c } \\right ) ^ { 3 / 5 } \\left ( \\frac { 1 } { \\beta } \\right ) ^ { 1 / 5 } , \\end{align*}"}
{"id": "718.png", "formula": "\\begin{align*} d _ 0 = \\end{align*}"}
{"id": "5613.png", "formula": "\\begin{align*} \\Xi _ \\perp : = \\{ X \\in \\Xi ~ | ~ \\mathbf { g } ( X , \\Xi _ t ) = 0 \\} \\Omega _ t : = \\{ \\omega \\in \\Omega ~ | ~ \\mathbf { g } ^ { - 1 } ( \\omega , \\Omega _ \\perp ) = 0 \\} , \\end{align*}"}
{"id": "9465.png", "formula": "\\begin{align*} f _ 2 ^ p + { \\bar f _ 2 } f _ 1 ^ p = f _ 2 ^ p + \\left ( \\tfrac { 2 c _ 1 } { c _ 0 } X ^ { p ^ s - 2 p ^ { s - 1 } } f _ 2 - f _ 1 ^ 2 ) \\right ) f _ 1 ^ p = f _ 2 ^ p - f _ 1 ^ p f _ 1 ^ 2 + \\tfrac { 2 c _ 1 } { c _ 0 } X ^ { p ^ s - 2 p ^ { s - 1 } } f _ 2 f _ 1 ^ p . \\end{align*}"}
{"id": "1021.png", "formula": "\\begin{align*} & \\bigg ( \\frac { \\theta d } { 6 4 } \\bigg ) ^ { - n - 2 } \\int _ { B _ { \\theta d / 6 4 } ^ + ( a ) } x _ 1 w ( x ) \\dd x \\\\ \\leqslant & \\ , C \\bigg ( \\partial _ 1 w ( 0 ) + ( r \\theta ) ^ { - n - 2 } + \\tau \\theta ^ { 2 s } d ^ { - n + 2 s - 2 } \\bigg ) \\\\ = & \\ , C \\tau d ^ { - n - 2 } \\bigg ( \\bigg ( 1 - \\frac \\theta 2 \\bigg ) ^ { - n - 2 } - 1 \\bigg ) \\partial _ 1 \\zeta ( 0 ) + C ( r \\theta ) ^ { - n - 2 } + C \\tau \\theta ^ { 2 s } d ^ { - n + 2 s - 2 } . \\end{align*}"}
{"id": "4258.png", "formula": "\\begin{align*} T ( t ) f ( x ) = \\begin{cases} f ( x - t ) & \\mbox { f o r } x > t \\\\ 0 & \\mbox { f o r } x \\leq t . \\end{cases} \\end{align*}"}
{"id": "7459.png", "formula": "\\begin{align*} \\| \\phi ( \\cdot , t ) \\| ^ 2 \\equiv \\| \\phi _ 0 \\| ^ 2 = 1 , \\frac { \\mathrm { d } } { \\mathrm { d } t } \\mathcal { F } ( t ) = - 2 \\eta ( t ) \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 , \\forall \\ , t > 0 . \\end{align*}"}
{"id": "9210.png", "formula": "\\begin{align*} \\begin{cases} \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) , \\\\ \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land p = _ X J ^ A _ { \\gamma } x \\rightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p ) \\right ) , \\end{cases} \\end{align*}"}
{"id": "8439.png", "formula": "\\begin{gather*} \\lim _ { \\eta \\downarrow 0 } \\left [ \\frac { \\eta } { 1 - ( 1 - \\eta ) ^ { \\alpha } } - \\frac { 1 } { \\alpha } \\right ] \\eta ^ { - 1 } = \\frac { \\alpha - 1 } { 2 \\alpha } \\quad \\lim _ { \\eta \\downarrow 0 } \\frac { \\eta ^ { 2 / \\alpha } } { 1 - ( 1 - \\eta ) ^ 2 } \\ , \\eta ^ { - 2 / \\alpha + 1 } = \\frac 1 2 . \\end{gather*}"}
{"id": "5466.png", "formula": "\\begin{align*} \\int _ \\Omega u ( t , x ; s , u _ 0 ) d x \\le \\max \\left \\{ \\int _ \\Omega u _ 0 ( x ) d x , M _ 0 ^ * \\right \\} = M _ 0 ^ * \\forall \\ , t \\ge s . \\end{align*}"}
{"id": "1587.png", "formula": "\\begin{align*} \\mathcal { R } : = \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } | v - v _ * | ^ \\gamma g _ * f ^ 2 d v _ * d v \\lesssim \\Vert g \\Vert _ { L ^ 2 _ { | \\gamma | + 2 } } \\Vert f \\Vert _ { H ^ s _ { \\gamma / 2 } } ^ 2 . \\end{align*}"}
{"id": "5187.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) & = \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ { N } ( C _ { \\ell , \\pi } ^ { a v } ( t ) + \\rho _ { \\ell } \\Delta _ { \\ell , \\pi } ^ { a v } ( t ) ) , \\end{align*}"}
{"id": "669.png", "formula": "\\begin{align*} \\begin{cases} \\ f ( x ) \\ = \\ \\max \\{ A ( n ) - E ( n ) : n \\leq x \\} \\\\ [ 1 1 p t ] \\ g ( x ) \\ = \\ \\min \\{ A ( n ) + E ( n ) : n \\leq x \\} \\end{cases} . \\end{align*}"}
{"id": "6742.png", "formula": "\\begin{align*} \\mathcal L = p ( t , S ) \\frac { \\partial ^ 2 } { \\partial S ^ 2 } + w ( t , S ) \\frac { \\partial } { \\partial S } + z ( t , S ) , \\end{align*}"}
{"id": "3676.png", "formula": "\\begin{align*} \\mathrm { s g n } _ X ( g ) = \\prod _ { \\{ u , v \\} \\in E ( X ) } \\frac { u ^ g - v ^ g } { u - v } . \\end{align*}"}
{"id": "9235.png", "formula": "\\begin{align*} \\begin{cases} \\forall \\gamma ^ 1 , r ^ 1 , x ^ X , y ^ X \\Big ( \\gamma > _ \\mathbb { R } 0 \\land r > _ \\mathbb { R } 0 _ \\mathbb { R } \\rightarrow \\norm { J ^ A _ \\gamma x - _ X J ^ A _ \\gamma y } _ X \\\\ \\qquad \\qquad \\qquad \\leq _ \\mathbb { R } \\norm { r ( x - _ X y ) + _ X ( 1 - r ) ( J ^ A _ \\gamma x - _ X J ^ A _ \\gamma y ) } _ X \\Big ) , \\\\ \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) , \\end{cases} \\end{align*}"}
{"id": "3859.png", "formula": "\\begin{align*} \\mathcal P ' ( x ) + \\mathcal P ( x ) \\tilde F ( x ) = F ( x ) \\mathcal P ( x ) . \\end{align*}"}
{"id": "5635.png", "formula": "\\begin{align*} H = 2 ( \\partial _ + \\star y ^ + - 1 - y ^ - \\star \\partial _ - ) , ~ ~ ~ ~ ~ ~ E ^ \\pm = \\displaystyle \\frac { 1 } { \\sqrt { a } } \\partial _ 0 \\star y ^ \\pm - 2 \\sqrt { a } y ^ 0 \\star \\partial _ \\mp . \\end{align*}"}
{"id": "824.png", "formula": "\\begin{align*} R > \\int _ { y _ 0 } ^ y \\rho d y = 1 - y _ 0 + \\frac { 1 } { \\beta - 1 } ( 1 - y ^ { 1 - \\beta } ) . \\end{align*}"}
{"id": "2902.png", "formula": "\\begin{align*} \\Big | \\partial _ r ^ k \\left ( ( f * g ) ( x ) \\right ) \\Big | \\lesssim \\sum _ { j = 1 } ^ k \\frac { \\left ( | D ^ j f | * | g | \\right ) ( x ) } { | x | ^ { k - j } } , \\forall x \\in \\mathbb { R } ^ N . \\end{align*}"}
{"id": "5841.png", "formula": "\\begin{align*} \\dot X _ { 0 , t } = b ( X _ { 0 , t } ) . \\end{align*}"}
{"id": "3124.png", "formula": "\\begin{align*} a _ i x _ 1 ^ 2 + f _ i ( x _ 2 , x _ 3 , x _ 4 , x _ 5 ) = 0 ( i = 1 , 2 , 3 ) \\end{align*}"}
{"id": "3772.png", "formula": "\\begin{align*} c ^ K _ r ( \\widetilde { W } , \\widetilde { W ' } ) = c ^ K _ { - r } \\big ( \\pi ( w _ n ) W , \\pi ' ( w _ { n - 1 } ) W ' \\big ) . \\end{align*}"}
{"id": "3558.png", "formula": "\\begin{align*} \\int n ( y ) \\Phi _ 1 ( y ) \\ , d y & = 2 \\int _ { 2 \\sqrt { a } v } ^ \\infty \\Phi _ 1 ( y ) \\ , d y \\\\ & \\le 2 \\int _ 0 ^ \\infty \\Phi _ 1 ( y ) \\ , d y \\\\ & = 1 . \\end{align*}"}
{"id": "8747.png", "formula": "\\begin{align*} \\begin{aligned} & w _ { 1 2 } = y _ { 1 2 3 } + y _ { 1 2 } , w _ { 1 3 } = y _ { 1 2 3 } + y _ { 1 3 } , w _ { 2 3 } = y _ { 1 2 3 } + y _ { 2 3 } , w _ { 1 2 3 } = y _ { 1 2 3 } , \\\\ & \\lambda _ { 1 } = y _ 1 + y _ { 1 2 } + y _ { 1 3 } + y _ { 1 2 3 } , \\lambda _ { 2 } = y _ 2 + y _ { 1 2 } + y _ { 2 3 } + y _ { 1 2 3 } , \\lambda _ { 3 } = y _ { 3 } + y _ { 2 3 } + y _ { 1 3 } + y _ { 1 2 3 } , \\\\ & \\sum _ { I \\subseteq \\{ 1 , 2 , 3 \\} } y _ I = 1 . \\end{aligned} \\end{align*}"}
{"id": "1400.png", "formula": "\\begin{align*} \\hat q = q + q _ d \\epsilon , \\end{align*}"}
{"id": "5033.png", "formula": "\\begin{align*} \\Delta _ G & = \\left \\{ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , 1 \\leq i \\leq p + q - 1 \\ , , \\ ; 2 \\ , \\widehat { x } _ { p + q } \\right \\} \\ , , \\\\ \\intertext { a n d } \\Delta _ K & = \\left \\{ \\begin{cases} \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , 1 \\leq i \\leq p - 1 \\ , , 2 \\ , \\widehat { x } _ { p } \\ , , \\\\ \\widehat { x } _ i - \\widehat { x } _ { i + 1 } \\ , , p + 1 \\leq i \\leq p + q - 1 \\ , , 2 \\ , \\widehat { x } _ { p + q } \\end{cases} \\right \\} \\ , , \\end{align*}"}
{"id": "4523.png", "formula": "\\begin{align*} ( e , b ) = ( e b , b ) = ( b , b - ( 1 / 2 ) a _ { 1 / 2 } + e _ 0 a _ { 1 / 2 } ) = 1 - ( 1 / 2 ) ( a _ { 1 / 2 } , a _ { 1 / 2 } ) + ( a _ { 1 / 2 } , e _ 0 a _ { 1 / 2 } ) . \\end{align*}"}
{"id": "891.png", "formula": "\\begin{align*} h _ q ( S _ q ) = \\begin{cases} - 1 & q = p 2 , \\\\ 1 & . \\end{cases} \\end{align*}"}
{"id": "8118.png", "formula": "\\begin{align*} T B ( f ) = 7 T B _ { 5 } ( f ) . \\end{align*}"}
{"id": "2626.png", "formula": "\\begin{align*} \\vartheta _ 3 ( \\tau ; z ) = \\sum _ { k \\in \\Z } e ^ { \\pi i \\tau k ^ 2 } e ^ { 2 \\pi i k z } , ( \\tau , z ) \\in \\mathbb { H } \\times \\C . \\end{align*}"}
{"id": "8610.png", "formula": "\\begin{align*} f ^ { \\# } ( t , k ) & = f ^ { \\# } ( 0 , k ) \\ , \\mp \\ , i \\int _ 0 ^ { t } \\big ( \\mathcal { N } _ { 0 } + \\mathcal { N } _ { L } + \\mathcal { N } _ { R , 1 } + \\mathcal { N } _ { R , 2 } \\big ) \\ , d s , \\end{align*}"}
{"id": "4565.png", "formula": "\\begin{align*} \\Bigg | \\ln \\frac { \\mathbf { P } \\left ( S _ n / ( \\sqrt { n } \\sigma ) > x \\right ) } { 1 - \\Phi \\left ( x \\right ) } \\Bigg | = O \\bigg ( ( 1 + x ) \\frac { \\ln n } { \\sqrt { n } } \\bigg ) \\end{align*}"}
{"id": "6843.png", "formula": "\\begin{align*} \\Phi ( X _ \\tau ) = 1 - \\lambda _ { i } ( M ) \\pm O _ { M , k , \\delta } ( \\gamma ) . \\end{align*}"}
{"id": "1174.png", "formula": "\\begin{align*} \\frac { d \\psi _ { 1 2 } } { d \\zeta } + \\frac { i \\zeta } { 2 } \\psi _ { 1 2 } = \\beta ^ { ( \\eta ) } _ { 1 2 } \\psi _ { 2 2 } , \\\\ \\frac { d \\psi _ { 2 2 } } { d \\zeta } - \\frac { i \\zeta } { 2 } \\psi _ { 2 2 } = \\beta ^ { ( \\eta ) } _ { 2 1 } \\psi _ { 1 2 } , . \\end{align*}"}
{"id": "1945.png", "formula": "\\begin{align*} 1 / \\underline { \\alpha _ 2 } & = \\max \\left ( 1 / \\alpha _ { 0 2 } , \\dfrac { M _ g } { 3 | 2 c _ 2 - 1 | \\varepsilon _ 2 } \\right ) \\\\ & \\leq \\varepsilon _ 2 ^ { - 1 } \\max \\left ( 1 / \\alpha _ { 0 2 } , \\dfrac { M _ g } { 3 | 2 c _ 2 - 1 | } \\right ) \\\\ & \\leq b _ 3 \\beta \\varepsilon _ 2 ^ { - 1 } . \\end{align*}"}
{"id": "4528.png", "formula": "\\begin{align*} Q ( x ) = C x + O ( x ^ { 1 1 / 1 2 + \\varepsilon } ) \\end{align*}"}
{"id": "8182.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) = \\frac { \\pi ^ 2 } { 2 } \\left \\{ \\prod _ { q \\mid d _ 0 } \\left ( 1 - \\frac { 1 } { q ^ 2 } \\right ) \\right \\} \\left ( 1 + \\frac { N _ { d _ 0 } ' ( p , H ) } { p } \\right ) , \\end{align*}"}
{"id": "6124.png", "formula": "\\begin{align*} S _ { m , n } = \\left \\{ ( s , t , s ^ { m + 2 } + \\varepsilon t ^ { n + 2 } ) , \\ \\ ( s , t ) \\in \\R ^ 2 \\right \\} \\ , . \\end{align*}"}
{"id": "8685.png", "formula": "\\begin{align*} P = F P F ^ T + G K _ { Z } G ^ T . \\end{align*}"}
{"id": "3376.png", "formula": "\\begin{align*} & \\sum _ { \\overset { { i + j + k = s } } { i , j , k \\geq 0 } } [ T _ i ( u ) , T _ j ( v ) , T _ k ( w ) ] \\\\ & = \\sum _ { \\overset { { i + j + k = s } } { i , j , k \\geq 0 } } T _ i \\Big ( D ( T _ j ( u ) , T _ k ( v ) ) w + \\theta ( T _ j ( v ) , T _ k ( w ) ) u - \\theta ( T _ j ( u ) , T _ k ( w ) ) v \\Big ) , \\end{align*}"}
{"id": "2284.png", "formula": "\\begin{align*} b _ 0 ( t ) = \\begin{cases} 1 , & - \\frac { 1 } { 2 } \\leq t \\leq \\frac { 1 } { 2 } \\\\ 0 , & \\end{cases} g _ 0 ( t ) = 2 ^ { 1 / 4 } e ^ { - \\pi t ^ 2 } . \\end{align*}"}
{"id": "1499.png", "formula": "\\begin{align*} D ( s , \\mathbf { f } , \\chi ) = \\sum _ { \\xi \\in K _ 1 ( \\mathfrak { n } ) \\backslash \\mathfrak { X } / K _ 1 ( \\mathfrak { n } ) } \\lambda _ { \\mathbf { f } } ( \\xi ) \\chi ^ { \\ast } ( l ( \\xi ) ) l ( \\xi ) ^ { - s } , \\mathrm { R e } ( s ) \\gg 0 . \\end{align*}"}
{"id": "4286.png", "formula": "\\begin{align*} u ( r , t ) = \\lambda ^ { - 1 } ( t ) Q \\left ( \\frac { r } { \\sqrt { \\lambda ( t ) } } \\right ) + \\tilde u ( r , t ) , t \\in [ 0 , T ) , \\end{align*}"}
{"id": "1797.png", "formula": "\\begin{align*} \\kappa _ G ( s , y , y ^ \\prime ) = \\int _ { { \\rm I m } \\lambda = r } s ^ { i \\lambda } G ( \\lambda ) ( y , y ^ \\prime ) d \\lambda \\end{align*}"}
{"id": "6831.png", "formula": "\\begin{align*} x _ { t _ { m ( k ) } } \\big ( s ( t _ { m ( k ) } ) \\big ) = x _ { t _ { m ( k ) } } \\big ( s ' ( t _ { m ( k ) } ) \\big ) = p ^ 1 _ { t _ { m ( k ) } } \\big ( s ' ( t _ { m ( k ) } ) \\big ) \\end{align*}"}
{"id": "5486.png", "formula": "\\begin{align*} \\bigg \\| \\int _ 0 ^ t S _ { t - s } \\alpha ( \\xi ( s ; x ) ) d s \\bigg \\| & \\leq \\int _ 0 ^ t \\| S _ { t - s } \\alpha ( \\xi ( s ; x ) ) \\| d s \\leq M \\int _ 0 ^ t e ^ { \\beta ( t - s ) } \\| \\alpha ( \\xi ( s ; x ) ) \\| d s \\\\ & \\leq M B \\int _ 0 ^ t e ^ { \\beta ( t - s ) } d s = M B \\varphi _ { \\beta } ( t ) \\end{align*}"}
{"id": "7494.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { \\tau } { 2 } \\eta ^ n + \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n + \\frac { \\tau ^ 2 } { 4 } \\Big ( ( \\varrho ^ { x } _ { p } ) ^ 2 + ( \\varrho ^ { y } _ { q } ) ^ 2 \\Big ) \\right ) \\widehat { ( \\tilde { \\phi } ^ { n + 1 } ) } _ { p q } = \\widehat { ( \\mathcal { H } ^ n ) } _ { p q } , p , q = - \\frac { M } { 2 } , \\ldots , \\frac { M } { 2 } - 1 . \\end{align*}"}
{"id": "4671.png", "formula": "\\begin{align*} \\int _ { y > ( x _ 1 + x _ 2 ) / 2 } R _ 1 R _ 2 ^ p \\lesssim \\frac { 1 } { | x _ 1 - x _ 2 | ^ { 2 p } } \\int R _ 1 = O \\bigg ( \\frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \\bigg ) , \\end{align*}"}
{"id": "5215.png", "formula": "\\begin{align*} ~ x \\notin \\cup _ i W _ i ~ ~ B f _ 1 ( x ) = 0 , \\end{align*}"}
{"id": "178.png", "formula": "\\begin{align*} \\tilde { f } = \\int _ { 0 } ^ { + \\infty } \\mathcal { P } _ t ( f ) d t = ( - \\mathcal { L } ) ^ { - 1 } ( f ) . \\end{align*}"}
{"id": "9206.png", "formula": "\\begin{align*} \\forall x ^ X , y ^ X \\left ( \\norm { x + _ X y } ^ 2 _ X + \\norm { x - _ X y } ^ 2 _ X = _ \\mathbb { R } 2 \\left ( \\norm { x } ^ 2 _ X + \\norm { y } ^ 2 _ X \\right ) \\right ) . \\end{align*}"}
{"id": "682.png", "formula": "\\begin{align*} K _ { \\alpha \\alpha } ^ { ( 1 ) } = K _ { \\beta \\beta } ^ { ( 1 ) } = K _ * , K _ { \\alpha \\beta } ^ { ( 1 ) } = C _ b + \\frac { C _ W } { n _ 0 } \\sum _ { j = 1 } ^ { n _ 0 } x _ { j ; \\alpha } x _ { j ; \\beta } = K _ * - \\delta K , \\end{align*}"}
{"id": "4871.png", "formula": "\\begin{align*} f ( z ) = C \\tan ( a z + b ) + D z + E , a , b , C , D , E \\in \\R . \\end{align*}"}
{"id": "1507.png", "formula": "\\begin{align*} \\varphi ( x , s ) = \\chi _ { \\mathbf { h } } ( \\det ( d _ p ) ) ^ { - 1 } j ( x , z _ 0 ) ^ { - l } | \\det ( d _ p ) | ^ { - s } _ { \\mathbf { h } } | j ( x , z _ 0 ) | ^ { l - s } , \\end{align*}"}
{"id": "4935.png", "formula": "\\begin{align*} \\frac { I _ t ( \\mu , R _ n ) \\mu ( B _ n ) } { \\mu ( R _ n ) ^ 2 } \\le \\beta _ 2 r _ n ^ { - a _ j t + a _ j \\sum _ { i = 1 } ^ j s _ i + \\sum _ { i = 1 } ^ d s _ i - \\sum _ { i = 1 } ^ j a _ i s _ i } \\le \\beta _ 2 . \\end{align*}"}
{"id": "6897.png", "formula": "\\begin{align*} \\bar { X } _ W = \\{ V \\in X ( k ) : V \\subset W \\} . \\end{align*}"}
{"id": "2084.png", "formula": "\\begin{align*} \\min \\{ v _ p ( u _ i ) , k \\} = \\min \\{ v _ p ( u _ n ) + ( n - i ) a , k \\} \\end{align*}"}
{"id": "8637.png", "formula": "\\begin{align*} a ( 0 ) = 0 \\mbox { o r } G ( 0 ) = 0 . \\end{align*}"}
{"id": "7084.png", "formula": "\\begin{align*} X ^ r : = \\underbrace { X \\times _ B \\ldots \\times _ B X } _ { } , \\end{align*}"}
{"id": "8823.png", "formula": "\\begin{align*} \\nabla f _ { \\mu } ( x ) = \\mathbb { E } _ U \\left [ \\frac { f \\left ( x + \\mu U \\right ) - f ( x ) } { \\mu } U \\right ] = \\mathbb { E } _ { U , \\xi } \\left [ \\frac { F \\left ( x + \\mu U , \\xi \\right ) - F ( x , \\xi ) } { \\mu } U \\right ] , \\end{align*}"}
{"id": "8633.png", "formula": "\\begin{align*} \\mathcal { N } ^ { ( 1 ) } _ { R , 1 } ( t , k ) & = e ^ { - i t k ^ 2 } \\int \\overline { \\mathcal { K } _ { R } ( x , k ) } u _ { M _ 1 } ( t , x ) \\overline { u _ { M _ 2 } } ( t , x ) u _ { M _ 3 } ( t , x ) \\ , d x . \\end{align*}"}
{"id": "992.png", "formula": "\\begin{align*} \\bar f ( x ) - f ( x ) = ( - \\Delta ) ^ s ( \\bar u - u ) ( x ) & = C \\int _ { \\R ^ n _ + \\setminus B _ R } \\bigg ( \\frac 1 { \\vert x - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\bigg ) ( u - \\bar u ) ( y ) \\dd y . \\end{align*}"}
{"id": "3661.png", "formula": "\\begin{align*} e _ { 1 } = \\frac { 1 } { \\sqrt { 2 } } ( \\nabla r - \\sqrt { - 1 } J \\nabla r ) , \\end{align*}"}
{"id": "5251.png", "formula": "\\begin{align*} B _ r \\cdot f ( A _ r ) = B \\cdot f ( A _ r ^ 2 ) = B \\cdot f ( A _ r ) = B _ r . \\end{align*}"}
{"id": "1273.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) + J _ { 0 } ( \\beta , \\gamma , Z ' ) = J _ { 0 } ( \\alpha , \\gamma , Z + Z ' ) . \\end{align*}"}
{"id": "9173.png", "formula": "\\begin{align*} \\mathcal { J } - j \\leq \\frac { \\log ( 1 / \\alpha _ { j } ) } { \\log 2 0 } , \\mbox { a n d } \\sum _ { X ^ { \\alpha _ { j } } < p \\leq X ^ { \\alpha _ { j + 1 } } } \\frac { 1 } { p } = \\log \\alpha _ { j + 1 } - \\log \\alpha _ { j } + o ( 1 ) = \\log 2 0 + o ( 1 ) \\leq 1 0 . \\end{align*}"}
{"id": "2801.png", "formula": "\\begin{align*} \\Phi _ 1 ( f _ * ) = \\frac { 1 } { 2 } \\int ( L _ + f _ * ) f _ * d x = - \\frac { \\lambda _ 0 ^ 2 } { 4 ( p - 1 ) } \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ { p } \\right ) Q ^ p d x = 0 , \\end{align*}"}
{"id": "2882.png", "formula": "\\begin{align*} i \\partial _ t U _ k ^ A + \\Delta U _ k ^ A + \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * \\big | U _ k ^ A \\big | ^ p \\right ) \\big | U _ k ^ A \\big | ^ { p - 2 } U _ k ^ A = O ( e ^ { - ( k + 1 ) e _ 0 t } ) \\mathcal { S } ( \\mathbb { R } ^ N ) . \\end{align*}"}
{"id": "6201.png", "formula": "\\begin{align*} \\begin{aligned} \\| C - C X X ^ { T } \\| _ F ^ 2 & = \\| C \\| _ F ^ 2 - \\| C X \\| _ F ^ 2 \\\\ & < \\| C \\| _ F ^ 2 - \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } + 2 \\xi \\| C \\| ^ 2 _ F \\\\ & = \\| C - C _ { ( k ) } \\| _ F ^ 2 + 2 \\xi \\| C \\| ^ 2 _ F , \\end{aligned} \\end{align*}"}
{"id": "1085.png", "formula": "\\begin{align*} m ( x , t , k ) : = \\left \\{ \\begin{aligned} & \\left ( \\frac { [ \\Phi _ L ( x , t , k ) ] _ 1 } { a ( k ) } , [ \\Phi _ R ( x , t , k ) ] _ 2 \\right ) e ^ { i t \\theta ( k , \\xi ) \\sigma _ 3 } , k \\in \\mathbb { C } _ { + } \\\\ & \\left ( [ \\Phi _ R ( x , t , k ) ] _ 1 , \\frac { [ \\Phi _ L ( x , t , k ) ] _ 2 } { a ^ { * } ( k ) } \\right ) e ^ { i t \\theta ( k , \\xi ) \\sigma _ 3 } , k \\in \\mathbb { C } _ { - } , \\end{aligned} \\right . \\end{align*}"}
{"id": "1866.png", "formula": "\\begin{align*} D \\left ( { 1 + x ^ 2 \\over a ^ 2 } \\right ) = 0 . \\end{align*}"}
{"id": "2078.png", "formula": "\\begin{align*} \\mathbb { Z } ^ n \\cap \\Omega = \\mathrm { v o l } ( \\Omega ) + O ( \\max _ { \\pi } \\mathrm { v o l } \\big ( \\pi ( \\Omega ) ) \\big ) , \\end{align*}"}
{"id": "9058.png", "formula": "\\begin{align*} m _ { i , j + 1 / 2 } = \\frac { 1 } { h } \\sum _ { l = 1 } ^ j ( \\rho _ { i l } - \\rho _ { i l } ^ n ) , \\end{align*}"}
{"id": "8664.png", "formula": "\\begin{align*} \\Lambda ( p , q ) : = & \\left ( p ^ { \\mu } q _ { \\mu } \\right ) ^ 2 \\big [ \\left ( p ^ { \\mu } q _ { \\mu } \\right ) ^ 2 - 1 \\big ] ^ { - \\frac { 3 } { 2 } } , \\\\ \\mathcal { S } ( p , q ) : = & \\big [ \\left ( p ^ { \\mu } q _ { \\mu } \\right ) ^ 2 - 1 \\big ] ^ 2 I _ 3 - \\big ( p - q \\big ) \\otimes \\big ( p - q \\big ) - \\big [ \\left ( p ^ { \\mu } q _ { \\mu } \\right ) - 1 \\big ] \\big ( p \\otimes q + q \\otimes p \\big ) . \\end{align*}"}
{"id": "4428.png", "formula": "\\begin{align*} ( u , v ) _ { H ^ 4 _ * } = \\int _ U u _ { x x x x } v _ { x x x x } d x . \\end{align*}"}
{"id": "1025.png", "formula": "\\begin{align*} \\tilde \\gamma \\subset \\bigcup _ { k = M + 1 } ^ N B ^ { ( k ) } \\Subset \\Omega ^ + . \\end{align*}"}
{"id": "6130.png", "formula": "\\begin{align*} u ^ { 1 } _ { s } + 2 j \\cdot s ^ { 2 j - 1 } w ^ { 1 } _ { s } = 0 , \\\\ u ^ { 1 } _ { t } + v ^ { 1 } _ { s } + 2 j \\cdot s ^ { 2 j - 1 } w ^ { 1 } _ { t } + 2 k \\cdot t ^ { 2 k - 1 } w ^ { 1 } _ { s } = 0 , \\\\ v ^ { 1 } _ { t } + 2 k \\cdot t ^ { 2 k - 1 } w ^ { 1 } _ { t } = 0 , \\\\ \\end{align*}"}
{"id": "2231.png", "formula": "\\begin{align*} \\mathbb { L } \\leq & \\Big \\| \\int _ 0 ^ t E ( t - s ) A ( P _ n - I ) P F ( X ^ { n } ( s ) ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\Big \\| \\int _ 0 ^ t E ( t - s ) A P \\big ( F ( X ^ { n } ( s ) ) - F ( X ( s ) ) \\big ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ) } = \\mathbb { L } _ 1 + \\mathbb { L } _ 2 . \\end{align*}"}
{"id": "6883.png", "formula": "\\begin{align*} N _ k ^ j = \\sum \\limits _ { i = 0 } ^ j q ^ { i ^ 2 } \\frac { { j \\choose i } _ q { k \\choose k - i } _ q } { { k + j \\choose k } _ q } S _ k ^ i \\end{align*}"}
{"id": "8646.png", "formula": "\\begin{align*} f ( X ) & = P ( X ) - 7 \\times 7 \\times 1 7 N Q ( X ) \\\\ & = X ^ 5 + ( 4 0 8 1 7 N + 4 ) X ^ 4 + ( 2 3 9 0 7 1 N - 5 ) X ^ 3 \\\\ & + ( 3 9 4 8 4 2 N - 2 8 ) X ^ 2 + ( 1 5 9 1 0 3 N - 1 8 ) X + ( 5 8 3 1 N - 2 ) \\end{align*}"}
{"id": "2046.png", "formula": "\\begin{align*} 2 \\operatorname { R e } \\left < T x , x \\right > = \\left < T x , x \\right > + \\left < x , T x \\right > = - 2 \\beta \\eta \\left | v \\left ( 1 \\right ) \\right | ^ 2 - 2 \\kappa \\ , | v ' \\left ( 1 \\right ) \\ ! | ^ 2 , \\end{align*}"}
{"id": "7264.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\alpha _ k = \\sum _ { n = 0 } ^ \\infty \\Bigl ( \\sum _ { k = 1 } ^ \\infty a _ { k , n } \\Bigr ) X ^ n \\end{align*}"}
{"id": "1080.png", "formula": "\\begin{align*} \\Phi _ { R , L + } = \\Phi _ { R , L - } \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "7069.png", "formula": "\\begin{align*} { d \\widetilde g ^ K _ { i } ( t ) \\over d t } < \\bar p \\log K \\sum _ { \\ell = - \\lfloor 1 / 2 \\delta _ K \\rfloor } ^ { 1 / \\delta _ K - 1 - \\lfloor 1 / 2 \\delta _ K \\rfloor } h _ { K } G ( h _ { K } \\ell ) \\left [ \\widetilde g ^ K _ { \\ell + i } ( t ) - \\widetilde g ^ K _ { i } ( t ) \\right ] ^ + \\ , e ^ { h _ { K } L | \\ell | } . \\end{align*}"}
{"id": "6572.png", "formula": "\\begin{align*} \\widetilde { V } ( s ) = - \\frac { 1 } { s } \\int _ 0 ^ { \\infty } V ' ( x ) x ^ s \\ , d x . \\end{align*}"}
{"id": "7252.png", "formula": "\\begin{align*} A _ 1 = & b ^ 2 - b ^ 2 \\theta _ 1 - 2 \\theta _ 2 , A _ 2 = c ^ 2 - c ^ 2 \\theta _ 1 - 2 t ^ 2 \\theta _ 2 , A _ 3 = 1 8 t \\theta _ 2 - 9 a d . \\end{align*}"}
{"id": "7355.png", "formula": "\\begin{align*} \\begin{aligned} & ( 1 - \\beta ) h _ q + v ( y _ q , t _ q ) ^ { \\beta } \\mu ( v ( y _ q , t _ q ) ^ { 1 - \\beta } ) \\geq - v ( y _ q , t _ q ) ^ { \\beta } \\omega ( | X _ q | ) , \\\\ & ( 1 - \\beta ) k _ q + v ( z _ q , t _ q ) ^ { \\beta } \\mu ( v ( z _ q , t _ q ) ^ { 1 - \\beta } ) \\geq - v ( z _ q , t _ q ) ^ { \\beta } \\omega ( | X _ q | ) . \\end{aligned} \\end{align*}"}
{"id": "104.png", "formula": "\\begin{align*} a ^ n - \\gamma _ { n - 1 } a ^ { n - 1 } - \\dots - \\gamma _ { 1 } a = 0 , \\end{align*}"}
{"id": "8814.png", "formula": "\\begin{align*} d \\Phi _ t = \\nabla \\log \\left ( \\frac { d \\gamma ^ { \\epsilon , L } } { d \\varphi } \\right ) ( \\Phi _ t ) d t + \\sqrt { 2 } d B _ t , \\Phi _ 0 \\sim \\nu ^ { \\epsilon , L } , \\end{align*}"}
{"id": "3359.png", "formula": "\\begin{align*} d _ T = \\begin{cases} \\delta _ T , n \\geq 1 , \\\\ \\partial _ T , \\ ; \\ ; \\ ; n = 0 . \\end{cases} \\end{align*}"}
{"id": "5798.png", "formula": "\\begin{align*} k [ P ] = k [ K ] \\quad \\textrm { a n d } \\mathcal { Z } _ P = \\mathcal { Z } _ K . \\end{align*}"}
{"id": "7549.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) = - \\frac { \\epsilon T } { 2 \\pi } \\log \\pi + \\frac { \\epsilon T } { 2 \\pi } \\log \\left ( \\frac { T } { 2 } \\right ) - \\frac { \\epsilon T } { 2 \\pi } + \\frac { \\epsilon } { \\pi } \\ \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "5540.png", "formula": "\\begin{align*} U ( x _ 0 , \\delta ) : = \\{ x \\in H : \\| x - x _ 0 \\| < \\eta \\} . \\end{align*}"}
{"id": "7854.png", "formula": "\\begin{align*} ( \\mathbf { v } \\mathrel { \\trianglelefteq } ( A , B , C ) \\ , \\& \\ , \\mid B \\cap \\mathbf { v } _ 2 \\mid = \\omega ) \\Rightarrow \\mid E \\cap \\mathbf { v } _ 2 \\mid = \\omega \\end{align*}"}
{"id": "371.png", "formula": "\\begin{align*} & d \\rho ( t ) + d i v _ G ^ { \\theta } ( \\rho ( t ) \\nabla _ G S ( t ) ) ( d t + d W _ t ^ { \\delta } ) = 0 , \\\\ & \\ < \\dot S ^ { \\textrm { a b s } } , \\rho \\ > + \\frac 1 4 \\sum _ { i j } ( S _ i - S _ j ) ^ 2 \\theta _ { i j } ( \\rho ) ( 1 + \\dot { W _ t ^ { \\delta } } ) ^ 2 = 0 , \\ ; \\mathcal L ^ 1 \\ ; \\\\ & \\ < \\frac { d \\dot S ^ { \\textrm { \\textrm { s i n g } } } } { d \\mu } , \\rho \\ > = 0 , \\ ; \\forall \\mu \\ ; , \\ ; \\mu \\ ; \\bot \\ ; \\mathcal L _ 1 . \\end{align*}"}
{"id": "5792.png", "formula": "\\begin{align*} \\operatorname { K o s } ^ { { k [ K ] } } ( v _ 1 , \\ldots , v _ m ) = \\big ( k [ K ] \\otimes \\Lambda ( u _ 1 , \\ldots , u _ m ) , \\ d \\big ) , d ( u _ i ) = v _ i , \\end{align*}"}
{"id": "2858.png", "formula": "\\begin{align*} u ( \\cdot + x ( t _ n ' ) , t _ n ' ) = u ( \\cdot + x ( t _ n ) + x ( t _ n ' ) - x ( t _ n ) , t _ n ' ) \\to e ^ { i t _ 0 } e ^ { i \\theta _ 0 } Q ( \\cdot - x _ 0 - x _ 1 ) H ^ 1 . \\end{align*}"}
{"id": "5084.png", "formula": "\\begin{align*} g ( t ) = \\sqrt { \\gamma } \\ , e ^ { - \\gamma | t | } , \\ ; \\gamma > 0 . \\end{align*}"}
{"id": "5761.png", "formula": "\\begin{align*} \\begin{cases} e ^ { - Q } \\delta _ { k l } \\leq \\hat { h } _ { \\delta , k l } = \\hat { h } _ { \\delta } ( \\frac { \\partial } { \\partial \\hat { f } ^ { k } } , \\frac { \\partial } { \\partial \\hat { f } ^ { l } } ) \\leq e ^ { Q } \\delta _ { k l } ; \\\\ R ^ { 1 + \\alpha } \\left \\| \\frac { \\partial \\hat { h } _ { \\delta , k l } } { \\partial \\hat { f } ^ { j } } \\right \\| _ { C ^ { 0 , \\alpha } ( B _ { R } ( \\hat x , \\hat h _ \\delta ) ) } \\leq \\Psi ( \\delta \\mid n , r , v , \\rho , \\alpha , Q ) \\end{cases} \\end{align*}"}
{"id": "7138.png", "formula": "\\begin{align*} R _ H ( s _ 1 , s _ 2 ) = \\int _ 0 ^ { s _ 1 \\wedge s _ 2 } K _ H ( s _ 1 , u ) K _ H ( s _ 2 , u ) d u . \\end{align*}"}
{"id": "2596.png", "formula": "\\begin{align*} | \\langle V _ g ^ * F , \\varphi \\rangle | & = | \\langle F , V _ g \\varphi \\rangle | \\\\ & \\leq \\norm { F } _ { L ^ { p , q } } \\norm { V _ g \\varphi } _ { L ^ { p ' , q ' } } \\\\ & \\leq \\norm { F } _ { L ^ { p , q } } \\norm { ( 1 + | z | ) ^ n V _ g \\varphi } _ \\infty \\norm { ( 1 + | z | ) ^ { - n } } _ { L ^ { p ' , q ' } } . \\end{align*}"}
{"id": "1861.png", "formula": "\\begin{align*} { \\rm G e n } ( a , t ) = { 1 \\over { \\rm G e n } ( a ^ { - 1 } , t ) } \\end{align*}"}
{"id": "4127.png", "formula": "\\begin{align*} \\bullet ' = f ( \\bullet ) , \\ \\ \\bullet = f ^ { - 1 } ( \\bullet ' ) ; \\widetilde { \\bullet } = \\tau ( \\bullet ' ) , \\ \\ \\bullet ' = \\tau ^ { - 1 } ( \\widetilde { \\bullet } ) . \\end{align*}"}
{"id": "8852.png", "formula": "\\begin{align*} ( \\partial _ q \\varphi ) ( x _ 0 , \\ldots , x _ { q + 1 } ) = \\sum _ { i = 0 } ^ { q + 1 } ( - 1 ) ^ i \\varphi ( x _ 0 , \\ldots , \\hat x _ i , \\ldots , x _ { q + 1 } ) . \\end{align*}"}
{"id": "7694.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | \\nabla u _ { \\ast } | ^ 2 \\le \\int _ { \\R ^ N } | \\nabla u | ^ 2 = \\tilde { \\lambda } _ 0 \\ ; , \\end{align*}"}
{"id": "1081.png", "formula": "\\begin{align*} S ( k ) = \\Phi _ { R } ^ { - 1 } ( k ) \\Phi _ { L } ( k ) \\end{align*}"}
{"id": "2169.png", "formula": "\\begin{align*} \\rho ( \\alpha ; u ) : = \\tilde { \\rho } ( u ) + \\overline { \\rho } ( \\alpha ; u ) \\ \\ \\Vert u \\Vert = \\inf \\left \\lbrace \\lambda > 0 : \\rho ( \\alpha ; \\frac { u } { \\lambda } ) \\leq 1 \\right \\rbrace . \\end{align*}"}
{"id": "2428.png", "formula": "\\begin{align*} \\pi ( \\gamma ) = M _ \\omega T _ x . \\end{align*}"}
{"id": "8435.png", "formula": "\\begin{gather*} C _ 1 : = \\frac { \\theta _ { 1 } \\beta } { 2 } \\left ( \\frac { 2 r | b ( 0 ) | ^ { 2 } } { \\theta _ { 1 } } + 2 \\eta ^ { 2 } | b ( 0 ) | ^ { 2 } + 1 + 2 \\eta K \\right ) + \\frac { \\beta | b ( 0 ) | ^ { 2 } } { \\theta _ { 1 } } + 2 \\beta r | b ( 0 ) | ^ { 2 } + \\beta K . \\end{gather*}"}
{"id": "8449.png", "formula": "\\begin{align*} L ' _ b \\wedge L ' _ { \\tilde { c } } = L ' _ { p u b } . \\end{align*}"}
{"id": "4916.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\| g _ t \\| ^ 2 & \\leq ( 2 B + \\frac { L } { 2 } ) ^ 2 \\\\ \\min _ { t \\in [ T ] } \\norm { g _ t } ^ 2 & \\leq \\frac { ( 2 B + L ^ 2 ) ^ 2 } { T } \\end{align*}"}
{"id": "2961.png", "formula": "\\begin{align*} \\Theta _ n & = \\frac { 1 6 } { n ^ 4 \\delta ^ 4 _ n ( k ) } \\sum _ { \\substack { \\mathbf { P } _ { k } \\in \\mathcal P ( d , k ) ^ 4 : \\\\ p _ { k , 1 } = p _ { k , 2 } = p _ { k , 3 } = p _ { k , 4 } } } \\sum _ { \\mathbf { i } \\in \\mathcal { J } ^ 2 } \\varphi _ { k - 1 } \\left ( \\mathbf { P } _ { k } , \\mathbf { i } \\right ) \\cdot \\varphi _ { 4 } \\left ( \\mathbf { i } \\right ) . \\end{align*}"}
{"id": "3543.png", "formula": "\\begin{align*} \\langle a p _ { 0 4 } ^ 2 + a p _ { 0 1 } p _ { 1 4 } + b p _ { 0 4 } p _ { 1 4 } & - d p _ { 1 4 } ^ 2 - p _ { 0 1 } p _ { 3 4 } - a u p _ { 0 4 } p _ { 1 4 } - b u p _ { 1 4 } ^ 2 - u p _ { 0 4 } p _ { 3 4 } + u ^ 2 p _ { 1 4 } p _ { 3 4 } , \\\\ \\ ; k _ 1 , \\ ; k _ 2 , \\ ; k _ 3 , & \\ ; k _ 4 , \\ ; k _ 5 , \\ ; k _ 6 \\rangle \\end{align*}"}
{"id": "708.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\sum _ { \\substack { I , J \\\\ \\abs { I , J } \\leq o ( f ) } } \\xi ^ I D ^ J u _ { I , J } ( \\xi ) , \\end{align*}"}
{"id": "2761.png", "formula": "\\begin{align*} \\| f \\| _ { S ( \\dot { H } ^ { s _ c } ) } = \\| f \\| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 1 } } \\lesssim \\big \\| | \\nabla | ^ { s _ c } f \\big \\| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 2 } } \\lesssim \\big \\| \\left \\langle \\nabla \\right \\rangle f \\| _ { L _ t ^ { q _ 2 } L _ x ^ { r _ 2 } } \\lesssim \\big \\| \\left \\langle \\nabla \\right \\rangle f \\big \\| _ { S ( L ^ 2 ) } . \\end{align*}"}
{"id": "6975.png", "formula": "\\begin{align*} \\frac { z _ k - g ( \\bar x ) } { \\norm { x _ k - \\bar x } } = \\frac { g ( x _ k ) - g ( \\bar x ) } { \\norm { x _ k - \\bar x } } - \\frac { y _ k } { \\norm { x _ k - \\bar x } } \\to v - 0 = v \\end{align*}"}
{"id": "3187.png", "formula": "\\begin{align*} \\varphi _ C ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( W _ k , \\emptyset ) . \\end{align*}"}
{"id": "2346.png", "formula": "\\begin{align*} \\left | \\int _ \\R x f ( x ) \\overline { f ' ( x ) } \\ , d x \\right | = | \\langle x f , f ' \\rangle | \\leq \\norm { x f } _ 2 \\norm { f ' } _ 2 = \\left ( \\int _ \\R x ^ 2 | f ( x ) | ^ 2 \\ , d x \\right ) ^ { 1 / 2 } \\left ( \\int _ \\R | f ' ( x ) | ^ 2 \\ , d x \\right ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "8458.png", "formula": "\\begin{align*} u b = v c , \\ ; u ^ + = v ^ + = ( v c ) ^ + \\mbox { a n d } b c ^ * = u ^ * b . \\end{align*}"}
{"id": "7756.png", "formula": "\\begin{gather*} \\left | \\left ( u _ x ( x ) \\right ) ^ { p } \\right | = | \\langle u _ x , p \\rangle p | = | \\langle u _ x , p - u \\rangle | \\leq 3 \\varepsilon | u _ x ( x ) | , \\\\ \\left | \\left ( u _ t ( x ) \\right ) ^ { p } \\right | \\leq 3 \\varepsilon | u _ t ( x ) | \\ ; \\textrm { a n d } \\ ; \\left | \\left ( u ( x ) - p \\right ) ^ { p } \\right | \\leq \\frac { 3 \\varepsilon } { 2 } \\left | u ( x ) - p \\right | . \\end{gather*}"}
{"id": "7714.png", "formula": "\\begin{align*} \\mathsf P _ k ( x ) = \\# \\{ a \\in H \\colon \\mathsf N ( a ) \\le x , a \\ \\ \\mathsf P _ k \\} \\ , . \\end{align*}"}
{"id": "6847.png", "formula": "\\begin{align*} \\pi _ { k } ( X _ x ) = \\sum \\limits _ { y \\in X ( k ) : y > x } \\pi _ { k } ( y ) = R ( k , i ) \\pi _ i ( x ) \\end{align*}"}
{"id": "631.png", "formula": "\\begin{align*} \\abs { A ( x , n ) - \\alpha ( n ) } \\ & = \\ \\abs { A ( n ) - A ( n ) } \\\\ [ 1 5 p t ] & = \\ 0 \\\\ [ 1 5 p t ] & \\leq \\ \\frac { 1 } { x + 1 } . \\end{align*}"}
{"id": "5406.png", "formula": "\\begin{align*} f ( X _ { t } , \\theta ) = - \\theta X ^ { \\gamma } _ { t } , \\end{align*}"}
{"id": "9307.png", "formula": "\\begin{align*} \\rho ( X ) = \\rho ( Y ) - n = \\rho ( \\mathcal { Y } _ { \\overline K } ) - n = \\rho ( \\mathcal { X } _ { \\overline K } ) , \\end{align*}"}
{"id": "2374.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } \\iint _ { R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "7887.png", "formula": "\\begin{align*} K _ R ( u - v ) = e ^ { - | | u - v | | ^ 2 / 2 } . \\end{align*}"}
{"id": "543.png", "formula": "\\begin{align*} P ' ( s ) s - P ( s ) = p ( s ) , \\ P '' ( s ) = \\frac { p ' ( s ) } { s } s \\in [ 0 , \\bar \\rho ) . \\end{align*}"}
{"id": "5487.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } ( t ) & = & \\beta \\xi ( t ) \\\\ \\xi ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "1252.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\vert B _ n \\vert ^ { \\varepsilon } \\mu ( B _ n ) \\leq \\sum _ { k \\geq 0 } \\sum _ { 1 \\leq j \\leq J _ k } 2 ^ { - k \\varepsilon } = \\sum _ { k \\geq 0 } J _ k 2 ^ { - k \\varepsilon } < + \\infty . \\end{align*}"}
{"id": "5146.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow 0 ^ + } \\frac { 2 } { t \\sinh ( 2 t ) } - \\frac { 1 } { t ^ 2 } = \\lim _ { t \\to 0 ^ + } - \\frac { \\frac { 4 } { 3 ! } + 4 \\sum \\limits _ { k = 1 } ^ \\infty \\frac { ( 2 t ) ^ { 2 k } } { ( 2 k + 1 ) ! } } { 1 + \\sum \\limits _ { k = 1 } ^ \\infty \\frac { ( 2 t ) ^ { 2 k } } { ( 2 k + 1 ) ! } } = - \\frac { 2 } { 3 } . \\end{align*}"}
{"id": "4673.png", "formula": "\\begin{align*} \\int R _ 1 R _ 2 ^ p = \\frac { \\kappa _ 0 } { ( x _ 2 - x _ 1 ) ^ 2 } \\int Q ^ p + O \\bigg ( \\frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "3805.png", "formula": "\\begin{align*} f ( \\tau ( g , m _ 1 ) ) = ( f \\circ \\gamma ) ( 1 ) = \\gamma _ 1 ( 1 ) = \\tau _ 1 ( g , n _ 1 ) = \\tau _ 1 ( g , f ( m _ 1 ) ) , \\end{align*}"}
{"id": "8524.png", "formula": "\\begin{align*} t ^ c _ j = \\prod _ { i \\in \\mathcal { N } _ c \\setminus \\{ j \\} } \\operatorname { s g n } ( t ^ v _ i ) \\min _ { i \\in \\mathcal { N } _ c \\setminus \\{ j \\} } | t ^ v _ i | , \\end{align*}"}
{"id": "4877.png", "formula": "\\begin{align*} H _ n ( w ) = \\frac { d ^ n } { d w ^ n } \\left ( ( w - 1 ) ^ { n - 1 } ( w + 1 ) ^ { n + 1 } \\right ) , \\end{align*}"}
{"id": "6232.png", "formula": "\\begin{align*} { \\bf D } & = ( \\delta { } _ { v _ 1 } ) \\otimes \\cdots \\otimes ( \\delta { } _ { v _ N } ) , \\\\ D ^ { \\prime } _ 1 & = ( \\delta { } _ { o ( e _ 1 ) } ) \\otimes \\cdots \\otimes ( \\delta { } _ { o ( e _ M ) } ) , \\\\ D ^ { \\prime } _ 2 & = ( \\delta { } _ { t ( e _ 1 ) } ) \\otimes \\cdots \\otimes ( \\delta { } _ { t ( e _ m ) } ) , \\\\ B ^ { \\prime } & = ( \\beta { } _ { e _ 1 } ) \\otimes \\cdots \\otimes ( \\beta { } _ { e _ m } ) . \\end{align*}"}
{"id": "8600.png", "formula": "\\begin{align*} \\mathcal { C } _ L = \\mathcal { C } \\smallsetminus \\{ 1 , - 1 \\} . \\end{align*}"}
{"id": "4781.png", "formula": "\\begin{align*} n = \\binom { m } { 2 } , \\theta = 5 - m , \\hat \\theta = | 5 - m | \\sqrt { \\frac { m + 1 } { m ( m - 1 ) ( m - 2 ) ( m - 3 ) } } . \\end{align*}"}
{"id": "6601.png", "formula": "\\begin{align*} \\frac { | m h \\pm n k | } { g \\ell } W \\left ( \\frac { c | m h \\pm n k | } { g \\ell Q } \\right ) = \\frac { Q } { 2 \\pi i c } \\int _ { ( \\varepsilon ) } \\ell ^ { - w } \\Upsilon _ { \\pm } ( w ; m h , n k ) \\ , d w , \\end{align*}"}
{"id": "5346.png", "formula": "\\begin{align*} & \\partial _ t \\eta + \\mathrm { d i v } ( \\eta u ) = \\nu \\Delta \\eta , \\\\ & \\partial _ t \\tau + \\mathrm { d i v } ( u \\tau ) - ( \\nabla u \\tau + \\tau \\nabla ^ T u ) - k \\eta ( \\nabla u + \\nabla ^ T u ) = \\nu \\Delta \\tau - \\frac { A _ 0 } { 2 } \\tau , \\end{align*}"}
{"id": "305.png", "formula": "\\begin{align*} & \\Theta \\in \\mathcal C ^ { \\infty } ( ( 0 , \\infty ) \\times ( 0 , \\infty ) ) ; \\\\ & \\Theta \\ ; \\ ; [ 0 , \\infty ) \\times [ 0 , \\infty ) ; \\\\ & \\Theta ( s , t ) = \\Theta ( t , s ) ; \\ ; \\Theta ( s , t ) > 0 , \\ ; \\ ; s , t > 0 , \\\\ & \\min ( s , t ) \\le \\Theta ( s , t ) \\le \\max ( s , t ) , s , t \\ge 0 . \\\\ & \\Theta \\ ; \\ ; ( 0 , \\infty ) \\times ( 0 , \\infty ) . \\end{align*}"}
{"id": "13.png", "formula": "\\begin{align*} ( \\xi , \\eta ) \\mapsto ( u , v ) : = ( \\xi + \\eta , - | \\xi | ^ { \\alpha } - | \\eta | ^ { \\alpha } ) . \\end{align*}"}
{"id": "970.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\frac { \\dd \\tilde z ' } { \\big ( 1 + \\vert \\tilde z ' \\vert ^ 2 \\big ) ^ { \\frac { n + 2 s } 2 } } & = \\frac { \\displaystyle \\pi ^ { \\frac { n - 1 } 2 } \\Gamma \\big ( \\frac { 1 + 2 s } 2 \\big ) } { \\displaystyle \\Gamma \\big ( \\frac { n + 2 s } 2 \\big ) } . \\end{align*}"}
{"id": "4201.png", "formula": "\\begin{align*} \\int _ I \\hat { V } ( 0 , \\gamma ( s ) ) \\ , d s = 0 , \\end{align*}"}
{"id": "2307.png", "formula": "\\begin{align*} R ( f , g ) ( x , \\omega ) = f ( x ) \\overline { \\widehat { g } ( \\omega ) } e ^ { - 2 \\pi i x \\cdot \\omega } = \\widehat { V _ g f } ( \\omega , - x ) . \\end{align*}"}
{"id": "6507.png", "formula": "\\begin{align*} M _ n ^ { ( 2 m ) } = M _ { j _ 0 + 1 } ^ { ( 2 m ) } \\ , \\bar { g } ^ { ( 2 m ) } _ n + F ^ { ( 2 m ) } _ n + H ^ { ( 2 m ) } _ n . \\end{align*}"}
{"id": "8049.png", "formula": "\\begin{align*} \\Psi _ { ( \\Sigma , U ) } ( f ) [ \\psi ] : = \\int _ { \\Sigma } f \\psi \\mathrm { d } V _ { \\Sigma } . \\end{align*}"}
{"id": "6293.png", "formula": "\\begin{align*} p _ { \\widehat { K } } ( \\hat { k } ) & = \\mathbb { P } \\big ( \\hat { k } - 1 / 2 \\le \\widehat { K } ^ \\mathbb { R } \\le \\hat { k } + 1 / 2 \\big ) = F _ { \\widehat { K } ^ \\mathbb { R } } ( \\hat { k } + 1 / 2 ) - F _ { \\widehat { K } ^ \\mathbb { R } } ( \\hat { k } - 1 / 2 ) . \\end{align*}"}
{"id": "501.png", "formula": "\\begin{align*} Y _ { \\delta _ { t } \\delta _ { s } } ^ { a _ { s } a _ { t } } + 1 = c _ { 1 } ^ { ( s , t ) } \\frac { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\delta _ { s } \\delta _ { t } } ^ { a _ { s } a _ { t } } ) } { \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\delta _ { s } } ^ { a _ { s } } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } _ { \\delta _ { t } } ^ { a _ { t } } ) } . \\end{align*}"}
{"id": "2300.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\norm { g } _ 2 ^ 2 } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) = \\frac { 1 } { \\norm { g } _ 2 ^ 2 } \\iint _ { \\R ^ { 2 d } } V _ g f ( \\l ) \\ , \\pi ( \\l ) g \\ , d \\l , \\l = ( x , \\omega ) \\in \\R ^ { 2 d } . \\end{align*}"}
{"id": "7806.png", "formula": "\\begin{align*} \\theta ( a _ x ) = \\theta \\left ( \\frac { f ( \\pi _ n ) } { [ \\omega _ x ] } \\right ) = \\frac { f ( \\theta ( \\pi _ n ) ) } { x } = \\frac { f ( \\zeta _ { 2 ^ n } - 1 ) } { x } = 1 . \\end{align*}"}
{"id": "6812.png", "formula": "\\begin{align*} \\Psi _ { r n j } \\left ( x \\right ) = p ^ { \\frac { - r } { 2 } } \\chi _ { p } \\left ( p ^ { - 1 } j \\left ( p ^ { r } x - n \\right ) \\right ) \\Omega \\left ( \\left \\vert p ^ { r } x - n \\right \\vert _ { p } \\right ) , \\end{align*}"}
{"id": "3413.png", "formula": "\\begin{align*} T ( f _ j ) ( x ) & = T ( g ) ( x ) + T ( h _ j ) ( x ) = T ( g ) ( x ) + \\int _ { 2 R \\leqslant d ( 0 , y ) \\le j } K ( x , y ) f ( y ) d \\omega ( y ) \\\\ & = T ( g ) ( x ) + \\int _ { 2 R \\leqslant d ( 0 , y ) \\le j } [ K ( x , y ) - K ( 0 , y ) ] f ( y ) d \\omega ( y ) + c _ j - C ( R ) , \\end{align*}"}
{"id": "8517.png", "formula": "\\begin{align*} ( a + b ) * ( x + y ) & = ( a + b ) * x + x + ( a * y ) - x \\\\ & = a * ( \\lambda _ a ^ { - 1 } ( b ) * x ) + \\lambda _ a ^ { - 1 } ( b ) * x + a * x + x + ( a * y ) - x \\end{align*}"}
{"id": "2756.png", "formula": "\\begin{align*} M [ v ] = M [ u ] , E [ v ] = E [ u ] - \\frac { 1 } { 2 } \\frac { P [ u ] ^ 2 } { M [ u ] } , \\| \\nabla v _ 0 \\| _ 2 ^ 2 = \\| \\nabla u _ 0 \\| _ 2 ^ 2 - \\frac { P [ u _ 0 ] ^ 2 } { M [ u _ 0 ] } . \\end{align*}"}
{"id": "8329.png", "formula": "\\begin{align*} \\int _ 0 ^ T h \\dd ( y - u ) = 0 \\int _ 0 ^ T ( v - y ) \\dd ( y - u ) \\geq 0 ~ ~ \\forall v \\in C ( [ 0 , T ] ; Z ) . \\end{align*}"}
{"id": "558.png", "formula": "\\begin{align*} | I _ { 3 , c } | \\leq & \\varrho \\overline P \\int _ 0 ^ \\tau \\int _ { \\Omega _ R } \\left | \\frac { \\varrho - \\rho } { \\varepsilon } \\right | | s | | \\Delta \\Psi | \\\\ \\leq & c \\left \\| \\frac { \\varrho - \\rho } { \\varepsilon } \\right \\| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \\Omega ) ) } \\| s \\| _ { L ^ { q _ 1 } ( 0 , T ; L ^ { q _ 1 } ( \\Omega ) ) } \\| \\Delta \\Psi \\| _ { L ^ { q _ 2 } ( 0 , T ; L ^ { q _ 2 } ( \\Omega ) ) } \\leq c ( D , T ) \\varepsilon ^ { 2 ( 1 - \\frac { 1 } { q _ 1 } - \\frac { 1 } { q _ 2 } ) } \\end{align*}"}
{"id": "3694.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } t ^ { \\frac { \\beta } { \\alpha } } \\| B ( t ) \\| _ { H ^ { \\beta } ( \\mathbb S ^ 1 ) } = 0 \\ \\ \\forall \\ \\ \\beta > 0 . \\end{align*}"}
{"id": "5876.png", "formula": "\\begin{align*} \\tau ^ 1 = & \\inf \\{ t \\ge 0 : \\ Y \\notin \\Pi _ { 1 } \\} , & \\\\ \\tilde \\tau ^ 2 = & \\inf \\{ t \\ge \\tau ^ 1 : \\ Y \\notin \\Pi _ 2 \\} , & \\tau ^ 2 = & \\inf \\{ t \\ge 0 : \\ Y \\notin \\Pi _ 2 \\} , \\\\ \\tilde \\tau ^ 3 = & \\inf \\{ t \\ge \\tau ^ 2 : \\ Y \\notin \\Pi _ 3 \\} , & \\tau ^ 3 = & \\inf \\{ t \\ge 0 : \\ Y \\notin \\Pi _ 3 \\} , \\end{align*}"}
{"id": "2393.png", "formula": "\\begin{align*} \\norm { f - \\sum _ { \\gamma \\in F _ { 2 n + 1 } } e _ \\gamma } _ \\mathcal { B } = \\norm { f - \\sum _ { k = 1 } ^ { N _ { 2 n + 1 } } e _ { \\pi ( k ) } } _ \\mathcal { B } < \\frac { \\varepsilon } { 2 } . \\end{align*}"}
{"id": "7582.png", "formula": "\\begin{align*} c _ { i , j } ^ { ( r , s ) } = \\dfrac { 4 } { \\pi ^ 2 } \\int _ { 0 } ^ { \\pi } \\int _ { 0 } ^ { \\pi } f _ { x ^ { r } y ^ s } ( \\cos \\theta _ x , \\cos \\theta _ y ) \\cos i \\theta _ x \\cos j \\theta _ y d \\theta _ x d \\theta _ y , \\end{align*}"}
{"id": "1389.png", "formula": "\\begin{align*} E _ i ( x ) = E _ i ( 1 ) - \\int _ x ^ 1 ( n _ i - b _ i ) ( y ) d y , i = 1 , 2 . \\end{align*}"}
{"id": "5141.png", "formula": "\\begin{align*} 1 + \\tfrac { 1 } { ( n ^ 2 - 1 ) } < \\Big ( 1 + \\tfrac { 1 } { n ^ 2 } \\Big ) ^ 2 = 1 + \\tfrac { 2 } { n ^ 2 } + \\tfrac { 1 } { n ^ 4 } \\Longleftrightarrow n ^ 2 < ( 2 + \\tfrac { 1 } { n ^ 2 } ) ( n ^ 2 - 1 ) . \\end{align*}"}
{"id": "1238.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\mu ( L _ { B , n } ) = + \\infty \\end{align*}"}
{"id": "2423.png", "formula": "\\begin{align*} S ^ { - 1 } f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , S ^ { - 1 } e _ \\gamma \\rangle S ^ { - 1 } e _ \\gamma . \\end{align*}"}
{"id": "6678.png", "formula": "\\begin{align*} _ r F _ s ( z ) \\begin{cases} z = 0 & , \\\\ z \\in \\mathbb { C } _ { \\infty } & , \\\\ z \\in \\mathbb { C } _ { \\infty } | z | _ { \\infty } < q ^ { \\sum _ { j = 1 } ^ s ( b _ j - 1 ) - \\sum _ { j = 1 } ^ r ( a _ j - 1 ) } & . \\end{cases} \\end{align*}"}
{"id": "1224.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\frac { 1 } { \\delta } } _ { \\infty } ( U _ { p , q , \\delta } ) \\geq ( \\mathcal { H } ^ { \\frac { \\log 2 } { \\log 3 } } _ { \\infty } ( U _ { p , q , \\delta } ) ) ^ { \\frac { \\log 3 } { \\delta \\log 2 } } \\geq \\widetilde { C } ( q ^ { - 2 \\delta \\frac { \\log 2 } { \\log 3 } } ) ^ { \\frac { \\log 3 } { \\delta \\log 2 } } = \\widetilde { C } \\mathcal { L } \\Big ( B \\Big ( \\frac { p } { q } , q ^ { - 2 } \\Big ) \\Big ) . \\end{align*}"}
{"id": "192.png", "formula": "\\begin{align*} \\mathcal { L } _ { d , \\delta } ( f ) ( x ) = \\Delta ( f ) ( x ) - \\delta \\| x \\| ^ { \\delta - 2 } \\langle x ; \\nabla ( f ) ( x ) \\rangle . \\end{align*}"}
{"id": "9551.png", "formula": "\\begin{align*} ( 1 - \\mu ^ k ) [ ( x ^ k ) ^ T w ^ k - ( x ^ k ) ^ T z _ 1 ^ k + ( m - 1 ) ( \\hat { \\mathcal { A } } ( x ^ k ) ^ { m - 1 } ) ^ T ( x ^ k - z _ 2 ^ k ) ] + \\mu ^ k ( x ^ k ) ^ T ( x ^ k - x ^ { ( 0 ) } ) = 0 . \\end{align*}"}
{"id": "9476.png", "formula": "\\begin{align*} | \\mathcal { D D } _ { ( s , t ) } | = \\binom { \\lfloor ( s - 1 ) / 2 \\rfloor + \\lfloor ( t - 1 ) / 2 \\rfloor } { \\lfloor ( s - 1 ) / 2 \\rfloor } , \\end{align*}"}
{"id": "1189.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\mu , s } _ { t } ( A ) = \\inf \\left \\{ \\mathcal { H } ^ { s } _ { t } ( E ) : \\ E \\subset A , \\ \\mu ( E ) = \\mu ( A ) \\right \\} . \\end{align*}"}
{"id": "5880.png", "formula": "\\begin{align*} x _ 0 = ( 0 , L ) , v = ( 1 , 0 ) , q _ \\pm = ( \\pm R , 0 ) , v _ \\pm = ( 0 , 1 ) \\end{align*}"}
{"id": "4287.png", "formula": "\\begin{align*} \\lambda ( t ) = C ( u _ 0 ) ( 1 + o ( 1 ) ) ( T - t ) ^ { \\frac { 2 } { \\alpha } } . \\end{align*}"}
{"id": "1878.png", "formula": "\\begin{align*} G = \\{ a \\rightarrow a v , \\ ; \\ ; v \\rightarrow u , \\ ; \\ ; u \\rightarrow 2 u v \\} . \\end{align*}"}
{"id": "7140.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( J ^ n _ 1 ( 1 , u ) ) ^ 2 d u & \\le \\Big ( \\int _ 0 ^ 1 \\Big ( \\exp \\Big ( \\int _ s ^ t \\tilde { f } ' ( r , x _ r ) d r \\Big ) \\Big ( \\tilde { f } ' _ n ( s , x _ s ^ n ) - K _ s \\Big ) d s \\Big ) ^ 2 \\\\ & = \\Big \\{ - \\exp \\Big ( \\int _ s ^ 1 \\tilde { f } ' ( r , x _ r ) d r \\Big ) \\Big \\vert _ { s = 0 } ^ 1 - \\int _ 0 ^ 1 K _ s \\exp \\Big ( \\int _ s ^ 1 \\tilde { f } ' ( r , x _ r ) d r \\Big ) d s \\Big \\} ^ 2 \\\\ & \\le \\Big ( \\exp ( \\int _ 0 ^ 1 K _ r d r ) + \\int _ 0 ^ 1 K _ s d s \\cdot \\exp ( \\int _ 0 ^ 1 K _ r d r ) \\Big ) ^ 2 . \\end{align*}"}
{"id": "1665.png", "formula": "\\begin{align*} R _ 0 : = \\sup _ { { \\bf y } \\in \\mathcal A } r ( { \\bf y } ) & \\le \\big ( 1 + \\alpha _ 0 + \\alpha ^ { - 1 } \\big ) \\sup _ { { \\bf y } \\in \\mathcal A } \\big ( | x | + | v | + | x ' | + | v ' | \\big ) \\\\ & \\le 2 \\big ( 1 + \\alpha _ 0 + \\alpha ^ { - 1 } \\big ) \\sup _ { ( x , v ) \\in \\mathcal A _ 0 } \\big ( | x | + | v | \\big ) \\\\ & \\le 2 R ^ * \\big ( 1 + \\alpha _ 0 + \\alpha ^ { - 1 } \\big ) . \\end{align*}"}
{"id": "1171.png", "formula": "\\begin{align*} \\beta ^ { m a t } : = \\left ( \\begin{array} { c c } 0 & \\beta ^ { ( \\eta ) } _ { 1 2 } \\\\ \\beta ^ { ( \\eta ) } _ { 2 1 } & 0 \\end{array} \\right ) = \\frac { i } { 2 } \\left [ \\sigma _ 3 , m ^ { P C } _ { \\eta , 1 } \\right ] = \\left ( \\begin{array} { c c } 0 & i [ m ^ { P C } _ { \\eta , 1 } ] _ { 1 2 } \\\\ - i [ m ^ { P C } _ { \\eta , 1 } ] _ { 2 1 } & 0 \\end{array} \\right ) . \\end{align*}"}
{"id": "544.png", "formula": "\\begin{align*} \\int _ \\Omega & \\left ( \\frac { 1 } { 2 } \\rho | u | ^ 2 + P ( \\rho ) \\right ) ( \\tau ) + \\int _ 0 ^ \\tau \\int _ \\Omega \\mathbb S ( \\nabla u ) \\cdot \\nabla u \\\\ & \\leq \\int _ \\Omega \\left ( \\frac { 1 } { 2 } \\rho _ 0 | u _ 0 | ^ 2 + P ( \\rho _ 0 ) \\right ) + \\int _ 0 ^ \\tau \\int _ \\Omega \\rho f \\cdot u . \\end{align*}"}
{"id": "6390.png", "formula": "\\begin{align*} \\mathbb G _ n = \\left \\{ \\left ( \\sum _ { 1 \\leq i \\leq n } z _ i , \\sum _ { 1 \\leq i < j \\leq n } z _ i z _ j , \\dots , \\prod _ { i = 1 } ^ n z _ i \\right ) : \\ , | z _ i | < 1 , i = 1 , \\dots , n \\right \\} . \\end{align*}"}
{"id": "4131.png", "formula": "\\begin{align*} | C \\cap L | = O _ d ( \\log | L | ) = O _ d ( \\log \\ell _ 1 ) , \\end{align*}"}
{"id": "7251.png", "formula": "\\begin{align*} \\theta _ 1 = \\dfrac { 3 ( 2 a c ^ 3 + 2 b ^ 3 d - b ^ 2 c ^ 2 - 3 a b c d ) } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } , ~ \\theta _ 2 = \\dfrac { c ^ 2 ( 3 a c - b ^ 2 ) ^ 2 } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } , ~ t = \\dfrac { b ( 3 b d - c ^ 2 ) } { c ( 3 a c - b ^ 2 ) } . \\end{align*}"}
{"id": "4019.png", "formula": "\\begin{align*} \\mod { \\frac { I _ 1 ( w ) } { G ( w ) } } & = \\mod { \\frac { O ( w ^ { - 1 } ) e ^ { - w ^ 2 + 1 + O ( w ^ { - 1 } ) } + O ( w ^ { - 1 } ) e ^ { O ( w ^ { - 1 } ) } } { e ^ { - w ^ 2 + 1 } - 1 } } \\leq \\frac { C } { \\mod { w } } \\frac { \\mod { e ^ { - w ^ 2 + 1 } } + 1 } { \\mod { e ^ { - w ^ 2 + 1 } - 1 } } , \\end{align*}"}
{"id": "8597.png", "formula": "\\begin{align*} \\prod _ \\ast a _ j ( k _ j ) : = \\overline { a _ 1 ( k _ 1 ) } a _ 2 ( k _ 2 ) \\overline { a _ 3 ( k _ 3 ) } a _ 4 ( k _ 4 ) . \\end{align*}"}
{"id": "933.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\big ( e _ { \\lambda } ( t ) - 1 \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } , ( k \\ge 0 ) . \\end{align*}"}
{"id": "6484.png", "formula": "\\begin{align*} E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n / ( 1 - 2 \\alpha ) } } \\right ) ^ 2 \\right ] - 1 \\sim - \\dfrac { 1 } { \\Gamma ( 2 \\alpha ) } \\cdot n ^ { - ( 1 - 2 \\alpha ) } \\qquad ( n \\to \\infty ) . \\end{align*}"}
{"id": "2917.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 ( k ) = \\frac 2 { 9 0 ^ k } \\{ 1 + O ( n ^ { - 1 } ) \\} . \\end{align*}"}
{"id": "7077.png", "formula": "\\begin{align*} K = \\langle R ( X _ 1 , X _ 2 ) X _ 2 , X _ 1 \\rangle = 2 \\langle \\xi , E _ 3 ^ { \\perp } \\rangle ^ 2 - 1 + 2 f ^ 2 - \\frac { 1 } { 2 } | A | ^ 2 , \\end{align*}"}
{"id": "2030.png", "formula": "\\begin{align*} N _ t ^ { h _ m } - N _ t ^ { h _ n } & = - \\int _ 0 ^ t h _ m ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } ^ m } + \\int _ 0 ^ t h _ n ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } ^ n } \\\\ & = - \\int _ 0 ^ t ( h _ m \\ 1 _ { K _ m } - h _ n \\ 1 _ { K _ n } ) ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } _ 1 ^ * } + \\int _ 0 ^ t ( h _ m - h _ n ) ( X _ s ) { \\rm d } A _ s ^ { \\mu _ 2 + N ( e ^ { F _ 1 } ( 1 - e ^ { - F _ 2 } ) \\mu _ H } . \\end{align*}"}
{"id": "8516.png", "formula": "\\begin{align*} [ x ^ e , g ] _ + = [ e x , g ] _ + = e [ x , g ] _ + = 0 g * ( x ^ e ) = g * ( e x ) = e ( g * x ) = 0 . \\end{align*}"}
{"id": "8924.png", "formula": "\\begin{align*} \\check H _ { c t } ^ q ( F _ n ; A ) = \\begin{cases} \\bigoplus _ \\N A & q = 0 \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "9382.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } \\frac { F _ { n , \\lambda } ( t ) } { t } d t = \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( k - 1 ) ! x ^ { k } , ( n \\ge 1 ) . \\end{align*}"}
{"id": "5950.png", "formula": "\\begin{align*} \\norm { | f | ^ 2 - | g | ^ 2 } _ 2 ^ 2 & = \\sum _ { k } | | a _ k | ^ 2 - | b _ k | ^ 2 | ^ 2 \\norm { s _ k } _ 2 ^ 2 + ( \\norm { f } _ 2 ^ 2 - \\norm { g } _ 2 ^ 2 ) ^ 2 + \\sum _ { i \\ne j } | a _ i \\overline { a _ j } - b _ i \\overline { b _ j } | ^ 2 \\norm { r _ i \\overline { r _ j } } _ 2 ^ 2 \\\\ & \\ge \\delta \\sum _ { k } | | a _ k | ^ 2 - | b _ k | ^ 2 | ^ 2 + ( \\norm { f } _ 2 ^ 2 - \\norm { g } _ 2 ^ 2 ) ^ 2 + \\delta \\sum _ { i \\ne j } | a _ i \\overline { a _ j } - b _ i \\overline { b _ j } | ^ 2 \\end{align*}"}
{"id": "9095.png", "formula": "\\begin{align*} \\alpha \\alpha ' = 1 + ( \\alpha _ m + \\alpha ' _ m ) t ^ m + \\cdots \\quad \\alpha ^ { - 1 } = 1 - \\alpha _ m t ^ m + \\cdots \\end{align*}"}
{"id": "7656.png", "formula": "\\begin{align*} \\tilde { \\Omega } _ { \\varepsilon } & \\coloneqq \\{ \\mathbf { x } \\in \\R ^ N : k _ { \\varepsilon } \\mathbf { x } + \\mathbf { x } _ { \\varepsilon } \\in \\Omega \\} \\ ; , \\\\ \\Omega _ { \\varepsilon } & \\coloneqq \\{ \\mathbf { x } \\in \\R ^ N : k _ { \\varepsilon } \\mathbf { x } + \\mathbf { q } \\in \\Omega \\} \\ ; . \\end{align*}"}
{"id": "4450.png", "formula": "\\begin{align*} x ^ { n } = \\sum _ { k = 0 } ^ { n } S _ { 2 } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 1 , 4 ] ) . \\end{align*}"}
{"id": "8831.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { | X | } | \\mathcal { X } ( L ^ { ( i ) } ) | \\end{align*}"}
{"id": "7872.png", "formula": "\\begin{align*} \\nu _ F = \\sum _ { H \\in H ( n - m ) } \\nu _ { F , H } . \\end{align*}"}
{"id": "8299.png", "formula": "\\begin{align*} \\begin{cases} E _ { \\text i } ^ + ( \\kappa _ x ) + E _ { \\text r } ^ - ( \\kappa _ x ) = E _ { \\text t } ^ + ( \\kappa _ x ) \\\\ E _ { \\text i } ^ + ( \\kappa _ x ) - E _ { \\text r } ^ - ( \\kappa _ x ) = \\alpha ( \\kappa _ x ) E _ { \\text t } ^ + ( \\kappa _ x ) \\end{cases} \\end{align*}"}
{"id": "1452.png", "formula": "\\begin{align*} \\lambda ( \\alpha _ 1 \\alpha _ 2 , z ) = \\lambda ( \\alpha _ 1 , \\alpha _ 2 z ) \\lambda ( \\alpha _ 2 , z ) \\alpha _ 1 , \\alpha _ 2 \\in \\mathcal { G } , z \\in \\mathcal { H } . \\end{align*}"}
{"id": "3911.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } e _ k x ^ k = \\sum _ { n = 0 } ^ n \\binom { n } { j } f _ j x ^ j ( 1 - x ) ^ { n - j } = ( 1 - x ) ^ n \\sum _ { m = 0 } ^ n \\binom { n } { j } f _ j \\left ( \\frac { x } { 1 - x } \\right ) ^ j . \\end{align*}"}
{"id": "3262.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } - i t \\cdot \\omega _ 2 ( i t ) & = \\lambda _ 1 ( \\mu ) ^ 2 - | \\lambda | ^ 2 . \\end{align*}"}
{"id": "8908.png", "formula": "\\begin{align*} \\eta ^ 2 _ { \\alpha ^ { - 1 } A _ Y } ( U ) : \\alpha ^ { - 1 } A _ Y ( U ) = A _ Y ( \\alpha ( U ) ) & \\to \\alpha ^ * A _ Y ( U ) = A _ X ( U ) \\\\ ( \\varphi : \\alpha ( U ) \\to A ) & \\mapsto ( \\varphi \\circ \\alpha : U \\to A ) \\end{align*}"}
{"id": "1235.png", "formula": "\\begin{align*} \\dim _ H ( \\limsup _ { n \\rightarrow + \\infty } R _ n ) = \\min _ { 1 \\leq i \\leq d } \\left \\{ \\frac { \\dim ( \\mu ) + \\sum _ { 1 \\leq j \\leq i } \\tau _ i - \\tau _ j } { \\tau _ i } \\right \\} . \\end{align*}"}
{"id": "8855.png", "formula": "\\begin{align*} \\tilde \\varphi : \\Z _ { \\ge 0 } & \\to A \\\\ x & \\mapsto \\sum _ { i = 0 } ^ { x - 1 } \\varphi ( i , i + 1 ) . \\end{align*}"}
{"id": "9455.png", "formula": "\\begin{align*} f _ 3 = - \\left ( \\tfrac { a _ 0 } { 2 b _ 2 } \\right ) \\cdot \\frac { f _ 2 ^ 2 - f _ 1 ^ p + ( 2 b _ 1 / a _ 0 ) \\cdot f _ 1 ^ { ( p + 1 ) / 2 } X ^ { p ^ s - p ^ { s - 1 } } } { X ^ { p ^ s - 2 p ^ { s - 2 } } } \\end{align*}"}
{"id": "2786.png", "formula": "\\begin{align*} \\left [ ( 1 - \\Delta ) ^ 2 + e _ 0 ^ 2 \\right ] \\mathcal { Y } _ 1 = F ( x ) , \\end{align*}"}
{"id": "1821.png", "formula": "\\begin{align*} D _ n ( u , v ) = A _ n ( x , y ) , \\end{align*}"}
{"id": "7020.png", "formula": "\\begin{align*} \\Big ( \\| R \\| ^ 2 + \\sum _ { j = 1 } ^ { L } \\| S _ j \\| ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\le \\frac { C } { \\delta ^ { \\max m _ i } } . \\end{align*}"}
{"id": "4554.png", "formula": "\\begin{align*} m _ 1 , m _ 4 & \\ge \\biggl ( k ' + \\frac { h - h _ 5 + 1 } { 2 } \\biggr ) - k \\ge \\frac { h - h _ 5 + 1 } { 2 } , \\\\ m _ 2 , m _ 3 & \\ge \\frac { 1 } { 2 } \\biggl ( \\frac { 3 } { 2 } ( h - 1 ) + \\frac { h - h _ 5 + 1 } { 2 } \\biggr ) = h - \\frac { h _ 5 } { 4 } - \\frac { 1 } { 2 } \\geq h - h _ 5 = h _ 1 + h _ 2 + h _ 3 + h _ 4 + 1 , \\end{align*}"}
{"id": "7500.png", "formula": "\\begin{align*} \\vartheta ^ n = \\frac 1 2 \\Big ( ( V + \\beta | \\phi ^ n | ^ 2 ) _ { \\max } + ( V + \\beta | \\phi ^ n | ^ 2 ) _ { \\min } \\Big ) , \\end{align*}"}
{"id": "6191.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\| W W ^ { T } \\bar { U } \\bar { \\Sigma } ^ { - 1 } \\| _ F - \\| ( S S ^ { T } - W W ^ { T } ) \\bar { U } \\bar { \\Sigma } ^ { - 1 } \\| _ F \\right | & \\leq \\| W W ^ { T } \\bar { U } \\bar { \\Sigma } ^ { - 1 } + ( S S ^ { T } - W W ^ { T } ) \\bar { U } \\bar { \\Sigma } ^ { - 1 } \\| _ F = \\| S \\hat { V } \\| _ F , \\end{aligned} \\end{align*}"}
{"id": "8249.png", "formula": "\\begin{align*} u \\leq v & \\Leftrightarrow \\ell ( v ) = \\ell ( u ) + \\ell ( v u ^ { - 1 } ) \\\\ & \\Leftrightarrow \\ell ( v \\xi ^ { - 1 } ) = \\ell ( u \\xi ^ { - 1 } ) + \\ell ( v u ^ { - 1 } ) \\\\ & \\Leftrightarrow \\ell ( v \\xi ^ { - 1 } ) = \\ell ( u \\xi ^ { - 1 } ) + \\ell ( v \\xi ^ { - 1 } ( u \\xi ^ { - 1 } ) ^ { - 1 } ) \\\\ & \\Leftrightarrow u \\xi ^ { - 1 } \\leq v \\xi ^ { - 1 } \\\\ & \\Leftrightarrow \\rho _ \\xi ( u ) \\leq \\rho _ \\xi ( v ) . \\end{align*}"}
{"id": "3677.png", "formula": "\\begin{align*} \\mathrm { s g n } _ X ( g h ) & = \\prod _ { \\{ u , v \\} \\in E ( X ) } \\frac { u ^ { g h } - v ^ { g h } } { u - v } \\\\ & = \\prod _ { \\{ u , v \\} \\in E ( X ) } \\frac { u ^ g - v ^ g } { u - v } \\cdot \\frac { ( u ^ g ) ^ h - ( v ^ g ) ^ h } { u ^ g - v ^ g } \\\\ & = \\prod _ { \\{ u , v \\} \\in E ( X ) } \\frac { u ^ g - v ^ g } { u - v } \\cdot \\prod _ { \\{ u , v \\} \\in E ( X ) } \\frac { ( u ^ g ) ^ h - ( v ^ g ) ^ h } { u ^ g - v ^ g } \\\\ & = \\mathrm { s g n } _ X ( g ) \\ \\mathrm { s g n } _ X ( h ) , \\end{align*}"}
{"id": "7346.png", "formula": "\\begin{align*} \\begin{aligned} & \\psi _ t ( x _ q , t _ q ) + \\lambda G _ \\beta ( a _ q , \\xi _ q , Y _ q , W _ \\star [ x _ 0 , t _ 0 ] \\} ) + ( 1 - \\lambda ) G _ \\beta ( b _ q , \\xi _ q , Z _ q , W _ \\star [ x _ 0 , t _ 0 ] ) ) \\\\ & \\geq \\lambda \\left ( G _ \\beta ( a _ q , \\xi _ q , Y _ q , W _ \\star [ x _ 0 , t _ 0 ] ) - G _ \\beta ( a _ q , \\xi _ q , Y _ q , U [ y _ q , t _ q ] ) \\right ) \\\\ & \\qquad + ( 1 - \\lambda ) \\left ( G _ \\beta ( b _ q , \\xi _ q , Z _ q , W _ \\star [ x _ 0 , t _ 0 ] ) - G _ \\beta ( b _ q , \\xi _ q , Z _ q , U [ z _ q , t _ q ] ) \\right ) , \\end{aligned} \\end{align*}"}
{"id": "7390.png", "formula": "\\begin{align*} c _ { x , y } ( \\eta ) : = \\tilde { c } _ { x , y } ( \\eta ) \\xi _ { x , y } ( \\eta ) , \\end{align*}"}
{"id": "51.png", "formula": "\\begin{align*} f ( t z ) & = t ^ { - k } f ( z ) , \\\\ f ( \\gamma z ) & = \\chi ( \\gamma ) f ( z ) , \\end{align*}"}
{"id": "9435.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } H _ \\mu u \\ ! + \\ ! B _ q ( w , u ) & = & f & ( x , t ) \\in { \\mathbb R } ^ n \\times ( 0 , T ) , \\\\ u & = & u _ 0 , & ( x , t ) \\in \\mathbb { R } ^ n \\times \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
{"id": "8885.png", "formula": "\\begin{align*} \\psi \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) : = \\psi _ { \\max ( y _ i ) } \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) \\end{align*}"}
{"id": "2923.png", "formula": "\\begin{align*} \\tilde M _ { n , A } = \\frac 2 n \\sum _ { i < j } \\prod _ { p \\in A } \\tilde I _ { i , j } ^ { ( p ) } , \\tilde N _ { n , A } = \\frac 1 n \\sum _ { i = 1 } ^ n \\prod _ { p \\in A } \\tilde I _ { i , i } ^ { ( p ) } . \\end{align*}"}
{"id": "30.png", "formula": "\\begin{align*} \\nabla _ U U = \\nabla _ V U = 0 , \\nabla _ U V = \\nabla _ V V = 0 . \\end{align*}"}
{"id": "1228.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\zeta } ( E ) = \\lim _ { t \\to 0 ^ + } \\mathcal { H } ^ { \\zeta } _ t ( E ) . \\end{align*}"}
{"id": "1214.png", "formula": "\\begin{align*} 4 B ^ { [ \\widetilde { U } ] } \\cap 4 B ^ { [ \\widehat { U } ] } = \\emptyset ; \\end{align*}"}
{"id": "5239.png", "formula": "\\begin{align*} V \\cdot W = \\left \\{ \\sum _ i v _ i w _ i \\mid v _ i \\in V , w _ i \\in W \\right \\} . \\end{align*}"}
{"id": "6208.png", "formula": "\\begin{align*} \\begin{aligned} & ~ ~ ~ ~ \\epsilon + 2 ( 2 + \\xi ) ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) ^ 2 \\\\ & = \\epsilon + \\xi ^ 4 + 6 \\xi ^ 3 + 8 \\xi ^ 2 + 4 \\xi ^ 3 \\sqrt { 1 + \\xi } + 4 \\xi ^ 2 \\sqrt { 1 + \\xi } + 8 \\xi \\sqrt { 1 + \\xi } \\\\ & < \\epsilon + 1 5 \\xi + 1 6 \\xi \\sqrt { 1 + \\xi } < \\epsilon + 3 6 \\xi = \\frac { \\epsilon ^ 2 + 2 0 \\epsilon } { \\epsilon + 2 } < 1 0 \\epsilon . \\end{aligned} \\end{align*}"}
{"id": "8099.png", "formula": "\\begin{align*} \\widehat { ( \\partial _ { \\Sigma _ 0 , \\epsilon } ^ { * } ) ^ { \\otimes 2 } \\overline { u } _ n } ( \\xi , \\eta ) = - 4 \\xi _ u \\eta _ u \\hat { \\delta } _ { \\epsilon } ( 2 \\xi _ v ) \\hat { \\delta } _ { \\epsilon } ( 2 \\eta _ v ) \\int _ { \\mathbb { R } ^ { 2 } } \\overline { u } _ n ( s - s ' ) e ^ { i ( \\xi _ u - \\xi _ v ) s } e ^ { i ( \\eta _ u - \\eta _ v ) s } \\ , \\mathrm { d } s \\ , \\mathrm { d } s ' . \\end{align*}"}
{"id": "9475.png", "formula": "\\begin{align*} p _ 1 ( { \\bf \\underline { f } } ) = - \\left ( \\frac { 2 b _ 1 } { a _ 0 } \\right ) \\cdot f _ 1 ^ { ( p + 1 ) / 2 } X ^ { p ^ s - p ^ { s - 1 } } - \\left ( \\frac { 2 b _ 2 } { a _ 0 } \\right ) f _ 3 X ^ { p ^ s - 2 p ^ { s - 2 } } , \\end{align*}"}
{"id": "5329.png", "formula": "\\begin{align*} \\psi | q _ i | \\rightarrow 0 , \\ ; \\ ; \\nabla _ { q _ i } \\psi \\cdot \\frac { q _ i } { | q _ i | } \\rightarrow 0 , \\ ; \\ ; \\ ; \\ ; | q _ i | \\rightarrow \\infty , \\ ; \\ ; \\ ; \\ ; ( x , t ) \\in \\mathbb { R } ^ 3 \\times ( 0 , T ] , \\ ; \\ ; i = 1 , \\ldots , K . \\end{align*}"}
{"id": "2301.png", "formula": "\\begin{align*} f = \\iint _ { \\R ^ { 2 d } } V _ g f ( \\l ) \\ , \\pi ( \\l ) g \\ , d \\l . \\end{align*}"}
{"id": "3208.png", "formula": "\\begin{align*} { \\rm S I N R } _ { D _ 2 , S } ^ { x _ 2 } = \\frac { | h _ 2 | ^ 2 P _ t \\alpha _ 2 } { | h _ 2 | ^ 2 P _ t \\alpha _ 1 + \\sigma ^ 2 } , \\end{align*}"}
{"id": "7320.png", "formula": "\\begin{align*} \\nabla \\varphi ( y _ \\varepsilon , t _ \\varepsilon ) = 0 , \\nabla ^ 2 \\varphi ( y _ \\varepsilon , t _ \\varepsilon ) \\leq 0 . \\end{align*}"}
{"id": "6715.png", "formula": "\\begin{align*} 0 = { \\rm t r . d e g } _ { \\overline { k } ( t ) } \\overline { k } ( t ) \\bigl \\{ 1 , \\ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d } } \\bigr \\} = { \\rm t r . d e g } _ { \\overline { k } } \\overline { k } \\bigl \\{ 1 , \\ _ { s + 1 } F _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d } } \\bigr \\} . \\end{align*}"}
{"id": "1382.png", "formula": "\\begin{align*} E _ i ( x ) = \\alpha + \\int _ 0 ^ x \\left ( n _ i ( y ) - b _ i ( y ) \\right ) d y , i = 1 , 2 . \\end{align*}"}
{"id": "831.png", "formula": "\\begin{align*} \\mu _ \\omega ( B _ \\rho ) & \\ge \\mu _ \\omega ( X \\times [ y _ 0 ( 1 - \\Delta ) , y _ 0 ( 1 + \\delta ) ] ) \\\\ & = \\frac { \\mu ( Z ) } { ( 2 \\beta - 1 - a ) y _ 0 ^ { 2 \\beta - 1 - a } } \\left [ ( 1 - \\Delta ) ^ { 1 + a - 2 \\beta } - ( 1 + \\Delta ) ^ { 1 + a - 2 \\beta } \\right ] \\\\ & \\approx r ^ { ( 2 \\beta - 1 - a ) / ( \\beta - 1 ) } . \\end{align*}"}
{"id": "7062.png", "formula": "\\begin{align*} \\theta _ K ( L ) = \\tau _ K ( L ) \\wedge \\tau ' _ { K } . \\end{align*}"}
{"id": "2494.png", "formula": "\\begin{align*} F _ r ( x , \\omega , e ^ { 2 \\pi i \\tau } ) = e ^ { - 2 \\pi i \\tau } F ( x , \\omega ) . \\end{align*}"}
{"id": "4211.png", "formula": "\\begin{align*} u _ \\varepsilon = \\varepsilon + \\varepsilon ^ m v _ m ( x ) + r _ \\varepsilon ( x ) \\forall \\ , x \\in M , \\end{align*}"}
{"id": "674.png", "formula": "\\begin{align*} \\begin{cases} \\ A ( x ) - f ( n ) > 2 ^ { - x } \\implies x > f ( n ) \\\\ [ 8 p t ] \\ A ( x ) - f ( n ) < 2 ^ { - x } \\implies x < f ( n ) \\end{cases} . \\end{align*}"}
{"id": "8060.png", "formula": "\\begin{align*} E ^ { i j } _ { 0 } ( x , x ' ) = \\sum _ { k = 0 } ^ n a _ k \\left ( \\frac { \\partial } { \\partial x } \\right ) ^ k \\delta ( x - x ' ) , \\end{align*}"}
{"id": "5568.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d Z ( t ) & = & ( B Z _ t + b ) d t + c \\ , d W ( t ) \\medskip \\\\ Z ( 0 ) & = & z . \\end{array} \\right . \\end{align*}"}
{"id": "1124.png", "formula": "\\begin{align*} G : = \\left \\{ \\begin{aligned} & \\begin{pmatrix} 1 & 0 \\\\ - r e ^ { 2 i t g } & 1 \\end{pmatrix} , & k \\in U _ 1 \\cup U _ 2 , \\\\ & \\begin{pmatrix} 1 & - r ^ { * } e ^ { - 2 i t g } \\\\ 0 & 1 \\end{pmatrix} , & k \\in U _ 1 ^ { * } \\cup U _ 2 ^ { * } , \\\\ & I , & \\textnormal { e l s e w h e r e } \\end{aligned} \\right . \\end{align*}"}
{"id": "3579.png", "formula": "\\begin{align*} r & = ( s _ 1 - c _ 0 + p ) + ( p - 1 ) ( t - 1 ) - 1 + { \\sum } _ { i = t } ^ { e - 1 } ( s _ { i + 1 } - c _ i ) \\\\ & = p - 1 - c _ 0 + { \\sum } _ { i = 1 } ^ e s _ i - { \\sum } _ { j = t } ^ { e - 1 } c _ j \\le n . \\end{align*}"}
{"id": "7861.png", "formula": "\\begin{align*} ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cup C _ 1 \\cup C _ 2 & = ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cup ( C \\setminus \\{ \\bar { a } _ n \\} ) \\\\ & = ( A \\cup C ) \\setminus \\{ \\bar { a } _ n \\} = \\mathbb { N } \\setminus \\{ \\bar { a } _ n \\} \\end{align*}"}
{"id": "7672.png", "formula": "\\begin{align*} - k _ { \\varepsilon } \\Delta v _ { \\varepsilon } ( \\mathbf { x } ) + | \\nabla v _ { \\varepsilon } ( \\mathbf { x } ) | ^ 2 - \\tilde { \\lambda } _ 0 \\underline { m } = - k _ { \\varepsilon } \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\frac { ( N - 1 ) } { | \\mathbf { x } - \\mathbf { y } _ { \\varepsilon } | } + \\sigma _ 3 ^ 2 \\tilde { \\lambda } _ 0 \\underline { m } - \\tilde { \\lambda } _ 0 \\underline { m } < 0 \\end{align*}"}
{"id": "3327.png", "formula": "\\begin{align*} e '' _ { f _ { \\lambda + 1 } } = ( n - 1 ) f _ { \\lambda + 1 } + g _ { \\lambda + 1 } = ( n - 1 ) f _ { \\lambda + 1 } + ( f _ { \\lambda + 1 } - f _ \\lambda ) = f _ { \\lambda + 1 } + f _ { \\lambda + 2 } \\end{align*}"}
{"id": "2831.png", "formula": "\\begin{align*} \\varphi ( x ) = \\begin{cases} | x | ^ 2 , & | x | \\le 1 , \\\\ 0 , & | x | \\ge 2 , \\end{cases} \\varphi ( r ) \\ge 0 , \\varphi '' ( r ) \\le 2 , \\end{align*}"}
{"id": "2733.png", "formula": "\\begin{align*} m _ { i , \\lambda } ( S ( \\underline { P } , B ) ) = 0 . \\end{align*}"}
{"id": "5322.png", "formula": "\\begin{align*} ( a \\omega ) ^ * = \\omega ^ * a ^ * , ( \\omega a ) ^ * = a ^ * \\omega ^ * . \\end{align*}"}
{"id": "4295.png", "formula": "\\begin{align*} \\Lambda _ \\xi = 2 + \\xi \\partial _ \\xi \\end{align*}"}
{"id": "3224.png", "formula": "\\begin{align*} F _ { \\mu } ( \\omega _ 1 ( z ) ) = F _ { \\nu } ( \\omega _ 2 ( z ) ) = F _ { \\mu \\boxplus \\nu } ( z ) , z \\in \\mathbb { C } ^ + . \\end{align*}"}
{"id": "1432.png", "formula": "\\begin{align*} \\sum _ { v _ i , v _ j \\notin V ( C ) } \\beta _ i \\beta _ j - \\sum _ { v _ i \\in V ( C ) , v _ j \\notin V ( C ) } \\alpha _ i \\beta _ j = \\sum _ { v _ j \\notin V ( C ) } \\beta _ j ( \\sum _ { v _ r \\notin V ( C ) } \\beta _ r - \\sum _ { v _ l \\in V ( C ) } \\alpha _ l ) \\end{align*}"}
{"id": "7305.png", "formula": "\\begin{align*} ( \\alpha ( \\beta ) ) ^ { ( s ) } & = \\sum _ { t = 1 } ^ s \\sum _ { A _ 1 \\dot \\cup \\ldots \\dot \\cup A _ t } \\beta ^ { ( | A _ 1 | ) } \\ldots \\beta ^ { ( | A _ t | ) } \\alpha ^ { ( t ) } ( \\beta ) \\\\ & = \\sum _ { ( 1 ^ { a _ 1 } , \\ldots , s ^ { a _ s } ) \\in P ( s ) } \\frac { s ! } { ( 1 ! ) ^ { a _ 1 } \\ldots ( s ! ) ^ { a _ s } a _ 1 ! \\ldots a _ s ! } ( \\beta ' ) ^ { a _ 1 } \\ldots ( \\beta ^ { ( s ) } ) ^ { a _ s } ( \\alpha ( \\beta ) ) ^ { ( a _ 1 + \\ldots + a _ s ) } , \\end{align*}"}
{"id": "6121.png", "formula": "\\begin{align*} \\alpha _ 0 ( x ) = \\sum _ { r = 0 } ^ \\infty a _ { r ( m + 2 ) } x ^ { r ( m + 2 ) } + \\sum _ { r = 0 } ^ \\infty a _ { r ( m + 2 ) + 1 } x ^ { r ( m + 2 ) + 1 } \\ , . \\end{align*}"}
{"id": "2093.png", "formula": "\\begin{align*} \\sum _ { [ K _ p : \\mathbb { Q } _ p ] = n } \\frac { | \\mathrm { D i s c } ( K _ p ) | _ { p } ^ { 1 / 2 } } { | \\mathrm { A u t } ( K _ p ) | } \\int _ { \\mathcal { O } _ { K _ p } } | \\mathrm { D i s c } ( f _ \\alpha ) | _ p ^ { 1 / 2 } \\psi _ { p ^ { 2 k } } ( f _ \\alpha ) e _ { p ^ { 2 k } } ( - u _ 1 \\sigma _ 1 ( \\pmb { \\lambda } ) + u _ 2 \\sigma _ 2 ( \\pmb { \\lambda } ) ) d \\mu ( \\alpha ) , \\end{align*}"}
{"id": "4507.png", "formula": "\\begin{align*} - \\int _ { H _ { \\sigma _ 2 } } ^ T N ( \\sigma _ 2 , t ) { d } \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t } } } { t } \\Big ) \\le - N ( \\sigma _ 2 , T ) \\int _ { t _ 0 } ^ T { d } \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t } } } { t } \\Big ) = N ( \\sigma _ 2 , T ) \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t _ 0 } } } { t _ 0 } - \\frac { x ^ { - \\frac { 1 } { R \\log T } } } { T } \\Big ) . \\end{align*}"}
{"id": "376.png", "formula": "\\begin{align*} & \\inf _ { \\rho , \\widehat v } \\ ; [ \\int _ 0 ^ 1 \\frac 1 2 \\frac 1 { ( 1 + \\epsilon \\dot W ^ { \\delta } ) ^ 2 } \\ < \\widehat v _ t , \\widehat v _ t \\ > _ { \\theta ( \\rho _ t ) } d t ] \\\\ & \\ ; \\ ; d \\rho ( t ) + d i v _ G ^ { \\theta } ( \\rho ( t ) \\widehat v ( t ) ) = 0 \\\\ & \\ ; \\ ; \\rho ( 0 ) = \\rho _ a , \\ ; \\rho ( 1 ) = \\rho _ b . \\end{align*}"}
{"id": "9241.png", "formula": "\\begin{align*} & \\forall \\gamma ^ 1 , { \\gamma ' } ^ 1 , x ^ X , { x ' } ^ X \\big ( \\gamma > _ \\mathbb { R } 0 \\land { \\gamma ' } > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\\\ & \\qquad \\qquad \\qquad \\qquad \\land x = _ X x ' \\land \\gamma = _ \\mathbb { R } \\gamma ' \\rightarrow J ^ A _ \\gamma x = _ X J ^ A _ { \\gamma ' } x ' \\big ) . \\end{align*}"}
{"id": "22.png", "formula": "\\begin{align*} I _ 1 \\leq C \\int _ 0 ^ { 4 \\epsilon } \\int _ { \\Gamma _ t } t ^ { - p s } d S d t \\leq C \\int _ { 0 } ^ { 4 \\epsilon } t ^ { - p s } d t = C \\epsilon ^ { 1 - p s } . \\end{align*}"}
{"id": "1649.png", "formula": "\\begin{align*} \\dot { W } _ { s , p } ^ \\nu ( x ) : = \\sum _ { j = - \\infty } ^ \\infty 2 ^ { j s q } \\int _ { B ( x , 2 ^ { - j } ) } \\left ( \\nu ( B ( y , 2 ^ { - j } ) ) \\right ) ^ { q - 1 } \\mu ( d y ) , x \\in M , \\end{align*}"}
{"id": "8811.png", "formula": "\\begin{align*} \\gamma ^ { \\epsilon , L } ( d \\varphi ) \\propto \\exp \\left [ - \\frac { 1 } { 2 } \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\left ( \\varphi _ x ( - \\Delta ^ 1 + \\epsilon ^ 2 m ) \\varphi _ x \\right ) \\right ] d \\varphi = \\exp \\left [ - \\frac { 1 } { 2 } ( \\varphi , A _ { \\epsilon } \\varphi ) \\right ] d \\varphi , \\end{align*}"}
{"id": "653.png", "formula": "\\begin{align*} \\abs { \\frac { f ( x ) - g ( x ) } { x + 1 } - \\alpha } \\ & \\leq \\ \\abs { \\frac { f ( x ) - g ( x ) } { x + 1 } - \\frac { f _ 0 ( x ) - g _ 0 ( x ) } { h _ 0 ( x ) + 1 } } + \\abs { \\frac { f _ 0 ( x ) - g _ 0 ( x ) } { h _ 0 ( x ) + 1 } - \\alpha } \\\\ [ 1 1 p t ] & \\leq \\frac { 1 } { 2 ( x + 1 ) } + \\abs { \\frac { f _ 0 ( x ) - g _ 0 ( x ) } { h _ 0 ( x ) + 1 } - \\alpha } \\\\ [ 1 1 p t ] & < \\ \\frac { 1 } { 2 ( x + 1 ) } + \\frac { 1 } { 2 ( x + 1 ) } \\\\ [ 1 1 p t ] & = \\ \\frac { 1 } { x + 1 } . \\end{align*}"}
{"id": "899.png", "formula": "\\begin{align*} a ( \\widetilde E _ 2 ^ { ( 4 ) } , T ) = \\frac { 1 1 5 2 } { ( 1 - p ) ^ 2 } \\prod _ { q \\not = p } F _ q ( T , q ^ { - 3 } ) . \\end{align*}"}
{"id": "4452.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( e ^ { t } - 1 \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 2 } ( n , k ) \\frac { t ^ { n } } { n ! } , ( k \\ge 0 ) , ( \\mathrm { s e e } \\ [ 4 ] ) . \\end{align*}"}
{"id": "3609.png", "formula": "\\begin{align*} S _ n = \\begin{cases} 0 & R _ { n + 2 } - R _ { n + 1 } \\ , , \\\\ 1 & R _ { n + 2 } - R _ { n + 1 } \\ , . \\end{cases} \\end{align*}"}
{"id": "4883.png", "formula": "\\begin{align*} 0 = ( 1 - w ^ 2 ) h '' ( w ) - 2 ( w + 1 ) h ' ( w ) + n ( n + 1 ) h ( w ) . \\end{align*}"}
{"id": "8086.png", "formula": "\\begin{align*} ( \\Psi _ { \\Sigma } ( f ) \\star _ { H , \\ell } \\Psi _ { \\Sigma } ( g ) ) [ \\psi ] = ( \\Psi _ { \\Sigma } ( f ) \\cdot \\Psi _ { \\Sigma } ( g ) ) [ \\psi ] - \\frac { \\hbar } { 4 \\pi } \\int _ { \\mathbb { R } ^ { 2 } } \\frac { f ( s ) g ( s ' ) } { ( s - s ' ) ^ 2 } \\ , \\mathrm { d } s \\ , \\mathrm { d } s ' , \\end{align*}"}
{"id": "8560.png", "formula": "\\begin{align*} S ( k ) : = \\left ( \\begin{array} { c c } T ( k ) & R _ { + } ( k ) \\\\ R _ { - } ( k ) & T ( k ) \\end{array} \\right ) , \\ \\ S ^ { - 1 } ( k ) : = \\left ( \\begin{array} { c c } \\overline { T ( k ) } & \\overline { R _ { - } ( k ) } \\\\ \\overline { R _ { + } ( k ) } & \\overline { T ( k ) } \\end{array} \\right ) . \\end{align*}"}
{"id": "3266.png", "formula": "\\begin{align*} \\omega _ 2 ( i t ) = i \\left [ t - s + \\frac { ( s - t ) ^ 2 + | \\lambda | ^ 2 } { s - t } \\right ] = i \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , t ) - t } = \\frac { | \\lambda | ^ 2 } { i t - \\omega _ 1 ( i t ) } . \\end{align*}"}
{"id": "2136.png", "formula": "\\begin{align*} G _ { k , l } ( x ) = \\sum _ { s = 1 } ^ \\infty \\frac { x ^ { k ( s + 1 ) } } { k ( s + 1 ) - l } . \\end{align*}"}
{"id": "7512.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { T } \\left [ \\log \\left | \\pi ^ { - \\left ( \\frac { \\frac { 1 } { 2 } - \\epsilon + i t } { 2 } \\right ) } \\right | - \\log \\left | \\pi ^ { - \\left ( \\frac { \\frac { 1 } { 2 } + \\epsilon + i t } { 2 } \\right ) } \\right | \\right ] \\ d t + \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\left [ \\arg \\pi ^ { - \\left ( \\frac { \\sigma + i T } { 2 } \\right ) } - \\arg \\pi ^ { - \\frac { \\sigma } { 2 } } \\right ] \\ d \\sigma = 0 \\end{align*}"}
{"id": "584.png", "formula": "\\begin{align*} X _ 1 \\ \\cap \\ X _ n \\ = \\ \\emptyset , Y _ 1 \\ \\cap \\ Y _ 2 = \\emptyset . \\end{align*}"}
{"id": "5488.png", "formula": "\\begin{align*} \\frac { 1 } { t } d _ K ( x + t \\beta x ) = 0 . \\end{align*}"}
{"id": "6950.png", "formula": "\\begin{align*} W _ { i } \\upharpoonright t ' = \\Phi _ i ^ { \\Gamma _ A } \\upharpoonright t ' , 0 = \\Gamma _ A ( \\langle x _ m , y _ m \\rangle ) = \\Psi _ i ^ { W _ i \\upharpoonright t ' } ( \\langle x _ m , y _ m \\rangle ) . \\end{align*}"}
{"id": "4656.png", "formula": "\\begin{align*} u ( t , x ) \\sim \\sum _ { i = 1 } ^ n Q ( x - t - x _ i ( t ) ) , \\end{align*}"}
{"id": "3542.png", "formula": "\\begin{align*} \\langle b c p _ { 0 1 } ^ 2 + c ^ 2 p _ { 0 1 } p _ { 0 2 } & + c d p _ { 0 1 } p _ { 0 4 } - a p _ { 0 1 } ^ 2 + b p _ { 0 1 } p _ { 0 4 } + c p _ { 0 2 } p _ { 0 4 } + d p _ { 0 4 } ^ 2 + d p _ { 0 1 } p _ { 1 4 } - p _ { 0 2 } p _ { 1 4 } , \\\\ \\ ; h _ 1 , \\ ; h _ 2 , \\ ; h _ 3 , & \\ ; h _ 4 , \\ ; h _ 5 , \\ ; h _ 6 \\rangle \\end{align*}"}
{"id": "670.png", "formula": "\\begin{align*} \\begin{cases} \\ f ( x ) = q - 2 ^ { - x } \\\\ [ 8 p t ] \\ g ( x ) = q + 2 ^ x \\end{cases} . ] \\end{align*}"}
{"id": "6736.png", "formula": "\\begin{align*} g _ 1 ( \\theta ^ { q ^ N } ) \\bigr ( ( \\theta ^ { q ^ N } - \\theta ^ q ) ^ n \\cdots ( \\theta ^ { q ^ N } - \\theta ^ { q ^ { N - 1 } } ) ^ n \\bigr ) \\alpha ^ { q ^ N } + g _ 2 ( \\theta ^ { q ^ N } ) L i _ { C , n } ( \\alpha ) ^ { q ^ N } - g _ 2 ( \\theta ^ { q ^ N } ) \\beta ^ { q ^ N } = 0 . \\end{align*}"}
{"id": "8938.png", "formula": "\\begin{align*} u ( x ) = \\int _ 0 ^ { T } e ^ { - s } L \\left ( \\xi ( s ) , - \\dot { \\xi } ( s ) \\right ) d s + e ^ { - T } u ( \\xi ( T ) ) . \\end{align*}"}
{"id": "7141.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( K _ H ( 1 , u ) ) ^ 2 d u = \\mathbb { E } [ ( B _ 1 ^ H ) ^ 2 ] = 1 . \\end{align*}"}
{"id": "2242.png", "formula": "\\begin{align*} \\tilde { E } _ { k , N } ( t ) = E _ { k , N } ^ { i } = E _ { k , N } ^ { j } E _ { k , N } ^ { i - j } = E _ { k , N } ^ j \\tilde { E } _ { k , N } ( t - t _ j ) , 0 \\leq t _ j \\leq t _ { i - 1 } < t \\leq t _ { i } . \\end{align*}"}
{"id": "308.png", "formula": "\\begin{align*} L ( \\rho ) = \\sum _ { i = 1 } ^ N ( \\log ( \\rho _ i ) \\rho _ i - \\rho _ i ) , \\end{align*}"}
{"id": "1738.png", "formula": "\\begin{align*} \\sup _ { f \\in M } \\| f \\| _ { Y _ q ( \\tilde \\Omega _ { [ \\hat t ( n ) ] } ) } \\stackrel { ( \\ref { n u 2 } ) , ( \\ref { e m b _ n u } ) } { \\underset { \\mathfrak { Z } _ 0 } { \\lesssim } } 2 ^ { ( ( 1 \u2010 \\lambda ) \\mu _ * \u2010 \\lambda \\alpha _ * ) k _ * \\hat t ( n ) } \\stackrel { ( \\ref { h a t _ m t } ) } { = } n ^ { ( ( 1 \u2010 \\lambda ) \\mu _ * \u2010 \\lambda \\alpha _ * ) / \\gamma _ * } = n ^ { \u2010 \\theta _ 3 } \\end{align*}"}
{"id": "3822.png", "formula": "\\begin{align*} \\chi _ { f , G } ^ I ( \\sigma , \\sigma ' ) = \\chi _ { f , G } ^ I ( s ^ { m ^ * } , 1 ) \\end{align*}"}
{"id": "3337.png", "formula": "\\begin{align*} [ x + u , y + v , z + w ] _ { L \\oplus V } = [ x , y , z ] + \\theta ( y , z ) u - \\theta ( x , z ) v + D ( x , y ) w , \\end{align*}"}
{"id": "6084.png", "formula": "\\begin{align*} g ( \\lambda - \\overline { \\lambda } ) L h = P h + Q \\overline { h } + g ( \\lambda - \\overline { \\lambda } ) \\left ( g ^ { 2 } F ^ { j } + g \\lambda G ^ { j } + \\lambda ^ { 2 } H ^ { j } \\right ) , \\end{align*}"}
{"id": "4476.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { t ^ { k } } { k ! } \\bigg ( \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) | z | ^ { 2 l } \\bigg ) = \\sum _ { l = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = l } ^ { \\infty } S _ { 2 , \\lambda } ( k , l ) \\frac { t ^ { k } } { k ! } \\bigg ) | z | ^ { 2 l } . \\end{align*}"}
{"id": "169.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\mu } ( g , \\tilde { f } _ \\mu ) = \\langle f ; g \\rangle _ { L ^ 2 ( \\mu ) } , \\tilde { f } _ \\mu = \\int _ 0 ^ { + \\infty } P ^ { \\mu } _ t ( f ) d t , \\end{align*}"}
{"id": "3167.png", "formula": "\\begin{align*} \\epsilon _ k = \\max _ { j \\in \\lbrace 1 , \\ldots , \\epsilon \\rbrace } \\frac { \\norm { E _ j ^ \\intercal ( A x _ k - b ) } _ 2 ^ 2 } { 2 \\norm { A x _ k - b } _ 2 ^ 2 \\norm { A ^ \\intercal E _ j } _ F ^ 2 } + \\frac { 1 } { 2 \\norm { A } _ F ^ 2 } , \\end{align*}"}
{"id": "4691.png", "formula": "\\begin{align*} r ( t , y ) = & \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\frac { A _ { i j } ( t , y - x _ { i } ( t ) ) } { x ^ 2 _ { i j } ( t ) } + \\sum _ { \\substack { i , j = 1 \\\\ j \\not = i } } ^ n \\frac { B _ { i j } ( t , y - x _ { i } ( t ) ) } { x _ { i j } ^ 3 ( t ) } \\varphi _ { i j } ( t , y ) , \\end{align*}"}
{"id": "6556.png", "formula": "\\begin{align*} \\| \\nabla ^ { 2 } K ( t ) \\| _ { L ^ { 1 } } & = \\| \\mathcal { F } ( \\xi ^ { 2 } \\check { K } ( \\xi , t ) ) \\| _ { L ^ { 1 } } \\\\ & = \\| \\mathcal { F } ( \\xi ^ { 2 } \\widehat { K } ( \\xi , t ) ) \\| _ { L ^ { 1 } } \\\\ & \\leq C \\| \\xi ^ { 2 } \\widehat { K } ( \\xi , t ) \\| _ { L ^ { 2 } } ^ { \\frac { 1 } { 2 } } \\| \\nabla _ { \\xi } ^ { 2 } ( \\xi ^ { 2 } \\widehat { K } ( \\xi , t ) ) \\| _ { L ^ { 2 } } ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "347.png", "formula": "\\begin{align*} \\mathcal A ( \\rho ^ * , m ^ * ) & = \\sup _ { S \\in \\mathbb H ^ 1 _ R } \\mathcal L ( \\rho ^ * , m ^ * , S ) \\ge \\inf _ { \\rho , m } \\sup _ { S \\in \\mathbb H ^ 1 _ R } \\mathcal L ( \\rho , m , S ) \\\\ & = \\mathcal A ( \\rho ^ { * , R } , m ^ { * , R } ) + R \\mathcal E ( \\rho ^ { * , R } , m ^ { * , R } ) \\\\ & \\ge \\mathcal A ( \\rho ^ { * , R } , m ^ { * , R } ) . \\end{align*}"}
{"id": "8945.png", "formula": "\\begin{align*} D u \\left ( \\xi ( t ) \\right ) = - C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } \\dot { \\xi } ( t ) . \\end{align*}"}
{"id": "2762.png", "formula": "\\begin{align*} M [ u ] = M [ Q ] , \\ ; E [ u ] = E [ Q ] , \\end{align*}"}
{"id": "7775.png", "formula": "\\begin{align*} e _ k & = \\phi _ k \\left ( \\langle 2 \\phi _ { k , x } + \\varphi _ { k , x } , \\varphi _ { k , x } \\rangle - \\langle 2 \\phi _ { k , t } + \\varphi _ { k , t } , \\varphi _ { k , t } \\rangle \\right ) \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; - \\varphi _ k \\left ( | \\tilde \\phi _ { k + 1 , t } | ^ 2 - | \\tilde \\phi _ { k + 1 , x } | ^ 2 \\right ) + \\mathbf { 1 } _ { \\omega } \\left ( h _ k ^ { p ^ { \\perp } } + f _ { k - 1 } ^ { \\phi _ k ^ { \\perp } } - f _ { k } ^ { \\tilde \\phi ^ { \\perp } _ { k + 1 } } \\right ) . \\end{align*}"}
{"id": "7605.png", "formula": "\\begin{align*} P = \\frac { | X | ^ { 2 } } { 2 } - \\frac { \\alpha } { \\alpha + 1 } \\langle X , \\nu \\rangle ^ { \\frac { \\alpha + 1 } { \\alpha } } , \\end{align*}"}
{"id": "2496.png", "formula": "\\begin{align*} F \\natural G ( x , \\omega ) = \\int _ { \\R ^ { 2 d } } F ( x ' , \\omega ' ) G ( x - x ' , \\omega - \\omega ' ) e ^ { \\pi i ( x ' \\cdot \\omega - x \\cdot \\omega ' ) } \\ , d ( x ' , \\omega ' ) . \\end{align*}"}
{"id": "1976.png", "formula": "\\begin{align*} \\Phi = \\varepsilon _ { \\mathbb C } + \\kappa \\prec \\Phi \\end{align*}"}
{"id": "5723.png", "formula": "\\begin{align*} \\gamma _ { ( M , g ) } : = \\inf _ { u \\in H ^ 1 ( M ) \\setminus \\{ 0 \\} } \\frac { \\displaystyle \\int _ M | \\nabla _ g u | ^ 2 { \\rm d } v _ g } { \\displaystyle \\int _ M u ^ 2 { \\rm d } v _ g } \\geq \\frac { ( n - 1 ) ^ 2 } { 4 } \\kappa . \\end{align*}"}
{"id": "284.png", "formula": "\\begin{align*} \\mu _ { 0 } : = \\left ( c _ { \\alpha } ^ { + } - c _ { \\alpha } ^ { - } \\right ) \\Gamma \\biggl ( \\frac { 3 - \\alpha } { 2 } \\biggl ) + \\frac { \\left ( c _ { \\alpha } ^ { + } + c _ { \\alpha } ^ { - } \\right ) \\beta \\chi _ { * } ( 0 ) } { 2 - \\alpha } \\Gamma \\biggl ( 2 - \\frac { \\alpha } { 2 } \\biggl ) , \\mu _ { 1 } : = \\frac { c _ { \\alpha } ^ { + } + c _ { \\alpha } ^ { - } } { 2 } - \\kappa d . \\end{align*}"}
{"id": "6842.png", "formula": "\\begin{align*} X ^ k _ \\sigma = \\{ y \\in X ( k ) : y > \\sigma \\} . \\end{align*}"}
{"id": "1098.png", "formula": "\\begin{align*} m ^ { G P } = \\Delta _ L ( k ) = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c } \\chi _ L ( k ) + \\chi _ L ^ { - 1 } ( k ) & i \\left ( \\chi _ L ( k ) - \\chi _ L ^ { - 1 } ( k ) \\right ) \\\\ - i \\left ( \\chi _ L ( k ) - \\chi _ L ^ { - 1 } ( k ) \\right ) & \\chi _ L ( k ) + \\chi _ L ^ { - 1 } ( k ) \\end{array} \\right ) , \\end{align*}"}
{"id": "2931.png", "formula": "\\begin{align*} \\sup _ { \\mathbf i \\in \\mathcal J : | \\mathbf i | = \\mu } | \\varphi _ 2 \\left ( \\mathbf { i } \\right ) | & = O \\left ( n ^ { 2 - \\mu } \\right ) . \\end{align*}"}
{"id": "8400.png", "formula": "\\begin{align*} \\mathbf { v } _ A ^ { \\pm } ( x , t ) = \\mathbf { v } _ 0 ^ { \\pm } ( x , t ) + \\varepsilon \\mathbf { v } _ 1 ^ { \\pm } ( x , t ) + \\varepsilon ^ 2 \\mathbf { v } _ 2 ^ { \\pm } ( x , t ) , p _ A ^ { \\pm } ( x , t ) = p _ 0 ^ { \\pm } ( x , t ) + \\varepsilon p _ 1 ^ { \\pm } ( x , t ) , c _ { \\pm } ^ { o u t } = \\pm 1 . \\end{align*}"}
{"id": "4469.png", "formula": "\\begin{align*} ( m ) _ { k , \\lambda } = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( m ) _ { l } , ( k \\ge 1 ) . \\end{align*}"}
{"id": "8950.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( e ^ { - t } C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } \\dot { \\xi } ( t ) \\right ) = e ^ { - t } D f \\left ( \\xi ( t ) \\right ) . \\end{align*}"}
{"id": "5849.png", "formula": "\\begin{align*} \\alpha ' = \\alpha \\rho \\wedge 1 , \\end{align*}"}
{"id": "5939.png", "formula": "\\begin{align*} J _ L ( g , g \\cdot \\xi _ 1 , \\dots , g \\cdot \\xi _ k ) = \\mu \\iff \\pmb { F } \\ell ( \\xi _ 1 , \\dots , \\xi _ k ) = \\big ( { \\rm A d } ^ * _ { g } \\big ) ^ k \\mu = ( { \\rm A d } ^ * _ { g } \\mu _ 1 , \\dots , { \\rm A d } ^ * _ { g } \\mu _ k ) . \\end{align*}"}
{"id": "7183.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ s g _ { R , X _ \\ast , V _ \\ast } + v \\cdot \\nabla _ x g _ { R , X _ \\ast , V _ \\ast } - \\nabla ( \\phi * _ x \\rho [ g _ { R , X _ \\ast , V _ \\ast } ] ) \\cdot \\nabla _ v \\mu & = - e _ 0 \\nabla \\Phi ( ( x - X _ \\ast + ( R - s ) V _ \\ast ) \\cdot \\nabla _ v \\mu , \\\\ g _ { R , X _ \\ast , V _ \\ast } ( 0 , \\cdot ) & = 0 . \\end{aligned} \\end{align*}"}
{"id": "1399.png", "formula": "\\begin{align*} \\ln q = { \\theta \\over 2 } x . \\end{align*}"}
{"id": "7046.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\| b _ j \\| _ b ^ 2 \\le \\frac { C } { \\delta ^ { 2 \\max m _ j } } \\Big ( 1 + \\frac { 1 } { \\delta ^ 2 } \\log \\frac { 1 } { \\delta } \\Big ) ^ 2 . \\end{align*}"}
{"id": "4311.png", "formula": "\\begin{align*} \\varepsilon _ \\ell ( \\tau ) = - \\frac { 2 } { \\alpha } m _ 0 b ^ { \\frac { \\alpha } { 2 } } ( \\tau ) . \\end{align*}"}
{"id": "334.png", "formula": "\\begin{align*} \\sum _ { j \\in N ( i ) } m _ { i j } ( t ) = \\sum _ { j \\in N ( i ) } ( \\Sigma _ i - \\Sigma _ j ) \\theta ( \\rho _ i , \\rho _ j ) \\dot W ^ { \\delta } ( t ) . \\end{align*}"}
{"id": "9160.png", "formula": "\\begin{align*} \\sum _ { p \\le x } \\frac { 1 } { p } = & \\log \\log x + b _ 1 + O \\Big ( \\frac { 1 } { \\log x } \\Big ) , \\mbox { a n d } \\\\ \\sum _ { p \\le x } \\frac { \\lambda ^ 2 _ f ( p ) } { p } = & \\log \\log x + b _ 2 + O \\Big ( \\frac { 1 } { \\log x } \\Big ) . \\end{align*}"}
{"id": "7984.png", "formula": "\\begin{align*} \\vec { B _ 0 } ( \\mathbf { 0 } ) = ( \\tilde { g } _ u ^ { i j } - g ^ { i j } ) \\partial _ j \\psi \\cdot \\partial _ i | _ { \\mathbf { 0 } } ; & \\ & B _ 1 ( \\mathbf { 0 } ) = \\frac 1 2 \\tilde { g } _ u ^ { i j } \\partial _ i ( \\log \\det [ \\tilde { g } _ u ] ) \\partial _ j \\psi ( \\mathbf { 0 } ) \\ , , \\end{align*}"}
{"id": "145.png", "formula": "\\begin{align*} c _ { \\alpha } = \\left ( \\dfrac { - \\alpha ( \\alpha - 1 ) } { 4 \\Gamma ( 2 - \\alpha ) \\cos \\left ( \\frac { \\alpha \\pi } { 2 } \\right ) } \\right ) . \\end{align*}"}
{"id": "7721.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } E ( t ) = - \\int _ { \\mathbb { S } ^ 1 } a ( x ) | \\phi _ t | ^ 2 ( t , x ) d x \\leq 0 , \\end{align*}"}
{"id": "6365.png", "formula": "\\begin{align*} v = \\frac { 1 } { 4 k + 1 } \\left ( 4 \\rho _ 1 + \\rho _ 2 + \\rho _ 3 \\right ) \\end{align*}"}
{"id": "4037.png", "formula": "\\begin{align*} p _ k = \\sqrt { | k \\pi | } + O ( | k | ^ { - \\frac { 1 } { 2 } } ) , q _ k = \\pm \\sqrt { | k \\pi | } + O ( | k | ^ { - \\frac { 1 } { 2 } } ) , \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "6595.png", "formula": "\\begin{align*} \\mathcal { L } ^ 0 ( h , k ) = - \\sum _ { \\substack { 1 \\leq c \\le C \\\\ ( c , h k ) = 1 } } \\mu ( c ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , c ) = 1 } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { 1 \\le d < \\infty \\\\ ( d , m h n k ) = 1 } } W \\left ( \\frac { c d } { Q } \\right ) . \\end{align*}"}
{"id": "383.png", "formula": "\\begin{align*} a _ j ( X _ j + \\overline { X _ j } ) = 0 . \\end{align*}"}
{"id": "110.png", "formula": "\\begin{align*} U ' = \\{ u + 2 u _ 0 u \\ ; | \\ ; u \\in U \\} , \\ ; \\ ; V ' = \\{ v - 2 ( u _ 0 + u _ 0 ^ 2 ) v \\ ; | \\ ; v \\in V \\} . \\end{align*}"}
{"id": "3146.png", "formula": "\\begin{align*} x _ 0 ^ \\perp = \\{ x _ 0 + r _ 1 n ( 0 ) + r _ 2 b ( 0 ) : \\sqrt { r _ 1 ^ 2 + r _ 2 ^ 2 } < \\delta \\} . \\end{align*}"}
{"id": "2034.png", "formula": "\\begin{align*} p _ t ^ A ( x , y ) = h ( x ) h ( y ) p _ t ^ { Y , h } ( x , y ) . \\end{align*}"}
{"id": "364.png", "formula": "\\begin{align*} m _ { 2 1 } ( t ) & = 0 , \\ ; t \\in [ 0 , 1 - \\delta ] , \\\\ m _ { 2 1 } ( t ) ( 1 + \\dot W ^ { \\delta } ( t ) ) & = ( \\rho _ 1 ^ b - \\rho _ 1 ^ a ) \\frac { 1 } { \\delta } , \\ ; t \\in [ 1 - \\delta , 1 ] , \\ ; \\mathcal L ^ 1 \\ ; , \\end{align*}"}
{"id": "7866.png", "formula": "\\begin{align*} G \\cup H \\cup I & = ( \\mathbb { N } \\setminus \\{ \\bar { b } _ n \\} ) \\cup \\{ \\bar { b } _ n \\} = \\mathbb { N } , \\end{align*}"}
{"id": "7203.png", "formula": "\\begin{align*} \\zeta _ { \\xi / t } ( | \\xi | ) & = | \\xi | \\left ( \\hat { \\psi } _ { \\xi / t } ( | \\xi | ) - \\hat { \\psi } _ { \\xi / t } ( 0 ) \\right ) \\eta ( | \\xi | ^ 2 ) + | \\xi | \\hat { \\psi } _ { \\xi / t } ( 0 ) \\eta ( | \\xi | ^ 2 ) + | \\xi | \\hat { \\psi } _ { \\xi / t } ( | \\xi | ) ( 1 - \\eta ( | \\xi | ^ 2 ) ) \\\\ & = R ^ { ( a ) } _ { \\xi / t } ( | \\xi | ) + R ^ { ( b ) } _ { \\xi / t } ( | \\xi | ) + R ^ { ( c ) } _ { \\xi / t } ( | \\xi | ) . \\end{align*}"}
{"id": "533.png", "formula": "\\begin{gather*} \\begin{matrix} \\Xi _ { k + 1 } = \\mathcal { A } \\Xi _ k + \\mathcal { B } \\mathbf { v } _ k , & \\mathbf { y } _ k = \\mathcal { C } \\Xi _ k , \\end{matrix} \\\\ \\mathbf { v } _ k \\coloneqq \\begin{bmatrix} v _ { 1 , k } \\\\ v _ { 2 , k } \\end{bmatrix} = \\begin{bmatrix} 2 \\xi _ { 2 , k } - \\xi _ { 1 , k } + T _ s ^ 2 z _ { 1 , k } \\\\ 2 \\xi _ { 4 , k } - \\xi _ { 3 , k } + T _ s ^ 2 z _ { 2 , k } \\end{bmatrix} . \\end{gather*}"}
{"id": "231.png", "formula": "\\begin{align*} U _ \\Sigma ( \\gamma _ \\Sigma ) = 1 . \\end{align*}"}
{"id": "14.png", "formula": "\\begin{align*} u _ n = \\sum _ { \\beta = 1 } ^ { N } f _ n ^ { \\beta } + q _ n ^ N \\end{align*}"}
{"id": "9178.png", "formula": "\\begin{align*} \\sum _ { \\substack { 2 < p \\leq X } } \\log p \\sim X . \\end{align*}"}
{"id": "6527.png", "formula": "\\begin{align*} \\frac { 1 } { t _ n ^ { ( 2 m ) } } \\sum ^ { n - 1 } _ { j = 1 } d _ j ^ { ( 2 \\ell ) } = O \\left ( \\dfrac { 1 } { t _ n ^ { ( 2 m ) } } \\right ) = O ( ( \\log n ) ^ { - m } ) , \\end{align*}"}
{"id": "1057.png", "formula": "\\begin{align*} & q _ t ( x , t ) - 6 q ^ 2 ( x , t ) q _ { x } ( x , t ) + q _ { x x x } ( x , t ) = 0 , ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { + } , \\\\ & q ( x , 0 ) = q _ { 0 } ( x ) = \\left \\{ \\begin{aligned} & C _ { L } , x < 0 , \\\\ & C _ { R } , x > 0 , \\end{aligned} \\right . , \\end{align*}"}
{"id": "5898.png", "formula": "\\begin{align*} X ^ T Q _ S ^ { ' } X & = \\sum _ { i = 1 } ^ k s _ i ^ 2 + 2 \\sum _ { \\substack { 1 \\leq i < j \\leq k } } s _ i s _ j - \\sum _ { i = 1 } ^ k s _ i + \\sum _ { i = 1 } ^ k t _ i ^ 2 + 2 \\sum _ { \\substack { 1 \\leq i < j \\leq k } } \\ , t _ i t _ j - \\sum _ { i = 1 } ^ k t _ i + 2 \\sum _ { i = 1 } ^ k s _ i t _ i \\\\ & = \\Big ( \\sum _ { i = 1 } ^ k \\ , s _ i \\Big ) ^ 2 + \\Big ( \\sum _ { i = 1 } ^ k \\ , t _ i \\Big ) ^ 2 - \\sum _ { i = 1 } ^ k ( s _ i + t _ i ) + 2 \\sum _ { i = 1 } ^ k s _ i t _ i . \\end{align*}"}
{"id": "263.png", "formula": "\\begin{align*} i _ { v } \\omega = \\sqrt { - 1 } \\overline { \\partial } \\theta _ { v } ^ { ( \\omega ) } \\quad \\int _ { M } \\theta _ { v } ^ { ( \\omega ) } \\omega ^ { n } = 0 . \\end{align*}"}
{"id": "7124.png", "formula": "\\begin{align*} f _ 1 ( s , y ) : = - \\alpha _ { - 1 } \\tilde { \\theta } ( 2 \\tilde { \\theta } + 1 ) y ^ { 2 \\tilde { \\theta } } + \\alpha _ 0 ( \\tilde { \\theta } + 1 ) y ^ { \\tilde { \\theta } } - \\frac { \\alpha _ 1 } { \\tilde { \\theta } } \\end{align*}"}
{"id": "4220.png", "formula": "\\begin{align*} ( \\partial _ t - \\sigma \\Delta ) \\ \\eta _ t = - \\div \\left ( V ( t , \\cdot ) \\ , \\eta _ t \\right ) , \\end{align*}"}
{"id": "2672.png", "formula": "\\begin{align*} \\beta ( z , \\tau ) F ( w ) = e ^ { 2 \\pi i \\tau } e ^ { \\pi w \\cdot z } e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } F ( w - \\overline { z } ) , \\end{align*}"}
{"id": "3755.png", "formula": "\\begin{align*} ( d _ { \\alpha } T ) ( \\phi ) : = ( - 1 ) ^ { k - 1 } T ( d _ { \\alpha } \\phi ) , \\ \\forall \\phi \\in \\mathcal { E } ^ { k - 1 } ( M , \\R ) , \\end{align*}"}
{"id": "134.png", "formula": "\\begin{align*} x ^ k = a _ k ^ { ( k ) } v ^ k + b _ { k - 1 } ^ { ( k ) } L _ v ^ { k - 1 } ( u ) + c _ { k - 2 } ^ { ( k ) } L _ v ^ { k - 2 } ( u ^ 2 ) , \\end{align*}"}
{"id": "467.png", "formula": "\\begin{align*} \\det ( \\mathbf { L } \\cdot \\mathbf { R } ) = \\sum _ { \\mathcal { I } \\in \\wp _ { k } [ n ] } \\Delta _ { \\mathbf { L } } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } } ( \\mathcal { I } ) \\end{align*}"}
{"id": "5330.png", "formula": "\\begin{align*} \\eta _ t + \\mathrm { d i v } _ x ( u \\eta ) = \\nu \\Delta _ x \\eta . \\end{align*}"}
{"id": "8678.png", "formula": "\\begin{align*} { \\bf P } _ { e r r o r } ^ { ( n ) } \\triangleq & \\frac { 1 } { M ^ { ( n ) } } \\sum _ { m \\in { \\cal M } ^ { ( n ) } } { \\bf P } ^ g \\Big \\{ d _ { n } ( Y ^ { n } ) \\neq m \\big | M = m \\Big \\} \\leq \\epsilon _ n . \\end{align*}"}
{"id": "9196.png", "formula": "\\begin{align*} s = _ \\rho t : = \\forall y _ 1 ^ { \\rho _ 1 } , \\dots , y _ k ^ { \\rho _ k } ( s y _ 1 \\dots y _ k = _ 0 t y _ 1 \\dots y _ k ) \\end{align*}"}
{"id": "2641.png", "formula": "\\begin{align*} X M _ l T _ k g ( x ) = ( k + x - k ) e ^ { 2 \\pi i l \\cdot x } g ( x - k ) = k M _ l T _ k g ( x ) + M _ l T _ k X g ( x ) , \\end{align*}"}
{"id": "2432.png", "formula": "\\begin{align*} \\L ^ \\perp = M ^ { - T } \\Z ^ d . \\end{align*}"}
{"id": "2996.png", "formula": "\\begin{align*} \\langle a ( x ) x - a ( y ) y , x - y \\rangle = & \\langle ( a ( x ) - a ( y ) ) x , x - y \\rangle + a ( y ) \\langle x , x - y \\rangle \\\\ = & ( a ( x ) - a ( y ) ) \\langle x , x - y \\rangle + a ( y ) | x - y | ^ { 2 } \\end{align*}"}
{"id": "6295.png", "formula": "\\begin{align*} F _ { \\widehat { K } ^ \\mathbb { R } _ { \\mathrm { u - c p t } } | K } ( \\hat { k } ) = 1 - e ^ { - \\frac { 1 + N \\bar { \\gamma } \\hat { k } } { 1 + N \\bar { \\gamma } K } } , \\ \\ f _ { \\widehat { K } ^ \\mathbb { R } _ { \\mathrm { u - c p t } } | K } ( \\hat { k } ) = \\frac { N \\bar { \\gamma } } { 1 + N \\bar { \\gamma } K } e ^ { - \\frac { 1 + N \\bar { \\gamma } \\hat { k } } { 1 + N \\bar { \\gamma } K } } , \\ \\ \\hat { k } \\ge - 1 / ( N \\bar { \\gamma } ) . \\end{align*}"}
{"id": "3788.png", "formula": "\\begin{align*} \\int _ { X _ E } r _ l ( \\widetilde { V } ) \\begin{pmatrix} g & 0 \\\\ 0 & 1 \\end{pmatrix} r _ l ( \\widetilde { \\mathcal { S } } ) ( g ) \\ , d g = \\int _ { X _ F } r _ l ( \\widetilde { V } ) \\begin{pmatrix} g & 0 \\\\ 0 & 1 \\end{pmatrix} r _ l ( \\widetilde { \\mathcal { S } } ) ( g ) \\ , d g , \\end{align*}"}
{"id": "4646.png", "formula": "\\begin{align*} S : = { \\sqrt { \\Delta ( { \\mathbb K } ) } } \\sum _ { d \\vert m } \\mu ( d ) \\lambda ^ { ( n / d ) ^ s } . \\end{align*}"}
{"id": "1162.png", "formula": "\\begin{align*} m ^ { ( 1 ) } _ { + } ( x , t , k ) = m ^ { ( 1 ) } _ { - } ( x , t , k ) J ^ { ( 1 ) } , k \\in \\mathbb { R } , \\ \\Gamma _ 1 , \\ \\Gamma _ 1 ^ * , \\end{align*}"}
{"id": "1917.png", "formula": "\\begin{align*} & \\gamma ^ - f ( t , x , v ) = g ( t , x , v ) , ( t , x , v ) \\in ( 0 , T ) \\times \\Sigma ^ - , \\\\ & \\rho ( t , x ) = \\rho _ B ( x ) , ( t , x ) \\in ( 0 , T ) \\times \\Gamma _ { \\rm { i n } } , \\\\ & u ( t , x ) = u _ B ( x ) , ( t , x ) \\in ( 0 , T ) \\times \\partial \\Omega , \\end{align*}"}
{"id": "119.png", "formula": "\\begin{align*} L ( A ) = \\sum \\limits _ { i = 1 } ^ r ( L _ { v _ 1 } , \\ldots , L _ { v _ n } ) . u _ i , \\end{align*}"}
{"id": "1901.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & - 1 & & \\\\ & 1 & - 1 & \\\\ & & 1 & - 1 \\\\ & & 1 & 1 \\end{pmatrix} \\quad W = \\begin{pmatrix} 1 & 1 & 1 / 2 & 1 / 2 \\\\ & 1 & 1 / 2 & 1 / 2 \\\\ & & 1 / 2 & 1 / 2 \\\\ & & - 1 / 2 & 1 / 2 \\end{pmatrix} \\end{align*}"}
{"id": "1553.png", "formula": "\\begin{align*} G : = G ( T ) : = \\{ g \\in \\mathrm { S L } _ n ( \\mathbb { B } ) : g ^ { \\ast } T g = T \\} . \\end{align*}"}
{"id": "8216.png", "formula": "\\begin{align*} P _ { h + 1 } = \\frac { x _ { h + 1 } z } { P _ { k + 1 - h } + \\frac { x _ h z } { P _ h } } . \\end{align*}"}
{"id": "5877.png", "formula": "\\begin{align*} \\overline Y ^ 1 _ t & = e ^ { \\lambda t } Y ^ 1 _ 0 , \\\\ \\overline Y ^ 2 _ t & = e ^ { - \\mu t } Y ^ 2 _ 0 = e ^ { - \\mu t } L , \\end{align*}"}
{"id": "7945.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { t } i a _ i ^ { \\ell } = \\sum _ { a \\in S } a ^ { \\ell } \\sum _ { \\substack { b \\in S \\\\ b \\leqslant a } } 1 = \\sum _ { a \\in S } a ^ \\ell \\left ( \\frac { a } { n } t + E _ a \\right ) = \\frac { t } { n } \\sum _ { a \\in S } a ^ { \\ell + 1 } + \\sum _ { a \\in S } a ^ \\ell E _ a \\end{align*}"}
{"id": "5094.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\tanh ( x ) = 1 \\end{align*}"}
{"id": "1957.png", "formula": "\\begin{align*} x _ { i _ 1 } \\cdots x _ { i _ n } \\triangleleft x _ { j _ 1 } \\cdots x _ { j _ m } = \\sum \\limits _ { k = 0 } ^ n x _ { i _ 1 } \\cdots x _ { i _ k } x _ { j _ 1 } \\cdots x _ { j _ m } x _ { i _ { k + 1 } } \\cdots x _ { i _ n } . \\end{align*}"}
{"id": "397.png", "formula": "\\begin{align*} u _ { t } + u _ { x } + v _ { x } = f _ { 1 } ( w , z , z _ { x } ) , \\end{align*}"}
{"id": "5636.png", "formula": "\\begin{align*} 0 = & f _ c ( y ) \\equiv \\frac { 1 } { 2 } y ^ - \\star y ^ + + \\frac { a } { 2 } y ^ 0 \\star y ^ 0 - c , \\\\ [ 6 p t ] 0 = & \\mathrm { d } f _ c = \\frac { 1 } { 2 } ( y ^ - \\star \\eta ^ + + \\eta ^ - \\star y ^ + ) + a y ^ 0 \\star \\eta ^ 0 , \\\\ [ 6 p t ] 0 = & y ^ - \\star E ^ + - y ^ + \\star E ^ - - \\sqrt { a } \\ , y ^ 0 \\star H + \\mathrm { i } \\nu y ^ + \\star H - 2 \\mathrm { i } \\nu ( 1 + \\mathrm { i } \\nu ) y ^ + \\star E ^ + . \\end{align*}"}
{"id": "3355.png", "formula": "\\begin{align*} & [ T u , T v , z ] = - [ T v , z , T u ] - [ z , T u , T v ] = [ z , T v , T u ] - [ z , T u , T v ] . \\end{align*}"}
{"id": "4139.png", "formula": "\\begin{align*} \\| f \\| _ { b , p } : = \\| J ^ { b } f \\| _ { p } \\simeq \\| f \\| _ { p } + \\| D ^ { b } f \\| _ { p } \\simeq \\| f \\| _ { p } + \\| \\mathcal { D } ^ { b } f \\| _ { p } . \\end{align*}"}
{"id": "82.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 3 \\cdot 2 ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) . \\end{align*}"}
{"id": "3383.png", "formula": "\\begin{align*} [ u , v ] _ T = \\rho ( T u ) v - \\rho ( T v ) u \\ ; \\forall u , v \\in V . \\end{align*}"}
{"id": "7167.png", "formula": "\\begin{align*} 0 \\leq \\frac { d } { d r } \\Big | _ { r = 1 } r ^ { - 1 } v ^ \\gamma ( r x ) = - v ^ \\gamma ( x ) + x \\cdot D v ^ \\gamma ( x ) . \\end{align*}"}
{"id": "3483.png", "formula": "\\begin{align*} C _ 2 & \\ll \\sum _ { n = N + 1 } ^ \\infty \\frac { 1 } { n ^ { \\sigma _ 2 } } \\int _ n ^ \\infty \\frac { 1 } { u ^ { \\sigma _ 1 + 1 } ( u + n ) ^ { \\sigma _ 3 } } d u \\\\ & \\ll \\sum _ { n = N + 1 } ^ \\infty \\frac { 1 } { n ^ { \\sigma _ 2 } } \\int _ n ^ \\infty \\frac { 1 } { u ^ { \\sigma _ 1 + \\sigma _ 3 + 1 } } d u . \\end{align*}"}
{"id": "5402.png", "formula": "\\begin{align*} \\hat { \\theta } _ { n } = \\arg \\min _ { \\theta \\in \\Theta } \\Phi _ { n } ( \\theta ) . \\end{align*}"}
{"id": "5344.png", "formula": "\\begin{align*} \\mathrm { d i v } u = 0 \\ ; \\ ; \\ ; \\ ; ( x , t ) \\in \\mathbb { R } ^ 3 \\times [ 0 , T ] . \\end{align*}"}
{"id": "6820.png", "formula": "\\begin{align*} \\nabla _ { k l } ^ F ( V ) \\equiv \\nabla _ k ^ { F } \\nabla _ l ^ { F } ( V ) : = \\nabla ^ F _ k ( \\nabla ^ F _ l ( V ) ) - ( \\Gamma _ F ) ^ m _ { k l } \\ , \\nabla ^ F _ m ( V ) . \\end{align*}"}
{"id": "5847.png", "formula": "\\begin{align*} U ^ { 1 } _ t & = \\int _ 0 ^ t e ^ { - \\lambda s } d W ^ 1 _ s , \\\\ U ^ { 2 } _ t & = \\int _ 0 ^ t e ^ { \\mu s } d W ^ 2 _ s , \\\\ N ^ 2 _ t & = e ^ { - \\mu t } U _ t ^ 2 = \\int _ 0 ^ t e ^ { - \\mu ( t - s ) } d W ^ 2 _ s . \\end{align*}"}
{"id": "2630.png", "formula": "\\begin{align*} Z f ( x , \\omega ) & = \\sum _ { k \\in \\Z ^ d } g ( k ) = \\sum _ { k \\in \\Z ^ d } \\widehat { g } ( k ) = \\sum _ { k \\in \\Z ^ d } T _ { - \\omega } M _ x \\widehat { f } ( k ) = \\sum _ { k \\in \\Z ^ d } \\widehat { f } ( k + \\omega ) e ^ { 2 \\pi i x \\cdot ( k + \\omega ) } \\\\ & = e ^ { 2 \\pi i x \\cdot \\omega } Z \\widehat { f } ( \\omega , - x ) . \\end{align*}"}
{"id": "3530.png", "formula": "\\begin{align*} W _ 1 & = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) - ( U _ 1 + U _ 3 ) - ( U _ 2 + U _ 3 ) + U _ 3 \\end{align*}"}
{"id": "8857.png", "formula": "\\begin{align*} \\check H _ { c t } ^ 1 ( T ; A ) = 0 . \\end{align*}"}
{"id": "147.png", "formula": "\\begin{align*} P ^ { \\gamma } _ t ( f ) ( x ) = \\int _ { \\mathbb { R } ^ d } f ( x e ^ { - t } + \\sqrt { 1 - e ^ { - 2 t } } y ) \\gamma ( d y ) , \\end{align*}"}
{"id": "3731.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\Lambda ^ { \\frac { \\alpha } 2 } B \\| ^ 2 _ { L ^ 2 } + \\mu \\| \\Lambda ^ { \\alpha } B \\| ^ 2 _ { L ^ 2 } = - \\int _ { \\mathbb S ^ 1 } B J _ x \\Lambda ^ { \\alpha } B \\ , d x + \\int _ { \\mathbb S ^ 1 } B _ x J \\Lambda ^ { \\alpha } B \\ , d x . \\end{align*}"}
{"id": "6199.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { l } \\bar { \\sigma } ^ 2 _ { t } = \\sum _ { t = 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } > \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } - 2 \\sqrt { k } \\theta \\| S \\| ^ 2 _ F . \\end{align*}"}
{"id": "1358.png", "formula": "\\begin{align*} \\norm { u } _ { L ^ \\frac { d } { d - \\zeta } ( W _ R ) } = \\sup _ { g \\in L ^ r ( W _ R ) , \\ ; g \\geq 0 , \\ ; \\norm { g } _ { L ^ r ( W _ R ) } = 1 } \\norm { g u } _ { L ^ 1 ( W _ R ) } \\leq C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { \\nabla u } _ { L ^ 1 ( W _ R ) } , \\end{align*}"}
{"id": "1141.png", "formula": "\\begin{align*} E _ 1 = t ^ { - 1 } \\left ( \\textnormal { R e s } _ { k = \\eta } T _ { \\eta , 1 } + \\textnormal { R e s } _ { k = - \\eta } T _ { - \\eta , 1 } \\right ) + \\mathcal { O } ( t ^ { - 2 } ) , \\end{align*}"}
{"id": "326.png", "formula": "\\begin{align*} \\inf \\{ S ( \\rho _ t , \\Phi _ t ) : ( - \\Delta _ { \\rho _ t } ) ^ \\dagger \\Phi _ t \\in \\mathcal T _ { \\rho _ t } \\mathcal P _ o ( G ) , \\rho ( 0 ) = \\rho ^ a , \\rho ( 1 ) = \\rho ^ b \\} . \\end{align*}"}
{"id": "2774.png", "formula": "\\begin{align*} u ( x , t ) = e ^ { i t } \\left ( Q ( x ) + h ( t , x ) \\right ) . \\end{align*}"}
{"id": "6.png", "formula": "\\begin{align*} \\kappa _ 1 ( y ) & = \\psi \\left ( \\frac { 1 + \\lvert y \\rvert } { 2 } \\right ) + \\ln { \\frac { 2 } { \\lvert y \\rvert } } . \\end{align*}"}
{"id": "6837.png", "formula": "\\begin{align*} U _ i f ( x ) & = \\underset { y \\lessdot x } { \\mathbb { E } } [ f ( y ) ] , \\\\ D _ i f ( y ) & = \\underset { x \\gtrdot y } { \\mathbb { E } } [ f ( x ) ] . \\end{align*}"}
{"id": "1219.png", "formula": "\\begin{align*} \\Omega _ { k } = \\bigcup _ { I \\in \\mathcal { K } _ k } \\overset { \\circ } { I } . \\end{align*}"}
{"id": "2534.png", "formula": "\\begin{align*} \\langle \\varphi , \\rho ( x , \\omega ) \\varphi \\rangle = A \\varphi = \\Phi ( x , \\omega ) . \\end{align*}"}
{"id": "7238.png", "formula": "\\begin{align*} \\partial _ s h + ( v - V _ \\ast ) \\cdot \\nabla _ x h - \\nabla ( \\phi * _ x \\rho [ h ] ) \\cdot \\nabla _ v \\mu & = - e _ 0 \\nabla \\Phi ( x ) \\cdot \\nabla _ v \\mu , h ( 0 , \\cdot ) = 0 . \\end{align*}"}
{"id": "3745.png", "formula": "\\begin{align*} V = V _ 0 + V _ 1 + \\ldots + V _ s + V _ 0 ' \\end{align*}"}
{"id": "1578.png", "formula": "\\begin{align*} \\frac { c _ k ( \\mu ) L ( \\mu , \\mathbf { f } , \\chi ) } { \\pi ^ { \\beta } } \\frac { \\mathbf { f } ( g _ { \\mathbf { h } } \\cdot g _ { \\infty } ) } { \\mathfrak { P } _ k ( w ) } = \\langle \\mathbf { h } _ w , \\mathbf { f } \\rangle . \\end{align*}"}
{"id": "1974.png", "formula": "\\begin{align*} \\big ( g \\curvearrowleft f \\big ) ( x ) : = ( g - 1 ) f _ { x g } ( x ) = ( g ( x ) - 1 ) f ( x g ( x ) ) = \\sum _ { \\substack { w \\in \\N ^ * \\\\ u \\in \\N ^ * } } f _ w g _ u x _ u ( x g ( x ) ) _ w \\in G ^ 0 . \\end{align*}"}
{"id": "7558.png", "formula": "\\begin{align*} \\mu : \\prod _ { i = 1 } ^ { m } G _ { b _ i } \\to G . \\end{align*}"}
{"id": "6495.png", "formula": "\\begin{align*} F ^ { ( 2 m - 1 ) } _ n : = \\bar { g } _ n ^ { ( 2 m - 1 ) } \\sum _ { j = j _ 0 + 1 } ^ { n - 1 } \\frac { f _ j ^ { ( 2 m - 1 ) } } { \\bar { g } _ { j + 1 } ^ { ( 2 m - 1 ) } } . \\end{align*}"}
{"id": "816.png", "formula": "\\begin{align*} \\lim _ { ( x , y ) \\to ( x _ 0 , y _ 0 ) } \\frac { d _ \\rho ( ( x , y ) , ( x _ 0 , y _ 0 ) ) } { d ( ( x , y ) , ( x _ 0 , y _ 0 ) ) } = \\rho ( x _ 0 , y _ 0 ) , \\end{align*}"}
{"id": "6004.png", "formula": "\\begin{align*} f ( z ) & : = \\mathrm { e } _ { \\pi _ { \\lambda , \\beta } } ( z ; 0 ) = \\frac { 1 } { l _ { \\pi _ { \\lambda , \\beta } } ( z ) } = \\frac { 1 } { E _ { \\beta } ( \\lambda ( \\mathrm { e } ^ { z } - 1 ) ) } , \\\\ g ( z ) & : = \\mathrm { \\mathrm { e } } ^ { z x } . \\end{align*}"}
{"id": "332.png", "formula": "\\begin{align*} \\rho _ 1 ( t ) - \\rho _ 1 ( 0 ) = \\int _ 0 ^ t m _ { 2 1 } ( s ) d s + \\int _ 0 ^ t \\frac 1 2 ( \\Sigma _ 1 - \\Sigma _ 2 ) d W ^ { \\delta } ( s ) . \\end{align*}"}
{"id": "7546.png", "formula": "\\begin{align*} \\Im \\left \\{ \\log \\zeta ( \\sigma + i T ) \\right \\} = \\mathcal { O } ( \\log T ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( - 1 \\leq \\sigma \\leq 2 ) \\end{align*}"}
{"id": "8327.png", "formula": "\\begin{align*} \\int _ 0 ^ s \\delta _ + \\dd ( \\delta - h ) = 0 \\forall s \\in ( 0 , T ] . \\end{align*}"}
{"id": "7016.png", "formula": "\\begin{align*} \\left | \\frac { \\beta _ k } { \\alpha _ i - \\beta _ k } \\right | \\geqslant \\frac { | \\beta _ k | } { | \\beta _ k | + M } \\geqslant \\frac { | \\beta _ k | } { \\frac { 3 } { 2 } | \\beta _ k | } = \\frac { 2 } { 3 } , \\end{align*}"}
{"id": "5459.png", "formula": "\\begin{align*} | \\Omega | = \\int _ \\Omega u ^ { \\frac { p } { p + 1 } } u ^ { - \\frac { p } { p + 1 } } \\le \\Big ( \\int _ \\Omega u \\Big ) ^ { \\frac { p } { p + 1 } } \\Big ( \\int _ \\Omega u ^ { - p } \\Big ) ^ { \\frac { 1 } { p + 1 } } \\forall \\ , t > s . \\end{align*}"}
{"id": "3021.png", "formula": "\\begin{align*} n ^ g _ { m , n } = \\sum _ { \\pi ( \\beta ) = m \\ell + n \\gamma } n ^ g _ \\beta , \\end{align*}"}
{"id": "1044.png", "formula": "\\begin{align*} & y _ { i j } = g _ { i 1 } g _ { j 2 } - g _ { i 2 } g _ { j 1 } , & & \\eta _ { k n } = g _ { k 1 } \\gamma _ { n 2 } - g _ { k 2 } \\gamma _ { n 1 } \\\\ & x _ { 5 5 } = \\gamma _ { 5 1 } \\gamma _ { 5 2 } , & & x _ { 6 6 } = \\gamma _ { 6 1 } \\gamma _ { 6 2 } & & x _ { 5 6 } = \\gamma _ { 5 1 } \\gamma _ { 6 2 } + \\gamma _ { 6 1 } \\gamma _ { 5 2 } \\end{align*}"}
{"id": "4617.png", "formula": "\\begin{gather*} F ( \\psi _ { X , Y } ) = \\psi _ { F ( X ) , F ( Y ) } \\end{gather*}"}
{"id": "6605.png", "formula": "\\begin{align*} \\mathcal { U } ^ 0 ( h , k ) = \\mathcal { U } ^ 1 ( h , k ) + \\mathcal { U } ^ 2 ( h , k ) , \\end{align*}"}
{"id": "2586.png", "formula": "\\begin{align*} f = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) , \\end{align*}"}
{"id": "6651.png", "formula": "\\begin{align*} \\tau _ E ( p ^ m ) = \\tau _ { E \\smallsetminus \\{ \\gamma \\} } ( p ^ m ) + p ^ { - \\gamma } \\tau _ E ( p ^ { m - 1 } ) . \\end{align*}"}
{"id": "1785.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ^ { P } _ { g } ( f _ 0 , f _ 1 , & \\dots , f _ m ) \\\\ : = & \\int _ { h \\in M / Z _ M ( x ) } \\int _ { K N } \\int _ { G ^ { \\times m } } C \\big ( H ( g _ 1 . . . g _ m k ) , H ( g _ 2 . . . g _ m k ) , \\cdots , H ( g _ m k ) \\big ) \\\\ & f _ 0 \\big ( k h g h ^ { - 1 } n k ^ { - 1 } ( g _ 1 \\dots g _ m ) ^ { - 1 } \\big ) f _ 1 ( g _ 1 ) \\dots f _ m ( g _ m ) d g _ 1 \\cdots d g _ m d k d n d h , \\end{aligned} \\end{align*}"}
{"id": "4890.png", "formula": "\\begin{align*} f ( z ) - 1 \\sim c z ^ { - s } , L ( z ) = \\frac { f ' ( z ) } { f ( z ) } \\sim f ' ( z ) \\sim - c s z ^ { - 1 - s } , F ( z ) \\sim \\frac { z ^ { 1 + s } } { c s } , \\end{align*}"}
{"id": "2083.png", "formula": "\\begin{align*} \\min \\{ v _ i ( \\mathbf { c } ) , k \\} = \\min \\{ v _ 1 ( \\mathbf { c } ) + ( i - 1 ) b , k \\} . \\end{align*}"}
{"id": "8342.png", "formula": "\\begin{align*} p = \\Lambda ( e ^ r - 1 ) \\end{align*}"}
{"id": "3233.png", "formula": "\\begin{align*} n ( s ) = \\int _ \\mathbb { R } \\frac { u ^ 2 } { s ^ 2 + u ^ 2 } d \\mu ( u ) \\qquad d ( s ) = \\int _ \\mathbb { R } \\frac { 1 } { s ^ 2 + u ^ 2 } d \\mu ( u ) . \\end{align*}"}
{"id": "5894.png", "formula": "\\begin{align*} N ^ j _ t = \\int _ 0 ^ t R ^ j ( s ) u ^ j _ l ( s ) d W ^ l _ s , j = 1 , 2 , \\dots , d , \\end{align*}"}
{"id": "9020.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( f ( x ) + \\sum _ { i = 1 } ^ { s } z _ i \\rho ^ { i n } _ i \\right ) d x + \\int _ { \\partial \\Omega } \\epsilon ( x ) \\phi ^ b d s = 0 , \\end{align*}"}
{"id": "5478.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } ( t ) & = & A \\xi ( t ) + \\alpha ( \\xi ( t ) ) \\\\ \\xi ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "1667.png", "formula": "\\begin{align*} \\mathcal W _ 0 ( x , v ) = 1 + 2 U ( x ) + \\theta _ 0 | x | ^ 2 + | v | ^ 2 + \\theta ^ * \\ < x , v \\ > , x , v \\in \\R ^ d , \\end{align*}"}
{"id": "4367.png", "formula": "\\begin{align*} | \\partial _ y ^ k \\tilde \\phi _ { i , b , \\beta } ( y ) | \\le C \\left ( y ^ { - \\gamma + 2 - k } I _ { y \\in [ 0 , y _ 0 ] } + y ^ { - \\gamma + 2 i + 2 - k } I _ { y \\in [ y _ 0 , + \\infty ) } \\right ) b ^ { 1 - \\frac { \\epsilon } { 2 } } , y \\in \\R k = 0 , 1 . \\end{align*}"}
{"id": "4002.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\overline { \\xi ( x ) } \\xi ^ { \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\eta ^ { \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\xi ( x ) } \\eta ^ { \\prime } ( x ) d x + \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\xi ^ \\prime ( x ) d x \\\\ \\quad + \\int _ { 0 } ^ { 1 } \\overline { \\eta ( x ) } \\eta ^ { \\prime \\prime } ( x ) d x = \\lambda \\int _ { 0 } ^ { 1 } | \\xi ( x ) | ^ 2 d x + \\lambda \\int _ { 0 } ^ { 1 } | \\eta ( x ) | ^ 2 d x . \\end{align*}"}
{"id": "8081.png", "formula": "\\begin{align*} { ( z - w ) } ^ N [ a ( z ) , b ( w ) ] = 0 . \\end{align*}"}
{"id": "4092.png", "formula": "\\begin{align*} \\begin{aligned} d Z _ t & = \\widetilde \\mu ( Z _ t ) \\ , d t + \\widetilde \\sigma ( Z _ t ) \\ , d W _ t , t \\in [ 0 , \\infty ) , \\\\ Z _ 0 & = G ( x _ 0 ) , \\end{aligned} \\end{align*}"}
{"id": "8963.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { t _ 1 } ^ { T _ { x , \\xi } } p \\left ( f \\left ( \\xi ( s ) \\right ) - u \\left ( \\xi ( s ) \\right ) \\right ) d s & \\leq \\int _ { t _ 1 } ^ { T _ { x , \\xi } } p ( 1 - c _ 0 ) f \\left ( \\xi ( s ) \\right ) d s \\\\ & \\leq \\left ( T _ { x , \\xi } - t _ 1 \\right ) p ( 1 - c _ 0 ) f ( \\xi ( t _ 1 ) ) \\end{aligned} \\end{align*}"}
{"id": "5148.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\left ( \\sinh ( x ) - x ^ 2 \\coth ( x ) \\right ) = 0 . \\end{align*}"}
{"id": "3620.png", "formula": "\\begin{align*} \\nu _ 2 ( t ) = \\frac { 1 } { 3 . 3 5 9 \\log | t | } \\left ( 1 - \\frac { 8 . 0 2 \\log \\log | t | } { \\log | t | } \\right ) . \\end{align*}"}
{"id": "6449.png", "formula": "\\begin{align*} T _ 0 ( u ) ( { x } ) = \\frac { \\eta _ 0 ^ 1 } { 4 \\pi } \\int _ { B _ 1 } \\frac { 1 } { | x - y | } u ( y ) d y + \\frac { \\eta _ 0 ^ 2 } { 4 \\pi } \\int _ { B _ 2 } \\frac { 1 } { | x - y | } u ( y ) d y \\end{align*}"}
{"id": "6807.png", "formula": "\\begin{align*} C : = C _ { 0 } ^ { \\prime } \\left ( T _ { 0 } ^ { \\prime } + C _ { 0 } \\left ( \\sqrt { \\frac { 2 T _ { 0 } ^ { \\prime } } { \\gamma } } \\right ) \\right ) < 1 , \\end{align*}"}
{"id": "8455.png", "formula": "\\begin{align*} ( \\alpha ^ * \\beta ^ * \\delta ) ^ * = ( \\delta ( \\alpha \\delta ) ^ * ( \\beta \\delta ) ^ * ) ^ * = ( \\delta ^ * ( \\alpha \\delta ) ^ * ( \\beta \\delta ) ^ * ) ^ * = \\delta ^ * ( \\alpha \\delta ) ^ * ( \\beta \\delta ) ^ * . \\end{align*}"}
{"id": "163.png", "formula": "\\begin{align*} X _ t ^ x = ( 1 - e ^ { - t } ) X _ { \\alpha - \\frac { 1 } { 2 } , 1 } + \\left ( e ^ { - \\frac { t } { 2 } } \\sqrt { x } + \\sqrt { \\frac { 1 - e ^ { - t } } { 2 } } Z \\right ) ^ 2 , \\end{align*}"}
{"id": "3669.png", "formula": "\\begin{align*} \\{ z ' = z _ i ' \\} \\subseteq \\{ \\log | z ' | \\leq \\log | z _ i ' | \\} \\subseteq \\{ \\psi _ V ( z ) \\leq \\log | z _ i ' | + C \\} , \\end{align*}"}
{"id": "9135.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { \\mu _ n } { \\lambda _ n } \\leq A < \\infty \\end{align*}"}
{"id": "5196.png", "formula": "\\begin{align*} T f ( x ) = \\lim \\limits _ { k \\to - \\infty } \\int _ { | y | > q ^ { k } } f ( x - y ) \\dfrac { \\Omega ( y ) } { | y | } d y , ~ ~ ~ ~ ~ k \\in \\mathbb { Z } , \\end{align*}"}
{"id": "9097.png", "formula": "\\begin{align*} \\alpha ^ p = 1 + \\alpha _ m ^ p t ^ { p m } + \\cdots \\end{align*}"}
{"id": "5771.png", "formula": "\\begin{align*} u _ { p , \\tau } = ( u ^ 1 , \\dots , u ^ m ) : B _ { 2 0 0 } ( p , g _ \\tau ) \\to \\mathbb R ^ m . \\end{align*}"}
{"id": "3742.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\Lambda ^ { \\frac { \\alpha } { 2 } } B \\| _ { L ^ 2 } ^ 2 + \\mu \\| \\Lambda ^ { \\alpha } B \\| _ { L ^ 2 } ^ 2 = \\int _ { \\mathbb S ^ 1 } B _ x J \\Lambda ^ \\alpha B \\ , d x . \\end{align*}"}
{"id": "3484.png", "formula": "\\begin{align*} B _ 2 , B _ 3 , B _ 4 \\ll \\left ( 1 + ( \\abs { s _ 3 } / \\sigma _ 3 ) \\right ) \\begin{cases} N ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ N ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log N & ( \\sigma _ 2 = 1 ) \\\\ N ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "5967.png", "formula": "\\begin{align*} \\Delta t = \\frac { \\operatorname { C r } \\Delta x _ { \\min } } { u _ { \\max } } , \\end{align*}"}
{"id": "1964.png", "formula": "\\begin{align*} g ( x ) & + \\sum _ { \\substack { u , u _ 1 , \\dotsc , u _ k \\in \\{ \\ 1 \\} \\cup \\N ^ * \\\\ v = i _ 1 \\dotsm i _ k \\in \\N ^ * } } f _ v g _ u g _ { u _ 1 } \\dotsm g _ { u _ k } x _ { u } x _ { i _ 1 } x _ { u _ 1 } \\dotsm x _ { i _ k } x _ { u _ k } \\\\ & = 1 + \\sum _ { w \\in \\N ^ * } ( \\phi * \\psi ) ( w ) x _ w = \\Lambda _ { \\mathrm { g r } } ( \\phi * \\psi ) ( x ) . \\end{align*}"}
{"id": "1402.png", "formula": "\\begin{align*} | \\hat q | : = \\left \\{ \\begin{aligned} | q | + { ( q q _ d ^ * + q _ d q ^ * ) \\over 2 | q | } \\epsilon , & \\ { \\rm i f } \\ q \\not = 0 , \\\\ | q _ d | \\epsilon , & \\ { \\rm o t h e r w i s e } , \\end{aligned} \\right . \\end{align*}"}
{"id": "778.png", "formula": "\\begin{align*} W _ { n + 2 } & = \\sigma ( \\sigma ^ { n + 1 } ( a ) ) '' \\\\ & = \\sigma ( W _ { n + 1 } ) a \\\\ & = \\sigma ( W _ n ) , \\sigma ( \\varepsilon _ n ) , \\sigma ( W _ n ) a \\\\ & = W _ { n + 1 } , \\varepsilon _ { n + 1 } , W _ { n + 1 } . \\end{align*}"}
{"id": "1151.png", "formula": "\\begin{align*} \\textnormal { R e s } _ { k = - \\eta } T _ { - \\eta , 1 } = - \\frac { 3 } { 4 \\left ( D _ { 0 } ^ { - 2 } r ^ { - 1 } ( \\eta ) - 1 \\right ) } \\begin{pmatrix} \\tilde { T } _ { - \\eta , 1 } ^ { ( 1 1 ) } & \\tilde { T } _ { - \\eta , 1 } ^ { ( 1 2 ) } \\\\ \\tilde { T } _ { - \\eta , 1 } ^ { ( 2 1 ) } & \\tilde { T } _ { - \\eta , 1 } ^ { ( 2 2 ) } \\end{pmatrix} \\end{align*}"}
{"id": "2739.png", "formula": "\\begin{align*} m _ { i , \\lambda } ( \\Z ( P , B ) ) = 0 , \\end{align*}"}
{"id": "8850.png", "formula": "\\begin{align*} E [ B ] : = \\{ x \\in X \\mid ( x , y ) \\in E , y \\in B \\} \\end{align*}"}
{"id": "3949.png", "formula": "\\begin{align*} \\begin{cases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ \\ u ( t , 1 ) = q ( t ) & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
{"id": "1129.png", "formula": "\\begin{align*} f ( k ) : = - \\left ( \\frac { 3 } { 2 } g ( k ) \\right ) ^ { \\frac { 2 } { 3 } } , k \\in \\mathbb { C } \\backslash ( - \\infty , \\eta ] . \\end{align*}"}
{"id": "2099.png", "formula": "\\begin{align*} E _ i = \\left \\lbrace ( [ x _ 0 : \\cdots : x _ n ] , v ) : \\ \\left | \\frac { x _ 0 } { x _ i } \\right | _ { v } \\leq 1 , \\cdots , \\left | \\frac { x _ n } { x _ i } \\right | _ { v } \\leq 1 \\right \\rbrace , \\end{align*}"}
{"id": "1611.png", "formula": "\\begin{align*} \\Theta ^ { Q , j } : = \\Theta _ { a , k , \\zeta , \\mu , \\sigma } , \\\\ W ^ { Q , j } : = W _ { a , k , \\zeta , \\mu , \\sigma } . \\end{align*}"}
{"id": "5120.png", "formula": "\\begin{align*} A ' ( \\gamma ) = - 2 a n \\ , \\frac { e ^ { 2 a \\gamma } ( 2 a \\gamma - 1 ) + 1 } { \\left ( e ^ { 2 a \\gamma } - 1 \\right ) ^ 2 } < 0 . \\end{align*}"}
{"id": "5607.png", "formula": "\\begin{align*} \\mathfrak { k } : = \\{ Z \\in \\Xi ~ | ~ \\mathcal { L } _ Z \\mathbf { g } ( X , Y ) = \\mathbf { g } ( [ Z , X ] , Y ) + \\mathbf { g } ( X , [ Z , Y ] ) X , Y \\in \\Xi \\} . \\end{align*}"}
{"id": "8870.png", "formula": "\\begin{align*} ( h \\varphi ) ( x _ 0 , \\ldots , x _ { q - 1 } ) = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( \\alpha ( x _ 0 ) , \\ldots , \\alpha ( x _ i ) , \\beta ( x _ i ) , \\ldots , \\beta ( x _ { q - 1 } ) ) . \\end{align*}"}
{"id": "8699.png", "formula": "\\begin{align*} \\prod _ { i ' = 1 } ^ d a _ { i ' 0 } + \\sum _ { i = 1 } ^ d \\sum _ { t : \\omega _ t = i } \\biggl ( \\prod _ { i ' \\neq i } a _ { i ' p ^ t _ { i ' } } \\bigl ( s _ { i p ^ t _ i } - s _ { i p ^ { t - 1 } _ { i } } \\bigr ) \\biggr ) . \\end{align*}"}
{"id": "2398.png", "formula": "\\begin{align*} C : \\mathcal { H } \\to \\ell ^ 2 ( \\Gamma ) , C f = ( \\langle f , e _ \\gamma \\rangle ) _ { \\gamma \\in \\Gamma } . \\end{align*}"}
{"id": "6393.png", "formula": "\\begin{align*} \\Lambda = \\{ ( s _ 1 , \\dots , s _ { n - 1 } , p ) \\in \\mathbb G _ n \\ , : \\ ; ( s _ 1 , \\dots , s _ { n - 1 } ) \\in \\sigma _ T ( F _ 1 ^ * + p F _ { n - 1 } \\ , , \\ , F _ 2 ^ * + p F _ { n - 2 } \\ , , \\ , \\dots \\ , , F _ { n - 1 } ^ * + p F _ 1 ) \\} , \\end{align*}"}
{"id": "7799.png", "formula": "\\begin{align*} \\mu \\otimes k ( A \\times B ) = \\int _ A k ( x , B ) \\ , \\mu ( d x ) A \\in \\mathcal B ( X _ { 1 : t } ) , B \\in \\mathcal B ( X _ { t + 1 } ) . \\end{align*}"}
{"id": "3288.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{bmatrix} i \\delta _ 1 & \\lambda \\\\ \\overline { \\lambda } & \\delta _ 2 \\end{bmatrix} ^ { - 1 } = \\begin{bmatrix} - i \\delta _ 2 ( | \\lambda | ^ 2 + \\delta _ 1 \\delta _ 2 ) ^ { - 1 } & \\lambda ( | \\lambda | ^ 2 + \\delta _ 1 \\delta _ 2 ) ^ { - 1 } \\\\ \\overline { \\lambda } ( | \\lambda | ^ 2 + \\delta _ 1 \\delta _ 2 ) ^ { - 1 } & - i \\delta _ 1 ( | \\lambda | ^ 2 + \\delta _ 1 \\delta _ 2 ) ^ { - 1 } \\end{bmatrix} . \\end{aligned} \\end{align*}"}
{"id": "1950.png", "formula": "\\begin{align*} f ( x ) = f _ 0 + \\sum _ { k = 1 } ^ \\infty \\sum _ { ( i _ 1 , \\dotsc , i _ k ) \\in \\N ^ k } f _ { i _ 1 \\dotsm i _ k } x _ { i _ 1 } \\dotsm x _ { i _ k } \\end{align*}"}
{"id": "8592.png", "formula": "\\begin{align*} 2 i t \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } \\phi ( k ) \\mathcal { Q } ( x , k ) e ^ { i k ^ { 2 } t } h ( k ) \\ , d k = \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } \\phi ( k ) k ^ { - 1 } \\mathcal { Q } ( x , k ) h ( k ) \\ , \\partial _ k e ^ { i k ^ { 2 } t } \\ , d k \\\\ = \\phi ' ( 0 ) h ( 0 ) \\mathcal { Q } ( x , 0 ) - \\int \\mathbf { 1 } _ { \\{ k \\geq 0 \\} } e ^ { i k ^ { 2 } t } \\partial _ { k } \\big ( \\phi ( k ) k ^ { - 1 } \\mathcal { Q } ( x , k ) h ( k ) \\big ) \\ , d k . \\end{align*}"}
{"id": "7527.png", "formula": "\\begin{align*} \\frac { T } { 4 } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) \\ d \\sigma = \\frac { T } { 4 } \\left [ \\sigma \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) - 2 \\sigma + 2 T \\arctan \\left ( \\frac { \\sigma } { T } \\right ) \\right ] \\Biggr | _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\end{align*}"}
{"id": "4960.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i \\partial _ t u - \\Delta u = \\rho ^ 2 | u | ^ 2 + \\dot B \\ , , t \\in [ 0 , T ] \\ , , \\ , x \\in \\mathbb { R } ^ d \\ , , \\\\ u _ 0 = \\phi \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "2572.png", "formula": "\\begin{align*} - \\lim _ { \\varepsilon \\to 0 } \\int _ { | x | > \\varepsilon } \\log | x | \\overline { f ' ( x ) } \\ , d x = - \\int _ \\R \\log | x | \\overline { f ' ( x ) } \\ , d x , \\end{align*}"}
{"id": "1818.png", "formula": "\\begin{align*} G = \\{ x \\rightarrow x y , \\ ; \\ ; y \\rightarrow x y \\} . \\end{align*}"}
{"id": "2894.png", "formula": "\\begin{align*} \\| \\left \\langle \\nabla \\right \\rangle ( R ( h ^ A ) - R ( h ) ) \\| _ { S ' ( ( t , + \\infty ) , L ^ 2 ) } & \\lesssim \\sum _ { n = 0 } ^ \\infty \\| \\left \\langle \\nabla \\right \\rangle ( R ( h ^ A ) - R ( h ) ) \\| _ { S ' ( ( t + n , t + n + 1 ) , L ^ 2 ) } \\\\ & \\lesssim \\sum _ { n = 0 } ^ \\infty e ^ { - \\lambda e _ 0 ^ - ( t + n ) } \\lesssim e ^ { - \\lambda e _ 0 ^ - t } , \\forall t \\ge t _ 0 , \\end{align*}"}
{"id": "4506.png", "formula": "\\begin{align*} \\Sigma _ { \\sigma _ 2 } ^ { 1 } \\le 2 \\int _ { H _ { \\sigma _ 2 } } ^ T \\frac { x ^ { - \\frac { 1 } { R \\log t } } } { t } d N ( \\sigma _ 2 , t ) = 2 \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log T } } } { T } N ( \\sigma _ 2 , T ) - \\int _ { H _ { \\sigma _ 2 } } ^ T N ( \\sigma _ 2 , t ) { d } \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t } } } { t } \\Big ) \\Big ) . \\end{align*}"}
{"id": "8719.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { i n } ( x ) = f _ n ( x ) , u _ { i j } ( x ) \\leq f _ { i } ( x ) , & & i = 1 , \\ldots , d , \\ j = 0 , \\ldots , n , \\\\ & u ( x ) \\leq a ( x ) , f ^ L _ i = u _ { i 0 } ( x ) = a _ { i 0 } ( x ) \\leq \\cdots \\leq a _ { i n } ( x ) = f ^ U _ { i } & & i = 1 , \\ldots , d , \\\\ & \\deg ( u _ { i j } ) \\leq \\tau , \\deg ( a _ { i j } ) \\leq \\tau & & i = 1 , \\ldots , d , \\ j = 0 , \\ldots , n . \\end{aligned} \\end{align*}"}
{"id": "5005.png", "formula": "\\begin{align*} q _ { ( k + m ) n } | \\lambda _ { k n } ^ 2 | & \\le \\prod _ { i = 1 } ^ { ( k + m ) n } { q _ { i } \\over q _ { i - 1 } } \\ \\prod _ { l = 0 } ^ { k n - 1 } \\theta _ l ^ 2 \\\\ & \\le \\left ( \\prod _ { i = k n + 1 } ^ { ( k + m ) n } ( r _ { i - 1 } + 1 ) \\right ) \\ \\prod _ { j = 1 } ^ { k n } { q _ j \\over q _ { j - 1 } } \\theta _ { j - 1 } ^ 2 \\\\ & \\le A _ { m , n } \\ \\prod _ { j = 1 } ^ { k n } { q _ j \\over q _ { j - 1 } } \\theta _ { j - 1 } ^ 2 , \\end{align*}"}
{"id": "2077.png", "formula": "\\begin{align*} | \\mathrm { D i s c } ( \\alpha ) | _ v = \\prod _ { i = 1 } ^ n \\prod _ { j \\ne i } | \\lambda _ i - \\lambda _ j | _ v . \\end{align*}"}
{"id": "2866.png", "formula": "\\begin{align*} \\beta _ j ( t ) = \\left \\langle h ( t ) , Q _ j \\right \\rangle - \\alpha _ + ( t ) \\left \\langle \\mathcal { Y } _ + , Q _ j \\right \\rangle - \\alpha _ - ( t ) \\left \\langle \\mathcal { Y } _ - , Q _ j \\right \\rangle . \\end{align*}"}
{"id": "9155.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( k , n ) : = \\Phi \\left ( \\left \\lceil 2 C ( k + 1 ) \\right \\rceil - 1 , \\max \\{ \\theta ( M \\varpi ( k ) + M - 1 ) , n \\} \\right ) . \\end{align*}"}
{"id": "9516.png", "formula": "\\begin{align*} \\langle u , y \\rangle : = E [ u \\cdot y ] . \\end{align*}"}
{"id": "5905.png", "formula": "\\begin{align*} D _ i ( c ) = \\begin{cases} \\dfrac { \\alpha ^ { i + 1 } - \\beta ^ { i + 1 } } { \\alpha - \\beta } & c \\neq \\pm 2 , \\\\ [ 3 m m ] i + 1 & c = 2 , \\\\ [ 2 m m ] ( - 1 ) ^ i ( i + 1 ) & c = - 2 , \\end{cases} \\end{align*}"}
{"id": "9430.png", "formula": "\\begin{align*} v ( r , t ) = \\frac { 1 } { 2 \\varkappa ( T - t ) } w \\Big ( \\frac { r } { \\sqrt { 2 \\varkappa ( T - t ) } } \\Big ) \\end{align*}"}
{"id": "1744.png", "formula": "\\begin{align*} \\int _ G c _ X ( g ^ { - 1 } x ) d g = 1 , \\quad \\mbox { f o r a l l } ~ x \\in X \\end{align*}"}
{"id": "111.png", "formula": "\\begin{align*} L ( A ) ( U \\oplus U ^ 2 ) = 0 , \\ ; V ^ 2 \\subseteq L ( A ) , \\ ; v ( v u ) \\in L ( A ) \\mbox { f o r a l l } u \\in U \\mbox { a n d } v \\in V \\end{align*}"}
{"id": "1849.png", "formula": "\\begin{align*} x M _ n ( x ) = W _ n ( x ) , \\end{align*}"}
{"id": "7107.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & ( \\lambda ^ 2 - 4 ) ^ { n - 1 } P _ { 1 } - 2 ( n - 1 ) ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\end{align*}"}
{"id": "1104.png", "formula": "\\begin{align*} m ^ { L C } ( x , t , k ) : = m ^ { G P } ( x , t , k ) \\cdot m ^ { m o d } ( k , \\xi ) . \\end{align*}"}
{"id": "5854.png", "formula": "\\begin{align*} I _ 1 = I _ { 1 1 } + I _ { 1 2 } \\end{align*}"}
{"id": "9091.png", "formula": "\\begin{align*} & L ( x _ { \\rm t h } ) = \\sqrt { \\frac { \\pi } { 2 } } \\sigma f _ D \\\\ [ - 0 . 2 e m ] & \\times \\sum _ { i = 1 } ^ N \\underbrace { \\int _ 0 ^ { x _ { \\rm t h } } \\ ! \\ ! \\cdots \\int _ 0 ^ { x _ { \\rm t h } } } _ { ( N - 1 ) - { \\rm f o l d } } p _ { | h _ { 1 } | , \\cdots , | h _ { N } | } ( x _ 1 , \\dots , x _ i = x _ { \\rm t h } , \\cdots , x _ N ) \\\\ [ - 0 . 2 e m ] & \\times \\underbrace { d x _ 1 \\cdots d x _ k \\cdots d x _ N } _ { \\substack { ( N - 1 ) - { \\rm f o l d } \\\\ k \\neq i } } . \\end{align*}"}
{"id": "3870.png", "formula": "\\begin{align*} \\Phi _ H = \\{ \\pm t _ i \\pm t _ j \\mid 1 \\leq i \\neq j \\leq n \\} \\subset X ^ * ( T ) . \\end{align*}"}
{"id": "5274.png", "formula": "\\begin{align*} \\check { A } = \\{ \\varphi ( - a ) \\mid a \\in A \\} = \\{ \\psi ( a - ) \\mid a \\in A \\} = \\{ \\psi ( - a ) \\mid a \\in A \\} . \\end{align*}"}
{"id": "3995.png", "formula": "\\begin{align*} \\widetilde { U } ^ p ( t ) = U ^ p ( t ) - U ^ p ( t - 1 ) = \\sum _ { k \\geq N } a _ k \\left ( 1 - e ^ { \\tilde { \\lambda } _ k } \\right ) e ^ { \\tilde { \\lambda } _ k ( T - t ) } , \\\\ \\widetilde { U } ^ h ( t ) = U ^ h ( t ) - U ^ h ( t - 1 ) = \\sum _ { \\mod { k } \\geq N } b _ k \\left ( 1 - e ^ { \\tilde { \\gamma } _ k } \\right ) e ^ { \\tilde { \\gamma } _ k ( T - t ) } , \\end{align*}"}
{"id": "4371.png", "formula": "\\begin{align*} \\varepsilon = \\varepsilon _ l ( \\phi _ l - \\phi _ 0 ) + \\sum _ { j = 1 , j \\ne l } ^ M \\varepsilon _ j \\phi _ j + \\varepsilon _ - , \\end{align*}"}
{"id": "7810.png", "formula": "\\begin{align*} ( - 1 ) ^ { 2 + 1 } \\frac { ( a v ) ^ 2 } { 2 } = - \\frac { a ^ 2 ( \\pi + 2 \\alpha ) } { 2 } = - \\frac { \\pi } { 2 } a ^ 2 - \\alpha . \\end{align*}"}
{"id": "3423.png", "formula": "\\begin{align*} T f _ 1 ( x ) & = \\int _ { \\R ^ N } K ( x , y ) u ( y ) [ f _ 1 ( y ) - f _ 1 ( x ) ] d \\omega ( y ) \\\\ & \\qquad + \\int _ { \\R ^ N } K ( x , y ) v ( y ) f _ 1 ( y ) d \\omega ( y ) + \\int _ { \\R ^ N } K ( x , y ) u ( y ) f _ 1 ( x ) d \\omega ( y ) , \\end{align*}"}
{"id": "5093.png", "formula": "\\begin{align*} c _ { ( k , l ) , ( k ' , l ' ) } = e ^ { - 2 \\pi i k ( l ' - l ) / ( a b ) } \\langle g , \\ , M _ { ( l ' - l ) / a } T _ { ( k ' - k ) / b } \\ , g \\rangle \\end{align*}"}
{"id": "7164.png", "formula": "\\begin{align*} v ^ \\mu ( x ' , x _ d ) - u ^ \\mu ( x ' , x _ d ) & = v ^ \\mu ( x ' , H _ u ^ \\mu ( x ' , y _ d ) ) - y _ d \\\\ & = v ^ \\mu ( x ' , H _ u ^ \\mu ( x ' , y _ d ) ) - v ^ \\mu ( x ' , H _ v ^ \\mu ( x ' , y _ d ) ) \\\\ & = \\int _ { H _ v ^ \\mu ( x ' , y _ d ) } ^ { H _ u ^ \\mu ( x ' , y _ d ) } D _ d v ^ \\mu ( x ' , t ) \\ , d t \\ , . \\end{align*}"}
{"id": "7432.png", "formula": "\\begin{align*} [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( u ) = & c _ { \\gamma } \\int _ { 0 } ^ { \\infty } \\frac { G ( u + w ) + G ( u - w ) - 2 G ( u ) } { w ^ { \\gamma + 1 } } d w . \\end{align*}"}
{"id": "9133.png", "formula": "\\begin{align*} \\norm { x _ { n + l } - x ^ * } & \\leq \\prod _ { k = 0 } ^ { l - 1 } \\left ( 1 + \\frac { \\mu _ { n + k } } { \\lambda _ { n + k } } \\right ) \\norm { x _ n - x ^ * } \\\\ & \\qquad + ( \\norm { T ^ \\circ x ^ * } + \\norm { y ^ * } ) \\sum _ { k = 0 } ^ { l - 1 } \\mu _ { n + k } \\prod _ { j = k + 1 } ^ { l - 1 } \\left ( 1 + \\frac { \\mu _ { n + j } } { \\lambda _ { n + j } } \\right ) \\\\ & \\qquad + \\sum _ { k = 1 } ^ l \\frac { 1 } { r + 1 } \\prod _ { j = k } ^ { l - 1 } \\left ( 1 + \\frac { \\mu _ { n + j } } { \\lambda _ { n + j } } \\right ) . \\end{align*}"}
{"id": "7294.png", "formula": "\\begin{align*} \\beta ' & = \\Bigl ( \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 2 k } ) ( 1 + X ^ { 2 k - 1 } ) ^ 2 \\Bigr ) ' = \\beta \\sum _ { k = 1 } ^ \\infty \\Bigl ( 2 \\frac { ( 2 k - 1 ) X ^ { 2 k - 2 } } { 1 + X ^ { 2 k - 1 } } - \\frac { 2 k X ^ { 2 k - 1 } } { 1 - X ^ { 2 k } } \\Bigr ) \\\\ \\gamma ' & = \\Bigl ( \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 2 k } ) ( 1 + X ^ { 2 k } ) ^ 2 \\Bigr ) ' = \\gamma \\sum _ { k = 1 } ^ \\infty \\Bigl ( 2 \\frac { 2 k X ^ { 2 k - 1 } } { 1 + X ^ { 2 k } } - \\frac { 2 k X ^ { 2 k - 1 } } { 1 - X ^ { 2 k } } \\Bigr ) \\end{align*}"}
{"id": "166.png", "formula": "\\begin{align*} \\operatorname { C o v } ( f ( X _ { \\alpha , \\beta } ) , g ( X _ { \\alpha , \\beta } ) ) = \\int _ { [ - 1 , 1 ] } ( 1 - x ^ 2 ) g ' ( x ) \\tilde { f } ' ( x ) \\mu _ { \\alpha , \\beta } ( d x ) , \\tilde { f } = \\int _ { 0 } ^ { + \\infty } Q _ t ^ { \\alpha , \\beta } ( f ) d t , \\end{align*}"}
{"id": "554.png", "formula": "\\begin{align*} \\nabla \\Psi ( t , x ) = \\nabla \\Psi _ 0 ( x ) , \\ \\Delta \\Psi ( t , x ) = s ( t , x ) = 0 t \\geq 0 , \\ | x | > D + \\frac { \\sqrt { p ' ( \\varrho ) } } { \\varepsilon } t . \\end{align*}"}
{"id": "1254.png", "formula": "\\begin{align*} P _ \\gamma ^ { - 1 } ( [ x , y ] ) = \\left ( H \\setminus U _ a ( x ) \\right ) \\cap \\left ( H \\setminus U _ b ( y ) \\right ) . \\end{align*}"}
{"id": "9250.png", "formula": "\\begin{align*} \\forall p ^ X , x ^ X , \\gamma ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) . \\end{align*}"}
{"id": "522.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq w \\leq h \\\\ 1 \\leq z \\leq s } } ( - 1 ) ^ { w + z } \\sum _ { \\substack { y _ 1 + . . . + y _ w = h \\\\ x _ 1 + . . . + x _ z = s } } \\sum _ { \\substack { 0 \\leq p _ w \\leq . . . \\leq p _ 1 \\leq i - h d \\\\ 0 \\leq v _ z \\leq . . . \\leq v _ 1 \\leq r - i - s d } } \\prod _ { a = 1 } ^ w i _ a \\alpha ^ { i _ a } _ { r - ( h - y _ 1 - . . . - y _ a ) d - p _ a } \\prod _ { b = 1 } ^ z x _ b \\alpha ^ { x _ b } _ { r - ( s - x _ 1 - . . . - x _ b ) d - v _ b } . \\end{align*}"}
{"id": "4586.png", "formula": "\\begin{align*} \\overline { \\lambda } = x - c _ { 2 } \\theta ( x ) \\big ( x ^ 2 ( \\epsilon _ n + \\delta _ n ) + x \\delta _ n \\sqrt { | \\ln \\delta _ n | } \\big ) \\in [ 0 , \\ , o ( \\min \\{ \\epsilon _ n ^ { - 1 } , \\delta _ n ^ { - 1 } \\} ) ) , \\end{align*}"}
{"id": "9458.png", "formula": "\\begin{align*} N _ { 2 0 } + L _ { 2 0 } \\cdot \\tfrac { B ( 0 , 1 ) } { B ( 0 ) } + \\tfrac { B ( 0 , 3 ) ^ p } { B ( 0 ) ^ p } - \\tfrac { B ( - 1 , 3 ) } { B ( - 1 ) } = 0 . \\end{align*}"}
{"id": "4109.png", "formula": "\\begin{align*} \\begin{aligned} | | y | ^ { \\ell _ \\mu } - | x | ^ { \\ell _ \\mu } | & \\leq \\ell _ \\mu \\cdot | x - y | \\cdot ( | x | ^ { \\ell _ \\mu - 1 } + | y | ^ { \\ell _ \\mu - 1 } ) . \\end{aligned} \\end{align*}"}
{"id": "979.png", "formula": "\\begin{align*} I : = \\{ \\tau \\geqslant 0 u ( x ) \\geqslant \\tau \\varphi ( x ) x \\in U \\} . \\end{align*}"}
{"id": "7257.png", "formula": "\\begin{align*} A _ 1 = & b ^ 2 ( 1 - \\theta _ 1 ) - 2 \\theta _ 2 = \\dfrac { 9 a b ^ 3 c d - b ^ 4 c ^ 2 } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } - \\dfrac { 2 c ^ 2 ( 3 a c - b ^ 2 ) ^ 2 } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } \\\\ = & \\dfrac { 3 c ( 4 a b ^ 2 c ^ 2 + 3 a b ^ 3 d - 6 a ^ 2 c ^ 3 - b ^ 4 c ) } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } \\\\ = & \\dfrac { 3 a b ^ 2 c ^ 3 } { 2 c ^ 2 ( 3 a c - b ^ 2 ) + 2 b ^ 2 ( 3 b d - c ^ 2 ) } \\cdot ( 4 + \\dfrac { 3 b d } { c ^ 2 } - \\dfrac { 6 a c } { b ^ 2 } - \\dfrac { b ^ 2 } { a c } ) . \\end{align*}"}
{"id": "8359.png", "formula": "\\begin{align*} y _ n = g _ n y _ { n - 1 } + h _ n , \\end{align*}"}
{"id": "2170.png", "formula": "\\begin{align*} ( - \\triangle _ { g } ) ^ { \\alpha } u ( x ) & = \\int _ { \\mathbb { R } ^ { d } } g \\bigg { ( } \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \\alpha } } \\bigg { ) } \\frac { u ( x ) - u ( y ) } { | u ( x ) - u ( y ) | } \\frac { d y } { | x - y | ^ { d + \\alpha } } \\\\ & = \\int _ { \\mathbb { R } ^ { d } } g \\bigg { ( } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { \\alpha } } \\bigg { ) } \\frac { d y } { | x - y | ^ { d + \\alpha } } \\ \\ ( \\ g \\ ) . \\end{align*}"}
{"id": "140.png", "formula": "\\begin{align*} c ^ { - \\alpha } T _ c ( \\nu _ \\alpha ) ( d u ) = \\nu _ \\alpha ( d u ) , \\end{align*}"}
{"id": "1909.png", "formula": "\\begin{align*} s '' ( t + k ' \\bar { \\tau } ) = \\bar { \\nu } ^ { \\kappa + k ' - 1 } \\left ( \\bigwedge _ { i = \\bar { t } _ \\textup { p } } ^ { \\bar { t } _ { \\textup { p } } + \\bar { \\tau } - 1 } \\bar { s } ( i ) \\right ) \\wedge \\bar { \\nu } ^ { \\kappa + k ' } \\left ( \\bigwedge _ { i = \\bar { t } _ \\textup { p } } ^ { \\bar { t } _ \\textup { p } + h } s ( i ) \\right ) = \\bar { \\nu } ^ { k ' } s '' ( t ) , \\end{align*}"}
{"id": "6641.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq m < n < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( 2 - \\alpha - w ) } p ^ { n ( 1 - \\beta ) } } = \\frac { \\tau _ B ( p ) } { p ^ { 1 - \\beta } } - \\frac { \\tau _ B ( p ) } { p ^ { 3 - \\beta - w } } + O \\left ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\right ) . \\end{align*}"}
{"id": "2909.png", "formula": "\\begin{align*} & \\| \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * ( | u _ 1 | ^ { p - 2 } u _ 2 u _ 3 ) \\right ) | u _ 4 | ^ { p - 2 } | u _ 5 | \\| _ { S ' ( L ^ 2 ) } \\\\ & \\lesssim \\min _ { j = 2 , 3 , 5 } \\left \\lbrace \\| u _ 1 \\| _ { S ( \\dot { H } ^ { s _ c } ) } ^ { p - 2 } \\| u _ 4 \\| _ { S ( \\dot { H } ^ { s _ c } ) } ^ { p - 2 } \\| u _ j \\| _ { S ( L ^ 2 ) } \\prod _ { i , k \\not \\in \\{ 1 , 4 , j \\} } \\| u _ i \\| _ { S ( \\dot { H } ^ { s _ c } ) } \\| u _ k \\| _ { S ( \\dot { H } ^ { s _ c } ) } \\right \\rbrace \\end{align*}"}
{"id": "966.png", "formula": "\\begin{align*} c _ { n , s } \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } = \\frac { c _ { 1 , s } } { 2 s } \\end{align*}"}
{"id": "5936.png", "formula": "\\begin{align*} f _ r ^ * ( \\pi ' _ \\mu ) ^ * \\omega '^ a _ \\mu & = f _ r ^ * ( i ' _ \\mu ) ^ * \\omega '^ a = ( i _ \\mu ) ^ * f ^ * \\omega '^ a \\\\ & = ( i _ \\mu ) ^ * \\omega ^ a = ( \\pi _ \\mu ) ^ * \\omega ^ a _ \\mu . \\end{align*}"}
{"id": "2783.png", "formula": "\\begin{align*} L _ + \\mathcal { Y } _ 1 = e _ 0 \\mathcal { Y } _ 2 L _ - \\mathcal { Y } _ 2 = - e _ 0 \\mathcal { Y } _ 1 . \\end{align*}"}
{"id": "5627.png", "formula": "\\begin{align*} \\mathcal { Q } ^ \\bullet _ { M _ c \\star } = \\bigoplus _ { p = 0 } ^ { n - 1 } \\biguplus _ { q = 0 } ^ \\infty \\biguplus _ { r = 0 } ^ \\infty \\mathcal { Q } ^ { p q r } _ { M _ c \\star } . \\end{align*}"}
{"id": "4980.png", "formula": "\\begin{align*} | | ( A , B ) | | = \\frac { 1 } { 2 } ( | | A | | + | | B | | ) , | | . | | \\end{align*}"}
{"id": "1319.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , N ) } , \\hat { \\alpha } \\cup { ( \\delta , 1 ) \\cup { ( \\gamma , N - p _ { i } ) } } ) = 2 . \\end{align*}"}
{"id": "4302.png", "formula": "\\begin{align*} Q _ { b ( \\tau ) } ( y ) = \\frac { 1 } { b ( \\tau ) } Q \\left ( \\frac { | y | } { \\sqrt { b ( \\tau ) } } \\right ) . \\end{align*}"}
{"id": "2607.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\norm { g _ 0 } _ 2 ^ 2 } V _ { g _ 0 } ^ * F = \\frac { 1 } { \\norm { g _ 0 } _ 2 ^ 2 } \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x g _ 0 \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "6850.png", "formula": "\\begin{align*} \\langle f , U _ { k - 1 } g \\rangle _ { X ( k ) } = \\langle D _ k f , g \\rangle _ { X ( k - 1 ) } . \\end{align*}"}
{"id": "9025.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( \\rho , \\phi ) ( t ) = - \\int _ { \\Omega } \\sum _ { i = 1 } ^ s D _ i ( x ) \\rho _ i | \\nabla ( \\log \\rho _ i + z _ i \\phi ) | ^ 2 d x \\leq 0 , t > 0 , \\end{align*}"}
{"id": "2082.png", "formula": "\\begin{align*} \\min \\{ v _ i ( \\mathbf { c } ) , k \\} = \\min \\{ v _ n ( \\mathbf { c } ) + ( n - i ) a , k \\} \\end{align*}"}
{"id": "5340.png", "formula": "\\begin{align*} & ( \\phi ^ \\epsilon , u ^ \\epsilon , \\eta ^ \\epsilon - 1 , \\tau ^ \\epsilon ) \\rightarrow ( \\phi , u , \\eta - 1 , \\tau ) \\ ; \\ ; \\ ; \\ ; L ^ { \\infty } ( \\mathbb { R } ^ + ; H ^ 3 ) , \\\\ & ( \\nabla u ^ \\epsilon , \\nabla \\eta ^ \\epsilon , \\tau ^ \\epsilon , \\nabla \\tau ^ \\epsilon ) \\rightarrow ( \\nabla u , \\nabla \\eta , \\tau , \\nabla \\tau ) \\ ; \\ ; \\ ; \\ ; L ^ 2 ( \\mathbb { R } ^ + ; H ^ 3 ) , \\end{align*}"}
{"id": "3188.png", "formula": "\\begin{align*} \\chi _ { \\ell } = \\begin{cases} 1 & A x _ { \\ell + 1 } - b \\neq A x _ { \\ell } - b , \\\\ 0 & A x _ { \\ell + 1 } - b = A x _ { \\ell } - b . \\end{cases} \\end{align*}"}
{"id": "5458.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Omega } u & = \\int _ { \\Omega } \\Delta u - \\chi \\int _ { \\Omega } \\nabla \\cdot \\Big ( \\frac { u } { v } \\nabla v \\Big ) + \\int _ { \\Omega } a ( x , t ) u - b ( x , t ) u ^ 2 \\\\ & = \\int _ { \\Omega } a ( x , t ) u ( x , t ) d x - \\int _ { \\Omega } b ( x , t ) u ^ 2 ( x , t ) d x \\\\ & \\leq a _ { \\sup } \\int _ { \\Omega } u - \\frac { b _ { \\inf } } { | \\Omega | } \\Big ( \\int _ { \\Omega } u \\Big ) ^ 2 . \\end{align*}"}
{"id": "3732.png", "formula": "\\begin{align*} B _ t + ( B J ) _ x + \\mu \\Lambda ^ \\alpha B = & \\ 0 , \\\\ B _ x = & \\ \\mathcal H J . \\end{align*}"}
{"id": "8217.png", "formula": "\\begin{align*} A & = z \\prod _ { i = 1 } ^ h F _ { k , i } ( A ) \\prod _ { i = h + 1 } ^ { k + 1 } F _ { k , i } ( A ) \\prod _ { i = 1 } ^ { h - 1 } G _ { k , i } ( A ) ^ { - 1 } \\prod _ { i = h } ^ { k } G _ { k , i } ( A ) ^ { - 1 } \\\\ & = z \\prod _ { i = 1 } ^ { k + 1 } F _ { k , i } ( A ) \\prod _ { i = 1 } ^ { k } G _ { k , i } ( A ) ^ { - 1 } . \\end{align*}"}
{"id": "846.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { P } \\left ( R _ j = a \\right ) = \\left ( 1 - \\epsilon _ m \\right ) \\epsilon ^ a _ m , a \\in \\mathbb { N } , \\\\ & \\mathbb { P } \\left ( V _ j = i \\right ) = \\frac { \\epsilon _ { i - 1 } - \\epsilon _ i } { 1 - \\epsilon _ m } , i \\in [ m ] . \\end{aligned} \\end{align*}"}
{"id": "541.png", "formula": "\\begin{align*} p \\in C ^ 1 ( [ 0 , \\bar \\rho ) ) , \\ p ( 0 ) = 0 , \\ p ' > 0 ( 0 , \\bar \\rho ) , \\ \\lim _ { s \\to \\bar \\rho _ - } p ( s ) = + \\infty . \\end{align*}"}
{"id": "5511.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X ( t ) & = & \\big ( A X ( t ) + \\alpha ( X ( t ) ) \\big ) d t + \\sigma ( X ( t ) ) d W ( t ) \\medskip \\\\ X ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "9522.png", "formula": "\\begin{gather*} x _ t \\ge 0 , \\ p _ t + R _ t \\le y , \\ x _ t ( p _ t + R _ t - y ) = 0 t = 0 , \\ldots , T , \\\\ y \\ge 0 , \\ \\sum _ { t = 0 } ^ T x _ t \\le 1 , \\ y ( \\sum _ { t = 0 } ^ T x _ t - 1 ) = 0 \\end{gather*}"}
{"id": "1391.png", "formula": "\\begin{align*} | ( E _ 1 - E _ 2 ) _ x | ( x ) = & | n _ 1 - n _ 2 - ( b _ 1 - b _ 2 ) | ( x ) \\le | n _ 1 - n _ 2 | ( x ) + | b _ 1 - b _ 2 | ( x ) \\\\ \\le & \\frac { | n _ 1 - n _ 2 | ( x ) } { ( 1 - x ) ^ { \\frac { 1 } { 2 } } } + \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } \\\\ \\le & C \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } , x \\in ( 1 - \\delta _ 1 , 1 ] . \\end{align*}"}
{"id": "1323.png", "formula": "\\begin{align*} 2 = I ( \\hat { \\alpha } \\cup { ( \\gamma , M - 1 ) } , & \\hat { \\alpha } \\cup { ( \\delta , 1 ) } \\cup { ( \\gamma , M - p _ { i } - 1 ) } ) \\\\ = & c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z , Z ) + 2 ( M - p _ { i } - 1 ) Q _ { \\tau } ( Z , Z _ { \\gamma } ) \\\\ & + \\sum _ { k = M - p _ { i } } ^ { M - 1 } ( 2 \\lfloor k \\theta \\rfloor + 1 ) - \\mu _ { \\tau } ( \\delta ) . \\end{align*}"}
{"id": "2230.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } X ^ { n } ( \\cdot ) = X ( \\cdot ) \\ ; \\ ; C _ W ( [ 0 , T ] ; H ^ \\frac d 3 ) . \\end{align*}"}
{"id": "379.png", "formula": "\\begin{align*} d \\Omega = - \\frac 1 2 [ \\Omega , \\Omega ] . \\end{align*}"}
{"id": "1240.png", "formula": "\\begin{align*} { A ( r ) = \\left \\{ x \\in A \\ : \\ \\forall \\tilde { r } \\leq r , \\ m ( B ( x , \\tilde { r } ) \\cap A ) \\geq c . m ( B ( x , \\tilde { r } ) ) \\right \\} } \\end{align*}"}
{"id": "8235.png", "formula": "\\begin{align*} \\frac { 1 } { n - 1 } & \\sum _ { r = 0 } ^ { n - \\ell } \\binom { ( 4 h - 4 h + 3 ) ( n - 1 ) } { r } \\binom { n + r + \\ell - 2 } { \\ell } \\binom { ( 4 h - 2 k - 3 ) ( n - 1 ) - r - \\ell } { n - r - \\ell } \\\\ & - \\frac { 1 } { 2 n - 1 } \\sum _ { r = 0 } ^ { n - \\ell } \\binom { 2 ( k + 1 - h ) ( 2 n - 1 ) - n } { r } \\binom { n + r + \\ell - 1 } { \\ell } \\binom { ( 2 h - k - 2 ) ( 2 n - 1 ) + n - r - \\ell } { n - r - \\ell } \\end{align*}"}
{"id": "8807.png", "formula": "\\begin{align*} \\partial _ t p _ t ( \\varphi ) = \\nabla \\cdot \\left ( p _ t ( \\varphi ) \\nabla u _ t ( \\varphi ) \\right ) \\forall \\varphi \\in \\R ^ { \\Lambda _ { \\epsilon , L } } , t \\ge 0 , \\end{align*}"}
{"id": "7680.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta g _ k + \\tilde { \\lambda } _ 0 \\underline { m } g _ k = f _ 0 ( g _ k ) + \\frac { 1 } { M _ k } F _ k \\tilde { \\Omega } _ { \\varepsilon _ k } \\\\ g _ k = 0 \\partial \\tilde { \\Omega } _ { \\varepsilon _ k } \\end{cases} \\end{align*}"}
{"id": "7812.png", "formula": "\\begin{align*} \\varrho ( x , y ) = 2 ^ { 1 - \\nu } \\sigma ^ 2 [ \\Gamma ( \\nu ) ] ^ { - 1 } ( \\kappa \\norm { x - y } { \\R ^ d } ) ^ \\nu K _ \\nu ( \\kappa \\norm { x - y } { \\R ^ d } ) , x , y \\in \\mathcal D , \\end{align*}"}
{"id": "303.png", "formula": "\\begin{align*} u ( x , t ) - \\chi ( x , t ) - V ( x , t ) = U [ \\psi _ { 0 } ] ( x , t , 0 ) + v ( x , t ) - V ( x , t ) + N [ u , \\psi ] ( x , t ) , \\end{align*}"}
{"id": "8129.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 6 } ( P G ) } c ( g ( \\gamma ) ) = 5 \\equiv 1 \\pmod 2 \\end{align*}"}
{"id": "2948.png", "formula": "\\begin{align*} \\Lambda _ { n , 1 } = \\sum _ { h , h ' \\in \\{ 2 , 3 , 4 \\} } \\Lambda ^ { ( h : h ' ) } _ { n , 1 } , \\Lambda _ { n , 2 } = \\sum _ { h , h ' \\in \\{ 2 , 3 , 4 \\} } \\Lambda ^ { ( h : h ' ) } _ { n , 2 } , \\end{align*}"}
{"id": "6082.png", "formula": "\\begin{align*} g h _ { s } ^ { j } + \\lambda h _ { t } ^ { j } = g \\eta ^ t _ s \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} + \\lambda \\eta ^ t _ t \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} - \\eta ^ { t } \\Lambda \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} + g ^ { 2 } F ^ { j } + g \\lambda G ^ { j } + \\lambda ^ { 2 } H ^ { j } . \\end{align*}"}
{"id": "7400.png", "formula": "\\begin{align*} \\forall t \\in [ 0 , T ] , \\forall x \\in \\mathbb { Z } , \\Phi _ n ( t , \\tfrac { x } { n } ) : = n ^ { \\gamma } \\sum _ { y : | x - y | \\geq \\varepsilon n } F ( t , \\tfrac { x } { n } , \\tfrac { y } { n } ) p ( y - x ) . \\end{align*}"}
{"id": "8356.png", "formula": "\\begin{align*} g ( p ) = r \\sim \\log | \\log p | , \\qquad \\textrm { a s } p \\to 0 . \\end{align*}"}
{"id": "5583.png", "formula": "\\begin{align*} \\left \\| x - \\sum _ { n = 1 } ^ { \\lceil \\lambda m \\rceil } e ^ * _ { \\rho _ x ( n ) } ( x ) e _ { \\rho _ x ( n ) } \\right \\| & \\ \\leqslant \\ \\mathbf C _ \\ell \\left \\| x - \\sum _ { n = 1 } ^ { m } e ^ * _ { \\rho _ x ( n ) } ( x ) e _ { \\rho _ x ( n ) } \\right \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ \\ell \\mathbf C _ { 1 , r p } \\sigma _ m ^ { \\mathcal { U } _ { X } , R } ( x ) . \\end{align*}"}
{"id": "6342.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k d _ \\ell \\leq k ( k - 1 ) + ( n - k ) , \\end{align*}"}
{"id": "1722.png", "formula": "\\begin{align*} \\hat \\nu = \\frac 1 2 \\cdot \\frac { \\alpha _ * ( 1 / p _ 1 \u2010 1 / q ) + \\mu _ * ( 1 / p _ 0 \u2010 1 / q ) } { ( \\mu _ * + \\alpha _ * ) ( 1 / 2 \u2010 1 / q ) + \\gamma _ * ( 1 / p _ 1 \u2010 1 / p _ 0 ) / q } , \\end{align*}"}
{"id": "15.png", "formula": "\\begin{align*} f _ n ( x ) = \\sum _ { j = 1 } ^ { M } [ g _ n ^ j ] [ G _ n ] ^ { - 1 } \\phi ^ j ( x ) + e _ n ^ M ( x ) \\end{align*}"}
{"id": "1037.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x _ 1 } ( 0 ) & = \\int _ { \\R ^ n } y _ 1 \\psi _ s ( y ) u ( y ) \\dd y . \\end{align*}"}
{"id": "2659.png", "formula": "\\begin{align*} g _ 0 ( t ) = 2 ^ { d / 4 } e ^ { - \\pi t ^ 2 } . \\end{align*}"}
{"id": "282.png", "formula": "\\begin{align*} r _ { 0 } ( x ) : = \\eta _ { * } ( x ) ^ { - 1 } \\int _ { - \\infty } ^ { x } \\left ( u _ { 0 } ( y ) - \\chi _ { * } ( y ) \\right ) d y , \\eta _ { * } ( x ) : = \\exp \\left ( \\frac { \\beta } { 2 } \\int _ { - \\infty } ^ { x } \\chi _ { * } ( y ) d y \\right ) , \\end{align*}"}
{"id": "2154.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M ) } | A ^ { ( n ) } _ { M , j , l } ) = \\frac 1 { ( n - j ) } , \\ j \\in [ n - 1 ] . \\end{align*}"}
{"id": "6912.png", "formula": "\\begin{align*} r ( t ) = \\Re \\left [ \\left ( \\sum _ { m = 1 } ^ M h _ m ( t ) s _ m [ t ] \\textbf { a } _ m ^ T ( \\theta ) \\textbf { w } _ m \\right ) e ^ { j 2 \\pi f _ 0 t } \\right ] + n ( t ) , \\end{align*}"}
{"id": "1111.png", "formula": "\\begin{align*} E = I + k ^ { - 1 } E _ 1 + \\mathcal { O } ( k ^ { - 2 } ) . \\end{align*}"}
{"id": "6618.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 ^ * ( h , k ) = \\sum _ { \\substack { \\alpha \\in A \\\\ \\beta \\in B } } \\frac { 1 } { 2 ( 2 \\pi i ) ^ 2 } \\int _ { ( \\epsilon ) } \\int _ { ( \\epsilon ) } \\mathcal { J } \\ , d s _ 2 \\ , d s _ 1 + O \\big ( X ^ { \\varepsilon } Q ^ { \\frac { 3 } { 2 } + \\varepsilon } ( h k ) ^ { \\varepsilon } \\big ) , \\end{align*}"}
{"id": "911.png", "formula": "\\begin{align*} x ^ { n } = \\sum _ { k = 0 } ^ { n } S _ { 2 } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "2953.png", "formula": "\\begin{align*} \\mathcal { S } _ k & \\coloneqq \\left \\{ \\mathbf { P } _ k \\in \\mathcal { W } _ { k } ( 0 ) : \\mathbf { p } _ { k - 1 , 1 } = \\mathbf { p } _ { k - 1 , 2 } , ~ \\mathbf { p } _ { k - 1 , 3 } = \\mathbf { p } _ { k - 1 , 4 } \\right \\} , \\\\ & = \\big \\{ \\mathbf { P } _ k \\in \\mathcal { W } _ { k } ( 0 ) : ( \\mathbf p _ { k - 1 , 1 } \\cup \\mathbf p _ { k - 1 , 2 } ) \\cap ( \\mathbf p _ { k - 1 , 3 } \\cup \\mathbf p _ { k - 1 , 4 } ) = \\varnothing \\big \\} . \\end{align*}"}
{"id": "1657.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\varphi P _ t ^ \\mathcal { R } u = u \\end{align*}"}
{"id": "9527.png", "formula": "\\begin{align*} l ( x , y , \\omega ) & = \\sum _ { t = 0 } ^ T \\left [ \\Delta x _ t \\cdot y _ t + H _ t ( x _ { t } , y _ t , \\omega ) ) \\right ] \\\\ & = \\sum _ { t = 0 } ^ T \\left [ - x _ { t } \\cdot \\Delta y _ { t + 1 } + H _ t ( x _ { t } , y _ t , \\omega ) \\right ] , \\end{align*}"}
{"id": "9302.png", "formula": "\\begin{align*} K _ Z + g _ { * } ^ { - 1 } \\varepsilon D _ 1 + \\sum f ^ { - 1 } _ { * } a _ i E _ i + g _ { * } ^ { - 1 } \\varepsilon D _ 2 + ( 2 \\varepsilon - 1 ) F = g ^ { * } ( K _ X + \\varepsilon D ) . \\end{align*}"}
{"id": "2541.png", "formula": "\\begin{align*} \\pi ( x , \\omega , \\tau ) = \\rho ( \\omega , - x , \\tau ) = e ^ { 2 \\pi i \\tau } e ^ { \\pi i x \\cdot \\omega } M _ { - x } T _ \\omega . \\end{align*}"}
{"id": "4845.png", "formula": "\\begin{align*} \\alpha = P ^ { - 1 } \\Big ( \\frac n \\beta \\Big ) , \\gamma = 2 \\frac { n ^ 2 } \\alpha . \\end{align*}"}
{"id": "3740.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ 1 } B D ^ m J _ x D ^ m B \\ , d x = & - \\int _ { \\mathbb S ^ 1 } B _ x D ^ m J D ^ m B \\ , d x - \\int _ { \\mathbb S ^ 1 } B D ^ m J D ^ m B _ x \\ , d x \\\\ \\int _ { \\mathbb S ^ 1 } J D ^ m B _ x D ^ m B \\ , d x = & - \\frac 1 2 \\int _ { \\mathbb S ^ 1 } J _ x D ^ m B D ^ m B \\ , d x \\end{align*}"}
{"id": "4313.png", "formula": "\\begin{align*} \\left \\langle \\varepsilon _ - , \\phi _ { i , b , \\beta } \\right \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } = 0 , \\forall j = 0 , . . . , \\ell . \\end{align*}"}
{"id": "1418.png", "formula": "\\begin{align*} \\psi ( F _ 1 ' \\cap F _ 2 ' ) & = P ( V _ 1 \\cap V _ 2 ) - P ( F ) \\\\ & = ( P ( V _ 1 ) \\cap P ( V _ 2 ) ) - P ( F ) \\\\ & = ( P ( V _ 1 ) - P ( F ) ) \\cap ( P ( V _ 2 ) - P ( F ) ) \\\\ & = \\psi ( F _ 1 ' ) \\cap \\psi ( F _ 2 ' ) , \\end{align*}"}
{"id": "2349.png", "formula": "\\begin{align*} [ A , B ] = A B - B A . \\end{align*}"}
{"id": "5763.png", "formula": "\\begin{align*} e ^ { - Q } \\delta _ { s t } \\leq h _ { s t } = h ( \\frac { \\partial } { \\partial y ^ { s } } , \\frac { \\partial } { \\partial y ^ { t } } ) \\leq e ^ { Q } \\delta _ { s t } \\end{align*}"}
{"id": "3247.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } s ( r , t ) t = r ^ 2 - \\lambda _ 2 ( \\mu ) ^ 2 . \\end{align*}"}
{"id": "1987.png", "formula": "\\begin{align*} M _ t = 1 + \\sum _ { n = 1 } ^ \\infty m _ n ( t ) x ^ n . \\end{align*}"}
{"id": "6340.png", "formula": "\\begin{gather*} \\sum _ { \\ell = 1 } ^ n ( 2 + p _ \\ell - p _ { \\ell - 1 } ) ^ 2 \\leq 9 \\sum _ { \\ell = 1 } ^ n ( p _ \\ell - p _ { \\ell - 1 } ) ^ 2 \\leq p _ n ^ { 4 / 3 + o ( 1 ) } = n ^ { 4 / 3 + o ( 1 ) } , \\\\ 1 6 ( p _ { n + 1 } - p _ n ) \\sum _ { \\ell = 1 } ^ n ( p _ \\ell - p _ { \\ell - 1 } ) ^ 2 \\leq p _ n ^ { 5 / 8 + o ( 1 ) } p _ n ^ { 4 / 3 + o ( 1 ) } = p _ n ^ { 4 7 / 2 4 + o ( 1 ) } . \\end{gather*}"}
{"id": "7570.png", "formula": "\\begin{align*} x ' ( t ) = g ( x ( t - d ( L x _ t ) ) ) \\end{align*}"}
{"id": "4628.png", "formula": "\\begin{gather*} d _ { k [ S ] } : k [ S ] ^ * \\otimes k [ S ] \\to k , d _ { k [ S ] } ( f \\otimes x ) = f ( x ) , \\\\ b _ { k [ S ] } : k \\to k [ S ] \\otimes k [ S ] ^ * , b _ V ( 1 ) = \\sum _ { g \\in S } g \\otimes g ^ * , \\\\ d _ { k [ S ] ^ * } : k [ S ] \\otimes k [ S ] ^ * \\to k , d _ { k [ S ] ^ * } ( x \\otimes f ) = f ( x ) , \\\\ b _ { k [ S ] ^ * } : k \\to k [ S ] ^ * \\otimes k [ S ] , b _ { k [ S ] ^ * } ( 1 ) = \\sum _ { g \\in S } g ^ * \\otimes g , \\end{gather*}"}
{"id": "6476.png", "formula": "\\begin{align*} E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n / ( 1 - 2 \\alpha ) } } \\right ) ^ 2 \\right ] - 1 \\sim - \\dfrac { 1 } { \\Gamma ( 2 \\alpha ) } \\cdot n ^ { - ( 1 - 2 \\alpha ) } \\qquad ( n \\to \\infty ) . \\end{align*}"}
{"id": "8928.png", "formula": "\\begin{align*} d _ q : C ^ q ( U , A ) & \\to \\mathcal C ^ { q + 1 } ( U , A ) \\\\ \\varphi & \\mapsto ( ( x _ 0 , \\ldots , x _ { q + 1 } ) \\mapsto \\sum _ { i = 0 } ^ { q + 1 } ( - 1 ) ^ i \\varphi ( x _ 0 , \\ldots , \\hat x _ i , \\ldots , x _ { q + 1 } ) . \\end{align*}"}
{"id": "1918.png", "formula": "\\begin{align*} & \\int ^ { T } _ 0 \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f \\big ( \\partial _ t \\varphi + v \\cdot \\nabla _ x \\varphi + ( u - v ) \\cdot \\nabla _ v \\varphi + L [ f ] \\cdot \\nabla _ v \\varphi + \\Delta _ v \\varphi \\big ) \\ , d x d v d t \\\\ & + \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 \\varphi ( 0 , x , v ) \\ ; d x d v = \\int ^ { T } _ { 0 } \\int _ { \\Sigma ^ - } \\big ( v \\cdot \\nu ( x ) \\big ) g \\varphi \\ , d \\sigma ( x ) \\ , d v d t , \\end{align*}"}
{"id": "2605.png", "formula": "\\begin{align*} \\norm { f - f _ n } _ { M ^ { p , q } } = \\norm { V _ g ^ * ( V _ g f - F _ n ) } _ { M ^ { p , q } } \\leq C \\norm { V _ g f - F _ n } _ { L ^ { p , q } } . \\end{align*}"}
{"id": "8380.png", "formula": "\\begin{align*} X : = \\big ( x , \\sqrt { t ^ 2 - \\| x \\| ^ 2 } \\big ) , Y : = \\big ( y , \\sqrt { s ^ 2 - \\| y \\| ^ 2 } \\big ) , Y _ * : = \\big ( y , - \\sqrt { s ^ 2 - \\| y \\| ^ 2 } \\big ) . \\end{align*}"}
{"id": "2356.png", "formula": "\\begin{align*} \\norm { ( X - a ) f } _ 2 ^ 2 = \\int _ \\R ( x - a ) ^ 2 | f ( x ) | ^ 2 \\ , d x \\end{align*}"}
{"id": "2316.png", "formula": "\\begin{align*} A g _ 0 ( x , 0 ) = e ^ { - \\frac { \\pi } { 2 } x ^ 2 } . \\end{align*}"}
{"id": "1534.png", "formula": "\\begin{align*} \\mathbf { E } ( \\mathfrak { g } , s ) = \\sum _ { t = 0 } ^ m \\sum _ { \\gamma \\in P _ N \\backslash P _ N \\tilde { \\tau } _ t ( G \\times G ) } \\phi ( \\gamma \\mathfrak { g } , s ) = : \\sum _ { t = 0 } ^ m \\mathbf { E } _ t ( \\mathfrak { g } , s ) . \\end{align*}"}
{"id": "3756.png", "formula": "\\begin{align*} d ( \\phi \\wedge \\psi ) = ( d _ { \\alpha } \\phi ) \\wedge \\psi + ( - 1 ) ^ { k - 1 } \\phi \\wedge d _ { - \\alpha } \\psi . \\end{align*}"}
{"id": "1236.png", "formula": "\\begin{align*} \\mu ( \\Omega ) \\geq \\mu \\left ( \\bigcup _ { L \\in \\mathcal { F } } L \\right ) \\geq \\mu ( \\Omega ) \\sum _ { k \\geq 1 } C ( 1 - C ) ^ { k - 1 } = \\mu ( \\Omega ) , \\end{align*}"}
{"id": "448.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 1 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle \\leq \\frac { C \\epsilon } { 2 } \\| \\nabla v \\| _ { m } ^ { 2 } + C \\left \\lbrace \\| f _ { 1 } \\| _ { m } ^ { 2 } + ( \\mu _ { 0 } ( t ) + \\mu _ { 1 } ( t ) ) \\left ( \\| u \\| _ { m } ^ { 2 } + \\| v \\| _ { m } ^ { 2 } \\right ) \\right \\rbrace , \\end{align*}"}
{"id": "4149.png", "formula": "\\begin{align*} M _ 1 : = \\sup _ { [ - T , T ] } \\{ \\| u _ 1 ( t ) \\| _ { Z _ { s , 5 / 2 } } + \\| u _ 2 ( t ) \\| _ { Z _ { s , 5 / 2 } } + \\| w ( t ) \\| _ { Z _ { s , 4 + \\theta } } \\} < \\infty . \\end{align*}"}
{"id": "7247.png", "formula": "\\begin{align*} E _ 1 ( x ) = & \\dfrac { 1 } { 4 } ( 4 + 4 + \\dfrac { 1 } { 4 } + \\dfrac { 1 } { 4 } ) = \\dfrac { 1 7 } { 8 } , \\\\ E _ 2 ( x ) = & \\dfrac { 1 } { 6 } ( 4 \\times 4 + 4 \\times 4 \\times \\dfrac { 1 } { 4 } + \\dfrac { 1 } { 4 } \\times \\dfrac { 1 } { 4 } ) = \\dfrac { 1 0 7 } { 3 2 } , \\\\ E _ 3 ( x ) = & \\dfrac { 1 } { 4 } ( 2 \\times 4 \\times 4 \\times \\dfrac { 1 } { 4 } + 2 \\times 4 \\times \\dfrac { 1 } { 4 } \\times \\dfrac { 1 } { 4 } ) = \\dfrac { 1 7 } { 8 } , \\\\ E _ 4 ( x ) = & 4 \\times 4 \\times \\dfrac { 1 } { 4 } \\times \\dfrac { 1 } { 4 } = 1 , \\end{align*}"}
{"id": "184.png", "formula": "\\begin{align*} \\mathcal { A } _ \\delta ( f _ \\delta ) = g , \\end{align*}"}
{"id": "6052.png", "formula": "\\begin{align*} \\phi ( u - \\delta f _ J ) \\ = \\ \\phi ( u ) + \\delta ( n - | J | ) \\sigma _ J - \\delta | J | ( m - \\sigma _ J ) \\ = \\ \\phi ( u ) + \\delta \\left ( n \\sigma _ J - m | J | \\right ) \\end{align*}"}
{"id": "4676.png", "formula": "\\begin{align*} | \\mu _ { i } ( t ) | \\leq \\omega _ 0 , d ( t ) : = \\min _ { i \\in \\{ 1 , \\ldots , n - 1 \\} } [ x _ { i } ( t ) - x _ { i + 1 } ( t ) ] \\geq \\frac { 1 } { \\omega _ 0 } > 0 , \\end{align*}"}
{"id": "8720.png", "formula": "\\begin{align*} p ( x ) = \\phi \\bigl ( g _ { j ( 1 ) } ( x ) , \\ldots , g _ { j ( d ) } ( x ) \\bigr ) j ( 1 ) , \\ldots , j ( d ) \\in \\{ 1 , \\ldots , m \\} , \\deg ( p ) = \\delta . \\end{align*}"}
{"id": "8131.png", "formula": "\\begin{align*} \\kappa ( f ) = \\sum _ { ( d , e ) \\in D _ { 2 } ( H G ) } | f ( d ) \\cap f ( e ) | \\end{align*}"}
{"id": "2305.png", "formula": "\\begin{align*} R f ( x , \\omega ) = f ( x ) \\overline { \\widehat { f } ( \\omega ) } e ^ { - 2 \\pi i x \\cdot \\omega } . \\end{align*}"}
{"id": "4292.png", "formula": "\\begin{align*} u ( t , r ) = \\frac { 1 } { \\lambda ( t ) } v ( \\xi , s ) \\end{align*}"}
{"id": "7786.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to 0 } \\frac { \\log \\prod \\limits _ { i = 1 } ^ { q } \\frac { p _ { w _ i } } { C _ { w _ i } } } { \\log r } = \\frac { \\sum \\limits _ { k \\in T } p _ k \\log p _ k } { \\sum \\limits _ { k \\in T } p _ k \\log C _ k } - 1 . \\end{align*}"}
{"id": "2513.png", "formula": "\\begin{align*} \\lim _ { | \\tau _ n | \\to 0 } \\norm { e ^ { 2 \\pi i \\tau _ n } f - f } _ 2 = 0 , \\ , \\lim _ { | x _ n | \\to 0 } \\norm { T _ x f - f } _ 2 = 0 \\ , \\ , \\lim _ { | \\omega _ n | \\to 0 } \\norm { M _ \\omega f - f } _ 2 = 0 . \\end{align*}"}
{"id": "5106.png", "formula": "\\begin{align*} \\coth ( x ) < 1 + \\frac { 2 } { 2 1 - 1 } = 1 . 1 , \\forall x > \\log ( \\sqrt { 2 1 } ) , \\end{align*}"}
{"id": "3770.png", "formula": "\\begin{align*} \\frac { \\epsilon \\big ( X , { \\rm i n d } _ { \\mathcal { W } _ { K ' } } ^ { \\mathcal { W } _ { K } } ( \\rho ) , \\psi _ K \\big ) } { \\epsilon ( X , \\rho , \\psi _ { K ' } ) } = \\frac { \\epsilon \\big ( X , { \\rm i n d } _ { \\mathcal { W } _ { K ' } } ^ { \\mathcal { W } _ { K } } ( 1 _ { K ' } ) , \\psi _ K \\big ) } { \\epsilon ( X , 1 _ { K ' } , \\psi _ { K ' } ) } , \\end{align*}"}
{"id": "5525.png", "formula": "\\begin{align*} \\delta _ m & : = \\frac { T } { m } , m \\in \\mathbb { N } , \\\\ [ t ] _ m ^ - & : = k \\delta _ m , k \\delta _ m \\leq t < ( k + 1 ) \\delta _ m , k = 0 , \\ldots , m - 1 , \\\\ [ t ] _ m ^ + & : = ( k + 1 ) \\delta _ m , k \\delta _ m \\leq t < ( k + 1 ) \\delta _ m , k = 0 , \\ldots , m - 1 , \\end{align*}"}
{"id": "790.png", "formula": "\\begin{align*} \\mathcal { E } _ p ( u ) ^ p : = \\inf _ g \\int _ X g ^ p \\ , d \\mu , \\end{align*}"}
{"id": "5034.png", "formula": "\\begin{align*} \\Phi ^ { + } _ G & = \\left \\{ \\widehat { x } _ i - \\widehat { x } _ j \\ , , \\ ; 1 \\leq i < j \\leq n \\ , , \\ ; \\widehat { x } _ i + \\widehat { x } _ j \\ , , \\ ; 1 \\leq i \\leq j \\leq n \\right \\} \\\\ \\intertext { a n d } \\Phi ^ { + } _ K & = \\left \\{ \\widehat { x } _ i - \\widehat { x } _ j ) \\ , , \\ ; 1 \\leq i < j \\leq n \\right \\} \\ , , \\end{align*}"}
{"id": "8408.png", "formula": "\\begin{align*} Y _ 0 = x , Y _ { k + 1 } = Y _ { k } + \\eta b ( Y _ { k } ) + ( Z _ { ( k + 1 ) \\eta } - Z _ { k \\eta } ) , k = 0 , 1 , 2 , \\dots , \\end{align*}"}
{"id": "764.png", "formula": "\\begin{align*} b \\mathfrak { h } _ \\omega ( a _ 0 , a _ 1 , \\ldots , a _ { p + 1 } , a _ { p + 2 } ) = p \\mathrm { S T r } _ \\omega \\left ( [ F , a _ { p + 2 } ] a _ 0 [ F , a _ 1 ] \\cdots [ F , a _ { p + 1 } ] \\right ) = 0 . \\end{align*}"}
{"id": "9533.png", "formula": "\\begin{align*} p _ { t } + \\Delta y _ { t + 1 } & \\in \\partial _ { x } H _ t ( x _ { t } , y _ t ) , \\\\ \\Delta x _ t & \\in \\partial _ { y } [ - H _ t ] ( x _ { t } , y _ t ) , \\end{align*}"}
{"id": "7904.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { j } e _ j ( x ) h _ j . \\end{align*}"}
{"id": "3708.png", "formula": "\\begin{align*} \\int _ 0 ^ t ( t - \\tau ) ^ { \\frac { m - \\frac 5 2 } { \\alpha } } \\tau ^ { \\frac { \\frac 5 2 - \\alpha - m } { \\alpha } } \\ , d \\tau = \\int _ 0 ^ 1 ( 1 - \\tau ' ) ^ { \\frac { m - \\frac 5 2 } { \\alpha } } ( \\tau ' ) ^ { \\frac { \\frac 5 2 - \\alpha - m } { \\alpha } } \\ , d \\tau ' \\leq C \\end{align*}"}
{"id": "5913.png", "formula": "\\begin{align*} \\frac { 1 } { \\displaystyle { \\frac { 2 k ( 2 k - 1 ) } { 2 } } } \\ , \\sum \\limits _ { i < j } \\ , | \\lambda _ i | \\ , | \\lambda _ j | \\geq \\left ( \\prod ^ { 2 k } _ { i = 1 } \\ , | \\lambda _ i | \\right ) ^ { \\frac { 1 } { k } } , \\end{align*}"}
{"id": "9325.png", "formula": "\\begin{align*} & \\| \\mathcal { P } ( \\mathbf { x } _ k ) - \\mathbf { x } _ k \\| \\leq \\| \\mathcal { P } ( \\mathbf { x } _ k ) - \\mathcal { P } ( \\mathbf { x } _ { k - 1 } ) \\| + \\| \\mathcal { P } ( \\mathbf { x } _ { k - 1 } ) - \\mathbf { x } _ { k } \\| \\\\ & \\leq \\eta \\| \\mathbf { x } _ k - \\mathbf { x } _ { k - 1 } \\| + \\| \\mathcal { P } ( \\mathbf { x } _ { k - 1 } ) - \\mathbf { x } _ { k } \\| , \\end{align*}"}
{"id": "6211.png", "formula": "\\begin{align*} - \\sum _ { b = 1 } ^ { j - 1 } \\sum _ { a = 1 } ^ { j } t _ { a , b } \\geq 1 - j . \\end{align*}"}
{"id": "3184.png", "formula": "\\begin{align*} \\varphi _ C ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( e _ { \\mathrm { r e m } ( \\zeta _ { k - 1 } , d ) + 1 } , \\zeta _ { k - 1 } + 1 ) . \\end{align*}"}
{"id": "9375.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( n + 1 , k ) = S _ { 2 , \\lambda } ( n , k - 1 ) + ( k - n \\lambda ) S _ { 2 , \\lambda } ( n , k ) , \\end{align*}"}
{"id": "5728.png", "formula": "\\begin{align*} 0 \\leq | \\xi | \\leq \\epsilon t + l _ \\epsilon t ^ { q - 1 } , \\ \\forall t \\geq 0 , \\ \\xi \\in \\partial F ^ + ( t ) = \\partial F ( t ) . \\end{align*}"}
{"id": "5891.png", "formula": "\\begin{align*} \\mathcal { D } \\mathcal { X } _ t = \\left ( \\mathcal { D } ^ { j } _ { r } \\mathcal { X } _ t \\right ) _ { j \\in \\{ 1 , 2 , \\dots , d \\} , \\ , r \\in [ 0 , T ] } . \\end{align*}"}
{"id": "7267.png", "formula": "\\begin{align*} \\Bigl | \\Bigl ( \\prod ( 1 + \\alpha _ k ) \\Bigr ) \\circ \\beta - \\prod _ { k = 1 } ^ n ( 1 + \\alpha _ k ( \\beta ) ) \\Bigr | & = \\Bigl | \\Bigl ( \\prod _ { k = 1 } ^ \\infty ( 1 + \\alpha _ k ) - \\prod _ { k = 1 } ^ n ( 1 + \\alpha _ k ) \\Bigr ) \\circ \\beta \\Bigr | \\\\ & \\le \\Bigl | \\prod _ { k = 1 } ^ \\infty ( 1 + \\alpha _ k ) - \\prod _ { k = 1 } ^ n ( 1 + \\alpha _ k ) \\Bigr | \\xrightarrow { n \\to \\infty } 0 . \\end{align*}"}
{"id": "7108.png", "formula": "\\begin{align*} P _ { n } ( \\lambda ) = & ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\left [ \\lambda ^ 3 - 6 \\lambda - 4 \\right ] - 2 ( n - 1 ) ( \\lambda + 2 ) ( \\lambda ^ 2 - 4 ) ^ { n - 1 } \\end{align*}"}
{"id": "1938.png", "formula": "\\begin{align*} \\mu ( x , t ) : = \\dfrac { 1 } { 2 } \\norm { r ( x , t ) } ^ 2 : = \\dfrac { 1 } { 2 } \\norm { \\begin{pmatrix} h ( x ) \\\\ f ( x ) - t \\end{pmatrix} } ^ 2 , \\end{align*}"}
{"id": "5781.png", "formula": "\\begin{align*} | P _ 1 \\rho | \\leq ( | p _ 1 | + | p _ 2 | ) . \\end{align*}"}
{"id": "3963.png", "formula": "\\begin{align*} \\Phi _ { \\lambda ^ h _ k } : = ( \\xi _ { \\lambda ^ h _ k } , \\eta _ { \\lambda ^ h _ k } ) \\ \\ \\lambda ^ h _ k , \\ \\ \\forall | k | \\geq k _ 0 . \\end{align*}"}
{"id": "1772.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) ( \\xi ) = \\int _ { \\mathbb { R } ^ n } f ( x ) \\exp ^ { - 2 \\pi i \\xi \\cdot x } d x , \\end{align*}"}
{"id": "3135.png", "formula": "\\begin{align*} P = 1 1 - \\delta \\ , , \\end{align*}"}
{"id": "8024.png", "formula": "\\begin{align*} ( \\rho ^ { * } _ { ( 1 ) } \\otimes \\rho ^ { * } _ { ( 1 ) } ) E _ { \\Sigma _ 0 } & = ( \\rho ^ { * } _ { ( 1 ) } \\otimes \\rho ^ { * } _ { ( 1 ) } ) ( \\partial _ { \\Sigma _ 0 } \\otimes \\partial _ { \\Sigma _ 0 } ) E _ { \\mathcal { M } _ 0 } \\\\ & = ( \\partial _ { \\Sigma } \\otimes \\partial _ { \\Sigma } ) ( \\chi ^ { * } \\otimes \\chi ^ { * } ) E _ { \\mathcal { M } _ 0 } \\\\ & = : E _ { \\Sigma } . \\end{align*}"}
{"id": "4634.png", "formula": "\\begin{align*} \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty ^ k } { ( q ; q ) _ \\infty ^ { 3 k + 1 } } = \\frac { ( - q ; q ) _ \\infty ^ k } { ( q ; q ) _ \\infty ^ k } \\frac { 1 } { ( q ; q ) _ \\infty ^ { k + 1 } } , \\end{align*}"}
{"id": "2217.png", "formula": "\\begin{align*} \\| g ( u ) v \\| _ { 1 } & \\leq C ( \\| g ( u ) v \\| + \\| v \\nabla g ( u ) \\| + \\| g ( u ) \\nabla v \\| ) \\\\ & \\leq C ( 1 + \\| u \\| ^ 2 _ V ) \\| v \\| + \\| v \\nabla g ( u ) \\| + C \\| v \\| _ 1 ( 1 + \\| u \\| _ V ^ 2 ) . \\end{align*}"}
{"id": "7070.png", "formula": "\\begin{align*} \\Delta _ K \\beta _ i ^ K ( t \\wedge \\theta _ K ) & = g _ i ^ K ( t ) - \\frac { \\| p \\| _ { } } { \\underline { p } } A _ { i + 1 } ^ K ( t \\wedge \\theta _ K ) + \\Delta _ K M _ i ^ K ( t \\wedge \\theta _ K ) \\\\ & \\leq \\max _ { 0 \\leq j \\leq 1 / \\delta _ K } g ^ K _ j ( 0 ) + 2 C ( K , L ) t + \\frac { \\| p \\| _ { } } { \\underline { p } } \\left ( \\max _ { 0 \\leq j \\leq 1 / \\delta _ K - 1 } \\beta ^ K _ i ( 0 ) + C _ 1 t \\right ) + \\varepsilon _ K . \\end{align*}"}
{"id": "1535.png", "formula": "\\begin{align*} \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { E } _ t ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h = 0 . \\end{align*}"}
{"id": "7434.png", "formula": "\\begin{align*} [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( u ) = & c _ { \\gamma } \\int _ { 0 } ^ { 3 b _ G } \\frac { [ G ( u + w ) - G ( u ) ] + [ G ( u - w ) - G ( u ) ] } { w ^ { \\gamma + 1 } } d w . \\end{align*}"}
{"id": "7456.png", "formula": "\\begin{align*} \\lambda _ { \\phi } ( t ) & = \\frac { \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi | ^ 2 + V | \\phi | ^ 2 + \\beta | \\phi | ^ 4 - \\Omega \\overline { \\phi } L _ z \\phi \\right ) \\mathrm { d } \\mathbf { x } - \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 } { \\| \\phi ( \\cdot , t ) \\| ^ 2 } . \\end{align*}"}
{"id": "6701.png", "formula": "\\begin{align*} \\mathcal { L i } _ { K , \\mathfrak { s } } ( { \\bf z } ) : = \\sum _ { i _ 1 > i _ 2 > \\cdots > i _ r > 0 } \\frac { z _ 1 ^ { q ^ { i _ 1 } } z _ 2 ^ { q ^ { i _ 2 } } \\cdots z _ r ^ { q ^ { i _ r } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { s _ 1 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { s _ 2 } \\cdots ( \\theta ^ { q ^ { i _ r } } - t ) ^ { s _ r } } \\end{align*}"}
{"id": "5789.png", "formula": "\\begin{align*} K _ I = \\{ \\sigma \\in K \\ , : \\ , \\sigma \\subseteq I \\} . \\end{align*}"}
{"id": "6446.png", "formula": "\\begin{align*} \\lambda _ h T _ h ( \\lambda _ h ) u _ h = u _ h \\end{align*}"}
{"id": "8232.png", "formula": "\\begin{align*} N _ { h + 1 } = \\frac { x _ { h + 1 } z } { \\frac { N _ 1 N _ { k + 1 - h } } { x _ 1 z } + \\frac { x _ h z } { N _ h } } . \\end{align*}"}
{"id": "3213.png", "formula": "\\begin{align*} \\theta _ { P _ H } = \\frac { \\frac { ( 1 + e ^ { a b } ) ^ { k _ 1 - 2 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { k _ 1 } } - \\frac { ( 1 + e ^ { a b } ) ^ { 2 k _ 1 - 2 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { 2 k _ 1 } } } { \\frac { ( 1 + e ^ { a b } ) ^ { k _ 1 - 1 } } { \\left ( 1 + e ^ { a b } \\left ( 1 - \\zeta \\right ) \\right ) ^ { k _ 1 } } - \\frac { 1 } { 1 + e ^ { a b } } } . \\end{align*}"}
{"id": "800.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\partial \\Omega } \\phi _ k \\ , f \\ , d \\nu = \\int _ { \\partial \\Omega } v \\ , f \\ , d \\nu . \\end{align*}"}
{"id": "1779.png", "formula": "\\begin{align*} H ^ \\infty _ L ( G ) = \\left \\{ f \\in L ^ 2 ( G ) | \\int _ G \\big ( 1 + L ( g ) \\big ) ^ { 2 k } | f ( g ) | ^ 2 d g < \\infty , \\ \\forall k \\in \\mathbb { N } \\right \\} . \\end{align*}"}
{"id": "1545.png", "formula": "\\begin{align*} \\int _ { \\mathfrak { Z } } \\delta ( w , z ) ^ { - k } | \\delta ( w , z ) | ^ { - 2 s } \\delta ( w ) ^ { s + k } f ( w ) d w = \\widetilde { c } _ k ( s ) f ( z ) \\delta ( z ) ^ { - s } , \\end{align*}"}
{"id": "6042.png", "formula": "\\begin{align*} P _ n ^ { ( \\alpha , \\beta ) , * } ( x ) = ( 1 - x ) ^ \\alpha ( 1 + x ) ^ \\beta P _ n ^ { ( \\alpha , \\beta ) } ( x ) , \\qquad \\hbox { f o r } ~ x \\in ( - 1 , 1 ) , ~ n \\geq 0 , \\end{align*}"}
{"id": "4606.png", "formula": "\\begin{align*} Q _ v ^ + & = \\langle F \\in \\mathcal { F } \\mid v \\cdot u _ F > 0 \\rangle , \\\\ Q _ v ^ - & = \\langle F \\in \\mathcal { F } \\mid v \\cdot u _ F < 0 \\rangle . \\end{align*}"}
{"id": "3223.png", "formula": "\\begin{align*} G _ \\mu ( z ) = \\int _ \\R \\frac { 1 } { z - x } d \\mu ( x ) , z \\in \\mathbb { C } ^ + ; \\end{align*}"}
{"id": "1182.png", "formula": "\\begin{align*} \\mathcal { K } _ j : = \\left ( \\Psi ^ { A i } _ { 0 } \\right ) ^ { - 1 } \\Psi ^ { A i } _ { j } \\left ( \\frac { 2 } { 3 } \\right ) ^ { - j } = \\frac { 3 ^ j } { 2 ^ { j + 1 } } e ^ { \\frac { i \\pi } { 4 } \\sigma _ 3 } \\begin{pmatrix} ( - 1 ) ^ { j } ( s _ j + \\nu _ j ) & s _ j - \\nu _ j \\\\ ( - 1 ) ^ { j } ( s _ j - \\nu _ j ) & s _ j + \\nu _ j \\end{pmatrix} e ^ { - \\frac { i \\pi } { 4 } \\sigma _ 3 } . \\end{align*}"}
{"id": "2511.png", "formula": "\\begin{align*} 0 = | \\langle f , \\rho ( \\mathbf { h } ) g \\rangle | = | A ( f , g ) ( x , \\omega ) | = | V _ g f ( x , \\omega ) | \\forall ( x , \\omega ) \\in \\R ^ { 2 d } . \\end{align*}"}
{"id": "5243.png", "formula": "\\begin{align*} \\sum _ i \\omega ( b _ i d ) a _ i c = 0 , \\forall c \\in A _ u ^ s , \\end{align*}"}
{"id": "636.png", "formula": "\\begin{align*} \\begin{cases} \\ i = ( x + 1 ) ( g _ 0 ( x , n ) - f _ 0 ( x , n ) ) \\\\ [ 8 p t ] \\ j = h _ 0 ( x , n ) \\end{cases} . \\end{align*}"}
{"id": "4731.png", "formula": "\\begin{align*} u ( t , y + t ) = \\Theta ( \\vec { \\mathfrak { q } } ( t ) , 0 , y ) + \\epsilon ( t , y ) , \\end{align*}"}
{"id": "7281.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 2 k } ) ( 1 + X ^ { 2 k - 1 } ) ^ 2 & = \\sum _ { k = - \\infty } ^ \\infty X ^ { k ^ 2 } , \\\\ \\prod _ { k = 1 } ^ \\infty \\frac { ( 1 - X ^ k ) ^ 2 } { 1 - X ^ { 2 k } } = \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 2 k } ) ( 1 - X ^ { 2 k - 1 } ) ^ 2 & = \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ k X ^ { k ^ 2 } , \\\\ \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 2 k } ) ( 1 + X ^ { 2 k } ) ^ 2 & = \\frac { 1 } { 2 } \\sum _ { k = - \\infty } ^ \\infty X ^ { k ^ 2 + k } = \\sum _ { k = 0 } ^ \\infty X ^ { k ^ 2 + k } , \\end{align*}"}
{"id": "3460.png", "formula": "\\begin{align*} \\omega ( B ( x _ 0 , 2 ^ { j _ { k + 1 } } 2 ^ { i _ 0 } \\sigma ) ) = \\omega ( B ( x _ 0 , 2 ^ { j _ k + 1 } 2 ^ { i _ 0 } \\sigma ) ) = \\omega \\Big ( \\bigcup \\limits _ { \\ell = j _ { k } + 1 } ^ { j _ { k + 1 } } \\chi _ \\ell \\Big ) \\geqslant 2 \\omega ( B ( x _ 0 , 2 ^ { j _ k } 2 ^ { i _ 0 } \\sigma ) ) \\end{align*}"}
{"id": "7086.png", "formula": "\\begin{align*} \\{ f _ { W _ { V , i } } : X _ { W _ { V , i } } \\to W _ V \\} _ { 1 \\leq i \\leq k } . \\end{align*}"}
{"id": "6458.png", "formula": "\\begin{align*} \\lambda _ h = \\lambda _ 0 - i \\frac { { \\lambda _ 0 } ^ \\frac { 5 } { 2 } } { 4 \\pi } \\left [ { U _ 0 \\sum _ { k = 1 } ^ { k = n } \\eta _ 0 ^ k U _ k } \\right ] h + \\mathcal { O } ( h ^ 2 ) \\end{align*}"}
{"id": "7833.png", "formula": "\\begin{align*} \\langle y \\rangle _ { T ^ { * 3 } } & = ~ \\{ y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( j - 1 - r ) } y \\} ~ , ~ ~ ~ \\langle y \\rangle _ { T ^ { * 3 } } = j - r , \\\\ \\langle z \\rangle _ { T ^ { * 3 } } & = ~ \\{ z , T ^ { * 3 } z , \\ldots , T ^ { * 3 ( n - 2 j - 2 + r ) } z \\} ~ , ~ ~ ~ \\langle z \\rangle _ { T ^ { * 3 } } = n - 2 j - 1 + r , \\end{align*}"}
{"id": "8772.png", "formula": "\\begin{align*} s _ { i j } = m _ { i j t _ i ( H ) } = a _ { i j } j \\leq \\tau ( i , t _ i ( H ) - 1 ) s _ { i j } = m _ { i j t _ i ( H ) } \\leq a _ { i j } \\tau ( i , t _ i ( H ) - 1 ) \\leq j < j _ i ' s _ { i j } = s _ { i n } \\end{align*}"}
{"id": "4645.png", "formula": "\\begin{align*} \\frac { \\sqrt { \\Delta ( { \\mathbb K } ) } } { n } \\sum _ { d \\vert n } \\mu ( n / d ) \\lambda ^ { d ^ s } = \\frac { { \\sqrt { \\Delta ( { \\mathbb K } ) } } } { n } \\sum _ { d \\vert n } \\mu ( d ) \\lambda ^ { ( n / d ) ^ s } \\end{align*}"}
{"id": "864.png", "formula": "\\begin{align*} { \\bar { \\Delta } ^ { \\rm P } _ { \\rm R a t e l e s s } } = - 1 - { \\tau _ { \\rm f } } + { 2 } \\left ( { { \\tau _ { \\rm c } } + { n _ 1 } + \\mathcal { T } + \\sum \\limits _ { i = 1 } ^ { \\infty } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) { \\epsilon _ i } } } \\right ) . \\end{align*}"}
{"id": "9127.png", "formula": "\\begin{align*} \\frac { 1 } { ( k + 1 ) ^ 2 } \\geq \\langle T ^ \\circ x - z , - z \\rangle + \\langle z - T ^ \\circ x , - T ^ \\circ x \\rangle = \\norm { T ^ \\circ x - z } ^ 2 \\end{align*}"}
{"id": "4208.png", "formula": "\\begin{align*} u _ t = u _ 0 + t \\ , v + t ^ 2 \\ , r _ t , \\forall \\ , t \\in [ 0 , 1 ] \\end{align*}"}
{"id": "4326.png", "formula": "\\begin{align*} \\int ( w ( \\tau _ 0 ) - Q _ b ) \\phi _ { 1 , b , \\beta } \\frac { y ^ 2 } { 2 } \\rho _ \\beta d y = \\varepsilon _ 0 ( \\tau _ 0 ) \\int \\phi _ { 0 , b , \\beta } \\phi _ { 1 , b , \\beta } \\frac { y ^ 2 } { 2 } \\rho _ \\beta d y + \\varepsilon _ 1 ( \\tau _ 0 ) \\int \\phi _ { 1 , b , \\beta } ^ 2 \\frac { y ^ 2 } { 2 } \\rho _ \\beta d y + o ( b ^ { \\frac { \\alpha } { 2 } } ) . \\end{align*}"}
{"id": "7973.png", "formula": "\\begin{align*} L _ { \\Sigma , g } \\hat { u } _ \\infty = \\frac { n } { 2 } \\nu _ { \\Sigma , g } ( \\hat { f } _ \\infty ) . \\end{align*}"}
{"id": "1541.png", "formula": "\\begin{align*} \\int _ { G _ { \\infty } / K _ { \\infty } } \\phi _ { \\infty } ( \\tilde { \\tau } _ m ( g _ { \\infty } \\times h _ { \\infty } ) , s ) \\mathbf { f } ( h _ { \\mathbf { h } } \\cdot h _ { \\infty } ) \\mathbf { d } h _ { \\infty } = c _ k ( s ) \\mathbf { f } ( h _ { \\mathbf { h } } \\cdot g _ { \\infty } ) , \\end{align*}"}
{"id": "4723.png", "formula": "\\begin{align*} & P _ { i , 1 } = ( \\dot { x } _ i - \\mu _ i ) \\sigma _ i \\partial _ y \\widetilde { R } _ i \\\\ & P _ { i , 2 } = - \\bigg ( \\dot { \\mu } _ i + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { x ^ 3 _ { i j } } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\mu _ k } { x ^ 3 _ { i j } } \\bigg ) \\frac { \\sigma _ i \\Lambda \\widetilde { R } _ i } { 1 + \\mu _ i } . \\end{align*}"}
{"id": "2331.png", "formula": "\\begin{align*} f ( t ) = e ^ { - \\pi t \\cdot A t + 2 \\pi b \\cdot t + c } , \\end{align*}"}
{"id": "4026.png", "formula": "\\begin{align*} \\eta ( x ) = ( e ^ { m _ 2 } - e ^ { m _ 3 } ) e ^ { m _ 1 x } + ( e ^ { m _ 3 } - e ^ { m _ 1 } ) e ^ { m _ 2 x } + ( e ^ { m _ 1 } - e ^ { m _ 2 } ) e ^ { m _ 3 x } , \\ \\ \\forall x \\in ( 0 , 1 ) . \\end{align*}"}
{"id": "782.png", "formula": "\\begin{align*} ( x \\cdot t ( t + 1 ) ) ' = x ^ 2 + 1 \\end{align*}"}
{"id": "2613.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x \\widetilde { g } \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "8822.png", "formula": "\\begin{align*} d \\tilde \\Phi _ t = - \\dot { C } _ { \\tau - t } \\nabla V _ { \\tau - t } ( \\tilde \\Phi _ t ) d t + \\dot { C } _ { \\tau - t } ^ { 1 / 2 } d B _ t , t \\in [ 0 , \\tau ] , \\end{align*}"}
{"id": "1140.png", "formula": "\\begin{align*} C _ { J ^ E } ( f ) ( k ) = C _ { - } { f ( J ^ E - I ) } = \\lim _ { s \\rightarrow k } \\int _ { \\Gamma _ E } \\frac { f ( s ) ( J ^ E ( s ) - I ) } { s - k } d s . \\end{align*}"}
{"id": "2470.png", "formula": "\\begin{align*} \\begin{pmatrix} I & 0 \\\\ Q & I \\end{pmatrix} \\begin{pmatrix} L & 0 \\\\ 0 & L ^ { - T } \\end{pmatrix} \\begin{pmatrix} I & P \\\\ 0 & I \\end{pmatrix} = \\begin{pmatrix} L & L P \\\\ Q L & Q L P + L ^ { - T } \\end{pmatrix} \\end{align*}"}
{"id": "1314.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha _ { k _ { n } } ) = \\sum _ { i = 0 } ^ { k _ { n } - 1 } J _ { 0 } ( \\alpha _ { i + 1 } , \\alpha _ { i } ) < 2 k _ { n } + C \\sqrt { k _ { n } } . \\end{align*}"}
{"id": "7788.png", "formula": "\\begin{align*} \\overline { \\dim } _ B \\big ( G ( h _ i ) \\big ) = \\varlimsup _ { \\delta \\rightarrow 0 } \\frac { \\log N _ { \\delta } ( G ( h _ i ) ) } { - \\log \\delta } \\le 2 - \\sigma ~ ~ ~ ~ ~ \\forall ~ ~ ~ i = 1 , 2 , \\cdots , M . \\end{align*}"}
{"id": "2112.png", "formula": "\\begin{align*} \\begin{aligned} & \\gamma ^ { ( k ) } ( c ) = c + ( 1 - c ) \\sum _ { i = 1 } ^ k \\frac { p _ c ^ { ( k ) } ( i ) } { k + 2 - i } = \\\\ & c + \\frac { 1 - c } { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } E \\frac 1 { k + 1 - X _ { k , c } } 1 _ { X _ { k , c } \\not \\in \\{ 0 , k \\} } + \\\\ & \\frac { 1 - c } { k + 1 } \\big ( 1 - \\frac 1 { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } ( 1 - c ^ k - ( 1 - c ) ^ k ) \\big ) , \\end{aligned} \\end{align*}"}
{"id": "5312.png", "formula": "\\begin{align*} \\check { A } = \\{ \\varphi ( g - h ) \\mid g , h \\in A \\} \\end{align*}"}
{"id": "8660.png", "formula": "\\begin{align*} P - \\alpha Q & = ( R + \\alpha S + \\alpha ^ 2 U ) ^ 2 \\\\ & = ( R ^ 2 - 2 m S U ) + \\alpha ( 2 R S - 2 k S U - m U ^ 2 ) + \\alpha ^ 2 ( S ^ 2 + 2 R U - k U ^ 2 ) \\end{align*}"}
{"id": "7435.png", "formula": "\\begin{align*} \\Big | [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( u ) \\Big | \\leq & c _ { \\gamma } \\int _ { 0 } ^ { 3 b _ G } \\frac { \\| \\Delta G \\| _ { \\infty } } { w ^ { \\gamma - 1 } } d w = H ^ G ( u ) , \\end{align*}"}
{"id": "3144.png", "formula": "\\begin{align*} \\partial _ z t > 0 , | \\partial _ z \\eta ^ L | : = | \\partial _ z \\eta ^ L ( t ( z ) , x _ 0 ) | = 1 \\quad t ( 0 ) = 0 . \\end{align*}"}
{"id": "2706.png", "formula": "\\begin{align*} B f ( i \\omega ) e ^ { - \\frac { \\pi } { 2 } \\omega ^ 2 } = e ^ { q ( i \\omega ) - \\frac { \\pi } { 2 } \\omega ^ 2 } = e ^ { - \\pi ( \\frac { 1 } { 2 } + A ' ) \\omega ^ 2 + 2 \\pi i b ' \\cdot \\omega + c ' } , \\end{align*}"}
{"id": "6267.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { l } | S _ i | \\right ) ^ 2 \\leq \\left ( \\sum _ { i = 1 } ^ { l } | S _ i | ^ 2 \\right ) l . \\end{align*}"}
{"id": "8136.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 8 } ( H G ) } c ( g ( \\gamma ) ) = 2 \\cdot 7 + 3 \\cdot 7 = 3 5 \\equiv 1 \\pmod 2 \\end{align*}"}
{"id": "378.png", "formula": "\\begin{align*} \\lambda _ \\alpha ^ \\gamma ( t ^ 1 , \\dots , t ^ s ) = \\lambda _ \\alpha ^ \\gamma ( t ^ 1 , \\dots , t ^ \\ell ) \\end{align*}"}
{"id": "5002.png", "formula": "\\begin{align*} i \\circ { \\mathcal { R } } \\circ i ^ { - 1 } = \\sigma \\end{align*}"}
{"id": "7199.png", "formula": "\\begin{align*} \\hat { G } ( t , \\xi ) & = \\eta ( | \\xi | ^ 2 ) \\hat { \\phi } ( \\xi ) | \\xi | \\left ( \\hat { \\psi } _ \\xi ( t | \\xi | ) - \\hat { \\psi } _ \\xi ( 0 ) \\right ) + \\eta ( | \\xi | ^ 2 ) \\hat { \\phi } ( \\xi ) | \\xi | \\hat { \\psi } _ \\xi ( 0 ) + ( 1 - \\eta ( | \\xi | ^ 2 ) ) \\hat { \\phi } ( \\xi ) | \\xi | \\hat { \\psi } _ \\xi ( t | \\xi | ) \\\\ & = R ^ { ( a ) } ( \\xi , t | \\xi | ) + R ^ { ( b ) } ( \\xi ) + R ^ { ( c ) } ( \\xi , t | \\xi | ) . \\end{align*}"}
{"id": "7204.png", "formula": "\\begin{align*} B _ { t , x } : = \\{ ( s , y ) \\in ( 0 , t ) \\times \\R ^ 3 : | t - s | + | x - y | \\leq \\frac 1 2 \\max \\{ 1 , \\check \\tau _ { t , x } , d _ { t , x } , | x ^ \\perp | \\} \\} . \\end{align*}"}
{"id": "2247.png", "formula": "\\begin{align*} \\| \\mathcal { O } _ { k , N } ( t _ m ) \\| _ \\mu ^ { 2 p _ 0 } \\leq & C \\left ( \\sum _ { j = 1 } ^ m \\int _ { t _ { j - 1 } } ^ { t _ j } \\| E _ { k , N } ^ { m - j } \\| ^ { \\frac { 2 p _ 0 } { 2 p _ 0 - 1 } } _ { \\mathcal { L } ( H ) } ( t _ m - s ) ^ { \\frac { 2 p _ 0 ( \\alpha - 1 ) } { 2 p _ 0 - 1 } } \\ , \\dd s \\right ) ^ { 2 p _ 0 - 1 } \\int _ 0 ^ { t _ m } \\| \\Theta _ { \\alpha } ( s ) \\| _ \\mu ^ { 2 p _ 0 } \\ , \\dd s \\\\ \\leq & C \\int _ 0 ^ { t _ m } \\| \\Theta _ { \\alpha } ( s ) \\| _ \\mu ^ { 2 p _ 0 } \\ , \\dd s . \\end{align*}"}
{"id": "8361.png", "formula": "\\begin{align*} I ^ { ( s + n h ) } \\subseteq \\prod _ { i = 1 } ^ n I ^ { ( s _ i + 1 ) } . \\end{align*}"}
{"id": "5324.png", "formula": "\\begin{align*} F _ i ( q _ i ) = q _ i \\ ; \\ ; \\ ; \\ ; q _ i \\in D _ i , \\ ; \\ ; U _ i ( s ) = s \\ ; \\ ; \\ ; \\ ; s \\geq 0 , \\ ; \\ ; i = 1 , \\ldots K . \\end{align*}"}
{"id": "2397.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum _ { k \\in F } s _ { k , N } = \\sum _ { k \\in F } g _ k . \\end{align*}"}
{"id": "3154.png", "formula": "\\begin{align*} f _ j ( x _ k ) = ( A x _ k - b ) ^ \\intercal S _ j ( S _ j ^ \\intercal A A ^ \\intercal S _ { i _ k } ) ^ \\dagger S _ { i _ k } ^ \\intercal ( A x _ k - b ) , ~ j = 1 , \\ldots . \\epsilon . \\end{align*}"}
{"id": "4417.png", "formula": "\\begin{align*} \\begin{array} { l l } R e ( z ^ * A d ^ j ) = c _ 1 , \\ ; \\ ; \\forall j \\in \\mathcal { J } \\\\ \\displaystyle \\sum _ { i = 1 } ^ { m } z _ i = 1 \\\\ \\lvert a r g z _ i \\rvert \\leq \\alpha , \\ ; \\ ; i = 1 , 2 , . . . , m \\end{array} \\end{align*}"}
{"id": "7762.png", "formula": "\\begin{gather*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } , \\ ; \\phi [ 0 ] = u [ 0 ] , \\end{gather*}"}
{"id": "6939.png", "formula": "\\begin{align*} a _ \\pi ( w , v ) = \\sum _ { E \\in { \\cal T } _ h } \\int _ E \\Pi _ { E , q } \\left ( \\mu \\nabla w \\right ) \\cdot \\nabla v + \\Pi _ { E , q - 1 } \\left ( \\boldsymbol { \\beta } \\cdot \\nabla w + \\sigma w \\right ) v \\ , , \\end{align*}"}
{"id": "2682.png", "formula": "\\begin{align*} \\widehat { g } ( \\omega ) = C \\ , e ^ { - \\gamma \\omega ^ 2 } \\prod _ { k = 1 } ^ M \\frac { e ^ { 2 \\pi i \\delta _ k \\omega } } { 1 + 2 \\pi i \\delta _ k \\omega } , M \\in \\N . \\end{align*}"}
{"id": "780.png", "formula": "\\begin{align*} X _ { n + 1 } + 1 = \\varepsilon ^ 4 _ n u _ n ^ 4 ( X _ n + 1 ) + \\varepsilon _ n ^ 3 u _ n ^ 4 ( a + b ) ( 1 + a b u _ n ^ 2 ) . \\end{align*}"}
{"id": "361.png", "formula": "\\begin{align*} \\rho \\in H ^ { 1 } ( [ 0 , 1 ] ; \\mathcal P ( G ) ) , m \\in L ^ 2 ( [ 0 , 1 ] ; \\mathcal S ^ { n \\times n } ) , \\ ; ( \\rho ( 0 ) , \\rho ( 1 ) ) = ( \\rho ^ a , \\rho ^ b ) \\end{align*}"}
{"id": "1110.png", "formula": "\\begin{align*} C _ { J ^ E } ( f ) ( k ) = C _ { - } { f ( J ^ E - I ) } = \\lim _ { s \\rightarrow k } \\int _ { \\Gamma _ E } \\frac { f ( s ) ( J ^ E ( s ) - I ) } { s - k } d s . \\end{align*}"}
{"id": "9429.png", "formula": "\\begin{align*} v ' _ t = v '' _ { r r } + \\frac { n + 1 } { r } v ' _ r + ( n + 2 ) v ^ 2 + 3 r v v ' _ r . \\end{align*}"}
{"id": "3056.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\pm y _ 1 \\ , , y _ 1 ^ \\prime = \\pm x _ 1 \\ , , x _ 2 ^ \\prime = y _ 2 \\ , , y _ 2 ^ \\prime = x _ 2 \\ , . \\end{align*}"}
{"id": "6078.png", "formula": "\\begin{align*} \\alpha _ { 3 } u ^ { j } & = \\alpha _ { 1 } w ^ { j } + \\varphi ^ { j } y _ { t } - \\psi ^ { j } y _ { s } , \\\\ \\alpha _ { 3 } v ^ { j } & = \\alpha _ { 2 } w ^ { j } - \\varphi ^ { j } x _ { t } + \\psi ^ { j } x _ { s } . \\end{align*}"}
{"id": "3441.png", "formula": "\\begin{align*} I \\ ! I _ { 1 2 } & = \\int _ { d ( y , u ) \\geqslant \\frac { 1 } { 2 } d ( x , y ) \\atop \\| u - y \\| > t } \\frac 1 { V ( x , u , t + d ( x , u ) ) } \\Big ( \\frac { t } { t + d ( x , u ) } \\Big ) ^ { \\varepsilon _ 0 } \\\\ & \\times \\frac 1 { V ( u , y , s + d ( u , y ) ) } \\bigg ( \\frac { s } { s + \\| u - y \\| } \\bigg ) ^ \\varepsilon \\bigg ( \\frac { s } { s + \\| u - y \\| } \\bigg ) ^ { \\varepsilon _ 0 - \\varepsilon } d \\omega ( u ) . \\end{align*}"}
{"id": "7436.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { x } \\sup _ { s \\in [ 0 , T ] } | [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x - 1 } { n } ) - [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] ( \\tfrac { x } { n } ) | = 0 . \\end{align*}"}
{"id": "182.png", "formula": "\\begin{align*} \\mu _ \\delta ( d x ) = C _ \\delta \\exp ( - | x | ^ \\delta ) d x = p _ \\delta ( x ) d x , \\end{align*}"}
{"id": "3490.png", "formula": "\\begin{align*} \\sum _ { y < m \\leq M } \\frac { 1 } { m ^ { s _ 1 } ( m + n ) ^ { s _ 3 } } & = \\frac { M ^ { - s _ 1 + 1 } ( M + n ) ^ { - s _ 3 } } { 1 - s _ 1 - s _ 3 } + \\frac { y ^ { - s _ 1 + 1 } ( y + n ) ^ { - s _ 3 } } { s _ 1 + s _ 3 - 1 } \\\\ & + \\frac { n s _ 3 } { s _ 1 + s _ 3 - 1 } \\int _ y ^ M \\frac { d u } { u ^ { s _ 1 } ( u + n ) ^ { s _ 3 + 1 } } + O \\left ( \\frac { 1 } { y ^ { \\sigma _ 1 } ( y + n ) ^ { \\sigma _ 3 } } \\right ) . \\end{align*}"}
{"id": "9279.png", "formula": "\\begin{align*} \\forall x ^ X , y ^ X \\left ( x = _ X y \\rightarrow \\forall k ^ 0 H ^ * \\left [ A x , A y , \\frac { 1 } { k + 1 } \\right ] \\right ) \\end{align*}"}
{"id": "744.png", "formula": "\\begin{align*} \\partial _ { x _ { 0 ; \\alpha } } = . \\end{align*}"}
{"id": "5956.png", "formula": "\\begin{align*} \\phi _ k = 0 , \\partial _ n \\phi _ k = 0 , \\Gamma ^ { s y m } , \\end{align*}"}
{"id": "7265.png", "formula": "\\begin{align*} \\Bigl ( \\sum \\alpha _ k \\Bigr ) \\circ \\beta & = \\sum \\alpha _ k ( \\beta ) , \\\\ \\Bigl ( \\prod ( 1 + \\alpha _ k ) \\Bigr ) \\circ \\beta & = \\prod ( 1 + \\alpha _ k ( \\beta ) ) , \\\\ \\alpha \\circ ( \\beta \\circ \\gamma ) & = ( \\alpha \\circ \\beta ) \\circ \\gamma . \\end{align*}"}
{"id": "5255.png", "formula": "\\begin{align*} S ( a ) _ { ( 1 ) } \\otimes S ( a ) _ { ( 2 ) } = S ( a _ { ( 2 ) } ) \\otimes S ( a _ { ( 1 ) } ) , a \\in A , \\end{align*}"}
{"id": "4771.png", "formula": "\\begin{align*} \\sum _ { \\xi , \\eta } \\langle X _ \\alpha , [ X _ \\xi , X _ \\eta ] \\rangle \\langle X _ \\alpha , [ \\tilde R X _ \\xi , \\tilde T X \\eta ] \\rangle = \\sum _ { \\xi \\triangle \\eta = \\alpha } r _ \\xi t _ \\eta \\langle X _ \\alpha , [ X _ \\xi , X _ \\eta ] \\rangle ^ 2 = 4 \\sum _ { \\xi \\triangle \\eta = \\alpha } r _ \\xi t _ \\eta . \\end{align*}"}
{"id": "8183.png", "formula": "\\begin{align*} N _ { d _ 0 } ' ( f , H ) : = - f + \\frac { 4 \\mu ( d _ 0 ) } { \\prod _ { q \\mid d _ 0 } ( q + 1 ) } \\sum _ { \\delta \\mid d _ 0 } \\frac { \\delta \\mu ( \\delta ) } { \\phi ( \\delta ) } S ( H _ \\delta , \\delta f ) . \\end{align*}"}
{"id": "3198.png", "formula": "\\begin{align*} \\frac { J _ \\infty ^ { \\mathsf { M P C } } - J _ \\infty ^ \\star } { J _ \\infty ^ \\star } J _ \\infty ^ { \\mathsf { M P C } } = \\sum _ { k = 0 } ^ \\infty \\ell \\left ( x _ k ^ \\mathsf { M P C } , \\tilde \\mu \\left ( x _ k ^ \\mathsf { M P C } \\right ) \\right ) , \\end{align*}"}
{"id": "1594.png", "formula": "\\begin{align*} \\nu _ { q + 1 } = a ( R _ q ) \\Theta _ { q + 1 } , w _ { q + 1 } = b ( R _ q ) W _ { q + 1 } \\end{align*}"}
{"id": "1146.png", "formula": "\\begin{align*} T _ { \\pm \\eta , j } : = f ^ { - 3 j / 2 } _ { \\pm \\eta } \\Delta _ { \\eta } Q ^ { - 1 } \\mathcal { K } _ j Q \\Delta _ { \\eta } ^ { - 1 } , j \\geqslant 1 , \\end{align*}"}
{"id": "399.png", "formula": "\\begin{align*} \\mathcal { P } _ { m } ( T ) : = \\left \\lbrace u \\in L ^ { \\infty } ( 0 , T ; H ^ { m } ) : u _ { t } \\in L ^ { 2 } ( 0 , T ; H ^ { m - 1 } ) , ~ \\nabla u \\in L ^ { 2 } ( 0 , T ; H ^ { m } ) \\right \\rbrace , \\end{align*}"}
{"id": "9056.png", "formula": "\\begin{align*} X ( \\theta ) = \\theta X ^ 0 + ( 1 - \\theta ) X ^ 1 , \\\\ Y ( \\theta ) = \\theta Y ^ 0 + ( 1 - \\theta ) Y ^ 1 , \\\\ Z ( \\theta ) = \\theta Z ^ 0 + ( 1 - \\theta ) Z ^ 1 , \\end{align*}"}
{"id": "3258.png", "formula": "\\begin{align*} 1 - 4 | \\lambda | ^ 2 h ( s ) ^ 2 = \\Big ( 1 - 2 h ( s ) ( s - t ) \\Big ) ^ 2 \\geq 0 . \\end{align*}"}
{"id": "7193.png", "formula": "\\begin{align*} \\check \\tau _ { t , x } = \\check \\tau _ { t , x _ 1 } : = [ t - \\tau _ { x _ 1 } ] _ + . \\end{align*}"}
{"id": "6539.png", "formula": "\\begin{align*} \\left | \\dfrac { \\sqrt { s _ n ^ 2 } } { \\sqrt { \\sigma _ n ^ 2 } } - 1 \\right | \\leq \\begin{cases} \\dfrac { C ' _ { \\alpha } } { n } & ( - 1 < \\alpha < 0 ) , \\\\ [ 4 m m ] \\dfrac { C ' _ { \\alpha } } { n ^ { 1 - 2 \\alpha } } & ( 0 < \\alpha < 1 / 2 ) , \\\\ [ 4 m m ] \\dfrac { C ' _ { 1 / 2 } } { \\log n } & ( \\alpha = 1 / 2 ) . \\end{cases} \\end{align*}"}
{"id": "2503.png", "formula": "\\begin{align*} \\iota ( \\mathbf { h } \\bullet \\mathbf { h } ' ) & = \\iota \\left ( x + x ' , \\omega + \\omega ' , \\tau + \\tau ' + \\tfrac { 1 } { 2 } ( x ' \\cdot \\omega - x \\cdot \\omega ' ) \\right ) \\\\ & = \\left ( x + x ' , \\omega + \\omega ' , \\tau + \\tau ' + \\tfrac { 1 } { 2 } ( x ' \\cdot \\omega - x \\cdot \\omega ' ) + \\tfrac { 1 } { 2 } ( x + x ' ) \\cdot ( \\omega + \\omega ' ) \\right ) . \\end{align*}"}
{"id": "9394.png", "formula": "\\begin{align*} \\tau _ t ( a _ 1 \\cdots a _ n ) = \\sum _ { j = 1 } ^ n t \\cdot \\tau ( a _ 1 \\cdots a _ { j - 1 } \\tau ' ( a _ j ) a _ { j + 1 } \\cdots a _ n ) + o ( t ) \\end{align*}"}
{"id": "7826.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { 2 n - 2 j - 3 } \\beta _ { i } e _ { i } , \\quad \\beta _ { 2 n - 2 j - 4 } , \\beta _ { 2 n - 2 j - 3 } \\neq 0 . \\end{align*}"}
{"id": "8023.png", "formula": "\\begin{align*} E _ { \\Sigma _ 0 } ( s , s ' ) = \\frac { 1 } { 2 } \\delta ' ( s - s ' ) . \\end{align*}"}
{"id": "2052.png", "formula": "\\begin{align*} f _ { r 1 } ' + \\frac { i \\beta \\eta } { 2 } \\omega ^ 2 _ r f _ { r 1 } - \\frac { i \\beta \\eta } { 4 } \\omega _ r f ' _ r - \\frac { i ( \\gamma - \\eta ^ 2 ) } { 2 \\beta \\eta } \\omega _ r f ' _ r = 0 \\end{align*}"}
{"id": "777.png", "formula": "\\begin{align*} W _ { n + 1 } = W _ n , \\varepsilon _ n , W _ n \\end{align*}"}
{"id": "6665.png", "formula": "\\begin{align*} R _ { 2 2 } = J _ { 2 1 } + O \\big ( ( X h k ) ^ { \\varepsilon } ( h , k ) ^ { 1 / 2 } Q ^ { - 9 6 } \\big ) . \\end{align*}"}
{"id": "8046.png", "formula": "\\begin{align*} \\chi ( u , v ) = ( - \\rho ( - u ) , \\widetilde { \\gamma } \\rho \\gamma ^ { - 1 } ( v ) ) , \\end{align*}"}
{"id": "7829.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { 3 j } \\alpha _ { i } e _ { i } , \\quad \\alpha _ { 3 j } \\neq 0 \\quad ~ \\langle x \\rangle _ { T ^ { * 3 } } = j + 1 . \\end{align*}"}
{"id": "6628.png", "formula": "\\begin{align*} 1 + \\frac { p ^ { w - 1 } - 1 } { p ( p - 1 ) } = 1 + O \\bigg ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\bigg ) . \\end{align*}"}
{"id": "6667.png", "formula": "\\begin{align*} V \\left ( \\frac { g _ 2 g _ 3 u } { H X } \\right ) = \\Psi \\Big ( \\frac { u } { X Q ^ { \\vartheta } } \\Big ) V \\left ( \\frac { g _ 2 g _ 3 u } { H X } \\right ) \\end{align*}"}
{"id": "2877.png", "formula": "\\begin{align*} \\Phi ( h ) = B ( h , h ) = 2 \\alpha _ + \\alpha _ - + B ( h _ \\perp , h _ \\perp ) , \\end{align*}"}
{"id": "738.png", "formula": "\\begin{align*} z _ { \\alpha } ^ { ( L + 1 ) } = J _ { x _ \\alpha } z _ { \\alpha } ^ { ( L + 1 ) } x _ \\alpha . \\end{align*}"}
{"id": "2431.png", "formula": "\\begin{align*} \\delta ( \\L ) = D ^ - ( \\L ) = D ^ + ( \\L ) . \\end{align*}"}
{"id": "1301.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda _ { ( \\infty , m ) } ( \\frac { M } { 2 } , \\Gamma ) | } { | \\Lambda ( \\frac { M } { 2 } ) | } = & \\lim _ { M \\to \\infty } \\frac { | \\Lambda _ { ( \\infty , m ) } ( \\frac { M } { 2 } ) | } { | \\Lambda ( M ) | } \\frac { | \\Lambda ( M ) | } { | \\Lambda ( \\frac { M } { 2 } ) | } \\\\ = & \\lim _ { M \\to \\infty } \\frac { | \\Lambda _ { ( \\infty , m ) } ( \\frac { M } { 2 } , \\Gamma ) | } { | \\Lambda ( M ) | } \\frac { M ^ 2 } { ( \\frac { M } { 2 } ) ^ 2 } = 0 \\end{align*}"}
{"id": "5812.png", "formula": "\\begin{align*} \\tilde { x } _ { m , n } = \\sqrt { \\eta _ { m k } P _ d } \\hat { g } _ { m k , n } ^ * s _ { k , n } , \\end{align*}"}
{"id": "5740.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( w '' ) \\cap V ( Q _ { i } ) | & = d _ { G } ( w '' ) - | N _ { G } ( w '' ) \\cap V ( B ) | - | N _ { G } ( w '' ) \\cap V ( H ) | \\\\ & - | N _ { G } ( w '' ) \\cap ( V ( T ' ) \\setminus V ( Q _ { i } ) ) | \\\\ & \\geq ( t + 2 ) - 1 - \\lceil \\frac { | V ( Q ' _ { i } ) | - 1 } { 2 } \\rceil - ( t - \\lfloor \\frac { | V ( Q _ { i } ) | } { 2 } \\rfloor ) \\\\ & \\geq 1 + \\lfloor \\frac { | V ( Q _ { i } ) | } { 2 } \\rfloor - \\lceil \\frac { | V ( Q ' _ { i } ) | } { 2 } \\rceil . \\\\ \\end{aligned} \\end{align*}"}
{"id": "5156.png", "formula": "\\begin{align*} H ( \\eta ) = h ( \\eta ) h '' ( \\eta ) - h ' ( \\eta ) ^ 2 , \\end{align*}"}
{"id": "6305.png", "formula": "\\begin{align*} p _ { K '' | K } ( k '' ) \\ ! = \\ ! \\binom { K } { k '' } \\xi ^ { k '' } ( 1 \\ ! - \\ ! \\xi ) ^ { K - k '' } \\ ! \\ ! , \\ ! \\ k '' \\ ! = \\ ! 0 , \\cdots , K , \\end{align*}"}
{"id": "3645.png", "formula": "\\begin{align*} A _ 2 & = A _ 1 \\left ( 1 + \\log ^ { 1 - B - \\alpha } x _ 0 + \\left ( \\frac { 2 } { \\log 2 } + 5 . 4 3 + 7 . 8 7 \\cdot 1 0 ^ { 1 2 } \\right ) \\frac { ( u ( x _ 0 ) ) ^ C \\log ^ { 1 - B } x _ 0 } { A _ 1 x _ 0 } \\right ) \\\\ & \\le 0 . 0 2 8 , \\end{align*}"}
{"id": "2610.png", "formula": "\\begin{align*} \\norm { \\widehat { f } } _ { M ^ { p } } ^ p & = \\norm { V _ { g _ 0 } \\widehat { f } } _ p ^ p \\leq C \\norm { V _ { \\widehat { g _ 0 } } \\widehat { f } } _ p ^ p \\\\ & = C \\iint _ { \\R ^ { 2 d } } | V _ { \\widehat { g _ 0 } } \\widehat { f } ( x , \\omega ) | ^ p \\ , d ( x , \\omega ) \\\\ & = C \\iint _ { \\R ^ { 2 d } } | V _ { g _ 0 } f ( - \\omega , x ) | ^ p \\ , d ( x , \\omega ) \\\\ & = C \\iint _ { \\R ^ { 2 d } } | V _ { g _ 0 } f ( x , \\omega ) | ^ p \\ , d ( x , \\omega ) \\leq C ' \\norm { f } _ { M ^ p } ^ p . \\end{align*}"}
{"id": "3142.png", "formula": "\\begin{align*} \\ell : = \\bigcup _ { t \\in ( - \\epsilon , \\epsilon ) } \\eta ^ L ( t , x _ * ) \\quad x _ * \\in \\mathbb { R } ^ 3 . \\end{align*}"}
{"id": "5284.png", "formula": "\\begin{align*} \\varphi = \\varphi ^ { \\vee \\vee } , \\psi = \\psi ^ { \\vee \\vee } . \\end{align*}"}
{"id": "7332.png", "formula": "\\begin{align*} \\frac { \\lambda } { T ^ 2 } \\le \\frac { \\lambda } { ( T - \\tilde { t } ) ^ 2 } = h - k \\le F ^ \\ast ( v ( \\tilde { y } , \\tilde { t } ) , p _ 2 , Y , K ) - F _ \\ast ( u ( \\tilde { x } , \\tilde { t } ) , p _ 1 , X , \\emptyset ) . \\end{align*}"}
{"id": "6468.png", "formula": "\\begin{align*} \\dfrac { S _ n } { \\sqrt { n \\log n } } \\stackrel { } { \\to } N ( 0 , 1 ) ( n \\to \\infty ) . \\end{align*}"}
{"id": "640.png", "formula": "\\begin{align*} \\begin{cases} \\ u ( x ) \\ = \\ F ( 2 x + 1 , \\xi ( 2 x + 1 ) ) \\\\ [ 8 p t ] \\ v ( x ) \\ = \\ G ( 2 x + 1 , \\xi ( 2 x + 1 ) ) \\\\ [ 8 p t ] \\ w ( x ) \\ = \\ H ( 2 x + 1 , \\xi ( 2 x + 1 ) ) \\end{cases} . \\end{align*}"}
{"id": "2734.png", "formula": "\\begin{align*} m _ { i , \\lambda } ( \\pi _ 1 ^ { - 1 } ( \\omega ) ) = 0 . \\end{align*}"}
{"id": "5453.png", "formula": "\\begin{align*} \\mathcal { M } ( \\tilde u _ 2 , a _ 2 , b _ 2 ) u - \\mathcal { M } ( \\tilde u _ 1 , a _ 1 , b _ 1 ) u = & [ \\mathcal { M } ( \\tilde u _ 2 , a _ 2 , b _ 2 ) u - \\mathcal { M } ( \\tilde u _ 1 , a _ 2 , b _ 2 ) u ] \\\\ & + [ \\mathcal { M } ( \\tilde u _ 1 , a _ 2 , b _ 2 ) u - \\mathcal { M } ( \\tilde u _ 1 , a _ 1 , b _ 2 ) u ] \\\\ & + [ \\mathcal { M } ( \\tilde u _ 1 , a _ 1 , b _ 2 ) u - \\mathcal { M } ( \\tilde u _ 1 , a _ 1 , b _ 1 ) u ] . \\end{align*}"}
{"id": "4973.png", "formula": "\\begin{align*} \\| u \\| _ { L _ t ^ q L _ x ^ r } = \\big ( \\int \\| u ( t , \\cdot ) \\| _ { L _ x ^ r } ^ q d t \\big ) ^ { \\frac { 1 } { q } } . \\end{align*}"}
{"id": "5520.png", "formula": "\\begin{align*} \\| S _ r \\Sigma - \\Sigma \\| _ { L _ 2 ^ 0 ( H ) } ^ 2 = \\sum _ { j > r } \\| \\Sigma ^ j \\| ^ 2 \\to \\infty \\end{align*}"}
{"id": "6728.png", "formula": "\\begin{align*} g _ 1 ( t ) \\mathcal { L } _ { K , \\mathfrak { s } _ 1 } ( \\alpha ) + \\cdots + g _ r ( t ) \\mathcal { L } _ { K , \\mathfrak { s } _ r } ( \\alpha ) = 0 \\end{align*}"}
{"id": "4195.png", "formula": "\\begin{align*} \\left ( \\psi ^ { \\beta } _ { z _ 0 + \\frac { 1 } { 2 } \\sum _ i x ^ 0 _ i y ^ 0 _ i } \\circ \\psi ^ H _ 1 \\right ) ( x ^ 0 _ i , y ^ 0 _ i , z ^ 0 ) = 0 \\end{align*}"}
{"id": "9026.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( \\rho , \\phi ) ( t ) & = - \\int _ { \\Omega } \\sum _ { i = 1 } ^ s D _ i ( x ) \\rho _ i | \\nabla ( \\log \\rho _ i + z _ i \\phi ) | ^ 2 d x \\\\ & + \\frac { 1 } { 2 } \\int _ { \\partial \\Omega } \\epsilon ( x ) \\left [ \\phi ( \\partial _ n \\phi ) _ t - ( \\partial _ n \\phi ) \\phi _ t \\right ] d s + \\frac { d } { d t } B ( \\phi ) . \\end{align*}"}
{"id": "8485.png", "formula": "\\begin{align*} f _ 0 & = 1 \\quad \\mbox { a n d } f _ k = z f _ { k - 1 } + z g _ { k - 1 } + z h _ { k - 1 } , k \\geq 1 , \\\\ g _ k & = z \\sum \\limits _ { \\ell \\geq k + 1 } f _ \\ell + z \\sum \\limits _ { \\ell \\geq k + 1 } g _ \\ell + z \\sum \\limits _ { \\ell \\geq k + 1 } h _ \\ell , k \\geq 0 , \\\\ h _ k & = z f _ k + z g _ k + z h _ k , k \\geq 0 . \\end{align*}"}
{"id": "1387.png", "formula": "\\begin{align*} h ( n _ 1 , E _ 1 ) - h ( n _ 2 , E _ 2 ) & = \\frac { n _ 1 ^ 2 } { n _ 1 + 1 } ( n _ 1 E _ 1 - \\alpha ) - \\frac { n _ 2 ^ 2 } { n _ 2 + 1 } ( n _ 2 E _ 2 - \\alpha ) \\\\ & = \\alpha \\left ( \\frac { n _ 2 ^ 2 } { n _ 2 + 1 } - \\frac { n _ 1 ^ 2 } { n _ 1 + 1 } \\right ) + \\left ( \\frac { n _ 1 ^ 3 E _ 1 } { n _ 1 + 1 } - \\frac { n _ 2 ^ 3 E _ 2 } { n _ 2 + 1 } \\right ) \\\\ & = : R _ 1 + R _ 2 . \\end{align*}"}
{"id": "9339.png", "formula": "\\begin{align*} H _ { 0 } = 0 , H _ { n } = 1 + \\frac { 1 } { 2 } + \\frac { 1 } { 3 } + \\cdots + \\frac { 1 } { n } , ( n \\in \\mathbb { N } ) , ( \\mathrm { s e e } \\ [ 4 , 5 , 1 5 ] ) . \\end{align*}"}
{"id": "8.png", "formula": "\\begin{align*} 0 & = \\lim _ { y \\to 0 } \\left ( \\gamma + \\ln \\left ( \\frac { 4 \\lvert y \\rvert } { \\epsilon _ 2 } \\right ) + \\psi \\left ( \\frac { 1 + \\lvert y \\rvert } { 2 } \\right ) + \\ln { \\frac { 2 } { \\lvert y \\rvert } } \\right ) , \\end{align*}"}
{"id": "7664.png", "formula": "\\begin{align*} \\begin{cases} - k _ { \\varepsilon } \\Delta \\tilde { \\Psi } _ { \\varepsilon } + | \\nabla \\tilde { \\Psi } _ { \\varepsilon } | ^ 2 - \\tilde { \\lambda } _ 0 \\underline { m } = 0 & \\Omega \\\\ \\tilde { \\Psi } _ { \\varepsilon } = - k _ { \\varepsilon } \\log ( w ) ( \\frac { \\mathbf { x } - \\mathbf { x } _ { \\varepsilon } } { k _ { \\varepsilon } } ) & \\partial \\Omega \\end{cases} \\end{align*}"}
{"id": "5783.png", "formula": "\\begin{align*} P = \\sum _ { i , j = 1 } ^ 3 \\Big ( \\mathcal { R } _ i \\mathcal { R } _ j ( u _ i u _ j ) + \\mathcal { R } _ i \\mathcal { R } _ j ( b _ i b _ j ) \\Big ) , \\end{align*}"}
{"id": "7781.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\sum \\limits _ { i = 0 } ^ { n - 1 } g _ 1 ( S _ i ( w ) ) & = \\int _ { \\Omega } g _ 1 ( w ) d \\mu , \\\\ \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\sum \\limits _ { i = 0 } ^ { n - 1 } g _ 2 ( S _ i ( w ) ) & = \\int _ { \\Omega } g _ 2 ( w ) d \\mu , \\end{align*}"}
{"id": "4749.png", "formula": "\\begin{align*} S _ 3 \\leq \\sum _ { \\gamma \\in \\mathcal { I } } | \\gamma | \\sum \\limits _ { k = 1 } ^ { | \\gamma | } \\tilde { M } ( T ) ^ { 2 k } \\bigg ( \\sum _ { \\substack { 0 \\leq | \\beta | \\leq | \\gamma | - k \\\\ \\beta < \\gamma } } a _ \\beta ( T ) \\bigg ( \\sum _ { \\substack { \\theta _ 1 + \\dots + \\theta _ k = \\gamma - \\beta \\\\ \\theta _ i \\not = \\mathbf { 0 } } } \\prod _ { i = 1 } ^ { k } \\| q _ { \\theta _ i } \\| _ { L ^ \\infty } \\bigg ) \\bigg ) ^ 2 ( 2 \\N ) ^ { - p \\gamma } . \\end{align*}"}
{"id": "7611.png", "formula": "\\begin{align*} 1 = f ( 1 , . . . , 1 ) \\leq f ( \\kappa ) + \\frac { \\partial f } { \\partial \\kappa _ { i } } ( \\kappa ) ( 1 - \\kappa _ { i } ) = \\sum _ { i } \\frac { \\partial f } { \\partial \\kappa _ { i } } ( \\kappa ) . \\end{align*}"}
{"id": "8215.png", "formula": "\\begin{align*} P _ { k + 1 - h } = \\frac { x _ { h + 1 } z } { P _ { h + 1 } } - \\frac { x _ h z } { P _ h } . \\end{align*}"}
{"id": "6675.png", "formula": "\\begin{align*} I _ { B \\cup \\{ - \\alpha \\} , \\{ \\beta \\} } ( n ) = { \\tau } _ { B \\smallsetminus \\{ \\beta \\} \\cup \\{ - \\alpha \\} } ( n ) \\end{align*}"}
{"id": "90.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 \\end{align*}"}
{"id": "1956.png", "formula": "\\begin{align*} f \\bullet ( g \\bullet h ) ( x ) & = ( g \\bullet h ) ( x ) f ( x ( { g \\bullet h } ) ( x ) ) = h ( x ) g ( x h ( x ) ) f ( x ( g \\bullet h ) ( x ) ) . \\end{align*}"}
{"id": "1381.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\left ( \\frac { \\tilde { E } _ 1 } { n _ 1 - 1 } - \\frac { \\tilde { E } _ 2 } { n _ 2 - 1 } \\right ) ( x ) = & \\frac { b _ 2 ( 0 ) - 1 } { A _ 2 } - \\frac { b _ 1 ( 0 ) - 1 } { A _ 1 } \\\\ = & \\frac { 1 } { 2 } \\left ( \\sqrt { \\alpha ^ 2 - 8 ( b _ 2 ( 0 ) - 1 ) } - \\sqrt { \\alpha ^ 2 - 8 ( b _ 1 ( 0 ) - 1 ) } \\right ) \\\\ = & \\frac { 2 } { \\sqrt { \\alpha ^ 2 - 8 ( \\eta - 1 ) } } \\left ( b _ 1 ( 0 ) - b _ 2 ( 0 ) \\right ) \\\\ \\leq & \\tilde { C } _ 0 \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } , \\end{align*}"}
{"id": "7519.png", "formula": "\\begin{align*} \\arg \\pi ^ { - \\frac { \\sigma + i T } { 2 } } = - \\left ( \\frac { \\log \\pi } { 2 } \\right ) T \\end{align*}"}
{"id": "8569.png", "formula": "\\begin{align*} \\mbox { f o r $ k \\geq 0 $ , } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = \\chi _ + ( x ) T ( k ) \\psi _ + ( x , k ) + \\chi _ - ( x ) \\left [ \\psi _ - ( x , - k ) + R _ - ( k ) \\psi _ - ( x , k ) \\right ] , \\\\ \\mbox { f o r $ k < 0 $ , } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = \\chi _ - ( x ) T ( - k ) \\psi _ - ( x , - k ) + \\chi _ + ( x ) \\left [ \\psi _ + ( x , k ) + R _ + ( - k ) \\psi _ + ( x , - k ) \\right ] . \\end{align*}"}
{"id": "234.png", "formula": "\\begin{align*} f _ j ( x ) = g _ j ( x ) + \\varepsilon f ( x ) , \\end{align*}"}
{"id": "5954.png", "formula": "\\begin{align*} \\nabla ^ { 2 } \\phi _ k & = 0 , \\Omega , \\\\ \\phi _ k & = \\phi _ k , \\Gamma ^ { F S } , \\\\ \\partial _ { n } \\phi _ k & = 0 , \\Gamma ^ b , \\\\ \\partial _ { n } \\phi _ k & = \\dot { x } _ k n _ k , \\Gamma ^ { b o d y } , \\end{align*}"}
{"id": "6560.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\frac { d } { d t } \\| \\omega ( t ) \\| _ { L ^ { p } } ^ { p } = & \\int _ { \\mathbb { R } ^ { 2 } } ( b \\cdot \\nabla j ) \\omega | \\omega | ^ { p - 2 } \\ , d x \\\\ \\leq & \\| b \\| _ { L ^ { \\infty } } \\| \\nabla j \\| _ { L ^ { p } } \\| \\omega \\| _ { L ^ { p } } ^ { p - 1 } . \\end{align*}"}
{"id": "4823.png", "formula": "\\begin{align*} \\| B _ { \\tilde { h } } Q u \\| _ { H ^ N _ { h \\tilde { h } } } = \\mathcal { O } ( h ^ \\infty \\tilde { h } ^ \\infty ) \\| u \\| _ { H ^ { - N } _ { h \\tilde { h } } } . \\end{align*}"}
{"id": "3913.png", "formula": "\\begin{align*} \\delta ( h ( S ) ) = f ( \\delta ( S ) ) , \\end{align*}"}
{"id": "4620.png", "formula": "\\begin{gather*} \\gamma _ { X \\otimes Y } = \\gamma _ X \\otimes \\gamma _ Y \\end{gather*}"}
{"id": "2060.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow + \\infty } \\| f \\| ^ 2 _ { q , \\lambda , t } = \\int _ { \\{ q \\Psi < - t \\} } | f | ^ 2 e ^ { - \\varphi _ 0 } , \\ \\forall t \\in [ 0 , + \\infty ) . \\end{align*}"}
{"id": "2993.png", "formula": "\\begin{align*} \\Theta \\mapsto S = - ( \\chi ^ + - \\Theta \\chi ^ - ) ^ { - 1 } ( \\chi ^ - - \\Theta \\chi ^ + ) , S \\mapsto \\Theta = ( \\chi ^ - + \\chi ^ + S ) ( \\chi ^ + + \\chi ^ - S ) ^ { - 1 } , \\end{align*}"}
{"id": "1765.png", "formula": "\\begin{align*} H P ^ { k } ( A ) : = \\lim _ { n \\to \\infty } H C _ \\lambda ^ { k + 2 n } ( A ) , \\ k = 0 , 1 . \\end{align*}"}
{"id": "395.png", "formula": "\\begin{align*} G _ { \\alpha } ( \\xi , w ) : = \\partial _ { x } ^ { \\alpha } ( \\xi w ) - \\xi \\partial _ { x } ^ { \\alpha } w , \\end{align*}"}
{"id": "3561.png", "formula": "\\begin{align*} u \\leqslant v & \\ , : \\Longleftrightarrow \\ , u _ i \\leq v _ i , \\ ; i = 1 , \\ldots , K \\ ; \\ ; \\ ; \\ ; u \\neq v , \\end{align*}"}
{"id": "7226.png", "formula": "\\begin{align*} K _ { s , t , x } ' & : = \\{ v : s < { \\mathcal T } _ { t , x _ 1 , v _ 1 } - 2 , | v | < \\frac { \\delta ^ { - \\beta } } 2 , \\check \\tau _ { t , x } \\langle v ^ \\perp \\rangle < \\langle x ^ \\perp \\rangle / 5 \\} , \\\\ K '' _ { s , t , x } & : = \\{ v : s < { \\mathcal T } _ { t , x _ 1 , v _ 1 } - 3 , | v | < \\frac { \\delta ^ { - \\beta } } 3 , \\check \\tau _ { t , x } \\langle v ^ \\perp \\rangle < \\langle x ^ \\perp \\rangle / 6 \\} . \\end{align*}"}
{"id": "7649.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ 0 = \\frac { \\int _ { \\R ^ N } | \\nabla \\bar { u } | ^ 2 } { \\int _ { \\R ^ N } \\tilde { m } _ 0 \\bar { u } ^ 2 } \\ ; . \\end{align*}"}
{"id": "1769.png", "formula": "\\begin{align*} H P ^ \\bullet ( \\mathbb { C } \\Gamma ) = \\prod _ { \\hat { \\gamma } \\in \\langle \\Gamma \\rangle ' } \\big ( H ^ \\bullet ( N _ \\gamma , \\mathbb { C } ) \\otimes H P ^ \\bullet ( \\mathbb { C } ) \\big ) \\times \\prod _ { \\hat { \\gamma } \\in \\langle \\Gamma \\rangle '' } H ^ \\bullet ( N _ \\gamma , \\mathbb { C } ) , \\end{align*}"}
{"id": "1379.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } n _ { i x } ( x ) = \\frac { 1 } { 4 } \\left ( \\alpha - \\sqrt { \\alpha ^ 2 - 8 \\left ( b _ i ( 0 ) - 1 \\right ) } \\right ) = : A _ i > 0 , i = 1 , 2 . \\end{align*}"}
{"id": "3644.png", "formula": "\\begin{align*} \\log t - C t \\log t u ' ( t ) & \\ge ( 1 - 4 \\cdot 1 0 ^ { - 6 } ) \\log t \\ge \\log ^ { 0 . 9 9 1 } t + 0 . 1 9 = \\log ^ { B + \\alpha - 1 } t + \\alpha . \\end{align*}"}
{"id": "6092.png", "formula": "\\begin{align*} m ^ { \\star } & = \\max \\left \\{ m _ { i } , \\ i = 1 , 2 , \\ldots , n \\right \\} , \\\\ \\mu ^ { \\star } & = \\max \\left \\{ \\mu _ { i } , \\ i = 1 , 2 , \\ldots , n \\right \\} , \\\\ \\mu _ \\star & = \\min \\left \\{ \\mu _ { i } , \\ i = 1 , 2 , \\ldots , n \\right \\} . \\end{align*}"}
{"id": "6562.png", "formula": "\\begin{align*} M _ k ( T ) : = \\int _ { 0 } ^ { T } \\left | \\zeta \\left ( \\tfrac { 1 } { 2 } + i t \\right ) \\right | ^ { 2 k } \\ , d t \\end{align*}"}
{"id": "3992.png", "formula": "\\begin{align*} \\widehat h ( t ) = \\rho ( t , 1 ) + \\widetilde p ( t ) , \\forall t \\in ( 0 , T ) , \\end{align*}"}
{"id": "4553.png", "formula": "\\begin{align*} f _ 2 & \\ge \\frac { h - 1 } { 2 } \\ge 1 + \\frac { 3 h - 3 } { 3 2 } \\ge 1 + h _ 1 = 1 + e _ 2 , \\\\ f _ 3 & \\ge \\frac { h + 5 } { 2 } \\ge 1 + \\frac { 3 h - 3 } { 3 2 } + \\frac { 3 h - 2 7 } { 8 } \\ge 1 + h _ 1 + \\max \\{ h _ 1 , \\dots , h _ 4 \\} \\ge 1 + e _ 2 + e _ 3 . \\end{align*}"}
{"id": "4285.png", "formula": "\\begin{align*} u _ { \\lambda } ( x , t ) = \\frac { 1 } { \\lambda } u \\left ( \\frac { x } { \\sqrt { \\lambda } } , \\frac { t } { \\lambda } \\right ) \\end{align*}"}
{"id": "8040.png", "formula": "\\begin{align*} \\left \\{ \\Phi ( f ) , \\Pi ( g ) \\right \\} ^ \\Sigma _ \\mathrm { c a n } = \\int _ \\Sigma f g \\ , \\mathrm { d } V _ \\Sigma . \\end{align*}"}
{"id": "4743.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - { \\mathcal L } \\right ) u ( t , x ) + { q ( x ) } \\cdot u ( t , x ) = f ( t , x ) , u ( 0 , x ) = g ( x ) , \\end{align*}"}
{"id": "9060.png", "formula": "\\begin{align*} \\mathcal { J } ( \\phi ( \\theta ) ) - \\theta \\mathcal { J } ( \\phi ^ 0 ) - ( 1 - \\theta ) \\mathcal { J } ( \\phi ^ 1 ) = & - \\frac { \\theta ( 1 - \\theta ) \\beta _ a } { 8 h ^ 2 } ( \\xi _ 0 - \\xi _ 1 ) ^ 2 \\\\ & - \\frac { \\theta ( 1 - \\theta ) } { 8 h } \\sum _ { j = 1 } ^ N \\epsilon _ j ( \\xi _ { j - 1 } - \\xi _ { j + 1 } ) ^ 2 \\\\ & - \\frac { \\theta ( 1 - \\theta ) } { 8 \\beta _ b } ( \\xi _ N + \\xi _ { N + 1 } ) ^ 2 = 0 . \\end{align*}"}
{"id": "8009.png", "formula": "\\begin{align*} \\widetilde { \\pi } _ { \\ell / r } \\circ \\chi = \\chi _ { \\ell / r } \\circ \\pi _ { \\ell / r } . \\end{align*}"}
{"id": "7019.png", "formula": "\\begin{align*} A ( z ) = \\prod _ { j = 1 } ^ { n } ( z - \\alpha _ j ) ^ { m _ k } , \\end{align*}"}
{"id": "114.png", "formula": "\\begin{align*} a ^ { n + 1 } = \\gamma _ 2 a ^ 2 + \\dots + \\gamma _ { n } a ^ { n } , \\end{align*}"}
{"id": "1973.png", "formula": "\\begin{align*} \\big ( g \\curvearrowright f \\big ) ( x ) : = f ( x g ( x ) ) = \\sum _ { w \\in \\N ^ * } f _ w ( x g ( x ) ) _ w \\in G ^ 0 \\end{align*}"}
{"id": "2122.png", "formula": "\\begin{align*} \\binom { c _ n k n } l \\sum _ { j = 1 } ^ { n - N } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } \\le C \\sum _ { j = 1 } ^ { n - N } ( \\frac j n ) ^ { c ' k n } . \\end{align*}"}
{"id": "2615.png", "formula": "\\begin{align*} f = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma \\pi ( \\gamma ) g , \\end{align*}"}
{"id": "2864.png", "formula": "\\begin{align*} h ( t ) = \\alpha _ + ( t ) \\mathcal { Y } _ + + \\alpha _ - ( t ) \\mathcal { Y } _ - + \\sum _ { j = 0 } ^ N \\beta _ j ( t ) Q _ j + h _ \\perp ( t ) , h _ \\perp \\in G _ \\perp ' , \\end{align*}"}
{"id": "5702.png", "formula": "\\begin{align*} w ( \\alpha ) = w ( \\beta ) + \\# ( \\mathbb { R } \\times \\gamma _ { 0 } \\cap { v } ) \\end{align*}"}
{"id": "357.png", "formula": "\\begin{align*} & \\dot { \\widetilde \\rho _ 2 } + \\widetilde m _ { 2 3 } = ( \\Sigma _ 2 - \\Sigma _ 3 ) \\dot W ^ { \\delta } , \\\\ & \\dot { \\widetilde \\rho _ 3 } + \\widetilde m _ { 3 2 } = ( \\Sigma _ 3 - \\Sigma _ 2 ) \\dot W ^ { \\delta } . \\end{align*}"}
{"id": "3797.png", "formula": "\\begin{align*} A = Z \\ , D _ A \\ , W ^ T , B = Z \\ , D _ B \\ , U ^ T , G = V \\ , D _ G \\ , W ^ T , \\end{align*}"}
{"id": "8942.png", "formula": "\\begin{align*} \\begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \\leq C | h | ^ 2 \\int _ 0 ^ T e ^ { - s } \\left ( 1 - \\frac { s } { T } \\right ) ^ 2 d s + C | h | ^ 2 \\int _ 0 ^ T \\frac { e ^ { - s } } { T } d s + C | h | ^ 2 \\int _ 0 ^ T \\frac { e ^ { - s } } { T ^ 2 } d s \\\\ & \\leq C | h | ^ 2 \\int _ 0 ^ T e ^ { - s } d s + C \\frac { | h | ^ 2 } { T } + C \\frac { | h | ^ 2 } { T } \\int _ 0 ^ 1 e ^ { - s T } d s \\\\ & \\leq C \\left ( 1 + \\frac { 1 } { T } \\right ) | h | ^ 2 , \\\\ \\end{aligned} \\end{align*}"}
{"id": "4978.png", "formula": "\\begin{align*} \\hat { 0 } _ { k - 1 } = \\{ \\bar { 0 } _ { 0 } , \\dots , \\bar { 0 } _ { k - 1 } \\} . \\end{align*}"}
{"id": "7588.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k - 2 s - 1 , l ) } | & \\leq \\dfrac { 1 } { 2 i } \\left ( | c _ { i - 1 , j } ^ { ( k - 2 s , l ) } | + | c _ { i + 1 , j } ^ { ( k - 2 s , l ) } | \\right ) \\\\ & \\le \\dfrac { 1 } { 2 i } \\frac { 4 V _ { k , l } } { \\pi ^ 2 j } \\left ( \\Gamma _ { 0 , - 1 } [ s ] ( i ) + \\Gamma _ { 0 , 1 } [ s ] ( i ) \\right ) = \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 j } \\Gamma _ { 1 , - 1 } [ s ] ( i ) . \\end{align*}"}
{"id": "3744.png", "formula": "\\begin{align*} ( A . R ) ( x , y ) = [ A , R ( x , y ) ] - R ( A ( x ) , y ) - R ( x , A ( y ) = 0 , \\ ; \\forall ( x , y ) \\in V ^ 2 , \\end{align*}"}
{"id": "2269.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\langle \\widehat { f } , \\widehat { g } \\rangle . \\end{align*}"}
{"id": "4102.png", "formula": "\\begin{align*} Y ^ { ( n ) } = \\Bigl ( \\int _ 0 ^ { v \\wedge \\tau _ n } X _ u | X _ u | ^ { p _ 0 - 2 } \\sigma ( X _ u ) \\ , d W _ u \\Bigr ) _ { v \\ge 0 } , n \\in \\N , \\end{align*}"}
{"id": "7037.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { L } | p _ { j } ( \\xi _ k ) | ^ 2 & = \\sum _ { j = 1 } ^ { \\infty } | p _ j ( \\xi _ k ) | ^ 2 - \\sum _ { j = L + 1 } ^ { \\infty } | p _ j ( \\xi _ k ) | ^ 2 \\\\ & \\geqslant \\delta ^ 2 - \\sum _ { j = \\ell _ k + 1 } ^ { \\infty } | p _ j ( \\xi _ k ) | ^ 2 \\\\ & \\geqslant \\delta ^ 2 - \\frac { \\delta ^ 2 } { 2 } = \\frac { \\delta ^ 2 } { 2 } > 0 . \\end{align*}"}
{"id": "2842.png", "formula": "\\begin{align*} x ( t ) = X ( t ) , \\forall t \\in D _ { \\delta _ 0 } , \\end{align*}"}
{"id": "7552.png", "formula": "\\begin{align*} N _ 0 ( T ) = \\frac { T } { 2 \\pi } \\log \\left ( \\frac { T } { 2 \\pi } \\right ) - \\frac { T } { 2 \\pi } + \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "826.png", "formula": "\\begin{align*} \\left | \\int _ { y _ 0 } ^ y y ^ { - \\beta } d y \\right | = \\frac { 1 } { \\beta - 1 } \\left | y _ 0 ^ { 1 - \\beta } - y ^ { 1 - \\beta } \\right | < R . \\end{align*}"}
{"id": "183.png", "formula": "\\begin{align*} f _ \\delta ( x ) = \\frac { 1 } { p _ \\delta ( x ) } \\int _ x ^ { + \\infty } g ( y ) p _ \\delta ( y ) d y = - \\frac { 1 } { p _ \\delta ( x ) } \\int _ { - \\infty } ^ x g ( y ) p _ \\delta ( y ) d y . \\end{align*}"}
{"id": "6475.png", "formula": "\\begin{align*} E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n \\log n } } \\right ) ^ { 2 m - 1 } \\right ] \\sim \\dfrac { 2 \\beta } { \\sqrt { \\pi } } \\cdot ( 2 m - 1 ) ! ! \\cdot ( \\log n ) ^ { - 1 / 2 } . \\end{align*}"}
{"id": "793.png", "formula": "\\begin{align*} \\| f \\| ^ p _ { \\theta , p } = \\int _ Z \\int _ Z \\frac { | f ( y ) - f ( x ) | ^ p } { \\nu ( B ( y , d ( x , y ) ) ) d ( x , y ) ^ { \\theta p } } d \\nu ( y ) d \\nu ( x ) < \\infty , \\end{align*}"}
{"id": "4545.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j } w _ H ( e _ i ) \\le w _ G ( \\phi ( e _ j ) ) . \\end{align*}"}
{"id": "6574.png", "formula": "\\begin{align*} \\mathcal { H } ( z , w ) = \\sqrt { \\pi } \\frac { \\Gamma ( \\tfrac { 1 - w } { 2 } ) \\Gamma ( \\tfrac { z } { 2 } ) \\Gamma ( \\tfrac { w - z } { 2 } ) } { \\Gamma ( \\tfrac { w } { 2 } ) \\Gamma ( \\tfrac { 1 - z } { 2 } ) \\Gamma ( \\tfrac { 1 - w + z } { 2 } ) } . \\end{align*}"}
{"id": "6991.png", "formula": "\\begin{align*} \\widehat { y _ 0 } = \\widehat { \\mu } + \\widehat { \\beta } ' x _ 0 . \\end{align*}"}
{"id": "514.png", "formula": "\\begin{align*} \\bar { \\psi } _ { 2 } ( \\alpha ; \\omega ) : = \\sum _ { i = 1 } ^ { \\kappa - 1 } \\psi _ { 2 } ( \\delta _ { i } ; \\delta _ { i + 1 } ) . \\end{align*}"}
{"id": "279.png", "formula": "\\begin{align*} \\chi _ { t } + \\beta \\chi \\chi _ { x } - \\chi _ { x x } = 0 , \\int _ { \\R } \\chi ( x , t ) d x = M . \\end{align*}"}
{"id": "6073.png", "formula": "\\begin{align*} h ^ { j } = L R \\cdot U ^ { j } \\end{align*}"}
{"id": "7538.png", "formula": "\\begin{align*} \\Re \\left ( \\frac { 1 } { \\rho } \\right ) = \\frac { \\beta } { \\beta ^ 2 + \\gamma ^ 2 } > 0 \\end{align*}"}
{"id": "8613.png", "formula": "\\begin{align*} \\mu ^ { \\# } _ { R , 1 } ( k , \\ell , m , n ) : = \\sum _ { ( A , B , C , D ) \\in \\mathcal { X } _ { R } } \\int \\overline { \\mathcal { K } ^ { \\# } _ { A } ( x , k ) } \\mathcal { K } ^ { \\# } _ { B } ( x , \\ell ) \\overline { \\mathcal { K } ^ { \\# } _ { C } ( x , m ) } \\mathcal { K } ^ { \\# } _ { D } ( x , n ) \\ , d x , \\\\ \\mathcal { X } _ { R } = : = \\{ S , R \\} ^ 4 \\smallsetminus \\{ ( S , S , S , S ) \\} , \\end{align*}"}
{"id": "2766.png", "formula": "\\begin{align*} \\| \\nabla u _ n \\| _ 2 ^ 2 = \\sum _ { j = 1 } ^ l \\| \\nabla U ^ j \\| _ 2 ^ 2 + \\| \\nabla r _ n ^ l \\| _ 2 ^ 2 + o _ n ( 1 ) . \\end{align*}"}
{"id": "137.png", "formula": "\\begin{align*} u _ { i + 1 } = f _ { i + 3 } ( a ) = ( a ^ 3 - a ^ 2 ) \\left ( a - \\frac 1 2 \\right ) ^ { i } \\ ; \\ ; ( 0 \\leq i \\leq n - 3 ) . \\end{align*}"}
{"id": "2675.png", "formula": "\\begin{align*} \\mathfrak { F } _ { ( \\alpha , \\beta ) } ( g ) = \\{ \\alpha \\Z ^ d \\times \\beta \\Z ^ d \\mid G ( g , \\alpha \\Z ^ d \\times \\beta \\Z ^ d ) \\} \\end{align*}"}
{"id": "5098.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial x } \\left [ \\sinh ( x ) - x - x ^ 3 \\right ] & = \\cosh ( x ) - 1 - 3 x ^ 2 , \\\\ \\tfrac { \\partial ^ 2 } { \\partial x ^ 2 } \\left [ \\sinh ( x ) - x - x ^ 3 \\right ] & = \\sinh ( x ) - 6 x = x \\left ( \\tfrac { \\sinh ( x ) } { x } - 6 \\right ) . \\\\ \\end{align*}"}
{"id": "9072.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\rho _ 1 = & \\partial _ x \\left ( \\partial _ x \\rho _ 1 + \\rho _ 1 \\partial _ x \\phi \\right ) , \\\\ \\partial _ t \\rho _ 2 = & \\partial _ x \\left ( \\partial _ x \\rho _ 2 - \\rho _ 2 \\partial _ x \\phi \\right ) , \\\\ - \\partial _ x ^ 2 \\phi = & \\rho _ 1 - \\rho _ 2 , \\end{aligned} \\end{align*}"}
{"id": "7992.png", "formula": "\\begin{align*} f _ i f _ j E _ n ^ { \\sigma } \\Pi _ n ^ { \\sigma } \\big ( f _ i ( ( 0 , 1 ) ^ d ) \\big ) \\cap \\Pi _ n ^ { \\sigma } \\big ( f _ j ( ( 0 , 1 ) ^ d ) \\big ) = \\emptyset . \\end{align*}"}
{"id": "6640.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { m } ) \\left ( 1 + \\frac { p ^ w } { p ^ { 2 } ( p - 1 ) } - \\frac { 1 } { p - 1 } \\right ) } { p ^ { m ( 3 - \\alpha - \\beta - w ) } } = \\frac { \\tau _ A ( p ) \\tau _ B ( p ) } { p ^ { 3 - \\alpha - \\beta - w } } + O \\left ( \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\right ) . \\end{align*}"}
{"id": "1800.png", "formula": "\\begin{align*} C C ^ k ( A , B ) : = C C ^ k ( A ) \\oplus C C ^ { k + 1 } ( B ) , \\end{align*}"}
{"id": "6557.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\omega + ( u \\cdot \\nabla ) \\omega = ( b \\cdot \\nabla ) j , \\\\ & \\partial _ { t } j + ( u \\cdot \\nabla ) j + \\mathcal { L } j = ( b \\cdot \\nabla ) \\omega + T ( \\nabla u , \\nabla b ) , \\endaligned \\right . \\end{align*}"}
{"id": "3414.png", "formula": "\\begin{align*} \\langle T \\phi , \\psi \\rangle & = \\langle K ( x , y ) , \\phi ( y ) \\psi ( x ) \\rangle = \\langle K ( x , y ) , \\chi _ 0 ( y ) \\phi ( y ) \\psi ( x ) \\rangle \\\\ & = \\langle K ( x , y ) , \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle + \\langle K ( x , y ) , \\chi _ 0 ( y ) \\phi ( x ) \\psi ( x ) \\rangle \\\\ & = : p + q , \\end{align*}"}
{"id": "676.png", "formula": "\\begin{align*} z _ { \\alpha } ^ { ( \\ell + 1 ) } : = \\begin{cases} b ^ { ( \\ell + 1 ) } + W ^ { ( \\ell + 1 ) } \\sigma ( z _ { \\alpha } ^ { ( \\ell ) } ) , & \\ell \\geq 1 \\\\ b ^ { ( 1 ) } + W ^ { ( 1 ) } x _ { \\alpha } , & \\ell = 0 \\end{cases} , W ^ { ( \\ell + 1 ) } \\in \\R ^ { n _ { \\ell + 1 } \\times n _ { \\ell } } , \\ , b ^ { ( \\ell + 1 ) } \\in \\R ^ { n _ { \\ell + 1 } } . \\end{align*}"}
{"id": "2210.png", "formula": "\\begin{align*} | v | _ \\alpha : = \\big ( \\sum _ { j = 1 } ^ \\infty \\lambda _ j ^ \\alpha | \\big < v , e _ j \\big > | ^ 2 \\big ) ^ { \\frac 1 2 } , \\ ; \\| v \\| _ \\alpha : = \\big ( | v | _ \\alpha ^ 2 + | \\big < v , e _ 0 \\big > | ^ 2 \\big ) ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "4644.png", "formula": "\\begin{align*} \\left . \\begin{aligned} M & = { } [ \\Delta ( { \\mathbb K } ) , \\Delta ( F ) ] \\mbox { a n d } \\\\ s & \\equiv 0 \\pmod { e ( G ) } \\mbox { w i t h } s \\ge N . \\end{aligned} \\right \\} \\end{align*}"}
{"id": "5760.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { h } _ { i j } = h \\left ( \\frac { \\partial } { \\partial x ^ { i } } , \\frac { \\partial } { \\partial x ^ { j } } \\right ) = h \\left ( \\frac { \\partial z ^ { k } } { \\partial x ^ i } \\frac { \\partial } { \\partial z ^ { k } } , \\frac { \\partial z ^ { l } } { \\partial x ^ j } \\frac { \\partial } { \\partial z ^ { l } } \\right ) = \\frac { \\partial z ^ { k } } { \\partial x ^ i } \\frac { \\partial z ^ { l } } { \\partial x ^ j } h _ { k l } . \\end{aligned} \\end{align*}"}
{"id": "9330.png", "formula": "\\begin{align*} \\mathcal { N } _ { C } ( \\Theta ) : = \\begin{bmatrix} H + \\rho I + \\Theta ^ { - 1 } & A ^ T \\\\ A & - \\delta I \\end{bmatrix} , \\end{align*}"}
{"id": "2048.png", "formula": "\\begin{align*} \\operatorname { R e } { \\lambda } \\left \\| x \\right \\| ^ 2 - \\operatorname { R e } \\left < T x , x \\right > = 0 . \\end{align*}"}
{"id": "9052.png", "formula": "\\begin{align*} \\rho _ { i j } - \\rho ^ n _ { i j } + d _ h ( m _ i ) _ j = 0 , \\end{align*}"}
{"id": "9128.png", "formula": "\\begin{align*} \\langle T ^ \\circ x - T ^ \\circ x ' , - T ^ \\circ x ' \\rangle & \\leq \\langle T ^ \\circ x - y ' , - T ^ \\circ x ' \\rangle + \\langle y ' - T ^ \\circ x ' , - T ^ \\circ x ' \\rangle \\\\ & = \\langle T ^ \\circ x - y ' , - T ^ \\circ x ' \\rangle \\\\ & \\leq \\norm { T ^ \\circ x - y ' } \\norm { T ^ \\circ x ' } \\\\ & \\leq \\frac { 1 } { ( k + 1 ) ^ 2 } . \\end{align*}"}
{"id": "931.png", "formula": "\\begin{align*} a ^ { + } ( \\hat { n } ) _ { k , \\lambda } a & = a ^ { + } \\bigg ( \\sum _ { p = 0 } ^ { k } S _ { 2 , \\lambda } ( k , p ) ( a ^ { + } ) ^ { p } a ^ { p } \\bigg ) a \\\\ & = \\sum _ { p = 0 } ^ { k } S _ { 2 , \\lambda } ( k , p ) ( a ^ { + } ) ^ { p + 1 } a ^ { p + 1 } . \\end{align*}"}
{"id": "8147.png", "formula": "\\begin{align*} D _ { d _ 0 } ( p , H ) = 1 + o ( 1 ) \\end{align*}"}
{"id": "1867.png", "formula": "\\begin{align*} a ^ 2 D ^ { n + 1 } ( x ) = ( 1 + x ^ 2 ) \\sum _ { k = 0 } ^ n { n \\choose k } D ^ k ( a ) D ^ { n - k } ( a ) . \\end{align*}"}
{"id": "2189.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } K ( x ) f ( x , w ^ - ) w ^ - d x & = A ^ - ( w ) . \\end{align*}"}
{"id": "8124.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } | \\{ \\lambda \\in \\Lambda \\mid \\lambda \\supset e \\cup e _ { i } \\} | \\equiv 0 \\pmod m , \\end{align*}"}
{"id": "3022.png", "formula": "\\begin{align*} N ^ 0 _ \\beta = \\sum _ { k | \\beta } \\frac { n ^ 0 _ { \\beta / k } } { k ^ 3 } \\end{align*}"}
{"id": "5365.png", "formula": "\\begin{align*} \\frac { \\left \\langle \\tilde a _ i , g \\right \\rangle - b _ i } { \\left \\langle \\tilde a _ i , c \\right \\rangle } = \\min \\left \\lbrace \\sigma \\in \\mathbb { R } _ { \\geqslant 0 } \\ ; \\middle | \\ ; x - \\sigma c \\in H _ i ^ + \\right \\rbrace . \\end{align*}"}
{"id": "4534.png", "formula": "\\begin{align*} T ( x ; k , r ) & = \\sum _ { u ( n r _ { 1 } + k _ { 1 } ) \\leq x } a ( u ( n r _ { 1 } + k _ { 1 } ) ) = \\sum _ { \\stackrel { n \\leq x / u } { n \\equiv k _ { 1 } ( m o d r _ { 1 } ) } } a ( u n ) \\\\ & = \\frac { 1 } { \\varphi ( r _ { 1 } ) } \\sum _ { \\chi \\ , ( m o d \\ , r _ { 1 } ) } \\bar { \\chi } ( k _ { 1 } ) \\left ( \\sum _ { n \\leq x / u } \\chi ( n ) a ( u n ) \\right ) , \\end{align*}"}
{"id": "3802.png", "formula": "\\begin{align*} A - A _ k = \\widehat { Z } \\ , \\widehat { D } _ A \\ , \\widehat { W } ^ T , B - B _ k = \\widehat { Z } \\ , \\widehat { D } _ B \\ , \\widehat { U } ^ T , G - G _ k = \\widehat { V } \\ , \\widehat { D } _ G \\ , \\widehat { W } ^ T . \\end{align*}"}
{"id": "3350.png", "formula": "\\begin{align*} & \\phi \\circ T = T ^ { ' } \\circ \\psi , \\\\ & \\psi \\theta ( x , y ) = \\theta ( \\phi ( x ) , \\phi ( y ) ) \\psi , \\forall x , y \\in L . \\end{align*}"}
{"id": "683.png", "formula": "\\begin{align*} K _ * \\geq 0 C _ b = 0 , \\ , C _ W = \\frac { 2 } { a _ + ^ 2 + a _ - ^ 2 } . \\end{align*}"}
{"id": "6579.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) = \\frac { 1 } { 2 } \\sum _ { \\substack { 1 \\leq q < \\infty \\\\ ( q , h k ) = 1 } } W \\left ( \\frac { q } { Q } \\right ) \\sum _ { \\substack { 1 \\leq m , n < \\infty \\\\ ( m n , q ) = 1 } } \\frac { \\tau _ A ( m ) \\tau _ B ( n ) } { \\sqrt { m n } } V \\left ( \\frac { m } { X } \\right ) V \\left ( \\frac { n } { X } \\right ) \\sum _ { \\substack { c , d \\geq 1 \\\\ c d = q \\\\ d | m h \\pm n k } } \\phi ( d ) \\mu ( c ) . \\end{align*}"}
{"id": "5096.png", "formula": "\\begin{align*} \\frac { \\sinh ( x ) } { x } = \\frac { \\sum \\limits _ { k = 0 } ^ \\infty \\tfrac { x ^ { 2 k + 1 } } { ( 2 k + 1 ) ! } } { x } = \\sum \\limits _ { k = 0 } ^ \\infty \\underbrace { \\tfrac { x ^ { 2 k } } { ( 2 k + 1 ) ! } } _ { \\nearrow } . \\end{align*}"}
{"id": "8765.png", "formula": "\\begin{align*} \\bar { \\Delta } ' _ i : = \\bigl \\{ ( z _ i , \\delta _ i , w ) \\bigm | ( z _ i , \\delta _ i ) \\in \\bar { \\Delta } _ i , \\ z _ { i ' \\tau ( i ' , t ) - 1 } \\geq w _ { i ' j ' } ^ t \\geq z _ { i ' \\tau ( i ' , t ) } t < t ' \\bigr \\} . \\end{align*}"}
{"id": "9472.png", "formula": "\\begin{align*} I _ { H _ s } = ( p _ 1 ( { \\bf \\underline { t } } ) , \\ldots , p _ s ( { \\bf \\underline { t } } ) ) , \\end{align*}"}
{"id": "5297.png", "formula": "\\begin{align*} \\kappa ^ n ( a ) \\otimes b ^ * \\rho ^ { n } ( b ) = \\sum _ { i = 1 } ^ n ( ( 1 \\otimes b ^ * ) \\Delta ( p _ i ) ) ( 1 \\otimes \\rho ^ n ( q _ i ) ) . \\end{align*}"}
{"id": "1330.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , p _ { i + 1 } ) } , \\hat { \\alpha } \\cup { ( \\gamma , p _ { i + 1 } - p _ { i } ) } \\cup { ( \\delta _ { 1 } , 1 ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 4 . \\end{align*}"}
{"id": "3551.png", "formula": "\\begin{align*} F ( x , t ) = ( x , ( a + t ) v ) \\end{align*}"}
{"id": "109.png", "formula": "\\begin{align*} U ^ i V ^ j = 0 \\mbox { f o r a l l } i \\geq 1 \\mbox { a n d } j \\geq 2 \\end{align*}"}
{"id": "6006.png", "formula": "\\begin{align*} A _ { n } ^ { \\lambda , \\beta } ( x ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\Bigg [ \\sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ { i } i ! B _ { n - k , i } ( ( M _ { \\lambda , \\beta } ( j ) ) _ { j = 1 } ^ { n - k - i + 1 } ) \\Bigg ] x ^ { k } . \\end{align*}"}
{"id": "7769.png", "formula": "\\begin{align*} \\Box \\tilde \\phi _ 1 = \\left ( | \\tilde \\phi _ { 1 , t } | ^ 2 - | \\tilde \\phi _ { 1 , x } | ^ 2 \\right ) \\tilde \\phi _ { 1 } + \\mathbf { 1 } _ { \\omega } f _ { 0 } ^ { \\tilde \\phi ^ { \\perp } _ 1 } + e _ 0 , \\ ; \\tilde \\phi _ 1 [ 0 ] = ( a , b ) \\end{align*}"}
{"id": "3871.png", "formula": "\\begin{align*} S _ H = \\{ t _ 1 - t _ 2 , \\dots , t _ { n - 1 } - t _ n , t _ { n - 1 } + t _ n \\} . \\end{align*}"}
{"id": "906.png", "formula": "\\begin{align*} ( a ^ { + } a ) _ { k , \\lambda } = \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) ( a ^ { + } ) ^ { l } a ^ { l } , \\end{align*}"}
{"id": "6790.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\ & f ( x ) \\\\ \\enspace & A x = b . \\end{aligned} \\end{align*}"}
{"id": "1089.png", "formula": "\\begin{align*} m ^ { ( 1 ) } _ { + } ( x , t , k ) = m ^ { ( 1 ) } _ { - } ( x , t , k ) J ^ { ( 1 ) } ( x , t , k ) , k \\in \\mathbb { R } , \\end{align*}"}
{"id": "390.png", "formula": "\\begin{align*} A ^ { 0 } ( U ) V _ { t } + A ^ { i } ( U ) \\partial _ { i } V - B ^ { i j } ( U ) \\partial _ { i } \\partial _ { j } V + D ( U ) V = F ( U ; D _ { x } U ) \\end{align*}"}
{"id": "4294.png", "formula": "\\begin{align*} Q '' ( \\xi ) + \\frac { d + 1 } { \\xi } Q ' _ \\xi - 3 ( d - 2 ) Q ^ 2 - ( d - 2 ) \\xi ^ 2 Q ^ 3 = 0 \\end{align*}"}
{"id": "7177.png", "formula": "\\begin{align*} \\nabla _ v \\mu ( v ) = - v \\psi ( v ) , \\end{align*}"}
{"id": "3322.png", "formula": "\\begin{align*} a _ i & = \\lfloor i \\overline { s } \\rfloor - \\lceil i \\overline { u } \\rceil + 1 \\\\ b _ { - i } & = \\lfloor - i \\overline { t } \\rfloor - \\lceil - i \\overline { u } \\rceil + 1 \\end{align*}"}
{"id": "605.png", "formula": "\\begin{align*} ( n , m ) \\ = \\ n ^ m \\end{align*}"}
{"id": "5902.png", "formula": "\\begin{align*} \\psi _ { Q _ { J - 2 A } } ( x ) = - \\frac { x } { 2 } \\ , \\Big ( \\det M _ 1 - x \\det M _ 2 \\Big ) , \\end{align*}"}
{"id": "8853.png", "formula": "\\begin{align*} H X ^ q ( \\R ^ n , \\R ) = \\begin{cases} \\R & q = n \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "2583.png", "formula": "\\begin{align*} \\widetilde { f } = \\norm { g } _ 2 ^ { - 2 } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "7025.png", "formula": "\\begin{align*} \\| R \\| ^ 2 + \\sum _ { j = 1 } ^ L \\| S _ j \\| ^ 2 = \\| R \\| ^ 2 + \\sum _ { j = 1 } ^ L | y _ j | ^ 2 \\| S \\| ^ 2 = \\| R \\| ^ 2 + \\| S \\| ^ 2 . \\end{align*}"}
{"id": "4880.png", "formula": "\\begin{align*} z ^ 2 ( z - 1 ) y '' ( z ) = - n ( n + 1 ) y ( z ) . \\end{align*}"}
{"id": "5733.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( v ) \\cap ( V ( G ) \\setminus ( V ( B ) \\cup \\{ u _ { l } \\} ) ) | = d _ { G } ( v ) - | N _ { G } ( v ) \\cap V ( B ) | - | N _ { G } ( v ) \\cap \\{ u _ { l } \\} | \\geq t \\\\ \\end{aligned} \\end{align*}"}
{"id": "573.png", "formula": "\\begin{align*} \\phi ( x ) ^ 2 \\ - \\ ( 1 - x ^ 2 ) \\ = \\ 0 , \\end{align*}"}
{"id": "6783.png", "formula": "\\begin{align*} \\tau _ { r _ { q _ y } + 1 } ^ { v _ { q _ { y - 1 } } + 1 } \\cdot \\tau _ { r _ { q _ y + 1 } } ^ { - ( v _ { q _ y } + 1 ) } = \\tau _ { r _ { q _ 1 } + 1 } ^ { v _ { q _ { y - 1 } } - v _ { q _ y } } i _ { k _ { p _ 0 + 1 } } = v _ { q _ { y - 1 } } - v _ { q _ y } \\end{align*}"}
{"id": "1176.png", "formula": "\\begin{align*} m ^ { P C } _ { \\eta } = I + \\frac { 1 } { \\zeta } \\left ( \\begin{array} { c c } 0 & - i \\beta ^ { ( \\eta ) } _ { 1 2 } \\\\ i \\beta ^ { ( \\eta ) } _ { 2 1 } & 0 \\end{array} \\right ) + \\mathcal { O } ( \\zeta ^ { - 2 } ) . \\end{align*}"}
{"id": "8396.png", "formula": "\\begin{align*} \\Gamma _ t ( \\delta ) \\triangleq \\{ y \\in \\Omega : \\hbox { d i s t } ( y , \\Gamma _ t ) < \\delta \\} , \\Gamma ( \\delta ) = \\bigcup _ { t \\in [ 0 , T _ 0 ] } \\Gamma _ t ( \\delta ) \\times \\{ t \\} , \\end{align*}"}
{"id": "5834.png", "formula": "\\begin{align*} E _ 1 ( H , r , w ) : = & \\left \\{ \\exists \\ , i \\in \\left \\{ 0 , \\dots , m \\right \\} : \\ y _ i \\big ( \\tfrac { H } { r } , r \\big ) \\right \\} , \\\\ E _ 2 ( H , r , w ) : = & \\bigcup _ { \\substack { 0 \\leq i \\leq m \\\\ 0 \\leq j \\leq r - 1 } } \\left \\{ X _ { ( j + 1 ) \\frac { H } { r } } ^ { y _ i } - X _ { j \\frac { H } { r } } ^ { y _ i } \\geq \\left ( v _ + + \\frac { \\theta } { 2 r } \\right ) \\frac { H } { r } \\right \\} . \\end{align*}"}
{"id": "2502.png", "formula": "\\begin{align*} \\iota ( x , \\omega , \\tau ) = ( x , \\omega , \\tau + \\tfrac { 1 } { 2 } x \\cdot \\omega ) \\end{align*}"}
{"id": "7975.png", "formula": "\\begin{align*} \\mathrm { P C } ( \\tilde T ) : = \\bigotimes _ { \\tilde u } \\mathrm { P C } _ { \\tilde u } \\otimes \\bigotimes _ { \\tilde w } \\mathrm { P C } _ { \\tilde w } \\ , . \\end{align*}"}
{"id": "1270.png", "formula": "\\begin{align*} \\mathrm { E C H } ( Y , \\lambda , \\Gamma ) : = \\bigoplus _ { * : \\ , \\ , \\mathbb { Z } \\ , \\ , \\mathrm { g r a d i n g } } \\mathrm { E C H } _ { * } ( Y , \\lambda , \\Gamma ) . \\end{align*}"}
{"id": "8221.png", "formula": "\\begin{align*} P _ 1 + P _ 2 + \\cdots + P _ k = 1 - \\frac { z F _ { k , 1 } ( A ) } { A } \\end{align*}"}
{"id": "9208.png", "formula": "\\begin{align*} J ^ A _ \\gamma : = ( I d + \\gamma A ) ^ { - 1 } . \\end{align*}"}
{"id": "8786.png", "formula": "\\begin{align*} \\lambda _ { i j } = z _ { i j } - z _ { i j + 1 } j = 0 , \\ldots , n _ i - 1 \\lambda _ { i n _ i } = z _ { i n _ i } , \\end{align*}"}
{"id": "5605.png", "formula": "\\begin{align*} \\mathfrak { e } : = \\{ Z \\in \\Xi ~ | ~ \\mathcal { L } _ Z \\nabla _ X Y = \\nabla _ { [ Z , X ] } Y + \\nabla _ X [ Z , Y ] X , Y \\in \\Xi \\} \\end{align*}"}
{"id": "2557.png", "formula": "\\begin{align*} \\widehat { S } _ { W , m } ^ { - 1 } = \\widehat { S } _ { W ^ * , m ^ * } W ^ * ( x , x ' ) = - W ( x ' , x ) , m ^ * = d - m . \\end{align*}"}
{"id": "4800.png", "formula": "\\begin{align*} \\phi ^ 2 = s ^ 2 \\chi _ A + t ^ 2 \\chi _ B + \\chi _ C . \\end{align*}"}
{"id": "3082.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 3 + x _ 1 ^ 3 y _ 2 ^ 3 + x _ 2 ^ 3 y _ 1 ^ 3 - x _ 2 ^ 3 y _ 2 ^ 3 = 0 \\ , , \\end{align*}"}
{"id": "96.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 3 / 2 } \\cdot ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) ( 2 ^ { n _ { 2 , \\nu _ 2 } - 2 } - 1 ) . \\end{align*}"}
{"id": "3116.png", "formula": "\\begin{align*} z ^ 3 y ^ 2 + a z y ^ 4 + x ^ 5 + b y ^ 5 = 0 \\ , . \\end{align*}"}
{"id": "8535.png", "formula": "\\begin{align*} u \\left ( t , x \\right ) = \\frac { e ^ { i \\frac { x ^ { 2 } } { 4 t } } } { \\sqrt { - 2 i t } } \\exp \\left ( - \\frac { i } { 2 } \\left | W _ { + \\infty } \\left ( - \\frac { x } { 2 t } \\right ) \\right | ^ { 2 } \\log t \\right ) W _ { + \\infty } \\left ( - \\frac { x } { 2 t } \\right ) + \\mathcal { O } \\left ( \\varepsilon t ^ { - \\frac { 1 } { 2 } - \\delta } \\right ) , t \\geq 1 . \\end{align*}"}
{"id": "7132.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } K _ H ( t , s ) = c _ H \\Big ( H - \\frac { 1 } { 2 } \\Big ) \\Big ( \\frac { 1 } { 2 } \\Big ) ^ { H - \\frac { 1 } { 2 } } \\Big ( t - s \\Big ) ^ { H - \\frac { 3 } { 2 } } . \\end{align*}"}
{"id": "6473.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n / ( 1 - 2 \\alpha ) } } \\right ) ^ k \\right ] = \\mu _ k ( k \\in \\mathbb { N } ) , \\end{align*}"}
{"id": "2386.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi 2 k / N ) & = \\Re \\left ( \\sum _ { k = 0 } ^ { N - 1 } e ^ { 2 \\pi i 2 k / N } \\right ) \\\\ & = \\frac { 1 - \\left ( e ^ { 4 \\pi i / N } \\right ) ^ N } { 1 - e ^ { 4 \\pi i / N } } = 0 , \\end{align*}"}
{"id": "7243.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } f ( t ) = \\lim _ { z \\downarrow 0 } z \\tilde f ( - i z ) , \\end{align*}"}
{"id": "2944.png", "formula": "\\begin{align*} \\forall \\ell \\in \\{ 1 , \\ldots , k - 1 \\} ~ \\exists \\ell ' \\neq \\ell : p _ { \\ell , 1 } = p _ { \\ell ' , 2 } , \\end{align*}"}
{"id": "7522.png", "formula": "\\begin{align*} \\log \\Gamma \\left ( \\frac { \\sigma + i T } { 2 } \\right ) = \\left ( \\frac { \\sigma - 1 } { 2 } + \\frac { i T } { 2 } \\right ) \\log \\left ( \\frac { \\sigma + i T } { 2 } \\right ) - \\frac { \\sigma + i T } { 2 } - \\frac { 1 } { 2 } \\log ( 2 \\pi ) + o ( 1 ) \\end{align*}"}
{"id": "4192.png", "formula": "\\begin{align*} H _ i ( x ) : = \\begin{cases} | \\phi ( x _ i ) | H ( R _ i \\phi ( x ) ) & ; \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "3587.png", "formula": "\\begin{align*} \\| u \\| = \\sup \\left \\{ \\left | \\sum _ { i = 1 } ^ n x ^ * ( x _ i ) y ^ * ( y _ i ) \\right | : x ^ * \\in B _ { X ^ * } , y ^ * \\in B _ { Y ^ * } \\right \\} , \\end{align*}"}
{"id": "3888.png", "formula": "\\begin{align*} w _ 1 ( M ) + ( a - b ) ( s _ 1 - t _ { b - 1 } ) & = O ( \\geq 0 ) - \\beta _ { b - 1 } - \\dots - \\beta _ { n - 2 } - \\frac { 1 } { 2 } ( \\beta _ { n - 1 } + \\beta _ n ) \\\\ & + O ( \\geq 0 ) - \\gamma _ { a - 1 } - \\dots - \\gamma _ { n - 2 } - \\frac { 1 } { 2 } ( \\gamma _ { n - 1 } + \\gamma _ n ) . \\end{align*}"}
{"id": "4079.png", "formula": "\\begin{align*} \\| \\xi - ( x , g ( x ) ) \\| ^ 2 = & \\| \\xi _ x - x \\| ^ 2 + | \\xi _ y - g ( \\xi _ x ) | ^ 2 + \\\\ & + | g ( \\xi _ x ) - g ( x ) | ^ 2 + 2 ( \\xi _ y - g ( \\xi _ x ) ) \\cdot ( g ( \\xi _ x ) - g ( x ) ) . \\end{align*}"}
{"id": "5732.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( v ) \\cap ( V ( T ' ) \\backslash \\{ u _ { 1 } , \\cdots , u _ { s } \\} ) | & = | N _ { G } ( v ) \\cap ( Y \\backslash \\{ u _ { 1 } , \\cdots , u _ { s } \\} ) | \\\\ & \\leq | Y | - \\lfloor \\frac { s } { 2 } \\rfloor \\\\ & \\leq t - \\lfloor \\frac { s } { 2 } \\rfloor \\end{aligned} \\end{align*}"}
{"id": "2456.png", "formula": "\\begin{align*} ( J ) = ( S ^ T J S ) = \\det ( S ) ( J ) \\Longrightarrow \\det ( S ) = 1 . \\end{align*}"}
{"id": "1247.png", "formula": "\\begin{align*} \\mu ( E _ { \\mu } ^ { [ \\alpha _ 1 , \\gamma _ 1 ] , \\varepsilon } ) = 1 . \\end{align*}"}
{"id": "2348.png", "formula": "\\begin{align*} \\Delta _ f x _ k \\Delta _ f p _ k \\geq \\frac { \\hbar } { 2 } , k = 1 , \\ldots , d . \\end{align*}"}
{"id": "803.png", "formula": "\\begin{align*} \\int _ { B ( x , r ) } | \\nabla v | ^ { p - 2 } \\nabla v \\cdot \\nabla ( u - v ) \\ , d \\mu = 0 . \\end{align*}"}
{"id": "3255.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } s ( r , t ) t & = r ^ 2 - \\lim _ { s \\rightarrow \\infty } \\frac { s ^ 2 \\Big ( 1 - s h ( s ) \\Big ) } { s h ( s ) } \\\\ & = r ^ 2 - \\lambda _ 2 ( \\mu ) ^ 2 . \\end{align*}"}
{"id": "3458.png", "formula": "\\begin{align*} \\langle f , g \\rangle & : = \\langle - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) \\psi _ Q ( \\cdot , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , g \\rangle \\\\ & = - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) \\psi _ Q g ( x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \\end{align*}"}
{"id": "2137.png", "formula": "\\begin{align*} G _ { k , l } ( x ) = \\begin{cases} - \\frac { x ^ k } { k - l } + x ^ l \\int _ 0 ^ x \\frac { y ^ { k - l - 1 } } { 1 - y ^ k } d y , \\ l = 1 , \\cdots , k - 1 ; \\\\ - \\frac { x ^ k } k \\log ( 1 - x ^ k ) , \\ l = k . \\end{cases} \\end{align*}"}
{"id": "1066.png", "formula": "\\begin{align*} q _ { 0 } ( x ) = \\left \\{ \\begin{aligned} & C _ { L } , x < 0 , \\\\ & C _ { R } , x > 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "8506.png", "formula": "\\begin{align*} A = I _ 0 \\supseteq I _ 1 \\supseteq I _ 2 \\supseteq \\cdots \\supseteq I _ n = 0 \\end{align*}"}
{"id": "8518.png", "formula": "\\begin{gather*} a * ( \\lambda _ a ^ { - 1 } ( b ) * x ) + \\lambda _ a ^ { - 1 } ( b ) * x = 0 \\shortintertext { a n d h e n c e } ( a + b ) * ( x + y ) = ( a * x ) + x + ( a * y ) - x . \\end{gather*}"}
{"id": "5764.png", "formula": "\\begin{align*} \\varphi _ { i , \\delta } = \\delta ^ { - 1 / 2 } \\varphi _ i : B _ { \\delta ^ { - 1 / 2 } r _ 1 } ( \\hat x _ i , \\hat g _ { i , \\delta } ) \\to \\mathbb R ^ n \\end{align*}"}
{"id": "245.png", "formula": "\\begin{align*} G ^ { \\mu } _ \\delta ( f ) = \\int _ 0 ^ { + \\infty } e ^ { - \\delta t } P ^ { \\mu } _ t ( f ) d t . \\end{align*}"}
{"id": "3280.png", "formula": "\\begin{align*} 1 + S ^ { \\langle - 1 \\rangle } _ { \\mu _ { T ^ * T } } \\left ( \\frac { 1 } { r ^ 2 } \\right ) & = 1 + \\psi _ { \\mu _ { T ^ * T } } \\left ( - \\frac { 1 } { s ( r , 0 ) ^ 2 } \\right ) \\\\ & = \\int _ 0 ^ \\infty \\frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + u ^ 2 } d \\mu _ { | T | } ( u ) \\\\ & = s ( r , 0 ) h ( s ( r , 0 ) ) \\\\ & = \\frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + r ^ 2 } . \\end{align*}"}
{"id": "3628.png", "formula": "\\begin{align*} s _ 3 ( x ) = \\frac { 4 . 3 1 2 8 } { T } \\log ^ { 0 . 6 } x . \\end{align*}"}
{"id": "3953.png", "formula": "\\begin{align*} A = \\begin{pmatrix} - \\partial _ x & - \\partial _ x \\\\ [ 4 p t ] - \\partial _ x & \\partial _ { x x } - \\partial _ x \\end{pmatrix} , \\end{align*}"}
{"id": "1693.png", "formula": "\\begin{align*} H _ n ( s ) & = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } } F _ s \\left ( \\left \\{ 1 \\right \\} ^ { s } , 1 - k ; \\left \\{ 2 \\right \\} ^ { s } ; 1 \\right ) , \\\\ H _ n ( \\overline { s } ) = & \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } } F _ s \\left ( \\left \\{ 1 \\right \\} ^ { s } , 1 - k ; \\left \\{ 2 \\right \\} ^ { s } ; - 1 \\right ) , \\end{align*}"}
{"id": "1951.png", "formula": "\\begin{align*} f g ( x ) : = \\sum _ { w \\in \\N ^ * \\cup \\{ \\mathbf { 1 } \\} } ( f g ) _ w x _ w . \\end{align*}"}
{"id": "8676.png", "formula": "\\begin{align*} \\| \\mathbf { m } _ n \\| _ \\infty = \\Theta \\left ( \\frac { 1 } { \\sqrt { n } } \\| \\mathbf { m } _ n \\| _ 2 \\right ) . \\end{align*}"}
{"id": "4368.png", "formula": "\\begin{align*} \\tilde T _ { j } ( \\xi ) = T _ j ( \\xi ) - C _ j \\xi ^ { 2 j - \\gamma } , \\end{align*}"}
{"id": "2565.png", "formula": "\\begin{align*} \\partial ^ \\alpha ( f g ) = \\sum _ { \\beta \\leq \\alpha } \\frac { \\alpha ! } { \\beta ! ( \\alpha - \\beta ) ! } ( \\partial ^ \\beta f ) ( \\partial ^ { \\alpha - \\beta } g ) . \\end{align*}"}
{"id": "4248.png", "formula": "\\begin{align*} \\rho _ c '' ( x ) = { } & \\rho '' ( x ) - a \\left ( \\frac { \\zeta '' ( x / c ) } { c ^ 2 } x ^ 4 - 1 2 \\zeta ( x / c ) x ^ 2 - \\frac { 8 \\zeta ' ( x / c ) } { c } x ^ 3 \\right ) \\\\ \\ge { } & 2 A ^ 2 - C c ^ 2 \\ge A ^ 2 > 0 , \\end{align*}"}
{"id": "8906.png", "formula": "\\begin{align*} U \\mapsto A _ \\ast ( \\rho ( U ) ) = \\begin{cases} A & U \\mbox { n o t b o u n d e d } \\\\ 0 & U \\mbox { b o u n d e d } \\end{cases} . \\end{align*}"}
{"id": "8088.png", "formula": "\\begin{align*} T _ \\Sigma ( f ) [ \\psi ] : = \\frac { 1 } { 2 } \\int _ \\Sigma f ( s ) \\psi { ( s ) } ^ 2 \\sqrt { \\gamma ' ( s ) } \\ , \\mathrm { d } s . \\end{align*}"}
{"id": "2125.png", "formula": "\\begin{align*} \\begin{aligned} & k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { j = n - N + 1 } ^ { n - 1 } \\frac { ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( n - j + 1 ) - l } = \\\\ & k \\sum _ { l = 1 } ^ k \\binom { c _ n k n } l ( k ) _ l \\sum _ { s = 1 } ^ { N - 1 } \\frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac 1 { k ( s + 1 ) - l } . \\end{aligned} \\end{align*}"}
{"id": "3757.png", "formula": "\\begin{align*} d _ { \\alpha } T _ { \\psi } = T _ { ( d _ { - \\alpha } \\psi ) } , \\ \\ d _ { - \\alpha } T _ { \\psi } = - T _ { ( d _ { \\alpha } \\psi ) } . \\end{align*}"}
{"id": "8116.png", "formula": "\\begin{align*} T B _ { 8 } ( f ) = 2 T B _ { 5 } ( f ) , \\end{align*}"}
{"id": "4769.png", "formula": "\\begin{align*} R ( e _ i \\wedge e _ j ) = r _ { i , j } e _ i \\wedge e _ j . \\end{align*}"}
{"id": "5551.png", "formula": "\\begin{align*} \\| h \\| _ H : = \\bigg ( | h ( 0 ) | ^ 2 + \\int _ 0 ^ { \\infty } | h ' ( x ) | ^ 2 w ( x ) d x \\bigg ) ^ { 1 / 2 } < \\infty . \\end{align*}"}
{"id": "7419.png", "formula": "\\begin{align*} f ( \\eta ) - f ( \\eta _ { 1 , x , r } ) = \\sum _ { i \\in I _ { 1 , x } ^ { N N } } [ f ( \\eta ^ { ( i - 1 ) } ) - f ( \\eta ^ { ( i ) } ) ] , \\end{align*}"}
{"id": "652.png", "formula": "\\begin{align*} \\abs { \\frac { f _ 0 ( x ) - g _ 0 ( x ) } { h _ 0 ( x ) + 1 } - \\alpha } \\ & = \\ \\abs { \\frac { u ( 2 ( x + 1 ) + 1 ) - v ( 2 ( x + 1 ) + 1 ) } { h _ 0 ( 2 ( x + 1 ) + 1 ) } - \\alpha } \\\\ [ 1 1 p t ] & \\leq \\ \\frac { 1 } { 2 ( x + 1 ) + 1 + 1 } \\\\ [ 1 1 p t ] & = \\ \\frac { 1 } { 2 x + 4 ) } \\\\ [ 1 1 p t ] & = \\ \\frac { 1 } { 2 ( x + 2 ) } \\\\ [ 1 1 p t ] & < \\ \\frac { 1 } { 2 ( x + 1 ) } . \\end{align*}"}
{"id": "915.png", "formula": "\\begin{align*} [ a , a ^ { + } ] = a a ^ { + } - a ^ { + } a = 1 , ( \\mathrm { s e e } \\ [ 4 , 5 , 8 ] ) . \\end{align*}"}
{"id": "8723.png", "formula": "\\begin{align*} s _ { i 0 } ( x ) = 1 , s _ { i 1 } ( x ) = 1 - 2 x _ i , s _ { i 2 } ( x ) = p _ i ( x ) . \\end{align*}"}
{"id": "6757.png", "formula": "\\begin{align*} D ^ 2 f _ 0 = D ^ 2 v _ 0 + \\frac { 2 } { h _ 0 ( h _ 0 + h _ 1 ) } ( u _ { - 1 } - v _ { - 1 } ) . \\end{align*}"}
{"id": "1656.png", "formula": "\\begin{align*} \\mathrm { C h } ( u , v ) = D ( u , v ) , u \\in C _ c ^ \\infty ( \\mathcal { R } ) , v \\in C _ c ( M ) \\cap W ^ { 1 } ( M ) . \\end{align*}"}
{"id": "974.png", "formula": "\\begin{align*} I _ 1 & = c _ { n , s } \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { B \\setminus B _ \\varepsilon ( x ) } \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { n + 2 s } } \\dd y - \\int _ B \\frac { u ( x ) - u ( y ) } { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\dd y . \\end{align*}"}
{"id": "3156.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal e _ { i _ k } \\frac { e _ { i _ k } ^ \\intercal ( A x _ k - b ) } { \\norm { A ^ \\intercal e _ j } _ 2 ^ 2 } . \\end{align*}"}
{"id": "4115.png", "formula": "\\begin{align*} & | \\sigma _ n ^ 2 ( y ) \\cdot ( G '' ( x ) - G '' ( y ) ) | \\leq \\begin{cases} c \\cdot ( 1 + | y | ^ { 2 \\ell _ \\sigma + 2 } ) \\cdot | x - y | , & ( x , y ) \\in B ^ c , \\\\ c \\cdot ( 1 + | y | ^ { 2 \\ell _ \\sigma + 2 } ) , & ( x , y ) \\in B . \\end{cases} \\end{align*}"}
{"id": "6876.png", "formula": "\\begin{align*} M ( v , X ( k ) \\setminus S ) = \\sum \\limits _ { y \\in X ( k ) \\setminus S } M ( v , y ) \\end{align*}"}
{"id": "2004.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ 1 ( t + 2 \\bar { T } ) \\\\ u _ 2 ( t + 2 \\bar { T } ) \\\\ u _ 3 ( t + 2 \\bar { T } ) \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & \\cos ( \\frac { 4 \\pi p } { n } ) & - \\sin ( \\frac { 4 \\pi p } { n } ) \\\\ 0 & \\sin ( \\frac { 4 \\pi p } { n } ) & \\cos ( \\frac { 4 \\pi p } { n } ) \\end{array} \\right ) \\left ( \\begin{array} { c } u _ 1 ( t ) \\\\ u _ 2 ( t ) \\\\ u _ 3 ( t ) \\end{array} \\right ) , \\end{align*}"}
{"id": "7283.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 5 k } ) ( 1 - X ^ { 5 k - 2 } ) ( 1 - X ^ { 5 k - 3 } ) = \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ k X ^ { \\frac { 5 k ^ 2 + k } { 2 } } , \\\\ \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 5 k } ) ( 1 - X ^ { 5 k - 1 } ) ( 1 - X ^ { 5 k - 4 } ) = \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ k X ^ { \\frac { 5 k ^ 2 + 3 k } { 2 } } . \\end{align*}"}
{"id": "8492.png", "formula": "\\begin{gather*} D _ t = ( - 1 ) ^ { t + 3 } \\frac { 1 - u ^ { t + 3 } } { ( 1 - u ) ( 1 + u ) ^ { t + 2 } } , \\\\ N _ 2 ^ t = \\frac { ( - 1 ) ^ { t + 1 } u ^ 2 ( 1 - u ^ t ) } { ( 1 + u ) ^ { t + 3 } ( 1 - u ) } , \\\\ N _ 3 ^ t = \\frac { ( - 1 ) ^ { t + 1 } u ( 1 - u ^ { t + 2 } ) } { ( 1 - u ) ( 1 + u ) ^ { t + 3 } } . \\end{gather*}"}
{"id": "4950.png", "formula": "\\begin{align*} z = [ g _ 1 \\cdot ( g ^ + ) ^ { - 1 } \\cdot g _ 2 ^ { - 1 } ] \\cdot g _ 0 ^ { - 1 } \\cdot \\phi ( g ^ + ) \\cdot g _ 0 \\end{align*}"}
{"id": "9506.png", "formula": "\\begin{align*} \\| H + r f \\| & = 1 , \\\\ p ( H + r f ) & = \\frac { p ( f ) } { | p ( f ) | } , \\\\ \\| H \\| & \\le 1 - r | p ( f ) | , \\end{align*}"}
{"id": "6968.png", "formula": "\\begin{align*} \\forall k \\in \\N \\colon \\lambda _ k : = y _ k ^ * \\norm { x _ k - \\bar x } / \\norm { y _ k - \\bar y } \\end{align*}"}
{"id": "9189.png", "formula": "\\begin{align*} \\Theta ( I , x , y ) = \\sum _ { \\tilde y \\in \\phi ^ { - 1 } ( y ) } \\Theta ( \\tilde I , \\tilde x , \\tilde y ) ; \\end{align*}"}
{"id": "8604.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ 2 \\mu ^ \\# ( { \\bf k } ) - \\mu ^ \\# _ { R , 1 } ( { \\bf k } ) & = \\int _ { \\R } \\prod _ \\ast \\mathcal { K } ^ \\# _ S ( k _ j , x ) \\ , d x \\\\ & = \\sum _ { ( \\iota _ 1 , \\iota _ 2 , \\iota _ 3 , \\iota _ 4 ) \\in \\{ 0 , + , - \\} ^ 4 } \\int _ { \\R } \\prod _ \\ast \\chi _ { \\iota _ j } ( x ) \\mathcal { K } ^ \\# _ { \\iota _ j } ( k _ j , x ) \\ , d x . \\end{align*}"}
{"id": "8178.png", "formula": "\\begin{align*} A ( d _ 0 , f ) = \\sum _ { a \\mod d _ 0 \\atop \\gcd ( d _ 0 , a ) = 1 } \\sum _ { b \\mod d _ 0 \\atop { \\gcd ( d _ 0 , b ) = 1 \\atop b \\neq a } } \\cot \\left ( \\frac { \\pi ( b - a ) } { d _ 0 } \\right ) \\left ( \\cot \\left ( \\frac { \\pi f a } { d _ 0 } \\right ) - \\cot \\left ( \\frac { \\pi f b } { d _ 0 } \\right ) \\right ) , \\end{align*}"}
{"id": "9266.png", "formula": "\\begin{align*} \\norm { w } \\leq \\frac { 2 L } { 2 ^ { - \\alpha _ n } } = L \\cdot 2 ^ { \\alpha _ n + 1 } . \\end{align*}"}
{"id": "2843.png", "formula": "\\begin{align*} P [ u ] = \\Im \\int \\bar { u } \\nabla u d x = 0 , \\end{align*}"}
{"id": "8174.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) = \\kappa _ { d _ 0 } \\times \\left ( 1 + \\frac { N _ { d _ 0 } ( p , H ) } { p } \\right ) , \\hbox { w h e r e } \\kappa _ { d _ 0 } : = \\frac { \\pi ^ 2 } { 6 } \\prod _ { q \\mid d _ 0 } \\left ( 1 - \\frac { 1 } { q ^ 2 } \\right ) . \\end{align*}"}
{"id": "1050.png", "formula": "\\begin{align*} \\Delta ( D _ { n n } ) = \\sum _ { 1 \\leq k < l \\leq 6 } a _ { n k } a _ { n l } \\otimes D _ { k l } + \\sum _ { 5 \\leq k \\leq 6 } a _ { n k } ^ { 2 } \\otimes D _ { k k } \\ , . \\end{align*}"}
{"id": "3585.png", "formula": "\\begin{align*} { S } ( x ) ( t ) : = z _ 1 ^ * ( ( \\widetilde { S } ( x ) ) ( t ) ) ( x \\in X , \\ , t \\in K ) . \\end{align*}"}
{"id": "1353.png", "formula": "\\begin{align*} \\langle M \\rangle _ t = \\int _ 0 ^ t f ( X _ s ) d s , \\ ; \\ ; t > 0 , \\ ; \\ ; P _ { x _ 0 } \\end{align*}"}
{"id": "534.png", "formula": "\\begin{align*} \\begin{matrix} \\mathbf { x } _ { k + 1 } = \\boldsymbol { f } ( \\mathbf { x } _ k , \\mathbf { u } _ k ) , & \\quad & \\mathbf { y } _ k = \\boldsymbol { h } ( \\mathbf { x } _ k ) , \\end{matrix} \\end{align*}"}
{"id": "1677.png", "formula": "\\begin{align*} ( 2 k _ 1 + 1 ) ^ { t _ 1 - s _ 1 } \\cdots ( 2 k _ l + 1 ) ^ { t _ l - s _ l } & \\leq ( 2 k _ l + 1 ) ^ { t _ 1 + \\cdots + t _ l - ( s _ 1 + \\cdots + s _ l ) } \\\\ & = ( 2 k _ l + 1 ) ^ { t _ 1 + \\cdots + t _ r - ( s _ 1 + \\cdots + s _ r ) } \\cdot ( 2 k _ l + 1 ) ^ { s _ { l + 1 } - t _ { l + 1 } + \\cdots + s _ r - t _ r } \\\\ & \\leq ( 2 k _ l + 1 ) ^ { s _ { l + 1 } - t _ { l + 1 } + \\cdots + s _ r - t _ r } \\\\ & \\leq ( 2 k _ { l + 1 } + 1 ) ^ { s _ { l + 1 } - t _ { l + 1 } } \\cdots ( 2 k _ r + 1 ) ^ { s _ r - t _ r } . \\end{align*}"}
{"id": "3296.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ m = E \\end{align*}"}
{"id": "7509.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { T } \\left [ \\log \\left | \\frac { \\frac { 1 } { 2 } - \\epsilon + i t } { 2 } \\right | - \\log \\left | \\frac { \\frac { 1 } { 2 } + \\epsilon + i t } { 2 } \\right | \\right ] \\ d t + \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\left [ \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) - \\arg \\frac { \\sigma } { 2 } \\right ] \\ d \\sigma = 0 \\end{align*}"}
{"id": "4596.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } > x \\Big ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) \\ \\ \\ \\textrm { a n d } \\ \\ \\ \\frac { \\mathbf { P } \\Big ( \\frac { 1 } { \\sigma } ( \\hat { \\theta } _ n - \\theta ) \\sqrt { \\Sigma _ { k = 1 } ^ n X _ { k - 1 } ^ 2 } < - x \\Big ) } { \\Phi \\left ( - x \\right ) } = 1 + o ( 1 ) . \\end{align*}"}
{"id": "740.png", "formula": "\\begin{align*} D ^ { ( L ) } W ^ { ( L ) } \\cdots D ^ { ( 1 ) } W ^ { ( 1 ) } \\stackrel { d } { = } A \\widehat { D } ^ { ( L ) } W ^ { ( L ) } \\cdots \\widehat { D } ^ { ( 1 ) } W ^ { ( 1 ) } , \\end{align*}"}
{"id": "1126.png", "formula": "\\begin{align*} J ^ { G P } = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , k \\in ( - \\eta , \\eta ) \\end{align*}"}
{"id": "1802.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { c _ Y } \\left ( [ \\Phi _ { A _ { 1 } } , \\Phi _ { A _ { 2 } } ] \\right ) = \\frac { i } { 2 \\pi } \\int _ { \\mathbb { R } } \\int _ G { \\rm T r } _ { \\partial Z } \\left ( \\frac { \\partial I ( \\Phi _ { A _ 1 } , h ^ { - 1 } , \\lambda ) } { \\partial \\lambda } \\circ I ( \\Phi _ { A _ 2 } , h , \\lambda ) \\right ) d h d \\lambda . \\end{align*}"}
{"id": "6765.png", "formula": "\\begin{align*} \\mathbf L \\tilde { \\mathbf V } + \\mathbf b + \\rho \\mathcal I _ { \\tilde { \\mathbf V } } ( \\mathbf V ^ * - \\tilde { \\mathbf V } ) + \\mathbf a _ 2 + \\mathbf a _ 1 = \\mathbf 0 , \\end{align*}"}
{"id": "7457.png", "formula": "\\begin{align*} \\lambda _ { \\phi } ( t ) = \\frac { E ( \\phi ) + \\frac { \\beta } { 2 } \\int _ { \\mathbb { R } ^ d } | \\phi ( \\mathbf { x } , t ) | ^ 4 \\mathrm { d } \\mathbf { x } - \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 } { \\| \\phi ( \\cdot , t ) \\| ^ 2 } = \\mu _ { \\phi } ( t ) - \\frac { \\| \\dot { \\phi } ( \\cdot , t ) \\| ^ 2 } { \\| \\phi ( \\cdot , t ) \\| ^ 2 } , \\end{align*}"}
{"id": "6958.png", "formula": "\\begin{align*} \\mathcal N _ Q ( \\bar x ; u ) = \\left \\{ \\eta \\in \\mathbb X \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ u _ k \\} _ { k \\in \\N } \\subset \\mathbb X , \\ , \\exists \\{ t _ k \\} _ { k \\in \\N } \\subset \\R _ + , \\ , \\exists \\{ \\eta _ k \\} _ { k \\in \\N } \\subset \\mathbb X \\colon \\\\ & u _ k \\to u , \\ , t _ k \\searrow 0 , \\ , \\eta _ k \\to \\eta , \\ , \\eta _ k \\in \\mathcal N _ Q ( \\bar x + t _ k u _ k ) \\ , \\forall k \\in \\N \\end{aligned} \\right \\} . \\end{align*}"}
{"id": "9283.png", "formula": "\\begin{align*} [ \\chi _ A ] _ \\mathcal { M } : = \\lambda x , y \\in X . \\begin{cases} 0 ^ 0 & y \\in A x , \\\\ 1 ^ 0 & y \\not \\in A x , \\end{cases} \\end{align*}"}
{"id": "3762.png", "formula": "\\begin{align*} P ( u ) = w . \\end{align*}"}
{"id": "5696.png", "formula": "\\begin{align*} \\mathrm { E C H } ( S ^ { 3 } , \\lambda , 0 ) = \\mathbb { F } [ U ^ { - 1 } , U ] / U \\mathbb { F } [ U ] \\end{align*}"}
{"id": "8423.png", "formula": "\\begin{align*} L _ { t } ( w , l ) : = w _ { l _ { t } } . \\end{align*}"}
{"id": "4929.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ d } f ( x ) : = \\mathbb E _ { z \\sim \\mathcal D } \\left [ f ( x ; z ) \\right ] , \\end{align*}"}
{"id": "2745.png", "formula": "\\begin{align*} S ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , \\widehat { z _ j } , \\dots , z _ n ) = 0 , \\end{align*}"}
{"id": "2055.png", "formula": "\\begin{align*} C ( \\Psi , \\varphi _ 0 , J , f ) = \\sup _ { \\substack { \\xi \\in A ^ 2 ( \\{ \\Psi < 0 \\} , e ^ { - \\varphi _ 0 } ) ^ * \\setminus \\{ 0 \\} \\\\ \\xi | _ { A ^ 2 ( \\{ \\Psi < 0 \\} , e ^ { - \\varphi _ 0 } ) \\cap J } \\equiv 0 } } \\frac { | \\xi \\cdot f | ^ 2 } { K ^ { \\varphi _ 0 } _ { \\xi , \\Psi , \\lambda } ( 0 ) } . \\end{align*}"}
{"id": "2238.png", "formula": "\\begin{align*} \\| A ^ \\mu E _ { k , N } ^ m v \\| & \\leq C t _ m ^ { - \\frac \\mu 2 } \\| v \\| , \\quad \\forall v \\in H , \\ ; m = 1 , 2 , 3 , \\cdots , M . \\end{align*}"}
{"id": "4556.png", "formula": "\\begin{align*} | S | & \\ge \\frac { ( 2 k ' - 3 h + 3 h _ 5 - 8 \\delta k ' ) n - 1 2 k + 4 ( 1 + 2 \\delta ) k ' + 6 ( h - h _ 5 ) } { 3 ( 3 h _ 5 - h + 1 ) } \\\\ & > \\frac { ( 2 k ' - 3 h + 3 h _ 5 - 1 ) n } { 3 ( 3 h _ 5 - h + 1 ) } = \\frac { n } { 3 } + \\frac { ( 2 k ' - 2 h - 2 ) n } { 3 ( 3 h _ 5 - h + 1 ) } \\\\ & \\geq \\frac { n } { 3 } + \\frac { ( 2 k ' - 2 h - 2 ) n } { 3 ( 2 h - 1 4 ) } . \\end{align*}"}
{"id": "469.png", "formula": "\\begin{align*} \\mathbf { L } ( \\mathbf { t } ) \\in \\mathbb { C } ^ { k \\times n } , \\mathbf { R } ( \\mathbf { t } ) = \\mathrm { d i a g } \\left ( \\mathbf { d } ( \\mathbf { t } ) \\right ) \\cdot \\mathbf { R } ( \\mathbf { 1 } ) \\end{align*}"}
{"id": "5904.png", "formula": "\\begin{align*} D _ i = a _ i D _ { i - 1 } - b _ { i - 1 } ^ 2 D _ { i - 2 } , \\end{align*}"}
{"id": "8075.png", "formula": "\\begin{align*} ( \\mathfrak { A } _ { \\ell } ( \\rho , \\chi ) F ) _ { \\widetilde { H } } = F _ { \\chi ^ { * } \\widetilde { H } } \\circ \\rho ^ { * } _ { ( 1 ) } . \\end{align*}"}
{"id": "4330.png", "formula": "\\begin{align*} \\| \\phi _ { 1 , b , \\beta } \\| _ { L ^ 2 _ { \\rho _ \\beta } } ^ 2 = \\| \\phi _ { 0 , b , \\beta } \\| _ { L ^ 2 _ { \\rho _ \\beta } } ^ 2 1 6 \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) , \\end{align*}"}
{"id": "7960.png", "formula": "\\begin{align*} \\sqrt { \\epsilon } = \\begin{cases} T ^ { - 1 / 2 } & \\\\ ( \\log ( T ) / T ) ^ { 1 / 2 } & \\\\ T ^ { - \\frac { 1 } { d - 1 / H } } & \\end{cases} \\end{align*}"}
{"id": "8163.png", "formula": "\\begin{align*} p ^ 3 - p \\sum _ { x \\bmod p } \\max ( q _ 1 x , q _ 2 x ) = \\frac { ( p - 1 ) p ( 2 p - 1 ) } { 6 } + \\frac { p ^ 3 } { 2 \\pi ^ 2 } M _ { q _ 1 , q _ 2 } ( p ) + o ( p ^ 3 ) . \\end{align*}"}
{"id": "7331.png", "formula": "\\begin{align*} \\begin{pmatrix} X ^ \\varepsilon & 0 \\\\ 0 & - Y ^ \\varepsilon \\end{pmatrix} \\le 0 , \\end{align*}"}
{"id": "4791.png", "formula": "\\begin{align*} n = | G | , \\quad \\hat \\theta = \\frac { | \\theta | } { \\sqrt { n } | | \\phi | | _ 2 } . \\end{align*}"}
{"id": "2934.png", "formula": "\\begin{align*} F _ { n 1 } = n \\ell ^ 2 - n ^ 2 \\ell , F _ { n 2 } = 6 \\ell ^ 2 + Z _ { \\{ 3 , 4 \\} } ( n ^ 2 - 6 n \\ell + n ^ 2 Z _ { \\{ 3 , 4 \\} } ) . \\end{align*}"}
{"id": "6696.png", "formula": "\\begin{align*} P _ { { \\bf b } , d } ^ { ( - 1 ) } & = \\prod _ { \\substack { j = 1 \\\\ b _ j \\geq 2 } } ^ { n } \\Bigl \\{ ( \\theta - t ) ^ { q ^ { d - 2 } } \\mathbb { D } _ { b _ j - 2 } ^ { q ^ { d - b _ j } } \\Bigr \\} P _ { { \\bf b } , d } = \\prod _ { \\substack { j = 1 \\\\ b _ j \\geq 2 } } ^ { n } \\biggl \\{ ( \\theta - t ) ^ { q ^ { d - 2 } } \\Bigl ( ( \\theta ^ q - t ) ^ { q ^ { b _ j - 3 } } \\cdots ( \\theta ^ { q ^ { b _ j - 2 } } - t ) \\Bigr ) ^ { q ^ { d - b _ j } } \\biggr \\} P _ { { \\bf b } , d } . \\end{align*}"}
{"id": "3419.png", "formula": "\\begin{align*} T f ( x ) & = \\langle K ( x , y ) , ( \\xi ( y ) + \\eta ( y ) ) f ( y ) \\rangle \\\\ & = \\int _ { \\R ^ N } K ( x , y ) \\xi ( y ) ( f ( y ) - f ( x ) ) d \\omega ( y ) + f ( x ) \\langle K ( x , y ) , \\xi ( y ) \\rangle \\\\ & + \\int _ { \\R ^ N } K ( x , y ) \\eta ( y ) f ( y ) d \\omega ( y ) \\\\ & = : I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
{"id": "396.png", "formula": "\\begin{align*} \\mathcal { C } ^ { 1 } ( [ 0 , T ] ; X ) : = \\left \\lbrace u \\in \\mathcal { C } ( [ 0 , T ] ; X ) : u _ { t } \\in \\mathcal { C } ( [ 0 , T ] ; X ) \\right \\rbrace . \\end{align*}"}
{"id": "7245.png", "formula": "\\begin{align*} \\sigma _ 1 ( x ) = x _ 1 + x _ 2 + x _ 3 , \\sigma _ 2 ( x ) = x _ 1 x _ 2 + x _ 1 x _ 3 + x _ 2 x _ 3 , \\sigma _ n ( x ) = x _ 1 x _ 2 x _ 3 . \\end{align*}"}
{"id": "7937.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } E _ k \\ll \\begin{cases} n ^ { \\ell + 3 / 8 } , ~ & ~ t \\geqslant n ^ { 1 / 2 } - n ^ { 1 / 4 } \\\\ n ^ { \\ell + 1 / 4 } \\sqrt { n ^ { 1 / 2 } - t } , ~ & t < n ^ { 1 / 2 } - n ^ { 1 / 4 } . \\end{cases} \\end{align*}"}
{"id": "9253.png", "formula": "\\begin{align*} \\forall x ^ X , p ^ X , \\gamma ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land x \\in \\mathrm { d o m } ( J ^ A _ \\gamma ) \\land p = _ X J ^ A _ \\gamma x \\rightarrow \\vert \\gamma \\vert ^ { - 1 } ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "4864.png", "formula": "\\begin{align*} \\sigma = \\frac \\alpha { q _ \\alpha n } , \\end{align*}"}
{"id": "5801.png", "formula": "\\begin{align*} f _ n ( x ) = \\begin{cases} x - \\frac { n - 1 } { n } & x < - \\frac { 1 } { n } \\\\ n x & x \\in [ - \\frac { 1 } { n } , \\frac { 1 } { n } ) \\\\ x + \\frac { n - 1 } { n } & x \\geq \\frac { 1 } { n } , \\end{cases} \\end{align*}"}
{"id": "3851.png", "formula": "\\begin{gather*} r ' _ { l , s } + r _ { l - 1 , s } + r _ { l , s - 1 } = q _ { l , s } - \\tilde q _ { l , s } , l , s = \\overline { 0 , m } , ( l , s ) \\ne ( m , m ) , \\\\ r _ { m , s } = r _ { s , m } = 0 , s = \\overline { 0 , m - 1 } . \\end{gather*}"}
{"id": "8337.png", "formula": "\\begin{align*} \\int _ a ^ b f \\dd g = \\int _ a ^ c f \\dd g + \\int _ c ^ b f \\dd g \\end{align*}"}
{"id": "2495.png", "formula": "\\begin{align*} F _ r * _ { \\mathbf { H } _ r } G _ r ( x , \\omega ) = ( F _ r * G _ r ) ( x , \\omega , e ^ { 2 \\pi i \\tau } ) \\end{align*}"}
{"id": "3031.png", "formula": "\\begin{align*} ( \\pi ^ { s + 1 , s } ) ^ { * } \\rho = \\Theta _ { \\lambda } + d \\mu + \\eta , \\end{align*}"}
{"id": "3549.png", "formula": "\\begin{align*} \\tilde \\Gamma ( t ) : = \\frac { \\Gamma ( t ) } { | t | ^ { 1 / 2 } } \\end{align*}"}
{"id": "4816.png", "formula": "\\begin{align*} ( \\phi ^ 2 + \\phi * \\phi ) ( 0 ) = c ^ 2 ( \\alpha * \\alpha ) ( 0 ) + \\bar c ^ 2 ( \\bar \\alpha * \\bar \\alpha ) ( 0 ) + 2 ( \\bar \\alpha * \\alpha ) = 2 ( q - 1 ) . \\end{align*}"}
{"id": "4453.png", "formula": "\\begin{align*} e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "233.png", "formula": "\\begin{align*} \\Gamma _ { \\Sigma } ( f , f ) ( x ) = \\langle \\nabla ( f ) ( x ) ; \\Sigma ( \\nabla ( f ) ( x ) ) \\rangle = \\left \\| \\sqrt { \\Sigma } ( \\nabla ( f ) ( x ) ) \\right \\| ^ 2 . \\end{align*}"}
{"id": "4118.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ d c _ i p _ i = 0 \\sum _ { i = 0 } ^ d c _ i = 0 c _ 0 = \\dotsb = c _ d = 0 \\end{align*}"}
{"id": "5765.png", "formula": "\\begin{align*} \\pi _ 0 ( \\hat x ^ 1 , \\dots , \\hat x ^ m , \\dots , \\hat x ^ n ) = ( \\hat x ^ 1 , \\dots , \\hat x ^ m ) , \\end{align*}"}
{"id": "879.png", "formula": "\\begin{align*} a _ i = n _ i - n _ { \\min } + 1 , i = 1 , 2 , \\cdots , m . \\end{align*}"}
{"id": "8607.png", "formula": "\\begin{align*} & \\sum _ { ( \\iota _ 3 , \\iota _ 4 ) \\in \\{ + , - , 0 \\} } \\int _ { \\R } \\overline { \\chi _ { + } ( x ) \\mathcal { K } ^ \\# _ { + } ( k _ 1 , x ) } \\ , \\chi _ { - } ( x ) \\mathcal { K } ^ \\# _ { - } ( k _ 2 , x ) \\ , \\overline { \\chi _ { \\iota _ 3 } ( x ) \\mathcal { K } ^ \\# _ { \\iota _ 3 } ( k _ 3 , x ) } \\ , \\chi _ { \\iota _ 4 } ( x ) \\mathcal { K } ^ \\# _ { \\iota _ 4 } ( k _ 4 , x ) \\ , d x . \\end{align*}"}
{"id": "8882.png", "formula": "\\begin{align*} \\tilde \\psi \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) = \\sum _ { t = 0 } ^ { \\max ( y _ j ) - 1 } ( h _ t ^ * \\psi _ 1 ) \\left ( \\begin{pmatrix} x _ 0 \\\\ y _ 0 \\end{pmatrix} , \\ldots , \\begin{pmatrix} x _ { q - 1 } \\\\ y _ { q - 1 } \\end{pmatrix} \\right ) . \\end{align*}"}
{"id": "5914.png", "formula": "\\begin{align*} S E ( G ) \\leq E ( J - I ) + 2 E ( A ( G ) ) = 2 ( n - 1 ) + 2 E ( A ( G ) ) , \\end{align*}"}
{"id": "6481.png", "formula": "\\begin{align*} E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n } } \\right ) ^ { 2 } \\right ] - 1 = 0 . \\end{align*}"}
{"id": "3026.png", "formula": "\\begin{align*} y _ { j _ { 1 } \\ldots j _ { k } } ^ { \\sigma } ( J _ { x } ^ { r } \\gamma ) = D _ { j _ { 1 } } \\ldots D _ { j _ { k } } ( y ^ { \\sigma } \\gamma \\varphi ^ { - 1 } ) ( \\varphi ( x ) ) , \\quad 0 \\leq k \\leq r . \\end{align*}"}
{"id": "7529.png", "formula": "\\begin{align*} T \\log \\left ( \\frac { a ^ 2 + T ^ 2 } { 4 } \\right ) = 2 T \\log \\left ( \\frac { T } { 2 } \\right ) + T \\log \\left ( 1 + \\frac { a ^ 2 } { T ^ 2 } \\right ) \\end{align*}"}
{"id": "8806.png", "formula": "\\begin{align*} d \\Phi _ t = \\nabla \\log \\left ( \\frac { d \\mu ^ { \\epsilon , L } } { d \\varphi } \\right ) ( \\Phi _ t ) d t + \\sqrt { 2 } d B _ t , \\Phi _ 0 \\sim \\nu ^ { \\epsilon , L } , \\end{align*}"}
{"id": "6901.png", "formula": "\\begin{align*} { k \\choose i } _ q \\alpha _ i = 1 + o _ q ( 1 ) \\end{align*}"}
{"id": "2095.png", "formula": "\\begin{align*} \\widehat \\psi _ { d ^ 2 } ( \\mathbf { u } ) = \\frac { 1 } { d ^ { 2 n } } \\sum _ { f \\in ( \\mathbb { Z } / d ^ 2 \\mathbb { Z } ) ^ { n } } \\psi _ d ( f ) e ^ { \\frac { 2 \\pi i \\langle f , \\mathbf { u } \\rangle } { d ^ 2 } } . \\end{align*}"}
{"id": "8194.png", "formula": "\\begin{align*} S ( a , b , 3 \\delta f ) = s ( a , b , 3 \\delta f ) + s ( a + 2 f , b , 3 \\delta f ) . \\end{align*}"}
{"id": "2426.png", "formula": "\\begin{align*} \\norm { f } _ \\mathcal { H } ^ 2 = \\langle f , f \\rangle = \\sum _ { \\gamma \\in \\Gamma } | \\langle f , S ^ { - 1 / 2 } e _ \\gamma \\rangle | ^ 2 . \\end{align*}"}
{"id": "2673.png", "formula": "\\begin{align*} \\beta ( z , \\tau ) = \\beta ( x , \\omega , \\tau ) = B \\rho ( x , \\omega , \\tau ) B ^ { - 1 } . \\end{align*}"}
{"id": "2955.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } ( \\mathcal { A } ) & = \\frac { 1 6 \\cdot \\left ( 1 + o ( 1 ) \\right ) } { 9 0 ^ 2 n ^ 4 \\delta ^ 4 _ n ( k ) } \\cdot \\sum \\limits _ { \\mathbf { P } _ { k } \\in \\mathcal { A } } \\sum \\limits _ { \\mathbf { i } \\in \\mathcal I _ 2 \\times \\mathcal I _ 2 } \\varphi _ { k - 1 } \\left ( \\mathbf { P } _ { k } , \\mathbf { i } \\right ) . \\end{align*}"}
{"id": "7540.png", "formula": "\\begin{align*} \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } = \\sum _ { \\rho } \\left ( \\frac { 1 } { s - \\rho } - \\frac { 1 } { 2 + i t - \\rho } \\right ) + \\mathcal { O } ( \\log t ) \\end{align*}"}
{"id": "2998.png", "formula": "\\begin{align*} | \\langle \\partial _ { 3 } u \\nabla u , \\partial _ { 3 } u \\rangle | & \\leq \\sum _ { i = 1 } ^ { 3 } \\int _ { \\R ^ 3 } | \\partial _ { 3 } u _ { i } \\partial _ { i } u \\partial _ { 3 } u | \\end{align*}"}
{"id": "3646.png", "formula": "\\begin{align*} \\int _ { t ^ k } ^ { t _ { k + 1 } } d N ( \\sigma , t ) = N ( \\sigma , t _ { k + 1 } ) - N ( \\sigma , t _ k ) . \\end{align*}"}
{"id": "4510.png", "formula": "\\begin{align*} & \\log x = R ( \\log t _ 0 ) ^ 2 , \\ \\log t _ 0 = \\sqrt { \\frac { \\log x } { R } } , \\ \\log t _ k = ( \\log t _ 0 ) w _ k , \\\\ \\ & \\frac { x ^ { - \\frac { 1 } { R ( \\log t _ { 0 } ) } } } { t _ 0 } = t _ 0 ^ { - 2 } , \\ \\frac { x ^ { - \\frac { 1 } { R \\log t _ { k } } } } { t _ k } = e ^ { - ( \\log t _ 0 ) \\big ( \\frac { 1 } { w _ k } + w _ k \\big ) } = t _ 0 ^ { - 2 - \\frac { ( w _ k - 1 ) ^ 2 } { w _ k } } . \\end{align*}"}
{"id": "5677.png", "formula": "\\begin{align*} \\mu _ 1 y \\cdot \\mu _ 2 x = \\mu _ 3 ( x \\cdot y ) \\end{align*}"}
{"id": "2573.png", "formula": "\\begin{align*} - \\int _ \\R \\log | x | \\overline { f ' ( x ) } \\ , d x = \\langle \\log ( | x | ) , - f ' \\rangle = \\langle \\log ( | x | ) ' , f \\rangle , \\end{align*}"}
{"id": "7188.png", "formula": "\\begin{align*} f ( t , x , v ) = & \\int _ 0 ^ t ( \\nabla \\phi \\ast \\rho [ f ] ) ( s , x - ( t - s ) v ) \\cdot \\nabla \\mu ( v ) \\dd s \\\\ + & \\int _ 0 ^ t E ( s , x - ( t - s ) v ) \\cdot \\nabla \\mu ( v ) \\dd s - \\int _ 0 ^ t E ( s , X _ { s , t } ( x , v ) ) \\cdot \\nabla _ v \\mu ( V _ { s , t } ( x , v ) ) \\\\ - & \\int _ 0 ^ t e _ 0 \\nabla \\Phi ( X _ { s , t } ( x , v ) - X ( s ) ) \\cdot \\nabla _ v \\mu ( V _ { s , t } ( x , v ) ) , \\end{align*}"}
{"id": "6643.png", "formula": "\\begin{align*} \\Sigma _ 1 : = p ^ { ( w - 1 ) h _ p } \\sum _ { m = 0 } ^ { k _ p - h _ p - 1 } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) \\left ( 1 - \\frac { p ^ w } { p ^ { 2 } } \\right ) } { p ^ { m ( 2 - \\alpha - w ) } p ^ { n ( 1 - \\beta ) } } \\end{align*}"}
{"id": "5693.png", "formula": "\\begin{align*} J _ { 0 } ( u ) = - \\chi ( u _ { 1 } ) \\end{align*}"}
{"id": "8768.png", "formula": "\\begin{align*} \\mathcal { Q } _ H : = \\left \\{ s \\left | \\begin{aligned} & s _ { i n } = s _ { i j } & & j \\geq \\tau ( i , t _ i ( H ) ) \\\\ & \\frac { s _ { i j + 1 } - s _ { i j } } { m _ { i ( j + 1 ) t _ i ( H ) } - m _ { i j t _ i ( H ) } } \\leq \\frac { s _ { i j } - s _ { i j - 1 } } { m _ { i j t _ i ( H ) } - m _ { i ( j - 1 ) t _ i ( H ) } } & & \\tau ( i , t _ i ( H ) - 1 ) \\leq j \\leq \\tau ( i , t _ i ( H ) ) \\\\ & s _ { i j } = m _ { i j t _ i ( H ) } & & j \\leq \\tau ( i , t _ i ( H ) - 1 ) \\end{aligned} \\right . \\right \\} \\end{align*}"}
{"id": "346.png", "formula": "\\begin{align*} \\mathcal A ( \\rho ^ * , m ^ * ) = \\sup _ { S \\in H _ R ^ 1 } \\mathcal L ( \\rho ^ * , m ^ * , S ) \\ge \\inf _ { ( \\rho , m ) } \\sup _ { S \\in H _ R ^ 1 } \\mathcal L ( \\rho , m , S ) . \\end{align*}"}
{"id": "5868.png", "formula": "\\begin{align*} m _ 0 = \\frac { \\rho _ 1 \\rho _ 2 \\rho _ 3 } { \\lambda _ 0 Z } , m _ 1 = \\frac { 1 } { \\lambda _ 1 Z } , m _ 2 = \\frac { \\rho _ 1 } { \\lambda _ 2 Z } , m _ 3 = \\frac { \\rho _ 1 \\rho _ 2 } { \\lambda _ 3 Z } , \\end{align*}"}
{"id": "151.png", "formula": "\\begin{align*} \\left ( E - \\mathcal { A } \\right ) ^ { - \\frac { r } { 2 } } f = \\frac { 1 } { \\Gamma ( \\frac { r } { 2 } ) } \\int _ 0 ^ { + \\infty } \\dfrac { e ^ { - t } } { t ^ { 1 - \\frac { r } { 2 } } } P _ t ( f ) d t , \\end{align*}"}
{"id": "4810.png", "formula": "\\begin{align*} ( \\phi ^ 2 + \\phi * \\phi ) ( a ) = ( 1 + J ' ( \\alpha ^ 2 , \\alpha ^ 2 ) ) \\alpha ^ 4 ( a ) . \\end{align*}"}
{"id": "5543.png", "formula": "\\begin{align*} d _ K ( x ) = \\inf _ { z \\in K } \\| x - z \\| = \\inf _ { z \\in K } \\| x - ( z - y ) \\| = \\inf _ { z \\in K } \\| ( x + y ) - z \\| = d _ K ( x + y ) , \\end{align*}"}
{"id": "2529.png", "formula": "\\begin{align*} S ^ * \\pi ( \\mathbf { h } ) = \\left ( \\pi ( \\mathbf { h } ^ { - 1 } ) S \\right ) ^ * = \\left ( S \\pi ( \\mathbf { h } ^ { - 1 } ) \\right ) ^ * = \\pi ( \\mathbf { h } ) S ^ * , \\end{align*}"}
{"id": "5988.png", "formula": "\\begin{align*} \\Phi = \\sum _ { k = 0 } ^ { \\infty } \\Phi _ { k } Q _ { k } ^ { \\pi _ { \\lambda , \\beta } } . \\end{align*}"}
{"id": "4385.png", "formula": "\\begin{align*} \\theta ( \\tau ) = \\beta ( \\tau ) \\left ( b ' ( \\tau ) - b ( \\tau ) \\right ) . \\end{align*}"}
{"id": "5852.png", "formula": "\\begin{align*} D f ( x ) b ( x ) = A f ( x ) . \\end{align*}"}
{"id": "5802.png", "formula": "\\begin{align*} \\mathbf { h } _ { m k } [ t ] = \\Bigl [ h _ { m k , 0 } [ t ] , \\ldots , h _ { m k , L _ { m k } - 1 } [ t ] \\Bigr ] ^ T , \\end{align*}"}
{"id": "772.png", "formula": "\\begin{align*} [ W ] = [ w _ 1 , \\ldots , w _ n ] = w _ 1 + \\cfrac { 1 } { w _ 2 + \\cfrac { 1 } { w _ 3 + \\cfrac { 1 } { \\ddots + \\cfrac { 1 } { w _ n } } } } \\end{align*}"}
{"id": "7482.png", "formula": "\\begin{align*} \\mathcal { H } ^ n = \\ ! \\left ( - 1 + \\frac { \\tau } { 2 } \\eta ^ n - \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n \\ ! \\right ) \\ ! \\phi ^ { n - 1 } + \\frac { \\tau ^ 2 } { 4 } \\Delta \\phi ^ { n - 1 } + 2 \\phi ^ n + \\tau ^ 2 \\left ( \\ ! ( \\vartheta ^ n + \\lambda ^ n ) \\phi ^ n - \\frac { 1 } { 2 } \\Delta \\phi ^ n - G ( \\phi ^ n ) \\ ! \\right ) \\end{align*}"}
{"id": "8339.png", "formula": "\\begin{align*} \\int _ a ^ b g \\dd g = \\frac { 1 } { 2 } ( g ( b ) ^ 2 - g ( a ) ^ 2 ) + \\frac { 1 } { 2 } \\sum _ { t \\in [ a , b ] } ( g ( t ) - g ( t - ) ) ^ 2 . \\end{align*}"}
{"id": "5767.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { g _ { i , \\delta } } f _ { i } ^ { j } = 0 , & \\ B _ { 4 R } ( x _ { i } , g _ { i , \\delta } ) ; \\\\ f _ { i } ^ { j } = b _ { i } ^ { j } , & \\ \\partial B _ { 4 R } ( x _ { i } , g _ { i , \\delta } ) , \\end{cases} \\end{align*}"}
{"id": "8340.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ a ^ b f _ n \\dd g = \\int _ a ^ b f \\dd g . \\end{align*}"}
{"id": "679.png", "formula": "\\begin{align*} d _ j : = \\sum _ { i = 1 } ^ { n _ 0 } c _ { i , j } \\partial _ { x _ { j } } \\bigg | _ { x = x _ { \\alpha } } , j = 1 , 2 . \\end{align*}"}
{"id": "164.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\alpha , \\beta } ( f ) ( x ) = ( 1 - x ^ 2 ) f '' ( x ) + ( ( \\beta - \\alpha ) - ( \\alpha + \\beta ) x ) f ' ( x ) . \\end{align*}"}
{"id": "384.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; f ( x ) : = ( g \\circ F ) ( x ) \\mbox { s u b j e c t t o } \\ ; \\ ; G ( x ) \\in X \\end{align*}"}
{"id": "9100.png", "formula": "\\begin{align*} v _ i \\big [ u ^ { i - 1 } \\beta _ i ( x y ) - u ^ { i - 1 } x \\beta _ i ( y ) - u ^ { i - 2 } \\beta _ m ( x ) \\beta _ { i - m } ( y ) - \\dots - u ^ { i - 1 } \\beta _ i ( x ) y \\big ] = 0 \\end{align*}"}
{"id": "7327.png", "formula": "\\begin{align*} Z : = \\frac { 4 | \\tilde { x } - \\tilde { y } | ^ 2 } { \\varepsilon ^ 4 } ( I + 2 ( \\tilde { x } - \\tilde { y } ) \\otimes ( \\tilde { x } - \\tilde { y } ) ) + 2 \\alpha \\begin{pmatrix} I & I \\\\ I & I \\end{pmatrix} . \\end{align*}"}
{"id": "4682.png", "formula": "\\begin{align*} & M _ 1 ( t ) = \\sum _ { i = 1 } ^ n | \\mu _ i ( t ) | ^ 2 , M _ 2 ( t ) = \\sup _ { 1 \\leq i \\leq n } | \\dot { x } _ i ( t ) - \\mu _ i ( t ) | , \\\\ & M _ 3 ( t ) = \\sup _ { 1 \\leq i \\leq n } \\bigg | \\dot { \\mu } _ i ( t ) + \\sum ^ n _ { j = 1 , j \\not = i } \\frac { a _ { i j } } { x ^ 3 _ { i j } ( t ) } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\mu _ k ( t ) } { x ^ 3 _ { i j } ( t ) } \\bigg | . \\end{align*}"}
{"id": "4966.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } d x \\ , \\mathbb { E } \\bigg [ \\Big | \\mathcal { F } ^ { - 1 } \\Big ( \\big ( 1 + | . | ^ 2 \\big ) ^ { - s } \\mathcal { F } \\Big ( \\rho [ \\ < P s i 2 > _ { n } ( t , . ) - \\ < P s i 2 > _ { n } ( s , . ) ] \\Big ) \\Big ) ( x ) \\Big | ^ { 2 p } \\bigg ] \\lesssim \\Bigg ( \\sum _ { k = 1 } ^ { 8 } I _ k \\Bigg ) ^ p , \\end{align*}"}
{"id": "8076.png", "formula": "\\begin{align*} \\theta _ { \\Sigma , \\epsilon } \\left ( F _ H \\right ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } : = \\left ( \\partial ^ { * } _ { \\Sigma , \\epsilon } F _ { H } \\right ) _ { H \\in \\mathrm { H a d } ( \\mathcal { M } ) } \\end{align*}"}
{"id": "2651.png", "formula": "\\begin{align*} Z f ( x + k , \\omega + l ) = e ^ { 2 \\pi i k \\cdot \\omega } Z f ( x , \\omega ) \\end{align*}"}
{"id": "421.png", "formula": "\\begin{align*} C _ { A ^ { 0 } _ { j } } ( g _ { 2 } , M ) , C _ { ( A ^ { 0 } _ { j } ) ^ { - 1 } } ( g _ { 2 } , M ) , \\sum _ { i , j = 1 } ^ { d } C _ { B ^ { i j } } ( g _ { 2 } , M ) \\leq g . \\end{align*}"}
{"id": "7101.png", "formula": "\\begin{align*} \\displaystyle \\lambda _ 1 \\leq & \\sqrt { \\left ( \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { m } 2 ^ 2 ( 2 ^ 2 - 1 ) - 0 \\right ) \\left ( \\frac { n - 1 } { n } \\right ) } \\\\ \\displaystyle \\lambda _ 1 \\leq & \\sqrt { 2 m \\left ( \\frac { n - 1 } { n } \\right ) } \\end{align*}"}
{"id": "3885.png", "formula": "\\begin{align*} w _ 1 ( M ) & = ( a - 3 ) t _ 1 - t _ 2 - \\dots - t _ b \\\\ & + ( 2 n - 2 - b ) s _ 1 - s _ 2 - \\dots - s _ { a - 1 } . \\end{align*}"}
{"id": "1362.png", "formula": "\\begin{align*} \\norm { u } _ { W ( \\sigma ' R ) , \\infty } \\leq C \\left ( \\frac { 1 \\vee \\norm { \\lambda ^ { - 1 } } _ { W ( R ) , q } \\norm { \\Lambda } _ { W ( R ) , p } } { ( \\sigma - \\sigma ' ) ^ 2 } \\right ) ^ \\kappa \\norm { u } _ { W ( \\sigma R ) , \\rho } ^ { \\prod _ { k = 1 } ^ \\infty \\gamma _ k } , \\end{align*}"}
{"id": "9370.png", "formula": "\\begin{align*} \\mathrm { L i } _ { 2 , \\lambda } ( t ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } ( 1 ) _ { n , \\frac { 1 } { \\lambda } } } { ( n - 1 ) ! n ^ { 2 } } ( - 1 ) ^ { n - 1 } t ^ { n } , ( \\mathrm { s e e } \\ [ 1 3 ] ) . \\end{align*}"}
{"id": "5875.png", "formula": "\\begin{align*} S _ t ( ( Y _ 0 , 0 , 0 ) , d W _ \\cdot ) = S _ t ( Y _ 0 , d W _ \\cdot ) , t \\ge 0 . \\end{align*}"}
{"id": "8962.png", "formula": "\\begin{align*} \\left | \\dot { \\xi } ( s ) \\right | = \\left ( \\frac { 1 } { C _ p q } \\right ) ^ \\frac { 1 } { q - 1 } \\left | D u ( \\xi ( s ) ) \\right | ^ \\frac { 1 } { q - 1 } = \\left ( \\frac { 1 } { q ^ { - 1 } p ^ { - \\frac { q } { p } } q } \\right ) ^ \\frac { 1 } { q - 1 } \\left | D u ( \\xi ( s ) ) \\right | ^ \\frac { 1 } { q - 1 } = p \\left | D u ( \\xi ( s ) ) \\right | ^ \\frac { 1 } { q - 1 } . \\end{align*}"}
{"id": "7255.png", "formula": "\\begin{align*} f _ 1 ( k ) : = - 2 k ^ 2 + 2 ( n - 2 ) k + ( n - 3 ) \\end{align*}"}
{"id": "6845.png", "formula": "\\begin{align*} \\pi _ i ( x ) = \\frac { 1 } { R ( i + 1 , i ) } \\sum \\limits _ { y \\gtrdot x } \\pi _ { i + 1 } ( y ) . \\end{align*}"}
{"id": "4333.png", "formula": "\\begin{align*} b ( \\tau ) = \\Psi _ \\infty \\exp { \\left ( \\left ( 1 - \\frac { 2 } { \\alpha } \\right ) \\left ( \\int _ { \\tau _ 0 } ^ \\tau 2 \\beta ( \\zeta ) d \\zeta + \\tau _ 0 \\right ) \\right ) } \\left [ 1 + O \\left ( I ^ \\eta ( \\tau ) \\right ) \\right ] \\tau \\to + \\infty . \\end{align*}"}
{"id": "1010.png", "formula": "\\begin{align*} \\varphi ^ { ( 3 ) } ( x ) & = C \\int _ { \\R ^ n \\setminus B _ 1 } \\bigg ( \\frac { 1 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - 1 } \\bigg ) ^ s \\frac { \\varphi ( y _ 1 ) } { \\vert x - y \\vert ^ n } \\dd y \\\\ & = C \\int _ { \\R ^ n _ + \\setminus B _ 1 ^ + } \\bigg ( \\frac { 1 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - 1 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert x - y \\vert ^ n } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ n } \\bigg ) \\varphi ( y _ 1 ) \\dd y . \\end{align*}"}
{"id": "150.png", "formula": "\\begin{align*} \\mathbf { D } ^ { \\alpha - 1 } ( f ) ( x ) : = \\frac { 1 } { 2 } \\left ( D ^ { \\alpha - 1 } \\left ( f \\right ) ( x ) - ( D ^ { \\alpha - 1 } ) ^ * \\left ( f \\right ) ( x ) \\right ) . \\end{align*}"}
{"id": "8875.png", "formula": "\\begin{align*} H X _ b ^ 1 ( X , A ) = \\begin{cases} 0 & e ( X ) = 0 \\\\ A ^ { e ( X ) - 1 } & 0 < e ( X ) < \\infty \\\\ \\bigoplus _ \\N A & e ( X ) = \\infty \\end{cases} \\end{align*}"}
{"id": "7735.png", "formula": "\\begin{align*} \\int _ { \\bar x } ^ { x _ 1 } \\int _ { \\R } \\phi _ { x x } \\cdot \\phi _ x \\psi ( t ) \\ , d t d x & = \\int _ { \\bar x } ^ { x _ 1 } \\int _ { \\R } ( \\phi _ { t t } \\cdot \\phi _ x \\psi ( t ) + a ( x ) \\phi _ t \\cdot \\phi _ x \\psi ( t ) ) \\ , d t d x \\\\ & = \\int _ { \\bar x } ^ { x _ 1 } \\int _ { \\R } ( - \\phi _ { t } \\cdot \\phi _ x \\psi ' ( t ) - \\phi _ t \\cdot \\phi _ { x t } \\psi ( t ) + a ( x ) \\phi _ t \\cdot \\phi _ x \\psi ( t ) ) \\ , d t d x . \\end{align*}"}
{"id": "7543.png", "formula": "\\begin{align*} \\log \\zeta ( s ) - \\log \\zeta ( 2 + i t ) = \\sum _ { | \\gamma - t | \\leq 1 } \\left [ \\log ( s - \\rho ) - \\log ( 2 + i t - \\rho ) \\right ] + \\mathcal { O } ( \\log t ) \\end{align*}"}
{"id": "2159.png", "formula": "\\begin{align*} ( - \\triangle _ { g } ) ^ { \\alpha } u + g ( u ) = K ( x ) f ( x , u ) , \\ \\ \\ \\mathbb { R } ^ { d } , \\end{align*}"}
{"id": "8281.png", "formula": "\\begin{align*} A ( N ( x ) , x ) = A ( x , N ( x ) ) = 0 , \\ \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "5842.png", "formula": "\\begin{align*} a \\cdot b = \\sum _ { i = 1 } ^ m a ^ i b ^ i . \\end{align*}"}
{"id": "1627.png", "formula": "\\begin{align*} \\mathcal { D } ( \\mathcal { L } ) = \\left \\lbrace f \\in \\mathcal { D } ( \\mathcal { E } ) : \\mathcal { L } f \\in L ^ 2 ( M ) \\right \\rbrace . \\end{align*}"}
{"id": "9539.png", "formula": "\\begin{align*} y & \\in \\partial V ( u - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } - \\bar c \\cdot \\bar x + S _ 0 ( \\bar x ) ) , \\\\ p _ t + y \\Delta s _ { t + 1 } & \\in N _ { D _ t } ( x _ t ) t = 0 , \\dots , T , \\\\ p _ { - 1 } + y \\bar c & \\in \\partial ( y S _ 0 ) ( \\bar x ) \\end{align*}"}
{"id": "7497.png", "formula": "\\begin{align*} V ( \\mathbf { x } ) = V _ { \\mathrm { h o } } ( \\mathbf { x } ) + \\frac { \\kappa } { 2 } \\begin{cases} \\sin ^ { 2 } \\left ( q _ { x } x \\right ) , & d = 1 , \\\\ \\sin ^ { 2 } \\left ( q _ { x } x \\right ) + \\sin ^ { 2 } \\left ( q _ { y } y \\right ) , & d = 2 , \\\\ \\end{cases} \\end{align*}"}
{"id": "9410.png", "formula": "\\begin{align*} \\varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots c _ { n - 1 } a _ n b _ n ) = \\varphi ( b _ 0 ) \\varphi ( a _ 1 ) \\varphi ( c _ 1 ) \\varphi ( a _ 2 ) \\cdots \\varphi ( c _ { n - 1 } ) \\varphi ( a _ n ) \\varphi ( b _ n ) = 0 . \\end{align*}"}
{"id": "8966.png", "formula": "\\begin{align*} C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ q = C _ p ( q - 1 ) p ^ q \\left [ D f ( \\xi ( t ) ) - D u ( \\xi ( t ) ) \\right ] \\cdot \\dot { \\xi } ( t ) - D f ( \\xi ( t ) ) \\cdot \\dot { \\xi } ( t ) . \\end{align*}"}
{"id": "6153.png", "formula": "\\begin{align*} V _ { n , m } ( x ) = 1 + \\sum _ { k = m + 1 } ^ { n + m + 1 } d _ k x ^ k \\end{align*}"}
{"id": "7039.png", "formula": "\\begin{align*} & q a _ 1 + q _ 1 p _ 1 + \\dots + q _ L p _ L \\equiv 1 \\quad \\\\ & \\Big ( \\| q \\| _ { H ^ 2 } ^ 2 + \\sum _ { j = 1 } ^ L \\| q _ j \\| _ { H ^ 2 } ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\le \\frac { C } { \\delta ^ { \\max m _ i } } \\end{align*}"}
{"id": "4625.png", "formula": "\\begin{gather*} \\Delta ( H ) = H \\otimes 1 + 1 \\otimes H , \\Delta ( E ) = E \\otimes 1 + K \\otimes E , \\Delta ( F ) = F \\otimes K ^ { - 1 } + 1 \\otimes F , \\\\ \\epsilon ( H ) = \\epsilon ( E ) = \\epsilon ( F ) = 0 , \\\\ S ( H ) = - H , S ( E ) = - K ^ { - 1 } E , S ( F ) = - F K . \\end{gather*}"}
{"id": "7768.png", "formula": "\\begin{gather*} \\| \\tilde \\phi _ 1 [ T ] - ( p , 0 ) \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } = \\| \\left ( \\phi _ 0 [ T ] - ( p , 0 ) \\right ) ^ p \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } \\leq 9 \\varepsilon \\| \\phi _ 0 [ T ] - ( p , 0 ) \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } , \\end{gather*}"}
{"id": "4789.png", "formula": "\\begin{align*} \\iota ( \\phi ) = \\sum _ { g \\in G } \\phi ( g ) g . \\end{align*}"}
{"id": "8208.png", "formula": "\\begin{align*} F _ { k , i + 1 } ( t ) = G _ { k , i } ( t ) + x _ { k + 1 - i } t G _ { k , i } ( t ) = F _ { k , i } ( t ) - x _ i t , \\end{align*}"}
{"id": "626.png", "formula": "\\begin{align*} \\phi ( x ) = \\min \\bigg \\{ j \\in \\{ 0 , 1 , 2 , \\ldots , x \\} : \\abs { P _ x ( a + ( b - a ) \\frac { 2 j + 1 } { 2 x + 2 } } - \\frac { q ^ \\prime } { x + 1 } \\leq 0 \\bigg \\} , \\ \\bigg ( q ^ \\prime = \\frac { d ( b - a ) } { 2 } + q \\bigg ) . \\end{align*}"}
{"id": "7136.png", "formula": "\\begin{align*} J _ 2 ^ n ( t , u ) & : = \\int _ u ^ t \\exp \\Big ( \\int _ s ^ t \\tilde { f } ' ( r , x _ r ) d r \\Big ) K _ s \\cdot K _ H ( s , u ) d s . \\end{align*}"}
{"id": "7463.png", "formula": "\\begin{align*} \\lambda _ { \\phi } ( 0 ) = \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi _ g | ^ 2 + V | \\phi _ g | ^ 2 + \\beta | \\phi _ g | ^ 4 - \\Omega \\ , \\overline { \\phi } _ g L _ z \\phi _ g \\right ) \\mathrm { d } \\mathbf { x } = \\mu ( \\phi _ g ) = : \\mu _ g , \\end{align*}"}
{"id": "6689.png", "formula": "\\begin{align*} z \\bigl ( ~ _ { s + 1 } F _ s ( 1 , \\ldots , 1 ; 2 , \\ldots , 2 ) ( z ) \\bigr ) = L i _ s ( z ) . \\end{align*}"}
{"id": "126.png", "formula": "\\begin{align*} u ^ 3 = u ( u v ) = u ^ 2 ( u v ) = ( u v ) ^ 2 = u ^ 2 v ^ 2 = 0 , \\end{align*}"}
{"id": "4447.png", "formula": "\\begin{align*} e _ { \\lambda } ^ { x } ( t ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( x ) _ { n , \\lambda } } { k ! } t ^ { k } , ( \\mathrm { s e e } \\ [ 2 , 4 , 5 , 6 ] ) , \\end{align*}"}
{"id": "3011.png", "formula": "\\begin{align*} R ^ { x y , z } _ { x ^ 3 y z } + R ^ { y z , x } _ { x ^ 3 y z } + R ^ { z x , y } _ { x ^ 3 y z } = 0 . \\end{align*}"}
{"id": "2836.png", "formula": "\\begin{align*} \\Bigg | \\int _ { \\mathbb { R } ^ 5 } \\left ( \\Delta ^ 2 \\varphi _ R \\right ) | u | ^ 2 d x \\Bigg | & = \\Bigg | \\frac { 1 } { R ^ 2 } \\int \\left ( \\Delta ^ 2 \\varphi \\right ) \\left ( \\frac { x } { R } \\right ) u ( t , x ) ^ 2 d x \\Bigg | \\\\ & \\le \\frac { C } { R ^ 2 } \\| u \\| _ 2 ^ 2 = \\frac { C } { R ^ 2 } \\| Q \\| _ 2 ^ 2 \\le 2 s _ c ( p - 1 ) \\delta _ 1 \\le 2 s _ c ( p - 1 ) \\delta ( t ) \\end{align*}"}
{"id": "9001.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ n ^ * ( k , F ^ * ) \\triangleq ( F ^ * _ 1 , F ^ * _ 2 , & \\dots , F ^ * _ { k - 1 } , ( U + n ) _ N , \\\\ & F ^ * _ { k } , F ^ * _ { k + 1 } , \\dots , F ^ * _ { K - 1 } ) , \\end{aligned} \\end{align*}"}
{"id": "627.png", "formula": "\\begin{align*} \\phi ( x ) \\ = \\ \\min \\{ j \\in \\ \\N : \\Phi ( x , j ) = 0 \\vee \\ j = x \\} , \\end{align*}"}
{"id": "8889.png", "formula": "\\begin{align*} \\begin{aligned} \\iota _ 0 : X & \\to X \\ast I \\\\ x & \\mapsto ( x , ( d ( x , x _ 0 ) , 0 ) ) \\end{aligned} & \\begin{aligned} \\iota _ 1 : X & \\to X \\ast I \\\\ x & \\mapsto ( x , ( 0 , d ( x , x _ 0 ) ) ) \\end{aligned} \\end{align*}"}
{"id": "5303.png", "formula": "\\begin{align*} \\langle v , a w \\rangle = \\langle a ^ * v , w \\rangle , \\forall a \\in A , v , w \\in V . \\end{align*}"}
{"id": "924.png", "formula": "\\begin{align*} a ^ { + } ( \\hat { n } + 1 - \\lambda ) _ { k , \\lambda } a & = a ^ { + } \\sum _ { l = 0 } ^ { k } \\binom { k } { l } ( \\hat { n } ) _ { l , \\lambda } ( 1 - \\lambda ) _ { k - l , \\lambda } a \\\\ & = \\sum _ { l = 0 } ^ { k } \\binom { k } { l } ( 1 - \\lambda ) _ { k - l , \\lambda } \\sum _ { m = 0 } ^ { l } S _ { 2 , \\lambda } ( l , m ) ( a ^ { + } ) ^ { m + 1 } a ^ { m + 1 } \\\\ & = \\sum _ { m = 0 } ^ { k } \\bigg ( \\sum _ { l = m } ^ { k } \\binom { k } { l } ( 1 - \\lambda ) _ { k - l , \\lambda } S _ { 2 , \\lambda } ( l , m ) \\bigg ) ( a ^ { + } ) ^ { m + 1 } a ^ { m + 1 } . \\end{align*}"}
{"id": "4542.png", "formula": "\\begin{align*} \\sum _ { n \\leq x / d ^ { 2 } s , \\ , ( n , s ) = 1 } a ( d ^ { 2 } n s + k ) = \\sum _ { \\delta \\mid s } \\mu ( \\delta ) \\ \\ \\sum _ { \\delta n _ { 1 } \\leq x / d ^ { 2 } s } a ( d ^ { 2 } \\ , \\delta \\ , n _ { 1 } \\ , s + k ) . \\end{align*}"}
{"id": "6596.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\le d < \\infty \\\\ ( d , m n h k ) = 1 } } W \\left ( \\frac { c d } { Q } \\right ) = W \\left ( \\frac { c } { Q } \\right ) + \\frac { \\phi ( m n h k ) } { m n h k } \\int _ { 1 } ^ { \\infty } W \\left ( \\frac { c x } { Q } \\right ) \\ , d x + \\int _ { 1 } ^ { \\infty } W \\left ( \\frac { c x } { Q } \\right ) \\ , d E . \\end{align*}"}
{"id": "4640.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 5 & 2 & 0 & 0 \\\\ 7 & 4 & \\overline { 4 } & 2 \\\\ 1 1 & 3 & 2 & 1 \\\\ \\overline { 6 } & 5 & \\overline { 3 } & 0 \\\\ 8 & 8 & 2 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "1232.png", "formula": "\\begin{align*} \\sum _ { s , t = 1 } ^ { Q } \\mu ( L _ { B , s } \\cap L _ { B , t } ) \\leq \\frac { C } { \\mu ( B ) } \\left ( \\sum _ { n = 1 } ^ { Q } \\mu ( L _ { B , n } ) \\right ) ^ 2 . \\end{align*}"}
{"id": "1039.png", "formula": "\\begin{align*} \\frac 1 { n + 2 } \\frac { \\partial u } { \\partial x _ 1 } ( 0 ) = \\int _ 0 ^ 1 r ^ { n + 1 } \\frac { \\partial u } { \\partial x _ 1 } ( 0 ) \\dd r = n \\gamma _ { n , s } \\int _ 0 ^ 1 \\int _ { \\R ^ n \\setminus B _ r } \\frac { r ^ { 2 s + n + 1 } y _ 1 u ( y ) } { ( \\vert y \\vert ^ 2 - r ^ 2 ) ^ s \\vert y \\vert ^ { n + 2 } } \\dd y \\dd r . \\end{align*}"}
{"id": "867.png", "formula": "\\begin{align*} \\varsigma _ { j , i } = s i g n \\left ( \\mathcal { X } _ j - \\epsilon _ i \\right ) i \\in [ m ] , \\end{align*}"}
{"id": "8844.png", "formula": "\\begin{align*} F _ 1 ^ { \\mathbb { P } ^ 1 } ( { \\bf v } _ 0 , { \\bf v } _ 1 ) = \\frac 1 { 2 4 } \\log \\bigl ( ( v _ 1 ) ^ 2 - e ^ { u } ( u _ 1 ) ^ 2 \\bigr ) - \\frac 1 { 2 4 } u . \\end{align*}"}
{"id": "6328.png", "formula": "\\begin{align*} \\langle \\nabla ^ 2 T , U \\otimes V \\otimes S \\rangle \\coloneqq \\langle \\nabla \\nabla T , U \\otimes V \\otimes S \\rangle = \\langle \\nabla _ U \\nabla T , V \\otimes S \\rangle \\end{align*}"}
{"id": "4630.png", "formula": "\\begin{align*} E _ r = \\big \\{ x \\in \\R ^ d : \\ 2 \\langle ( \\widehat { A } + ( \\widehat { A } ) ^ T ) ^ { - 1 } x , x \\rangle < r ^ 2 \\big \\} \\end{align*}"}
{"id": "5121.png", "formula": "\\begin{align*} B ' ( \\gamma ) = 2 a n \\ , \\frac { e ^ { 2 a \\gamma } \\left ( e ^ { 2 a \\gamma } - 1 - 2 a \\gamma \\right ) } { \\left ( e ^ { 2 a \\gamma } - 1 \\right ) ^ 2 } > 0 . \\end{align*}"}
{"id": "3449.png", "formula": "\\begin{align*} R _ M f ( x ) & = - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\int _ { Q } \\big ( \\psi _ { Q } ( x , y ) - \\psi _ { Q } ( x , x _ { Q } ) \\big ) q _ { Q } f ( y ) d \\omega ( y ) \\\\ & \\quad - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\int _ { Q } \\psi _ { Q } ( x , x _ Q ) \\big ( q _ { Q } f ( x _ Q ) - q _ { Q } f ( y ) \\big ) d \\omega ( y ) \\\\ & = R ^ 1 _ M f ( x ) + R ^ 2 _ M f ( x ) . \\end{align*}"}
{"id": "2552.png", "formula": "\\begin{align*} V _ P D _ L J V _ { - Q } J ^ { - 1 } = \\begin{pmatrix} L & L Q \\\\ P L & P L Q + L ^ { - T } \\end{pmatrix} \\end{align*}"}
{"id": "5227.png", "formula": "\\begin{align*} \\| B f \\| _ { F _ { r t } ^ s ( K ) } & \\leq \\sum _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ \\| f \\| _ { F _ { r t } ^ s ( K ) } \\\\ & = q ^ { - k } ( q - 1 ) ^ { - 1 } \\| f \\| _ { F _ { r t } ^ s ( K ) } . \\\\ \\end{align*}"}
{"id": "4793.png", "formula": "\\begin{align*} \\alpha = \\alpha ^ \\sigma , \\epsilon ( \\alpha ) = 0 , \\alpha ^ 2 = \\alpha , \\alpha * \\alpha = ( \\lambda - \\mu ) \\alpha + \\mu + ( r - \\mu ) \\delta _ e . \\end{align*}"}
{"id": "5674.png", "formula": "\\begin{align*} \\rho _ 1 ( x \\cdot y ) \\cdot \\rho _ 2 y = \\rho _ 3 x \\end{align*}"}
{"id": "6418.png", "formula": "\\begin{align*} \\mathcal { R } ( R , L ) & : = - 2 \\sum _ { i } \\beta _ i \\otimes R ( L ( \\beta _ i ) ) , \\\\ \\mathcal { T } ( R , L ) & : = - \\sum _ i \\big ( L ^ * ( \\alpha _ i ) \\otimes 1 + 1 \\otimes \\alpha _ i \\big ) \\circ \\big ( L ^ * ( R ( \\alpha _ i ) ) \\otimes 1 + 1 \\otimes R ( \\alpha _ i ) \\big ) , \\end{align*}"}
{"id": "3683.png", "formula": "\\begin{align*} B _ t + \\nabla \\times ( ( \\nabla \\times B ) \\times B ) = & \\ \\mu \\Delta B , \\\\ \\nabla \\cdot B = & \\ 0 \\end{align*}"}
{"id": "3038.png", "formula": "\\begin{align*} m = 6 \\ , , p = 4 \\end{align*}"}
{"id": "3085.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\varepsilon x _ 1 \\ , , x _ 2 ^ \\prime = \\varepsilon ^ 4 x _ 2 \\ , , y _ 1 ^ \\prime = \\varepsilon ^ 2 y _ 1 \\ , , y _ 2 ^ \\prime = \\varepsilon ^ 3 y _ 2 \\ , . \\end{align*}"}
{"id": "5989.png", "formula": "\\begin{align*} \\| I _ { p , q } \\| _ { H S } ^ { 2 } = \\sum _ { n = 0 } ^ { \\infty } \\| I _ { p , q } \\bar { C } _ { n } ^ { \\lambda , \\beta } \\| _ { q , \\kappa , \\pi _ { \\lambda , \\beta } } ^ { 2 } & = \\sum _ { n = 0 } ^ { \\infty } ( n ! ) ^ { 1 + \\kappa } 2 ^ { n q } \\left ( ( n ! ) ^ { 1 + \\kappa } 2 ^ { n p } \\right ) ^ { - 1 } = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 2 ^ { p - q } } \\right ) ^ { n } < \\infty . \\end{align*}"}
{"id": "6380.png", "formula": "\\begin{align*} \\dim W > 0 , \\Omega = G \\cap W \\overline { \\Omega } \\cap \\partial \\overline { G } = \\overline { \\Omega } \\cap b \\overline { G } , \\end{align*}"}
{"id": "6936.png", "formula": "\\begin{align*} \\tilde { r } _ { h , i } ( w ) = F _ h ( \\varphi _ i ) - a _ h ( w _ H , \\varphi _ i ) \\ , , i \\in I _ h \\ , ; \\end{align*}"}
{"id": "5498.png", "formula": "\\begin{align*} \\Phi _ { \\gamma , \\delta } ( \\Phi _ { \\gamma , \\delta } ( d , e , s - r ) , e , t - s ) = \\Phi _ { \\gamma , \\delta } ( d , e , t - r ) . \\end{align*}"}
{"id": "3524.png", "formula": "\\begin{align*} W _ 1 T = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + O \\left ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } \\right ) . \\end{align*}"}
{"id": "6253.png", "formula": "\\begin{align*} q \\cdot \\mathcal { E } ( q , \\lambda ( q , t ) ) = \\frac { 1 - q } { 1 - q e ^ { \\lambda ( q , t ) \\cdot ( 1 - q ) } } - 1 \\end{align*}"}
{"id": "7191.png", "formula": "\\begin{align*} X ^ T ( t ) : = \\begin{cases} X ( t ) & [ 0 , T ] , \\\\ X ( T ) + ( t - T ) V ( T ) & [ T , \\infty ) , \\\\ t V _ 0 & ( - \\infty , 0 ] . \\end{cases} \\end{align*}"}
{"id": "6099.png", "formula": "\\begin{align*} U ^ { 1 } ( r , \\theta ) = \\left ( r ^ { \\lambda _ { p } } a ^ { 1 } ( \\theta ) , r ^ { \\lambda _ { p } } b ^ { 1 } ( \\theta ) , r ^ { \\lambda _ { p } + 1 - m } c ^ { 1 } ( \\theta ) \\right ) , \\end{align*}"}
{"id": "927.png", "formula": "\\begin{align*} ( a ^ { + } ) ^ { k + 1 } a ^ { k + 1 } & = \\sum _ { m = 0 } ^ { k + 1 } S _ { 1 , \\lambda } ( k + 1 , m ) ( a ^ { + } a ) _ { m , \\lambda } \\\\ & = \\sum _ { m = 1 } ^ { k + 1 } S _ { 1 , \\lambda } ( k + 1 , m ) ( \\hat { n } ) _ { m , \\lambda } \\\\ & = \\sum _ { m = 0 } ^ { k } S _ { 1 , \\lambda } ( k + 1 , m + 1 ) ( \\hat { n } ) _ { m + 1 , \\lambda } , ( k \\ge 0 ) . \\end{align*}"}
{"id": "4193.png", "formula": "\\begin{align*} H ( v ) : = \\beta ( v ) ( d \\alpha _ 0 ) ( v , u ) , \\end{align*}"}
{"id": "1711.png", "formula": "\\begin{align*} \\frac 1 q = \\frac { 1 \u2010 \\lambda } { p _ 1 } + \\frac { \\lambda } { p _ 0 } , \\frac 1 2 = \\frac { 1 \u2010 \\tilde \\lambda } { p _ 1 } + \\frac { \\tilde \\lambda } { p _ 0 } . \\end{align*}"}
{"id": "5350.png", "formula": "\\begin{align*} \\left \\langle \\tilde a _ i , x \\right \\rangle \\leqslant b _ i , i = 1 , \\ldots , m . \\end{align*}"}
{"id": "3516.png", "formula": "\\begin{align*} & \\sum _ { n \\leq a t _ 3 } \\frac { E ( s _ 1 , s _ 3 ; a t _ 3 , n , M ) } { n ^ { s _ 2 } } + \\sum _ { a t _ 3 < n \\leq N } \\frac { E ( s _ 1 , s _ 3 ; n , M ) } { n ^ { s _ 2 } } \\\\ & - B _ 2 - B _ 3 - B _ 4 - C _ 2 - C _ 3 - C _ 4 \\\\ & \\ll \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 & ( \\sigma _ 2 = 1 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "5552.png", "formula": "\\begin{align*} \\| g - h \\| ^ 2 = | g ( \\infty ) + \\eta | ^ 2 + \\int _ 0 ^ { \\infty } | g ' ( x ) | ^ 2 e ^ { \\gamma x } d x \\end{align*}"}
{"id": "3336.png", "formula": "\\begin{align*} & \\theta ( z , t ) \\theta ( x , y ) - \\theta ( y , t ) \\theta ( x , z ) - \\theta ( x , [ y , z , t ] ) + D ( y , z ) \\theta ( x , t ) = 0 , \\\\ & \\theta ( z , t ) D ( x , y ) - D ( x , y ) \\theta ( z , t ) + \\theta ( [ x , y , z ] , t ) + \\theta ( z , [ x , y , t ] ) = 0 , \\end{align*}"}
{"id": "4508.png", "formula": "\\begin{align*} - \\int _ { H _ { \\sigma _ 2 } } ^ T N ( \\sigma _ 2 , t ) \\frac { d } { d t } \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t } } } { t } \\Big ) \\le \\sum _ { k = 0 } ^ { K - 1 } N ( \\sigma _ 2 , t _ { k + 1 } ) \\Big ( \\frac { x ^ { - \\frac { 1 } { R \\log t _ k } } } { t _ k } - \\frac { x ^ { - \\frac { 1 } { R \\log t _ { k + 1 } } } } { t _ { k + 1 } } \\Big ) . \\end{align*}"}
{"id": "2190.png", "formula": "\\begin{align*} \\mu _ w ( 0 , 0 ) = J ( 0 ) = 0 < m _ 1 \\leq \\mu _ w ( 1 , 1 ) = J ( w ) . \\end{align*}"}
{"id": "8705.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { 5 + x _ 2 ^ 2 } - \\sqrt { 5 } & = \\sqrt { 5 + s _ { 2 1 } ( x ) } - \\sqrt { 5 } + \\sqrt { 5 + s _ { 2 2 } ( x ) } - \\sqrt { 5 + s _ { 2 1 } ( x ) } \\\\ & \\geq \\frac { \\sqrt { 8 } - \\sqrt { 5 } } { 3 } s _ { 2 1 } ( x ) + \\frac { \\sqrt { 9 } - \\sqrt { 8 } } { 1 } \\bigl ( s _ { 2 2 } ( x ) - s _ { 2 1 } ( x ) \\bigr ) , \\end{aligned} \\end{align*}"}
{"id": "1798.png", "formula": "\\begin{align*} P ^ b _ { Q } : = \\left ( \\begin{array} { c c } { } ^ b S _ { + } ^ 2 & { } ^ b S _ { + } ( I + { } ^ b S _ { + } ) Q ^ b \\\\ { } ^ b S _ { - } D ^ + & I - { } ^ b S _ { - } ^ 2 \\end{array} \\right ) \\end{align*}"}
{"id": "9036.png", "formula": "\\begin{align*} d ^ 2 ( \\rho ^ 0 , \\rho ^ 1 ) : = \\min _ { ( \\rho , u ) \\in K } \\sum _ { i = 1 } ^ s \\int _ 0 ^ 1 \\int _ { \\Omega } D _ i ^ { - 1 } | u _ i | ^ 2 \\rho _ i d x d t . \\end{align*}"}
{"id": "6060.png", "formula": "\\begin{align*} \\Phi _ p : \\ , D _ \\rho \\ , \\longrightarrow \\ , U ^ p _ \\rho = \\Phi _ p ( D _ \\rho ) \\subset \\mathcal { O } ; \\Phi _ p ( 0 ) = p \\end{align*}"}
{"id": "2674.png", "formula": "\\begin{align*} \\mathfrak { F } _ \\alpha ( g ) = \\{ \\alpha \\Z ^ { 2 d } \\mid G ( g , \\alpha \\Z ^ { 2 d } ) \\} \\end{align*}"}
{"id": "1925.png", "formula": "\\begin{align*} u _ N ( t , x ) = \\sum ^ N _ { i = 1 } \\zeta _ i ( t ) e _ i ( x ) , \\end{align*}"}
{"id": "8402.png", "formula": "\\begin{align*} \\Theta _ 2 = \\nabla c _ 0 ^ { i n } \\otimes \\nabla g + \\nabla g \\otimes \\nabla c _ 0 ^ { i n } \\| ( \\Theta _ 3 , \\Theta _ 4 ) \\| _ { L ^ \\infty ( 0 , T ; L ^ 2 ( \\Omega ) ) } \\le C \\varepsilon ^ 2 , \\end{align*}"}
{"id": "7918.png", "formula": "\\begin{align*} \\partial ^ 2 _ { t _ i t _ j } ( \\Phi \\circ \\varphi ) = \\sum _ { l , k } ( \\partial ^ 2 _ { x _ l x _ k } \\Phi ) ( \\partial _ { t _ j } \\varphi ^ l ) ( \\partial _ { t _ i } \\varphi ^ k ) + \\sum _ l ( \\partial _ { x _ l } \\Phi ) ( \\partial ^ 2 _ { t _ i t _ j } \\varphi ^ l ) . \\end{align*}"}
{"id": "2713.png", "formula": "\\begin{align*} \\mathbf { F } _ d = \\left ( \\R [ X _ 1 , \\ldots , X _ n ] _ { \\leq d } \\right ) _ { n > 0 } . \\end{align*}"}
{"id": "8127.png", "formula": "\\begin{align*} \\kappa ( g ) = \\sum _ { ( d , e ) \\in D _ { 1 } ( P G ) } | g ( d ) \\cap g ( e ) | = 5 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "8734.png", "formula": "\\begin{align*} p ( x ) = \\phi \\bigl ( g _ { j ( 1 ) } ( x ) , \\ldots , g _ { j ( d ) } ( x ) \\bigr ) j ( 1 ) , \\ldots , j ( d ) \\in \\{ 1 , \\ldots , m \\} , \\deg ( p ) = \\delta . \\end{align*}"}
{"id": "827.png", "formula": "\\begin{align*} d _ \\rho ( ( x _ 0 , y _ 0 ) , ( x , y ) ) = \\int _ 0 ^ L \\gamma _ { \\R } ( t ) ^ { - \\beta } \\ , \\max \\{ | \\gamma _ Z ^ \\prime | ( t ) , | \\gamma _ { \\R } ^ \\prime | ( t ) \\} \\ , d t & \\ge \\int _ 0 ^ L \\gamma _ { \\R } ( t ) ^ { - \\beta } | \\gamma _ { \\R } ^ \\prime | ( t ) \\ , d t \\\\ & \\ge \\frac { 1 } { \\beta - 1 } \\bigg \\vert \\frac { 1 } { y _ 0 ^ { \\beta - 1 } } - \\frac { 1 } { y ^ { \\beta - 1 } } \\bigg \\vert . \\end{align*}"}
{"id": "258.png", "formula": "\\begin{align*} \\prod _ { i \\in I ^ * } c _ i = 1 - 1 ) \\Longleftrightarrow \\bigg ( | I _ \\kappa ^ * | - \\sum _ { i \\in I _ \\kappa ^ * } c _ i \\bigg ) \\equiv 0 2 ) \\mod 4 . \\end{align*}"}
{"id": "1991.png", "formula": "\\begin{align*} & M _ t = \\exp ^ { \\triangleleft } ( t h ) = 1 + \\sum _ { \\tau \\in \\mathcal T } \\operatorname { c m } ( \\tau ) P _ h ( \\tau ) \\frac { t ^ { | \\tau | } } { | \\tau | ! } = 1 + \\sum _ { \\tau \\in \\mathcal T } \\frac { 1 } { \\tau ! \\sigma ( \\tau ) } P _ h ( \\tau ) t ^ { | \\tau | } , \\end{align*}"}
{"id": "5766.png", "formula": "\\begin{align*} b _ { i } ^ { j } ( \\cdot ) = d _ { g _ { i , \\delta } } ( p _ { i } ^ { j } , \\cdot ) - d _ { g _ { i , \\delta } } ( p _ { i } ^ { j } , x _ { i } ) , \\end{align*}"}
{"id": "2960.png", "formula": "\\begin{align*} \\Lambda ^ { ( h : h ' ) } _ { n , 2 , 0 } ( \\mathcal { W } _ { k } ( 0 ) \\setminus \\mathcal S _ k ) = \\frac { c \\left ( 1 + o ( 1 ) \\right ) } { n ^ { h + h ' } \\delta ^ 4 _ n ( k ) } \\cdot | \\mathcal { W } _ { k } ( 0 ) \\setminus \\mathcal S _ k | \\cdot | \\mathcal { I } _ h \\times \\mathcal { I } _ { h ' } | \\cdot O ( n ^ { - 1 } ) = O ( n ^ { - 1 } ) \\end{align*}"}
{"id": "3710.png", "formula": "\\begin{align*} B ^ { k } _ t + a B ^ { k - 1 } J _ x ^ { k } + b J ^ { k - 1 } B _ x ^ { k } + \\mu \\Lambda ^ \\alpha B ^ { k } = & \\ 0 , \\\\ B _ x ^ k = & \\ \\mathcal H J ^ k \\end{align*}"}
{"id": "2492.png", "formula": "\\begin{align*} ( F _ 1 * F _ 2 ) ( \\mathbf { h } _ 0 ) = \\int _ { \\mathbf { H } } F _ 1 ( \\mathbf { h } ) F _ 2 ( \\mathbf { h } ^ { - 1 } \\mathbf { h } _ 0 ) \\ , d \\mathbf { h } . \\end{align*}"}
{"id": "1177.png", "formula": "\\begin{align*} & \\Gamma ^ { A i } _ 1 = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta = 2 \\pi / 3 \\right \\} , \\Gamma ^ { A i } _ 2 = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta = \\pi \\right \\} , \\\\ & \\Gamma ^ { A i } _ 3 = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta = - 2 \\pi / 3 \\right \\} , \\Gamma ^ { A i } _ 4 = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta = 0 \\right \\} , \\end{align*}"}
{"id": "638.png", "formula": "\\begin{align*} \\begin{cases} \\ i = ( x + 1 ) ( f _ 0 ( x , n ) - g _ 0 ( x , n ) ) \\\\ [ 8 p t ] \\ j = h _ 0 ( x , n ) \\end{cases} . \\end{align*}"}
{"id": "6083.png", "formula": "\\begin{align*} ( \\lambda - \\overline { \\lambda } ) g \\varphi ^ { j } & = \\lambda \\overline { h } ^ { j } - \\overline { \\lambda } h ^ { j } , \\\\ ( \\lambda - \\overline { \\lambda } ) \\psi ^ { j } & = h ^ { j } - \\overline { h } ^ { j } . \\end{align*}"}
{"id": "6775.png", "formula": "\\begin{align*} \\tau _ { r _ j + 1 } ^ { v _ j + 1 } \\cdot \\tau _ { r _ { j + 1 } } ^ { - ( v _ { j + 1 } + 1 ) } = \\tau _ { r _ j + 1 } ^ { v _ j + 1 } \\cdot \\tau _ { r _ j + 1 } ^ { - ( v _ j + 1 ) } = 1 . \\end{align*}"}
{"id": "6716.png", "formula": "\\begin{align*} P _ { { \\bf b } _ s , d _ s } { g _ 0 ( t ) } + P _ { { \\bf b } _ s , d _ s } { g _ 1 ( t ) } ~ _ { 2 } \\mathcal { F } _ { 1 } ( { \\bf a } _ 1 ; { \\bf b } _ 1 ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + P _ { d _ s , { \\bf b } _ s } { g _ s ( t ) } ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } _ s ; { \\bf b } _ s ) ( \\alpha _ s ) ^ { q ^ d } = 0 . \\end{align*}"}
{"id": "2689.png", "formula": "\\begin{align*} | V _ g f ( x , \\omega ) | = \\mathcal { O } ( e ^ { - \\frac { \\pi } { 2 } ( a x ^ 2 + b \\omega ^ 2 ) } ) , \\end{align*}"}
{"id": "3354.png", "formula": "\\begin{align*} \\theta _ T ( u , v ) x = [ x , T u , T v ] + T \\Big ( \\theta ( x , T v ) u - D ( x , T u ) v \\Big ) , \\ \\forall x \\in L , \\ u , v \\in V . \\end{align*}"}
{"id": "9293.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\hat { L } u , \\ , \\hat { L } = \\frac { \\partial } { \\partial x } + \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } , \\ ; u ( x , 0 ) = x \\end{align*}"}
{"id": "8653.png", "formula": "\\begin{align*} & A = 3 a ^ 2 - a d + b c + 9 c ^ 2 \\\\ & B = 6 a b + 1 8 c d \\\\ & C = - 3 a d + 3 b ^ 2 + 3 b c + 9 d ^ 2 \\end{align*}"}
{"id": "7653.png", "formula": "\\begin{align*} \\int _ { R _ k \\le | \\mathbf { x } | \\le R _ k + 1 } \\tilde { u } _ n ^ 2 \\eta _ k ^ 2 \\le C e ^ { - \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } R _ k } R _ k = o ( 1 ) k \\to + \\infty \\ ; , \\end{align*}"}
{"id": "488.png", "formula": "\\begin{align*} k _ { 0 } ( u , \\omega ) : = \\mathcal { J } \\setminus \\{ u \\} \\cup \\{ \\omega \\} , k _ { 0 } ( u , \\mathfrak { g } _ { c } ) = k _ { 0 } ( \\mathfrak { g } _ { c } , u ) : = \\mathcal { J } \\setminus \\mathfrak { g } _ { r } \\cup \\{ \\alpha _ { 1 } , i _ { 3 } , u \\} \\end{align*}"}
{"id": "5485.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\bigg \\| \\alpha ( x ) - \\frac { 1 } { t } \\int _ 0 ^ t S _ { t - s } \\alpha ( \\xi ( s ; x ) ) d s \\bigg \\| = 0 . \\end{align*}"}
{"id": "6584.png", "formula": "\\begin{align*} \\ell = \\frac { | m h \\pm n k | } { d } . \\end{align*}"}
{"id": "1985.png", "formula": "\\begin{align*} \\dot M _ t ( x ) = M _ t ( x ) h ( x M _ t ( x ) ) = h ( x ) + ( ( M _ t - 1 ) \\triangleleft h ) ( x ) . \\end{align*}"}
{"id": "2143.png", "formula": "\\begin{align*} \\begin{aligned} & P _ { n , k } ( J _ { n , k } ^ { \\mathcal { S } } = l | W _ { \\mathcal { S } ( n , k ; M _ n ) } ) = P _ { n , k } ( A ^ { ( n ) } _ { M _ n , n , l - 1 } | W _ { \\mathcal { S } ( n , k ; M _ n ) } ) = \\frac { P _ { n , k } ( A ^ { ( n ) } _ { M _ n , n , l - 1 } ) } { P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } = \\\\ & \\frac 1 { P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } \\frac { \\binom { M _ n } { l - 1 } ( k ) _ { l - 1 } ( k ( n - 1 ) ) _ { M _ n - l + 1 } } { ( k n ) _ { M _ n } } , \\ l \\in \\{ 2 , \\cdots , k \\} . \\end{aligned} \\end{align*}"}
{"id": "1377.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 A ( n _ { i j } , n _ { i j x } ) \\varphi _ x d x + \\int _ 0 ^ 1 ( n _ { i j } - b _ i ) \\varphi d x = 0 , \\quad \\forall \\varphi \\in H _ 0 ^ 1 ( 0 , 1 ) , \\end{align*}"}
{"id": "766.png", "formula": "\\begin{align*} j ^ * [ \\mathfrak { h } _ \\omega ] & = 0 \\in H H ^ { p + 1 } ( \\mathfrak { A } _ 0 ) \\\\ j ^ * [ \\mathfrak { c } _ \\omega ] & = 0 \\in H C _ \\lambda ^ { p } ( \\mathfrak { A } _ 0 ) . \\end{align*}"}
{"id": "3036.png", "formula": "\\begin{align*} d Z _ { \\lambda } = E _ { \\lambda } + F _ { 0 } + F _ { 1 } + F _ { 2 } , \\end{align*}"}
{"id": "75.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq \\frac { q _ v ^ { n _ { v , \\nu _ v } / 2 } + 1 } { q _ v ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "2407.png", "formula": "\\begin{align*} f = \\sum _ { k , l \\in \\Z ^ d } c _ { k , l } M _ { \\beta l } T _ { \\alpha k } g \\end{align*}"}
{"id": "5125.png", "formula": "\\begin{align*} \\frac { f ' ( \\gamma ) } { f ( \\gamma ) } = - \\frac { h ' ( \\gamma ) } { h ( \\gamma ) } . \\end{align*}"}
{"id": "3482.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) & = \\sum _ { n = 1 } ^ \\infty \\sum _ { m > n } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } \\\\ & = \\left ( \\sum _ { n \\leq N } \\sum _ { n < m \\leq M } + \\sum _ { n \\leq N } \\sum _ { m > M } + \\sum _ { n = N + 1 } ^ \\infty \\sum _ { m > n } \\right ) \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } \\\\ & = A + B + C , \\end{align*}"}
{"id": "2250.png", "formula": "\\begin{align*} J ( Y _ m ^ { M , N } ) - J ( Y _ { m - 1 } ^ { M , N } ) = \\big < A Y _ m ^ { M , N } , \\overline { \\partial } Y _ m ^ { M , N } \\big > + \\big < F ( Y _ m ^ { M , N } ) , \\overline { \\partial } Y _ m ^ { M , N } \\big > + \\big < Y _ m ^ { M , N } - Y _ { m - 1 } ^ { M , N } , \\overline { \\partial } Y _ m ^ { M , N } \\big > . \\end{align*}"}
{"id": "8672.png", "formula": "\\begin{align*} 1 - F _ n ( x s _ n ) = \\exp \\left ( - \\frac { 1 } { 2 } x ^ 2 Q _ N ( x ) \\right ) \\left ( 1 - \\Phi ( x ) + \\vartheta \\lambda _ N e ^ { - x ^ 2 / 2 } \\right ) . \\end{align*}"}
{"id": "590.png", "formula": "\\begin{align*} S \\ = \\ \\bigcup \\limits _ { i = 1 } ^ { m _ + } \\ L _ i ^ + , T \\ = \\ \\bigcup \\limits _ { i = 1 } ^ { m _ - } \\ L _ i ^ - . \\end{align*}"}
{"id": "2788.png", "formula": "\\begin{align*} \\mathcal { Y } _ 1 = \\int _ { \\mathbb { R } ^ N } G _ + ( x - z ) \\left ( \\int _ { \\mathbb { R } ^ N } G _ - ( z - y ) F ( y ) d y \\right ) d z . \\end{align*}"}
{"id": "2629.png", "formula": "\\begin{align*} Z f ( x , \\omega ) = e ^ { 2 \\pi i x \\cdot \\omega } Z \\widehat { f } ( \\omega , - x ) , \\end{align*}"}
{"id": "8417.png", "formula": "\\begin{gather*} \\lim _ { k \\to \\infty } W _ { 1 } \\big ( \\mu , \\mathrm { l a w } ( X _ { \\eta k } ) \\big ) = \\lim _ { k \\to \\infty } W _ { 1 } \\big ( \\mathrm { l a w } ( \\tilde { Y } _ { k } ) , \\tilde { \\mu } _ { \\eta } \\big ) = 0 . \\end{gather*}"}
{"id": "1676.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } H _ n ( s _ 1 , \\ldots , s _ r ) = \\zeta ( s _ 1 , \\ldots , s _ r ) \\quad \\lim \\limits _ { n \\rightarrow \\infty } \\overline { H } _ n ( s _ 1 , \\ldots , s _ r ) = t ( s _ 1 , \\ldots , s _ r ) , \\end{align*}"}
{"id": "4003.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\overline { \\xi ( x ) } \\xi ^ \\prime ( x ) d x = \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\frac { d } { d x } | \\xi ( x ) | ^ 2 d x + i \\int _ 0 ^ 1 \\Im ( \\overline { \\xi ( x ) } \\xi ^ \\prime ( x ) ) d x = i \\int _ 0 ^ 1 \\Im ( \\overline { \\xi ( x ) } \\xi ^ \\prime ( x ) ) d x , \\end{align*}"}
{"id": "7858.png", "formula": "\\begin{align*} J \\cup K \\cup L & = ( \\mathbb { N } \\setminus \\{ \\bar { a } _ n \\} ) \\cup \\{ \\bar { a } _ n \\} = \\mathbb { N } , \\end{align*}"}
{"id": "2967.png", "formula": "\\begin{align*} \\sum _ { B \\subset S _ 6 } E ^ { ( S _ 3 , B ) } _ { n , \\ell } & = \\frac { - \\ell _ 1 \\left ( n - \\ell _ 3 \\right ) \\left ( \\sum _ { j = 0 } ^ 4 n ^ j p _ j \\left ( \\ell _ 1 , \\ell _ 2 , \\ell _ 3 \\right ) \\right ) } { n ^ 6 ( n - 1 ) ( n - 2 ) ( n - 3 ) ( n - 4 ) ( n - 5 ) } , \\end{align*}"}
{"id": "6760.png", "formula": "\\begin{align*} \\mathbf L = \\begin{bmatrix} \\mathbf L _ { 1 1 } & \\mathbf L _ { 1 2 } \\\\ \\mathbf L _ { 2 1 } & \\mathbf L _ { 2 2 } \\end{bmatrix} , \\end{align*}"}
{"id": "9216.png", "formula": "\\begin{align*} 0 & = \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x + ( ( x - J ^ A _ \\gamma x ) - ( x - J ^ A _ { \\gamma ' } x ) ) } \\\\ & = \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x + \\vert \\gamma \\vert ( \\gamma ^ { - 1 } ( x - J ^ A _ \\gamma x ) - \\gamma '^ { - 1 } ( x - J ^ A _ { \\gamma ' } x ) ) } \\\\ & \\geq \\norm { J ^ A _ \\gamma x - J ^ A _ { \\gamma ' } x } . \\end{align*}"}
{"id": "6811.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l } u _ { t } = - \\gamma \\boldsymbol { D } _ { x } ^ { \\alpha } u + F ( u ) + \\boldsymbol { D } _ { x } ^ { \\alpha _ { 1 } } u ^ { 3 } , & x \\in \\mathbb { Q } _ { p } ^ { n } , \\ t \\in \\left [ 0 , T \\right ] ; \\\\ & \\\\ u ( 0 ) = f _ { 0 } \\in \\mathcal { H } _ { \\infty } , & \\end{array} \\right . \\end{align*}"}
{"id": "9377.png", "formula": "\\begin{align*} \\phi _ { n + 1 , \\lambda } ( x ) & = x \\phi _ { n , \\lambda } ( x ) + \\bigg ( x \\frac { d } { d x } - n \\lambda \\bigg ) \\phi _ { n , \\lambda } ( x ) \\\\ & = x \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( 1 - \\lambda ) _ { n - k , \\lambda } \\phi _ { k , \\lambda } ( x ) . \\end{align*}"}
{"id": "7417.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega _ 1 ( x ) } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta _ { 2 , x , r } ) - f ( \\eta _ { 3 , x , r } ) ] d \\nu _ { b } \\Big | , \\end{align*}"}
{"id": "2802.png", "formula": "\\begin{align*} 0 \\ge \\Phi ( f ) = B ( f , f ) = \\frac { 1 } { 2 } \\int \\left ( L _ + f _ 1 \\right ) f _ 1 + \\frac { 1 } { 2 } \\int \\left ( L _ - f _ 2 \\right ) f _ 2 . \\end{align*}"}
{"id": "909.png", "formula": "\\begin{align*} ( x ) _ { 0 , \\lambda } = 1 , ( x ) _ { k , \\lambda } = x ( x - \\lambda ) \\cdots \\big ( x - ( k - 1 ) \\lambda \\big ) , ( k \\ge 1 ) . \\end{align*}"}
{"id": "1582.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } ( \\mathcal { S } _ L ( \\tau ) L f , \\mathcal { S } _ L ( \\tau ) f ) _ { L ^ 2 _ v } d \\tau = - \\int _ 0 ^ { + \\infty } \\frac d { d \\tau } \\Vert \\mathcal { S } _ L ( \\tau ) f \\Vert _ { L ^ 2 } ^ 2 d \\tau = \\Vert f \\Vert _ { L ^ 2 } ^ 2 - \\lim _ { \\tau \\to \\infty } \\Vert \\mathcal { S } _ L ( \\tau ) f \\Vert _ { L ^ 2 } ^ 2 = \\Vert f \\Vert _ { L ^ 2 } ^ 2 , \\end{align*}"}
{"id": "18.png", "formula": "\\begin{align*} \\{ ( \\beta , j ) < ( \\gamma , k ) \\} : = \\{ \\beta + j < \\gamma + k \\ ; \\ ; \\ ; \\ ; \\beta + j = \\gamma + k \\ ; \\ ; \\ ; \\ ; \\beta < \\gamma \\} , \\end{align*}"}
{"id": "1484.png", "formula": "\\begin{align*} = | \\det ( B ( z _ 1 , z _ 2 ) ) | ^ { - 2 } \\delta ( z _ 1 ) \\delta ( z _ 2 ) . \\end{align*}"}
{"id": "6713.png", "formula": "\\begin{align*} \\Bigl ( ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d } } \\Bigr ) ^ { ( - 1 ) } & = \\Bigl ( \\sum _ { n \\geq 0 } \\prod ^ { d - 1 } _ { l = 1 } ( \\theta ^ { q ^ { n + d } } - t ) ^ { c ( l ) q ^ { d - l } } \\alpha ^ { q ^ { n + d } } \\Bigr ) ^ { ( - 1 ) } \\\\ & = \\prod _ { l = 1 } ^ { d - 1 } ( \\theta ^ { q ^ { d - 1 } } - t ) ^ { c ( l ) q ^ { d - l } } \\alpha ^ { q ^ { d - 1 } } + ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d - 1 } } . \\end{align*}"}
{"id": "5489.png", "formula": "\\begin{align*} \\frac { 1 } { t } d _ K ( a + t \\beta a ) = 0 , t > 0 , \\end{align*}"}
{"id": "840.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 ^ + } y ^ a \\partial _ y u = f . \\end{align*}"}
{"id": "8779.png", "formula": "\\begin{align*} \\begin{aligned} & s _ { i j } = a _ { i j } j \\leq \\min ( J _ i ) , \\\\ & 0 \\leq \\frac { s _ { i \\max ( J _ i ) } - s _ { i \\max ( J _ i ) - 1 } } { a _ { i \\max ( J _ i ) } - a _ { i \\max ( J _ i ) - 1 } } \\leq \\cdots \\frac { s _ { i \\min ( J _ i ) + 1 } - s _ { i \\min ( J _ i ) } } { a _ { i \\min ( J _ i ) + 1 } - a _ { i \\min ( J _ i ) } } \\leq 1 , \\\\ & s _ { i j } = s _ { i n } j \\geq \\max ( J _ i ) , \\end{aligned} \\end{align*}"}
{"id": "9442.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } H _ \\mu u \\ ! + \\ ! \\varPhi _ q d _ q B _ q ( w , u ) & = & \\varPhi _ q d _ q f & ( x , t ) \\in { \\mathbb R } ^ n \\times ( 0 , T ) , \\\\ u & = & u _ 0 , & ( x , t ) \\in \\mathbb { R } ^ n \\times \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
{"id": "2133.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } k \\sum _ { j = 1 } ^ { n - 1 } \\sum _ { l = 1 } ^ k \\frac { \\binom { M _ n } l ( k ) _ l ( k ( j - 1 ) ) _ { M _ n - l } } { ( k n ) _ { M _ n } } \\frac 1 { k ( n - j + 1 ) - l } = \\\\ & k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\sum _ { s = 1 } ^ \\infty ( 1 - c ) ^ { k ( s + 1 ) } \\frac 1 { k ( s + 1 ) - l } , \\ \\ M _ n \\sim c n , \\ c \\in ( 0 , 1 ) . \\end{aligned} \\end{align*}"}
{"id": "177.png", "formula": "\\begin{align*} \\mathcal { L } ( f ) = \\frac { 1 } { \\alpha } \\left ( \\mathcal { L } ^ \\alpha + ( \\mathcal { L } ^ { \\alpha } ) ^ * \\right ) ( f ) , \\end{align*}"}
{"id": "6031.png", "formula": "\\begin{align*} & ( 1 - t ) ^ 2 \\frac { \\partial w } { \\partial t } + \\big ( ( 1 - t ) ( \\alpha - 1 ) - x \\big ) w = 0 , \\\\ & ( 1 - t ) \\frac { \\partial w } { \\partial x } - t w = 0 . \\end{align*}"}
{"id": "5315.png", "formula": "\\begin{align*} \\Delta ( a ) ( b \\chi \\otimes c \\theta ) = ( \\Delta ( a ) ( b \\otimes c ) ) ( \\chi \\otimes \\theta ) , ( \\chi b \\otimes \\theta c ) \\Delta ( a ) = ( \\chi \\otimes \\theta ) ( ( b \\otimes c ) \\Delta ( a ) ) . \\end{align*}"}
{"id": "8413.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N - 1 } \\sup _ { h \\in \\mathrm { L i p } ( 1 ) } \\left | \\tilde { Q } _ { i - 1 } \\big ( P _ { \\eta } - \\tilde { Q } _ { 1 } \\big ) P _ { ( N - i ) \\eta } h ( x ) \\right | & \\leq C ( 1 + | x | ) \\eta ^ { 2 / \\alpha } \\sum _ { i = 1 } ^ { N - 1 } [ ( N - i ) \\eta ] ^ { - 1 / \\alpha } \\\\ & \\leq C \\frac { \\alpha } { \\alpha - 1 } \\ , ( 1 + | x | ) \\eta ^ { 2 / \\alpha - 1 } . \\end{align*}"}
{"id": "5532.png", "formula": "\\begin{align*} Y _ m ^ j = \\sup _ { t \\in [ 0 , T ] } | \\dot { B } _ m ^ j ( t ) | \\leq \\frac { 1 } { \\delta _ m } \\sum _ { k = 1 } ^ m | ( B ^ j ( k \\delta _ m ) - B ^ j ( ( k - 1 ) \\delta _ m ) ) | = \\sum _ { k = 1 } ^ m | \\eta _ m ^ { j , k } | , \\end{align*}"}
{"id": "274.png", "formula": "\\begin{align*} u _ { t } - u _ { x x t } - u _ { x x } + \\gamma u _ { x x x } + \\beta u u _ { x } & = 0 , \\ \\ x \\in \\R , \\ t > 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) , \\ \\ x \\in \\R , \\end{align*}"}
{"id": "6662.png", "formula": "\\begin{align*} h ^ { - \\frac { 1 } { 2 } + \\alpha } k ^ { - \\frac { 1 } { 2 } - \\alpha } \\mathcal { G } ( 2 , \\alpha , \\beta ) = \\mathcal { K } ( 0 , - \\alpha - \\beta , 2 ) . \\end{align*}"}
{"id": "5101.png", "formula": "\\begin{align*} h _ A ( x ) = h _ A ( y ) \\Longleftrightarrow x = y \\ ; \\vee \\ ; x = y ^ { - 1 } . \\end{align*}"}
{"id": "1260.png", "formula": "\\begin{align*} \\mathrm { E C C } ( Y , \\lambda , \\Gamma ) : = \\bigoplus _ { \\alpha : \\mathrm { a d m i s s i b e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } { [ \\alpha ] = \\Gamma } } \\mathbb { Z } _ { 2 } \\langle \\alpha \\rangle . \\end{align*}"}
{"id": "1275.png", "formula": "\\begin{align*} A ( \\alpha ) = \\sum m _ { i } A ( \\alpha _ { i } ) = \\sum m _ { i } \\int _ { \\alpha _ { i } } \\lambda . \\end{align*}"}
{"id": "5166.png", "formula": "\\begin{align*} B ( b ) = \\left | Z _ { b ^ { - 1 } } \\phi ( 0 , 0 ) \\right | ^ 2 = \\frac { \\pi } { 2 } \\left | b \\ , \\vartheta ' _ 1 ( 0 , i b ) \\frac { \\vartheta _ 3 ( 0 , i b ) } { \\vartheta _ 4 ( 0 , i b ) \\vartheta _ 4 ( 0 , i b ^ { - 1 } ) } \\right | ^ 2 . \\end{align*}"}
{"id": "3547.png", "formula": "\\begin{align*} \\rho _ m ( x , t ) & = \\frac 1 { ( 4 \\pi | t | ) ^ { m / 2 } } \\exp \\left ( - \\frac { | x | ^ 2 } { 4 | t | } \\right ) , \\\\ \\Phi _ m ( x ) & = \\rho ( x , - 1 ) = ( 2 \\pi ) ^ { - m / 2 } \\exp \\left ( - \\frac 1 4 | x | ^ 2 \\right ) . \\end{align*}"}
{"id": "3129.png", "formula": "\\begin{align*} F _ i = a _ i x _ 1 ^ 2 + \\varphi _ i ( x _ 2 , x _ 3 ) + \\psi _ i ( x _ 4 , x _ 5 ) = 0 \\ , . \\end{align*}"}
{"id": "6399.png", "formula": "\\begin{align*} V = \\{ ( z , w ) \\in \\mathbb D ^ 2 \\ , : \\ , \\det ( \\psi ( z ) - w I ) = 0 \\} , \\end{align*}"}
{"id": "8017.png", "formula": "\\begin{align*} \\rho ^ { * } _ { ( \\mu ) } \\psi : = \\omega ^ { \\mu } \\rho ^ { * } \\psi \\end{align*}"}
{"id": "2647.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\langle ( P X - X P ) g _ n , g _ n \\rangle = \\frac { 1 } { 2 \\pi i } \\lim _ { n \\to \\infty } \\norm { g _ n } _ 2 ^ 2 = \\frac { 1 } { 2 \\pi i } \\norm { g } _ 2 ^ 2 . \\end{align*}"}
{"id": "8754.png", "formula": "\\begin{align*} & \\begin{aligned} z _ i \\in \\Delta _ i , \\ & \\delta _ { i t } \\in \\{ 0 , 1 \\} , \\\\ & z _ { i \\tau ( i , t ) } \\geq \\delta _ { i t } \\geq z _ { i \\tau ( i , t ) + 1 } i = 1 , \\ldots , d , \\ t = 1 , \\ldots , l _ i - 1 , \\end{aligned} \\\\ & f _ i = a _ { i 0 } z _ { i 0 } + \\sum _ { j = 1 } ^ { n } ( a _ { i j } - a _ { i j - 1 } ) z _ { i j } = : F _ i ( z _ i ) i = 1 , \\ldots , d . \\end{align*}"}
{"id": "2525.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\langle \\pi ( x , \\omega ) f , g \\rangle e ^ { - 2 \\pi i ( x \\cdot \\eta - \\xi \\cdot \\omega ) } \\ , d ( x , \\omega ) = 0 . \\end{align*}"}
{"id": "7472.png", "formula": "\\begin{align*} \\mu ^ n : = \\left \\langle G ( \\phi ^ { n } ) , \\phi ^ n \\right \\rangle = \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi ^ n | ^ 2 + V | \\phi ^ n | ^ 2 + \\beta | \\phi ^ n | ^ 4 - \\Omega \\overline { \\phi ^ n } L _ z \\phi ^ n \\right ) \\mathrm { d } \\mathbf { x } = E ( \\phi ^ n ) + \\frac { \\beta } { 2 } \\int _ { \\mathbb { R } ^ d } | \\phi ^ n | ^ 4 \\mathrm { d } \\mathbf { x } . \\end{align*}"}
{"id": "4801.png", "formula": "\\begin{align*} \\phi * \\phi = s ^ 2 \\chi _ A * \\chi _ A + t ^ 2 \\chi _ B * \\chi _ B + \\chi _ C * \\chi _ C + 2 s t \\chi _ A * \\chi _ B + 2 t \\chi _ B * \\chi _ C + 2 s \\chi _ C * \\chi _ A . \\end{align*}"}
{"id": "4088.png", "formula": "\\begin{align*} G _ { z , \\alpha , \\nu } ( x ) = x + \\sum _ { i = 1 } ^ k \\alpha _ i \\cdot ( x - z _ i ) \\cdot | x - z _ i | \\cdot \\phi \\Bigl ( \\frac { x - z _ i } { \\nu } \\Bigr ) . \\end{align*}"}
{"id": "7577.png", "formula": "\\begin{align*} L \\chi = L \\phi , \\end{align*}"}
{"id": "5848.png", "formula": "\\begin{align*} \\rho = \\mu / \\lambda . \\end{align*}"}
{"id": "1064.png", "formula": "\\begin{align*} q | _ { \\mathcal { R } _ { \\xi , I I I } } ( x , t ) = C _ R + \\mathcal { O } ( e ^ { - \\gamma t } ) . \\end{align*}"}
{"id": "1258.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) - I ( \\alpha , \\beta , Z ' ) = < c _ { 1 } ( \\xi ) + 2 \\mathrm { P D } ( \\Gamma ) , Z - Z ' > . \\end{align*}"}
{"id": "1454.png", "formula": "\\begin{align*} \\eta ( z _ 1 , z _ 2 ) : = i U ( z _ 1 ) ^ { \\ast } H U ( z _ 2 ) , \\ , \\ , \\delta ( z _ 1 , z _ 2 ) : = \\det ( \\eta ( z _ 1 , z _ 2 ) ) \\ , \\ , \\ , \\ , \\eta ( z ) : = \\eta ( z , z ) , \\delta ( z ) : = \\delta ( z , z ) . \\end{align*}"}
{"id": "3737.png", "formula": "\\begin{align*} ( D ^ m B ) _ t + [ D ^ m , B ] J _ x + B D ^ m J _ x - [ D ^ m , J ] B _ x - J D ^ m B _ x + \\mu \\Lambda ^ \\alpha D ^ m B = 0 \\end{align*}"}
{"id": "1248.png", "formula": "\\begin{align*} \\mu \\left ( F _ { \\mu } ^ { [ \\alpha _ 2 , \\gamma _ 2 ] , \\varepsilon } \\right ) = 1 . \\end{align*}"}
{"id": "6362.png", "formula": "\\begin{align*} h ^ { ( k ) } ( x , y _ 1 , y _ 2 , z ) = a x ^ { 4 k + 2 } + h _ 4 ( y _ 1 , y _ 2 ) + b x z ^ 2 + x ^ { 2 k + 1 } h _ 2 ( y _ 1 , y _ 2 ) + x ^ { k + 1 } h _ 1 ( y _ 1 , y _ 2 ) z \\end{align*}"}
{"id": "2385.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi k / N ) ^ 2 = \\frac { 1 } { 2 } \\sum _ { k = 0 } ^ { N - 1 } \\left ( 1 + \\cos ( 2 \\pi 2 k / N ) \\right ) = \\frac { N } { 2 } + \\frac { 1 } { 2 } \\sum _ { k = 0 } ^ { N - 1 } \\cos ( 2 \\pi 2 k / N ) . \\end{align*}"}
{"id": "641.png", "formula": "\\begin{align*} f ( x , n ) \\ = \\ \\bigg [ \\frac { x } { n ! } \\bigg ] . \\end{align*}"}
{"id": "833.png", "formula": "\\begin{align*} | \\nabla u ( x , y ) | _ { \\rho , ( x , y ) } = \\rho ( x , y ) ^ { - 1 } | \\nabla u ( x , y ) | _ { ( x , y ) } . \\end{align*}"}
{"id": "3898.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < a < p \\\\ \\gcd ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } \\ll p ^ { 1 - \\varepsilon } . \\end{align*}"}
{"id": "7074.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\bar \\nabla _ { E _ 1 } E _ 1 = - E _ 3 , & \\bar \\nabla _ { E _ 1 } E _ 2 = 0 , & \\bar \\nabla _ { E _ 1 } E _ 3 = E _ 1 \\\\ \\\\ \\bar \\nabla _ { E _ 2 } E _ 1 = 0 , & \\bar \\nabla _ { E _ 2 } E _ 2 = E _ 3 , & \\bar \\nabla _ { E _ 2 } E _ 3 = - E _ 2 \\\\ \\\\ \\bar \\nabla _ { E _ 3 } E _ 1 = 0 , & \\bar \\nabla _ { E _ 3 } E _ 2 = 0 , & \\bar \\nabla _ { E _ 3 } E _ 3 = 0 . \\end{array} \\end{align*}"}
{"id": "7779.png", "formula": "\\begin{align*} r C _ { \\min } { \\beta } ^ { - 1 } & \\leq \\prod _ { i = 1 } ^ { q } C _ { \\sigma _ i } < r { \\beta } ^ { - 1 } \\end{align*}"}
{"id": "2504.png", "formula": "\\begin{align*} \\iota ( \\mathbf { h } ) \\odot \\iota ( \\mathbf { h } ' ) & = \\left ( x , \\omega , \\tau + \\tfrac { 1 } { 2 } x \\cdot \\omega \\right ) \\odot \\left ( x ' , \\omega ' , \\tau ' + \\tfrac { 1 } { 2 } x ' \\cdot \\omega ' \\right ) \\\\ & = \\left ( x + x ' , \\omega + \\omega ' , \\tau + \\tau ' + \\tfrac { 1 } { 2 } ( x \\cdot \\omega + x ' \\cdot \\omega ' ) + x ' \\cdot \\omega \\right ) \\\\ & = \\iota ( \\mathbf { h } \\bullet \\mathbf { h } ' ) . \\end{align*}"}
{"id": "7939.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } | E _ { n , k } | \\ll ( n ^ { 1 / 4 } + L _ n ^ { 1 / 2 } n ^ { 1 / 8 } ) \\sum _ { k = 1 } ^ { n } ( 1 + k ^ { 1 / 2 } n ^ { - 3 / 8 } ) \\ll n ^ { 1 1 / 8 } + L _ n ^ { 1 / 2 } n ^ { 5 / 4 } . \\end{align*}"}
{"id": "2640.png", "formula": "\\begin{align*} \\langle X g , P g \\rangle = \\sum _ { k , l \\in \\Z } \\langle X g , M _ l T _ k g \\rangle \\langle M _ l T _ k g , P g \\rangle . \\end{align*}"}
{"id": "3993.png", "formula": "\\begin{align*} U ^ p ( t ) = \\sum _ { k \\geq N } a _ k e ^ { \\tilde { \\lambda } _ k ( T - t ) } , \\ \\ U ^ h ( t ) = \\sum _ { \\mod { k } \\geq N } b _ k e ^ { \\tilde { \\gamma } _ k ( T - t ) } , \\ \\ t \\geq 0 , \\end{align*}"}
{"id": "6865.png", "formula": "\\begin{align*} \\norm { Y ^ b _ { j } g _ \\ell - \\prod \\limits _ { s = 1 } ^ { j } \\left ( 1 - \\delta ^ { i _ s + \\ell - 2 s } _ { i _ s - 2 s } \\right ) Y _ 0 ^ b g _ \\ell } \\leq \\norm { g _ \\ell } \\sum \\limits _ { s = 1 } ^ j \\gamma _ { i _ s - 2 s } ^ { i _ s + \\ell - 2 s } \\prod \\limits _ { t = 1 } ^ { s - 1 } \\left ( 1 - \\delta ^ { i _ t + \\ell - 2 t } _ { i _ t - 2 t } \\right ) , \\end{align*}"}
{"id": "3289.png", "formula": "\\begin{align*} \\delta = \\Im \\omega _ 2 ( i \\varepsilon ) = \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , \\varepsilon ) - \\varepsilon } \\end{align*}"}
{"id": "3932.png", "formula": "\\begin{align*} H _ \\Lambda ( \\Phi ) = \\sum _ { X \\subset \\Lambda } \\Phi ( X ) . \\end{align*}"}
{"id": "7213.png", "formula": "\\begin{align*} \\tilde { W } _ { s , t } ( x , v ) & = W _ { s , t } ( x - \\check { \\mathcal { T } } _ { t , x , v } v , v ) , \\\\ \\tilde { Y } _ { s , t } ( x , v ) & = Y _ { s , t } ( x - \\check { \\mathcal { T } } _ { t , x , v } v , v ) . \\end{align*}"}
{"id": "2837.png", "formula": "\\begin{align*} \\frac { p } { 2 } - \\frac { 1 } { s ' } + \\frac { N p } { 2 } - N = \\frac { 1 } { s } + \\left ( \\frac { p } { 2 } - 1 \\right ) \\left ( N + 1 \\right ) \\in \\Bigg [ \\left ( \\frac { p } { 2 } - 1 \\right ) \\left ( N + 1 \\right ) , \\frac { N - \\gamma } { N } + \\left ( \\frac { p } { 2 } - 1 \\right ) \\left ( N + 1 \\right ) \\Bigg ) , \\end{align*}"}
{"id": "4799.png", "formula": "\\begin{align*} \\langle \\phi , 1 \\rangle = s ( \\ell - 1 ) + t ( m - 1 ) + ( \\ell - 1 ) ( m - 1 ) . \\end{align*}"}
{"id": "6095.png", "formula": "\\begin{align*} z _ { s s } z _ { t t } - z _ { s t } ^ { 2 } = r ^ { 2 m - 4 } ( m - 1 ) \\left [ m ^ { 2 } P ^ { 2 } ( \\theta ) + m P ( \\theta ) P '' ( \\theta ) - ( m - 1 ) P ' ( \\theta ) ^ { 2 } \\right ] . \\end{align*}"}
{"id": "8761.png", "formula": "\\begin{align*} \\begin{aligned} s _ { i k } & = G _ { i k } ( z , \\delta ) : = \\sum _ { t = 1 } ^ { l _ i } \\Bigl ( m ^ t _ { i 0 } ( \\delta _ { i t - 1 } - \\delta _ { i t } ) + \\sum _ { j = 1 } ^ k ( m ^ t _ { i j } - m ^ t _ { i j - 1 } ) ( z _ { i j } - \\delta _ { i t } ) \\Bigr ) , \\end{aligned} \\end{align*}"}
{"id": "8543.png", "formula": "\\begin{align*} f ^ { \\# } ( t , k ) & = f ^ { \\# } ( 0 , k ) \\pm i \\int _ { 0 } ^ { t } \\mathcal { N } _ { \\mu ^ \\sharp } [ f , f , f ] ( s , k ) \\ , d s , \\\\ \\mathcal { N } _ { \\mu ^ \\sharp } [ f , f , f ] ( s , k ) & : = \\ ! \\ ! \\iiint e ^ { i s ( - k ^ 2 + \\ell ^ 2 - m ^ 2 + n ^ 2 ) } f ^ { \\# } ( s , \\ell ) \\overline { f ^ { \\# } ( s , m ) } f ^ { \\# } ( s , n ) \\ , \\mu ^ \\sharp ( k , \\ell , m , n ) \\ , d n d m d \\ell , \\end{align*}"}
{"id": "6117.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } ( j + 2 ) ( j + 1 ) x ^ m \\alpha _ { j + 2 } ( x ) y ^ j + \\sum _ { j = n } ^ { \\infty } \\left [ ( j + 2 ) ( j + 1 ) x ^ m \\alpha _ { j + 2 } ( x ) + \\varepsilon \\alpha '' _ { j - n } ( x ) \\right ] y ^ j = 0 . \\end{align*}"}
{"id": "8740.png", "formula": "\\begin{align*} \\sum _ { v \\in V _ 1 } b _ v = \\sum _ { v \\in V _ 2 } b _ v = \\cdots = \\sum _ { v \\in V _ d } b _ v . \\end{align*}"}
{"id": "1544.png", "formula": "\\begin{align*} f ( w ) = j ( h _ { \\infty } , z _ 0 ) ^ { k } \\mathbf { f } ( h _ { \\infty } ) , j ( \\tilde { \\tau } _ m , z \\times w ) = \\delta ( w , z ) , \\end{align*}"}
{"id": "1305.png", "formula": "\\begin{align*} \\hat { I } _ { ( n , m ) } ( k ) : = \\{ \\ , \\ , \\alpha _ { k ' } \\ , \\ , | \\ , \\ , k ' \\leq k , \\ , \\ , ( E ( \\alpha _ { k ' } ) , H ( \\alpha _ { k ' } ) ) = ( n , m ) \\ , \\ , \\} \\end{align*}"}
{"id": "8349.png", "formula": "\\begin{align*} 2 \\sqrt { e ^ r - 1 } = | \\log p | ( 1 + o ( 1 ) ) \\end{align*}"}
{"id": "8172.png", "formula": "\\begin{align*} N _ { d _ 0 } ( f , H ) = - f + \\frac { 1 2 \\mu ( d _ 0 ) } { \\prod _ { q \\mid d _ 0 } ( q ^ 2 - 1 ) } \\sum _ { \\delta \\mid d _ 0 } \\delta \\mu ( \\delta ) \\sum _ { h \\in H _ { d _ 0 } } s ( h , \\delta f ) . \\end{align*}"}
{"id": "9088.png", "formula": "\\begin{align*} L ( x _ { \\rm t h } ) = \\frac { \\sqrt { 2 \\pi } } { \\sigma } f _ D x _ { \\rm t h } e ^ { - \\frac { x _ { \\rm t h } ^ 2 } { \\sigma ^ 2 } } . \\end{align*}"}
{"id": "7662.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\tilde { h } _ { \\varepsilon } + \\tilde { \\lambda } _ 0 \\underline { m } \\ , \\tilde { h } _ { \\varepsilon } = 0 & \\tilde { \\Omega } _ { \\varepsilon } \\ ; , \\\\ \\tilde { h } _ { \\varepsilon } = w & \\partial \\tilde { \\Omega } _ { \\varepsilon } \\ ; . \\end{cases} \\end{align*}"}
{"id": "42.png", "formula": "\\begin{align*} g _ { t } = g + d \\zeta _ { t } \\circ ( J \\otimes \\mathrm { I d } ) + \\zeta _ { t } \\otimes \\theta ^ { c } + \\theta ^ { c } \\otimes \\zeta _ { t } + \\zeta _ { t } \\otimes \\zeta _ { t } . \\end{align*}"}
{"id": "6425.png", "formula": "\\begin{align*} B = \\sum _ { i = 2 } ^ 4 e _ 1 e _ i \\nabla ^ \\partial _ { e _ i } , A = \\sum _ { i = 2 } ^ 4 e _ 1 e _ i \\left ( \\nabla _ { e _ i } - \\nabla ^ \\partial _ { e _ i } \\right ) . \\end{align*}"}
{"id": "9012.png", "formula": "\\begin{align*} & \\partial _ t \\rho _ i = \\nabla \\cdot \\left [ D _ i ( x ) \\left ( \\nabla \\rho _ i + z _ i \\rho _ i \\nabla \\phi \\right ) \\right ] , x \\in \\Omega \\subset \\R ^ d , t > 0 , \\\\ & - \\nabla \\cdot ( \\epsilon ( x ) \\nabla \\phi ) = f ( x ) + \\sum _ { i = 1 } ^ { s } z _ i \\rho _ i , \\end{align*}"}
{"id": "3235.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 ^ + } k ( s , 0 ) = \\frac { 1 } { \\int _ \\mathbb { R } \\frac { 1 } { u ^ 2 } d \\mu ( u ) } \\qquad \\lim _ { s \\rightarrow \\infty } k ( s , 0 ) = \\int _ \\mathbb { R } u ^ 2 d \\mu ( u ) . \\end{align*}"}
{"id": "1281.png", "formula": "\\begin{align*} I ( \\alpha _ { l } ^ { \\Gamma } ) - I ( \\alpha _ { k } ^ { \\Gamma } ) = 2 ( l - k ) . \\end{align*}"}
{"id": "3217.png", "formula": "\\begin{align*} P _ { \\rm o u t } ^ { D _ 2 } = 1 - e ^ { - \\frac { 1 } { \\beta _ { s s } \\beta _ { s 2 } N } \\left ( \\frac { \\sigma ^ 2 \\gamma _ { t h 2 } } { P _ t ( \\alpha _ 2 - \\alpha _ 1 \\gamma _ { t h 2 } ) } \\right ) } . \\end{align*}"}
{"id": "4491.png", "formula": "\\begin{align*} B _ 1 ( U , V ) = \\Big ( \\frac { 1 } { 2 \\pi } + \\frac { b _ 1 \\log U + b _ 2 } { U \\log U \\log ( U / 2 \\pi ) } \\Big ) \\big ( \\log ( V / U ) \\ , \\log ( \\sqrt { V U } / ( 2 \\pi ) ) \\big ) + \\frac { 2 R ( U ) } { U } . \\end{align*}"}
{"id": "530.png", "formula": "\\begin{align*} \\Xi _ { k + 1 } = \\mathcal { A } \\Xi _ k + \\mathcal { B } \\mathcal { G } \\hat { \\Psi } _ k ( \\mathbf { u } , \\Xi ) , \\mathbf { y } _ k = \\mathcal { C } \\Xi _ k . \\end{align*}"}
{"id": "8582.png", "formula": "\\begin{align*} \\big ( ( \\mathcal { F } ^ \\# ) ^ { - 1 } \\circ \\mathcal { F } ^ \\# g \\big ) ( x ) & = \\int _ { \\R _ k } \\mathcal { K } ^ \\# ( x , k ) \\Big ( \\int _ { \\R _ y } \\overline { \\mathcal { K } ^ \\# ( y , k ) } g ( y ) \\ , d y \\Big ) d k \\\\ & = \\int _ { \\R _ k } \\mathcal { K } ( x , k ) \\Big ( \\int _ { \\R _ y } \\overline { \\mathcal { K } ( y , k ) } g ( y ) \\ , d y \\Big ) d k = g ( x ) \\end{align*}"}
{"id": "1778.png", "formula": "\\begin{align*} ( \\partial c ) ( g _ 0 , \\ldots , g _ { k + 1 } ) : = \\sum _ { i = 0 } ^ { k + 1 } ( - 1 ) ^ i c ( g _ 1 , \\ldots , \\hat { g } _ i , \\ldots , g _ { k + 1 } ) . \\end{align*}"}
{"id": "980.png", "formula": "\\begin{align*} \\tau \\leqslant \\inf _ { B _ { 1 / 2 } } \\tilde u = \\inf _ { B _ { \\rho / 2 } ( e _ 1 ) } u . \\end{align*}"}
{"id": "5066.png", "formula": "\\begin{align*} [ \\varphi ( a ) , a ^ 2 \\cdot v _ 0 ] = [ \\varphi ( a ^ 2 \\cdot v _ 0 ) , a ] . \\end{align*}"}
{"id": "6289.png", "formula": "\\begin{align*} \\widehat { K } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } & = \\widehat { K ' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } + \\widehat { K '' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } , \\end{align*}"}
{"id": "2556.png", "formula": "\\begin{align*} \\widehat { S } _ { W , m } = \\widehat { V } _ P \\widehat { D } _ { L , m } \\widehat { J } \\widehat { V } _ Q . \\end{align*}"}
{"id": "938.png", "formula": "\\begin{align*} ( C E ) ( r , s ) & = \\sum _ { u \\in { \\bf m } } C ( r , u ) E ( u , s ) = \\sum _ { u \\in { \\bf m } } D ( r , u ) \\Big ( D ^ { - 1 } ( u , s ) - \\frac { D ^ { - 1 } ( u , \\ell ) D ^ { - 1 } ( \\ell , s ) } { D ^ { - 1 } ( \\ell , \\ell ) } \\Big ) \\\\ & = \\delta _ { r s } - D ( r , \\ell ) D ^ { - 1 } ( \\ell , s ) - \\big ( \\delta _ { r \\ell } - D ( r , \\ell ) D ^ { - 1 } ( \\ell , \\ell ) \\big ) \\frac { D ^ { - 1 } ( \\ell , s ) } { D ^ { - 1 } ( \\ell , \\ell ) } . \\end{align*}"}
{"id": "9449.png", "formula": "\\begin{align*} G _ j ( t _ 1 , \\ldots , t _ m ) = { \\bf t } ^ { { \\bf a } _ j } - { \\bf t } ^ { { \\bf b } _ j } - \\sum _ { { \\bf a } } c _ { j , { \\bf a } } { \\bf t } ^ { { \\bf a } } , j = 1 , \\ldots , r , \\end{align*}"}
{"id": "4159.png", "formula": "\\begin{align*} \\int x w ( x , t ) d x = 0 , \\end{align*}"}
{"id": "957.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + c u = 0 \\qquad B _ { \\rho } ( e _ 1 ) . \\end{align*}"}
{"id": "4463.png", "formula": "\\begin{align*} | z \\rangle = e ^ { - \\frac { | z | ^ { 2 } } { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { \\sqrt { n ! } } | n \\rangle . \\end{align*}"}
{"id": "2893.png", "formula": "\\begin{align*} \\| \\left \\langle \\nabla \\right \\rangle R ( h ) \\| _ { S ' ( ( t , + \\infty ) , L ^ 2 ) } \\lesssim \\sum _ { n = 0 } ^ \\infty \\| \\left \\langle \\nabla \\right \\rangle R ( h ) \\| _ { S ' ( ( t + n , t + n + 1 ) , L ^ 2 ) } \\lesssim \\begin{cases} e ^ { - ( p - 1 ) c t } , & p \\in ( 2 , 3 ) , \\\\ e ^ { - 2 c t } , & p = 2 p \\ge 3 . \\end{cases} \\end{align*}"}
{"id": "2375.png", "formula": "\\begin{align*} f = \\mathop { \\sum \\sum } _ { k , l \\in \\Z ^ d } \\langle f , M _ { \\beta l } T _ { \\alpha k } \\ , g \\rangle M _ { \\beta l } T _ { \\alpha k } \\ , \\widetilde { g } , \\end{align*}"}
{"id": "2709.png", "formula": "\\begin{align*} b ( S ) = \\sum _ { i \\geq 0 } b _ i ( S ) . \\end{align*}"}
{"id": "887.png", "formula": "\\begin{align*} a ( \\widetilde E _ 2 ^ { ( 3 ) } , T ) = a ( \\mathrm { g e n u s } \\ \\Theta ^ { ( 3 ) } ( S ^ { ( p ) } ) , T ) . \\end{align*}"}
{"id": "9167.png", "formula": "\\begin{align*} L ( s , \\operatorname { s y m } ^ 2 f ) \\ll & \\left ( 1 + | s | \\right ) ^ { ( 3 ( 1 - \\Re ( s ) ) ) / 2 + \\varepsilon } , 0 \\leq \\Re ( s ) \\leq 1 . \\end{align*}"}
{"id": "3313.png", "formula": "\\begin{align*} \\bullet \\ \\ & \\mbox { $ { \\frak p } $ i s n o t c o m p l e t e i n t e r s e c t i o n , a n d } \\\\ \\bullet \\ \\ & \\mbox { $ z ^ u - x ^ { s _ 3 } y ^ { t _ 3 } $ i s t h e n e g a t i v e c u r v e , i . e . , $ u c < \\sqrt { a b c } $ . } \\end{align*}"}
{"id": "1836.png", "formula": "\\begin{align*} D _ { n + 1 } ( x ^ 2 , y ) = x ^ 2 \\sum _ { k = 0 } ^ { n } { n \\choose k } L _ { k } ( x , y ) L _ { n - k } ( x , y ) . \\end{align*}"}
{"id": "1136.png", "formula": "\\begin{align*} E ( x , t , k ) : = \\left \\{ \\begin{aligned} & m ^ { ( 3 ) } ( x , t , k ) \\left ( m _ { \\pm \\eta } \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , & k \\in U _ { \\delta } ( \\pm \\eta ) , \\\\ & m ^ { ( 3 ) } ( x , t , k ) \\left ( \\Delta _ \\eta \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , & e l s e w h e r e . \\end{aligned} \\right . \\end{align*}"}
{"id": "5953.png", "formula": "\\begin{align*} \\partial _ { t } \\eta _ k & = \\partial _ z \\phi _ k , \\partial _ { t } \\phi _ k = - g \\eta _ k , \\Gamma ^ { F S } \\times \\mathcal { T } , \\end{align*}"}
{"id": "5061.png", "formula": "\\begin{align*} [ a + t a \\cdot v _ 0 , b + t b \\cdot v _ 0 ] = 0 \\end{align*}"}
{"id": "6866.png", "formula": "\\begin{align*} Y ^ b _ j g _ \\ell = \\left ( 1 - \\delta ^ { i _ 1 + \\ell - 2 } _ { i _ 1 - 2 } \\right ) Y ^ { b } _ { j - 1 } g _ { \\ell } + \\Gamma g _ \\ell , \\end{align*}"}
{"id": "7770.png", "formula": "\\begin{align*} e _ 0 ( t , x ) & = \\phi _ 0 \\left ( \\langle 2 \\phi _ { 0 , x } + \\varphi _ { 0 , x } , \\varphi _ { 0 , x } \\rangle - \\langle 2 \\phi _ { 0 , t } + \\varphi _ { 0 , t } , \\varphi _ { 0 , t } \\rangle \\right ) \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; - \\varphi _ 0 \\left ( | \\tilde \\phi _ { 1 , t } | ^ 2 - | \\tilde \\phi _ { 1 , x } | ^ 2 \\right ) + \\mathbf { 1 } _ { \\omega } \\left ( f _ 0 ^ { p ^ { \\perp } } - f _ { 0 } ^ { \\tilde \\phi ^ { \\perp } _ 1 } \\right ) . \\end{align*}"}
{"id": "6791.png", "formula": "\\begin{align*} p ( \\theta ) = P \\theta + q , \\end{align*}"}
{"id": "7420.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\rightarrow 0 ^ { + } } \\limsup _ { n \\rightarrow \\infty } \\mathbb { E } _ { \\mu _ n } \\Big [ \\Big | \\int _ { 0 } ^ { t } \\frac { 1 } { n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) [ \\overleftarrow { \\eta } _ { s } ^ { \\ell } ( x ) - \\overleftarrow { \\eta } _ { s } ^ { \\varepsilon n } ( x ) ] \\overrightarrow { \\eta } _ { s } ^ { \\varepsilon n } ( x + 1 ) d s \\Big | \\Big ] = 0 . \\end{align*}"}
{"id": "7744.png", "formula": "\\begin{gather*} \\| \\psi \\| _ { L ^ { \\infty } _ x L ^ 2 _ t ( P _ { \\alpha , \\beta } ^ l ( y ) ) } : = \\sup _ { x \\in [ y , y + l ] } \\| \\psi ( t , x ) \\| _ { L ^ 2 _ t ( \\alpha + x - y , \\beta - x + y ) } . \\end{gather*}"}
{"id": "644.png", "formula": "\\begin{align*} & \\xi ( 0 ) \\ = \\ 1 , \\\\ & \\xi ( 1 ) \\ = \\ 2 , \\\\ & \\xi ( x ) \\ = \\ x ( x \\geq 2 ) . \\end{align*}"}
{"id": "3973.png", "formula": "\\begin{align*} \\eta ^ { \\prime \\prime } ( 0 ) = \\eta ^ { \\prime \\prime } ( 1 ) = 0 . \\end{align*}"}
{"id": "5548.png", "formula": "\\begin{align*} P ( t , T ) = \\exp \\bigg ( - \\int _ t ^ T f ( t , s ) d s \\bigg ) , \\end{align*}"}
{"id": "6237.png", "formula": "\\begin{align*} g _ k ( B ) = g ( B ) - ( \\dim M - \\dim B ) + k . \\end{align*}"}
{"id": "7241.png", "formula": "\\begin{align*} E ( t , X ( t ) ) & = - \\nabla ( \\phi * \\rho ( t , \\cdot ) ) ( X ( t ) ) \\\\ & = - \\nabla ( \\phi * ( G * _ { s , x } \\mathcal S + \\mathcal S ) ) ( X ( t ) ) \\\\ & = \\mathcal { F } ^ R ( t ) + \\mathcal { E } ^ R _ 1 ( t ) + \\mathcal { E } _ 2 ( t ) + \\mathcal { E } _ 3 ( t ) , \\end{align*}"}
{"id": "8037.png", "formula": "\\begin{align*} \\rho ( x ) = 2 \\int _ { x ' = 0 } ^ x \\left ( \\frac { 1 } { t _ { + } ( x ) } + \\frac { 1 } { t _ { - } ( x ) } \\right ) \\mathrm { d } x ' \\end{align*}"}
{"id": "2498.png", "formula": "\\begin{align*} \\widetilde { \\pi } ( \\l ) \\widetilde { \\pi } ( \\l ' ) = e ^ { 2 \\pi i x ' \\cdot \\omega } \\widetilde { \\pi } ( \\l + \\l ' ) . \\end{align*}"}
{"id": "5845.png", "formula": "\\begin{align*} D = \\{ ( x ^ 1 , x ^ 2 ) : | x ^ 1 | < R \\} . \\end{align*}"}
{"id": "5258.png", "formula": "\\begin{align*} \\omega ( a S ( b _ { ( 1 ) } ) c ) b _ { ( 2 ) } = 0 . \\end{align*}"}
{"id": "7883.png", "formula": "\\begin{align*} J = ( j _ 0 , \\dots , j _ n ) , \\ | J | = \\sum _ { s = 0 } ^ n j _ s , \\ \\binom { d } { J } = \\frac { d ! } { j _ 0 ! \\cdot \\ldots \\cdot j _ n ! } \\end{align*}"}
{"id": "5287.png", "formula": "\\begin{align*} \\delta _ { \\varphi } ^ * = \\delta _ { \\varphi } . \\end{align*}"}
{"id": "9267.png", "formula": "\\begin{align*} & \\forall n ^ 0 , x ^ X \\preceq _ X L 1 _ X , y ^ X \\exists z ^ X \\preceq _ X L 1 _ X \\\\ & \\qquad \\qquad \\qquad \\exists w ^ X \\preceq _ X L 2 ^ { \\alpha _ n + 1 } 1 _ X \\left ( y \\in A x \\rightarrow ( w \\in A z \\land x = _ X z + _ X \\gamma _ n w ) \\right ) . \\end{align*}"}
{"id": "6272.png", "formula": "\\begin{align*} n ' & = 3 0 h ^ 2 \\alpha n ^ { \\frac { 1 } { s + 1 } } + 1 6 0 h ^ 3 \\alpha n ^ { \\frac { 3 } { 2 } - \\frac { s } { s + 1 } } + 3 \\cdot ( 6 h ) ^ { s + 3 } \\alpha ^ { \\frac { s + 1 } { 2 } } n \\\\ [ 5 p t ] & \\leq \\left ( 3 0 h ^ 2 + 1 6 0 h ^ 3 + 3 \\cdot ( 6 h ) ^ { s + 3 } \\alpha ^ { \\frac { s - 1 } { 2 } } \\right ) \\alpha n \\\\ [ 5 p t ] & \\leq 4 \\cdot ( 6 h ) ^ { s + 3 } \\alpha n \\\\ [ 5 p t ] & \\leq 4 \\cdot ( 1 2 s k ) ^ { s + 3 } \\alpha n . \\end{align*}"}
{"id": "4962.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i \\partial _ t u - \\Delta u = \\rho ^ 2 | u | ^ 2 + C _ { \\rho } + \\langle \\nabla \\rangle ^ { - \\alpha } \\dot { W } \\ , , t \\in [ 0 , T ] \\ , , \\ , x \\in \\R ^ d \\ , , \\\\ u _ 0 = \\phi \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "8976.png", "formula": "\\begin{align*} \\begin{aligned} & u ' + 2 u ' u '' - 1 = 0 \\\\ \\Longrightarrow & u '' = \\frac { 1 - u ' } { 2 u ' } = \\frac { 1 } { 2 u ' } - \\frac { 1 } { 2 } \\to + \\infty \\end{aligned} \\end{align*}"}
{"id": "8218.png", "formula": "\\begin{align*} [ z ^ n ] f ( A ) = \\frac { 1 } { n } [ t ^ { n - 1 } ] f ' ( t ) \\Phi ( t ) ^ n . \\end{align*}"}
{"id": "3450.png", "formula": "\\begin{align*} \\mathcal S f ( x ) & = \\Big ( \\sum \\limits _ { Q } | q _ { Q } f ( x _ { Q } ) | ^ 2 \\chi _ Q ( x ) \\Big ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\Big ( \\sum \\limits _ { Q } | \\int _ { \\R ^ N } \\chi _ Q ( x ) q _ { Q } ( x _ Q , y ) f ( y ) d \\omega ( y ) | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "196.png", "formula": "\\begin{align*} q _ \\alpha ( t ) = \\dfrac { e ^ { - t } } { ( 1 - e ^ { - \\alpha t } ) ^ { 1 - \\frac { 1 } { \\alpha } } } \\left ( ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } + e ^ { - \\alpha t } \\right ) ^ { \\frac { 1 } { \\alpha } } . \\end{align*}"}
{"id": "9234.png", "formula": "\\begin{align*} \\begin{cases} \\forall x ^ X , y ^ X , \\gamma ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\rightarrow \\norm { x - _ X y } _ X \\geq _ \\mathbb { R } \\norm { J ^ A _ { \\gamma } x - _ X J ^ A _ { \\gamma } y } _ X \\right ) , \\\\ \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) , \\end{cases} \\end{align*}"}
{"id": "4264.png", "formula": "\\begin{align*} C ^ { ( w ) } ( t , x ) ~ = ~ E ^ { ( w ) } _ 1 ( t ) - ( c _ 1 - 1 ) \\cdot \\left [ { \\dot { \\sigma } ^ { ( w ) } ( t ) t \\over 2 } - ( w ( t , x ) ) - w ( t , 0 - ) ) \\right ] \\cdot \\phi _ x \\left ( x , { \\sigma ^ { ( w ) } ( t ) t \\over 2 } \\right ) , \\end{align*}"}
{"id": "9537.png", "formula": "\\begin{align*} \\lambda \\frac { d Q } { d P } & \\in \\partial V ( u - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } - \\bar c \\cdot \\bar x + S _ 0 ( \\bar x ) ) , \\\\ E ^ Q _ t [ \\Delta s _ { t + 1 } ] & \\in N _ { D _ t } ( x _ t ) t = 0 , \\dots , T , \\\\ E ^ Q [ \\bar c ] & \\in \\partial S _ 0 ( \\bar x ) , \\end{align*}"}
{"id": "6533.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ n & = \\dfrac { L ^ { ( 2 ) } _ n } { t ^ { ( 2 ) } _ n } - 1 = \\dfrac { 1 } { \\log n } \\left ( \\sum _ { j = 1 } ^ n \\dfrac { 1 } { j } - \\log n \\right ) \\sim \\gamma ( \\log n ) ^ { - 1 } ( n \\to \\infty ) . \\end{align*}"}
{"id": "8418.png", "formula": "\\begin{align*} \\nabla _ { v } X _ { t } ^ { x } : = \\lim _ { \\epsilon \\to 0 } \\frac { X _ { t } ^ { x + \\epsilon v } - X _ { t } ^ { x } } { \\epsilon } , t \\geq 0 . \\end{align*}"}
{"id": "6343.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k ( 2 + d _ \\ell ) \\leq k ^ 2 + n . \\end{align*}"}
{"id": "6196.png", "formula": "\\begin{align*} \\| C X \\| _ F ^ 2 \\geq ( { 1 - \\xi } ) \\left [ \\sum ^ { l } _ { t = 1 } \\bar { \\sigma } ^ 2 _ { t } - \\frac { 2 \\theta \\| S \\| ^ 2 _ F } { \\sqrt { \\alpha } } - \\theta ( k + \\sqrt { k } \\xi ) \\| C \\| ^ 2 _ F \\right ] . \\end{align*}"}
{"id": "7058.png", "formula": "\\begin{align*} \\beta _ { i } ^ K ( t ) = M ^ K _ { i } ( t ) + A ^ K _ { i } ( t ) \\end{align*}"}
{"id": "1106.png", "formula": "\\begin{align*} E _ { + } ( x , t , k ) = E _ { - } ( x , t , k ) J ^ { E } ( x , t , k ) , \\end{align*}"}
{"id": "8378.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\zeta ( x , t , y , s ; v ) = - v _ 1 \\frac { \\sqrt { s } } { 4 \\sqrt { t } ^ 3 } \\ , \\Psi + v _ 2 \\frac { \\sqrt { 1 - s } } { 4 \\sqrt { 1 - t } ^ 3 } . \\end{align*}"}
{"id": "3854.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } ( r _ { k , s } + r _ { s , k } ) ( 0 ) = 0 , k = \\overline { 0 , m - 1 } , s = \\overline { k , k + i _ { 2 k } - 1 } \\\\ ( r _ { k , s } - r _ { s , k } ) ( 0 ) = 0 , k = \\overline { 0 , m + \\tau - 2 } , s = \\overline { k + 1 , k + i _ { 2 k + 1 } } \\end{array} \\right \\} \\end{align*}"}
{"id": "5751.png", "formula": "\\begin{align*} \\begin{cases} e ^ { - Q } \\delta _ { s t } \\leq g _ { s t } \\leq e ^ { Q } \\delta _ { s t } ; \\\\ r ^ { 1 + \\alpha } \\| \\partial _ { j } g _ { s t } \\| _ { C ^ { 0 , \\alpha } ( B _ { r } ( p ) ) } \\leq e ^ { Q } , \\end{cases} \\end{align*}"}
{"id": "8409.png", "formula": "\\begin{align*} \\tilde { Y } _ 0 = x , \\tilde { Y } _ { k + 1 } = \\tilde { Y } _ { k } + \\eta b ( \\tilde { Y } _ { k } ) + \\frac { \\eta ^ { 1 / \\alpha } } { \\sigma } \\ , \\widetilde { Z } _ { k + 1 } , k = 0 , 1 , 2 , \\dots , \\end{align*}"}
{"id": "6600.png", "formula": "\\begin{align*} \\mathcal { U } ( h , k ) = \\mathcal { U } ^ 0 ( h , k ) + \\mathcal { U } ^ r ( h , k ) , \\end{align*}"}
{"id": "6405.png", "formula": "\\begin{align*} | z _ i | = \\left | ( P _ i ^ { \\perp } + z P _ i ) \\xi , U _ i ^ * \\xi \\rangle \\right | \\leq \\| ( P _ i ^ { \\perp } + z P _ i ) \\xi \\| \\| U _ i ^ * \\xi \\| \\leq \\| P _ i ^ { \\perp } + z P _ i \\| < 1 . \\end{align*}"}
{"id": "6149.png", "formula": "\\begin{align*} \\mu _ { n , m } = ( n + 1 ) { n + m + 1 \\choose m } . \\end{align*}"}
{"id": "8698.png", "formula": "\\begin{align*} P _ i = \\left \\{ u _ i \\in \\R ^ { n + 1 } \\ , \\middle | \\ , \\begin{aligned} a _ { i 0 } \\leq u _ { i j } \\leq \\min \\{ a _ { i j } , u _ { i n } \\} , \\ u _ { i 0 } = a _ { i 0 } , \\ ; a _ { i 0 } \\leq u _ { i n } \\leq a _ { i n } \\end{aligned} \\right \\} . \\end{align*}"}
{"id": "1280.png", "formula": "\\begin{align*} \\mathrm { E C H } ( Y , \\lambda , \\Gamma ) \\cong \\bigoplus _ { i = 0 } ^ { \\infty } \\mathbb { F } \\langle \\alpha _ { i } ^ { \\Gamma } \\rangle \\bigoplus \\bigoplus _ { j = 1 } ^ { m _ { \\Gamma } } \\mathbb { F } \\langle \\beta _ { j } ^ { \\Gamma } \\rangle \\end{align*}"}
{"id": "7085.png", "formula": "\\begin{align*} \\alpha _ { s _ m } ( f ) = \\alpha _ { s ' _ m } ( f _ { U ' } ) \\end{align*}"}
{"id": "5046.png", "formula": "\\begin{align*} \\varphi ( \\theta ) = \\sum _ { n \\not = 0 } \\varphi _ n e ^ { i n \\theta } \\end{align*}"}
{"id": "8185.png", "formula": "\\begin{align*} S ( H _ 3 , 3 f ) = \\frac { 5 f - 6 d + 1 } { 6 } \\hbox { a n d } N _ 3 ' ( f , H ) = - d , \\end{align*}"}
{"id": "6490.png", "formula": "\\begin{align*} \\prod _ { j = j _ 0 ( \\delta ) + 1 } ^ { n - 1 } \\left ( 1 + \\dfrac { \\delta } { j } \\right ) \\sim A _ { \\delta } \\cdot n ^ { \\delta } ( n \\to \\infty ) . \\end{align*}"}
{"id": "5270.png", "formula": "\\begin{align*} \\varphi ( S ^ 2 ( a ) ) = \\varphi ( S ( a ) \\delta _ { \\varphi } ) = \\varphi ( S ( \\delta _ { \\varphi } ^ { - 1 } a ) ) = \\varphi ( \\delta _ { \\varphi } ^ { - 1 } a \\delta _ { \\varphi } ) , a \\in A , \\end{align*}"}
{"id": "6285.png", "formula": "\\begin{align*} y [ n ] = \\sum _ { i \\in \\mathcal { K } ' } \\sqrt { \\rho } \\hat { h } _ i ^ * h _ i s [ n ] / | \\hat { h } _ i | ^ 2 + w [ n ] = \\sqrt { \\rho } K ' s [ n ] \\ ! + \\ ! \\varphi _ { \\mathcal { K } ' } s [ n ] \\ ! + \\ ! w [ n ] , \\ n \\ ! = \\ ! 1 , 2 , \\cdots , N _ 2 , \\end{align*}"}
{"id": "6817.png", "formula": "\\begin{align*} \\big { ( } \\partial _ t f _ t ( x ) \\big { ) } ^ { \\perp _ { f _ t } } = - \\frac { 1 } { 2 } \\frac { 1 } { | A ^ 0 _ { f _ t } ( x ) | ^ 4 } \\ , \\Big { ( } \\triangle _ { f _ t } ^ { \\perp } \\vec H _ { f _ t } ( x ) + Q ( A ^ { 0 } _ { f _ t } ) ( \\vec H _ { f _ t } ) ( x ) \\Big { ) } \\end{align*}"}
{"id": "1571.png", "formula": "\\begin{align*} \\langle \\mathbf { T } ( g , h ) , \\mathbf { f } ( h ) \\rangle = \\langle \\mathfrak { E } ( g \\times h , \\mu ) , \\mathbf { f } ( h ) \\rangle . \\end{align*}"}
{"id": "3446.png", "formula": "\\begin{align*} R _ M f ( x ) & = - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\int _ { Q } \\big ( \\psi _ { Q } ( x , y ) - \\psi _ { Q } ( x , x _ { Q } ) \\big ) q _ { Q } f ( y ) d \\omega ( y ) \\\\ & + \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\int _ { Q } \\psi _ { Q } ( x , x _ { Q } ) \\big ( q _ { Q } f ( x _ Q ) - q _ { Q } f ( y ) \\big ) d \\omega ( y ) \\\\ & = R ^ 1 _ M f ( x ) + R ^ 2 _ M f ( x ) , \\end{align*}"}
{"id": "6479.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 m - 1 ) ! ! } E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n \\log n } } \\right ) ^ { 2 m } \\right ] - 1 \\sim m \\gamma \\cdot ( \\log n ) ^ { - 1 } \\end{align*}"}
{"id": "5086.png", "formula": "\\begin{align*} \\pi ( \\l ) g ( t ) = M _ \\omega T _ x g ( t ) = g ( t - x ) e ^ { 2 \\pi i \\omega t } , \\l = ( x , \\omega ) . \\end{align*}"}
{"id": "9342.png", "formula": "\\begin{align*} H _ { n } ^ { ( r ) } = \\binom { n + r - 1 } { r - 1 } ( H _ { n + r - 1 } - H _ { r - 1 } ) , ( r \\ge 2 ) , H _ { 0 } ^ { ( r ) } = 0 , ( \\mathrm { s e e } \\ [ 5 ] ) . \\end{align*}"}
{"id": "599.png", "formula": "\\begin{align*} f \\ = \\ g \\circ \\big ( P _ 2 ^ 2 , P _ 1 ^ 2 \\big ) \\end{align*}"}
{"id": "4589.png", "formula": "\\begin{align*} \\Big ( \\lambda - c _ \\alpha \\ , \\lambda ^ 2 \\epsilon _ n \\Big ) \\Big ( 1 - \\delta _ n ( \\lambda ) \\Big ) = x \\sqrt { 1 + \\delta _ n ( \\lambda ) } . \\end{align*}"}
{"id": "7710.png", "formula": "\\begin{align*} \\mathsf c ( H ) = \\sup \\{ \\mathsf c ( a ) \\colon a \\in H \\} \\ , . \\end{align*}"}
{"id": "4488.png", "formula": "\\begin{align*} E _ { \\psi } ( x ) \\le \\varepsilon _ { } ( x _ 0 , x ) = A ( x _ 0 ) \\Big ( \\frac { \\log x } { R } \\Big ) ^ { 3 / 2 } \\exp \\Big ( - 2 \\sqrt { \\frac { \\log x } { R } } \\Big ) \\ \\ x \\ge x _ 0 . \\end{align*}"}
{"id": "1942.png", "formula": "\\begin{align*} K ( \\beta ) = ( g ( x _ 0 ) - \\underline { g } ) \\ ! \\left [ \\min \\ ! \\left ( c _ 1 \\tau _ 1 \\min ( \\underline { \\alpha _ 1 } , \\underline { t _ 1 } ) \\varepsilon _ 1 ^ 2 , c _ 2 \\tau _ 2 ^ 2 \\min \\ ! \\left ( \\min \\ ! \\left ( \\alpha _ { 0 2 } , \\dfrac { 3 | 2 c _ 2 - 1 | \\varepsilon _ 2 } { M _ g } \\right ) \\ ! , \\underline { t _ 2 } \\right ) ^ 2 \\ ! \\varepsilon _ 2 \\right ) \\ ! \\right ] ^ { - 1 } \\ ! . \\end{align*}"}
{"id": "6037.png", "formula": "\\begin{align*} \\delta _ { \\alpha , \\beta } g ( x ) & = \\frac { d } { d x } \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ( x ) \\big ) - \\frac { \\beta - \\alpha - ( \\alpha + \\beta + 2 ) x } { 2 ( 1 - x ^ 2 ) } \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ( x ) \\big ) \\\\ \\delta _ { \\alpha , \\beta } ^ * g ( x ) & = \\frac { d } { d x } \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ( x ) \\big ) + \\frac { \\beta - \\alpha - ( \\alpha + \\beta ) x } { 2 ( 1 - x ^ 2 ) } \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } g ( x ) \\big ) . \\end{align*}"}
{"id": "5091.png", "formula": "\\begin{align*} T = \\langle \\pi ( \\l _ 1 ^ \\circ ) g , \\pi ( \\l _ 2 ^ \\circ ) g \\rangle , \\l _ 1 ^ \\circ , \\ , \\l _ 2 ^ \\circ \\in \\L ^ \\circ _ { a , b } . \\end{align*}"}
{"id": "4829.png", "formula": "\\begin{align*} { \\rm W F } _ h ' ( e ^ { - i t _ 0 h ^ { - 1 } \\widetilde { P } _ h ( z ) } \\widetilde { R } _ h ( z ) ) \\cap S ^ * ( X \\times X ) & \\subset \\\\ \\kappa ( \\{ ( x , \\xi , y , \\eta ) \\ , : \\ , ( e ^ { - t _ 0 H _ p } ( x , \\xi ) , y , \\eta ) & \\in \\Delta ( T ^ * X ) \\cup \\Omega _ + \\cup ( E ^ * _ u \\times E ^ * _ s ) \\setminus \\{ 0 \\} \\xi = 0 , \\eta \\neq 0 \\} ) . \\end{align*}"}
{"id": "6955.png", "formula": "\\begin{align*} F ( k ) = \\psi ^ { - 1 } ( \\mathrm { S u c c } ^ { \\sigma } ( \\psi ( k ) ) ) . \\end{align*}"}
{"id": "2111.png", "formula": "\\begin{align*} p _ c ^ { ( k ) } ( i ) = \\begin{cases} \\frac 1 { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } ( 1 - c ) ^ { k - i + 1 } c ^ { i - 1 } \\binom k { i - 1 } , \\ i \\in \\{ 2 , \\cdots , k \\} ; \\\\ 1 - \\frac 1 { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } \\big ( 1 - c ^ k - ( 1 - c ) ^ k \\big ) , \\ i = 1 . \\end{cases} \\end{align*}"}
{"id": "5017.png", "formula": "\\begin{align*} G _ { k n } ( x , y ) = \\left ( { g _ { k n } ( x ) + \\delta _ 1 ( x , y ) \\atop g _ { k n } ( y ) + \\delta _ 2 ( x , y ) } \\right ) , \\end{align*}"}
{"id": "5556.png", "formula": "\\begin{align*} d _ K ( h ) \\leq \\| h ^ - \\| = \\sqrt { \\int _ 0 ^ { x _ 0 } | h ' ( x ) | ^ 2 e ^ { \\gamma x } d x } . \\end{align*}"}
{"id": "4135.png", "formula": "\\begin{align*} I _ 3 ( u ) = \\int _ { \\R } \\Big ( | D ^ { 1 / 2 } u | ^ 2 - \\frac { u ^ 3 } { 3 } \\Big ) ( x , t ) d x , \\end{align*}"}
{"id": "8028.png", "formula": "\\begin{align*} \\left \\langle ( \\partial _ \\Sigma \\otimes \\partial _ \\Sigma ) E , f \\otimes g \\right \\rangle = - \\frac { 1 } { 2 } \\int _ \\Sigma f ( * d _ \\Sigma g ) \\ , \\mathrm { d } V _ \\Sigma . \\end{align*}"}
{"id": "1811.png", "formula": "\\begin{align*} D ^ n ( f g ) = \\sum _ { k = 0 } ^ n { n \\choose k } D ^ k ( f ) D ^ { n - k } ( g ) , \\end{align*}"}
{"id": "4443.png", "formula": "\\begin{align*} \\dfrac { ( 2 n + \\nu ) ! } { n ! ^ 2 } = \\dfrac { 2 ^ { 2 n + \\nu } n ^ { \\nu - \\frac { 1 } { 2 } } } { \\sqrt { \\pi } } \\left ( 1 + \\dfrac { ( - 1 ) ^ { \\nu + 1 } ( 2 \\nu + 1 ) } { 8 n } + \\dfrac { ( - 1 ) ^ \\nu ( 6 \\nu + 1 ) } { 1 2 8 n ^ 2 } + \\cdots \\right ) . \\end{align*}"}
{"id": "5052.png", "formula": "\\begin{align*} \\mathbf { L } _ n = \\mathbf { L } _ n ^ 0 + \\frac { \\mu } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } e ^ { \\gamma \\varphi ( \\theta ) } e ^ { i n \\theta } d \\theta , \\widetilde { \\mathbf { L } } _ n = \\widetilde { \\mathbf { L } } _ n ^ 0 + \\frac { \\mu } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } e ^ { \\gamma \\varphi ( \\theta ) } e ^ { - i n \\theta } d \\theta . \\end{align*}"}
{"id": "6023.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial x } + 2 t w = 0 . \\end{align*}"}
{"id": "4337.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\varepsilon _ 1 ( \\tau ) & = & c _ { 1 , 0 } \\| \\phi _ { \\ell , b , \\beta } \\| ^ { - 2 } _ { L ^ 2 _ { \\rho _ \\beta } } \\langle \\varepsilon , \\phi _ { 1 , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } , \\\\ \\varepsilon _ 1 ( \\tau ) & = & - \\| \\phi _ { 0 , b , \\beta } \\| ^ { - 2 } _ { L ^ 2 _ { \\rho _ \\beta } } \\langle \\varepsilon , \\phi _ { 0 , b , \\beta } \\rangle _ { L ^ 2 _ { \\rho _ \\beta } } . \\end{array} \\right . \\end{align*}"}
{"id": "8933.png", "formula": "\\begin{align*} \\ker d _ 0 & = \\{ \\varphi : X \\to A : d _ 0 \\varphi = 0 \\} \\\\ & = \\{ \\varphi : X \\to A : \\varphi ( x _ 1 ) - \\varphi ( x _ 0 ) = 0 \\forall x _ 1 , x _ 0 \\in X \\} \\\\ & = \\{ \\varphi : X \\to A \\mbox { c o n s t a n t } \\} \\\\ & = A \\end{align*}"}
{"id": "1495.png", "formula": "\\begin{align*} \\langle \\mathbf { f , h } \\rangle = \\sum _ { j } \\langle f _ j , h _ j \\rangle . \\end{align*}"}
{"id": "6045.png", "formula": "\\begin{align*} ( 1 - x ^ 2 ) y '' + [ - \\beta + \\alpha + ( \\alpha + \\beta - 2 ) x ] \\ , y ' + n ( n - \\alpha - \\beta + 1 ) y = 0 \\end{align*}"}
{"id": "4569.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } ( U _ n > x ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) , \\ \\ \\ \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "5589.png", "formula": "\\begin{align*} \\mu & = C \\cos ( \\omega \\Delta ) \\\\ \\omega & = - C \\sin ( \\omega \\Delta ) \\end{align*}"}
{"id": "9306.png", "formula": "\\begin{align*} H ^ 0 ( Y , \\mathcal { O } _ Y ) \\to H ^ 0 ( F _ i , \\mathcal { O } _ { F _ i } ) \\to H ^ 1 ( Y , \\mathcal { O } _ Y ( - F _ i ) ) \\to H ^ 1 ( Y , \\mathcal { O } _ Y ) = 0 . \\end{align*}"}
{"id": "5306.png", "formula": "\\begin{align*} \\langle w \\otimes b , v \\otimes a \\rangle _ A = \\langle w , v \\rangle b ^ * a , v , w \\in V , a , b \\in A , \\end{align*}"}
{"id": "1577.png", "formula": "\\begin{align*} \\mathcal { V } = \\{ \\mathbf { f } \\in \\mathcal { S } _ { k } ( K _ 1 ( \\mathfrak { n } ) ) : \\mathbf { f } | T _ { \\xi } = \\lambda ( \\xi ) \\mathbf { f } \\} , \\mathcal { V } ( \\overline { \\Q } ) = \\mathcal { V } \\cap \\mathcal { S } _ k ( K _ 1 ( \\mathfrak { n } ) ) \\end{align*}"}
{"id": "4739.png", "formula": "\\begin{align*} & y ^ 2 Q ( y ) - \\kappa _ 0 = \\int _ { \\mathbb { R } } F ( x ) \\bigg ( y ^ 2 G ( y - x ) - C _ 0 \\bigg ) \\ , \\dd x \\\\ & = \\int _ { \\{ | y | < 2 | y - x | \\} } F ( x ) \\bigg ( y ^ 2 G ( y - x ) - C _ 0 \\bigg ) \\ , \\dd x + \\int _ { \\{ | y | > 2 | y - x | \\} } F ( x ) \\bigg ( y ^ 2 G ( y - x ) - C _ 0 \\bigg ) \\ , \\dd x \\\\ & : = { \\rm I } + { \\rm I I } . \\end{align*}"}
{"id": "8803.png", "formula": "\\begin{align*} t _ { i j } = \\begin{cases} s _ { i j } & j \\in K _ i \\backslash \\{ 0 , n \\} \\\\ a _ { i j } & j = 0 \\\\ s _ { i k _ i ^ * } & j = n \\\\ ( 1 - \\gamma _ { i j } ) s _ { i l ( i , j ) } + \\gamma _ { i j } s _ { i r ( i , j ) } & j \\notin K _ i \\cup \\{ 0 \\} \\cup \\{ n \\} , \\end{cases} \\end{align*}"}
{"id": "6682.png", "formula": "\\begin{align*} L i _ { K , s } ( z ) : = \\sum _ { i \\geq 1 } \\frac { z ^ { q ^ i } } { ( \\theta ^ { q ^ i } - \\theta ) ^ s } \\in \\mathbb { C } _ { \\infty } \\end{align*}"}
{"id": "3835.png", "formula": "\\begin{align*} { \\vec y } \\ , ' = ( F ( x ) + \\Lambda ) \\vec y , \\end{align*}"}
{"id": "6168.png", "formula": "\\begin{align*} z _ 3 = & \\frac { - \\frac { 1 } { 3 } - \\frac { 2 } { 3 } ( z _ 1 + z _ 2 ) - z _ 1 z _ 2 } { z _ 1 + z _ 2 + \\frac { 2 } { 3 } + z _ 1 z _ 2 } . \\end{align*}"}
{"id": "7642.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\nabla \\bar { u } \\cdot \\nabla v = \\tilde { \\lambda } _ 0 \\int _ { \\R ^ n } \\tilde { m } _ 0 \\bar { u } v \\forall v \\in H ^ 1 ( \\R ^ N ) \\ ; , \\end{align*}"}
{"id": "7449.png", "formula": "\\begin{align*} \\mu ( \\phi ) = \\int _ { \\mathbb { R } ^ d } \\left ( \\frac 1 2 | \\nabla \\phi | ^ 2 + V | \\phi | ^ 2 + \\beta | \\phi | ^ 4 - \\Omega \\overline { \\phi } L _ z \\phi \\right ) \\mathrm { d } \\mathbf { x } = E ( \\phi ) + \\frac { \\beta } { 2 } \\int _ { \\mathbb { R } ^ d } | \\phi ( \\mathbf { x } ) | ^ 4 \\mathrm { d } \\mathbf { x } . \\end{align*}"}
{"id": "4069.png", "formula": "\\begin{align*} f = ( P _ E | _ { W ^ \\perp } ) ^ { - 1 } f _ 1 + ( P _ W | _ { E ^ \\perp } ) ^ { - 1 } f _ 2 , \\end{align*}"}
{"id": "5371.png", "formula": "\\begin{align*} S _ c = \\left \\lbrace \\sum _ { i = 1 } ^ { n - 1 } \\lambda _ i c ^ { ( i ) } \\middle | \\lambda _ i \\in \\mathbb { R } \\right \\rbrace . \\end{align*}"}
{"id": "3695.png", "formula": "\\begin{align*} B _ t - B _ x J + \\mu \\Lambda ^ \\alpha B = 0 , \\ \\ B _ x = \\mathcal H J . \\end{align*}"}
{"id": "2949.png", "formula": "\\begin{align*} \\forall \\ , ( h , h ' , i ) \\ne ( 2 , 2 , 2 ) : \\lim _ { n \\to \\infty } \\Lambda ^ { ( h : h ' ) } _ { n , i } = 0 , \\lim _ { n \\to \\infty } \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 } = \\frac 1 2 . \\end{align*}"}
{"id": "6355.png", "formula": "\\begin{align*} c _ 1 ( \\widetilde { M } _ l ) \\cdot E _ t = \\delta _ { l , i _ t } . \\end{align*}"}
{"id": "4836.png", "formula": "\\begin{align*} F _ n ^ D : = I _ n ^ D ( \\overline { \\mathbf { a } } ) - \\beta \\sum _ { i = 1 } ^ n Q ( \\overline { a } _ i ) , F _ n ^ C : = I _ n ^ C ( \\overline { \\mu } ) - \\beta \\int _ \\R Q ( x ) \\ , d \\overline { \\mu } ( x ) \\end{align*}"}
{"id": "8134.png", "formula": "\\begin{align*} \\Gamma ( H G ) = \\bigcup _ { k = 3 } ^ { 7 } \\Gamma _ { 2 k } ( H G ) . \\end{align*}"}
{"id": "6585.png", "formula": "\\begin{align*} & \\ll \\sum _ { q \\ll Q } \\sum _ { \\ell \\ll X } \\frac { ( H K \\ell ) ^ \\varepsilon } { \\ell \\sqrt { H K } } \\sum _ { \\substack { c > C , d \\geq 1 \\\\ c d = q } } \\phi ( d ) \\ \\ll \\frac { Q ^ 2 } { C } \\frac { ( X H K ) ^ { \\varepsilon } } { \\sqrt { H K } } . \\end{align*}"}
{"id": "8970.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( x ) + | u ' ( x ) | ^ p - f ( x ) \\leq 0 & ( a , b ) , \\\\ u ( x ) + | u ' ( x ) | ^ p - f ( x ) \\geq 0 & [ a , b ] . \\end{aligned} \\right . \\end{align*}"}
{"id": "4392.png", "formula": "\\begin{align*} v ^ + ( \\xi , \\tau ) = \\theta ^ + ( \\tau ) H _ 1 ( \\xi ) - M ( \\eta ) b ^ \\frac { \\eta } { 4 } ( \\tau ) H _ 0 \\left ( \\xi \\right ) v ^ { - } = \\theta ^ - ( \\tau ) H _ 1 ( \\xi ) + M ( \\eta ) b ^ { \\frac { \\eta } { 4 } } ( \\tau ) H _ 0 ( \\xi ) , \\end{align*}"}
{"id": "983.png", "formula": "\\begin{align*} \\vert x - y \\vert ^ { n + 2 s + 2 } = R ^ { n + 2 s + 2 } \\vert \\tilde x - \\tilde y \\vert ^ { n + 2 s + 2 } \\geqslant C R ^ { n + 2 s + 2 } \\big ( 1 + R ^ { - n - 2 s - 2 } \\vert y - a \\vert ^ { n + 2 s + 2 } \\big ) . \\end{align*}"}
{"id": "9184.png", "formula": "\\begin{align*} \\Delta u = f ~ ~ ~ ~ \\mbox { i n } ~ B _ 1 , \\end{align*}"}
{"id": "1433.png", "formula": "\\begin{align*} b _ n & = \\sum _ { s _ 1 , \\ldots , s _ n \\in S _ n } { n \\choose { s _ 1 , \\ldots , s _ k } } 2 ^ { \\sum _ { v _ i , v _ j \\in E ( D ) } s _ i s _ j } \\\\ & \\leq 2 ^ { n ^ 2 d ( 1 - 1 / n + 1 / n ^ 2 ) } \\sum _ { s _ 1 , \\ldots , s _ n \\in S _ n } { n \\choose { s _ 1 , \\ldots , s _ k } } \\leq 2 ^ { n ^ 2 d _ n ( 1 - 1 / n + 1 / n ^ 2 ) } n ^ k \\end{align*}"}
{"id": "1536.png", "formula": "\\begin{align*} = \\int _ { P ^ t _ n \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\sum _ { \\xi \\in G _ { 2 t + r } } \\sum _ { \\gamma \\in P ^ t _ n \\backslash G _ n } \\phi ( \\tilde { \\tau } _ t ( ( \\xi \\times 1 _ { 2 n - 2 t } ) g \\times \\gamma h ) , s ) \\mathbf { f } ( h ) \\mathbf { d } h \\end{align*}"}
{"id": "2424.png", "formula": "\\begin{align*} 1 = \\norm { e _ { \\gamma ' } } _ \\mathcal { H } ^ 2 = \\sum _ { \\gamma \\in \\Gamma } | \\langle e _ { \\gamma ' } , e _ \\gamma \\rangle | ^ 2 = 1 + \\sum _ { \\gamma \\neq \\gamma ' } | \\langle e _ { \\gamma ' } , e _ \\gamma \\rangle | ^ 2 . \\end{align*}"}
{"id": "4604.png", "formula": "\\begin{align*} \\biggl [ { a \\choose c } + { b \\choose c } \\biggr ] - \\biggl [ { a - 1 \\choose c } + { b + 1 \\choose c } \\biggr ] = { a - 1 \\choose c - 1 } - { b \\choose c - 1 } \\geq 0 . \\end{align*}"}
{"id": "386.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; L ( W , b ) : = \\frac { 1 } { M } \\ ; \\sum _ { i = 1 } ^ { M } \\| f ( W , b \\ ; ; \\ ; x ^ i ) \\ ; - \\ ; y ^ i \\| ^ 2 \\mbox { o v e r a l l } \\ ; \\ ; W \\ ; , \\ ; b \\end{align*}"}
{"id": "2911.png", "formula": "\\begin{align*} \\oint [ [ x , e ] , d ] & = [ [ x , e ] , d ] + [ [ e , d ] , x ] + [ [ d , x ] , e ] \\\\ & = 0 + [ v + t e , x ] + 0 \\\\ & = \\overline { [ v , x ] } + \\theta ( v , x ) e , \\end{align*}"}
{"id": "9544.png", "formula": "\\begin{align*} \\tilde S _ 0 ( \\bar x ) - \\bar x \\cdot \\bar c - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } \\le - S _ 0 ^ \\infty ( - \\bar x ) - \\bar x \\cdot \\bar c - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } < 0 \\end{align*}"}
{"id": "3300.png", "formula": "\\begin{align*} Q ( x ) = \\frac { x } { 1 - e ^ { - x } } = 1 + \\frac { x } { 2 } + \\sum _ { l \\geq 1 } \\frac { ( - 1 ) ^ { l - 1 } B _ { 2 l } } { ( 2 l ) ! } x ^ { 2 l } = 1 + \\frac { x } { 2 } + \\frac { x ^ 2 } { 1 2 } - \\frac { x ^ 4 } { 7 2 0 } + \\ldots \\end{align*}"}
{"id": "8021.png", "formula": "\\begin{align*} \\left \\{ \\partial _ \\Sigma ^ * \\Psi ( f ) , \\partial _ \\Sigma ^ * \\Psi ( g ) \\right \\} = \\left \\langle ( \\partial _ \\Sigma \\otimes \\partial _ \\Sigma ) E , f \\otimes g \\right \\rangle _ { \\Sigma ^ 2 } . \\end{align*}"}
{"id": "6604.png", "formula": "\\begin{align*} R ( s ; u , v ) = \\prod _ { p | v } \\left ( 1 - \\frac { 1 } { p ^ { s + 1 } } \\right ) \\prod _ { p \\nmid u v } \\left ( 1 + \\frac { 1 } { p ^ { s + 1 } ( p - 1 ) } \\right ) \\end{align*}"}
{"id": "4725.png", "formula": "\\begin{align*} \\| V \\partial _ y \\Phi _ 2 \\| _ { L ^ \\infty } \\lesssim \\frac { 1 } { t ^ 2 } , \\quad \\bigg \\| \\Phi _ 2 \\partial _ y V - \\sum _ { i = 1 } ^ n \\sigma _ i \\mu _ i \\partial _ y \\R _ i \\bigg \\| \\lesssim \\frac { 1 } { t ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
{"id": "8692.png", "formula": "\\begin{align*} \\begin{aligned} \\phi ( f ^ L ) & + \\sum _ { i = 1 } ^ d \\Biggl [ \\phi \\Biggl ( \\sum _ { j = 1 } ^ { i - 1 } e _ { \\omega _ j } f _ { \\omega _ j } ^ U + e _ { \\omega _ i } f _ { \\omega _ i } + \\sum _ { j = i + 1 } ^ d e _ { \\omega _ j } f _ { \\omega _ j } ^ L \\Biggr ) \\\\ & - \\phi \\Biggl ( \\sum _ { j = 1 } ^ { i - 1 } e _ { \\omega _ j } f _ { \\omega _ j } ^ U + \\sum _ { j = i } ^ d e _ { \\omega _ j } f _ { \\omega _ j } ^ L \\Biggr ) \\Biggr ] . \\end{aligned} \\end{align*}"}
{"id": "1912.png", "formula": "\\begin{align*} s '' ( t + k \\bar { \\tau } ) & = \\bigoplus _ { i = t + k \\bar { \\tau } } ^ { t + ( k + 1 ) \\bar { \\tau } - 1 } \\bar { s } ( i ) = \\bar { \\nu } ^ { k } \\left ( \\bigoplus _ { i = t } ^ { t + \\bar { \\tau } - 1 } \\bar { s } ( i ) \\right ) = \\bar { \\nu } ^ { k } s '' ( t ) , \\end{align*}"}
{"id": "3347.png", "formula": "\\begin{align*} & \\{ x _ 5 , x _ 1 , [ x _ 2 , x _ 3 , x _ 4 ] _ C \\} = \\{ \\{ x _ 5 , x _ 1 , x _ 2 \\} , x _ 3 , x _ 4 \\} - \\{ \\{ x _ 5 , x _ 1 , x _ 3 \\} , x _ 2 , x _ 4 \\} + \\{ x _ 2 , x _ 3 , \\{ x _ 5 , x _ 1 , x _ 4 \\} \\} ^ { * } , \\\\ & \\{ x _ 1 , x _ 2 , \\{ x _ 5 , x _ 3 , x _ 4 \\} \\} ^ { * } \\ ; = \\{ \\{ x _ 1 , x _ 2 , x _ 5 \\} ^ { * } , x _ 3 , x _ 4 \\} + \\{ x _ 5 , [ x _ 1 , x _ 2 , x _ 3 ] _ C , x _ 4 \\} + \\{ x _ 5 , x _ 3 , [ x _ 1 , x _ 2 , x _ 4 ] _ C \\} , \\end{align*}"}
{"id": "1630.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ { \\mathcal { D } ( \\mathcal { L } ^ { ( p ) } ) } : = \\left \\| ( \\lambda - \\mathcal { L } ^ { ( p ) } ) f \\right \\| _ { L ^ p ( M ) } . \\end{align*}"}
{"id": "302.png", "formula": "\\begin{align*} w ( x , t ) = U [ \\psi _ { 0 } ] ( x , t , 0 ) + N [ u , \\psi ] ( x , t ) , \\end{align*}"}
{"id": "3079.png", "formula": "\\begin{align*} x _ 1 ^ \\prime & = j x _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 1 ^ \\prime = \\phantom { j } y _ 1 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , ; \\\\ x _ 1 ^ \\prime & = \\phantom { j } x _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 1 ^ \\prime = j y _ 1 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , . \\end{align*}"}
{"id": "8331.png", "formula": "\\begin{align*} \\sup _ { s \\in [ 0 , T ] } \\left | z ( s ) - z _ j ( s ) \\right | \\leq \\sup _ { s \\in [ 0 , T ] } \\left ( \\sum _ { k \\colon z ( t _ k ) = z ( t _ k - ) } \\frac { \\psi _ k ( s ) } { j } + \\sum _ { k \\colon z ( t _ k ) \\neq z ( t _ k - ) } \\frac { \\psi _ k ( s ) } { j } \\right ) = \\frac { 1 } { j } . \\end{align*}"}
{"id": "9424.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { \\mathbf { s } ( 0 , \\lambda , \\delta ) } _ T } = \\sup _ { t \\in [ 0 , T ] } \\| u ( \\cdot , t ) \\| _ { C ^ { 0 , \\lambda , \\delta } } + \\sup _ { t ' , t '' \\in [ 0 , T ] \\atop t ' \\neq t '' } \\frac { \\| u ( \\cdot , t ' ) - u ( \\cdot , t '' ) \\| _ { C ^ { 0 , 0 , \\delta } } } { | t ' - t '' | ^ { \\lambda / 2 } } . \\end{align*}"}
{"id": "1654.png", "formula": "\\begin{align*} | \\nabla u | _ w = | \\nabla u | \\end{align*}"}
{"id": "8667.png", "formula": "\\begin{align*} \\partial _ t ( n _ 0 u ^ 0 ) + \\nabla _ x ( n _ 0 u ) = 0 , \\end{align*}"}
{"id": "3302.png", "formula": "\\begin{align*} \\alpha _ { d - j } ( E | _ D ) = \\sum _ { l = 1 } ^ { j } \\frac { ( - 1 ) ^ { l - 1 } a ^ { l - 1 } } { l ! } \\beta _ { d - j + l - 1 } ( E | _ D ) . \\end{align*}"}
{"id": "1288.png", "formula": "\\begin{align*} \\Lambda ( M , \\Gamma ) = \\{ \\ , \\alpha \\ , | \\ , \\alpha \\ , \\ , \\mathrm { i s \\ , \\ , a n \\ , \\ , a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , s u c h \\ , \\ , t h a t \\ , \\ , } [ \\alpha ] = \\Gamma \\mathrm { \\ , \\ , a n d } \\ , \\ , A ( \\alpha ) < M \\ , \\} . \\end{align*}"}
{"id": "130.png", "formula": "\\begin{align*} L ( A ) ( U \\oplus U ^ 2 ) = 0 , \\ ; V ^ 2 \\subseteq L ( A ) , \\ ; v ( v u ) \\in L ( A ) \\mbox { f o r a l l } u \\in U \\mbox { a n d } v \\in V \\end{align*}"}
{"id": "8274.png", "formula": "\\begin{align*} F _ { \\alpha } = \\sum _ { \\alpha \\leq \\beta } M _ { \\beta } M _ { \\alpha } = \\sum _ { \\alpha \\leq \\beta } ( - 1 ) ^ { \\ell ( \\beta ) - \\ell ( \\alpha ) } M _ { \\beta } . \\end{align*}"}
{"id": "2205.png", "formula": "\\begin{align*} \\ , \\dd X ( t ) + A ( A X ( t ) + F ( X ( t ) ) ) \\ , \\dd t = \\ , \\dd W ( t ) , \\ ; X ( 0 ) = X _ 0 , \\end{align*}"}
{"id": "7083.png", "formula": "\\begin{align*} | A | ^ 2 - 4 f ^ 2 - 8 f \\langle \\xi , E _ 3 ^ { \\perp } \\rangle ( \\langle \\xi , E _ 1 ^ { \\perp } \\rangle ^ 2 - \\langle \\xi , E _ 2 ^ { \\perp } \\rangle ^ 2 ) = 0 . \\end{align*}"}
{"id": "3147.png", "formula": "\\begin{align*} \\Phi ( t , x ) & = : \\Phi ( z , r _ 1 , r _ 2 ) \\\\ & = \\eta ^ L ( t ( z ) , x _ 0 ) + Z ( z , r _ 1 , r _ 2 ) \\tau ( z ) + R _ 1 ( z , r _ 1 , r _ 2 ) n ( z ) + R _ 2 ( z , r _ 1 , r _ 2 ) b ( z ) , \\end{align*}"}
{"id": "3005.png", "formula": "\\begin{align*} \\log ^ m | u _ i | \\Big [ \\sum _ { p = 0 } ^ m ( - 1 ) ^ { k + p } \\frac { 2 ^ m } { p ! ( m - p ) ! } \\Big ] & = \\log ^ m | u _ i | \\Big [ \\sum _ { p = 0 } ^ m ( - 1 ) ^ { k + p } \\frac { 2 ^ m } { m ! } { m \\choose p } \\Big ] \\\\ & = \\log ^ m | u _ i | \\ , \\frac { 2 ^ m } { m ! } ( - 1 ) ^ k \\Bigg [ \\sum _ { p = 0 } ^ m ( - 1 ) ^ p { m \\choose p } \\Bigg ] . \\end{align*}"}
{"id": "1978.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( x ) = 1 + \\sum _ { w \\in \\N ^ * } m ( w ) x _ w , \\widehat { \\kappa } ( x ) = \\sum _ { w \\in \\N ^ * } \\kappa ( w ) x _ w , \\end{align*}"}
{"id": "8521.png", "formula": "\\begin{align*} \\phi _ { \\Delta } ( t ) { = } \\operatorname { s g n } ( \\phi ( t ) ) \\operatorname { m i n } \\left ( \\left \\lfloor \\frac { 1 } { \\Delta } | \\phi ( t ) | { + } \\frac { 1 } { 2 } \\right \\rfloor { , } 2 ^ { w _ { \\phi } { - } 1 } { - } 1 \\right ) \\end{align*}"}
{"id": "3183.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k + e _ i \\frac { e _ i ^ \\intercal A ^ \\intercal ( b - A x _ k ) } { \\norm { A e _ i } _ 2 ^ 2 } . \\end{align*}"}
{"id": "6878.png", "formula": "\\begin{align*} \\Phi ( S ) \\geq 1 - \\alpha - ( 1 - \\alpha ) \\eta - \\sum \\limits _ { i = 1 } ^ { r } ( \\lambda _ i ( M ) - \\eta ) R ( k , i ) \\varepsilon _ i - c \\gamma \\end{align*}"}
{"id": "4323.png", "formula": "\\begin{align*} w ( \\tau ) = Q _ { b ( \\tau ) } + \\varepsilon ( \\tau ) , \\end{align*}"}
{"id": "5412.png", "formula": "\\begin{align*} 1 . & \\textrm { t h e i n t e r i o r s o f $ T _ i $ a r e p a i r w i s e d i s j o i n t a n d $ X = \\cup _ i T _ i $ , } \\\\ 2 . & \\textrm { t h e i n t e r i o r o f e a c h t r i a n g l e a d m i t s a b i j e c t i v e l o c a l h a l f - d i l a t i o n t o a t r i a n g l e i n $ \\R ^ 2 $ , } \\\\ 3 . & \\textrm { a l l v e r t i c e s a r e p o i n t s o f s i n g u l a r i t y , } \\\\ 4 . & \\textrm { t h e c o l l e c t i o n o f p o i n t s o f s i n g u l a r i t y d o n o t m e e t a t r i a n g l e $ T _ i $ o f f i t s v e r t i c e s . } \\end{align*}"}
{"id": "791.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { N ^ { 1 , p } ( X ) } : = \\Vert u \\Vert _ { L ^ p ( X ) } + \\mathcal { E } _ p ( u ) . \\end{align*}"}
{"id": "4636.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 5 & 5 & 4 & 3 \\\\ 7 & 4 & 1 & 0 \\\\ - 5 & - 9 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
{"id": "7301.png", "formula": "\\begin{align*} \\rho _ 1 = \\sigma _ 1 , & & \\rho _ 2 = \\sigma _ 1 ^ 2 - 2 \\sigma _ 2 , & & \\rho _ 3 = \\sigma _ 1 ^ 3 - 3 \\sigma _ 1 \\sigma _ 2 + 3 \\sigma _ 3 . \\end{align*}"}
{"id": "7347.png", "formula": "\\begin{align*} & W _ \\star [ x , t ] : = K \\cap \\{ w _ \\star ( \\cdot , t ) \\leq w _ \\star ( x , t ) \\} , \\\\ & U [ x , t ] : = K \\cap \\{ u ( \\cdot , t ) \\leq u ( x , t ) \\} = K \\cap \\{ v ( \\cdot , t ) \\leq v ( x , t ) \\} . \\end{align*}"}
{"id": "1843.png", "formula": "\\begin{align*} L _ n \\left ( \\sqrt { \\bar { x } \\bar { y } } , \\frac { \\bar { x } + \\bar { y } } { 2 } \\right ) = \\sqrt { \\bar { x } \\bar { y } ^ { - 1 } } \\sum _ { k = 0 } ^ n { n \\choose k } A _ k ( \\bar { x } , \\bar { y } ) \\frac { ( \\bar { y } - \\bar { x } ) ^ { n - k } } { 2 ^ { n - k } } . \\end{align*}"}
{"id": "1246.png", "formula": "\\begin{align*} \\mu ( E _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\varepsilon } ) & = \\mu ( \\left \\{ x : \\underline { \\dim } ( \\mu , x ) \\in [ \\alpha , \\gamma ] \\right \\} ) \\\\ \\mu ( F _ { \\mu } ^ { [ \\alpha , \\gamma ] , \\varepsilon } ) & = \\mu ( \\left \\{ x : \\overline { \\dim } ( \\mu , x ) \\in [ \\alpha , \\gamma ] \\right \\} ) \\end{align*}"}
{"id": "409.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & D _ { 0 } \\\\ \\end{array} \\right ) \\in \\mathbb { M } _ { N \\times N } . \\end{align*}"}
{"id": "6828.png", "formula": "\\begin{align*} ( f _ j \\circ \\psi _ j ) ^ * ( g _ { \\textnormal { e u c } } ) = \\psi _ j ^ * ( e ^ { 2 \\tilde u _ j } \\ , g _ { \\textnormal { p o i n } , j } ) = e ^ { 2 \\tilde u _ j \\circ \\psi _ j } \\ , \\psi _ j ^ * g _ { \\textnormal { p o i n } , j } = e ^ { 2 \\tilde u _ j \\circ \\psi _ j + 2 v _ j } \\ , g _ { \\textnormal { e u c } } \\textnormal { o n } \\ , \\ , \\ , B _ 1 ^ 2 ( 0 ) . \\end{align*}"}
{"id": "3783.png", "formula": "\\begin{align*} \\cfrac { n ( \\widetilde { \\chi } _ 0 \\pi _ E , \\psi _ E ) } { e } = n ( \\chi r _ l ( \\pi _ F ) ^ { ( l ) } , \\overline { \\psi } _ F ^ l ) \\end{align*}"}
{"id": "8553.png", "formula": "\\begin{align*} & \\left | \\partial _ { k } \\left ( m _ { \\pm } ( x , k ) - 1 \\right ) \\right | \\lesssim \\frac { 1 } { | k | } \\mathcal { W } _ { \\pm } ^ { 1 } ( x ) , \\pm x \\geq - 1 . \\end{align*}"}
{"id": "4132.png", "formula": "\\begin{align*} \\begin{cases} u _ { t } + D ^ { a + 1 } _ x \\partial _ { x } u + u u _ { x } = 0 , \\ ; \\ ; x , t \\in \\R , a \\in ( 0 , 1 ) , \\\\ u ( x , 0 ) = \\phi ( x ) , \\end{cases} \\end{align*}"}
{"id": "1282.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 2 | \\Lambda ( A ( \\alpha _ { k } ^ { \\Gamma } ) , \\Gamma ) | } { I ( \\alpha _ { k } ^ { \\Gamma } ) } = 1 . \\end{align*}"}
{"id": "7659.png", "formula": "\\begin{align*} w = P _ { \\Omega _ { \\varepsilon } } w + e ^ { - \\beta _ { \\varepsilon } \\Psi _ { \\varepsilon } ( \\mathbf { q } ) } V _ { \\varepsilon } \\Omega _ { \\varepsilon } \\ ; , \\end{align*}"}
{"id": "5669.png", "formula": "\\begin{align*} ( x \\cdot y ) \\cdot J x = y \\end{align*}"}
{"id": "3386.png", "formula": "\\begin{align*} & \\omega ( x , y , z ) = \\varphi ( [ x , y ] , z ) - \\rho ( z ) \\varphi ( x , y ) \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\quad = - \\Big ( \\varphi ( [ y , x ] , z ) - \\rho ( z ) \\varphi ( y , z ) \\Big ) = - \\omega ( y , x , z ) . \\end{align*}"}
{"id": "2697.png", "formula": "\\begin{align*} m ( t ) = \\sum _ { k \\in \\Z } m _ k e ^ { 2 \\pi i k t } , \\end{align*}"}
{"id": "4352.png", "formula": "\\begin{align*} v '' ( x ) + ( d - 4 ) v ' ( x ) - ( d - 2 ) v ( v - 1 ) ( v - 2 ) = 0 , x \\in ( - \\infty , + \\infty ) . \\end{align*}"}
{"id": "9070.png", "formula": "\\begin{align*} w ^ T \\mathcal { I } _ { \\phi \\phi } w = & \\frac { 1 } { 4 \\beta _ a } ( w _ 0 + w _ 1 ) ^ 2 + \\frac { 1 } { 4 h } \\sum _ { j = 1 } ^ { N } \\epsilon _ { j } ( w _ { j - 1 } - w _ { j + 1 } ) ^ 2 + \\frac { 1 } { 4 \\beta _ b } ( w _ N + w _ { N + 1 } ) ^ 2 \\geq 0 . \\end{align*}"}
{"id": "2297.png", "formula": "\\begin{align*} \\ell ( h ) = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\overline { \\langle h , M _ \\omega T _ x g \\rangle } \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "3831.png", "formula": "\\begin{align*} \\ell _ n ( y ) = y ^ { ( n ) } + \\sum _ { k = 0 } ^ { n - 2 } p _ k ( x ) y ^ { ( k ) } . \\end{align*}"}
{"id": "962.png", "formula": "\\begin{align*} u ( x _ \\ast ) = - u ( x ) x \\in \\R ^ n . \\end{align*}"}
{"id": "6491.png", "formula": "\\begin{align*} \\prod _ { j = j _ 0 ( \\delta ) + 1 } ^ { n - 1 } \\left ( 1 + \\dfrac { \\delta } { j } \\right ) = \\prod _ { j = 1 } ^ { n - 1 } \\dfrac { j + \\delta } { j } = \\dfrac { \\Gamma ( n + \\delta ) } { \\Gamma ( n ) \\Gamma ( 1 + \\delta ) } . \\end{align*}"}
{"id": "8325.png", "formula": "\\begin{align*} \\int _ { t } ^ { t + \\varepsilon } z \\dd ( y - u ) \\geq c \\int _ { t } ^ { t + \\varepsilon } \\dd ( u - y ) = c \\left ( ( u - y ) ( t + \\varepsilon ) - ( u - y ) ( t ) \\right ) > 0 . \\end{align*}"}
{"id": "1076.png", "formula": "\\begin{align*} \\Phi _ { R , L } ( x , t , k ) = \\Phi ^ { \\pm \\infty } ( x , t , k ) + \\int _ { \\pm \\infty } ^ { x } \\Phi ^ { \\pm \\infty } ( x , t , k ) \\left ( \\Phi ^ { \\pm \\infty } \\right ) ^ { - 1 } ( y , t , k ) \\left ( Q - Q ^ { \\pm \\infty } \\right ) ( y , t , k ) \\Phi _ { R , L } ( y , t , k ) d y . \\end{align*}"}
{"id": "9086.png", "formula": "\\begin{align*} L ( x _ { \\rm t h } ) = N \\sqrt { 2 \\pi } f _ D \\frac { x _ { \\rm t h } } { \\sigma } e ^ { - \\frac { x _ { \\rm t h } ^ 2 } { \\sigma ^ 2 } } \\left ( 1 - e ^ { - \\frac { x _ { \\rm t h } ^ 2 } { \\sigma ^ 2 } } \\right ) ^ { N - 1 } . \\end{align*}"}
{"id": "6550.png", "formula": "\\begin{align*} & g ( r ) = r ^ { \\mu _ { 1 } } \\ \\ \\mbox { w i t h } \\ \\mu _ { 1 } > 0 ; \\\\ & g ( r ) = \\big ( \\ln ( 1 + r ) \\big ) ^ { \\mu _ { 2 } } \\ \\ \\mbox { w i t h } \\ \\mu _ { 2 } > 1 ; \\\\ & g ( r ) = r ^ { \\mu _ { 3 } } \\big ( \\ln ( 1 + r ) \\big ) ^ { \\mu _ { 4 } } \\ \\ \\mbox { w i t h } \\ \\mu _ { 3 } > 0 , \\ , \\mu _ { 4 } \\geq 0 ; \\\\ & g ( r ) = \\ln ( 1 + r ) \\Big ( \\ln \\big ( 1 + \\ln ( 1 + r ) \\big ) \\Big ) ^ { \\mu _ { 5 } } \\ \\ \\mbox { w i t h } \\ \\mu _ { 5 } > 1 . \\end{align*}"}
{"id": "2638.png", "formula": "\\begin{align*} X g ( x ) = x g ( x ) P g ( x ) = \\frac { 1 } { 2 \\pi i } \\ , g ' ( x ) . \\end{align*}"}
{"id": "6065.png", "formula": "\\begin{align*} S = \\left \\{ R ( s , t ) = \\left ( x ( s , t ) , y ( s , t ) , z ( s , t ) \\right ) ; \\ ( s , t ) \\in \\Omega \\right \\} , \\end{align*}"}
{"id": "7115.png", "formula": "\\begin{align*} \\tilde { f } ( s , y ) & = \\alpha _ { - 1 } ( - \\tilde { \\theta } y ^ { 2 \\tilde { \\theta } + 1 } ) + \\alpha _ 0 y ^ { \\tilde { \\theta } + 1 } - \\alpha _ 1 \\frac { y } { \\tilde { \\theta } } + \\alpha _ 2 s ^ { 2 H - 1 } \\frac { 1 } { \\tilde { \\theta } ^ { \\rho } } y ^ { - \\tilde { \\theta } \\rho + \\tilde { \\theta } + 1 } - \\tilde { \\sigma } H s ^ { 2 H - 1 } y ^ { - 1 } ( \\tilde { \\theta } + 1 ) , \\end{align*}"}
{"id": "9376.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n + 1 , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } & = \\frac { d } { d t } \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } \\\\ & = \\frac { d } { d t } e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } = x e _ { \\lambda } ^ { 1 - \\lambda } ( t ) e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( x \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( 1 - \\lambda ) _ { n - k , \\lambda } \\phi _ { k , \\lambda } ( x ) \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "7423.png", "formula": "\\begin{align*} \\Omega _ 2 ^ j ( x ) : = \\Big \\{ \\eta \\in \\Omega : \\overleftarrow { \\eta } ^ { \\ell } ( x ) \\geq \\frac { 2 } { \\ell } \\Big \\} \\cup \\Big \\{ \\eta \\in \\Omega : \\overleftarrow { \\eta } ^ { \\ell } ( x - j \\ell ) \\geq \\frac { 2 } { \\ell } \\Big \\} . \\end{align*}"}
{"id": "2714.png", "formula": "\\begin{align*} \\R [ \\mathcal { B } _ n ] = \\R [ X _ 1 , \\ldots , X _ n ] / ( X _ 1 ( X _ 1 - 1 ) , \\ldots , X _ n ( X _ n - 1 ) ) , \\end{align*}"}
{"id": "5962.png", "formula": "\\begin{align*} \\phi _ k ^ n ( x ^ n , z ^ n ) \\approx \\sum _ { m = 1 } ^ { N _ { e p } } \\hat { \\phi } ^ n _ { k , m } \\psi _ { m } ( x ^ n , z ^ n ) = \\sum _ { i = 1 } ^ { N _ { e p } } \\phi _ k ^ n ( x _ i ^ n , z _ i ^ n ) h _ { i } ( x ^ n , z ^ n ) , ( x ^ n , z ^ n ) \\in E ^ n , \\end{align*}"}
{"id": "8478.png", "formula": "\\begin{align*} \\sigma : = \\left ( \\begin{smallmatrix} 0 & 1 \\\\ 1 & 0 \\end{smallmatrix} \\right ) \\qquad \\sigma ' : = \\mathbf { d ' } ( F ) \\ , ; \\end{align*}"}
{"id": "1193.png", "formula": "\\begin{align*} t ( \\mu , \\delta , \\mathcal { B } ) = \\lim _ { \\varepsilon \\to 0 } t ( \\mu , \\delta , \\varepsilon , \\mathcal { B } ) . \\end{align*}"}
{"id": "9144.png", "formula": "\\begin{align*} \\chi _ k ( r , n , m ) : = \\max \\{ \\delta ( k ) , \\chi ( r , n , m ) \\} \\delta ( k ) : = 2 k + 1 \\end{align*}"}
{"id": "8103.png", "formula": "\\begin{align*} \\mathrm { W F } ( u ) = \\left \\{ ( s , s ; \\xi _ s , - \\xi _ s ) \\in \\dot { T } ^ { * } \\Sigma _ 0 ^ 2 \\ , | \\ , \\xi _ s > 0 \\right \\} . \\end{align*}"}
{"id": "2364.png", "formula": "\\begin{align*} \\widehat { F } ( x , \\omega ) = F ( - \\omega , x ) . \\end{align*}"}
{"id": "2116.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M _ n ) } ) = - \\big ( 1 - ( 1 - c ) ^ k \\big ) \\log ( 1 - ( 1 - c ) ^ k ) . \\end{aligned} \\end{align*}"}
{"id": "5335.png", "formula": "\\begin{align*} \\| \\rho ^ { \\epsilon } \\| _ { L ^ { \\infty } ( \\mathbb { R } ^ + ; L ^ { \\infty } ) } = \\| 1 + \\epsilon \\phi ^ { \\epsilon } \\| _ { L ^ { \\infty } ( \\mathbb { R } ^ + ; L ^ { \\infty } ) } \\approx 1 , \\end{align*}"}
{"id": "3792.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ { - r } \\big ( r _ l ( \\pi _ F ) ^ { ( l ) } ( w _ n ) W , \\sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \\big ) q _ F ^ { - \\frac { r } { 2 } } X ^ { - l r } = \\gamma \\big ( X ^ l , r _ l ( \\pi _ F ) ^ { ( l ) } , \\sigma _ F ^ { ( l ) } , \\overline { \\psi } _ F ^ l \\big ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( W , W ' ) q _ F ^ { \\frac { r } { 2 } } X ^ { l r } . \\end{align*}"}
{"id": "2118.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , j , l } ) = \\frac k { k ( n - j + 1 ) - l } , \\ j \\in [ n - 1 ] . \\end{align*}"}
{"id": "5271.png", "formula": "\\begin{align*} \\Delta \\circ S ^ 2 = ( \\sigma ^ { \\varphi } \\otimes ( \\sigma ^ { \\varphi _ S } ) ^ { - 1 } ) \\circ \\Delta , \\end{align*}"}
{"id": "5435.png", "formula": "\\begin{align*} \\sigma _ k = \\begin{cases} 1 & \\mbox { f o r } k < k _ c , \\\\ 2 & \\mbox { f o r } k \\ge k _ c , \\end{cases} \\end{align*}"}
{"id": "8466.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( \\mathcal { L } ) = \\| \\mathcal { L } \\| \\leq 1 - \\frac { 2 \\sigma _ { \\min } ^ 2 } { \\| A \\| _ F ^ 2 } , \\end{aligned} \\end{align*}"}
{"id": "5736.png", "formula": "\\begin{align*} \\begin{aligned} | N _ { G } ( h ) \\backslash V ( T ' \\cup B ) | & \\geq d _ { G } ( h ) - | N _ { G } ( h ) \\cap V ( B ) | \\\\ & - | N _ { G } ( h ) \\cap ( V ( T ' ) \\backslash \\{ u _ { l + 1 } , \\cdots , u _ { j } \\} ) | \\\\ & \\geq ( t + 3 ) - 2 - ( t - \\lfloor \\frac { j - l } { 2 } \\rfloor ) \\\\ & \\geq 1 + \\lfloor \\frac { j - l } { 2 } \\rfloor , \\end{aligned} \\end{align*}"}
{"id": "4785.png", "formula": "\\begin{align*} n = q , \\theta = \\hat \\theta = 0 . \\end{align*}"}
{"id": "5657.png", "formula": "\\begin{align*} \\varphi _ { \\sigma ^ { - 1 } i } = \\alpha ^ { - 1 } _ i , i = 1 , 2 , 3 . \\end{align*}"}
{"id": "1962.png", "formula": "\\begin{align*} \\phi * \\psi : = m _ A ( \\phi \\otimes \\psi ) \\Delta , \\end{align*}"}
{"id": "6734.png", "formula": "\\begin{align*} \\mathcal { L } _ { K , ( i s _ 1 , i s _ 2 + j s _ 2 ) } ( \\alpha _ 1 , \\alpha _ 2 \\alpha _ 3 ) = \\sum _ { \\substack { i _ 1 > i _ 2 > 0 \\\\ i _ 2 \\neq N } } \\frac { \\alpha _ 1 ^ { q ^ { i _ 1 } } \\alpha _ 2 ^ { q ^ { i _ 2 } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { i s _ 1 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { i s _ 2 + j s _ 2 } } + \\sum _ { i _ 1 > N = i _ 2 > 0 } \\frac { \\alpha _ 1 ^ { q ^ { i _ 1 } } \\alpha _ 2 ^ { q ^ { i _ 2 } } } { ( \\theta ^ { q ^ { i _ 1 } } - t ) ^ { i s _ 1 } ( \\theta ^ { q ^ { i _ 2 } } - t ) ^ { i s _ 2 + j s _ 2 } } . \\end{align*}"}
{"id": "8168.png", "formula": "\\begin{align*} M _ { 2 d _ 0 } ( f , \\{ 1 \\} ) = M _ { d _ 0 } ( 2 f , \\{ 1 \\} ) . \\end{align*}"}
{"id": "4254.png", "formula": "\\begin{align*} R ( \\lambda ) f = \\sigma \\lim _ { \\mathclap { n \\to \\infty } } R _ n ( \\lambda ) f . \\end{align*}"}
{"id": "8161.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in X _ p } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) \\sum _ { k = 1 } ^ { p - 1 } \\vert S ( k , \\chi ) \\vert ^ 2 & = \\sum _ { \\chi \\in X _ p } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) \\frac { p ^ 2 - 1 } { 1 2 } + \\frac { ( p - 1 ) p ( 2 p - 1 ) } { 6 } \\\\ & + \\frac { p ^ 3 } { 2 \\pi ^ 2 } M _ { q _ 1 , q _ 2 } ( p ) + O ( p ^ 2 ) . \\end{align*}"}
{"id": "7724.png", "formula": "\\begin{gather*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } , \\ ; \\phi [ 0 ] = u [ 0 ] , \\end{gather*}"}
{"id": "1552.png", "formula": "\\begin{align*} \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { E } ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h = c _ k ( s ) D ( s , \\mathbf { f } , \\chi ) \\mathbf { f } ( g ) . \\end{align*}"}
{"id": "5013.png", "formula": "\\begin{align*} { B _ { ( k + 1 ) n } \\over B _ { k n } } = { 1 + \\rho _ * [ r _ { k n - 1 } , r _ { k n - 2 } , \\ldots , r _ 0 ] \\over 1 + \\rho _ * [ r _ { ( k + 1 ) n - 1 } , r _ { ( k + 1 ) n - 2 } , \\ldots , r _ 0 ] } { 1 + { p _ { ( k + 1 ) n } \\over q _ { ( k + 1 ) n } } \\over 1 + { p _ { k n } \\over q _ { k n } } } \\le ( 1 + C \\theta ^ { 2 k n } ) . \\end{align*}"}
{"id": "5293.png", "formula": "\\begin{align*} \\delta _ { \\varphi } ^ { n / 2 } ( ( \\sigma ^ { \\varphi } ) ^ { n / 2 } S ^ n ( b ) ) = 0 , \\end{align*}"}
{"id": "1378.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left ( A ( n _ { 2 j } , n _ { 2 j x } ) - A ( n _ { 1 j } , n _ { 1 j x } ) \\right ) \\varphi _ x d x + \\int _ 0 ^ 1 \\left ( n _ { 2 j } - n _ { 1 j } \\right ) \\varphi d x = \\int _ 0 ^ 1 \\left ( b _ 2 - b _ 1 \\right ) \\varphi d x \\le 0 , \\end{align*}"}
{"id": "62.png", "formula": "\\begin{align*} L \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 L _ { p _ i , j } } ( \\pi ^ j ) & ( v = p _ i \\ \\mathrm { f o r } \\ i = 1 , \\cdots , k ) , \\\\ L _ { v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
{"id": "3365.png", "formula": "\\begin{align*} \\Big ( \\phi _ { t } = I d _ L + t \\mathcal { L } ( \\mathfrak { X } ) - = I d _ L + t [ \\mathfrak { X } , - ] , \\ ; \\psi _ { t } = I d _ V + t D ( \\mathfrak { X } ) ( - ) \\Big ) , \\end{align*}"}
{"id": "9285.png", "formula": "\\begin{align*} [ J ^ { \\chi _ A } ] _ \\mathcal { M } : = \\lambda \\alpha \\in \\mathbb { N } ^ \\mathbb { N } , x \\in X . \\begin{cases} J ^ A _ { r _ \\alpha } x & r _ \\alpha > 0 \\rho > - r _ \\alpha / 2 , \\\\ 0 & , \\end{cases} \\end{align*}"}
{"id": "1493.png", "formula": "\\begin{align*} \\langle \\mathbf { f , h } \\rangle = \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { f } ( g ) \\overline { \\mathbf { h } ( g ) } \\mathbf { d } g . \\end{align*}"}
{"id": "4090.png", "formula": "\\begin{align*} G '' ( \\xi _ i ) = 2 \\alpha _ i + 2 \\ , \\frac { \\mu ( \\xi _ i + ) - \\mu ( \\xi _ i ) } { \\sigma ^ 2 ( \\xi _ i ) } , \\end{align*}"}
{"id": "2164.png", "formula": "\\begin{align*} G ( t ) = \\int _ { 0 } ^ { t } g ( s ) \\ d s , \\end{align*}"}
{"id": "6711.png", "formula": "\\begin{align*} g _ 1 ( t ) + g _ 2 ( t ) _ { s + 1 } \\mathcal { F } _ s ( a _ 1 , \\ldots , a _ { s + 1 } ; b _ 1 , \\ldots , b _ s ) ( \\alpha ) ^ { q ^ d } = 0 . \\end{align*}"}
{"id": "2433.png", "formula": "\\begin{align*} \\pi ( \\l ) \\pi ( \\l ^ \\circ ) = \\pi ( \\l ^ \\circ ) \\pi ( \\l ) \\end{align*}"}
{"id": "6217.png", "formula": "\\begin{align*} \\epsilon ^ 2 _ { v } \\geq \\| V _ l - \\hat { V } \\| ^ 2 _ F \\geq \\| \\mathbf { V } _ { 1 1 } - \\hat { \\mathbf { V } } _ { 1 1 } \\| ^ 2 _ F + \\| \\mathbf { v } _ { 2 1 } - \\hat { \\mathbf { v } } _ { 2 1 } \\| ^ 2 _ { 2 } = \\| \\check { E } \\| ^ 2 _ F + \\| e \\| ^ 2 _ { 2 } . \\end{align*}"}
{"id": "7956.png", "formula": "\\begin{align*} \\overline { \\frac { \\partial u } { \\partial Z _ { j } } } = \\sum _ { i = 1 } ^ { n } \\overline { g _ { i } } \\overline { \\frac { \\partial h _ { i } } { \\partial Z _ { j } } } , j = 1 , \\cdots , n . \\end{align*}"}
{"id": "2584.png", "formula": "\\begin{align*} \\norm { V _ g f } _ { L ^ \\infty _ { ( 1 + | z | ) ^ s } } = \\sup _ { z \\in \\R ^ { 2 d } } ( 1 + | z | ) ^ s | V _ g f ( z ) | , s \\geq 0 , \\end{align*}"}
{"id": "1024.png", "formula": "\\begin{align*} \\tilde \\Omega \\cap \\{ x _ 1 = 0 \\} \\subset \\bigcup _ { k = 1 } ^ M B ^ { ( k ) } \\Subset \\Omega . \\end{align*}"}
{"id": "7234.png", "formula": "\\begin{align*} \\omega = \\check x _ { t , x , v } = x - \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } v \\Leftrightarrow v = \\tfrac { x - \\omega } { t - \\tau _ \\omega } , \\end{align*}"}
{"id": "7492.png", "formula": "\\begin{align*} & ( \\partial _ { x } ^ { h } \\phi ) _ { j k } = \\sum _ { p = - M / 2 } ^ { M / 2 - 1 } \\sum _ { q = - M / 2 } ^ { M / 2 - 1 } i \\varrho ^ { x } _ { p } \\widehat { \\phi } _ { p q } \\ , e ^ { i \\frac { 2 j p \\pi } { M } } e ^ { i \\frac { 2 k q \\pi } { M } } , ( \\partial _ { y } ^ { h } \\phi ) _ { j k } = \\sum _ { p = - M / 2 } ^ { M / 2 - 1 } \\sum _ { q = - M / 2 } ^ { M / 2 - 1 } i \\varrho ^ { y } _ { q } \\widehat { \\phi } _ { p q } \\ , e ^ { i \\frac { 2 j p \\pi } { M } } e ^ { i \\frac { 2 k q \\pi } { M } } , \\end{align*}"}
{"id": "4893.png", "formula": "\\begin{align*} \\mathfrak { T } ( r , L ) = O ( \\log r ) \\hbox { a s $ r \\to + \\infty $ . } \\end{align*}"}
{"id": "8159.png", "formula": "\\begin{align*} M _ { q _ 1 , q _ 2 } ( p ) & = \\frac { \\pi ^ 2 } { 4 p ^ 2 } \\sum _ { a = 1 } ^ { p - 1 } \\sum _ { b = 1 } ^ { p - 1 } \\epsilon ( q _ 1 a , q _ 2 b ) \\cot \\left ( \\frac { \\pi a } { p } \\right ) \\cot \\left ( \\frac { \\pi b } { p } \\right ) \\\\ & = \\frac { \\pi ^ 2 } { 2 p ^ 2 } \\sum _ { a = 1 } ^ { p - 1 } \\cot \\left ( \\frac { \\pi q _ 1 a } { p } \\right ) \\cot \\left ( \\frac { \\pi q _ 2 a } { p } \\right ) = \\frac { 2 \\pi ^ 2 } { p } s ( q _ 1 , q _ 2 , p ) . \\end{align*}"}
{"id": "2443.png", "formula": "\\begin{align*} S ^ T J S = J . \\end{align*}"}
{"id": "5572.png", "formula": "\\begin{align*} \\| x \\| \\ = \\ \\left \\| x - P _ A ( x ) + \\sum _ { n \\in A } e _ n ^ * ( x ) e _ n \\right \\| & \\ \\leqslant \\ \\sup _ { \\delta } \\| x - P _ A ( x ) + 1 _ { \\delta A } \\| \\\\ & \\ \\leqslant \\ \\mathbf C _ { \\lambda , p l } \\| x - P _ A ( x ) + 1 _ { \\varepsilon B } \\| . \\end{align*}"}
{"id": "792.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { \\theta , p } ^ p : = \\int _ { Z } \\int _ { Z } \\frac { | u ( y ) - u ( x ) | ^ p } { d ( x , y ) ^ { \\theta p } \\ , \\nu ( B ( x , d ( x , y ) ) ) } \\ , d \\nu ( y ) \\ , d \\nu ( x ) , \\end{align*}"}
{"id": "4082.png", "formula": "\\begin{align*} \\dim ( X \\cap Y ) + \\dim ( Y \\cap V ) = \\dim ( X \\cap Y + Y \\cap V ) - \\dim ( X \\cap Y \\cap V ) \\leq \\dim ( Y ) - \\dim ( X \\cap Y \\cap V ) . \\end{align*}"}
{"id": "5716.png", "formula": "\\begin{align*} ( \\delta _ { 1 2 } , \\delta _ { 1 3 } , \\delta _ { 1 4 } , \\delta _ { 1 5 } ) = ( \\gamma _ { 1 } \\cup { \\gamma _ { 3 } } , \\gamma _ { 1 } \\cup { \\gamma _ { 4 } } , \\gamma _ { 2 } \\cup { \\gamma _ { 3 } } , \\gamma _ { 2 } \\cup { \\gamma _ { 4 } } ) \\end{align*}"}
{"id": "4010.png", "formula": "\\begin{align*} \\begin{aligned} m _ 1 & = - \\frac { 1 } { 3 } \\left ( a + C + \\frac { D _ 0 } { C } \\right ) , \\\\ m _ 2 & = - \\frac { 1 } { 3 } \\left ( a + \\frac { ( - 1 + i \\sqrt { 3 } ) } { 2 } \\ , C + \\frac { ( - 1 - i \\sqrt { 3 } ) } { 2 } \\ , \\frac { D _ 0 } { C } \\right ) , \\\\ m _ 3 & = - \\frac { 1 } { 3 } \\left ( a + \\frac { ( - 1 - i \\sqrt { 3 } ) } { 2 } \\ , C + \\frac { ( - 1 + i \\sqrt { 3 } ) } { 2 } \\ , \\frac { D _ 0 } { C } \\right ) , \\end{aligned} \\end{align*}"}
{"id": "7524.png", "formula": "\\begin{align*} \\log \\Gamma \\left ( \\frac { \\sigma + i T } { 2 } \\right ) & = \\left ( \\frac { \\sigma - 1 } { 2 } + \\frac { i T } { 2 } \\right ) \\left [ \\frac { 1 } { 2 } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) + i \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) \\right ] - \\\\ & \\frac { \\sigma + i T } { 2 } - \\frac { 1 } { 2 } \\log ( 2 \\pi ) + o ( 1 ) \\end{align*}"}
{"id": "3660.png", "formula": "\\begin{align*} h ( s ) : = s + A s ^ { 1 + \\delta } . \\end{align*}"}
{"id": "6142.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 0 } ^ n p _ k x ^ k \\end{align*}"}
{"id": "2335.png", "formula": "\\begin{align*} \\mathcal { P } f ( t ) = \\sum _ { k \\in \\Z ^ d } f ( t + k ) = \\sum _ { l \\in \\Z ^ d } \\widehat { f } ( l ) e ^ { 2 \\pi i l \\cdot t } . \\end{align*}"}
{"id": "2187.png", "formula": "\\begin{align*} m _ 1 = \\inf _ { \\mathcal { M } } J . \\end{align*}"}
{"id": "6761.png", "formula": "\\begin{align*} \\mathbf e \\approx \\sum _ { j = m + 1 } ^ M r _ j \\begin{bmatrix} \\mathbf 0 \\\\ [ \\mathbf L _ { 2 2 } ^ { - 1 } ] _ { : , j } \\end{bmatrix} \\approx \\sum _ { j = m + 1 } ^ M r _ j \\frac { h } { 2 } \\begin{bmatrix} \\mathbf 0 \\\\ G ( \\mathbf S _ 2 , S _ j ) \\end{bmatrix} , \\end{align*}"}
{"id": "7925.png", "formula": "\\begin{align*} \\tilde { \\theta } _ { T } = \\frac { \\int _ { 0 } ^ { T } f ( X _ { t } ) ( \\mathrm { d } X _ { t } - \\mathrm { d } L _ { t } + \\mathrm { d } R _ { t } ) } { \\int _ { 0 } ^ { T } f ^ { 2 } ( X _ { t } ) \\mathrm { d } t } = \\theta _ { 0 } + \\frac { \\sigma \\int _ { 0 } ^ { T } f ( X _ { t } ) \\mathrm { d } W _ { t } } { \\int _ { 0 } ^ { T } f ^ { 2 } ( X _ { t } ) \\mathrm { d } t } . \\end{align*}"}
{"id": "3234.png", "formula": "\\begin{align*} \\frac { d } { d s } \\frac { s } { h ( s ) } = \\frac { d } { d s } \\frac { 1 } { \\int _ \\mathbb { R } \\frac { 1 } { s ^ 2 + u ^ 2 } d \\mu ( u ) } = 2 s \\cdot \\frac { \\int _ \\mathbb { R } \\frac { 1 } { ( s ^ 2 + u ^ 2 ) ^ 2 } d \\mu ( u ) } { \\left ( \\int _ \\mathbb { R } \\frac { 1 } { s ^ 2 + u ^ 2 } d \\mu ( u ) \\right ) ^ 2 } . \\end{align*}"}
{"id": "4329.png", "formula": "\\begin{align*} \\| \\phi _ { 1 , b , \\beta } \\| _ { L ^ 2 _ { \\rho _ \\beta } } ^ 2 = \\frac { \\left ( \\frac { 2 } { \\sqrt { 2 \\beta } } \\right ) ^ { - 2 \\gamma } } { 2 \\pi ^ { \\frac { d + 2 } { 2 } } } \\Gamma \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) \\left [ \\left ( \\frac { 4 } { \\beta ^ 2 } - \\frac { 1 6 } { \\beta } + 1 6 \\right ) \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) ^ 2 + \\frac { 4 } { \\beta ^ 2 } \\left ( \\frac { d } { 2 } - \\gamma + 1 \\right ) \\right ] \\end{align*}"}
{"id": "1735.png", "formula": "\\begin{align*} \\varphi ( t , \\ , m , \\ , n ) = \\varphi _ i ( t , \\ , m , \\ , n ) ( t , \\ , m ) \\in A _ i . \\end{align*}"}
{"id": "4376.png", "formula": "\\begin{align*} \\varepsilon ^ + = e ^ { \\int _ { \\tau _ 0 } ^ \\tau \\lambda _ { M , b } d \\tau ' } \\phi _ { M , b } f ( y ) \\end{align*}"}
{"id": "1716.png", "formula": "\\begin{align*} \\nu _ * = \\frac { \\alpha _ * ( 1 / p _ 1 \u2010 1 / q ) + \\mu _ * ( 1 / p _ 0 \u2010 1 / q ) } { 1 / p _ 1 \u2010 1 / p _ 0 } . \\end{align*}"}
{"id": "4418.png", "formula": "\\begin{align*} \\begin{array} { l l } R e ( ( d ^ i ) ^ * A w ) = c _ 2 , \\ ; \\ ; \\forall i \\in \\mathcal { I } \\\\ \\displaystyle \\sum _ { j = 1 } ^ { n } w _ j = 1 \\\\ \\lvert a r g w _ j \\rvert \\leq \\beta , \\ ; \\ ; j = 1 , 2 , . . . , n \\end{array} \\end{align*}"}
{"id": "1820.png", "formula": "\\begin{align*} u = x y , \\ ; \\ ; 2 v = x + y . \\end{align*}"}
{"id": "4185.png", "formula": "\\begin{align*} 0 = \\omega ( u ^ k ) = k \\omega ( u ) , \\end{align*}"}
{"id": "6175.png", "formula": "\\begin{align*} \\{ x _ { \\rm T L S } , E _ { \\rm T L S } , f _ { \\rm T L S } \\} : = \\mathop { \\rm a r g m a x } \\limits _ { x , E , f } \\| E , F \\| _ { F } \\ \\ \\ \\ s . t . \\ \\ ( A + E ) x = b + f , \\end{align*}"}
{"id": "1233.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\mu , s } _ { t } ( A ) = \\inf \\left \\{ \\mathcal { H } ^ { s } _ { t } ( E ) : \\ E \\subset A , \\ \\mu ( E ) = \\mu ( A ) \\right \\} . \\end{align*}"}
{"id": "217.png", "formula": "\\begin{align*} B = \\partial _ k ( p _ \\alpha ) ( x ) D _ { k } ^ { \\alpha - 1 } ( f ) ( x ) + p _ \\alpha ( x ) \\partial _ k D _ k ^ { \\alpha - 1 } ( f ) ( x ) . \\end{align*}"}
{"id": "6319.png", "formula": "\\begin{align*} \\rho _ 3 ( \\mathbf { c } ) \\Sigma \\rho _ 2 ( A ) \\Sigma ^ { - 1 } \\rho _ 1 ( a ) = \\left ( \\begin{array} { c c c } a \\iota ( \\alpha _ { 1 1 } ) & \\iota ( \\alpha _ { 1 2 } ) \\sigma ^ { - 1 } & a C _ 1 \\\\ a \\sigma \\iota ( \\alpha _ { 2 1 } ) & \\sigma \\iota ( \\alpha _ { 2 2 } ) \\sigma ^ { - 1 } & a C _ 2 \\\\ 0 & 0 & a I _ 2 \\end{array} \\right ) , \\end{align*}"}
{"id": "6531.png", "formula": "\\begin{align*} c _ n & : = \\frac { ( n + m - 1 ) ! } { n ^ m ( n - 1 ) ! } = \\prod ^ { m - 1 } _ { k = 1 } \\left ( 1 + \\frac { k } { n } \\right ) = 1 + O \\left ( \\dfrac { 1 } { n } \\right ) ( n \\to \\infty ) , \\\\ c _ n ' & : = \\frac { n ^ m n ! } { ( n + m ) ! } = \\prod ^ { m } _ { k = 1 } \\left ( 1 + \\frac { k } { n } \\right ) ^ { - 1 } = 1 + O \\left ( \\dfrac { 1 } { n } \\right ) ( n \\to \\infty ) . \\end{align*}"}
{"id": "133.png", "formula": "\\begin{align*} a ^ { n + 1 } = \\gamma _ 2 a ^ 2 + \\dots + \\gamma _ { n } a ^ { n } , \\end{align*}"}
{"id": "4274.png", "formula": "\\begin{align*} \\begin{cases} { \\bf D } ^ { ( z ) } ( t , x ) ~ = ~ { \\bf D } _ 1 ^ { ( z ) } ( t , x ) + { \\bf D } _ 2 ^ { ( z ) } ( t , x ) + { \\bf D } _ 3 ^ { ( z ) } ( t , x ) \\\\ [ 2 m m ] { \\bf D } _ 1 ^ { ( z ) } ( t , x ) ~ = ~ I ^ { ( z ) } _ { 1 } ( t , x ) \\cdot \\phi _ 0 ' ( x - t ) , { \\bf D } _ 2 ^ { ( z ) } ( t , x ) ~ = ~ I ^ { ( z ) } _ 2 ( t , x ) \\cdot \\phi _ 0 ' ( x ) \\ , , \\\\ [ 2 m m ] { \\bf D } _ 3 ^ { ( z ) } ( t , x ) ~ = ~ I ^ { ( z ) } _ { 3 1 } ( t ) \\cdot g _ { | t | } ( t - x ) + I ^ { ( z ) } _ { 3 2 } ( t ) \\cdot \\phi _ 0 ' ( t ) . \\end{cases} \\end{align*}"}
{"id": "7830.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { 3 j - 1 } \\beta _ { i } e _ { i } , \\quad ~ \\beta _ { 3 j - 1 } , \\beta _ { 3 j - 2 } \\neq 0 . \\end{align*}"}
{"id": "3515.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) & = \\sum _ { m \\leq a t _ 3 } \\sum _ { n < m } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + D _ 1 + D _ 2 \\\\ & + \\sum _ { n \\leq a t _ 3 } \\frac { E ( s _ 1 , s _ 3 ; a t _ 3 , n , M ) } { n ^ { s _ 2 } } + \\sum _ { a t _ 3 < n \\leq N } \\frac { E ( s _ 1 , s _ 3 ; n , M ) } { n ^ { s _ 2 } } \\\\ & - B _ 2 - B _ 3 - B _ 4 - C _ 2 - C _ 3 - C _ 4 \\end{align*}"}
{"id": "8520.png", "formula": "\\begin{align*} t ^ v _ j = Q ^ v \\left ( \\phi ^ { c h } ( t ^ { c h } ) + \\sum _ { i \\in \\mathcal { N } _ v \\setminus \\{ j \\} } \\phi ^ c _ i ( t ^ c _ i ) \\right ) , \\end{align*}"}
{"id": "7845.png", "formula": "\\begin{align*} X _ { 1 } & = ~ \\{ x , T ^ { * 3 } x , \\ldots , T ^ { * 3 ( 3 j - n + 1 - r ) } x , y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( 3 j - n - 2 r ) } y \\} \\\\ X _ { 2 } & = ~ \\{ T ^ { * 3 ( 3 j - n + 2 - r ) } x , \\ldots , T ^ { * 3 j } x , T ^ { * 3 ( 3 j - n - 2 r + 1 ) } y , \\ldots , T ^ { * 3 ( j - 1 - r ) } y , z , \\ldots , T ^ { * 3 ( n - 2 j - 2 + r ) } z \\} . \\end{align*}"}
{"id": "5354.png", "formula": "\\begin{align*} x = \\frac { b _ i \\tilde a _ i } { \\| \\tilde a _ i \\| ^ 2 } \\in H _ i . \\end{align*}"}
{"id": "1650.png", "formula": "\\begin{align*} \\mathcal { M } _ s \\nu ( x ) : = \\sup _ { r > 0 } r ^ { - s } \\nu ( B ( x , r ) ) , x \\in M , \\end{align*}"}
{"id": "4782.png", "formula": "\\begin{align*} n = 3 0 6 9 3 6 , \\theta = 2 7 1 , \\hat \\theta = \\frac { 2 7 1 } { 3 0 2 4 } \\sqrt { \\frac { 7 8 5 } { 7 4 8 3 7 4 } } \\approx 2 . 9 0 2 4 3 8 2 \\cdot 1 0 ^ { - 3 } . \\end{align*}"}
{"id": "3952.png", "formula": "\\begin{align*} c _ m = \\left \\langle \\psi ( \\cdot ) , e ^ { \\frac { 2 \\pi i m \\cdot } { | I | } } \\right \\rangle _ { H ^ { - s } _ { p e r } ( I ) , H ^ s _ { p e r } ( I ) } , \\ \\ \\ \\forall m \\in \\mathbb { Z } . \\end{align*}"}
{"id": "4847.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 Q _ \\alpha ( x ) \\ , d x = \\frac 1 { P ( \\alpha ) } \\int _ 0 ^ \\alpha Q ( z ) \\ , d z = 1 \\alpha > q _ 2 , \\end{align*}"}
{"id": "3580.png", "formula": "\\begin{align*} x ' - h - 1 & = ( p ^ e - p ^ t ) - \\Big ( 1 + p ^ e \\\\ & \\ \\ \\ \\ + p + { \\sum } _ { i = 1 } ^ { t - 1 } ( p - 1 ) p ^ i + ( s _ { t + 1 } - 2 ) p ^ t + { \\sum } _ { i = t + 1 } ^ { e - 1 } s _ { i + 1 } p ^ i \\Big ) \\in \\Upsilon _ x , \\end{align*}"}
{"id": "3409.png", "formula": "\\begin{align*} | K ( x , y ) - K ( x , y ' ) | & \\lesssim \\int _ 0 ^ \\infty | S _ t ( x , y ) - S _ t ( x , y ' ) | \\frac { d t } { t } \\\\ & = \\int _ 0 ^ { \\| y - y ' \\| } | S _ t ( x , y ) - S _ t ( x , y ' ) | \\frac { d t } { t } + \\int _ { \\| y - y ' \\| } ^ { \\| x - y \\| } | S _ t ( x , y ) - S _ t ( x , y ' ) | \\frac { d t } { t } \\\\ & + \\int _ { \\| x - y \\| } ^ \\infty | S _ t ( x , y ) - S _ t ( x , y ' ) | \\frac { d t } { t } \\\\ & = : I \\ ! I _ 1 + I \\ ! I _ 2 + I \\ ! I _ 3 . \\end{align*}"}
{"id": "9170.png", "formula": "\\begin{align*} | \\lambda ^ 2 _ f ( p ) - 2 | \\leq | \\lambda ^ 2 _ f ( p ) | + 2 \\leq d ( p ^ 2 ) + 2 \\ll 1 . \\end{align*}"}
{"id": "3662.png", "formula": "\\begin{align*} B _ { 1 } = 2 \\left [ 1 - \\frac { k } { m } + \\left ( 1 - \\frac { k } { m } \\right ) \\delta + \\frac { \\delta ^ { 2 } } { 4 } \\right ] \\end{align*}"}
{"id": "6522.png", "formula": "\\begin{align*} d _ j ^ { ( 2 \\ell ) } : = \\binom { j + m } { j } ^ { - 1 } \\binom { j + \\ell - 1 } { j - 1 } t _ j ^ { ( 2 \\ell ) } . \\end{align*}"}
{"id": "6617.png", "formula": "\\begin{align*} \\sum _ { m = 2 } ^ { \\infty } \\frac { \\tau _ { A _ { s _ 1 } \\smallsetminus \\{ \\alpha + s _ 1 \\} \\cup \\{ - \\beta - s _ 2 \\} } ( p ^ m ) \\tau _ { B _ { s _ 2 } \\smallsetminus \\{ \\beta + s _ 2 \\} \\cup \\{ - \\alpha - s _ 1 \\} } ( p ^ m ) } { p ^ m } \\ll \\sum _ { m = 2 } ^ { \\infty } \\frac { p ^ { m ( \\frac { 1 } { 2 } - \\epsilon + \\varepsilon ) } } { p ^ m } \\ll \\frac { 1 } { p ^ { 1 + \\varepsilon } } \\end{align*}"}
{"id": "1560.png", "formula": "\\begin{align*} p _ { \\mathcal { S } } ( \\psi _ i , \\Phi ) = \\prod _ { j = 1 } ^ n p _ { \\mathcal { S } _ j } ( \\psi _ { i j } , \\Phi _ j ) = \\prod _ { j = 1 } ^ n p _ { K _ j } ( _ { \\mathcal { S } _ j / K _ j } ( \\psi _ { i j } ) , \\Phi ' _ j ) = p _ Y ( _ { \\mathcal { S } / Y } ( \\psi _ i ) , \\Phi ' ) , \\end{align*}"}
{"id": "1055.png", "formula": "\\begin{align*} & \\lim _ { \\varepsilon \\to 0 } \\Big ( \\int _ { \\{ \\vert f \\vert \\geq \\varepsilon \\} } \\vert f \\vert ^ { 2 \\alpha } f ^ { - N } \\xi + 2 i \\pi \\sum _ { m = m ' + N } ^ { \\alpha + r + m ' < 0 } \\tilde { T } _ { m , m ' } ^ { r , j } ( \\xi ) \\frac { \\varepsilon ^ { 2 ( \\alpha + r + m ' ) } } { 2 ( \\alpha + r + m ' ) } ( - L o g \\varepsilon ) ^ j \\Big ) \\end{align*}"}
{"id": "2728.png", "formula": "\\begin{align*} \\omega \\underline { P } = \\omega _ 1 P _ 1 + \\cdots + \\omega _ q P _ q , \\end{align*}"}
{"id": "4780.png", "formula": "\\begin{align*} n = m \\ell , \\theta = \\ell - 1 , \\hat \\theta = \\frac { 1 } { \\ell } \\sqrt { \\frac { ( \\ell - 1 ) ( m \\ell - 1 ) } { m ( m - 1 ) } } . \\end{align*}"}
{"id": "748.png", "formula": "\\begin{align*} Y ^ { ( \\ell + 1 ) } [ ( \\partial _ \\mu z ) ^ 2 ; f ] = Y ^ { = ( \\ell + 1 ) } [ ( \\partial _ \\mu z ) ^ 2 ; f ] + Y ^ { < ( \\ell + 1 ) } [ ( \\partial _ \\mu z ) ^ 2 ; f ] , \\end{align*}"}
{"id": "509.png", "formula": "\\begin{align*} h ( \\mathcal { A } _ { \\beta } ^ { i } ) = 0 , h ( \\mathcal { A } _ { \\omega \\beta } ^ { i v } ) = h ( \\mathcal { B } _ { \\beta } ^ { i } ) = 0 \\end{align*}"}
{"id": "3330.png", "formula": "\\begin{align*} a _ { f _ { \\lambda + 1 } } \\ge ( n - 1 ) f _ { \\lambda + 1 } + g _ { \\lambda + 1 } = f _ { \\lambda + 1 } + f _ { \\lambda + 2 } \\end{align*}"}
{"id": "2177.png", "formula": "\\begin{align*} \\langle J ^ { ' } ( u ) , u \\rangle = 0 \\end{align*}"}
{"id": "9404.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { n + m + 1 } ( - 1 ) ^ { r + 1 } \\sum _ { 1 \\leq k _ 1 < \\ldots < k _ r \\leq n + m + 1 } \\left ( \\prod _ { i = 1 } ^ { r } \\tau _ t ( c _ { k _ i } ) \\right ) \\tau _ t ( c _ 1 \\cdots \\hat { c } _ { k _ 1 } \\cdots \\hat { c } _ { k _ r } \\cdots c _ { n + m + 1 } ) + o ( t ) , \\end{align*}"}
{"id": "2141.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { E \\tau ^ { ( M A , A ) } } { M A } = \\frac 1 { A + 1 } . \\end{align*}"}
{"id": "3800.png", "formula": "\\begin{align*} U & = [ U _ k \\ \\ , \\widehat { U } ] ~ , \\ \\ , V = [ V _ k \\ \\ , \\widehat { V } ] ~ , \\ \\ , W = [ W _ k \\ \\ , \\widehat { W } ] ~ , \\ \\ , Z = [ Z _ k \\ \\ , \\widehat { Z } ] ~ , \\\\ [ 1 m m ] D _ A & = { \\sf d i a g } ( D _ { A _ k } , \\ , \\widehat D _ A ) ~ , \\ \\ , D _ B = { \\sf d i a g } ( D _ { B _ k } , \\ , \\widehat D _ B ) ~ , \\ \\ , D _ G = { \\sf d i a g } ( D _ { G _ k } , \\ , \\widehat D _ G ) , \\end{align*}"}
{"id": "4592.png", "formula": "\\begin{align*} M _ n = a _ n S _ n . \\end{align*}"}
{"id": "1940.png", "formula": "\\begin{align*} U = V ( P ^ \\top ) ^ { - 1 } W = X P ^ { - 1 } ( P ^ \\top ) ^ { - 1 } W = X ( X ^ \\top X ) ^ { - 1 } W . \\end{align*}"}
{"id": "4572.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } ( X _ n / \\sqrt { \\langle X \\rangle _ n } > x ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + O \\bigg ( x ^ 3 ( \\epsilon _ n + \\delta _ n ) + ( 1 + x ) \\big ( \\delta _ n | \\ln \\delta _ n | + \\epsilon _ n | \\ln \\epsilon _ n | \\big ) \\bigg ) . \\end{align*}"}
{"id": "7840.png", "formula": "\\begin{align*} z = \\sum _ { i = 0 } ^ { 3 n - 6 j - 7 } \\gamma _ { i } e _ { i } ~ ~ \\gamma _ { 3 n - 6 j - 7 } \\neq 0 . \\end{align*}"}
{"id": "952.png", "formula": "\\begin{align*} \\Gamma _ { 2 , y , ( r , s ) } ^ { \\delta } = \\{ n \\in \\Gamma _ { 2 , y , ( r , s ) } : A ^ { \\delta } \\subset ( H ^ { \\delta } ) ^ c \\} , \\delta \\in \\{ 1 , \\ldots , d \\} . \\end{align*}"}
{"id": "3277.png", "formula": "\\begin{align*} s ^ 2 + r ^ 2 - \\frac { s } { h ( s ) } = 0 . \\end{align*}"}
{"id": "8166.png", "formula": "\\begin{align*} M ( f , H ) = { 2 \\pi ^ 2 \\over f } \\sum _ { \\delta \\mid f } \\frac { \\mu ( \\delta ) } { \\delta } \\sum _ { h \\in H } s ( h , f / \\delta ) . \\end{align*}"}
{"id": "2038.png", "formula": "\\begin{align*} \\ell _ { \\beta } '' ( x ) ( 1 - \\phi _ R ( x ) ) { \\rm d } x = \\left \\{ \\begin{array} { l l } \\frac { 1 } { x } ( 1 - \\phi _ R ( x ) ) { \\rm d } x , & \\beta = - 1 \\\\ | x | ^ { \\beta } ( { \\rm s g n } ( x ) ) ( 1 - \\phi _ R ( x ) ) { \\rm d } x , & \\beta \\in ] - 3 / 2 , \\ , - 1 \\ , [ \\ , \\cup \\ , ] - 1 , \\ , 0 \\ , ] \\end{array} \\right . \\in S _ { K } ^ 1 ( { \\bf X } ) . \\end{align*}"}
{"id": "7778.png", "formula": "\\begin{align*} \\mu _ * ( \\cup _ { k \\in T } W _ k ( ~ \\mu _ * ) ) & = \\sum _ { k \\in T } p _ k \\mu _ * \\circ W _ k ^ { - 1 } ( \\cup _ { k \\in T } W _ k ( ~ \\mu _ * ) ) ) \\\\ & = \\sum _ { k \\in T } p _ k \\mu _ * ( W _ k ^ { - 1 } ( \\cup _ { k \\in T } W _ k ( ~ \\mu _ * ) ) ) \\\\ & \\ge \\sum _ { k \\in T } p _ k \\mu _ * ( W _ k ^ { - 1 } ( W _ k ( ~ \\mu _ * ) ) ) \\\\ & \\ge \\sum _ { k \\in T } p _ k \\mu _ * ( ~ \\mu _ * ) \\\\ & = 1 . \\end{align*}"}
{"id": "6964.png", "formula": "\\begin{align*} D ^ * \\Phi ( x , y ) ( y ^ * ) \\subset \\begin{cases} D ^ * g ( x ) ( y ^ * ) & y ^ * \\in \\mathcal N _ D ( g ( x ) - y ) , \\\\ \\varnothing & . \\end{cases} \\end{align*}"}
{"id": "8549.png", "formula": "\\begin{align*} \\partial _ { t } ^ 2 u - \\partial _ x ^ 2 u + u + V ( x ) u + a ( x ) u ^ 2 + \\pm u ^ 3 = 0 , \\ , u ( 0 ) = u _ 0 , \\ , \\partial _ t u ( 0 ) = u _ 1 . \\end{align*}"}
{"id": "4570.png", "formula": "\\begin{align*} X _ { 0 } = 0 , \\ \\ \\ \\ \\ X _ k = \\sum _ { i = 1 } ^ k \\xi _ i , k = 1 , . . . , n . \\end{align*}"}
{"id": "4662.png", "formula": "\\begin{align*} \\mathcal { L } f = | D | f + f - p Q ^ { p - 1 } f . \\end{align*}"}
{"id": "7012.png", "formula": "\\begin{align*} | \\det { \\mathfrak { S } } | = | A ( \\beta _ 1 ) \\cdots A ( \\beta _ k ) | \\geqslant \\frac { 1 } { 2 ^ K } | \\beta _ 1 | ^ N \\dots | \\beta _ k | ^ N . \\end{align*}"}
{"id": "894.png", "formula": "\\begin{align*} & \\lim _ { m \\to \\infty } F _ p ( T , p ^ { k _ m - n - 1 } ) = 1 \\end{align*}"}
{"id": "3523.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) & = \\sum _ { m \\leq a t _ 3 } \\sum _ { n < m } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } + \\begin{cases} O ( t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } ) & ( \\sigma _ 2 > 2 ) \\\\ O ( t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 ) & ( \\sigma _ 2 = 2 ) \\\\ O ( t _ 3 ^ { \\frac { 3 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) & ( \\sigma _ 2 < 2 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "5775.png", "formula": "\\begin{align*} L ^ { p } ( \\mathbb { R } ^ 3 ) = L ^ { p , p } ( \\mathbb { R } ^ 3 ) \\hookrightarrow L ^ { p , q } ( \\mathbb { R } ^ 3 ) \\hookrightarrow L ^ { p , \\infty } ( \\mathbb { R } ^ 3 ) \\ , \\ p \\leq q < \\infty . \\end{align*}"}
{"id": "6171.png", "formula": "\\begin{align*} \\theta _ { r + 1 } ( z ) : = ( z - z _ 1 ) \\cdots ( z - z _ { r + 1 } ) = \\phi _ { r + 1 } ( z ) + b _ 1 \\phi _ r ( z ) + \\cdots + b _ r \\phi _ 1 ( z ) + 0 \\cdot \\phi _ 0 ( z ) . \\end{align*}"}
{"id": "3096.png", "formula": "\\begin{align*} z ^ 3 + z ^ 2 f _ 2 ( \\lambda , 1 ) + z f _ 4 ( \\lambda , 1 ) + f _ 6 ( \\lambda , 1 ) = 0 = \\varphi _ 3 ( z ) \\ , , \\end{align*}"}
{"id": "796.png", "formula": "\\begin{align*} v _ k ( x , y ) = u _ k ( x ) - u _ k ( y ) . \\end{align*}"}
{"id": "67.png", "formula": "\\begin{align*} \\mathrm { r k } ( L ' _ { 2 , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( L ' _ { 2 , 1 } ) = 2 , \\\\ \\mathrm { r k } ( K _ { \\ell , v , 0 } ) & = n - 1 , \\ \\mathrm { r k } ( K _ { \\ell , v , 1 } ) = 1 ( v = 2 , p _ { k ' + 1 } , \\cdots , p _ k ) , \\\\ \\mathrm { r k } ( L ' _ { v , 0 } ) & = n , \\ \\mathrm { r k } ( L ' _ { v , 1 } ) = 1 ( v \\neq 2 , p _ { k ' } , \\cdots , p _ k ) . \\end{align*}"}
{"id": "9239.png", "formula": "\\begin{align*} \\alpha ^ 2 \\norm { x } _ X ^ 2 - \\norm { ( \\alpha - 1 ) x + _ X y } _ X ^ 2 = 2 \\alpha \\langle x - _ X y , y \\rangle _ X - ( 1 - 2 \\alpha ) \\norm { x - _ X y } _ X ^ 2 . \\end{align*}"}
{"id": "2598.png", "formula": "\\begin{align*} V _ { \\widetilde { g } } ( V _ g ^ * F ) ( \\xi , \\eta ) & = \\langle V _ g ^ * F , M _ \\eta T _ \\xi \\widetilde { g } \\rangle \\\\ & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\overline { V _ g ( M _ \\eta T _ \\xi \\widetilde { g } ) ( x , \\omega ) } \\ , d ( x , \\omega ) \\\\ & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) V _ { \\widetilde { g } } g ( \\xi - x , \\eta - \\omega ) e ^ { - 2 \\pi i x \\cdot ( \\eta - \\omega ) } \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "4345.png", "formula": "\\begin{align*} \\varepsilon _ { e x t , e } ( \\tau ) = \\mathcal { K } _ \\beta ( \\tau , \\tau ' ) \\varepsilon _ { e x t , e } ( \\tau ' ) + \\int _ { \\tau ' } ^ \\tau \\mathcal { K } _ \\beta ( \\tau , \\tilde \\tau ) \\left [ \\mathcal { C } ( \\varepsilon _ { e x t } ) + \\mathcal { N } ( \\varepsilon _ { e x t } ) \\right ] ( \\tilde \\tau ) d \\tilde \\tau . \\end{align*}"}
{"id": "6725.png", "formula": "\\begin{align*} h _ { 1 , j ' - 1 } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + h _ { r - j ' + 1 , j ' - 1 } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ { r - j ' + 1 } ) ^ { q ^ d } + R _ { j ' - 1 } = 0 \\end{align*}"}
{"id": "1726.png", "formula": "\\begin{align*} m _ t ' = \\overline { m } _ t , \\ ; \\ ; m _ t ' = m _ t ; m _ t ' = \\hat m _ t , \\ ; \\ ; m _ t ' = \\tilde { m } _ t ; \\end{align*}"}
{"id": "2599.png", "formula": "\\begin{align*} \\norm { V _ g ^ * F } _ { M ^ { p , q } } = \\norm { V _ { g _ 0 } ( V _ g ^ * F ) } _ { L ^ { p , q } } \\leq C \\norm { F } _ { L ^ { p , q } } \\norm { V _ { g _ 0 } g } _ { L ^ 1 } < \\infty . \\end{align*}"}
{"id": "666.png", "formula": "\\begin{align*} \\abs { q - \\alpha } \\ \\leq \\ \\frac { 1 } { 4 ^ { k + 1 } - 1 + 1 } \\ = \\ \\frac { 1 } { 4 ^ { k + 1 } } , \\end{align*}"}
{"id": "5481.png", "formula": "\\begin{align*} d _ K ( x ) : = \\inf _ { y \\in K } \\| x - y \\| , x \\in X . \\end{align*}"}
{"id": "7972.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 } \\limsup _ { j \\to \\infty } \\theta _ { | \\Sigma _ j | } ( p _ j , r ) - \\theta _ { | \\Sigma _ j | } ( p _ j ) = 0 \\ , . \\end{align*}"}
{"id": "3194.png", "formula": "\\begin{align*} [ n ] = \\{ 0 \\leq 1 \\leq \\ldots \\leq n \\} n \\geq 0 , \\end{align*}"}
{"id": "8070.png", "formula": "\\begin{align*} F \\star _ { H , \\ell } G [ \\psi ] = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\hbar ^ { n } } { n ! } \\left \\langle W _ { \\Sigma } ^ { \\otimes n } , F ^ { ( n ) } [ \\psi ] \\otimes G ^ { ( n ) } [ \\psi ] \\right \\rangle \\end{align*}"}
{"id": "3402.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } f ( x ) d \\omega ( x ) = 0 . \\end{align*}"}
{"id": "8946.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { \\xi } ( s ) & = e ^ \\frac { s } { q - 1 } p \\left | \\eta ( s ) \\right | ^ { p - 2 } \\eta ( s ) \\\\ \\dot { \\eta } ( s ) & = e ^ { - s } D f \\left ( \\xi ( s ) \\right ) \\end{aligned} \\right . \\end{align*}"}
{"id": "1119.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & D _ { + } ( k ) D _ { - } ( k ) = r _ { + } ( k ) , & k \\in I _ \\eta \\backslash I _ R , \\\\ & D _ { + } D _ { - } = 1 , & k \\in I _ R , \\\\ & D _ { + } = D _ { - } , & \\textnormal { e l s e w h e r e } . \\end{aligned} \\right . \\end{align*}"}
{"id": "602.png", "formula": "\\begin{align*} f ( x + 1 , x _ 2 , \\ldots , x _ m ) \\ = \\ h ( x , f ( x , x _ 2 , \\ldots , x _ m ) , x _ 2 , \\ldots , x _ m ) \\end{align*}"}
{"id": "117.png", "formula": "\\begin{align*} x ^ i x ^ j = \\frac 1 2 \\left [ y ^ i x ^ j + y ^ j x ^ i \\right ] , \\end{align*}"}
{"id": "8389.png", "formula": "\\begin{align*} \\P ( \\exists S \\in \\mathcal S : \\eqref { e q - e n c l o s e - L a m b d a - 2 r } ~ h o l d s ) \\leq \\sum _ { t \\geq 1 } \\sum _ { t ' \\geq t } n ^ { - 8 ^ { - d } t ' } ( 1 0 0 t ) ^ { d } = o ( 1 ) \\ , . \\end{align*}"}
{"id": "865.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum \\limits _ { i = 1 } ^ { \\infty } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) { \\epsilon _ i } } , \\\\ & \\sum \\limits _ { i = 1 } ^ { \\infty } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) \\left ( { 2 { \\tau _ { \\rm c } } + 2 \\mathcal { T } + { n _ { i + 1 } } + { n _ i } } \\right ) { \\epsilon _ i } } . \\end{aligned} \\end{align*}"}
{"id": "9535.png", "formula": "\\begin{align*} y & \\in \\partial V ( u - \\sum _ { t = 0 } ^ { T - 1 } x _ t \\Delta s _ { t + 1 } - \\bar c \\cdot \\bar x + S _ 0 ( \\bar x ) ) , \\\\ E _ t [ y \\Delta s _ { t + 1 } ] & \\in N _ { D _ t } ( x _ t ) t = 0 , \\dots , T , \\\\ \\frac { E [ y \\bar c ] } { E [ y ] } & \\in \\partial S _ 0 ( \\bar x ) \\end{align*}"}
{"id": "4921.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { t } \\frac { \\alpha _ k } { \\Gamma _ k } = \\frac { 1 } { \\Gamma _ t } ~ ~ ~ ~ ~ \\implies ~ ~ ~ ~ ~ \\Gamma _ t \\sum _ { k = 1 } ^ t \\frac { \\alpha _ k } { \\Gamma _ k } = 1 \\end{align*}"}
{"id": "6999.png", "formula": "\\begin{align*} A ( z ) = \\prod _ { j = 1 } ^ { n } ( z - \\alpha _ j ) ^ { m _ j } , \\end{align*}"}
{"id": "3917.png", "formula": "\\begin{align*} D T \\left [ f ( \\eta ) f '' ( \\eta ) \\right ] = \\sum _ { l = 0 } ^ { k } ( l + 1 ) ( l + 2 ) F ( l ) F ( k - l ) \\end{align*}"}
{"id": "5072.png", "formula": "\\begin{align*} | A ( x ) - A ( x + r v ) | & = | A ( x ) - f ( x ) | + | f ( x ) - f ( x + r v ) | + | f ( x + r v ) - A ( x + r v ) | \\\\ & \\leq \\delta r + r + \\delta r \\\\ & = r ( 1 + 2 \\delta ) . \\end{align*}"}
{"id": "9045.png", "formula": "\\begin{align*} E ( \\rho ( \\theta ) , \\phi ( \\theta ) ) - \\theta E ( \\rho ^ 0 , \\phi ^ 0 ) - ( 1 - \\theta ) E ( \\rho ^ 1 , \\phi ^ 1 ) = - \\theta ( 1 - \\theta ) I \\end{align*}"}
{"id": "1585.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } b ( \\cos \\theta ) | v - v _ * | ^ \\gamma f ( v ' ) d \\sigma d v _ * = \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } b ( \\cos \\theta ) \\frac 1 { \\sin ^ { 3 + \\gamma } ( \\theta / 2 ) } | v - v _ * | ^ \\gamma f ( v _ * ) d \\sigma d v _ * . \\end{align*}"}
{"id": "6212.png", "formula": "\\begin{align*} - \\sum _ { a = 1 } ^ { j - 1 } \\sum _ { b = 1 } ^ { j } t _ { a , b } \\geq 1 - j . \\end{align*}"}
{"id": "5857.png", "formula": "\\begin{align*} \\bar \\alpha _ i = \\begin{cases} 1 , & i \\in H , \\\\ \\bar \\alpha _ { i + 1 } / \\rho _ { i + 1 } , & i \\notin H . \\end{cases} \\end{align*}"}
{"id": "3862.png", "formula": "\\begin{align*} \\tilde \\sigma _ 0 \\in A C [ 0 , \\infty ) , \\tilde \\sigma _ 0 ( x ) = \\tilde \\sigma _ 0 ( 0 ) - \\int _ 0 ^ x \\sigma _ 0 ( t ) \\ , d t . \\end{align*}"}
{"id": "3990.png", "formula": "\\begin{align*} \\begin{dcases} - \\int _ 0 ^ T e ^ { \\lambda _ 0 ( T - t ) } p ( t ) \\ , d t & = m _ { 2 , 0 } , \\ \\ \\ - \\int _ 0 ^ T p ( t ) \\ , d t = m _ { 2 , 0 } , \\\\ - \\int _ { 0 } ^ { T } \\overline { e ^ { \\lambda ^ h _ k ( T - t ) } } p ( t ) \\ , d t & = m _ { 2 , k } , \\forall | k | \\geq k _ 0 , \\\\ - \\int _ { 0 } ^ { T } \\overline { e ^ { \\widehat \\lambda _ { n _ l } ( T - t ) } } p ( t ) \\ , d t & = m _ { 1 , l } , \\forall \\ , 1 \\leq l \\leq l _ 0 , \\end{dcases} \\end{align*}"}
{"id": "7258.png", "formula": "\\begin{align*} ( 4 + \\dfrac { 3 b d } { c ^ 2 } - \\dfrac { 6 a c } { b ^ 2 } - \\dfrac { b ^ 2 } { a c } ) = & \\dfrac { 2 ( n + 1 ) [ - k ^ 3 + ( n - 5 ) k ^ 2 + ( 3 n - 2 ) k - n - 1 ] } { k ( k + 1 ) ( k + 2 ) ( n - k + 1 ) ( n - k ) } . \\end{align*}"}
{"id": "3503.png", "formula": "\\begin{align*} D _ { 2 1 } & = \\frac { ( a t _ 3 ) ^ { 2 - s _ 1 - s _ 2 - s _ 3 } } { ( s _ 1 + s _ 2 + s _ 3 - 2 ) ( s _ 1 + s _ 3 - 1 ) } \\\\ & \\quad + \\frac { ( a t _ 3 ) ^ { 2 - s _ 1 - s _ 2 - s _ 3 } } { 2 \\pi i ( s _ 1 + s _ 2 + s _ 3 - 2 ) \\Gamma ( s _ 3 ) } \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { \\Gamma ( s _ 3 + z ) \\Gamma ( - z ) } { s _ 1 + s _ 3 - 1 + z } d z . \\end{align*}"}
{"id": "7815.png", "formula": "\\begin{align*} \\frac { \\mathrm d ^ k } { \\mathrm d t ^ k } \\ , \\Phi _ { a , b } ( t ) = \\sum _ { j = 0 } ^ k C _ { a , j , k } t ^ { a - ( k - j ) } A ^ { b + j } S ( t ) Q ^ \\frac { 1 } { 2 } = \\sum _ { j = 0 } ^ k C _ { a , j , k } \\Phi _ { a - ( k - j ) , b + j } ( t ) , \\end{align*}"}
{"id": "1599.png", "formula": "\\begin{align*} \\tilde { \\varphi } = \\alpha _ 1 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 1 ) } + \\alpha _ 2 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 2 ) } + \\alpha _ 3 \\chi _ { B _ { \\frac { 1 } { 8 } } ( P _ 3 ) } \\end{align*}"}
{"id": "2000.png", "formula": "\\begin{align*} z \\cdot ( x _ 1 , x _ 2 ) = ( z x _ 1 , z x _ 2 ) , \\ z \\in S ^ 1 , ( x _ 1 , x _ 2 ) \\in \\mathbb { C } ^ 2 . \\end{align*}"}
{"id": "7403.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\log ( a _ n + b _ n ) = \\max \\Big \\{ \\limsup _ { n \\rightarrow \\infty } \\log ( a _ n ) , \\limsup _ { n \\rightarrow \\infty } \\log ( b _ n ) \\Big \\} . \\end{align*}"}
{"id": "8304.png", "formula": "\\begin{align*} \\kappa _ { i z } = \\sqrt { \\kappa _ i ^ 2 - \\kappa _ x ^ 2 - \\kappa _ y ^ 2 } \\end{align*}"}
{"id": "2702.png", "formula": "\\begin{align*} \\langle W f , W ( M _ { - \\omega } T _ x g _ 0 ) \\rangle & = | \\langle f , M _ { - \\omega } T _ x g _ 0 \\rangle | ^ 2 \\\\ & = | V _ { g _ 0 } f ( x , - \\omega ) | ^ 2 \\\\ & = | B f ( z ) | ^ 2 e ^ { - \\pi | z | ^ 2 } . \\end{align*}"}
{"id": "4429.png", "formula": "\\begin{align*} < u '' ( t ) , v > + ( F _ 1 ( u ' ( t ) ) , v ) _ { L ^ 2 } + ( u ( t ) , v ) _ { H ^ 2 _ * } + ( F _ 2 ( u ( t ) ) , v ) _ { L ^ 2 } = ( f ( t ) , v ) _ { L ^ 2 } . \\end{align*}"}
{"id": "9282.png", "formula": "\\begin{align*} S _ { \\tau ( \\xi ) } : = S _ { \\tau } ^ { S _ { \\xi } } . \\end{align*}"}
{"id": "7176.png", "formula": "\\begin{align*} \\hat { \\phi } ( \\xi ) = \\frac { 1 } { 1 + | \\xi | ^ 2 } . \\end{align*}"}
{"id": "4322.png", "formula": "\\begin{align*} w ( y , \\tau _ 0 ) = Q _ { b ( \\tau _ 0 ) } ( y ) + \\varepsilon ( \\tau _ 0 ) . \\end{align*}"}
{"id": "7689.png", "formula": "\\begin{align*} \\partial _ { x _ i x _ j } f ( \\mathbf { x } ) = 0 \\end{align*}"}
{"id": "6721.png", "formula": "\\begin{align*} & h _ 1 ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ 1 ) ^ { q ^ d } + \\cdots + h _ { r - 1 } ( t ) ~ _ { s + 1 } \\mathcal { F } _ { s } ( { \\bf a } ; { \\bf b } ) ( \\alpha _ { r - 1 } ) ^ { q ^ d } + R = 0 \\end{align*}"}
{"id": "6483.png", "formula": "\\begin{align*} E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n / 2 } } \\right ) ^ 2 \\right ] - 1 = 0 \\end{align*}"}
{"id": "5238.png", "formula": "\\begin{align*} m : A \\times A \\rightarrow A , ( a , b ) \\mapsto a \\cdot b = a b . \\end{align*}"}
{"id": "7028.png", "formula": "\\begin{align*} f = a _ 1 \\widetilde { f } + p , \\mbox { w h e r e $ \\widetilde { f } \\in H ^ 2 $ a n d $ p \\in \\P _ { N - 1 } $ } , \\end{align*}"}
{"id": "5200.png", "formula": "\\begin{align*} \\phi _ 0 ( \\xi ) & = \\Phi _ { \\Gamma ^ 0 } ( \\xi ) , \\\\ \\phi _ j ( \\xi ) & = \\Phi _ { \\Gamma ^ j \\setminus \\Gamma ^ { j - 1 } } ( \\xi ) , ~ ~ ~ j \\in \\mathbb { N } , \\end{align*}"}
{"id": "9240.png", "formula": "\\begin{align*} \\forall p ^ X , x ^ X , \\gamma ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) . \\end{align*}"}
{"id": "4194.png", "formula": "\\begin{align*} \\psi ^ H _ t ( x _ i , y _ i , z ) = \\left ( x _ i - t x ^ 0 _ i , y _ i - t y ^ 0 _ i , z + \\sum _ i \\left ( \\frac { x ^ 0 _ i y ^ 0 _ i } { 2 } t - y ^ 0 _ i x _ i \\right ) t \\right ) \\end{align*}"}
{"id": "540.png", "formula": "\\begin{align*} H _ L ( \\hat { \\Psi } ( \\mathbf { u } , \\tilde { \\Xi } ) ) \\alpha ^ * = \\hat { \\Psi } ( \\mathbf { \\bar { u } } , \\hat { \\Xi } ) + c . \\end{align*}"}
{"id": "475.png", "formula": "\\begin{align*} \\alpha \\triangledown _ { \\mathcal { H } } \\beta \\overset { { \\scriptstyle \\mathrm { d e f } } } { \\Leftrightarrow } \\mathcal { H } _ { \\alpha } ^ { \\beta } \\in \\mathfrak { G } ( \\mathbf { L } ) \\mathcal { H } _ { \\beta } ^ { \\alpha } \\in \\mathfrak { G } ( \\mathbf { L } ) , \\triangledown : = \\bigcup _ { \\mathcal { H } \\in \\mathfrak { G } ( \\mathbf { L } ) } \\triangledown _ { \\mathcal { H } } . \\end{align*}"}
{"id": "1137.png", "formula": "\\begin{align*} E _ { + } ( x , t , k ) = E _ { - } ( x , t , k ) J ^ { E } ( x , t , k ) , \\end{align*}"}
{"id": "3839.png", "formula": "\\begin{align*} \\sigma ^ { ( i ) } y & = \\sum _ { s = 0 } ^ i ( - 1 ) ^ s C _ i ^ s ( \\sigma y ^ { ( s ) } ) ^ { ( i - s ) } , \\\\ ( \\sigma ^ { ( i ) } y ^ { ( k ) } ) ^ { ( k ) } & = \\sum _ { s = 0 } ^ i ( - 1 ) ^ s C _ i ^ s ( \\sigma y ^ { ( s + k ) } ) ^ { ( i - s + k ) } , \\\\ ( ( \\sigma ^ { ( i ) } y ^ { ( k ) } ) ^ { ( k ) } , z ) & = \\sum _ { s = 0 } ^ i ( - 1 ) ^ { i + k } C _ i ^ s ( \\sigma y ^ { ( s + k ) } , z ^ { ( i - s + k ) } ) , z \\in \\mathfrak D . \\end{align*}"}
{"id": "7143.png", "formula": "\\begin{align*} x ^ n _ { t \\wedge \\tau _ n } - x _ { t \\wedge \\tau _ n } & = \\int _ 0 ^ { t \\wedge \\tau _ n } \\Big \\{ \\tilde { f } _ n ( s , x _ s ^ n ) - \\tilde { f } ( s , x _ s ) \\Big \\} d s \\\\ & = \\int _ 0 ^ t \\chi _ { _ { [ 0 , \\tau _ n ) } } ( s ) \\Big \\{ \\tilde { f } _ n ( s , x _ { s \\wedge \\tau _ n } ^ n ) - \\tilde { f } _ n ( s , x _ { s \\wedge \\tau _ n } ) \\Big \\} d s . \\end{align*}"}
{"id": "7844.png", "formula": "\\begin{align*} \\langle y \\rangle _ { T ^ { * 3 } } & = ~ \\{ y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( j - 1 - r ) } y \\} ~ , ~ ~ ~ \\langle y \\rangle _ { T ^ { * 3 } } = j - r , \\\\ \\langle z \\rangle _ { T ^ { * 3 } } & = ~ \\{ z , T ^ { * 3 } z , \\ldots , T ^ { * 3 ( n - 2 j - 2 + r ) } z \\} ~ , ~ ~ ~ \\langle z \\rangle _ { T ^ { * 3 } } = n - 2 j - 1 + r . \\end{align*}"}
{"id": "9345.png", "formula": "\\begin{align*} ( x ) _ { n } = \\sum _ { k = 0 } ^ { n } S _ { 1 , \\lambda } ( n , k ) ( x ) _ { k , \\lambda } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 8 ] ) , \\end{align*}"}
{"id": "3764.png", "formula": "\\begin{align*} H ^ 1 _ 0 ( S , \\C ^ { * } ) = { \\rm P i c } ^ 0 ( S ) \\cong \\C ^ * , \\end{align*}"}
{"id": "4549.png", "formula": "\\begin{align*} h _ 1 = \\max \\{ h _ 1 , \\dots , h _ 4 \\} \\ge ( h - 1 ) / 2 h _ 2 + h _ 3 + h _ 4 + h _ 5 \\le ( h - 1 ) / 2 . \\end{align*}"}
{"id": "6900.png", "formula": "\\begin{align*} \\lambda _ j = \\frac { q ^ { k - j } - 1 } { q ^ { k } - 1 } = q ^ { - j } - O ( q ^ { - k } ) \\end{align*}"}
{"id": "5493.png", "formula": "\\begin{align*} \\xi ( t ; r , x ) = \\xi ( t ; s , \\xi ( s ; r , x ) ) . \\end{align*}"}
{"id": "4012.png", "formula": "\\begin{align*} m _ 2 & = - \\frac { 1 } { 3 } \\left [ \\mu + \\frac { - 1 + i \\sqrt { 3 } } { 2 } \\left ( \\mu + \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ) + \\frac { - 1 - i \\sqrt { 3 } } { 2 } \\left ( \\mu - \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ) \\right ] \\\\ & = - \\frac { 1 } { 3 } \\left [ \\frac { 3 } { 2 } + 3 i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ] \\\\ & = - \\frac { 1 } { 2 } - i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) , \\end{align*}"}
{"id": "2547.png", "formula": "\\begin{align*} \\rho ( S z , \\tau ) = \\rho _ S ( z , \\tau ) = \\mu ( S ) \\rho ( z , \\tau ) \\mu ( S ) ^ { - 1 } . \\end{align*}"}
{"id": "8801.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { 5 + x _ 2 ^ 2 } - \\sqrt { 5 } & = \\sqrt { 5 + s _ { 2 1 } ( x ) } - \\sqrt { 5 } + \\sqrt { 5 + s _ { 2 2 } ( x ) } - \\sqrt { 5 + s _ { 2 1 } ( x ) } \\\\ & \\geq \\frac { \\sqrt { 8 } - \\sqrt { 5 } } { 3 } s _ { 2 1 } ( x ) + \\frac { \\sqrt { 9 } - \\sqrt { 8 } } { 1 } \\bigl ( s _ { 2 2 } ( x ) - s _ { 2 1 } ( x ) \\bigr ) , \\end{aligned} \\end{align*}"}
{"id": "5053.png", "formula": "\\begin{align*} \\Big \\{ \\alpha = Q + i p \\in \\C \\ , \\Big | \\ , { \\rm R e } ( \\alpha ) < Q , { \\rm I m } \\sqrt { p ^ 2 - 2 \\ell } < \\beta \\Big \\} \\cup \\Big \\{ Q + i p \\in Q + i \\R \\ , \\Big | \\ , | p | \\in \\bigcup _ { j \\geq \\ell } ( \\sqrt { 2 j } , \\sqrt { 2 ( j + 1 ) } ) \\Big \\} , \\end{align*}"}
{"id": "8294.png", "formula": "\\begin{align*} C ^ { \\gamma } ( \\mathbb { R } ^ { N - 1 } ) = \\bigg \\{ u \\in C _ b ( \\mathbb { R } ^ { N - 1 } ) : \\ , [ u ] _ { C ^ { \\gamma } } : = \\sup _ { \\stackrel { x , y \\in \\mathbb { R } ^ { N - 1 } } { x \\neq y } } \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \\gamma } } < \\infty \\bigg \\} \\end{align*}"}
{"id": "6296.png", "formula": "\\begin{align*} \\sqrt { 2 } \\widehat { K } _ \\mathrm { a - c p t - f } ^ \\mathbb { R } / \\upsilon = \\sum \\nolimits _ { i \\in \\mathcal { K } } U _ i + R , \\end{align*}"}
{"id": "4997.png", "formula": "\\begin{align*} B _ { k n } ( x , y ) = \\begin{bmatrix} \\xi _ { k n } ( x ) + b ^ { ( 1 + O ( \\alpha ^ { n } ) ) q _ { k n } } f ( x ) y ( 1 + O ( \\alpha ^ { k } ) ) \\\\ x \\end{bmatrix} \\end{align*}"}
{"id": "2520.png", "formula": "\\begin{align*} \\rho ( \\Phi ) & = \\iint _ { \\R ^ { 2 d } } \\int _ 0 ^ 1 \\sum _ { k \\in \\Z } \\Phi _ k ( x , \\omega ) e ^ { 2 \\pi i k \\tau } e ^ { 2 \\pi i \\tau } \\rho ( x , \\omega ) \\ , d \\tau \\ , d ( x , \\omega ) \\\\ & = \\iint _ { \\R ^ { 2 d } } \\Phi _ { - 1 } ( x , \\omega ) \\rho ( x , \\omega ) \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "1272.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) : = - c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z ) + \\sum _ { i } \\sum _ { k = 1 } ^ { m _ { i } - 1 } \\mu _ { \\tau } ( \\alpha _ { i } ^ { k } ) - \\sum _ { j } \\sum _ { k = 1 } ^ { n _ { j } - 1 } \\mu _ { \\tau } ( \\beta _ { j } ^ { k } ) . \\end{align*}"}
{"id": "7218.png", "formula": "\\begin{align*} v _ * ^ \\perp ( t , x , v _ 1 ) = \\frac { x ^ \\perp } { \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } } . \\end{align*}"}
{"id": "1251.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\vert B _ n \\vert ^ { \\varepsilon } \\mu ( B _ n ) = \\sum _ { k \\geq 0 } \\sum _ { B \\in \\mathcal { T } _ k ( \\mathcal { B } ) } \\vert B \\vert ^ { \\varepsilon } \\mu ( B ) \\leq \\sum _ { k \\geq 0 } \\sum _ { 1 \\leq j \\leq J _ k } 2 ^ { - k \\varepsilon } \\sum _ { B \\in T _ { k , j } ( \\mathcal { B } ) } \\mu ( B ) . \\end{align*}"}
{"id": "8861.png", "formula": "\\begin{align*} \\varphi = \\begin{array} { c | c c c c } & 0 & 1 & 2 & 3 \\\\ \\hline 0 & 0 & 1 & 2 & 1 \\\\ 1 & 2 & 0 & 1 & 2 \\\\ 2 & 2 & 2 & 0 & 1 \\\\ 3 & 2 & 2 & 2 & 0 \\end{array} . \\end{align*}"}
{"id": "3362.png", "formula": "\\begin{align*} & [ T u , T _ 1 ( v ) , T _ 1 ( w ) ] + [ T _ 1 ( u ) , T v , T _ 1 ( w ) ] + [ T _ 1 ( u ) , T _ 1 ( v ) , T w ] \\\\ = & T \\Big ( D ( T _ 1 ( u ) , T _ 1 ( v ) ) w + \\theta ( T _ 1 ( v ) , T _ 1 ( w ) ) u - \\theta ( T _ 1 ( u ) , T _ 1 ( w ) ) v \\Big ) \\\\ & + T _ 1 \\Big ( D ( T u , T _ 1 ( v ) ) w + D ( T _ 1 ( u ) , T v ) w + \\theta ( T v , T _ 1 ( w ) ) u \\\\ & + \\theta ( T _ 1 ( v ) , T w ) u - \\theta ( T u , T _ 1 ( w ) ) v - \\theta ( T _ 1 ( u ) , T w ) v \\Big ) , \\end{align*}"}
{"id": "8538.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi + \\phi + V ( x ) \\phi = a ( x ) \\phi ^ 2 + b ( x ) \\phi ^ 3 , ( \\phi , \\phi _ t ) ( 0 ) = ( \\phi _ 0 , \\phi _ 1 ) , \\end{align*}"}
{"id": "4027.png", "formula": "\\begin{align*} \\eta ^ \\prime ( x ) = m _ 1 ( e ^ { m _ 2 } - e ^ { m _ 3 } ) e ^ { m _ 1 x } + m _ 2 ( e ^ { m _ 3 } - e ^ { m _ 1 } ) e ^ { m _ 2 x } + m _ 3 ( e ^ { m _ 1 } - e ^ { m _ 2 } ) e ^ { m _ 3 x } , \\end{align*}"}
{"id": "8845.png", "formula": "\\begin{align*} B _ 1 ( { \\bf z } ) = \\frac { z _ 1 } { 2 z _ 0 ^ { 3 / 2 } } , B _ 2 ( { \\bf z } ) = \\frac { z _ 1 ^ 2 } { 2 z _ 0 ^ 3 } - \\frac { z _ 2 } { 4 z _ 0 ^ 2 } , B _ 3 ( { \\bf z } ) = \\frac 1 8 \\frac { z _ 3 } { z _ 0 ^ { 5 / 2 } } - \\frac { 1 5 } { 1 6 } \\frac { z _ 1 z _ 2 } { z _ 0 ^ { 7 / 2 } } + \\frac { 3 5 } { 3 2 } \\frac { z _ 1 ^ 3 } { z _ 0 ^ { 9 / 2 } } \\end{align*}"}
{"id": "2984.png", "formula": "\\begin{align*} \\Theta _ { n } = \\Theta _ { n } ^ { ( 1 ) } + \\Theta _ { n } ^ { ( 2 ) } + 3 \\Theta _ { n } ^ { ( 3 ) } + 3 \\Theta _ { n } ^ { ( 4 ) } + 6 \\Theta _ { n } ^ { ( 5 ) } , \\end{align*}"}
{"id": "5893.png", "formula": "\\begin{align*} R ^ j _ t & = R ^ j _ 0 + M ^ j _ t + A ^ j _ t + B ^ j _ t \\\\ & = R ^ j _ 0 + \\int _ 0 ^ t u ^ j _ l ( s ) d W ^ l _ s + \\int _ 0 ^ t a ^ j ( s ) d s + \\int _ 0 ^ t b ^ j ( s ) d s , j = 1 , 2 , \\dots , d , \\end{align*}"}
{"id": "7898.png", "formula": "\\begin{align*} P \\cap \\tau _ v P = P ( x _ 1 , \\dots , x _ k , x _ 1 - v , \\dots , x _ k - v ; B \\times B ) . \\end{align*}"}
{"id": "1660.png", "formula": "\\begin{align*} \\Psi _ \\xi ( u ) : = \\varphi ( u ) - \\psi _ \\xi ( u ) \\end{align*}"}
{"id": "6026.png", "formula": "\\begin{align*} \\widetilde { O } \\widetilde { H } _ n = 2 n \\widetilde { H } _ n . \\end{align*}"}
{"id": "3697.png", "formula": "\\begin{align*} \\varphi _ q ( \\xi ) = \\begin{cases} \\varphi ( \\lambda _ q ^ { - 1 } \\xi ) , \\ \\ \\ \\mbox { f o r } \\ \\ q \\geq 0 \\\\ \\chi ( \\xi ) , \\ \\ \\ \\ \\ \\ \\ \\ \\mbox { f o r } \\ \\ q = - 1 . \\end{cases} \\end{align*}"}
{"id": "673.png", "formula": "\\begin{align*} \\alpha \\ = \\ \\sum \\limits _ { n \\in A } 2 ^ { - n } \\end{align*}"}
{"id": "4008.png", "formula": "\\begin{align*} m ^ 3 - \\lambda m ^ 2 - 2 \\lambda m + \\lambda ^ 2 = 0 . \\end{align*}"}
{"id": "304.png", "formula": "\\begin{align*} d X _ i ( t ) = b ( t , X _ i ( t ) , \\mu ( t ) , \\alpha _ i ) d t + \\sigma ( t , X _ i ( t ) , \\mu ( t ) ) d B _ i ( t ) + \\sigma _ 0 ( t , X _ i ( t ) , \\mu ( t ) ) d W ( t ) , \\end{align*}"}
{"id": "299.png", "formula": "\\begin{align*} D _ { 1 . 2 } & \\le C \\int _ { t / 2 } ^ { t } \\| G ( \\cdot , t - \\tau ) \\| _ { L ^ { 1 } } \\left \\| \\psi ^ { 2 } ( \\cdot , \\tau ) \\right \\| _ { L ^ { \\infty } } d \\tau \\le C \\int _ { t / 2 } ^ { t } \\| \\psi ( \\cdot , \\tau ) \\| _ { L ^ { \\infty } } ^ { 2 } d \\tau \\\\ & \\le C \\int _ { t / 2 } ^ { t } ( 1 + \\tau ) ^ { - \\alpha } d \\tau \\le C ( 1 + t ) ^ { - \\alpha + 1 } , \\ \\ t \\ge 0 . \\end{align*}"}
{"id": "3928.png", "formula": "\\begin{align*} \\zeta _ 2 ( - z ) = - \\zeta _ 2 ( z ) , \\wp _ 2 ( - z ) = \\wp _ 2 ( z ) , f _ 2 ( - z ) = f _ 2 ( z ) , f _ 2 ' ( - z ) = - f _ 2 ' ( z ) . \\end{align*}"}
{"id": "7963.png", "formula": "\\begin{align*} v ( r , \\omega ) = \\sum _ { j \\geq 1 } ( v _ j ^ + ( r ) + v _ j ^ - ( r ) ) \\varphi _ j ( \\omega ) , \\end{align*}"}
{"id": "7695.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | \\nabla u _ { \\ast } | ^ 2 = \\int _ { \\R ^ N } | \\nabla u | ^ 2 \\ ; . \\end{align*}"}
{"id": "6989.png", "formula": "\\begin{align*} y = \\mu + X \\beta + e , \\end{align*}"}
{"id": "4579.png", "formula": "\\begin{align*} X _ { n + 1 } = \\theta X _ n + \\varepsilon _ { n + 1 } , \\end{align*}"}
{"id": "8992.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Ddot { \\xi } ( s ) & = \\dot { \\xi } ( s ) + f ^ \\prime ( \\xi ( s ) ) , s \\in ( 0 , b ) , \\\\ \\xi ( 0 ) & = x _ 0 , \\\\ \\dot { \\xi } ( 0 ) & = y _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "574.png", "formula": "\\begin{align*} \\big \\{ ( x , x ^ \\prime , y , y ^ \\prime ) \\ \\ y = f ( x ) , \\ y ^ \\prime = f ^ \\prime ( x ^ \\prime ) \\big \\} \\end{align*}"}
{"id": "8284.png", "formula": "\\begin{align*} W e _ i - W e _ j & = ( e _ { v } + w _ j e _ j ) - ( e _ { v } + w _ i e _ i ) = w _ j e _ j - w _ i e _ i , \\end{align*}"}
{"id": "3786.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r \\big ( \\widetilde { U } , \\widetilde { W ' } \\big ) q _ F ^ { - \\frac { r } { 2 } } X ^ { - r } = \\gamma \\big ( X , r _ l ( \\pi _ F ) ^ { ( l ) } , \\sigma _ F ^ { ( l ) } , \\overline { \\psi } _ F ^ l \\big ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( U , W ' ) q _ F ^ { \\frac { r } { 2 } } X ^ { r } , \\end{align*}"}
{"id": "1456.png", "formula": "\\begin{align*} i U ( \\alpha z _ 1 ) ^ { \\ast } H U ( \\alpha z _ 2 ) = \\left [ \\begin{array} { c c } \\eta ( \\alpha z _ 1 , \\alpha z _ 2 ) & \\ast \\\\ \\ast & \\ast \\end{array} \\right ] , i U ( z _ 1 ) ^ { \\ast } H U ( z _ 2 ) = \\left [ \\begin{array} { c c } \\eta ( z _ 1 , z _ 2 ) & \\ast \\\\ \\ast & \\ast \\end{array} \\right ] . \\end{align*}"}
{"id": "6653.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq n < m < \\infty } \\frac { \\tau _ A ( p ^ { m } ) \\tau _ B ( p ^ { n } ) } { p ^ { m ( 1 - \\alpha ) } p ^ { n \\alpha } } = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\tau _ { A } ( p ^ m ) \\tau _ { B \\cup \\{ - \\alpha \\} } ( p ^ m ) } { p ^ { m } } - \\sum _ { m = 0 } ^ { \\infty } \\frac { \\tau _ { A } ( p ^ m ) \\tau _ B ( p ^ m ) } { p ^ { m } } . \\end{align*}"}
{"id": "3133.png", "formula": "\\begin{align*} x _ 1 : x _ 2 : x _ 3 : x _ 4 : x _ 5 = U _ { i 1 } : U _ { i 2 } : U _ { i 3 } : U _ { i 4 } : U _ { i 5 } \\end{align*}"}
{"id": "3606.png", "formula": "\\begin{align*} 2 - R _ { n + 2 } = 1 + ( 1 - R _ { n + 2 } ) \\ge d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] + d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] > 2 e + S _ { n } - R _ { n + 2 } \\ , , \\end{align*}"}
{"id": "6352.png", "formula": "\\begin{align*} A _ \\rho \\coloneqq \\left [ 0 \\to \\tau ( M _ \\rho ) \\to N _ \\rho \\to M _ \\rho \\to 0 \\right ] , \\end{align*}"}
{"id": "2971.png", "formula": "\\begin{align*} X _ { n , r } & : = \\begin{dcases} \\delta ^ { - 1 } _ n ( 3 ) \\cdot \\sum _ { q = 1 } ^ { r - 1 } \\left [ \\frac { \\delta _ n ( 3 ) } { \\delta _ n ( 2 ) } \\cdot \\tilde M _ { n , \\{ q , r \\} } + \\sum _ { p = 1 } ^ { q - 1 } \\tilde M _ { n , \\{ p , q , r \\} } \\right ] & , r \\ge 2 \\\\ 0 & , r = 1 , \\end{dcases} \\end{align*}"}
{"id": "7976.png", "formula": "\\begin{align*} \\Delta ( K ; \\alpha , \\beta ) : = \\Big [ \\frac { ( K ^ { \\alpha + \\beta } - 1 ) ^ 3 } { \\alpha + \\beta } \\Big ] ^ 2 - \\frac { ( K ^ { 2 \\alpha } - 1 ) ^ 3 } { 2 \\alpha } \\cdot \\frac { ( K ^ { 2 \\beta } - 1 ) ^ 3 } { 2 \\beta } < 0 . \\end{align*}"}
{"id": "757.png", "formula": "\\begin{align*} t ^ \\star = \\begin{cases} \\bar t & \\mbox { \\ i f \\ } \\bar t \\mbox { \\ e x i s t s } \\\\ \\tfrac { 3 } { \\mu + L + \\sqrt { \\mu ^ 2 - L \\mu + L ^ 2 } } & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
{"id": "4350.png", "formula": "\\begin{align*} \\partial _ \\xi ^ k \\Lambda Q = \\partial _ \\xi ^ k \\left ( a _ 0 \\xi ^ { - \\gamma } \\right ) + O ( \\xi ^ { - \\gamma - g - k } ) \\xi \\to + \\infty . \\end{align*}"}
{"id": "7496.png", "formula": "\\begin{align*} V ( \\mathbf { x } ) = V _ { \\mathrm { h o } } ( \\mathbf { x } ) : = \\frac { 1 } { 2 } \\begin{cases} \\gamma _ { x } ^ { 2 } x ^ { 2 } , & d = 1 , \\\\ \\gamma _ { x } ^ { 2 } x ^ { 2 } + \\gamma _ { y } ^ { 2 } y ^ { 2 } , & d = 2 , \\\\ \\end{cases} \\end{align*}"}
{"id": "545.png", "formula": "\\begin{align*} b , b ' [ \\bar \\rho - \\alpha _ 0 , \\bar \\rho ) \\alpha _ 0 \\in ( 0 , \\bar \\rho ) , \\\\ | b ' | ^ \\frac { 5 } { 2 } + | b | ^ \\frac { 5 } { 2 } \\leq c ( 1 + p ) [ 0 , \\bar \\rho ) c > 0 . \\end{align*}"}
{"id": "9082.png", "formula": "\\begin{align*} \\mu _ k = J _ 0 \\left ( \\frac { 2 \\pi ( k - 1 ) } { N - 1 } W \\right ) , k = 2 , { \\dots } , N , \\end{align*}"}
{"id": "5387.png", "formula": "\\begin{align*} \\hat { b } ( x ) = \\int _ { - \\infty } ^ { \\infty } K _ { h } ( y - x ) b ( y ) \\mathrm { d } y . \\end{align*}"}
{"id": "5331.png", "formula": "\\begin{align*} \\mathbb { T } _ t + \\mathrm { d i v } _ x ( u \\mathbb { T } ) - ( \\nabla _ x u \\mathbb { T } + \\mathbb { T } \\nabla _ x ^ T u ) = \\nu \\Delta _ x \\mathbb { T } + \\frac { k A _ 0 } { 2 } \\eta \\mathbb { I } - \\frac { A _ 0 } { 2 } \\mathbb { T } . \\end{align*}"}
{"id": "6614.png", "formula": "\\begin{align*} | 1 + r | ^ { - \\omega } + | 1 - r | ^ { - \\omega } = \\frac { 1 } { 2 \\pi i } \\int _ { ( c ) } \\mathcal { H } ( z , \\omega ) r ^ { - z } \\ , d z . \\end{align*}"}
{"id": "7827.png", "formula": "\\begin{align*} p _ { 1 } & = T ^ { * 2 ( j - l - 1 ) } x - \\sum _ { i = 0 } ^ { 2 l + 1 } \\alpha _ { i } e _ { i } = \\alpha _ { 2 l + 2 } e _ { 2 l + 2 } + \\alpha _ { 2 l + 3 } e _ { 2 l + 3 } \\\\ q _ { 1 } & = T ^ { * 2 ( n - j - l - 3 ) } y - \\sum _ { i = 0 } ^ { 2 l + 1 } \\beta _ { i } e _ { i } = \\beta _ { 2 l + 2 } e _ { 2 l + 2 } + \\beta _ { 2 l + 3 } e _ { 2 l + 3 } . \\end{align*}"}
{"id": "6411.png", "formula": "\\begin{align*} D _ { T ^ * } T _ i ^ * = P _ i ^ { \\perp } U _ i ^ * D _ { T ^ * } + P _ i U _ i ^ * D _ { T ^ * } T ^ * 1 \\leq i \\leq n , \\end{align*}"}
{"id": "9553.png", "formula": "\\begin{align*} H ( v ( s ) , \\mu ( s ) ) = 0 , \\ v ( 0 ) = v ^ { ( 0 ) } , \\ \\mu ( 0 ) = 1 . \\end{align*}"}
{"id": "5245.png", "formula": "\\begin{align*} a \\otimes b = \\sum _ { i j } x _ { i j } y _ { i j } . \\end{align*}"}
{"id": "3794.png", "formula": "\\begin{align*} \\gamma ( X , \\pi _ E , Q ( \\widehat { D _ i } ) , \\psi _ E ) = \\prod _ { j = 0 } ^ { m _ i - 1 } \\gamma ( X , \\pi _ E , \\widehat { \\sigma _ i } \\nu _ E ^ j , \\psi _ E ) . \\end{align*}"}
{"id": "6563.png", "formula": "\\begin{align*} \\sum _ { q \\le Q } \\ , \\sideset { } { ^ * } \\sum _ { \\chi \\bmod q } \\left | L \\left ( \\tfrac { 1 } { 2 } , \\chi \\right ) \\right | ^ { 2 k } \\sim c _ k \\sum _ { q \\le Q } \\ , \\sideset { } { ^ * } \\sum _ { \\chi \\bmod q } \\prod _ { p | q } \\Bigg ( \\sum _ { m = 0 } ^ { \\infty } \\frac { \\binom { m + k - 1 } { k - 1 } ^ 2 } { p ^ m } \\Bigg ) ^ { - 1 } ( \\log q ) ^ { k ^ 2 } , Q \\to \\infty \\end{align*}"}
{"id": "4759.png", "formula": "\\begin{align*} R _ { 1 } & = ( \\underbrace { 0 , 1 , \\cdots , 1 } _ { k + 2 } , \\underbrace { 0 , \\cdots , 0 } _ { r } , \\underbrace { 0 , \\cdots , 0 } _ { t } ) \\\\ R _ { 2 } & = ( \\underbrace { 1 , 0 , \\cdots , 0 } _ { k + 2 } , \\underbrace { 1 , \\cdots , 1 } _ { r } , \\underbrace { 0 , \\cdots , 0 } _ { t } ) . \\end{align*}"}
{"id": "943.png", "formula": "\\begin{align*} D _ 1 ^ j = \\frac { N _ { n , \\nu _ j } ^ j } { \\| N _ { n , \\nu _ j } ^ j \\| _ 2 } , j \\geq 2 , \\end{align*}"}
{"id": "3994.png", "formula": "\\begin{align*} U ( t ) = U ^ p ( t ) + U ^ h ( t ) , \\ \\ t \\geq 0 . \\end{align*}"}
{"id": "628.png", "formula": "\\begin{align*} A ( x , n ) \\ = \\ \\frac { f ( x , n ) - g ( x , n ) } { h ( x , n ) + 1 } ( x , n = 0 , 1 , 2 , \\ldots ) , \\end{align*}"}
{"id": "616.png", "formula": "\\begin{align*} \\abs { A ( c x + c ) - \\alpha } \\ \\leq \\ \\frac { c } { c x + c } \\ = \\ \\frac { 1 } { x + 1 } . \\end{align*}"}
{"id": "9003.png", "formula": "\\begin{align*} & \\log \\frac { ( K - i ) x _ { K - i } + i } { K } = \\sum _ { j = 0 } ^ { i - 1 } ( 1 - N ) ^ j \\log \\frac { ( K - 1 ) x _ { K - 1 } + 1 } { K } \\\\ & \\ , \\ , \\quad \\qquad \\qquad \\qquad \\qquad - \\sum _ { j = 1 } ^ { i - 1 } ( 1 - N ) ^ j \\log x _ { K - i + j } . \\end{align*}"}
{"id": "5934.png", "formula": "\\begin{align*} \\phi ( t ) = g ( t ) \\cdot \\overline { \\phi } _ h ( t ) . \\end{align*}"}
{"id": "5254.png", "formula": "\\begin{align*} \\Delta ( a ) = a _ { ( 1 ) } \\otimes a _ { ( 2 ) } \\in \\widetilde { M } ( A \\otimes ^ I A ) , \\end{align*}"}
{"id": "2036.png", "formula": "\\begin{align*} H _ t ^ { \\beta } : = \\lim _ { \\varepsilon \\to 0 } \\int _ 0 ^ t | X _ s | ^ { \\beta } ( ( X _ s ) ) \\ 1 _ { \\{ | X _ s | \\geq \\varepsilon \\} } { \\rm d } s . \\end{align*}"}
{"id": "5590.png", "formula": "\\begin{align*} \\mathbb { E } [ X | X < a ] & = \\alpha - \\sigma h \\left ( \\frac { a - \\alpha } { \\sigma } \\right ) \\\\ \\mathbb { E } [ X | X > a ] & = \\alpha + \\sigma h \\left ( \\frac { a - \\alpha } { \\sigma } \\right ) \\end{align*}"}
{"id": "6655.png", "formula": "\\begin{align*} D _ { 1 , m } = \\tau _ { A \\smallsetminus \\{ \\alpha \\} \\cup \\{ - \\beta \\} } ( p ^ m ) \\tau _ { B \\smallsetminus \\{ \\beta \\} } ( p ^ { m } ) . \\end{align*}"}
{"id": "5168.png", "formula": "\\begin{align*} \\sqrt { b } \\ , \\vartheta ' _ 1 ( 0 , i b ) = \\sqrt { b } \\ , \\vartheta _ 2 ( 0 , i b ) \\vartheta _ 3 ( 0 , i b ) \\vartheta _ 4 ( 0 , i b ) = \\vartheta _ 4 ( 0 , i / b ) \\vartheta _ 3 ( 0 , i b ) \\vartheta _ 4 ( 0 , i b ) , \\end{align*}"}
{"id": "4895.png", "formula": "\\begin{align*} z L ( z ) \\to 0 \\hbox { a n d } \\frac { f _ 1 ' ( z ) } { f _ 1 ( z ) } = O \\left ( | z | \\right ) \\hbox { a s $ z \\to \\infty $ w i t h $ \\delta < \\arg z < \\pi - \\delta $ } . \\end{align*}"}
{"id": "7233.png", "formula": "\\begin{align*} \\nabla _ x \\int f _ 0 ( x - t v ) g ( v ) \\dd v & = \\frac { 1 } { t } \\nabla _ x \\frac 1 { t ^ 3 } \\int f _ 0 ( w ) g ( \\frac { x - w } { t } ) \\dd w \\\\ & = \\frac 1 { t ^ 4 } \\int f _ 0 ( w ) \\nabla g ( \\frac { x - w } { t } ) \\dd w = \\frac 1 { t } \\int f _ 0 ( x - t v ) \\nabla g ( v ) \\dd v . \\end{align*}"}
{"id": "4975.png", "formula": "\\begin{align*} A & = 2 a _ { 0 } ^ { ( 2 ) } ( b _ { 0 } ( 0 ) ) b _ { 0 } ^ { ( 1 ) } ( 0 ) + 2 \\partial _ { 2 } a _ { 0 } ^ { ( 1 ) } ( b _ { 0 } ( 0 ) ) - a _ { 0 } ^ { ( 2 ) } ( 0 ) \\\\ B & = 2 a _ { 0 } ^ { ( 1 ) } ( b _ { 0 } ( 0 ) ) - 2 a _ { 0 } ^ { ( 2 ) } ( 0 ) a _ { 0 } ( 0 ) - 2 a _ { 0 } ^ { ( 1 ) } ( 0 ) ^ { 2 } \\\\ C & = - 3 a _ { 0 } ( 0 ) \\left ( a _ { 0 } ( 0 ) a _ { 0 } ^ { ( 2 ) } ( 0 ) + 2 a _ { 0 } ^ { ( 1 ) } ( 0 ) ^ { 2 } \\right ) \\end{align*}"}
{"id": "8102.png", "formula": "\\begin{align*} \\mathrm { W F } ( W ) = \\Gamma _ { \\ell } \\sqcup \\Gamma _ r , \\end{align*}"}
{"id": "3230.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } \\frac { s } { s ^ 2 + u ^ 2 } d \\mu ( u ) = \\int _ 0 ^ \\infty \\frac { s } { s ^ 2 + u ^ 2 } d \\mu _ { | T | } ( u ) . \\end{align*}"}
{"id": "3847.png", "formula": "\\begin{align*} p _ { m + 1 , j } = \\tilde f _ { m , j } - f _ { m , j } , j = \\overline { 1 , m } . \\end{align*}"}
{"id": "8848.png", "formula": "\\begin{align*} & 2 F ^ { \\rm W K } _ 1 ( z _ 0 , z _ 1 ) = F ^ { \\rm e v e n } _ 1 ( z _ 1 ) - \\zeta ' ( - 1 ) = \\frac 1 { 1 2 } \\log z _ 1 , \\\\ & 2 ^ g F ^ { \\rm W K } _ g ( z _ 1 , \\dots , z _ { 3 g - 2 } ) = F _ g ^ { { \\rm e v e n } , [ 1 - g ] } ( z _ 1 , \\dots , z _ { 3 g - 2 } ) , g \\geq 2 . \\end{align*}"}
{"id": "7531.png", "formula": "\\begin{align*} \\frac { T ^ 2 } { 2 } \\arctan \\left ( \\frac { \\frac { 1 } { 2 } + \\epsilon } { T } \\right ) - \\frac { T ^ 2 } { 2 } \\arctan \\left ( \\frac { \\frac { 1 } { 2 } - \\epsilon } { T } \\right ) = \\epsilon T + \\epsilon \\ \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "1309.png", "formula": "\\begin{align*} \\hat { I } _ { = 2 , < \\epsilon } ( k ) : = \\{ \\ , \\ , \\alpha _ { k ' } \\ , \\ , | \\ , \\ , k ' \\leq k , \\ , \\ , J ( \\alpha _ { k ' + 1 } , \\alpha _ { k ' } ) = 2 , \\ , \\ , \\ , \\alpha _ { k ' + 1 } , \\ , \\ , \\alpha _ { k ' } \\ , \\ , \\mathrm { s a t i s f y \\ , \\ , 2 , 3 , 4 , 5 , 6 \\ , \\ , i n \\ , \\ , L e m m a \\ , \\ , \\ref { m a i n l e m m a } } \\} \\end{align*}"}
{"id": "6040.png", "formula": "\\begin{align*} \\delta _ { \\alpha , \\beta } ( \\delta _ { \\alpha , \\beta } ^ * g ) & = \\big ( ( 1 - x ^ 2 ) ^ { 1 / 2 } ( ( 1 - x ^ 2 ) ^ { 1 / 2 } ) ' \\big ) ' + x g ' \\\\ & - \\Big ( \\frac { \\alpha } { 2 } + \\frac { \\beta } { 2 } \\Big ) g - \\frac { [ ( \\frac { \\beta } { 2 } - \\frac { \\alpha } { 2 } ) - ( \\frac { \\alpha } { 2 } + \\frac { \\beta } { 2 } + 1 ) x ] ^ 2 } { 1 - x ^ 2 } , \\end{align*}"}
{"id": "6432.png", "formula": "\\begin{align*} \\Psi ( a , b , c , d ) & : = \\sum _ { \\pi \\in D } \\omega _ \\pi ( a , b , c , d ) = \\frac { ( - a , - a b c ; Q ) _ \\infty } { ( a b ; Q ) _ \\infty } , \\\\ \\Phi ( a , b , c , d ) & : = \\sum _ { \\pi \\in U } \\omega _ \\pi ( a , b , c , d ) = \\frac { ( - a , - a b c ; Q ) _ \\infty } { ( a b , a c , Q ; Q ) _ \\infty } . \\end{align*}"}
{"id": "1593.png", "formula": "\\begin{align*} \\div ( \\beta ( \\rho ) u ) = T \\end{align*}"}
{"id": "3069.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\pm i ( x _ 1 \\pm x _ 2 ) \\ , , x _ 2 ^ \\prime = x _ 1 \\pm x _ 2 \\ , , y _ 1 ^ \\prime = \\pm i ( y _ 1 \\pm y _ 2 ) \\ , , y _ 2 ^ \\prime = y _ 1 \\pm y _ 2 \\ , , \\end{align*}"}
{"id": "5044.png", "formula": "\\begin{align*} \\nu _ i = \\lambda _ i \\ \\ 1 \\leq i \\leq n - s . \\end{align*}"}
{"id": "7266.png", "formula": "\\begin{align*} \\Bigl ( \\sum \\alpha _ k \\Bigr ) \\circ \\beta = \\sum _ { n = 0 } ^ \\infty \\Bigl ( \\sum _ { k = 1 } ^ \\infty a _ { k , n } \\Bigr ) \\beta ^ n = \\sum _ { k = 1 } ^ \\infty \\Bigl ( \\sum _ { n = 0 } ^ \\infty a _ { k , n } \\beta ^ n \\Bigr ) = \\sum \\alpha _ k ( \\beta ) . \\end{align*}"}
{"id": "5071.png", "formula": "\\begin{align*} A ( t ) = L ( t ) + C , \\end{align*}"}
{"id": "1102.png", "formula": "\\begin{align*} \\zeta ( k ) = t ^ { 1 / 2 } \\sqrt { - 2 g '' ( - \\eta ) } ( k + \\eta ) . \\end{align*}"}
{"id": "1120.png", "formula": "\\begin{align*} D _ { \\infty } ( \\xi ) : = \\exp \\left \\{ - \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { - \\eta } ^ { - C _ R } + \\int _ { C _ R } ^ { \\eta } \\right ) \\frac { \\log r _ { + } ( s ) } { X _ { \\eta + } ( s ) ( s - k ) } d s \\right \\} . \\end{align*}"}
{"id": "7215.png", "formula": "\\begin{align*} \\begin{aligned} | \\tilde Y _ { s , t } ( x , v ) | & \\lesssim \\frac { \\delta ( t - s ) } { \\langle d _ { t , x } \\rangle ^ 2 + \\langle x ^ \\perp \\rangle ^ 2 } , \\\\ | \\nabla _ v \\tilde Y _ { s , t } ( x , v ) | & \\lesssim \\frac { \\delta ( t - s ) } { \\langle d _ { t , x } \\rangle + \\langle x ^ \\perp \\rangle } . \\end{aligned} \\end{align*}"}
{"id": "7780.png", "formula": "\\begin{align*} \\mu _ * ( B ( ( t , h ( t ) ) , r ) ) \\geq \\frac { \\prod \\limits _ { i = 1 } ^ { q } p _ { \\sigma _ i } } { \\prod \\limits _ { i = 1 } ^ { q } C _ { \\sigma _ i } } r C _ { \\min } { \\beta } ^ { - 1 } = C _ * \\cdot r \\frac { \\prod \\limits _ { i = 1 } ^ { q } p _ { \\sigma _ i } } { \\prod \\limits _ { i = 1 } ^ { q } C _ { \\sigma _ i } } , \\end{align*}"}
{"id": "6429.png", "formula": "\\begin{align*} \\int _ N \\langle \\nabla ^ * \\nabla \\xi , \\xi \\rangle = \\int _ N \\| \\nabla \\xi \\| ^ 2 + \\int _ { \\partial N } \\langle \\nabla _ \\nu \\xi , \\xi \\rangle , \\end{align*}"}
{"id": "3408.png", "formula": "\\begin{align*} | K ( x , y ) | & \\lesssim \\int _ 0 ^ \\infty | S _ t ( x , y ) | \\frac { d t } { t } = \\int _ 0 ^ { d ( x , y ) } \\ , { 1 \\over V ( x , y , t + d ( x , y ) ) } \\frac { t } { t + \\| x - y \\| } \\frac { d t } { t } \\\\ & + \\int _ { d ( x , y ) } ^ { \\| x - y \\| } \\ , { 1 \\over V ( x , y , t + d ( x , y ) ) } \\frac { t } { t + \\| x - y \\| } \\frac { d t } { t } \\\\ & + \\int _ { \\| x - y \\| } ^ \\infty { 1 \\over V ( x , y , t + d ( x , y ) ) } \\frac { t } { t + \\| x - y \\| } \\frac { d t } { t } \\\\ & = : I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
{"id": "41.png", "formula": "\\begin{align*} \\theta _ { t } ^ { c } \\otimes \\theta _ { t } ^ { c } = \\theta ^ { c } \\otimes \\theta ^ { c } + \\zeta _ { t } \\otimes \\theta ^ { c } + \\theta ^ { c } \\otimes \\zeta _ { t } + \\zeta _ { t } \\otimes \\zeta _ { t } . \\end{align*}"}
{"id": "7392.png", "formula": "\\begin{align*} c _ { x , x + 1 } ( \\eta ) = & [ \\eta ( x - 1 ) + \\eta ( x + 1 ) + \\eta ( x ) + \\eta ( x + 2 ) ] \\big ( \\eta ( x ) [ 1 - \\eta ( x + 1 ) ] + \\eta ( x + 1 ) [ 1 - \\eta ( x ) ] \\big ) \\\\ = & \\xi _ { x , x + 1 } ( \\eta ) [ \\eta ( x - 1 ) + \\eta ( x + 2 ) + 1 ] . \\end{align*}"}
{"id": "4321.png", "formula": "\\begin{align*} \\chi _ 0 \\in C ^ \\infty , \\chi _ 0 ( x ) = 1 , \\forall x \\in [ 0 , 1 ] , \\chi _ 0 ( x ) = 0 , \\forall x \\ge 2 . \\end{align*}"}
{"id": "8197.png", "formula": "\\begin{align*} M _ 2 ( p , H ) = { \\pi ^ 2 \\over 8 } \\left ( 1 + \\frac { N _ 2 ( p , H ) } { p } \\right ) = { \\pi ^ 2 \\over 8 } \\left ( 1 + O ( p ^ { - 1 / 2 } ) \\right ) \\hbox { a n d } h _ K ^ - \\leq 2 \\left ( \\frac { p + o ( p ) } { 3 2 } \\right ) ^ { ( p - 1 ) / 1 2 } , \\end{align*}"}
{"id": "4828.png", "formula": "\\begin{align*} T ( z ) = \\mathcal { O } ( h ^ { - 2 n - 2 } ) . \\end{align*}"}
{"id": "7120.png", "formula": "\\begin{align*} 0 = x _ { \\tau _ 0 } = x _ t + \\int _ t ^ { \\tau _ 0 } b ( s , x _ s ) d s + B ^ H _ { \\tau _ 0 } - B ^ H _ t , t \\in ( \\hat { \\tau } _ 0 , \\tau _ 0 ] . \\end{align*}"}
{"id": "9546.png", "formula": "\\begin{align*} H ( v , v ^ { ( 0 ) } , \\mu ) = \\left [ \\begin{array} { c } ( 1 - \\mu ) ( w - z _ 1 + ( m - 1 ) ( \\hat { \\mathcal { A } } x ^ { m - 2 } ) ^ T ( x - z _ 2 ) ) + \\mu ( x - x ^ { ( 0 ) } ) \\\\ Z _ 1 x - \\mu Z _ 1 ^ { ( 0 ) } x ^ { ( 0 ) } \\\\ Z _ 2 w - \\mu Z _ 2 ^ { ( 0 ) } w ^ { ( 0 ) } + ( 1 - \\mu ) X w \\\\ w - ( 1 - \\mu ) ( \\mathcal { A } x ^ { m - 1 } + q ) - \\mu w ^ { ( 0 ) } \\\\ \\end{array} \\right ] = 0 \\end{align*}"}
{"id": "3907.png", "formula": "\\begin{align*} 0 = \\sum _ { x \\leq p \\leq 2 x } \\Psi ( z ) \\Psi ( f ( z ) ) . \\end{align*}"}
{"id": "5746.png", "formula": "\\begin{align*} \\sup _ { J _ j } | \\psi _ \\lambda | \\leq C _ 7 \\left ( 2 ^ { - 4 / 3 \\ell } \\right ) ^ { \\frac { c _ 0 } { 2 } m \\sqrt { \\lambda } } e ^ { \\sqrt { \\lambda } } = C _ 7 e ^ { ( 1 - \\frac { 2 } { 3 } ( \\log 2 ) { c _ 0 } m \\ell ) \\sqrt { \\lambda } } \\leq C _ 7 e ^ { ( 1 - { c _ 0 } m \\ell ) \\sqrt { \\lambda } } . \\end{align*}"}
{"id": "1266.png", "formula": "\\begin{align*} P _ { \\gamma } ^ { \\mathrm { i n } } ( M ) = P _ { \\gamma } ^ { \\mathrm { o u t } } : = ( 1 , . . . , \\ , 1 ) \\end{align*}"}
{"id": "9272.png", "formula": "\\begin{align*} & \\exists { a ^ * } ^ { 0 ( 0 ) } \\forall a ^ { X ( X ) } \\Big ( \\forall x ^ X ( x \\in \\mathrm { d o m } A \\rightarrow a x \\in A x ) \\\\ & \\qquad \\qquad \\qquad \\land \\forall x ^ X ( x \\not \\in \\mathrm { d o m } A \\rightarrow \\norm { a x } = 0 ) \\rightarrow a ^ * \\gtrsim a \\Big ) . \\end{align*}"}
{"id": "1049.png", "formula": "\\begin{align*} \\Delta ( D _ { 5 6 } ) & = \\sum _ { \\substack { k < 5 \\\\ k < l } } a _ { 5 k } a _ { 6 l } \\otimes D _ { k l } - q ^ { - 1 } \\sum _ { \\substack { k < 5 \\\\ l < k } } a _ { 5 k } a _ { 6 l } \\otimes D _ { l k } \\\\ & + ( a _ { 5 5 } a _ { 6 6 } + q ^ { - 1 } a _ { 5 6 } a _ { 6 5 } ) \\otimes D _ { 5 6 } \\\\ & + ( 1 + q ^ { - 2 } ) \\sum _ { 5 \\leq k \\leq 6 } a _ { 5 k } a _ { 6 k } \\otimes D _ { k k } + q ^ { - 1 } \\sum _ { \\substack { k \\geq 5 \\\\ l < 5 } } a _ { 5 k } a _ { 6 l } \\otimes D _ { l k } \\ , , \\end{align*}"}
{"id": "2330.png", "formula": "\\begin{align*} | W f ( x , \\omega ) | = 2 ^ d | V _ { f ^ \\vee } f ( 2 x , 2 \\omega ) | = 2 ^ d | \\widehat { f } * M _ { - 2 x } \\widehat { \\overline { f } } ^ \\vee ( 2 \\omega ) | . \\end{align*}"}
{"id": "4689.png", "formula": "\\begin{align*} ( \\mathcal { L } f ) ' = g , \\lim _ { y \\rightarrow + \\infty } f ( y ) = 0 , \\lim _ { y \\rightarrow - \\infty } f ( y ) = - \\int g . \\end{align*}"}
{"id": "430.png", "formula": "\\begin{align*} \\begin{aligned} V \\in L ^ { \\infty } ( 0 , T _ { 0 } ; H ^ { s } ) , & v _ { 2 } \\in L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s + 1 } ) , \\\\ V _ { t } \\in L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { s - 1 } ) , & V ( x , t ) \\in \\mathcal { O } _ { g _ { 2 } } \\quad \\forall ( x , t ) \\in Q _ { T _ { 0 } } , \\end{aligned} \\end{align*}"}
{"id": "8394.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varepsilon \\nabla \\mathbf { v } _ { \\varepsilon } \\nabla c _ { \\varepsilon } \\otimes \\nabla c _ { \\varepsilon } \\ , \\mathrm { d } x = \\int _ { \\Omega } \\Big ( \\frac { 1 } { 2 } \\partial _ { t } ( \\mathbf { v } _ { \\varepsilon } ) ^ 2 + ( \\nabla \\mathbf { v } _ { \\varepsilon } ) ^ 2 \\Big ) \\mathrm { d } x . \\end{align*}"}
{"id": "1256.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) : = c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z ) + \\sum _ { i } \\sum _ { k = 1 } ^ { m _ { i } } \\mu _ { \\tau } ( \\alpha _ { i } ^ { k } ) - \\sum _ { j } \\sum _ { k = 1 } ^ { n _ { j } } \\mu _ { \\tau } ( \\beta _ { j } ^ { k } ) . \\end{align*}"}
{"id": "8739.png", "formula": "\\begin{align*} T P ( S ) : = \\biggl \\{ ( \\lambda , y ) \\in \\R _ + ^ { | V | + | E | } \\biggm | \\lambda \\in S , \\ \\sum _ { e \\in E : v \\in e } y _ e = \\lambda _ v , \\ v \\in V \\biggr \\} . \\end{align*}"}
{"id": "1839.png", "formula": "\\begin{align*} A _ n ( y + \\sqrt { y ^ 2 - x ^ 2 } , y - \\sqrt { y ^ 2 - x ^ 2 } ) = D _ n ( x ^ 2 , y ) . \\end{align*}"}
{"id": "829.png", "formula": "\\begin{align*} \\mu _ \\omega ( B _ \\rho ( \\infty , R ) ) = \\mu ( Z ) \\ , \\int _ { H _ R } ^ \\infty y ^ { a - 2 \\beta } \\ , d y & = \\frac { \\mu ( Z ) } { 2 \\beta - 1 - a } \\ , H _ R ^ { 1 + a - 2 \\beta } \\\\ & = \\mu ( Z ) \\ , \\frac { ( \\beta - 1 ) ^ { \\tfrac { 2 \\beta - 1 - a } { \\beta - 1 } } } { 2 \\beta - 1 - a } \\ , R ^ { \\tfrac { 2 \\beta - 1 - a } { \\beta - 1 } } . \\end{align*}"}
{"id": "3897.png", "formula": "\\begin{align*} T ( u , p ) = \\sum _ { \\substack { 0 < a < p \\\\ g c d ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } . \\end{align*}"}
{"id": "2992.png", "formula": "\\begin{align*} S ( z ) = I _ E + i \\alpha ( { L ^ { \\phantom { s } } \\ ! \\ ! \\ ! } ^ { \\scriptscriptstyle { - | | } } - z I ) ^ { - 1 } \\alpha , z \\in \\rho ( { L ^ { \\phantom { s } } \\ ! \\ ! \\ ! } ^ { \\scriptscriptstyle { - | | } } ) , { L ^ { \\phantom { s } } \\ ! \\ ! \\ ! } ^ { \\scriptscriptstyle { - | | } } : = ( { L ^ { \\phantom { s } } \\ ! \\ ! } ^ { \\scriptscriptstyle { | | } } ) ^ * . \\end{align*}"}
{"id": "1883.png", "formula": "\\begin{align*} ( g \\pi ^ { - 1 } , h k ^ { - 1 } ) = ( \\gamma , \\tau ( \\gamma ) ) \\in ( g \\Pi ^ { - 1 } \\times h K ^ { - 1 } ) \\cap \\Gamma \\subseteq ( g \\Pi ^ { - 1 } \\times I K ^ { - 1 } ) \\cap \\Gamma \\end{align*}"}
{"id": "2263.png", "formula": "\\begin{align*} | x | = \\sqrt { x \\cdot x } \\ . \\end{align*}"}
{"id": "581.png", "formula": "\\begin{align*} Y _ 1 \\ = \\ ] 0 , 1 [ \\ \\times \\ X _ 1 , Y _ 2 \\ = \\ ] - 1 , 0 [ \\ \\times X _ 2 \\end{align*}"}
{"id": "2995.png", "formula": "\\begin{align*} N _ 0 ^ i ( L ) : = K \\ominus \\{ N _ - ( L ^ * ) \\vee N _ + ( L ^ * ) \\} \\subset N _ i ( L ) , \\end{align*}"}
{"id": "6482.png", "formula": "\\begin{align*} E \\left [ \\left ( \\frac { S _ { n + 1 } } { \\sqrt { ( n + 1 ) / 2 } } \\right ) ^ 2 \\right ] - 1 = \\frac { n - 1 } { n + 1 } \\left ( E \\left [ \\left ( \\frac { S _ { n } } { \\sqrt { n / 2 } } \\right ) ^ 2 \\right ] - 1 \\right ) , \\end{align*}"}
{"id": "7325.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { | \\tilde { x } - \\tilde { y } | ^ 4 } { \\varepsilon ^ 4 } + \\alpha ( | \\tilde { x } | ^ 2 + | \\tilde { y } | ^ 2 ) \\leq L ( | \\tilde { x } - \\tilde { y } | + 1 ) + M ( \\tilde { t } + 1 ) \\\\ & - u ( x _ 1 , t _ 1 ) + v ( x _ 1 , t _ 1 ) + 2 \\alpha | x _ 1 | ^ 2 + { \\lambda \\over T - t _ 1 } - { \\lambda \\over T - \\tilde { t } } . \\end{aligned} \\end{align*}"}
{"id": "5542.png", "formula": "\\begin{align*} x : = B h + \\alpha ( h ) - \\rho ( h ) \\eta : = \\sigma ( h ) u . \\end{align*}"}
{"id": "8980.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f _ 2 ( x ) & \\leq 0 \\ , ( - 1 , 1 ) , \\\\ u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f _ 2 ( x ) & \\geq 0 [ - 1 , 1 ] , \\end{aligned} \\right . \\end{align*}"}
{"id": "4732.png", "formula": "\\begin{align*} \\big ( \\mathcal { Z } _ 1 ( t ) , \\epsilon ( t ) \\big ) = \\big ( \\mathcal { Z } _ 2 ( t ) , \\epsilon ( t ) \\big ) = 0 , \\end{align*}"}
{"id": "1033.png", "formula": "\\begin{align*} u ( x ) & = \\frac 1 2 \\gamma _ { n , s } \\int _ { \\R ^ n \\setminus B _ r } \\bigg ( \\frac { r ^ 2 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - r ^ 2 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert x - y \\vert ^ n } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ n } \\bigg ) u ( y ) \\dd y . \\end{align*}"}
{"id": "8860.png", "formula": "\\begin{align*} d _ 0 \\tilde \\varphi ( s , t ) - \\varphi ( s , t ) = \\varphi ( a _ k , c _ 0 ) + \\sum _ { i = 0 } ^ { m - 1 } \\varphi ( c _ i , c _ { i + 1 } ) - \\varphi ( a _ k , b _ 0 ) - \\sum _ { i = 0 } ^ { l - 1 } \\varphi ( b _ i , b _ { i + 1 } ) - \\varphi ( s , t ) \\end{align*}"}
{"id": "1342.png", "formula": "\\begin{align*} & \\mathbb { G } _ \\delta = \\{ G ( z , \\delta ) \\mid z \\in \\mathbb { Q } ^ d \\} , \\ ; \\ ; \\mathbb { G } = \\bigcup _ { \\delta \\in ( 0 , 1 ) \\cap \\mathbb { Q } } \\mathbb { G } _ \\delta , \\ ; \\ ; \\mathcal { Q } _ = \\left \\{ \\bigcup _ { i = 1 } ^ N Q _ i \\mid Q _ i \\in \\mathbb { G } \\right \\} . \\end{align*}"}
{"id": "1830.png", "formula": "\\begin{align*} W _ n ( x ) = \\sum _ { k = 1 } ^ { \\lfloor ( n + 1 ) / 2 \\rfloor } W ( n , k ) x ^ { k } . \\end{align*}"}
{"id": "4583.png", "formula": "\\begin{align*} Y _ k ( \\lambda ) = \\sum _ { i = 1 } ^ k \\eta _ i ( \\lambda ) \\quad \\quad \\textrm { a n d } \\quad \\quad B _ k ( \\lambda ) = \\sum _ { i = 1 } ^ k b _ i ( \\lambda ) \\end{align*}"}
{"id": "3479.png", "formula": "\\begin{align*} \\abs * { \\sum _ { k / 2 < m \\leq k - 1 } \\frac { 1 } { m ^ { s _ 1 } ( k - m ) ^ { s _ 2 } } } & \\ll \\sum _ { k / 2 < m \\leq k - 1 } \\frac { 1 } { m ^ { \\sigma _ 1 } ( k - m ) ^ { \\sigma _ 2 } } \\\\ & \\ll k ^ { - \\sigma _ 1 } \\sum _ { k / 2 < m \\leq k - 1 } \\frac { 1 } { ( k - m ) ^ { \\sigma _ 2 } } \\\\ & \\ll k ^ { - \\sigma _ 1 } \\times \\begin{cases} 1 & ( \\sigma _ 2 > 1 ) \\\\ \\log k & ( \\sigma _ 2 = 1 ) \\\\ k ^ { 1 - \\sigma _ 2 } & ( \\sigma _ 2 < 1 ) . \\end{cases} \\end{align*}"}
{"id": "3878.png", "formula": "\\begin{align*} S _ H & = \\{ t _ 1 - s _ 1 , s _ 1 - t _ 2 , \\dots , s _ { n - 1 } - t _ n , t _ n - s _ n , t _ n + s _ n \\} , \\\\ S _ G & = \\{ t _ 1 - t _ 2 , \\dots , t _ { n - 1 } - t _ n , t _ { n - 1 } + t _ n \\} \\cup \\{ s _ 1 - s _ 2 , \\dots , s _ { n - 1 } - s _ n , s _ { n - 1 } + s _ n \\} . \\end{align*}"}
{"id": "2047.png", "formula": "\\begin{align*} \\left \\| \\mathcal { U } \\left ( t \\right ) x _ 0 \\right \\| ^ 2 & \\leq M _ 2 \\sum ^ \\infty _ { n = 1 } \\bigl | e ^ { \\lambda _ n t } \\left < x _ 0 , z _ n \\right > \\bigr | ^ 2 \\\\ & \\leq M _ 2 e ^ { - 2 \\varepsilon t } \\sum ^ \\infty _ { n = 1 } \\left | \\left < x _ 0 , z _ n \\right > \\right | ^ 2 \\leq \\frac { M _ 2 } { M _ 1 } e ^ { - 2 \\varepsilon t } \\left \\| x _ 0 \\right \\| ^ 2 , \\end{align*}"}
{"id": "3668.png", "formula": "\\begin{align*} r _ { i j } = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & \\frac { 1 } { r } \\cdot \\mathrm { I d } _ { k - 1 } & 0 \\\\ 0 & 0 & h _ { i j } \\end{pmatrix} + o ( 1 ) , \\end{align*}"}
{"id": "4605.png", "formula": "\\begin{align*} \\rho ( d , k ) = \\frac { 1 } { 2 } \\biggl [ { \\lceil \\frac { d } { 2 } \\rceil \\choose k } + { \\lfloor \\frac { d } { 2 } \\rfloor \\choose k } \\biggr ] . \\end{align*}"}
{"id": "5594.png", "formula": "\\begin{align*} ( h \\rhd s ) ^ * = S ( h ) ^ * \\rhd s ^ * \\end{align*}"}
{"id": "2585.png", "formula": "\\begin{align*} f = \\norm { g } _ 2 ^ { - 2 } \\iint _ { \\R ^ { 2 d } } V _ g f ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "7644.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\nabla \\bar { u } \\cdot \\nabla v = \\tilde { \\lambda } _ 0 \\int _ { \\R ^ N } \\tilde { m } _ 0 \\bar { u } v \\ ; . \\end{align*}"}
{"id": "8809.png", "formula": "\\begin{align*} \\partial _ t V _ t = \\frac { 1 } { 2 } \\Delta _ { \\dot { C } _ t } V _ t - \\frac { 1 } { 2 } ( \\nabla V _ t ) _ { \\dot { C } _ t } ^ 2 , \\end{align*}"}
{"id": "4451.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( e _ { \\lambda } ( t ) - 1 \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } , ( k \\ge 0 ) , ( \\mathrm { s e e } \\ [ 4 ] ) . \\end{align*}"}
{"id": "6422.png", "formula": "\\begin{align*} \\big ( D _ E | _ { S ^ + ( T N ) \\otimes E } \\big ) & = \\left < \\hat { A } ( T N ) \\cdot f ^ * \\mathrm { c h } ( S ^ + ( T M ) ) , [ N ] \\right > \\\\ & = \\left < - \\tfrac { 1 } { 1 2 } p _ 1 ( T N ) + \\tfrac 1 4 f ^ * p _ 1 ( T M ) + \\tfrac 1 2 f ^ * e ( T M ) , [ N ] \\right > \\\\ & = - \\tfrac 1 4 \\sigma ( N ) + ( f ) \\left ( \\tfrac 3 4 \\sigma ( M ) + \\tfrac 1 2 \\chi ( M ) \\right ) > 0 , \\end{align*}"}
{"id": "1822.png", "formula": "\\begin{align*} E _ n ( u , v ) = v ^ { n + 1 } E _ n \\left ( { u \\over v ^ 2 } \\right ) . \\end{align*}"}
{"id": "8910.png", "formula": "\\begin{align*} \\check H ^ q _ { c t } ( X , \\mathcal F ) = H ^ q ( 0 \\to \\mathcal F _ 0 ( X ) \\to \\mathcal F _ 1 ( X ) \\to \\cdots ) , \\end{align*}"}
{"id": "4487.png", "formula": "\\begin{align*} E _ { \\psi } ( x ) = \\mathcal { O } \\big ( \\exp \\big ( - c \\sqrt { \\log x } \\big ) \\big ) , \\end{align*}"}
{"id": "8033.png", "formula": "\\begin{align*} \\mathrm { W F } ( K ) = \\left \\{ ( - u , u , v ; \\xi , \\xi , 0 ) \\in \\dot { T } ^ { * } ( \\Sigma _ 0 \\times \\mathbb { M } ^ 2 ) \\right \\} . \\end{align*}"}
{"id": "2633.png", "formula": "\\begin{align*} \\int _ \\mathcal { Q } \\left ( \\int _ \\mathcal { Q } | Z f ( x , \\omega ) | ^ 2 \\ , d \\omega \\right ) \\ , d x = \\sum _ { k \\in \\Z ^ d } \\int _ \\mathcal { Q } | f ( x + k ) | ^ 2 \\ , d x = \\norm { f } _ 2 ^ 2 . \\end{align*}"}
{"id": "2402.png", "formula": "\\begin{align*} D ^ * : \\mathcal { H } \\to \\ell ^ 2 ( \\Gamma ) , D ^ * f = ( \\langle f , e _ \\gamma \\rangle ) _ { \\gamma \\in \\Gamma } \\end{align*}"}
{"id": "4918.png", "formula": "\\begin{align*} X = \\frac { 3 \\sigma { \\log ( 1 / \\delta ) } \\pm \\sqrt { ( 3 \\sigma { \\log ( 1 / \\delta ) } ) ^ 2 + 4 ( \\kappa + { L } ) \\sqrt { \\sum _ { t = 1 } ^ T \\| g _ t \\| ^ 2 } + 1 2 ( G ^ 2 + G \\tilde G ) { \\log ( 1 / \\delta ) } } } { 2 } \\end{align*}"}
{"id": "2448.png", "formula": "\\begin{align*} S = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} , \\end{align*}"}
{"id": "2747.png", "formula": "\\begin{align*} a _ t = \\frac { - R ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , z _ { j - 1 } , t , z _ { j + 1 } , \\dots z _ n ) } { Q ( z _ 1 , \\dots , \\widehat { z _ i } , \\dots , z _ { j - 1 } , t , z _ { j + 1 } , \\dots , z _ n ) } \\end{align*}"}
{"id": "977.png", "formula": "\\begin{align*} \\varphi ^ { ( 1 ) } ( x ) : = \\zeta ( x ) - \\zeta \\big ( Q _ \\rho ( x ) \\big ) x \\in \\R ^ n . \\end{align*}"}
{"id": "1524.png", "formula": "\\begin{align*} P _ { n } ^ t \\times P _ { n } ^ t = \\bigcup _ { \\xi \\in G _ { 2 t + r } } \\tilde { V } _ t ( ( \\xi \\times 1 _ { 2 m - 2 t } ) \\times 1 _ { 2 n } ) = \\bigcup _ { \\xi \\in G _ { 2 t + r } } \\tilde { V } _ t ( 1 _ { 2 n } \\times ( \\xi \\times 1 _ { 2 m - 2 t } ) ) , \\end{align*}"}
{"id": "4788.png", "formula": "\\begin{align*} \\langle \\phi , \\psi \\rangle = \\sum _ { g \\in G } \\phi ( g ) \\psi ( g ) \\end{align*}"}
{"id": "1850.png", "formula": "\\begin{align*} M _ 0 ( x ) = x ^ { - 1 } \\mbox { a n d } W _ 0 ( x ) = 1 . \\end{align*}"}
{"id": "2418.png", "formula": "\\begin{align*} f = S ^ { - 1 } S f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , e _ \\gamma \\rangle S ^ { - 1 } e _ \\gamma . \\end{align*}"}
{"id": "6344.png", "formula": "\\begin{align*} C ( \\rho ) : = \\frac { 1 - ( 1 + \\delta ) ^ { \\rho } } { \\rho } . \\end{align*}"}
{"id": "5726.png", "formula": "\\begin{align*} \\int _ M | \\nabla _ g u ( x ) | ^ 2 { \\rm d } v _ g - \\mu \\int _ M \\frac { u ^ 2 ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \\rm d } v _ g + \\int _ M u ^ 2 ( x ) { \\rm d } v _ g \\leq | \\lambda | C _ 0 \\| \\alpha \\| _ { L ^ \\infty } \\int _ M u ^ 2 ( x ) { \\rm d } v _ g . \\end{align*}"}
{"id": "2696.png", "formula": "\\begin{align*} L = \\begin{pmatrix} \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots \\\\ \\ddots & m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & m _ { - 3 } & \\ddots \\\\ \\ddots & m _ 2 & m _ 1 & m _ 0 & m _ { - 1 } & m _ { - 2 } & \\ddots \\\\ \\ddots & m _ 3 & m _ 2 & m _ 1 & m _ 0 & m _ { - 1 } & \\ddots \\\\ \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots \\end{pmatrix} \\end{align*}"}
{"id": "4550.png", "formula": "\\begin{align*} f _ 2 & \\ge \\frac { h - 1 } { 2 } \\ge 1 + \\frac { h - 1 } { 8 } \\ge 1 + h _ 1 = 1 + e _ 2 \\quad \\\\ f _ 3 & \\ge \\frac { 3 h - 3 } { 4 } \\ge 1 + \\frac { h - 1 } { 8 } + \\frac { h - 7 } { 2 } \\ge 1 + h _ 1 + \\max \\{ h _ 1 , \\dots , h _ 4 \\} \\ge 1 + e _ 2 + e _ 3 . \\end{align*}"}
{"id": "1934.png", "formula": "\\begin{align*} - \\left ( F _ h ( x ) [ \\dot x ] \\right ) [ \\lambda ( x ) - 2 \\beta h ( x ) ] = - \\sum _ { i = 1 } ^ m ( \\lambda _ i ( x ) - 2 \\beta h _ i ( x ) ) \\nabla ^ 2 h _ i ( x ) [ \\dot x ] , \\end{align*}"}
{"id": "2663.png", "formula": "\\begin{align*} P _ R = \\{ z \\in \\C ^ d \\mid | z _ k | \\leq R , \\ , k = 1 , \\ldots d \\} : \\end{align*}"}
{"id": "6120.png", "formula": "\\begin{align*} a _ j = 0 \\ \\textrm { i f } \\ j \\ne q ( m + 2 ) \\ \\textrm { a n d } \\ j \\ne q ( m + 2 ) + 1 \\ \\textrm { w i t h } \\ q \\in \\Z \\ , . \\end{align*}"}
{"id": "7944.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } L _ n ^ { 1 / 2 } n ^ { 5 / 4 } \\ll N ^ { 1 5 / 8 } . \\end{align*}"}
{"id": "4263.png", "formula": "\\begin{align*} { d \\over d b } g _ b ( x ) ~ = ~ { 2 \\eta ( x ) \\over \\pi } [ \\ln ( x + b ) - \\ln ( b ) ] \\forall x > 0 . \\end{align*}"}
{"id": "3563.png", "formula": "\\begin{align*} \\mu _ i \\coloneqq \\frac { \\nu ( \\eta _ i ) } { \\sum _ { j = 1 } ^ K \\nu ( \\eta _ j ) } , i = 1 , \\dots , K . \\end{align*}"}
{"id": "859.png", "formula": "\\begin{align*} T ^ { \\rm P r o a c } _ j = \\tau ^ { \\rm P r o a c } _ { m } R _ j + \\tau ^ { \\rm P r o a c } _ { V _ j } + \\tau _ { \\rm f } . \\end{align*}"}
{"id": "3006.png", "formula": "\\begin{align*} t _ 1 ^ 5 + t _ 2 ^ 5 + t _ 3 ^ 5 + t _ 4 ^ 5 + t _ 5 ^ 5 = 5 \\psi t _ 1 t _ 2 t _ 3 t _ 4 t _ 5 . \\end{align*}"}
{"id": "7746.png", "formula": "\\begin{align*} - \\phi _ { t , u } ( u , v ) = - \\phi _ { t , u } ( u , u - 2 x _ 0 ) + \\int _ { u - 2 x _ 0 } ^ v G ( u , v _ 0 ) d v _ 0 , \\end{align*}"}
{"id": "9204.png", "formula": "\\begin{align*} k _ 0 : = \\max k \\left [ \\frac { k } { 2 ^ { n + 1 } } \\leq r \\right ] . \\end{align*}"}
{"id": "2508.png", "formula": "\\begin{align*} \\rho ' _ \\kappa ( x , \\omega , \\tau ) = \\rho _ \\kappa ( \\iota ^ { - 1 } ( x , \\omega , \\tau ) ) = e ^ { 2 \\pi i \\kappa \\tau } T _ { \\kappa x } M _ \\omega , \\end{align*}"}
{"id": "9538.png", "formula": "\\begin{align*} x \\in \\N , \\ \\sum _ { t = 0 } ^ { T - 1 } x _ t \\cdot \\Delta s _ { t + 1 } \\ge 0 \\ a . s . \\implies \\sum _ { t = 0 } ^ { T - 1 } x _ t \\cdot \\Delta s _ { t + 1 } = 0 \\ a . s . ; \\end{align*}"}
{"id": "3403.png", "formula": "\\begin{align*} \\begin{aligned} | f ( x ) - f ( x ' ) | & \\leqslant C \\Big ( \\frac { \\| x - x ' \\| } { r } \\Big ) ^ \\beta \\Big \\{ \\frac { 1 } { V ( x , x _ 0 , r + d ( x , x _ 0 ) ) } \\Big ( \\frac { r } { r + \\boldsymbol { d ( x , x _ 0 ) } } \\Big ) ^ \\gamma \\\\ & \\qquad + \\frac { 1 } { V ( x ' , x _ 0 , r + d ( x ' , x _ 0 ) ) } \\Big ( \\frac { r } { r + \\boldsymbol { d ( x ' , x _ 0 ) } } \\Big ) ^ \\gamma \\Big \\} ; \\end{aligned} \\end{align*}"}
{"id": "4388.png", "formula": "\\begin{align*} H '' + \\frac { d + 1 } { \\xi } H ' - 3 ( d - 2 ) ( 2 Q + \\xi ^ 2 Q ^ 2 ) H = - \\Lambda _ \\xi Q : = T ( \\xi ) . \\end{align*}"}
{"id": "7116.png", "formula": "\\begin{align*} x _ t = x _ 0 + \\int ^ t _ 0 b ( s , x _ s ) d s + \\sigma B ^ H _ t , 0 \\le t \\le 1 , H \\in ( 0 , \\frac { 1 } { 2 } ) , \\sigma > 0 . \\end{align*}"}
{"id": "1346.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } U _ i ( \\tau _ x \\omega ) \\partial _ j \\eta ( x ) d x = \\int _ { \\mathbb { R } ^ d } U _ j ( \\tau _ x \\omega ) \\partial _ i \\eta ( x ) d x \\end{align*}"}
{"id": "7126.png", "formula": "\\begin{align*} \\tilde { f } ' ( s , y ) \\le f _ 1 ( s , y ) \\le K \\le s ^ { 2 H - 1 } K = K _ s \\end{align*}"}
{"id": "662.png", "formula": "\\begin{align*} E ( x ) \\ = \\ \\frac { 1 } { f ( x ) + 1 } . \\end{align*}"}
{"id": "518.png", "formula": "\\begin{align*} \\mathbf { r } _ { ( \\alpha ) } ( \\tau ) = \\tau ^ { - 1 } \\cdot \\mathbf { s } \\cdot \\mathbf { s } ^ { \\mathtt { T } } + \\mathbf { r } _ { 0 } ( 0 ) _ { \\kappa \\left ( \\mathcal { I } \\setminus \\{ \\alpha \\} \\right ) } ^ { \\kappa \\left ( \\mathcal { I } \\setminus \\{ \\alpha \\} \\right ) } + ( \\mathbf { 1 } \\cdot \\mathbf { s } ^ { \\mathtt { T } } - \\mathbf { s } \\cdot \\mathbf { 1 } ^ { \\mathtt { T } } ) . \\end{align*}"}
{"id": "6621.png", "formula": "\\begin{align*} J _ 2 : = \\sum _ { \\substack { \\alpha \\in A \\\\ \\beta \\in B } } \\frac { 1 } { 4 \\pi i } \\int _ { ( \\epsilon ) } \\underset { s _ 2 = - s _ 1 - \\alpha - \\beta } { } \\ \\mathcal { J } \\ , d s _ 1 . \\end{align*}"}
{"id": "994.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ \\varepsilon + k u _ \\varepsilon & = f _ \\varepsilon B _ { 7 / 8 } . \\end{align*}"}
{"id": "4641.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 9 & 6 & \\overline { 4 } & 2 & 1 \\\\ \\overline { 1 3 } & 8 & \\overline { 7 } & 3 & \\overline { 1 } \\\\ 1 1 & \\overline { 7 } & 5 & \\overline { 2 } & 0 \\\\ \\end{pmatrix} . \\end{align*}"}
{"id": "1514.png", "formula": "\\begin{align*} \\mathbf { E } _ l ^ { \\ast } ( g , s ) = j ( k _ { \\infty } , z _ 0 ) ^ { - l } \\sum _ { h } c ( h , q , s ) e _ { \\mathbb { A } } ( \\lambda ( h \\sigma ) ) , \\end{align*}"}
{"id": "7628.png", "formula": "\\begin{align*} F ^ { i j } \\nabla _ { i } \\log u \\nabla _ { j } P = \\lambda F ^ { i j } r _ { j } \\nabla _ { i } \\log u - u ^ { \\frac { \\alpha + 1 } { \\alpha } } F ^ { i j } \\nabla _ { i } \\log u \\nabla _ { j } \\log u . \\end{align*}"}
{"id": "5000.png", "formula": "\\begin{align*} \\zeta ^ { \\bar { s } } = \\xi ^ { b _ { k } } \\circ \\eta ^ { a _ { k } } \\circ \\dots \\circ \\xi ^ { b _ { 1 } } \\circ \\eta ^ { a _ { 1 } } . \\end{align*}"}
{"id": "2067.png", "formula": "\\begin{align*} g _ { \\alpha } ^ { ( \\beta ) } ( o ) = \\int _ D g _ { \\alpha } \\overline { { g } _ { \\beta } } e ^ { - \\varphi } = \\overline { \\int _ D g _ { \\beta } \\overline { { g } _ { \\alpha } } e ^ { - \\varphi } } = 0 . \\end{align*}"}
{"id": "4826.png", "formula": "\\begin{align*} \\int \\psi ( x ) e ^ { - i x \\cdot \\rho \\eta _ 0 / h \\Tilde { h } } \\widetilde { R } _ h ( z ) ^ * ( \\chi ( x ) e ^ { i x \\cdot \\rho \\xi _ 0 / h \\Tilde { h } } ) d x = \\mathcal { O } ( h ^ \\infty \\Tilde { h } ^ \\infty ) , \\end{align*}"}
{"id": "441.png", "formula": "\\begin{align*} A ^ { 0 } ( U ) V _ { t } + A ^ { i } ( U ) \\partial _ { i } V + D ( U ) V = 0 \\end{align*}"}
{"id": "5626.png", "formula": "\\begin{align*} \\mathcal { Q } ^ \\bullet _ { M _ c } = \\bigoplus _ { p = 0 } ^ { n - 1 } \\biguplus _ { q = 0 } ^ \\infty \\biguplus _ { r = 0 } ^ \\infty \\mathcal { Q } ^ { p q r } _ { M _ c } \\end{align*}"}
{"id": "6699.png", "formula": "\\begin{align*} & \\Bigl ( P _ { { \\bf b } , d } ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d - 1 } } \\Bigr ) ^ { ( - 1 ) } \\\\ & = \\prod _ { \\substack i = 1 } ^ { s + 1 } ( \\theta - t ) ^ { q ^ { d - 1 } } \\mathbb { D } _ { a _ i - 2 } ^ { q ^ { d - a _ i } } \\alpha ^ { q ^ { d - 2 } } P _ { { \\bf b } , d } + \\prod _ { j = 1 } ^ { s } ( \\theta - t ) ^ { q ^ { d - 1 } } \\mathbb { D } _ { b _ j - 2 } ^ { q ^ { d - b _ j } } P _ { { \\bf b } , d } ~ _ { s + 1 } \\mathcal { F } _ s ( { \\bf a } ; { \\bf b } ) ( \\alpha ) ^ { q ^ { d - 1 } } . \\end{align*}"}
{"id": "6709.png", "formula": "\\begin{align*} \\Bigl ( ( \\theta ^ { q ^ N } - t ) ^ { c ( N - n _ 1 ) q ^ { n _ 1 + d - N } } \\bigl ( ~ _ r \\mathcal { F } _ s ( \\alpha ) \\bigr ) ^ { q ^ d } \\Bigr ) | _ { t = \\theta ^ { q ^ N } } = \\sum _ { i = 1 } ^ { l } \\prod _ { \\substack { m = 1 \\\\ m \\neq N } } ^ { n _ i + d - 1 } ( \\theta ^ { q ^ m } - \\theta ^ { q ^ N } ) ^ { c ( m - n _ i ) q ^ { n _ i + d - m } } \\alpha ^ { q ^ { n _ i + d } } \\end{align*}"}
{"id": "695.png", "formula": "\\begin{align*} \\theta _ \\mu ( t + 1 ) = \\theta _ \\mu ( t ) - \\eta \\partial _ \\mu \\mathcal L ( \\theta ( t ) ) , \\mu \\in [ \\# ] \\end{align*}"}
{"id": "1692.png", "formula": "\\begin{align*} & { _ { s + 1 } } F _ s \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - n ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; 1 \\right ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } \\overline { H } _ k ( s ) , \\\\ & { _ { s + 1 } } F _ s \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - n ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; - 1 \\right ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } \\overline { H } _ k ( \\overline { s } ) , \\end{align*}"}
{"id": "4650.png", "formula": "\\begin{align*} \\zeta _ k ( s ) = \\prod _ { \\pi } L ( \\C [ G ] ( \\pi ) , s ) = \\prod _ { \\pi } L ( V _ { \\pi } , s ) ^ { \\dim V _ { \\pi } } \\end{align*}"}
{"id": "3638.png", "formula": "\\begin{align*} h ' ( t ) = \\frac { A _ 1 \\exp ( - C u ( t ) ) } { \\log ^ \\alpha t } \\left ( \\frac { \\log t - \\alpha } { \\log t } - C t u ' ( t ) \\right ) \\ge A _ 1 \\log ^ { B - 2 } t \\exp ( - C u ( t ) ) \\end{align*}"}
{"id": "1525.png", "formula": "\\begin{align*} P ^ N \\tilde { \\tau } _ t ( G _ { n } \\times G _ { n } ) = \\bigcup _ { \\xi , \\beta , \\gamma } P ^ N \\tilde { \\tau } _ t ( ( \\xi \\times 1 _ { 2 m - 2 t } ) \\beta \\times \\gamma ) = \\bigcup _ { \\xi , \\beta , \\gamma } P ^ N \\tilde { \\tau } _ t ( \\beta \\times ( \\xi \\times 1 _ { 2 m - 2 t } ) \\gamma ) . \\end{align*}"}
{"id": "9354.png", "formula": "\\begin{align*} H _ { n , \\lambda } ^ { ( 1 ) } = H _ { n , \\lambda } , H _ { n , \\lambda } ^ { ( r ) } = \\sum _ { k = 1 } ^ { n } H _ { k , \\lambda } ^ { ( r - 1 ) } , ( r \\ge 2 ) . \\end{align*}"}
{"id": "8212.png", "formula": "\\begin{align*} P _ 1 = \\frac { x _ 1 A } { F _ { k , 1 } ( A ) } \\end{align*}"}
{"id": "2261.png", "formula": "\\begin{align*} x = ( x _ 1 , \\ \\ldots , \\ x _ d ) . \\end{align*}"}
{"id": "4837.png", "formula": "\\begin{align*} \\min _ \\R Q = Q ( 0 ) = 0 . \\end{align*}"}
{"id": "1762.png", "formula": "\\begin{align*} C ^ k ( A ) \\colon = \\mathrm { H o m } _ { \\mathbb { C } } \\big ( A ^ { \\otimes ( k + 1 ) } , \\mathbb { C } \\big ) \\end{align*}"}
{"id": "1626.png", "formula": "\\begin{align*} \\mathcal { L } f ( g ) : = - \\mathcal { E } ( f , g ) , g \\in \\mathcal { D } ( \\mathcal { E } ) . \\end{align*}"}
{"id": "4335.png", "formula": "\\begin{align*} \\tau = ( 2 \\beta _ \\infty ) ^ { - 1 } \\tilde \\tau ( 1 + O ( I ^ \\eta ( \\tilde \\tau ) ) ) , \\tilde \\tau \\to + \\infty . \\end{align*}"}
{"id": "7100.png", "formula": "\\begin{align*} \\displaystyle \\lambda ^ { 2 } _ 1 \\leq & \\left ( \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { m } r ^ 2 _ i ( r ^ 2 _ i - 1 ) - \\frac { 1 5 } { 8 } m _ 2 - \\frac { 3 } { 4 } m _ 3 - \\frac { 1 } { 2 } m _ 4 \\right ) \\left ( \\frac { n - 1 } { n } \\right ) \\\\ \\displaystyle \\lambda _ 1 \\leq & \\sqrt { \\left ( \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ { m } r ^ 2 _ i ( r ^ 2 _ i - 1 ) - \\frac { 1 5 } { 8 } m _ 2 - \\frac { 3 } { 4 } m _ 3 - \\frac { 1 } { 2 } m _ 4 \\right ) \\left ( \\frac { n - 1 } { n } \\right ) } \\end{align*}"}
{"id": "6893.png", "formula": "\\begin{align*} b _ i = { k + i \\choose k } _ q \\left ( N _ k ^ i ( V , V ' ) - \\frac { 1 } { { k + i \\choose k } _ q } I ( V , V ' ) \\right ) , \\end{align*}"}
{"id": "8442.png", "formula": "\\begin{align*} \\nabla \\phi ( X _ { t } ^ { x ; l ^ \\epsilon } ) \\nabla _ { x } X _ { t } ^ { x ; l ^ \\epsilon } = D _ { s } \\phi ( X _ { t } ^ { x ; l ^ \\epsilon } ) \\nabla _ { x } X _ { \\gamma ^ { \\epsilon } _ { s } } ^ { x ; l ^ \\epsilon } . \\end{align*}"}
{"id": "3915.png", "formula": "\\begin{align*} f ( \\eta ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left . f ^ { k } ( \\eta ) \\right | _ { \\eta = \\eta _ i } } { k ! } ( \\eta - \\eta _ i ) ^ k = \\sum _ { k = 0 } ^ { \\infty } F ( k ) ( \\eta - \\eta _ i ) ^ k \\end{align*}"}
{"id": "4905.png", "formula": "\\begin{align*} \\sqrt { \\sum _ { i = 1 } ^ { n } a _ i } \\leq \\sum _ { i = 1 } ^ { n } \\frac { a _ i } { \\sqrt { \\sum _ { k = 1 } ^ { i } a _ k } } \\leq 2 \\sqrt { \\sum _ { i = 1 } ^ { n } a _ i } \\end{align*}"}
{"id": "6519.png", "formula": "\\begin{align*} M ^ { ( 4 ) } _ n \\sim H _ n ^ { ( 4 ) } \\sim \\dfrac { ( 1 - 4 \\alpha ) c _ 4 } { 2 ( 1 - 2 \\alpha ) - 1 } \\cdot n ^ { - 1 } = c _ 4 n ^ { - 1 } . \\end{align*}"}
{"id": "5561.png", "formula": "\\begin{align*} \\epsilon = \\frac { 1 } { z _ 6 } \\| e ^ { - 2 z _ 6 \\cdot } - e ^ { - z _ 7 \\cdot } \\| . \\end{align*}"}
{"id": "4768.png", "formula": "\\begin{align*} \\langle ( R \\# T ) ( x \\wedge y ) , u \\wedge v \\rangle = \\langle ( T \\# R ) ( u \\wedge v ) , x \\wedge y \\rangle = \\langle ( R \\# T ) ( u \\wedge v ) , x \\wedge y \\rangle . \\end{align*}"}
{"id": "8707.png", "formula": "\\begin{align*} \\begin{aligned} U ^ 0 ( x ) \\preceq \\cdots \\preceq U ^ n ( x ) = F ( x ) & V ^ 0 ( x ) \\preceq \\cdots \\preceq V ^ n ( x ) = G ( x ) , \\\\ U ^ 0 ( x ) = A ^ 0 ( x ) \\preceq \\cdots \\preceq A ^ n ( x ) & V ^ 0 ( x ) = B ^ 0 ( x ) \\preceq \\cdots \\preceq B ^ n ( x ) , \\\\ U ^ i ( x ) \\preceq A ^ i ( x ) & U ^ i ( x ) \\preceq B ^ i ( x ) i = 0 , \\ldots , n . \\end{aligned} \\end{align*}"}
{"id": "5684.png", "formula": "\\begin{align*} \\partial _ { J } \\langle \\alpha \\rangle = \\sum _ { \\beta : \\mathrm { a d m i s s i b l e \\ , \\ , o r b i t \\ , \\ , s e t \\ , \\ , w i t h \\ , \\ , } [ \\beta ] = \\Gamma } \\# ( \\mathcal { M } _ { 1 } ^ { J } ( \\alpha , \\beta ) / \\mathbb { R } ) \\cdot \\langle \\beta \\rangle . \\end{align*}"}
{"id": "286.png", "formula": "\\begin{align*} \\| u ( \\cdot , t ) - \\chi ( \\cdot , t ) \\| _ { L ^ { \\infty } } \\sim \\begin{cases} ( 1 + t ) ^ { - \\frac { \\alpha } { 2 } } , & 1 < \\alpha < 2 , \\\\ ( 1 + t ) ^ { - 1 } \\log ( 1 + t ) , & \\alpha \\ge 2 \\end{cases} \\end{align*}"}
{"id": "9054.png", "formula": "\\begin{align*} \\begin{aligned} & ( h + 2 \\beta _ a \\epsilon ( a ) ) \\phi _ 0 + ( h - 2 \\beta _ a \\epsilon ( a ) ) \\phi _ 1 - 2 h \\phi ^ b ( a ) = 0 , \\\\ & - d _ h ( \\epsilon D _ h \\phi ) _ j - f _ j - \\sum _ { i = 1 } ^ s z _ i \\rho _ { i j } = 0 , j = 1 , \\cdots , N , \\\\ & ( h - 2 \\beta _ b \\epsilon ( b ) ) \\phi _ N + ( h + 2 \\beta _ b \\epsilon ( b ) ) \\phi _ { N + 1 } - 2 h \\phi ^ b ( b ) = 0 . \\end{aligned} \\end{align*}"}
{"id": "7523.png", "formula": "\\begin{align*} \\log \\left ( \\frac { \\sigma + i T } { 2 } \\right ) = \\frac { 1 } { 2 } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) + i \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) \\end{align*}"}
{"id": "5539.png", "formula": "\\begin{align*} C : = \\phi _ { \\delta } ^ { x _ 0 } ( T ) \\eta _ 0 : = \\frac { \\delta } { 4 C } . \\end{align*}"}
{"id": "3974.png", "formula": "\\begin{align*} \\vartheta ( x ) = \\begin{cases} \\eta ( x ) , & \\ \\ x \\in ( 0 , 1 ) , \\\\ 0 , & \\ \\ x \\in \\mathbb { R } \\setminus ( 0 , 1 ) . \\end{cases} \\end{align*}"}
{"id": "4601.png", "formula": "\\begin{align*} \\rho ( d , k ) = \\frac { 1 } { 2 } \\biggl [ { \\lceil \\frac { d } { 2 } \\rceil \\choose k } + { \\lfloor \\frac { d } { 2 } \\rfloor \\choose k } \\biggr ] . \\end{align*}"}
{"id": "6456.png", "formula": "\\begin{align*} T _ h ( \\lambda ) ( u ) ( { x } ) = \\sum _ { k = 1 } ^ { k = n } \\frac { \\eta _ 0 ^ k } { 4 \\pi } \\int _ { B _ k } \\frac { \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } } { | x - y | } u ( y ) d y \\end{align*}"}
{"id": "4879.png", "formula": "\\begin{align*} F _ 4 ( z ) = H _ n \\left ( 1 - \\frac 2 { z } \\right ) , \\end{align*}"}
{"id": "2311.png", "formula": "\\begin{align*} | A f ( x , \\omega ) | = | \\langle f , M _ \\omega T _ x f \\rangle | \\leq \\norm { f } _ 2 ^ 2 \\end{align*}"}
{"id": "5971.png", "formula": "\\begin{align*} f _ r = \\frac { \\omega _ r } { 2 \\pi } , \\omega _ r = \\sqrt { g k _ { r } \\tanh \\left ( k _ { r } h \\right ) } , k _ r = \\frac { 2 \\pi } { L _ r } , L _ r = \\alpha ~ \\Delta x _ { \\max } , \\end{align*}"}
{"id": "630.png", "formula": "\\begin{align*} A ( x , n ) \\ = \\ A ( n ) \\end{align*}"}
{"id": "1528.png", "formula": "\\begin{align*} \\tilde { \\tau } _ m ( \\xi \\times 1 ) \\tilde { \\tau } _ m ^ { - 1 } = \\left [ \\begin{array} { c c c c c c } & & & & & \\\\ & & & & & \\\\ & & & & & \\\\ - h & - l & - d + 1 & d & \\frac { l \\zeta } { 2 } & 0 \\\\ - \\zeta ^ { - 1 } g & - \\zeta ^ { - 1 } ( e - 1 ) & - \\zeta ^ { - 1 } f & \\zeta ^ { - 1 } f & \\frac { e + 1 } { 2 } & 0 \\\\ a - 1 & b & c & - c & \\frac { - b \\zeta } { 2 } & 1 \\end{array} \\right ] . \\end{align*}"}
{"id": "6695.png", "formula": "\\begin{align*} _ { s + 1 } \\mathcal { F } _ s ( 1 , \\ldots , 1 ; 2 , \\ldots , 2 ) ( \\alpha ) ^ q = \\mathcal { L i } _ { K , s } ( \\alpha ) ( \\alpha \\in \\overline { k } \\ \\ | \\alpha | _ { \\infty } < q ^ s ) . \\end{align*}"}
{"id": "8091.png", "formula": "\\begin{align*} \\Psi ^ i _ { \\Sigma _ 0 } ( \\Lambda s ) = \\Lambda ^ { \\mu - 1 } \\mathfrak { A } _ { \\ell } m _ { \\Lambda } ( \\Psi ^ i _ { \\Sigma _ 0 } ( s ) ) , \\\\ \\Psi ^ i _ { \\Sigma _ 0 } ( s + c ) = \\mathfrak { A } _ { \\ell } t _ c ( \\Psi ^ i _ { \\Sigma _ 0 } ( s ) ) . \\end{align*}"}
{"id": "1375.png", "formula": "\\begin{align*} E ( x ) = \\alpha + \\int _ 0 ^ x \\left ( n ( y ) - b ( y ) \\right ) d y . \\end{align*}"}
{"id": "223.png", "formula": "\\begin{align*} D ^ { \\alpha - 1 } ( p _ \\alpha ) ( x ) = - x p _ \\alpha ( x ) . \\end{align*}"}
{"id": "7639.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ { n } = \\inf _ { B ^ 1 ( \\mathbf { x } ) \\subset \\tilde { \\Omega } _ n } \\lambda ^ 1 ( B ^ 1 ( \\mathbf { x } ) , \\tilde { \\Omega } _ n ) \\ ; , \\end{align*}"}
{"id": "2775.png", "formula": "\\begin{align*} \\partial _ t h + \\mathcal { L } h = i R ( h ) , \\mathcal { L } = \\left ( \\begin{matrix} 0 & - L _ - \\\\ L _ + & 0 \\end{matrix} \\right ) , \\end{align*}"}
{"id": "3937.png", "formula": "\\begin{align*} \\Psi _ \\tau = \\Phi _ 0 + \\lambda ( \\tau ) ( \\Phi _ 1 - \\Phi _ 0 ) , \\end{align*}"}
{"id": "8257.png", "formula": "\\begin{align*} \\ell ( w ' ) - \\ell ( w ) \\geq \\ , & \\sum _ { i = 1 } ^ k \\left ( \\ell ( u '^ { ( i ) } ) - \\ell ( u ^ { ( i ) } ) \\right ) + 2 ( p _ 1 + \\cdots + p _ { j - 1 } ) ( | L ' _ j | - | L _ j | ) \\\\ \\geq \\ , & \\ell ( u '^ { ( j ) } ) - \\ell ( u ^ { ( j ) } ) + 2 ( p _ 1 + \\cdots + p _ { j - 1 } ) \\\\ \\geq \\ , & 3 . \\end{align*}"}
{"id": "7981.png", "formula": "\\begin{align*} \\partial _ i \\Phi ^ u ( x ) & = ( \\partial _ i + u _ i ( x ) \\partial _ z ) | _ { ( x , u ( x ) ) } , \\ \\ \\ \\forall 1 \\leq i \\leq n \\ , ; \\\\ \\nu ^ { u , f } ( x , u ( x ) ) & = \\frac { 1 } { \\sqrt { ( 1 + f ) ( 1 + g _ u ^ { i j } u _ i u _ j ) } } \\cdot ( \\partial _ z - g _ u ^ i j u _ i ( x ) \\partial _ j ) \\ , . \\end{align*}"}
{"id": "7295.png", "formula": "\\begin{align*} \\alpha & = \\beta \\gamma \\Bigl ( 1 + 8 \\sum _ { k = 1 } ^ \\infty \\Bigl ( \\frac { 2 k X ^ { 2 k } } { 1 + X ^ { 2 k } } - \\frac { ( 2 k - 1 ) X ^ { 2 k - 1 } } { 1 + X ^ { 2 k - 1 } } \\Bigr ) \\Bigr ) . \\end{align*}"}
{"id": "2564.png", "formula": "\\begin{align*} C ^ \\infty ( \\Omega ) = \\bigcap _ { k = 0 } ^ \\infty C ^ k ( \\Omega ) . \\end{align*}"}
{"id": "9039.png", "formula": "\\begin{align*} G _ \\tau ( \\rho , \\phi ) = \\frac { 1 } { 2 \\tau } d ^ 2 ( \\rho ^ * , \\rho ) + E ( \\rho , \\phi ) , ( \\rho , \\phi ) \\in \\mathcal { A } . \\end{align*}"}
{"id": "2399.png", "formula": "\\begin{align*} D : \\ell ^ 2 ( \\Gamma ) \\to \\mathcal { H } , D c = \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma e _ \\gamma \\in \\mathcal { H } . \\end{align*}"}
{"id": "197.png", "formula": "\\begin{align*} X e ^ { - t } - \\frac { e ^ { - \\alpha t } Y } { ( 1 - e ^ { - \\alpha t } ) ^ { 1 - \\frac { 1 } { \\alpha } } } = _ { \\mathcal { L } } \\left ( e ^ { - \\alpha t } \\dfrac { ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } + e ^ { - \\alpha t } } { ( 1 - e ^ { - \\alpha t } ) ^ { \\alpha - 1 } } \\right ) ^ { \\frac { 1 } { \\alpha } } X . \\end{align*}"}
{"id": "5057.png", "formula": "\\begin{align*} { \\bf S } _ \\ell ( Q + i p ) \\sum _ { j = 0 } ^ { \\ell } F _ j : = \\sum _ { j = 0 } ^ { \\ell } F _ j ^ + ( Q + i p ) \\end{align*}"}
{"id": "4440.png", "formula": "\\begin{align*} F ( q ) ` ` = \" - \\frac { 1 } { 2 } \\sum _ { n = 1 } ^ { \\infty } n \\Bigl ( \\frac { 1 2 } { n } \\Bigr ) q ^ { \\frac { n ^ 2 - 1 } { 2 4 } } \\end{align*}"}
{"id": "3828.png", "formula": "\\begin{align*} E ^ { [ 1 ] } ( f , G \\rtimes S ) ( t , \\overline { t } ) = ( - 1 ) ^ n E ^ { [ 1 ] } ( \\widetilde { f } , \\widetilde { G } \\rtimes S ) ( t ^ { - 1 } , \\overline { t } ) \\ , . \\end{align*}"}
{"id": "5161.png", "formula": "\\begin{align*} \\vartheta _ 1 ( z , \\tau ) = \\sum _ { k \\in \\Z } ( - 1 ) ^ { ( k - 1 / 2 ) } e ^ { \\pi i ( k + 1 / 2 ) ^ 2 \\tau } e ^ { ( 2 k + 1 ) \\pi i z } , \\end{align*}"}
{"id": "6933.png", "formula": "\\begin{align*} a _ h ( u ^ { \\cal N \\ ! N } , v _ h ) = F _ h ( v _ h ) \\forall v _ h \\in V _ h \\ , . \\end{align*}"}
{"id": "3130.png", "formula": "\\begin{align*} F _ i = a _ i x _ 1 ^ 2 + b _ i x _ 2 ^ 2 + c _ i x _ 3 ^ 2 + \\psi _ i ( x _ 4 , x _ 5 ) = 0 \\ , . \\end{align*}"}
{"id": "3007.png", "formula": "\\begin{align*} ( \\mathbb { Z } _ 5 ) ^ 4 = \\{ ( \\zeta _ 1 , \\zeta _ 2 , \\zeta _ 3 , \\zeta _ 4 , \\zeta _ 5 ) \\in ( \\mathbb { Z } _ 5 ) ^ 5 \\ | \\ \\prod _ { j = 1 } ^ 5 \\zeta _ j = 1 \\} . \\end{align*}"}
{"id": "9130.png", "formula": "\\begin{align*} \\vert \\norm { a } - \\norm { T ^ \\circ p } \\vert & \\leq \\vert \\norm { a } - \\norm { T ^ \\circ q } \\vert + \\vert \\norm { T ^ \\circ p } - \\norm { T ^ \\circ q } \\vert \\\\ & \\leq \\frac { 1 } { \\delta ( k ) + 1 } + \\norm { T ^ \\circ p - T ^ \\circ q } \\\\ & \\leq \\frac { 1 } { 2 ( k + 1 ) } + \\frac { 1 } { 2 ( k + 1 ) } \\\\ & = \\frac { 1 } { k + 1 } \\end{align*}"}
{"id": "9005.png", "formula": "\\begin{align*} p _ d & \\triangleq N ( p ' _ 0 + p _ 0 ) \\\\ D & = p _ d + \\frac { N } { N - 1 } ( 1 - p _ d ) , \\end{align*}"}
{"id": "4186.png", "formula": "\\begin{align*} \\rho \\mu ( u ) = \\rho \\mu ( v _ i \\# u ) = \\omega ( v _ i \\# u ) = \\omega ( v _ i ) + \\omega ( u ) . \\end{align*}"}
{"id": "8547.png", "formula": "\\begin{align*} \\partial _ { t } ^ 2 u - \\partial _ x ^ 2 u + u + V ( x ) u \\pm u ^ 3 = 0 , \\ , u ( 0 ) = u _ 0 , \\ , \\partial _ t u ( 0 ) = u _ 1 . \\end{align*}"}
{"id": "860.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } T ^ { \\rm P r o a c } _ j = \\tau ^ { \\rm P r o a c } _ { m } \\mathbb { E } R _ j + \\mathbb { E } \\tau ^ { \\rm P r o a c } _ { V _ j } + \\tau _ { \\rm f } , \\\\ & \\mathbb { E } \\left ( T ^ { \\rm P r o a c } _ j \\right ) ^ 2 = \\mathbb { E } \\left ( \\tau ^ { \\rm P r o a c } _ { m } R _ j + \\tau ^ { \\rm P r o a c } _ { V _ j } + \\tau _ { \\rm f } \\right ) ^ 2 . \\\\ \\end{aligned} \\end{align*}"}
{"id": "3951.png", "formula": "\\begin{align*} \\norm { \\psi } _ { { H } ^ { - s } _ { { p e r } } ( I ) } = \\left ( \\sum _ { m \\in \\mathbb Z } \\left ( 1 + \\frac { 4 \\pi ^ 2 m ^ 2 } { \\mod { I } ^ 2 } \\right ) ^ { - s } \\mod { c _ m } ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "7670.png", "formula": "\\begin{align*} \\nabla v _ { \\varepsilon } ( \\mathbf { x } ) = \\sigma _ 3 \\sqrt { \\tilde { \\lambda } _ 0 \\underline { m } } \\frac { \\mathbf { x } - \\mathbf { y } _ { \\varepsilon } } { | \\mathbf { x } - \\mathbf { y } _ { \\varepsilon } | } \\end{align*}"}
{"id": "1854.png", "formula": "\\begin{align*} M _ { n + 1 } ( x , y ) = \\sum _ { k = 0 } ^ n { n \\choose k } M _ { k } ( x , y ) M _ { n - k } ( x , y ) . \\end{align*}"}
{"id": "6992.png", "formula": "\\begin{align*} \\Sigma _ X = \\Phi \\Delta \\Phi ' + \\Phi _ 0 \\Delta _ 0 \\Phi _ 0 ' . \\end{align*}"}
{"id": "8769.png", "formula": "\\begin{align*} s _ { i j } : = \\min \\bigl \\{ f _ i ( x ) , m _ { i j t _ i ( H ) } \\bigr \\} i \\in \\{ 1 , \\ldots , d \\} \\ ; j \\in \\{ 0 , \\ldots , n \\} . \\end{align*}"}
{"id": "8795.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ i \\in \\Lambda _ i , \\ \\delta _ { i } \\in \\{ 0 , 1 \\} ^ { \\lceil \\log _ 2 ( n ) \\rceil } , \\ & \\sum _ { j \\notin L _ { i k } } \\lambda _ { i j } \\leq \\delta _ { i k } \\leq 1 - \\sum _ { j \\notin R _ { i k } } \\lambda _ { i j } , \\ \\\\ & k = 1 , \\ldots , \\lceil \\log _ 2 ( n ) \\rceil , \\end{aligned} \\end{align*}"}
{"id": "1237.png", "formula": "\\begin{align*} \\sum _ { s , t = 1 } ^ { Q } \\mu ( L _ { B , s } \\cap L _ { B , t } ) \\leq \\frac { C } { \\mu ( B ) } \\left ( \\sum _ { n = 1 } ^ { Q } \\mu ( L _ { B , n } ) \\right ) ^ 2 . \\end{align*}"}
{"id": "2555.png", "formula": "\\begin{align*} \\widehat { V } _ Q f ( x ) = \\mu ( V _ Q ) f ( x ) = e ^ { \\pi i Q x ^ 2 } f ( x ) , Q = Q ^ T . \\end{align*}"}
{"id": "3088.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 2 y _ 2 + x _ 1 ^ 2 x _ 2 y _ 2 ^ 3 + x _ 1 x _ 2 ^ 2 y _ 1 ^ 3 - x _ 2 ^ 3 y _ 1 y _ 2 ^ 2 = 0 \\end{align*}"}
{"id": "1520.png", "formula": "\\begin{align*} g _ 1 \\mapsto g _ 1 \\left [ \\begin{array} { c c c c c } 0 & 0 & 0 & 1 _ t & 0 \\\\ 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 \\\\ - 1 _ t & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 \\end{array} \\right ] x g _ 1 = \\left [ \\begin{array} { c c c c c } 0 & 0 & 0 & - 1 _ t & 0 \\\\ 0 & 0 & 0 & 0 & u \\\\ 0 & 0 & 1 & 0 & 0 \\\\ 1 _ t & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & v \\end{array} \\right ] . \\end{align*}"}
{"id": "7169.png", "formula": "\\begin{align*} \\prod _ { \\mu = 0 } ^ \\infty c ^ { \\chi ^ { - \\mu } } \\chi ^ { \\mu \\chi ^ { - \\mu } } \\leq c ( d , \\chi ) , \\end{align*}"}
{"id": "8467.png", "formula": "\\begin{align*} \\mathrm { e x } ( n , F ) = \\left ( 1 - \\frac { 1 } { \\chi ( F ) - 1 } + o ( 1 ) \\right ) \\frac { n ^ 2 } { 2 } , \\end{align*}"}
{"id": "9256.png", "formula": "\\begin{align*} \\forall x ^ X , { x ' } ^ X , \\gamma ^ 1 , { \\gamma ' } ^ 1 \\left ( \\gamma > _ \\mathbb { R } 0 \\land x \\in \\mathrm { d o m } ( J ^ A _ \\gamma ) \\land \\gamma = _ \\mathbb { R } \\gamma ' \\land x = _ X x ' \\rightarrow J ^ A _ \\gamma x = _ X J ^ A _ { \\gamma ' } x ' \\right ) . \\end{align*}"}
{"id": "9438.png", "formula": "\\begin{align*} ( d _ q v ( \\cdot , t ) , d _ q h ) _ { L ^ 2 ( { \\mathbb R } ^ n , \\varLambda ^ { q + 1 } ) } = ( v ( \\cdot , t ) , ( d _ q ^ * d _ q h ) _ { L ^ 2 ( { \\mathbb R } ^ n , \\varLambda ^ { q + 1 } ) } = 0 \\end{align*}"}
{"id": "6844.png", "formula": "\\begin{align*} J _ q ( n , k , t ) = \\frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \\choose t } _ q } \\sum \\limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \\choose 2 } } { k - t \\choose i } _ q { k + i \\choose i } _ q N _ k ^ { i } . \\end{align*}"}
{"id": "5267.png", "formula": "\\begin{align*} \\varphi _ S ( a ) = \\varphi ( a \\delta _ { \\varphi } ^ + ) , \\varphi _ { S ^ { - 1 } } ( a ) = \\varphi ( \\delta _ { \\varphi } ^ - a ) , \\forall a \\in A . \\end{align*}"}
{"id": "7202.png", "formula": "\\begin{align*} \\hat { G } ( t , \\xi ) = \\frac { \\hat { \\phi } ( \\xi ) } { t } \\zeta _ \\xi ( t | \\xi | ) , \\zeta _ \\xi ( p ) = | p | \\hat { \\psi } _ \\xi ( | p | ) . \\end{align*}"}
{"id": "4540.png", "formula": "\\begin{align*} \\int _ j : = \\frac { 1 } { 2 \\pi i } \\int _ { C _ j } L ( z , \\chi ) L ( 2 z , \\chi ^ { 2 } ) G ( z , \\chi ) F _ { u } ( z , \\chi ) \\frac { X ^ { z } } { z } d z . \\end{align*}"}
{"id": "4728.png", "formula": "\\begin{align*} \\vec { A } = T _ 1 ^ { \\frac { 1 } { 4 } - \\delta _ 0 } \\big ( z _ { k _ 1 } ( T _ 1 ) , \\ldots , z _ { k _ { n _ 0 } } ( T _ 1 ) \\big ) \\in \\mathbb { B } _ 0 , \\ ; \\ ; \\vec { B } = T _ 1 ^ { \\frac { 3 } { 2 } } \\big ( a _ 1 ^ - ( T _ 1 ) , \\ldots , a _ n ^ - ( T _ 1 ) \\big ) \\in \\mathbb { B } _ 1 . \\end{align*}"}
{"id": "7568.png", "formula": "\\begin{align*} u \\rhd _ { n } v = v ^ { - n } u v ^ n \\in G . \\end{align*}"}
{"id": "832.png", "formula": "\\begin{align*} \\mu _ \\omega ( 2 B _ \\rho ) \\le \\mu _ \\omega ( B _ \\rho ( \\infty , 8 r ) ) = \\mu ( Z ) \\ , C _ { \\beta , a } \\ r ^ { ( 2 \\beta - 1 - a ) / ( \\beta - 1 ) } . \\end{align*}"}
{"id": "8426.png", "formula": "\\begin{align*} \\nabla _ { v } Y _ { t } ^ { x ; l ^ { \\epsilon } } = J _ { l _ { 0 } ^ { \\epsilon } , t } ^ { x ; l ^ { \\epsilon } } v , \\end{align*}"}
{"id": "2237.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq m \\leq M } \\| X _ m ^ { M , N } \\| _ { L ^ p ( \\Omega ; H ^ \\kappa ) } < \\infty , \\ ; \\kappa = \\min \\big \\{ \\gamma , \\tfrac d 2 + \\tfrac 1 4 \\big \\} . \\end{align*}"}
{"id": "1871.png", "formula": "\\begin{align*} P _ n ( x ) = \\sum _ { k = 1 } ^ { \\lfloor ( n + 1 ) / 2 \\rfloor } \\beta _ { n , k } ( 2 x ) ^ { n + 1 - 2 k } ( 1 + x ^ 2 ) ^ { k } , \\end{align*}"}
{"id": "512.png", "formula": "\\begin{align*} \\psi _ { 2 } ( \\alpha ; \\beta ) : = \\Psi \\left ( h ( \\mathcal { J } ) ^ { - 1 } \\cdot h ( \\mathcal { J } _ { \\beta } ^ { \\alpha } ) \\right ) \\end{align*}"}
{"id": "8311.png", "formula": "\\begin{align*} Z : = \\left \\{ v \\in H _ 0 ^ 1 ( \\Omega ) \\colon \\psi _ 1 \\leq v \\leq \\psi _ 2 \\Omega \\right \\} \\end{align*}"}
{"id": "294.png", "formula": "\\begin{align*} r _ { 0 } ( x ) : = \\eta _ { * } ( x ) ^ { - 1 } \\int _ { - \\infty } ^ { x } z _ { 0 } ( y ) d y , \\eta _ { * } ( x ) : = \\exp \\left ( \\frac { \\beta } { 2 } \\int _ { - \\infty } ^ { x } \\chi _ { * } ( y ) d y \\right ) , \\end{align*}"}
{"id": "2621.png", "formula": "\\begin{align*} Z f ( x , \\omega + l ) = Z f ( x , \\omega ) \\end{align*}"}
{"id": "4824.png", "formula": "\\begin{align*} \\int \\chi ( x ) e ^ { - i x \\cdot \\rho \\xi _ 0 / h \\Tilde { h } } \\widetilde { R } _ h ( z ) ( \\psi ( x ) e ^ { i x \\cdot \\rho \\eta _ 0 / h \\Tilde { h } } ) d x = \\mathcal { O } ( h ^ \\infty \\Tilde { h } ^ \\infty ) \\end{align*}"}
{"id": "9311.png", "formula": "\\begin{align*} \\sum _ { n = 0 } \\alpha _ n | f _ n | ^ 2 \\leq C \\| f \\| ^ 2 _ \\omega \\end{align*}"}
{"id": "8424.png", "formula": "\\begin{align*} l _ { \\gamma ^ { \\epsilon } _ { t } } ^ { \\epsilon } = t , t \\geq l _ { 0 } ^ { \\epsilon } \\quad \\gamma _ { l _ { t } ^ { \\epsilon } } ^ { \\epsilon } = t , t \\geq 0 . \\end{align*}"}
{"id": "3653.png", "formula": "\\begin{align*} \\log \\log t = \\log B _ 0 + \\frac { 3 } { 5 } \\log \\log x - \\frac { 1 } { 5 } \\log \\log \\log x \\le \\frac { 3 } { 5 } \\log \\log x . \\end{align*}"}
{"id": "4688.png", "formula": "\\begin{align*} & \\partial _ y ( \\mathcal { L } \\mathfrak { B } _ { i j k } ) = - b _ { i j k } \\sigma _ i \\Lambda Q + G _ { i j k } , \\end{align*}"}
{"id": "6248.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\lambda = \\frac { 1 - q } { e ^ { q \\lambda } - q e ^ { \\lambda } } . \\end{align*}"}
{"id": "5377.png", "formula": "\\begin{align*} { L _ { m a x } } = \\frac { 1 } { 2 } \\sqrt { { { \\left ( { \\frac { { { t _ c } } } { { { t _ a } \\ln 2 } } } \\right ) } ^ 2 } + \\frac { { { t _ { M a p } } } } { { { t _ a } } } + 4 m } - \\frac { { { t _ c } } } { { { t _ a } \\ln 2 } } . \\end{align*}"}
{"id": "3718.png", "formula": "\\begin{align*} & \\left \\| \\lambda _ q ^ { s } e ^ { - \\mu \\lambda _ q ^ \\alpha t } \\| B _ q ( 0 ) \\| _ { L ^ { 2 } } \\right \\| _ { L ^ { \\frac { \\alpha } { s } } ( 0 , T ; l ^ 2 ) } \\\\ \\leq & \\ \\left \\| e ^ { - \\mu \\lambda _ q ^ \\alpha t } B _ q ( 0 ) \\right \\| _ { L ^ { \\frac { \\alpha } { s } } ( 0 , T ; H ^ { s } ) } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { L ^ { 2 } } \\end{align*}"}
{"id": "9366.png", "formula": "\\begin{align*} \\beta _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } \\frac { \\lambda ^ { k } ( 1 ) _ { k + 1 , \\frac { 1 } { \\lambda } } } { k + 1 } S _ { 2 , \\lambda } ( n , k ) , ( n \\ge 0 ) . \\end{align*}"}
{"id": "2990.png", "formula": "\\begin{align*} \\Theta ( z ) = I _ E + i J \\alpha ( L ^ * - z I ) ^ { - 1 } \\alpha : E \\to E , z \\in \\rho ( L ^ * ) . \\end{align*}"}
{"id": "9387.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - x } F _ { n , \\lambda } \\bigg ( \\frac { x } { 1 - x } \\bigg ) = \\sum _ { k = 0 } ^ { \\infty } x ^ { k } ( k ) _ { n , \\lambda } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "7759.png", "formula": "\\begin{align*} \\Box \\phi = f , \\ ; \\phi [ 0 ] = ( 0 , 0 ) , \\end{align*}"}
{"id": "4375.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d \\tau } \\| \\varepsilon _ - \\| _ { L ^ 2 _ { \\rho } } ^ 2 = ( \\partial _ \\tau \\varepsilon _ - , \\varepsilon _ - ) \\end{align*}"}
{"id": "2484.png", "formula": "\\begin{align*} e ^ { 2 \\pi i \\tau } \\rho ( \\l ) e ^ { 2 \\pi i \\tau ' } \\rho ( \\l ' ) = e ^ { 2 \\pi i ( \\tau + \\tau ' ) } e ^ { - \\pi i \\sigma ( \\l , \\l ' ) } \\rho ( \\l + \\l ' ) . \\end{align*}"}
{"id": "9156.png", "formula": "\\begin{align*} \\kappa ( k ) : = 4 ( M + 1 ) ( B ( 4 k + 4 ) - 1 ) ^ 2 - 1 . \\end{align*}"}
{"id": "4853.png", "formula": "\\begin{align*} E ^ \\alpha ( \\rho ) = \\frac 1 2 \\int _ \\R ( K _ \\alpha * \\rho ) \\rho + \\int _ \\R Q _ \\alpha \\rho . \\end{align*}"}
{"id": "2430.png", "formula": "\\begin{align*} M = \\begin{pmatrix} v _ { 1 , 1 } & v _ { 2 , 1 } & & v _ { d , 1 } \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ v _ { 1 , d } & v _ { 2 , d } & & v _ { d , d } \\end{pmatrix} \\end{align*}"}
{"id": "4162.png", "formula": "\\begin{align*} \\norm { A \\oplus A } _ { p } = 2 ^ { \\frac { 1 } { p } } \\norm { A } _ { p } . \\end{align*}"}
{"id": "9063.png", "formula": "\\begin{align*} \\rho _ { i , j } ^ * \\geq \\delta + r _ p ( h ) , j _ p < j < j _ { p + 1 } , p = 1 , \\cdots , k , \\end{align*}"}
{"id": "6230.png", "formula": "\\begin{align*} d \\Theta ^ { ( d ) } _ { u n i f } = \\frac { d \\theta _ 1 } { 2 \\pi } \\cdots \\frac { d \\theta _ d } { 2 \\pi } . \\end{align*}"}
{"id": "619.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { A ( x ) } - \\frac { 1 } { \\alpha } } \\ = \\ \\frac { \\abs { \\alpha - A ( x ) } } { \\abs { A ( x ) } \\abs { \\alpha } } \\ \\leq \\ \\frac { a } { x + 1 } , \\end{align*}"}
{"id": "3672.png", "formula": "\\begin{align*} g _ E = \\left ( \\{ 1 , 2 \\} , \\{ 2 , 3 \\} , \\{ 3 , 4 \\} , \\{ 1 , 4 \\} \\right ) \\ \\left ( \\{ 1 , 3 \\} , \\{ 2 , 4 \\} \\right ) . \\end{align*}"}
{"id": "4719.png", "formula": "\\begin{align*} & \\phi _ 1 ^ + ( y ) = \\mathbf { 1 } _ { [ \\alpha _ 1 , + \\infty ) } ( y ) , \\quad \\phi _ n ^ - ( y ) = \\mathbf { 1 } _ { ( - \\infty , \\alpha _ n ] } ( y ) , \\\\ & \\phi _ i ^ - ( y ) = \\psi \\bigg ( \\frac { 3 ( y - \\alpha _ i ) } { \\alpha _ i - \\alpha _ { i + 1 } } \\bigg ) \\mathbf { 1 } _ { [ \\alpha _ { i + 1 } , \\alpha _ i ] } ( y ) , \\forall i = 1 , \\ldots , n - 1 , \\\\ & \\phi _ i ^ + ( y ) = [ 1 - \\phi _ { i - 1 } ^ - ( y ) ] \\mathbf { 1 } _ { [ \\alpha _ i , \\alpha _ { i - 1 } ] } ( y ) , \\forall i = 2 , \\ldots , n . \\end{align*}"}
{"id": "8744.png", "formula": "\\begin{align*} ( \\lambda , y ) = \\sum _ { e \\in E } \\chi ( e ) y _ e \\end{align*}"}
{"id": "8826.png", "formula": "\\begin{align*} \\alpha _ t = \\frac { 1 } { 2 } \\min \\left \\{ \\frac { 1 } { \\rho } , \\sqrt { \\frac { \\Delta } { ( n ^ 2 + 2 n ) \\rho L _ { f , 0 } ^ 2 ( T + 1 ) } } \\right \\} , \\end{align*}"}
{"id": "66.png", "formula": "\\begin{align*} L ' \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } L ' _ { v , j } ( \\pi ^ j ) & ( v = 2 , p _ 1 , \\cdots , p _ k ) , \\\\ L ' _ { v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\\\ \\end{cases} \\\\ K _ { \\ell } \\otimes \\Z _ v & = \\begin{cases} \\displaystyle { \\bigoplus _ { j = 0 } ^ 1 } K _ { \\ell , v , j } ( \\pi ^ j ) & ( v = 2 , p _ { k ' + 1 } , \\cdots , p _ k ) , \\\\ K _ { \\ell , v , 0 } & ( \\mathrm { o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
{"id": "5325.png", "formula": "\\begin{align*} \\tilde { \\tau } ( \\psi ) ( x , t ) & : = \\frac { 1 } { } \\tau _ 1 ( \\psi ) ( x , t ) - \\frac { 1 } { ^ 2 } \\Big ( \\int _ { D \\times D } \\tilde { \\gamma } ( q , q ' ) \\psi ( x , q , t ) \\psi ( x , q ' , t ) \\ ; \\mathrm { d } q \\mathrm { d } q ' \\Big ) \\mathbb { I } \\\\ & = \\frac { 1 } { } \\tau _ 1 ( \\psi ) ( x , t ) - \\frac { \\tilde { \\mathfrak { z } } } { ^ 2 } \\eta ^ 2 ( x , t ) \\mathbb { I } , \\end{align*}"}
{"id": "3711.png", "formula": "\\begin{align*} & \\| B ^ k \\| _ { L ^ { \\frac { \\alpha } { ( s + \\alpha - \\frac 5 2 ) } } ( 0 , T ; H ^ s ) } \\\\ \\leq & \\ C ( T ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } + C \\| B ^ { k - 1 } \\| _ { L ^ { \\frac { \\alpha } { ( s + \\alpha - \\frac 5 2 ) } } ( 0 , T ; H ^ s ) } \\| B ^ { k } \\| _ { L ^ { \\frac { \\alpha } { ( s + \\alpha - \\frac 5 2 ) } } ( 0 , T ; H ^ s ) } \\end{align*}"}
{"id": "3.png", "formula": "\\begin{align*} \\cosh { x } : = \\prod _ { k = 0 } ^ \\infty \\left ( 1 + \\frac { 4 x ^ 2 } { \\pi ^ 2 \\left ( 2 k + 1 \\right ) ^ 2 } \\right ) , x \\in \\mathbb { C } , \\end{align*}"}
{"id": "1885.png", "formula": "\\begin{align*} d ' ( \\lambda _ 1 , \\lambda _ 2 ) = \\max \\{ d _ G ( \\lambda _ 1 , \\lambda _ 2 ) , d _ H ( w _ 1 , w _ 2 ) \\} . \\end{align*}"}
{"id": "6417.png", "formula": "\\begin{align*} D _ E ( \\phi \\otimes \\varepsilon ) = \\sum _ { i = 1 } ^ { 2 n } ( e _ i \\nabla ^ S _ { e _ i } \\phi ) \\otimes \\varepsilon + ( e _ i \\phi ) \\otimes \\nabla ^ E _ { e _ i } \\varepsilon . \\end{align*}"}
{"id": "6515.png", "formula": "\\begin{align*} F _ n ^ { ( 2 m ) } & \\sim - ( 1 - 2 \\alpha ) \\frac { m ( m - 1 ) } { \\Gamma ( 2 \\alpha ) } n ^ { - m ( 1 - 2 \\alpha ) } \\sum ^ { n - 1 } _ { j = 1 } j ^ { - 2 ( 1 - \\alpha ) + m ( 1 - 2 \\alpha ) } \\\\ & \\sim - \\frac { m } { \\Gamma ( 2 \\alpha ) } n ^ { - ( 1 - 2 \\alpha ) } \\end{align*}"}
{"id": "6126.png", "formula": "\\begin{align*} ( m + 2 ) ( m + 1 ) s ^ m w _ { t t } + \\varepsilon ( n + 2 ) ( n + 1 ) t ^ n w _ { s s } = 0 \\ , , \\end{align*}"}
{"id": "7230.png", "formula": "\\begin{align*} G _ { 4 , 2 } ^ 1 : = G _ { 4 , 2 } \\cap \\left \\{ ( s , v ) \\in [ 0 , t ] \\times \\R ^ 3 : | v | \\geq \\frac { | x ^ \\perp | } { 2 ( \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } + 5 ) } \\right \\} . \\end{align*}"}
{"id": "7726.png", "formula": "\\begin{align*} \\gamma ^ t ( x ) : \\S & \\rightarrow \\mathcal { M } , \\\\ x & \\mapsto \\gamma ^ t ( x ) : = \\phi ( t \\cdot T , x ) . \\end{align*}"}
{"id": "6361.png", "formula": "\\begin{align*} ( f ^ * x ^ 5 = 0 ) = E _ 1 + 3 E _ 2 + 5 A , & & D _ 1 \\equiv B \\equiv E _ 2 + 2 A , \\\\ ( f ^ * y ^ 5 = 0 ) = 5 B + 2 E _ 1 + E _ 2 , & & D _ 2 \\equiv 3 B + E _ 1 \\equiv A . \\end{align*}"}
{"id": "660.png", "formula": "\\begin{align*} s ( x ) \\ = \\ \\min \\bigg \\{ n : E ( n ) \\leq \\frac { 1 } { x + 1 } \\bigg \\} . \\end{align*}"}
{"id": "5644.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta \\Phi & = \\varepsilon ^ { - 2 } ( 1 - \\left ( x ^ { 2 } + y ^ { 2 } \\right ) ) & \\Omega , \\\\ \\Phi & = g & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
{"id": "3713.png", "formula": "\\begin{align*} & \\| B ^ k \\| _ { L ^ { \\infty } ( 0 , T ; H ^ { \\frac 5 2 - \\alpha } ) } \\\\ \\leq & \\ \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } + C \\| B ^ { k - 1 } \\| _ { L ^ { 2 } ( 0 , T ; H ^ { \\frac 5 2 - \\frac { \\alpha } { 2 } } ) } \\| B ^ { k } \\| _ { L ^ { 2 } ( 0 , T ; H ^ { \\frac 5 2 - \\frac { \\alpha } { 2 } } ) } . \\end{align*}"}
{"id": "2549.png", "formula": "\\begin{align*} \\mathcal { V } _ Q \\ , \\rho ( x , \\omega ) \\ , \\mathcal { V } _ Q ^ { - 1 } f ( t ) & = e ^ { \\pi i Q t \\cdot t } e ^ { \\pi i x \\cdot \\omega } e ^ { 2 \\pi i \\omega \\cdot ( t - x ) } e ^ { - \\pi i Q ( t - x ) \\cdot ( t - x ) } f ( t - x ) \\\\ & = e ^ { \\pi i ( \\omega + Q x ) \\cdot x } e ^ { 2 \\pi i ( \\omega + Q x ) \\cdot ( t - x ) } f ( t - x ) \\\\ & = \\rho ( x , Q x + \\omega , \\tau ) f ( t ) = \\rho ( V _ Q ( x , \\omega ) , \\tau ) f ( t ) , \\end{align*}"}
{"id": "4524.png", "formula": "\\begin{align*} G _ 2 ( a _ 1 ) = \\{ b _ 1 = x _ { a _ 1 } ( a _ 2 ) , \\ldots , b _ { n - 1 } = x _ { a _ 1 } ( a _ n ) \\} . \\end{align*}"}
{"id": "5064.png", "formula": "\\begin{align*} [ \\varphi ( a \\cdot v _ 0 ) , a ^ 2 \\cdot v _ 0 ] = 2 [ [ \\zeta '' , a ] , a ] a \\cdot v _ 0 = 2 { \\rm I I } ( [ [ \\zeta '' , a ] , a ] , a ) \\cdot v _ 0 , \\end{align*}"}
{"id": "7823.png", "formula": "\\begin{align*} ( a + b i ) ( x , y ) & : = ( a x - b y , b x + a y ) , & & x , y \\in H ; \\ ; a , b \\in \\R , \\\\ ( ( x , y ) , ( u , v ) ) _ { H _ \\C } & : = ( x , u ) _ H + ( y , v ) _ H + i [ ( y , u ) _ H - ( x , v ) _ H ] , & & x , y , u , v \\in H . \\end{align*}"}
{"id": "3041.png", "formula": "\\begin{align*} \\gamma ^ \\prime & = r n + 1 2 r - 1 4 n - 6 m - 8 \\Theta - 4 \\Delta \\ , ; \\\\ t & = \\frac { 1 } { 6 } \\left ( r ^ 3 - 3 r ^ 2 - 5 8 r - 3 r \\left ( n + 3 m + 3 \\Theta \\right ) + 4 2 n + 7 8 m + 7 8 \\Theta \\right ) \\ , ; \\\\ t ^ \\prime & = \\frac { 1 } { 6 } \\left ( r ^ 3 - 3 r ^ 2 - 5 8 r - 3 r \\left ( m + 3 n + 3 \\Theta \\right ) + 4 2 m + 7 8 n + 7 8 \\Theta \\right ) \\ , , \\end{align*}"}
{"id": "9510.png", "formula": "\\begin{align*} \\norm { \\Gamma ( f ) } & = \\norm { [ \\gamma ( f ) ] } \\\\ & \\le \\norm { \\gamma ( f ) } \\\\ & = \\norm { f _ - ( - M _ x ) + f _ + ( H ) - f ( 0 ) } \\\\ & \\le \\norm { f _ - ( - M _ x ) } + \\norm { f _ + ( H ) } + | f ( 0 ) | \\\\ & = \\max _ { t \\in [ - 1 , 0 ] } | f _ - ( t ) | + \\max _ { z \\in \\overline { \\d ( 1 , 1 ) } } | f _ + ( z ) | + | f ( 0 ) | \\\\ & \\le 3 \\norm { f } , \\end{align*}"}
{"id": "8320.png", "formula": "\\begin{align*} B _ - ( y , u ) : = \\{ t \\in [ 0 , T ] \\colon y ( t ) = - r \\exists \\varepsilon > 0 y - u = \\mathrm { c o n s t } [ t , t + \\varepsilon ) \\} , \\end{align*}"}
{"id": "6185.png", "formula": "\\begin{align*} \\| W \\| _ F ^ 2 = \\sum _ { t = 1 } ^ { p } \\| W _ { : , t } \\| ^ 2 _ { 2 } = \\sum _ { t = 1 } ^ { p } \\left \\| \\frac { S _ { : , j _ { t } } } { \\sqrt { p P ' _ { j _ { t } } } } \\right \\| ^ 2 _ { 2 } = \\sum _ { t = 1 } ^ { p } \\frac { \\| S _ { : , j _ { t } } \\| ^ 2 _ { 2 } } { p \\frac { \\| S _ { : , j _ { t } } \\| ^ 2 _ { 2 } } { \\| S \\| _ F ^ 2 } } = \\sum _ { t = 1 } ^ { p } \\frac { \\| S \\| _ F ^ 2 } { p } = \\| S \\| _ F ^ 2 . \\end{align*}"}
{"id": "2602.png", "formula": "\\begin{align*} \\norm { V _ g f } _ { L ^ { p , q } } \\leq C \\norm { V _ g g _ 0 } _ { L ^ 1 } \\norm { V _ { g _ 0 } f } _ { L ^ { p , q } } = C _ 1 \\norm { f } _ { M ^ { p , q } } . \\end{align*}"}
{"id": "949.png", "formula": "\\begin{align*} \\Big ( \\sum _ j u _ j ( t ) \\Big ) ^ 2 = \\big ( u _ { j _ 1 } ( t ) + \\cdots + u _ { j _ m } ( t ) \\big ) ^ 2 \\lesssim \\big ( \\max _ \\ell u _ { j _ \\ell } ( t ) \\big ) ^ 2 \\leq \\sum _ { \\ell } u _ { j _ \\ell } ( t ) ^ 2 \\leq \\sum _ j u _ j ( t ) ^ 2 . \\end{align*}"}
{"id": "372.png", "formula": "\\begin{align*} \\dot \\rho _ 1 + m _ { 1 2 } ( 1 + \\dot W ^ { \\delta } ) & = 0 , \\\\ \\dot \\rho _ 2 + ( m _ { 2 1 } + m _ { 2 3 } ) ( 1 + \\dot W ^ { \\delta } ) & = 0 , \\\\ \\dot \\rho _ 3 + m _ { 3 2 } ( 1 + \\dot W ^ { \\delta } ) & = 0 . \\end{align*}"}
{"id": "1445.png", "formula": "\\begin{align*} \\mathfrak { i } : M _ n ( \\mathbb { H } ) \\stackrel { \\sim } \\longrightarrow \\{ x \\in M _ { 2 n } ( \\C ) : \\overline { x } J _ n ' = J _ n ' x \\} . \\end{align*}"}
{"id": "5217.png", "formula": "\\begin{align*} g _ j \\ast f _ 1 ( x ) & = \\int \\limits _ { y \\in ( x - W _ i ) \\cap \\mathfrak { D } ^ * } a ( y ) f _ 1 ( x - y ) d y . \\end{align*}"}
{"id": "7354.png", "formula": "\\begin{align*} \\begin{aligned} & h _ q + G _ \\beta ^ \\ast ( v ( y _ q , t _ q ) , 0 , Y _ q , K ) \\geq 0 , \\\\ & k _ q + G _ \\beta ^ \\ast ( v ( z _ q , t _ q ) , 0 , Z _ q , K ) \\geq 0 . \\end{aligned} \\end{align*}"}
{"id": "4887.png", "formula": "\\begin{align*} H ^ + = \\{ z \\in \\C : \\ , { \\rm I m } \\ , z > 0 \\} , W ^ { \\pm } = \\{ z \\in H ^ + : \\ , \\pm F ( z ) \\in H ^ + \\} . \\end{align*}"}
{"id": "1780.png", "formula": "\\begin{align*} A ( g _ 0 , g _ 1 , g _ 2 ) : = { \\rm A r e a } _ { \\mathbb { H } } ( \\Delta ( g _ 0 [ e ] , g _ 1 [ e ] , g _ 2 [ e ] ) ) , \\end{align*}"}
{"id": "8187.png", "formula": "\\begin{align*} S ( H _ { 1 5 } , 1 5 f ) = \\frac { 1 4 f - \\left ( 1 2 + 3 \\varepsilon _ d \\right ) d + 1 } { 3 } \\hbox { a n d } N _ { 1 5 } ' ( f , H ) = - \\frac { 3 2 + 1 5 \\varepsilon _ d } { 4 8 } d + \\frac { 1 - 2 \\varepsilon _ d } { 4 8 } . \\end{align*}"}
{"id": "896.png", "formula": "\\begin{align*} F _ q ( T , q ^ { - 2 } ) = - F _ q ( T , q ^ { - 2 } ) . \\end{align*}"}
{"id": "2186.png", "formula": "\\begin{align*} 0 = \\langle J ^ { ' } ( \\widehat { u } ) , \\widehat { u } \\rangle + \\lambda _ 0 \\langle \\varphi ^ { * } _ { \\widehat { u } } , \\widehat { u } \\rangle = \\lambda _ 0 \\langle \\varphi ^ { * } _ { \\widehat { u } } , \\widehat { u } \\rangle , \\ \\ \\ \\varphi ^ * _ { \\widehat { u } } \\in \\partial \\varphi ( \\widehat { u } ) \\end{align*}"}
{"id": "5081.png", "formula": "\\begin{align*} \\frac { B } { A } = e ^ { 2 a \\gamma } . \\end{align*}"}
{"id": "8264.png", "formula": "\\begin{align*} F _ { 1 2 } F _ { 2 1 } = F _ { 1 2 4 3 } + F _ { 1 4 2 3 } + F _ { 1 4 3 2 } + F _ { 4 1 2 3 } + F _ { 4 1 3 2 } + F _ { 4 3 1 2 } \\end{align*}"}
{"id": "8805.png", "formula": "\\begin{align*} \\nu ^ { \\epsilon , L } ( d \\varphi ) = e ^ { - V _ 0 ^ { \\epsilon , L } ( \\varphi ) } \\gamma ^ { \\epsilon , L } ( d \\varphi ) , \\end{align*}"}
{"id": "1048.png", "formula": "\\begin{align*} \\Delta ( D _ { i m } ) & = \\sum _ { \\substack { k < 5 \\\\ k < l } } a _ { i k } a _ { m l } \\otimes D _ { k l } - q ^ { - 1 } \\sum _ { \\substack { k < 5 \\\\ l < k } } a _ { i k } a _ { m l } \\otimes D _ { l k } \\\\ & + ( a _ { i 5 } a _ { m 6 } + q ^ { - 1 } a _ { i 6 } a _ { m 5 } ) \\otimes D _ { 5 6 } \\\\ & + ( 1 + q ^ { - 2 } ) \\sum _ { 5 \\leq k \\leq 6 } a _ { i k } a _ { m k } \\otimes D _ { k k } + q ^ { - 1 } \\sum _ { \\substack { k \\geq 5 \\\\ l < 5 } } a _ { i k } a _ { m l } \\otimes D _ { l k } \\ , . \\end{align*}"}
{"id": "300.png", "formula": "\\begin{align*} v _ { t } + ( \\beta \\chi v ) _ { x } - v _ { x x } & = - \\gamma \\chi _ { x x x } , \\ \\ x \\in \\R , \\ t > 0 , \\\\ v ( x , 0 ) & = 0 , \\ \\ x \\in \\R . \\end{align*}"}
{"id": "4543.png", "formula": "\\begin{align*} J _ { 1 1 } = C + O ( E _ y ) , \\end{align*}"}
{"id": "220.png", "formula": "\\begin{align*} \\tilde { \\Gamma } ( f , g ) ( x ) = \\frac { 1 } { \\alpha } \\left ( \\Gamma ( f , g ) ( x ) + \\Gamma ^ * ( f , g ) ( x ) \\right ) . \\end{align*}"}
{"id": "5320.png", "formula": "\\begin{align*} X = \\omega _ { ( 1 ) } \\theta ( S ^ { - 1 } ( a _ { ( 3 ) } ) - a _ { ( 1 ) } ) \\otimes a _ { ( 2 ) } \\otimes \\omega _ { ( 2 ) } \\otimes a _ { ( 4 ) } b , \\end{align*}"}
{"id": "8376.png", "formula": "\\begin{align*} \\Delta _ 0 = \\sum _ { 1 \\le i < j \\le d } D _ { i , j } ^ 2 . \\end{align*}"}
{"id": "5701.png", "formula": "\\begin{align*} w ( \\alpha ) - w ( \\beta ) = \\# ( \\gamma _ { 0 } \\cap { \\pi ( v \\cap { [ - s _ { * } , s _ { * } ] \\times S ^ { 3 } } ) } ) . \\end{align*}"}
{"id": "1227.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\zeta } _ t ( E ) = \\inf \\left \\{ \\sum _ { n \\in \\mathbb { N } } \\zeta ( \\vert B _ n \\vert ) : \\ ( B _ n ) _ { n \\in \\N } \\mbox { c l o s e d b a l l s , } \\vert B _ n \\vert \\leq t E \\subset \\bigcup _ { n \\in \\mathbb { N } } B _ n \\right \\} . \\end{align*}"}
{"id": "8574.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\# } ( x , k ) = ( k ) \\mathcal { K } ( x , k ) . \\end{align*}"}
{"id": "736.png", "formula": "\\begin{align*} C _ b = 0 , C _ W = \\frac { 2 } { a _ + ^ 2 + a _ - ^ 2 } \\end{align*}"}
{"id": "7470.png", "formula": "\\begin{align*} G ( \\phi ^ { n } ) = \\frac { \\delta E } { \\delta \\overline { \\phi } } ( \\phi ^ { n } ) = - \\frac 1 2 \\Delta { \\phi } ^ { n } + V \\phi ^ n + \\beta | \\phi ^ n | ^ 2 \\phi ^ n - \\Omega L _ z \\phi ^ n , n = 0 , 1 , \\ldots . \\end{align*}"}
{"id": "1781.png", "formula": "\\begin{align*} \\chi ( A ) ( f _ 0 , f _ 1 , f _ 2 ) = & \\int _ { S L _ 2 ( \\mathbb { R } ) } \\int _ { S L _ 2 ( \\mathbb { R } ) } f _ 0 \\big ( ( g _ 1 g _ 2 ) ^ { - 1 } \\big ) f _ 1 ( g _ 1 ) f _ 2 ( g _ 2 ) \\\\ & \\operatorname { A r e a } _ { \\mathbb { H } } \\big ( \\Delta ( [ e ] , g _ 1 [ e ] , g _ 1 g _ 2 [ e ] ) \\big ) d g _ 1 d g _ 2 . \\end{align*}"}
{"id": "5184.png", "formula": "\\begin{align*} T _ { \\ell i } ^ m = \\sum _ { j = 1 } ^ { K _ { \\ell i } ^ { m } } X _ { \\ell i j } , \\end{align*}"}
{"id": "3245.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } s ( r , t ) = s ( r , 0 ) , r > 0 . \\end{align*}"}
{"id": "1685.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { n - 1 } \\frac { x ^ { 2 k } } { ( 2 k + 1 ) ^ s } = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } F _ s } \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - k ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; x ^ 2 \\right ) . \\end{align*}"}
{"id": "5823.png", "formula": "\\begin{align*} \\lambda = 2 \\Lambda \\end{align*}"}
{"id": "3241.png", "formula": "\\begin{align*} - ( \\omega ( z ) - z ) ^ 2 + \\frac { \\omega ( z ) - z } { G _ { { \\mu } } ( \\omega ( z ) ) } + | \\lambda | ^ 2 = 0 . \\end{align*}"}
{"id": "1776.png", "formula": "\\begin{align*} C ( x _ 1 , \\cdots , x _ n ) : = \\left | \\begin{array} { l l l } x _ 1 ^ 1 & \\cdots & x _ 1 ^ n \\\\ & \\cdots & \\\\ x _ n ^ 1 & \\cdots & x _ n ^ n \\end{array} \\right | . \\end{align*}"}
{"id": "7760.png", "formula": "\\begin{gather*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } , \\ ; \\phi [ 0 ] = ( a , b ) , \\end{gather*}"}
{"id": "8482.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { 4 | S _ { \\ell } | } { | N | } = \\frac { 2 } { | T _ { \\ell ' } | } & \\mbox { i f } a = 1 , \\mbox { a n d } \\\\ \\frac { 2 | S _ { \\ell } | } { | N | } \\left ( \\alpha ( a ) + \\alpha ( a ^ { - 1 } ) \\right ) = \\frac { 1 } { | T _ { \\ell ' } | } \\left ( \\alpha ( a ) + \\alpha ( a ^ { - 1 } ) \\right ) & \\mbox { i f } 1 \\neq a \\in ( \\mu _ { q - 1 } ) _ { \\ell ' } . \\end{array} \\right . \\end{align*}"}
{"id": "5140.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial n } \\left [ \\underbrace { ( n ^ 2 - 1 ) ^ { 1 / 2 } - n + 1 / n } _ { = ( * ) } \\right ] & = \\Big ( 1 + \\frac { 1 } { ( n ^ 2 - 1 ) } \\Big ) ^ { 1 / 2 } - 1 - \\frac { 1 } { n ^ 2 } < 0 , \\end{align*}"}
{"id": "4255.png", "formula": "\\begin{align*} ( t , x ) \\mapsto ( T ( t ) f ) ( x ) = \\lim _ { n \\to \\infty } \\big ( T _ n ( t ) f \\big ) ( x ) \\end{align*}"}
{"id": "1815.png", "formula": "\\begin{align*} u = x y , \\ ; \\ ; 2 v = x + y . \\end{align*}"}
{"id": "2016.png", "formula": "\\begin{align*} N _ t ^ { R \\nu } = - A _ t ^ { \\nu } , t \\in [ 0 , \\zeta [ \\end{align*}"}
{"id": "6860.png", "formula": "\\begin{align*} \\lambda _ { n - k + 1 } = \\min _ { U } \\left \\{ \\max _ { f \\in U } \\left \\{ \\frac { \\langle f , A f \\rangle } { \\langle f , f \\rangle } \\right \\} ~ \\middle | ~ ( U ) = k \\right \\} . \\end{align*}"}
{"id": "5897.png", "formula": "\\begin{align*} x _ i = a _ i x _ 1 + b _ i x _ 2 ~ ~ ~ \\mbox { f o r } 3 \\leq i \\leq 2 k . \\end{align*}"}
{"id": "6934.png", "formula": "\\begin{align*} r _ { h , i } ( w ) = F _ h ( \\varphi _ i ) - a _ h ( w , \\varphi _ i ) \\ , , i \\in I _ h \\ , , \\end{align*}"}
{"id": "7624.png", "formula": "\\begin{align*} F ^ { i j } \\bar { R } _ { \\nu j l i } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) & = - \\lambda \\left ( \\frac { \\lambda '' } { \\lambda } + \\frac { c - \\lambda '^ { 2 } } { \\lambda ^ { 2 } } \\right ) F ^ { i j } ( \\delta _ { i j } r _ { l } - \\delta _ { j l } r _ { i } ) r _ { \\nu } r _ { l } \\\\ & = - u \\left ( \\frac { \\lambda '' } { \\lambda } + \\frac { c - \\lambda '^ { 2 } } { \\lambda ^ { 2 } } \\right ) \\Big ( ( \\sum _ { l } r _ { l } ^ { 2 } ) ( \\sum _ { i } F ^ { i i } ) - F ^ { i j } r _ { i } r _ { j } \\Big ) . \\end{align*}"}
{"id": "9220.png", "formula": "\\begin{align*} A _ \\gamma : = \\frac { 1 } { \\gamma } ( I d - J ^ A _ \\gamma ) . \\end{align*}"}
{"id": "4312.png", "formula": "\\begin{align*} \\lambda _ { i , \\infty , \\beta } = 2 \\beta \\left ( \\frac { \\alpha } { 2 } - i \\right ) , \\end{align*}"}
{"id": "7059.png", "formula": "\\begin{align*} & N ^ K _ { j } = N ^ K _ { j - \\lfloor j \\delta _ K \\rfloor / \\delta _ K } \\quad p ( j \\delta _ K ) = p ( ( j - \\lfloor j \\delta _ K \\rfloor / \\delta _ K ) \\delta _ K ) , \\mbox { w h e n } j \\geq 1 / \\delta _ K , \\\\ & N ^ K _ { j } = N ^ K _ { j + \\lceil | j | \\delta _ K \\rceil / \\delta _ K } \\quad p ( j \\delta _ K ) = p ( ( j + \\lceil | j | \\delta _ K \\rceil / \\delta _ K ) \\delta _ K ) , \\mbox { w h e n } j < 0 . \\end{align*}"}
{"id": "8115.png", "formula": "\\begin{align*} T B _ { 6 } ( f ) = T B _ { 5 } ( f ) , \\end{align*}"}
{"id": "6216.png", "formula": "\\begin{align*} x _ { \\rm T T L S } - x _ { \\rm Q i T T L S } = - ( \\hat { \\mathbf { V } } _ { 1 1 } ^ T ) ^ { \\dag } \\check { E } ^ T x _ { \\rm T T L S } - x _ { \\rm T T L S } + { ( \\hat { \\mathbf { V } } _ { 1 1 } ^ T ) } ^ { \\dag } \\hat { \\mathbf { V } } _ { 1 1 } ^ T x _ { \\rm T T L S } - ( \\hat { \\mathbf { V } } _ { 1 1 } ^ T ) ^ { \\dag } e . \\end{align*}"}
{"id": "1016.png", "formula": "\\begin{align*} \\int _ { B _ { r } ^ + ( a ) } x _ 1 ^ 2 \\dd x \\ge \\int _ { B _ { r / 4 } ( a + ( 3 r ) / 4 e _ 1 ) } x _ 1 ^ 2 \\dd x \\ge \\frac { r ^ 2 } 4 \\ , | B _ { r / 4 } ( a + ( 3 r ) / 4 e _ 1 ) | = C r ^ { n + 2 } , \\end{align*}"}
{"id": "4218.png", "formula": "\\begin{align*} u ^ i ( t , \\mu ) = h ^ i ( t ) g ^ i ( \\mu ) \\end{align*}"}
{"id": "6044.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { G } } _ { \\alpha , \\beta } \\widetilde { P } _ n ^ { \\alpha , \\beta } = - n ( n - \\alpha - \\beta + 1 ) \\widetilde { P } _ n ^ { \\alpha , \\beta } . \\end{align*}"}
{"id": "9139.png", "formula": "\\begin{align*} \\norm { x _ N - J ^ S _ { \\mu _ i } ( x _ N + \\mu _ i T _ { \\lambda _ N } x _ N ) } & \\leq \\norm { x _ N - x _ { N + 1 } } + \\vert \\mu _ N - \\mu _ i \\vert \\frac { \\norm { x _ N - x _ { N + 1 } } } { \\mu _ N } \\\\ & < \\frac { 1 } { 2 ( k + 1 ) } + C \\frac { 1 } { 2 C ( k + 1 ) } \\\\ & = \\frac { 1 } { k + 1 } \\end{align*}"}
{"id": "7675.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ { \\varepsilon } \\le \\frac { \\int _ { \\tilde { \\Omega } _ { \\varepsilon } } | \\nabla P _ { \\tilde { \\Omega } _ { \\varepsilon } } w | ^ 2 } { \\int _ { \\tilde { \\Omega } _ { \\varepsilon } } \\tilde { m } _ 0 \\ , | P _ { \\tilde { \\Omega } _ { \\varepsilon } } w | ^ 2 } = \\frac { \\tilde { \\lambda } _ 0 + B } { 1 - A } \\end{align*}"}
{"id": "8988.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f ( x ) & \\leq 0 \\ , ( - 1 , 1 ) , \\\\ u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f ( x ) & \\geq 0 [ - 1 , 1 ] , \\end{aligned} \\right . \\end{align*}"}
{"id": "2732.png", "formula": "\\begin{align*} m _ { i , \\lambda } ( \\pi _ 1 ^ { - 1 } ( \\omega ) ) = 0 . \\end{align*}"}
{"id": "4986.png", "formula": "\\begin{align*} p ( Z ) & = \\left ( ( B \\circ A ^ { r _ 0 } ) ^ { r _ 1 } \\circ A , B \\circ A ^ { r _ 0 } \\right ) . \\end{align*}"}
{"id": "459.png", "formula": "\\begin{align*} | X ( K _ q ) - \\mu _ d ( X ) q ^ d | = O ( q ^ { d - 1 / 2 } ) . \\end{align*}"}
{"id": "1032.png", "formula": "\\begin{align*} u ( x ) & = \\gamma _ { n , s } \\int _ { \\R ^ n _ + \\setminus B _ r ^ + } \\bigg ( \\frac { r ^ 2 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - r ^ 2 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert x - y \\vert ^ n } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ n } \\bigg ) u ( y ) \\dd y \\end{align*}"}
{"id": "6917.png", "formula": "\\begin{align*} \\gamma = \\frac { P _ t \\Bigl | \\sum _ { n = 1 } ^ N g _ n c _ n h _ n B _ r ( \\theta _ I ) + h _ d B _ d ( \\theta _ d ) \\Bigr | ^ 2 } { \\sigma _ n ^ 2 } . \\end{align*}"}
{"id": "2921.png", "formula": "\\begin{align*} \\tilde T _ n ( k ) : = \\sum \\limits _ { \\substack { A \\subset \\{ 1 , \\dots , d \\} \\\\ | A | = k } } \\tilde S _ { n , A } ^ { \\rm M } , \\tilde S _ { n , A } ^ { \\rm M } : = \\frac 1 n \\sum _ { i , j = 1 } ^ n \\prod _ { p \\in A } \\tilde I _ { i , j } ^ { ( p ) } , \\end{align*}"}
{"id": "3727.png", "formula": "\\begin{align*} K _ 1 < & - \\frac { 1 } { 2 \\pi X ( x _ 0 , t ) } \\int _ { - 1 } ^ { - X ( x _ 0 , t ) } B _ y ( y , t ) \\ , d y \\\\ = & - \\frac { 1 } { 2 \\pi X ( x _ 0 , t ) } \\left ( B ( - X ( x _ 0 , t ) , t ) - B ( - 1 , t ) \\right ) \\\\ = & \\ \\frac { 1 } { 2 \\pi X ( x _ 0 , t ) } . \\end{align*}"}
{"id": "6068.png", "formula": "\\begin{align*} \\begin{cases} x _ { s } u ^ { 1 } _ { s } + y _ { s } v ^ { 1 } _ { s } + z _ { s } w ^ { 1 } _ { s } = 0 , \\\\ x _ { s } u ^ { 1 } _ { t } + y _ { s } v ^ { 1 } _ { t } + z _ { s } w ^ { 1 } _ { t } + x _ { t } u ^ { 1 } _ { s } + y _ { t } v ^ { 1 } _ { s } + z _ { t } w ^ { 1 } _ { s } = 0 , \\\\ x _ { t } u ^ { 1 } _ { t } + y _ { t } v ^ { 1 } _ { t } + z _ { t } w ^ { 1 } _ { t } = 0 , \\\\ \\end{cases} \\end{align*}"}
{"id": "4762.png", "formula": "\\begin{align*} \\Gamma \\in \\mathcal { M } ( D _ 6 , & D _ { 1 0 } , D _ { 1 4 } , D _ { 1 5 } , D _ { 1 7 } , D _ { 1 8 } , D _ { 2 1 } , D _ { 2 2 } , D _ { 2 4 } ) \\\\ & = \\mathcal { M } ( D _ { 1 0 } , D _ { 2 1 } , D _ { 2 2 } , D _ { 2 4 } ) \\cup \\mathcal { M } ( D _ 6 , D _ { 1 4 } , D _ { 1 5 } , D _ { 1 7 } , D _ { 1 8 } ) . \\end{align*}"}
{"id": "2360.png", "formula": "\\begin{align*} \\mathcal { P } ( M _ { - \\omega } f ) ( x ) = \\sum _ { k \\in \\Z ^ d } f ( k + x ) e ^ { - 2 \\pi i \\omega \\cdot ( k + x ) } = \\sum _ { k \\in \\Z ^ d } T _ { - x } M _ { - \\omega } f ( k ) , \\end{align*}"}
{"id": "9243.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , p ^ X , x ^ X ( \\gamma > _ \\mathbb { R } 0 \\land \\tilde { \\rho } > _ \\mathbb { R } - \\gamma \\land p \\in A ( x - _ X \\gamma p ) \\rightarrow p = _ X A _ \\gamma x ) . \\end{align*}"}
{"id": "4225.png", "formula": "\\begin{align*} & \\widehat { R } _ j ^ i ( s ) = \\left | \\int _ 0 ^ s \\left ( h _ j ^ i ( z ) - h ^ i ( z ) \\right ) g ^ i \\left ( \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\delta _ { X ^ n ( z ) } \\right ) \\ , d z \\right | \\\\ & + M \\int _ 0 ^ T \\left | g ^ i _ j \\left ( \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\delta _ { X ^ n ( z ) } \\right ) - g ^ i \\left ( \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\delta _ { X ^ n ( z ) } \\right ) \\right | \\ , d z . \\end{align*}"}
{"id": "3896.png", "formula": "\\begin{align*} E _ 2 ( x ) = \\sum _ { x \\leq p \\leq 2 x } \\frac { 1 } { p } \\sum _ { \\substack { 0 < a < p \\\\ g c d ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 < b < p \\\\ \\gcd ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } \\ll x ^ { 1 - 2 \\varepsilon } . \\end{align*}"}
{"id": "5472.png", "formula": "\\begin{align*} \\int _ \\Omega g ( u ( x ) ) d x = \\int _ { \\Omega \\setminus \\Omega _ 0 } g ( u ( x ) ) d x = \\int _ { \\Omega \\setminus \\Omega _ 0 } \\lim _ { n \\rightarrow \\infty } g ( u _ n ( x ) ) d x \\leq \\lim _ { n \\rightarrow \\infty } \\inf \\int _ { \\Omega \\setminus \\Omega _ 0 } g ( u _ n ( x ) ) d x . \\end{align*}"}
{"id": "5969.png", "formula": "\\begin{align*} \\frac { \\partial \\phi _ k ( x , z , t _ i ) } { \\partial t } \\approx \\sum _ { n = \\alpha } ^ { \\beta } a _ n \\phi ( x , z , t _ i + n \\Delta t ) , i = 0 , 1 , . . . N , \\end{align*}"}
{"id": "7581.png", "formula": "\\begin{align*} V _ { 1 , 1 } = \\int _ { D } \\left | f _ { x y } ( x , y ) \\right | \\omega ( x , y ) d x d y , \\end{align*}"}
{"id": "1926.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ x f + { \\rm { d i v } } _ v \\big ( ( \\alpha u _ N \\chi _ { \\{ \\alpha | u _ N | \\leq N \\} } - v ) f \\big ) - \\Delta _ v f + \\alpha { \\rm { d i v } } _ v ( f L [ f ] ) = 0 . \\end{align*}"}
{"id": "3231.png", "formula": "\\begin{align*} k ( s , t ) = ( s - t ) \\bigg ( \\frac { 1 } { h ( s ) } - s + t \\bigg ) , s > 0 , \\ , t \\in \\R . \\end{align*}"}
{"id": "6369.png", "formula": "\\begin{align*} & \\Sigma ^ \\prime _ i = ( \\Sigma ^ \\prime _ { i - 1 } ) ^ \\ast ( v _ { i - 1 } + v ) i = 1 , \\dots , k \\\\ \\end{align*}"}
{"id": "3293.png", "formula": "\\begin{align*} s - t > | \\lambda | ^ 2 h ( s ) = | \\lambda | ^ 2 \\int _ \\mathbb { R } \\frac { t } { t ^ 2 + u ^ 2 } d \\mu \\boxplus \\nu ( t ) > \\frac { | \\lambda | ^ 2 t } { t ^ 2 + M } \\end{align*}"}
{"id": "2295.png", "formula": "\\begin{align*} \\langle g _ 0 , M _ \\omega T _ x g _ 0 \\rangle = e ^ { - \\pi i x \\cdot \\omega } e ^ { - \\frac { \\pi } { 2 } ( x ^ 2 + \\omega ^ 2 ) } \\end{align*}"}
{"id": "3857.png", "formula": "\\begin{align*} a _ { k , l } ( \\rho ) : = ( - 1 ) ^ { k + l } \\frac { \\det [ U _ j ( y _ r ) ] _ { j = \\overline { 1 , k - 1 } , \\ , r = \\overline { 1 , k } \\setminus l } } { \\det [ U _ j ( y _ r ) ] _ { j , r = \\overline { 1 , k } } } . \\end{align*}"}
{"id": "4239.png", "formula": "\\begin{align*} \\frac { 9 ( \\rho '' ) ^ 3 } { 9 ( \\rho '' ) ^ 4 + ( \\rho ''' ) ^ 2 } = K ^ { - 1 } \\left ( 1 + \\frac { ( \\partial _ s K ) ^ 2 } { 9 K ^ 4 } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "703.png", "formula": "\\begin{align*} g _ j ( X _ n ) = p _ j ( X _ n ) + O ( n ^ { - k + 1 } ) , \\end{align*}"}
{"id": "877.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum \\limits _ { i = 1 } ^ { N } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) } = n _ { N + 1 } - n _ 1 , \\\\ & \\sum \\limits _ { i = 1 } ^ { N } { \\left ( { { n _ { i + 1 } } - { n _ i } } \\right ) \\left ( { 2 { \\tau _ { \\rm c } } + 2 \\mathcal { T } + { n _ { i + 1 } } + { n _ i } } \\right ) } = \\left ( n _ { N + 1 } + \\tau _ { \\rm c } + \\mathcal { T } \\right ) ^ 2 - \\left ( n _ 1 + \\tau _ { \\rm c } + \\mathcal { T } \\right ) ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "8681.png", "formula": "\\begin{align*} & \\lim _ { n \\longrightarrow \\infty } ( A _ n , B _ n , C _ n , N _ n , K _ { W _ n } ) = ( A , B , C , N , K _ { W } ) , \\ : K _ W \\succ 0 , \\\\ & \\lim _ { n \\longrightarrow \\infty } ( F _ n , G _ n , \\Gamma _ n , D _ n , K _ { Z _ n } ) = ( F , G , \\Gamma , D , K _ { Z } ) , \\ : K _ Z \\succeq 0 \\end{align*}"}
{"id": "8300.png", "formula": "\\begin{align*} R _ - ( \\kappa _ x ) = \\frac { E _ { \\text r } ^ - ( \\kappa _ x ) } { E _ { \\text i } ^ + ( \\kappa _ x ) } \\qquad T _ + ( \\kappa _ x ) = \\frac { E _ { \\text t } ^ + ( \\kappa _ x ) } { E _ { \\text i } ^ + ( \\kappa _ x ) } , \\end{align*}"}
{"id": "9450.png", "formula": "\\begin{align*} { g _ 1 } : = f _ { ( 1 , 2 , \\ldots , r , r + 1 ) } \\quad \\mbox { a n d } { g _ 2 } : = f _ { ( 1 , 2 , \\ldots , r , r + 2 ) } . \\end{align*}"}
{"id": "2536.png", "formula": "\\begin{align*} \\pi ( 0 , 0 , \\tau ) \\pi ( x , \\omega , \\tau ' ) = \\pi ( x , \\omega , \\tau + \\tau ' ) = \\pi ( x , \\omega , \\tau ' ) \\pi ( 0 , 0 , \\tau ) , \\end{align*}"}
{"id": "8639.png", "formula": "\\begin{align*} I _ 1 ( t , K ) = a ( K ) \\int _ \\R e ^ { - i 2 y K t + i y ^ 2 t } G ( y + K ) \\frac { \\psi ( y ) } { y } \\chi _ 2 \\big ( y t ^ \\gamma \\big ) \\chi _ 1 ( ( y + K ) t ^ { 3 \\alpha } ) \\ , d y . \\end{align*}"}
{"id": "94.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 3 / 2 } \\cdot ( 2 ^ { ( n _ { 2 , \\nu _ 2 } + 1 ) / 2 } + 1 ) ( 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 ) . \\end{align*}"}
{"id": "7207.png", "formula": "\\begin{align*} \\check { \\mathcal T } _ { t , x , v } : = \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } : = [ t - \\mathcal T _ { t , x _ 1 , v _ 1 } ] _ + . \\end{align*}"}
{"id": "6570.png", "formula": "\\begin{align*} | \\alpha _ { \\nu } | & = \\frac { 2 ^ { \\nu } C _ 0 } { \\log Q } \\ \\ \\ \\nu = 1 , 2 , \\dots , j , \\\\ | \\beta _ { \\nu } | & = \\frac { 2 ^ { j + \\nu } C _ 0 } { \\log Q } \\ \\ \\ \\nu = 1 , 2 , \\dots , \\ell . \\end{align*}"}
{"id": "6650.png", "formula": "\\begin{align*} h ^ { - \\frac { 1 } { 2 } + \\alpha } k ^ { - \\frac { 1 } { 2 } + \\beta } \\mathcal { G } ( 2 - \\alpha - \\beta , \\alpha , \\beta ; A , B , h , k ) = \\mathcal { K } ( 0 , 0 , 2 - \\alpha - \\beta ; A , B , \\alpha , \\beta , h , k ) . \\end{align*}"}
{"id": "5386.png", "formula": "\\begin{align*} \\hat { b } _ { t } = \\frac { 1 } { h } \\int _ { 0 } ^ { T } K \\left ( \\frac { \\tau - t } { h } \\right ) \\mathrm { d } X _ { \\tau } , \\end{align*}"}
{"id": "4721.png", "formula": "\\begin{align*} \\partial _ y \\Phi _ 1 ( t , y ) & = - \\frac { 2 \\mu _ i } { \\sqrt { t } } [ \\phi _ i ^ - ] ' ( y ) - \\frac { 2 \\mu _ { i + 1 } } { \\sqrt { t } } [ \\phi _ { i + 1 } ^ + ] ' ( y ) \\\\ & = - \\frac { 2 \\mu _ i } { \\alpha _ i - \\alpha _ { i + 1 } } \\psi ' \\bigg ( \\frac { 3 ( y / \\sqrt { t } - \\alpha _ i ) } { \\alpha _ i - \\alpha _ { i + 1 } } \\bigg ) + \\frac { 2 \\mu _ { i + 1 } } { \\alpha _ i - \\alpha _ { i + 1 } } \\psi ' \\bigg ( \\frac { 3 ( y / \\sqrt { t } - \\alpha _ i ) } { \\alpha _ i - \\alpha _ { i + 1 } } \\bigg ) \\end{align*}"}
{"id": "7943.png", "formula": "\\begin{align*} p _ { \\ell } = p _ { \\ell - 1 } + ( p _ { \\ell } - p _ { \\ell - 1 } ) \\leqslant \\sqrt { N + 1 } + O ( p _ { \\ell - 1 } ^ { 0 . 5 2 5 } ) \\ll N ^ { 1 / 2 } . \\end{align*}"}
{"id": "7023.png", "formula": "\\begin{align*} \\| B \\| \\le \\Big ( \\sum _ { j = 1 } ^ { L } | y _ j | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\Big ( \\sum _ { j = 1 } ^ { L } \\| B _ j \\| ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\le 1 . \\end{align*}"}
{"id": "6222.png", "formula": "\\begin{align*} A _ { n } x = b _ { n } . \\end{align*}"}
{"id": "6063.png", "formula": "\\begin{align*} \\abs { Z _ 0 ( s , t ) - w } ^ 2 = \\abs { Z _ 0 ( s , t ) - Z _ 0 ( a , b ) } ^ 2 \\ge C _ \\rho \\left [ ( s - a ) ^ 2 + ( t - b ) ^ { 2 \\mu _ 0 } \\right ] \\end{align*}"}
{"id": "8447.png", "formula": "\\begin{align*} t v c = t u b = s x b = s y \\tilde { b } . \\end{align*}"}
{"id": "4868.png", "formula": "\\begin{align*} g : = f - \\tilde f . \\end{align*}"}
{"id": "2683.png", "formula": "\\begin{align*} \\mathfrak { F } _ { ( \\alpha , \\beta ) } ( g ) = \\{ \\alpha \\Z \\times \\beta \\Z \\mid \\alpha \\beta < 1 \\} . \\end{align*}"}
{"id": "8112.png", "formula": "\\begin{align*} { \\mathcal L } ( \\varphi ) = \\sum _ { ( d , e ) \\in D _ { 1 } ( H G ) \\cup D _ { 2 } ( H G ) } \\varepsilon ( d , e ) \\ell _ { D } ( d , e ) . \\end{align*}"}
{"id": "5499.png", "formula": "\\begin{align*} & \\Phi _ { \\gamma , \\delta } ( \\Phi _ { \\gamma , \\delta } ( d , e , s - r ) , e , t - s ) = e ^ { \\gamma ( t - s ) } \\Phi _ { \\gamma , \\delta } ( d , e , s - r ) + \\varphi _ { \\gamma } ( t - s ) ( e + \\delta ) \\\\ & = e ^ { \\gamma ( t - s ) } \\big ( e ^ { \\gamma ( s - r ) } d + \\varphi _ { \\gamma } ( s - r ) ( e + \\delta ) \\big ) + \\varphi _ { \\gamma } ( t - s ) ( e + \\delta ) \\\\ & = e ^ { \\gamma ( t - r ) } d + \\varphi _ { \\gamma } ( t - r ) ( e + \\delta ) = \\Phi _ { \\gamma , \\delta } ( d , e , t - r ) , \\end{align*}"}
{"id": "5506.png", "formula": "\\begin{align*} \\alpha ( t , x ) = a _ n ( x ) , ( t , x ) \\in [ t _ n , t _ { n + 1 } ) \\times X . \\end{align*}"}
{"id": "1059.png", "formula": "\\begin{align*} q | _ { \\mathcal { R } _ { \\xi , I } } ( x , t ) = D ^ { - 2 } _ { \\infty } ( \\xi ) C _ { L } + t ^ { - 1 / 2 } f _ { R _ { \\xi , I } } ( x , t ) + \\mathcal { O } ( t ^ { - 1 } ) , \\end{align*}"}
{"id": "5465.png", "formula": "\\begin{align*} | \\Omega | ^ 2 = \\left ( \\int _ { \\Omega } u _ 0 ^ { - p / 2 } ( x ) u _ 0 ^ { p / 2 } ( x ) d x \\right ) ^ 2 \\le \\int _ \\Omega u _ 0 ^ { - p } ( x ) d x \\cdot \\int _ \\Omega u _ 0 ^ p ( x ) d x \\le M _ 1 ^ * \\int _ \\Omega u _ 0 ^ p ( x ) d x . \\end{align*}"}
{"id": "8874.png", "formula": "\\begin{align*} H X _ b ^ 0 ( X , A ) = \\begin{cases} A & e ( X ) = 0 \\\\ 0 & e ( X ) > 0 ; \\end{cases} \\end{align*}"}
{"id": "7757.png", "formula": "\\begin{align*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } \\end{align*}"}
{"id": "6680.png", "formula": "\\begin{align*} L i _ { C , \\mathfrak { s } } ( { \\bf z } ) : = \\sum _ { i _ 1 > \\cdots > i _ r \\geq 0 } \\frac { z _ 1 ^ { q ^ { i _ 1 } } \\cdots z _ r ^ { q ^ { i _ r } } } { L _ { i _ 1 } ^ { s _ 1 } \\cdots L _ { i _ r } ^ { s _ r } } \\in \\mathbb { C } _ { \\infty } . \\end{align*}"}
{"id": "3134.png", "formula": "\\begin{align*} x _ i x _ k = \\rho U _ { i k } \\ , , \\end{align*}"}
{"id": "9548.png", "formula": "\\begin{align*} Z _ 1 ^ k x ^ k - \\mu ^ k Z _ 1 ^ { ( 0 ) } x ^ { ( 0 ) } = 0 \\end{align*}"}
{"id": "1526.png", "formula": "\\begin{align*} P ^ N \\tilde { \\tau } _ { m } ( G _ { n } \\times G _ { n } ) = \\bigcup _ { \\xi \\in G _ { n } } P ^ N \\tilde { \\tau } _ { m } ( \\xi \\times 1 _ { n } ) = \\bigcup _ { \\xi \\in G _ { n } } P ^ N \\tilde { \\tau } _ { m } ( 1 _ n \\times \\xi ) . \\end{align*}"}
{"id": "6904.png", "formula": "\\begin{align*} U ^ k _ i f ( x _ k ) = \\frac { 1 } { \\prod \\limits _ { j = i + 1 } ^ k R ( j ) } \\sum \\limits _ { x _ { k - 1 } < x _ k } \\ldots \\sum \\limits _ { x _ { i } < x _ { i + 1 } } f ( x _ i ) \\end{align*}"}
{"id": "2146.png", "formula": "\\begin{align*} X _ { n , k } ^ { \\mathcal { S } } = M + \\tau ( A ^ { ( n ) } _ { M , j , l } ) \\ \\ W _ { \\mathcal { S } ( n , k ; M ) } \\cap A ^ { ( n ) } _ { M , j , l } . \\end{align*}"}
{"id": "9.png", "formula": "\\begin{align*} \\kappa ( y ) = - \\psi \\left ( \\frac { 1 } { 2 } \\right ) + \\psi \\left ( \\frac { 1 + \\lvert y \\rvert } { 2 } \\right ) . \\end{align*}"}
{"id": "3016.png", "formula": "\\begin{align*} [ \\sigma _ { y , z } ] = [ D _ { y ^ 4 z } \\cap D _ { y ^ 3 z ^ 2 } ] + [ \\gamma _ { v y ^ 3 z , y ^ 3 z ^ 2 } ] + [ \\gamma _ { w y ^ 3 z , y ^ 3 z ^ 2 } ] - [ \\gamma _ { x y ^ 2 z ^ 2 , y ^ 3 z ^ 2 } ] \\end{align*}"}
{"id": "1966.png", "formula": "\\begin{align*} \\Delta _ \\succ ( a _ 1 \\dotsm a _ n ) : = \\ 1 \\otimes a _ 1 \\dotsm a _ n + \\sum _ { \\substack { 1 \\not \\in S \\subsetneq [ n ] \\\\ S \\neq \\varnothing } } w _ S \\otimes w _ { J ^ S _ 1 } \\vert \\dotsm \\vert w _ { J ^ S _ m } . \\end{align*}"}
{"id": "1796.png", "formula": "\\begin{align*} { \\rm s p e c } _ { L ^ 2 } ( D _ { \\partial } ) \\cap [ - \\alpha , \\alpha ] = \\emptyset . \\end{align*}"}
{"id": "811.png", "formula": "\\begin{align*} I _ f ( u ) = \\int _ { \\overline \\Omega } g _ u ^ p \\ , d \\mu - p \\int _ { \\partial \\Omega } u f \\ , d \\nu & \\le \\int _ { \\overline \\Omega } g ^ p \\ , d \\mu - p \\int _ { \\partial \\Omega } u f \\ , d \\nu \\\\ & = \\lim _ { k \\to \\infty } \\left ( \\int _ { \\overline \\Omega } g _ k ^ p \\ , d \\mu - p \\int _ { \\partial \\Omega } v _ k f \\ , d \\nu \\right ) . \\end{align*}"}
{"id": "7986.png", "formula": "\\begin{align*} \\mathrm { d i s t } \\big ( \\{ ( x _ 1 , \\ldots , x _ d ) \\in [ 0 , 1 ] ^ d : \\ , x _ n = u \\} , f _ { k _ { u } ^ { ( n ) } } ( [ 0 , 1 ] ^ d ) \\big ) \\geq r _ 0 . \\end{align*}"}
{"id": "3754.png", "formula": "\\begin{align*} \\nabla _ { e _ i } R ( X , Y ) = - 2 \\mu _ i R ( X , Y ) , \\end{align*}"}
{"id": "6928.png", "formula": "\\begin{align*} a : V \\times V \\to \\mathbb { R } \\ , , a ( w , v ) = \\int _ \\Omega \\mu \\nabla w \\cdot \\nabla v + \\boldsymbol { \\beta } \\cdot \\nabla w \\ , v + \\sigma w \\ , v \\ , , \\end{align*}"}
{"id": "3823.png", "formula": "\\begin{align*} \\chi _ { f , G } ^ I ( \\sigma , 1 ) = ( - 1 ) ^ n \\chi _ { \\widetilde { f } , \\widetilde { G } } ^ { \\overline { I } } ( \\sigma , 1 ) \\end{align*}"}
{"id": "5536.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { \\xi } _ { r , m } ( t ) & = & A \\xi _ { r , m } ( t ) + b ( t , \\xi _ { r , m } ( t ) ) \\\\ \\xi _ { r , m } ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "591.png", "formula": "\\begin{align*} \\deg ( \\pi ) = 2 . \\end{align*}"}
{"id": "6071.png", "formula": "\\begin{align*} e = R _ { s s } \\cdot N , \\ \\ \\ \\ \\ \\ f = R _ { s t } \\cdot N , \\ \\ \\ \\ \\ g = R _ { t t } \\cdot N . \\end{align*}"}
{"id": "9483.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b c _ { s } ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ { s } ; q ^ { s } ) ^ { ( s - 2 ) / 2 } _ \\infty } { ( - q ^ { s / 2 } ; q ^ { s / 2 } ) _ \\infty } \\sum _ { n \\geq 0 } q ^ { s n ^ 2 / 2 } . \\end{align*}"}
{"id": "5796.png", "formula": "\\begin{align*} ( D ^ 2 , S ^ 1 ) ^ \\sigma = \\left \\{ ( z _ 1 , \\ldots , z _ m ) \\in ( D ^ 2 ) ^ m \\ , : \\ , z _ i \\in S ^ 1 i \\notin \\sigma \\right \\} . \\end{align*}"}
{"id": "4839.png", "formula": "\\begin{align*} P ( x ) : = \\int _ 0 ^ x Q ( y ) \\ , d y x \\in \\R \\end{align*}"}
{"id": "2292.png", "formula": "\\begin{align*} \\langle f , M _ \\omega T _ x g \\rangle = 0 , \\ , \\forall ( x , \\omega ) \\in \\R ^ { 2 d } \\Longrightarrow f = 0 . \\end{align*}"}
{"id": "1671.png", "formula": "\\begin{align*} c ^ * \\theta ^ * - \\Theta _ { 2 , \\rho } > \\frac { ( C ^ { * * } _ 0 + \\Theta _ { 1 , \\rho } ) ^ 2 } { 4 ( \\ll _ 1 + 2 \\gamma - \\theta ^ * - \\Theta _ { 3 , \\rho } ) } = : c _ { 0 , \\rho } ^ * , \\end{align*}"}
{"id": "4805.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in Y } \\alpha ( a ) = \\begin{cases} | Y | & Y \\subseteq a ^ \\perp \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "7987.png", "formula": "\\begin{align*} \\boldsymbol { \\varphi } = \\{ \\varphi _ n ^ { \\sigma } : \\ , \\mathcal { I } _ n ^ { \\sigma } \\to \\mathbb { R } \\ , | \\ , \\sigma \\in \\mathcal { A } , \\ , 1 \\leq n \\leq d \\} \\end{align*}"}
{"id": "125.png", "formula": "\\begin{align*} U ^ 2 \\subseteq V , \\ ; U V \\subseteq U , \\ ; V ^ 2 \\subseteq U , \\ ; U V ^ 2 = 0 . \\end{align*}"}
{"id": "1508.png", "formula": "\\begin{align*} \\mathbf { E } _ l ^ { \\ast } \\left ( \\left [ \\begin{array} { c c } q & \\sigma \\hat { q } \\\\ 0 & \\hat { q } \\end{array} \\right ] , s \\right ) = \\sum _ { h \\in S } c ( h , q , s ) e _ { \\mathbb { A } } ( \\lambda ( h \\sigma ) ) , \\end{align*}"}
{"id": "4235.png", "formula": "\\begin{align*} w = { } & \\frac { 1 } { 2 ( a ^ 2 + c ^ 2 ) } \\bigg ( 1 - a ^ 2 \\left ( x + \\frac { b } { a \\sqrt { D } } \\right ) ^ 2 \\\\ { } & - b \\left ( x + \\frac { b } { a \\sqrt { D } } \\right ) \\left ( y - \\frac { 2 a } { \\sqrt { D } } \\right ) - c ^ 2 \\left ( y - \\frac { 2 a } { \\sqrt { D } } \\right ) ^ 2 \\bigg ) . \\end{align*}"}
{"id": "9024.png", "formula": "\\begin{align*} E ( \\rho , \\phi ) = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } \\epsilon ( x ) | \\nabla \\phi | ^ 2 \\bigg ) d x - \\int _ { \\Gamma _ D } \\epsilon ( x ) \\phi _ D ^ b \\partial _ n \\phi d s + \\frac { 1 } { 2 \\beta _ R } \\int _ { \\Gamma _ R } | \\phi | ^ 2 d s . \\end{align*}"}
{"id": "5178.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ N \\left ( \\frac { \\rho _ \\ell \\beta _ \\ell ^ \\pi } { 2 T _ { \\ell , \\pi } ^ { a v } } + \\frac { \\rho _ \\ell T _ { \\ell , \\pi } ^ { a v } } { 2 } + \\rho _ \\ell \\gamma _ \\ell + \\frac { c _ \\ell } { T _ { \\ell , \\pi } ^ { a v } } \\right ) , \\end{align*}"}
{"id": "6603.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\ell = 1 \\\\ ( \\ell , v ) = 1 } } ^ { \\infty } \\frac { 1 } { \\phi ( u \\ell ) \\ell ^ s } = \\frac { 1 } { \\phi ( u ) } \\zeta ( 1 + s ) R ( s ; u , v ) , \\end{align*}"}
{"id": "7290.png", "formula": "\\begin{align*} \\alpha _ 1 ^ 2 = - \\alpha _ 0 \\alpha _ 2 . \\end{align*}"}
{"id": "8285.png", "formula": "\\begin{align*} P ( n ) : = \\prod _ { v \\in V _ { \\ge 4 } } \\frac { ( n _ { v } + \\deg ( v ) - 3 ) ( n _ { v } + \\deg ( v ) - 4 ) \\cdots ( n _ v + 1 ) } { ( \\deg ( v ) - 3 ) ! } . \\end{align*}"}
{"id": "5601.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle _ \\star : = \\langle \\cdot , \\cdot \\rangle \\circ \\overline { \\mathcal { F } } \\rhd \\colon \\Xi _ \\star \\otimes _ { \\mathcal { X } _ \\star } \\Omega _ \\star \\rightarrow \\mathcal { X } _ \\star . \\end{align*}"}
{"id": "1071.png", "formula": "\\begin{align*} Q ^ { \\pm \\infty } = \\begin{pmatrix} 0 & C \\\\ C & 0 \\end{pmatrix} , V ^ { \\pm \\infty } = 4 k ^ 2 Q ^ { \\pm \\infty } - 2 i k \\sigma _ 3 ( Q ^ { \\pm \\infty } ) ^ 2 + 2 ( Q ^ { \\pm \\infty } ) ^ { 3 } . \\end{align*}"}
{"id": "5750.png", "formula": "\\begin{align*} \\| \\operatorname { H e s s } y ^ j \\| _ { C ^ { 0 , \\alpha } ( B _ { r } ( p ) ) } \\leq \\tau , j = 1 , \\dots , m . \\end{align*}"}
{"id": "3462.png", "formula": "\\begin{align*} B _ l = \\left \\{ Q : \\omega ( Q \\cap \\Omega _ l ) > { 1 \\over 2 } \\omega ( Q ) \\omega ( Q \\cap \\Omega _ { l + 1 } ) \\leqslant { 1 \\over 2 } \\omega ( Q ) \\right \\} \\end{align*}"}
{"id": "7256.png", "formula": "\\begin{align*} 2 a c ^ 3 + 2 b ^ 3 d - b ^ 2 c ^ 2 - 3 a b c d = & b ^ 2 c ^ 2 ( \\dfrac { 2 a c } { b ^ 2 } + \\dfrac { 2 b d } { c ^ 2 } - 1 - \\dfrac { 3 a d } { b c } ) \\\\ = & b ^ 2 c ^ 2 \\cdot \\dfrac { 2 ( n + 1 ) ( - k ^ 2 + k n - k - 1 ) } { ( k + 1 ) ( k + 2 ) ( n - k + 1 ) ( n - k ) } . \\end{align*}"}
{"id": "2553.png", "formula": "\\begin{align*} S J ^ { - 1 } = \\begin{pmatrix} B & - A \\\\ D & - C \\end{pmatrix} , \\end{align*}"}
{"id": "1977.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( x ) = 1 + \\widehat { \\kappa } \\big ( x \\ , \\widehat { \\Phi } ( x ) \\big ) . \\end{align*}"}
{"id": "531.png", "formula": "\\begin{align*} \\bar { \\Xi } _ L & \\stackrel { \\eqref { X i } } { = } \\begin{bmatrix} \\bar { y } _ { 1 , [ L , L + d _ 1 - 1 ] } \\\\ \\vdots \\\\ \\bar { y } _ { m , [ L , L + d _ m - 1 ] } \\end{bmatrix} \\stackrel { \\eqref { l a s t _ d i _ o u t p u t s } } { = } \\begin{bmatrix} H _ { d _ 1 } ( y _ { 1 , [ L , N + d _ 1 - 1 ] } ) \\\\ \\vdots \\\\ H _ { d _ m } ( y _ { m , [ L , N + d _ m - 1 ] } ) \\end{bmatrix} \\alpha \\\\ & = H _ 1 ( \\Xi _ { [ L , N ] } ) \\alpha . \\end{align*}"}
{"id": "1307.png", "formula": "\\begin{align*} \\hat { I } _ { ( \\infty , m ) } ( k ) : = \\bigcup _ { n = 0 } ^ { \\infty } \\hat { I } _ { ( n , m ) } ( k ) \\end{align*}"}
{"id": "2477.png", "formula": "\\begin{align*} \\L ^ \\circ = J \\L ^ \\perp . \\end{align*}"}
{"id": "6979.png", "formula": "\\begin{align*} D ^ * \\Phi ( ( x , 0 ) ; ( u , 0 ) ) ( \\lambda , \\mu ) \\subset \\begin{cases} \\bigl ( D ^ * S ( x ; u ) ( \\mu ) - \\lambda , - \\mu \\bigr ) & - \\lambda \\in \\mathcal N _ \\Omega ( x ^ 1 ; u ^ 1 ) , \\\\ \\varnothing & \\end{cases} \\end{align*}"}
{"id": "301.png", "formula": "\\begin{align*} w ( x , t ) : = u ( x , t ) - \\chi ( x , t ) - v ( x , t ) = \\psi ( x , t ) - v ( x , t ) . \\end{align*}"}
{"id": "3624.png", "formula": "\\begin{align*} \\nu ( t ) & = \\nu _ 1 ( t ) 3 \\le | t | \\le \\exp ( 9 1 . 2 ) , \\\\ \\nu ( t ) & = \\nu _ 2 ( t ) \\exp ( 9 1 . 3 ) \\le | t | \\le \\exp ( 5 4 5 6 3 ) , \\\\ \\nu ( t ) & = \\nu _ 3 ( t ) | t | \\ge \\exp ( 5 4 5 6 3 . 1 ) . \\\\ \\end{align*}"}
{"id": "2977.png", "formula": "\\begin{align*} \\Lambda _ { n , 2 } ^ { ( 1 , 3 ) } & = \\frac { 1 6 } { n ^ 4 \\delta ^ 2 _ n ( 2 ) \\delta ^ 2 _ n ( 3 ) } \\sum _ { r \\ne r ' } ^ d \\sum _ { q _ 1 , q _ 2 = 1 } ^ { r - 1 } \\sum _ { q _ 3 , q _ 4 = 1 } ^ { r ' - 1 } \\sum _ { p _ 1 = 1 } ^ { q _ 1 - 1 } \\sum _ { p _ 2 = 1 } ^ { q _ 2 - 1 } A ( p _ 1 , q _ 1 , p _ 2 , q _ 2 , q _ 3 , q _ 4 ) \\end{align*}"}
{"id": "5025.png", "formula": "\\begin{align*} B _ { k n } = ( \\Phi _ { Z } ^ { k } ) ^ { - 1 } \\circ p B _ { k n } \\circ \\Phi _ { Z } ^ { k } \\end{align*}"}
{"id": "2365.png", "formula": "\\begin{align*} F _ { ( \\xi , \\eta ) } ( x , \\omega ) = e ^ { 2 \\pi i x \\cdot \\omega } V _ g \\left ( M _ \\eta T _ \\xi f \\right ) ( x , \\omega ) \\ , V _ g \\left ( M _ \\eta T _ \\xi f \\right ) ( - x , - \\omega ) \\end{align*}"}
{"id": "2622.png", "formula": "\\begin{align*} Z f ( x + l , \\omega ) = e ^ { 2 \\pi i l \\cdot \\omega } Z f ( x , \\omega ) . \\end{align*}"}
{"id": "5467.png", "formula": "\\begin{align*} ( u ( t , x ; s , u _ 0 ) , v ( t , x ; s , u _ 0 ) ) = ( u ( t , x ; \\tilde s , \\tilde u _ 0 ) , v ( t , x ; \\tilde s , \\tilde u _ 0 ) ) \\forall \\ , t \\ge \\tilde s , \\end{align*}"}
{"id": "3858.png", "formula": "\\begin{align*} \\mathcal P ( 0 ) = U ^ { - 1 } \\tilde U . \\end{align*}"}
{"id": "8316.png", "formula": "\\begin{align*} ( v - y _ i ( t ) ) ( y _ i ' ( t ) - u _ i ' ( t ) ) \\geq 0 \\forall v \\in Z ~ t \\in ( 0 , T ) i = 1 , 2 . \\end{align*}"}
{"id": "2309.png", "formula": "\\begin{align*} | \\langle e , M _ \\omega T _ x f \\rangle | = | V _ f f ( x - \\Delta t , \\omega - \\Delta \\omega ) | = | A f ( x - \\Delta t , \\omega - \\Delta \\omega ) | . \\end{align*}"}
{"id": "7034.png", "formula": "\\begin{align*} \\| \\phi _ j f \\| _ b ^ 2 & = \\| a _ 1 \\widetilde { f } \\phi _ j + \\phi _ j p \\| ^ 2 _ b \\\\ & \\le 2 ( \\| a _ 1 \\widetilde { f } \\phi _ j \\| ^ 2 _ b + \\| \\phi _ j p \\| ^ 2 _ b ) \\\\ & = 2 ( \\| \\widetilde { f } \\phi _ j \\| ^ 2 _ { H ^ 2 } + \\| \\phi _ j p \\| ^ 2 _ b ) . \\end{align*}"}
{"id": "5036.png", "formula": "\\begin{align*} \\Phi _ K ^ { + } = \\Big \\{ & e _ 3 \\ , , e _ 4 \\ , , e _ 1 + e _ 2 \\ , , e _ 1 - e _ 2 \\ , , e _ 3 + e _ 4 \\ , , e _ 3 - e _ 4 \\ , , \\frac { 1 } { 2 } \\ , ( e _ 1 - e _ 2 - e _ 3 - e _ 4 ) \\ , , \\\\ & \\frac { 1 } { 2 } \\ , ( e _ 1 - e _ 2 - e _ 3 + e _ 4 ) \\ , , \\frac { 1 } { 2 } \\ , ( e _ 1 - e _ 2 + e _ 3 - e _ 4 ) \\ , , \\frac { 1 } { 2 } \\ , ( e _ 1 - e _ 2 + e _ 3 + e _ 4 ) \\Big \\} \\ , . \\end{align*}"}
{"id": "4356.png", "formula": "\\begin{align*} \\epsilon ( x ) = h _ + e ^ { \\lambda _ 1 x } + h _ - e ^ { \\lambda _ 2 x } - \\frac { 1 } { \\lambda _ 1 - \\lambda _ 2 } \\int _ { x } ^ \\infty \\left ( e ^ { \\lambda _ 1 ( x - x ' ) } - e ^ { \\lambda _ 2 ( x - x ' ) } \\right ) g ( \\epsilon ( x ' ) ) d x ' , \\end{align*}"}
{"id": "4179.png", "formula": "\\begin{align*} \\left [ ( \\psi _ 1 \\circ f ) ^ * \\lambda \\right ] = f ^ * \\left [ \\lambda + \\mathrm { F l u x } ( \\{ \\psi _ t \\} ) \\right ] = [ \\sigma ' ] - f ^ * \\pi ^ * [ \\sigma ] = 0 , \\end{align*}"}
{"id": "3666.png", "formula": "\\begin{align*} H _ m ( G _ m ( t ) ) = t ^ 2 . \\end{align*}"}
{"id": "6978.png", "formula": "\\begin{align*} \\widehat D ^ * \\Phi ( x , y ) ( \\lambda , \\mu ) = \\begin{cases} \\bigl ( \\widehat D ^ * S ( x ^ 1 , x ^ 2 + y ^ 2 ) ( \\mu ) - \\lambda , - \\mu \\bigr ) & - \\lambda \\in \\widehat { \\mathcal N } _ \\Omega ( x ^ 1 + y ^ 1 ) , \\\\ \\varnothing & \\end{cases} \\end{align*}"}
{"id": "5704.png", "formula": "\\begin{align*} w ( \\alpha ) + ( w i n d ( \\eta _ { \\kappa _ { - } } ) - 1 ) - w ( \\beta ) = \\# ( \\gamma _ { 0 } \\cap { \\pi ( v \\cap { [ - s _ { * } , s _ { * } ] \\times S ^ { 3 } } ) } ) = \\# ( \\mathbb { R } \\times \\gamma _ { 0 } \\cap v ) . \\end{align*}"}
{"id": "1368.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } P _ 0 ^ \\omega \\left ( \\sup _ { 0 \\leq t \\leq T } | \\epsilon \\chi ( X _ { t / \\epsilon ^ 2 } ^ \\omega , \\omega ) | > \\delta \\right ) = 0 . \\end{align*}"}
{"id": "7422.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\sum _ { z = x - \\ell } ^ { x - 1 } \\int _ { \\Omega } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta ) - f ( \\eta ^ { z , z - j \\ell } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "5640.png", "formula": "\\begin{align*} \\mathrm { R } ^ \\mathcal { F } _ { t \\star } ( X , Y , Z ) = \\frac { ( \\overline { \\mathcal { R } } _ 1 \\rhd Y ) \\star \\mathbf { g } _ { t \\star } ( \\overline { \\mathcal { R } } _ 2 \\rhd X , Z ) - X \\star \\mathbf { g } _ { t \\star } ( Y , Z ) } { 2 c } , \\mathrm { R i c } ^ \\mathcal { F } _ { t \\star } ( Y , Z ) = - \\frac { \\mathbf { g } _ { t \\star } ( Y , Z ) } { 2 c } \\end{align*}"}
{"id": "5082.png", "formula": "\\begin{align*} \\frac { B } { A } = e ^ { 2 a \\gamma } \\coth \\left ( \\frac { \\gamma n a } { 2 } \\right ) ^ 2 . \\end{align*}"}
{"id": "1870.png", "formula": "\\begin{align*} \\sum _ { n = 0 } P _ n ( x ) { t ^ { n + 1 } \\over ( n + 1 ) ! } = \\log \\left ( \\sum _ { n = 0 } ^ \\infty Q _ n ( x ) { t ^ n \\over n ! } \\right ) = \\log \\left ( { 1 \\over \\cos ( t ) - x \\sin ( t ) } \\right ) , \\end{align*}"}
{"id": "941.png", "formula": "\\begin{align*} \\langle N _ { i } ^ { { \\bf m } * } , N _ \\ell \\rangle = 0 , \\ell \\in { \\bf m _ 1 } . \\end{align*}"}
{"id": "2027.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } h ( X _ t ) - h ( X _ 0 ) = M _ t ^ { h } + N _ t ^ { h } , \\\\ N _ t ^ { h } = - \\int _ 0 ^ t h ( X _ s ) { \\rm d } A _ s ^ { \\overline { \\mu } } \\end{array} \\right . \\end{align*}"}
{"id": "4937.png", "formula": "\\begin{align*} \\eta _ n = \\lambda ^ { - 1 } _ n \\phi ( \\lambda _ n ) \\cdot h ^ { - 1 } , \\lambda _ { n + 1 } = \\lambda _ n \\cdot \\eta _ n , \\end{align*}"}
{"id": "4544.png", "formula": "\\begin{align*} \\frac { 2 + 2 / r ^ 2 } { ( r - 1 ) ( r - 2 ) } ( h - 1 ) = \\frac { 1 } { \\binom { r - 1 } { 2 } } ( h - 1 ) + O \\biggl ( \\frac { h } { r ^ 4 } \\biggr ) . \\end{align*}"}
{"id": "7674.png", "formula": "\\begin{align*} | 2 w - e ^ { - \\beta _ { \\varepsilon } \\Psi _ { \\varepsilon } ( \\mathbf { q } ) } V _ { \\varepsilon } | = w + P _ { \\Omega _ { \\varepsilon } } w \\le 2 | w | . \\end{align*}"}
{"id": "2945.png", "formula": "\\begin{align*} \\zeta ^ { ( h ) } _ n = \\frac { 4 } { n ^ 2 \\delta ^ 2 _ n ( k ) } \\cdot | \\mathcal P ( d , k ) | \\cdot | \\mathcal I _ h | \\cdot O ( n ^ { k ( 2 - h ) } ) = O ( n ^ { ( k - 1 ) ( 2 - h ) } ) = o ( 1 ) , \\end{align*}"}
{"id": "4911.png", "formula": "\\begin{align*} \\Gamma _ t & = \\prod _ { i = 1 } ^ { t } ( 1 - \\alpha _ i ) = \\prod _ { i = 2 } ^ { t } \\frac { k - 1 } { k } = \\frac { 1 } { k } \\\\ \\end{align*}"}
{"id": "1720.png", "formula": "\\begin{align*} \\tilde \\theta = s _ * \\frac { \\alpha _ * + \\gamma _ * / p _ 0 \u2010 \\gamma _ * / q } { \\mu _ * + \\alpha _ * + \\gamma _ * ( s _ * + 1 / p _ 0 \u2010 1 / p _ 1 ) } , \\end{align*}"}
{"id": "8509.png", "formula": "\\begin{align*} \\alpha * \\beta & = ( s _ 1 \\circ a _ { j _ 1 } \\circ \\cdots a _ { j _ f } ) * ( s _ 2 + y _ { l _ 1 } + \\cdots + y _ { l _ g } ) \\\\ & = ( a _ { j _ 1 } \\circ \\cdots a _ { j _ f } ) * ( y _ { l _ 1 } + \\cdots + y _ { l _ g } ) \\end{align*}"}
{"id": "7526.png", "formula": "\\begin{align*} \\arg \\Gamma \\left ( \\frac { \\sigma + i T } { 2 } \\right ) & = \\frac { T } { 4 } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) + \\left ( \\frac { \\sigma - 1 } { 2 } \\right ) \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) - \\frac { T } { 2 } + o ( 1 ) \\end{align*}"}
{"id": "5040.png", "formula": "\\begin{align*} \\mathcal G _ 0 ( X + \\xi , Y + \\eta ) : = \\frac { 1 } { 2 } ( \\xi ( Y ) + \\eta ( X ) ) . \\end{align*}"}
{"id": "4150.png", "formula": "\\begin{align*} N _ 1 : = \\sup _ { [ - T , T ] } \\| u ( t ) \\| _ { Z _ { s , 3 + \\theta } } \\end{align*}"}
{"id": "381.png", "formula": "\\begin{align*} V + a ^ j Y _ j = W + b ^ k \\overline { Y } _ k , \\end{align*}"}
{"id": "5419.png", "formula": "\\begin{align*} x & = \\sqrt { ( b '' ) ^ 2 + ( c ' ) ^ 2 - 2 ( b '' ) ( c ' ) \\cos \\gamma } \\\\ y & = \\sqrt { ( a ' ) ^ 2 + ( c '' ) ^ 2 - 2 ( a ' ) ( c '' ) \\cos \\alpha } \\\\ z & = \\sqrt { ( a '' ) ^ 2 + ( b ' ) ^ 2 - 2 ( a '' ) ( b ' ) \\cos \\beta } \\end{align*}"}
{"id": "6097.png", "formula": "\\begin{align*} \\begin{array} { l } \\alpha _ { 1 1 } ( r , \\theta ) = { \\left ( z _ { t t } - z _ { s s } \\right ) \\sin \\theta \\cos \\theta + 2 z _ { s t } \\cos ^ { 2 } \\theta } , \\\\ \\alpha _ { 1 2 } ( r , \\theta ) = { - z _ { s s } } , \\\\ \\alpha _ { 2 1 } ( r , \\theta ) = { z _ { t t } } , \\\\ \\alpha _ { 2 2 } ( r , \\theta ) = { \\left ( z _ { t t } - z _ { s s } \\right ) \\sin \\theta \\cos \\theta - 2 z _ { s t } \\sin ^ { 2 } \\theta } , \\end{array} \\end{align*}"}
{"id": "7814.png", "formula": "\\begin{align*} \\Phi _ { a , b } ( t ) : = t ^ { a } A ^ b S ( t ) Q ^ \\frac { 1 } { 2 } , t \\in ( 0 , \\infty ) . \\end{align*}"}
{"id": "4872.png", "formula": "\\begin{align*} f _ 4 ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ { k + 1 } } { k ! ( k + 1 ) ! } = z + \\frac { z ^ 2 } 2 + \\frac { z ^ 3 } { 1 2 } + \\ldots \\end{align*}"}
{"id": "292.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { i \\gamma t \\xi ^ { 3 } } { 1 + \\xi ^ { 2 } } \\right ) - 1 & = \\frac { i \\gamma t \\xi ^ { 3 } } { 1 + \\xi ^ { 2 } } \\exp \\left ( \\frac { i \\gamma \\theta _ { 1 } t \\xi ^ { 3 } } { 1 + \\xi ^ { 2 } } \\right ) , \\\\ 1 - \\exp \\left ( \\frac { - t \\xi ^ { 4 } } { 1 + \\xi ^ { 2 } } \\right ) & = \\frac { t \\xi ^ { 4 } } { 1 + \\xi ^ { 2 } } \\exp \\left ( \\frac { - \\theta _ { 2 } t \\xi ^ { 4 } } { 1 + \\xi ^ { 2 } } \\right ) . \\end{align*}"}
{"id": "4078.png", "formula": "\\begin{align*} g ( x ) - y = g ( z + t ) - Q _ z ( t ) = \\int _ z ^ { z + t } ( g '' ( u ) + A ) ( z + t - u ) \\ , d u \\in \\ge C M | t | ^ 2 . \\end{align*}"}
{"id": "6150.png", "formula": "\\begin{align*} \\sum _ { \\nu = 0 } ^ p ( - 1 ) ^ { \\nu } { x + \\nu \\choose \\nu + 1 } { x \\choose p - \\nu } = { x \\choose p + 1 } . \\end{align*}"}
{"id": "4686.png", "formula": "\\begin{align*} B _ { i j } ( t , y ) = B _ { i j , 0 } ( y ) + \\sum _ { k = 1 } ^ n \\mu _ k ( t ) \\mathfrak { B } _ { i j k } ( y ) . \\end{align*}"}
{"id": "7181.png", "formula": "\\begin{align*} g ( k ) = g ( | k | ) , \\end{align*}"}
{"id": "730.png", "formula": "\\begin{align*} C _ b = 0 , C _ W = \\sigma _ 1 ^ { - 2 } , \\end{align*}"}
{"id": "4934.png", "formula": "\\begin{align*} & F _ { 1 , j } = \\Big \\{ x \\in \\prod _ { i = 1 } ^ d B _ { i } ( y _ i , 2 r _ n ^ { a _ i } ) \\colon \\max _ { 1 \\le i \\le j } \\rho _ i ( x _ i , y _ i ) < 4 r _ n ^ { a _ j } \\Big \\} , \\\\ & F _ { 2 , j } = \\Big \\{ x \\in \\prod _ { i = 1 } ^ d B _ { i } ( y _ i , 2 r _ n ^ { a _ i } ) \\colon \\max _ { 1 \\le i \\le j - 1 } \\rho _ i ( x _ i , y _ i ) \\ge r _ n ^ { a _ j } \\Big \\} . \\end{align*}"}
{"id": "5462.png", "formula": "\\begin{align*} \\tilde M _ 2 ( p , q , \\tau , s , u _ 0 ) = & C _ 6 \\Big ( \\int _ \\Omega u ^ { - p } ( \\tau , x ) d x + M _ 1 ( p ) + \\tilde M _ 1 ( p , \\tau , s , u _ 0 ) \\Big ) ^ { \\frac { 2 ( q + 1 ) ( q + 1 - \\varepsilon _ 0 / 2 ) } { \\varepsilon _ 0 p } } \\\\ & - C _ 6 \\Big ( 2 M _ 1 ( p ) \\Big ) ^ { \\frac { 2 ( q + 1 ) ( q + 1 - \\varepsilon _ 0 / 2 ) } { \\varepsilon _ 0 p } } . \\end{align*}"}
{"id": "8641.png", "formula": "\\begin{align*} b ^ + _ + ( k ) + b _ - ^ + ( k ) = \\mathbf { 1 } _ { + } ( k ) \\big ( T ( k ) + 1 \\big ) + \\mathbf { 1 } _ { - } ( k ) \\big ( 1 + T ( - k ) \\big ) , \\end{align*}"}
{"id": "4591.png", "formula": "\\begin{align*} 1 - \\Phi \\left ( \\underline { \\lambda } \\right ) = \\Big ( 1 - \\Phi ( x ) \\Big ) \\exp \\bigg \\{ - \\theta ( x ) c _ 1 ( 1 + x ) \\Big ( x ^ 2 ( \\epsilon _ n + \\delta _ n ) + x \\delta _ n \\sqrt { | \\ln \\delta _ n | } \\Big ) \\bigg \\} , \\end{align*}"}
{"id": "5933.png", "formula": "\\begin{align*} [ X _ a , X _ b ] & = { \\rm H o r } ( [ X _ a , X _ b ] ) + { \\rm V e r } ( [ X _ a , X _ b ] ) = [ \\overline { X } _ a , \\overline { X } _ b ] ^ h + { \\rm V e r } ( [ X _ a , X _ b ] ) \\\\ & = { \\rm V e r } ( [ X _ a , X _ b ] ) . \\end{align*}"}
{"id": "5404.png", "formula": "\\begin{align*} G ( \\theta ) = \\int _ { a } ^ { b } \\big ( \\partial _ { \\theta } f ( x , \\theta ) \\big ) ^ { 2 } \\pi _ { a , b } ( x ) \\mathrm { d } x . \\end{align*}"}
{"id": "8465.png", "formula": "\\begin{align*} \\mathbb { E } _ { i _ 1 , i _ 2 , \\ldots , i _ { j } } \\langle y , Q _ j x \\rangle = \\mathbb { E } _ { i _ 1 , i _ 2 , \\ldots , i _ { j - 1 } } \\langle \\mathcal { L } y , Q _ { j - 1 } x \\rangle = \\mathbb { E } _ { i _ 1 , i _ 2 , \\ldots , i _ { j - 2 } } \\langle \\mathcal { L } ^ 2 y , Q _ { j - 2 } x \\rangle = \\cdots = \\langle \\mathcal { L } ^ j y , x \\rangle . \\end{align*}"}
{"id": "9087.png", "formula": "\\begin{align*} Q _ 1 ( 0 , b ) = \\int _ b ^ { \\infty } x e ^ { - \\frac { x ^ 2 } { 2 } } d x = e ^ { - \\frac { b ^ 2 } { 2 } } b \\geq 0 . \\end{align*}"}
{"id": "5292.png", "formula": "\\begin{align*} ( ( \\sigma ^ { \\varphi } ) ^ { n / 2 } S ^ { n } ( b ) ) ^ * \\delta _ { \\varphi } ^ n ( ( \\sigma ^ { \\varphi } ) ^ { n / 2 } S ^ { n } ( b ) ) = 0 . \\end{align*}"}
{"id": "3110.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + a z y ^ 3 x ^ 2 + x ^ 6 + b x ^ 3 y ^ 3 + y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "7684.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ { \\varepsilon _ { n _ k } } = \\tilde { \\lambda } _ 0 + \\Phi \\ , e ^ { - \\beta _ { \\varepsilon _ { n _ k } } \\tilde { \\Psi } _ { \\varepsilon _ { n _ k } } ( \\mathbf { x } _ { \\varepsilon _ { n _ k } } ) } + o \\biggl ( e ^ { - \\beta _ { \\varepsilon _ { n _ k } } \\tilde { \\Psi } _ { \\varepsilon _ { n _ k } } ( \\mathbf { x } _ { \\varepsilon _ { n _ k } } ) } \\biggr ) k \\to + \\infty \\ ; , \\end{align*}"}
{"id": "6474.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n \\log n } } \\right ) ^ k \\right ] = \\mu _ k ( k \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "936.png", "formula": "\\begin{align*} P _ n ^ { k } X _ { n + 1 } = X _ n , n \\geq 1 . \\end{align*}"}
{"id": "88.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 \\end{align*}"}
{"id": "149.png", "formula": "\\begin{align*} \\mathcal { P } _ t = ( P ^ { \\nu _ \\alpha } _ { \\frac { t } { \\alpha } } ) ^ * \\circ P ^ { \\nu _ \\alpha } _ { \\frac { t } { \\alpha } } = P ^ { \\nu _ \\alpha } _ { \\frac { t } { \\alpha } } \\circ ( P ^ { \\nu _ \\alpha } _ { \\frac { t } { \\alpha } } ) ^ * . \\end{align*}"}
{"id": "1191.png", "formula": "\\begin{align*} \\mathcal { N } _ \\mu ( \\mathcal { B } , \\mathcal { U } , s ) & = \\left \\{ n \\in \\mathbb { N } \\ : s _ \\mu ( B _ n , U _ n ) \\geq s \\right \\} . \\end{align*}"}
{"id": "6398.png", "formula": "\\begin{align*} \\Lambda = & \\{ ( s _ 1 , \\dots , s _ { n - 1 } , p ) \\in \\mathbb G _ n \\ , : \\\\ & \\ ; ( s _ 1 , \\dots , s _ { n - 1 } ) \\in \\sigma _ T ( F _ 1 ^ * + p F _ { n - 1 } \\ , , \\ , F _ 2 ^ * + p F _ { n - 2 } \\ , , \\ , \\dots \\ , , F _ { n - 1 } ^ * + p F _ 1 ) \\} , \\end{align*}"}
{"id": "5757.png", "formula": "\\begin{align*} \\begin{cases} e ^ { - 2 Q } \\delta _ { s t } \\leq \\bar { h } _ { s t } \\leq e ^ { 2 Q } \\delta _ { s t } ; \\\\ ( r / 2 ) ^ { 1 + \\alpha } \\left \\| \\frac { \\partial } { \\partial x ^ { j } } \\bar { h } _ { s t } \\right \\| _ { C ^ { 0 , \\alpha } ( B _ { r / 2 } ( \\bar { p } ) ) } \\leq e ^ { 2 Q } , \\end{cases} \\end{align*}"}
{"id": "1205.png", "formula": "\\begin{align*} { A ( r ) = \\left \\{ x \\in A \\ : \\ \\forall \\tilde { r } \\leq r , \\ m ( B ( x , \\tilde { r } ) \\cap A ) \\geq \\frac { 3 } { 4 } m ( B ( x , \\tilde { r } ) ) \\right \\} } \\end{align*}"}
{"id": "311.png", "formula": "\\begin{align*} W ( \\rho ^ 0 , \\rho ^ 1 ) : = \\inf _ { v } \\Big \\{ \\sqrt { \\int _ { 0 } ^ 1 \\ < v , v \\ > _ { \\theta ( \\rho ) } } d t \\ , \\ : \\ , \\frac { d \\rho } { d t } + d i v _ G ^ { \\theta } ( \\rho v ) = 0 , \\ ; \\rho ( 0 ) = \\rho ^ 0 , \\ ; \\rho ( 1 ) = \\rho ^ 1 \\Big \\} . \\end{align*}"}
{"id": "1186.png", "formula": "\\begin{align*} \\mathcal { H } ^ { s } _ { t } ( E ) = \\inf \\left \\{ \\sum _ { n \\in \\mathbb { N } } \\vert B _ n \\vert ^ s : \\ , \\vert B _ n \\vert \\leq t , \\ B _ n E \\subset \\bigcup _ { n \\in \\mathbb { N } } B _ n \\right \\} . \\end{align*}"}
{"id": "1393.png", "formula": "\\begin{align*} ( n _ 1 - n _ 2 ) _ x = & \\frac { n _ 1 ^ 3 E _ 1 - \\alpha n _ 1 ^ 2 } { n _ 1 ^ 2 - 1 } - \\frac { n _ 2 ^ 3 E _ 2 - \\alpha n _ 2 ^ 2 } { n _ 2 ^ 2 - 1 } \\\\ = & E _ 1 \\left ( f ( n _ 1 ) - f ( n _ 2 ) \\right ) + f ( n _ 2 ) ( E _ 1 - E _ 2 ) - \\alpha \\left ( g ( n _ 1 ) - g ( n _ 2 ) \\right ) \\\\ = & \\left ( E _ 1 f ' ( \\bar { \\eta } ) - \\alpha g ' ( \\tilde { \\eta } ) \\right ) ( n _ 1 - n _ 2 ) + f ( n _ 2 ) ( E _ 1 - E _ 2 ) , \\quad \\exists \\bar { \\eta } , \\tilde { \\eta } \\in ( n _ 2 , n _ 1 ) , \\end{align*}"}
{"id": "8872.png", "formula": "\\begin{align*} ( \\phi \\vee \\psi ) ( x _ 0 , \\ldots , x _ { p + q } ) = \\phi ( x _ 0 , \\ldots , x _ p ) \\psi ( x _ p , \\ldots , x _ { p + q } ) . \\end{align*}"}
{"id": "6930.png", "formula": "\\begin{align*} a ( u , v ) = F ( v ) \\forall v \\in V \\ , . \\end{align*}"}
{"id": "3989.png", "formula": "\\begin{align*} \\begin{dcases} - \\int _ { 0 } ^ { T } \\overline { e ^ { \\lambda ^ p _ k ( T - t ) } } p ( t ) \\ , d t & = m _ { 1 , k } , \\forall k \\geq k _ 0 , \\\\ - \\int _ { 0 } ^ { T } \\overline { e ^ { \\widehat \\lambda _ { n _ j } ( T - t ) } } p ( t ) \\ , d t & = m _ { 1 , j } , \\forall \\ , 1 \\leq j \\leq j _ 0 , \\end{dcases} \\end{align*}"}
{"id": "6181.png", "formula": "\\begin{align*} \\mathcal { D } _ { M _ { i , : } } ( j ) = \\frac { | M _ { i , j } | ^ 2 } { | | M _ { i , : } | | ^ 2 _ { 2 } } . \\end{align*}"}
{"id": "4080.png", "formula": "\\begin{align*} J _ n ' ( y ) = ( n / 2 + \\beta + 1 / 2 ) J _ { n - 1 } ^ { ( \\beta + 1 , \\beta + 1 ) } ( y ) . \\end{align*}"}
{"id": "3746.png", "formula": "\\begin{align*} V = E _ 0 \\oplus E _ 1 \\oplus \\ldots \\oplus E _ r , \\end{align*}"}
{"id": "1459.png", "formula": "\\begin{align*} \\lambda ( \\alpha , z ) ^ { \\ast } \\eta ( \\alpha z ) \\lambda ( \\alpha , z ) = \\eta ( z ) , \\ , \\ , \\ , \\delta ( \\alpha z ) = | j ( \\alpha , z ) | ^ { - 2 } \\delta ( z ) . \\end{align*}"}
{"id": "3931.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\omega _ { V ( 0 ) } \\circ \\alpha _ T ^ { 0 \\to \\tau } ( \\dot V ( \\tau ) ) = \\omega _ { V ( \\tau ) } ( \\dot V ( \\tau ) ) . \\end{align*}"}
{"id": "3566.png", "formula": "\\begin{align*} \\langle \\mathfrak { x } _ 1 , \\mathfrak { y } _ 1 , \\ldots , \\mathfrak { x } _ b , \\mathfrak { y } _ b , \\mathfrak { z } _ 1 , \\ldots , \\mathfrak { z } _ { s } \\rangle & = G , \\\\ \\mathfrak { z } _ 1 + \\cdots + \\mathfrak { z } _ { s } & = 0 , \\\\ b - 1 + \\sum \\limits _ { i = 1 } ^ { s } \\Big ( 1 - \\frac { 1 } { | \\mathfrak { z } _ { i } | } \\Big ) & = \\frac { g - 1 } { | G | } . \\end{align*}"}
{"id": "1169.png", "formula": "\\begin{align*} \\left ( \\frac { d \\psi } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 \\psi \\right ) _ { + } = \\left ( \\frac { d \\psi } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 \\psi \\right ) _ { - } J ^ { \\psi } . \\end{align*}"}
{"id": "9495.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { a - 1 } \\binom { a + b - 2 } { \\lfloor i / 2 \\rfloor } \\left ( \\binom { a + b - 2 - \\lfloor i / 2 \\rfloor } { a - i - 1 } + \\binom { a + b - 2 - \\lfloor i / 2 \\rfloor } { a - i - 2 } \\right ) , \\end{align*}"}
{"id": "8562.png", "formula": "\\begin{align*} a : = \\psi _ + ( - \\infty , 0 ) \\in \\R \\setminus \\{ 0 \\} . \\end{align*}"}
{"id": "8907.png", "formula": "\\begin{align*} \\eta ^ 1 _ { A _ Y } ( V ) : A _ Y ( V ) & \\to \\alpha _ * \\alpha ^ { - 1 } A _ Y ( V ) = A _ Y ( \\alpha \\circ \\alpha ^ { - 1 } ( V ) ) \\\\ \\varphi & \\mapsto \\varphi | _ { \\alpha \\circ \\alpha ^ { - 1 } ( W ) } \\end{align*}"}
{"id": "2318.png", "formula": "\\begin{align*} A ( M _ \\eta T _ \\xi f , M _ \\eta T _ \\xi g ) ( x , \\omega ) = e ^ { 2 \\pi i ( x \\cdot \\eta - \\omega \\cdot \\xi ) } A ( f , g ) ( x , \\omega ) . \\end{align*}"}
{"id": "984.png", "formula": "\\begin{align*} u ( a ) = \\tau d ^ { - n - 2 } , \\end{align*}"}
{"id": "5720.png", "formula": "\\begin{align*} \\int _ M \\nabla _ g u ( x ) \\nabla _ g w ( x ) { \\rm d } v _ g - \\mu \\int _ M \\frac { u ( x ) w ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } { \\rm d } v _ g + \\int _ M u ( x ) w ( x ) { \\rm d } v _ g = \\lambda \\int _ M \\alpha ( x ) \\xi _ x w ( x ) { \\rm d } v _ g . \\end{align*}"}
{"id": "2809.png", "formula": "\\begin{align*} \\| u - Q \\| _ { H ^ 1 } < \\varepsilon _ 0 \\Rightarrow \\exists ! \\ ; ( \\sigma , X ) , \\ ; | \\sigma | + | X | \\le \\eta _ 0 J ( \\sigma , X , u ) = 0 . \\end{align*}"}
{"id": "7260.png", "formula": "\\begin{align*} ( 4 + \\dfrac { 3 a c } { b ^ 2 } - \\dfrac { 6 b d } { c ^ 2 } - \\dfrac { c ^ 2 } { b d } ) = & \\dfrac { 2 ( n + 1 ) [ k ^ 3 - ( 2 n + 2 ) k ^ 2 + ( n ^ 2 + 3 n - 5 ) k - n ^ 2 + 2 n - 3 ] } { ( k + 1 ) ( k + 2 ) ( n - k + 1 ) ( n - k ) ( n - k - 1 ) } . \\end{align*}"}
{"id": "195.png", "formula": "\\begin{align*} \\tilde { f } _ { \\delta } = \\int _ 0 ^ { + \\infty } P ^ { \\delta } _ t ( g ) d t , \\end{align*}"}
{"id": "7166.png", "formula": "\\begin{align*} w ^ \\mu = \\lambda _ \\mu ^ { - 1 } ( v ^ \\mu - u ^ \\mu ) , k ^ \\mu = \\lambda _ \\mu ^ { - 1 } ( H _ u ^ \\mu - H _ v ^ \\mu ) , \\tau ^ \\mu = \\lambda _ \\mu ^ { - 1 } ( \\xi ^ \\mu - \\eta ^ \\mu ) . \\end{align*}"}
{"id": "4427.png", "formula": "\\begin{align*} ( u , v ) _ { H ^ 2 _ * } = \\int _ U u _ { x x } v _ { x x } d x . \\end{align*}"}
{"id": "6077.png", "formula": "\\begin{align*} \\varphi _ { s } ^ { j } & = R _ { s s } \\cdot U ^ { j } + F ^ { j } = x _ { s s } u ^ { j } + y _ { s s } v ^ { j } + z _ { s s } w ^ { j } + F ^ { j } , \\\\ \\psi _ { t } ^ { j } & = R _ { t t } \\cdot U ^ { j } + H ^ { j } = x _ { t t } u ^ { j } + y _ { t t } v ^ { j } + z _ { t t } w ^ { j } + H ^ { j } , \\\\ \\varphi _ { t } ^ { j } + \\psi _ { s } ^ { j } & = 2 R _ { s t } \\cdot U ^ { j } + G ^ { j } = 2 \\left [ x _ { s t } u ^ { j } + y _ { s t } v ^ { j } + z _ { s t } w ^ { j } \\right ] + G ^ { j } . \\end{align*}"}
{"id": "4568.png", "formula": "\\begin{align*} \\mathbf { P } \\bigg ( \\Big | \\frac 1 n \\sum _ { i = 1 } ^ n \\mathbf { E } ( \\eta _ i ^ 2 | \\mathcal { F } _ { i - 1 } ) - \\sigma ^ 2 \\Big | \\geq x \\bigg ) \\leq \\frac { 1 } { H } \\exp \\bigg \\{ - H x ^ 2 n \\bigg \\} . \\end{align*}"}
{"id": "1274.png", "formula": "\\begin{align*} J _ { 0 } ( \\alpha , \\beta , Z ) - J _ { 0 } ( \\alpha , \\beta , Z ' ) = < - c _ { 1 } ( \\xi ) + 2 \\mathrm { P D } ( \\Gamma ) , Z - Z ' > . \\end{align*}"}
{"id": "3340.png", "formula": "\\begin{align*} [ T u , T v , T w ] = T \\Big ( D ( T u , T v ) w + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v \\Big ) , \\forall u , v , w \\in V . \\end{align*}"}
{"id": "2313.png", "formula": "\\begin{align*} A b _ 0 ( x , \\omega ) = \\int _ \\R b _ 0 ( t + \\tfrac { x } { 2 } ) b _ 0 ( t - \\tfrac { x } { 2 } ) e ^ { - 2 \\pi i \\omega \\cdot t } \\ , d t . \\end{align*}"}
{"id": "3935.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } s ( \\nu _ 0 \\circ \\alpha _ { T \\Psi } ^ { 0 \\to \\tau } | \\nu _ \\tau ) = 0 . \\end{align*}"}
{"id": "5279.png", "formula": "\\begin{align*} \\check { \\psi } ( \\omega ) = \\varepsilon ( a ) , \\omega = \\psi ( a - ) , a \\in A . \\end{align*}"}
{"id": "2327.png", "formula": "\\begin{align*} \\mathcal { T } _ s F ( x , t ) = F ( x + \\tfrac { t } { 2 } , x - \\tfrac { t } { 2 } ) . \\end{align*}"}
{"id": "6494.png", "formula": "\\begin{align*} \\bar { g } _ n ^ { ( 2 m - 1 ) } : = \\prod _ { k = j _ 0 + 1 } ^ { n - 1 } g ^ { ( 2 m - 1 ) } _ k ( n = j _ 0 + 1 , j _ 0 + 2 , \\ldots ) . \\end{align*}"}
{"id": "6542.png", "formula": "\\begin{align*} h ^ 0 ( X , \\iota _ * F ) \\leq \\dfrac { \\chi ( X , \\iota _ * F ) } { 2 } + \\dfrac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } \\lVert \\overline { p _ i p _ { i - 1 } } \\rVert \\end{align*}"}
{"id": "5641.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = \\varepsilon ^ { - 2 } \\left ( e ^ { u } - \\left ( x ^ { 2 } + y ^ { 2 } \\right ) e ^ { - u } \\right ) & B _ 1 ( 0 ) , \\\\ u & = 0 & \\partial B _ 1 ( 0 ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "8869.png", "formula": "\\begin{align*} H o m ( X \\times \\R , Y ) = H o m ( X , \\Omega Y ) . \\end{align*}"}
{"id": "489.png", "formula": "\\begin{align*} k _ { 1 } ( \\alpha _ { 1 } , \\alpha _ { 2 } ) : = \\mathcal { I } , k _ { 1 } ( i _ { u } , \\alpha _ { w } ) : = \\mathcal { I } _ { \\alpha _ { w } } ^ { i _ { u } } , k _ { 1 } ( i _ { s } , i _ { u } ) : = \\mathcal { I } _ { \\alpha _ { 1 } \\alpha _ { 2 } } ^ { i _ { s } i _ { u } } \\end{align*}"}
{"id": "4897.png", "formula": "\\begin{align*} \\frac { f ' } { f } = P \\psi , \\end{align*}"}
{"id": "1895.png", "formula": "\\begin{align*} \\Sigma = \\dot \\bigcup _ { F \\in \\mathcal { F } _ P } N _ F \\end{align*}"}
{"id": "908.png", "formula": "\\begin{align*} e _ { \\lambda } ^ { x } ( t ) = \\sum _ { k = 0 } ^ { \\infty } ( x ) _ { n , \\lambda } \\frac { t ^ { k } } { k ! } , ( \\mathrm { s e e } \\ [ 6 , 7 ] ) , \\end{align*}"}
{"id": "7504.png", "formula": "\\begin{align*} \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ n } { \\tau } = \\left ( \\frac { 1 } { 2 } \\Delta - V - \\beta | \\phi ^ n | ^ 2 + \\Omega L _ z + \\mu ^ n \\right ) \\phi ^ n , \\end{align*}"}
{"id": "5863.png", "formula": "\\begin{align*} H ' \\cap \\{ 0 , \\dots , k - 1 \\} = \\left ( H ' \\cap \\{ 0 , \\dots , j - 1 \\} \\right ) \\cup \\{ j \\} . \\end{align*}"}
{"id": "1091.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & D _ { + } ( k ) = D _ { - } ( k ) ( 1 - r r ^ * ) , & k \\in ( - \\eta , - C _ L ) \\cup ( C _ L , \\eta ) , \\\\ & D _ { + } ( k ) D _ { - } ( k ) = r _ { + } ( k ) , & k \\in I _ L \\backslash I _ R , \\\\ & D _ { + } D _ { - } = 1 , & k \\in I _ R . \\end{aligned} \\right . \\end{align*}"}
{"id": "4154.png", "formula": "\\begin{align*} \\tilde M : = \\sup _ { [ - T , T ] } \\{ \\| u ( t ) \\| _ { Z _ { s , r } } + \\| v ( t ) \\| _ { Z _ { s , r } } \\} < \\infty . \\end{align*}"}
{"id": "4411.png", "formula": "\\begin{align*} \\tilde \\psi _ { i , \\beta } ( y ) = - \\phi _ { i , \\infty , \\beta } \\int _ 1 ^ y \\frac { ( y ' ) ^ { - ( d + 1 ) } e ^ { 2 \\beta \\frac { ( y ' ) ^ 2 } { 4 } } } { \\phi ^ 2 _ { i , \\infty , \\beta } ( y ' ) } d y ' . \\end{align*}"}
{"id": "7017.png", "formula": "\\begin{align*} | \\widehat B ( \\alpha _ i ) | = \\frac { | \\beta _ k | | B ( \\alpha _ i ) | } { 2 D | \\alpha _ i - \\beta _ k | } \\geqslant \\frac { 1 } { 3 D } | B ( \\alpha _ i ) | \\geqslant \\frac { \\delta } { 3 D } . \\end{align*}"}
{"id": "8628.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ 2 \\mu ( k , \\ell , m , n ) = \\mu _ S ( k , \\ell , m , n ) + \\mu _ R ( k , \\ell , m , n ) ; \\end{align*}"}
{"id": "1714.png", "formula": "\\begin{align*} d _ n ( \\nu _ 0 B _ { p _ 0 } ^ N \\cap \\nu _ 1 B _ { p _ 1 } ^ N , \\ , l _ q ^ N ) = d _ n ( \\nu _ 0 B _ { p _ 0 } ^ N , \\ , l _ q ^ N ) . \\end{align*}"}
{"id": "3421.png", "formula": "\\begin{align*} T f _ 1 ( x ) & = \\langle K ( x , y ) u ( y ) f _ 1 ( y ) \\rangle \\\\ & = \\int _ { \\R ^ N } K ( x , y ) u ( y ) [ f _ 1 ( y ) - f _ 1 ( x ) ] d \\omega ( y ) + f _ 1 ( x ) \\langle K ( x , \\cdot ) , u ( \\cdot ) \\rangle \\\\ & = : I + I \\ ! I . \\end{align*}"}
{"id": "4413.png", "formula": "\\begin{align*} \\lambda w _ i + ( 1 - \\lambda ) h _ i = \\frac { 1 } { 2 } \\pm b i \\end{align*}"}
{"id": "3508.png", "formula": "\\begin{align*} D _ { 1 3 } & = \\frac { s _ 3 } { ( a t _ 3 ) ^ { s _ 1 + s _ 3 } ( s _ 1 + s _ 3 ) } \\int _ 1 ^ { a t _ 3 } \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 } } d v \\\\ & \\quad + \\frac { s _ 3 } { 2 \\pi i \\Gamma ( s _ 3 + 1 ) } \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { ( a t _ 3 ) ^ { s _ 1 + s _ 3 + z } ( s _ 1 + s _ 3 + z ) } \\left ( \\int _ 1 ^ { a t _ 3 } \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 - z } } d v \\right ) d z \\\\ & = D _ { 1 3 1 } + D _ { 1 3 2 } , \\end{align*}"}
{"id": "7885.png", "formula": "\\begin{align*} \\bigg \\{ \\sqrt { \\binom { d } { J } } x ^ J \\bigg \\} _ { | J | = d } \\end{align*}"}
{"id": "73.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & = \\frac { q _ v ^ { 1 / 2 } } { q _ v ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "971.png", "formula": "\\begin{align*} \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } & = \\frac { \\pi ^ { \\frac { n - 1 } 2 } \\Gamma \\big ( \\frac { 1 + 2 s } 2 \\big ) } { \\Gamma \\big ( \\frac { n + 2 s } 2 \\big ) } \\int _ 0 ^ \\infty \\frac { \\dd z _ 1 } { ( z _ 1 + 1 ) ^ { 1 + 2 s } } = \\frac { \\pi ^ { \\frac { n - 1 } 2 } \\Gamma \\big ( \\frac { 1 + 2 s } 2 \\big ) } { 2 s \\Gamma \\big ( \\frac { n + 2 s } 2 \\big ) } . \\end{align*}"}
{"id": "8222.png", "formula": "\\begin{align*} [ z ^ n x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] ( P _ 1 + P _ 2 + \\cdots + P _ k ) & = \\frac { 1 } { n - 1 } [ t ^ { n - 2 } x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] t ^ { - 2 } \\Phi ( t ) ^ { n - 1 } \\\\ & = \\frac { 1 } { n - 1 } [ t ^ { n } x _ 1 ^ { \\ell _ 1 } \\cdots x _ k ^ { \\ell _ k } ] \\Phi ( t ) ^ { n - 1 } , \\end{align*}"}
{"id": "5138.png", "formula": "\\begin{align*} g _ \\gamma ( t ) = \\sqrt { \\gamma } \\ , e ^ { - \\gamma | t | } , t \\in \\R , \\end{align*}"}
{"id": "5790.png", "formula": "\\begin{align*} K _ 1 \\ast K _ 2 = \\{ \\sigma \\cup \\tau \\ , : \\ , \\sigma \\in K _ 1 , \\ , \\tau \\in K _ 2 \\} . \\end{align*}"}
{"id": "8414.png", "formula": "\\begin{align*} \\sum _ { i = \\lfloor N - \\eta ^ { - 1 } \\rfloor + 1 } ^ { N - 1 } [ ( N - i ) \\eta ] ^ { - 1 / \\alpha } \\leq \\frac { \\alpha } { \\alpha - 1 } \\ , \\eta ^ { - 1 } . \\end{align*}"}
{"id": "6552.png", "formula": "\\begin{align*} W ( t ) = K ( t ) \\ast W _ { 0 } + \\int _ { 0 } ^ { t } { K ( t - \\tau ) \\ast f ( \\tau ) \\ , d \\tau } , \\end{align*}"}
{"id": "7351.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\lambda u ( y _ q , t _ q ) ^ { q - 1 } + ( 1 - \\lambda ) u ( z _ q , t _ q ) ^ { q - 1 } } { \\varphi ( x _ q , t _ q ) ^ { q - 1 } } = \\frac { \\lambda u ( y _ q , t _ q ) ^ { q - 1 } + ( 1 - \\lambda ) u ( z _ q , t _ q ) ^ { q - 1 } } { w _ q ( x _ q , t _ q ) ^ { q - 1 } } \\\\ & \\leq \\frac { \\lambda u ( y _ q , t _ q ) ^ { q } + ( 1 - \\lambda ) u ( z _ q , t _ q ) ^ { q } } { c _ 0 w _ q ( x _ q , t _ q ) ^ { q - 1 } } = \\frac { w _ q ( x _ q , t _ q ) ^ q } { c _ 0 w _ q ( x _ q , t _ q ) ^ { q - 1 } } = \\frac { w _ q ( x _ q , t _ q ) } { c _ 0 } , \\end{aligned} \\end{align*}"}
{"id": "7982.png", "formula": "\\begin{align*} \\partial _ \\xi F ^ f ( x , z , \\xi ) & = F ^ f ( x , z , \\xi ) \\cdot g ^ 0 ( x ) G ( x , z , \\xi ) ^ { - 1 } \\xi , \\\\ \\partial _ z F ^ f ( x , z , \\xi ) & = F ^ f ( x , z , \\xi ) \\cdot \\left ( \\frac { n \\partial _ z f } { 2 ( 1 + f ) } \\Big | _ { ( x , z ) } + t r ( G ^ { - 1 } \\partial _ z G ) | _ { ( x , z , \\xi ) } \\right ) \\ , , \\end{align*}"}
{"id": "2546.png", "formula": "\\begin{align*} \\rho _ S ( z , \\tau ) = \\rho ( S z , \\tau ) . \\end{align*}"}
{"id": "3429.png", "formula": "\\begin{align*} | G _ k ( f ) ( x ) | & = \\Big | \\int _ { \\R ^ N } D _ k ( x , y ) D ^ M _ K ( f ) ( y ) d \\omega ( y ) \\Big | = \\Big | \\int _ { \\R ^ N } D _ k ( x , y ) [ D ^ M _ K ( f ) ( y ) - D ^ M _ K ( f ) ( x ) ] d \\omega ( y ) \\Big | \\\\ & \\lesssim \\| f \\| _ \\eta \\int _ { \\R ^ N } | D _ k ( x , y ) | \\| x - y \\| ^ \\eta d \\omega ( y ) \\lesssim r ^ { - k \\eta } \\| f \\| _ \\eta . \\end{align*}"}
{"id": "7632.png", "formula": "\\begin{align*} \\lambda _ { \\varepsilon } = \\lambda ^ 1 ( B ^ { \\varepsilon } ( \\mathbf { x } _ { \\varepsilon } ) ) , \\end{align*}"}
{"id": "2072.png", "formula": "\\begin{align*} f _ { \\alpha } ^ { ( \\gamma ) } ( o ) = f ^ { ( \\gamma ) } ( o ) \\Leftrightarrow f ^ { ( \\gamma ) } ( o ) = \\sum _ { \\beta < \\gamma } a _ { \\beta } g _ { \\beta } ^ { ( \\gamma ) } ( o ) + a _ { \\gamma } g _ { \\gamma } ^ { ( \\gamma ) } ( o ) . \\end{align*}"}
{"id": "6668.png", "formula": "\\begin{align*} V \\left ( \\frac { g _ 2 g _ 3 u } { H X } \\right ) = \\frac { 1 } { 2 \\pi i } \\Psi \\Big ( \\frac { u } { X Q ^ { \\vartheta } } \\Big ) \\int _ { ( \\varepsilon ) } \\left ( \\frac { X H } { g _ 2 g _ 3 u } \\right ) ^ { s _ 3 } \\widetilde { V } ( s _ 3 ) \\ , d s _ 3 . \\end{align*}"}
{"id": "187.png", "formula": "\\begin{align*} \\int _ x ^ { + \\infty } p _ \\delta ( y ) d y = \\frac { 1 } { \\delta } \\int _ { x ^ { \\delta } } ^ { + \\infty } z ^ { \\frac { 1 } { \\delta } - 1 } e ^ { - z } d z = \\frac { 1 } { \\delta } \\Gamma \\left ( \\frac { 1 } { \\delta } , x ^ { \\delta } \\right ) , \\end{align*}"}
{"id": "4442.png", "formula": "\\begin{align*} n ! = \\sqrt { 2 \\pi n } \\left ( \\dfrac { n } { e } \\right ) ^ n \\left ( 1 + \\dfrac { 1 } { 1 2 n } + \\frac { 1 } { 2 8 8 n ^ { 2 } } + \\cdots \\right ) \\end{align*}"}
{"id": "6395.png", "formula": "\\begin{align*} \\Omega = \\{ ( z _ 1 , \\dots , z _ n ) \\in \\mathbb D ^ n \\ , : \\ , f _ i \\circ \\pi _ n ( z _ 1 , \\dots , z _ n ) = 0 \\ , , \\ , 1 \\leq i \\leq n - 1 \\} . \\end{align*}"}
{"id": "8295.png", "formula": "\\begin{align*} u \\mapsto \\| u \\| _ { C ^ { \\gamma } } = \\| u \\| _ { L ^ \\infty } + [ u ] _ { C ^ { \\gamma } } . \\end{align*}"}
{"id": "5576.png", "formula": "\\begin{align*} \\| 1 _ B \\| \\ \\geqslant \\ \\| 1 _ { B ' } \\| \\ \\geqslant \\ \\ln | B ' | & \\ \\geqslant \\ \\ln | B | - \\ln 2 \\\\ & \\ \\geqslant \\ \\ln ( ( \\lambda - 1 ) 2 ^ { n _ j } ) - \\ln ( 2 ) \\\\ & \\ = \\ n _ j \\ln 2 + \\ln ( ( \\lambda - 1 ) / 2 ) \\\\ & \\ \\geqslant \\ j \\ln 2 + \\ln ( ( \\lambda - 1 ) / 2 ) \\\\ & \\ \\gtrsim \\ \\| 1 _ F \\| \\ln 2 + \\ln ( ( \\lambda - 1 ) / 2 ) . \\end{align*}"}
{"id": "9409.png", "formula": "\\begin{align*} \\varphi ( b _ 0 a _ 1 b _ 1 ) = \\varphi ( b _ 0 ) \\varphi ( a _ 1 ) \\varphi ( b _ 1 ) . \\end{align*}"}
{"id": "3550.png", "formula": "\\begin{align*} N ( T ) = \\cup _ { T \\le t < 0 } \\tilde \\Gamma ( t ) . \\end{align*}"}
{"id": "4947.png", "formula": "\\begin{align*} = ( \\dot { \\gamma } k ^ o _ 1 \\dot { \\gamma } ^ { - 1 } ) ( \\dot { \\gamma } \\omega \\dot { \\gamma } ^ { - 1 } ) ( \\dot { \\gamma } k \\phi ( { \\dot \\gamma } ^ { - 1 } ) ) . \\end{align*}"}
{"id": "5646.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = \\sigma \\varepsilon ^ { - 2 } \\left ( e ^ { u } - \\left ( x ^ { 2 } + y ^ { 2 } \\right ) e ^ { - u } \\right ) & \\Omega , \\\\ u & = \\sigma g & \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
{"id": "806.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla \\phi \\ , d \\mu = \\int _ { \\partial \\Omega } \\phi f d \\nu \\int _ \\Omega | \\nabla v | ^ { p - 2 } \\nabla v \\cdot \\nabla \\phi \\ , d \\mu = \\int _ { \\partial \\Omega } \\phi g d \\nu . \\end{align*}"}
{"id": "7880.png", "formula": "\\begin{align*} N ( R ; F ) = \\sum _ { H \\in H ( n - m ) } N ( R ; F , H ) . \\end{align*}"}
{"id": "3060.png", "formula": "\\begin{align*} c = - b \\ , . \\end{align*}"}
{"id": "8955.png", "formula": "\\begin{align*} \\dot { \\xi } ( t ) = \\frac { e ^ t } { C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } } \\left ( \\int _ 0 ^ t e ^ { - s } D f \\left ( \\xi ( s ) \\right ) d s + \\tilde { c } \\right ) , \\end{align*}"}
{"id": "7955.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial Z _ { j } } = \\sum _ { i = 1 } ^ { n } \\frac { \\partial g _ { i } } { \\partial Z _ { j } } h _ { i } + \\sum _ { i = 1 } ^ { n } g _ { i } \\frac { \\partial h _ { i } } { \\partial Z _ { j } } , j = 1 , \\cdots , n . \\end{align*}"}
{"id": "2245.png", "formula": "\\begin{align*} \\mathcal { O } _ { k , N } ( t _ m ) = & \\frac { \\sin \\pi \\alpha } { \\pi } \\int _ 0 ^ { t _ m } \\int _ \\sigma ^ { t _ m } \\tilde { E } _ { k , N } ( t _ m - \\sigma ) ( t _ m - s ) ^ { \\alpha - 1 } ( s - \\sigma ) ^ { - \\alpha } \\ , \\dd s \\ , \\dd W ( \\sigma ) \\\\ = & \\frac { \\sin \\pi \\alpha } { \\pi } \\int _ 0 ^ { t _ m } \\int _ 0 ^ s \\tilde { E } _ { k , N } ( t _ m - \\sigma ) ( t _ m - s ) ^ { \\alpha - 1 } ( s - \\sigma ) ^ { - \\alpha } \\ , \\dd W ( \\sigma ) \\ , \\dd s . \\end{align*}"}
{"id": "8258.png", "formula": "\\begin{align*} w \\xi ^ { - 1 } = \\left ( ( u ^ { ( 1 ) } \\times \\cdots \\times u ^ { ( k - 1 ) } ) \\zeta '^ { - 1 } \\times u ^ { ( k ) } \\right ) \\eta ^ { - 1 } . \\end{align*}"}
{"id": "2886.png", "formula": "\\begin{align*} w ( t ) = M ( w ) ( t ) , \\end{align*}"}
{"id": "4720.png", "formula": "\\begin{align*} \\Phi _ 1 ( t , y ) = \\sum _ { i = 1 } ^ n \\phi _ i ( y / \\sqrt { t } ) [ 1 - \\mu _ i ( t ) ] , \\quad \\Phi _ 2 ( t , y ) = \\sum _ { i = 1 } ^ n \\frac { \\mu _ i ( t ) \\phi _ i ( y / \\sqrt { t } ) } { 1 + [ \\mu _ i ( t ) ] ^ 2 } . \\end{align*}"}
{"id": "495.png", "formula": "\\begin{align*} \\{ \\Psi ^ { ( 1 ) } ( Y _ { + , 1 } + 1 ) , \\Psi ^ { ( 1 ) } ( Y _ { + , 2 } + 1 ) \\} = \\{ - \\Psi ^ { ( 1 ) } ( Y _ { - , 1 } + 1 ) , - \\Psi ^ { ( 1 ) } ( Y _ { - , 2 } + 1 ) \\} . \\end{align*}"}
{"id": "2193.png", "formula": "\\begin{align*} \\langle J ^ { ' } ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ { + } + s _ { \\widehat { w } ^ - } \\widehat { w } ^ { - } ) , \\widehat { w } ^ + \\rangle = 0 \\ \\ \\langle J ^ { ' } ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ { + } + s _ { \\widehat { w } ^ - } \\widehat { w } ^ { - } ) , \\widehat { w } ^ - \\rangle = 0 . \\end{align*}"}
{"id": "7697.png", "formula": "\\begin{align*} \\mathcal { L } ( \\mathcal { A } ) = 0 , \\mathcal { A } \\coloneqq \\{ \\mathbf { x } \\in \\R ^ N : u _ { \\ast } > 0 , \\nabla u _ { \\ast } = 0 \\} \\ ; . \\end{align*}"}
{"id": "5927.png", "formula": "\\begin{align*} L ( q ^ i , v ^ i _ a ) = \\left ( \\frac { \\lambda } { 2 } + \\nu \\right ) \\left [ ( v ^ 1 _ 1 ) ^ 2 + ( v ^ 2 _ 2 ) ^ 2 \\right ] + \\frac { \\nu } { 2 } \\left [ ( v ^ 1 _ 2 ) ^ 2 + ( v ^ 2 _ 1 ) ^ 2 \\right ] + ( \\lambda + \\nu ) v ^ 1 _ 1 v ^ 2 _ 2 \\end{align*}"}
{"id": "2475.png", "formula": "\\begin{align*} \\pi ( ( x _ 1 , - \\omega _ 1 ) ; ( x _ 2 , \\omega _ 2 ) ) = M _ { ( x _ 2 , \\omega _ 2 ) } T _ { ( x _ 1 , - \\omega _ 1 ) } \\end{align*}"}
{"id": "9131.png", "formula": "\\begin{align*} \\norm { x ^ * - J ^ S _ { \\mu _ 0 } ( x ^ * + \\mu _ 0 T ^ \\circ x ^ * ) } = 0 \\end{align*}"}
{"id": "4346.png", "formula": "\\begin{align*} I _ \\ell ( \\tau ) = e ^ { \\left ( \\frac { 2 \\ell } { \\alpha } - 1 \\right ) \\tau } . \\end{align*}"}
{"id": "1107.png", "formula": "\\begin{align*} J ^ E ( x , t , k ) = \\left \\{ \\begin{aligned} & m ^ { G P } ( x , t , k ) J ^ { ( 3 ) } ( x , t , k ) \\left ( m ^ { G P } \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , & k \\in \\Gamma ^ { ( 3 ) } \\backslash U _ { \\xi } , \\\\ & m ^ { G P } ( x , t , k ) m ^ { m o d } ( x , t , k ) \\left ( m ^ { G P } \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , & k \\in \\partial U _ { \\xi } , \\end{aligned} \\right . \\end{align*}"}
{"id": "2758.png", "formula": "\\begin{align*} \\| u \\| _ { S \\left ( \\dot { H } ^ { s _ c } \\right ) } = \\sup \\left \\lbrace \\| u \\| _ { L _ t ^ q L _ x ^ r } : ( q , r ) ( \\ref { S t r i c h a r t z p a i r } ) ( \\ref { r e s t r i c t i o n o n S t r i c h a r t z p a i r } ) \\right \\rbrace . \\end{align*}"}
{"id": "7881.png", "formula": "\\begin{align*} \\limsup _ { L \\to \\infty } \\max _ { 0 \\leq | \\alpha | \\leq 3 } \\sup _ { x \\in Q } L ^ { - 2 | \\alpha | } | D _ x ^ { \\alpha } D _ y ^ { \\alpha } ( K _ L ) _ { i _ 1 i _ 2 } ( x , y ) | _ { y = x } | \\leq M , \\ i _ 1 , i _ 2 = 1 , 2 , \\dots , m . \\end{align*}"}
{"id": "2700.png", "formula": "\\begin{align*} W g _ 0 ( x , \\omega ) = 2 ^ d e ^ { - 2 \\pi ( x ^ 2 + \\omega ^ 2 ) } , \\end{align*}"}
{"id": "6441.png", "formula": "\\begin{align*} ( x , b ) + ( x ' , b ' ) = ( \\rho ( x + \\varphi ( b , x ' ) + \\gamma ( b , b ' ) , b + b ' ) , b + b ' ) \\end{align*}"}
{"id": "580.png", "formula": "\\begin{align*} f ( x , y ) \\ = \\ \\frac { x y } { ( x ^ 2 + y ^ 2 ) ^ 2 } \\end{align*}"}
{"id": "8556.png", "formula": "\\begin{align*} \\frac { 1 } { T ( k ) } = 1 - \\frac { 1 } { 2 i k } \\int V ( x ) m _ { \\pm } ( x , k ) \\ , d x , \\end{align*}"}
{"id": "4495.png", "formula": "\\begin{align*} J _ { a , b } ( T ) = \\int _ { T } ^ { \\infty } ( \\log y ) ^ a y ^ { - b } d y = \\int _ { ( \\log T ) ( b - 1 ) } ^ { \\infty } t ^ a ( b - 1 ) ^ { - a } e ^ { - t } \\frac { d t } { b - 1 } = \\frac { 1 } { ( b - 1 ) ^ { a + 1 } } \\Gamma ( a + 1 , ( b - 1 ) \\log T ) , \\end{align*}"}
{"id": "7505.png", "formula": "\\begin{align*} \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ n } { \\tau } = \\left ( \\frac { 1 } { 2 } \\Delta - \\vartheta ^ n \\right ) \\tilde { \\phi } ^ { n + 1 } + \\Big ( \\vartheta ^ n - V - \\beta | \\phi ^ n | ^ { 2 } + \\Omega L _ z + \\mu ^ n \\Big ) \\phi ^ n , \\end{align*}"}
{"id": "7855.png", "formula": "\\begin{align*} ( \\mathbf { v } \\mathrel { \\trianglelefteq } ( D , E , F ) \\ , \\& \\ , \\mid E \\cap \\mathbf { v } _ 2 \\mid = \\omega ) \\Rightarrow \\mid H \\cap \\mathbf { v } _ 2 \\mid = \\omega \\end{align*}"}
{"id": "8443.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\nabla f ( X _ { t } ^ { x ; l ^ \\epsilon } ) \\nabla _ { x } X _ { t } ^ { x ; l ^ \\epsilon } G \\right ] = \\mathbb { E } \\left [ D _ { s } f ( X _ { t } ^ { x ; l ^ \\epsilon } ) \\nabla _ { x } X _ { \\gamma ^ { \\epsilon } _ { s } } ^ { x ; l ^ \\epsilon } G \\right ] . \\end{align*}"}
{"id": "5715.png", "formula": "\\begin{align*} 1 \\geq \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 0 } \\cup { \\gamma _ { 2 } } ) = \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 3 } ) - 1 0 . \\end{align*}"}
{"id": "8266.png", "formula": "\\begin{align*} M _ { u } : = \\sum _ { u \\leq v } \\mu _ { \\mathfrak { S } _ n } ( u , v ) F _ { v } , \\end{align*}"}
{"id": "4913.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } Z _ t \\leq \\sigma ^ 2 \\log ( 1 / \\delta ) \\end{align*}"}
{"id": "6129.png", "formula": "\\begin{align*} \\varphi ^ { 1 } ( s , t ) & = u ^ { 1 } + 2 j \\cdot s ^ { 2 j - 1 } w ^ { 1 } , \\\\ \\psi ^ { 1 } ( s , t ) & = v ^ { 1 } + 2 k \\cdot t ^ { 2 k - 1 } w ^ { 1 } , \\end{align*}"}
{"id": "1284.png", "formula": "\\begin{align*} \\epsilon _ { ( t _ { 1 } , t _ { 2 } , n ) } : = ( \\gamma _ { 1 } , t _ { 1 } s _ { 1 } ) \\cup { ( \\gamma _ { 2 } , t _ { 2 } s _ { 2 } ) } \\cup { \\epsilon _ { n } } \\end{align*}"}
{"id": "1795.png", "formula": "\\begin{align*} X = \\partial Y _ 0 \\cong G \\times _ K S . \\end{align*}"}
{"id": "4669.png", "formula": "\\begin{gather*} \\partial _ y ( \\mathcal { L } Y ^ - ) = - e _ 0 Y ^ - , \\partial _ y ( \\mathcal { L } \\textbf { } Y ^ + ) = e _ 0 Y ^ + , \\\\ Y ^ { + } ( y ) = Y ^ - ( - y ) , \\quad \\int _ { \\mathbb { R } } Y ^ \\pm = 0 , \\\\ ( Y ^ - , \\mathcal { L } Y ^ - ) = ( Y ^ + , \\mathcal { L } Y ^ + ) = 0 , \\\\ ( Y ^ - , \\mathcal { L } Y ^ + ) = ( Y ^ + , \\mathcal { L } Y ^ - ) \\not = 0 . \\end{gather*}"}
{"id": "6085.png", "formula": "\\begin{align*} \\begin{array} { l l } e & = r ^ { m _ { j } - 2 } \\left ( e _ 1 ( { \\theta } ) + r e _ { 2 } ( r , \\theta ) \\right ) , \\\\ f & = r ^ { m _ { j } - 2 } \\left ( f _ 1 ( { \\theta } ) + r f _ { 2 } ( r , \\theta ) \\right ) , \\\\ g & = r ^ { m _ { j } - 2 } \\left ( g _ 1 ( { \\theta } ) + r g _ { 2 } ( r , \\theta ) \\right ) . \\end{array} \\end{align*}"}
{"id": "6306.png", "formula": "\\begin{align*} p _ { K '' | K ' } ( k '' ) = \\binom { K ' + k '' } { k '' } \\xi ^ { k '' } ( 1 - \\xi ) ^ { K ' } \\end{align*}"}
{"id": "5931.png", "formula": "\\begin{align*} \\Phi ^ { T ^ * Q } _ g ( \\alpha _ q ) & = T ^ * _ { g \\cdot q } \\Phi _ { g ^ { - 1 } } ( \\alpha _ q ) , \\\\ \\Phi ^ { ( T ^ 1 _ k ) ^ * Q } _ g ( \\alpha ^ 1 _ { q } , \\dots , \\alpha ^ k _ { q } ) & = ( T ^ * _ { g \\cdot q } \\Phi _ { g ^ { - 1 } } ( \\alpha ^ 1 _ q ) , \\dots , T ^ * _ { g \\cdot q } \\Phi _ { g ^ { - 1 } } ( \\alpha ^ k _ q ) ) . \\end{align*}"}
{"id": "499.png", "formula": "\\begin{align*} \\varepsilon _ { \\delta _ { 1 } \\delta _ { 2 } \\delta _ { 3 } } ^ { a _ { 1 } a _ { 2 } a _ { 3 } } : = - \\mathrm { s i g n } \\left [ \\prod _ { u < w } ( a _ { u } - a _ { w } ) \\cdot \\prod _ { x < z } ( \\delta _ { x } - \\delta _ { z } ) \\cdot \\prod _ { r \\neq s } ( a _ { r } - \\delta _ { s } ) \\right ] . \\end{align*}"}
{"id": "8727.png", "formula": "\\begin{align*} \\begin{aligned} l _ 1 ( z ) & : = z _ { 2 , 2 } - 2 z _ { 2 , 1 } - 2 z _ { 1 , 2 } + z _ { 2 , 0 } + 4 z _ { 1 , 1 } + z _ { 0 , 2 } - 2 z _ { 1 , 0 } - 2 z _ { 0 , 1 } + 1 , \\\\ l _ 2 ( z ) & : = 0 . 2 5 ( - z _ { 1 , 0 } + 0 . 7 5 ) + 0 . 7 5 ( z _ { 2 , 0 } - 2 z _ { 1 , 0 } + 1 ) + 0 . 2 5 ( - z _ { 0 , 1 } + 0 . 7 5 ) + 0 . 7 5 ( z _ { 0 , 2 } - 2 z _ { 0 , 1 } + 1 ) - 0 . 9 3 5 . \\end{aligned} \\end{align*}"}
{"id": "172.png", "formula": "\\begin{align*} \\nu _ m ( d u ) = \\frac { 2 } { \\| u \\| ^ d } \\left ( \\int _ 0 ^ { + \\infty } g _ m ( 2 w ) L _ { \\frac { d } { 2 } } \\left ( \\sqrt { 2 w } \\| u \\| \\right ) d w \\right ) d u , \\end{align*}"}
{"id": "8603.png", "formula": "\\begin{align*} \\mu ^ \\# _ { R , 1 } ( { \\bf k } ) = \\sum _ { ( A _ 1 , A _ 2 , A _ 3 , A _ 4 ) \\in \\mathcal { X } _ R } \\int _ { \\R } \\prod _ \\ast \\mathcal { K } ^ \\# _ { A _ j } ( k _ j , x ) \\ , d x , \\ , \\mathcal { X } _ R : = \\{ S , R \\} ^ 4 \\smallsetminus \\{ ( S , S , S , S ) \\} , \\end{align*}"}
{"id": "8834.png", "formula": "\\begin{align*} | N _ { F } ( s ) - N _ { H _ { 1 } } ( v _ { t } ) | & = | N _ { F } ( s ) \\cap N _ { F } ( v _ { t } ) | + | N _ { F } ( s ) \\cap N _ { H _ { 2 } } ( v _ { t } ) | \\\\ & \\geq | N _ { F } ( s ) \\cap N _ { F } ( v _ { t } ) | + | X | + | N _ { H _ { 2 } } ( s ) \\cap N _ { F } ( v _ { t } ) | \\\\ & \\geq | N _ { F } ( s ) | - | N _ { H _ { 1 } } ( s ) \\cap N _ { F } ( v _ { t } ) | + | X | \\\\ & = r - | N _ { H _ { 1 } } ( s ) \\cap N _ { F } ( v _ { t } ) | + | X | \\end{align*}"}
{"id": "945.png", "formula": "\\begin{align*} P _ { n , m } f ( x ) = P _ { n , 0 } f ( x ) + \\sum _ { \\mu = 1 } ^ m \\langle f , f _ { n , \\mu } \\rangle f _ { n , \\mu } . \\end{align*}"}
{"id": "9143.png", "formula": "\\begin{align*} \\omega ( k ) : = \\max \\{ \\varpi ( 2 k + 1 ) , 4 k + 3 , \\varpi ( 4 M ( ( k + 1 ) ^ 2 ) - 1 ) ) \\} \\end{align*}"}
{"id": "5425.png", "formula": "\\begin{align*} M _ { 3 , 2 } = \\begin{bmatrix} 1 & - \\lambda _ 3 ( \\lambda _ 2 + 1 ) \\\\ 0 & \\lambda _ 3 \\lambda _ 2 \\end{bmatrix} . \\end{align*}"}
{"id": "375.png", "formula": "\\begin{align*} \\mathcal A ( \\widetilde \\rho , \\widetilde m ) = \\frac 1 2 \\int _ 0 ^ 1 \\frac { m _ { 2 3 } ^ 2 } { \\theta _ { 2 3 } ( \\widetilde \\rho ) } d t = \\frac 1 2 \\int _ 0 ^ 1 m _ { 2 3 } ^ 2 d t . \\end{align*}"}
{"id": "4977.png", "formula": "\\begin{align*} \\psi _ { \\bar w } ^ Z \\vert _ { \\Gamma _ i } = Z ^ { \\bar w } \\circ L _ Z , \\bar w \\prec \\bar t _ { n } ^ i . \\end{align*}"}
{"id": "5407.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } Y _ { t } = ( R _ { t } + \\theta _ { 1 } ) \\mathrm { d } t + \\sigma \\mathrm { d } W ^ { 1 } _ { t } + \\mathrm { d } L ^ { 1 } _ { t } + \\mathrm { d } R _ { t } , \\\\ & \\mathrm { d } R _ { t } = \\theta _ { 2 } ( 1 - R _ { t } ) \\mathrm { d } t + \\sigma \\mathrm { d } W ^ { 2 } _ { t } + \\mathrm { d } L ^ { 2 } _ { t } , \\\\ & Y _ { 0 } = y \\in [ a , b ] , R _ { 0 } = r > 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "8930.png", "formula": "\\begin{align*} \\varphi = \\varphi | _ { U _ 1 } + \\cdots + \\varphi | _ { U _ n } + \\varphi | _ V \\end{align*}"}
{"id": "9552.png", "formula": "\\begin{align*} w - z _ 1 + ( m - 1 ) ( \\hat { \\mathcal { A } } x ^ { m - 2 } ) ^ T ( x - z _ 2 ) = 0 \\\\ Z _ 1 x = 0 \\\\ Z _ 2 w + X w = 0 \\\\ w - ( \\mathcal { A } x ^ { m - 1 } + q ) = 0 \\\\ \\end{align*}"}
{"id": "6985.png", "formula": "\\begin{align*} \\begin{aligned} y ^ * & \\in \\mathcal N _ { \\mathbf T ( u ) } ( w _ s ( u , v ) ) \\cap \\ker \\nabla g ( \\bar x ) ^ * , \\\\ z ^ * & \\in \\mathcal T _ { \\mathcal N _ { \\mathbf T ( u ) } ( w _ s ( u , v ) ) } ( y ^ * ) , \\end{aligned} \\end{align*}"}
{"id": "5088.png", "formula": "\\begin{align*} V _ g f ( x , \\omega ) = \\langle f , \\pi ( \\l ) g \\rangle = \\int _ \\R f ( t ) \\overline { g ( t - x ) } e ^ { - 2 \\pi i \\omega t } \\ , d t . \\end{align*}"}
{"id": "1108.png", "formula": "\\begin{align*} \\vert J ^ E - I \\vert = \\vert m ^ { G P } ( m ^ { m o d } - I ) ( m ^ { G P } ) ^ { - 1 } \\vert \\in \\mathcal { O } ( t ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "4706.png", "formula": "\\begin{align*} & \\mathcal { L } \\mathfrak { A } _ { i j k , 1 } = \\mu _ k E _ { i j k } , \\\\ & \\mathcal { L } \\mathfrak { A } _ { i j , 2 } = F _ { i j } + \\sum _ { k = 1 } ^ n \\mu _ k ( p - 1 ) \\Lambda Q Q ^ { p - 2 } \\mathfrak { A } _ { i j k , 1 } . \\end{align*}"}
{"id": "52.png", "formula": "\\begin{align*} \\chi ( \\P ^ n , \\omega _ { \\P ^ n } ^ { 1 - k } ) & = h ^ 0 ( \\P ^ n , \\omega _ { \\P ^ n } ^ { 1 - k } ) \\\\ & = \\frac { ( n + 1 ) ^ n } { n ! } k ^ n + O ( k ^ { n - 1 } ) \\end{align*}"}
{"id": "8195.png", "formula": "\\begin{align*} S ( a , b , 3 \\delta f ) = s ( a - \\delta f , b , 3 \\delta f ) + s ( a + \\delta f , b , 3 \\delta f ) . \\end{align*}"}
{"id": "7964.png", "formula": "\\begin{align*} v _ j ^ + ( r ) & = c _ j ^ + \\cdot r ^ { \\gamma _ j ^ + } ; \\\\ v _ j ^ - ( r ) & = \\begin{cases} c _ j ^ - \\cdot r ^ { \\gamma _ j ^ - } , & \\ \\mu _ j > - \\frac { ( n - 2 ) ^ 2 } { 4 } ; \\\\ c _ j ^ - \\cdot r ^ { \\gamma _ j ^ - } \\log r , & \\ \\mu _ j = - \\frac { ( n - 2 ) ^ 2 } { 4 } . \\end{cases} \\end{align*}"}
{"id": "2355.png", "formula": "\\begin{align*} \\tfrac { 1 } { 4 \\pi } \\norm { f } _ 2 ^ 2 = \\tfrac { 1 } { 2 } | \\langle [ X , P ] f , f \\rangle | \\leq \\norm { ( X - a ) f } _ 2 ^ 2 \\norm { ( P - b ) f } _ 2 ^ 2 . \\end{align*}"}
{"id": "8119.png", "formula": "\\begin{align*} T B _ { 8 } ( f ) = T B _ { 6 } ( f ) , \\end{align*}"}
{"id": "2461.png", "formula": "\\begin{align*} C B ^ T - D A ^ T = - I \\ ; \\Leftrightarrow \\ ; ( D A ^ T - C B ^ T ) ^ T = I ^ T \\ ; \\Leftrightarrow \\ ; A D ^ T - B C ^ T = I . \\end{align*}"}
{"id": "8479.png", "formula": "\\begin{align*} \\pi _ { q , i } : = \\frac { ( q - 1 ) _ \\ell \\cdot \\ell ^ { - i } - 1 } { 2 } \\ , , \\end{align*}"}
{"id": "7156.png", "formula": "\\begin{align*} J _ U ( u ) : = \\int _ { U } \\Big ( | D u | ^ 2 + 1 _ { \\{ u > 0 \\} } \\Big ) \\ , d x . \\end{align*}"}
{"id": "2887.png", "formula": "\\begin{align*} M ( w ) ( t ) = - i \\int _ t ^ { + \\infty } e ^ { i ( t - s ) ( \\Delta - 1 ) } \\left [ S ( \\mathcal { V } _ { l ( k ) } ^ A + w ) - S ( \\mathcal { V } _ { l ( k ) } ^ A ) + \\varepsilon _ { l ( k ) } \\right ] ( s ) d s . \\end{align*}"}
{"id": "365.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 m _ { 2 1 } ^ 2 ( s ) d s \\\\ & = \\int _ { 1 - \\delta } ^ 1 \\frac 1 { \\delta ^ 2 } \\frac { ( \\rho _ 1 ^ b - \\rho _ 1 ^ a ) ^ 2 } { ( 1 + \\dot { W ^ { \\delta } } ( s ) ) ^ 2 } d s \\\\ & \\le \\int _ { 1 - \\delta } ^ 1 \\frac { ( \\rho _ 1 ^ b - \\rho _ 1 ^ a ) ^ 2 } { ( \\delta + { W _ { t _ { K } } - W _ { t _ { K - 1 } } } ) ^ 2 } d s \\\\ & \\le C ( \\delta ) < \\infty , \\ ; \\end{align*}"}
{"id": "3000.png", "formula": "\\begin{align*} \\lhd \\ , T ^ + y , x \\ , \\rhd = \\lhd \\ , y , T x \\rhd . \\end{align*}"}
{"id": "3679.png", "formula": "\\begin{align*} S = \\{ ( X , g ) : g X \\} . \\end{align*}"}
{"id": "2643.png", "formula": "\\begin{align*} \\langle P M _ l T _ k g , g \\rangle & = \\langle X T _ l M _ { - k } \\widehat { g } , \\widehat { g } \\rangle \\\\ & = l \\langle T _ l M _ { - k } \\widehat { g } , \\widehat { g } \\rangle + \\langle T _ l M _ { - k } X \\widehat { g } , \\widehat { g } \\rangle \\\\ & = \\langle M _ l T _ k P g , g \\rangle . \\end{align*}"}
{"id": "5675.png", "formula": "\\begin{align*} x y \\cdot \\rho y = x \\end{align*}"}
{"id": "5807.png", "formula": "\\begin{align*} \\mathbf { y } _ k ^ { c p } [ t ] = \\sum _ { m = 1 } ^ M \\mathbf { g } _ { m k } [ t ] \\ast \\textbf { x } _ m ^ { c p } [ t ] + \\mathbf { z } _ k [ t ] , \\end{align*}"}
{"id": "1729.png", "formula": "\\begin{align*} 2 ^ { \\mu _ * k _ * t } \\cdot 2 ^ { \u2010 \\tilde m _ t ( s _ * + 1 / q \u2010 1 / p _ 1 ) } n ^ { 1 / q \u2010 1 / p _ 1 } = 2 ^ { \u2010 \\alpha _ * k _ * t } \\cdot 2 ^ { \u2010 \\tilde m _ t ( 1 / q \u2010 1 / p _ 0 ) } n ^ { 1 / q \u2010 1 / p _ 0 } \\stackrel { \\eqref { t i l d e _ t h e t a _ d e f } } { = } n ^ { \u2010 \\tilde \\theta } ; \\end{align*}"}
{"id": "6241.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } A _ n ( q ) \\frac { t ^ n } { n ! } = \\frac { 1 - q } { 1 - q e ^ { t ( 1 - q ) } } . \\end{align*}"}
{"id": "6074.png", "formula": "\\begin{align*} C L h ^ { j } = A h ^ { j } - B \\overline { h } ^ { j } + C \\left [ g ^ { 2 } F ^ { j } + g \\lambda G ^ { j } + \\lambda ^ { 2 } H ^ { j } \\right ] , \\end{align*}"}
{"id": "2635.png", "formula": "\\begin{align*} Z ( M _ l T _ k g ) ( x , \\omega ) = e ^ { 2 \\pi i ( l \\cdot x - k \\cdot \\omega ) } Z g ( x , \\omega ) . \\end{align*}"}
{"id": "20.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } h _ n ^ j \\xi _ n ^ j = \\infty \\end{align*}"}
{"id": "8777.png", "formula": "\\begin{align*} u _ { i j } ( x ; J ) = a _ { i j } j \\leq \\min ( J _ i ) u _ { i j } ( x ; J ) = f _ i ( x ) j \\geq \\max ( J _ i ) u _ { i j } ( x ; J ) = u _ { i j } ( x ) . \\end{align*}"}
{"id": "549.png", "formula": "\\begin{align*} P ( \\rho ) - P ( r ) - P ' ( r ) ( \\rho - r ) \\geq \\begin{cases} c ( \\rho - r ) ^ 2 , & \\rho \\in ( \\alpha _ 1 , \\bar \\rho - \\alpha _ 1 ) , \\\\ \\frac { p ( r ) } { 2 } , & \\rho \\in [ 0 , \\alpha _ 1 ] , \\\\ \\frac { P ( \\rho ) } { 2 } > 1 , & \\rho \\in [ \\bar \\rho - \\alpha _ 1 , \\bar \\rho ) . \\end{cases} \\end{align*}"}
{"id": "123.png", "formula": "\\begin{align*} a ^ n - \\gamma _ { n - 1 } a ^ { n - 1 } - \\dots - \\gamma _ { 1 } a = 0 , \\end{align*}"}
{"id": "592.png", "formula": "\\begin{align*} \\deg ( \\pi ) \\ = \\ 1 ] \\end{align*}"}
{"id": "8785.png", "formula": "\\begin{align*} \\begin{aligned} & f _ i = a _ { i 0 } z _ { i 0 } + \\sum _ { j = 1 } ^ { n _ i } ( a _ { i j } - a _ { i j - 1 } ) z _ { i j } , z _ i \\in \\Delta _ i , \\\\ & z _ { i j } \\geq \\delta _ { i j } \\geq z _ { i j + 1 } j \\in \\{ 1 , \\ldots , n _ i - 1 \\} , \\\\ & \\delta _ { i } \\in \\{ 0 , 1 \\} ^ { n _ i - 1 } , \\end{aligned} \\end{align*}"}
{"id": "1052.png", "formula": "\\begin{align*} P ( z , \\partial _ z , \\lambda ) f ^ { \\lambda + 1 } = b ( \\lambda ) f ^ \\lambda \\end{align*}"}
{"id": "8207.png", "formula": "\\begin{align*} P _ r = x _ r z \\sum _ { j \\geq 0 } ( P _ 1 + P _ 2 + \\cdots + P _ { k + 1 - r } ) ^ j = \\frac { x _ r z } { 1 - P _ 1 - P _ 2 - \\cdots - P _ { k + 1 - r } } \\end{align*}"}
{"id": "1619.png", "formula": "\\begin{align*} \\psi ( x ) : = \\mu ^ { \\frac { d - 1 } { p } } \\varphi ( \\mu x ) \\\\ \\tilde { \\psi } ( x ) : = \\mu ^ { \\frac { d - 1 } { p ' } } \\tilde { \\varphi } ( \\mu x ) \\end{align*}"}
{"id": "785.png", "formula": "\\begin{align*} A \\alpha _ \\mathbf { p } ^ 4 + B \\alpha _ \\mathbf { p } ^ 3 + C \\alpha _ \\mathbf { p } ^ 2 + 1 = 0 \\end{align*}"}
{"id": "5358.png", "formula": "\\begin{align*} \\pi _ c ( x ) = x - \\frac { \\left \\langle c , x - z \\right \\rangle } { \\| c \\| ^ 2 } c . \\end{align*}"}
{"id": "26.png", "formula": "\\begin{gather*} \\omega ( \\dot { x } _ i ^ \\pm ( u ) ) = \\dot { x } _ { i } ^ \\mp ( u ) , \\omega ( \\dot { h } _ i ( u ) ) = \\dot { h } _ i ( u ) , \\\\ \\varsigma ( \\dot { x } _ i ^ \\pm ( u ) ) = \\dot { x } _ i ^ \\pm ( - u ) , \\varsigma ( \\dot { h } _ i ( u ) ) = \\dot { h } _ i ( - u ) . \\end{gather*}"}
{"id": "2782.png", "formula": "\\begin{align*} \\sigma _ { } ( \\mathcal { L } ) = \\{ i \\xi : \\xi \\in \\mathbb { R } , | \\xi | \\ge 1 \\} , \\sigma ( \\mathcal { L } ) \\cap \\mathbb { R } = \\{ - e _ 0 , 0 , e _ 0 \\} e _ 0 > 0 . \\end{align*}"}
{"id": "4360.png", "formula": "\\begin{align*} \\partial _ \\xi ^ k T _ i ( \\xi ) = \\partial _ \\xi ^ k \\left ( C _ i \\xi ^ { - \\gamma + 2 i } \\right ) + O \\left ( \\xi ^ { - \\gamma + 2 i - 2 - k } \\ln \\xi \\right ) , \\xi \\to + \\infty . \\end{align*}"}
{"id": "5294.png", "formula": "\\begin{align*} \\kappa = ( \\sigma ^ { \\varphi } ) ^ { - 1 } \\circ S ^ 2 , \\rho = \\sigma ^ { \\varphi _ S } \\circ S ^ 2 . \\end{align*}"}
{"id": "186.png", "formula": "\\begin{align*} \\left | f _ \\delta ( x ) \\right | \\leq \\frac { 1 } { p _ \\delta ( x ) } \\left ( \\int _ x ^ { + \\infty } p _ \\delta ( y ) d y \\right ) ^ { \\frac { 1 } { q } } \\| g \\| _ { L ^ p ( \\mu _ \\delta ) } = G _ \\delta ( x ) \\| g \\| _ { L ^ p ( \\mu _ \\delta ) } , \\end{align*}"}
{"id": "1648.png", "formula": "\\begin{align*} U ^ s \\nu ( x ) : = \\int _ M \\varrho ( x , y ) ^ { - s } \\nu ( d y ) , x \\in M , \\end{align*}"}
{"id": "5678.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) : = c _ { 1 } ( \\xi | _ { Z } , \\tau ) + Q _ { \\tau } ( Z ) + \\sum _ { i } \\sum _ { k = 1 } ^ { m _ { i } } \\mu _ { \\tau } ( \\alpha _ { i } ^ { k } ) - \\sum _ { j } \\sum _ { k = 1 } ^ { n _ { j } } \\mu _ { \\tau } ( \\beta _ { j } ^ { k } ) . \\end{align*}"}
{"id": "354.png", "formula": "\\begin{align*} & \\ ; \\ < \\bar S ( 1 ) , \\rho ^ b \\ > - \\ < \\bar S ( 0 ) , \\rho ^ a \\ > - \\int _ { \\mathcal O } H ( \\dot { \\bar S } , \\nabla _ G { \\bar S } ) d t \\\\ & = \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > - \\int _ { \\mathcal O } H ( \\dot S , \\nabla _ G S ) d t \\\\ & \\ge \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > , \\end{align*}"}
{"id": "1775.png", "formula": "\\begin{align*} \\Psi ( \\hat { f } _ 0 , \\cdots , \\hat { f } _ n ) = \\int _ { \\mathbb { R } ^ n } \\hat { f } _ 0 d \\hat { f } _ 1 \\cdots d \\hat { f } _ n . \\end{align*}"}
{"id": "9033.png", "formula": "\\begin{align*} d ^ 2 ( \\rho ^ 0 , \\rho ^ 1 ) : = \\min _ { ( \\rho , u ) \\in K } \\sum _ { i = 1 } ^ 2 \\int _ 0 ^ 1 \\int _ { \\Omega } | u _ i | ^ 2 \\rho _ i d x d t . \\end{align*}"}
{"id": "2276.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k \\in \\Z ^ d } c _ k e ^ { 2 \\pi i k \\cdot t } , \\end{align*}"}
{"id": "2889.png", "formula": "\\begin{align*} & B ^ { k } \\triangleq \\left \\lbrace w \\in X ^ k : \\| w \\| _ { X ^ k } \\le 1 \\right \\rbrace , \\\\ & X ^ k \\triangleq \\Big \\{ w : \\left \\langle \\nabla \\right \\rangle w \\in S ( [ t _ k , + \\infty ) , L ^ 2 ) : \\| w \\| _ { X ^ k } = \\sup _ { t \\ge t _ k } \\| \\left \\langle \\nabla \\right \\rangle w \\| _ { S ( [ t , + \\infty ) , L ^ 2 ) } e ^ { ( k + \\frac { 1 } { 2 } ) e _ 0 t } < \\infty \\Big \\} , \\end{align*}"}
{"id": "2304.png", "formula": "\\begin{align*} C ( c _ 1 f + c _ 2 h ) = | c _ 1 | ^ 2 \\ , C f + | c _ 2 | ^ 2 \\ , C h + c _ 1 \\overline { c _ 2 } \\ , G ( f , h ) + \\overline { c _ 1 } c _ 2 \\ , G ( h , f ) , c _ 1 , c _ 2 \\in \\C . \\end{align*}"}
{"id": "8828.png", "formula": "\\begin{align*} \\underset { x \\in \\mathbb { R } ^ n , w \\in \\mathbb { R } ^ n } { } \\ c ^ \\top x + \\frac { 1 } { 2 } x ^ \\top Q x + \\| D w \\| _ 1 + \\delta _ { \\mathcal { K } } ( w ) , \\ A x = b , w - x = 0 . \\end{align*}"}
{"id": "8599.png", "formula": "\\begin{align*} \\mathcal { C } : = \\big \\{ f : { \\bf k } : = ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) \\in \\R ^ 4 \\rightarrow \\C , \\ , \\ , \\mbox { s . t . } \\ , f ( { \\bf k } ) = \\prod _ \\ast a _ j ( k _ j ) , \\ , \\ , a _ j \\in A \\big \\} , \\end{align*}"}
{"id": "7329.png", "formula": "\\begin{align*} \\begin{aligned} & \\liminf _ { \\alpha \\to 0 } \\left ( - \\frac { | \\tilde { x } - \\tilde { y } | ^ 4 } { \\varepsilon ^ 4 } + 2 \\alpha \\max _ { x \\in K } | x | ^ 2 - \\alpha ( | \\tilde { x } | ^ 2 + | \\tilde { y } | ^ 2 ) \\right ) \\\\ \\leq & \\liminf _ { \\alpha \\to 0 } \\left ( - \\frac { | \\tilde { x } - \\tilde { y } | ^ 4 } { \\varepsilon ^ 4 } + 2 \\alpha \\max _ { x \\in K } | x | ^ 2 \\right ) < 0 , \\end{aligned} \\end{align*}"}
{"id": "3724.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int | B | \\ , d x = 0 . \\end{align*}"}
{"id": "3185.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k + E _ i ( E _ i ^ \\intercal A ^ \\intercal A E _ i ) ^ \\dagger E _ i ^ \\intercal A ^ \\intercal ( b - A x _ k ) . \\end{align*}"}
{"id": "1970.png", "formula": "\\begin{gather*} ( \\phi \\prec \\psi ) \\prec \\rho = \\phi \\prec ( \\psi * \\rho ) , \\\\ ( \\phi \\succ \\psi ) \\prec \\rho = \\phi \\succ ( \\psi \\prec \\rho ) , \\\\ \\phi \\succ ( \\psi \\succ \\rho ) = ( \\phi * \\psi ) \\succ \\rho . \\end{gather*}"}
{"id": "8125.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } | \\{ \\lambda \\in \\Lambda \\mid \\lambda \\supset e \\cup e _ { i } \\} | = 2 \\sum _ { 1 \\leq i < j \\leq k } | \\{ \\lambda \\in \\Lambda \\mid \\lambda \\supset e \\cup e _ { i } \\cup e _ { j } \\} | \\equiv 0 \\pmod 2 . \\end{align*}"}
{"id": "3151.png", "formula": "\\begin{align*} z _ { k + 1 } = z _ k - B ^ { - 1 } \\bar { A } ^ \\intercal S _ { i _ k } ( S _ { i _ k } ^ \\intercal \\bar A B ^ { - 1 } \\bar { A } ^ \\intercal S _ { i _ k } ) ^ \\dagger S _ { i _ k } ^ \\intercal ( \\bar A z _ k - b ) , \\end{align*}"}
{"id": "4525.png", "formula": "\\begin{align*} G _ 3 ( a _ 1 , b _ 1 ) = \\{ x _ { b _ 1 } ( b _ 2 ) , \\ldots , x _ { b _ 1 } ( b _ { n - 1 } ) \\} , \\end{align*}"}
{"id": "1046.png", "formula": "\\begin{align*} & D _ { i j } D _ { k l } = q ^ { - 2 } D _ { k l } D _ { i j } , & & 1 \\leq i < j < k < l \\leq 6 , \\\\ & D _ { i j } D _ { k l } = q ^ { - 2 } D _ { k l } D _ { i j } - ( q ^ { - 1 } - q ) D _ { i k } D _ { j l } & & 1 \\leq i < k < j < l \\leq 6 , \\\\ & D _ { i j } D _ { k l } = D _ { k l } D _ { i j } & & 1 \\leq i < k < l < j \\leq 6 , \\end{align*}"}
{"id": "5102.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial x } \\big ( x ^ 3 e ^ { - x } \\big ) = ( 3 - x ) \\ , x ^ 2 e ^ { - x } , \\end{align*}"}
{"id": "7867.png", "formula": "\\begin{align*} G \\cap H & = ( ( D \\setminus \\{ \\bar { b } _ n \\} ) \\cap ( E \\setminus \\{ \\bar { b } _ n \\} ) ) \\cup \\\\ & \\qquad \\qquad \\cup ( ( D \\setminus \\{ \\bar { b } _ n \\} ) \\cap [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( K ) ) \\\\ & \\qquad \\qquad \\cup ( ( E \\setminus \\{ \\bar { b } _ n \\} ) \\cap [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) ) \\\\ & \\qquad \\qquad \\cup ( [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) \\cap [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( K ) ) \\end{align*}"}
{"id": "6637.png", "formula": "\\begin{align*} I _ { 1 1 } = R _ 0 + O \\bigg ( \\frac { ( X C h k ) ^ { \\varepsilon } Q ^ { 2 } ( h , k ) } { C \\sqrt { h k } } \\bigg ) , \\end{align*}"}
{"id": "3173.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal W _ k ( W _ k ^ \\intercal A A ^ \\intercal W _ k ) ^ \\dagger W _ k ^ \\intercal ( A x _ k - b ) , \\end{align*}"}
{"id": "7948.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ t ( n + 1 - i ) a _ i & = ( n + 1 ) \\sum _ { i = 1 } ^ t a _ i - \\sum _ { i = 1 } ^ t i a _ i \\\\ & = \\begin{cases} \\frac { 1 } { 2 } n ^ 2 t + O \\left ( n ^ { 1 9 / 8 } \\right ) , ~ & ~ t \\geqslant n ^ { 1 / 2 } - n ^ { 1 / 4 } \\\\ \\frac { 1 } { 2 } n ^ 2 t + O \\left ( n ^ { 9 / 4 } \\sqrt { n ^ { 1 / 2 } - t } \\right ) , ~ & t < n ^ { 1 / 2 } - n ^ { 1 / 4 } . \\end{cases} \\end{align*}"}
{"id": "2008.png", "formula": "\\begin{align*} M _ { ( k _ 1 , \\ell _ 1 , \\ldots , k _ r , \\ell _ r ) } = \\left ( \\begin{matrix} 1 & 1 \\\\ 0 & 1 \\end{matrix} \\right ) ^ { k _ 1 } \\left ( \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\right ) ^ { \\ell _ 1 } \\dots \\left ( \\begin{matrix} 1 & 1 \\\\ 0 & 1 \\end{matrix} \\right ) ^ { k _ r } \\left ( \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\right ) ^ { \\ell _ r } . \\end{align*}"}
{"id": "1725.png", "formula": "\\begin{gather*} 2 ^ { \\gamma _ * k _ * t } \\cdot 2 ^ { \\hat m _ t } = n , \\\\ 2 ^ { \\gamma _ * k _ * t } \\cdot 2 ^ { \\overline { m } _ t } = n ^ { q / 2 } , \\\\ 2 ^ { \\tilde m _ t s _ * } = 2 ^ { ( \\mu _ * + \\alpha _ * + \\gamma _ * / p _ 0 - \\gamma _ * / p _ 1 ) k _ * t } , \\\\ 2 ^ { m _ t ( s _ * + 1 / p _ 0 - 1 / p _ 1 ) } = 2 ^ { ( \\mu _ * + \\alpha _ * ) k _ * t } , \\end{gather*}"}
{"id": "3941.png", "formula": "\\begin{align*} \\tan ^ 2 ( 2 r \\pi ) = \\frac { 4 \\tan ^ 2 ( r \\pi ) } { ( 1 - \\tan ^ 2 ( r \\pi ) ) ^ 2 } \\end{align*}"}
{"id": "2023.png", "formula": "\\begin{align*} h ( x ) = R ^ { Y } ( h \\overline { \\mu } ) ( x ) + 1 , x \\in E . \\end{align*}"}
{"id": "1127.png", "formula": "\\begin{align*} m ^ { G P } = \\Delta _ \\eta ( k ) : = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c } \\chi _ \\eta ( k ) + \\chi ^ { - 1 } _ \\eta ( k ) & i \\left ( \\chi _ \\eta ( k ) - \\chi _ \\eta ^ { - 1 } ( k ) \\right ) \\\\ - i \\left ( \\chi _ \\eta ( k ) - \\chi _ { \\eta } ^ { - 1 } ( k ) \\right ) & \\chi _ \\eta ( k ) + \\chi _ \\eta ^ { - 1 } ( k ) \\end{array} \\right ) , \\end{align*}"}
{"id": "1948.png", "formula": "\\begin{align*} f ( x ) = g ( x ) \\end{align*}"}
{"id": "4964.png", "formula": "\\begin{align*} & \\ < P s i > _ { m , n } ( t , y ) - \\ < P s i > _ { m , n } ( s , y ) = - i \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ d } a _ { m , n } ( t - t ' , y - x ' ) W ( d t ' , d x ' ) + i \\int _ 0 ^ s \\int _ { \\mathbb { R } ^ d } a _ { m , n } ( s - t ' , y - x ' ) W ( d t ' , d x ' ) \\\\ & = - i \\int _ s ^ t \\int _ { \\mathbb { R } ^ d } a _ { m , n } ( t - t ' , y - x ' ) W ( d t ' , d x ' ) - i \\int _ 0 ^ s \\int _ { \\mathbb { R } ^ d } \\Big [ a _ { m , n } ( t - t ' , y - x ' ) - a _ { m , n } ( s - t ' , y - x ' ) \\Big ] W ( d t ' , d x ' ) \\end{align*}"}
{"id": "6895.png", "formula": "\\begin{align*} w ( S ^ j _ k ) & = \\frac { 1 } { q ^ { j ^ 2 } { k \\choose k - j } _ q } \\sum \\limits _ { i = 0 } ^ j q ^ { { j - i \\choose 2 } } { j \\choose i } _ q { k + i \\choose i } _ q \\\\ & \\leq \\frac { q ^ { j k } } { q ^ { j ^ 2 } { k \\choose k - j } _ q } \\sum \\limits _ { i = 0 } ^ j q ^ { - i } \\\\ & \\leq \\left ( \\frac { q } { q - 1 } \\right ) ^ { \\min ( j , k - j ) + 1 } \\end{align*}"}
{"id": "3729.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| B \\| ^ 2 _ { L ^ 2 } + \\mu \\| \\Lambda ^ { \\frac { \\alpha } { 2 } } B \\| ^ 2 _ { L ^ 2 } = - \\int _ { \\mathbb S ^ 1 } B J _ x B \\ , d x + \\int _ { \\mathbb S ^ 1 } B _ x J B \\ , d x . \\end{align*}"}
{"id": "5961.png", "formula": "\\begin{align*} \\Omega \\simeq \\bigcup _ { n = 1 } ^ { N _ { e l m } } E ^ n , \\phi _ k ( x , z ) = \\bigoplus _ { n = 1 } ^ { N _ { e l m } } \\phi _ k ^ n ( x ^ n , z ^ n ) , \\left ( x ^ n , z ^ n \\right ) \\in E ^ n , \\end{align*}"}
{"id": "795.png", "formula": "\\begin{align*} \\nu _ 0 ( A ) = \\iint _ A \\ \\frac { 1 } { d ( x , y ) ^ { \\theta p + Q - 1 } } \\ , d \\nu \\times \\nu ( x , y ) , \\end{align*}"}
{"id": "3439.png", "formula": "\\begin{align*} I \\ ! I _ 1 = \\int _ { \\| u - y \\| > t } \\frac 1 { V ( x , u , t + d ( x , u ) ) } \\Big ( \\frac { t } { t + d ( x , u ) } \\Big ) ^ { \\varepsilon _ 0 } \\frac 1 { V ( u , y , s + d ( u , y ) ) } \\Big ( \\frac { s } { s + \\| u - y \\| } \\Big ) ^ { \\varepsilon _ 0 } d \\omega ( u ) \\end{align*}"}
{"id": "6349.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } f + O ^ * ( A ( x ) + B ( x ) + 6 \\log x ) . \\end{align*}"}
{"id": "3143.png", "formula": "\\begin{align*} \\omega _ \\theta ^ S ( t , \\eta ^ L _ r ( t ) ) = e ^ t \\omega ^ S _ { 0 , \\theta } ( e ^ { - t } r _ 0 ) \\quad \\omega ^ S _ { 0 , \\theta } = \\omega ^ S _ \\theta ( t ) | _ { t = 0 } . \\end{align*}"}
{"id": "2108.png", "formula": "\\begin{align*} \\begin{aligned} & P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M ) } ) = \\sum _ { l = 1 } ^ { k - 1 } \\frac { \\binom M l ( k ) _ l ( k ( n - 1 ) ) _ { M - l } } { ( k n ) _ M } + \\\\ & k \\sum _ { j = 1 } ^ { n - 1 } \\sum _ { l = 1 } ^ k \\frac { \\binom M l ( k ) _ l ( k ( j - 1 ) ) _ { M - l } } { ( k n ) _ M } \\frac 1 { k ( n - j + 1 ) - l } . \\end{aligned} \\end{align*}"}
{"id": "1006.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s \\partial _ 1 v ( 0 ) & = - c _ { n , s } \\int _ { \\R ^ n } \\frac { \\partial _ 1 v ( y ) } { \\vert y \\vert ^ { n + 2 s } } \\dd y \\\\ & = c _ { n , s } \\int _ { \\R ^ n } v ( y ) \\partial _ 1 \\vert y \\vert ^ { - n - 2 s } \\dd y \\\\ & = - c _ { n , s } ( n + 2 s ) \\int _ { \\R ^ n } \\frac { y _ 1 v ( y ) } { \\vert y \\vert ^ { n + 2 s + 2 } } \\dd y \\\\ & = - 2 c _ { n , s } ( n + 2 s ) \\int _ { \\R ^ n _ + } \\frac { y _ 1 v ( y ) } { \\vert y \\vert ^ { n + 2 s + 2 } } \\dd y \\end{align*}"}
{"id": "3052.png", "formula": "\\begin{align*} & x _ 1 ^ 2 y _ 1 ^ 2 ( x _ 1 y _ 2 + x _ 2 y _ 1 ) + x _ 1 y _ 1 ( x _ 1 ^ 2 y _ 2 ^ 2 + x _ 2 ^ 2 y _ 1 ^ 2 + a x _ 1 x _ 2 y _ 1 y _ 2 ) + \\\\ + & b ( x _ 1 ^ 3 y _ 2 ^ 3 + x _ 2 ^ 3 y _ 1 ^ 3 ) + c x _ 1 x _ 2 y _ 1 y _ 2 ( x _ 1 y _ 2 + x _ 2 y _ 1 ) + \\\\ + & x _ 2 y _ 2 ( d ( x _ 1 ^ 2 y _ 2 ^ 2 + x _ 2 ^ 2 y _ 1 ^ 2 ) + e x _ 1 x _ 2 y _ 1 y _ 2 ) + f x _ 2 ^ 2 y _ 2 ^ 2 ( x _ 1 y _ 2 + x _ 2 y _ 1 ) = 0 \\ , . \\end{align*}"}
{"id": "8545.png", "formula": "\\begin{align*} \\mu ( k , \\ell , m , n ) = \\sqrt { 2 \\pi } \\delta _ 0 ( k - \\ell + m - n ) + \\mu ^ { \\# } _ L ( k , \\ell , m , n ) + \\mu ^ { \\# } _ R ( k , \\ell , m , n ) . \\end{align*}"}
{"id": "1408.png", "formula": "\\begin{align*} \\ln \\hat q = { 1 \\over 2 } ( \\theta x + \\epsilon p ^ b ) . \\end{align*}"}
{"id": "173.png", "formula": "\\begin{align*} g _ m ( w ) = \\frac { 2 } { \\pi ^ 2 w } \\dfrac { 1 } { J _ m ^ 2 ( \\sqrt { w } ) + Y _ m ^ 2 ( \\sqrt { w } ) } , L _ { \\frac { d } { 2 } } ( w ) = \\frac { 1 } { ( 2 \\pi ) ^ { \\frac { d } { 2 } } } w ^ { \\frac { d } { 2 } } K _ { \\frac { d } { 2 } } ( w ) , \\end{align*}"}
{"id": "6619.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 ^ * ( h , k ) = J _ 1 + J _ 2 + J _ 3 + O \\big ( X ^ { - \\frac { 1 } { 2 } + \\varepsilon } Q ^ { \\frac { 5 } { 2 } } ( h k ) ^ { \\varepsilon } \\big ) + O \\big ( X ^ { \\varepsilon } Q ^ { \\frac { 3 } { 2 } + \\varepsilon } ( h k ) ^ { \\varepsilon } \\big ) . \\end{align*}"}
{"id": "5024.png", "formula": "\\begin{align*} l _ { Z _ * , n } ' ( 0 ) = { \\lambda _ { Z _ * , n } ^ 3 } \\prod _ { i = 0 } ^ { n - 1 } { \\xi _ { k } ^ { ''' } ( 0 ) \\over \\xi _ { k + 1 } ^ { ''' } ( 0 ) } = { \\lambda _ { Z _ * , n } ^ 3 } { \\xi _ { 0 } ^ { ''' } ( 0 ) \\over \\xi _ n ^ { ''' } ( 0 ) } = { \\lambda _ { Z _ * , n } ^ 3 } . \\end{align*}"}
{"id": "5672.png", "formula": "\\begin{align*} \\lambda _ 1 x \\cdot \\lambda _ 2 ( x \\cdot y ) = \\lambda _ 3 y \\end{align*}"}
{"id": "733.png", "formula": "\\begin{align*} \\sigma ( t ) = ( a _ + { \\bf 1 } _ { t > 0 } + a _ - { \\bf 1 } _ { t < 0 } ) t \\end{align*}"}
{"id": "1845.png", "formula": "\\begin{align*} D ^ n ( y ) = W _ n ( x , y ) . \\end{align*}"}
{"id": "7259.png", "formula": "\\begin{align*} A _ 2 = & c ^ 2 ( 1 - \\theta _ 1 ) - 2 t ^ 2 \\theta _ 2 = \\dfrac { 9 a b c ^ 3 d - b ^ 2 c ^ 4 } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } - \\dfrac { 2 b ^ 2 ( 3 b d - c ^ 2 ) ^ 2 } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } \\\\ = & \\dfrac { 3 b ( 4 b ^ 2 c ^ 2 d + 3 a c ^ 3 d - 6 b ^ 3 d ^ 2 - b c ^ 4 ) } { 6 a c ^ 3 + 6 b ^ 3 d - 4 b ^ 2 c ^ 2 } \\\\ = & \\dfrac { 3 b ^ 3 c ^ 2 d } { 2 c ^ 2 ( 3 a c - b ^ 2 ) + 2 b ^ 2 ( 3 b d - c ^ 2 ) } \\cdot ( 4 + \\dfrac { 3 a c } { b ^ 2 } - \\dfrac { 6 b d } { c ^ 2 } - \\dfrac { c ^ 2 } { b d } ) . \\end{align*}"}
{"id": "289.png", "formula": "\\begin{align*} T ( x , t ) : = \\mathcal { F } ^ { - 1 } \\left [ \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left ( \\frac { - t \\xi ^ { 2 } + i \\gamma t \\xi ^ { 3 } } { 1 + \\xi ^ { 2 } } \\right ) \\right ] ( x ) . \\end{align*}"}
{"id": "2081.png", "formula": "\\begin{align*} D W _ j & = \\prod _ { i < k , i \\neq j , k \\neq j } ( \\lambda _ i - \\lambda _ k ) ^ 2 \\cdot \\prod _ { k \\neq j } ( \\lambda _ j - \\lambda _ k ) \\cdot \\sum _ { k \\neq j } ( \\lambda _ j - \\lambda _ k ) ^ { - 1 } \\end{align*}"}
{"id": "2405.png", "formula": "\\begin{align*} \\langle f , D ( c _ \\gamma ) \\rangle _ \\mathcal { H } = \\langle ( \\langle f , e _ \\gamma \\rangle _ \\mathcal { H } ) , ( c _ \\gamma ) \\rangle _ { \\ell ^ 2 ( \\Gamma ) } . \\end{align*}"}
{"id": "3442.png", "formula": "\\begin{align*} f ( x ) & = \\int _ 0 ^ \\infty \\psi _ t \\ast q _ t \\ast f ( x ) \\frac { d t } { t } \\\\ & = - \\sum \\limits _ { j = - \\infty } ^ \\infty \\int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \\psi _ { j } \\ast q _ { j } \\ast f ( x ) \\frac { d t } { t } + \\sum \\limits _ { j = - \\infty } ^ \\infty \\int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \\Big [ \\psi _ { j } \\ast q _ { j } \\ast f ( x ) - \\psi _ t \\ast q _ t \\ast f ( x ) \\Big ] \\frac { d t } { t } \\\\ & = - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\psi _ { j } \\ast q _ { j } \\ast f ( x ) + R _ 1 ( f ) , \\end{align*}"}
{"id": "789.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ { \\theta } u = f \\in L ^ { p ' } ( Z ) . \\end{align*}"}
{"id": "902.png", "formula": "\\begin{align*} \\alpha _ p ( U _ 0 \\bot p U _ 0 , T ) = 2 ( 1 + p ^ { - 1 } ) ^ 2 p ^ m . \\end{align*}"}
{"id": "2158.png", "formula": "\\begin{align*} ( - \\triangle _ { g } ) ^ { \\alpha } u + g ( u ) = K ( x ) f ( x , u ) , \\ \\ \\ \\mathbb { R } ^ { d } , \\end{align*}"}
{"id": "4155.png", "formula": "\\begin{align*} P = \\sup _ { [ - T , T ] } \\{ \\| u _ 1 ( t ) \\| _ { H ^ s } + \\| u _ 2 ( t ) \\| _ { H ^ s } + \\| w ( t ) \\| _ { Z _ { s , 4 } } \\} < \\infty . \\end{align*}"}
{"id": "3464.png", "formula": "\\begin{align*} \\| f ( x ) \\| _ p ^ p \\leq \\sum \\limits _ { l = - \\infty } ^ \\infty \\| \\sum \\limits _ { Q \\in B _ l } \\omega ( Q ) D _ k ( x , x _ Q ) { \\widetilde D } _ k ( f ) ( x _ Q ) \\| _ p ^ p . \\end{align*}"}
{"id": "7198.png", "formula": "\\begin{align*} \\hat { G } ( t , \\xi ) & = \\hat { \\phi } ( \\xi ) | \\xi | \\hat { \\psi } _ \\xi ( t | \\xi | ) . \\end{align*}"}
{"id": "3751.png", "formula": "\\begin{align*} B _ { i } ^ j ( X ) = \\varepsilon _ j g ( \\nabla _ { X } e _ i , e _ j ) = - \\varepsilon _ j g ( e _ i , \\nabla _ { X } e _ j ) = - \\varepsilon _ j \\varepsilon _ i B ^ { i } _ j ( X ) . \\end{align*}"}
{"id": "470.png", "formula": "\\begin{align*} \\varphi _ { \\mathbf { L } } : \\mathbb { C } [ y _ { \\mathcal { I } } : \\ , \\mathcal { I } \\in \\mathfrak { G } ( \\mathbf { L } ) ] \\longrightarrow \\mathbb { C } [ x _ { \\alpha } : \\ , \\alpha \\in [ n ] ] , \\varphi _ { \\mathbf { L } } ( y _ { \\mathcal { I } } ) : = \\prod _ { \\alpha \\in \\mathcal { I } } x _ { \\alpha } . \\end{align*}"}
{"id": "2906.png", "formula": "\\begin{align*} \\Bigg | \\partial _ r ^ k \\left ( \\frac { 1 } { r } \\partial _ r Q \\right ) \\Bigg | \\lesssim \\sum _ { j = 0 } ^ k r ^ { - ( k - j + 1 ) } | \\partial _ r ^ { j + 1 } Q | \\lesssim Q ( r ) , \\forall \\ ; r \\ge 1 \\end{align*}"}
{"id": "8227.png", "formula": "\\begin{align*} N _ r = x _ r z \\sum _ { j \\geq 0 } \\Big ( \\frac { N _ 1 } { x _ 1 z } ( N _ 1 + N _ 2 + \\cdots + N _ { k + 1 - r } ) \\Big ) ^ j = \\frac { x _ r z } { 1 - \\frac { N _ 1 } { x _ 1 z } ( N _ 1 + N _ 2 + \\cdots + N _ { k + 1 - r } ) } \\end{align*}"}
{"id": "1954.png", "formula": "\\begin{align*} ( f \\bullet g ) ( x ) : = q g ( x ) f ( x g ( x ) ) . \\end{align*}"}
{"id": "3833.png", "formula": "\\begin{align*} y ^ { [ 0 ] } : = y , y ^ { [ k ] } = ( y ^ { [ k - 1 ] } ) ' - \\sum _ { j = 1 } ^ k f _ { k , j } y ^ { [ j - 1 ] } , k = \\overline { 1 , n } , \\end{align*}"}
{"id": "3312.png", "formula": "\\begin{align*} \\mbox { $ r _ i = r _ { i 1 } / r _ { i 2 } $ w i t h $ { \\rm G C D } ( r _ { i 1 } , r _ { i 2 } ) = 1 $ a n d $ r _ { i 2 } > 0 $ f o r $ i = 1 , 2 , 3 $ . } \\end{align*}"}
{"id": "3748.png", "formula": "\\begin{align*} T _ m M = V _ 0 ( m ) + V _ 1 ( m ) + \\ldots + V _ s ( m ) + V _ 0 ' ( m ) \\end{align*}"}
{"id": "7580.png", "formula": "\\begin{align*} c _ { i , j } \\approx \\dfrac { 4 } { n _ x n _ y } \\sum _ { l _ x = 1 } ^ { n _ x } \\sum _ { l _ y = 1 } ^ { n _ y } f ( x _ { l _ x } , y _ { l _ y } ) T _ i ( x _ { l _ x } ) T _ j ( y _ { l _ y } ) = : \\tilde { c } _ { i , j } , \\end{align*}"}
{"id": "5171.png", "formula": "\\begin{align*} A ( b ) ^ 2 = 4 \\left | Z _ { b ^ { - 1 } } \\phi \\left ( \\tfrac { 1 } { 4 } , \\tfrac { 1 } { 2 } \\right ) \\right | ^ 4 = 8 \\pi ^ 2 \\ , \\frac { b ^ { 3 } \\vartheta _ 1 ' ( 0 , i b ) ^ 4 } { 2 ^ 4 } \\ , e ^ { - \\pi \\frac { b } { 4 } } \\frac { \\vartheta _ 3 ( \\frac { 1 } { 2 } - \\frac { i b } { 4 } , i b ) ^ 4 } { \\vartheta _ 4 ( \\frac { 1 } { 2 } , i b ) ^ 4 \\vartheta _ 4 ( \\frac { 1 } { 4 } , \\frac { i } { b } ) ^ 4 } . \\end{align*}"}
{"id": "8244.png", "formula": "\\begin{align*} W ^ J : = \\{ w \\in W \\ , | \\ , \\ell ( w ) < \\ell ( w s ) \\ \\ s \\in J \\} \\end{align*}"}
{"id": "4065.png", "formula": "\\begin{align*} ( \\rho , u ) : = ( \\tilde \\rho , \\tilde u ) + ( \\check \\rho , \\check u ) , \\end{align*}"}
{"id": "8937.png", "formula": "\\begin{align*} u ( x ) = \\inf \\left \\lbrace \\int _ 0 ^ \\infty e ^ { - s } \\left ( L \\left ( \\gamma ( s ) , - \\dot { \\gamma } ( s ) \\right ) \\right ) d s : \\gamma \\in \\mathrm { A C } ( [ 0 , \\infty ) ; \\overline { \\Omega } ) , \\gamma ( 0 ) = x \\right \\rbrace , \\end{align*}"}
{"id": "5925.png", "formula": "\\begin{align*} \\langle ( F L ) ^ a ( \\pmb { v } ) , w _ q \\rangle = \\left . \\frac { d } { d s } \\right | _ { s = 0 } L ( v _ { 1 q } , \\dots , v _ { a q } + s w _ q , \\dots , v _ { k q } ) , w _ q \\in T Q . \\end{align*}"}
{"id": "5152.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial t ^ 2 } \\left [ \\log ( \\cosh ( t ) ^ 2 + 1 ) \\right ] & = \\frac { \\partial } { \\partial t } \\frac { 2 \\sinh ( 2 t ) } { \\cosh ( 2 t ) + 3 } \\\\ & = \\frac { 4 \\cosh ( 2 t ) \\ , ( \\cosh ( 2 t ) + 3 ) - 4 \\sinh ( 2 t ) \\sinh ( 2 t ) } { ( \\cosh ( 2 t ) + 3 ) ^ 2 } \\\\ & = \\frac { 4 + 1 2 \\cosh ( 2 t ) } { ( \\cosh ( 2 t ) + 3 ) ^ 2 } > 0 . \\end{align*}"}
{"id": "4372.png", "formula": "\\begin{align*} b ( \\tau ) \\sim e ^ { \\frac { 1 } { \\alpha } ( \\frac { \\alpha } { 2 } - 1 ) \\tau } . \\end{align*}"}
{"id": "1035.png", "formula": "\\begin{align*} \\frac { u ( h e _ 1 ) } h & = \\frac 1 { h } \\gamma _ { n , s } \\int _ { \\R ^ n _ + \\setminus B _ r ^ + } \\bigg ( \\frac { r ^ 2 - h ^ 2 } { \\vert y \\vert ^ 2 - r ^ 2 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert h e _ 1 - y \\vert ^ n } - \\frac 1 { \\vert h e _ 1 + y \\vert ^ n } \\bigg ) u ( y ) \\dd y \\end{align*}"}
{"id": "510.png", "formula": "\\begin{align*} \\forall \\delta \\in \\mathfrak { C } _ { c } , \\ , a \\in \\mathfrak { C } _ { r } : h ( \\mathcal { A } _ { \\delta } ^ { a } ) \\neq 0 \\Leftrightarrow a = \\varrho ( \\delta ) . \\end{align*}"}
{"id": "2839.png", "formula": "\\begin{align*} \\Im \\int x \\cdot \\nabla u ( x , t ) \\bar { u } ( x , t ) d x = \\frac { 1 } { 4 } \\dot { y } ( t ) > 0 \\end{align*}"}
{"id": "7185.png", "formula": "\\begin{align*} F ( t , x , v ) = \\mu ( v ) + f ( t , x , v ) . \\end{align*}"}
{"id": "9153.png", "formula": "\\begin{align*} \\kappa ( k ) : = 4 ( M + 1 ) ( B ( 4 k + 4 ) - 1 ) ^ 2 - 1 . \\end{align*}"}
{"id": "3051.png", "formula": "\\begin{align*} x ^ \\prime = \\alpha _ 1 \\alpha _ 2 \\beta _ 1 \\beta _ 2 x \\ , , y ^ \\prime = \\alpha _ 1 ^ 2 \\beta _ 1 ^ 2 y \\ , , z ^ \\prime = \\alpha _ 1 \\beta _ 1 \\alpha _ 2 \\beta _ 2 z \\ , , w ^ \\prime = \\alpha _ 2 ^ 2 \\beta _ 2 ^ 2 w \\ , , \\end{align*}"}
{"id": "9411.png", "formula": "\\begin{align*} & \\varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) \\\\ = & \\varphi ( b _ 0 a _ 1 ( c _ 1 + \\tau ( b _ 1 ) ) a _ 2 \\cdots ( c _ { n - 1 } + \\tau ( b _ { n - 1 } ) ) a _ n b _ n ) \\\\ = & \\varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots c _ { n - 1 } a _ n b _ n ) \\\\ & + \\sum _ { r = 1 } ^ n \\sum _ { 1 \\leq k _ 1 < \\cdots < k _ r \\leq n - 1 } \\varphi ( b _ 0 a _ 1 c _ 1 a _ 2 \\cdots \\hat { c } _ { k _ 1 } \\cdots \\hat { c } _ { k _ r } \\cdots c _ { n - 1 } a _ n b _ n ) \\tau ( b _ { k _ 1 } ) \\cdots \\tau ( b _ { k _ r } ) \\end{align*}"}
{"id": "5992.png", "formula": "\\begin{align*} | \\varphi ( z ) | & \\leq C _ { \\varepsilon } \\mathrm { e } ^ { \\varepsilon | z | } \\left ( \\sum _ { n = 0 } ^ { \\infty } ( n ! ) ^ { 2 } 2 ^ { n q } | \\varphi _ { n } | ^ { 2 } \\right ) ^ { 1 / 2 } \\left ( \\sum _ { n = 0 } ^ { \\infty } 2 ^ { - n q } \\sigma _ { \\varepsilon } ^ { - 2 n } \\right ) ^ { 1 / 2 } \\\\ & = C _ { \\varepsilon } ( 1 - 2 ^ { - q } \\sigma _ { \\varepsilon } ^ { - 2 } ) ^ { - 1 / 2 } \\| \\varphi \\| _ { q , 1 , \\pi _ { \\lambda , \\beta } } \\mathrm { e } ^ { \\varepsilon | z | } , \\end{align*}"}
{"id": "4265.png", "formula": "\\begin{align*} u ( 0 , x ) ~ = ~ \\overline { w } ( x ) + \\left ( c _ 1 \\cdot \\chi _ { \\strut ] - \\infty , 0 [ } + c _ 2 \\cdot \\chi _ { \\strut ] 0 , + \\infty [ } \\right ) \\cdot \\phi ( x , 0 ) . \\end{align*}"}
{"id": "254.png", "formula": "\\begin{align*} G _ R ( y ) = ( - R ) \\vee \\left ( y \\wedge R \\right ) . \\end{align*}"}
{"id": "3959.png", "formula": "\\begin{align*} \\lambda ^ p _ k & = - k ^ 2 \\pi ^ 2 - 2 c _ k k \\pi - 2 i d _ k k \\pi + O ( 1 ) , k \\geq k _ 0 , \\\\ \\lambda ^ h _ k & = - 1 - \\alpha _ { 1 , k } - i ( 2 k \\pi + \\alpha _ { 2 , k } ) , | k | \\geq k _ 0 , \\end{align*}"}
{"id": "252.png", "formula": "\\begin{align*} W _ 1 ( \\mu _ n , \\gamma _ { \\Sigma } ) \\leq \\dfrac { \\| \\Sigma ^ { - \\frac { 1 } { 2 } } \\| _ { o p } \\| \\Sigma \\| _ { H S } } { n } \\left ( \\sum _ { k = 1 } ^ n ( U _ { \\Sigma , \\mu _ k } - 1 ) \\right ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "6553.png", "formula": "\\begin{align*} \\int _ { | \\xi | \\leq R } { | \\xi | ^ { 2 s } g ( \\xi ) ^ { k } e ^ { - 2 t | \\xi | ^ { 2 } g ( \\xi ) } \\ , d \\xi } = & \\int _ { | \\xi | \\leq R } { | \\xi | ^ { 2 s } \\big ( 2 t | \\xi | ^ { 2 } \\big ) ^ { - k } \\big ( 2 t | \\xi | ^ { 2 } g ( \\xi ) \\big ) ^ { k } e ^ { - 2 t | \\xi | ^ { 2 } g ( \\xi ) } \\ , d \\xi } \\\\ \\leq & C t ^ { - k } \\int _ { | \\xi | \\leq R } { | \\xi | ^ { 2 s - 2 k } \\ , d \\xi } \\\\ \\leq & C t ^ { - k } \\int _ { 0 } ^ { R } { r ^ { 2 s - 2 k + 1 } \\ , d r } \\\\ \\leq & C t ^ { - k } R ^ { 2 ( s - k + 1 ) } , \\end{align*}"}
{"id": "6543.png", "formula": "\\begin{align*} \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s + r \\right ) ^ 2 \\leq \\ & \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s + r \\right ) ^ 2 + 2 k ^ 2 H ^ 2 - 4 r s + 4 ( r - k ) ^ 2 \\\\ = \\ & \\left ( - k H ^ 2 + \\frac { r } { 2 } H ^ 2 + s - r \\right ) ^ 2 + ( 2 H ^ 2 + 4 ) ( r - k ) ^ 2 . \\end{align*}"}
{"id": "9550.png", "formula": "\\begin{align*} w ^ k - ( 1 - \\mu ^ k ) ( \\mathcal { A } ( x ^ k ) ^ { m - 1 } + q ) - \\mu ^ k w ^ { ( 0 ) } = 0 \\end{align*}"}
{"id": "3087.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\varepsilon ^ 2 y _ 1 \\ , , y _ 1 ^ \\prime = - \\varepsilon ^ 4 x _ 2 \\ , , x _ 2 ^ \\prime = \\varepsilon ^ 3 y _ 2 \\ , , y _ 2 ^ \\prime = \\varepsilon x _ 1 \\ , , \\end{align*}"}
{"id": "5756.png", "formula": "\\begin{align*} \\operatorname { H e s s } z ^ { j } ( \\bar { \\nabla } z ^ { k } , \\bar { \\nabla } z ^ { l } ) = \\operatorname { H e s s } y ^ { j } ( \\nabla y ^ { k } , \\nabla y ^ { l } ) \\end{align*}"}
{"id": "1686.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } \\cdot x \\cdot { _ { s + 2 } F _ { s + 1 } } \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s + 1 } , 1 - k ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s + 1 } ; x ^ 2 \\right ) . \\end{align*}"}
{"id": "330.png", "formula": "\\begin{align*} C _ F ( \\rho ^ a , \\rho ^ b ) & = \\Bigg \\{ \\rho \\in H ^ { 1 } ( [ 0 , 1 ] ; \\mathcal P ( G ) ) , m \\in L ^ 2 ( [ 0 , 1 ] ; \\mathcal S ^ { N \\times N } ) \\Bigg | ( \\rho ( 0 ) , \\rho ( 1 ) ) = ( \\rho ^ a , \\rho ^ b ) , \\\\ & d \\rho _ i ( t ) + \\underset { j \\in N ( i ) } { \\sum } m _ { i j } d t + \\underset { j \\in N ( i ) } { \\sum } ( \\Sigma _ j - \\Sigma _ i ) \\theta _ { i j } ( \\rho ) d W ^ { \\delta } ( t ) = 0 . \\Bigg \\} \\end{align*}"}
{"id": "1221.png", "formula": "\\begin{align*} S _ { n + k ^ { \\prime } } \\phi ( x ) = S _ { n } \\phi ( x ) \\times \\frac { n } { n + k ^ { \\prime } } . \\end{align*}"}
{"id": "6079.png", "formula": "\\begin{align*} - g \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} _ { s } - \\begin{pmatrix} 0 & - e \\\\ g & - 2 f \\end{pmatrix} \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} _ { t } = \\Lambda \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} + \\begin{pmatrix} e H ^ { j } - g F ^ { j } \\\\ 2 f H ^ { j } - g G ^ { j } \\end{pmatrix} \\end{align*}"}
{"id": "7717.png", "formula": "\\begin{align*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + f ^ { \\phi ^ { \\perp } } , \\ ; \\phi [ 0 ] = u [ 0 ] \\end{align*}"}
{"id": "919.png", "formula": "\\begin{align*} ( a ^ { + } a ) _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( a ^ { + } ) ^ { k } a ^ { k } . \\end{align*}"}
{"id": "6291.png", "formula": "\\begin{align*} \\mathrm { v a r } [ \\widehat { K } _ { \\mathrm { a - c p t - d } } ^ \\mathbb { R } ] & = \\frac { \\psi } { 2 } \\Big ( \\frac { 1 } { N _ 2 \\bar { \\gamma } _ c ' } \\ ! - \\ ! \\frac { \\mathrm { l i } ( 1 - \\xi ) K ' } { ( 1 + N _ 1 \\bar { \\gamma } ' ) ( 1 - \\xi ) } \\Big ) . \\end{align*}"}
{"id": "2110.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , 2 } ( W _ { \\mathcal { S } ( n , 2 ; M _ n ) } ) = - 2 c ( 1 - c ) + ( 2 c - c ^ 2 ) \\log ( 2 - c ) - ( 2 c + c ^ 2 ) \\log c ; \\\\ & \\lim _ { n \\to \\infty } P _ { n , 3 } ( W _ { \\mathcal { S } ( n , 3 ; M _ n ) } ) = - \\frac 3 2 ( 1 - c ) c ( 1 + 3 c ) - ( 3 c + 3 c ^ 2 + c ^ 3 ) \\log c + \\\\ & ( \\frac 3 2 c + \\frac 3 2 c ^ 2 - c ^ 3 ) \\log ( c ^ 2 - 3 c + 3 ) + 3 \\sqrt 3 ( - c + c ^ 2 ) \\arctan ( \\frac { 3 - 2 c } { \\sqrt 3 } ) + \\frac { \\sqrt 3 \\pi } 2 ( c - c ^ 2 ) . \\end{aligned} \\end{align*}"}
{"id": "2080.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n - 1 } ( x - a _ i ) = \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ k \\sigma _ i ( \\pmb { a } ) x ^ { n - 1 - k } \\end{align*}"}
{"id": "3424.png", "formula": "\\begin{align*} q ( x ) - q ( x ' ) & = \\int _ { d ( x , y ) \\geqslant 2 \\delta } [ K ( x , y ) - K ( x ' , y ) ] v ( y ) [ f _ 1 ( y ) - f _ 1 ( x ) ] d \\omega ( y ) \\\\ & \\qquad + [ f _ 1 ( x ) - f _ 1 ( x ' ) ] \\int _ { \\R ^ N } K ( x ' , y ) u ( y ) d \\omega ( y ) = : I + I \\ ! I . \\end{align*}"}
{"id": "2592.png", "formula": "\\begin{align*} \\norm { f } _ { M ^ 2 } ^ 2 = \\iint _ { \\R ^ { 2 d } } | V _ { g _ 0 } f ( x , \\omega ) | ^ 2 \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "9545.png", "formula": "\\begin{align*} x \\geq 0 , ~ ~ ~ \\omega = \\mathcal { A } x ^ { m - 1 } + q \\geq 0 , ~ ~ ~ \\mbox { a n d } ~ ~ x ^ { T } ( \\mathcal { A } x ^ { m - 1 } + q ) = 0 \\end{align*}"}
{"id": "9000.png", "formula": "\\begin{align*} F ^ * : = ( F ^ * _ 1 , F ^ * _ 2 , \\dots , F ^ * _ { K - 1 } , U ) , \\end{align*}"}
{"id": "8674.png", "formula": "\\begin{align*} W _ n = \\prod _ { i = 1 } ^ { k _ n } I _ 2 ( m _ { n , i } ) \\end{align*}"}
{"id": "5591.png", "formula": "\\begin{align*} g ( x ) & = \\sum _ { k = 1 } ^ { m } p _ k \\theta _ k e ^ { - \\theta _ k x } \\\\ \\bar { G } ( x ) & = \\sum _ { k = 1 } ^ { m } p _ k e ^ { - \\theta _ k x } \\end{align*}"}
{"id": "7743.png", "formula": "\\begin{gather*} P _ { \\alpha , \\beta } ^ l ( y ) : = \\{ ( t , x ) : x \\in [ y , y + l ] , t \\in [ \\alpha + x - y , \\beta - x + y ] \\} , \\end{gather*}"}
{"id": "3251.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 ^ + } s ^ 2 \\Big ( 1 - s h ( s ) \\Big ) = \\lim _ { s \\rightarrow 0 ^ + } \\int _ 0 ^ \\infty \\frac { s ^ 2 u ^ 2 } { s ^ 2 + u ^ 2 } d \\mu ( u ) = \\lambda _ 2 ( \\mu ) ^ 2 . \\end{align*}"}
{"id": "4582.png", "formula": "\\begin{align*} d \\mathbf { P } _ \\lambda = Z _ n ( \\lambda ) d \\mathbf { P } . \\end{align*}"}
{"id": "8868.png", "formula": "\\begin{align*} \\check H ^ q _ { c t } ( X , A ) \\cong \\check H _ { c t } ^ q ( U _ 1 \\cap U _ 2 , A ) \\cong \\check H _ { c t } ^ { q + 1 } ( U _ 1 \\cup U _ 2 , A ) = \\check H _ { c t } ^ { q + 1 } ( X \\times \\R , A ) \\end{align*}"}
{"id": "6772.png", "formula": "\\begin{align*} f _ t - \\frac { 1 } { 2 \\sqrt { t } } f '' + \\frac { 1 } { 2 \\sqrt { t } } + \\rho \\max ( f ^ * - f , 0 ) = 0 , \\end{align*}"}
{"id": "3349.png", "formula": "\\begin{align*} \\{ u , v , w \\} = \\theta ( T v , T w ) u , \\forall u , v , w \\in V . \\end{align*}"}
{"id": "4855.png", "formula": "\\begin{align*} 1 = \\int _ 0 ^ 1 Q _ \\alpha ( x ) \\ , d x \\leq \\int _ 0 ^ 1 Q _ \\alpha ( 1 ) x \\ , d x = \\frac 1 2 Q _ \\alpha ( 1 ) . \\end{align*}"}
{"id": "3010.png", "formula": "\\begin{align*} R ^ { x y , z } _ { x ^ 3 y z } = [ \\gamma _ { x ^ 4 y , x ^ 3 y z } ] + [ \\gamma _ { x ^ 3 y ^ 2 , x ^ 3 y z } ] - [ \\gamma _ { x ^ 3 y z , x ^ 3 z ^ 2 } ] - [ \\gamma _ { x ^ 3 y z , x ^ 2 y z ^ 2 } ] . \\end{align*}"}
{"id": "8620.png", "formula": "\\begin{align*} i k \\mathcal { K } ^ { \\# } _ { S } ( x , k ) = \\partial _ x \\mathcal { K } _ { S , \\# } ' ( x , k ) - \\mathcal { K } _ { S , \\# } '' ( x , k ) \\end{align*}"}
{"id": "7381.png", "formula": "\\begin{align*} u _ t ( x , t ) - | \\nabla u ( x , t ) | m ( \\{ u ( \\cdot , t ) < u ( x , t ) \\} ) = 0 \\end{align*}"}
{"id": "1244.png", "formula": "\\begin{align*} F _ { \\mu } ^ { [ \\alpha , \\beta ] , \\rho , \\varepsilon } = \\left \\{ x \\in \\mathbb { R } ^ d : \\overline { \\dim } ( \\mu , x ) \\in [ \\alpha , \\beta ] \\forall r < \\rho , \\ \\mu ( B ( x , r ) ) \\geq r ^ { \\overline { \\dim } ( \\mu , x ) + \\varepsilon } \\right \\} . \\end{align*}"}
{"id": "6094.png", "formula": "\\begin{align*} z = r ^ m P ( \\theta ) , \\end{align*}"}
{"id": "1333.png", "formula": "\\begin{align*} A ( \\delta _ { 1 } ) \\approx \\frac { 2 } { 3 } p _ { i } R , \\ , \\ , A ( \\delta _ { 2 } ) \\approx \\frac { 1 } { 3 } p _ { i } R , \\ , \\ , p _ { i + 1 } = \\frac { 4 } { 3 } p _ { i } , \\ , \\ , A ( \\eta ) \\approx \\frac { 1 } { 6 } p _ { i } R . \\end{align*}"}
{"id": "8914.png", "formula": "\\begin{align*} E = E | _ { U _ 1 ^ 2 } \\cup \\cdots \\cup E | _ { U _ n ^ 2 } \\cup E | _ { B ^ 2 } \\end{align*}"}
{"id": "5632.png", "formula": "\\begin{align*} f _ c ( y ) = \\frac { 1 } { 2 } y ^ + y ^ - + \\frac { a } { 2 } ( y ^ 0 ) ^ 2 - c , \\\\ H = 2 y ^ + \\partial _ + - 2 y ^ - \\partial _ - ~ , ~ ~ ~ ~ ~ E ^ \\pm = \\frac { 1 } { \\sqrt { a } } y ^ \\pm \\partial _ 0 - 2 \\sqrt { a } y ^ 0 \\partial _ \\mp . \\end{align*}"}
{"id": "2925.png", "formula": "\\begin{align*} X _ k = \\begin{cases} X _ { k - 2 } + X _ { k - 1 } & , X _ { k - 2 } + X _ { k - 1 } \\le 1 \\\\ X _ { k - 2 } + X _ { k - 1 } - 1 & , X _ { k - 2 } + X _ { k - 1 } > 1 . \\end{cases} \\end{align*}"}
{"id": "4547.png", "formula": "\\begin{align*} d _ { U _ 1 } ( v ) \\le d _ { U _ i } ( v ) & \\le k | M _ i | + \\frac { k - 1 } { 2 } | U _ i - M _ i | = \\frac { k - 1 } { 2 } | U _ i | + \\frac { k + 1 } { 2 } | M _ i | \\\\ & < \\frac { k - 1 } { 2 } \\biggl ( \\frac { n } { r - 1 } + \\delta ^ { 1 / 5 } n \\biggr ) + \\frac { k + 1 } { 2 } \\cdot \\frac { n } { 4 k ^ 3 } \\\\ & < \\frac { k - 1 / 8 } { 2 ( r - 1 ) } n , \\end{align*}"}
{"id": "6436.png", "formula": "\\begin{align*} \\hat { \\omega } ( \\pi ) : = ( N - \\lambda _ 1 ) \\cdot \\lambda _ { \\# ( \\pi ) } \\cdot \\prod _ { i = 1 } ^ { \\# ( \\pi ) - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } - 1 ) \\ \\ \\ \\ \\hat { \\omega } ( \\emptyset ) = N + 1 . \\end{align*}"}
{"id": "6961.png", "formula": "\\begin{align*} \\forall u \\in \\mathbb S _ { \\mathbb X } \\colon \\ker D ^ * \\Phi ( ( \\bar x , \\bar y ) ; ( u , 0 ) ) = \\{ 0 \\} \\end{align*}"}
{"id": "4914.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } ( \\norm { \\xi _ t } ^ 2 - \\sigma ^ 2 ) \\leq \\sigma ^ 2 \\log ( 1 / \\delta ) \\end{align*}"}
{"id": "4996.png", "formula": "\\begin{align*} A ( x , y ) & = \\left ( a _ { 0 } ( x ) , h _ { 0 } ( x ) \\right ) , \\\\ B ( x , y ) & = \\left ( b _ { 0 } ( x ) , x \\right ) . \\end{align*}"}
{"id": "8555.png", "formula": "\\begin{align*} T ( k ) \\psi _ { + } ( x , k ) & = R _ { - } ( k ) \\psi _ { - } ( x , k ) + \\psi _ { - } \\left ( x , - k \\right ) , \\\\ T ( k ) \\psi _ { - } ( x , k ) & = R _ { + } ( k ) \\psi _ { + } ( x , k ) + \\psi _ { + } \\left ( x , - k \\right ) . \\end{align*}"}
{"id": "3876.png", "formula": "\\begin{align*} \\begin{pmatrix} A ^ * & C ^ * \\\\ B ^ * & D ^ * \\end{pmatrix} . \\end{align*}"}
{"id": "7005.png", "formula": "\\begin{align*} R ( z ) = - \\frac { 1 } { \\delta } \\mbox { a n d } S ( z ) = \\frac { 1 } { \\delta } . \\end{align*}"}
{"id": "5637.png", "formula": "\\begin{align*} f _ c ( x ) = \\frac { 1 } { 2 } ( ( x ^ 1 ) ^ 2 + ( x ^ 2 ) ^ 2 - ( x ^ 3 ) ^ 2 ) - c , \\end{align*}"}
{"id": "706.png", "formula": "\\begin{align*} \\mathbb P ( S _ n ^ c ) = O ( n ^ { - \\infty } ) . \\end{align*}"}
{"id": "3664.png", "formula": "\\begin{align*} h ' ( s ) = 1 + A ( 1 + \\delta ) s ^ { \\delta } , \\ \\ \\ \\ h '' ( s ) = A \\delta ( 1 + \\delta ) s ^ { \\delta - 1 } . \\end{align*}"}
{"id": "612.png", "formula": "\\begin{align*} \\frac { 1 } { A ( x ) } \\ = \\ \\frac { ( h ( x ) + 1 ) f ( x ) - ( h ( x ) + 1 ) g ( x ) } { \\abs { \\abs { f ( x ) + g ( x ) } ^ 2 - 1 } + 1 } . ] \\end{align*}"}
{"id": "5673.png", "formula": "\\begin{align*} \\lambda x \\cdot x y = y \\end{align*}"}
{"id": "4884.png", "formula": "\\begin{align*} H ( w ) = \\sum _ { k = 0 } ^ { n - 1 } b _ k ( w + 1 ) ^ { k + 1 } , b _ k = \\frac { ( n - 1 ) ! ( n + 1 + k ) ! } { k ! ( n - k - 1 ) ! ( k + 1 ) ! } ( - 2 ) ^ { n - 1 - k } , \\end{align*}"}
{"id": "6430.png", "formula": "\\begin{align*} \\nabla _ \\nu \\xi = \\textstyle \\sum \\limits _ { i = 2 } ^ 4 \\nu e _ i \\nabla _ { e _ i } \\xi = - B \\xi - A \\xi . \\end{align*}"}
{"id": "7882.png", "formula": "\\begin{align*} \\langle P _ d , Q _ d \\rangle = \\sum _ { | J | = d } \\binom { d } { J } ^ { - 1 } p _ J q _ J , \\end{align*}"}
{"id": "3380.png", "formula": "\\begin{align*} & T ^ { ' } _ t ( u ) = ( I d _ L + t [ \\mathfrak { X } , - ] ) ( T _ t \\Big ( I d _ V - t D ( \\mathfrak { X } ) + t ^ { 2 } D ^ { 2 } ( \\mathfrak { X } ) + \\cdots + ( - 1 ) ^ { i } t ^ { i } D ^ { i } ( \\mathfrak { X } ) + \\cdots ) ( u ) \\Big ) ) \\\\ & \\quad \\ ; \\ ; \\ ; \\ ; \\ ; = T ( u ) + t ( T _ 1 ( u ) - T D ( \\mathfrak { X } ) ( u ) + [ \\mathfrak { X } , T ( u ) ] ) + t ^ { 2 } T ^ { ' } _ { 2 } ( u ) + \\cdots \\\\ & \\quad \\ ; \\ ; \\ ; \\ ; \\ ; = T ( u ) + t ^ { 2 } T ^ { ' } _ { 2 } ( u ) + \\cdots ( a s \\ ; T _ 1 ( u ) = d _ T ( \\mathfrak { X } ) ( u ) ) . \\end{align*}"}
{"id": "7793.png", "formula": "\\begin{align*} { _ { 0 } \\mathfrak { I } ^ { \\beta } f ( t ) } = \\frac { 1 } { \\Gamma ( \\beta ) } \\int _ o ^ t ( t - \\eta ) ^ { \\beta - 1 } f ( \\eta ) ~ \\mathrm { d } \\eta = \\frac { t ^ \\beta } { \\Gamma { ( \\beta + 1 ) } } . \\end{align*}"}
{"id": "3588.png", "formula": "\\begin{align*} B _ T ( e , y ^ * ) ( x ^ * ) = \\langle ( T ( e ) ) ( x ^ * ) , y ^ * \\rangle \\ , \\ , ( e , y ^ * ) \\in E \\times Y ^ * x ^ * \\in X ^ * , \\end{align*}"}
{"id": "4694.png", "formula": "\\begin{align*} \\partial _ y ( \\R \\eta _ i ) = & \\partial _ y \\Bigg [ \\bigg | 1 + \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n \\frac { \\sigma _ i \\sigma _ j \\R _ j } { \\R _ i } \\bigg | ^ { p - 1 } \\bigg ( 1 + \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n \\frac { \\sigma _ i \\sigma _ j \\R _ j } { \\R _ i } \\bigg ) \\sigma _ i \\R _ i ^ p \\eta _ i \\Bigg ] \\\\ = & \\partial _ y \\bigg ( \\sigma _ i \\R _ i ^ p \\eta _ i + \\sum _ { \\substack { j = 1 , \\\\ j \\not = i } } ^ n p \\sigma _ j \\R _ i ^ { p - 1 } \\R _ j \\eta _ i \\bigg ) + C , \\end{align*}"}
{"id": "4985.png", "formula": "\\begin{align*} \\lambda _ { Z } = \\eta ^ { r _ 0 } \\circ \\xi ( 0 ) . \\end{align*}"}
{"id": "4701.png", "formula": "\\begin{align*} J _ 4 = \\sum _ { \\substack { i , j = 1 , \\\\ i \\not = j } } ^ n \\frac { \\partial _ y ( \\tau _ i F _ { i j , 3 } ) } { x _ { i j } ^ 2 } + O ( \\Gamma ) , \\end{align*}"}
{"id": "8791.png", "formula": "\\begin{align*} \\begin{aligned} V ^ { - 1 } ( s ) _ { i 2 } & \\leq \\delta _ { i 1 } , V ^ { - 1 } ( s ) _ { i 0 } + V ^ { - 1 } ( s ) _ { i 4 } \\leq 1 - \\delta _ { i 1 } & & \\forall i \\in \\{ 1 , 2 \\} \\\\ V ^ { - 1 } ( s ) _ { i 0 } + V ^ { - 1 } ( s ) _ { i 1 } & \\leq \\delta _ { i 2 } , V ^ { - 1 } ( s ) _ { i 3 } + V ^ { - 1 } ( s ) _ { i 4 } \\leq 1 - \\delta _ { i 2 } & & \\forall i \\in \\{ 1 , 2 \\} . \\end{aligned} \\end{align*}"}
{"id": "8305.png", "formula": "\\begin{align*} e ( \\vect { r } ) = \\int _ { V _ S } j ( \\vect { s } ) \\ , h ( \\vect { r } , \\vect { s } ) \\ , d \\vect { s } \\end{align*}"}
{"id": "4483.png", "formula": "\\begin{align*} E _ { \\psi } ( x ) = \\Big | \\frac { \\psi ( x ) - x } { x } \\Big | . \\end{align*}"}
{"id": "8068.png", "formula": "\\begin{align*} \\left \\{ T ( h ) , \\Psi ( f ) \\right \\} _ \\ell ^ \\Sigma [ \\varphi ] & = - \\int _ \\mathcal { I } T ( h ) ^ { ( 1 ) } [ \\varphi ] ( s ) \\frac { d \\Psi ( f ) ^ { ( 1 ) } [ \\phi ] ( s ) } { d s } \\ , \\mathrm { d } s \\\\ & = - \\int _ \\mathcal { I } h ( s ) \\varphi ( s ) \\sqrt { \\gamma ' ( s ) } \\frac { d f } { d s } ( s ) \\ , \\mathrm { d } s \\\\ & = - \\Psi \\left ( h f ' \\right ) [ \\varphi ] . \\end{align*}"}
{"id": "7135.png", "formula": "\\begin{align*} J _ 1 ^ n ( t , u ) & : = \\int _ u ^ t \\exp \\Big ( \\int _ s ^ t \\tilde { f } ' ( r , x _ r ) d r \\Big ) \\Big ( \\tilde { f } ' _ n ( s , x _ s ^ n ) - K _ s \\Big ) K _ H ( s , u ) d s \\end{align*}"}
{"id": "8629.png", "formula": "\\begin{align*} \\mu _ { R } ( k , \\ell , m , n ) = \\mu _ { R , 1 } ( k , \\ell , m , n ) + \\mu _ { R , 2 } ( k , \\ell , m , n ) \\end{align*}"}
{"id": "2694.png", "formula": "\\begin{align*} \\norm { U ^ { - 1 } - \\sum _ { k = 1 } ^ n ( I - U ) ^ k } _ { o p } \\to 0 , n \\to \\infty . \\end{align*}"}
{"id": "5334.png", "formula": "\\begin{align*} ( \\phi ^ { \\epsilon } , u ^ { \\epsilon } , \\eta ^ { \\epsilon } , \\tau ^ { \\epsilon } ) \\big | _ { t = 0 } = ( \\phi _ 0 ^ { \\epsilon } , u _ 0 ^ { \\epsilon } , \\eta _ 0 ^ { \\epsilon } , \\tau _ 0 ^ { \\epsilon } ) . \\end{align*}"}
{"id": "2166.png", "formula": "\\begin{align*} W ^ { \\alpha , G } ( \\mathbb { R } ^ { d } ) = \\bigg { \\{ } u \\in L ^ { G } ( \\mathbb { R } ^ { d } ) : \\ \\overline { \\rho } ( \\alpha ; u ) < \\infty \\bigg { \\} } , \\end{align*}"}
{"id": "5805.png", "formula": "\\begin{align*} \\mathbf { D } = \\left [ \\begin{aligned} \\Omega _ N ^ { 0 0 } & & \\cdots & & \\Omega _ N ^ { 0 ( N - 1 ) } \\\\ \\vdots & & \\ddots & & \\vdots \\\\ \\Omega _ N ^ { ( N - 1 ) 0 } & & \\cdots & & \\Omega _ N ^ { ( N - 1 ) ( N - 1 ) } \\end{aligned} \\right ] \\end{align*}"}
{"id": "2188.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } K ( x ) f ( x , w ^ + ) w ^ + d x & = A ^ + ( w ) \\end{align*}"}
{"id": "8330.png", "formula": "\\begin{align*} 0 \\leq \\varphi ( t ) \\leq 1 ~ \\forall t \\in \\R , \\varphi ( t ) = 1 ~ \\forall t \\in ( - \\infty , - 1 ] , \\varphi ( t ) = 0 ~ \\forall t \\in [ 0 , \\infty ) . \\end{align*}"}
{"id": "6402.png", "formula": "\\begin{align*} ( M _ { z } ^ * ) ^ j k _ { \\overline y } = y ^ j k _ { \\overline y } \\rightarrow 0 \\ ; \\textup { a s } j \\rightarrow \\infty , \\end{align*}"}
{"id": "5942.png", "formula": "\\begin{align*} L ( v _ 1 , \\dots , v _ k ) = \\frac { 1 } { 2 } \\mathcal { G } ( v _ 1 , v _ 1 ) + \\dots + \\frac { 1 } { 2 } \\mathcal { G } ( v _ k , v _ k ) , \\end{align*}"}
{"id": "4129.png", "formula": "\\begin{align*} P = \\left \\{ \\sum _ { i = 1 } ^ { d - 1 } \\lambda _ i b _ i : \\lambda _ i \\in [ \\alpha _ i , \\beta _ i ] \\cap \\Z \\right \\} . \\end{align*}"}
{"id": "8393.png", "formula": "\\begin{align*} \\P ( \\exists v \\in \\Lambda _ n : v \\mbox { i s u n d e t e r m i n e d } ) & \\leq \\sum _ { v \\in \\Lambda _ n } \\sum _ { 1 \\leq \\ell \\leq r } \\sum _ { I _ 1 , \\ldots , I _ \\ell } \\sum _ { B ' _ { I _ 1 } , \\ldots , B ' _ { I _ \\ell } } \\P ( \\sigma | _ { B _ { I _ j } ( v ) } = \\sigma | _ { B ' _ { I _ j } } \\mbox { f o r } 1 \\leq j \\leq \\ell ) \\\\ & \\leq n ^ d r \\binom { r - 1 } { \\ell - 1 } n ^ { d \\ell } q ^ { - ( r - 1 ) ^ { d } ( 1 + \\ell ) } = o ( 1 ) \\ , , \\end{align*}"}
{"id": "9159.png", "formula": "\\begin{align*} \\Lambda ( s , f \\otimes \\chi _ d ) = \\left ( \\frac { | d | } { 2 \\pi } \\right ) ^ s \\Gamma ( s + \\tfrac { \\kappa - 1 } { 2 } ) L ( s , f \\otimes \\chi _ d ) = i ^ \\kappa \\epsilon ( d ) \\Lambda ( 1 - s , f \\otimes \\chi _ d ) . \\end{align*}"}
{"id": "5944.png", "formula": "\\begin{align*} [ e _ x , e _ y ] = - 2 e _ z , [ e _ x , e _ \\theta ] = e _ y , [ e _ y , e _ \\theta ] = - e _ x . \\end{align*}"}
{"id": "732.png", "formula": "\\begin{align*} \\epsilon ^ { ( \\ell + 1 ) } & \\leq \\frac { C } { \\ell ^ 3 } + \\epsilon ^ { ( \\ell ) } \\left [ 1 - \\frac { 2 - \\delta } { \\ell } \\right ] + C ( \\epsilon ^ { ( \\ell ) } ) ^ 3 . \\end{align*}"}
{"id": "6586.png", "formula": "\\begin{align*} \\prod _ { p \\nmid h k } \\sum _ { \\substack { 0 \\le q , \\ell < \\infty \\\\ \\min ( \\ell , q ) = 0 } } p ^ { - q w } \\phi ^ { \\star } ( p ^ q ) \\frac { \\tau _ A ( p ^ { \\ell } ) \\tau _ B ( p ^ { \\ell } ) } { p ^ { \\ell ( 1 + s _ 1 + s _ 2 ) } } = \\prod _ { p \\nmid h k } \\left ( 1 + \\sum _ { q = 1 } ^ { \\infty } p ^ { - q w } \\phi ^ { \\star } ( p ^ q ) + \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { \\tau _ A ( p ^ { \\ell } ) \\tau _ B ( p ^ { \\ell } ) } { p ^ { \\ell ( 1 + s _ 1 + s _ 2 ) } } \\right ) \\end{align*}"}
{"id": "6498.png", "formula": "\\begin{align*} f _ n ^ { ( 2 m - 1 ) } & \\sim \\binom { 2 m - 1 } { 2 m - 3 } M ^ { ( 2 m - 3 ) } _ n \\\\ & \\sim ( 2 m - 1 ) ( m - 1 ) C ' _ { m - 1 } n ^ { m - 3 / 2 } ( \\log n ) ^ { m - 2 } . \\end{align*}"}
{"id": "3374.png", "formula": "\\begin{align*} & [ T _ t ( u ) , T _ t ( v ) , T _ t ( w ) ] = T _ t \\Big ( D ( T _ t ( u ) , T _ t ( v ) ) w + \\theta ( T _ t ( v ) , T _ t ( w ) ) u - \\theta ( T _ t ( u ) , T _ t ( w ) ) v \\Big ) , \\end{align*}"}
{"id": "9177.png", "formula": "\\begin{align*} X ^ 2 \\ll & \\left ( \\sum _ { 2 < p \\leq X } ( \\log p ) L ( \\tfrac { 1 } { 2 } , f \\otimes \\chi _ { 8 p } ) \\mathcal { M } ( p ) \\right ) ^ 2 \\leq \\ \\mathcal { N } \\sum _ { 2 < p \\leq X } ( \\log p ) \\left | L ( \\tfrac { 1 } { 2 } , f \\otimes \\chi _ { 8 p } ) \\right | ^ 2 | \\mathcal { M } ( p ) | ^ 2 \\ll \\mathcal { N } X . \\end{align*}"}
{"id": "4712.png", "formula": "\\begin{align*} & \\alpha _ i = \\sum _ { j = i } ^ k \\theta _ j , \\forall i = 1 , \\ldots , k , \\\\ & \\alpha _ { k + 1 } = 0 , \\quad \\alpha _ { n + 1 - i } = - \\alpha _ i , \\forall i = 1 , \\ldots , k . \\end{align*}"}
{"id": "5780.png", "formula": "\\begin{align*} \\nabla P = \\nabla ( a \\rho ^ \\gamma ) = \\frac { a \\gamma } { \\gamma - 1 } \\rho \\nabla ( \\rho ^ { \\gamma - 1 } ) = \\frac { a \\gamma } { \\gamma - 1 } \\rho \\nabla P _ 1 \\end{align*}"}
{"id": "1705.png", "formula": "\\begin{align*} \\nu _ { t , m } = \\lceil 2 \\cdot 2 ^ { \\gamma _ * k _ * t } \\cdot 2 ^ m \\rceil , \\end{align*}"}
{"id": "2157.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ^ + ( n , k ; M _ n ) } ) = \\Big ( k \\sum _ { l = 1 } ^ k \\binom k l ( \\frac c { 1 - c } ) ^ l \\Big ) \\Big ( ( - \\frac { ( 1 - c ) ^ k } k \\log ( 1 - ( 1 - c ) ^ k ) \\Big ) = \\\\ & \\Big ( ( \\frac c { 1 - c } + 1 ) ^ k - 1 \\Big ) \\Big ( - ( 1 - c ) ^ k \\log ( 1 - ( 1 - c ) ^ k ) \\Big ) = \\\\ & - \\big ( 1 - ( 1 - c ) ^ k \\big ) \\log ( 1 - ( 1 - c ) ^ k ) . \\end{aligned} \\end{align*}"}
{"id": "9094.png", "formula": "\\begin{align*} \\alpha = 1 + \\alpha _ m t ^ m + \\alpha _ { m + 1 } t ^ { m + 1 } + \\cdots + \\alpha _ { n - 1 } t ^ { n - 1 } \\end{align*}"}
{"id": "2981.png", "formula": "\\begin{align*} \\delta _ n ^ 2 ( 2 ) \\delta _ n ^ 2 ( 3 ) = \\frac { 4 d ^ 5 } { 9 0 ^ 5 \\cdot 1 2 } ( 1 + o ( 1 ) ) \\end{align*}"}
{"id": "1082.png", "formula": "\\begin{align*} S ( k ) = \\begin{pmatrix} a ( k ) & b ^ * ( k ) \\\\ b ( k ) & a ^ * ( k ) \\end{pmatrix} . \\end{align*}"}
{"id": "6890.png", "formula": "\\begin{align*} J _ q ( n , k , t ) = S _ k ^ { k - t } = \\frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \\choose t } _ q } \\sum \\limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \\choose 2 } } { k - t \\choose i } _ q { k + i \\choose i } _ q N _ k ^ { i } \\end{align*}"}
{"id": "7337.png", "formula": "\\begin{align*} x = \\lambda y _ j + ( 1 - \\lambda ) z _ j , \\big ( \\lambda u ( y _ j , t ) ^ q + ( 1 - \\lambda ) u ( z _ j , t ) ^ q \\big ) ^ { 1 \\over q } \\le u _ { q , \\lambda } ( x , t ) + \\frac { 1 } { j } . \\end{align*}"}
{"id": "1147.png", "formula": "\\begin{align*} I _ 1 & = - \\frac { 1 } { 2 \\pi i } \\oint _ { \\partial U _ { \\delta } ( \\pm \\eta ) } \\left ( \\sum _ { j = 1 } ^ { \\infty } \\frac { T _ { \\pm \\eta , j } ( \\xi , k ) } { t ^ j } \\right ) d s = t ^ { - 1 } \\textnormal { R e s } _ { k = \\eta } T _ { \\eta , 1 } + t ^ { - 1 } \\textnormal { R e s } _ { k = - \\eta } T _ { - \\eta , 1 } + \\mathcal { O } ( t ^ { - 2 } ) , \\end{align*}"}
{"id": "1687.png", "formula": "\\begin{align*} \\overline { H } _ n ( s ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } F _ s } \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - k ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; 1 \\right ) . \\end{align*}"}
{"id": "6035.png", "formula": "\\begin{align*} - x \\widetilde { L } _ n ' + ( \\alpha - n ) \\widetilde { L } _ n ' + n \\widetilde { L } _ n + ( n - \\alpha ) \\widetilde { L } _ { n - 1 } ' = 0 . \\end{align*}"}
{"id": "7487.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { \\tau ^ 2 } + \\frac { \\eta ^ n } { 2 \\tau } + \\frac { \\vartheta ^ n } { 2 } - \\frac 1 4 \\Delta \\right ) \\Big ( { \\phi } ^ { n + 1 } - \\phi ^ n \\Big ) = 0 , \\end{align*}"}
{"id": "6529.png", "formula": "\\begin{align*} J ^ { ( 2 m ) } _ n = \\frac { 1 } { t _ n ^ { ( 2 m ) } } \\sum ^ { n - 1 } _ { j = 1 } \\binom { 2 m } { 2 m - 2 } d _ j ^ { ( 2 m - 2 ) } - 1 + O ( ( \\log n ) ^ { - m } ) . \\end{align*}"}
{"id": "4702.png", "formula": "\\begin{align*} J _ 5 = D - \\sum _ { \\substack { i , j , k = 1 , \\\\ i \\not = j } } ^ n \\frac { \\mu _ k \\partial _ y [ \\tau _ i ( \\mathcal { L } \\mathfrak { B } _ { i j k } ) ( \\varphi _ { i j } - 1 ) ] } { x ^ 3 _ { i j } } - \\sum _ { \\substack { i , j , k = 1 , \\\\ i \\not = j } } ^ n \\partial _ y \\bigg ( \\frac { \\mu _ k \\tau _ i ( [ | D | , \\varphi _ { i j } ] \\mathfrak { B } _ { i j k } ) } { x _ { i j } ^ 3 } \\bigg ) , \\end{align*}"}
{"id": "608.png", "formula": "\\begin{align*} & f _ 0 ( x , y ) \\ = \\ x + 1 , \\\\ [ 3 p t ] & f _ 1 ( x , y ) \\ = \\ x + y , \\\\ [ 3 p t ] & f _ 2 ( x , y ) \\ = \\ x y , \\\\ [ 3 p t ] & f _ { n + 1 } ( x , 0 ) \\ = \\ 1 , \\\\ [ 3 p t ] & f _ { n + 1 } ( x , y + 1 ) \\ = \\ f _ n ( x , f _ { n + 1 } ( x , y ) ) ( n \\geq 2 ) . \\end{align*}"}
{"id": "7518.png", "formula": "\\begin{align*} \\arg \\pi ^ { - \\frac { \\sigma + i T } { 2 } } = \\arg \\left ( \\pi ^ { - \\frac { \\sigma } { 2 } } e ^ { - \\left ( \\frac { i \\log \\pi } { 2 } \\right ) T } \\right ) \\end{align*}"}
{"id": "6259.png", "formula": "\\begin{align*} { \\bf E } \\left ( { \\bf A } , { \\mathbb V } \\right ) = \\left \\{ { \\bf A } _ { \\theta } : \\theta \\left ( x , y \\right ) = \\sum _ { i = 1 } ^ { s } \\theta _ { i } \\left ( x , y \\right ) e _ { i } \\ \\ \\ \\ \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in { \\bf T } _ { s } ( { \\bf A } ) \\right \\} . \\end{align*}"}
{"id": "270.png", "formula": "\\begin{align*} & k _ { \\tau _ { 0 } } ^ { ( 2 ) } ( z ) = - \\frac { 3 1 } { 1 9 } z + \\frac { 1 2 } { 1 9 } , \\\\ & k _ { \\tau _ { 0 } } ^ { ( 2 ) } ( 1 ) = - 1 , \\\\ & k _ { \\tau _ { 0 } } ^ { ( 2 ) } ( - 1 ) = \\frac { 4 3 } { 1 9 } . \\end{align*}"}
{"id": "2051.png", "formula": "\\begin{align*} f _ r \\left ( s \\right ) = - 1 + e ^ { - \\frac { i \\beta \\eta } { 2 } \\omega ^ 2 _ r s } , f _ { r 1 } \\left ( s \\right ) = \\frac { \\omega ^ 3 _ r } { 4 } \\left ( \\frac { \\beta ^ 2 \\eta ^ 2 } { 2 } + \\gamma - \\eta ^ 2 \\right ) s e ^ { - \\frac { i \\beta \\eta } { 2 } \\omega ^ 2 _ r s } \\end{align*}"}
{"id": "3868.png", "formula": "\\begin{align*} \\int _ { v \\in G _ 0 \\cdot \\Lambda \\cdot s _ i ( L _ i ) } f ( v ) m _ { \\infty } ( v ) \\ , d v = | W _ 0 | \\int _ { b \\in \\Lambda \\cdot L _ i } \\int _ { g \\in G _ 0 } f ( g \\cdot s _ i ( b ) ) \\ , d g \\ , d b , \\end{align*}"}
{"id": "8398.png", "formula": "\\begin{align*} \\rho = \\frac { d _ { \\Gamma ( x , t ) } } { \\varepsilon } - h _ 1 ( S ( x , t ) , t ) - \\varepsilon h _ { 2 , \\varepsilon } ( S ( x , t ) , t ) , \\end{align*}"}
{"id": "1980.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( x ) = 1 + \\widehat { \\beta } ( x ) \\widehat { \\Phi } ( x ) . \\end{align*}"}
{"id": "7978.png", "formula": "\\begin{align*} \\frac { ( \\alpha + \\beta ) ^ 2 } { | \\alpha \\beta | } \\geq \\begin{cases} \\frac { \\sigma } { 2 | \\beta | } , & \\ \\alpha + \\beta > \\alpha / 2 ; \\\\ \\frac { \\sigma ^ 2 } { 2 | \\beta | ^ 2 } , & \\ \\alpha + \\beta \\leq \\alpha / 2 . \\end{cases} \\end{align*}"}
{"id": "3436.png", "formula": "\\begin{align*} T _ 2 & \\lesssim \\int _ { d ( y , z ) \\geqslant \\ 1 2 d ( x , y ) } \\frac 1 { V ( x , z , t + d ( x , z ) ) } \\Big ( \\frac { t } { t + d ( x , z ) } \\Big ) ^ { \\varepsilon _ 1 } \\frac 1 { \\omega ( B ( y , d ( x , y ) ) ) } d \\omega ( z ) \\\\ & = \\frac 1 { \\omega ( B ( y , d ( x , y ) ) ) } \\int _ { \\R ^ N } \\frac 1 { V ( x , z , t + d ( x , z ) ) } \\Big ( \\frac { t } { t + d ( x , z ) } \\Big ) ^ { \\varepsilon _ 1 } d \\omega ( z ) . \\end{align*}"}
{"id": "6532.png", "formula": "\\begin{align*} L ^ { ( 2 ) } _ n & = 1 + \\sum _ { j = 1 } ^ { n - 1 } \\dfrac { 1 } { j + 1 } = \\sum _ { j = 1 } ^ n \\dfrac { 1 } { j } , \\end{align*}"}
{"id": "9147.png", "formula": "\\begin{align*} \\forall x , y \\in X \\left ( x = y \\rightarrow \\forall k \\in \\mathbb { N } \\left ( H ^ * \\left [ T x , T y , \\frac { 1 } { k + 1 } \\right ] \\right ) \\right ) . \\end{align*}"}
{"id": "6049.png", "formula": "\\begin{align*} d _ \\triangle ( a , b ) \\ = \\ \\sum _ { i \\in [ n ] } ( b _ i - a _ i ) - n \\min _ { i \\in [ n ] } ( b _ i - a _ i ) \\ = \\ \\sum _ { i \\in [ n ] } ( b _ i - a _ i ) + n \\max _ { i \\in [ n ] } ( a _ i - b _ i ) \\enspace , \\end{align*}"}
{"id": "3309.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F ) \\geq { } & \\alpha _ { n - j } ( F _ 1 / T _ 1 ) + \\alpha _ { n - j } ( F _ 2 / T _ 2 ) - \\\\ { } & \\sum _ { l = 1 } ^ { j - 1 } \\frac { ( - 1 ) ^ { l - 1 } } { 2 l ! } ( a _ 2 ^ l \\alpha _ { n - j + l } ( F _ 1 / T _ 1 ) + a _ 1 ^ l \\alpha _ { n - j + l } ( F _ 2 / T _ 2 ) ) . \\end{align*}"}
{"id": "3388.png", "formula": "\\begin{align*} \\delta ^ { 1 } ( \\alpha ) ( x , y , z ) = \\partial _ { \\rho } ( \\alpha ) ( [ x , y ] , z ) - \\rho ( z ) \\partial _ { \\rho } ( \\alpha ) ( x , y ) . \\end{align*}"}
{"id": "5116.png", "formula": "\\begin{align*} c _ l = \\int _ 0 ^ { 1 / b } C _ { b , \\gamma } ^ 2 \\ , e ^ { - 2 \\gamma t } e ^ { - 2 \\pi i \\frac { l } { a } t } \\ , d t = C _ { b , \\gamma } ^ 2 \\ , \\frac { a \\left ( 1 - e ^ { - \\frac { 2 \\gamma } { b } } \\right ) } { 2 a \\gamma + 2 \\pi i l } . \\end{align*}"}
{"id": "2815.png", "formula": "\\begin{align*} i \\partial _ t h + \\Delta h + i \\dot { \\alpha } Q - i \\dot { X } \\cdot \\nabla Q - \\dot { \\theta } Q = O ( \\delta + \\delta \\delta ^ * ) L ^ 2 . \\end{align*}"}
{"id": "3717.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\| t ^ { \\frac { \\beta } { \\alpha } } B \\| _ { H ^ { \\frac 5 2 - \\alpha + \\beta } } = 0 , \\ \\ \\ \\beta > 0 . \\end{align*}"}
{"id": "1502.png", "formula": "\\begin{align*} ( 1 - p ^ { 2 n - 2 } \\chi ^ 2 ( p ) ) ^ { - 1 } \\prod _ { i = 1 } ^ n \\left ( ( 1 - \\alpha _ { i , p } \\chi ( p ) p ^ { n - 1 - s } ) ( 1 - \\alpha _ { i , p } ^ { - 1 } \\chi ( p ) p ^ { n - 1 - s } ) \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "7429.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\int _ { \\Omega _ 2 ^ j ( x ) } \\sum _ { z = x - \\ell } ^ { x - 1 } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta _ { 1 , x , j , z } ) - f \\big ( \\eta _ { 2 , x , j , z } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "2176.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { F ( x , t u ) } { \\vert t u \\vert ^ { g ^ + } } \\vert u \\vert ^ { g ^ + } = + \\infty , \\ \\ x \\in A . \\end{align*}"}
{"id": "7856.png", "formula": "\\begin{align*} [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( I ) & = [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G \\cup H \\cup I ) \\\\ & = [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( \\mathbb { N } ) = \\{ \\bar { a } _ n \\} \\end{align*}"}
{"id": "719.png", "formula": "\\begin{align*} \\frac { \\partial z _ { q ; \\alpha } ^ { ( L + 1 ) } } { \\partial W _ { i j } ^ { ( 1 ) } } = x _ { j ; \\alpha } \\sigma ' ( z _ { i ; \\alpha } ^ { ( 1 ) } ) \\frac { \\partial z _ { q ; \\alpha } ^ { ( L + 1 ) } } { \\partial \\sigma ( z _ { i ; \\alpha } ^ { ( 1 ) } ) } . \\end{align*}"}
{"id": "4797.png", "formula": "\\begin{align*} A & = \\{ ( a , 0 ) \\in G \\mid a \\neq 0 \\} \\\\ B & = \\{ ( 0 , b ) \\in G \\mid b \\neq 0 \\} \\\\ C & = \\{ ( a , b ) \\in G \\mid a , b \\neq 0 \\} \\\\ \\end{align*}"}
{"id": "1515.png", "formula": "\\begin{align*} \\det ( q _ { \\infty } ) ^ { - l } c ( h , q , l ) = C \\cdot e _ { \\infty } ( i \\lambda ( q ^ { \\ast } h q ) ) . \\end{align*}"}
{"id": "9074.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ 1 ^ { i n } ( x ) = \\frac { 1 0 } { 3 } \\chi _ { _ { [ - 0 . 5 , 0 . 5 ] } } , \\rho _ 2 ^ { i n } ( x ) = 2 + s i n ( \\pi x ) , \\\\ & \\phi ( 0 , t ) = - 1 , \\ \\phi ( 1 , t ) = 1 . \\end{aligned} \\end{align*}"}
{"id": "2178.png", "formula": "\\begin{align*} \\langle J ^ { ' } ( t _ 2 u ) , u \\rangle = 0 . \\end{align*}"}
{"id": "8939.png", "formula": "\\begin{align*} \\Tilde { \\xi } ( s ) : = \\left \\{ \\begin{aligned} & \\xi ( s ) + \\left ( 1 - \\frac { s } { T } \\right ) h , 0 \\leq s \\leq T , \\\\ & \\xi ( s ) , s \\geq T , \\end{aligned} \\right . \\end{align*}"}
{"id": "6602.png", "formula": "\\begin{align*} \\Upsilon _ { \\pm } ( w ; m h , n k ) = \\Upsilon _ { \\pm } ( w ; m h , n k ; c , Q ) : = \\int _ { 0 } ^ { \\infty } \\frac { c | m h \\pm n k | } { g x Q } W \\left ( \\frac { c | m h \\pm n k | } { g x Q } \\right ) x ^ { w - 1 } \\ , d x . \\end{align*}"}
{"id": "84.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 1 / 2 } ( 2 ^ { ( n _ { 2 , \\nu _ 2 } + 1 ) / 2 } + 1 ) ( 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 ) . \\end{align*}"}
{"id": "280.png", "formula": "\\begin{align*} \\widetilde { f } _ { M } ( x ) & : = - \\frac { \\gamma ( f _ { M } ( x ) - x ) e ^ { - \\frac { x ^ { 2 } } { 4 } } } { 3 2 \\sqrt { \\pi } H ( x ) } \\int _ { \\R } H ( y ) ( f _ { M } ( y ) ) ^ { 3 } d y , \\\\ H ( x ) & : = \\cosh \\frac { M } { 4 } - \\sinh \\left ( \\frac { M } { 4 } \\right ) \\frac { 2 } { \\sqrt { \\pi } } \\int _ { 0 } ^ { x / 2 } e ^ { - y ^ { 2 } } d y . \\end{align*}"}
{"id": "7128.png", "formula": "\\begin{align*} \\tilde { f } ' ( s , y ) \\le K _ 1 + s ^ { 2 H - 1 } K _ 2 \\le s ^ { 2 H - 1 } K = K _ s \\end{align*}"}
{"id": "707.png", "formula": "\\begin{align*} \\forall q \\geq 1 ~ \\exists C = C ( q ) I _ C = O ( n ^ { - q } ) . \\end{align*}"}
{"id": "1592.png", "formula": "\\begin{align*} \\div u = \\div ( \\rho u ) = 0 , \\end{align*}"}
{"id": "6749.png", "formula": "\\begin{align*} \\mathbf L \\tilde { \\mathbf V } + \\mathbf b + \\mathbf q ( \\tilde { \\mathbf V } ) = 0 . \\end{align*}"}
{"id": "1311.png", "formula": "\\begin{align*} \\epsilon | \\hat { I } _ { \\geq \\epsilon } ( k _ { n } ) | < A ( \\alpha _ { k _ { n } } ) - A ( \\alpha _ { 0 } ) = s n R - A ( \\alpha _ { 0 } ) < C \\sqrt { k _ { n } } \\end{align*}"}
{"id": "3390.png", "formula": "\\begin{align*} & \\omega _ 2 ( x , y , z ) - \\omega _ 1 ( x , y , z ) = ( \\varphi _ 2 - \\varphi _ 1 ) ( [ x , y ] , z ) - \\rho ( z ) ( \\varphi _ 2 - \\varphi _ 1 ) ( x , y ) \\\\ & \\quad \\ ; \\ ; \\ ; = \\partial _ { \\rho } ( \\alpha ) ( [ x , y ] , z ) - \\rho ( z ) \\partial _ { \\rho } ( x , y ) \\\\ & \\quad \\ ; \\ ; \\ ; = \\delta ^ { 1 } ( \\alpha ) ( x , y , z ) , \\ ; \\alpha \\in C ^ { 1 } _ { L i e } ( L , V ) , \\end{align*}"}
{"id": "6090.png", "formula": "\\begin{align*} \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\\\ \\varphi _ { s } ^ { j } - F ^ { j } \\end{pmatrix} = \\begin{pmatrix} x _ { s } & y _ { s } & z _ { s } \\\\ x _ { t } & y _ { t } & z _ { t } \\\\ x _ { s s } & y _ { s s } & z _ { s s } \\end{pmatrix} \\begin{pmatrix} u ^ { j } \\\\ v ^ { j } \\\\ w ^ { j } \\end{pmatrix} = M ( s , t ) \\begin{pmatrix} u ^ { j } \\\\ v ^ { j } \\\\ w ^ { j } \\end{pmatrix} \\ , . \\end{align*}"}
{"id": "9418.png", "formula": "\\begin{align*} \\tau ' ( a _ 1 \\cdots a _ n ) & = \\tau ( a _ 1 \\cdots a _ { k - 1 } \\tau ' ( a _ k ) a _ { k + 1 } \\cdots a _ n ) \\\\ & = \\sum _ { i = 1 } ^ n \\tau ( a _ 1 \\cdots a _ { j - 1 } \\tau ' ( a _ j ) a _ { j + 1 } \\cdots a _ n ) \\end{align*}"}
{"id": "1479.png", "formula": "\\begin{align*} = R ^ { - 1 } \\left [ \\begin{array} { c c } g _ 1 & 0 \\\\ 0 & g _ 2 \\end{array} \\right ] R R ^ { - 1 } \\left [ \\begin{array} { c c } U ( z _ 1 ) & 0 \\\\ 0 & \\mathfrak { J } \\overline { U ( z _ 2 ) } \\end{array} \\right ] \\left [ \\begin{array} { c c } \\lambda ( g _ 1 , z _ 1 ) & 0 \\\\ 0 & \\overline { \\lambda ( g _ 2 , z _ 2 ) } \\end{array} \\right ] ^ { - 1 } B ( g _ 1 z _ 1 , g _ 2 z _ 2 ) ^ { - 1 } S \\end{align*}"}
{"id": "8626.png", "formula": "\\begin{align*} & \\mathcal { K } _ - ( x , k ) = b _ - ^ + ( k ) e ^ { i k x } + b _ - ^ - ( k ) e ^ { - i k x } , \\\\ & \\mbox { w i t h } b _ - ^ + ( k ) = \\mathbf { 1 } _ { + } ( k ) + \\mathbf { 1 } _ { - } ( k ) T ( - k ) , b ^ - _ - ( k ) = \\mathbf { 1 } _ { + } ( k ) R _ - ( k ) . \\end{align*}"}
{"id": "113.png", "formula": "\\begin{align*} a ^ { n + 1 } = \\gamma _ 1 a + \\gamma _ 2 a ^ 2 + \\dots + \\gamma _ { n } a ^ { n } \\end{align*}"}
{"id": "8507.png", "formula": "\\begin{align*} \\lambda _ { q _ n } ( w ) * a = \\lambda _ { q _ { n + 1 } } ( w ) * a . \\end{align*}"}
{"id": "1051.png", "formula": "\\begin{align*} & \\psi ^ * ( g _ { i j } ) = a _ { i j } \\\\ & \\psi ^ * ( u _ { i 1 } ) = - d _ { 2 i } d _ { 1 2 } ^ { - 1 } , & & \\psi ^ * ( u _ { i 2 } ) = d _ { 1 i } d _ { 1 2 } ^ { - 1 } . \\\\ & \\psi ^ * ( \\nu _ { k 1 } ) = - d _ { 2 k } d _ { 1 2 } ^ { - 1 } & & \\psi ^ * ( \\nu _ { k 2 } ) = d _ { 1 k } d _ { 1 2 } ^ { - 1 } \\end{align*}"}
{"id": "2561.png", "formula": "\\begin{align*} V _ g f ( S ( x , \\omega ) ) = e ^ { \\pi i ( x \\cdot \\omega - x ' \\cdot \\omega ' ) } V _ { \\widehat { S } ^ { - 1 } g } ( \\widehat { S } ^ { - 1 } f ) ( x , \\omega ) , \\end{align*}"}
{"id": "1856.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } L _ k ( x ) L _ { n - k } ( x ) = \\sum _ { k = 0 } ^ n { n \\choose k } M _ { k } ( x ) M _ { n - k } ( x ) . \\end{align*}"}
{"id": "6810.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in \\lbrack 0 , T ] } \\left \\Vert u _ { n } ( t ) - u ( t ) \\right \\Vert _ { s } = 0 . \\end{align*}"}
{"id": "8817.png", "formula": "\\begin{align*} u _ t ( \\varphi ) : = \\log U _ t e ^ { - V _ 0 ( \\varphi ) } , \\end{align*}"}
{"id": "6884.png", "formula": "\\begin{align*} d i m ( V \\cap V ' ) = k - i . \\end{align*}"}
{"id": "5076.png", "formula": "\\begin{align*} L ( w _ 1 ) & = L ( x - y _ 1 ) \\\\ & = L ( x ) - L ( y _ 1 ) \\\\ & \\geq f ( x ) - C - \\delta r - ( f ( x ) - r + r ( \\epsilon + \\delta ) - C ) \\\\ & = - 2 \\delta r - \\epsilon r + r \\\\ & = r ( 1 - ( 2 \\delta + \\epsilon ) ) . \\end{align*}"}
{"id": "7623.png", "formula": "\\begin{align*} \\Delta P - \\frac { n } { \\alpha } \\bar { g } ( \\nabla P , \\nabla \\log u ) & = u ^ { \\frac { \\alpha + 1 } { \\alpha } } ( | h | ^ { 2 } - n H ^ { 2 } ) - u ^ { \\frac { 1 } { \\alpha } } \\overline { \\mathrm { R i c } } ( \\nu , \\lambda \\partial _ { r } ^ { T } ) \\\\ & + \\frac { n - 1 } { \\alpha } u ^ { \\frac { \\alpha + 1 } { \\alpha } } | \\nabla \\log u | ^ { 2 } . \\end{align*}"}
{"id": "7384.png", "formula": "\\begin{align*} K ( x , y ) = c ( x , y ) \\vert x - y \\vert ^ { - d - \\alpha } , ~ ~ 0 < \\lambda ^ { - 1 } \\le c ( x , y ) = c ( y , x ) \\le \\lambda , ~ ~ \\forall x , y \\in \\R ^ d , \\end{align*}"}
{"id": "8704.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { x _ 1 + x _ 2 ^ 2 } & = \\sqrt { 0 + 0 } + \\bigl ( \\sqrt { x _ 1 + 0 } - \\sqrt { 0 + 0 } \\bigr ) + \\bigl ( \\sqrt { x _ 1 + s _ { 2 1 } ( x ) } - \\sqrt { x _ 1 + 0 } \\bigr ) \\\\ & + \\bigl ( \\sqrt { x _ 1 + s _ { 2 2 } ( x ) } - \\sqrt { x _ 1 + s _ { 2 1 } ( x ) } \\bigr ) \\\\ & \\geq \\sqrt { x _ 1 } + \\sqrt { 5 + s _ { 2 1 } ( x ) } - \\sqrt { 5 } + \\sqrt { 5 + s _ { 2 2 } ( x ) } - \\sqrt { 5 + s _ { 2 1 } ( x ) } \\\\ & \\geq \\frac { \\sqrt { 5 } } { 5 } x _ 1 + \\sqrt { 5 + x _ 2 ^ 2 } - \\sqrt { 5 } = : \\varphi _ 1 ( x ) , \\end{aligned} \\end{align*}"}
{"id": "404.png", "formula": "\\begin{align*} X = & L ^ { 2 } , ~ U = u , ~ U _ { 0 } = u _ { 0 } , \\\\ A ( t ) = & ( A ^ { 0 } ) ^ { - 1 } \\left \\lbrace B ^ { i j } ( t ) \\partial _ { i } \\partial _ { j } \\cdot - A ^ { i } ( t ) \\partial _ { i } \\cdot - D ( t ) \\cdot \\right \\rbrace , \\\\ D ( A ( t ) ) = & Y = H ^ { 2 } , S ( t ) = \\lambda _ { 0 } I - A ( t ) , ~ B ( t ) = 0 , \\end{align*}"}
{"id": "5710.png", "formula": "\\begin{align*} 0 = I ( \\gamma _ { 2 } , \\gamma _ { 1 } ) - ( \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 2 } ) - \\mu _ { \\mathrm { d i s c } } ( \\gamma _ { 1 } ) ) = J _ { 0 } ( \\gamma _ { 2 } , \\gamma _ { 1 } ) = J _ { 0 } ( v ^ { z } ) . \\end{align*}"}
{"id": "2021.png", "formula": "\\begin{align*} e _ A ( t ) = e ^ { u ( { X } _ t ) - u ( X _ 0 ) } Y _ t \\exp \\left ( A _ t ^ { \\overline { \\mu } } \\right ) t \\in [ \\ , 0 , \\ , + \\infty \\ , [ , \\end{align*}"}
{"id": "3286.png", "formula": "\\begin{align*} U ^ { \\mu _ T } ( \\lambda ) = - \\int _ \\mathbb { C } \\log | \\lambda - z | d \\mu _ T ( z ) . \\end{align*}"}
{"id": "8397.png", "formula": "\\begin{align*} \\zeta ( s ) = 1 \\ ; \\hbox { f o r } \\ ; | s | \\leq \\delta ; \\zeta ( s ) = 0 \\ ; \\hbox { f o r } \\ ; | s | > 2 \\delta ; 0 \\leq - s \\zeta ' ( s ) \\leq 4 \\ ; \\hbox { f o r } \\ ; \\delta \\leq | s | \\leq 2 \\delta . \\end{align*}"}
{"id": "4734.png", "formula": "\\begin{align*} & \\dot { \\mathfrak { q } } _ 1 ( \\epsilon , \\partial _ y \\mathcal { Z } _ 1 ) - \\big ( \\partial _ y ( \\mathcal { L } _ 1 \\epsilon ) , \\mathcal { Z } _ 1 \\big ) - p \\big ( [ | V _ 0 | ^ { p - 1 } - R _ 1 ^ { p - 1 } ] \\epsilon , \\partial _ y \\mathcal { Z } _ 1 \\big ) \\\\ & = \\Big ( \\big [ | V _ 0 + \\epsilon | ^ { p - 1 } ( V _ 0 + \\epsilon ) - | V _ 0 | ^ { p - 1 } V _ 0 - p | V _ 0 | ^ { p - 1 } \\epsilon \\big ] , \\partial _ y \\mathcal { Z } _ 1 \\Big ) - ( \\Psi _ { V _ 0 } , \\mathcal { Z } _ 1 ) , \\end{align*}"}
{"id": "8888.png", "formula": "\\begin{align*} X \\ast I = \\{ ( x , i ) \\in X \\times I \\mid d ( x , x _ 0 ) = d ( i , ( 0 , 0 ) ) \\} \\end{align*}"}
{"id": "838.png", "formula": "\\begin{align*} \\int _ { A _ 1 } | u | \\ , d \\mu _ \\omega & \\le C H _ R ^ { 1 - 2 \\beta } \\ , \\left ( \\int _ { A _ 1 } g _ { u , \\rho } ^ 2 \\ , d \\mu _ \\omega \\right ) ^ { 1 / 2 } H _ R ^ \\beta \\ , \\mu _ \\omega ( A _ 1 ) ^ { 1 / 2 } \\\\ & = C ( \\beta - 1 ) \\ , R \\ , \\left ( \\int _ { A _ 1 } g _ { u , \\rho } ^ 2 \\ , d \\mu _ \\omega \\right ) ^ { 1 / 2 } \\ , \\mu _ \\omega ( A _ 1 ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "7777.png", "formula": "\\begin{gather*} \\Box \\phi = \\left ( | \\phi _ { t } | ^ 2 - | \\phi _ { x } | ^ 2 \\right ) \\phi + \\mathbf { 1 } _ { \\omega } f ^ { \\phi ^ { \\perp } } , \\\\ \\phi [ 0 ] = ( a , b ) \\ ; \\textrm { a n d } \\ ; \\phi [ T ] = ( 0 , 0 ) . \\end{gather*}"}
{"id": "3001.png", "formula": "\\begin{align*} b _ { i j } & = ( - 1 ) ^ { j + 1 } \\lhd \\ , T ^ + ( u _ i ) , u _ { j + ( - 1 ) ^ { j + 1 } } \\ , \\rhd = ( - 1 ) ^ { j + 1 } ( - 1 ) \\lhd \\ , T ( u _ { j + ( - 1 ) ^ { j + 1 } } ) , u _ { i } \\ , \\rhd \\\\ & = ( - 1 ) ^ j \\lhd \\ , T ( u _ { j + ( - 1 ) ^ { j + 1 } } ) , u _ { i } \\ , \\rhd = ( - 1 ) ^ { j + i } a _ { j + ( - 1 ) ^ { j + 1 } \\ , i + ( - 1 ) ^ { i + 1 } } \\end{align*}"}
{"id": "2357.png", "formula": "\\begin{align*} f ' - 2 \\pi i b f = - 2 \\pi c ( x - a ) f . \\end{align*}"}
{"id": "5383.png", "formula": "\\begin{align*} \\mathcal { P } ^ \\star = \\mathcal { P } \\cup \\{ ( n - 1 , 1 , 2 , 3 , \\ldots , n - 2 , n ) , ( 2 , n , n - 1 , \\ldots , 4 , 3 , 1 ) \\} . \\end{align*}"}
{"id": "9018.png", "formula": "\\begin{align*} D _ i ( x ) \\left ( \\nabla \\rho _ i + z _ i \\rho _ i \\nabla \\phi \\right ) \\cdot \\mathbf { n } = 0 , x \\in \\partial \\Omega , i = 1 , \\cdots , s . \\end{align*}"}
{"id": "7433.png", "formula": "\\begin{align*} \\Big | [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( u ) \\Big | = c _ { \\gamma } \\Big | \\int _ { 0 } ^ { \\infty } \\frac { G ( u - w ) } { w ^ { \\gamma + 1 } } d w \\Big | = c _ { \\gamma } \\Big | \\int _ { u - b _ G } ^ { u + b _ G } \\frac { G ( u - w ) } { w ^ { \\gamma + 1 } } d w \\Big | \\leq H ^ G ( u ) . \\end{align*}"}
{"id": "5919.png", "formula": "\\begin{align*} ( | \\lambda _ 1 | + | \\lambda _ 2 | + | \\lambda _ 3 | ) ^ 2 & = ( \\lambda _ 1 + \\lambda _ 2 - \\lambda _ 3 ) ^ 2 \\\\ [ 1 m m ] & = ( \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 3 ) ^ 2 - 4 \\ , ( \\lambda _ 1 \\lambda _ 2 + \\lambda _ 2 \\lambda _ 3 + \\lambda _ 3 \\lambda _ 1 ) + 4 \\ , \\lambda _ 1 \\lambda _ 2 \\\\ [ 1 m m ] & = ( n - 3 ) ^ 2 + 4 \\ , ( 2 n - 3 ) + 2 \\ , [ n - 4 + \\sqrt { n ^ 2 + 4 n - 1 2 } ] \\\\ [ 1 m m ] & > n ^ 2 + 4 n - 1 1 + 2 \\ , \\sqrt { n ^ 2 + 4 n - 1 2 } = ( \\sqrt { n ^ 2 + 4 n - 1 2 } + 1 ) ^ 2 , \\end{align*}"}
{"id": "5183.png", "formula": "\\begin{align*} \\underset { t \\to \\infty } { \\lim } \\Delta _ { \\ell , s r } ^ { a v } ( t ) = \\lim _ { t \\to \\infty } \\frac { \\sum _ { i = 1 } ^ { R _ \\ell ^ m ( t ) } \\left ( \\frac { ( T _ { \\ell i } ^ m ) ^ 2 } { 2 } + T _ { \\ell i } ^ m Z _ { \\ell i } ^ m \\right ) } { t } , \\end{align*}"}
{"id": "4674.png", "formula": "\\begin{align*} & \\frac { 1 } { p + 1 } \\bigg | \\int \\big ( | R _ 1 + \\sigma R _ 2 | ^ { p + 1 } - R _ 1 ^ { p + 1 } - R _ 2 ^ { p + 1 } - ( p + 1 ) \\sigma R _ 1 R _ 2 ^ p - ( p + 1 ) \\sigma R _ 2 R _ 1 ^ p \\big ) \\bigg | \\\\ & \\lesssim \\int R _ 1 ^ 2 R _ 2 ^ { p - 1 } + R _ 2 ^ 2 R _ 1 ^ { p - 1 } = O \\bigg ( \\frac { 1 } { | x _ 1 - x _ 2 | ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "7594.png", "formula": "\\begin{align*} | c _ { i , j } | \\le \\dfrac { 4 V _ k } { \\pi ^ 2 } \\begin{cases} \\Gamma _ { 0 , 0 } [ s ] ( i ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , \\\\ \\Gamma _ { 1 , - 1 } [ s ] ( i ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 . \\end{cases} \\end{align*}"}
{"id": "8345.png", "formula": "\\begin{align*} \\Lambda ( e ^ r - 1 ) = \\exp \\left ( - ( e ^ r - 1 ) / c \\right ) \\end{align*}"}
{"id": "1095.png", "formula": "\\begin{align*} m ^ { ( 3 ) } ( x , t , k ) = m ^ { ( 2 ) } ( x , t , k ) D ^ { \\sigma _ 3 } ( k , \\xi ) G ( x , t , k ) D ^ { - \\sigma _ 3 } ( k , \\xi ) , \\end{align*}"}
{"id": "1929.png", "formula": "\\begin{align*} \\partial _ t H _ N + v \\cdot \\nabla _ x H _ N = { \\rm { d i v } } _ v ^ \\alpha G _ N , H _ N | _ { t = 0 } = H _ 0 \\in L ^ p ( \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ 3 ) , \\end{align*}"}
{"id": "6778.png", "formula": "\\begin{align*} \\pi = \\tau _ { k _ { 1 } } ^ { i _ { k _ { 1 } } } \\cdot \\tau _ { k _ { 2 } } ^ { i _ { k _ { 2 } } } \\cdots \\tau _ { k _ { m } } ^ { i _ { k _ { m } } } \\end{align*}"}
{"id": "6308.png", "formula": "\\begin{align*} \\widehat { K '' } _ \\mathrm { a - c p t - d } ^ \\mathbb { R } & = \\sum _ { k '' = 0 } ^ { K _ l } k '' p _ { K '' | \\widehat { K ' } } ( k '' ) = \\sum _ { k '' = 0 } ^ { K _ l } k '' \\binom { \\widehat { K ' } + k '' } { k '' } \\xi ^ { k '' } ( 1 - \\xi ) ^ { \\widehat { K ' } } , \\end{align*}"}
{"id": "3448.png", "formula": "\\begin{align*} | R _ 1 ( x , y ) | & \\lesssim \\sum \\limits _ { j = - \\infty } ^ \\infty \\int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } | S _ j ( x , y ) | \\frac { d t } { t } \\\\ & \\lesssim ( r - 1 ) \\sum \\limits _ { j = - \\infty } ^ \\infty \\int _ { r ^ { - j } } ^ { r ^ { - j + 1 } } \\frac 1 { V ( x , y , t + d ( x , y ) ) } \\Big ( \\frac { t } { t + \\| x - y \\| } \\Big ) ^ { \\varepsilon } \\frac { d t } { t } \\\\ & \\lesssim ( r - 1 ) \\int _ { 0 } ^ \\infty \\frac 1 { V ( x , y , t + d ( x , y ) ) } \\Big ( \\frac { t } { t + \\| x - y \\| } \\Big ) ^ { \\varepsilon } \\frac { d t } { t } . \\end{align*}"}
{"id": "5883.png", "formula": "\\begin{gather*} \\chi _ 0 = x _ 0 + [ - c _ 0 , c _ 0 ] v , \\chi _ 1 = f ^ { - 1 } \\left ( [ - R , R ] \\times \\{ L \\} \\right ) , \\\\ \\chi _ { 2 , \\pm } = f ^ { - 1 } \\left ( \\{ \\pm R \\} \\times [ - L ' , L ' ] \\right ) , \\chi _ 2 = \\chi _ { 2 , + } \\cup \\ \\chi _ { 2 , - } , \\\\ \\chi _ { 3 , \\pm } = q _ \\pm + [ - 1 , 1 ] v _ + , \\chi _ 3 = \\chi _ { 3 , + } \\cup \\ \\chi _ { 3 , - } , \\end{gather*}"}
{"id": "7566.png", "formula": "\\begin{align*} ( u \\rhd _ s v ) \\rhd _ t w = ( u \\rhd _ t w ) \\rhd _ s ( v \\rhd _ t w ) \\end{align*}"}
{"id": "7679.png", "formula": "\\begin{align*} \\hat { \\phi } _ { \\varepsilon } ( \\mathbf { 0 } ) = 0 \\end{align*}"}
{"id": "3939.png", "formula": "\\begin{align*} \\bar \\omega _ T = \\frac { 1 } { T } \\int _ 0 ^ T \\omega \\circ \\tau _ \\Phi ^ t \\d t . \\end{align*}"}
{"id": "4618.png", "formula": "\\begin{gather*} c ^ R _ X = c ^ L _ X \\end{gather*}"}
{"id": "2994.png", "formula": "\\begin{align*} \\Theta _ 1 ( z ) = \\chi ^ - + S ( z ) \\chi ^ + , \\Theta _ 2 ( z ) = \\chi ^ + + S ( z ) \\chi ^ - , \\Theta ' _ 1 ( z ) = \\chi ^ - + S ^ * ( \\bar z ) \\chi ^ + , \\Theta ' _ 2 ( z ) = \\chi ^ + + S ^ * ( \\bar z ) \\chi ^ - , \\end{align*}"}
{"id": "6971.png", "formula": "\\begin{align*} \\forall x \\in \\R \\colon \\Phi ( x ) : = \\{ 0 , x ^ 2 \\} . \\end{align*}"}
{"id": "8940.png", "formula": "\\begin{align*} \\hat { \\xi } ( s ) : = \\left \\{ \\begin{aligned} & \\xi ( s ) - \\left ( 1 - \\frac { s } { T } \\right ) h , 0 \\leq s \\leq T , \\\\ & \\xi ( s ) , s \\geq T . \\end{aligned} \\right . \\end{align*}"}
{"id": "8711.png", "formula": "\\begin{align*} ~ \\begin{aligned} \\tilde { \\upsilon } _ { i 0 } \\preceq _ i \\cdots \\preceq _ i \\tilde { \\upsilon } _ { i n } = \\upsilon _ { i n } \\upsilon _ { i j } \\preceq _ { i } \\tilde { \\upsilon } _ { i j } \\preceq _ i \\alpha _ { i j } j \\in \\{ 0 , \\ldots , n \\} . \\end{aligned} \\end{align*}"}
{"id": "1679.png", "formula": "\\begin{align*} p ' ( x ) = 2 - \\frac { 3 2 7 5 e } { 2 x - 1 } > 0 \\end{align*}"}
{"id": "7431.png", "formula": "\\begin{align*} \\forall ( t , u ) \\in [ 0 , T ] \\times \\mathbb { R } , [ - ( - \\Delta ) ^ { \\gamma / 2 } G ] ( t , u ) = \\sum _ { j = 0 } ^ { k } t ^ { j } [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ j ] ( u ) . \\end{align*}"}
{"id": "5395.png", "formula": "\\begin{align*} M _ \\mu ( n , \\alpha , \\varepsilon , \\delta , T , f ) = \\inf \\{ \\sum _ { i } e ^ { - \\alpha n _ i + f _ { n _ i } ( x _ i ) } : \\mu \\big ( \\cup _ { i } B _ { n _ i } ( x _ i , \\varepsilon ) \\big ) \\geq 1 - \\delta \\} , \\end{align*}"}
{"id": "4320.png", "formula": "\\begin{align*} I ( \\tau ) = e ^ { \\left ( 1 - \\frac { 2 } { \\alpha } \\right ) \\tau } . \\end{align*}"}
{"id": "78.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 1 / 2 } ( 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 ) . \\end{align*}"}
{"id": "3100.png", "formula": "\\begin{align*} x ^ \\prime : y ^ \\prime : z ^ \\prime = \\alpha x : \\beta y : \\gamma z \\ , . \\end{align*}"}
{"id": "5164.png", "formula": "\\begin{align*} P _ 2 ( t ) = \\vartheta _ 2 ( 0 , i t ) \\vartheta _ 2 ( 0 , i / t ) P _ 4 ( t ) = \\vartheta _ 4 ( 0 , i t ) \\vartheta _ 4 ( 0 , i / t ) , \\end{align*}"}
{"id": "1207.png", "formula": "\\begin{align*} m ( E _ { m } ^ { [ \\alpha , \\beta ] , \\varepsilon } ) = 1 . \\end{align*}"}
{"id": "701.png", "formula": "\\begin{align*} g _ \\epsilon ( x ) : = ( g * \\psi _ \\epsilon ) ( x ) , \\psi _ \\epsilon ( y ) = \\exp \\left [ - \\frac { \\norm { y } ^ 2 } { 2 \\epsilon } - \\frac { 1 } { 2 } \\log ( 2 \\pi \\epsilon ) \\right ] . \\end{align*}"}
{"id": "2826.png", "formula": "\\begin{align*} \\ddot { y } ( t ) & = 1 6 ( s _ c ( p - 1 ) + 1 ) E [ Q ] - 8 s _ c ( p - 1 ) \\| \\nabla u \\| _ 2 ^ 2 \\\\ & = 8 s _ c ( p - 1 ) \\left ( \\| \\nabla Q \\| _ 2 ^ 2 - \\| \\nabla u \\| _ 2 ^ 2 \\right ) = - 8 s _ c ( p - 1 ) \\delta ( t ) < 0 , \\end{align*}"}
{"id": "5886.png", "formula": "\\begin{align*} h ( a , b ; z ) = | ( b - z ) \\vee 0 | ^ \\frac { 1 } { \\rho } - | ( a - z ) \\vee 0 | ^ \\frac { 1 } { \\rho } \\end{align*}"}
{"id": "756.png", "formula": "\\begin{align*} \\beta = ( L t _ 1 - 1 ) ^ 2 + \\mu _ p t _ 1 ( 2 - L t _ 1 ) . \\end{align*}"}
{"id": "4552.png", "formula": "\\begin{align*} h _ 1 = \\min \\{ h _ 1 , \\dots , h _ 4 \\} \\le \\frac { 3 h - 3 } { 3 2 } , \\max \\{ h _ 1 , \\dots , h _ 4 \\} \\le \\frac { 3 h - 2 7 } { 8 } . \\end{align*}"}
{"id": "6730.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\geq i \\geq 0 \\\\ m \\geq j \\geq 0 } } f _ { i j } X ^ i Y ^ j = 0 \\end{align*}"}
{"id": "7831.png", "formula": "\\begin{align*} z = \\sum _ { i = 0 } ^ { 3 n - 6 j - 4 } \\gamma _ { i } e _ { i } , \\quad ~ \\gamma _ { 3 n - 6 j - 4 } , \\gamma _ { 3 n - 6 j - 5 } \\neq 0 . \\end{align*}"}
{"id": "6156.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m { n + k \\choose k } B _ a ( \\alpha + k , \\beta + n + 1 ) + \\sum _ { k = 0 } ^ n { m + k \\choose k } B _ a ( \\alpha + m + 1 , \\beta + k ) = B _ a ( \\alpha , \\beta ) , \\end{align*}"}
{"id": "6268.png", "formula": "\\begin{align*} R _ 1 ^ * \\cup R _ 2 ^ * = \\left \\{ a , b \\right \\} \\cup \\left \\{ c _ 1 ^ i \\colon \\phi _ { x y } ( c _ 1 ^ i ) \\in U \\cup V \\right \\} \\cup \\left \\{ c _ { 2 k - 1 } ^ i \\colon \\phi _ { x y } ( c _ { 2 k - 1 } ^ i ) \\in U \\cup V \\right \\} . \\end{align*}"}
{"id": "542.png", "formula": "\\begin{align*} P ( s ) = s \\int _ { \\frac { \\bar \\rho } { 2 } } ^ s \\frac { p ( z ) } { z ^ 2 } \\end{align*}"}
{"id": "7700.png", "formula": "\\begin{align*} H = \\mathcal F ( \\{ p \\} ) \\times T \\ , , T = \\{ a \\in H \\colon p \\nmid a \\} \\ , . \\end{align*}"}
{"id": "989.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s w ( x ) + c ( x ) w ( x ) & = ( - \\Delta ) ^ s v ( x ) + c ( x ) v ( x ) + ( - \\Delta ) ^ s ( w - v ) ( x ) \\\\ & \\geqslant - C \\big ( 1 + u ( a ) + ( \\theta d ) ^ { - n - 2 s - 2 } \\big ) \\\\ & \\geqslant - C \\big ( ( \\theta d ) ^ { - n - 2 s - 2 } + u ( a ) \\big ) \\end{align*}"}
{"id": "2814.png", "formula": "\\begin{align*} | \\dot { \\alpha } | + | \\dot { X } | + | \\dot { \\theta } | = O ( \\delta ) , \\ ; \\forall t \\in D _ { \\delta _ 0 } . \\end{align*}"}
{"id": "1903.png", "formula": "\\begin{align*} R ( x ) & = \\{ \\theta \\in \\Theta : f ( \\theta | x ) \\geq k \\} . \\end{align*}"}
{"id": "7704.png", "formula": "\\begin{align*} A + D + E = C + B + E \\ , . \\end{align*}"}
{"id": "1474.png", "formula": "\\begin{align*} T = \\left [ \\begin{array} { c c c c c } 1 _ { m _ 1 } & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 _ r & 0 & 0 \\\\ 0 & 0 & 0 & 1 _ { m _ 1 } & 0 \\\\ 0 & - 1 _ { m _ 2 } & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 _ { m _ 2 } \\end{array} \\right ] . \\end{align*}"}
{"id": "4526.png", "formula": "\\begin{align*} w ' = ( ( \\ldots ( ( ( a _ i ) c _ 1 ) c _ 2 ) \\ldots ) c _ k ) ( ( \\ldots ( ( ( a _ j ) d _ 1 ) d _ 2 ) \\ldots ) d _ r ) \\end{align*}"}
{"id": "1746.png", "formula": "\\begin{align*} D = D _ { G , K } \\hat { \\otimes } 1 + 1 \\hat { \\otimes } D _ { Z } , \\end{align*}"}
{"id": "3461.png", "formula": "\\begin{align*} \\begin{aligned} \\omega ( B ( x _ 0 , 2 ^ { j _ k } 2 ^ { i _ 0 } \\sigma ) ) & \\geqslant 2 ^ k \\omega ( B ( x _ 0 , 2 ^ { i _ 0 } \\sigma ) ) \\\\ & \\geqslant { C _ \\omega ^ { - 1 } } 2 ^ { - \\bf N } 2 ^ k \\omega ( B ( x _ 0 , 2 ^ { i _ 0 + 1 } \\sigma ) ) \\geqslant { C _ \\omega ^ { - 1 } } 2 ^ { - \\bf N } 2 ^ k \\sigma ^ { \\bf N } . \\end{aligned} \\end{align*}"}
{"id": "5769.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| \\varphi _ { i , \\delta } ^ j + \\hat { f } _ i ^ { j } \\right \\| _ { C ^ { 2 , \\alpha } \\left ( B _ { R } ( \\hat { x } , \\hat { g } _ { i , \\delta } ^ * ) \\right ) } & \\leq C _ { 5 } ( n , r , v , \\rho , \\alpha , Q ) \\left \\| \\varphi _ { i , \\delta } ^ j + \\hat { f } _ i ^ { j } \\right \\| _ { C ^ { 0 } \\left ( B _ { R } ( \\hat { x } , \\hat { g } _ { i , \\delta } ^ * ) \\right ) } \\\\ & \\leq \\Psi ( i ^ { - 1 } , \\delta \\ , | \\ , n , r , v , \\rho , \\alpha , Q ) , \\end{aligned} \\end{align*}"}
{"id": "4013.png", "formula": "\\begin{align*} m _ 3 & = - \\frac { 1 } { 3 } \\left [ \\mu + \\frac { - 1 - i \\sqrt { 3 } } { 2 } \\left ( \\mu + \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ) + \\frac { - 1 + i \\sqrt { 3 } } { 2 } \\left ( \\mu - \\sqrt { 3 } \\mu ^ { 1 / 2 } - \\frac { 3 } { 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ) \\right ] \\\\ & = - \\frac { 1 } { 3 } \\left [ \\frac { 3 } { 2 } - 3 i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) \\right ] \\\\ & = - \\frac { 1 } { 2 } + i \\mu ^ { 1 / 2 } + O ( \\mu ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "141.png", "formula": "\\begin{align*} \\varphi _ \\alpha ( \\xi ) = \\exp \\left ( - \\int _ { \\mathbb { S } ^ { d - 1 } } | \\langle y ; \\xi \\rangle | ^ \\alpha \\lambda _ 1 ( d y ) \\right ) . \\end{align*}"}
{"id": "2854.png", "formula": "\\begin{align*} \\dot { y } _ R ( t ) & = 2 R \\Im \\int _ { \\mathbb { R } ^ N } \\nabla \\varphi \\left ( \\frac { x + X ( t ) } { R } \\right ) \\cdot \\nabla Q \\bar { v } d x + 2 R \\Im \\int _ { \\mathbb { R } ^ N } \\nabla \\varphi \\left ( \\frac { x + X ( t ) } { R } \\right ) \\cdot \\nabla v Q d x \\\\ & + 2 R \\Im \\int _ { \\mathbb { R } ^ N } \\nabla \\varphi \\left ( \\frac { x + X ( t ) } { R } \\right ) \\cdot \\nabla v \\bar { v } d x , v = \\alpha Q + h , \\end{align*}"}
{"id": "6366.png", "formula": "\\begin{align*} v _ 0 = \\frac { 1 } { 2 k + 1 } ( \\rho _ 1 + k \\rho _ 4 ) \\end{align*}"}
{"id": "9085.png", "formula": "\\begin{align*} L ( r _ { \\rm t h } ) = \\int _ 0 ^ { \\infty } \\dot { r } p _ { \\dot { R } R } ( \\dot { r } , r _ { \\rm t h } ) d \\dot { r } , \\end{align*}"}
{"id": "7168.png", "formula": "\\begin{align*} \\Psi ( \\nu ) = \\left ( \\int _ \\Omega | w | ^ { 4 \\chi ^ \\nu } \\phi ^ { 4 \\alpha \\chi ^ \\nu } d \\eta \\right ) ^ { 1 / 4 \\chi ^ \\nu } \\end{align*}"}
{"id": "1900.png", "formula": "\\begin{align*} C _ { \\Sigma , \\tau } = \\{ w \\in V \\mid \\langle w , v \\rangle \\geq 0 v \\tau \\} . \\end{align*}"}
{"id": "445.png", "formula": "\\begin{align*} \\langle B ^ { i j } \\partial _ { i } \\partial _ { j } ( \\partial _ { x } ^ { \\alpha } u ) , \\partial _ { x } ^ { \\alpha } u _ { t } \\rangle = - \\langle B ^ { i j } \\partial _ { j } \\partial _ { x } ^ { \\alpha } u , \\partial _ { i } \\partial _ { x } ^ { \\alpha } u _ { t } \\rangle - \\langle \\left ( \\partial _ { i } B ^ { i j } \\right ) \\partial _ { j } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u _ { t } \\rangle , \\end{align*}"}
{"id": "7041.png", "formula": "\\begin{align*} \\Big ( 1 - \\sum _ { k = 1 } ^ { L } \\phi _ k q _ k \\Big ) e _ j & = \\Big ( 1 - \\sum _ { k = 1 } ^ { L } q _ k ( a _ 1 \\widetilde { \\phi _ k } + p _ k ) \\Big ) e _ j \\\\ & = \\Big ( 1 - \\sum _ { k = 1 } ^ { L } q _ k p _ k - a _ 1 \\sum _ { k = 1 } ^ { L } q _ k \\widetilde { \\phi _ k } \\Big ) e _ j \\\\ & = a _ 1 \\Big ( q - \\sum _ { k = 1 } ^ { L } q _ k \\widetilde { \\phi _ k } \\Big ) e _ j . \\end{align*}"}
{"id": "3291.png", "formula": "\\begin{align*} C _ 1 \\min \\Big \\{ 1 , \\frac { 1 } { t } \\Big \\} \\leq \\Im \\omega _ 1 ( i t ) - t = s ( | \\lambda | , t ) - t \\leq C _ 2 \\min \\Big \\{ 1 , \\frac { 1 } { t } \\Big \\} . \\end{align*}"}
{"id": "2619.png", "formula": "\\begin{align*} Z f ( x , \\omega ) = \\sum _ { k \\in \\Z ^ d } f ( x + k ) e ^ { - 2 \\pi i k \\cdot \\omega } . \\end{align*}"}
{"id": "4898.png", "formula": "\\begin{align*} \\lim _ { y \\to + \\infty , y \\in \\R } \\frac { \\log | P ( i y ) | } { \\log y } = - \\infty , \\lim _ { y \\to + \\infty , y \\in \\R } y L ( i y ) = 0 . \\end{align*}"}
{"id": "8019.png", "formula": "\\begin{align*} \\phi ( u , v ) = \\int _ 0 ^ u \\psi _ 0 ( - u ' ) \\ , \\mathrm { d } u ' + \\frac { 1 } { 2 \\pi } \\left ( \\int _ 0 ^ { 2 \\pi } \\psi ( x ) \\ , \\mathrm { d } x \\right ) ( u + v ) , \\end{align*}"}
{"id": "8774.png", "formula": "\\begin{align*} P _ i = \\left \\{ u _ i \\in \\R ^ { n + 1 } \\ , \\middle | \\ , \\begin{aligned} a _ { i 0 } \\leq u _ { i j } \\leq \\min \\{ a _ { i j } , u _ { i n } \\} , \\ u _ { i 0 } = a _ { i 0 } , \\ ; a _ { i 0 } \\leq u _ { i n } \\leq a _ { i n } \\end{aligned} \\right \\} . \\end{align*}"}
{"id": "4438.png", "formula": "\\begin{align*} H ' ( t ) \\leq & - \\epsilon E ( t ) - \\frac { 1 } { 4 } \\epsilon | | u _ { x x } | | ^ 2 _ 2 + a _ 2 \\epsilon \\gamma \\delta | | u | | ^ 2 _ 2 \\\\ & - [ a _ 1 - ( \\frac { 3 } { 2 } + a _ 2 ^ 2 B ) \\epsilon ] | | u _ t | | _ 2 ^ 2 - a _ 1 [ 1 - \\frac { a _ 2 } { a _ 1 } \\epsilon c ( \\delta ) ] | | u _ t | | _ m ^ m , \\end{align*}"}
{"id": "362.png", "formula": "\\begin{align*} d \\rho _ i ( t ) + \\sum _ { j \\in N ( i ) } m _ { i j } d t + \\sum _ { j \\in N ( i ) } m _ { i j } d W ^ { \\delta } ( t ) = 0 . \\end{align*}"}
{"id": "8559.png", "formula": "\\begin{align*} T ( k ) W ( \\psi _ + ( \\cdot , k ) , \\psi _ - ( \\cdot , k ) ) = - 2 i k . \\end{align*}"}
{"id": "1360.png", "formula": "\\begin{align*} & \\bar { d } C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { u ^ { \\bar { d } - 1 } \\nabla u } _ { L ^ 1 ( W _ R ) } \\leq \\bar { d } C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { u ^ { \\bar { d } - 1 } } _ { L ^ r ( W _ R ) } \\norm { \\nabla u } _ { L ^ { q ^ \\# } ( W _ R ) } \\\\ & = \\bar { d } C _ S ^ { - 1 } | W _ R | ^ \\frac { 1 - \\zeta } { d } \\norm { \\nabla u } _ { L ^ { q ^ \\# } ( W _ R ) } \\left ( \\int _ { W _ R } | u | ^ \\frac { d q ^ \\# } { d - q ^ \\# \\zeta } d x \\right ) ^ \\frac { q ^ \\# - 1 } { q ^ \\# } . \\end{align*}"}
{"id": "1117.png", "formula": "\\begin{align*} m ^ { ( 1 ) } _ { + } ( x , t , k ) = m ^ { ( 1 ) } _ { - } ( x , t , k ) J ^ { ( 1 ) } ( x , t , k ) , k \\in \\mathbb { R } , \\end{align*}"}
{"id": "2451.png", "formula": "\\begin{align*} S ^ T S + I = S ^ T ( S + S ^ { - T } ) = S ^ T ( S + J S J ^ { - 1 } ) . \\end{align*}"}
{"id": "797.png", "formula": "\\begin{align*} \\langle \\nabla u ( x ) , \\nabla v ( x ) \\rangle _ x = \\nabla u ( x ) \\cdot \\nabla v ( x ) \\end{align*}"}
{"id": "4205.png", "formula": "\\begin{align*} 0 & = \\int _ M ( - \\Delta _ g r + V r + V \\dot u ) r \\ , d V _ g = \\int _ M | \\nabla r | ^ 2 \\ , d V _ g + \\int _ M V r ^ 2 \\ , d V _ g + \\int _ M V \\dot u r \\ , d V _ g . \\end{align*}"}
{"id": "8358.png", "formula": "\\begin{align*} \\tau _ n = ( \\tau _ 0 - 1 ) \\sum _ { i = 1 } ^ n k _ { i - 1 } + \\tau _ 0 . \\end{align*}"}
{"id": "2901.png", "formula": "\\begin{align*} - \\Delta Q + Q = \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * | Q | ^ p \\right ) | Q | ^ { p - 2 } Q , \\mathbb { R } ^ N . \\end{align*}"}
{"id": "3960.png", "formula": "\\begin{align*} c _ k = O ( k ^ { - 1 } ) , \\ \\ d _ k = O ( k ^ { - 1 } ) , \\alpha _ { 1 , k } = O ( \\mod { k } ^ { - 1 } ) , \\ \\ \\alpha _ { 2 , k } = O ( | k | ^ { - 1 } ) , \\ \\ k . \\end{align*}"}
{"id": "8063.png", "formula": "\\begin{align*} \\Lambda ^ { \\mu _ i - 1 } \\Psi ^ i \\left ( { m _ { \\Lambda } } _ { * } \\left ( f \\right ) \\right ) = \\mathfrak { P } _ { \\ell } m _ { \\Lambda } \\Psi ^ i ( f ) , \\end{align*}"}
{"id": "1411.png", "formula": "\\begin{align*} \\hat q _ e = \\hat q _ t ^ * \\hat q = q _ e + { \\epsilon \\over 2 } q _ e p _ e ^ b , \\end{align*}"}
{"id": "2754.png", "formula": "\\begin{align*} \\ker L _ + = \\Big \\{ \\partial _ { x _ 1 } Q , \\partial _ { x _ 2 } Q , . . . , \\partial _ { x _ N } Q \\Big \\} , \\end{align*}"}
{"id": "2601.png", "formula": "\\begin{align*} \\norm { f } _ { M ^ { p , q } } = \\norm { V _ { g _ 0 } f } _ { L ^ { p , q } } = \\norm { V _ { g _ 0 } ( V _ g ^ * V _ g f ) } _ { L ^ { p , q } } \\leq C \\norm { V _ { g _ 0 } g } _ { L ^ 1 } \\norm { V _ g f } _ { L ^ { p , q } } . \\end{align*}"}
{"id": "5393.png", "formula": "\\begin{align*} M ^ P ( n , \\alpha , \\varepsilon , Z , T , f ) = \\sup \\{ \\sum _ { i } e ^ { - s n _ i + f _ { n _ i } ( x _ i ) } \\} , \\end{align*}"}
{"id": "7837.png", "formula": "\\begin{align*} x _ { 1 } & = T ^ { * 3 ( j - l - 2 ) } x - \\sum _ { i = 0 } ^ { 3 l + 3 } \\alpha _ { i } e _ { i } \\in M _ { 3 l + 6 } \\setminus M _ { 3 l + 3 } , \\\\ y _ { 1 } & = T ^ { * 3 ( j - r - l - 2 ) } y - \\sum _ { i = 0 } ^ { 3 l + 3 } \\beta _ { i } e _ { i } \\in M _ { 3 l + 6 } \\setminus M _ { 3 l + 3 } , \\\\ z _ { 1 } & = T ^ { * 3 ( n - 2 j + r - l - 3 ) } z - \\sum _ { i = 0 } ^ { 3 l + 3 } \\gamma _ { i } e _ { i } \\in M _ { 3 l + 6 } \\setminus M _ { 3 l + 3 } . \\end{align*}"}
{"id": "3527.png", "formula": "\\begin{align*} \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & = \\begin{cases} O \\left ( \\int _ 2 ^ T t _ 3 ^ { - 2 \\sigma _ 1 - 2 \\sigma _ 3 } d t _ 3 \\right ) & ( \\sigma _ 2 > 2 ) \\\\ O \\left ( \\int _ 2 ^ T t _ 3 ^ { - 2 \\sigma _ 1 - 2 \\sigma _ 3 } ( \\log T ) ^ 2 d t _ 3 \\right ) & ( \\sigma _ 2 = 2 ) \\\\ O \\left ( \\int _ 2 ^ T t _ 3 ^ { 3 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } d t _ 3 \\right ) & ( \\sigma _ 2 < 2 ) \\\\ \\end{cases} \\\\ & = O ( 1 ) , \\end{align*}"}
{"id": "582.png", "formula": "\\begin{align*} \\begin{cases} \\ \\\\ [ 8 p t ] \\ \\end{cases} . \\end{align*}"}
{"id": "6221.png", "formula": "\\begin{align*} A = \\tilde { A } + \\eta \\| \\tilde { A } \\| _ F \\frac { G } { \\| G \\| _ F } , \\ \\ b = \\tilde { b } + \\eta \\| \\tilde { b } \\| _ 2 \\frac { \\zeta } { \\| \\zeta \\| _ 2 } , \\end{align*}"}
{"id": "6383.png", "formula": "\\begin{align*} \\sigma _ T ( { \\underline T } , \\mathcal X ) = \\{ \\lambda \\in \\mathbb { C } ^ n : K ( { \\underline T } - \\lambda , \\mathcal X ) \\} . \\end{align*}"}
{"id": "1755.png", "formula": "\\begin{align*} K _ i ( \\mathcal { C } ( G ) ) \\cong K _ i ( C ^ * _ r ( G ) ) , \\ i = 0 , 1 . \\end{align*}"}
{"id": "3867.png", "formula": "\\begin{align*} E = \\O _ C ( - e _ 1 \\cdot \\infty _ 1 ) \\oplus \\cdots \\oplus \\O _ C ( - e _ m \\cdot \\infty _ m ) \\oplus \\O _ C ^ { \\oplus r - m } . \\end{align*}"}
{"id": "5182.png", "formula": "\\begin{align*} \\hat { p } _ \\ell = \\frac { ( p _ \\ell / \\mu _ \\ell ) } { \\sum _ { i = 1 } ^ { N } ( p _ i / \\mu _ i ) } . \\end{align*}"}
{"id": "5048.png", "formula": "\\begin{align*} { \\bf H } = { \\bf H } ^ 0 + \\mu e ^ { \\gamma c } V \\end{align*}"}
{"id": "2454.png", "formula": "\\begin{align*} \\det \\left ( \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} I & - i I \\\\ I & i I \\end{pmatrix} \\right ) = \\left ( \\tfrac { 1 } { \\sqrt { 2 } } \\right ) ^ { 2 d } \\det ( 2 i I ) = 2 ^ { - d } ( 2 i ) ^ { d } = i ^ d . \\end{align*}"}
{"id": "7499.png", "formula": "\\begin{align*} \\phi _ { 0 } ( \\mathbf { x } ) = \\frac { \\phi ^ { \\mathrm { T F } } ( \\mathbf { x } ) } { \\| \\phi ^ { \\mathrm { T F } } \\| } \\phi ^ { \\mathrm { T F } } ( \\mathbf { x } ) = \\begin{cases} \\sqrt { \\left ( \\mu ^ { \\mathrm { T F } } - V ( \\mathbf { x } ) \\right ) / \\beta } , & \\mbox { i f } V ( \\mathbf { x } ) \\leq \\mu ^ { \\mathrm { T F } } , \\\\ 0 , & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
{"id": "5368.png", "formula": "\\begin{align*} q = g - \\frac { \\left \\langle \\tilde a _ i , g \\right \\rangle - b _ i } { \\left \\langle \\tilde a _ i , c \\right \\rangle } c . \\end{align*}"}
{"id": "9471.png", "formula": "\\begin{align*} \\phi ' : T ' = K [ t _ 1 , \\ldots , t _ { m + 1 } ] \\rightarrow K [ H ' ] \\end{align*}"}
{"id": "2265.png", "formula": "\\begin{align*} z = \\Re ( z ) + i \\ \\Im ( z ) . \\end{align*}"}
{"id": "8025.png", "formula": "\\begin{align*} \\left \\{ F , G \\right \\} _ { \\ell } ^ { \\Sigma } [ \\psi ] = \\left \\langle E _ { \\Sigma } , F ^ { ( 1 ) } [ \\psi ] \\otimes G ^ { ( 1 ) } [ \\psi ] \\right \\rangle . \\end{align*}"}
{"id": "8057.png", "formula": "\\begin{align*} \\left \\langle E ^ { i j } _ 0 , { t _ { c } } _ { * } f \\otimes { t _ { c } } _ { * } g \\right \\rangle = \\left \\langle E ^ { i j } _ 0 , f \\otimes g \\right \\rangle \\end{align*}"}
{"id": "4391.png", "formula": "\\begin{align*} H _ 1 '' + \\frac { d + 1 } { \\xi } H _ 1 ' - 3 ( d - 2 ) ( 2 Q _ \\sigma + \\xi ^ 2 Q _ \\sigma ^ 2 ) H _ 1 = T ( \\xi ) , \\end{align*}"}
{"id": "3412.png", "formula": "\\begin{align*} T ( a ) ( x ) & = \\int _ Q K ( x , y ) a ( y ) d \\omega ( y ) = \\int _ Q [ K ( x , y ) - K ( x , x _ Q ) ] a ( y ) d \\omega ( y ) \\\\ & \\leqslant C \\Big ( \\frac { \\l ( Q ) } { \\| x - x _ Q \\| } \\Big ) ^ \\varepsilon \\frac { 1 } { \\omega ( x , d ( x , x _ Q ) ) } \\| a \\| _ 1 . \\end{align*}"}
{"id": "8229.png", "formula": "\\begin{align*} N _ h & = x _ h \\sqrt { \\frac { z B } { F _ { k , 1 } ( B ) } } \\prod _ { i = 2 } ^ { h } F _ { k , i } ( B ) ^ { - 1 } \\prod _ { i = 1 } ^ { h - 1 } G _ { k , i } ( B ) , \\\\ N _ { k + 1 - h } & = x _ { k + 1 - h } z \\prod _ { i = 1 } ^ { h } F _ { k , i } ( B ) \\prod _ { i = 1 } ^ h G _ { k , i } ( B ) ^ { - 1 } . \\end{align*}"}
{"id": "1291.png", "formula": "\\begin{align*} \\Lambda _ { ( \\infty , m ) } ( M , \\Gamma ) : = \\bigcup _ { n = 0 } ^ { \\infty } \\Lambda _ { ( n , m ) } ( M , \\Gamma ) \\end{align*}"}
{"id": "6677.png", "formula": "\\begin{align*} ( a ) _ n : = \\begin{cases} D _ { n + a - 1 } ^ { q ^ { - ( a - 1 ) } } & , \\\\ 1 / L _ { - a - n } ^ { q ^ n } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "3722.png", "formula": "\\begin{align*} I _ 1 = \\frac 2 3 \\Re \\left ( i \\sum _ { \\substack { m + n + k = 0 , \\\\ | m | , | n | , | k | \\leq N } } ( m | n | | k | ^ 8 + m | k | | n | ^ 8 + k | n | | m | ^ 8 ) \\xi ^ N ( m ) \\xi ^ N ( n ) \\xi ^ N ( k ) \\right ) . \\end{align*}"}
{"id": "474.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\alpha _ { 1 } \\alpha _ { 2 } \\dots } ^ { i _ { 1 } i _ { 2 } \\dots } : = \\mathcal { I } \\backslash \\{ i _ { 1 } , i _ { 2 } , \\dots \\} \\cup \\{ \\alpha _ { 1 } , \\alpha _ { 2 } , \\dots \\} , i _ { 1 } , i _ { 2 } , \\dots \\in \\mathcal { I } , \\ , \\alpha _ { 1 } , \\alpha _ { 2 } , \\dots \\in \\mathcal { I } ^ { \\mathtt { C } } \\end{align*}"}
{"id": "6010.png", "formula": "\\begin{align*} \\begin{aligned} \\mathfrak { D } _ { { \\rm l e f t } , a } u ( x ) & = D _ { { \\rm l e f t } } u ( x ) + a ( x ) u ( x ) \\\\ \\mathfrak { D } _ { { \\rm r i g h t } , a } u ( x ) & = D _ { { \\rm r i g h t } } u ( x ) + a ( x ) u ( x ) , \\end{aligned} \\end{align*}"}
{"id": "1898.png", "formula": "\\begin{align*} \\beta _ { v _ i } | _ \\Sigma & = \\beta _ { v _ i ' } | _ { \\Sigma ' } \\circ \\iota & & \\\\ \\mu ^ { c } _ i | _ \\Sigma & = \\mu ^ { c ' } _ i | _ { \\Sigma ' } \\circ \\iota & & i \\end{align*}"}
{"id": "8354.png", "formula": "\\begin{align*} \\Lambda ( \\rho ) = \\int _ 0 ^ 1 \\exp ( - \\rho y ^ { - 1 } ) d y \\sim \\exp \\left ( - \\rho \\right ) / \\rho , \\textrm { a s } \\rho \\to \\infty . \\end{align*}"}
{"id": "93.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ 2 ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot | M _ 2 ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( 2 ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 ^ { 5 / 2 } \\cdot 3 \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "2429.png", "formula": "\\begin{align*} S _ { g , \\Gamma } f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , \\pi ( \\gamma ) g \\rangle \\ \\pi ( \\gamma ) g . \\end{align*}"}
{"id": "9121.png", "formula": "\\begin{align*} x _ { n + 1 } = J ^ S _ { \\mu _ n } ( x _ n + \\mu _ n T _ { \\lambda _ n } x _ n ) \\end{align*}"}
{"id": "6081.png", "formula": "\\begin{align*} h ^ { j } _ { s } = \\eta ^ t { \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} } _ { s } + \\eta ^ t _ s \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} , \\ \\ \\ \\ \\ \\ \\ \\ h ^ { j } _ { t } = \\eta ^ t { \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} } _ { t } + \\eta ^ t _ t \\begin{pmatrix} \\varphi ^ { j } \\\\ \\psi ^ { j } \\end{pmatrix} , \\end{align*}"}
{"id": "8403.png", "formula": "\\begin{align*} \\partial _ t \\ , c _ A + \\mathbf { \\tilde { v } } _ A \\cdot \\nabla c _ A - \\Delta c _ A + \\frac { 1 } { \\varepsilon ^ 2 } f ' ( c _ A ) + \\mathbf { w } | _ \\Gamma \\ , \\nabla c _ { A , 0 } = \\mathcal { C } + \\mathbf { w } | _ \\Gamma \\ , \\mathbf { Q } , \\end{align*}"}
{"id": "4474.png", "formula": "\\begin{align*} \\log f ( t ) = \\int _ { 0 } ^ { t } \\frac { f ^ { \\prime } ( t ) } { f ( t ) } d t = \\int _ { 0 } ^ { t } | z | ^ { 2 } e _ { \\lambda } ^ { 1 - \\lambda } ( t ) d t = \\big ( e _ { \\lambda } ( t ) - 1 \\big ) | z | ^ { 2 } . \\end{align*}"}
{"id": "8863.png", "formula": "\\begin{align*} \\begin{aligned} \\pi : X \\times \\R _ { \\ge 0 } & \\to \\R _ { \\ge 0 } \\\\ ( x , i ) & \\mapsto d ( x , x _ 0 ) + i \\end{aligned} & \\begin{aligned} \\iota : \\R _ { \\ge 0 } & \\to X \\times \\R _ { \\ge 0 } \\\\ i & \\mapsto ( x _ 0 , i ) \\end{aligned} \\end{align*}"}
{"id": "6741.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ t \\hat { V } - \\mathcal L \\hat { V } - g \\geq 0 , \\\\ & \\hat { V } - V ^ * \\geq 0 , \\\\ & ( \\partial _ t \\hat { V } - \\mathcal L \\hat { V } - g ) \\cdot ( \\hat { V } - V ^ * ) = 0 , \\end{cases} \\end{align*}"}
{"id": "634.png", "formula": "\\begin{align*} C ( i , j ) \\ = \\ \\bigg [ \\frac { i } { j + 1 } + \\frac { 1 } { 2 } \\bigg ] , \\end{align*}"}
{"id": "6261.png", "formula": "\\begin{align*} | A \\cup B | & \\leq n - t ^ s n ^ { \\frac { 1 } { s + 1 } } = n - \\alpha ^ s n = \\left ( 1 - \\alpha ^ s \\right ) n . \\end{align*}"}
{"id": "7685.png", "formula": "\\begin{align*} \\alpha = \\inf _ { a \\ge 0 } \\{ a : \\mathcal { L } ( \\{ u ^ 2 \\ge a \\} ) < \\mathcal { L } ( B _ 1 ( \\mathbf { 0 } ) ) \\} \\ ; . \\end{align*}"}
{"id": "7758.png", "formula": "\\begin{gather*} \\left | \\int _ { \\S } \\phi ( t , x ) - \\phi ( 0 , x ) \\ , d x \\right | = \\left | \\int _ { \\S } \\int _ 0 ^ t \\phi _ t ( s , x ) d s d x \\right | \\lesssim \\int _ 0 ^ t \\| \\phi _ t ( s , x ) \\| _ { L ^ 2 _ x } d s , \\\\ \\left | \\int _ { \\S } \\phi ( 0 , x ) - p \\ , d x \\right | \\lesssim \\| \\phi ( 0 , x ) - p \\| _ { L ^ 2 _ x } , \\end{gather*}"}
{"id": "3353.png", "formula": "\\begin{align*} [ u , v , w ] _ T : = D ( T u , T v ) w + \\theta ( T v , T w ) u - \\theta ( T u , T w ) v . \\end{align*}"}
{"id": "6710.png", "formula": "\\begin{align*} ( g _ 1 , g _ 2 ) \\begin{pmatrix} P _ { { \\bf b } , d } \\\\ P _ { { \\bf b } , d } ~ _ { s + 1 } \\mathcal { F } _ s ( a _ 1 , \\ldots , a _ { s + 1 } ; b _ 1 , \\ldots , b _ s ) ( \\alpha ) ^ { q ^ d } \\end{pmatrix} = 0 \\end{align*}"}
{"id": "7705.png", "formula": "\\begin{align*} G = \\{ [ ( A , B ) ] \\colon A , B \\in H \\} \\end{align*}"}
{"id": "3811.png", "formula": "\\begin{align*} \\tau ^ i ( a , b ) = \\begin{cases} ( a , b + \\frac { i } { 4 k } ) & \\\\ ( - a , a + b + \\frac { i } { 4 k } ) & . \\end{cases} \\end{align*}"}
{"id": "9334.png", "formula": "\\begin{align*} \\eta = 1 + \\frac { \\mathbf { u } _ 1 ^ T ( ( \\delta I + ( \\widehat { \\Theta } ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } - ( \\delta I + ( { \\Theta } ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } ) \\mathbf { u } _ 1 } { \\mathbf { u } _ 1 ^ T ( ( H + \\rho I + \\frac { 1 } { \\delta } A ^ T A + ( \\delta I + ( { \\Theta } ^ { - 1 } + \\rho I ) ^ { - 1 } ) ^ { - 1 } ) \\mathbf { u } _ 1 } . \\end{align*}"}
{"id": "2063.png", "formula": "\\begin{align*} f ( o ) = \\int _ D f \\overline { { g } _ { ( 0 , \\cdots , 0 ) } } e ^ { - \\varphi } . \\end{align*}"}
{"id": "7938.png", "formula": "\\begin{align*} | E _ { n , k } | \\leqslant 5 2 n ^ { 1 / 4 } ( 1 + ( k / n ) ^ { 1 / 2 } n ^ { 1 / 8 } ) ( 1 + L _ { n } ^ { 1 / 2 } n ^ { - 1 / 8 } ) , ~ ~ L _ { n } = \\max \\{ 0 , n ^ { 1 / 2 } - S _ n \\} . \\end{align*}"}
{"id": "7094.png", "formula": "\\begin{align*} \\psi ( a , b , c , d ) & = \\psi ( a + b L _ 1 + c L _ 2 + d L _ 3 , 0 , 0 , 0 ) + \\psi ( - b L _ 1 - c L _ 2 - d L _ 3 , b , c , d ) \\\\ & = \\psi ( a + b L _ 1 + c L _ 2 + d L _ 3 , 0 , 0 , 0 ) . \\end{align*}"}
{"id": "9055.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ^ { n + 1 } = \\arg & \\min _ { u \\in V _ { h , \\delta } ^ n } \\bigg \\{ \\mathcal { F } _ h ( u ) \\bigg \\} , u : = ( \\rho , m , \\phi ) . \\end{aligned} \\end{align*}"}
{"id": "9264.png", "formula": "\\begin{align*} \\forall n ^ 0 , x ^ X \\preceq _ X L 1 _ X , y ^ X \\exists z ^ X \\preceq _ X L 1 _ X \\exists w ^ X \\left ( y \\in A x \\rightarrow ( w \\in A z \\land x = _ X z + _ X \\gamma _ n w ) \\right ) . \\end{align*}"}
{"id": "8899.png", "formula": "\\begin{align*} \\check H _ { c t } ^ q ( T ; A ) = \\begin{cases} \\bigoplus _ \\N A & q = 0 \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "465.png", "formula": "\\begin{align*} | \\frac { | \\rho _ u W ^ * ( u ) \\cap W '' ( u ) | } { | \\rho _ u W ^ * ( u ) | } - \\frac { | W '' ( u ) | } { | W ^ * ( u ) ^ - | } | = O ( q ^ { - 2 ^ { - | u | - 1 } } ) \\end{align*}"}
{"id": "3636.png", "formula": "\\begin{align*} I _ 1 = \\int _ 2 ^ { 5 9 9 } \\frac { | \\theta ( t ) - t | } { t \\log ^ 2 t } t , I _ 2 = \\int _ { 5 9 9 } ^ { \\exp ( 5 8 ) } \\frac { | \\theta ( t ) - t | } { t \\log ^ 2 t } t , I _ 3 = \\int _ { \\exp ( 5 8 ) } ^ x \\frac { | \\theta ( t ) - t | } { t \\log ^ 2 t } t . \\end{align*}"}
{"id": "5873.png", "formula": "\\begin{align*} d S _ t & = - \\mu S _ t d t + F ^ 2 ( Y _ t ) d W _ t , \\\\ d \\widetilde W _ t & = d W _ t . \\end{align*}"}
{"id": "3348.png", "formula": "\\begin{align*} & \\{ x , y , z \\} ^ { * } = \\{ z , y , x \\} - \\{ z , x , y \\} , \\\\ & [ x , y , z ] _ C \\ ; = \\{ z , y , x \\} - \\{ z , x , y \\} + \\{ x , y , z \\} - \\{ y , x , z \\} \\\\ & \\quad \\ ; \\ ; \\ ; \\ ; = \\{ x , y , z \\} ^ { * } + \\{ x , y , z \\} - \\{ y , x , z \\} , \\end{align*}"}
{"id": "8390.png", "formula": "\\begin{align*} \\P ( \\exists B , B ' \\in \\mathfrak B _ { r - 1 } : \\min _ { u \\in B , u ' \\in B ' } | u - u ' | _ { \\infty } \\leq 4 r \\mbox { a n d } \\sigma | _ B = \\sigma | _ { B ' } ) \\leq n ^ { - \\epsilon / 4 } \\ , . \\end{align*}"}
{"id": "4177.png", "formula": "\\begin{align*} D ^ * _ R L : = \\{ ( x , v ) \\in T ^ * L \\ | \\ | v | \\leq R \\} \\end{align*}"}
{"id": "5399.png", "formula": "\\begin{align*} \\mu _ i ( \\overline { B } ( x , \\gamma _ i ) ) = e ^ { - m _ i ( x ) s } \\leq \\sum _ { y \\in E _ { i + 1 } ( x ) } e ^ { - m _ { i + 1 } ( y ) s + f _ { m _ { i + 1 } } ( y ) } \\leq ( 1 + 2 ^ { - i - 1 } ) \\mu _ i ( \\overline { B } ( x , \\gamma _ i ) ) , \\end{align*}"}
{"id": "1573.png", "formula": "\\begin{align*} c _ k ( \\mu ) L ( \\mu , \\mathbf { f } , \\chi ) \\mathbf { f } ( g ) = \\langle \\mathbf { T } ( g , h ) , \\mathbf { f } ( h ) \\rangle . \\end{align*}"}
{"id": "3618.png", "formula": "\\begin{align*} T _ { j } = R _ { n + 3 } - R _ { j } + d ( - a _ { j } a _ { j + 1 } ) \\ge R _ { n + 3 } + 2 e > T _ { 0 } \\ , . \\end{align*}"}
{"id": "729.png", "formula": "\\begin{align*} K _ * = 0 , C _ b = 0 , C _ W = \\sigma _ 1 ^ { - 2 } . \\end{align*}"}
{"id": "7239.png", "formula": "\\begin{align*} ( \\tau + k \\cdot ( v - V _ \\ast ) ) \\tilde { h } - \\hat { \\phi } ( k ) \\rho [ \\tilde { h } ] ( \\tau , k ) k \\cdot \\nabla _ v \\mu = \\frac { - e _ 0 \\hat { \\Phi } ( k ) k \\cdot \\nabla _ v \\mu } { i \\tau } , \\end{align*}"}
{"id": "72.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & = 1 . \\end{align*}"}
{"id": "7686.png", "formula": "\\begin{align*} m ' ( \\mathbf { x } ) \\coloneqq \\begin{cases} \\overline { m } & \\mathbf { x } \\in E _ u \\ ; , \\\\ \\underline { m } & \\mathbf { x } \\in \\R ^ N \\setminus E _ u \\ ; , \\end{cases} \\end{align*}"}
{"id": "1642.png", "formula": "\\begin{align*} \\mathcal { L } F ( u ) ( v ) = \\int _ M ( \\mathcal { L } u ) F ' ( u ) v d \\mu + \\int _ { \\{ u > 0 \\} } F '' ( u ) \\Gamma ( u ) v d \\mu , v \\in \\mathcal { D } ( \\mathcal { E } ) \\cap C _ c ( M ) . \\end{align*}"}
{"id": "5521.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X _ r ( t ) & = & \\big ( A X _ r ( t ) + \\alpha ( X _ r ( t ) ) \\big ) d t + \\sigma _ r ( X _ r ( t ) ) d W ( t ) \\medskip \\\\ X _ r ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "4112.png", "formula": "\\begin{align*} \\begin{aligned} & \\bigl | | u + v | ^ { p _ 0 - 2 } - | u | ^ { p _ 0 - 2 } \\bigr | \\cdot | \\mu _ n \\sigma _ n ( u ) | \\cdot | w | \\\\ & \\qquad \\qquad \\le c \\cdot \\bigl ( | v | ^ { p _ 0 } \\cdot ( 1 + ( n ^ { 3 / 4 } | w | ) ^ { p _ 0 } ) + 1 + | w | ^ { p _ 0 / 2 } + | u | ^ { p _ 0 } \\bigr ) . \\end{aligned} \\end{align*}"}
{"id": "6501.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ 1 = - 2 \\alpha , M ^ { ( 2 ) } _ { n + 1 } = \\frac { n } { n + 1 } \\left ( 1 + \\frac { 2 \\alpha } { n } \\right ) M _ n ^ { ( 2 ) } ( n \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "6254.png", "formula": "\\begin{align*} \\mathcal { E } ( q , \\lambda ( q , t ) ) = \\frac { - 1 + e ^ { \\lambda ( q , t ) \\cdot ( 1 - q ) } } { 1 - q e ^ { \\lambda ( q , t ) \\cdot ( 1 - q ) } } = \\frac { e ^ { \\lambda ( q , t ) } - e ^ { \\lambda ( q , t ) \\cdot q } } { e ^ { \\lambda ( q , t ) \\cdot q } - q e ^ { \\lambda ( q , t ) } } ; \\end{align*}"}
{"id": "5285.png", "formula": "\\begin{align*} \\varphi ( a ^ * ) = \\overline { \\varphi ( a ) } , a \\in A . \\end{align*}"}
{"id": "9378.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { p + 1 } S _ { 2 , \\lambda } ( p + 1 , k ) x ^ { k } = \\phi _ { p + 1 , \\lambda } ( x ) = e ^ { - x } \\sum _ { n = 1 } ^ { \\infty } ( n ) _ { p + _ 1 , \\lambda } \\frac { x ^ { n } } { n ! } \\\\ & = e ^ { - x } \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) _ { p + 1 , \\lambda } \\frac { x ^ { n + 1 } } { ( n + 1 ) ! } = e ^ { - x } \\sum _ { n = 0 } ^ { \\infty } ( n + 1 - \\lambda ) _ { p , \\lambda } \\frac { x ^ { n + 1 } } { n ! } . \\end{align*}"}
{"id": "3943.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + ( \\rho u ) _ x = 0 \\ & ( 0 , + \\infty ) \\times ( 0 , L ) , \\\\ \\rho \\big ( u _ t + u u _ x \\big ) + ( p ( \\rho ) ) _ x - \\nu u _ { x x } = 0 \\ & ( 0 , + \\infty ) \\times ( 0 , L ) , \\end{dcases} \\end{align*}"}
{"id": "8012.png", "formula": "\\begin{align*} \\chi ( x ) = ( - \\rho \\circ \\pi _ { \\ell } ^ { \\Sigma } ( x ) , \\rho \\circ \\pi _ r ^ { \\Sigma } ( x ) ) , \\end{align*}"}
{"id": "4410.png", "formula": "\\begin{align*} W ( \\xi ) = e ^ { \\int ^ { \\xi } a ( \\xi ' ) d \\xi ' } = \\xi ^ { - ( d + 1 ) } e ^ { 2 \\beta \\frac { \\xi ^ 2 } { 4 } } \\end{align*}"}
{"id": "1316.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { | \\hat { I } _ { = 2 , < \\epsilon } ( k _ { n } ) | + | \\hat { I } _ { > 2 , < \\epsilon } ( k _ { n } ) | } { k _ { n } } = 1 . \\end{align*}"}
{"id": "4624.png", "formula": "\\begin{gather*} [ H , E ] = 2 E , [ H , F ] = - 2 F , [ E , F ] = \\frac { K - K ^ { - 1 } } { v - v ^ { - 1 } } , \\end{gather*}"}
{"id": "2624.png", "formula": "\\begin{align*} Z ( M _ l T _ k f ) ( x , \\omega ) & = e ^ { 2 \\pi i l \\cdot x } Z f ( x - k , \\omega - l ) \\\\ & = e ^ { 2 \\pi i l \\cdot x } e ^ { - 2 \\pi i k \\cdot \\omega } Z f ( x , \\omega ) . \\end{align*}"}
{"id": "3389.png", "formula": "\\begin{align*} \\omega _ i ( x , y , z ) = \\varphi _ i ( [ x , y ] , z ) - \\rho ( z ) \\varphi _ i ( x , y ) , \\ ; i = 1 , 2 \\end{align*}"}
{"id": "4444.png", "formula": "\\begin{align*} C _ { n , f } = ( - 1 ) ^ \\nu G _ f ^ { ( \\nu ) } ( k _ \\nu ) \\left ( \\dfrac { M } { 2 \\pi } \\right ) ^ { 2 n + \\nu + 1 } \\dfrac { ( 2 n + \\nu ) ! } { k _ \\nu ^ { 2 n + \\nu + 1 } \\sqrt { M } } \\left ( 1 + \\frac { 1 } { G _ f ^ { ( \\nu ) } ( k _ \\nu ) } \\sum _ { k = k _ { \\nu } + 1 } ^ \\infty \\dfrac { G _ f ^ { ( \\nu ) } ( k ) } { k ^ { 2 n + \\nu + 1 } } \\right ) . \\end{align*}"}
{"id": "1468.png", "formula": "\\begin{align*} g z = ( a z + b ) ( c z + d ) ^ { - 1 } , \\lambda ( g , z ) = c z + d , g = \\left [ \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ] . \\end{align*}"}
{"id": "8165.png", "formula": "\\begin{align*} M _ { q _ 1 , q _ 2 } ( p , H ) : = \\frac { 1 } { \\# X _ p ^ - ( H ) } \\sum _ { \\chi \\in X _ p ^ - ( H ) } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) | L ( 1 , \\chi ) | ^ 2 . \\end{align*}"}
{"id": "7991.png", "formula": "\\begin{align*} P ( \\boldsymbol { \\varphi } ) = \\max _ { \\sigma \\in \\mathcal { A } } \\ , T _ d ^ { \\sigma } . \\end{align*}"}
{"id": "1392.png", "formula": "\\begin{align*} 1 < l : = 1 + m ( \\alpha , \\underline { b } _ 2 ) \\sin ( \\pi \\delta ) \\le n _ 2 ( x ) \\le n _ 1 ( x ) \\le \\overline { b } _ 1 , x \\in [ \\delta , 1 - \\delta ] . \\end{align*}"}
{"id": "6784.png", "formula": "\\begin{align*} - k _ { p _ 0 } \\leq \\sum _ { j = 1 } ^ { p _ 0 } i _ { k _ j } \\leq 0 , \\end{align*}"}
{"id": "8998.png", "formula": "\\begin{align*} \\hat { W } _ k : = \\psi ( A ^ { [ k ] } _ { 1 : N } , k , F ) . \\end{align*}"}
{"id": "4612.png", "formula": "\\begin{gather*} f _ { k - \\ell ^ + - 1 } ( Q / X ^ + ) \\geq { \\ell ^ - + 1 \\choose k - \\ell ^ + } = { \\ell ^ - + 1 \\choose d - k - 1 } , \\\\ f _ { k - \\ell ^ - - 1 } ( Q / X ^ - ) \\geq { \\ell ^ + + 1 \\choose k - \\ell ^ - } = { \\ell ^ + + 1 \\choose d - k - 1 } . \\end{gather*}"}
{"id": "3629.png", "formula": "\\begin{align*} B = \\frac { 5 - 2 \\sigma } { 2 } \\quad C _ k = \\frac { K + k } { K } + \\frac { K } { K + k } - \\frac { 8 } { 3 } ( 1 - \\sigma ) \\left ( 1 + \\frac { k + 1 } { K } \\right ) . \\end{align*}"}
{"id": "1385.png", "formula": "\\begin{gather*} | E _ i ( x ) | \\le \\alpha + 2 \\bar { b } _ i , x \\in [ 0 , 1 ] , i = 1 , 2 , \\\\ | E _ 1 ( 1 ) - E _ 2 ( 1 ) | \\le C \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } \\end{gather*}"}
{"id": "2566.png", "formula": "\\begin{align*} p _ { l , m } ( f ) = \\sup \\{ | \\partial ^ \\alpha f ( x ) | \\mid x \\in \\mathbf { K } _ l , | \\alpha | \\leq m \\} , \\end{align*}"}
{"id": "1487.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { \\tau \\in S _ + } c ( \\tau ; v , w ) e ( \\lambda ( \\mathfrak { i } ( \\tau ) u ) ) , \\end{align*}"}
{"id": "8253.png", "formula": "\\begin{align*} w _ i = w ( \\xi _ r ) = ( u \\times v ) ( r ) = u _ r \\quad w _ j = w ( \\xi _ s ) = ( u \\times v ) ( s ) = u _ s . \\end{align*}"}
{"id": "7310.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n & ( a _ { i 1 } X _ 1 + \\ldots + a _ { i n } X _ n ) ^ { k _ i } \\\\ & = \\sum _ { k _ { 1 1 } + \\ldots + k _ { 1 n } = k _ 1 } \\ldots \\sum _ { k _ { n 1 } + \\ldots + k _ { n n } = k _ n } \\frac { k _ 1 ! \\ldots k _ n ! } { \\prod _ { i , j } k _ { i j } ! } a _ { 1 1 } ^ { k _ { 1 1 } } a _ { 1 2 } ^ { k _ { 1 2 } } \\ldots a _ { n n } ^ { k _ { n n } } X _ 1 ^ { k _ { 1 1 } + \\ldots + k _ { n 1 } } \\ldots X _ n ^ { k _ { 1 n } + \\ldots + k _ { n n } } . \\end{align*}"}
{"id": "1569.png", "formula": "\\begin{align*} \\mathbf { f } ( g \\times h ) = j ( g _ { \\infty } \\times h _ { \\infty } , z _ 0 \\times z _ 0 ) ^ { - k } \\mathbf { f } ( g _ { \\mathbf { h } } \\times h _ { \\mathbf { h } } , z \\times w ) \\end{align*}"}
{"id": "7157.png", "formula": "\\begin{align*} v ' ( x ) = \\left \\{ \\begin{array} { l l } u ( x ) & x \\not \\in \\Omega ' \\\\ \\min \\{ u ( x ) , v ( x ) \\} & x \\in \\Omega ' \\end{array} \\right . \\end{align*}"}
{"id": "1808.png", "formula": "\\begin{align*} \\rho _ g ^ P ( \\mathbf { h } ) : = \\eta _ g ( D _ { X _ M } ) \\end{align*}"}
{"id": "9415.png", "formula": "\\begin{align*} \\tau ' ( b _ 0 a _ 1 b _ 1 ) = \\tau ( b _ 0 \\tau ' ( a _ 1 ) b _ 1 ) . \\end{align*}"}
{"id": "8450.png", "formula": "\\begin{align*} p u b = p v c = q x c = q y \\tilde { c } . \\end{align*}"}
{"id": "4881.png", "formula": "\\begin{align*} P ( z ) y '' ( z ) = y ( z ) \\end{align*}"}
{"id": "7282.png", "formula": "\\begin{align*} \\boxed { \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ k ) = \\prod _ { k = 1 } ^ \\infty ( 1 - X ^ { 3 k } ) ( 1 - X ^ { 3 k - 1 } ) ( 1 - X ^ { 3 k - 2 } ) = \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ k X ^ { \\frac { 3 k ^ 2 + k } { 2 } } . } \\end{align*}"}
{"id": "4318.png", "formula": "\\begin{align*} \\hat { \\varepsilon } _ - ( \\tau ) = \\varepsilon _ - ( \\tau ) + \\varepsilon _ + - \\hat { \\varepsilon } _ + \\end{align*}"}
{"id": "7401.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon _ 1 \\rightarrow 0 ^ { + } } \\limsup _ { n \\rightarrow \\infty } \\mathbb { E } _ { \\mu _ n } \\Big [ \\int _ 0 ^ T n ^ { \\gamma - 1 } \\sum _ { x , y : | x - y | \\geq \\varepsilon n } & [ \\overleftarrow { \\eta } _ t ^ { \\varepsilon _ 1 n } ( x ) \\overrightarrow { \\eta } _ t ^ { \\varepsilon _ 1 n } ( x + 1 ) \\\\ - & \\eta _ t ^ n ( x ) \\eta _ t ^ n ( x + 1 ) ] F ( t , \\tfrac { x } { n } , \\tfrac { y } { n } ) ( c _ { \\gamma } ) ^ { - 1 } p ( x - y ) d t \\Big ] = 0 . \\end{align*}"}
{"id": "2359.png", "formula": "\\begin{align*} P f = \\pm i X f \\Longleftrightarrow f ' ( x ) = \\pm 2 \\pi x f ( x ) \\end{align*}"}
{"id": "7617.png", "formula": "\\begin{align*} \\mathcal { L } P & = \\lambda ' ( \\sum _ { i } F ^ { i i } - 1 ) + u ^ { \\frac { \\alpha + 1 } { \\alpha } } ( F ^ { i j } h _ { i l } h _ { j l } - F ^ { 2 } ) - u ^ { \\frac { 1 } { \\alpha } } F ^ { i j } \\bar { R } _ { \\nu j l i } \\bar { g } ( \\lambda \\partial _ { r } , e _ { l } ) \\\\ & + \\frac { 1 } { \\alpha } \\bar { g } ( \\lambda \\partial _ { r } , \\nabla \\log u ) - \\frac { 1 } { \\alpha } u ^ { \\frac { \\alpha + 1 } { \\alpha } } F ^ { i j } \\nabla _ { i } \\log u \\nabla _ { j } \\log u . \\end{align*}"}
{"id": "1339.png", "formula": "\\begin{align*} W ' ( \\omega ) \\equiv W _ { \\rho ' } ( \\omega ) = \\bigcup _ { x \\in I ( \\omega ) } B ( x , \\rho ' ) . \\end{align*}"}
{"id": "9289.png", "formula": "\\begin{align*} \\frac { \\partial u ( \\mathbf { x } , t ) } { \\partial t } = - \\left [ \\sum _ { i = 1 } ^ { N } \\frac { \\partial } { \\partial x _ { i } } A _ { i } ( \\mathbf { x } , t ) + \\sum _ { i , j = 1 } ^ { N } \\frac { \\partial ^ { 2 } } { \\partial x _ { i } \\partial x _ { j } } B ( \\mathbf { x } , t ) \\right ] u ( \\mathbf { x } , t ) , \\ ; \\ , u ( \\mathbf { x } , 0 ) = f ( \\mathbf { x } ) \\end{align*}"}
{"id": "2049.png", "formula": "\\begin{align*} w ^ { ( 4 ) } - ( \\gamma - \\eta ^ 2 ) \\ , w '' + 2 i \\rho ^ 2 \\beta \\eta w ' - \\rho ^ 4 w = 0 . \\end{align*}"}
{"id": "7553.png", "formula": "\\begin{align*} N ( T ) = \\frac { T } { 2 \\pi } \\log \\left ( \\frac { T } { 2 \\pi } \\right ) - \\frac { T } { 2 \\pi } + \\frac { 7 } { 8 } + \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "6193.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { k } \\bar { \\sigma } ^ 2 _ { t } \\geq \\sum _ { t = 1 } ^ { k } \\sigma ^ 2 _ { t } - 2 \\sqrt { k } \\theta \\| S \\| ^ 2 _ F . \\end{align*}"}
{"id": "1132.png", "formula": "\\begin{align*} m _ { \\eta } ( \\xi , t , k ) : = P ( \\xi , t , k ) \\Psi ^ { A i } ( \\zeta ( \\xi , t , k ) ) Q ( \\xi , k ) , \\end{align*}"}
{"id": "8931.png", "formula": "\\begin{align*} ( U _ 1 ^ { q + 1 } \\cup \\cdots \\cup U _ n ^ { q + 1 } ) ^ c \\cap U ^ { q + 1 } & = \\{ ( x _ 0 , \\ldots , x _ q ) \\in U ^ { q + 1 } : \\forall i \\in \\{ 1 , \\ldots , n \\} \\exists j \\in \\{ 0 , \\ldots , q \\} x _ j \\not \\in U _ i \\} \\\\ & = \\bigcup _ { f : \\{ 1 , \\ldots , n \\} \\to \\{ 0 , \\ldots , q \\} } A _ f . \\end{align*}"}
{"id": "1069.png", "formula": "\\begin{align*} q ^ { \\pm \\infty } : \\mathbb { R } \\times \\mathbb { R } ^ { + } \\rightarrow \\mathbb { C } , q ^ { \\pm \\infty } ( x , t ) = C . \\end{align*}"}
{"id": "5060.png", "formula": "\\begin{align*} { \\bf L } _ n ^ 0 ( \\chi _ T e ^ { i P ' c } h ) = i e ^ { i P ' c } \\chi _ T ' { \\bf A } _ n h + \\chi _ T { \\bf L } _ { n } ^ 0 ( e ^ { i P ' c } h ) . \\end{align*}"}
{"id": "2469.png", "formula": "\\begin{align*} V _ Q D _ L U _ P = \\left \\{ \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\in S p ( \\R , 2 d ) \\mid \\det ( A ) \\neq 0 \\right \\} . \\end{align*}"}
{"id": "8043.png", "formula": "\\begin{align*} U ( w ) \\Phi ( f ) U ( w ) ^ { - 1 } = \\Phi ( w _ { * } ^ { ( \\mu - 1 ) } f ) , \\end{align*}"}
{"id": "2253.png", "formula": "\\begin{align*} J \\leq & \\Big \\| \\sum _ { j = 1 } ^ m \\int _ { t _ { j - 1 } } ^ { t _ j } E ( t _ m - s ) A P \\big ( F ( X ( s ) ) - F ( X ( t _ j ) ) \\big ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\Big \\| \\int _ 0 ^ { t _ m } \\Psi _ k ^ { M , N } ( t _ m - s ) A P F ( X ( t _ m ) ) \\ , \\dd s \\Big \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\sum _ { j = 1 } ^ m \\int _ { t _ { j - 1 } } ^ { t _ j } \\| \\Psi _ k ^ { M , N } ( t _ m - s ) A P \\big ( F ( X ( t _ j ) ) - F ( X ( t _ m ) ) \\big ) \\| _ { L ^ p ( \\Omega ; H ) } \\ , \\dd s \\\\ = : & J _ 1 + J _ 2 + J _ 3 . \\end{align*}"}
{"id": "7274.png", "formula": "\\begin{align*} \\sin ( X ) & : = \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n } { ( 2 n + 1 ) ! } X ^ { 2 n + 1 } , & \\cos ( X ) & : = \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n } { ( 2 n ) ! } X ^ { 2 n } , \\\\ \\tan ( X ) & : = \\frac { \\sin ( X ) } { \\cos ( X ) } , & \\sinh ( X ) & : = \\sum _ { k = 0 } ^ \\infty \\frac { X ^ { 2 k + 1 } } { ( 2 k + 1 ) ! } , \\\\ \\arcsin ( X ) & : = \\sum _ { n = 0 } ^ \\infty \\frac { ( 2 n ) ! } { ( 2 ^ n n ! ) ^ 2 } \\frac { X ^ { 2 n + 1 } } { 2 n + 1 } , & \\arctan ( X ) & : = \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { 2 k + 1 } X ^ { 2 k + 1 } . \\end{align*}"}
{"id": "2153.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } \\sum _ { j = 1 } ^ { n - 1 } \\sum _ { l = 1 } ^ k \\frac { \\binom { c _ n k n } l ( k ) _ l ( k ( j - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } \\frac k { k ( n - j + 1 ) - l } \\frac { E _ { n , k } ( \\tau ( A ^ { ( n ) } _ { c _ n k n , j , l } ) | \\mathcal { W A } ^ { ( n , k ) } _ { M _ n , j , l } ) } { n k } = \\\\ & ( 1 - c ) \\frac { p _ c ^ { ( k ) } ( 1 ) } { k + 1 } . \\end{aligned} \\end{align*}"}
{"id": "7237.png", "formula": "\\begin{align*} \\mathcal R _ { N L } ( t , x ) = \\frac { 1 } { R ^ 3 } \\int _ 0 ^ { t } \\int _ { \\R ^ 3 } E ( s , \\omega - ( t - R - s ) \\tfrac { x - \\omega } { R } + \\tilde { Y } ( x , \\tfrac { x - \\omega } { R } ) ) \\nabla _ v \\mu ( \\tfrac { x - \\omega } { R } + \\tilde { W } ( x , \\tfrac { x - \\omega } { R } ) ) \\dd \\omega \\dd s . \\end{align*}"}
{"id": "6731.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\geq i \\geq 0 \\\\ m \\geq j \\geq 0 } } f _ { i j } \\bigl ( \\sum _ { r _ { i j } \\geq h \\geq 1 } L i _ { K , \\mathfrak { s } _ { i j , h } } ( { \\boldsymbol \\alpha _ { i j , h } } ) \\bigr ) = 0 . \\end{align*}"}
{"id": "6385.png", "formula": "\\begin{align*} \\mathcal K = \\overline { \\textup { s p a n } } \\{ f ( \\underline N ) h \\ , : \\ ; h \\in \\mathcal H \\textup { a n d } f \\in \\mathcal R ( K ) \\} . \\end{align*}"}
{"id": "745.png", "formula": "\\begin{align*} S _ { ( i j ) } ^ { ( \\ell ) } : = K _ { ( i j ) } ^ { ( \\ell ) } - \\kappa _ { ( i j ) } ^ { ( \\ell ) } , \\end{align*}"}
{"id": "4779.png", "formula": "\\begin{align*} \\mu ( n - r - 1 ) = r ( r - \\lambda - 1 ) . \\end{align*}"}
{"id": "2354.png", "formula": "\\begin{align*} [ X , P ] f ( x ) = \\tfrac { 1 } { 2 \\pi i } ( x f ' ( x ) - ( x f ) ' ( x ) ) = \\tfrac { 1 } { 2 \\pi i } \\left ( x f ' ( x ) - ( f ( x ) + x f ' ( x ) ) \\right ) = - \\tfrac { 1 } { 2 \\pi i } f ( x ) . \\end{align*}"}
{"id": "246.png", "formula": "\\begin{align*} \\tau _ \\mu ( x ) = \\Sigma \\nabla \\left ( G ^ \\mu _ { 0 ^ + } ( g ) \\right ) ( x ) . \\end{align*}"}
{"id": "7270.png", "formula": "\\begin{align*} \\Bigl | \\exp \\Bigl ( \\sum \\alpha _ k \\Bigr ) - \\prod _ { k = 1 } ^ n \\exp ( \\alpha _ k ) \\Bigr | & = \\Bigl | \\exp \\Bigl ( \\sum _ { k = 1 } ^ n \\alpha _ k + \\sum _ { k = n + 1 } ^ \\infty \\alpha _ k \\Bigr ) - \\exp \\Bigl ( \\sum _ { k = 1 } ^ n \\alpha _ k \\Bigr ) \\Bigr | \\\\ & = \\Bigl | \\exp \\Bigl ( \\sum _ { k = 1 } ^ n \\alpha _ k \\Bigr ) \\Bigr | \\Bigl | \\exp \\Bigl ( \\sum _ { k = n + 1 } ^ \\infty \\alpha _ k \\Bigr ) - 1 \\Bigr | \\xrightarrow { n \\to \\infty } 0 . \\end{align*}"}
{"id": "3777.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( \\widetilde { W _ v ^ l } , \\widetilde { U _ v ^ l } ) q _ F ^ { - r / 2 } X ^ { - l r } = \\omega _ { \\sigma ^ { ( l ) } } ( - 1 ) ^ { n - 2 } \\gamma ( X ^ l , \\pi ^ { ( l ) } , \\sigma ^ { ( l ) } , \\psi ^ l ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r ( W _ v ^ l , U _ v ^ l ) q _ F ^ { r / 2 } X ^ { l r } . \\end{align*}"}
{"id": "1315.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 2 | \\hat { I } _ { = 2 , < \\epsilon } ( k _ { n } ) | + 3 | \\hat { I } _ { > 2 , < \\epsilon } ( k _ { n } ) | } { k _ { n } } \\leq 2 . \\end{align*}"}
{"id": "5541.png", "formula": "\\begin{align*} \\partial V = \\{ y \\in V : y _ 1 = 0 \\} . \\end{align*}"}
{"id": "6905.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod \\limits _ { j = i + 1 } ^ k R ( j ) } \\sum \\limits _ { x _ { k - 1 } < x _ k } \\ldots \\sum \\limits _ { x _ { i + 1 } < x _ i } f ( x _ i ) & = \\frac { m ( k , i ) } { \\prod \\limits _ { j = i + 1 } ^ k R ( j ) } \\sum \\limits _ { x _ { i } < x _ { k } } f ( x _ i ) \\\\ & = \\frac { 1 } { R ( k , i ) } \\sum \\limits _ { x _ { i } < x _ { k } } f ( x _ i ) . \\end{align*}"}
{"id": "5827.png", "formula": "\\begin{align*} L _ 0 : = 1 0 ^ { 1 0 } L _ { k + 1 } : = \\ell _ k L _ k , k \\geq 0 , \\end{align*}"}
{"id": "3767.png", "formula": "\\begin{align*} I ( X , r _ l ( \\pi _ E ) ( w _ n ) W , \\sigma ( w _ { n - 1 } ) W ' ) = I ( X , r _ l ( \\pi _ F ) ^ { ( l ) } ( w _ n ) W , \\sigma ( w _ { n - 1 } ) W ' ) , \\end{align*}"}
{"id": "5901.png", "formula": "\\begin{align*} \\psi _ S ( x ) = \\frac { ( x + 1 ) } { 2 } ^ { n - 2 k + 1 } \\Big [ 2 ( x + 1 ) ^ { 2 k - 1 } + ( - 1 ) ^ { k - 1 } ( x + 1 ) ^ 2 \\det U _ { x + 1 } + ( - 1 ) ^ k \\det V _ { x + 1 } \\Big ] \\end{align*}"}
{"id": "7153.png", "formula": "\\begin{align*} ( K _ { H } ^ { \\ast } \\varphi ) ( s ) = & \\ , c _ { H } \\Gamma \\left ( H + \\frac { 1 } { 2 } \\right ) \\left ( D _ { T ^ { - } } ^ { \\frac { 1 } { 2 } - H } \\varphi ( s ) \\right ) ( s ) \\\\ & + c _ { H } \\left ( \\frac { 1 } { 2 } - H \\right ) \\int _ { s } ^ { T } \\varphi ( t ) ( t - s ) ^ { H - \\frac { 3 } { 2 } } \\left ( 1 - \\left ( \\frac { t } { s } \\right ) ^ { H - \\frac { 1 } { 2 } } \\right ) d t . \\end{align*}"}
{"id": "583.png", "formula": "\\begin{align*} \\begin{cases} \\ X _ 1 \\ \\subset \\ \\R ^ { d _ 1 } , \\ X _ 2 \\ \\subset \\ \\R ^ { d _ 1 } \\\\ [ 8 p t ] \\ Y _ 1 \\ \\subset \\ \\R ^ { d _ 2 } , \\ Y _ 2 \\ \\subset \\ \\R ^ { d _ 2 } \\end{cases} \\end{align*}"}
{"id": "8773.png", "formula": "\\begin{align*} s _ { i j } = \\min \\bigl \\{ b _ { i j } ( \\delta ) , s _ { i n } \\bigr \\} i \\in \\{ 1 , \\ldots , d \\} \\ ; j \\in \\{ 0 , \\ldots , n \\} . \\end{align*}"}
{"id": "7096.png", "formula": "\\begin{align*} & ( j - 1 ) + \\cdots + 2 + 1 + 1 + 2 + \\cdots + ( r _ i - j ) \\\\ = & \\frac { j ( j - 1 ) } { 2 } + \\frac { ( r _ i - j ) ( r _ i - j + 1 ) } { 2 } \\\\ = & \\frac { 1 } { 2 } [ 2 j ^ 2 - 2 j - 2 j r _ i + r _ i ( r _ i + 1 ) ] \\\\ = & j ^ 2 - j - j r _ i + \\frac { r _ i ( r _ i + 1 ) } { 2 } \\end{align*}"}
{"id": "3518.png", "formula": "\\begin{align*} \\frac { ( a t _ 3 ) ^ { 1 - s _ 1 } } { s _ 1 + s _ 3 - 1 } \\sum _ { n \\leq a t _ 3 } \\frac { 1 } { n ^ { s _ 2 } ( a t _ 3 + n ) ^ { s _ 3 } } & \\ll \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 & ( \\sigma _ 2 = 1 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "2777.png", "formula": "\\begin{align*} L _ - = - \\Delta + 1 - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 2 } , \\end{align*}"}
{"id": "5450.png", "formula": "\\begin{align*} \\mathcal { S } ( T ) = \\left \\{ u \\in \\mathcal { X } _ T : \\| u \\| _ { \\mathcal { X } _ T } \\leq R , \\ ; \\ ; { \\nu A ^ { - 1 } u } \\ge \\varepsilon \\ ; \\ ; t \\in [ s , s + T ] \\right \\} , \\end{align*}"}
{"id": "6378.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 0 & 1 & 1 & 1 \\\\ 0 & 1 & 0 & 1 & 1 & 1 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "6170.png", "formula": "\\begin{align*} & \\frac { 2 } { 3 } f ( - 1 ) + \\frac { 3 + \\sqrt { - 3 } } { 1 8 } f \\left ( \\frac { - 1 + \\sqrt { - 3 } } { 2 } \\right ) + \\frac { 3 - \\sqrt { - 3 } } { 1 8 } f \\left ( \\frac { - 1 - \\sqrt { - 3 } } { 2 } \\right ) \\\\ = & - \\frac { 1 } { 4 \\pi \\sqrt { - 1 } } \\int _ { S ^ 1 } f ( z ) e ^ { - \\frac { 2 } { z } } d z \\end{align*}"}
{"id": "1804.png", "formula": "\\begin{align*} \\tau ^ { Y , r } _ \\varphi ( A _ 0 , \\ldots , A _ k ) : = & \\int _ { G ^ { \\times k } } { } ^ b { \\rm T r } _ Z \\left ( \\Phi _ { A _ 0 } ( ( g _ 1 \\cdots g _ k ) ^ { - 1 } ) \\circ \\Phi _ { A _ 1 } ( g _ 1 ) \\circ \\ldots \\circ \\Phi _ { A _ k } ( g _ k ) \\right ) \\\\ & \\qquad \\varphi ( e , g _ 1 , g _ 1 g _ 2 , \\ldots , g _ 1 \\cdots g _ k ) d g _ 1 \\cdots d g _ k . \\end{align*}"}
{"id": "8298.png", "formula": "\\begin{align*} \\kappa _ { x , { \\text r } } = \\kappa _ { x , { \\text t } } = \\kappa _ x \\end{align*}"}
{"id": "2975.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { i } \\in \\mathcal { J } ^ 2 } | \\Psi _ { \\mathbf { i } } | = \\sum _ { h , h ' = 2 } ^ 4 \\sum _ { \\mathbf { i } \\in \\mathcal I _ h \\times \\mathcal I _ { h ' } } | \\Psi _ { \\mathbf { i } } | = O ( n ^ 4 ) . \\end{align*}"}
{"id": "663.png", "formula": "\\begin{align*} \\abs { A ( x ) - \\alpha } \\ & = \\ \\abs { A ^ \\prime ( f ( x ) ) - \\alpha } \\\\ [ 1 1 p t ] & \\leq \\frac { 1 } { f ( x ) + 1 } \\\\ [ 1 1 p t ] & = \\ E ( x ) . \\end{align*}"}
{"id": "3535.png", "formula": "\\begin{align*} S _ 1 T = \\zeta _ { M T , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + O \\left ( T ^ { 4 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } \\right ) . \\end{align*}"}
{"id": "5887.png", "formula": "\\begin{align*} B _ { a , b } = c ^ { - \\frac { 1 } { \\rho } } \\left [ ( a \\vee 0 ) ^ \\frac { 1 } { \\rho } , \\ , ( b \\vee 0 ) ^ \\frac { 1 } { \\rho } \\right ] . \\end{align*}"}
{"id": "4778.png", "formula": "\\begin{align*} r ^ c + r = n - 1 , \\lambda ^ c - \\mu = n - 2 ( r + 1 ) , \\mu ^ c - \\lambda = n - 2 r , \\end{align*}"}
{"id": "7311.png", "formula": "\\begin{align*} ( 1 + X ) ^ n & = \\sum _ { I \\subseteq N } X ^ { | I | } = \\sum _ { k = 0 } ^ n \\binom { n } { k } X ^ k , \\\\ ( 1 - X ) ^ { - n } & = \\sum _ { k _ 1 , \\ldots , k _ n \\ge 0 } X ^ { k _ 1 + \\ldots + k _ n } = \\sum _ { k = 0 } ^ \\infty \\binom { n + k - 1 } { k } X ^ k , \\end{align*}"}
{"id": "2025.png", "formula": "\\begin{align*} R ^ Y ( h \\overline { \\mu } _ 1 ) = R ^ Y ( h \\overline { \\mu } _ 2 ) + h - 1 \\end{align*}"}
{"id": "3938.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\nu _ 0 \\circ \\alpha _ { T \\Psi } ^ { 0 \\to \\tau } ( E _ { \\Phi _ 1 - \\Phi _ 0 } ) = \\nu _ \\tau ( E _ { \\Phi _ 1 - \\Phi _ 0 } ) . \\end{align*}"}
{"id": "3872.png", "formula": "\\begin{align*} \\begin{cases} t _ i = \\alpha _ i + \\dots + \\alpha _ { n - 2 } + \\frac { 1 } { 2 } ( \\alpha _ { n - 1 } + \\alpha _ n ) , & \\ , ( 1 \\leq i \\leq n - 2 ) \\\\ t _ { n - 1 } = \\frac { 1 } { 2 } ( \\alpha _ { n - 1 } + \\alpha _ { n } ) , & \\\\ t _ n = \\frac { 1 } { 2 } ( - \\alpha _ { n - 1 } + \\alpha _ { n } ) . & \\end{cases} \\end{align*}"}
{"id": "1178.png", "formula": "\\begin{align*} S _ { 1 } = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta \\in ( 0 , 2 \\pi / 3 ) \\right \\} , S _ { 2 } = \\left \\{ \\zeta \\in \\mathbb { C } \\vert \\textnormal { a r g } \\zeta \\in ( 2 \\pi / 3 , \\pi ) \\right \\} , \\end{align*}"}
{"id": "8925.png", "formula": "\\begin{align*} C ^ q _ { A _ 1 , \\ldots , A _ n } ( U , A ) = \\{ \\varphi : U ^ { q + 1 } \\to A \\mid \\varphi | _ { A _ { i _ 0 } \\times \\cdots \\times A _ { i _ q } } \\mbox { c o n s t a n t } \\forall i _ 0 , \\ldots , i _ q \\in \\{ 1 , \\ldots , q \\} \\} \\end{align*}"}
{"id": "1672.png", "formula": "\\begin{align*} H _ n ( \\mathbf { s } ) = H _ n ( s _ 1 , s _ 2 , \\ldots , s _ r ) : = \\sum \\limits _ { 1 \\leq k _ 1 < k _ 2 < \\cdots < k _ r \\leq n } \\prod _ { j = 1 } ^ { r } \\frac { 1 } { k _ j ^ { s _ j } } , \\end{align*}"}
{"id": "7723.png", "formula": "\\begin{gather*} \\phi [ T ] = u [ T ] . \\end{gather*}"}
{"id": "19.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( h _ n ^ j h _ n ^ k ) ^ { - \\frac { 1 + \\alpha } { 2 } } \\Big | B _ n ^ j ( R ) \\cap B _ n ^ k ( R ) \\Big | = 0 \\end{align*}"}
{"id": "9436.png", "formula": "\\begin{align*} F \\ ! = \\ ! ( f , u _ 0 ) \\in \\mathcal { F } ^ { k , \\mathbf { s } ( s , \\lambda , \\lambda ' , \\delta ) } _ { T , \\varLambda ^ q , { \\mathcal D } _ { d \\oplus d ^ * } } \\times C ^ { 2 s + k + 1 , \\lambda , \\delta } _ { \\varLambda ^ q } \\cap \\mathcal { S } _ { d ^ \\ast } \\end{align*}"}
{"id": "7773.png", "formula": "\\begin{gather*} \\| \\tilde \\phi _ { k + 1 } [ T ] - ( p , 0 ) \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } = \\| \\left ( \\phi _ k [ T ] - ( p , 0 ) \\right ) ^ p \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } \\lesssim \\varepsilon \\| \\phi _ k [ T ] - ( p , 0 ) \\| _ { H ^ 1 _ x \\times L ^ 2 _ x } , \\end{gather*}"}
{"id": "8151.png", "formula": "\\begin{align*} s ( c , d ) = { 1 \\over 4 d } \\sum _ { n = 1 } ^ { \\vert d \\vert - 1 } \\cot \\left ( { \\pi n \\over d } \\right ) \\cot \\left ( { \\pi n c \\over d } \\right ) \\ \\ \\ \\ \\ ( c \\in { \\mathbb Z } , \\ d \\in { \\mathbb Z } \\setminus \\{ 0 \\} , \\ \\gcd ( c , d ) = 1 ) , \\end{align*}"}
{"id": "725.png", "formula": "\\begin{align*} S _ { ( 0 0 ) } ^ { ( \\ell + 1 ) } = - \\frac { 1 } { 3 a n } ( 1 + O ( \\ell ^ { - 1 } ) ) + O ( n ^ { - 2 } ) , \\end{align*}"}
{"id": "8616.png", "formula": "\\begin{align*} & A _ { 1 , \\phi _ { 1 } } ( t , k ) = ( \\sqrt { 2 \\pi } ) ^ 3 \\int _ { 0 } ^ { t } \\phi _ { 1 } ( k ) \\ , i k \\ , e ^ { - i s k ^ { 2 } } \\Big ( \\int \\overline { \\mathcal { K } } ^ { \\# } _ { R } ( x , k ) \\ , s \\ , u _ { M _ 1 } ( s , x ) u _ { M _ 2 } ( s , x ) u _ { M _ 3 } ( s , x ) \\ , d x \\Big ) d s . \\end{align*}"}
{"id": "216.png", "formula": "\\begin{align*} A = R _ k ^ { \\alpha } \\left ( \\partial _ k ( p _ \\alpha ) , f \\right ) ( x ) + R _ k ^ { \\alpha } \\left ( \\partial _ k ( f ) , p _ \\alpha \\right ) ( x ) , \\end{align*}"}
{"id": "7747.png", "formula": "\\begin{align*} \\int _ { 2 x _ 0 + v } ^ u ( \\phi _ u \\cdot \\phi _ t ) \\phi ( u _ 0 , u _ 0 - 2 x _ 0 ) d u _ 0 & = \\int _ { t - d } ^ { t + d } ( \\phi _ u \\cdot \\phi _ t ) \\phi ( s , x _ 0 ) d s \\\\ & = \\frac { 1 } { 2 } \\int _ { t - d } ^ { t + d } ( \\phi _ x \\cdot \\phi _ t ) \\phi ( s , x _ 0 ) d s + \\frac { 1 } { 2 } \\int _ { t - d } ^ { t + d } | \\phi _ t | ^ 2 \\phi ( s , x _ 0 ) d s . \\end{align*}"}
{"id": "77.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq q _ v ^ { 1 / 2 } \\cdot \\frac { q _ v ^ { ( n _ { v , \\nu _ v } - 1 ) / 2 } + 1 } { q _ v ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "6179.png", "formula": "\\begin{align*} \\sigma ^ { A } _ q > \\sigma _ { q + 1 } = \\sigma _ { q + 2 } = \\cdots = \\sigma _ { n + 1 } , \\ \\ q \\leq n , \\end{align*}"}
{"id": "588.png", "formula": "\\begin{align*} I \\bigg ( X , \\frac { P } { Q } \\bigg ) \\ = \\ \\sum \\limits _ { i = 1 } ^ m \\ I \\bigg ( K _ i , \\frac { P _ i } { Q _ i } \\bigg ) . \\end{align*}"}
{"id": "4557.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j - 1 } w ( v _ i v _ j ) \\geq ( j - 1 ) \\max \\biggl \\{ ( h - 1 ) , \\frac { r - 2 } { r - 1 } k \\biggr \\} . \\end{align*}"}
{"id": "7583.png", "formula": "\\begin{align*} | c _ { i , j } | \\le \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 } \\begin{cases} \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 0 , 0 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , l = 2 r , \\\\ \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 1 , - 1 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s , l = 2 r + 1 , \\\\ \\Gamma _ { 1 , - 1 } [ s ] ( i ) \\Gamma _ { 0 , 0 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 , l = 2 r , \\\\ \\Gamma _ { 1 , - 1 } [ s ] ( i ) \\Gamma _ { 1 , - 1 } [ r ] ( j ) , ~ ~ ~ ~ ~ ~ \\mbox { i f } k = 2 s + 1 , l = 2 r + 1 , \\end{cases} \\end{align*}"}
{"id": "2650.png", "formula": "\\begin{align*} Z f ( x , \\omega ) = | Z f ( x , \\omega ) | e ^ { 2 \\pi i \\varphi ( x , \\omega ) } . \\end{align*}"}
{"id": "2562.png", "formula": "\\begin{align*} \\sum _ { \\l \\in S \\L } | \\langle f , \\rho ( \\l ) \\widehat { S } g \\rangle | ^ 2 & = \\sum _ { \\l \\in \\L } | \\langle f , \\rho ( S \\l ) \\widehat { S } g \\rangle | ^ 2 \\\\ & = \\sum _ { \\l \\in \\L } | \\langle f , \\widehat { S } \\rho ( \\l ) \\widehat { S } ^ { - 1 } \\widehat { S } g \\rangle | ^ 2 \\\\ & = \\sum _ { \\l \\in \\L } | \\langle \\widehat { S } ^ { - 1 } f , \\rho ( \\l ) g \\rangle | ^ 2 . \\end{align*}"}
{"id": "9043.png", "formula": "\\begin{align*} X ( \\theta ) \\log X ( \\theta ) = \\theta X ^ 0 \\log ( X ( \\theta ) ) + ( 1 - \\theta ) X ^ 1 \\log ( X ( \\theta ) ) . \\end{align*}"}
{"id": "3417.png", "formula": "\\begin{align*} & \\langle K ( x , y ) , \\lambda _ \\delta ( x , y ) \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle \\\\ & = \\sum \\limits _ { j \\in \\Gamma } \\langle K ( x , y ) , \\lambda _ \\delta ( x , y ) \\bar \\eta _ j ( y ) \\chi _ 0 ( y ) [ \\phi ( y ) - \\phi ( x ) ] \\psi ( x ) \\rangle . \\end{align*}"}
{"id": "7753.png", "formula": "\\begin{align*} C _ { S _ 0 } : = \\left ( \\frac { 2 4 C _ H } { S _ 0 } \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "7929.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } S ( k ) & = \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } \\sum _ { \\substack { a \\in S \\\\ 1 \\leqslant a \\leqslant k } } 1 = \\sum _ { a \\in S } \\sum _ { a \\leqslant k \\leqslant n } k ^ { \\ell - 1 } . \\end{align*}"}
{"id": "667.png", "formula": "\\begin{align*} [ 2 \\cdot 4 ^ k q + 1 / 2 ] \\ = \\ [ 2 \\cdot 4 ^ k \\alpha ] + d ( d = 0 \\ \\ d = 1 ) \\end{align*}"}
{"id": "1616.png", "formula": "\\begin{align*} R ^ { 2 , \\operatorname { l i n } } & = \\lbrack \\beta ( \\rho _ 0 + \\nu + \\nu _ c ) - \\beta ( \\rho _ 0 ) \\rbrack u _ 0 + [ \\beta ( \\rho _ 0 + \\nu + \\nu _ c ) - \\beta ( \\nu ) ] w , \\\\ R ^ { 2 , \\operatorname { c o r r } } & = \\beta ( \\rho _ 0 + \\nu + \\nu _ c ) w _ c , \\\\ R ^ h & = h _ 0 - h ^ \\ast . \\end{align*}"}
{"id": "7321.png", "formula": "\\begin{align*} \\psi ( x , y ) = L ( | x - y | ^ 2 + 1 ) ^ { 1 \\over 2 } . \\end{align*}"}
{"id": "1516.png", "formula": "\\begin{align*} \\tau _ i = \\left [ \\begin{array} { c c c c c c } 1 _ { m } & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 _ r & 0 & 0 & 0 _ r & 0 \\\\ 0 & 0 & 1 _ { m } & 0 & 0 & 0 \\\\ 0 & 0 & e _ t & 1 _ { m } & 0 & 0 \\\\ 0 & 0 _ r & 0 & 0 & 1 _ r & 0 \\\\ e _ t ^ { \\ast } & 0 & 0 & 0 & 0 & 1 _ { m } \\end{array} \\right ] , e _ t = \\left [ \\begin{array} { c c } 1 _ t & 0 \\\\ 0 & 0 \\end{array} \\right ] \\in \\mathbb { B } ^ { m } _ { m } . \\end{align*}"}
{"id": "7481.png", "formula": "\\begin{align*} & \\frac { \\tilde { \\phi } ^ { n + 1 } - 2 \\phi ^ n + \\phi ^ { n - 1 } } { \\tau ^ 2 } + \\eta ^ n \\frac { \\tilde { \\phi } ^ { n + 1 } - \\phi ^ { n - 1 } } { 2 \\tau } \\\\ & = \\left ( \\frac 1 2 \\Delta - \\vartheta ^ n \\right ) \\frac { \\tilde { \\phi } ^ { n + 1 } + \\phi ^ { n - 1 } } { 2 } + \\vartheta ^ n \\phi ^ n - V \\phi ^ n - \\beta | \\phi ^ n | ^ 2 \\phi ^ n + \\Omega L _ z \\phi ^ n + \\lambda ^ n \\phi ^ n , \\\\ & \\phi ^ { n + 1 } = \\tilde { \\phi } ^ { n + 1 } / \\| \\tilde { \\phi } ^ { n + 1 } \\| , n \\geq 1 , \\end{align*}"}
{"id": "3081.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = y _ 1 \\ , , y _ 1 ^ \\prime = x _ 1 \\ , , x _ 2 ^ \\prime = y _ 2 \\ , , y _ 2 ^ \\prime = x _ 2 \\ , . \\end{align*}"}
{"id": "4275.png", "formula": "\\begin{align*} { \\bf D } ^ { ( z ) } ( t , x ) ~ = ~ z _ x ( t , 0 - ) \\cdot ( t - x ) \\cdot \\phi _ 0 ' ( x - t ) + ( z _ 1 ( t , 0 ) - z _ 1 ( t , x ) ) \\cdot \\phi ' _ 0 ( x ) + \\left [ a ^ { ( z ) } _ 1 ( t ) - a ^ { ( z ) } _ 2 ( t ) \\right ] \\cdot \\phi _ 0 ' ( t ) , \\end{align*}"}
{"id": "963.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s _ \\delta u ( x ) : = c _ { n , s } \\int _ { \\R ^ n \\setminus B _ \\delta ( x ) } \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { n + 2 s } } \\dd y . \\end{align*}"}
{"id": "8680.png", "formula": "\\begin{align*} { C } _ { n } ( \\kappa , { \\bf P } _ { Y _ 1 } ) = \\sup _ { ( \\Lambda _ t , K _ { \\overline { Z } _ t } ) , t = 1 , \\ldots , n , \\frac { 1 } { n } { \\bf E } \\big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\big \\} \\leq \\kappa } \\sum _ { t = 1 } ^ n \\Big \\{ H ( I _ t ) - H ( \\hat { I } _ t ) \\Big \\} \\end{align*}"}
{"id": "8522.png", "formula": "\\begin{align*} Q ( y ) = \\operatorname { s g n } ( y ) \\begin{dcases} 1 & | y | { < } \\tau _ { 0 } \\\\ i & \\tau _ { i } { < } | y | { < } \\tau _ { i + 1 } , 0 { < } i { < } 2 ^ { w - 1 } { - } 2 \\\\ 2 ^ { w - 1 } & | y | { < } \\tau _ { 2 ^ { w - 1 } - 2 } \\end{dcases} . \\end{align*}"}
{"id": "5923.png", "formula": "\\begin{align*} \\flat _ { \\omega _ Q } ( \\pmb { X } ) = d H , \\end{align*}"}
{"id": "6205.png", "formula": "\\begin{align*} \\begin{aligned} \\| C - C Z _ { j } Z _ { j } ^ { T } \\| _ F & = \\| C - C ( \\hat { V } _ j - \\bar { E } ) ( \\hat { V } _ j - \\bar { E } ) ^ { T } \\| _ F \\\\ & \\leq \\| C - C \\hat { V } _ { j } \\hat { V } ^ { T } _ { j } \\| _ F + \\| C \\bar { E } \\hat { V } ^ { T } _ { j } \\| _ F + \\| C \\hat { V } _ { j } \\bar { E } ^ { T } \\| _ F + \\| C \\bar { E } \\bar { E } ^ { T } \\| _ F \\\\ & < \\| C - C \\hat { V } _ j \\hat { V } _ j ^ { T } \\| _ F + ( 2 \\xi \\sqrt { 1 + \\xi } + \\xi ^ 2 ) \\| C \\| _ F . \\end{aligned} \\end{align*}"}
{"id": "4505.png", "formula": "\\begin{align*} \\Sigma _ { \\sigma _ 2 } ^ { 1 } = 2 \\sum _ { \\substack { 0 < \\gamma < T \\\\ \\sigma _ 2 \\le \\beta < 1 } } \\frac { x ^ { \\beta - 1 } } { \\gamma } \\le 2 N ( \\sigma _ 2 , T ) \\frac { x ^ { - \\frac { 1 } { R \\log t _ 0 } } } { t _ 0 } , \\end{align*}"}
{"id": "2236.png", "formula": "\\begin{align*} X _ m ^ { M , N } = E _ { k , N } ^ m P _ N X _ 0 - k \\sum _ { j = 1 } ^ m E _ { k , N } ^ { m - j + 1 } P _ N A F ( X _ j ^ { M , N } ) + \\sum _ { j = 1 } ^ m E _ { k , N } ^ { m - j + 1 } P _ N \\Delta W _ j . \\end{align*}"}
{"id": "895.png", "formula": "\\begin{align*} F _ q ( T , q ^ { - 4 } X ^ { - 1 } ) & = \\eta _ q ( T ) ( q ^ 2 X ) ^ { \\mathrm { o r d } _ q ( 2 ^ { 2 } \\det T ) } F _ q ( T , X ) \\\\ & = - ( q ^ 2 X ) ^ { \\mathrm { o r d } _ q ( 2 ^ { 2 } \\det T ) } F _ q ( T , X ) . \\end{align*}"}
{"id": "7554.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { N _ 0 ( T ) } { N ( T ) } = 1 \\end{align*}"}
{"id": "312.png", "formula": "\\begin{align*} & \\mathcal H _ 0 ( \\rho , S ) = \\mathcal K ( S , \\rho ) + \\mathcal F ( \\rho ) - \\alpha L ( \\rho ) , \\ ; \\\\ & \\mathcal H _ 1 ( \\rho , S ) = \\eta _ 1 \\mathcal K ( S , \\rho ) + \\eta _ 2 I ( \\rho ) + \\eta _ 3 \\mathcal V ( \\rho ) + \\eta _ 4 \\mathcal W ( \\rho ) - \\eta _ 5 L ( \\rho ) \\end{align*}"}
{"id": "6907.png", "formula": "\\begin{align*} r = \\sqrt { P _ t } \\Biggl ( \\sum _ { n = 1 } ^ N g _ n c _ n \\mathbf { h } _ n ^ T + \\mathbf { h } _ d ^ T \\Biggr ) \\mathbf { w } s + n , \\end{align*}"}
{"id": "9359.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( \\log _ { \\lambda } ( 1 + t ) \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 1 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } , ( k \\ge 0 ) . \\end{align*}"}
{"id": "1374.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left ( \\frac { 1 } { n } - \\frac { 1 } { n ^ 3 } \\right ) n _ x \\varphi _ x d x + \\alpha \\int _ 0 ^ 1 \\frac { \\varphi _ x } { n } d x + \\int _ 0 ^ 1 \\left ( n - b \\right ) \\varphi d x = 0 , \\end{align*}"}
{"id": "3071.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = j x _ 1 \\ , , x _ 2 ^ \\prime = x _ 2 \\ , , y _ 1 ^ \\prime = y _ 1 \\ , , y _ 2 ^ \\prime = y _ 2 \\ , , \\end{align*}"}
{"id": "774.png", "formula": "\\begin{align*} [ W ] = \\frac { \\langle W \\rangle } { \\langle W ' \\rangle } . \\end{align*}"}
{"id": "8265.png", "formula": "\\begin{align*} \\Delta ( F _ { 1 4 2 3 } ) = \\iota \\otimes F _ { 1 4 2 3 } + F _ { 1 } \\otimes F _ { 3 1 2 } + F _ { 1 2 } \\otimes F _ { 1 2 } + F _ { 1 3 2 } \\otimes F _ { 1 } + F _ { 1 4 2 3 } \\otimes \\iota . \\end{align*}"}
{"id": "1887.png", "formula": "\\begin{align*} \\abs { x } = \\prod _ { s \\in V } \\abs { x _ s } _ s \\end{align*}"}
{"id": "3432.png", "formula": "\\begin{align*} | H _ k ( x , y ) | & = \\bigg | \\int _ { \\Bbb R ^ N } D _ k ( x , z ) \\big [ S _ { k - M - 1 } ( z , y ) - S _ { k - M - 1 } ( x , y ) \\big ] d \\omega ( z ) \\bigg | \\\\ & \\leqslant C \\int _ { | x - z | \\leqslant r ^ { 2 - k } } ( V _ k ( x ) ) ^ { - 1 } \\frac { r ^ { k - M - 1 } \\| x - z \\| } { V _ { k - M - 1 } ( x ) } d \\omega ( z ) \\\\ & \\leqslant C r ^ { - M } ( V _ { k - M - 1 } ( x ) ) ^ { - 1 } . \\end{align*}"}
{"id": "5233.png", "formula": "\\begin{align*} & ( w ( b _ r - d + e ) + k + 1 / ( d - 1 ) ) ( d k + e ) - ( b _ r + d k - d + e ) ( k + 1 ) \\\\ = & \\frac { ( b _ r - d ) ( d k ( d - 4 ) + 2 k - 2 ) + ( b _ r + d k - d + e ) ( d - 2 ) ( e - 2 ) ) } { ( 2 d - 2 ) } \\ge 0 . \\end{align*}"}
{"id": "4754.png", "formula": "\\begin{align*} A ( \\Gamma \\bigodot m ) = \\left ( \\begin{array} { c c c c } A _ { 1 1 } & A _ { 1 2 } & \\cdots & A _ { 1 n } \\\\ A _ { 2 1 } & A _ { 2 2 } & \\cdots & A _ { 2 n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ { n 1 } & A _ { n 2 } & \\cdots & A _ { n n } \\end{array} \\right ) , \\end{align*}"}
{"id": "5111.png", "formula": "\\begin{align*} \\psi ( x ) > q _ 1 ( 1 . 3 ) + 6 \\ , q _ 2 ( 1 . 3 1 ) = 0 . 0 5 6 6 > 0 . \\end{align*}"}
{"id": "7453.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\ddot { u } ( t ) + \\eta ( t ) \\dot { u } ( t ) = - \\nabla f ( u ( t ) ) , t > 0 , \\\\ & u ( 0 ) = u _ 0 , \\dot { u } ( 0 ) = v _ 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "3619.png", "formula": "\\begin{align*} T _ { j } + G _ { n } & = ( R _ { n + 3 } - R _ { j } + d ( - a _ { j } a _ { j + 1 } ) ) + ( 2 e - R _ { n + 2 } + R _ { n + 1 } - t ) \\\\ & = ( R _ { n + 3 } - R _ { n + 2 } + 2 e ) + ( R _ { n + 1 } - R _ { j } + d ( - a _ { j } a _ { j + 1 } ) - t ) \\\\ & = 2 T _ { 0 } + R _ { n + 1 } - R _ { j } + d ( - a _ { j } a _ { j + 1 } ) - t \\ , . \\end{align*}"}
{"id": "2361.png", "formula": "\\begin{align*} \\widehat { V _ g f } ( \\xi , \\eta ) = e ^ { 2 \\pi i \\xi \\cdot \\eta } f ( - \\eta ) \\overline { \\widehat { g } ( \\xi ) } . \\end{align*}"}
{"id": "6350.png", "formula": "\\begin{align*} X = X _ { 8 k + 4 } \\subset \\P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) . \\end{align*}"}
{"id": "4393.png", "formula": "\\begin{align*} \\langle \\Phi , \\phi _ i \\rangle _ { L ^ 2 _ \\rho } = \\left \\{ \\begin{array} { r c l } & & \\mathcal { I } _ 0 \\left [ \\frac { b _ \\tau } { b } - 1 \\right ] b ^ \\alpha + O \\left ( b ^ { \\alpha + 1 - \\frac { \\epsilon } { 2 } } \\right ) , \\\\ & & O ( b ^ { \\alpha + 1 - \\frac { \\epsilon } { 2 } } ) . \\end{array} \\right . \\end{align*}"}
{"id": "7999.png", "formula": "\\begin{align*} N ^ { t - T _ 3 ^ { \\sigma } } \\big ( N a ^ t + c ^ t \\big ) = 1 \\ ; \\ ; N ^ { 1 - T _ 3 ^ { \\omega } } \\big ( N b + d \\big ) = 1 , \\end{align*}"}
{"id": "7446.png", "formula": "\\begin{align*} L _ z = - i ( x \\partial _ y - y \\partial _ x ) \\end{align*}"}
{"id": "5139.png", "formula": "\\begin{align*} & \\ , ( n ^ 2 - 1 ) ^ { 1 / 2 } \\cosh ( \\rho ( n ) ) - n - \\tfrac { \\sinh ( 2 \\rho ( n ) ) } { 4 } > ( n ^ 2 - 1 ) ^ { 1 / 2 } \\Big ( 1 + \\tfrac { \\rho ( n ) ) ^ 2 } { 2 } \\Big ) - n - \\tfrac { 2 \\rho ( n ) + ( 2 \\rho ( n ) ) ^ 3 } { 4 } \\\\ = & \\ , ( n ^ 2 - 1 ) ^ { 1 / 2 } - n + ( n ^ 2 - 1 ) ^ { 1 / 2 } \\tfrac { \\rho ( n ) ) ^ 2 } { 2 } - \\rho ( n ) - 4 \\rho ( n ) ^ 3 . \\end{align*}"}
{"id": "8061.png", "formula": "\\begin{align*} E ^ { i j } _ 0 ( x , x ' ) = a \\left ( \\frac { \\partial } { \\partial x } \\right ) ^ { \\mu _ i + \\mu _ j - 1 } \\delta ( x - x ' ) . \\end{align*}"}
{"id": "4666.png", "formula": "\\begin{align*} \\dot { x } _ i \\sim \\mu _ i , \\quad \\dot { \\mu } _ i + \\sum ^ n _ { \\substack { j = 1 , \\\\ j \\not = i } } \\frac { a _ { i j } } { ( x _ i - x _ j ) ^ 3 } + \\sum _ { \\substack { j , k = 1 , \\\\ j \\not = i } } ^ n \\frac { b _ { i j k } \\mu _ k } { ( x _ i - x _ j ) ^ 3 } \\sim 0 , \\end{align*}"}
{"id": "8733.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { i n } ( x ) = f _ n ( x ) , u _ { i j } ( x ) \\leq f _ { i } ( x ) , & & i = 1 , \\ldots , d , \\ j = 0 , \\ldots , n , \\\\ & u ( x ) \\leq a ( x ) , f ^ L _ i = u _ { i 0 } ( x ) = a _ { i 0 } ( x ) \\leq \\cdots \\leq a _ { i n } ( x ) = f ^ U _ { i } & & i = 1 , \\ldots , d , \\\\ & \\deg ( u _ { i j } ) \\leq \\tau , \\deg ( a _ { i j } ) \\leq \\tau & & i = 1 , \\ldots , d , \\ j = 0 , \\ldots , n . \\end{aligned} \\end{align*}"}
{"id": "7575.png", "formula": "\\begin{align*} L A ( \\phi ) = L \\phi \\quad \\mbox { f o r a l l } \\quad \\phi \\in C ^ 1 . \\end{align*}"}
{"id": "3111.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + a z y ^ 3 x ^ 2 + x ^ 6 + y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "9228.png", "formula": "\\begin{align*} ( ( y + x ) - x ) \\in A x & \\leftrightarrow x = _ X J ^ A _ \\gamma ( y + x ) \\\\ & \\leftrightarrow x ' = _ X J ^ A _ \\gamma ( y ' + x ' ) \\\\ & \\leftrightarrow ( ( y ' + x ' ) - x ' ) \\in A x ' \\end{align*}"}
{"id": "2071.png", "formula": "\\begin{align*} f _ { \\alpha } ^ { ( \\gamma ) } ( o ) = \\sum _ { \\beta < \\gamma } a _ { \\beta } g _ { \\beta } ^ { ( \\gamma ) } ( o ) + a _ { \\gamma } g _ { \\gamma } ^ { ( \\gamma ) } ( o ) . \\end{align*}"}
{"id": "8184.png", "formula": "\\begin{align*} S ( H , f ) = \\frac { f - 2 d + 1 } { 4 } \\hbox { a n d } N ' ( f , H ) = - 2 d + 1 , \\end{align*}"}
{"id": "8375.png", "formula": "\\begin{align*} H = - ( h _ { i j } \\dot { M } ^ i _ k \\dot { ) } M ^ j _ l x ^ k x ^ l \\end{align*}"}
{"id": "3017.png", "formula": "\\begin{align*} [ \\sigma _ { y , z } ] = [ \\ell _ y ] + [ \\gamma _ { v y ^ 4 , y ^ 4 z } ] + [ \\gamma _ { w y ^ 4 , y ^ 4 z } ] + [ \\gamma _ { x y ^ 4 , y ^ 4 z } ] + [ \\gamma _ { v y ^ 3 z , y ^ 3 z ^ 2 } ] + [ \\gamma _ { w y ^ 3 z , y ^ 3 z ^ 2 } ] - [ \\gamma _ { x y ^ 2 z ^ 2 , y ^ 3 z ^ 2 } ] . \\end{align*}"}
{"id": "7517.png", "formula": "\\begin{align*} \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) + \\arg \\left ( \\sigma - 1 + i T \\right ) = \\pi + \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "3925.png", "formula": "\\begin{align*} R = R ^ { ( 1 ) } + R ^ { ( 2 ) } , \\end{align*}"}
{"id": "2618.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k \\in \\Z } f ( k ) g ( t - k ) , \\forall t \\in \\R \\forall f \\in B _ I ^ 1 ( \\R ) , \\end{align*}"}
{"id": "4062.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "6062.png", "formula": "\\begin{align*} \\Phi _ { p , q } ( \\sigma , \\tau ) = \\gamma ( \\sigma ) + \\tau \\eta ( \\sigma ) \\ , . \\end{align*}"}
{"id": "4690.png", "formula": "\\begin{align*} \\tau _ i f ( t , y ) = f ( t , y - x _ i ( t ) ) ; \\end{align*}"}
{"id": "5906.png", "formula": "\\begin{align*} \\det Q _ S = \\frac { 1 } { 2 } \\Big [ 2 + ( - 1 ) ^ { k } ( \\det V - \\det U ) \\Big ] , \\end{align*}"}
{"id": "79.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } \\Bigr \\} \\Bigl \\{ 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } \\Bigr \\} ^ { - 1 } \\\\ & \\leq 2 \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "8098.png", "formula": "\\begin{align*} \\hat { \\overline { u } } _ n ( \\xi ) = \\frac { ( - i \\xi ) ^ n } { n ! } \\left [ \\alpha - i \\pi \\ , \\mathrm { s g n } ( \\xi ) \\right ] . \\end{align*}"}
{"id": "5090.png", "formula": "\\begin{align*} \\L ^ \\circ _ { a , b } = \\{ \\l ^ \\circ \\in \\R ^ 2 \\mid \\pi ( \\l ^ \\circ ) \\pi ( \\l ) = \\pi ( \\l ) \\pi ( \\l ^ \\circ ) \\forall \\l \\in \\L _ { a , b } \\} . \\end{align*}"}
{"id": "6111.png", "formula": "\\begin{align*} M ( X , Y ) = \\sum _ { j \\ge 0 } H _ j ( X ^ { m + 2 } ) Y ^ { j ( n + 2 ) } \\ , , \\end{align*}"}
{"id": "5198.png", "formula": "\\begin{align*} \\sup _ { | y | = 1 } \\sum \\limits _ { j = 1 } ^ { \\infty } \\int _ { \\mathfrak { D } ^ * } ~ | \\Omega ( x + \\mathfrak { p } ^ j y ) - \\Omega ( x ) | ~ d x < \\infty . \\end{align*}"}
{"id": "8793.png", "formula": "\\begin{align*} \\bigl \\{ ( s _ { 1 n _ 1 } , s _ { 2 1 } , \\phi ) \\bigm | \\phi = s _ { 1 n _ 1 } s _ { 2 1 } , \\ s _ { 1 n _ 1 } \\in \\{ a _ { 1 0 } , \\ldots , a _ { 1 n _ 1 } \\} , \\ s _ { 2 1 } \\in [ a _ { 2 0 } , a _ { 2 1 } ] \\bigr \\} . \\end{align*}"}
{"id": "8368.png", "formula": "\\begin{align*} C _ { v } ( u , w , z ) = \\frac { 1 } { 4 } \\left . \\frac { \\partial ^ 3 } { \\partial r \\partial s \\partial t } L ( v + t u + s w + r z ) \\right | _ { t = s = r = 0 } , \\end{align*}"}
{"id": "3369.png", "formula": "\\begin{align*} & T _ 1 ( u ) - T ^ { ' } _ 1 ( u ) = T D ( \\mathfrak { X } ) u - [ \\mathfrak { X } , T u ] = ( d _ T \\mathfrak { X } ) ( u ) . \\end{align*}"}
{"id": "9412.png", "formula": "\\begin{align*} \\varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) & = 0 + \\varphi ( b _ 0 a _ 1 a _ 2 \\cdots a _ n b _ n ) \\tau ( b _ 1 ) \\cdots \\tau ( b _ { n - 1 } ) \\\\ & \\varphi ( b _ 0 a _ 1 a _ 2 \\cdots a _ n b _ n ) \\varphi ( b _ 1 ) \\cdots \\varphi ( b _ { n - 1 } ) . \\end{align*}"}
{"id": "837.png", "formula": "\\begin{align*} \\int _ { A _ 1 } | u | \\ , d \\mu _ X = \\int _ { A _ 1 } | u - u _ { A _ 1 } | \\ , d \\mu _ X \\le C \\ , H _ R \\ , \\mu _ X ( A _ 1 ) ^ { 1 / 2 } \\ , \\left ( \\int _ { A _ 1 } g _ u ^ 2 \\ , d \\mu _ X \\right ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "3884.png", "formula": "\\begin{align*} w _ 1 ( M ) = \\begin{cases} O ( \\geq 0 ) - \\frac { 1 } { 2 } \\beta _ n - \\frac { 1 } { 2 } \\gamma _ n & ( a , b ) = ( n , n ) , \\\\ O ( \\geq 0 ) - \\beta _ n & ( a , b ) = ( n + 1 , n ) , \\\\ O ( \\geq 0 ) - \\frac { 1 } { 2 } ( \\beta _ { n - 1 } + \\beta _ n + \\gamma _ { n - 1 } + \\gamma _ n ) & ( a , b ) = ( n + 1 , n + 1 ) . \\end{cases} \\end{align*}"}
{"id": "2303.png", "formula": "\\begin{align*} ( u \\otimes \\overline { v } ) ( h ) = \\langle h , v \\rangle _ \\mathcal { H } u , u , v , h \\in \\mathcal { H } . \\end{align*}"}
{"id": "1684.png", "formula": "\\begin{align*} _ { s + 1 } F _ { s } ( a _ 1 , \\ldots , a _ { s + 1 } ; b _ 1 , \\ldots , b _ s ; x ) : = \\sum \\limits _ { i = 0 } ^ \\infty \\frac { ( a _ 1 ) _ i \\cdots ( a _ { s + 1 } ) _ i } { ( b _ 1 ) _ i \\cdots ( b _ s ) _ i } \\frac { x ^ i } { i ! } , \\end{align*}"}
{"id": "6838.png", "formula": "\\begin{align*} ( M ) \\in \\{ 1 \\} \\cup \\bigcup _ { i = 1 } ^ k \\left [ \\lambda _ i ( M ) - e , \\lambda _ i ( M ) + e \\right ] \\end{align*}"}
{"id": "2657.png", "formula": "\\begin{align*} S _ { g , \\widetilde { g } , \\Gamma } f = \\sum _ { \\gamma \\in \\Gamma } \\langle f , \\pi ( \\gamma ) g \\rangle \\pi ( \\gamma ) \\widetilde { g } . \\end{align*}"}
{"id": "2727.png", "formula": "\\begin{align*} m _ { 0 , \\lambda } ( \\Z ( \\{ P _ 1 , \\ldots , P _ q \\} , B ) ) = 0 . \\end{align*}"}
{"id": "776.png", "formula": "\\begin{align*} \\langle W \\rangle \\langle ( W ' ) '' \\rangle - \\langle W ' \\rangle \\langle W '' \\rangle = ( - 1 ) ^ { | W | } \\mbox { f o r } | W | \\geq 2 . \\end{align*}"}
{"id": "8415.png", "formula": "\\begin{align*} & \\sum _ { i = \\lfloor N - \\eta ^ { - 1 } \\rfloor + 1 } ^ { N - 1 } \\sup _ { h \\in \\mathrm { L i p } ( 1 ) } \\left | \\tilde { Q } _ { i - 1 } \\big ( P _ { \\eta } - \\tilde { Q } _ { 1 } \\big ) P _ { ( N - i ) \\eta } h ( x ) \\right | \\\\ & \\qquad \\leq C ( 1 + | x | ) \\ , \\eta ^ { 2 / \\alpha } \\sum _ { i = \\lfloor N - \\eta ^ { - 1 } \\rfloor + 1 } ^ { N - 1 } [ ( N - i ) \\eta ] ^ { - 1 / \\alpha } \\leq C \\frac { \\alpha } { \\alpha - 1 } \\ , ( 1 + | x | ) \\ , \\eta ^ { 2 / \\alpha - 1 } . \\end{align*}"}
{"id": "8282.png", "formula": "\\begin{align*} A ( N ( x ) , x ) = A ( x , N ( x ) ) = 1 , \\ \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "684.png", "formula": "\\begin{align*} \\chi _ { | | ; \\alpha } ^ { ( \\ell ) } : = \\chi _ { | | } ( K _ { \\alpha \\alpha } ^ { ( \\ell ) } ) = 1 = \\chi _ { \\perp ; } ( K _ { \\alpha \\alpha } ^ { ( \\ell ) } ) = : \\chi _ { \\perp ; \\alpha } ^ { ( \\ell ) } , \\forall \\ell \\geq 1 , \\ , x _ \\alpha \\in \\R ^ { n _ 0 } . \\end{align*}"}
{"id": "8302.png", "formula": "\\begin{align*} 1 + R _ - ( \\kappa _ x ) = T _ + ( \\kappa _ x ) . \\end{align*}"}
{"id": "6039.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\rm l e f t } ( x ) & = \\exp \\bigg [ - \\int _ 0 ^ x \\frac { ( \\alpha + \\beta + 2 ) y - ( \\beta - \\alpha ) } { 2 ( 1 - y ^ 2 ) } \\ , d y \\bigg ] = ( 1 - x ) ^ { ( \\alpha + 1 ) / 2 } ( 1 + x ) ^ { ( \\beta + 1 ) / 2 } \\\\ \\mathcal { E } _ { \\rm r i g h t } ( x ) & = { \\exp } \\bigg [ - \\int _ 0 ^ x - \\frac { ( \\alpha + \\beta ) y - ( \\beta - \\alpha ) } { 2 ( 1 - y ^ 2 ) } \\ , d y \\bigg ] = ( 1 - x ) ^ { - \\alpha / 2 } ( 1 + x ) ^ { - \\beta / 2 } . \\end{align*}"}
{"id": "3936.png", "formula": "\\begin{align*} 0 \\le s ( \\nu _ + | \\nu _ 1 ) \\le \\liminf _ { T \\to \\infty } s ( \\nu _ 0 \\circ \\alpha _ { T \\Psi } ^ { 0 \\to 1 } | \\nu _ 1 ) = 0 . \\end{align*}"}
{"id": "1173.png", "formula": "\\begin{align*} \\frac { d \\psi _ { 1 1 } } { d \\zeta } + \\frac { i \\zeta } { 2 } \\psi _ { 1 1 } = \\beta ^ { ( \\eta ) } _ { 1 2 } \\psi _ { 2 1 } , \\\\ \\frac { d \\psi _ { 2 1 } } { d \\zeta } - \\frac { i \\zeta } { 2 } \\psi _ { 2 1 } = \\beta ^ { ( \\eta ) } _ { 2 1 } \\psi _ { 1 1 } , \\end{align*}"}
{"id": "2725.png", "formula": "\\begin{align*} P _ n = \\sum _ { i = 1 } ^ n X _ i ^ 2 ( X _ i - 1 ) ^ 2 \\end{align*}"}
{"id": "1681.png", "formula": "\\begin{align*} \\overline { H } _ n ( s _ 1 , \\ldots , s _ r ) = \\sum \\limits _ { k _ 1 = 0 } ^ { n - r } \\frac { c _ { k _ 1 } } { ( 2 k _ 1 + 1 ) ^ { s _ 1 } } \\quad c _ { k _ 1 } = \\sum \\limits _ { k _ 1 < k _ 2 < \\cdots < k _ r \\leq n - 1 } \\prod \\limits _ { j = 2 } ^ r \\frac { 1 } { ( 2 k _ j + 1 ) ^ { s _ j } } . \\end{align*}"}
{"id": "4017.png", "formula": "\\begin{align*} m _ 1 ^ 2 \\left ( 1 - e ^ { m _ 1 } \\right ) \\left ( e ^ { m _ 3 } - e ^ { m _ 2 } \\right ) + m _ 2 ^ 2 \\left ( 1 - e ^ { m _ 2 } \\right ) \\left ( e ^ { m _ 1 } - e ^ { m _ 3 } \\right ) + m _ 3 ^ 2 \\left ( 1 - e ^ { m _ 3 } \\right ) \\left ( e ^ { m _ 2 } - e ^ { m _ 1 } \\right ) = 0 . \\end{align*}"}
{"id": "7508.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in R _ \\epsilon } ( \\rho ) & = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { T } \\left [ \\log \\left | \\frac { \\frac { 1 } { 2 } - \\epsilon + i t } { 2 } \\right | - \\log \\left | \\frac { \\frac { 1 } { 2 } + \\epsilon + i t } { 2 } \\right | \\right ] \\ d t + \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\left [ \\arg \\left ( \\frac { \\sigma + i T } { 2 } \\right ) - \\arg \\frac { \\sigma } { 2 } \\right ] \\ d \\sigma \\end{align*}"}
{"id": "2076.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\prod _ { j \\ne i } | \\alpha _ i - \\alpha _ j | _ v \\leq \\delta . \\end{align*}"}
{"id": "5278.png", "formula": "\\begin{align*} \\check { \\varphi } ( \\omega ) = \\varepsilon ( a ) , \\omega = \\varphi ( - a ) , a \\in A , \\end{align*}"}
{"id": "2720.png", "formula": "\\begin{align*} \\Lambda _ n = \\mathrm { R } [ X _ 1 , \\ldots , X _ n ] ^ { \\mathfrak { S } _ n } \\end{align*}"}
{"id": "7559.png", "formula": "\\begin{align*} s B s _ i & = s X _ i U _ i C s _ i \\\\ & = { } ^ s X _ { i } s s _ i U _ i C \\\\ & \\subseteq B s s _ i B , \\end{align*}"}
{"id": "2531.png", "formula": "\\begin{align*} \\Phi ( x , \\omega ) = e ^ { - \\tfrac { \\pi } { 2 } ( x ^ 2 + \\omega ^ 2 ) } \\Phi ^ { \\xi , \\eta } ( x , \\omega ) = e ^ { \\pi i ( x \\cdot \\eta - \\xi \\cdot \\omega ) } \\Phi ( x - \\xi , \\omega - \\eta ) \\end{align*}"}
{"id": "8048.png", "formula": "\\begin{align*} \\partial _ { \\Sigma , \\epsilon } ^ { * } \\circ \\Psi _ { ( \\Sigma , U ) } \\circ \\eta _ { ( \\Sigma , U ) } = \\partial _ { \\widetilde { \\Sigma } , \\widetilde { \\epsilon } } ^ { * } \\circ \\Psi _ { ( \\widetilde { \\Sigma } , U ) } \\circ \\eta _ { ( \\widetilde { \\Sigma } , U ) } , \\end{align*}"}
{"id": "9445.png", "formula": "\\begin{align*} \\varPhi _ q \\ , d _ q \\ , \\varPsi _ { \\mu , q } { \\mathcal N } _ q u = \\varPhi _ q \\ , d _ q \\ , \\varPsi _ \\mu M _ 1 ^ { ( q ) } ( ( d _ q \\oplus d _ { q - 1 } ^ * u , u ) , \\end{align*}"}
{"id": "9248.png", "formula": "\\begin{align*} x \\in \\mathrm { d o m } ( J ^ A _ \\gamma ) : = \\exists y ^ X \\left ( \\gamma ^ { - 1 } ( x - _ X y ) \\in A y \\right ) . \\end{align*}"}
{"id": "9513.png", "formula": "\\begin{align*} B & = \\Big ( [ f _ - ( - M _ x ) ] [ g _ + ( H ) ] \\Big ) + \\Big ( [ g _ - ( - M _ x ) ] [ f _ + ( H ) ] \\Big ) \\\\ & = \\Big ( g ( 0 ) [ f _ - ( - M _ x ) ] + [ f ( 0 ) g _ + ( H ) ] - f ( 0 ) g ( 0 ) \\Big ) + \\Big ( f ( 0 ) [ g _ - ( - M _ x ) ] + [ g ( 0 ) f _ + ( H ) ] - f ( 0 ) g ( 0 ) \\Big ) \\\\ & = C - 2 f ( 0 ) g ( 0 ) . \\end{align*}"}
{"id": "319.png", "formula": "\\begin{align*} \\phi _ c ^ 1 ( S , t ) = \\theta ^ 1 _ c ( \\| S \\| _ { \\mathcal C ( [ 0 , t ] ; \\mathbb R ^ N ) } ) , \\ ; \\ ; \\phi _ c ^ 2 ( \\rho , t ) = \\theta ^ 2 _ c \\Big ( \\min _ { i = 1 } ^ N \\min _ { s \\in [ 0 , t ] } \\rho _ i ( s ) \\Big ) , \\end{align*}"}
{"id": "2731.png", "formula": "\\begin{align*} \\omega ^ { ( j ) } = ( \\delta _ { 1 , j } , \\ldots , \\delta _ { q , j } ) \\in \\Omega . \\end{align*}"}
{"id": "3954.png", "formula": "\\begin{align*} D ( A ) = \\Big \\{ \\Phi = ( \\xi , \\eta ) \\in { H } ^ 1 ( 0 , 1 ) \\times H ^ 2 ( 0 , 1 ) : \\xi ( 0 ) = \\xi ( 1 ) , \\ \\eta ( 0 ) = \\eta ( 1 ) = 0 \\Big \\} . \\end{align*}"}
{"id": "7300.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ k \\frac { \\rho _ k } { k } Y ^ k & = \\sum _ { l = 1 } ^ \\infty \\frac { ( - 1 ) ^ l } { l } \\sum _ { a _ 1 + \\ldots + a _ n = l } \\frac { l ! } { a _ 1 ! \\ldots a _ n ! } \\sigma _ 1 ^ { a _ 1 } \\ldots \\sigma _ n ^ { a _ n } Y ^ { a _ 1 + 2 a _ 2 + \\ldots + n a _ n } \\\\ & = \\sum _ { k = 1 } ^ \\infty \\sum _ { ( 1 ^ { a _ 1 } , \\ldots , k ^ { a _ k } ) \\in P ( k ) } ( - 1 ) ^ { a _ 1 + \\ldots + a _ k } \\frac { ( a _ 1 + \\ldots + a _ k - 1 ) ! } { a _ 1 ! \\ldots a _ k ! } \\sigma _ 1 ^ { a _ 1 } \\ldots \\sigma _ k ^ { a _ k } Y ^ k . \\end{align*}"}
{"id": "5946.png", "formula": "\\begin{align*} \\norm { \\sum _ n a _ n r _ n } _ 6 ^ 6 & = \\sum _ { i _ 1 , i _ 2 , i _ 3 } \\sum _ { j _ 1 , j _ 2 , j _ 3 } \\prod _ { k = 1 } ^ 3 a _ { i _ k } \\ , \\prod _ { l = 1 } ^ 3 \\overline { a _ { j _ l } } \\ , \\ , \\Big \\langle r _ { i _ 1 } r _ { i _ 2 } r _ { i _ 3 } , \\ , { r _ { j _ 1 } } { r _ { j _ 2 } } { r _ { j _ 3 } } \\Big \\rangle \\\\ & \\le \\sum _ n | a _ n | ^ 6 A ^ 6 + \\binom { 6 } { 2 } A ^ 6 \\sum _ m \\sum _ n | a _ m | ^ 4 | a _ n | ^ 2 + \\binom { 6 } { 3 } A ^ 6 \\sum _ m \\sum _ n | a _ m | ^ 3 | a _ n | ^ 3 \\end{align*}"}
{"id": "7899.png", "formula": "\\begin{align*} \\gamma _ F ( P \\cap \\tau _ v P ) = ( 2 \\pi ) ^ { - m k } ( \\det \\tilde { \\Lambda } ) ^ { - \\frac { 1 } { 2 } } \\int _ { B \\times B } e ^ { - \\frac { 1 } { 2 } \\tilde { \\Lambda } ^ { - 1 } t \\cdot t } \\ d t , \\end{align*}"}
{"id": "9349.png", "formula": "\\begin{align*} \\mathrm { L i } _ { k , \\lambda } ( t ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } ( 1 ) _ { n , 1 / \\lambda } } { ( n - 1 ) ! n ^ { k } } ( - 1 ) ^ { n - 1 } t ^ { n } , ( \\mathrm { s e e } \\ [ 8 , 1 3 ] ) . \\end{align*}"}
{"id": "4290.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ r ^ 2 u + \\frac { d + 1 } { r } \\partial _ r u - ( d - 1 ) \\frac { \\sin ( 2 u ) } { 2 r ^ 2 } , ( r , t ) \\in \\R _ + \\times \\R _ + . \\end{align*}"}
{"id": "8684.png", "formula": "\\begin{align*} & { C } _ n ( \\kappa , { \\bf P } _ { Y _ 1 } ) = \\sup _ { \\frac { 1 } { n } { \\bf E } \\big \\{ \\sum _ { t = 1 } ^ n | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\big \\} \\leq \\kappa } \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ n \\ln \\Big \\{ \\frac { \\det ( K _ { I _ t } ) } { \\det ( K _ { \\hat { I } _ t } ) } \\Big \\} ^ + \\end{align*}"}
{"id": "5281.png", "formula": "\\begin{align*} \\check { \\varphi } ( \\varphi ( - a ) \\varphi ( - b ) ) = \\varphi ( S ^ { - 1 } ( b ) a ) . \\end{align*}"}
{"id": "16.png", "formula": "\\begin{align*} P _ n ( x ) = \\sum _ { j = 1 } ^ { M } [ G _ n ] [ g _ n ^ j ] [ G _ n ] ^ { - 1 } \\phi ^ j ( x ) + p _ n ^ M ( x ) , \\end{align*}"}
{"id": "4560.png", "formula": "\\begin{align*} | E _ 1 \\cup E _ 2 | \\le \\varepsilon n ^ 2 . \\end{align*}"}
{"id": "4105.png", "formula": "\\begin{align*} \\begin{aligned} Y & \\le c \\cdot \\ , \\int _ s ^ { s + \\delta } | X _ u - X _ s | ^ { p - 2 } \\cdot \\bigl ( 1 + X _ u ^ 2 + | X _ s | \\cdot ( 1 + | X _ u | ^ { \\ell _ \\mu + 1 } ) \\bigr ) \\ , d u \\\\ & + \\sup _ { s \\le t \\le s + \\delta } \\Bigl | \\int _ s ^ t p \\cdot ( X _ u - X _ s ) \\cdot | X _ u - X _ s | ^ { p - 2 } \\cdot \\sigma ( X _ u ) \\ , d W _ u \\Bigr | \\end{aligned} \\end{align*}"}
{"id": "4189.png", "formula": "\\begin{align*} \\mathrm { V o l } ( N ) & \\leq \\mathrm { V o l } \\left ( B ^ { M ^ k ( - \\lambda ) } _ { \\mathrm { D i a m } ( N ) } ( x ' ) \\right ) = \\frac { 2 \\pi ^ { \\frac { k } { 2 } } } { \\Gamma ( \\frac { k } { 2 } ) } \\int _ 0 ^ { \\mathrm { D i a m } ( N ) } \\left ( \\frac { \\sinh ( t \\sqrt { \\lambda } ) } { \\sqrt { \\lambda } } \\right ) ^ { k - 1 } d t . \\end{align*}"}
{"id": "2528.png", "formula": "\\begin{align*} P \\pi ( \\mathbf { h } ) f = P \\pi ( \\mathbf { h } ) P f = \\pi ( \\mathbf { h } ) P f , \\forall \\mathbf { h } \\in \\mathbf { H } . \\end{align*}"}
{"id": "6625.png", "formula": "\\begin{align*} J _ 3 : = \\sum _ { \\substack { \\alpha \\in A \\\\ \\beta \\in B } } \\sum _ { \\substack { \\alpha ' \\neq \\alpha \\\\ \\beta ' \\neq \\beta } } \\frac { 1 } { 4 \\pi i } \\int _ { ( \\epsilon ) } \\underset { s _ 2 = - s _ 1 - \\alpha ' - \\beta ' } { } \\ \\mathcal { J } \\ , d s _ 1 . \\end{align*}"}
{"id": "356.png", "formula": "\\begin{align*} & \\widetilde m _ { 1 2 } = ( \\Sigma _ 1 - \\Sigma _ 2 ) \\theta _ { 1 2 } ( \\rho ) \\dot W ^ { \\delta } , \\ ; \\widetilde m _ { 2 3 } - ( \\Sigma _ 2 - \\Sigma _ 3 ) \\dot W ^ { \\delta } = m _ { 2 3 } - ( \\Sigma _ 2 - \\Sigma _ 3 ) \\theta _ { 2 3 } ( \\rho ) \\dot W ^ { \\delta } . \\end{align*}"}
{"id": "548.png", "formula": "\\begin{align*} \\int _ 0 ^ \\tau \\int _ \\Omega p ( \\rho ) b _ \\alpha ( \\rho ) - I ^ \\alpha _ 1 = - \\int _ 0 ^ \\tau \\langle \\nabla p ( \\rho ) , \\mathcal { B } ( b _ \\alpha ( \\rho ) - \\langle b _ \\alpha ( \\rho ) \\rangle \\rangle = \\sum _ { j = 2 } ^ 6 I ^ { \\alpha } _ { j } + J ^ { \\alpha } ( \\tau ) , \\end{align*}"}
{"id": "6944.png", "formula": "\\begin{align*} \\eta _ { { \\rm r h s } , 2 } ( E ) = h _ E \\Vert f - \\Pi _ { E , q - 1 } f \\Vert _ { 0 , E , \\omega } + \\Vert f - \\Pi _ { E , q } f \\Vert _ { 0 , E , \\omega } \\ , . \\end{align*}"}
{"id": "7206.png", "formula": "\\begin{align*} X _ 1 ^ T ( \\tau ) = x _ 1 - ( t - \\tau ) v _ 1 . \\end{align*}"}
{"id": "7222.png", "formula": "\\begin{align*} \\zeta _ { s , t , x } ( v ) : = v - \\frac { \\tilde Y _ { s , t } ( x , v ) } { t - s } . \\end{align*}"}
{"id": "4344.png", "formula": "\\begin{align*} \\mathcal { L } _ \\beta = \\Delta - \\beta ( \\tau ) z \\cdot \\nabla - \\beta ( \\tau ) I d \\end{align*}"}
{"id": "4224.png", "formula": "\\begin{align*} \\mathcal { F } _ N ( \\mathbf { h } , \\mathbf { g } ) = \\mathbb { E } \\left [ \\int _ 0 ^ T \\mathcal { L } \\left ( t , \\mathbf { Y } ( t ) , \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\delta _ { X ^ n ( t ) } \\right ) d t + \\int _ 0 ^ T \\Psi \\left ( \\mathbf { h } ( t ) , \\mathbf { g } \\left ( \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\delta _ { X ^ n ( t ) } \\right ) \\right ) d t \\right ] , \\end{align*}"}
{"id": "3153.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - A ^ \\intercal S _ { i _ k } ( S _ { i _ k } ^ \\intercal A B ^ { - 1 } A ^ \\intercal S _ { i _ k } ) ^ \\dagger S _ { i _ k } ^ \\intercal ( A x _ k - b ) , \\end{align*}"}
{"id": "61.png", "formula": "\\begin{align*} L ' \\otimes \\Z _ 2 & = \\bigoplus _ { j = 0 } ^ 1 L ' _ { 2 , j } ( \\pi ^ j ) , \\\\ K _ { \\ell } \\otimes \\Z _ 2 & = K _ { \\ell , 2 , 0 } , \\end{align*}"}
{"id": "2411.png", "formula": "\\begin{align*} \\langle S f , f \\rangle = \\sum _ { \\gamma \\in \\Gamma } | \\langle f , e _ \\gamma \\rangle | ^ 2 , \\end{align*}"}
{"id": "875.png", "formula": "\\begin{align*} \\bar { \\Delta } _ { \\rm R e a c t i v e } = \\frac { \\mathbb { E } G ^ { \\rm R e a c } _ j } { \\mathbb { E } T ^ { \\rm R e a c } _ j } = \\frac { \\mathbb { E } T ^ { \\rm P r o a c } _ j } { 2 \\mathbb { E } \\left ( T ^ { \\rm R e a c } _ j \\right ) ^ 2 } + \\mathbb { E } \\tau ^ { \\rm R e a c } _ { V _ j } - \\tau _ { \\rm f } - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "7147.png", "formula": "\\begin{align*} x _ t = x _ 0 + \\int _ 0 ^ t \\big ( \\alpha _ { - 1 } x ^ { - 1 } _ s - \\alpha _ 0 + \\alpha _ 1 x _ s - \\alpha _ 2 s ^ { 2 H - 1 } x _ s ^ { \\rho } \\big ) d s + \\int _ 0 ^ t \\sigma x _ s ^ { \\theta } d B _ s ^ H , \\end{align*}"}
{"id": "3808.png", "formula": "\\begin{align*} F ( x ) = ( f _ { g _ 1 } ( x ) , f _ { g _ 2 } ( x ) , \\ldots , f _ { g _ n } ( x ) ) \\end{align*}"}
{"id": "5687.png", "formula": "\\begin{align*} \\mathrm { E C H } ( Y , \\lambda , \\Gamma ) : = \\bigoplus _ { * : \\ , \\ , \\mathbb { Z } \\mathrm { - g r a d i n g } } \\mathrm { E C H } _ { * } ( Y , \\lambda , \\Gamma ) . \\end{align*}"}
{"id": "3019.png", "formula": "\\begin{align*} \\pi : M \\to \\mathbf { H } , \\pi ( C ) = \\left ( C \\cdot D ^ x \\right ) \\ell + \\left ( C \\cdot \\mathbb { D } \\right ) \\gamma . \\end{align*}"}
{"id": "8446.png", "formula": "\\begin{align*} L ' _ { \\tilde { b } } \\wedge L ' _ { c } = L ' _ { s y \\tilde { b } } . \\end{align*}"}
{"id": "6465.png", "formula": "\\begin{align*} E [ ( S _ n ) ^ 2 ] & = \\dfrac { \\Gamma ( n + 2 \\alpha ) } { \\Gamma ( n ) } \\sum _ { \\ell = 1 } ^ n \\dfrac { \\Gamma ( \\ell ) } { \\Gamma ( \\ell + 2 \\alpha ) } \\\\ & = \\begin{cases} \\dfrac { n } { 1 - 2 \\alpha } + \\dfrac { 1 } { 2 \\alpha - 1 } \\cdot \\dfrac { \\Gamma ( n + 2 \\alpha ) } { \\Gamma ( n ) \\Gamma ( 2 \\alpha ) } & ( \\alpha \\neq 1 / 2 ) , \\\\ \\displaystyle n \\sum _ { \\ell = 1 } ^ n \\dfrac { 1 } { \\ell } & ( \\alpha = 1 / 2 ) . \\end{cases} \\end{align*}"}
{"id": "8570.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow 0 ^ + } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = \\chi _ + ( x ) T ( 0 ) m _ + ( x , 0 ) + \\chi _ - ( x ) [ m _ - ( x , 0 ) + R _ - ( 0 ) m _ - ( x , 0 ) ] , \\end{align*}"}
{"id": "2232.png", "formula": "\\begin{align*} \\mathbb { L } _ 1 \\leq & \\int _ 0 ^ t \\| E ( t - s ) A ^ { \\frac 3 2 } \\| _ { \\mathcal { L } ( H ) } \\| ( I - P _ n ) A ^ { - \\frac 1 2 } \\| _ { \\mathcal { L } ( H ) } \\| P F ( X ^ { n } ( s ) ) \\| _ { L ^ p ( \\Omega ; H ) } \\ , \\dd s \\\\ \\leq & C \\lambda _ n ^ { - \\frac 1 2 } \\int _ 0 ^ t ( t - s ) ^ { - \\frac 3 4 } \\ , \\dd s \\big ( 1 + \\sup _ { t \\in [ 0 , T ] } \\| X ^ n ( t ) \\| _ { L ^ { 3 p } ( \\Omega ; L ^ 6 ) } ^ 3 \\big ) \\\\ \\leq & C \\lambda _ n ^ { - \\frac 1 2 } \\rightarrow 0 . \\end{align*}"}
{"id": "4959.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i \\partial _ t u - \\Delta u = \\rho ^ 2 | u | ^ 2 + \\langle \\nabla \\rangle ^ { - \\alpha } \\dot { W } \\ , , t \\in [ 0 , T ] \\ , , \\ , x \\in \\R ^ d \\ , , \\\\ u _ 0 = \\phi \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "7342.png", "formula": "\\begin{align*} \\lambda h _ q + ( 1 - \\lambda ) k _ q = \\psi _ t ( x _ q , t _ q ) , \\eta _ q = \\zeta _ q = \\nabla \\psi ( x _ q , t _ q ) , \\end{align*}"}
{"id": "9421.png", "formula": "\\begin{align*} \\kappa ^ N _ { \\sigma } ( \\mathbf { M } ) = \\sum _ { g \\ge 0 } N ^ { - 2 g } \\kappa _ { \\sigma } ^ { ( 2 g ) } ( \\mathbf { M } ) . \\end{align*}"}
{"id": "8252.png", "formula": "\\begin{align*} a [ p ] : = a + { \\rm s g n } ( a ) \\cdot p = \\begin{cases} a + p , & a > 0 , \\\\ a - p , & a < 0 . \\end{cases} \\end{align*}"}
{"id": "2290.png", "formula": "\\begin{align*} ( f \\otimes g ) ( x , \\omega ) = f ( x ) g ( \\omega ) \\end{align*}"}
{"id": "6106.png", "formula": "\\begin{align*} l \\lambda _ p - m s _ \\sigma + t _ \\sigma + 1 = \\lambda _ q \\ , , \\end{align*}"}
{"id": "5550.png", "formula": "\\begin{align*} \\alpha _ { \\rm H J M } ( h ) = \\sum _ { j = 1 } ^ { \\infty } \\sigma ^ j ( h ) \\int _ 0 ^ { \\bullet } \\sigma ^ j ( h ) ( \\eta ) d \\eta , h \\in H . \\end{align*}"}
{"id": "7452.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ { k } = w _ { k - 1 } - s \\nabla f \\left ( w _ { k - 1 } \\right ) , \\\\ & w _ { k } = u _ { k } + \\frac { k - 1 } { k + 2 } \\left ( u _ { k } - u _ { k - 1 } \\right ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "6938.png", "formula": "\\begin{align*} \\int _ E \\Pi _ { E , k } \\varphi = \\int _ E \\varphi \\forall \\varphi \\in L ^ 2 ( E ) \\ , . \\end{align*}"}
{"id": "1028.png", "formula": "\\begin{align*} \\inf _ { B _ 1 ( 2 e _ 1 ) } \\tilde u _ { \\bar M } = 0 . \\end{align*}"}
{"id": "3480.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , s _ 3 ) & \\ll \\sum _ { k = 3 } ^ \\infty \\begin{cases} k ^ { - 2 \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ k ^ { - 2 \\sigma _ 1 - \\sigma _ 3 } ( \\log k ) ^ 2 & ( \\sigma _ 2 = 1 ) \\\\ k ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "5899.png", "formula": "\\begin{align*} 2 \\sum _ { i = 1 } ^ k \\ , s _ i t _ i - \\sum _ { i = 1 } ^ k \\ , ( s _ i + t _ i ) & = \\sum _ { i = 1 } ^ k \\ , \\Big [ s _ i \\ , ( t _ i - 1 ) + t _ i \\ , ( s _ i - 1 ) \\Big ] \\\\ & \\geq \\sum _ { i = 1 } ^ k \\ , ( s _ i + t _ i - 2 ) = n - 2 k . \\end{align*}"}
{"id": "1011.png", "formula": "\\begin{align*} u ( a ) & = \\tau \\zeta ( a ) ( \\rho - \\vert a \\vert ) ^ { - n - 2 } . \\end{align*}"}
{"id": "112.png", "formula": "\\begin{align*} x ^ r + \\gamma _ 1 \\omega ( x ) x ^ { r - 1 } + \\cdots + \\gamma _ { r - 1 } \\omega ( x ) ^ { r - 1 } x = 0 , \\end{align*}"}
{"id": "3592.png", "formula": "\\begin{align*} & \\Gamma : = \\bigcup _ { i \\in I } P _ i ^ * ( \\Gamma _ i ) , \\\\ & \\rho ( P _ i ^ * ( x ^ * ) ) : = \\rho _ i ( x ^ * ) \\ , \\ , i \\in I x ^ * \\in \\Gamma _ i . \\end{align*}"}
{"id": "6673.png", "formula": "\\begin{align*} \\mathcal { E } ( h , k ) = \\mathcal { L } ^ r ( h , k ) + \\mathcal { U } ^ r ( h , k ) + O \\bigg ( \\bigg ( Q + \\frac { Q ^ 2 } { C } \\bigg ) \\frac { ( X C Q h k ) ^ { \\varepsilon } ( h , k ) } { \\sqrt { h k } } \\bigg ) \\\\ + O \\Big ( ( X C Q h k ) ^ { \\varepsilon } \\big ( X C + X ^ { - \\frac { 1 } { 2 } } Q ^ { \\frac { 5 } { 2 } } + Q ^ { \\frac { 3 } { 2 } } + X ^ 2 h k Q ^ { - 9 6 } \\big ) \\Big ) . \\end{align*}"}
{"id": "856.png", "formula": "\\begin{align*} { \\bar { \\Delta } _ { \\rm T A R Q } } = - \\frac { 1 } { 2 } + { n _ 1 } \\left ( { \\frac { 2 } { { 1 - { \\epsilon _ 1 } } } - \\frac { 1 } { 2 } - \\frac { { m \\epsilon _ 1 ^ m } } { { 1 - \\epsilon _ 1 ^ m } } } \\right ) . \\end{align*}"}
{"id": "3533.png", "formula": "\\begin{align*} \\int _ 2 ^ T \\abs { E ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & = O \\left ( \\int _ 2 ^ T t _ 3 ^ { 3 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } d t _ 3 \\right ) \\\\ & \\ll \\begin{cases} T ^ { 4 - 2 \\sigma _ 1 - 2 \\sigma _ 2 - 2 \\sigma _ 3 } & ( \\frac { 3 } { 2 } < \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 < 2 ) \\\\ \\log T & ( \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 = 2 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "9388.png", "formula": "\\begin{align*} F _ { n , \\lambda } ( 1 ) = \\sum _ { k = 0 } ^ { \\infty } ( k ) _ { n , \\lambda } \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { k + 1 } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "5089.png", "formula": "\\begin{align*} S _ { g , \\L _ { a , b } } f = \\sum _ { \\l \\in \\L _ { a , b } } \\langle f , \\pi ( \\l ) g \\rangle \\ , \\pi ( \\l ) g . \\end{align*}"}
{"id": "6013.png", "formula": "\\begin{align*} e ^ { - t \\mathfrak { D } _ { { \\rm l e f t } , a } } u ( x ) & = \\exp \\bigg [ - \\int _ { x - t } ^ { x } a ( y ) \\ , d y \\bigg ] u ( x - t ) \\\\ e ^ { - t \\mathfrak { D } _ { { \\rm r i g h t } , a } } v ( x ) & = \\exp \\bigg [ - \\int _ { x } ^ { x + t } a ( y ) \\ , d y \\bigg ] v ( x + t ) , \\end{align*}"}
{"id": "8602.png", "formula": "\\begin{align*} \\mu ^ \\# _ R ( { \\bf k } ) = \\mu ^ \\# _ { R , 1 } ( { \\bf k } ) + \\mu ^ \\# _ { R , 2 } ( { \\bf k } ) , \\end{align*}"}
{"id": "5122.png", "formula": "\\begin{align*} \\frac { B } { A } = e ^ { 2 a \\gamma } . \\end{align*}"}
{"id": "8214.png", "formula": "\\begin{align*} 1 - P _ 1 - P _ 2 - \\cdots - P _ { k + 1 - h } = \\frac { x _ h z } { P _ h } , \\\\ 1 - P _ 1 - P _ 2 - \\cdots - P _ { k - h } = \\frac { x _ { h + 1 } z } { P _ { h + 1 } } . \\end{align*}"}
{"id": "4260.png", "formula": "\\begin{align*} { \\bf H } [ g _ b ] ( x ) ~ = ~ { 1 \\over \\pi } \\cdot \\int _ { 0 } ^ { 2 } g _ b ' ( y ) \\cdot \\ln | x - y | ~ d y . \\end{align*}"}
{"id": "3172.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k + w _ k \\frac { w _ k ^ \\intercal A ^ \\intercal ( b - A x _ k ) } { \\norm { A w _ k } _ 2 ^ 2 } , \\end{align*}"}
{"id": "4798.png", "formula": "\\begin{align*} \\phi = s \\chi _ A + t \\chi _ B + \\chi _ C \\end{align*}"}
{"id": "1927.png", "formula": "\\begin{align*} & \\int ^ { T } _ 0 \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ N \\big ( \\partial _ t \\varphi + v \\cdot \\nabla _ x \\varphi + ( u _ { N } \\chi _ { \\{ | u _ N | \\leq N \\} } - v ) \\cdot \\nabla _ v \\varphi + L [ f _ N ] \\cdot \\nabla _ v \\varphi + \\Delta _ v \\varphi \\big ) \\ , d x d v d t \\\\ & = - \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ { 0 , N } \\varphi ( 0 , x , v ) \\ , d x d v + \\int _ 0 ^ T \\int _ { \\Sigma ^ - } ( v \\cdot \\nu ( x ) ) g _ N \\varphi \\ , d \\sigma ( x ) d v d t ; \\end{align*}"}
{"id": "4138.png", "formula": "\\begin{align*} w ( t ) = \\ & U ( t ) \\sigma - \\int _ 0 ^ t U ( t - \\tau ) z ( \\tau ) d \\tau . \\end{align*}"}
{"id": "4113.png", "formula": "\\begin{align*} \\bigl | | u + v | ^ { p _ 0 - 2 } - | u | ^ { p _ 0 - 2 } \\bigr | & \\le \\frac { \\bigl | | u + v | ^ { \\alpha ( p _ 0 - 2 ) } - | u | ^ { \\alpha ( p _ 0 - 2 ) } \\bigr | } { | u + v | ^ { ( \\alpha - 1 ) ( p _ 0 - 2 ) } + | u | ^ { ( \\alpha - 1 ) ( p _ 0 - 2 ) } } \\\\ & \\le \\frac { \\bigl | | u + v | ^ { \\alpha ( p _ 0 - 2 ) } - | u | ^ { \\alpha ( p _ 0 - 2 ) } \\bigr | } { | u | ^ { ( \\alpha - 1 ) ( p _ 0 - 2 ) } } \\\\ & = \\frac { \\bigl | | u + v | ^ { 2 } - | u | ^ { 2 } \\bigr | } { | u | ^ { 4 - p _ 0 } } \\le \\frac { 2 | v | ( | u | + | v | ) } { | u | ^ { 4 - p _ 0 } } . \\end{align*}"}
{"id": "8042.png", "formula": "\\begin{align*} U ( w ) \\Phi ( z ) U ( w ) ^ { - 1 } = \\left ( \\frac { d w } { d z } \\right ) ^ { \\mu } \\Phi ( w ( z ) ) \\ , , \\end{align*}"}
{"id": "7621.png", "formula": "\\begin{align*} \\nabla P = \\lambda \\partial _ { r } ^ { T } - u ^ { \\frac { 1 } { \\alpha } } \\nabla u . \\end{align*}"}
{"id": "7791.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { l - 1 } \\| ( S f - S g ) & ( t _ j ) - ( S f - S g ) ( t _ { j + 1 } ) \\| \\\\ & \\leq \\alpha _ { \\max } \\sum \\limits _ { j = 0 } ^ { l - 1 } \\| ( f - g ) ( P _ k ^ { - 1 } ( t _ j ) ) - ( f - g ) ( P _ k ^ { - 1 } ( t _ { j + 1 } ) ) \\| \\\\ & \\leq \\alpha _ { \\max } \\| f - g \\| _ { \\mathcal { B V } } . \\end{align*}"}
{"id": "9015.png", "formula": "\\begin{align*} \\rho ^ { n + 1 } = _ { \\rho \\in K } \\left \\{ \\frac { 1 } { 2 \\tau } W ^ 2 _ 2 ( \\rho ^ n , \\rho ) + E ( \\rho ) \\right \\} , \\rho ^ 0 = \\rho ^ { i n } ( x ) , \\end{align*}"}
{"id": "7382.png", "formula": "\\begin{align*} D _ { 1 , q } : = | \\nabla \\varphi ( x _ q , t _ q ) | \\left ( m ( \\{ u ( \\cdot , t _ q ) < u ( y _ q , t _ q ) \\} ) - m ( \\{ u _ { \\star , \\lambda } ( \\cdot , t _ 0 ) < u _ { \\star , \\lambda } ( x _ 0 , t _ 0 ) \\} ) \\right ) , \\end{align*}"}
{"id": "1857.png", "formula": "\\begin{align*} R _ { n + 1 } ( x , y ) = \\sum _ { k = 0 } ^ n { n \\choose k } L _ { k } ( x , y ) R _ { n - k } ( x , y ) . \\end{align*}"}
{"id": "7460.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\mathcal { F } ( t ) & = 2 \\ , \\mathrm { R e } \\langle \\dot { \\phi } , \\ddot { \\phi } \\rangle + 2 \\ , \\mathrm { R e } \\left \\langle - \\frac 1 2 \\Delta \\phi + V \\phi + \\beta | \\phi | ^ 2 \\phi - \\Omega L _ z \\phi , \\ , \\dot { \\phi } \\right \\rangle \\\\ & = 2 \\ , \\mathrm { R e } \\left \\langle \\dot { \\phi } , - \\eta ( t ) \\dot { \\phi } + \\lambda _ { \\phi } ( t ) \\phi \\right \\rangle \\\\ & = - 2 \\ , \\eta ( t ) \\| \\dot { \\phi } \\| ^ 2 . \\end{align*}"}
{"id": "6265.png", "formula": "\\begin{align*} e _ { \\bar { G } } ( X , Y ) & \\geq | S | | Y | - h ( | S | + | Y | ) \\\\ & = | Y | ( | S | - h ) - h | S | \\\\ & \\geq | S | ^ 2 - 2 h | S | \\\\ & \\geq \\frac { | S | ^ 2 } { 2 } > \\frac { | S | ^ 2 } { 1 6 h ^ 2 } . \\end{align*}"}
{"id": "1384.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 ^ - } ( 1 - x ) ^ { \\frac { 1 } { 2 } } n _ { i x } ( x ) = - \\frac { 1 } { 2 } \\sqrt { \\int _ 0 ^ 1 ( b _ i - n _ i ) d x } = : B _ i < 0 , i = 1 , 2 . \\end{align*}"}
{"id": "3407.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { Q } ( x , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \\end{align*}"}
{"id": "9406.png", "formula": "\\begin{align*} \\varphi ( a _ 1 \\cdots a _ n ) = \\frac { 1 } { \\tau ' ( p ) } \\tau ' ( ( p a _ 1 ) a _ 2 \\cdots a _ n ) = \\frac { 1 } { \\tau ' ( p ) } \\tau ' ( p a _ 1 ) \\tau ( a _ 2 \\cdots a _ n ) = 0 . \\end{align*}"}
{"id": "6576.png", "formula": "\\begin{align*} \\sideset { } { ^ { \\flat } } \\sum _ { \\chi \\bmod q } \\chi ( m ) \\overline { \\chi ( n ) } = \\frac { 1 } { 2 } \\Bigg ( \\sum _ { \\substack { d | q \\\\ d | ( m \\pm n ) } } \\phi ( d ) \\mu \\left ( \\frac { q } { d } \\right ) \\Bigg ) , \\end{align*}"}
{"id": "5401.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } X _ { t } = f ( X _ { t } , \\theta ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ { t } + \\mathrm { d } L _ { t } - \\mathrm { d } R _ { t } , \\\\ & X _ { 0 } = x \\in [ a , b ] , \\end{aligned} \\right . \\end{align*}"}
{"id": "9365.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\log _ { \\lambda } ( 1 + t ) & = \\frac { 1 } { t } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } ( 1 ) _ { n , \\frac { 1 } { \\lambda } } } { n ! } t ^ { n } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\lambda ^ { n } ( 1 ) _ { n + 1 , \\frac { 1 } { \\lambda } } } { ( n + 1 ) ! } t ^ { n } . \\end{align*}"}
{"id": "7591.png", "formula": "\\begin{align*} | c _ { i , j } ^ { ( k - 2 s , l - 2 r - 2 ) } | & \\leq \\dfrac { 1 } { 2 j } \\left ( | c _ { i , j - 1 } ^ { ( k - 2 s , l - 2 r - 1 ) } | + | c _ { i , j + 1 } ^ { ( k - 2 s , l - 2 r - 1 ) } | \\right ) \\\\ & \\leq \\dfrac { 1 } { 2 j } \\frac { 4 V _ { k , l } } { \\pi ^ 2 } \\Gamma _ { 0 , 0 } [ s ] ( i ) \\left ( \\Gamma _ { 1 , - 2 } [ r ] ( j ) + \\Gamma _ { 1 , 0 } [ r ] ( j ) \\right ) = \\dfrac { 4 V _ { k , l } } { \\pi ^ 2 } \\Gamma _ { 0 , 0 } [ s ] ( i ) \\Gamma _ { 0 , - 2 } [ r ] ( j ) . \\end{align*}"}
{"id": "487.png", "formula": "\\begin{align*} \\mathcal { N } ( \\mathfrak { c } _ { c } ) = \\mathcal { I } \\setminus \\mathfrak { c } _ { r } , \\mathcal { N } ( \\mathfrak { c } _ { r } ) = \\mathcal { I } ^ { \\mathtt { C } } \\setminus \\mathfrak { c } _ { c } . \\end{align*}"}
{"id": "1355.png", "formula": "\\begin{align*} \\norm { u } _ { L ^ \\alpha ( E ) } : = \\left ( \\int _ E | u ( x ) | ^ \\alpha d x \\right ) ^ \\frac { 1 } { \\alpha } , \\ ; \\norm { u } _ { E , \\alpha } : = \\left ( \\frac { 1 } { | E | } \\int _ E | u ( x ) | ^ \\alpha d x \\right ) ^ \\frac { 1 } { \\alpha } . \\end{align*}"}
{"id": "484.png", "formula": "\\begin{align*} Y _ { \\alpha \\beta } ^ { i j } Y _ { \\beta \\gamma } ^ { i j } = - Y _ { \\alpha \\gamma } ^ { i j } , Y _ { \\alpha \\beta } ^ { i m } \\cdot Y _ { \\alpha \\beta } ^ { m j } = - Y _ { \\alpha \\beta } ^ { i j } . \\end{align*}"}
{"id": "3521.png", "formula": "\\begin{align*} & = \\frac { s _ 3 } { ( s _ 1 + s _ 3 - 1 ) ( s _ 1 + s _ 3 ) ( a t _ 3 ) ^ { s _ 1 + s _ 3 } } \\sum _ { n \\leq a t _ 3 } \\frac { 1 } { n ^ { s _ 2 - 1 } } \\\\ & + \\frac { s _ 3 } { 2 \\pi i ( s _ 1 + s _ 3 - 1 ) \\Gamma ( s _ 3 + 1 ) } \\int _ { \\left ( \\frac { 1 } { 2 } \\right ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { ( s _ 1 + s _ 3 + z ) ( a t _ 3 ) ^ { s _ 1 + s _ 3 + z } } \\left ( \\sum _ { n \\leq a t _ 3 } \\frac { 1 } { n ^ { s _ 2 - 1 - z } } \\right ) d z . \\end{align*}"}
{"id": "4268.png", "formula": "\\begin{align*} w _ t + a _ n ( t , x ) \\cdot w _ x ~ = ~ F ^ { ( k ) } ( t , x ) , w ( 0 , \\cdot ) ~ = ~ \\overline { w } ( \\cdot ) \\end{align*}"}
{"id": "9556.png", "formula": "\\begin{align*} R _ { 0 0 } + R _ { 0 1 } + R _ { 0 2 } + R _ { 0 3 } & = R _ { 0 } , \\\\ R _ { 1 1 } + R _ { 1 2 } + R _ { 1 3 } & = \\log | \\mathcal { D } _ 1 ( K _ 0 ) | , \\\\ R _ { 2 1 } + R _ { 2 2 } + R _ { 2 3 } & = \\log | \\mathcal { D } _ 2 ( K _ 0 ) | , \\\\ R _ { 3 1 } + R _ { 3 2 } + R _ { 3 3 } & = \\log | \\mathcal { D } _ 3 ( K _ 0 ) | . \\end{align*}"}
{"id": "4522.png", "formula": "\\begin{align*} x _ a ( b ) = \\frac { 2 a b - ( a , b ) a - b } { ( a , b ) - 1 } = \\frac { a _ 0 ( b ) } { 1 - ( a , b ) } . \\end{align*}"}
{"id": "6391.png", "formula": "\\begin{align*} \\pi _ n ( z ) = ( s _ 1 ( z ) , \\dots , s _ { n - 1 } ( z ) , p ( z ) ) , z = ( z _ 1 , \\dots , z _ n ) , \\end{align*}"}
{"id": "8726.png", "formula": "\\begin{align*} \\sum _ { \\gamma } c _ \\gamma z _ \\gamma = \\sum _ { \\gamma } c _ \\gamma x ^ \\gamma = p ( x ) \\geq 0 z \\in M _ { ( n , d ) } , \\end{align*}"}
{"id": "3821.png", "formula": "\\begin{align*} E ^ { [ \\sigma ] } ( f , G \\rtimes S ) ( t , \\overline { t } ) = ( - 1 ) ^ n E ^ { [ \\sigma ] } ( \\widetilde { f } , \\widetilde { G } \\rtimes S ) ( t ^ { - 1 } , \\overline { t } ) \\ , . \\end{align*}"}
{"id": "451.png", "formula": "\\begin{align*} \\mathcal { E } _ { m } ^ { 2 } ( u , v , w ) : = \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 1 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle + \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 2 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle + \\sum _ { | \\alpha | = 0 } ^ { m } \\langle A _ { 3 } ^ { 0 } \\partial _ { x } ^ { \\alpha } u , \\partial _ { x } ^ { \\alpha } u \\rangle . \\end{align*}"}
{"id": "2929.png", "formula": "\\begin{align*} \\mathcal { I } _ \\ell : = \\mathcal { I } _ { \\ell , n } : = \\left \\{ \\mathbf { i } = ( i _ 1 , i _ 2 , i _ 3 , i _ 4 ) \\in \\mathcal J : | \\mathbf i | = \\ell \\right \\} , \\ell \\in \\{ 2 , 3 , 4 \\} , \\end{align*}"}
{"id": "2789.png", "formula": "\\begin{align*} B ( g , h ) = \\frac { 1 } { 2 } \\int ( L _ + g _ 1 ) h _ 1 d x + \\frac { 1 } { 2 } \\int ( L _ - g _ 2 ) h _ 2 d x . \\end{align*}"}
{"id": "1100.png", "formula": "\\begin{align*} m ^ { m o d } ( k ) = m ^ { P C } _ { \\eta } ( \\zeta ( k ) , \\xi ) + m ^ { P C } _ { - \\eta } ( \\zeta ( k ) , \\xi ) + \\mathcal { O } ( t ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "3778.png", "formula": "\\begin{align*} \\overline { \\mathbb { K } } _ { \\psi _ E } ^ { \\pi _ E } ( w ) \\xi \\{ \\chi , 0 \\} = r _ l \\big ( \\epsilon ( \\widetilde { \\chi } _ 0 ^ { - 1 } \\pi _ E , \\psi _ E ) \\big ) \\xi \\bigg \\{ \\chi , \\cfrac { - n ( \\widetilde { \\chi } _ 0 ^ { - 1 } \\pi _ E , \\psi _ E ) } { e } \\bigg \\} \\end{align*}"}
{"id": "6000.png", "formula": "\\begin{align*} \\sum _ { n = m } ^ { \\infty } S ( n , m ) \\frac { z ^ { n } } { n ! } & = \\frac { ( \\mathrm { e } ^ { z } - 1 ) ^ { m } } { m ! } \\\\ & = \\frac { 1 } { m ! } \\left ( \\sum _ { l = 1 } ^ { \\infty } \\frac { z ^ { l } } { l ! } \\right ) ^ { m } \\\\ & = \\frac { 1 } { m ! } \\sum _ { n = m } ^ { \\infty } \\frac { z ^ { n } } { n ! } \\underbrace { \\sum _ { l _ { 1 } + \\dots + l _ { m } = n } \\frac { n ! } { l _ { 1 } ! \\dots l _ { m } ! } } _ { = : B _ { n } ^ { m } } \\\\ & = \\sum _ { n = m } ^ { \\infty } \\frac { B _ { n } ^ { m } } { m ! } \\frac { z ^ { n } } { n ! } . \\end{align*}"}
{"id": "5957.png", "formula": "\\begin{align*} F _ { j k } = \\int _ { \\Gamma ^ { b o d y } } p _ k n _ { j } \\ d \\Gamma , \\mathcal { T } , \\end{align*}"}
{"id": "4858.png", "formula": "\\begin{align*} ( K _ \\alpha * f ) '' ( x ) = \\int _ { \\R } \\big [ f ( x + z ) - f ( x ) - z f ' ( x ) \\big ] \\ , K _ \\alpha '' ( z ) d z . \\end{align*}"}
{"id": "562.png", "formula": "\\begin{align*} \\int _ { \\Omega _ R } | b ' ( \\rho ) | ^ { \\beta _ 0 - 1 } \\leq & c \\int _ { \\{ \\rho > \\overline \\rho - \\alpha _ 1 \\} } P ( \\rho ) \\leq c \\int _ { \\{ \\rho > \\overline \\rho - \\alpha _ 1 \\} } \\left ( P ( \\rho ) - P ( r ) - P ' ( r ) ( \\rho - r ) \\right ) , \\\\ \\int _ { \\Omega _ R } | b ' ( \\rho ) | ^ { \\beta _ 0 } \\leq & c \\int _ { \\Omega _ R } p ( \\rho ) . \\end{align*}"}
{"id": "9019.png", "formula": "\\begin{align*} \\alpha \\phi + \\beta \\epsilon ( x ) \\frac { \\partial \\phi } { \\partial \\mathbf { n } } = \\phi ^ b , x \\in \\partial \\Omega . \\end{align*}"}
{"id": "8407.png", "formula": "\\begin{align*} J _ 4 = \\int _ 0 ^ t \\ ! \\int _ { \\Gamma _ { t } ( 2 \\delta ) } ( \\zeta \\circ d _ \\Gamma ) \\mathbf { w } | _ \\Gamma \\theta _ { 0 } ^ { \\prime } ( \\rho ) ( \\mathbf { n } - \\varepsilon \\nabla _ \\tau h _ { \\varepsilon } ) ( \\mu _ { \\varepsilon } - \\mu _ A ) \\ , \\mathrm { d } x \\ , \\mathrm { d } \\varsigma \\triangleq \\ ; J _ { 4 1 } + J _ { 4 2 } , \\end{align*}"}
{"id": "7646.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\nabla \\bar { u } \\cdot \\nabla v = \\tilde { \\lambda } _ 0 \\int _ { \\R ^ n } \\tilde { m } _ 0 \\bar { u } v \\forall v \\in H _ 0 ^ 1 ( T ) \\ ; . \\end{align*}"}
{"id": "4856.png", "formula": "\\begin{align*} Q _ \\alpha ( - 1 ) = 2 \\int _ { - 1 } ^ 0 Q _ \\alpha ( - 1 ) x \\ , d x \\geq 2 \\int _ { - 1 } ^ 0 Q _ \\alpha ( x ) \\ , d x = 2 \\frac { - P ( - \\alpha ) } { P ( \\alpha ) } \\geq 2 , \\end{align*}"}
{"id": "4558.png", "formula": "\\begin{align*} ( m _ { 1 , i } - 1 ) \\alpha _ r & = ( i - 1 ) \\left ( ( r - 1 ) - \\frac { i } { 2 } \\right ) \\frac { 1 } { r - 2 } \\left ( \\frac { 2 } { r - 1 } + \\frac { 2 } { r ^ 2 ( r - 1 ) } \\right ) \\\\ & = \\frac { i - 1 } { r - 2 } \\left ( 2 - \\frac { i } { r - 1 } + \\frac { 2 } { r ^ 2 } - \\frac { i } { r ^ 2 ( r - 1 ) } \\right ) . \\end{align*}"}
{"id": "1556.png", "formula": "\\begin{align*} \\Lambda ( \\alpha , w ) p ( x , w ) = p ( x \\alpha , w ) , \\ , \\ , \\ , \\ , \\ , \\ , \\alpha \\in h ( Y ^ 1 ) , \\ , \\ , \\ , x \\in \\C ^ { 2 n } \\end{align*}"}
{"id": "7424.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\sum _ { z = x - \\ell } ^ { x - 1 } \\int _ { \\Omega - \\Omega _ 2 ^ j ( x ) } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta ) - f ( \\eta ^ { z , z - j \\ell } ) ] d \\nu _ { b } \\Big | \\end{align*}"}
{"id": "3674.png", "formula": "\\begin{align*} g _ A = \\left ( ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 4 ) , ( 4 , 1 ) \\right ) \\left ( ( 2 , 1 ) , ( 3 , 2 ) , ( 4 , 3 ) , ( 1 , 4 ) \\right ) \\left ( ( 1 , 3 ) , ( 2 , 4 ) , ( 3 , 1 ) , ( 4 , 2 ) \\right ) . \\end{align*}"}
{"id": "168.png", "formula": "\\begin{align*} \\| f \\| _ { 1 , p , \\mu } = \\| f \\| _ { L ^ p ( \\mu ) } + \\sum _ { k = 1 } ^ d \\| \\partial _ k ( f ) \\| _ { L ^ p ( \\mu ) } , \\end{align*}"}
{"id": "6877.png", "formula": "\\begin{align*} R _ \\eta ( M ) = | \\{ W _ k ^ i : \\exists f \\in V ^ i , M f = \\lambda f , \\lambda > \\eta \\} | . \\end{align*}"}
{"id": "678.png", "formula": "\\begin{align*} \\kappa _ { 2 k ; \\alpha } ^ { ( \\ell ) } : = \\kappa _ { ( \\alpha \\alpha ) \\cdots ( \\alpha \\alpha ) } ^ { ( 0 0 ) \\cdots ( 0 0 ) , ( \\ell ) } = \\frac { 1 } { ( 2 k - 1 ) ! ! } \\kappa \\bigg ( \\underbrace { z _ { i ; \\alpha } ^ { ( \\ell + 1 ) } , \\ldots , z _ { i ; \\alpha } ^ { ( \\ell + 1 ) } } _ { 2 k } \\bigg ) \\end{align*}"}
{"id": "553.png", "formula": "\\begin{align*} s ( 0 , \\cdot ) = \\rho ^ { ( 1 ) } _ 0 , \\ \\nabla \\Psi ( 0 , \\cdot ) = \\nabla \\Psi _ 0 = u _ 0 - H ( u _ 0 ) . \\end{align*}"}
{"id": "7765.png", "formula": "\\begin{gather*} \\Box \\tilde \\phi = \\left ( | \\tilde \\phi _ { t } | ^ 2 - | \\tilde \\phi _ { x } | ^ 2 \\right ) \\tilde \\phi + \\mathbf { 1 } _ { \\omega } \\tilde f ^ { \\tilde \\phi ^ { \\perp } } , \\\\ \\tilde \\phi [ 0 ] = ( a , b ) \\textrm { a n d } \\tilde \\phi [ T ] = ( p , 0 ) . \\end{gather*}"}
{"id": "1664.png", "formula": "\\begin{align*} R _ 0 = R _ 0 ( \\alpha , \\alpha _ 0 ) : = \\sup \\big \\{ r ( { \\bf y } ) : { \\bf y } \\in \\mathcal A \\big \\} . \\end{align*}"}
{"id": "4316.png", "formula": "\\begin{align*} \\hat { \\varepsilon } _ j ( \\tau ) = \\| \\phi _ { j , \\infty } \\| ^ { - 2 } _ \\rho \\langle \\varepsilon , \\phi _ { j , \\infty } \\rangle _ { L ^ 2 _ \\rho } , \\end{align*}"}
{"id": "9270.png", "formula": "\\begin{align*} x \\in ( 0 , \\pi / 2 ) \\mapsto \\partial \\varphi ( x ) : = \\left \\{ \\frac { 1 } { \\cos ^ 2 x } \\right \\} . \\end{align*}"}
{"id": "7004.png", "formula": "\\begin{align*} \\delta = \\min \\{ | A ( \\eta ) | , | A ( 1 - \\eta ) | , | B ( 0 ) | , | B ( 1 ) | \\} = \\eta ( 1 - \\eta ) . \\end{align*}"}
{"id": "1329.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , N ) } , \\hat { \\alpha } \\cup { ( \\gamma , N - p _ { i } ) } \\cup { ( \\delta _ { 1 } , 1 ) } \\cup { ( \\delta _ { 2 } , 1 ) } ) = 2 . \\end{align*}"}
{"id": "1643.png", "formula": "\\begin{align*} G _ \\lambda f ( x ) = \\int _ M g _ \\lambda ( x , y ) f ( y ) \\mu ( d y ) , x \\in M , \\end{align*}"}
{"id": "7073.png", "formula": "\\begin{align*} S = & \\sum _ { \\ell = - \\lfloor 1 / 2 \\delta _ K \\rfloor } ^ { 1 / \\delta _ K - 1 - \\lfloor 1 / 2 \\delta _ K \\rfloor } h _ K G \\big ( h _ K \\ell \\big ) e ^ { h _ K \\ell \\partial _ x \\varphi ( t _ K , x _ K ) + C \\frac { ( h _ K \\ell ) ^ 2 } { \\log K } } . \\end{align*}"}
{"id": "6089.png", "formula": "\\begin{align*} \\widetilde { h } ^ { j } = D ( \\zeta ) ^ q g ^ j ( \\zeta ) \\quad \\textrm { w i t h } D ( \\zeta ) = { \\prod _ { i = 1 } ^ { n } ( \\zeta - \\zeta _ { i } ) } . \\end{align*}"}
{"id": "208.png", "formula": "\\begin{align*} T _ t ^ { \\alpha } ( f ) \\geq 0 , T _ t ^ { \\alpha } ( 1 ) = e ^ { t d } , T _ t ^ { \\alpha } ( p _ \\alpha ) = p _ \\alpha . \\end{align*}"}
{"id": "146.png", "formula": "\\begin{align*} \\nu _ { \\alpha , d } ( d u ) = \\sum _ { k = 1 } ^ d \\delta _ 0 ( d u _ 1 ) \\otimes \\dots \\otimes \\delta _ 0 ( d u _ { k - 1 } ) \\otimes \\nu _ { \\alpha , 1 } ( d u _ k ) \\otimes \\delta _ 0 ( d u _ { k + 1 } ) \\otimes \\dots \\otimes \\delta _ 0 ( d u _ d ) , \\end{align*}"}
{"id": "6729.png", "formula": "\\begin{align*} g _ 1 ( \\theta ^ { q ^ N } ) \\sum _ { N > i _ 2 > 0 } \\frac { \\alpha ^ { q ^ N } } { ( \\theta ^ { q ^ { i _ 2 } } - \\theta ^ { q ^ N } ) ^ { w - s } } + g _ 2 ( \\theta ^ { q ^ N } ) L i _ { K , w - s } ( \\alpha ) ^ { q ^ N } = 0 . \\end{align*}"}
{"id": "105.png", "formula": "\\begin{align*} U = \\{ u \\in A \\ ; | \\ ; 2 e u = u \\} , \\ ; \\ ; V = \\{ v \\in A \\ ; | \\ ; e v = 0 \\} . \\end{align*}"}
{"id": "265.png", "formula": "\\begin{align*} & - \\infty < \\alpha = \\min _ { M } \\theta _ { V } ^ { ( \\omega ) } \\leqq \\max _ { M } \\theta _ { V } ^ { ( \\omega ) } < \\beta \\leqq + \\infty \\\\ & \\dot { \\sigma } \\leqq { 0 } \\leqq \\ddot { \\sigma } , \\\\ & \\lim _ { t \\to \\alpha + 0 } \\sigma ( t ) = + \\infty . \\end{align*}"}
{"id": "7150.png", "formula": "\\begin{align*} \\tilde { f } ( s , y ) & = \\alpha _ { - 1 } ( - \\tilde { \\theta } y ^ { 2 \\tilde { \\theta } + 1 } ) + \\alpha _ 0 y ^ { \\tilde { \\theta } + 1 } - \\alpha _ 1 \\frac { y } { \\tilde { \\theta } } + \\alpha _ 2 \\frac { 1 } { \\tilde { \\theta } ^ { \\rho } } y ^ { - \\tilde { \\theta } \\rho + \\tilde { \\theta } + 1 } \\end{align*}"}
{"id": "6535.png", "formula": "\\begin{align*} \\dfrac { S _ n } { I _ n } = 1 + O ( ( I _ n ) ^ { - 1 } ) \\mbox { a s $ n \\to \\infty $ . } \\end{align*}"}
{"id": "1253.png", "formula": "\\begin{align*} U _ { x } ( y ) : = \\{ z \\in H : P _ { [ x , y ] } z \\neq y \\} , \\end{align*}"}
{"id": "1715.png", "formula": "\\begin{align*} \\nu _ * = \u2010 \\mu _ * \u2010 \\left ( \\frac { \\gamma _ * } { q } \u2010 \\frac { \\gamma _ * } { p _ 1 } \\right ) _ + . \\end{align*}"}
{"id": "3084.png", "formula": "\\begin{align*} x _ 1 ^ 3 y _ 1 ^ 2 y _ 2 + x _ 1 ^ 2 x _ 2 y _ 2 ^ 3 + x _ 1 x _ 2 y _ 2 ^ 3 + x _ 1 x _ 2 ^ 2 y _ 1 ^ 3 + a ^ 5 x _ 2 ^ 3 y _ 1 y _ 2 ^ 2 = 0 \\ , , \\end{align*}"}
{"id": "6622.png", "formula": "\\begin{align*} J _ { 2 } = J _ { 2 1 } + J _ { 2 2 } + J _ { 2 3 } . \\end{align*}"}
{"id": "3621.png", "formula": "\\begin{align*} a _ 1 ( t ) & : = \\frac { 0 . 0 1 9 6 } { 0 . 0 4 9 6 2 ( J ( t ) + 1 . 1 5 ) } \\leq \\frac { 6 \\cdot 0 . 0 1 9 6 } { 0 . 0 4 9 6 2 \\log t } \\leq \\frac { 0 . 5 2 6 \\log \\log t } { \\log t } , \\\\ a _ 2 ( t ) & : = \\frac { 6 ( 1 . 1 5 5 \\log \\log t + \\log ( 0 . 7 7 ) + 0 . 6 8 5 ) } { \\log t } \\leq \\frac { 7 . 4 9 4 \\log \\log t } { \\log t } . \\end{align*}"}
{"id": "9273.png", "formula": "\\begin{align*} p = P _ C x \\left ( p \\in C \\forall q \\in C \\left ( \\langle q - p , x - p \\rangle \\leq 0 \\right ) \\right ) . \\end{align*}"}
{"id": "6505.png", "formula": "\\begin{align*} & h ^ { ( 2 m ) } _ n : = \\frac { 1 } { s ^ { ( 2 m ) } _ { n + 1 } } \\left \\{ 1 + \\ , \\sum ^ m _ { \\ell = 1 } \\left \\{ \\binom { 2 m } { 2 \\ell } + \\frac { \\alpha } { n } \\binom { 2 m } { 2 \\ell - 1 } \\right \\} s ^ { ( 2 \\ell ) } _ n \\right \\} - 1 , \\end{align*}"}
{"id": "7180.png", "formula": "\\begin{align*} x = ( x _ 1 , x ^ \\perp ) . \\end{align*}"}
{"id": "6902.png", "formula": "\\begin{align*} R ( k , i ) = \\frac { \\prod _ { j = i + 1 } ^ k R ( j ) } { m ( k , i ) } . \\end{align*}"}
{"id": "7610.png", "formula": "\\begin{align*} 1 - \\frac { \\sigma _ { k - 1 } ( \\kappa | i ) \\kappa _ { i } } { \\sigma _ { k } ( \\kappa ) } = \\frac { \\sigma _ { k } ( \\kappa | i ) } { \\sigma _ { k } ( \\kappa ) } \\geq 0 . \\end{align*}"}
{"id": "2593.png", "formula": "\\begin{align*} V ^ * _ g F = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "9434.png", "formula": "\\begin{align*} u \\ ! + \\ ! \\varPsi _ { \\mu , q } B _ q ( w , u ) = v ^ { ( 0 ) } . \\end{align*}"}
{"id": "8056.png", "formula": "\\begin{align*} \\Psi ^ { P } ( f ) [ \\psi ] = \\int _ { \\mathbb { R } } f ( x ) P ( \\phi ( x ) , \\partial _ x \\phi ( x ) , \\ldots , \\partial _ x ^ n \\phi ( x ) ) \\ , \\mathrm { d } x , \\end{align*}"}
{"id": "1990.png", "formula": "\\begin{align*} & m _ 1 ( t ) = h _ 1 t , \\\\ & m _ 2 ( t ) = h _ 2 t + h _ 1 ^ 2 t ^ 2 , \\\\ & m _ 3 ( t ) = h _ 3 t + 5 h _ 1 h _ 2 \\frac { t ^ 2 } { 2 } + h _ 1 ^ 3 t ^ 3 , \\\\ & m _ 4 ( t ) = h _ 4 t + \\left ( 3 h _ 1 h _ 3 + \\frac { 3 } { 2 } h _ 2 ^ 2 \\right ) t ^ 2 + \\frac { 1 3 } { 3 } h _ 1 ^ 2 h _ 2 t ^ 3 + h _ 1 ^ 4 t ^ 4 . \\end{align*}"}
{"id": "3332.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\mid n \\\\ m \\neq n / d _ p } } \\nu _ p ( \\Phi _ m ( \\beta ) ) \\le \\sum _ { m \\mid n } \\nu _ p ( m ) + \\sum ^ 7 _ { m = 1 } \\nu _ p ( \\Phi _ m ( \\beta ) ) . \\end{align*}"}
{"id": "8541.png", "formula": "\\begin{align*} w ( t , k ) : = \\mathcal { F } f ( t , k ) \\exp \\Big ( i \\int _ 0 ^ t \\big | \\mathcal { F } f ( s , k ) \\big | ^ 2 \\ , \\frac { d s } { 2 ( s + 1 ) } \\Big ) , | \\mathcal { F } f ( t , k ) | = | w ( t , k ) | . \\end{align*}"}
{"id": "7315.png", "formula": "\\begin{align*} \\begin{aligned} u _ { \\star , \\lambda } ( x , t ) = \\inf \\bigg \\{ \\max \\{ u ( y , t ) , u ( z , t ) \\} : x = \\lambda y + & ( 1 - \\lambda ) z \\bigg \\} , \\\\ & ( x , t ) \\in \\R ^ n \\times ( 0 , \\infty ) , \\end{aligned} \\end{align*}"}
{"id": "4067.png", "formula": "\\begin{align*} ( \\lambda I - A ^ * ) X _ n = Y _ n . \\end{align*}"}
{"id": "4121.png", "formula": "\\begin{align*} \\pi ( p ' ) = \\sum _ { i = 1 } ^ d \\alpha _ i \\pi ( p _ i ) , h ( p ' ) < \\sum _ { i = 1 } ^ d \\alpha _ i h ( p _ i ) . \\end{align*}"}
{"id": "8748.png", "formula": "\\begin{align*} \\begin{aligned} & w _ { 1 2 3 } \\geq 0 , w _ { 2 3 } - w _ { 1 2 3 } \\geq 0 , w _ { 1 3 } - w _ { 1 2 3 } \\geq 0 , w _ { 1 2 } - w _ { 1 2 3 } \\geq 0 , \\\\ & \\lambda _ 1 - w _ { 1 2 } - w _ { 1 3 } + w _ { 1 2 3 } \\geq 0 , \\lambda _ 2 - w _ { 1 2 } - w _ { 2 3 } + w _ { 1 2 3 } \\geq 0 , \\lambda _ { 3 } - w _ { 1 3 } - w _ { 2 3 } + w _ { 1 2 3 } \\geq 0 , \\\\ & \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 3 - w _ { 1 2 } - w _ { 2 3 } - w _ { 1 3 } + w _ { 1 2 3 } \\leq 0 . \\end{aligned} \\end{align*}"}
{"id": "6056.png", "formula": "\\begin{align*} \\texttt { ( C p : 0 . 5 7 0 3 3 3 , } & \\texttt { E t : 0 . 5 7 0 3 3 3 , T g : 0 . 5 7 0 3 3 3 , T t : 0 . 5 7 0 3 3 3 , } \\\\ & \\texttt { ( P f : 0 . 4 3 8 6 2 , P v : 0 . 4 3 8 6 2 ) : 0 . 1 3 1 7 1 3 , ( B b : 0 . 5 7 0 3 3 , T a : 0 . 5 7 0 3 3 ) : 0 . 0 0 0 0 0 3 ) } \\enspace , \\end{align*}"}
{"id": "2527.png", "formula": "\\begin{align*} f = \\sum _ { k = 1 } ^ n c _ k \\pi ( \\mathbf { h _ k } ) g \\in \\mathcal { H } _ g , \\end{align*}"}
{"id": "3688.png", "formula": "\\begin{align*} \\mathcal H f = \\frac { 1 } { \\pi } P . V . \\int _ { - \\infty } ^ { \\infty } \\frac { f ( y ) } { x - y } \\ , d y . \\end{align*}"}
{"id": "8154.png", "formula": "\\begin{align*} s ( 1 , d ) = \\frac { d ^ 2 - 3 \\vert d \\vert + 2 } { 1 2 d } s ( 2 , d ) = \\frac { d ^ 2 - 6 \\vert d \\vert + 5 } { 2 4 d } \\ \\ \\ \\ \\ ( d \\in { \\mathbb Z } \\setminus \\{ 0 \\} ) . \\end{align*}"}
{"id": "2037.png", "formula": "\\begin{align*} \\ell _ { \\beta } ( x ) : = \\left \\{ \\begin{array} { l l } x \\log | x | - x , & \\beta = - 1 \\\\ \\frac { 1 } { \\beta + 1 } \\int _ 0 ^ x | y | ^ { \\beta + 1 } { \\rm d } y , & \\beta \\in \\ , ] \\ , - 3 / 2 , \\ , + \\infty \\ , [ \\ , \\setminus \\ , \\{ \\pm 1 \\} \\end{array} \\right . , \\end{align*}"}
{"id": "6229.png", "formula": "\\begin{align*} \\overline { \\zeta } \\left ( A , T ^ d _ N , u \\right ) = \\det \\Big ( I _ { d _ c N ^ d } - u M _ A \\Big ) ^ { - 1 / N ^ d } . \\end{align*}"}
{"id": "8209.png", "formula": "\\begin{align*} F _ { k , i + 1 } ( t ) = F _ { k , i } ( t ) + ( x _ { k + 1 - i } - x _ i ) t G _ { k , i + 1 } ( t ) = G _ { k , i } ( t ) + ( x _ { k + 1 - i } - x _ { i + 1 } ) t . \\end{align*}"}
{"id": "5337.png", "formula": "\\begin{align*} \\partial _ t P ' ( \\rho ^ { \\epsilon } ) = - u ^ { \\epsilon } \\cdot \\nabla P ' ( \\rho ^ { \\epsilon } ) - P '' ( \\rho ^ { \\epsilon } ) \\rho ^ { \\epsilon } \\mathrm { d i v } u ^ { \\epsilon } . \\end{align*}"}
{"id": "2213.png", "formula": "\\begin{align*} \\big < \\Gamma _ 1 , \\Gamma _ 2 \\big > _ { \\mathcal { L } _ 2 ( H ) } = \\sum _ { j = 1 } ^ \\infty \\big < \\Gamma _ 1 \\phi _ j , \\Gamma _ 2 \\phi _ j \\big > , \\| \\Gamma \\| ^ 2 _ { \\mathcal { L } _ 2 ( H ) } = \\sum _ { j = 1 } ^ \\infty \\big \\| \\Gamma \\phi _ j \\| ^ 2 , \\end{align*}"}
{"id": "5517.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X ( t ) & = & \\mu X ( t ) d t + \\sigma X ( t ) d W ( t ) \\\\ X ( 0 ) & = & x \\end{array} \\right . \\end{align*}"}
{"id": "8030.png", "formula": "\\begin{align*} ( \\rho ^ { * } _ { ( 1 ) } \\otimes \\rho ^ { * } _ { ( 1 ) } ) E _ { \\widetilde { \\Sigma } } = ( \\partial _ { \\Sigma } \\otimes \\partial _ { \\Sigma } ) E _ { \\mathcal { M } } = E _ { \\Sigma } , \\end{align*}"}
{"id": "8384.png", "formula": "\\begin{align*} \\P ( | U _ { \\mathsf k } | \\geq N \\mbox { f o r a l l } \\mathsf k = \\mathsf 1 , \\ldots , \\mathsf q ) \\geq 1 - n ^ { - 4 } \\ , , \\mbox { w h e r e } N \\stackrel { \\Delta } { = } \\frac { n ^ d } { q ( 1 6 r ) ^ d } \\ , . \\end{align*}"}
{"id": "3893.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq a < p \\\\ \\gcd ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } = 0 \\end{align*}"}
{"id": "713.png", "formula": "\\begin{align*} d _ i z _ { i ; \\alpha } ^ { ( \\ell ) } : = \\partial _ { x _ { i ; \\alpha } } z _ { i ; \\alpha } ^ { ( \\ell ) } , i = 1 , \\ldots , n _ 0 . \\end{align*}"}
{"id": "3678.png", "formula": "\\begin{align*} C = ( a _ 1 , a _ 2 , \\ldots , a _ k , \\overline { a _ 1 } , \\overline { a _ 2 } , \\ldots , \\overline { a _ k } ) \\end{align*}"}
{"id": "2653.png", "formula": "\\begin{align*} \\varphi ( k , l ) & = \\varphi ( 0 , l ) + k \\cdot l + \\kappa ( k , 0 ) \\\\ & = \\left ( \\varphi ( 0 , 0 ) + \\kappa ( 0 , l ) \\right ) + k \\cdot l + \\kappa ( k , 0 ) \\end{align*}"}
{"id": "8203.png", "formula": "\\begin{align*} N _ 3 ( f , H ) = - f - 3 S ( H , f ) + \\frac { 9 } { 2 } S ( H _ 3 , 3 f ) = \\frac { c _ a '' + 1 } { 4 } = \\begin{cases} - 2 a - 1 & \\hbox { i f $ a \\equiv 0 \\pmod 3 $ } , \\\\ 2 a + 1 & \\hbox { i f $ a \\equiv 2 \\pmod 3 $ } , \\end{cases} \\end{align*}"}
{"id": "6234.png", "formula": "\\begin{align*} & \\det ( I _ { n + 2 M } - u { \\cal A } ) = ( 1 + 2 ( \\beta + 2 \\delta ) u + 4 \\delta ( \\delta + 1 ) u ^ 2 ) ^ { M - N } \\\\ & \\times \\det ( ( 1 + \\delta u ) ( 1 + 2 ( \\beta + 2 \\delta ) u + 4 \\delta ( \\delta + 1 ) u ^ 2 ) I _ N \\\\ & - ( 1 + 2 \\delta u ) \\beta u ( 1 + ( \\beta + 2 \\delta ) u ) { \\bf A } ( G ) + \\beta { } ^ 2 ( 1 + 2 \\delta u ) u ^ 2 { \\bf D } _ G ) . \\end{align*}"}
{"id": "5986.png", "formula": "\\begin{align*} l _ { \\pi _ { \\lambda , \\beta } } ( f ( z ) ) & = E _ { \\beta } ( \\lambda z ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { ( \\lambda z ) ^ { m } } { \\Gamma ( m \\beta + 1 ) } \\\\ & = \\sum _ { m = 0 } ^ { \\infty } \\frac { z ^ { m } } { m ! } \\underbrace { \\frac { \\lambda ^ { m } m ! } { \\Gamma ( m \\beta + 1 ) } } _ { = : \\tilde { M } _ { \\lambda , \\beta } ( m ) } = \\sum _ { m = 0 } ^ { \\infty } \\frac { z ^ { m } } { m ! } \\tilde { M } _ { \\lambda , \\beta } ( m ) . \\end{align*}"}
{"id": "9112.png", "formula": "\\begin{align*} y _ { i } = \\left \\langle \\mathcal { X } _ { i } , \\mathcal { A } ^ { * } \\right \\rangle + \\varepsilon _ { i } \\quad \\quad \\mathbb { E } \\left ( \\varepsilon _ { i } \\mid \\mathcal { X } _ { i } \\right ) = 0 \\quad M _ { \\delta } = \\sqrt [ k ] { \\mathbb { E } \\left ( \\mathbb { E } \\left ( | \\varepsilon _ { i } | ^ { 1 + \\delta } | \\mathcal { X } _ { i } \\right ) ^ k \\right ) } < \\infty \\end{align*}"}
{"id": "3552.png", "formula": "\\begin{align*} S = \\{ ( r x , ( r a - r ^ { - 1 } ) v ) : r > 0 , \\ , x \\in \\Sigma \\} . \\end{align*}"}
{"id": "120.png", "formula": "\\begin{align*} x ^ i x ^ j = 0 \\mbox { f o r a l l } x \\in N \\mbox { a n d } i , j \\geq 2 . \\end{align*}"}
{"id": "8204.png", "formula": "\\begin{align*} S ( H _ 6 , 6 f ) = \\frac { 1 0 f + c _ a ''' } { 1 8 } \\hbox { w h e r e } c _ a ''' : = \\begin{cases} - 1 9 a - 1 0 & \\hbox { i f $ a \\equiv 0 \\pmod 6 $ } , \\\\ a - 1 8 & \\hbox { i f $ a \\equiv 2 \\pmod 6 $ } , \\\\ - a - 1 9 & \\hbox { i f $ a \\equiv 3 \\pmod 6 $ } , \\\\ 1 9 a + 9 & \\hbox { i f $ a \\equiv 5 \\pmod 6 $ } , \\end{cases} \\end{align*}"}
{"id": "798.png", "formula": "\\begin{align*} \\mu _ \\beta ( A ) : = \\int _ A e ^ { - \\beta d ( x , v _ 0 ) } \\nu ( B _ Z ( \\zeta _ x , \\alpha ^ { - n _ x } ) ) \\ , d \\mathcal { H } ^ 1 ( x ) , \\end{align*}"}
{"id": "866.png", "formula": "\\begin{align*} Q \\left ( x \\right ) = \\int _ { x } ^ { \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left ( - \\frac { t ^ 2 } { 2 } \\right ) d t . \\end{align*}"}
{"id": "8002.png", "formula": "\\begin{align*} T _ n ^ { \\sigma } + \\sum _ { k = n + 1 } ^ { d } C _ { k } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\bigg ( H ( \\mathbf { p } _ { \\sigma _ { k } } ) + \\int \\ ! \\varphi _ k ^ { \\sigma } \\ , \\mathrm { d } \\mathbf { p } _ { \\sigma _ k } - \\sum _ { \\ell = 1 } ^ { n } \\chi _ { \\ell } ^ { \\sigma } ( \\mathbf { p } _ { \\sigma _ k } ) \\big ( T _ { \\ell } ^ { \\sigma } - T _ { \\ell - 1 } ^ { \\sigma } \\big ) \\bigg ) , \\end{align*}"}
{"id": "5372.png", "formula": "\\begin{align*} e ^ { ( i ) } = \\frac { c ^ { ( i ) } } { \\| c ^ { ( i ) } \\| } \\ ; ( i = 1 , \\ldots , n - 1 ) . \\end{align*}"}
{"id": "4807.png", "formula": "\\begin{align*} J ' ( \\alpha , \\beta ) = \\sum _ { t \\neq 0 , 1 } \\alpha ( t ) \\beta ( 1 - t ) \\end{align*}"}
{"id": "3630.png", "formula": "\\begin{align*} C ' = \\min _ { k \\in \\{ 0 , \\ldots , K - 1 \\} } C _ k \\end{align*}"}
{"id": "6050.png", "formula": "\\begin{align*} \\begin{array} { l l @ { } l l } & \\displaystyle n \\cdot ( t _ { 1 } + \\dots + t _ { m } ) & \\\\ & \\displaystyle v _ { i j } - x _ j \\ \\leq \\ t _ i \\ , , & i \\in [ m ] j \\in [ n ] \\\\ & \\displaystyle x _ { 1 } + \\dots + x _ { n } \\ = \\ 0 & \\end{array} \\end{align*}"}
{"id": "1015.png", "formula": "\\begin{align*} U = \\bigg \\{ x \\in B _ \\rho ^ + u ( x ) > \\frac 1 2 \\tau d ^ { - n - 2 } \\zeta ( x ) \\bigg \\} \\end{align*}"}
{"id": "2144.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } P _ { n , k } ( J _ { n , k } ^ { \\mathcal { S } } = l | W _ { \\mathcal { S } ( n , k ; M _ n ) } ) = \\frac 1 { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } \\binom k { l - 1 } ( \\frac c { 1 - c } ) ^ { l - 1 } ( 1 - c ) ^ { k } = \\\\ & \\frac 1 { \\lim _ { n \\to \\infty } P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M _ n ) } ) } \\binom k { l - 1 } c ^ { l - 1 } ( 1 - c ) ^ { k - l + 1 } , \\ l \\in \\{ 2 , \\cdots , k \\} , \\end{aligned} \\end{align*}"}
{"id": "2279.png", "formula": "\\begin{align*} M _ \\omega T _ x = e ^ { 2 \\pi i \\omega \\cdot x } \\ , T _ x M _ \\omega , \\end{align*}"}
{"id": "5937.png", "formula": "\\begin{align*} v ^ 1 _ 1 = \\frac { \\mu _ 1 - ( \\lambda + \\nu ) v ^ 2 _ 2 } { \\lambda + 2 \\nu } , v ^ 1 _ 2 = - \\frac { \\mu _ 2 } { \\nu } . \\end{align*}"}
{"id": "3465.png", "formula": "\\begin{align*} \\| f ( x ) \\| _ p ^ p \\leq \\sum \\limits _ { l = - \\infty } ^ \\infty \\mu ( \\widetilde { \\Omega } _ l ) ^ { 1 - \\frac { p } { 2 } } \\| \\sum \\limits _ { Q \\in B _ l } \\omega ( Q ) D _ k ( x , x _ Q ) { \\widetilde D } _ k ( f ) ( x _ Q ) \\| _ 2 ^ p . \\end{align*}"}
{"id": "4717.png", "formula": "\\begin{align*} & ( \\widetilde { x } _ 1 ( t ) , \\ldots , \\widetilde { x } _ n ( t ) ) ^ \\intercal = T ( x _ 1 ( t ) , \\ldots , x _ n ( t ) ) ^ \\intercal , \\\\ & ( \\widetilde { \\mu } _ 1 ( t ) , \\ldots , \\widetilde { \\mu } _ n ( t ) ) ^ \\intercal = T ( \\mu _ 1 ( t ) , \\ldots , \\mu _ n ( t ) ) ^ \\intercal , \\\\ & ( \\widetilde { \\alpha } _ 1 , \\ldots , \\widetilde { \\alpha } _ n ) ^ \\intercal = T ( \\alpha _ 1 , \\ldots , \\alpha _ n ) ^ \\intercal . \\end{align*}"}
{"id": "537.png", "formula": "\\begin{align*} \\mathcal { D } ( \\mathbf { x } _ k , { \\mathbf { u } } _ k ) _ { \\{ i , j \\} } = { \\partial h _ i ( \\boldsymbol { f } _ O ^ { d _ i - 1 } \\left ( \\boldsymbol { f } ( \\mathbf { x } _ k , { \\mathbf { u } } _ k ) \\right ) ) } \\big / { \\partial { u } _ { j , k } } . \\end{align*}"}
{"id": "4414.png", "formula": "\\begin{align*} \\lambda w _ j + ( 1 - \\lambda ) h _ j = \\frac { 1 } { 2 } \\mp b i \\end{align*}"}
{"id": "8251.png", "formula": "\\begin{align*} u \\times v : = ( u _ 1 , \\ldots , u _ p , v _ 1 [ p ] , \\ldots , v _ { q } [ p ] ) . \\end{align*}"}
{"id": "8987.png", "formula": "\\begin{align*} \\begin{aligned} & e ^ { - t } C _ p \\left | \\dot { \\xi } ( t ) \\right | ^ { q } \\leq e ^ { - t } \\phi \\left ( \\xi ( t ) \\right ) - e ^ { - t } D \\phi \\left ( \\xi ( t ) \\right ) \\cdot \\dot { \\xi } ( t ) \\\\ \\Longrightarrow & C _ p \\left | \\dot { \\xi } ( t ) \\right | ^ { q } \\leq \\phi \\left ( \\xi ( t ) \\right ) - D \\phi \\left ( \\xi ( t ) \\right ) \\cdot \\dot { \\xi } ( t ) . \\end{aligned} \\end{align*}"}
{"id": "9222.png", "formula": "\\begin{align*} \\frac { F \\rightarrow s = _ X s ' F \\rightarrow t = _ X t ' } { F \\rightarrow ( s \\in A t \\leftrightarrow s ' \\in A t ' ) } \\end{align*}"}
{"id": "8836.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { t + 1 } e _ { F _ { 0 } } ( D _ { i } , S ) \\leq k ' ( t - 1 ) . \\end{align*}"}
{"id": "4094.png", "formula": "\\begin{align*} G ^ { - 1 } ( x ) = x , \\ , G ' ( x ) = 1 , \\ , G '' ( x ) = 0 , \\end{align*}"}
{"id": "7643.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\nabla \\tilde { u } _ n \\cdot \\nabla v = \\tilde { \\lambda } _ n \\int _ { \\R ^ N } \\tilde { m } _ 0 \\tilde { u } _ n v \\ ; . \\end{align*}"}
{"id": "2787.png", "formula": "\\begin{align*} G _ - ( x ) \\le ( 4 \\pi ) ^ { - \\frac { N } { 2 } } \\int _ 0 ^ \\infty e ^ { - \\frac { | x | ^ 2 } { 4 \\delta } } \\delta ^ { - \\frac { N } { 2 } } e ^ { - \\delta } d \\delta = \\begin{cases} C | x | ^ { - N + 2 } + o ( | x | ^ { - N + 2 } ) , & N \\ge 3 , \\\\ C \\log \\frac { 1 } { | x | } + o \\left ( \\log \\frac { 1 } { | x | } \\right ) , & N = 2 , \\\\ C + o ( 1 ) , & N = 1 . \\end{cases} \\end{align*}"}
{"id": "4738.png", "formula": "\\begin{align*} G ( y ) - \\frac { C _ 0 } { y ^ 2 } = \\Bigg [ \\int _ 0 ^ { \\infty } t e ^ { - t } \\bigg ( \\frac { C _ 0 } { t ^ 2 + y ^ 2 } - \\frac { C _ 0 } { y ^ 2 } \\bigg ) \\ , \\dd t \\Bigg ] \\lesssim \\frac { 1 } { y ^ 4 } \\int _ 0 ^ \\infty t ^ 3 e ^ { - t } \\ , \\dd t = O \\bigg ( \\frac { 1 } { y ^ 4 } \\bigg ) , \\end{align*}"}
{"id": "71.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m - 1 } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq \\frac { q _ v ^ { n _ { v , _ n u _ v } } - 1 } { q _ v ^ { n + 1 } - 1 } . \\end{align*}"}
{"id": "8108.png", "formula": "\\begin{align*} F \\star G [ j ] : = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\hbar ^ n } { n ! } \\left \\langle ( \\tfrac { i } { 2 } E _ { \\mathcal { I } } + H _ c ) ^ { \\otimes n } , F ^ { ( n ) } [ j ] \\otimes G ^ { ( n ) } [ j ] \\right \\rangle . \\end{align*}"}
{"id": "1689.png", "formula": "\\begin{align*} \\overline { H } _ n ( \\overline { s } ) = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } F _ s } \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - k ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; - 1 \\right ) . \\end{align*}"}
{"id": "7641.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ { n } \\le \\lambda ^ 1 ( B ^ 1 ( \\mathbf { y } _ n ) , B ^ { \\delta / k _ n } ( \\mathbf { y } _ n ) ) = \\lambda ^ 1 ( B ^ 1 ( \\mathbf { 0 } ) , B ^ { \\delta / k _ n } ( \\mathbf { 0 } ) ) . \\end{align*}"}
{"id": "1419.png", "formula": "\\begin{align*} ( P \\circ \\sigma ^ { \\# } ) ( F _ 1 \\vee F _ 2 ) & = ( P \\circ \\sigma ^ { \\# } ) ( F _ 1 ) \\vee ( P \\circ \\sigma ^ { \\# } ) ( F _ 2 ) \\\\ & = P ( \\sigma ^ { \\# } ( F _ 1 ) \\vee \\sigma ^ { \\# } ( F _ 2 ) ) . \\end{align*}"}
{"id": "9002.png", "formula": "\\begin{align*} D ^ * = \\frac { N } { N - 1 } \\left ( 1 - \\frac { 1 } { N ^ { K - 1 } } \\right ) + \\frac { 1 } { N ^ { K - 1 } } = \\frac { 1 - N ^ { - K } } { 1 - N ^ { - 1 } } , \\end{align*}"}
{"id": "7450.png", "formula": "\\begin{align*} u _ { k } = u _ { k - 1 } - s \\nabla f \\left ( u _ { k - 1 } \\right ) , k = 1 , 2 , \\ldots , \\end{align*}"}
{"id": "8708.png", "formula": "\\begin{align*} A \\otimes B = \\begin{pmatrix} a _ { 1 1 } B & \\cdots & a _ { 1 m } B \\\\ \\vdots & \\ddots & \\vdots \\\\ a _ { m 1 } B & \\cdots & a _ { m n } B \\end{pmatrix} . \\end{align*}"}
{"id": "8296.png", "formula": "\\begin{align*} L _ 1 u ( x ) & = b ( x ) \\Bigl ( P . V . \\displaystyle \\int _ { \\mathbb { R } ^ { N - 1 } } \\frac { ( u ( x ) - u ( x + y ) ) } { | y | ^ { N + \\alpha } } A _ K ^ e ( x , 0 , y / | y | ) \\ ; d y \\Bigr ) \\\\ & = \\frac { b ( x ) } { 2 } \\int _ { \\mathbb { R } ^ { N - 1 } } \\frac { ( 2 u ( x ) - u ( x + y ) - u ( x - y ) ) } { | y | ^ { N + \\alpha } } A _ K ^ e ( x , 0 , y / | y | ) \\ ; d y \\end{align*}"}
{"id": "2686.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 d } } | V _ g f ( x , \\omega ) | ^ p \\ , d ( x , \\omega ) \\begin{cases} \\leq \\left ( \\tfrac { 2 } { p } \\right ) ^ d ( \\norm { f } _ 2 \\norm { g _ 2 } ) ^ p & 2 \\leq p < \\infty \\\\ \\geq \\left ( \\tfrac { 2 } { p } \\right ) ^ d ( \\norm { f } _ 2 \\norm { g _ 2 } ) ^ p & 1 \\leq p \\leq 2 \\\\ \\end{cases} \\end{align*}"}
{"id": "1640.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\int _ { \\left \\lbrace u > \\varepsilon \\right \\rbrace } \\frac { 1 } { u ^ p } ( P _ s u ) ^ p \\Big ( \\sup _ { t > 0 } P _ t f \\Big ) ^ p \\ : d \\mu = \\int _ { \\left \\lbrace u > \\varepsilon \\right \\rbrace } \\Big ( \\sup _ { t > 0 } P _ t f \\Big ) ^ p \\ : d \\mu . \\end{align*}"}
{"id": "2145.png", "formula": "\\begin{align*} \\sum _ { l = 2 } ^ k \\binom k { l - 1 } c ^ { l - 1 } ( 1 - c ) ^ { k - l + 1 } = 1 - c ^ k - ( 1 - c ) ^ k , \\end{align*}"}
{"id": "3400.png", "formula": "\\begin{align*} \\langle T ( 1 - \\eta ) b , f \\rangle & = \\int _ { \\R ^ N } \\int _ { \\R ^ N } K ( x , y ) ( 1 - \\eta ( y ) ) b ( y ) f ( x ) d \\omega ( y ) d \\omega ( x ) \\\\ & = \\int _ { \\R ^ N } \\int _ { \\R ^ N } [ K ( x , y ) - K ( x _ 0 , y ) ] ( 1 - \\eta ( y ) ) b ( y ) f ( x ) d \\omega ( y ) d \\omega ( x ) . \\end{align*}"}
{"id": "4940.png", "formula": "\\begin{align*} ( ( \\phi ^ { - 1 } ) ^ * ( \\phi ^ * ( r _ { g } ) \\circ \\sigma ) ) ^ { - 1 } \\circ ( r _ g \\circ \\psi ) = ( r _ g \\circ ( \\phi ^ { - 1 } ) ^ * \\sigma ) ^ { - 1 } \\circ r _ g \\circ \\psi = ( ( \\phi ^ { - 1 } ) ^ * \\sigma ) ^ { - 1 } \\circ \\psi . \\end{align*}"}
{"id": "4141.png", "formula": "\\begin{align*} \\psi ( \\xi , t ) : = e ^ { - i t \\xi | \\xi | ^ { 1 + a } } . \\end{align*}"}
{"id": "6470.png", "formula": "\\begin{align*} D ( X ) : = \\sup _ { x \\in \\mathbb { R } } | P ( X \\leq x ) - P ( Z \\leq x ) | . \\end{align*}"}
{"id": "143.png", "formula": "\\begin{align*} \\varphi ^ { \\operatorname { r o t } } _ \\alpha ( \\xi ) = \\hat { \\mu } _ { \\alpha } ^ { \\operatorname { r o t } } \\left ( \\xi \\right ) = \\exp \\left ( - \\frac { \\| \\xi \\| ^ \\alpha } { 2 } \\right ) . \\end{align*}"}
{"id": "1320.png", "formula": "\\begin{align*} I ( \\hat { \\alpha } \\cup { ( \\gamma , p _ { i + 1 } ) } , \\hat { \\alpha } \\cup { ( \\delta , 1 ) \\cup { ( \\gamma , p _ { i + 1 } - p _ { i } ) } } ) = 4 . \\end{align*}"}
{"id": "1602.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d - 1 } } \\tilde \\varphi \\ , d x = 0 . \\end{align*}"}
{"id": "4180.png", "formula": "\\begin{align*} d _ H ( N , N ' ) : = \\inf \\left \\{ | | \\phi | | _ H \\ \\middle | \\ \\phi \\in \\mathrm { H a m } ( M ) , \\ \\phi ( N ) = N ' \\right \\} , \\end{align*}"}
{"id": "8270.png", "formula": "\\begin{align*} \\mathbf { M } _ { 1 } \\mathbf { M } _ { \\bar { 2 } \\ , \\bar 1 } = { } & \\mathbf { M } _ { 1 \\bar { 3 } \\ , \\bar 2 } + \\mathbf { M } _ { \\bar 3 1 \\bar 2 } + \\mathbf { M } _ { \\bar { 3 } \\ , \\bar 2 1 } . \\end{align*}"}
{"id": "1475.png", "formula": "\\begin{align*} R = R _ { \\infty } = \\left [ \\begin{array} { c c c c c c c c } 1 _ { 2 m } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 / 2 & 0 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 1 / 2 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 _ { 2 m } & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 _ { 2 m } & 0 & 0 & 0 & 0 \\\\ 0 & - 1 / 2 & 0 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & - 1 / 2 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 _ { 2 m } \\end{array} \\right ] , \\end{align*}"}
{"id": "5872.png", "formula": "\\begin{align*} S _ { \\tau ' } ( Y _ 0 , d W _ \\cdot ) = e ^ { - \\mu ( \\tau ' - \\tau ) } S _ { \\tau } ( Y _ 0 , d W _ \\cdot ) + S _ { \\tau ' - \\tau } ( Y _ { \\tau } , d ( \\Theta ^ { \\tau } W ) _ \\cdot ) . \\end{align*}"}
{"id": "360.png", "formula": "\\begin{align*} & \\inf _ { \\rho , v } \\ ; [ \\int _ 0 ^ 1 \\frac 1 2 \\ < v _ t , v _ t \\ > _ { \\theta ( \\rho _ t ) } d t ] \\\\ & \\ ; \\ ; d \\rho ( t ) + d i v _ G ^ { \\theta } ( \\rho ( t ) v ( t ) ) + d i v _ G ^ { \\theta } ( \\rho ( t ) v ( t ) ) d W _ t ^ { \\delta } = 0 \\\\ & \\ ; \\ ; \\rho ( 0 ) = \\rho _ a , \\ ; \\rho ( 1 ) = \\rho _ b . \\end{align*}"}
{"id": "8491.png", "formula": "\\begin{align*} f _ k & = [ u ^ k ] F ( u ) = \\frac { N _ 1 ^ { t - k } } { D _ t } ( - 1 ) ^ { k } z ^ k , \\\\ g _ k & = [ u ^ k ] G ( u ) = \\frac { N _ 2 ^ { t - k } } { D _ t } ( - 1 ) ^ { k } z ^ k , \\mbox { a n d } \\\\ h _ k & = [ u ^ k ] H ( u ) = \\frac { N _ 3 ^ { t - k } } { D _ t } ( - 1 ) ^ { k } z ^ k . \\end{align*}"}
{"id": "7891.png", "formula": "\\begin{align*} \\frac { 1 } { L ^ n } \\int _ U \\varphi ( y ) \\ d n _ L ( y ) - \\int _ U \\varphi ( x ) \\bar { \\nu } ( x ) d x & \\geq - ( \\sup _ U \\varphi ) \\cdot \\int _ { Q _ 1 } \\big | \\nu _ { R , L } ( x , w ) - \\bar { \\nu } ( x ) \\big | \\ d x \\\\ & - \\omega _ \\varphi ( \\delta ) \\cdot | | \\bar { \\nu } | | _ { L ^ \\infty ( Q _ 1 ) } \\cdot | Q _ 1 | . \\end{align*}"}
{"id": "3604.png", "formula": "\\begin{align*} & R _ { 1 } ( N _ { 1 } ) = u _ { 1 } = 0 \\ , , R _ { 2 } ( N _ { 1 } ) = 2 r _ { 1 } - u _ { 1 } = 2 - 2 e \\quad R _ { 1 } ( N _ { 2 } ) = r _ { 2 } = u _ { 2 } = 0 \\ , . \\end{align*}"}
{"id": "3121.png", "formula": "\\begin{align*} \\lambda _ 1 F _ 1 + \\lambda _ 2 F _ 2 + \\lambda _ 3 F _ 3 = 0 \\end{align*}"}
{"id": "4904.png", "formula": "\\begin{align*} \\sqrt { \\sum _ { i = 1 } ^ { n } a _ i } \\leq \\sum _ { i = 1 } ^ { n } \\frac { a _ i } { \\sqrt { \\sum _ { k = 1 } ^ { i } a _ k } } \\leq 2 \\sqrt { \\sum _ { i = 1 } ^ { n } a _ i } \\end{align*}"}
{"id": "7732.png", "formula": "\\begin{align*} \\phi ( u , v ) & = \\int _ { - v } ^ u \\int _ { - u _ 0 } ^ v - F ( u _ 0 , v _ 0 ) d v _ 0 d u _ 0 + \\int _ { - v } ^ u \\phi _ u ( u _ 0 , - u _ 0 ) d u _ 0 + \\phi ( - v , v ) , \\\\ & = 2 \\int _ { - v } ^ u \\int _ { - u _ 0 } ^ v F ( s , y ) d y d s + \\int _ { - v } ^ u \\phi _ u ( u _ 0 , - u _ 0 ) d u _ 0 + \\phi ( - v , v ) , \\\\ & = 2 \\int _ 0 ^ t \\int _ { x - t + s } ^ { x + t - s } F ( s , y ) d y d s + \\int _ { - v } ^ u \\phi _ u ( u _ 0 , - u _ 0 ) d u _ 0 + \\phi ( - v , v ) . \\end{align*}"}
{"id": "3474.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , s _ 3 ) = \\sum _ { k = 3 } ^ \\infty \\abs * { \\sum _ { k / 2 < m \\leq k - 1 } \\frac { 1 } { m ^ { s _ 1 } ( k - m ) ^ { s _ 2 } } } ^ 2 \\frac { 1 } { k ^ { s _ 3 } } . \\end{align*}"}
{"id": "4957.png", "formula": "\\begin{align*} \\omega ( z \\cdot q ) = \\kappa _ G ( z ) + \\omega ( q ) \\end{align*}"}
{"id": "8627.png", "formula": "\\begin{align*} \\mu ( k , \\ell , m , n ) : = \\int \\overline { \\mathcal { K } ( x , k ) } \\mathcal { K } ( x , \\ell ) \\overline { \\mathcal { K } ( x , m ) } \\mathcal { K } ( x , n ) \\ , d x . \\end{align*}"}
{"id": "1105.png", "formula": "\\begin{align*} E ( x , t , k ) : = \\left \\{ \\begin{aligned} & m ^ { ( 3 ) } ( x , t , k ) \\left ( m ^ { L C } \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , k \\in U _ { \\xi } \\\\ & m ^ { ( 3 ) } ( x , t , k ) \\left ( m ^ { G P } \\left ( x , t , k \\right ) \\right ) ^ { - 1 } , e l s e w h e r e , \\end{aligned} \\right . \\end{align*}"}
{"id": "7813.png", "formula": "\\begin{align*} ( \\partial _ t + A ) ^ \\gamma X ( t ) = \\dot { W } ^ { Q } ( t ) , t \\in [ 0 , T ] , X ( 0 ) = 0 . \\end{align*}"}
{"id": "6257.png", "formula": "\\begin{align*} W _ { 1 } = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle , W _ { 2 } = \\left \\langle \\left [ \\vartheta _ { 1 } \\right ] , \\left [ \\vartheta _ { 2 } \\right ] , \\dots , \\left [ \\vartheta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) , \\end{align*}"}
{"id": "9090.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\ ! \\ ! \\dot { x } p _ { | \\dot { h _ i } | } ( \\dot { x } ) d \\dot { x } = \\frac { \\sigma _ { \\dot { X } } } { \\sqrt { 2 \\pi } } = \\sqrt { \\frac { \\pi } { 2 } } \\sigma f _ D , \\ : \\ : \\forall i = 1 , \\cdots , N . \\end{align*}"}
{"id": "44.png", "formula": "\\begin{align*} \\left \\Vert \\theta \\right \\Vert _ { g _ t } = \\left \\Vert \\theta _ { t } ^ { c } \\right \\Vert _ { g _ t } = 1 . \\end{align*}"}
{"id": "5865.png", "formula": "\\begin{align*} \\tilde \\theta = \\sum _ { j \\in \\tilde J } \\left ( \\frac { \\bar \\alpha _ j } { \\rho _ j } - 1 \\right ) = \\sum _ { j \\in J \\cap \\{ i + 1 , \\dots , n \\} } \\left ( \\frac { \\bar \\alpha _ j } { \\rho _ j } - 1 \\right ) . \\end{align*}"}
{"id": "3270.png", "formula": "\\begin{align*} h _ \\lambda ( v ) = ( h \\circ W ) ( v ) = \\frac { W ( v ) - v } { ( W ( v ) - v ) ^ 2 + | \\lambda | ^ 2 } . \\end{align*}"}
{"id": "6450.png", "formula": "\\begin{align*} \\lambda _ h = \\lambda _ 0 - i \\frac { { \\lambda _ 0 } ^ \\frac { 5 } { 2 } } { 4 \\pi } \\left [ { \\eta _ 0 ^ 1 U _ 1 ^ 2 + \\eta _ 0 ^ 2 U _ 2 ^ 2 + ( \\eta _ 0 ^ 1 + \\eta _ 0 ^ 2 ) U _ 1 U _ 2 } \\right ] h + \\mathcal { O } ( h ^ 2 ) . \\end{align*}"}
{"id": "4423.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\sum _ { j = 1 } ^ { n } a _ { i j } w _ j = \\theta , \\ ; \\ ; \\ ; i = 1 , 2 , . . . , m \\\\ \\displaystyle \\sum _ { j = 1 } ^ { n } w _ j = 1 \\end{array} \\end{align*}"}
{"id": "7808.png", "formula": "\\begin{align*} ( - 1 ) ^ { m + 1 } \\frac { ( a v ) ^ m } { m } = \\frac { ( - 1 ) ^ { m + 1 } } { s } a ^ m \\cdot \\frac { ( v ^ 2 ) ^ { 2 ^ { \\ell - 1 } s } } { 2 ^ { \\ell } } = \\frac { ( - 1 ) ^ { m + 1 } } { s } a ^ m \\cdot \\frac { ( \\pi + 2 \\alpha ) ^ { 2 ^ { \\ell - 1 } s } } { 2 ^ { \\ell } } . \\end{align*}"}
{"id": "552.png", "formula": "\\begin{align*} \\rho ( 0 , \\cdot ) = \\rho _ { 0 , \\varepsilon } = \\varrho + \\varepsilon \\rho ^ { ( 1 ) } _ { 0 , \\varepsilon } , \\ u ( 0 , \\cdot ) = u _ { 0 , \\varepsilon } \\end{align*}"}
{"id": "2119.png", "formula": "\\begin{align*} P _ { n , k } ( W _ { \\mathcal { S } ( n , k ; M ) } | A ^ { ( n ) } _ { M , n , l } ) = \\begin{cases} 1 , \\ \\ l \\in [ k - 1 ] ; \\\\ 0 , \\ \\ l = k . \\end{cases} \\end{align*}"}
{"id": "6573.png", "formula": "\\begin{align*} \\underset { s = 0 } { } \\ \\widetilde { V } ( s ) = \\lim _ { s \\rightarrow 0 } s \\widetilde { V } ( s ) = 1 \\end{align*}"}
{"id": "7406.png", "formula": "\\begin{align*} \\Omega _ { j , x } ^ 1 : = \\{ \\eta \\in \\Omega _ x : \\ ; \\eta ( x ) = 1 , \\eta ( z _ { 1 j } ) = 0 , \\eta ( x + r ) = 0 \\} . \\end{align*}"}
{"id": "844.png", "formula": "\\begin{align*} \\bar { \\Delta } \\triangleq \\mathop { \\lim } \\limits _ { N \\to \\infty } \\frac { 1 } { N } \\sum \\nolimits _ { t = 1 } ^ N { \\Delta \\left ( t \\right ) } . \\end{align*}"}
{"id": "6066.png", "formula": "\\begin{align*} S _ \\varepsilon = \\left \\{ R _ { \\varepsilon } ( s , t ) = R ( s , t ) + 2 \\varepsilon U ^ { 1 } ( s , t ) + \\ldots + 2 \\varepsilon ^ { m } U ^ { m } ( s , t ) \\right \\} , \\end{align*}"}
{"id": "8089.png", "formula": "\\begin{align*} \\left ( T _ { \\Sigma } ( s ) \\star _ { H , \\ell } T _ { \\Sigma } ( s ' ) \\right ) [ \\psi ] = \\frac { 1 } { 4 } \\psi { ( s ) } ^ 2 \\psi { ( s ' ) } ^ 2 + \\frac { \\hbar } { 4 \\pi } \\frac { \\psi ( s ) \\psi ( s ' ) } { { ( s - s ' ) } ^ 2 } + \\frac { \\hbar ^ 2 } { 3 2 \\pi ^ 2 } \\frac { 1 } { { ( s - s ' ) } ^ 4 } . \\end{align*}"}
{"id": "275.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { \\frac { 1 } { 2 } } \\left \\| u ( \\cdot , t ) - t ^ { - \\frac { 1 } { 2 } } f _ { M } \\left ( ( \\cdot ) t ^ { - \\frac { 1 } { 2 } } \\right ) \\right \\| _ { L ^ { \\infty } } = 0 , \\end{align*}"}
{"id": "8085.png", "formula": "\\begin{align*} W _ { \\Sigma } ( s , s ' ) = \\frac { - 1 } { 4 \\pi } \\frac { 1 } { \\sqrt { \\gamma ' ( s ) \\gamma ' ( s ' ) } } \\left ( \\frac { 1 } { ( s - s ' ) ^ 2 } + i \\delta ' ( s - s ' ) \\right ) . \\end{align*}"}
{"id": "1924.png", "formula": "\\begin{align*} G = \\Big ( \\int _ { \\mathbb { R } ^ 3 } | v | ^ { \\kappa _ 0 } f \\ , d v \\Big ) ^ \\frac { 1 } { \\kappa _ 0 + 3 } , \\end{align*}"}
{"id": "4137.png", "formula": "\\begin{align*} u _ 1 ( 0 ) = \\phi , u _ 2 ( 0 ) = \\varphi , \\phi \\not = \\varphi , \\end{align*}"}
{"id": "3577.png", "formula": "\\begin{align*} \\Lambda _ e & = \\Big \\{ { \\sum } _ { i = 0 } ^ { e - 1 } a _ i ( p ^ e - p ^ i ) \\colon 0 \\le a _ i < p \\Big \\} \\\\ & \\ \\ \\ \\ - \\Big \\{ 1 + p ^ e \\delta _ { n - r } + { \\sum } _ { i = 1 } ^ r p ^ { d ' _ i } \\colon ( d ' _ 1 , \\ldots , d ' _ r ) \\le \\theta _ e , \\ d ' _ r = 0 \\Big \\} . \\end{align*}"}
{"id": "2627.png", "formula": "\\begin{align*} \\vartheta _ 3 ( i s ; \\omega ) = Z \\varphi _ s ( 0 , - \\omega ) . \\end{align*}"}
{"id": "8213.png", "formula": "\\begin{align*} P _ h & = x _ h A \\prod _ { i = 1 } ^ { h } F _ { k , i } ( A ) ^ { - 1 } \\prod _ { i = 1 } ^ { h - 1 } G _ { k , i } ( A ) , \\\\ P _ { k + 1 - h } & = x _ { k + 1 - h } z \\prod _ { i = 1 } ^ { h } F _ { k , i } ( A ) \\prod _ { i = 1 } ^ h G _ { k , i } ( A ) ^ { - 1 } . \\end{align*}"}
{"id": "7330.png", "formula": "\\begin{align*} p _ 1 : = p + 2 \\alpha \\tilde { y } \\neq 0 , p _ 2 : = p - 2 \\alpha \\tilde { x } \\neq 0 \\quad . \\end{align*}"}
{"id": "525.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j { r \\choose { i - 1 } } { { r - i } \\choose { j - i } } ( - 1 ) ^ { j - i } = 1 \\end{align*}"}
{"id": "2089.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { v } \\in P _ { \\mathbf { c } } } \\exp { \\Big ( 2 \\pi i \\frac { \\langle \\pmb { \\xi } , \\mathbf { v } - \\mathbf { c } \\rangle } { p ^ { 2 k } } \\Big ) } = \\exp { \\Big ( 2 \\pi i \\frac { \\langle \\pmb { \\xi } , \\mathbf { u } - \\mathbf { c } \\rangle } { p ^ { 2 k } } \\Big ) } \\sum _ { \\mathbf { v } \\in P _ { \\mathbf { c } } } \\exp { \\Big ( 2 \\pi i \\frac { \\langle \\pmb { \\xi } , \\mathbf { v } - \\mathbf { c } \\rangle } { p ^ { 2 k } } \\Big ) } \\end{align*}"}
{"id": "8341.png", "formula": "\\begin{align*} R _ \\pi = \\sum _ { i = 1 } ^ { n } w _ { i \\pi ( i ) } . \\end{align*}"}
{"id": "717.png", "formula": "\\begin{align*} d _ j = \\partial _ { x _ { j ; \\alpha } } , j = 1 , 2 . \\end{align*}"}
{"id": "4820.png", "formula": "\\begin{align*} \\begin{aligned} y _ { t + L - 1 } & = \\sum _ { k = 0 } ^ { L - 2 } a _ k y _ { t + k } + \\sum _ { k = 0 } ^ { L - 1 } b _ k u _ { t + k } , \\\\ y _ { t + L - 1 } & = \\sum _ { k = 0 } ^ { L - 2 } \\tilde { a } _ k y _ { t + k } + \\sum _ { k = 0 } ^ { L - 1 } \\tilde { b } _ k u _ { t + k } . \\end{aligned} \\end{align*}"}
{"id": "5067.png", "formula": "\\begin{align*} f _ { s + t } ( u , u ' ) = \\begin{array} { l } f _ t ( u , u ' ) + s A \\cdot f _ t ( u , u ' ) \\\\ - s f _ t ( u , A \\cdot u ' ) - s f _ t ( A \\cdot u , u ' ) \\\\ + s ^ 2 f _ t ( A \\cdot u , A \\cdot u ' ) - s ^ 2 A \\cdot f _ t ( u , A \\cdot u ' ) \\\\ - s ^ 2 A \\cdot f _ t ( A \\cdot u , u ' ) + s ^ 3 A \\cdot f _ t ( A \\cdot u , A \\cdot u ' ) . \\end{array} \\end{align*}"}
{"id": "4465.png", "formula": "\\begin{align*} \\Big ( x \\frac { d } { d x } \\Big ) _ { n , \\lambda } f ( x ) = \\sum _ { k = 1 } ^ { n } S _ { 2 , \\lambda } ( n , k ) x ^ { k } \\Big ( \\frac { d } { d x } \\Big ) ^ { k } f ( x ) , \\end{align*}"}
{"id": "2710.png", "formula": "\\begin{align*} \\mathbf { F } = \\left ( \\R [ X _ 1 , \\ldots , X _ n ] _ { \\leq 2 } \\right ) _ { n > 0 } \\end{align*}"}
{"id": "1617.png", "formula": "\\begin{align*} R ^ 2 _ 1 = - \\left ( R ^ { 2 , \\operatorname { q u a d } } + R ^ { 2 , \\operatorname { m e a n } } + R ^ { \\chi } + R ^ { 2 , \\operatorname { l i n } } + R ^ { 2 , \\operatorname { c o r r } } + R ^ h \\right ) , \\end{align*}"}
{"id": "6646.png", "formula": "\\begin{align*} \\Sigma _ 2 \\ll p ^ { ( \\frac { 3 } { 2 } - \\epsilon ) h _ p } \\sum _ { m = \\max \\{ 0 , k _ p - h _ p \\} } ^ { \\infty } \\frac { p ^ { \\frac { 1 } { 2 } - \\epsilon } } { p ^ { m ( \\frac { 1 } { 2 } - \\varepsilon + \\epsilon ) } p ^ { ( h _ p - k _ p + 1 ) ( 1 - \\varepsilon ) } } . \\end{align*}"}
{"id": "2041.png", "formula": "\\begin{align*} \\psi _ * ( r ) : = \\psi ( r ) \\ 1 _ { \\{ r \\leq 1 \\} } + r ^ 2 \\ 1 _ { \\{ r > 1 \\} } \\hbox { f o r } r \\geq 0 ; \\end{align*}"}
{"id": "1414.png", "formula": "\\begin{align*} \\hat q _ { j i } = ( \\hat q _ j ) ^ { - 1 } \\hat q _ i = ( \\hat q _ { i j } ) ^ { - 1 } . \\end{align*}"}
{"id": "3899.png", "formula": "\\begin{align*} E _ 0 ( f , p ) = \\sum _ { z \\leq p } \\frac { 1 } { p } \\sum _ { \\substack { 0 \\leq a < p \\\\ \\gcd ( n , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi a ( \\tau ^ { n } - u ) } { p } } \\cdot \\frac { 1 } { p } \\sum _ { \\substack { 0 < \\leq b < p \\\\ g c d ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { m } - v ) } { p } } = 0 . \\end{align*}"}
{"id": "9149.png", "formula": "\\begin{align*} \\forall \\delta > 0 \\forall n , m \\geq \\theta ( \\delta ) : = \\tau \\left ( \\phi \\left ( \\alpha _ G \\left ( \\frac { \\beta _ H ( \\delta / 2 ) } { 2 } \\right ) \\right ) , \\xi \\left ( \\frac { \\beta _ H ( \\delta / 2 ) } { 2 } \\right ) \\right ) ( d ( x _ n , x _ m ) < \\delta ) . \\end{align*}"}
{"id": "7539.png", "formula": "\\begin{align*} \\Re \\left ( \\frac { 1 } { s - \\rho } \\right ) = \\frac { 2 - \\beta } { ( 2 - \\beta ) ^ 2 + ( T - \\gamma ) ^ 2 } \\geq \\frac { 1 } { 4 + ( T - \\gamma ) ^ 2 } \\geq \\frac { 1 } { 4 ( 1 + ( T - \\gamma ) ^ 2 ) } \\end{align*}"}
{"id": "91.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 ^ { 1 / 2 } \\cdot \\frac { 2 ^ { ( n _ { 2 , \\nu _ 2 } - 1 ) / 2 } + 1 } { 2 ^ { n + 1 } - 1 } \\end{align*}"}
{"id": "2440.png", "formula": "\\begin{align*} \\sigma ( \\l ' , \\l ) = \\l ' \\cdot J \\l , \\end{align*}"}
{"id": "1749.png", "formula": "\\begin{align*} \\mathcal { A } ^ c _ G ( X ) : = \\big ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\Psi ^ { - \\infty } ( S ) \\big ) ^ { K \\times K } \\subset \\mathcal { A } _ G ( X ) . \\end{align*}"}
{"id": "3715.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ) } \\| B ^ k ( t ) \\| _ { L ^ 2 } \\leq ( 1 + C ( T ) ) \\| B _ 0 \\| _ { L ^ 2 } , \\\\ \\inf _ { t \\in [ 0 , T ) } \\| B ^ k ( t ) \\| _ { L ^ 2 } \\leq ( 1 - C ( T ) ) \\| B _ 0 \\| _ { L ^ 2 } . \\end{align*}"}
{"id": "8755.png", "formula": "\\begin{align*} f _ 1 = F _ 1 ( z _ 1 ) , \\ z _ 1 \\in \\Delta _ 1 , \\ \\delta _ 1 \\in \\{ 0 , 1 \\} ^ { n - 1 } , \\ z _ { 1 j } \\geq \\delta _ { 1 j } \\geq z _ { 1 j + 1 } j = 1 , \\ldots , n - 1 . \\end{align*}"}
{"id": "6264.png", "formula": "\\begin{align*} e ( A , B ) = \\frac { 1 } { 2 } & \\sum _ { x \\in A \\cup B } \\deg _ { G [ A , B ] } ( x ) \\\\ & \\geq \\frac { 1 } { 2 } \\left ( \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 h } \\right ) n - h \\right ) ( | A | + | B | ) \\\\ & \\geq \\frac { 1 } { 2 } \\left ( \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 h } \\right ) 1 0 h - h \\right ) ( | A | + | B | ) \\\\ & > \\frac { h } { 2 } ( | A | + | B | ) \\\\ & > \\frac { l - 1 } { 2 } ( | A | + | B | ) . \\end{align*}"}
{"id": "6235.png", "formula": "\\begin{align*} & \\det ( I _ { n + 2 M } - u { \\cal A } ) = ( 1 + 2 ( \\beta + 2 \\delta ) u + 4 \\delta ( \\delta + 1 ) u ^ 2 ) ^ { M - N } \\\\ & \\ \\ \\ \\times \\det ( \\{ 1 + ( 2 \\beta + 5 \\delta ) u + ( 2 \\delta ( \\beta + 4 \\delta + 2 ) + d \\beta { } ^ 2 ) u ^ 2 + 2 \\delta ( 2 \\delta ( \\delta + 1 ) + d \\beta { } ^ 2 ) u ^ 3 \\} I _ N \\\\ & \\ \\ \\ - \\beta ( 1 + 2 \\delta u ) ( 1 + ( \\beta + 2 \\delta ) u ) { \\bf A } ( G ) ) . \\end{align*}"}
{"id": "4551.png", "formula": "\\begin{align*} f _ 2 & \\ge 1 + h _ 5 = 1 + e _ 2 \\quad \\\\ f _ 3 & \\ge 1 + h _ 5 + h _ 1 = 1 + e _ 2 + e _ 3 . \\end{align*}"}
{"id": "4533.png", "formula": "\\begin{align*} T ( x ; k , r ) = \\frac { c ( r , k ) } { r } x + O ( d ( r ) x ^ { 1 / 2 } \\log ^ { 2 . 5 } x ) , \\end{align*}"}
{"id": "124.png", "formula": "\\begin{align*} U = \\{ u \\in A \\ ; | \\ ; 2 e u = u \\} , \\ ; \\ ; V = \\{ v \\in A \\ ; | \\ ; e v = 0 \\} . \\end{align*}"}
{"id": "9205.png", "formula": "\\begin{align*} x ^ X = _ X y ^ X : = \\norm { x - _ X y } _ X = _ \\mathbb { R } 0 \\end{align*}"}
{"id": "3752.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } B ^ i ( X , Y ) & : = & - g ( A _ i ( X ) , Y ) , & & \\\\ B ^ i ( X , e _ j ) & : = & B ^ i ( e _ j , X ) & = & B ^ i ( e _ i , e _ l ) = 0 , \\end{array} \\end{align*}"}
{"id": "7248.png", "formula": "\\begin{align*} [ E _ 1 ( x ) + E _ 3 ( x ) ] ^ 2 - [ 1 + E _ 2 ( x ) ] [ E _ 2 ( x ) + E _ 4 ( x ) ] = - \\dfrac { 8 2 5 } { 1 0 2 4 } < 0 . \\end{align*}"}
{"id": "8128.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma _ { 5 } ( P G ) } c ( g ( \\gamma ) ) = 5 \\equiv 1 \\pmod 2 . \\end{align*}"}
{"id": "2883.png", "formula": "\\begin{align*} J ( z ) = \\sum _ { j _ 1 + j _ 2 \\ge 2 } a _ { j _ 1 j _ 2 } z ^ { j _ 1 } \\bar { z } ^ { j _ 2 } , K ( z ) = \\sum _ { k _ 1 + k _ 2 \\ge 2 } b _ { k _ 1 k _ 2 } z ^ { k _ 1 } \\bar { z } ^ { k _ 2 } \\end{align*}"}
{"id": "4291.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ r ^ 2 u + \\frac { d + 1 } { r } \\partial _ r u - 3 ( d - 2 ) u ^ 2 - ( d - 2 ) r ^ 2 u ^ 3 , ( r , t ) \\in \\R _ + \\times \\R _ + . \\end{align*}"}
{"id": "2523.png", "formula": "\\begin{align*} \\pi ( F \\natural G ) = \\pi ( ( F \\natural G ) _ r ) = \\pi ( F _ r * _ { \\mathbf { H } _ r } G _ r ) = \\pi ( F _ r ) \\pi ( G _ r ) = \\pi ( F ) \\pi ( G ) . \\end{align*}"}
{"id": "5829.png", "formula": "\\begin{align*} X ^ { y } _ { h L _ { k + 1 } } - \\pi _ { 1 } ( y ) & = \\sum _ { j = 0 } ^ { \\ell _ k - 1 } \\left [ X ^ { Y ^ { y } _ { j h L _ { k } } } _ { h L _ { k } } - X ^ { y } _ { j h L _ { k } } \\right ] \\leq \\left [ \\ell _ k - N \\right ] h L _ k v _ { \\min } + N h L _ { k } v _ { \\max } \\\\ & = h L _ { k + 1 } \\left \\{ v _ { \\min } + ( v _ { \\max } - v _ { \\min } ) \\frac { N } { \\ell _ k } \\right \\} < h L _ { k + 1 } \\bar { v } , \\end{align*}"}
{"id": "2597.png", "formula": "\\begin{align*} V ^ * _ g F = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "9188.png", "formula": "\\begin{align*} F _ w ( t ) = \\sum _ { n = 0 } ^ \\infty \\# \\{ \\} t ^ n . \\end{align*}"}
{"id": "6141.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 0 } ^ n { n + m + 1 \\choose k } x ^ { n - k } ( 1 - x ) ^ k \\quad \\mbox { a n d } Q ( x ) = \\sum _ { k = 0 } ^ m { n + m + 1 \\choose k } x ^ k ( 1 - x ) ^ { m - k } . \\end{align*}"}
{"id": "8612.png", "formula": "\\begin{align*} \\mu ^ { \\# } _ { 0 } ( k , \\ell , m , n ) = \\sqrt { 2 \\pi } \\ , \\delta _ 0 ( k - \\ell + m - n ) . \\end{align*}"}
{"id": "4298.png", "formula": "\\begin{align*} b = \\frac { \\lambda } { T - t } . \\end{align*}"}
{"id": "4060.png", "formula": "\\begin{align*} \\Lambda _ 1 ( f , g ) = \\sigma ( t , 1 ) , \\end{align*}"}
{"id": "3316.png", "formula": "\\begin{align*} F / x ^ \\ell F \\simeq L / x ^ \\ell L = L / y ^ \\ell L \\simeq G / y ^ \\ell G \\end{align*}"}
{"id": "9284.png", "formula": "\\begin{align*} [ J ^ { \\chi _ A } ] _ \\mathcal { M } : = \\lambda \\alpha \\in \\mathbb { N } ^ \\mathbb { N } , x \\in X . \\begin{cases} J ^ A _ { r _ \\alpha } x & r _ \\alpha > 0 , \\\\ 0 & , \\end{cases} \\end{align*}"}
{"id": "515.png", "formula": "\\begin{align*} \\psi ( \\alpha ) : = \\psi ( \\overline { i _ { c } } ) + \\bar { \\psi } _ { 2 } ( \\overline { i _ { c } } ; \\alpha ) . \\end{align*}"}
{"id": "7464.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\sigma } ( \\phi , \\chi ) : = E ( \\phi ) - \\chi ( \\| \\phi \\| ^ 2 - 1 ) + \\sigma R ( \\| \\phi \\| ^ 2 - 1 ) , \\end{align*}"}
{"id": "755.png", "formula": "\\begin{align*} & a _ 1 = \\frac { \\mu t _ 1 - 1 } { \\left ( L + \\mu - \\mu _ p \\right ) t _ 1 - 2 } , \\ \\ \\ \\ & & a _ 2 = \\frac { 1 - L t _ 1 } { \\left ( L + \\mu - \\mu _ p \\right ) t _ 1 - 2 } \\\\ & a _ 3 = - \\frac { \\left ( ( L t _ 1 - 2 ) ( \\mu t _ 1 - 2 ) - 1 \\right ) \\mu _ p t _ 1 } { \\left ( L + \\mu - \\mu _ p \\right ) t _ 1 - 2 } \\ \\ \\ \\ & & a _ 4 = - \\frac { \\mu _ p t _ 1 } { \\left ( L + \\mu - \\mu _ p \\right ) t _ 1 - 2 } . \\end{align*}"}
{"id": "4894.png", "formula": "\\begin{align*} \\lim _ { y \\to + \\infty , y \\in \\R } \\frac { \\log | f ( i y ) / f '' ( i y ) | } { \\log y } = + \\infty . \\end{align*}"}
{"id": "3852.png", "formula": "\\begin{align*} ( r _ s ^ { \\pm } ) ' + r _ { s - 1 } ^ { \\pm } = \\hat q _ s ^ { \\pm } , s = \\overline { 0 , m } , r _ m ^ { \\pm } = 0 . \\end{align*}"}
{"id": "7192.png", "formula": "\\begin{align*} X _ 1 ^ T ( \\tau _ { x _ 1 } ) = x _ 1 , \\end{align*}"}
{"id": "6848.png", "formula": "\\begin{align*} U ^ k _ i f ( y ) \\coloneqq U _ { k - 1 } \\circ \\ldots \\circ U _ i f ( y ) = \\frac { 1 } { R ( k , i ) } \\sum \\limits _ { x \\in X ( i ) : x < y } f ( x ) \\end{align*}"}
{"id": "8391.png", "formula": "\\begin{align*} \\P ( \\sigma | _ A = \\sigma | _ { A ' } \\mbox { a n d } \\sigma | _ { B } = \\sigma | _ { B ' } ) = q ^ { - 2 ( r - 1 ) ^ d } \\ , . \\end{align*}"}
{"id": "4354.png", "formula": "\\begin{align*} Q ( \\xi ) = - \\frac { ( \\epsilon ( x ) + 1 ) } { e ^ { 2 x } } , \\xi = e ^ x . \\end{align*}"}
{"id": "672.png", "formula": "\\begin{align*} A \\ = \\ \\{ n : f ( n ) = 1 \\} . \\end{align*}"}
{"id": "5298.png", "formula": "\\begin{align*} ( 1 \\otimes b ^ * ) \\Delta ( p _ i ) = \\sum _ { j = 1 } ^ { m _ i } a _ { i j } \\otimes b _ { i j } , \\end{align*}"}
{"id": "7179.png", "formula": "\\begin{align*} \\langle a \\rangle & : = \\sqrt { 1 + | a | ^ 2 } . \\end{align*}"}
{"id": "5653.png", "formula": "\\begin{align*} ( \\sigma , R ) ^ { - 1 } = ( \\sigma ^ { - 1 } , ( R ^ { - 1 } ) ^ { \\sigma ^ { - 1 } } ) = ( \\sigma ^ { - 1 } , ( \\alpha ^ { - 1 } _ { \\sigma 1 } , \\alpha ^ { - 1 } _ { \\sigma 2 } , \\alpha ^ { - 1 } _ { \\sigma 2 } ) ) \\end{align*}"}
{"id": "6262.png", "formula": "\\begin{align*} e ( A , B ) = \\frac { 1 } { 2 } & \\sum _ { x \\in A \\cup B } \\deg _ { G [ A , B ] } ( x ) \\\\ & \\geq \\frac { 1 } { 2 } \\left ( \\left ( \\frac { 1 } { 2 } - \\frac { 3 } { 1 0 h } \\right ) n - h - 1 \\right ) ( | A | + | B | ) \\\\ & \\geq \\frac { 1 } { 2 } \\left ( \\left ( \\frac { 1 } { 2 } - \\frac { 3 } { 1 0 h } \\right ) 1 0 h - h - 1 \\right ) ( | A | + | B | ) \\\\ & > \\frac { h } { 2 } ( | A | + | B | ) \\\\ & > \\frac { ( l - 2 ) - 1 } { 2 } ( | A | + | B | ) . \\end{align*}"}
{"id": "5454.png", "formula": "\\begin{align*} \\| ( \\mathcal { M } ( \\tilde u _ 0 , a _ n , b ) u ) ( t , \\cdot ) - ( \\mathcal { M } ( \\tilde u _ 0 , a _ 0 , b ) u ) ( t , \\cdot ) \\| _ { C ^ 0 ( \\bar \\Omega ) } & = \\| \\int _ s ^ t e ^ { - A ( t - \\tau ) } u ( \\tau , \\cdot ) \\big [ a _ n ( \\tau , \\cdot ) - a _ 0 ( \\tau , \\cdot ) \\big ] d \\tau \\| _ { C ^ 0 ( \\bar \\Omega ) } \\\\ & \\le R \\| a _ n - a _ 0 \\| _ { \\mathcal { X } _ T } T . \\end{align*}"}
{"id": "3047.png", "formula": "\\begin{align*} x _ 1 ^ \\prime & = \\alpha _ { 1 1 } y _ 1 + \\alpha _ { 1 2 } y _ 2 \\ , , x _ 2 ^ \\prime = \\alpha _ { 2 1 } y _ 1 + \\alpha _ { 2 2 } y _ 2 \\ , ; \\\\ y _ 1 ^ \\prime & = \\beta _ { 1 1 } x _ 1 + \\beta _ { 1 2 } x _ 2 \\ , , y _ 2 ^ \\prime = \\beta _ { 2 1 } x _ 1 + \\beta _ { 2 2 } x _ 2 \\ , . \\end{align*}"}
{"id": "4374.png", "formula": "\\begin{align*} \\partial _ s v = \\Delta v + \\frac { 1 } { 2 } \\left ( \\frac { b _ s } { b } - b \\right ) \\Lambda _ \\xi v - 3 ( n - 4 ) v ^ 2 - ( n - 4 ) | \\xi | ^ 3 v ^ 3 \\end{align*}"}
{"id": "2197.png", "formula": "\\begin{align*} 0 = J ^ { ' } ( \\widehat { u } ) + \\lambda _ + \\varphi _ { w ^ + } ^ * + \\lambda _ - \\varphi _ { w ^ - } ^ * \\ \\ \\ ( W ^ { \\alpha , G } , ( \\mathbb { R } ^ d ) ) ^ * , \\ \\ \\varphi ^ * _ { \\widehat { w } ^ + } \\in \\partial \\varphi _ + ( \\widehat { w } ) \\ \\ \\varphi ^ * _ { \\widehat { w } ^ - } \\in \\partial \\varphi _ - ( \\widehat { w } ) . \\end{align*}"}
{"id": "2009.png", "formula": "\\begin{align*} \\hat { \\sigma } ( \\beta _ { n , p } ) = \\hat { \\sigma } ( u _ n ^ p v _ n ^ p ) = ( 1 , 3 , \\dots , 2 n - 1 ) ^ p ( 2 , 4 , \\dots , 2 n ) ^ { - p } = \\sigma ( h _ { n , p } ^ { - 1 } ) . \\end{align*}"}
{"id": "3430.png", "formula": "\\begin{align*} { R } _ { M } f & = \\sum \\limits _ { k = - \\infty } ^ \\infty \\sum \\limits _ { \\{ j \\in \\Bbb Z : \\ , | k - j | > { M } \\} } { D } _ k { D } _ j = \\sum \\limits _ { \\{ \\ell \\in \\Bbb Z : \\ , | \\ell | > M \\} } \\sum \\limits _ { k = - \\infty } ^ \\infty D _ k D _ { k + \\ell } f \\\\ & = \\sum \\limits _ { k = - \\infty } ^ \\infty D _ k ( I - S _ { k + M } ) f + \\sum \\limits _ { k = - \\infty } ^ \\infty D _ k S _ { k - M - 1 } f . \\end{align*}"}
{"id": "2203.png", "formula": "\\begin{align*} 2 m _ 0 & \\leq J ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ + ) + J ( s _ { \\widehat { w } ^ - } \\widehat { w } ^ - ) \\\\ & < J ( t _ { \\widehat { w } ^ + } \\widehat { w } ^ + + s _ { \\widehat { w } ^ - } \\widehat { w } ^ - ) \\\\ & \\leq J ( \\widehat { w } ) \\\\ & = \\inf \\limits _ { \\mathcal { M } } J = m _ 1 . \\end{align*}"}
{"id": "3568.png", "formula": "\\begin{align*} \\frac { 2 ( g - 1 ) } { | G | } = 2 h + u - \\delta _ u - \\frac { 1 } { p ^ { e ( \\alpha ) } } + \\sum \\limits _ { i = 1 } ^ { r } \\Big ( 1 - \\frac { 1 } { p ^ { d _ { i } } } \\Big ) + \\sum \\limits _ { t \\in \\Omega } \\Big ( 1 - \\frac { 1 } { p ^ { c _ t } } \\Big ) \\end{align*}"}
{"id": "8760.png", "formula": "\\begin{align*} a _ { i \\tau ( i , t - 1 ) } \\leq m ^ t _ { i j } \\leq a _ { i \\max \\{ \\tau ( i , t - 1 ) , \\min \\{ j , \\tau ( i , t ) \\} \\} } m ^ { t } _ { i \\tau ( i , t ) } = a _ { \\tau ( i , t ) } , \\end{align*}"}
{"id": "5947.png", "formula": "\\begin{align*} \\min _ { | z | = 1 } \\norm { f - z g } _ 2 \\le \\min _ { | z | = 1 } \\norm { f - z g } _ 4 \\le C ' \\norm { | f | - | g | } _ 4 \\le C '' \\norm { | f | - | g | } _ 2 ^ { \\theta } ( \\norm { f } _ 2 + \\norm { g } _ 2 ) ^ { 1 - \\theta } . \\end{align*}"}
{"id": "7931.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k ^ { \\ell - 1 } S ( k ) = \\frac { 1 } { \\ell } B _ \\ell ( n + 1 ) t - \\frac { 1 } { \\ell } \\sum _ { a \\in S } B _ \\ell ( a ) . \\end{align*}"}
{"id": "1350.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\chi ^ k ( x , \\omega ) \\partial _ i \\eta ( x ) d x = \\int _ { \\mathbb { R } ^ d } U _ i ^ k ( \\tau _ y \\omega ) \\int _ 0 ^ 1 \\sum _ { j \\neq i } \\partial _ j \\left ( \\frac { y _ j } { t ^ d } \\eta \\left ( \\frac { y } { t } \\right ) \\right ) + \\frac { y _ i } { t ^ { d + 1 } } \\partial _ i \\eta \\left ( \\frac { y } { t } \\right ) d t d y . \\end{align*}"}
{"id": "5658.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } = ( \\sigma , ( \\varphi ^ { - 1 } _ { \\sigma ^ { - 1 } 1 } , \\varphi ^ { - 1 } _ { \\sigma ^ { - 1 } 2 } , \\varphi ^ { - 1 } _ { \\sigma ^ { - 1 } 3 } ) ) = ( \\sigma , ( \\varphi ^ { - 1 } _ { \\sigma ^ { - 1 } 1 } , \\varepsilon , \\varphi ^ { - 1 } _ { \\sigma ^ { - 1 } 3 } ) ) \\in \\prescript { \\sigma } { } N _ l ^ A . \\end{align*}"}
{"id": "1430.png", "formula": "\\begin{align*} c _ n : = \\sum _ { s _ 1 , \\dots , s _ k \\in { \\mathbb N } s _ 1 + \\cdots + s _ k = n } { n \\choose s _ 1 , \\dots , s _ k } \\ ; \\prod _ { ( v _ i , v _ j ) \\in E ( D ) } 2 ^ { s _ i s _ j } . \\end{align*}"}
{"id": "9385.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n + 1 } S _ { 2 , \\lambda } ( n + 1 , k ) k ! x ^ { k } = - n \\lambda F _ { n , \\lambda } ( x ) + x F _ { n , \\lambda } ( x ) + ( x + x ^ { 2 } ) F _ { n , \\lambda } ^ { \\prime } ( x ) . \\end{align*}"}
{"id": "2983.png", "formula": "\\begin{align*} \\tilde \\Lambda _ { n , 2 } ^ { ( 1 , 3 ) } = 1 + o ( 1 ) , \\end{align*}"}
{"id": "5668.png", "formula": "\\begin{align*} x \\cdot J ( y \\cdot x ) = J y \\end{align*}"}
{"id": "1786.png", "formula": "\\begin{align*} \\langle \\operatorname { t r } _ e , \\operatorname { I n d } _ { C ^ * _ r ( \\Gamma ) } ( \\widetilde { D } ) \\rangle = \\operatorname { i n d } ( D ) . \\end{align*}"}
{"id": "659.png", "formula": "\\begin{align*} A ( x ) \\ = \\ \\frac { f ( x ) - g ( x ) } { h ( x ) + 1 } ( x = 0 , 1 , 2 , \\ldots ) , \\end{align*}"}
{"id": "7335.png", "formula": "\\begin{align*} \\max \\{ u ( y _ i , t ) , u ( z _ i , t ) \\} < u ( x _ i , t ) + \\varepsilon < h + \\varepsilon \\ i = 1 , 2 . \\end{align*}"}
{"id": "1003.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\frac { v ( y ) + v ( - y ) } { \\vert y \\vert ^ { n + 2 s } } \\dd y & = 0 . \\end{align*}"}
{"id": "2173.png", "formula": "\\begin{align*} \\langle J ^ { ' } ( u ) , v \\rangle & = \\int _ { \\mathbb { R } ^ { d } } \\int _ { \\mathbb { R } ^ { d } } g \\left ( \\frac { u ( x ) - u ( y ) } { \\vert x - y \\vert ^ { \\alpha } } \\right ) \\frac { v ( x ) - v ( y ) } { \\vert x - y \\vert ^ { \\alpha + d } } d x d y + \\int _ { \\mathbb { R } ^ { d } } g ( u ) v \\ d x - \\int _ { \\mathbb { R } ^ { d } } K ( x ) f ( x , u ) v \\ d x , \\end{align*}"}
{"id": "9249.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) . \\end{align*}"}
{"id": "1469.png", "formula": "\\begin{align*} \\phi _ { \\infty } '' = \\left [ \\begin{array} { c c } i _ n & 0 \\\\ 0 & - i _ n \\end{array} \\right ] , \\psi _ { \\infty } '' = \\left [ \\begin{array} { c c } 0 & - i _ n \\\\ - i _ n & 0 \\end{array} \\right ] . \\end{align*}"}
{"id": "4498.png", "formula": "\\begin{align*} \\varepsilon _ 1 ( x , T ) = 2 \\frac { ( \\log x ) ^ 2 } { T } . \\end{align*}"}
{"id": "424.png", "formula": "\\begin{align*} X : = X _ { T _ { 0 } } ^ { s } ( g _ { 2 } , M ) . \\end{align*}"}
{"id": "2394.png", "formula": "\\begin{align*} \\norm { \\sum _ { k = N _ { 2 n } + 1 } ^ { N _ { 2 n + 1 } } e _ { \\sigma ( k ) } } _ \\mathcal { B } & = \\norm { \\sum _ { \\gamma \\in F _ { 2 n + 1 } } e _ \\gamma - \\sum _ { \\gamma \\in F _ { 2 n } } e _ \\gamma } _ \\mathcal { B } \\\\ & \\geq \\norm { f - \\sum _ { \\gamma \\in F _ { 2 n } } e _ \\gamma } _ \\mathcal { B } - \\norm { f - \\sum _ { \\gamma \\in F _ { 2 n + 1 } } e _ \\gamma } _ \\mathcal { B } > \\varepsilon - \\frac { \\varepsilon } { 2 } = \\frac { \\varepsilon } { 2 } . \\end{align*}"}
{"id": "7035.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\| \\phi _ j p \\| _ b ^ 2 \\le C _ 1 ^ 2 \\| p \\| _ { H ^ 2 } ^ 2 \\Big ( \\sum _ { j = 1 } ^ \\infty \\| \\phi _ j \\| _ b ^ 2 \\Big ) = C _ 1 ^ 2 C _ 2 ^ 2 \\| p \\| _ { H ^ 2 } ^ 2 . \\end{align*}"}
{"id": "5020.png", "formula": "\\begin{align*} H _ { Z _ { * } } \\circ \\Lambda _ { Z _ { * } } ( 1 , 0 ) & = H _ { Z _ { * } } ( \\lambda _ { Z _ { * } } , 0 ) \\\\ & = ( \\eta ^ { - r _ 0 } _ * ( \\lambda _ { Z _ { * } } ) , 0 ) \\\\ & = ( \\eta ^ { - r _ 0 } _ * ( \\eta ^ { r _ 0 } _ { * } ( 1 ) ) , 0 ) \\\\ & = ( 1 , 0 ) \\end{align*}"}
{"id": "1947.png", "formula": "\\begin{align*} f ( \\bar { x } ) = g ( \\bar { x } ) \\leq f ( x ) . \\end{align*}"}
{"id": "8372.png", "formula": "\\begin{align*} h _ { i j } ( \\tilde { x } _ 0 ) M ^ i _ k ( \\tilde { x } ^ 0 ) M ^ j _ l ( \\tilde { x } ^ 0 ) = \\delta _ { k l } \\ , , \\end{align*}"}
{"id": "6896.png", "formula": "\\begin{align*} \\bar { X } _ W = \\{ V \\in X ( k ) : V \\subseteq W \\} . \\end{align*}"}
{"id": "159.png", "formula": "\\begin{align*} \\tilde { f } _ \\gamma = \\int _ { 0 } ^ { + \\infty } P _ t ^ \\gamma ( f ) d t = ( - \\mathcal { L } ^ \\gamma ) ^ { - 1 } ( f ) . \\end{align*}"}
{"id": "2098.png", "formula": "\\begin{align*} H _ k ( x ) = H _ K ( x ) ^ { 1 / [ K : k ] } , h _ k ( x ) = \\frac { 1 } { [ K : k ] } h _ K ( x ) , \\end{align*}"}
{"id": "7154.png", "formula": "\\begin{align*} W _ { t } : = B ^ { H } ( ( K _ { H } ^ { \\ast } ) ^ { - 1 } ( 1 _ { [ 0 , t ] } ) ) \\end{align*}"}
{"id": "6286.png", "formula": "\\begin{align*} \\widehat { K ' } _ { \\mathrm { a - c p t - d } } ^ \\mathbb { R } = \\Re \\{ \\mathbf { s } ^ H \\mathbf { y } \\} / ( N _ 2 \\sqrt { \\rho } ) \\end{align*}"}
{"id": "2211.png", "formula": "\\begin{align*} F ( v ) ( x ) = f ( v ( x ) ) = v ^ 3 ( x ) - v ( x ) , x \\in D , v \\in L ^ 6 ( D ; \\mathbb { R } ) . \\end{align*}"}
{"id": "760.png", "formula": "\\begin{align*} \\| \\delta _ \\alpha ( f ) \\| _ { C _ b ( \\Omega _ X ) } & = \\max ( | f | _ { C ^ \\alpha ( X ) } , \\| f \\| _ { C ( X ) } ) , f , f _ 1 , f _ 2 \\in C ^ \\alpha ( X ) . \\end{align*}"}
{"id": "8527.png", "formula": "\\begin{align*} y \\coloneqq \\left ( \\prod _ { i = 1 } ^ { d _ c - 1 } \\operatorname { s g n } ( \\psi _ i ) \\right ) \\sum _ { i = 1 } ^ { d _ c - 1 } ( - \\log | \\psi _ i | ) . \\end{align*}"}
{"id": "3492.png", "formula": "\\begin{align*} E ( s _ 1 , s _ 3 ; n , M ) = O \\left ( \\frac { 1 } { n ^ { \\sigma _ 1 + \\sigma _ 3 } } \\right ) \\end{align*}"}
{"id": "4784.png", "formula": "\\begin{align*} n = m ^ 2 , \\theta = m - 3 , \\hat \\theta = \\frac { | m - 3 | } { m ( m - 1 ) } \\sqrt { \\frac { m + 1 } { 2 } } . \\end{align*}"}
{"id": "7454.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\left ( \\| \\phi \\| ^ 2 \\right ) = 2 \\mathrm { R e } \\langle \\phi , \\dot { \\phi } \\rangle = 0 \\quad \\mbox { a n d } \\frac { \\mathrm { d } ^ 2 } { \\mathrm { d } t ^ 2 } \\left ( \\| \\phi \\| ^ 2 \\right ) = 2 \\left ( \\mathrm { R e } \\ , \\langle \\phi , \\ddot { \\phi } \\rangle + \\| \\dot { \\phi } \\| ^ 2 \\right ) = 0 , \\forall t > 0 . \\end{align*}"}
{"id": "8344.png", "formula": "\\begin{align*} g ( p ) : = \\inf \\big \\{ r : \\P ( w \\ge r ) \\le p \\big \\} = \\inf \\big \\{ r : \\Lambda ( e ^ r - 1 ) \\le p \\big \\} . \\end{align*}"}
{"id": "7951.png", "formula": "\\begin{align*} \\begin{cases} L _ { \\alpha _ j } < k _ 1 & \\alpha = 1 \\pi _ { 1 } ^ { \\circ } = 1 \\\\ L _ { \\alpha _ j } \\geq k _ { m - 1 } & \\alpha = \\mu - 1 \\pi _ { \\mu } ^ { \\circ } = 1 \\\\ k _ { r _ { \\alpha } } \\leq L _ { \\alpha _ j } < k _ { r _ { \\alpha } + 1 } & \\end{cases} \\end{align*}"}
{"id": "6824.png", "formula": "\\begin{align*} f ^ * ( g _ { \\textnormal { e u c } } ) = e ^ { 2 u } \\ , g _ { \\textnormal { p o i n } } , \\end{align*}"}
{"id": "9319.png", "formula": "\\begin{align*} \\| ( \\mathbf { x } _ { k + 1 } , \\mathbf { y } _ { k + 1 } ) - \\mathcal { P } ( \\mathbf { x } _ { k } , \\mathbf { y } _ { k } ) \\| \\leq \\varepsilon _ k \\min \\{ 1 , \\| ( \\mathbf { x } _ { k + 1 } , \\mathbf { y } _ { k + 1 } ) - ( \\mathbf { x } _ { k } , \\mathbf { y } _ { k } ) \\| \\} , \\hbox { w h e r e } \\sum _ { k = 0 } ^ { + \\infty } \\varepsilon _ k < \\infty . \\end{align*}"}
{"id": "7287.png", "formula": "\\begin{align*} p ( 1 ) & = p ( 0 ) = 1 , & p ( 4 ) & = p ( 3 ) + p ( 2 ) = 3 + 2 = 5 , \\\\ p ( 2 ) & = p ( 1 ) + p ( 0 ) = 2 , & p ( 5 ) & = p ( 4 ) + p ( 3 ) - p ( 0 ) = 5 + 3 - 1 = 7 , \\\\ p ( 3 ) & = p ( 2 ) + p ( 1 ) = 3 , & p ( 6 ) & = p ( 5 ) + p ( 4 ) - p ( 1 ) = 7 + 5 - 1 = 1 1 \\end{align*}"}
{"id": "914.png", "formula": "\\begin{align*} ( x ) _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 6 ] ) . \\end{align*}"}
{"id": "5262.png", "formula": "\\begin{align*} \\varphi _ S = \\varphi \\circ S , \\varphi _ { S ^ { - 1 } } = \\varphi \\circ S ^ { - 1 } . \\end{align*}"}
{"id": "3651.png", "formula": "\\begin{align*} \\frac { 2 \\log \\log t _ 0 + 1 } { \\log ^ { 5 / 3 } t _ 0 ( \\log \\log t _ 0 ) ^ { 4 / 3 } } = \\frac { 3 c } { \\log x } \\end{align*}"}
{"id": "8053.png", "formula": "\\begin{align*} \\partial \\Phi ^ n _ { U } ( f ) [ \\phi ] & = \\int _ U f ( u , v ) ( \\partial _ u \\phi ) ^ n ( u ) \\ , \\mathrm { d } u \\mathrm { d } v \\\\ & = \\int _ { \\mathcal { I } } \\left ( \\int _ { \\mathbb { R } } f ( u , v ) \\ , \\mathrm { d } v \\right ) ( \\partial _ u \\phi ) ^ n ( u ) \\ , \\mathrm { d } u \\\\ & = \\Psi ^ n ( \\eta f ) [ \\partial _ { \\Sigma } \\phi ] \\end{align*}"}
{"id": "6444.png", "formula": "\\begin{align*} \\P \\Big [ \\ , \\bigcap _ { j = 1 } ^ n F _ { z _ j , r _ j } \\Big ] \\le c ' ( \\xi ) ^ { K } + \\P [ | \\mathcal S | < K ] \\end{align*}"}
{"id": "2463.png", "formula": "\\begin{align*} S = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} S ^ { - 1 } = \\begin{pmatrix} d & - b \\\\ - c & a \\end{pmatrix} . \\end{align*}"}
{"id": "564.png", "formula": "\\begin{align*} \\gamma = & c \\mathcal E ( \\rho _ 0 , u _ 0 | r ( 0 , \\cdot ) , U ( 0 , \\cdot ) ) + c ( D , T ) ( \\varepsilon ^ \\alpha + R ^ { - 1 } + ( \\varepsilon R ) ^ { - 1 } + \\nu + \\varepsilon ^ 2 \\nu ^ 2 ( 1 + R ^ { - 2 } ) + \\varepsilon \\nu ^ { - 1 } ) , \\\\ \\delta ( \\tau ) = & c ( D , T ) \\Bigl ( \\frac { 1 } { | \\Omega _ R | } \\int _ { \\Omega _ R } p ( \\rho ) + \\varepsilon ^ \\frac { 4 } { 3 } \\nu ^ { - 1 } ( 1 + R ^ { - 4 } ) + ( 1 + \\varepsilon ^ 2 ) ( 1 + R ^ { - 2 } ) + \\varepsilon ^ 2 \\nu + 1 \\Bigr ) . \\end{align*}"}
{"id": "882.png", "formula": "\\begin{align*} \\Delta _ { g _ M } \\xi + \\sum _ { j = 1 } ^ d a _ j e ^ { ( \\iota ^ \\ast u ^ j \\xi ) } \\iota ^ \\ast u ^ j = w . \\end{align*}"}
{"id": "2019.png", "formula": "\\begin{align*} Y _ { t } = \\exp \\left ( M _ { t } ^ { F ^ u } + M _ t ^ { - u , c } - \\int _ { 0 } ^ { t } N ( G ^ { u } - F ^ { u } ) ( X _ { s } ) { \\rm d } H _ { s } - \\frac { \\ , 1 \\ , } { 2 } \\ < M ^ { u , c } \\ > _ t \\right ) \\end{align*}"}
{"id": "3120.png", "formula": "\\begin{align*} F _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) = 0 \\ , , F _ 2 = 0 \\ , , F _ 3 = 0 \\ , . \\end{align*}"}
{"id": "1747.png", "formula": "\\begin{align*} [ D _ { G , K } \\hat { \\otimes } 1 , 1 \\hat { \\otimes } D _ { Z } ] = 0 , D ^ 2 = D _ { G , K } ^ 2 + D _ { Z } ^ 2 . \\end{align*}"}
{"id": "8498.png", "formula": "\\begin{align*} f _ { \\rho , L } ( x , y ) = 1 + \\rho \\sin ( 2 ^ { L + 1 } \\pi x ) \\sin ( 2 ^ { L + 1 } \\pi x ) \\end{align*}"}
{"id": "8488.png", "formula": "\\begin{align*} f ^ { t } _ k = \\frac { N ^ t _ { 3 k + 1 } } { D _ t } , g ^ { t } _ k = \\frac { N ^ t _ { 3 k + 2 } } { D _ t } , h ^ { t } _ k = \\frac { N ^ t _ { 3 k + 3 } } { D _ t } , \\end{align*}"}
{"id": "4516.png", "formula": "\\begin{align*} \\begin{gathered} A _ 0 ( a _ i ) ^ 2 \\subseteq A _ 0 ( a _ i ) , A _ { \\eta } ( a _ i ) ^ 2 \\subseteq A _ 0 ( a _ i ) + A _ 1 ( a _ i ) , \\\\ ( A _ 0 ( a _ i ) + A _ 1 ( a _ i ) ) A _ { \\eta } ( a _ i ) \\subseteq A _ { \\eta } ( a _ i ) , A _ 0 ( a _ i ) A _ 1 ( a _ i ) = ( 0 ) . \\end{gathered} \\end{align*}"}
{"id": "4116.png", "formula": "\\begin{align*} d \\theta ^ 1 & \\equiv \\theta ^ 1 \\wedge \\omega _ 1 { } ^ 1 \\mod \\theta \\wedge \\theta ^ { \\bar 1 } , \\\\ \\omega _ { 1 \\bar 1 } + \\omega _ { \\bar 1 1 } & = d h _ { 1 \\bar 1 } , \\end{align*}"}
{"id": "5232.png", "formula": "\\begin{align*} & ( ( b _ r - d ) / 2 + k + 1 ) ( d k + e ) - ( b _ r + d k - d + e ) ( k + 1 ) \\\\ = & ( b _ r - d ) ( ( d - 2 ) k + e - 2 ) / 2 \\ge 0 . \\end{align*}"}
{"id": "1701.png", "formula": "\\begin{align*} \\widehat M = \\left \\{ f : \\Omega \\rightarrow \\R , \\ ; \\ ; \\left \\| \\frac { \\nabla ^ r f } { g } \\right \\| ^ { p _ 1 } _ { L _ { p _ 1 } ( \\Omega ) } + \\left \\| \\frac { f } { g _ 0 } \\right \\| ^ { p _ 1 } _ { L _ { p _ 1 } ( \\Omega ) } \\le 1 , \\ ; \\ ; \\| w f \\| _ { L _ { p _ 0 } ( \\Omega ) } \\le 1 \\right \\} , \\end{align*}"}
{"id": "9408.png", "formula": "\\begin{align*} \\varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) = \\varphi ( a _ 1 a _ 2 \\cdots a _ n ) \\varphi ( b _ 0 ) \\varphi ( b _ 1 ) \\cdots \\varphi ( b _ { n - 1 } ) \\varphi ( b _ n ) . \\end{align*}"}
{"id": "3152.png", "formula": "\\begin{align*} f _ j ( z _ k ) = ( \\bar A z _ k - b ) ^ \\intercal S _ j ( S _ j ^ \\intercal \\bar A B ^ { - 1 } \\bar { A } ^ \\intercal S _ { i _ k } ) ^ \\dagger S _ { i _ k } ^ \\intercal ( \\bar A z _ k - b ) , ~ j = 1 , \\ldots , \\epsilon . \\end{align*}"}
{"id": "741.png", "formula": "\\begin{align*} \\widehat { D } ^ { ( \\ell ) } = \\mathrm { D i a g } \\bigg ( \\underbrace { a _ + \\xi _ i ^ { ( \\ell ) } + a _ - ( 1 - \\xi _ { i } ^ { ( \\ell ) } ) } _ { = : d _ { i } ^ { ( \\ell ) } } , \\ , i = 1 , \\ldots , n _ \\ell \\bigg ) , \\xi _ i ^ { ( \\ell ) } \\sim \\mathrm { B e r n o u l l i } ( 1 / 2 ) \\ , \\ , i i d . \\end{align*}"}
{"id": "3815.png", "formula": "\\begin{align*} 0 & = \\varepsilon \\circ \\psi ( u v u v ^ { - 1 } [ a _ 1 , a _ 2 ] \\cdots [ a _ { 2 m - 1 } , a _ { 2 m } ] ) = \\varepsilon \\circ \\psi ( u v u v ^ { - 1 } ) \\\\ & = \\varepsilon \\bigl ( w _ u \\ldotp g ^ { 2 k + 1 } w _ v g ^ { - 2 k - 1 } \\ldotp g ^ { 2 k + 1 + b } w _ u g ^ { - 2 k - 1 - b } \\ldotp \\Delta _ { 4 k + 2 } ^ 2 \\ldotp w _ v ^ { - 1 } \\bigr ) = 2 \\varepsilon ( w _ u ) + \\varepsilon ( \\Delta _ { 4 k + 2 } ^ 2 ) . \\end{align*}"}
{"id": "6433.png", "formula": "\\begin{align*} H _ { N } ( z q , q ^ 2 ) = \\Psi _ { N } ( z q , z q , q / z , q / z ) . \\end{align*}"}
{"id": "253.png", "formula": "\\begin{align*} p _ { \\varepsilon } ( y ) = \\frac { 1 } { ( 2 \\pi \\varepsilon ) ^ { \\frac { d } { 2 } } } \\exp \\left ( - \\dfrac { \\| y \\| ^ 2 } { 2 \\varepsilon } \\right ) . \\end{align*}"}
{"id": "5023.png", "formula": "\\begin{align*} ( \\eta _ k ^ { r _ { k } } ) ' ( \\xi _ k ( 0 ) ) \\cdot \\xi _ k ^ { ''' } ( 0 ) \\cdot \\lambda _ { Z _ * ^ k } ^ 2 = \\xi _ { k + 1 } ^ { ''' } ( 0 ) , \\end{align*}"}
{"id": "9446.png", "formula": "\\begin{align*} d _ { q - 1 } \\tilde p = ( I - \\varPhi _ q \\ , d _ q ) \\Big ( \\varPsi _ { \\mu , q } f + \\varPsi _ { \\mu , q , 0 } u _ 0 - \\varPsi _ \\mu { \\mathcal N } _ q u \\Big ) . \\end{align*}"}
{"id": "3077.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\pm x _ 2 ( \\pm j x _ 2 , \\pm j ^ 2 x _ 2 ) \\ , , x _ 2 ^ \\prime = x _ 1 \\ , , y _ 1 ^ \\prime = \\pm y _ 2 \\ , , y _ 2 ^ \\prime = y _ 1 \\ , . \\end{align*}"}
{"id": "2952.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , \\lambda } & = \\frac { 1 6 \\cdot \\left ( 1 + o ( 1 ) \\right ) } { 9 0 ^ 2 n ^ 4 \\delta ^ 4 _ n ( k ) } \\cdot \\sum \\limits _ { \\mathbf { P } _ { k } \\in \\mathcal { W } _ { k } ( \\lambda ) } \\sum \\limits _ { \\mathbf { i } \\in \\mathcal { I } _ 2 \\times \\mathcal I _ 2 } \\varphi _ { k - 1 } \\left ( \\mathbf { P } _ { k } , \\mathbf { i } \\right ) . \\end{align*}"}
{"id": "1706.png", "formula": "\\begin{align*} W _ { t , m } = 2 ^ { \\mu _ * k _ * t } \\cdot 2 ^ { \u2010 m ( s _ * + 1 / q \u2010 1 / p _ 1 ) } B _ { p _ 1 } ^ { \\nu _ { t , m } } \\cap 2 ^ { \u2010 \\alpha _ * k _ * t } \\cdot 2 ^ { \u2010 m ( 1 / q \u2010 1 / p _ 0 ) } B _ { p _ 0 } ^ { \\nu _ { t , m } } . \\end{align*}"}
{"id": "4261.png", "formula": "\\begin{align*} | I _ 3 ( x ) | ~ \\leq ~ \\O ( 1 ) , \\big | I ' _ 3 ( x ) \\big | ~ = ~ \\left | \\int _ { 3 x / 2 } ^ { 2 } { g _ b ' ( y ) \\over y - x } ~ d y + { 3 \\over 2 } \\cdot g ' _ b \\left ( { 3 x \\over 2 } \\right ) \\cdot \\ln \\left ( { x \\over 2 } \\right ) \\right | ~ \\leq ~ \\O ( 1 ) \\cdot \\ln ^ 2 x , \\end{align*}"}
{"id": "6477.png", "formula": "\\begin{align*} & \\frac { 1 } { ( 2 m - 1 ) ! ! } E \\left [ \\left ( \\dfrac { S _ n } { \\sqrt { n / ( 1 - 2 \\alpha ) } } \\right ) ^ { 2 m } \\right ] - 1 \\sim \\frac { m ( m - 1 ) } { 2 } \\cdot c ( \\alpha ) \\cdot n ^ { - 1 } , \\end{align*}"}
{"id": "4866.png", "formula": "\\begin{align*} F ( z ) : = f ( x + z ) - f ( x ) - z f ' ( x ) . \\end{align*}"}
{"id": "1783.png", "formula": "\\begin{align*} g = \\kappa ( g ) \\mu ( g ) e ^ { H ( g ) } n \\in K M A N = G . \\end{align*}"}
{"id": "248.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\psi } ( g _ i ) ( x ) = \\Delta ( g _ i ) ( x ) + \\left \\langle \\frac { \\nabla ( \\psi ) ( x ) } { \\psi ( x ) } ; \\nabla ( g _ i ) ( x ) \\right \\rangle . \\end{align*}"}
{"id": "7368.png", "formula": "\\begin{align*} u _ t + \\big \\{ m ( K \\cap \\{ u ( \\cdot , t ) < u ( x , t ) \\} ) - 1 \\big \\} | \\nabla u | = 0 \\end{align*}"}
{"id": "575.png", "formula": "\\begin{align*} \\big \\{ ( x , x ^ \\prime , y , y ^ \\prime ) \\ \\ y = f ( x ) \\big \\} \\approx \\Gamma _ f \\times \\R ^ { n ^ \\prime } \\times \\R ^ { m ^ \\prime } \\end{align*}"}
{"id": "9444.png", "formula": "\\begin{align*} ( I \\ ! + \\ ! \\varPhi _ q \\ , d _ q \\ , \\varPsi _ { \\mu , q } { \\mathcal N } _ q ) u = \\varPhi _ q \\ , d _ q \\ , ( \\varPsi _ { \\mu , q } f + \\varPsi _ { \\mu , q , 0 } u _ 0 ) \\end{align*}"}
{"id": "9309.png", "formula": "\\begin{gather*} \\int _ 0 ^ 1 x ^ { 2 N + 1 } \\Lambda ( x ) \\ , d x \\geq \\int _ { 1 - 1 / N } ^ 1 x ^ { 2 N + 1 } \\Lambda ( x ) \\ , d x \\geq \\alpha \\sum _ { n = N } ^ \\infty | I _ n | c _ n . \\end{gather*}"}
{"id": "1691.png", "formula": "\\begin{align*} G _ { m , n } & = \\sum \\limits _ { k = 1 } ^ n \\frac { ( - 1 ) ^ { k - 1 } } { ( m - 1 ) ! } \\binom { n } { k } \\sum \\limits _ { l = 0 } ^ { k - 1 } \\binom { k - 1 } { l } \\frac { l ! ( m + k - 2 ) ! } { ( m + k + l - 1 ) ! } \\\\ & = \\sum \\limits _ { k = 1 } ^ n \\frac { ( - 1 ) ^ { k - 1 } } { ( m - 1 ) ! ( m + k - 1 ) } \\binom { n } { k } \\sum \\limits _ { l = 0 } ^ { k - 1 } \\frac { ( 1 ) _ l ( 1 - k ) _ l ( - 1 ) ^ l } { l ! ( m + k ) _ l } \\\\ & = \\sum \\limits _ { k = 1 } ^ n \\frac { ( - 1 ) ^ { k - 1 } } { ( m - 1 ) ! ( m + k - 1 ) } \\binom { n } { k } { _ 2 } F _ 1 ( 1 , 1 - k ; m + k ; - 1 ) . \\end{align*}"}
{"id": "21.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\| \\phi ^ j \\| _ { L _ x ^ 2 } ^ 2 \\leq \\lim _ { n \\to \\infty } \\| u _ n \\| ^ 2 _ { L _ x ^ 2 } = 1 . \\end{align*}"}
{"id": "7901.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\gamma _ F ( P \\cap \\tau _ { v _ l } P ) = ( \\gamma _ F ( P ) ) ^ 2 . \\end{align*}"}
{"id": "9514.png", "formula": "\\begin{align*} \\Gamma ( f ) \\Gamma ( g ) & = A + B - C + f ( 0 ) g ( 0 ) \\\\ & = ( \\Gamma ( f g ) + f ( 0 ) g ( 0 ) ) + ( C - 2 f ( 0 ) g ( 0 ) ) - C + f ( 0 ) g ( 0 ) \\\\ & = \\Gamma ( f g ) . \\end{align*}"}
{"id": "2646.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\langle X g _ n , P g _ n \\rangle - \\langle P g _ n , X g _ n \\rangle \\right ) = \\langle X g , P g \\rangle - \\langle P g , X g \\rangle = 0 . \\end{align*}"}
{"id": "5179.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) \\ge \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ N \\left ( \\frac { \\rho _ \\ell T _ { \\ell , \\pi } ^ { a v } } { 2 } + \\rho _ \\ell \\gamma _ \\ell + \\frac { c _ \\ell } { T _ { \\ell , \\pi } ^ { a v } } \\right ) . \\end{align*}"}
{"id": "7114.png", "formula": "\\begin{align*} y _ t = x + \\int _ 0 ^ t \\tilde { f } ( s , y _ s ) d s - \\tilde { \\sigma } B _ t ^ H , 0 \\le t \\le 1 , H \\in ( 0 , \\frac { 1 } { 2 } ) , \\end{align*}"}
{"id": "4635.png", "formula": "\\begin{align*} \\lambda _ 1 + \\lambda _ { k + 1 } + \\lambda _ { 2 k + 1 } + \\cdots + \\lambda _ { ( t - 1 ) k + 1 } = \\sum \\limits _ { j = 1 } ^ { t } \\sum \\limits _ { i = 1 } ^ k a _ { i , j } , \\end{align*}"}
{"id": "4936.png", "formula": "\\begin{align*} B ( G ) _ { \\rm b a s i c } \\simeq \\pi _ 1 ( G ) _ \\Gamma = ( \\pi _ 1 ( G ) _ I ) _ \\phi = \\Omega _ \\phi , \\end{align*}"}
{"id": "1631.png", "formula": "\\begin{align*} \\Gamma ( f , g ) = \\frac 1 2 \\left \\lbrace \\mathcal { L } ^ { ( 1 ) } ( f g ) - f \\mathcal { L } ^ { ( 1 ) } g - g \\mathcal { L } ^ { ( 1 ) } f \\right \\rbrace \\end{align*}"}
{"id": "4668.png", "formula": "\\begin{align*} | g ' ( y ) | = O \\bigg ( \\frac { 1 } { | y | } \\bigg ) \\ ; \\ ; | y | \\rightarrow + \\infty . \\end{align*}"}
{"id": "439.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { 0 } U _ { t } + A ^ { i } \\partial _ { i } U - \\epsilon B ^ { i j } \\partial _ { i } \\partial _ { j } U + D U & = F , \\\\ \\left . V \\right \\rvert _ { t = 0 } & = U _ { 0 } , \\end{aligned} \\end{align*}"}
{"id": "6012.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - t D _ { { \\rm l e f t } } } u ( x ) & = u ( x - t ) \\\\ e ^ { - t D _ { { \\rm r i g h t } } } v ( x ) & = v ( x + t ) \\end{aligned} \\end{align*}"}
{"id": "4903.png", "formula": "\\begin{align*} | p - q | \\leq 1 , p + q = 1 + s \\geq 2 , 2 p + r = 2 + s = p + q + 1 , r = q - p + 1 , 0 \\leq r \\leq 2 . \\end{align*}"}
{"id": "6251.png", "formula": "\\begin{align*} \\mathcal { L } ( q , \\lambda ( q , t ) ) = \\frac { \\partial } { \\partial t } ( e ^ { \\lambda ( q , t ) } - 1 ) - 1 . \\end{align*}"}
{"id": "7640.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\tilde { \\lambda } _ { n } = \\tilde { \\lambda } _ { 0 } . \\end{align*}"}
{"id": "6210.png", "formula": "\\begin{align*} \\sum _ { a , b = 1 } ^ { j - 1 } t _ { a , b } + \\sum _ { a , b = 1 } ^ { j } t _ { a , b } \\geq 2 j - 1 - \\left ( \\frac { 1 } { \\eta _ j } + \\frac { 1 } { \\eta _ { j - 1 } } \\right ) 1 0 \\epsilon \\| C \\| _ F ^ 2 . \\end{align*}"}
{"id": "9452.png", "formula": "\\begin{align*} f _ 1 = \\Delta ^ { p ^ { s - 1 } } + \\tfrac { a _ 1 } { a _ 0 } X ^ { p ^ { s - 1 } } Y ^ { p ^ { s - 1 } } + \\tfrac { a _ 2 } { a _ 0 } \\Delta ^ { p ^ { s - 2 } } X ^ { 2 p ^ { s - 1 } - 2 p ^ { s - 2 } } \\ldots + \\tfrac { a _ r } { a _ 0 } Y X ^ { 2 p ^ { s - 1 } - 1 } \\end{align*}"}
{"id": "2819.png", "formula": "\\begin{align*} t _ n < t _ n ' , \\ ; \\forall n \\in \\mathbb { N } , \\delta ( t _ n ) = \\varepsilon _ 1 , \\delta ( t ) < \\varepsilon _ 1 , \\ ; \\forall t \\in [ t _ n , t _ n ' ) . \\end{align*}"}
{"id": "5012.png", "formula": "\\begin{align*} [ r _ 0 , r _ 1 , \\ldots , r _ { ( k + 1 ) n + 1 } ] - [ r _ 0 , r _ 1 , \\ldots , r _ { n + 1 } ] & < C \\sum _ { i = k n } ^ { ( k + 1 ) n } { 1 \\over { \\rm F i b } _ { i } \\ { \\rm F i b } _ { i + 1 } } < C \\theta ^ { 2 k n } , \\\\ \\left | [ r _ { ( k + 1 ) n - 1 } , r _ { ( k + 1 ) n - 2 } , \\ldots r _ 0 ] - [ r _ { k n - 1 } , r _ { k n - 2 } , \\ldots r _ 0 ] \\right | & < { 1 \\over { \\rm F i b } _ n \\ { \\rm F i b } _ { n + 1 } } < C \\theta ^ { 2 k n } , \\\\ \\end{align*}"}
{"id": "5021.png", "formula": "\\begin{align*} { T _ { 1 } ^ { - 1 } \\circ L _ { Z } \\circ T _ { 1 } ( x , y ) } & = ( \\ell _ { Z , n } ( x , y ) , \\lambda _ { Z , n } y ) , \\\\ l _ { Z , n } ( x ) & = \\ell _ { Z , n } ( x , 0 ) , \\end{align*}"}
{"id": "1589.png", "formula": "\\begin{align*} \\| \\Pi ^ \\perp \\mathcal { S } _ L ( t ) \\| _ { H ^ 2 _ x L ^ 2 ( \\mu ^ { - ( 1 / 2 + \\varepsilon ) } ) \\rightarrow H ^ 2 _ x L ^ 2 ( \\mu ^ { - 1 / 2 } ) } \\leq C e ^ { - \\lambda t ^ b } , \\Pi ^ \\perp = I - \\Pi \\end{align*}"}
{"id": "7488.png", "formula": "\\begin{align*} & e _ { r } ^ { n } : = \\left \\| G ( \\phi ^ { n } ) - \\mu ^ { n } \\phi ^ { n } \\right \\| _ { \\infty } < \\varepsilon _ { r } , \\\\ & d _ { v } ^ { n } : = \\frac { \\left \\| \\phi ^ { n } - \\phi ^ { n - 1 } \\right \\| _ { \\infty } } { \\tau } < \\varepsilon _ { v } , \\\\ & e _ { c } ^ { n } : = | \\| \\phi ^ n \\| ^ 2 - 1 | < \\varepsilon _ { c } , \\end{align*}"}
{"id": "9362.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } S _ { 1 , \\lambda } ( n , k ) \\mathcal { E } _ { k , \\lambda } ( x ) = n ! \\sum _ { k = 0 } ^ { n } \\binom { x } { k } \\bigg ( - \\frac { 1 } { 2 } \\bigg ) ^ { n - k } . \\end{align*}"}
{"id": "5423.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - 1 \\\\ - 1 & 0 \\end{bmatrix} ^ n \\cdot \\begin{bmatrix} \\lambda & 0 \\\\ 0 & \\lambda \\end{bmatrix} \\cdot \\cal { M } ^ { v _ 1 } _ { v _ 2 } ( \\psi _ { j , i , \\Delta } ) . \\end{align*}"}
{"id": "715.png", "formula": "\\begin{align*} S _ { ( i j ) } ^ { ( \\ell ) } : = K _ { ( i j ) } ^ { ( \\ell ) } - \\kappa _ { ( i j ) } ^ { ( \\ell ) } . \\end{align*}"}
{"id": "8767.png", "formula": "\\begin{align*} b _ { i j } ( \\delta ) = m _ { i j 1 } + \\sum _ { t = 1 } ^ { l _ i - 1 } ( m _ { i j t + 1 } - m _ { i j t } ) \\delta _ { i t } i \\in \\{ 1 , \\ldots , d \\} \\ ; j \\in \\{ 0 , \\ldots , n \\} . \\end{align*}"}
{"id": "610.png", "formula": "\\begin{align*} \\frac { f ( x ) - g ( x ) } { h ( x ) + 1 } \\ = \\ r . \\end{align*}"}
{"id": "340.png", "formula": "\\begin{align*} \\min _ { ( \\rho , m ) \\in C _ F ( \\rho ^ a , \\rho ^ b ) } \\mathcal A ( \\rho , m ) & = \\sup _ { S } \\Big \\{ \\ < S ( 1 ) , \\rho ^ b \\ > - \\ < S ( 0 ) , \\rho ^ a \\ > : \\sup _ { \\rho } \\{ \\ < \\dot S , \\rho \\ > + \\frac 1 4 \\sum _ { i j } v _ { i j } ^ 2 \\theta _ { i j } ( \\rho ) \\\\ & \\quad + \\frac 1 2 \\sum _ { i j } ( \\Sigma _ i - \\Sigma _ j ) ( S _ { i } - S _ { j } ) \\theta _ { i j } ( \\rho ) d W ^ { \\delta } ( t ) \\} = 0 \\Big \\} . \\end{align*}"}
{"id": "3055.png", "formula": "\\begin{align*} d = e \\ , . \\end{align*}"}
{"id": "4059.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) + p ( t ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = u ( 0 , x ) = 0 & x \\in ( 0 , 1 ) , \\end{dcases} \\end{align*}"}
{"id": "3190.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & - 1 & 1 \\\\ 1 & - 1 & 1 + 1 0 ^ { - 5 } \\\\ 3 & - 1 & 3 \\\\ 0 & 1 & 6 \\end{bmatrix} \\end{align*}"}
{"id": "7733.png", "formula": "\\begin{align*} - \\phi _ { t , t t } + \\phi _ { t , x x } = ( | \\phi _ t | ^ 2 - | \\phi _ x | ^ 2 ) \\phi _ t + 2 ( \\phi _ t \\cdot \\phi _ { t t } - \\phi _ x \\cdot \\phi _ { t x } ) \\phi + a \\phi _ { t t } . \\end{align*}"}
{"id": "2590.png", "formula": "\\begin{align*} \\varphi = \\frac { 1 } { \\langle g , \\widetilde { g } \\rangle } \\iint _ { \\R ^ { 2 d } } V _ { \\widetilde { g } } \\varphi ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "1007.png", "formula": "\\begin{align*} \\tau \\leqslant \\inf _ { B _ { 1 / 2 } ^ + } \\frac { \\tilde u ( x ) } { x _ 1 } = \\rho \\inf _ { B _ { \\rho / 2 } ^ + } \\frac { u ( x ) } { x _ 1 } . \\end{align*}"}
{"id": "9493.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor a / 2 \\rfloor } \\binom { a + b - 1 } { k , b + k - 1 , a - 2 k } + \\sum _ { k = 0 } ^ { \\lfloor ( a - 1 ) / 2 \\rfloor } \\binom { a + b - 1 } { k , b + k , a - ( 2 k + 1 ) } , \\end{align*}"}
{"id": "905.png", "formula": "\\begin{align*} & a ( \\ , \\Theta ^ { ( 3 ) } ( S ^ { ( 1 1 ) } ) , T ) = \\\\ & [ a ( \\theta ^ { ( 3 ) } ( S _ 1 ^ { ( 1 1 ) } ; Z ) , T ) / 3 2 + a ( \\theta ^ { ( 3 ) } ( S _ 2 ^ { ( 1 1 ) } ; Z ) , T ) / 7 2 + a ( \\theta ^ { ( 3 ) } ( S _ 3 ^ { ( 1 1 ) } ; Z ) , T ) / 2 4 ] \\\\ & \\cdot [ ( 1 / 3 2 ) + ( 1 / 7 2 ) + ( 1 / 2 4 ) ] ^ { - 1 } . \\end{align*}"}
{"id": "8084.png", "formula": "\\begin{align*} W _ { \\mathbb { M } ^ 2 } ( u , v , u ' , v ' ) = \\lim _ { \\epsilon \\searrow 0 } \\frac { - 1 } { 4 \\pi } \\ln \\left ( \\frac { - ( u - u ' ) ( v - v ' ) + i \\epsilon t } { \\Lambda ^ 2 } \\right ) . \\end{align*}"}
{"id": "6948.png", "formula": "\\begin{align*} u ( x , y ) = \\tanh \\left [ 2 \\left ( x ^ 3 - y ^ 4 \\right ) \\right ] . \\end{align*}"}
{"id": "8606.png", "formula": "\\begin{align*} \\mathcal { I } : = \\big \\{ ( \\iota _ 1 , \\iota _ 2 , \\iota _ 3 , \\iota _ 4 ) \\in \\{ + , - , 0 \\} ^ 4 \\ , \\mbox { s . t . } \\ , \\exists \\ , a , b \\in \\{ 1 , 2 , 3 , 4 \\} \\ , \\ , \\mbox { w i t h } \\ , \\ , \\iota _ a = + , \\iota _ b = - \\big \\} . \\end{align*}"}
{"id": "4708.png", "formula": "\\begin{align*} \\alpha _ i + \\alpha _ { n + 1 - i } = 0 . \\end{align*}"}
{"id": "7863.png", "formula": "\\begin{align*} J \\cap K & = ( ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) ) \\cap ( C _ 1 \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) ) \\\\ & \\subseteq ( ( A \\setminus \\{ \\bar { a } _ n \\} ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( G ) ) \\cap ( ( C \\setminus \\{ \\bar { a } _ n \\} ) \\cup [ \\bar { b } _ n \\mapsto \\bar { a } _ n ] ( H ) ) \\end{align*}"}
{"id": "6160.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } x _ i f ( z _ i ) = \\int _ { \\gamma } f ( z ) w ( z ) d z \\end{align*}"}
{"id": "5739.png", "formula": "\\begin{align*} \\begin{aligned} \\lfloor \\frac { j - 1 } { 2 } \\rfloor - \\lceil \\frac { s - 1 } { 2 } \\rceil + \\frac { l } { 2 } + q + 1 & \\geq \\lceil \\frac { j } { 2 } \\rceil + \\frac { l } { 2 } + \\lfloor \\frac { q } { 2 } \\rfloor \\\\ & \\geq \\lceil \\frac { j } { 2 } \\rceil + \\frac { l } { 2 } + \\lfloor \\frac { j - l } { 2 } \\rfloor \\\\ & \\geq j - \\frac { 1 } { 2 } . \\end{aligned} \\end{align*}"}
{"id": "7547.png", "formula": "\\begin{align*} M = \\mathcal { O } ( \\log T ) \\end{align*}"}
{"id": "7493.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { \\tau } { 2 } \\eta ^ n + \\frac { \\tau ^ 2 } { 2 } \\vartheta ^ n \\right ) \\tilde { \\phi } ^ { n + 1 } _ { j k } - \\frac { \\tau ^ 2 } { 4 } \\left ( \\Delta _ h \\tilde { \\phi } ^ { n + 1 } \\right ) _ { j k } = \\mathcal { H } ^ n _ { j k } , j , k = 0 , 1 , \\ldots , M - 1 , \\end{align*}"}
{"id": "8153.png", "formula": "\\begin{align*} s ( c , d ) + s ( d , c ) = { c ^ 2 + d ^ 2 - 3 \\vert c d \\vert + 1 \\over 1 2 c d } , \\ \\ \\ \\ \\ ( c , d \\in { \\mathbb Z } \\setminus \\{ 0 \\} , \\ \\gcd ( c , d ) = 1 ) . \\end{align*}"}
{"id": "507.png", "formula": "\\begin{align*} \\frac { Y ( \\mathcal { I } _ { \\alpha _ { 1 } } ^ { i _ { 1 } } ) _ { \\omega _ { 2 } \\omega _ { 1 } } ^ { i _ { 2 } m _ { 2 } } } { Y ( \\mathcal { I } ) _ { \\omega _ { 2 } \\omega _ { 1 } } ^ { i _ { 2 } m _ { 2 } } } = g _ { \\alpha _ { 1 } \\omega _ { 1 } \\omega _ { 2 } } ^ { i _ { 1 } i _ { 2 } m _ { 2 } } = \\frac { Y ( \\mathcal { I } _ { \\omega _ { 2 } } ^ { m _ { 2 } } ) _ { \\omega _ { 1 } \\alpha _ { 1 } } ^ { i _ { 1 } i _ { 2 } } } { Y ( \\mathcal { I } ) _ { \\omega _ { 1 } \\alpha _ { 1 } } ^ { i _ { 1 } i _ { 2 } } } \\in \\mathbb { C } \\end{align*}"}
{"id": "7447.png", "formula": "\\begin{align*} E _ g : = E ( \\phi _ g ) = \\inf _ { \\phi \\in S } E ( \\phi ) , \\end{align*}"}
{"id": "7843.png", "formula": "\\begin{align*} z = \\sum _ { i = 0 } ^ { 3 n - 6 j - 4 } \\gamma _ { i } e _ { i } , \\quad \\gamma _ { 3 n - 6 j - 4 } \\neq 0 . \\end{align*}"}
{"id": "5352.png", "formula": "\\begin{align*} H _ i = \\left \\lbrace x \\in \\mathbb { R } ^ n \\middle | \\left \\langle \\tilde a _ i , x \\right \\rangle = b _ i \\right \\rbrace . \\end{align*}"}
{"id": "6681.png", "formula": "\\begin{align*} P _ { { \\bf b } , d } & : = ( - \\theta ) ^ { \\frac { - \\sum _ { j = 1 } ^ s ( b _ j - 1 ) q ^ { d - 1 } } { q - 1 } } \\prod _ { l = 1 } ^ { \\infty } \\prod _ { \\substack { j = 1 \\\\ b _ j \\geq 2 } } ^ s \\Biggl \\{ \\biggl ( 1 - \\frac { t } { \\theta ^ { q ^ l } } \\biggr ) ^ { q ^ { b _ j - 2 } } \\biggl ( 1 - \\frac { t } { \\theta ^ { q ^ { l + 1 } } } \\biggr ) ^ { q ^ { b _ j - 3 } } \\cdots \\biggl ( 1 - \\frac { t } { \\theta ^ { q ^ { l + b _ j - 2 } } } \\biggr ) \\Biggr \\} ^ { q ^ { d - b _ j } } \\end{align*}"}
{"id": "4841.png", "formula": "\\begin{align*} F _ { K , Q } ^ { \\rm C } ( n ) : = F _ n ^ C F _ { K , Q } ^ { \\rm D } ( n ) : = \\frac { n - 1 } n F _ n ^ D . \\end{align*}"}
{"id": "7442.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ 0 ^ T \\int _ { \\mathbb { R } } \\rho _ { 3 } ( s , u ) \\partial _ s G _ { k } ( s , u ) d u d s = - \\int _ 0 ^ T \\int _ { \\mathbb { R } } [ \\rho _ { 3 } ( s , u ) ] ^ 2 \\rho _ { 4 } ( s , u ) d u d s . \\end{align*}"}
{"id": "4317.png", "formula": "\\begin{align*} \\langle \\hat \\varepsilon _ - , \\phi _ { j , \\infty } \\rangle _ { L ^ 2 _ \\rho } = 0 , \\forall j \\le \\ell . \\end{align*}"}
{"id": "4573.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } ( X _ n / \\sqrt { \\langle X \\rangle _ n } > x ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) . \\end{align*}"}
{"id": "2376.png", "formula": "\\begin{align*} f = \\mathop { \\sum \\sum } _ { k , l \\in \\Z ^ d } c _ { k , l } M _ { \\beta l } T _ { \\alpha k } g , \\end{align*}"}
{"id": "2092.png", "formula": "\\begin{align*} \\widehat { \\psi _ { p ^ { 2 k } } } ( \\mathbf { u } ) = \\int _ { V ( \\mathbb { Z } _ p ) } \\psi _ { p ^ { 2 k } } ( f ) e _ { p ^ { 2 k } } ( u _ 1 c _ 1 ( f ) + u _ 2 c _ 2 ( f ) ) \\ , d \\nu ( f ) , \\end{align*}"}
{"id": "7152.png", "formula": "\\begin{align*} K _ { H } ( t , s ) = c _ { H } \\left [ \\left ( \\frac { t } { s } \\right ) ^ { H - \\frac { 1 } { 2 } } ( t - s ) ^ { H - \\frac { 1 } { 2 } } + \\left ( \\frac { 1 } { 2 } - H \\right ) s ^ { \\frac { 1 } { 2 } - H } \\int _ { s } ^ { t } u ^ { H - \\frac { 3 } { 2 } } ( u - s ) ^ { H - \\frac { 1 } { 2 } } d u \\right ] . \\end{align*}"}
{"id": "8912.png", "formula": "\\begin{align*} H o m _ { S h } ( \\Z _ { V , X } ^ \\# , \\mathcal I ) = H o m _ { P S h } ( \\Z _ { V , X } , \\mathcal I ) = \\mathcal I ( V ) \\end{align*}"}
{"id": "8677.png", "formula": "\\begin{align*} Y _ t = H _ t X _ t + V _ t , \\ \\ t = 1 , \\ldots , n , \\ \\ \\frac { 1 } { n } { \\bf E } \\Big \\{ \\sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \\mathbb R } ^ { n _ x } } ^ 2 \\Big \\} \\leq \\kappa \\end{align*}"}
{"id": "6658.png", "formula": "\\begin{align*} \\Sigma _ 1 = \\sum _ { \\substack { 0 \\leq m , n < \\infty \\\\ m + h _ p = n + k _ p } } \\Big ( D _ { 1 , m , n } + D _ { 2 , m , n } + D _ { 3 , m , n } \\Big ) \\frac { 1 } { p ^ { \\frac { m } { 2 } + \\frac { n } { 2 } } } , \\end{align*}"}
{"id": "1896.png", "formula": "\\begin{align*} \\bigcup _ { t \\in \\R } Y _ t = \\Sigma , \\bigcap _ { t \\in \\R } Y _ t = \\emptyset Y _ s \\subset \\mathring { Y } _ t s < t . \\end{align*}"}
{"id": "3099.png", "formula": "\\begin{align*} x ^ \\prime : y ^ \\prime = ( a _ 1 x + b _ 1 y ) : ( a _ 2 x + b _ 2 y ) \\end{align*}"}
{"id": "5083.png", "formula": "\\begin{align*} \\gamma a = \\frac { \\log \\left ( n + \\sqrt { n ^ 2 + 4 } \\right ) - \\log ( 2 ) } { n } . \\end{align*}"}
{"id": "6769.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 ^ - } f ( x ) = \\lim _ { x \\rightarrow 0 ^ + } f ( x ) = f ( 0 ) = 0 , \\lim _ { x \\rightarrow 0 ^ - } f ' ( x ) = \\lim _ { x \\rightarrow 0 ^ + } f ' ( x ) = f ' ( 0 ) = 1 . \\end{align*}"}
{"id": "5099.png", "formula": "\\begin{align*} h _ A ( t ) = t \\ , f _ A ' ( t ) h _ B ( t ) = t \\ , f _ B ' ( t ) . \\end{align*}"}
{"id": "5347.png", "formula": "\\begin{align*} \\bar x = \\arg \\max \\left \\{ \\left \\langle c , x \\right \\rangle \\middle | A x \\leqslant b , x \\in \\mathbb { R } ^ n \\right \\} , \\end{align*}"}
{"id": "3747.png", "formula": "\\begin{align*} T _ m M = E _ 0 ( m ) \\oplus E _ 1 ( m ) \\oplus . . . \\oplus E _ r ( m ) . \\end{align*}"}
{"id": "578.png", "formula": "\\begin{align*} X \\ = \\ \\big \\{ x \\in \\R \\ ( x ^ 2 - 1 ) ( x - 2 ) ^ 2 < 0 \\big \\} . \\end{align*}"}
{"id": "972.png", "formula": "\\begin{align*} \\tilde c _ { n , s } & = \\frac { 2 ^ { 2 s - 1 } \\Gamma \\big ( \\frac { 1 + 2 s } 2 \\big ) } { \\pi ^ { 1 / 2 } \\Gamma ( 1 - s ) } = \\frac { c _ { 1 , s } } { 2 s } . \\end{align*}"}
{"id": "8471.png", "formula": "\\begin{align*} ( k - 0 . 1 ) n c < \\sum _ { u \\sim z } \\sum _ { \\substack { w \\sim u \\\\ w \\in N _ 2 ( z ) } } \\mathbf { \\mathrm { v } } _ w = \\sum _ { u \\sim z } \\sum _ { \\substack { w \\sim u \\\\ w \\in M _ 2 } } \\mathbf { \\mathrm { v } } _ w + \\sum _ { u \\sim z } \\sum _ { \\substack { w \\sim u \\\\ w \\in S _ 2 \\setminus M _ 2 } } \\mathbf { \\mathrm { v } } _ w < 0 . 9 n c + e ( N _ 1 , S _ 2 \\setminus M _ 2 ) \\frac { \\alpha } { 3 } , \\end{align*}"}
{"id": "7868.png", "formula": "\\begin{align*} [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( K ) \\cup [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( L ) & = [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( J \\cup K \\cup L ) \\\\ & = [ \\bar { a } _ n \\mapsto \\bar { b } _ n ] ( \\mathbb { N } ) = \\{ \\bar { b } _ n \\} \\end{align*}"}
{"id": "4822.png", "formula": "\\begin{align*} \\Omega _ + = \\{ ( e ^ { t H _ p } ( y , \\eta ) , y , \\eta ) \\ , : \\ , p ( y , \\eta ) = 0 \\} \\subset T ^ \\ast ( X \\times X ) \\simeq T ^ \\ast X \\times T ^ \\ast X , \\end{align*}"}
{"id": "5630.png", "formula": "\\begin{align*} f _ c ( x ) = \\frac { 1 } { 2 } ( ( x ^ 1 ) ^ 2 + a ( x ^ 2 ) ^ 2 - b ( x ^ 3 ) ^ 2 ) - c = 0 \\end{align*}"}
{"id": "6940.png", "formula": "\\begin{align*} F _ \\pi ( v ) = \\sum _ { E \\in { \\cal T } _ h } \\int _ E \\left ( \\Pi _ { E , q - 1 } f \\right ) v \\ , , \\end{align*}"}
{"id": "1564.png", "formula": "\\begin{align*} \\mathfrak { D } ^ d f = \\mathfrak { D D } ^ { d - 1 } f , \\ , \\ , \\overline { \\mathfrak { D } } ^ d f = \\overline { \\mathfrak { D } } \\overline { \\mathfrak { D } } ^ { d - 1 } f , \\ , \\ , \\mathfrak { C } ^ d f = \\mathfrak { C C } ^ { d - 1 } f , \\ , \\ , \\mathfrak { D } ^ 0 f = \\overline { \\mathfrak { D } } ^ 0 f = \\mathfrak { C } f = f . \\end{align*}"}
{"id": "6091.png", "formula": "\\begin{align*} r = \\min \\left \\{ [ q \\mu _ { i } - ( m _ { i } - 1 ) ] , 2 \\ell - 2 \\right \\} - ( m _ { i } - 2 ) . \\end{align*}"}
{"id": "2703.png", "formula": "\\begin{align*} B f ( z ) = e ^ { q ( z ) } . \\end{align*}"}
{"id": "5041.png", "formula": "\\begin{align*} \\Omega _ 0 ( X + \\xi , Y + \\eta ) : = \\frac { 1 } { 2 } ( \\xi ( Y ) - \\eta ( X ) ) . \\end{align*}"}
{"id": "9073.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ 1 ^ { i n } ( x ) = 2 - x ^ 2 , \\rho _ 2 ^ { i n } ( x ) = 2 + s i n ( \\pi x ) , \\\\ & \\phi ( 0 , t ) = - 1 , \\ \\phi ( 1 , t ) = 1 . \\end{aligned} \\end{align*}"}
{"id": "7928.png", "formula": "\\begin{align*} B _ n ( x ) = \\sum _ { i = 0 } ^ { n } \\binom n i B _ i x ^ { n - i } , \\end{align*}"}
{"id": "8228.png", "formula": "\\begin{align*} N _ 1 = x _ 1 \\sqrt { \\frac { z B } { F _ { k , 1 } ( B ) } } . \\end{align*}"}
{"id": "4736.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ 2 [ \\mathfrak { a } _ i ^ + ( t ) ] ^ 2 \\leq c _ 0 \\ , \\sup _ { \\tau \\geq t } \\bigg ( \\frac { 1 } { [ \\mathfrak { q } _ 1 ( \\tau ) - \\mathfrak { q } _ 2 ( \\tau ) ] ^ 4 } + \\sum _ { i = 1 } ^ 2 [ \\mathfrak { a } _ i ^ - ( \\tau ) ] ^ 2 \\bigg ) , \\\\ & \\| \\epsilon ( t ) \\| _ { H ^ 1 } ^ 2 \\leq C _ 0 \\ , \\sup _ { \\tau \\geq t } \\bigg ( \\frac { 1 } { [ \\mathfrak { q } _ 1 ( \\tau ) - \\mathfrak { q } _ 2 ( \\tau ) ] ^ 4 } + \\sum _ { i = 1 } ^ 2 [ \\mathfrak { a } _ i ^ - ( \\tau ) ] ^ 2 \\bigg ) . \\end{align*}"}
{"id": "1607.png", "formula": "\\begin{align*} \\tilde { \\Theta } _ { a , k , \\zeta , \\mu , \\sigma } ( x ) & = \\Psi _ { a , \\zeta , \\mu , \\sigma } ( x _ 1 , \\dots , x _ { k - 1 } , x _ { k + 1 } , \\dots , x _ d ) , \\\\ \\tilde { W } _ { a , k , \\zeta , \\mu , \\sigma } ( x ) & = \\tilde { \\Psi } _ { a , \\zeta , \\mu , \\sigma } ( x _ 1 , \\dots , x _ { k - 1 } , x _ { k + 1 } , \\dots , x _ d ) e _ k . \\end{align*}"}
{"id": "5319.png", "formula": "\\begin{align*} ( m \\otimes m ) X = ( m \\otimes m ) ( Y ) , \\end{align*}"}
{"id": "7821.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\norm { u ( \\ , \\cdot \\ , + h ) - u } { L ^ 2 ( J _ h ; U ) } = 0 . \\end{align*}"}
{"id": "2499.png", "formula": "\\begin{align*} ( x , \\omega , \\tau ) \\odot ( x ' , \\omega ' , \\tau ' ) = ( x + x ' , \\omega + \\omega ' , \\tau + \\tau ' + x ' \\cdot \\omega ) . \\end{align*}"}
{"id": "6797.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\ & f ( x ) \\\\ \\enspace & h ( x ) = 0 , \\end{aligned} \\end{align*}"}
{"id": "8309.png", "formula": "\\begin{align*} C ( { \\sf S N R } ) = \\max _ { \\rho \\in \\{ 1 , 2 , \\ldots , N \\} } \\rho \\log _ 2 \\left ( 1 + \\frac { { \\sf S N R } } { \\rho } \\frac { N ^ 2 } { \\rho } \\right ) . \\end{align*}"}
{"id": "9236.png", "formula": "\\begin{align*} \\begin{cases} \\forall r ^ 1 , x ^ X , y ^ X \\Big ( r > _ \\mathbb { R } 0 _ \\mathbb { R } \\rightarrow \\norm { J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y } _ X \\\\ \\qquad \\qquad \\qquad \\leq _ \\mathbb { R } \\norm { r ( x - _ X y ) + _ X ( 1 - r ) ( J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y ) } _ X \\Big ) , \\\\ \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) , \\end{cases} \\end{align*}"}
{"id": "2208.png", "formula": "\\begin{align*} ( I - P ) v = | D | ^ { - 1 } \\int _ D v \\ , \\dd x . \\end{align*}"}
{"id": "6173.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 \\\\ b _ 1 \\\\ \\vdots \\\\ b _ r \\\\ 0 \\end{pmatrix} = A ^ { - 1 } \\begin{pmatrix} 1 \\\\ Z _ 1 ( z _ 1 , \\ldots , z _ { r + 1 } ) \\\\ \\vdots \\\\ Z _ { r + 1 } ( z _ 1 , \\ldots , z _ { r + 1 } ) \\end{pmatrix} . \\end{align*}"}
{"id": "8651.png", "formula": "\\begin{align*} P ( x , y ) - \\sqrt { - 3 } Q ( x , y ) & = ( x - \\sqrt { - 3 } y ) ^ 3 ( L _ 1 ( x , y ) + \\sqrt { - 3 } L _ 2 ( x , y ) ) \\\\ & = ( x ^ 3 - 9 x y ^ 2 ) L _ 1 ( x , y ) - 9 ( y ^ 3 - x ^ 2 y ) L _ 2 ( x , y ) \\\\ & + ( ( x ^ 3 - 9 x y ^ 2 ) L _ 2 ( x , y ) + 3 ( y ^ 3 - x ^ 2 y ) L _ 1 ( x , y ) ) \\sqrt { - 3 } \\end{align*}"}
{"id": "926.png", "formula": "\\begin{align*} a ^ { + } ( \\hat { n } + 1 - \\lambda ) _ { k , \\lambda } a & = ( \\hat { n } ) _ { k + 1 , \\lambda } = ( a ^ { + } a ) _ { k + 1 , \\lambda } \\\\ & = \\sum _ { m = 0 } ^ { k + 1 } S _ { 2 , \\lambda } ( k + 1 , m ) ( a ^ { + } ) ^ { m } a ^ { m } \\\\ & = \\sum _ { m = 1 } ^ { k + 1 } S _ { 2 , \\lambda } ( k + 1 , m ) ( a ^ { + } ) ^ { m } a ^ { m } \\\\ & = \\sum _ { m = 0 } ^ { k } S _ { 2 , \\lambda } ( k + 1 , m + 1 ) ( a ^ { + } ) ^ { m + 1 } a ^ { m + 1 } . \\end{align*}"}
{"id": "4715.png", "formula": "\\begin{align*} & | \\dot { x } _ i ( t ) - \\mu _ i ( t ) | \\lesssim \\frac { 1 } { t ^ { 5 / 4 - \\delta _ 0 } } , \\\\ & \\bigg | \\dot { \\mu } _ i ( t ) + \\frac { \\alpha _ i } { t ^ { \\frac { 3 } { 2 } } } - \\sum ^ n _ { j = 1 } \\frac { m _ { i j } x _ j ( t ) } { t ^ 2 } + \\sum ^ n _ { \\substack { k , j = 1 , \\\\ j \\not = i } } \\frac { b _ { i j k } \\alpha _ k } { ( \\alpha _ i - \\alpha _ j ) ^ 3 t ^ 2 } \\bigg | \\lesssim \\frac { 1 } { t ^ { 9 / 4 - \\delta _ 0 } } , \\end{align*}"}
{"id": "2652.png", "formula": "\\begin{align*} \\varphi ( x + k , \\omega + l ) = \\varphi ( x , \\omega ) + k \\cdot \\omega + \\kappa ( k , l ) . \\end{align*}"}
{"id": "8101.png", "formula": "\\begin{align*} \\Gamma \\mapsto \\big \\{ ( s , \\xi _ s ) & \\in \\dot { T } ^ { * } \\Sigma _ 0 \\ , | \\ , \\\\ & \\exists ( u , v ; \\xi _ u , \\xi _ v ) \\in \\Gamma , ( s , u , v ; \\xi _ s , \\xi _ u , \\xi _ v , ) \\in \\mathrm { W F } ( K ) \\big \\} \\end{align*}"}
{"id": "8995.png", "formula": "\\begin{align*} L : = H ( W _ 1 ) = H ( W _ 2 ) = \\cdots = H ( W _ K ) , \\end{align*}"}
{"id": "6137.png", "formula": "\\begin{align*} x ^ { m + 1 } P ( x ) + ( 1 - x ) ^ { n + 1 } Q ( x ) = 1 , \\end{align*}"}
{"id": "930.png", "formula": "\\begin{align*} & S _ { 1 , \\lambda } ( k + 1 , m + 1 ) \\\\ & = \\sum _ { l = m } ^ { k } S _ { 1 , \\lambda } ( k , l ) \\bigg \\{ \\binom { l } { m } \\langle 1 \\rangle _ { l - m , \\lambda } ( - 1 ) ^ { l - m } + l \\lambda \\binom { l - 1 } { m } ( - 1 ) ^ { l - m - 1 } \\langle 1 \\rangle _ { l - m - 1 , \\lambda } \\bigg \\} . \\end{align*}"}
{"id": "1431.png", "formula": "\\begin{align*} \\sum _ { i \\in \\{ 1 , \\dots , \\ell \\} } \\alpha _ i = \\sum _ { j \\in \\{ \\ell + 1 , \\dots , k \\} } \\beta _ j . \\end{align*}"}
{"id": "2950.png", "formula": "\\begin{align*} \\mathcal { U } _ { k } ( \\lambda ) & = \\big \\{ \\mathbf { P } _ k \\in \\mathcal { U } _ k : c _ { k - 1 } ( \\mathbf P _ k ) = 2 k - 2 - \\lambda \\big \\} , \\\\ \\mathcal { W } _ { k } ( \\lambda ) & = \\big \\{ \\mathbf { P } _ k \\in \\mathcal { W } _ k : c _ { k - 1 } ( \\mathbf P _ k ) = 2 k - 2 - \\lambda \\big \\} , \\end{align*}"}
{"id": "3451.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } \\omega ( Q ) \\psi _ { Q } ( x , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \\end{align*}"}
{"id": "8978.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f ( x ) & \\leq 0 \\ , ( - 1 , 1 ) , \\\\ u ( x ) + \\frac { 1 } { 2 } | u ^ \\prime ( x ) | ^ 2 - f ( x ) & \\geq 0 [ - 1 , 1 ] . \\end{aligned} \\right . \\end{align*}"}
{"id": "4995.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\norm { \\mathcal { R } ^ { n } Z } _ { y } = 0 . \\end{align*}"}
{"id": "7397.png", "formula": "\\begin{align*} F ( t , \\rho , G , g ) : = & \\langle \\rho _ t , G _ t \\rangle - \\langle g , G _ 0 \\rangle - \\int _ 0 ^ t \\langle \\rho _ s , \\partial _ s G _ s \\rangle d s - \\int _ 0 ^ t \\langle \\rho ^ 2 _ s , [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ s ] \\rangle d s ; \\end{align*}"}
{"id": "883.png", "formula": "\\begin{align*} \\Delta _ B \\xi + \\sum _ { j = 1 } ^ d a _ j e ^ { ( \\iota ^ \\ast u ^ j , \\xi ) } \\iota ^ \\ast u ^ j = w . \\end{align*}"}
{"id": "7528.png", "formula": "\\begin{align*} T \\log \\left ( \\frac { a ^ 2 + T ^ 2 } { 4 } \\right ) = T \\log \\left ( \\frac { T ^ 2 } { 4 } \\left ( 1 + \\frac { a ^ 2 } { T ^ 2 } \\right ) \\right ) \\end{align*}"}
{"id": "8729.png", "formula": "\\begin{align*} u _ { i 0 } ( x ) = 0 , u _ { i 1 } ( x ) = - 0 . 7 5 x _ 1 + 0 . 5 , u _ { i 2 } ( x ) = p _ i ( x ) i = 1 , 2 . \\end{align*}"}
{"id": "8623.png", "formula": "\\begin{align*} \\mathcal { K } _ { S } ( x , k ) = \\chi _ + ( x ) \\mathcal { K } _ + ( x , k ) + \\chi _ - ( x ) \\mathcal { K } _ - ( x , k ) \\end{align*}"}
{"id": "3015.png", "formula": "\\begin{align*} [ \\sigma _ { y , z } ] = [ D _ { y ^ 4 z } \\cap D _ { y ^ 3 z ^ 2 } ] + 2 [ \\varphi _ { y ^ 3 z ^ 2 } ] - [ \\gamma _ { v y ^ 2 z ^ 2 , y ^ 3 z ^ 2 } ] - [ \\gamma _ { w y ^ 2 z ^ 2 , y ^ 3 z ^ 2 } ] - [ \\gamma _ { x y ^ 2 z ^ 2 , y ^ 3 z ^ 2 } ] , \\end{align*}"}
{"id": "2240.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\sup _ { 1 \\leq m \\leq M } \\Big \\| \\sum _ { i = 1 } ^ m E _ { k , N } ^ { m - i + 1 } P _ N \\Delta W _ i \\Big \\| _ \\mu ^ p \\Big ] \\leq C ( \\| A ^ { \\frac { \\gamma - 2 } 2 } P Q ^ { \\frac 1 2 } \\| ^ p _ { \\mathcal { L } _ 2 ( H ) } + \\| ( I - P ) Q ^ { \\frac 1 2 } \\| ^ p _ { \\mathcal { L } _ 2 ( H ) } ) < \\infty . \\end{align*}"}
{"id": "2795.png", "formula": "\\begin{align*} \\int \\left ( \\partial _ { x _ j } Q \\right ) f _ * = \\int ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p ) Q ^ { p - 1 } f _ * = 0 , \\forall 1 \\le j \\le N . \\end{align*}"}
{"id": "8350.png", "formula": "\\begin{align*} e ^ r - 1 = \\frac { | \\log p | ^ 2 } { 4 } \\ ( 1 + o ( 1 ) ) , \\end{align*}"}
{"id": "1153.png", "formula": "\\begin{align*} g ( k , \\xi ) : = ( 4 k ^ 2 + 1 2 \\xi + 2 C _ { R } ^ 2 ) X _ { R } ( k ) , X _ { R } ( k ) = \\sqrt { k ^ 2 - C _ { R } ^ 2 } . \\end{align*}"}
{"id": "1167.png", "formula": "\\begin{align*} J ^ { \\mathcal { \\psi } } ( \\xi ) = \\left ( \\begin{array} { c c } 1 - r _ \\eta r ^ * _ \\eta & - r ^ * _ { \\eta } \\\\ r _ { \\eta } & 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "4900.png", "formula": "\\begin{align*} 2 p + r = n + 1 = p + q + 1 , r = q - p + 1 , \\end{align*}"}
{"id": "4382.png", "formula": "\\begin{align*} \\langle \\varepsilon , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } = \\sum _ { j = 0 } ^ { M } \\varepsilon _ j \\langle \\phi _ j , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } + \\langle \\varepsilon _ { - } , \\partial _ \\tau \\phi _ i \\rangle _ { L ^ 2 _ \\rho } \\end{align*}"}
{"id": "3074.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = x _ 2 ( j x _ 2 , j ^ 2 x _ 2 ) \\ , , x _ 2 ^ \\prime = x _ 1 ( j ^ 2 x _ 1 , j x _ 1 ) \\ , , y _ 1 ^ \\prime = y _ 1 \\ , , y _ 2 ^ \\prime = - y _ 2 \\ , . \\end{align*}"}
{"id": "5826.png", "formula": "\\begin{align*} v _ - : = \\sup \\big \\{ v \\in \\mathbb { R } \\colon \\liminf _ { H \\to \\infty } \\tilde { p } _ H ( v ) = 0 \\big \\} . \\end{align*}"}
{"id": "9525.png", "formula": "\\begin{align*} \\partial _ { ( X _ t , U _ t ) } H _ t ( X _ t , U _ t , y _ { t + 1 } ) = \\partial _ { X _ t } H _ t ( X _ t , U _ t , y _ { t + 1 } ) \\times \\partial _ { U _ t } H _ t ( X _ t , U _ t , y _ { t + 1 } ) , \\end{align*}"}
{"id": "2954.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } & = \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } ( \\mathcal { S } _ { k } ) + \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 , 0 } ( \\mathcal { W } _ { k } ( 0 ) \\setminus \\mathcal { S } _ k ) , \\end{align*}"}
{"id": "2134.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { l = 1 } ^ { k - 1 } \\frac { \\binom { M _ n } l ( k ) _ l ( k ( n - 1 ) ) _ { M _ n - l } } { ( k n ) _ { M _ n } } = ( 1 - c ) ^ k \\sum _ { l = 1 } ^ { k - 1 } \\binom k l ( \\frac c { 1 - c } ) ^ l . \\end{align*}"}
{"id": "2904.png", "formula": "\\begin{align*} - \\partial _ r ^ 2 Q - \\frac { N - 1 } { r } \\partial _ r Q + Q - \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ { p - 1 } = 0 . \\end{align*}"}
{"id": "2609.png", "formula": "\\begin{align*} \\norm { M _ \\omega T _ x f } _ { M ^ { p , q } } = \\norm { T _ { ( x , \\omega ) } V _ { g _ 0 } f } _ { L ^ { p , q } } \\leq C \\norm { V _ { g _ 0 } f } _ { L ^ { p , q } } = C \\norm { f } _ { M ^ { p , q } } . \\end{align*}"}
{"id": "3166.png", "formula": "\\begin{align*} \\varphi _ R ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( w _ k , \\emptyset ) . \\end{align*}"}
{"id": "8387.png", "formula": "\\begin{align*} B _ s ( v ) = ( v _ 1 + 1 , v _ 2 - s , v _ 3 , \\ldots , v _ d ) + [ 0 , r - 2 ] \\times [ 0 , r - 1 ] ^ { d - 1 } \\end{align*}"}
{"id": "3290.png", "formula": "\\begin{align*} \\delta = \\Im \\omega _ 2 ( i \\varepsilon ) = \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , \\varepsilon ) - \\varepsilon } . \\end{align*}"}
{"id": "5163.png", "formula": "\\begin{align*} Z _ { b ^ { - 1 } } \\phi ( x , \\omega ) & = \\sqrt { \\tfrac { \\pi } { 2 } } \\ , b \\ , \\vartheta _ 1 ' ( 0 , i b ) \\ , e ^ { - \\pi b x ^ 2 } \\frac { \\vartheta _ 3 ( \\omega - i b x , i b ) } { \\vartheta _ 4 ( \\omega , i b ) \\vartheta _ 4 ( x , i / b ) } \\\\ & = \\sqrt { \\tfrac { \\pi } { 2 ^ { 3 / 2 } } } \\ , b ^ { 3 / 4 } \\ , \\vartheta _ 1 ' ( 0 , i b ) \\ , \\frac { Z _ { b ^ { - 1 / 2 } } \\ , \\varphi ( x , \\omega ) } { \\vartheta _ 4 ( \\omega , i b ) \\vartheta _ 4 ( x , i / b ) } , \\end{align*}"}
{"id": "5364.png", "formula": "\\begin{align*} \\gamma _ i ( g ) = g - \\frac { \\left \\langle \\tilde a _ i , g \\right \\rangle - b _ i } { \\left \\langle \\tilde a _ i , c \\right \\rangle } c . \\end{align*}"}
{"id": "9500.png", "formula": "\\begin{align*} \\frac { \\lambda _ p + \\lambda _ q } { 2 } + i \\frac { \\mu _ p + \\mu _ q } { 2 } = \\frac { \\lambda _ p + i \\mu _ p } { 2 } + \\frac { \\lambda _ q + i \\mu _ q } { 2 } = 1 . \\end{align*}"}
{"id": "3836.png", "formula": "\\begin{align*} U _ s ( y ) : = y ^ { [ p _ s ] } ( 0 ) + \\sum _ { j = 1 } ^ { p _ s } u _ { s , j } y ^ { [ j - 1 ] } ( 0 ) = 0 , s = \\overline { 1 , n } . \\end{align*}"}
{"id": "8421.png", "formula": "\\begin{align*} W _ { t } ( w ) = w _ { t } \\end{align*}"}
{"id": "5915.png", "formula": "\\begin{align*} \\psi _ S ( x ) & = ( x + 1 ) ^ { n - 3 } \\ , \\Big [ x ^ 3 - ( s _ 1 + t _ 2 - 1 ) \\ , x ^ 2 - ( 2 s _ 1 + 2 t _ 2 + 1 ) \\ , x + 3 s _ 1 + 3 t _ 2 - 1 \\Big ] \\\\ & = ( x + 1 ) ^ { n - 3 } \\ , ( x - 1 ) \\ , \\Big [ x ^ 2 - ( s _ 1 + t _ 2 - 2 ) \\ , x - 3 s _ 1 - 3 t _ 2 + 1 \\Big ] \\\\ & = ( x + 1 ) ^ { n - 3 } \\ , ( x - 1 ) \\ , \\Big [ x ^ 2 - ( n - 4 ) \\ , x - 3 n + 7 \\Big ] , \\end{align*}"}
{"id": "8032.png", "formula": "\\begin{align*} K ( s , u , v ) = - \\partial _ u \\left ( \\delta ( u + s ) \\delta _ { \\epsilon } ( \\tfrac { u + v } { 2 } ) \\right ) , \\end{align*}"}
{"id": "2404.png", "formula": "\\begin{align*} \\langle f , D ( c _ \\gamma ) \\rangle _ \\mathcal { H } = \\langle f , \\sum _ { \\gamma \\in \\Gamma } c _ \\gamma e _ \\gamma \\rangle _ \\mathcal { H } = \\sum _ { \\gamma \\in \\Gamma } \\overline { c _ \\gamma } \\langle f , e _ \\gamma \\rangle _ \\mathcal { H } . \\end{align*}"}
{"id": "8883.png", "formula": "\\begin{align*} \\psi _ 2 ( \\bar z ) + h _ 2 ^ * \\psi _ 1 ( \\bar z ) & = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 2 ( z _ i ) , \\ldots , h _ 2 ( z _ { q - 1 } ) ) \\\\ & + \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 2 ( z _ 0 ) , \\ldots , h _ 2 ( z _ i ) , h _ 3 ( z _ i ) , \\ldots , h _ 3 ( z _ { q - 1 } ) ) \\\\ & = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( h _ 0 ( z _ 0 ) , \\ldots , h _ 0 ( z _ i ) , h _ 3 ( z _ i ) , \\ldots , h _ 3 ( z _ { q - 1 } ) ) + d _ { q - 2 } \\chi _ 2 ( \\bar z ) \\\\ & = \\psi _ 3 ( \\bar z ) + d _ { q - 2 } \\chi _ 2 ( \\bar z ) . \\end{align*}"}
{"id": "7736.png", "formula": "\\begin{gather*} 2 \\int _ { \\bar x } ^ { x _ 1 } \\phi _ { x x } \\cdot \\phi _ x ( t , x ) \\ , d x = | \\phi _ x | ^ 2 ( t , x _ 1 ) - | \\phi _ x | ^ 2 ( t , \\bar x ) , \\\\ 2 \\int _ { \\bar x } ^ { x _ 1 } \\phi _ { x t } \\cdot \\phi _ t ( t , x ) \\ , d x = | \\phi _ t | ^ 2 ( t , x _ 1 ) - | \\phi _ t | ^ 2 ( t , \\bar x ) , \\end{gather*}"}
{"id": "8832.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { | X | } | \\mathcal { X } ( L ^ { ( i ) } ) | \\end{align*}"}
{"id": "4745.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - \\mathcal L \\right ) \\ , u _ { \\mathbf { 0 } } ( t , x ) + \\ , q _ { \\mathbf { 0 } } ( x ) \\ , u _ { \\mathbf { 0 } } ( t , x ) = f _ { \\mathbf { 0 } } ( t , x ) , u _ { \\mathbf { 0 } } ( 0 , x ) = g _ { \\mathbf { 0 } } ( x ) . \\end{align*}"}
{"id": "107.png", "formula": "\\begin{align*} u ^ 3 = u ( u v ) = u ^ 2 ( u v ) = ( u v ) ^ 2 = u ^ 2 v ^ 2 = 0 . \\end{align*}"}
{"id": "3177.png", "formula": "\\begin{align*} \\varphi _ R ( A , b , \\lbrace x _ j : j \\leq k \\rbrace , \\lbrace W _ j : j < k \\rbrace , \\lbrace \\zeta _ j : j < k \\rbrace ) = ( E _ { \\pi ( k ) } , \\emptyset ) . \\end{align*}"}
{"id": "7462.png", "formula": "\\begin{align*} \\lambda _ { \\phi } ( t _ n ) = \\ ! \\int _ { \\mathbb { R } ^ d } \\ ! \\left ( \\frac 1 2 | \\nabla \\phi ( \\cdot , t _ n ) | ^ 2 + V | \\phi ( \\cdot , t _ n ) | ^ 2 + \\beta | \\phi ( \\cdot , t _ n ) | ^ 4 - \\Omega \\overline { \\phi } ( \\cdot , t _ n ) L _ z \\phi ( \\cdot , t _ n ) \\ ! \\right ) \\ ! \\mathrm { d } \\mathbf { x } - \\| \\dot { \\phi } ( \\cdot , t _ n ) \\| ^ 2 . \\end{align*}"}
{"id": "4403.png", "formula": "\\begin{align*} S _ j = 2 \\beta b H ^ { - 1 } \\left [ \\left ( \\frac { \\alpha } { 2 } - i + \\tilde \\lambda + \\frac { 1 } { 2 } \\Lambda \\right ) \\left ( S _ j + c _ { i , j } ( 2 \\beta ) ^ { j + 1 } T _ { j + 1 } \\right ) \\right ] . \\end{align*}"}
{"id": "1743.png", "formula": "\\begin{align*} \\gamma _ * = \\theta , \\ ; \\ ; s _ * = \\frac r d , \\ ; \\ ; \\mu _ * = \\beta + \\lambda \u2010 r \u2010 \\frac d q + \\frac { d } { p _ 1 } , \\ ; \\ ; \\alpha _ * = \\sigma \u2010 \\lambda + \\frac d q \u2010 \\frac { d } { p _ 0 } . \\end{align*}"}
{"id": "7149.png", "formula": "\\begin{align*} x _ t & = \\frac { x ^ { - \\tilde { \\theta } } } { \\tilde { \\theta } } + \\int _ 0 ^ t f ( s , \\frac { y _ s ^ { - \\tilde { \\theta } } } { \\tilde { \\theta } } ) d s + \\int _ 0 ^ t \\tilde { \\sigma } y _ s ^ { - ( \\tilde { \\theta } + 1 ) } d B _ s ^ H \\\\ & = \\frac { x ^ { - \\tilde { \\theta } } } { \\tilde { \\theta } } + \\int _ 0 ^ t f ( s , \\frac { y _ s ^ { - \\tilde { \\theta } } } { \\tilde { \\theta } } ) d s + \\int _ 0 ^ t \\tilde { \\sigma } \\tilde { \\theta } ^ { \\theta } ( x _ s ) ^ { \\theta } d B _ s ^ H , \\end{align*}"}
{"id": "1873.png", "formula": "\\begin{align*} P _ n ( x ) = 2 ^ n E _ n \\left ( { 1 + x ^ 2 \\over 2 } , x \\right ) . \\end{align*}"}
{"id": "3317.png", "formula": "\\begin{align*} v ^ \\alpha w ^ \\beta x ^ n = v ^ \\alpha w ^ { \\beta - 1 } x ^ n + v ^ \\alpha w ^ { \\beta - 1 } ( v - 1 ) x ^ { n + 1 } = v ^ \\alpha w ^ { \\beta - 1 } x ^ n + v ^ { \\alpha + 1 } w ^ { \\beta - 1 } x ^ { n + 1 } - v ^ \\alpha w ^ { \\beta - 1 } x ^ { n + 1 } . \\end{align*}"}
{"id": "5495.png", "formula": "\\begin{align*} \\xi ( t ; r , x ) & = S _ { t - r } x + \\int _ r ^ t S _ { t - u } \\alpha ( u , \\xi ( u ; r , x ) ) d u \\\\ & = S _ { t - s } S _ { s - r } x + S _ { t - s } \\int _ r ^ s S _ { s - u } \\alpha ( u , \\xi ( u ; r , x ) ) d u + \\int _ s ^ t S _ { t - u } \\alpha ( u , \\xi ( u ; r , x ) ) d u \\\\ & = S _ { t - s } \\xi ( s ; r , x ) + \\int _ s ^ t S _ { t - u } \\alpha ( u , \\xi ( u ; r , x ) ) d u , t \\in [ s , \\infty ) . \\end{align*}"}
{"id": "410.png", "formula": "\\begin{align*} \\left ( u ( x , 0 ) , v ( x , 0 ) , w ( x , 0 ) \\right ) ^ { \\top } = \\left ( u _ { 0 } , v _ { 0 } , w _ { 0 } \\right ) ^ { \\top } ( x ) = U _ { 0 } ( x ) x \\in \\mathbb { R } ^ { d } , \\end{align*}"}
{"id": "6321.png", "formula": "\\begin{align*} \\zeta ^ \\wedge _ { \\mathcal { L } _ f , p } ( s ) & = & & \\sum _ { v = 0 } ^ \\infty p ^ { ( 4 n - ( n + 2 ) s ) v } \\prod _ { i = 1 } ^ r \\frac { 1 - p ^ { e _ i f _ i v + f _ i } } { 1 - p ^ { f _ i } } = \\\\ & & & \\frac { \\sum _ { I \\subseteq [ r ] } ( - 1 ) ^ { | I | } \\sum _ { v = 0 } ^ \\infty p ^ { ( 4 n - ( n + 2 ) s ) v } \\cdot p ^ { ( \\sum _ { i \\in I } e _ i f _ i ) v + \\sum _ { i \\in I } f _ i } } { \\prod _ { i = 1 } ^ r ( 1 - p ^ { f _ i } ) } , \\end{align*}"}
{"id": "7410.png", "formula": "\\begin{align*} \\eta ( x + 1 ) - \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) = \\displaystyle \\frac { 1 } { \\varepsilon n } \\sum _ { r = 1 } ^ { \\varepsilon n } [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] , \\end{align*}"}
{"id": "8148.png", "formula": "\\begin{align*} M _ { d _ 0 } ( p , H ) = \\frac { \\pi ^ 2 } { 6 } \\left \\{ \\prod _ { q \\mid d _ 0 } \\left ( 1 - \\frac { 1 } { q ^ 2 } \\right ) \\right \\} \\left ( 1 + \\frac { N _ { d _ 0 } ( p , H ) } { p } \\right ) , \\end{align*}"}
{"id": "6558.png", "formula": "\\begin{align*} H _ { 1 } ( t ) = C \\| \\nabla K ( t ) \\| _ { L _ { t } ^ { 1 } L ^ { 2 } } \\| j _ { 0 } \\| _ { L ^ { 2 } } ; H _ { 3 } ( t ) = C \\| b \\| _ { L _ { t } ^ { \\infty } L ^ { \\infty } } \\| \\nabla ^ { 2 } K ( t ) \\| _ { L _ { t } ^ { 1 } L ^ { 1 } } ; \\end{align*}"}
{"id": "1591.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\nabla \\rho \\cdot u = 0 . \\end{align*}"}
{"id": "3714.png", "formula": "\\begin{align*} \\| B ^ k \\| _ { L ^ 2 ( 0 , T ; H ^ { \\frac 5 2 - \\frac { \\alpha } 2 } ) } \\leq & \\ 2 C ( T ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } \\\\ \\| B ^ k \\| _ { L ^ \\infty ( 0 , T ; H ^ { \\frac 5 2 - \\alpha } ) } \\leq & \\ ( 1 + C ( T ) ) \\| B ( 0 ) \\| _ { H ^ { \\frac { 5 } { 2 } - \\alpha } } \\end{align*}"}
{"id": "5509.png", "formula": "\\begin{align*} Q e _ j = \\lambda _ j e _ j \\end{align*}"}
{"id": "4481.png", "formula": "\\begin{align*} \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) & = \\sum _ { l = 0 } ^ { k } | z | ^ { 2 l } S _ { 2 , \\lambda } ( k , l ) = e ^ { - | z | ^ { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\frac { | z | ^ { 2 n } } { n ! } ( n ) _ { k , \\lambda } \\\\ & = e ^ { - | z | ^ { 2 } } \\sum _ { n = 1 } ^ { \\infty } \\frac { | z | ^ { 2 n } } { ( n - 1 ) ! } ( n - \\lambda ) _ { k - 1 , \\lambda } , ( k \\in \\mathbb { N } ) . \\end{align*}"}
{"id": "3434.png", "formula": "\\begin{align*} \\begin{aligned} & | H _ k ( x , y ) - H _ k ( x ' , y ) | \\\\ & = \\bigg | \\int _ { \\Bbb R ^ N } \\big [ D _ k ( x , z ) - D _ k ( x ' , z ) \\big ] S _ { k - M - 1 } ( z , y ) d \\omega ( z ) \\bigg | \\\\ & \\leqslant C \\int _ { \\{ \\| x - z \\| \\leqslant r ^ { 2 - k } \\ \\ \\| x ' - z \\| \\leqslant r ^ { 2 - k } \\} } \\frac { r ^ k \\| x - x ' \\| } { V _ k ( x ) + V _ k ( z ) } ( V _ { k - M - 1 } ( y ) ) ^ { - 1 } d \\omega ( z ) \\\\ & \\leqslant C r ^ k \\| x - x ' \\| ( V _ { k - M - 1 } ( y ) ) ^ { - 1 } . \\end{aligned} \\end{align*}"}
{"id": "2485.png", "formula": "\\begin{align*} ( x , \\omega , e ^ { 2 \\pi i \\tau } ) \\circ ( x ' , \\omega ' , e ^ { 2 \\pi i \\tau ' } ) = ( x + x ' , \\omega + \\omega ' , e ^ { 2 \\pi i ( \\tau + \\tau ' ) } e ^ { \\pi i ( x ' \\cdot \\omega - x \\cdot \\omega ' ) } ) . \\end{align*}"}
{"id": "8373.png", "formula": "\\begin{align*} h _ { i j } \\dot { M } ^ i _ k M ^ j _ l = h _ { i j } M ^ i _ k \\dot { M } ^ j _ l \\ , , \\end{align*}"}
{"id": "7428.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 m \\ell n } \\sum _ { x } \\Phi _ { n } ( s , \\tfrac { x } { n } ) \\sum _ { j = 1 } ^ { m - 1 } \\int _ { \\Omega _ 2 ^ j ( x ) } \\sum _ { z = x - \\ell } ^ { x - 1 } [ \\eta ( z ) - \\eta ( z - j \\ell ) ] \\overrightarrow { \\eta } ^ { \\varepsilon n } ( x + 1 ) [ f ( \\eta _ { 2 , x , j , z } ) - f ( \\eta _ { 3 , x , j , z } ) ] d \\nu _ { b } \\Big | , \\end{align*}"}
{"id": "6226.png", "formula": "\\begin{align*} \\Psi _ { n } ( { \\bf x } ) = \\begin{bmatrix} \\Psi ^ { 1 } _ { n } ( { \\bf x } ) \\\\ \\Psi ^ { 2 } _ { n } ( { \\bf x } ) \\\\ \\vdots \\\\ \\Psi ^ { 2 d } _ { n } ( { \\bf x } ) \\end{bmatrix} \\in \\mathbb { C } ^ { 2 d } . \\end{align*}"}
{"id": "1040.png", "formula": "\\begin{align*} u ( x ) & = C \\int _ { \\R ^ n \\setminus B _ 1 } \\bigg ( \\frac { 1 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - 1 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert x - y \\vert ^ n } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ n } \\bigg ) u ( y ) \\dd y \\\\ & \\geqslant C x _ 1 \\int _ { \\R ^ n \\setminus B _ 1 } \\frac { y _ 1 u ( y ) } { \\big ( \\vert y \\vert ^ 2 - 1 \\big ) ^ s \\vert y \\vert ^ { n + 2 } } \\dd y \\end{align*}"}
{"id": "7313.png", "formula": "\\begin{align*} \\begin{pmatrix} X _ 1 & 0 \\\\ 0 & - X _ 2 \\end{pmatrix} \\le \\begin{pmatrix} Z & - Z \\\\ - Z & Z \\end{pmatrix} + \\alpha \\begin{pmatrix} I & I \\\\ I & I \\end{pmatrix} \\end{align*}"}
{"id": "3687.png", "formula": "\\begin{align*} \\mathcal H f ( x ) = \\frac { 1 } { 2 \\pi } P . V . \\int _ { - \\pi } ^ { \\pi } f ( y ) \\cot \\left ( \\frac { x - y } { 2 } \\right ) \\ , d y \\end{align*}"}
{"id": "3832.png", "formula": "\\begin{align*} & f _ { k , j } ( x ) \\equiv 0 , k + 1 < j , \\\\ & f _ { k , k + 1 } ( x ) \\equiv 1 , k = \\overline { 1 , n - 1 } , \\end{align*}"}
{"id": "1067.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Phi _ x + i k \\sigma _ 3 \\Phi = Q \\Phi , \\\\ & \\Phi _ t + 4 i k ^ 3 \\sigma _ 3 \\Phi = V \\Phi , \\end{aligned} \\right . \\end{align*}"}
{"id": "1417.png", "formula": "\\begin{align*} \\hat u _ i = \\sum _ { ( i , j ) \\in E } \\hat l _ { i j } ( \\hat z _ j - \\hat z _ i ) , \\end{align*}"}
{"id": "1696.png", "formula": "\\begin{align*} f ( x ) = x ^ n + a _ 1 x ^ { n - 1 } + a _ 2 x ^ { n - 2 } + \\cdots + a _ n , \\end{align*}"}
{"id": "524.png", "formula": "\\begin{align*} ( - 1 ) ^ \\ell \\prod _ { a = 1 } ^ \\ell i _ a \\alpha ^ { i _ a } _ { r - ( j - i _ 1 - . . . - i _ a ) d - u _ a } \\end{align*}"}
{"id": "2964.png", "formula": "\\begin{align*} | \\mathcal { S } _ k | & = \\binom { d } { 2 k } \\cdot \\binom { 2 k - 1 } { k - 1 } . \\end{align*}"}
{"id": "8589.png", "formula": "\\begin{align*} I ( s , x ) : = \\int _ 0 ^ \\infty \\mathcal { Q } ( x , k ) \\phi ( k ) e ^ { i k ^ { 2 } s } h ( k ) \\ , d k & = \\int _ 0 ^ { \\infty } \\mathcal { Q } ( x , \\sqrt { \\lambda } ) \\phi ( \\sqrt { \\lambda } ) e ^ { i s \\lambda } h ( \\sqrt { \\lambda } ) \\frac { 1 } { 2 \\sqrt { \\lambda } } \\ , d \\lambda . \\end{align*}"}
{"id": "5721.png", "formula": "\\begin{align*} - \\Delta _ g u ( x ) - \\mu \\frac { u ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } + u ( x ) = \\lambda \\alpha ( x ) f ( u ( x ) ) , \\ \\ x \\in M , \\end{align*}"}
{"id": "3790.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } c ^ F _ { - r } \\big ( \\overline { \\pi _ E ( w _ n ) } W , \\sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \\big ) q _ F ^ { - \\frac { r } { 2 } } X ^ { - l r } = r _ l \\big ( \\gamma \\big ( X , \\pi _ E , \\widehat { \\sigma _ E } , \\psi _ E \\big ) \\big ) \\sum _ { r \\in \\mathbb { Z } } c ^ F _ r \\big ( W , W ' \\big ) q _ F ^ { \\frac { r } { 2 } } X ^ { l r } , \\end{align*}"}
{"id": "1989.png", "formula": "\\begin{align*} m _ n ( t ) = \\sum _ { k = 1 } ^ n \\sum _ { i _ 1 + \\dotsb + i _ k = n } ( i _ 1 + 1 ) \\dotsm ( i _ 1 + \\dotsb + i _ { k - 1 } + 1 ) h _ { i _ 1 } \\dotsm h _ { i _ k } \\frac { t ^ k } { k ! } . \\end{align*}"}
{"id": "1519.png", "formula": "\\begin{align*} x g _ 1 = \\left [ \\begin{array} { c c c c c } 1 _ t & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & u \\\\ 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 _ t & 0 \\\\ 0 & 0 & 0 & 0 & v \\end{array} \\right ] , y g _ 2 = \\left [ \\begin{array} { c c c c c } 1 _ t & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & u ' \\\\ 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 _ t & 0 \\\\ 0 & 0 & 0 & 0 & v ' \\end{array} \\right ] . \\end{align*}"}
{"id": "7990.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { I } _ { n } ^ { \\sigma } } \\underbrace { e ^ { \\varphi _ { n } ^ { \\sigma } ( i ) } \\prod _ { \\ell = 1 } ^ n \\big ( \\lambda _ i ^ { ( \\sigma _ { \\ell } ) } \\big ) ^ { T _ { \\ell } ^ { \\sigma } - T _ { \\ell - 1 } ^ \\sigma } } _ { = : \\ , p _ { \\sigma _ n } ^ { \\ast } ( i ) } = 1 , \\end{align*}"}
{"id": "6164.png", "formula": "\\begin{align*} a _ k = \\begin{cases} t ( 2 r + 1 ) ! ! & ( k = 0 ) , \\\\ ( t ( 2 r + 1 ) + s ) ( 2 r - 1 ) ! ! & ( k = 1 ) , \\\\ \\left ( t \\binom { 2 ( r + 1 ) - k } { 2 ( r + 1 - k ) } + s \\binom { 2 r + 1 - k } { 2 ( r + 1 - k ) } \\right ) ( 2 ( r + 1 - k ) - 1 ) ! ! & ( 2 \\leq k \\leq r + 1 ) . \\end{cases} \\end{align*}"}
{"id": "1390.png", "formula": "\\begin{align*} | I _ 2 | & = \\left | \\left ( h ( n _ 1 , E _ 1 ) - h ( n _ 2 , E _ 2 ) \\right ) \\frac { ( 1 - x ) ^ \\frac { 1 } { 2 } } { n _ 2 - 1 } \\right | \\\\ & \\le C \\| b _ 1 - b _ 2 \\| _ { C [ 0 , 1 ] } + C \\left ( \\frac { | n _ 1 - n _ 2 | ( x ) } { ( 1 - x ) ^ { \\frac { 1 } { 2 } } } + \\frac { | n _ 1 - n _ 2 | ( \\xi ) } { ( 1 - \\xi ) ^ { \\frac { 1 } { 2 } } } \\right ) , \\quad \\exists \\xi \\in [ x , 1 ] . \\end{align*}"}
{"id": "7236.png", "formula": "\\begin{align*} \\omega ^ \\perp = \\check x ^ \\perp = x ^ \\perp - \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } v ^ \\perp , \\end{align*}"}
{"id": "2074.png", "formula": "\\begin{align*} \\mathrm { i n d e x } ( f ) ^ 2 \\cdot \\mathrm { D i s c } ( K ) = \\mathrm { d i s c } ( f ) > H ^ { n ^ 2 - n - 2 } . \\end{align*}"}
{"id": "1848.png", "formula": "\\begin{align*} x M _ n \\left ( { 4 x \\over ( 1 + x ) ^ 2 } \\right ) = { 2 ^ { n - 1 } \\over ( 1 + x ) ^ { n - 1 } } \\ , A _ n ( x ) , \\end{align*}"}
{"id": "1601.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d - 1 } } \\psi \\tilde \\psi \\ , d x & = 1 \\\\ \\int _ { \\mathbb { R } ^ { d - 1 } } \\beta ( | a | ^ \\frac { 1 } { p } \\psi ) \\tilde \\psi \\ , d x & = \\sigma | a | ^ \\frac { 1 } { p } . \\end{align*}"}
{"id": "9373.png", "formula": "\\begin{align*} \\phi _ { n , \\lambda } ( x ) & = \\frac { 1 } { e ^ { x } } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( k ) _ { n , \\lambda } } { k ! } x ^ { k } = \\frac { 1 } { e ^ { x } } \\bigg ( x \\frac { d } { d x } \\bigg ) _ { n , \\lambda } e ^ { x } , ( n \\ge 1 ) . \\end{align*}"}
{"id": "7324.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { | \\tilde { x } - \\tilde { y } | ^ 4 } { \\varepsilon ^ 4 } + \\alpha ( | \\tilde { x } | ^ 2 + | \\tilde { y } | ^ 2 ) \\\\ & \\leq u ( \\tilde { x } , \\tilde { t } ) - v ( \\tilde { y } , \\tilde { t } ) - u ( x _ 1 , t _ 1 ) + v ( x _ 1 , t _ 1 ) + 2 \\alpha | x _ 1 | ^ 2 + { \\lambda \\over T - t _ 1 } - { \\lambda \\over T - \\tilde { t } } . \\end{aligned} \\end{align*}"}
{"id": "3853.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } I C ( 2 k ) \\colon & p _ { n - s , k + 1 } ( 0 ) + p _ { n - k , s + 1 } ( 0 ) = 0 , s = \\overline { k , k + i _ { 2 k } - 1 } , k = \\overline { 0 , m - 1 } \\\\ I C ( 2 k + 1 ) \\colon & p _ { n - s , k + 1 } ( 0 ) - p _ { n - k , s + 1 } ( 0 ) = 0 , s = \\overline { k + 1 , k + i _ { 2 k + 1 } } , k = \\overline { 0 , m + \\tau - 2 } \\end{array} \\right \\} \\end{align*}"}
{"id": "4073.png", "formula": "\\begin{align*} \\begin{cases} - \\sigma _ t - \\sigma _ x - v _ { x } = f & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ - v _ { t } - v _ { x x } - v _ x - \\sigma _ x = g & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\sigma ( t , 0 ) = \\sigma ( t , 1 ) & t \\in ( 0 , T ) , \\\\ v ( t , 0 ) = v ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\sigma ( T , x ) = 0 , \\ \\ v ( T , x ) = 0 & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
{"id": "9298.png", "formula": "\\begin{align*} \\Sigma = \\bigcup _ { g \\in \\mathbb { Z } / d \\mathbb { Z } } g ( \\tilde { p } ) = \\bigcup _ { g \\in ( \\mathbb { Z } / l \\mathbb { Z } ) } g ( \\tilde { p } ) , \\end{align*}"}
{"id": "1890.png", "formula": "\\begin{align*} b \\cdot \\beta ' ( x ) = b \\cdot \\beta ' ( u . x ) = \\beta ( u . x ) = \\beta ( x ) \\end{align*}"}
{"id": "7161.png", "formula": "\\begin{align*} v ' ( T ) & = V ( T ) T ^ { 1 / ( n - 1 ) } - V ( T ) ^ { ( n - 1 ) / n } \\left ( \\int _ 0 ^ { T } V ( t ) ^ { ( n - 1 ) / n } \\ , d t \\right ) ^ { 1 / ( n - 1 ) } \\\\ & \\le V ( T ) T ^ { 1 / ( n - 1 ) } - V ( T ) ^ { ( n - 1 ) / n } \\left ( T V ( T ) ^ { ( n - 1 ) / n } \\right ) ^ { 1 / ( n - 1 ) } = 0 , \\end{align*}"}
{"id": "1501.png", "formula": "\\begin{align*} L ( s , \\mathbf { f } , \\chi ) = \\prod _ { p \\nmid \\mathfrak { n } } L _ p ( s , \\mathbf { f } , \\chi ) , \\end{align*}"}
{"id": "1504.png", "formula": "\\begin{align*} P _ { n } ^ t = \\{ x \\in G : a _ 2 = g _ 2 = h _ 2 = h _ 3 = h _ 4 = l _ 2 = d _ 3 = 0 \\} . \\end{align*}"}
{"id": "8902.png", "formula": "\\begin{align*} \\check H ^ q _ { c t } ( \\Z ^ n ; A ) = \\begin{cases} A \\oplus A & n = 1 , q = 0 \\\\ A & n \\not = 1 , q = 0 \\vee q = n - 1 \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "9269.png", "formula": "\\begin{align*} & \\forall n ^ 0 , x ^ X , y ^ X \\exists z ^ X \\preceq _ X L 1 _ X \\exists w ^ X \\preceq _ X L 2 ^ { \\alpha _ n + 1 } 1 _ X \\\\ & \\qquad \\qquad \\qquad \\left ( y \\in A ( \\tilde { x } ^ L ) \\rightarrow ( w \\in A z \\land \\tilde { x } ^ L = _ X z + _ X \\gamma _ n w ) \\right ) \\end{align*}"}
{"id": "4207.png", "formula": "\\begin{align*} \\int _ M | \\nabla \\dot u | ^ 2 \\ , d V _ g & = \\int _ M V \\dot u \\ , r \\ , d V _ g + \\int _ M V \\dot u ^ 2 \\ , d V _ g . \\end{align*}"}
{"id": "9333.png", "formula": "\\begin{align*} ( D _ A ) _ { i i } = \\frac { \\widehat { \\Theta } _ { i i } ^ { - 1 } - \\Theta _ { i i } ^ { - 1 } } { 1 + \\delta ^ 2 ( { \\Theta } _ { i i } ^ { - 1 } + \\rho ) ( \\widehat { \\Theta } _ { i i } ^ { - 1 } + \\rho ) + \\delta ( { \\Theta } _ { i i } ^ { - 1 } + \\rho ) + \\delta ( \\widehat { \\Theta } _ { i i } ^ { - 1 } + \\rho ) } . \\end{align*}"}
{"id": "3919.png", "formula": "\\begin{align*} f _ { s _ i } ( \\eta _ i ) & = f _ { s _ { i - 1 } } ( \\eta _ i ) \\\\ f _ { s _ i } ' ( \\eta _ i ) & = f ' _ { s _ { i - 1 } } ( \\eta _ i ) \\\\ f _ { s _ i } '' ( \\eta _ i ) & = f '' _ { s _ { i - 1 } } ( \\eta _ i ) \\end{align*}"}
{"id": "2957.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Lambda ^ { ( 2 : 2 ) } _ { n , 2 } = \\frac 1 { 2 } , \\end{align*}"}
{"id": "4304.png", "formula": "\\begin{align*} u ( r , t ) = \\frac { 1 } { \\mu ( t ) } w ( y , \\tau ) , y = \\frac { r } { \\sqrt { \\mu ( t ) } } \\frac { d \\tau } { d t } = \\frac { 1 } { \\mu ( t ) } , \\tau | _ { t = 0 } = \\tau _ 0 , \\end{align*}"}
{"id": "7056.png", "formula": "\\begin{align*} \\lim _ { K \\rightarrow + \\infty } \\mathbb { P } \\Big ( \\sup _ { i \\neq j } \\frac { | \\beta ^ K _ i ( 0 ) - \\beta ^ K _ j ( 0 ) | } { \\rho ( i \\ , \\delta _ K , j \\ , \\delta _ K ) } > A \\Big ) = 0 . \\end{align*}"}
{"id": "4953.png", "formula": "\\begin{align*} g _ 0 g _ 1 = \\phi ( h ^ + ) ^ { - 1 } \\cdot g _ 0 g _ 2 \\cdot h ^ + . \\end{align*}"}
{"id": "5621.png", "formula": "\\begin{align*} \\mathcal { B } : = \\{ \\beta ^ { \\vec { p } , \\vec { q } , \\vec { r } } : = ( \\xi ^ 1 ) ^ { p _ 1 } \\ldots ( \\xi ^ n ) ^ { p _ n } ( x ^ 1 ) ^ { q _ 1 } \\ldots ( x ^ n ) ^ { q _ n } \\partial _ 1 ^ { r _ 1 } \\ldots \\partial _ n ^ { r _ n } ~ | ~ \\vec { p } \\in \\{ 1 , 0 \\} ^ n , \\vec { q } , \\vec { r } \\in \\mathbb { N } _ 0 ^ n \\} . \\end{align*}"}
{"id": "925.png", "formula": "\\begin{align*} ( \\hat { n } ) _ { k + 1 , \\lambda } & = ( \\hat { n } - \\lambda ) _ { k , \\lambda } \\hat { n } = a ^ { + } \\big ( ( \\hat { n } + 1 - \\lambda ) \\cdots ( \\hat { n } + 1 - k \\lambda ) \\big ) a \\\\ & = a ^ { + } ( \\hat { n } + 1 - \\lambda ) _ { k , \\lambda } a . \\end{align*}"}
{"id": "2559.png", "formula": "\\begin{align*} \\pi ^ { M p } ( \\widehat { S } _ 1 \\widehat { S } _ 2 ) = \\pi ^ { M p } ( \\widehat { S } _ 1 ) \\pi ^ { M p } ( \\widehat { S } _ 2 ) . \\end{align*}"}
{"id": "3685.png", "formula": "\\begin{align*} B _ t + ( B \\cdot \\nabla ) J - ( J \\cdot \\nabla ) B + \\mu \\Lambda ^ \\alpha B = 0 , \\ \\ \\ \\Lambda = ( - \\Delta ) ^ { \\frac 1 2 } \\end{align*}"}
{"id": "288.png", "formula": "\\begin{align*} \\tilde { u } _ { t } - \\tilde { u } _ { x x t } - \\tilde { u } _ { x x } + \\gamma \\tilde { u } _ { x x x } & = 0 , \\ \\ x \\in \\R , \\ t > 0 , \\\\ \\tilde { u } ( x , 0 ) & = u _ { 0 } ( x ) , \\ \\ x \\in \\R . \\end{align*}"}
{"id": "5276.png", "formula": "\\begin{align*} \\check { S } ( \\omega ) = \\omega \\circ S ^ { - 1 } . \\end{align*}"}
{"id": "6459.png", "formula": "\\begin{align*} ( T _ 0 - T _ h ( \\lambda ) ) ( u _ 0 ) ( x ) & = \\sum _ { k = 1 } ^ { k = n } \\frac { \\eta _ 0 ^ k } { 4 \\pi } \\int _ { B _ k } \\big ( 1 - \\exp ( { i \\sqrt { \\lambda } h | x - y | ) } \\big ) \\frac { u _ 0 ( y ) } { | x - y | } d y \\\\ & = - i \\frac { { \\lambda _ 0 } ^ \\frac { 1 } { 2 } } { 4 \\pi } ( \\sum _ { k = 1 } ^ { k = n } \\eta _ 0 ^ k U _ k ) h + \\mathcal { O } ( h ^ 2 ) \\end{align*}"}
{"id": "3125.png", "formula": "\\begin{align*} \\varphi _ i ( x _ 1 , x _ 2 ) + \\psi _ i ( x _ 3 , x _ 4 , x _ 5 ) = 0 \\ , . \\end{align*}"}
{"id": "3520.png", "formula": "\\begin{align*} & \\frac { s _ 3 } { s _ 1 + s _ 3 - 1 } \\sum _ { n \\leq a t _ 3 } \\frac { 1 } { n ^ { s _ 2 - 1 } } \\int _ { a t _ 3 } ^ \\infty \\frac { d u } { u ^ { s _ 1 } ( u + n ) ^ { s _ 3 + 1 } } \\\\ & = \\frac { s _ 3 } { 2 \\pi i ( s _ 1 + s _ 3 - 1 ) \\Gamma ( s _ 3 + 1 ) } \\sum _ { n \\leq a t _ 3 } \\int _ { a t _ 3 } ^ \\infty \\int _ { ( c ) } \\frac { \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { u ^ { s _ 1 + s _ 3 + 1 + z } n ^ { s _ 2 - 1 - z } } d z d u , \\\\ \\end{align*}"}
{"id": "7731.png", "formula": "\\begin{align*} \\phi _ u ( u , v ) = - \\int _ { - u } ^ v F ( u , v _ 0 ) d v _ 0 + \\phi _ u ( u , - u ) = - 2 \\int _ { 0 } ^ { t } F ( s , x + t - s ) d s + \\phi _ u ( u , - u ) , \\end{align*}"}
{"id": "5978.png", "formula": "\\begin{align*} M _ { \\lambda , \\beta } ( n ) : = \\int _ { \\mathbb { R } } x ^ { n } \\ , \\mathrm { d } \\pi _ { \\lambda , \\beta } ( x ) = \\sum _ { m = 0 } ^ { n } \\frac { m ! } { \\Gamma ( m \\beta + 1 ) } S ( n , m ) \\lambda ^ { m } , \\end{align*}"}
{"id": "1603.png", "formula": "\\begin{align*} \\Phi & : = \\eta _ { \\ell } \\ast \\varphi , \\\\ \\tilde { \\Phi } & : = \\eta _ { \\ell } \\ast \\tilde { \\varphi } . \\end{align*}"}
{"id": "3971.png", "formula": "\\begin{align*} \\eta ^ { \\prime \\prime \\prime } ( x ) + \\lambda \\xi ^ \\prime ( x ) - \\lambda \\eta ^ \\prime ( x ) = 0 , \\ \\ \\forall x \\in ( 0 , 1 ) . \\end{align*}"}
{"id": "5338.png", "formula": "\\begin{align*} \\sum \\limits _ { \\ell = 0 } ^ { 2 } \\frac { d } { d t } \\langle \\nabla ^ { \\ell } u ^ { \\epsilon } , \\epsilon \\nabla ^ { \\ell + 1 } \\phi ^ { \\epsilon } \\rangle + \\frac { 3 } { 4 } \\| \\nabla \\phi ^ { \\epsilon } \\| _ { H ^ 2 _ { \\frac { P ' ( \\rho ^ { \\epsilon } ) } { \\rho ^ { \\epsilon } } } } ^ 2 \\lesssim D ( u ^ { \\epsilon } , \\eta ^ { \\epsilon } , \\tau ^ { \\epsilon } ) + \\| \\mathrm { d i v } u ^ { \\epsilon } \\| _ { H ^ 2 } ^ 2 . \\end{align*}"}
{"id": "1688.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { n - 1 } \\frac { ( - 1 ) ^ k x ^ { 2 k } } { ( 2 k + 1 ) ^ s } = \\sum \\limits _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } \\binom { n } { k } { _ { s + 1 } F _ s } \\left ( \\left \\{ \\frac { 1 } { 2 } \\right \\} ^ { s } , 1 - k ; \\left \\{ \\frac { 3 } { 2 } \\right \\} ^ { s } ; - x ^ 2 \\right ) . \\end{align*}"}
{"id": "7263.png", "formula": "\\begin{align*} \\frac { 1 } { ( a + X ) ( b + X ) } = \\frac { 1 } { b - a } \\Bigl ( \\frac { 1 } { a + X } - \\frac { 1 } { b + X } \\Bigr ) , \\end{align*}"}
{"id": "4277.png", "formula": "\\begin{align*} v _ t + a _ n ( t , x ) \\cdot v _ x ~ = ~ 0 , v ( \\tau _ 0 , \\cdot ) ~ = ~ \\bar { v } ~ \\in ~ H ^ 2 ( \\R \\backslash I ^ { \\tau } _ { t } ) , \\end{align*}"}
{"id": "9492.png", "formula": "\\begin{align*} | \\mathcal { F } ( a + b , - b \\ , ; \\ , \\{ U \\} , \\{ U \\} ) | = \\sum _ { i = 0 } ^ { a } \\binom { a + b - 2 } { \\lfloor i / 2 \\rfloor } \\binom { a + b - 1 - \\lfloor i / 2 \\rfloor } { a - i } . \\end{align*}"}
{"id": "139.png", "formula": "\\begin{align*} \\gamma _ 1 ( x ^ 2 ) - \\gamma _ 2 ( x ) = \\omega ( x ) ^ 2 \\ ; \\ ; \\forall x \\in A \\end{align*}"}
{"id": "2497.png", "formula": "\\begin{align*} F _ r * _ { \\mathbf { H } _ r } G _ r = ( F \\natural G ) _ r . \\end{align*}"}
{"id": "7888.png", "formula": "\\begin{align*} p _ { ( F ( u _ 1 ) , F ( u _ 2 ) ) } ( 0 ) \\leq \\begin{cases} \\frac { k _ 2 } { | u _ 1 - u _ 2 | ^ m } , & \\ | u _ 1 - u _ 2 | \\leq 1 , \\\\ & \\\\ k _ 2 , & \\ | u _ 1 - u _ 2 | > 1 , \\end{cases} \\end{align*}"}
{"id": "7846.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { 3 j + 2 } \\alpha _ { i } e _ { i } , \\quad \\alpha _ { 3 j + 2 } \\neq 0 \\quad ~ \\langle x \\rangle _ { T ^ { * 3 } } = j + 1 . \\end{align*}"}
{"id": "4514.png", "formula": "\\begin{align*} \\left ( \\frac { R } { b _ { i + 1 } } \\right ) ^ { \\frac { 3 } { 2 } } \\exp \\left ( \\frac { 2 } { \\sqrt { R } } \\sqrt { b _ { i + 1 } } \\right ) \\left ( \\frac { b _ { i } } { R } \\right ) ^ { \\frac { 3 } { 2 } } \\exp \\left ( \\frac { - 2 } { \\sqrt { R } } \\sqrt { b _ { i } } \\right ) = \\left ( \\frac { b _ i } { b _ { i + 1 } } \\right ) ^ { 3 / 2 } \\exp \\left ( \\frac { 2 } { \\sqrt { R } } ( \\sqrt { b _ { i + 1 } } - \\sqrt { b _ { i } } ) \\right ) \\end{align*}"}
{"id": "6316.png", "formula": "\\begin{align*} W _ { \\mathbf { 1 } , \\mathbf { f } } ( X ^ { - 1 } , Y ^ { - 1 } ) = ( - 1 ) ^ { r + 1 } p ^ { 9 n - ( 2 n + 4 ) s } W _ { \\mathbf { 1 } , \\mathbf { f } } ( X , Y ) . \\end{align*}"}
{"id": "3895.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq b < p \\\\ \\gcd ( m , p - 1 ) = 1 } } e ^ { \\frac { i 2 \\pi b ( \\tau ^ { e m } - v ) } { p } } = 0 \\end{align*}"}
{"id": "1498.png", "formula": "\\begin{align*} \\mathbf { f } | [ K _ 1 ( \\mathfrak { n } ) \\xi K _ 1 ( \\mathfrak { n } ) ] = \\lambda _ { \\mathbf { f } } ( \\xi ) \\mathbf { f } \\xi \\in \\mathfrak { X } . \\end{align*}"}
{"id": "9191.png", "formula": "\\begin{align*} T = ( 1 - \\alpha ) I d + \\alpha N \\end{align*}"}
{"id": "144.png", "formula": "\\begin{align*} \\nu _ { \\alpha , 1 } ( d u ) = c _ { \\alpha } \\frac { d u } { | u | ^ { \\alpha + 1 } } , \\end{align*}"}
{"id": "6326.png", "formula": "\\begin{align*} \\alpha ( \\alpha \\beta ) \\gamma \\delta \\varepsilon ( \\zeta \\zeta \\eta ) = \\alpha ( \\beta \\alpha ) \\gamma \\delta \\varepsilon ( \\zeta \\eta \\zeta ) \\neq \\alpha ( \\beta \\alpha ) \\gamma \\varepsilon \\delta ( \\zeta \\eta \\zeta ) , \\alpha , \\ldots , \\eta \\in [ d ] \\end{align*}"}
{"id": "8205.png", "formula": "\\begin{align*} N _ 6 ( f , H ) = - f + S ( H , f ) - 2 S ( H _ 2 , 2 f ) - \\frac { 3 } { 2 } S ( H _ 3 , 3 f ) + 3 S ( H _ 6 , 6 f ) = \\begin{cases} - 2 a - 1 & \\hbox { i f $ a \\equiv 0 \\pmod 6 $ , } \\\\ - 3 & \\hbox { i f $ a \\equiv 2 , 3 \\pmod 6 $ , } \\\\ 2 a + 1 & \\hbox { i f $ a \\equiv 5 \\pmod 6 $ . } \\end{cases} \\end{align*}"}
{"id": "1647.png", "formula": "\\begin{align*} \\int _ M V ^ \\nu _ { 2 , p } \\ d \\nu = \\left \\| G _ \\lambda \\nu \\right \\| _ { L ^ q ( M ) } ^ q . \\end{align*}"}
{"id": "1063.png", "formula": "\\begin{align*} & g ( k , \\xi ) : = ( 4 k ^ 2 + 1 2 \\xi + 2 C _ { R } ^ 2 ) X _ R ( k ) , \\eta ( \\xi ) = \\sqrt { - \\xi + \\frac { C _ { R } ^ 2 } { 2 } } \\in \\mathbb { R } , \\end{align*}"}
{"id": "4370.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ M | \\varepsilon _ j | + \\| \\varepsilon _ - \\| _ { L ^ 2 _ \\rho } \\le 1 . \\end{align*}"}
{"id": "4034.png", "formula": "\\begin{align*} p ^ 2 _ k - q ^ 2 _ k = a _ k \\Longrightarrow 4 p _ k ^ 4 - 4 a _ k p _ k ^ 2 - b ^ 2 _ k = 0 , \\end{align*}"}
{"id": "8968.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { t } ^ { T _ { x , \\xi } } C _ p q \\left | \\dot { \\xi } ( s ) \\right | ^ q d s = - C _ p ( q - 1 ) p ^ q \\left ( f ( \\xi ( t ) ) - u ( \\xi ( t ) ) \\right ) + f ( \\xi ( t ) ) \\\\ \\Longrightarrow & \\int _ { t } ^ { T _ { x , \\xi } } p \\left ( f ( \\xi ( s ) ) - u ( \\xi ( s ) ) \\right ) d s = u ( \\xi ( t ) ) . \\end{aligned} \\end{align*}"}
{"id": "3855.png", "formula": "\\begin{align*} \\Gamma _ k = \\left \\{ \\rho \\colon \\frac { \\pi ( k - 1 ) } { n } < \\arg \\rho < \\frac { \\pi k } { n } \\right \\} , k = \\overline { 1 , 2 n } . \\end{align*}"}
{"id": "5380.png", "formula": "\\begin{align*} c _ { F _ k } = 2 m n + 1 1 n - 3 . \\end{align*}"}
{"id": "2958.png", "formula": "\\begin{align*} \\frac { c \\left ( 1 + o ( 1 ) \\right ) } { n ^ { h + h ' } \\delta ^ 4 _ n ( k ) } \\cdot \\sum _ { \\lambda = 0 } ^ { k - 1 } | \\mathcal { U } _ k ( \\lambda ) | \\cdot | \\mathcal { I } _ { h } \\times \\mathcal { I } _ { h ' } | . \\end{align*}"}
{"id": "920.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( k + 1 , l ) = S _ { 2 , \\lambda } ( k , l - 1 ) + ( l - k \\lambda ) S _ { 2 , \\lambda } ( k , l ) , \\end{align*}"}
{"id": "9207.png", "formula": "\\begin{align*} \\langle x ^ X , y ^ X \\rangle _ X : = _ 1 \\frac { 1 } { 4 } \\left ( \\norm { x + _ X y } ^ 2 _ X - \\norm { x - _ X y } ^ 2 _ X \\right ) . \\end{align*}"}
{"id": "4664.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow 0 ^ + } \\delta ( \\alpha ) = 0 . \\end{align*}"}
{"id": "1394.png", "formula": "\\begin{align*} ( E _ 1 - E _ 2 ) _ x = ( n _ 1 - n _ 2 ) - ( b _ 1 - b _ 2 ) , \\end{align*}"}
{"id": "6738.png", "formula": "\\begin{align*} & g _ 1 ( t ) ^ { ( - 1 ) } ( - 1 ) ^ { s _ r } ( t - \\theta ) ^ { w - s _ r } \\Omega ^ w \\alpha _ r \\mathcal { L i } _ { K , ( s _ 1 , \\ldots , s _ { r - 1 } ) } ( \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } ) + g _ 1 ( t ) ^ { ( - 1 ) } ( t - \\theta ) ^ w \\Omega ^ w \\mathcal { L i } _ { K , \\mathfrak { s } } ( { \\boldsymbol \\alpha } ) \\\\ & + g _ 2 ( t ) ^ { ( - 1 ) } = 0 . \\end{align*}"}
{"id": "2062.png", "formula": "\\begin{align*} \\eta \\cdot f : = \\sum _ { \\alpha \\in \\mathbb { N } ^ n } \\eta _ { \\alpha } \\frac { f ^ { ( \\alpha ) } ( z _ 0 ) } { \\alpha ! } . \\end{align*}"}
{"id": "8621.png", "formula": "\\begin{align*} & \\mathcal { K } _ { S , \\# } '' ( x , k ) : = \\chi _ 0 ( x ) T ( 0 ) \\partial _ x m _ { + } ( x , 0 ) e ^ { i k x } + \\partial _ x \\chi _ + ( x ) \\mathcal { K } ' _ { + , \\# } ( x , k ) + \\partial _ x \\chi _ - ( x ) \\mathcal { K } ' _ { - , \\# } ( x , k ) . \\end{align*}"}
{"id": "8655.png", "formula": "\\begin{align*} ( a , b , c , d ) = \\left ( \\frac { p r + 3 q s } { 8 } , \\frac { 9 r s - p q } { 8 } , \\frac { p q - r s } { 8 } , \\frac { 3 p r + q s } { 8 } \\right ) \\end{align*}"}
{"id": "2755.png", "formula": "\\begin{align*} \\mathcal { M } \\mathcal { E } [ u ] = 1 . \\end{align*}"}
{"id": "5242.png", "formula": "\\begin{align*} \\sum _ i a _ i c \\otimes b _ i d = 0 , \\forall u \\in I , \\forall c \\in A _ u ^ s , \\forall d \\in B ^ u _ w . \\end{align*}"}
{"id": "7214.png", "formula": "\\begin{align*} V _ { s ' , t } ( x , v ) = v ' , X _ { s ' , t } ( x , v ) = x ' , V _ { s , t } ( x , v ) = V _ { s , s ' } ( x ' , v ' ) . \\end{align*}"}
{"id": "2965.png", "formula": "\\begin{align*} \\mathcal { \\tilde M } _ k = \\big \\{ \\{ 0 , 1 \\} ^ { ( 2 k - 1 ) \\times 4 } : ~ M _ { 1 : 2 k - 2 , 1 : 4 } \\in \\mathcal { N } _ { k - 1 } , ~ M _ { 2 k - 1 , 1 : 4 } = ( 1 , 1 , 1 , 1 ) \\big \\} . \\end{align*}"}
{"id": "2005.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ 1 ( - t + \\bar { T } ) \\\\ u _ 2 ( - t + \\bar { T } ) \\\\ u _ 3 ( - t + \\bar { T } ) \\end{array} \\right ) = M ( 2 p - 1 ) \\left ( \\begin{array} { c } u _ 1 ( t + \\bar { T } ) \\\\ u _ 2 ( t + \\bar { T } ) \\\\ u _ 3 ( t + \\bar { T } ) \\end{array} \\right ) . \\end{align*}"}
{"id": "37.png", "formula": "\\begin{align*} d \\theta ^ c = \\omega + \\theta \\wedge \\theta ^ c . \\end{align*}"}
{"id": "49.png", "formula": "\\begin{align*} J _ t = & J - \\frac \\partial { \\partial x } \\otimes d ^ c \\left ( \\frac 1 2 t ^ 2 \\left ( \\frac { 1 - \\left | w \\right | ^ 2 } { 1 + \\left | w \\right | ^ 2 } \\right ) ^ 2 \\right ) - \\frac \\partial { \\partial y } \\otimes d ^ c \\left ( \\frac 1 2 t ^ 2 \\left ( \\frac { 1 - \\left | w \\right | ^ 2 } { 1 + \\left | w \\right | ^ 2 } \\right ) ^ 2 \\right ) \\circ J . \\end{align*}"}
{"id": "3560.png", "formula": "\\begin{align*} \\Phi _ m ( x , y ) & = \\frac 1 { ( 4 \\pi ) ^ { m / 2 } } \\exp \\left ( \\frac { - | x | ^ 2 - | y | ^ 2 } { 4 } \\right ) \\\\ & = \\frac 1 { ( 4 \\pi ) ^ { ( m - 1 ) / 2 } } \\exp \\left ( \\frac { - | x | } { 4 } \\right ) \\frac 1 { ( 4 \\pi ) ^ { 1 / 2 } } \\exp \\left ( \\frac { - | y | ^ 2 } { 4 } \\right ) \\\\ & = \\Phi _ { m - 1 } ( x ) \\Phi _ 1 ( y ) . \\end{align*}"}
{"id": "7293.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { a _ { n + 1 } } { n ! } X ^ n & = \\alpha ' = \\exp ( X ) \\exp ( \\exp ( X ) - 1 ) \\\\ & = \\Bigl ( \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } X ^ k \\Bigr ) \\Bigl ( \\sum _ { k = 0 } ^ \\infty \\frac { a _ k } { k ! } X ^ k \\Bigr ) = \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n \\frac { a _ k } { k ! ( n - k ) ! } X ^ n . \\end{align*}"}
{"id": "6523.png", "formula": "\\begin{align*} I ^ { ( 2 m ) } _ n & = O ( ( \\log n ) ^ { - m } ) , \\\\ J ^ { ( 2 m ) } _ n & = O ( ( \\log n ) ^ { - m } ) . \\end{align*}"}
{"id": "271.png", "formula": "\\begin{align*} B ( \\varepsilon { x _ k } , { \\delta _ 0 } { a _ \\varepsilon } ) \\subset T _ { \\varepsilon , k } = \\varepsilon { x _ k } + a _ \\varepsilon T _ { 0 } \\subset \\subset B ( \\varepsilon { x _ k } , { \\delta _ 1 } { a _ \\varepsilon } ) \\subset \\subset B ( \\varepsilon { x _ k } , { \\delta _ 2 } { a _ \\varepsilon } ) \\subset B ( \\varepsilon { x _ k } , \\delta _ 3 \\varepsilon ) \\subset \\varepsilon { Q _ k } , \\end{align*}"}
{"id": "6155.png", "formula": "\\begin{align*} U _ { n , m } ( x ) + V _ { n , m } ( x ) = 1 + \\sum _ { k = m + 1 } ^ { n + m + 1 } ( a _ { k - m - 1 } + d _ k ) x ^ k . \\end{align*}"}
{"id": "3782.png", "formula": "\\begin{align*} r _ l \\big ( \\epsilon ( X , \\widetilde { \\chi } _ 0 \\pi _ E , \\psi _ E ) \\big ) = \\epsilon \\big ( X , \\chi _ 0 r _ l ( \\pi _ F ) , \\overline { \\psi } _ F \\big ) ^ l = \\epsilon ( X ^ l , \\chi r _ l ( \\pi _ F ) ^ { ( l ) } , \\overline { \\psi } _ F ^ l \\big ) \\end{align*}"}
{"id": "6163.png", "formula": "\\begin{align*} x _ 1 & = \\int _ { S ^ 1 } \\frac { z - z _ 2 } { z _ 1 - z _ 2 } w ( z ) d z = \\frac { 1 1 + \\sqrt { - 1 1 } } { 2 2 } , \\\\ x _ 2 & = \\int _ { S ^ 1 } \\frac { z - z _ 1 } { z _ 2 - z _ 1 } w ( z ) d z = \\frac { 1 1 - \\sqrt { - 1 1 } } { 2 2 } . \\end{align*}"}
{"id": "6661.png", "formula": "\\begin{align*} h ^ { - \\frac { 1 } { 2 } + \\alpha } k ^ { - \\frac { 1 } { 2 } + \\beta - \\alpha ^ * - \\beta ^ * } \\mathcal { G } ( 2 - \\alpha - \\beta + \\alpha ^ * + \\beta ^ * , \\alpha , \\beta - \\alpha ^ * - \\beta ^ * ; A , B _ { - \\alpha ^ * - \\beta ^ * } , h , k ) \\\\ = \\mathcal { K } ( 0 , 0 , 2 - \\alpha - \\beta + \\alpha ^ * + \\beta ^ * ; A , B _ { - \\alpha ^ * - \\beta ^ * } , \\alpha , \\beta - \\alpha ^ * - \\beta ^ * , h , k ) . \\end{align*}"}
{"id": "7749.png", "formula": "\\begin{align*} \\int _ { t - d } ^ { t + d } \\int _ { \\lambda > 1 } e ^ { i s \\lambda } \\hat { g } ( \\lambda ) d \\lambda d s & = \\int _ { \\lambda > 1 } \\hat { g } ( \\lambda ) \\int _ { t - d } ^ { t + d } e ^ { i s \\lambda } d s d \\lambda \\\\ & = - i \\int _ { \\lambda > 1 } \\hat { g } ( \\lambda ) \\left ( \\frac { e ^ { i ( t + d ) \\lambda } } { \\lambda } - \\frac { e ^ { i ( t - d ) \\lambda } } { \\lambda } \\right ) d \\lambda . \\end{align*}"}
{"id": "3929.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\ , \\sup _ { \\tau \\in [ 0 , 1 ] } S \\left ( \\omega _ { V ( 0 ) } \\circ \\alpha _ T ^ { 0 \\to \\tau } \\bigg | \\omega _ { V ( \\tau ) } \\right ) = 0 . \\end{align*}"}
{"id": "8595.png", "formula": "\\begin{align*} \\mu ^ { \\# } ( k , \\ell , m , n ) : = \\int \\overline { \\mathcal { K } ^ { \\# } ( x , k ) } \\mathcal { K } ^ { \\# } ( x , \\ell ) \\overline { \\mathcal { K } ^ { \\# } ( x , m ) } \\mathcal { K } ^ { \\# } ( x , n ) \\ , d x . \\end{align*}"}
{"id": "1424.png", "formula": "\\begin{align*} & \\int _ { [ 0 , 1 ) } \\frac { ( 1 - | a | ) ^ t } { ( 1 - x ) ^ r ( 1 - | a | x ) ^ { s + t - r } } d \\mu ( x ) \\\\ = & \\int _ { S _ 1 ( a ) } \\frac { ( 1 - | a | ) ^ t } { ( 1 - x ) ^ r ( 1 - | a | x ) ^ { s + t - r } } d \\mu ( x ) \\\\ & + \\sum ^ { n _ a } _ { n = 2 } \\int _ { S _ n ( a ) \\backslash S _ { n - 1 } ( a ) } \\frac { ( 1 - | a | ) ^ t } { ( 1 - x ) ^ r ( 1 - | a | x ) ^ { s + t - r } } d \\mu ( x ) \\\\ = : & J _ 1 ( a ) + J _ 2 ( a ) . \\end{align*}"}
{"id": "7572.png", "formula": "\\begin{align*} L \\psi _ { \\eta } = \\psi _ { \\eta } ( 0 ) = 0 . \\end{align*}"}
{"id": "5155.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial \\eta } A ( \\tfrac { \\eta } { 3 } , \\tfrac { 1 } { \\eta } ) \\Bigr | _ { \\substack { \\eta = 3 . 9 } } > 0 \\tfrac { \\partial } { \\partial \\eta } B ( \\tfrac { \\eta } { 3 } , \\tfrac { 1 } { \\eta } ) \\Bigr | _ { \\substack { \\eta = 3 . 9 } } > 0 , \\end{align*}"}
{"id": "7398.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 ^ { + } } | \\langle \\pi _ s , \\overleftarrow { i _ { \\varepsilon } ^ { u } } \\rangle - \\rho ( s , u ) | = \\lim _ { \\varepsilon \\rightarrow 0 ^ { + } } | \\langle \\pi _ s , \\overrightarrow { i _ { \\varepsilon } ^ { u } } \\rangle - \\rho ( s , u ) | = 0 , \\end{align*}"}
{"id": "953.png", "formula": "\\begin{align*} \\Lambda _ m = \\{ n \\in \\Gamma _ { 2 , y , ( r , s ) } ^ { \\delta } : A _ n ^ \\delta \\subseteq A _ m ^ \\delta \\} \\end{align*}"}
{"id": "4161.png", "formula": "\\begin{align*} \\norm { A \\oplus B } _ { p } = ( \\norm { A } _ { p } ^ { p } + \\norm { B } _ { p } ^ { p } ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
{"id": "3064.png", "formula": "\\begin{align*} x ^ \\prime = z ( j ^ 2 z , j z ) \\ , , y ^ \\prime = y \\ , , z ^ \\prime = x ( j x , j ^ 2 x ) \\ , , w ^ \\prime = w \\ , . \\end{align*}"}
{"id": "665.png", "formula": "\\begin{align*} q \\ = \\ A ( 4 ^ { k + 1 } - 1 ) , \\end{align*}"}
{"id": "8649.png", "formula": "\\begin{align*} n a + ( n - 2 ) b = 0 \\end{align*}"}
{"id": "5225.png", "formula": "\\begin{align*} \\| B f \\| _ { F _ { r t } ^ s ( K ) } \\leq \\sum _ { j = k } ^ { \\infty } q ^ { - ( j + 1 ) } ~ \\| g _ j \\ast f ( x ) \\| _ { F _ { r t } ^ s ( K ) } . \\end{align*}"}
{"id": "8430.png", "formula": "\\begin{gather*} \\nabla V _ { \\beta } ( x ) = \\frac { \\beta x } { ( 1 + | x | ^ { 2 } ) ^ { \\frac { 2 - \\beta } { 2 } } } , \\nabla ^ { 2 } V _ { \\beta } ( x ) = \\frac { \\beta I _ d } { ( 1 + | x | ^ { 2 } ) ^ { 1 - \\frac { \\beta } { 2 } } } + \\frac { \\beta ( \\beta - 2 ) x x ^ { \\top } } { ( 1 + | x | ^ { 2 } ) ^ { 2 - \\frac { \\beta } { 2 } } } , \\end{gather*}"}
{"id": "4958.png", "formula": "\\begin{align*} V _ s = V _ s ( { r _ v - 1 } ) \\oplus V _ s ( { r _ v } ) . \\end{align*}"}
{"id": "39.png", "formula": "\\begin{align*} \\| ( d \\theta _ { t } ^ { c } ) ^ { n } \\wedge \\theta \\wedge \\theta _ { t } ^ { c } \\| & \\ge 1 - \\sum ^ n _ { k = 2 } \\binom n k \\| d \\theta _ { t } ^ { c } \\| ^ { n - k } \\cdot \\| d \\zeta ^ k _ t \\| \\cdot \\| \\theta \\| \\cdot \\| \\theta ^ c \\| \\\\ & - \\sum ^ n _ { k = 1 } \\binom n k \\| d \\theta _ { t } ^ { c } \\| ^ { n - k } \\cdot \\| d \\zeta ^ k _ t \\| \\cdot \\| \\theta \\| \\cdot \\| \\zeta _ t \\| \\\\ & > 1 - ( 2 ^ { n + 1 } - 1 ) \\frac 1 { 2 ^ { n + 1 } - 1 } > 0 . \\end{align*}"}
{"id": "5888.png", "formula": "\\begin{align*} A ^ { \\pm h } _ { a , b } = c ^ { - \\frac { 1 } { \\rho } } \\left [ ( a \\vee 0 ) ^ \\frac { 1 } { \\rho } \\mp h , \\ , ( b \\vee 0 ) ^ \\frac { 1 } { \\rho } \\pm h \\right ] , \\end{align*}"}
{"id": "4705.png", "formula": "\\begin{align*} \\partial _ y ( \\mathcal { L } \\mathfrak { B } _ { i j k } ) = G _ { i j k } - b _ { i j k } \\sigma _ i \\Lambda Q . \\end{align*}"}
{"id": "8523.png", "formula": "\\begin{align*} Q ( y ) = \\operatorname { s g n } ( y ) \\min \\left ( \\lfloor | y | / 2 ^ r \\rfloor + 1 , 2 ^ { w - 1 } \\right ) . \\end{align*}"}
{"id": "7811.png", "formula": "\\begin{align*} ( g - 1 ) \\left ( \\frac { c _ x } { \\pi _ 1 } - \\frac { a ^ 2 } { 2 } \\right ) = - \\kappa ( x ) ( g ) . \\end{align*}"}
{"id": "1902.png", "formula": "\\begin{align*} \\beta _ v ( \\rho ( t ) ) = a _ { v , \\xi } t + b _ { v , x , \\xi } \\end{align*}"}
{"id": "6932.png", "formula": "\\begin{align*} F _ h ( v ) = \\sum _ { E \\in { \\cal T } _ h } \\sum _ { \\iota \\in I ^ E } [ f v ] ( \\xi ^ E _ \\iota ) \\ , \\omega ^ E _ \\iota \\ , . \\end{align*}"}
{"id": "7541.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { s - \\rho } - \\frac { 1 } { 2 + i t - \\rho } \\right | = ( 2 - \\sigma ) \\left | \\frac { 1 } { ( s - \\rho ) ( 2 + i t - \\rho ) } \\right | \\leq \\frac { 3 } { | \\gamma - t | ^ 2 } \\end{align*}"}
{"id": "2229.png", "formula": "\\begin{align*} X ( \\cdot ) = \\lim _ { n \\rightarrow \\infty } X ^ { n } ( \\cdot ) \\ ; \\ ; C _ W ( [ 0 , T ] ; H ^ { \\frac d 3 } ) , d = 1 , 2 , 3 . \\end{align*}"}
{"id": "3768.png", "formula": "\\begin{align*} J _ \\psi : = { \\rm i n d } _ { N _ 2 ( K ) } ^ { P _ 2 ( K ) } ( \\psi ) , \\end{align*}"}
{"id": "3499.png", "formula": "\\begin{align*} D _ { 2 3 } & = s _ 3 \\int _ { a t _ 3 } ^ \\infty \\frac { v - [ v ] - \\frac { 1 } { 2 } } { v ^ { s _ 2 } } \\int _ v ^ \\infty \\frac { 1 } { u ^ { s _ 1 + s _ 3 + 1 } ( 1 + \\frac { v } { u } ) ^ { s _ 3 + 1 } } d u d v \\\\ & = \\frac { s _ 3 } { 2 \\pi i \\Gamma ( s _ 3 + 1 ) } \\int _ { a t _ 3 } ^ \\infty \\int _ v ^ \\infty \\int _ { ( c ) } \\frac { ( v - [ v ] - \\frac { 1 } { 2 } ) \\Gamma ( s _ 3 + 1 + z ) \\Gamma ( - z ) } { v ^ { s _ 2 - z } u ^ { s _ 1 + s _ 3 + 1 + z } } d z d u d v \\end{align*}"}
{"id": "4509.png", "formula": "\\begin{align*} t _ 0 = t _ 0 ( x ) = \\exp \\Big ( \\sqrt { \\frac { \\log x } R } \\Big ) , \\ T = t _ 0 ^ c , \\ \\ \\sigma _ 2 = 1 - \\frac { 2 } { R \\log t _ 0 } . \\end{align*}"}
{"id": "9083.png", "formula": "\\begin{align*} | h _ { \\mathrm { F A S } } | = \\max \\{ | h _ { 1 } | , | h _ { 2 } | , { \\dots } , | h _ { N } | \\} . \\end{align*}"}
{"id": "4982.png", "formula": "\\begin{align*} \\mathcal { R } ^ { 2 } ( \\eta , \\xi ) = \\Lambda ( ( \\eta ^ { r _ 0 } \\circ \\xi ) ^ { r _ 1 } \\circ \\eta , \\eta ^ { r _ 0 } \\circ \\xi ) \\end{align*}"}
{"id": "7947.png", "formula": "\\begin{align*} \\sum _ { a \\in S } a ^ { \\ell + 1 } = \\begin{cases} \\frac { 1 } { \\ell + 2 } t n ^ { \\ell + 1 } + O _ \\ell \\left ( n ^ { \\ell + 1 1 / 8 } \\right ) , ~ & ~ t \\geqslant n ^ { 1 / 2 } - n ^ { 1 / 4 } \\\\ \\frac { 1 } { \\ell + 2 } t n ^ { \\ell + 1 } + O _ \\ell \\left ( n ^ { \\ell + 5 / 4 } \\sqrt { n ^ { 1 / 2 } - t } \\right ) , ~ & t < n ^ { 1 / 2 } - n ^ { 1 / 4 } . \\end{cases} \\end{align*}"}
{"id": "8909.png", "formula": "\\begin{align*} \\check H _ { c t } ^ 0 ( X ; A _ X ) = A _ X ( X ) = \\begin{cases} A ^ { e ( X ) } & e ( X ) < \\infty \\\\ \\bigoplus _ { \\N } A & e ( X ) = \\infty . \\end{cases} \\end{align*}"}
{"id": "8873.png", "formula": "\\begin{align*} d _ { p + q } ( \\phi \\vee \\psi ) = d _ p \\phi \\vee \\psi + ( - 1 ) ^ p \\phi \\vee d _ q \\psi \\end{align*}"}
{"id": "3507.png", "formula": "\\begin{align*} D _ { 1 2 2 } \\ll \\begin{cases} t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } & ( \\sigma _ 2 > 1 ) \\\\ t _ 3 ^ { - \\sigma _ 1 - \\sigma _ 3 } \\log t _ 3 & ( \\sigma _ 2 = 1 ) \\\\ t _ 3 ^ { 1 - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } & ( \\sigma _ 2 < 1 ) . \\\\ \\end{cases} \\end{align*}"}
{"id": "1072.png", "formula": "\\begin{align*} \\Phi ^ { \\pm \\infty } ( x , t , k ) = \\Delta _ { R , L } ( k ) e ^ { - i ( X _ { R , L } ( k ) x + \\Omega _ { R , L } ( k ) t ) \\sigma _ 3 } , \\Phi ^ { \\pm \\infty } : \\mathbb { R } \\times [ 0 , + \\infty ) \\times ( \\mathbb { C } \\backslash [ - C , C ] ) \\rightarrow \\mathbb { C } , \\end{align*}"}
{"id": "8280.png", "formula": "\\begin{align*} \\sum _ { y \\leq q , \\ , g ( y ) = p } \\mu _ { Q } ( y , q ) = \\begin{cases} \\mu _ { Q } ( y , q ) , & y \\in Q \\ a n d \\ p = g ( y ) , \\\\ 0 , & o t h e r w i s e . \\end{cases} \\end{align*}"}
{"id": "6608.png", "formula": "\\begin{align*} \\ll Q \\sum _ { 1 \\leq c \\leq C } \\frac { 1 } { c } \\sum _ { \\substack { 1 \\leq m , n \\ll X \\\\ m h = n k } } \\frac { ( m n ) ^ \\varepsilon } { \\sqrt { m n } } . \\end{align*}"}
{"id": "5068.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } D ^ + \\psi = \\Phi & \\Omega \\\\ P ^ \\pm \\psi _ { | \\Sigma } = P ^ \\pm \\Psi & \\Sigma \\end{array} \\right . \\end{align*}"}
{"id": "3226.png", "formula": "\\begin{align*} F _ { \\mu \\boxplus \\nu } ( z ) = F _ \\mu ( \\omega _ 1 ( z ) ) = F _ \\nu ( \\omega _ 2 ( z ) ) . \\end{align*}"}
{"id": "9062.png", "formula": "\\begin{align*} \\frac { 1 } { b - a } < \\rho ^ * _ { i , j _ { k + 1 } } < \\frac { 1 } { h } = \\frac { N } { b - a } . \\end{align*}"}
{"id": "7824.png", "formula": "\\begin{align*} X & : M _ { k } \\rightarrow { M _ { k } } X e _ { n } = \\delta _ { n } e _ { n } \\ ~ \\delta _ { 0 } = 1 , ~ \\delta _ { n } = w _ { 0 } w _ { 1 } \\cdots w _ { n - 1 } ~ ~ 1 \\leq n \\leq k ; \\\\ T _ { 1 } ^ { * } & : M _ { k } \\rightarrow { M _ { k } } T _ { 1 } ^ { * } e _ { 0 } = 0 , \\ T _ { 1 } ^ { * } e _ { n } = e _ { n - 1 } ~ ~ 1 \\leq n \\leq k . \\end{align*}"}
{"id": "7713.png", "formula": "\\begin{align*} X h = g _ 1 \\cdot \\ldots \\cdot g _ { n + 1 } \\ , . \\end{align*}"}
{"id": "7407.png", "formula": "\\begin{align*} \\sqrt { f \\left ( \\eta ^ { x , x + r } \\right ) } - \\sqrt { f \\left ( \\eta \\right ) } = \\sum _ { k = 1 } ^ { 6 } [ \\sqrt { f \\left ( \\eta _ { k } \\right ) } - \\sqrt { f \\left ( \\eta _ { k - 1 } \\right ) } ] . \\end{align*}"}
{"id": "3214.png", "formula": "\\begin{align*} R _ 2 ^ { \\infty } = \\mathbb { E } \\left [ \\log _ 2 \\left ( 1 + \\frac { \\alpha _ 2 } { \\alpha _ 1 } + \\frac { P _ { t h } | h _ { 1 2 } | ^ 2 } { \\sigma ^ 2 } \\right ) \\right ] , \\end{align*}"}
{"id": "968.png", "formula": "\\begin{align*} \\int _ { \\R ^ n _ + } \\frac { \\dd z } { \\vert e _ 1 + z \\vert ^ { n + 2 s } } & = \\int _ 0 ^ \\infty \\int _ { \\R ^ { n - 1 } } \\frac { \\dd z _ 1 \\dd z ' } { \\big ( ( z _ 1 + 1 ) ^ 2 + \\vert z ' \\vert ^ 2 \\big ) ^ { \\frac { n + 2 s } 2 } } \\end{align*}"}
{"id": "3228.png", "formula": "\\begin{align*} s ( | \\lambda | , t ) = W ( t ) = \\Im \\omega _ 1 ( i t ) . \\end{align*}"}
{"id": "4741.png", "formula": "\\begin{align*} Q ( y ) = \\frac { \\kappa _ 0 } { y ^ 2 } + O \\bigg ( \\frac { 1 } { y ^ 4 } \\bigg ) , \\ ; \\ ; y \\rightarrow + \\infty . \\end{align*}"}
{"id": "6512.png", "formula": "\\begin{align*} h _ n ^ { ( 2 m ) } \\sim - \\frac { m ( m - 1 ) } { 3 } \\cdot ( 2 \\alpha ^ 2 + 1 ) \\cdot n ^ { - 2 } ( n \\to \\infty ) . \\end{align*}"}
{"id": "5137.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\rightarrow \\infty } \\psi ' ( x ) = 1 - 1 + 0 = 0 . \\end{align*}"}
{"id": "794.png", "formula": "\\begin{align*} C : = \\int _ Z \\frac { | f ( y ) - f ( x _ 0 ) | ^ p } { \\nu ( B ( y , d ( x _ 0 , y ) ) ) d ( x _ 0 , y ) ^ { \\theta p } } d \\nu ( y ) < \\infty . \\end{align*}"}
{"id": "9457.png", "formula": "\\begin{align*} L _ { 2 0 } + \\tfrac { B ( 0 , 2 ) ^ p } { B ( 0 ) ^ p } - \\tfrac { B ( - 1 , 2 ) } { B ( - 1 ) } = 0 . \\end{align*}"}
{"id": "8897.png", "formula": "\\begin{align*} \\psi ( z _ 0 , \\ldots , z _ { q - 1 } ) = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( z _ 0 , \\ldots , z _ i , p ( z _ i ) , \\ldots , p ( z _ { q - 1 } ) ) . \\end{align*}"}
{"id": "6708.png", "formula": "\\begin{align*} \\bigl ( ~ _ r \\mathcal { F } _ s ( \\alpha ) \\bigr ) ^ { q ^ d } & = \\sum _ { n = N } ^ { \\infty } \\biggl ( \\prod _ { m = 1 } ^ { n + d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \\biggr ) \\alpha ^ { q ^ { n + d } } + \\sum _ { n = N - d + 1 } ^ { N - 1 } \\biggl ( \\prod _ { m = 1 } ^ { n + d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \\biggr ) \\alpha ^ { q ^ { n + d } } \\\\ & + \\sum _ { n = 0 } ^ { N - d } \\biggl ( \\prod _ { m = 1 } ^ { n + d - 1 } ( \\theta ^ { q ^ m } - t ) ^ { c ( m - n ) q ^ { n + d - m } } \\biggr ) \\alpha ^ { q ^ { n + d } } . \\end{align*}"}
{"id": "944.png", "formula": "\\begin{align*} J _ { n , m } = J \\times J _ { n , m } ^ { 2 } \\cdots \\times J _ { n , m } ^ d , \\end{align*}"}
{"id": "5188.png", "formula": "\\begin{align*} \\Gamma ( \\pi ) & = \\lim _ { t \\to \\infty } \\frac { 1 } { N } \\sum _ { \\ell = 1 } ^ { N } \\frac { \\sum _ { i = 1 } ^ { R _ \\ell ^ \\pi ( t ) } \\rho _ \\ell ( \\frac { ( T _ { \\ell i } ^ \\pi ) ^ 2 } { 2 } + T _ { \\ell i } ^ \\pi Z _ { \\ell i } ^ \\pi ) + c _ \\ell \\tilde { R } _ \\ell ^ \\pi ( t ) } { t } . \\end{align*}"}
{"id": "8472.png", "formula": "\\begin{align*} 2 e ( S _ 1 ) \\alpha + 2 e ( L _ 1 ) \\leq 2 e ( N _ 1 ) \\alpha + 2 \\binom { | L | } { 2 } \\leq ( 2 k - 1 ) n \\alpha + | L | ( | L | - 1 ) < 2 k n \\alpha , \\end{align*}"}
{"id": "7635.png", "formula": "\\begin{align*} - \\Delta w = \\tilde { \\lambda } _ 0 \\tilde { m } _ 0 w \\R ^ N \\ , . \\end{align*}"}
{"id": "3591.png", "formula": "\\begin{align*} ( \\overline { R } ( e ) ) ( x ^ * ) = R ( e \\otimes x ^ * \\vert _ { Q ( X ) } ) \\ , \\ , e \\in E x ^ * \\in X ^ * . \\end{align*}"}
{"id": "6693.png", "formula": "\\begin{align*} & \\mathbb { D } _ { i } : = \\prod _ { j = 1 } ^ i ( \\theta ^ { q ^ j } - t ) ^ { q ^ { i - j } } \\mathbb { D } _ { i } : = 1 , \\\\ & \\mathbb { L } _ { i } : = \\prod _ { j = 1 } ^ i ( t - \\theta ^ { q ^ j } ) \\mathbb { L } _ { i } : = 1 , \\end{align*}"}
{"id": "4419.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\sum _ { i = 1 } ^ { m } \\overline { z _ i } a _ { i j } = \\eta , \\ ; \\ ; \\ ; j = 1 , 2 , . . . , n \\\\ \\displaystyle \\sum _ { i = 1 } ^ { m } \\overline { z _ i } = 1 \\\\ \\lvert a r g \\overline { z _ i } \\rvert \\leq \\alpha , \\ ; \\ ; \\ ; i = 1 , 2 , . . . , m \\end{array} \\end{align*}"}
{"id": "9115.png", "formula": "\\begin{align*} \\binom { n + 1 } { 2 } - t \\geq \\frac { n ( n + 1 ) - ( s + 1 ) ( 2 n - s ) } { 2 } & = \\frac { n ^ 2 + n - 2 n s + s ^ 2 - 2 n + s } { 2 } \\\\ & = \\frac { ( n - s ) ^ 2 - ( n - s ) } { 2 } = \\binom { n - s } { 2 } \\end{align*}"}
{"id": "8840.png", "formula": "\\begin{align*} x : = n \\epsilon . \\end{align*}"}
{"id": "1285.png", "formula": "\\begin{align*} A ( \\epsilon _ { ( t _ { 1 } , t _ { 2 } , n ) } ) = t _ { 1 } R _ { 1 } + t _ { 2 } R _ { 2 } + T _ { n } \\end{align*}"}
{"id": "8051.png", "formula": "\\begin{align*} \\Psi ^ n _ { ( \\Sigma , U ) } ( f ) [ \\psi ] : = \\int _ { \\mathcal { I } } f ( s ) \\psi ^ n ( s ) \\gamma ' ( s ) ^ { \\tfrac { n } { 2 } } \\mathrm { d } s \\end{align*}"}
{"id": "2244.png", "formula": "\\begin{align*} \\int _ \\sigma ^ t ( t - s ) ^ { \\alpha - 1 } ( s - \\sigma ) ^ { - \\alpha } \\ , \\dd s = \\frac { \\pi } { \\sin \\pi \\alpha } , \\ ; 0 \\leq \\sigma \\leq t , \\ ; 0 < \\alpha < 1 . \\end{align*}"}
{"id": "8292.png", "formula": "\\begin{align*} \\partial _ t X _ t \\cdot \\nu ( X _ t ) & = \\frac { - \\dot { x } ( t ) \\cdot \\nabla u ( t , x ( t ) ) } { \\sqrt { 1 + | \\nabla u ( t , x ( t ) ) | ^ 2 } } + \\frac { \\dot { x } ( t ) \\cdot \\nabla u ( t , x ( t ) ) } { \\sqrt { 1 + | \\nabla u ( t , x ( t ) ) | ^ 2 } } + \\frac { \\partial _ t u ( t , x ( t ) ) } { \\sqrt { 1 + | \\nabla u ( t , x ( t ) ) | ^ 2 } } \\\\ & = \\frac { \\partial _ t u ( t , x ( t ) ) } { \\sqrt { 1 + | \\nabla u ( t , x ( t ) ) | ^ 2 } } . \\end{align*}"}
{"id": "7585.png", "formula": "\\begin{align*} c _ { i , j } ^ { ( r , s ) } & = \\dfrac { 1 } { 2 j } \\left ( c _ { i , j - 1 } ^ { ( r , s + 1 ) } - c _ { i , j + 1 } ^ { ( r , s + 1 ) } \\right ) \\end{align*}"}
{"id": "1635.png", "formula": "\\begin{align*} \\sup _ { t > 0 } | t ^ { i - 1 } F ^ { ( i ) } ( t ) | \\leq L , i = 0 , 1 , 2 , \\end{align*}"}
{"id": "9507.png", "formula": "\\begin{align*} \\| H ' + r f \\| & = 1 , \\\\ p ( H ' + r f & ) = - \\frac { p ( f ) } { | p ( f ) | } , \\\\ \\| H ' \\| & \\le 1 + | p ( f ) | . \\end{align*}"}
{"id": "3104.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + z y ( a x ^ 4 + b x ^ 2 y ^ 2 + c y ^ 4 ) + d x ^ 6 + e x ^ 4 y ^ 2 + f x ^ 2 y ^ 4 + g y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "3856.png", "formula": "\\begin{align*} G _ k = \\Bigl \\{ \\rho \\in \\mathbb C \\colon \\arg \\rho \\in \\left ( ( - 1 ) ^ { n - k } - 1 ) \\tfrac { \\pi } { 2 n } , ( ( - 1 ) ^ { n - k } + 3 ) \\tfrac { \\pi } { 2 n } \\right ) \\Bigr \\} , \\end{align*}"}
{"id": "1041.png", "formula": "\\begin{align*} u ( x ) & = C \\int _ { \\R ^ n \\setminus B _ 1 } \\bigg ( \\frac { 1 - \\vert x \\vert ^ 2 } { \\vert y \\vert ^ 2 - 1 } \\bigg ) ^ s \\bigg ( \\frac 1 { \\vert x - y \\vert ^ n } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ n } \\bigg ) u ( y ) \\dd y \\\\ & \\leqslant C x _ 1 \\int _ { \\R ^ n \\setminus B _ 1 } \\frac { y _ 1 u ( y ) } { \\big ( \\vert y \\vert ^ 2 - 1 \\big ) ^ s \\vert y \\vert ^ { n + 2 } } \\dd y . \\end{align*}"}
{"id": "559.png", "formula": "\\begin{align*} b ( s ) = \\begin{cases} 0 & s \\leq \\overline \\rho - \\alpha _ 1 \\\\ - \\log ( \\overline \\rho - s ) & \\overline \\rho - \\alpha _ 2 \\leq s < \\overline { \\rho } \\end{cases} \\end{align*}"}
{"id": "2820.png", "formula": "\\begin{align*} \\delta ( t ) \\simeq \\| h ( t ) \\| _ { H ^ 1 } \\simeq | \\alpha ( t ) | = \\Big | \\int _ t ^ \\infty \\dot { \\alpha } ( s ) d s \\Big | \\lesssim \\int _ t ^ \\infty \\delta ( s ) d s \\le C e ^ { - c t } , \\ ; \\forall t \\ge 0 , \\end{align*}"}
{"id": "5108.png", "formula": "\\begin{align*} \\psi ( x ) = q _ 1 ( x ) + 6 \\ , q _ 2 ( x \\ , \\coth ( x ) ) , q _ 1 ( x ) = 1 - 2 x ^ 2 q _ 2 ( x ) = x ( x - 1 ) . \\end{align*}"}
{"id": "5304.png", "formula": "\\begin{align*} \\langle a , b \\rangle = \\varphi ( a ^ * b ) . \\end{align*}"}
{"id": "747.png", "formula": "\\begin{align*} Y ^ { ( \\ell + 1 ) } [ \\partial _ \\mu z , \\partial _ \\mu z ; f , g ] = Y ^ { = ( \\ell + 1 ) } [ \\partial _ \\mu z , \\partial _ \\mu z ; f , g ] + Y ^ { < ( \\ell + 1 ) } [ \\partial _ \\mu z , \\partial _ \\mu z ; f , g ] , \\end{align*}"}
{"id": "5010.png", "formula": "\\begin{align*} D _ { k n } \\le { \\log 2 \\over \\pi ( K + 1 ) } + { 1 \\over \\pi | \\bar { u } _ { k n } | } \\sum _ { k = 1 } ^ K { 1 \\over k } \\min \\left ( | \\bar { u } _ { k n } | , { 1 \\over \\{ k \\rho _ * \\} } \\right ) , \\end{align*}"}
{"id": "4086.png", "formula": "\\begin{align*} \\begin{aligned} d X _ t & = \\mu ( X _ t ) \\ , d t + \\sigma ( X _ t ) \\ , d W _ t , t \\in [ 0 , \\infty ) , \\\\ X _ 0 & = x _ 0 , \\end{aligned} \\end{align*}"}
{"id": "3405.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ \\infty \\psi _ { t } \\ast q _ { t } \\ast f ( x ) \\frac { d t } { t } , \\end{align*}"}
{"id": "6331.png", "formula": "\\begin{align*} \\partial _ { \\alpha \\beta \\gamma } g = \\nabla _ \\lambda g S ^ \\lambda _ { \\alpha \\beta \\gamma } + \\nabla _ { \\mu \\nu } g S ^ { \\mu \\nu } _ { \\alpha \\beta \\gamma } + \\nabla _ { \\lambda \\mu \\nu } g S ^ { \\lambda \\mu \\nu } _ { \\alpha \\beta \\gamma } \\end{align*}"}
{"id": "6666.png", "formula": "\\begin{align*} R _ { 2 3 } = J _ { 3 1 } + O \\big ( ( X h k ) ^ { \\varepsilon } ( h , k ) ^ { 1 / 2 } Q ^ { - 9 6 } \\big ) . \\end{align*}"}
{"id": "1045.png", "formula": "\\begin{align*} & D _ { i j } : = a _ { i 1 } a _ { j 2 } - q ^ { - 1 } a _ { i 2 } a _ { j 1 } & & D _ { i n } : = a _ { i 1 } a _ { n 2 } - q ^ { - 1 } a _ { i 2 } a _ { n 1 } & & \\\\ & D _ { 5 5 } : = a _ { 5 1 } a _ { 5 2 } & & D _ { 6 6 } : = a _ { 6 1 } a _ { 6 2 } \\\\ & D _ { 5 6 } = a _ { 5 1 } a _ { 6 2 } - q ^ { - 1 } a _ { 5 2 } a _ { 6 1 } \\\\ \\end{align*}"}
{"id": "70.png", "formula": "\\begin{align*} \\frac { \\lambda _ v ^ { K _ { \\ell } } } { \\lambda _ v ^ L } & = \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ { K _ { \\ell } } - m ) / 2 } \\cdot | M _ v ^ { K _ { \\ell } } | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} \\Bigl \\{ q _ v ^ { ( \\dim M _ v ^ L - m ) / 2 } \\cdot | M _ v ^ L | ^ { - 1 } \\cdot \\prod _ { i = 1 } ^ { m } ( q _ v ^ { 2 i } - 1 ) \\Bigr \\} ^ { - 1 } \\\\ & \\leq q _ v ^ { - 1 / 2 } ( q _ v ^ { n _ { v , _ n u _ v } } - 1 ) . \\end{align*}"}
{"id": "1799.png", "formula": "\\begin{align*} V _ { { \\rm C M } } ( D ) : = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) , \\end{align*}"}
{"id": "650.png", "formula": "\\begin{align*} \\begin{cases} \\ \\\\ [ 5 p t ] \\ \\end{cases} \\in \\ \\R _ { \\ell } . \\end{align*}"}
{"id": "560.png", "formula": "\\begin{align*} \\lim _ { s \\to \\overline \\rho _ - } \\frac { p ( s ) } { ( b ( s ) ) ^ \\gamma } = \\lim _ { s \\to \\overline \\rho _ - } \\frac { P ( s ) } { ( b ( s ) ) ^ \\gamma } = \\lim _ { s \\to \\overline \\rho _ - } \\frac { p ( s ) } { ( b ' ( s ) ) ^ \\beta } = \\lim _ { s \\to \\overline \\rho _ - } \\frac { P ( s ) } { ( b ( s ) ) ^ { \\beta - 1 } } = + \\infty . \\end{align*}"}
{"id": "8632.png", "formula": "\\begin{align*} \\mu _ { R , 1 } ^ { ( 1 ) } ( k , \\ell , m , n ) & : = \\int \\overline { \\mathcal { K } } _ { R } ( x , k ) \\mathcal { K } _ { M _ { 1 } } ( x , \\ell ) \\overline { \\mathcal { K } _ { M _ { 2 } } ( x , n ) } \\mathcal { K } _ { M _ { 3 } } ( x , m ) \\ , d x , \\\\ \\mu _ { R , 1 } ^ { ( 2 ) } ( k , \\ell , m , n ) & : = \\int \\overline { \\mathcal { K } } _ { S } ( x , k ) \\mathcal { K } _ { M _ { 1 } } ( x , \\ell ) \\overline { \\mathcal { K } _ { M _ { 2 } } ( x , n ) } \\mathcal { K } _ { R } ( x , m ) \\ , d x , M _ { i } \\in \\{ S , R \\} . \\end{align*}"}
{"id": "2379.png", "formula": "\\begin{align*} \\sigma _ 1 ^ 2 \\norm { f } ^ 2 \\leq \\norm { C _ \\mathbf { F } f } ^ 2 = \\langle S _ \\mathbf { F } f , f \\rangle = \\sum _ { k = 1 } ^ K | \\langle f , v _ k \\rangle | ^ 2 \\leq \\sigma _ d ^ 2 \\norm { f } ^ 2 . \\end{align*}"}
{"id": "2018.png", "formula": "\\begin{align*} M _ { t } ^ { G ^ u } = M _ { t } ^ { F ^ u } + \\sum _ { 0 < s \\le t } ( G ^ { u } - F ^ { u } ) ( X _ { s - } , X _ { s } ) - \\int _ { 0 } ^ { t } N ( G ^ { u } - F ^ { u } ) ( X _ { s } ) { \\rm d } H _ { s } , t < \\zeta . \\end{align*}"}
{"id": "9462.png", "formula": "\\begin{align*} f _ { s + 2 } = \\left ( \\tfrac { 2 ^ { p ^ { s - 1 } } B ( s - 2 ) ^ p } { B ( s - 3 ) ^ p } \\right ) \\frac { F _ { s + 2 } ( { \\bf \\underline { f } } ) } { X ^ { 2 p } } . \\end{align*}"}
{"id": "4518.png", "formula": "\\begin{gather*} a _ 0 ( b ) ^ 2 = ( 1 - \\alpha ) a _ 0 ( b ) , \\\\ a _ { 1 / 2 } ( b ) ^ 2 = \\alpha a _ 0 ( b ) + ( \\alpha - \\alpha ^ 2 ) a , \\\\ a _ 0 ( b ) a _ { 1 / 2 } ( b ) = \\frac { 1 } { 2 } ( 1 - \\alpha ) a _ { 1 / 2 } ( b ) . \\end{gather*}"}
{"id": "1919.png", "formula": "\\begin{align*} \\int ^ { T } _ 0 \\int _ { \\Omega } ( \\rho \\partial _ t \\psi + \\rho u \\cdot \\nabla _ x \\psi ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 \\psi ( 0 , x ) \\ , d x = 0 ; \\end{align*}"}
{"id": "975.png", "formula": "\\begin{align*} I _ 1 & = ( - \\Delta ) ^ s \\bar u ( x ) - \\frac 1 { 2 s } c _ { n , s } n \\vert B _ 1 \\vert r ^ { - 2 s } u ( x ) - c _ { n , s } \\int _ B \\frac { u ( x ) - u ( y ) } { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\dd y \\end{align*}"}
{"id": "1690.png", "formula": "\\begin{align*} G _ { m , n } & = \\sum \\limits _ { k = 1 } ^ n \\frac { ( - 1 ) ^ { k - 1 } } { ( m - 1 ) ! ( m + k - 1 ) } \\binom { n } { k } { _ 2 } F _ 1 ( 1 , 1 - k ; m + k ; - 1 ) . \\end{align*}"}
{"id": "8984.png", "formula": "\\begin{align*} e ^ { - t } C _ p q \\left | \\dot { \\xi } ( t ) \\right | ^ { q - 2 } \\dot { \\xi } ( t ) = \\int _ 0 ^ t e ^ { - s } D f \\left ( \\xi ( s ) \\right ) d s + \\tilde { c } \\end{align*}"}
{"id": "8273.png", "formula": "\\begin{align*} M _ { \\alpha } : = \\sum _ { 0 < i _ 2 < \\cdots < i _ n } x _ 0 ^ { \\alpha _ 1 } x _ { i _ 2 } ^ { \\alpha _ 2 } \\cdots x _ { i _ k } ^ { \\alpha _ k } F _ { \\alpha } : = \\sum _ { { 0 \\leq i _ 1 \\leq i _ 2 \\leq \\cdots \\leq i _ n } \\atop { j \\in D ( \\alpha ) \\Rightarrow i _ j < i _ { j + 1 } } } x _ { i _ 1 } x _ { i _ 2 } \\cdots x _ { i _ n } , \\end{align*}"}
{"id": "1629.png", "formula": "\\begin{align*} \\mathcal { D } _ p : = \\left \\lbrace f \\in \\mathcal { D } ( \\mathcal { L } ) \\cap L ^ p ( M ) : \\mathcal { L } f \\in L ^ p ( M ) \\right \\rbrace . \\end{align*}"}
{"id": "4709.png", "formula": "\\begin{align*} & \\alpha _ i = \\sum _ { j = i } ^ k \\theta _ j , \\forall i = 1 , \\ldots , k , \\\\ & \\alpha _ { n + 1 - i } = - \\alpha _ i , \\forall i = 1 , \\ldots , k . \\end{align*}"}
{"id": "5286.png", "formula": "\\begin{align*} S ( S ( a ) ^ * ) ^ * = a , \\forall a \\in A , \\end{align*}"}
{"id": "765.png", "formula": "\\begin{align*} B [ \\mathfrak { h } _ \\omega ] & = p [ \\mathfrak { c } _ \\omega ] \\in H C _ \\lambda ^ { p } ( \\mathfrak { A } ) \\\\ S [ \\mathfrak { c } _ \\omega ] & = 0 \\in H C _ \\lambda ^ { p + 2 } ( \\mathfrak { A } ) . \\end{align*}"}
{"id": "2623.png", "formula": "\\begin{align*} Z ( M _ \\eta T _ \\xi f ) ( x , \\omega ) = e ^ { 2 \\pi i \\eta \\cdot x } Z f ( x - \\xi , \\omega - \\eta ) , \\end{align*}"}
{"id": "4513.png", "formula": "\\begin{align*} \\varepsilon _ { } ( \\exp ( 4 0 ) ) & = 1 . 9 3 3 7 8 \\cdot 1 0 ^ { - 8 } < 2 ( 1 \\ , 0 0 0 ) ^ { 3 / 2 } \\exp ( - 0 . 8 4 7 6 8 3 6 \\sqrt { 1 \\ , 0 0 0 } ) , \\\\ \\ \\varepsilon _ { } ( \\exp ( 1 \\ , 0 0 0 ) ) & = 1 . 9 4 7 5 1 \\cdot 1 0 ^ { - 1 2 } < 2 ( 2 \\ , 1 0 0 ) ^ { 3 / 2 } \\exp ( - 0 . 8 4 7 6 8 3 6 \\sqrt { 2 \\ , 1 0 0 } ) . \\end{align*}"}
{"id": "5749.png", "formula": "\\begin{align*} ( r / t ) ^ \\perp _ p ( G ) = \\sup \\left \\{ \\left . \\| r ^ \\perp _ p ( g ) \\| / d ( p , g ( p ) ) \\ , \\right | \\ , g \\neq e \\in G , \\ ; \\| r _ p ^ \\perp ( g ) \\| \\right \\} , \\end{align*}"}
{"id": "205.png", "formula": "\\begin{align*} ( P ^ { \\nu _ \\alpha } _ t ) ^ * ( g ) = ( ( M _ \\alpha ) ^ { - 1 } \\circ T ^ { \\alpha } _ t \\circ M _ \\alpha ) ( g ) . \\end{align*}"}
{"id": "1437.png", "formula": "\\begin{align*} \\mathbb { B } = \\Q \\oplus \\Q \\zeta \\oplus \\Q \\xi \\oplus \\Q \\zeta \\xi , \\end{align*}"}
{"id": "1138.png", "formula": "\\begin{align*} J ^ E ( x , t , k ) = \\left \\{ \\begin{aligned} & \\Delta _ \\eta J ^ { ( 3 ) } ( x , t , k ) { \\Delta _ \\eta } ^ { - 1 } , & k \\in \\Gamma ^ { ( 3 ) } \\backslash \\left ( U _ { \\delta } ( \\eta ) \\cup U _ { \\delta } ( - \\eta ) \\right ) , \\\\ & m _ { \\pm \\eta } ( x , t , k ) \\Delta _ \\eta ^ { - 1 } , & k \\in \\partial U _ { \\delta } ( \\pm \\eta ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "8683.png", "formula": "\\begin{align*} H ( Y ^ n ) = \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ n \\ln \\big ( ( 2 \\pi e ) ^ { n _ y } \\det \\big ( { \\bf C } _ t \\Pi _ t { \\bf C } _ t ^ T + { \\bf D } _ t K _ { \\overline { W } _ t } { \\bf D } _ t ^ T \\big ) \\big ) \\end{align*}"}
{"id": "7391.png", "formula": "\\begin{align*} \\tilde { c } _ { x , y } ( \\eta ) : = \\eta ( x - 1 ) + \\eta ( x + 1 ) + \\eta ( y - 1 ) + \\eta ( y + 1 ) \\end{align*}"}
{"id": "1996.png", "formula": "\\begin{align*} g \\cdot ( ( x _ 1 , \\dots , x _ { 2 n } ) ( t ) ) = ( \\rho ( g ) x _ { \\sigma ( g ^ { - 1 } ) ( 1 ) } , \\dots , \\rho ( g ) x _ { \\sigma ( g ^ { - 1 } ) ( 2 n ) } ) ( \\tau ( g ^ { - 1 } ) ( t ) ) \\end{align*}"}
{"id": "269.png", "formula": "\\begin{align*} & I _ { 0 } ^ { ( 2 ) } \\left ( - \\frac { 3 1 } { 1 9 } \\right ) = \\frac { 1 9 } { 9 \\cdot { 3 1 ^ { 4 } } } \\left ( - 2 2 6 8 2 1 4 \\cdot { e ^ { - \\frac { 3 1 } { 1 9 } } } + 8 0 0 4 8 \\cdot { e ^ { \\frac { 3 1 } { 1 9 } } } \\right ) < 0 , \\\\ & I _ { 1 } ^ { ( 2 ) } \\left ( - \\frac { 3 1 } { 1 9 } \\right ) = \\frac { 6 2 } { 8 5 5 } > 0 . \\end{align*}"}
{"id": "2874.png", "formula": "\\begin{align*} | \\alpha _ - ( t ) | \\lesssim e ^ { e _ 0 t } \\sum _ { n = [ t ] } ^ \\infty \\Bigg | \\int _ { n } ^ { n + 1 } e ^ { - e _ 0 s } B ( R , \\mathcal { Y } _ + ) ( s ) d s \\Bigg | \\lesssim e ^ { e _ 0 t } \\sum _ { n = [ t ] } ^ \\infty e ^ { - ( e _ 0 + c _ 1 ) n } \\lesssim e ^ { - c _ 1 t } . \\end{align*}"}
{"id": "3264.png", "formula": "\\begin{align*} i s ( | \\lambda | , t ) + \\omega _ 2 ( i t ) = i t + \\frac { 1 } { - i h ( s ( | \\lambda | , t ) ) } . \\end{align*}"}
{"id": "6188.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\hat { V } ^ { T } \\hat { V } \\| _ { 2 } & \\leq 1 + \\xi , \\\\ \\| \\hat { V } \\| ^ 2 _ F & \\leq k + \\sqrt { k } \\xi , \\\\ \\| \\hat { V } \\| _ { 2 } & \\leq \\sqrt { 1 + \\xi } . \\end{aligned} \\end{align*}"}
{"id": "3801.png", "formula": "\\begin{align*} A _ k : = Z _ k D _ { A _ k } W _ k ^ T , B _ k : = Z _ k D _ { B _ k } U _ k ^ T , G _ k : = V _ k D _ { G _ k } W _ k ^ T , \\end{align*}"}
{"id": "761.png", "formula": "\\begin{align*} \\| \\delta _ \\alpha ( f ) \\| _ { C _ b ( \\Omega _ X ) } & = \\max \\left ( \\sup _ { x \\in X } \\sup _ { y \\in \\tilde { X } \\setminus \\{ x \\} } | \\delta _ \\alpha ( f ) ( x , y ) | , \\sup _ { y \\in X } | \\delta _ \\alpha ( f ) ( \\infty , y ) | \\right ) \\\\ & = \\max \\left ( \\sup _ { x , y \\in X , \\ , x \\neq y } | \\delta _ \\alpha ( f ) ( x , y ) | , \\sup _ { y \\in X } | f ( y ) | \\right ) = \\max ( | f | _ { C ^ \\alpha ( X ) } , \\| f \\| _ { C ( X ) } ) . \\end{align*}"}
{"id": "8943.png", "formula": "\\begin{align*} \\begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \\leq C \\left ( 1 + \\frac { 1 } { \\mathrm { d i s t } \\left ( x , \\partial \\Omega \\right ) } \\right ) | h | ^ 2 \\\\ & \\leq \\frac { C } { \\mathrm { d i s t } \\left ( x , \\partial \\Omega \\right ) } | h | ^ 2 , \\end{aligned} \\end{align*}"}
{"id": "9209.png", "formula": "\\begin{align*} \\forall \\gamma ^ 1 , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\rightarrow ( p = _ X J ^ A _ { \\gamma } x \\leftrightarrow \\gamma ^ { - 1 } ( x - _ X p ) \\in A p ) \\right ) \\end{align*}"}
{"id": "4938.png", "formula": "\\begin{align*} \\eta _ { n + 1 } = h \\cdot \\phi ( \\eta _ n ) \\cdot h ^ { - 1 } . \\end{align*}"}
{"id": "7772.png", "formula": "\\begin{align*} \\| \\phi \\| _ { \\mathcal { W } } : = \\| ( \\phi _ { x } , \\phi _ { t } ) \\| _ { L ^ 2 _ t \\times L ^ { \\infty } _ x \\cap L ^ { \\infty } _ t \\times L ^ { 2 } _ x ( [ 0 , T ] \\times \\S ) } . \\end{align*}"}
{"id": "1286.png", "formula": "\\begin{align*} \\{ \\ , \\ , ( t _ { 1 } , t _ { 2 } ) \\ , \\ , | \\ , \\ , A ( \\epsilon _ { ( t _ { 1 } , t _ { 2 } , n ) } ) < M \\ , \\ , \\} = \\{ \\ , \\ , ( t _ { 1 } , t _ { 2 } ) \\in \\mathbb { Z } _ { \\geq 0 } \\times \\mathbb { Z } _ { \\geq 0 } \\ , \\ , | \\ , \\ , t _ { 1 } R _ { 1 } + t _ { 2 } R _ { 2 } < M - T _ { n } \\ , \\ , \\} \\end{align*}"}
{"id": "2482.png", "formula": "\\begin{align*} \\rho ( \\l ) = M _ { \\omega / 2 } T _ x M _ { \\omega / 2 } = T _ { x / 2 } M _ \\omega T _ { x / 2 } = e ^ { - \\pi i \\omega \\cdot x } \\pi ( \\l ) . \\end{align*}"}
{"id": "2897.png", "formula": "\\begin{align*} \\int \\left ( | \\cdot | ^ { - ( N - \\gamma ) } * Q ^ p \\right ) Q ^ p d x = \\frac { 2 p } { \\gamma + 2 p - N ( p - 1 ) } \\| Q \\| _ 2 ^ 2 = \\frac { 2 p } { N ( p - 1 ) - \\gamma } \\| \\nabla Q \\| _ 2 ^ 2 . \\end{align*}"}
{"id": "7897.png", "formula": "\\begin{align*} \\gamma _ F ( P ) = ( 2 \\pi ) ^ { - \\frac { m k } { 2 } } ( \\det \\Lambda ) ^ { - \\frac { 1 } { 2 } } \\int _ B e ^ { - \\frac { 1 } { 2 } \\langle \\Lambda ^ { - 1 } t , t \\rangle } \\ d t , \\end{align*}"}
{"id": "5585.png", "formula": "\\begin{align*} \\bar { G } ( x ) = M _ Y ( x ) & = \\mathbb { E } [ e ^ { Y x } ] \\\\ & = \\mathbb { E } [ e ^ { - Z x } ] \\\\ & = \\int _ { 0 } ^ { \\infty } e ^ { - z x } f _ Z ( z ) d z \\\\ & = \\mathcal { L } [ f _ Z ] ( x ) \\end{align*}"}
{"id": "8578.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\# [ f ] ( k ) = f ^ \\# ( k ) = \\int \\overline { \\mathcal { K } ^ \\# ( x , k ) } f ( x ) \\ , d x . \\end{align*}"}
{"id": "9238.png", "formula": "\\begin{align*} \\alpha ^ 2 ( \\norm { x } _ X ^ 2 - \\norm { ( 1 - \\alpha ^ { - 1 } ) x + _ X \\alpha ^ { - 1 } y } _ X ^ 2 ) = \\alpha ( \\norm { x } _ X ^ 2 - \\alpha ^ { - 1 } ( 1 - \\alpha ) \\norm { x - _ X y } _ X ^ 2 - \\norm { y } _ X ^ 2 ) . \\end{align*}"}
{"id": "1853.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } L _ k ( x , y ) L _ { n - k } ( x , y ) = \\sum _ { k = 0 } ^ n { n \\choose k } M _ { k } ( x , y ) M _ { n - k } ( x , y ) . \\end{align*}"}
{"id": "5580.png", "formula": "\\begin{align*} \\| x - P _ A ( x ) \\| _ \\omega & \\ = \\ \\left \\| x - P _ A ( x ) - P _ B ( x ) + \\sum _ { n \\in B } e _ n ^ * ( x ) e _ n \\right \\| _ \\omega \\\\ & \\ \\leqslant \\ \\left \\| P _ { ( A \\cup B ) ^ c } ( x ) + t 1 _ { B } \\right \\| _ \\omega \\\\ & \\ \\leqslant \\ \\left \\| P _ { ( A \\cup B ) ^ c } ( x ) + t 1 _ F \\right \\| _ \\omega \\\\ & \\ \\leqslant \\ \\left \\| P _ { ( A \\cup B ) ^ c } ( x ) + P _ F ( x ) \\right \\| _ \\omega \\ = \\ \\left \\| x - S _ k ( x ) \\right \\| _ \\omega , \\end{align*}"}
{"id": "6508.png", "formula": "\\begin{align*} \\bar { g } _ n ^ { ( 2 m ) } & : = \\prod _ { j = j _ 0 + 1 } ^ { n - 1 } g ^ { ( 2 m ) } _ j = \\left ( \\frac { j _ 0 + 1 } { n } \\right ) ^ m \\prod ^ { n - 1 } _ { j = j _ 0 + 1 } \\left ( 1 + \\frac { 2 m \\alpha } { j } \\right ) , \\end{align*}"}
{"id": "3463.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { l = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in B _ l } \\omega ( Q ) D _ k ( x , x _ Q ) { \\widetilde D } _ k ( f ) ( x _ Q ) , \\end{align*}"}
{"id": "7911.png", "formula": "\\begin{align*} \\eta ^ T \\cdot k _ w ( t ) \\cdot \\eta = \\frac { t ^ 2 } { 2 } \\eta ^ T \\cdot ( k _ w '' ( 0 ) + \\sqrt { t } R ) \\cdot \\eta \\geq \\frac { t ^ 2 } { 2 } \\cdot k _ 1 . \\end{align*}"}
{"id": "7441.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ 0 ^ T \\int _ { \\mathbb { R } } \\rho _ 5 ( s , u ) [ - ( - \\Delta ) ^ { \\gamma / 2 } G _ k ] ( s , u ) d u d s = - \\frac { c _ { \\gamma } } { 4 } \\iint _ { \\mathbb { R } ^ 2 } \\frac { [ \\int _ 0 ^ T \\rho _ 5 ( r , u ) d r - \\int _ 0 ^ T \\rho _ 5 ( s , v ) d s ] ^ 2 } { | u - v | ^ { 1 + \\gamma } } d u d v . \\end{align*}"}
{"id": "9226.png", "formula": "\\begin{align*} \\forall x ^ X , p ^ X \\left ( p = _ X J ^ A _ { 1 } x \\rightarrow ( x - _ X p ) \\in A p \\right ) . \\end{align*}"}
{"id": "1829.png", "formula": "\\begin{align*} M _ n ( x ) = \\sum _ { k = 0 } ^ { \\lfloor ( n - 1 ) / 2 \\rfloor } M ( n , k ) x ^ { k } . \\end{align*}"}
{"id": "5018.png", "formula": "\\begin{align*} z _ k ^ { l + k } = \\Phi ^ { k } _ { Z } ( \\zeta _ k ^ { l + k } ) , \\xi _ k = \\Phi ^ { k } _ { Z } ( c _ k ) . \\end{align*}"}
{"id": "6058.png", "formula": "\\begin{align*} P ( x , y ) = y ^ 2 ( x ^ 2 + y ^ 2 ) ( y - 1 + x ^ 2 ) ^ 6 ( x ^ 2 + ( y + 1 ) ^ 2 ) ( x ^ 2 + ( y - 3 ) ^ 2 - 1 ) ^ 2 \\ , . \\end{align*}"}
{"id": "8512.png", "formula": "\\begin{align*} [ x + y , z ] _ + = x + [ y , z ] _ + - x + [ x , z ] _ + \\end{align*}"}
{"id": "3066.png", "formula": "\\begin{align*} b = c \\ , . \\end{align*}"}
{"id": "8066.png", "formula": "\\begin{align*} \\left \\{ T _ \\Sigma ( h ) , \\Psi _ \\Sigma ( f ) \\right \\} _ \\ell ^ \\Sigma = - \\Psi _ \\Sigma ( h f ' ) = \\frac { d } { d t } \\left ( \\mathfrak { P } _ { \\ell } \\rho ^ { ( t ) } \\Psi _ \\Sigma ( f ) \\right ) \\big | _ { t = 0 } . \\end{align*}"}
{"id": "7477.png", "formula": "\\begin{align*} \\frac { ( c - 1 ) \\phi ^ n } { \\tau ^ 2 } + \\eta ^ n \\frac { ( c - 1 ) \\phi ^ n } { 2 \\tau } = - G ( \\phi ^ n ) + \\mu ^ n \\phi ^ n . \\end{align*}"}
{"id": "4850.png", "formula": "\\begin{align*} \\int _ \\R Q _ \\alpha \\ , d \\rho & : = \\int _ \\R Q _ \\alpha ( x ) \\ , d \\rho ( x ) \\\\ \\int _ { - 1 } ^ 1 K _ \\alpha & : = \\int _ { - 1 } ^ 1 K _ \\alpha ( x ) \\ , d x \\\\ ( K _ \\alpha * \\rho ) ( x ) & : = \\int _ \\R K _ \\alpha ( x - y ) \\ , d \\rho ( y ) , \\end{align*}"}
{"id": "9294.png", "formula": "\\begin{align*} { i } & \\longmapsto \\begin{cases} L & i \\le \\alpha ( 0 ) , \\\\ j & \\exists j \\colon i \\in [ \\alpha ( j - 1 ) + 1 , \\alpha ( j ) ] , \\\\ R & i > \\alpha ( p ) . \\end{cases} \\end{align*}"}
{"id": "1354.png", "formula": "\\begin{align*} \\frac { \\mathcal { H } _ { d - 1 } ( W \\cap \\partial O ) } { | O | ^ \\frac { d - \\zeta } { d } } \\geq \\frac { c _ H } { | O | ^ \\frac { d - \\zeta } { d } } \\geq \\frac { c _ H } { R ^ { \\theta \\cdot \\frac { d - \\zeta } { d } } } = \\frac { c _ H } { R ^ { 1 - \\zeta } } . \\end{align*}"}
{"id": "5546.png", "formula": "\\begin{align*} \\eta : = \\frac { \\delta } { 2 \\sqrt { K _ 1 e ^ { K _ 2 T } } } \\wedge \\frac { \\delta } { 2 } , \\end{align*}"}
{"id": "1542.png", "formula": "\\begin{align*} c _ k ( s ) = \\alpha ( s ) \\pi ^ { \\frac { n ( n - 1 ) } { 2 } } \\frac { \\Gamma ( s + k - 2 n + 3 ) \\Gamma ( s + k - 2 n + 5 ) \\ldots \\Gamma ( s + k - 1 ) } { \\Gamma ( s + k - n + 2 ) \\Gamma ( s + k - n + 3 ) \\ldots \\Gamma ( s + k ) } , \\end{align*}"}
{"id": "4307.png", "formula": "\\begin{align*} \\Lambda _ y w = 2 w + y \\partial _ y w . \\end{align*}"}
{"id": "1265.png", "formula": "\\begin{align*} P _ { \\theta } ^ { \\mathrm { o u t } } ( M ) : = P _ { - \\theta } ^ { \\mathrm { i n } } ( M ) . \\end{align*}"}
{"id": "9549.png", "formula": "\\begin{align*} Z _ 2 ^ k w ^ k - \\mu ^ k Z _ 2 ^ { ( 0 ) } w ^ { ( 0 ) } + ( 1 - \\mu ^ k ) X ^ k w ^ k = 0 \\end{align*}"}
{"id": "7338.png", "formula": "\\begin{align*} \\begin{aligned} \\max \\left \\{ u ( y _ j , t ) , u ( z _ j , t ) \\right \\} & \\leq \\left ( { 1 \\over \\min \\{ \\lambda , 1 - \\lambda \\} } \\right ) ^ { 1 \\over q } ( u _ { q , \\lambda } ( x , t ) + 1 ) \\\\ & \\leq \\left ( { 1 \\over \\min \\{ \\lambda , 1 - \\lambda \\} } \\right ) ^ { 1 \\over q } ( u _ { \\star , \\lambda } ( x , t ) + 1 ) \\end{aligned} \\end{align*}"}
{"id": "260.png", "formula": "\\begin{align*} \\lim _ { t \\mapsto 0 } \\frac { D ^ G ( x \\cdot \\exp ( t X ) ) } { D ^ { G _ x } ( t X ) } = D ^ G ( x ) . \\end{align*}"}
{"id": "5523.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } d X _ r ( t ) & = & \\big ( A X _ r ( t ) + \\alpha ( X _ r ( t ) ) \\big ) d t + \\sum _ { j = 1 } ^ r \\sigma ^ j ( X _ r ( t ) ) d B ^ j ( t ) \\medskip \\\\ X _ r ( 0 ) & = & x , \\end{array} \\right . \\end{align*}"}
{"id": "1054.png", "formula": "\\begin{align*} & \\langle V ( T _ { \\alpha , N } ) , \\xi \\rangle = \\langle T _ { \\alpha , N } , V ^ * ( \\xi ) \\rangle = \\lim _ { \\varepsilon \\to 0 } \\int _ { \\vert f \\vert \\geq \\varepsilon } \\vert f \\vert ^ { 2 \\alpha } f ^ { - N } V ^ * ( \\xi ) \\\\ & = \\lim _ { \\varepsilon \\to 0 } \\int _ { \\vert f \\vert \\geq \\varepsilon } V ( \\vert f \\vert ^ { 2 \\alpha } f ^ { - N } ) \\xi \\end{align*}"}
{"id": "8830.png", "formula": "\\begin{align*} m ( \\pi ) + m ( \\overline { \\pi } ) = \\begin{cases} n + 1 , & d _ { m ( \\pi ) } = m ( \\pi ) - 1 \\\\ n , & \\end{cases} \\end{align*}"}
{"id": "1812.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } A _ n ( x ) { t ^ n \\over n ! } = { 1 - x \\over 1 - x e ^ { ( 1 - x ) t } } . \\end{align*}"}
{"id": "6231.png", "formula": "\\begin{align*} F \\left ( { \\bf w } , u \\right ) = I _ { 2 d } - u \\widehat { M } _ A ( { \\bf w } ) \\ \\ \\ a n d \\ \\ \\ \\widehat { M } _ A ( { \\bf w } ) = \\sum _ { j = 1 } ^ { d } \\Big ( e ^ { i w _ j } P _ { 2 j - 1 } A + e ^ { - i w _ j } P _ { 2 j } A \\Big ) , \\end{align*}"}
{"id": "2090.png", "formula": "\\begin{align*} \\widehat { \\psi _ { p ^ { 2 k } } } ( \\mathbf { u } ) = \\frac { 1 } { p ^ { 2 n k } } \\sum _ { \\mathbf { c } \\in ( \\mathbb { Z } / p ^ { 2 k } \\mathbb { Z } ) ^ n } \\psi _ { p ^ { 2 k } } ( f _ \\mathbf { c } ) \\exp \\left ( \\frac { 2 \\pi i c _ 1 u _ 1 } { p ^ { 2 k } } \\right ) . \\end{align*}"}
{"id": "3478.png", "formula": "\\begin{align*} & \\int _ 2 ^ T \\abs * { \\zeta _ { M T , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 \\\\ & = \\zeta _ { M T , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T + \\begin{cases} O ( T ^ { \\frac { 5 } { 2 } - \\sigma _ 1 - \\sigma _ 2 - \\sigma _ 3 } ) & ( \\frac { 3 } { 2 } < \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 < 2 ) \\\\ O ( ( T \\log T ) ^ \\frac { 1 } { 2 } ) & ( \\sigma _ 1 + \\sigma _ 2 + \\sigma _ 3 = 2 ) \\\\ \\end{cases} \\end{align*}"}
{"id": "3127.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = - x _ 1 \\ , , x _ 2 ^ \\prime = - x _ 2 \\ , , x _ i ^ \\prime = x _ i ( i = 3 , 4 , 5 ) \\end{align*}"}
{"id": "3236.png", "formula": "\\begin{align*} k ( s , t ) = \\bigg ( 1 - \\frac { t } { s } \\bigg ) s \\bigg ( \\frac { 1 } { h ( s ) } - s \\bigg ) + ( s - t ) t = k ( s , 0 ) \\bigg ( 1 - \\frac { t } { s } \\bigg ) + ( s - t ) t . \\end{align*}"}
{"id": "6959.png", "formula": "\\begin{align*} \\partial \\varphi ( \\bar x ; u ) : = \\left \\{ \\eta \\in \\mathbb X \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ u _ k \\} _ { k \\in \\N } \\subset \\mathbb X , \\ , \\exists \\{ t _ k \\} _ { k \\in \\N } \\subset \\R _ + , \\ , \\exists \\{ \\eta _ k \\} _ { k \\in \\N } \\subset \\mathbb X \\colon \\\\ & u _ k \\to u , \\ , t _ k \\searrow 0 , \\ , \\eta _ k \\to \\eta , \\ , \\eta _ k \\in \\widehat \\partial \\varphi ( \\bar x + t _ k u _ k ) \\ , \\forall k \\in \\N \\end{aligned} \\right \\} \\end{align*}"}
{"id": "647.png", "formula": "\\begin{align*} \\alpha _ n ( m ) \\ = \\ \\frac { ( - 1 ) ^ m } { ( m + 2 ) ( n + 1 ) ^ { m + 2 } } \\end{align*}"}
{"id": "3043.png", "formula": "\\begin{align*} x : y : z : w = x _ 2 y _ 1 : x _ 1 y _ 1 : x _ 1 y _ 2 : x _ 2 y _ 2 \\end{align*}"}
{"id": "5212.png", "formula": "\\begin{align*} | E _ 2 | = \\int _ K \\Phi _ { E _ 2 } ( x ) d x = \\int _ { E _ 2 } 1 ~ d x & ~ \\leq ~ \\dfrac { 4 } { { \\lambda } ^ 2 } \\| B f _ 2 ( x ) \\| ^ 2 _ { L ^ 2 ( K ) } \\\\ & ~ \\leq ~ \\dfrac { 4 } { { \\lambda } ^ 2 } \\| f _ 2 \\| ^ 2 _ { L ^ 2 ( K ) } \\\\ & ~ \\leq ~ \\dfrac { 4 } { { \\lambda } ^ 2 } \\lambda q \\| f \\| _ { L ^ 1 ( K ) } \\\\ & ~ = ~ \\dfrac { 4 q } { \\lambda } \\| f \\| _ { L ^ 1 ( K ) } . \\end{align*}"}
{"id": "1960.png", "formula": "\\begin{align*} T ( V ) : = \\bigoplus _ { n \\ge 0 } V ^ { \\otimes n } \\end{align*}"}
{"id": "4574.png", "formula": "\\begin{align*} \\frac { \\mathbf { P } ( U _ n > x ) } { 1 - \\Phi \\left ( x \\right ) } = 1 + o ( 1 ) . \\end{align*}"}
{"id": "6857.png", "formula": "\\begin{align*} \\forall V \\in X ( i ) : \\pi _ i ( V ) = \\frac { 1 } { { d \\choose i } _ q } \\sum \\limits _ { W \\in X ( d ) : W \\supset V } \\pi _ d ( W ) , \\end{align*}"}
{"id": "4740.png", "formula": "\\begin{align*} \\bigg | { \\rm I } - C _ 0 \\int _ { \\{ | y | < 2 | y - x | \\} } \\frac { 2 x F ( x ) } { y - x } \\ , \\dd x \\bigg | \\lesssim \\frac { 1 } { y ^ 2 } \\int _ { \\mathbb { R } } ( 1 + | x | ^ 2 ) F ( x ) \\ , \\dd x = O \\bigg ( \\frac { 1 } { y ^ 2 } \\bigg ) . \\end{align*}"}
{"id": "9419.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } } x ^ k \\ d \\mu _ { A _ N + U _ N B _ N U _ N ^ * } ( x ) \\\\ & = \\tau _ { A _ N , U _ N B _ N U _ N ^ * } ( ( X + Y ) ^ k ) = \\tau ( ( X + Y ) ^ k ) + o ( 1 ) = \\int _ { \\mathbb { R } } x ^ k \\ d \\mu _ 1 \\boxplus \\mu _ 2 ( x ) \\end{align*}"}
{"id": "1713.png", "formula": "\\begin{align*} d _ n ( \\nu _ 0 B _ { p _ 0 } ^ N \\cap \\nu _ 1 B _ { p _ 1 } ^ N , \\ , l _ q ^ N ) = d _ n ( \\nu _ 1 B _ { p _ 1 } ^ N , \\ , l _ q ^ N ) . \\end{align*}"}
{"id": "813.png", "formula": "\\begin{align*} \\nu ( B ( x , r ) ) \\approx \\ , \\frac { \\mu ( B ( x , r ) ) } { r ^ \\Theta } \\ , \\theta = \\frac { p - \\Theta } { p } , \\end{align*}"}
{"id": "9138.png", "formula": "\\begin{align*} \\norm { x _ N - x _ { N + 1 } } = \\mu _ N \\frac { \\norm { x _ N - x _ { N + 1 } } } { \\mu _ N } < C \\frac { 1 } { 2 C ( k + 1 ) } = \\frac { 1 } { 2 ( k + 1 ) } . \\end{align*}"}
{"id": "7298.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty k \\tau _ k Y ^ k & = Y \\tau ' = \\tau \\sum _ { k = 1 } ^ n \\frac { X _ k Y } { 1 - X _ k Y } = \\tau \\sum _ { k = 1 } ^ n \\sum _ { i = 1 } ^ \\infty ( X _ k Y ) ^ i \\\\ & = \\tau \\sum _ { i = 1 } ^ \\infty \\rho _ i Y ^ i = \\sum _ { k = 1 } ^ \\infty \\Bigl ( \\sum _ { i = 1 } ^ k \\rho _ i \\tau _ { k - i } \\Bigr ) Y ^ k . \\end{align*}"}
{"id": "4521.png", "formula": "\\begin{align*} ( c , d ) = \\begin{cases} 1 , & c = d , \\\\ 0 , & | c d | = 2 , \\\\ \\frac { \\eta } { 2 } , & | c d | = 3 . \\end{cases} \\end{align*}"}
{"id": "6839.png", "formula": "\\begin{align*} ( M ) \\subseteq \\bigcup _ { i = 1 } ^ k \\left [ \\lambda _ i - c _ i , \\lambda _ i + c _ i \\right ] . \\end{align*}"}
{"id": "3308.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F ) \\geq \\alpha _ { n - j } ( F _ 1 / T _ 1 ) + \\alpha _ { n - j } ( F _ 2 / T _ 2 ) - \\sum _ { l = 1 } ^ { j - 1 } \\frac { ( - 1 ) ^ { l - 1 } a _ 2 ^ l } { l ! } \\alpha _ { n - j + l } ( F _ 1 / T _ 1 ) . \\end{align*}"}
{"id": "7734.png", "formula": "\\begin{align*} \\left | \\int _ { \\R } \\int _ { \\S } ( | \\phi _ t | ^ 2 + | \\phi _ x | ^ 2 ) ( t , x ) \\psi ( t ) d x d t - E ( 0 ) \\right | = \\left | \\int _ { \\R } \\psi ( t ) \\big ( E ( t ) - E ( 0 ) \\big ) d t \\right | \\lesssim \\delta E ( 0 ) . \\end{align*}"}
{"id": "4722.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t W = W _ 1 + W _ 2 + W _ 3 + W _ 4 , \\end{align*}"}
{"id": "3261.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 ^ + } \\omega _ 2 ( i t ) = \\begin{cases} \\infty , & 0 < | \\lambda | \\leq \\lambda _ 1 ( \\mu ) ; \\\\ i \\frac { | \\lambda | ^ 2 } { s ( | \\lambda | , 0 ) } , & | \\lambda | \\in ( \\lambda _ 1 ( \\mu ) , \\lambda _ 2 ( \\mu ) ) ; \\\\ 0 , & , | \\lambda | \\geq \\lambda _ 2 ( \\mu ) . \\end{cases} \\end{align*}"}
{"id": "5326.png", "formula": "\\begin{align*} \\tau _ 1 : = \\frac { \\beta } { } \\bigg ( \\sum _ { i = 1 } ^ K \\mathbf { C } _ i ( \\psi ) - L \\int _ D \\psi \\ ; \\mathrm { d } q \\ , \\mathbb { I } \\bigg ) , \\end{align*}"}
{"id": "3049.png", "formula": "\\begin{align*} x _ 1 ^ \\prime = \\alpha _ 1 y _ 1 \\ , , x _ 2 ^ \\prime = \\alpha _ 2 y _ 2 \\ , , y _ 1 ^ \\prime = \\beta _ 1 x _ 1 \\ , , y _ 2 ^ \\prime = \\beta _ 2 x _ 2 \\ , . \\end{align*}"}
{"id": "4259.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d a _ { i j } ( x ) \\xi _ i \\xi _ j \\geq \\eta ( x ) | \\xi | ^ 2 x , \\xi \\in \\R ^ d , \\end{align*}"}
{"id": "6504.png", "formula": "\\begin{align*} s ^ { ( 2 m ) } _ n = \\left ( \\frac { n } { 1 - 2 \\alpha } \\right ) ^ m ( 2 m - 1 ) ! ! , \\end{align*}"}
{"id": "4989.png", "formula": "\\begin{align*} H ^ { - 1 } \\circ B \\circ A ^ { r _ 0 } \\circ H ( x , y ) & = ( a _ { x } ^ { r _ 0 } ( b _ { 0 } ( x ) + O ( \\varepsilon ) ) , x ) \\\\ & = ( a _ { 0 } ^ { r _ 0 } ( b _ { 0 } ( x ) ) + O ( \\varepsilon ) , x ) \\\\ & = ( \\eta ^ { r _ 0 } \\circ \\xi ( x ) + O ( \\varepsilon ) , x ) . \\end{align*}"}
{"id": "1257.png", "formula": "\\begin{align*} I ( \\alpha , \\beta , Z ) + I ( \\beta , \\gamma , Z ' ) = I ( \\alpha , \\gamma , Z + Z ' ) . \\end{align*}"}
{"id": "5015.png", "formula": "\\begin{align*} A _ { k n } = G _ { k n } \\circ B _ { k n } , \\end{align*}"}
{"id": "318.png", "formula": "\\begin{align*} & \\theta ^ 1 _ c ( x ) : = 1 , \\ , x \\in [ 0 , c ] , \\ ; \\theta ^ 1 _ c ( x ) = 0 , x \\in [ 2 c , \\infty ) , \\\\ & \\theta ^ 2 _ c ( x ) : = 1 , \\ ; x \\in [ 1 / c , 1 ] , \\ ; \\theta ^ 2 _ c ( x ) = 0 , x \\in [ 0 , 1 / { 2 c } ] . \\end{align*}"}
{"id": "5533.png", "formula": "\\begin{align*} \\| a ( h ) - a ( g ) \\| & \\leq L \\| h - g \\| , \\\\ \\| \\sigma ^ j ( h ) - \\sigma ^ j ( g ) \\| & \\leq L \\| h - g \\| , j = 1 , \\ldots , r , \\end{align*}"}
{"id": "472.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { L } ( \\mathbf { t } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { t } ) } ( \\mathcal { I } ) = \\Delta _ { \\mathbf { L } ( \\mathbf { 1 } ) } ( \\mathcal { I } ) \\cdot \\Delta _ { \\mathbf { R } ( \\mathbf { 1 } ) } ( \\mathcal { I } ) \\cdot \\mathbf { t } ^ { \\mathbf { \\Psi } ( \\mathcal { I } ) } , \\mathcal { I } \\in \\mathfrak { G } ( \\mathbf { L } ( \\mathbf { 1 } ) ) . \\end{align*}"}
{"id": "323.png", "formula": "\\begin{align*} d \\rho _ i = \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ i - S _ j ) \\theta _ { i j } ( \\rho ) d t ; & \\\\ d S _ i + ( \\sum _ { j \\in N ( i ) } \\frac { \\omega _ { i j } } { 2 } ( S _ i - S _ j ) ^ 2 \\frac { \\partial \\theta _ { i j } } { \\partial \\rho _ i } + \\frac 1 8 \\frac { \\partial } { \\partial \\rho _ i } I ( \\rho ) + \\mathbb V _ i & + \\sum _ { j \\in N ( i ) } \\mathbb W _ { i j } \\rho _ j - \\log ( \\rho _ i ) ) d t + \\sigma _ i d W _ t = 0 . \\end{align*}"}
{"id": "1149.png", "formula": "\\begin{align*} T _ { \\eta , 1 } = \\tilde { P } f ^ { - \\frac { \\sigma _ 3 } { 4 } } \\Psi _ 0 ^ { A i } \\frac { \\mathcal { K } _ 1 } { f ^ { 3 / 2 } } ( \\Psi _ 0 ^ { A i } ) ^ { - 1 } f ^ { \\frac { \\sigma _ 3 } { 4 } } { \\tilde { P } } ^ { - 1 } , f = f _ { \\eta } = - \\left ( \\frac { 3 } { 2 } g \\right ) ^ { \\frac { 2 } { 3 } } . \\end{align*}"}
{"id": "4355.png", "formula": "\\begin{align*} \\Lambda Q = 2 Q + y \\partial _ y Q = - \\frac { \\epsilon ' _ x ( x ) } { e ^ { 2 x } } < 0 \\forall x \\in ( - \\infty , \\infty ) . \\end{align*}"}
{"id": "2329.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 d } } W f ( x , \\omega ) \\ , d ( x , \\omega ) = 1 . \\end{align*}"}
{"id": "8050.png", "formula": "\\begin{align*} \\Psi _ { ( \\Sigma , U ) } ( \\eta _ { ( \\Sigma , U ) } h ) [ \\partial _ { \\Sigma , \\epsilon } \\phi ] = \\Psi _ { ( \\widetilde { \\Sigma } , U ) } ( \\eta _ { ( \\widetilde { \\Sigma } , U ) } h ) [ \\partial _ { \\widetilde { \\Sigma } , \\widetilde { \\epsilon } } \\phi ] . \\end{align*}"}
{"id": "3999.png", "formula": "\\begin{align*} ( \\sigma , v ) = \\sum _ { k \\geq K _ 0 } a _ k e ^ { \\lambda ^ p _ k ( T - t ) } ( \\xi _ { \\lambda ^ p _ k } , \\eta _ { \\lambda ^ p _ k } ) + \\sum _ { | k | \\geq K _ 0 } b _ k e ^ { \\lambda ^ h _ k ( T - t ) } ( \\xi _ { \\lambda ^ h _ k } , \\eta _ { \\lambda ^ h _ k } ) , \\end{align*}"}
{"id": "437.png", "formula": "\\begin{align*} S ( U ) = \\left ( \\begin{array} { c c c c } \\frac { p _ { \\rho } } { \\rho } & & & \\\\ & \\mathbb { I } _ { 3 \\times 3 } & & \\\\ & & \\frac { 1 } { \\theta } & \\\\ & & & \\frac { 1 } { \\kappa \\theta } \\mathbb { I } _ { 3 \\times 3 } \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "2595.png", "formula": "\\begin{align*} f = \\frac { 1 } { \\langle g , \\widetilde { g } \\rangle } \\iint _ { \\R ^ { 2 d } } V _ { \\widetilde { g } } f ( x , \\omega ) M _ \\omega T _ x g \\ , d ( x , \\omega ) \\end{align*}"}
{"id": "2676.png", "formula": "\\begin{align*} \\mathfrak { F } _ { ( \\alpha , \\beta ) } ( \\cosh ( \\pi t ) ^ { - 1 } ) = \\{ \\alpha \\Z \\times \\beta \\Z \\mid \\alpha \\beta < 1 \\} \\end{align*}"}
{"id": "2403.png", "formula": "\\begin{align*} D _ n : \\ell ^ 2 ( \\Gamma ) \\to \\mathcal { H } , D _ n ( c _ \\gamma ) _ { \\gamma \\in \\Gamma } = \\sum _ { | \\gamma | \\leq n } c _ \\gamma e _ \\gamma . \\end{align*}"}
{"id": "2224.png", "formula": "\\begin{align*} \\| v ^ { n } ( t ) - v ^ { n } ( s ) \\| _ { L ^ p ( \\Omega ; H ^ \\beta ) } \\leq & \\| ( E ( t - s ) - I ) v ^ { n } ( s ) \\| _ { L ^ p ( \\Omega ; H ^ \\beta ) } + \\Big \\| \\int _ s ^ t E ( t - r ) P _ n A P F ( X ^ { n } ( r ) ) \\ , \\dd r \\Big \\| _ { L ^ p ( \\Omega ; H ^ \\beta ) } \\\\ \\leq & C ( t - s ) ^ { \\frac { \\gamma - \\beta } 4 } \\| v ^ { n } ( s ) \\| _ { L ^ p ( \\Omega ; H ^ \\gamma ) } + C ( t - s ) ^ { \\frac { \\gamma - \\beta } 4 } . \\\\ \\leq & C ( T , \\gamma , X _ 0 , p ) ( t - s ) ^ { \\frac { \\gamma - \\beta } 4 } . \\end{align*}"}
{"id": "2524.png", "formula": "\\begin{align*} 0 & = \\langle \\pi ( F ) \\pi ( \\xi , \\eta ) f , \\pi ( \\xi , \\eta ) g \\rangle \\\\ & = \\iint _ { \\R ^ { 2 d } } F ( x , \\omega ) \\langle \\pi ( \\xi , \\eta ) ^ { - 1 } \\pi ( x , \\omega ) \\pi ( \\xi , \\eta ) f , g \\rangle \\ , d ( x , \\omega ) . \\end{align*}"}
{"id": "3375.png", "formula": "\\begin{align*} \\quad \\ ; [ u , v , w ] _ { T _ t } & = \\displaystyle \\sum _ { k = 0 } ^ { + \\infty } \\sum _ { i + j = k } \\Big ( D ( T _ i ( u ) , T _ j ( v ) ) w + \\theta ( T _ i ( v ) , T _ j ( w ) ) u - \\theta ( T _ i ( u ) , T _ j ( w ) ) v \\Big ) t ^ { k } , \\end{align*}"}
{"id": "1038.png", "formula": "\\begin{align*} \\psi _ s ( y ) & : = n ( n + 2 ) \\gamma _ { n , s } \\int _ 0 ^ { \\min \\{ 1 / \\vert y \\vert , 1 \\} } \\frac { r ^ { 2 s + n + 1 } } { ( 1 - r ^ 2 ) ^ s } \\dd r . \\end{align*}"}
{"id": "9467.png", "formula": "\\begin{align*} L _ { 2 j } + \\tfrac { C ( j , 3 ) ^ p } { C ( j ) ^ p } - \\tfrac { C ( j - 1 , 3 ) } { C ( j - 1 ) } = 0 . \\end{align*}"}
{"id": "3200.png", "formula": "\\begin{align*} \\sigma ( y ) & = \\sup \\{ 0 < \\varepsilon ' < c \\ ; | \\ ; \\exists \\delta > 0 \\ \\ ( \\mathcal B _ d ( x , \\delta ) \\times \\mathcal B _ { n - d } ( y , \\varepsilon ' ) ) \\cap X \\\\ & \\} . \\end{align*}"}
{"id": "7601.png", "formula": "\\begin{align*} H ^ { - \\alpha } = \\langle X , \\nu \\rangle , \\end{align*}"}
{"id": "6325.png", "formula": "\\begin{align*} i \\neq j \\Rightarrow \\langle ( \\mathcal H ^ A _ \\emph { C K } ) _ { i } , ( \\mathcal H ^ A _ \\emph { G L } ) ^ { j } \\rangle = 0 , \\langle x , ( \\mathcal H ^ A _ \\emph { G L } ) ^ { | x | } \\rangle = 0 \\Rightarrow x = 0 , \\langle ( \\mathcal H ^ A _ \\emph { C K } ) _ { | y | } , y \\rangle = 0 \\Rightarrow y = 0 \\end{align*}"}
{"id": "6659.png", "formula": "\\begin{align*} D _ { 1 , m , n } = \\tau _ { A \\smallsetminus \\{ \\alpha \\} \\cup \\{ - \\beta \\} } ( p ^ m ) \\tau _ { B \\smallsetminus \\{ \\beta \\} } ( p ^ n ) . \\end{align*}"}
{"id": "239.png", "formula": "\\begin{align*} \\langle \\xi ; \\nabla ( \\varphi _ \\mu ) ( \\xi ) \\rangle = - \\langle \\xi ; \\Sigma ( \\xi ) \\rangle \\varphi _ \\mu ( \\xi ) . \\end{align*}"}
{"id": "7443.png", "formula": "\\begin{align*} & \\int _ 0 ^ T \\int _ { \\mathbb { R } } [ \\rho _ { 3 } ( s , u ) ] ^ 2 \\rho _ { 4 } ( s , u ) d u d s + \\frac { c _ { \\gamma } } { 4 } \\iint _ { \\mathbb { R } ^ 2 } \\frac { [ \\int _ 0 ^ T \\rho _ { 5 } ( r , u ) d r - \\int _ 0 ^ T \\rho _ { 5 } ( s , v ) d s ] ^ 2 } { | u - v | ^ { 1 + \\gamma } } d u d v = 0 . \\end{align*}"}
{"id": "7372.png", "formula": "\\begin{align*} \\inf _ { \\gamma ( x ) \\ge R } u _ 0 ( x ) \\ge \\sigma ( R ) \\ge c _ 0 \\ ; \\ ; R \\in [ 0 , \\infty ) . \\end{align*}"}
{"id": "8022.png", "formula": "\\begin{align*} E ( u , v , u ' , v ' ) = - \\frac { 1 } { 4 } \\left ( \\mathrm { s g n } ( u - u ' ) + \\mathrm { s g n } ( v - v ' ) \\right ) . \\end{align*}"}
{"id": "8072.png", "formula": "\\begin{align*} \\beta _ { H ' - H } F = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\hbar ^ n } { 2 ^ n n ! } \\left \\langle ( H ' _ { \\Sigma } - H _ { \\Sigma } ) ^ { \\otimes n } , F ^ { ( 2 n ) } \\right \\rangle \\end{align*}"}
{"id": "5602.png", "formula": "\\begin{align*} [ X , Y ] _ \\star : = \\mathcal { L } ^ \\mathcal { F } _ X Y = [ \\overline { \\mathcal { F } } _ 1 \\rhd X , \\overline { \\mathcal { F } } _ 2 \\rhd Y ] = X \\star Y - ( \\overline { \\mathcal { R } } _ 1 \\rhd Y ) \\star ( \\overline { \\mathcal { R } } _ 2 \\rhd X ) , \\end{align*}"}
{"id": "6256.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Phi ^ H ( q , t ) - 1 = \\Phi ^ T ( q , t ) , \\end{align*}"}
{"id": "1097.png", "formula": "\\begin{align*} J ^ { G P } = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , k \\in ( - C _ L , C _ L ) \\end{align*}"}
{"id": "8326.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } z \\dd ( y - u ) = \\int _ { 0 } ^ { t } z \\dd ( y - u ) + \\int _ { t } ^ { t + \\varepsilon } z \\dd ( y - u ) + \\int _ { t + \\varepsilon } ^ { T } z \\dd ( y - u ) > 0 . \\end{align*}"}
{"id": "1496.png", "formula": "\\begin{align*} ( \\mathbf { f } | [ K _ 1 ( \\mathfrak { n } ) \\xi K _ 1 ( \\mathfrak { n } ) ] ) ( g ) = \\sum _ { y \\in Y } \\mathbf { f } ( g y ^ { - 1 } ) , \\end{align*}"}
{"id": "1760.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
{"id": "6771.png", "formula": "\\begin{align*} \\begin{aligned} f _ t - \\frac { 1 } { 2 \\sqrt { t } } f '' + \\frac { 1 } { 2 \\sqrt { t } } & \\geq 0 , \\\\ f - f ^ * & \\geq 0 , \\\\ \\left ( f _ t - \\frac { 1 } { 2 \\sqrt { t } } f '' + \\frac { 1 } { 2 \\sqrt { t } } = 0 \\right ) \\vee ( f - f ^ * & = 0 ) , \\end{aligned} \\end{align*}"}
{"id": "8460.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\| x _ \\ast - \\frac { 1 } { M } \\sum _ { k = 1 } ^ { M } x _ { k } \\right \\| \\leq \\frac { 1 + \\| A \\| _ { F } \\left \\| A ^ { - 1 } \\right \\| } { \\sqrt { M } } \\left \\| x _ \\ast - x _ { 0 } \\right \\| . \\end{align*}"}
{"id": "3798.png", "formula": "\\begin{align*} \\begin{aligned} U ^ T \\ ! A X & = \\Gamma = ( \\gamma _ 1 , \\dots , \\gamma _ n ) , & \\gamma _ i \\in [ 0 , 1 ] , \\\\ V ^ T \\ ! G X & = \\Sigma = ( \\sigma _ 1 , \\dots , \\sigma _ n ) , & \\sigma _ i \\in [ 0 , 1 ] , \\end{aligned} \\end{align*}"}
{"id": "5105.png", "formula": "\\begin{align*} h _ B ( x ) = h _ B ( y ) \\Longleftrightarrow x = y . \\end{align*}"}
{"id": "5144.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\eta ^ 2 } \\left [ \\log \\left ( \\tfrac { 1 } { n } A ( \\tfrac { \\eta } { n } , \\tfrac { 1 } { \\eta } ) \\right ) \\right ] = \\frac { \\tanh '' ( \\tfrac { \\eta } { 2 } ) \\tanh ( \\tfrac { \\eta } { 2 } ) - \\tanh ' ( \\tfrac { \\eta } { 2 } ) ^ 2 } { 4 \\tanh ( \\tfrac { \\eta } { 2 } ) ^ 2 } + \\frac { \\psi '' ( \\eta ) \\psi ( \\eta ) - \\psi ' ( \\eta ) ^ 2 } { \\psi ( \\eta ) ^ 2 } . \\end{align*}"}
{"id": "1077.png", "formula": "\\begin{align*} & \\Phi _ { R , L } ( x , t , k ) \\sim \\Phi ^ { \\pm \\infty } ( x , t , k ) , & x \\rightarrow \\pm \\infty , \\\\ & \\Phi _ { R , L } ( x , k ) e ^ { i X _ { R , L } ( k ) x \\sigma _ 3 } = I + \\mathcal { O } ( k ^ { - 1 } ) , & k \\rightarrow \\infty . \\end{align*}"}
{"id": "4163.png", "formula": "\\begin{align*} n ^ { 2 - p } \\norm { T } _ { p } ^ { p } \\leq \\overset { n } { \\underset { i , j = 1 } { \\sum } } \\norm { T _ { i j } } _ { p } ^ { p } \\leq \\norm { T } _ { p } ^ { p } , \\end{align*}"}
{"id": "4133.png", "formula": "\\begin{align*} I _ 1 ( u ) = \\int _ { \\R } u ( x , t ) d x , \\end{align*}"}
{"id": "9182.png", "formula": "\\begin{align*} \\exp \\Big ( 2 \\sum ^ j _ { i = 0 } { \\mathcal M } _ { i , j } ( p ) \\Big ) \\ll \\prod ^ j _ { i = 1 } \\Big | E _ { e ^ 2 \\alpha ^ { - 3 / 4 } _ i } ( { \\mathcal M } _ { i , j } ( p ) ) \\Big | ^ 2 , \\end{align*}"}
{"id": "4380.png", "formula": "\\begin{align*} \\langle B ( \\epsilon ) , \\phi _ 0 \\rangle _ { L ^ 2 _ \\rho } \\lesssim ( \\sum _ { j = 0 } ^ M \\varepsilon _ j ^ 2 + \\epsilon _ - ^ 2 ) . \\end{align*}"}
{"id": "5413.png", "formula": "\\begin{align*} \\operatorname { h r m } ( P ) = \\sum _ { s \\in S _ p } \\frac { l ( s ) ^ 2 } { A ( P ) } \\end{align*}"}
{"id": "2757.png", "formula": "\\begin{align*} \\frac { 2 } { q } + \\frac { N } { r } = \\frac { N } { 2 } - s , 2 \\le q , r \\le \\infty , ( q , r , N ) \\not = ( 2 , \\infty , 2 ) . \\end{align*}"}
{"id": "1122.png", "formula": "\\begin{align*} m ^ { ( 2 ) } _ { + } ( x , t , k ) = m ^ { ( 2 ) } _ { - } ( x , t , k ) J ^ { ( 2 ) } ( x , t , k ) , k \\in \\mathbb { R } , \\end{align*}"}
{"id": "7988.png", "formula": "\\begin{align*} C _ { n } ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\coloneqq \\bigg ( \\ ! 1 - \\ ! \\sum _ { m = n + 1 } ^ { d } C _ m ^ { ( d ) , \\sigma } ( \\mathbf { P } _ { \\ ! \\sigma } ) \\cdot \\chi _ n ^ { \\sigma } ( \\mathbf { p } _ { \\sigma _ m } ) \\ ! \\bigg ) \\frac { 1 } { \\chi _ n ^ { \\sigma } ( \\mathbf { p } _ { \\sigma _ n } ) } . \\end{align*}"}
{"id": "9386.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = 0 } ^ { \\infty } x ^ { k } ( k ) _ { n , \\lambda } \\bigg ) \\frac { t ^ { n } } { n ! } = \\sum _ { k = 0 } ^ { \\infty } x ^ { k } e _ { \\lambda } ^ { k } ( t ) = \\frac { 1 } { 1 - x e _ { \\lambda } ( t ) } \\\\ & = \\frac { 1 } { 1 - x - x ( e _ { \\lambda } ( t ) - 1 ) } = \\frac { 1 } { 1 - x } \\frac { 1 } { 1 - \\frac { x } { 1 - x } ( e _ { \\lambda } ( t ) - 1 ) } = \\frac { 1 } { 1 - x } \\sum _ { n = 0 } ^ { \\infty } F _ { n , \\lambda } \\bigg ( \\frac { x } { 1 - x } \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "7232.png", "formula": "\\begin{align*} \\int \\nabla _ x f _ 0 ( x - t v ) g ( v ) \\dd v & = \\frac { 1 } { t } \\int f _ 0 ( x - t v ) \\nabla g ( v ) \\dd v . \\end{align*}"}
{"id": "8856.png", "formula": "\\begin{align*} d _ 0 \\tilde \\varphi ( x , x + n ) & = \\sum _ { i = 0 } ^ { x + n - 1 } \\varphi ( i , i + 1 ) - \\sum _ { i = 0 } ^ { x - 1 } \\varphi ( i , i + 1 ) \\\\ & = \\sum _ { i = x } ^ { x + n - 1 } \\varphi ( i , i + 1 ) \\\\ & = \\begin{cases} \\varphi ( x , x + n ) - \\varphi _ n ( x ) & x = x _ 1 , \\ldots , x _ { r _ n } \\\\ \\varphi ( x , x + n ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "2785.png", "formula": "\\begin{align*} \\sigma _ { } ( \\mathcal { P } ) = [ 0 , + \\infty ) . \\end{align*}"}
{"id": "9448.png", "formula": "\\begin{align*} X \\cdot e _ j = X , Y \\cdot e _ j = x _ { 1 j } X + Y \\quad \\mbox { a n d } Z \\cdot e _ j = ( x _ { 1 j } ^ 2 + x _ { 2 j } ) X + 2 x _ { 1 j } Y + Z . \\end{align*}"}
{"id": "4288.png", "formula": "\\begin{align*} u ( r , t ) = \\lambda _ { \\ell } ^ { - 1 } ( t ) Q \\left ( \\frac { r } { \\sqrt { \\lambda _ { \\ell } ( t ) } } \\right ) + \\tilde u ( r , t ) , \\end{align*}"}
{"id": "1963.png", "formula": "\\begin{align*} f \\bullet g ( x ) & = g ( x ) f ( x g ( x ) ) = \\sum _ { u , v \\in \\{ \\ 1 \\} \\cup \\N ^ * } g _ u f _ v x _ { u } ( x g ( x ) ) _ { v } \\\\ & = g ( x ) + \\sum _ { \\substack { u , u _ 1 , \\dotsc , u _ k \\in \\{ \\ 1 \\} \\cup \\N ^ * \\\\ v = i _ 1 \\dotsm i _ k \\in \\N ^ * } } f _ v g _ u g _ { u _ 1 } \\dotsm g _ { u _ k } x _ { u } x _ { i _ 1 } x _ { u _ 1 } \\dotsm x _ { i _ k } x _ { u _ k } . \\end{align*}"}
{"id": "3454.png", "formula": "\\begin{align*} & \\bigg | D _ k T _ M ( f ) ( x ) \\bigg | = \\bigg | D _ k \\bigg ( - \\ln r \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) \\psi _ { Q } ( \\cdot , x _ { Q } ) q _ { Q } f ( x _ { Q } ) \\bigg ) ( x ) \\bigg | \\\\ & \\lesssim \\sum \\limits _ { j = - \\infty } ^ \\infty \\sum \\limits _ { Q \\in Q ^ j } w ( Q ) | D _ k \\psi _ { Q } ( x , x _ { Q } ) | | q _ { Q } f ( x _ { Q } ) | . \\end{align*}"}
{"id": "5310.png", "formula": "\\begin{align*} \\omega g = g _ { ( 2 ) } \\omega ( g _ { ( 3 ) } - S ^ { - 1 } ( g _ { ( 1 ) } ) ) . \\end{align*}"}
{"id": "9511.png", "formula": "\\begin{align*} \\Gamma ( f ) \\Gamma ( g ) & = [ \\gamma ( f ) ] \\ [ \\gamma ( g ) ] \\\\ \\\\ & = [ f _ - ( - M _ x ) + f _ + ( H ) - f ( 0 ) ] \\ [ g _ - ( - M _ x ) + g _ + ( H ) - g ( 0 ) ] \\\\ \\\\ & = \\Big ( [ f _ - ( - M _ x ) ] [ g _ - ( - M _ x ) ] + [ f _ + ( H ) ] [ g _ + ( H ) ] \\Big ) \\\\ \\\\ & \\ \\ + \\Big ( [ f _ - ( - M _ x ) ] [ g _ + ( H ) ] + [ g _ - ( - M _ x ) ] [ f _ + ( H ) ] \\Big ) \\\\ \\\\ & \\ \\ - \\Big ( f ( 0 ) [ g _ - ( - M _ x ) + g _ + ( H ) ] + g ( 0 ) [ f _ - ( - M _ x ) + f _ + ( H ) ] \\Big ) \\\\ \\\\ & \\ \\ + f ( 0 ) g ( 0 ) \\\\ \\\\ & = \\ \\ A + B - C + f ( 0 ) g ( 0 ) . \\end{align*}"}
{"id": "8379.png", "formula": "\\begin{align*} 2 ( \\xi _ i \\eta _ j - \\xi _ j \\eta _ i ) = ( \\xi _ i - \\eta _ i ) ( \\xi _ j + \\eta _ j ) - ( \\xi _ j - \\eta _ j ) ( \\xi _ i + \\eta _ i ) \\end{align*}"}
{"id": "1450.png", "formula": "\\begin{align*} \\mathcal { H } = \\{ z \\in \\C _ n ^ n : U ( z ) \\in \\Omega \\} , \\ , \\ , \\ , \\ , U ( z ) : = \\left [ \\begin{array} { c } z \\\\ u _ 0 \\end{array} \\right ] , \\end{align*}"}
{"id": "2054.png", "formula": "\\begin{align*} \\cos { \\rho } _ n & = \\cos \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi \\cos \\xi _ n - \\sin \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi \\sin \\xi _ n = - \\left ( - 1 \\right ) ^ n \\sin \\xi _ n , \\\\ [ 0 . 2 e m ] \\sin { \\rho } _ n & = \\sin \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi \\cos \\xi _ n + \\cos \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi \\sin \\xi _ n = \\left ( - 1 \\right ) ^ n \\cos \\xi _ n . \\end{align*}"}
{"id": "4056.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } u _ x \\bar { \\sigma } - \\int _ { 0 } ^ { 1 } \\rho \\bar { \\sigma } _ x + \\rho \\bar { \\sigma } | _ { 0 } ^ { 1 } + \\lambda \\int _ { 0 } ^ { 1 } \\rho \\bar { \\sigma } = \\int _ { 0 } ^ { 1 } f \\bar { \\sigma } . \\end{align*}"}
{"id": "7567.png", "formula": "\\begin{align*} ( u \\rhd _ s v ) \\rhd _ s w = ( u \\rhd _ s w ) \\rhd _ s ( v \\rhd _ s w ) . \\end{align*}"}
{"id": "3304.png", "formula": "\\begin{align*} \\alpha _ { n - j } ( F _ i ) = \\alpha _ { n - j } ( T _ i ) + \\alpha _ { n - j } ( F _ i / T _ i ) . \\end{align*}"}
{"id": "1561.png", "formula": "\\begin{align*} \\mathfrak { A } _ { \\omega } = \\bigcup _ e \\{ g ^ { - 1 } f : f \\in M _ { \\tau _ e } , 0 \\neq g \\in M _ e \\} , \\end{align*}"}
{"id": "6240.png", "formula": "\\begin{align*} A _ n ( q ) = \\begin{cases} { \\displaystyle \\sum _ { k = 1 } ^ { n } A ( n , k ) q ^ k , } & n \\geq 1 , \\\\ 1 , & n = 0 . \\end{cases} \\end{align*}"}
{"id": "3473.png", "formula": "\\begin{align*} \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) = \\sum _ { m = 1 } ^ \\infty \\sum _ { n < m } \\frac { 1 } { m ^ { s _ 1 } n ^ { s _ 2 } ( m + n ) ^ { s _ 3 } } \\end{align*}"}
{"id": "1984.png", "formula": "\\begin{align*} \\dot \\Phi _ t = \\rho \\ast \\Phi _ t = \\Phi _ t * \\rho . \\end{align*}"}
{"id": "5240.png", "formula": "\\begin{align*} a ( m b ) = ( a m ) b , \\forall a , b \\in A . \\end{align*}"}
{"id": "2241.png", "formula": "\\begin{align*} \\tilde { E } _ { k , N } ( t ) = E _ { k , N } ^ { i } , t \\in ( t _ { i - 1 } , t _ { i } ] . \\end{align*}"}
{"id": "1972.png", "formula": "\\begin{align*} \\Lambda ( \\phi \\succ \\gamma ) = \\sum _ { \\substack { u _ 1 , \\dotsc , u _ k \\in \\{ \\ 1 \\} \\cup \\N ^ * \\\\ { \\substack { u \\in \\N ^ * \\\\ v = i _ 1 \\dotsm i _ k \\in \\N ^ * } } } } f _ v g _ u g _ { u _ 1 } \\dotsm g _ { u _ k } x _ { u } x _ { i _ 1 } x _ { u _ 1 } \\dotsm x _ { i _ k } x _ { u _ k } = ( g ( x ) - 1 ) f ( x g ( x ) ) . \\end{align*}"}
{"id": "8693.png", "formula": "\\begin{align*} \\begin{aligned} \\phi ( f _ 1 , f _ 2 ) & = \\phi ( 1 , 1 ) + \\bigl ( \\phi ( f _ 1 , 1 ) - \\phi ( 1 , 1 ) \\bigr ) + \\bigl ( \\phi ( f _ 1 , f _ 2 ) - \\phi ( f _ 1 , 1 ) \\bigr ) \\\\ & \\leq \\phi ( f _ 1 , 1 ) + \\bigl ( \\phi ( 2 , f _ 2 ) - \\phi ( 2 , 1 ) \\bigr ) . \\end{aligned} \\end{align*}"}
{"id": "1823.png", "formula": "\\begin{align*} D _ n ( 2 u , v ) = 2 ^ n E _ n ( u , v ) . \\end{align*}"}
{"id": "8972.png", "formula": "\\begin{align*} u ( x ) = - \\int _ x ^ b ( f ( t ) - u ( t ) ) ^ \\frac { 1 } { p } d t < 0 \\end{align*}"}
{"id": "2637.png", "formula": "\\begin{align*} \\norm { g } _ 2 ^ 2 = \\iint _ { \\mathcal { Q } \\times \\mathcal { Q } } | Z g ( x , \\omega ) | ^ 2 \\ , d ( x , \\omega ) = 1 . \\end{align*}"}
{"id": "6697.png", "formula": "\\begin{align*} \\mathbb { D } _ i ^ { ( - 1 ) } = \\begin{cases} ( \\theta - t ) ^ { q ^ { i - 1 } } \\mathbb { D } _ { i - 1 } \\ & , \\\\ 1 \\ & . \\end{cases} \\end{align*}"}
{"id": "9423.png", "formula": "\\begin{align*} d _ { q + 1 } \\circ d _ q = 0 , d ^ * _ q d _ q + d _ { q - 1 } d ^ * _ { q - 1 } = - E _ { m ( q ) } \\varDelta , \\ , 0 \\leq q \\leq n , \\end{align*}"}
{"id": "2614.png", "formula": "\\begin{align*} V _ { \\widetilde { g } } f = \\frac { 1 } { \\langle \\widetilde { g } , g \\rangle } V _ { \\widetilde { g } } V ^ * _ { \\widetilde { g } } ( V _ g f ) . \\end{align*}"}
{"id": "8077.png", "formula": "\\begin{align*} \\alpha _ { H ' - H } \\partial _ { \\Sigma , \\epsilon } ^ { * } = \\partial _ { \\Sigma , \\epsilon } ^ { * } \\beta _ { H ' - H } , \\end{align*}"}
{"id": "6416.png", "formula": "\\begin{align*} \\Xi ^ { - 1 } _ 0 ( x \\wedge y ) = \\tfrac 1 2 x y , \\end{align*}"}
{"id": "1847.png", "formula": "\\begin{align*} D _ { n } ( u , v ) = M _ n ( x , y ) , \\end{align*}"}
{"id": "7576.png", "formula": "\\begin{align*} L B ( \\chi ) = L \\chi \\quad \\mbox { f o r a l l } \\quad \\chi \\in C ^ 1 . \\end{align*}"}
{"id": "1539.png", "formula": "\\begin{align*} \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { E } ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h = \\int _ { G ( \\Q ) \\backslash G ( \\mathbb { A } ) / K _ 1 ( \\mathfrak { n } ) K _ { \\infty } } \\mathbf { E } _ m ( g \\times h , s ) \\mathbf { f } ( h ) \\mathbf { d } h . \\end{align*}"}
{"id": "3334.png", "formula": "\\begin{align*} & [ x , y , z ] + [ y , z , x ] + [ z , x , y ] = 0 , \\\\ & [ x , y , [ z , t , e ] ] = [ [ x , y , z ] , t , e ] + [ z , [ x , y , t ] , e ] + [ z , t , [ x , y , e ] ] , \\end{align*}"}
{"id": "3528.png", "formula": "\\begin{align*} \\int _ 2 ^ T \\abs { \\zeta _ { A V , 2 } ( s _ 1 , s _ 2 , s _ 3 ) } ^ 2 d t _ 3 & = \\zeta _ { A V , 2 } ^ { [ 2 ] } ( s _ 1 , s _ 2 , 2 \\sigma _ 3 ) T \\\\ & + \\begin{cases} O ( T ^ { 2 - 2 \\sigma _ 1 - 2 \\sigma _ 3 } \\log T ) & ( \\frac { 1 } { 2 } < \\sigma _ 1 + \\sigma _ 3 \\leq \\frac { 3 } { 4 } ) \\\\ O ( T ^ \\frac { 1 } { 2 } ) & ( \\frac { 3 } { 4 } < \\sigma _ 1 + \\sigma _ 3 \\leq 1 ) , \\\\ \\end{cases} \\end{align*}"}
{"id": "6101.png", "formula": "\\begin{align*} \\begin{pmatrix} \\varphi ^ { 2 } \\\\ \\psi ^ { 2 } \\end{pmatrix} = \\rho ^ { 2 \\lambda _ { p } - 1 } X _ { 1 } ( \\theta ) + \\rho ^ { 2 \\lambda _ { p } - 2 m + 1 } X _ { 2 } ( \\theta ) , \\end{align*}"}
{"id": "1568.png", "formula": "\\begin{align*} \\mathbf { f } ( g \\times h ) = \\sum _ { j = 1 } ^ e \\mathbf { g } _ j ( g ) \\overline { \\mathbf { h } _ j ( h ) } . \\end{align*}"}
{"id": "418.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } \\left ( h ( U ^ { 1 } ) - h ( U ^ { 0 } ) \\right ) \\partial _ { i } \\phi d x & = \\int _ { 0 } ^ { 1 } \\left ( \\int _ { \\mathbb { R } ^ { d } } D h ( U _ { r } ) ( U ^ { 1 } - U ^ { 0 } ) \\partial _ { i } \\phi d x \\right ) d r , \\\\ & = - \\int _ { \\mathbb { R } ^ { d } } \\int _ { 0 } ^ { 1 } \\partial _ { i } \\left ( D h ( U _ { r } ) ( U ^ { 1 } - U ^ { 0 } ) \\right ) d r \\phi d x \\end{align*}"}
{"id": "1000.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { ( - \\Delta ) ^ s v ( h e _ 1 ) } h = ( - \\Delta ) ^ s \\partial _ 1 v ( 0 ) . \\end{align*}"}
{"id": "8433.png", "formula": "\\begin{align*} Y _ { 1 } = x + \\eta b ( x ) + Z _ { \\eta } , \\end{align*}"}
{"id": "5492.png", "formula": "\\begin{align*} \\xi ( t ; s , x ) = S _ { t - s } x + \\int _ s ^ t S _ { t - u } \\alpha ( u , \\xi ( u ; s , x ) ) d u , t \\in [ s , \\infty ) . \\end{align*}"}
{"id": "6611.png", "formula": "\\begin{align*} \\int _ { - \\delta } ^ { \\delta } f ( v | x - e ^ { \\xi } y | ) \\ , d \\xi - \\int _ { - \\delta } ^ { \\delta } f ( v | x - y | ) \\ , d \\xi & = \\int _ { - \\delta } ^ { \\delta } \\int _ 1 ^ { e ^ { \\xi } } \\frac { d } { d t } f ( v | x - t y | ) \\ , d t \\ , d \\xi \\\\ & \\ll | v y | \\int _ { - \\delta } ^ { \\delta } | \\xi | \\ , d \\xi \\ll | v y | \\delta ^ 2 . \\end{align*}"}
{"id": "8210.png", "formula": "\\begin{align*} F _ { k , i } ( t ) = F _ { k , k + 2 - i } ( t ) G _ { k , i } ( t ) = G _ { k , k + 1 - i } ( t ) . \\end{align*}"}
{"id": "6853.png", "formula": "\\begin{align*} \\widehat { N } _ k ^ j = D ^ { k + j } _ { k } U ^ { k + j } _ { k } , \\end{align*}"}
{"id": "272.png", "formula": "\\begin{align*} \\Omega _ \\varepsilon : = \\Omega \\backslash \\bigcup \\limits _ { k \\in K _ \\varepsilon } { T _ { \\varepsilon , k } } , ~ ~ ~ ~ { K _ \\varepsilon } : = \\{ k \\in \\mathbb { Z } ^ 3 : \\varepsilon \\overline { Q _ k } \\subset \\Omega \\} . \\end{align*}"}
{"id": "6690.png", "formula": "\\begin{align*} _ { s + 1 } F _ s ( 1 , \\ldots , 1 ; 2 , \\ldots , 2 ) ( z ) ^ q = L i _ { K , s } ( z ) . \\end{align*}"}
{"id": "9113.png", "formula": "\\begin{align*} \\operatorname { P } \\left ( \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\psi _ { \\tau } ( y _ { i } ) z _ { i } - \\mathbb { E } \\psi _ { \\tau } ( y _ { i } ) z _ { i } \\right | \\leq C \\sqrt { \\frac { \\tau ^ { 1 - \\delta } M _ { \\delta } ^ { 1 / k } t } { n } } + C \\frac { \\tau t } { n } \\right ) \\geq 1 - 2 e ^ { - t } . \\end{align*}"}
{"id": "2123.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { j = 1 } ^ { n - N } ( \\frac j n ) ^ { c ' k n } = n \\sum _ { j = 1 } ^ { n - N } \\frac 1 n ( \\frac j n ) ^ { c ' k n } \\le n \\int _ 0 ^ { 1 - \\frac { N - 1 } n } x ^ { c ' k n } d x \\le \\\\ & \\frac n { c ' k n + 1 } ( 1 - \\frac { N - 1 } n ) ^ { c ' k n + 1 } \\le \\frac n { c ' k n + 1 } e ^ { - \\frac { N - 1 } n ( c ' k n + 1 ) } . \\end{aligned} \\end{align*}"}
{"id": "7049.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty | e _ j ( z ) | ^ 2 \\le \\Big ( \\frac { C } { \\delta ^ 2 } \\log \\frac { 1 } { \\delta } \\Big ) ^ 2 , \\end{align*}"}
{"id": "2371.png", "formula": "\\begin{align*} \\norm { Q _ \\Omega P _ T } _ { H . S . } ^ 2 = \\iint _ { \\R ^ { 2 d } } | k ( x , t ) | ^ 2 \\ , d ( x , t ) . \\end{align*}"}
{"id": "4744.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - { \\mathcal L } \\right ) U ( t , x , \\omega ) + { Q ( x , \\omega ) } \\lozenge U ( t , x , \\omega ) & = F ( t , x , \\omega ) , t \\in ( 0 , T ] , \\ , x \\in \\mathbb R ^ d , \\ , \\omega \\in \\Omega \\\\ U ( 0 , x , \\omega ) & = G ( x , \\omega ) x \\in \\mathbb R ^ d , \\ , \\omega \\in \\Omega . \\end{align*}"}
{"id": "6789.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\ & f ( x ) \\\\ \\enspace & A x = b \\\\ & g ( x ) \\le 0 , \\end{aligned} \\end{align*}"}
{"id": "7352.png", "formula": "\\begin{align*} \\begin{aligned} & F ( u ( y _ q , t _ q ) , \\nabla \\varphi ( x _ q , t _ q ) , Y _ q ' , W _ \\star [ x _ 0 , t _ 0 ] ) \\\\ & \\geq F ( u ( y _ q , t _ q ) , \\nabla \\varphi ( x _ q , t _ q ) , Y _ q ' , W _ \\star [ x _ 0 , t _ 0 ] \\cap U [ y _ q , t _ q ] ) , \\end{aligned} \\end{align*}"}
{"id": "1613.png", "formula": "\\begin{align*} \\nabla \\psi _ Q & \\neq 0 \\Rightarrow \\chi _ Q = 0 \\\\ \\nabla \\chi _ Q & \\neq 0 \\Rightarrow \\psi _ Q = 1 . \\end{align*}"}
{"id": "9179.png", "formula": "\\begin{align*} { \\mathcal M } _ { i , j } ( p ) = & \\sum _ { \\substack { X ^ { \\alpha _ { i - 1 } } < q \\leq X ^ { \\alpha _ { i } } \\\\ q } } \\frac { \\chi _ { 8 p } ( q ) } { q ^ { 1 / 2 + 1 / ( \\log X ^ { \\alpha _ { j } } ) } } \\frac { \\log ( X ^ { \\alpha _ { j } } / q ) } { \\log X ^ { \\alpha _ { j } } } , 1 \\leq i \\leq j \\leq \\mathcal { J } . \\end{align*}"}
{"id": "1287.png", "formula": "\\begin{align*} | \\Lambda ( M , 0 ) | > & | \\{ \\ , \\ , ( t _ { 1 } , t _ { 2 } , n ) \\ , \\ , | \\ , \\ , A ( \\epsilon _ { ( t _ { 1 } , t _ { 2 } , n ) } ) < M \\ , \\ , , \\ , \\ , 1 \\leq n \\leq N \\ , \\ , \\} | \\\\ = & \\sum _ { k = 1 } ^ { N } | \\{ \\ , \\ , ( t _ { 1 } , t _ { 2 } ) \\ , \\ , | \\ , \\ , t _ { 1 } R _ { 1 } + t _ { 2 } R _ { 2 } < M - T _ { k } \\ , \\ , \\} | \\\\ = & \\sum _ { k = 1 } ^ { N } \\frac { ( M - T _ { k } ) ^ { 2 } } { 2 R _ { 1 } R _ { 2 } } + O ( M - T _ { k } ) \\\\ = & \\frac { ( M ) ^ { 2 } N } { 2 R _ { 1 } R _ { 2 } } + O ( M ) \\end{align*}"}
{"id": "7970.png", "formula": "\\begin{align*} \\liminf _ { j \\to \\infty } \\tilde { d } _ j / d _ j = 0 . \\end{align*}"}
{"id": "1396.png", "formula": "\\begin{align*} | E _ i ( x ) | = \\left | \\alpha + \\int _ 0 ^ x ( n _ i - b _ i ) d y \\right | \\le \\alpha + \\int _ 0 ^ 1 ( n _ i + b _ i ) d y \\le \\alpha + 2 \\bar { b } _ i , \\quad \\forall x \\in [ 0 , 1 ] , i = 1 , 2 . \\end{align*}"}
{"id": "700.png", "formula": "\\begin{align*} \\abs { \\xi _ \\ell - C _ 1 \\ell ^ { - \\psi } } \\leq C _ 1 ' \\ell ^ { - 1 - \\psi } , \\abs { \\zeta _ \\ell - C _ 2 \\ell ^ { - 1 } } \\leq C _ 2 ' \\ell ^ { - 2 } . \\end{align*}"}
{"id": "6419.png", "formula": "\\begin{align*} \\sum _ { i < j } ( 1 - \\lambda _ i ^ 2 \\lambda _ j ^ 2 ) \\ , \\sec _ R ( v _ i \\wedge v _ j ) = 0 . \\end{align*}"}
{"id": "5202.png", "formula": "\\begin{align*} \\| f \\| _ { { B _ { r t } ^ s } ( K ) } & = \\| q ^ { s j } \\Delta _ j f \\| _ { \\ell _ t ( L ^ r ( K ) ) } \\\\ & = \\Bigg \\{ \\sum \\limits _ { j = 0 } ^ { \\infty } \\| q ^ { s j } \\Delta _ j f \\| ^ t _ { L ^ r ( K ) } \\Bigg \\} ^ { \\frac { 1 } { t } } \\\\ & = \\Bigg \\{ \\sum _ { j = 0 } ^ { \\infty } q ^ { s j t } \\bigg \\{ \\int _ { K } | \\Delta _ j f | ^ r d x \\bigg \\} ^ { \\frac { t } { r } } \\Bigg \\} ^ { \\frac { 1 } { t } } . \\end{align*}"}
{"id": "8496.png", "formula": "\\begin{align*} H ( u ) = \\frac { z \\left ( z H ( 0 ) ( u z - u + 1 ) - u ^ { 2 } z + u ^ { 2 } - u \\right ) } { ( 1 + z ^ 2 ) ( u - s _ 1 ) ( u - s _ 2 ) } \\end{align*}"}
{"id": "1637.png", "formula": "\\begin{align*} g \\mapsto \\left \\| g \\right \\| _ \\varphi : = \\int _ M \\sqrt { \\Gamma ( g ) } ( x ) \\varphi ( x ) \\mu ( d x ) \\end{align*}"}
{"id": "414.png", "formula": "\\begin{align*} K _ { 0 } ^ { 2 } ( t ) : = \\| u _ { 0 } \\| _ { m } ^ { 2 } + \\| v _ { 0 } \\| _ { m } ^ { 2 } + \\| w _ { 0 } \\| _ { m } ^ { 2 } + \\int _ { 0 } ^ { t } \\mathcal { F } _ { m } ^ { 2 } ( f _ { 1 } ( \\tau ) , f _ { 2 } ( \\tau ) , f _ { 3 } ( \\tau ) ) d \\tau \\end{align*}"}
{"id": "5316.png", "formula": "\\begin{align*} \\check { \\Delta } ( \\omega ) ( \\chi b \\otimes \\theta c ) = ( \\check { \\Delta } ( \\omega ) ( \\chi \\otimes \\theta ) ) ( b \\otimes c ) , ( b \\chi \\otimes c \\theta ) \\check { \\Delta } ( \\omega ) = ( b \\otimes c ) ( ( \\chi \\otimes \\theta ) \\check { \\Delta } ( \\omega ) . \\end{align*}"}
{"id": "8800.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { x _ 1 + x _ 2 ^ 2 } & = \\sqrt { 0 + 0 } + \\bigl ( \\sqrt { x _ 1 + 0 } - \\sqrt { 0 + 0 } \\bigr ) + \\bigl ( \\sqrt { x _ 1 + s _ { 2 1 } ( x ) } - \\sqrt { x _ 1 + 0 } \\bigr ) \\\\ & + \\bigl ( \\sqrt { x _ 1 + s _ { 2 2 } ( x ) } - \\sqrt { x _ 1 + s _ { 2 1 } ( x ) } \\bigr ) \\\\ & \\geq \\sqrt { x _ 1 } + \\sqrt { 5 + s _ { 2 1 } ( x ) } - \\sqrt { 5 } + \\sqrt { 5 + s _ { 2 2 } ( x ) } - \\sqrt { 5 + s _ { 2 1 } ( x ) } \\\\ & \\geq \\frac { \\sqrt { 5 } } { 5 } x _ 1 + \\sqrt { 5 + x _ 2 ^ 2 } - \\sqrt { 5 } = : \\varphi _ 1 ( x ) , \\end{aligned} \\end{align*}"}
{"id": "8997.png", "formula": "\\begin{align*} A ^ { ( q ) } _ n : = \\varphi _ n ( q , W _ { 1 : K } ) , \\end{align*}"}
{"id": "9532.png", "formula": "\\begin{align*} \\Delta a _ { t + 1 } & \\in \\partial _ { x } H _ t ( x _ { t } , m _ t + a _ t ) , \\\\ \\Delta x _ t & \\in \\partial _ { y } [ - H _ t ] ( x _ { t } , m _ t + a _ t ) , \\end{align*}"}
{"id": "315.png", "formula": "\\begin{align*} & \\frac { d \\rho } { d t } = \\frac { \\partial } { \\partial S } \\mathcal H _ 0 ( \\rho , S ) + \\frac { \\partial } { \\partial S } \\mathcal H _ 1 ( \\rho , S ) d W _ { \\delta } ( t ) , \\\\ & \\frac { d S } { d t } = - \\frac { \\partial } { \\partial \\rho } \\mathcal H _ 0 ( \\rho , S ) - \\frac { \\partial } { \\partial \\rho } \\mathcal H _ 1 ( \\rho , S ) d W _ { \\delta } ( t ) . \\end{align*}"}
{"id": "4021.png", "formula": "\\begin{align*} \\mathcal { S } _ k = \\left \\{ z = x + i y \\in \\mathbb { C } \\ : \\ 1 - \\frac { \\pi } { 2 } \\leq x \\leq 1 + \\frac { \\pi } { 2 } , \\ \\ 2 k \\pi - \\frac { \\pi } { 2 } \\leq y \\leq 2 k \\pi + \\frac { \\pi } { 2 } \\right \\} . \\end{align*}"}
{"id": "870.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( V _ j = i \\right ) = \\frac { \\mathbb { P } \\left ( \\left \\{ \\varsigma _ { Q _ j , m } = 1 , \\varsigma _ { Q _ j , i } = 1 \\right \\} \\bigcap _ { r \\in [ i - 1 ] } \\left \\{ \\varsigma _ { Q _ j , r } = 0 \\right \\} \\right ) } { \\mathbb { P } \\left ( \\varsigma _ { Q _ j , m } = 1 \\right ) } . \\end{align*}"}
{"id": "4215.png", "formula": "\\begin{align*} \\zeta _ f ( z ) = \\zeta _ f ^ { ( 1 ) } ( z ) - \\zeta _ f ^ { ( 2 ) } ( z ) . \\end{align*}"}
{"id": "2294.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k = 1 } ^ n c _ k M _ { \\omega _ k } T _ { x _ k } g _ 0 . \\end{align*}"}
{"id": "6822.png", "formula": "\\begin{align*} ( F \\circ \\psi ) ^ * ( g _ { \\textnormal { e u c } } ) = \\psi ^ * ( e ^ { 2 u } \\ , g _ { \\textnormal { p o i n } } ) = e ^ { 2 u \\circ \\psi + 2 v } \\ , g _ { \\textnormal { e u c } } \\textnormal { o n } \\ , \\ , B ^ 2 _ 1 ( 0 ) . \\end{align*}"}
{"id": "7235.png", "formula": "\\begin{align*} \\mathcal R _ c = : \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\mathfrak r _ c ( s , x , v ) \\dd v \\dd s . \\end{align*}"}
{"id": "310.png", "formula": "\\begin{align*} & \\frac { d \\rho _ i } { d t } + \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ j - S _ i ) \\theta _ { i j } ( \\rho ) = 0 , \\\\ & \\frac { d S _ i } { d t } + \\frac 1 2 \\sum _ { j \\in N ( i ) } \\omega _ { i j } ( S _ i - S _ j ) ^ 2 \\frac { \\partial \\theta _ { i j } ( \\rho ) } { \\partial \\rho _ i } + \\beta \\frac { \\partial I ( \\rho ) } { \\partial \\rho _ i } - \\alpha \\log ( \\rho _ i ) + \\mathbb V _ i + \\sum _ { j = 1 } ^ N \\mathbb W _ { i j } \\rho _ j = 0 . \\end{align*}"}
{"id": "8335.png", "formula": "\\begin{align*} 0 \\leq \\int _ 0 ^ T ( h - \\eta ) \\dd \\bar p = \\int _ 0 ^ T h \\dd \\bar p - \\int _ 0 ^ T \\eta \\dd \\bar p , \\end{align*}"}
{"id": "4473.png", "formula": "\\begin{align*} \\frac { \\partial f ( t ) } { \\partial t } = e _ { \\lambda } ^ { 1 - \\lambda } ( t ) | z | ^ { 2 } f ( t ) \\ \\Longleftrightarrow \\ \\frac { f ^ { \\prime } ( t ) } { f ( t ) } = e _ { \\lambda } ^ { 1 - \\lambda } ( t ) | z | ^ { 2 } , \\bigg ( f ^ { \\prime } ( t ) = \\frac { d } { d t } f ( t ) \\bigg ) . \\end{align*}"}
{"id": "1194.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ m p _ i \\mu \\circ f _ i ^ { - 1 } . \\end{align*}"}
{"id": "7231.png", "formula": "\\begin{align*} B _ { 4 , 1 } : = B _ { 4 } \\cap \\{ ( s , v ) \\in [ 0 , t ] \\times \\R ^ 3 : | v | \\geq \\frac { \\langle x ^ \\perp \\rangle } { 2 \\langle \\check { \\mathcal T } _ { t , x _ 1 , v _ 1 } \\rangle } \\} . \\end{align*}"}
{"id": "1852.png", "formula": "\\begin{align*} L _ { n + 1 } ( x ) = x \\sum _ { k = 0 } ^ n { n \\choose k } L _ { k } ( x ) M _ { n - k } ( x ) . \\end{align*}"}
{"id": "5563.png", "formula": "\\begin{align*} g _ n ' ( 0 ) = f _ n ( 0 ) = \\sqrt { n } , \\end{align*}"}
{"id": "3269.png", "formula": "\\begin{align*} ( s - t ) ^ 2 - \\frac { s - t } { h ( s ) } + | \\lambda | ^ 2 = 0 . \\end{align*}"}
{"id": "2796.png", "formula": "\\begin{align*} \\inf _ { f \\in \\mathcal { K } } \\Phi _ 1 ( f ) = \\inf _ { f \\in \\mathcal { K } } \\frac { 1 } { 2 } \\int \\left ( L _ + f \\right ) f d x , \\end{align*}"}
{"id": "2408.png", "formula": "\\begin{align*} s _ { K , L } = \\sum _ { \\substack { k \\in \\Z ^ d \\\\ | k _ j | \\leq K } } \\sum _ { \\substack { l \\in \\Z ^ d \\\\ | l _ j | \\leq L } } c _ { k , l } M _ { \\beta l } T _ { \\alpha k } g \\end{align*}"}
{"id": "1559.png", "formula": "\\begin{align*} p _ { \\mathcal { S } } ( \\psi _ i , \\Phi ) = p _ Y ( _ { \\mathcal { S } / Y } ( \\psi _ i ) , \\Phi ' ) = p _ Y ( _ { \\mathcal { S } / Y } ( \\phi _ i ) , \\Phi ' ) = p _ { \\mathcal { S } } ( \\phi _ i , \\Phi ) , \\end{align*}"}
{"id": "4249.png", "formula": "\\begin{align*} R ( \\lambda ) - R ( \\mu ) = ( \\mu - \\lambda ) R ( \\lambda ) R ( \\mu ) ( \\Re \\lambda , \\ , \\Re \\mu > 0 ) . \\end{align*}"}
{"id": "8481.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { ( q + 1 ) ^ 2 | S _ { \\ell } | } { | G | } = \\frac { q + 1 } { q | T _ { \\ell ' } | } & \\mbox { i f } a = 1 , \\mbox { a n d } \\\\ \\frac { ( q + 1 ) | S _ { \\ell } | } { | G | } \\left ( \\alpha ( a ) + \\alpha ( a ^ { - 1 } ) \\right ) = \\frac { 1 } { q | T _ { \\ell ' } | } \\left ( \\alpha ( a ) + \\alpha ( a ^ { - 1 } ) \\right ) & \\mbox { i f } 1 \\neq a \\in ( \\mu _ { q - 1 } ) _ { \\ell ' } . \\end{array} \\right . \\end{align*}"}
{"id": "1357.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\norm { 1 _ { U _ t } } _ { L ^ \\frac { d } { d - 1 } ( W _ R ) } d t \\geq \\int _ 0 ^ \\infty \\norm { g 1 _ { U _ t } } _ { L ^ 1 ( W _ R ) } \\ ; d t = \\int _ { W _ R } g ( x ) \\int _ 0 ^ \\infty 1 _ { U _ t } ( x ) \\ ; d t d x = \\norm { g u } _ { L ^ 1 ( W _ R ) } . \\end{align*}"}
{"id": "7418.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { 2 \\varepsilon n ^ 2 } \\sum _ { x } \\Phi _ n ( s , \\tfrac { x } { n } ) \\sum _ { r = 1 } ^ { \\varepsilon n } \\int _ { \\Omega _ 1 ( x ) } \\overleftarrow { \\eta } ^ { \\ell } ( x ) [ \\eta ( x + 1 ) - \\eta ( x + 1 + r ) ] [ f ( \\eta _ { 1 , x , r } ) - f ( \\eta _ { 2 , x , r } ) ] d \\nu _ { b } \\Big | . \\end{align*}"}
{"id": "2740.png", "formula": "\\begin{align*} \\frac { 4 ^ m } { 2 m + 1 } \\leq \\binom { 2 m } { m } , \\end{align*}"}
{"id": "6239.png", "formula": "\\begin{align*} H = G _ m \\cap G _ { s ( K _ j ) } = G _ m \\cap G _ { s ( K _ 1 ) } \\cap \\dotsb \\cap G _ { s ( K _ t ) } . \\end{align*}"}
{"id": "6566.png", "formula": "\\begin{align*} \\mathcal { S } ( h , k ) = \\mathcal { I } _ 0 ( h , k ) + \\mathcal { I } _ 1 ( h , k ) + \\mathcal { E } ( h , k ) , \\end{align*}"}
{"id": "333.png", "formula": "\\begin{align*} m _ { 2 1 } ( t ) & = \\frac 1 2 ( \\Sigma _ 2 - \\Sigma _ 1 ) \\dot W ^ { \\delta } ( t ) , \\ ; t \\in [ 0 , 1 - \\delta ] , \\\\ m _ { 2 1 } ( t ) & = ( \\rho _ 1 ^ b - \\rho _ 1 ^ a ) \\frac { 1 } { \\delta } + \\frac 1 2 ( \\Sigma _ 2 - \\Sigma _ 1 ) \\dot W ^ { \\delta } ( t ) , \\ ; t \\in [ 1 - \\delta , 1 ] , \\end{align*}"}
{"id": "6846.png", "formula": "\\begin{align*} U _ i f ( y ) = & ~ \\frac { 1 } { R ( i + 1 , i ) } \\sum \\limits _ { x \\lessdot y } f ( x ) , \\\\ D _ { i + 1 } f ( x ) = & ~ \\frac { 1 } { \\pi _ { i + 1 } ( X _ { x } ) } \\sum \\limits _ { y \\gtrdot x } \\pi _ { i + 1 } ( y ) f ( y ) , \\end{align*}"}
{"id": "2701.png", "formula": "\\begin{align*} \\langle W f , W ( M _ { - \\omega } T _ x g _ 0 ) \\rangle = \\iint _ { \\R ^ { 2 d } } W f ( \\xi , \\eta ) W ( M _ { - \\omega } T _ x g _ 0 ) ( \\xi , \\eta ) \\ , d ( \\xi , \\eta ) > 0 , \\forall ( x , \\omega ) \\in \\R ^ { 2 d } . \\end{align*}"}
{"id": "3398.png", "formula": "\\begin{align*} f = \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\sum \\limits _ { Q } \\mu ( Q ) { \\widetilde { \\widetilde { D } } } _ { k } ( x , x _ Q ) D _ { k } ( f ) ( x _ Q ) = \\sum \\limits _ { k = - \\infty } ^ { \\infty } \\sum \\limits _ { Q } \\mu ( Q ) D _ { k } ( x , x _ Q ) { \\bar { \\bar { D } } } _ { k } ( f ) ( x _ Q ) , \\end{align*}"}
{"id": "2104.png", "formula": "\\begin{align*} [ x _ 0 : x _ 1 : x _ 2 : x _ 3 ] \\to [ y _ 0 : y _ 1 : y _ 2 : y _ 3 ] = \\left [ x _ 0 ^ 2 : x _ 1 ^ 2 : x _ 2 ^ 2 : x _ 3 ^ 2 \\right ] \\end{align*}"}
{"id": "6255.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } ( e ^ { \\lambda ( q , t ) } - 1 ) - 1 & { } \\stackrel { \\phantom { \\eqref { e q _ l a m b d a _ p r i m o } } } { = } e ^ { \\lambda ( q , t ) } \\frac { \\partial } { \\partial t } ( \\lambda ( q , t ) ) - 1 = \\\\ & { } \\stackrel { \\eqref { e q _ l a m b d a _ p r i m o } } { = } \\frac { e ^ { \\lambda ( q , t ) } ( 1 - q ) } { e ^ { \\lambda ( q , t ) \\cdot q } - q e ^ { \\lambda ( q , t ) } } - 1 = \\frac { e ^ { \\lambda ( q , t ) } - e ^ { \\lambda ( q , t ) \\cdot q } } { e ^ { \\lambda ( q , t ) \\cdot q } - q e ^ { \\lambda ( q , t ) } } . \\end{align*}"}
{"id": "2321.png", "formula": "\\begin{align*} W ( f , g ) ( x , \\omega ) = 2 ^ d e ^ { 4 \\pi i x \\cdot \\omega } V _ { g ^ \\vee } f ( 2 x , 2 \\omega ) = 2 ^ d A ( f , g ^ \\vee ) ( 2 x , 2 \\omega ) . \\end{align*}"}
{"id": "5558.png", "formula": "\\begin{align*} \\sigma = e ^ { - z _ 6 \\cdot } . \\end{align*}"}
{"id": "1828.png", "formula": "\\begin{align*} L _ n ( x , y ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } L ( n , k ) x ^ { 2 k + 1 } y ^ { n - 2 k } . \\end{align*}"}
{"id": "6159.png", "formula": "\\begin{align*} y _ n ( x ) : = \\sum _ { k = 0 } ^ n \\frac { ( n + k ) ! } { ( n - k ) ! k ! } \\left ( \\frac { x } { 2 } \\right ) ^ k . \\end{align*}"}
{"id": "1093.png", "formula": "\\begin{align*} m ^ { ( 2 ) } ( x , t , k ) = D _ { \\infty } ^ { \\sigma _ 3 } ( \\xi ) m ^ { ( 1 ) } ( x , t , k ) D ^ { - \\sigma _ 3 } ( k , \\xi ) . \\end{align*}"}
{"id": "2336.png", "formula": "\\begin{align*} \\theta ( \\tau ) = \\sqrt { \\tfrac { i } { \\tau } } \\ \\theta ( - \\tfrac { 1 } { \\tau } ) \\end{align*}"}
{"id": "4077.png", "formula": "\\begin{align*} g ( \\bar { x } ) - Q _ { z _ i } ( t _ i ) = g ( z _ i + t _ i ) - Q _ { z _ i } ( t _ i ) = \\int _ 0 ^ { t _ i } [ g '' ( \\bar { x } - v ) + A ] v \\ , d v , \\end{align*}"}
{"id": "3820.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ n ) = \\sum _ { i = 1 } ^ n c _ i \\prod _ { j = 1 } ^ n x _ j ^ { E _ { i j } } \\end{align*}"}
{"id": "5846.png", "formula": "\\begin{align*} \\partial _ \\pm = \\{ x = ( x ^ 1 , x ^ 2 ) \\in \\R ^ 2 : x ^ 1 = \\pm R \\} . \\end{align*}"}
{"id": "6920.png", "formula": "\\begin{align*} B _ r ( \\theta _ I ) = \\sqrt { \\frac { 1 } { N _ s } } \\textbf { a } _ 1 ^ T ( \\theta _ I ) \\textbf { a } _ 1 ^ * ( \\theta _ I ) = \\sqrt { N _ s } . \\end{align*}"}
{"id": "9340.png", "formula": "\\begin{align*} H _ { 0 , \\lambda } = 0 , H _ { n , \\lambda } = \\sum _ { k = 1 } ^ { n } \\frac { 1 } { \\lambda } \\binom { \\lambda } { k } ( - 1 ) ^ { k - 1 } , ( n \\in \\mathbb { N } ) , ( \\mathrm { s e e } \\ [ 9 ] ) . \\end{align*}"}
{"id": "711.png", "formula": "\\begin{align*} \\kappa _ { \\alpha \\alpha } ^ { ( \\ell ) } = K _ { \\alpha \\alpha } ^ { ( \\ell ) } + O ( n ^ { - 1 } ) . \\end{align*}"}
{"id": "8564.png", "formula": "\\begin{align*} & T ( k ) = \\alpha k + \\mathcal { O } ( k ^ 2 ) , \\alpha \\neq 0 , \\ k \\rightarrow 0 , \\\\ & 1 + R _ { \\pm } ( k ) = \\alpha _ { \\pm } k + \\mathcal { O } ( k ^ 2 ) , k \\rightarrow 0 . \\end{align*}"}
{"id": "7110.png", "formula": "\\begin{align*} d x ( t ) = ( \\alpha _ { - 1 } x ( t ) ^ { - 1 } - \\alpha _ { 0 } + \\alpha _ { 1 } x ( t ) - \\alpha _ { 2 } x ( t ) ^ { 2 } ) d t + \\sigma x ( t ) ^ { \\theta } d B _ t \\end{align*}"}
{"id": "5104.png", "formula": "\\begin{align*} h _ A ( \\tfrac { \\eta } { n } ) = h _ A ( \\tfrac { 1 } { \\eta } ) { \\Longleftrightarrow } \\frac { \\eta } { n } = \\frac { 1 } { \\eta } \\ ; \\vee \\ ; \\frac { \\eta } { n } = \\eta \\Longleftrightarrow \\eta = \\sqrt { n } \\ ; \\vee \\ ; n = 1 . \\end{align*}"}
{"id": "2712.png", "formula": "\\begin{align*} P _ n = \\sum _ { i = 1 } ^ n X _ i ^ 2 - n . \\end{align*}"}
{"id": "7532.png", "formula": "\\begin{align*} \\frac { T } { 4 } \\int _ { \\frac { 1 } { 2 } - \\epsilon } ^ { \\frac { 1 } { 2 } + \\epsilon } \\log \\left ( \\frac { \\sigma ^ 2 + T ^ 2 } { 4 } \\right ) \\ d \\sigma & = \\epsilon T \\log \\left ( \\frac { T } { 2 } \\right ) - \\epsilon T + \\epsilon T + \\epsilon \\ \\mathcal { O } \\left ( \\frac { 1 } { T } \\right ) \\end{align*}"}
{"id": "3164.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { Q _ s \\in \\mathfrak { Q } _ s , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) \\geq \\sup _ { Q _ s \\in \\mathfrak { Q } _ s ' , s \\in \\lbrace 1 , \\ldots , k \\rbrace } \\min _ { G \\in \\mathcal { G } ( Q _ 0 , \\ldots , Q _ k ) } \\det ( G ^ \\intercal G ) , \\end{aligned} \\end{align*}"}
{"id": "8035.png", "formula": "\\begin{align*} \\rho ( x ) : = 2 \\int _ { x ' = 0 } ^ x \\frac { \\mathrm { d } x ' } { t _ + ( x ' ) } . \\end{align*}"}
{"id": "3632.png", "formula": "\\begin{align*} T : = \\frac { 1 } { \\max _ { t \\ge H } \\frac { x ^ { - \\nu _ 2 ( t ) } } { t } } = \\min _ { t \\ge H } t x ^ { \\nu _ 2 ( t ) } . \\end{align*}"}
{"id": "5032.png", "formula": "\\begin{align*} T : = \\left \\{ \\begin{pmatrix} \\mathrm { e } ^ { \\mathbf { i } \\beta _ { 1 } } & & \\\\ & \\ddots & \\\\ & & \\mathrm { e } ^ { \\mathbf { i } \\beta _ { p + q } } \\end{pmatrix} \\ ; , \\ ; \\beta _ { 1 } , \\ldots , \\beta _ { p + q } \\ , \\in \\mathbb { R } \\right \\} \\ ; , \\end{align*}"}
{"id": "6247.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\lambda = 1 + \\frac { \\frac { \\partial } { \\partial t } \\lambda } { q - 1 } ( e ^ { q \\lambda } - q e ^ { \\lambda } + q - 1 ) . \\end{align*}"}
{"id": "7927.png", "formula": "\\begin{align*} \\sum _ { a \\in S } a = \\frac { 1 } { 2 } n ^ { 3 / 2 } + O ( n ^ { 1 1 1 / 8 0 + \\varepsilon } ) . \\end{align*}"}
{"id": "3497.png", "formula": "\\begin{align*} D _ 1 + D _ 2 = D _ { 1 1 } - D _ { 1 2 } - D _ { 1 3 } + \\frac { 1 } { 2 } D _ { 1 4 } + ( D _ { 2 1 } - D _ { 2 2 } - D _ { 2 3 } - \\frac { 1 } { 2 ^ { s _ 3 } } D _ { 2 4 } ) . \\end{align*}"}
{"id": "2437.png", "formula": "\\begin{align*} \\widetilde { g } = S _ { g , \\L } ^ { - 1 } g \\end{align*}"}
{"id": "8045.png", "formula": "\\begin{align*} ( \\eta _ { ( \\Sigma , U ) } h ) ( s ) : = \\int _ { \\mathbb { R } } h ( - s , v ) \\mathrm { d } v , \\end{align*}"}
{"id": "6396.png", "formula": "\\begin{align*} \\tilde { g } _ i ( z _ 1 \\dots , z _ n ) = \\prod _ { ( \\sigma ( z _ 1 ) , \\dots , \\sigma ( z _ n ) ) \\in S _ n } \\ ; g _ i ( \\sigma ( z _ 1 ) , \\dots , \\sigma ( z _ n ) ) \\ , , \\end{align*}"}
{"id": "6190.png", "formula": "\\begin{align*} \\| C \\hat { V } \\| _ F ^ 2 = { \\rm T r } \\left ( \\hat { V } ^ { T } S ^ { T } S \\hat { V } \\right ) + { \\rm T r } \\left [ \\hat { V } ^ { T } ( C ^ { T } C - S ^ { T } S ) \\hat { V } \\right ] \\geq \\| S \\hat { V } \\| ^ 2 _ F - \\theta ( k + \\sqrt { k } \\xi ) \\| C \\| ^ 2 _ F . \\end{align*}"}
{"id": "1668.png", "formula": "\\begin{align*} \\theta _ 0 : = \\frac { 1 } { 4 } ( \\lambda _ 1 + \\gamma ) ^ 2 , \\theta ^ * : = \\frac { ( \\lambda _ 1 + \\gamma ) ^ 2 } { 2 ( \\lambda _ 2 + \\gamma ) } \\end{align*}"}
{"id": "7395.png", "formula": "\\begin{align*} b _ { G } : = \\inf \\Big \\{ \\forall s \\in [ 0 , T ] , \\tilde { b } \\geq 0 : \\sup _ { | u | \\geq \\tilde { b } } | G ( s , u ) | = 0 \\Big \\} . \\end{align*}"}
{"id": "3914.png", "formula": "\\begin{align*} f ( \\eta ) = \\frac { \\kappa } { 2 } { \\eta } ^ 2 - \\frac { { \\kappa } ^ 2 } { 2 4 0 } { \\eta } ^ 5 + \\frac { 1 1 } { 1 6 1 2 8 0 } { \\kappa } ^ 3 { \\eta } ^ 8 - \\frac { 5 } { 4 2 5 7 7 9 2 } { \\kappa } ^ 4 { \\eta } ^ { 1 1 } + . . . \\end{align*}"}
{"id": "3112.png", "formula": "\\begin{align*} z ^ 3 y ^ 3 + z y ^ 2 ( a x ^ 3 + b y ^ 3 ) + x ^ 6 + c x ^ 3 y ^ 3 + d y ^ 6 = 0 \\ , . \\end{align*}"}
{"id": "1995.png", "formula": "\\begin{gather*} \\varphi = \\exp \\left ( \\log \\varphi \\right ) = \\exp f , \\\\ w = \\exp \\left ( \\log w \\right ) = \\exp g , \\end{gather*}"}
{"id": "8528.png", "formula": "\\begin{align*} \\lambda + \\mu X = \\lambda ^ { 2 ^ h } + \\mu ^ { 2 ^ h } X ^ { 2 ^ h } \\end{align*}"}
{"id": "3608.png", "formula": "\\begin{align*} 1 - R _ { n + 1 } = d [ ( - 1 ) ^ { ( n + 1 ) / 2 } a _ { 1 , n + 1 } ] \\ , \\in \\ , \\{ d ( ( - 1 ) ^ { ( n + 1 ) / 2 } a _ { 1 , n + 1 } ) \\ , , \\ , \\alpha _ { n + 1 } \\} \\end{align*}"}
{"id": "6764.png", "formula": "\\begin{align*} & C '' _ 1 = C '' _ { 1 , 0 } + C '' _ { 1 , 1 } + C '' _ { 1 , 2 } , \\\\ & C '' _ 2 = C '' _ { 2 , 0 } + C '' _ { 2 , 1 } + C '' _ { 2 , 2 } , \\\\ & C '' _ 3 = C '' _ { 3 , 0 } + C '' _ { 3 , 1 } + C '' _ { 3 , 2 } , \\\\ & C '' _ 4 = C '' _ { 4 , 0 } + C '' _ { 4 , 1 } + C '' _ { 4 , 2 } . \\end{align*}"}
{"id": "4336.png", "formula": "\\begin{align*} \\lambda ( t ) = C ( u _ 0 ) ( T - t ) ^ { \\frac { 2 } { \\alpha } } ( 1 + O ( \\left | \\ln ( T - t ) \\right | ^ { - 1 } ) ) t \\to T . \\end{align*}"}
{"id": "1264.png", "formula": "\\begin{align*} P _ { \\theta } ^ { \\mathrm { i n } } ( M ) : = P _ { \\theta } ^ { \\mathrm { i n } } ( M - a ) \\cup { ( a ) } \\end{align*}"}
{"id": "5502.png", "formula": "\\begin{align*} \\eta _ s ( t ; x ) & = \\xi ( s + t ; s , x ) \\\\ & = S _ t x + \\int _ s ^ { s + t } S _ { s + t - v } \\alpha ( v , \\xi ( v ; s , x ) ) d v \\\\ & = S _ t x + \\int _ s ^ { s + t } S _ { s + t - v } a ( \\xi ( v ; s , x ) ) d v \\\\ & = S _ t x + \\int _ 0 ^ { t } S _ { t - v } a ( \\xi ( s + v ; s , x ) ) d v \\\\ & = S _ t x + \\int _ 0 ^ t S _ { t - v } a ( \\eta _ s ( v ; x ) ) d v . \\end{align*}"}
{"id": "8010.png", "formula": "\\begin{align*} \\pi _ { 3 , \\ell / r } \\circ ( \\chi _ 2 \\circ \\chi _ 1 ) = : ( \\chi _ 2 \\circ \\chi _ 1 ) _ { \\ell / r } \\circ \\pi _ { 1 , \\ell / r } . = ( \\chi _ { 2 , \\ell / r } \\circ \\chi _ { 1 , \\ell / r } ) \\circ \\pi _ { 1 , \\ell / r } \\end{align*}"}
{"id": "1530.png", "formula": "\\begin{align*} \\mathbf { E } ( \\mathfrak { g } , s ) : = E ^ n _ k ( \\mathfrak { g } \\ , \\sigma ^ { - 1 } , s ; \\chi ) , \\ , \\ , \\ , \\ , \\mathfrak { g } \\in G _ N ( \\mathbb { A } ) , \\end{align*}"}
{"id": "4517.png", "formula": "\\begin{align*} ( a b ) b = \\frac { 1 } 2 ( \\alpha b + a b ) , ( a b ) a = \\frac { 1 } 2 ( \\alpha a + a b ) , ( a b ) ( a b ) = \\frac { \\alpha } 4 ( a + b + 2 a b ) . \\end{align*}"}
{"id": "2167.png", "formula": "\\begin{align*} \\| u \\| _ { \\alpha , G } = \\| u \\| _ { ( G ) } + [ u ] _ { ( \\alpha , G ) } , \\end{align*}"}
{"id": "5269.png", "formula": "\\begin{align*} \\sigma ^ { \\varphi } ( \\delta _ { \\varphi } ) = \\delta _ { \\varphi } \\nu ^ { - 1 } . \\end{align*}"}
{"id": "6322.png", "formula": "\\begin{align*} W _ { \\mathbf { e } , \\mathbf { f } } ( X , Y ) = \\prod _ { i = 1 } ^ r \\left ( \\frac { 1 } { 1 - X ^ { f _ i } } \\right ) \\sum _ { I \\subseteq [ r ] } ( - 1 ) ^ { | I | } \\frac { X ^ { \\sum _ { i \\in I } f _ i } } { 1 - X ^ { 4 n + \\sum _ { i \\in I } e _ i f _ i } Y ^ { n + 2 } } , \\end{align*}"}
{"id": "2678.png", "formula": "\\begin{align*} M _ g = \\left ( g ( x _ j - y _ k ) \\right ) _ { j , k = 1 } ^ N \\end{align*}"}
{"id": "7650.png", "formula": "\\begin{align*} \\rho _ n \\ge 0 \\ ; \\R ^ N , \\int _ { \\R ^ N } \\rho _ n = \\lambda \\end{align*}"}
{"id": "3250.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s h ( s ) = \\lim _ { s \\rightarrow \\infty } \\int _ 0 ^ \\infty \\frac { s ^ 2 } { s ^ 2 + u ^ 2 } d \\mu ( u ) = 1 . \\end{align*}"}
{"id": "1299.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\frac { | \\Lambda ( M ) | } { | \\Lambda ( M , \\Gamma ) | } = \\lim _ { M \\to \\infty } \\frac { | \\Lambda ( M ) | } { M ^ { 2 } } \\frac { M ^ { 2 } } { | \\Lambda ( M , \\Gamma ) | } = | H _ { 1 } ( Y ) | . \\end{align*}"}
{"id": "4842.png", "formula": "\\begin{align*} \\chi _ A ( x ) : = \\left \\{ \\begin{array} { l l } 1 & x \\in A \\\\ 0 & \\end{array} \\right . \\end{align*}"}
{"id": "4475.png", "formula": "\\begin{align*} f ( t ) = e ^ { | z | ^ { 2 } ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { l = 0 } ^ { \\infty } | z | ^ { 2 l } \\frac { 1 } { l ! } ( e _ { \\lambda } ( t ) - 1 ) ^ { l } . \\end{align*}"}
{"id": "4776.png", "formula": "\\begin{align*} | | S | | _ 2 = \\sqrt { 2 } | | W ( R ) | | _ 2 . \\end{align*}"}
{"id": "6894.png", "formula": "\\begin{align*} \\sum \\limits _ { Y \\in \\mathcal Y } \\alpha _ Y \\lambda _ { Y , \\delta , \\ell } & = \\frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \\choose t } _ q } \\sum \\limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \\choose 2 } } { k - t \\choose i } _ q { k + i - \\ell \\choose i } _ q \\\\ & = \\frac { { k - j \\choose \\ell } _ q } { { k \\choose \\ell } _ q } . \\end{align*}"}
{"id": "8123.png", "formula": "\\begin{align*} T B ( f ) = 1 3 T B _ { 6 } ( f ) . \\end{align*}"}
{"id": "2840.png", "formula": "\\begin{align*} 0 < \\Im \\int x \\cdot \\nabla v ( x , t ) \\bar { v } ( x , t ) d x = - \\Im \\int x \\cdot \\nabla u ( x , - t ) \\bar { u } ( x , - t ) d x , \\ ; \\forall t > 0 , \\end{align*}"}
{"id": "1361.png", "formula": "\\begin{align*} \\norm { u } _ { W ( \\sigma _ { j + 1 } R ) , 2 \\alpha _ { j + 1 } p ^ * } \\leq \\prod _ { k = 1 } ^ j \\left ( C \\frac { ( \\rho / p ^ * ) ^ { 2 k } } { ( \\sigma - \\sigma ' ) ^ 2 } \\norm { \\lambda ^ { - 1 } } _ { W ( R ) , q } \\norm { \\Lambda } _ { W ( R ) , p } \\right ) ^ { 1 / ( 2 \\alpha _ k ) } \\norm { u } _ { W ( \\sigma R ) , \\rho } ^ { \\prod _ { k = 1 } ^ j \\gamma _ k } . \\end{align*}"}
{"id": "4942.png", "formula": "\\begin{align*} \\beta = ( 1 - \\phi ) \\gamma \\end{align*}"}
{"id": "2275.png", "formula": "\\begin{align*} C _ 1 = I ( 0 ) = \\int _ { \\R } e ^ { - \\pi M x ^ 2 } \\ , d x = M ^ { - 1 / 2 } , \\end{align*}"}
{"id": "3352.png", "formula": "\\begin{align*} & \\psi ( \\{ u , v , w \\} _ { T } ) = \\psi ( \\theta ( T ( v ) , T ( w ) ) u ) = \\theta ( \\phi ( T ( v ) ) , \\phi ( T ( w ) ) ) \\psi ( u ) \\\\ & \\ ; \\ ; \\ ; = \\theta ( T ^ { ' } ( \\psi ( v ) ) , T ^ { ' } ( \\psi ( w ) ) ) \\psi ( u ) = \\{ \\psi ( u ) , \\psi ( v ) , \\psi ( w ) \\} _ { T ^ { ' } } . \\end{align*}"}
{"id": "5248.png", "formula": "\\begin{align*} \\widetilde { m } x : = \\sum _ i ( m y _ i ) z _ i , x \\in A \\otimes B , E x = \\sum _ i y _ i z _ i \\textrm { f o r } y _ i \\in A \\ , { } ^ I \\ ! \\otimes ^ I B , z _ i \\in A \\otimes B , \\end{align*}"}
{"id": "4802.png", "formula": "\\begin{align*} ( \\phi ^ 2 + \\phi * \\phi ) ( 0 , 0 ) & = s ^ 2 ( \\ell - 1 ) + t ^ 2 ( m - 1 ) + ( \\ell - 1 ) ( m - 1 ) \\\\ ( \\phi ^ 2 + \\phi * \\phi ) ( a , 0 ) & = s ^ 2 ( \\ell - 1 ) + ( \\ell - 2 ) ( m - 1 ) + 2 t ( m - 1 ) \\\\ ( \\phi ^ 2 + \\phi * \\phi ) ( 0 , b ) & = t ^ 2 ( m - 1 ) + ( \\ell - 1 ) ( m - 2 ) + 2 s ( \\ell - 1 ) \\\\ ( \\phi ^ 2 + \\phi * \\phi ) ( a , b ) & = 1 + ( \\ell - 2 ) ( m - 2 ) + 2 s t + 2 s ( \\ell - 2 ) + 2 t ( m - 2 ) , \\end{align*}"}
{"id": "7809.png", "formula": "\\begin{align*} c _ m : = \\frac { ( - 1 ) ^ { m + 1 } } { s } a ^ m \\cdot \\left ( \\alpha ^ { 2 ^ { \\ell - 1 } s } 2 ^ { 2 ^ { \\ell - 1 } s - \\ell } + 2 ^ { 2 ^ { \\ell - 1 } s - 2 } s \\alpha ^ { 2 ^ { \\ell - 1 } s - 1 } \\pi \\right ) . \\end{align*}"}
{"id": "536.png", "formula": "\\begin{align*} h _ i ( \\boldsymbol { f } _ O ^ { d _ i - 1 } \\left ( \\boldsymbol { f } ( \\mathbf { x } _ k , \\tilde { \\mathbf { u } } _ k ) \\right ) ) = 0 , \\forall i \\in \\mathbb { Z } _ { [ 1 , m ] } . \\end{align*}"}
{"id": "3292.png", "formula": "\\begin{align*} s - t = \\frac { 1 - \\sqrt { 1 - 4 | \\lambda | ^ 2 h ( s ) ^ 2 } } { 2 h ( s ) } = \\frac { 2 | \\lambda | ^ 2 h ( s ) } { 1 + \\sqrt { 1 - 4 | \\lambda | ^ 2 h ( s ) ^ 2 } } . \\end{align*}"}
{"id": "5213.png", "formula": "\\begin{align*} | E _ 1 | ~ & = ~ | \\{ x \\in K : | B f _ 1 ( x ) | > \\lambda / 2 \\} | \\\\ & = ~ \\bigg | \\bigg ( \\bigcup _ i W _ i \\bigg ) \\bigcap \\{ x \\in K : | B f _ 1 ( x ) | > \\lambda / 2 \\} \\bigg | ~ + ~ \\bigg | \\bigg ( \\bigcup _ i W _ i \\bigg ) ^ \\complement \\bigcap \\{ x \\in K : | B f _ 1 ( x ) | > \\lambda / 2 \\} \\bigg | . \\end{align*}"}
{"id": "9451.png", "formula": "\\begin{align*} g _ 2 = \\frac { c _ 0 { \\bar g _ 2 } + g _ 1 ^ 2 } { c _ 0 c _ 1 X ^ { p ^ s - 2 p ^ { s - 1 } } } . \\end{align*}"}
{"id": "6394.png", "formula": "\\begin{align*} \\Lambda = \\{ ( s _ 1 , \\dots , s _ { n - 1 } , p ) \\in \\mathbb G _ n \\ , : \\ , f _ i ( s _ 1 , \\dots , s _ { n - 1 } , p ) = 0 \\ , , \\ , 1 \\leq i \\leq n - 1 \\} . \\end{align*}"}
{"id": "5349.png", "formula": "\\begin{align*} M = \\left \\{ x \\in \\mathbb { R } ^ n \\middle | A x \\leqslant b \\right \\} . \\end{align*}"}
{"id": "5362.png", "formula": "\\begin{align*} \\gamma _ i ( x ) = x - \\sigma _ i ( x ) c , \\end{align*}"}
{"id": "6011.png", "formula": "\\begin{align*} \\begin{aligned} D _ { { \\rm l e f t } } u ( x ) = \\lim _ { t \\rightarrow 0 ^ + } \\frac { u ( x ) - u ( x - t ) } { t } \\\\ D _ { { \\rm r i g h t } } u ( x ) = \\lim _ { t \\rightarrow 0 ^ + } \\frac { u ( x ) - u ( x + t ) } { t } , \\end{aligned} \\end{align*}"}
{"id": "3886.png", "formula": "\\begin{align*} w _ 1 ( M ) + ( 2 n - a - b ) ( t _ 1 - s _ { a - 1 } ) & = ( 2 n - b - 3 ) t _ 1 - t _ 2 - \\dots - t _ b + O ( \\geq 0 ) \\end{align*}"}
{"id": "964.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) & = c _ { n , s } \\lim _ { \\delta \\to 0 ^ + } \\int _ { \\R ^ n _ + \\setminus B _ \\delta ( x ) } \\left ( \\frac 1 { \\vert x - y \\vert ^ { n + 2 s } } - \\frac 1 { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } \\right ) \\big ( u ( x ) - u ( y ) \\big ) \\dd y \\\\ & + 2 c _ { n , s } u ( x ) \\int _ { \\R ^ n _ + } \\frac { \\dd y } { \\vert x _ \\ast - y \\vert ^ { n + 2 s } } . \\end{align*}"}
{"id": "4987.png", "formula": "\\begin{align*} H ( x , y ) = ( ( \\pi _ { 1 } A ^ { r _ 0 } ( \\cdot , y ) ) ^ { - 1 } ( x ) , y ) . \\end{align*}"}
{"id": "3824.png", "formula": "\\begin{align*} C _ { G \\rtimes S } ( \\underline { \\lambda } , \\sigma ) = \\langle { \\rm K e r } \\ , A _ \\sigma \\cap G , ( \\underline { \\lambda } , \\sigma ) \\rangle \\ , . \\end{align*}"}
{"id": "3013.png", "formula": "\\begin{align*} [ D _ { y ^ 4 z } \\cap D _ { y ^ 3 z ^ 2 } ] = [ \\ell _ y ] + 3 [ \\varphi _ { y ^ 4 z } ] - [ \\gamma _ { v y ^ 3 z , y ^ 4 z } ] - [ \\gamma _ { w y ^ 3 z , y ^ 4 z } ] - [ \\gamma _ { x y ^ 3 z , y ^ 4 z } ] , \\end{align*}"}
{"id": "6187.png", "formula": "\\begin{align*} N _ { : , t } = \\frac { M _ { : , j _ t } } { \\sqrt { p P ' _ { j _ t } } } , \\ \\ t \\in [ p ] . \\end{align*}"}
{"id": "6486.png", "formula": "\\begin{align*} M ^ { ( 2 m - 1 ) } _ n & : = E [ ( S _ n ) ^ { 2 m - 1 } ] , \\\\ f ^ { ( 2 m - 1 ) } _ n & : = { \\displaystyle \\sum _ { \\ell = 1 } ^ { m - 1 } \\left \\{ \\binom { 2 m - 1 } { 2 \\ell - 1 } + \\frac { \\alpha } { n } \\binom { 2 m - 1 } { 2 \\ell - 2 } \\right \\} M ^ { ( 2 \\ell - 1 ) } _ n } , \\\\ g ^ { ( 2 m - 1 ) } _ n & : = 1 + \\frac { ( 2 m - 1 ) \\alpha } { n } . \\end{align*}"}
{"id": "4401.png", "formula": "\\begin{align*} { L } f ( \\xi ) = \\frac { 1 } { \\xi ^ { d + 1 } \\Lambda Q ( \\xi ) } \\int _ 0 ^ \\xi f ( \\xi ' ) \\Lambda Q ( \\xi ' ) ( \\xi ' ) ^ { d + 1 } d \\xi ' . \\end{align*}"}
{"id": "6051.png", "formula": "\\begin{align*} t ^ * = ( 5 , 4 , 5 , 7 , 3 ) x ^ * = ( 9 , - 6 , - 3 ) \\enspace , \\end{align*}"}
{"id": "2117.png", "formula": "\\begin{align*} P _ { n , k } ( A ^ { ( n ) } _ { M , j , l } ) = \\frac { \\binom M l ( k ) _ l ( k ( j - 1 ) ) _ { M - l } } { ( k n ) _ M } . \\end{align*}"}
{"id": "4042.png", "formula": "\\begin{align*} \\begin{cases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = \\rho ( t , 1 ) & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ \\ u ( t , 1 ) = 0 & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
{"id": "1180.png", "formula": "\\begin{align*} s _ 0 = \\nu _ 0 = 1 , s _ j = \\frac { \\Gamma ( 3 j + \\frac { 1 } { 2 } ) } { 5 4 ^ j j ! \\Gamma ( j + \\frac { 1 } { 2 } ) } , \\nu _ j = \\frac { 6 j + 1 } { 1 - 6 j } s _ j , j \\geqslant 1 . \\end{align*}"}
{"id": "2824.png", "formula": "\\begin{align*} \\ddot { y } ( t ) = 1 6 ( s _ c ( p - 1 ) + 1 ) E [ u ] - 8 s _ c ( p - 1 ) \\| \\nabla u \\| _ 2 ^ 2 . \\end{align*}"}
{"id": "3946.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + b u _ x = 0 & ( 0 , + \\infty ) \\times ( 0 , \\delta L ) , \\\\ u _ t - u _ { x x } + u _ x + b \\rho _ x = 0 \\ & ( 0 , + \\infty ) \\times ( 0 , \\delta L ) , \\end{dcases} \\end{align*}"}
{"id": "3203.png", "formula": "\\begin{align*} h _ \\ell = \\sqrt { \\beta _ { s s } \\beta _ { s \\ell } } \\mathbf { h } _ { s s } ^ H \\mathbf { \\Theta } \\mathbf { h } _ { s \\ell } , \\end{align*}"}
{"id": "8444.png", "formula": "\\begin{align*} u ^ { - 1 } v = u ^ { - 1 } v c c ^ { - 1 } = u ^ { - 1 } u b c ^ { - 1 } = u ^ { - 1 } u b b ^ { - 1 } b c ^ { - 1 } = b b ^ { - 1 } u ^ { - 1 } u b c ^ { - 1 } , \\end{align*}"}
{"id": "7131.png", "formula": "\\begin{align*} ( K _ H ^ * y ) ( s ) = K _ H ( T , s ) y ( s ) + \\int _ s ^ T ( y ( t ) - y ( s ) ) \\frac { \\partial } { \\partial t } K _ H ( t , s ) d t \\end{align*}"}
{"id": "9077.png", "formula": "\\begin{align*} \\partial _ t \\rho = & \\nabla \\cdot \\left ( \\nabla \\rho + \\rho \\nabla \\phi \\right ) , \\\\ - \\Delta \\phi = & \\rho + f ( x , y ) , \\end{align*}"}
{"id": "2088.png", "formula": "\\begin{align*} \\widehat { \\mathbf { 1 } _ { P _ { \\textbf { c } } } } ( \\pmb { \\xi } ) = \\begin{cases} \\exp ( 2 \\pi i \\frac { \\pmb { \\xi } \\cdot \\textbf { c } } { p ^ { 2 k } } ) \\cdot \\frac { p ^ { w - k } } { p ^ { k n } } , & \\pmb { \\xi } \\equiv \\alpha \\textbf { D } _ { \\textbf { c } } ( p ^ k ) \\alpha \\in \\mathbb { Z } / p ^ { k } \\mathbb { Z } \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "2460.png", "formula": "\\begin{align*} S ^ { - 1 } = \\begin{pmatrix} D ^ T & - B ^ T \\\\ - C ^ T & A ^ T \\end{pmatrix} . \\end{align*}"}
{"id": "3137.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + 2 x _ 4 ^ 2 = 0 \\ , , x _ 4 ^ 2 ( x _ 2 ^ 2 + x _ 4 ^ 2 ) - x _ 3 ^ 4 = 0 \\ , . \\end{align*}"}
{"id": "2043.png", "formula": "\\begin{align*} T = A + B \\end{align*}"}
{"id": "6299.png", "formula": "\\begin{align*} f _ { Z _ i } ( z ) = \\frac { \\Gamma \\big ( 3 / 2 , - ( z ^ 2 / 2 + 1 ) \\ln ( 1 - \\xi ) \\big ) } { ( 1 - \\xi ) \\sqrt { 2 \\pi ( z ^ 2 / 2 + 1 ) ^ 3 } } , \\ \\forall i . \\end{align*}"}
{"id": "8142.png", "formula": "\\begin{align*} M ( p , \\{ 1 \\} ) = { \\pi ^ 2 \\over 6 } \\left ( 1 - { 1 \\over p } \\right ) \\left ( 1 - { 2 \\over p } \\right ) \\leq \\frac { \\pi ^ 2 } { 6 } \\ \\ \\ \\ \\ \\hbox { ( $ p \\geq 3 $ ) } . \\end{align*}"}
{"id": "8881.png", "formula": "\\begin{align*} \\psi _ t ( z _ 0 , \\ldots , z _ { q - 1 } ) = \\sum _ { i = 0 } ^ { q - 1 } ( - 1 ) ^ i \\varphi ( z _ 0 , \\ldots , z _ i , h _ t ( z _ i ) , \\ldots , h _ t ( z _ { q - 1 } ) ) . \\end{align*}"}
{"id": "1944.png", "formula": "\\begin{align*} \\underline { \\alpha _ 2 } = \\min \\ ! \\left ( \\alpha _ { 0 2 } , \\dfrac { 3 | 2 c _ 2 - 1 | \\varepsilon _ 2 } { M _ g } \\right ) . \\end{align*}"}
{"id": "3073.png", "formula": "\\begin{align*} & x _ 1 ^ 3 ( y _ 1 - y _ 2 ) ( y _ 1 - a y _ 2 ) ( y _ 1 - b y _ 2 ) + \\\\ + & x _ 2 ^ 3 ( y _ 1 + y _ 2 ) ( y _ 1 + a y _ 2 ) ( y _ 1 + b y _ 2 ) = 0 \\ , . \\end{align*}"}
{"id": "160.png", "formula": "\\begin{align*} \\mathcal { E } _ \\alpha ( f , g ) = \\int _ { 0 } ^ { + \\infty } x f ' ( x ) g ' ( x ) \\gamma _ { \\alpha , 1 } ( d x ) . \\end{align*}"}
{"id": "2367.png", "formula": "\\begin{align*} F ( x , \\omega ) = \\left ( V _ g f \\ , \\overline { V _ f g } \\right ) ( x , \\omega ) . \\end{align*}"}
{"id": "7076.png", "formula": "\\begin{align*} \\begin{array} { c l } \\langle \\bar R ( X , Y ) Z , V \\rangle = & \\langle \\nabla ^ { \\perp } _ X \\sigma ( Y , Z ) , V \\rangle - \\langle \\sigma ( \\nabla _ X Y , Z ) , V \\rangle - \\langle \\sigma ( Y , \\nabla _ X Z ) , V \\rangle \\\\ \\\\ & - \\langle \\nabla ^ { \\perp } _ Y \\sigma ( X , Z ) , V \\rangle + \\langle \\sigma ( \\nabla _ Y X , Z ) , V \\rangle + \\langle \\sigma ( X , \\nabla _ Y Z ) , V \\rangle . \\end{array} \\end{align*}"}
{"id": "8673.png", "formula": "\\begin{align*} 1 - F _ n ( \\mu _ n + s _ n y ) \\sim 1 - \\Phi ( y ) , y = o ( N _ n ^ { 1 / 6 } ) . \\end{align*}"}
{"id": "3346.png", "formula": "\\begin{align*} [ x , y , z ] _ N = & [ N x , N y , z ] + [ x , N y , N z ] + [ N x , y , N z ] \\\\ - & N \\Big ( [ N x , y , z ] + [ x , N y , z ] + [ x , y , N z ] - N [ x , y , z ] \\Big ) , \\end{align*}"}
{"id": "8157.png", "formula": "\\begin{align*} M _ { q _ 1 , q _ 2 } ( p ) : = \\frac { 2 } { \\phi ( p ) } \\sum _ { \\chi \\in X _ p ^ - } \\chi ( q _ 1 ) \\overline { \\chi } ( q _ 2 ) \\vert L ( 1 , \\chi ) \\vert ^ 2 . \\end{align*}"}
{"id": "4471.png", "formula": "\\begin{align*} f ( t ) & = \\langle z | e _ { \\lambda } ^ { a ^ { \\dagger } a } ( t ) | z \\rangle = \\sum _ { k = 0 } ^ { \\infty } \\frac { t ^ { k } } { k ! } \\langle z | ( a ^ { \\dagger } a ) _ { k , \\lambda } | z \\rangle \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\frac { t ^ { k } } { k ! } \\sum _ { l = 0 } ^ { k } S _ { 2 , \\lambda } ( k , l ) | z | ^ { 2 l } = \\sum _ { k = 0 } ^ { \\infty } \\phi _ { k , \\lambda } ( | z | ^ { 2 } ) \\frac { t ^ { k } } { k ! } . \\end{align*}"}
{"id": "290.png", "formula": "\\begin{align*} \\psi ( x , t ) : = u ( x , t ) - \\chi ( x , t ) , \\psi _ { 0 } ( x ) : = u _ { 0 } ( x ) - \\chi _ { * } ( x ) . \\end{align*}"}
{"id": "2483.png", "formula": "\\begin{align*} A ( f , g ) ( \\l ) = \\langle \\pi ( - \\l / 2 ) f , \\pi ( \\l / 2 ) g \\rangle = \\langle f , \\rho ( \\l ) g \\rangle . \\end{align*}"}
{"id": "9346.png", "formula": "\\begin{align*} ( x ) _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 8 ] ) . \\end{align*}"}
{"id": "7954.png", "formula": "\\begin{align*} \\pi = w _ { 1 1 } \\cdot t _ { 7 } ^ { 2 } \\cdot t _ { 9 } ^ { 2 } \\cdot t _ { 1 2 } ^ { 3 } | w _ { 1 4 } \\cdot w _ { 1 5 } \\cdot t _ { 1 2 } \\cdot t _ { 1 4 } ^ { 7 } \\cdot t _ { 1 6 } ^ { 4 } | w _ { 2 0 } \\cdot t _ { 1 9 } ^ { 1 0 } \\cdot t _ { 2 2 } ^ { 5 } \\cdot t _ { 2 3 } ^ { 4 } \\end{align*}"}
{"id": "1210.png", "formula": "\\begin{align*} E _ { U } = U \\cap E _ { \\mu } ^ { [ \\alpha , \\beta ] , \\varepsilon _ 2 } \\cap \\limsup _ { n \\to + \\infty } B _ n . \\end{align*}"}
{"id": "494.png", "formula": "\\begin{align*} Y _ { \\alpha \\beta } ^ { i j } \\cdot ( Y _ { \\alpha \\omega } ^ { m i } Y _ { \\beta \\omega } ^ { m j } ) = - ( Y _ { \\alpha \\omega } ^ { m j } Y _ { \\beta \\omega } ^ { m i } ) . \\end{align*}"}
{"id": "4464.png", "formula": "\\begin{align*} \\langle x | y \\rangle & = e ^ { - \\frac { | x | ^ { 2 } } { 2 } } \\sum _ { m = 0 } ^ { \\infty } \\frac { ( \\overline { x } ) ^ { m } } { \\sqrt { m ! } } e ^ { - \\frac { | y | ^ { 2 } } { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\frac { y ^ { n } } { \\sqrt { n } ! } \\langle m | n \\rangle \\\\ & = e ^ { - \\frac { | x | ^ { 2 } } { 2 } - \\frac { | y | ^ { 2 } } { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\overline { x } y ) ^ { n } } { n ! } = e ^ { - \\frac { 1 } { 2 } ( | x | ^ { 2 } + | y | ^ { 2 } ) + \\overline { x } y } . \\end{align*}"}
{"id": "5403.png", "formula": "\\begin{align*} \\pi _ { a , b } ( x ) = \\left ( \\int _ { a } ^ { b } m ( x , \\theta ) \\mathrm { d } x \\right ) ^ { - 1 } = \\frac { e ^ { - \\frac { 2 } { \\sigma ^ { 2 } } \\int _ { a } ^ { x } f ( y , \\theta ) \\mathrm { d } y } } { \\int _ { a } ^ { b } e ^ { - \\frac { 2 } { \\sigma ^ { 2 } } \\int _ { a } ^ { x } f ( y , \\theta ) \\mathrm { d } y } \\mathrm { d } x } . \\end{align*}"}
{"id": "1158.png", "formula": "\\begin{align*} m ^ { ( 2 ) } _ { + } ( x , t , k ) = m ^ { ( 2 ) } _ { - } ( x , t , k ) J ^ { ( 2 ) } ( x , t , k ) , k \\in \\Gamma ^ { ( 2 ) } , \\end{align*}"}
{"id": "2160.png", "formula": "\\begin{align*} ( - \\triangle _ { g } ) ^ { \\alpha } u ( x ) & = \\int _ { \\mathbb { R } ^ { d } } g \\bigg { ( } \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \\alpha } } \\bigg { ) } \\frac { u ( x ) - u ( y ) } { | u ( x ) - u ( y ) | } \\frac { d y } { | x - y | ^ { d + \\alpha } } , \\end{align*}"}
{"id": "601.png", "formula": "\\begin{align*} f ( 0 , x _ 2 , \\ldots , x _ m ) \\ = \\ g ( x _ 2 , \\ldots , x _ m ) \\end{align*}"}
{"id": "3882.png", "formula": "\\begin{align*} \\beta _ { n - 1 } = t _ { n - 1 } - t _ n & = \\boxed { ( t _ { n - 1 } - s _ n ) } + ( s _ n - t _ n ) \\\\ & = \\boxed { ( t _ { n - 1 } + s _ n ) } + ( - s _ n - t _ n ) \\\\ & = ( t _ { n - 1 } - s _ { n - 1 } ) + \\boxed { ( s _ { n - 1 } - t _ n ) } . \\end{align*}"}
{"id": "5160.png", "formula": "\\begin{align*} \\vartheta _ 3 ( z , \\tau ) = \\sum _ { k \\in \\Z } e ^ { \\pi i \\tau k ^ 2 } e ^ { 2 \\pi i k z } , \\ , z \\in \\C , \\tau \\in \\mathbb { H } . \\end{align*}"}
{"id": "2228.png", "formula": "\\begin{align*} X ( t ) = E ( t ) X _ 0 - \\int _ 0 ^ t E ( t - s ) A P F ( X ( s ) ) \\ , \\mathrm { d } s + \\int _ 0 ^ t E ( t - s ) \\mathrm { d } W ( s ) . \\end{align*}"}
{"id": "1065.png", "formula": "\\begin{align*} q | _ { \\mathcal { R } _ { \\xi , I V } } ( x , t ) = C _ R + \\mathcal { O } ( t ^ { - 1 / 2 } e ^ { - 1 6 t \\xi ^ { 3 / 2 } } ) . \\end{align*}"}
{"id": "7677.png", "formula": "\\begin{align*} \\frac { \\int _ { \\tilde { \\Omega } _ { \\varepsilon } } | \\nabla P _ { \\tilde { \\Omega } _ { \\varepsilon } } | ^ 2 } { \\int _ { \\tilde { \\Omega } _ { \\varepsilon } } \\tilde { m } _ 0 \\ , | P _ { \\tilde { \\Omega } _ { \\varepsilon } } | ^ 2 } = \\tilde { \\lambda } _ 0 + \\Phi e ^ { - \\beta _ { \\varepsilon } \\tilde { \\Psi } _ { \\varepsilon } ( \\mathbf { x } _ { \\varepsilon } ) } + o ( e ^ { - \\beta _ { \\varepsilon } \\tilde { \\Psi } _ { \\varepsilon } ( \\mathbf { x } _ { \\varepsilon } ) } ) . \\end{align*}"}
{"id": "5022.png", "formula": "\\begin{align*} \\lambda _ { Z , n } = \\lambda _ { Z } \\cdot \\lambda _ { \\mathcal { R } Z } \\cdot \\ldots \\cdot \\lambda _ { \\mathcal { R } ^ { n - 1 } Z } . \\end{align*}"}
{"id": "6075.png", "formula": "\\begin{align*} \\begin{cases} A & = ( L R \\times \\overline { L } R ) \\cdot ( L ^ { 2 } R \\times \\overline { L } R ) , \\\\ B & = ( L R \\times \\overline { L } R ) \\cdot ( L ^ { 2 } R \\times L R ) , \\\\ C & = ( L R \\times \\overline { L } R ) \\cdot ( L R \\times \\overline { L } R ) . \\end{cases} \\end{align*}"}
{"id": "6541.png", "formula": "\\begin{align*} 0 = E _ { 0 } \\subset E _ { 1 } \\subset . . . . \\subset E _ { n - 1 } \\subset E _ n = E , \\end{align*}"}
{"id": "5804.png", "formula": "\\begin{align*} x _ { m , n ' } [ t ] = \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\tilde { x } _ { m , n } [ t ] e ^ { \\frac { 2 \\pi j n ' n } { N } } , \\ : \\ : \\ : \\ : \\forall n ' . \\end{align*}"}
{"id": "1763.png", "formula": "\\begin{align*} \\begin{aligned} & b \\Phi ( a _ 0 \\otimes \\dots \\otimes a _ { k + 1 } ) \\\\ = & \\sum _ { i = 0 } ^ k ( - 1 ) ^ i \\Phi ( a _ 0 \\otimes \\dots \\otimes a _ i a _ { i + 1 } \\otimes \\dots \\otimes a _ { k + 1 } ) + ( - 1 ) ^ { k + 1 } \\Phi ( a _ { k + 1 } a _ 0 \\otimes a _ 1 \\otimes \\dots \\otimes a _ { k } ) . \\end{aligned} \\end{align*}"}
{"id": "1152.png", "formula": "\\begin{align*} q ( x , t ) = 2 i D ^ { - 2 } _ { \\infty } ( \\xi ) \\left [ E _ { 1 } ^ { ( 1 2 ) } + \\lim _ { k \\rightarrow \\infty } k \\Delta _ { \\eta } ^ { ( 1 2 ) } ( k ) \\right ] . \\end{align*}"}
{"id": "2474.png", "formula": "\\begin{align*} \\L _ I = I \\Z ^ { 2 d } \\text a n d \\ \\L _ { 2 , 3 } = P _ { 2 , 3 } \\Z ^ { 2 d } \\end{align*}"}
{"id": "5214.png", "formula": "\\begin{align*} \\bigg | \\bigg ( \\bigcup _ i W _ i \\bigg ) \\bigcap \\{ x \\in K : | B f _ 1 ( x ) | > \\lambda / 2 \\} \\bigg | & ~ = ~ \\bigg | \\bigcup _ i W _ i \\cap \\{ x \\in K : | B f _ 1 ( x ) | > \\lambda / 2 \\} \\bigg | \\\\ & ~ \\leq ~ \\bigg | \\bigcup _ i W _ i \\bigg | \\\\ & ~ \\leq ~ \\sum \\limits _ i | W _ i | \\\\ & ~ \\leq ~ \\| f \\| _ { L ^ 1 ( K ) } ~ \\lambda ^ { - 1 } . \\end{align*}"}
{"id": "6406.png", "formula": "\\begin{align*} \\Omega = \\{ ( z _ 1 , \\dots , z _ n ) \\in \\mathbb D ^ n \\ , : \\ , f _ i ( z _ 1 , \\dots , z _ n ) = 0 \\ , , \\ , i = 1 , \\dots , n \\} . \\end{align*}"}
{"id": "4910.png", "formula": "\\begin{align*} \\Gamma _ t = \\prod _ { i = 1 } ^ { t } ( 1 - \\alpha _ i ) = \\prod _ { i = 1 } ^ { t } ( 1 - \\frac { i } { A _ i } ) = \\prod _ { i = 1 } ^ { t } \\frac { A _ { i - 1 } } { A _ i } = \\frac { 1 } { A _ t } \\\\ \\end{align*}"}
{"id": "4140.png", "formula": "\\begin{align*} u ( t ) = U ( t ) \\phi - \\int _ 0 ^ t U ( t - \\tau ) \\vartheta ( \\tau ) d \\tau , \\end{align*}"}
{"id": "2771.png", "formula": "\\begin{align*} U ^ 1 ( x ) = \\lambda _ 2 Q ( \\mu _ 2 x + x _ 2 ) , \\end{align*}"}
{"id": "1958.png", "formula": "\\begin{align*} x ^ n \\triangleleft x ^ m = ( n + 1 ) x ^ { n + m } , \\end{align*}"}
{"id": "2723.png", "formula": "\\begin{align*} U ( w ) T ( w ) = V ( w ) S ( w ) . \\end{align*}"}
{"id": "897.png", "formula": "\\begin{align*} a ( \\widetilde E _ 2 ^ { ( 3 ) } , T ) = \\frac { 5 7 6 } { ( 1 - p ) ^ 2 } \\prod _ { q \\not = p } F _ q ( T , q ^ { - 2 } ) , \\end{align*}"}
{"id": "6953.png", "formula": "\\begin{align*} g ( k ) = \\alpha \\ \\Leftrightarrow \\ \\exists t ( \\forall s \\geq t ) ( g _ s ( k ) = \\alpha ) . \\end{align*}"}
{"id": "1784.png", "formula": "\\begin{align*} H = ( H _ 1 , \\dots , H _ m ) : G \\to \\mathfrak { a } . \\end{align*}"}
{"id": "8202.png", "formula": "\\begin{align*} S ( H _ 3 , 3 f ) = \\frac { 5 f + c _ a '' } { 1 8 } \\hbox { w h e r e } c _ a '' : = \\begin{cases} - 8 a - 5 & \\hbox { i f $ a \\equiv 0 \\pmod 3 $ } , \\\\ 8 a + 3 & \\hbox { i f $ a \\equiv 2 \\pmod 3 $ } , \\end{cases} \\end{align*}"}
{"id": "5007.png", "formula": "\\begin{align*} | I _ { k n } | | \\bar u _ { k n } | = | I _ { k n } | ( q _ { k n } + p _ { k n } ) = { q _ { k n } + p _ { k n } \\over R _ { k n } q _ { k n - 1 } + q _ { k n - 2 } } = B _ { k n } . \\end{align*}"}
{"id": "3567.png", "formula": "\\begin{align*} \\frac { g - 1 } { | G | } = b - 1 \\ge \\Big [ \\frac { { \\rm r k } ( G ) + 1 } { 2 } \\Big ] - 1 . \\end{align*}"}
{"id": "550.png", "formula": "\\begin{align*} p ( \\rho ) - p ( r ) - p ' ( r ) ( \\rho - r ) \\leq \\begin{cases} c ( \\rho - r ) ^ 2 , & \\rho \\in ( \\alpha _ 1 , \\bar \\rho - \\alpha _ 1 ) , \\\\ 1 + p ' ( r ) r - p ( r ) , & \\rho \\in [ 0 , \\alpha _ 1 ] , \\\\ 2 p ( \\rho ) , & \\rho \\in [ \\bar \\rho - \\alpha _ 1 , \\bar \\rho ) . \\end{cases} \\end{align*}"}
{"id": "8007.png", "formula": "\\begin{align*} \\phi ( u , v ) = \\phi _ \\ell ( u ) + \\phi _ r ( v ) + \\frac { p } { 2 \\pi } ( u + v ) , \\end{align*}"}
{"id": "5050.png", "formula": "\\begin{align*} \\mu _ h : = \\dd c \\otimes \\P _ h \\end{align*}"}
{"id": "8838.png", "formula": "\\begin{align*} 0 \\geq \\sum _ { i = 2 } ^ { t + 1 } ( k ' - 2 l _ { i } - e _ { F _ { 0 } } ( X _ { i } , S ) ) = k ' t - 2 \\sum _ { i = 2 } ^ { t + 1 } l _ { i } - \\sum _ { i = 2 } ^ { t + 1 } e _ { F _ { 0 } } ( X _ { i } , S ) . \\end{align*}"}
{"id": "6029.png", "formula": "\\begin{align*} \\widetilde { \\mathfrak { L } } _ \\alpha f & : = \\Big ( \\sqrt { x } \\Big ( \\frac { d } { d x } - \\Big ( \\frac { \\alpha - 1 / 2 } { x } - 1 \\Big ) \\Big ) \\Big ( \\sqrt { x } \\frac { d f } { d x } \\Big ) \\\\ & = x \\frac { d ^ 2 f } { d x ^ 2 } + ( - \\alpha + x + 1 ) \\frac { d f } { d x } . \\end{align*}"}
{"id": "4933.png", "formula": "\\begin{align*} \\gamma _ 1 \\colon & = - \\limsup _ { n \\to \\infty } \\frac { \\max _ { Q \\in \\mathcal { Q } _ n } \\log _ { b ^ { - 1 } } P _ n ( Q ) } { n } , \\\\ \\gamma _ 2 \\colon & = - \\limsup _ { n \\to \\infty } \\frac { \\min _ { Q \\in \\mathcal { Q } _ n } \\log _ { b ^ { - 1 } } P _ n ( Q ) } { n } . \\end{align*}"}
{"id": "1548.png", "formula": "\\begin{align*} j ( g _ { \\infty } , z _ 0 ) ^ { - k } \\delta ( z ) ^ { \\frac { s - k } { 2 } } \\int _ { \\mathfrak { Z } } \\delta ( w , z ) ^ { - k } | \\delta ( w , z ) | ^ { k - s } \\delta ( w ) ^ { \\frac { k + s } { 2 } } f ( w ) d w = \\end{align*}"}
{"id": "1928.png", "formula": "\\begin{align*} & \\int ^ { T } _ 0 \\int _ { \\Omega } ( \\rho _ { N } \\partial _ t \\psi + \\rho _ { N } u _ { N } \\cdot \\nabla \\psi - \\varepsilon \\nabla \\rho _ { N } \\cdot \\nabla \\psi ) \\ , d x d t + \\int _ { \\Omega } \\rho _ 0 \\psi ( 0 , x ) \\ , d x \\\\ & \\qquad \\qquad \\qquad = \\int _ 0 ^ T \\int _ { \\Gamma _ { \\rm { i n } } } \\rho _ B u _ B \\cdot \\nu ( x ) \\psi \\ , d \\sigma ( x ) d t ; \\end{align*}"}
{"id": "8843.png", "formula": "\\begin{align*} & \\sum _ { i , j \\geq 1 } \\frac { \\omega _ { i , j } ^ { [ 0 ] } } { \\lambda ^ { i + 1 } \\mu ^ { j + 1 } } = \\frac { B ( \\lambda ) B ( \\mu ) ( ( \\lambda - v ) ( \\mu - v ) - 4 w ) - 1 } { 2 \\ , ( \\lambda - \\mu ) ^ 2 } , \\\\ & \\frac 1 \\lambda + \\sum \\limits _ { i \\geq 1 } \\frac { \\varphi _ i ^ { [ 0 ] } } { \\lambda ^ { i + 1 } } = B ( \\lambda ) , \\end{align*}"}
{"id": "7609.png", "formula": "\\begin{align*} f - \\frac { \\partial f } { \\partial \\kappa _ { i } } \\kappa _ { i } = H _ { k } ^ { \\frac { 1 } { k } } \\left ( 1 - \\frac { \\sigma _ { k - 1 } ( \\kappa | i ) \\kappa _ { i } } { k \\sigma _ { k } ( \\kappa ) } \\right ) \\geq \\frac { 1 } { k } H _ { k } ^ { \\frac { 1 } { k } } \\left ( 1 - \\frac { \\sigma _ { k - 1 } ( \\kappa | i ) \\kappa _ { i } } { \\sigma _ { k } ( \\kappa ) } \\right ) . \\end{align*}"}
{"id": "8515.png", "formula": "\\begin{align*} [ x ^ e , g ] _ { \\circ } = [ x , g ] _ { \\circ } ^ e = 0 . \\end{align*}"}
{"id": "6945.png", "formula": "\\begin{align*} \\eta _ { \\rm r e s } ^ 2 & = \\sum _ { E \\in { \\cal T } _ h } \\eta _ { \\rm r e s } ^ 2 ( E ) \\ , , \\eta _ { \\rm c o e f } ^ 2 = \\sum _ { E \\in { \\cal T } _ h } \\sum _ { k = 1 } ^ 6 \\eta _ { { \\rm c o e f } , k } ^ 2 ( E ) \\ , , \\eta _ { \\rm r h s } ^ 2 = \\sum _ { E \\in { \\cal T } _ h } \\sum _ { k = 1 } ^ 2 \\eta _ { { \\rm r h s } , k } ^ 2 ( E ) \\ , . \\end{align*}"}
{"id": "6112.png", "formula": "\\begin{align*} H _ j ( X ^ { m + 2 } ) = \\frac { 1 } { A _ j ^ n } D ^ j \\left [ h ( X ^ { m + 2 } ) \\right ] \\ , . \\end{align*}"}
{"id": "1971.png", "formula": "\\begin{align*} \\Lambda ( \\phi \\prec \\gamma ) = \\sum _ { \\substack { u _ 1 , \\dotsc , u _ k \\in \\{ \\ 1 \\} \\cup \\N ^ * \\\\ v = i _ 1 \\dotsm i _ k \\in \\N ^ * } } f _ v g _ { u _ 1 } \\dotsm g _ { u _ k } x _ { i _ 1 } x _ { u _ 1 } \\dotsm x _ { i _ k } x _ { u _ k } = f ( x g ( x ) ) , \\end{align*}"}
{"id": "5615.png", "formula": "\\begin{align*} \\mathbf { g } _ t ^ \\mathcal { F } : = ( \\mathrm { p r } _ { t \\star } \\otimes _ { \\mathcal { X } _ \\star } \\mathrm { p r } _ { t \\star } ) ( \\mathbf { g } ) = ( \\mathrm { p r } _ { t } \\otimes _ { \\mathcal { X } } \\mathrm { p r } _ { t } ) ( \\mathbf { g } ) = \\mathbf { g } _ t \\end{align*}"}
{"id": "5434.png", "formula": "\\begin{align*} \\begin{gathered} \\phi _ k : = \\begin{bmatrix} - y _ { k - 1 } & - y _ { k - 2 } & u _ { k - 1 } & u _ { k - 2 } \\end{bmatrix} ^ \\top , \\\\ \\theta ^ * _ 1 : = \\begin{bmatrix} 0 . 5 & - 0 . 1 & 1 & - 0 . 4 \\end{bmatrix} ^ \\top , \\\\ \\theta ^ * _ 2 : = \\begin{bmatrix} - 1 . 4 & 0 . 3 & 1 & - 1 . 3 \\end{bmatrix} ^ \\top , \\end{gathered} \\end{align*}"}
{"id": "7690.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ 0 = \\frac { 1 } { \\int _ { \\R ^ N } m u ^ 2 } \\ ; . \\end{align*}"}
{"id": "202.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\alpha ( f ) ( x ) = - \\langle x ; \\nabla ( f ) ( x ) \\rangle + \\sum _ { j = 1 } ^ d \\partial _ j D ^ { \\alpha - 1 } _ j ( f ) ( x ) , \\end{align*}"}
{"id": "7832.png", "formula": "\\begin{align*} \\langle y \\rangle _ { T ^ { * 3 } } & = ~ \\{ y , T ^ { * 3 } y , \\ldots , T ^ { * 3 ( j - 2 ) } y \\} ~ , ~ ~ ~ \\langle y \\rangle _ { T ^ { * 3 } } = j - 1 , \\\\ \\langle z \\rangle _ { T ^ { * 3 } } & = ~ \\{ z , T ^ { * 3 } z , \\ldots , T ^ { * 3 ( n - 2 j - 1 ) } z \\} ~ , ~ ~ ~ \\langle z \\rangle _ { T ^ { * 3 } } = n - 2 j , \\end{align*}"}
{"id": "1036.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac 1 h \\bigg ( \\frac 1 { \\vert h e _ 1 - y \\vert ^ n } - \\frac 1 { \\vert h e _ 1 + y \\vert ^ n } \\bigg ) & = \\frac { 2 n y _ 1 } { \\vert y \\vert ^ { n + 2 } } . \\end{align*}"}
{"id": "3339.png", "formula": "\\begin{align*} & f ( x _ 1 , x _ 1 , x _ 2 ) = 0 , \\\\ & f ( x _ 1 , x _ 2 , x _ 3 ) + f ( x _ 2 , x _ 3 , x _ 1 ) + f ( x _ 3 , x _ 1 , x _ 2 ) = 0 , \\\\ & f ( x _ 1 , x _ 2 , [ y _ 1 , y _ 2 , y _ 3 ] ) + D ( x _ 1 , x _ 2 ) f ( y _ 1 , y _ 2 , y _ 3 ) \\\\ & = f ( [ x _ 1 , x _ 2 , y _ 1 ] , y _ 2 , y _ 3 ) + f ( y _ 1 , [ x _ 1 , x _ 2 , y _ 2 ] , y _ 3 ) + f ( y _ 1 , y _ 2 , [ x _ 1 , x _ 2 , y _ 3 ] ) \\\\ & + \\theta ( y _ 2 , y _ 3 ) f ( x _ 1 , x _ 2 , y _ 1 ) - \\theta ( y _ 1 , y _ 3 ) f ( x _ 1 , x _ 2 , y _ 2 ) + D ( y _ 1 , y _ 2 ) f ( x _ 1 , x _ 2 , y _ 3 ) . \\end{align*}"}
{"id": "1949.png", "formula": "\\begin{align*} H & = \\nabla ^ 2 f ( x ) - \\sum _ { i = 1 } ^ { m } \\lambda _ i ( x ) \\nabla ^ 2 h _ i ( x ) . \\end{align*}"}
{"id": "5194.png", "formula": "\\begin{align*} T _ \\Omega f ( x ) = \\lim _ { \\epsilon \\to 0 } \\int _ { | y | > \\epsilon } \\dfrac { \\Omega ( y / | y | ) } { | y | ^ n } f ( x - y ) d y . \\end{align*}"}
{"id": "4361.png", "formula": "\\begin{align*} \\partial _ \\xi ^ k \\Theta _ i ( \\xi ) = O \\left ( \\xi ^ { - \\gamma + 2 i - k - 2 } \\ln \\xi \\right ) \\xi \\to + \\infty . \\end{align*}"}
{"id": "4049.png", "formula": "\\begin{align*} \\begin{dcases} \\rho _ t + \\rho _ x + u _ { x } = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { t } - u _ { x x } + u _ x + \\rho _ x = 0 & ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\rho ( t , 0 ) = 0 & t \\in ( 0 , T ) , \\\\ u ( t , 0 ) = 0 , \\ u ( t , 1 ) = q ( t ) & t \\in ( 0 , T ) , \\\\ \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\ u ( 0 , x ) = u _ 0 ( x ) & x \\in ( 0 , 1 ) . \\end{dcases} \\end{align*}"}
{"id": "570.png", "formula": "\\begin{align*} \\big \\{ ( x , y ) \\in \\ \\R ^ 2 \\ y ^ 2 = x , y \\geq 0 \\big \\} , \\end{align*}"}
{"id": "285.png", "formula": "\\begin{align*} \\| V ( \\cdot , t ) \\| _ { L ^ { \\infty } } = | \\kappa d | \\| V _ { * } ( \\cdot ) \\| _ { L ^ { \\infty } } ( 1 + t ) ^ { - 1 } \\log ( 1 + t ) . \\end{align*}"}
{"id": "873.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( V _ j = i \\right ) = \\frac { \\epsilon _ { i - 1 } - \\epsilon _ i } { 1 - \\epsilon _ m } . \\end{align*}"}
{"id": "9326.png", "formula": "\\begin{align*} \\rho = \\max \\{ \\frac { 1 } { \\max \\{ \\| A \\| _ { \\infty } , \\| H \\| _ { \\infty } \\} } , 1 0 ^ { - 1 0 } \\} \\end{align*}"}
{"id": "25.png", "formula": "\\begin{gather*} [ x _ { i , r + 1 } ^ \\pm , x _ { j s } ^ \\pm ] - [ x _ { i r } ^ \\pm , x _ { j , s + 1 } ^ \\pm ] = \\pm \\hbar d _ { i j } ( x _ { i r } ^ \\pm x _ { j s } ^ \\pm + x _ { j s } ^ \\pm x _ { i r } ^ \\pm ) \\\\ \\sum _ { \\pi \\in S _ { m } } \\left [ x _ { i , r _ { \\pi ( 1 ) } } ^ { \\pm } , \\left [ x _ { i , r _ { \\pi ( 2 ) } } ^ { \\pm } , \\cdots , \\left [ x _ { i , r _ { \\pi ( m ) } } ^ { \\pm } , x _ { j s } ^ { \\pm } \\right ] \\cdots \\right ] \\right ] = 0 , \\end{gather*}"}
{"id": "2848.png", "formula": "\\begin{align*} | \\dot { y } _ R ( t ) | = \\Big | 2 R \\Im \\int \\nabla \\varphi \\left ( \\frac { x } { R } \\right ) \\cdot \\nabla u \\bar { u } d x \\Big | \\le R \\| \\nabla \\varphi \\| _ \\infty \\| \\nabla u \\| _ 2 \\| u \\| _ 2 \\le C R t > 0 . \\end{align*}"}
{"id": "5272.png", "formula": "\\begin{align*} \\sigma ^ { \\varphi _ S } ( a ) = \\delta _ { \\varphi } \\sigma ^ { \\varphi } ( a ) \\delta _ { \\varphi } ^ { - 1 } \\end{align*}"}
{"id": "8858.png", "formula": "\\begin{align*} S = \\{ t \\in T \\mid d ( t _ 0 , t ) \\in \\N _ 0 \\} . \\end{align*}"}
{"id": "7396.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho ( t , u ) = [ - ( - \\Delta ) ^ { \\gamma / 2 } \\rho ^ 2 ] ( t , u ) , & ( t , u ) \\in [ 0 , T ] \\times \\mathbb { R } , \\\\ \\rho ( 0 , u ) = g ( u ) , & u \\in \\mathbb { R } , \\end{cases} \\end{align*}"}
{"id": "8352.png", "formula": "\\begin{align*} \\Lambda ( \\rho ) = \\int _ 0 ^ \\infty \\exp \\left ( - \\rho y ^ { - 1 } \\right ) \\ , q ( y ) \\ , d y \\sim b \\ , \\rho ^ { - ( \\alpha - 1 ) } , \\textrm { a s } \\rho \\to \\infty , \\end{align*}"}
{"id": "8141.png", "formula": "\\begin{align*} M ( p , H ) = \\frac { \\pi ^ 2 } { 6 } + o ( 1 ) \\end{align*}"}
{"id": "4066.png", "formula": "\\begin{align*} H ( \\rho _ 0 , u _ 0 , p ) = ( \\rho , u ) . \\end{align*}"}
{"id": "5510.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { U _ 0 } : = \\langle Q ^ { - 1 / 2 } u , Q ^ { - 1 / 2 } v \\rangle _ U \\end{align*}"}
{"id": "8410.png", "formula": "\\begin{align*} \\nabla _ v f ( x ) & = \\left \\langle \\nabla f ( x ) , v \\right \\rangle = \\lim _ { \\varepsilon \\to 0 } \\frac { f ( x + \\varepsilon v ) - f ( x ) } { \\varepsilon } , \\\\ \\nabla _ { v _ 2 } \\nabla _ { v _ 1 } f ( x ) & = \\left \\langle \\nabla ^ 2 f ( x ) , v _ 1 v ^ \\top _ 2 \\right \\rangle _ { \\mathrm { H S } } = \\lim _ { \\varepsilon \\to 0 } \\frac { \\nabla _ { v _ 1 } f ( x + \\varepsilon v _ 2 ) - \\nabla _ { v _ 1 } f ( x ) } { \\varepsilon } , \\end{align*}"}
{"id": "6169.png", "formula": "\\begin{align*} ( s , t , u , v ) & = \\left ( \\frac { - 2 8 - \\sqrt { - 1 1 } } { 3 0 } , - \\frac { \\sqrt { 1 2 7 - 5 6 \\sqrt { - 1 1 } } } { 3 0 } , \\frac { - 2 8 + \\sqrt { - 1 1 } } { 3 0 } , \\frac { \\sqrt { 1 2 7 + 5 6 \\sqrt { - 1 1 } } } { 3 0 } \\right ) , \\\\ \\end{align*}"}
{"id": "6840.png", "formula": "\\begin{align*} \\Phi ( S ) = \\underset { v \\sim S } { \\mathbb { E } } \\left [ M ( v , X ( k ) \\setminus S ) \\right ] , \\end{align*}"}
{"id": "8657.png", "formula": "\\begin{align*} d = q ( u , c ) \\end{align*}"}
{"id": "7663.png", "formula": "\\begin{align*} \\begin{cases} - k _ { \\varepsilon } \\Delta \\Psi _ { \\varepsilon } + | \\nabla \\Psi _ { \\varepsilon } | ^ 2 - \\tilde { \\lambda } _ 0 \\underline { m } = 0 & \\Omega \\\\ \\Psi _ { \\varepsilon } = - k _ { \\varepsilon } \\log ( w ) ( \\frac { \\mathbf { x } - \\mathbf { q } } { k _ { \\varepsilon } } ) & \\partial \\Omega \\end{cases} \\end{align*}"}
{"id": "1702.png", "formula": "\\begin{align*} g _ 0 ( x ) = { \\rm d i s t } ^ { r \u2010 \\beta } ( x , \\ , \\Gamma ) \\end{align*}"}
{"id": "7109.png", "formula": "\\begin{align*} & \\sigma \\tau | _ E \\sigma ^ { - 1 } ( a , b , c ) \\\\ & = \\left ( a + ( 1 - e ) b , e b , - 3 \\frac { ( a + b ) ^ 2 } { e b } + 3 ( a + b ) - e b + e ^ { - 2 } \\left ( 3 \\frac { a ^ 2 } { b } + 3 a + b + c \\right ) \\right ) . \\end{align*}"}
{"id": "8745.png", "formula": "\\begin{align*} M : = \\biggl \\{ ( \\lambda , w ) \\biggm | \\lambda \\in [ 0 , 1 ] ^ d , \\ w _ I = \\prod _ { i \\in I } \\lambda _ i , \\ I \\subseteq \\{ 1 , \\ldots , d \\} , \\ \\lvert I \\rvert \\geq 2 \\biggr \\} . \\end{align*}"}
{"id": "5296.png", "formula": "\\begin{align*} \\kappa ^ n ( a ) \\otimes \\rho ^ { n } ( b ) = \\sum _ { i = 1 } ^ n \\Delta ( p _ i ) ( 1 \\otimes \\rho ^ n ( q _ i ) ) \\end{align*}"}
{"id": "7742.png", "formula": "\\begin{align*} \\eta _ { \\alpha } ^ { \\beta } [ \\tau ] ( t ) : = \\begin{cases} \\mu ( \\frac { t - \\beta } { \\tau } ) , \\ ; \\ ; \\forall t \\in ( \\beta , + \\infty ) , \\\\ 1 , \\ ; \\ ; \\forall t \\in [ \\alpha , \\beta ] , \\\\ \\mu ( \\frac { t - \\alpha } { \\tau } ) , \\ ; \\ ; \\forall t \\in ( - \\infty , \\alpha ) , \\end{cases} \\end{align*}"}
{"id": "2472.png", "formula": "\\begin{align*} \\L = r S \\Z ^ { 2 d } , S \\in S p ( \\R , 2 d ) \\end{align*}"}
{"id": "4614.png", "formula": "\\begin{align*} \\sum _ { \\substack { F \\in \\mathcal { F } \\\\ { G \\in \\mathcal { G } } } } \\varphi ( F , G ) \\leq | \\mathcal { G } | = f _ k ( P ) , \\end{align*}"}
{"id": "4538.png", "formula": "\\begin{align*} D _ { u } ( z , \\chi ) = L ( z , \\chi ) L ( 2 z , \\chi ^ { 2 } ) G ( z , \\chi ) F _ { u } ( z , \\chi ) . \\end{align*}"}
{"id": "9485.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } c s _ s ( n ) q ^ n = \\frac { ( - q ; q ) _ \\infty ( q ^ { s } ; q ^ { s } ) ^ { ( s - 2 ) / 2 } _ \\infty } { ( - q ^ s ; q ^ { s / 2 } ) _ \\infty } . \\end{align*}"}
{"id": "8441.png", "formula": "\\begin{align*} D _ { s } \\phi ( X _ { t } ^ { x ; l ^ \\epsilon } ) = \\nabla \\phi ( X _ { t } ^ { x ; l ^ \\epsilon } ) D _ { s } X _ { t } ^ { x ; l ^ \\epsilon } = \\nabla \\phi ( X _ { t } ^ { x ; l ^ \\epsilon } ) \\nabla _ { x } X _ { t } ^ { x ; l ^ \\epsilon } ( \\nabla _ { x } X _ { \\gamma ^ { \\epsilon } _ { s } } ^ { x ; l ^ \\epsilon } ) ^ { - 1 } . \\end{align*}"}
{"id": "6635.png", "formula": "\\begin{align*} I _ 2 = I _ { 2 1 } + I _ { 2 2 } , \\end{align*}"}
{"id": "4143.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int x ^ 2 u ( x , t ) d x = \\int x u ^ 2 ( x , t ) d x , \\end{align*}"}
{"id": "9413.png", "formula": "\\begin{align*} \\omega ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \\cdots a _ n b _ n ) = \\omega ( a _ 1 a _ 2 \\cdots a _ n ) \\tau ( b _ 1 ) \\tau ( b _ 2 ) \\cdots \\tau ( b _ { n - 1 } ) \\tau ( b _ 0 b _ n ) . \\end{align*}"}
{"id": "162.png", "formula": "\\begin{align*} \\partial _ \\sigma ( f ) ( x ) : = \\sqrt { x } f ' ( x ) , \\end{align*}"}
{"id": "5085.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int _ \\R f ( t ) \\ , \\overline { g ( t ) } \\ , d t \\norm { f } _ 2 ^ 2 = \\langle f , f \\rangle . \\end{align*}"}
{"id": "7950.png", "formula": "\\begin{align*} w _ k = s _ k \\cdot s _ { k - 1 } \\cdots s _ 2 \\cdot s _ 1 \\cdot s _ { 1 ^ { \\prime } } \\cdot s _ 2 \\cdots s _ { k - 1 } \\cdot s _ k . \\end{align*}"}
{"id": "9027.png", "formula": "\\begin{align*} E ( \\rho , \\phi ) = \\int _ { \\Omega } \\bigg ( \\sum _ { i = 1 } ^ s \\rho _ i \\log \\rho _ i + \\frac { 1 } { 2 } \\epsilon ( x ) | \\nabla \\phi | ^ 2 \\bigg ) d x + \\frac { 1 } { 2 \\beta _ R } \\int _ { \\Gamma _ R } \\phi ^ 2 d s \\end{align*}"}
{"id": "5468.png", "formula": "\\begin{align*} \\liminf _ { t - s \\to \\infty } \\inf _ { x \\in \\Omega } u ( t , x ; s , u _ n , a , b ) \\le m _ n : = \\frac { 1 } { n } . \\end{align*}"}
{"id": "819.png", "formula": "\\begin{align*} \\int _ X \\left ( \\frac { g } { \\rho } \\right ) ^ 2 \\ , d \\mu _ \\omega = \\int _ X g ^ 2 \\ , d \\mu _ X , \\end{align*}"}
{"id": "7498.png", "formula": "\\begin{align*} V ( \\mathbf { x } ) = ( 1 - \\theta ) \\frac { x ^ 2 + y ^ 2 } { 2 } + \\frac { \\kappa \\left ( x ^ { 2 } + y ^ { 2 } \\right ) ^ { 2 } } { 4 } \\end{align*}"}
{"id": "1412.png", "formula": "\\begin{align*} \\hat \\Sigma : = { \\rm d i a g } \\left ( \\lambda _ 1 + \\lambda _ { 1 , 1 } \\epsilon , \\cdots , \\lambda _ 1 + \\lambda _ { 1 , k _ 1 } \\epsilon , \\lambda _ 2 + \\lambda _ { 2 , 1 } \\epsilon , \\cdots , \\lambda _ r + \\lambda _ { r , k _ r } \\epsilon \\right ) , \\end{align*}"}
{"id": "4250.png", "formula": "\\begin{align*} R ( \\lambda ) g = \\sigma \\lim _ { \\mathclap { n \\to \\infty } } R ( \\lambda ) T ( t _ n ) f = \\sigma \\lim _ { \\mathclap { n \\to \\infty } } T ( t _ n ) R ( \\lambda ) f = T ( t ) R ( \\lambda ) f = R ( \\lambda ) T ( t ) f ; \\end{align*}"}
{"id": "621.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ k , j ) \\ = \\ 0 \\end{align*}"}
{"id": "8548.png", "formula": "\\begin{align*} i \\partial _ t v + \\sqrt { H + 1 } v \\pm \\big ( \\frac { v - \\bar { v } } { - 2 i \\sqrt { H + 1 } } \\big ) ^ 3 = 0 . \\end{align*}"}
{"id": "121.png", "formula": "\\begin{align*} x ^ 2 ( x y ) = 0 , \\end{align*}"}
{"id": "6180.png", "formula": "\\begin{align*} P _ { i } = \\frac { \\| M _ { i , : } \\| ^ 2 _ { 2 } } { | | M | | _ F ^ 2 } . \\end{align*}"}
{"id": "8567.png", "formula": "\\begin{align*} \\mbox { f o r $ k \\geq 0 $ , } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = T ( k ) m _ + ( x , k ) e ^ { i x k } , \\\\ \\mbox { f o r $ k < 0 $ , } \\sqrt { 2 \\pi } \\mathcal { K } ( x , k ) & = T ( - k ) m _ - ( x , - k ) e ^ { i x k } . \\end{align*}"}
{"id": "8121.png", "formula": "\\begin{align*} T B _ { 1 2 } ( f ) = 4 T B _ { 6 } ( f ) , \\end{align*}"}
{"id": "6270.png", "formula": "\\begin{align*} l & \\leq \\sum _ { p + f \\leq s } \\binom { | T | } { p } \\binom { | T | - p } { f } + \\sum _ { q + g \\leq s } \\binom { | T | } { q } \\binom { | T | - q } { g } \\\\ & = 2 \\sum _ { p + f \\leq s } \\binom { | T | } { p } \\binom { | T | - p } { f } \\\\ & \\leq 2 \\sum _ { p + f \\leq s } | T | ^ s \\leq 2 s ^ 2 | T | ^ s . \\end{align*}"}
{"id": "7902.png", "formula": "\\begin{align*} \\gamma _ F ( A ) = \\limsup _ { l \\to \\infty } \\gamma _ F ( A \\cap \\tau _ { v _ l } A ) \\leq \\limsup _ { l \\to \\infty } \\gamma _ F ( P \\cap \\tau _ { v _ l } P ) + 2 \\epsilon \\leq ( \\gamma _ F ( A ) + \\epsilon ) ^ 2 + 2 \\epsilon . \\end{align*}"}
{"id": "87.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 ^ { K _ { \\ell } } } { \\lambda _ 2 ^ L } & = ( 2 ^ { ( \\dim M _ 2 ^ { K _ { \\ell } } - m + 1 ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m - 1 } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ { K _ { \\ell } } | } ) ( 2 ^ { ( \\dim M _ 2 ^ L - m ) / 2 } \\cdot \\frac { \\displaystyle { \\prod _ { i = 1 } ^ { m } } ( 2 ^ { 2 i } - 1 ) } { | M _ 2 ^ L | } ) ^ { - 1 } \\\\ & \\leq 2 \\cdot \\frac { 2 ^ { n _ { 2 , \\nu _ 2 } } - 1 } { 2 ^ { n + 1 } - 1 } \\end{align*}"}
{"id": "3642.png", "formula": "\\begin{align*} A ( x , \\sigma ) : = \\left [ s _ 1 '' ( x , \\sigma ) + s _ 2 '' ( x , \\sigma ) + s _ 3 '' ( x , \\sigma ) \\right ] \\left [ \\exp \\left ( \\frac { B _ 2 ( 5 - 8 \\sigma ) } { 3 } r \\right ) ( B _ 2 r ) ^ { 5 - 2 \\sigma } \\right ] ^ { - 1 } \\end{align*}"}
{"id": "5882.png", "formula": "\\begin{align*} Q _ 3 ( z , y ) = D b ( z + y ) - D b ( z ) , z , y \\in \\R ^ 2 , \\end{align*}"}