diff --git "a/data_tmp/process_22/tokenized_finally.jsonl" "b/data_tmp/process_22/tokenized_finally.jsonl"
deleted file mode 100644--- "a/data_tmp/process_22/tokenized_finally.jsonl"
+++ /dev/null
@@ -1,9834 +0,0 @@
-{"id": "2636.png", "formula": "\\begin{align*} \\| [ w _ n , a ] \\| & = \\| w _ n a w _ n ^ * - a \\| \\\\ & = \\| w _ n a w _ n ^ * - z _ n a w _ n ^ * + z _ n a w _ n ^ * - z _ n a z _ n ^ * + z _ n a z _ n ^ * - z _ n z _ n ^ * a + z _ n z _ n ^ * a - a \\| \\\\ & \\leq \\| w _ n a - z _ n a \\| + \\| z _ n \\| \\| w _ n a ^ * - z _ n a ^ * \\| + \\| z _ n \\| \\| a z _ n ^ * - z _ n ^ * a \\| + \\| z _ n z _ n ^ * a - a \\| \\\\ & \\to 0 \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} y _ { i _ { 1 } . . . i _ { m } } = \\left \\{ \\begin{array} [ c ] { c } 1 ; i _ { 1 } , . . . , i _ { m } \\in \\left \\{ 1 , 2 \\right \\} \\\\ 0 ; i _ { 1 } , . . . , i _ { m } \\in \\left \\{ 3 , N \\right \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 f ( t ) x ( t , \\rho ) d t = \\rho \\sum _ { k = 0 } ^ { \\infty } \\int _ { - 1 } ^ 1 f ( t ) H ^ k ( t , \\rho ) d t \\rho ^ { k } = 0 , \\ , \\ , | \\rho | < \\rho _ 0 . \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\mathbb E \\langle ( F \\mathbf 1 _ { A } ) \\star \\bar { \\mu } , ( G \\mathbf 1 _ { A } ) \\star \\bar { \\mu } \\rangle = \\mathbb E ( \\langle F , G \\rangle \\mathbf 1 _ { A } ) \\star \\nu . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\mathcal { J } _ n ( \\gamma ) & = \\left \\{ ( \\boldsymbol { j } , \\boldsymbol { k } ) : | \\theta _ { \\boldsymbol { j } , \\boldsymbol { k } } ^ { 0 } | > \\prod _ { l = 1 } ^ d \\min \\left \\{ 2 ^ { - \\alpha _ l j _ l \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha ^ { * } } \\right ) } , \\gamma \\left ( \\frac { \\log { n } } { n } \\right ) ^ { \\frac { 1 } { 2 d } } \\right \\} \\right \\} , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\partial _ a ( \\Phi _ a ( x ) ) & = & Y ( \\Phi _ a ( x ) ) \\\\ \\partial _ a ( B _ a ( x ) ) & = & R _ { B _ a ( x ) * } ( C ( \\Phi _ a ( x ) ) ) \\\\ \\partial _ a ( \\eta _ a ( x ) ) & = & \\tau ( \\Phi _ a ( x ) ) \\eta _ a ( x ) , \\end{array} \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} { E } _ 2 ( t , \\tau ) = - i \\int _ 0 ^ { \\tau } e ^ { i ( \\tilde { \\tau } - \\tau ) A ( t ) ^ { 1 / 2 } } F ( t ) \\bigl ( A ( t ) ^ { 1 / 2 } F ( t ) - ( t ^ 2 S ) ^ { 1 / 2 } P \\bigr ) e ^ { - i \\tilde { \\tau } ( t ^ 2 S ) ^ { 1 / 2 } P } P \\ , d \\tilde { \\tau } . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} \\int _ D | A _ { f _ k } | ^ 2 \\ , d \\mu _ { f _ k } \\leq \\gamma < \\gamma _ n = \\begin{cases} 8 \\pi & \\mbox { f o r } n = 3 , \\\\ 4 \\pi & \\mbox { f o r } n \\geq 4 . \\end{cases} \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} { { E } } _ P [ \\varphi ( X _ { \\tau + t _ 1 } ^ x , \\cdots , X _ { \\tau + t _ m } ^ x ) | \\mathcal { F } _ { { \\tau + } } ] = { { E } } _ P [ \\varphi ( X _ { t _ 1 } ^ y , \\cdots , X _ { t _ m } ^ y ) ] _ { y = X _ \\tau ^ x } \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\binom { t } { r } ( \\delta / T ^ r ) \\lceil n / t \\rceil ^ 2 \\le \\delta n ^ 2 \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\tilde { Z } ( \\eta _ 1 ^ \\alpha ) < \\Phi ^ { - 1 } ( \\alpha ) \\right ) = \\alpha + O ( n ^ { - 1 / 2 } ) \\ , . \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\epsilon _ { \\min } \\stackrel { \\triangle } { = } \\max \\left \\{ K , L \\right \\} \\cdot ( t + 1 ) \\leq \\epsilon \\leq K \\cdot L . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} H ^ { 3 1 , T } ( \\tau , t , x ) = \\begin{cases} H ( t , x ) & \\\\ H ^ { 2 1 } ( \\tau + T , t , x ) = H ( t , x ) & \\\\ H ^ 2 ( t , x ) = H ( t , x ) & \\\\ H ^ { 3 2 } ( \\tau - T , t , x ) = H _ { \\tau - T , t } ( x ) & \\\\ H ^ 3 ( t , x ) = H ' ( t , x ) & . \\end{cases} \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} ( a ' _ { 1 } , \\ldots , a ' _ { r } ) = ( b _ 1 , \\ldots , b _ r ) + \\delta ' , \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\begin{cases} \\psi = - d \\varphi _ 0 + \\pi ^ * \\theta _ { U _ 0 } \\ , \\ , i n \\ , \\ , U _ 0 \\\\ \\psi = - d \\varphi _ 1 + \\pi ^ * \\theta _ { U _ 1 } \\ , \\ , i n \\ , \\ , U _ 1 . \\end{cases} \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak h _ { a b } ^ { i + 1 } + \\frak h _ { a b } ^ { i + 1 } \\circ \\hat d _ { 1 } ^ { i + 1 } ) _ { \\alpha ' _ 2 \\alpha _ 1 } = ( \\frak n _ b ^ { i + 1 } - \\frak n _ a ^ { i + 1 } ) _ { \\alpha ' _ 2 \\alpha _ 1 } . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} E ( y ) & = 2 0 \\pi \\alpha _ 3 \\varepsilon ^ { \\frac { 1 } { 2 } } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 4 \\Bigl ( - \\mu _ i ^ { \\frac { 1 } { 2 } } g _ \\lambda ( \\zeta _ i ) + \\sum _ { j \\not = i } \\mu _ j ^ { \\frac { 1 } { 2 } } G _ \\lambda ( \\zeta _ i , \\zeta _ j ) \\Bigr ) \\\\ & + O ( w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 3 \\varepsilon ^ 2 ) + O ( \\varepsilon ^ 5 ) , y \\in B _ { \\delta / \\varepsilon } ( \\zeta _ i ' ) ) , \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} w ( \\lambda ; L ) = \\frac { ( 1 - \\lambda ^ 2 ) ^ { L / 2 - 1 } } { \\sqrt { 2 \\pi } } \\left ( L \\frac { \\Gamma ( 1 / 2 ) \\Gamma \\left ( ( L + 1 ) / 2 \\right ) } { \\Gamma ( L / 2 ) } \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} U _ i ( x ) = w _ i ( x ) + \\pi _ i ( x ) . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} a \\left ( x , t \\right ) \\geq 0 D _ { T } ^ { 4 } a \\left ( x , t \\right ) = 0 x \\in \\mathbb { R } ^ { 3 } \\diagdown \\Omega _ { 2 } . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\limsup _ \\gamma \\limsup _ \\delta x _ \\gamma \\mathbf { d } x _ \\delta & \\leq \\limsup _ \\gamma \\inf _ { y \\in Y } \\limsup _ \\delta ( x _ \\gamma \\mathbf { d } y + y \\mathbf { d } x _ \\delta ) \\\\ & = \\limsup _ \\gamma \\inf _ { y \\in Y } ( x _ \\gamma \\mathbf { d } y + \\limsup _ \\delta y \\mathbf { d } x _ \\delta ) \\\\ & = \\limsup _ \\gamma x _ \\gamma \\mathbf { d } Y \\quad Y \\leq ^ \\mathbf { d } ( x _ \\lambda ) \\\\ & = 0 \\quad ( x _ \\lambda ) \\leq ^ \\mathbf { d } Y . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} { \\| I _ { a ^ { + } } ^ { \\alpha } f \\| } _ { C _ { 1 - \\gamma } [ a , b ] } & = { \\| ( x - a ) ^ { 1 - \\gamma } I _ { a ^ { + } } ^ { \\alpha } f \\| } _ { C [ a , b ] } \\\\ & \\leq { \\| f \\| } _ { C _ { 1 - \\gamma } [ a , b ] } { \\| I _ { a ^ { + } } ^ { \\alpha } ( x - a ) ^ { \\gamma - 1 } \\| } _ { C _ { 1 - \\gamma } [ a , b ] } , \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} ( 1 - c _ 0 ) e ( G _ S ) / \\binom { r } { 2 } \\ge ( 1 - c _ 0 ) ( 1 - 2 \\xi ) \\alpha _ S ( n / t ) ^ 2 \\ge ( 1 - 3 \\xi ) \\alpha _ S ( n / t ) ^ 2 \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\| z e _ 1 \\| ^ 2 = \\langle z e _ 1 , z e _ 1 \\rangle = \\langle f _ 1 , f _ 1 \\rangle = 1 = \\| e _ 1 \\| ^ 2 . \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} V ( t , x ) = \\frac 1 2 \\langle P _ V ( t ) x , x \\rangle _ { [ R ( Q _ { t } ^ { 1 / 2 } ) ] ^ { * H } , R ( Q _ { t } ^ { 1 / 2 } ) } \\forall t > 0 , \\forall x \\in R ( Q _ { t } ^ { 1 / 2 } ) ; \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} \\prod _ { j = 2 } ^ n \\left ( 1 + \\sum _ { i = 1 } ^ { j - 1 } c _ { i , j } \\cdot x _ { n + i } \\right ) \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} R _ { q , \\ell } \\ge \\frac { \\log \\left ( f _ { 3 , 2 } \\prod _ { j = 4 } ^ q \\left ( j ! { j ^ 2 - j + 1 \\choose j } \\right ) \\cdot \\left ( \\prod _ { i = 3 } ^ { \\ell } ( q ! ) ^ { q ^ { i - 1 } - 2 q ^ { i - 2 } + q ^ { i - 3 } } \\right ) \\right ) } { \\log ( q ^ \\ell ! ) } , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} p ( v , \\xi ) = \\frac { m ^ { - 1 } ( v ) - \\xi } { v - m ( \\xi ) } , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} - t ^ { 1 - 2 \\alpha } u ' ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) 4 ^ { \\alpha } } \\Bigl ( - 2 \\alpha \\int \\limits _ 0 ^ { \\infty } r ^ { - \\alpha - 1 } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } T ( r ) x \\ , d r + \\frac { t ^ 2 } { 2 } \\int \\limits _ 0 ^ { \\infty } r ^ { - \\alpha - 2 } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } T ( r ) x \\ , d r \\Bigr ) . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} K _ { h } ( ( x _ 1 , x _ 2 ) , A \\times B ) : = \\left \\{ \\begin{array} { r @ { \\quad \\quad } l } K _ h ^ 2 ( x _ 2 , B ) \\cdot \\delta _ { x _ 1 } ( A ) & h \\in D _ 2 \\\\ \\omega _ { X _ 2 } ( B ) \\cdot K _ { h } ^ 1 ( x _ 1 , A ) & h \\in D _ 1 \\setminus \\{ e _ 1 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} d : = \\inf \\{ \\L ( u ) : u \\in \\mathcal { N } \\} . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} S _ i S _ j = \\sum _ { k \\in D } \\frac { \\omega _ k } { \\omega _ i \\omega _ j } p _ { i , j } ^ k S _ k \\quad ( i , j \\in D ) \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\widehat { \\Gamma } _ l = \\left \\{ u \\in \\widehat { \\Gamma } : \\ \\textnormal { I n d } ( v _ u , c _ i ) \\equiv l _ i \\mod 2 , \\ \\forall \\ i = 1 , \\ldots , N \\right \\} , \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} Q _ t x = Q _ \\infty x - Q _ { \\infty } e ^ { 2 t A } x = Q _ \\infty ( x - e ^ { 2 t A } x ) . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty ( n , 1 ) ^ { \\dagger } z ^ n = 1 - \\frac { 1 } { G ( z ) } , \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} y '' ( z ) + \\left ( 2 \\frac { \\phi ' ( z ) } { \\phi ( z ) } + \\frac { \\tilde { \\tau } ( z ) } { \\sigma ( z ) } \\right ) y ' ( z ) + \\left ( \\frac { \\phi '' ( z ) } { \\phi ( z ) } + \\frac { \\phi ' ( z ) } { \\phi ( z ) } \\frac { \\tilde { \\tau } ( z ) } { \\sigma ( z ) } + \\frac { \\tilde { \\sigma } ( z ) } { \\sigma ^ 2 ( z ) } \\right ) y ( z ) = 0 . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} z _ 5 ( t ) = \\dfrac { \\varepsilon e ^ { 2 \\pi i t } i - 1 } { \\varepsilon e ^ { 2 \\pi i t } - i } , \\ z _ 6 ( t ) = - \\dfrac { \\varepsilon e ^ { 2 \\pi i t } i + 1 } { \\varepsilon e ^ { 2 \\pi i t } + i } . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\begin{aligned} ( e ^ { k _ E ( \\tau _ s - t ) } - 1 ) \\frac { b _ E } { b _ M k _ M } > - ( \\frac { a _ M } { a _ E } - 1 ) ( T - \\tau _ s ) + ( \\tau _ s - t ) \\end{aligned} \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\mu ( t ; a , b ) & = \\binom { | [ s ] | } { | [ s ] \\cap T | } \\binom { | [ r ] \\backslash S | } { | \\bar { [ s ] } \\cap \\bar { T } | } \\\\ & = \\binom { s } { a } \\binom { r - s } { b } . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} ( I \\otimes \\Delta ^ e ) \\Delta ^ e m ( a ) & = ( I \\otimes \\Delta ^ e ) ( m \\otimes m ) \\Delta ( a ) = m ( a _ { ( 1 ) } ) \\otimes \\Delta ^ e m ( a _ { ( 2 ) } ) \\\\ & = m ( a _ { ( 1 ) } ) \\otimes ( m \\otimes m ) \\Delta ( a _ { ( 2 ) } ) = m ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 1 ) } ) \\otimes m ( a _ { ( 2 2 ) } ) , \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\frac { \\partial ^ 4 u } { \\partial x _ j ^ 4 } - \\triangle u + u = | u | ^ { q - 2 } u , \\ ; \\ ; u \\in H ^ 2 ( \\mathbb { R } ^ { n } ) , \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = a x ( t ) , x ( 0 ) = x _ 0 > 0 , \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} X ^ { t } ( x , \\xi ) = \\beta _ { t } ( x , \\xi ) X ( x , \\xi ) , \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\begin{aligned} m _ N ( w ) & = \\tfrac { 1 - \\rho } { \\rho } N - 2 w ( 1 - \\rho ) \\chi ^ { - 1 / 3 } N ^ { 2 / 3 } , \\\\ n _ N ( w ) & = \\tfrac { \\rho } { 1 - \\rho } N + 2 w \\rho \\chi ^ { - 1 / 3 } N ^ { 2 / 3 } , \\end{aligned} \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} \\int _ { \\Gamma ^ 0 } \\frac { ( N \\cdot \\nu ^ t _ { \\theta } ) } { \\frac { \\partial \\sigma } { \\partial s } } ( x ) f ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) = \\lim _ { \\varepsilon \\downarrow 0 } \\frac 1 \\varepsilon \\int _ { A ^ { \\varepsilon } } f ( x ) d \\mathcal { H } ^ n ( x ) . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} P _ 3 ( x ) = & - 2 ^ { r - 4 } \\left [ ( | S | - S ( x ) ) ( | S | - S ( x ) - 1 ) ( | S | + S ( x ) + 2 ) \\right . \\\\ & \\left . + 4 ( S ( x ) ) ( ( | S | - S ( x ) ) + ( | S | - S ( x ) ) ^ 2 ) + 8 S ( x ) ( | S | + S ( x ) ) \\right ] . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} \\log | Q _ N | = \\sum _ { p \\in \\mathcal { S } _ N } \\nu _ p ( Q _ N ) \\log p + \\sum _ { p \\in \\mathcal { S } _ N ^ \\prime } \\nu _ p ( Q _ N ) \\log p , \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} \\sum _ { p _ 1 , \\dots , p _ k \\leq z } g _ 1 ( p _ 1 ) \\cdots g _ k ( p _ k ) 2 ^ { \\omega ( p _ 1 \\cdots p _ k ) } \\ll \\sum _ { p _ 1 , \\dots , p _ k \\leq z } \\prod _ { j = 1 } ^ k \\log z = ( \\pi ( z ) \\log z ) ^ k \\ll z ^ k = \\sqrt x , \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\overline { K ( w ) } = K ( \\overline { w } ) \\ , , \\big | K ( w ) \\big | = \\big | K ( \\overline { w } ) \\big | \\ , , \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} { n + k - r \\brack k } _ q = { n + k - r - 1 \\brack k } _ q + q ^ { n - r } \\cdot { n + k - r - 1 \\brack k - 1 } _ q , \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} \\omega _ { 2 } = \\omega _ { 1 } ^ { 2 } , \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} G \\left ( { u , \\xi } \\right ) = \\frac { 1 } { u ^ { 2 } } \\sum \\limits _ { s = 0 } ^ { n - 1 } { \\frac { G _ { s } \\left ( \\xi \\right ) } { u ^ { 2 s } } } + \\varepsilon _ { n } \\left ( { u , \\xi } \\right ) , \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\left \\{ u + \\sum _ { j = 0 } ^ { \\infty } \\frac { { F _ { 2 j + 1 } ( \\xi ) } } { { u ^ { 2 j + 1 } } } \\right \\} \\exp \\left \\{ 2 \\sum _ { j = 1 } ^ { \\infty } \\frac { { E _ { 2 j } ( \\xi ) } } { { u ^ { 2 j } } } \\right \\} \\sim C , \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} k ( t , x ) = \\frac 1 { \\sqrt { 2 \\pi } } \\lim \\limits _ { \\epsilon \\downarrow 0 } \\int _ { | \\xi | > \\epsilon } | \\xi | ^ { \\frac 1 2 } e ^ { i t ( \\xi ^ 2 \\coth \\xi - \\xi ) + i x \\xi } \\ , d \\xi . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} \\Delta _ p ( f ) : = d i v _ \\mathfrak { m } ( F ( \\nabla f ) ^ { p - 2 } \\nabla f ) : = d i v _ \\mathfrak { m } ( | \\nabla f | ^ { p - 2 } \\nabla f ) , \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} C _ { M , M ' } = ( \\det X X ' ) ^ { 1 / 4 } \\det \\left [ \\tfrac { 1 } { 2 } ( M + \\overline { M ' } ) \\right ] ^ { - 1 / 2 } \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\ell \\left ( ( - 1 ) ^ { m _ j } \\sum _ { k = 1 } ^ n a _ { j k } \\frac { \\partial ^ { 2 m _ j } u } { \\partial x _ k ^ { 2 m _ j } } \\right ) = | u | ^ { q - 2 } u , a _ { j k } > 0 , \\ ; m _ j \\in \\mathbb N _ 0 , \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\mathbb { P } ( L _ n ^ { * } = \\ell ) = \\frac { \\ell \\ , ( \\ell , 1 ) ^ { \\dagger } } { n \\cdot n ! } \\sum _ { k = 1 } ^ n k \\ , ( n - l , k - 1 ) ^ { \\dagger } , \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} f ( \\lambda , 0 ) = 0 \\textrm { a n d } \\quad \\frac { \\partial f } { \\partial x } ( \\lambda , 0 ) = f ' _ \\lambda ( 0 ) = 0 , \\forall \\lambda \\in \\R . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} S ( z , z ) = \\frac { 2 \\mathrm { I m } ( z ) } { \\pi | 1 - z ^ 2 | ( 1 - | z | ^ 2 ) ^ 2 } \\left ( 1 - N | z | ^ { 2 N - 2 } + ( N - 1 ) | z | ^ { 2 N } \\right ) \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} N _ f \\xi = f \\xi . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} u ^ \\# ( x ) = u ^ * ( \\kappa _ n ( F ^ o ( x ) ) ^ n ) . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } k ( t ) e ^ { i w t } d t = ( 1 + e ^ { i w T } ) K ( w ) \\ , . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} F _ T ( x ) - 1 - x F _ T ( x ) & = x C ( x ) ( F _ T ( x ) - 1 ) + \\frac { 1 } { 1 - x } \\left ( \\frac { ( x ^ 4 + x ^ 5 C ^ 4 ( x ) + x ^ 5 C ^ 3 ( x ) ) C ^ 2 ( x ) } { 1 - x } + x ^ 3 C ^ 2 ( x ) \\right ) \\\\ & + x ^ 3 C ^ 3 ( x ) ( F _ T ( x ) - 1 ) + \\frac { x ^ 5 C ^ 3 ( x ) } { ( 1 - x ) ^ 2 } . \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{gather*} \\rho = \\sqrt { \\lambda ^ 2 R ^ 2 - ( n - 1 ) ^ 2 / 4 } . \\end{gather*}"}
-{"id": "1865.png", "formula": "\\begin{align*} R _ Z ( p , p + n ) & = \\frac { a ^ 2 + b ^ 2 } { 2 } \\bigl [ ( n + 1 ) ^ { 2 H } - 2 n ^ { 2 H } + ( n - 1 ) ^ { 2 H } \\bigr ] \\\\ & - a b \\bigl [ ( 2 p + n + 2 ) ^ { 2 H } - 2 ( 2 p + n + 1 ) ^ { 2 H } + ( 2 p + n ) ^ { 2 H } \\bigr ] \\\\ & = \\bigl ( a ^ 2 + b ^ 2 \\bigr ) R _ B ( 0 , n ) - a b f _ p ( n ) , \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} D ^ { - r } = c _ r \\int _ 0 ^ \\infty t ^ { - 1 + r } e ^ { - t D } d t \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} | f | = | f | \\chi _ { B ^ c } \\leq s \\chi _ { B ^ c } = s \\chi _ { [ 0 , \\alpha ) } , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} p _ t ( x , y ) ~ = ~ \\int _ 0 ^ \\infty e ^ { - \\lambda t } \\ , d e _ \\lambda ( x , y ) \\ , . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} \\nabla ^ { ^ M } _ { X } V = \\mathcal { V } ( \\nabla ^ { ^ M } _ { X } V ) + \\mathcal { A } _ { X } V ; \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\tilde { X _ 0 } _ { \\vert \\mathcal { E } } : \\begin{cases} \\dot x & = 0 \\\\ \\dot u & = - E ( 1 + u ^ 2 ) \\\\ \\dot \\varphi & = 1 \\end{cases} \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\dim D ^ { ( p - 1 , p - 1 , 2 ) } & = \\dim D ^ { ( p - 1 , p - 2 , 2 ) } \\cr & = \\dim ( p - 1 , p - 2 , 2 ) - \\dim ( p , p - 2 , 1 ) + \\dim ( 2 p - 3 , 1 , 1 ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} M : = \\max _ { 1 \\le j \\le m } \\ , \\max _ { 0 \\le \\ell \\le n + 1 } \\ , \\max _ { 0 \\le i \\le n } \\ , \\sup _ { x _ j \\in U _ j } | f ^ { ( \\ell ) } _ { j , i } ( x _ j ) | < \\infty \\ , . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\alpha _ { j , k } & = \\int _ { - 1 } ^ { 1 } \\mathrm { d } x \\ , w _ r ^ { ( m ) } ( x ) \\int _ { - 1 } ^ { 1 } \\mathrm { d } y \\ , w _ r ^ { ( m ) } ( y ) p _ { j - 1 } ( x ) p _ { k - 1 } ( y ) \\mathrm { s g n } ( y - x ) , \\\\ \\beta _ { j , k } & = 2 \\mathrm { i } \\int _ { D _ { + } } \\mathrm { d } x \\mathrm { d } y \\ , w _ c ^ { ( m ) } ( x , y ) \\left ( p _ { j - 1 } ( x + \\mathrm { i } y ) p _ { k - 1 } ( x - \\mathrm { i } y ) - p _ { k - 1 } ( x + \\mathrm { i } y ) p _ { j - 1 } ( x - \\mathrm { i } y ) \\right ) , \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\sum _ { m = M } ^ { n / ( \\log n ) ^ 2 } \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\leq O \\ ( \\eta ^ M \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) + O \\ ( e ^ { - c n ^ { 1 / 2 } } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } ^ x & = \\binom { | S \\backslash \\{ x \\} | } { a - | S \\cap \\{ x \\} | } \\binom { | \\bar { S } \\backslash \\{ x \\} | } { b } \\\\ & = \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - 1 + S ( S ) } { b } . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} - q = : \\cos ( C ) = \\frac { 1 + y _ 1 ^ 2 - ( y _ 1 ^ 2 - 1 ) ^ 2 } { 2 y _ 1 } = \\frac { y _ 1 ( 3 - y _ 1 ^ 2 ) } { 2 } \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\gamma ( \\{ a , b \\} ) & = \\delta ( a ) \\gamma ( b ) - ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\delta ( a ) , \\\\ \\delta ( a b ) & = \\gamma ( a ) \\delta ( b ) + ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\delta ( a ) , \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} R _ { 2 , \\epsilon } ^ { - 1 } ( x + \\alpha ) ( P _ { 1 , \\epsilon } + \\epsilon ^ 2 \\widetilde { P } _ { 1 , \\epsilon } ( x ) ) R _ { 2 , \\epsilon } ( x ) = P _ { 2 , \\epsilon } + \\epsilon ^ 3 \\widetilde { P } _ { 2 , \\epsilon } ( x ) , \\\\ \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\omega ( A , B ) : = T ^ { \\ast } T = ( R ^ { \\ast [ - 1 ] ) } B ) ^ { \\ast } R ^ { \\ast [ - 1 ] } B \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} n ^ { - 1 } \\tilde { l } ^ { G E L } _ { \\gamma } ( \\theta ) = \\sum _ { k , l } \\bar { \\psi } ^ k \\omega ^ { k l } \\bar { \\psi } ^ l + \\frac { 2 } { 3 } \\sum _ { k , l , m } \\bar { \\psi } ^ k \\bar { \\psi } ^ l \\bar { \\psi } ^ m \\omega ^ { k l } \\omega ^ { k m } \\omega ^ { l m } \\alpha _ { k l m } \\ , , \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} e ^ z & = e ^ { x + \\epsilon y + \\epsilon ^ 2 z } \\\\ & = e ^ x e ^ { \\epsilon y } e ^ { \\epsilon ^ 2 z } \\\\ & = e ^ x \\left ( 1 + \\epsilon y + \\frac { \\epsilon ^ 2 y ^ 2 } { 2 } \\right ) \\left ( 1 + \\epsilon ^ 2 z \\right ) = e ^ x \\left ( 1 + \\epsilon y + \\epsilon ^ 2 \\left ( z + \\frac { y ^ 2 } { 2 } \\right ) \\right ) \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} P ( \\psi ( X _ { n + 1 } ) \\in A | \\ > \\sigma ( \\psi ( X _ n ) ) ) = K _ { \\psi ( X _ n ) } \\circ K ^ X _ { \\mu _ 1 } ( x _ 0 , \\psi ^ { - 1 } ( A ) ) = P ( \\psi ( X _ { n + 1 } ) \\in A | \\ > \\cal F _ n ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\overline { G _ T } ( z , w ) = \\sum _ { \\alpha \\in \\N ^ n } ( - 1 ) ^ { \\alpha } T ( z ^ \\alpha ) \\frac { w ^ \\alpha } { \\alpha ! } \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} A ' ( x , v ) & = \\frac { x } { 1 - v } \\left ( A ' ( x , v ) - \\frac { 1 } { v } A ' ( v x , 1 ) \\right ) + \\frac { x ( 1 + v ) } { v } ( A ( x v , 1 ) - 1 ) - x ^ 2 A ( x v , 1 ) , \\textrm { a n d } \\\\ A '' ( x , v ) & = x A '' ( x , v ) + ( 1 - x ) x \\big ( g ( x ) - 1 \\big ) + B ( x , v ) - B ( x , 0 ) + ( 1 - x ) ( B ' ( x , v ) - B ' ( x , 0 ) ) \\ , . \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} I ( M _ 1 ; Y _ { 1 , 0 } ^ T | M _ 2 ) & \\leq \\lim _ { n \\to \\infty } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\log ( 2 \\pi e ( \\alpha P \\delta _ n ^ 2 + N _ 1 \\delta _ n ) ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\log ( 2 \\pi e N _ 1 \\delta _ n ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\log \\left ( 1 + \\frac { \\alpha P \\delta _ n } { N _ 1 } \\right ) \\\\ & = \\frac { \\alpha P T } { 2 N _ 1 } . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\dot { \\hat { v } } ( t ) = \\hat { A } \\hat { v } ( t ) , \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\widehat { ( \\mathfrak { a } ^ i ) } _ { \\mathfrak { a } } = \\varphi _ { \\mathfrak { a } ^ i } ( \\mathfrak { a } ^ i ) \\cdot \\hat \\Lambda _ { \\mathfrak { a } } = \\hat \\Lambda _ { \\mathfrak { a } } \\cdot \\varphi ( \\mathfrak { a } ^ i ) = \\hat { \\mathfrak { a } } ^ i . \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} { \\cal R } _ 0 : = \\begin{pmatrix} R _ 1 ^ { ( 0 ) } & R _ 2 ^ { ( 0 ) } \\\\ \\overline R _ 2 ^ { ( 0 ) } & \\overline R _ 1 ^ { ( 0 ) } \\end{pmatrix} \\ , , R _ i ^ { ( 0 ) } : H _ { { \\mathbb S } _ 0 } ^ \\bot \\to H _ { { \\mathbb S } _ 0 } ^ \\bot \\ , , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\varphi ( \\bar x , \\kappa ^ j r ) \\le \\kappa ^ { j \\beta } \\varphi ( \\bar x , r ) + C \\left ( \\norm { D u } _ { L ^ \\infty ( B ^ { + } ( \\bar x , 2 r ) ) } \\sum _ { i = 1 } ^ { j } \\kappa ^ { ( i - 1 ) \\beta } \\omega _ { \\mathbf { A } } ( 2 \\kappa ^ { j - i } r ) + \\sum _ { i = 1 } ^ { j } \\kappa ^ { ( i - 1 ) \\beta } \\omega _ { \\vec g } ( 2 \\kappa ^ { j - i } r ) \\right ) . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} \\left ( \\sum _ { r \\geq 0 } ( - 1 ) ^ r e _ r ( \\mathbf { x } ^ { ( \\mathbf { 1 } ) } , \\mathbf { y } ^ { ( \\mathbf { 1 } ) } ) \\right ) \\prod _ { l = 1 } ^ { \\infty } \\exp \\left ( T _ l \\left ( \\log \\left ( 1 + \\sum _ { U } p _ l \\left ( \\mathbf { x } ^ { ( U ) } , \\mathbf { y } ^ { ( U ) } , \\mathbf { z } ^ { ( U ) } \\right ) [ U ] \\right ) \\right ) \\right ) \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} r _ { m + 1 / 2 } & = \\frac { 1 } { 2 } ( r _ { m + 1 } + r _ m ) , & r _ { m , \\theta } = \\theta r _ { m + 1 } + ( 1 - 2 \\theta ) r _ m + \\theta r _ { m - 1 } , \\\\ \\partial _ t r _ { m + 1 / 2 } & = ( r _ { m + 1 } - r _ m ) / \\Delta t , & \\partial ^ 2 _ t r _ m = ( r _ { m + 1 } - 2 r _ m + r _ { m - 1 } ) / ( \\Delta t ) ^ 2 , \\\\ \\delta _ t r _ m & = ( r _ { m + 1 } - r _ { m - 1 } ) / ( 2 \\Delta t ) . & \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} I _ { q ^ { 2 } } - A ^ { 2 } \\frac { \\left ( A - Z ''' \\right ) \\left ( A - { Z '' } ^ { t } \\right ) } { \\left ( I _ { q } - A Z ''' \\right ) \\left ( I _ { q } - A { Z '' } ^ { t } \\right ) } = \\frac { I _ { q ^ { 2 } } - A ^ { 4 } + A Z ''' + A { Z '' } ^ { t } - A ^ { 3 } Z ''' - A ^ { 3 } { Z '' } ^ { t } } { \\left ( I _ { q } - A Z ''' \\right ) \\left ( I _ { q } - A { Z '' } ^ { t } \\right ) } , \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} 2 \\ , h \\ , \\Re \\ , G ( 1 / 2 ) = - \\ , h ^ 3 \\ , \\Re \\ , G ^ { [ 2 ] } ( 1 / 2 ) \\ , . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} \\widetilde { \\Pi } ^ T \\ , A \\ , \\widetilde { \\Pi } = \\widehat { \\Pi } ^ T L D L ^ T \\widehat { \\Pi } . \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} ( Q _ n x ) ( \\tau _ i ^ k ) = x ( \\tau _ i ^ k ) , \\ ; \\ ; \\ ; i = 1 , \\ldots , r , \\ ; \\ ; \\ ; k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } H ( 1 , \\rho ) | _ { \\rho = 0 } = 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ 2 ( t ) d t + 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) F ( t ) d t . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} [ h _ { i , \\pm 1 } , h _ { i , \\pm 2 } ] = 0 , \\ [ h _ { j , \\mp 1 } , h _ { i , \\pm 2 } ] = 0 . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\| f \\| _ X = \\sup \\{ | \\langle f , p \\rangle | \\colon p \\in \\mathcal { P } , \\ \\| p \\| _ { X ' } \\le 1 \\} . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\| ( 1 , 1 ) \\| _ { \\Omega } ^ { * } = \\max \\{ \\| ( 1 , 0 ) \\| _ { \\Omega } ^ { * } , \\| ( 0 , 1 ) \\| _ { \\Omega } ^ { * } \\} = b , [ ( 1 , 1 ) ] _ { \\Omega } = \\min \\{ [ ( 1 , 2 ) ] _ { \\Omega } , [ ( 2 , 1 ) ] _ { \\Omega } = a \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} d ( \\omega \\wedge f ^ * \\eta ) = d \\omega \\wedge f ^ * \\eta + ( - 1 ) ^ { k } \\omega \\wedge f ^ * d \\eta . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} D R _ 1 : = D R [ 3 q ^ 3 z + \\cdots + 2 1 q ^ { 2 5 } z ^ 5 ] \\leq 1 0 ^ { - 2 } ~ ~ { \\rm a n d } ~ ~ D I _ 1 : = D I [ 3 q ^ 3 z + \\cdots + 2 1 q ^ { 2 5 } z ^ 5 ] \\leq 1 0 ^ { - 2 } ~ . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} v ^ { R ' } ( D , E , F ) = ( - r ( \\beta ) D , r ( \\beta ) D , r ( \\beta ) D ) . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{gather*} ( N - 1 ) ( | \\lambda | - | \\mu | ) - ( n ( \\lambda ) - n ( \\mu ) ) \\\\ \\qquad { } = ( N - 1 ) ( | \\lambda | - | \\mu | ) - ( ( N - 1 ) \\lambda _ N + ( N - 2 ) ( \\lambda _ { N - 1 } - \\mu _ { N - 1 } ) + \\dots + ( \\lambda _ 2 - \\mu _ 2 ) ) \\\\ { } \\geq ( N - 1 ) ( | \\lambda | - | \\mu | ) - ( N - 1 ) ( \\lambda _ N + \\lambda _ { N - 1 } - \\mu _ { N - 1 } + \\dots + \\lambda _ 2 - \\mu _ 2 ) \\\\ \\qquad { } = ( N - 1 ) ( \\lambda _ 1 - \\mu _ 1 ) \\geq 0 . \\end{gather*}"}
-{"id": "3106.png", "formula": "\\begin{align*} a _ { T - 1 , T } = - a _ { T , T } \\frac { \\det { \\overline C ^ { T - 1 } _ T } } { \\det { \\overline C ^ { T - 1 } } } , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\tau ' = \\min \\left \\{ t : G \\left ( t \\right ) f + 2 \\right \\} . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} | \\div z _ k | ( B _ R \\setminus S _ k ) = r _ k ^ { 1 - n } | \\div z | ( B _ { R r _ k } ( x _ 0 ) \\setminus S ) \\to 0 k \\to \\infty \\ , , \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} f _ A ( n _ 1 + n _ 2 ) & = \\left | A \\cap [ u ^ * _ 1 , u ^ * _ 1 + n _ 1 + n _ 2 - 1 ] \\right | \\\\ & = \\left | A \\cap [ u ^ * _ 1 , u ^ * _ 1 + n _ 1 - 1 ] \\right | + \\left | A \\cap [ u ^ * _ 1 + n _ 1 , u ^ * _ 1 + n _ 1 + n _ 2 - 1 ] \\right | \\\\ & = \\left | A \\cap [ u ^ * _ 1 , u ^ * _ 1 + n _ 1 - 1 ] \\right | + \\left | A \\cap [ u ^ * _ 2 , u ^ * _ 2 + n _ 2 - 1 ] \\right | \\\\ & \\leq f _ A ( n _ 1 ) + f _ A ( n _ 2 ) . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\frac { 1 } { r } = \\frac { 1 } { p } - \\frac { s } { d } \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\Lambda _ 2 & = 2 l \\log \\beta _ 1 - 2 n \\log \\beta _ 3 + \\log \\beta _ 4 , \\\\ \\Lambda _ 3 & = 2 m \\log \\beta _ 2 - 2 n \\log \\beta _ 3 + \\log \\beta _ 5 . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\boldsymbol { w } _ { k + 1 } = \\boldsymbol { w } _ k - \\mu \\left ( \\boldsymbol { g } _ k - \\dfrac { \\big | \\boldsymbol { b } _ k ^ H \\boldsymbol { g } _ k \\big | } { \\| \\boldsymbol { b } _ k \\| ^ 2 } \\boldsymbol { b } _ k \\right ) , \\\\ \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 4 } \\| c _ i ( t ) - c _ { i , \\infty } \\| _ { L ^ { p } ( \\Omega ) } \\leq C e ^ { - \\lambda ' t } , t \\ge 0 , \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} \\bar B = \\left [ \\begin{array} { c c } \\times \\\\ 0 \\\\ 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} c _ q & \\le q ( c _ { q - 1 } ( q + 1 ) + q + 1 ) = q ( q + 1 ) ( c _ { q - 1 } + 1 ) \\\\ & \\le 2 q ( q + 1 ) c _ { q - 1 } \\le \\ldots \\le 2 ^ { q - 3 } \\cdot \\frac { q ! } { 6 } \\cdot \\frac { ( q + 1 ) ! } { 2 4 } \\cdot c _ 3 . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} & ~ \\partial _ { \\lambda } \\Gamma ( \\lambda ) \\\\ = & ~ i \\int _ \\real u ( x ) e ^ { - i \\lambda x } \\left ( - \\frac { 1 } { 2 \\pi \\lambda } \\int _ \\real { u ( y ) m _ e ( y , \\lambda + ) } ~ d y \\right ) m _ 1 ( x , \\lambda + ) ~ d x \\\\ = & f ( \\lambda ) \\beta ( \\lambda ) \\Gamma ( \\lambda ) . \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\lim _ { \\Delta t \\to 0 } \\sup _ { 0 \\le m < M } \\left \\| \\partial _ t u _ { 0 , m + 1 / 2 } - { \\partial u _ 0 \\over \\partial t } ( t _ m ) \\right \\| _ H = 0 ; \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\mathrm { d } X _ { t } & = \\beta X _ { t } \\mathrm { d } t + C C ^ { T } \\nabla \\log \\phi ( X _ { t } ) \\mathrm { d } t + C \\mathrm { d } B _ { t } , \\\\ \\mathrm { d } Y _ { t } & = G X _ { t } \\mathrm { d } t + \\Gamma \\mathrm { d } W _ { t } , \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ { ( \\eta ) } > x ) = \\Biggl ( \\sum \\limits _ { n = 1 } ^ { K } + \\sum \\limits _ { n = K + 1 } ^ { \\infty } \\Biggr ) \\mathbb { P } ( S _ { ( n ) } > x ) \\mathbb { P } ( \\eta = n ) . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\Pi ( | \\sigma ^ 2 - \\sigma _ 0 ^ 2 | \\leq \\widetilde { W } _ 3 \\epsilon _ n ) \\geq 2 \\widetilde { W } _ 3 \\epsilon _ n \\inf _ { | u - \\sigma _ 0 ^ 2 | \\leq \\widetilde { W _ 3 } \\epsilon _ n } \\pi _ { \\sigma } ( u ) = 2 \\widetilde { W } _ 3 \\epsilon _ n \\pi _ { \\sigma } ( \\sigma _ 0 ^ 2 ) [ 1 + o ( 1 ) ] , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} \\tau = \\left [ \\begin{array} { c c c c } 1 & & & \\\\ & & & 1 \\\\ & & r & \\\\ & - 1 & & \\end{array} \\right ] , \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\aligned & \\partial \\mathcal N ( \\alpha _ - , \\alpha _ + ) \\\\ & \\supseteq ( \\mathcal M ( \\alpha _ - , \\alpha _ + ) \\times _ { R _ { \\alpha _ + } } \\mathcal N ( \\alpha _ + , \\alpha _ + ) ) \\cup ( \\mathcal N ( \\alpha _ - , \\alpha _ - ) \\times _ { R _ { \\alpha _ - } } \\mathcal M ( \\alpha _ - , \\alpha _ + ) ) \\\\ & = \\mathcal M ( \\alpha _ - , \\alpha _ + ) \\sqcup \\mathcal M ( \\alpha _ - , \\alpha _ + ) \\endaligned \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} t = t ' + 1 . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} x \\sqsubset \\quad & = \\quad \\{ y \\in Y : x \\sqsubset y \\} . \\\\ \\sqsubset y \\quad & = \\quad \\{ x \\in X : x \\sqsubset y \\} . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\big | K ^ * ( u ) \\big | = \\big | K ^ * ( \\overline { u } ) \\big | \\ , , \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } g ( s , W _ 0 ^ { ( n ) , t _ { n , i } } , Y _ 0 ^ { ( n ) , t _ { n , i } } ) d s + B ( t _ { n , i + 1 } ) - B ( t _ { n , i } ) , \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\begin{cases} { \\displaystyle g _ n : = g _ 0 + \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { d ^ { i + 1 } } ( f ^ i ) ^ * ( \\lambda ) } & ( n \\geqslant 1 ) \\\\ [ 1 . 5 e x ] { \\displaystyle \\varphi _ 0 = 1 , \\varphi _ n = \\prod _ { i = 0 } ^ { n - 1 } ( f ^ i ) ^ * ( \\varphi ) ^ { 1 / d ^ { i + 1 } } } & ( n \\geqslant 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\frac { d x } { d t } = a x - b x y , \\frac { d y } { d t } = - c y + d x y , \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} C : = \\int _ { B _ 1 ( 0 ) } \\phi ( h ) \\ , | h | ^ 2 \\ , d h \\ , . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\frac { d + \\varepsilon \\sqrt { a d } } { d + \\varepsilon \\sqrt { b d } - 1 } & < \\frac { d + \\sqrt { b d } } { d - \\sqrt { b d } } = 1 + \\frac { 2 \\sqrt { b d } } { d - \\sqrt { b d } } = 1 + \\frac { 2 } { \\frac { d } { \\sqrt { b d } } - 1 } \\\\ & < 1 + \\frac { 2 } { \\sqrt { B _ 0 } - 1 } = \\frac { \\sqrt { B _ 0 } + 1 } { \\sqrt { B _ 0 } - 1 } , \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} M _ h ( x ) & \\ll \\sum _ { \\substack { h _ 0 , h _ 1 , h _ 2 \\ge 0 \\\\ h _ 0 + h _ 1 + h _ 2 = h } } \\binom h { h _ 0 , h _ 1 , h _ 2 } x ( \\log \\log x ) ^ { 3 h / 2 - 1 / 4 } ( \\log \\log \\log x ) ^ { 2 h } \\\\ & \\ll x ( \\log \\log x ) ^ { 3 h / 2 - 1 / 4 } ( \\log \\log \\log x ) ^ { 2 h } , \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} z = \\sqrt { \\frac { t } { 2 } } \\pm \\sqrt { \\frac { | w | } { \\sqrt { 8 t } } - \\frac { t } { 2 } } \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\int _ { \\Omega } A _ \\tau \\nabla ( U _ \\tau - u _ 0 ) \\nabla \\varphi + \\int _ \\Omega ( g ( U _ \\tau ) - g ( u _ 0 ) ) J _ \\tau \\varphi & = \\int _ \\Omega ( f _ \\tau J _ \\tau - f ) \\varphi + \\\\ & + \\int _ \\Omega ( I - A _ \\tau ) \\nabla u _ 0 \\nabla \\varphi \\\\ & + \\int _ \\Omega ( J _ \\tau - 1 ) g ( u _ 0 ) \\varphi . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} & \\mathbb { H } _ n = \\{ \\ , Z \\in _ n ( \\mathbb { C } ) \\ , \\mid \\ , ( Z ) > 0 \\ , \\} \\ ; \\ ; ( ) , \\\\ & \\Lambda _ n : = \\{ \\ , T = ( t _ { j l } ) \\in _ n ( \\mathbb { Q } ) \\ , \\mid \\ , t _ { j j } \\in \\mathbb { Z } , \\ , 2 t _ { j l } \\in \\mathbb { Z } \\ , \\} . \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} Q [ \\varphi ] = \\sum _ { i = 1 } ^ { n } Q [ J _ i \\varphi ] + R _ n , \\varphi \\in H ^ 1 _ \\mu \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\mathcal { D } _ 0 \\Bigl ( \\frac { x - \\zeta _ i } { \\mu _ i } \\Bigr ) = O ( \\mu _ i \\log \\mu _ i ) , \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\big ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z _ 0 ) ) - 1 \\big ) \\Omega _ B ( z _ 0 ) - \\Omega _ A ( z _ 0 ) & = r _ 1 ( z _ 0 ) + O \\left ( | \\Omega _ B ( z _ 0 ) | ^ 2 \\right ) \\ , , \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} Y ( t _ { n , i + 1 } ) - Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ( t _ { n , i } ) - Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } ( g ( s , W _ 0 ^ s , Y _ 0 ^ s ) - g ( s , W _ 0 ^ { t _ { n , i } } , Y _ 0 ^ { ( n ) , t _ { n , i } } ) ) d s . \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{gather*} \\frac { \\big ( x \\cdot t \\cdot t ^ 2 \\cdots t ^ { N - 1 } \\big ) ^ m \\cdot P _ { \\lambda } \\big ( x , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } { \\big ( 1 \\cdot t \\cdot t ^ 2 \\cdots t ^ { N - 1 } \\big ) ^ m \\cdot P _ { \\lambda } \\big ( 1 , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } = \\frac { P _ { \\widetilde { \\lambda } } \\big ( x , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } { P _ { \\widetilde { \\lambda } } \\big ( 1 , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } . \\end{gather*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\pi i I _ { \\nu } \\left ( \\nu z \\right ) = K _ { \\nu } \\left ( \\nu z e ^ { - \\pi i } \\right ) - e ^ { \\nu \\pi i } K _ { \\nu } \\left ( \\nu z \\right ) , \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} F _ i ( 1 , X _ 1 , \\ldots , X _ n ) = f _ i ( X _ 1 , \\ldots , X _ n ) \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} P _ k ( \\mathbf { a } ) : = u _ 1 ^ { a _ 1 } \\cdots u _ k ^ { a _ k } . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\Phi ( f ) & \\geq e ^ { - \\frac { x a _ u } { \\tau | v _ 1 | } } f _ { L } ( v ) 1 _ { v _ 1 > 0 } + e ^ { - \\frac { ( 1 - x ) a _ u } { \\tau | v _ 1 | } } f _ { R } ( v ) 1 _ { v _ 1 < 0 } \\cr & \\geq e ^ { - \\frac { a _ u } { \\tau | v _ 1 | } } f _ { L } ( v ) 1 _ { v _ 1 > 0 } + e ^ { - \\frac { a _ u } { \\tau | v _ 1 | } } f _ { R } ( v ) 1 _ { v _ 1 < 0 } \\cr & = e ^ { - \\frac { a _ u } { \\tau | v _ 1 | } } f _ { L R } . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} m \\dot { v } = - d v + u _ 1 , \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} y _ n ( z ) = \\sum _ { m = 0 } ^ n ( - 1 ) ^ { n - m } e _ { n - m } z ^ m , \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} f = z _ i ^ { n _ i - 1 } f _ 1 + f _ 2 \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\min \\| L x \\| \\mbox { \\ \\ s u b j e c t t o \\ } \\| A x - b \\| = \\min , \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\dot { \\hat { x } } & = A \\hat { x } + B u + L \\left ( y - \\frac { 1 } { h } C \\int _ { t - h } ^ t \\hat { x } ( s ) d s \\right ) , \\\\ \\dot { \\epsilon } & = A \\epsilon - \\frac { 1 } { h } L C \\int _ { t - h } ^ t \\epsilon ( s ) d s , \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} k _ { u , l } = n ^ { - } \\left ( L _ { 0 } + l ^ { 2 } \\alpha ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\rho : = \\frac { 1 } { 2 } \\underset { \\alpha \\in \\sum ^ + } { \\sum } m _ { \\alpha } \\alpha \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } { k } \\ , . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\omega _ { \\phi _ { \\epsilon } } : = \\omega _ 0 + \\sqrt { - 1 } \\partial \\bar { \\partial } \\phi = \\omega _ { \\varphi _ { \\epsilon } } . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} N _ { 3 , h } ( R ) = C ' R ^ { \\frac { 1 } { 2 } } \\log R + C R ^ { \\frac { 1 } { 2 } } + O ( R ^ { \\frac { 1 } { 2 } - \\frac { 1 } { 4 4 } + \\epsilon } ) \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\{ a , b \\} = \\sum _ { \\alpha \\in \\Lambda } \\psi _ { \\alpha } ( a ) \\{ x _ { \\alpha } , b \\} , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} \\tilde G ^ \\varepsilon ( y ' ) = G _ 0 ^ \\varepsilon ( \\varepsilon y ' , x ) y ' \\in \\frac { 1 } { \\varepsilon } ( \\Omega _ \\varepsilon \\cap B _ \\rho ( 0 ) ) . \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} I ^ m f ( \\gamma ) = m ! ( - 1 ) ^ m I ^ 0 v _ m ( \\gamma ) = 0 \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\mathtt { B } _ { b , p , c } ( z ) : = \\sum _ { k = 0 } ^ \\infty \\frac { ( - c ) ^ k } { k ! \\ ; \\mathrm { \\Gamma } { \\left ( k + p + \\frac { b + 1 } { 2 } \\right ) } } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 k + p } . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\Phi ' ( v ) = m ^ { - 1 } ( v ) , \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\Upsilon = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N a _ i Q _ i \\ , . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\hat { \\psi } ( 0 ) = \\hat { \\psi } ( 1 ) = 0 , \\hat { \\psi } ( r ) > 0 , \\mbox { f o r } 0 < r < 1 . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\aligned & - ( \\psi _ 2 ^ { i + 1 i } \\circ \\frak h _ { a b } ^ i ) \\vert _ { E ^ i } + ( \\frak h _ { a b } ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } ) \\vert _ { E ^ i } - ( \\frak h _ { a } ^ { i + 1 i } ) \\vert _ { E ^ i } + ( \\frak h _ { b } ^ { i + 1 i } ) \\vert _ { E ^ i } \\\\ & = \\left ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak H _ { a b } ^ { i + 1 i } - \\frak H _ { a b } ^ { i + 1 i } \\circ \\hat d _ { 1 } ^ { i } \\right ) \\vert _ { E ^ i } . \\endaligned \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\pi _ i ( \\varepsilon y ) & = - 4 \\pi \\alpha _ 3 \\mu _ i ^ { \\frac { 1 } { 2 } } H _ \\lambda ( \\varepsilon y , \\zeta _ i ) + O ( \\mu _ i ^ { \\frac { 3 } { 2 } } ) = - 4 \\pi \\alpha _ 3 \\mu _ i ^ { \\frac { 1 } { 2 } } g _ \\lambda ( \\zeta _ i ) + O ( \\varepsilon ^ { \\frac { 3 } { 2 } } ) . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\det \\left ( \\det X \\begin{pmatrix} ( J \\setminus ( j _ \\alpha ) ) \\sqcup K \\\\ ( J \\setminus ( j ' _ \\beta ) ) \\sqcup K ' \\end{pmatrix} \\right ) _ { 1 \\le \\alpha , \\ , \\beta \\le q } \\\\ = \\left ( \\det X \\begin{pmatrix} J \\sqcup K \\\\ J ' \\sqcup K ' \\end{pmatrix} \\right ) ^ { q - 1 } \\cdot \\det X \\begin{pmatrix} K \\\\ K ' \\end{pmatrix} . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} d ^ { 2 } W / d \\xi ^ { 2 } = \\left \\{ { u ^ { 2 } + u \\phi \\left ( \\xi \\right ) + \\psi \\left ( \\xi \\right ) } \\right \\} W , \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} u ^ \\epsilon ( x , t ) = u ( \\hat { x } | x | ^ { 1 / \\epsilon } , t / \\epsilon ) , \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\Phi _ t \\vert _ { \\{ L \\} \\times E ^ s _ \\sigma } = \\Phi ^ * _ t \\vert _ { \\{ L \\} \\times E ^ s _ \\sigma } . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} L ^ { \\rm r e s c , h } _ n ( u ) : = \\frac { L _ { ( 0 , 0 ) \\to ( \\gamma ^ 2 n + \\beta _ 1 u n ^ { 2 / 3 } , n ) } - \\left ( ( 1 + \\gamma ) ^ 2 n + 2 u ( 1 + \\gamma ) ^ { 5 / 3 } \\gamma ^ { 1 / 3 } n ^ { 2 / 3 } - \\beta _ 2 u ^ 2 n ^ { 1 / 3 } \\right ) } { \\beta _ 2 n ^ { 1 / 3 } } , \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\forall ( Y _ 1 , \\dots , Y _ { n - 1 } ) \\beta _ t ^ A ( \\nabla _ t Y _ 1 , \\dots , \\nabla _ t Y _ { n - 1 } ) = \\beta _ 0 ^ A ( \\widetilde { \\nabla } _ t ^ A Y _ 1 , \\dots , \\widetilde { \\nabla } _ t ^ A Y _ { n - 1 } ) . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\partial _ { \\boldsymbol e } \\eta ^ { t , \\bar t } = \\left ( \\frac { \\partial _ N ( \\partial _ { \\boldsymbol e } u ^ { t , \\bar t } ) } { ( N \\cdot \\nu ^ { t , \\bar t } ) ^ 2 \\ , \\Delta h ^ { t , \\bar t } } \\right ) ( z , \\eta ^ { t , \\bar t } ( x ) ) , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} P = P _ { 4 , 1 } \\cdot P _ 4 ( \\mathcal { O } ) \\cdot P _ { 4 , 2 } ( \\mathcal { O } ) \\cdot u _ 3 ^ 2 u _ 4 ^ 3 . \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} h ^ i ( Y , V ) = \\sum _ { s \\in \\Sigma } \\dim \\ , \\mathbb { H } ^ { i - 1 } ( w ^ { - 1 } ( s ) , \\phi _ { w - s } \\mathbb { C } ) . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} O b j ( \\bold { P _ i } ) = \\sum _ { k } \\omega _ k C R T _ k ( \\bold { P _ i } ) , \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ { \\chi , j } = \\frac { \\gamma _ { \\chi , j } } { 2 \\pi } \\log X \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\mathbb { E } ( \\# \\mathrm { r e a l s } ) = 2 \\sum _ { j = 0 } ^ { N - 2 } ( - 1 ) ^ j \\prod _ { i = 1 } ^ m \\binom { L _ i + j } { L _ i } a _ { 2 \\lceil \\frac j 2 + 1 \\rceil - 1 , 2 \\lfloor \\frac j 2 + 1 \\rfloor } , \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } \\omega Z ) & = g _ { 1 } ( \\mathcal { T } _ { U } \\omega \\phi Z , V ) \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} f ( x ) = x ^ { 2 ^ t + 3 } , \\ \\ \\ t = \\frac { n - 1 } { 2 } \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\norm { T _ \\lambda } _ 2 \\leq \\begin{cases} C \\lambda ^ { - ( n _ X + n _ Y ) / ( 2 d ) } & \\ d > n _ { X } + n _ { Y } , \\\\ C \\lambda ^ { - 1 / 2 } \\log { \\lambda } & \\ d = n _ { X } + n _ { Y } , \\\\ C \\lambda ^ { - 1 / 2 } & \\ 2 \\leq d < n _ { X } + n _ { Y } . \\end{cases} \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} h ( j , t ) = \\begin{cases} 2 J ( t ) + \\sum ^ j _ { i = 1 } ( 1 - 2 \\eta _ i ( t ) ) & \\textrm { f o r } j \\geq 1 , \\\\ 2 J ( t ) & \\textrm { f o r } j = 0 , \\\\ 2 J ( t ) - \\sum ^ 0 _ { i = j + 1 } ( 1 - 2 \\eta _ i ( t ) ) & \\textrm { f o r } j \\leq - 1 , \\end{cases} \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} N = \\sum _ { i = 0 } ^ { \\min ( r - k , \\ell ) } \\frac { ( r - 1 ) ! ( d - r - 1 ) ! } { ( k + i ) ! ( r - k - i ) ! ( \\ell - i ) ! ( d - r - \\ell + i ) ! } \\left ( ( k + i ) d - ( k + \\ell ) r \\right ) ; \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\tfrac { d } { d t } ( \\mathrm { e x p } ( Y ( t ) ) ) \\mathrm { e x p } ( Y ( t ) ) ^ { - 1 } & = \\frac { \\mathrm e ^ { \\mathrm { a d } ( Y ( t ) ) } - 1 } { \\mathrm { a d } ( Y ( t ) ) } \\dot Y ( t ) , \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} w ^ t _ 2 ( x ) = - \\int \\frac 1 t \\bigg ( \\frac { \\partial _ N v } { N \\cdot \\nu ^ 0 } \\ , \\mathcal { H } ^ { n - 1 } \\restriction _ { \\Gamma ^ 0 } - \\frac 1 t \\Delta h ^ 0 \\ , \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } \\bigg ) ( d y ) P ( x - y ) . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\textnormal { ( I ) } & : = \\frac { 4 ( p - 2 ) } { p } \\int u _ k ^ { p / 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u _ k ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x , \\\\ \\textnormal { ( I I ) } & : = \\int u _ k ^ p ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x + ( p - 1 ) \\int u \\eta ^ 2 u _ k ^ p \\ ; d x + p k \\int u \\eta ^ 2 u _ k ^ { p - 1 } \\ ; d x . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\left ( C \\mathcal { E } \\right ) \\left ( x \\right ) = \\left ( C _ { i j } ^ { \\ : \\ ; \\ ; k l } \\left ( x \\right ) \\mathcal { E } _ { k l } \\left ( x \\right ) \\right ) _ { i , j } . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} S ( \\mathfrak { C } ) = \\mathbb { G } \\cap Q \\cap \\tilde { Q } \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} G _ 1 & = C _ 8 \\rtimes D _ 8 = \\langle x , y , z \\mid x ^ 8 , y ^ 4 , z ^ 2 , z y z ^ { - 1 } = y ^ { - 1 } , y x y ^ { - 1 } = x ^ 5 , z x z ^ { - 1 } = x \\rangle . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\partial _ t | w ( t ) | ^ 2 = \\mathcal { O } ( t ^ { - \\frac 5 4 + \\beta } \\| w \\| _ { X _ T } ^ 3 ) | w ( t ) | , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\delta _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - \\frac { 1 } { n } r ^ { n } , \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\frac { d y ( t ) } { d t } = - c y ( t ) , y ( 0 ) = y _ 0 > 0 , \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} f \\ = \\ c _ u z ^ u \\ + \\ \\sum _ { v \\prec u } c _ v z ^ v c _ u \\neq 0 \\ , , \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} f _ { S _ M } ( s ) = \\sum _ { i = 1 } ^ { M } \\frac { \\prod _ { j = 1 } ^ M \\lambda _ j } { \\prod _ { j = 1 , j \\neq i } ^ M ( \\lambda _ j - \\lambda _ i ) } e ^ { - s \\lambda _ i } . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( | Z _ { T + h } | = P _ Y , | Z _ T | = P _ Y \\right ) , \\mathbb { E } Y _ T ^ 2 Y _ { T + h } ^ 2 \\ : \\mbox { f o r } \\ : T \\geq 0 , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\nabla w ( x ) = ( w ^ * ) ' ( \\mu _ u ( u ( x ) ) ) \\ , ( \\mu _ u ) ' ( u ( x ) ) \\ , \\nabla u ( x ) \\ , . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { M } } = ( \\phi _ 1 , \\ldots , \\phi _ m ) : ( S ^ d ) ^ k \\rightarrow U _ k ^ { \\oplus m } \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} N _ { d , h } ( R ) & = \\delta _ { [ d = 3 ] } \\delta _ { [ h = a ^ 2 ] } C ' _ 3 R ^ { \\frac { 1 } { 2 } } \\log R + C _ d R ^ { \\frac { d } { 2 } - 1 } + O ( R ^ { \\frac { d } { 2 } - 1 - \\lambda ( d ) + \\epsilon } ) . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} B ' ( x , 1 ) = \\frac { x ( x - 1 ) ^ 2 ( 2 x - 1 ) ( x ^ 2 - x + 1 ) A ( x , 1 ) + x ( 3 x ^ 4 - 7 x ^ 3 + 9 x ^ 2 - 5 x + 1 ) } { ( x - 1 ) ^ 2 ( 2 x - 1 ) } . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} c ( x , z ) c ( y , z ) \\alpha ( x , z , y ) & = c ( x + y , z ) \\alpha ( x , y , z ) \\alpha ( z , x , y ) \\\\ c ( x , z ) c ( x , y ) \\alpha ( x , y , z ) & = c ( x , y + z ) \\alpha ( y , x , z ) \\alpha ( y , z , x ) \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} r _ j = r \\sqrt { 8 h ( q _ j ) G _ j ( q _ j ) } \\ \\ \\textrm { f o r } \\ \\ j = 1 , \\cdots , m . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} & \\Delta ( x ) = x \\otimes 1 + 1 \\otimes x , \\\\ & \\Delta ( y ) = y \\otimes 1 + 1 \\otimes y , \\\\ & \\Delta ( z ) = z \\otimes 1 + 1 \\otimes z - 2 x \\otimes y , \\\\ & S ( x ) = - x , S ( y ) = - y , S ( z ) = - z - 2 x y , \\\\ & \\{ x , y \\} = y , \\{ y , z \\} = y ^ 2 , \\{ x , z \\} = z . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} N _ k \\ , : = \\ , \\inf \\{ n > N _ { k - 1 } : X _ n ^ 1 = X _ n ^ 2 \\} , k = 1 , 2 , \\dots \\ , . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\mathcal { F } _ q ( H , q ^ { - m } ) = 0 . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\| g \\| _ { E ^ \\times } = \\sup \\left \\{ \\int _ I \\mu ( f ) \\mu ( g ) : \\| g \\| _ E \\le 1 \\right \\} , \\ \\ \\ g \\in E ^ \\times . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} Z ( t ) = X ( 0 ) + \\int _ 0 ^ t F ( Z ( s ) , \\alpha _ s ) d s + \\int _ 0 ^ t \\sigma ( Z ( s ^ - ) , \\alpha _ s ) d B _ s . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} C ^ n _ { x y } : = \\begin{cases} C ^ n _ { e , \\tilde { e } } x \\in e ^ n , \\ , y \\in \\tilde { e } ^ n x \\sim y \\\\ 0 x y \\Gamma ^ n . \\end{cases} \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} a ' ( n ; i + 1 ) - a ' ( n ; i ) & = a ( n - 1 ; i ) + a ( n ; i + 1 , i - 1 ) - a ( n ; i , i - 1 ) + \\sum _ { j = 1 } ^ { i - 2 } a ( n ; i + 1 , j ) - a ( n ; i , j ) \\\\ & = a ( n - 1 ; i ) - \\sum _ { i ' = i - 2 } ^ { n - 5 } w _ { i ' , i - 2 } + \\sum _ { j = 0 } ^ { i - 2 } w _ { i - 3 , j } , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\| \\Delta ( h ) \\| _ 2 ^ 2 = \\iiint e ^ { i A O _ { \\tau } ( u , v ) } H _ { \\tau } ( u ) \\Theta _ { \\tau } ( v ) d u d v \\ , d \\tau , \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} X _ L : \\begin{cases} \\dot w & = i l w \\\\ \\dot \\varphi & = 1 \\end{cases} \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\sigma _ 1 ( u _ 1 , u _ 2 , e _ 1 , e _ 2 ) & : = \\frac { 1 } { 2 } \\left \\langle \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\\\ e _ 1 \\\\ e _ 2 \\end{bmatrix} , \\begin{bmatrix} 0 & - I & 0 & 0 \\\\ - I & 0 & I & 0 \\\\ 0 & I & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\\\ e _ 1 \\\\ e _ 2 \\end{bmatrix} \\right \\rangle . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A Q ( \\overline { \\partial } \\eta ) = \\widetilde { Q } ( \\overline { \\partial } \\eta ) \\overline { \\partial } _ A \\overline { \\partial } \\eta . \\end{gather*}"}
-{"id": "6222.png", "formula": "\\begin{align*} b _ n = \\chi _ { - n } A \\chi _ n . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} { ( T ) } = \\lambda { ( T _ 1 ) } . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } F _ { \\omega _ 0 } ( n ) ^ { h _ 0 } \\prod _ { i = 1 } ^ { h _ 1 } F _ { \\omega _ { q _ i } } ( n ) ^ 2 \\prod _ { i = h _ 1 + 1 } ^ { h _ 1 + h _ 2 } F _ { \\omega _ { q _ i } } ( n ) \\ll x ( \\log \\log x ) ^ { ( 3 h _ 0 + 2 h _ 1 + h _ 2 - 1 ) / 2 } . \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} L _ z ^ \\pm P _ \\pm = \\frac 1 { 2 t } L _ z P _ \\pm , \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} P ^ { ( a ) } ( u ) = \\frac { P ( u ) } { \\prod _ { k = 0 } ^ { a - 1 } ( u - \\alpha + k m ) ( u - l + \\alpha - k m ) } \\in \\C ( u ) , \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} A _ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { \\partial Y _ 0 } a _ { i } d y , B _ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { \\partial Y _ 0 } b _ { i } d y . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} ( 1 - v + x v ^ 2 ) H ( x ; v ) = H _ 2 ( x ) v ^ 2 + H _ 1 ( x ) v ( 1 - v ) \\ , . \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} J _ 0 = \\frac { 2 \\sqrt { 2 } } { | G ^ T \\nu | ^ 2 } . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} z ' & = w \\\\ w ' & \\geq c _ 2 \\left ( \\frac { c _ 1 } { c _ 2 } z ^ 3 - w \\right ) \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} b ^ \\prime = \\lceil ( k - r + 1 ) b / r \\rceil . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} \\textnormal { ( I I ) } & = p \\int ( u ^ p ) ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x - \\int ( ( p - 1 ) u ^ p ) ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x \\\\ & \\ ; \\ ; \\ ; \\ ; + \\int u \\eta ^ 2 ( ( p - 1 ) u ^ p ) \\ ; d x \\\\ & = \\int u ^ p ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x + \\int u \\eta ^ 2 \\left ( ( p - 1 ) u ^ p \\right ) \\ ; d x . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\mu } F _ \\lambda ( \\mu , \\zeta ) = 0 \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} R _ { n } ^ { 2 } - g R _ { n - 1 } ^ { 2 } - f R _ { n } R _ { n - 1 } = \\left ( - g h ^ { 2 } + k ^ { 2 } - f h k \\right ) ( - g ) ^ { n - 1 } , \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\sigma _ i . f : = \\frac { x _ i f - x _ { i + 1 } s _ i ( f ) } { x _ i - x _ { i + 1 } } . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} M _ G ^ { p , 0 } ( 0 , T ) : = & \\{ \\eta = \\sum _ { j = 0 } ^ { N - 1 } \\xi _ j ( \\omega ) I _ { [ t _ j , t _ { j + 1 } ) } ( t ) : N \\in \\mathbb { N } , \\ 0 \\leq t _ 0 \\leq t _ 1 \\leq \\cdots \\leq t _ N \\leq T , \\\\ & \\ \\xi _ j \\in L _ { G } ^ p ( \\Omega _ { t _ j } ) , \\ j = 0 , 1 \\cdots , N \\} . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\widehat { u } ^ n _ t ( y ) & : = \\sum _ e \\Big ( Y ^ e _ t ( \\phi ) + Z ^ e _ t ( \\phi ) + E ^ { ( 1 , e ) } _ t ( \\phi ) + E ^ { ( 2 , e ) } _ t ( \\phi ) \\Big ) . \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\frac { ( 2 q k ) ! } { ( 2 q ) ! ^ k } \\cdot \\prod _ { i = 2 q k + 1 } ^ { q ^ \\ell } i = \\frac { q ^ \\ell ! } { ( 2 q ) ! ^ k } \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma \\le T } } \\frac 2 { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } \\ge \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma \\le T } } \\frac 2 \\gamma - \\frac 1 4 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma \\le T } } \\frac { 1 } { \\gamma ^ 3 } > W ( 1 ) - \\frac 1 4 \\sum _ { \\gamma \\in \\Gamma ^ S ( \\chi ) } \\frac { 1 } { \\gamma ^ 3 } = \\max \\{ 8 , | 2 - n | \\} . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} c _ 1 ( T ) \\sum _ { n = - \\infty } ^ { \\infty } | C _ n | ^ 2 \\le \\int _ { 0 } ^ { T } \\Big | \\sum _ { n = - \\infty } ^ { \\infty } \\big ( C _ n e ^ { i \\omega _ n t } + R _ n e ^ { r _ n t } \\big ) \\Big | ^ 2 d t \\ , , \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\dfrac { \\partial ^ { ( k ) } } { \\partial a _ j } \\left ( a _ { i _ 1 } \\otimes \\ldots \\otimes a _ { i _ { m } } \\right ) = \\left ( - 1 \\right ) ^ { | a _ j | ( | a _ { i _ 1 } | + \\ldots | a _ { i _ { k - 1 } } | ) } \\delta _ j ^ { i _ k } \\left ( a _ { i _ 1 } \\otimes \\ldots \\otimes \\widehat { a _ { i _ k } } \\otimes \\ldots \\otimes a _ { i _ { m } } \\right ) \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } j a _ { i j } r ^ j & = c _ i r + \\sum _ { j = 1 } ^ n c _ { i j } \\left ( \\sum _ { q = 1 } ^ { \\infty } a _ { j q } r ^ q \\right ) \\\\ & + \\sum _ { k _ 1 + \\cdots + k _ { n + 1 } \\geq 2 } c _ { i k _ 1 \\cdots k _ { n + 1 } } \\left ( \\sum _ { j = 1 } ^ { \\infty } a _ { k _ 1 j } r ^ j \\right ) ^ { k _ 1 } \\cdots \\left ( \\sum _ { j = 1 } ^ { \\infty } a _ { k _ n j } r ^ j \\right ) ^ { k _ n } r ^ { k _ { n + 1 } } \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\sigma _ F = \\left ( \\frac { 1 + \\mathbf { b } } { 1 - \\mathbf { b } } \\right ) ^ 2 = \\kappa _ F ^ 2 . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} f ^ { \\boxplus \\tau } _ { ! } ( h ^ { \\boxplus \\tau } ; \\frak S ^ { \\boxplus \\tau } ) = f _ { ! } ( h ; \\frak S ) . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\nu _ \\mu ^ c ( T _ \\lambda ) = \\sum _ { \\substack { \\nu \\in X ^ + ( T ) \\\\ \\mu \\preccurlyeq \\nu \\prec \\lambda } } { a _ \\nu \\chi _ \\mu ( \\nu ) } , \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\tau _ { \\boldsymbol { \\zeta } } \\doteq \\inf \\{ t \\in [ 0 , T ] : r ( \\boldsymbol { \\zeta } ( t ) ) = 0 \\} \\wedge T . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} C _ 1 ( \\delta _ d , M ( G ) ) : = \\begin{cases} 1 / 2 , & \\textrm { i f $ \\delta _ d + M ( G ) \\leq 1 $ } \\\\ \\displaystyle 1 - \\frac { \\delta _ d + M ( G ) } { 2 } , & \\textrm { i f $ 1 < \\delta _ d + M ( G ) < 2 $ } . \\end{cases} \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} [ f ] ^ { ( 1 , \\frac { 1 } { 2 } ) } _ { Q } = | | \\nabla f | | _ { L ^ { \\infty } ( Q ) } + \\sup _ { x \\in \\Omega } [ f ] _ { ( \\delta , T ] } ^ { ( \\frac { 1 } { 2 } ) } ( x ) . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} C _ { \\theta } \\left ( u , v \\right ) = \\left ( u ^ { - \\theta } + v ^ { - \\theta } - 1 \\right ) ^ { - 1 / \\theta } , u , v \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} u _ i ( T , x ) = u _ { i } ^ { 0 } ( x ) \\ , , u _ { i t } ( T , x ) = u _ { i } ^ { 1 } ( x ) \\ , , x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} [ Y _ 1 , Y _ 2 ] & = 0 & [ Y _ 1 , Y _ 3 ] & = 0 & [ Y _ 1 , Y _ 4 ] & = 0 & [ Y _ 1 , Y _ 5 ] & = 0 \\\\ [ Y _ 2 , Y _ 3 ] & = 0 & [ Y _ 2 , Y _ 4 ] & = 0 & [ Y _ 2 , Y _ 5 ] & = Y _ 1 & [ Y _ 3 , Y _ 4 ] & = Y _ 1 \\\\ [ Y _ 3 , Y _ 5 ] & = \\alpha Y _ 1 + Y _ 2 & [ Y _ 4 , Y _ 5 ] & = \\beta Y _ 1 + \\gamma Y _ 2 \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} | U ' | \\sum _ a ( - 1 ) ^ { a } ( a - | U ' | ) \\binom { | S | - | U ' | } { a - | U ' | } = 0 . \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\Omega _ 0 ^ s & : = s , \\\\ \\Omega _ { n + 1 } ^ s & : = \\Omega ^ { \\Omega _ n ^ s } . \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} F _ { s + 1 } ^ { \\pm } \\left ( \\xi \\right ) = { \\dfrac { 1 } { 2 } } \\psi _ { s } \\left ( \\xi \\right ) - { \\dfrac { 1 } { 2 } } \\frac { d F _ { s } ^ { \\pm } \\left ( \\xi \\right ) } { d \\xi } - { \\dfrac { 1 } { 2 } } \\sum \\limits _ { j = 0 } ^ { s } { F _ { j } ^ { \\pm } \\left ( \\xi \\right ) F _ { s - j } ^ { \\pm } \\left ( \\xi \\right ) } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\left < R _ { i } \\left ( Z _ { i } \\right ) , \\displaystyle R _ { j } \\left ( Z _ { j } \\right ) \\right > = \\left < Z _ { i } , Z _ { j } \\right > , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} f _ { 2 } ( z _ { n } ) z _ { n } - f _ { 2 } ( t _ { n } z _ { n } ) t _ { n } z _ { n } = ( f _ { 2 } ' ( \\sigma ^ { i } _ { n } ) \\sigma ^ { i } _ { n } + f _ { 2 } ( \\sigma ^ { i } _ { n } ) ) z _ { n } ( 1 - t _ { n } ) , \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} 2 \\deg ( \\mathcal { L } _ 1 ) - d _ 1 = \\deg ( \\mathcal { L } _ 3 ) = 2 \\deg ( \\mathcal { L } _ 2 ) - d _ 2 , \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\left < R _ { k } \\left ( Z _ { i } \\right ) , \\displaystyle R _ { l } \\left ( Z _ { j } \\right ) \\right > = 0 , \\quad \\mbox { f o r a l l $ k , l , i , j = 1 , \\dots , q $ w i t h $ k \\neq i $ a n d $ l \\neq j $ . } \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} E ( f ) : = \\frac { \\int _ M | \\nabla f | ^ p \\ d \\mathfrak { m } } { \\int _ M | f | ^ p \\ d \\mathfrak { m } } . \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\partial _ { t } u _ { i } ^ { \\varepsilon } + \\mathcal { A } _ { d _ { i } } ^ { \\varepsilon } u _ { i } ^ { \\varepsilon } = \\rho _ { i } ^ { \\varepsilon } \\nabla ^ { \\delta } \\theta ^ { \\varepsilon } \\cdot \\nabla u _ { i } ^ { \\varepsilon } + R _ { i } \\left ( u ^ { \\varepsilon } \\right ) \\quad \\mbox { i n } \\ ; Q _ { T } ^ { \\varepsilon } , \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} \\min _ { \\scriptstyle \\omega \\in \\mathcal D ^ s _ { \\rm S r } ( \\R ^ n _ + ) \\atop \\scriptstyle \\omega | _ { \\R ^ n _ + } = u } \\mathcal E _ s ( \\omega ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) ; \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\phi ( t , x , y ) = t \\Phi ( t ^ { - 1 } z ) , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{gather*} f _ { 4 , 1 , 0 } ( x _ 1 , x _ 2 , x _ 3 ; q , 2 ) = - ( q - 1 ) \\frac { ( x _ 2 + x _ 3 ) ( x _ 1 - q x _ 3 ) ( x _ 1 - q x _ 2 ) } { ( q x _ 2 - x _ 3 ) ( q x _ 1 - x _ 3 ) ( q x _ 1 - x _ 2 ) } , \\\\ f _ { 2 , 1 , 1 } ( x _ 1 , x _ 2 , x _ 3 ; q , 2 ) = - ( q - 1 ) ^ 2 \\frac { ( q + 1 ) ( x _ 2 - x _ 3 ) \\big ( x _ 1 ^ 2 + x _ 2 x _ 3 + 2 x _ 1 x _ 2 + 2 x _ 1 x _ 3 \\big ) } { ( q x _ 1 - x _ 2 ) ( q x _ 1 - x _ 3 ) ( q x _ 2 - x _ 3 ) } . \\end{gather*}"}
-{"id": "949.png", "formula": "\\begin{align*} r _ { \\langle \\rangle } ( \\mathcal C _ t P ) & = ( \\exists _ z , \\mathbb L _ { \\vartheta ( t + \\Omega ^ { o _ { \\langle \\rangle } ( n ( P , 0 ) ) } ) } ^ u , \\forall _ { x \\in a } \\exists _ { y \\in z } \\theta ) , \\\\ n ( \\mathcal C _ t P , a ) & = \\mathcal B _ { \\exists , \\forall _ { x \\in a } \\exists _ y \\theta } ^ { \\vartheta ( t + \\Omega ^ { o _ { \\langle \\rangle } ( n ( P , 0 ) ) } ) } \\mathcal C _ t n ( P , 0 ) . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} \\widetilde { v } _ { i } \\mid _ { \\Gamma } = \\widetilde { p } _ { 0 , i } \\left ( x \\right ) , \\partial _ { n } \\widetilde { v } _ { i } \\mid _ { \\Gamma } = \\widetilde { p } _ { 1 , i } \\left ( x \\right ) , i = 0 , . . . , N - 1 , \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} W \\left ( x \\right ) = \\left ( w _ { 0 } , w _ { 1 } , . . . , w _ { N - 1 } \\right ) ^ { T } \\left ( x \\right ) = V \\left ( x \\right ) - p \\left ( x \\right ) . \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} K _ X = ( \\det E ) ^ { - 5 } . \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\phi = - \\frac { z ^ 2 } { 4 t } . \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\textrm { t r } ^ { \\flat } \\ ( \\prod _ { j = 1 } ^ J \\ ( \\mathcal { K } _ t ^ { m _ j } \\ ) _ b \\ ) = 0 . \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} \\biggl \\| \\sum _ { l = 1 } ^ { n } \\left ( \\cos ( \\varepsilon ^ { - 1 } \\tau \\sqrt { \\mathstrut \\lambda _ l ( t ) } ) - \\cos ( \\varepsilon ^ { - 1 } \\tau | t | \\sqrt { \\mathstrut \\gamma _ l } ) \\right ) ( \\cdot , \\omega _ l ) \\omega _ l \\biggr \\| \\varepsilon ^ { s } ( t ^ 2 + \\varepsilon ^ 2 ) ^ { - s / 2 } \\le \\widehat { C } ( \\tau ) \\varepsilon \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} D _ t - { A } ( t , x , D _ x ) = \\big ( D _ t - \\mu ( t , x , D _ x ) \\big ) I - { \\hat A } ( t , x , D _ x ) \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\tilde { \\mathcal { W } } \\left ( Z ' , Z '' \\right ) = \\left ( \\frac { \\left ( I _ { q } - A { Z ''' } \\right ) { Z ' } ^ { t } } { \\sqrt { I _ { q } + A } \\left ( I _ { q } + A ^ { 2 } - A { Z ''' } - A { Z '' } ^ { t } \\right ) } , \\frac { \\left ( A - Z ''' \\right ) { Z ' } ^ { t } } { \\sqrt { I _ { q } + A } \\left ( I _ { q } + A ^ { 2 } - A { Z ''' } - A { Z '' } ^ { t } \\right ) } , \\frac { A { Z '' } ^ { t } + A { Z ''' } - \\left ( I _ { q } + A ^ { 2 } \\right ) { Z ''' } { Z '' } ^ { t } } { I _ { q } + A ^ { 2 } - A { Z ''' } - A { Z '' } ^ { t } } \\right ) . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} \\frac { 1 } { C } m ^ n \\leq A _ m ( x ) : = S _ { m } ( x ) + S _ { k _ { 1 } m } ( x ) + \\cdot \\cdot \\cdot + S _ { k _ { t - 1 - j _ 0 } m } ( x ) \\leq C m ^ { n } , \\ \\ \\forall x \\in X ^ { j _ 0 } _ { { \\rm s i n g \\ , } } , \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\Im m _ { \\mu _ A \\boxplus \\mu _ B } ( z ) \\sim \\begin{cases} \\sqrt { \\kappa + \\eta } \\ , , \\quad & E \\in \\mathrm { s u p p } \\ , \\mu _ A \\boxplus \\mu _ B \\ , , \\\\ \\frac { \\eta } { \\sqrt { \\kappa + \\eta } } \\ , , & E \\not \\in \\mathrm { s u p p } \\ , \\mu _ A \\boxplus \\mu _ B \\ , , \\end{cases} \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} B = \\frac { 1 } { 2 } \\left ( r + \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right | ^ 2 - 2 \\Re \\left ( \\frac { P _ { r + 2 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) \\right ) \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\bar { \\pi } : = \\pi | _ { \\pi ^ { - 1 } ( U ' ) } : \\bar { U } : = \\pi ^ { - 1 } ( U ' ) \\to U ' \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} F _ \\lambda ( \\zeta , \\mu ) & : = \\ , k \\ , a _ 0 + a _ 1 \\sum _ { i = 1 } ^ k \\Bigl ( \\mu _ i \\ , g _ { \\lambda } ( \\zeta _ i ) - \\sum _ { j \\neq i } \\mu _ i ^ { 1 / 2 } \\ , \\mu _ j ^ { 1 / 2 } \\ , G _ { \\lambda } ( \\zeta _ i , \\zeta _ j ) \\Bigr ) + a _ 2 \\ , \\lambda \\ , \\sum _ { i = 1 } ^ k \\mu _ i ^ 2 \\\\ & - a _ 3 \\ , \\sum _ { i = 1 } ^ k \\Bigl ( \\mu _ i \\ , g _ { \\lambda } ( \\zeta _ i ) - \\sum _ { j \\neq i } \\mu _ i ^ { 1 / 2 } \\mu _ j ^ { 1 / 2 } \\ , G _ { \\lambda } ( \\zeta _ i , \\zeta _ j ) \\Bigr ) ^ 2 , \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} x _ { g h } = x _ { g } ( \\sigma _ { g } \\circ x _ h ) . \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} l _ { j , 1 } = \\left \\{ \\begin{array} { l l } a _ { j , r } / a _ { r , r } , & \\mbox { f o r } 2 \\le j \\le n , ~ j \\not = r , \\\\ a _ { 1 , r } / a _ { r , r } , & \\mbox { f o r } j = r . \\end{array} \\right . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{gather*} P _ { ( \\lambda _ 1 + 1 , \\dots , \\lambda _ N + 1 ) } ( x _ 1 , \\dots , x _ N ; q , t ) = ( x _ 1 \\cdots x _ N ) \\cdot P _ { \\lambda } ( x _ 1 , \\dots , x _ N ; q , t ) . \\end{gather*}"}
-{"id": "6051.png", "formula": "\\begin{align*} J _ 0 ( x ) = \\frac { x ^ 2 } { 1 - x } \\big ( K ( x ) - 1 \\big ) \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} \\inf _ { x \\in [ - R _ 0 , R _ 0 ] } p ( x ) = p _ { \\mathrm { m i n } } > 0 . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\varepsilon ^ { 1 ' 0 ' } = - \\varepsilon ^ { 0 ' 1 ' } = 1 , \\varepsilon ^ { 0 ' 0 ' } = \\varepsilon ^ { 1 ' 1 ' } = 0 \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ k ^ { ( i ) } \\frac { \\partial \\| \\mathbf { g } _ i \\| ^ { - 1 } } { \\partial g _ { i k } } \\mathbf { e } _ k ^ * X _ i G \\mathbf { e } _ i = - \\frac { 1 } { 2 N } \\frac { 1 } { \\| \\mathbf { g } _ i \\| ^ 3 } \\sum _ { k } ^ { ( i ) } \\bar { g } _ { i k } \\mathbf { e } _ k ^ * X _ i \\mathbf { e } _ i = - \\frac { 1 } { 2 N } \\frac { 1 } { \\| \\mathbf { g } _ i \\| ^ 2 } \\mathring { \\mathbf { h } } _ i ^ * X _ i G \\mathbf { e } _ i = O _ \\prec ( \\frac { 1 } { N } ) , \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( X _ { \\tau + t _ { 1 } } ^ { x } , \\cdots , X _ { \\tau + t _ { m } } ^ { x } ) ] = \\hat { \\mathbb { E } } [ \\varphi ( X _ { t _ { 1 } } ^ { y } , \\cdots , X _ { t _ { m } } ^ { y } ) ] _ { y = X _ { \\tau } ^ { x } } . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} N _ 0 = \\sum _ { l = 1 } ^ { n } \\mu _ l ( \\cdot , \\omega _ l ) \\omega _ l , N _ * = \\sum _ { l = 1 } ^ { n } \\gamma _ l \\left ( ( \\cdot , \\widetilde { \\omega } _ l ) \\omega _ l + ( \\cdot , \\omega _ l ) \\widetilde { \\omega } _ l \\right ) . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} u \\left ( x , x _ { 0 } , k \\right ) = A \\left ( x , y \\right ) e ^ { i k \\tau \\left ( x , y \\right ) } \\left ( 1 + h \\left ( x , y \\left ( x _ { 0 } \\right ) , k \\right ) \\right ) , k \\rightarrow \\infty , \\forall x \\in \\overline { \\Omega } , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} S _ j u \\ , : = \\ , \\chi ( 2 ^ { - j } D ) \\ , = \\ , \\sum _ { k \\leq j - 1 } \\Delta _ { k } \\qquad \\mbox { f o r } j \\geq 0 \\ , . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\sum _ { n _ 1 + n _ 2 = a } { b + n _ 1 \\choose n _ 1 } { n _ 2 \\choose c } = { a + b + 1 \\choose a - c } . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} ( \\lambda , m - d ( \\lambda ) ) ( \\mu , m - d ( \\mu ) ) ^ * & = ( \\lambda , m - d ( \\lambda ) ) \\Big ( \\sum _ { \\alpha \\in s ( \\lambda ) \\Lambda ^ { n - m } } ( \\alpha , m ) ( \\alpha , m ) ^ * \\Big ) ( \\mu , m - d ( \\mu ) ) ^ * \\\\ & = \\sum _ { \\alpha \\in s ( \\lambda ) \\Lambda ^ { n - m } } ( \\lambda \\alpha , m - d ( \\lambda ) ) ( \\mu \\alpha , m - d ( \\mu ) ) ^ * \\in B _ n . \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} p ( z ) = \\sum _ { j = r } ^ { d } P _ j ( z ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\epsilon ^ { \\frac { p ^ k - 1 } { 2 } } = - 1 . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial z ^ j } ( \\log { \\rm t r } _ { \\omega _ { \\epsilon } } \\omega _ { \\varphi _ { \\epsilon } } + \\Psi _ { \\epsilon , \\rho } - B \\varphi _ { \\epsilon } ) = \\frac { 1 } { { \\rm t r } _ { \\omega _ { \\epsilon } } \\omega _ { \\varphi _ { \\epsilon } } } \\varphi _ { \\epsilon i \\bar { i } j } + \\Psi _ { \\epsilon , \\rho , j } - B \\varphi _ { \\epsilon j } . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\theta _ 0 = \\min _ { 1 \\le j \\le m } \\theta _ { j , 0 } \\qquad \\theta _ \\infty = \\max _ { \\substack { 1 \\le j \\le m } } \\theta _ { j , \\ell _ j } \\ , . \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} P _ { 1 1 } ( 0 ) = \\sum _ { j , k \\in S _ { + } } p _ { j k } ( 0 ) | x _ j | | x _ k | , P _ { 1 2 } ( 0 ) = \\sum _ { j \\in S _ { + } , k \\in S _ { - } } p _ { j k } ( 0 ) | x _ j | | x _ k | , P _ { 2 2 } ( 0 ) = \\sum _ { j , k \\in S _ { - } } p _ { j k } ( 0 ) | x _ j | | x _ k | . \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} f _ { \\pm } ( z ) & = \\frac { ( \\alpha + 1 ) } { 2 } \\frac { c _ 1 ^ { \\ast } \\pm \\epsilon } { c _ 2 ^ { \\ast } \\mp \\epsilon } z ^ { \\frac { \\alpha - 1 } { \\alpha + 1 } } \\equiv \\frac { ( \\alpha + 1 ) } { 2 } \\gamma _ { \\pm } z ^ { \\frac { \\alpha - 1 } { \\alpha + 1 } } . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} { \\cal P } _ \\bot : = \\Pi _ { { \\mathbb S } _ 0 } ^ \\bot { \\cal P } \\Pi _ { { \\mathbb S } _ 0 } ^ \\bot \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\sigma _ j = 2 \\pi \\left ( \\frac { j } { \\omega _ n | \\Sigma | } \\right ) ^ { \\frac { 1 } { n } } + O ( 1 ) , j \\rightarrow \\infty , \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 1 & x _ 2 & \\cdots & x _ { n - 1 } & x _ n \\\\ y _ 1 & y _ 2 & \\cdots & y _ { n - 1 } & y _ n \\\\ 1 & 0 & \\cdots & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} | E ( y ) | \\leq C \\varepsilon ^ 5 , y \\in \\widetilde \\Omega _ \\varepsilon : = \\Omega _ \\varepsilon \\setminus \\bigcup _ { j = 1 } ^ k B _ { \\delta / \\varepsilon } ( \\zeta _ j ' ) , \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\omega \\left ( t \\right ) x ^ { i } r ^ { - 4 } = 0 . ~ \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} O ( X ^ { k + \\epsilon - \\lambda ( k ) } ) , \\lambda ( k ) = \\frac { 1 } { 6 + \\frac { 1 9 } { k } } . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} d \\varGamma _ { t , s } = \\varGamma _ { t , s - } \\biggl [ \\alpha _ s d s + \\beta _ s d W _ s + \\int _ U \\gamma _ s ( e ) \\widehat { \\pi } ( d e , d s ) \\biggr ] , \\quad \\varGamma _ { t , t } = 1 . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\sum _ { \\mu ^ { ( 1 ) } \\vdash q _ 1 } \\sum _ { \\mu ^ { ( 2 ) } \\vdash q _ 2 } \\cdots \\sum _ { \\mu ^ { ( k ) } \\vdash q _ k } \\frac { \\varepsilon _ { { \\cup _ i \\mu ^ { ( i ) } } } } { z _ { \\cup _ i \\mu ^ { ( i ) } } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( \\sum _ { i = 1 } ^ { k } m _ j ( \\mu ^ { ( i ) } ) \\right ) ! } { \\prod _ { i = 1 } ^ { k } m _ j ( \\mu ^ { ( i ) } ) ! } \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} P _ { r , k } ( A ) = 2 ^ { k + r } m ! ( r ! ) ^ 2 \\prod _ { j = 1 } ^ k ( A + r + j - 1 ) ( - 2 A + ( n - 2 j ) ) \\prod _ { j = 1 } ^ r ( A - n - j + 2 ) . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} u _ { i } ^ { \\varepsilon } \\left ( 0 , x \\right ) = u _ { i } ^ { \\varepsilon , 0 } \\left ( x \\right ) \\quad \\mbox { f o r } \\ ; x \\in \\Omega ^ { \\varepsilon } , \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} f _ * ( x ) : = \\inf \\bigg \\{ f ( x ) \\ : \\ f \\in C ( \\R ^ 2 ) , \\ f \\ge h ^ t , \\ \\Delta f \\le 0 , \\ \\lim _ { x \\to \\infty } \\frac { f } { - \\log | x | } = c ^ t \\bigg \\} \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} ( x ) _ { 0 , \\lambda } = 1 , \\ , \\ , ( x ) _ { l , \\lambda } = x ( x - \\lambda ) \\cdots ( x - ( l - 1 ) \\lambda ) , \\ , \\ , ( l \\geq 1 ) , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} F ( z ) \\stackrel { 1 } { = } 1 + 3 ( z - 1 ) + \\frac { 1 7 } { 2 } ( z - 1 ) ^ 2 + \\frac { 1 } { 2 } ( 4 7 + \\pi ^ 2 ) ( z - 1 ) ^ 3 + \\cdots , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { \\lambda T } \\Big \\{ \\sum _ { t = 0 } ^ { T - 1 } \\sum _ { i = 1 } ^ { 2 } u _ l ^ i ( t ) D _ { l t } ^ i ( t ) + u _ c ^ i ( t ) D _ { c t } ^ i ( t ) \\Big \\} . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} | \\nabla h ( x ) | ~ = ~ \\Big | \\int h ( y ) \\nabla _ x { F _ x ( y ) } \\ , d \\mu ( y ) \\Big | ~ \\le ~ \\int | h ( y ) | \\cdot | \\nabla _ x F _ x ( y ) | \\ , d \\mu ( y ) ~ \\le ~ \\norm { h } \\cdot \\norm { | \\nabla _ x F _ x | } \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} f ^ { 3 - p , q } ( Y _ \\Delta , w _ \\Delta ) = h ^ { p , q } ( \\hat { X } _ \\Delta ) . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\mathcal D ^ t _ 1 : = - \\frac 1 t \\Delta \\dot h ^ 0 \\ , \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } - \\frac 1 2 \\Delta \\delta _ t ^ 2 h ^ 0 \\chi _ { \\Omega ^ t } \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { \\infty } \\ln \\left ( \\lambda _ q \\right ) & \\leq - \\sum _ { q = 1 } ^ { \\infty } \\frac { 1 } { n ^ { \\ell _ q } } . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\mathcal { R } _ { i , j } ^ { s , t } : = \\int _ { \\mathcal { O _ \\rho } } U _ i ^ s \\ , U _ j ^ t = O ( \\mu ^ 3 ) . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} | d | ^ 2 - | b | ^ 2 = 1 \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda & 1 & 0 & 0 & \\dots & 0 \\\\ 0 & \\lambda & 1 & 0 & \\dots & 0 \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & \\lambda & 1 & 0 \\\\ 0 & 0 & \\dots & 0 & \\lambda & 1 \\\\ 0 & 0 & \\dots & 0 & 0 & \\lambda \\end{bmatrix} . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\frac { L ^ \\alpha [ e ^ a ] } { e ^ a } = \\begin{cases} \\frac { L ^ \\alpha [ e ^ g ] } { e ^ { g } } + \\frac { 1 } { 2 } \\int _ { | y | < R } \\left ( H _ \\epsilon ( a _ 0 , x , y ) + H _ \\epsilon ( a _ 0 , x , - y ) \\right ) K d y \\\\ [ 2 . 5 m m ] + \\int _ { | y | \\geq R } G _ \\epsilon ( x , y ) \\left ( 1 - \\frac { \\exp ( a _ 0 ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( a _ 0 / \\epsilon ) } \\right ) K d y , \\end{cases} \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\Xi _ { B } ( z _ { ( 1 ) } , z _ { ( 2 ) } ) = ( B \\cdot z _ { ( 1 ) } \\cdot B ^ T , B \\cdot z _ { ( 2 ) } ) , \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\theta _ 2 ( q ) & = 2 \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { ( 2 n + 1 ) ^ 2 } { 8 } } , & \\theta _ 3 ( q ) & = 1 + 2 \\sum _ { n \\geq 1 } q ^ { n ^ 2 / 2 } , & \\theta _ 4 ( q ) & = 1 + 2 \\sum _ { n \\geq 1 } ( - 1 ) ^ n q ^ { n ^ 2 / 2 } . \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} a = t _ 0 < t _ 1 < \\cdots < t _ n = b . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} D _ i ^ 2 = \\begin{cases} 2 l ^ * ( \\psi _ i ) - 2 & \\textnormal { i f $ v _ i $ i s a v e r t e x . } \\\\ - 2 & \\textnormal { i f $ v _ i $ i s i n t h e i n t e r i o r o f a n e d g e . } \\end{cases} \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\rho _ i ( x ) \\leq a \\sin \\sqrt { a } d ( x ) \\frac { \\dfrac { \\sqrt { a } } { \\kappa _ + } \\tan \\sqrt { a } d _ 0 + 1 } { \\dfrac { \\sqrt { a } } { \\kappa _ + } - \\tan \\sqrt { a } d _ 0 } = \\frac { a \\sin \\sqrt { a } d ( x ) } { \\tan \\sqrt { a } ( \\bar { h } - d _ 0 ( x ) ) } \\leq a \\cos \\sqrt { a } d ( x ) . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\| \\mathcal { R } ( \\mathbf { k } , \\varepsilon ) ^ { s / 2 } ( I - \\widehat { P } ) \\| _ { L _ 2 ( \\Omega ) \\to L _ 2 ( \\Omega ) } = \\sup _ { 0 \\ne \\mathbf { b } \\in \\widetilde { \\Gamma } } \\varepsilon ^ s ( | \\mathbf { b } + \\mathbf { k } | ^ 2 + \\varepsilon ^ 2 ) ^ { - s / 2 } \\le r _ 0 ^ { - s } \\varepsilon ^ s , \\varepsilon > 0 , \\ \\mathbf { k } \\in \\widetilde { \\Omega } . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } p _ { j l } = 0 , \\ \\forall 1 \\leq l \\leq r . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} W _ { \\boldsymbol { \\mathrm { T } } \\circ \\Theta ^ { ( n , \\sigma ) } } & = \\frac { 1 } { n } C _ { \\boldsymbol { \\mathrm { T } } \\circ \\Theta ^ { ( n , \\sigma ) } } = \\frac { 1 } { n } \\left ( \\sum _ { i , j = 1 } ^ { n } E _ { i j } \\otimes \\boldsymbol { \\mathrm { T } } \\circ \\Theta ^ { ( n , \\sigma ) } ( E _ { i j } ) \\right ) \\\\ & = \\frac { 1 } { n } \\left ( \\sum _ { i , j = 1 } ^ { n } E _ { i j } \\otimes W _ { j i } ^ { ( n , \\sigma ) } \\right ) , \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | x - a _ i | ^ { 2 \\beta } e ^ { - \\sum _ { i = 1 } ^ n \\frac { | A ^ { \\frac { 1 } { 2 } } ( x - a _ i ) | ^ 2 } { 2 } } \\ , d x \\le C _ 2 \\ , 2 ^ { 2 \\beta + N - 1 } \\tilde \\alpha _ 1 ^ { - \\beta - \\frac { N } { 2 } } \\sigma _ N \\Gamma \\left ( \\beta + \\frac { N } { 2 } \\right ) . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} Q ' & = \\sum _ b ( - 1 ) ^ b \\binom { b } { 2 } \\binom { r - | S | - | U | + | U ' | } { b } \\\\ & = \\binom { r - | S | - | U | + | U ' | } { 2 } \\sum _ b ( - 1 ) ^ b \\binom { r - | S | - | U | + | U ' | - 2 } { b - 2 } . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ s \\left ( 1 - \\frac 1 { p _ i } \\right ) < \\frac { c _ 1 } { 3 } . \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} J = e ^ { u t } , \\ \\phi = \\frac { \\chi } { u ^ { n } } , \\ \\psi _ { 0 } = \\frac { 2 T } { u } + \\frac { \\chi } { u ^ { n } } , \\ \\psi _ { 1 } = - \\frac { 2 T } { u ^ { 2 } } . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} ( A _ { d } ( \\delta _ { n } ) ) & \\leq C e ^ { - c ( \\delta _ { n } v _ { n } ) ^ \\tau } . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} R = R _ { 0 } + R _ { 1 } + \\cdots + R _ { n } , \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} m _ { 2 q + 1 } = \\bar { \\beta } _ { 2 q + 1 } = 0 \\ { \\rm a n d } \\ m _ { 2 q } = \\bar { \\beta } _ { 2 q } , \\forall \\ q \\in \\N _ 0 . { } \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} & \\left ( 1 - v u x - \\frac { x v } { 1 - v } \\right ) F ( x , u ; v ) \\\\ & = 1 - x + x ( 1 - x ) u K + x u F ( x , u ; 0 ) - x u - v u x + \\frac { x } { 1 - v } F ( x , u ; 1 ) \\\\ & + \\frac { u x ^ 3 } { ( 1 - x ) ^ 2 } \\left ( \\frac { F ( x , u ; 0 ) + F ( x , u ; 1 ) - 1 } { ( 1 - u x ) ( 1 - u v x ) } - 1 \\right ) \\\\ & + \\frac { u x ^ 3 } { ( 1 - x ) ^ 2 ( 1 - u x ) ( 1 - v u x ) } \\left ( \\frac { 1 } { 1 - x } F \\Big ( x , u ; \\frac { 1 } { 1 - x } \\Big ) - \\frac { x } { 1 - x } - F ( x , u ; 1 ) \\right ) . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\begin{aligned} y _ t - d \\Delta y & = f , & & x \\in \\Omega , \\\\ \\nabla y \\cdot \\nu & = 0 , & & x \\in \\partial \\Omega , \\\\ y ( x , 0 ) & = y _ 0 ( x ) , & & x \\in \\Omega \\end{aligned} \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} u ^ \\ast ( t ) = \\left \\{ \\begin{aligned} \\begin{pmatrix} 0 \\\\ k _ E \\end{pmatrix} & \\quad \\textnormal { f o r } t < \\tau _ 1 \\\\ \\begin{pmatrix} k _ M \\\\ 0 \\end{pmatrix} & \\quad \\textnormal { f o r } \\tau _ 1 < t < \\tau _ s \\\\ 0 & \\quad \\textnormal { f o r } t > \\tau _ s , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\overline { \\pmb e } m _ e ( \\lambda \\pm ) = 1 + \\overline { \\pmb e } G _ { \\lambda \\pm 0 i } * ( u \\pmb e \\overline { \\pmb e } m _ e ( \\lambda \\pm ) ) , \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { ( 1 - x ) ^ 2 ( 1 - 4 x + 6 x ^ 2 - 5 x ^ 3 + x ^ 4 ) C ( x ) - 1 + 6 x - 1 4 x ^ 2 + 1 5 x ^ 3 - 8 x ^ 4 + x ^ 5 } { x ( 1 - 3 x + x ^ 2 ) ( 1 - x + x ^ 3 ) } \\ , . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} B ' _ n ( v ) & = C _ { n - 2 } ( v ^ { n - 2 } + v ^ { n - 3 } ) + 2 ( B ' _ { n - 1 } ( v ) - C _ { n - 3 } v ^ { n - 3 } ) \\\\ & + v ( B _ n ( v ) - b ( n ; n - 3 ) v ^ { n - 4 } - b ( n ; n - 2 ) v ^ { n - 3 } - b ( n ; n - 1 ) v ^ { n - 2 } ) . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} f _ s ( \\lambda , \\mu ) = \\mu ^ 2 + R _ { 2 s + 1 } ( \\lambda ) \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} | \\Gamma _ r | = | \\Gamma _ 4 | + \\sum _ { i = 4 } ^ { r - 1 } ( i - 1 ) = \\sum _ { i = 0 } ^ { r - 1 } ( i - 1 ) = \\binom { r } { 2 } - r . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{gather*} \\eta ^ 0 ( H _ v \\Sigma _ p ) = 0 , \\eta ^ 0 ( v ) = 1 , \\end{gather*}"}
-{"id": "9014.png", "formula": "\\begin{align*} I _ { u , n } ^ { ( q + 1 ) } = [ u + q d , u + n - 1 ] \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\pmb e \\partial _ { \\lambda } ( \\overline { \\pmb e } m _ e ( \\lambda \\pm ) ) = - \\frac { 1 } { 2 \\pi \\lambda } \\int _ { \\mathbb { R } } u ( y ) m _ e ( y , \\lambda \\pm ) ~ d y + G _ { \\lambda \\pm 0 i } * ( u \\pmb e \\partial _ { \\lambda } ( \\overline { \\pmb e } m _ e ( \\lambda \\pm ) ) ) . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d W _ \\varepsilon } { d t } ( t ) = G ( W _ \\varepsilon ( t ) ) , \\\\ v _ \\varepsilon ( 0 ) = \\varepsilon . \\end{cases} \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} F : = \\lbrace y \\in B _ { r } ( 0 ) \\ , \\colon \\ , \\theta ( y , 4 r ) - \\theta ( y , 2 r ) < \\delta \\rbrace . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in \\mathfrak { M } _ m } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) & = \\frac 1 { ( m / 2 ) ! } \\ ( - \\frac 1 { n ^ 2 } \\sum _ { x \\in G } | f ^ { - 1 } ( x ) | ^ 2 \\ ) ^ { m / 2 } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\\\ & \\qquad + O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} P ( \\pi _ 1 ^ m ( i _ 1 ) , \\ldots , \\pi _ d ^ m ( i _ d ) ) = \\pi ^ m ( P ( i _ 1 , \\ldots , i _ d ) ) . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} g ( ( 1 + i ) k , n ) = \\overline { \\chi } ( 1 + i ) g ( k , n ) , \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} e _ { \\theta } \\cdot \\mathbb { P } u \\cdot \\nabla u & = e _ { \\theta } \\cdot ( h - \\nabla \\Phi ) \\\\ & = e _ { \\theta } \\cdot ( h - e _ { r } \\partial _ r \\Phi - e _ { z } \\partial _ z \\Phi ) \\\\ & = v \\cdot \\nabla u ^ { \\theta } + \\frac { u ^ { r } } { r } u ^ { \\theta } . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} z ^ { ( j ) } _ i = \\left ( \\frac { ( \\theta - 1 ) z ^ { ( j + 1 ) } _ i + \\sum _ { l = 1 } ^ { q - 1 } z ^ { ( j + 1 ) } _ l + 1 } { \\theta + \\sum _ { l = 1 } ^ { q - 1 } z ^ { ( j + 1 ) } _ l } \\right ) ^ k , \\ \\ 1 \\leq i \\leq q - 1 , \\ \\ 0 \\leq j \\leq m - 1 \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k a _ j ^ 2 \\le C _ 3 \\| \\nabla _ \\theta \\phi \\| _ { L ^ 2 ( \\partial B _ 1 ) } ^ { 2 ( 1 - \\gamma ) } . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} a _ \\mu ( u , v ) = \\int _ { \\R ^ N } \\nabla u \\cdot \\nabla v \\ , d \\mu \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} d ( x ) : = d i s t ( x , \\Sigma _ h ) . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\kappa = \\beta \\norm { x } + \\xi \\beta / \\alpha . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} W _ { 1 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ { S _ { n } \\left ( { u , \\xi } \\right ) - S _ { n } \\left ( { u , } \\alpha _ { 1 } \\right ) } \\right \\} \\left \\{ { e ^ { u \\xi } + \\varepsilon _ { n , 1 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} { \\rm d i v } b ^ 0 { \\partial ^ 2 u _ 0 \\over \\partial t ^ 2 } = { \\rm d i v } f \\in L ^ \\infty ( 0 , T ; L ^ 2 ( D ) ) . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\lambda v + A _ 2 v = g \\hbox { i n } \\ : D _ 2 , \\lambda v + B _ { 2 , + } v = g _ { + } \\hbox { i n } \\ : D _ 2 ^ + , \\lambda v + B _ { 2 , - } v = g _ { - } \\hbox { i n } \\ : D _ 2 ^ - , \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} x V ( \\abs { b _ n } ) = V ( S ( x ) ) V ( \\abs { b _ n } ) = V ( S ( x ) \\abs { b _ n } ) \\ge 0 , \\ \\ \\ \\ n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{gather*} \\mathbb { P } - \\lim _ { i \\rightarrow \\infty } { M ^ { \\nu ^ { ( i ) } } } = M \\Longrightarrow \\mathbb { P } - \\lim _ { k \\rightarrow \\infty } { M ^ { \\nu ^ { ( i _ k ) } } } = M , \\end{gather*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ n ( x ) - \\int _ M \\zeta _ n \\mathrm { d } \\mu & = \\int _ M G ( y , x ) f _ n ^ * ( y ) \\mathrm { d } \\mu ( y ) \\\\ & = \\sum _ { j = 1 } ^ m A _ { n , j } G ( x _ { n , j } ^ { ( 1 ) } , x ) + \\sum _ { j = 1 } ^ m \\int _ { M _ j } ( G ( y , x ) - G ( x _ { n , j } ^ { ( 1 ) } , x ) ) f _ n ^ * ( y ) \\mathrm { d } \\mu ( y ) . \\end{aligned} \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\partial _ x e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } = - \\partial _ \\xi e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} a _ { u } & = 2 \\int _ { \\mathbb { R } ^ 3 } f _ { L R } d v , \\ , a _ { \\ell } = \\int _ { \\mathbb { R } ^ 3 } e ^ { - \\frac { a _ u } { \\tau | v _ 1 | } } f _ { L R } d v , \\ , a _ s = \\int _ { \\mathbb { R } ^ 3 } \\frac { 1 } { | v _ 1 | } f _ { L R } d v , \\cr c _ { u } & = 2 \\int _ { \\mathbb { R } ^ 3 } f _ { L R } | v | ^ 2 d v , \\ , c _ { \\ell } = \\int _ { \\mathbb { R } ^ 3 } e ^ { - \\frac { a _ u } { \\tau | v _ 1 | } } f _ { L R } | v | ^ 2 d v , \\ , c _ s = \\int _ { \\mathbb { R } ^ 3 } \\frac { 1 } { | v _ 1 | } f _ { L R } | v | ^ 2 d v , \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\theta _ { h } ( 0 , 2 ) - \\theta _ { h } ( 0 , 1 ) = 0 \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} \\Delta L ( Q ( t ) ) = \\mathbb { E } \\Big \\{ L ( Q ( t + 1 ) ) - L ( Q ( t ) ) | Q ( t ) \\Big \\} . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} H _ { d , 0 } ( x ) = \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ { d - 1 } } J ( x ) \\ , . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} G _ k ^ 0 ( x ) - G _ 0 ^ 0 ( x ) = \\frac 1 { 2 \\pi } \\int _ 0 ^ 1 \\frac { e ^ { i x \\xi } - 1 } { \\xi } \\frac { k } { \\xi - k } ~ d \\xi + \\frac 1 { 2 \\pi } \\int _ 1 ^ { \\infty } \\frac { e ^ { i x \\xi } - \\chi ( \\xi ) } { \\xi } \\frac { k } { \\xi - k } ~ d \\xi . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\int _ { B ( 0 , R C _ { 2 } ^ { 1 / 2 } t ^ { 1 / 2 } ) ^ { c } } \\frac { C _ { 1 } } { t ^ { n / 2 } } \\exp \\left ( - \\frac { 1 } { C _ { 2 } } \\left ( \\frac { \\vert \\xi \\vert ^ { 2 } } { t } \\right ) \\right ) d \\xi = C \\int _ { B ( 0 , R ) ^ { c } } \\exp \\left ( - \\vert \\xi \\vert ^ { 2 } \\right ) d \\xi , \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k \\P ( T _ 1 \\le t _ { j 1 } - t _ { j - 1 } , \\ldots , T _ { \\ell _ j } \\le t _ { j \\ell _ j } - t _ { j - 1 } , N ( 0 , t _ j - t _ { j - 1 } ] = \\ell _ j ) . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} F _ T ( x ) & = 1 + x F _ T ( x ) + x C ( x ) \\big ( F _ T ( x ) - 1 \\big ) + x ^ 2 \\big ( F _ T ( x ) - 1 \\big ) \\\\ & + \\frac { x } { 1 - x } \\Big ( x C ( x ) \\big ( F _ T ( x ) - 1 \\big ) - x \\big ( F _ T ( x ) - 1 \\big ) \\Big ) \\\\ & + \\frac { x ^ 2 } { ( 1 - x ) ^ 2 } \\Big ( x ^ 2 C ( x ) \\big ( F _ T ( x ) - 1 \\big ) + x ^ 2 \\Big ) + x \\Big ( - x F _ T ( x ) - x C ( x ) \\big ( F _ T ( x ) - 1 \\big ) + F _ T ( x ) - 1 \\Big ) \\ , . \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } + \\frac { \\partial v } { \\partial r } + \\frac { v } { r } = 0 \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\big | \\widetilde { \\Psi } _ { p , k } \\big | = \\min _ { 1 \\leq j \\leq r } \\big | \\Psi _ p ( z _ { k + j } ) \\big | . \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 2 & \\cdots & x _ { n - 1 } & x _ n \\\\ y _ 2 & \\cdots & y _ { n - 1 } & y _ n \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} F _ { 1 } \\left ( \\xi \\right ) = { \\tfrac { 1 } { 2 } } \\psi \\left ( \\xi \\right ) , \\ F _ { 2 } \\left ( \\xi \\right ) = - { \\tfrac { 1 } { 4 } } { \\psi } ^ { \\prime } \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} \\mathcal { T } ( E _ 1 , E _ 2 ) = \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { \\mathcal { V } E _ 1 } \\mathcal { V } E _ 2 + \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { \\mathcal { V } E _ 1 } \\mathcal { H } E _ 2 \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} h _ R ( x _ 1 ^ { i _ 1 } x _ 2 ^ { i _ 2 } \\cdots x _ n ^ { i _ n } ) = \\sum _ { k = 1 } ^ n ( - 1 ) ^ { | x _ 1 ^ { i _ 1 } \\cdots x _ { k - 1 } ^ { i _ { k - 1 } } | | x _ k | } \\bigtriangleup ( i _ k ) y _ k x _ 1 ^ { i _ 1 } \\cdots x _ { k - 1 } ^ { i _ { k - 1 } } x _ k ^ { i _ k - 1 } x _ { k + 1 } ^ { i _ { k + 1 } } \\cdots x _ n ^ { i _ n } . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} F _ { \\omega _ q } ( a ) = \\sum _ { p \\leq x } \\omega _ q ( p ) f _ p ( a ) . \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\mathcal { W } _ n = \\{ \\widetilde { \\Theta } : | \\widetilde { \\beta } _ n ( \\widetilde { \\Theta } ) | \\leq \\tau _ n \\sqrt { n \\log { n } } \\} , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\widehat { c } ^ { \\circ } : = \\min _ { ( k , l ) \\in \\widehat { \\mathcal { K } } } \\widehat { c } ^ { \\circ } _ { k l } . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l \\varkappa ^ T _ { t + 1 } + \\varkappa ^ T _ { t - 1 } = 0 , t = 0 , \\ldots , T , \\\\ \\varkappa ^ T _ { T } = 0 , \\ , \\ , \\varkappa ^ T _ { T - 1 } = 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} \\norm { x - p } = \\norm { x - P _ { K } x } = d _ { K } { \\left ( x \\right ) } . \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\bar { \\eta } : = a \\eta , \\bar { \\xi } : = \\frac { 1 } { a } \\xi , \\bar { g } = a g + a ( a - 1 ) \\eta \\otimes \\eta \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} \\int \\phi \\mathcal D _ 2 = \\frac 1 2 \\int _ { \\Gamma ^ 0 } d \\mathcal { H } ^ { n - 1 } ( \\dot \\eta ^ 0 \\circ \\pi _ 1 ) ^ 2 \\ , \\frac { N \\cdot \\nu ^ 0 } { J } \\partial _ N \\ , \\big ( J \\Delta h ^ 0 \\ , \\phi \\big ) - \\frac 1 2 \\int _ { \\Gamma ^ 0 } d \\mathcal { H } ^ { n - 1 } ( N \\cdot \\nu ^ 0 ) \\ , ( \\ddot \\eta ^ 0 \\circ \\pi _ 1 ) \\ , \\Delta h ^ 0 \\phi . \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\Gamma _ { T } ^ \\lambda ( f , g ) \\right ) = u ( x , y , 0 ) . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} \\psi _ n ( x ) = \\frac { 1 } { \\varepsilon _ n ^ \\nu } \\ , \\phi _ n \\Bigl ( \\frac { x } { \\varepsilon _ n } \\Bigr ) , x \\in \\Omega \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} \\tilde \\lambda ^ { - 1 } = \\lim _ { k _ 0 < k \\to \\infty } \\frac { D ^ { \\bf a } _ k + E ^ { \\bf a } _ k } { D ^ { \\bf a } _ { k + 1 } } = \\lim _ { k _ 0 < k \\to \\infty } \\frac { D ^ { \\bf a } _ k } { D ^ { \\bf a } _ { k + 1 } } . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\Delta _ j ( z ) = \\psi _ { 1 , p } ( z ) \\bigg ( \\sum ^ \\mu _ { s = 1 } \\widehat { e } ^ { ( p ) } _ s ( z ) \\frac { \\psi _ { 1 , j } ( z _ s ) } { \\Psi _ { p } ( z _ s ) } + \\Theta [ z , \\xi _ { j + 1 } , \\ldots , \\xi _ { p } ] \\bigg ) , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\overline { D } _ n = a _ { n 1 } \\overline { H } _ 1 + \\cdots + a _ { n r } \\overline { H } _ r \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} & n = 3 , d = 4 , b _ 1 = 2 h , b _ 2 = - 2 j , b _ 3 = 1 ; \\\\ & \\alpha _ 1 = \\frac { r + \\sqrt { a b } } { 2 } , \\alpha _ 2 = \\frac { s + \\sqrt { a c } } { 2 } , \\alpha _ 3 = \\frac { \\sqrt { c } ( \\sqrt { a } + \\sqrt { b } ) } { \\sqrt { b } ( \\sqrt { a } + \\sqrt { c } ) } . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} H ( t , x , a , p , g ) : = \\int _ U \\left [ g ( x + f ( t , x , u , a , p ) ) - g ( x ) \\right ] \\nu ( d u ) + c ( t , x , a , p ) \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} F _ { r } ( - z ) = \\frac { 1 - u ^ { r } } { ( 1 - u ) ( 1 + u ) ^ { r - 1 } } . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} V = \\operatorname { R e } \\left \\{ E \\odot \\Bigl [ \\widetilde { D } _ x Q - i Q \\widetilde { D } _ y ( t ) ^ T \\Bigr ] \\right \\} . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} p _ a ( \\bar { C } _ 1 ) = 5 , \\ : 4 , \\ : 3 , \\ : 2 . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} N ( \\phi ) = 2 0 \\int _ { 0 } ^ 1 ( 1 - t ) \\ , [ V + t \\phi ] ^ 3 \\ , d t \\ , \\phi ^ 2 , \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} \\overline F ( x ) & = \\frac 1 { 2 } x ^ { - 2 } ( 1 + x ^ { { - 1 } } ) , x \\ge 1 . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\iota ( \\beta _ 1 ) = \\beta _ 1 , \\iota ( \\beta _ 2 ) = - \\beta _ 2 , \\iota \\circ \\rho = \\rho ^ { - 1 } \\circ \\iota . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } u _ { 1 } ( x , t , y , s ) v _ { 1 } ( x ) c _ { 1 } ( t ) c _ { 2 } ( s ) \\nabla _ { y } \\cdot ( \\nabla _ { y } \\rho ( y ) ) d y d s d x d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } u _ { 1 } ( x , t , y , s ) v _ { 1 } ( x ) c _ { 1 } ( t ) c _ { 2 } ( s ) v _ { 2 } ( y ) d y d s d x d t \\end{gather*}"}
-{"id": "1185.png", "formula": "\\begin{align*} F = ( 1 + e ^ { 2 \\pi i \\frac 1 6 } ) \\theta _ 2 ^ 4 - e ^ { 2 \\pi i \\frac 5 6 } ( \\theta _ 3 ^ 4 + \\theta _ 4 ^ 4 ) \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } ( y _ { r , i } ' y _ { r , i } - y _ { r , i } ^ 2 ) \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} F ( q , p , \\dot { q } ) = p _ 1 \\dot { q } ^ 1 + p _ 2 \\dot { q } ^ 2 - \\frac { 1 } { 2 } ( \\dot { q } ^ 1 ) ^ 2 - q ^ 2 ( q ^ 1 ) ^ 2 . \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} L _ z \\partial _ x = - \\left ( J + \\frac { 1 } { 4 t } L _ y ^ 2 \\partial _ x + \\frac 1 2 \\right ) , \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} h ^ 2 = I , \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\det \\left ( N _ m ^ T N _ m \\right ) = 1 + O ( 1 / m ) . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} ( f ^ { \\mathrm { a n } } ) ^ * ( g _ 0 ) = d g _ 0 - \\log | \\varphi | + \\lambda , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} R _ { k \\rightarrow k } ^ { \\rm N O M A } ( \\nu ) = \\log _ 2 \\left ( 1 + \\frac { p _ k ( \\nu ) g _ k ( \\nu ) } { p _ { \\bar k } ( \\nu ) g _ k ( \\nu ) + 1 } \\right ) . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\mathfrak { a } [ \\mathbf { v } , \\mathbf { v } ] = \\int _ { \\widetilde { \\Omega } } \\mathfrak { a } ( \\mathbf { k } ) [ \\widetilde { \\mathbf { v } } ( \\mathbf { k } , \\cdot ) , \\widetilde { \\mathbf { v } } ( \\mathbf { k } , \\cdot ) ] \\ , d \\mathbf { k } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} r \\frac { d u _ i } { d r } & = P _ i ( u , r , c ) \\\\ u _ i ( 0 , c ) & = 0 i = 1 , 2 , 3 , 4 , \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\begin{pmatrix} x s \\\\ w y \\end{pmatrix} = \\bar { \\mu } ( z ) e \\ , . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\begin{array} { r c l } H ( t , \\rho ) & = & \\displaystyle \\int _ { - 1 } ^ t [ f ( s ) x ( s , \\rho ) + g ( s ) ] d s \\\\ & = & \\displaystyle \\left ( \\int _ { - 1 } ^ t f ( s ) d s \\right ) \\rho + \\sum _ { k = 2 } ^ { \\infty } \\left ( \\int \\limits _ { - 1 } ^ t f ( s ) r _ k ( s ) d s \\right ) \\rho ^ k + \\int _ { - 1 } ^ t g ( s ) d s \\\\ & = & \\displaystyle G ( t ) + F ( t ) \\rho + \\sum _ { k = 2 } ^ { \\infty } \\left ( \\int \\limits _ { - 1 } ^ t f ( s ) r _ k ( s ) d s \\right ) \\rho ^ k . \\end{array} \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} K '' : = \\mathbb C ( X _ 1 , \\dots , X _ N , Z _ 1 , \\dots , Z _ r ) \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\mathbf { d } ( x _ \\lambda ) = \\overline { \\mathbf { d } } ( x _ \\lambda ) \\qquad \\qquad ( x _ \\lambda ) \\mathbf { d } = ( x _ \\lambda ) \\underline { \\mathbf { d } } . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} \\langle { a , \\beta , \\gamma , r , s , \\tau , v } \\mid [ r , a ] = \\gamma ^ { - 1 } \\beta { r } \\beta ^ { - 1 } r ^ { - 1 } = r \\gamma ^ { - 1 } r ^ { - 1 } , ~ \\gamma { a } \\gamma ^ { - 1 } a ^ { - 1 } = \\beta , ~ s r \\tau { s } = r \\tau { s r \\tau } , \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\Gamma ( u ) \\ ; = \\ ; \\frac 1 2 \\Big \\{ \\log \\frac { \\gamma ( u ) } { 1 - \\gamma ( u ) } \\ , - \\ , \\log \\frac { \\alpha } { 1 - \\alpha } \\Big \\} \\ ; \\cdot \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} X ( t ) = \\xi + \\int _ 0 ^ t \\int _ U f ( s , X ( s ^ - ) , u , \\gamma ( s , X ( s ^ - ) ) , m ( s ) ) \\N ( d s , d u ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\lim _ { X \\rightarrow \\infty } \\frac { S ( X , Y ; \\hat { \\phi } , \\Phi ) } { X \\log X } = 0 . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} N _ L = \\{ L \\} \\times L ^ \\perp = \\{ L \\} \\times E ^ s _ \\sigma \\oplus \\{ L \\} \\times ( E ^ u _ \\sigma \\cap L ^ \\perp ) , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} \\Theta ( d t , d s , d a ) = \\Theta _ 1 ( d t , d x ) [ \\widehat { \\gamma } ( t , x ) ] ( d a ) \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\tau _ - ^ * d y _ i = \\rho ^ { - 1 } \\left ( b \\ , \\rho d \\rho + \\sum _ { j = 1 } ^ n b _ { i j } d y _ j \\right ) \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} t \\mapsto \\exp ( t D ) = \\begin{pmatrix} e ^ t & 0 & 0 \\\\ 0 & e ^ { - \\frac { t } { 2 } } \\cos t \\delta & e ^ { - \\frac { t } { 2 } } \\sin t \\delta \\\\ 0 & - e ^ { - \\frac { t } { 2 } } \\sin t \\delta & e ^ { - \\frac { t } { 2 } } \\cos t \\delta \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} u = 1 - \\frac { \\vartheta _ { [ 0 , 1 ] } ^ 3 ( p ) } { \\vartheta _ { [ 0 , 2 ] } ^ 3 ( p ) } = 1 - \\frac { \\vartheta _ { ( - 1 , 0 , 0 , 1 ) / 3 } ^ 3 ( \\tau ( \\eta ) , \\zeta ( p ) ) } { \\vartheta _ { ( - 2 , 0 , 0 , 2 ) / 3 } ^ 3 ( \\tau ( \\eta ) , \\zeta ( p ) ) } . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { l = 0 } ^ k a _ l z ^ l , Y ( z ) = \\sum _ { l = 0 } ^ { k - 1 } b _ l z ^ l , Z ( z ) = \\sum _ { l = 0 } ^ { k - 2 } c _ l z ^ l , \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\left ( \\frac { b - a } { a + b \\omega } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { a + b \\omega } { a - b } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { a + b \\omega + ( a - b ) \\omega } { a - b } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { a } { a - b } \\right ) \\left ( \\frac { 1 + \\omega } { a - b } \\right ) . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} f ( x ) \\le \\frac { 1 } { 2 } \\langle Q ' x , x \\rangle + \\langle q ' , x \\rangle + \\theta ' , \\forall x \\in X . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\vert g _ n ( x ) \\vert \\leq o ( 1 ) \\ , \\varepsilon _ n ^ { 2 + \\nu } \\ , \\biggl ( \\sum _ { i = 1 } ^ k \\frac { 1 } { \\varepsilon _ n + \\vert x - \\zeta _ { i , n } \\vert } \\biggr ) ^ { 2 + \\nu } x \\in \\Omega . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c c } \\cos \\omega _ { 1 } T _ { 0 } & - 1 & \\sin \\omega _ { 1 } T _ { 0 } & 0 \\\\ 1 & - \\cos \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) & 0 & - \\sin \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) \\end{array} \\right ) \\left ( \\begin{array} { c } a _ { 1 } \\\\ a _ { 2 } \\\\ b _ { 1 } \\\\ b _ { 2 } \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\varepsilon _ d = \\frac { 1 } { 2 } \\big ( \\chi _ { - 1 } ^ 2 ( d ) + \\chi _ { - 1 } ( d ) \\big ) + \\frac { i } { 2 } \\big ( \\chi _ { - 1 } ^ 2 ( d ) - \\chi _ { - 1 } ( d ) \\big ) , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} s ( \\Tilde g ) = d t + e \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} I = \\bigcup _ { j = 1 } ^ { q + 1 } I _ j . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\lefteqn { \\{ ( r _ 0 , r _ 1 , r _ 2 ) \\in { \\cal R } _ { \\mathtt { G W } } ^ * ( P _ { X Y } ) : r _ 0 + r _ 1 = H ( X ) \\} } \\\\ & = \\{ ( r _ 0 , r _ 1 , r _ 2 ) : ( r _ 0 , r _ 2 ) \\in { \\cal R } _ { \\mathtt { W A K } } ^ * ( P _ { X Y } ) , r _ 0 + r _ 1 = H ( X ) \\} . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\widehat { N } ( \\boldsymbol { \\theta } ) & = b ( \\boldsymbol { \\theta } ) ^ * L ( \\boldsymbol { \\theta } ) b ( \\boldsymbol { \\theta } ) \\widehat { P } , \\\\ L ( \\boldsymbol { \\theta } ) & : = | \\Omega | ^ { - 1 } \\int _ { \\Omega } \\bigl ( \\Lambda ( \\mathbf { x } ) ^ * b ( \\boldsymbol { \\theta } ) ^ * \\widetilde { g } ( \\mathbf { x } ) + \\widetilde { g } ( \\mathbf { x } ) ^ * b ( \\boldsymbol { \\theta } ) \\Lambda ( \\mathbf { x } ) \\bigr ) \\ , d \\mathbf { x } . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} \\tilde \\lambda = \\lim _ { k _ 0 < k \\to \\infty } \\frac { F ^ { \\bf a } _ { k + 1 } } { F ^ { \\bf a } _ k } = \\lim _ { k _ 0 < k \\to \\infty } \\frac { D ^ { \\bf a } _ { k + 1 } + E ^ { \\bf a } _ { k + 1 } } { D ^ { \\bf a } _ k + E ^ { \\bf a } _ k } = \\lim _ { k _ 0 < k \\to \\infty } \\frac { D ^ { \\bf a } _ { k + 1 } } { D ^ { \\bf a } _ k + E ^ { \\bf a } _ k } . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\frac { P _ { \\lambda ( N ) } \\big ( x _ 1 t ^ { 1 - m } , \\dots , x _ { m - 1 } t ^ { - 1 } , x _ m , t ^ { - m } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , t ^ { - 2 } , \\dots , t ^ { 1 - N } \\big ) } } \\\\ \\qquad { } = \\Phi ^ { \\nu } \\big ( x _ 1 t ^ { 1 - m } , \\dots , x _ { m - 1 } t ^ { - 1 } , x _ m ; q , t \\big ) \\end{gather*}"}
-{"id": "1214.png", "formula": "\\begin{align*} T _ { n p } ^ c = \\sqrt { p ( p - 1 ) } T ^ * _ { n p } + \\frac { 1 } { 2 } p ( p - 1 ) = \\sqrt { \\frac { n - 6 } { n - 3 } } ( n - 4 ) T _ { n p } + \\frac { 1 } { 2 } p ( p - 1 ) \\left ( 1 - \\sqrt { \\frac { n - 6 } { n - 3 } } \\right ) . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\jmath _ p \\ , : = \\ , \\jmath _ { b _ n } \\circ \\ldots \\circ \\jmath _ { b _ 1 } \\ , : B _ a \\to B _ o \\ \\ , \\ \\ \\forall p = b _ n * \\ldots * b _ 1 \\ , , \\ , p : a \\to o \\ . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} ( Y \\mathbf { d } ) Z & = ( \\sup _ { y \\in Y } y \\mathbf { d } ) Z = \\inf _ { z \\in Z } \\sup _ { y \\in Y } y \\mathbf { d } z . \\\\ Y ( \\mathbf { d } Z ) & = Y ( \\inf _ { z \\in Z } \\mathbf { d } z ) = \\sup _ { y \\in Y } \\inf _ { z \\in Z } y \\mathbf { d } z . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} V ( x ) = \\sum _ { i = 1 } ^ n \\frac { c } { | x - a _ i | ^ 2 } = c \\ , V _ n , \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\mu ( V ( g ) ) = \\mu ( U ^ { - 1 } ( g \\circ \\sigma ) ) = \\mu ( g \\circ \\sigma ) = \\mu ( g ) . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\partial _ { t } \\omega = \\left ( I - P _ { 1 } \\right ) \\sin y \\partial _ { x } \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\omega . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\lambda _ { k , \\textrm { o p t i m a l } } = \\dfrac { \\big | \\boldsymbol { b } _ k ^ H \\boldsymbol { g } _ k \\big | } { \\| \\boldsymbol { b } _ k \\| ^ 2 } \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\| Y _ 0 ^ T \\| & \\leq ( L _ 1 T + L _ 1 T \\| W _ 0 ^ T \\| + \\| B _ 0 ^ T \\| ) e ^ { \\int _ 0 ^ T L _ 1 d t } \\\\ & = e ^ { L _ 1 T } ( L _ 1 T + L _ 1 T \\| W _ 0 ^ T \\| + \\| B _ 0 ^ T \\| ) \\\\ & = L _ 2 + L _ 2 \\| W _ 0 ^ T \\| + L _ 2 \\| B _ 0 ^ T \\| . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } & = \\sum _ { l = 0 } ^ \\infty { \\frac { x } { \\lambda } \\choose l } \\lambda ^ l t ^ l = \\sum _ { l = 0 } ^ \\infty \\left ( \\frac { x } { \\lambda } \\right ) _ l \\lambda ^ l \\frac { t ^ l } { l ! } \\\\ & = \\sum _ { l = 0 } ^ \\infty ( x ) _ { l , \\lambda } \\frac { t ^ l } { l ! } = \\sum _ { l = 0 } ^ \\infty { x \\choose l } _ \\lambda t ^ l , \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} O _ { \\nu | _ { K ' } } = \\left \\{ f \\in K ' \\ , : \\ , \\nu ( f ) \\geqslant 0 \\right \\} = O _ { \\nu } \\cap K ' , \\quad \\mathfrak m _ { \\nu | _ { K ' } } = \\mathfrak m _ { \\nu } \\cap K ' . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} s _ 1 s _ 2 s _ 1 ^ { - 1 } = t _ 1 t _ 2 u _ 1 t _ 2 ^ { - 1 } s _ 2 u _ 1 ^ { - 1 } t _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} X ^ { \\boxplus \\tau } : = \\left ( \\coprod _ { p \\in X } ( s _ p ^ { \\boxplus \\tau } ) ^ { - 1 } ( 0 ) / \\Gamma _ { p } \\right ) / \\sim . \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} P ( g ) = \\lim \\limits _ { n \\to \\infty } \\frac 1 n \\log \\sum _ { ( \\xi _ 1 , . . . , \\xi _ n ) \\in \\mathcal { A } ^ n } \\exp { \\sup \\{ S _ n g ( \\eta ) : \\eta \\in [ \\xi _ 1 . . . \\xi _ n ] \\} } , \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{gather*} g ( x ) \\geq \\min \\left \\{ g \\left ( \\frac { 1 } { \\sqrt { 2 } } - \\frac { 1 } { \\sqrt { 2 k } } \\right ) , g \\left ( \\frac { 1 } { \\sqrt { 2 } } + \\frac { 1 } { \\sqrt { 2 k } } \\right ) \\right \\} = \\frac { 1 } { \\sqrt { 2 } } . \\end{gather*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\mathsf { K } \\left ( { \\xi , t } \\right ) = { \\tfrac { 1 } { 2 } } \\left \\{ { e ^ { u \\left ( { \\xi - t } \\right ) } - e ^ { u \\left ( { t - \\xi } \\right ) } } \\right \\} , \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { ( 1 - 5 x + 9 x ^ 2 - 8 x ^ 3 + 4 x ^ 4 ) C ( x ) - ( 1 - 5 x + 9 x ^ 2 - 6 x ^ 3 + x ^ 4 ) } { x ( 1 - 2 x ) ^ 2 } \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\beta ^ { \\Gamma ^ { ( j ) } _ i } _ { i + 1 } = \\frac { p ^ { ( j ) } _ { i + 1 } D ^ { ( j ) } _ { i + 1 } } { p } \\Big ( \\sum _ { i ' = 1 } ^ { i } \\big ( k _ { i ' } ^ { ( j ) } \\cdot a ^ { ( j ) } _ { i ' } p ^ { ( j ) } _ { i ' } \\dots p ^ { ( j ) } _ { i } - x ^ { ( j ) } _ { v _ { i ' } } \\cdot a ^ { ( j ) } _ { i ' } p ^ { ( j ) } _ { i ' + 1 } \\dots p ^ { ( j ) } _ { i } \\big ) - x ^ { ( j ) } _ { v _ { 0 } } \\cdot p ^ { ( j ) } _ { 1 } \\dots p ^ { ( j ) } _ { i } \\Big ) . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} B [ u ^ + , v ^ + ] = Q [ \\partial _ x ^ { - 1 } u ^ + , v ^ + ] + Q [ v ^ + , \\partial _ x ^ { - 1 } u ^ + ] , \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } { \\eta _ { k } } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta _ { k } } ( \\theta ) + { \\eta _ { k } } ( \\xi _ { j } ) , & \\forall \\ 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k , l ' } = ( p _ { k } ) { \\eta _ { k } } ( \\theta ) + { l ' \\over | p _ { k } | } , & \\forall \\ 0 \\leq l ' \\leq | p _ { k } | - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} \\min _ { m \\in \\Xi _ { g . x , d } } \\bigg \\langle \\eta ( \\gamma ) \\otimes \\frac { 1 } { \\Vert \\gamma \\Vert } , m \\bigg \\rangle = \\bigg \\langle \\eta ( \\gamma ) \\otimes \\frac { 1 } { \\Vert \\gamma \\Vert } , m _ { \\alpha , d } \\bigg \\rangle \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} J _ { d , e } ( x ) & = J ' _ { d , e } ( x ) + \\sum _ { j = 1 } ^ { d + e - 1 } \\frac { x ^ { d + e + 3 } } { ( 1 - x ) ^ { j + e } } ( C ( x ) - 1 ) , \\textrm { \\ w h e r e } \\\\ J ' _ { d , e } ( x ) & = J '' _ { d , e } ( x ) + \\frac { x ^ { d + e + 2 } } { ( 1 - x ) ^ e } ( C ( x ) - 1 ) , \\textrm { \\ a n d } \\\\ J '' _ { d , e } ( x ) & = \\sum _ { j = 1 } ^ e \\frac { x ^ { d + e + 4 } } { ( 1 - x ) ^ { j } ( 1 - 2 x ) } C ( x ) \\ , . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} h _ x = h _ 0 + h _ 2 x ^ 2 + h _ 4 x ^ 4 ~ . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} s _ { 0 , m } = & \\frac { 1 } { 4 } \\int _ 0 ^ { \\Delta t } \\left ( 1 - 2 \\left ( 1 - \\frac { | \\tau | } { \\Delta t } \\right ) ^ 2 \\right ) \\frac { \\partial ^ 3 u _ 0 } { \\partial t ^ 3 } ( t _ m + \\tau ) d \\tau - \\frac { 1 } { 4 } \\int ^ 0 _ { - \\Delta t } \\left ( 1 - 2 \\left ( 1 - \\frac { | \\tau | } { \\Delta t } \\right ) ^ 2 \\right ) \\frac { \\partial ^ 3 u _ 0 } { \\partial t ^ 3 } ( t _ m + \\tau ) d \\tau \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log | \\tilde { { \\cal M } } _ i ^ { ( n ) } | \\le r _ i , ~ ~ ~ i = 0 , 2 \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\tilde { { E } } _ { s } \\left ( p \\right ) = - \\int _ { 0 } ^ { p } { \\frac { \\tilde { { F } } _ { s } \\left ( q \\right ) } { q ^ { 2 } \\left ( { 1 - q ^ { 2 } } \\right ) } d q } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) , \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} & \\frac { ( 1 - v + x v ^ 2 ) ^ 2 } { ( 1 - x v ) ( 1 - v ) ^ 2 } C ( x , t ) \\\\ & \\qquad + \\frac { x v ^ 2 ( 1 - v + x v ^ 2 ) ( 2 v ^ 2 x ^ 2 - 2 x ^ 2 v + 2 x - 1 ) } { ( 1 - v ) ^ 2 ( 1 - 2 x ) ( 1 - x v ) } C ( x , 1 ) \\\\ & \\qquad - \\frac { ( 2 v ^ 2 x ^ 2 - 2 x v + 1 ) x ^ 2 v ^ 3 } { ( 1 - x v ) ( 1 - 2 x ) ( 1 - v ) } A ( x , 1 ) - \\frac { ( 2 v ^ 2 x ^ 2 - v + 1 ) v ^ 3 x ^ 3 } { ( 1 - x v ) ( 1 - 2 x ) ( 1 - v ) } = 0 \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\Omega _ { n } \\left ( \\nu , p \\right ) : = \\frac { 2 } { \\nu ^ { n } } \\int _ { p } ^ { 1 } \\frac { { \\left \\vert \\tilde { { F } } _ { n } \\left ( q \\right ) \\right \\vert } d q } { q ^ { 2 } \\left ( { 1 - q ^ { 2 } } \\right ) } . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} { \\rm d i m } \\ , { \\rm K e r } \\ , \\big ( { \\tilde \\tau } - C ^ { - 1 } \\sum _ { j = 1 } ^ d A ^ * _ j ( t , x ) { \\tilde \\xi } _ j \\big ) = m _ j \\mbox { o n } { \\tilde \\Sigma } _ { ( j ) } . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\xi ( t ) = \\left [ \\begin{array} { c c c c } \\displaystyle x ^ { \\top } ( t ) \\ \\ \\ x ^ { \\top } ( t - h ) \\ \\ \\ \\dot { x } ^ { \\top } ( t ) \\ \\ \\ \\frac { 1 } { h } \\int _ { t - h } ^ t x ^ { \\top } ( s ) d s \\end{array} \\right ] ^ { \\top } , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 } \\langle A x , x \\rangle + \\langle b , x \\rangle + \\gamma , x \\in X , \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} c ( q , a ) = - 1 + \\# \\{ 0 \\le b \\le q - 1 \\colon b ^ 2 \\equiv a \\mod q \\} . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\sup _ { x \\in X } \\frac { | \\langle B y , x \\rangle | ^ { 2 } } { \\langle A x , x \\rangle } = \\langle \\omega ( A , B ) y , y \\rangle , y \\in Y . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} f ( \\{ a , b \\} ) & = f ( a ) g ( b ) - ( - 1 ) ^ { | a | | b | } g ( b ) f ( a ) , \\\\ g ( a b ) & = g ( a ) f ( b ) + ( - 1 ) ^ { | a | | b | } g ( b ) f ( a ) . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} { \\tilde L } = D _ { { \\tilde t } } - C ^ { - 1 } \\sum A ^ * _ j ( t , x ) D _ { { \\tilde x } _ j } \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} ( N \\cdot \\nu ) \\partial _ \\nu w _ { \\rm i m p . } ( x ) = \\frac { \\Theta ( x ) } { 2 } + \\tilde \\Theta ( x ) \\mbox { o n } \\Gamma ^ 0 _ { \\rm o u t } . \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ( z ) = c _ { \\alpha , \\beta } \\int _ 0 ^ 1 ( 1 - t ^ { 1 / \\alpha } ) ^ { \\beta - \\alpha - 1 } W _ { \\alpha , \\alpha } ( z t ) d t , \\ ; z \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\partial _ { \\frak C ^ h } ( X , \\widehat { \\mathcal U } ) = \\partial ( X , \\widehat { \\mathcal U } ) \\setminus \\partial _ { \\frak C ^ v } ( X , \\widehat { \\mathcal U } ) . \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\Phi _ H ( t , \\omega , h ) : = \\Phi ( t , \\omega ) h , \\ ; \\ ; \\ ; t \\geq 0 , \\omega \\in \\Omega , h \\in H , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v + \\beta ' ( w _ \\Omega ) v = 0 & \\Omega \\setminus N _ \\Omega , \\\\ v = 0 & \\partial N _ \\Omega , \\\\ v = - \\nabla w _ \\Omega \\cdot \\theta & \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial x } = \\frac { G _ x + G _ y u } { x } - \\frac { G } { x ^ 2 } . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} _ F ^ \\textup { P R F } ( A ) = | \\P ( A ( f _ s ) = 1 ) - \\P ( A ( f ) = 1 ) | , \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} G _ \\sigma \\to \\C \\ , , ( x , t ) \\mapsto \\langle \\lambda , \\rho _ \\sigma ( x , t ) u \\rangle = \\overline t \\langle \\lambda , \\rho ( x ) u \\rangle \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} 0 = \\int _ { \\Omega } \\frac { d u _ \\tau } { d \\tau } \\Big | _ { \\tau = 0 } \\left ( - \\Delta \\varphi + g ^ { \\prime } ( u _ 0 ) \\varphi \\right ) , \\varphi \\in \\mathcal C _ c ^ \\infty ( \\Omega ) . \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\frac { d } { d v } ( K ( x / v , v ) ^ 2 A ( v / x , v ) ) & = K ' ( x / v , v ) - \\frac { x ^ 2 ( 2 v - x ) } { v ^ 2 ( v - x ) ^ 2 } ( K ( x / v , v ) - 1 ) - \\frac { x ^ 2 } { v ( v - x ) } K ' ( x / v , v ) \\\\ & - \\frac { x } { ( 1 - v ) ^ 2 } K ( x / v , v ) A ( x , 1 ) - \\frac { x } { 1 - v } K ' ( x / v , v ) A ( x , 1 ) \\\\ & - \\frac { x ^ 3 ( x - 2 v + 3 v ^ 2 - 2 x v ) } { v ^ 2 ( 1 - v ) ^ 2 ( v - x ) ^ 2 } C ( x ) , \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} & \\le \\sum _ { n = 1 } ^ \\infty \\sum _ { p \\in S ( A _ n , A _ { n + 1 } ) } \\exp \\bigg ( - \\frac { 1 } { n } d ( x , p x ) - \\delta _ \\P d ( x , p x ) \\bigg ) \\\\ & \\le \\sum _ { n = 1 } ^ \\infty \\sum _ { p \\in S ( A _ n , \\infty ) } \\exp \\bigg ( - \\bigg ( \\frac { 1 } { n } + \\delta _ \\P \\bigg ) d ( x , p x ) \\bigg ) \\\\ & \\le \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ 2 } , \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} j _ * ( v ( c ) + i _ * ( I ) ) \\ ; + \\ ; { j ' } _ * ( v ' ( c ' ) - { o ' } _ * ( O ' ) ) = 0 . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} \\left \\Vert \\omega _ { s 1 } ^ { \\nu } \\right \\Vert _ { L ^ { 2 } } \\left ( t \\right ) + \\left \\Vert \\omega _ { n 1 } ^ { \\nu } \\right \\Vert _ { L ^ { 2 } } & = e \\left ( t \\right ) \\\\ & \\leq e ^ { - \\frac { 1 } { 2 } \\nu t } e \\left ( 0 \\right ) + C d \\nu \\int _ { 0 } ^ { t } e ^ { - \\frac { 1 } { 2 } \\nu \\left ( t - s \\right ) } \\left \\Vert \\omega _ { n 2 } ^ { \\nu } \\right \\Vert _ { L ^ { 2 } } \\left ( s \\right ) d s \\leq C d \\nu . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} u ( z , w ) : = \\sum _ k v _ k ( z ) v _ k ( w ) . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} c _ \\chi ( n ) = \\delta _ { \\frac { r } { r _ 0 } \\mid n } \\frac { r } { r _ 0 } \\tau ( \\chi _ * ) \\overline { \\chi _ * \\left ( \\tfrac { n } { r / r _ 0 } \\right ) } \\chi _ * \\left ( \\tfrac { q } { r } \\right ) c _ { \\frac { q } { r } } ( n ) . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\| \\varTheta ( t _ \\gamma ) - \\varTheta ( \\tau ) \\| & = { \\sup } _ { x \\in X } | \\vartheta ( t _ \\gamma x ) - \\vartheta ( \\tau x ) | = { \\sup } _ { x \\in X } | \\vartheta ( t _ \\gamma ( a _ j x ) ) - \\vartheta ( \\tau ( a _ j x ) ) | \\\\ & = \\| \\varTheta ( t _ \\gamma a _ j ) - \\varTheta ( \\tau a _ j ) \\| \\\\ & < 2 \\varepsilon \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} W _ { \\alpha } ( x _ t , \\dot { x } _ t ) = \\dot { V } ( x _ t , \\dot { x } _ t ) + 2 \\alpha V ( x _ t , \\dot { x } _ t ) \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ { d c } } ( \\frac { \\beta n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , c } } } { P _ { c } G _ 0 } ) = \\exp \\bigg ( - 2 \\pi \\lambda _ d \\bigg ( \\sum _ { j \\in \\{ L , N \\} } \\sum _ { i = 1 } ^ 3 p _ { G _ i } \\times \\\\ & \\bigg ( \\int _ 0 ^ { \\infty } \\bigg ( 1 - 1 / \\bigg ( 1 + \\frac { \\beta n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , c } } P _ d G _ i } { P _ c G _ 0 N _ j t ^ { \\alpha _ { j , c } } } \\bigg ) ^ { N _ j } \\bigg ) p _ { j , c } ( t ) t d t \\bigg ) \\bigg ) \\bigg ) \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} A ( G , \\alpha ) = \\left \\{ \\left . ( z , c ) \\in Z ( G ) \\times C ^ { 1 } ( G , k ^ * ) \\ \\right | c ( x ) c ( y ) = \\alpha ( x , y | z ) c ( x y ) \\forall x , y \\in G \\right \\} \\ . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\partial g ( x ) = \\begin{cases} 1 , & ~ ~ x \\in ( - 1 / \\theta , 1 / \\theta ) , \\\\ 0 , & ~ ~ x \\in ( - \\infty , - 1 / \\theta ) \\cup ( 1 / \\theta , + \\infty ) , \\\\ [ 0 , 1 ] , & ~ ~ x = - 1 / \\theta , 1 / \\theta . \\end{cases} \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} \\Gamma ( 2 , x ; 0 , \\xi ) & \\geq \\int _ { B _ { 1 } } \\cdots \\int _ { B _ { k } } \\prod _ { m = 1 } ^ { k + 1 } \\Gamma ( \\frac { 2 m } { k + 1 } , z _ { m } ; \\frac { 2 ( m - 1 ) } { k + 1 } , z _ { m - 1 } ) d z _ { 1 } \\cdots d z _ { k } \\\\ & \\geq ( k ^ { n / 2 } e ^ { - 2 C } ) ^ { k + 1 } ( \\vert B ( 1 ) \\vert k ^ { - n / 2 } ) ^ { k } . \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\nu ( t ) x ^ \\ast ( t ) = 0 . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\phi _ 0 = \\frac { \\mu _ 0 + \\alpha ( \\lambda _ 0 ) } { \\varphi ( \\lambda _ 0 ) } \\in K . \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} & E [ g ( X ( t ) ) ] - E [ g ( \\xi ) ] \\\\ & = E \\int _ 0 ^ t \\int _ U \\int _ A \\left [ g ( X ( s ) + f ( s , X ( s ) , u , a , m ( s ) ) ) - g ( X ( s ) ) \\right ] \\nu ( d u ) \\rho _ s ( d a ) d s . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} G _ n ^ t = \\\\ \\begin{cases} 1 , ~ ~ ~ ~ ~ M + 1 \\le n \\le N , ~ ~ h _ { k _ t } ( t ) = n \\\\ 0 , ~ ~ ~ ~ . \\end{cases} \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} \\binom { n } { m } O _ m ( n ^ { m - k } ) O _ m \\ ( \\frac 1 { n ^ { m - k } } \\frac { n ! } { n ^ n } \\ ) ^ 3 = O _ m \\ ( \\frac 1 { n ^ { m - 2 k } } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) = O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{gather*} a = ( \\nabla \\eta ) ^ { - 1 } , \\end{gather*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\tau _ s = \\{ t : A _ t = s \\} , \\ ; \\ ; \\ ; s \\geq 0 . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\delta { x } \\beta = \\beta { x } \\delta , ~ \\alpha { x } \\gamma = \\beta { x } \\alpha , ~ \\gamma { x } \\alpha = x \\gamma , ~ x \\alpha = \\beta { x } \\rangle \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} \\rho _ i ( x ) = \\nabla ^ 2 \\eta ( x ) ( E _ i , E _ i ) \\leq - d ( x ) \\frac { - \\kappa _ + } { 1 - \\kappa _ + d _ 0 ( x ) } . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} f ( 0 ) = 0 , D f ( 0 ) = I , \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} k ( t ) : = \\left \\{ \\begin{array} { l } \\displaystyle \\sin \\frac { \\pi t } { T } \\ , \\qquad \\mbox { i f } \\ , \\ , t \\in \\ [ 0 , T ] \\ , , \\\\ \\\\ 0 \\ , \\qquad \\qquad \\quad \\ \\ \\ \\ \\mbox { o t h e r w i s e } \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\kappa ^ { \\chi _ n } _ { \\ell } ( x _ { 1 } , \\dots , x _ { \\ell } ) \\ n ^ { - \\frac { \\deg x _ 1 + \\cdots + \\deg x _ \\ell - 2 ( \\ell - 1 ) } { 2 } } \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} T _ l \\left ( \\sum _ { \\mu \\in \\mathcal { P } , U \\in I ( \\mathcal { C } ) } c _ \\mu ^ { ( U ) } p _ \\mu ^ { ( U ) } [ U ] \\right ) = \\sum _ { \\mu \\in \\mathcal { P } , U \\in I ( \\mathcal { C } ) } c _ \\mu ^ { ( U ) } p _ \\mu ^ { ( U ) } \\otimes T _ l ( U ) \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\sup _ { { \\bar { \\tau } } < t \\le T } | { \\bar { B } } ^ n _ k ( t ) - { \\bar { B } } ^ n _ k ( { \\bar { \\tau } } ) | = { \\bar { B } } ^ n _ k ( T ) - { \\bar { B } } ^ n _ k ( { \\bar { \\tau } } ) = { \\bar { X } } ^ n _ k ( { \\bar { \\tau } } ) - { \\bar { X } } ^ n _ k ( T ) \\le { \\bar { X } } ^ n _ k ( { \\bar { \\tau } } ) , \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} - T _ 0 ^ 3 T _ 2 - 2 T _ 0 ^ 2 T _ 1 T _ 2 + 6 T _ 0 ^ 2 T _ 2 ^ 2 - 2 T _ 0 T _ 1 ^ 3 + 9 T _ 0 T _ 1 ^ 2 T _ 2 + 1 2 T _ 0 T _ 1 T _ 2 ^ 2 - 3 T _ 0 T _ 2 ^ 3 + T _ 1 ^ 4 + 6 T _ 1 ^ 3 T _ 2 \\\\ { } + 5 T _ 1 ^ 2 T _ 2 ^ 2 - 4 T _ 1 T _ 2 ^ 3 - T _ 2 ^ 4 = 0 \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} E ( z , s ) = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma } \\Im ( \\gamma z ) ^ s . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} T R _ 2 = H ( M _ 2 ) \\leq I ( M _ 2 ; Y _ { 2 , 0 } ^ T ) + T \\varepsilon _ { 2 , T } . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\sum _ { j _ 1 = N _ 1 } ^ { J _ { n , 1 } } \\sum _ { k _ 1 = 0 } ^ { 2 ^ { j _ 1 } - 1 } \\cdots \\sum _ { j _ d = N _ d } ^ { J _ { n , d } } \\sum _ { k _ d = 0 } ^ { 2 ^ { j _ d } - 1 } 2 ^ { \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } \\sum _ { l = 1 } ^ d j _ l } 2 \\omega _ { \\boldsymbol { j } , n } \\lesssim n ^ { \\frac { 1 } { 2 \\sigma ^ 2 } \\kappa ( \\underline { \\gamma } ) - \\frac { 1 } { 2 } } \\sigma \\prod _ { l = 1 } ^ d \\sum _ { j _ l = N _ l } ^ { J _ { n , l } } 2 ^ { j _ l \\left ( \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } - \\mu _ l \\right ) } . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} F ( \\mathfrak { C } ) = \\tilde { \\Delta } = \\mathbb { G } \\cap Q \\cap \\tilde { Q } \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} h ( x , t ) = \\exp \\left [ - \\frac { \\lambda _ { h } ( t ) } { 2 } \\| x \\| ^ { 2 } \\right ] \\tilde { h } ( x , t ) , \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} M = \\left [ \\begin{array} { c c c } 1 & \\frac { 1 } { 3 } & \\frac { 1 } { 5 } \\\\ \\\\ \\frac { 1 } { 1 + n } & \\frac { 1 } { 3 ( 3 + n ) } & \\frac { 1 } { 5 ( 5 + n ) } \\\\ \\\\ \\frac { 1 } { ( 1 + 2 n ) ( 1 + n ) } & \\frac { 1 } { 3 ( 3 + 2 n ) ( 3 + n ) } & \\frac { 1 } { 5 ( 5 + 2 n ) ( 5 + n ) } \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\Delta = t ^ 4 ( t ^ 2 - e / 3 ) ^ 4 Q ( t ^ 2 ) ^ 2 , \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} & U _ K = \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty k B ^ n _ k ( T ) = \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty k \\left ( p _ k - \\zeta ^ n _ k ( T ) \\right ) \\le \\sum _ { k = K } ^ \\infty k p _ k , \\\\ & \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty k \\| B ^ n _ k \\| _ \\infty \\le \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty k \\| \\zeta ^ n _ k \\| _ \\infty = \\sum _ { k = K } ^ \\infty k p _ k . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\left ( s \\pm F v - { \\partial ^ 2 \\over \\partial v ^ 2 } \\right ) \\psi _ { s , F } ( \\pm v ) = 0 \\ , , \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} \\bar \\partial F & = \\bar \\partial ( e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } a _ { 1 } + e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } a _ { 2 } ) \\\\ & = e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\bar \\partial a _ { 1 } + e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\bar \\partial a _ { 2 } \\\\ & + 2 m _ { 1 } a _ { 1 } e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\bar \\partial ( \\varphi - \\eta ) + 2 m _ { 2 } a _ { 2 } e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\bar \\partial ( \\varphi - \\eta ) . \\\\ \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon } & = - \\int _ { 0 } ^ { t } e ^ { ( t - s ) L _ 1 } ( b \\cdot \\nabla \\Gamma - b _ { \\varepsilon } \\cdot \\nabla \\Gamma _ { \\varepsilon } ) \\dd s \\\\ & = - \\int _ { 0 } ^ { t } e ^ { ( t - s ) L _ 1 } ( a _ { \\varepsilon } \\cdot \\nabla \\Gamma + b _ { \\varepsilon } \\cdot \\nabla \\rho _ { \\varepsilon } ) \\dd s . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} M _ { 1 , 2 2 } \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & \\left ( 1 + Q Q ^ { * } + \\alpha \\partial _ { 0 } ^ { - 1 } \\right ) ^ { - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\frac { x _ { k + 1 } - x _ k } { \\phi _ 1 ( a , h ) } = a x _ k \\Longrightarrow x _ k = x _ 0 e ^ { a h k } , \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} M ( r ) = \\max _ { Q ( ( t _ { 0 } , x _ { 0 } ) , r ) } u , m ( r ) = \\min _ { Q ( ( t _ { 0 } , x _ { 0 } ) , r ) } u , \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} & ( M _ r { V } ) _ T = \\\\ & \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\left [ \\binom { | S | - 3 } { a - 2 } \\binom { r - | S | - 1 } { b - 1 } + \\binom { | S | - 3 } { a - 1 } \\binom { r - | S | - 1 } { b - 2 } \\right . \\\\ & \\left . - \\binom { | S | - 3 } { a - 1 } \\binom { r - | S | - 1 } { b - 2 } - \\binom { | S | - 3 } { a - 2 } \\binom { r - | S | - 1 } { b - 1 } \\right ] = 0 . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\int _ \\R \\overline { K ( x , \\xi ) } \\phi ( x ) \\ , d x & = \\int _ + \\bar T ( \\xi ) e ^ { - i x \\xi } \\phi ( x ) \\ , d x + \\int _ - [ e ^ { - i x \\xi } + \\bar R ( \\xi ) e ^ { i x \\xi } ] \\phi ( x ) \\ , d x \\\\ & = \\int _ \\R e ^ { - i x \\xi } \\phi ( x ) \\ , d x + \\bar R ( \\xi ) \\int _ \\R e ^ { - i | x | \\xi } \\phi ( x ) \\ , d x , \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\tilde { z } _ n ^ M - \\varphi _ m = \\mathcal { K } _ m ( { z } _ n ^ M ) - \\mathcal { K } _ m ( \\varphi _ m ) . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} y ( x ) = y _ { 0 } ^ { * } ( x ) + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { x _ 1 } ^ { x } ( x - t ) ^ { \\alpha - 1 } f ( t , y ( t ) ) d t , x \\in [ x _ 1 , x _ 2 ] , \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} - \\Delta v & = \\textrm { c u r l } \\ \\textrm { c u r l } \\ v - \\nabla \\textrm { d i v } \\ v \\\\ & = \\textrm { c u r l } \\ ( \\omega ^ { \\theta } e _ { \\theta } ) , \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} \\min _ { Q \\in \\mathcal { P } ( A ) } \\widetilde { H } ( t , x , Q , p , g ) = \\min _ { a \\in A } H ( t , x , a , p , g ) \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} F _ r D _ k = \\bigoplus _ { 1 \\leq p \\leq r } H _ k \\left ( \\frac { F _ p C _ * } { F _ { p - 1 } C _ * } \\right ) \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} = & m ( B _ { n - t , x , y , c } ^ { ( 0 ) } - v , k ) + \\sum \\limits _ { r \\in N _ { B _ { 3 } } ( v ) } m ( B _ { n - t , x , y , c } ^ { ( 0 ) } - v - r , k - 1 ) + t m ( T , k - 1 ) \\\\ = & m ( T , k ) + \\sum \\limits _ { r \\in N _ { B _ { 3 } } ( v ) } m ( T - r , k - 1 ) + t m ( T , k - 1 ) . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} \\mathcal { N } = \\varGamma _ { 1 } \\cup \\varGamma _ { 2 } , \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\nabla _ { r } V = - | r | ^ { n - 1 } \\int _ 0 ^ z [ \\nabla _ r f ( r , s ) + ( n - 1 ) f ( r , s ) | r | ^ { - 2 } r ] \\ , d s \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} z ^ 3 - | w | z + \\frac { \\lambda } { 4 } = 0 \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\Phi _ \\omega ^ R H _ n ( G , K ) \\subseteq { \\rm K e r } ( H _ n ( G , K ) \\to H _ n ( \\hat G _ R , K ) ) \\subseteq \\ \\bigcap _ { i = 1 } ^ \\infty \\ : \\Phi _ i ^ R H _ n ( G , K ) . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} u ^ n _ t ( y ) = \\ , & P ^ n _ { t } u ^ n _ 0 ( y ) + \\sum _ e \\frac { 1 } { L ^ e } \\int _ 0 ^ t u ^ n _ s ( x ^ e _ 1 ) \\partial _ s \\phi _ s ( x ^ e _ 1 ) \\ , d s \\\\ & + \\sum _ e \\Big ( Y ^ e _ t ( \\phi ) + Z ^ e _ t ( \\phi ) + E ^ { ( 1 , e ) } _ t ( \\phi ) + E ^ { ( 2 , e ) } _ t ( \\phi ) \\Big ) \\\\ & + \\sum _ e \\Big ( T ^ e _ t ( \\phi ) + U ^ e _ t ( \\phi ) + V ^ e _ t ( \\phi ) + E ^ { ( 3 , e ) } _ t ( \\phi ) + E ^ { ( 4 , e ) } _ t ( \\phi ) \\Big ) \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} v ^ { ( k ) } : = \\mathcal { Q } _ - \\omega ^ { ( k ) } \\mathrm { a n d } \\tilde u ^ { ( k ) } : = ( \\Phi _ - ) ^ { \\lambda _ 0 } \\sum _ { \\ell = 1 } ^ k \\frac { ( \\log \\Phi _ - ) ^ { k - \\ell } } { ( k - \\ell ) ! } \\tilde v ^ { ( \\ell ) } . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} f ^ { 1 , 2 } ( Y _ \\Delta , w _ \\Delta ) = f ^ { 2 , 1 } ( Y _ \\Delta , w _ \\Delta ) = p h ( Y _ \\Delta , w _ \\Delta ) - 2 + h ^ { 2 , 1 } ( Z _ \\Delta ) . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\bar { \\Lambda } _ 1 } \\left [ - 2 a _ 1 \\bar \\Lambda _ 1 + \\frac { \\partial } { \\partial \\bar \\Lambda _ 1 } \\mathcal P o l y _ 4 ( 0 , \\zeta _ 0 , \\bar \\Lambda ) \\right ] ( \\bar { \\Lambda } ^ 0 ) & = - 2 a _ 1 \\ , + 1 2 a _ 2 \\lambda _ 0 \\Bigl ( \\sum _ { i = 1 } ^ k P _ { i 1 } ( 0 , \\zeta ^ 0 ) ^ 4 \\Bigr ) ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 \\\\ & = 4 a _ 1 , \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} U _ m = \\max _ { \\chi \\in X _ m } | \\hat { 1 _ S } ( \\chi ) | . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\left \\vert \\sum _ { j = 1 } ^ { n } a _ { j } r _ { j } \\left ( t \\right ) \\right \\vert ^ { p } d t = \\int _ { 0 } ^ { 1 } \\left \\vert \\sum _ { j = 1 } ^ { n } \\delta _ { j } a _ { j } r _ { j } \\left ( t \\right ) \\right \\vert ^ { p } d t , \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} & ( p - 1 ) \\eta ^ 2 u ^ { p - 2 } ( a \\nabla u , \\nabla u ) + 2 u ^ { p - 1 } \\eta ( a \\nabla u , \\nabla \\eta ) \\\\ & = \\frac { 4 ( p - 1 ) } { p ^ 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla ( \\eta u ^ { p / 2 } ) ) \\\\ & \\ ; \\ ; \\ ; \\ ; - \\frac { ( 2 p - 4 ) } { p } u ^ { p - 1 } \\eta ( a \\nabla u , \\nabla \\eta ) - \\frac { 4 ( p - 1 ) } { p ^ 2 } u ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} \\| F ( g _ { i _ n } ) \\| _ { L _ 2 } ^ 2 & = \\int _ 0 ^ { \\alpha } \\frac { 1 } { m ( A _ { i _ n } ) } \\abs { F ( \\chi _ { A _ { i _ n } } ) } ^ 2 = \\int _ 0 ^ { \\alpha } \\frac { 1 } { m ( A _ { i _ n } ) } \\abs { F ( \\chi _ { A _ { i _ n } } \\chi _ { [ i _ n - 1 , i _ n ) } ) } ^ 2 \\\\ & = \\int _ 0 ^ { \\alpha } \\frac { 1 } { m ( A _ { i _ n } ) } \\abs { F ( \\chi _ { [ i _ n - 1 , i _ n ) } ) } ^ 2 \\chi _ { A _ { i _ n } } \\geq n ^ 2 , \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} H ( f ) - H ( \\mathcal { M } _ { \\nu , 0 } ) & \\geq H ( f | \\mathcal { M } _ { 0 , 0 } ) + \\max \\{ \\nu , - 2 \\nu \\} \\{ H ( \\mathcal { M } _ { 0 , 0 } ) - H ( \\mathcal { M } _ { \\Theta } ) \\} \\cr & \\geq H ( f | \\mathcal { M } _ { 0 , 0 } ) + \\max \\{ \\nu , - 2 \\nu \\} \\{ H ( \\mathcal { M } _ { 0 , 0 } ) - H ( f ) \\} \\cr & = H ( f | \\mathcal { M } _ { 0 , 0 } ) - \\max \\{ \\nu , - 2 \\nu \\} H ( f | \\mathcal { M } _ { 0 , 0 } ) \\cr & = \\min \\big \\{ 1 - \\nu , 1 + 2 \\nu \\big \\} H ( f | \\mathcal { M } _ { 0 , 0 } ) , \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} r \\mathtt { J } ' _ \\nu ( r ) - \\beta \\nu \\mathtt { J } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} T = \\min \\left \\{ t : F \\leq \\sum _ { k = 1 } ^ { t } T _ 0 C [ k ] \\right \\} \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} P _ 0 : = \\{ 2 ^ { k - 1 } < \\mathbb E ( h | \\mathcal { F } _ i ) \\leq 2 ^ k \\} \\cap \\Omega _ 0 , \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } ( L _ s - \\lambda _ 0 ) ( y ) = 0 , \\\\ ( A _ { 2 s + 1 } - \\mu _ 0 ) ( y ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} w ^ { \\gamma } _ i ( \\theta ) = \\frac { \\left ( 1 + \\lambda ^ T _ { \\gamma } \\psi ( x _ i , \\theta ) \\right ) ^ { - \\frac { 1 } { \\gamma + 1 } } } { \\sum _ { i = 1 } ^ n \\left ( 1 + \\lambda ^ T _ { \\gamma } \\psi ( x _ i , \\theta ) \\right ) ^ { - \\frac { 1 } { \\gamma + 1 } } } , \\textrm { f o r } \\gamma \\neq \\{ - 1 , 0 \\} , \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\Pi _ { K , l } : \\boldsymbol { L } ^ { 2 } ( K ) \\rightarrow \\mathbb { P } _ { l } ( K ) ^ { d } , \\left ( \\mathbf { t } - \\Pi _ { K , l } ( \\mathbf { t } ) , \\mathbf { v } \\right ) _ { \\boldsymbol { L } ^ { 2 } ( K ) } = 0 \\quad \\forall \\mathbf { v } \\in \\mathbb { P } _ { l } ( K ) ^ { d } . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} G ( s ) L ( s , f ) = \\epsilon ( f ) G ( \\delta - s ) L ( \\delta - s , g ) \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\rho _ i & \\equiv \\ \\underset { j = 0 } { \\stackrel { d } { \\textstyle { \\prod } } } [ x _ i ^ { - 1 } , x _ j ^ { - 1 } ] ^ { c _ { i j } } \\cdot [ x _ i , x _ 0 , x _ 0 ] ^ { c _ { i 0 } ( c _ { i 0 } - 1 ) / 2 } \\cdot \\underset { a < b } { \\textstyle { \\prod } } [ x _ i , x _ a , x _ b ] ^ { c _ { i a } c _ { i b } } [ x _ a , x _ b , x _ i ] ^ { c _ { i a b } } \\\\ & \\bmod { F _ 3 ^ 2 F _ 4 ( F _ 2 ) _ 2 N } \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} \\Upsilon ^ { \\gamma } _ { g , \\alpha } K ^ { \\alpha \\beta } \\Upsilon _ { g , \\beta } ^ { \\delta } = K ^ { \\gamma \\delta } \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} E [ X _ { i } ( 2 ) ] , . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\sum _ { k = 1 } ^ { K + 1 } \\zeta _ k \\int _ { [ 0 , t ] \\times [ 0 , \\infty ) } 1 _ { [ q _ { k - 1 } ( s - ) , q _ k ( s - ) ) } ( x ) N ( d s \\times d x ) , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} h ' _ { \\bar { \\zeta } } + \\begin{pmatrix} 0 & 6 \\zeta ^ 2 / ( 3 - \\zeta ^ 2 \\bar { \\zeta } ^ 2 ) \\\\ - 1 & 0 \\end{pmatrix} h ' = 0 . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} S ( \\mathcal { N } , \\eta ) = \\{ N _ f : \\ , f \\in L ^ 0 [ 0 , \\alpha ) \\exists A , m ( A ^ c ) < \\infty , f \\chi _ A \\in L _ { \\infty } [ 0 , \\alpha ) \\} , \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} f ' ( y ^ n ; \\mu , \\Sigma ) \\pi ( \\mu , \\Sigma ; \\nu , \\sigma ^ 2 , \\rho ^ 2 ) & = f ( y ^ n ; A ^ { - 1 } \\mu + \\mu _ 0 , A ^ { - 1 } \\Sigma ( A ^ \\top ) ^ { - 1 } ) \\pi ( \\mu , \\Sigma ; \\nu , \\sigma ^ 2 , \\rho ^ 2 ) \\\\ & = f ( y ^ n ; \\mu ' , \\Sigma ' ) \\pi ( A ( \\mu ' - \\mu _ 0 ) , A \\Sigma ' A ^ \\top ; \\nu , \\sigma ^ 2 , \\rho ^ 2 ) \\\\ & = f ( y ^ n ; \\mu ' , \\Sigma ' ) \\pi ( \\mu ' , \\Sigma ' ; \\nu , \\mu _ 0 , \\Sigma _ 0 , \\rho ^ 2 ) , \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\abs { f ^ { * } ( s ) t ^ { - s } } = O \\Big ( \\abs { \\Im ( t ) } ^ { 4 } \\abs { t } ^ { - \\Re ( s ) } \\exp \\Big ( - \\frac { \\pi } { 1 0 } \\abs { \\Im ( s ) } \\Big ) \\Big ) \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} i _ X \\omega _ 1 = d h _ 1 , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} \\partial _ t \\lambda _ j & = 0 , \\\\ \\partial _ t \\gamma _ j & = 2 \\lambda _ j , \\\\ \\partial _ t \\Gamma ( \\lambda ) & = 0 , \\\\ \\partial _ t \\beta ( \\lambda ) & = i \\lambda ^ 2 \\beta ( \\lambda ) . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} u _ \\tau = U _ \\tau \\circ ( I + \\tau \\theta ) ^ { - 1 } \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} g _ 1 ( \\phi W _ 1 , \\phi W _ 2 ) & = \\cos ^ { 2 } \\theta g _ 1 ( W _ 1 , W _ 2 ) , \\\\ g _ 1 ( \\omega W _ 1 , \\omega W _ 2 ) & = \\sin ^ { 2 } \\theta g _ 1 ( W _ 1 , W _ 2 ) , \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} \\gamma = t ( \\ , | \\ , , 1 - W ) = 1 - t ( \\ , | \\ , , W ) \\eta = 1 - \\sqrt { 1 - \\gamma } \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\ < f , g \\ > _ { e } : = \\frac { 1 } { L ^ e } \\sum _ { x \\in e ^ { n } } f ( x ) g ( x ) \\quad \\ < f , g \\ > : = \\ < f , g \\ > _ { m _ n } : = \\sum _ { e } \\ < f , g \\ > _ { e } . \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\mu _ i ( u ) = g _ { i i } + 2 g _ { i i } \\mu _ i u ^ { - 1 } + ( \\mu _ 1 ^ 2 - \\mu _ 2 ^ 2 ) u ^ { - 2 } 1 \\leq i \\leq 2 . \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} T _ x M = E _ x ^ s \\oplus E _ x ^ u \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha } f ( x ) = \\frac { 1 } { \\Gamma ( n - \\alpha ) } \\frac { d ^ { n } } { d x ^ { n } } \\int _ { a } ^ { x } \\frac { f ( t ) d t } { ( x - t ) ^ { \\alpha - n + 1 } } , \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} t + z ( t ) = \\frac { \\gamma + t ^ 2 } { 2 t } = 1 - \\frac { 1 - \\gamma - ( 1 - t ) ^ 2 } { 2 t } \\le 1 \\ , , \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} V _ { , k } \\eta ^ { k } + V \\xi _ { , t } + \\xi V _ { , t } & = - f _ { , t } \\\\ \\frac { \\partial \\eta ^ { i } } { \\partial t } g _ { i j } - \\xi _ { , j } V & = f _ { , j } \\\\ L _ { \\eta } g _ { i j } & = 2 \\left ( \\frac { 1 } { 2 } \\frac { \\partial \\xi } { \\partial t } \\right ) g _ { i j } \\\\ \\frac { \\partial \\xi } { \\partial x ^ { k } } & = 0 . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} h ( q ) = 0 \\omega q \\in R _ 0 ^ s . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , X ) & = g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega \\phi W ) , \\pi _ { \\ast } ( X ) ) + g _ { 1 } ( \\mathcal { T } _ { Z } \\omega W , \\mathcal { B } X ) \\\\ & - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega W ) , \\pi _ { \\ast } ( X ) ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) , \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{gather*} \\begin{bmatrix} g _ { 1 1 } & g _ { 1 2 } \\\\ g _ { 2 1 } & g _ { 2 2 } \\end{bmatrix} \\left ( \\begin{matrix} a ^ 1 \\\\ a ^ 2 \\end{matrix} \\right ) = \\left ( \\begin{matrix} \\partial _ 1 \\eta \\cdot \\overline { \\partial } _ A \\hat { n } \\\\ \\partial _ 2 \\eta \\cdot \\overline { \\partial } _ A \\hat { n } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "9127.png", "formula": "\\begin{align*} I _ { T } ( \\boldsymbol { \\zeta } , \\psi ) \\doteq \\inf _ { \\boldsymbol { \\varphi } \\in \\mathcal { S } _ { T } ( \\boldsymbol { \\zeta } , \\psi ) } \\left \\{ \\sum _ { k = 0 } ^ { \\infty } \\int _ { [ 0 , T ] \\times \\lbrack 0 , 1 ] } \\ell ( \\varphi _ { k } ( s , y ) ) \\ , d s \\ , d y \\right \\} . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} d \\leq n - k + 2 - \\sum _ { j = 1 } ^ { s ^ * - 1 } \\left \\lceil \\frac { n _ j } { r _ j + 1 } \\right \\rceil - \\left \\lceil \\frac { k - \\sum _ { j = 1 } ^ { s ^ * - 1 } \\left \\lceil \\frac { n _ j } { r _ j + 1 } \\right \\rceil r _ j } { r _ { s ^ * } } \\right \\rceil \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i Q _ i \\Big | \\prec \\hat { \\Pi } ^ 2 + \\Psi \\hat { \\Upsilon } + \\Psi \\hat { \\Pi } \\prec \\Psi \\hat { \\Upsilon } + \\Psi \\hat { \\Pi } \\ , , \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} p ^ { \\circ } _ { j } = \\frac { \\mu ^ \\circ } { \\mu } \\frac { 1 } { j } \\sum _ { \\ell = 1 } ^ \\infty \\ell p _ { \\ell } \\mathbb { P } ( L _ { \\ell } ^ * = j ) \\mbox { f o r } j \\in \\mathbb { N } _ { + } , \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} & \\mathbb P \\{ \\exists u \\geq 0 : \\Delta M _ u \\neq 0 , u \\notin \\{ \\tau _ 0 , \\tau _ 1 , \\ldots \\} \\} \\\\ & = \\mathbb P \\{ \\exists u \\geq 0 : \\Delta M _ u \\neq 0 , u \\notin \\{ \\tau ' _ 0 , \\tau ' _ 1 , \\ldots \\} \\} = 0 . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} f ( x ) : = \\frac { 1 } { 2 } \\langle A x , x \\rangle + \\frac { \\beta + \\frac { 1 } { 2 } \\tau \\langle c , A ^ { - 1 } c \\rangle } { \\tau + 1 } , x \\in X . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\Phi _ { V ( S ( y ) ) } ( x ) = \\tau ( x V ( S ( y ) ) ) = \\| V ( S ( x ) S ( y ) ) \\| _ { C _ 1 } = \\| S ( x ) S ( y ) \\| _ { \\ell _ 1 } = \\sum _ { i = 1 } ^ \\infty s _ i ( x ) s _ i ( y ) = 1 . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} \\Gamma _ { 1 2 } & = \\frac { 2 \\sqrt { 2 } } { J _ 0 } - \\frac { 1 } { 2 } \\left ( \\frac { 2 } { J _ 1 } + \\frac { 2 } { J _ 2 } \\right ) = \\frac { 2 \\sqrt { 2 } } { J _ 0 } - \\frac { 1 } { J _ 1 } - \\frac { 1 } { J _ 2 } . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 [ a _ 0 + a _ 2 t ^ 2 + a _ 4 t ^ 4 ] d t = 0 , \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} \\operatorname { D i s c } ( \\operatorname { T r } _ { \\mathbb { F } _ { q ^ a } / \\mathbb { F } _ { q } } ( \\langle u \\rangle ) ) = & \\operatorname { N o r m } ( u ) \\cdot \\operatorname { D i s c } ( \\operatorname { T r } _ { \\mathbb { F } _ { q ^ a } / \\mathbb { F } _ { q } } ( \\langle 1 \\rangle ) ) \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\begin{cases} & D _ { 0 + } ^ { \\frac { 1 } { 2 } , \\frac { 1 } { 2 } } x ( t ) = t ^ { - \\frac { 1 } { 3 } } [ 1 + t ( x ( t ) ) ^ { \\frac { 4 } { 3 } } ] , t > { 0 } , \\\\ & \\lim _ { t \\to { 0 } } { t ^ { \\frac { 1 } { 4 } } } x ( t ) = 3 . \\end{cases} \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\pi _ { i } \\varphi ( \\mathfrak { P } ) = f _ { i } ^ { - 1 } ( \\mathfrak { P } ) = f _ { i } ^ { - 1 } ( \\mathfrak { P } ' ) = \\mathfrak { P } _ { i } \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} C _ { j , i } = \\frac { d _ { j , i } } { D _ i } , \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} & \\operatorname { C o v } \\bigl ( Z _ t ^ H ( a , b ) , Z _ s ^ H ( a , b ) \\bigr ) \\\\ & = \\frac { 1 } { 2 } ( a + b ) ^ 2 \\bigl ( s ^ { 2 H } + t ^ { 2 H } \\bigr ) - a b ( t + s ) ^ { 2 H } - \\frac { a ^ 2 + b ^ 2 } { 2 } | t - s | ^ { 2 H } . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\left ( N _ { g ( v , m ) } ( u , 1 ) , 0 \\le u \\le v \\right ) \\stackrel { ( d ) } { = } \\left ( Y _ m \\left ( \\log \\left ( \\frac { 1 - v } { 1 - u } \\right ) \\right ) , 0 \\le u \\le v \\right ) , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\geq 0 \\\\ k \\geq 1 } } \\frac { u ^ { n + k } x ^ { k } ( 1 - x ) ^ { n } ( 1 - u ) ^ { 2 k } } { ( 1 + u ) ^ { n + 2 k } ( 1 - u x ) ^ { n + 2 k } } n ^ { \\underline { d } } C _ { k - 1 } \\binom { n + 2 k - 2 } { n } 2 ^ { n } = \\frac { ( 1 - x ) ^ { d } u ^ { d } 2 ^ { d } d ! \\ , \\tilde N _ { d - 1 } ( u x ) } { ( 1 - u ) ^ { d - 1 } ( 1 + u ) ( 1 - u x ) ^ { 2 d } } \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} & | R m | ( x , t ) \\\\ \\le & C _ 0 \\left \\{ \\frac { \\rho ^ { \\frac { 4 } { 5 } } e ^ { C ( \\rho + t K ) } } { K ^ { \\frac { 2 } { 5 } } V _ { x _ 0 } \\left ( \\rho / \\sqrt { 2 K } \\right ) ^ { \\frac { 2 } { 5 } } \\min ( t , \\rho ^ 2 / K ) } \\left [ ( K + ( 4 \\rho / \\sqrt { K } ) ^ { \\frac { 1 } { p } } E _ p ( t ) ^ { \\frac { 1 } { p } } ) t + 1 \\right ] \\right \\} ^ { \\frac { 5 } { 2 p } } \\cdot \\\\ & \\qquad \\cdot ( 1 + ( 4 \\rho / \\sqrt { K } ) \\sqrt { t V _ { x _ 0 } \\left ( 2 \\rho / \\sqrt { K } \\right ) } E _ p ( t ) ^ { \\frac { 1 } { 2 } } ) ^ { \\frac { 7 } { 2 p } } \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\boldsymbol { \\zeta } ( t ) = { \\tilde { \\boldsymbol { \\zeta } } } ( t ) , t \\in [ \\tau , \\tau + \\delta ] \\mbox { f o r s o m e } \\delta > 0 . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\omega _ p ( n ) = \\pi ( V ; p , 1 ) \\ge \\frac { \\theta ( V ; p , 1 ) } { \\log V } \\ge \\frac { \\log x } { 2 \\log \\log x } \\bigg ( 1 + O \\bigg ( \\frac { \\log \\log \\log x } { \\log \\log x } \\bigg ) \\bigg ) \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} c _ n = \\max _ { \\lambda \\in \\{ \\pm 1 , \\pm i \\} } \\| x + \\lambda x _ n \\| . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} \\psi ( t ) : = \\frac { G \\left ( \\frac { 1 } { 2 } \\right ) } { G \\left ( \\frac { 1 } { 2 } + \\frac t 2 \\right ) G \\left ( 1 + \\frac t 2 \\right ) } \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\int ( a \\nabla u , \\nabla ( \\eta ^ 2 u ^ { p - 1 } ) ) \\ ; d x = \\int ( p - 1 ) \\eta ^ 2 u ^ { p - 2 } ( a \\nabla u , \\nabla u ) + 2 u ^ { p - 1 } \\eta ( a \\nabla u , \\nabla \\eta ) \\ ; d x . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} \\{ \\tau , b _ 1 \\} \\neq 0 , \\{ \\tau , b _ j \\} = \\{ b _ 1 , b _ j \\} = 0 , \\ ; \\ ; j = 2 , \\ldots , k , \\ ; \\ ; . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} & T ^ e _ t ( \\phi ) : = \\int _ 0 ^ t \\alpha _ e \\ , u ^ n _ { s - } ( x ^ e _ 2 ) \\ , \\nabla _ L \\phi _ s ( x ^ e _ 1 ) \\ ; d s \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{gather*} \\| \\phi - \\psi \\| = { \\sup } _ { x \\in X } d ( \\phi ( x ) , \\psi ( x ) ) \\forall \\phi , \\psi \\in C ( X , Z ) . \\end{gather*}"}
-{"id": "9297.png", "formula": "\\begin{align*} - \\frac { c _ 0 } { d + 2 } \\int _ { \\partial B _ 1 ^ + } \\phi & \\le \\frac { c _ 0 ^ 2 } { 4 } \\Theta + \\Big ( \\frac { 1 } { ( d + 2 ) \\Theta } \\int _ { \\partial B _ 1 ^ + } \\phi \\Big ) ^ 2 \\\\ & \\le \\frac { c _ 0 ^ 2 } { 4 } \\Theta + \\frac { | \\partial B _ 1 | } { ( d + 2 ) \\Theta } \\int _ { \\partial B _ 1 } \\phi ^ 2 = \\frac { c _ 0 ^ 2 } { 4 } \\Theta + 8 d ^ 2 \\int _ { \\partial B _ 1 } \\phi ^ 2 . \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} \\overline { \\delta } ( B ) ^ \\perp = \\{ ( a , a ' ) \\in A \\times A ' | \\ q ( a + \\iota ( x ) ) q ' ( a ' - \\iota ' ( x ) ) = q ( a ) q ' ( a ' ) \\quad \\forall x \\in B \\} \\ . \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} \\begin{aligned} T ( U , V ) T ( X , Y , Z ) = & T ( T ( U , V ) X , Y , Z ) - T ( X , T ( U , V ) Y , Z ) \\\\ & + T ( X , Y , T ( U , V ) Z ) , \\end{aligned} \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} v ( z ) = z - \\left ( y _ { 0 } + \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ( y _ { i } - y _ { 0 } ) \\right ) . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} s ( k ) = \\# \\{ ( c , d ) \\in \\Z ^ 2 : d \\mid n , \\ ; 0 < c < d , \\ ; c \\equiv d k \\bmod p \\} ; \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n { } _ N \\langle \\xi , a _ i \\rangle a _ i & = J _ \\phi \\sum _ { i = 1 } ^ n J _ \\phi ( { } _ N \\langle \\xi , a _ i \\Omega \\rangle a _ i \\Omega ) = J _ \\phi \\sum _ { i = 1 } ^ n ( J _ \\phi a _ i \\Omega ) { } _ N \\langle a _ i \\Omega , \\xi \\rangle \\\\ & = J _ \\phi \\sum _ { i = 1 } ^ n ( J _ \\phi a _ i \\Omega ) \\langle J _ \\phi a _ i \\Omega | J _ \\phi \\xi \\rangle _ N = J _ \\phi \\sum _ { i = 1 } ^ n b _ i \\langle b _ i | J _ \\phi \\xi \\rangle _ N = J _ \\phi J _ \\phi \\xi = \\xi . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\begin{aligned} - \\frac { 1 } { 2 } b _ M k _ M k _ E ( \\tau _ s - t ) ^ 2 + ( b _ M k _ M - b _ E k _ E ) ( \\tau _ s - t ) \\\\ + ( \\frac { a _ E } { a _ M } - 1 ) b _ E k _ E ( T - \\tau _ s ) > 0 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} Q & = 2 ^ { r - 5 } [ ( r - 2 ) ( r - 3 ) + ( r - 2 ) ( r - 3 ) + 8 ( r - 2 ) + 8 ] \\\\ & = 2 ^ { r - 4 } ( r ^ 2 - r + 2 ) . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} H ( x ; 1 ) & = H _ 1 ( x ) + E ( x ; 1 ) - H _ { 1 , 0 } ( x ) + J ( x ; 1 ) - J _ 1 ( x ) + x \\left ( H \\big ( x ; \\frac { 1 } { 1 - x } \\big ) - E \\big ( x ; \\frac { 1 } { 1 - x } \\big ) \\right ) . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} T = ( \\Phi ( x ) , B ( x ) ) = \\left ( \\left ( \\begin{array} { c } x ^ 1 \\\\ x ^ 2 \\end{array} \\right ) , \\left ( \\begin{array} { c c } \\frac { x ^ 1 } { \\sqrt { ( x ^ 1 ) ^ 2 + ( x ^ 2 ) ^ 2 } } & \\frac { x ^ 2 } { \\sqrt { ( x ^ 1 ) ^ 2 + ( x ^ 2 ) ^ 2 } } \\\\ \\frac { - x ^ 2 } { \\sqrt { ( x ^ 1 ) ^ 2 + ( x ^ 2 ) ^ 2 } } & \\frac { x ^ 1 } { \\sqrt { ( x ^ 1 ) ^ 2 + ( x ^ 2 ) ^ 2 } } \\end{array} \\right ) \\right ) \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} K ( x ) = \\frac { 1 } { x } K _ 1 ( x ) \\ , . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\left \\| p / q \\right \\| = \\sum _ { j = 1 } ^ { m } w _ { j } \\Delta \\left ( p / q , b _ j / c _ j \\right ) = \\sum _ { j = 1 } ^ { N } w _ { j } ( b _ { j } q - p c _ { j } ) + \\sum _ { j = N + 1 } ^ { m } w _ { j } ( - b _ { j } q + p c _ { j } ) . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} X = Q \\cap \\mathrm { G r } ( 2 , W ) \\cap H \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} K ( ( x H , H h H ) , A ) : = p _ G ( \\delta _ x * \\omega _ H * \\delta _ h * \\omega _ H ) ( A ) \\quad \\quad ( x , h \\in G , \\ > A \\in { \\mathcal B } ( X ) ) \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\partial _ t g _ { i j } = - 2 \\mathrm { R i c } _ { i j } \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\left ( v ^ { l k ' } _ { i k } \\left ( W ' , Z ' , \\overline { W } ' , \\overline { Z } ' \\right ) \\right ) _ { 1 \\leq k \\leq N ' } ^ { 1 \\leq k ' \\leq N ' } \\overline { \\left ( \\left ( v ^ { l ' k ' } _ { j k } \\left ( W ' , Z ' , \\overline { W } ' , \\overline { Z } ' \\right ) \\right ) _ { 1 \\leq k \\leq N ' } ^ { 1 \\leq k ' \\leq N ' } \\right ) ^ { t } } = a _ { i j } ^ { l l ' } \\left ( W ' , Z ' , \\overline { W } ' , \\overline { Z } ' \\right ) I _ { N ' } , \\quad \\mbox { f o r a l l $ i , j , l , l ' = 1 , 2 $ , } \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} \\| u ^ { * } ( \\varphi ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) } ^ m ) u \\\\ & - \\psi ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) } ^ m \\| < \\varepsilon \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\mathrm { r e c } _ S ( \\iota _ v ( \\beta _ v ) \\widetilde { \\mathcal { U } } _ S \\overline { k ^ { \\times } } ) = \\sigma _ v ^ { h _ k } [ \\widetilde { G } _ S , \\widetilde { G } _ S ] , \\mathrm { r e c } _ S ( \\iota _ v ( \\alpha _ v ) \\widetilde { \\mathcal { U } } _ S \\overline { k ^ { \\times } } ) = \\tau _ v ^ { h _ k } [ \\widetilde { G } _ S , \\widetilde { G } _ S ] . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} { \\cal E } ^ { ( \\nu ) } = \\hat { c } R _ { \\rm h y b } \\ , \\nu ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} ( ( \\omega _ H * \\delta _ x * \\omega _ H ) & * ( \\omega _ H * \\delta _ y * \\omega _ H ) ) ( H g H ) = \\\\ & = \\frac { 1 } { \\omega _ { H x ^ { - 1 } H } \\cdot \\omega _ { H y H } } \\sum _ { k = 1 } ^ { \\omega _ { H x ^ { - 1 } H } } \\sum _ { l = 1 } ^ { \\omega _ { H y H } } ( \\omega _ H * \\delta _ { \\tilde x _ k ^ { - 1 } } * \\delta _ { y _ l } * \\omega _ H ) ( H g H ) . \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\Omega _ { d } \\subset \\Omega , \\Omega _ { d } \\cap \\left \\{ t = T / 4 \\right \\} = \\Omega _ { d } \\cap \\left \\{ t = T \\right \\} = \\varnothing . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} 2 ^ { 1 - p } ( \\| u + v \\| _ p ^ p + \\| u - v \\| _ p ^ p ) + \\sum _ { i = 1 } ^ \\infty 2 ^ { 1 - p _ i } ( | u _ i + v _ i | ^ { p _ i } + | u _ i - v _ i | ^ { p _ i } ) \\leq 2 \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla U _ i \\cdot \\nabla U _ j = \\int _ { \\Omega } ( - \\Delta U _ i ) U _ j + \\int _ { \\partial \\Omega } \\frac { \\partial U _ i } { \\partial \\eta } U _ j = \\int _ { \\Omega } ( - \\Delta U _ i ) U _ j , \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\| u \\| _ { \\ell ^ q \\ell ^ p L ^ 2 } ^ q = \\sum \\limits _ { N \\in 2 ^ \\Z } N ^ { \\frac 1 4 q } \\left ( \\sum \\limits _ { k \\in \\Z } \\| u _ { N , k } \\| _ { L ^ 2 } ^ p \\right ) ^ { \\frac q p } . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\begin{array} { c } \\varGamma _ { 2 } = \\left \\{ \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\left | \\omega _ { 1 } T _ { 0 } = 2 k _ { 1 } \\pi , \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) = 2 k _ { 2 } \\pi , k _ { 1 } \\in \\mathbb { Z } , k _ { 2 } \\in \\mathbb { Z } \\right . \\right \\} . \\end{array} \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{gather*} | a ^ T N | = \\sqrt { g } , \\end{gather*}"}
-{"id": "7641.png", "formula": "\\begin{align*} A _ 1 : = A \\cap [ 1 , v _ 1 ' ] , \\ , \\ A _ 2 : = A \\cap [ v _ 1 ' + 1 , v _ 1 ' + v _ 2 ' ] , \\ , \\ B _ 1 : = B \\cap [ 1 , v _ 1 ' ] , \\ , \\ B _ 2 : = B \\cap [ v _ 1 ' + 1 , v _ 1 ' + v _ 2 ' ] . \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} a & \\in \\sum _ { j \\in \\{ n + 1 \\} \\cup J } [ b _ { j } , b _ { j } + \\ell _ j - 1 ] \\\\ & = \\left [ b _ { n + 1 } , b _ { n + 1 } + \\ell _ { n + 1 } - 1 \\right ] + \\sum _ { j \\in J } [ b _ { j } , b _ { j } + \\ell _ j - 1 ] \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} \\langle \\phi ( E _ { i i } ) e _ { j } , e _ { j } \\rangle & = \\langle ( \\varphi ( E _ { i i } ) + \\psi ( E _ { i i } ) ) e _ { j } , e _ { j } \\rangle \\\\ & = e _ { j } ^ { * } \\phi ( E _ { i i } ) e _ { j } = 0 . \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} ( \\imath _ X d \\eta ) ( Y ) = d \\eta ( X , Y ) = - \\eta ( [ X , Y ] ) = 0 \\ , . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} d X ^ i _ t = X ^ j _ { t _ - } ( Z ^ { - 1 } _ { ( 1 ) } ) ^ k _ { j , t _ - } { d Z ^ i _ { k , ( 1 ) , t } + d Z ^ i _ { ( 2 ) , t } } , \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} d ( f _ 2 ) = & \\sum \\limits _ { \\substack { d | N \\\\ 1 \\leq d \\leq \\sqrt { N } } } \\frac { \\varphi ( \\gcd ( d , N / d ) ) } { \\gcd ( d , N / d ) } \\max { \\left ( \\frac { N } { d } ( N - 2 ) , \\left ( \\frac { N } { d } - d \\right ) ( N - 1 ) \\right ) } \\\\ & + \\sum \\limits _ { \\substack { d | N \\\\ d \\geq \\sqrt { N } } } \\frac { \\varphi ( \\gcd ( d , N / d ) ) } { \\gcd ( d , N / d ) } \\frac { N } { d } ( N - 2 ) . \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\phi ^ { k } \\left ( ( a _ { i j } ) _ { 1 \\leq i , j \\leq k } \\right ) = \\left ( \\phi ( a _ { i j } ^ { t } ) \\right ) _ { 1 \\leq i , j \\leq k } , \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} b ^ { i j } _ { k l } = \\overline { c ^ { j i } _ { l k } } , \\mbox { f o r a l l $ k , l = 1 , \\dots , q $ a n d c o r r e s p o n d i n g $ i , j $ . } \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} L _ { k } : = \\Delta + 5 w _ { \\mu _ k ^ { \\prime } , \\zeta _ k ^ { \\prime } } ^ 4 , \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\ , V _ { X _ 2 } ( - ( i + 1 ) A - l B ) = \\ , V _ { X _ 2 } ( - i A - l B ) - 1 \\\\ \\ , \\ , \\ , \\ , V _ { X _ 2 } ( - i A - ( l + 1 ) B ) = \\ , V _ { X _ 2 } ( - i A - l B ) - 1 , \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} c _ { \\mu } L _ { r } ( k ) \\xi ( \\eta , H _ { v } ' , k , 0 ) = ( - 1 ) ^ { r ' } 2 ^ { r } & D _ { \\boldsymbol { K } } ^ { r ' ( k - r ) } \\left ( H ' _ { \\infty } \\right ) ^ { k - r } \\\\ & \\times \\prod _ { i = 1 } ^ { r } L ( i - k , \\chi _ { \\boldsymbol { K } } ^ { i - 1 } ) ^ { - 1 } \\textbf { e } ( i H Y ) . \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} B _ i = \\bigcup _ { j \\in X _ i } [ b _ { j } , b _ { j } + \\ell _ j - 1 ] . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\mbox { f o r $ i = 1 , 2 $ , t h e r e e x i s t s $ P _ i > 0 $ s u c h t h a t f o r a l l $ t > 0 $ , } \\frac { 1 } { t } \\int _ 0 ^ t E [ X _ i ^ 2 ( s ) ] d s = P _ i , \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\varphi = \\big ( z _ 0 g _ { d - 1 } + g _ d : ( z _ 0 q _ { d - 2 } + q _ { d - 1 } ) z _ 1 : ( z _ 0 q _ { d - 2 } + q _ { d - 1 } ) z _ 2 \\big ) \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} g ( V ) = \\sum _ { v \\in V } g ( v ) & = 2 \\sum _ { v \\in V } p ( v ) + \\sum _ { v \\in V } \\sum _ { y \\in N ^ o ( v ) } g _ y ( v ) p ( y ) \\\\ & = 2 + \\sum _ { y \\in V } p ( y ) \\sum _ { v \\in N ^ o ( y ) } g _ y ( v ) \\\\ & \\le 2 + \\sum _ { y \\in V } p ( y ) ( 2 \\alpha ( G ) - 2 ) \\\\ & = 2 \\alpha ( G ) . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} [ ( 1 + b ) ^ { \\alpha + 1 } - ( 1 + a ) ] ^ 2 = [ ( 1 + a ) - ( 1 + b ) ^ { \\alpha + 1 } ] ^ 2 \\geq ( a - b ) ^ 2 \\geq a ^ 2 + b ^ 2 . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} T _ h \\tilde \\alpha ( x ) = T _ h ( \\alpha \\circ \\pi _ z ) ( x ) = \\alpha ( \\pi _ z ( x ) * h ) = \\alpha ( \\pi _ z ( x ) ) \\alpha ( h ) = \\tilde \\alpha ( x ) \\alpha ( h ) \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\left | \\int _ a ^ b f ( t ) d t - \\tilde h \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; f ( \\zeta _ i ^ j ) \\right | \\leq C _ 1 \\| f ^ { ( d ) } \\| _ \\infty \\tilde { h } ^ { d } , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\| \\ , \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { q } ( \\mathbb { R } ^ { n } ) } } \\ ; \\ ! = \\ ; \\Bigl \\{ \\ , \\sum _ { i \\ , = \\ , 1 } ^ { n } \\int _ { \\mathbb { R } ^ { n } } \\ ! | \\ : u _ { i } ( x , t ) \\ , | ^ { q } \\ , d x \\ , \\Bigr \\} ^ { \\ ! \\ ! \\ : \\ ! 1 / q } \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\int _ X f _ 1 ( \\pi _ 1 ( x _ 1 , x _ 2 ) ) & f _ 2 ( \\pi _ 2 ( y _ 1 , y _ 2 ) ) \\ > g ( y _ 1 ) g ( y _ 2 ) \\ > d ( \\omega _ { X _ 1 } \\times \\omega _ { X _ 2 } ) ( y _ 1 , y _ 2 ) = \\\\ & = \\int _ D \\int _ X g ( y _ 1 ) g ( y _ 2 ) \\ > K _ { h _ 1 , h _ 2 } ( ( x _ 1 , x _ 2 ) , d ( y _ 1 , y _ 2 ) ) \\ > f _ 1 ( h _ 1 ) f _ 2 ( h _ 2 ) d \\omega _ D ( h _ 1 , h _ 2 ) . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} o s c _ r f = g _ 1 - g _ 2 \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\# \\{ X ^ 2 + Y ^ 2 = Z ^ 2 + 1 \\ ; \\lvert Z \\rvert \\leq R \\} = \\sum _ { n \\leq R } 4 d _ o ( n ^ 2 + 1 ) . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} N : = \\inf \\{ n > 0 : X _ n ^ 1 = X _ n ^ 2 \\} \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\langle f _ { l m } W _ { \\boldsymbol { \\mathrm { T } } \\circ \\Theta ^ { ( n , \\sigma ) } } , f _ { l m } \\rangle = & \\frac { 1 } { n } \\Bigg ( \\sum _ { i = 1 } ^ { n } ( e _ { l } \\otimes e _ { m } ) ^ { * } \\Big ( E _ { i i } \\otimes ( ( n - c - 1 ) E _ { i i } + c E _ { \\sigma ^ { - 1 } ( i ) , \\sigma ^ { - 1 } ( i ) } ) \\Big ) ( e _ { l } \\otimes e _ { m } ) \\\\ & - \\sum _ { 1 \\leq i \\neq j \\leq n } ( e _ { l } \\otimes e _ { m } ) ^ { * } E _ { i j } \\otimes E _ { j i } ( e _ { l } \\otimes e _ { m } ) \\Bigg ) \\\\ = & 0 . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} \\textrm { i n d } ( G ) : = \\min \\{ \\textrm { i n d } ( g ) : ~ 1 \\neq g \\in G \\} \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} d _ { M } ( \\tilde { x } , \\tilde { y } ) = \\min _ { \\gamma \\in \\Gamma } d ( x , \\gamma y ) . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} E _ { \\max } ^ { \\varphi } = \\left \\{ x \\in [ 0 , 1 ] : \\liminf _ { n \\rightarrow \\infty } \\frac { r _ n ( x , \\beta ) } { \\varphi ( n ) } = 0 , \\ \\limsup _ { n \\rightarrow \\infty } \\frac { r _ n ( x , \\beta ) } { \\varphi ( n ) } = + \\infty \\right \\} . \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} & | E [ g ( X ( t ) ) ] - E [ g ( X ( s ) ) ] | \\\\ & \\leq \\int _ s ^ t \\int _ U \\int _ A E | g ( X ( r ) + f ( r , X ( r ) , u , a , m ( r ) ) ) - g ( X ( r ) ) | \\rho _ r ( d a ) \\nu ( d u ) d r \\\\ & \\leq \\int _ s ^ t \\int _ U \\int _ A 2 | g | _ { \\infty } \\rho _ r ( d a ) \\nu ( d u ) d r = 2 \\nu ( U ) | g | _ { \\infty } ( t - s ) \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} a ^ { - } ( n ) : = \\# \\{ \\textrm { $ E / F $ t o t a l l y i m a g i n a r y q u a d r a t i c } , ~ \\mathcal { N } _ { F / \\Q } ( \\frak { D } _ { E / F } ) = n \\} . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\hat T _ 0 \\ , \\hat y _ 0 = 0 \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\rightarrow \\infty } \\frac { \\overline { F * F } ( x ) } { \\overline { F } ( x ) } = 2 . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ \\xi _ { \\theta } \\otimes \\xi _ { \\theta } : \\xi _ { \\theta } = \\sum _ { j = 1 } ^ { n } e ^ { i \\theta _ { j } } e _ { j } \\} , \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} \\delta _ X ( \\epsilon ) = \\inf \\{ 1 - \\| x + y \\| / 2 : \\ , x , y \\in B _ X \\| x - y \\| \\geq \\epsilon \\} , \\ \\ \\ \\ 0 \\leq \\epsilon \\leq 2 . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\langle M \\rangle _ t = \\int _ 0 ^ t e ^ { - A s } a ( 0 ) e ^ { - A ^ T s } d s . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} I _ j = [ a + ( j - 1 ) d + 1 , a + j d ] \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} H _ 0 ^ 1 ( \\Omega ) \\ni \\frac { d U _ \\tau } { d \\tau } \\Big | _ { \\tau = 0 } = \\frac { d u _ \\tau } { d \\tau } \\Big | _ { \\tau = 0 } + \\nabla u _ 0 \\cdot \\theta . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\Gamma _ { 1 1 } + 2 \\Gamma _ { 1 2 } + \\Gamma _ { 2 2 } = \\frac { 4 \\sqrt { 2 } } { J _ 0 } , \\end{align*}"}
-{"id": "9330.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": "2277.png", "formula": "\\begin{align*} u ^ { ( \\rho ) } ( t , x ) = u ( \\rho ^ { 2 } t , \\rho x ) , a ^ { ( \\rho ) } ( t , x ) = a ( \\rho ^ { 2 } t , \\rho x ) , b ^ { ( \\rho ) } ( t , x ) = \\rho b ( \\rho ^ { 2 } t , \\rho x ) \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} & \\sum _ { \\lvert 2 m ^ 2 + h - X \\rvert \\leq X } r _ { 2 k + 1 } ( m ^ 2 + h ) \\\\ & \\quad = \\delta _ { [ k = \\frac { 1 } { 2 } ] } \\delta _ { [ h = a ^ 2 ] } \\bigg ( 2 R ' _ h X ^ { \\frac { 1 } { 2 } } \\log X - 4 R ' _ h X ^ { \\frac { 1 } { 2 } } \\bigg ) + \\tfrac { 1 } { k } R _ { k , h } ^ k X ^ { k } + O ( X ^ { k + \\epsilon - \\lambda ( k ) } ) , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\mu _ i ^ { - \\frac { 1 } { 2 } } U _ i ( x ) = 4 \\pi \\ , \\alpha _ 3 \\ , G _ { \\lambda } ( x , \\zeta _ i ) + \\mu _ i ^ { 2 - \\sigma } \\ , \\hat { \\theta } ( \\mu _ i , x , \\zeta _ i ) \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} \\mathcal { T } _ { V } W = g _ 1 ( V , W ) H , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} I ( M ; Y ^ { ( n ) } ( \\Delta _ n ) ) & = \\sum _ { i = 1 } ^ { n } h ( Y ^ { ( n ) } ( t _ { n , i } ) - Y ^ { ( n ) } ( t _ { n , i - 1 } ) | Y ^ { ( n ) , t _ { n , i - 2 } } _ { t _ { n , 0 } } ) - \\sum _ { i = 1 } ^ { n } h ( B ( t _ { n , i } ) - B ( t _ { n , i - 1 } ) ) \\\\ & \\leq \\sum _ { i = 1 } ^ { n } h ( Y ^ { ( n ) } ( t _ { n , i } ) - Y ^ { ( n ) } ( t _ { n , i - 1 } ) ) - \\sum _ { i = 1 } ^ { n } h ( B ( t _ { n , i } ) - B ( t _ { n , i - 1 } ) ) . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} C H _ { X } ^ d \\rightarrow { \\rm N S } _ { a l g , X } ^ d = { \\rm N S } _ { a l g } ^ d ( X ) \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} f ^ { 1 , 2 } ( Y _ \\Delta , w _ \\Delta ) & = f ^ { 2 , 1 } ( Y _ \\Delta , w _ \\Delta ) \\\\ & = 2 4 - \\ell ( \\Delta ^ \\circ ) + \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) - \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\widetilde z '' ( \\omega _ \\beta ( z ) ) = - F '' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) + \\frac { 1 } { F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) } F '' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) - \\frac { 1 } { ( F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) ) ^ 3 } F '' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) \\cdot ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) ) ^ 2 \\ , . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} u ^ e ( S ^ e \\otimes I ) \\Delta ^ e h ( a ) = & u ^ e ( S ^ e \\otimes I ) ( m \\otimes h + h \\otimes m ) \\Delta ( a ) = S ^ e m ( a _ { ( 1 ) } ) h ( a _ { ( 2 ) } ) + S ^ e h ( a _ { ( 1 ) } ) m ( a _ { ( 2 ) } ) \\\\ = & m S ( a _ { ( 1 ) } ) h ( a _ { ( 2 ) } ) + h S ( a _ { ( 1 ) } ) m ( a _ { ( 2 ) } ) = h ( S ( a _ { ( 1 ) } ) a _ { ( 2 ) } ) + m ( \\{ S ( a _ 1 ) , a _ 2 \\} ) \\\\ = & h ( \\eta \\varepsilon ( a ) ) + m ( \\{ S ( a _ 1 ) , a _ 2 \\} ) = \\varepsilon ( a ) h ( 1 _ A ) + m ( \\{ S ( a _ 1 ) , a _ 2 \\} ) = m ( \\{ S ( a _ 1 ) , a _ 2 \\} ) , \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\nabla _ { 0 \\bar { k } } U _ { j l } ^ i = \\nabla _ { \\phi \\bar { k } } U _ { j l } ^ i = \\partial _ { \\bar { k } } U _ { j l } ^ i = - R _ { \\phi } { } ^ i { } _ { l \\bar { k } j } + R _ 0 { } ^ i { } _ { l \\bar { k } j } , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\alpha ( \\{ a , b \\} ) & = \\beta ( a ) \\alpha ( b ) - ( - 1 ) ^ { ( | a | + p ) | b | } \\alpha ( b ) \\beta ( a ) , \\\\ \\beta ( a b ) & = \\alpha ( a ) \\beta ( b ) + ( - 1 ) ^ { | a | | b | } \\alpha ( b ) \\beta ( a ) , \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\Vert \\nabla ( u _ { \\varepsilon } - u _ { \\varepsilon , h } ) \\Vert _ { A _ { \\varepsilon } } ^ { 2 } \\ , & \\leq \\ , \\mathcal { M } _ { \\Omega } ^ { 2 } ( u _ { \\varepsilon , h } , y , \\gamma ) \\\\ & : = ( 1 + \\gamma ) \\Vert A _ { \\varepsilon } \\nabla u _ { \\varepsilon , h } - y \\Vert _ { A _ { \\varepsilon } ^ { - 1 } } ^ { 2 } + \\left ( 1 + \\frac { 1 } { \\gamma } \\right ) C _ { \\Omega } ^ { 2 } \\Vert \\operatorname { d i v } y + f \\Vert _ { 2 , \\Omega } ^ { 2 } . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} \\frac { 2 } { q } + \\frac { d } { r } = \\frac { d } { 2 } . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} V _ { \\frak r } ^ + ( p ; A ) = ( V ^ + _ { \\frak r , S _ A } ) ^ { \\boxplus \\tau } \\times [ - \\tau , 0 ) ^ { A } . \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} ( C ^ { \\prime } ) = ( C ) \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\partial _ { t } \\hat n _ { \\mu } = - g ^ { k l } \\partial _ { k } v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} b ( \\mathbf { D } ) ^ * \\mathbf { g } _ k ( \\mathbf { x } ) = 0 , k = 1 , \\ldots , m , \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} b ( \\boldsymbol { \\theta } ) ^ * g ^ 0 b ( \\boldsymbol { \\theta } ) \\zeta _ l ( \\boldsymbol { \\theta } ) = \\gamma _ l ( \\boldsymbol { \\theta } ) \\overline { Q } \\zeta _ l ( \\boldsymbol { \\theta } ) , l = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\deg ( C ) = \\deg ( \\Gamma ) = \\sum _ i \\lambda _ i ( \\Gamma ) = \\sum _ i \\lambda _ i ( C ) \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} h ^ { 2 d - 2 } ( Y , V ) + k = h ^ { 2 d - 2 } ( \\widetilde { Y } , V ) = \\sum _ { s \\in \\Sigma } ( \\widetilde { \\rho } _ s - 1 ) . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} P ( \\ | \\ \\mathcal { F } _ t ) & = \\lambda _ k ( t , X ( t ) ) h + o ( h ) \\\\ P ( \\ | \\ \\mathcal { F } _ t ) & = o ( h ) \\\\ P ( \\ | \\ \\mathcal { F } _ t ) & = 1 - \\sum _ { k } \\lambda _ k ( t , X ( t ) ) h + o ( h ) , \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{gather*} \\Phi _ x ( \\xi ) \\coloneqq \\frac { 1 } { | x _ 0 | + | { \\bf x } | } \\log \\left | \\frac { - x _ 0 \\sqrt { \\sum \\limits _ { i = 1 } ^ { n } ( \\xi _ i ) ^ { 2 } } + \\sum \\limits _ { i = 1 } ^ { n } x _ i \\xi _ i } { \\mu R } \\right | . \\end{gather*}"}
-{"id": "7207.png", "formula": "\\begin{align*} x ( t + 1 ) = W ( t : 0 ) x ( 0 ) , \\cr y ( t + 1 ) = W ( t : 0 ) y ( 0 ) , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} D ' : = \\begin{cases} ( 1 + y / x ) & , \\\\ ( 1 ) & . \\end{cases} \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} H ^ { 3 1 , T } ( \\tau , t , x ) = \\begin{cases} H ^ 1 ( t , x ) & \\\\ H ^ { 2 1 } ( \\tau + T , t , x ) & \\\\ H ^ 2 ( t , x ) & \\\\ H ^ { 3 2 } ( \\tau - T , t , x ) & \\\\ H ^ 3 ( t , x ) & . \\end{cases} \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} \\mathbb { P } ( L _ { n , 1 } = \\ell ) = \\frac { ( \\ell , 1 ) ^ { \\dagger } ( n - \\ell ) ! } { n ! } \\mbox { f o r } ~ 1 \\le \\ell \\le n , \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} f \\chi _ A = N _ f ( \\chi _ A ) = | x | e ^ { | x | } ( s , \\infty ) ( \\xi ) \\neq s e ^ { | x | } ( s , \\infty ) ( \\xi ) = s \\chi _ A . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} \\mathcal { D } _ { m - 2 } = \\{ I \\subset [ m ] : m \\in I | I | \\ge 2 \\} \\cup \\{ [ m - 1 ] \\} . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} ( H \\phi ) _ n & = a _ { n } \\phi _ { n + 1 } + a _ { n - 1 } \\phi _ { n - 1 } + b _ n \\phi _ n , n \\geqslant 1 , \\\\ ( H \\phi ) _ 0 & = a _ 0 \\phi _ 1 , n = 0 . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} t = \\dfrac { 1 - x _ 1 - x _ 2 } { ( 1 - x _ 1 ) ( 1 - x _ 2 ) } \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\int | z _ k | ^ 2 f \\ , d x \\ge \\left | \\int z _ k f \\ , d x \\right | ^ 2 = \\left ( \\int z _ { k , 1 } f \\ , d x \\right ) ^ 2 + \\left ( \\int z _ { k , 2 } f \\ , d x \\right ) ^ 2 \\ , . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} \\left . \\left . - \\int _ 0 ^ T g ( s , W _ 0 ^ s , Y _ 0 ^ s ) d Y ( s ) + \\frac { 1 } { 2 } \\int _ 0 ^ T g ( s , W _ 0 ^ s , Y _ 0 ^ s ) ^ 2 d s \\right | \\right ] = O ( \\delta ^ { \\frac { 1 } { 2 } } _ { \\Delta _ n } ) . \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} Z ^ { A A ' } = \\overline { Z _ { A A ' } } , \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} C ^ { i j k l } \\left ( x \\right ) \\coloneqq \\sum _ { s , t = 1 } ^ { 3 } g ^ { i s } \\left ( x \\right ) g ^ { j t } \\left ( x \\right ) C _ { s t } ^ { \\ : \\ ; \\ ; k l } \\left ( x \\right ) \\ : a . e . \\ , , \\ ; i , j , k , l = 1 , 2 , 3 , \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\norm { w _ k } _ { L ^ 1 ( B _ 3 ) } = 1 \\quad w _ k ( x _ k ) \\le 1 / k \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} W _ 1 & = \\tilde { W } _ 1 / \\{ c \\in \\tilde { W } _ 1 : \\langle c , \\tilde { W } _ 2 \\rangle = 0 \\} \\\\ W _ 2 & = \\tilde { W } _ 2 / \\{ c \\in \\tilde { W } _ 2 : \\langle c , \\tilde { W } _ 1 \\rangle = 0 \\} , \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} E ( x ; q , a ) = \\frac { \\log x } { \\sqrt x } \\big ( \\phi ( q ) \\pi ( x ; q , a ) - \\pi ( x ) \\big ) \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} R ( x ) = \\left [ \\begin{array} { c c } R _ { 1 1 } ( x ) & R _ { 1 2 } ( x ) \\\\ R _ { 2 1 } ( x ) & R _ { 2 2 } ( x ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\tau = \\min \\left \\{ t : G \\left ( t \\right ) f + 1 \\right \\} . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\abs { \\vec q _ { x , 2 ^ { - k } r } - \\vec q _ { x , r } } \\lesssim \\sum _ { j = 0 } ^ k \\phi ( x , 2 ^ { - j } r ) . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} ( F _ { A } ) ^ { o } ( \\xi ) = \\sup _ { x \\in \\mathbb R ^ { n } \\setminus \\{ 0 \\} } \\frac { \\langle x , \\xi \\rangle } { F _ { A } ( x ) } = \\sup _ { y \\in \\mathbb R ^ { n } \\setminus \\{ 0 \\} } \\frac { \\langle A ^ { T } y , \\xi \\rangle } { F ( y ) } = \\sup _ { y \\in \\mathbb R ^ { n } \\setminus \\{ 0 \\} } \\frac { \\langle y , A \\xi \\rangle } { F ( y ) } = ( F ^ { o } ) _ { A } ( \\xi ) . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} N _ a = \\frac { a } { ( T ) } , \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} P = \\lambda ^ { n } - b _ 1 \\lambda ^ { n - 1 } - \\cdots - b _ { n } , b _ n \\ ! \\neq \\ ! 0 . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\mathbb { E } ( \\# \\mathrm { r e a l s } ) = \\int _ { - 1 } ^ 1 \\ , \\rho _ { ( 1 ) } ^ r ( x ) \\mathrm { d } x = 2 \\sum _ { j = 0 } ^ { N - 2 } \\prod _ { i = 1 } ^ m \\binom { L _ i + j } { L _ i } a _ { j + 1 , j + 2 } \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\varphi _ { i l } ^ { \\star \\star } \\left ( Z \\right ) = \\displaystyle \\sum _ { i _ { 1 } , i _ { 2 } = 1 } ^ { q } \\varphi _ { i l } ^ { i _ { 1 } i _ { 2 } } \\left ( Z _ { i _ { 1 } } , Z _ { i _ { 2 } } \\right ) , \\quad \\mbox { f o r a l l $ i = 1 , \\dots , q $ a n d $ l = 1 , \\dots , p - q $ } , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\widehat { a } ( n ) = \\langle a , \\chi _ n \\rangle = \\langle A \\chi _ 0 , \\chi _ n \\rangle = a _ n , n \\in \\Z . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} - ( p _ i - 2 ) F ^ { p _ i - 4 } ( \\nabla \\phi ( x _ i ) ) \\mathcal { Q } _ \\infty \\phi ( x _ i ) - F ^ { p _ i - 2 } ( \\nabla \\phi ( x _ i ) ) \\Delta _ F ( \\phi ( x _ i ) ) \\ge \\qquad \\\\ \\Lambda _ { p _ i } ^ { p _ i } ( \\Omega ) | u _ { p _ i } ( x _ i ) | ^ { p _ i - 2 } u _ { p _ i } ( x _ i ) . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} S ( x , y ) = 2 \\sum _ { j = 0 } ^ { ( N - 1 ) / 2 - 1 } \\frac { 1 } { h _ j } \\left ( \\hat { q } _ { 2 j } ( x ) \\hat { \\tau } _ { 2 j + 1 } ( y ) - \\hat { q } _ { 2 j + 1 } ( x ) \\hat { \\tau } _ { 2 j } ( y ) \\right ) + \\frac { q _ { N - 1 } ( x ) } { \\mu _ N } . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} g \\left ( ( M _ { 0 } , M _ { 1 } , M _ { 2 } ) \\right ) = \\sum _ { i = 0 } ^ { 2 } f ( M _ { i } ) + c , \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} t ^ \\pm _ { i i } [ 0 ] t ^ \\mp _ { i i } [ 0 ] = 1 \\ \\ \\mathrm { f o r } \\ 1 \\leq i \\leq n , \\ t ^ + _ { i j } [ 0 ] = t ^ - _ { j i } [ 0 ] = 0 \\ \\ \\mathrm { f o r } \\ j < i , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} ( a , { \\boldsymbol { v } } ) & \\mapsto ( a - 2 , { \\boldsymbol { v } } ) \\mbox { w i t h p r o b a b i l i t y } \\frac { a - 1 } { \\sum _ { i = 1 } ^ \\infty i v _ i + a - 1 } , \\\\ ( a , { \\boldsymbol { v } } ) & \\mapsto ( a + k - 2 , { \\boldsymbol { v } } - { \\boldsymbol { e } } _ k ) \\mbox { w i t h p r o b a b i l i t y } \\frac { k v _ k } { \\sum _ { i = 1 } ^ \\infty i v _ i + a - 1 } , \\ : k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} A _ m ( X ) : = \\bigcup ^ { t - 1 } _ { j = 0 } H ^ 0 _ { b , k _ j \\alpha m } ( X ) . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\mathbb { P } ( N _ j > 0 \\ , | \\ , N > 0 ) = \\frac { 1 } { q } \\left [ 1 - \\exp \\left ( - \\frac { q ^ j } { j } \\right ) \\right ] \\mbox { f o r } j \\ge 1 . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} [ z ] T ^ - _ { i i } ( z ) = \\tilde { g } ^ { ( - 1 ) } _ i + \\sum _ { j < i } \\tilde { f } ^ { ( 0 ) } _ { i j } \\tilde { g } ^ - _ j \\tilde { e } ^ { ( - 1 ) } _ { j i } . \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} \\nu _ 1 = 0 \\nu _ 3 = 0 . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\bar J _ \\lambda \\Bigl ( \\sum _ { j = 1 } ^ k V _ j + \\phi \\Bigr ) = J _ \\lambda \\Bigl ( \\sum _ { j = 1 } ^ k U _ j \\Bigr ) + o ( \\varepsilon ^ { 2 } ) \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{gather*} 0 \\to H ^ 2 ( Y _ 0 ) \\to H ^ 2 ( \\widetilde { X } _ 0 ) \\oplus \\bigoplus _ i H ^ 2 ( E _ i ) \\to \\bigoplus _ i H ^ 2 ( Q _ i ) \\to \\\\ \\to H ^ 3 ( Y _ 0 ) \\to H ^ 3 ( \\widetilde { X } _ 0 ) = H ^ 3 ( X _ 0 ) \\to 0 . \\end{gather*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\varphi _ { \\lambda } \\left ( x \\right ) = \\exp \\left ( \\lambda \\xi \\left ( x \\right ) \\right ) . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} ( - 1 ) ^ { m _ 1 } \\sum _ { j = 1 } ^ N ( X _ j ^ { 2 m _ 1 } + Y _ j ^ { 2 m _ 1 } ) u + ( - 1 ) ^ { m _ 2 } \\sum _ { j = 1 } ^ N \\alpha _ j ( X _ j ^ { 2 m _ 2 } + Y _ j ^ { 2 m _ 2 } ) u = | u | ^ { q - 2 } u . \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} v ( x ) = | x | ^ { \\beta - n } \\ast u ^ p = \\int _ { R ^ n } \\frac { u ^ p ( y ) } { | x - y | ^ { n - \\beta } } d y , \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\Delta ^ e d ^ e m ( a ) = & ( m \\otimes m ) \\Delta d ( a ) \\\\ = & ( m \\otimes m ) ( d ( a _ { ( 1 ) } ) \\otimes a _ { ( 2 ) } + ( - 1 ) ^ { | a _ { ( 1 ) } | } a _ { ( 1 ) } \\otimes d ( a _ { ( 2 ) } ) ) \\\\ = & m d ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) + ( - 1 ) ^ { | a _ { ( 1 ) } | } m ( a _ { ( 1 ) } ) \\otimes m d ( a _ { ( 2 ) } ) , \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\sum _ p x _ p \\in \\ker \\delta _ k \\Leftrightarrow ( \\forall p , r ) ( \\partial _ { p , k - p } ^ { r } x _ p = 0 ) . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} y ( x ) = \\frac { y _ a } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } ( x - t ) ^ { \\alpha - 1 } f ( t , y ( t ) ) d t , x > a . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } a ( n ) e ^ { - n / X } = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } D ( s ) X ^ s \\Gamma ( s ) d s . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\widehat { T } ( z _ { 0 } + z _ 1 ) = e ^ { - \\tfrac { i } { 2 \\hbar } \\sigma ( z _ 0 , z _ 1 ) } \\widehat { T } ( z _ { 0 } ) \\widehat { T } ( z _ { 1 } ) . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla U _ i \\cdot \\nabla U _ j - \\lambda \\int _ { \\Omega } U _ i \\ , U _ j = \\int _ { \\Omega } w _ i ^ 5 \\ , U _ j . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} y ^ { ( \\alpha ) } _ n = \\sum _ { \\ell = 1 } ^ n \\frac { 1 } { ( n - \\ell + 1 ) ! } B _ { n , \\ell } ( 1 ! \\alpha , 2 ! a _ 1 , 0 , 4 ! a _ 2 , 0 , \\dots ) \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} f ( v ) \\exp \\left [ - \\frac { 1 } { 2 } ( v - a - B u ) ^ { T } S ( v - a - B u ) \\right ] = \\exp \\left ( - \\frac { 1 } { 2 } u ^ { T } B ^ { T } C B u \\right ) \\tilde { f } ( v ) \\exp \\left [ - \\frac { 1 } { 2 } z ^ { T } ( F + S ) z \\right ] , \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\begin{psmallmatrix} x ( \\beta ) s ( \\beta ) \\\\ w ( \\beta ) y ( \\beta ) \\end{psmallmatrix} - \\bar \\mu ( z ( \\beta ) ) e = ( 1 - \\beta ) \\Bigl [ \\begin{psmallmatrix} x ^ k s ^ k \\\\ w ^ k y ^ k \\end{psmallmatrix} - \\mu ^ k e \\Bigr ] + \\beta ^ 2 \\begin{psmallmatrix} \\Delta x \\Delta s \\\\ \\Delta w \\Delta y \\end{psmallmatrix} \\ , . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} m _ \\alpha ( r ) \\ ; = \\ ; \\begin{cases} m ( r ) & \\\\ \\alpha & \\end{cases} \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\widetilde { X } ( t ) = \\int _ { \\R _ + } \\int _ { \\R } e ^ { - \\xi ( t - s ) } \\mathbf { 1 } _ { [ 0 , \\infty ) } ( t - s ) \\widetilde { \\Lambda } ( d \\xi , d s ) , \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} | \\dot A _ R ( t ) | = | x _ { R } ( t ) \\wedge \\nabla W ( x _ { R } ( t ) ) | \\le \\frac { C _ { + } } { r _ { R } ^ { \\beta } ( t ) } \\le C _ { + } \\left [ \\frac { 2 \\alpha } { ( 2 - \\alpha ) m } \\right ] ^ { \\beta / ( \\alpha + 2 ) } \\dfrac { 1 } { ( t - t ^ { + } _ { R } ) ^ { 2 \\beta / ( \\alpha + 2 ) } } \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} F ^ { - 1 } \\left ( - \\frac { i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } \\right ) = \\epsilon \\chi _ { \\mathbb { R } ^ + } ( x ) e ^ { i ( \\lambda + i \\epsilon ) x } , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\real \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } & \\geq a ^ { a / 2 } + \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\sum _ { n = 1 } ^ \\infty \\frac { 2 | z | ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } + | z | ^ 2 } = \\frac { i | z | \\mathtt { f } ' _ { a , \\nu } ( i | z | ) } { \\mathtt { f } _ { a , \\nu } ( i | z | ) } . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} C _ 3 ( \\delta _ d , \\delta ^ { \\prime } ) : = \\min _ { G \\leq S _ d } C _ 2 ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) > 0 . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} | \\lambda _ { n , 1 } ^ { ( 1 ) } - \\lambda _ { n , 1 } ^ { ( 2 ) } | = O \\left ( \\sum _ { i = 1 } ^ 2 \\frac { 1 } { \\lambda _ { n , 1 } ^ { ( i ) } } \\right ) . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} J _ \\lambda \\Bigl ( \\sum _ { i = 1 } ^ k U _ i \\Bigr ) = \\sum _ { i = 1 } ^ k J _ \\lambda ( U _ i ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ k \\sum _ { j \\neq i } \\int _ { \\Omega } w _ i ^ 5 \\ , U _ j - \\frac { 1 } { 6 } \\int _ { \\Omega } \\Bigl [ \\Bigl ( \\sum _ { i = 1 } ^ k U _ i \\Bigr ) ^ 6 - \\sum _ { i = 1 } ^ k U _ i ^ 6 \\Bigr ] . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} a _ 0 * \\ldots * a _ { k - 1 } = \\left ( \\frac { \\det { \\overline C ^ k } } { \\det { \\overline C ^ { k - 1 } } } \\right ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} q _ i \\ge \\binom { k + 2 } { i } - \\binom { k + 1 } { i } - \\binom { k - 1 } { i - 2 } = \\binom { k } { i - 1 } + \\binom { k - 1 } { i - 3 } \\ge 0 \\ , . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} | \\beta ( V ( x + v ) ) - \\beta ( V ( x ) ) | ^ { 2 / \\theta } & \\le ( M | V ( x + v ) - V ( x ) | ^ \\theta ) ^ { 2 / \\theta } \\\\ & = M ^ { 2 / \\theta } | V ( x + v ) - V ( x ) | ^ 2 . \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} G _ { m , n } ( z ) = \\sum ^ n _ { i = m } D _ { m , i } \\ , \\psi _ { m , i - 1 } ( z ) . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} d = \\frac { - 3 \\epsilon ( 1 - \\epsilon ^ 2 ) ^ { 1 / 2 } - \\epsilon ^ 2 } { ( 1 - \\epsilon ^ 2 ) ^ { 1 / 2 } + \\epsilon } \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} c x _ { 0 } ^ { 2 } x _ { 1 } ^ { 2 } + c x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + d x _ { 1 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 0 } ^ { 4 } + x _ { 0 } x _ { 2 } ^ { 3 } - x _ { 1 } ^ { 3 } x _ { 3 } + x _ { 1 } x _ { 3 } ^ { 3 } = 0 , \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\mathfrak { X } _ { k } ^ + ( u , \\lambda ) : = \\sum _ { i = 0 } ^ { \\infty } g ^ { ( i ) } _ { \\lambda _ k } ( z _ k ) u ^ { i } = g _ { \\lambda _ k } ( u + z _ k ) = \\frac { \\vartheta ( u + z _ k + \\lambda _ k ) } { \\vartheta ( u + z _ k ) \\vartheta ( \\lambda _ k ) } , \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} \\mathrm { E } N ^ { ( i ) } N ^ { ( j ) } = \\sum _ { p = 0 } ^ { d - i } \\sum _ { q = 0 } ^ { d - j } \\frac { ( - 1 ) ^ { p + q } } { p ! q ! } a _ { p + i , q + j } ( 0 ) \\int _ 0 ^ \\infty e ^ { - 2 \\lambda s } s ^ { p + q } d s . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\partial ( \\Upsilon ) = \\tilde { \\phi } _ + \\Upsilon \\ , \\ , \\ , \\mbox { a n d } \\ , \\ , \\ , \\partial ( \\Upsilon ) = \\tilde { \\phi } _ - \\Upsilon . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} \\norm { D _ { k } u } _ { L ^ \\infty ( B _ { 1 } ^ + ) } + \\norm { D D _ { k } u } _ { L ^ \\infty ( B _ { 1 } ^ + ) } \\le C \\norm { D _ { k } u } _ { L ^ 2 ( B _ { 3 / 2 } ^ + ) } , k = 1 , 2 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} \\int _ { v _ 1 > 0 } F _ + ( x , v _ 1 ) d v _ 1 & = C _ { \\ell , u } \\int _ { 0 } ^ { x } \\int _ { v _ 1 > 0 } \\frac { 1 } { \\tau | v _ 1 | } e ^ { - \\frac { a _ { \\ell } ( x - y ) } { \\tau | v _ 1 | } } \\mathcal { M } _ 1 ( v _ 1 ) d v _ 1 d y \\cr & \\leq C _ { \\ell , u } \\left ( \\frac { \\ln \\tau + 1 } { \\tau } \\right ) . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} \\partial _ t X ' = \\Delta X ' - \\frac { 1 } { m + 2 } : | X ' | ^ 2 X ' : + \\xi \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\frac { C _ { k - 2 - l , n } } { l ! } & = - 2 \\sum _ { m = 2 } ^ { k - 1 - l } a _ { l + m + 1 } \\biggl [ ( n - 1 ) m _ { ( m - 1 , \\dot { 0 } } + \\sum _ { p = 1 } ^ { [ ( m - 1 ) / 2 ] } m _ { ( m - 1 - p , p , \\dot { 0 } ) } \\biggr ] \\\\ & - \\sum _ { m = 1 } ^ { k - 2 - l } b _ { l + m + 1 } m _ { ( m , \\dot { 0 } ) } - \\frac { n ( n - 1 ) } { k ( k - 1 ) } ( k - l - 2 ) ( k + l + 1 ) a _ { l + 2 } \\\\ & - \\frac { n } { k - 1 } ( k - l - 2 ) b _ { l + 1 } , l = 0 , 1 , \\ldots , k - 3 . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ s a _ j = \\log | \\L _ 1 ( Y ) | + . . . + \\log L _ { n - 1 } + \\frac { k } { n } \\log | \\L _ n ( Y ) | + \\sum \\limits _ { j = 1 } ^ s \\frac { c } { \\sqrt { j } } . \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} g ( A ( X , Y ) , Z ) + g ( Y , A ( X , Z ) ) = 0 . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\dim { \\mathcal N } ( { \\alpha _ 1 } , { \\alpha _ 2 } ) = \\mu ( \\alpha _ 2 ) - \\mu ( \\alpha _ 1 ) + \\dim R _ { \\alpha _ 2 } . \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} - \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = C \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} u _ \\lambda ( x ) = w _ { \\mu , \\zeta } \\ , ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 1 + x } } = \\sum \\limits _ { k = 0 } ^ { \\infty } \\begin{pmatrix} - 1 / 2 \\cr k \\end{pmatrix} x ^ k = \\sum \\limits _ { k = 0 } ^ { \\infty } ( - 1 ) ^ k \\begin{pmatrix} 2 k \\cr k \\end{pmatrix} \\frac { x ^ k } { 2 ^ { 2 k } } , \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\deg h _ { ( i + 1 ) \\nu } & = ( d ^ { \\nu } - \\delta ) ( d ^ { ( i - 1 ) \\nu } - \\delta ^ { i - 1 } ) \\delta + d ^ { i \\nu } ( d ^ { \\nu } - \\delta ) \\\\ & = d ^ { i \\nu } - \\delta ^ i . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\Upsilon ( \\zeta , \\widehat \\Lambda ) = A ( \\zeta , \\widehat { \\Lambda } ) + \\mathcal R ( \\zeta , \\widehat { \\Lambda } ) , \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\begin{psmallmatrix} x ( \\beta ) s ( \\beta ) \\\\ w ( \\beta ) y ( \\beta ) \\end{psmallmatrix} = \\beta \\mu e + ( 1 - \\beta ) \\begin{psmallmatrix} x ^ k s ^ k \\\\ w ^ k y ^ k \\end{psmallmatrix} + \\beta ^ 2 \\begin{psmallmatrix} \\Delta x \\Delta s \\\\ \\Delta w \\Delta y \\end{psmallmatrix} \\ , . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} E ^ \\sharp = \\tilde Z E ' \\tilde Z ^ { - 1 } = \\tilde Z ^ { - 1 } E ' \\tilde Z \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ ( X + k ) I _ { D _ k } ] = \\lim _ { n \\rightarrow \\infty } \\hat { \\mathbb { E } } [ ( X _ n + k ) I _ { D _ k } ] = \\lim _ { n \\rightarrow \\infty } \\hat { \\mathbb { E } } [ ( \\hat { \\mathbb { E } } _ { \\tau + } [ X _ n ] + k ) I _ { D _ k } ] = \\hat { \\mathbb { E } } [ ( \\eta + k ) I _ { D _ k } ] . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\div T u = H \\Omega \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\L ^ { - \\alpha } [ \\Lambda ^ \\alpha , \\nabla \\phi ] \\psi = [ \\nabla \\phi , \\L ^ { - \\alpha } ] \\L ^ \\alpha \\psi \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} & a c = q \\ , c a , a c ^ * = q \\ , c ^ * a , c c ^ * = c ^ * c , \\\\ & a ^ * a + c ^ * c = 1 , a a ^ * + q ^ 2 c ^ * c = 1 . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} 1 = \\beta ^ n = \\beta ^ { 2 t } = ( - 1 - \\beta ) ^ 2 = 1 - \\beta + \\beta ^ 2 , \\ \\ \\mbox { t h a t , i s , } \\ \\ \\beta ( \\beta - 1 ) = 0 , \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\limsup _ { r \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( L [ 0 ] _ n - 4 n \\leq - \\frac { r ^ 2 n ^ { 1 / 3 } } { c } \\right ) = 0 \\ , . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( G _ { i i } + T _ i ) \\Upsilon = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N P _ i = O _ \\prec ( \\Psi ) \\ , . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\psi ( ( \\tilde { a } * h ^ * ) * h ) = \\psi ( h * \\tau _ { i \\beta } ( \\tilde { a } * h ^ * ) ) . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega W ) , \\pi _ { \\ast } ( X ) ) - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega \\phi W ) , \\pi _ { \\ast } ( X ) ) & = g _ { 1 } ( \\mathcal { T } _ { Z } \\omega W , \\mathcal { B } X ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} L ( x ) = \\frac { ( 1 - x ) ^ 3 } { 1 - 4 x + 5 x ^ 2 - 3 x ^ 3 } \\ , . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} \\widetilde { H } ( t , x , Q , p , g ) : = \\int _ A H ( t , x , a , p , g ) Q ( d a ) . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\widetilde z '' ( \\omega _ \\beta ( E _ - ) ) = \\frac { F '' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) } { F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) } \\big ( 1 - F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) \\big ) - \\frac { F '' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) } { F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) } \\big ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) - 1 \\big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} F ( t ) = \\varphi ( \\alpha _ t ( A ) B ) , F ( t + i \\beta ) = \\varphi ( B \\alpha _ t ( A ) ) . \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} F _ x ( y ) ~ = ~ \\int _ { M } F _ y ( z ) \\cdot F _ x ( z ) \\ , d \\mu ( z ) \\ , . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} g = \\frac { d \\rho ^ 2 + h } { \\rho ^ 2 } \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} f _ 2 ( l ) = \\min \\{ l , h ( l ) \\} , \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} G _ 2 ( x ; r ) = x \\ , G _ 2 ( x ; r - 1 ) + \\frac { x ^ { r + 1 } \\big ( F _ T ( x ) - 1 \\big ) } { ( 1 - x ) ^ { r } } \\ , . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = K + ( l _ 2 ) , \\\\ ( 0 ) : l _ 2 & = K + ( l _ 1 ) , \\\\ K \\cap ( l _ 1 ) & = K \\cap ( l _ 2 ) = ( 0 ) . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\mathrm { T r } _ { Y _ 3 / X } ( \\xi ) = \\xi + \\xi ^ { \\sigma _ 3 } \\in H ^ 0 ( X _ S , \\Delta ) . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} A _ i = \\left [ 4 ^ { i } , 4 ^ { i } + i - 1 \\right ] . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\int _ A \\varphi ( t , X ( t ) , a ) [ \\widehat { \\gamma } ( t , X ( t ) ) ] ( d a ) = E \\left [ \\int _ A \\varphi ( t , X ( t ) , a ) \\rho _ t ( d a ) \\big | X ( t ) \\right ] \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} Y _ n : = \\sum _ { - n 2 ^ n } ^ { n 2 ^ n - 1 } \\frac { k } { 2 ^ { n } } I _ { \\{ \\frac { k } { 2 ^ { n } } \\leq Y < \\frac { k + 1 } { 2 ^ { n } } \\} } + n I _ { \\{ Y \\geq n \\} } - n I _ { \\{ Y < - n \\} } . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} G ( s , z ) & = \\frac { \\Gamma ( s - \\tfrac { 1 } { 2 } + z ) \\Gamma ( s - \\tfrac { 1 } { 2 } - z ) } { \\Gamma ( s ) \\Gamma ( s + k - 1 ) } \\\\ \\mathcal { Z } ( s , w , z ) & = \\frac { \\zeta ( s + w - \\frac { 1 } { 2 } + z ) \\zeta ( s + w - \\frac { 1 } { 2 } - z ) } { \\zeta ^ * ( 1 + 2 z ) } . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = K + ( l _ 2 ) , \\\\ ( 0 ) : l _ 2 & = K + ( l _ 1 ) , \\\\ K \\cap ( l _ 1 ) & = K \\cap ( l _ 2 ) = ( 0 ) . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} { \\bar { B } } _ { k } ^ { n } ( t ) = { \\tilde { B } } _ { k } ^ { n } ( t ) + { \\hat { B } } _ { k } ^ { n } ( t ) , \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} d ^ \\Delta = d ^ 1 - d ^ 2 \\ , . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} G ' & = \\left \\{ \\begin{array} { l l } G [ y \\rightarrow x ; A ] & \\quad { G [ y \\rightarrow x ; A ] } \\\\ G [ y \\rightarrow x ; A ] + y x ' & \\quad { } . \\end{array} \\right . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} F _ z ~ = ~ a F _ x \\ , + \\ , h \\ \\mbox { w i t h } h \\perp F _ x \\ , . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\norm { T _ \\lambda ^ { z } } _ { 2 } \\lesssim \\begin{cases} C _ z \\abs { \\lambda } ^ { - 1 / 2 } , & \\sigma > \\sigma _ 2 ; \\\\ C _ z \\abs { \\lambda } ^ { - 1 / 2 } \\log ( \\lambda ) , & \\sigma = \\sigma _ 2 ; \\\\ C _ z \\abs { \\lambda } ^ { - [ ( d - 2 ) \\sigma + n ] / d } , & \\sigma _ 1 < \\sigma < \\sigma _ 2 . \\end{cases} \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\widetilde { H } ( t , x , Q ^ \\ast , p , g ) = \\min _ { Q \\in \\mathcal { P } ( A ) } \\widetilde { H } ( t , x , Q , p , g ) = \\min _ { a \\in A } H ( t , x , a , p , g ) = H ( t , x , a ^ \\ast , p , g ) \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} r a ^ { a / 2 } \\mathtt { I } ' _ \\nu ( r ) - \\left ( ( \\nu - 1 ) ( 1 - a ) a ^ { a / 2 } + \\beta ( a \\nu - a + 1 ) \\right ) \\mathtt { I } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} X = \\sum _ i X _ i \\otimes f _ i , f _ i \\in L o o p ( \\mathbb C ) \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\Delta f ( x ) = f ( x + 1 ) - f ( x ) , ( \\textnormal { s e e } \\ , \\ , [ 3 , 7 ] ) . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\widehat S _ { \\ell } ( X _ 1 \\times _ { M _ 1 } \\dots \\times _ { M _ { n - 1 } } X _ n ) = \\coprod _ { \\ell _ 1 + \\dots + \\ell _ n = \\ell } ( \\widehat S _ { \\ell _ 1 } ( X _ 1 ) \\times _ { M _ 1 } \\dots \\times _ { M _ { n - 1 } } \\widehat S _ { \\ell _ n } ( X _ n ) ) . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} D _ { f , \\chi } ( s ) = ( q / q _ * ) ^ { 1 - 2 s } \\xi ( q / q _ * ) D _ { g , \\overline \\chi } ( 1 - s ) . \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} \\bar B = \\left [ \\begin{array} { c c } 0 & 0 \\\\ \\times & 0 \\\\ 0 & \\times \\end{array} \\right ] , \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\mathcal { V } _ { i } ^ { ( n , \\sigma ) } = \\{ e _ { i } \\otimes e _ { j } : j \\neq i \\ ; \\ ; j \\neq \\sigma ^ { - 1 } ( i ) , \\ ; \\ ; 1 \\leq j \\leq n \\} . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\bar { \\epsilon } = K \\cdot L - \\epsilon . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\bar n ( P , \\langle \\rangle ) & = P , \\\\ \\bar n ( P , \\sigma ^ \\frown a ) & = n ( \\bar n ( P , \\sigma ) , a ) . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\phi V _ { 2 } , \\varphi Z ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\omega V _ { 2 } , \\mathcal { B } Z ) \\\\ & - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\omega V _ { 2 } , \\mathcal { C } Z ) - g _ { 1 } ( V _ { 2 } , \\phi U _ { 2 } ) \\eta ( Z ) . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} ( 1 - | y ^ { - 1 } x \\cdot o | ^ 2 ) ^ { ( s - \\epsilon ) / 2 } = & | d _ { y ^ { - 1 } x } | ^ { - ( s - \\epsilon ) } \\\\ = & ( 1 - | x \\cdot o | ^ 2 ) ^ { ( s - \\epsilon ) / 2 } \\ , ( 1 - | y \\cdot o | ^ 2 ) ^ { ( s - \\epsilon ) / 2 } ( 1 - ( x \\cdot o , y \\cdot o ) ) ^ { - ( s - \\epsilon ) } . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\log g _ { k } ( x , y ) = - \\frac { p } { 2 } \\log 2 \\pi - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { p } x ^ { ( i ) } - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { p } ( y ^ { ( i ) } ) ^ { 2 } \\exp ( - x ^ { ( i ) } ) , \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} a ( n ) : = \\# \\{ ( E , F ) : ~ F \\in \\mathcal { F } _ G ^ { + } , ~ \\textrm { $ E / F $ t o t a l l y i m a g i n a r y q u a d r a t i c } , ~ | d _ E | = n \\} . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ i = n _ i , 1 \\le i \\le k ) : = \\lim _ { n \\to \\infty } \\mathbb { P } ( \\Pi ^ { [ n ] } _ i = n _ i , 1 \\le i \\le k ) , \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} s _ k + \\sum _ j \\varphi _ j \\varphi _ { j k } \\quad , k = 2 , \\dots , d \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} u _ n = \\int _ 0 ^ { \\infty } e ^ { - x } \\mathbb { E } \\prod _ { i = 1 } ^ n \\left ( 1 - e ^ { - x P _ i / T _ n } \\right ) d x . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ X ] I _ { \\{ \\tau = \\sigma \\} } = \\hat { \\mathbb { E } } _ { \\sigma + } [ X I _ { \\{ \\tau = \\sigma \\} } ] . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\xi _ n & = T _ n ( \\xi ) , f _ n ( t , y ) = f _ n ( t , y , \\varUpsilon _ t , \\varPsi _ t ) = f ( t , y ) - f ( t , 0 ) + T _ n \\bigl ( f ( t , 0 ) \\bigr ) , \\\\ R ^ n _ t & = \\int _ 0 ^ t \\ 1 _ { \\{ | R | _ s \\le n \\} } d R _ s . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} I _ 2 & = \\int _ { E _ 2 } \\big \\lvert H \\big ( u + \\mathrm { i } a ( u ) + \\mathrm { i } \\tau \\big ) \\big \\rvert ^ q \\cdot \\lvert 1 + \\mathrm { i } a ' ( u ) \\rvert \\ , \\mathrm { d } u \\\\ & \\leqslant M _ 2 ^ q \\cdot \\sqrt { 1 + M ^ 2 } \\cdot 2 \\delta _ 0 . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} 2 = 2 F ( x ) = F ( y _ 1 + y _ 2 ) = F ( y _ 1 ) + F ( y _ 2 ) \\leq 2 , \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\bar \\mu ^ T \\ ; = \\ ; \\int _ 0 ^ 1 \\mu ^ { 1 , T } ( \\eta _ s ) \\ , d s \\ ; . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} ( s ^ 2 - t ^ 2 ) ^ 2 + 2 ( 2 s t ) ^ 2 + 2 i ( 2 s t ) ( i ( s ^ 2 + t ^ 2 ) ) = ( s - t ) ^ 4 \\quad , \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\mathcal N ( \\alpha _ - , \\alpha _ + ) = \\begin{cases} R _ { \\alpha _ - } & \\alpha _ - = \\alpha _ + , \\\\ \\emptyset & \\alpha _ - \\ne \\alpha _ + . \\end{cases} \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} { \\rm d i v } \\left ( b ^ 0 { \\partial ^ 2 u _ 0 \\over \\partial t ^ 2 } \\right ) = { \\rm d i v } f , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\left ( z - z _ { 0 } \\right ) ^ { - m } \\sum \\limits _ { s = 0 } ^ { \\infty } { f } _ { s } \\left ( z - z _ { 0 } \\right ) ^ { s } , \\ g \\left ( z \\right ) = \\left ( z - z _ { 0 } \\right ) ^ { - p } \\sum \\limits _ { s = 0 } ^ { \\infty } { g } _ { s } \\left ( z - z _ { 0 } \\right ) ^ { s } , \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} \\lambda ^ * = \\inf \\left \\{ \\frac { \\int | \\nabla u | ^ p d x } { \\int | u | ^ p d x } : ~ ~ \\int f | u | ^ { \\gamma } d x \\geq 0 , ~ u \\in W \\setminus 0 \\right \\} , \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} \\frak h ^ { \\epsilon } _ { \\alpha _ 2 , \\alpha _ 1 } ( h ) = { \\rm e v } _ { + } ! ( { \\rm e v } _ { - } ^ * h ; \\widehat { \\frak S ^ { + \\epsilon } } ( \\alpha _ 1 , \\alpha _ 2 ; [ 1 , 2 ] ) ) . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\frac { d a } { d s } \\Big | _ { s = 0 } & = \\frac { 1 } { p ( r ) } \\frac { d a } { d r } \\Big | _ { r = r _ b } = \\frac { 1 } { 2 L } \\frac { d a ^ 2 } { d r } \\Big | _ { r = r _ b } = - \\frac { ( n + 1 ) L } { r _ b } \\\\ b | _ { s = 0 } & = ( L ^ 2 - r _ b ^ 2 ) . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} A ' _ n ( v ) & = \\frac { 1 } { 1 - v } ( A ' _ { n - 1 } ( v ) - A ' _ { n - 1 } ( 1 ) v ^ { n - 2 } ) + ( A _ { n - 1 } ( 1 ) - A _ { n - 2 } ( 1 ) ) v ^ { n - 2 } + A _ { n - 1 } ( 1 ) v ^ { n - 1 } \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} & ( a ) \\ \\ \\phi D _ 1 = D _ 1 , \\ , \\ , \\ , \\ , \\ , \\ , ( b ) \\ \\ \\omega D _ 1 = 0 , \\\\ & ( c ) \\ \\ \\phi D _ 2 \\subset D _ 2 , \\ , \\ , \\ , \\ , \\ , \\ , ( d ) \\ \\ \\mathcal { B } ( k e r \\pi _ * ) ^ \\perp = D _ 2 , \\\\ & ( e ) \\ \\ \\mathcal { T } _ { U _ 1 } \\xi = \\phi U _ 1 , \\ , \\ , \\ , \\ , ( f ) \\ \\ \\hat { \\nabla } _ { U _ 1 } \\xi = - \\omega U _ 1 , \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\| f \\| _ { X ^ { 1 - \\theta } Y ^ { \\theta } } = \\inf \\{ \\max \\{ \\| g \\| _ X , \\| h \\| _ Y \\} : | f | = g ^ { 1 - \\theta } h ^ { \\theta } , \\ g \\in X , \\ h \\in Y \\} , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\left ( \\mathbf { X } _ { 0 } , \\mathbf { Q } _ { 0 } \\right ) = \\left ( x _ { 0 } ^ { c } , \\mathbf { q } _ { 0 } \\right ) \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} J _ { \\mathcal { G } _ { f , g _ \\diamond } , g _ \\diamond } ( x ) = \\left \\{ \\begin{array} { @ { } l l } \\mu _ g \\cdot \\frac { I _ f ( x ) } { 1 - x } & \\ \\mbox { i f } \\ x \\in [ 0 , 1 ) , \\\\ \\infty & \\ \\mbox { o t h e r w i s e , } \\end{array} \\right . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} \\| f \\| _ { M _ G } : = \\sup _ { t > 0 } \\frac { \\int _ 0 ^ t \\mu ( f ) } { \\int _ 0 ^ t \\mu ( g ) } \\leq 1 . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\l ( H _ i ) = i + 2 > \\log _ 3 l ( H _ i ) + 1 . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\mu _ 0 D _ v H _ { \\lambda ^ * } ( \\phi ^ * _ 1 ) = \\mu _ 1 D _ v F ( \\phi ^ * _ 1 ) . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} \\lVert A ^ { n + m } ( x ) P ( x ) \\rVert & = \\lVert A ^ n ( f ^ m ( x ) ) A ^ m ( x ) P ( x ) ^ 2 \\rVert \\\\ & = \\lVert A ^ n ( f ^ m ( x ) ) P ( f ^ m ( x ) ) A ^ m ( x ) P ( x ) \\rVert \\\\ & \\le \\lVert A ^ n ( f ^ m ( x ) ) P ( f ^ m ( x ) \\rVert \\cdot \\lVert A ^ m ( x ) P ( x ) \\rVert , \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} h _ l = \\prod _ { i = 1 } ^ m \\frac { L _ i ! ( 2 l ) ! } { ( L _ i + 2 l ) ! } . \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} q = - \\sigma \\Delta _ { g } \\eta ^ { \\alpha } \\hat n _ { \\alpha } , \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} B ^ { \\delta , \\varrho } _ { T , \\gamma } \\ ; = \\ ; \\Big \\{ \\eta : \\int _ 0 ^ 1 W ^ { \\delta } _ { T , \\gamma } ( \\eta _ s ) \\ , d s \\ , \\le \\varrho \\Big \\} \\ ; , \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} | w | ^ 2 - 4 ( 2 t ) \\bigg ( t ^ 2 - \\frac { \\lambda } { 3 } \\bigg ) = 0 ~ ~ \\Rightarrow ~ ~ \\bigg ( t ^ 2 - \\frac { \\lambda } { 3 } \\bigg ) = \\frac { | w | ^ 2 } { 8 t } \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} \\dfrac { 1 } { 2 } | \\dot { x } | ^ 2 = U ( x ) ; \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} X _ H ^ { \\gamma } = T \\pi \\circ X _ H \\circ \\gamma . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\langle I ' ( u _ { 0 } , v _ { 0 } ) , ( u _ { 0 } , v _ { 0 } ) \\rangle + \\liminf _ { n \\rightarrow + \\infty } \\langle I ' ( w _ { n } , z _ { n } ) , ( w _ { n } , z _ { n } ) \\rangle = 0 . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\aligned & \\sum _ { k _ 1 + k _ 2 = k + 1 } \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\sum _ { i = 1 } ^ { k - k _ 2 + 1 } \\\\ & ( - 1 ) ^ * { \\frak m } ^ { \\epsilon , \\rho , [ 0 , 1 ] ^ { \\boxplus \\tau _ 0 } } _ { k _ 1 , \\beta _ 1 } ( h _ 1 , \\ldots , { \\frak m } ^ { \\epsilon , \\rho , [ 0 , 1 ] ^ { \\boxplus \\tau _ 0 } } _ { k _ 2 , \\beta _ 2 } ( h _ i , \\ldots , h _ { i + k _ 2 - 1 } ) , \\ldots , h _ { k } ) = 0 , \\endaligned \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} H ( t , \\rho ) = \\int _ { - 1 } ^ t ( f ( s ) x ( s , \\rho ) + g ( s ) ) d s . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d W } { d t } ( t ) \\le G ( W ( t ) ) , \\\\ W ( 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\omega ^ { A _ { \\Delta } } ( V ) \\leq \\sum _ { j \\in J } \\omega ^ { A _ { \\Delta } } ( 5 \\Delta _ { j } ) \\le C \\sum _ { j \\in J } \\omega ^ { A _ { \\Delta } } ( \\Delta _ { j } ) \\le C \\sum _ { j \\in J } \\frac { \\omega ( \\Delta _ { j } ) } { \\omega ( \\Delta ) } = C \\frac { \\omega \\left ( \\bigcup _ { j \\in J } \\Delta _ { j } \\right ) } { \\omega ( \\Delta ) } \\le C \\frac { \\omega ( V ) } { \\omega ( \\Delta ) } . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\phi ( x , \\kappa ^ i r ) \\lesssim \\sum _ { i = 0 } ^ { i _ 0 } \\varphi ( \\bar x , 2 \\kappa ^ i r ) + \\norm { D u } _ { L ^ \\infty ( B ( x , r ) \\cap B ^ + _ 4 ) } \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\mathbf { A } } ( t ) } t \\ , d t + \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\vec g } ( t ) } t \\ , d t . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\varphi ^ { i , m } _ { l } \\cdot \\widetilde { \\varphi } ^ { i , m } _ { l } = 0 , \\ \\ \\ \\ \\ l \\geq 1 . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\Psi _ + \\Psi _ - = \\frac { \\varphi w ( \\Psi _ + , \\Psi _ - ) } { 2 \\mu } \\in K ( \\Gamma _ s ) , \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} \\mathcal { L } _ \\omega ( a ) : = \\{ t \\geq 0 : B _ t ( \\omega ) = a \\} . \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} \\mathrm { b i d e g } ( \\phi ) = \\mathrm { b i d e g } ( \\varepsilon _ { r - ( n + p ) , s - ( n + p ) , 0 } ) = ( \\ , r - s , r + s - 2 ( n + p ) \\ , ) . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\nu : T \\longrightarrow X _ * ( T ) = P ( \\Phi ^ \\vee ) \\subseteq V \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\| r ( t ) \\| _ { L ^ q } \\ , = \\ , \\| r _ 0 \\| _ { L ^ q } \\qquad \\qquad \\qquad \\mbox { f o r a l l } \\ ; q \\ , \\in \\ , [ 2 , + \\infty ] \\ , . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{gather*} \\partial _ t \\eta = v . \\end{gather*}"}
-{"id": "1380.png", "formula": "\\begin{align*} a d + 4 = x ^ 2 , & b d + 4 = y ^ 2 , c d + 4 = z ^ 2 , \\\\ x = \\frac { a t + r s } { 2 } , & y = \\frac { r t + b s } { 2 } , z = \\frac { c r + s t } { 2 } . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\int _ \\Omega u ^ * \\ , d \\mu = \\lim _ { h \\to + \\infty } \\int _ \\Omega u ^ * \\ , d \\mu _ h , \\int _ \\Omega u ^ \\pm \\ , d \\mu = \\lim _ { h \\to + \\infty } \\int _ \\Omega u ^ \\pm \\ , d \\mu _ h . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\underset { E \\searrow 0 } { { \\lim } } ~ ~ \\frac { \\ln | \\ln N ( E ) | } { \\ln E } ~ = ~ - \\frac { d } { 2 } \\ , . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} C F ( \\mathcal F ^ i ; \\epsilon ) = ( C F ( \\mathcal F ) , \\{ \\frak m ^ { i , \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } \\} ) . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} X = \\left ( 2 a _ { 1 } \\psi _ { Y } t + d _ { 1 } \\right ) \\partial _ { t } + a _ { 1 } Y ^ { i } \\partial _ { i } ~ , ~ ~ f = c _ { 2 } \\int \\omega d t , \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} Q _ 1 \\sqrt { \\frac { 2 ^ { \\sum _ { l = 1 } ^ d h _ { n , l } ( \\boldsymbol { \\alpha } ) } \\log { n } } { n } } + \\sqrt { 2 Q _ 2 C _ { I } } 2 ^ { \\sum _ { l = 1 } ^ d h _ { n , l } ( \\boldsymbol { \\alpha } ) / 2 } \\epsilon _ n \\leq 2 \\sqrt { 2 Q _ 2 C _ I } \\epsilon _ n ^ { 2 \\alpha ^ { * } / ( 2 \\alpha ^ { * } + d ) } , \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\mathcal D ^ { t } : = \\Delta w ^ { t } = \\frac 1 t \\frac { \\partial _ N \\ , v } { N \\cdot \\nu ^ 0 } \\ , \\mathcal { H } ^ { n - 1 } \\restriction _ { \\Gamma ^ 0 } - \\left ( \\frac { 1 } { t ^ 2 } \\Delta h ^ 0 + \\frac { 1 } { t } \\Delta \\dot h ^ 0 \\right ) \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } - \\frac 1 2 \\Delta \\delta _ t ^ 2 h ^ 0 \\chi _ { \\Omega ^ t } . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\tilde { A } \\left ( W ' , \\overline { W } ' \\right ) = \\frac { 1 } { \\left ( I _ { q '^ { 2 } } + R \\left ( W ' \\right ) \\right ) \\cdot \\left ( \\overline { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) ^ { t } } . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} W _ { B , \\beta } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { k ! \\Gamma ( \\beta _ 1 + k B _ 1 ) ) } , \\ ; z \\in \\mathbb { C } . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { n p } ^ 4 } \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 4 ) = O \\left ( \\frac { 1 } { p _ n } \\right ) \\to 0 ~ ~ \\mbox { a s } n \\to \\infty , \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} d ( f _ 2 ) = N ( N - 2 ) + 1 + \\sum \\limits _ { \\substack { d | N \\\\ 1 < d \\leq N } } \\frac { \\varphi ( \\gcd ( d , N / d ) ) } { \\gcd ( d , N / d ) } \\frac { N } { d } ( N - 2 ) = ( N - 2 ) d ( f _ 1 ) + 1 . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\langle A \\chi _ j , \\chi _ k \\rangle = a _ { k - j } \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} G \\left ( t \\right ) = \\begin{cases} G \\left ( t - 1 \\right ) & G \\left ( t - 1 \\right ) + e \\left ( t \\right ) k ; \\\\ G \\left ( t - 1 \\right ) + e \\left ( t \\right ) & \\end{cases} \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} V ' : = \\{ V \\setminus \\{ t , u \\} \\} \\cup \\{ u ' \\} . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\lambda _ i : = \\min \\{ j \\ , | \\ , f ( i , j ) \\not = \\circ \\} . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\Pi \\begin{pmatrix} u _ + \\\\ u _ - \\end{pmatrix} \\ ! \\ ! = \\ ! \\ ! \\begin{pmatrix} v ( P _ Y , 0 ^ - ) - v ( P _ Y , 0 ^ + ) \\\\ v ( - P _ Y , 0 ^ - ) - v ( - P _ Y , 0 ^ + ) \\end{pmatrix} , \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\begin{aligned} | \\eta _ { n , j } | + | \\nabla \\eta _ { n , j } | = O ( \\lambda _ { n , j } ^ 2 e ^ { - \\lambda _ { n , j } } ) . \\end{aligned} \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\mathbb E \\langle f \\star \\bar { \\mu } , g \\star \\bar { \\mu } \\rangle = \\mathbb E \\langle f , g \\rangle \\star \\nu . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} f ( \\overline y , ( t _ 1 , \\dots , t _ k ) ) = \\sum _ { I \\subseteq \\{ 1 , \\dots , k \\} } f _ I ( \\overline y , t _ { I } ) . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} ~ d ( \\mathbf { x } ) = \\hat f _ { } - \\hat f ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{gather*} \\partial _ t \\eta = v . \\end{gather*}"}
-{"id": "9820.png", "formula": "\\begin{align*} ( F ( M ) \\circ F ( D ' ) \\circ F ( M ' ) \\circ F ( D ) ) ^ 2 & = F ( M ) \\circ F ( D ' ) \\circ F ( M ' ) \\circ F ( D ) . \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} A ^ + _ { j i } = \\tilde { e } ^ { ( 0 ) } _ { j i } - \\sum _ { j < j ' < i } \\tilde { e } ^ { ( 0 ) } _ { j j ' } A ^ + _ { j ' i } . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\frac { x _ { k + 1 } - x _ k } h = a x _ k - b x _ k y _ k , \\frac { y _ { k + 1 } - y _ k } h = - c y _ k + d x _ k y _ k , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} L ( t , x , D _ t , D _ x ) = D _ { t } - \\sum _ { j = 1 } ^ d A _ j ( t , x ) D _ { x _ j } = D _ t - A ( t , x , D _ x ) \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ m a ( j ) y _ j \\right \\| \\le C \\| a \\| _ p , \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} \\partial ( p f _ s ) = \\partial ( p ) f _ s + p \\partial ( f _ s ) = \\partial ( p ) f _ s \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} & A ( t ) ^ { 1 / 2 } F ( t ) = | t | S ^ { 1 / 2 } P + t | t | G + \\Phi ( t ) , | t | \\le t ^ 0 , \\\\ & G : = Z S ^ { 1 / 2 } P + S ^ { 1 / 2 } P Z ^ * + \\frac { 1 } { 2 } N _ 0 S ^ { - 1 / 2 } P + S ^ { - 1 / 2 } N _ * P + N _ * S ^ { - 1 / 2 } P + \\mathcal { I } _ * ( 1 ) . \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} T : = \\min \\Big \\{ \\big ( \\tfrac { \\eta } { M } \\big ) ^ { \\frac { d + 2 } { d - 2 } } , \\big ( \\tfrac { \\eta } { M } \\big ) ^ { \\frac { 4 } { d - 2 } } \\Big \\} ^ \\frac { 1 } { \\theta } . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} A \\leq \\lambda _ 1 , B = C \\geq \\lambda _ 2 , D \\geq \\lambda _ 4 , E \\geq \\lambda _ 5 + 2 t . \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\begin{aligned} L ^ { \\rm r e s c , h } _ n ( u ) - L ^ { \\rm r e s c , h } _ n ( v ) & \\geq B ^ { \\rho _ - } _ n ( u ) - B ^ { \\rho _ - } _ n ( v ) + ( u ^ 2 - v ^ 2 ) \\\\ & + \\left ( \\frac { \\beta _ 1 } { 1 - \\rho _ - } - 2 ( 1 + \\gamma ) ^ { 5 / 3 } \\gamma ^ { 1 / 3 } \\right ) \\frac { ( u - v ) } { \\beta _ 2 } n ^ { 1 / 3 } , \\end{aligned} \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\kappa ( L ( I _ n - \\widetilde { V } _ k \\widetilde { V } _ k ^ T ) ) & = \\kappa ( L \\widetilde { V } _ { n - k } \\widetilde { V } _ { n - k } ^ T ) = \\kappa ( L \\widetilde { V } _ { n - k } ) . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} e ^ { r e g } = \\sum _ i v _ i \\frac { - \\lambda } { s _ i ^ 2 + \\lambda } s _ i ( u _ i ^ T z ) \\ , , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} H ( X ^ { ( i ) } | \\bar { X } | ^ { 2 n } , \\mathfrak { c } I _ m ) = e ^ { - \\frac { \\mathfrak { c } } { 2 } \\Delta } X ^ { ( i ) } | \\bar { X } | ^ { 2 n } = \\sum _ { k = 1 } ^ { n } b _ k ( \\mathfrak { c } ) X ^ { ( i ) } | \\bar { X } | ^ { 2 k } \\ ; , \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} C ^ { \\delta ; \\frac { \\delta } { 2 } } ( \\overline { Q } ) : = \\{ g \\in C ( \\overline { Q } ) : \\mathrm { p h \\ddot { o } l } _ \\delta ( g , \\overline { Q } ) < \\infty \\} . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\ell \\left ( ( - 1 ) ^ { m _ j } \\sum _ { k = 1 } ^ n a _ { j k } \\frac { \\partial ^ { 2 m _ j } u } { \\partial x _ k ^ { 2 m _ j } } \\right ) = | u | ^ { q - 2 } u , \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} h ^ { p , q } ( Y , V ) : = \\dim H ^ q ( Z , \\Omega _ Z ^ p ( \\log D _ \\infty , \\mathrm { r e l } \\ , V ) ) . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} L ( \\vert y - \\zeta _ { i , n } ^ { \\prime } \\vert ^ { - \\nu } ) & = - \\Bigl ( \\nu \\ , ( 1 - \\nu ) - \\bigl ( \\varepsilon _ n ^ 2 \\ , \\lambda + 5 ( V _ 1 + V _ 2 ) ^ 4 \\bigr ) \\ , \\vert y - \\zeta _ { i , n } ^ { \\prime } \\vert ^ 2 \\Bigr ) \\ , \\vert y - \\zeta _ { i , n } ^ { \\prime } \\vert ^ { - ( 2 + \\nu ) } \\\\ & \\leq - \\frac { \\nu \\ , ( 1 - \\nu ) } { 2 } \\ , \\vert y - \\zeta _ { i , n } ^ { \\prime } \\vert ^ { - ( 2 + \\nu ) } \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k \\int _ { [ 0 , t ] \\times [ 0 , \\infty ) } 1 _ { [ q _ { k - 1 } ( s - ) , q _ k ( s - ) ) } ( x ) N ( d s \\times d x ) \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = 0 } ^ \\infty \\int _ { [ 0 , T ] \\times [ 0 , 1 ] } \\ell ( \\varphi _ k ^ n ( s , y ) ) \\ , d s \\ , d y \\le 2 M _ h \\| h \\| _ { \\infty } / \\sigma \\doteq K _ 0 , \\mbox { a . s . } { P } . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} L ( \\phi ) = \\Delta \\phi + \\varepsilon ^ 2 \\ , \\lambda \\ , \\phi + 5 V ^ 4 \\ , \\phi , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} 0 & = - 2 a _ 1 \\bar \\Lambda _ 1 + \\frac { \\partial } { \\partial \\bar \\Lambda _ 1 } \\mathcal P o l y _ 4 ( 0 , \\zeta _ 0 , \\bar \\Lambda ) , \\\\ 0 & = - 2 a _ 1 \\sigma _ l \\bar \\Lambda _ l , l = 2 , \\ldots , k . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} S = P M ^ * \\widehat { S } M | _ \\mathfrak { N } . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} 1 r _ 1 + 2 r _ 2 + \\cdots + ( m - k + 1 ) r _ { m - k + 1 } = m . \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} x \\cdot ( x - 2 ) ( x - 3 ) \\cdots ( x - k + 1 ) \\cdot ( x - 1 ) ^ { n - k + 1 } = x ^ { \\underline { k } } \\ , ( x - 1 ) ^ { n - k } \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} A _ { n , i } : = \\left \\{ \\min _ { j > n } \\Pi ^ { - 1 } _ j < \\Pi ^ { - 1 } _ i \\right \\} , \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\left < L , V \\right > = L \\overline { V } ^ { t } , \\quad \\mbox { f o r $ L \\in \\mathcal { M } _ { m , n } \\left ( \\mathbb { C } \\right ) $ a n d $ V \\in \\mathcal { M } _ { n , p } \\left ( \\mathbb { C } \\right ) $ , f o r $ m , n , p \\in \\mathbb { N } ^ { \\star } $ , } \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} D _ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } d _ { i } \\left ( y \\right ) d y , \\quad \\mathbb { D } _ { 0 } ^ { i } : = \\left ( \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } d _ { i } \\left ( y \\right ) \\frac { \\partial \\bar { u } _ { i } ^ { j } } { \\partial y _ { k } } d y \\right ) _ { j k } , \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} & K _ i : = \\Lambda ( y ^ 0 _ i , r _ i , \\lfloor \\frac { n } { 2 } \\rfloor ) , \\\\ & t _ i : = m o d ( K _ i ( 1 , \\lfloor \\frac { n } { 2 } \\rfloor ) \\times 1 0 ^ 5 , m \\times n ) . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} & \\sum _ { \\lvert 2 m ^ 2 + h - X \\rvert \\leq X } r _ { 2 k + 1 } ( m ^ 2 + h ) \\\\ & = \\delta _ { [ h = a ^ 2 ] } \\bigg ( 2 R ' _ h X ^ { \\frac { 1 } { 2 } } \\log X - 4 R ' _ h X ^ { \\frac { 1 } { 2 } } \\bigg ) + 2 R _ { 1 / 2 , h } ^ { 1 / 2 } X ^ { \\frac { 1 } { 2 } } + O ( X ^ { \\frac { 1 } { 2 } - \\frac { 1 } { 4 4 } + \\epsilon } ) , \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\varphi = ( z _ 1 , P ( z _ 1 ) - \\delta z _ 0 ) = ( z _ 1 , z _ 0 ) \\circ \\big ( P ( z _ 1 ) - \\delta z _ 0 , z _ 1 \\big ) . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\vartheta } - \\boldsymbol { \\vartheta } _ 0 \\| _ \\infty \\left \\| \\sum _ { m _ 1 = 0 } ^ { 2 ^ { N _ 1 } - 1 } \\cdots \\sum _ { m _ d = 0 } ^ { 2 ^ { N _ d } - 1 } | D ^ { \\boldsymbol { r } } \\varphi _ { \\boldsymbol { N } , \\boldsymbol { m } } | \\right \\| _ \\infty \\lesssim \\| \\boldsymbol { \\vartheta } - \\boldsymbol { \\vartheta } _ 0 \\| \\lesssim \\epsilon _ { n , \\boldsymbol { r } } , \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\overline { \\partial } { } ^ 2 g ^ { - 1 } & = Q ( \\overline { \\partial } \\eta ) ( \\overline { \\partial } { } ^ 2 \\eta ) ^ 2 + Q ( \\overline { \\partial } \\eta ) \\overline { \\partial } { } ^ 3 \\eta , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} ( \\gamma _ 1 , g _ 1 ) \\circ ( \\gamma _ 2 , g _ 2 ) = ( \\gamma _ 1 h _ { g _ 1 } ( \\gamma _ 2 ) \\gamma _ { g _ 1 , g _ 2 } , g _ 1 g _ 2 ) . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} x ( t , \\rho ) = \\rho + \\sum _ { k = 2 } ^ { \\infty } r _ k ( t ) \\rho ^ k . \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\| ( x _ 0 , \\ldots , x _ n ) \\| = \\max \\{ ( 1 / a _ 0 ) | x _ 0 | , \\ldots , ( 1 / a _ d ) | x _ d | \\} . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} ( Q h ) ( x ) = \\int _ 0 ^ { \\infty } \\int _ 0 ^ { \\infty } \\chi ( s , t ) \\ J _ { ( s , t ) } ( x ) \\ h ( \\Psi ^ { ( s , t ) } ( x ) ) \\ d s \\ d t . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} u ( t , x ) \\ \\longmapsto \\ u _ { \\mu } ( t , x ) : = \\mu ^ { \\frac { d - 2 } { 2 } } u ( \\mu ^ 2 t , \\mu x ) \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} B ' + C ' = ( B + t ) + ( C - t ) = B + C . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} A _ 0 = \\sup \\big \\{ 2 + 2 | | b | | _ { L ^ { \\infty } ( \\Pi \\times [ 0 , T ] ) } | x | - | x | ^ { 2 } \\ | \\ x \\in \\Pi \\ \\big \\} > 0 . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} S : = \\lim _ { t \\to \\infty } \\Delta _ { H _ 1 } ^ { i t } \\Delta _ { H _ 2 } ^ { - i t } = W ^ 1 ( - 1 ) W ^ 2 ( 1 ) , \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} v \\ \\sim \\ w \\ \\Longleftrightarrow \\ P _ v = P _ w \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} ( \\alpha _ { 2 \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } ) ( s ) = \\alpha _ { 2 \\underline { d } } ( s ) \\in V _ { X _ 2 } ( - ( i - 1 ) A - l B ) . \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\Sigma _ 0 = \\sum _ { A ' , B ' } \\left ( \\sum _ { A } \\delta _ { A ' } ^ { A } f _ { B ' A } , \\sum _ { B } \\delta ^ { B } _ { A ' } f _ { B ' B } \\right ) _ { \\varphi } = \\sum _ { A ' , B ' } \\left \\| \\sum _ { A } \\delta _ { A ' } ^ { A } f _ { B ' A } \\right \\| _ \\varphi ^ 2 \\geq 0 , \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\frac 1 { 1 - \\omega _ 1 h _ 1 - \\cdots - \\omega _ p h _ p } \\cdot \\prod _ { i = 1 } ^ p \\frac { ( 1 - h _ i ) ^ { m _ i } } { 1 - 2 \\omega _ i h _ i } \\quad . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} ( 1 - v ) A ' _ n ( v ) & = v A _ { n - 1 } ( v ) - A _ { n - 1 } ( 1 ) v ^ n - v ^ 2 \\sum _ { j = 0 } ^ { n - 5 } W _ j ( v ) + \\sum _ { j = 0 } ^ { n - 5 } W _ j ( 1 ) v ^ { j + 3 } . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} V _ \\Delta + \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) & = \\ell ( \\Delta ) - 1 - \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) \\\\ V _ { \\Delta ^ \\circ } + \\sum _ { F ^ \\circ \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ^ \\circ ) & = \\ell ( \\Delta ^ \\circ ) - 1 - \\sum _ { F ^ \\circ \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ^ \\circ ) . \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} H ( \\chi ) \\approx \\sum _ { i = 1 } ^ k \\frac { a _ i } n \\log \\frac { n } { a _ i } . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} \\Phi _ h \\bigg ( \\bigg ( \\sum _ { i = 0 } ^ \\ell r _ i x _ i \\bigg ) ^ h \\bigg ) = \\frac 1 { h ! } \\sum _ { \\sigma \\in \\Sigma _ h } \\sum _ { j _ 1 = 0 } ^ \\ell \\cdots \\sum _ { j _ h = 0 } ^ \\ell r _ { j _ 1 } \\cdots r _ { j _ h } z _ { j _ 1 j _ 2 } \\cdots z _ { j _ { h - 1 } j _ h } . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} S ( u , x ) = \\sum _ { \\substack { x < q \\le V \\\\ q | P ( x ) } } \\frac { \\mu ( q ) } { \\varphi ( q ) } A ( q , u ) + \\sum _ { \\substack { q > V \\\\ q | P ( x ) } } \\frac { \\mu ( q ) } { \\varphi ( q ) } A ( q , u ) . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} ( m + 2 ) \\int _ { \\Omega } u = m c ^ { 2 } | \\Omega | , \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\eta _ n : = \\sum _ { - 2 ^ n } ^ { 2 ^ n } \\frac { k C _ \\eta } { 2 ^ { n } } I _ { \\{ \\frac { k C _ \\eta } { 2 ^ { n } } \\leq \\eta < \\frac { ( k + 1 ) C _ \\eta } { 2 ^ { n } } \\} } \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} E ( z , w ) & = y ^ w + \\phi ( w ) y ^ { 1 - w } \\\\ & + \\frac { 2 \\pi ^ w \\sqrt { y } } { \\Gamma ( w ) \\zeta ( 2 w ) } \\sum _ { m \\neq 0 } \\vert m \\vert ^ { w - \\frac { 1 } { 2 } } \\sigma _ { 1 - 2 w } ( \\vert m \\vert ) K _ { w - \\frac { 1 } { 2 } } ( 2 \\pi \\vert m \\vert y ) e ^ { 2 \\pi i m x } , \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} f ^ { i } ( x , \\zeta _ i , ( \\zeta _ j ) _ { j \\neq i } ) = \\partial _ { \\zeta _ i } F ( x , \\zeta ) \\ ; \\ ; \\ ; \\mbox { f o r a . e . $ x \\in \\Omega $ , $ \\forall \\ ; \\zeta = ( \\zeta _ 1 , . . . , \\zeta _ N ) \\in \\R ^ { N } $ . } \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} g V _ \\ell = V _ { \\sigma _ g ( \\ell ) } . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\mathcal { C } = \\oplus _ { m , n } A [ m , n ] , \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m { | V _ i | } = \\left | \\bigcup _ { i = 1 } ^ m { V _ i } \\right | + \\sum _ { j = 1 } ^ { m - 2 } { | V _ { \\mathcal { C } _ j } | } + \\left | \\bigcap _ { i = 1 } ^ m { V _ i } \\right | . \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} { \\bf C } ^ { \\bullet } _ { { \\bf b } , J } : ~ 0 \\to B \\to \\prod _ { i = 1 } ^ s B _ { b _ i , J } \\to \\prod _ { i < j } \\big ( B _ { b _ i , J } \\big ) _ { b _ j , J } \\to \\cdots \\to \\big ( \\cdots \\big ( B _ { b _ , J } \\big ) \\cdots \\big ) _ { b _ s , J } \\to 0 \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\mathcal { K } ^ m _ { \\overrightarrow { \\omega } , t } \\phi = \\ ( \\sum _ { \\omega ' \\in \\omega } A _ { \\omega , \\omega ' , t } \\phi _ { \\omega ' } \\ ) _ { \\omega \\in \\Omega } . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} \\sum _ { A , B } h _ { A B } \\overline { H _ { A B } } = \\frac 1 2 \\sum _ { A , B } h _ { A B } \\overline { ( H _ { A B } - H _ { B A } ) } = \\sum _ { A , B } h _ { A B } \\overline { H _ { [ A B ] } } \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\left ( \\frac { b - a } { a + b \\omega } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { - a } { a - b } \\right ) \\left ( \\frac { \\omega ^ 2 } { a - b } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { - a } { a - b } \\right ) . \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} y _ 1 ^ { j _ 1 } y _ 2 ^ { j _ 2 } \\cdots y _ { n - 1 } ^ { j _ { n - 1 } - 1 } y _ i y _ { n - 1 } & = ( k _ 1 ' Y _ 1 ' a _ 1 ' + k _ 2 ' Y _ 2 ' a _ 2 ' + \\cdots + k _ s ' Y _ s ' a _ s ' ) y _ { n - 1 } \\\\ & = \\sum _ { i = 1 } ^ s ( - 1 ) ^ { | x _ { n - 1 } | | a _ i ' | } k _ i ' Y _ i ' ( y _ { n - 1 } a _ i ' - \\{ x _ { n - 1 } , a _ i ' \\} ) \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } p _ 1 ^ \\ast = 0 , \\ p _ 2 ^ \\ast = \\hat P , \\ \\alpha _ 1 ^ \\ast = 0 , & { \\rm i f } \\ \\frac { 1 + \\mu } { 1 + \\delta } > \\frac { \\log _ 2 ( 1 + \\hat P g _ 1 ) } { \\log _ 2 ( 1 + \\hat P g _ 2 ) } , \\\\ p _ 1 ^ \\ast = \\hat P , \\ p _ 2 ^ \\ast = 0 , \\ \\alpha _ 1 ^ \\ast = 1 , & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ { d d } } ( \\frac { n \\eta _ L \\Gamma r _ 0 ^ { \\alpha _ { L , d } } } { P _ { d } G _ 0 } ) = \\exp \\bigg ( - 2 \\pi \\lambda _ d \\bigg ( \\sum _ { j \\in \\{ L , N \\} } \\sum _ { i = 1 } ^ 3 p _ { G _ i } \\times \\\\ & \\bigg ( \\int _ 0 ^ { \\infty } \\bigg ( 1 - 1 / \\bigg ( 1 + \\frac { n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , d } } G _ i } { G _ 0 N _ j t ^ { \\alpha _ { j , d } } } \\bigg ) ^ { N _ j } \\bigg ) p _ { j , d } ( t ) t d t \\bigg ) \\bigg ) \\bigg ) \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\big \\{ \\sigma ^ 1 > n \\big \\} \\ , & \\subseteq \\ , \\big \\{ S ^ 1 _ { K ^ 1 _ m } - R ^ 1 _ { K ^ 1 _ m + 1 } > d m + z \\ , : \\ , m = 0 , \\dots , T ^ 1 _ n \\big \\} \\ , , \\\\ \\big \\{ \\sigma ^ 2 > n \\big \\} \\ , & \\subseteq \\ , \\big \\{ S ^ 2 _ { K ^ 2 _ m } + R ^ 2 _ { K ^ 2 _ m + 1 } < d m + z \\ , : \\ , m = 0 , \\dots , T ^ 2 _ n \\big \\} \\ , . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} L _ { S _ A \\to E } = \\max _ { \\begin{subarray} { c } \\pi : A \\to E \\\\ A \\in S _ A \\end{subarray} } \\sum _ { 1 \\leq k \\leq \\ell ( \\pi ) } \\omega _ { \\pi ( k ) } . \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} f ^ { ( k - 1 ) } ( X ) & = b _ 1 X ^ { e _ 1 } + \\ldots + b _ t X ^ { e _ t } ; \\\\ t & > 1 , \\ : d ^ { k - 1 } = e _ 1 > \\ldots > e _ t \\geq 0 , \\ : b _ 1 , \\ldots , b _ t \\in \\mathbb { F } \\setminus \\left \\lbrace 0 \\right \\rbrace . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} & \\limsup _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { | t - s | \\le \\delta } | \\psi ^ n ( t ) - \\psi ^ n ( s ) | = 0 , \\\\ & \\limsup _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { | t - s | \\le \\delta } | B ^ n _ k ( t ) - B ^ n _ k ( s ) | = 0 , k \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} \\Delta \\tilde w = D _ i ( \\tilde b ^ i \\tilde u ) \\ ; \\ ; \\vec \\Phi ( \\Omega ) , \\tilde w = 0 \\ ; \\ ; \\vec \\Phi ( \\partial \\Omega ) , \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} \\varpi _ { n , 1 } \\left ( { \\nu , p } \\right ) = \\sum \\limits _ { s = 0 } ^ { n - 2 } { \\frac { 4 } { \\nu ^ { s } } \\int _ { 1 } ^ { p } { \\left \\vert { \\frac { \\tilde { { F } } _ { s + 1 } \\left ( q \\right ) d q } { q ^ { 2 } \\left ( { 1 - q ^ { 2 } } \\right ) } } \\right \\vert . } } \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left \\langle A _ { \\scriptscriptstyle 0 } \\nabla u , \\nabla v \\right \\rangle = \\int _ { \\Omega } f v = : F ( v ) \\qquad \\forall v \\in H _ { 0 } ^ { 1 } \\left ( \\Omega \\right ) . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} A ^ { ' } _ { 1 } = A _ { 1 } + \\frac { 1 } { 2 } \\nabla \\psi , \\quad \\mbox { a n d } A ^ { ' } _ { 2 } = A _ { 2 } - \\frac { 1 } { 2 } \\nabla \\psi , \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} \\mu ( \\omega _ q ) = \\sum _ { p \\leq x } \\frac { \\omega _ q ( p ) } { p } = \\sum _ { \\substack { p \\leq x \\\\ p \\equiv 1 \\mod q } } \\frac { 1 } { p } = \\frac { \\log \\log x } { \\phi ( q ) } + O \\bigg ( \\frac { \\log q } { \\phi ( q ) } \\bigg ) \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} \\chi _ i ( \\epsilon ^ 2 + u ) : = \\frac { 1 } { 1 - ( 1 - \\beta ) \\tau _ i } \\int _ 0 ^ u \\frac { ( \\epsilon ^ 2 + r ) ^ { 1 - ( 1 - \\beta ) \\tau _ i } - \\epsilon ^ { 2 ( 1 - ( 1 - \\beta ) \\tau _ i ) } } { r } d r \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} 1 = \\left ( s - { \\partial ^ 2 \\over \\partial v ^ 2 } \\right ) I ( v ) \\ , , I ( v ) \\equiv \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 3 / 2 } \\left [ \\psi _ { s , F } ( - v ) + \\psi _ { s , F } ( v ) \\right ] , \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\omega _ { a b , 1 } ( [ y , z ] , x ) & = \\omega ( a b [ y , z ] , x ) = ( - 1 ) ^ { | y | | b | } \\omega ( [ a y , b z ] , x ) \\\\ & = ( - 1 ) ^ { | y | | b | } \\omega ( a y , b [ z , x ] ) - ( - 1 ) ^ { | a | | b | + | a | | z | + | y | | z | } \\omega ( b z , a [ y , x ] ) \\\\ & = \\omega _ { a , b } ( y , [ z , x ] ) - ( - 1 ) ^ { | a | | b | + | y | | z | } \\omega _ { b , a } ( z , [ y , x ] ) \\\\ & = \\omega _ { a , b } ( y , [ z , x ] ) - ( - 1 ) ^ { | y | | z | } \\omega _ { a , b } ( z , [ y , x ] ) \\\\ & = \\omega _ { a , b } ( [ y , z ] , x ) . \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} \\frac { F ( b ) - F ( y ) } { b - y } - \\frac { F ( y ) - F ( a ) } { y - a } = F ' ( y ) - F ' ( a ) \\ , , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\left < R _ { i } \\left ( Z \\right ) , R _ { j } \\left ( Z \\right ) \\right > = \\left < Z _ { i } , Z _ { j } \\right > , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} f _ 1 ( x ) \\ = \\ f _ 2 ( x ) \\ = \\ \\dotsb \\ = \\ f _ n ( x ) \\ = \\ 0 \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} \\left ( { \\rm s i n g } _ { H } ( u ) \\cap B _ { \\frac { r } { 2 } } ( 0 ) \\right ) \\setminus B _ { \\rho r } ( L ) = \\emptyset ; \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ u $ } _ { t } \\ ; \\ ! + \\ , \\mbox { \\boldmath $ u $ } \\ ! \\cdot \\ ! \\nabla \\mbox { \\boldmath $ u $ } \\ , + \\ , \\nabla p \\ ; = \\ ; \\nu \\ , \\Delta \\ : \\ ! \\mbox { \\boldmath $ u $ } , \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} 0 \\subseteq W _ { - 2 } M = [ 0 \\to T ] \\subseteq W _ { - 1 } M = [ 0 \\to G ] \\subseteq W _ 0 M = M \\ . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { 2 \\epsilon n } \\sum _ { \\substack { m ~ \\chi \\\\ H ( \\chi ) \\leq \\epsilon } } \\hat { 1 _ S } ( \\chi ) ^ 3 = ( e ^ { - 1 / 2 } + o ( 1 ) ) \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\theta } ^ { * } - \\boldsymbol { \\theta } _ 0 \\| + d / \\sqrt { n } = R 2 ^ { - \\sum _ { l = 1 } ^ d \\alpha _ l J _ { n , l } ( \\boldsymbol { \\alpha } ) \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha ^ { * } } \\right ) } + d / \\sqrt { n } \\lesssim \\sqrt { \\log { n } / n } . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\mu _ { \\mathrm { o p t } } = \\frac 1 2 \\nu \\lambda _ 1 ( \\Omega ) , \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\mathrm { d } X _ { t } = ( \\alpha + \\beta X _ { t } ) \\mathrm { d } t + \\sigma \\mathrm { d } B _ { t } , \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} M _ t ^ u : = u ( X _ t ) - u ( x ) - \\int _ 0 ^ t L _ r u ( X _ s ) \\ , d s , t \\ge 0 , \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} ( J - i I ) ^ T ( F + I ) ^ { - 1 } ( J - i I ) = 0 . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\sum _ { n \\leq X } a ( n ) \\Big ( 1 - \\frac { n } { X } \\Big ) ^ k & = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } D ( s ) \\frac { X ^ s } { s ( s + 1 ) \\cdots ( s + k ) } d s \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } D ( s ) \\frac { X ^ s \\Gamma ( s ) } { \\Gamma ( s + k + 1 ) } d s , \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\prod _ { q \\mid \\gamma ( H ) } \\mathcal { F } _ q ( H , q ^ { k - 2 m } ) & = \\prod _ { q \\mid \\gamma ( H ) } \\mathcal { F } _ q ( H , q ^ { - m + ( p - 1 ) \\cdot t } ) \\\\ & \\equiv \\prod _ { q \\mid \\gamma ( H ) } \\mathcal { F } _ q ( H , q ^ { - m } ) = 0 \\pmod { p } . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} f _ 0 = ( \\overline { Q } ) ^ { - 1 / 2 } = ( \\underline { f f ^ * } ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\partial ( \\Psi _ + ) = \\phi _ + \\Psi _ + \\mbox { a n d } \\partial ( \\Psi _ - ) = \\phi _ - \\Psi _ - \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\ , K \\ ! \\ ; \\ ! ( \\alpha , m ) \\ , = \\ ; \\mbox { $ { \\displaystyle \\min _ { \\delta \\ , > \\ , 0 } } $ } \\ : \\big \\{ \\ ; \\ ! \\delta ^ { \\mbox { } ^ { \\scriptstyle \\ : \\ ! - \\ , 1 / 2 } } \\ , \\ ! \\prod _ { \\ : \\ ! j \\ , = \\ , 0 } ^ { \\ ; \\ ! m } \\ : \\ ! \\bigl ( \\ : \\ ! \\alpha + j / 2 + \\delta \\ ; \\ ! \\bigr ) ^ { \\ ! \\ : \\ ! 1 / 2 } \\ : \\ ! \\bigr \\} . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{gather*} O p _ h \\circ \\sigma ( A ) = A + O _ { \\Psi ^ { m - 1 } } ( h ) , A \\in \\Psi ^ m , \\qquad \\qquad \\sigma \\circ O p _ h = \\pi : S ^ m \\to S ^ m / h S ^ { m - 1 } , \\end{gather*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\mathcal { W } ^ 0 = b _ \\wedge ( \\mathbf { D } ) ^ * g _ \\wedge ^ 0 b _ \\wedge ( \\mathbf { D } ) . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = ( l _ 2 , a _ 3 a _ 9 ) , \\\\ ( 0 ) : l _ 2 & = ( l _ 1 , a _ 4 a _ { 1 0 } ) , \\\\ ( l _ 2 ) \\cap ( a _ 3 a _ 9 ) & = ( l _ 1 ) \\cap ( a _ 4 a _ { 1 0 } ) = ( 0 ) . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } k ( t ) e ^ { i u t } d t = ( 1 + e ^ { i u T } ) K ( u ) \\ , , \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} Y ^ i ( \\Psi ( x , z ) ) - Y ^ j ( x ) \\partial _ { x ^ j } ( \\Psi ^ i ) ( x , z ) - \\tau ( x ) H ^ { \\alpha } ( z ) \\partial _ { z ^ { \\alpha } } ( \\Psi ^ i ) ( x , z ) - C ^ { \\ell } ( x ) K ^ { \\alpha } _ { \\ell } ( z ) \\partial _ { z ^ { \\alpha } } ( \\Psi ^ i ) ( x , z ) = 0 , \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} p = s ( \\abs { x } - \\mu ( \\infty , x ) s ( x ) ) = e ^ { \\abs { x } } ( \\mu ( \\infty , x ) , \\infty ) . \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\partial _ { t } u ( t , x ) - \\sum _ { i , j = 1 } ^ { n } \\partial _ { x _ { i } } ( a _ { i j } ( T - t , x ) \\partial _ { x _ { j } } u ( t , x ) ) + \\sum _ { i = 1 } ^ { n } b _ { i } ( T - t , x ) \\partial _ { x _ { i } } u ( t , x ) = 0 \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\sum _ { \\substack { m = n _ 0 \\\\ m \\not = n } } ^ \\infty \\ | K ( \\sigma _ { n } - \\overline { \\sigma _ m } ) | + \\sum _ { m = n _ 0 } ^ \\infty | K ( \\sigma _ n + \\sigma _ m ) | \\le \\frac { 4 \\pi } { T \\gamma ^ 2 ( 1 - \\varepsilon ) } \\bigg ( 1 + \\sum _ { m = n _ 0 } ^ \\infty \\frac { 1 } { 4 m ^ { 2 } - 1 } \\bigg ) \\ , , \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} \\gamma _ { m k } ( L ) : = \\ \\sum _ { i = 1 } ^ m ( l _ { m i } + m - 1 ) ^ k \\prod _ { j \\ne i } \\left ( 1 - \\frac { 1 } { l _ { m i } - l _ { m j } } \\right ) . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\rho ( x y , \\psi \\varphi ) & = T _ { x y } M _ { \\psi \\varphi } \\\\ & = T _ x T _ y M _ \\psi M _ \\varphi \\\\ & = \\overline { \\psi ( y ) } T _ x M _ \\psi T _ y M _ \\varphi \\\\ & = \\sigma ( ( x , \\psi ) , ( y , \\varphi ) ) \\rho ( x , \\psi ) \\rho ( y , \\varphi ) \\ , . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} d _ \\varphi \\mathcal { F } ( h ) = h _ { \\bar { \\zeta } } + B _ 1 ^ \\varphi h + B _ 2 ^ \\varphi \\bar { h } , \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\frac { \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x y ) } { \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) } & = \\frac { \\sum _ { n = 1 } ^ { K } \\mathbb { P } \\big ( \\bigcup _ { k = 1 } ^ { n } \\{ \\xi _ k > x y \\} \\big ) \\mathbb { P } ( \\eta = n ) } { \\mathbb { P } ( \\xi _ { ( \\eta ) } > x ) } \\\\ & + \\frac { \\sum _ { n = K + 1 } ^ { \\infty } \\mathbb { P } \\big ( \\bigcup _ { k = 1 } ^ { n } \\{ \\xi _ k > x y \\} \\big ) \\mathbb { P } ( \\eta = n ) } { \\mathbb { P } ( \\xi _ { ( \\eta ) } > x ) } \\\\ & = : \\mathcal { J } _ 1 + \\mathcal { J } _ 2 , \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} A _ n ( y ) & = A _ { n - 1 } ( y ) + ( y - 1 ) B _ { n - 1 } ( y ) \\\\ B _ n ( y ) & = A _ { n - 1 } ( y ) + ( y - 2 ) B _ { n - 1 } ( y ) \\\\ C _ n ( y ) & \\le B _ { n - 1 } ( y ) + ( y - 2 ) C _ { n - 1 } ( y ) . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\tilde { \\rho } ^ { - 1 } ( \\alpha , x ) = \\hat { \\theta } ^ M _ 1 + \\frac { 1 } { \\sqrt { n } } \\Phi ^ { - 1 } ( \\alpha ) \\sqrt { \\hat { \\nu } ^ { 1 1 } } + O _ p ( n ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ d 2 ^ { ( a _ l + b _ l ) / 2 } \\left ( \\int _ { \\mathcal { I } _ { \\boldsymbol { b } } } \\prod _ { l = 1 } ^ d 2 ^ { b _ l } \\| \\psi _ { k _ l } \\| _ { \\infty } \\| \\psi _ { k _ l } ^ { ' } \\| _ { \\infty } d \\boldsymbol { x } + \\int _ { \\mathcal { I } _ { \\boldsymbol { a } } } \\prod _ { l = 1 } ^ d 2 ^ { a _ l } \\| \\psi _ { k _ l } \\| _ { \\infty } \\| \\psi _ { k _ l } ^ { ' } \\| _ { \\infty } d \\boldsymbol { x } \\right ) , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\overline q _ k = \\sum _ { \\ell = 1 } ^ k \\overline \\lambda _ \\ell , \\overline \\lambda _ \\ell \\ge \\sup _ { s \\in I } \\{ \\lambda _ l ^ X ( s , X ( s ) ) , \\lambda _ l ^ Z ( s , Z ( s ) ) \\} , \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} G _ m ( x ) = \\frac { x ^ { m - 2 } } { ( 1 - x ) ^ { 2 m - 4 } } G _ 2 ( x ) \\ , . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} M ^ { m , i } : = \\sum _ { k = 0 } ^ m N ^ { k , i } , \\ ; \\ ; \\ ; m \\geq 0 , \\ ; \\ ; \\ ; i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} f ( u ) = \\frac { 1 } { 2 } ( | u + 1 | - | u - 1 | ) . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\beta ^ * ( T M ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } = \\beta ^ * ( E ^ s _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } \\oplus \\beta ^ * ( E ^ u _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} ( T _ 0 \\ , T _ 1 \\ , T _ 2 ) \\ , M ( s , t ) \\ ! \\left ( \\begin{array} { c } T _ 0 \\\\ T _ 1 \\\\ T _ 2 \\end{array} \\right ) = 0 \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\mathcal { D I F F } _ { A / R } ( M ) ( n ) = \\mathcal { D I F F } _ { A / R } ( \\overbrace { M , \\ldots , M } ^ n ; M ) . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\frac { z \\ ; { } _ { a } \\mathtt { B } ' _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } = \\frac { z \\mathtt { J } _ { \\nu - 1 } \\left ( z \\right ) } { \\mathtt { J } _ { \\nu } \\left ( z \\right ) } - ( 2 - a ) \\nu + 1 - a . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} f ^ { ( k ) } ( X ) & = f ( f ^ { ( k - 1 ) } ( X ) ) \\\\ & = a _ 1 ( b _ 1 X ^ { e _ 1 } + \\ldots + b _ t X ^ { e _ t } ) ^ { d _ 1 } + \\ldots + a _ s ( b _ 1 X ^ { e _ 1 } + \\ldots + b _ t X ^ { e _ t } ) ^ { d _ s } \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\tau = \\left [ \\begin{array} { c c c c } & p & & \\\\ 1 & & & \\\\ & & & p \\\\ & & 1 & \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} I _ { m - 1 } ( t ) = \\int _ 0 ^ \\infty \\left ( a _ { m - 1 } + t \\xi + \\sum _ { k = 1 } ^ { m - 1 } ( - 1 ) ^ { k - 1 } { m - 1 \\choose k } \\frac { 1 } { k } e ^ { - k t \\xi } \\right ) \\xi ^ { - m } \\pi ( d \\xi ) \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} R _ i \\mathbf { e } _ i = - \\mathbf { h } _ i \\ , , R _ i \\mathbf { h } _ i = - \\mathbf { e } _ i \\ , , \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i , j \\leq n } \\| \\mathbf x _ i - \\mathbf y _ j \\| ^ 2 = \\sum _ { 1 \\leq i < j \\leq n } \\| \\mathbf x _ i - \\mathbf x _ j \\| ^ 2 + \\sum _ { 1 \\leq i < j \\leq n } \\| \\mathbf y _ i - \\mathbf y _ j \\| ^ 2 + n ^ 2 \\| \\mathbf x - \\mathbf y \\| ^ 2 , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} Z = \\mathcal { V } Z + \\mathcal { H } Z \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} f _ 1 & = q ^ { \\frac 1 { 1 2 } } \\left ( 1 + 2 q - 5 q ^ 2 - 1 0 q ^ 3 + 9 q ^ 4 + 1 4 q ^ 5 - 1 0 q ^ 6 + 1 4 q ^ 8 + \\cdots \\right ) , \\\\ f _ 2 & = q ^ { \\frac { 1 } { 3 } } \\left ( 1 - 4 q ^ 2 + 2 q ^ 4 + 8 q ^ 6 - 5 q ^ 8 + \\cdots \\right ) , \\\\ f _ 3 & = q ^ { \\frac { 7 } { 1 2 } } \\left ( 1 - 2 q + q ^ 2 - 2 q ^ 3 + 4 q ^ 5 + \\cdots \\right ) . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N } \\lbrack f \\rbrack _ { s _ i , p _ { i } } ( B _ { i } ) \\leq c _ { 3 } \\lbrack f \\rbrack _ { s , p ( \\cdot , \\cdot ) } ( \\Omega ) . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} S = | U | _ { \\omega _ { \\phi } } ^ 2 , \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} \\frac { \\sigma _ j ( B ( p , r ) ) } { r ^ { n - 1 } } = \\frac { \\sigma ( B ( \\widetilde p , r r _ j ) ) } { ( r r _ j ) ^ { n - 1 } } \\sim 1 \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} G _ m ( x ) = x ^ { m - 2 } C ( x ) ^ { m - 2 } G _ 2 ( x ) \\ , . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} p _ { N , N } ^ { P _ 1 } = \\prod _ { j = 0 } ^ { N - 1 } \\frac { \\Gamma ( L + j ) \\Gamma ( ( L + j ) / 2 ) } { \\Gamma ( L + ( N + j - 1 ) / 2 ) \\Gamma ( L / 2 ) } . \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } E [ X _ { 1 } ^ \\theta ( t ) ] \\big | _ { \\theta = 1 5 } \\approx \\frac 1 h E \\big [ X _ { 1 } ^ { 1 5 + h / 2 } ( t ) - X _ { 1 } ^ { 1 5 - h / 2 } ( t ) \\big ] \\quad \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} m _ B ( \\{ b , b ' \\} ) & = h _ B ( b ) m _ B ( b ' ) - ( - 1 ) ^ { ( | b | + p ) | b ' | } m _ B ( b ' ) h _ B ( b ) , \\\\ h _ B ( b b ' ) & = m _ B ( b ) h _ B ( b ' ) + ( - 1 ) ^ { | b | | b ' | } m _ B ( b ' ) h _ B ( b ) . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} ( a \\cdot \\delta _ u ) * ( b \\cdot \\delta _ v ) = a b \\cdot \\delta _ { u v } ( a \\cdot \\delta _ u ) ^ * = a ^ * \\cdot \\delta _ { u ^ { - 1 } } . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} L _ N ^ { \\rm r e s c , B } ( v ) : = \\frac { L _ { B ( v ) \\to E _ N ( w ) } - \\mu ( E _ N ( w ) - B ( v ) ) } { \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } } , \\mu ( m , n ) = ( \\sqrt { m } + \\sqrt { n } ) ^ 2 \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{gather*} c _ 1 = 2 n , c _ 2 = n - 1 , c _ i = n - i , \\mathrm { f o r } \\ 3 \\le i \\le n - 1 . \\end{gather*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\bigcap _ { \\substack { N _ l \\leq j _ l \\leq J _ { n , l } - 1 \\\\ 0 \\leq k _ l \\leq 2 ^ { j _ l } - 1 \\\\ l = 1 , \\dotsc , d } } \\left \\{ \\frac { | \\beta _ n ( \\widetilde { \\Theta } ) - \\widetilde { \\beta } _ n ( \\widetilde { \\Theta } ) | } { \\sqrt { \\sigma _ 0 ^ 2 ( \\boldsymbol { \\Psi } _ { \\boldsymbol { j } } ^ T \\boldsymbol { \\Psi } _ { \\boldsymbol { j } } ) _ { \\boldsymbol { k } , \\boldsymbol { k } } } } \\leq \\left ( 2 \\log { \\prod _ { l = 1 } ^ d 2 ^ { j _ l } } + c \\log { n } \\right ) ^ { 1 / 2 } \\right \\} . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} F ( q , p , \\dot { q } ) = p _ i \\dot { q } ^ i - L ( q , \\dot { q } ) \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { \\infty } \\frac { 1 } { n ^ { \\ell _ q } } = \\infty , \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} L _ u \\varphi & = \\frac { 1 } { i } \\varphi _ x - C _ + ( u \\varphi ) , \\\\ B _ u \\varphi & = \\frac { 1 } { i } \\varphi _ { x x } + 2 [ ( C _ + u _ x ) \\varphi - C _ + ( ( u \\varphi ) _ x ) ] . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\ , \\P \\ , \\bigg ( \\sup _ { \\substack { t _ 1 - t _ 2 < \\delta \\\\ 0 \\leq t _ 2 \\leq t _ 1 \\leq T } } \\big \\| U ^ n ( t _ 1 ) - U ^ n ( t _ 2 ) \\big \\| \\ , > \\epsilon \\bigg ) = 0 . \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} \\varDelta _ i ^ { ( n ) } = C _ i n ^ { - 1 / 2 } , i = 1 , 2 , \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} L _ F ( \\chi , s ) : = \\prod _ { \\frak { p } } \\left ( 1 - \\chi ( \\frak { p } ) \\mathcal { N } _ { F / \\Q } ( \\frak { p } ) ^ { - s } \\right ) ^ { - 1 } , \\textrm { R e } ( s ) > 1 . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} & f _ m ( x _ 1 , x _ 2 , \\cdots , x _ m ) = \\sum _ { j = 1 } ^ { m } \\big [ \\log ( h ( x _ j ) ) + 4 \\pi R ( x _ j , x _ j ) \\big ] + 4 \\pi \\sum _ { l \\neq j } ^ { 1 , \\cdots , m } G ( x _ l , x _ j ) , \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\partial _ t u _ t \\ , = \\ , - u _ t + g X u _ t \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\aligned & \\int _ { ( ( X _ 1 , \\widehat { \\mathcal U _ 1 } ) \\times _ { R \\times P } ( X _ 2 , \\widehat { \\mathcal U _ 2 } ) , ( \\widehat { \\frak S _ 1 } \\times _ { R \\times P } \\widehat { \\frak S _ 2 } ) ^ { \\epsilon } ) } h _ 1 \\wedge h _ 2 \\\\ & = \\int _ { ( ( X _ 2 , \\widehat { \\mathcal U _ 2 } ) , \\widehat { \\frak S _ 2 } ) } \\widehat f _ 2 ^ * \\widehat f _ 1 ! ( h _ 1 ; ( \\widehat { \\frak S _ 1 } ) ^ { \\epsilon } ) \\wedge h _ 2 . \\endaligned \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\lim _ { x _ 1 \\to \\pm \\infty } \\eta _ 1 ( x _ 1 , t ) = 0 \\ , . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } ( k - 1 - i ) b ^ i \\le \\frac { ( k - 1 ) k } { 2 } + \\frac { k - 2 } 3 \\Biggl \\{ \\sum _ { i = 1 } ^ { k - 1 } b ^ i - ( k - 1 ) \\Biggr \\} \\ , . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} S _ { T ^ * Q } = \\{ p \\in T ^ * Q : \\pi _ Q ( p ) \\in N , \\langle Z , \\theta _ Q ( p ) \\rangle = \\langle T \\pi ( Z ) , d F \\rangle \\} \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} \\Phi ( v ) = ( m ^ { - 1 } ( v ) ) ^ 2 \\coth ( m ^ { - 1 } ( v ) ) - ( 1 + v ) m ^ { - 1 } ( v ) . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\frac { \\bar { \\sigma } ( z ) } { \\sigma ( z ) } = h ( z ) , \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { m } , \\mathcal { O } ; k ) = U ( m _ 1 ; k ) \\ , \\ , \\ , \\ , k \\Delta ( \\mathbf { m } , \\mathcal { O } ; k ) = C ( \\mathbf { m } , \\mathbf { a } , \\mathcal { O } ; k ) , \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\mu = \\int _ \\Omega \\mu _ \\omega \\ , d P ( \\omega ) , \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\iota _ { \\xi _ L } \\omega _ L = d E _ L , \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\varphi _ j ( t ) = t ^ j + p ^ c t ^ { k + 1 } \\psi _ j ( t ) ( 1 \\le j \\le k ) . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\Lambda _ \\alpha ( x , y ) : = j _ \\alpha ( x y ) \\quad j _ \\alpha ( z ) : = \\ > _ 0 F _ 1 ( \\alpha + 1 ; - z ^ 2 / 4 ) \\quad ( y \\in \\mathbb C ) \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} H = F \\otimes G . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} C = \\left ( \\int _ { \\R ^ N } e ^ { - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } \\langle A ( x - a _ i ) , x - a _ i \\rangle } \\ , d x \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\sum _ { I \\subseteq \\{ 2 , 3 , 4 , 5 \\} } ( - 1 ) ^ { | I | } \\Sigma _ I & = y ^ 5 + y ^ 2 - 4 ( y ^ 4 - y ^ 3 + y ^ 2 ) + y ^ 3 + 4 ( y ^ 3 - y ^ 2 + y ) \\\\ & \\qquad + ( y - 1 ) ^ 3 + y ^ 2 - 2 y ^ 2 - 2 ( y ^ 2 - y + 1 ) + y \\\\ [ 2 m m ] & = y ^ 5 - 4 y ^ 4 + 1 0 y ^ 3 - 1 3 y ^ 2 + 1 0 y - 3 . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\sup _ { l \\geq 2 } \\frac { \\sqrt [ l ] { | a _ l | } } { l ^ m } < \\infty , m = \\begin{cases} 2 & g \\neq 0 , \\\\ 1 & g = 0 , \\end{cases} \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\real \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } & = a ^ { a / 2 } - \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\real \\sum _ { n = 1 } ^ \\infty \\frac { 2 z ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - z ^ 2 } \\\\ & \\geq a ^ { a / 2 } - \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\sum _ { n = 1 } ^ \\infty \\frac { 2 | z | ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - | z | ^ 2 } = \\frac { | z | \\mathtt { f } ' _ { a , \\nu } ( | z | ) } { \\mathtt { f } _ { a , \\nu } ( | z | ) } . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} d _ 0 \\circ o ( \\alpha _ 1 , \\alpha ' _ 2 ) + o ( \\alpha _ 1 , \\alpha ' _ 2 ) \\circ d _ 0 = 0 . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\hat { { F } } _ { s + 1 } \\left ( z \\right ) = - { \\dfrac { 1 } { 2 f ^ { 1 / 2 } \\left ( z \\right ) } } \\frac { d \\hat { { F } } _ { s } \\left ( z \\right ) } { d z } - { \\dfrac { 1 } { 2 } } \\sum \\limits _ { j = 1 } ^ { s - 1 } { \\hat { { F } } _ { j } \\left ( z \\right ) \\hat { { F } } _ { s - j } \\left ( z \\right ) } \\ \\left ( { s = 2 , 3 , 4 \\cdots } \\right ) . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} \\Delta ( X ) = \\langle \\{ i _ 1 , \\dots , i _ m \\} \\mid [ i _ 1 , \\dots , i _ m ] \\ne 0 \\rangle . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} k ^ * ( t ) : = \\left \\{ \\begin{array} { l } \\cos \\frac { \\pi t } { 2 T } \\ , \\qquad \\mbox { i f } \\ | t | \\le T \\ , , \\\\ \\\\ 0 \\ , \\qquad \\qquad \\quad \\ \\ \\ \\ \\mbox { i f } \\ | t | > T \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} l ( \\rho _ 1 \\oplus \\rho _ 2 ) = l ( \\rho _ 1 ) + l ( \\rho _ 2 ) \\ , . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} X = \\left \\{ \\omega \\in L ^ { 2 } | \\ \\omega = \\sum _ { 0 \\neq k \\in \\mathbf { Z } } \\omega _ { k } \\left ( y \\right ) e ^ { i k \\alpha x } \\right \\} . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} Q _ t e _ n = \\frac 1 { 2 \\lambda _ n } ( 1 - e ^ { - 2 \\lambda _ n t } ) b _ n e _ n , Q _ \\infty e _ n = \\frac 1 { 2 \\lambda _ n } b _ n e _ n \\forall n \\in \\N . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} P _ j ( i ) & = \\frac { 2 ( q - 1 ) } { q ^ 2 } g _ \\frac { 1 } { 2 } ( j ) \\Sigma ( i ) = \\frac { 2 ( q - 1 ) } { q ^ 2 } g _ \\frac { 1 } { 2 } ( j ) \\frac { - q ^ 2 q ^ { - i } } { ( q - 1 ) ^ 2 } \\left ( 1 - q ^ { - f ( i ) } \\right ) \\\\ & = \\frac { - 2 q ^ { - i } } { q - 1 } g _ \\frac { 1 } { 2 } ( j ) ( 1 - q ^ { - f ( i ) } ) , \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} a ( \\tau ) = \\int _ { i + 2 } ^ \\tau \\eta ^ 4 ( z ) d z . \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\dot { x } ( t , p ) = A ( p ) x ( t , p ) + s ( t , p ) , \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} & \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\prec \\frac { \\sqrt { ( \\Im m _ { \\mu _ A \\boxplus \\mu _ B } + \\hat { \\Lambda } ) ( | \\mathcal { S } | + \\hat { \\Lambda } ) } } { N \\eta } + \\frac { 1 } { ( N \\eta ) ^ 2 } , \\iota = A , B \\ , . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\max _ { i = 1 , \\dots , v } \\{ d ^ \\mathtt { T } \\nabla ^ 2 L ( x ^ * , \\lambda ^ i , \\mu ^ i ) d \\} \\geq 0 , \\forall d \\in K . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} f ( \\l , x ) = 0 \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} f ' ( j ) = \\begin{cases} f ( j ) & \\mbox { i f } f ( j ) < i , \\\\ f ( j ) - 1 & \\mbox { i f } i < f ( j ) < i + m , \\\\ f ( j ) - 2 & \\mbox { i f } i + m < f ( j ) . \\end{cases} \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} u & = d ( d + 1 ) - \\max _ { 0 \\le m < d } \\frac { d ( d + 1 ) - ( d - m ) ( d - m + 1 ) } { m + 1 } \\\\ & = d ( d + 1 ) - \\max _ { 0 \\le m < d } \\frac { 2 d m - m ( m - 1 ) } { m + 1 } = s _ 0 . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\sum _ { c \\geq 1 } \\frac { g _ h ( 4 c ) } { ( 4 c ) ^ { 2 w } } = \\frac { L ^ { ( 2 ) } ( 2 w - \\frac { 1 } { 2 } , \\chi _ { k , h } ) } { \\zeta ^ { ( 2 h ) } ( 4 w - 1 ) } \\widetilde { D } _ \\infty ^ k ( h , w ) , \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\sum _ { j _ i \\geq J _ { n , i } ( \\boldsymbol { \\alpha } ) } 2 ^ { - j _ i \\mu _ i } \\prod _ { l \\neq i } ^ d \\sum _ { j _ l < J _ { n , l } ( \\boldsymbol { \\alpha } ) } 2 ^ { - j _ l \\mu _ l } \\lesssim \\sum _ { i = 1 } ^ d 2 ^ { - J _ { n , i } ( \\boldsymbol { \\alpha } ) / 2 } , \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\mathcal { K } _ m ' ( x ) v ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\frac { \\partial \\kappa } { \\partial u } ( s , \\zeta _ i ^ j , x ( \\zeta _ i ^ j ) ) v ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} u ( x _ 1 , \\dots , x _ n ) = f ( x _ 1 , \\dots , x _ n ) e ^ { i k x _ 1 } \\begin{pmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} , p ( x _ 1 , \\dots , x _ n ) = f ( x _ 1 , \\dots , x _ n ) e ^ { i k x _ 1 } , \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} ( f , g ) _ { 0 } = \\sum _ { k = 1 } ^ { d } ( f _ { k } , g _ { k } ) _ { L ^ { 2 } } = \\sum _ { k = 1 } ^ { d } \\int _ { \\mathbb { R } } f _ { k } ( x ) g _ { k } ( x ) \\ , d x , \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} g _ 1 ( \\omega U _ 1 , X ) = - g _ 1 ( U _ 1 , \\mathcal { B } X ) \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} C _ { \\gamma } [ a , b ] & = \\{ f : ( a , b ] \\to \\R : ( x - a ) ^ { \\gamma } f ( x ) \\in C [ a , b ] \\} , \\quad 0 \\leq \\gamma < 1 , \\\\ C _ { 1 - \\gamma } [ a , b ] & = \\{ f : ( a , b ] \\to \\R : ( x - a ) ^ { 1 - \\gamma } f ( x ) \\in C [ a , b ] \\} , \\quad 0 \\leq \\gamma < 1 , \\\\ C _ { \\gamma } ^ { n } [ a , b ] & = \\{ f : ( a , b ] \\to \\R , f \\in C ^ { n - 1 } [ a , b ] : f ^ { ( n ) } ( x ) \\in C _ { \\gamma } [ a , b ] \\} , \\ , n \\in \\N . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} \\kappa ^ \\mp ( \\sigma , x ) : = \\left \\{ \\begin{array} { l } \\kappa ( \\sigma , \\mp x ) \\ , , \\ x \\in { \\mathbf R } ^ + \\ , , \\\\ 0 \\ , , \\ x \\in { \\mathbf R } ^ - \\ , , \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} \\widetilde z ' ( \\omega _ \\beta ( z ) ) = - F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) + 1 + \\frac { 1 } { F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) } F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) \\ , , \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} k ( k + 1 ) - r ( r + 1 ) & = ( k + r ) ( k - r ) + ( k - r ) \\\\ & = \\frac { ( k - r ) ( k - r + 1 ) } { k - r + 1 } + \\left ( k ( k + 1 ) - r ( r - 1 ) \\right ) \\cdot \\frac { k - r } { k - r + 1 } . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} T ^ { d } = & P ^ \\pi \\sum _ { j = 0 } ^ { n - 1 } P ^ j \\left [ \\begin{array} { c c } 0 & 0 \\\\ 0 & ( Z ^ d ) ^ { j + 1 } \\end{array} \\right ] , \\\\ X = & \\sum _ { k = 0 } ^ { g - 1 } ( P ^ { d } ) ^ { k + 2 } \\left [ \\begin{array} { c c } 0 & 0 \\\\ C A ^ \\pi & 0 \\end{array} \\right ] T ^ { k } T ^ { \\pi } , \\\\ \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } g _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - V \\left ( t , x ^ { k } \\right ) . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} H ( \\chi ) = \\frac 1 n \\log \\binom { n } { a _ 1 , \\dots , a _ k } . \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} { \\displaystyle \\partial _ { t } v _ { i } ^ { 0 } = A _ { i } u _ { i } ^ { 0 } - B _ { i } v _ { i } ^ { 0 } } \\quad \\mbox { i n } \\ ; Q _ { T } , \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} A _ 0 = \\frac { 1 } { 4 } \\sum _ { p } \\frac { p ^ 2 \\log p } { ( p - 1 ) ^ 3 ( p + 1 ) } \\quad B = \\frac 1 4 \\sum _ p \\frac { p ^ 3 ( p ^ 4 - p ^ 3 - p ^ 3 - p - 1 ) ( \\log p ) ^ 2 } { ( p - 1 ) ^ 6 ( p + 1 ) ^ 2 ( p ^ 2 + p + 1 ) } . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} & X = X _ { \\vec \\ell } : z ^ 3 = p ^ { - 2 } q ^ { - 2 } ( 1 - p ) ^ { - 2 } ( 1 - q ) ^ { - 2 } ( 1 - x _ 1 p - x _ 2 q ) ^ { - 2 } . \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} T = \\left ( \\frac { X } { N ( b ) ^ { n / 2 } N ( d ) ^ { ( n - 2 ) / 4 } } \\right ) ^ { 1 / R } \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { 2 } \\sum _ { k < l } u r _ { k l } \\wedge u i _ { k l } . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} J ( s , t ) : = H ^ { - 1 } _ s \\cdot f _ { * } ( t ) H ( s ) \\cdot H _ s , \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} Q & : = | \\nabla u | ^ 2 \\Delta \\eta - 2 \\nabla ^ 2 \\eta ( \\nabla u , \\nabla u ) \\\\ & = | \\nabla u | ^ 2 \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) - 2 \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) \\alpha _ i ( x ) ^ 2 . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\Gamma _ { j k } ^ { i } \\dot { x } ^ { j } \\dot { x } ^ { k } + \\sum \\limits _ { m = 0 } ^ { n } P _ { j _ { 1 } . . . j _ { m } } ^ { i } \\dot { x } ^ { j _ { 1 } } \\ldots \\dot { x } ^ { j _ { m } } = 0 , \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\left \\vert \\left \\vert \\left \\vert M \\right \\vert \\right \\vert \\right \\vert _ { \\infty , \\Omega } = \\underset { x \\in \\Omega } { \\operatorname * { e s s } \\sup } \\left ( \\sup _ { \\zeta \\in \\mathbb { R } ^ { d } \\backslash \\left \\{ 0 \\right \\} } \\frac { \\left \\Vert M \\left ( x \\right ) \\zeta \\right \\Vert } { \\left \\Vert \\zeta \\right \\Vert } \\right ) . \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\left [ Z ^ { A ' } _ { B } , \\delta _ { B ' } ^ { A } \\right ] = - 2 Z ^ { A ' } _ { B } Z _ { B ' } ^ { A } \\varphi = 2 Z ^ { A ' } _ { B } \\overline { Z ^ { B ' } _ { A } } \\varphi , \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} v \\left ( x , x _ { 0 } \\right ) = \\displaystyle \\sum \\limits _ { k = 0 } ^ { N - 1 } v _ { k } \\left ( x \\right ) \\psi _ { k } \\left ( x _ { 0 } \\right ) , \\forall x \\in \\overline { \\Omega } , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\mathbb { P } ( \\widehat { Q } _ 0 = m ) = \\frac { \\mathbb { E } W ^ m } { m \\ , \\mathbb { E } \\left [ - \\log ( 1 - W ) \\right ] } \\mbox { f o r } ~ m \\ge 1 , \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\alpha _ i F ( d _ i ) \\le \\max \\left ( \\bigl ( 1 - \\sqrt { \\gamma } \\bigr ) F ( 0 ) + \\sqrt { \\gamma } F ( \\sqrt { \\gamma } ) , ( 1 - \\eta ) F ( \\eta ) + \\eta F ( 1 ) \\right ) \\ , , \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\rho _ { ( 1 ) } ^ c ( z ) = \\frac { ( 1 - \\alpha ) } { \\pi \\alpha } \\frac { 1 } { ( 1 - | z | ^ 2 ) ^ 2 } \\ , \\chi _ { | z | < \\sqrt \\alpha } . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\langle { s , t , v , w , x , y } \\mid { y v = w y } , ~ t y t ^ { - 1 } x = w t y t ^ { - 1 } \\ ! , ~ x ^ 2 t y ^ { - 1 } t ^ { - 1 } y s ^ { - 1 } w ^ { - 1 } x v ^ { - 1 } = 1 , \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} ^ { C } D _ { a ^ + } ^ { \\alpha } f ( t ) = \\frac { 1 } { \\Gamma ( n - \\alpha ) } \\int _ { a } ^ { t } \\frac { f ^ { ( n ) } ( s ) d s } { ( t - s ) ^ { \\alpha - n + 1 } } , \\ , \\ , n - 1 < \\alpha \\leq n , \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} = 1 / \\| P \\| _ { Z \\to Z } \\| f \\| _ { H [ Z ] ^ { \\wedge } } . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} d _ 0 ( x ) : = d i s t ( x , \\Sigma ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\sum _ { k \\geq 2 } \\phi ^ e _ k \\Delta _ L U ^ { e } _ { k } = - ( L ^ e ) ^ 2 \\ , [ \\phi ^ e _ 2 \\ , ( U ^ e _ 2 - U ^ e _ 1 ) + ( \\phi ^ e _ 3 - \\phi ^ e _ 2 ) \\ , U ^ e _ 2 ] + \\sum _ { j \\geq 3 } U ^ e _ j \\Delta _ L \\phi ^ { e } _ { j } \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} { \\lim } _ { n \\to + \\infty } x ( t _ n ) = x ^ { * } . \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial u } { \\partial x } \\left ( x , r , t \\right ) d r + 2 \\pi a \\left ( x , t \\right ) \\frac { \\partial a } { \\partial x } \\left ( x , t \\right ) u \\left ( x , a \\left ( x , t \\right ) , t \\right ) . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\Psi = U _ { \\phi } \\psi = ( 2 \\pi \\hbar ) ^ { n / 2 } W ( \\psi , \\phi ) \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} [ A , B ] : = A B - B A . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} B ( p ) = T ( p ) A ( p ) T ( p ) ^ { - 1 } \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} - \\frac { \\partial P } { \\partial x } \\left ( x , t \\right ) = \\mathcal { L } \\left ( x , t \\right ) \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) + \\mathcal { R } \\left ( x , t \\right ) Q \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} u = u _ { < 1 } + \\sum \\limits _ { N \\geq 1 } u _ N . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\aligned & d _ 0 \\circ \\psi ^ { \\epsilon } _ { \\alpha _ 2 , \\alpha _ 1 } - \\psi ^ { \\epsilon } _ { \\alpha _ 2 , \\alpha _ 1 } \\circ d _ 0 \\\\ & + \\sum _ { \\alpha ' _ 2 } \\frak m ^ { 2 , \\epsilon } _ { 1 ; \\alpha _ 2 , \\alpha ' _ 2 } \\circ \\psi ^ { \\epsilon } _ { \\alpha ' _ 2 , \\alpha _ 1 } - \\sum _ { \\alpha ' _ 1 } \\psi ^ { \\epsilon } _ { \\alpha _ 2 , \\alpha ' _ 1 } \\circ \\frak m ^ { 1 , \\epsilon } _ { 1 ; \\alpha ' _ 1 , \\alpha _ 1 } = 0 . \\endaligned \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n - 2 \\} , \\ , \\forall g \\in \\mathcal { E } _ f , \\ \\kappa ( s _ i , s _ { i + 1 } ( g ) ) = \\kappa ( s _ i , g ) \\ \\mathrm { a n d } \\ \\kappa ( s _ { i + 1 } , s _ i ( g ) ) = \\kappa ( s _ { i + 1 } , g ) . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} X + t = \\{ x + t : x \\in X \\} . \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\partial _ { \\lambda } G _ { \\lambda \\pm 0 i } ( x ) = - \\frac { 1 } { 2 \\pi \\lambda } + i x G _ { \\lambda \\pm 0 i } ( x ) . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} F ( y ) = F ( b ) + ( y - b ) F ' ( b ) + \\tfrac 1 2 ( y - b ) ^ 2 F '' ( b ) + \\tfrac 1 6 ( y - b ) ^ 3 F ''' ( \\xi ) \\ , . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} { \\displaystyle \\partial _ { t } \\theta ^ { 0 } + \\nabla \\cdot \\left ( - \\mathbb { K } \\nabla \\theta ^ { 0 } \\right ) + g _ { 0 } \\frac { \\left | \\Gamma _ { R } \\right | } { \\left | Y _ { 1 } \\right | } \\theta ^ { 0 } = \\sum _ { i = 1 } ^ { N } \\left ( \\mathbb { T } ^ { i } \\nabla ^ { \\delta } u _ { i } ^ { 0 } \\right ) \\cdot \\nabla \\theta ^ { 0 } } \\quad \\mbox { i n } \\ ; Q _ { T } , \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} v ' ( f - f _ 0 ) = v ' f - v ' f _ 0 = f '' v - f '' v = 0 , \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 3 } u ^ p \\ ; d x & = \\int _ { \\mathbb { R } ^ 3 } u ^ { p \\theta } u ^ { p ( 1 - \\theta ) } ( 1 + | x | ) ^ m ( 1 + | x | ) ^ { - m } \\ ; d x \\\\ & \\le \\left ( \\int _ { \\mathbb { R } ^ 3 } u ^ { p p _ 1 \\theta } ( 1 + | x | ) ^ { p _ 1 m } \\ ; d x \\right ) ^ { \\frac { 1 } { p _ 1 } } \\left ( \\int _ { \\mathbb { R } ^ 3 } u ^ { p ( 1 - \\theta ) p _ 2 } ( 1 + | x | ) ^ { - m p _ 2 } \\ ; d x \\right ) ^ { \\frac { 1 } { p _ 2 } } , \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} B _ { r } ( z ) = 1 + z B _ { r - 1 } ( z ) ^ { 2 } \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } | [ W ( t : 0 ) ] _ { i j } - \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } [ W ( t : 0 ) ] _ { i k } | < { \\delta \\varepsilon } . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} W _ 1 ( \\mu , \\nu ) = \\sup _ { f \\in \\mathrm { L i p } ( X ) } \\left | \\int f \\mathrm d \\mu - \\int f \\mathrm d \\nu \\right | \\ , , \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} ( A Q _ \\infty ) x = - ( A Q _ \\infty ) ^ * x - B B ^ * x \\forall x \\in D ( ( A Q _ \\infty ) ^ * ) . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} \\geq 1 / \\| P \\| _ { Z \\to Z } \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| g _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } g _ k , g _ k \\in H [ Z ] , n \\in \\N \\} \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} y ^ * ( t , p ) = \\sum _ { j = 1 } ^ r w _ j ^ * ( t ) \\Psi _ j ( p ) \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\ , K _ { b _ { 0 } , l _ { 0 } - 1 } ^ { X _ { 2 } ^ { c } , 0 } - \\ , K _ { b _ { 0 } , l _ { 0 } } ^ { X _ { 2 } ^ { c } , 0 } = 1 - 0 = 1 \\ , \\ , \\ , \\ , l _ { 0 } > 0 . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} L _ s - \\chi _ 1 ( \\tau ) = ( - \\partial - \\tilde { \\phi } _ s ) ( \\partial - \\tilde { \\phi } _ s ) , \\mbox { i n } K ( \\tau ) [ \\partial ] . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} V _ { n , l } \\ ( x , y \\ ) = \\int _ { \\R ^ 3 } e ^ { i \\ ( x - w \\ ) \\xi + i \\ ( F \\ ( w \\ ) - y \\ ) \\eta } \\tilde { \\psi _ n } \\ ( \\xi \\ ) \\frac { \\psi _ l \\ ( \\eta \\ ) } { \\ ( 1 + \\eta ^ 2 \\ ) ^ { \\frac { 1 + \\epsilon } { 2 } } } \\Phi \\ ( w , \\xi , \\eta \\ ) \\mathrm { d } w \\mathrm { d } \\xi \\mathrm { d } \\eta , \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { M _ h ( x ) } { C ^ { h / 2 } x ( \\log \\log x ) ^ { 3 h / 2 } } = \\begin{cases} \\frac { h ! } { ( h / 2 ) ! 2 ^ { h / 2 } } , & \\mbox { i f } h \\\\ 0 , & \\mbox { i f } h \\end{cases} \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} z \\mapsto \\exp \\ ( - \\sum _ { n \\geqslant 1 } \\frac { 1 } { n } \\sum _ { T _ t ^ n x = x } \\frac { \\exp \\ ( g _ { n , t } \\ ( x \\ ) \\ ) } { 1 - \\ ( \\ ( T _ t ^ n \\ ) ' \\ ( x \\ ) \\ ) ^ { - 1 } } z ^ n \\ ) \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n b _ { i , j } \\equiv 0 \\bmod 2 . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} A ' ( x , 1 ) = x ( 1 - x ) C ( x ) A ( x , 1 ) - x C ( x ) - x . \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} Z ( s , w , f \\times f ) = \\frac { ( 4 \\pi ) ^ { s + k - 1 } } { \\Gamma ( s + k - 1 ) } \\sum _ { h \\geq 1 } \\frac { \\langle \\lvert f \\rvert ^ 2 \\Im ( \\cdot ) ^ k , P _ h \\rangle } { h ^ w } \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} J _ \\lambda ( u ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\vert \\nabla u \\vert ^ 2 - \\frac { \\lambda } { 2 } \\int _ { \\Omega } u ^ 2 - \\frac { 1 } { 6 } \\int _ { \\Omega } | u | ^ { 6 } . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\| z _ n ^ M - \\varphi \\| _ \\infty = O ( \\max \\{ \\tilde { h } ^ { d } , h ^ { 3 r } \\} ) . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\binom { k } { 2 } \\binom { n } { k - x } = \\binom { n } { 2 } \\binom { n - 2 } { k - x - 2 } + x ( k - x ) \\binom { n } { k - x } + \\binom { x } { 2 } \\binom { n } { k - x } \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} P _ j ( i ) = \\frac { 2 ( q - 1 ) } { q ^ 2 } g _ \\frac { 1 } { 2 } ( j ) \\left ( \\Delta ( i ) - \\frac { q } { q - 1 } f ( i ) \\right ) = \\frac { 2 ( q - 1 ) } { q ^ 2 } g _ \\frac { 1 } { 2 } ( j ) \\Sigma ( i ) . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\left ( \\sum \\limits _ { k = 1 } ^ { p } D _ { i _ 1 } \\cdots D _ { i _ { k - 1 } } S _ { i _ k } D _ { i _ { k + 1 } } \\cdots D _ { i _ p } \\right ) \\cdot \\left ( \\sum \\limits _ { k = 1 } ^ { q } D _ { j _ 1 } \\cdots D _ { j _ { k - 1 } } S _ { j _ k } D _ { j _ { k + 1 } } \\cdots D _ { j _ p } \\right ) . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} B ( x , y ) = \\frac { x ^ 2 ( 1 - x ) y } { ( 1 - 2 x ) ^ 2 } A ( x , y ) \\ , . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V - g _ { 1 } ( U , V ) \\xi - \\eta ( V ) U , \\varphi Z ) \\\\ & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V , \\varphi Z ) . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\mathbb { P } \\Biggl ( \\max _ { 1 \\leqslant k \\leqslant n } \\sum \\limits _ { i = 1 } ^ { k } X _ i > x \\Biggr ) \\underset { x \\to \\infty } \\sim \\mathbb { P } \\Biggl ( \\sum \\limits _ { i = 1 } ^ { n } X _ i > x \\Biggr ) . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} h _ n ( z ) = - \\tfrac { 1 } { 6 } n ( n - 1 ) \\sigma '' ( z ) - \\tfrac { 1 } { 2 } n \\tau ' ( z ) + C _ n \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\beta _ { [ X ^ { 1 / 2 } Y '^ { 1 / 2 } ] ' } = \\beta _ { X '^ { 1 / 2 } Y ''^ { 1 / 2 } } \\leq \\frac { 1 } { 2 } \\beta _ { X ' } + \\frac { 1 } { 2 } \\beta _ { Y '' } < 1 , \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\Phi ( \\pi ) & = \\prod _ { i = 1 } ^ { n } x _ { n - i + 1 } ^ { d _ i } \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} k ( x , y ) : = \\nu ( x ) \\frac { x - y } { | x - y | ^ n } \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\Gamma ^ S ( \\chi ) & = \\{ \\gamma \\in \\Gamma ( \\chi ) \\colon \\gamma \\} , \\\\ \\Gamma ^ S ( q ) & = \\bigcup _ { \\substack { \\chi \\mod q \\\\ \\chi \\ne \\chi _ 0 } } \\Gamma ^ S ( \\chi ) . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\widetilde { A } _ { { \\mathcal { G } } } W ( \\psi , \\phi ) = W ( \\widehat { A } _ { { \\mathcal { G } } } \\psi , \\phi ) . \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} P _ { T - t } u _ t ( x ) = P _ T u _ 0 ( x ) & + \\int _ 0 ^ t P _ { T - s } \\big ( \\beta \\ , u _ s ( 1 - u _ s ) \\big ) ( x ) \\ , d s \\\\ & + \\int _ { [ 0 , t ] \\times \\Gamma } p ( T - s , x , y ) \\ , \\ell ( y ) \\sqrt { \\gamma ( y ) \\ , u _ s ( y ) \\big ( 1 - u _ s ( y ) \\big ) } \\ , d W ( s , y ) \\\\ & + \\frac { 1 } { 2 } \\int _ 0 ^ t \\sum _ { v \\in V } p ( T - s , x , v ) \\ , \\ell ( v ) \\ , \\hat { \\beta } ( v ) \\ , u _ s ( v ) ( 1 - u _ s ( v ) ) \\ , d s \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} p ( x _ i , y _ i , z _ i , w _ i ( \\epsilon ) ) + \\sum _ { j \\ne i } p ( x _ j , y _ j , z _ j , w _ j ) = - \\epsilon \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} & \\widehat { c } _ * = \\alpha _ 0 \\| g ^ { - 1 } \\| _ { L _ \\infty } ^ { - 1 } , \\\\ & \\widehat { \\delta } = \\frac { 1 } { 4 } \\alpha _ 0 \\| g ^ { - 1 } \\| _ { L _ \\infty } ^ { - 1 } r ^ 2 _ 0 , \\\\ & \\widehat { t } ^ { \\ , 0 } = \\frac { r _ 0 } { 2 } \\alpha _ 0 ^ { 1 / 2 } \\alpha _ 1 ^ { - 1 / 2 } \\left ( \\| h \\| _ { L _ { \\infty } } \\| h ^ { - 1 } \\| _ { L _ { \\infty } } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} g _ j ( t ) : = \\exp \\biggl \\{ i \\lambda \\int _ 1 ^ t | f _ j ( s ) | ^ 2 \\tfrac { d s } { s } \\biggr \\} f _ j ( t ) , \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} \\overline { F } _ { S _ \\eta } ( x ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } \\mathbb { P } ( \\eta = n ) \\mathbb { P } ( S _ n > x ) , \\\\ \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } \\mathbb { P } ( \\eta = n ) \\mathbb { P } ( \\xi _ { ( n ) } > x ) , \\\\ \\overline { F } _ { S _ { ( \\eta ) } } ( x ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } \\mathbb { P } ( \\eta = n ) \\mathbb { P } ( S _ { ( n ) } > x ) . \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\sum _ { p \\leq t } \\omega _ 0 ( p ) ^ 2 = \\sum _ { p \\le t } \\omega ( p - 1 ) ^ 2 = \\frac { t ( \\log \\log t ) ^ 2 } { \\log t } + O \\bigg ( \\frac { t | \\ ! \\log \\log t | } { \\log t } \\bigg ) \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\varphi ' ( \\Gamma _ i ) \\frac { \\partial | G _ { i i } | ^ 2 } { \\partial g _ { i k } } = \\varphi ' ( \\Gamma _ i ) \\overline { G _ { i i } } \\frac { \\partial G _ { i i } } { \\partial g _ { i k } } + \\varphi ' ( \\Gamma _ i ) G _ { i i } \\frac { \\partial \\overline { G _ { i i } } } { \\partial g _ { i k } } \\ , . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} f ( \\gamma z ) = j ( \\gamma , z ) ^ { 2 k } f ( z ) \\gamma \\in \\Gamma _ 0 ( 4 ) \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\Omega ( { \\mathbf X } , x ) = \\overline { \\big \\{ S ^ n ( x , x , \\ldots , x ) : n \\in \\Z \\big \\} } . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} c ^ { t , \\bar t } : = t \\phi _ - \\bigg ( \\frac 1 t ( c ^ t - c ^ 0 ) \\bigg ) + \\bar t \\phi _ + \\bigg ( \\frac { 1 } { \\bar t } ( c ^ { \\bar t } - c ^ 0 ) \\bigg ) . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ t } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} ( p - 1 ) ( 1 + p + \\cdots + p ^ { n - 1 } ) - p ^ j = p ^ n - p ^ j - 1 , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} I + I I = I I I \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} A = u ( x , y , 0 ) \\mbox { a n d } A ' = ( u v + w ) ( x , y , 0 ) . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\mathcal { Q } _ h ( 1 ) = \\widetilde { \\mathfrak { s w } } ^ { n o r m } _ { h } ( M ) . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} c _ { m k } \\ = \\ \\displaystyle { \\sum _ { ( i _ 1 , \\ldots , i _ k ) \\in \\{ 1 , \\ldots , m \\} ^ k } } E _ { i _ 1 i _ 2 } E _ { i _ 2 i _ 3 } \\ldots E _ { i _ k i _ 1 } , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} h ^ { p , q } ( W ) = h ^ { d - q , p } ( W ^ \\vee ) \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} \\sup _ x \\| f \\| _ { L ^ 1 _ 2 } & = \\sup _ x \\Big \\{ \\int _ { \\mathbb { R } ^ 3 } | f ( x , v ) | ( 1 + | v | ^ 2 ) d v \\Big \\} , \\cr \\| f \\| _ { L ^ { \\infty } _ 2 } & = \\sup _ { x , v } | f ( x , v ) | ( 1 + | v | ^ 2 ) . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} Z \\mathbf { d } & = \\sup _ { z \\in Z } z \\mathbf { d } . \\\\ \\mathbf { d } Z & = \\inf _ { z \\in Z } \\mathbf { d } z . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} q _ 1 ( 0 ) ( i + 1 ) / 2 + q _ 2 ( 0 ) ( i - 1 ) / 2 = \\nabla p ( 0 ) \\cdot ( x + i \\tilde { y } ) \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\bigcap _ { k = 1 } ^ j A _ { n , i _ k } \\right ) & = \\mathbb { P } \\left ( \\min _ { 1 \\leq k \\leq j } \\left ( \\frac { \\varepsilon _ { i _ k } } { P _ { i _ k } } \\right ) > \\frac { \\varepsilon } { T _ n } \\right ) \\\\ & = \\mathbb { E } \\exp \\left ( - \\frac { \\varepsilon ( P _ { i _ 1 } + \\cdots + P _ { i _ j } ) } { T _ n } \\right ) \\\\ & = \\mathbb { E } \\left ( \\frac { T _ n } { T _ n + P _ { i _ 1 } + \\cdots + P _ { i _ j } } \\right ) , \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\mathbf V _ { \\alpha } : = \\langle \\mathbf v _ i , \\mathbf v _ j \\rangle - \\cos \\alpha , \\{ \\mathbf v _ 1 , \\dots , \\mathbf v _ n \\} , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} D ' ( 1 / x ) = \\sum _ { r = 0 } ^ { n - 2 } D ' _ r x ^ { n - 1 - r } = u ^ 2 ( x ) C ' ( x ) \\bmod { ( x ^ { n - 1 } - 1 ) } , \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\widehat { u \\varphi } \\chi _ { \\mathbb { R } ^ + } = ( \\xi - \\lambda ) \\hat { \\varphi } . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} 0 = \\sum _ { i } b _ { n i } A _ i = \\sum _ { j = 1 } ^ s \\left ( \\sum _ { i = 1 } ^ r b _ { n i } \\alpha _ { i j } \\right ) E _ j , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} u ^ { r } = 0 , \\partial _ r u ^ { \\theta } - u ^ { \\theta } = 0 , \\partial _ r u ^ { z } = 0 \\textrm { o n } \\ \\partial \\Pi \\times ( 0 , T ) , \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} X ( z ) y '' ( z ) + Y ( z ) y ' ( z ) + Z ( z ) y ( z ) = 0 , \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\kappa _ { n } ^ { 2 } = \\zeta _ { n } ^ { 2 } s i n \\left ( \\sqrt { C L \\left ( \\omega _ { n } ^ { 2 } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x _ { 0 } \\right ) + \\eta _ { n } ^ { 2 } c o s \\left ( \\sqrt { C L \\left ( \\omega _ { n } ^ { 2 } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x _ { 0 } \\right ) . \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} ( \\lambda I + M ) \\boldsymbol { u } = \\boldsymbol { f } . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty } R _ Z ( p , p + n ) = \\bigl ( a ^ 2 + b ^ 2 \\bigr ) R _ B ( 0 , n ) . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} B _ m ( x ) & = x \\big ( L ( x ) - 1 \\big ) A _ { m - 1 } ( x ) \\\\ & + \\sum _ { d \\geq 1 } \\left ( \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ d } L ( x ) + \\left ( x ^ { d + 1 } + \\sum _ { p = 1 } ^ { d - 1 } \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ p } \\right ) \\big ( L ( x ) - 1 / ( 1 - x ) \\big ) \\right ) A _ { m - 1 } ( x ) \\\\ & = x \\big ( L ( x ) - 1 \\big ) A _ { m - 1 } ( x ) + \\frac { x ^ 3 } { ( 1 - 2 x ) ^ 2 } A _ { m - 1 } ( x ) , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\| X \\| _ * = \\sum _ i \\sigma _ i , \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} N ^ k & = ( B _ 1 + B _ 2 + B _ 3 + B _ 4 ) ( P + Q + R + S ) \\\\ & = B _ 1 P + B _ 1 R + B _ 2 Q + B _ 2 S + B _ 3 Q + B _ 3 S + B _ 4 P + B _ 4 R \\\\ & = A _ 1 + A _ 2 + A _ 3 + A _ 4 . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } \\left ( s _ 1 \\right ) = 1 , \\quad P _ n ^ { ( a , b ) } \\left ( s _ 0 \\right ) = ( 1 - b ) ^ { - n } \\quad \\quad ( n \\ge 0 ) . \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} & \\Sigma : = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\ , , \\Pi _ 0 : = \\frac 1 2 \\begin{pmatrix} \\pi _ 0 & \\pi _ 0 \\\\ - \\pi _ 0 & - \\pi _ 0 \\end{pmatrix} \\ , , { \\mathbb I } _ 2 : = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } _ T = \\mu ( w , y ) - \\mu ( w , z ) - \\mu ( x , y ) + \\mu ( x , z ) , \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} m _ 3 = \\int \\limits _ 0 ^ { 2 \\pi } ( \\sin \\theta - \\sin 2 \\theta + \\sin 3 \\theta ) \\left ( \\int \\limits _ 0 ^ { \\theta } ( \\cos \\ , s + 2 \\cos 2 s ) d s \\right ) ^ 3 d \\theta = \\frac { \\pi } { 2 } \\neq 0 , \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} p _ { 4 , 0 } ^ { P _ 1 } = 1 - \\frac { 3 8 5 \\ , 0 2 4 } { 1 3 5 \\ , 1 3 5 } \\frac 1 \\pi + \\frac { 1 6 \\ , 7 7 7 \\ , 2 1 6 } { 1 8 \\ , 7 2 9 \\ , 7 1 1 } \\frac 1 { \\pi ^ 2 } . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\lambda _ { K + 1 } ( s , X ( s ) ) = \\overline \\lambda _ 0 - \\lambda _ 0 ( s , X ( s ) ) , \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\| f \\| _ { H [ Z ^ { \\wedge } ] } = \\| f \\| _ { Z ^ { \\wedge } } = \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| f _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in Z , n \\in \\N \\} \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} L = - 6 \\Delta + \\Phi , \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} ( M _ r V ) _ S = 2 ^ { r - 3 } ( S ( x ) - S ( y ) ) ( r - 2 | S | ) . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } \\langle b ( t , x ) , \\nabla f _ { t } ( x ) \\rangle f _ { t } ^ { 2 p - 1 } ( x ) \\ ; d x = 0 \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} ( \\operatorname { P r o j } _ m ) _ * A _ k M = A _ k M _ m . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\pi _ 4 & = ( 2 , 3 , 4 , 1 ) , \\\\ t _ 4 & = ( 0 , 0 , 1 , 1 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 ) , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} R ^ { ( 1 ) } ( x ) = \\left [ \\begin{array} { c c } \\mathcal { V } ( x ) & T \\frac { \\mathcal { V } ( x ) } { | | \\mathcal { V } ( x ) | | ^ 2 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 3 } ( \\bar q ( R ) ) = \\bar q ' ( R ) \\partial _ { t } ^ { 3 } R + 3 \\bar q '' ( R ) \\partial _ { t } ^ { 2 } R \\partial _ { t } R + \\bar q ''' ( R ) ( \\partial _ { t } R ) ^ { 3 } , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\omega _ { \\varphi } } { \\partial t } = - { \\rm R i c } ( \\omega _ { \\varphi } ) + \\gamma \\omega _ { \\varphi } + ( 1 - \\beta ) [ D ] + L _ X \\omega _ { \\varphi } \\\\ \\omega _ { \\varphi } | _ { t = 0 } = \\omega ^ { \\ast } \\end{cases} \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} w _ t + \\int ( w ( x , t ) - w ( x + y , t ) ) \\frac { u ^ + ( x + y ) } { u ^ + ( x ) } K ( x , y ) d y = w \\left ( \\frac { f ( x , u ^ + w ) } { u ^ + w } - \\frac { f ( x , u ^ + ) } { u ^ + } \\right ) . \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} T _ k & : = \\inf \\{ n > T _ { k - 1 } : \\ : Q _ n = q _ 0 \\} \\ , , \\\\ \\zeta _ k & : = X _ { T _ k } - X _ { T _ { k - 1 } } \\ , , \\\\ \\nu _ k & : = T _ k - T _ { k - 1 } \\ , . \\\\ \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\mathcal { R } ^ + _ { } = \\left \\{ Q \\in \\mathcal { L } ^ + : \\begin{matrix} q _ { 1 3 } \\ ! = \\ ! q _ { 1 4 } , q _ { 2 3 } \\ ! = \\ ! q _ { 2 4 } , q _ { 3 1 } \\ ! = \\ ! q _ { 3 2 } , q _ { 4 1 } \\ ! = \\ ! q _ { 4 2 } \\\\ q _ { 1 2 } q _ { 2 3 } \\ ! = \\ ! q _ { 2 1 } q _ { 1 3 } , q _ { 3 4 } q _ { 1 3 } \\ ! = \\ ! q _ { 1 2 } q _ { 3 1 } , q _ { 4 3 } q _ { 1 3 } \\ ! = \\ ! q _ { 1 2 } q _ { 4 1 } , \\ldots \\end{matrix} \\right \\} , \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} H _ 3 ^ f = \\arg \\min \\{ S _ 3 ( \\tilde { H } ^ { 0 , k } ) , k = 1 \\ldots N _ g \\} . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\psi ( \\rho , y ) : = ( \\otimes ^ m { \\tau _ - } _ { ( x , \\xi _ - ) } ) \\Psi ( y ) \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\bigl ( \\bigl ( A ( \\psi , \\phi ) | A ( \\psi ^ { \\prime } , \\phi ^ { \\prime } ) \\bigr ) \\bigr ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } ( \\psi | \\psi ^ { \\prime } ) \\overline { ( \\phi | \\phi ^ { \\prime } ) } . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} f _ s ( r _ c ) = 2 \\pi \\lambda _ B r _ c p _ { s , c } ( r _ c ) e ^ { - 2 \\pi \\lambda _ B \\psi _ s ( r _ c ) } / \\mathcal { B } _ { s , c } \\ ; s \\in \\{ L , N \\} \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\mu _ 0 ( \\lambda , r ) = \\frac { - a _ 1 f _ \\lambda ( r ) } { k a _ 2 \\lambda - a _ 3 f _ \\lambda ( r ) ^ 2 } > 0 . \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} T _ { \\alpha } x : = - \\lim \\limits _ { t \\rightarrow 0 + } t ^ { 1 - 2 \\alpha } u ' ( t ) \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } w _ { 0 } ( x , t , y , s ) \\cdot v ( x , y ) c _ { 1 } ( t ) c _ { 2 } ( s ) d y d x d s d t \\\\ = - \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } u ( x , t ) \\nabla _ { x } \\cdot v ( x , y ) c _ { 1 } ( t ) c _ { 2 } ( s ) d y d x d s d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } \\nabla u ( x , t ) \\cdot v ( x , y ) c _ { 1 } ( t ) c _ { 2 } ( s ) d y d x d s d t . \\end{gather*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\phi ( x , t ) = \\varphi ( x + t \\omega ) , \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\int _ { N _ 1 \\ , { } _ { f _ 1 } \\times _ { f _ 2 } N _ 2 } h _ 1 \\wedge h _ 2 = \\pm \\int _ { N _ 2 } f _ 2 ^ * ( f _ 1 ! ( h _ 1 ) ) \\wedge h _ 2 . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\mathrm { T r d } _ { M _ 2 ( \\mathbb { O } ) } ( \\alpha ( x ) \\alpha ( y ) ^ \\iota ) = \\mathrm { T r } _ { R / \\mathbb { O } } ( \\xi x \\overline { y } ) \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} w '' _ { R } = \\frac { w _ { R } } { | w _ { R } | } | w _ { R } ' | ^ { 2 } - \\frac { m _ { 1 } w _ { R } } { 2 | w _ { R } | ^ { 2 } } + \\frac { \\overline { w } _ { R } } { 2 } | w _ { R } | ^ { 2 } \\nabla U _ { 1 } ( c _ { 1 } + w _ { R } ^ { 2 } ) \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } & = a ^ { a / 2 } - \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\sum _ { n = 1 } ^ \\infty \\frac { 2 z ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - z ^ 2 } . \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} D _ \\mathfrak { a } ^ k ( h , w ) = \\frac { L ( 2 w - \\frac { 1 } { 2 } , \\chi _ { k , h } ) } { \\zeta ( 4 w - 1 ) } \\widetilde { D } _ \\mathfrak { a } ( h , w ) , \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} L \\phi : = \\Delta \\phi + \\frac { 8 } { ( 1 + | z | ^ 2 ) ^ 2 } \\phi \\quad \\textrm { i n } \\ \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} r v r ^ { - 1 } = t u t ^ { - 1 } , ~ s t s ^ { - 1 } = u , ~ u s u ^ { - 1 } = t , ~ v s v ^ { - 1 } = r \\rangle , \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\psi _ { \\alpha } ( d ( a ) ) - d \\psi _ { \\alpha } ( a ) - \\sum _ { \\beta \\in \\Lambda } ( - 1 ) ^ { | \\psi _ { \\beta } ( a ) | } \\psi _ { \\beta } ( a ) \\psi _ { \\alpha } ( d ( x _ { \\beta } ) ) = 0 . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 & \\leq e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) } ( 1 - \\delta ) ^ { m / 2 } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\\\ & + 2 ^ m n ^ { \\delta m + 1 } \\binom { n } { m } ^ { - 1 / 2 } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 . \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\mathrm { o r d } _ x ( D ) = \\lim _ { t ( \\xi ) \\rightarrow + \\infty } \\frac { g ( \\xi ) } { t ( \\xi ) } = \\lim _ { t ( \\xi ) \\rightarrow + \\infty } \\frac { g ( \\xi ) - g ( \\eta _ 0 ) } { t ( \\xi ) } . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} | B ^ T e _ 3 | ^ 2 = \\epsilon ^ 2 + ( 1 - \\epsilon ^ 2 ) = 1 , \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\partial _ { ( \\zeta ^ \\prime ) _ n } \\phi = \\sum _ { m , l } b _ { m l } \\ , { \\bf z } _ { m l } + T ( f ) \\quad \\Vert \\partial _ { ( \\zeta ^ \\prime ) _ n } \\phi \\Vert _ { \\ast } \\leq C \\ , \\Vert h \\Vert _ { \\ast \\ast } . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} { J } _ 1 ( t , \\tau ) & : = M \\cos ( \\tau A ( t ) ^ { 1 / 2 } ) M ^ { - 1 } \\widehat { P } - M _ 0 \\cos ( \\tau ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { 1 / 2 } ) M _ 0 ^ { - 1 } \\widehat { P } , \\\\ \\widetilde { J } _ 2 ( t , \\tau ) & : = M A ( t ) ^ { - 1 / 2 } \\sin ( \\tau A ( t ) ^ { 1 / 2 } ) P M ^ * - M _ 0 ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { 1 / 2 } ) M _ 0 \\widehat { P } . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } + L u - \\lambda u = - g \\ : \\mbox { i n } \\ : D _ T , u ( \\pm 1 , y ) = u ( \\pm 1 , - e y ) \\ : \\mbox { i n } \\ : D _ T ^ \\pm , u ( T ) = f \\ : \\mbox { i n } \\ : D . \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) ( 1 - 2 \\tilde { \\alpha } _ i ( x ) ^ 2 ) & \\leq n . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\chi _ { \\rho } ( \\epsilon ^ 2 + u ) = \\frac { 1 } { \\rho } \\int _ 0 ^ u \\frac { ( \\epsilon ^ 2 + r ) ^ { \\rho } - \\epsilon ^ { 2 \\rho } } { r } d r \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} P ( x , D ) = \\sum _ { j = 1 } ^ { N } X _ { j } ( x , D ) ^ { 2 } , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} I _ j = n ^ { - 1 / 2 } I _ 0 = 2 \\int _ { T _ 0 } | v | ^ 2 d y . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} m _ j ( \\xi , \\eta ) = \\int \\rho ( t ) e ^ { - 2 \\pi i \\left ( \\frac { \\xi } { 2 ^ { a j } } ( t ^ a + \\epsilon _ P ( t ) ) + \\frac { \\eta } { 2 ^ { b j } } ( t ^ b + \\epsilon _ Q ( t ) ) \\right ) } \\ , d t , \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} s \\rightarrow f ( x , s ) / s \\ s , \\ f ( x , 0 ) = 0 , \\ \\ f ( x , \\cdot ) \\leq 0 \\ \\ [ M , \\infty ) . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\pi _ { U } ( 1 _ { B } ) ( u ) = f _ { B } ( u ) \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} x _ N ( 0 ) - a _ E b _ E x _ E ( 0 ) ( e ^ { k _ E \\tau _ 1 } - 1 ) - a _ M b _ M k _ M ( \\tau _ s - \\tau _ 1 ) & = 0 \\\\ \\frac { 1 } { 2 } b _ M k _ M k _ E ( \\tau _ s - \\tau _ 1 ) ^ 2 + ( b _ E k _ E - b _ M k _ M ) ( \\tau _ s - \\tau _ 1 ) & \\\\ + ( 1 - \\frac { a _ E } { a _ M } ) b _ E k _ E ( T - \\tau _ s ) & = 0 , \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} P _ h ^ k ( z , s ) = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma _ 0 ( 4 ) } \\Im ( \\gamma z ) ^ s e ^ { 2 \\pi i h \\gamma z } J ( \\gamma , z ) ^ { - 2 k } \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\mu ( \\mathtt b ) : = \\aleph ( M , s _ 0 + \\mathtt b ) \\ , , \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} ( k _ 1 , h _ 1 ) ( k _ 2 , h _ 2 ) : = ( k _ 1 \\theta _ { h _ 1 } ( k _ 2 ) , h _ 1 h _ 2 ) . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l v _ { n , t + 1 } + v _ { n , t - 1 } - a _ n v _ { n + 1 , t } - a _ { n - 1 } v _ { n - 1 , t } - b _ n v _ { n , t } = 0 , t \\in \\mathbb { N } _ 0 , \\ , \\ , n \\in 0 , \\ldots , N + 1 \\\\ v _ { n , - 1 } = v _ { n , 0 } = 0 , n = 1 , 2 , \\ldots , N + 1 \\\\ v _ { 0 , t } = f _ t , v _ { N + 1 , t } + h v _ { N , t } = 0 , t \\in \\mathbb { N } _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{gather*} g _ { i j } = \\partial _ i \\eta _ \\mu \\partial _ j \\eta _ \\mu \\end{gather*}"}
-{"id": "9562.png", "formula": "\\begin{align*} E _ P [ \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { t _ i } [ \\xi _ i ] I _ { A _ i } ] \\leq \\lim _ { m \\rightarrow \\infty } \\lim _ { l \\rightarrow \\infty } \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\xi _ i { \\varphi ^ { i , m } _ { l } } \\prod _ { j = 1 } ^ { i - 1 } { \\widetilde { \\varphi } ^ { j , m } _ l } ] . \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} o ( \\alpha _ 1 , \\alpha ' _ 2 ) = \\left ( ( \\widehat \\psi ' _ { 2 1 } \\circ \\widehat \\psi _ { 1 } - \\widehat \\psi _ { 2 } \\circ \\widehat \\psi ' _ { 2 1 } ) - ( \\widehat d ' _ { 2 } \\circ \\widehat { \\frak h } + \\widehat { \\frak h } \\circ \\widehat d _ { 1 } ) \\right ) _ { \\alpha ' _ 2 \\alpha _ 1 } . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} & U R U = R ^ { \\tau } ( R = T ^ { - 1 } ) . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} [ [ a , _ { 2 n } b ] , a ] = \\left [ \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ { i } [ [ a , _ { 2 n - 1 - i } b ] , [ a , _ { i } b ] ] \\ , \\ b \\ \\right ] . \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\psi _ { j , k } : = 2 ^ { - \\frac j 2 } \\psi ( 2 ^ j t + k ) \\mbox { a n d } \\phi _ { j , k } : = 2 ^ { - \\frac j 2 } \\phi ( 2 ^ j t + k ) , j = 1 , \\ldots , n , \\ , \\ , k = 1 , \\ldots , 2 ^ j , \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} n \\times H _ { t } - n \\times Q ^ { * } v + E _ { t } & = 0 \\partial \\Omega , \\\\ T n - Q \\left ( n \\times E _ { t } \\right ) + \\left ( 1 + \\alpha \\partial _ { 0 } ^ { - 1 } \\right ) v & = 0 \\partial \\Omega , \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ M \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) = \\ ( \\exp \\ ( - \\frac 1 { 2 n ^ 2 } \\sum _ { x \\in G } | f ^ { - 1 } ( x ) | ^ 2 \\ ) + o ( 1 ) \\ ) \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} 2 \\pi ( 3 r + 1 ) { { 5 r - 2 } \\choose { 3 r + 1 } } = O \\bigg ( \\sqrt { r } \\Big ( \\frac { 3 1 2 5 } { 1 0 8 } \\Big ) ^ r \\bigg ) \\ , . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\begin{pmatrix} \\hat { u } \\\\ \\hat { v } \\end{pmatrix} = \\begin{pmatrix} 0 . 1 x ^ 2 + 0 . 1 x y + 0 . 2 y ^ 2 \\\\ 0 . 0 5 x ^ 2 + 0 . 1 5 x y + 0 . 1 y ^ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} ( \\widehat { R } _ { \\epsilon _ m } ( x + \\alpha ) { Q } ) ^ { - 1 } A ^ { E _ m ^ { + } + \\epsilon _ m } ( x ) \\widehat { R } _ { \\epsilon _ m } ( x ) { Q } = e ^ { \\sqrt { \\Delta } \\left ( \\left [ \\begin{array} { c c } 0 & - 1 \\\\ 1 & 0 \\end{array} \\right ] + \\epsilon _ m ^ 3 \\mathfrak { S } ( x ) \\right ) } , \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\langle A \\chi _ j , \\chi _ k \\rangle = a _ { k + j + 1 } { \\rm \\ f o r \\ a l l \\ } j , k \\geq 0 \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} ( z ^ 2 + t ) ^ 2 = ( 2 t ) z ^ 2 + | w | z + \\bigg ( t ^ 2 - \\frac { \\lambda } { 3 } \\bigg ) \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} c _ 1 & = \\max \\left \\{ ( \\chi M L ) ^ { 2 / 3 } , \\sqrt { 2 M L / A } \\right \\} , \\\\ c _ 2 & = \\max \\left \\{ 2 ^ { 1 / 3 } ( M L ) ^ { 2 / 3 } , \\sqrt { M / A } L \\right \\} , \\\\ c _ 3 & = ( 6 M ^ 2 ) ^ { 1 / 3 } L , \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} Q : \\ , q _ 1 x _ 1 ^ 2 + q _ 2 x _ 2 ^ 2 + \\cdots + q _ 6 x _ 6 ^ 2 = 0 \\quad , \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} G _ 0 ^ 0 ( x ) = & ~ - \\frac 1 { 2 \\pi } \\int _ 1 ^ x \\frac 1 { \\xi } ~ d \\xi + \\frac 1 { 2 \\pi } \\int _ 0 ^ 1 \\frac { e ^ { i \\xi } - 1 } { \\xi } ~ d \\xi + \\frac { 1 } { 2 \\pi } \\int _ 1 ^ { \\infty } \\frac { e ^ { i \\xi } } { \\xi } ~ d \\xi \\\\ & - \\frac 1 { 2 \\pi } \\int _ 1 ^ { 2 } \\frac { \\chi ( \\xi ) } { \\xi } ~ d \\xi \\\\ = & ~ - \\frac 1 { 2 \\pi } \\log | x | + \\frac { c _ 1 } { 2 \\pi } . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\left \\langle \\kappa ( z ) , Z _ { N } \\right \\rangle = \\left \\langle d F , Z _ { P } \\right \\rangle , \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\mathcal { B } = \\mathcal { W } \\mathcal { A } \\mathcal { V } , \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\underline w _ U = \\inf _ { U } w > 0 . \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\eta = - d \\varphi _ 1 \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\varphi _ { \\epsilon } } } \\right ) \\varphi _ { \\epsilon } = \\dot { \\varphi } _ { \\epsilon } - { \\rm t r } _ { \\omega _ { \\varphi _ { \\epsilon } } } ( \\omega _ { \\varphi _ { \\epsilon } } - \\omega _ { \\epsilon } ) = \\dot { \\varphi } - n + { \\rm t r } _ { \\omega _ { \\varphi _ { \\epsilon } } \\omega _ { \\epsilon } } \\geq { \\rm t r } _ { \\omega _ { \\varphi _ { \\epsilon } } } \\omega _ { \\epsilon } - ( C + n ) . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} S ( z ) = \\prod ^ k _ { i = 1 } ( z - z _ i ) , \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log D ( f ^ n ( x ) ) = - a . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\beta ^ 1 _ p \\circ \\eta ' _ a \\circ \\eta _ a \\ = \\ \\beta ^ 2 _ p \\circ \\{ \\eta ' \\circ \\eta \\} _ a \\ = \\ \\beta ^ 2 _ p \\circ \\eta ' _ a \\circ \\eta _ a \\ , . \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\beta ( \\alpha ) = \\limsup _ { k \\rightarrow \\infty } \\frac { - \\ln { | | k \\alpha | | _ { \\mathbb { R } / \\mathbb { Z } } } } { | k | } , \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} ( d f _ \\tau ) ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n + 1 } ) & = x _ 1 f _ \\tau ( x _ 2 \\otimes x _ 3 \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i f _ \\tau ( x _ 1 \\otimes \\dots \\otimes x _ i x _ { i + 1 } \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f _ \\tau ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n } ) x _ { n + 1 } . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} ( 1 + \\lambda t ) ^ { \\frac { x + y } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty { x + y \\choose n } _ \\lambda t ^ n . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\mathcal { Y } = L ^ { 2 } \\left ( \\Omega , \\mathbb { C } ^ { 3 } \\right ) \\oplus \\left ( L ^ { 2 } \\left ( \\Omega , \\mathrm { s y m } \\left [ \\mathbb { C } ^ { 3 \\times 3 } \\right ] \\right ) \\oplus \\tilde { Y } \\right ) \\oplus L ^ { 2 } \\left ( \\Omega , \\mathbb { C } ^ { 3 } \\right ) \\oplus \\left ( L ^ { 2 } \\left ( \\Omega , \\left [ \\mathbb { C } ^ { 3 } \\right ] \\right ) \\oplus Y \\right ) \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} N _ \\alpha : = \\{ x \\in B ^ + _ 1 \\ , : \\ , | \\xi ( x ) + e _ 2 | \\ge \\alpha \\} \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} p = P _ { K } x \\iff \\left [ \\ ; p \\in K , ~ x - p \\perp p , ~ x - p \\in K ^ { \\ominus } \\ ; \\right ] . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\mu ( X , q y ) = \\mu ( X , q ) y , \\ X \\in V , \\ q , y \\in H , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} ( \\epsilon _ { A B } ) = \\left ( \\begin{array} { r r r r r } 0 & 1 & & & \\\\ - 1 & 0 & & & \\\\ & & \\ddots & & \\\\ & & & 0 & 1 \\\\ & & & - 1 & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} | a _ { i j } | & = \\left | \\sum _ { q = 1 } ^ n D ( j ) ^ { - 1 } _ { i q } M _ j ( c _ { q k _ 1 k _ 2 \\cdots k _ { n + 1 } } ; \\{ a _ { p q } \\ ; | \\ ; q \\leq j - 1 , 1 \\leq p \\leq n \\} ) \\right | \\\\ & \\leq B \\sqrt { n } M _ j ( C _ { 1 k _ 1 k _ 2 \\cdots k _ { n + 1 } } ; \\{ A _ { p q } \\ ; | \\ ; q \\leq j - 1 , 1 \\leq p \\leq n \\} ) \\\\ & = A _ { i j } \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( X _ { \\tau + t _ 1 } ^ x , \\cdots , X _ { \\tau + t _ m } ^ x ) ] & = \\mathbb { L } ^ 1 \\lim _ { n \\rightarrow \\infty } \\hat { \\mathbb { E } } _ { \\tau _ n } [ \\varphi ( X _ { \\tau _ n + t _ 1 } ^ x , \\cdots , X _ { \\tau _ n + t _ m } ^ x ) ] \\\\ & = \\mathbb { L } ^ 1 \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { d _ n } \\hat { \\mathbb { E } } _ { t ^ n _ i } [ \\varphi ( X _ { { t ^ n _ i } + t _ 1 } ^ x , \\cdots , X _ { { t ^ n _ i } + t _ m } ^ x ) ] I _ { \\{ \\tau _ n = { t ^ n _ i } \\} } . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S & = 2 ^ { r - | S | } \\sum _ a \\left [ \\binom { | S | - 1 } { 2 } \\binom { | S | - 3 } { a - 3 } + ( a - 1 ) \\binom { | S | - 1 } { a - 1 } \\right ] \\\\ & + 2 ^ { | S | - 1 } \\sum _ b \\binom { r - | S | } { 2 } \\binom { r - | S | - 2 } { b - 2 } . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } \\bigl ( \\frac { z + z ^ { - 1 } } { 2 } \\bigr ) = \\frac { c ( z ) z ^ n + c ( z ^ { - 1 } ) z ^ { - n } } { ( ( a - 1 ) ( b - 1 ) ) ^ { n / 2 } } \\quad \\quad \\ > \\ > z \\in \\mathbb C \\setminus \\{ 0 , \\pm 1 \\} \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} F ( t ) X ^ t = x _ k + y _ k ( t - ( \\tfrac a 8 + k ) ) + O ( ( t - ( \\tfrac a 8 + k ) ) ^ 2 ) , \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} P \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) = \\underset { b _ { 1 } , b _ { 2 } , \\bar { p } } { m i n i m i z e } \\ , \\left \\Vert \\mathbf { Q } + \\bar { p } \\mathbf { 1 } - \\mathbf { f } \\right \\Vert _ { 2 } ^ { 2 } \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d B } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 } { \\lambda _ 4 B ^ 2 } \\left ( \\frac { 1 } { \\omega } ( B ^ 2 - k ) \\right ) ^ { - 1 / 2 } \\\\ B ^ 2 ( B ^ 2 - k ) ^ { 1 / 2 } \\frac { d B } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 \\omega ^ { 1 / 2 } } { \\lambda _ 4 } . \\end{aligned} \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\textrm { e } _ 1 = ( 1 , 0 , 0 ) , \\textrm { e } _ 2 = ( 0 , 1 , 0 ) \\textrm { e } _ 3 = ( 0 , 0 , 1 ) . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\frac { K _ { N , L } } { k ! \\left ( ( N - k ) / 2 \\right ) ! } \\left | \\Delta \\left ( \\{ \\lambda _ l \\} _ { l = 1 } ^ k \\cup \\{ x _ j \\pm \\mathrm { i } y _ j \\} _ { j = 1 } ^ { ( N - k ) / 2 } \\right ) \\right | \\prod _ { j = 1 } ^ k w ( \\lambda _ j ; L ) \\prod _ { j = 1 } ^ { ( N - k ) / 2 } 2 \\left ( w \\left ( ( x _ j , y _ j ) ; L \\right ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} \\Upsilon _ 1 = \\displaystyle { \\frac { \\left ( \\tau - 1 \\right ) z ^ 2 + \\tau + 1 } { z ^ 2 + 1 } } { e } ^ { x \\tau } , \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} | D | ^ m : = { \\rm O p } \\big ( \\chi ( \\xi ) | \\xi | ^ m \\big ) \\ , , \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\Delta _ K = c ^ 2 m ^ 2 a _ 1 ^ 2 b _ 1 ^ 2 \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} a ^ 2 & = - ( 2 n + 2 ) L ^ 2 r ( r ^ 2 - L ^ 2 ) ^ { - n } \\int _ { r _ b } ^ r \\frac { ( s ^ 2 - L ^ 2 ) ^ n } { s ^ 2 } \\ , \\mathrm { d } s \\\\ b ^ 2 & = ( L ^ 2 - r ^ 2 ) \\\\ p ^ 2 & = \\frac { L ^ 2 } { a ^ 2 } , \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} ( \\psi ^ \\epsilon _ { i , - \\epsilon b ^ \\epsilon _ i } ) ^ { \\pm 1 } = ( \\phi ^ \\epsilon _ i ) ^ { \\pm 2 } \\cdot \\prod _ { j - i } ( \\phi ^ \\epsilon _ j ) ^ { \\pm c _ { j i } } , \\ ( \\phi ^ \\epsilon _ i ) ^ { \\pm 1 } \\cdot ( \\phi ^ \\epsilon _ i ) ^ { \\mp 1 } = 1 , \\ [ \\phi ^ \\epsilon _ i , \\phi ^ { \\epsilon ' } _ j ] = 0 , \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} l _ { j , 1 } & = a _ { j , 1 } / a _ { 1 , 1 } , \\mbox { f o r } j > 1 , \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} i ^ { 3 , 3 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) & = 1 , i ^ { 2 , 2 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) = p h ( Y , w ) - 3 , \\\\ i ^ { 2 , 1 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) & = i ^ { 1 , 2 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) = h ^ { 1 , 2 } ( Z ) . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\lambda _ 1 ( L _ g ) = \\inf _ { \\phi \\in C ^ { \\infty } , \\phi \\neq 0 } \\dfrac { \\int _ X \\phi L _ g \\phi \\ d V _ g } { \\int _ X \\phi ^ 2 \\ d V _ g } , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} & \\frac { 1 } { k ! } e ^ { x t } \\big ( e ^ t - 1 \\big ) ^ k = \\frac { 1 } { k ! } \\big ( e ^ t - 1 \\big ) ^ k e ^ { x t } \\\\ & = \\left ( \\sum _ { l = k } ^ \\infty S _ 2 ( l , k ) \\frac { t ^ l } { l ! } \\right ) \\left ( \\sum _ { m = 0 } ^ \\infty \\frac { 1 } { m ! } x ^ m t ^ m \\right ) \\\\ & = \\sum _ { n = k } ^ \\infty \\left \\{ \\sum _ { l = k } ^ n { n \\choose l } S _ 2 ( l , k ) x ^ { n - l } \\right \\} \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} V _ { n , l } \\ ( x , y \\ ) = i ^ { r } \\int _ { \\R ^ 3 } \\frac { e ^ { i \\ ( x - w \\ ) \\xi + i \\ ( F \\ ( w \\ ) - y \\ ) \\eta } } { \\ ( x - w \\ ) ^ { r } } f \\ ( w \\ ) \\tilde { \\psi _ n } ^ { \\ ( r \\ ) } \\ ( \\xi \\ ) \\frac { \\psi _ l \\ ( \\eta \\ ) } { \\ ( 1 + \\eta ^ 2 \\ ) ^ { \\frac { 1 + \\epsilon } { 2 } } } \\mathrm { d } w \\mathrm { d } \\xi \\mathrm { d } \\eta \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\kappa ( x ^ { - 1 } , g ) = \\kappa ( x , x ^ { - 1 } ( g ) ) . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} C _ { \\alpha } : = \\mathrm { t o p } ( P _ { \\alpha } ) = P _ { \\alpha } / \\mathrm { r a d } ( P _ { \\alpha } ) \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} D _ \\zeta \\sigma _ 1 ( 0 , \\zeta ^ 0 ) = 0 , D ^ 2 _ { \\zeta \\zeta } \\sigma _ 1 ( 0 , \\zeta ^ 0 ) \\ , , \\frac { \\partial \\sigma _ 1 } { \\partial \\lambda } ( 0 , \\zeta ^ 0 ) < 0 . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { w ( x ) } { - \\log | x | } = \\frac 1 2 \\ddot c ^ 0 \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} & A _ p T - T A _ p ^ * = Q _ p ( p = 1 , 2 ) , \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} { \\bf 1 } : = M ( { \\rm S p e c } \\ , k ) , { \\bf L } : = { \\bf 1 } ( 1 ) [ 2 ] . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} \\alpha = \\frac { \\gamma } { \\sqrt { 3 } } \\beta = \\frac { \\bar { \\gamma } } { \\sqrt { 3 } } . \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} F _ \\varepsilon ( u ) = \\frac 1 p \\int _ { \\Omega } ( \\varepsilon ^ 2 + | \\nabla u | ^ 2 ) ^ { p / 2 } + \\int _ \\Omega \\langle u , f _ \\varepsilon \\rangle _ { \\R ^ N } , \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} \\left \\langle x _ t , y _ t \\right \\rangle = \\int _ { - h } ^ 0 \\int _ { \\theta } ^ 0 e ^ { 2 \\alpha s } x _ t ^ { \\top } ( s ) R y _ t ( s ) d s d \\theta . \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} c _ { k - 2 } & = - n ( n - 1 ) a _ k - n b _ { k - 1 } , \\\\ c _ l & = - n ( n - 1 ) a _ { l + 2 } - n b _ { l + 1 } - 2 \\sum _ { m = 2 } ^ { k - 1 - l } a _ { l + m + 1 } \\biggl [ ( n - 1 ) m _ { ( m - 1 , \\dot { 0 } ) } \\\\ & + \\sum _ { p = 1 } ^ { [ ( m - 1 ) / 2 ] } m _ { ( m - 1 - p , p , \\dot { 0 } ) } \\biggr ] - \\sum _ { m = 1 } ^ { k - 2 - l } b _ { l + m + 1 } m _ { ( m , \\dot { 0 } ) } , \\\\ & l = 0 , 1 , \\ldots , k - 3 . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} = - \\left [ F _ { 2 } \\left ( x , \\displaystyle \\sum \\limits _ { k = 0 } ^ { N - 1 } \\nabla v _ { k } \\left ( x \\right ) \\psi _ { k } \\left ( x _ { 0 } \\right ) , \\displaystyle \\sum \\limits _ { k = 0 } ^ { N - 1 } \\nabla v _ { k } \\left ( x \\right ) \\psi _ { k } ^ { \\prime } \\left ( x _ { 0 } \\right ) \\right ) , \\psi _ { m } \\left ( x _ { 0 } \\right ) \\right ] , \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} g _ 1 ( U , V ) \\xi & = \\mathcal { T } _ { U } \\phi V + \\hat { \\nabla } _ { U } \\phi V + \\mathcal { T } _ { U } \\omega V + \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\omega V \\\\ & - \\mathcal { B } \\mathcal { T } _ { U } V - \\mathcal { C } \\mathcal { T } _ { U } V - \\phi \\hat { \\nabla } _ { U } V - \\omega \\hat { \\nabla } _ { U } V . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} \\left ( W ^ T f \\right ) _ n = u ^ f _ { n , T } = \\prod _ { k = 0 } ^ { n - 1 } a _ k f _ { T - n } + \\sum _ { s = n } ^ { T - 1 } w _ { n , s } f _ { T - s - 1 } , n = 1 , \\ldots , T . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\mathcal { R } ( \\zeta , \\widehat { \\Lambda } ) = ( \\mathcal { R } _ 0 ( \\zeta , \\widehat { \\Lambda } ) , \\mathcal { R } _ 1 ( \\zeta , \\widehat { \\Lambda } ) , \\ldots , \\mathcal { R } _ k ( \\zeta , \\widehat { \\Lambda } ) ) ) \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ { 2 } } { d y ^ { 2 } } + \\alpha ^ { 2 } + \\frac { U ^ { \\prime \\prime } } { U - c } \\right ) \\psi = 0 , \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\lambda y _ q = \\sum \\limits _ { i , j \\in N } a _ { i j q } x _ i y _ j , \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\rho _ { \\frac { 1 } { 2 } } ( s ) = - \\frac { L ( s , f \\times \\overline { g } ) } { \\zeta ( 2 s ) } . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} I _ { 1 2 } & = \\int _ 0 ^ t \\int _ { \\Gamma _ 1 } \\frac { 1 } { \\sqrt { g } } ( \\partial _ t \\det A ^ 1 - \\det A ^ 2 - \\det A ^ 3 ) , \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} a _ { 2 } = \\frac { b _ { 1 } \\sin \\omega _ { 1 } T _ { 0 } + b _ { 2 } \\cos \\omega _ { 1 } T _ { 0 } \\sin \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) } { 1 - \\cos \\omega _ { 1 } T _ { 0 } \\cos \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) } . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} X _ { n } \\left ( x \\right ) = \\zeta _ { n } s i n \\left ( \\sqrt { C L \\left ( \\omega _ { n } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x \\right ) + \\eta _ { n } c o s \\left ( \\sqrt { C L \\left ( \\omega _ { n } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x \\right ) , \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} \\zeta _ K \\left ( \\frac 1 2 + i t \\right ) = O ( x ^ { n / 6 + \\varepsilon } ) \\ t \\rightarrow \\infty . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\gamma _ { a , \\pm } & = \\frac { 4 n Q _ { \\infty } ^ 4 \\pm \\epsilon } { - f ' _ { \\infty } \\mp \\epsilon } \\\\ \\gamma _ { b , \\pm } & = \\frac { 4 ( n + 1 - Q _ { \\infty } ^ 2 ) \\pm \\epsilon } { - f ' _ { \\infty } \\mp \\epsilon } . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\alpha _ { L ^ { \\varphi } } = a _ { \\varphi } \\ { \\rm a n d \\ } \\beta _ { L ^ { \\varphi } } = b _ { \\varphi } , \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} f ^ { \\ast } _ 0 = \\sup \\{ f _ 0 \\in \\mathbb { R } \\ ; | \\ ; f '' ( 0 ) \\leq f _ 0 \\} \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} = \\int _ { \\R _ + } \\int _ { \\R } \\kappa _ L \\left ( \\sum _ { j = 1 } ^ m \\mathbf { 1 } _ { [ 0 , \\infty ) } ( \\xi t _ j - s ) \\zeta _ j e ^ { - \\xi t _ j + s } \\right ) d s \\ , \\pi ( d \\xi ) . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} P _ 2 = 2 ^ { r - | S | - 4 } \\sum _ a \\binom { a } { 2 } \\binom { | S | - 2 } { a } + r ^ { | S | - 2 } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 2 } { b - 2 } . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } \\frac { Y ^ s } { s ( s + 1 ) \\cdots ( s + k ) } d s = \\begin{cases} \\frac { 1 } { k ! } \\big ( 1 - \\frac { 1 } { Y } \\big ) ^ k & Y \\geq 1 , \\\\ 0 & Y < 1 . \\end{cases} \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\mathcal { Q } _ \\tau ( v ) = \\mathcal { Q } _ \\tau ^ 1 ( v ) + \\mathcal { Q } _ \\tau ^ 2 ( v ) , \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\Phi _ { \\mathcal { M } } ( \\lambda \\ , x ) = ( \\lambda _ 1 \\cdots \\lambda _ k ) \\phi _ { \\mathcal { M } } ( x ) \\oplus \\left ( \\lambda _ 1 - \\frac { 1 } { k } , \\ldots , \\lambda _ k - \\frac { 1 } { k } \\right ) . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\forall i = 1 , 2 \\ , , \\ , n \\in [ N _ { k - 1 } , N _ k ] : \\ : \\max \\left \\{ \\left \\| X ^ i _ n - Y _ { k - 1 } \\right \\| , \\left \\| X ^ i _ n - Y _ k \\right \\| \\right \\} \\ , \\leq \\ , R _ { k } \\ , . \\footnote { U n l e s s s t a t e d o t h e r w i s e , t h e n o t a t i o n $ \\| \\cdot \\| $ i s u s e d t o d e n o t e t h e $ { \\ell } _ { \\infty } $ n o r m . } \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\theta ^ 0 _ 1 & = - \\frac { a ' } { a } e ^ 1 & & \\theta ^ 0 _ i = - \\frac { b ' } { b } e ^ i \\\\ \\theta ^ 1 _ i & = - \\frac { a } { b ^ 2 } \\omega _ { i j } e ^ j & & \\theta ^ i _ j = \\hat { \\theta } ^ i _ j + \\frac { a } { b ^ 2 } \\omega _ { i j } e ^ 1 . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} f ^ * ( y + b ) \\geq \\langle y + b , t y \\rangle - \\left ( \\frac { 1 } { 2 } \\langle A ( t y ) , t y \\rangle + \\langle b , t y \\rangle + \\gamma \\right ) = t \\| y \\| ^ 2 - \\gamma . \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} & b \\in C ^ { \\mu , \\mu / 2 } ( \\overline { \\Pi } \\times [ 0 , T ] ) , \\mu \\in ( 0 , 1 ) , \\\\ & \\Gamma _ 0 \\in C ^ { 2 + \\mu } ( \\overline { \\Pi } ) \\textrm { a n d } \\partial _ n \\Gamma + 2 \\Gamma = 0 \\textrm { o n } \\ \\partial \\Pi . \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} c _ { 2 , \\infty } ( \\mu _ 2 ^ 2 + 2 \\mu _ 2 ) + c _ { 3 , \\infty } ( \\mu _ 3 ^ 2 + 2 \\mu _ 3 ) + c _ { 4 , \\infty } ( \\mu _ 4 ^ 2 + 2 \\mu _ 4 ) = 0 . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} w ( t ) = w ( 1 ) - i \\lambda \\int _ 1 ^ t s ^ { - 1 } V ( s ) ^ { - 1 } \\bigl [ | V ( s ) w ( s ) | ^ 2 V ( s ) w ( s ) \\bigr ] \\ , d s . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\smash { E E ' + E \\widetilde { E } ' = E ( E ' + \\widetilde { E } ' ) = E \\cdot \\pi ^ * L = \\pi _ * E \\cdot L = 1 } \\ , , \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} | u ( y ) \\psi ( y ) | & \\le w ^ { - s _ 1 } ( y ) u _ 1 ( y ) \\| \\psi \\| _ { L ^ { \\infty } _ { - ( s - s _ 1 ) } } \\\\ & = w ^ { - p _ 2 } ( y ) w ^ { - \\frac 1 2 - p _ 2 } u _ 1 ( y ) \\| \\psi \\| _ { L ^ { \\infty } _ { - ( s - s _ 1 ) } } \\\\ & = w ^ { - p _ 2 } ( y ) u _ 2 ( y ) \\| \\psi \\| _ { L ^ { \\infty } _ { - ( s - s _ 1 ) } } , \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\begin{cases} g _ \\lambda ( \\zeta _ i ) & i = j \\\\ - G _ \\lambda ( \\zeta _ i , \\zeta _ j ) & i \\not = j . \\end{cases} \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} s ^ i _ 0 : = S ^ i _ 0 \\ , \\ \\ \\zeta ^ i _ k : = S ^ i _ k - S ^ i _ { k - 1 } \\ , \\ \\ \\nu _ k ^ i : = T _ k ^ i - T _ { k - 1 } ^ i \\ , \\ \\ R _ k ^ i = 2 \\nu _ k ^ i \\ , . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} \\min _ { \\rho \\in \\mathcal { R } } J ( \\rho , m ) = \\inf _ { \\alpha \\in \\mathcal { A } } J ( \\alpha , m ) . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty f _ { n , q } z ^ n = 1 - \\frac { 1 } { U _ q ( z ) } \\mbox { w h e r e } U _ q ( z ) = 1 + \\sum _ { n = 1 } ^ \\infty u _ { q , n } z ^ n , \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} Z _ 1 ( t ) : = \\int _ 0 ^ t \\int _ U \\int _ A | f ( s , X ( s ^ - ) , u , a ) - f ( s , Y ( s ^ - ) , u , a ) | \\N _ { \\rho ^ { \\widehat { \\gamma } , Y } } ( d s , d u , d a ) \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} { \\displaystyle \\partial _ { t } u _ { i } ^ { 0 } + \\nabla \\cdot \\left ( - \\mathbb { D } ^ { i } \\nabla u _ { i } ^ { 0 } \\right ) + A _ { i } u _ { i } ^ { 0 } - B _ { i } v _ { i } ^ { 0 } } = \\left ( \\mathbb { F } ^ { i } \\nabla u _ { i } ^ { 0 } \\right ) \\cdot \\nabla ^ { \\delta } \\theta ^ { 0 } + R _ { i } \\left ( u ^ { 0 } \\right ) \\quad \\mbox { i n } \\ ; Q _ { T } , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\Omega ( X ) : = \\{ Q : X \\to 2 ^ X : \\ , Q ( x ) \\neq \\emptyset \\ , \\ , \\forall x \\in X \\ , \\ , \\textnormal { a n d } \\ , \\ , Q \\ , \\ , \\textnormal { i s m o n o t o n e } \\} , \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\int _ X g \\ > d \\omega _ X = \\int _ X T _ h g \\ > d \\omega _ X \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\ , \\varphi _ { \\underline { d } ''' , \\underline { d } } ( V _ { \\underline { d } ''' } ) = & r + 1 - \\ , V _ { \\underline { d } ''' } ^ { X _ { 3 } ^ { c } , 0 } \\\\ = & r + 1 - ( \\ , V _ { \\underline { d } '' } ^ { X _ 1 , 0 } - \\ , V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } ) . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) v \\rVert = \\lambda _ j , \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\norm { A } _ { H S } & = ( \\mathrm { t r } ( A \\cdot A ^ T ) ) ^ { 1 / 2 } \\\\ & = ( \\sum _ { i , j } \\abs { a _ { i j } } ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\operatorname { V o l } ( \\rho ) = \\int _ { M } ^ { } D ^ \\ast ( \\varpi ) - \\sum _ { r = 1 } ^ k \\int _ { T _ r } ^ { } D _ r ^ \\ast ( \\beta _ r ) . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} d \\mu ( z ) = \\frac { d x \\ , d y } { y ^ 2 } . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} M _ g ^ X ( t ) & = g ( X ( t ) ) - g ( X ( 0 ) ) \\\\ & - \\int _ 0 ^ t \\int _ U \\int _ A \\left [ g ( X ( s ) + f ( s , X ( s ) , u , a , m ( s ) ) ) - g ( X ( s ) ) \\right ] \\nu ( d u ) \\rho _ s ( d a ) d s \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} c _ { 2 } \\left ( \\nu \\right ) = \\left ( { \\frac { \\pi } { 2 \\nu } } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} W _ { n , 2 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ - { u \\xi + S _ { n } \\left ( - { u , \\xi } \\right ) + \\delta _ { n , 2 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} { J } _ 2 ( t , \\tau ) : = M A ( t ) ^ { - 1 / 2 } \\sin ( \\tau A ( t ) ^ { 1 / 2 } ) M ^ * \\widehat { P } - M _ 0 ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { 1 / 2 } ) M _ 0 \\widehat { P } . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} d ( t , x ) = \\tau ( e ^ { | x | } ( t , \\infty ) ) , t \\geq 0 . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} p ( z _ 1 , z _ 2 ) = q ( \\phi ^ { - 1 } ( z _ 1 ) , \\phi ^ { - 1 } ( z _ 2 ) ) ( z _ 1 + i ) ^ n ( z _ 2 + i ) ^ m . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( | u _ 1 ( t ) | ^ 2 + | u _ 2 ( t ) | ^ 2 \\big ) \\ d t \\asymp \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\ , . \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\frac { \\lambda _ { i - 1 } ( u ) } { \\lambda _ i ( u ) } = \\frac { P _ i ( u + 1 ) } { P _ i ( u ) } \\textit { f o r a l l } 2 \\leq i \\leq n , \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} m = \\inf _ { x \\in \\Omega _ + } ( g _ { \\varepsilon , \\gamma } ( x ) - u ( x ) ) = \\inf _ { x \\in \\partial \\Omega _ + \\cap \\Omega } ( g _ { \\varepsilon , \\gamma } ( x ) - u ( x ) ) \\ge 0 . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} x ' ( \\theta ) = g ( \\theta ) x ^ 2 ( \\theta ) + \\epsilon f ( \\theta ) x ^ 3 ( \\theta ) , \\ , \\ , \\theta \\in [ 0 , 2 \\pi ] , \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} s ( p ) E ^ { \\le n } = \\{ q \\in E ^ * : s ( q ) = s ( p ) , | q | = n | q | < n r ( q ) \\} . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} u \\ , v \\ ; = \\ ; T _ u v \\ , + \\ , T _ v u \\ , + \\ , R ( u , v ) \\ , , \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\ll \\frac { x ( \\log \\log x ) ^ 2 } { \\log ^ 2 x } \\max _ { \\substack { b \\pmod * { q } \\\\ ( b , q ) = 1 } } \\sum _ { \\chi _ 0 \\neq \\chi _ 1 , \\chi _ 2 \\pmod * { q } } \\frac { \\chi _ 1 ( b ) \\overline \\chi _ 2 ( b ) } { \\varphi ( q ) ^ 2 } \\sum _ { n \\leq \\frac { x } { r ( x ) } } \\frac { d _ { \\chi _ 1 } ( n ) d _ { \\chi _ 2 } ( n ) } { \\varphi ( n ) } . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\frac { d ^ { \\ , 2 } z ( t ) } { d t ^ 2 } + ( a c ) z ( t ) = 0 , z ( t ) = \\alpha ( t ) \\mbox { o r } \\beta ( t ) , \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} f ( \\Psi , \\dots , \\Psi ) = ( \\Psi , \\dots , \\Psi ) . \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} \\sum _ { 2 \\le q _ i \\leq X } \\Lambda ( q _ i ) \\ll X = ( \\log \\log x ) ^ { 1 / 2 } ( \\log \\log \\log x ) ^ 2 \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{gather*} \\mathbb { P } - \\lim _ { N \\rightarrow \\infty } { \\Lambda ^ N _ m \\delta _ { \\lambda ( N ) } } = M _ m \\forall \\ , m = 0 , 1 , 2 , \\dots . \\end{gather*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\frac { 1 } { i } ( P _ n \\varphi ) _ x - P _ n ( u \\varphi ) = \\lambda P _ n \\varphi . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\int _ { \\O ^ \\# } f ^ \\# ( x , y ) \\ , W ( x , y ) d x d y = \\int _ { \\O ^ \\# } \\nabla v ( x , y ) \\cdot \\nabla W ( x , y ) d x d y \\ , . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\psi \\left ( \\xi \\right ) = \\frac { z ^ { 2 } \\left ( { 4 - z ^ { 2 } } \\right ) } { 4 \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 3 } } . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{gather*} \\big | 1 - q ^ { z - \\theta - j } \\big | \\leq \\big | 1 - q ^ { z - j } \\big | , \\forall \\ , j = 0 , 1 , \\dots , M . \\end{gather*}"}
-{"id": "6720.png", "formula": "\\begin{align*} N ( \\star _ { i = 1 } ^ { k + l } G , \\alpha ^ { k + 1 } \\star \\alpha ^ { l + 1 } ) = & N ( \\star _ { i = 1 } ^ { k + l } G , \\alpha ^ { k + l + 1 } ) = N ( \\star _ { i = 1 } ^ { k + l } G , \\star _ { i = 1 } ^ { k + l } \\alpha ^ { 2 } ) \\\\ \\le & N ( \\star _ { i = 1 } ^ { k } G , \\alpha ^ { k + 1 } ) N ( \\star _ { i = 1 } ^ { l } G , \\alpha ^ { l + 1 } ) \\\\ = & N ( \\star _ { i = 1 } ^ { k } G , \\star _ { i = 1 } ^ { k } \\alpha ^ { 2 } ) N ( \\star _ { i = 1 } ^ { l } G , \\star _ { i = 1 } ^ { l } \\alpha ^ { 2 } ) . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\pi ^ \\varrho _ o : G _ \\bullet \\to U ( H ^ \\varrho _ o ) \\ \\ \\ , \\ \\ \\ \\pi ^ \\varrho _ o ( g ) : = \\{ \\bar { v } _ { o a } \\cdot g \\cdot \\bar { v } _ { a o } \\} ^ { \\varrho _ o } \\ . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} C \\| G \\mathbf { h } _ i \\| \\| G \\mathbf { e } _ i \\| = \\frac { C } { \\eta } \\sqrt { \\Im \\mathbf { h } _ i ^ * G \\mathbf { h } _ i } \\sqrt { \\Im G _ { i i } } = \\frac { C } { \\eta } \\sqrt { \\Im \\mathcal { G } _ { i i } } \\sqrt { \\Im G _ { i i } } \\leq C ' \\frac { \\Im ( G _ { i i } + \\mathcal { G } _ { i i } ) } { \\eta } , \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} \\langle { a , \\beta , \\gamma , r , \\sigma , \\tau , \\xi } \\mid [ r , a ] = \\gamma ^ { - 1 } \\beta . { r } \\beta ^ { - 1 } r ^ { - 1 } = r \\gamma ^ { - 1 } r ^ { - 1 } , ~ \\gamma . { a } \\gamma ^ { - 1 } a ^ { - 1 } = \\beta , ~ \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} D ' _ { - s } = \\sum _ { r = 0 } ^ { n - 2 } u _ { s - r } \\sum _ { t = 0 } ^ { n - 2 } u _ { r - t } C ' _ { t } , \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} [ \\ker Q _ t ] ^ \\perp \\supseteq [ \\ker Q _ s ] ^ \\perp \\supseteq [ \\ker B ^ * ] ^ \\perp = [ \\ker B B ^ * ] ^ \\perp , \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} E | \\mu ^ N ( t ) - \\widetilde { \\mu } _ N ( t ) | + \\frac 1 N \\sum _ { i = 2 } ^ N E | X ^ N _ i ( t ) - \\widetilde { X } ^ N _ i ( t ) | \\leq \\frac d N e ^ { 2 K _ 1 T } \\leq \\frac C N . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} H ^ { ( 1 ) } _ { \\zeta } ( x , s , t ) = & - \\lambda H _ { \\zeta } ( x , s , 0 ) ( e _ i \\cdot e _ 2 ) , \\\\ H ^ { ( 2 ) } _ { \\zeta } ( x , s , t ) = & - \\lambda H _ { \\zeta } ( x , 0 , t ) ( e _ i \\cdot e _ 1 ) . \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\ < u ^ n _ t , \\ , \\phi _ t \\ > _ e - & \\ < u ^ n _ 0 , \\ , \\phi _ 0 \\ > _ e - \\int _ 0 ^ t \\ < u ^ n _ s , \\partial _ s \\phi _ s \\ > _ e \\ , d s \\\\ & = ( M ^ e L ^ e ) ^ { - 1 } \\sum _ { z \\in \\Lambda ^ { e } _ n } \\sum _ { w \\sim z } \\int _ 0 ^ t ( \\xi _ { s - } ( w ) - \\xi _ { s - } ( z ) ) \\phi _ s ( z ) \\ , d P ^ { z , w } _ s \\\\ & + ( M ^ e L ^ e ) ^ { - 1 } \\sum _ { z \\in \\Lambda ^ { e } _ n } \\sum _ { w \\sim z } \\int _ 0 ^ t \\xi _ { s - } ( w ) ( 1 - \\xi _ { s - } ( z ) ) \\phi _ s ( z ) \\ , d \\tilde P ^ { z , w } _ s . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} y ' ( r ) = ( A + B B ^ * Q _ { \\infty } ^ { - 1 } ) y ( r ) , r \\in \\ , ] - \\infty , 0 ] \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p } _ { a , \\R } ( \\mathbb { G } ) } : = \\left ( \\int _ { \\mathbb { G } } ( | \\mathcal { R } ^ { \\frac { a } { \\nu } } u ( x ) | ^ { p } + | u ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\Delta ( \\beta , t ) = \\frac { 1 } { t } \\big ( \\beta _ t ^ A ( \\nabla _ t Y _ 1 , \\dots , \\nabla _ t Y _ { n - 1 } ) - \\beta _ 0 ^ A ( \\nabla _ 0 Y _ 1 , \\dots , \\nabla _ 0 Y _ { n - 1 } ) \\big ) ( \\ast ) \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | \\varphi | ^ 2 ~ d x + \\frac { 1 } { 2 \\pi } \\sum _ n \\int _ 0 ^ { \\infty } \\left ( \\frac { \\psi _ n ^ 2 } { 2 } \\right ) ' \\widehat { u \\varphi } \\bar { \\hat \\varphi } ( 1 - \\chi ) ~ d \\xi = 0 . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} E = V ^ 5 - \\sum _ { i = 1 } ^ k w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 5 . \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{gather*} Q = Q ( \\overline { \\partial } \\eta ) . \\end{gather*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty ( | u _ i | ^ p + | u _ i | ^ { p _ i } ) = | | u | | _ p ^ p + \\sum _ { i = 1 } ^ \\infty | u _ i | ^ { p _ i } = 1 \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} a ^ { a / 2 } r \\mathtt { J } _ \\nu ' ( r ) + \\left ( ( a ^ { a / 2 } - 1 ) ( 1 - a + a \\nu ) - a ^ { a / 2 } \\nu + 2 ( 1 - \\beta ) \\right ) \\mathtt { J } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X ( s , M , Y _ 0 ^ s ) d s + B ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} z = - \\left ( x + \\frac 1 { 4 t } y ^ 2 \\right ) , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} | { \\cal M } _ 1 | = \\max _ { m \\in { \\cal M } _ 0 } | \\varphi _ 0 ^ { - 1 } ( m ) \\cap { \\cal T } _ { \\bar { X } } ^ n | , \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\widetilde G _ { M } = \\Big \\{ \\max _ { | v | \\leq M } L _ { \\widetilde A ( v ) \\to E _ N ( w ) } - \\alpha _ 2 v N ^ { 2 / 3 } \\leq S ( s ) \\Big \\} , \\alpha _ 2 = \\frac { 2 } { \\rho ^ { 4 / 3 } ( 1 - \\rho ) ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} d _ { K } { \\left ( x \\right ) } = \\sqrt { \\norm { x } ^ { 2 } - \\sum _ { i \\in I } \\left ( \\max \\left \\{ \\xi _ { i } , 0 \\right \\} \\right ) ^ { 2 } } = \\sqrt { \\sum _ { i \\in I } \\xi _ { i } ^ { 2 } - \\sum _ { i \\in I \\smallsetminus I _ { - } } \\xi _ { i } ^ { 2 } } = \\sqrt { \\sum _ { i \\in I _ { - } } \\xi _ { i } ^ { 2 } } \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} | \\hat { \\mathbb { E } } _ { \\tau + } [ X ] - \\hat { \\mathbb { E } } _ { \\sigma + } [ X ] | & = | \\sum _ { i = 1 } ^ { n } \\hat { \\mathbb { E } } _ { t _ i } [ X ] I _ { \\{ \\tau = t _ i \\} } - \\sum _ { j = 1 } ^ { m } \\hat { \\mathbb { E } } _ { s _ j } [ X ] I _ { \\{ \\sigma = s _ j \\} } | \\\\ & \\leq \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { m } | \\hat { \\mathbb { E } } _ { t _ i } [ X ] - \\hat { \\mathbb { E } } _ { s _ j } [ X ] | I _ { \\{ \\tau = t _ i \\} \\cap \\{ \\sigma = s _ j \\} } . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} x ( t ) & = \\frac c d + \\alpha ( t ) , | \\alpha ( 0 ) | \\ll \\frac c d , \\\\ y ( t ) & = \\frac a b + \\beta ( t ) , | \\beta ( 0 ) | \\ll \\frac a b , \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} R _ { \\alpha } \\big ( ( \\alpha - L _ r ) u \\big ) ( x ) = u ( x ) , \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{gather*} \\dot { I } _ { i j k } = 0 . \\end{gather*}"}
-{"id": "9357.png", "formula": "\\begin{align*} r _ { x } ( y ) : = \\frac { y - x } { | y - x | } \\mbox { f o r e v e r y } y \\in \\R ^ { m } \\setminus \\{ x \\} . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} T F ( z ) = F \\big ( \\Phi ( z ) \\big ) \\cdot \\big ( \\Phi ' ( z ) \\big ) ^ \\frac { 1 } { p } , F \\in H ^ p ( \\Omega _ + ) , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} { \\bar J } _ { \\nu } ( x ) = \\frac { \\sin \\pi \\nu } { \\pi } \\sum \\limits _ { l = - \\infty } ^ { \\infty } ( - 1 ) ^ l \\frac { J _ l ( x ) } { \\nu - l } . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\Delta : = \\Delta ( X ) = \\langle \\{ 1 , 2 \\} , \\{ 2 , 3 \\} , \\{ 3 , 4 \\} , \\{ 4 , 5 \\} , \\{ 1 , 4 \\} , \\{ 2 , 5 \\} \\rangle . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\phi ( w ) = \\sqrt { \\pi } \\frac { \\Gamma ( w - \\tfrac { 1 } { 2 } ) \\zeta ( 2 w - 1 ) } { \\Gamma ( w ) \\zeta ( 2 w ) } . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} u ( 0 ) = u ' ( 0 ) = 0 , \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} g _ 0 ( s ) = C _ 0 u _ 0 ( s ) + C _ 1 s ^ { a _ 1 } u _ 1 ( s ) + C _ 2 s ^ { a _ 2 } u _ 2 ( s ) + C _ 3 s ^ { a _ 3 } u _ 3 ( s ) \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} K _ { H h _ 1 H } & \\circ K _ { H h _ 2 H } ( x H , A ) = \\int _ X K _ { H h _ 2 H } ( w , A ) \\ > K _ { H h _ 1 H } ( x H , d w ) \\\\ & = \\int _ G ( \\delta _ y * \\omega _ H * \\delta _ { h _ 2 } * \\omega _ H ) ( p _ G ^ { - 1 } ( A ) ) \\ > d ( \\delta _ x * \\omega _ H * \\delta _ { h _ 1 } * \\omega _ H ) ( y ) \\\\ & = ( \\delta _ x * \\omega _ H * \\delta _ { h _ 1 } * \\omega _ H * \\delta _ { h _ 2 } * \\omega _ H ) ( p _ G ^ { - 1 } ( A ) ) \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} f ^ { ( k ) } ( X ) & = f ( f ^ { ( k - 1 ) } ( X ) ) \\\\ & = a _ 1 ( b _ 1 X ^ { e _ 1 } + b _ 2 X ^ { e _ 2 } + \\ldots + b _ t ) ^ { d _ 1 } + \\ldots + a _ s . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} ( \\hat d ^ 1 \\circ \\hat d ^ 1 ) \\circ \\hat d ^ 1 - \\hat d ^ 1 \\circ ( \\hat d ^ 1 \\circ \\hat d ^ 1 ) = 0 . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} & ~ \\Phi ( u _ i ( ( L _ i / R ) ^ 3 x ) + \\eta ( R , s , 1 ; x ) ) \\\\ \\leq & ~ \\Phi ( u _ i ) ( R / L _ i ) ^ { 3 N } + C _ 0 R ^ { 3 N - 1 } + \\left ( \\frac { 1 } { 2 } b C _ 3 L ^ { 3 N - 5 } _ i \\int _ { \\mathbb { R } ^ N } | \\nabla u _ i | ^ 2 \\right ) ( R / L _ i ) ^ { 6 N - 1 1 } . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} u \\left ( x , k _ { 0 } \\right ) \\mid _ { \\Gamma _ { b } = } g _ { 0 } \\left ( x _ { 1 } , x _ { 2 } , x _ { 0 } \\right ) , \\partial _ { n } u \\left ( x , k _ { 0 } \\right ) \\mid _ { \\Gamma _ { b } = } g _ { 1 } \\left ( x _ { 1 } , x _ { 2 } , x _ { 0 } \\right ) , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ { n } } \\frac { 1 } { | D _ { n } | } \\frac { 1 } { \\prod _ { k < n } | D _ { k } | } > \\sum _ { m = n + 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { m } } \\frac { | D _ { m } | } { | D _ { m } | } \\frac { 1 } { \\prod _ { k < n } | D _ { k } | } . \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} ( \\zeta _ { n , j } ^ * ) ' ( r ) = \\frac { o \\Big ( \\frac { 1 } { \\lambda _ { n , j } ^ { ( 1 ) } } \\Big ) } { r } + O \\left ( \\frac { { e ^ { - \\lambda _ { n , j } ^ { ( 1 ) } } } } { r ^ 3 } \\right ) \\ \\ \\textrm { f o r a l l } \\ \\ r \\in ( \\Lambda _ { n , j , R } ^ { - } , { \\delta } ) . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} L _ { \\varepsilon } = \\sup _ { 0 < t \\leq T _ 1 } t ^ { \\frac { 1 } { 2 } - \\frac { 3 } { 2 p } } | | b _ { \\varepsilon } | | _ { p } & \\leq \\sup _ { 0 < t \\leq T _ 1 } t ^ { \\frac { 1 } { 2 } - \\frac { 3 } { 2 p } } | | b _ { \\varepsilon } - b | | _ { p } + \\sup _ { 0 < t \\leq T _ 1 } t ^ { \\frac { 1 } { 2 } - \\frac { 3 } { 2 p } } | | b | | _ { p } \\\\ & \\leq 2 \\delta \\textrm { f o r } \\ \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} a ( - 1 ) = b ( - 1 ) = - 1 , a ( 1 ) = b ( 1 ) = 1 . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\sup _ { 0 \\leq t \\leq T } t ^ { \\frac { 1 } { 2 } - \\frac { 3 } { 2 p } } | | b - b _ { \\varepsilon } | | _ { p } ( t ) = 0 \\textrm { f o r } \\ 3 < p < \\infty . \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} U \\ ; = \\ ; U ( N ) \\cdots U ( 1 ) \\ ; . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} s _ { 0 , m } = \\frac { 1 } { 1 2 } \\int _ { - \\Delta t } ^ { \\Delta t } ( \\Delta t - | \\tau | ) \\left ( 3 - 2 \\left ( 1 - \\frac { | \\tau | } { \\Delta t } \\right ) ^ 2 \\right ) \\frac { \\partial ^ 4 u _ 0 } { \\partial t ^ 4 } ( t _ m + \\tau ) d \\tau . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\min \\| L x \\| { \\rm s u b j e c t \\ \\ t o } x \\in \\mathcal { S } _ k = \\{ x \\mid \\| \\widetilde { A } _ k x - b \\| = \\min \\} \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\partial _ a ( E _ { T _ a } ( \\Psi ) ) | _ { a = 0 } = 0 . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} P = u _ 1 ^ d \\cdots u _ i ^ d \\cdot P _ { k , i + 1 } ^ { 2 ^ { q + i } + 2 ^ q - t - a _ { i + 1 } } \\cdot u _ { i + 1 } ^ { a _ { i + 1 } } \\cdot h _ { i + 1 } \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\kappa _ { L } ( \\zeta ) = \\zeta \\kappa _ X ' ( \\zeta ) . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} H ^ 2 _ \\tau ( \\Z _ 2 , Z ( G ) ) \\cong Z ( G ) ^ \\tau / Z ( G ) _ \\tau , \\mbox { w h e r e } Z ( G ) _ \\tau : = \\{ \\tau ( z ) z \\ : z \\in Z ( G ) \\} \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} { \\bf \\Phi } _ { M } : = \\Phi _ 0 \\circ \\ldots \\circ \\Phi _ { 2 M - 1 } \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} B ( z , s - z ) = \\int _ 0 ^ \\infty \\frac { x ^ z } { ( 1 + x ) ^ { s } } \\ ; \\frac { d x } { x } . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\sum _ { i \\ge 0 } | \\frac 1 { A _ i } \\sum _ { j = 0 } ^ i a _ { i - j } u _ j | ^ p \\le C _ p ^ p \\sum _ { i \\ge 0 } | u _ i | ^ p \\ , . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} \\tilde \\beta ( h ) = \\int _ S \\beta ( h ) \\ > d \\mu ( \\beta ) \\quad \\quad ( h \\in D ) \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial t } + L v - \\mu v = - \\psi \\ : \\mbox { i n } \\ : D _ { T + h } , \\frac { \\partial v } { \\partial t } + L _ \\pm v - \\mu v = - \\psi \\ : \\mbox { i n } \\ : D _ { T + h } ^ \\pm , v ( T + h ) = \\phi \\ : \\mbox { i n } \\ : D . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} f \\left ( u , z \\right ) \\sim \\sum \\limits _ { s = 0 } ^ { \\infty } \\frac { f { _ { s } \\left ( z \\right ) } } { u ^ { s } } . \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\partial ^ { j } } { \\partial \\rho ^ { j } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = j ! \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ { j - 1 } ( t ) d t = j ! \\int _ { - 1 } ^ 1 f ( t ) r _ j ( t ) d t = 0 , \\mbox { f o r e a c h } j = 1 , 2 \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} ( 1 5 u ^ 2 - 1 0 ) - 4 \\sqrt { 5 } = ( - 1 ) ^ { b _ 0 } \\epsilon ^ { b _ 1 } 5 ^ { b _ 2 } w ^ b , \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} t = \\tfrac { n + 1 } N \\ c ^ { \\frac 1 { n + 1 } } \\ z ^ { \\frac N { n + 1 } } . \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} [ A ^ { \\epsilon } _ i ( z ) , A ^ { \\epsilon ' } _ j ( w ) ] = 0 , \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\begin{array} { c c l } j _ * ( v ( c ) ) + j ' _ * ( v ' ( c ' ) ) & = & j _ * ( v ( b \\circ j ) ) + j ' _ * ( v ' ( b \\circ j ' ) ) \\\\ & = & ( j _ * \\circ v \\circ j ^ * + j ' _ * \\circ v ' \\circ j '^ * ) ( b ) \\\\ & = & u ( b ) . \\end{array} \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} d + o ( 1 ) & = \\L ( \\omega _ { k } ) = \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) \\int _ { \\mathbb { G } } | \\omega _ { k } ( x ) | ^ { q } d x \\\\ & \\geq \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) \\int _ { \\mathbb { G } } | \\psi _ { k } ( x ) | ^ { q } d x \\\\ & = \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) \\mu _ { k } ^ { - q } \\int _ { \\mathbb { G } } | \\mu _ { k } \\psi _ { k } ( x ) | ^ { q } d x + o ( 1 ) \\\\ & = \\mu _ { k } ^ { - q } \\L ( \\mu _ { k } \\psi _ { k } ) + o ( 1 ) . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} \\Vert \\psi ( \\cdot , t ) \\Vert _ { D ( \\L ^ { \\frac { \\alpha } { 2 } } ) } = \\Vert \\psi ( \\cdot , 0 ) \\Vert _ { D ( \\L ^ { \\frac { \\alpha } { 2 } } ) } \\quad \\forall t > 0 . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\frac { \\Delta _ { \\omega _ { \\epsilon } } \\varphi _ { \\epsilon } } { { \\rm t r } _ { \\omega _ { \\epsilon } } \\omega _ { \\varphi _ { \\epsilon } } } = \\frac { \\sum _ i \\varphi _ { \\epsilon i \\bar { i } } } { \\sum _ i ( 1 + \\varphi _ { \\epsilon i \\bar { i } } ) } \\leq 1 , \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\tanh ( \\mathtt h | D _ x | ) = { \\rm I d } + { \\rm O p } ( r _ { \\mathtt h } ) , r _ { \\mathtt h } ( \\xi ) : = \\ , - \\frac { 2 } { 1 + e ^ { 2 \\mathtt h | \\xi | \\chi ( \\xi ) } } \\in S ^ { - \\infty } , \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\mu _ i ^ { \\prime } = \\frac { \\mu _ i } { \\varepsilon } , \\zeta _ i ^ { \\prime } = \\frac { \\zeta _ i } { \\varepsilon } . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\dfrac { \\partial \\tau _ { i } } { \\partial \\omega _ { i } } = 0 . \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( \\log ( 1 + t ) \\big ) ^ k = \\sum _ { n = k } ^ \\infty S _ 1 ( n , k ) \\frac { t ^ n } { n ! } , \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\pi _ k ^ { f , \\mathrm { i d } } = \\frac { 1 } { k } \\cdot p _ { k - 1 } ^ { * k } , \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} u _ \\tau = u _ { ( I + \\tau \\theta ) \\Omega } . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} G _ t ( X ) : = \\xi + \\int _ 0 ^ t \\int _ U \\int _ A f ( s , X ( s ^ - ) , u , a ) \\N _ \\rho ( d s , d u , d a ) \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ^ { \\gamma , \\sigma } ( z ) = \\frac { \\Gamma ( \\sigma ) } { \\Gamma ( \\gamma ) \\Gamma ( \\sigma - \\gamma ) } \\int _ 0 ^ 1 t ^ { \\gamma - 1 } ( 1 - t ) ^ { \\sigma - \\gamma - 1 } W _ { \\alpha , \\beta } ( z t ) d t , \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } ( h ) = { \\rm e v } _ { + } ! ( { \\rm e v } _ { - } ^ * h ; \\widehat { \\frak S ^ { + \\epsilon } } ( \\alpha _ - , \\alpha _ + ) ) . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\begin{pmatrix} F _ { 1 } \\left ( Z , W \\right ) \\\\ F _ { 2 } \\left ( Z , W \\right ) \\end{pmatrix} = \\begin{pmatrix} Z + \\mbox { O } ( 2 ) & \\mbox { O } ( 2 ) \\\\ \\mbox { O } ( 2 ) & \\mbox { O } ( 2 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{gather*} R ^ a { } _ { n b n } = W ^ a { } _ { n b n } + p _ { n n } \\delta ^ a { } _ b , R ^ a { } _ { n b c } = W ^ a { } _ { n b c } + \\delta ^ a { } _ c p _ { n b } + \\delta ^ a { } _ b p _ { n c } , \\\\ R ^ a { } _ { n 0 b } = - p ^ a { } _ b + \\delta ^ a { } _ b p _ { n 0 } , R ^ a { } _ { n 0 n } = p ^ a { } _ n . \\end{gather*}"}
-{"id": "6725.png", "formula": "\\begin{align*} & N ( \\star _ { i = 1 } ^ { m + k } H , \\alpha ^ { m N + k N + 1 } ) = N ( \\star _ { i = 1 } ^ { m + k } H , ( \\star _ { i = 1 } ^ { m } \\beta ) \\star ( \\star _ { i = m + 1 } ^ { m + k } \\beta ) ) = \\\\ & N ( \\star _ { i = 1 } ^ { m + k } H , ( \\star _ { i = 1 } ^ { m } \\beta ) \\star ( \\star _ { i = 1 } ^ { k } \\beta ) ) \\le N ( \\star _ { i = 1 } ^ { m } H , \\star _ { i = 1 } ^ { m } \\beta ) N ( \\star _ { i = 1 } ^ { k } H , \\star _ { i = 1 } ^ { k } \\beta ) , \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} { \\mathcal I } _ * ( t ) : = I _ * ( t ) - t | t | N _ * S ^ { - 1 / 2 } P . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ n \\frac { a _ i } n \\log \\frac { n } { a _ i } + O \\ ( \\frac { H \\log \\log n } { \\log n } \\ ) , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\theta _ 1 ^ 2 = \\frac { E - \\frac { 1 } { 4 } C } { A + E - \\frac { 1 } { 2 } C } \\approx 0 . 5 3 9 4 . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{gather*} \\psi _ { a b } = \\phi _ { b a } + ( \\phi _ 0 + \\phi _ n ) \\delta _ { a b } + E _ { a i b } \\omega ^ i + F _ a { } ^ { c } { } _ b \\theta _ c , \\end{gather*}"}
-{"id": "4636.png", "formula": "\\begin{align*} R = \\mathbb { O } _ 0 \\otimes _ \\mathbb { O } \\mathbb { O } _ 3 \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} a K _ e + b H + c = 0 ~ , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} 0 = \\left \\| p / q \\right \\| - \\left \\| p / q ' \\right \\| & = \\sum _ { j = 1 } ^ { m } w _ j ( ( p c _ j - q b _ j ) - ( q ' b _ j - p c _ j ) ) \\\\ & = \\sum _ { j = 1 } ^ { m } w _ j ( 2 p c _ j - ( q + q ' ) b _ j ) \\\\ & = 2 p \\sum _ { j = 1 } ^ { m } w _ j c _ j - ( q + q ' ) \\sum _ { j = 1 } ^ { m } w _ j b _ j \\ ; . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\hat { E } _ { s } \\left ( z \\right ) = \\int \\hat { { F } } _ { s } \\left ( z \\right ) f ^ { 1 / 2 } \\left ( z \\right ) d z \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\begin{aligned} p ( z ) & = c \\det \\left ( \\begin{pmatrix} 0 & 0 \\\\ 0 & I - K \\end{pmatrix} U ^ * \\Delta ( z ) U + i \\begin{pmatrix} 2 I & 0 \\\\ 0 & I + K \\end{pmatrix} \\right ) \\\\ & = c \\det ( I - K ) \\det \\left ( \\begin{pmatrix} 0 & 0 \\\\ 0 & I \\end{pmatrix} U ^ * \\Delta ( z ) U + \\begin{pmatrix} 2 i I & 0 \\\\ 0 & A \\end{pmatrix} \\right ) \\end{aligned} \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\boldsymbol { r } ( t ) = \\boldsymbol { h } ( t ) \\otimes \\boldsymbol { s } ( t ) + \\boldsymbol { w } ( t ) , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} ( 1 \\pm \\eta ) \\left ( \\prod _ { \\substack { \\{ k , \\ell \\} \\in \\binom { [ r ] } { 2 } \\\\ \\{ k , \\ell \\} \\neq \\{ 1 , 2 \\} } } p _ { k \\ell } \\right ) \\delta m ^ { r - 2 } \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} A ( v ) = A ^ { \\rm f l a t } ( v ) + \\lambda ( v ) \\mathbf { e } _ \\rho \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} c _ \\chi ( n q _ 2 ) = q _ 2 \\chi _ * ( q _ 0 ) c _ { \\chi _ * } ( n ) c _ { q _ 0 } ( n ) = q _ 2 \\chi _ * ( q _ 0 ) \\tau ( \\chi _ * ) \\mu ( q _ 0 ) \\overline { \\chi _ * ( n ) } \\mu ( \\gcd ( q _ 0 , n ) ) \\varphi ( \\gcd ( q _ 0 , n ) ) . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} & \\sum _ { p = 0 } ^ { r - 2 } ( - 1 ) ^ p \\left [ a _ { k - r + 2 + p } ( n - p ) ( n - p - 1 ) + b _ { k - r + 1 + p } ( n - p ) + c _ { k - r + p } \\right ] e _ p = 0 , \\\\ & r = 2 , 3 , \\ldots , k . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\log I ( n ) = \\sum _ { p \\mid \\phi ( n ) } \\log I _ p ( n ) < \\pi \\sqrt { \\frac 2 3 } \\sum _ { p \\mid \\phi ( n ) } \\sqrt { k _ p } . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} X = \\phi ^ i ( q , p ) \\frac { \\partial } { \\partial q ^ i } + \\phi _ i ( q , p ) \\frac { \\partial } { \\partial p _ i } , \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} u _ n = f _ 1 u _ { n - 1 } + f _ 2 u _ { n - 2 } + \\cdots + f _ n u _ 0 \\mbox { f o r a l l } n > 0 , \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} I _ { 0 } & = X _ { I J } ^ { i } \\dot { x } _ { j } \\\\ I _ { 1 } & = 2 t \\left ( \\frac { 1 } { 2 } \\delta _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - r ^ { - 1 } \\right ) - H ^ { i } \\dot { x } _ { i } . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} c ^ 6 + 1 9 \\ , c ^ 4 + 3 6 \\ , c ^ 3 + 1 9 \\ , c ^ 2 + 1 = 0 . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} f ( z ) = \\sum _ j \\langle f , \\mu _ j \\rangle \\mu _ j ( z ) + \\sum _ { \\mathfrak { a } } \\frac { 1 } { 4 \\pi } \\int _ \\mathbb { R } \\langle f , E _ \\mathfrak { a } ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle E _ \\mathfrak { a } ( a , \\tfrac { 1 } { 2 } + i t ) \\ ; d t , \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\partial _ { t } \\omega = - \\sin y \\partial _ { x } \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\omega = J L \\omega . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} C = ( c _ { i , j } ) \\in M _ n ( \\mathbb Z _ 2 ) \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\bar { A } = T _ { \\rm l } ^ \\top \\hat { A } T _ { \\rm r } , \\bar { B } = T _ { \\rm l } ^ \\top \\hat { B } , \\bar { C } = \\hat { C } T _ { \\rm r } , \\bar { E } = T _ { \\rm l } ^ \\top \\hat { A } T _ { \\rm r } , \\bar { F } ( \\bar { v } ) = T _ { \\rm l } ^ \\top \\hat { F } ( T _ { \\rm r } \\bar { v } ) . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ { 2 } ( t ) d t = \\int _ { - 1 } ^ 1 f ( t ) r _ 3 ( t ) d t = 0 . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} a _ { 1 } L _ { Y } V ^ { , i } + d _ { 1 } V ^ { , i } = 0 . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} o \\left ( \\sqrt { n } \\right ) + O \\left ( n ^ { 3 } p ^ { 2 } \\left ( 1 - p \\right ) ^ { n } \\right ) = o \\left ( \\sqrt { n } \\log n \\right ) . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} & ~ m _ 1 ( \\lambda + 0 i ) = ( I - T _ { \\lambda + 0 i } ) ^ { - 1 } 1 \\\\ = & ~ ( I + R _ { \\lambda + 0 i } ) 1 - ( I + R _ { \\lambda + 0 i } ) \\widetilde { T } _ { \\lambda + 0 i } ( I + R _ { \\lambda + 0 i } ) 1 + O \\left ( \\frac 1 { \\lambda } \\right ) , \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\begin{cases} D _ { a ^ + } ^ { \\alpha , \\beta } x ( t ) = f ( t , x ( t ) ) , & 0 < \\alpha < 1 , \\ , 0 \\leq \\beta \\leq 1 , t > { a } , \\\\ \\lim _ { t \\to { a ^ { + } } } { ( t - a ) } ^ { 1 - \\gamma } x ( t ) = x _ 0 , & \\gamma = \\alpha + \\beta ( 1 - \\alpha ) , \\end{cases} \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} h _ 1 x _ 1 + h _ 2 x _ 2 + \\ldots + h _ { d _ c } x _ { d _ c } = 0 , \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { \\left [ x , y \\right ) } \\left ( t \\right ) = \\left \\{ \\begin{array} { c c } 1 , & x \\leq t < y , \\\\ 0 , & e l s e . \\end{array} \\right . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} P & = 2 ^ { r - 5 } [ ( r - 2 ) ( r - 3 ) + 8 ( r - 2 ) + 8 - ( r - 2 ) ( r - 3 ) ] \\\\ & = 2 ^ { r - 2 } ( r - 1 ) \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} m _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) = \\int _ { \\R } \\frac { \\dd \\mu _ \\alpha ( x ) } { x - \\omega _ \\beta ( E _ - ) } \\ge c _ 1 > 0 \\ , . \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} U = \\sum _ { i = 1 } ^ { N _ u } U _ i = \\sum _ { i = 1 } ^ { N _ u } \\sum _ { j = 1 } ^ { \\mu _ i } C _ { j , i } . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\tau _ { f } ^ { S } = \\min \\left \\{ t \\le N : S ^ X ( t ) > f \\right \\} \\land \\left ( N + 1 \\right ) , \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} W _ { i j } ( t ) = \\frac { R _ { i j } ( t ) } { \\sum _ { i = 1 } ^ { n } R _ { i j } ( t ) } , \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\prod _ { \\pi | 2 a } ( 1 + N ( \\pi ) ^ { - 1 } ) ^ { - 1 } = \\sum _ { ( d ) | ( 2 a ) } \\frac { \\mu _ { [ i ] } ( d ) } { \\sigma ( d ) } , \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\lambda + \\sum _ { i = 1 } ^ { k } \\lambda _ i + ( n - k - 1 ) = \\lambda ^ 3 + \\sum _ { i = 1 } ^ { k } \\lambda _ i ^ 3 + ( n - k - 1 ) = 0 . \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} \\int _ 0 ^ X \\lvert S _ f ( r ) \\rvert ^ 2 d r = c X ^ { k + \\frac { 3 } { 2 } } + O ( X ^ { k + \\epsilon } ) , \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} g \\mapsto F _ g , \\ ; \\ ; \\ ; F _ g ( f ) = \\mathbb E \\Bigl ( \\sum _ { k = 0 } ^ { n } \\langle f _ k , g _ k \\rangle \\Bigr ) \\ ; \\ ; \\ ; \\bigl ( f \\in H ^ { s _ q ^ n } _ p ( X ) , g \\in H ^ { s _ { q ' } ^ n } _ { p ' } ( X ^ * ) \\bigr ) . \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\frac { 1 } { b _ 0 } + \\frac { 1 } { b _ 0 ^ 2 } + \\frac { 1 } { b _ 0 + \\Delta } + \\frac { 1 } { b _ 0 ^ 2 + \\Delta ^ 2 } + \\Delta = 0 . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} f ''' = \\left ( \\left ( \\frac { a ' } { a } \\right ) ^ 2 + 2 n \\left ( \\frac { b ' } { b } \\right ) ^ 2 \\right ) f ' < 0 \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} X ^ N _ i ( t ) = \\xi ^ N _ i + \\int _ 0 ^ t \\int _ U f ( s , X ^ N _ i ( s ^ - ) , u , \\gamma ^ N _ i ( s , X ^ N ( s ^ - ) ) , \\mu ^ N ( s ^ - ) ) \\N ^ N _ i ( d s , d u ) \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} \\frac { C _ { 2 , n } } { ( k - 4 ) ! } & = - [ 2 ( n - 1 ) a _ k + b _ { k - 1 } ] m _ { ( 2 , \\dot { 0 } ) } - [ 2 ( n - 1 ) a _ { k - 1 } + b _ { k - 2 } ] m _ { ( 1 , \\dot { 0 } ) } \\\\ & - 2 a _ k m _ { ( 1 ^ 2 , \\dot { 0 } ) } - \\frac { 2 n ( n - 1 ) } { k ( k - 1 ) } ( 2 k - 3 ) a _ { k - 2 } - \\frac { 2 n } { k - 1 } b _ { k - 3 } , \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} h ^ { 0 , 0 } ( \\hat { X } _ \\Delta ) = h ^ { 3 , 3 } ( \\hat { X } _ \\Delta ) = f ^ { 3 , 0 } ( Y _ \\Delta , w _ \\Delta ) = f ^ { 0 , 3 } ( Y _ \\Delta , w _ \\Delta ) = 1 \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} h = t _ { k } - t _ { k - 1 } = \\frac { b - a } { n } , \\ ; \\ ; \\ ; k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} \\partial _ { w _ { b b } } \\left ( \\partial _ { w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot G \\left ( W , Z \\right ) \\right ) = \\partial ^ { 2 } _ { w _ { b b } w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot G \\left ( W , Z \\right ) + \\partial _ { w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot \\partial _ { w _ { b b } } \\left ( G \\left ( W , Z \\right ) \\right ) , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\Pr ( E _ e \\cap E _ { e ' } ) = O ( \\Pr ( \\overline { Q } ) ) + \\Pr ( E _ e \\cap E _ { e ' } \\cap Q ) = O \\left ( \\frac 1 n \\right ) + \\Pr ( E _ e \\cap E _ { e ' } \\cap Q ) , \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} 0 \\longrightarrow G _ { \\hat { \\Sigma } ( \\Delta ) } = \\mathrm { H o m } _ \\mathbb { Z } ( \\mathrm { P i c } ( \\hat { X } _ \\Delta ) , \\mathbb { C } ^ \\times ) \\longrightarrow ( \\mathbb { C } ^ \\times ) ^ { \\partial \\Delta \\cap M } \\longrightarrow T _ M = M \\otimes \\mathbb { C } ^ \\times \\longrightarrow 0 . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} \\tilde { f } _ { i + 1 / 2 } = f _ { i + 1 / 2 } ^ { } ( q _ { i + 1 / 2 } ^ { L } , q _ { i + 1 / 2 } ^ { R } ) \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\alpha ^ { m ^ i } = f ^ { ( i ) } ( \\alpha ) , \\ : i \\geq 0 . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\aligned ( \\ref { c o m b i n e 1 7 7 5 } ) = & \\left ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\left ( - \\frak h _ { a b } ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } + \\frak h _ { a } ^ { i + 1 i } - \\frak h _ { b } ^ { i + 1 i } \\right ) \\right ) _ { \\alpha ' _ 2 \\alpha _ 1 } \\\\ & \\quad + \\left ( \\left ( - \\frak h _ { a b } ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } + \\frak h _ a ^ { i + 1 i } - \\frak h _ b ^ { i + 1 i } \\right ) \\circ \\hat d _ { 1 } ^ { i } \\right ) _ { \\alpha ' _ 2 \\alpha _ 1 } . \\endaligned \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ m \\prod _ { j = 1 } ^ { k - 1 } { \\big ( x _ i t ^ { N - 1 } - q ^ { j - \\theta } \\big ) } = ( - 1 ) ^ { m ( k - 1 ) } q ^ { m \\left ( { k \\choose 2 } - \\theta ( k - 1 ) \\right ) } \\prod _ { i = 1 } ^ m { \\big ( x _ i q ^ { \\theta N - k + 1 } ; q \\big ) _ { k - 1 } } . \\end{gather*}"}
-{"id": "3507.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N } \\Vert f \\Vert _ { \\scriptstyle L ^ { p ^ { * } _ { i } } ( \\overline { B _ { i } } \\cap \\partial \\Omega ) } \\geq c _ { 1 } \\Vert f \\Vert _ { \\scriptstyle L ^ { q ( \\cdot ) } ( \\partial \\Omega ) } . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} \\partial _ { 0 } M _ { 0 } U = F - M _ { 1 } \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) U - A U \\in H _ { \\nu , 0 } \\left ( \\mathbb { R } , D \\left ( A ^ { * } \\right ) ^ { \\prime } \\right ) \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\beta ( \\delta _ d , \\delta ^ { \\prime } ) : = \\max _ { G \\leq S _ d } \\beta ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) < 1 . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} K _ { \\boldsymbol { W } } ( P ) = P , \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} h _ n = \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { d ^ { i + 1 } } ( ( f ^ { \\mathrm { a n } } ) ^ i ) ^ * ( \\lambda ) \\quad ( n \\geqslant 1 ) . \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\phi _ s ( x ) : = \\phi ^ { t , y } _ s ( x ) : = \\begin{cases} p ^ { n } ( t - s , x , y ) & s \\in [ 0 , t ] , \\\\ 0 & \\hbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} \\max _ { \\chi \\notin \\hat { G } _ d } | \\hat { 1 _ S } ( \\chi ) | \\leq \\binom { n } { 2 } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } = O \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ n } \\ ) \\ ) . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\norm { ( \\omega _ i - \\zeta _ i ) - ( \\omega _ j - \\zeta _ j ) } _ p & = \\norm { ( \\omega _ i - \\omega _ j ) - ( \\zeta _ i - \\zeta _ j ) } _ p \\\\ & \\leq C \\norm { d ( \\omega _ i - \\omega _ j ) } _ q = C \\norm { \\tau _ i - \\tau _ j } _ q . \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\lambda & = \\mu ' = ( \\mu ^ 1 \\bigcup \\cdots \\bigcup \\mu ^ t ) ' \\cr & = \\lambda ^ 1 + \\cdots + \\lambda ^ t \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\psi ( t ) = \\frac { 1 } { \\sqrt { 2 } } \\int _ t ^ \\infty \\frac { d s } { \\sqrt { F ( s ) } } \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\nabla _ { \\phi \\bar { u } } \\nabla _ { \\phi l } \\nabla _ { \\phi j } X ^ i = - \\nabla _ { \\phi l } X ^ k \\cdot R _ { \\phi j } { } ^ i { } _ { k \\bar { u } } - X ^ k \\nabla _ { \\phi l } R _ { \\phi j } { } ^ i { } _ { k \\bar { u } } - \\nabla _ { \\phi j } X ^ p \\cdot R _ { \\phi p } { } ^ i { } _ { l \\bar { u } } + \\nabla _ { \\phi s } X ^ i \\cdot R _ { \\phi l } { } ^ s { } _ { j \\bar { u } } . \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} f _ n ^ * ( x ) = \\frac { \\rho _ n h ( x ) } { \\| \\tilde { u } _ n ^ { ( 1 ) } - \\tilde { u } _ n ^ { ( 2 ) } \\| _ { L ^ { \\infty } ( M ) } } \\left ( e ^ { \\tilde { u } _ n ^ { ( 1 ) } ( x ) } - e ^ { \\tilde { u } _ n ^ { ( 2 ) } ( x ) } \\right ) , \\ ; x \\in M , \\ \\ \\textrm { a n d } \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} ( \\mathcal { F } \\otimes _ { t _ 1 , t _ 2 } ( \\mathcal { G } \\otimes _ { t _ 1 , t _ 2 } \\mathcal { H } ) ) _ { v } = \\sum _ { v _ 1 + v _ 2 + v _ 3 = v } ( \\mathbb { S } _ { 1 2 3 } ) _ * \\Big ( \\mathcal { F } _ { v _ 1 } \\boxtimes \\mathcal { G } _ { v _ 2 } \\boxtimes \\mathcal { H } _ { v _ 3 } \\otimes \\mathcal { L } _ { v _ 2 , v _ 3 } \\otimes \\mathbb { S } _ { 2 3 } ^ * \\mathcal { L } _ { v _ 1 , v _ 2 + v _ 3 } \\Big ) . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} u ^ { T } C u + z ^ { T } ( F + S ) z & = u ^ { T } S u + v ^ { T } ( F + S ) v - 2 u ^ { T } S v \\\\ & = v ^ { T } F v + ( u - v ) ^ { T } S ( u - v ) . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\{ ( x _ 1 , y _ 1 , z _ 1 , w _ 1 , & \\hdots , x _ g , y _ g , z _ g , w _ g , x , y ) \\ , | \\ , x _ i , z _ i \\in \\mathbb { R } - \\{ 0 \\} \\ \\mbox { a n d } \\ y _ i , w _ i \\in \\mathbb { R } \\ \\mbox { f o r a l l } \\ i , \\ \\\\ & x , y \\in \\mathbb { R } , \\ x ^ 2 = 1 , \\ \\mbox { a n d } \\ \\sum _ { i = 1 } ^ g p ( x _ i , y _ i , z _ i , w _ i ) + y ( x + 1 / x ) = 0 \\} \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} & p \\int \\eta ^ 2 u ^ { p - 1 } _ k \\partial _ t u _ k \\ ; d x \\\\ & = - p \\int ( a \\nabla u , \\nabla ( \\eta ^ 2 u _ k ^ { p - 1 } ) ) \\ ; d x + p \\int ( u \\nabla a , \\nabla ( \\eta ^ 2 u _ k ^ { p - 1 } ) ) \\ ; d x \\\\ & = \\widetilde { \\textnormal { ( I ) } } + \\textnormal { ( I I ) } . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} & 0 = \\sum _ { j = 1 } ^ { N } w _ { j } ( b _ { j } q - p c _ { j } ) + \\sum _ { j = N + 1 } ^ { m } w _ { j } ( - b _ { j } q + p c _ { j } ) - \\sum _ { j = 1 } ^ { m } w _ { j } ( - b _ { j } q ' + p c _ { j } ) . \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 5 \\ , U _ j \\ , d y = & 4 \\pi \\alpha _ 3 \\ , \\mu _ i ^ { \\frac { 1 } { 2 } } \\ , \\mu _ j ^ { \\frac { 1 } { 2 } } \\ , G _ \\lambda ( \\zeta _ i , \\zeta _ j ) \\int _ { \\R ^ 3 } U ^ 5 ( z ) \\ , d z + \\mathcal { R } _ { i , j } \\\\ = & 2 a _ 1 \\ , \\mu _ i ^ { \\frac { 1 } { 2 } } \\ , \\mu _ j ^ { \\frac { 1 } { 2 } } \\ , G _ \\lambda ( \\zeta _ i , \\zeta _ j ) + \\mathcal { R } _ { i , j } ^ 2 , \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} K _ z ( \\zeta , \\zeta _ 0 ) = \\frac 1 { \\pi \\mathrm { i } } \\cdot \\frac { z } { ( \\zeta - \\zeta _ 0 ) ^ 2 - z ^ 2 } , \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\bigcup _ { ( \\ell , j , j ' ) \\neq ( 0 , j , j ) } R _ { \\ell j j ' } ^ { ( I I ) } = \\bigcup _ { \\begin{subarray} { c } \\ell \\neq 0 \\\\ | \\sqrt { j } - \\sqrt { j ' } | \\leq C \\langle \\ell \\rangle \\end{subarray} } R _ { \\ell j j ' } ^ { ( I I ) } . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\frac { d } { d t } ( A B C ) = \\frac { d } { d t } ( A ^ 2 B D ^ 2 E ) = 0 . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\psi ^ { \\epsilon } _ { \\alpha _ 2 , \\alpha _ 1 } ( h ) = { \\rm e v } _ { + } ! ( { \\rm e v } _ { - } ^ * h ; \\widehat { \\frak S ^ { + \\epsilon } } ( { \\rm m o r } ; \\alpha _ 1 , \\alpha _ 2 ) ) . \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\left \\| \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) ^ { \\frac { 1 } { p } } \\right \\| _ X \\leq M \\left ( \\sum _ { i = 1 } ^ n \\| x _ i \\| _ X ^ p \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} y _ { 0 } ^ { * } ( x ) = \\frac { y _ a } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x _ 1 } ( x - t ) ^ { \\alpha - 1 } f ( t , y ( t ) ) d t \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} J _ { \\ell } ( \\alpha , z ) & = \\sum _ { h \\in \\mathbb { Z } _ q ^ * } e \\left ( \\frac { a h ^ { \\ell } } { q } \\right ) \\sum _ { \\substack { z ^ { 1 / \\ell } < m \\le 2 z ^ { 1 / \\ell } \\\\ m \\equiv h \\bmod q } } \\Lambda ( m ) e ( m ^ { \\ell } \\beta ) + O ( \\log q \\log z ) \\\\ & = \\frac { 1 } { \\varphi ( q ) } \\sum _ { h \\in \\mathbb { Z } _ q ^ * } e \\left ( \\frac { a h ^ { \\ell } } { q } \\right ) \\sum _ { z ^ { 1 / \\ell } < m \\le 2 z ^ { 1 / \\ell } } e \\left ( m ^ { \\ell } \\beta \\right ) + R _ { \\ell } ( \\alpha , z ) , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\| \\mathcal { K } ' _ m ( x ) - \\mathcal { K } ' _ m ( y ) \\| \\leq \\gamma \\| x - y \\| _ \\infty , \\ ; \\ ; \\ ; x , y \\in ( \\varphi , \\delta ) , \\ ; \\ ; \\ ; \\gamma = C _ { 6 } ( b - a ) \\left ( \\sum _ { i = 1 } ^ \\rho | w _ i | \\right ) . \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} \\frac { 2 } { | \\rho | } \\bigg | \\sum _ { j = 1 } ^ { r } t _ j \\overline \\chi ( a _ j ) \\bigg | \\le 4 \\bigg | \\sum _ { j = 1 } ^ { r } t _ j \\overline \\chi ( a _ j ) \\bigg | \\le 4 r \\max _ { 1 \\le j \\le r } | t _ j | < \\frac { 1 2 } 5 \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\Big | \\int _ \\R h ( x ) \\dd \\mu _ A ( x ) - \\int _ \\R h ( x ) \\dd \\mu _ \\alpha ( x ) \\Big | = \\Big | \\int _ \\R h ' ( x ) ( \\mathcal { F } _ { \\mu _ A } ( x ) - \\mathcal { F } _ { \\mu _ \\alpha } ( x ) ) \\dd x \\Big | \\ , , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} \\langle A ^ { - 1 } \\xi _ { 0 } , \\xi _ { 0 } \\rangle & = \\frac { | x _ 1 | ^ { 2 } } { a | x _ 1 | ^ { 2 } + c _ 1 | x _ { \\sigma ( 1 ) } | ^ { 2 } } + \\frac { | x _ 2 | ^ { 2 } } { a | x _ 2 | ^ { 2 } + c _ 2 | x _ { \\sigma ( 2 ) } | ^ { 2 } } + \\cdots + \\frac { | x _ n | ^ { 2 } } { a | x _ n | ^ { 2 } + c _ n | x _ { \\sigma ( n ) } | ^ { 2 } } \\\\ & \\leq 1 . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\| \\theta \\| _ { \\sup } = \\| ( f ^ { \\mathrm { a n } } ) ^ * ( \\theta ) \\| _ { \\sup } = \\| d \\theta \\| _ { \\sup } = d \\| \\theta \\| _ { \\sup } , \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\sup _ { 0 < t \\leq T _ 1 } \\Big \\{ | | \\Gamma - \\Gamma _ { \\varepsilon } | | _ { \\infty } + t ^ { \\frac { 1 } { 2 } } | | \\nabla ( \\Gamma - \\Gamma _ { \\varepsilon } ) | | _ { \\infty } \\Big \\} = 0 . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} \\bar B = \\left [ \\begin{array} { c c } 0 & 0 \\\\ \\times & 0 \\\\ 0 & \\times \\end{array} \\right ] , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} G _ { r } ( z , v _ { I } , v _ { L } ) & = \\Phi ^ { r } ( T ( z v _ { I } , t v _ { L } ) ) \\vert _ { t = z } \\\\ & = \\frac { 1 - u ^ { 2 ^ { r + 1 } } } { ( 1 - u ^ { 2 ^ { r + 1 } - 1 } ) ( 1 + u ) } T \\Bigg ( \\frac { u ( 1 - u ^ { 2 ^ { r + 1 } - 1 } ) ^ { 2 } } { ( 1 - u ^ { 2 ^ { r + 1 } } ) ^ { 2 } } v _ { I } , \\frac { u ^ { 2 ^ { r + 1 } - 1 } ( 1 - u ) ^ { 2 } } { ( 1 - u ^ { 2 ^ { r + 1 } } ) ^ { 2 } } v _ { L } \\bigg ) , \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\beta ( x , y , z ) = \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} V ( t , x ) = J _ { [ 0 , t ] } ( \\hat u _ { t , x } ) = \\frac 1 2 \\| Q ^ { - 1 / 2 } _ t x \\| _ X ^ 2 , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\limsup _ { p \\to \\infty } \\big \\| F ( z ) - R _ { p , k } ( z ) \\big \\| ^ { 1 / p } \\leq \\frac { \\Phi ( z ) } { \\Phi ( z _ { k + 1 } ) } , z \\in \\widetilde { K } = K \\setminus \\{ z _ 1 , \\ldots , z _ \\mu \\} , \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\begin{aligned} & { \\frac { 8 ( e ^ { - \\lambda _ { n , 1 } ^ { ( 1 ) } } - e ^ { - \\lambda _ { n , 1 } ^ { ( 2 ) } } ) } { h ^ 2 ( q _ 1 ) e ^ { G _ 1 ^ * ( q _ 1 ) } \\pi m } \\Big ( D ( \\mathbf { q } ) + O ( \\delta ^ \\sigma ) \\Big ) } { = O ( \\sum _ { i = 1 } ^ 2 ( \\lambda _ { n , 1 } ^ { ( i ) } ) ^ 2 e ^ { - \\frac { 3 } { 2 } \\lambda _ { n , 1 } ^ { ( i ) } } ) + O ( \\sum _ { i = 1 } ^ 2 ( e ^ { - ( 1 + \\frac { \\sigma } { 2 } ) \\lambda _ { n , 1 } ^ { ( i ) } } ) , } \\end{aligned} \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} \\lim _ { k _ 0 < k \\to \\infty } \\frac { F ^ { \\bf a } _ { k + 1 } } { F ^ { \\bf a } _ k } = \\Psi , \\quad { \\rm w h e r e } F ^ { \\bf a } _ k \\ne 0 \\quad { \\rm f o r } k > k _ 0 , \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} A ( t ) = M ^ * \\widehat { A } ( t ) M . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\textrm { d e t } ^ { \\flat } \\ ( I - z A \\ ) = \\exp \\ ( - \\sum _ { n \\geqslant 1 } \\frac { \\textrm { t r } ^ { \\flat } \\ ( A ^ n \\ ) } { n } z ^ n \\ ) = \\sum _ { n \\geqslant 0 } a _ n z ^ n \\in \\C \\left [ \\left [ z \\right ] \\right ] \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} Q = \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) ( 1 - 2 \\tilde { \\alpha } _ i ( x ) ^ 2 ) | \\nabla u ( x ) | ^ 2 . \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\sum _ { \\substack { m ~ \\chi \\\\ \\chi \\notin X _ m } } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\leq 2 ^ m n ^ { \\delta m + 1 } \\binom { n } { m } ^ { - 1 / 2 } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c c } \\vline & \\vline \\\\ v _ 1 & v _ 2 \\\\ \\vline & \\vline \\end{array} \\right ) , \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} L _ 1 ( T _ 1 , \\dots , T _ N ) & = a _ { 1 , 1 } T _ 1 + \\dots + a _ { 1 , N } T _ N , \\\\ & \\vdots \\\\ L _ r ( T _ 1 , \\dots , T _ N ) & = a _ { r , 1 } T _ 1 + \\dots + a _ { r , N } T _ N . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} f _ 2 ' ( x ) = \\frac 1 \\theta \\cdot \\frac { \\theta + 2 x \\theta ^ 2 } { 1 + x \\theta + x ^ 2 \\theta ^ 2 } = \\frac { 1 + 2 x \\theta } { 1 + x \\theta + x ^ 2 \\theta ^ 2 } . \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} r ^ { - m } \\int _ { B _ { r } ( 0 ) } \\left | D _ { v } u ( z ) \\right | ^ { 2 } \\ , \\mathrm { d } z \\leq C ( m , \\rho ) \\sum _ { i = 0 } ^ { k } \\left ( \\theta ( y _ i , 4 r ) - \\theta ( y _ i , 2 r ) \\right ) , \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} u ^ { r } = 0 , \\partial _ r u ^ { \\theta } - u ^ { \\theta } = 0 , \\partial _ r u ^ { z } = 0 \\textrm { o n } \\ \\{ r = 1 \\} , \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S & = \\sum _ a \\binom { a } { 2 } \\binom { | S | - 1 } { a - 1 } \\sum _ b \\binom { r - | S | } { b } \\\\ & + \\sum _ a \\binom { | S | - 1 } { a - 1 } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | } { b } . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ M { \\frac { ( q ^ { - 1 } ; q ^ { - 1 } ) _ M } { ( q ^ { - 1 } ; q ^ { - 1 } ) _ n ( q ^ { - 1 } ; q ^ { - 1 } ) _ { M - n } } ( - 1 ) ^ n q ^ { - { n \\choose 2 } } z ^ n } = \\big ( z ; q ^ { - 1 } \\big ) _ M . \\end{gather*}"}
-{"id": "5516.png", "formula": "\\begin{align*} P _ { k , i } ( \\mathcal { O } ) : = \\sum _ { \\psi \\in \\mathfrak { S } _ { k - i + 1 } } u _ { \\psi ( i ) } ^ { k - 1 } \\cdots u _ { \\psi ( k ) } ^ { i - 1 } \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} \\begin{aligned} ( a , { \\boldsymbol { v } } ) & \\mapsto ( a - 2 , { \\boldsymbol { v } } ) \\mbox { w i t h p r o b a b i l i t y } \\frac { ( a - 1 ) ^ + } { \\sum _ { i = 1 } ^ \\infty i v _ i + ( a - 1 ) ^ + } , \\\\ ( a , { \\boldsymbol { v } } ) & \\mapsto ( a + k - 2 , { \\boldsymbol { v } } - { \\boldsymbol { e } } _ k ) \\mbox { w i t h p r o b a b i l i t y } \\frac { k v _ k } { \\sum _ { i = 1 } ^ \\infty i v _ i + ( a - 1 ) ^ + } , \\ : k \\in \\mathbb { N } . \\end{aligned} \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\deg A = \\deg P - \\deg D _ { i j } = \\deg g _ { i , e _ i } + \\deg g _ { j , e _ j } + e _ j \\deg u - \\deg D _ { i j } , \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\sum _ { i = 1 } ^ n X _ i I _ { A _ i } ] & = \\hat { \\mathbb { E } } _ { \\tau + } [ \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m \\xi _ j ^ i I _ { B _ j ^ i } I _ { A _ i } ] \\\\ & = \\hat { \\mathbb { E } } _ { \\tau + } [ \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m \\xi _ j ^ i I _ { A _ i \\cap B _ j ^ i } ] \\\\ & = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m \\hat { \\mathbb { E } } _ { \\tau + } [ \\xi _ j ^ i ] I _ { A _ i \\cap B _ j ^ i } . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} ( 1 5 u ^ 2 - 1 0 ) ^ 2 - 8 0 = 5 p ^ b . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} W : = W _ 0 \\circ W _ 1 ^ { ( 0 ) } \\circ W _ 1 ^ { ( 1 ) } \\circ \\ldots \\circ W _ { 2 M - 1 } ^ { ( 0 ) } \\circ W _ { 2 M - 1 } ^ { ( 1 ) } \\ , , \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} Z _ { \\alpha } ( G ) = \\left \\{ z \\in Z ( G ) \\left | \\ \\exists c \\in C ^ 1 ( G , k ^ * ) \\ \\ c ( x ) c ( y ) = \\alpha ( x , y | z ) c ( x y ) \\ \\forall x , y \\in G \\right . \\right \\} \\ . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} u '' ( t ) + \\frac { 1 - 2 \\alpha } { t } u ' ( t ) & = A u ( t ) \\big ( t \\in ( 0 , \\infty ) \\big ) , \\\\ u ( 0 ) & = x , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\pi _ 1 + \\pi _ 2 + \\pi _ 3 = f , \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} P ( x , t ; 1 / 2 ) = \\frac { c _ n t } { ( t ^ 2 + | x | ^ 2 ) ^ { ( n + 1 ) / 2 } } \\ , , \\mbox { w i t h p r o f i l e } \\ F _ { 1 / 2 } ( x ) = \\frac { c _ { n } } { ( 1 + | x | ^ 2 ) ^ { ( n + 1 ) / 2 } } , \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} | B ^ T e _ 1 | ^ 2 = a ^ 2 + d ^ 2 + \\epsilon ^ 2 = 1 , \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\Delta _ { \\omega _ 0 } F = - { \\rm t r } _ { \\omega _ 0 } { \\rm R i c } ( \\omega _ 0 ) + \\gamma + { \\rm t r } _ { \\omega _ 0 } \\widetilde { \\eta } . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\sum _ { i = 1 } ^ { q } \\ln \\left ( 1 - \\frac { 1 } { n ^ { l _ q } } \\right ) \\leq \\ln \\left ( 1 - \\frac { 1 } { n ^ { \\frac { t } { q } } } \\right ) , \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} | \\hat F ' _ 1 ( x _ 1 ) | & \\ll \\sum _ { i = 1 } ^ { N _ 1 } | \\hat F _ 1 ^ { ( i ) } ( y _ 1 ) | \\cdot | x _ 1 - y _ 1 | ^ { i - 1 } \\\\ [ 0 e x ] & \\stackrel { \\eqref { e q 8 7 } } { \\ll } ~ \\sum _ { i = 1 } ^ { N _ 1 } H ^ { \\ell _ { 1 , i } \\delta } \\cdot | \\sigma _ 1 ( F , t _ 1 ) | ^ { i - 1 } \\\\ [ 0 e x ] & \\stackrel { \\eqref { h j } } { \\ll } ~ \\sum _ { i = 1 } ^ { N _ 1 } H ^ { \\ell _ { 1 , i } \\delta } \\cdot H ^ { ( \\ell _ { 1 , 1 } - \\ell _ { 1 , i } + 1 ) \\delta } \\\\ [ 0 e x ] & \\ll ~ H ^ { ( \\ell _ { 1 , 1 } + 1 ) \\delta } \\ , . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} \\mathbb E \\langle M ^ q _ t , M ^ a _ t \\rangle = \\sum _ { i = 1 } ^ d \\mathbb E M ^ { q , i } _ t \\langle x _ i , M ^ a _ t \\rangle = \\sum _ { i = 1 } ^ d \\mathbb E M ^ { q , i } _ 0 \\langle x _ i , M ^ a _ 0 \\rangle = 0 . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 2 } { m } \\frac { \\lambda _ { n , 1 } ^ { ( 1 ) } e ^ { - \\lambda _ { n , 1 } ^ { ( 1 ) } } } { h ^ 2 ( x _ { n , 1 } ^ { ( 1 ) } ) e ^ { G _ 1 ^ * ( x _ { n , 1 } ^ { ( 1 ) } ) } } \\ell ( \\mathbf { q } ) + O ( e ^ { - \\lambda _ { n , 1 } ^ { ( 1 ) } } ) = \\frac { 2 } { m } \\frac { \\lambda _ { n , 1 } ^ { ( 2 ) } e ^ { - \\lambda _ { n , 1 } ^ { ( 2 ) } } } { h ^ 2 ( x _ { n , 1 } ^ { ( 2 ) } ) e ^ { G _ 1 ^ * ( x _ { n , 1 } ^ { ( 2 ) } ) } } \\ell ( \\mathbf { q } ) + O ( e ^ { - \\lambda _ { n , 1 } ^ { ( 2 ) } } ) , \\end{aligned} \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\frac { 2 } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } + 1 } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty \\mathcal { E } _ { n , \\lambda } ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} S _ { \\mu _ { \\beta } } \\Phi ( \\varphi ) : = \\frac { \\langle \\ ! \\langle \\Phi , e ^ { \\langle \\cdot , \\varphi \\rangle } \\rangle \\ ! \\rangle _ { \\mu _ { \\beta } } } { \\mathbb { E } \\big ( e ^ { \\langle \\cdot , \\varphi \\rangle } \\big ) } = \\frac { 1 } { E _ { \\beta } ( \\frac { 1 } { 2 } \\langle \\varphi , \\varphi \\rangle ) } \\langle \\ ! \\langle \\Phi , e ^ { \\langle \\cdot , \\varphi \\rangle } \\rangle \\ ! \\rangle _ { \\mu _ { \\beta } } . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} & \\{ ( x _ 1 , y _ 1 , z _ 1 , w _ 1 , \\hdots , x _ { g + 1 } , y _ { g + 1 } , z _ { g + 1 } , w _ { g + 1 } ) \\ , | \\ , x _ i , z _ i \\in \\mathbb { R } - \\{ 0 \\} \\ \\mbox { a n d } \\ y _ i , w _ i \\in \\mathbb { R } \\ \\forall \\ i , \\\\ & x _ { g + 1 } ^ 2 = 1 , \\ \\mbox { a n d } \\ \\sum _ { i = 1 } ^ g p ( x _ i , y _ i , z _ i , w _ i ) + x _ { g + 1 } y _ { g + 1 } ( z _ { g + 1 } ^ 2 + 1 ) = 0 \\} \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} l ( \\langle \\rangle ) & = \\langle \\forall _ { \\vec v } \\forall _ x \\exists _ y ( \\forall _ { z \\in y } ( z \\in x \\land \\theta ( x , z , \\vec v ) ) \\land \\forall _ { z \\in x } ( \\theta ( x , z , \\vec v ) \\rightarrow z \\in y ) ) \\rangle , \\\\ o ( \\langle \\rangle ) & = \\Omega + \\omega \\cdot ( k + 1 ) , \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} v = ( c _ 1 , c _ 2 , c _ 3 ) \\begin{pmatrix} B _ 1 \\\\ B _ 2 \\\\ B _ 3 \\end{pmatrix} , c _ 1 , c _ 2 , c _ 3 \\in \\bold Z [ \\rho ] . \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\gamma _ w = \\prod _ { i = 1 } ^ n e ^ { \\frac 1 2 a _ i ^ 2 \\psi _ i } \\prod _ { ( h , h ' ) \\in E } \\frac { 1 - e ^ { - \\frac 1 2 w ( h ) w ( h ' ) ( \\psi _ h + \\psi _ { h ' } ) } } { \\psi _ h + \\psi _ { h ' } } , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} S _ { T ^ { \\ast } N } = \\left \\{ w \\in T ^ { \\ast } N : T ^ { \\ast } \\pi ( w ) = d F \\left ( z \\right ) \\right \\} \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\Omega _ d ( x + 1 ) = \\sum _ { m = 0 } ^ d { d \\choose m } ( d - m + 1 ) ^ d x ^ m . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } \\overline { F } _ { \\xi _ { ( n ) } } ( x ) \\mathbb { P } ( \\eta = n ) \\\\ & = \\Biggl ( \\sum \\limits _ { n = 1 } ^ { K } + \\sum \\limits _ { n = K + 1 } ^ { \\infty } \\Biggr ) \\mathbb { P } \\Biggl ( \\bigcup \\limits _ { k = 1 } ^ { n } \\{ \\xi _ k > x \\} \\Biggr ) \\mathbb { P } ( \\eta = n ) . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } X _ n = X \\ \\textrm { a n d } \\ \\lim \\limits _ { n \\to \\infty } p _ n = p . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} \\tilde { Q } ^ { ( 3 ) } _ p ( 0 , 0 , s ) \\equiv \\epsilon \\thinspace { p ^ 4 \\over s ^ 5 } \\ , , \\epsilon = 0 . 0 0 0 \\thinspace 8 7 2 \\thinspace 0 7 3 \\thinspace 2 \\ , , \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} ( N _ r V ) _ s & = \\sum _ t \\left [ \\binom { s } { 2 } \\binom { r - 2 } { r - t } + \\binom { r - s } { 2 } \\binom { r - 2 } { t } \\right ] \\left [ \\binom { t } { 2 } - \\binom { r - t } { 2 } \\right ] \\\\ & = \\binom { s } { 2 } P + \\binom { r - s } { 2 } Q , \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\min _ { 1 \\le j \\le m } | F ' _ j ( x _ j ) | < H ^ { \\frac 1 2 } \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\lambda ^ { \\frac { 2 } { r } } \\| u \\| _ { \\Phi } ^ { 2 } + \\sum _ { j = 1 } ^ { N } \\| X _ { j } ^ { \\Omega } u \\| _ { \\Phi } ^ { 2 } \\leq C \\left ( \\langle P ^ { \\Omega } u , u \\rangle _ { \\Phi } + \\lambda ^ { \\alpha } \\| u \\| _ { \\Phi , \\Omega \\setminus \\Omega _ { 1 } } ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\Delta = t ^ 4 P ( t ^ 2 ) ^ 2 , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} f ^ { 3 , 0 } ( Y , w ) & = f ^ { 0 , 3 } ( Y , w ) = 1 , \\\\ f ^ { 1 , 1 } ( Y , w ) & = f ^ { 2 , 2 } ( Y , 2 ) = k ( Y , w ) , \\\\ f ^ { 2 , 1 } ( Y , w ) & = f ^ { 1 , 2 } ( Y , w ) = p h ( Y , w ) - 2 + h ^ { 2 , 1 } ( Z ) , \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\partial ^ 2 R } { \\partial z ^ 2 } - ( p + h D _ x ) ^ 2 R = 0 , \\ - D ( x ) \\leq z \\leq 0 \\cr \\ & R | _ { z = 0 } = 1 \\cr & \\partial _ z R + i h \\langle \\nabla D ( x ) , ( p + h D _ x ) R \\rangle | _ { z = - H ( x ) } = 0 \\cr \\end{aligned} \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\mu ( x ) = \\mu ( x ) \\chi _ { [ 0 , \\tau ( r ) ) } + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } = \\mu ( x _ 0 ) + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } , \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { k , k ' , u ' = 1 \\atop k ' \\neq u ' } ^ { q } \\left ( \\overline { D ^ { i j } _ { k k k ' u ' } } \\left ( \\Re w _ { k k } + \\sqrt { - 1 } \\left < Z _ { k } , Z _ { k } \\right > \\right ) \\left < Z _ { u ' } , Z _ { k ' } \\right > \\right ) . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} \\ , K _ { i \\tilde { l } } = \\ , K _ { i - 1 , \\tilde { l } } = \\ , K _ { b _ { 0 } , \\tilde { l } } = r + 1 \\ , \\ , \\ , l _ { 0 } < \\tilde { l } \\leq b ' _ { r } \\ , \\ , \\ , \\ , i + \\tilde { l } \\leq d . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} x ( t ) = \\frac { x _ 0 } { \\Gamma ( \\gamma ) } t ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } f ( s , x ( s ) ) d s , t \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\prod _ { i = 1 } ^ m { \\frac { P _ { \\lambda ( N ) } \\big ( x _ i , t ^ { - 1 } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , \\dots , t ^ { 1 - N } \\big ) } ( x _ i q ; q ) _ { \\theta N - 1 } } } = \\prod _ { i = 1 } ^ m { \\Phi ^ { \\nu } ( x _ i ; q , t ) ( x _ i q ; q ) _ { \\infty } } . \\end{gather*}"}
-{"id": "4613.png", "formula": "\\begin{align*} ( \\mathcal { L } _ 1 , \\mathcal { L } _ 2 , \\mathcal { L } _ 3 , \\phi ) \\otimes \\mathcal { L } = ( \\mathcal { L } _ 1 \\otimes \\mathcal { L } , \\mathcal { L } _ 2 \\otimes \\mathcal { L } , \\mathcal { L } _ 3 \\otimes f _ 3 ^ * \\mathcal { L } , \\phi \\otimes \\mathrm { i d } ) , \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{gather*} D ^ { k + 1 } = D ^ k + \\big [ D ^ k , D \\big ] . \\end{gather*}"}
-{"id": "9318.png", "formula": "\\begin{align*} c ^ * _ - = \\inf _ { \\mu > 0 } \\Phi ^ - ( \\mu ) = \\inf _ { \\mu > 0 } \\frac { \\ln \\lambda ( - \\mu ) } { \\mu } = \\inf _ { \\mu > 0 } \\frac { a - 1 + \\alpha e ^ { - \\mu } + \\beta e ^ { \\mu } } { \\mu } , \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\delta ( x ^ { - 1 } , s ^ { - 1 } ) = ( t ^ { - 1 } , y ^ { - 1 } ) \\mbox { i f $ \\lambda ( s , x ) = ( y , t ) $ } , \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\frac { t ^ \\frac { p - 1 } { 2 } \\phi ( t ) } { \\sqrt { F ( t ) } } = 0 , \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} { J } _ 1 ( t , \\tau ) = M \\bigl ( \\cos ( \\tau A ( t ) ^ { 1 / 2 } ) P - \\cos ( \\tau ( t ^ 2 S ) ^ { 1 / 2 } P ) P \\bigr ) M ^ { - 1 } \\widehat { P } . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} T N = ( T N \\cap J ( T N ) ) \\oplus ( T N \\cap J ( \\nu N ) ) \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} A = \\bigcup _ { i = 1 } ^ { \\infty } \\left [ 4 ^ { i } , 4 ^ { i } + i - 1 \\right ] . \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} A _ g = \\langle e _ 1 \\rangle \\oplus \\langle e _ { \\tau + 4 } , e _ { t + \\tau + 4 } , . . . , e _ { ( r - 1 ) t + \\tau + 4 } \\rangle \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} | | u - e ^ { t A } u _ 0 | | _ { \\tilde { L } ^ { 3 } } & \\leq \\int _ { 0 } ^ { t } | | e ^ { ( t - s ) A } \\mathbb { P } u \\cdot \\nabla u | | _ { \\tilde { L } ^ { 3 } } \\dd s \\\\ & \\leq C K ^ { 2 } _ { 1 } \\int _ { 0 } ^ { t } \\frac { \\dd s } { ( t - s ) ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { q } - \\frac { 1 } { 3 } ) } s ^ { \\frac { 3 } { 2 } ( 1 - \\frac { 1 } { q } ) } } = C ' K _ 1 ^ { 2 } . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ j ( U ) } ~ = ~ \\sup _ { U } \\ , | u | _ { C ^ j } \\ , , \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} T _ 1 f & = \\sum _ i f ( x _ i ) \\sigma ( x , x ^ { - 1 } x _ i ) \\psi _ i \\# \\phi \\\\ T _ 2 f & = \\sum _ i \\lambda _ i ( f ) \\ell _ { x _ i } ^ \\sigma \\phi \\\\ T _ 3 f & = \\sum _ i c _ i f ( x _ i ) \\ell _ { x _ i } ^ \\sigma \\phi \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\left ( \\ln \\omega \\right ) _ { , t } \\left ( 2 \\psi _ { Y } t + d _ { 2 } \\right ) = d _ { 1 } \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } r \\left ( \\frac { \\partial ^ { 2 } u } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial u } { \\partial r } \\right ) d r = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } \\frac { \\partial \\left ( r \\frac { \\partial u } { \\partial r } \\right ) } { \\partial r } d r , \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\sum _ { n \\ge 1 } \\frac { a ( t n ^ 2 ) } { n ^ s } = \\dfrac { 1 } { L ( s - k + 1 , \\chi _ { t , N } ) } L ( s , \\mathrm { S h } _ t ( f ) ) , \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\chi ^ - _ 1 + \\cdots + \\chi ^ - _ { k _ 1 } \\leq \\sum _ { i = 1 } ^ { k _ 1 } \\int \\log r ^ - _ i d \\mu \\ \\textrm { a n d } \\ \\chi ^ + _ 1 + \\cdots + \\chi ^ + _ { k _ 2 } \\geq \\sum _ { i = 1 } ^ { k _ 2 } \\int \\log r ^ + _ i d \\mu . \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} w = \\Big [ \\mu _ { i _ { 0 1 } } , \\ldots , \\mu _ { i _ { 0 p _ 0 } } , [ \\ldots [ \\mu _ { i _ { n 1 } } , \\ldots , \\mu _ { i _ { n p _ n } } ] \\ldots \\Big ] , \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} \\frac { d g _ { \\phi k \\bar { l } } } { d t } = - R _ { \\phi k \\bar { l } } + \\gamma g _ { \\phi k \\bar { l } } + \\widetilde { \\eta } _ { k \\bar { l } } + \\nabla _ { \\phi k } X _ { \\bar { l } } . \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} ( P _ { \\Pi _ 1 } , P _ { \\Pi _ 2 } , \\ldots ) \\stackrel { ( d ) } { = } ( P _ 1 , P _ 2 , \\ldots ) \\mbox { f o r $ \\Pi $ a $ P $ - b i a s e d p e r m u t a t i o n o f $ \\mathbb { N } _ { + } $ } . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} w ( T ( 4 , 5 , 1 , 2 ) ) & = [ 3 ] \\sqcup \\mathcal { W } ( 1 , 2 ) \\sqcup \\mathcal { W } ( 2 , 1 ) = [ 3 ] \\sqcup [ 1 ] \\sqcup \\mathcal { W } ( 1 , 1 ) \\sqcup [ 1 ] \\sqcup \\mathcal { W } ( 1 , 1 ) \\\\ & = [ 3 , 1 , 1 , 1 , 1 ] \\sqcup \\mathcal { W } ( 1 , 0 ) \\sqcup \\mathcal { W } ( 1 , 0 ) = [ 3 , 1 , 1 , 1 , 1 ] . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\dot { \\tilde { v } } ( t ) = \\tilde { F } ( \\tilde { v } ( t ) ) \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} 4 \\cdot \\chi = \\begin{pmatrix} 2 0 & 4 0 & 1 0 0 & 4 8 \\\\ 6 1 & 1 2 0 & 1 4 0 & 5 1 \\\\ 8 1 & 1 6 0 & 1 8 0 & 8 3 \\\\ 4 6 & 5 2 & 8 4 & 4 4 \\end{pmatrix} , \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} 1 = v \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 1 / 2 } \\left [ \\psi _ { s , F } ( - v ) - \\psi _ { s , F } ( v ) \\right ] \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { m } , ( 1 , 2 ) ^ C ; k ) = U ( m _ 1 ; k ) \\ , \\ , \\ , \\ , \\ , \\ , k \\Delta ( \\mathbf { m } , ( 1 , 2 ) ^ C ; k ) = C ( \\mathbf { m } , \\mathcal { O } ; k ) ) , \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} \\widehat { q } ( m , n ) \\stackrel { \\theta } { = } \\frac { ( m ) _ { n - m } ( \\theta ) _ m } { ( 1 + \\theta ) _ { n } } \\stackrel { 1 } { = } \\frac { m } { n ( n + 1 ) } \\mbox { f o r } ~ m \\le n , \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} \\mathbb { P } ( \\widehat { Q } _ 0 < \\widehat { Q } _ 1 < \\cdots ) \\stackrel { 1 } { = } \\prod _ { j = 1 } ^ \\infty \\mathbb { P } ( G _ j \\le 1 ) \\stackrel { 1 } { = } \\prod _ { j = 1 } ^ \\infty \\frac { j ( j + 2 ) } { ( j + 1 ) ^ 2 } = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\frac { \\ 1 g _ { i j } } { \\ 1 t } = - 2 R _ { i j } \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} g _ { 3 , V } ( U ) : = f _ { V } ( U ) + h _ { 2 , V } ( U ) , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} d X ^ 0 = Y ^ 0 d t , d Y ^ 0 = - ( c _ 0 Y ^ 0 + k X ^ 0 ) d t + d W \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t c _ 1 - d _ 1 \\Delta c _ 1 = - c _ 1 c _ 2 - c _ 1 c _ 4 + 2 c _ 3 , & x \\in \\Omega , t > 0 , \\\\ \\partial _ t c _ 2 - d _ 2 \\Delta c _ 2 = - c _ 1 c _ 2 + c _ 3 , & x \\in \\Omega , t > 0 , \\\\ \\partial _ t c _ 3 - d _ 3 \\Delta c _ 3 = c _ 1 c _ 2 + c _ 1 c _ 4 - 2 c _ 3 , & x \\in \\Omega , t > 0 , \\\\ \\partial _ t c _ 4 - d _ 4 \\Delta c _ 4 = - c _ 1 c _ 4 + c _ 3 , & x \\in \\Omega , t > 0 , \\end{cases} \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} H = \\Lambda _ 1 \\partial ( p _ N ) - \\partial ( \\Lambda _ 1 ) p _ N . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} 1 \\ ! \\ ! 1 _ { [ 0 , t ) } : = ( 1 \\ ! \\ ! 1 _ { [ 0 , t ) } \\otimes e _ { 1 } , \\ldots , 1 \\ ! \\ ! 1 _ { [ 0 , t ) } \\otimes e _ { d } ) \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} \\aligned \\sum _ { k _ 1 + k _ 2 = k + 1 } & \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\sum _ { i = 1 } ^ { k - k _ 2 + 1 } \\\\ & ( - 1 ) ^ * { \\frak m } _ { k _ 1 , \\beta _ 1 } ( h _ 1 , \\ldots , { \\frak m } _ { k _ 2 , \\beta _ 2 } ( h _ i , \\ldots , h _ { i + k _ 2 - 1 } ) , \\ldots , h _ { k } ) = 0 \\endaligned \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} X \\odot Y ' = L ^ { p _ 1 , 1 } \\odot L ^ { p ' , 1 } = L ^ { p _ 2 , 1 / 2 } , \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } U _ i ^ 5 \\ , U _ j = & 2 \\ , a _ 1 \\ , \\mu _ i ^ { \\frac { 1 } { 2 } } \\mu _ j ^ { \\frac { 1 } { 2 } } G _ \\lambda ( \\zeta _ i , \\zeta _ j ) - 2 \\ , a _ 3 \\ , \\mu _ i ^ { \\frac { 3 } { 2 } } \\ , \\mu _ j ^ { \\frac { 1 } { 2 } } g _ \\lambda ( \\zeta _ i ) \\ , G _ \\lambda ( \\zeta _ i , \\zeta _ j ) \\\\ & + \\mathcal { R } _ { i , j } ^ 1 + \\mathcal { R } _ { i , j } ^ 2 + 5 \\ , \\mathcal { R } _ { i , j } ^ 3 , \\mathcal { R } _ { i , j } ^ { 5 , 1 } . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} \\mathrm { d i s t } ( G _ m , G _ { m ' } ) = \\inf \\limits _ { x \\in G _ m , x ' \\in G _ { m ' } } { | x - x ' | } \\geq c _ { \\star } e ^ { - 8 \\beta ( \\alpha ) | m ' | } , \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} & - 2 a _ 1 ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 ( D _ \\zeta \\sigma _ 1 ) + 2 ( D _ { \\zeta } \\sigma _ 1 ) \\ , \\mathcal P o l y _ 4 ( 0 , \\zeta ^ 0 , \\bar \\Lambda _ 0 ) \\\\ & = - a _ 1 ( \\bar \\Lambda _ 1 ^ 0 ) ^ 2 ( D _ \\zeta \\sigma _ 1 ) \\\\ & = - a _ 1 ( \\bar \\Lambda _ 1 ^ 0 ) ^ 2 D ^ 2 _ { \\zeta \\zeta } \\sigma _ 1 ( 0 , \\zeta ^ 0 ) ( \\zeta - \\zeta ^ 0 ) - a _ 1 ( \\bar \\Lambda _ 1 ^ 0 ) ^ 2 \\bigl ( ( D _ \\zeta \\sigma _ 1 ) - D ^ 2 _ { \\zeta \\zeta } \\sigma _ 1 ( 0 , \\zeta ^ 0 ) ( \\zeta - \\zeta ^ 0 ) \\bigr ) . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} y ^ 2 ( y - 3 ) - ( \\theta + 2 ) ( \\theta - 1 ) y ^ 2 - ( 2 \\theta + 1 ) ( 1 - \\theta - q ) y - ( 1 - \\theta - q ) ^ 2 = 0 . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\psi & \\ll N ( d ) ^ { ( n - 2 ) ( 3 / 2 - \\Re ( s ) + \\varepsilon ) / 4 } N ( b ) ^ { 3 n / 4 - 1 - n \\Re ( s ) / 2 + 2 \\varepsilon } ( 1 + | s | ^ 2 ) ^ { \\sum _ { \\infty } ( k ) \\cdot ( n / 2 - 1 / 2 ) ( 3 / 2 - \\Re ( s ) + \\varepsilon ) } , \\\\ _ { s = 1 + 1 / n } \\psi & \\ll N ( d ) ^ { ( n / 4 - 1 / 2 ) ( 1 / 2 - 1 / n + \\varepsilon ) } N ( b ) ^ { n / 4 - 3 / 2 + \\varepsilon } . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} L _ { ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) \\to ( \\gamma ^ 2 n , n ) } \\simeq \\left ( \\sqrt { n } + \\sqrt { \\gamma ^ 2 n - \\xi _ 0 n ^ { 2 / 3 } } \\right ) ^ 2 = ( 1 + \\gamma ) ^ 2 n - \\tfrac { 1 + \\gamma } { \\gamma } \\xi _ 0 n ^ { 2 / 3 } - \\frac { \\xi _ 0 ^ 2 } { 4 \\gamma ^ 3 } n ^ { 1 / 3 } + O ( 1 ) . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\omega _ { t } + u \\omega _ { x } + v \\omega _ { y } - \\nu \\bigtriangleup \\omega = 0 , \\ \\ \\omega = v _ { x } - u _ { y } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\l ( { \\rm L } _ 2 ( p ^ k ) ) = 1 + \\l ( { \\rm L } _ 2 ( p ^ { k / s } ) ) = \\O ( k ) + \\l ( { \\rm L } _ 2 ( p ) ) , \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} & \\xi _ 1 = \\xi _ 2 = \\cdots = \\xi _ { r _ 1 } = a _ 1 \\\\ & \\xi _ { r _ 1 + 1 } = \\xi _ { r _ 1 + 2 } = \\cdots = \\xi _ { r _ 1 + r _ 2 } = a _ 2 \\\\ & \\xi _ { r _ 1 + r _ 2 + 1 } = \\xi _ { r _ 1 + r _ 2 + 2 } = \\cdots = \\xi _ { r _ 1 + r _ 2 + r _ 3 } = a _ 3 \\\\ & \\mbox { a n d s o o n . } \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} \\cos ( \\tau \\mathcal { A } _ \\varepsilon ^ { 1 / 2 } ) = T _ { \\varepsilon } ^ * \\cos ( \\varepsilon ^ { - 1 } \\tau \\mathcal { A } ^ { 1 / 2 } ) T _ { \\varepsilon } , \\mathcal { A } _ \\varepsilon ^ { - 1 / 2 } \\sin ( \\tau \\mathcal { A } _ \\varepsilon ^ { 1 / 2 } ) = \\varepsilon \\ , T _ { \\varepsilon } ^ * \\mathcal { A } ^ { - 1 / 2 } \\sin ( \\varepsilon ^ { - 1 } \\tau \\mathcal { A } ^ { 1 / 2 } ) T _ { \\varepsilon } . \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} x = \\pi ^ { - 1 } \\ ( \\pi ( i _ 1 ) + \\cdots + \\pi ( i _ d ) \\ ) . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} L _ s - \\lambda = ( - \\partial - \\phi _ s ) ( \\partial - \\phi _ s ) , \\mbox { i n } K ( \\Gamma _ s ) [ \\partial ] , \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{gather*} \\sum _ { n \\ge 1 } \\frac { \\max _ { 1 \\le i \\le n } V _ n ^ q } { n ^ p } \\le \\sum _ { r \\ge 0 } \\sum _ { n = 2 ^ r } ^ { 2 ^ { r + 1 } - 1 } \\frac { \\max _ { 1 \\le i \\le n } V _ n ^ q } { n ^ p } \\le \\sum _ { r \\ge 0 } \\frac 1 { 2 ^ { ( p - 1 ) r } } ( \\sum _ { k = 0 } ^ r V _ { 2 ^ k } ) ^ q \\end{gather*}"}
-{"id": "519.png", "formula": "\\begin{align*} ( k e r F _ { \\ast } ) ^ { \\perp } = s p a n \\{ & H _ { 1 } = \\frac { \\partial } { \\partial x ^ { 1 } } + \\frac { \\partial } { \\partial x ^ { 2 } } , \\ H _ { 2 } = \\frac { \\partial } { \\partial y ^ { 1 } } + \\frac { \\partial } { \\partial y ^ { 2 } } , \\ H _ { 3 } = \\sin \\alpha \\frac { \\partial } { \\partial x ^ { 3 } } - \\cos \\alpha \\frac { \\partial } { \\partial x ^ { 4 } } , \\\\ & H _ { 4 } = \\frac { \\partial } { \\partial y ^ { 4 } } , \\ H _ { 5 } = \\frac { \\partial } { \\partial z } = \\xi \\} . \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{gather*} \\underline \\vartheta ( p , g ) = { } ^ t \\big ( \\underline \\omega ^ 0 , \\dots , \\underline \\theta _ { n - 1 } \\big ) : = g ^ { - 1 } \\cdot \\vartheta _ p , \\end{gather*}"}
-{"id": "4736.png", "formula": "\\begin{align*} U ^ { \\nu } \\cdot \\nabla \\omega ^ { \\nu } & = \\left ( U _ { s 1 } ^ { \\nu } + U _ { n 1 } ^ { \\nu } \\right ) \\cdot \\nabla \\left ( \\omega _ { n 2 } ^ { \\nu } - \\psi _ { n 2 } ^ { \\nu } \\right ) + U _ { n 1 } ^ { \\nu } \\cdot \\nabla \\left ( \\omega _ { s 2 } ^ { \\nu } - \\psi _ { s 2 } ^ { \\nu } \\right ) \\\\ & \\ \\ \\ \\ \\ \\ \\ + U _ { s 2 } ^ { \\nu } \\cdot \\nabla \\omega _ { n 2 } ^ { \\nu } + U _ { n 2 } ^ { \\nu } \\cdot \\nabla \\omega _ { s 2 } ^ { \\nu } + U _ { n 2 } ^ { \\nu } \\cdot \\nabla \\omega _ { n 2 } ^ { \\nu } , \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} X ^ { T } = X ^ i \\frac { \\partial } { \\partial q ^ i } + \\dot { q } ^ i \\frac { \\partial X ^ i } { \\partial q ^ i } \\frac { \\partial } { \\partial \\dot { q } ^ i } . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} & \\ell _ 1 = p , \\ell _ 2 = q , \\ell _ 3 = 1 - p , \\ell _ 4 = 1 - q , \\\\ & \\ell _ 5 = 1 - x _ 1 p - x _ 2 q , \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\| f _ y \\| _ { L ^ 4 } ^ 4 = \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\int f _ { N _ 1 , y } f _ { N _ 2 , y } ( f _ { \\leq N _ 2 , y } ) ^ 2 \\ , d x d y . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\textnormal { ( I I ) } & = p \\int ( u _ k ^ p + k u _ k ^ { p - 1 } ) ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x - \\int ( ( p - 1 ) u _ k ^ p + p k u _ k ^ { p - 1 } ) ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x \\\\ & \\ ; \\ ; \\ ; \\ ; + \\int u \\eta ^ 2 ( ( p - 1 ) u _ k ^ p + p k u _ k ^ { p - 1 } ) \\ ; d x \\\\ & = \\int u _ k ^ p ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x + \\int u \\eta ^ 2 \\left ( ( p - 1 ) u _ k ^ p + p k u _ k ^ { p - 1 } \\right ) \\ ; d x . \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} { F _ { 0 } ^ { \\pm } } \\left ( \\xi \\right ) = \\pm { \\tfrac { 1 } { 2 } } \\phi \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\bar a ^ { i j } D _ { i j } v = D _ i ( \\bar a ^ { i j } D _ j v ) = \\bar g \\ ; \\mbox { i n } \\ ; B ^ { + } ( \\bar x , r ) ; v = 0 \\ ; \\mbox { o n } \\ ; T ( \\bar x , r ) , \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} D ( \\ln \\omega ) _ { , t } + 2 D _ { , t } & = d _ { 0 } T \\\\ T \\ , _ { , t t } & = m \\omega T \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\Psi : A \\times ( 0 , h _ 0 ) \\to \\R ^ n , \\Psi ( q , h ) = \\Phi ( T ( p , h ) , p ) \\ , , \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{gather*} W _ { i j k } + W _ { k i j } + W _ { j k i } = 0 , W _ { i j k } = - W _ { i k j } \\end{gather*}"}
-{"id": "7227.png", "formula": "\\begin{align*} t \\mapsto \\{ x _ { 2 } = - x _ { 0 } - x _ { 1 } - t { \\left ( x _ { 1 } + x _ { 3 } + x _ { 0 } / p \\right ) } \\} . \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{gather*} a = m _ { n 0 } n _ { i } = m _ { i 0 } + m _ { i n } . \\end{gather*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\mathfrak { m } = \\ker ( \\lambda _ { \\pi , \\mathfrak { l } } ) \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} { \\rm C o r r } _ { ( \\frak X _ { 2 3 } , \\widehat { \\frak S } _ { 2 3 } ^ { \\epsilon } ) } \\circ { \\rm C o r r } _ { ( \\frak X _ { 1 2 } , \\widehat { \\frak S } _ { 1 2 } ^ { \\epsilon } ) } = { \\rm C o r r } _ { ( \\frak X _ { 1 3 } , \\widehat { \\frak S } _ { 1 3 } ^ { \\epsilon } ) } \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } { \\eta _ { k } } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta _ { k } } ( \\theta ) + { \\eta _ { k } } ( \\xi _ { j } ) , \\ \\forall 1 \\leq j \\leq k - 1 , \\\\ { \\eta _ { k } } ( \\hat { \\theta } _ { k } ) = p _ { k } { \\eta _ { k } } ( \\theta ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} b _ 1 D _ { m _ 1 } ( X _ 1 , a _ 1 ) + \\dots + b _ n D _ { m _ n } ( X _ n , a _ n ) = c , \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t g ( s , W _ 0 ^ s , Y _ 0 ^ { s } ) d s + B ( t ) , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { M } _ i } : = ( S ^ d ) ^ k \\rightarrow U _ { k , i } ^ { \\oplus m _ i } \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} s _ { K , N } ( r ) : = \\begin{cases} \\sqrt { \\frac { N - 1 } { K } } \\sin ( r \\sqrt { \\frac { K } { N - 1 } } ) , \\qquad & K > 0 , \\\\ r , & K = 0 , \\\\ \\sqrt { \\frac { N - 1 } { | K | } } \\sinh ( r \\sqrt { \\frac { | K | } { N - 1 } } ) , & K < 0 . \\end{cases} \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , \\phi ^ { 2 } Z ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , \\omega \\phi Z ) + g _ { 1 } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V , \\omega Z ) \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\boldsymbol { r } _ f = \\boldsymbol { h } _ f \\boldsymbol { s } _ f + \\boldsymbol { w } _ f = \\boldsymbol { x } + \\boldsymbol { w } _ f , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} X ^ N _ i ( t ) = \\xi _ i ^ N + \\int _ 0 ^ t \\int _ U f ( s , X ^ N _ i ( s ^ - ) , u , \\alpha ^ N _ i ( s ) , \\mu _ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\# \\{ X ^ 2 + Y ^ 2 = ( 2 Z ) ^ 2 + 1 \\ ; \\lvert Z \\rvert \\leq R \\} & = \\sum _ { n \\leq R } 4 d _ o ( 4 n ^ 2 + 1 ) \\\\ & = \\sum _ { n \\leq R } 4 d ( 4 n ^ 2 + 1 ) . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\begin{aligned} c ( \\mathbb { M } _ { g } , \\mathbb { M } _ { b } ) : = \\sup \\bigg \\{ \\big | \\langle \\boldsymbol { g } _ k , \\boldsymbol { b } _ k \\rangle \\big | : \\boldsymbol { g } _ k \\in \\mathbb { M } _ { g } \\cap ( \\mathbb { M } _ { g } \\cap \\mathbb { M } _ b ) ^ \\bot , & \\\\ \\boldsymbol { b } _ k \\in \\mathbb { M } _ { b } \\cap ( \\mathbb { M } _ { g } \\cap \\mathbb { M } _ b ) ^ \\bot , \\| \\boldsymbol { g } _ k \\| ^ 2 = \\| \\boldsymbol { b } _ k \\| ^ 2 = 1 \\bigg \\} & \\end{aligned} \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} u ^ \\ast ( t ) = \\left \\{ \\begin{aligned} \\begin{pmatrix} 0 \\\\ k _ E \\end{pmatrix} & \\quad \\textnormal { f o r } t < \\tau _ 1 \\\\ \\begin{pmatrix} k _ M \\\\ 0 \\end{pmatrix} & \\quad \\textnormal { f o r } \\tau _ 1 < t < T \\end{aligned} \\right . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} z ^ { ( p ) } _ m - z _ m = O \\bigg ( \\frac { \\Psi _ p ( z _ m ) } { \\widetilde { \\Psi } _ { p , k } } \\bigg ) \\quad \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} x _ { g h , \\sigma _ { g h } ( i ) } = x _ { g , \\sigma _ { g h } ( i ) } x _ { h , \\sigma _ h ( i ) } \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } \\mathcal { C } X ) & = g _ { 1 } ( V , \\hat { \\nabla } _ { U } \\phi \\mathcal { B } X + \\mathcal { T } _ { U } \\omega \\mathcal { B } X ) + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\int _ U \\sum _ { y = 1 } ^ d \\varphi _ y ( u _ y ) \\nu ( d u ) = \\sum _ { y = 1 } ^ d \\int _ { U _ y } \\varphi _ y ( u _ y ) d u _ y \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} { } _ a \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { { a } / { 2 } } z ) & = ( 2 \\pi ) ^ { \\tfrac { a - 1 } { a } } a ^ { \\tfrac { 1 } { 2 } ( a ^ 2 \\nu - 2 a \\nu - a ^ 2 + a - 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { - \\tfrac { 1 } { 2 } ( a - 1 ) } \\\\ & \\times \\mathtt { J } _ { ( \\nu - 1 ) + 1 / a } ( z ) \\mathtt { J } _ { ( \\nu - 1 ) + 2 / a } ( z ) \\ldots \\mathtt { J } _ { \\nu } ( z ) . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} c _ n = \\frac { b } { \\beta ( \\eta + i \\omega _ n ) } \\ , , \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l } P _ { 6 } [ X ] & = 1 + X + X ^ 6 \\\\ P _ { 7 } [ X ] & = 1 + X ^ 3 + X ^ 7 \\\\ P _ { 8 } [ X ] & = 1 + X ^ 2 + X ^ 3 + X ^ 4 + X ^ 8 \\\\ P _ { 9 } [ X ] & = 1 + X ^ 5 + X ^ 9 \\\\ P _ { 1 0 } [ X ] & = 1 + X ^ 4 + X ^ { 1 0 } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} & \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\leq N ^ \\varepsilon \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { ( N \\eta ) ^ { \\frac 1 3 } } , \\iota = A , B . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} S _ n = \\{ a ^ { n _ 1 } b ^ { m _ 1 } a ^ { n _ 2 } b ^ { m _ 2 } \\dots a ^ { n _ k } b ^ { m _ k } : n _ i , m _ i \\in \\Z \\} . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} H = \\left [ \\partial ( p _ N ) \\Lambda _ 1 - p _ N \\partial ( \\Lambda _ 1 ) \\right ] \\varphi - n p _ N \\Lambda _ 1 ( \\mu + \\alpha ) . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\gamma _ m ( p , n ) = \\gamma _ m ( p , n - 1 ) + \\gamma _ m ( p - 1 , n - m ) . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} f ^ { 1 , 2 } ( Y _ \\Delta , w _ \\Delta ) & = f ^ { 2 , 1 } ( Y _ \\Delta , w _ \\Delta ) \\\\ & = 2 4 - \\ell ( \\Delta ^ \\circ ) + \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) - \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} E | \\mu ^ N ( t ) & - \\widetilde { \\mu } _ N ( t ) | + \\frac 1 N \\sum _ { i = 2 } ^ N E | X ^ N _ i ( t ) - \\widetilde { X } ^ N _ i ( t ) | \\\\ & \\leq \\frac d N + 2 K _ 1 \\frac 1 N \\sum _ { i = 2 } ^ N \\int _ 0 ^ t \\left [ E | X ^ N _ i ( s ) - \\widetilde { X } ^ N _ i ( s ) | + E | \\mu ^ N ( s ) - \\widetilde { \\mu } _ N ( s ) | \\right ] d s \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ k { \\big ( t ^ { N - 1 } x - q ^ { j - \\theta } \\big ) } = ( - 1 ) ^ k q ^ { { k + 1 \\choose 2 } - \\theta k } \\prod _ { i = 1 } ^ k { \\big ( 1 - x q ^ { \\theta N - i } \\big ) } , k \\in \\N . \\end{gather*}"}
-{"id": "7028.png", "formula": "\\begin{align*} ( \\psi ( a ) - 1 ) H ^ 2 ( a ) + a \\psi ( a ) H ( a ) + \\psi ( a ) + 1 = 0 . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} 1 ^ k + 2 ^ k + \\cdots + x ^ k = y ^ n . \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\begin{pmatrix} G _ { 1 1 } \\left ( Z , W \\right ) & G _ { 1 2 } \\left ( Z , W \\right ) \\\\ G _ { 2 1 } \\left ( Z , W \\right ) & G _ { 2 2 } \\left ( Z , W \\right ) \\end{pmatrix} = \\begin{pmatrix} W + \\mbox { O } ( 2 ) & \\mbox { O } ( 2 ) \\\\ \\mbox { O } ( 2 ) & \\mbox { O } ( 2 ) \\end{pmatrix} , \\quad \\begin{pmatrix} F _ { 1 } \\left ( Z , W \\right ) \\\\ F _ { 2 } \\left ( Z , W \\right ) \\end{pmatrix} = \\begin{pmatrix} \\mbox { O } ( 1 ) \\\\ \\mbox { O } ( 2 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} g ( u ) = \\begin{pmatrix} u _ M \\\\ u _ E \\\\ 1 - \\frac { u _ M } { k _ M } - \\frac { u _ E } { k _ E } , \\end{pmatrix} \\geq 0 \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\| z _ n ^ M - \\varphi \\| _ \\infty = O ( \\max \\{ \\tilde { h } ^ { 2 r } , h ^ { 3 r } \\} ) . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} H ( x , u , \\lambda _ 0 , \\lambda ) = \\lambda _ 0 K ( x , u ) + \\lambda f ( x , u ) , \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} \\mathsf { K } \\left ( { \\xi , t } \\right ) = \\frac { \\exp \\left \\{ { u \\xi + E _ { 0 } ^ { + } \\left ( \\xi \\right ) - u t - E _ { 0 } ^ { + } \\left ( t \\right ) } \\right \\} - \\exp \\left \\{ { u t + E _ { 0 } ^ { + } \\left ( t \\right ) - u \\xi - E _ { 0 } ^ { + } \\left ( \\xi \\right ) } \\right \\} } { 2 + u ^ { - 1 } \\phi \\left ( t \\right ) } . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\lambda x _ p = \\sum \\limits _ { j , k \\in N } a _ { p j k } y _ j y _ k , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} m _ 1 = 2 ^ { q + 1 } - t , m _ 2 = t , m _ 3 = 2 ^ q + t - 2 , \\ , \\ , \\ , \\ , m _ i = 2 ^ q \\cdot [ 2 ^ { i - 2 } - 1 ] + t + i - 3 \\ , \\ , \\ , \\ , i \\geq 4 . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} H \\sigma ( V ) = \\sigma ( V ) H = \\mu ( V , H ) . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} m ( t , x ) = - \\frac { ( C _ { 1 } \\Lambda t ^ { \\nu } - \\vert x \\vert ) ^ { 2 } } { 4 C _ { 1 } t } . \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} \\tau _ s = \\frac { x _ N ( 0 ) } { a _ M b _ M k _ M x _ E ( 0 ) } \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} A _ 1 ^ { n p } ( x ) = L ( x ) ^ { - 1 } A _ 2 ^ { n p } ( h ( x ) ) L ( x ) , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\sup _ { n \\in \\N } A _ k ( n ) : = \\sup _ { n \\in \\N } \\frac { w _ { 2 k } ( n ) } { n } \\sum _ { m = 1 } ^ n \\frac { 1 } { w _ k ( m ) } < \\infty . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\int f ( x ) \\cdot g ( y ) \\ , d \\sigma ( x , y ) = 0 . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} V ( t , x ) = \\frac 1 2 \\langle P _ V ( t ) x , x \\rangle _ H \\forall t \\ge T _ 0 , \\forall x \\in H ; \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} B ( \\mathbf { t } _ { \\mathcal { N } } ) = C ( \\mathbf { t } _ { \\mathcal { N } } ) \\cdot A ( \\mathbf { t } _ { \\mathcal { N } } ) + R ( \\mathbf { t } _ { \\mathcal { N } } ) \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} ( \\phi ^ k , \\phi ^ l ) = \\delta _ { k l } \\frac { \\rho _ k } { a _ 0 } , \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} g _ { \\varphi _ { \\epsilon } } ^ { p \\bar { q } } g _ { \\varphi _ { \\epsilon } j \\bar { m } } R _ { \\omega _ { \\epsilon } } ^ { \\bar { m } j } { } _ { p \\bar { q } } = \\frac { 1 + \\varphi _ { \\epsilon i \\bar { i } } } { 1 + \\varphi _ { \\epsilon j \\bar { j } } } R _ { \\omega _ { \\epsilon } } ^ { i \\bar { i } } { } _ { j \\bar { j } } , \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\check { \\mathrm { T } } ^ 2 : \\check { Z } ^ 2 ( \\Delta , { } ^ 2 G ) \\to \\check { Z } ^ 2 ( X , { } ^ 2 G ) \\ \\ , \\ \\ \\check { \\mathrm { T } } ^ 2 : = S \\circ { \\mathrm { T } } ^ 1 \\circ \\mu _ 2 \\ . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} A _ { \\overline { 0 } } = \\langle e _ { t + 2 } , e _ { 2 t + 2 } , e _ { 3 t + 2 } , . . . , e _ { r t + 2 } \\rangle \\oplus \\langle e _ 2 \\rangle \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} V _ { \\underline { d } '' } ^ { X _ { 1 } , 0 } = ( V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } \\oplus V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } ) \\oplus \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle = V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } \\oplus \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle . \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } A ^ { \\epsilon } ( \\cdot , \\cdot ) \\stackrel { d i s t . } { = } \\sqrt { 2 } \\left ( B _ 1 ( \\cdot ) + B _ 2 ( \\cdot ) \\right ) \\ , , \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} F _ p ( p , a , 2 ) & = F _ p ( p , a , 3 ) - \\sum _ { i = 3 } ^ { a - 1 } F _ p ( p , i , 2 ) \\cr & = F _ p ( p , a , 3 ) - \\sum _ { i = 3 } ^ { a - 1 } \\dim ( i , 2 ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( \\dagger ) \\cr & = F _ p ( p , a , 3 ) - \\dim ( a - 1 , 3 ) . \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} g _ 2 ( x ) : = \\inf _ { y \\in B _ r ( x ) } f ( y ) . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} d ( x , y ) = d ( x _ { 0 } , \\exp H x _ { 0 } ) = \\left \\Vert H \\right \\Vert . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} d e ^ 1 = e ^ { 2 3 } , d e ^ 2 = - e ^ { 1 3 } \\textrm { a n d } d e ^ 3 = e ^ { 1 2 } \\ , . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} p ^ { \\ast } \\left ( t , P \\right ) = \\left \\{ \\begin{array} [ c ] { l l } \\left ( \\frac { 1 } { 2 } - \\dfrac { \\log \\left ( \\dfrac { 1 } { 1 - t } \\right ) } { \\log C _ { P } } \\left ( \\dfrac { 1 } { 2 } - \\dfrac { 1 } { P } \\right ) \\right ) ^ { - 1 } & 0 \\leq t \\leq 1 - C _ { P } ^ { - 1 } , \\\\ & \\quad \\\\ P & 1 - C _ { P } ^ { - 1 } < t \\leq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} H _ { d , e } ( x ) = x H _ { d - 1 , e } ( x ) + \\sum _ { j = 0 } ^ e \\frac { x ^ 2 } { ( 1 - x ) ^ { j + 1 } } H _ { d - 1 + j , e - j } ( x ) \\ , . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} u _ 1 ( t , 0 ) = u _ 2 ( t , 0 ) = 0 \\ , , u _ 1 ( t , \\pi ) = g _ 1 ( t ) \\ , , u _ 2 ( t , \\pi ) = g _ 2 ( t ) t \\in ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\alpha x } J _ 0 \\bigl ( \\beta \\sqrt { x ^ 2 + 2 \\gamma x } \\bigr ) \\ , { \\rm d } x = \\frac { 1 } { \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } } e ^ { \\gamma ( \\alpha - \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } ) } . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} w ^ t _ { 2 2 } ( x ) = - \\int _ 0 ^ 1 ( 1 - \\theta ) d \\theta I _ \\theta ( x ) \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} I ( M _ 1 , M _ 2 ; Y _ { 2 , 0 } ^ T ) = I ( M _ 2 ; Y _ { 2 , 0 } ^ T ) + I ( M _ 1 ; Y _ { 2 , 0 } ^ T | M _ 2 ) \\geq I ( M _ 2 ; Y _ { 2 , 0 } ^ T ) + \\frac { s n r _ 2 } { s n r _ 1 } I ( M _ 1 ; Y _ { 1 , 0 } ^ T | M _ 2 ) , \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} p ( z ) = c \\det ( I - K ) \\det \\begin{pmatrix} 2 i I & 0 \\\\ * & A + \\sum _ j z _ j B _ j \\end{pmatrix} = c _ 0 \\det ( A + \\sum _ j z _ j B _ j ) \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} A _ { \\omega , \\omega ' , t } = \\ ( A _ { \\omega , \\omega ' , t } \\ ) _ b + \\ ( A _ { \\omega , \\omega ' , t } \\ ) _ c \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} A _ 1 ^ n ( x ) = L ( f _ 1 ^ n ( x ) ) ^ { - 1 } A _ 2 ^ n ( h ( x ) ) L ( x ) , \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} [ z ^ { - 1 } ] T ^ + _ { i i } ( z ) = \\tilde { g } ^ { ( 1 ) } _ i + \\sum _ { j < i } \\tilde { f } ^ { ( 1 ) } _ { i j } \\tilde { g } ^ + _ j \\tilde { e } ^ { ( 0 ) } _ { j i } . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\displaystyle { \\begin{array} { c c l } A ( \\theta ) & = & \\cos \\theta P _ n ( \\cos \\theta , \\sin \\theta ) + \\sin \\theta Q _ n ( \\cos \\theta , \\sin \\theta ) , \\\\ B ( \\theta ) & = & \\cos \\theta Q _ n ( \\cos \\theta , \\sin \\theta ) - \\sin \\theta P _ n ( \\cos \\theta , \\sin \\theta ) . \\\\ \\end{array} } \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} L _ \\lambda : = \\dfrac { \\partial ^ 2 f } { \\partial x ^ 2 } ( \\l , 0 ) : H \\to H , \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\tau ( \\varpi _ { i _ 0 } ) = \\tau ( \\tilde { \\alpha } ^ \\vee ) = s _ { \\tilde { \\alpha } , 1 } s _ { \\tilde { \\alpha } } . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} g = \\sigma \\otimes \\sigma + \\pi ^ { \\ast } g _ { S ^ 2 ( \\frac { 1 } { 2 } ) } , \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } E [ f ( X ^ \\theta ) ] \\approx \\frac { E [ f ( X ^ { \\theta + h / 2 } ) ] - E [ f ( X ^ { \\theta - h / 2 } ) ] } { h } = E \\bigg [ \\frac { f ( X ^ { \\theta + h / 2 } ) - f ( X ^ { \\theta - h / 2 } ) } { h } \\bigg ] , \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} 1 - \\Q _ k = \\prod _ { i = 1 } ^ { k } ( 1 - H _ i ) , k = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\Delta _ f ( \\widetilde { H } _ { \\lambda } ) : = f [ B _ { \\lambda } ] \\cdot \\widetilde { H } _ { \\lambda } . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\mathbb { P } ( L _ { n , 1 } = \\ell \\ , | \\ , K _ n = k ) = \\frac { ( \\ell , 1 ) ^ { \\dagger } ( n - \\ell , k - 1 ) ^ { \\dagger } } { ( n , k ) ^ { \\dagger } } \\mbox { f o r } ~ 1 \\le \\ell \\le n , \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} P _ { 0 } \\left ( { \\xi } \\right ) = \\left \\vert { e ^ { u \\xi } } \\right \\vert , \\ P _ { 1 } \\left ( { \\xi } \\right ) = \\left \\vert { u e ^ { u \\xi } } \\right \\vert , \\ Q \\left ( { t } \\right ) = \\left \\vert { e ^ { - u t } } \\right \\vert . \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} F u t _ f ( X ) = \\int _ M S ( J , f ) \\ , u _ X \\ , \\frac { \\omega ^ m } { f ^ { 2 m + 1 } } \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} ( - \\Delta _ g - \\lambda _ j ^ 2 ) u _ { \\lambda _ j } = 0 , \\| u _ { \\lambda _ j } \\| _ { L ^ 2 } = 1 \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\tilde { W } ' = \\begin{pmatrix} \\Re w _ { 1 1 } & w _ { 1 2 } \\\\ \\overline { w } _ { 1 2 } & \\Re w _ { 2 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\ker ( T _ 2 ) = \\begin{cases} H _ 2 / H _ 2 ^ \\prime = \\langle x , s _ 2 , \\ldots , s _ { n - 1 } \\rangle / \\langle s _ 3 , \\ldots , s _ { n - 1 } \\rangle \\simeq \\langle x , s _ 2 \\rangle \\simeq C _ 3 \\times C _ 3 w = 0 , \\\\ G ^ \\prime / H _ 2 ^ \\prime = \\langle s _ 2 , \\ldots , s _ { n - 1 } \\rangle / \\langle s _ 3 , \\ldots , s _ { n - 1 } \\rangle \\simeq \\langle s _ 2 \\rangle \\simeq C _ 3 w = \\pm 1 . \\end{cases} \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\begin{aligned} L ^ { \\rho _ + } _ { - } ( \\gamma ^ 2 n , n ) & \\geq L ^ { \\rho _ + } ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) + L ^ { \\rho _ + } _ { ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) \\to ( \\gamma ^ 2 n , n ) } \\\\ & \\geq L ^ { \\rho _ + } ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) + L _ { ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) \\to ( \\gamma ^ 2 n , n ) } , \\end{aligned} \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} x ( t + 1 ) & = W ( t ) x ( t ) , \\cr y ( t + 1 ) & = W ( t ) y ( t ) , \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} \\mathbb { P } ( \\widehat { Q } _ 0 = m ) = \\frac { 1 } { m ( m + 1 ) } \\mbox { f o r } ~ m \\ge 1 . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\mathcal { U } = & \\left \\{ \\alpha _ { k _ { 0 } } \\times \\alpha _ { k _ { 1 } } \\times \\ldots \\times \\alpha _ { k _ { M } } \\times \\alpha _ { k _ { M + 1 } } \\times I ^ { \\infty } : \\{ k _ { j } \\} _ { j = 0 } ^ { M + 1 } \\right . \\\\ & \\left . \\{ 1 , \\ldots , n \\} M + 2 \\right . \\big \\} . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\theta = C \\eta ^ { \\frac { 1 } { n + 1 } } \\ , . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { m } ( m + 1 - i ) ( 3 i - m ) x ^ i \\ge 0 \\ , . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\psi ^ { \\boxplus \\tau } _ p \\left ( ( s ^ { \\boxplus \\tau } _ p ) ^ { - 1 } ( 0 ) \\cap \\{ ( \\overline y , ( t _ 1 , \\dots , t _ k ) ) \\mid t _ i \\le 0 , \\ , \\ , i = 1 , \\dots , k \\} \\right ) \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} \\| u ^ { * } ( \\varphi ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) } ^ m ) u \\\\ & - \\psi ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma ( a ) } ^ m \\| < \\varepsilon \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} u ( t ) \\leq - \\beta ^ { 2 ^ { m } - 1 } C _ { 3 } ^ { 2 ^ { m } } ( T - t ) ^ { 2 ^ { m } - 1 } \\prod _ { k = 1 } ^ { m } ( \\frac { 1 } { 2 ^ { k } - 1 } ) ^ { 2 ^ { m - k } } \\leq - C \\left [ \\beta C _ { 3 } ( T - t ) \\prod _ { k = 1 } ^ { m } ( 2 ^ { - \\frac { k } { 2 ^ { k } } } ) \\right ] ^ { 2 ^ { m } } \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\omega \\left ( s \\right ) = \\left ( \\frac { d S ^ { - 1 } \\left ( s \\right ) } { d s } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} x \\mathbf { f } y & = x ( 1 - y ) . \\\\ x \\mathbf { q } y & = ( x - y ) _ + . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} \\mathbf { r } _ i = \\ell _ i ( \\mathbf { e } _ i + \\mathbf { h } _ i ) \\ , . \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { U } \\left ( \\frac { 1 } { 2 } \\langle A \\nabla \\left ( \\rho P _ { \\cdot } g _ n \\right ) , \\nabla \\varphi \\rangle + ( \\rho P _ { \\cdot } g _ n ) \\langle \\beta , \\nabla \\varphi \\rangle - ( \\rho P _ { \\cdot } g _ n ) \\partial _ t \\varphi \\right ) d x d t = 0 . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} h - w + \\frac { \\lambda } { 3 } \\frac { \\textrm { s i g n } ( h ) } { | h | ^ { 1 / 3 } } = 0 \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} & M _ { 2 1 } ^ { \\tau } = M _ { 3 1 } , U M _ { 3 1 } U _ m = M _ { 3 1 } . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ s K _ 2 \\right ) \\times H \\cong \\bigoplus _ { i = 1 } ^ { 2 ^ { s - 1 } } ( K _ 2 \\times H ) . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{gather*} F _ { a b c } = F _ { ( a b c ) } , F ^ { a } { } _ { a b } = 0 , K ^ { a b } = K ^ { b a } , K ^ a { } _ a = 0 , f _ { a b c } = F _ { a b c ; n } , \\\\ W _ { a n b n } = W _ { b n a n } , W ^ a { } _ { n a n } = 0 , W _ { a n b c } = - W _ { 0 a c b } , W _ { [ a | n | b c ] } = 0 . \\end{gather*}"}
-{"id": "1987.png", "formula": "\\begin{align*} u ( b ) + ( j i ) _ * ( I ) - ( j ' o ' ) _ * ( O ' ) = 0 . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} - & \\left ( 2 g + 1 - \\sum b _ j \\right ) ! \\sum _ { c _ 2 , \\ldots , c _ n } ( 2 k - 1 ) ! ! \\cdot ( 2 c _ i + 1 ) ! ! \\cdot \\psi _ 1 ^ k ( \\psi ' ) ^ { c _ i } \\\\ & \\cdot \\prod _ { j \\neq i } \\frac { \\psi _ j ^ { c _ j } } { ( 2 c _ j ) ! ! ( b _ j - 2 c _ j ) ! } \\cdot \\\\ & \\cdot \\sum _ { d = 0 } ^ { 2 c _ i + 2 } { 4 g - 1 + n - \\sum b _ j \\choose 2 g + 1 - \\sum b _ j - d } \\sum _ { e _ { n + 1 } = 0 } ^ 1 \\frac { 1 } { b _ i ! } { b _ i \\choose 2 c i _ i + 2 - d - e _ { n + 1 } } . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\nu ^ c ( T _ \\lambda ) = \\sum _ { \\nu \\in X ^ + ( T ) } { a _ \\nu \\chi ( \\nu ) } . \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} w _ { k } ( Y _ n ) = w _ { k } ( Y _ { n - 1 } ) + w _ { k - 1 } ( Y _ { n - 1 } ) \\cdot ( \\sum _ { i = 1 } ^ { n - 1 } c _ { i , n } x _ { n + i } ) \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\Pi _ i ^ 2 \\leq 2 \\Pi ^ 2 , \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} p _ t ( x , y ) = t ^ { - n \\alpha / 2 } \\phi ( \\frac { x } { t ^ { \\alpha / 2 } } , \\frac { y } { t ^ { \\alpha / 2 } } ) \\theta ( \\frac { y } { t ^ { \\alpha / 2 } } ) \\exp \\{ - \\frac { \\rho ( x ) + \\rho ( y ) } { 2 t } \\} , \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\left < \\alpha _ { u } \\left ( i \\right ) , \\alpha _ { l } \\left ( j \\right ) \\right > = \\delta _ { u } ^ { l } \\cdot \\delta _ { i } ^ { j } , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ a n d $ u , l = 1 , \\dots , p - q $ . } \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{gather*} ( K + \\mu ) ( D _ 1 \\Lambda _ { 1 1 } + 1 ) = \\underline { ( K + \\mu ) } , \\mu \\left ( \\frac { 1 } { 2 } D _ 1 \\Lambda _ { 2 2 } + 1 \\right ) = \\underline { \\mu } , \\\\ ( K + \\mu ) D _ 1 \\Lambda _ { 1 3 } + K - \\mu = \\underline { ( K + \\mu ) } \\overline { \\left ( \\frac { K - \\mu } { K + \\mu } \\right ) } . \\end{gather*}"}
-{"id": "3472.png", "formula": "\\begin{align*} f \\# g ( x ) = \\lim _ { n \\to \\infty } \\int _ G \\psi _ n ( y ) f ( y ) \\ell _ y ^ \\sigma g ( x ) \\ , d y \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\partial _ { t } \\omega = \\nu \\Delta \\omega - e ^ { - \\nu t } \\left [ \\sin y \\partial _ { x } \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\right ] \\omega = L \\left ( t \\right ) \\omega , \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { h } \\oplus \\mathfrak { m } , \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} u _ j \\left ( A _ j \\left ( p _ j , \\frac { \\epsilon } { 2 } \\right ) \\right ) & = \\dfrac { r _ j ^ { n - 2 } G \\left ( X _ 0 , A \\left ( \\widetilde p _ j , \\frac { \\epsilon } { 2 } r _ j \\right ) \\right ) } { \\omega ( B ( q _ j , r _ j ) ) } \\\\ & \\geq C \\dfrac { r _ j ^ { n - 2 } G \\left ( X _ 0 , A ( q _ j , r _ j ) \\right ) } { \\omega ( B ( q _ j , r _ j ) ) } = C u _ j ( A _ j ( 0 , 1 ) ) \\geq C ' > 0 , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ a p ^ { ( a - b ) b } = p ^ { c ^ 2 } + 2 \\sum _ { b = 0 } ^ { c - 1 } p ^ { ( 2 c - b ) b } = p ^ { c ^ 2 } + O \\bigg ( \\sum _ { b = 0 } ^ { c - 1 } p ^ { c ^ 2 - ( c - b ) } \\bigg ) = p ^ { c ^ 2 } + O \\big ( p ^ { c ^ 2 - 1 } \\big ) , \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\Delta ^ \\perp ( \\mathbf { m } , \\mathbf { a } ; k ) = 2 ^ q \\cdot [ 2 ^ { k - 1 } + 1 ] - t \\ , \\ , \\ , \\ , \\ , \\ , k \\Delta ^ \\perp ( \\mathbf { m } , \\mathbf { a } ; k ) = C ( \\mathbf { m } , \\mathbf { a } , \\mathcal { O } ; k ) ) , \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( V , \\hat { \\nabla } _ { U } \\phi \\mathcal { B } X ) + g _ { 1 } ( V , \\mathcal { T } _ { U } \\omega \\mathcal { B } X ) + g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } \\mathcal { C } X ) \\\\ & + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) , \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } b ^ i = \\sum _ { i = 1 } ^ { k - 1 } a ^ i + k ( 1 - a ^ { k - 1 } ) \\ , . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} & \\widetilde { T } ( z _ { 0 } ) : \\mathcal { S } ( \\mathbb { R } ^ { 2 n } ) \\to \\mathcal { S } ( \\mathbb { R } ^ { 2 n } ) , \\\\ & \\widetilde { T } ( z _ { 0 } ) \\Psi ( z ) = e ^ { - \\frac { i } { \\hbar } \\sigma ( z , z _ { 0 } ) } \\Psi ( z - \\tfrac { 1 } { 2 } z _ { 0 } ) . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\eqref { v a 4 } = T ( x ) \\psi ( x ) + \\mathcal { O } ( t ^ { - \\frac 1 4 } \\| \\psi \\| _ { H ^ 1 ( \\R ) } ) , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} A _ { t } ^ { \\psi } u ( x ) & = \\exp ( - \\psi ( x ) ) \\sum _ { i , j = 1 } ^ { n } \\partial _ { x _ { i } } ( a _ { i j } ( t , x ) \\partial _ { x _ { j } } [ \\exp ( \\psi ( x ) ) u ( x ) ] ) \\\\ & \\quad - \\exp ( - \\psi ( x ) ) \\sum _ { i = 1 } ^ { n } b _ { i } ( t , x ) \\partial _ { x _ { i } } [ \\exp ( \\psi ( x ) ) u ( x ) ] . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} \\norm { v ^ { j + 1 } _ V } _ { j + 1 } = \\norm { A ( f ^ j ( x ) ) u ^ j _ V } _ { j + 1 } \\leq e ^ { \\lambda _ 2 + \\delta } \\norm { u ^ j _ V } _ j . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} Y ^ i _ n : = S ^ i _ n - d T ^ i _ n = s ^ i _ 0 + \\sum _ { k = 1 } ^ n \\big ( \\zeta ^ i _ k - d \\nu ^ i _ k \\big ) \\ , . \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} ( e _ { i } f _ { j } - f _ { j } e _ { i } ) T ( R ) = & \\sum _ { r = 1 } ^ { j } \\sum _ { s = 1 } ^ { i } \\Phi ( R , - \\delta ^ { j r } ) f _ { j r } ( R ) e _ { i s } ( R + \\delta ^ { j r } ) T ( R - \\delta ^ { j r } + \\delta ^ { i s } ) \\\\ - & \\sum _ { r = 1 } ^ { j } \\sum _ { s = 1 } ^ { i } \\Phi ( R , \\delta ^ { i s } ) e _ { i s } ( R ) f _ { j r } ( R + \\delta ^ { i s } ) T ( R - \\delta ^ { j r } + \\delta ^ { i s } ) . \\\\ \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} H = \\Bigl \\{ \\phi \\in H _ 0 ^ 1 ( \\Omega _ \\varepsilon ) : \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , \\phi = 0 , \\ , i = 1 , \\ldots , k , \\ , j = 1 , 2 , 3 , 4 \\Bigr \\} \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} | \\mathcal { S } ' ( z ) | = \\Big | F '' _ B ( \\omega _ A ) ( F ' _ A ( \\omega _ B ) - 1 ) \\omega _ A ' ( z ) + F '' _ A ( \\omega _ B ) ( F ' _ B ( \\omega _ A ) - 1 ) \\omega _ B ' ( z ) \\Big | \\leq C | \\mathcal { S } ^ { - 1 } ( z ) | \\ , , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\displaystyle \\sum \\limits _ { k = 0 } ^ { N - 1 } A _ { 0 } \\left ( x , v _ { k } \\right ) \\psi _ { k } ^ { \\prime } \\left ( x _ { 0 } \\right ) + F _ { 2 } \\left ( x , \\displaystyle \\sum \\limits _ { m = 0 } ^ { N - 1 } \\nabla v _ { m } \\left ( x \\right ) \\psi _ { m } \\left ( x _ { 0 } \\right ) , \\displaystyle \\sum \\limits _ { k = 0 } ^ { N - 1 } \\nabla v _ { k } \\left ( x \\right ) \\psi _ { k } ^ { \\prime } \\left ( x _ { 0 } \\right ) \\right ) = 0 , \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} & \\| U ( t ) \\| + \\gamma \\int _ 0 ^ t e ^ { \\gamma ( t - s ) } \\ , \\| U ( s ) \\| d s \\\\ & \\leq \\frac { 1 } { \\sqrt { 1 - \\mu ^ 2 } } \\Big ( e ^ { \\gamma t } \\ , \\| U ( 0 ) \\| + \\int _ 0 ^ t e ^ { \\gamma ( t - s ) } \\ , \\| { \\tilde L } _ a U ( s ) \\| d s \\Big ) . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} \\Phi ( m ) : = \\left \\{ F l o w ( X _ { \\rho , m } ) : J ( \\rho , m ) \\leq J ( \\sigma , m ) \\forall \\sigma \\in \\mathcal { R } \\right \\} , m \\in \\mathcal { L } . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} X \\boxplus Y = ( X ^ c + Y ^ c ) ^ c \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} T ( z , t ) = \\sum _ { n \\geq 1 } z ^ { n } \\tilde { N } _ { n - 1 } \\Bigl ( \\frac t z \\Bigr ) . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k \\int _ { [ 0 , t ] \\times [ 0 , \\infty ) } 1 _ { [ q _ { k - 1 } ( s - ) , q _ k ( s - ) ) } ( x ) N ( d s \\times d x ) , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} x _ i x _ j & = ( - 1 ) ^ { | x _ i | | x _ j | } x _ j x _ i , \\\\ y _ i x _ j & = ( - 1 ) ^ { | x _ i | | x _ j | } x _ j y _ i + \\{ x _ i , x _ j \\} , \\\\ y _ i y _ j & = ( - 1 ) ^ { | x _ i | | x _ j | } y _ j y _ i + \\psi ( \\{ x _ i , x _ j \\} ) , \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\| z _ n ^ M - \\varphi \\| _ \\infty = O ( \\max \\{ \\tilde { h } ^ { d } , h ^ { 3 r } \\} ) . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} J ( \\gamma , z ) = \\frac { j ( \\gamma , z ) } { \\lvert j ( \\gamma , z ) \\rvert } \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{gather*} A _ \\vartheta ( x , s ) = \\vartheta ( s ) \\ , A ( x , s ) \\ , , B _ \\vartheta ( x , s ) = \\vartheta ' ( s ) \\ , A ( x , s ) + \\vartheta ( s ) \\ , B ( x , s ) \\ , , \\\\ g _ \\vartheta ( x , s ) = \\vartheta ( s ) \\ , g ( x , s ) \\ , , \\end{gather*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( X _ { \\tau + t _ 1 } ^ x , \\cdots , X _ { \\tau + t _ m } ^ x ) ] = \\hat { \\mathbb { E } } [ \\varphi ( X _ { t _ 1 } ^ y , \\cdots , X _ { t _ m } ^ y ) ] _ { y = X _ \\tau ^ x } . \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\phi _ { n + 1 } ( t ) = x _ 0 ( t - a ) ^ { \\gamma - 1 } + \\int _ { a } ^ { t } \\frac { ( t - s ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } f ( s , \\phi _ { n } ( s ) ) d s . \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} \\Vert T \\Vert _ p = ( \\tau \\vert T \\vert ^ p ) ^ { \\frac { 1 } { p } } = \\bigg ( \\sum _ { j = 1 } ^ { \\infty } s _ j ^ p ( T ) \\bigg ) ^ { \\frac { 1 } { p } } < \\infty , \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\frac { S ( z ) } { z } = \\frac { ( \\log \\log z ) ^ 2 } { \\log z } + O \\bigg ( \\frac { | \\ ! \\log \\log z | } { \\log z } \\bigg ) \\ll 1 \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\mu _ i = \\min \\left \\{ l _ i , \\lfloor r _ i / I _ p \\rfloor \\right \\} , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\int _ { \\varepsilon Y _ { 1 } } \\left ( p ^ { \\varepsilon } - \\bar { p } \\right ) \\phi d x = \\varepsilon ^ { d } \\left | Y _ { 1 } \\right | ^ { - 1 } \\int _ { Y _ { 1 } } \\int _ { Y _ { 1 } } \\left ( p \\left ( y \\right ) \\phi \\left ( \\varepsilon y \\right ) - p \\left ( y \\right ) \\phi \\left ( \\varepsilon z \\right ) \\right ) d z d y . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p } _ { a _ { 1 } , a _ { 2 } , \\R _ { 1 } , \\R _ { 2 } } ( \\mathbb { G } ) } : = \\left ( \\int _ { \\mathbb { G } } ( | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } + | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} \\mathbf { w } _ { 0 } ^ { \\varGamma _ { 2 } } = \\left ( \\begin{array} { c } \\begin{array} { c } \\cos \\omega _ { 1 } \\mathbf { t } _ { 1 } \\end{array} \\\\ \\cos \\omega _ { 2 } \\mathbf { t } _ { 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} d ( f _ 1 ) = \\sum \\limits _ { \\substack { d | N \\\\ 1 \\leq d \\leq N } } \\frac { N } { d } \\frac { \\varphi ( \\gcd ( d , N / d ) ) } { \\gcd ( d , N / d ) } = \\psi ( N ) , \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} a _ { i j } ^ { l l ' } \\left < Z _ { i } , Z _ { j } \\right > = \\displaystyle \\sum _ { k ' = 1 } ^ { N } \\left ( \\displaystyle \\sum _ { k = 1 } ^ { N } v ^ { l k ' } _ { i k } z _ { i k } \\right ) \\overline { \\left ( \\displaystyle \\sum _ { k = 1 } ^ { N } v ^ { l ' k ' } _ { j k } z _ { j k } \\right ) } , \\quad \\quad \\mbox { f o r a l l $ l , l ' , i , j = 1 , 2 $ . } \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + g ( u ) = \\widehat f , & \\Omega , \\\\ u = 0 , & \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d s ^ { 2 } } + \\Gamma _ { j k } ^ { i } \\frac { d x ^ { j } } { d s } \\frac { d x ^ { k } } { d s } + \\left ( \\frac { d S ^ { - 1 } \\left ( s \\right ) } { d s } \\right ) ^ { 2 } V ^ { , i } = 0 . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s ( \\phi _ j ( u _ i , v _ i ) - \\phi _ j ( z _ i , w _ i ) ) & = 0 , \\\\ \\sum _ { i = 1 } ^ s ( \\psi _ j ( u _ i , v _ i ) - \\psi _ j ( z _ i , w _ i ) ) & = 0 , \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\overline { \\mathbf { d } } & = \\mathbf { d } / \\mathbf { d } \\in [ 0 , \\infty ] ^ { X \\times X } & & & x \\overline { \\mathbf { d } } z & = \\sup _ { y \\in Y } ( x \\mathbf { d } y - z \\mathbf { d } y ) _ + . \\\\ \\underline { \\mathbf { d } } & = \\mathbf { d } \\backslash \\mathbf { d } \\in [ 0 , \\infty ] ^ { Y \\times Y } & & & z \\underline { \\mathbf { d } } y & = \\sup _ { x \\in X } ( x \\mathbf { d } y - x \\mathbf { d } z ) _ + . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} [ \\lambda , W ] ( x , y ) = \\begin{cases} 0 & 0 \\le x < \\lambda 0 \\le y < \\lambda , \\\\ W \\left ( \\frac { x - \\lambda } { 1 - \\lambda } , \\frac { y - \\lambda } { 1 - \\lambda } \\right ) & \\lambda \\le x \\le 1 \\lambda \\le y \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\mathbb { A } : = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\gamma , \\xi , \\N ) \\right \\} \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} f = \\sum _ { \\iota , \\kappa } g _ \\iota \\otimes h _ \\kappa \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} \\mathcal E ^ { \\boxplus \\tau } _ x = \\mathcal R _ x ^ * ( \\mathcal E _ x ) = ( E _ x \\times V ^ { \\boxplus \\tau } _ x ) / \\Gamma _ x , \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\sigma _ { g } \\circ x _ h : = ( x _ { h , \\sigma _ { g } ^ { - 1 } ( 1 ) } , \\ldots , x _ { h , \\sigma _ { g } ^ { - 1 } ( d ) } ) . \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} \\varphi = \\big ( z _ 0 g _ { d - 1 } + g _ d : ( z _ 0 q _ { \\ell - 1 } + q _ \\ell ) t _ 1 : ( z _ 0 q _ { \\ell - 1 } + q _ \\ell ) t _ 2 : \\ldots : ( z _ 0 q _ { \\ell - 1 } + q _ \\ell ) t _ n \\big ) \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} h ( \\sigma , \\alpha ^ { M + 1 } \\times I ^ { \\infty } ) & = \\lim _ { l \\to \\infty } \\frac { \\log ( N ( \\mathbf { G } , ( \\vee _ { i = 0 } ^ { l } \\sigma ^ { - i } \\mathcal { V } ) ^ { * } ) ) } { l } = \\lim _ { l \\to \\infty } \\frac { \\log ( N ( \\star _ { i = 1 } ^ { M + l } G , \\alpha ^ { M + l + 1 } ) ) } { l } , \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} & \\mathbb { E } [ L ( Q ( T ) ) ] - \\mathbb { E } [ L ( Q ( 1 ) ) ] + V \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ D ^ { A l g } ( t ) | Q ( t ) ] \\\\ & \\leq T ( \\frac { 5 } { 2 } + V \\bar D ^ { o p t } ) . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} L _ F ( \\chi _ { 0 , \\frak { c } } , s ) = \\zeta _ F ( s ) \\prod _ { \\frak { p } | \\frak { c } } \\left ( 1 - \\mathcal { N } _ { F / \\Q } ( \\frak { p } ) ^ { - s } \\right ) \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\sum _ { s \\in \\Sigma } ( \\widetilde { \\rho } _ s - 1 ) = k + \\sum _ { s \\in \\Sigma } ( \\rho _ s - 1 ) . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} \\left ( Q ^ { * } _ { n ( k , m ) } ( k - j ) , 0 \\le j \\le k \\right ) \\stackrel { ( d ) } { = } \\left ( Y _ m ( \\tau _ j ) , 0 \\le j \\le k \\right ) , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} & A = c i r c s h i f t ( A , [ \\lfloor \\frac { y _ 1 } { 4 } \\rfloor ~ ~ - \\lfloor \\frac { y _ 2 } { 4 } \\rfloor ~ ~ \\lfloor \\frac { y _ 3 } { 4 } \\rfloor ] ) , \\\\ & A = c i r c s h i f t ( A , [ \\lfloor \\frac { y _ 1 + y _ 2 } { 1 0 } \\rfloor ~ ~ \\lfloor \\frac { y _ 2 + y _ 3 } { 1 0 } \\rfloor ~ ~ - \\lfloor \\frac { y _ 3 + y _ 1 } { 1 0 } \\rfloor ] ) , \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} c ^ { \\circ } : = \\min _ { ( k , l ) \\in \\mathcal { K } } c ^ { \\circ } _ { k l } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} \\sup _ { j \\geq 1 } \\sup _ { \\theta \\in [ 0 , 2 \\pi ) } \\left | \\sum _ { n = 1 } ^ j \\frac { \\omega _ n } { n \\psi ( n ) } e ^ { i \\gamma _ { T ^ n } ( \\theta ) } \\right | \\leq C ( \\omega ) . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} | D ^ j u | \\left ( \\partial B _ r ( x ) \\right ) = 0 . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\mu ( x ) = \\frac { 1 } { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } f _ i , f _ i \\in S _ E , \\ , i = 1 , 2 , \\dots , k + 1 . \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Bigg | \\frac 1 N \\sum _ { n = 1 } ^ N F ( S _ X ^ n ( x , \\ldots , x ) ) & G ( S _ Y ^ n ( y , \\ldots , y ) ) \\\\ & - \\Bigg ( \\frac 1 N \\sum _ { n = 1 } ^ N F ( S _ X ^ n ( x , \\ldots , x ) ) \\Bigg ) \\Bigg ( \\frac 1 N \\sum _ { n = 1 } ^ N G ( S _ Y ^ n ( y , \\ldots , y ) ) \\Bigg ) \\Bigg | = 0 . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} ( o s c _ \\delta \\bar { g } _ 1 ) ( x ) = \\sup _ { y \\in B _ \\delta ( x ) } \\sup _ { z \\in \\overline { B _ r ( y ) } } f ( z ) - \\inf _ { y \\in B _ \\delta ( x ) } \\sup _ { z \\in \\overline { B _ r ( y ) } } f ( z ) . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\mathcal { P } ^ \\mathcal { C } = \\bigcup _ { n \\geq 0 } \\mathcal { P } _ n ^ \\mathcal { C } = \\{ \\lambda : I ( \\mathcal { C } ) \\to \\mathcal { P } \\mid \\sum _ { U \\in I ( C ) } | \\lambda ( U ) | < \\infty \\} \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\Phi _ j ( t ) = t ^ j + p ^ c t ^ { k + 1 } \\Psi _ j ( t ) . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} a _ { m + k } & = \\log ( N ( \\star _ { i = 1 } ^ { m + k } G , \\alpha ^ { m + k + 1 } ) ) \\le \\log ( N ( \\star _ { i = 1 } ^ { m } G , \\star _ { i = 1 } ^ { m } \\alpha ^ { 2 } ) N ( \\star _ { i = 1 } ^ { k } G , \\star _ { i = 1 } ^ { k } \\alpha ^ { 2 } ) ) \\\\ & = \\log ( N ( \\star _ { i = 1 } ^ { m } G , \\star _ { i = 1 } ^ { m } \\alpha ^ { 2 } ) + \\log ( N ( \\star _ { i = 1 } ^ { k } G , \\star _ { i = 1 } ^ { k } \\alpha ^ { 2 } ) ) = a _ { m } + a _ { k } . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{gather*} H _ 4 ( a , b ; c , d ; x , y ) = \\sum _ { m , n = 0 } ^ \\infty \\frac { ( a ) _ { 2 m + n } ( b ) _ n } { ( c ) _ m ( d ) _ n } \\frac { x ^ m y ^ n } { m ! n ! } , \\end{gather*}"}
-{"id": "5255.png", "formula": "\\begin{align*} \\aleph _ s ( \\tau ) = ( \\chi _ 1 ( \\tau ) , \\chi _ 2 ( \\tau ) ) = \\left ( - \\tau ^ 2 , - \\tau \\prod _ { \\kappa = 1 } ^ s ( \\tau ^ 2 - \\kappa ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\sigma ^ i ( q , \\gamma ( q ) ) \\frac { \\partial \\gamma _ j } { \\partial q ^ i } ( q ) = \\sigma _ j ( q , \\gamma ( q ) ) . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\xi _ S ( O ) = { \\rm T r } ( J ( O ) S ) , \\ { \\rm f o r } \\ O \\in { \\rm S O } ( 2 n ) . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\rho ( z ) = \\sqrt { \\det \\Sigma ^ { - 1 } } e ^ { - \\pi \\ , \\Sigma ^ { - 1 } z ^ { 2 } } , \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{gather*} \\gamma ^ 0 { } _ n = M _ { n i } \\omega ^ i + L ^ a \\theta _ a . \\end{gather*}"}
-{"id": "7629.png", "formula": "\\begin{align*} H _ 1 ( t ) = \\varrho \\min \\left \\{ 1 ; \\left [ \\frac { h ( t ) } { c _ { 2 , \\infty } } \\right ] ^ { \\alpha } \\right \\} = \\varrho \\min \\left \\{ 1 ; \\frac { 1 } { c _ { 2 , \\infty } ^ { \\alpha } } \\left [ \\frac { 1 } { \\| 1 / c _ { 2 , 0 } ^ \\alpha \\| _ { L ^ { \\infty } ( \\Omega ) } } + \\alpha ( \\alpha + 1 ) k _ 3 t \\right ] ^ { - 1 } \\right \\} \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} f ( t , x , y ) = u _ { < 1 } ( t , x - \\frac { 1 } { 4 t } y ^ 2 , y ) , \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} \\begin{cases} - \\mbox { \\rm d i v } \\left ( | \\nabla v | ^ { p - 2 } \\nabla v \\right ) = f ^ \\star & \\mbox { i n } \\O ^ \\star \\ , , \\\\ v = 0 & \\mbox { o n } \\partial \\O ^ \\star \\ , . \\end{cases} \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} ( f ^ { \\mathrm { a n } } ) ^ * ( g ) = ( d g _ 0 - \\log | \\varphi | + \\lambda ) + ( d h - \\lambda ) = d g - \\log | \\varphi | , \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} B ( x , v ) & = \\frac { ( 3 v ^ 2 x ^ 2 - 3 v x + 1 ) x ^ 3 } { ( 1 - x ) ^ 2 ( 1 - v x ) ^ 2 ( 1 - 2 v x ) } , \\textrm { a n d } \\\\ B ' ( x , v ) & = \\frac { x ^ 2 } { 1 - x } + \\frac { x } { 1 - x } A ' ( x , v ) . \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} & \\left ( \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) | v _ 1 | d v \\right ) ^ 2 - \\Big | \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) v d v \\Big | ^ 2 \\cr & \\geq \\left ( \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) | v _ 1 | d v \\right ) ^ 2 - \\Big \\{ \\sum _ { 1 \\leq i \\leq 3 } \\Big | \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) v _ i d v \\Big | \\Big \\} ^ 2 \\cr & \\qquad = \\left ( \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) | v _ 1 | d v \\right ) ^ 2 - \\Big ( \\int _ { \\mathbb { R } ^ 3 } \\Phi ( f ) v _ 1 d v \\Big ) ^ 2 - R \\cr & \\qquad \\equiv I - R . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} W ^ T = \\left ( A + K \\right ) J ^ T \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} a _ 2 s _ 2 b _ 2 - a _ 1 s _ 1 b _ 1 & = a _ 1 u _ 1 s _ 2 u _ 2 b _ 1 - a _ 1 s _ 1 b _ 1 \\\\ & = a _ 1 ( u _ 1 s _ 2 u _ 2 - s _ 1 ) b _ 2 = a _ 1 ( s _ 2 , s _ 1 ) _ { \\overline { s _ 1 } } b _ 1 . \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} f ( t , x , y ) = \\chi _ M ( t ^ { - 1 } x ) u _ { < ( t M ) ^ { - 1 } } ( t , x - \\frac { 1 } { 4 t } y ^ 2 , y ) , f ( t , x , y ) = \\chi _ M ( t ^ { - 1 } x ) u _ N ( t , x - \\frac { 1 } { 4 t } y ^ 2 , y ) , \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} X _ { \\lambda } = ( \\alpha _ 1 ( \\lambda ) S _ { \\lambda } , \\alpha _ 2 ( \\lambda ) X _ { \\mu } ) \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} X : = L ^ { p } ( \\Omega ; \\gamma ( L ^ 2 ( \\mathbb R _ + , [ M ^ { c } ] ; H ) , L ^ q ( S ) ) ) \\times \\mathcal I _ { p , q } \\times \\mathcal A _ { p , q } ^ { \\mathcal T } \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} B ( \\phi ^ * h , \\phi ^ * h ' ) = \\phi ^ * B ( h , h ' ) ~ . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\| S ( y ) - \\abs { b _ n } \\| _ { E ^ { \\times } } & = \\| V ( S ( y ) ) - V ( \\abs { b _ n } ) \\| _ { C _ { E ^ { \\times } } } = \\| y - V ( \\abs { b _ n } ) \\| _ { C _ { E ^ { \\times } } } \\\\ & = \\| \\Phi _ y - \\Phi _ { V ( \\abs { b _ n } ) } \\| _ { E ^ * } \\to 0 . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} L & = \\partial _ t + b \\cdot \\nabla - \\Delta + \\frac { 2 } { r } \\partial _ r , \\\\ N & = n \\cdot \\nabla . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} D ^ { \\bf a } _ k = k ^ { j _ { 0 } } \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ { l j _ { 0 } } \\lambda ^ k _ l . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\tilde { Q } _ p ( 0 , v , s ) = { 1 \\over s } - { p \\over s } \\thinspace \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 3 / 2 } \\psi _ { s , F } ( - v ) + { \\rm O } \\left ( p ^ 2 \\right ) . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ N { \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i + 1 ) - z ) } } \\\\ \\qquad { } = \\frac { \\Gamma _ q ( \\lambda _ 1 + \\theta ( N - 1 ) - z ) } { \\Gamma _ q ( \\lambda _ 2 + \\theta ( N - 1 ) - z ) } \\cdots \\frac { \\Gamma _ q ( \\lambda _ { N - 1 } + \\theta - z ) } { \\Gamma _ q ( \\lambda _ N + \\theta - z ) } \\frac { \\Gamma _ q ( \\lambda _ N - z ) } { \\Gamma _ q ( \\lambda _ 1 + \\theta N - z ) } , \\end{gather*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\| \\ + F \\| \\leq \\sum _ { i < n } y _ i = ( k - 1 ) \\frac { 1 } { k - 1 } + ( n - k + 1 ) \\frac { 1 } { k } = \\frac { n + 1 } { k } = 1 / p . \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} w _ { \\xi , \\eta } ( x ) = \\langle x \\xi , \\eta \\rangle , \\xi , \\eta \\in H , \\ , x \\in B ( H ) . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\mathrm { o p t } ( \\alpha _ 1 , \\alpha _ 2 , R ) = \\begin{cases} 1 & \\mbox { i f } \\alpha _ 1 ( \\mathbb { O } _ 1 ) \\cup \\alpha _ 2 ( \\mathbb { O } _ 2 ) \\subset R \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\tau _ { i } = \\bigg ( q ^ { W } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) \\bigg ) ^ 2 + \\bigg ( q ^ { W } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) - q _ { i } ( x _ { i + \\frac { 1 } { 2 } } ) \\bigg ) ^ 2 , \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} C _ n = R _ n = 0 \\qquad \\mbox { f o r a n y } n \\in { \\cal F } \\ , , \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\int _ X & f ( \\pi ( x , y ) ) \\ > \\overline { g ( \\pi ( z , y ) ) } \\ > d \\omega _ X ( y ) = T ^ { f \\circ \\pi } ( \\overline { g \\circ \\pi _ z } ) ( x ) = T _ f ( \\overline { g \\circ \\pi _ z } ) ( x ) \\\\ & = ( ( f ^ - * \\bar g ) \\circ \\pi _ z ) ( x ) = ( f ^ - * \\bar g ) ( \\pi ( z , x ) ) = \\int _ D f ( h ) \\ > \\overline { g ( \\pi ( z , x ) * h ) } \\ > d \\omega _ D ( h ) \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} H ^ { - 2 \\eta } ( x , y ) \\geq \\phi _ 1 ( x ) \\ , \\phi _ 2 ( y ) : = \\ , U ^ { - 2 \\eta } ( x ) \\ ; \\left ( 1 + \\frac 1 2 \\ , | y | ^ 2 \\right ) ^ { - 2 \\eta } \\ , . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} ( x - d ) ^ k + x ^ k + ( x + d ) ^ k = y ^ n , \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\frac { 1 } { a _ i - \\omega _ B ( z ) } = m _ { \\mu _ A } ( \\omega _ B ( z ) ) = m _ { \\mu _ A \\boxplus \\mu _ B } ( z ) . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\| \\sum _ { n = 2 ^ p } ^ { 2 ^ q - 1 } a _ n P ^ n f \\| _ { L ^ 2 ( m ) } ^ 2 \\le \\sum _ { n \\in \\N } \\Big ( \\sum _ { k = 2 ^ p } ^ { 2 ^ q - 1 } | a _ k | u _ { n + k } \\Big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} B ^ \\perp = \\{ x \\in A \\ | \\ q ( x + y ) = q ( x ) q ( y ) \\ \\forall y \\in B \\} \\ . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} { \\cal D } _ 8 : = \\begin{pmatrix} D _ 8 & 0 \\\\ 0 & - D _ 8 \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\xi ^ T \\xi = \\eta ^ T B B ^ T \\eta . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\left | m - \\left ( \\log n - \\log \\left ( f + 1 \\right ) - h \\right ) \\frac { n } { 2 } \\right | & = o \\left ( n \\right ) , \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | \\varphi | ^ 2 ~ d x + \\frac { 1 } { 2 \\pi } \\sum _ n \\int _ 0 ^ { \\infty } \\left ( \\frac { \\psi _ n ^ 2 } { 2 } \\right ) ' \\widehat { u \\varphi } \\bar { \\hat \\varphi } ~ d \\xi + \\frac { 1 } { 2 \\pi } \\int _ { \\real } \\widehat { u \\varphi } ' \\bar { \\hat \\varphi } ~ d \\xi = 0 . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} { p \\over s ( s + p ) } = \\int _ 0 ^ \\infty d F \\thinspace \\left [ - a ( F ) \\thinspace \\psi _ { s , F } ( - v ) + b ( F ) \\thinspace \\psi _ { s + p , F } ( v ) \\right ] . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathfrak { m } ^ { ( p , p ) } \\big ] = & \\mathbb { E } \\big [ ( O _ \\prec ( \\hat { \\Pi } ^ 2 ) + O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ) ) \\mathfrak { m } ^ { ( p - 1 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 2 , p ) } \\big ] \\\\ & \\qquad + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 1 , p - 1 ) } \\big ] . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} Z _ { \\bold F _ 3 } & = \\ker ( [ P _ 0 ] \\bold F _ 3 \\oplus [ P _ 1 ] \\bold F _ 3 \\oplus [ P _ t ] \\bold F _ 3 \\oplus [ P _ \\infty ] \\bold F _ 3 \\to \\bold F _ 3 ) \\\\ & a _ 0 [ P _ 0 ] + a _ 1 [ P _ 1 ] + a _ t [ P _ t ] + a _ { \\infty } [ P _ \\infty ] \\mapsto a _ 0 + a _ 1 + a _ t + a _ { \\infty } , \\\\ H _ { \\bold F _ 3 } & = Z _ { \\bold F _ 3 } / ( [ P _ 0 ] - [ P _ 1 ] - [ P _ t ] + [ P _ \\infty ] ) . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} V ( r , z ) : = - | r | ^ { n - 1 } \\int _ 0 ^ z f ( r , s ) \\ , d s \\ , . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle v '' ( t ) + A v ( t ) - \\int _ 0 ^ t \\ H ( t - s ) A v ( s ) d s = 0 \\ , , t \\in [ 0 , T ] \\ , , \\\\ \\\\ v ( 0 ) = z _ 0 \\ , , v ' ( 0 ) = - z _ 1 \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\int f ( x ' , w ) \\ , g ( y ) \\ , d \\sigma ( x ' , w , y ) = 0 \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} z = f ( \\tilde { z } ) = \\tanh \\left ( \\frac { \\tilde { z } } { 2 } \\right ) , \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\begin{aligned} u & = K ^ h _ { ( \\Lambda , d \\mu ) } \\bigl [ { ( 1 - \\rho ) A \\over H } \\bigr ] + \\cr & \\bigl ( { 2 i \\pi \\over h } \\bigr ) ^ { 1 / 2 } K ^ h _ { ( \\Lambda _ + , d \\mu _ + ) } \\bigl [ ( 1 - \\theta ( t _ 0 ) ) A ( t _ 0 ) \\bigr ] \\cr & + { i \\over h } \\int _ 0 ^ { t _ 0 } \\exp \\bigl [ i \\int _ { \\alpha ^ * } ^ { g _ H ^ t ( \\alpha ^ * ) } p \\ , d x \\bigr ] \\cr & K ^ h _ { ( \\Lambda _ t , d \\mu _ t ) } \\rho \\theta ( t ) A ( t ) \\ , d t + { \\cal O } ( h ^ { 1 / 2 } ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\frac { | b | } { a } < \\nu _ { n , e } a n d \\gcd ( a , b ) = 1 . \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} F ( t \\xi ) = | t | F ( \\xi ) , t \\in \\R , \\ , \\xi \\in \\R ^ { n } , \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} C _ { 6 } = \\max _ { \\stackrel { s , t \\in [ a , b ] } { | u | \\leq \\| \\varphi \\| _ \\infty + \\delta } } \\left | \\frac { \\partial ^ { 2 } \\kappa } { \\partial u ^ 2 } ( s , t , u ) \\right | . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\alpha ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) = \\frac { 1 } { a d - b c } \\left ( \\begin{matrix} a d x _ 1 - b c x _ 3 & a b ( x _ 3 - x _ 1 ) \\\\ c d ( x _ 4 - x _ 2 ) & a d x _ 2 - b c x _ 4 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\nabla \\theta _ { 1 } ^ { \\varepsilon } = \\nabla _ x \\theta ^ { 0 } + \\left ( \\nabla _ { y } \\bar { \\theta } \\right ) ^ { \\varepsilon } \\nabla _ { x } \\theta ^ { 0 } + \\varepsilon \\bar { \\theta } ^ { \\varepsilon } \\nabla _ { x } \\nabla \\theta ^ { 0 } + \\varepsilon \\left ( \\nabla _ { x } \\bar { \\theta } \\right ) ^ { \\varepsilon } \\nabla _ { x } \\theta ^ { 0 } . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t c _ 1 - d _ 1 \\Delta c _ 1 = - k _ 1 c _ 1 + k _ 3 c _ 2 ^ { \\alpha + 1 } , & x \\in \\Omega , t > 0 , \\\\ \\partial _ t c _ 2 - d _ 2 \\Delta c _ 2 = k _ 1 \\alpha c _ 1 + k _ 2 c _ 2 ^ { \\alpha } c _ 3 - k _ 3 ( \\alpha + 1 ) c _ 2 ^ { \\alpha + 1 } , & x \\in \\Omega , t > 0 , \\\\ \\partial _ t c _ 3 - d _ 3 \\Delta c _ 3 = k _ 1 c _ 1 - k _ 2 c _ 2 ^ { \\alpha } c _ 3 , & x \\in \\Omega , t > 0 , \\end{cases} \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} T _ { \\rho , p , q } : = \\inf \\left \\{ \\| u \\| _ { L ^ { p } _ { a _ { 1 } , a _ { 2 } , \\R _ { 1 } , \\R _ { 2 } } ( \\mathbb { G } ) } ^ { p } : u \\in L ^ { p } _ { a _ { 1 } , a _ { 2 } } ( \\mathbb { G } ) \\ ; \\ ; { \\rm a n d } \\ ; \\ ; \\int _ { \\mathbb { G } } | u ( x ) | ^ { q } d x = \\rho \\right \\} , \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} s x ( y ^ { - 1 } x ) ^ n = y t ( y ^ { - 1 } x ) ^ n = y ( x ^ { - 1 } y ) ^ n t \\ \\mbox { a n d } \\ s y ( x ^ { - 1 } y ) ^ n = x t ' ( x ^ { - 1 } y ) ^ n = x ( y ^ { - 1 } x ) ^ n t ' . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} F _ { \\epsilon } : = F _ 0 + \\log \\left ( \\frac { \\omega _ { \\epsilon } ^ n } { \\omega _ 0 ^ n } \\cdot \\prod _ { i = 1 } ^ d ( \\epsilon ^ 2 + | s _ i | _ { H _ i } ^ 2 ) ^ { ( 1 - \\beta ) \\tau _ i } \\right ) . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\mathcal { O } _ \\rho = \\Omega \\setminus \\cup _ { j = 1 } ^ k B _ { \\rho } ( \\zeta _ j ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , \\ \\forall j \\in \\{ 1 , 2 , \\dots , k \\} \\backslash \\{ j ' , j '' \\} . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\beta ^ { ( j ) } _ { i } \\leq a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\cdots p ^ { ( j ) } _ { r _ j } \\beta _ { + } + ( \\mu _ { \\mathfrak { f } ^ { ( j ) } _ i } - 1 ) + \\sum _ { i ' = i + 2 } ^ { r _ j } \\frac { a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\cdots p ^ { ( j ) } _ { i ' - 1 } } { a ^ { ( j ) } _ { i ' } } ( \\mu _ { \\mathfrak { f } ^ { ( j ) } _ { i ' } } - 1 ) - \\frac { a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\cdots p ^ { ( j ) } _ { i ' } } { a ^ { ( j ) } _ { i ' } } ( \\mu _ { \\mathfrak { f } ^ { ( j ) } _ { i ' - 1 } } - 1 ) . \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\frac { x _ { k + 1 } - x _ k } { \\phi _ 1 ( a , h ) } & = a x _ k - b x _ { k + 1 } y _ k , \\\\ \\frac { y _ { k + 1 } - y _ k } { \\phi _ 2 ( c , h ) } & = - c y _ k + d x _ { k + 1 } y _ k , \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} u ( X ) = \\int f ( q ) d \\omega ^ X ( q ) . \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\begin{pmatrix} \\overline c _ { 1 , 1 } & . . & . . & \\overline c _ { 1 , T } \\\\ . . & . . & . . & . . \\\\ \\cdot & \\cdot & \\cdot & \\cdot \\\\ \\overline c _ { T - 1 , 1 } & . . & & \\overline c _ { T - 1 , T - 1 } \\end{pmatrix} \\begin{pmatrix} a _ { 1 , T } \\\\ a _ { 2 , T } \\\\ \\cdot \\\\ a _ { T - 1 , T } \\end{pmatrix} = - a _ { T , T } \\begin{pmatrix} a _ { 1 , T } \\\\ a _ { 2 , T } \\\\ \\cdot \\\\ a _ { T - 1 , T } \\end{pmatrix} \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} m _ { ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ n ) } ( z _ 1 , z _ 2 , \\ldots , z _ n ) = \\sum _ { \\pi \\in S _ { \\lambda } } z _ { \\pi ( 1 ) } ^ { \\lambda _ 1 } z _ { \\pi ( 2 ) } ^ { \\lambda _ 2 } \\ldots z _ { \\pi ( n ) } ^ { \\lambda _ n } , \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\| A ^ { \\frac { r } { 2 } } f \\| = \\| f \\| _ { \\mathbb { V } ^ r } \\le \\ , \\| f \\| ^ { 1 - r } \\| f \\| ^ r _ { \\mathbb { V } } \\le \\ , \\| f \\| ^ { 1 - r } \\| \\nabla f \\| ^ r . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\Delta u = - K - e ^ { 2 u } ~ , \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} & T _ w T _ { w ' } = T _ { w w ' } , \\hbox { i f } \\l ( w w ' ) = \\l ( w ) + \\l ( w ' ) , \\hbox { a n d } \\cr & ( T _ s + 1 ) ( T _ s - q ) = 0 \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { \\tau , t } } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} V + V ' = \\phi _ 8 - \\phi _ 7 & & & I = I _ 7 = - I _ 8 . \\\\ \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\mathsf { M o n } ( C ) = \\phi _ { v _ { n } , e _ { n - 1 } } ^ { - 1 } \\cdot \\phi _ { v _ { n - 1 } , e _ { n - 1 } } \\cdot \\dots \\cdot \\phi _ { v _ 2 , e _ 1 } ^ { - 1 } \\cdot \\phi _ { v _ 1 , e _ 1 } \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} c _ { J , f } = \\int _ M S ( J , f ) \\ , f ^ { - 2 m - 1 } \\omega ^ m / \\int _ M f ^ { - 2 m - 1 } \\omega ^ m \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\tilde { l } ^ { G E L } ( \\theta ) = - 2 \\log \\left [ \\left ( \\frac { 1 / 2 } { F _ n ( \\theta ) } \\right ) ^ { n F _ n ( \\theta ) } \\left ( \\frac { 1 - 1 / 2 } { 1 - F _ n ( \\theta ) } \\right ) ^ { n \\left ( 1 - F _ n ( \\theta ) \\right ) } \\right ] \\ , . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\frac { \\omega _ \\beta ( z ) } { \\omega _ \\alpha ( z ) } = O \\left ( ( \\omega _ \\alpha ( z ) ^ { - 1 } \\right ) \\ , . \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} y _ k ( \\nu ) = \\sqrt { p _ k ( \\nu ) } h _ k ( \\nu ) s _ k + \\sqrt { p _ { \\bar k } ( \\nu ) } h _ k ( \\nu ) s _ { \\bar k } + n _ k , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} \\nu _ { T , \\gamma } \\{ \\eta , \\ , \\eta ( x ) = 1 \\} \\ ; = \\ ; \\gamma ( \\sigma _ T ( x ) ) \\ ; . \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\Phi ^ { A } \\left ( q ^ i , \\gamma _ i ( q ) , \\dot { q } ^ i , \\frac { \\partial \\gamma _ j } { \\partial q ^ i } \\dot { q } ^ i \\right ) = 0 , \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ \\xi _ { \\theta } \\otimes \\xi _ { \\theta } : \\xi _ { \\theta } = \\sum _ { j = 1 } ^ { n } e ^ { i \\theta _ { j } } e _ { j } \\} . \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\int _ { - 1 } ^ 1 f ( t ) ( t ^ n - 1 ) d t = 0 \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\gamma ( [ J X , Z ] , J Y ) = \\gamma ( [ X , Z ] , Y ) , \\ ; \\forall X , Y \\in P , Z \\in Q . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} V ( x ) = \\int _ { \\R ^ n } d y \\left ( \\frac { 1 } { t } F \\ , \\chi _ { \\Omega ^ t \\setminus \\Omega ^ 0 } \\right ) ( y ) P ( x - y ) . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} ( \\pi , \\eta ) \\ , : = \\ , \\{ t : H \\to K \\ , : \\ , \\eta ( g ) \\circ t = t \\circ \\pi ( g ) \\ , , \\ , \\forall g \\in G \\} \\ . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} \\| \\mathcal { K } _ m ' ( \\varphi _ m ) ( I - Q _ n ) \\mathcal { K } _ m ' ( \\varphi _ m ) ( I - Q _ n ) x \\| _ \\infty = O ( h ^ { 4 r } ) . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} | k | x \\left | \\int _ 0 ^ { \\frac 1 2 } \\frac { e ^ { i \\xi } - 1 } { \\xi } \\frac { 1 } { \\xi - k x } ~ d \\xi \\right | & \\le C | k | x \\int _ 0 ^ { \\frac 1 2 } \\frac { 1 } { | k | x - \\xi } ~ d \\xi \\\\ & = C | k | x \\log \\left ( \\frac { | k | x } { | k | x - \\frac 1 2 } \\right ) \\\\ & = C | k | x \\log \\left ( 1 + \\frac { 1 } { 2 | k | x - 1 } \\right ) \\\\ & \\le C | k | x \\log \\left ( 1 + \\frac { 1 } { | k | x } \\right ) \\le C | k | ^ { \\epsilon } | x | ^ { \\epsilon } . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} r _ d ( G ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { R _ d ( F ) } { d _ F ^ { 2 } } , \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} T _ { 0 } ^ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\tau _ { i } \\left ( y \\right ) d y , T _ { j k } ^ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\tau _ { i } \\left ( y \\right ) \\frac { \\partial \\bar { \\theta } ^ { j } } { \\partial y _ { i } } d y , \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\min \\{ - \\Delta u ^ { t , \\bar t } , u ^ { t , \\bar t } - h ^ { t , \\bar t } \\} = 0 \\mbox { i n } \\R ^ n , \\lim _ { | x | \\to \\infty } u ^ { t , \\bar t } ( x ) = c ^ { t , \\bar t } \\lim _ { | x | \\to \\infty } \\frac { u ^ { t , \\bar t } ( x ) } { - \\log | x | } = c ^ { t , \\bar t } \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} d \\mathfrak { m } _ { B H } : = \\frac { \\omega _ n } { V o l ( B _ x M ) } d x ^ 1 \\wedge \\dots \\wedge d x ^ n , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ 2 } \\int _ { - 1 } ^ 1 f ( t ) ( t ^ n - 1 ) ^ 2 d t = 0 . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} L ( P ( \\lambda ) , \\dotsc , P ( \\lambda ) ) \\ = \\ \\sum _ { a _ 1 , \\dotsc , a _ n = 1 } ^ { n } L ( P _ { a _ 1 } , P _ { a _ 2 } , \\dotsc , P _ { a _ n } ) \\lambda _ { a _ 1 } \\lambda _ { a _ 2 } \\dotsb \\lambda _ { a _ n } \\ , . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} F ( z + \\zeta _ 0 ) & = \\frac 1 { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma } F ( \\zeta ) \\bigg ( \\frac { 1 } { \\zeta - \\zeta _ 0 - z } - \\frac { 1 } { \\zeta - \\zeta _ 0 + z } \\bigg ) \\ , \\mathrm { d } \\zeta \\\\ & = \\int _ \\Gamma K _ z ( \\zeta , \\zeta _ 0 ) F ( \\zeta ) \\ , \\mathrm { d } \\zeta . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\alpha ( x | y , z ) = \\frac { \\alpha ( x , y , z ) \\alpha ( x y x ^ { - 1 } , x z x ^ { - 1 } , x ) } { \\alpha ( x y x ^ { - 1 } , x , z ) } \\ . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\mathbf { w } _ { 0 } ^ { \\varGamma _ { 1 } } = \\left ( \\begin{array} { c } \\begin{array} { c } \\phantom { - } \\cos \\omega _ { 1 } \\mathbf { t } _ { 1 } \\end{array} \\\\ - \\cos \\omega _ { 2 } \\mathbf { t } _ { 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\omega & = \\omega ^ { r } e _ { r } + \\omega ^ { \\theta } e _ { \\theta } + \\omega ^ { z } e _ { z } \\\\ & = ( - \\partial _ z u ^ { \\theta } ) e _ { r } + ( \\partial _ { z } u ^ { r } - \\partial _ { r } u ^ { z } ) e _ { \\theta } + \\Big ( \\partial _ r u ^ { \\theta } + \\frac { u ^ { \\theta } } { r } \\Big ) e _ { z } , \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} \\tilde { L } ( \\tilde { F } , x ) = \\prod _ { i = 1 } ^ n \\left \\{ \\tilde { F } ( x _ i ) - \\tilde { F } ( x _ i - ) \\right \\} \\ , , \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} B _ \\xi ( r ) = \\{ x \\in \\widetilde { M } : b _ \\xi ( x _ 0 , x ) \\ge r \\} , \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} f ( u ) = \\min \\{ f ( U ) \\mid U \\subset V , t \\not \\in U u \\in U \\} . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\phi _ t ( \\bar { x } , \\bar { t } ) \\leq \\psi _ t ^ \\epsilon ( \\bar { x } , \\bar { t } ) = \\theta _ t ( \\bar { x } , \\bar { t } ) + \\delta < | \\lambda _ 1 | - A _ { k + 1 } - \\sigma / 3 . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} G ( 1 , x ) \\geq \\min \\left \\{ - C _ { 1 } - 2 \\sqrt { \\frac { C _ { 1 } } { C _ { 2 } } } , - \\frac { 8 } { 3 C _ { 2 } } \\right \\} = - C . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\begin{aligned} A ( t ) & \\sim k _ A t ^ { - 1 / 3 } \\\\ C ( t ) \\lesssim B ( t ) & \\sim k _ B t ^ { 1 / 3 } \\\\ D ( t ) & \\rightarrow K _ 1 \\\\ E ( t ) & \\sim k _ E t . \\end{aligned} \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\mathfrak { g } = \\bar { \\mathfrak { h } } \\oplus \\mathfrak { b } . \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} b ' ( n ; i ) & = \\sum _ { j = 1 } ^ i b ' ( n - 1 ; j ) , \\\\ b ( n ; i ) & = b ' ( n - 1 ; i ) + \\binom { n - 3 } { i } , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} \\lambda \\pi ^ + + A _ 2 \\pi ^ + = 0 \\hbox { i n } \\ : D _ 2 , \\lambda \\pi ^ + + B _ { 2 , + } \\pi ^ + = 0 \\hbox { i n } \\ : D _ 2 ^ + , \\lambda \\pi ^ + + B _ { 2 , - } \\pi ^ + = 0 \\hbox { i n } \\ : D _ 2 ^ - , \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} a _ { \\xi ' } : = - \\min \\{ 0 , \\nu _ { \\xi ' } ( f _ 1 ) , \\dots , \\nu _ { \\xi ' } ( f _ r ) \\} , \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} D ( s ) = \\sum _ { n \\geq 1 } \\frac { a ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} \\aligned & \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } ( [ r _ - ] ) \\\\ & = \\sum _ { r _ + \\in R _ { \\alpha _ + } } [ ( e v _ - , e v _ + ) ^ { - 1 } ( ( r _ - , r _ + ) ) \\\\ & \\qquad \\cap ( \\mathcal M ( \\alpha _ - , \\alpha _ + ) ^ { \\boxplus \\tau _ 0 } , \\widehat { \\mathcal U ^ + } ( \\alpha _ - , \\alpha _ + ) , \\widehat { \\frak O } ( \\alpha _ - , \\alpha _ + ) , \\widehat { { \\frak s ^ { + \\epsilon } } } ( \\alpha _ - , \\alpha _ + ) ) ] [ r _ + ] . \\endaligned \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\Phi _ Y ( s ) = \\int _ 0 ^ \\infty t ^ s \\phi _ Y ( t ) \\frac { d t } { t } , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} g = \\frac { d \\rho ^ 2 + h } { \\rho ^ 2 } \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\frac { x ^ { d + 2 } } { 1 - x } \\big ( K ( x ) - 1 \\big ) \\sum _ { j = 1 } ^ { d + 1 } \\frac { 1 } { ( 1 - x ) ^ { j - 1 } } \\ , . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} { q } _ { i } ( x ) = \\left \\{ \\begin{array} { l } { q } ^ { T } _ { i } , ~ ~ ~ { \\rm i f } \\ \\ m T B V ^ { T } _ i < m T B V ^ { W } _ i , \\\\ { q } ^ { W } _ { i } ~ ~ ~ ~ \\mathrm { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} G ( s , z ) = \\frac { \\Gamma ( s - \\tfrac { 1 } { 2 } + z ) \\Gamma ( s - \\tfrac { 1 } { 2 } - z ) } { \\Gamma ( s ) \\Gamma ( s + k - 1 ) } , \\ ; \\mathcal { Z } ( s , w , z ) = \\frac { \\zeta ( s + w - \\frac { 1 } { 2 } + z ) \\zeta ( s + w - \\frac { 1 } { 2 } - z ) } { \\zeta ^ * ( 1 + 2 z ) } . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{gather*} F ^ a : = \\varepsilon _ { b c } F ^ { a b c } , \\end{gather*}"}
-{"id": "5158.png", "formula": "\\begin{align*} Q \\left ( x , t \\right ) = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r u \\left ( x , r , t \\right ) d r . \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} l ( H _ i ) = 1 + l ( B _ i ) \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} h ^ 0 ( N _ Y ( - 1 ) & ( y ) [ x \\to y ] [ 2 y \\to x ] [ w _ 1 \\to x ] [ w _ 2 \\to x ] [ z _ 3 \\to w _ 3 ] [ w _ 3 \\to z _ 3 ] ) \\\\ & = h ^ 0 ( N _ X ( - 1 ) [ z _ 1 \\to w _ 1 ] [ w _ 1 \\to z _ 1 ] [ z _ 2 \\to w _ 2 ] [ w _ 2 \\to z _ 2 ] [ z _ 3 \\to w _ 3 ] [ w _ 3 \\to z _ 3 ] ) \\\\ & = h ^ 0 ( N _ { C } ( - 1 ) ( - q _ 1 - \\cdots - q _ 5 ) ) . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\sigma _ j = \\inf _ { \\substack { V \\subset H ^ 1 ( \\Omega ) , \\\\ d i m V = j + 1 } } \\sup _ { \\substack { 0 \\neq u \\in V , \\\\ \\int _ \\Sigma u ^ 2 d \\sigma = 1 } } \\int _ \\Omega | \\nabla u | ^ 2 d v , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\mathcal { D } = \\left \\{ \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\left | 0 . 5 \\leqslant \\frac { \\omega _ { 1 } T _ { 0 } } { \\pi } \\leqslant 1 . 5 , \\ , 0 . 5 \\leqslant \\frac { \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) } { \\pi } \\leqslant 3 \\right . \\right \\} . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\Gamma _ k : = \\left \\{ \\gamma \\in C ( \\mathbb { D } _ k , H ^ 1 _ r ( \\mathbb { R } ^ N ) ) ~ | ~ \\gamma ~ ~ \\gamma ( \\sigma ) = \\gamma _ { 0 k } ( \\sigma ) ~ ~ \\sigma \\in \\mathbb { S } ^ { k - 1 } \\right \\} . \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} f ( \\Psi , \\dots , \\Psi ) = \\big ( \\Psi , \\dots , \\Psi , b _ 1 + b _ 2 \\Psi ^ { - 1 } + \\dots + b _ n \\Psi ^ { 1 - n } ) . \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\Xi _ g ( \\Gamma _ r ( z ) ) = \\Gamma _ r ( \\Xi _ g ( z ) ) , \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} { } _ a \\mathtt { B } _ { b , p , c } ( x ) : = \\sum _ { k = 0 } ^ \\infty \\frac { ( - c ) ^ k } { k ! \\ ; \\mathrm { \\Gamma } { \\left ( a k + p + \\frac { b + 1 } { 2 } \\right ) } } \\left ( \\frac { x } { 2 } \\right ) ^ { 2 k + p } \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ 1 ( t ) & = \\gamma _ 1 \\\\ \\lambda _ 2 ( t ) & = \\lambda _ 0 b _ M ( T - t ) \\\\ \\lambda _ 3 ( t ) & = \\lambda _ 0 \\bigl ( b _ E ( T - \\tau _ s ) + \\frac { 1 } { 2 } b _ M k _ M ( \\tau _ s - t ) ^ 2 + ( b _ E \\\\ & \\mbox { } - a _ M b _ M k _ M \\frac { \\gamma _ 1 } { \\lambda _ 0 } + b _ M k _ M ( T - \\tau _ s ) ) ( \\tau _ s - t ) \\bigr ) \\end{aligned} \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} \\left ( \\chi _ { _ { I } } \\left ( m _ { 0 } \\right ) f \\right ) \\left ( t \\right ) = \\begin{cases} f \\left ( t \\right ) & t \\in I \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\begin{cases} * _ { \\varepsilon } & : S ^ 1 \\times P \\rightarrow P \\\\ \\theta * _ { \\varepsilon } p & = \\phi _ { \\varepsilon } ^ { \\theta } \\end{cases} \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{gather*} \\partial ^ 3 _ t Q ( \\overline { \\partial } \\eta ) = Q ( \\overline { \\partial } \\eta ) ( \\partial _ t \\overline { \\partial } \\eta ) ^ 3 + Q ( \\overline { \\partial } \\eta ) \\partial _ t \\overline { \\partial } \\eta \\partial ^ 2 _ t \\overline { \\partial } \\eta + Q ( \\overline { \\partial } \\eta ) \\partial ^ 3 _ t \\overline { \\partial } \\eta \\end{gather*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\Upsilon ( \\zeta , \\widehat \\Lambda ) = 0 . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\mbox { d i v } \\left ( | \\nabla u | ^ { p - 2 } \\nabla u \\right ) = f & & \\ \\Omega \\\\ u = 0 & & \\ \\partial \\Omega \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} g s _ { \\pi } = \\prod _ { \\pi _ i > \\pi _ { i + 1 } } x _ { \\pi _ 1 } \\cdots x _ { \\pi _ i } \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\alpha _ { j } = \\sum _ { l = 1 } ^ { r } p _ { j l } \\beta _ { l } + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq m . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 0 } ^ n \\left ( \\sum \\limits _ { j = 1 } ^ { p _ k } D _ { i _ { k 1 } } \\cdots D _ { i _ { k ( j - 1 ) } } S _ { i _ { k j } } D _ { i _ { k ( j + 1 ) } } \\cdots D _ { i _ { k p _ k } } \\right ) . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} k \\langle \\langle T _ { 1 1 } , \\dotsc , T _ { n ( m + n ) } , T \\rangle \\rangle / ( \\operatorname { d e t } _ J ) & \\longrightarrow k \\langle \\langle T _ { 1 1 } , \\dotsc , T _ { n m } , T ' \\rangle \\rangle \\\\ T _ { i j } & \\longmapsto T ' T _ { i j } \\ j = 1 , \\dotsc , m , \\\\ T _ { i m + j } & \\longmapsto 0 i \\neq j , \\\\ T _ { i m + i } & \\longmapsto ( 1 - T ' ) T _ { i m + i } . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} q ( z ) = c \\det ( I - D \\Delta ( z ) ) \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\rho ^ { \\alpha _ n } = \\rho P , \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} p _ a ( \\bar { C } _ 1 ) = \\frac { 1 } { 2 } \\left ( 9 - \\frac { 1 } { 2 } \\sum _ { i \\in I } b _ i ^ 2 \\right ) + 1 = \\frac { 1 1 } { 2 } - \\frac { 1 } { 4 } \\sum _ { i \\in I } b _ i ^ 2 . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} L _ i v = \\nabla f _ i ( u _ { \\infty } ) \\cdot v = \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } \\sum _ { j = 1 } ^ { N } ( y _ { r , i } ' - y _ { r , i } ) y _ { r , j } \\frac { v _ j } { u _ { j , \\infty } } . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} v _ { x _ { 0 } } \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = \\partial _ { x _ { 0 } } \\widetilde { g } _ { 0 } \\left ( x , x _ { 0 } \\right ) , \\partial _ { n } v _ { x _ { 0 } } \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = \\partial _ { x _ { 0 } } \\widetilde { g } _ { 1 } \\left ( x , x _ { 0 } \\right ) , \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\bigoplus _ { 1 \\leq r \\leq t } \\left ( B _ { m , k - 1 - m } ^ { r + 1 } \\cap H _ { m , k - 1 - m } ^ { r } \\right ) = B _ { m , k - 1 - m } ^ { t + 1 } : \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { Z } _ 1 } { \\partial g _ { i k } } & = \\frac { \\partial \\Phi _ 1 ^ c } { \\partial g _ { i k } } + ( F _ A ' ( \\omega _ B ) - 1 ) \\frac { \\partial \\Phi _ 2 ^ c } { \\partial g _ { i k } } \\\\ & = \\Big ( \\big ( F _ A ' ( \\omega _ B ) - 1 \\big ) \\big ( F _ B ' ( \\omega _ A ^ c ) - 1 \\big ) - 1 \\Big ) \\frac { \\partial \\omega _ A ^ c } { \\partial g _ { i k } } + \\big ( F _ A ' ( \\omega _ B ^ c ) - F _ A ' ( \\omega _ B ) \\big ) \\frac { \\partial \\omega _ B ^ c } { \\partial g _ { i k } } . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} ( 1 \\pm \\gamma / 2 ) \\left ( \\prod _ { \\substack { \\{ k , \\ell \\} \\in \\binom { [ r ] } { 2 } \\\\ \\{ k , \\ell \\} \\neq \\{ i , j \\} } } d _ { k \\ell } \\right ) \\prod _ { s \\in [ r ] , s \\neq i , j } | V _ s | \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} G = G _ 0 > G _ 1 > \\cdots > G _ { t - 1 } > G _ t = 1 , \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\{ S ( a _ { ( 1 ) } ) , a _ { ( 2 ) } \\} = 0 , \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} R ^ { - 1 } ( x + l \\alpha ) A _ l ^ { E } ( x ) R ( x ) = \\left [ \\begin{array} { c c } \\pm 1 & l \\mu \\\\ 0 & \\pm 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} \\tau _ Y ( q ) = H q . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} ( M _ r { V } ) _ S = \\sum _ { a , b } \\left ( \\binom { a } { 2 } + \\binom { b } { 2 } \\right ) \\left ( \\mu ( w , y ) - \\mu ( w , z ) - \\mu ( x , y ) + \\mu ( x , z ) \\right ) \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac { h ( 2 v \\chi ^ { 1 / 3 } L ^ { 2 / 3 } , 0 ) - 2 v ( 1 - 2 \\rho ) \\chi ^ { 1 / 3 } L ^ { 2 / 3 } } { 2 \\chi ^ { 2 / 3 } L ^ { 1 / 3 } } = { \\cal R } ( v ) = \\sqrt { 2 } \\sigma { \\cal B } ( v ) , \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m ( \\ell _ j + 1 ) = n + 1 ~ ( = k ) \\ , . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\frac { 2 } { z _ i - z _ j } + \\frac { \\sum _ { l = 0 } ^ { k - 1 } b _ l z _ i ^ l } { \\sum _ { l = 0 } ^ k a _ l z _ i ^ l } = 0 , i = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} ( \\nabla \\pi _ * ) ( X , Y ) = \\nabla ^ { ^ \\pi } _ { X } \\pi _ * Y - \\pi _ * ( \\nabla _ { X } Y ) \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} y _ 0 = 1 , y _ n = \\sum _ { k = 1 } ^ n \\frac { 1 } { ( n - k + 1 ) ! } B _ { n , k } ( 1 ! x _ 1 , 2 ! x _ 2 , \\dots ) \\ n \\ge 1 , \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} J _ \\epsilon = \\sum _ { m \\geqslant 0 } l _ m \\otimes x _ m \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} V + V ' = \\phi _ 6 - \\phi _ 5 & & & I = I ' = I _ 5 = - I _ 6 \\\\ V '' = \\phi _ 6 - \\phi _ 5 & & & I '' = I _ 5 = - I _ 6 . & & & \\\\ \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} K _ { h _ 1 } \\circ K _ { h _ 2 } & ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) = \\delta _ { x _ 1 } ( A _ 1 ) \\cdot K _ { h _ 1 } ^ 2 \\circ K _ { h _ 2 } ^ 2 ( x _ 2 , A _ 2 ) \\\\ & = \\delta _ { x _ 1 } ( A _ 1 ) \\cdot \\int _ { D _ 2 } K _ { h } ^ 2 ( x _ 2 , A _ 2 ) \\ > d ( \\delta _ { h _ 1 } * _ 2 \\delta _ { h _ 2 } ) ( h ) \\\\ & = \\int _ { D _ 2 } K _ { h } ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) \\ > d ( \\delta _ { h _ 1 } * _ 2 \\delta _ { h _ 2 } ) ( h ) \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} ( x ^ 2 - y ^ 2 ) ^ 2 - 2 x ^ 2 w ^ 2 - 2 y ^ 2 w ^ 2 - 1 6 z ^ 2 w ^ 2 + w ^ 4 \\ = \\ 0 \\ , . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\Delta _ N ( v ) : = \\frac { \\mu ( E _ N ( w ) - A ( v ) ) - \\mu ( E _ N ( w ) - A ^ { \\rm f l a t } ( v ) ) } { \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} g \\mapsto F _ g , \\ ; \\ ; \\ ; F _ g ( f ) & = \\mathbb E \\sum _ { t \\in \\mathcal T } \\langle \\Delta g _ t , \\Delta f _ t \\rangle \\stackrel { ( * ) } = \\mathbb E \\langle g _ { \\infty } , f _ { \\infty } \\rangle , \\\\ \\| F _ g \\| _ { ( \\mathcal A ^ { \\mathcal T } _ { p , q } ) ^ { * } } & \\eqsim _ { p , q } \\| g \\| _ { \\mathcal A ^ { \\mathcal T } _ { p ' , q ' } } . \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\inf \\{ n \\geq 0 \\ , : \\ , X _ n = x \\} \\geq \\inf \\{ n \\geq 0 \\ , : \\ , | S _ n - x | \\leq R _ { n + 1 } \\} \\ , . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{gather*} F _ 3 = F ^ { a b c } \\theta _ a \\circ \\theta _ b \\circ \\theta _ c , \\end{gather*}"}
-{"id": "9466.png", "formula": "\\begin{align*} L _ z = z - t A ^ 2 . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} E _ 2 ( t , \\tau ) = - i \\int _ 0 ^ { \\tau } e ^ { i ( \\tilde { \\tau } - \\tau ) A ( t ) ^ { 1 / 2 } } F ( t ) ( t | t | G + \\Phi ( t ) ) e ^ { - i \\tilde { \\tau } ( t ^ 2 S ) ^ { 1 / 2 } P } P \\ , d \\tilde { \\tau } . \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\beta _ 2 ( r ) \\le \\beta _ 2 ( m s ) + \\beta _ 2 ( t ) + 6 \\le c m s + d \\le \\left ( c + \\frac { d } { r } \\right ) r = ( c + o ( 1 ) ) r \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 1 } , V _ { 1 } ) , \\pi _ * Z ) & = - g _ { 1 } ( \\hat { \\nabla } _ { U _ { 1 } } \\varphi V _ { 1 } , \\mathcal { B } Z ) - g _ { 1 } ( \\mathcal { T } _ { U _ { 1 } } \\varphi V _ { 1 } , \\mathcal { C } Z ) - g _ { 1 } ( V _ { 1 } , \\phi U _ { 1 } ) \\eta ( Z ) \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} S = \\sum _ { \\substack { p \\leq x \\\\ p \\equiv p _ 0 \\pmod * { n } \\\\ \\frac { p - p _ 0 } { n } } } \\ll \\frac { x ( \\log \\log x ) ^ 3 } { n \\log ^ 2 x } . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} M ( X , Y ) \\odot M ( X , Y ) ' \\equiv L ^ 1 \\equiv Y \\odot Y ' = X \\odot M ( X , Y ) \\odot Y ' . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} S ( \\chi , \\phi ) = \\sum _ { j } \\phi ( \\tilde { \\gamma } _ { \\chi , j } ) . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} z '' ( t ) + A z ( t ) - \\int _ t ^ T \\ H ( s - t ) A z ( s ) d s = 0 \\ , , t \\in [ 0 , T ] \\ , , \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) \\in \\hat { G } ^ n , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\Lambda ( s , \\chi ) = ( | D _ K | N ( m ) ) ^ { s / 2 } ( 2 \\pi ) ^ { - s } \\Gamma ( s ) L ( s , \\chi ) , \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\iota _ C ^ * ( u ) & = \\frac { 1 - u } { 1 - t u } , \\iota _ C ^ * ( w ) = ( 1 - t ) ^ { 1 / 3 } \\frac { u ( 1 - u ) } { w ( 1 - t u ) } . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ n x _ i \\right \\| _ { X } \\le M \\left ( \\sum _ { i = 1 } ^ n \\| x _ i \\| _ { X } ^ p \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\int _ { \\Lambda } | f ( x - j ) | ~ d x \\leq & C \\int _ { \\Lambda } \\frac { 1 } { | x - j | ^ { \\alpha } } ~ d x \\\\ \\leq & C L ^ d \\frac { 1 } { \\underset { x \\in \\Lambda } { \\inf } ~ | x - j | ^ { \\alpha } } \\\\ \\leq & C ^ { ' } \\frac { L ^ { d } } { | j | ^ { \\alpha } } \\\\ = : & \\sigma _ j \\ , . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} | \\zeta - \\zeta _ 0 | ^ 2 + \\tau ^ 2 & \\leqslant \\frac 4 9 \\tau ^ 2 + \\tau ^ 2 = \\frac { 1 3 } { 9 } \\tau ^ 2 , \\\\ | ( \\zeta - \\zeta _ 0 ) ^ 2 + \\tau ^ 2 | & \\geqslant \\tau ^ 2 - | \\zeta - \\zeta _ 0 | ^ 2 \\geqslant \\frac 5 9 \\tau ^ 2 . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} K _ n ( u , v ) = \\int _ { \\R _ + } d \\lambda H _ n ( u , \\lambda ) J _ n ( \\lambda , v ) , \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} u ( x ) = u \\circ \\gamma ( r ( x ) ) = y ( r ( x ) ) = - \\dfrac { 1 } { 2 m } r ( x ) ^ 2 + \\alpha r ( x ) + \\beta \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { { \\bf { I } } ( x , t ) } { A _ { 1 } ( b _ { 1 } ( t ) ) } = \\lim _ { t \\to \\infty } \\frac { A _ d ( b _ { d } ( t ) ) } { A _ 1 ( b _ { 1 } ( t ) ) } \\times \\frac { { \\bf { I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } = 0 . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} m ( T - r _ { 3 } , k - 1 ) = & m ( P _ { x - 2 } \\cup P _ { y - 2 } \\cup P _ { c - 2 } , k - 1 ) + m ( P _ { x - 3 } \\cup P _ { y - 3 } \\cup P _ { c - 2 } , k - 2 ) \\\\ & + m ( P _ { x - 2 } \\cup P _ { y - 3 } \\cup P _ { c - 3 } , k - 2 ) . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\norm { ( I _ { X ^ * } - F ^ * ) ( x ^ * ) } & = \\norm { z ^ * - f _ 0 ( t ) F _ Z ^ * z ^ * } \\\\ & \\le | 1 - f _ 0 ( t ) | \\norm { z ^ * } + | f _ 0 ( t ) | \\cdot \\norm { ( I _ { Z ^ * } - F _ Z ^ * ) ( z ^ * ) } \\\\ & = | f _ 0 ( t ) | \\cdot \\norm { ( I _ { Z ^ * } - F _ Z ^ * ) ( z ^ * ) } + | 1 - f _ 0 ( t ) | . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{gather*} \\widehat S : = \\{ y \\in T _ x M \\ , | \\ , [ y ] \\in S \\} \\subset T _ x M . \\end{gather*}"}
-{"id": "6444.png", "formula": "\\begin{align*} { J } _ 1 ( \\tau ) & : = f \\cos ( \\tau \\mathcal { A } ^ { 1 / 2 } ) f ^ { - 1 } - f _ 0 \\cos ( \\tau ( \\mathcal { A } ^ 0 ) ^ { 1 / 2 } ) f _ 0 ^ { - 1 } , \\\\ { J } _ 2 ( \\tau ) & : = f { \\mathcal { A } } ^ { - 1 / 2 } \\sin ( \\tau { \\mathcal { A } } ^ { 1 / 2 } ) f ^ * - f _ 0 ( { \\mathcal { A } } ^ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) f _ 0 . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{gather*} C _ { \\tau _ 1 , \\dots , \\tau _ n } ^ { ( q , t ) } = ( - 1 ) ^ { \\tau _ 1 + \\dots + \\tau _ n } . \\end{gather*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\kappa _ { r F } ( A , \\lambda b ) = \\lim _ { \\delta \\rightarrow 0 } \\sup _ { \\left \\| [ \\Delta A \\ ; \\lambda \\Delta b ] \\right \\| _ F < \\delta } \\frac { \\frac { \\left \\| F \\left ( A + \\Delta A , \\lambda ( b + \\Delta b ) \\right ) - F ( A , \\lambda b ) \\right \\| _ 2 } { \\| F ( A , \\lambda b ) \\| _ 2 } } { \\frac { \\left \\| [ \\Delta A \\ ; \\lambda \\Delta b ] \\right \\| _ F } { \\left \\| [ A \\ ; \\lambda b ] \\right \\| _ F } } . \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} a & b \\\\ b & c \\end{pmatrix} = 0 \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} \\Delta _ \\zeta : = \\min \\{ | \\zeta _ 1 | , | \\zeta _ 2 | \\} - 1 , \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} 2 & ( y - a ) \\left [ a ^ k + k ( y - a ) a ^ { k - 1 } + \\binom { k } { 2 } ( y - a ) ^ 2 a ^ { k - 2 } - y ^ k \\right ] \\\\ + & ( b - y ) \\left [ a ^ k + k ( y - a ) y ^ { k - 1 } + k ( k - 1 ) ( y - a ) ^ 2 y ^ { k - 2 } - y ^ k \\right ] > 0 \\ , . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} o \\left ( \\frac { r ( f + 1 ) \\sqrt { n } } { n ^ { 2 } } \\right ) = o \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\| a \\| _ { L ^ { \\infty } } = \\| T _ a \\| _ { H ^ 2 \\to H ^ 2 } . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\ , K _ { i - 1 , l } ^ { X _ 1 , 0 } = \\ , V _ { X _ 2 } ( - ( i - 1 ) A - l B ) + \\ , V _ { X _ 3 } ( - ( d - l ) B ) - \\ , ( e v _ { 1 } ^ { i - 1 , l } ) , \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] ^ 2 } Q ( x , y ) \\mathrm { d } x \\mathrm { d } y = \\alpha _ i \\alpha _ r \\alpha _ s \\bigl ( ( 1 + \\delta _ { i s } ) ( 2 - \\delta _ { i s } ) - ( 1 + \\delta _ { i r } ) ( 2 - \\delta _ { i r } ) \\bigr ) = 0 \\ , , \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\Lambda _ f ( s , c _ q ) = i ^ k \\xi ( q ) ( N q ^ 2 ) ^ { \\frac 1 2 - s } \\Lambda _ g ( 1 - s , c _ q ) . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} P _ 1 ( u ) : = \\partial , \\ , \\ , \\ , P _ { 2 n + 1 } ( u ) : = v _ { n } \\partial - \\frac { 1 } { 2 } \\partial ( v _ { n } ) + P _ { 2 n - 1 } ( u ) L ( u ) , \\mbox { f o r } n \\geq 1 . \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\pi ( x ; q , a ) - \\pi ( x ; q , b ) = \\frac 1 { \\phi ( q ) } \\sum _ { \\chi \\mod q } \\big ( \\overline { \\chi } ( a ) - \\overline { \\chi } ( b ) \\big ) \\pi ( x , \\chi ) , \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\Delta _ t = \\int _ 0 ^ t \\mathbf { 1 } _ { \\{ | Z _ s | = P _ Y \\} } Y _ s \\hbox { d } s . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} A ( \\phi ) = - T ( N ( \\phi ) + E ) , \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\left ( R ^ T \\delta \\right ) _ t = u ^ \\delta _ { 1 , t } = r _ { t - 1 } . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} R _ d ( F ) = \\frac { \\textrm { R e s } _ { s = 1 } \\zeta _ F ( s ) } { 2 ^ d \\zeta _ F ( 2 ) } . \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\frac { 1 } { m ^ k k ! } e ^ { r t } ( e ^ { m t } - 1 ) ^ k = \\sum _ { n = k } ^ \\infty W _ { m , r } ( n , k ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\| P _ { \\tilde { W } \\tilde { X } \\tilde { Y } } - P _ { W X Y } \\| _ 1 = \\| P _ { \\tilde { X } \\tilde { Y } } - P _ { X Y } \\| _ 1 \\le \\epsilon . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha f ( x _ { i - 1 } ( t ) ) + a f ( x _ { i } ( t ) ) + \\beta f ( x _ { i + 1 } ( t ) ) , \\ , \\ , i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\Delta u = 0 \\ ; \\ ; B ^ + ( 0 , 2 ) , u = \\ ; \\ ; T ( 0 , 2 ) . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} u _ \\infty = \\mathbb { P } ( \\widehat { Q } _ 0 < \\widehat { Q } _ 1 < \\cdots \\ , | \\ , \\widehat { Q } _ 0 = 1 ) . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } \\omega ( x ) | ^ { p } d x = \\lambda ^ { p } \\mu ^ { a _ { 1 } p - Q } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } d x , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\beta ( \\lambda ) = & ~ i \\int _ { \\real } u ( x ) e ^ { - i \\lambda x } [ ( I + R _ { \\lambda + 0 i } ) 1 ] ( x ) ~ d x \\\\ & ~ - i \\int _ { \\real } u ( x ) e ^ { - i \\lambda x } [ ( I + R _ { \\lambda + 0 i } ) \\widetilde T _ { \\lambda + 0 i } ( I + R _ { \\lambda + 0 i } ) 1 ] ( x ) ~ d x + O \\left ( \\frac 1 { \\lambda } \\right ) . \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} 0 \\leq \\sup _ { x \\in M } \\| ( \\beta _ 1 ) _ x \\| _ g : = b _ 1 \\leq b _ 2 : = \\sup _ { x \\in M } \\| ( \\beta _ 2 ) _ x \\| _ g < 1 . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} [ \\Phi _ i ( u ) , \\Phi _ j ( v ) ] = 0 \\ , \\ \\ , \\ [ h , \\Phi _ i ( u ) ] = 0 . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} 0 = p ( m _ 1 - m _ 2 ) f ' ( x ) + p ^ 2 ( m _ 1 ^ 2 - m _ 2 ^ 2 ) f '' ( x ) + \\cdots + p ^ { k - 1 } ( m _ 1 ^ { k - 1 } - m _ 2 ^ { k - 1 } ) f ^ { ( k - 1 ) } ( x ) . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\widetilde { W } ( 0 ) = \\pi + O ( \\frac 1 { U } ) . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} v ^ f _ { n , t } = \\left \\{ \\begin{array} l \\sum _ { k = 1 } ^ N c _ t ^ k \\phi ^ k _ n , n = 1 , \\ldots , N \\\\ f _ t , n = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\int _ X f _ 1 \\cdot \\tilde T _ { h } f _ 2 \\ > d \\tilde \\omega _ X & = \\int _ X \\int _ X f _ 1 ( x ) f _ 2 ( y ) \\tilde K _ { h } ( x , d y ) \\ > d \\tilde \\omega _ X ( x ) \\\\ & = \\int _ X \\int _ X \\frac { \\phi _ 0 ( y ) \\ > f _ 1 ( x ) \\ > f _ 2 ( y ) } { \\alpha _ 0 ( h ) \\ > \\phi _ 0 ( x ) } K _ { h } ( x , d y ) \\ > \\phi _ 0 ( x ) ^ 2 \\ > d \\omega _ X ( x ) \\\\ & = \\frac { 1 } { \\alpha _ 0 ( h ) } \\int _ X \\phi _ 0 f _ 1 \\cdot T _ { h } ( \\phi _ 0 f _ 2 ) \\ > d \\omega _ X . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\sum _ { \\mu ^ { ( 1 ) } \\vdash q _ 1 } \\sum _ { \\mu ^ { ( 2 ) } \\vdash q _ 2 } \\cdots \\sum _ { \\mu ^ { ( k ) } \\vdash q _ k } \\prod _ { i = 1 } ^ { k } \\frac { \\varepsilon _ { { \\mu ^ { ( i ) } } } } { z _ { \\mu ^ { ( i ) } } } = \\prod _ { i = 1 } ^ { k } \\left ( \\sum _ { \\mu ^ { ( i ) } \\vdash q _ i } \\frac { \\varepsilon _ { \\mu ^ { ( i ) } } } { z _ { \\mu ^ { ( i ) } } } \\right ) \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} u _ { + } ( t ) = ( \\l _ { + } ( t ) , y _ { + } ( t ) ) , u _ { - } ( t ) = ( \\l _ { - } ( t ) , y _ { - } ( t ) ) . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\widehat { \\gamma } ( t , X ( t ) ) = E [ \\rho _ t | X ( t ) ] \\ell \\otimes P ( t , \\omega ) \\in [ 0 , T ] \\times \\Omega . \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\mathbb { T } } ^ { \\varepsilon } : = \\nabla \\cdot \\left ( - \\mathbb { T } \\left ( \\frac { x } { \\varepsilon } \\right ) \\nabla \\right ) = \\frac { \\partial } { \\partial x _ { i } } \\left [ - \\tau _ { i j } ^ { \\alpha \\beta } \\left ( \\frac { x } { \\varepsilon } \\right ) \\frac { \\partial } { \\partial x _ { j } } \\right ] . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} \\int _ \\Omega A _ \\tau \\nabla U _ \\tau \\nabla \\varphi + \\int _ \\Omega g ( U _ \\tau ) \\varphi J _ \\tau = \\int _ \\Omega f _ \\tau \\varphi J _ \\tau , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} M _ { \\beta } ( z ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - z ) ^ { n } } { n ! \\Gamma ( - \\beta n + 1 - \\beta ) } . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\Psi \\left ( Z ' , Z '' \\right ) = U _ { A } \\circ \\tilde { \\varphi } _ { A ^ { 2 } } \\circ \\mathcal { W } \\circ \\varphi _ { A } \\left ( Z ' , Z '' \\right ) , \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{gather*} \\Lambda ^ { N + 1 } _ N ( \\lambda , \\mu ) = t ^ { | \\mu | } \\psi _ { \\lambda / \\mu } ( q , t ) \\frac { P _ { \\mu } \\big ( 1 , t , \\dots , t ^ { N - 1 } \\big ) } { P _ { \\lambda } \\big ( 1 , t , \\dots , t ^ N \\big ) } = \\psi _ { \\lambda / \\mu } ( q , t ) \\frac { P _ { \\mu } \\big ( t , t ^ 2 , \\dots , t ^ N \\big ) } { P _ { \\lambda } ( 1 , t , \\dots , t ^ N \\big ) } , \\end{gather*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t v + \\Delta v = \\mathcal { N } ( v + f ) , \\\\ v | _ { t = 0 } = v _ 0 \\in H ^ 1 ( \\R ^ d ) , \\end{cases} \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} - \\sum _ { j \\ne i } | a _ i - a _ j | ^ 2 + \\frac { n + 1 } { 2 } | x - a _ i | ^ 2 \\le & \\sum _ { i = 1 } ^ n | x - a _ i | ^ 2 \\\\ & \\le ( 2 n - 1 ) | x - a _ i | ^ 2 + 2 \\sum _ { j \\ne i } | a _ i - a _ j | ^ 2 . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi _ m ( X ^ x _ { \\tau + \\cdot } ) ] = \\hat { \\mathbb { E } } [ \\varphi _ m ( X ^ y _ \\cdot ) ] _ { y = X ^ x _ { \\tau } } . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} s ^ { ( \\nu ) } _ { \\rm h y b } = s _ { \\rm s h a r p } + \\nu ^ { 1 / 2 } \\hat { c } \\ , R _ { \\rm h y b } . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} r _ { \\langle \\rangle } ( \\mathcal B _ { \\forall , \\psi } ^ \\beta P ) & = ( \\forall _ x , x \\notin \\mathbb L _ \\beta ^ u \\lor \\theta ) , \\\\ n ( \\mathcal B _ { \\forall , \\psi } ^ \\beta P , a ) & = \\textstyle \\bigvee _ { a \\notin \\mathbb L _ \\beta ^ u , \\theta ( a ) } ^ 1 \\mathcal B _ { \\forall , \\psi } ^ \\beta n ( P , a ) . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\begin{array} { l } Q = \\displaystyle { \\frac { 2 \\tau } { \\tau + 1 } } E , \\\\ q = \\displaystyle { \\frac { \\tau } { \\tau + 1 } } \\left ( \\displaystyle { \\frac { 1 } { \\tau } } w + c \\right ) , \\\\ \\theta = \\displaystyle { \\frac { \\beta } { \\tau + 1 } } . \\end{array} \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} S = \\begin{pmatrix} L & 0 \\\\ 0 & L ^ { - T } \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ P & 1 \\end{pmatrix} = \\begin{pmatrix} L & 0 \\\\ L ^ { - T } P & L ^ { - T } \\end{pmatrix} \\in S p ( n ) , \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\partial _ t m _ e ( \\lambda + 0 i ) & = \\Gamma \\partial _ t m _ e ( \\lambda - 0 i ) + ( \\partial _ t \\Gamma ) m _ e ( \\lambda - 0 i ) , \\\\ B _ u m _ e ( \\lambda + 0 i ) & = \\Gamma B _ u m _ e ( \\lambda - 0 i ) . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} s a t ( n , k , r ) = \\alpha ( k , r ) n + o ( n ) \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 1 & x _ 2 & \\cdots & x _ n \\\\ y _ 1 & y _ 2 & \\cdots & y _ n \\\\ u _ 1 & u _ 2 & \\cdots & u _ n \\end{pmatrix} \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} S _ t ( t , x ) = - f \\big ( { \\cal F } ( t , x ) \\big ) . \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} ( 1 - x ) A '' ( x , 1 ) & = x ( 1 - x ) \\big ( g ( x ) - 1 \\big ) + B ( x , 1 ) - B ( x , 0 ) + ( 1 - x ) \\big ( B ' ( x , 1 ) - B ' ( x , 0 ) \\big ) , \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\widetilde { e } \\left ( \\frac { c _ 3 } { \\varpi ^ { l - h - 1 } } \\right ) \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - h - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 } { \\varpi ^ { l - h - 1 } } \\right ) = \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - h - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 + c _ 3 } { \\varpi ^ { l - h - 1 } } \\right ) = \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - h - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 } { \\varpi ^ { l - h - 1 } } \\right ) . \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\alpha _ 1 = \\rho ( \\beta _ 2 ) , \\alpha _ 2 = \\rho ( \\beta _ 1 ) , \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} c _ { \\alpha } \\left ( \\left ( - \\Delta \\right ) ^ { \\alpha } f \\right ) ( x ) = - \\lim \\limits _ { t \\rightarrow 0 + } t ^ { 1 - 2 \\alpha } \\partial _ t u ( t , x ) ( x \\in \\R ^ n ) , \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} g _ { 1 } ( \\mathcal { A } _ { X } \\omega W , \\mathcal { B } Y ) + \\eta ( Y ) g _ { 1 } ( X , \\omega W ) & = g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , Y ) , \\pi _ * \\omega \\phi W ) - g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , \\mathcal { C } Y ) , \\pi _ * \\omega W ) \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} b _ 1 X _ 1 ^ { m _ 1 } + \\dots + b _ n X _ n ^ { m _ n } = c , \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{gather*} x _ 1 = 1 , \\\\ x _ { 2 n + 1 } = 0 x _ { 2 n } = a _ n n \\ge 1 , \\end{gather*}"}
-{"id": "3148.png", "formula": "\\begin{align*} ( B x ) ( t ) = \\frac { x _ 0 } { \\Gamma ( \\gamma ) } t ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } f ( s , x ( s ) ) d s , t \\in [ 0 , h ] . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} q _ { s , t } ^ { ( x ) } ( a , b ) = t ^ { - n \\alpha } \\frac { p _ { \\frac 1 t } ( x , \\frac { b } { t ^ { \\alpha } } ) p _ { \\frac 1 s - \\frac 1 t } ( \\frac { b } { t ^ { \\alpha } } , \\frac { a } { s ^ { \\alpha } } ) } { p _ { \\frac 1 s } ( x , \\frac { a } { s ^ { \\alpha } } ) } . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\psi ^ { \\frak C \\boxplus \\tau } _ p \\left ( ( s ^ { \\frak C \\boxplus \\tau } _ p ) ^ { - 1 } ( 0 ) \\cap \\{ ( \\overline y , ( t _ 1 , \\dots , t _ k ) ) \\mid t _ i \\le 0 , \\ , \\ , i = k ' - k + 1 , \\dots , k ' \\} \\right ) . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} & N ( \\star _ { i = 1 } ^ { m + k } G , \\alpha ^ { m + k + 1 } ) = N ( \\star _ { i = 1 } ^ { m + k } G , ( \\star _ { i = 1 } ^ { m } \\alpha ^ { 2 } ) \\star ( \\star _ { i = m + 1 } ^ { m + k } \\alpha ^ { 2 } ) ) = \\\\ & N ( \\star _ { i = 1 } ^ { m + k } G , ( \\star _ { i = 1 } ^ { m } \\alpha ^ { 2 } ) \\star ( \\star _ { i = 1 } ^ { k } \\alpha ^ { 2 } ) ) \\le N ( \\star _ { i = 1 } ^ { m } G , \\star _ { i = 1 } ^ { m } \\alpha ^ { 2 } ) N ( \\star _ { i = 1 } ^ { k } G , \\star _ { i = 1 } ^ { k } \\alpha ^ { 2 } ) , \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } k ^ * ( t ) \\Big | \\sum _ { n = n _ 0 } ^ \\infty F _ { n } e ^ { i \\sigma _ { n } t } + \\overline { F _ { n } } e ^ { - i \\overline { \\sigma _ n } t } \\Big | ^ 2 \\ d t \\le c ( T ) \\sum _ { n = n _ 0 } ^ \\infty | F _ { n } | ^ 2 \\ , . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\mathcal { R } _ { T } ^ * ( \\alpha ) = \\left \\{ T _ { n p } \\ge \\frac { p ( p - 1 ) } { 2 ( n - 4 ) } + z _ { \\alpha } \\sqrt { \\frac { p ( p - 1 ) ( n - 3 ) } { ( n - 4 ) ^ 2 ( n - 6 ) } } \\right \\} . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} & R _ C = \\{ ( w , u ) \\mid u = 0 , 1 , \\infty , \\dfrac { 1 } { t } \\} \\subset C , C ^ 0 = C - R _ C , \\\\ & R _ { E } = \\{ ( w ' , r ) \\mid r = 0 , 1 , \\infty \\} \\subset E , E ^ 0 = E - R _ { E } , \\\\ & { X ' } ^ 0 = { \\pi ' } ^ { - 1 } ( \\bold U ) . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\langle x ^ * , x \\rangle = \\langle x ^ * , y \\rangle + \\langle x ^ * , z \\rangle ( x ^ * \\in X ^ * \\cap Y ^ * , \\ x = y + z \\in X + Y ) , \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\begin{aligned} & \\delta < C < 1 - \\delta , \\\\ & ( \\bar { q } _ { i + 1 } - \\bar { q } _ { i } ) ( \\bar { q } _ { i } - \\bar { q } _ { i - 1 } ) > 0 , \\end{aligned} \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} \\tilde { \\hat { \\varphi } } ^ { \\frak r , p } _ { 1 2 } ( s _ { \\frak r _ 1 } ( y , \\xi ) ) = s _ { \\frak r _ 2 } ( \\tilde \\varphi ^ { \\frak r , p } _ { 1 2 } ( y ) , \\xi ) . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} S : = s u p p \\ > \\rho ^ { ( a , b ) } = \\left \\{ \\begin{array} { r @ { \\quad \\quad } l } [ - 1 , 1 ] & a \\ge b \\ge 2 \\\\ \\{ s _ 0 \\} \\cup [ - 1 , 1 ] & b > a \\ge 2 \\end{array} \\right . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} h _ t ( x , y , \\varphi , \\varepsilon ) = \\left ( \\begin{array} { c } x \\\\ y \\\\ \\varphi \\end{array} \\right ) + t X _ 0 ( x , y , \\varphi ) + \\underline F ( t , x , y , \\varphi , \\varepsilon ) \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} z ' ( t ) = - \\frac { z ( t ) } { t } - 1 \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t Z _ { \\epsilon } ^ { ( j ) } ( t ) & = \\Delta Z _ { \\epsilon } ^ { ( j ) } ( t ) + \\sqrt { 2 } d W _ { \\epsilon } ^ { ( j ) } ( t ) \\\\ Z _ { \\epsilon } ^ { ( j ) } ( t ) & = 0 \\end{cases} \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\int _ { B ' } \\sum _ { i = 1 } ^ d \\Big ( \\partial _ i \\varphi \\Big ( \\sum _ { j = 1 } ^ d \\frac { a _ { i j } } { 2 } \\partial _ j u + h _ i u \\Big ) \\Big ) + \\varphi ( c u + e ) \\ , d x = 0 , \\forall \\varphi \\in C _ 0 ^ { \\infty } ( B ' ) , \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} x ( t + 1 ) = A ( t ) x ( t ) , t \\geq 0 . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\epsilon _ P ( t ) : = 2 ^ { a j } P _ { \\epsilon } ( 2 ^ { - a j } t ) ; \\\\ \\epsilon _ Q ( t ) : = 2 ^ { b j } Q _ { \\epsilon } ( 2 ^ { - b j } t ) . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} f ( x + m p ) = f ( x ) + p ^ { k - 1 } f ' ( x ) = f ( x ) \\in \\Z / p ^ k . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} p _ n ( z ) = \\left ( 1 + \\frac { c _ 1 z } { n } \\right ) ^ n \\left ( 1 - \\frac { d _ n z ^ 2 } { n } \\right ) ^ n \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ X I _ { \\{ \\tau = \\sigma \\} } ] = \\hat { \\mathbb { E } } _ { \\tau + } [ X ] I _ { \\{ \\tau = \\sigma \\} } . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\mathcal { A } _ { k u } ^ { i } = \\left ( a _ { k u } ^ { i 1 } , a _ { k u } ^ { i 2 } , \\dots , a _ { k u } ^ { i N } \\right ) , \\quad \\quad \\mbox { f o r a l l $ k , u , i = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\Delta ( z ) = \\sum _ { n \\geq 1 } \\tau ( n ) e ( n z ) . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} f = x _ { 1 } \\cdot P _ 1 + x _ { 2 } \\cdot P _ 2 \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} L y = x ^ 2 \\frac { d ^ 2 y } { d x ^ 2 } + x \\frac { d y } { d x } - \\bigl ( x ^ 2 + \\nu ^ 2 \\bigr ) y = \\frac { 4 ( x / 2 ) ^ { \\nu + 1 } } { \\sqrt { \\pi } \\varGamma ( \\nu + 1 / 2 ) } . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} C = 2 ( \\sum _ { j = 1 } ^ { n } | a ( e _ j ) | ) ^ 2 + 2 \\sum _ { j , k = 1 } ^ { n } | \\Re [ a ( e _ j + e _ k ) ] | ) . \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\left < \\displaystyle \\sum _ { k = 1 } ^ { q } R _ { k } \\left ( Z _ { i } \\right ) , \\displaystyle \\sum _ { l = 1 } ^ { q } R _ { l } \\left ( Z _ { j } \\right ) \\right > = \\displaystyle \\sum _ { k , l = 1 } ^ { q } \\left < R _ { k } \\left ( Z _ { j } \\right ) , \\displaystyle R _ { l } \\left ( Z _ { j } \\right ) \\right > = \\left < Z _ { i } , Z _ { j } \\right > , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} Z _ 0 & : = X _ 1 \\cdots X _ N , \\\\ Z _ 1 & : = Z _ 0 L _ 1 \\left ( \\frac { Y _ 1 } { X _ 1 } , \\dots , \\frac { Y _ N } { X _ N } \\right ) , \\\\ & \\vdots \\\\ Z _ r & : = Z _ 0 L _ r \\left ( \\frac { Y _ 1 } { X _ 1 } , \\dots , \\frac { Y _ N } { X _ N } \\right ) , \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\Theta ^ E _ { W , \\bar W } : = \\bar W \\rfloor ( W \\rfloor \\Theta ^ E ) , \\ D ' _ W : = W \\rfloor D ' , \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\infty } ( \\mathcal { C } ) = \\bigotimes _ { i = 1 } ^ { \\infty } U ( \\mathcal { G } ( \\mathcal { C } ) _ i ) \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{gather*} \\lim _ { k \\rightarrow \\infty } { \\frac { P _ { \\lambda ( N _ k ) } \\big ( x _ 1 t ^ { 1 - m } , x _ 2 t ^ { 2 - m } , \\dots , x _ m , t ^ { - m } , \\dots , t ^ { 1 - N _ k } \\big ) } { P _ { \\lambda ( N _ k ) } \\big ( 1 , t ^ { - 1 } , \\dots , t ^ { 1 - N _ k } \\big ) } } \\\\ \\qquad { } = \\Phi ^ { \\nu } \\big ( x _ 1 t ^ { 1 - m } , x _ 2 t ^ { 2 - m } , \\dots , x _ m ; q , t \\big ) \\end{gather*}"}
-{"id": "7096.png", "formula": "\\begin{align*} { q } _ { i } ( x ) = \\omega _ { i } { q } _ { i } ( x ) ^ { T } + ( 1 - \\omega _ { i } ) { q } _ { i } ( x ) ^ { W } , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\min ( I _ n ) = u + v ^ * > v ^ * = \\max \\left ( \\bigcup _ { i = 1 } ^ h V _ i \\right ) \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\aligned & \\int _ { ( ( X _ 1 , \\widehat { \\mathcal U _ 1 } ) \\times _ { R \\times P } ( X _ 2 , \\widehat { \\mathcal U _ 2 } ) , ( \\widehat { \\frak S _ 1 } \\times _ { R \\times P } \\widehat { \\frak S _ 2 } ) ^ { \\epsilon } ) } h _ 1 \\wedge h _ 2 \\\\ & = \\int _ { ( ( X _ 2 , \\widehat { \\mathcal U _ 2 } ) , \\widehat { \\frak S _ 2 } ) } ( \\pi _ { R } \\circ \\widehat f _ 2 ) ^ * ( \\pi _ { R } \\circ \\widehat f _ 1 ) ! ( h _ 1 ; ( \\widehat { \\frak S _ 1 } ) ^ { \\epsilon } ) \\wedge h _ 2 . \\endaligned \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} T x : = x + 2 \\delta \\frac { \\pi ( x ) - x } { | \\pi ( x ) - x | } . \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\zeta ( X , s ) = \\frac { \\zeta ( s ) \\zeta ( s - 1 ) } { L ( X , s ) } , \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} ( t ^ + _ { 1 1 } [ 0 ] ) ^ { \\pm 1 } ( t ^ + _ { 1 1 } [ 0 ] ) ^ { \\mp 1 } = 1 , \\ ( t ^ - _ { 1 1 } [ n ] ) ^ { \\pm 1 } ( t ^ - _ { 1 1 } [ n ] ) ^ { \\mp 1 } = 1 , \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} a & = ( 1 2 ) ( 3 4 ) & A = ( 1 , 1 , - 1 , - 1 ) \\\\ b & = ( 1 3 ) ( 2 4 ) & B = ( 1 , - 1 , 1 , - 1 ) \\\\ c & = ( 1 4 ) ( 2 3 ) & C = ( 1 , - 1 , - 1 , 1 ) \\\\ & & Z = ( - 1 , - 1 , - 1 , - 1 ) \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\varphi _ i ^ { ( n ) } : { \\cal X } ^ n \\times { \\cal Y } ^ n \\to { \\cal M } _ i ^ { ( n ) } , ~ ~ ~ i = 0 , 1 , 2 \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\tilde { { F } } _ { s + 1 } \\left ( p \\right ) = \\frac { 1 } { 2 } p ^ { 2 } \\left ( { 1 - p ^ { 2 } } \\right ) \\frac { d \\tilde { { F } } _ { s } \\left ( p \\right ) } { d p } - \\frac { 1 } { 2 } \\sum \\limits _ { j = 1 } ^ { s - 1 } { \\tilde { { F } } _ { j } \\left ( p \\right ) \\tilde { { F } } _ { s - j } \\left ( p \\right ) } \\ \\left ( { s = 2 , 3 , 4 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\varphi V _ { 2 } , \\varphi Z ) - \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } V _ { 2 } ) \\eta ( Z ) \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} q _ { 1 2 } ( 0 ) = \\sum _ { j , k } p _ { j k } ( 0 ) ( \\tilde { y } _ j + x _ j ) ( \\tilde { y } _ k - x _ k ) \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} R _ { N _ 1 } ( y , \\xi ) = N _ 1 \\sum _ { | \\ell | = N _ 1 } \\frac { y ^ { \\ell } } { \\ell ! } \\int _ 0 ^ 1 \\ ! ( 1 - \\gamma ) ^ { N _ 1 - 1 } ( \\partial ^ { \\ell } \\phi _ k ) ( \\xi + \\gamma y ) \\ , \\mathrm d \\gamma \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} ( q ! ) ^ 2 \\cdot 2 \\cdot C _ q = \\frac { 2 } { q + 1 } \\cdot ( 2 q ) ! . \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{gather*} 0 \\leq s _ { k , l } : = \\sum _ { m = 0 } ^ k \\sum _ { n = 0 } ^ l d _ { m , n } ^ { q - 2 } \\leq \\lim _ { r \\to 1 ^ - } \\sum _ { m , n \\geq 0 } d _ { m , n } ^ { q - 2 } S _ { m , n } ^ { q } ( r ) = g ( 1 ) , k , l \\in \\mathbb { Z } _ + . \\end{gather*}"}
-{"id": "1978.png", "formula": "\\begin{align*} v ^ R ( c ) = \\sum _ { \\tau \\in T } r ( \\tau ) \\ , ( t ( \\tau ) - s ( \\tau ) ) c ^ { s ( \\tau ) } \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} \\varphi _ { \\underline { d } , \\underline { d } '' } ( \\varphi _ { \\underline { d } ''' , \\underline { d } } ( V _ { \\underline { d } ''' } ) ) = \\varphi _ { \\underline { d } ''' , \\underline { d } '' } ( V _ { \\underline { d } ''' } ) = V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} F _ { L , + } ( v _ 1 ) = 0 , \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} \\lambda _ 1 ( t ) & = \\lambda _ 1 ( \\tau _ 1 ) \\\\ \\lambda _ 2 ( t ) & = \\lambda _ 2 ( \\tau _ 1 ) + \\lambda _ 0 b _ M ( \\tau _ 1 - t ) + \\int _ t ^ { \\tau _ 1 } \\nu _ 2 ( s ) d s \\\\ \\lambda _ 3 ( t ) & = e ^ { k _ E ( \\tau _ 1 - t ) } \\bigl ( \\lambda _ 3 ( \\tau _ 1 ) - a _ E b _ E \\lambda _ 1 ( \\tau _ 1 ) + \\lambda _ 0 \\frac { b _ E } { k _ E } \\bigr ) \\\\ & + a _ E b _ E \\lambda _ 1 ( \\tau _ 1 ) - \\lambda _ 0 \\frac { b _ E } { k _ E } . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} M _ d = \\left [ \\begin{array} { c c c c } \\alpha _ 0 & \\alpha _ { n - 1 } & \\cdots & \\alpha _ 1 \\\\ \\alpha _ 1 & \\alpha _ 0 & \\cdots & \\alpha _ 2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ { n - 1 } & \\alpha _ { n - 2 } & \\cdots & \\alpha _ 0 \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} \\gamma ( A ) = \\sup \\{ \\langle A \\chi _ 0 , p q \\rangle \\colon \\| p \\| _ { H [ X ] } \\leq 1 , \\| q \\| _ { H [ Y ' ] } \\leq 1 , p , q \\in \\mathcal { P } _ A \\} . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } x ' ( t ) = a _ 0 x ( t ) + a _ 1 x ( t - d ) + b _ 0 u ( t ) , t \\in [ 0 , T ] \\\\ x ( 0 ) = x _ 0 , x ( s ) = x _ 1 ( s ) , \\ ; s \\in [ - d , 0 [ \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\rho _ { ( 1 ) } ^ c ( z ) = \\frac { 1 - \\alpha } { m \\pi \\alpha } \\frac { | z | ^ { 2 / m - 2 } } { ( 1 - | z | ^ { 2 / m } ) ^ 2 } \\ , \\chi _ { | z | < \\alpha ^ { m / 2 } } . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\| \\varphi - \\tilde { z } _ n ^ M \\| _ \\infty = O \\left ( h ^ r \\max \\left \\{ \\tilde { h } ^ d , h ^ { 3 r } \\right \\} \\right ) . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} W \\psi _ { j } ( x , p ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\R ^ { n } } e ^ { - \\frac { i } { \\hbar } p y } \\psi _ { j } ( x + \\tfrac { 1 } { 2 } y ) \\overline { \\psi _ { j } ( x - \\tfrac { 1 } { 2 } y ) } \\ , d ^ { n } y . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\hat W _ i ^ L = \\bigoplus _ { l _ 0 + \\ldots + l _ { i - 1 } \\le L } { \\cal V } ^ { l _ 0 } \\otimes { \\cal V } ^ { l _ 1 } _ \\# \\otimes \\ldots \\otimes { \\cal V } ^ { l _ { i - 1 } } _ \\# \\otimes W ^ { L - ( l _ 0 + \\ldots + l _ { i - 1 } ) } _ \\# ; \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\epsilon _ { r e l } ( t ; u _ 0 ) = \\frac { | u ( x , t ) - M P _ t ( x ) | } { P _ t ( x ) } \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\left . \\frac { \\partial f ^ { \\star \\star } _ { i l } \\left ( Z , W \\right ) } { \\partial w _ { a b } } \\right \\vert _ { Z = O _ { q \\times N } \\atop { W = O _ { q \\times q } } } = 0 , \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\| A \\| _ { H [ X ] \\to H [ Y ] } = \\sup \\{ \\varphi ( A g ) \\colon \\| g \\| _ { H [ X ] } \\leq 1 , \\| \\varphi \\| _ { H [ Y ] ^ * } \\leq 1 \\} \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in [ 0 , 2 \\pi ) } \\sup _ { j \\geq 0 } \\left | \\sum _ { n = 1 } ^ j \\alpha _ n e ^ { i \\gamma _ n ( \\theta ) } \\right | = \\infty . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} \\frac { r _ { \\varepsilon } } { \\hat { \\varepsilon } _ { i } } = \\frac { \\check { \\varepsilon } _ { \\lambda } } { \\left ( \\hat { \\varepsilon } _ { i } \\right ) ^ { 2 } } \\rightarrow 0 \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} b + \\frac { 1 } { w d } + \\frac { 1 } { w ^ 2 d ^ 2 } = c , \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} K _ 2 = \\frac 1 2 K _ 3 \\min \\left \\{ \\frac 1 2 ; C _ P ( N | \\mathcal R | \\max _ { i } \\{ \\mathcal { H } _ i \\} ) ^ { - 1 } \\right \\} \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } x ( t , \\rho ) & = & \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } ( H ( t , \\rho ) \\rho ^ 2 ) + \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } ( H ^ 2 ( t , \\rho ) \\rho ^ 3 ) + \\dots \\\\ & = & 2 H ( t , \\rho ) + R _ 1 ( t , \\rho ) , \\end{array} \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l c ^ k _ { t + 1 } + c ^ k _ { t - 1 } - \\lambda _ k c ^ k _ { t } = \\frac { a _ 0 } { \\rho _ k } f _ t , \\\\ c ^ k _ { - 1 } = c ^ k _ 0 = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\tilde { \\mathcal { K } } _ n ^ M ( x ) = Q _ n { \\mathcal { K } } _ m ( x ) + { \\mathcal { K } } _ m ( Q _ n x ) - Q _ n { \\mathcal { K } } _ m ( Q _ n x ) . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} p _ 0 = \\frac { d ( \\mu - 1 ) } { 2 } . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} g ( z ) = \\sum _ { n \\geq 1 } b ( n ) e ( n z ) . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\int _ { S ' _ { d } } e ^ { i \\langle w , \\varphi \\rangle _ { 0 } } \\ , d \\mu _ { \\beta } ( w ) = E _ { \\beta } \\left ( - \\frac { 1 } { 2 } | \\varphi | _ { 0 } ^ { 2 } \\right ) , \\quad \\varphi \\in S _ { d } . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\zeta _ 0 ( t ) = { \\tilde { \\zeta } } _ 0 ( t ) \\mbox { f o r a l l } t \\in [ \\tau , \\tau + \\delta ] . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} F _ \\lambda ( \\mu ( \\lambda , r ) , r ) + R _ \\lambda ( \\mu ( \\lambda , r ) , r ) = - \\frac { a _ 1 ^ 2 f _ \\lambda ( r ) ^ 2 } { k a _ 2 \\lambda - a _ 3 f _ \\lambda ( r ) } + O ( | f _ \\lambda ( r ) | ^ { 3 - \\sigma } ) . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\left \\| p / q \\right \\| - \\left \\| p / q ' \\right \\| \\right ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { m } w _ j \\left ( \\left | p c _ j - q b _ j \\right | - \\left | p c _ j - q ' b _ j \\right | \\right ) . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{gather*} E ( t , \\tau ) = E _ 1 ( t , \\tau ) + E _ 2 ( t , \\tau ) , \\\\ E _ 1 ( t , \\tau ) : = e ^ { - i \\tau A ( t ) ^ { 1 / 2 } } F ( t ) ^ { \\perp } P - F ( t ) ^ { \\perp } e ^ { - i \\tau ( t ^ 2 S ) ^ { 1 / 2 } P } P , \\\\ E _ 2 ( t , \\tau ) : = e ^ { - i \\tau A ( t ) ^ { 1 / 2 } } F ( t ) P - F ( t ) e ^ { - i \\tau ( t ^ 2 S ) ^ { 1 / 2 } P } P . \\end{gather*}"}
-{"id": "9363.png", "formula": "\\begin{align*} T _ { x , r } ^ { u } ( y ) : = u ( x + r y ) . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} E _ { ( B , \\beta ) _ n } ( z ) = { } _ 1 \\Psi _ n \\Big [ _ { ( \\beta _ 1 , B _ 1 ) , . . . , ( \\beta _ q , B _ q ) } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; ( 1 , 1 ) \\ ; \\ ; \\ ; \\ ; } \\Big | z \\Big ] , \\ ; z \\in \\mathbb { C } . \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} X = D \\left ( t \\right ) \\partial _ { t } + T ( t ) Y ^ { i } \\partial _ { i } , \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} c ( \\theta ) = \\sum _ { j = 1 } ^ \\infty c _ j \\phi _ j ( \\theta ) = \\ ; c _ 1 \\phi _ 1 ( \\theta ) \\quad + \\sum _ { \\{ j \\ , : \\ , \\lambda _ j = d - 1 \\} } c _ j \\ , \\phi _ j ( \\theta ) \\quad + \\sum _ { \\{ j \\ , : \\ , \\lambda _ j = 2 d \\} } c _ j \\ , \\phi _ j ( \\theta ) \\quad + \\sum _ { \\{ j \\ , : \\ , \\lambda _ j > 2 d \\} } c _ j \\phi _ j ( \\theta ) \\ , . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} w _ 1 ( Y _ n ) = \\sum _ { i = 1 } ^ { n - 1 } ( \\sum _ { j = i + 1 } ^ n c _ { i , j } ) \\cdot x _ { n + i } = \\sum _ { i = 1 } ^ { n - 1 } ( \\sum _ { j = 1 } ^ n c _ { i , j } ) \\cdot x _ { n + i } \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\lambda \\vert \\xi \\vert ^ { 2 } \\leq \\sum _ { i , j = 1 } ^ { n } a _ { i j } \\xi _ { i } \\xi _ { j } \\leq \\frac { 1 } { \\lambda } \\vert \\xi \\vert ^ { 2 } \\tag * { \\textbf { E } } \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} q _ { 1 1 } ( 0 ) + 2 q _ { 1 2 } ( 0 ) + q _ { 2 2 } ( 0 ) = 4 \\sum _ { j , k } p _ { j k } ( 0 ) \\tilde { y } _ j \\tilde { y } _ k . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\prod _ { i = 1 } ^ m { \\big ( t ^ { N - 1 + 1 } ; q \\big ) } _ { \\theta ( i - 1 ) } } = 1 , \\lim _ { N \\rightarrow \\infty } { \\frac { 1 } { \\prod \\limits _ { i = 1 } ^ m { \\big ( x _ i t ^ { m - 1 } q ; q \\big ) _ { \\theta ( N - m + 1 ) - 1 } } } } = \\frac { 1 } { \\prod \\limits _ { i = 1 } ^ m { \\big ( x _ i t ^ { m - 1 } q ; q \\big ) _ { \\infty } } } . \\end{gather*}"}
-{"id": "6156.png", "formula": "\\begin{align*} G ( x , y ) = 1 + A ( x , y ) + B ( x , y ) \\ , . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} g ( z ) = ( \\sqrt { N } z ) ^ { - k } f \\ ! \\left ( - \\frac 1 { N z } \\right ) . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = - \\cos ^ { 2 } \\theta g _ { 1 } ( V , \\nabla ^ { ^ { M _ 1 } } _ { U } X ) + g _ { 1 } ( \\pi _ * ( \\omega \\phi V ) , \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } X ) ) - g _ { 1 } ( \\omega V , \\mathcal { T } _ { U } \\mathcal { B } X ) \\\\ & - g _ { 1 } ( \\pi _ * ( \\omega V ) , \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\mathcal { C } X ) ) - g _ { 1 } ( V , \\phi U ) \\eta ( X ) \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\phi ( x , 2 ^ { - k } r ) = 0 , \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\phi _ { 0 } ( x , y ) = \\frac { i } { 2 } ( x - y ) ^ { 2 } , x \\in \\C ^ { n } , y \\in \\R ^ { n } . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} C _ 2 ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) : = \\min \\{ C _ 1 ( \\delta _ d , M ( G ) ) , 1 - \\beta ( \\delta _ d , \\delta ' ) \\} > 0 . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} H ( t , x , a , p , g ) & = \\sum _ { y = 1 } ^ d \\lambda ( t , x , y , a , p ) [ g ( y ) - g ( x ) ] + c ( t , x , a , p ) \\\\ & = \\sum _ { y = 1 } ^ d ( a _ y + \\zeta ( p ) ) [ g ( y ) - g ( x ) ] + c ( t , x , a , p ) . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} a _ { i j } ^ { l l ' } \\left < Z _ { i } , Z _ { j } \\right > = \\left < v _ { i l } Z _ { i } ^ { t } , v _ { j l ' } Z _ { j } ^ { t } \\right > , \\quad \\quad \\mbox { f o r a l l $ l , l ' , i , j = 1 , 2 $ . } \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\lambda _ p ( n ) = \\max \\big \\{ \\nu _ p ( n ) - 1 , \\max \\{ j \\colon \\omega _ { p ^ j } ( n ) \\ge 1 \\} \\big \\} ; \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} f _ { L } ( x ) = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ \\left \\{ p _ { j } x + \\xi _ { j } \\right \\} + \\bar { p } L \\hat { \\theta } _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ \\left \\{ p _ { j } x \\right \\} + \\bar { p } L \\hat { \\theta } _ { j } \\right \\} , \\ \\forall x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} m _ B \\phi ( a a ' ) & = m _ B ( \\phi ( a ) \\phi ( a ' ) ) = m _ B \\phi ( a ) m _ B \\phi ( a ' ) , \\\\ h _ B \\phi ( \\{ a , a ' \\} ) & = h _ B ( \\{ \\phi ( a ) , \\phi ( a ' ) \\} ) = [ h _ B \\phi ( a ) , h _ B \\phi ( a ' ) ] , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\left | \\int _ a ^ b f ( t ) d t - \\tilde h \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; f ( \\zeta _ i ^ j ) \\right | \\leq C _ 1 \\| f ^ { ( d ) } \\| _ \\infty \\tilde { h } ^ { d } , \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} F ' _ { \\mu _ \\beta } ( \\omega ) = 1 + \\int _ { \\R } \\frac { \\dd \\widehat \\mu _ \\beta ( x ) } { ( x - \\omega ) ^ 2 } \\ , , \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} L ( z , w ) = \\frac { 1 - \\sqrt { 1 - 4 z - 4 w + 4 z ^ { 2 } } \\ , } { 2 } . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} \\bigcap _ { j = 0 } ^ N \\Lambda _ j \\frac { 2 \\pi + \\frac { \\pi } { 6 } } { T ^ { N + 1 } + 1 } . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} & x _ 1 ^ { p ^ n } = x _ 2 ^ { p ^ 2 } = x _ 3 ^ { p ^ 2 } = x _ 4 ^ p = 1 , \\\\ & [ x _ 1 , x _ 2 ] = x _ 2 ^ p , \\ ; [ x _ 1 , x _ 3 ] = [ x _ 1 , x _ 4 ] = x _ 3 ^ p , \\\\ & [ x _ 2 , x _ 3 ] = x _ 1 ^ { p ^ { n - 1 } } , \\ ; [ x _ 2 , x _ 4 ] = x _ 2 ^ p , \\ ; [ x _ 3 , x _ 4 ] = 1 . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial v } { \\partial t } ( t , x ) + \\inf _ { a \\in A } \\left \\{ \\Lambda ^ { a , m ( t ) } _ t v ( t , x ) + c ( t , x , a , m ( t ) ) \\right \\} = 0 & [ 0 , T [ \\times \\Sigma \\\\ v ( T , x ) = \\psi ( x , m ( T ) ) & \\Sigma \\end{cases} \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\omega _ { a , b } ( x , y ) = & ( - 1 ) ^ { | x | | y | } \\omega _ { a , b } ( y , x ) = ( - 1 ) ^ { | b | | y | + | x | | y | } \\omega ( a y , b x ) \\\\ = & - ( - 1 ) ^ { | a | | b | + | a | | x | } \\omega ( b x , a y ) = - ( - 1 ) ^ { | a | | b | } \\omega _ { b , a } ( x , y ) \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} 0 & = \\lim _ { k \\rightarrow \\infty } E \\left [ h ( X _ { n _ k } ( t _ i ) ; i \\leq j ) ( M _ g ^ { n _ k } ( t + s ) - M _ g ^ { n _ k } ( t ) ) \\right ] \\\\ & = E \\left [ h ( X ( t _ i ) ; i \\leq j ) ( M _ g ( t + s ) - M _ g ( t ) ) \\right ] \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} 2 \\rho ( \\alpha , A ) - 2 \\rho ( \\alpha , B ) = \\deg { ( R ) } \\alpha . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} [ M ] _ t = \\sum _ { n = 1 } ^ { \\infty } [ M ^ n ] _ t . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\left | \\int _ 0 ^ T g ( X _ t , Y _ t ) e ^ { - \\mathbf { i } \\omega t } \\textup { d } t \\right | ^ 2 \\right ) = v ( x , y , 0 ) + | u ( x , y , 0 ) | ^ 2 . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} \\partial _ N v ^ t ( z , \\lambda ^ t ( z ) ) = ( \\partial _ N v ^ t _ 1 + \\partial _ N v ^ t _ 2 ) ( z , \\lambda ^ t ( z ) ) \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} - \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = \\mathcal { C } \\left ( x , t \\right ) \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} S ( 1 ) & = 1 , \\\\ S ( a ) & = - a , ~ ~ ~ ~ \\qquad \\forall a \\in L , \\\\ S ( u v ) & = ( - 1 ) ^ { | u | | v | } S ( v ) S ( u ) \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\delta _ j ( B _ k ) \\geq d \\min \\{ \\delta _ { \\infty } ( X ) , \\delta _ { \\infty } ( Y ) \\} k = 1 , 2 , \\cdots , K . \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} s _ { \\lambda / \\mu } = ( - 1 ) ^ q \\det \\begin{pmatrix} \\Bigl ( s _ { ( \\alpha _ i | \\beta _ j ) } \\Bigr ) _ { 1 \\le i , j \\le p } & \\Bigl ( h _ { \\alpha _ i - \\gamma _ j } \\Bigr ) _ { \\substack { 1 \\le i \\le p \\\\ 1 \\le j \\le q } } \\\\ \\Bigl ( e _ { \\beta _ j - \\delta _ i } \\Bigr ) _ { \\substack { 1 \\le i \\le q \\\\ 1 \\le j \\le p } } & O \\end{pmatrix} , \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\frac { L ^ \\alpha [ e ^ a ] } { e ^ a } = = \\frac { L ^ \\alpha [ e ^ g ] } { e ^ { g } } + \\int G _ \\epsilon ( x , y ) \\left ( 1 - \\frac { \\exp ( a _ 0 ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( a _ 0 / \\epsilon ) } \\right ) K d y , \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\Omega ( z ) : = \\Omega _ 1 ( z ) \\cap \\Omega _ 3 ( z ) . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\sigma ( P _ { \\rho } ^ * ) ( \\xi ) & = \\left [ \\sum _ { | \\ell | < N _ 1 } \\sum _ m \\frac { ( - m ) ^ { \\ell } } { \\ell ! } ( \\partial ^ { \\ell } \\rho _ m ) ( \\xi ) \\prod _ { j = 1 } ^ m U _ j ^ { m _ j } + T _ { N _ 1 } ( \\xi , y ) \\right ] ^ * \\\\ & = \\sum _ { | \\ell | < N _ 1 } \\frac { \\partial ^ { \\ell } \\delta ^ { \\ell } [ ( \\rho ( \\xi ) ) ^ * ] } { \\ell ! } + T _ { N _ 1 } ( \\xi , y ) ^ * \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} U ( z ) = ( 1 - F ( z ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\| R ^ { \\ast [ - 1 ] } _ { A } B y \\| ^ { 2 } = \\langle \\omega ( A , B ) y , y \\rangle \\leq \\langle D y , y \\rangle = \\| R _ { D } y \\| ^ { 2 } , y \\in Y , \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ 1 = k ) = \\frac { 1 - q } { q } \\left ( \\lambda _ 1 ( q ) - \\sum _ { h = 1 } ^ { k - 1 } \\frac { q ^ h } { h } \\right ) \\mbox { f o r } k \\in \\mathbb { N } _ { + } , \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} q _ { 2 2 } ( 0 ) = \\sum _ { j , k } p _ { j k } ( 0 ) ( \\tilde { y } _ j - x _ j ) ( \\tilde { y } _ k - x _ k ) \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} 0 \\leq \\rho _ i ( x ) \\leq a \\cos \\sqrt { a } d ( x ) , \\ : 1 \\leq i \\leq n ; \\ : \\rho _ { n + 1 } ( x ) = a \\cos \\sqrt { a } d ( x ) . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} { ^ { \\tau } ( m \\overline { M } _ 2 ) } = \\overline { x _ d } m \\overline { x _ d ^ { - 1 } } \\overline { M } _ 2 = ( m \\overline { M } _ 2 ) ^ { - 1 } \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} [ e _ { i , 0 } , F ^ { ( 1 ) } _ { n 1 } ] = 0 \\ \\mathrm { f o r } \\ 1 < i < n - 1 , \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} D _ { 0 ^ + } ^ { \\alpha , \\beta } f ( t ) = I _ { 0 ^ + } ^ { \\beta ( n - \\alpha ) } D ^ { n } I _ { 0 ^ + } ^ { ( 1 - \\beta ) ( n - \\alpha ) } f ( t ) , \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} [ X _ i , Y _ j ] = Z \\delta _ { i j } I , \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = - 2 e _ 3 , [ e _ 1 , e _ 3 ] = 2 e _ 2 \\textrm { a n d } [ e _ 2 , e _ 3 ] = 2 e _ 1 \\ , . \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d C } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 } { \\lambda _ 5 C ^ 2 } ( \\omega C ^ 2 + k ) ^ { - 1 / 2 } \\\\ C ^ 2 ( \\omega C ^ 2 + k ) ^ { 1 / 2 } \\frac { d C } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ^ 2 } { \\lambda _ 5 } . \\end{aligned} \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\bar q ( R ) = \\frac { q ( R ) } { R } . \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\theta ( z ) = \\sum _ { n \\in \\mathbb { Z } } e ^ { 2 \\pi i n ^ 2 z } \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} Q ^ { \\Omega } u ( x , \\lambda ) = \\left ( \\frac { \\lambda } { \\pi } \\right ) ^ { n } \\int _ { \\Omega } e ^ { 2 \\lambda \\psi ( x , \\bar { y } ) } \\tilde { q } ( x , \\bar { y } , \\lambda ) u ( y ) e ^ { - 2 \\lambda \\Phi ( y ) } L ( d y ) , \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} A ' ( x , v ) = \\frac { v x } { 1 - v } A ( x , v ) - \\frac { x v } { 1 - v } A ( v x , 1 ) - \\frac { x ^ 5 v ^ 2 } { ( 1 - x ) ( 1 - v ) } W ( x , v ) + \\frac { v ^ 3 x ^ 5 } { ( 1 - x ) ( 1 - v ) } W ( v x , 1 ) . \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\lambda ( \\mu ) = \\lambda _ 1 ( \\mu ) > \\lambda _ 2 ( \\mu ) > \\ldots > \\lambda _ i ( \\mu ) > \\ldots > \\kappa ( \\mu ) . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} & \\lim _ { v \\rightarrow l } \\left ( \\sum _ { r = 1 } ^ { k } \\sum _ { s = 1 } ^ { k } f _ { k r } ( v ) e _ { k s } ( v + \\delta ^ { k t } ) - \\sum _ { r = 1 } ^ { k } \\sum _ { s = 1 } ^ { k } e _ { k s } ( v ) f _ { k r } ( v + \\delta ^ { k s } ) \\right ) \\\\ & = \\lim _ { v \\rightarrow R } h _ { k } ( v ) = h _ { k } ( R ) . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} H _ 1 = \\langle y , G ^ \\prime \\rangle , H _ 2 = \\langle x , G ^ \\prime \\rangle , H _ 3 = \\langle x y , G ^ \\prime \\rangle , H _ 4 = \\langle x y ^ 2 , G ^ \\prime \\rangle , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} J ( \\rho , m ) : = E \\left [ \\int _ 0 ^ T \\int _ A c ( s , X _ { \\rho , m } ( s ) , a , m ( s ) ) \\rho _ s ( d a ) d s + \\psi ( X _ { \\rho , m } ( T ) , m ( T ) ) \\right ] . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} Q _ { h } ( x ) = \\chi _ { K + 2 r _ h } ( x ) + \\mathfrak { s w } _ h ^ { n o r m } ( M ) , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} v _ { 3 } ( J ( \\ell - 2 , m ) - v _ { 3 } ( J ( \\ell , m ) ) & = \\frac { 1 } { 8 } \\left ( a _ { 2 } ( J ( \\ell - 2 , m ) ) + a _ { 2 } ( J ( \\ell , m ) ) + \\frac { m ^ { 2 } } { 4 } \\right ) \\\\ & = \\frac { 1 } { 8 } \\left ( \\frac { ( \\ell - 2 ) m } { 4 } + \\frac { \\ell m } { 4 } + \\frac { m ^ { 2 } } { 4 } \\right ) = \\frac { m } { 3 2 } ( 2 \\ell - 2 + m ) . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} b _ { l + 1 } = b ' _ { l + 1 } + \\gamma \\sum _ { i \\geq 1 } \\sum _ { k _ 1 + \\cdots + k _ i = l } \\frac { l } { i } ( k _ 1 - 1 ) \\cdots ( k _ i - 1 ) a _ { k _ 1 } \\cdots a _ { k _ i } . \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} a _ 1 = q _ 1 \\ \\mbox { a n d } \\ a _ { i + 1 } = q _ { i + 1 } + a _ i p _ i p _ { i + 1 } \\ \\mbox { f o r } \\ i \\geq 1 . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} ( d _ 0 \\circ \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } + \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } \\circ d _ 0 ) ( h ) = { \\rm e v } _ { + } ! ( { \\rm e v } _ { - } ^ * h ; \\partial \\widehat { \\frak S ^ { + \\epsilon } } ( \\alpha _ - , \\alpha _ + ) ) . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\begin{aligned} H ( x ; v ) & = H _ 1 ( x ) v + E ( x ; v ) - H _ { 1 , 0 } ( x ) v + \\frac { x v } { 1 - x } \\big ( H ( x ; v ) - E ( x ; v ) \\big ) + J ( x ; v ) - J _ 1 ( x ) v \\\\ & + \\frac { v x ^ 2 \\left ( - H ( x ; v ) + v ( 1 - x ) H \\big ( x ; \\frac { 1 } { 1 - x } \\big ) + E ( x ; v ) - v ( 1 - x ) E \\big ( x ; \\frac { 1 } { 1 - x } \\big ) \\right ) } { ( 1 - x ) ( 1 - v + x v ) } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t } ^ { x } = b ( X _ { t } ^ { x } ) d t + \\sum _ { i , j = 1 } ^ { d } h _ { i j } ( X _ { t } ^ { x } ) d \\langle B ^ { i } , B ^ { j } \\rangle _ { t } + \\sum _ { j = 1 } ^ { d } \\sigma _ { j } ( X _ { t } ^ { x } ) d B _ { t } ^ { j } , \\ \\ \\ \\ t \\in \\lbrack 0 , T ] , \\\\ X _ { 0 } ^ { x } = x , \\end{cases} \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} f ( \\l , x ) = x - \\l C ( x ) , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} k = \\sqrt { \\lambda _ 1 ( \\Omega ) } . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} T _ k ( s ) : = \\max \\{ \\min \\{ s , k \\} , - k \\} \\ , , s \\in \\R . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\lambda ( v ) } { \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } } = { \\cal R } ( v ) = \\sqrt { 2 } \\sigma { \\cal B } ( v ) , \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} { \\cal R } _ { [ 0 , t ] } ^ 0 : = { \\cal L } _ { t } \\left ( L ^ 2 ( 0 , t ; U ) \\right ) , \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } \\int _ { \\mathcal { Y } _ { i , \\lambda } } - u _ { i } \\left ( x , t , y ^ { i } , s ^ { \\lambda } \\right ) v _ { 1 } \\left ( x \\right ) v _ { 2 } \\left ( y _ { 1 } \\right ) \\cdots v _ { i + 1 } \\left ( y _ { i } \\right ) \\\\ \\times c _ { 1 } \\left ( t \\right ) c _ { 2 } \\left ( s _ { 1 } \\right ) \\cdots \\partial _ { s _ { \\lambda } } c _ { \\lambda + 1 } \\left ( s _ { \\lambda } \\right ) d y ^ { i } d s ^ { \\lambda } d x d t = 0 \\end{gather*}"}
-{"id": "5289.png", "formula": "\\begin{align*} & \\mathcal { E } _ { k } ^ { ( m ) } ( Z , s ) = \\sum _ { M = \\binom { * \\ ; * } { C \\ , D } \\in \\Gamma _ { \\boldsymbol { K } , \\infty } ^ { ( m ) } \\backslash \\Gamma _ { \\boldsymbol { K } } ^ { ( m ) } } ( C Z + D ) ^ { - k } | ( C Z + D ) | ^ { - s } , \\\\ & ( Z , s ) \\in \\mathcal { H } _ m \\times \\mathbb { C } . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\widehat { \\mathcal { A } } ^ 0 = b ( \\mathbf { D } ) ^ * g ^ 0 b ( \\mathbf { D } ) \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} | \\hat { \\mathbb { E } } _ { \\tau + } [ X ] - \\hat { \\mathbb { E } } _ { \\sigma + } [ X ] | & \\leq \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { m } C ( \\sup _ { ( u _ 1 , u _ 2 ) \\in \\Lambda _ { | t _ i - s _ j | , T } } ( | B _ { u _ 2 } - B _ { u _ 1 } | \\wedge 1 ) + \\sqrt { | t _ i - s _ j | } ) I _ { \\{ \\tau = t _ i \\} \\cap \\{ \\sigma = s _ j \\} } \\\\ & \\leq C ( \\sup _ { ( u _ 1 , u _ 2 ) \\in \\Lambda _ { \\delta , T } } ( | B _ { u _ 2 } - B _ { u _ 1 } | \\wedge 1 ) + \\sqrt { \\delta } ) . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\theta _ { u } ( x , r ) : = r ^ { 2 - m } \\int \\varphi \\left ( \\frac { | x - y | } { r } \\right ) | D u ( y ) | ^ { 2 } \\ , \\mathrm { d } y . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{gather*} \\frac { \\partial _ 1 r \\times \\partial _ 2 r } { | \\partial _ 1 r \\times \\partial _ 2 r | } = \\frac { 1 } { \\sqrt { g } } \\partial _ 1 r \\times \\partial _ 2 r \\end{gather*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\xi \\left ( x \\right ) = \\left [ x _ { n } + \\left ( x _ { 1 } - 1 / 2 \\right ) ^ { 2 } / \\omega ^ { 2 } + \\displaystyle \\sum \\limits _ { k = 2 } ^ { n - 1 } x _ { k } ^ { 2 } + \\frac { 1 } { 4 } \\right ] ^ { - \\nu } , \\varphi _ { \\lambda } \\left ( x \\right ) = \\exp \\left ( \\lambda \\xi \\left ( x \\right ) \\right ) , \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} F _ 2 ( x ) = \\sum _ n \\langle z e _ n , x e _ n \\rangle + \\langle z \\xi _ 0 , x \\xi _ 0 \\rangle = \\sum _ n \\langle f _ n , s _ n ( x ) f _ n \\rangle + \\langle \\eta _ 0 , 0 \\rangle = \\sum _ n s _ n ( x ) = 1 . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\Omega ( R _ { \\alpha } ) = \\bigoplus _ { r \\in \\Omega ( R _ { \\alpha } ) } \\Q [ r ] . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\rho \\left ( \\frac { \\partial u } { \\partial t } + u \\frac { \\partial u } { \\partial x } + v \\frac { \\partial u } { \\partial r } \\right ) + \\frac { \\partial P } { \\partial x } = \\mu \\left ( \\frac { \\partial ^ { 2 } u } { \\partial x ^ { 2 } } + \\frac { \\partial ^ { 2 } u } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial u } { \\partial r } \\right ) , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} M ^ * : = \\sup _ { t \\in [ 0 , T ^ * + 1 ] } M ( t ) < \\infty . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} S = \\left \\{ \\left ( ( q ^ i , \\frac { \\partial W } { \\partial q ^ i } ( \\bar { q } , q ) \\right ) \\in T ^ * Q : \\frac { \\partial W } { \\partial \\bar { q } ^ i } ( \\bar { q } , q ) = 0 \\right \\} . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\varphi = ( \\varphi _ { 1 } \\otimes e _ { 1 } , \\ldots , \\varphi _ { d } \\otimes e _ { d } ) , \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} S ^ i _ n : = s ^ i _ 0 + \\sum _ { k = 1 } ^ n \\zeta ^ i _ k \\ , \\ \\ T ^ i _ n : = \\sum _ { k = 1 } ^ n \\nu ^ i _ k ; n = 0 , 1 , \\dots \\ , . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\| ( 1 - \\chi _ { \\Sigma } ( x , h D ) ) u \\| _ { L ^ \\infty } = o ( h ^ { \\frac { 2 - n } { 2 } } ) . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\tilde W = f \\quad & \\mbox { i n } \\R ^ n \\setminus U \\\\ \\tilde W = 0 & \\mbox { o n } \\partial U \\\\ \\tilde W ( \\infty ) = 0 , & \\mbox { r e s p . } \\lim _ { x \\to \\infty } \\frac { \\tilde W ( x ) } { - \\log | x | } = 0 \\ . \\end{cases} \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\lambda ( k ) = \\frac { 1 } { 6 + \\frac { 1 9 } { k } } . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} \\chi ^ { \\pm } = \\left ( \\pm u \\right ) ^ { n - 1 } \\left [ \\psi \\mp \\frac { 1 } { 2 } { \\phi } ^ { \\prime } - \\frac { 1 } { 4 } \\phi ^ { 2 } - 2 T ^ { \\pm } - \\frac { \\phi T ^ { \\pm } } { u } \\mp \\frac { 1 } { u } \\frac { d T ^ { \\pm } } { d \\xi } - \\frac { \\left \\{ T ^ { \\pm } \\right \\} ^ { 2 } } { u ^ { 2 } } \\right ] . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} | \\hat { { \\cal M } } _ 0 ( i ) | = 2 ^ i | \\tilde { { \\cal M } } _ 0 ( i ) | \\le 2 | \\tilde { { \\cal M } } _ 0 | , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} I _ { a ^ + } ^ { \\alpha } f ( x ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } \\frac { f ( t ) d t } { ( x - t ) ^ { 1 - \\alpha } } , x > a , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\overline { K ^ * ( u ) } = K ^ * ( \\overline { u } ) \\ , , \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\widehat { u } ( \\xi , t ) = \\widehat { u _ 0 } ( \\xi ) \\ , e ^ { - | \\xi | ^ { 2 s } t } \\ , . \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\displaystyle { \\begin{array} { c c l } \\dot { r } & = & A ( \\theta ) r ^ n \\\\ \\dot { \\theta } & = & 1 + B ( \\theta ) r ^ { n - 1 } , \\\\ \\end{array} } \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} d \\Omega ( X , Y , Z ) = d \\Omega ( J X , J Y , Z ) , \\ ; \\gamma ( b ( F X , F Y ) - b ( X , Y ) , J Z ) = 0 , \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} \\mathbb { E } z ^ { Y _ 1 } = \\sum _ { n = 1 } ^ \\infty f _ n z ^ n = 1 - \\frac { 1 } { U ( z ) } \\mbox { w h e r e } U ( z ) : = 1 + \\sum _ { n = 1 } ^ \\infty u _ { n } z ^ n . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\xi _ i { \\varphi ^ { i , m } _ { l } } \\prod _ { j = 1 } ^ { i - 1 } { \\widetilde { \\varphi } ^ { j , m } _ l } ] & = \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\xi _ i I _ { K ^ { i } _ m } \\prod _ { j = 1 } ^ { i - 1 } I _ { \\widetilde { K } ^ { j } _ m } ] \\\\ & \\leq \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\xi _ i I _ { K ^ { i } _ m } ] \\\\ & \\leq \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n \\xi _ i I _ { A _ i } ] . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} [ u ' , u ' + d ' - 1 ] = [ u ' , u ' + u + 2 d - 1 ] \\subseteq A . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} - \\mathbb { K } \\nabla \\theta ^ { 0 } \\cdot \\mbox { n } = 0 \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\langle A \\chi _ j , \\chi _ k \\rangle = a _ { k - j } \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} & \\limsup _ { y \\uparrow 1 } \\limsup _ { x \\to \\infty } \\mathcal { I } _ 2 \\\\ & \\leqslant \\frac { 2 c _ 3 } { \\mathbb { P } ( \\eta = a ) } \\biggl ( \\limsup _ { y \\uparrow 1 } \\limsup _ { x \\to \\infty } \\frac { \\overline { F } _ { \\xi _ 1 } ( \\mathit { x y } ) } { \\overline { F } _ { \\xi _ 1 } ( x ) } \\biggr ) \\sum \\limits _ { n = K + 1 } ^ { \\infty } n ^ { p + 1 } \\mathbb { P } ( \\eta = n ) . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} & \\left ( \\frac { 1 } { 8 } \\right ) \\left ( ( B ^ 2 - k ) ^ { 1 / 2 } ( 2 B ^ 3 - k B ) - k ^ 2 \\ln \\left ( B + ( B ^ 2 - k ) ^ { 1 / 2 } \\right ) \\right ) \\\\ & \\qquad = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 \\omega ^ { 1 / 2 } } { \\lambda _ 4 } t + k _ 1 , \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} \\widehat { \\rho } = \\sum _ { j } \\lambda _ { j } \\widehat { \\rho } _ { j } , \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} x _ 0 ( x _ 0 ^ 3 - x _ 1 ^ 3 ) = x _ 2 ( x _ 2 ^ 3 - x _ 3 ^ 3 ) . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\phi ( t ) = \\int _ t ^ \\infty \\frac { d s } { F ( s ) } . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} Q [ u ^ + , v ^ + ] = \\sum \\limits _ { N } B [ u _ { < N } ^ + , v _ N ^ + ] . \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\left ( \\frac { 1 } { 2 } r _ R ( t ) ^ 2 \\right ) = 2 U ( x _ R ( t ) ) + \\nabla U ( x _ R ( t ) ) \\cdot x _ R ( t ) . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\frac { \\gamma ^ n } { 3 ^ { ( n - 1 ) / 2 } } - \\frac { \\bar { \\gamma } ^ n } { 3 ^ { ( n - 1 ) / 2 } } = 2 d \\sqrt { - 6 } , \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\partial _ { t } \\omega = \\nu \\Delta \\omega + e ^ { - \\nu t } \\left [ \\left ( \\sin y \\partial _ { x } - \\sin x \\partial _ { y } \\right ) \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\right ] \\omega . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { 2 ^ N } \\sum _ { k \\ge 2 ^ N + 1 } ( n + 1 ) ^ { - \\alpha } c _ { k + n } \\varepsilon _ k : = v _ { N } \\underset { N \\to \\infty } \\longrightarrow 0 \\mbox { $ m $ - a . e . } \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\min \\| x \\| \\quad { \\rm s u b j e c t \\ \\ t o } \\quad \\| A x - b \\| = \\min . \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} \\pm \\phi ( x + \\alpha ) \\mp \\phi ( x ) = \\nu ( x ) - [ \\nu ] . \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\frac { P _ { \\lambda ( N ) } \\big ( x _ 1 , \\dots , x _ m , t ^ { - m } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , t ^ { - 2 } , \\dots , t ^ { 1 - N } \\big ) } } = \\Phi ^ { \\nu } ( x _ 1 , \\dots , x _ m ; q , t ) , \\end{gather*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( [ X , Y ] , W ) & = g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Y , \\mathcal { C } X ) - ( \\nabla \\pi _ { \\ast } ) ( X , \\mathcal { C } Y ) , \\pi _ { \\ast } \\omega W ) + g _ { 1 } ( \\mathcal { A } _ { X } \\mathcal { B } Y + \\mathcal { A } _ { Y } \\mathcal { B } X , \\omega W ) \\\\ & + \\eta ( Y ) g _ { 1 } ( X , \\omega W ) - \\eta ( X ) g _ { 1 } ( Y , \\omega W ) , \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} Z ^ 2 _ 0 : = c i r c s h i f t ( Z ^ 1 _ 0 , - t _ 0 ) , \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} [ M ' ] _ t = \\sum _ { n \\geq 1 } [ \\langle M ' , h _ n \\rangle ] _ t = \\sum _ { n \\geq 1 } \\sum _ { 0 \\leq s \\leq t } \\Delta [ \\langle M ' , h _ n \\rangle ] _ s & = \\sum _ { 0 \\leq s \\leq t } \\sum _ { n \\geq 1 } \\Delta [ \\langle M ' , h _ n \\rangle ] _ s \\\\ & = \\sum _ { 0 \\leq s \\leq t } \\Delta [ M ' ] _ s . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\mathfrak { C } = \\mathbb { G } \\cap Q \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} & \\alpha ( B _ { n , 3 , 3 , 3 } ^ { ( n - 5 ) } , x ) = x ^ { n } - ( n + 1 ) x ^ { n - 2 } + ( 3 n - 9 ) x ^ { n - 4 } ; \\\\ & \\alpha ( B _ { n , 3 , 3 } ^ { ( n - 5 ) } , x ) = x ^ { n } - ( n + 1 ) x ^ { n - 2 } + ( 2 n - 5 ) x ^ { n - 4 } - ( n - 5 ) x ^ { n - 6 } . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} \\bar { u } = U , \\bar { p } = P , \\eta = 0 \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} h ^ { \\mathrm { a n } } _ { \\pi ^ * ( D , g ) } ( \\zeta ) = { \\pi ^ { \\mathrm { a n } } } ^ * ( \\vartheta _ f ) ( \\zeta ) = \\vartheta _ f ( \\pi ^ { \\mathrm { a n } } ( \\zeta ) ) = h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\pi ^ { \\mathrm { a n } } ( \\zeta ) ) . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} c _ \\lambda = \\inf _ { \\eta \\in \\Gamma _ \\lambda } \\max _ { t \\in [ 0 , 1 ] } \\Phi _ \\lambda ( \\eta ( t ) ) , \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\Delta ^ e m & = ( m \\otimes m ) \\Delta & \\Delta ^ e h & = ( m \\otimes h + h \\otimes m ) \\Delta \\\\ \\varepsilon ^ e m & = \\varepsilon & \\varepsilon ^ e h & = 0 . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} J _ h = L _ { x , h } \\partial _ x + L _ y \\partial _ y . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} J _ { \\lambda , \\gamma } \\left ( W \\right ) = e ^ { - 2 \\lambda \\left ( d + c \\right ) } \\displaystyle \\int \\limits _ { \\Omega } \\left [ L \\left ( x , p , W \\right ) \\right ] ^ { 2 } \\varphi _ { \\lambda } ^ { 2 } \\left ( x \\right ) d x + \\gamma \\left \\Vert W \\right \\Vert _ { H ^ { s } \\left ( \\Omega \\right ) } ^ { 2 } , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\| f \\| _ X = \\sup \\{ | \\langle f , g \\rangle | : g \\in X ' , \\ \\| g \\| _ { X ' } \\le 1 \\} , \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} N _ { 2 d } ^ { \\textrm { W e y l } } ( X ) : = \\sum _ { G \\leq S _ d } N _ { 2 d } ^ { \\textrm { W e y l } } ( X , G ) . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} | w ^ { s _ 1 - s } ( x ) ( e ^ { i ( \\lambda + h ) x } - e ^ { i \\lambda x } ) | & = | w ^ { s _ 1 - s } ( x ) ( e ^ { i h x } - 1 ) | \\\\ & \\le w ^ { s _ 1 - s } ( x ) \\min \\{ | h x | , 2 \\} \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{gather*} \\prod _ { s = 1 } ^ m { { g _ { n _ s + \\tau _ s ^ + - \\tau _ s ^ - } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } } , \\end{gather*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\begin{aligned} b _ { 2 j + 1 } = \\sharp D _ { 2 j + 1 } = \\prod _ { i = 1 } ^ { 2 j + 1 } \\sharp G _ i & \\geq c ( \\overline { \\beta } ) \\overline { \\beta } ^ { \\frac { p - 1 } { p } n _ { 2 j + 1 } - \\frac { 1 } { p } \\sum \\limits _ { i = 1 } ^ j n _ { 2 i - 1 } - \\sum \\limits _ { i = 1 } ^ { 2 j } d _ i } . \\end{aligned} \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} 2 s \\ , ( - \\Delta ) ^ s F = n F + r \\partial _ { r } F \\ , , r = | x | \\ , , \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\omega _ { \\rm m o d e l } : = \\sqrt { - 1 } \\sum _ { i = 1 } ^ r | z ^ i | ^ { 2 ( \\beta - 1 ) \\tau _ i } d z ^ i \\wedge d z ^ { \\bar { i } } + \\sqrt { - 1 } \\sum _ { i = r + 1 } ^ n d z ^ i \\wedge d z ^ { \\bar { i } } \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\det ( S _ 1 ^ 0 ) = - \\mu - \\alpha ( \\lambda ) \\mbox { a n d } \\det ( S _ 1 ^ 1 ) = \\varphi _ 2 ( \\lambda ) , \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\overline { c _ i } ( t ) = c _ { i , \\infty } ( 1 + \\mu _ i ( t ) ) ^ 2 , i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\begin{aligned} \\log | p ( z ) | & \\leq \\| z \\| _ { \\infty } \\sqrt { 2 } \\sum _ j | p _ j ( 0 ) | + \\| z \\| _ { \\infty } ^ 2 ( ( \\sum _ j | p _ j ( 0 ) | ) ^ 2 + \\sum | \\Re [ p _ { j k } ( 0 ) ] | ) \\\\ & \\leq \\| z \\| _ { \\infty } \\sqrt { 2 } \\sum _ j | a ( e _ j ) | + \\| z \\| _ { \\infty } ^ 2 ( ( \\sum _ j | a ( e _ j ) | ) ^ 2 + 2 \\sum | \\Re [ a ( e _ j + e _ k ) ] | ) \\\\ & \\leq \\frac { 1 } { 2 } + \\| z \\| _ { \\infty } ^ 2 ( 2 ( \\sum _ j | a ( e _ j ) | ) ^ 2 + 2 \\sum | \\Re [ a ( e _ j + e _ k ) ] | ) \\end{aligned} \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} a = ( 1 - d ^ 2 - \\epsilon ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\alpha ( x , y | z ) = \\frac { \\alpha ( x , y z y ^ { - 1 } , y ) } { \\alpha ( x , y , z ) \\alpha ( x y z ( x y ) ^ { - 1 } , x , y ) } \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\int _ { \\Theta - D _ \\theta } e ^ { - \\frac { 1 } { 2 } \\delta ^ T \\hat { K } \\delta + O _ p ( n ^ { - 1 / 2 } | | \\delta | | ^ 3 ) } d \\delta = O \\left ( e ^ { - n ^ { N } } \\right ) < O \\left ( n ^ { - N } \\right ) \\ , . \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} x ( t _ 0 , p ) = x _ 0 ( p ) \\mbox { f o r } \\ ; \\ ; p \\in \\Pi \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} \\| ( R ( Q _ n \\varphi - \\varphi ) ) ^ { ( r ) } \\| _ \\infty \\leq \\frac { C _ { 8 } ( b - a ) } { 2 } \\left ( \\sum _ { i = 1 } ^ \\rho | w _ i | \\right ) \\| Q _ n \\varphi - \\varphi \\| _ \\infty ^ 2 . \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\mathbb { A } ^ N : = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\gamma ^ N , \\xi ^ N , \\N ^ N ) \\right \\} \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} { \\dd ^ 2 \\over \\dd z ^ 2 } S _ { n , r } ( z ) \\Big | _ { z = 0 } & = \\frac { 1 } { 4 } \\sum _ { j = 1 } ^ { r } \\psi ^ { ( 1 ) } \\left ( \\frac { n - r + j } { 2 } \\right ) = \\frac 1 2 \\log n + c _ 1 - \\frac 1 2 \\log ( n - r ) - c _ 1 + o ( 1 ) \\\\ & = \\frac 1 2 \\log \\frac n { n - r } + o ( 1 ) = \\frac 1 2 \\log \\frac 1 { 1 - \\alpha } + o ( 1 ) . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} M ( X , Y ) = \\{ f \\in L ^ 0 \\colon f g \\in Y g \\in X \\} , \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\frac { d x } { d t } = g ( t ) x ^ 2 + f ( t ) x ^ 3 \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\L ^ { - \\alpha + \\delta } [ \\Lambda ^ \\alpha , \\nabla \\phi ] \\psi = [ \\nabla \\phi , \\L ^ { - \\alpha + \\delta } ] \\L ^ \\alpha \\psi + [ \\L ^ \\delta , \\nabla \\phi ] \\psi , \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} \\Gamma \\coloneqq \\{ x ^ * \\otimes y ^ * \\colon x ^ * \\in \\Gamma _ X , \\ , y ^ * \\in \\Gamma _ Y \\} = \\phi ( \\Gamma _ X \\times \\Gamma _ Y ) . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} u ( r ) = \\dfrac { 1 } { 2 m } ( \\rho ^ 2 - r ^ 2 ) \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\hat { F } _ { i } ( \\hat { v } ) : = \\mathbb { E } \\left [ F \\left ( \\sum _ { j = 1 } ^ m \\hat { v } _ j \\Phi _ j ( \\cdot ) , \\cdot \\right ) \\Phi _ i ( \\cdot ) \\right ] \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} U _ n : = & \\left ( \\frac { 2 ^ { n + 2 } } { T } + C ( \\varepsilon , p ) \\right ) \\int _ { T _ n } ^ { T } \\int \\eta _ n ^ 2 u _ n ^ p \\ ; d x d t \\\\ & \\ ; + ( C ( p ) + 1 ) 2 ^ { 2 n + 2 } \\int _ { T _ n } ^ { T } \\int _ { B _ n } a \\eta _ n ^ 2 u _ n ^ p \\ ; d x d t + 2 p k _ n ^ 2 \\int _ { T _ n } ^ { T } \\int \\eta _ n ^ 2 u _ n ^ { p - 1 } \\ ; d x d t . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} C _ 1 & = C _ 2 = \\cdots = C _ n \\\\ C _ { 1 k _ 1 \\cdots k _ { n + 1 } } & = C _ { 2 k _ 1 \\cdots k _ { n + 1 } } = \\cdots = C _ { n k _ 1 \\cdots k _ { n + 1 } } \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\mu ( t , x ) = \\mu \\left ( t , \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } f i \\right ) \\leq \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } \\mu \\left ( \\frac t { k + 1 } , f _ i \\right ) , \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} C = \\frac { \\pi + 3 ^ { 3 / 4 } } { 2 } \\ , . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V , \\varphi X ) + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} I _ \\lambda ( \\zeta ' , \\mu ' ) = J _ \\lambda ( \\sum _ { i = 1 } ^ k U _ i ) + \\theta _ \\lambda ^ { ( 2 ) } ( \\zeta ' , \\mu ' ) , \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\left < Z _ { i } , \\mathcal { B } _ { k u } ^ { j } \\left ( Z _ { j } \\right ) \\right > \\left < Z _ { k } , Z _ { u } \\right > = \\displaystyle \\sum _ { l ' = 1 } ^ { p - q } \\varphi _ { i l ' } ^ { i k } \\left ( Z _ { i } , Z _ { k } \\right ) \\overline { \\varphi _ { j l ' } ^ { j u } \\left ( Z _ { j } , Z _ { u } \\right ) } , \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\underset { Q ( ( t _ { 0 } , x _ { 0 } ) , R ) } { O s c } u = \\max _ { Q ( ( t _ { 0 } , x _ { 0 } ) , R ) } u - \\min _ { Q ( ( t _ { 0 } , x _ { 0 } ) , R ) } u . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} x ^ n = \\sum _ { l = 0 } ^ n S _ 2 ( n , l ) ( x ) _ l \\ , \\ , ( n \\geq 0 ) , ( \\textnormal { s e e } \\ , \\ , [ 1 , 2 , 4 ] ) . \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\tau _ s = \\frac { 1 } { k _ E } \\ln ( 1 + \\frac { x _ N ( 0 ) } { a _ E b _ E x _ E ( 0 ) } ) , \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\frac { N _ { 2 d } ^ { \\mathrm { W e y l } } ( X , G ) } { N _ { 2 d } ^ { \\mathrm { c m } } ( X , G ) } = 1 + O _ { d , G , \\epsilon } ( X ^ { - C _ 1 ( \\delta _ d , M ( G ) ) + \\epsilon } ) , \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} v _ { 1 } \\frac { \\partial f } { \\partial x } = \\frac { \\rho } { \\tau } \\big ( \\mathcal { M } _ { \\nu } ( f ) - f \\big ) , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\left ( m I - \\frac { \\partial P } { \\partial u } \\right ) ^ { - 1 } = \\begin{bmatrix} \\frac { 1 } { m } & - \\frac { n ( n + 1 ) } { a _ 0 ^ 4 m ( m + 1 ) } & - \\frac { n ( n + 1 ) } { a _ 0 ^ 2 m ( m + 1 ) } & - \\frac { 2 n } { m ^ 2 + m } \\\\ 0 & \\frac { 1 } { m } & 0 & 0 \\\\ 0 & 0 & \\frac { 1 } { m } & 0 \\\\ 0 & - \\frac { n + 1 } { 2 a _ 0 ^ 4 m ( m + 1 ) } & - \\frac { n + 1 } { 2 a _ 0 ^ 2 m ( m + 1 ) } & \\frac { 1 } { m + 1 } \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\phi _ j ^ { ( \\ell ) } ( u ) = \\sum _ { i = 1 } ^ \\ell ( - 1 ) ^ { i - 1 } \\binom { \\ell - 1 } { i - 1 } \\int _ { ( t _ { j - 1 } , t _ j ] \\times \\R _ + } y ^ i e ^ { - u y } \\rho ( d ( x , y ) ) \\cdot \\phi _ j ^ { ( \\ell - i ) } ( u ) . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} v _ { K , N } ( r ) : = \\omega _ N \\int _ 0 ^ r \\big | s _ { K , N } ( t ) \\big | ^ { N - 1 } \\ , \\d t , \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\liminf _ { r \\rightarrow 0 } \\frac { \\mu ( B ( x , r ) ) } { r ^ N } = \\kappa , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} \\mathbb { P } ( D ^ * _ z = d ) = \\frac { 1 } { \\mu } \\sum _ { k = 1 } ^ \\infty \\mathbb { P } ( \\Pi _ k = k + d , Y _ 1 \\ge k ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} - \\Delta v + V v = f \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} X _ k ^ { \\rm N O M A } ( \\nu ) = \\left \\{ \\begin{array} { l l } 1 , & { \\rm i f } \\ \\ R _ { k \\rightarrow k } ^ { \\rm N O M A } ( \\nu ) < \\bar R _ k , \\\\ 0 , & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} ( f ^ { \\mathrm { a n } } ) ^ * ( g - \\log | \\theta | ) & = d g - \\log | \\varphi | - \\log | f ^ * ( \\theta ) ) | \\\\ & = d ( g - \\log | \\theta | ) - \\log | f ^ * ( \\theta ) \\theta ^ { - d } \\varphi | , \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\lim _ { t \\to t _ { F _ n } ^ + } \\bigg ( h _ { \\mu _ t } ( g ) + t _ { F _ n } \\int F _ n d \\mu _ t \\bigg ) & = \\lim _ { t \\to t _ { F _ n } ^ + } \\bigg ( h _ { \\mu _ t } ( g ) + t \\int F _ n d \\mu _ t \\bigg ) \\\\ & = \\lim _ { t \\to t _ { F _ n } ^ + } P ( t F _ n ) = P ( t _ n F _ n ) . \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\ , V _ { \\underline { d } '' } ^ { X _ 1 , 0 } - \\ , V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } = \\ , V _ { \\underline { d } ' } ^ { X _ { 2 } ^ { c } , 0 } . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} I I ( X , Y ) : = \\langle \\nabla _ X \\nu , Y \\rangle , \\ : X , Y \\in T \\Sigma . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} ( a ' x ^ 2 + b ' y ^ 2 + c ' z ^ 2 ) ( a '' x ^ 2 + b '' y ^ 2 + c '' z ^ 2 ) = 0 \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ a p ^ { ( a - b ) b } = p ^ { a ^ 2 / 4 + O ( 1 ) } \\quad p ^ { ( a - \\lfloor \\frac a 2 \\rfloor ) \\lfloor \\frac a 2 \\rfloor } = p ^ { a ^ 2 / 4 + O ( 1 ) } . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { m - 1 } ( i + 1 ) ( m - i ) ( m + 1 - i ) x ^ i \\ge 0 \\ , . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} S _ 2 ( n + 1 , k ) = k S _ 2 ( n , k ) + S _ 2 ( n , k - 1 ) , \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } H ( t , \\rho ) = 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) H ( s , \\rho ) d s + 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) \\frac { \\partial } { \\partial \\rho } H ( s , \\rho ) d s + \\int \\limits _ { - 1 } ^ t R _ 2 ( s , \\rho ) d s \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s ( x _ i ^ j - y _ i ^ j ) = 0 ( 1 \\le j \\le k ) . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\int _ D K _ w ( x H , A ) \\ > & d ( \\delta _ { H h _ 1 H } * \\delta _ { H h _ 2 H } ) ( w ) = \\int _ H K _ { H h _ 1 h h _ 2 H } ( x H , A ) \\ > d \\omega _ H ( h ) \\\\ & = \\int _ H ( p _ G ( \\delta _ x * \\omega _ H * \\delta _ { h _ 1 h h _ 2 } * \\omega _ H ) ) ( A ) \\ > d \\omega _ H ( h ) \\\\ & = ( \\delta _ x * \\omega _ H * \\delta _ { h _ 1 } * \\omega _ H * \\delta _ { h _ 2 } * \\omega _ H ) ( p _ G ^ { - 1 } ( A ) ) , \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} S ( z , z ) = \\frac { 2 \\mathrm { I m } ( z ) L ( L - 1 ) } { \\pi } \\frac { | 1 - z ^ 2 | ^ { L - 2 } } { ( 1 - | z | ^ 2 ) ^ { L + 1 } } \\int _ { \\frac { 2 | \\mathrm { I m } ( z ) | } { | 1 - z ^ 2 | } } ^ 1 ( 1 - t ^ 2 ) ^ { ( L - 3 ) / 2 } \\mathrm { d } t \\left ( 1 - I _ { | z | ^ 2 } ( N - 1 , L + 1 ) \\right ) , \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} A _ 0 = \\frac { 1 } { 4 } \\sum _ { p } \\frac { p ^ 2 \\log p } { ( p - 1 ) ^ 3 ( p + 1 ) } \\quad A = A _ 0 + \\frac { \\log 2 } 2 \\approx 0 . 7 2 1 0 9 \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} U _ \\gamma ^ * \\omega ( y ) = T _ \\gamma ( y ) ^ { - \\lambda - m } \\omega ( y ) . \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} l _ i \\colon X _ 0 + a _ i X _ 1 + a _ i ^ 2 X _ 2 = 0 , a _ i \\in \\C , i = 0 , \\dots , 5 . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{gather*} J a ^ T N = \\partial _ 1 r \\times \\partial _ 2 r , \\end{gather*}"}
-{"id": "2507.png", "formula": "\\begin{align*} f _ { 1 n } ( t ) = R _ { n } e ^ { r _ { n } t } + C _ { n } e ^ { i \\omega _ { n } t } + \\overline { C _ { n } } e ^ { - i \\overline { \\omega _ { n } } t } + D _ { n } e ^ { i \\zeta _ { n } t } + \\overline { D _ { n } } e ^ { - i \\overline { \\zeta _ { n } } t } \\ , , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} Z ( s , w , f \\times \\overline { g } ) : = \\sum _ { m \\geq 1 } \\sum _ { \\ell \\geq 1 } \\frac { a ( m ) \\overline { b ( m - \\ell ) } + a ( m - \\ell ) \\overline { b ( m ) } } { m ^ { s + k - 1 } \\ell ^ w } . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\eta _ 1 ( t ) = \\Psi _ 1 \\big ( \\Delta _ 1 ( t ) \\big ) , \\eta _ 2 ( t ) = \\Psi _ 2 \\big ( \\Delta _ 2 ( t ) \\big ) , \\eta ( t ) = \\Psi \\big ( \\Delta ( t ) \\big ) . \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\frac 1 p + \\frac { d + s - \\gamma } { d } = 1 + \\frac 1 r . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\inf _ { \\overline { v } \\in \\mathbb R ^ N , \\ ; \\mathbb Q \\ , \\overline { v } = 0 } \\frac { \\frac { 1 } { 2 | \\Omega | } \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } \\left ( \\sum _ { i = 1 } ^ { N } ( y _ { r , i } ' - y _ { r , i } ) \\frac { \\overline v _ i } { u _ { i , \\infty } } \\right ) ^ 2 } { \\sum _ { i = 1 } ^ { N } \\frac { \\overline v _ i ^ 2 } { u _ { i , \\infty } } } = 0 . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} \\log \\ ( e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) } ( 1 - \\delta ) ^ { m / 2 } \\ ) \\leq O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) - \\delta m / 2 \\\\ \\qquad \\qquad = \\ ( - ( c / 2 ) \\frac { \\log ( n / m ) } { \\log n } + O ( ( m / n ) ^ { 1 / 2 } + m ^ { - 1 / 2 } ) \\ ) m . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} | \\hat { { \\cal M } } _ 0 | = \\sum _ { i = 0 } ^ { L _ n } | \\hat { { \\cal M } } _ 0 ( i ) | \\le ( 2 L _ n + 1 ) | \\tilde { { \\cal M } } _ 0 | \\le 4 L _ n | \\tilde { { \\cal M } } _ 0 | , \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} \\phi _ 1 ( t ) = x _ 0 ( t - a ) ^ { \\gamma - 1 } + \\int _ { a } ^ { t } \\frac { ( t - s ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } f ( s , \\phi _ { 0 } ( s ) ) d s . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\theta _ \\lambda ^ { ( 2 ) } ( \\zeta ' , \\mu ' ) = - \\int _ 0 ^ 1 s \\left [ \\int _ { \\Omega _ \\varepsilon } \\vert \\nabla \\phi \\vert ^ 2 - \\varepsilon ^ 2 \\lambda \\phi ^ 2 - 5 ( V + s \\phi ) ^ 4 \\phi ^ 2 \\right ] \\ , d s , \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} d ( [ x ] , [ x ' ] ) = \\min _ { g \\in G } d ( x , g x ' ) \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} \\sigma _ { X ^ * } ( m ) = \\lim _ { t \\to \\infty } \\frac { \\log \\left | \\kappa _ { X ^ * } ^ { ( m ) } ( t ) \\right | } { \\log t } = \\lim _ { t \\to \\infty } \\frac { \\log \\left | \\kappa _ X ^ { ( m ) } m I _ { m - 1 } ( t ) \\right | } { \\log t } = \\lim _ { t \\to \\infty } \\frac { \\log \\left | I _ { m - 1 } ( t ) \\right | } { \\log t } . \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\varphi ( t ) _ { \\Omega } = ( \\varphi _ 0 ) _ { \\Omega } , \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\frac 1 { 1 - \\omega _ 1 h _ 1 - \\cdots - \\omega _ p h _ p } \\cdot \\prod _ { i = 1 } ^ p \\frac { ( 1 - h _ i ) ^ { m _ i } } { 1 - 2 h _ i } \\quad . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } ^ x = \\sum _ { \\substack { T \\ni x \\\\ S \\thicksim T } } 1 \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} ( \\nabla a , u \\nabla ( \\eta ^ 2 u ^ { p - 1 } ) ) & = u u ^ { p - 1 } ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) + ( p - 1 ) u u ^ { p - 2 } \\eta ^ 2 ( \\nabla a , \\nabla u ) \\\\ & = u u ^ { p - 1 } ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) + ( p - 1 ) ( u ^ { p - 1 } ) \\eta ^ 2 ( \\nabla a , \\nabla u ) \\\\ & = ( u ^ p ) ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) + \\eta ^ 2 ( \\nabla a , \\nabla ( \\frac { p - 1 } { p } u ^ p ) ) , \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} C _ { \\operatorname * { r e g } , A _ { \\scriptscriptstyle 0 } } : = \\frac { 1 } { \\beta _ { \\scriptscriptstyle 0 } } \\frac { C _ { P } ^ { \\eta \\left ( p , P \\right ) } } { 1 - C _ { P } ^ { \\eta \\left ( p , P \\right ) } \\left ( 1 - \\frac { \\alpha _ { \\scriptscriptstyle 0 } } { \\beta _ { \\scriptscriptstyle 0 } } \\right ) } \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\langle f , g \\rangle : = \\iint _ { \\Gamma \\backslash \\mathcal { H } } f ( z ) \\overline { g ( z ) } d \\mu ( z ) . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\ , u = f & \\mbox { i n } \\O \\\\ u = 0 & \\mbox { o n } \\partial \\O \\ , , \\end{cases} \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} Z \\boxminus Y = \\bigcap _ { X \\in \\Delta ( Z , Y ) } X . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} \\sigma ^ 2 ( V _ \\chi ^ S ) = 4 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma > T } } \\frac { \\sigma ^ 2 ( Z _ \\gamma ) } { { \\frac 1 4 + \\gamma ^ 2 } } = 4 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma > T } } \\frac 1 { { \\frac 1 4 + \\gamma ^ 2 } } \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\tilde { l } ^ { G E L } _ { \\gamma } ( \\theta ) = \\sum _ { r , s } \\delta _ r \\delta _ s \\hat { \\nu } _ { r s } + \\frac { 1 } { \\sqrt { n } } \\sum _ { r , s , t } \\delta _ r \\delta _ s \\delta _ t \\hat { G } _ { r s t } + O _ p ( n ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} \\mathbf { l } _ k ( \\mathbf { x } ) = \\mathbf { l } ^ 0 _ k + b ( \\mathbf { D } ) \\mathbf { w } _ k ( \\mathbf { x } ) , \\mathbf { l } ^ 0 _ k \\in \\mathbb { C } ^ m , \\mathbf { w } _ k \\in \\widetilde { H } ^ 1 ( \\Omega ; \\mathbb { C } ^ n ) , k = 1 , \\ldots , m , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} u ( t ) = M ( t ) D ( t ) \\bigl [ e ^ { - i \\lambda | W | ^ 2 \\log t } W \\bigr ] + \\mathcal { O } ( t ^ { - \\frac 3 4 + \\beta } ) \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} z _ \\lambda : = \\prod _ { i \\geq 1 } m _ i ( \\lambda ) ! \\ i ^ { m _ i ( \\lambda ) } , \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | \\nabla \\varphi | ^ 2 \\ , d \\mu = \\int _ { \\R ^ N } \\sum _ { i = 1 } ^ { n + 1 } | \\nabla \\left ( J _ i \\varphi \\right ) | ^ 2 \\ , d \\mu - \\int _ { \\R ^ N } \\sum _ { i = 1 } ^ { n + 1 } | \\nabla J _ i | ^ 2 \\varphi ^ 2 \\ , d \\mu . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n A _ { i j k } ( v ) v ^ i = \\sum _ { j = 1 } ^ n A _ { i j k } ( v ) v ^ j = \\sum _ { k = 1 } ^ n A _ { i j k } ( v ) v ^ k = 0 \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} D _ s ^ { \\ell } \\alpha _ s ( a ) & = ( - i \\partial _ s ) ^ { \\ell } \\sum _ m e ^ { i s \\cdot m } a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\sum _ m m ^ { \\ell } e ^ { i s \\cdot m } a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\sum _ m \\delta ^ { \\ell } e ^ { i s \\cdot m } a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\delta ^ { \\ell } \\alpha _ s ( a ) \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\| x _ i \\| _ X ^ q \\right ) ^ { \\frac { 1 } { q } } \\leq M \\left \\| \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ q \\right ) ^ { \\frac { 1 } { q } } \\right \\| _ X \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} { \\rm R i c } ( \\omega ) = \\gamma \\omega + L _ X \\omega \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\infty } ( \\mathcal { C } ) = \\bigotimes _ { i = 1 } ^ { \\infty } U ( \\mathcal { G } ( C ) _ i ) \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\mathcal { P } _ { k , n } = \\langle { a , t } \\mid { a ^ k , ~ t a ^ n = a ^ n t } \\rangle , \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} 2 p k _ n ^ 2 \\int _ { T _ n } ^ { T } \\int \\eta _ n ^ 2 u _ n ^ { p - 1 } \\ ; d x d t \\le & \\ ; 2 p M ^ 2 \\int _ { T _ n } ^ { T } \\int _ { B _ n } u _ n ^ { p - 1 } \\ ; d x d t \\\\ = & \\ ; 2 p M ^ 2 \\int _ { T _ n } ^ { T } \\int _ { B _ n } u _ n ^ { p - 1 } \\chi _ { \\{ u _ { n - 1 } \\ge \\frac { M } { 2 ^ { n } } \\} } \\ ; d x d t \\\\ \\le & \\ ; 2 p M ^ 2 \\int _ { T _ n } ^ { T } \\int _ { B _ n } u _ { n - 1 } ^ { p - 1 } \\chi _ { \\{ \\eta _ { n - 1 } ^ { 2 / p } u _ { n - 1 } \\ge \\frac { M } { 2 ^ { n } } \\} } \\ ; d x d t . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} L ( t , x , \\tau , \\xi ) = \\tau I + \\sum _ { j = 1 } ^ k { \\tilde A } _ { 1 j } ( t , x , \\xi ) b _ j ( t , x , \\xi ) \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\langle P _ h ( \\cdot , s ) , E _ { \\mathfrak { a } } ^ k ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle = \\frac { \\overline { \\rho _ \\mathfrak { a } ( h , \\tfrac { 1 } { 2 } + i t ) } } { ( 4 \\pi h ) ^ { s - 1 } } \\frac { \\Gamma ( s - \\tfrac { 1 } { 2 } + i t ) \\Gamma ( s - \\tfrac { 1 } { 2 } - i t ) } { \\Gamma ( s - \\frac { k } { 2 } ) } . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} T _ f g ( x ) = K _ r ( x , \\{ z \\} ) \\ > \\omega _ D ( \\{ r \\} ) = K _ r ( x , \\{ z \\} ) \\ > \\omega _ D ( \\{ r \\} ) \\ > \\delta _ { r , \\pi ( x , z ) } . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} a ^ { i j } D _ { i j } u = g \\ ; \\mbox { i n } \\ ; B ^ + _ 4 \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} r ^ { - m } \\int _ { B _ { r } ( 0 ) } \\left ( r ^ { 2 } | D _ { \\hat { L } } u | ^ { 2 } + | D _ { \\hat { L } ^ { \\perp } } u | ^ { 2 } | v | ^ { 2 } \\right ) \\leq C ( m , \\rho ) \\sum _ { i = 0 } ^ { m - 2 } \\left ( \\theta ( y _ { i } , 8 r ) - \\theta ( y _ { i } , 4 r ) \\right ) , \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} M ( \\delta t , x , y ) \\triangleq \\begin{pmatrix} m _ 1 ( \\delta t , x , y ) \\\\ m _ 2 ( \\delta t , x , y ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} \\partial F & = \\partial ( e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } a _ { 1 } + e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } a _ { 2 } ) \\\\ & = e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\partial a _ { 1 } + e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\partial a _ { 2 } \\\\ & + 2 m _ { 1 } a _ { 1 } e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\partial ( \\varphi - \\eta ) + 2 m _ { 2 } a _ { 2 } e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\partial ( \\varphi - \\eta ) , \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} x _ i ^ 4 = 1 , \\ , \\ , ( 1 \\leq i \\leq n ) , \\ , \\ , [ x _ i , x _ n ] = x _ { i + 1 } ^ 2 , \\ , \\ , ( 1 \\leq i \\leq n - 1 ) , \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} \\tilde { W } = A ^ { - 1 } \\otimes W , \\tilde { Z } = V ^ { - 1 } \\otimes Z , \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} Y _ 1 ( T _ 0 + t ) - Y _ 1 ( T _ 0 ) & = \\sqrt { s n r } \\int _ { T _ 0 } ^ { T _ 0 + t } X _ 1 ( s ) d s + \\sqrt { i n r } \\int _ { T _ 0 } ^ { T _ 0 + t } X _ 2 ( s ) d s + B _ 1 ( T _ 0 + t ) - B _ 1 ( T _ 0 ) \\\\ & = \\sqrt { s n r } \\int _ 0 ^ t X _ 2 ( s ) d s - \\sqrt { i n r } \\int _ 0 ^ t X _ 1 ( s ) d s + B _ 1 ( T _ 0 + t ) - B _ 1 ( T _ 0 ) . \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\rho \\frac { \\partial u } { \\partial t } + \\frac { \\partial P } { \\partial x } = \\mu \\left ( \\frac { \\partial ^ { 2 } u } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial u } { \\partial r } \\right ) \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} I ( M ; \\overline { M } ) = D _ { K L } \\left ( P ( X ) | | P ( M ) P \\left ( \\overline { M } \\right ) \\right ) . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\langle P _ h ^ k ( \\cdot , s ) , V \\rangle = \\sum _ j \\langle P _ h ^ k ( \\cdot , s ) , \\mu _ j \\rangle \\langle \\mu _ j , V \\rangle + \\sum _ { \\frac { 1 } { 2 } \\leq \\ell \\leq k } \\sum _ j \\langle P _ h ^ k ( \\cdot , s ) , \\mu _ { j , \\ell } \\rangle \\langle \\mu _ { j , \\ell } , V \\rangle . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\aligned & ( \\hat d _ { 2 } ^ { i + 1 } \\circ ( \\frak h _ b ^ { i i + 1 } - \\frak h _ a ^ { i + 1 i } ) + ( \\frak h _ b ^ { i + 1 i } - \\frak h _ a ^ { i i + 1 } ) \\circ \\hat d _ { 1 } ^ { i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } \\\\ & = ( \\frak n _ b ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } - \\psi _ 2 ^ { i i + 1 } \\circ \\frak n _ b ^ { i } - \\frak n _ a ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } + \\psi _ 2 ^ { i i + 1 } \\circ \\frak n _ a ^ { i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } . \\endaligned \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\frac { N _ { 2 d } ^ { \\textrm { W e y l } } ( X , G ) } { N _ { 2 d } ^ { \\textrm { c m } } ( X ) } = C ( d , G ) + O ( X ^ { - \\alpha ( d , G ) } ) \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\Psi ( t ) = ( t - q _ 1 ) \\cdots ( t - q _ r ) , \\ ; \\ ; \\ ; t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty ( b _ n \\lambda _ n - c _ n ) a _ n = 0 \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} S = \\left \\{ y \\in R : \\tau \\left ( y \\right ) \\in N , \\left \\langle \\theta , u \\right \\rangle = \\left \\langle d F , T \\tau ( u ) \\right \\rangle , \\forall u \\in T _ { y } Y \\right \\} \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} \\tilde { R } ( t ) = C t ^ { ( 2 - \\gamma ) / 2 } ( 1 + ( \\frac { 1 } { 2 } - \\frac { 2 - \\gamma } { 2 } ) \\ln \\frac { 1 } { t } ) \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} \\lambda ( t _ f ^ - ) = \\gamma \\frac { \\partial h ^ \\ast } { \\partial x } [ t _ f ] , \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\sum _ { A , B } \\left ( h _ { A B } , { H _ { A B } } \\right ) _ { \\varphi } = \\sum _ { A , B } \\left ( h _ { A B } , { H _ { [ A B ] } } \\right ) _ { \\varphi } . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} C _ n = R _ n = 0 \\qquad \\mbox { f o r a n y } n \\in { \\cal F } \\ , , \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} a n n _ A ( X ) & = m _ A ^ { - 1 } ( a n n _ { A ^ e } ( X ) ) = m _ A ^ { - 1 } \\eta ( a n n _ { R ^ e } ( X ) ) \\\\ & = \\pi ' m _ R ^ { - 1 } ( a n n _ { R ^ e } ( X ) ) = \\pi ' ( a n n _ { R } ( X ) ) = \\pi ' \\pi '^ { - 1 } ( P ) = P , \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} x ' = t ^ { n - 1 } x ^ 2 + f ( t ) x ^ 3 , \\ , \\ , \\ , t \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} F _ T ( x ) & = 1 + x F _ T ( x ) + \\frac { x ^ 3 - 7 x ^ 2 + 5 x - 1 } { ( 1 - x ) ( 1 - 2 x ) } - \\frac { 3 x ^ 3 - 9 x ^ 2 + 6 x - 1 } { ( 1 - x ) ^ 2 } C ( x ) + x C ( x ) F _ T ( x ) \\\\ & + \\frac { x } { 1 - x } \\big ( C ( x ) - 1 - x C ( x ) \\big ) + \\frac { x ^ 4 } { ( 1 - x ) ^ 3 } \\frac { 1 - x } { 1 - 2 x } C ( x ) \\ , , \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} H ( q , p ) = E _ L \\circ \\mathbb { F } L ^ { - 1 } . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} L _ s - \\lambda = ( - \\partial - \\phi _ - ) ( \\partial - \\phi _ - ) , \\mbox { i n } K ( \\Gamma _ s ) [ \\partial ] , \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\hat { U } _ i = \\sum _ { j = 1 } ^ { \\hat { \\mu } _ i } C _ { j , i } = \\sum _ { j = 1 } ^ { \\min \\{ l _ i , \\lfloor \\hat { r } _ i / I _ p \\rfloor \\} } C _ { j , i } . \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} X _ B ( t ) = \\begin{cases} 1 & \\sum _ { t ' = t B } ^ { ( t + 1 ) B - 1 } W ( t ' ) , \\cr 0 & \\end{cases} \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} E _ { M } ( I ) : = \\big \\{ \\omega \\in \\Omega : \\| \\phi ^ { \\omega } \\| _ { H ^ s } + \\| S ( t ) \\phi ^ { \\omega } \\| _ { S ^ s ( I ) } \\le M \\big \\} . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\ , V _ { X _ 1 } ( - ( d - i + 1 ) A ) - \\ , \\alpha _ { 1 \\underline { d } '' } ( K _ { i - 1 , l } ) = \\ , ( e v ^ { i - 1 , l } ) - \\ , ( \\overline { e v } ^ { i - 1 , l } ) , \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\psi _ { m , n } ( z ) = \\prod ^ { n } _ { r = m } ( z - \\xi _ { r } ) , n \\geq m \\geq 1 ; \\psi _ { m , m - 1 } ( z ) = 1 , m \\geq 1 . \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} f \\bigl | \\bigl [ g _ i - \\xi ( g _ i ) \\bigr ] = \\sum _ { n \\in \\Z } \\lambda _ i ( n ) e \\ ! \\left ( n \\tfrac { z } { m } \\right ) . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} B : = L ^ \\top A L ^ { - \\top } , \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} V ^ { ( \\ell ) } : = H _ n ( 0 ) ^ { ( \\ell ) } v . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\frac { d } { d r } \\Big ( \\frac { - 1 } { \\gamma e ( r ) ^ \\gamma } - c \\log r \\Big ) = \\frac { 1 } { e ( r ) ^ { 1 + \\gamma } } \\frac { d } { d r } e ( r ) - \\frac { c } { r } \\geq \\frac { 1 } { e ( r ) ^ { 1 + \\gamma } } f ( r ) \\geq 0 \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\lambda _ r : = \\limsup _ { n \\to \\infty } \\frac { \\gamma _ { n + r } - \\gamma _ n } { 2 \\pi / \\log \\gamma _ n } \\mu _ r : = \\liminf _ { n \\to \\infty } \\frac { \\gamma _ { n + r } - \\gamma _ n } { 2 \\pi / \\log \\gamma _ n } . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\lambda ( k ) = \\frac { 1 } { 6 + \\frac { 1 9 } { k } } . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\begin{cases} \\Delta W = f \\ , \\chi _ { \\R ^ n \\setminus U } \\quad & \\mbox { i n } \\R ^ n \\\\ W ( \\infty ) = 0 & \\lim _ { x \\to \\infty } \\frac { W ( x ) } { - \\log | x | } = 2 \\pi \\int _ { \\R ^ 2 } f \\ , \\chi _ { \\R ^ n \\setminus U } \\ . \\end{cases} \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} \\| ( g - p _ { \\boldsymbol { k } } ) | _ { \\mathcal { I } _ { \\boldsymbol { k } } } \\| _ p \\leq C _ 2 \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l W _ l } \\left \\| \\frac { \\partial ^ { \\alpha _ l } } { \\partial x _ l ^ { \\alpha _ l } } g \\middle | _ { \\mathcal { I } _ { \\boldsymbol { k } } } \\right \\| _ p , \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\frac { \\overline { C _ i } } { C _ { i , \\infty } } = \\frac { \\sqrt { \\overline { C _ i ^ 2 } } } { C _ { i , \\infty } } - \\frac { \\| \\delta _ i \\| ^ 2 } { C _ { i , \\infty } \\Bigl ( \\sqrt { \\overline { C _ i ^ 2 } } + \\overline { C _ i } \\Bigr ) } = \\sqrt { \\frac { { \\overline { c _ i } } } { c _ { i , \\infty } } } - Q ( C _ i ) \\| \\delta _ i \\| \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} u = \\sum _ { j = 1 } ^ k w _ { \\mu _ j , \\zeta _ j } + O ( \\varepsilon ^ { \\frac { 1 } { 2 } } ) \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} q ( X _ { C , G ^ 0 } ) = & \\frac { g ( C ) - 1 } { G ^ 0 } + 1 , & \\chi ( { \\mathcal O } _ { X _ { C , G ^ 0 } } ) = & \\frac { | G ^ 0 | ( q - 1 ) ( q - 2 ) } { 2 } , & K ^ 2 _ { X _ { C , G ^ 0 } } = & | G ^ 0 | ( q - 1 ) ( 4 q - 9 ) . \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\xi ^ * \\sigma ^ 2 \\cap [ Z _ 2 , M ] = \\sigma ^ 2 \\cap \\xi _ * [ Z _ 2 , M ] = t \\sigma ^ 2 \\cap \\theta _ * [ P , S ^ 3 ] + \\sigma ^ 2 \\cap \\psi _ * [ Z _ 1 , M ] . \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} C H ^ L ( U , \\lambda ) : = \\lim _ { \\substack { \\longleftarrow \\\\ ( X , \\lambda | _ X ) \\textrm { t a m e d o m a i n } \\\\ X \\subset U \\\\ } } C H ^ L ( X , \\lambda ) . \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} P _ 1 + P _ 2 - 2 P _ 3 = 2 ^ { r - 3 } . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } \\Delta v + \\varepsilon ^ 2 \\ , \\lambda \\ , v & = & - v ^ 5 & \\Omega _ \\varepsilon , \\\\ v & > & 0 & \\Omega _ \\varepsilon , \\\\ v & = & 0 & \\partial \\Omega _ \\varepsilon , \\end{array} \\right . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} \\langle \\mathcal { V } _ { f , \\overline { g } } , \\mu _ j \\rangle = \\langle T _ { - 1 } \\mathcal { V } _ { f , \\overline { g } } , \\mu _ j \\rangle = \\langle \\mathcal { V } _ { f , \\overline { g } } , T _ { - 1 } \\mu _ j \\rangle = - \\langle \\mathcal { V } _ { f , \\overline { g } } , \\mu _ j \\rangle . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} a _ { n + 1 } = \\langle A \\chi _ 0 , \\chi _ n \\rangle = \\hat d ( n ) \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( P _ \\alpha ^ u \\sigma ) = l _ \\alpha ^ u ( \\sigma ) , \\\\ r _ { \\langle \\rangle } ( P _ \\alpha ^ u \\sigma ) = r _ \\alpha ^ u ( \\sigma ) , \\\\ o _ { \\langle \\rangle } ( P _ \\alpha ^ u \\sigma ) = o _ \\alpha ^ u ( \\sigma ) . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\left < Z _ { i } , \\mathcal { B } _ { k u } ^ { j } \\left ( Z _ { j } \\right ) \\right > = \\left < \\mathcal { B } _ { u k } ^ { i } \\left ( Z _ { i } \\right ) , Z _ { j } \\right > , \\quad \\quad \\quad \\mbox { f o r a l l $ k , u , i , j = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{gather*} \\operatorname { R e } \\sigma = - \\frac { n - 1 } { 2 } \\textrm { a n d } \\operatorname { I m } \\sigma = \\pm \\sqrt { \\mu ^ { 2 } R ^ { 2 } - ( n - 1 ) ^ { 2 } / 4 } \\eqqcolon \\pm \\mu ' . \\end{gather*}"}
-{"id": "7657.png", "formula": "\\begin{align*} & x _ 1 - x _ 2 = v + z _ j , & & x _ 2 - x _ 3 = - ( - u + v + a ) , & & x _ 1 - x _ 3 = u + z _ j - a \\\\ & x _ 1 - y _ 1 = v + z _ j + \\lambda , & & x _ 1 - y _ 2 = u + z _ j + a , & & x _ 1 - y _ 3 = - \\lambda - 2 a \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} x ^ 2 = \\frac { s } { k } + \\frac { k + 1 } { 2 k } \\leq - \\frac { 1 } { \\sqrt { k } } + \\frac { k + 1 } { 2 k } = \\left ( \\frac { 1 } { \\sqrt { 2 } } - \\frac { 1 } { \\sqrt { 2 k } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\sum a ^ - _ 1 \\star b ^ + _ 2 \\cdot ( a _ 2 , b _ 1 ) = \\sum b ^ + _ 1 \\star a ^ - _ 2 \\cdot ( b _ 2 , a _ 1 ) , \\ , \\ \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\omega ^ { \\nu } | ^ { 2 } - | \\bigtriangledown \\psi ^ { \\nu } | ^ { 2 } ) d x d y = - 2 \\nu \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\bigtriangledown \\omega ^ { \\nu } | ^ { 2 } - | \\omega ^ { \\nu } | ^ { 2 } ) d x d y , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} B & = C \\\\ A ^ 2 E ^ 2 ( B ^ 2 - C ^ 2 ) = \\lambda _ 1 ^ 2 \\lambda _ 5 ^ 2 ( & \\lambda _ 2 ^ 2 - \\lambda _ 3 ^ 2 ) = \\lambda _ 1 ^ 2 \\lambda _ 5 ^ 2 ( 0 ) = 0 \\\\ A ^ 2 B C D ^ 2 = \\lambda _ 1 ^ 2 \\lambda _ 2 & \\lambda _ 3 \\lambda _ 4 ^ 2 = \\lambda _ 1 ^ 2 \\lambda _ 2 ^ 2 \\lambda _ 4 ^ 2 . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} f = \\left ( \\begin{array} { c } f _ { 0 ' 0 ' \\ldots 0 ' 0 ' } \\\\ f _ { 1 ' 0 ' \\ldots 0 ' 0 ' } \\\\ \\vdots \\\\ f _ { 1 ' 1 ' \\ldots 1 ' 1 ' } \\end{array} \\right ) = \\left ( \\begin{array} { c } f _ { 0 } \\\\ f _ { 1 } \\\\ \\vdots \\\\ f _ { k } \\end{array} \\right ) , \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } \\frac { e ^ { f } } { b ^ { 2 n - 1 } } \\int _ { s _ 0 } ^ { s } b ^ { 2 n - 1 } e ^ { - f } \\left ( Q ^ 3 - Q \\right ) \\ ; \\mathrm { d } s & \\geq n \\sqrt { n + 1 } \\lim _ { s \\rightarrow \\infty } e ^ { f } \\int _ { s _ 0 } ^ s e ^ { - f } \\ ; \\mathrm { d } s \\\\ & = - \\frac { n \\sqrt { n + 1 } } { f ' _ { \\infty } } > 0 , \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\int _ { \\xi } ^ { \\alpha _ { 2 } } { e ^ { - u t } G _ { n } \\left ( t \\right ) d t } = \\frac { 1 } { u } e ^ { - u \\xi } G _ { n } \\left ( \\xi \\right ) + \\frac { 1 } { u } \\int _ { \\xi } ^ { \\alpha _ { 2 } } { e ^ { - u t } { G } _ { n } ^ { \\prime } \\left ( t \\right ) d t . } \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} \\partial _ { w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\cdot G \\left ( W , Z \\right ) \\right ) = \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\cdot \\partial _ { w _ { a a } } \\left ( G \\left ( W , Z \\right ) \\right ) + \\partial _ { w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot G \\left ( W , Z \\right ) , \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\begin{cases} \\rho ^ { n + \\lambda } F _ + ( \\rho ) , & \\lambda \\not \\in - \\frac n 2 + \\N ; \\\\ \\rho ^ { n + \\lambda } ( \\log ( \\rho ) F _ + ( \\rho ) + F _ + ' ( \\rho ) ) , & \\lambda \\in - \\frac n 2 + \\N , \\end{cases} \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} f ^ { ( e ) } ( X ) = X ^ S \\phi ( X ) , S \\geq 1 , \\ : \\phi ( 0 ) \\neq 0 . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\psi _ { c a } ^ { i } = \\psi _ { c b } ^ { i } \\circ \\psi _ { b a } ^ { i } , \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\mathbf { w } _ { 2 } = \\left ( \\begin{array} { c } \\mathbf { 0 } _ { 2 } \\\\ \\sin \\omega _ { 2 } \\mathbf { t } _ { 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in \\hat { G } ^ n } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) & = n \\sum _ { m = 0 } ^ M \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) \\\\ & \\qquad + O \\ ( n \\sum _ { m = M + 1 } ^ { 2 \\epsilon n } \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\ ) \\\\ & \\qquad + O \\ ( \\sum _ { H ( \\chi ) > \\epsilon } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\ ) , \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} [ x _ p ] = \\varphi _ { p r } ^ { \\frak C \\boxplus \\tau } ( [ x _ r ] ) , [ x _ q ] = \\varphi _ { q r } ^ { \\frak C \\boxplus \\tau } ( [ x _ r ] ) . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} W _ { n } = P _ { \\overline { B \\left ( R \\right ) } } \\left ( W _ { n - 1 } - \\varsigma J _ { \\lambda , \\gamma } ^ { \\prime } \\left ( W _ { n - 1 } \\right ) \\right ) ; n = 1 , 2 , . . . \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} P _ h ( 1 ) = \\mathfrak { s w } ^ { n o r m } _ h ( M ) . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\vartheta ( \\hbar ) [ \\mathfrak { X } _ i ^ { + } ( u , \\lambda _ 1 ) , \\mathfrak { X } _ i ^ { - } ( v , \\lambda _ 2 ) ] = \\frac { \\vartheta ( u - v + \\lambda _ { 1 , i } ) } { \\vartheta ( u - v ) \\vartheta ( \\lambda _ { 1 , i } ) } \\Phi _ i ( v ) + \\frac { \\vartheta ( u - v - \\lambda _ { 2 , i } ) } { \\vartheta ( u - v ) \\vartheta ( \\lambda _ { 2 , i } ) } \\Phi _ i ( u ) , \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) ^ { \\tau _ 1 } & = ( x _ 3 , x _ 4 , x _ 1 , x _ 2 ) \\\\ ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) ^ { \\tau _ 2 } & = ( x _ 4 , x _ 3 , x _ 2 , x _ 1 ) \\\\ ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) ^ { \\tau _ 3 } & = ( x _ 2 , x _ 1 , x _ 4 , x _ 3 ) , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} w ( \\boldsymbol { \\varphi _ { x } } ) & = \\sum _ { i = 1 } ^ { k } \\sum _ { \\substack { ( y , z ) \\in S _ { i } \\\\ x \\not \\neq y , z } } \\frac { 1 + b _ { k } - b _ { i } } { 4 } \\\\ w ( \\boldsymbol { \\varphi _ { \\{ x , y \\} } } ) & = 2 \\cdot \\frac { 1 + b _ { k } - b _ { i } } { 4 } i ( x , y ) \\in S _ { i } \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} \\overline { M } ^ { ( n ) } \\overline { M } _ 2 \\overline { R } = \\overline { M } ^ { ( n ) } \\overline { M } _ 2 ( \\mathfrak { M } ^ { ( n ) } \\overline { R } ) = \\overline { M } ^ { ( n ) } \\overline { M } _ 2 ( \\mathfrak { M } ^ { ( n ) } \\overline { R } ^ { ( n ) } ) = \\overline { M } ^ { ( n ) } \\overline { M } _ 2 \\overline { R } ^ { ( n ) } . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} v _ q ^ { V } = \\frac { d } { d t } | _ { t = 0 } \\left ( w _ q + t v _ q \\right ) \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\hat M _ { \\mathfrak { a } } = \\varphi _ M ( M ) \\cdot \\hat \\Lambda _ { \\mathfrak { a } } \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\rho ( \\tilde { \\theta } _ { 1 } ^ \\alpha , x ) < \\alpha \\right ) = \\alpha + O ( n ^ { - 1 / 2 } ) \\ , , \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} \\mu _ { j , \\ell } ( z ) = \\sum _ { n \\neq 0 } \\rho _ { j , \\ell } ( n ) W _ { \\frac { n } { \\lvert n \\rvert } \\frac { k } { 2 } , \\frac { \\ell - 1 } { 2 } } ( 4 \\pi \\lvert n \\rvert y ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\Lambda _ j ( s ) = \\pi ^ { - s } \\Gamma \\left ( \\tfrac { s + \\epsilon + i t _ j } { 2 } \\right ) \\Gamma \\left ( \\tfrac { s + \\epsilon - i t _ j } { 2 } \\right ) L ( s , \\mu _ j ) = ( - 1 ) ^ \\epsilon \\Lambda _ j ( 1 - s ) , \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\ + E _ s = \\left \\{ Y \\ , : \\ , \\Upsilon ^ Y _ s C _ s \\right \\} \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} 0 \\leq \\rho ( y ) \\left ( 1 - P _ s 1 _ { A } ( y ) \\right ) \\leq C _ 2 \\rho ( x _ 0 ) \\left ( 1 - P _ { t _ 0 } 1 _ { A } ( x _ 0 ) \\right ) = 0 . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\omega ^ { \\sharp } _ { \\bullet } ( t ) : = \\sup _ { s \\in [ t , 1 ] } \\ , \\left ( \\frac { t } { s } \\right ) ^ \\beta \\ , \\tilde \\omega _ { \\bullet } ( s ) ( 0 < t \\le 1 ) \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} & M _ { 4 2 } M _ { 3 1 } - M _ { 2 2 } U M _ { 4 1 } ^ { \\tau } U _ m + U _ n K _ { 1 2 } ^ { \\tau } U _ m = K _ { 1 1 } , \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} - 2 a _ 1 ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 ( D _ \\zeta \\sigma _ 1 ) + 2 ( D _ { \\zeta } \\sigma _ 1 ) \\ , \\mathcal P o l y _ 4 ( 0 , \\zeta ^ 0 , \\bar \\Lambda _ 0 ) & = 2 ( D _ { \\zeta } \\sigma _ 1 ) \\Bigl [ - a _ 1 ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 + \\mathcal P o l y _ 4 ( 0 , \\zeta ^ 0 , \\bar \\Lambda _ 0 ) \\Bigr ] . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac 1 N \\rho _ { ( 1 ) } ^ r \\Big ( 1 - \\frac x N \\Big ) = \\tilde \\rho _ { ( 1 ) } ^ r ( x ) \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} M _ r ( S , T ) = \\binom { | S \\cap T | } { 2 } + \\binom { | \\bar { S } \\cap \\bar { T } | } { 2 } . \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\tilde { z } _ n ^ M - \\varphi _ m = \\mathcal { K } _ m ' ( \\varphi _ m ) ( z _ n ^ M - \\varphi _ m ) + O ( \\max \\{ \\tilde { h } ^ { d } , h ^ { 3 r } \\} ^ 2 ) \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\left ( { \\partial \\over \\partial t } - v { \\partial \\over \\partial x } - { \\partial ^ 2 \\over \\partial v ^ 2 } + p \\theta ( x ) \\right ) Q _ p ( x , v , t ) = 0 \\ , , \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial ^ 2 v _ 0 ( \\mathbf { x } , \\tau ) } { \\partial \\tau ^ 2 } = - \\mathbf { D } ^ * g ^ 0 \\mathbf { D } v _ 0 ( \\mathbf { x } , \\tau ) + F ( \\mathbf { x } , \\tau ) , \\\\ & v _ 0 ( \\mathbf { x } , 0 ) = \\phi ( \\mathbf { x } ) , \\frac { \\partial v _ 0 } { \\partial \\tau } ( \\mathbf { x } , 0 ) = \\psi ( \\mathbf { x } ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} R _ { 1 1 } ( x + \\alpha ) \\hat { R } ( x ) - R _ { 1 1 } ( x ) \\hat { R } ( x + \\alpha ) = 1 + \\mu R _ { 1 1 } ( x + \\alpha ) R _ { 1 1 } ( x ) . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} [ L ^ 2 ; H ^ 1 _ 0 ] _ \\gamma = D ( \\L ^ \\gamma ) , [ H ^ { - 1 } , L ^ 2 ] _ \\gamma = \\big ( [ H ^ 1 , L ^ 2 ] _ \\gamma \\big ) ' = \\big ( [ L ^ 2 , H ^ 1 ] _ { 1 - \\gamma } \\big ) ' = ( D ( \\L ^ { 1 - \\gamma } ) ) ' = D ( \\L ^ { \\gamma - 1 } ) . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\| \\varphi - \\varphi _ n ^ M \\| _ \\infty = O ( h ^ { 3 r } ) , \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} ( \\omega ' \\sqcup \\omega '' ) ( i ) : = \\begin{cases} \\omega ' ( i ) , & \\hbox { i f $ 1 \\leq i \\leq n $ , a n d } \\cr \\omega '' ( i - n ) , & \\hbox { i f $ n < i \\leq n + m $ \\ . } \\end{cases} \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\xi ( t ) = \\left [ \\begin{matrix} x ^ { \\top } ( t ) & \\ x ^ { \\top } ( t - h ) \\ & \\frac { 1 } { h } \\int _ { t - h } ^ t x ^ { \\top } ( s ) d s \\end{matrix} \\right ] ^ { \\top } . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} \\{ \\rho \\in \\Delta ^ \\circ , \\rho ( x ) = - 1 : x \\in F \\} . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\Sigma = \\int _ { \\R ^ { 2 n } } z \\rho ( z ) z ^ { T } \\ , d ^ { 2 n } z \\mbox { f o r } z = \\binom { x } { p } . \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\| s ' \\| _ { ( f ^ { \\mathrm { a n } } ) ^ * ( h ^ n ) , \\hat { \\kappa } ( \\zeta ) } & = \\| f ^ * _ { \\hat { \\kappa } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) } ( s ) \\| _ { ( f ^ { \\mathrm { a n } } ) ^ * ( h ^ n ) , \\hat { \\kappa } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) } \\\\ & \\leqslant \\| s \\| _ { h ^ n , \\hat { \\kappa } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) } \\leqslant e ^ { n \\epsilon } | s | _ { h ^ n } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) \\\\ & = e ^ { n \\epsilon } | s ' | _ { ( f ^ { \\mathrm { a n } } ) ^ * ( h ^ n ) } ( \\zeta ) , \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} 0 = \\frac { \\partial } { \\partial \\lambda } \\L ( \\widetilde { \\phi } _ { \\lambda } ) | _ { \\lambda = 1 } = a _ { 1 } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } \\phi ( x ) | ^ { p } d x + a _ { 2 } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x - \\frac { Q ( q - p ) } { p q } \\int _ { \\mathbb { G } } | \\phi ( x ) | ^ { q } d x . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} M _ { \\varepsilon } & = \\sup \\big \\{ \\Gamma _ { \\varepsilon } ( x , t ) \\ | \\ x \\in \\Pi , \\ t \\in [ 0 , T ] \\ \\big \\} \\\\ & = \\Gamma _ { \\varepsilon } ( x _ 1 , t _ 1 ) > 0 . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\nu ( d x ) = \\ell ( x ) \\ , m ( d x ) , \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} E = \\left \\{ ( q ^ i , p _ j ; \\dot { q } ^ i , \\dot { p } _ i ) \\in T T ^ * Q : \\dot { q } ^ i - \\phi ^ i ( q , p ) = 0 , \\dot { p } _ i + \\phi _ i ( q , p ) = 0 \\right \\} . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , X ) , \\pi _ * Y ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\phi \\mathcal { B } X , Y ) + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\omega \\mathcal { B } X , Y ) - g _ { 1 } ( \\mathcal { T } _ { U } \\mathcal { C } X , \\mathcal { B } Y ) - g _ { 2 } ( \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\mathcal { C } X ) , \\pi _ * \\mathcal { C } Y ) \\\\ & + \\eta ( X ) g _ { 1 } ( Q U , \\varphi Y ) - U \\eta ( X ) \\eta ( Y ) - g _ { 1 } ( X , \\omega U ) \\eta ( Y ) . \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\int _ 0 ^ { 1 - x ^ 2 - y ^ 2 } \\frac { \\delta ( ( 1 - \\mu _ { + } ^ 2 ) ( 1 - \\mu _ { - } ^ 2 ) ) ^ { ( L - 3 ) / 2 } } { \\sqrt { \\delta ^ 2 + 4 y ^ 2 } } \\mathrm { d } \\delta = | 1 - z ^ 2 | ^ { L - 2 } \\int _ { 2 | \\mathrm { I m } \\ , z | / | 1 - z ^ 2 | } ^ 1 ( 1 - t ^ 2 ) ^ { ( L - 3 ) / 2 } \\mathrm { d } t , \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} g _ { k } ( x , y ) = ( 2 \\pi ) ^ { - n / 2 } \\det ( \\Sigma ) ^ { - 1 / 2 } \\exp \\left [ - \\frac { 1 } { 2 } ( y - A x ) ^ { T } \\Sigma ^ { - 1 } ( y - A x ) \\right ] , \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} E ( T ) = \\frac { 2 } { \\pi } \\int ^ { \\infty } _ { 0 } \\frac { 1 } { x ^ { 2 } } \\ln \\left [ \\sum \\limits _ { k \\geq 0 } m ( T , k ) x ^ { 2 k } \\right ] d x . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} { \\mathbb E } _ { \\mu } ( { \\| J \\colon ( O K ) ^ { \\circ } \\rightarrow ( O K ) \\| ) } \\geq R ( K ^ { \\circ } ) \\ , { \\mathbb E } _ { \\sigma _ { 2 n - 2 } } ( { \\| O ^ T J O \\widetilde v \\| _ { K } ) } = R ( K ^ { \\circ } ) M ( K \\cap L ) , \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} [ x , y ] = \\{ n \\in \\N : x \\leq n \\leq y \\} . \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} R ^ \\Sigma _ { i j k l } = R _ { i j k l } + I I _ { i k } I I _ { j l } - I I _ { i l } I I _ { j k } , \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} P _ k ( \\mathcal { O } _ j ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k - j + 1 } } u ^ { k - j } _ { \\sigma ( j ) } u ^ { k - j - 1 } _ { \\sigma ( j + 1 ) } \\cdots u ^ 0 _ { \\sigma ( k ) } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d t ^ { 2 } } + \\omega \\left ( t \\right ) x ^ { i } = 0 ~ , ~ \\omega \\left ( t \\right ) = \\frac { \\gamma ^ { 2 } } { t ^ { 2 } } . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} \\alpha ( O _ 1 K ) - \\alpha ( O _ 2 K ) \\leq \\| J ( O _ 1 ) - J ( O _ 2 ) \\| _ { K ^ { \\circ } \\to K } = \\sup _ { A \\in \\mathcal S } { \\rm T r } \\bigl ( ( J ( O _ 1 ) - J ( O _ 2 ) ) A \\bigr ) . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ r t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } = \\frac { ( q ; q ) _ r } { \\big ( t ^ N ; q \\big ) _ r } \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\oint _ C { \\frac { 1 } { y ^ { r + 1 } } \\prod _ { i = 1 } ^ N { \\frac { \\big ( q ^ { \\lambda _ i } t ^ { N - i + 1 } y ; q \\big ) _ { \\infty } } { \\big ( q ^ { \\lambda _ i } t ^ { N - i } y ; q \\big ) _ { \\infty } } { \\rm d } y } } , \\end{gather*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\varkappa _ s ( G ) _ i : = \\begin{cases} j & \\ker ( T _ { G , H _ i } ) = H _ j / G ^ \\prime j \\in \\lbrace 1 , \\ldots , p + 1 \\rbrace , \\\\ 0 & \\ker ( T _ { G , H _ i } ) = G / G ^ \\prime . \\end{cases} \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\partial ^ 2 _ t q ^ L _ { 0 , m } = ( \\Delta t ) ^ { - 2 } \\int _ { - \\Delta t } ^ { \\Delta t } ( \\Delta t - | \\tau | ) { \\partial ^ 2 q ^ L _ { 0 } \\over \\partial t ^ 2 } ( t _ m + \\tau ) d \\tau , \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c } 1 & 1 - F _ A ' ( \\omega _ B ( z ) ) \\\\ 1 - F _ B ' ( \\omega _ A ( z ) ) & 1 \\end{array} \\right ) \\left ( \\begin{array} { c c } \\omega _ A ' ( z ) \\\\ \\omega _ B ' ( z ) \\end{array} \\right ) = \\left ( \\begin{array} { c c } 1 \\\\ 1 \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} x = \\frac { A } { B E } , y = \\frac { A } { C D } , z = \\frac { B } { C E } , w = \\frac { C } { D E } , \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} [ B ^ { \\epsilon } _ i ( z ) , B ^ { \\epsilon ' } _ i ( w ) ] = [ C ^ { \\epsilon } _ i ( z ) , C ^ { \\epsilon ' } _ i ( w ) ] = [ D ^ { \\epsilon } _ i ( z ) , D ^ { \\epsilon ' } _ i ( w ) ] = 0 , \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} - 2 7 ( H z ( z ^ 2 - 1 ) & - i Q _ 0 ( z ) ) ^ 2 - 4 Q _ 1 ( z ) ^ 3 = \\\\ & - 2 7 ( z ( z ^ 2 - 1 ) ) ^ 2 H ^ 2 - ( 5 4 i z ( z ^ 2 - 1 ) Q _ 0 ( z ) ) H + 2 7 Q _ 0 ( z ) ^ 2 - 4 Q _ 1 ( z ) ^ 3 > 0 \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} B ' _ n ( v ) & = 2 B ' _ { n - 1 } ( v ) + v B _ n ( v ) - ( C _ { n - 3 } + C _ { n - 2 } v ^ 2 ) v ^ { n - 3 } . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} E ^ { 0 } \\cap T C ^ { 0 } = \\left \\{ \\dot { x } = \\frac { \\partial H } { \\partial p } ( x , p ) , \\dot { r } = \\dot { p } = - \\frac { \\partial H } { \\partial x } ( x , p ) , r = p , s = 0 \\right \\} \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\big ( 1 + ( b _ i - z + \\omega _ B ) m _ { \\mu _ A \\boxplus \\mu _ B } + O _ \\prec ( N ^ { - \\frac { \\gamma } { 4 } } ) \\big ) T _ { i } = O _ \\prec ( N ^ { - \\frac { \\gamma } { 4 } } ) + O _ \\prec ( \\Psi ) = O _ \\prec ( N ^ { - \\frac { \\gamma } { 4 } } ) \\ , , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} \\mu ^ { \\frac { 1 } { 2 } } = | \\sigma _ 1 ( \\varepsilon , \\zeta ) | ^ { \\frac { 1 } { 2 } } P ( \\varepsilon , \\zeta ) \\bar { \\Lambda } . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\frac { h _ K } { h _ { K ^ + } } = \\omega _ K \\nu _ \\infty ( f ) \\prod _ \\ell \\nu _ \\ell ( K ) . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} A ^ * : = \\lim _ { T \\to T ^ * } \\| v \\| _ { L ^ { q _ d } _ t ( [ 0 , T ) ; W _ x ^ { 1 , r _ d } ) } < \\infty . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\varepsilon > \\frac { 1 } { d } - \\frac { 1 } { | I | } \\geq \\frac { 1 } { d } - \\frac { 1 } { q d } = \\frac { q - 1 } { q d } \\geq \\frac { 1 } { 2 d } > \\varepsilon \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} p ^ 4 - p ^ 3 + 2 \\ , p ^ 2 + p + 1 = 0 . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( W ' , Z ' \\right ) \\cdot \\overline { \\mathcal { L } \\left ( W ' , Z ' \\right ) } ^ { t } = \\overline { \\mathcal { L } \\left ( W ' , Z ' \\right ) } ^ { t } \\cdot \\mathcal { L } \\left ( W ' , Z ' \\right ) , \\quad \\quad \\quad \\overline { \\mathcal { L } \\left ( W ' , Z ' \\right ) } ^ { t } \\otimes W ' = \\mathcal { L } \\left ( W ' , Z ' \\right ) \\otimes \\overline { W ' } ^ { t } . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\big ( \\nabla _ { o u t } u \\cdot [ \\alpha ] \\big ) ( v ) \\ , & = \\sum _ { e \\in E ( v ) } \\Big ( { \\bf 1 } _ { e _ - = v } \\nabla u _ { e + } ( v ) + { \\bf 1 } _ { e _ + = v } \\nabla u _ { e - } ( v ) \\Big ) \\ , \\alpha _ { e } . \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} f ( z _ 1 , \\dots , z _ n ) = \\biggr ( z _ 2 , \\dots , z _ n , b _ 1 + \\frac { b _ 2 } { z _ { n } } + \\frac { b _ 3 } { z _ { n - 1 } z _ { n } } + \\dots + \\frac { b _ n } { z _ 2 \\dots z _ { n } } \\biggr ) . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial t } + L w - ( \\lambda + \\mu ) w = - \\frac { \\partial u } { \\partial y } \\frac { \\partial v } { \\partial y } , w ( T ) = 0 \\mbox { i n } \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\frac { 1 } { X } \\sum _ { n \\leq X } \\frac { \\lvert S _ f ( n ) \\rvert ^ 2 } { n ^ { k - 1 } } = C X ^ { \\frac { 1 } { 2 } } + O ( \\log ^ 2 X ) . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\begin{cases} b ^ \\varepsilon ( x ) \\dfrac { \\partial ^ 2 u ^ \\varepsilon ( t , x ) } { \\partial t ^ 2 } + \\mathrm { c u r l } ( a ^ \\varepsilon ( x ) \\mathrm { c u r l } u ^ \\varepsilon ( t , x ) ) = f ( t , x ) , & ( 0 , T ) \\times D \\\\ u ^ \\varepsilon ( 0 , x ) = g _ 0 ( x ) & \\\\ u _ t ^ \\varepsilon ( 0 , x ) = g _ 1 ( x ) & \\\\ \\end{cases} \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ \\infty \\sum _ { s = 1 } ^ \\infty x ^ r x ^ s x ^ r x ^ s = \\bigg ( \\frac { x ^ 2 } { 1 - x ^ 2 } \\bigg ) ^ 2 \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\dot { q } ^ i = \\frac { \\partial H } { \\partial p _ i } ( q , p ) , \\dot { p } _ i = - \\frac { \\partial H } { \\partial q ^ i } ( q , p ) , \\Phi ^ { \\alpha } ( q , p ) = 0 , \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\| A - A _ k \\| = \\sigma _ { k + 1 } , \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} F _ q ( T , q ^ { - n - 1 } X ^ { - 1 } ) = \\eta _ q ( T ) ( q ^ { ( n + 1 ) / 2 } X ) ^ { - { \\rm o r d } _ q ( D ( T ) ) } F _ q ( T , X ) , \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} A & = \\frac { 2 } { \\ell } ( D ^ 3 + K _ 1 D ) ^ { - 1 } \\\\ & = m ( D ^ 3 + K _ 1 D ) ^ { - 1 } \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\delta = \\frac { 1 } { 4 } c _ * r ^ 2 _ 0 = \\frac { 1 } { 4 } \\alpha _ 0 \\| f ^ { - 1 } \\| _ { L _ \\infty } ^ { - 2 } \\| g ^ { - 1 } \\| _ { L _ \\infty } ^ { - 1 } r ^ 2 _ 0 . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} \\hat { J } _ \\lambda ^ + ( \\mu ) = \\inf \\{ J _ \\lambda ^ + ( v ) : \\ v \\in \\Theta _ { \\mu } ^ + \\} \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} I & = \\int _ { E _ 1 } \\lvert H ( w ) \\rvert ^ q \\lvert \\mathrm { d } w \\rvert + \\int _ { E _ 2 } \\lvert H ( w ) \\rvert ^ q \\lvert \\mathrm { d } w \\rvert \\\\ & = I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , X ) & = \\cos ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , X ) - g _ { 1 } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { Z } \\omega \\phi W , X ) \\\\ & + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } \\omega W , \\mathcal { B } X ) + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } \\omega W , \\mathcal { C } X ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} h _ { r } = \\frac { u ^ { 2 ^ { r + 1 } - 1 } ( 1 - u ) } { 1 - u ^ { 2 ^ { r + 2 } - 1 } } = \\frac { z ^ { 2 ^ { r + 1 } - 1 } } { F _ { 2 ^ { r + 2 } - 1 } ( - z ) } , \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\pi _ { E } ( v ) = 1 _ { Z ( v ) } , \\quad \\pi _ { E } ( e ) = 1 _ { Z ( e , r ( e ) ) } , \\quad \\pi _ { E } ( e ^ { * } ) = 1 _ { Z ( r ( e ) , e ) } , \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} w ^ t _ 1 ( x ) = \\int \\left ( \\frac 1 t \\Delta \\dot h ^ 0 \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } - \\frac { 1 } { 2 } \\Delta \\delta ^ 2 _ t h ^ 0 \\chi _ { \\Omega ^ t } \\right ) ( d y ) P ( x - y ) w ^ t _ 1 = 0 \\ \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\int _ { \\alpha _ { 1 } } ^ { \\xi } { e ^ { u t } G _ { n } \\left ( t \\right ) d t } = \\frac { 1 } { u } e ^ { u \\xi } G _ { n } \\left ( \\xi \\right ) - \\frac { 1 } { u } \\int _ { \\alpha _ { 1 } } ^ { \\xi } { e ^ { u t } { G } _ { n } ^ { \\prime } \\left ( t \\right ) d t , } \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} z b \\otimes ( u ^ { - 1 } s ) = x \\otimes 1 \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} c _ { \\mathcal { N } } \\leq \\max _ { t \\geq 0 } \\left \\{ \\frac { t ^ { 2 } } { 2 } S _ { q } ( \\psi , \\phi ) ^ { 2 } \\| ( \\psi , \\phi ) \\| _ { L ^ { q } } ^ { 2 } - \\vartheta t ^ { q } \\| ( \\psi , \\phi ) \\| _ { L ^ { q } } ^ { q } \\right \\} = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { q } \\right ) \\frac { S _ { q } ( \\psi , \\phi ) ^ { 2 q / ( q - 2 ) } } { ( q \\vartheta ) ^ { 2 / ( q - 2 ) } } . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} { \\eta } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta } ( \\theta ) + { \\eta } ( \\xi _ { j } ) , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\bar { C } = U \\begin{pmatrix} \\Sigma \\\\ 0 \\\\ \\end{pmatrix} Q ^ \\top \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\mathbf { S } \\left ( \\omega _ { 1 } , \\omega _ { 1 } , b _ { 1 } , b _ { 2 } , \\bar { p } ; \\mathbf { t } \\right ) = \\mathbf { Q } \\left ( \\omega _ { 1 } , \\omega _ { 1 } , b _ { 1 } , b _ { 2 } ; \\mathbf { t } \\right ) + \\bar { p } \\mathbf { 1 } , \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} h = R _ { X } x + R _ { Y } y = R _ { X } x + ( I - P ) R _ { Y } y + P R _ { Y } y \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} X _ { \\leqslant 0 } ^ { \\mathrm { a n } } : = \\{ \\xi \\in X ^ { \\mathrm { a n } } \\mid h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) \\leqslant 0 \\} \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} E _ r ^ { p , q } \\to E _ r ^ { p + r , q - r + 1 } \\mbox { i s z e r o } \\Rightarrow E _ { r + 1 } ^ { p , q } = E _ r ^ { p , q } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} V \\mu ( x ) = V \\mu ( x _ 0 ) + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = x _ 0 + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = \\abs { x } r + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\begin{aligned} \\bigl | [ V ( t ) w ( t ) ] _ { x = 0 } \\bigr | & \\lesssim t ^ { - \\frac 1 4 } \\| w ( t ) \\| _ { H ^ 1 } , \\\\ \\| V ( t ) w ( t ) \\| _ { L ^ \\infty } & \\lesssim \\| w ( t ) \\| _ { L ^ \\infty } + t ^ { - \\frac 1 4 } \\| w ( t ) \\| _ { H ^ 1 } , \\\\ \\| V ( t ) w ( t ) \\| _ { \\dot H ^ 1 } & \\lesssim \\| w ( t ) \\| _ { H ^ 1 } . \\end{aligned} \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\omega _ { t } + U \\left ( y \\right ) \\partial _ { x } \\omega + U ^ { \\prime \\prime } \\left ( y \\right ) \\partial _ { x } \\psi = 0 \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\Gamma \\left ( \\frac { p _ { 0 } + 1 } { 2 } \\right ) = \\frac { \\sqrt { \\pi } } { 2 } . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\sum _ { k = 1 } ^ K Y _ k \\big ( \\int _ 0 ^ t \\lambda _ k ( s , X ( s ) ) d s \\big ) \\zeta _ k , \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} x ' ( t ) = f ( t ) x ^ 3 + g ( t ) x ^ 2 , ~ t \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} & \\varphi ( a _ 1 \\pi _ { \\chi } ( B _ 1 ) + a _ 2 \\pi _ { \\chi } ( B _ 2 ) ) = \\dfrac { 1 } { 3 } ( a _ 2 , a _ 1 ) , \\\\ & \\varphi ' ( a _ 1 \\pi _ { \\overline { \\chi } } ( B _ 1 ) + a _ 2 \\pi _ { \\overline { \\chi } } ( B _ 2 ) ) = \\dfrac { 1 } { 3 } ( a _ 2 , a _ 1 ) . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} 2 4 = \\sum _ { F \\in \\Delta [ 1 ] } ( \\ell ( F ) - 1 ) ( \\ell ( F ^ \\circ ) - 1 ) . \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\searrow 0 } \\ ; \\frac { \\mu ^ * _ \\beta ( f \\ , \\widetilde \\Psi _ \\epsilon ) } { \\mu ^ * _ \\beta ( \\widetilde \\Psi _ \\epsilon ) } \\ ; = \\ ; \\mu ^ * _ \\beta ( f ) \\ , . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\min \\{ F ( \\nabla u ) - \\varepsilon , - \\mathcal { Q } _ \\infty u \\} = 0 , \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} I _ p ( n ) \\leq \\sum _ { j = 0 } ^ { k _ p } P ( j ) \\leq ( k _ p + 1 ) P ( k _ p ) . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} \\exp \\left ( - \\beta _ t ^ 2 \\left ( \\frac { t } { B } - 2 \\right ) \\right ) \\leq & \\exp \\left ( - \\beta _ t ^ 2 \\frac { t } { B } + \\frac { 1 } { 2 } \\right ) \\cr = & \\exp \\left ( - \\frac { p ^ 2 t } { 4 B } + 2 p ^ 2 + \\frac { 1 } { 2 } - \\frac { 4 p ^ 2 B } { t } \\right ) \\cr \\leq & \\exp \\left ( - \\frac { p ^ 2 t } { 4 B } + \\frac { 5 } { 2 } \\right ) \\cr \\leq & 1 3 \\exp \\left ( - \\frac { p ^ 2 t } { 4 B } \\right ) \\cr = & 1 3 \\left ( \\exp \\left ( - \\frac { p n } { 2 } \\right ) \\right ) ^ { \\frac { p t } { 2 n B } } . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\mathcal { J } _ 1 \\leqslant \\frac { 1 } { \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) } \\sum \\limits _ { n = 1 } ^ { K } \\sum \\limits _ { k = 1 \\atop k \\notin \\mathcal { K } } ^ { n } \\overline { F } _ { \\xi _ { k } } ( x y ) \\mathbb { P } ( \\eta = n ) + \\sum \\limits _ { n = 1 } ^ { K } \\sum \\limits _ { k = 1 \\atop k \\in \\mathcal { K } } ^ { n } \\frac { \\overline { F } _ { \\xi _ { k } } ( x / 2 ) } { \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) } \\mathbb { P } ( \\eta = n ) . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} T _ { H h H } ( f \\circ \\pi _ { x H } ) ( y H ) & = \\int _ X f ( H x ^ { - 1 } z H ) \\ > K _ { H h H } ( y , d ( z H ) ) ) \\\\ & = \\int _ H f ( H x ^ { - 1 } y w h H ) \\ > d \\omega _ H ( w ) . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} S _ 2 ( n , k | x ) = \\sum _ { l = k } ^ n { n \\choose l } S _ 2 ( l , k ) x ^ { n - l } , \\ , \\ , ( n , k \\geq 0 ) . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} | S _ { n , k - 1 , r } | + | S _ { n , k , r + 1 } | = | S _ { n , k , r } | , \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\Phi _ 1 ^ c & = - ( F ' _ A ( \\omega _ B ) - 1 ) \\Phi _ 2 ^ c + \\mathcal { S } \\Lambda _ A + \\mathcal { T } _ A \\Lambda _ A ^ 2 + O ( ( \\Phi _ 2 ^ c ) ^ 2 ) + O ( \\Phi _ 2 ^ c \\Lambda _ A ) + O ( \\Lambda _ A ^ 3 ) \\ , , \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\begin{cases} P _ { i m a x } \\bar { z } _ { i i } / ( P _ { j m a x } \\bar { z } _ { j i } ) \\geq \\gamma \\\\ P _ { j m a x } \\bar { z } _ { j j } / ( P _ { i m a x } \\bar { z } _ { i j } ) \\geq \\gamma \\end{cases} \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\vert \\mathcal C _ 1 \\cap \\mathcal C _ 2 \\vert \\geq \\vert C _ 1 \\vert - 2 \\alpha \\vert C _ 1 \\vert = ( 1 - 2 \\alpha ) \\vert \\mathcal C _ 1 \\vert . \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\theta _ 1 = & \\ x _ 1 d x ^ 2 - x _ 2 d x ^ 1 + x _ 3 d x ^ 4 - x _ 4 d x ^ 3 \\\\ \\theta _ 2 = & \\ x _ 1 d x ^ 3 - x _ 3 d x ^ 1 + x _ 4 d x ^ 2 - x _ 2 d x ^ 4 \\\\ \\theta _ 3 = & \\ x _ 1 d x ^ 4 - x _ 4 d x ^ 1 + x _ 2 d x ^ 3 - x _ 3 d x ^ 2 . \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} - 2 T _ 0 ^ 3 T _ 2 + 3 7 T _ 0 ^ 2 T _ 1 T _ 2 + 6 7 T _ 0 ^ 2 T _ 2 ^ 2 + 2 T _ 0 T _ 1 ^ 3 - 1 0 T _ 0 T _ 1 ^ 2 T _ 2 + 1 1 4 T _ 0 T _ 1 T _ 2 ^ 2 + 1 6 6 T _ 0 T _ 2 ^ 3 + 4 T _ 1 ^ 4 \\\\ { } - 1 6 8 T _ 1 ^ 3 T _ 2 + 4 2 T _ 1 ^ 2 T _ 2 ^ 2 + 3 6 9 T _ 1 T _ 2 ^ 3 - 4 5 T _ 2 ^ 4 = 0 \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} k e r F _ { \\ast } = s p a n \\{ & Z _ { 1 } = \\frac { \\partial } { \\partial x ^ { 1 } } - \\frac { \\partial } { \\partial x ^ { 2 } } , \\ Z _ { 2 } = \\frac { \\partial } { \\partial y ^ { 1 } } - \\frac { \\partial } { \\partial y ^ { 2 } } , \\\\ & Z _ { 3 } = - \\cos \\alpha \\frac { \\partial } { \\partial x ^ { 3 } } - \\sin \\alpha \\frac { \\partial } { \\partial x ^ { 4 } } , Z _ { 4 } = \\frac { \\partial } { \\partial y ^ { 3 } } \\} \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\theta ^ * = ( a ^ * , b ^ * , g ^ * , k ^ * ) ^ \\intercal = ( 1 . 1 7 , 1 . 5 0 , 0 . 4 1 , 0 . 2 3 ) ^ \\intercal . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n \\xi _ i I _ { A _ i } ] = \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { t _ i } [ \\xi _ i ] I _ { A _ i } ] . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\frac { f _ { i } ( t s ) s } { t } \\right ) = \\frac { f _ { i } ' ( t s ) t s ^ { 2 } - f _ { i } ( t s ) s } { t ^ { 2 } } = \\frac { f _ { i } ' ( t s ) t ^ { 2 } s ^ { 2 } - f _ { i } ( t s ) t s } { t ^ { 3 } } > 0 , \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} ( N _ r V ) _ s & = 2 ^ { r - 4 } ( r ^ 2 - r + 2 ) \\left [ \\binom { s } { 2 } + \\binom { r - s } { 2 } \\right ] \\\\ & = 2 ^ { r - 4 } ( r ^ 2 - r + 2 ) V _ s . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} v = \\frac { d w _ \\tau } { d \\tau } \\Big | _ { \\tau = 0 } \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} f _ 1 ( x ) \\ = \\ f _ 2 ( x ) \\ = \\ \\dotsb \\ = \\ f _ n ( x ) \\ = \\ 0 \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\delta _ t c ^ 0 = \\int _ { \\R ^ 2 } \\Delta v ^ t = \\int _ { \\R ^ 2 } - \\frac { \\Delta h ^ 0 } { t } \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } + \\Delta \\delta _ t h ^ 0 \\chi _ { \\Omega ^ t } . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} \\norm { S } = \\norm { S ^ * } = \\sup \\left \\{ \\norm { S ^ * y ^ * } : y ^ * \\in \\Gamma \\right \\} . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} A _ { 0 } = - \\Delta - K _ { 2 } \\left ( y \\right ) : H ^ { 2 } \\rightarrow L ^ { 2 } \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} ( m ^ { - 1 } ( t ^ { - 1 } z ) + \\xi ) ( m ^ { - 1 } ( t ^ { - 1 } z ) - \\xi ) = t ^ { - 1 } ( z - t m ( \\xi ) ) q ( t ^ { - 1 } z , \\xi ) , \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} \\mathbf { x } _ k = ( x _ { k 1 } , \\cdots , x _ { k p } ) ' , ~ ~ ~ ~ k = 1 , \\cdots , n . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\widetilde { f } ( X ) = a _ 2 ^ 2 X ^ 2 - \\Delta _ f , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\mathcal { R } _ 0 & = \\mathcal { R } \\setminus ( \\mathcal { R } _ i \\sqcup \\mathcal { R } _ j ) \\\\ \\mathcal { R } _ 1 & = \\mathcal { R } \\cap \\mathcal { G } _ { l _ 1 } \\\\ \\mathcal { R } _ 2 & = \\mathcal { R } \\cap \\mathcal { G } _ { l _ 2 } \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} m = \\underset { a \\in \\Sigma { } _ { 0 } { } ^ { + } } { \\sum } \\left ( \\frac { m { } _ { \\alpha } + m { } _ { 2 \\alpha } } { 2 } - 1 \\right ) A = \\underset { a \\in \\Sigma _ { 0 } ^ { + } } { \\sum } \\frac { m _ { \\alpha } + m _ { 2 \\alpha } } { 2 } . \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} ( 2 q ) ! ^ { \\alpha ^ * } - \\left ( \\frac { q - 1 } { q + 1 } \\cdot ( 2 q ) ! \\right ) ^ { \\alpha ^ * } = \\left ( 1 - \\left ( \\frac { q - 1 } { q + 1 } \\right ) ^ { \\alpha ^ * } \\right ) ( 2 q ) ! ^ { \\alpha ^ * } \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\vec { a } _ j = \\ln \\left ( \\frac { \\vec { f } ^ { ( j ) } _ - ( x ^ { ( j ) } , v ^ { ( j ) } ) } { \\phi ^ { ( j ) } _ { R , 1 } ( x _ * ^ { ( j ) } , v ^ { ( j ) } _ * ) } \\right ) \\ , , \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } \\Delta U _ i + \\lambda U _ i & = & - w _ i ^ 5 & \\Omega , \\\\ U _ i & = & 0 & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} \\| e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) \\| _ { L _ 2 } ^ 2 = \\langle e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) , e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) \\rangle = \\langle e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) , \\xi \\rangle = 0 , \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} D ( s , S _ f ) = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) } { n ^ { s + \\frac { k - 1 } { 2 } } } = \\sum _ { \\substack { n \\geq 1 \\\\ m \\geq 0 } } \\frac { a ( n - m ) } { n ^ { s + \\frac { k - 1 } { 2 } } } . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} ( W ^ { \\pm } _ { \\eta } ( T ) , U ( T ) ) = ( W ^ { \\pm } _ { \\eta } ( 0 ) , U ( 0 ) ) . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} R _ { M } = \\{ L _ { M ^ c } > S ( s ) \\} , G _ { M } = \\{ L _ { M } \\leq S ( s ) \\} . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} \\frac 1 2 \\sum _ { j = 1 } ^ n \\frac 1 { d _ j ^ - } + \\left ( \\frac 1 2 + \\frac 1 { q - 1 } \\right ) \\sum _ { j = 1 } ^ n \\frac 1 { d _ j ^ + } & \\ge \\frac { e _ 1 + \\dots + e _ n } { 2 ( q - 1 ) } + \\left ( \\frac 1 2 + \\frac 1 { q - 1 } \\right ) \\cdot \\frac n 2 \\\\ & \\ge \\frac n { 2 ( q - 1 ) } + \\frac n 4 + \\frac n { 2 ( q - 1 ) } > 1 + \\frac n { q - 1 } \\\\ & \\ge 1 + \\frac { n - \\delta _ 1 - \\dots - \\delta _ n } { q - 1 } . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} J f ( t ) = t ^ { - 1 } f ( t ^ { - 1 } ) . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\hat { S } = \\left \\{ \\left ( \\bar { q } , q ; \\frac { \\partial W } { \\partial \\bar { q } } , \\frac { \\partial W } { \\partial q } \\right ) \\in T ^ * ( \\bar { Q } \\times Q ) \\right \\} . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\left \\| \\varphi ^ s ( a ) \\right \\| _ { { c ( s ) } } ^ 2 & = \\left \\| \\varphi ^ s ( a ) ^ \\ast \\varphi ^ s ( a ) \\right \\| _ { { c ( s ) } } \\\\ & \\leq \\| \\varphi ^ s ( \\Re a ) ^ 2 + \\varphi ^ s ( \\Im a ) ^ 2 \\| _ { { c ( s ) } } \\\\ & \\leq 2 D ( s ) ^ 2 \\| a \\| _ { { c ( s ) } } ^ 2 \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} f ^ * ( x ) = \\inf \\{ \\lambda \\colon \\mu _ f ( \\lambda ) \\le x \\} , x \\ge 0 . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\xi = \\tau \\sigma . { r ^ 2 } \\tau { r ^ { - 2 } } . r \\sigma ^ { - 1 } \\tau ^ { - 1 } r ^ { - 1 } = \\beta ^ { - 1 } . r ^ { - 1 } \\tau { r } . \\sigma . { r ^ 2 } \\tau { r ^ { - 2 } } . r \\sigma ^ { - 1 } r ^ { - 1 } , ~ [ [ \\ , , \\ , ] , [ \\ , , \\ , ] ] = 1 \\rangle , \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} L ^ \\alpha e ^ g - \\mu e ^ g = \\lambda _ 1 e ^ g . \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\eta ^ { 1 2 } F & = u \\left ( E _ 6 F + 6 E _ 4 D F \\right ) , & \\eta ^ { 1 2 } D F & = - \\frac { 1 } { 6 } u \\left ( E _ 4 ^ 2 F + 6 E _ 6 D F \\right ) , \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} & \\langle P _ h ^ k ( \\cdot , s ) , E _ \\infty ^ k ( z , \\tfrac { k + 1 } { 2 } ) + E _ 0 ^ k ( z , \\tfrac { k + 1 } { 2 } ) \\rangle \\\\ & = \\frac { \\overline { \\rho _ \\infty ^ k ( h , \\frac { k + 1 } { 2 } ) + \\rho _ 0 ^ k ( h , \\frac { k + 1 } { 2 } ) } } { ( 4 \\pi h ) ^ { s - 1 } } \\frac { \\Gamma ( s + \\frac { k } { 2 } - \\frac { 1 } { 2 } ) \\Gamma ( s - \\frac { k } { 2 } - \\frac { 1 } { 2 } ) } { \\Gamma ( s - \\frac { k } { 2 } ) } . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { d e t } \\left [ [ b _ { j , k } ( \\zeta ) ] _ { \\substack { j = 1 , \\ldots , ( N + 1 ) / 2 \\\\ k = 1 , \\ldots , ( N - 1 ) / 2 } } [ \\mu _ { 2 j - 1 } ] _ { j = 1 , \\ldots , ( N + 1 ) / 2 } \\right ] . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ X I _ { \\{ \\tau \\leq \\sigma \\} } ] & = \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { \\tau + } [ \\xi _ i ] I _ { A _ i } I _ { \\{ \\tau \\leq \\sigma \\} } \\\\ & = \\hat { \\mathbb { E } } _ { \\tau + } [ X ] I _ { \\{ \\tau \\leq \\sigma \\} } . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} 0 ~ = ~ \\int _ M \\frac { F ^ H _ x ( y ) } { F ^ H _ x ( x ) } \\ , d \\mu ( y ) ~ \\ge ~ \\int _ 0 ^ D | \\partial B ( x , r ) | \\ , \\cos ( \\sqrt \\lambda _ 1 r ) \\ , d r \\ , . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} M = \\sup \\big \\{ \\Gamma ( x , t ) \\ | \\ x \\in \\Pi , \\ t \\in [ 0 , T ] \\ \\big \\} > 0 . \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} < ^ \\mathbf { d } \\circ \\leq ^ \\mathbf { d } \\ & \\subseteq \\ < ^ \\mathbf { d } . \\\\ \\leq ^ \\mathbf { d } \\circ < ^ \\mathbf { d } \\ & \\subseteq \\ < ^ \\mathbf { d } . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\deg { ( \\widehat { R } _ { \\epsilon } ) } = \\deg { ( R ) } . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} & X _ { t } = x _ { 0 } + \\int _ { 0 } ^ { t } \\alpha + \\beta X _ { s } + \\sigma ^ { 2 } \\nabla _ { x } \\log h ( X _ { s } , s ) \\mathrm { d } s + \\sigma B _ { t } , \\\\ & X _ { t } ^ { \\prime } = x _ { 0 } ^ { \\prime } + \\int _ { 0 } ^ { t } \\alpha + \\beta X _ { s } ^ { \\prime } + \\sigma ^ { 2 } \\nabla _ { x } \\log h ( X _ { s } ^ { \\prime } , s ) \\mathrm { d } s + \\sigma B _ { t } . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} X ^ * = L ^ { p ' } ( \\Omega ; \\gamma ( L ^ 2 ( \\mathbb R _ + , [ M ^ { c } ] ; H ) , L ^ { q ' } ( S ) ) ) \\times \\mathcal I _ { p ' , q ' } \\times \\mathcal A _ { p ' , q ' } ^ { \\mathcal T } . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} P _ 1 \\ , _ { \\frak C _ 1 } \\cup _ { \\frak C _ 2 } P _ 2 = ( P _ 1 \\cup P _ 2 ) / \\sim . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} n _ i = n _ { i - 1 } + q = n _ 1 + ( i - 1 ) q \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} f ( x ) = \\langle E x + c , ( f ' ) ^ { - 1 } ( E x + c ) \\rangle - f ( ( f ' ) ^ { - 1 } ( E x + c ) ) + \\langle w , x \\rangle + \\beta , x \\in X . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{gather*} A ( t ) = [ J _ 1 ( t ) , \\dots , J _ { n - 1 } ( t ) ] , \\end{gather*}"}
-{"id": "1092.png", "formula": "\\begin{align*} C ^ { i j k l } \\left ( x \\right ) = C ^ { i j l k } \\left ( x \\right ) = C ^ { j i k l } \\left ( x \\right ) \\ : a . e . \\ , , \\ ; i , j , k , l = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} { } ^ t \\Gamma _ { \\iota _ Y } = - \\Delta _ Y + \\gamma \\ \\ \\ \\hbox { i n } \\ H ^ 8 ( Y \\times Y ) \\ , \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} q = - \\sigma g ^ { i j } \\hat n _ { \\mu } \\partial _ { i j } ^ 2 \\eta ^ { \\mu } , \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} P ( i ) = \\frac { 2 ( q - 1 ) } { q } \\norm { T _ i f \\cdot g _ \\frac { 1 } { 2 } } _ { \\ell ^ 1 ( \\N ^ * ) } . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} 0 & \\leq E \\left [ \\psi ( X ' ( T ) , m ( T ) ) - \\psi ( X ( T ) , m ( T ) ) + \\psi ( X ( T ) , m ' ( T ) ) - \\psi ( X ' ( T ) , m ' ( T ) ) \\right ] \\\\ & + E \\left [ \\int _ 0 ^ T [ c _ 1 ( X ' ( t ) , m ( t ) ) - c _ 1 ( X ( t ) , m ( t ) ) \\right . \\\\ & \\left . + c _ 1 ( X ( t ) , m ' ( t ) ) - c _ 1 ( X ' ( t ) , m ' ( t ) ) ] d t \\right ] \\\\ & = \\sum _ { x \\in \\Sigma } ( \\psi ( x , m ( T ) ) - \\psi ( x , m ' ( T ) ) ) ( m ' _ x ( T ) - m _ x ( T ) ) \\\\ & + \\int _ 0 ^ T \\left [ \\sum _ { x \\in \\Sigma } ( c _ 1 ( x , m ( t ) ) - c _ 1 ( x , m ' ( t ) ) ) ( m ' _ x ( t ) - m _ x ( t ) ) \\right ] d t . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\tau _ k = \\begin{cases} \\inf \\{ t \\leq T : \\int _ 0 ^ t g ^ 2 ( s , W _ 0 ^ s , Y _ 0 ^ s ) d s \\geq k \\} , \\mbox { i f } \\int _ 0 ^ T g ^ 2 ( s , W _ 0 ^ s , Y _ 0 ^ s ) d s \\geq k \\\\ T , \\mbox { i f } \\int _ 0 ^ T g ^ 2 ( s , W _ 0 ^ s , Y _ 0 ^ s ) d s < k . \\end{cases} \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\| x _ { i } ( \\widetilde { w } _ { 2 n } - \\widetilde { w } _ { 2 n - 2 } ) \\| & = \\| x _ i w _ { 2 n } v _ { 2 n } \\widetilde { w } _ { 2 n - 2 } v _ { 2 n } ^ * - x _ i \\widetilde { w } _ { 2 n - 2 } v _ { 2 n } v _ { 2 n } ^ * \\| \\\\ & \\leq \\| x _ i w _ { 2 n } [ v _ { 2 n } , \\widetilde { w } _ { 2 n - 2 } ] \\| + \\| x _ { i } ( w _ { 2 n } - 1 ) \\widetilde { w } _ { 2 n - 2 } v _ { 2 n } \\| \\\\ & < \\frac { 1 } { 2 ^ { 2 n - 1 } } \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} & H _ d ( T , \\mathbb { Z } ) = 0 , \\\\ & | H _ { d - 1 } ( T , \\mathbb { Z } ) | < \\infty , \\quad \\\\ & f _ d ( T ) = f _ d ( \\Sigma ) - \\beta _ d ( \\Sigma ) . \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} h ( q ) = \\infty \\omega q \\in R _ 0 ^ b , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} B _ 1 ^ + = B _ 1 \\cap \\R ^ 2 _ + \\ , . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\left ( C ^ T f , g \\right ) _ { \\mathcal { F } ^ T } = \\left ( u ^ f _ { \\cdot , T } , u ^ g _ { \\cdot , T } \\right ) _ { \\mathcal { H } ^ T } = \\left ( W ^ T f , W ^ T g \\right ) _ { \\mathcal { H } ^ T } . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} R _ { i j } : = E _ { i j } - E _ { j i } , P _ k : = E _ { 0 k } + E _ { k 0 } \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\int _ { B _ { r } ( x ) } \\sum _ { i = 1 } ^ { m } \\sum _ { \\ell = 1 } ^ { Q } \\left ( \\langle D _ { i } u _ { \\ell } , D _ { i } ( Y ( y , u _ { \\ell } ) ) \\rangle + \\langle A _ { u _ { \\ell } } ( D _ { i } u _ { \\ell } , D _ { i } u _ { \\ell } ) , Y ( y , u _ { \\ell } ) \\rangle \\right ) \\ , \\mathrm { d } y = 0 . \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} \\L ^ \\circ = J \\mathcal { M } ^ { - T } \\Z ^ { 2 n } . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} 0 & \\le \\sum _ { i , j = 1 } ^ n \\overline { c _ i } \\ > c _ j \\ > \\Phi _ { \\tilde \\alpha } ( i , j ) \\beta ( \\pi ( x _ i , x _ j ) ) \\\\ & = \\sum _ { i , j = 1 } ^ n \\overline { c _ i } \\ > c _ j \\ > \\int _ D \\int _ { A _ i } \\tilde K _ h ( x , A _ j ) \\ > d \\tilde \\omega _ X ( x ) \\cdot \\tilde \\alpha ( h ) \\ > \\beta ( \\pi ( x _ i , x _ j ) ) \\ > d \\tilde \\omega _ D ( h ) . \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} I _ { u , n } ^ { ( j ) } = [ u + ( j - 1 ) d , u + j d - 1 ] \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} s ( x , y ) = & \\left ( \\left ( \\sqrt { x \\left ( 1 - \\frac { x } { 4 } \\right ) } , \\sqrt { y \\left ( 1 - \\frac { x } { 2 } - \\frac { y } { 4 } \\right ) } , 1 - \\frac { x + y } { 2 } \\right ) , \\right . \\\\ & \\phantom { c o u c o u } \\left . \\left ( - \\sqrt { x \\left ( 1 - \\frac { x } { 4 } \\right ) } , \\sqrt { y \\left ( 1 - \\frac { x } { 2 } - \\frac { y } { 4 } \\right ) } , 1 - \\frac { x + y } { 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} Q u ( x , \\lambda ) = \\left ( \\frac { \\lambda } { 2 i \\pi } \\right ) ^ { n } \\int e ^ { 2 \\lambda ( \\psi ( x , \\theta ) - \\psi ( y , \\theta ) ) } \\tilde { q } ( x , \\theta , \\lambda ) u ( y ) d y d \\theta . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} f ( x ) = p ( x ) + g _ { f , p } ( x ) , x \\in X , \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\min \\left \\| \\begin{bmatrix} E & r \\\\ \\end{bmatrix} \\right \\| _ F , \\textrm { s u b j e c t t o } \\lambda ( b + \\Delta b ) - r \\in \\mathcal { R } \\left ( ( A + \\Delta A ) + E \\right ) . \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} z \\ ; \\frac { d } { d z } { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( z ) = \\frac { z } { a } { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a , 1 } ( z ) - \\left ( \\nu ( 2 - a ) + a - 1 \\right ) { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( z ) . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac 1 n \\log \\rho ( A ^ n ( x ) ) = \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) \\rVert = \\lambda _ 1 ( \\mu ) \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\Phi ^ { r } ( z ^ { n } t ^ { k } ) | _ { t = z } & = \\frac { z ^ { n + ( 2 ^ { r + 1 } - 1 ) k } F _ { 2 ^ { r + 1 } - 1 } ( - z ) ^ { 2 n - 1 } } { F _ { 2 ^ { r + 1 } } ( - z ) ^ { 2 n + 2 k - 1 } } \\\\ & = \\frac { 1 - u ^ { 2 ^ { r + 1 } } } { ( 1 - u ^ { 2 ^ { r + 1 } - 1 } ) ( 1 + u ) } \\Big ( \\frac { u ( 1 - u ^ { 2 ^ { r + 1 } - 1 } ) ^ { 2 } } { ( 1 - u ^ { 2 ^ { r + 1 } } ) ^ { 2 } } \\Big ) ^ { n } \\Big ( \\frac { u ^ { 2 ^ { r + 1 } - 1 } ( 1 - u ) ^ { 2 } } { ( 1 - u ^ { 2 ^ { r + 1 } } ) ^ { 2 } } \\Big ) ^ { k } , \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\begin{pmatrix} \\frac { 1 } { A ^ * } & * & * \\\\ 0 & 1 & * \\\\ 0 & 0 & A \\end{pmatrix} \\quad \\textrm { a n d } \\begin{pmatrix} \\frac { 1 } { B ^ * } & * & * \\\\ 0 & 1 & * \\\\ 0 & 0 & B \\end{pmatrix} , \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} P _ h ( z , s ) & = \\sum _ j \\langle P _ h ( \\cdot , s ) , \\mu _ j \\rangle \\mu _ j ( z ) \\\\ & + \\sum _ \\mathfrak { a } \\frac { 1 } { 4 \\pi } \\int _ \\mathbb { R } \\langle P _ h ( \\cdot , s ) , E _ \\mathfrak { a } ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle E _ \\mathfrak { a } ( z , \\tfrac { 1 } { 2 } + i t ) \\ d t . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { k ! \\Gamma ( \\alpha k + \\beta ) } , \\ ; \\alpha > - 1 , \\beta \\in \\mathbb { C } . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} L _ n : = \\log | \\tilde { { \\cal M } } _ 0 | \\le n \\log | { \\cal X } | . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{gather*} \\mathbb { P } - \\lim _ { N \\rightarrow \\infty } { \\Lambda ^ N _ m \\delta _ { \\lambda ( N ) } } = M ^ { \\nu } _ m \\forall \\ , m \\in \\N , \\end{gather*}"}
-{"id": "7925.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\R ^ n } P ( x - y ) d \\mu ( y ) , \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p } _ { a _ { 1 } , . . . , a _ { \\ell } , \\R _ { 1 } , . . . , \\R _ { \\ell } } ( \\mathbb { G } ) } : = \\left ( \\int _ { \\mathbb { G } } ( \\sum _ { j = 1 } ^ { \\ell } | \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } u ( x ) | ^ { p } ) d x \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{gather*} \\theta ' = \\operatorname { t r } ( B ' ) = - \\operatorname { t r } ( \\overset { \\mathrm { s f } } R ) - \\operatorname { t r } ( B ^ 2 ) = - \\operatorname { t r } ( \\overset { \\mathrm { s f } } R ) - \\operatorname { t r } \\left [ \\left ( \\omega + \\sigma + \\frac { \\theta } { n - 1 } \\mathrm { I d } \\right ) ^ 2 \\right ] . \\end{gather*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\gamma ( l , m ) \\cdot \\gamma ( l m , n ) \\ , = \\ , \\nu _ l ( \\gamma ( m , n ) ) \\cdot \\gamma ( l , m n ) \\ \\ \\ , \\ \\ \\ \\gamma ( l , 1 ) \\ , = \\ , \\gamma ( 1 , l ) \\ , = \\ , 1 \\ , \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} \\widehat { \\mu } _ \\pi ^ R ( t ) = \\prod _ { \\gamma \\in \\Gamma ^ S ( \\chi _ 0 ) } J _ 0 \\Bigg ( \\frac { 2 t } { \\sqrt { \\frac { 1 } { 4 } + \\gamma ^ 2 } } \\Bigg ) . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\bigl [ \\varphi _ n ( \\alpha , \\beta ) \\bigr ] ^ n = \\exp { \\biggl ( - \\frac { C _ 1 ^ 2 } { 2 } \\alpha ^ 2 - \\frac { C _ 2 ^ 2 } { 2 } \\beta ^ 2 \\biggr ) } \\exp { \\bigl ( - b \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } \\bigr ) } , \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\mathcal { R ^ { \\geq } } & : = \\{ ( i , j ) \\geq ( i - 1 , j ' ) \\ | \\ 1 \\leq j \\leq i \\leq n , \\ 1 \\leq j ' \\leq i - 1 \\} , \\\\ \\mathcal { R ^ { > } } & : = \\{ ( i - 1 , j ' ) > ( i , j ) \\ | \\ 1 \\leq j \\leq i \\leq n , \\ 1 \\leq j ' \\leq i - 1 \\} , \\\\ \\mathcal { R } ^ { 0 } & : = \\{ ( n , i ) \\geq ( n , j ) \\ | \\ 1 \\leq i \\neq j \\leq n \\} . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\lambda _ i | _ { T } = \\lambda _ { 2 n - i } | _ { T } = \\varpi _ i , \\mbox { } \\lambda _ { n - 1 } | _ { T } = \\lambda _ { n + 1 } | _ { T } = \\varpi _ { n - 1 } + \\varpi _ n , \\mbox { } \\lambda _ n | _ { T } = 2 \\varpi _ n \\mbox { f o r $ 1 \\leqslant i \\leqslant n - 2 . $ } \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\lambda ( \\mu ) = \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) \\rVert \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} L _ s \\Psi = \\lambda \\Psi , A _ { 2 s + 1 } \\Psi = \\mu \\Psi \\ , \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} y _ m ( x ) = y _ 0 ( x ) + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } ( x - t ) ^ { \\alpha - 1 } f ( t , y _ { m - 1 } ( t ) ) d t , m \\in \\N . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} D _ s ^ { \\ell } \\alpha _ s ( \\vec { a } ) & = ( - i \\partial _ s ) ^ { \\ell } \\sum _ { j , m } e ^ { i s \\cdot m } a _ { j , m } \\prod _ { g = 1 } ^ n U _ g ^ { m _ g } f _ j \\\\ & = \\sum _ { j , m } m ^ { \\ell } e ^ { i s \\cdot m } a _ { j , m } \\prod _ { g = 1 } ^ n U _ g ^ { m _ g } f _ j \\\\ & = \\sum _ { j , m } \\delta ^ { \\ell } e ^ { i s \\cdot m } a _ { j , m } \\prod _ { g = 1 } ^ n U _ g ^ { m _ g } f _ j \\\\ & = \\delta ^ { \\ell } \\alpha _ s ( \\vec { a } ) \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} x _ k = \\bar x + \\alpha _ k , y _ k = \\bar y + \\beta _ k , \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} G ( t ) = \\frac { 1 } { n } ( t ^ n - 1 ) = u \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} x _ { i } = x _ { n + i } + \\sum _ { j = 1 } ^ { i - 1 } c _ { j , i } \\cdot x _ { n + j } . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} r a ^ { a / 2 } \\mathtt { J } ' _ \\nu ( r ) - \\left ( ( \\nu - 1 ) ( 1 - a ) a ^ { a / 2 } + \\beta ( a \\nu - a + 1 ) \\right ) \\mathtt { J } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\left ( \\sum _ { r = 0 } ^ { \\infty } h _ r ^ { \\perp } \\right ) s _ { \\lambda } \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} Q _ { n p } = \\frac { ( n - 3 ) \\sum _ { 1 \\le j < i \\le p } z _ { i j } ^ 2 - \\frac 1 2 p ( p - 1 ) } { \\sqrt { p ( p - 1 ) } } , \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i Z _ i \\Big | \\prec \\Psi ^ 2 \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} \\rho e ^ { i \\varphi } \\mapsto \\begin{cases} ( 1 - \\rho + \\rho e ^ { 2 i \\varphi } , 1 ) \\\\ ( 1 , 1 - \\rho + \\rho e ^ { 2 i \\varphi } ) . \\end{cases} \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X ( s , M , Y _ 0 ^ { s } ) d s + B ( t ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\tilde \\delta \\circ d ^ { i } _ { \\mathcal K ^ { \\bullet } } = \\delta . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\mathcal { D } _ j = \\mathcal { C } _ j \\cup \\{ I \\cup \\{ m \\} : I \\in \\mathcal { C } _ j \\} \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} \\Lambda _ F ( \\chi , s ) = \\varepsilon ( \\chi ) \\Lambda _ F ( \\overline { \\chi } , 1 - s ) , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\mathfrak { X } _ i ^ { + } ( u , \\lambda + \\frac { \\hbar } { 2 } \\alpha _ j ) = g _ { ( \\lambda + \\frac { \\hbar } { 2 } \\alpha _ j , \\alpha _ i ) } ( u + z _ i ) = g _ { l + a } ( u + z _ i ) = \\frac { \\vartheta ( u + z _ i + l + a ) } { \\vartheta ( u + z _ i ) \\vartheta ( l + a ) } \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} | \\theta _ { r } - \\theta _ { ( 4 ) } | \\leq \\sum _ { j = 5 } ^ { n } | q | ^ { j ( j + 1 ) / 2 } | q | ^ { - 2 j } = \\sum _ { j = 5 } ^ { n } | q | ^ { j ( j - 3 ) / 2 } \\leq \\sum _ { j = 5 } ^ { \\infty } | q | ^ { j ( j - 3 ) / 2 } < 0 . 0 2 ~ . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} g _ { 1 } ( [ X , Y ] , V ) & = g _ { 1 } ( \\varphi V , \\mathcal { V } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { B } Y - \\nabla ^ { ^ { M _ 1 } } _ { Y } \\mathcal { B } X ) ) + g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { C } Y ) , ( \\nabla \\pi _ { \\ast } ) ( X , \\varphi V ) ) \\\\ & - g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { C } X ) , ( \\nabla \\pi _ { \\ast } ) ( Y , \\varphi V ) ) , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\frac { L ^ \\alpha [ e ^ { \\psi ^ \\epsilon } ] } { e ^ { \\psi ^ \\epsilon } } = - | \\lambda _ 1 | + \\mu ( \\hat { \\bar { x } } | \\bar { x } | ^ { 1 / \\epsilon } ) + \\mathcal { I } _ { \\epsilon , \\alpha } + \\int _ { | y | \\geq r } G _ \\epsilon ( \\bar { x } , y ) \\left ( 1 - \\frac { \\exp ( \\theta ( \\eta ^ \\epsilon ( \\bar { x } , y ) , \\bar { t } ) / \\epsilon ) } { \\exp ( \\theta / \\epsilon ) } \\right ) K d y \\ \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\min \\| L x \\| { \\rm s u b j e c t \\ \\ t o } x \\in \\mathcal { S } _ k = \\left \\{ x \\mid \\| A _ k x - b \\| = \\min \\right \\} \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N f ( T ^ n x ) e ( n \\alpha ) \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\nu ( K ^ { \\prime \\times } ) = e ( K ' , \\nu ) \\nu ( K ^ { \\times } ) . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} Y _ { n } ^ { ( \\omega ) } = X _ { n } ^ { ( \\omega ) } + Z _ { n } ^ { ( \\omega ) } , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\texttt { C } _ { K } = \\left \\{ \\begin{array} { l l } \\frac { h _ K } { \\pi \\sqrt { \\varepsilon } } , & \\textrm { i f } ~ \\kappa = 0 , \\\\ \\min \\left \\{ \\frac { h _ K } { \\pi \\sqrt { \\varepsilon } } , \\frac { 1 } { \\sqrt { \\kappa } } \\right \\} , & \\textrm { i f } ~ \\kappa \\neq 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\Delta ( ( 2 ^ { q + 1 } - t , t , 2 ^ q + t , 3 \\cdot 2 ^ q + t , \\ldots , 2 ^ q \\cdot [ 2 ^ { k - 2 } - 1 ] + t ) ; k ) = U ( 2 ^ { q + 1 } - t ; k ) . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} & | \\psi _ 1 ( x _ 1 , \\cdots , x _ { i } , x _ { i + 1 } ) - \\psi _ 2 ( x ' _ 1 , \\cdots , x ' _ { i } , x ' _ { i + 1 } ) | \\\\ & \\ \\ \\leq ( L _ \\varphi ( \\sum _ { j = 1 } ^ { i + 1 } | x _ j - x ' _ j | + \\hat { \\mathbb { E } } [ | B _ { s _ 2 } - B _ { s _ 1 } | ] ) ) \\wedge ( 2 C _ \\varphi ) \\\\ & \\ \\ \\leq C _ 1 ( \\sum _ { j = 1 } ^ { i + 1 } | x _ j - x ' _ j | \\wedge 1 + \\sqrt { s _ 2 - s _ 1 } ) , \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\gamma _ m ( p , n ) = \\sum _ { k = 1 } ^ { n } { \\gamma _ m ( p - 1 , k - m ) } . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} w _ { \\rm s o l . } ( x ) = \\int _ { \\R ^ n } d \\mathcal H ^ n ( y ) \\big ( \\Delta \\ddot h ^ 0 \\ , \\chi _ { \\Omega _ 0 } \\big ) ( y ) P ( x - y ) , \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} C ^ T = a _ 0 C ^ T _ { i j } , C ^ T _ { i j } = \\sum _ { k = 0 } ^ { T - \\max { i , j } } r _ { | i - j | + 2 k } , r _ 0 = a _ 0 . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon } ( x _ 0 , t _ 0 ) = \\Gamma ( x _ 0 , t _ 0 ) - \\varepsilon ( A t _ 0 + | x _ 0 | ^ { 2 } ) , \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} \\left \\vert e ^ { z } - 1 \\right \\vert \\leq \\sum \\limits _ { k = 1 } ^ { \\infty } { \\dfrac { \\left \\vert z \\right \\vert ^ { k } } { k ! } . } \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} & ( \\xi _ { s - } ( w ) - \\xi _ { s - } ( z ) ) \\ , \\phi _ s ( z ) \\\\ = \\ ; & [ \\xi _ { s - } ( w ) \\ , \\xi ^ c _ { s - } ( z ) - \\xi _ { s - } ( z ) \\ , \\xi ^ c _ { s - } ( w ) ] \\ , \\phi _ s ( z ) \\\\ = \\ ; & \\xi _ { s - } ( w ) \\ , \\xi ^ c _ { s - } ( z ) \\phi _ s ( w ) - \\xi _ { s - } ( z ) \\ , \\xi ^ c _ { s - } ( w ) \\phi _ s ( z ) \\\\ & + \\xi _ { s - } ( w ) \\ , \\xi ^ c _ { s - } ( z ) \\ , \\big ( \\phi _ s ( z ) - \\phi _ s ( w ) \\big ) . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} A _ n ^ - ( n - 1 ) = \\sum _ { j = 0 } ^ { n - 3 } a ( j ) + 2 ^ { n - 2 } - n + 1 \\ , . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\| u \\| _ { E _ a } ^ 2 \\leq C \\sum _ { j = 0 } ^ \\infty \\| R _ j u \\| _ { E _ a } ^ 2 \\forall a \\geq 0 , \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x = \\frac { a _ { 1 } p q - Q ( q - p ) } { ( a _ { 1 } - a _ { 2 } ) ( q - p ) } d . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} A f ( x ) = \\sum _ k \\lambda _ k ( x ) ( f ( x + \\zeta _ k ) - f ( x ) ) , \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i } { a \\alpha _ i + c _ i \\alpha _ { \\sigma ( i ) } } & = \\frac { 1 } { a + c _ { 1 } \\frac { \\alpha _ { \\sigma ( 1 ) } } { \\alpha _ 1 } } + \\frac { 1 } { a + c _ { 2 } \\frac { \\alpha _ { \\sigma ( 2 ) } } { \\alpha _ 2 } } + \\cdots + \\frac { 1 } { a + c _ { n } \\frac { \\alpha _ { \\sigma ( n ) } } { \\alpha _ n } } \\\\ & = \\frac { 1 } { a + c _ { 1 } \\lambda ^ { n - 1 } } + \\frac { 1 } { a + \\frac { c _ { 2 } } { \\lambda } } + \\cdots + \\frac { 1 } { a + \\frac { c _ { n } } { \\lambda } } \\leq 1 . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} A = \\langle e _ 3 \\rangle _ { \\overline { 0 } } \\oplus \\langle e _ 1 , e _ 2 \\rangle _ { \\overline { 1 } } . \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\mathfrak { X } _ j ^ + ( v , \\zeta ) * \\mathfrak { X } _ i ^ + ( u , \\lambda ) = & ( - 1 ) ^ { a + 1 } g _ { \\zeta _ j } ( z _ j - v ) g _ { \\lambda _ i } ( z _ i - u ) \\prod _ { m \\in S } \\vartheta ( z _ { i } - z _ j + m \\frac { \\hbar } { 2 } ) \\\\ = & - g _ { \\zeta _ j } ( z _ j - v ) g _ { \\lambda _ i } ( z _ i - u ) \\prod _ { m \\in S } \\vartheta ( z _ j - z _ i - m \\frac { \\hbar } { 2 } ) \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} \\Phi ( x , t ) = ( x _ 1 , x _ 2 , x _ 3 + \\bar { \\eta } ( x , t ) ( 1 + x _ 3 / b ) ) \\in \\Omega ( t ) . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\phi ( u ( r + 1 , r ) ( \\xi ) v ) & = \\phi ( \\sum _ { i \\geq 0 } \\xi ^ i X ( r + 1 , r ) _ i v ) = \\sum _ { i \\geq 0 } \\xi ^ i X ( r + 1 , r ) _ i \\phi ( v ) \\cr & = u ( r + 1 , r ) ( \\xi ) \\phi ( v ) . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} ( \\partial _ t L _ u + [ L _ u , B _ u ] ) \\varphi & = \\frac { 2 } { i } ( C _ + u _ { x x } ) \\varphi - \\frac 2 i C _ + ( ( C _ + u _ { x x } ) \\varphi ) \\\\ & = \\frac { 2 } { i } C _ - \\left [ ( C _ + u _ { x x } ) \\varphi \\right ] . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} \\frac { \\partial F _ j } { \\partial \\nu _ i } ( e _ d / 2 ) & = \\int _ { \\partial B _ 1 ^ + } \\ ! \\ ! \\ ! x _ i x _ d x _ { j - 1 } \\phi ( x _ d ) \\ , d x = \\delta _ { i ( j - 1 ) } \\int _ { \\partial B _ 1 ^ + } \\ ! \\ ! \\ ! x _ i ^ 2 x _ d \\phi ( x _ d ) \\ , d x = \\frac { \\delta _ { i ( j - 1 ) } } { d - 1 } \\int _ { \\partial B _ 1 ^ + } \\ ! \\ ! \\ ! ( 1 - x _ d ^ 2 ) x _ d \\phi ( x _ d ) \\ , d x \\ , , \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\Delta ( \\{ a b , c \\} ) & = \\Delta ( a \\{ b , c \\} + ( - 1 ) ^ { | a | | b | } b \\{ a , c \\} ) \\\\ & = \\Delta ( a ) \\Delta ( \\{ b , c \\} ) + ( - 1 ) ^ { | a | | b | } \\Delta ( b ) \\Delta ( \\{ a , c \\} ) \\\\ & = \\Delta ( a ) \\{ \\Delta ( b ) , \\Delta ( c ) \\} + ( - 1 ) ^ { | a | | b | } \\Delta ( b ) \\{ \\Delta ( a ) , \\Delta ( c ) \\} \\\\ & = \\{ \\Delta ( a b ) , \\Delta ( c ) \\} . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\mathbb E \\langle M ^ { a , 1 } _ t , ( F _ 2 \\star \\bar { \\mu } ^ { M ^ { q } } ) _ t \\rangle = \\mathbb E \\langle M ^ { a , 2 } _ t , ( F _ 1 \\star \\bar { \\mu } ^ { M ^ { q } } ) _ t \\rangle = 0 , \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\vartheta ( z + \\gamma ) = e _ { \\gamma } ( z ) \\vartheta ( z ) , \\ , \\ \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} & \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathbb { E } \\big [ \\mathfrak { d } _ { i , 1 } Q _ i \\mathfrak { l } ^ { ( p - 1 , p ) } \\big ] = \\mathbb { E } \\big [ O _ \\prec ( \\hat { \\Pi } ^ 2 ) \\mathfrak { l } ^ { ( p - 1 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { l } ^ { ( p - 2 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { l } ^ { ( p - 1 , p - 1 ) } \\big ] , \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\Gamma & = \\{ g \\in G L ( 2 , \\bold Z [ \\rho ] ) \\mid g U \\ ^ t \\overline { g } = U \\} , \\\\ \\Gamma ( 1 - \\rho ) & = \\{ g \\in \\Gamma \\mid g \\equiv I _ 2 ( 1 - \\rho ) \\} , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} L ^ \\alpha [ u ] ( x ) : = \\int ( u ( x ) - u ( x + y ) ) K ( x , y ) d y \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\psi _ { x } ( W _ p ) = \\sigma _ c \\big ( \\eta , ( x , p , x ) \\big ) \\sigma _ c \\big ( ( y , m + p , x ) , \\eta ^ { - 1 } \\big ) \\overline { \\sigma _ c ( \\eta ^ { - 1 } , \\eta ) } \\psi _ { y } ( W _ p ) . \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\frac { d } { d r } \\psi \\left ( \\frac { \\abs { x - y } } { r } \\right ) = - \\frac { 1 } { r } \\varphi ' \\left ( \\frac { \\abs { x - y } } { r } \\right ) \\left ( \\frac { \\abs { x - y } } { r } \\right ) ^ { m - 1 } , \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\omega _ Q ( X _ 1 ^ { T } , \\dots , X _ n ^ { T } ) = \\omega _ Q ( X _ 1 , \\dots , X _ n ) ^ { T } \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } I ( W ^ { ( n ) } ( \\Delta _ n ) ; Y ^ { ( n ) } ( \\Delta _ n ) ) = I ( W _ 0 ^ T ; Y _ 0 ^ T ) . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { s _ j } [ X ] = \\psi _ j ( { B _ { t _ 1 } , \\cdots , B _ { t _ { i } } - B _ { t _ { i - 1 } } , B _ { s _ j } - B _ { t _ { i } } } ) , \\ \\ \\ \\ \\ j = 1 , 2 , \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ 2 } \\int _ { - 1 } ^ 1 [ a _ 0 + a _ 2 t ^ 2 + a _ 4 t ^ 4 ] ( t ^ n - 1 ) ^ 2 d t = 0 . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} J _ l x \\cdot y x = y , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} \\Lambda ( \\mu ) = \\lim _ { n \\to \\infty } \\frac { F _ n ( x ) } { n } \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} - d _ { i } ^ { \\varepsilon } \\nabla u _ { i } ^ { \\varepsilon } \\cdot \\mbox { n } = \\varepsilon \\left ( a _ { i } ^ { \\varepsilon } u _ { i } ^ { \\varepsilon } - b _ { i } ^ { \\varepsilon } v _ { i } ^ { \\varepsilon } \\right ) \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\Gamma ^ { \\varepsilon } , \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\overline { c _ i } \\leq \\widetilde { K } : = 2 \\left ( K + \\sum _ { i = 1 } ^ { N } c _ { i , \\infty } \\right ) , i = 1 , 2 , \\ldots , N . \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\psi \\left ( \\mu \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) \\right ) \\triangleleft \\mu \\left ( \\sum _ { i = 1 } ^ n \\psi ( \\mu ( | x _ i | ^ p ) ) \\right ) , \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\lambda _ i = p ( y _ i | c _ i ) = \\begin{cases} p _ i & y _ i \\neq c _ j \\\\ 1 - p _ i & y _ i = c _ i \\end{cases} . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} { \\cal P } ( T _ + \\vert x _ 0 , v _ 0 , t ) \\equiv \\left \\langle \\delta \\left ( T _ + - \\int _ 0 ^ t d t ' \\thinspace \\theta \\left ( x ( t ' ) \\right ) \\right ) \\right \\rangle = { \\cal L } ^ { - 1 } _ { p \\to T _ + } Q _ p ( x _ 0 , v _ 0 , t ) \\ , , \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\begin{aligned} 1 & = 3 k _ 1 + k _ 2 + k _ 4 \\\\ 1 & = 3 k _ 2 + k _ 1 + k _ 3 \\\\ 1 & = 3 k _ 3 + k _ 2 \\\\ 1 & = 3 k _ 4 + k _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\chi = \\chi _ { n } \\left ( { u , \\xi } \\right ) = u ^ { n - 1 } \\left \\{ { \\psi - 2 T - \\frac { 1 } { u } \\frac { d T } { d \\xi } - \\frac { T ^ { 2 } } { u ^ { 2 } } } \\right \\} . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\sum _ { ( e , \\tilde { e } ) } \\frac { C ^ n _ { e , \\tilde { e } } M ^ { \\tilde { e } } } { L ^ e M ^ e } \\Big ( \\frac { 1 } { M ^ e } + \\frac { 1 } { M ^ { \\tilde { e } } } \\Big ) \\leq C \\ , \\sum _ { ( e , \\tilde { e } ) } \\frac { M ^ { \\tilde { e } } } { M ^ e } \\Big ( \\frac { 1 } { M ^ e } + \\frac { 1 } { M ^ { \\tilde { e } } } \\Big ) \\to 0 , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} h ^ { p , q } ( \\hat { X } _ \\Delta ) = f ^ { 3 - p , q } ( Y _ \\Delta , w _ \\Delta ) \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\| U ^ { * } ( \\Phi ^ { \\prime } ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) } ^ m ) U \\\\ & - \\Psi ^ { \\prime } ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) } ^ m \\| < \\frac { \\varepsilon } { 7 } \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} h ( r , z ) = - \\int _ 0 ^ z \\big ( r \\cdot \\nabla _ r f ( r , s ) + ( n - 1 ) f ( r , s ) \\big ) \\ , d s \\ , . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} p ^ { \\circ } _ { j } = \\frac { \\nu _ j } { \\sum _ { j = 1 } ^ { \\infty } \\nu _ j } \\mbox { f o r } j \\in \\mathbb { N } _ { + } , \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\Phi ^ { r } ( z ^ { n } t ^ { k } ) | _ { t = z } = \\Psi ^ { r } ( z ^ { n } t ^ { k } ) | _ { t = z } \\prod _ { j = 0 } ^ { r - 1 } ( 1 - f _ { j } ) = \\frac { z ^ { n + k ( r + 1 ) } F _ { r + 1 } ( - z ) ^ { 2 n - 1 } } { F _ { r + 2 } ( - z ) ^ { 2 n + 2 k - 1 } } . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\rho \\frac { \\partial u } { \\partial t } d r + \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial P } { \\partial x } d r = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\mu \\left ( \\frac { \\partial ^ { 2 } u } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial u } { \\partial r } \\right ) d r . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\int \\chi _ R ( \\alpha ) \\alpha ^ { n - 1 } d \\alpha = 1 . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} f ( t ) = \\frac { 1 } { 2 \\pi i } \\int _ { 2 - i \\infty } ^ { 2 + i \\infty } \\Gamma ( s ) \\zeta ( s ) A ( s ) t ^ { - s } ~ d s , \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\begin{cases} d X ^ { t , \\xi } _ { s } = b ( X _ { s } ^ { t , \\xi } ) d s + \\sum _ { i , j = 1 } ^ d h _ { i j } ( X _ { s } ^ { t , \\xi } ) d \\langle B ^ i , B ^ j \\rangle _ s + \\sum _ { j = 1 } ^ { d } \\sigma _ j ( X _ { s } ^ { t , \\xi } ) d B ^ j _ s , \\ \\ \\ \\ s \\in [ t , T ] , \\\\ X _ { t } ^ { t , \\xi } = \\xi , \\end{cases} \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] & = 0 & [ X _ 1 , X _ 3 ] & = 0 & [ X _ 1 , X _ 4 ] & = 0 & [ X _ 1 , X _ 5 ] & = 0 & [ X _ 2 , X _ 3 ] & = X _ 1 \\\\ [ X _ 2 , X _ 4 ] & = 0 & [ X _ 2 , X _ 5 ] & = X _ 2 & [ X _ 3 , X _ 4 ] & = 0 & [ X _ 3 , X _ 5 ] & = - X _ 3 & [ X _ 4 , X _ 5 ] & = X _ 1 . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} a ( i + 1 ) - a ( i ) \\geq m , \\ : i = 1 , 2 , \\ldots , p - 1 \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\delta _ { [ k = \\frac { 1 } { 2 } ] } \\delta _ { [ h = a ^ 2 ] } = \\begin{cases} 1 & k = \\frac { 1 } { 2 } \\ ; h \\ ; , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} G = S ^ T S , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} C _ 1 = \\frac { \\sqrt { 2 } } { | P _ r ( \\vec { 1 } ) | } \\left ( \\| \\nabla P _ r ( \\vec { 1 } ) \\| _ 1 \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right | + \\| \\nabla P _ { r + 1 } ( \\vec { 1 } ) \\| _ { 1 } \\right ) \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} y ^ 3 + a _ 2 y + a _ 1 = 0 . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} W _ \\varepsilon ( t ) = \\Psi ^ { - 1 } ( t + \\Psi ( \\varepsilon ) ) . \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} \\mbox { s u m o f r o w } i & = ( x _ { i , 0 } + \\varepsilon ) + \\sum _ { j = 1 } ^ { q - 2 } x _ { i , j } + ( y _ i - \\varepsilon ) = \\sum _ { j = 0 } ^ { q - 2 } x _ { j , i } + y _ { i } = \\mbox { s u m o f c o l u m n } i . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\Delta w + w ^ 5 = 0 \\quad \\mathbb { R } ^ 3 \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\mathbf V : = \\| \\mathbf v _ i - \\mathbf v _ j \\| ^ 2 . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} V _ i ( y ) & = \\varepsilon ^ { \\frac { 1 } { 2 } } U _ i ( \\varepsilon y ) = w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) + \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ i ( \\varepsilon \\ , y ) y \\in \\Omega _ \\varepsilon , \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\widehat { f } ( n ) = \\langle f , \\chi _ n \\rangle , n \\in \\Z , \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} 2 ( - 1 ) ^ { | a | | b | } g ( a b ) & = ( - 1 ) ^ { | a | | b | } g ( a ) f ( b ) + ( - 1 ) ^ { | a | | b | } f ( a ) g ( b ) + g ( b ) f ( a ) + f ( b ) g ( a ) \\\\ & = ( - 1 ) ^ { | a | | b | } g ( a ) f ( b ) + g ( b ) f ( a ) + ( - 1 ) ^ { | a | | b | } g ( a b ) . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} h ( z ) - \\pi ' ( z ) = g ( z ) , \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{gather*} n \\circ \\eta = \\frac { \\partial _ 1 r \\times \\partial _ 2 r } { | \\partial _ 1 r \\times \\partial _ 2 r | } \\end{gather*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\mathcal { H } = \\{ H _ { ( 1 , 1 ) } ^ 1 , H _ { ( 1 , 1 ) } ^ 2 , H _ { ( 1 , 2 ) } ^ 1 , H _ { ( 1 , 2 ) } ^ 2 , H _ { ( 2 , 1 ) } ^ 1 , H _ { ( 2 , 1 ) } ^ 2 , H _ { ( 2 , 2 ) } ^ 1 , H _ { ( 2 , 2 ) } ^ 2 , \\dots , H _ { ( 4 , 4 ) } ^ 1 , H _ { ( 4 , 4 ) } ^ 2 \\} \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} r _ { i j } = \\frac { \\sum _ { k = 1 } ^ n ( x _ { k i } - \\bar { x } _ i ) ( x _ { k j } - \\bar { x } _ j ) } { \\sqrt { \\sum _ { k = 1 } ^ n ( x _ { k i } - \\bar { x } _ i ) ^ 2 \\cdot \\sum _ { k = 1 } ^ n ( x _ { k j } - \\bar { x } _ j ) ^ 2 } } , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} Z '^ { \\alpha } _ t = \\int _ 0 ^ t { B ^ { \\alpha } _ { \\beta , s } d Z ^ { \\beta } _ s } \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} ( T ) = \\lambda ^ 2 ( T _ 1 ) . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} \\frac { 2 ^ { 1 \\ , - \\ , z } - 1 } { ( 4 \\ , \\pi ) ^ { \\frac { 1 - z } { 2 } } \\ , \\Gamma \\left ( \\frac { 1 + ( 1 - z ) } { 2 } \\right ) } \\ , G ( 1 - z ) : = \\frac { 2 ^ z - 1 } { ( 4 \\ , \\pi ) ^ { \\frac { z } { 2 } } \\ , \\Gamma \\left ( \\frac { 1 + z } { 2 } \\right ) } \\ , G ( z ) \\ , . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\tilde \\omega _ D : = \\alpha _ 0 ^ 2 \\cdot \\omega _ D \\in M ^ + ( D ) \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} 0 \\leq u _ n ( y , s ) \\leq u _ n ( x _ 0 , t _ 0 ) \\ ; \\underbrace { \\exp \\left ( C \\Big ( \\frac { \\| x _ 0 - y \\| ^ 2 } { t _ 0 - s } + \\frac { t _ 0 - s } { \\min ( 1 , s ) } + 1 \\Big ) \\right ) } _ { = : C _ 2 } . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} I ( M ; Y _ 0 ^ T ) = \\lim _ { n \\to \\infty } I ( M ; Y ^ { ( n ) } ( \\Delta _ n ) ) . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} \\log G ( n ) = P _ n ( x ) + O \\bigg ( \\frac { ( \\log \\log x ) ^ { 3 / 2 } } { \\log \\log \\log x } \\bigg ) . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma _ { j } } { \\partial q ^ i } \\frac { \\partial F } { \\partial p _ j } ( q , \\gamma ( q ) , \\lambda ) + \\frac { \\partial F } { \\partial q ^ i } ( q , \\gamma ( q ) , \\lambda ) = 0 , \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} 2 ( 1 - a ) \\Biggl \\{ \\sum _ { i = 0 } ^ { k - 2 } ( i + 1 ) a ^ { i } - \\binom { k } { 2 } a ^ { k - 2 } \\Biggr \\} < ( b - 1 ) \\Biggl \\{ ( k - 1 ) k + \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) a ^ { i } \\Biggr \\} \\ , . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} d = p ^ { i _ i } d ' , \\ \\ \\mbox { w h e r e } \\ \\ d ' = 1 + p ^ { i _ 2 - i _ 1 } + \\cdots + p ^ { i _ p - i _ 1 } , \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\norm { v + i v ' } = \\sup _ { \\theta \\in [ 0 , 2 \\pi ] } \\norm { \\cos ( \\theta ) v + \\sin ( \\theta ) v ' } \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} A Q _ \\infty x = - B B ^ * x - Q _ \\infty A ^ * x \\forall x \\in D ( A ^ * ) . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\iota _ { X _ H } \\omega _ Q = d H , \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\ + U _ { k , s } = \\bigcup _ { t \\in [ k , s ] } \\ + V _ { k , t } . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} u _ s = \\frac { - s ( s + 1 ) } { \\cosh ^ 2 ( x ) } , \\ , \\ , \\ , s \\geq 1 , \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} \\chi _ 1 ( \\tau ) = \\dfrac { g _ 1 ( \\tau ) } { h _ 1 ( \\tau ) } , \\chi _ 2 ( \\tau ) = \\dfrac { g _ 2 ( \\tau ) } { h _ 2 ( \\tau ) } \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} b ^ { 2 q _ 1 + 2 } + b ^ { q _ 1 + 1 } + b ^ { 2 q _ 1 } = b ^ 2 c . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\frac { p ( r | c _ i ) } { p ( r | c _ j ) } = \\frac { p ( c _ i | r ) } { p ( c _ j | r ) } p ( c _ i | r ) = \\prod _ { i = 1 } ^ { n } q _ i \\\\ q _ i = \\begin{cases} p _ i & r _ i \\neq c _ i \\\\ 1 - p _ i & r _ i = c _ i \\end{cases} . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} \\aligned \\delta & = ( i - 1 ) ( 1 + \\dim L \\dim { \\mathcal M } _ { k _ 2 + 1 } ( \\beta _ 2 ) + \\dim L ) \\\\ & \\equiv ( i - 1 ) \\Big ( 1 + ( \\mu ( \\beta _ 2 ) + k _ 2 ) \\dim L \\Big ) \\mod 2 , \\endaligned \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} & P _ j ( u , v ) - P _ j ( w , v ) \\\\ & = \\sum _ { | \\alpha | \\geq 1 , | \\beta | = l } c _ { \\alpha , \\beta } [ ( x _ Q + 2 ^ j r u ) ^ { \\alpha } - ( x _ Q + 2 ^ j r w ) ^ { \\alpha } ] ( x _ Q + d _ Q v ) ^ { \\beta } + \\tilde { Q } ( u , w , v ) . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\geq 0 \\\\ k \\geq 1 } } \\frac { u ^ { n + k } x ^ { k } ( 1 - x ) ^ { n } ( 1 - u ) ^ { 2 k } } { ( 1 + u ) ^ { n + 2 k } ( 1 - u x ) ^ { n + 2 k } } k ^ { \\underline { d } } C _ { k - 1 } \\binom { n + 2 k - 2 } { n } 2 ^ { n } = ( 2 d - 2 ) ^ { \\underline { d - 1 } } \\frac { u ^ { d } x ^ { d } ( 1 - u ) } { ( 1 - u x ) ^ { 2 d } ( 1 + u ) } \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\triangle u - \\partial _ t u = e ^ { - 2 z } \\big ( \\partial ^ 2 _ { z } u + ( \\partial _ { z } \\ln \\sqrt { \\gamma } + ( n - 2 ) ) \\partial _ { z } u + \\triangle _ { \\theta } u \\big ) - \\partial _ t u . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} \\mathcal F _ { a a b b } ( v ) & = \\mathcal F _ { * * 0 0 } ( v ) \\cup \\mathcal F _ { 0 0 * * } ( v ) , \\\\ \\mathcal F _ { a b a b } ( v ) & = \\mathcal F _ { * 0 * 0 } ( v ) \\cup \\mathcal F _ { 0 * 0 * } ( v ) , \\\\ \\mathcal F _ { a b b a } ( v ) & = \\mathcal F _ { * 0 0 * } ( v ) \\cup \\mathcal F _ { 0 * * 0 } ( v ) . \\\\ \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\prod _ { j = 1 } ^ k F _ { g _ j } ( n ) & = \\sum _ { n \\leq x } \\prod _ { j = 1 } ^ k \\sum _ { p \\leq z } g _ j ( p ) f _ p ( n ) \\\\ & = \\sum _ { p _ 1 , \\dots , p _ k \\leq z } g _ 1 ( p _ 1 ) \\cdots g _ k ( p _ k ) \\sum _ { n \\leq x } f _ { p _ 1 \\cdots p _ k } ( n ) \\\\ & = \\sum _ { p _ 1 , \\dots , p _ k \\leq z } g _ 1 ( p _ 1 ) \\cdots g _ k ( p _ k ) \\big ( H ( p _ 1 \\cdots p _ k ) x + O \\big ( 2 ^ { \\omega ( p _ 1 \\cdots p _ k ) } \\big ) \\big ) , \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ i ( x ) : = \\frac { \\alpha _ i ( x ) } { \\sqrt { \\sum _ { i = 1 } ^ { n + 1 } \\alpha _ i ( x ) ^ 2 } } = \\frac { \\alpha _ i ( x ) } { | \\nabla u ( x ) | } . \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\C ( A , q ) \\odot _ { \\C ( { B } ) } \\C ( A ' , q ' ) = ( \\C ( A , q ) \\boxtimes \\C ( A ' , q ' ) ) ^ { l o c } _ R \\simeq \\C ( A \\times A ' , q \\times q ' ) ^ { { l o c } } _ { R ( \\overline { \\delta } ( B ) ) } \\simeq \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} s ( J ( Q \\cap X ) , X ) = \\frac { h \\cdot [ X ] } { 1 + h } \\quad . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\left ( X ^ H \\right ) _ { t \\ge 0 } \\stackrel { ( d ) } { = } \\left ( X _ { \\tau _ t } \\right ) _ { t \\ge 0 } . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\| u \\| _ { X _ \\infty } ^ 2 = \\| u \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ x ^ 4 u \\| _ { L ^ 2 } ^ 2 + \\| L _ y ^ 2 \\partial _ x u \\| _ { L ^ 2 } ^ 2 + \\| J _ \\infty u \\| _ { L ^ 2 } ^ 2 , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} F P _ 1 = ( 0 , 0 ) , F P _ 2 = \\left ( \\frac c d , \\frac a b \\right ) . \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} \\triangle u + \\partial _ t u = V ( x , t ) u \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{gather*} Q = Q ( \\partial _ 1 \\eta ^ 1 , \\partial _ 2 \\eta ^ 1 , \\partial _ 1 \\eta ^ 2 , \\partial _ 2 \\eta ^ 2 , \\partial _ 1 \\eta ^ 3 , \\partial _ 2 \\eta ^ 3 ) . \\end{gather*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\kappa _ { F 1 } ( A , \\lambda b ) = \\left \\| M ^ { - 1 } \\left ( ( 1 + \\| x _ S \\| ^ 2 _ 2 ) A ^ T A - A ^ T r x _ S ^ T - x _ S r ^ T A + \\| r \\| _ 2 ^ 2 I _ n \\right ) M ^ { - 1 } \\right \\| _ 2 ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} C ^ { ( \\omega ) } = \\omega \\log \\left ( 1 + \\frac { P } { 2 \\omega } \\right ) . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} L ^ \\alpha [ u ] ( x ) = \\frac { 1 } { 2 } \\int ( 2 u ( x ) - u ( x + y ) - u ( x - y ) ) K ( x , y ) d y . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\bar { u } _ 1 \\ , = \\ , 1 \\ \\ \\ , \\ \\ \\ \\bar { u } _ l \\ , \\bar { u } _ m \\ , \\bar { u } _ { l m } ^ { - 1 } \\ , = : \\ , \\gamma ( l , m ) \\ , \\in i ( G ) \\ . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} M _ t ^ { \\mu , \\phi ' } = \\phi ' ( x , y , 0 ) + \\int _ 0 ^ t e ^ { - \\mu \\tau } \\dfrac { \\partial \\phi ' } { \\partial y } ( X _ \\tau , Y _ \\tau , \\tau ) \\d W _ \\tau . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} P _ 1 = \\begin{pmatrix} I _ n & 0 \\\\ 0 & O _ m \\end{pmatrix} P _ 2 = \\begin{pmatrix} O _ n & 0 \\\\ 0 & I _ m \\end{pmatrix} . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} P _ 3 = 2 ^ { r - | S | - 2 } \\sum _ a \\binom { a } { 2 } \\binom { | S | - 2 } { a - 1 } + 2 ^ { | S | - 2 } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 2 } { b - 1 } . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} \\partial _ { \\nu , \\ , { \\rm i n } } w ( x ) = \\partial _ { \\nu , 0 } w ( x ) + \\frac { 1 } { 2 } f ( x ) \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} ( a \\otimes b ) \\bigstar ( a ' \\otimes b ' ) : = ( - 1 ) ^ { | a ' | | b | } ( a \\cdot a ' ) \\otimes ( b * b ' ) & , \\\\ d ( a \\otimes b ) : = d _ A ( a ) \\otimes b + ( - 1 ) ^ { | a | } a \\otimes d _ B ( b ) & , \\\\ \\{ a \\otimes b , a ' \\otimes b ' \\} : = ( - 1 ) ^ { ( | a ' | + p ) | b | } \\{ a , a ' \\} _ A \\otimes ( b * b ' ) + ( - 1 ) ^ { ( | b | + p ) | a ' | } ( a \\cdot a ' ) \\otimes \\{ b , b ' \\} _ B & , \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} S _ \\mathfrak { a } ( m , n ; c ) = \\sum _ { \\left ( \\begin{smallmatrix} a & \\cdot \\\\ c & d \\end{smallmatrix} \\right ) \\in \\Gamma _ \\infty \\backslash \\sigma _ \\alpha ^ { - 1 } \\Gamma _ 0 ( N ) / \\Gamma _ \\infty } e \\left ( m \\frac { d } { c } + n \\frac { a } { c } \\right ) \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} r _ { i j } = w _ i ' w _ j , ~ ~ 1 \\le i , j \\le p . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} J ( \\rho , m ) & = E \\left [ \\int _ 0 ^ T \\int _ A c ( t , X ( t ) , a , m ( t ) ) \\rho _ t ( d a ) d t + \\psi ( X ( T ) , m ( T ) ) \\right ] \\\\ & = E \\left [ \\int _ 0 ^ T \\int _ A c ( t , X ( t ) , a , m ( t ) ) [ \\widehat { \\gamma } ( s , X ( s ) ) ] ( d a ) d t + \\psi ( X ( T ) , m ( T ) ) \\right ] \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} \\omega _ i : = p _ { i , \\bar i } ^ e = | \\{ z \\in X : \\ > ( x , z ) \\in R _ i \\} | \\in \\mathbb N \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\langle f , \\varphi \\rangle _ { 0 } = \\sum _ { k = 1 } ^ { d } ( f _ { k } , \\varphi _ { k } ) _ { L ^ { 2 } } , \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} \\varphi ^ \\hbar ( x ) = ( \\tfrac { 1 } { \\pi \\hbar } ) ^ { n / 4 } e ^ { - \\tfrac { 1 } { 2 \\hbar } x ^ 2 } \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} ( H _ { \\lambda f , \\alpha , \\theta } x ) _ n = x _ { n + 1 } + x _ { n - 1 } + \\lambda f ( \\theta + n \\alpha ) x _ n , \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} | \\alpha _ p | = p ^ { k - 1 / 2 } , \\quad | \\beta _ p | = p ^ { k - 1 / 2 } \\cdot \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{gather*} \\mathcal { A } _ { \\varepsilon } = f ^ { \\varepsilon } ( \\mathbf { x } ) ^ * b ( \\mathbf { D } ) ^ * g ^ { \\varepsilon } ( \\mathbf { x } ) b ( \\mathbf { D } ) f ^ { \\varepsilon } ( \\mathbf { x } ) , \\\\ \\widehat { \\mathcal { A } } _ { \\varepsilon } = b ( \\mathbf { D } ) ^ * g ^ { \\varepsilon } ( \\mathbf { x } ) b ( \\mathbf { D } ) . \\end{gather*}"}
-{"id": "8449.png", "formula": "\\begin{align*} L ( f , s ) = \\sum _ { n = 1 } ^ \\infty \\frac { a _ n } { n ^ s } . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\phi \\mathcal { T } _ { U } X + \\mathcal { B } \\nabla ^ { ^ { M _ 1 } } _ { U } X - \\eta ( X ) U & = \\hat { \\nabla } _ { U } \\mathcal { B } X + \\mathcal { T } _ { U } \\mathcal { C } X , \\\\ \\omega \\mathcal { T } _ { U } X + \\mathcal { C } \\nabla ^ { ^ { M _ 1 } } _ { U } X & = \\mathcal { T } _ { U } \\mathcal { B } X + \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\mathcal { C } X , \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} b _ q ( x , m , w ) = \\sup _ { \\beta \\in \\R ^ { d } } \\left \\{ \\beta \\cdot w - m H ( x , - \\beta ) \\right \\} , \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\varphi _ { Z _ t ^ H - Z _ s ^ H } ( u ) = { \\Bbb E } \\bigl [ \\exp \\bigl ( i u \\bigl ( Z _ t ^ H - Z _ s ^ H \\bigr ) \\bigr ) \\bigr ] = \\exp \\biggl ( - \\frac { u ^ 2 } { 2 } { \\Bbb E } \\bigl ( Z _ t ^ H - Z _ s ^ H \\bigr ) ^ 2 \\biggr ) , \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} ( g \\circ f ) ( i ) & = g ( \\sigma _ f ( i ) + a _ i p ) \\\\ & = ( \\sigma _ g \\circ \\sigma _ f ) ( i ) + p ( b _ { \\sigma _ f ( i ) } + a _ i g _ { \\sigma _ f ( i ) } ) . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\Psi _ \\chi = \\sum _ { x \\in \\{ \\pm 1 \\} ^ { 2 d } } \\chi ( x ) \\Phi _ x . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} ( 1 - x y ) A ( x , y ) = x y + \\frac { x y } { 1 - y } ( F _ T ( x ) - G ( x , y ) ) \\ , . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty n ^ 2 \\Big ( | C _ n | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\asymp \\| z _ { 1 } ^ { 0 } \\| ^ 2 _ { H ^ 1 _ 0 } + \\| z _ { 1 } ^ { 1 } \\| ^ 2 _ { L ^ 2 } + \\| z _ { 2 } ^ { 0 } \\| ^ 2 _ { H ^ 1 _ 0 } + \\| z _ { 2 } ^ { 1 } \\| _ { L ^ 2 } ^ 2 \\ , . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\begin{array} { l } 0 = - \\ , \\frac { ( 2 \\ , h ) ^ 2 } { 1 2 } \\ , \\Re \\ , G ^ { ( 2 \\ , n + 3 ) } ( 1 / 2 ) \\\\ \\\\ 0 = - \\ , \\frac { ( 2 \\ , h ) ^ 2 } { 1 2 } \\ , \\Im \\ , G ^ { ( 2 \\ , n + 3 ) } ( 1 / 2 ) . \\end{array} \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 3 } f v _ 2 d v = \\int _ { \\mathbb { R } ^ 3 } f v _ 3 d v = 0 . \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] & = 0 & [ X _ 1 , X _ 3 ] & = 0 & [ X _ 1 , X _ 4 ] & = 0 & [ X _ 1 , X _ 5 ] & = 0 & [ X _ 2 , X _ 3 ] & = 0 \\\\ [ X _ 2 , X _ 4 ] & = X _ 1 & [ X _ 2 , X _ 5 ] & = 0 & [ X _ 3 , X _ 4 ] & = 0 & [ X _ 3 , X _ 5 ] & = X _ 1 & [ X _ 4 , X _ 5 ] & = 0 . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} g _ v \\left ( \\sum _ { i = 1 } ^ n a _ i \\frac { \\partial } { \\partial x ^ i } , \\sum _ { j = 1 } ^ n b _ j \\frac { \\partial } { \\partial x ^ j } \\right ) : = \\sum _ { i , j = 1 } ^ n g _ { i j } ( v ) a _ i b _ j . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} b ^ 2 = L ^ 2 - r ^ 2 . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\int _ X & \\alpha ( \\pi ( x , y ) ) \\ > \\overline { \\beta ( \\pi ( z , y ) ) } \\ > d \\omega _ X ( y ) = \\int _ D \\alpha ( h ) \\ > \\overline { \\beta ( \\pi ( z , x ) * h ) } \\ > d \\omega _ D ( h ) \\\\ & = \\int _ D \\alpha ( h ) \\ > \\overline { \\beta ( h ) } \\ > d \\omega _ D ( h ) \\cdot \\overline { \\beta ( \\pi ( z , x ) ) } . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} ( f _ { x } ^ { \\boxplus \\tau } ) ! ( h _ x ^ { \\boxplus \\tau } ; \\mathcal S _ x ^ { \\boxplus \\tau } ) = f _ { x } ! ( h _ x ; \\mathcal S _ x ) . \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\varpi ' _ 1 | _ T = \\varpi _ 1 = \\varpi ' _ 3 | _ T , \\varpi ' _ 2 | _ T = \\varpi _ 2 , \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} g _ { 1 } ( \\omega V , \\mathcal { T } _ { U } \\mathcal { B } X ) + g _ { 1 } ( V , \\phi U ) \\eta ( X ) & = g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , \\mathcal { C } X ) , \\pi _ * \\omega V ) - g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , X ) , \\pi _ * \\omega \\phi V ) \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} \\Omega = \\Big \\{ f \\in L ^ 1 _ 2 ( [ 0 , 1 ] _ x \\times \\mathbb { R } ^ 3 _ v ) ~ | & ~ f \\mbox { s a t i s f i e s } ( \\mathcal { A } ) , ( \\mathcal { B } ) , ( \\mathcal { C } ) \\Big \\} \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} \\sup _ { \\tau < t \\le T } | B ^ n _ 0 ( t ) - B ^ n _ 0 ( \\tau ) | \\le \\sup _ { \\tau < t \\le T } | \\zeta ^ n _ 0 ( t ) - \\zeta ^ n _ 0 ( \\tau ) | + 2 \\sum _ { k = 1 } ^ \\infty | k - 2 | \\sup _ { \\tau < t \\le T } | B ^ n _ k ( t ) - B ^ n _ k ( \\tau ) | . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\Phi ( u + n v ) = u + n v \\quad \\textnormal { a n d } \\Phi ( u - n v ) = - u + n v , \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} 1 = \\mathrm { T r } _ { K _ 3 / F } ( \\xi ) = 2 \\xi . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\frac { k - t + q - 2 } { | B | - 2 } \\leq 1 + \\frac { q - 2 } { ( | B | - 1 ) ( | B | - 2 ) } + \\frac { q } { | B | - 2 } \\leq 1 + \\frac { q - 2 } { 1 2 } + \\frac { q } { 3 } = 1 + \\frac { 5 q - 2 } { 1 2 } \\leq \\frac { 5 ( k - t ) } { 1 2 } , \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\Gamma ^ i _ { j k } ( v ) : = \\gamma ^ i _ { j k } ( v ) - \\sum _ { l , m = 1 } ^ n \\frac { g ^ { i l } } { F } ( A _ { l k m } N ^ m _ j + A _ { j l m } N ^ m _ k - A _ { j k m } N ^ m _ l ) ( v ) , v \\in T M \\setminus 0 , \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\det ( ( a - b + c ) B ^ * + ( c - a ) E ) = b ^ 2 - 4 a c ~ . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\int _ M L _ X \\omega \\wedge \\zeta = - \\int _ M \\omega \\wedge L _ X \\zeta \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\delta ( A ) = \\limsup _ { n \\rightarrow \\infty } \\frac { f _ A ( n ) } { n } . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} V = \\begin{pmatrix} \\left ( v ^ { 1 k ' } _ { 1 k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } & \\left ( v ^ { 1 k ' } _ { 2 k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } \\\\ \\left ( v ^ { 2 k ' } _ { 1 k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } & \\left ( v ^ { 2 k ' } _ { 2 k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } \\end{pmatrix} \\in \\mathcal { M } _ { 2 N ^ { 2 } \\times 2 N ^ { 2 } } \\left ( \\mathbb { C } \\right ) . \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ 4 ^ 4 + C \\sum _ { i = 1 } ^ { 4 } \\| v _ i ^ 2 \\| _ { H ^ 1 ( \\Omega ) } ^ 2 \\leq C \\sum _ { i = 1 } ^ { N } \\| v _ i ^ 2 \\| _ { H ^ 1 ( \\Omega ) } ^ 2 ( \\| v _ i \\| _ 2 ^ 2 - 1 ) + C . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} G _ { 1 1 } \\left ( \\tilde { W } \\right ) = \\tilde { W } . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} & \\P \\big ( \\cap _ { j = 1 } ^ k \\{ T _ { s _ { j - 1 } + 1 } \\le t _ { j 1 } , \\ldots , T _ { s _ j } \\le t _ { j \\ell _ j } \\} \\mid \\cap _ { j = 1 } ^ k N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\big ) \\\\ & = \\prod _ { j = 1 } ^ k \\P \\big ( T _ { 1 } \\le t _ { j 1 } - t _ { j - 1 } , \\ldots , T _ { \\ell _ j } \\le t _ { j \\ell _ j } - t _ { j - 1 } \\mid N ( 0 , t _ j - t _ { j - 1 } ] = \\ell _ j \\big ) . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} & \\sim \\sum _ { r _ 2 , \\dots , r _ { m - k + 1 } } L _ { r _ 2 , \\dots , r _ { m - k + 1 } } ( t ) t ^ { ( 2 - \\alpha ) r _ 2 } \\cdots t ^ { ( m - k + 1 - \\alpha ) r _ { m - k + 1 } } \\\\ & = \\sum _ { r _ 2 , \\dots , r _ { m - k + 1 } } L _ { r _ 2 , \\dots , r _ { m - k + 1 } } ( t ) t ^ { 2 r _ 2 + \\cdots + ( m - k + 1 ) r _ { m - k + 1 } - \\alpha \\left ( r _ 2 + \\cdots + r _ { m - k + 1 } \\right ) } \\\\ & = \\sum _ { r _ 2 , \\dots , r _ { m - k + 1 } } L _ { r _ 2 , \\dots , r _ { m - k + 1 } } ( t ) t ^ { m - \\alpha k } , \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} Y ^ N _ i ( t ) = \\xi _ i ^ N + \\int _ 0 ^ t \\int _ U f ( s , Y ^ N _ i ( s ^ - ) , u , \\alpha ^ N _ i ( s ) , m ( s ) ) \\N _ i ^ N ( d s , d u ) i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} d _ * : H ^ 1 ( \\Delta , N ) \\to \\check { H } ^ 2 ( \\Delta , { } ^ 2 G ) \\ \\ , \\ \\ d _ * [ u ] : = [ d u ] \\ . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Vert f _ { t } \\Vert _ { L _ { x } ^ { 2 p } } ^ { 2 p } = 2 p \\langle A _ { t } ^ { \\psi } f _ { t } , f _ { t } ^ { 2 p - 1 } \\rangle _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} 0 = h _ \\mu + \\int - c \\rho d \\mu \\geq h _ \\nu + \\int - c \\rho d \\nu \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} v \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = \\widetilde { g } _ { 0 } \\left ( x , x _ { 0 } \\right ) , \\partial _ { n } v \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = \\widetilde { g } _ { 1 } \\left ( x , x _ { 0 } \\right ) , \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\Psi _ { \\epsilon , \\rho } : = \\widetilde { C } \\sum _ { i = 1 } ^ d \\chi _ { \\rho } ( \\epsilon ^ 2 + | s _ i | _ { H _ i } ^ 2 ) , \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ x | u ( y ) I ( y ) | ~ d y & \\le C \\int _ { - \\infty } ^ x w ^ { - s - r } ( y ) | w ^ s u ( y ) | ~ d y \\\\ & = C \\int _ { - \\infty } ^ x w ^ { - r - \\frac 1 2 ( s - \\frac 1 2 ) } ( y ) w ^ { - \\frac 1 2 ( s + \\frac 1 2 ) } ( y ) | w ^ s u ( y ) | ~ d y \\\\ & \\le C w ^ { - r - \\frac 1 2 ( s - \\frac 1 2 ) } ( x ) \\int _ { \\real } w ^ { - \\frac 1 2 ( s + \\frac 1 2 ) } ( y ) | w ^ s u ( y ) | ~ d y \\\\ & \\le C w ^ { - ( r + \\frac 1 2 ( s - \\frac 1 2 ) ) } ( x ) \\| w ^ { - \\frac 1 2 ( s + \\frac 1 2 ) } \\| _ 2 \\| u \\| _ { L ^ 2 _ s } \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} S _ 1 = \\tfrac { 1 } { 2 } n ( n - 1 ) , T _ 0 = n . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\frac { t - t _ { n , i } } { t _ { n , i + 1 } - t _ { n , i } } ( Y ^ { ( n ) } ( t _ { n , i + 1 } ) - Y ^ { ( n ) } ( t _ { n , i } ) ) , t _ { n , i } \\leq t \\leq t _ { n , i + 1 } . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { m } , \\mathbf { a } ; k ) = 2 ^ q \\cdot [ 2 ^ { k - 1 } + 1 ] - t \\ , \\ , \\ , \\ , k \\Delta ( \\mathbf { m } , \\mathbf { a } ; k ) = C ( \\mathbf { m } , \\mathbf { a } ; k ) , \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} I _ { a ^ + } ^ { 1 - \\gamma } y ( a ) = y _ a , \\gamma = \\alpha + \\beta ( 1 - \\alpha ) , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} \\| f \\| _ { X ^ { \\wedge } } = \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| f _ k \\| _ X \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in X , n \\in \\N \\} . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in \\mathfrak { M } _ m } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) & = \\sum _ { \\chi \\in \\mathfrak { M } _ m } \\ ( \\frac { ( - 1 ) ^ { m / 2 } } { n ^ { m / 2 } } + O _ m \\ ( \\frac 1 { n ^ { m / 2 + 1 } } \\ ) \\ ) ^ 3 \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\chi ( f ) \\\\ & = \\frac { ( - 1 ) ^ { m / 2 } } { n ^ { 3 m / 2 } } \\ ( \\sum _ { \\chi \\in \\mathfrak { M } _ m } \\chi ( f ) \\ ) \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 + O _ m \\ ( \\frac { | \\mathfrak { M } _ m | } { n ^ { 3 m / 2 + 1 } } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} x ^ { ( m ) } ( t , p ) = \\sum _ { i = 1 } ^ m v _ i ( t ) \\Phi _ i ( p ) \\mbox { a n d } y ^ { ( m ) } ( t , p ) = \\sum _ { i = 1 } ^ m w _ i ( t ) \\Phi _ i ( p ) \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} g \\cdot \\pi ( t ) = g \\cdot L a w ( \\xi ) + \\int _ 0 ^ t \\int _ \\Sigma \\int _ A \\Lambda _ s ^ a g ( x ) [ \\widehat { \\gamma } ( s , x ) ] ( d a ) [ \\pi ( t ) ] ( d x ) d s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} ( - D _ x ) ^ { \\ell } \\alpha _ x ( a ) & = ( - D _ x ) ^ { \\ell } \\alpha _ x \\left ( \\sum _ m a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\right ) \\\\ & = ( - D _ x ) ^ { \\ell } \\sum _ m a _ m e ^ { i x \\cdot m } \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\sum _ m a _ m ( - m ) ^ { \\ell } e ^ { i x \\cdot m } \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = ( - \\delta ) ^ { \\ell } \\alpha _ x ( a ) . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{gather*} z ^ { ( p ) } _ m - z _ m \\sim C _ m \\frac { \\Psi _ p ( z _ m ) } { \\Psi _ p ( z _ { k + 1 } ) } \\quad \\\\ C _ m = ( - 1 ) ^ { k - m } \\frac { T _ { 1 , \\ldots , m - 1 , m + 1 , \\ldots , k + 1 } } { T _ { 1 , \\ldots , k } } ( z _ { k + 1 } - z _ m ) \\prod ^ k _ { \\substack { i = 1 \\\\ i \\neq m } } \\frac { z _ { k + 1 } - z _ i } { z _ m - z _ i } . \\end{gather*}"}
-{"id": "6266.png", "formula": "\\begin{align*} \\operatorname { d i v } ( | \\nabla u | ^ { p - 2 } \\nabla u ) = f , f \\in W ^ { 1 , p ' } ( \\Omega , \\mathbb { R } ^ N ) , \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} & x _ 2 - y _ 1 = \\lambda , & & x _ 2 - y _ 2 = u - v + a , & & x _ 2 - y _ 3 = - \\lambda - z _ j - v - 2 a \\\\ & x _ 3 - y _ 1 = - u + v + \\lambda + a , & & x _ 3 - y _ 2 = 2 a , & & x _ 3 - y _ 3 = - \\lambda - z _ j - u - a . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} \\omega _ 0 ( n ) = \\omega ( \\phi ( n ) ) . \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{gather*} K \\coloneqq \\begin{pmatrix} 1 & 0 \\\\ 0 & \\mathrm { S O } ( n ) \\end{pmatrix} , \\textrm { a n d } M \\coloneqq \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\mathrm { S O } ( n - 1 ) & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} . \\end{gather*}"}
-{"id": "8060.png", "formula": "\\begin{align*} x & \\left ( 1 - \\frac { x } { 4 } \\right ) + y \\left ( 1 - \\frac { x } { 2 } - \\frac { y } { 4 } \\right ) + \\left ( 1 - \\frac { x + y } { 2 } \\right ) ^ 2 \\\\ & = x + y - \\frac { x ^ 2 + y ^ 2 } { 4 } - \\frac { x y } { 2 } + 1 - x - y + \\frac { x ^ 2 + 2 x y + y ^ 2 } { 4 } = 1 . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} \\frac { 2 ^ { p - p n / 2 } } { \\Gamma _ { p } ( n / 2 ) } \\mbox { e t r } ( - \\frac { 1 } { 2 } { \\bf T } { \\bf T } ^ { ' } ) \\prod _ { i = 1 } ^ { p } t ^ { n - i } _ { i i } , \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\tilde { { \\cal M } } _ 0 ( i ) = \\bigg \\{ m : \\frac { | { \\cal T } _ { \\bar { X } } ^ n | } { | \\tilde { { \\cal M } } _ 0 | } 2 ^ { ( i - 1 ) } < | \\tilde { \\varphi } _ 0 ^ { - 1 } ( m ) \\cap { \\cal T } _ { \\bar { X } } ^ n | \\le \\frac { | { \\cal T } _ { \\bar { X } } ^ n | } { | \\tilde { { \\cal M } } _ 0 | } 2 ^ i \\bigg \\} \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} \\Lambda ^ T M _ \\lambda ( \\zeta ) \\Lambda = \\sigma _ 1 ( \\varepsilon , \\zeta ) | \\sigma _ 1 ( \\varepsilon , \\zeta ) | \\bar \\Lambda _ 1 ^ 2 + | \\sigma _ 1 ( \\varepsilon , \\zeta ) | ( \\bar \\Lambda ' ) ^ T Q ( \\varepsilon , \\zeta ) \\bar \\Lambda ' , \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\frac { ( \\sqrt { \\mu ^ 2 + \\alpha ^ 2 + \\beta ^ 2 } - \\mu ) ^ k } { \\sqrt { \\mu ^ 2 + \\alpha ^ 2 + \\beta ^ 2 } } = \\bigl ( \\sqrt { \\mu ^ 2 + \\alpha ^ 2 + \\beta ^ 2 } \\bigr ) ^ { k - 1 } \\biggl ( 1 - \\frac { \\mu } { \\sqrt { \\mu ^ 2 + \\alpha ^ 2 + \\beta ^ 2 } } \\biggr ) ^ k , \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\gamma _ m ( 2 , n ) = \\binom { 2 } { n - m + 1 } = \\frac { ( n - m + 1 ) ( n - m ) } { 2 } . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\mu ^ { \\frac { 1 } { 2 } } = \\left [ \\begin{matrix} \\mu _ 1 ^ { \\frac { 1 } { 2 } } \\\\ \\vdots \\\\ \\mu _ k ^ { \\frac { 1 } { 2 } } \\end{matrix} \\right ] \\in \\R ^ k . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{gather*} \\Phi ( \\gamma ) = \\int _ I \\gamma ^ * \\phi , \\end{gather*}"}
-{"id": "6078.png", "formula": "\\begin{align*} A ^ + ( x ) = x A ( x ) + ( 1 - 2 x ) C ( x ) - 1 \\ , . \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{gather*} D _ { q , \\theta } = \\left ( \\frac { 1 } { x _ 1 - x _ 2 } \\circ ( T _ { q , x _ 2 } - T _ { q , x _ 1 } ) \\right ) ^ { \\theta } = \\sum _ { n = 0 } ^ { \\theta } { b _ n ^ { ( \\theta ) } T _ { q , x _ 1 } ^ n T _ { q , x _ 2 } ^ { \\theta - n } } \\end{gather*}"}
-{"id": "3693.png", "formula": "\\begin{gather*} ( a _ i , a _ j ) = 0 = ( b _ i , b _ j ) \\\\ ( a _ i , b _ j ) = \\delta _ { i j } = - ( b _ j , a _ i ) \\end{gather*}"}
-{"id": "200.png", "formula": "\\begin{gather*} f ( x ) = \\sum _ { k \\geq 0 } a _ k ^ d c _ k ( d , x ) , \\sum _ { k \\geq 0 } a _ k ^ d < \\infty , a _ k ^ d \\geq 0 , \\forall \\ , k \\geq 0 , \\end{gather*}"}
-{"id": "3913.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ A \\varphi ( t , a ) [ \\delta _ { \\gamma _ n ( t , X _ { n } ( t ^ - ) ) } - \\widehat { \\gamma } ( t , X ( t ^ - ) ) ] ( d a ) d t = Y _ n + Z _ n , \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} & ~ ( T _ { \\lambda + 0 i + h } - T _ { \\lambda + 0 i } ) \\psi ( x ) \\\\ = & ~ i \\int _ { - \\infty } ^ x e ^ { i \\lambda ( x - y ) } [ e ^ { i h ( x - y ) } - 1 ] u ( y ) \\psi ( y ) ~ d y - ( \\widetilde { G } _ { \\lambda + h } - \\widetilde { G } _ { \\lambda } ) * ( u \\psi ) ( x ) \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} P _ 1 ( z ) = \\sum _ { j \\in S _ { + } } p _ j ( z _ 1 x _ + + z _ 2 x _ { - } ) | x _ j | P _ 2 ( z ) = \\sum _ { j \\in S _ { - } } p _ j ( z _ 1 x _ + + z _ 2 x _ { - } ) | x _ j | \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} & | \\mathcal { M } _ { \\nu } ( f ) - \\mathcal { M } _ { \\nu } ( g ) | \\cr & \\qquad \\leq C _ { \\ell , u } \\Big \\{ | \\rho _ { f } - \\rho _ { g } | + | U _ { f } - U _ { g } | + | \\mathcal { T } _ { f } - \\mathcal { T } _ { g } | \\Big \\} e ^ { - C _ { \\ell , u } | v | ^ 2 } . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} S : = \\lim _ { t \\to - \\infty } \\Delta _ { H _ 1 } ^ { i t } \\Delta _ { H _ 2 } ^ { - i t } = W ^ 1 ( 1 ) W ^ 2 ( - 1 ) , \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} & h ( \\lambda , \\delta , \\mu ) = 0 , \\\\ & c _ 1 \\ge 0 , \\\\ & c _ 2 \\ge 0 , \\\\ & \\left \\{ \\begin{array} { l l } \\frac { \\hat P - c _ 2 } { c _ 1 - c _ 2 } \\ge 0 , & { \\rm i f } \\ c _ 1 > c _ 2 , \\\\ \\frac { \\hat P - c _ 2 } { c _ 1 - c _ 2 } \\le 1 , & { \\rm o t h e r w i s e , } \\end{array} , \\right . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\mathcal { O } ( \\mathbb { A } ^ 1 _ \\mathrm { r i g } ) ^ \\times = k ^ \\times . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} | m _ H - m _ { \\mu _ A \\boxplus \\mu _ B } | & = | m _ { H } m _ { \\mu _ A \\boxplus \\mu _ B } | \\Big | \\frac { 1 } { m _ H ( z ) } - \\frac { 1 } { m _ { \\mu _ A \\boxplus \\mu _ B } ( z ) } \\Big | \\\\ & \\leq C | m _ { H } m _ { \\mu _ A \\boxplus \\mu _ B } | \\ , \\Lambda \\ , \\leq C | m _ { H } - m _ { \\mu _ A \\boxplus \\mu _ B } | \\ , \\Lambda \\ , + C | m _ { \\mu _ A \\boxplus \\mu _ B } | ^ 2 \\ , \\Lambda \\ , . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} M ( X , Y ) ' = X \\odot Y ' . \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} T & = \\frac { ( 3 \\rho T + \\rho | U | ^ 2 ) - | \\rho U | ^ 2 \\rho ^ { - 1 } } { 3 \\rho } \\cr & = \\frac { \\int _ { \\mathbb { R } ^ 3 } f | v | ^ 2 d v - \\Big | \\int _ { \\mathbb { R } ^ 3 } f v d v \\Big | ^ 2 \\Big ( \\int _ { \\mathbb { R } ^ 3 } f d v \\Big ) ^ { - 1 } } { 3 \\int _ { \\mathbb { R } ^ 3 } f d v } . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} d \\rho ^ { ( a , b ) } ( x ) = w ^ { ( a , b ) } ( x ) d x \\Bigr | _ { [ - 1 , 1 ] } \\quad \\quad { \\rm f o r } a \\ge b \\ge 2 \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\mathbf { A } _ { k } & : = \\mathbf { Q } _ { N _ { k } } \\ , , \\\\ Y _ k & : = X ^ 1 _ { N _ k } = X ^ 2 _ { N _ k } \\ , , \\\\ R _ { k + 1 } & : = N _ { k + 1 } - N _ k \\ , , R _ { 0 } : = 0 \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} - ( E _ v ^ * , E ^ * _ w ) = \\frac { \\det _ { \\Gamma \\setminus [ v , w ] } } { \\det _ { \\Gamma } } . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} f = f _ { m } \\circ f _ { e m } \\circ f _ { e } , \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} L _ z = z - t m ( D _ x ) , \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} g : = \\lim _ { n \\to \\infty } { | E _ n | } ^ { 1 / n } , \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d s ^ { 2 } } - \\frac { 1 } { \\gamma } \\frac { d x ^ { i } } { d s } + x ^ { i } = 0 \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} H _ \\omega : = H _ 0 + V _ \\omega \\ , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} & x _ 0 = R \\sinh \\beta , \\\\ & x _ 1 = R \\cosh \\beta \\sin \\varphi _ 1 \\cdots \\sin \\varphi _ { n - 2 } \\sin \\phi , \\\\ & x _ 2 = R \\cosh \\beta \\sin \\varphi _ 1 \\cdots \\sin \\varphi _ { n - 2 } \\cos \\phi , \\\\ & \\vdotswithin { = } \\\\ & x _ { n - 1 } = R \\cosh \\beta \\sin \\varphi _ 1 \\cos \\varphi _ { 2 } , \\\\ & x _ n = R \\cosh \\beta \\cos \\varphi _ 1 , \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} e _ 1 & = ( - u E _ 6 + \\eta ^ { 1 2 } , - 6 u E _ 4 ) , & e _ 2 & = ( u E _ 4 ^ 2 , 6 u E _ 6 + 6 \\eta ^ { 1 2 } ) . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} ( d f ) ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n + 1 } ) & = x _ 1 f ( x _ 2 \\otimes x _ 3 \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i f ( x _ 1 \\otimes \\dots \\otimes x _ i x _ { i + 1 } \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n } ) x _ { n + 1 } , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} c ^ * _ + - c ^ * _ - = & \\inf \\limits _ { \\mu > 0 } \\Phi ^ + ( \\mu ) - \\inf \\limits _ { \\mu > 0 } \\Phi ^ - ( \\mu ) \\\\ \\geq & \\inf \\limits _ { \\mu > 0 } [ \\Phi ^ + ( \\mu ) - \\Phi ^ - ( \\mu ) ] \\\\ = & ( \\alpha - \\beta ) \\inf \\limits _ { \\mu > 0 } \\frac { ( e ^ { \\mu } - e ^ { - \\mu } ) } { \\mu } > 0 . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} A _ n \\equiv _ { \\mathrm { T } } \\Phi _ { A _ n } ( A _ n ) = \\Phi _ { A _ n } ( X _ n ) \\leq _ { \\mathrm { T } } \\Psi ^ n _ { A _ { < n } } ( X _ { < n } ) \\leq A _ { < n } , \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n c _ { i , j } \\equiv 0 \\bmod 2 \\qquad 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} H ^ \\tau [ u ] = \\int _ { \\{ u < 1 \\} } u ( x ) \\log ( u ( x ) ) d x + \\int _ { \\{ 1 \\leq u < \\tau ^ { - \\alpha } \\} } u ( x ) \\log ( u ( x ) ) d x . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\frac { y _ { k + 1 } - y _ k } { \\phi _ 2 ( c , h ) } = - c y _ k \\Longrightarrow y _ k = y _ 0 e ^ { - c h k } , \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} D _ k = \\left \\{ ( u _ 1 , \\ldots , u _ k ) : u _ i \\in G _ i , \\ { \\rm f o r \\ a l l } \\ 1 \\leq i \\leq k \\right \\} . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} \\frac { r ^ { 4 } } { t ^ { \\left ( a + 2 \\right ) } } = . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} e ^ { - \\frac { i \\lambda } { 2 } x ^ 2 } = \\frac { \\partial _ x \\bigl [ x e ^ { - \\frac { i \\lambda } { 2 } x ^ 2 } \\bigr ] } { 1 - i \\lambda x ^ 2 } \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} H _ 2 ( f ) = - \\log \\sum _ { x \\in G } \\P ( f = x ) ^ 2 . \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\| \\varphi - z _ n ^ M \\| _ \\infty = O \\left ( \\max \\left \\{ \\tilde { h } ^ d , h ^ { 3 r } \\right \\} \\right ) , \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} - \\triangle u + W ( x ) \\cdot \\nabla u + V ( x ) u = 0 \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} F _ { s + 1 } \\left ( \\xi \\right ) = - { \\dfrac { 1 } { 2 } } \\frac { d F _ { s } \\left ( \\xi \\right ) } { d \\xi } - { \\dfrac { 1 } { 2 } } \\sum \\limits _ { j = 1 } ^ { s - 1 } { F _ { j } \\left ( \\xi \\right ) F _ { s - j } \\left ( \\xi \\right ) } \\ \\left ( { s = 2 , 3 , 4 \\cdots } \\right ) . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} S _ Q ( q ^ i , \\dot { q } ^ i , \\delta q ^ i , \\delta \\dot { q } ^ i ) = ( q ^ i , \\delta q ^ i , \\dot { q } ^ i , \\delta \\dot { q } ^ i ) \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\widehat \\mu ( \\R ) = \\int _ \\R x ^ 2 \\ , \\dd \\mu ( x ) - \\Big ( \\int _ \\R x \\ , \\dd \\mu ( x ) \\Big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\omega \\Bigl ( \\omega \\bigl ( \\omega ( A , B ) , B ^ { \\sim } \\bigr ) , B \\Bigr ) = \\omega ( A , B ) , \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + x ^ { i } \\frac { \\omega \\left ( t \\right ) } { r ^ { 2 } } = 0 \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} \\chi ' = \\begin{pmatrix} 1 & 2 & 5 \\\\ 3 & 6 & 7 \\\\ 4 & 8 & 9 \\end{pmatrix} , \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { f ( n ) } { n } = \\inf _ { n \\in \\N } \\frac { f ( n ) } { n } . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Lambda _ { n , j , r } ^ { - } = r e ^ { - \\lambda _ { n , j } ^ { ( 1 ) } / 2 } , \\\\ \\Lambda _ { n , j , r } ^ { + } = r e ^ { \\lambda _ { n , j } ^ { ( 1 ) } / 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} g ( \\alpha _ i ) = \\alpha _ j , \\ \\ \\ g ( \\sqrt { \\alpha _ i } ) = \\pm \\sqrt { \\alpha _ j } \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\mathcal { W } = \\left ( Z _ { 1 } , Z _ { 2 } \\odot Z \\right ) ^ { t } , \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} R _ 1 [ u , v , w ] = \\sum \\limits _ N C [ u , \\partial _ x ^ { - 1 } v _ N , w _ { \\leq N } ] , R _ 2 [ u , v , w ] = \\sum \\limits _ N C [ u , \\partial _ x ^ { - 1 } v _ N , w _ { > N } ] . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} ( \\ker \\pi _ * ) ^ \\perp = \\omega D _ 2 \\oplus \\mu , \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\phi _ t ( \\bar { x } , \\bar { t } ) \\geq \\psi _ t ^ \\epsilon ( \\bar { x } , \\bar { t } ) = \\theta _ t ( \\bar { x } , \\bar { t } ) - \\delta > \\psi _ t ( \\bar { x } , \\bar { t } ) - \\sigma / 3 > B _ { k + 1 } + \\sigma / 3 . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\frac { d } { d t } \\boldsymbol { u } = \\mathbf { R } ( \\boldsymbol { u } ) , \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} | \\widetilde { \\beta } _ n ( \\widetilde { \\Theta } ) | = O _ P \\left ( \\sqrt { n } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } / 2 } \\epsilon _ n ^ { 1 / 2 } \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l J _ { n , l } / 2 } \\right ) . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} C _ { 1 } ( d , p ) : = \\begin{cases} C ( d , p ) & 1 < p \\leq \\frac { 3 } { 2 } \\\\ C ^ { \\frac { 3 } { p } ( 2 - p ) } \\left ( d , \\frac { 3 } { 2 } \\right ) & \\frac { 3 } { 2 } < p \\leq 2 \\\\ C ^ { \\frac { 3 } { p ^ { \\prime } } ( 2 - p ^ { \\prime } ) } ( d , \\frac { 3 } { 2 } ) & 2 \\leq p < 3 \\\\ C ( d , p ^ { \\prime } ) & 3 \\leq p < \\infty \\end{cases} \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} ( x , \\sigma ) ( y , \\tau ) = ( x ( \\sigma \\circ y ) , \\sigma \\tau ) \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ | \\hat { \\mathbb { E } } _ { \\tau + } [ X _ n ] - \\hat { \\mathbb { E } } _ { \\tau + } [ X _ m ] | ] \\leq \\hat { \\mathbb { E } } [ \\hat { \\mathbb { E } } _ { \\tau + } [ | X _ n - X _ m | ] ] = \\hat { \\mathbb { E } } [ | X _ n - X _ m | ] \\rightarrow 0 , \\ \\ \\ \\ \\ n , m \\rightarrow \\infty . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\Delta _ j ( z ) = \\psi _ { 1 , j } ( z ) \\big [ F ( z ) - G _ { j + 1 , p } ( z ) \\big ] = \\psi _ { 1 , p } ( z ) F [ z , \\xi _ { j + 1 } , \\ldots , \\xi _ p ] , j = 0 , 1 , \\ldots \\ . \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} E = \\{ ( q , p , \\dot { q } , \\dot { p } ) \\in T T ^ { * } Q , q ^ 2 + p ^ 2 + ( \\dot { q } + 1 ) ^ 2 + \\dot { p } = k \\} \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\left [ u _ n , u _ n + 2 \\sum _ { i = 1 } ^ n b _ { i , 1 } \\right ] \\subseteq A . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} T \\gamma \\left ( X _ { \\sigma } ^ { \\gamma } \\right ) = \\sigma ^ i \\left ( \\frac { \\partial } { \\partial q ^ i } + \\frac { \\partial \\gamma _ j } { \\partial q ^ i } \\frac { \\partial } { \\partial p _ j } \\right ) \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} h _ { k } ( L ) = \\left \\{ \\begin{array} { c c } 0 , & T ( L ) \\notin \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\\\ q ^ { 2 \\sum _ { i = 1 } ^ { k } l _ { k i } - \\sum _ { i = 1 } ^ { k - 1 } l _ { k - 1 , i } - \\sum _ { i = 1 } ^ { k + 1 } l _ { k + 1 , i } - 1 } , & T ( L ) \\in \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} c ^ k = \\frac { a _ 0 } { \\rho _ k } T \\left ( \\lambda _ k \\right ) * f , \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} I _ \\lambda ( \\zeta , \\bar { \\Lambda } ) & = k a _ 0 + a _ 1 \\Big [ - \\sigma _ 1 ( \\varepsilon , \\zeta ) ^ 2 \\bar \\Lambda _ 1 ^ 2 + | \\sigma _ 1 ( \\varepsilon , \\zeta ) | ( \\bar \\Lambda ' ) ^ T Q ( \\varepsilon , \\zeta ) \\bar \\Lambda ' \\Bigr ] \\\\ & + \\sigma _ 1 ( \\varepsilon , \\zeta ) ^ 2 \\ , \\mathcal P o l y _ 4 ( \\varepsilon , \\zeta , \\bar \\Lambda ) + \\theta _ \\lambda ( \\zeta , \\bar \\Lambda ) , \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} \\beta = \\sum _ { i = 0 } ^ { 2 n - 1 } { a _ i 3 ^ i } \\textup { w h e r e } a _ i = \\begin{cases} 1 & i \\textup { e v e n } \\\\ 2 & i \\textup { o d d } \\\\ \\end{cases} \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\varphi _ { j , k } ( x ) = \\exp \\left [ - \\frac { \\lambda _ { g } ( j , y _ { j } ) } { 2 } \\| x \\| ^ { 2 } \\right ] \\tilde { \\varphi } _ { j , k } ( x ) . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { \\ast } & = \\sup _ { y \\in \\Omega _ \\epsilon } \\Bigl ( \\omega ( y ) ^ { - \\nu } \\ , \\vert f ( y ) \\vert + \\omega ( y ) ^ { - \\nu - 1 } \\ , \\vert \\nabla f ( y ) \\vert \\Bigr ) \\\\ \\Vert f \\Vert _ { \\ast \\ast } & = \\sup _ { y \\in \\Omega _ \\epsilon } \\omega ( y ) ^ { - ( 2 + \\nu ) } \\ , \\vert f ( y ) \\vert , \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} & u _ \\lambda ( x , y ) = \\mathbb { E } \\int _ 0 ^ \\infty \\exp ( - \\lambda t ) g ( X _ t , Y _ t ) \\d t , v _ \\mu ( x , y ) = \\mathbb { E } \\int _ 0 ^ \\infty \\exp ( - \\mu t ) \\psi ( X _ t , Y _ t ) \\d t \\\\ & w _ { \\lambda + \\mu } ( x , y ) = \\mathbb { E } \\int _ 0 ^ \\infty \\exp ( - ( \\lambda + \\mu ) t ) \\frac { \\partial u _ \\lambda } { \\partial y } ( X _ t , Y _ t ) \\frac { \\partial u _ \\mu } { \\partial y } ( X _ t , Y _ t ) \\d t \\\\ \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\varepsilon _ n ^ { - \\nu } \\ , \\sum _ { j = 1 } ^ 4 c _ { i j } ^ n \\ , w _ { \\mu _ { i , n } , \\zeta _ { i , n } } ^ 4 \\ , Z _ { i j } ^ n = o ( 1 ) \\varepsilon _ n \\rightarrow 0 , \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} F _ n ( x ) = \\log \\lVert A ^ n ( x ) P ( x ) \\rVert , x \\in M . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} w ( Y _ { n - 1 } ) = \\prod _ { i = 1 } ^ { n - 1 } ( 1 + x _ i ) \\cdot \\prod _ { i = n + 1 } ^ { 2 n - 1 } ( 1 + x _ i ) \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { c } } = \\sigma _ { \\mathrm { a b } } = \\sigma _ { \\mathrm { h o l } } , \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} h ^ { p , q } ( X ) = f ^ { d - p , q } ( Y , w ) \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\omega _ { a , b } ( [ x , y ] , z ) & = \\omega _ { a , b } ( x , [ y , z ] ) = - ( - 1 ) ^ { | y | | z | } \\omega _ { a , b } ( x , [ z , y ] ) \\\\ & = - ( - 1 ) ^ { | y | | z | } \\omega _ { a , b } ( [ x , z ] , y ) = ( - 1 ) ^ { | y | | z | + | x | | z | } \\omega _ { a , b } ( z , [ x , y ] ) . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} p _ { N , N } ^ { P _ 1 } = \\frac { 1 } { ( \\Gamma ( L / 2 ) ) ^ N } \\frac { G ( ( 2 L + N ) / 2 ) } { G ( L ) } \\frac { G ( ( N + L ) / 2 ) } { G ( L / 2 ) } \\frac { G ( ( N + L + 1 ) / 2 ) } { G ( ( L + 1 ) / 2 ) } \\frac { G ( ( 2 L + N - 1 ) / 2 ) } { G ( N + L - 1 / 2 ) } . \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} & [ f ] _ { Q } ^ { ( m , \\frac { m } { 2 } ) } = \\sum _ { 2 s + | k | = [ m ] } [ \\partial _ t ^ { s } \\partial _ x ^ { k } f ] ^ { ( \\mu , \\frac { \\mu } { 2 } ) } _ { Q } , \\\\ & | f | _ { Q } ^ { ( m , \\frac { m } { 2 } ) } = \\sum _ { 2 s + | k | \\leq [ m ] } | | \\partial _ t ^ { s } \\partial _ x ^ { k } f | | _ { L ^ { \\infty } ( Q ) } + [ f ] _ { Q } ^ { ( m , \\frac { m } { 2 } ) } . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\frac { \\partial \\bar { q } ( t ) } { \\partial t } = - \\frac { 1 } { \\Delta x } ( \\tilde { f } _ { i + 1 / 2 } - \\tilde { f } _ { i - 1 / 2 } ) , \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\sum _ { d ' = 0 } ^ { 2 ^ { \\alpha - 3 } - 1 } e \\bigg ( \\frac { h d ' } { 2 ^ { \\alpha - 3 } } \\bigg ) , \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\int _ M u \\ , S ( \\sigma J , f ) \\ , f ^ { - 2 m - 1 } \\omega ^ m = \\int _ M u \\circ \\sigma ^ { - 1 } \\ , S _ { J , f } \\ , f ^ { - 2 m - 1 } \\omega ^ m . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} F '' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) = \\int _ \\R \\frac { \\dd \\widehat \\mu _ \\beta ( x ) } { ( x - \\omega _ \\alpha ( E _ - ) ) ^ 3 } > 0 \\ , , F '' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) = \\int _ \\R \\frac { \\dd \\widehat \\mu _ \\alpha ( x ) } { ( x - \\omega _ \\beta ( E _ - ) ) ^ 3 } > 0 \\ , . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\varphi = \\Big ( A _ 0 \\circ \\sigma _ 2 \\circ A _ 0 ^ { - 1 } \\Big ) \\circ \\Big ( A _ 1 \\circ \\sigma _ 2 \\circ A _ 1 ^ { - 1 } \\Big ) \\circ \\ldots \\circ \\Big ( A _ k \\circ \\sigma _ 2 \\circ A _ k ^ { - 1 } \\Big ) \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} Q _ 0 = \\mathcal { E } _ 0 \\cup \\bigcup _ i Q _ 0 ^ * ( \\epsilon _ i ) \\qquad \\hbox { w i t h } \\sigma ( \\mathcal { E } _ 0 ) = 0 . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} J _ { d , e } ( x ) & = J ' _ { d , e } ( x ) + \\sum _ { j = 1 } ^ { d + e - 1 } \\frac { x ^ { d + e + 3 } } { ( 1 - x ) ^ { j + e } } ( C ( x ) - 1 ) \\ , . \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} G _ 3 ( x ) = \\frac { x ^ 3 } { ( 1 - x ) ^ 5 } \\ , , \\textrm { a n d } \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\frac { r ^ { \\left ( 2 - n \\right ) } } { t ^ { \\left ( a + 2 \\right ) } } = \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} S ( x , y ) = 2 \\sum _ { j = 0 } ^ { N / 2 - 1 } \\frac { 1 } { h _ j } \\left ( q _ { 2 j } ( x ) \\tau _ { 2 j + 1 } ( y ) - q _ { 2 j + 1 } ( y ) \\tau _ { 2 j } ( x ) \\right ) . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\jmath ^ { \\ , '' } _ p \\circ \\eta ' _ a \\circ \\eta _ a \\ , = \\ , \\beta ^ 1 _ p \\circ \\eta ' _ a \\circ \\eta _ a \\circ \\jmath _ p \\ , , \\ \\ \\ , \\ \\ \\ \\jmath ^ { \\ , '' } _ p \\circ \\{ \\eta ' \\circ \\eta \\} _ a \\ , = \\ , \\beta ^ 2 _ p \\circ \\{ \\eta ' \\circ \\eta \\} _ a \\circ \\jmath _ p \\ , , \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} f ^ * ( l ) \\ , : = \\ , \\bar \\gamma ( l , l ^ { - 1 } ) ^ * \\ , \\bar \\nu _ l ( f ( l ^ { - 1 } ) ) ^ * \\ . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{gather*} g = 2 \\omega ^ 0 \\circ \\omega ^ n - \\varepsilon _ { a b } \\omega ^ a \\circ \\omega ^ b , \\end{gather*}"}
-{"id": "6930.png", "formula": "\\begin{align*} f | S = & \\sum a _ { \\underline { i } } \\phi _ { 1 } ^ { i _ 1 } \\phi _ { 2 } ^ { i _ 2 } \\phi _ { 3 } ^ { i _ { 3 } } \\phi _ { 4 } ^ { i _ 4 } \\\\ = & \\sum a _ { \\underline { i } } ( x \\widetilde { \\phi } _ { 3 } + x ' \\widetilde { \\phi } _ { 4 } ) ^ { i _ { 1 } } ( y \\widetilde { \\phi } _ { 3 } + y ' \\widetilde { \\phi } _ { 4 } ) ^ { i _ { 2 } } \\widetilde { \\phi } _ { 3 } ^ { i _ 3 } \\widetilde { \\phi } _ { 4 } ^ { i _ 4 } . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} h ^ { p , q } ( \\hat { X } _ \\Delta ) = \\begin{cases} 1 & \\mathrm { i f } ( p , q ) = ( 0 , 0 ) , ( 3 , 3 ) \\\\ \\ell ( \\Delta ) - 4 & \\mathrm { i f } ( p , q ) = ( 1 , 1 ) , ( 2 , 2 ) \\\\ 0 & \\mathrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} y = B x . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\widehat { J } _ 1 ( \\tau ) & : = \\cos ( \\tau \\widehat { \\mathcal { A } } ^ { 1 / 2 } ) - \\cos ( \\tau ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) , \\\\ \\widehat { J } _ 2 ( \\tau ) & : = \\widehat { \\mathcal { A } } ^ { - 1 / 2 } \\sin ( \\tau \\widehat { \\mathcal { A } } ^ { 1 / 2 } ) - ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} E \\left [ \\left . | X ^ n ( \\tau + h ) - X ^ n ( \\tau ) | ^ 2 \\right | \\mathcal { F } _ \\tau \\right ] = O ( h ) \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} z \\frac { d } { d z } { } _ { a } \\mathtt { B } _ { b , p , c } ( z ) = \\frac { z } { a } { } _ { a } \\mathtt { B } _ { b , p - 1 , c } ( z ) - \\left ( \\frac { 2 p + b - 1 } { a } - p \\right ) { } _ { a } \\mathtt { B } _ { b , p , c } ( z ) , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\frac { \\sin ( \\pi c _ r ( 1 - \\delta ) v ) } { \\pi v } ( 1 - v ) ^ { \\ell ^ 2 } \\ , d v = I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} e ( G ) & \\ge \\sum _ { i = 1 } ^ { k } \\sum _ { v \\in V _ i ^ - \\setminus U } d ( v ) \\ge \\sum _ { i = 1 } ^ { k } | V _ i ^ - \\setminus U | d ( v _ i ) \\ge ( n - | V _ i ^ + | - | U | ) \\sum _ { i = 1 } ^ { k } d ( v _ i ) \\\\ & \\ge \\alpha ( k , r ) \\left ( n - \\frac { 2 \\alpha ( k , r ) } { d } n - C _ { k , d } \\right ) = \\alpha ( k , r ) n - \\left ( \\frac { 2 \\alpha ( k , r ) ^ 2 } { d } + \\frac { \\alpha ( k , r ) C _ { k , d } } { n } \\right ) n \\ge \\alpha ( k , r ) n - \\varepsilon n \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} X ( t ) = \\xi + \\int _ 0 ^ t \\int _ U \\int _ A f ( s , X ( s ^ - ) , u , a , m ( s ) ) \\N _ \\rho ( d s , d u , d a ) \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} T = \\left \\{ \\sum _ { j = 0 } ^ 3 t _ j p _ j : \\sum _ { j = 0 } ^ 3 t _ j = 1 , t _ j \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} m = \\max _ { \\overline { \\Omega _ { d } } } \\xi \\left ( x \\right ) \\Rightarrow \\max _ { \\overline { \\Omega _ { d } } } \\varphi _ { \\lambda } \\left ( x \\right ) = e ^ { \\lambda m } . \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} C ' = C ' ( N , \\gamma , \\omega _ 0 , X , \\| \\phi ( \\cdot , 0 ) \\| _ { C ^ 3 ( B _ r ( p ) ) } , \\| \\widetilde { \\eta } \\| _ { C ^ 1 ( B _ r ( p ) ) } ) \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} u ^ { L } _ { i + \\frac { 1 } { 2 } } = q _ i ( x _ { i + \\frac { 1 } { 2 } } ) \\ \\ { \\rm a n d } \\ \\ u ^ { R } _ { i - \\frac { 1 } { 2 } } = q _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} S _ k ( X ^ { \\boxplus \\tau } ) : = \\left ( \\coprod _ { p \\in X } S _ k ( U _ p ^ { \\boxplus \\tau } ) \\cap \\left ( ( s _ p ^ { \\boxplus \\tau } ) ^ { - 1 } ( 0 ) / \\Gamma _ { p } \\right ) \\right ) / \\sim . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} R _ { 0 0 } & = - \\frac { a '' } { a } - 2 n \\frac { b '' } { b } \\\\ R _ { 1 1 } & = - \\frac { a '' } { a } + 2 n \\left ( \\frac { a ^ 2 } { b ^ 4 } - \\frac { a ' b ' } { a b } \\right ) \\\\ R _ { i i } & = - \\frac { b '' } { b } + \\frac { 2 n + 2 } { b ^ 2 } - 2 \\frac { a ^ 2 } { b ^ 4 } - \\frac { a ' b ' } { a b } - ( 2 n - 1 ) \\left ( \\frac { b ' } { b } \\right ) ^ 2 \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\alpha ' _ { \\phi ( \\gamma ( l , m ) ) } \\circ \\eta \\ = \\ \\alpha ' _ { \\gamma ' ( l , m ) } \\circ \\eta \\ \\ \\ , \\ \\ \\ l , m \\in \\Pi \\ ; \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\Vert \\tt \\Vert _ { H ^ { m + 1 } } \\le C \\Vert f \\Vert _ { H ^ { m - 1 } } \\le C \\Vert \\Delta \\tt \\Vert _ { m - 1 , D } = C \\Vert \\tt \\Vert _ { m + 1 , D } \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} 1 \\otimes \\lambda a = \\overline { \\beta } ( a ) \\otimes 1 \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} q n + r = p r x + q ( n - r y ) \\in \\{ p m + q n \\ , : \\ , m , n \\in \\mathbb N \\} . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\varphi ^ { 2 } = - I + \\eta \\otimes \\xi , \\ \\ \\ \\eta ( \\xi ) = 1 \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\tilde { \\theta } ^ \\ell = \\theta ^ \\ell + \\left \\{ \\hat m ( \\eta ( \\mathbf y ) ) - \\hat m ( \\eta ( \\mathbf z ^ \\ell ) ) \\right \\} , \\ell = 1 , \\dots , L . \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} A _ t = \\nu ( [ 0 , t ] \\times J ) + t , \\ ; \\ ; \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} X \\odot Y = \\{ g h \\colon g \\in X , \\ h \\in Y \\} \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\hat { \\mu } _ { \\mathrm { e s s } } ( D , g ) = g ( \\eta _ 0 ) = \\log \\max \\{ a _ 0 , \\ldots , a _ d \\} , \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty y ^ { s + \\frac { k - 1 } { 2 } } \\sum _ { m \\in \\mathbb { Z } } r _ { 2 k + 1 } ( m ^ 2 + h ) e ^ { - 2 \\pi y ( 2 m ^ 2 + h ) } \\frac { d y } { y } = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 2 k + 1 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s + \\frac { k - 1 } { 2 } } } \\frac { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} k = \\left \\{ \\frac { C _ 1 ( A _ 1 t + K t + 1 ) \\rho ^ { \\frac { 2 n } { n + 2 } } e ^ { C ( \\rho + t K ) } b ^ { \\beta } } { K ^ { \\frac { n } { n + 2 } } V _ { x _ 0 } \\left ( \\rho / \\sqrt { K } \\right ) ^ { \\frac { 2 } { n + 2 } } \\min ( t , \\rho ^ 2 / K ) } \\right \\} ^ { \\frac { n + 2 } { 2 p } } \\left ( 1 + \\int _ { t / 4 } ^ t \\int _ { B _ { 2 \\rho / \\sqrt { K } } } v ^ p \\ , d v _ 0 \\ , d t \\right ) ^ { \\frac { n + 4 } { 2 p } } . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} D [ f ] = 0 \\Longleftrightarrow f = f _ { \\infty } . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\sum _ { \\substack { N ( c ) \\leq X / N ( d ^ 2 ) \\\\ ( c , 1 + i ) = 1 } } 1 = \\frac { \\pi X } { 2 N ( d ^ 2 ) } + O \\left ( \\left ( \\frac { X } { N ( d ^ 2 ) } \\right ) ^ { \\theta } \\right ) . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} { \\bar { Y } } ^ { n } ( t ) & = { \\bar { X } } _ { 0 } ^ { n } ( 0 ) + \\sum _ { k = 0 } ^ { \\infty } ( k - 2 ) { \\bar { B } } _ { k } ^ { n } ( t ) , \\\\ { \\bar { X } } _ { 0 } ^ { n } ( t ) & = { \\bar { Y } } ^ { n } ( t ) + { \\bar { \\eta } } ^ { n } ( t ) , \\\\ { \\bar { X } } _ { k } ^ { n } ( t ) & = { \\bar { X } } _ { k } ^ { n } ( 0 ) - { \\bar { B } } _ { k } ^ { n } ( t ) , k \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\| \\Delta ( w _ m - w ) \\| _ { L ^ 2 } & = \\| \\beta ( w ) - \\beta _ m ( w _ m ) \\| _ { L ^ 2 } \\\\ & \\le \\| \\beta ( w ) - \\beta ( w _ m ) \\| _ { L ^ 2 } + \\| \\beta ( w _ m ) - \\beta _ m ( w _ m ) \\| _ { L ^ 2 } \\\\ & \\le \\| \\beta ' \\| _ { L ^ \\infty } \\| w _ m - w \\| _ { L ^ 2 } + \\| \\beta _ m - \\beta \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( \\tilde { u } _ 2 ( t ) - \\epsilon y _ 2 ( t ) ) ^ T y _ 2 ( t ) \\ , d t = \\int _ 0 ^ \\infty u _ 2 ( t ) ^ T y _ 2 ( t ) \\ , d t \\geq 0 . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{gather*} \\lim _ { n \\rightarrow \\infty } { T _ { q , x } ^ s f _ n ( x ) } = \\lim _ { n \\rightarrow \\infty } { f _ n ( q ^ s x ) } = f ( q ^ s x ) = T _ { q , x } ^ s f ( x ) \\end{gather*}"}
-{"id": "3887.png", "formula": "\\begin{align*} v ( t , x ) = J ( t , x , \\gamma _ m , m ) = V _ m ( t , x ) \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} & ~ \\Phi ( u _ i ( ( L _ i / R ) ^ 3 x ) + \\eta ( R , 1 , t ; x ) ) \\\\ \\leq & ~ \\Phi ( u _ i ) ( R / L _ i ) ^ { 3 N } - \\left [ C _ 2 R ^ { 3 N } - \\left ( \\frac { 1 } { 2 } b C _ 3 L ^ { 3 N - 5 } _ i \\int _ { \\mathbb { R } ^ N } | \\nabla u _ i | ^ 2 \\right ) ( R / L _ i ) ^ { 6 N - 1 1 } \\right ] t ^ N . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} w _ 0 ( t ) + w _ 1 ( t ) \\ & = \\ - \\ s _ 1 ^ \\R \\ 2 ^ { - \\frac 3 4 } \\ \\pi ^ { - \\frac 1 2 } \\vert t \\vert ^ { - \\frac 1 2 } \\ e ^ { - 2 \\sqrt 2 \\vert t \\vert } + O ( \\vert t \\vert ^ { - 1 } e ^ { - 2 \\sqrt 2 \\vert t \\vert } ) \\\\ w _ 0 ( t ) - w _ 1 ( t ) \\ & = \\ \\ \\ s _ 2 ^ \\R \\ 2 ^ { - \\frac 3 2 } \\ \\pi ^ { - \\frac 1 2 } \\vert t \\vert ^ { - \\frac 1 2 } \\ e ^ { - 4 \\vert t \\vert } + O ( \\vert t \\vert ^ { - 1 } e ^ { - 4 \\vert t \\vert } ) \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\ell _ { \\omega o } \\ , : = \\ , p _ { a \\omega } * ( \\omega o ) * p _ { o a } : a \\to a \\ \\ \\ \\Rightarrow \\ \\ \\ \\ell _ { \\xi \\omega } * \\ell _ { \\omega o } = \\ell _ { \\xi o } \\ , \\ \\forall o \\leq \\omega \\leq \\xi \\ . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} D _ { F , C _ 2 } ^ - ( s ) = A ( s ) + B ( s ) , \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} u _ k ^ { p - 1 } \\eta ( a \\nabla u _ k , \\nabla \\eta ) = \\frac { 2 } { p } u _ k ^ { p / 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla \\eta ) - \\frac { 2 } { p } u _ k ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} K _ { \\nu } \\left ( { \\nu z e } ^ { - \\pi i } \\right ) = i \\left ( { \\frac { \\pi } { 2 \\nu } } \\right ) ^ { 1 / 2 } \\frac { 1 } { \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 4 } } \\exp \\left \\{ { \\nu \\xi + \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { \\tilde { { E } } _ { s } \\left ( p \\right ) } { \\nu ^ { s } } } } \\right \\} \\left \\{ { 1 + \\eta _ { n , 3 } \\left ( { \\nu , \\xi } \\right ) } \\right \\} . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | = \\frac { ( m - 1 ) ( m - 3 ) \\cdots 1 } { ( n - m + 1 ) ( n - m + 3 ) \\cdots n } \\frac { n ! } { n ^ n } \\approx \\binom { n } { m } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} x = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } w V ( f _ i ) , \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} I _ { u , n } = \\bigcup _ { j = 1 } ^ { q + 1 } I _ { u , n } ^ { ( j ) } . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} u ^ e ( S ^ e \\otimes I ) \\Delta ^ e m ( a ) & = u ^ e ( S ^ e \\otimes I ) ( m \\otimes m ) \\Delta ( a ) \\\\ & = S ^ e m ( a _ { ( 1 ) } ) m ( a _ { ( 2 ) } ) = m S ( a _ { ( 1 ) } ) m ( a _ { ( 2 ) } ) \\\\ & = m ( S ( a _ { ( 1 ) } ) a _ { ( 2 ) } ) = m ( \\eta \\varepsilon ( a ) ) = \\varepsilon ( a ) 1 _ { A ^ e } \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} u _ n ( z , w ) : = v _ { N _ n } ( z ) + | p _ n ( w ) | . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} b _ m & = \\sum _ { 1 \\leqslant i _ 1 < i _ 2 < \\cdots < i _ m \\leqslant d } i _ 1 ( i _ 2 - 1 ) ( i _ 3 - 2 ) \\cdots ( i _ m - m + 1 ) \\\\ & \\qquad \\qquad { } \\times ( d + 1 ) ^ { i _ 1 - 1 } d ^ { i _ 2 - i _ 1 - 1 } ( d - 1 ) ^ { i _ 3 - i _ 2 - 1 } \\cdots ( d - m + 1 ) ^ { d - i _ m } \\\\ & = \\sum _ { 1 \\leqslant k _ 1 \\leqslant k _ 2 \\leqslant \\cdots \\leqslant k _ m \\leqslant d - m + 1 } \\prod _ { i = 0 } ^ { m } k _ { i } ( d + 1 - i ) ^ { k _ { i + 1 } - k _ { i } } , \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\Delta _ { e _ n } \\mid _ { \\Lambda _ n } = \\nabla . \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} D \\left ( \\ln \\omega \\right ) _ { , t } + 2 D _ { , t } + \\frac { 1 } { \\omega } T _ { , t t } & = 0 \\\\ 2 T _ { , t } - D _ { , t t } & = 0 . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\widehat { \\phi _ { Y } } ( 1 ) r _ d ( G ) X = r _ d ( G ) X + O ( X Y ^ { - 1 } ) . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\mathcal { R } & : = \\mathcal { R } ^ { \\geq } \\cup \\mathcal { R } ^ { > } \\cup \\mathcal { R } ^ { 0 } , \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\bar J _ \\lambda ( v ) = \\frac { 1 } { 2 } \\int _ { \\Omega _ \\varepsilon } \\vert \\nabla v \\vert ^ 2 - \\varepsilon ^ 2 \\ , \\frac { \\lambda } { 2 } \\int _ { \\Omega _ \\varepsilon } v ^ 2 - \\frac { 1 } { 6 } \\int _ { \\Omega _ \\varepsilon } v ^ 6 , \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla \\widehat v \\nabla \\varphi + \\int _ \\Omega \\beta ' ( w ) \\widehat v \\varphi = 0 \\forall \\varphi \\in H _ 0 ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} S ( x , y ) = \\int _ { - 1 } ^ { 1 } \\ , ( x - v ) \\mathrm { s g n } ( y - v ) w _ r ^ { ( m ) } ( x ) w _ r ^ { ( m ) } ( v ) \\sum _ { j = 0 } ^ { N - 2 } \\bigg ( \\prod _ { i = 1 } ^ m \\frac { ( L _ i + j ) ! } { L _ i ! j ! } \\bigg ) ( x v ) ^ j \\mathrm { d } v , \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\varepsilon _ n ^ { - \\nu } \\vert \\phi _ n ( y ) \\vert \\leq \\eta _ n ^ 2 \\vert y - \\zeta _ { i , n } ^ \\prime \\vert = \\delta / 4 \\varepsilon _ n , \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathbb { C } T ^ * X = T ^ { * 1 , 0 } X \\oplus T ^ { * 0 , 1 } X \\oplus \\left \\{ \\lambda \\omega _ 0 : \\lambda \\in \\mathbb { C } \\right \\} , \\\\ \\mathbb { C } T X = T ^ { 1 , 0 } X \\oplus T ^ { 0 , 1 } X \\oplus \\left \\{ \\lambda T : \\lambda \\in \\mathbb { C } \\right \\} . \\end{array} \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\| ( f _ n ) _ { n \\geq 0 } \\| _ { Q _ q ^ p } : = \\Bigl ( \\mathbb E \\Bigl \\| \\Bigl ( \\sum _ { n = 0 } ^ { \\infty } | f _ n | ^ 2 \\Bigr ) ^ { \\frac 1 2 } \\Bigr \\| ^ { p } _ { L ^ q ( S ) } \\Bigr ) ^ { \\frac 1 p } < \\infty . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\mathcal { C } ^ k ( e ) : = \\left \\{ g \\in \\mathcal { C } ^ k ( \\mathring { e } ) : \\ , \\nabla ^ r g \\mathring { e } r \\leq k \\right \\} . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} y _ \\varepsilon ( t ) = \\varepsilon \\ , x _ \\infty \\left ( \\frac { t } { \\varepsilon ^ { \\frac { 2 + \\alpha } { 2 } } } \\right ) , t \\in \\mathbb { R } ; \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ \\varphi ( 2 B _ { t _ 1 \\wedge \\tau _ k } - B _ { t _ 1 } , \\cdots , 2 B _ { t _ n \\wedge \\tau _ k } - B _ { t _ n } - ( 2 B _ { t _ { n - 1 } \\wedge \\tau _ k } - B _ { t _ { n - 1 } } ) ) ] = \\hat { \\mathbb { E } } [ \\varphi ( B _ { t _ 1 } , \\cdots , B _ { t _ n } - B _ { t _ { n - 1 } } ) ] . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\mathbf { v } _ 0 ( \\cdot , \\tau ) = \\cos ( \\tau ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) \\boldsymbol { \\phi } + ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) \\boldsymbol { \\psi } + \\int _ { 0 } ^ { \\tau } ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { - 1 / 2 } \\sin ( ( \\tau - \\tilde { \\tau } ) ( \\widehat { \\mathcal { A } } ^ 0 ) ^ { 1 / 2 } ) \\mathbf { F } ( \\cdot , \\tilde { \\tau } ) \\ , d \\tilde { \\tau } . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\begin{array} { r c r } f ( Z _ t ) - f ( Z _ 0 ) & = & \\int _ 0 ^ t { \\partial _ { k ^ i } ( \\tilde { f } ) ( Z _ { s _ - } ) d \\tilde { Z } ^ i _ s } + \\frac { 1 } { 2 } \\int _ 0 ^ t { \\partial _ { k ^ i k ^ j } ( \\tilde { f } ) ( Z _ { s _ - } ) d [ \\tilde { Z } ^ i , \\tilde { Z } ^ j ] ^ c _ s } + \\\\ & & + \\sum _ { 0 \\leq s \\leq t } ( f ( Z _ s ) - f ( Z _ { s _ - } ) - \\Delta { Z } ^ i _ s \\partial _ { k ^ i } ( \\tilde { f } ) ( Z _ { s _ - } ) ) \\end{array} \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\partial _ { t } \\omega = \\nu \\Delta \\omega - e ^ { - \\nu t } \\sin y \\partial _ { x } \\omega = \\tilde { L } \\left ( t \\right ) \\omega , \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} F \\star G ( z ) = \\int _ { [ - 1 , 1 ] ^ 2 } F ( x - y ) G ( y ) d y \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} ( o s c _ \\delta g _ 1 ) ( x ) = \\sup _ { y \\in B _ \\delta ( x ) } \\sup _ { z \\in B _ r ( y ) } f ( z ) - \\inf _ { y \\in B _ \\delta ( x ) } \\sup _ { z \\in B _ r ( y ) } f ( z ) . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} N : = \\{ x \\in X \\ ; : \\ ; \\langle A x , x \\rangle = 0 \\} \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} K ( x / v , v ) ^ 2 A ( v / x , v ) & = K ( x / v , v ) - \\frac { x ^ 2 } { v ( v - x ) } \\big ( K ( x / v , v ) - 1 \\big ) \\\\ & - \\frac { x } { 1 - v } K ( x / v , v ) A ( x , 1 ) - \\frac { x ^ 3 } { v ( 1 - v ) ( v - x ) } C ( x ) , \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} d \\mu _ m = \\frac { d S ^ { 2 a _ m - 1 } } { { \\rm v o l \\ , } ( S ^ { 2 a _ m - 1 } ) } , \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\dot { q } ^ i \\frac { \\partial \\gamma _ i } { \\partial q ^ j } ( q ) - \\frac { \\partial L } { \\partial q ^ j } ( q , \\dot { q } ) = 0 , \\gamma _ i ( q ) - \\frac { \\partial L } { \\partial \\dot { q } ^ i } ( q , \\dot { q } ) = 0 . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ { p ( x ) - 2 } \\nabla u \\nabla \\varphi = \\int _ { \\Omega } \\biggl ( \\frac { f ( x , T u ) | T u | _ { L ^ { q ( x ) } } ^ { \\alpha ( x ) } } { \\mathcal { A } ( x , | T u | _ { L ^ { r ( x ) } } ) } + \\frac { g ( x , T u ) | T u | _ { L ^ { s ( x ) } } ^ { \\gamma ( x ) } } { \\mathcal { A } ( x , | T u | _ { L ^ { r ( x ) } } ) } \\biggl ) \\varphi \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} b = a ^ { \\tau _ 1 } , c = a ^ { \\tau _ 2 } , d = a ^ { \\tau _ 3 } . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\overline { x _ d } ( w _ { 0 1 } w _ { 0 2 } ^ { - 1 } ) \\overline { x _ d ^ { - 1 } } & = ( w _ { 0 1 } w _ { 0 2 } ^ { - 1 } ) ^ { - 1 } = [ \\overline { x _ d } , w _ { 0 1 } ^ { - 1 } ] , \\\\ \\overline { x _ d } w _ i \\overline { x _ d ^ { - 1 } } & = w _ i ^ { - 1 } \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\mathbb { C } ( \\Gamma _ s ) \\overset { \\rho _ 1 } { \\simeq } \\mathbb { C } ( \\tau ) \\mbox { w i t h } \\rho _ 1 ( \\lambda ) = \\chi _ 1 ( \\tau ) \\mbox { a n d } \\rho _ 1 ( \\mu ) = \\chi _ 2 ( \\tau ) \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\Xi ( A ) \\geq _ { \\mathrm { t t } } \\bigoplus _ { i = 0 } ^ { n - 1 } \\Phi _ { A _ i } ( A _ i ) \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} K _ { X ^ { [ n + 1 , n ] } } & = - 3 H ^ b - 3 H ^ { } + \\frac { 1 } { 2 } B ^ { } \\\\ & = - 3 H ^ b - 3 H ^ { } + ( n H ^ a - D ^ a ) - ( ( n - 1 ) H ^ b - D ^ b ) \\\\ & = - 2 H ^ b + ( n - 3 ) H ^ { } - D ^ a + D ^ b \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\eta _ \\omega \\circ \\jmath _ { \\omega o } \\ , = \\ , { \\jmath \\ , } ' _ { \\omega o } \\circ \\eta _ o \\ \\ , \\ \\ \\forall o \\leq \\omega \\ . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} { \\cal L } _ \\bot : = { \\cal P } _ \\bot ^ { - 1 } { \\cal L } _ \\omega { \\cal P } _ \\bot = \\Pi _ { { \\mathbb S } _ 0 } ^ \\bot { \\cal L } _ 8 \\Pi _ { { \\mathbb S } _ 0 } ^ \\bot + R _ M \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} \\mathfrak { L } ( u ) = \\frac { 1 } { p } \\int \\limits _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } d x + \\frac { 1 } { p } \\int \\limits _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } d x - \\frac { 1 } { q } \\int \\limits _ { \\mathbb { G } } | u ( x ) | ^ { q } d x \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} \\varphi _ i = \\varphi _ i ( z _ i , z _ { i + 1 } , \\ldots , z _ { n - 1 } ) \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} K \\subset \\bigcup _ { j = 1 } ^ L B _ { r _ j } ^ { \\R ^ N } ( y _ j ) \\subset \\bigcup _ { j = 1 } ^ L B _ { \\lambda r _ j } ^ { \\R ^ N } ( y _ j ) \\subset \\bigcup _ { i = 1 } ^ M B _ { 3 \\lambda r _ { j _ i } } ^ { \\R ^ N } ( y _ { j _ i } ) , \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\mathcal { L } : = \\left \\{ m : [ 0 , T ] \\longrightarrow S : | m ( t ) - m ( s ) | \\leq K | t - s | , m ( 0 ) = m _ 0 \\right \\} \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} & h ' \\geq \\max \\left \\{ D \\left ( \\log \\left ( \\frac { b _ 1 } { a _ 2 ' } + \\frac { b _ 2 } { a _ 1 ' } \\right ) + \\log \\lambda ' + 1 . 7 5 \\right ) + 0 . 0 6 , \\lambda ' , \\frac { D \\log 2 } { 2 } \\right \\} , \\\\ & a _ i ' \\geq \\max \\{ 1 , \\varrho | \\log \\gamma _ i | - \\log | \\gamma _ i | + 2 D h ( \\gamma _ i ) \\} , i = 1 , 2 , \\\\ & a _ 1 ' a _ 2 ' \\geq \\lambda '^ 2 . \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} & T = \\frac { \\partial } { \\partial \\theta } \\\\ & Z _ j = \\frac { \\partial } { \\partial z _ j } + i \\frac { \\partial \\varphi ( z ) } { \\partial z _ j } \\frac { \\partial } { \\partial \\theta } , j = 1 , \\cdots , n , \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} W ^ { n } ( y ^ { n } | x ^ { n } ) = \\prod \\limits _ { j = 1 } ^ { n } W ( y _ j | x _ j ) , x ^ { n } \\in \\mathcal { X } ^ { n } , y ^ { n } \\in \\mathcal { Y } ^ { n } . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} w _ { \\mu , \\zeta } ( x ) = \\frac { \\alpha _ 3 \\ , \\mu ^ { 1 / 2 } } { ( \\mu ^ 2 + | x - \\zeta | ^ 2 ) ^ { 1 / 2 } } , \\alpha _ 3 = 3 ^ { 1 / 4 } , \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} 0 \\leq u ( x , t ) \\leq u ( x _ 0 , t _ 0 ) \\ ; \\exp \\left ( C \\Big ( \\frac { \\| x _ 0 - x \\| ^ 2 } { t _ 0 - t } + \\frac { t _ 0 - t } { \\min ( 1 , t ) } + 1 \\Big ) \\right ) = 0 . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} u _ n : = \\mathbb { P } ( [ n ] \\mbox { i s a b l o c k o f } \\Pi ) = \\int _ 0 ^ { \\infty } e ^ { - x } \\mathbb { E } \\prod _ { i = 1 } ^ n \\left ( 1 - \\exp \\left ( - \\frac { x W _ i } { T _ { i } } \\right ) \\right ) d x , \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} \\langle y _ 2 - y _ 1 , x _ 2 - x _ 1 \\rangle \\geq 0 \\quad \\ , \\forall ( x _ 1 , x _ 2 ) \\in X ^ 2 , \\ , \\forall y _ i \\in A x _ i , \\ , i = 1 , 2 . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} 1 - \\gamma = ( x + y + z ) ^ 2 - ( y ^ 2 + 2 y z ) \\ge ( x + z ) ^ 2 \\ , , \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} [ X \\odot M ( X , Y ) \\odot Y ' ] ^ { ( 4 ) } = [ M ( X , Y ) ^ { ( 2 ) } ] ^ { 1 / 2 } [ X ^ { 1 / 2 } ( Y ' ) ^ { 1 / 2 } ] ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\dim \\sharp _ { i = 1 } ^ m M _ i = \\sum _ { i = 1 } ^ m d _ i - ( m - 1 ) \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\lambda _ 0 = \\sup \\ , \\{ \\lambda \\in ( 0 , \\lambda _ 1 ) : \\sigma _ j ( \\lambda ' , r ) > 0 \\forall r \\in ( a , 1 ) , \\ j = 1 , \\ldots , k , \\ \\lambda ' \\in ( 0 , \\lambda ) \\} . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\beta _ { k , c _ 1 ^ 2 } = \\frac { \\tau _ k } { \\tau _ { \\bar k } ( \\tau _ k + 1 ) } \\\\ \\beta _ { k , c _ 2 ^ 2 } = \\frac { 1 } { \\tau _ { \\bar k } } \\ , \\mbox { o r } \\ , + \\infty \\\\ \\beta _ { k , c _ 3 ^ 2 } = \\min \\left \\{ \\frac { \\tau _ k ( \\tau _ { \\bar k } + 1 ) } { \\tau _ { \\bar k } ( \\tau _ k + 1 ) } , \\frac { 1 } { \\tau _ { \\bar k } } \\right \\} \\ , \\mbox { o r } \\ , \\min \\left \\{ \\tau _ k , \\frac { 1 } { \\tau _ { \\bar k } } \\right \\} \\end{array} . \\right . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\mathbf { T } N = ( V _ + \\cap \\mathbf { T } N ) \\oplus ( V _ - \\cap \\mathbf { T } N ) \\subseteq \\mathbf { T } M \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} ( x - d ) ^ 2 + x ^ 2 + ( x + d ) ^ 2 = y ^ n . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} v ^ R ( A , B , C ) = ( - r ( \\alpha ) A B , - r ( \\alpha ) A B , 2 r ( \\alpha ) A B ) . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} \\Delta ( t ) = \\frac { ( 1 - t ^ { a _ 1 p _ 1 p _ 2 \\cdots p _ r } ) ( 1 - t ^ { a _ 2 p _ 2 \\cdots p _ r } ) \\cdots ( 1 - t ^ { a _ r p _ r } ) ( 1 - t ) } { ( 1 - t ^ { a _ 1 p _ 2 \\cdots p _ r } ) ( 1 - t ^ { a _ 2 p _ 3 \\cdots p _ r } ) \\cdots ( 1 - t ^ { a _ r } ) ( 1 - t ^ { p _ 1 \\cdots p _ r } ) } . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\blacksquare ( F ) = \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} k & : = \\frac { n + 1 } { 2 } + ( p - 1 ) \\cdot t , \\\\ \\alpha _ p ( n , k ) & : = _ p ( A _ { n , k } ) = _ p \\left ( \\zeta ( 1 - k ) ^ { - 1 } \\prod _ { i = 1 } ^ { ( n - 1 ) / 2 } \\zeta ( 1 + 2 i - 2 k ) ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} P \\left ( \\lim _ { n \\rightarrow \\infty } X _ { n } ( t ) = X ( t ) \\mbox { f o r a n y } t \\notin \\overline { \\eta } \\right ) = 1 , \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\frac { d } { d t } k ( \\gamma ( t ) , y ) = \\nu ' ( \\xi ) \\frac { \\xi - y } { | \\xi - y | ^ n } + \\nu _ i ( \\xi ) \\gamma ' _ j ( t ) \\frac { | z | ^ 2 \\delta _ { i j } - n z _ i z _ j } { | z | ^ { n + 2 } } \\mbox { f o r } z = \\xi - y . \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\lambda _ m ^ { ( n ) } = P r ( \\mathcal { G } ( y ^ n ) \\neq m | X ^ n = c ^ { ( n ) } ( m ) ) , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} { x \\choose l } _ \\lambda = \\frac { ( x ) _ { l , \\lambda } } { l ! } = \\frac { x ( x - \\lambda ) \\cdots ( x - ( l - 1 ) \\lambda ) } { l ! } . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} I _ 1 ( z ) : = \\partial _ { N } V ^ \\theta ( z , \\lambda ^ t ( z ) ) - \\partial _ { N , { \\rm o u t } } V ^ \\theta ( z , \\theta \\lambda ^ t ( z ) ) \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} A _ g = \\langle e _ 1 \\rangle \\oplus \\langle e _ 3 , e _ { t + 3 } , e _ { 2 t + 3 } , . . . , e _ { ( r - 1 ) t + 3 } \\rangle \\oplus \\langle e _ n \\rangle \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\mathbb E \\| f \\star \\bar { \\mu } \\| ^ 2 = \\mathbb E \\| f \\| ^ 2 \\star \\nu . \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} B = \\frac 1 4 \\sum _ p \\frac { p ^ 3 ( p ^ 4 - p ^ 3 - p ^ 3 - p - 1 ) ( \\log p ) ^ 2 } { ( p - 1 ) ^ 6 ( p + 1 ) ^ 2 ( p ^ 2 + p + 1 ) } , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} N _ { 2 d } ^ { \\mathrm { c m } } ( X ) = \\left ( \\sum _ { G \\leq S _ d } r _ d ( G ) \\right ) X + O _ { d , \\epsilon } ( X ^ { \\beta ( \\delta _ d , \\delta ^ { \\prime } ) + \\epsilon } ) \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( \\tilde W - W ) = \\partial _ { \\nu , \\ , { \\rm o u t } } \\tilde W \\ , H ^ { n - 1 } \\restriction _ { \\partial U } \\quad \\mbox { i n } \\R ^ n \\\\ ( \\tilde W - W ) ( \\infty ) = 0 , \\\\ \\mbox { r e s p . } \\lim _ { x \\to \\infty } \\frac { ( \\tilde W - W ) ( x ) } { - \\log | x | } = - 2 \\pi \\int _ { \\R ^ 2 } f \\ , \\chi _ { \\R ^ n \\setminus U } = 2 \\pi \\int _ { \\partial U } \\partial _ { \\nu , \\ , { \\rm o u t } } \\tilde W \\mbox { a t } \\infty . \\end{cases} \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} \\pi ^ 2 ( z ) + [ \\tilde { \\tau } ( z ) - \\sigma ' ( z ) ] \\pi ( z ) + \\tilde { \\sigma } ( z ) - g ( z ) \\sigma ( z ) = 0 . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} { \\eta } ( \\theta ) = \\theta + \\eta , \\ { \\eta } ( \\hat { \\theta } _ { j } ) = \\hat { \\theta } _ { j } - \\left [ \\xi _ { j } - p _ { j } \\eta \\right ] \\ \\ { \\eta } ( { \\xi } _ { j } ) = \\left \\{ \\xi _ { j } - p _ { j } \\eta \\right \\} , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\frac { d } { d t } \\langle R ( t ) x , y \\rangle _ X = - \\langle A x , R ( t ) y \\rangle _ X - \\langle R ( t ) x , A y \\rangle _ X - \\langle B ^ * R ( t ) x , B ^ * R ( t ) y \\rangle _ U \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\widetilde { f } ( X ) : = a _ k ^ { 2 k - 2 } \\prod _ { \\substack { 1 \\leq i , j \\leq k \\\\ i \\neq j } } ( X - ( \\alpha _ i - \\alpha _ j ) ) , \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { n p _ n } ^ 2 } \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 2 | \\mathcal { F } _ { n ( \\ell - 1 ) } ) \\to 1 ~ ~ ~ \\mbox { i n p r o b a b i l i t y } . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} I ( W \\wedge X , Y ) + \\mu H ( Y | W ) = R _ \\mu ( P _ { X Y } ) . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\bar { x } ( p ) : = \\sum _ { j = 1 } ^ m \\hat { v } _ j ^ * \\Phi _ j ( p ) \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} E _ \\mathfrak { a } ( z , w ) = \\delta _ \\mathfrak { a } y ^ w + \\varphi _ \\mathfrak { a } ( 0 , w ) y ^ { 1 - w } + \\sum _ { m \\neq 0 } \\varphi _ \\mathfrak { a } ( m , w ) W _ w ( \\lvert m \\rvert z ) , \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} K i G ( f ) \\alpha _ k & = \\alpha _ n \\alpha _ n ^ \\dagger K i G ( f ) \\alpha _ k \\\\ K i G ( g ) \\alpha _ m & = \\alpha _ k \\alpha _ k ^ \\dagger K i G ( g ) \\alpha _ m \\\\ \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\underset { n \\to + \\infty } { \\lim } \\int _ { \\mathbb { R } ^ N } f _ 1 ( u _ n ) u _ n = \\int _ { \\mathbb { R } ^ N } f _ 1 ( u ) u . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} F _ { r + 1 } ( z ) F _ { s } ( z ) - F _ { r } ( z ) F _ { s + 1 } ( z ) = ( - z ) ^ { r } F _ { s - r } ( z ) . \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\lambda _ { \\psi } ( A ) \\le \\lambda _ 1 ( A ) \\norm { \\psi } { \\infty } = ( \\mu _ X \\otimes \\mu _ Z ) ( A ) \\norm { \\psi } { \\infty } . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} A ^ { \\rm f l a t } ( v ) = \\left ( - 2 ( 1 - \\rho ) \\chi ^ { - 1 / 3 } v N ^ { 2 / 3 } , 2 \\rho \\chi ^ { - 1 / 3 } v N ^ { 2 / 3 } \\right ) \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } & \\frac { 1 } { n ^ { s + k - 1 } } \\bigg ( \\sum _ { m \\geq 0 } a ( n - m ) \\overline { b ( n - m ) } \\\\ & + \\sum _ { \\substack { \\ell \\geq 1 \\\\ m \\geq 0 } } a ( n - m ) \\overline { b ( n - m - \\ell ) } + \\sum _ { \\substack { \\ell \\geq 1 \\\\ m \\geq 0 } } a ( n - m - \\ell ) \\overline { b ( n - m ) } \\bigg ) . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} x - \\mathcal { K } x = f \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 } = \\int _ { \\Omega ^ { \\varepsilon } } \\kappa ^ { \\varepsilon } \\nabla \\left ( \\theta ^ { \\varepsilon } - \\theta _ { 1 } ^ { \\varepsilon } \\right ) \\cdot \\nabla \\varphi d x + \\int _ { \\Omega ^ { \\varepsilon } } \\left ( \\kappa ^ { \\varepsilon } \\nabla \\theta _ { 1 } ^ { \\varepsilon } - \\mathbb { K } \\nabla \\theta ^ { 0 } \\right ) \\cdot \\nabla \\varphi d x . \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} B _ j = ( I _ N , 0 , \\dots , 0 ) P _ j ( I _ N , 0 , \\dots , 0 ) ^ t . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\frac { c _ o } { n } \\int _ { \\R ^ N } \\sum _ { i = 1 } ^ n \\frac { \\varphi ^ 2 } { | x - a _ i | ^ 2 } \\ , d x \\le \\int _ { { \\R } ^ N } | \\nabla \\varphi | ^ 2 \\ , d x \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} a '' ( n ; i ) & = \\sum _ { j = i + 1 } ^ n a ( n ; i , j ) = b ( n ; i ) + b ' ( n ; i ) + \\sum _ { j = i + 1 } ^ { n - 2 } a ( n - 1 ; i , j ) , \\\\ a '' ( n - 1 ; i ) & = \\sum _ { j = i + 1 } ^ { n - 1 } a ( n - 1 ; i , j ) = b ' ( n - 1 ; i ) + \\sum _ { j = i + 1 } ^ { n - 2 } a ( n - 1 ; i , j ) , \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} f _ j ( \\alpha ( b _ i ) ) = a _ j , \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} f ^ * ( x ^ * ) = h ^ * ( x ^ * - b ) - \\gamma = \\langle x ^ * - b , \\frac { 1 } { 2 } A ^ { - 1 } ( x ^ * - b ) \\rangle - \\gamma , \\quad \\forall x ^ * \\in X . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} \\| f _ x \\| _ { L ^ \\infty } = \\| u _ x \\| _ { L ^ \\infty } , \\qquad \\| f _ y \\| _ { L ^ 4 } = \\| L _ y \\partial _ x u \\| _ { L ^ 4 } , \\| \\partial _ x ^ { - 1 } f _ { y y } \\| _ { L ^ 2 } = \\| L _ y ^ 2 \\partial _ x u \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} [ E _ m ^ - , E _ m ^ + ] = \\{ E \\in \\mathbb { R } : 2 \\rho _ { \\lambda f , \\alpha } ( E ) = m \\alpha \\mathbb { Z } \\} . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} C = 2 ( \\sum _ j | a ( e _ j ) | ) ^ 2 + 2 \\sum | \\Re [ a ( e _ j + e _ k ) ] | . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} T _ { n p } = \\sum _ { 1 \\le j < i \\le p } \\frac { r _ { i j } ^ 2 } { 1 - r _ { i j } ^ 2 } . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\begin{cases} ( 2 r ( 2 r - 1 ) + ( 1 - r + 2 r m ) 2 r ) [ \\lambda ] + ( 2 r + ( 1 - r + 2 r m ) ) [ \\mu ] = p [ \\lambda ] + q [ \\mu ] \\\\ ( 2 r ( 2 r - 1 ) + ( 1 - 3 r - 2 r m ) 2 r ) [ \\lambda ] + ( 2 r + ( 1 - 3 r - 2 r m ) ) [ \\mu ] = p [ \\lambda ] + q ' [ \\mu ] . \\end{cases} \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\frac { ( q ; q ) _ { \\theta N - 1 } } { ( x q ; q ) _ { \\theta N - 1 } } } = \\frac { ( q ; q ) _ { \\infty } } { ( x q ; q ) _ { \\infty } } , \\end{gather*}"}
-{"id": "2564.png", "formula": "\\begin{align*} u ( T , x ) = u _ 0 ( x ) \\ , , u _ { t } ( T , x ) = u _ 1 ( x ) \\ , , x \\in ( 0 , \\pi ) \\ , . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} \\eta = \\dfrac { p _ { 1 1 } } { p _ { 2 1 } } , z = ( z _ 1 , z _ 2 ) = ( \\frac { p _ { 3 1 } } { p _ { 2 1 } } , \\frac { p _ { 3 2 } } { p _ { 2 2 } } ) \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} \\textstyle c ( z ) : = \\frac { ( a - 1 ) z - z ^ { - 1 } + ( b - 2 ) ( a - 1 ) ^ { 1 / 2 } ( b - 1 ) ^ { - 1 / 2 } } { a ( z - z ^ { - 1 } ) } . \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} x = _ A u \\Longrightarrow s ( x ) = _ { \\bar { A } } s ( u ) . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\textrm { t r } ^ { \\flat } \\ ( \\mathcal { K } _ t ^ m \\ ) = \\sum _ { T _ t ^ m x = x } \\frac { \\exp \\ ( g _ { m , t } \\ ( x \\ ) \\ ) } { 1 - \\ ( \\ ( T _ t ^ m \\ ) ' \\ ( x \\ ) \\ ) ^ { - 1 } } . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} \\dot { x } ( t ) = & u _ i ( x ( t ) ) , \\\\ x ( 0 ) = & x \\ , , \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\lim ( g C ) = \\lim ( a C ) = a \\lim ( C ) . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\norm { v } _ 3 + \\norm { \\partial _ t v } _ { 2 } + \\norm { \\partial ^ 2 _ t v } _ { 1 } + \\norm { \\partial ^ 3 _ t v } _ 0 + \\norm { R } _ 3 + \\norm { \\partial _ t R } _ { 2 } + \\norm { \\partial ^ 2 _ t R } _ 1 + \\norm { \\partial ^ 3 _ t R } _ 0 \\leq C _ * . \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{gather*} \\Delta _ g ( \\cdot ) = \\frac { 1 } { \\sqrt { g } } \\partial _ i ( \\sqrt { g } g ^ { i j } \\partial _ j ( \\cdot ) ) , \\end{gather*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\{ a \\cdot P _ { 1 } ( 0 , 0 , x _ { 3 } , x _ { 4 } ) + b \\cdot P _ { 2 } ( 0 , 0 , x _ { 3 } , x _ { 4 } ) = 0 \\} \\subset \\ell \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} n _ { p , X } = \\min \\bigg \\{ t \\in \\mathbb { N } \\bigg \\vert x _ { 0 } ^ { d - t } | f \\bigg \\} . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} f ( x ) & = \\sigma _ f ( x ) + a _ x p \\\\ f ( x + m p ) & = f ( x ) + m p f _ x \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} \\mathcal B = \\{ ( \\zeta , \\widehat \\Lambda ) \\in \\R ^ { 3 k } \\times \\R ^ k : | ( \\zeta - \\zeta ^ 0 , \\widehat \\Lambda ) | < \\varepsilon ^ { 1 - \\sigma } \\} \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} G _ 0 ^ { 0 ( 2 ) } * ( u m _ 1 ^ { 0 ( 1 ) } ( 0 ) ) & = G _ 0 ^ { 0 ( 1 ) } * ( u m _ 1 ^ { 0 ( 1 ) } ( 0 ) ) + c \\langle m _ 1 ^ { 0 ( 1 ) } ( 0 ) , u \\rangle \\\\ & = G _ 0 ^ { 0 ( 1 ) } * ( u m _ 1 ^ { 0 ( 1 ) } ( 0 ) ) \\\\ & = m _ 1 ^ { 0 ( 1 ) } ( 0 ) - 1 . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} C ^ { i j k l } \\left ( x \\right ) = C ^ { k l i j } \\left ( x \\right ) \\ : a . e . \\ , , \\ ; i , j , k , l = 1 , 2 , 3 , \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\mathtt { B } _ { b , p , 1 } ( z ) = \\left ( \\frac { 2 } { z } \\right ) ^ { \\tfrac { b - 1 } { 2 } } \\mathtt { J } _ { p + \\tfrac { b - 1 } { 2 } } ( z ) , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\tau _ { k , l } = \\inf \\{ t \\in \\mathbb R _ + : \\# \\{ s \\in [ 0 , t ] : \\| \\Delta M _ s \\| \\in [ 1 / k , k ] \\} = l \\} . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} G ( z , w ) \\delta \\left ( \\frac { \\nu _ 1 } { z } \\right ) \\delta \\left ( \\frac { \\nu _ 2 } { w } \\right ) = G ( \\nu _ 1 , \\nu _ 2 ) \\delta \\left ( \\frac { \\nu _ 1 } { z } \\right ) \\delta \\left ( \\frac { \\nu _ 2 } { w } \\right ) , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{gather*} \\min \\{ A ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\le 0 \\ u ( x _ 0 ) > 0 \\\\ \\min \\{ B ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\le 0 \\ u ( x _ 0 ) < 0 \\\\ \\min \\{ - \\mathcal { Q } _ \\infty \\phi ( x _ 0 ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\le 0 \\ u ( x _ 0 ) = 0 \\end{gather*}"}
-{"id": "469.png", "formula": "\\begin{align*} g _ { 1 } ( [ X , Y ] , V ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Y , \\varphi V ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Y } \\varphi X , \\varphi V ) \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} \\langle X _ { \\alpha _ q } , \\theta _ Q ( \\alpha _ q ) \\rangle = \\langle T \\pi _ Q ( X _ { \\alpha _ q } ) , \\alpha _ q \\rangle , \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\| \\phi \\| _ { ( \\overline { H [ X \\odot Y ' ] } ) ^ * } \\approx \\| \\phi \\| _ { ( \\overline { H [ ( X \\odot Y ' ) ^ { \\wedge } ] } ) ^ * } = \\inf \\| \\tilde \\phi \\| _ { ( ( X \\odot Y ' ) ^ { \\wedge } ) ^ * } , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\psi ( t ) & = \\sum _ { k = 1 } ^ \\infty ( k - 2 ) ( p _ k - \\zeta _ k ( t ) ) - 2 \\int _ { [ 0 , t ] \\times [ 0 , 1 ] } { { 1 } } _ { [ 0 , r _ 0 ( \\boldsymbol { \\zeta } ( s ) ) ) } ( y ) \\ , \\varphi _ 0 ^ \\varepsilon ( s , y ) d s \\ , d y , \\\\ { \\tilde { \\psi } } ( t ) & = \\sum _ { k = 1 } ^ \\infty ( k - 2 ) ( p _ k - { \\tilde { \\zeta } } _ k ( t ) ) - 2 \\int _ { [ 0 , t ] \\times [ 0 , 1 ] } { { 1 } } _ { [ 0 , r _ 0 ( { \\tilde { \\boldsymbol { \\zeta } } } ( s ) ) ) } ( y ) \\ , \\varphi _ 0 ^ \\varepsilon ( s , y ) d s \\ , d y . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} P _ k ( \\mathbf { m } ) : = P _ { k , 1 } ^ { m _ 1 } \\cdots P _ k ^ { m _ k } . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} n _ \\alpha ( \\zeta ) = x _ \\alpha ( \\zeta ) x _ { - \\alpha } ( - \\zeta ^ { - 1 } ) x _ \\alpha ( \\zeta ) , \\alpha ^ \\vee ( \\zeta ) = n _ \\alpha ( \\zeta ) n _ \\alpha ( - 1 ) . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 = O \\Big ( N ^ { \\varepsilon } \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { ( N \\eta ) ^ { \\frac 1 3 } } + N ^ { - \\frac { \\gamma } { 4 } } \\hat { \\Lambda } ^ 2 \\Big ) , \\iota = A , B . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} M _ m ^ { \\nu } ( n _ m \\geq \\dots \\geq n _ 2 \\geq n _ 1 ) = \\lim _ { N \\rightarrow \\infty } { \\Lambda ^ N _ m \\delta _ { \\lambda ( N ) } ( \\lambda ( N ) _ { N - m + 1 } \\geq \\dots \\geq \\lambda ( N ) _ N ) } , \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} J ' _ { d , e } ( x ) & = J '' _ { d , e } ( x ) + \\frac { x ^ { d + e + 2 } } { ( 1 - x ) ^ e } ( C ( x ) - 1 ) . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} F _ { h _ 1 , n } : = S ^ { - d n } ( \\chi \\circ \\tau _ { d ( n + h _ 1 ) + 1 } \\cdot \\overline \\chi \\circ \\tau _ { d n + 1 } ) \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} \\langle e _ 1 , y _ 1 \\rangle & = \\langle u _ 1 + y _ 2 , y _ 1 \\rangle = \\langle u _ 1 , y _ 1 \\rangle + \\langle y _ 2 , y _ 1 \\rangle \\\\ & = \\langle u _ 1 , y _ 1 \\rangle + \\langle u _ 2 , y _ 2 \\rangle \\geq \\varepsilon \\| y _ 1 \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} { \\cal D } _ \\bot : = \\begin{pmatrix} D _ \\bot & 0 \\\\ 0 & - D _ \\bot \\end{pmatrix} , D _ \\bot : = \\Pi _ { \\mathbb S _ 0 } ^ \\bot D _ 8 \\Pi _ { \\mathbb S _ 0 } ^ \\bot \\ , , \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\mu \\circ \\tau ( ( s , t ] \\times B ) = \\mu ( ( \\tau _ s , \\tau _ t ] \\times A ) , \\ ; \\ ; \\ ; t \\geq s \\geq 0 , A \\in \\mathcal J . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} T _ m ( a , a , \\ldots , a ) = \\sum _ { \\lambda \\vdash n } \\frac { \\varepsilon _ \\lambda n ! } { z _ \\lambda } T _ m ( a ) ^ { m _ 1 } T _ m ( a ^ 2 ) ^ { m _ 2 } \\cdots T _ m ( a ^ n ) ^ { m _ n } \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} & \\tau = \\frac { 1 } { 2 } \\begin{pmatrix} \\sqrt { - 3 } \\eta ^ { - 1 } & - 1 \\\\ - 1 & \\sqrt { - 3 } \\eta \\\\ \\end{pmatrix} , \\\\ & \\zeta = \\frac { 1 } { 2 } \\left ( ( \\dfrac { z _ 1 } { 1 - \\omega } - \\dfrac { z _ 2 } { 1 - \\omega ^ 2 } ) \\eta ^ { - 1 } , \\dfrac { z _ 1 } { 1 - \\omega } + \\dfrac { z _ 2 } { 1 - \\omega ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} & \\left \\| A \\right \\| _ { 1 , 2 } = { \\bf m a x } _ { 1 \\le i \\le n } \\left \\| A ( : , i ) \\right \\| _ 2 , \\left \\| A \\right \\| _ { 1 , \\infty } = { \\bf m a x } _ { i , j } | a _ { i , j } | , \\\\ & \\left \\| A B \\right \\| _ { 1 , 2 } \\le \\left \\| A \\right \\| _ 2 \\left \\| B \\right \\| _ { 1 , 2 } , \\left \\| A B \\right \\| _ { 1 , 2 } \\le \\left \\| A \\right \\| _ { 1 , 2 } \\left \\| B \\right \\| _ 1 , \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{gather*} \\tilde \\omega ^ 0 = \\omega ^ 0 , \\tilde \\omega ^ a = \\omega ^ a + A ^ a \\omega ^ 0 , \\tilde \\omega ^ 3 = \\omega ^ 3 + B \\omega ^ 0 , \\tilde \\theta _ a = \\theta _ a , \\end{gather*}"}
-{"id": "356.png", "formula": "\\begin{align*} ( \\nabla _ X h ) Y = A ( X , h Y ) - h A ( X , Y ) . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} d _ 4 = c _ 4 - 4 c _ 3 \\frac { 1 } { n - 1 } + 6 c _ 2 \\frac { 1 } { ( n - 1 ) ^ 2 } - 4 c _ 1 \\frac { 1 } { ( n - 1 ) ^ 4 } + \\frac { 1 } { ( n - 1 ) ^ 4 } = O \\left ( \\frac { 1 } { n ^ 4 } \\right ) . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} z ( T ) = z _ 0 \\ , , z ' ( T ) = z _ 1 \\ , . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} i \\partial _ t \\vec w ( t ) = \\lambda t ^ { - 1 } A ( t ) \\vec w ( t ) + \\mathcal { O } \\bigl ( t ^ { - \\frac 5 4 } \\bigl [ \\| w ( t ) \\| _ { L ^ \\infty } + t ^ { - \\frac 1 4 } \\| w ( t ) \\| _ { H ^ 1 } \\bigr ] ^ 2 \\| w ( t ) \\| _ { H ^ 1 } \\bigr ) \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} ( F ^ * z _ 1 ^ * ) ( x \\otimes y ) = ( x _ 1 ^ * \\otimes y _ 1 ^ * ) ( F _ X x \\otimes F _ Y y ) = ( F _ X ^ * x _ 1 ^ * ) ( x ) ( F _ Y ^ * y _ 1 ^ * ) ( y ) , \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\int _ { - T } ^ { T } \\Big | \\sum _ { n = - \\infty } ^ { \\infty } \\big ( C _ n e ^ { i \\omega _ n t } + R _ n e ^ { r _ n t } \\big ) \\Big | ^ 2 d t \\le c _ 2 \\sum _ { n = - \\infty } ^ { \\infty } | C _ n | ^ 2 \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\begin{aligned} & \\varkappa _ d ( F ) _ i = 3 & \\Longleftrightarrow & \\quad \\mathcal { G } _ i \\simeq \\langle 2 4 3 , 2 7 \\rangle & \\varkappa _ s ( \\mathcal { G } _ i ) = ( 1 , 0 , 0 , 0 ) , \\\\ & \\varkappa _ d ( F ) _ i = 9 & \\Longleftrightarrow & \\quad \\mathcal { G } _ i \\simeq \\langle 2 4 3 , 2 6 \\rangle & \\varkappa _ s ( \\mathcal { G } _ i ) = ( 0 , 0 , 0 , 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ u $ } ( \\cdot , 0 ) \\ , = \\ , \\mbox { \\boldmath $ u $ } _ 0 \\in \\mbox { \\boldmath $ L $ } ^ { 2 } _ { \\sigma } ( \\mathbb { R } ^ { n } ) , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 f ( t ) d t = 0 , \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\mu _ j ^ { - \\frac { 1 } { 2 } } \\ , U _ j ( \\zeta _ i + \\mu _ i z ) = 4 \\pi \\alpha _ 3 \\ , G _ { \\lambda } ( \\zeta _ i + \\mu _ i z , \\zeta _ j ) + \\mu _ j ^ { 2 - \\sigma } \\hat { \\theta } ( \\mu _ j , \\zeta _ i + \\mu _ i z , \\zeta _ j ) . \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} d \\mu = \\mu ( x ) d x = C \\ , e ^ { - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } \\langle A ( x - a _ i ) , x - a _ i \\rangle } \\ , d x \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\log m ! = m \\log m - m + O ( \\log ( m + 2 ) ) ( m \\geq 0 ) . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} & \\| \\mathcal { J } ( t , \\zeta ) \\| \\le C _ 6 t ^ 4 ( c _ * t ^ 2 + \\zeta ) ^ { - 2 } + C _ 7 t ^ 2 ( c _ * t ^ 2 + \\zeta ) ^ { - 1 } , | t | \\le t ^ 0 , \\ \\zeta > 0 ; \\\\ & C _ 6 = \\beta _ 6 \\delta ^ { - 1 } \\bigl ( \\| X _ 1 \\| ^ 4 + c _ * ^ { - 1 } \\| X _ 1 \\| ^ 6 \\bigr ) , C _ 7 = \\beta _ 7 \\delta ^ { - 1 } \\bigl ( \\| X _ 1 \\| ^ 2 + c _ * ^ { - 1 } \\| X _ 1 \\| ^ 4 \\bigr ) . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\mathcal { U } ( x ) = \\left ( \\begin{array} { c c } e ^ { 2 \\pi i \\theta } u ( x ) \\\\ u ( x - \\alpha ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\Psi ( x , \\tau ) = c _ 1 ( \\tau ) \\Upsilon _ 1 ( x , \\tau ) + c _ 2 ( \\tau ) \\Upsilon _ { - } ( x , \\tau ) \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} & \\eta ( 0 ) = 0 , \\eta ( t ) \\mbox { i s n o n - d e c r e a s i n g a n d } \\int _ 0 ^ T \\zeta _ 0 ( t ) \\ , \\eta ( d t ) = 0 , \\\\ & { \\tilde { \\eta } } ( 0 ) = 0 , { \\tilde { \\eta } } ( t ) \\mbox { i s n o n - d e c r e a s i n g a n d } \\int _ 0 ^ T { \\tilde { \\zeta } } _ 0 ( t ) \\ , { \\tilde { \\eta } } ( d t ) = 0 . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ i = \\pi _ i , \\ , 1 \\le i \\le n ) = \\mathbb { E } \\left ( P _ { \\pi _ 1 } \\prod _ { i = 2 } ^ { n } \\frac { P _ { \\pi _ i } } { 1 - \\sum _ { j = 1 } ^ { i - 1 } P _ { \\pi _ j } } \\right ) . \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } \\langle b ( s , x ) , \\nabla f _ { s } ( x ) \\rangle f _ { s } ( x ) \\ ; d x = 0 . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} I : = ( T _ 1 ^ { d _ 1 } - f _ 1 T ^ { d _ 1 } , \\dots , T _ r ^ { d _ r } - f _ r T ^ { d _ r } ) \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} k ( t ) : = \\left \\{ \\begin{array} { l } \\displaystyle \\sin \\frac { \\pi t } { T } \\ , \\qquad \\mbox { i f } \\ , \\ , t \\in \\ [ 0 , T ] \\ , , \\\\ \\\\ 0 \\ , \\qquad \\qquad \\quad \\ \\ \\ \\ \\mbox { o t h e r w i s e } \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\vec q _ { x , \\kappa ^ i r } = D u ( x ) \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | \\leq \\binom { n } { m } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { Y ^ { \\ast } } \\eta _ { 0 } ( x , t , y ) \\cdot v ( x , t , y ) d y d x d t \\\\ = - \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { Y ^ { \\ast } } u ( x , t ) \\nabla _ { x } \\cdot v ( x , t , y ) d y d x d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ( x , t ) \\left ( \\nabla _ { x } \\cdot \\int _ { Y ^ { \\ast } } v ( x , t , y ) d y \\right ) d x d t \\end{gather*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } P _ k ( n ) ^ 2 e ^ { - n / X } & = \\delta _ { [ k = 3 ] } C ' X ^ 2 \\log X + C X ^ { k - 1 } + \\delta _ { [ k = 4 ] } C '' X ^ { \\frac { 5 } { 2 } } + O ( X ^ { k - 2 + \\epsilon } ) , \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} m _ { u v } : = \\left \\{ \\begin{array} { l l } 0 , & u = v ; \\\\ | [ u ] _ h | , & \\rho ( [ u ] , [ v ] ) = 1 ; \\\\ \\max \\left \\{ \\sum _ { i = 0 } ^ { j - 1 } m _ { z _ { i } z _ { i + 1 } } \\colon [ z _ 0 ] , [ z _ 1 ] , \\dots , [ z _ j ] \\textrm { ~ i s a ~ } [ u ] [ v ] \\textrm { - c h a i n } \\right \\} , & \\end{array} \\right . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} q _ r : = P \\bigl ( V _ 0 ^ { f , g } = r \\bigr ) ( \\mbox { f o r a l l i n t e g e r } \\ r \\geq 0 ) ; \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} \\lambda = \\lambda ( N ) : = \\frac { \\sqrt N + 1 } { \\bar \\delta } . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} B ( \\varphi , \\delta ) = \\{ x : \\| x - \\varphi \\| _ \\infty \\leq \\delta \\} . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} ( U \\cup \\{ u ' \\} ) + V = ( U + V ) \\cup ( \\{ u ' \\} + V ) \\subseteq A \\cup [ u ' , u ' + \\max ( V ) ] \\subseteq A . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} L _ s - \\lambda _ 0 = ( - \\partial - \\phi _ 0 ) ( \\partial - \\phi _ 0 ) , \\ , \\ , \\mbox { i n } K [ \\partial ] . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} L _ 1 ( I ) \\cap L _ { \\infty } ( I ) \\hookrightarrow E \\hookrightarrow L _ 1 ( I ) + L _ { \\infty } ( I ) \\ \\ \\ \\ \\ I = [ 0 , \\alpha ) , \\ \\ \\ \\ \\ell _ 1 \\hookrightarrow E \\hookrightarrow \\ell _ \\infty \\ \\ \\ \\ \\ I = \\mathbb { N } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\widehat \\Lambda _ j = \\bar \\Lambda _ j - \\bar \\Lambda _ j ^ 0 , 1 \\leq j \\leq k . \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} r _ { \\langle \\rangle } ( \\mathcal B _ { \\exists , \\varphi } ^ \\beta P ) & = r _ { \\langle \\rangle } ( P ) , \\\\ n ( \\mathcal B _ { \\exists , \\varphi } ^ \\beta P , a ) & = \\mathcal B _ { \\exists , \\varphi } ^ \\beta n ( P , a ) . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} b _ { 2 , 1 } = u _ 1 + b _ { 1 , 1 } . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} P \\left [ X ( t + h ) = y | X ( t ) = x \\right ] = \\lambda ( t , x , y , \\alpha , m ) \\cdot h + o ( h ) \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d A } { d t } & = \\frac { - A ^ 3 } { \\lambda _ 1 \\lambda _ 2 } \\left ( \\frac { A D ^ 2 } { \\lambda _ 1 \\lambda _ 4 ^ 2 \\lambda _ 5 } + \\frac { B } { \\lambda _ 3 D } \\right ) \\\\ \\frac { d B } { d t } & = \\frac { A ^ 3 D ^ 2 } { \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 4 ^ 2 \\lambda _ 5 } \\left ( \\frac { - B ^ 4 } { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 } + B \\right ) \\\\ \\frac { d D } { d t } & = \\frac { A ^ 2 B } { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 } . \\end{aligned} \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\begin{aligned} p ( z ) & = c \\det ( ( z _ 1 + i ) P _ 1 + ( z _ 2 + i ) P _ 2 - D ( ( z _ 1 - i ) P _ 1 + ( z _ 2 - i ) P _ 2 ) ) \\\\ & = c \\det ( ( I - D ) \\Delta ( z ) + i ( I + D ) ) \\end{aligned} \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} 0 & = \\left \\| p / q \\right \\| - \\left \\| p / q ' \\right \\| = \\sum _ { j = 1 } ^ { m } w _ j ( ( q b _ j - p c _ j ) - ( q ' b _ j - p c _ j ) ) = \\sum _ { j = 1 } ^ { m } w _ j ( q - q ' ) b _ j \\\\ & = ( q - q ' ) \\sum _ { j = 1 } ^ { m } w _ j b _ j = ( q - q ' ) \\sum _ { j = 1 } ^ { m } w _ j | 1 \\cdot b _ j - 0 \\cdot c _ j | \\\\ & = ( q - q ' ) \\left \\| 0 / 1 \\right \\| . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\overline { F } _ { \\xi _ k } ( x / 2 ) = o \\bigl ( \\ , \\overline { F } _ { \\xi _ { ( \\eta ) } } ( x ) \\bigr ) , \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} & ( M _ r { V } ) _ S = \\\\ & \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\left [ \\binom { | S | - 1 } { a - 1 } \\binom { r - | S | - 3 } { b } + \\binom { | S | - 1 } { a } \\binom { r - | S | - 3 } { b - 1 } \\right . \\\\ & \\left . \\qquad - \\binom { | S | - 1 } { a - 1 } \\binom { r - | S | - 3 } { b } - \\binom { | S | - 1 } { a } \\binom { r - | S | - 3 } { b - 1 } \\right ] = 0 . \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} h ( x + \\lambda v ) = h ( x + v ) \\mbox { f o r a l l } \\lambda > 0 , \\mbox { f o r e v e r y } v \\in \\R ^ { m } , \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} \\norm { p } ^ { 2 } = \\norm * { \\sum _ { i \\in I } \\alpha _ { i } e _ { i } } ^ { 2 } = \\sum _ { i \\in I } \\alpha _ { i } ^ { 2 } . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\tilde { \\Omega } = \\{ \\omega = ( \\omega ( t ) ) _ { t \\ge 0 } \\in C ( [ 0 , \\infty ) , \\R ^ d _ { \\Delta } ) \\ , : \\ , \\omega ( t ) = \\Delta \\forall t \\ge \\zeta ( \\omega ) \\} \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 ^ N } \\sum _ { n \\ge 2 ^ N + 1 } ( n + 1 ) ^ { - \\alpha } c _ { k + n } \\varepsilon _ k : = w _ { N } \\underset { N \\to \\infty } \\longrightarrow 0 \\mbox { $ m $ - a . e . } \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} f ^ { 3 , 0 } ( Y , w ) = f ^ { 0 , 3 } ( Y , w ) = 1 , f ^ { 2 , 1 } ( Y , w ) = f ^ { 1 , 2 } ( Y , w ) = p h ( Y , w ) - 2 + h ^ { 1 , 2 } ( Z ) . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} a _ k & = \\sqrt { 2 \\ln ( k ) } \\ , \\ \\ \\forall k \\in \\mathbb { N } \\\\ b _ k & = \\sqrt { 2 \\ln ( k ) } - \\frac { \\ln \\ln ( k ) - \\ln ( 4 \\pi ) } { 2 \\sqrt { 2 \\ln ( k ) } } \\ , \\ \\ \\forall k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\chi _ 3 ( 0 1 0 ) & = T ( 1 1 ) + \\frac { P ( 1 1 ) ( 0 ) } { q ^ { q + 2 } } \\\\ \\chi _ 3 ( 1 0 1 ) & = T ( 1 1 ) + \\frac { P ( 1 1 ) ( 1 ) } { q ^ { q + 2 } } \\\\ \\chi _ 3 ( 2 2 2 ) & = T ( 1 1 ) + \\frac { P ( 1 1 ) ( 2 ) } { q ^ { q + 2 } } , \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} \\forall x , y \\in \\R : V _ { n , l } \\ ( x , y \\ ) = \\int _ { \\R ^ 3 } e ^ { i \\ ( x - w \\ ) \\xi + i \\ ( F \\ ( w \\ ) - y \\ ) \\eta } f \\ ( w \\ ) \\tilde { \\psi _ n } \\ ( \\xi \\ ) \\frac { \\psi _ l \\ ( \\eta \\ ) } { \\ ( 1 + \\eta ^ 2 \\ ) ^ { \\frac { 1 + \\epsilon } { 2 } } } \\mathrm { d } w \\mathrm { d } \\xi \\mathrm { d } \\eta . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\bar { r } _ { a o } \\ , : = \\ , r _ { p _ { a o } } \\ , = \\ , r _ { a | b _ n | } \\circ r _ { | b _ n | \\partial _ 0 b _ { n - 1 } } \\circ \\ldots \\circ r _ { | b _ 1 | o } \\ , \\\\ \\\\ \\bar { v } _ { a o } \\ , : = \\ , v _ { | b _ n | a } ^ { - 1 } \\cdot v _ { | b _ n | \\partial _ 0 b _ { n - 1 } } \\cdot \\ldots \\cdot v _ { | b _ 1 | o } \\ . \\end{array} \\right . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\operatorname * { O p } \\nolimits ^ { \\mathrm { B } } ( a ) \\Psi ( z ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\mathbb { R } ^ { 2 n } } a _ { \\sigma } ( z _ { 0 } ) \\widetilde { T } ( z _ { 0 } ) \\Psi ( z ) d ^ { 2 n } z _ { 0 } . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ \\frac { 2 \\sqrt 2 } { ( 2 \\pi ) ^ { ( d + 2 ) / 2 } } \\int _ r ^ \\infty \\frac { \\sinh s } { \\sqrt { \\cosh s - \\cosh r } } \\cdot \\left ( \\frac { - 1 } { \\sinh s } \\partial _ s \\right ) ^ { \\frac { d } { 2 } } \\frac { \\sin ( s \\cdot \\sqrt { \\lambda - b _ d } ) } { s } \\ , d s \\ , . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} L ^ T \\Gamma L = \\left ( \\begin{array} { c c } \\lambda _ 1 ^ 2 & 0 \\\\ 0 & \\lambda _ 2 ^ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\nu _ 1 \\ , = \\ , i d \\ \\ \\ , \\ \\ \\ \\nu _ l \\circ \\nu _ m \\ , = \\ , \\alpha _ { \\gamma ( l , m ) } \\circ \\nu _ { l m } \\ \\ \\ , \\ \\ \\ \\forall l , m \\in \\Pi \\ . \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} ( c _ 1 , c _ 2 ) & = ( 1 / e , ( s _ 2 - s _ 1 ) / e - 1 / 2 ) = ( 1 / e , ( 2 k - 1 ) / 2 - n / e + ( 1 + a ) / e ) \\\\ & = ( r / n , ( 2 k - 1 ) / 2 - r + r ( 1 + a ) / n ) = ( r / n , p / 2 - r + ( r s ) / n ) \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} B V ( q ) _ { i - \\frac { 1 } { 2 } } = | q ^ { \\xi ' } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q ^ { \\eta ' } _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) | , \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{gather*} \\prod _ { 1 \\leq i < j \\leq m } { ( - 1 ) ^ { \\tau _ { i , j } } T _ { q , x _ i } ^ { \\tau _ { i , j } } T _ { q , x _ j } ^ { 1 - \\tau _ { i , j } } } = ( - 1 ) ^ { \\sum \\limits _ { 1 \\leq i < j \\leq m } { \\tau _ { i , j } } } \\prod _ { 1 \\leq i < j \\leq m } { T _ { q , x _ i } ^ { \\tau _ { i , j } } T _ { q , x _ j } ^ { 1 - \\tau _ { i , j } } } . \\end{gather*}"}
-{"id": "2309.png", "formula": "\\begin{align*} M = \\sup _ { s \\in [ 0 , t ] , z \\in B ( x _ { 0 } , R ) ^ { c } } \\Gamma _ { s } f ( z ) . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\lambda ^ { q } \\mu ^ { - Q } \\int _ { \\mathbb { G } } | u ( x ) | ^ { q } d x = \\int _ { \\mathbb { G } } | \\phi ( x ) | ^ { q } d x . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} f ^ { T } ( v _ q ) = d f ( q ) ( v _ q ) \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} q ( a , b ) = \\frac { a ^ 2 } { 2 ^ { 2 \\ell } } - \\frac { b ^ 2 } { 4 } \\ . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} P _ { j \\ , n - 1 } = P _ { j n } = Q _ { j \\ , n - 1 } = Q _ { j n } = 0 . \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} b _ { n + 1 , 1 } = u _ n + \\sum _ { i = 1 } ^ n b _ { i , 1 } . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} b _ M ( x ) \\ , : = \\ , b \\bigl ( 2 ^ { M / 2 } \\ , \\nabla \\rho _ 0 ( x ) \\bigr ) \\ , . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\tilde \\Theta ( x ) : = \\int _ { \\Gamma _ 0 } d \\mathcal { H } ^ { n - 1 } ( y ) \\left ( - \\frac { \\Theta } { ( N \\cdot \\nu ) } \\right ) ( y ) \\ \\nu ( x ) \\cdot \\nabla P ( x - y ) . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} \\| s \\| _ g : = \\sup \\{ | s | _ g ( x ) \\mid x \\in X ^ { \\mathrm { a n } } \\} . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} w _ G ( s _ 1 ^ { - 1 } t _ 1 u _ 1 , \\ldots , s _ d ^ { - 1 } t _ d u _ d ) & = w _ G ( \\vec { s } ^ { - 1 } \\vec { t } \\vec { u } ) = \\varphi ( \\vec { s } ) ^ { - 1 } \\varphi ( \\vec { t } ) \\varphi ( \\vec { u } ) = w _ G ( \\vec { s } ) ^ { - 1 } w _ G ( \\vec { t } ) w _ G ( \\vec { u } ) \\\\ & = w _ G ( s _ 1 , \\ldots , s _ d ) ^ { - 1 } w _ G ( t _ 1 , \\ldots , t _ d ) w _ G ( u _ 1 , \\ldots , u _ d ) . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\widehat { q } ( m , n ) : = { n - 1 \\choose m - 1 } \\mathbb { E } W ^ { n - m } ( 1 - W ) ^ m \\mbox { f o r } ~ m \\le n . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\ker R = \\ker A = \\{ x \\in X \\ ; { : } \\ ; \\langle A x , x \\rangle = 0 \\} . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} T \\gamma ( X _ { \\sigma } ^ { \\gamma } ) = X _ { \\sigma } \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} D ( \\mathcal { A } ) = \\left \\lbrace u \\in D ( a _ c ) : \\exists \\ , v \\in L ^ 2 _ \\mu \\quad \\textit { s . t . } a _ c ( u , \\phi ) = \\int _ { \\R ^ N } v \\phi \\ , d \\mu \\forall \\phi \\in D ( a _ c ) \\right \\rbrace , \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{gather*} \\sum _ { N \\in \\N } \\| \\ , \\max _ { 2 ^ N \\le n \\le 2 ^ { N + 1 } - 1 } | \\sum _ { k = 2 ^ N } ^ n a _ n P ^ k f | \\ , \\| _ { L ^ 2 ( m ) } ^ 2 \\le \\sum _ { N \\in \\N } A _ { 2 ^ { N + 1 } } ^ 2 \\| P ^ { 2 ^ N } f \\| _ { L ^ 2 ( m ) } ^ 2 < \\infty \\ , . \\end{gather*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathbf { k } , \\varepsilon ) : = \\varepsilon ^ 2 ( \\mathcal { H } _ 0 ( \\mathbf { k } ) + \\varepsilon ^ 2 I ) ^ { - 1 } . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} Q _ \\infty : = \\lim _ { t \\to \\infty } Q _ t \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho } H ( t , \\rho ) | _ { \\rho = 0 } = \\int \\limits _ { - 1 } ^ t f ( s ) d s = F ( t ) . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} n \\le \\sum _ { x \\in V } g ( N ^ - [ x ] ) & = g ( V ) + \\sum _ { x \\in V } g ( N ^ - ( x ) ) \\\\ & = g ( V ) + \\sum _ { u x \\in E } g ( u ) \\\\ & = g ( V ) + \\sum _ { u \\in V } g ( u ) | N ^ + ( u ) | \\\\ & \\leq g ( V ) + d \\cdot g ( V ) \\\\ & = ( d + 1 ) g ( V ) < n , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} x _ { n + 1 } = T ( r , x _ n ) : = \\left \\{ \\begin{array} { l l } r x _ n / 2 , & w h e n ~ x _ n < 0 . 5 , \\\\ \\\\ r ( 1 - x _ n ) / 2 , & w h e n ~ x _ n \\geq 0 . 5 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{gather*} \\frac { \\big ( x t ^ N ; q \\big ) _ { \\infty } } { ( x q ; q ) _ { \\infty } } \\times \\sum _ { n = 0 } ^ { \\infty } { \\frac { ( - 1 ) ^ n q ^ { { n + 1 \\choose 2 } } ( q ^ { \\theta N - n } ; q ) _ n } { ( q ; q ) _ n } x ^ n } . \\end{gather*}"}
-{"id": "4768.png", "formula": "\\begin{align*} a = m a _ 1 , b = m b _ 1 ( a _ 1 , m ) = ( a _ 1 , b _ 1 ) = ( b _ 1 , m ) = 1 . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\log d \\arg \\sum _ { i , j = 1 } ^ n ( \\kappa ^ { - 1 } ) _ { i j } l _ i \\wedge m _ j . \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} \\Phi _ h \\big ( ( r _ 0 x _ 0 + \\cdots + r _ \\ell x _ \\ell ) ^ h \\big ) = \\bigg ( \\sum _ { 0 \\le i , j \\le \\ell } r _ i r _ j z _ { i j } \\bigg ) ^ { h / 2 } . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} ( I + \\delta t M ) \\boldsymbol u ^ { n + 1 } = \\boldsymbol u ^ n + \\delta t \\boldsymbol g , \\boldsymbol u ^ 0 = \\boldsymbol f , \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} & y _ k : = \\sum _ { i , j } A ( i , j , k ) , ~ k = 1 , 2 , 3 , \\\\ & y ^ 0 _ k : = \\frac { y _ k } { m \\times n \\times 2 5 5 } , ~ k = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\begin{array} { l l l } | s ^ \\Delta _ 0 | > \\rho & & d ^ \\Delta = 0 \\ , , \\\\ s ^ \\Delta _ 0 > \\rho & & d ^ \\Delta > 0 \\ , , \\\\ s ^ \\Delta _ 0 < - \\rho & & d ^ \\Delta < 0 \\ , , \\end{array} \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} \\nabla ( \\varpi _ 2 ) \\otimes ( - \\varpi _ 2 ) = \\begin{tabular} { | c | } \\hline $ \\varepsilon $ \\\\ \\hline $ \\nabla ^ { P } ( 3 \\varpi _ 1 - 2 \\varpi _ 2 ) $ \\\\ \\hline $ \\nabla ^ { \\alpha _ 1 } ( 2 \\varpi _ 1 - 2 \\varpi _ 2 ) $ \\\\ \\hline $ - \\varpi _ 2 $ \\\\ \\hline $ \\nabla ^ { \\alpha _ 1 } ( 3 \\varpi _ 1 - 3 \\varpi _ 2 ) $ \\\\ \\hline $ - 2 \\varpi _ 2 $ \\\\ \\hline \\end{tabular} . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} f _ { \\theta , q , 3 } ( \\bar { x } ) - f _ { \\theta , q , 3 } ( \\bar { \\bar { x } } ) = \\frac { ( \\theta - 1 ) ( 1 - \\theta - q ) ( \\bar { \\bar { x } } - \\bar { x } ) } { ( \\bar { x } - \\mathbf { x } ^ { \\infty } ) ( \\bar { \\bar { x } } - \\mathbf { x } ^ { \\infty } ) } F _ { \\theta , q , 3 } \\left ( \\bar { x } , \\bar { \\bar { x } } \\right ) \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} d _ { p , q ; E } ( u ) = ( { \\rm p r } _ t ) _ ! ( { \\rm p r } _ s ^ * ( u ) ) . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\frac { A \\otimes W - \\left ( \\overline { A \\otimes W } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } = \\left ( V \\otimes Z \\right ) \\left ( \\overline { V \\otimes Z } \\right ) ^ { t } , \\quad \\mbox { f o r s o m e i n v e r t i b l e m a t r i x $ V \\in \\mathcal { M } _ { q N \\times q N } \\left ( \\mathbb { C } \\right ) $ . } \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} f ( x , h ) = \\bigl [ K ^ h _ { ( \\Lambda , d \\mu ) } A \\bigr ] ( x ; h ) \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} K = \\sup _ { 0 \\leq t \\leq T } t ^ { \\gamma } \\{ | | u | | _ { \\tilde { L } ^ { p } } + t ^ { \\frac { 1 } { 2 } } | | \\nabla u | | _ { \\tilde { L } ^ { p } } \\} \\leq 2 K _ 1 . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} \\nabla ^ { ^ M } _ { X } Y = \\mathcal { A } _ { X } Y + \\mathcal { H } ( \\nabla ^ { ^ M } _ { X } Y ) , \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{gather*} \\| v _ { N } \\| _ { L ^ 2 ( m ) } ^ 2 = \\sum _ { k \\ge 2 ^ N + 1 } ( \\sum _ { n = 0 } ^ { 2 ^ N } ( n + 1 ) ^ { - \\alpha } c _ { k + n } ) ^ 2 \\le C 2 ^ { 2 N ( 1 - \\alpha ) } \\sum _ { k \\ge 2 ^ N + 1 } c _ { k } ^ 2 \\le \\frac { C ' } { N ( \\log ( N + 1 ) ) ^ 3 } \\ , . \\end{gather*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\varpi \\left ( \\xi \\right ) = f ^ { - 3 / 4 } \\left ( z \\right ) p \\left ( z \\right ) , \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} | { \\cal E } ^ { ( \\nu ) } | = \\hat { c } | R _ { \\rm h y b } | \\nu ^ { 1 / 2 } \\leq C \\nu ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} L ^ { \\rm r e s } _ \\ell : = \\frac { L _ { ( 0 , 0 ) \\to ( \\eta \\ell , \\ell ) } - \\mu } { \\sigma \\ell ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\sup _ { \\tau < t \\le T } | B ^ n _ 0 ( t ) - B ^ n _ 0 ( \\tau ) | & \\le 4 r ( \\boldsymbol { \\zeta } ^ n ( \\tau ) ) , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} W _ { 0 , \\nu } ( y ) = \\big ( \\frac { y } { \\pi } \\big ) ^ { \\frac { 1 } { 2 } } K _ \\nu \\big ( \\frac { y } { 2 } \\big ) , K _ \\nu ( y ) = \\frac { 1 } { 2 } \\int _ 0 ^ \\infty e ^ { - \\frac { 1 } { 2 } y ( u + u ^ { - 1 } ) } u ^ \\nu \\frac { d u } { u } . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\frac { \\| x _ { L , k } \\| } { \\| x _ { t r u e } \\| } = { \\cal O } ( 1 ) \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} ( 1 + \\mu _ 2 ( t ) ) ^ 2 = \\frac { \\overline { c _ 2 } ( t ) } { c _ { 2 , \\infty } } \\geq \\frac { h ( t ) } { c _ { 2 , \\infty } } \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} ( \\phi _ 1 , \\dots , \\phi _ n ) \\mapsto \\sum _ { i = 1 } ^ n ( - 1 ) ^ { n - i } \\Big [ \\alpha ( \\phi _ 1 ( 0 ) , \\dots , \\phi _ { i - 1 } ( 0 ) , \\widehat { \\phi _ i ( 0 ) } , \\phi _ { i + 1 } ( 0 ) , \\dots , \\phi _ n ( 0 ) ) \\Big ] \\big ( \\dot { \\phi _ i } ( 0 ) \\big ) \\ , . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\| R _ j g \\| _ { F _ { \\b + c } } ^ 2 \\leq A _ c ^ 2 \\| g \\| _ { F _ { \\b + c } } ^ 2 \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} \\frac { P ( X ) } { Q ( X ) } & = \\frac { a _ l \\left ( \\frac { G ( X ) } { H ( X ) } \\right ) ^ l + \\ldots + a _ s \\left ( \\frac { G ( X ) } { H ( X ) } \\right ) ^ s } { b _ m \\left ( \\frac { G ( X ) } { H ( X ) } \\right ) ^ m + \\ldots + b _ t \\left ( \\frac { G ( X ) } { H ( X ) } \\right ) ^ t } \\\\ & = H ( X ) ^ { m - l } G ( X ) ^ { s - t } \\frac { q ( X ) } { r ( X ) } , \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} F _ { i } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\rho _ { i } \\left ( y \\right ) d y , \\quad \\mathbb { F } ^ { i } : = \\left ( \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\rho _ { i } \\left ( y \\right ) \\frac { \\partial \\bar { u } _ { i } ^ { j } } { \\partial y _ { k } } d y \\right ) _ { j k } , \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} ( g \\rtimes \\sigma ) \\cdot ( x _ 1 , \\ldots , x _ k ) = ( ( - 1 ) ^ { g _ 1 } x _ { \\sigma ^ { - 1 } ( 1 ) } , \\ldots , ( - 1 ) ^ { g _ k } x _ { \\sigma ^ { - 1 } ( k ) } ) \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} S _ { n } ^ { \\pm } \\left ( { u , \\xi } \\right ) = \\sum \\limits _ { s = 1 } ^ { n - 1 } \\left ( \\pm 1 \\right ) ^ { s } { \\frac { E _ { s + 1 } ^ { \\pm } \\left ( \\xi \\right ) } { u ^ { s } } , } \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} h ( x , \\xi , \\lambda ) = \\lambda ^ { - \\frac { r - 1 } { r } } \\left [ | x - x _ { 0 } | ^ { 2 } + | \\xi - \\xi _ { 0 } | ^ { 2 } \\right ] . \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} \\log ( 1 + x ) = \\frac { \\log ( 1 + x ) ^ { \\rho } } { \\rho } \\le \\frac { \\log ( 1 + x ^ { \\rho } ) } { \\rho } \\le \\frac { x ^ { \\rho } } { \\rho } \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} m _ { i i } ( \\zeta ) = g _ 0 ( \\zeta _ i ) , m _ { i j } ( \\zeta ) = - G _ 0 ( \\zeta _ i , \\zeta _ j ) , i \\not = j . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\sum _ { \\substack { N ( a ) \\leq x \\\\ a \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } 1 = \\frac { \\pi } { 8 } x + O ( x ^ { \\theta } ) . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{gather*} g x = \\tau _ 1 \\tau _ 2 \\dotsm \\tau _ n x \\ \\forall x \\in X \\textrm { a n d } g y = \\tau _ 1 \\tau _ 2 \\dotsm \\tau _ n y \\ \\forall y \\in Y . \\end{gather*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , X ) & = - g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { H } \\nabla \\nabla ^ { ^ { M _ 1 } } _ { Z } \\omega \\phi W ) , \\pi _ { \\ast } ( X ) ) + g _ { 2 } ( \\mathcal { T } _ { Z } \\omega W , \\mathcal { B } X ) \\\\ & + g _ { 2 } ( \\pi _ * ( \\nabla ^ { ^ { M _ 1 } } _ { Z } \\omega \\phi W ) , \\pi _ * ( X ) ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} [ x _ p ] = \\varphi _ { p r } ^ { \\boxplus \\tau } ( [ x _ r ] ) , [ x _ q ] = \\varphi _ { q r } ^ { \\boxplus \\tau } ( [ x _ r ] ) . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} V \\otimes Z = \\left ( \\displaystyle \\sum _ { l = 1 } ^ { N } \\displaystyle \\sum _ { k = 1 } ^ { q } v _ { k l } ^ { i j } z _ { k l } \\right ) _ { 1 \\leq i \\leq q \\atop 1 \\leq j \\leq N } . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} X = a _ { 1 } \\left ( \\frac { 4 } { a + 2 } t \\right ) \\partial _ { t } + \\left ( a _ { 0 } X _ { I J } + a _ { 1 } H ^ { i } \\right ) \\partial _ { i } . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} m _ k = \\int \\limits _ { - 1 } ^ 1 f ( t ) ( G ( t ) ) ^ k d t = 0 , ~ ~ k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\phi _ { k } \\left ( ( a _ { i j } ) _ { 1 \\leq i , j \\leq k } \\right ) = \\left ( \\phi ( a _ { i j } ) \\right ) _ { 1 \\leq i , j \\leq k } \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{gather*} \\overline { \\partial } { } ^ 2 \\eta = \\int _ 0 ^ t \\overline { \\partial } { } ^ 2 v . \\end{gather*}"}
-{"id": "2803.png", "formula": "\\begin{align*} C = \\frac { \\Gamma ( \\frac { 3 } { 2 } ) } { 4 \\pi ^ 2 } \\frac { L ( \\frac { 3 } { 2 } , f \\times \\overline { g } ) } { \\zeta ( 3 ) } , C ' = \\frac { \\Gamma ( \\frac { 3 } { 2 } ) } { 4 \\pi ^ 2 } \\frac { L ( \\frac { 3 } { 2 } , f \\times g ) } { \\zeta ( 3 ) } . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} T N ^ { \\bot } = \\{ u \\in T M | \\omega ( u , v ) = 0 , \\forall v \\in T N \\} . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\partial _ t v + { \\mathcal L } v = \\nabla \\cdot ( v \\nabla V ) \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} P ( \\mathcal { N } ) = \\{ N _ { \\chi _ A } : \\ , A [ 0 , \\alpha ) \\} . \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\Psi ( h ) = o \\Big ( h ^ { - \\frac { n - m } m } \\Big ) h \\to \\infty . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} I _ { 1 1 2 2 1 } & = - \\int _ { 0 } ^ { t } \\int _ { \\Gamma _ 1 } \\sqrt { g } g ^ { i j } \\hat n ^ { \\sigma } g ^ { k l } \\partial _ { k } v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } \\partial _ { j } \\partial _ { t } ^ { 2 } v ^ { \\mu } \\Pi _ { \\sigma } ^ { \\alpha } \\partial _ { i } \\partial _ { t } ^ { 2 } v _ { \\alpha } . \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\frac { a } { b } < \\nu _ e \\qquad { a n d } \\gcd ( a , b ) = 1 . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } W ( t ) + F ( t , W ( t ) ) = 0 , & t \\in [ 0 , T [ , \\\\ W ( T ) = \\Psi , & \\end{cases} \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} \\mathcal H ( \\tau , t , x ) = \\begin{cases} H ^ 1 ( t , x ) & \\\\ H ^ 2 ( t , x ) & \\end{cases} \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} | I | = n = q d + r . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} L a w ( X _ { \\rho _ 3 , m } ) = \\theta L a w ( X _ { \\rho _ 1 , m } ) + ( 1 - \\theta ) L a w ( X _ { \\rho _ 2 , m } ) \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} V \\left ( t , x ^ { k } \\right ) = \\omega \\left ( t \\right ) V \\left ( x ^ { k } \\right ) ~ \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} e _ { \\rm n u m } = ( E F ) ^ { - p } . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\tilde { u } _ n ( x ) = u _ n ( x ) - \\log \\left ( \\int _ M h e ^ { u _ n } \\mathrm { d } \\mu \\right ) . \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\tilde { x } ( t ) & = x _ 1 ( t ) + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { \\nu } ^ { t } ( t - s ) ^ { \\alpha - 1 } f ( s , \\tilde { x } ( s ) ) d s \\\\ & = \\frac { x _ 0 } { \\Gamma ( \\gamma ) } t ^ { 1 - \\gamma } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } f ( s , \\bar { x } ( s ) ) d s \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha - \\delta , \\beta } \\hat { y } ( x ) = f ( x , \\hat { y } ) , 0 < \\alpha < 1 , \\ , 0 \\leq \\beta \\leq 1 , \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\varGamma ( \\rho ^ { 2 } t , \\rho x + z ; 0 , \\rho \\xi + z ) = \\rho ^ { - n } \\varGamma ^ { ( a _ { \\rho , z } , b _ { \\rho , z } ) } ( t , x ; 0 , \\xi ) \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} ( 1 \\pm \\gamma ) \\prod _ { \\substack { \\{ k , \\ell \\} \\in S \\\\ \\{ k , \\ell \\} \\neq \\{ i , j \\} } } \\frac { \\alpha _ S } { d _ { k \\ell } } ( 1 \\pm \\gamma ) \\left ( \\prod _ { \\substack { \\{ k , \\ell \\} \\in S \\\\ \\{ k , \\ell \\} \\neq \\{ i , j \\} } } d _ { k \\ell } \\right ) \\prod _ { a \\in S , a \\neq i , j } | V _ a | = ( 1 \\pm 2 \\gamma ) \\alpha _ S ^ { \\binom { r } { 2 } - 1 } ( n / t ) ^ { r - 2 } \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 = O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\lambda _ 1 = & l + \\mu _ 1 \\cr \\geq & l + ( l - k _ 1 ) + l ( k _ 1 - 1 ) + \\cdots + ( l - k _ { t - 1 } ) + l ( k _ { t - 1 } - 1 ) + l ( k _ t - 1 ) \\cr & = l + ( t - 1 ) l - ( m - k _ t ) + l m - l t = m ( l - 1 ) + k _ t \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\tilde { \\rho } ( \\tilde { \\theta } _ { 1 } ^ { \\alpha } , x ) = P _ { \\tilde { \\pi } ^ { G E L } } ( \\theta _ 1 < \\tilde { \\theta } ^ { \\alpha } _ 1 | x ) = \\frac { \\int ^ { \\tilde { \\theta } ^ { \\alpha } _ 1 } \\ldots \\int e ^ { - \\frac { 1 } { 2 } \\tilde { l } _ \\gamma ^ { G E L } ( \\theta ) + \\xi ( \\theta ) } d \\theta _ d \\ldots d \\theta _ 1 } { \\int \\ldots \\int e ^ { - \\frac { 1 } { 2 } \\tilde { l } _ \\gamma ^ { G E L } ( \\theta ) + \\xi ( \\theta ) } d \\theta _ d \\ldots d \\theta _ 1 } = \\alpha \\ , . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\frac { 1 } { X } \\int _ 0 ^ X \\Big \\lvert \\sum _ { n \\leq t } r _ 2 ( n ) - \\pi t \\Big \\rvert ^ 2 d t = c X ^ { 1 / 2 } + O ( X ^ { \\frac { 1 } { 4 } + \\epsilon } ) , \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\left ( \\delta _ { A ' } ^ A u , v \\right ) _ \\varphi = \\left ( u , Z _ { A } ^ { A ' } v \\right ) _ \\varphi . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} I : = \\varphi ( I _ { \\Delta } ) = ( ( y _ 2 - y _ 3 - y _ 5 ) y _ 3 , ( y _ 1 - y _ 4 ) y _ 4 , ( y _ 2 - y _ 3 - y _ 5 ) y _ 5 , y _ 3 y _ 5 ) \\subseteq S . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} \\Delta ^ n f ( x ) = \\sum _ { k = 0 } ^ n { n \\choose k } ( - 1 ) ^ { n - k } f ( x + k ) , \\ , \\ , ( n \\in \\mathbb { N } \\cup \\{ 0 \\} ) . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} 1 ^ 2 + 2 ^ 2 + \\cdots + x ^ 2 = y ^ 2 , \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } - u \\left ( x , t \\right ) v \\left ( x \\right ) \\partial _ { t } c \\left ( t \\right ) + \\left ( \\int _ { \\mathcal { Y } _ { n , m } } a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) d y ^ { n } d s ^ { m } \\right ) \\cdot \\nabla v \\left ( x \\right ) c \\left ( t \\right ) d x d t \\\\ = \\int _ { \\Omega _ { T } } f \\left ( x , t \\right ) v \\left ( x \\right ) c \\left ( t \\right ) d x d t \\end{gather*}"}
-{"id": "7321.png", "formula": "\\begin{align*} B _ { j , m } ( f , g ) ( x ) = 2 ^ { - \\frac { ( b - a ) j } { 2 } } \\iint \\hat { f } ( \\xi ) \\hat { g } ( \\eta ) e ^ { 2 \\pi i \\left ( \\frac { \\xi } { 2 ^ { ( b - a ) j } } + \\eta \\right ) x } I _ { \\rho , m } \\hat { \\Phi } ( \\xi ) \\hat { \\Phi } ( \\eta ) \\ , d \\xi \\ , d \\eta , \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} C ( d , p ) : = 2 \\left ( \\frac { p } { p - 1 } + \\frac { p } { 2 - p } \\right ) ^ { 1 / p } C ( d ) ^ { 2 / p - 1 } . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} \\lambda \\langle e ^ { \\abs { x } } ( \\lambda , \\infty ) ( \\xi ) , \\xi \\rangle & \\leq \\langle | x | e ^ { \\abs { x } } ( \\lambda , \\infty ) ( \\xi ) , \\xi \\rangle = \\langle e ^ { \\abs { x } } ( \\lambda , \\infty ) | x | e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) , \\xi \\rangle \\\\ & = \\langle e ^ { \\abs { x } } ( \\lambda , \\infty ) s e ^ { \\abs { x } } ( s , \\infty ) ( \\xi ) , \\xi \\rangle = s \\langle e ^ { \\abs { x } } ( \\lambda , \\infty ) ( \\xi ) , \\xi \\rangle . \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r ( x _ i ^ j - y _ i ^ j ) = \\sum _ { l = 1 } ^ { s - r } \\left ( ( p ^ b u _ l + \\eta ) ^ j - ( p ^ b v _ l + \\eta ) ^ j \\right ) ( 1 \\le j \\le k ) , \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{gather*} \\Lambda _ F [ x _ 1 , \\dots , x _ N ] = \\bigoplus _ { m \\geq 0 } { \\Lambda _ F ^ m [ x _ 1 , \\dots , x _ N ] } . \\end{gather*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\log | p ( z ) / P _ r ( \\vec { 1 } ) | \\leq - r / 2 + \\Re ( \\sum _ { j = 1 } ^ { n } c _ j z _ j ) + B \\| z \\| _ \\infty ^ 2 \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} S _ n : = x _ 0 + \\sum _ { k = 1 } ^ n \\zeta _ k = X _ { T _ n } \\ , , \\ , \\ , n = 0 , \\dots \\ , , \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\partial ( P _ 1 \\ , _ { \\frak C _ 1 } \\cup _ { \\frak C _ 2 } P _ 2 ) = \\partial _ { \\frak C ^ c _ 1 } P _ 1 \\ , \\ , { } _ { \\frak C _ 1 } \\cup _ { \\frak C _ 2 } \\ , \\ , \\partial _ { \\frak C ^ c _ 2 } P _ 2 . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{gather*} D ( x ) = : D ^ 1 ( x ) \\subset D ^ 2 ( x ) \\subset \\cdots \\subset D ^ \\mu ( x ) = T _ x M . \\end{gather*}"}
-{"id": "9182.png", "formula": "\\begin{align*} \\langle f ^ { * } , g ^ { * } \\rangle _ { \\mathcal { C } } = \\langle f , g \\rangle _ { \\mathcal { C } } ^ { * } \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} N _ { 2 d } ^ { \\mathrm { c m } } ( X , G ) = r _ d ( G ) X + O _ { \\epsilon } ( X ^ { \\beta ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) + \\epsilon } ) , \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} K _ { j } = \\sup _ { 0 \\leq t \\leq T } t ^ { \\gamma } \\{ | | u _ j | | _ { \\tilde { L } ^ { p } } + t ^ { \\frac { 1 } { 2 } } | | \\nabla u _ j | | _ { \\tilde { L } ^ { p } } \\} , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} H _ \\omega = - \\Delta + V _ \\omega = - \\Delta + { U _ \\omega } ^ { 2 } \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\Lambda _ f ( s , \\chi ) = i ^ k \\xi ( q ) \\chi ( N ) q ^ { - 1 } \\tau ( \\chi ) ^ 2 ( N q ^ 2 ) ^ { \\frac 1 2 - s } \\Lambda _ g ( 1 - s , \\overline { \\chi } ) , \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} \\alpha _ i = \\pi _ * \\circ \\tilde \\alpha _ i \\circ \\pi ^ * . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} | S _ { n , k , r } | = { n + k - r \\choose k } \\cdot | S _ n | = { n + k - r \\choose k } \\cdot n ! . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} a ^ { ( e , v ) } _ j + a ^ { ( e , v ' ) } _ { r - j } \\geq d j = 0 , \\dots , r . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} D ^ k _ h ( s ) : = \\frac { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\langle P _ h ^ k ( \\cdot , s ) , V \\rangle = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 2 k + 1 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s + \\frac { k - 1 } { 2 } } } - \\mathfrak { E } _ h ^ k ( s ) . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\Phi _ k = \\frac { \\prod _ { i = 1 } ^ v \\vartheta ( z - t _ i + \\hbar ) \\prod _ { i = v + 1 } ^ w \\vartheta ( z - t _ i - \\hbar ) } { \\prod _ { i = 1 } ^ w \\vartheta ( z - t _ i ) } \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\Theta ( j , 2 ^ b , X ) = \\Theta ( g ( j , 0 ) , b , X _ 0 ) \\oplus \\Theta ( g ( j , 1 ) , 2 , X _ 1 ) \\leq _ { \\mathrm { T } } A _ 0 \\oplus A _ 1 = A . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} g _ \\lambda ( z ; \\tau ) : = \\frac { \\vartheta ( z ; \\tau ) \\vartheta ( \\lambda ; \\tau ) } { \\vartheta ( z + \\lambda ; \\tau ) } , \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\gamma _ { m k } ( L ) = ( k ) _ { q ^ { - 2 } } ! ( m - k ) _ { q ^ { - 2 } } ! q ^ { k ( k + 1 ) + \\frac { m ( m - 3 ) } { 2 } } \\sum _ { \\tau } q ^ { \\sum _ { i = 1 } ^ { k } l _ { m \\tau ( i ) } - \\sum _ { i = k + 1 } ^ { m } l _ { m \\tau ( i ) } } \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} X ^ { \\Delta } = \\{ ( y , z ) \\in Y \\times \\R ^ n : z \\in c o n v ( i ( f ^ { - 1 } ( y ) ) ) \\} . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\tilde { N } _ { K , L } ^ \\epsilon = F ( \\epsilon , K ) . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\sigma ^ 2 ( \\Delta _ t ) } { t } = \\mu \\gamma ^ 2 , \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} ( n + 1 ) \\log \\left ( { \\Gamma \\Big ( { n + \\nu \\over 2 } \\Big ) \\over \\Gamma \\Big ( { n \\over 2 } + { \\nu + t \\over 2 } \\Big ) } \\right ) = { ( n + 1 ) t \\over 2 } \\log { 2 \\over n } - { t \\over 4 } ( t - 2 + 2 \\nu ) + o ( 1 ) \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} Z ^ e _ t ( \\phi ) : = ( M ^ e L ^ e ) ^ { - 1 } \\sum _ { z \\in \\Lambda ^ e _ n } \\sum _ { w \\sim z } { \\bf 1 } _ { \\{ w , z \\notin x _ 1 \\} } \\int _ 0 ^ t \\xi _ { s - } ( z ) \\xi ^ c _ { s - } ( w ) \\phi _ s ( z ) \\ , d Q ^ { z , w } _ s . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} A = \\frac { 1 } { P _ r ( \\vec { 1 } ) } \\left ( \\nabla P _ r ( \\vec { 1 } ) \\cdot ( x + i \\tilde { y } ) \\left ( \\frac { 1 } { 2 m } - \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) + \\nabla P _ { r + 1 } ( \\vec { 1 } ) \\cdot ( x + i \\tilde { y } ) \\right ) \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} \\sum _ { p \\mid \\phi ( n ) } \\log I _ p ( n ) \\leq \\sum _ { p \\mid \\phi ( n ) } \\log \\big ( 2 ^ { \\nu _ p ( \\phi ( n ) ) } \\big ) = \\sum _ { p \\mid \\phi ( n ) } { \\nu _ p ( \\phi ( n ) ) } \\log 2 = \\Omega ( \\phi ( n ) ) \\log 2 , \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } E ( u , - T ) = \\mathcal A _ H ( [ \\gamma _ - , w _ - ] ) = E ( \\alpha _ - ) . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\mathcal { M } ^ 1 ( \\mathbb { R } ^ d ) = \\{ \\mu \\in \\mathcal { M } ( \\mathbb { R } ^ d ) : \\int \\| x \\| \\mu ( d x ) < + \\infty \\} . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\psi \\left ( u , \\xi \\right ) = \\sum \\limits _ { s = 0 } ^ { n - 1 } \\frac { \\psi { _ { s } \\left ( \\xi \\right ) } } { u ^ { s } } { + } \\frac { { \\Psi } _ { n } \\left ( u , \\xi \\right ) } { u ^ { n } } , \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} T = T _ 1 + \\lim _ { n \\to \\infty } \\sum _ { k = 1 } ^ { n - 1 } ( T _ { k + 1 } - T _ k ) , \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} ( \\psi ^ + _ { i , 0 } ) ^ { \\pm 1 } \\cdot ( \\psi ^ + _ { i , 0 } ) ^ { \\mp 1 } = 1 , \\ \\psi ^ + _ { i , 0 } \\psi ^ + _ { j , 0 } = \\psi ^ + _ { j , 0 } \\psi ^ + _ { i , 0 } , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} | \\mathcal { Z } _ { K , L } ^ \\epsilon | = D ( \\epsilon , K , L , 0 ) \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , X ) , \\pi _ * Y ) & = - g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { U } X , \\varphi Y ) - \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { U } X ) \\eta ( Y ) . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} C _ 1 \\ , e ^ { - \\alpha _ 2 ( 2 n - 1 ) \\frac { | x - a _ i | ^ 2 } { 2 } } \\le e ^ { - \\sum _ { i = 1 } ^ n \\frac { | A ^ { \\frac { 1 } { 2 } } ( x - a _ i ) | ^ 2 } { 2 } } \\le C _ 2 \\ , e ^ { - \\alpha _ 1 \\frac { n + 1 } { 2 } \\frac { | x - a _ i | ^ 2 } { 2 } } \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} \\Delta ( z ) = \\sum _ { n \\geq 1 } \\tau ( n ) e ( n z ) , \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\widehat { A } _ \\omega \\ ; = \\ ; \\Pi \\ , A _ \\omega \\ , \\Pi ^ * \\ ; + \\ ; \\widetilde { A } _ \\omega \\ ; , \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\varrho _ g ( r ) : = \\sup _ { x \\in \\Omega } \\ , \\sup _ { y , y ' \\in \\Omega ( x , r ) } \\abs { g ( y ) - g ( y ' ) } \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} E _ { s } \\left ( \\xi \\right ) = \\int { F _ { s } \\left ( \\xi \\right ) d \\xi } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} A & = 0 , \\\\ c & = - \\frac { 4 ( m - 1 ) ( 2 m - 3 ) } { b - a } , \\\\ d & = - \\frac { 4 ( a + b ) ( m - 1 ) ( 2 m - 1 ) } { b - a } , \\\\ e & = \\frac { 4 a b m ( 2 m - 1 ) } { b - a } \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} 2 \\theta ( E ) = \\widetilde { n } \\alpha \\ \\mod { \\mathbb { Z } } , \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} [ x ] _ r = [ d _ r ] _ r r \\in ( \\{ 1 , \\ldots , t \\} \\setminus \\{ h \\} ) . \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\max _ { v \\in \\Omega } \\| P _ v \\| & = \\max _ { v \\in \\Omega } \\sum _ { j = 1 } ^ d | P _ j ( v ) | + | a _ 0 | \\ge \\max _ { v \\in \\Omega } | P _ { j _ 0 } ( v ) | \\ge \\frac { \\mu ( \\Omega ) ^ d } { c _ { 3 } } \\| P _ { j _ 0 } \\| \\\\ & \\ge \\frac { \\mu ( \\Omega ) ^ d } { c _ { 7 } c _ { 2 } ( d + 1 ) } = : c _ { 1 } \\mu ( \\Omega ) ^ d , \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} P \\phi = \\sum _ { j \\in J } \\langle \\psi _ j , \\phi \\rangle _ { \\mathcal { A } } \\ , \\psi _ j \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i Q _ i \\Big | \\prec \\Psi \\hat { \\Pi } \\ , . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} \\xi _ { x , y } ( O ) = \\langle J ( O ) x , y \\rangle , \\ { \\rm f o r } \\ O \\in { \\rm S O } ( 2 n ) . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\sum \\limits _ { n = K + 1 } ^ { \\infty } \\mathbb { P } ( S _ { ( n ) } > \\mathit { x y } ) \\mathbb { P } ( \\eta = n ) \\leqslant c _ 3 \\overline { F } _ { \\xi _ 1 } ( \\mathit { x y } ) \\sum \\limits _ { n = K + 1 } ^ { \\infty } n ^ { p + 1 } \\mathbb { P } ( \\eta = n ) , \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ s a _ j \\ge \\log | \\L _ s ( Y ) | + \\sum \\limits _ { j = 1 } ^ s \\frac { c } { \\sqrt { j } } . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] & = 0 & [ X _ 1 , X _ 3 ] & = 0 & [ X _ 1 , X _ 4 ] & = 0 & [ X _ 1 , X _ 5 ] & = 0 & [ X _ 2 , X _ 3 ] & = 0 \\\\ [ X _ 2 , X _ 4 ] & = 0 & [ X _ 2 , X _ 5 ] & = X _ 1 & [ X _ 3 , X _ 4 ] & = X _ 1 & [ X _ 3 , X _ 5 ] & = X _ 2 & [ X _ 4 , X _ 5 ] & = 0 . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} e = \\frac { 1 + c \\epsilon } { 2 } , f = \\frac { 1 - c \\epsilon } { 2 } \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\mathtt { C } _ { P , \\Omega } = \\frac { 1 } { \\pi } \\left ( \\sum _ { i = 1 } ^ d \\frac { 1 } { \\left | l _ i \\right | ^ 2 } \\right ) ^ { - 1 / 2 } \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} V a r ( \\tilde { \\theta } _ 1 ^ \\alpha | \\psi ^ { 0 } , \\tilde { F } _ { \\gamma = 0 } ^ { G E L } ) = \\underset { \\gamma } \\inf V a r ( \\tilde { \\theta } _ 1 ^ \\alpha | \\psi ^ { 0 } , \\tilde { F } _ { \\gamma } ^ { G E L } ) \\ , . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} F _ T ( x ) & = 1 + x F _ T ( x ) + x \\big ( F _ T ( x ) - 1 \\big ) \\ , C \\ ! \\left ( x + \\frac { x ^ 2 } { 1 - x } \\big ( K ( x ) - 1 \\big ) \\right ) \\\\ & + \\frac { x \\big ( K ( x ) - 1 \\big ) } { 1 - x } \\left ( x F _ T ( x ) + x \\big ( F _ T ( x ) - 1 \\big ) \\ , C \\ ! \\left ( x + \\frac { x ^ 2 } { 1 - x } \\big ( K ( x ) - 1 \\big ) \\right ) \\right ) \\ , . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} \\left \\| \\sum _ { \\boldsymbol { m } } | D ^ { \\boldsymbol { r } } \\varphi _ { \\boldsymbol { N } , \\boldsymbol { m } } | \\right \\| _ \\infty = O ( 1 ) , \\quad \\left \\| \\sum _ { \\boldsymbol { k } } | D ^ { \\boldsymbol { r } } \\psi _ { \\boldsymbol { j } , \\boldsymbol { k } } | \\right \\| _ \\infty \\lesssim \\prod _ { l = 1 } ^ d 2 ^ { ( 1 / 2 + r _ l ) j _ l } . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} x ^ { i + 1 } = \\pi ( x ^ i - \\frac { \\eta ^ i } { N ^ i } \\sum _ { j = 1 } ^ { N ^ i } \\nabla F ( x ^ i , y _ j ^ i ) ) . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } ( - \\Delta ) ^ { 1 / 2 } u + V _ { 1 } ( x ) u = f _ { 1 } ( u ) + \\lambda ( x ) v , & x \\in \\mathbb { R } , \\\\ ( - \\Delta ) ^ { 1 / 2 } v + V _ { 2 } ( x ) v = f _ { 2 } ( v ) + \\lambda ( x ) u , & x \\in \\mathbb { R } , \\end{array} \\right . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{gather*} A ^ a { } _ b { } = 0 , W _ { a n b n } { } = W _ { b n a n } , E _ { a n b c } { } = E _ { a n c b } , \\\\ K ^ { a b } = K ^ { b a } , B ^ { a b } { } _ c = 0 , W _ { [ a | n | b c ] } = 0 . \\end{gather*}"}
-{"id": "9819.png", "formula": "\\begin{align*} ( F ( M ' ) \\circ F ( D ) \\circ F ( M ) \\circ F ( D ' ) ) ^ 2 & = F ( M ' ) \\circ F ( D ) \\circ F ( M ) \\circ F ( D ' ) . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} s ( x _ 3 ) = \\frac { \\gamma } { 2 } ( b ^ 2 - x _ 3 ^ 2 ) . \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{gather*} \\psi _ { \\xi , \\sigma } ( x ) = \\left ( \\frac { x \\cdot \\xi } { \\mu R } \\right ) ^ { \\sigma } = \\exp \\left [ \\sigma \\log \\left ( \\frac { x \\cdot \\xi } { \\mu R } \\right ) \\right ] \\end{gather*}"}
-{"id": "9089.png", "formula": "\\begin{align*} c \\sum _ { a \\in A } u _ a \\ = \\ c \\sum _ { a \\in A } v _ a \\ , , \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\varphi ( t ) - k * \\varphi ( t ) = \\psi ( t ) , t \\ge 0 \\ , , \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\kappa ( f ) = \\begin{cases} \\frac { k - 1 } { 2 } & $ f $ \\ ; , \\\\ 0 & $ f $ \\ ; , \\end{cases} \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\overline { c _ 1 } ( t ) + \\overline { c _ 3 } ( t ) = \\overline { c _ { 1 , 0 } } + \\overline { c _ { 3 , 0 } } = : M _ { 1 3 } \\overline { c _ 2 } ( t ) + \\overline { c _ 3 } ( t ) + \\overline { c _ 4 } ( t ) = \\overline { c _ { 2 , 0 } } + \\overline { c _ { 3 , 0 } } + \\overline { c _ { 4 , 0 } } = : M _ { 2 3 4 } . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} \\partial _ { t } \\omega _ { 1 } = \\nu \\Delta \\omega _ { 1 } - e ^ { - \\nu t } \\left [ \\left ( I - P _ { 1 } \\right ) \\sin y \\partial _ { x } \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\right ] \\omega _ { 1 } . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\phi _ 0 ( t ) & = x _ 0 t ^ { - \\frac { 1 } { 4 } } , t \\in ( 0 , l ] , \\\\ \\phi _ n ( t ) = 3 t ^ { - \\frac { 1 } { 4 } } + & \\int _ { 0 } ^ { t } \\frac { ( t - s ) ^ { - \\frac { 1 } { 2 } } } { \\Gamma ( \\frac { 1 } { 2 } ) } s ^ { - \\frac { 1 } { 3 } } [ 1 + ( \\phi _ { n - 1 } ( s ) ) ^ { \\frac { 4 } { 3 } } ] d s , t \\in ( 0 , l ] , n = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} K _ { ( h _ 1 , h _ 2 ) } ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) : = K _ { h _ 1 } ^ 1 ( x _ 1 , A _ 1 ) \\cdot K _ { h _ 2 } ^ 2 ( x _ 2 , A _ 2 ) \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} { \\lim } _ { t \\to \\nu ^ { - } } \\sup { | M ( t ) | } = + \\infty , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\langle ( f _ n ) _ { n \\geq 0 } , ( g _ n ) _ { n \\geq 0 } \\rangle : = \\mathbb E \\sum _ { n = 0 } ^ { \\infty } \\langle f _ n , g _ n \\rangle ( ( g _ n ) _ { n \\geq 0 } \\in Q _ { q ' } ^ { p ' } , \\ ( f _ n ) _ { n \\geq 0 } \\in Q _ { q } ^ { p } ) . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = f ^ * ( x _ 2 , - x _ 1 ) , ( x _ 1 , x _ 2 ) \\in X . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} | \\Delta ( u - U ) ( x ) | & = | f ( u ( x ) ) - f ( U ( x ) ) | \\\\ & \\le C U ( x ) ^ q | u ( x ) - U ( x ) | \\le C , x \\in B . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\P _ x ( X _ t \\in d y ) = p ( t , x , y ) \\nu ( d y ) \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} 2 w _ { R } '' w _ { R } = 2 \\frac { w _ { R } ^ { 2 } | w _ { R } ' | ^ { 2 } } { | w _ { R } | ^ { 2 } } - \\frac { m _ { 1 } w _ { R } ^ { 2 } } { | w _ { R } | ^ { 2 } } + | w _ { R } | ^ { 4 } \\nabla U _ { 1 } ( c _ { 1 } + w _ { R } ^ { 2 } ) \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} ( ( \\Psi | \\Phi ) ) = \\int _ { \\R ^ { 2 n } } \\Psi ( z ) \\overline { \\Phi ( z ) } \\ , d ^ { 2 n } z \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\int _ q ^ z \\frac { S ( t ) } { t ^ 2 } \\ , d t & = \\int _ q ^ z \\bigg ( \\frac { \\log \\log t } { \\phi ( q ) \\ , t \\log t } + O \\bigg ( \\frac { 1 } { t \\log t } \\bigg ) \\bigg ) \\ , d t \\\\ & = \\frac { ( \\log \\log t ) ^ 2 } { 2 \\phi ( q ) } \\bigg | _ q ^ z + O \\big ( \\log \\log t \\big | _ q ^ z \\big ) = \\frac { ( \\log \\log z ) ^ 2 } { 2 \\phi ( q ) } + O ( \\log \\log z ) . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle u _ { t t } ( t , x ) - u _ { x x } ( t , x ) + \\beta \\int _ 0 ^ t \\ e ^ { - \\eta ( t - s ) } u _ { x x } ( s , x ) d s = 0 \\ , , t \\in ( 0 , T ) , \\ , \\ , x \\in ( 0 , \\pi ) , \\\\ u ( 0 , x ) = u _ { t } ( 0 , x ) = 0 , x \\in ( 0 , \\pi ) , \\\\ u ( t , 0 ) = 0 u ( t , \\pi ) = g ( t ) \\ , , t \\in ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\phi _ { F _ 1 } ^ { t } & \\bigl ( ( v _ 1 , v _ 2 , v _ 3 ) , ( w _ 1 , w _ 2 , w _ 3 ) \\bigr ) \\\\ & = \\Bigl ( \\bigl ( ( \\cos t ) v _ 1 - ( \\sin t ) v _ 2 , ( \\sin t ) v _ 1 + ( \\cos t ) v _ 2 , v _ 3 \\bigr ) , \\bigl ( ( \\cos t ) w _ 1 - ( \\sin t ) w _ 2 , ( \\sin t ) w _ 1 + ( \\cos t ) w _ 2 , w _ 3 \\bigr ) \\Bigr ) \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { U } \\left ( \\frac { 1 } { 2 } \\langle A \\nabla \\rho , \\nabla \\varphi \\rangle + \\rho \\langle \\beta , \\nabla \\varphi \\rangle - \\rho \\partial _ t \\varphi \\right ) d x d t = \\int _ 0 ^ T \\left ( \\int _ { U } \\langle \\mathbf { B } , \\nabla \\varphi \\rangle \\rho d x \\right ) d t = 0 , \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\frac { z \\mathtt { g } _ { a , \\nu } ' ( z ) } { \\mathtt { g } _ { a , \\nu } ( z ) } & = a ( 1 - \\nu ) + a ^ { a / 2 } \\frac { z \\ ; { } _ { a } \\mathtt { B } ' _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} d U _ x ( t ) = \\Big [ \\mathcal { L } _ n U _ x + \\beta _ e \\ , U _ x ( 1 - U _ x ) \\Big ] \\ , d t + \\sqrt { \\gamma _ e \\ , L ^ e \\ , U _ x ( 1 - U _ x ) } \\ , d B _ { x } ( t ) \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} \\{ a _ { ( 1 ) } , S ( a _ { ( 2 ) } ) \\} = \\{ S \\cdot S ( a _ { ( 1 ) } ) , S ( a _ { ( 2 ) } ) \\} = ( - 1 ) ^ { | a _ { ( 1 ) } | | a _ { ( 2 ) } | } S ( \\{ a _ { ( 2 ) } , S ( a _ { ( 1 ) } ) \\} ) = - S ( \\{ S ( a _ { ( 1 ) } ) , a _ { ( 2 ) } \\} ) \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\zeta _ 0 ( t ) & = \\Gamma ( \\psi ) ( t ) = \\psi ( t ) + \\eta ( t ) , \\\\ \\\\ \\zeta _ k ( t ) & = p _ k - B _ k ( t ) , \\ ; k \\in \\mathbb { N } , \\\\ \\psi ( t ) & = \\sum _ { k = 0 } ^ \\infty ( k - 2 ) B _ k ( t ) . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\frac 1 i \\partial _ x m _ 1 ( \\lambda + ) & = i \\lambda e ^ { i \\lambda x } \\int _ { - \\infty } ^ x e ^ { - i \\lambda y } u m _ 1 ( y , \\lambda + ) ~ d y + u m _ 1 ( \\lambda + ) - C _ - ( u m _ 1 ( \\lambda + ) ) \\\\ & - \\lambda \\widetilde { G } _ { \\lambda } * ( u m _ 1 ( \\lambda + ) ) \\\\ & = C _ + ( u m _ 1 ( \\lambda + ) ) + \\lambda G _ { \\lambda + 0 i } * ( u m _ 1 ( \\lambda + ) ) \\\\ & = C _ + ( u m _ 1 ( \\lambda + ) ) + \\lambda ( m _ 1 ( \\lambda + ) - 1 ) . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} S ^ t _ 3 ( H ) = \\sum _ { 1 \\leq i < j < k \\leq d _ c } S _ 3 ( \\{ h _ i , h _ j , h _ k \\} ) , \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\mathrm { s u p p } \\ , \\mu = \\mathrm { s u p p } \\ , \\widehat \\mu \\ , . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} W ' ( x , y ) = \\begin{cases} \\beta _ { i j } & \\ , , \\\\ 0 & \\ , , \\\\ 1 & \\ , . \\end{cases} \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\int _ { \\Omega } | ( \\mathbf { D } + \\mathbf { k } ) \\mathbf { u } | ^ 2 \\ , d \\mathbf { x } = \\sum _ { \\mathbf { b } \\in \\widetilde { \\Gamma } } | \\mathbf { b } + \\mathbf { k } | ^ 2 | \\hat { \\mathbf { u } } _ { \\mathbf { b } } | ^ 2 , \\mathbf { u } \\in \\widetilde { H } ^ 1 ( \\Omega ; \\mathbb { C } ^ n ) , \\ ; \\mathbf { k } \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\lambda _ m = P r ( \\mathcal { G } _ { 0 u } ( y ^ n ) = \\mathcal { E } | X ^ n = c _ { 0 u } ^ { ( n ) } ( m ) ) . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} { \\cal L } _ { t } \\left ( L ^ 2 ( 0 , t ; U ) \\right ) = { \\cal R } ^ 0 _ { [ 0 , t ] } = R ( Q _ { t } ^ { 1 / 2 } ) \\forall t \\ge 0 \\ , . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\Delta _ n : = \\sqrt { n } \\left ( \\omega _ { \\Lambda _ n } - \\omega _ { \\Lambda _ \\infty } \\right ) . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} ( E _ 1 , E _ 2 , E _ 3 ) = ( - \\mathrm { d i v } ( s _ 1 ) , - \\mathrm { d i v } ( s _ 2 ) , - \\mathrm { d i v } ( s _ 3 ) ) . \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} Y _ 2 ^ { ( n ) } ( t ) = Y _ 2 ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t } X ^ { ( n ) } ( t _ { n , i } ) d s + B _ 2 ( t ) - B _ 2 ( t _ { n , i } ) . \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\bigl ( \\# \\{ ( a , b , c , d ) & \\in B _ n ^ { c > 0 } : k _ { - d / c } = k _ * \\} - \\# \\{ ( a , b , c , d ) \\in B _ n ^ { c > 0 } : k _ { - d / c } = k \\} \\bigr ) \\\\ & = \\tfrac { 1 } { 2 } \\bigl ( \\# \\{ ( a , b , c , d ) \\in B _ n ^ { c > 0 } : k _ { c / d } = k \\} - \\# \\{ ( a , b , c , d ) \\in B _ n ^ { c > 0 } : k _ { c / d } = k _ * \\} \\bigr ) . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} d ( T _ \\Delta x , H ) = \\begin{cases} d ( x , H ) - \\Delta , & \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} \\operatorname { C o v } \\bigl ( Z ^ H _ s , Z ^ H _ u \\bigr ) \\operatorname { C o v } \\bigl ( Z ^ H _ t , Z ^ H _ t \\bigr ) = \\operatorname { C o v } \\bigl ( Z ^ H _ s , Z ^ H _ t \\bigr ) \\operatorname { C o v } \\bigl ( Z ^ H _ t , Z ^ H _ u \\bigr ) . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { x } ( t ) = ( A - B K ) x ( t ) + B K \\epsilon ( t ) , \\\\ \\displaystyle \\dot { \\epsilon } ( t ) = A \\epsilon ( t ) - \\frac { 1 } { h } L C \\int _ { t - h } ^ t \\epsilon ( s ) d s , \\\\ u ( t ) = - K \\hat { x } ( t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} K = \\{ ( \\lambda , c ) \\in I \\times J / \\exists x \\in X , f _ \\lambda ( x ) = c , f _ \\lambda ' ( x ) = 0 \\} \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} w ^ { E L } _ i ( \\theta ) = \\frac { 1 } { n } \\frac { 1 } { 1 + \\lambda ^ T _ { E L } \\psi ( x _ i , \\theta ) } \\ , , \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = 0 } ^ { \\infty } \\int _ { [ 0 , T ] \\times \\lbrack 0 , 1 ] } \\ell ( \\varphi _ { k } ^ { n } ( s , y ) ) \\ , d s \\ , d y \\leq M _ { 0 } , \\mbox { a . s . } { P } . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\begin{array} { r l l l } \\Delta \\pi _ i + \\lambda \\ , \\pi _ i & = & - \\lambda \\ , w _ i & \\Omega , \\\\ \\pi _ i & = & - w _ i & \\partial \\Omega , \\end{array} \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\textrm { t r } _ \\epsilon \\ ( \\sum _ { k = 0 } ^ { + \\infty } u _ k \\ ) = \\sum _ { k = 0 } ^ { + \\infty } \\textrm { t r } _ \\epsilon u _ k . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} u ^ { ( 1 ) } ( t , x , y ) = h u ^ { ( h ) } ( h ^ 2 t , h x , h ^ { \\frac 3 2 } y ) . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\textrm { t r } _ \\epsilon u _ k & = \\sum _ { k = 0 } ^ n \\sum _ { m \\geqslant 0 } l _ m \\ ( u _ k \\ ( x _ m \\ ) \\ ) \\\\ & = \\sum _ { m \\geqslant 0 } l _ m \\ ( \\ ( \\sum _ { k = 0 } ^ n u _ k \\ ) \\ ( x _ m \\ ) \\ ) . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\xi _ { ( , i | j } \\delta _ { r ) } ^ { k } = 0 . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} ( \\nabla _ j A ) _ \\omega \\ ; = \\ ; \\imath [ A _ { \\omega } , X _ j ] \\ ; . \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\mathcal M ( h _ * ; p , q ; E ) = \\bigcup _ { t \\in [ 0 , 1 ] } \\mathcal M ( h _ t ; p , q ; E ) \\times \\{ t \\} \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\delta _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } + r ^ { - 1 } . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k + 1 } C _ i V ( f _ i ) = 0 \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} B _ k ( t ) = \\int _ { [ 0 , t ] \\times [ 0 , 1 ] } { { 1 } } _ { [ 0 , r _ k ( \\boldsymbol { \\zeta } ( s ) ) ) } ( y ) \\ , \\varphi _ k ( s , y ) d s \\ , d y , k \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{gather*} a = ( \\nabla \\eta ) ^ { - 1 } , \\end{gather*}"}
-{"id": "9641.png", "formula": "\\begin{align*} c _ k : = \\underset { \\gamma \\in \\Gamma _ k } { \\inf } \\underset { \\sigma \\in \\mathbb { D } _ k } { \\max } ~ \\Phi ( \\gamma ( \\sigma ) ) \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\begin{bmatrix} p ' ( x _ 1 ) \\\\ \\vdots \\\\ p ' ( x _ N ) \\end{bmatrix} = D \\begin{bmatrix} p ( x _ 1 ) \\\\ \\vdots \\\\ p ( x _ N ) \\end{bmatrix} \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} & e A f \\cdot e A ^ { d } f = e A A ^ { d } f = e A ^ { d } f \\cdot e A f , \\\\ & e A f \\cdot ( e A ^ { d } f ) ^ { 2 } = e A f \\cdot e ( A ^ { d } ) ^ 2 f = e A ( A ^ { d } ) ^ 2 f = e A ^ { d } f , \\\\ & ( e A f ) ^ { k + 1 } \\cdot e A ^ { d } f = e A ^ { k + 1 } f \\cdot e A ^ { d } f = e A ^ { k + 1 } A ^ { d } f = e A ^ { k } f = ( e A f ) ^ { k } , \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} D ( \\ln \\omega ) _ { , t } + 2 D _ { , t } + \\frac { k } { \\omega } T & = 0 \\\\ D \\ , _ { , t t } & = 2 \\psi T _ { , t } \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} U _ { a _ { i } } = \\begin{pmatrix} \\frac { 1 } { \\sqrt { 1 + a ^ { 2 } } } I _ { p ' - q } & - \\frac { a } { \\sqrt { 1 + a ^ { 2 } } } I _ { p ' - q } & \\mbox { O } _ { p ' - q , 1 } \\\\ \\frac { 1 } { \\sqrt { 1 + a ^ { 2 } } } I _ { p ' - q } & \\frac { a } { \\sqrt { 1 + a ^ { 2 } } } I _ { p ' - q } & \\mbox { O } _ { p ' - q , 1 } \\\\ \\mbox { O } _ { 1 , p ' - q } & \\mbox { O } _ { 1 , p ' - q } & 1 \\end{pmatrix} , \\quad \\mbox { w h e r e $ i = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} m : = \\max \\{ m _ { x y } \\colon x , y \\in A \\} \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} A P = P A , S ( t ) P ( D ( A ) ) \\subseteq D ( A ) \\cap R ( P ) . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} [ \\delta ] = - ( ( x ^ 2 - x + 1 ) y + 2 - 2 x ) [ \\gamma ] = 2 ( x - 1 ) [ \\gamma ] . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\frac { N _ { 1 0 } ^ { \\mathrm { W e y l } } ( X ) } { N _ { 1 0 } ^ { \\mathrm { c m } } ( X ) } = 1 + O _ { \\epsilon } ( X ^ { - \\frac { 3 } { 2 0 } + \\epsilon } ) . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\partial U _ p = \\partial ^ 0 U _ p \\cup \\partial ^ 1 U _ p \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} ( f _ h \\circ \\pi _ x ) ( y ) & = \\int _ D { \\bf 1 } _ { \\{ r \\} } \\ > d ( \\delta _ { \\pi ( x , y ) } * \\delta _ h ) = ( \\delta _ { \\pi ( x , y ) } * \\delta _ h ) ( \\{ r \\} ) \\\\ & = \\frac { \\omega _ r } { \\omega _ h \\omega _ { \\pi ( x , y ) } } p _ { \\pi ( x , y ) , h } ^ r = \\frac { 1 } { \\omega _ h } p _ { r , \\bar h } ^ { \\pi ( x , y ) } , \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} V \\subset \\bigcup _ { j \\in J } 5 \\Delta _ { j } , \\quad 5 \\Delta _ { j } : = \\Delta ( x _ j , 5 r _ { x _ j } ) . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} F = Z _ 0 ^ m f ( Z _ 1 , Z _ 2 , Z _ 3 ) , \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} Y ( t ) = X ( t ) + Z ( t ) , t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{gather*} P ^ n f = \\sum _ { k \\ge 0 } ( U ^ { - k } P ^ { n + k } f - U ^ { - k - 1 } P ^ { n + k + 1 } f ) \\ , . \\end{gather*}"}
-{"id": "351.png", "formula": "\\begin{align*} T ( X , Y ) Z = T ( Z , Y ) X \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} K ( x , v ) A ( x , v ) - \\frac { x ^ 2 } { 1 - x } C ( x , v ) & = 1 - \\frac { x ^ 2 } { 1 - x } - \\frac { x v } { 1 - v } A ( x v , 1 ) , \\\\ K ( x , v ) C ( x , v ) & = 1 - \\frac { x v } { 1 - v } C ( x v , 1 ) \\ , . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\langle A \\chi _ j , \\chi _ k \\rangle = a _ { k - j } \\quad j , k \\in \\Z , \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} X ^ + ( t ) = \\sum _ { i = 1 } ^ { \\lfloor t \\rfloor } X ( i ) \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} y _ 0 ^ 2 = y _ i ^ 4 = y _ n ^ 2 = 1 , \\ , \\ , ( 1 \\leq i \\leq n - 1 ) , \\ , \\ , [ y _ i , y _ n ] = y _ { i + 1 } ^ 2 , \\ , \\ , ( 1 \\leq i \\leq n - 2 ) \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} | I \\cap A | \\leq | I | - q < | I | - \\frac { | I | } { d } + 1 = \\left ( 1 - \\frac { 1 } { d } \\right ) | I | + 1 \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\dim ( p , 1 ) + \\dim ( p - 2 , 2 , 1 ) = 1 & + \\dim ( p - 1 , 1 ) + \\dim ( p - 3 , 2 , 1 ) \\cr & + \\dim ( p - 2 , 1 , 1 ) + \\dim ( p - 2 , 2 ) . \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} C _ { 0 f a } = \\begin{cases} C & \\mbox { i f } C _ { 0 u } > 0 \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} v & = \\left ( \\frac { ( - 1 ) \\int e ^ { - \\int D ^ { - 1 } d D } ( - \\ell D ^ 2 ) d D } { e ^ { - \\int D ^ { - 1 } d D } } \\right ) \\\\ & = \\frac { \\ell } { 2 } D ^ 3 + K D \\\\ & = \\frac { \\ell } { 2 } ( D ^ 3 + K _ 1 D ) \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} & U M _ { 4 1 } ^ { \\tau } + M _ { 1 1 } U _ m = T M _ { 3 1 } U _ m , M _ { 1 1 } ^ { \\tau } U + U _ m M _ { 4 1 } = U _ m M _ { 2 1 } T . \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ 2 } \\sum _ { i = 1 } ^ N \\sum _ k ^ { ( i ) } \\tilde { d } _ i \\mathbf { e } _ k ^ * X _ i G \\mathbf { e } _ i \\frac { \\partial \\omega _ \\iota ^ c } { \\partial g _ { i k } } = O _ \\prec ( \\Psi ^ 2 \\Pi ^ 2 ) \\ , , \\iota = A , B \\ , . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} V _ { p _ 1 p _ 2 p _ 3 } & = \\left \\{ \\sigma \\in H ^ 0 ( N _ Y ( - 1 ) | _ C ) : \\sigma | _ { p _ i } = 0 , \\right \\} \\\\ & = \\left \\{ \\sigma \\in H ^ 0 ( N _ Y ( - 1 ) | _ C ( - p _ 1 - p _ 2 - p _ 3 ) ) : \\right \\} \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{gather*} a ( 0 , \\cdot ) = I , \\end{gather*}"}
-{"id": "2704.png", "formula": "\\begin{align*} K ^ i _ m : = \\sup \\{ n \\geq 0 : \\ : T ^ i _ n \\leq m \\} \\ , . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} a ( p ) = \\pm p ^ { ( k - 1 ) / 2 } \\upsilon _ { \\mu _ p } , \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\exists t \\in \\Lambda _ m \\mbox { s u c h t h a t } \\sum _ { i = 1 } ^ m t _ i A _ i \\mbox { i s p o s i t i v e s e m i d e f i n i t e o n } K . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\sum _ { h = 1 } ^ { \\infty } h ^ { n - m } \\Psi ^ m ( h ) < \\infty . \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} \\binom { | S | - | U ' | } { 2 } \\sum _ a ( - 1 ) ^ a \\binom { | S | - | U ' | - 2 } { a - | U ' | - 2 } = 0 . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} g _ { 1 } ( U _ 1 , U _ 2 ) g _ { 1 } ( \\xi , Z ) = g _ { 1 } ( U _ 1 , \\phi U _ 2 ) g _ { 1 } ( H , Z ) - g _ { 1 } ( U _ 1 , U _ 2 ) g _ { 1 } ( H , \\varphi Z ) . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} f ( \\overline { z } ) = \\overline { f ( z ) } \\ , . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathfrak { l } ^ { ( p , p ) } \\big ] = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathbb { E } \\big [ \\mathfrak { d } _ { i , 1 } Q _ i \\mathfrak { l } ^ { ( p - 1 , p ) } \\big ] + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathbb { E } \\big [ \\mathfrak { d } _ { i , 2 } \\mathcal { Q } _ i \\mathfrak { l } ^ { ( p - 1 , p ) } \\big ] . \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{gather*} K ^ { - 1 } A \\index { $ K ^ { - 1 } A $ } = { \\bigcup } _ { k \\in K } L _ k ^ { - 1 } [ A ] , \\textrm { w h e r e } L _ k ^ { - 1 } [ A ] = \\{ t \\in T \\ , | \\ , k t \\in A \\} . \\end{gather*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\frac { \\phi '' ( z ) } { \\phi ( z ) } + \\frac { \\phi ' ( z ) } { \\phi ( z ) } \\frac { \\tilde { \\tau } ( z ) } { \\sigma ( z ) } + \\frac { \\tilde { \\sigma } ( z ) } { \\sigma ^ 2 ( z ) } = \\frac { \\bar { \\sigma } ( z ) } { \\sigma ^ 2 ( z ) } , \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\beta ^ * ( T M ) \\vert _ { B _ \\sigma ( \\Lambda ) } = N ^ s \\oplus P \\oplus N ^ u . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} \\lambda _ { E T } : \\sum _ { i = 1 } ^ n e ^ { \\lambda ^ T _ { E T } \\psi ( x _ i , \\theta ) } \\psi ( x _ i , \\theta ) = 0 \\ , , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\sup _ { Q ^ { { 1 } / { 2 } } _ r } | u | = \\delta \\geq C r ^ C \\exp \\{ - C M ^ { \\frac { 1 } { 2 } } \\} . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\# \\{ q : q \\nmid N , \\ ; H _ q \\supseteq \\Gamma _ 1 ( N ) , \\ ; q \\le x \\} = ( 1 + o ( 1 ) ) \\pi ( x ) \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & | \\nabla \\phi _ m | \\le \\frac { C \\sqrt { K } 2 ^ m } { \\rho } \\quad \\mbox { i n } Q _ m \\\\ & 0 \\le \\phi _ { m , t } \\le \\frac { C 2 ^ m } { t } \\quad \\mbox { i n } Q _ m . \\end{aligned} \\right . \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\sum _ k \\binom { k } { 2 } \\binom { n } { k - x } = \\binom { n } { 2 } 2 ^ { n - 2 } + x n 2 ^ { n - 1 } + \\binom { x } { 2 } 2 ^ { n } \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\| \\varphi - \\tilde { \\varphi } _ n ^ M \\| _ \\infty = O ( h ^ { 4 r } ) , \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\phi _ - = \\frac { - \\mu + \\alpha ( \\lambda ) } { \\varphi ( \\lambda ) } . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} \\delta ( X - X _ 1 \\cdots X _ m ) \\prod _ { i = 1 } ^ m C _ { N , L _ i } \\ , \\mathrm { d e t } \\left ( \\mathbb { I } - X _ i ^ T X _ i \\right ) ^ { ( L _ i - N - 1 ) / 2 } ( \\mathrm { d } X _ i ) ( \\mathrm { d } X ) . \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} \\sum _ { c \\geq 1 } \\frac { g _ h ( 4 c ) } { ( 4 c ) ^ { 2 w } } = \\frac { L ^ { ( 2 ) } ( 2 w - \\frac { 1 } { 2 } , \\chi _ { k , h } ) } { \\zeta ^ { ( 2 h ) } ( 4 w - 1 ) } \\widetilde { D } _ \\infty ^ k ( h , w ) , \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} P _ h ( z , s ) : = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma } \\Im ( \\gamma z ) ^ s e \\left ( h \\gamma z \\right ) , \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\bar { y } ( t , p ) = \\displaystyle \\sum _ { i = 1 } ^ m \\bar { w } _ i ( t ) \\Phi _ i ( p ) \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} [ V ^ { - 1 } ( t ) \\phi ] ( \\xi ) = \\eqref { v i a 1 } + \\eqref { v i a 2 } + \\eqref { v i a 3 } . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} m ( H , k - 1 ) = & m ( H - v , k - 1 ) + \\sum \\limits _ { w ' \\in N _ { H } ( v ) } m ( H - v - w ' , k - 2 ) \\\\ = & m ( T ( a , y - 1 , c - 1 ) \\cup P _ { x - a - 2 } , k - 1 ) + m ( T ( a , y - 1 , c - 1 ) \\cup P _ { x - a - 3 } , k - 2 ) \\\\ & + m ( T ( a , y - 2 , c - 1 ) \\cup P _ { x - a - 2 } , k - 2 ) + m ( T ( a , y - 1 , c - 2 ) \\cup P _ { x - a - 2 } , k - 2 ) \\\\ = & m ( T ( a , y - 1 , c - 1 ) \\cup P _ { x - a - 1 } , k - 1 ) + m ( T ( a , y - 2 , c - 1 ) \\cup P _ { x - a - 2 } , k - 2 ) \\\\ & + m ( T ( a , y - 1 , c - 2 ) \\cup P _ { x - a - 2 } , k - 2 ) . \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} \\int _ T | w | ^ 2 d V = \\int _ T | u ( G x ) | ^ 2 d V = | G | ^ { - 1 } \\int _ { T _ 0 } | u | ^ 2 d V = \\frac { 1 } { \\lambda _ 1 \\lambda _ 2 } . \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{gather*} \\frac { \\partial _ 1 r \\times \\partial _ 2 r } { | \\partial _ 1 r \\times \\partial _ 2 r | } = \\frac { 1 } { \\sqrt { g } } \\partial _ 1 r \\times \\partial _ 2 r . \\end{gather*}"}
-{"id": "5506.png", "formula": "\\begin{align*} U ( 2 ^ { q + 1 } - t ; k ) = 2 ^ q \\cdot [ 2 ^ { k - 1 } + 1 ] - t \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} u ^ f _ { n , t } = \\prod _ { k = 0 } ^ { n - 1 } a _ k f _ { t - n } + \\sum _ { s = n } ^ { t - 1 } w _ { n , s } f _ { t - s - 1 } , n , t \\in \\mathbb { N } . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\int _ M | \\nabla f | ^ { p - 2 } d \\varphi ( \\nabla f ) \\ d \\mathfrak { m } = \\lambda _ { 1 , p } ( M ) \\int _ M | f | ^ { p - 2 } f \\varphi \\ d \\mathfrak { m } . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} & X ^ { \\downarrow k } = \\{ ( u , v - k ) : ( u , v ) \\in X , v \\geq k \\} , \\\\ & X ^ { \\leftarrow k } = \\{ ( u - k , v ) : ( u , v ) \\in X , u \\geq k \\} , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} f ( M ) & : = I \\left ( M ; \\overline { M } \\right ) , \\\\ & = H ( M ) + H \\left ( \\overline { M } \\right ) - H \\left ( M , \\overline { M } \\right ) , \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\| \\partial _ \\phi A ( \\phi ; \\mu ' , \\zeta ' ) [ D _ { \\zeta _ i ' } \\phi ] \\| _ * & = \\| T ( \\partial _ \\phi N ( \\phi ; \\mu ' , \\zeta ' ) [ D _ { \\zeta _ i ' } \\phi ] ) \\| _ * \\\\ & \\leq \\| \\partial _ \\phi N ( \\phi ; \\mu ' , \\zeta ' ) [ D _ { \\zeta _ i ' } \\phi ] \\| _ { * * } \\\\ & \\leq C \\ , \\| \\phi \\| _ * \\ , \\| D _ { \\zeta _ i ' } \\phi \\| _ * \\\\ & \\leq C \\ , \\| E \\| _ * \\ , \\| D _ { \\zeta _ i ' } \\phi \\| _ * . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} p ^ \\circ _ j = \\frac { \\sum _ { i = j } ^ { \\infty } p _ i } { j \\sum _ { i = 1 } ^ { \\infty } p _ i H _ i } \\mbox { f o r } j \\in \\mathbb { N } _ { + } , \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} \\frac { \\mathrm e ^ { \\mathrm { a d } ( Y ) } - 1 } { \\mathrm { a d } ( Y ) } \\dot Y & = h _ \\vartheta . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\lambda _ { \\max } ^ { \\mathrm { a s y } } ( D , g ) = \\hat { \\mu } _ { \\mathrm { e s s } } ( D , g ) = \\log \\max \\{ a _ 0 , \\ldots , a _ d \\} . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( | u _ 1 ( t ) | ^ 2 + | u _ 2 ( t ) | ^ 2 \\big ) \\ d t \\asymp \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\ , . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\Lambda _ n : = T _ { \\sqrt { \\frac { \\alpha } { n } } , \\sqrt { \\frac { 1 } { \\alpha n } } } \\lambda _ n \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\zeta = & \\left ( \\dfrac { 1 } { 1 - \\omega } y _ 3 , \\dfrac { 1 } { 1 - \\omega ^ 2 } y _ 4 \\right ) \\tau _ B ^ { - 1 } \\\\ = & \\frac { 1 } { 2 } \\left ( \\frac { y _ 3 } { ( 1 - \\omega ) y _ 1 } + \\frac { y _ 4 } { ( 1 - \\omega ^ 2 ) y _ 1 } , \\frac { y _ 3 } { ( 1 - \\omega ) y _ 2 } - \\frac { y _ 4 } { ( 1 - \\omega ^ 2 ) y _ 2 } \\right ) . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } c _ { i j } ^ n = 0 , i , j . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\omega _ i : = p _ { i , \\bar i } ^ e = | \\{ z \\in X : \\ > ( x , z ) \\in R _ i \\} | \\in \\mathbb N \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert \\phi _ n \\Vert _ { \\infty } = \\gamma , \\quad 0 < \\gamma \\leq 1 . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} & \\beta _ 1 = \\frac { x + \\sqrt { a d } } { 2 } , \\beta _ 2 = \\frac { y + \\sqrt { b d } } { 2 } , \\beta _ 3 = \\frac { z + \\sqrt { c d } } { 2 } \\\\ & \\beta _ 4 = \\frac { \\sqrt { c } ( \\varepsilon \\sqrt { a } + \\sqrt { d } ) } { \\sqrt { a } ( \\varepsilon \\sqrt { c } + \\sqrt { d } ) } , \\beta _ 5 = \\frac { \\sqrt { c } ( \\varepsilon \\sqrt { b } + \\sqrt { d } ) } { \\sqrt { b } ( \\varepsilon \\sqrt { c } + \\sqrt { d } ) } , \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\Delta \\left ( \\{ z _ l \\} _ { k = 1 } ^ p \\right ) = \\prod _ { 1 \\leq i < j \\leq p } ( z _ i - z _ j ) \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} P : = P ( u _ 1 , \\ldots , u _ k ) = P _ { k , 1 } ^ { m _ 1 } \\cdot u _ 1 ^ { a _ 1 } \\cdot P _ { k , 2 } ^ { m _ 2 } \\cdot u _ 2 ^ { a _ 2 } \\cdots P _ { k , k } ^ { m _ k } \\cdot u _ k ^ { a _ k } . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} A ( \\psi , \\phi ) ( z ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } ( \\psi | \\widehat { T } ( z ) \\phi ) . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{gather*} { \\rm d } \\nu _ { \\alpha } ( z ) = \\frac { \\alpha + 1 } { \\pi } \\big ( 1 - x ^ 2 - y ^ 2 \\big ) ^ { \\alpha } { \\rm d } x { \\rm d } y , z = x + i y . \\end{gather*}"}
-{"id": "2069.png", "formula": "\\begin{align*} P y = \\sum _ { n = 1 } ^ \\infty \\langle y e _ n , f _ n \\rangle \\langle \\cdot , e _ n \\rangle f _ n , y \\in C _ E . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\partial _ { \\frak C ^ v } ( X , \\widehat { \\mathcal U } ) = \\widehat f ^ { - 1 } ( \\partial P ) \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} \\alpha ^ { \\gamma } ( t ) : = \\gamma ( t , X _ { \\gamma , m } ( t ^ - ) ) . \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\| \\sup _ { n \\ge 0 } \\frac 1 { A _ { n } } | \\sum _ { k = 0 } ^ { n } a _ k f \\circ \\tau ^ k | \\ , \\| _ { L ^ p ( \\nu ) } \\le C \\| f \\| _ { L ^ p ( \\nu ) } \\forall f \\in L ^ p ( \\nu ) \\ , . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\left \\| p / q \\right \\| = \\sum _ { j = 1 } ^ { m } w _ { j } | p c _ { j } - q b _ { j } | . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} u _ j ( Z ) = \\frac { r _ j ^ { n - 2 } G ( X _ 0 , q _ j + r _ j Z ) } { \\omega ( B ( q _ j , r _ j ) ) } Z \\in \\Omega _ j \\hbox { a n d } u _ j = 0 \\hbox { i n } \\Omega _ j ^ c . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} ( C ) = \\{ ( A ) | h ^ 0 ( A ) \\ge 2 , h ^ 1 ( A ) \\ge 2 \\} . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} h = 2 T ^ { + } \\varepsilon + \\frac { 1 } { 2 } { \\phi } ^ { \\prime } \\varepsilon + \\frac { \\phi T ^ { + } \\varepsilon } { u } + \\frac { \\chi ^ { + } \\varepsilon } { u ^ { n - 1 } } + \\frac { 1 } { 4 } \\phi ^ { 2 } \\varepsilon - \\frac { 2 T ^ { + } } { u } \\frac { d \\varepsilon } { d \\xi } + \\frac { \\exp \\left ( u \\xi + E _ { 0 } ^ { + } \\right ) \\chi ^ { + } } { u ^ { n - 1 } } . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + ( 1 - \\epsilon ) \\alpha x _ { i - 1 } ( t ) + ( 1 - \\epsilon ) a x _ { i } ( t ) + ( 1 - \\epsilon ) \\beta x _ { i + 1 } ( t ) , \\ , \\ , i \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} \\nu = \\frac { k - 1 } { 2 } + \\frac { 1 } { 6 } - \\epsilon \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\widetilde \\Omega = \\left \\{ \\mathbf { k } \\in \\mathbb { R } ^ d \\colon | \\mathbf { k } | < | \\mathbf { k } - \\mathbf { b } | , \\ ; 0 \\ne \\mathbf { b } \\in \\widetilde \\Gamma \\right \\} . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\sum _ { \\substack { N ( a ) \\leq x \\\\ a \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\frac { 1 } { N ( a ) } \\prod _ { \\pi | 2 a } \\left ( 1 + N ( \\pi ) ^ { - 1 } \\right ) ^ { - 1 } = \\sum _ { \\substack { ( d ) \\\\ N ( d ) \\leq 2 x } } \\frac { \\mu _ { [ i ] } ( d ) } { \\sigma ( d ) } \\sum _ { \\substack { a \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } \\\\ \\frac { d } { ( 2 , d ) } | a , \\ N ( a ) \\leq x } } \\frac { 1 } { N ( a ) } . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} A ( t ) : = \\sum _ { n \\le X } \\frac { a ^ \\pm ( n ) } { n ^ { i t } } , \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{gather*} \\dim G ^ { ( 1 ) } _ 1 < \\sum ^ { n - 1 } _ { j = 1 } j c _ j , \\end{gather*}"}
-{"id": "3010.png", "formula": "\\begin{align*} O \\ ( n \\sum _ { m = 0 } ^ { M } \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\ ) & + O \\ ( n \\sum _ { m = M + 1 } ^ { 2 \\epsilon n } \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\ ) \\\\ & + O \\ ( \\sum _ { H ( \\chi ) > \\epsilon } | \\hat { 1 _ S } ( \\chi ) | ^ 3 \\ ) . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} S _ m = ( n - 1 ) m _ { ( m - 1 , \\dot { 0 } ) } + \\sum _ { p = 1 } ^ { [ ( m - 1 ) / 2 ] } m _ { ( m - 1 - p , p , \\dot { 0 } ) } , m \\ge 2 . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\widehat { \\mathcal U ' } \\vert _ { Z _ 1 } = \\widehat { \\mathcal U ^ + _ { Z _ 1 } } . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} \\gamma ( \\nabla ^ \\gamma _ X J _ \\pm ( Y ) , Z ) = \\mp \\frac { 1 } { 2 } [ d \\psi ( X , J _ \\pm Y , Z ) + d \\psi ( X , Y , J _ \\pm Z ) ] , \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} r _ 1 ' t _ 2 \\Lambda _ 1 = 2 h \\log \\left ( \\alpha _ 1 ^ { r _ 1 ' t _ 2 } \\cdot \\alpha _ 2 ^ { - s _ 1 ' t _ 1 } \\right ) - \\log \\left ( \\alpha _ 2 ^ { \\delta r _ 1 ' s _ 1 ' } \\cdot \\alpha _ 3 ^ { - r _ 1 ' t _ 2 } \\right ) , \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\widehat { \\mathbb { A } } : = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\widehat { \\gamma } , \\xi , \\N ) \\right \\} \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B _ t ( 0 ) } \\frac { 8 } { ( 1 + | z | ^ 2 ) ^ 2 } \\Big ( \\frac { | z | ^ 2 } { ( 1 + | z | ^ 2 ) ^ 2 } - \\frac { ( 1 - | z | ^ 2 ) ^ 2 } { 2 ( 1 + | z | ^ 2 ) ^ 2 } \\Big ) \\mathrm { d } z = \\frac { 4 \\pi t ^ 2 ( t ^ 2 - 1 ) } { ( t ^ 2 + 1 ) ^ 3 } . \\end{aligned} \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ N { \\frac { \\big ( q ^ { \\lambda _ i + \\theta ( N - i + 1 ) - z } ; q \\big ) _ { \\infty } } { \\big ( q ^ { \\lambda _ i + \\theta ( N - i ) - z } ; q \\big ) _ { \\infty } } } , \\end{gather*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\widetilde { G } & = \\widetilde { D } _ x H + i H \\widetilde { D } _ y ( t ) ^ T , \\\\ G & = \\overline { E } \\odot \\widetilde { G } , \\\\ Q & = - \\Psi \\odot G , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\mathfrak { S } _ k ^ \\pm = \\{ g \\rtimes \\sigma \\mid g \\in \\mathbb { Z } _ 2 ^ { \\oplus k } \\ , \\ , \\ , \\ , \\sigma \\in \\mathfrak { S } _ k \\} . \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} \\left ( W _ { i j } \\right ) = \\left ( \\frac { \\partial ^ 2 L } { \\partial \\dot { q } ^ i \\partial \\dot { q } ^ j } \\right ) \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m x _ i ^ 3 \\geq ( m + 3 l ) y ^ 3 . \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} I _ { m - 1 } ( t ) & = \\int _ 0 ^ \\infty \\left ( a _ { m - 1 } + t \\xi + \\sum _ { k = 1 } ^ { m - 1 } ( - 1 ) ^ { k - 1 } { m - 1 \\choose k } \\frac { 1 } { k } e ^ { - k t \\xi } \\right ) \\xi ^ { - m } \\pi ( d \\xi ) \\\\ & = \\int _ 0 ^ \\infty \\int _ 0 ^ { \\xi t } \\left ( 1 + \\sum _ { k = 1 } ^ { m - 1 } ( - 1 ) ^ { k } { m - 1 \\choose k } e ^ { - k w } \\right ) d w \\xi ^ { - m } \\pi ( d \\xi ) , \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\lim _ { \\ell \\rightarrow \\infty } \\gamma _ { \\ell } ^ i = \\left ( 1 - \\sqrt { 1 - \\lambda { } _ { \\epsilon } } \\right ) ^ i . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ \\ell \\frac 1 { 1 - p ^ { - j } } < \\frac 1 { 1 - p ^ { - 1 } } \\bigg ( 1 - \\frac 1 { p ( p - 1 ) } \\bigg ) ^ { - 1 } \\le 1 + 6 p ^ { - 1 } , \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\mathfrak { I } ( u ) : = \\int \\limits _ { \\mathbb { G } } ( | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } + | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } - | u ( x ) | ^ { q } ) d x . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} c _ { k - 2 } & = - n ( n - 1 ) a _ k - n b _ { k - 1 } , \\\\ c _ l & = - \\frac { n ( n - 1 ) } { k ( k - 1 ) } ( l + 2 ) ( l + 1 ) a _ { l + 2 } - \\frac { n } { k - 1 } ( l + 1 ) b _ { l + 1 } + \\frac { C _ { k - l - 2 , n } } { l ! } , \\\\ & l = 0 , 1 , \\ldots , k - 3 . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} \\partial _ t X = \\Delta X + a _ 1 X + a _ 2 X ^ { 2 } + \\dots + a _ { 2 n - 1 } X ^ { 2 n - 1 } + \\xi \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u = H u + \\lambda | u | ^ 2 u , \\\\ u ( 0 ) = u _ 0 . \\end{cases} \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} & ( g ^ { i j } g ^ { k l } - 2 g ^ { j l } g ^ { i k } ) \\partial _ k \\partial _ l \\partial ^ 2 _ t q \\partial ^ 2 _ { i j } \\eta ^ \\mu \\\\ & = ( g ^ { i j } g ^ { k l } - g ^ { j l } g ^ { i k } ) \\partial _ k \\partial _ l \\partial ^ 2 _ t q \\partial ^ 2 _ { i j } \\eta ^ \\mu - g ^ { j l } g ^ { i k } \\partial _ k \\partial _ l \\partial ^ 2 _ t q \\partial ^ 2 _ { i j } \\eta ^ \\mu , \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\begin{aligned} & K _ 0 ( x , p ; E ) = G ( x , E ) p ^ 2 - 1 = \\cr & C _ 0 ( x , p ; E ) ^ 2 ( L _ 0 ( x , p ) - E ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} A ( X , \\xi ) = \\tilde { \\nabla } _ X \\xi - \\nabla _ X \\xi = \\varphi X + \\varphi h X . \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} \\| A ( t ) ^ { - 1 / 2 } P - ( t ^ 2 S ) ^ { - 1 / 2 } P \\| \\le { C } _ { 1 4 } ; { C } _ { 1 4 } = { \\beta } _ { 1 4 } \\delta ^ { - 1 / 2 } c _ * ^ { - 1 / 2 } \\| X _ 1 \\| ( 1 + c _ * ^ { - 1 } \\| X _ 1 \\| ^ 2 ) . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\sum _ { q \\bmod c } e ^ { 2 \\pi i \\frac { h q } { c } } = \\begin{cases} 0 & c \\nmid h \\\\ c & c \\mid h . \\end{cases} \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} C _ { N , L _ i } = \\frac { \\left ( \\mathrm { v o l } \\ , O ( L _ i ) \\right ) ^ 2 } { \\mathrm { v o l } \\ , O ( L _ i + N ) \\mathrm { v o l } \\ , O ( L _ i - N ) } = \\frac { 1 } { \\pi ^ { N ^ 2 / 2 } } \\prod _ { j = 1 } ^ N \\frac { \\Gamma ( \\frac { L _ i + j } 2 ) } { \\Gamma ( \\frac { L _ i - N + j } 2 ) } \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} F ( s ) = \\int _ 0 ^ \\infty f ( x ) x ^ s \\frac { d x } { x } , \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} \\tilde \\eta = \\eta - x . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\binom { | U ' | } { 2 } \\sum _ a ( - 1 ) ^ { a } \\binom { | S | - | U ' | } { a - | U ' | } = 0 . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} G _ 2 ( x ) = \\frac { x ^ 3 ( 1 - 4 x + 3 x ^ 2 + x ^ 3 ) } { ( 1 - x ) ^ 2 ( 1 - 2 x ) ( 1 - 3 x ) ( 1 - 3 x + x ^ 2 ) } + \\frac { x ^ 2 } { 1 - x } F _ T ( x ) \\ , . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} H ' = \\sum _ { i = 1 } ^ n \\frac { a _ i } n \\log \\frac { n } { a _ i } . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\mathcal { K } _ t = \\mathcal { K } _ { 0 , t } + \\mathcal { K } _ { 1 , t } \\textrm { a n d } \\mathcal { K } _ { 1 , t } \\mathcal { K } _ { 0 , t } = \\mathcal { K } _ { 0 , t } \\mathcal { K } _ { 1 , t } = 0 \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} S _ { T ^ * Q } = \\left \\{ \\left ( q ^ i , \\frac { \\partial F } { \\partial q ^ i } ( q , \\lambda ) \\right ) \\in T ^ * Q : \\frac { \\partial F } { \\partial \\lambda ^ a } ( q , \\lambda ) = 0 \\right \\} . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\lim _ { k _ 0 < k \\to \\infty } \\biggr ( \\frac { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ { l j _ 0 } ( \\lambda _ l / R ) ^ { k + 1 } } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ { l j _ 0 } ( \\lambda _ l / R ) ^ k } \\biggr ) = \\tilde \\lambda / R . \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\mathbb E | \\mathbf 1 _ { [ 0 , 1 ] } \\star \\bar { \\mu } | ^ 4 = \\mathbb E | N - \\lambda | ^ 4 = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( k - \\lambda ) ^ 4 \\lambda ^ k e ^ { - \\lambda } } { k ! } = \\lambda ( 3 \\lambda + 1 ) , \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s x _ { i 1 } ^ j = \\sum _ { i = 1 } ^ s x _ { i 2 } ^ j = \\ldots = \\sum _ { i = 1 } ^ s x _ { i h } ^ j ( 1 \\le j \\le k ) . \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} 0 = ( ( - 1 ) ^ a + 1 ) \\kappa ( S _ 1 ) = ( ( - 1 ) ^ a + 1 ) \\kappa ( S _ 2 ) = ( \\zeta ^ { 2 a } + \\zeta ^ a + 1 ) \\kappa ( U ) \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { n = 1 } ^ \\infty a _ n q ^ n \\ , \\ , ( q = e ^ { 2 \\pi i \\tau } ) \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\overline { c _ 2 } ( t ) \\geq h ( t ) : = \\left [ \\frac { 1 } { \\| 1 / c _ { 2 , 0 } ^ \\alpha \\| _ { L ^ { \\infty } ( \\Omega ) } } + \\alpha ( \\alpha + 1 ) k _ 3 t \\right ] ^ { - 1 / \\alpha } t > 0 . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} F ( h ) & = \\lim _ n F ( h _ n ) = \\lim _ n F \\left ( \\sum _ { i = 1 } ^ n h \\chi _ { [ i - 1 , i ) } \\right ) = \\lim _ n \\sum _ { i = 1 } ^ n F ( h \\chi _ { [ i - 1 , i ) } ) \\\\ & = \\lim _ n \\sum _ { i = 1 } ^ n F ( \\chi _ { [ i - 1 , i ) } ) h = \\left ( \\sum _ { i = 1 } ^ \\infty F ( \\chi _ { [ i - 1 , i ) } ) \\right ) h . \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\begin{cases} g ( x , y ) = ( g x , \\varphi ( g ) y ) \\\\ \\tau ' g ( x , y ) = ( \\varphi ( g ) y , \\tau g x ) \\end{cases} \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\partial _ { t } v ^ { \\mu } \\partial _ { l } \\eta _ { \\mu } = - \\frac { J } { \\rho _ 0 } \\partial _ { l } q , \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} - \\int \\ , f \\ , L _ \\eta f \\ , \\dd \\mu \\ , = \\ , \\int \\ , ( H ^ { - 2 \\eta } \\ , | \\nabla _ x f | ^ 2 + | \\nabla _ y f | ^ 2 ) \\ , \\dd \\mu \\ , : = \\mathcal E _ \\eta ( f ) . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} & { J } _ 1 ( \\mathbf { k } , \\tau ) : = f \\cos ( \\tau \\mathcal { A } ( \\mathbf { k } ) ^ { 1 / 2 } ) f ^ { - 1 } - f _ 0 \\cos ( \\tau \\mathcal { A } ^ 0 ( \\mathbf { k } ) ^ { 1 / 2 } ) f _ 0 ^ { - 1 } , \\\\ & { J } _ 2 ( \\mathbf { k } , \\tau ) : = f \\mathcal { A } ( \\mathbf { k } ) ^ { - 1 / 2 } \\sin ( \\tau \\mathcal { A } ( \\mathbf { k } ) ^ { 1 / 2 } ) f ^ * - f _ 0 \\mathcal { A } ^ 0 ( \\mathbf { k } ) ^ { - 1 / 2 } \\sin ( \\tau \\mathcal { A } ^ 0 ( \\mathbf { k } ) ^ { 1 / 2 } ) f _ 0 . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} g _ { 1 } ( V , \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\phi \\mathcal { B } Y + \\mathcal { A } _ { X } \\omega \\mathcal { B } Y ) & = g _ { 2 } ( \\pi _ * ( \\mathcal { C } Y ) , ( \\nabla \\pi _ * ) ( X , \\varphi V ) ) , \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\tilde { L } ^ { p } ( \\Pi ) = L ^ { p } \\cap L ^ { 2 } ( \\Pi ) \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\int e ^ { i \\lambda z ^ \\prime ( x , \\theta ^ \\prime ) . \\xi ^ \\prime } a _ { N } ( x , \\theta ^ \\prime ) f _ { i _ { 1 } \\dots i _ { m } } ( x ) { b ^ { i _ { 1 } } } ( x , \\theta ^ \\prime ) \\cdots { b ^ { i _ m } } ( x , \\theta ^ \\prime ) d x = 0 . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\blacksquare ( F ' ) = \\{ ( D , I _ 4 , E , F , O _ 5 , O _ 6 ) : \\ ; I _ 4 = O _ 5 = O _ 6 = r ( \\beta ) D \\} . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} d _ { i , \\alpha } { \\cal F } ( i _ 0 ) & = D G _ \\delta ( { \\mathtt u } _ \\delta ) \\circ { \\mathbb D } \\circ D { \\widetilde G } _ \\delta ( { \\mathtt u } _ \\delta ) ^ { - 1 } + { \\cal E } + { \\cal E } _ \\omega + { \\cal E } _ \\omega ^ \\bot \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} I _ \\lambda ( \\zeta , \\mu ) = F _ \\lambda ( \\zeta , \\mu ) + \\theta _ \\lambda ( \\zeta , \\mu ) \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\mathcal W = \\{ \\xi \\in \\R ^ n \\colon F ^ o ( \\xi ) < 1 \\} \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} K _ 0 ^ \\times \\times K _ 3 ^ \\times = \\tilde { T } _ 0 ( F ) \\times \\tilde { T } _ 3 ( F ) \\to T _ 0 ( F ) \\times T _ 3 ( F ) \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} u _ { j + 1 } & = e ^ { t A } u _ 0 - \\int _ { 0 } ^ { t } e ^ { ( t - s ) A } \\mathbb { P } ( u _ { j } \\cdot \\nabla u _ j ) \\dd s , \\\\ u _ 1 & = e ^ { t A } u _ 0 , \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\aligned V _ { \\frak r } ( p ; A ) = & \\{ y = ( \\overline y , ( t _ 1 , \\dots , t _ k ) ) \\in \\overline V _ { \\frak r } \\times [ - \\tau , 1 ) ^ { k } ~ \\vert ~ \\\\ & \\Pi _ { A ^ c } ( y ) \\in ( S _ A ( V _ { \\frak r } ) ) ^ { \\boxplus \\tau } , i \\in A \\Rightarrow t _ i \\in [ - \\tau , 0 ) \\} . \\endaligned \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} y ^ 3 - ( 1 + \\theta + \\theta ^ 2 ) y ^ 2 - ( 2 \\theta + 1 ) ( 1 - \\theta - q ) y - ( 1 - \\theta - q ) ^ 2 = 0 . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} \\mu ( \\mathcal { R } _ g ) = \\frac { 1 } { 2 ^ k } \\mu ( \\mathbb { R } ^ d ) \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S = \\sum _ a \\binom { a } { 2 } \\binom { | S | } { a } \\sum _ b \\binom { r - | S | - 1 } { 2 } + \\sum _ a \\binom { | S | } { a } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 1 } { b } . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\frac { ( E ^ { * } _ { n _ { \\ell } } , E ^ { * } _ { n _ { i } } ) } { ( E ^ { * } _ { n _ { 1 } } , E ^ { * } _ { n _ { i } } ) } < \\frac { ( E ^ { * } _ { n _ { \\ell } } , E ^ { * } _ { n _ { i + 1 } } ) } { ( E ^ { * } _ { n _ { 1 } } , E ^ { * } _ { n _ { i + 1 } } ) } , \\forall \\ i < \\ell \\textnormal { a n d } \\frac { ( E ^ { * } _ { n _ { \\ell } } , E ^ { * } _ { n _ { i } } ) } { ( E ^ { * } _ { n _ { 1 } } , E ^ { * } _ { n _ { i } } ) } = \\frac { ( E ^ { * } _ { n _ { \\ell } } , E ^ { * } _ { n _ { i + 1 } } ) } { ( E ^ { * } _ { n _ { 1 } } , E ^ { * } _ { n _ { i + 1 } } ) } , \\forall \\ i \\geq \\ell . \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} U _ { \\mu _ { i , n } , \\zeta _ { i , n } } ( x ) = C \\ , \\varepsilon _ { n } ^ { 1 / 2 } \\ , ( 1 + o ( 1 ) ) \\ , G _ { \\lambda } ( x , \\zeta _ { i , n } ) . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\ddot { x } _ { R } = 2 w _ { R } '' \\dfrac { w _ { R } } { | w _ { R } | ^ { 4 } } = - 2 \\frac { w _ { R } ^ { 2 } | w _ { R } ' | ^ { 2 } } { | w _ { R } | ^ { 6 } } . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\alpha _ Q ( q ^ i , p _ i , \\dot { q } ^ i , \\dot { p } _ i ) = ( q ^ i , \\dot { q } ^ i , \\dot { p } ^ i , p _ i ) . \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\psi ( \\mathbf { D } ) = \\left \\{ \\{ n ( n - 1 ) \\} ^ { - 1 } \\sum _ { 1 \\leq i < j \\leq n } \\frac { 1 } { \\prod _ { k = 1 } ^ p ( x _ { i , k } - x _ { j , k } ) ^ 2 } \\right \\} ^ { 1 / p } . \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\int \\eta ^ 2 u _ k ^ p \\ ; d x + \\frac { 4 ( p - 1 ) } { p } \\int ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla ( \\eta u _ k ^ { p / 2 } ) ) \\ ; d x \\\\ & = \\frac { 4 ( p - 2 ) } { p } \\int u _ k ^ { p / 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u _ k ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x \\\\ & \\ ; \\ ; \\ ; \\ ; + p \\int ( u \\nabla a , \\nabla ( \\eta ^ 2 u _ k ^ { p - 1 } ) ) \\ ; d x . \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{gather*} A ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) \\ge 0 \\ u ( x _ 0 ) > 0 \\\\ B ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) \\ge 0 \\ u ( x _ 0 ) < 0 \\\\ - \\mathcal { Q } _ \\infty \\phi ( x _ 0 ) \\ge 0 \\ u ( x _ 0 ) = 0 \\end{gather*}"}
-{"id": "7700.png", "formula": "\\begin{align*} R _ k ^ { \\rm O M A I I } ( \\nu ) = \\alpha _ k ( \\nu ) \\log _ 2 \\left ( 1 + \\frac { p _ k ( \\nu ) g _ k ( \\nu ) } { \\alpha _ k ( \\nu ) } \\right ) . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} v = - \\left ( v _ x + \\frac 1 4 v _ y ^ 2 \\right ) \\in \\Sigma _ t , \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\sum _ { i , k , l = 1 } ^ { 2 } \\bigl ( g ^ { j i } g ^ { k l } - g ^ { i k } g ^ { l j } \\bigr ) = 0 \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\tilde h = s _ { k } - s _ { k - 1 } = \\frac { b - a } { m } , \\ ; k = 1 , \\ldots , m . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} ( X \\cap U _ x ) ^ { \\boxplus \\tau } : = ( s _ x ^ { \\boxplus \\tau } ) ^ { - 1 } ( 0 ) / \\Gamma _ x , \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\int _ \\R e ^ { - \\frac { i x ^ 2 } { 2 } } \\ , d x = \\sqrt { \\tfrac { 2 \\pi } { i } } . \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} F _ { \\mu _ 1 } ( \\omega _ { 2 } ( z ) ) = F _ { \\mu _ 2 } ( \\omega _ { 1 } ( z ) ) \\ , , \\qquad \\omega _ 1 ( z ) + \\omega _ 2 ( z ) - z = F _ { \\mu _ 1 } ( \\omega _ { 2 } ( z ) ) \\ , . \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\mathcal { L } _ r ( m , w ) = \\lim _ { k \\to \\infty } \\int _ { \\Omega } \\sup _ { ( a , b ) \\in A _ { r ' , k } ( x ) } [ a m ( x ) + b \\cdot w ( x ) ] \\ , \\dd x . \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} c _ E y _ 1 & = \\omega ( 1 - \\omega ^ 2 ) \\varphi _ 2 , c _ E y _ 2 = ( 1 - \\omega ^ 2 ) \\varphi _ 1 + ( 1 - \\omega ^ 2 ) \\varphi _ 2 , \\\\ c _ E y _ 3 & = ( 1 - \\omega ) \\varphi _ 3 , c _ E y _ 4 = ( 1 - \\omega ^ 2 ) \\varphi _ 4 - \\omega ^ 2 ( 1 - \\omega ^ 2 ) \\varphi _ 2 . \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\ e ^ g \\ \\ ( L ^ \\alpha - \\mu ) e ^ g = \\lambda _ 1 e ^ g \\ \\ \\lambda _ 1 < 0 . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} p _ k ( x ) : = \\sup _ { n \\in \\N } w _ k ( n ) | x _ n | , x \\in \\Lambda _ 0 ( \\alpha ) , k \\in \\N . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\L = \\delta ^ { - 1 } \\begin{pmatrix} L & 0 \\\\ 0 & L ^ { - 1 } \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ - P & 1 \\end{pmatrix} \\Z ^ 2 . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} Y ( t _ { n , i } ) = \\int _ 0 ^ { t _ { n , i } } g ( s , W _ 0 ^ s , Y _ 0 ^ s ) d s + B ( t _ { n , i } ) , i = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } \\epsilon ^ { - 1 / 2 } \\left ( A _ 1 ( \\epsilon x ) - A _ 1 ( 0 ) \\right ) \\stackrel { d i s t . } { = } \\sqrt { 2 } B ( x ) \\ , . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\sigma \\circ y : = ( y _ { \\sigma ^ { - 1 } ( 1 ) } , \\ldots , y _ { \\sigma ^ { - 1 } ( d ) } ) . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\widehat { \\deg } _ + ( D , g ) : = { \\int _ { 0 } ^ { + \\infty } \\mathrm { r k } _ K ( \\mathcal F ^ t ( H ^ 0 ( D ) ) ) } \\ , \\mathrm { d } t \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\eta ( a , b ) = e ^ { - b } \\frac { 1 - e ^ { - a b } } { 1 - e ^ { - b } } . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\begin{aligned} x ' & = - 3 x ^ 2 - x y - x w \\\\ y ' & = - 3 y ^ 2 - x y - y z \\\\ z ' & = - 3 z ^ 2 - y z \\\\ w ' & = - 3 w ^ 2 - x w . \\end{aligned} \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\forall t , \\forall | \\phi | \\leq P _ Y , \\ : \\d Y _ t = - ( c _ 0 Y _ t + k Z _ t ) \\d t + \\d W _ t , ( \\d Z _ t - Y _ t \\d t ) ( \\phi - Z _ t ) \\geq 0 \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\mathbf V = v ^ t r + r ^ t v - 2 \\langle \\mathbf v _ i , \\mathbf v _ j \\rangle , \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} G ( t , x , g ) : = \\inf _ { a \\in A } \\left \\{ \\int _ U [ g ( x + f ( t , x , u , a , m ( t ) ) ) - g ( x ) ] \\nu ( d u ) + c ( t , x , a , m ( t ) ) \\right \\} . \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\log n = o ( ( n / m ) ^ { 1 / 2 } \\log ( n / m ) ) . \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} D = a _ 1 A _ 1 + \\cdots + a _ r A _ r \\quad g = a _ 1 h _ 1 + \\cdots + a _ r h _ r \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\tilde { \\mathcal { R } } _ { i , j } ^ { s , t } : = \\int _ { B _ \\rho ( \\zeta _ i ) } U _ i ^ t \\ , U _ j ^ s = O ( \\mu ^ 3 ) . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} V ( x ) = \\sum _ { i = 1 } ^ n \\frac { c } { | x - a _ i | ^ 2 } , x \\in \\R ^ N , c > 0 , a _ 1 , \\dots , a _ n \\in \\R ^ N , \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} a ( i ) \\in \\{ 0 , 1 , \\ldots n - 1 \\} , \\ : i = 1 , 2 , \\ldots , p \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} c _ j = \\frac { 1 } { P _ { r } ( \\vec { 1 } ) } \\left [ \\frac { \\partial P _ r } { \\partial z _ j } ( \\vec { 1 } ) ( 1 - \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } ) + \\frac { \\partial P _ { r + 1 } } { \\partial z _ j } ( \\vec { 1 } ) \\right ] \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} p \\left ( x , t \\right ) = \\sum _ { n = 1 } ^ { \\infty } T _ { n } \\left ( t \\right ) X _ { n } \\left ( x \\right ) \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\big \\| \\Psi ^ { t + s } ( x ) - \\Psi ^ { t } ( x ) - s \\ , \\dot \\Psi ^ t ( x ) \\big \\| _ { C ^ { 0 } ( \\R ^ n ; \\R ^ n ) } = o ( s ) \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} \\nu _ j = j ^ { - 1 } \\mathbb { P } ( Y _ 1 \\ge j ) \\mbox { f o r } ~ j \\in \\mathbb { N } _ { + } , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\partial ^ { j } } { \\partial \\rho ^ { j } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = j ! \\int \\limits _ { - 1 } ^ 1 f ( t ) r _ j ( t ) d t \\mbox { f o r e a c h } j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} \\int _ { \\bar { Q } ^ T _ r ( x _ 0 ) } \\ d x d t = O ( r ^ N ) , \\mbox { a s } \\ r \\to 0 \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } g ( t _ { n , i } , W _ 0 ^ { ( n ) , t _ { n , i } } , Y _ 0 ^ { ( n ) , t _ { n , i } } ) d s + B ( t _ { n , i + 1 } ) - B ( t _ { n , i } ) , \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} f ^ { 3 , 0 } ( Y , w ) & = f ^ { 0 , 3 } ( Y , w ) = 1 , \\\\ f ^ { 1 , 1 } ( Y , w ) & = f ^ { 2 , 2 } ( Y , 2 ) = k ( Y , w ) , \\\\ f ^ { 2 , 1 } ( Y , w ) & = f ^ { 1 , 2 } ( Y , w ) = p h ( Y , w ) - 2 + h ^ { 2 , 1 } ( Z ) \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} H ( t , x , a , p , g ) : = \\Lambda ^ { a , p } _ t g ( x ) + c ( t , x , a , p ) . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} 2 ( g _ d - 1 ) = - 2 \\deg \\widetilde h _ d + m _ d + 4 \\deg S _ d = - 6 d ( d - 2 ) + m _ d + 1 2 ( d - 1 ) ^ 2 . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} \\bar A ^ { \\lambda } = \\left [ \\begin{array} { c c c } \\times & \\otimes & \\otimes \\\\ \\otimes & \\otimes & \\otimes \\\\ \\otimes & \\otimes & \\otimes \\end{array} \\right ] , \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\sigma _ { X ^ * } ( m ) = \\lim _ { t \\to \\infty } \\frac { \\log \\left | I _ { m - 1 } ( t ) \\right | } { \\log t } = m - \\alpha \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} a '' & = \\frac { 2 n Q ^ 4 } { a } - 2 n \\frac { ( a ' ) ^ 2 } { a } + \\left ( f ' + 2 n \\frac { Q ' } { Q } \\right ) a ' \\\\ b '' & = 2 \\frac { n + 1 - Q ^ 2 } { b } - 2 n \\frac { ( b ' ) ^ 2 } { b } + \\left ( f ' - \\frac { Q ' } { Q } \\right ) b ' . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\tilde { \\partial } ( q + ( f _ s ) ) = \\partial ( q ) + ( f _ s ) , \\ , \\ , \\ , q \\in K [ \\lambda , \\mu ] . \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial u } { \\partial x } \\left ( x , r , t \\right ) d r . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t \\eta = G ( 0 , { \\mathtt h } ) \\psi \\\\ & \\partial _ t \\psi = - \\eta \\end{aligned} \\right . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\partial ^ { j } } { \\partial \\rho ^ { j } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = j ! \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ { j - 1 } ( t ) d t \\mbox { f o r e a c h } j = 1 , 2 \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} \\Phi _ { n } \\left ( \\nu , p \\right ) : = \\dfrac { \\left \\vert \\tilde { { E } } _ { n } \\left ( p \\right ) - k _ { n } \\right \\vert } { \\nu ^ { n } } . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} S ^ { \\pm 1 } X ^ { \\pm 1 } = X ^ { \\pm 1 } S ^ { \\pm 1 } , \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{gather*} A '' + \\overset { \\mathrm { s f } } R A = 0 . \\end{gather*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\left . \\partial ^ 2 _ t ( \\Delta _ g \\eta ^ 3 ) \\right | _ { t = 0 } = & \\ , \\delta ^ { i j } \\partial ^ 2 _ { i j } \\partial _ t v ^ 3 ( 0 ) + \\partial _ t g ^ { i j } ( 0 ) \\partial ^ 2 _ { i j } v ^ 3 ( 0 ) - \\delta ^ { i j } \\partial ^ k v ^ 3 ( 0 ) \\partial ^ 2 _ { i j } v _ k ( 0 ) \\\\ = & \\ , \\delta ^ { i j } \\partial ^ 2 _ { i j } \\partial _ t v ^ 3 ( 0 ) + F _ 0 , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} _ C \\langle y \\cdot d , y ' \\rangle = _ C \\langle y , y ' \\cdot d ^ * \\rangle , \\quad & \\langle c \\cdot y , y ' \\rangle _ D = \\langle y , c ^ * \\cdot y ' \\rangle _ D \\\\ _ C \\langle y , y ' \\rangle \\cdot y '' & = y \\cdot \\langle y ' , y '' \\rangle _ D . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\dim ( R _ { n , k - 1 , r } ) + \\dim ( R _ { n , k , r + 1 } ) = \\dim ( R _ { n , k , r } ) . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu ( \\{ x \\in X : \\log D ( f ^ n ( x ) ) / n \\ge c \\} ) = \\lim _ { n \\to \\infty } \\mu ( \\{ x \\in X : \\log D ( x ) \\ge n c \\} ) = 0 , \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} ( \\ ! ( F , G ) \\ ! ) _ { L ^ { 2 } ( \\mu _ { \\beta } ) } : = \\int _ { S ' _ { d } } F ( w ) \\bar { G } ( w ) \\ , d \\mu _ { \\beta } ( w ) , F , G \\in L ^ { 2 } ( \\mu _ { \\beta } ) . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} \\log ( e ^ { \\epsilon Q } e ^ { \\epsilon Q ' } ) - \\left ( \\epsilon Q + \\epsilon Q ' \\right ) = \\textstyle { \\frac { 1 } { 2 } } \\epsilon ^ 2 \\left [ Q , Q ' \\right ] + \\ldots \\in \\mathcal { R } , \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} f ( x ) = f ^ * ( - x + c ) , x \\in X . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} g _ { 1 } ( \\mathcal { A } _ { X } \\omega W , \\mathcal { B } Y ) + \\eta ( Y ) g _ { 1 } ( X , \\omega W ) & = g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , Y ) , \\pi _ * \\omega \\phi W ) - g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , \\mathcal { C } Y ) , \\pi _ * \\omega W ) \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} V _ m ^ r ( x , K ) = \\sum _ { \\mathfrak { N } \\mathfrak { a } \\le x ^ { 1 / r } } \\mu ( \\mathfrak { a } ) I _ K \\left ( \\frac x { \\mathfrak { N } \\mathfrak { a } ^ r } \\right ) ^ m . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} F _ p ( p , a , 2 ) & = F _ p ( p , a , 3 ) - \\dim ( a - 1 , 3 ) \\cr & = \\dim ( a , 3 ) - \\dim ( a - 1 , 3 ) \\cr & = \\dim ( a , 2 ) \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} g _ 2 ( \\pi _ * X _ 1 , \\pi _ * X _ 2 ) & = g _ 1 ( X _ 1 , X _ 2 ) \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} ( \\overline { C } _ i + \\delta _ i ) ^ { y _ { r , i } } = \\overline { C _ i } ^ { y _ { r , i } } + \\widetilde { R } _ i \\delta _ i \\widetilde { R } _ i = y _ { r , i } ( \\theta \\overline { C _ i } + ( 1 - \\theta ) \\delta _ i ) ^ { y _ { r , i } - 1 } \\theta \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\left ( s + p \\theta ( x ) - v { \\partial \\over \\partial x } - { \\partial ^ 2 \\over \\partial v ^ 2 } \\right ) \\tilde { Q } _ p ( x , v , s ) = 1 \\ , , \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\left ( \\left ( \\overline W ^ T \\right ) ^ { - 1 } \\right ) ^ * \\overline C ^ T \\left ( \\overline W ^ T \\right ) ^ { - 1 } = I , \\overline C ^ T = J C ^ T J . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} x _ { k + 1 } = \\frac { e ^ { a h } x _ k } { 1 + ( b \\phi _ 1 ) y _ k } , y _ { k + 1 } = \\left [ e ^ { - c h } + d \\phi _ 2 x _ { k + 1 } \\right ] y _ k . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\dot { \\hat { v } } ( t ) = F ( \\hat { v } ( t ) ) \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ n \\Gamma ( k + z ) = \\frac { G ( z + n + 1 ) } { G ( z + 1 ) } . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} \\int f \\ { \\rm d } \\mu _ \\eta \\otimes { \\rm d } s _ \\eta ^ { \\alpha _ 1 } = \\int ( g - K ) j \\ { \\rm d } \\mu _ \\eta \\otimes { \\rm d } s _ \\eta ^ { \\alpha _ 1 } = \\int ( g - K ) \\ { \\rm d } \\mu _ \\eta \\otimes { \\rm d } s _ \\eta ^ { \\sf H } = 0 , \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} \\abs { r ^ { n + 2 } ( x _ 0 ) \\sum ^ { N _ { x _ 0 } } _ { j = 1 } \\abs { g _ j ( x _ 0 ) } ^ 2 } \\geq \\frac { 1 } { 2 } \\abs { F ( x _ 0 ) } , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} p ^ n ( t , x , y ) : = \\frac { \\P ( X ^ n _ t = y \\ , | X ^ n _ 0 = x ) } { m _ n ( y ) } . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} 2 4 = \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) + \\sum _ { F ^ \\circ \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ^ \\circ ) + V _ \\Delta + V _ { \\Delta ^ \\circ } - 2 . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} c _ { V , W } ( v \\otimes w ) = c ( x , y ) ( w \\otimes v ) , v \\in V _ x , \\ w \\in W _ y \\ . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\widetilde { D } _ { q , \\theta } ^ { ( m ) } \\big \\{ ( x _ 1 \\cdots x _ m ) ^ k f \\big \\} = t ^ { k { m \\choose 2 } } ( x _ 1 \\cdots x _ m ) ^ k \\widetilde { D } _ { q , \\theta } ^ { ( m ) } \\{ f \\} . \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} 1 ) & X ( \\cdot , \\varepsilon ) = X _ { \\varepsilon } ( \\cdot ) + \\frac { \\partial } { \\partial \\varepsilon } \\\\ 2 ) \\mathcal { F } _ { \\varepsilon } & \\ , i s \\ , a \\ , f o l i a t i o n \\ , b y \\ , c i r c l e s \\ , d e f i n e d \\ , b y \\ , X _ { \\varepsilon } : P \\rightarrow T P . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\partial _ { t } u ( t , x ) - \\sum _ { i , j = 1 } ^ { n } \\partial _ { x _ { i } } ( a _ { i j } ( t , x ) \\partial _ { x _ { j } } u ( t , x ) ) + \\sum _ { i = 1 } ^ { n } b _ { i } ( t , x ) \\partial _ { x _ { i } } u ( t , x ) = 0 , \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} u = \\sum _ { N \\in 2 ^ \\Z } \\sum _ { k \\in \\Z } u _ { N , k } , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\mathcal { \\overline { D } } _ { f r } = \\left \\{ \\left ( \\omega _ { 1 } ^ { l } , \\omega _ { 2 } ^ { m } \\right ) \\left | \\omega _ { 1 } = \\frac { l } { r } C , \\ , \\omega _ { 2 } = \\frac { m } { r } C ; \\ , l , m \\in \\left \\{ 0 , 1 , \\ldots , r \\right \\} \\right . \\right \\} . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{gather*} n \\circ \\eta = \\frac { a ^ T N } { | a ^ T N | } . \\end{gather*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( M ) : = \\inf _ { f \\in \\mathcal { H } ^ p _ 0 } E ( f ) , \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\sum _ { p ^ j \\le Y } \\omega _ { p ^ j } ( n ) \\log p = \\sum _ { n \\le x } \\sum _ { p ^ j \\le Y } \\log p \\sum _ { \\substack { q \\mid n \\\\ q \\equiv 1 \\mod { p ^ j } } } 1 & = \\sum _ { p ^ j \\le Y } \\log p \\sum _ { \\substack { q \\le x \\\\ q \\equiv 1 \\mod { p ^ j } } } \\sum _ { \\substack { n \\le x \\\\ q \\mid n } } 1 \\\\ & \\le \\sum _ { p ^ j \\le Y } \\log p \\sum _ { \\substack { q \\le x \\\\ q \\equiv 1 \\mod { p ^ j } } } \\frac x q . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { k _ E ( \\tau _ 1 - t ) } ( \\frac { 1 } { 2 } b _ M k _ M k _ E ( T - \\tau _ 1 ) ^ 2 + b _ E k _ E ( T - \\tau _ 1 ) + b _ E ) > \\\\ b _ M k _ M ( T - \\tau _ 1 ) + b _ M k _ M ( \\tau _ 1 - t ) + b _ E . \\end{aligned} \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} \\abs { x } p - \\mu ( \\infty , x ) p & = \\abs { x } - \\mu ( \\infty , x ) s ( x ) = V \\mu ( \\abs { x } - \\mu ( \\infty , x ) s ( x ) ) \\\\ & = V \\left ( \\mu ( x ) - \\mu ( \\infty , x ) \\right ) = V \\mu ( x ) - \\mu ( \\infty , x ) V ( \\chi _ { [ 0 , \\infty ) } ) \\\\ & = V \\mu ( x ) - \\mu ( \\infty , x ) p , \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v = - \\Delta \\dot h ^ 0 & \\mbox { i n } \\Omega ^ 0 \\\\ v = 0 & \\mbox { o n } \\Gamma ^ 0 \\\\ v ( \\infty ) = \\dot c ^ 0 , & \\lim _ { x \\to \\infty } \\frac { v ( x ) } { - \\log | x | } = \\dot c ^ 0 . \\end{cases} \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } x | P _ n \\varphi ( x ) | ^ 2 \\bigg | _ { - \\infty } ^ { \\infty } - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | P _ n \\varphi | ^ 2 ~ d x + \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { \\infty } ( \\psi _ n \\widehat { u \\varphi } ) ' \\psi _ n \\bar { \\hat \\varphi } ~ d \\xi = 0 . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} \\mathbb E \\bigl [ ( f ( s ) - \\widetilde f ( s ) ) g ( s ) \\bigr ] & = \\mathbb E \\Bigl [ \\mathbb E \\bigl ( ( f ( s ) - \\widetilde f ( s ) ) g ( s ) \\big | \\mathcal F _ { \\frac { n } { 2 ^ m } } \\bigr ) \\Bigr ] \\\\ & = \\mathbb E \\Bigl [ \\mathbb E \\bigl ( ( f ( s ) - \\widetilde f ( s ) ) \\big | \\mathcal F _ { \\frac { n } { 2 ^ m } } \\bigr ) g ( s ) \\Bigr ] = 0 . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\delta _ { A ' } ^ A : = Z ^ { A } _ { A ' } - 2 Z ^ { A } _ { A ' } \\varphi , \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} \\widehat { S } ( \\mathbf { k } ) : = t ^ 2 \\widehat { S } ( \\boldsymbol { \\theta } ) = b ( \\mathbf { k } ) ^ * g ^ 0 b ( \\mathbf { k } ) , \\mathbf { k } \\in \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} X ^ { V } ( w _ q ) = ( X ( q ) ) _ { w _ q } ^ V \\in T _ { w _ q } ( T Q ) \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} A \\left ( u \\right ) = - \\delta \\left ( x _ { 1 } - x _ { 0 } , \\overline { x } - \\overline { x } ^ { 0 } \\right ) , x \\in \\mathbb { R } ^ { n } , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} \\int _ { \\Omega _ 2 } g ( y ) \\ , d \\Psi _ * \\mu ( y ) = \\int _ { \\Omega _ 1 } g ( \\Psi ( x ) ) \\ , d \\mu ( x ) \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 0 ^ m , 0 ^ { n - m } ) , \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} b ^ \\prime & = b _ { r _ 2 } \\ge \\frac { k - r _ 2 + 1 } { r _ 2 } b _ { r _ 1 } \\ge \\frac { r _ 1 + 1 } { k - r _ 1 } b _ { r _ 1 } \\\\ & \\ge \\frac { r _ 1 + 1 } { k - r _ 1 } \\cdot \\frac { k - r _ 1 + 1 } { r _ 1 } b = \\frac { r _ 1 + 1 } { r _ 1 } \\cdot \\frac { k - r _ 1 + 1 } { k - r _ 1 } b \\\\ & \\ge ( k / ( k - 1 ) ) ^ 2 b > ( 1 + 2 / k ) b . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\hat { B } _ { i j } = \\mathbb { E } \\left [ B \\Phi _ i \\Phi _ j \\right ] \\approx \\tilde { B } _ { i j } : = \\sum _ { r = 1 } ^ k w _ r B ( p ^ { ( r ) } ) \\Phi _ i ( p ^ { ( r ) } ) \\Phi _ j ( p ^ { ( r ) } ) \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\frac { t - t _ { n , i } } { t _ { n , i + 1 } - t _ { n , i } } ( Y ^ { ( n ) } ( t _ { n , i + 1 } ) - Y ^ { ( n ) } ( t _ { n , i } ) ) , t _ { n , i } \\leq t \\leq t _ { n , i + 1 } . \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} b _ { 1 } = K ^ { 1 } \\kappa _ { 1 } ^ { 1 } \\mathcal { \\alpha } _ { 1 } ^ { 1 } , \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} K ( u ) : = \\frac { T \\pi } { \\pi ^ 2 - T ^ 2 u ^ 2 } \\ , , u \\in \\C \\ , , \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} S _ { p , q } \\ni \\mathcal { C } \\left ( W , Z \\right ) , \\left ( \\mathcal { C } \\left ( W , Z \\right ) \\right ) ^ { t } = \\frac { \\left [ W - \\sqrt { - 1 } I _ { q } , 2 Z \\right ] } { W + \\sqrt { - 1 } I _ { q } } , \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} ( X _ { 0 } , Q _ { 0 } , R _ { 0 } ) \\ , = \\ , ( x _ { 0 } , q _ { 0 } , 0 ) \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\sup \\{ S _ n g ( \\eta ) : \\eta \\in [ \\xi _ 1 , . . . , \\xi _ n ] \\} = - n h ( Y ) . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} f _ { \\cdots ( A _ 1 ' \\ldots A _ k ' ) \\cdots } : & = \\frac 1 { k ! } \\sum _ { \\sigma \\in S _ k } f _ { \\cdots A _ { \\sigma ( 1 ) } ' \\ldots A _ { \\sigma ( k ) } ' \\cdots } . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} ( P _ { 1 , 1 } ( t ) , P _ { 1 , 2 } ( t ) , \\ , P _ { 2 , 1 } ( t ) , P _ { 2 , 2 } ( t ) , \\ , P _ { 3 , 1 } ( t ) , P _ { 3 , 2 } ( t ) , \\ , Q ( t ) ) = ( 0 , \\infty , \\ , 1 , - 1 , \\ , i , - i , \\ , \\varepsilon \\ , e ^ { 2 \\pi i t } ) \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} \\varphi _ M ^ \\hbar ( x ) = ( \\tfrac { 1 } { \\pi \\hbar } ) ^ { n / 4 } \\det ( R e ( M ) ) ^ { 1 / 4 } e ^ { - \\tfrac { 1 } { 2 \\hbar } M x ^ 2 } , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\big \\lvert \\Psi \\big ( u + \\mathrm { i } a ( u ) + \\mathrm { i } \\tau \\big ) - \\alpha \\big \\rvert & = \\lvert h _ j ( u ) - z _ { 0 j } \\rvert \\\\ & \\geqslant \\frac 1 2 \\big ( \\lvert \\mathrm { R e } \\ , h _ j ( u ) - x _ { 0 j } \\rvert + \\varepsilon \\big ) , \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} a ( p ) = p + 1 - \\# E ( \\mathbb { F } _ p ) , \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} \\mathcal { K } _ m ' ( \\varphi _ m ) v ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\frac { \\partial \\kappa } { \\partial u } ( s , \\zeta _ i ^ j , \\varphi _ m ( \\zeta _ i ^ j ) ) v ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} F ( q , p , \\lambda ) = p _ i \\lambda ^ a X _ a ^ i ( q ) . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\begin{cases} T _ i ^ 2 = T _ i & 1 \\leq i \\leq n - 1 \\\\ T _ i T _ j = T _ j T _ i & | i - j | > 1 \\\\ T _ i T _ { i + 1 } T _ i = T _ { i + 1 } T _ i T _ { i + 1 } & 1 \\leq i \\leq n - 2 . \\end{cases} \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} ( 1 - \\chi _ { \\Sigma } ) ( x , h D ) q ( x , h D ) u = o _ { L ^ 2 } ( h ) . \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\psi _ x \\big ( ( a \\cdot \\delta _ u ) * ( a \\cdot \\delta _ u ) ^ * \\big ) & = \\psi _ x \\big ( a a ^ * \\cdot \\delta _ { u u ^ { - 1 } } \\big ) \\\\ & = \\| \\xi ( x ) \\| ^ { - 2 } \\big ( \\pi ( a a ^ * ) \\xi ( x ) \\big | \\xi ( x ) \\big ) \\\\ & = \\| \\xi ( x ) \\| ^ { - 2 } \\big ( \\pi ( a ^ * ) \\xi ( x ) \\big | \\pi ( a ^ * ) \\xi ( x ) \\big ) \\\\ & \\geq 0 \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\iint F ( x , z ) W ^ { \\rho _ \\sigma } _ v ( u ) ( ( x , z ) ^ { - 1 } ) \\ , d z d x & = \\iint F ( x , z ) W ^ { \\rho _ \\sigma } _ v ( u ) ( ( x ^ { - 1 } , \\overline { z } \\sigma ( x ^ { - 1 } , x ) ) ) \\ , d z d x \\\\ & = \\iint \\overline { z } f ( x ) W ^ { \\rho } _ v ( u ) ( x ^ { - 1 } ) z \\overline { \\sigma ( x , x ^ { - 1 } ) } \\ , d z d x \\\\ & = f \\# W ^ \\rho _ v ( u ) ( 1 ) . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} R i c ( W , W ) = - \\frac { 1 } { 2 } \\sum _ { i } | [ W , \\hat { Y _ i } ] | ^ 2 - \\frac { 1 } { 2 } \\sum _ { i } \\langle [ W , [ W , \\hat { Y _ i } ] ] , \\hat { Y _ i } \\rangle + \\frac { 1 } { 2 } \\sum _ { i < j } \\langle [ \\hat { Y _ i } , \\hat { Y _ j } ] , W \\rangle ^ 2 . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} T ' \\circ T = ( \\Phi ' \\circ \\Phi , ( B ' \\circ \\Phi ) \\cdot B , ( \\eta ' \\circ \\Phi ) \\eta ) . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} & P ^ { e } N P ^ { e } = P ^ { e } P , \\\\ & P ^ { e } N P ^ { \\pi } = P ^ { e } R , \\\\ & P ^ { \\pi } N P ^ { e } = S P ^ { e } , \\\\ & P ^ { \\pi } N P ^ { \\pi } = Q + P ^ { \\pi } R + ( S + P ) P ^ { \\pi } \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} c _ 0 = & \\ln \\left ( 2 \\| x ( 0 ) \\| _ 1 \\right ) + \\ln ( n ) \\left ( \\frac { n B } { p } + B \\right ) + \\ln ( 1 5 ) , \\cr c _ 1 = & - \\frac { p } { 2 n B } \\ln \\left ( 1 - \\frac { 1 } { n ^ { \\frac { 4 n B } { p } } } \\right ) . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\begin{cases} [ 1 ] \\dot { q } ^ 1 \\frac { \\partial \\gamma _ 1 ( q ) } { \\partial q ^ 1 } + \\dot { q } ^ 2 \\frac { \\partial \\gamma _ 2 ( q ) } { \\partial q ^ 1 } - 2 q ^ 2 q ^ 1 = 0 \\\\ [ 2 ] \\dot { q } ^ 1 \\frac { \\partial \\gamma _ 1 ( q ) } { \\partial q ^ 2 } + \\dot { q } ^ 2 \\frac { \\partial \\gamma _ 2 ( q ) } { \\partial q ^ 2 } - ( q ^ 1 ) ^ 2 = 0 \\\\ [ 3 ] \\gamma _ 1 ( q ) - \\dot { q } ^ 1 = 0 \\\\ [ 4 ] \\gamma _ 2 ( q ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} H _ d ( x ) & = x ^ d H _ 0 ( x ) + x ^ { d + 2 } \\left ( \\frac { 1 } { ( 1 - x ) ^ d } - 1 \\right ) L \\\\ & + \\left ( \\frac { 1 } { ( 1 - x ) ^ d } - 1 \\right ) \\sum _ { t \\ge 2 } x ^ { t + 1 + d } \\left ( 1 + ( L - 1 ) \\frac { 1 - 1 / ( 1 - x ) ^ t } { 1 - 1 / ( 1 - x ) } \\right ) \\ , . \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} l ( B _ i ) = \\O ( | B _ i | ) = \\O ( ( 3 ^ { 3 ^ i } - 1 ) / 2 ) + \\O ( | P _ i | ) = \\O ( ( 3 ^ { 3 ^ i } - 1 ) / 2 ) + 3 ^ i . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\frak m ^ { \\epsilon } _ { k , \\beta } ( h _ 1 , \\dots , h _ k ) : = { \\rm C o r r } _ { \\frak M _ { k + 1 } ( \\beta ) } \\left ( h _ 1 \\times \\dots \\times h _ k ; \\widehat { \\frak S } ^ { \\epsilon } _ { k + 1 } ( \\beta ) \\right ) . \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} W _ { n , 2 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ S _ { n } ^ { - } \\left ( { u , \\xi } \\right ) \\right \\} \\left [ \\exp \\left \\{ - u \\xi + { E _ { 0 } ^ { - } \\left ( \\xi \\right ) } \\right \\} { + \\varepsilon _ { n , 2 } \\left ( { u , \\xi } \\right ) } \\right ] , \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} \\lambda & = ( l + a _ 1 , \\ldots , l + a _ r , a _ { r + 1 } , \\ldots , a _ m ) + l \\nu \\cr & = ( l - m + r , \\ldots , l - m + r , a _ { r + 1 } , \\ldots , a _ m ) \\cr & + ( a _ 1 + m - r , \\ldots , a _ r + m - r ) + l \\nu . \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\phi ( [ X , Y ] , Z ) = \\phi ( [ X , Z ] , Y ) = - \\phi ( [ Z , X ] , Y ) \\forall X , Y , Z \\in \\mathfrak { s u } ( n ) \\ , . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\phi ( \\theta ) = \\sum _ { \\{ j \\ , : \\ , \\lambda _ j > 2 d \\} } c _ j \\phi _ j ( \\theta ) , \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} \\frac { \\partial W } { \\partial q ^ i } \\lambda ^ a X _ a ^ i ( q ) = . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} m ^ { - 1 / 2 } = o \\ ( \\frac { \\log ( n / m ) } { \\log n } \\ ) \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} X = \\int _ 0 ^ { \\infty } e ^ { - s } d L ( s ) , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { L _ { t } ^ { s } L _ { x } ^ { r } } \\leq C \\Vert f \\Vert _ { L _ { t } ^ { \\infty } L _ { x } ^ { 2 } } ^ { 1 - \\beta } \\Vert \\nabla f \\Vert _ { L _ { t } ^ { 2 } L _ { x } ^ { 2 } } ^ { \\beta } , \\qquad \\beta = \\frac { n } { 2 } - \\frac { n } { r } . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} X _ 0 ^ 2 = 1 , X _ 0 ^ 2 X _ 1 ^ 2 = 1 \\ ! - \\ ! 2 c _ 2 \\log \\vert z \\vert , \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} h _ { t } ^ { M } ( \\tilde { x } , \\tilde { y } ) = \\sum _ { \\gamma \\in \\Gamma } h _ { t } ( x , \\gamma y ) , \\ , \\ , \\ , x , y \\in X , t > 0 , \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\Bigl \\vert \\int _ { \\Omega _ { \\varepsilon _ n } } L ( { \\bf z } _ { k l } ^ n ) \\ , \\phi _ n \\Bigr \\vert = o ( 1 ) \\ , \\Vert \\phi _ n \\Vert _ { \\ast } \\quad l = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} m ( f g ) & = f g = m ( f ) m ( g ) , \\\\ \\partial m ( f ) & = \\partial ( f ) = d ( f ) = m d ( f ) \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} | L _ 1 [ h ] | \\lesssim \\begin{cases} e ^ { - \\alpha \\lambda t / ( d + \\alpha ) } ( 1 + e ^ { - \\lambda t } | x | ^ { d + \\alpha } ) ^ { - 1 - \\alpha / ( d + \\alpha ) } , & \\ e ^ { - \\lambda t } | x | ^ { d + \\alpha } > 1 \\\\ e ^ { - \\alpha \\lambda t / ( d + \\alpha ) } , & . \\end{cases} \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} t = \\frac { 2 } { 3 } \\sqrt { \\lambda } \\cosh \\left ( \\frac { 1 } { 3 } \\cosh ^ { - 1 } \\ ! \\left ( \\frac { 2 7 } { 1 6 } | w | ^ 2 \\lambda ^ { - 3 / 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} X _ { 0 } ^ { n } ( t ) & = Y ^ { n } ( t ) + \\eta ^ { n } ( t ) , \\\\ X _ { k } ^ { n } ( t ) & = X _ { k } ^ { n } ( 0 ) - \\frac { 1 } { n } \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { [ 0 , r _ { k } ( \\boldsymbol { X } ^ { n } ( s - ) ) ) } ( y ) \\ , N _ { k } ^ { n } ( d s \\ , d y ) , \\ : k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} & \\tau = \\frac { 1 } { 2 } \\begin{pmatrix} \\sqrt { - 3 } \\eta ^ { - 1 } & - 1 \\\\ - 1 & \\sqrt { - 3 } \\eta \\\\ \\end{pmatrix} , \\eta = \\dfrac { y _ 1 } { y _ 2 } \\in \\bold C . \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} h ^ { p , q } ( X ) = f ^ { d - p , q } ( Y , w ) . \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\frac { d } { d t } ( S U , U ) = \\frac { d } { d t } \\| S ^ { 1 / 2 } U \\| ^ 2 = 2 { \\mathsf { R e } } \\ , ( S U , F ) \\leq 2 \\| S ^ { 1 / 2 } U \\| \\| S ^ { 1 / 2 } F \\| \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\abs { \\sum _ { i = 1 } ^ n w _ { i 1 } } = \\left ( \\sum _ { i = 1 } ^ n | w _ { i 1 } | ^ 2 \\right ) ^ { \\frac 1 2 } = \\left ( \\sum _ { i = 1 } ^ n e _ 1 \\right ) ^ { \\frac 1 2 } = n ^ { \\frac 1 2 } e _ 1 , \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\Lambda _ f ( s , c _ \\chi ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n = 1 } ^ \\infty \\frac { f _ n c _ \\chi ( n ) } { n ^ { s + \\frac { k - 1 } 2 } } \\quad \\Lambda _ g ( s , c _ { \\overline { \\chi } } ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n = 1 } ^ \\infty \\frac { g _ n c _ { \\overline { \\chi } } ( n ) } { n ^ { s + \\frac { k - 1 } 2 } } . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\psi _ { k } ^ { \\prime } \\left ( x _ { 0 } \\right ) = \\psi _ { k } \\left ( x _ { 0 } \\right ) + \\displaystyle \\sum \\limits _ { j = 0 } ^ { k - 1 } b _ { j k } \\psi _ { j } \\left ( x _ { 0 } \\right ) . \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} d ( n ; a ) = \\frac 1 { \\varphi ( q ) } \\sum _ { \\chi \\pmod * { q } } \\overline \\chi ( a ) d _ \\chi ( n ) . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\lambda _ p = \\alpha _ p + \\beta _ p = \\pm p ^ { k - 1 / 2 } ( 1 + \\chi ^ 2 ( p ) ) . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} \\gamma ( a ) \\delta ( b ) ( c ) + ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\delta ( a ) ( c ) & = \\gamma ( a ) \\{ b , c \\} + ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\{ a , c \\} \\\\ & = a \\{ b , c \\} + ( - 1 ) ^ { | a | | b | } b \\{ a , c \\} \\\\ & = \\delta ( a b ) ( c ) , \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\gamma ^ { \\Delta r } ( t , x ) : = \\begin{cases} a _ 0 & t \\in [ 0 , \\Delta [ \\\\ a _ j ^ r \\quad & t \\in I _ { i j } ^ { \\Delta r } ( x ) \\end{cases} \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} [ L _ u , B _ u ] \\varphi = \\frac { 2 } { i } ( C _ + u _ { x x } ) \\varphi - \\frac { 1 } { i } C _ + ( u _ { x x } \\varphi ) - 2 C _ + ( u _ x u \\varphi ) . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\partial _ { t } \\omega _ { s } ^ { \\nu } = \\nu \\partial _ { y y } \\omega _ { s } ^ { \\nu } + P _ { 0 } \\left ( U _ { n } ^ { \\nu } \\cdot \\nabla \\omega _ { n } ^ { \\nu } \\right ) = \\nu \\partial _ { y y } \\omega _ { s } ^ { \\nu } + \\partial _ { y } P _ { 0 } \\left ( v _ { n } ^ { \\nu } \\omega _ { n } ^ { \\nu } \\right ) , \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} \\phi ^ T ( g ) = \\phi ( T ^ { - 1 } g T ) . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\phi { } _ { k , \\infty } ( x ) = \\exp \\left [ - \\frac { \\lambda _ { g } ( k , y _ { k } ) } { 2 } \\| x \\| ^ { 2 } \\right ] \\tilde { \\phi } _ { k , \\infty } ( x ) . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty k ^ * ( t ) \\bigg | \\sum _ { n = n _ 0 } ^ { \\infty } R _ { n } e ^ { r _ n t } \\bigg | ^ 2 \\ d t \\le c ( T ) \\sum _ { n = n _ 0 } ^ \\infty \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\ , . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} e r r _ 1 ( t ) = & \\int _ 0 ^ t \\sum _ { ( e , \\tilde { e } ) } \\ , 2 C ^ n _ { \\tilde { e } , e } \\ , \\big ( \\phi _ s ( x ^ e _ 1 ) - \\phi _ s ( x ^ { \\tilde { e } } _ 1 ) \\big ) \\ , \\big ( u ^ n _ { s - } ( x ^ { \\tilde { e } } _ 1 ) - u ^ n _ { s - } ( x ^ e _ 1 ) \\big ) \\ , d s . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} e _ { \\theta } \\cdot e ^ { t A } f & = e ^ { t L _ 0 } f ^ { \\theta } , \\\\ e _ { \\theta } \\cdot \\textrm { c u r l } \\ e ^ { t A } g & = e ^ { t L _ { 0 } ' } \\big ( e _ { \\theta } \\cdot \\textrm { c u r l } \\ g \\big ) , \\\\ r e ^ { t L _ 0 } \\gamma & = e ^ { t L _ 1 } ( r \\gamma ) , \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} B = \\frac { 1 } { 2 } \\left ( \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ { r } ( \\vec { 1 } ) } \\right | ^ 2 - 2 \\Re \\left ( \\frac { P _ { r + 2 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) + r \\right ) . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\int _ x ^ { h ^ n x } \\widetilde { F } + \\int _ x ^ { p x } \\widetilde { F } = \\int _ { p x } ^ { p h ^ n x } \\widetilde { F } + \\int _ x ^ { p x } \\widetilde { F } & \\ge - 2 L ( r , B ) - C _ { F , \\xi } ( x , p h ^ n x ) \\\\ & \\ge - 3 L ( r , B ) + \\int _ x ^ { p h ^ n x } \\widetilde { F } . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} U _ { \\phi } : L ^ { 2 } ( { \\mathbb { R } } ^ { n } ) & \\to L ^ { 2 } ( { \\mathbb { R } } ^ { 2 n } ) , \\\\ \\psi & \\mapsto \\Psi = ( 2 \\pi \\hbar ) ^ { n / 2 } W ( \\psi , \\phi ) . \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{gather*} \\det ( v ^ t r + r ^ t v - \\mu \\mathbf I _ n ) \\\\ = ( - 1 ) ^ n \\mu ^ { n - 2 } \\Bigl ( \\mu ^ 2 - \\Bigl ( 2 \\sum _ { i = 1 } ^ n \\| \\mathbf v _ i \\| ^ 2 \\Bigr ) \\mu - \\sum _ { 1 \\leq i < j \\leq n } \\left ( \\| \\mathbf v _ i \\| ^ 2 - \\| \\mathbf v _ j \\| ^ 2 \\right ) ^ 2 \\Bigr ) = 0 . \\end{gather*}"}
-{"id": "6576.png", "formula": "\\begin{align*} w _ r ^ { ( m ) } ( \\lambda ) = \\int _ { ( 0 , 1 ) ^ m } \\mathrm { d } \\lambda ^ { ( 1 ) } \\cdots \\mathrm { d } \\lambda ^ { ( m ) } \\delta ( \\lambda - \\lambda ^ { ( 1 ) } \\cdots \\lambda ^ { ( m ) } ) \\prod _ { l = 1 } ^ m w ( \\lambda ^ { ( l ) } ; L _ l ) \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\nu _ j = \\frac { q ^ { j - 1 } } { j } \\mbox { f o r } j \\in \\mathbb { N } _ { + } . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\frac { 1 } { A } \\frac { d A } { d t } & = - \\frac { A } { B E } - \\frac { A } { C D } \\\\ & = \\frac { - \\frac { 4 } { 1 1 } } { t + c } \\\\ \\ln A & = - \\frac { 4 } { 1 1 } \\ln ( t + c ) + \\ell _ 1 \\\\ A & = \\ell _ A ( t + c ) ^ { - 4 / 1 1 } \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\eta = 0 \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{gather*} a _ n ^ { ( \\theta ) } = \\frac { 1 } { x _ 1 - x _ 2 } \\big ( T _ { q , x _ 2 } a _ n ^ { ( \\theta - 1 ) } - T _ { q , x _ 1 } a _ { n - 1 } ^ { ( \\theta - 1 ) } \\big ) , 1 \\leq n \\leq \\theta - 1 , \\\\ a _ 0 ^ { ( \\theta ) } = \\frac { 1 } { \\prod \\limits _ { i = 0 } ^ { \\theta - 1 } { ( x _ 1 - q ^ i x _ 2 ) } } , a _ { \\theta } ^ { ( \\theta ) } = \\frac { 1 } { \\prod \\limits _ { i = 0 } ^ { \\theta - 1 } { ( x _ 2 - q ^ i x _ 1 ) } } . \\end{gather*}"}
-{"id": "3410.png", "formula": "\\begin{align*} { W } _ { \\eta } ^ { \\pm } ( t , x , y ) = e ^ { - i \\beta \\eta T } { \\widetilde W } _ { \\eta } ( t , x , y ) \\\\ = e ^ { \\pm i \\eta ^ 2 y - i \\beta \\eta ( T - t ) } e ^ { - \\eta ^ 2 x ^ 2 / 2 } \\big ( W _ 0 + \\eta x W _ 1 ^ { \\pm } \\big ) \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} ( \\alpha _ { 1 \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } ) ( s ) = ( \\alpha _ { 1 \\underline { d } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } ) ( s ) = ( \\alpha _ { 1 \\underline { d } } \\circ 0 ) ( s ) = 0 \\in V _ { X _ 1 } ( - ( d - i + 1 ) A ) . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{gather*} b \\left ( x , t , \\nabla u \\left ( x , t \\right ) \\right ) \\\\ = \\int _ { \\mathcal { Y } _ { 2 , 3 } } a \\left ( y ^ { 2 } , s ^ { 3 } , \\nabla u \\left ( x , t \\right ) + \\nabla _ { y _ { 1 } } u _ { 1 } \\left ( x , t , y _ { 1 } , s _ { 1 } \\right ) + \\nabla _ { y _ { 2 } } u _ { 2 } \\left ( x , t , y ^ { 2 } , s ^ { 3 } \\right ) \\right ) d y ^ { 2 } d s ^ { 3 } \\end{gather*}"}
-{"id": "7871.png", "formula": "\\begin{align*} x ' = 2 t x ^ 2 + \\epsilon f ( t ) x ^ 3 , \\ , \\ , \\ , t \\in [ - 1 , 1 ] \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} \\int _ { - r } ^ r | \\eta _ 2 ( x _ 1 , t ) | \\ , d x _ 1 = \\int _ { - r } ^ r \\eta _ 2 ( x _ 1 , t ) \\ , d x _ 1 = \\int _ 0 ^ t \\eta _ 1 ( - r , x _ 2 ) - \\eta _ 1 ( r , x _ 2 ) \\ , d x _ 2 \\ , \\le 2 t \\| \\eta \\| _ \\infty \\ , . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\phi ( B _ { \\tau + t _ 1 } - B _ { \\tau } , \\cdots , B _ { \\tau + t _ m } - B _ { \\tau } ) ] = \\hat { \\mathbb { E } } [ \\phi ( B _ { \\tau + t _ 1 } - B _ { \\tau } , \\cdots , B _ { \\tau + t _ m } - B _ { \\tau } ) ] = \\hat { \\mathbb { E } } [ \\phi ( B _ { t _ 1 } , \\cdots , B _ { t _ m } ) ] . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\boldsymbol { w } _ { k + 1 } = \\boldsymbol { w } _ k - \\mu \\ , \\partial L ( \\boldsymbol { w } _ k , \\lambda _ k ) \\big / \\partial \\boldsymbol { w } _ k ^ \\ast , \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ { 2 } } { d y ^ { 2 } } + \\alpha ^ { 2 } l ^ { 2 } + \\frac { U ^ { \\prime \\prime } } { U - c } \\right ) \\psi = 0 \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} e _ { \\gamma + \\delta } ( z ) = e _ { \\gamma } ( z + \\delta ) e _ { \\delta } ( z ) . \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\mathcal { S } ^ { \\mu } \\otimes \\mathcal { S } ^ \\nu = \\bigoplus _ { \\lambda } \\left ( \\mathcal { S } ^ \\lambda \\right ) ^ { \\oplus k _ { \\mu , \\nu } ^ { \\lambda } } \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} A ^ 2 + \\zeta B ^ 2 - ( \\zeta + 1 ) C ^ 2 = 0 , \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Bigg | \\frac 1 N \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ \\ell \\tilde { f } _ i ( R _ X ^ { i n } x ) \\tilde { g } _ j ( R _ Y ^ { j n } y ) - \\Bigg ( \\frac 1 N \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k \\tilde { f } _ i ( R _ X ^ { i n } x ) \\Bigg ) \\Bigg ( \\frac 1 N \\sum _ { n = 1 } ^ N \\prod _ { j = 1 } ^ \\ell \\tilde { g } _ j ( R _ Y ^ { j n } y ) \\Bigg ) \\Bigg | = 0 . \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} u ( t , x ) = 1 + \\sum _ { n \\geq 1 } I _ n ( f _ n ( \\cdot , t , x ) ) , \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} u _ i ( T , x ) = u _ { i } ^ { 0 } ( x ) \\ , , u _ { i t } ( T , x ) = u _ { i } ^ { 1 } ( x ) \\ , , x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } y } { d \\xi ^ { 2 } } - \\frac { { \\phi } ^ { \\prime } } { 2 u + \\phi } \\frac { d y } { d \\xi } - \\left \\{ { u ^ { 2 } + u \\phi + \\frac { 1 } { 4 } \\phi ^ { 2 } } \\right \\} y = 0 . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} { d x \\over d t } = v \\ , , { d v \\over d t } = \\eta ( t ) \\ , , \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\# \\{ q \\in \\N : ( q , N ) = 1 , \\ ; H _ q \\supseteq \\Gamma _ 1 ( N ) , \\ ; q \\le x \\} = \\bigl ( \\tfrac { \\varphi ( N ) } { N } + o ( 1 ) \\bigr ) x \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\nu = \\left ( \\nu _ { k l } \\right ) _ { 1 \\leq k , l \\leq q } , \\Xi = \\left ( \\xi _ { k l } \\right ) _ { 1 \\leq k \\leq q \\atop 1 \\leq l \\leq p - q } . \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} T ^ { d } = & P ^ \\pi \\sum _ { j = 0 } ^ { n - 1 } P ^ j \\left [ \\begin{array} { c c } 0 & 0 \\\\ 0 & ( Z ^ d ) ^ { j + 1 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} a = s _ 0 < s _ 1 < \\cdots < s _ m = b . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} c _ 1 = \\frac { 1 } { n - 1 } , ~ c _ 2 = \\frac { 3 } { ( n - 1 ) ( n + 1 ) } , ~ c _ 3 = \\frac { 1 5 } { ( n - 1 ) ( n + 1 ) ( n + 3 ) } , \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\langle w , T v \\rangle = \\lim _ { n \\to \\infty } \\langle w , T _ n v \\rangle = \\lim _ { n \\to \\infty } \\langle T _ n w , v \\rangle = \\langle T w , v \\rangle . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} q ( z _ 1 , z _ 2 ) = p ( \\phi ( z _ 1 ) , \\phi ( z _ 2 ) ) \\left ( \\frac { 1 - z _ 1 } { 2 i } \\right ) ^ n \\left ( \\frac { 1 - z _ 2 } { 2 i } \\right ) ^ m . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\left \\langle \\left ( ( X _ i ) _ { j k } \\right ) ^ 2 \\right \\rangle = \\frac { 1 } { L _ i + 2 n } . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\left ( v ^ { l k ' } _ { i k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } \\overline { \\left ( \\left ( v ^ { l ' k ' } _ { j k } \\right ) _ { 1 \\leq k \\leq N } ^ { 1 \\leq k ' \\leq N } \\right ) ^ { t } } = a _ { i j } ^ { l l ' } I _ { N } , \\quad \\mbox { f o r a l l $ i , j , l , l ' = 1 , 2 $ . } \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} d q / d \\theta = 1 / \\theta _ q ~ ~ , ~ ~ d z / d \\theta = - \\theta _ { q z } / ( 2 q / z ^ 2 ) ( \\theta _ q ) ^ 2 ~ . \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} x ' ( t ) = f ( t ) x ^ 3 + g ( t ) x ^ 2 , ~ t \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} i ^ { 3 , 3 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) & = 1 , i ^ { 2 , 2 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) = p h ( Y , w ) - 3 , \\\\ i ^ { 1 , 2 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) & = i ^ { 2 , 1 } ( H ^ 3 ( Y ; \\mathbb { Q } ) ) = h ^ { 1 , 2 } ( Z ) . \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} \\vec g ' : = \\vec g - u \\vec b + \\nabla v \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\Delta w _ m = \\beta _ m ( w _ m ) - f \\to \\beta ( w ) - f = \\Delta w \\textrm { i n } L ^ p ( \\Omega \\setminus \\overline U ) . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} w _ c ^ { ( m ) } ( ( x , y ) ) = \\int _ 0 ^ { 1 - x ^ 2 - y ^ 2 } \\mathrm { d } \\delta \\frac { \\delta } { \\sqrt { \\delta ^ 2 + 4 y ^ 2 } } W \\left ( \\begin{bmatrix} \\mu _ + & 0 \\\\ 0 & \\mu _ - \\end{bmatrix} \\right ) \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\| { \\mathcal E } ( t , \\tau ) \\| \\le { C } _ { 1 8 } + C _ { 1 9 } | \\tau | t ^ 2 , \\ 0 < | t | \\le t ^ { 0 0 } , \\ N _ 0 = 0 ; C _ { 1 8 } = { C } _ { 1 4 } + c _ * ^ { - 1 / 2 } C _ { 1 2 } , \\ C _ { 1 9 } = c _ * ^ { - 1 / 2 } C _ { 1 3 } . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\varphi _ n ^ S = \\mathcal { K } ( \\phi _ n ^ C ) + f . \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\tilde \\theta _ 1 & = C ( 2 ^ { - t v _ 1 } , 2 ^ { - t v _ 1 ' } , \\underbrace { 2 ^ t , \\dots , 2 ^ t } _ { n + 1 - m - \\phi } ) \\in \\R ^ { n + 3 - m - \\phi } \\ , , \\\\ [ 1 e x ] \\tilde \\theta _ i & = C ( 2 ^ { - t v _ i } , 2 ^ { - t v _ i ' } ) \\in \\R ^ 2 ( 2 \\le i \\le \\phi ) \\ , , \\\\ [ 3 e x ] \\tilde \\theta _ i & = C 2 ^ { - t v _ i } \\in \\R ( \\phi + 1 \\le i \\le m ) \\ , , \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} e ( G _ 1 - C ) + e ( G _ 1 - C , C ) & = e ( H ) + e ( H , C ) - e ( B _ 1 ) - e ( B _ 1 - b _ 1 , C ) \\\\ & > \\left ( \\left \\lfloor \\frac { c } { 2 } \\right \\rfloor - 1 \\right ) \\cdot ( n - r - c ) = \\left ( \\left \\lfloor \\frac { c } { 2 } \\right \\rfloor - 1 \\right ) \\cdot ( | V ( G _ 1 ) | - c ) . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} R _ n ^ { 2 } = \\frac { ( - g ) ^ { n } ( g h ^ { 2 } - k ^ { 2 } + f h k ) } { \\left ( f ^ { 2 } + 4 g \\right ) } \\left | M + ( - g ) ^ n B ^ { - 2 n } \\right | \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} | \\partial _ i d | \\leq | d | \\ , \\ \\partial _ { h k } d = \\partial _ { k , h + 1 } d \\ \\ , \\ \\ i , h = 0 , \\ldots , n \\ , , \\ , k \\leq h \\ . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\Lambda _ F ( \\chi , s ) : = q ( \\chi ) ^ { s / 2 } \\gamma ( \\chi , s ) L _ F ( \\chi , s ) , \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} a ^ 2 - e b ^ 2 = t , \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} | p ( z ) | \\leq | P _ r ( \\vec { 1 } ) | e ^ { - r / 2 } \\exp \\left [ \\Re ( \\sum _ { j = 1 } ^ { n } c _ j z _ j ) + B \\| z \\| _ { \\infty } ^ 2 \\right ] \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} K _ 1 = \\max _ { i = 1 , \\ldots , N } c _ { i , \\infty } \\Phi \\left ( \\frac { \\widetilde K } { c _ { i , \\infty } } \\right ) . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\delta _ t f ^ s : = \\frac { f ^ { s + t } - f ^ s } { t } \\mbox { a n d } \\dot f ^ s : = \\lim _ { t \\downarrow 0 } \\delta _ t f ^ s = \\partial _ t f ( s , y ) . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} & \\int _ { I ^ { m } } \\left \\vert \\sum _ { i _ { 1 } , . . . , i _ { m } = 1 } ^ { n } y _ { i _ { 1 } . . . i _ { m } } r _ { i _ { 1 } } ( t _ { 1 } ) \\cdots r _ { i _ { m } } ( t _ { m } ) \\right \\vert ^ { p } d t _ { 1 } \\cdots d t _ { m } \\\\ & = \\int _ { I ^ { m } } \\left \\vert \\sum _ { i _ { 1 } , . . . , i _ { m } = 1 } ^ { n } y _ { i _ { 1 } . . . i _ { m } } r _ { i _ { 1 } } ( t _ { 1 } ) \\cdots r _ { i _ { m } } ( t _ { m } ) \\varepsilon _ { i _ { 1 } } \\cdots \\varepsilon _ { i _ { m } } \\right \\vert ^ { p } d t _ { 1 } \\cdots d t _ { m } \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\chi _ 2 ( G ) : = \\lbrace g \\in G \\mid ( \\forall \\ h \\in G ^ \\prime ) \\ \\lbrack g , h \\rbrack \\in \\gamma _ 4 ( G ) \\rbrace \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} & e ^ + _ i ( z ) : = \\sum _ { r \\geq 0 } e _ { i , r } z ^ { - r } , \\ e ^ - _ i ( z ) : = - \\sum _ { r < 0 } e _ { i , r } z ^ { - r } , \\\\ & f ^ + _ i ( z ) : = \\sum _ { r > 0 } f _ { i , r } z ^ { - r } , \\ f ^ - _ i ( z ) : = - \\sum _ { r \\leq 0 } f _ { i , r } z ^ { - r } . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} E \\left ( \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 2 | \\mathcal { F } _ { n ( \\ell - 1 ) } ) \\right ) = \\tau _ { n p _ n } ^ 2 . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ X I _ { \\{ \\tau \\leq \\sigma \\} } ] & = \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ \\sum _ { i = 1 } ^ n \\xi _ i I _ { A _ i \\cap \\{ \\tau \\leq \\sigma \\} } ] \\\\ & = \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ \\xi _ i ] I _ { A _ i \\cap \\{ \\tau \\leq \\sigma \\} } \\\\ & = \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { ( \\tau \\wedge \\sigma ) + } [ \\xi _ i ] I _ { \\{ \\tau \\leq \\sigma \\} } I _ { A _ i } . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} A ( x , v ) & = a _ 3 ( x ) v ^ 3 + \\frac { x } { 1 - v } ( v ^ 4 A ( x , 1 ) - v ^ 2 A ( x , v ) ) , \\\\ B ( x , v ) & = b _ 3 ( x ) v ^ 3 + x v ( v ^ 3 A ( x , v ) + B ( x , v ) + C ( x , v ) - c _ 2 ( x ) v ^ 2 ) , \\\\ C ( x , v ) & = c _ 2 ( x ) v ^ 2 + \\frac { x } { 1 - v } ( v ^ 3 B ( x , 1 ) - v B ( x , v ) ) + \\frac { x } { 1 - v } ( v ^ 3 C ( x , 1 ) - v C ( x , v ) ) , \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\beta _ 2 = \\displaystyle ( 1 + h ^ 2 ) \\lambda _ { m a x } ( P ) + h \\lambda _ { m a x } ( S ) + \\frac { h ^ 3 } { 2 } \\lambda _ { m a x } ( R ) . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} \\dot \\lambda _ 1 & = 0 \\\\ \\dot \\lambda _ 2 & = - \\lambda _ 0 b _ M - \\nu _ 2 \\\\ \\dot \\lambda _ 3 & = - \\lambda _ 0 b _ E + a _ E b _ E k _ E \\lambda _ 1 - k _ E \\lambda _ 3 , \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} M ( \\xi ) = \\xi ^ 2 \\coth \\xi - \\xi , \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} f ( y , \\vartheta ) = e ^ { \\vartheta ^ T U ( y ) - \\Gamma ( \\vartheta ) } f _ 0 ( y ) , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\begin{cases} ( u , t ) \\in K \\times [ 0 , 1 ] \\ , , \\\\ \\noalign { \\medskip } \\displaystyle { \\int _ { \\Omega } \\bigl [ A ( x , u ) \\nabla u \\cdot \\nabla ( v - u ) + B ( x , u ) \\ , | \\nabla u | ^ 2 \\ , ( v - u ) \\bigr ] \\ , d x \\geq \\int _ \\Omega g _ t ( x , u ) ( v - u ) } \\\\ \\noalign { \\medskip } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\ , , \\end{cases} \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} S J _ { H _ 2 } = J _ { H _ 1 } S . \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} ( \\alpha + 1 ) c _ { 1 , \\infty } ( a + 1 ) ^ 2 + c _ { 2 , \\infty } ( b + 1 ) ^ 2 + c _ { 3 , \\infty } ( c + 1 ) ^ 2 = ( \\alpha + 1 ) c _ { 1 , \\infty } + c _ { 2 , \\infty } + c _ { 3 , \\infty } , \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} \\int _ 0 ^ X \\big \\lvert \\sum _ { n \\leq r } \\tau ( n ) \\big \\rvert ^ 2 d r = c X ^ { 1 1 + \\frac { 3 } { 2 } } + O ( X ^ { 1 2 + \\epsilon } ) . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} \\Omega _ i \\cap \\Omega _ j = \\emptyset \\quad i , j = 1 , \\dots , n , \\ , i \\neq j , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\| v f _ x \\| _ { L ^ 2 } ^ 2 - 2 \\int v | A f _ x | ^ 2 \\ , d x + \\| A ^ 2 f _ x \\| _ { L ^ 2 } ^ 2 = t ^ { - 2 } \\| L _ z \\partial _ x f \\| _ { L ^ 2 } ^ 2 , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\Gamma _ { T } ^ \\lambda ( f , g ) \\Gamma _ { T + h } ^ \\mu ( \\varphi , \\psi ) \\right ) = \\left ( u v + w \\right ) ( x , y , 0 ) . \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\Omega ( g ) = \\{ f \\in L _ 1 [ 0 , \\alpha ) + L _ { \\infty } [ 0 , \\alpha ) : \\ , f \\prec g \\} . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\mathfrak { V } ( n ) = \\{ ( x _ 1 , y _ 1 , \\hdots , x _ n , y _ n ) & \\ , | \\ , x _ i \\in \\mathbb { R } - \\{ 0 \\} \\ \\mbox { a n d } \\ y _ i \\in \\mathbb { R } \\ \\mbox { f o r a l l } \\ i , \\\\ & s _ 1 \\cdots s _ n = 1 , \\ \\mbox { a n d } \\ q _ n ( s _ 1 , t _ 1 , \\hdots , s _ n , t _ n ) = 0 \\ \\} \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} X ( 0 ) = ( 1 0 0 0 , 1 0 0 0 , 0 , 0 ) k _ 1 ( 0 ) = 0 . 5 . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} W : = \\kappa ^ 2 \\frac { S } { K - | X | _ { \\omega _ { \\phi } } ^ 2 } + A { \\rm t r } h , \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\dim ( p , 1 ) + \\dim ( p - 2 , 2 , 1 ) & - ( \\dim ( p - 2 , 2 ) + \\dim ( p - 1 , 1 , 1 ) + 2 ) \\cr & = \\dim ( p - 3 , 2 , 1 ) - 1 > 0 . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\prod _ { i \\in I } \\prod _ { j = 1 } ^ { v ^ i } \\frac { \\vartheta ( z ^ { ( i ) } _ j + \\lambda _ i ) } { \\vartheta ( z ^ { ( i ) } _ j ) \\vartheta ( \\lambda _ i ) } \\cdot \\prod _ { i \\in I } \\prod _ { j = 1 } ^ { v ^ i } \\frac { \\vartheta ( z ^ { ( i ) } _ j + \\frac { c _ { k i } } { 2 } \\hbar ) } { \\vartheta ( z ^ { ( i ) } _ j - \\frac { c _ { k i } } { 2 } \\hbar ) } . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} D ( s , S _ f ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) } { n ^ { s + \\frac { k - 1 } { 2 } } } = L ( s , f ) + \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } L ( s - z , f ) \\zeta ( z ) \\frac { \\Gamma ( z ) \\Gamma ( s + \\frac { k - 1 } { 2 } - z ) } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\ ; d z , \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\langle [ v = 0 ] , g \\rangle = \\frac { i } { 2 \\pi } \\int \\tilde \\partial \\bar { \\tilde \\partial } \\log | v | ^ { 2 } \\wedge g . \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} P ( \\lambda ) _ w \\ = \\ \\lambda _ 1 P _ { 1 , w } + \\dotsb + \\lambda _ r P _ { r , w } \\ , . \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} T R _ 1 = H ( M _ 1 ) = H ( M _ 1 | M _ 2 ) \\leq I ( M _ 1 ; Y _ { 1 , 0 } ^ T | M _ 2 ) + T \\varepsilon _ { 1 , T } , \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\limsup _ m y _ n \\mathbf { d } x _ m & \\leq \\liminf _ l \\limsup _ m ( y _ n \\mathbf { d } y _ l + y _ l \\mathbf { d } x _ m ) \\\\ & \\leq \\liminf _ l \\limsup _ m ( y _ n \\mathbf { d } y _ l + y _ l \\overline { \\mathbf { d } } x _ l + x _ l \\mathbf { d } x _ m ) = 0 \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\rho \\left ( \\frac { \\partial v } { \\partial t } + u \\frac { \\partial v } { \\partial x } + v \\frac { \\partial v } { \\partial r } \\right ) + \\frac { \\partial P } { \\partial r } = \\mu \\left ( \\frac { \\partial ^ { 2 } v } { \\partial x ^ { 2 } } + \\frac { \\partial ^ { 2 } v } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial v } { \\partial r } - \\frac { v } { r ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\begin{pmatrix} O _ s & B \\\\ 0 & C \\end{pmatrix} \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\mathcal { K } _ t \\phi = \\ ( P \\circ \\L _ t \\circ S \\ ) \\phi . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\ , K _ { i \\tilde { l } } = r + 1 \\ , \\ , \\ , 0 \\leq \\tilde { l } \\leq b ' _ { r } \\ , \\ , \\ , i + \\tilde { l } \\leq d . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\rho = \\frac { s + \\sqrt { s ^ 2 + 4 t } } { 2 } , \\sigma = \\frac { - s + \\sqrt { s ^ 2 + 4 t } } { 2 } . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\alpha \\triangleleft \\beta \\Longleftrightarrow \\left \\{ \\begin{array} { l l } \\mathrm { n d } ( \\alpha ) < _ { \\mathrm { l e x } } \\mathrm { n d } ( \\beta ) & \\textnormal { o r } \\\\ \\mathrm { n d } ( \\alpha ) = \\mathrm { n d } ( \\beta ) & \\textnormal { a n d } \\ \\alpha < _ { \\mathrm { l e x } } \\beta \\end{array} \\right . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\frak s _ x ^ { \\boxplus \\tau } ( y , \\xi ) = \\frak s _ x ( \\mathcal R _ x ( y ) , \\xi ) . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} d = \\sum _ { s = 0 } ^ { n - 1 } \\alpha _ s p ^ s \\ \\ \\ ( \\alpha _ s \\in \\mathbb { F } _ p , \\ \\alpha _ 0 + \\cdots + \\alpha _ { n - 1 } = p ) . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\tilde A ^ \\flat = \\tilde A z ^ { \\frac N 4 E ^ T } U ^ T , \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} ( \\underline X , \\underline A ) = \\{ ( X _ 1 , A _ 1 ) , \\dots , ( X _ m , A _ m ) \\} \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} & \\underline { \\rm H o m } ( { \\bf L } ^ { n - d } , M ( X ) ) \\xrightarrow { \\sim } \\tau _ { \\geq 0 } \\underline { \\rm H o m } ( { \\bf L } ^ { n - d } , M ( X ) ) ) \\\\ \\to & h _ 0 ( \\underline { \\rm H o m } ( { \\bf L } ^ { n - d } , M ( X ) ) ) \\xrightarrow { \\sim } C H _ { X } ^ d . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} \\norm { \\ ( \\frac { n ^ m ( n - m ) ! } { n ! } \\ ) ^ 2 1 _ { S _ m } * 1 _ { S _ m } - 1 } _ 2 ^ 2 = \\ ( \\frac { n ^ m ( n - m ) ! } { n ! } \\ ) ^ 4 \\sum _ { \\substack { \\chi \\in \\hat { G } ^ m \\\\ \\chi \\neq 0 } } | \\hat { 1 _ { S _ m } } ( \\chi ) | ^ 4 . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} a ^ { m } ( a ^ { * } ) ^ { m } = \\sum _ { i = 0 } ^ { m } \\Big [ \\begin{array} { c } m \\\\ i \\end{array} \\Big ] _ { q ^ { - 2 } } ( - 1 ) ^ { i } q ^ { i + 2 i m - i ^ 2 } c ^ { i } ( c ^ { * } ) ^ i , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\tilde { A } \\left ( W ' , \\overline { W ' } \\right ) \\otimes \\left ( \\left ( \\overline { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) ^ { t } \\otimes W ' - \\left ( \\overline { \\left ( \\overline { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) ^ { t } \\otimes W ' } \\right ) ^ { t } \\right ) = 2 \\sqrt { - 1 } \\left ( V \\otimes Z ' \\right ) \\overline { \\left ( V \\otimes Z ' \\right ) } ^ { t } , \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} & \\bar { \\mathcal { A } } _ { b , m } \\doteq \\{ ( \\varphi _ { k } ) _ { k \\in \\mathbb { N } _ { 0 } } : \\varphi _ { k } \\in \\bar { \\mathcal { A } } _ { + } \\mbox { f o r e a c h } k \\in \\mathbb { N } _ { 0 } \\mbox { s u c h t h a t f o r a l l } ( \\omega , t , y ) \\in \\Omega \\times \\lbrack 0 , T ] \\times \\lbrack 0 , 1 ] , \\\\ & \\frac { 1 } { m } \\leq \\varphi _ { k } ( \\omega , t , y ) \\leq m \\mbox { f o r } k \\leq m \\mbox { a n d } \\varphi _ { k } ( \\omega , t , y ) = 1 \\mbox { f o r } k > m \\} \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\Psi ^ { X } _ t = \\Pi ^ X \\circ \\Phi ^ { * , X } _ t , \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} [ Y _ 1 , Y _ 2 ] & = 0 & [ Y _ 1 , Y _ 3 ] & = 0 & [ Y _ 1 , Y _ 4 ] & = 0 \\\\ [ Y _ 1 , Y _ 5 ] & = 0 & [ Y _ 2 , Y _ 3 ] & = Y _ 1 & [ Y _ 2 , Y _ 4 ] & = \\alpha Y _ 1 \\\\ [ Y _ 2 , Y _ 5 ] & = \\beta Y _ 1 + Y _ 2 & [ Y _ 3 , Y _ 4 ] & = \\gamma Y _ 1 & [ Y _ 3 , Y _ 5 ] & = \\delta Y _ 1 + \\eta Y _ 2 - Y _ 3 \\\\ [ Y _ 4 , Y _ 5 ] & = \\mu Y _ 1 + \\rho Y _ 2 - \\alpha Y _ 3 \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\psi ^ { G L M r o b } ( y _ i , x _ i , \\beta _ j ) = \\frac { \\psi _ c ( r _ i ) } { \\tilde { V } ( \\mu _ i ) ^ { 1 / 2 } } \\frac { \\partial \\mu _ i } { \\partial \\beta _ j } - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { E _ { F } [ \\psi _ c ( r _ i ) ] } { \\tilde { V } ( \\mu _ i ) ^ { 1 / 2 } } \\frac { \\partial \\mu _ i } { \\partial \\beta _ j } , \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} { \\rm P I } _ { \\beta _ 2 ; \\beta _ 1 \\alpha } \\circ { \\rm P I } _ { \\beta _ 1 ; \\alpha } = { \\rm P I } _ { \\beta _ 2 \\beta _ 1 ; \\alpha } \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} D ^ 7 & \\approx ( M t ) \\\\ D & \\approx ( M t ) ^ { 1 / 7 } . \\intertext { F u r t h e r m o r e , } A & = m ( D ^ 3 + K _ 1 D ) ^ { - 1 } \\\\ & \\approx \\frac { m } { D ^ 3 } \\\\ & \\approx m ( M t ) ^ { - 3 / 7 } . \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\phi _ i ( x ) = \\sum _ { h \\in \\mathbb { Z } _ 2 ^ { \\oplus k } } \\left ( \\mu _ i ( \\mathcal { R } _ h ( x ) ) - \\frac { 1 } { 2 ^ k } \\mu _ i ( \\mathbb { R } ^ d ) \\right ) h \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\ \\ & \\frac { d } { d t } \\left ( \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } - | \\partial _ { y } \\psi _ { s 2 } ^ { \\nu } | ^ { 2 } ) d x d y + \\left \\Vert \\omega _ { n } ^ { \\nu } \\right \\Vert _ { X } ^ { 2 } \\right ) \\\\ & = - 2 \\nu \\left ( \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\partial _ { y } \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } - | \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } ) d x d y + \\left \\Vert \\omega _ { n } ^ { \\nu } \\right \\Vert _ { X ^ { 1 } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\lambda _ 2 = e ^ { \\gamma / a } = \\left \\{ \\begin{array} { l l } ( 2 ^ { 1 - \\alpha } \\rho ) ^ { 1 / ( 3 - \\alpha ) } & \\mbox { f o r e q u a t i o n \\eqref { H A M } } \\\\ \\rho ^ { 2 / ( 2 - \\alpha ) } & \\mbox { f o r e q u a t i o n \\eqref { P A M } } \\end{array} \\right . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S = 2 ^ { r - 4 } [ ( | S | - 1 ) ( | S | - 2 ) + 4 ( | S | - 1 ) + ( r - | S | ) ( r - | S | - 1 ) ] . \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\int _ M h e ^ { \\tilde { u } _ n } \\mathrm { d } \\mu = 1 . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X _ 1 ( s , M _ 1 , Y _ 0 ^ s ) d s + \\int _ 0 ^ t X _ 2 ( s , M _ 2 , Y _ 0 ^ s ) d s + \\cdots + \\int _ 0 ^ t X _ m ( s , M _ m , Y _ 0 ^ s ) d s + B ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} K _ { h _ 1 } \\circ K _ { h _ 2 } & ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) = \\int _ { X _ 1 } \\int _ { X _ 2 } K _ { h _ 2 } ^ 2 ( y _ 2 , A _ 2 ) \\delta _ { y _ 1 } ( A _ 1 ) \\ > d \\omega _ { X _ 2 } ( y _ 2 ) \\ > K _ { h _ 1 } ^ 1 ( x _ 1 , d y _ 1 ) \\\\ & = K _ { h _ 1 } ^ 1 ( x _ 1 , A _ 1 ) \\cdot \\omega _ { X _ 2 } ( A _ 2 ) = K _ { h _ 1 } ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} M _ h ( x ) = \\sum _ { n \\leq x } \\big ( P _ n ( x ) - D ( x ) \\big ) ^ h . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} ( x ) _ q = \\frac { q ^ x - 1 } { q - 1 } , [ x ] _ q = \\frac { q ^ x - q ^ { - x } } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\mathcal { R } _ { t } ^ * ( \\alpha ) = \\left \\{ t _ { n p } \\ge \\frac { p ( p - 1 ) } { 2 ( n - 1 ) } + z _ { \\alpha } \\sqrt { \\frac { p ( p - 1 ) ( n - 2 ) } { ( n - 1 ) ^ 2 ( n + 1 ) } } \\right \\} . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} h ( \\alpha _ 3 ) & \\leq \\frac { 1 } { 4 } \\left ( \\log ( b ^ 2 ( c - a ) ^ 2 ) + \\log \\frac { c ( \\sqrt { a } + \\sqrt { b } ) ^ 2 } { b ( c - a ) } \\right ) \\\\ & = \\frac { 1 } { 4 } \\log ( c b ( c - a ) ( \\sqrt { a } + \\sqrt { b } ) ^ 2 ) \\\\ & < \\frac { 1 } { 4 } \\log c ^ 4 = \\log c . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\omega ^ * _ \\bullet ( t ) : = \\hat \\omega _ \\bullet ( t ) + \\int _ 0 ^ t \\frac { \\tilde \\omega _ \\bullet ( s ) } s \\ , d s + \\tilde \\omega _ \\bullet ( 4 t ) + \\int _ 0 ^ t \\frac { \\tilde \\omega _ \\bullet ( 4 s ) } s \\ , d s . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} S \\circ \\tau _ r = \\tau _ r \\circ S ^ r . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} \\langle P _ h ( \\cdot , s ) , E ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle = \\frac { 2 \\sqrt { \\pi } \\sigma _ { 2 i t } ( h ) } { \\Gamma ( s ) ( 4 \\pi h ) ^ { s - \\frac { 1 } { 2 } } } \\frac { \\Gamma ( s - \\frac { 1 } { 2 } + i t ) \\Gamma ( s - \\frac { 1 } { 2 } - i t ) } { h ^ { i t } \\zeta ^ * ( 1 - 2 i t ) } , \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\int _ { \\O ^ \\# } \\int _ { B _ 1 ( 0 ) } D ^ 2 _ x u ^ \\# ( x , y , \\epsilon ) W _ \\delta ( x , y ) \\phi ( h ) & \\ , d x d y d h \\\\ = \\ ! \\int _ { \\Omega ^ \\# } \\ ! \\ ! \\int _ { B _ 1 ( 0 ) } \\ ! \\ ! \\frac { ( u ^ \\# ( x + \\varepsilon h , y ) - u ^ \\# ( x , y ) ) ( W _ \\delta ( x + \\varepsilon h , y ) - W _ \\delta ( x , y ) ) } { \\epsilon ^ 2 } & \\phi ( h ) \\ , d x d y d h \\ , . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} F ( 1 , a ) = F ( 1 , a 1 ) = F ( a , 1 ) - F ( a , 1 ) = 0 . \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{gather*} \\max _ { 2 ^ N \\le n \\le 2 ^ { N + 1 } - 1 } | \\sum _ { k = 2 ^ N } ^ n a _ n P ^ k f | \\le \\sum _ { k = 2 ^ N } ^ { 2 ^ { N + 1 } - 1 } | a _ k | \\ , | P ^ k f | \\ , . \\end{gather*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\sum _ { k , \\ell \\in \\Z } | \\varpi | ^ { 2 k s } f _ x \\big ( ( 1 , \\varpi ^ { - k } ) \\cdot \\gamma \\cdot ( e + \\varpi ^ \\ell f ) \\big ) \\ , \\eta _ x ( e + \\varpi ^ \\ell f ) = | c | ^ { 2 s } = | \\epsilon | _ x ^ { - 2 s } . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ n \\left ( \\sum \\limits _ { j = 1 } ^ { p _ k } D _ { i _ { k 1 } } \\cdots D _ { i _ { k ( j - 1 ) } } S _ { i _ { k j } } D _ { i _ { k ( j + 1 ) } } \\cdots D _ { i _ { k p _ k } } \\right ) . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} S _ 1 ( H ) = 0 . \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} J ( t ) = t F \\bigl ( t + z ( t ) \\bigr ) + z ( t ) F ( t ) \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} f \\biggr ( \\frac { F _ { k _ 0 + 2 } } { F _ { k _ 0 + 1 } } , \\dots , \\frac { F _ { k _ 0 + n + 1 } } { F _ { k _ 0 + n } } \\biggr ) \\ ! = \\biggr ( \\frac { F _ { k _ 0 + 3 } } { F _ { k _ 0 + 2 } } , \\dots , \\frac { F _ { k _ 0 + n + 1 } } { F _ { k _ 0 + n } } , \\frac { b _ 1 F _ { k _ 0 + n + 1 } + b _ 2 F _ { k _ 0 + n } + \\dots + b _ n F _ { k _ 0 + 2 } } { F _ { k _ 0 + n + 1 } } \\biggr ) . \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\prod _ { j = 2 } ^ \\ell ( 1 - p ^ { - j } ) \\ge 1 - \\sum _ { j = 2 } ^ \\ell p ^ { - j } > 1 - \\frac 1 { p ( p - 1 ) } , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} t ^ { - 2 / 3 } Y _ t = \\arg \\max _ { z \\in \\R } \\left \\{ A _ 2 ( z ) + \\epsilon _ t ^ { - 1 / 2 } \\left [ A _ 1 ( \\epsilon _ t z ) - A _ 1 ( 0 ) \\right ] - z ^ 2 \\right \\} \\ , . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\pi _ 1 + \\cdots + \\pi _ d = \\pi \\qquad ( \\pi _ 1 , \\dots , \\pi _ d \\in S ) . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\| E ( t , \\tau ) \\| \\le C _ { 1 2 } | t | + C _ { 1 3 } | \\tau | | t | ^ 3 , | t | \\le t ^ { 0 0 } , \\ N _ 0 = 0 ; C _ { 1 2 } = 2 C _ 1 + C _ { 1 1 } , \\ C _ { 1 3 } = C _ { 9 } + C _ { 1 0 } . \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} C \\left \\{ \\zeta \\ddagger \\Lambda ( A ) \\right \\} = M ( A ) \\kappa _ { L } ( \\zeta ) \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} B ^ { \\rho _ + } _ n ( u ) = \\frac { 1 } { \\beta _ 2 n ^ { 1 / 3 } } \\sum _ { m = 1 } ^ { \\beta _ 1 u n ^ { 2 / 3 } } ( X _ m - ( 1 - \\rho _ + ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\frac { a } { a _ 1 } = \\frac { ( T ) } { ( T _ 1 ) } , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} K \\wr _ { \\Omega } H : = K ^ { \\Omega } \\rtimes _ { \\theta } H . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} U e ^ { t A } f & = e ^ { t A } U f , \\\\ U \\mathbb { P } g & = \\mathbb { P } U g , \\\\ U ( h \\cdot \\nabla h ) & = ( U h ) \\cdot \\nabla ( U h ) , \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} \\log \\ ( 2 ^ m n ^ { \\delta m + 1 } \\binom { n } { m } ^ { - 1 / 2 } \\ ) \\leq - ( m / 4 ) \\log ( n / m ) . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} L _ \\lambda : = D _ X \\nabla f ( \\l , 0 ) \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} R ( d _ { 1 } , \\dotsc , d _ { n } ) = \\prod _ { i } d _ { i } + \\sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k } \\sum _ { 1 \\leqslant i _ { 1 } < \\dotsb < i _ { k } \\leqslant n } \\left ( \\gcd ( d _ { i _ { 1 } } , \\dotsc , d _ { i _ { k } } ) \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\dot { x } ( t ) = & u ( x ( t ) ) , \\\\ x ( 0 ) = & x , \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\hat { \\mathbb { E } } _ { t _ i } [ \\xi _ i ] { \\varphi ^ { i , m } _ { l } } \\prod _ { j = 1 } ^ { i - 1 } { \\widetilde { \\varphi } ^ { j , m } _ l } ] = \\hat { \\mathbb { E } } [ \\hat { \\mathbb { E } } _ { t _ 1 } [ \\sum _ { i = 1 } ^ { n } \\xi _ i { \\varphi ^ { i , m } _ { l } } \\prod _ { j = 1 } ^ { i - 1 } { \\widetilde { \\varphi } ^ { j , m } _ l } ] ] = \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ { n } \\xi _ i { \\varphi ^ { i , m } _ { l } } \\prod _ { j = 1 } ^ { i - 1 } { \\widetilde { \\varphi } ^ { j , m } _ l } ] . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} f ( u ) = ( f _ 1 ( u ) , \\ldots , f _ N ( u ) ) = \\sum _ { r = 1 } ^ { R } k _ r ( y _ r ' - y _ r ) u ^ { y _ r } \\in \\mathrm { r a n g e } ( W ) \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\nabla _ { \\phi m } \\widetilde { \\eta } _ { l \\bar { q } } = \\nabla _ { 0 m } \\widetilde { \\eta } _ { l \\bar { q } } - U _ { m l } ^ s \\widetilde { \\eta } _ { s \\bar { q } } , \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} \\Delta = t ^ 3 ( a ^ 3 t ^ 3 - 1 ) ^ 3 P , \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } | \\mathcal { I } _ { \\epsilon , \\alpha } | = 0 , \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\Vert \\psi ( \\cdot , t ) \\Vert _ { D ( \\L ^ { \\frac { \\alpha } { 2 } } ) } = \\Vert \\psi ( \\cdot , 0 ) \\Vert _ { D ( \\L ^ { \\frac { \\alpha } { 2 } } ) } \\forall t > 0 . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\zeta _ k ( t ) = { \\tilde { \\zeta } } _ k ( t ) \\mbox { f o r a l l } t \\in [ \\tau , \\tau + \\delta ] , k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} { } _ a \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { { a } / { 2 } } z ) & = ( 2 \\pi ) ^ { \\tfrac { a - 1 } { a } } a ^ { - ( a \\nu + \\tfrac { 1 } { 2 } ) } a ^ { \\tfrac { a } { 2 } ( a \\nu - a + 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { a \\nu - a + 1 } \\\\ & \\times \\prod _ { j = 1 } ^ { a } \\left ( \\frac { z } { 2 } \\right ) ^ { - ( \\nu - 1 ) - \\tfrac { j } { a } } \\mathtt { B } _ { 1 , ( \\nu - 1 ) + j / a , 1 } ( z ) . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} E ^ { S } = \\left \\{ \\left ( q ^ i ; \\frac { \\partial F } { \\partial p _ i } ( q , \\frac { \\partial W } { \\partial q } ( q , \\mu ) , \\lambda ) \\right ) \\in T Q : \\frac { \\partial F } { \\partial \\lambda ^ a } = 0 , \\frac { \\partial W } { \\partial \\mu ^ \\beta } = 0 \\right \\} . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\dot \\eta ^ { t } = \\left ( \\frac { \\partial _ N ( \\dot u ^ { t } - \\dot h ^ t ) } { ( N \\cdot \\nu ^ { t } ) ^ 2 \\ , \\Delta h ^ { t } } \\right ) ( z , \\eta ^ { t } ( x ) ) . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} W _ w ( z ) = 2 \\sqrt y K _ { w - \\frac { 1 } { 2 } } ( 2 \\pi y ) e ( x ) \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} S _ d ( X ) : = \\sum _ { n \\leq X } r _ d ( n ) . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\int _ { S ' _ { d } } \\big | X ^ { \\beta } ( \\varphi , w ) \\big | ^ { 2 n + 1 } \\ , d \\mu _ { \\beta } ( w ) & = 0 , \\\\ \\int _ { S ' _ { d } } \\big | X ^ { \\beta } ( \\varphi , w ) \\big | ^ { 2 n } \\ , d \\mu _ { \\beta } ( w ) & = \\frac { ( 2 n ) ! } { 2 ^ { n } \\Gamma ( \\beta n + 1 ) } | \\varphi | _ { 0 } ^ { 2 n } . \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} d & x \\leq 0 , \\\\ d - 2 x & 0 \\leq x \\leq d , \\\\ - d & d \\leq x . \\end{cases} \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\Phi _ H ( t , \\omega , h ) : = \\Phi ( t , \\omega ) h , \\ ; \\ ; \\ ; t \\geq 0 , \\omega \\in \\Omega , h \\in H , \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\vartheta > \\vartheta _ { 0 } = \\frac { S _ { q } ^ { q } } { q } \\left ( \\frac { 1 } { 1 - \\delta } \\frac { \\mu } { \\mu - 2 } \\frac { q - 2 } { q } \\frac { \\alpha _ { 0 } \\kappa ^ { - 1 } } { \\omega } \\right ) ^ { ( q - 2 ) / 2 } , \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i Q _ i \\Big | \\leq N ^ { \\frac { \\varepsilon } { 3 } } \\hat { \\Pi } \\ , , \\Theta \\Big ( z , { \\frac { N ^ { 3 \\varepsilon } } { ( N \\eta ) ^ { \\frac 1 3 } } } \\ , , { \\frac { N ^ { 3 \\varepsilon } } { \\sqrt { N \\eta } } } \\Big ) \\cap \\Omega _ 2 ( z ) \\ , . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} h _ \\rho ( y ) = \\frac { 1 } { y } \\ , , \\ \\forall | y | \\geq \\frac { 2 \\rho } { 3 } \\ , , | h _ \\rho ( y ) | \\leq \\frac { 3 } { \\rho } \\ , , \\ \\forall y \\in \\R \\ , . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} | \\frac { { t _ 2 } ^ { 1 - \\gamma } } { \\Gamma ( \\alpha ) } \\int _ { t _ 1 } ^ { t _ 2 } ( t _ 2 - s ) ^ { \\alpha - 1 } f _ { i } ( s , x _ { i } ( s ) ) d s | & \\leq \\frac { M _ i { ( \\frac { \\tilde { { \\delta } _ { i } } } { 2 } ) } ^ { - \\delta _ i } } { \\Gamma ( \\alpha ) } \\int _ { t _ 1 } ^ { t _ 2 } { t _ 2 } ^ { 1 - \\gamma } ( t _ 2 - s ) ^ { \\alpha - 1 } d s \\\\ & = \\frac { M _ i { ( \\frac { \\tilde { { \\delta } _ { i } } } { 2 } ) } ^ { - \\delta _ i } } { \\Gamma ( \\alpha + 1 ) } { t _ 2 } ^ { 1 - \\gamma } ( t _ 2 - t _ 1 ) ^ { \\alpha } . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{gather*} \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n \\ ! \\ ! \\times \\ ! \\R ^ n ) = 2 \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n _ + \\ ! \\ ! \\times \\ ! \\R ^ n _ + ) + 2 \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n _ + \\ ! \\ ! \\times \\ ! \\R ^ n _ - ) < 4 \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n _ + \\ ! \\ ! \\times \\ ! \\R ^ n _ + ) \\\\ \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) = \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n \\ ! \\ ! \\times \\ ! \\R ^ n ) - \\mathcal E _ s ( { U _ { \\ ! s } } ; \\R ^ n _ + \\ ! \\ ! \\times \\ ! \\R ^ n _ + ) \\end{gather*}"}
-{"id": "1480.png", "formula": "\\begin{align*} u _ k = x _ k ( 0 ) - x _ k ^ { \\rm f l a t } ( 0 ) \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} ( q _ d , r _ d ) : = \\big ( \\tfrac { 2 d } { d - 2 } , \\tfrac { 2 d ^ 2 } { d ^ 2 - 2 d + 4 } \\big ) . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} [ Y _ 1 , Y _ 2 ] & = 0 & [ Y _ 1 , Y _ 3 ] & = 0 & [ Y _ 1 , Y _ 4 ] & = 0 \\\\ [ Y _ 1 , Y _ 5 ] & = 0 & [ Y _ 2 , Y _ 3 ] & = Y _ 1 & [ Y _ 2 , Y _ 4 ] & = \\alpha Y _ 1 \\\\ [ Y _ 2 , Y _ 5 ] & = \\beta Y _ 1 + \\gamma Y _ 2 + Y _ 3 & [ Y _ 3 , Y _ 4 ] & = \\delta Y _ 1 & [ Y _ 3 , Y _ 5 ] & = \\eta Y _ 1 + ( - 1 - \\gamma ^ 2 ) Y _ 2 - \\gamma Y _ 3 \\\\ [ Y _ 4 , Y _ 5 ] & = \\kappa Y _ 1 + \\rho Y _ 2 + \\sigma Y _ 3 \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} W \\sim \\exp \\left \\{ \\pm u \\xi + \\sum \\limits _ { s = 0 } ^ { \\infty } \\left ( \\pm 1 \\right ) ^ { s } { \\frac { E _ { s } \\left ( \\xi \\right ) } { u ^ { s } } } \\right \\} . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\phi = A ( \\phi ) , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} 2 7 \\ , c ^ { 4 } d ^ { 2 } - 5 4 \\ , c ^ { 2 } d ^ { 3 } + 1 0 0 \\ , c ^ { 4 } + 2 7 \\ , d ^ { 4 } - 1 9 8 \\ , c ^ { 2 } d + 1 6 2 \\ , d ^ { 2 } + 2 4 3 = 0 . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\big ( D _ t - \\mu ( t , x , D ) I - { \\hat A } \\big ) S ( 0 , t ) = S ( 0 , t ) \\big ( D _ t - { \\tilde A } \\big ) + S ( 0 , t ) R . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} r _ d ( G ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { R _ d ( F ) } { d _ F ^ { 2 } } = \\sum _ { F \\in \\mathcal { F } ^ { + } _ { G } } \\frac { \\mathrm { R e s } _ { s = 1 } \\zeta _ F ( s ) } { 2 ^ d d _ F ^ 2 \\zeta _ F ( 2 ) } > 0 . \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} L ( z , w ) = z + \\frac { w + z ( L ( z , w ) - z ) } { 1 - L ( z , w ) } . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} f _ p ( n ) & = ( 2 p ) ^ { 2 H } \\biggl [ \\biggl ( 1 + \\frac { n + 2 } { 2 p } \\biggr ) ^ { 2 H } - 2 \\biggl ( 1 + \\frac { n + 1 } { 2 p } \\biggr ) ^ { 2 H } + \\biggl ( 1 + \\frac { n } { 2 p } \\biggr ) ^ { 2 H } \\biggr ] \\\\ & = \\bigl ( 2 ^ { 2 H - 1 } H ( 2 H - 1 ) \\bigr ) p ^ { 2 ( H - 1 ) } \\bigl ( 1 + o ( 1 ) \\bigr ) . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} Q _ p ( x , v , t ) = 1 - p \\int _ 0 ^ t d t ' \\int _ 0 ^ \\infty d x ' \\int _ { - \\infty } ^ \\infty d v ' \\ , Q _ p ( x ' , v ' , t ' ) G ( x ' , v ' ; x , v ; t - t ' ) \\ , . \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} v _ { i } ^ { 0 } \\left ( t = 0 \\right ) = v _ { i } ^ { 0 , 0 } \\quad \\mbox { o n } \\ ; \\Gamma . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\tilde { l } ^ { E T } ( \\theta ) = 2 n \\log \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n e ^ { \\lambda ^ T _ { E T } \\psi ( x _ i , \\theta ) } \\right ) - 2 \\lambda ^ T _ { E T } \\sum _ { i = 1 } ^ n \\psi ( x _ i , \\theta ) \\ , . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} x - z = L ( z ' ) - L ( x ' ) = - ( L ( x ' ) - L ( z ' ) ) . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} h _ { 1 } ( t , x ) = \\frac { C _ { 1 } } { t ^ { n / 2 } } \\exp \\left ( - \\frac { 1 } { C _ { 2 } } \\left ( \\frac { \\vert x \\vert ^ { 2 } } { t } \\right ) \\right ) , h _ { 2 } ( t , x ) = \\frac { C _ { 1 } } { t ^ { n / 2 } } \\exp \\left ( - \\frac { 1 } { C _ { 2 } } \\left ( \\frac { \\vert x \\vert ^ { \\mu } } { t ^ { \\nu } } \\right ) ^ { \\frac { 1 } { \\mu - 1 } } \\right ) . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} h ^ { \\mathrm { a n } } _ { a ( D , g ) + a ' ( D ' , g ' ) } ( \\xi ) = ( a \\vartheta _ f + a ' \\vartheta _ { f ' } ) ( \\xi ) = a h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) + a ' h ^ { \\mathrm { a n } } _ { ( D ' , g ' ) } ( \\xi ) . \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\left ( \\dfrac { f ' ( t ) } { t } \\right ) ' = t ^ { 2 m - 4 } \\{ 2 m ( 2 m - 2 ) \\alpha t + ( 2 m - 1 ) ( 2 m - 3 ) \\beta - \\dfrac { \\delta } { t ^ { 2 m - 2 } } \\} . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} g _ { 1 } ( V , \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\phi \\mathcal { B } Y + \\mathcal { A } _ { X } \\omega \\mathcal { B } Y ) & = g _ { 2 } ( \\pi _ * ( \\mathcal { C } Y ) , ( \\nabla \\pi _ * ) ( X , \\varphi V ) ) , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\bar V _ i ^ L = \\left ( \\bigoplus _ { 0 \\le l _ 0 , \\ldots , l _ { i - 1 } \\le L } { \\cal V } ^ { l _ 0 } \\otimes { \\cal V } ^ { l _ 1 } _ \\# \\otimes \\ldots \\otimes { \\cal V } ^ { l _ { i - 1 } } _ \\# \\right ) \\otimes V ^ L _ \\# . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\mathbb { E } [ D ( t ) | Q ( t ) ] = \\bar D ^ { A l g } \\leq \\bar D ^ { O p t } + \\gamma , \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} c _ 1 \\sum _ { n = - \\infty } ^ { \\infty } | C _ n | ^ 2 \\le \\int _ { 0 } ^ { T } \\Big | \\sum _ { n = - \\infty } ^ { \\infty } \\big ( C _ n e ^ { i \\omega _ n t } + R _ n e ^ { r _ n t } \\big ) \\Big | ^ 2 d t \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\big ( \\sum _ i ( s _ i ^ 1 ) ^ 2 \\big ) \\cdots \\big ( \\sum _ i ( s _ i ^ p ) ^ 2 \\big ) = 0 \\quad . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} S = \\begin{pmatrix} L & 0 \\\\ L ^ { - T } P & L ^ { - T } \\end{pmatrix} \\in S p ( n ) , \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\mathcal { F } _ G ^ { + } : = \\{ F / \\Q : ~ \\textrm { $ F $ t o t a l l y r e a l o f d e g r e e $ d $ } , ~ \\textrm { G a l } ( F ^ c / \\Q ) \\cong G \\} . \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} P _ 0 = G b ^ { - 1 } O _ 0 \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} c ^ k _ t = \\frac { a _ 0 } { \\rho _ k } \\sum _ { l = 0 } ^ { t } T _ l f _ { t - l } . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} D ( s , S _ f \\times S _ g ) = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) S _ g ( n ) } { n ^ { s + k - 1 } } , \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} F _ k ^ { \\bf a } = \\sum _ { i = 1 } ^ \\nu \\sum _ { j = 0 } ^ { \\mu _ i - 1 } c _ { i j } ^ { \\bf a } k ^ j \\lambda ^ k _ i . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\int _ X T _ { \\bar h } f \\cdot g \\ > d \\omega _ X = \\int _ X f \\cdot T _ { h } g \\ > d \\omega _ X . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} C _ r ( \\pi ) = ( \\log q ) ^ { - r } \\cdot \\frac { d ^ r } { d s ^ r } C ( \\pi , s ) \\big | _ { s = 0 } \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} T _ { n } ^ { f } ( M ) = \\frac { 1 } { n } \\sum _ { \\lambda \\vdash n } \\chi _ { ( n ) } ^ { \\lambda } [ M ^ { \\boxtimes | \\lambda | } \\otimes \\mathcal { S } ^ { \\lambda } ] \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} \\Phi ( \\eta ( R , s , t ; x ) ) & \\leq \\frac { 1 } { 2 } a C _ 3 t ^ { N - 2 } R ^ { 3 N - 5 } + \\frac { 1 } { 4 } b ( C _ 3 t ^ { N - 2 } R ^ { 3 N - 5 } ) ^ 2 \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - t ^ N ( C _ 4 s ^ N R ^ { 3 N } - C _ 5 R ^ { 3 N - 1 } ) \\\\ & \\leq t ^ N ( C _ 0 R ^ { 3 N - 1 } - C _ 4 s ^ N R ^ { 3 N } ) . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\left \\{ \\alpha _ { 1 } \\left ( i \\right ) , \\alpha _ { 2 } \\left ( i \\right ) , \\dots , \\alpha _ { N } \\left ( i \\right ) \\right \\} _ { i = 1 , \\dots , q } \\quad \\mbox { i n $ \\mathbb { C } ^ { N } , $ } \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} \\big | { y } _ { 2 } ( x ) - \\hat { y } _ { 2 } ( x ) \\big | & \\leq | { \\epsilon } | \\frac { ( x - a ) ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) } + \\frac { A } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } ( x - t ) ^ { \\alpha - 1 } | y _ { 1 } ( t ) - \\hat { y } _ { 1 } ( t ) | d t \\\\ & \\leq | { \\epsilon } | ( x - a ) ^ { \\gamma - 1 } \\sum _ { j = 0 } ^ { 2 } \\frac { A ^ j ( x - a ) ^ { \\alpha { j } } } { \\Gamma ( \\alpha { j } + \\gamma ) } . \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} Q ( x _ 0 , x _ 1 , \\ldots , x _ { \\rho ( X ) } ) = \\log 2 \\cdot x _ 0 + \\frac { 1 } { 4 } \\sum _ { i = 1 } ^ { \\rho ( X ) } \\Lambda ( q _ i ) x _ i ^ 2 . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\tilde { f } ^ { ( 1 ) } _ { i j } \\tilde { g } ^ + _ j = \\sum _ { s \\geq 1 } \\sum _ { j _ 1 < \\ldots < j _ s = j } ( - 1 ) ^ { s - 1 } \\left ( [ z ^ { - 1 } ] T ^ + _ { i j _ 1 } ( z ) \\right ) \\tilde { e } ^ { ( 0 ) } _ { j _ 1 j _ 2 } \\cdots \\tilde { e } ^ { ( 0 ) } _ { j _ { s - 1 } j _ s } . \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\widehat { S } ( \\boldsymbol { \\theta } ) = \\begin{pmatrix} A \\theta _ 1 ^ 2 + \\frac { 1 } { 4 } C \\theta _ 2 ^ 2 & \\left ( B + \\frac { 1 } { 4 } C \\right ) \\theta _ 1 \\theta _ 2 \\\\ \\left ( B + \\frac { 1 } { 4 } C \\right ) \\theta _ 1 \\theta _ 2 & E \\theta _ 2 ^ 2 + \\frac { 1 } { 4 } C \\theta _ 1 ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} S _ { a , b , k _ 1 } \\boxplus S _ { a , b , k _ 2 } = S _ { a , b , k _ 1 + k _ 2 } . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} P _ V ( t ) x = Q _ \\infty Q _ t ^ { - 1 } x \\forall x \\in R ( Q _ t ) , \\forall t \\ge T _ 0 . \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} ( 1 + E _ \\ell ) ^ { 1 - 1 / p } & \\leq ( 1 + E _ { 0 } ) ^ { 1 - 1 / p } + C _ p \\tau \\sum _ { k = 1 } ^ \\ell \\left ( 1 + \\frac { H ^ \\tau [ u ^ { k - 1 } ] - H ^ \\tau [ u ^ { k } ] } { \\tau } \\right ) ^ { 1 / p } \\\\ & \\leq ( 1 + E _ { 0 } ) ^ { 1 - 1 / p } + C _ p \\tau \\ell ^ { 1 - 1 / p } \\left ( \\ell + \\frac { H ^ \\tau [ u _ 0 ] - H ^ \\tau [ u ^ { \\ell } ] } { \\tau } \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} u _ - ( - 1 ) = ( - 1 , - \\varepsilon / 2 ) , u _ - ( 1 ) = ( 1 , - \\varepsilon / 2 ) , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} g _ 0 ( x ) = \\frac { 1 } { \\omega _ { 2 } } \\sum _ { m = 0 } ^ \\infty P _ m ( x ) G _ 0 ( x , - x ) = \\frac { 1 } { \\omega _ { 2 } } \\left [ \\frac { 1 } { 2 \\vert x \\vert } - \\sum _ { m = 0 } ^ \\infty ( - 1 ) ^ m P _ m ( x ) \\right ] , \\end{align*}"}
-{"id": "2383.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 ! } , ( \\textnormal { s e e } \\ , \\ , [ 9 ] ) . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\overline { Y } ^ { \\overline { \\bullet } } = \\{ x \\in X : x \\mathbf { d } Y = 0 \\} . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\phi = \\phi _ { n p r s } = ( - q ) ^ n B _ { n , q } B _ { p , 1 } \\ , Q ( a ^ { r - n } ( c ^ * ) ^ n c ^ { s - p } a ^ { * p } ) \\in \\mathcal { P } . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} \\hat u _ { \\infty , x } ( r ) = B ^ * e ^ { - r A ^ * } { Q } ^ { - 1 } _ { \\infty } x r \\in \\ , ] - \\infty , 0 ] , \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} & \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\prec \\frac { \\sqrt { ( \\Im m _ { \\mu _ A \\boxplus \\mu _ B } + \\hat { \\Lambda } ) ( | \\mathcal { S } | + \\hat { \\Lambda } ) } } { N \\eta } + \\frac { 1 } { ( N \\eta ) ^ 2 } , \\iota = A , B \\ , . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} \\tilde { l } ^ { G E L } _ { \\gamma } ( \\theta ) = n \\sum _ { k , l } \\bar { \\psi } ^ k \\omega ^ { k l } \\bar { \\psi } ^ l + O _ p \\left ( n ^ { - 1 / 2 } \\right ) \\ , , \\textrm { f o r a l l } k , l = 1 \\ldots d , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) & = g _ { 2 } ( \\nabla \\pi _ * ( U _ { 2 } , \\omega \\phi V _ { 2 } ) , \\pi _ * Z ) - g _ { 1 } ( \\mathcal { T } _ { U _ { 2 } } \\omega V _ { 2 } , \\mathcal { B } Z ) \\\\ & + g _ { 2 } ( \\nabla \\pi _ * ( U _ { 2 } , \\omega V _ { 2 } ) , \\pi _ * ( \\mathcal { C } Z ) ) - g _ { 1 } ( V _ { 2 } , \\phi U _ { 2 } ) \\eta ( Z ) \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} & \\Delta ^ { ( n , \\sigma ) } [ a ; c _ 1 , c _ { 2 } , \\ldots , c _ n ] ( X ) \\\\ = & \\left ( \\begin{array} { c c c c } a x _ { 1 1 } + c _ { 1 } x _ { \\sigma ( 1 ) , \\sigma ( 1 ) } & 0 & \\cdots & 0 \\\\ 0 & a x _ { 2 2 } + c _ { 2 } x _ { \\sigma ( 2 ) , \\sigma ( 2 ) } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & a x _ { n n } + c _ { n } x _ { \\sigma ( n ) , \\sigma ( n ) } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { m = 1 } ^ N a _ m x _ m \\Bigr \\| \\leq K \\| x \\| . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} y _ 0 = \\Psi ( P _ 0 , x , x _ 0 ) = \\exp \\left ( \\int _ { x _ 0 } ^ { x } { \\phi _ s ( P _ 0 , x ' ) d x ' } \\right ) , \\ , \\ , \\ , P _ 0 \\in \\Gamma _ s . \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} N _ r ( s , t ) & = \\sum _ { T : | T | = t } M _ r ( [ s ] , T ) \\\\ & = \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\left ( \\binom { a } { 2 } + \\binom { b } { 2 } \\right ) \\mu ( t ; a , b ) , \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} q = \\frac { n - r } { d } > \\frac { n } { d } - 1 . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\beta ^ { ( j ) } _ { i } \\leq a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\cdots p ^ { ( j ) } _ { r _ j } \\beta _ { + } + \\frac { q ^ { ( j ) } _ { i + 1 } } { a ^ { ( j ) } _ { i + 1 } } ( \\mu _ { \\mathfrak { f } ^ { ( j ) } _ i } - 1 ) + \\sum _ { i ' = i + 1 } ^ { r _ j } \\frac { a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\cdots p ^ { ( j ) } _ { i ' - 1 } q ^ { ( j ) } _ { i ' + 1 } } { a ^ { ( j ) } _ { i ' } a ^ { ( j ) } _ { i ' + 1 } } ( \\mu _ { \\mathfrak { f } ^ { ( j ) } _ { i ' } } - 1 ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} 2 s \\ , ( - \\Delta ) ^ s F = n F + r \\partial _ { r } F . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} H _ p ^ X ( \\epsilon ) = \\inf \\left \\{ \\left ( \\int _ 0 ^ { 2 \\pi } \\| x + e ^ { i \\theta } y \\| ^ p d \\theta \\right ) ^ { 1 / p } - 1 : \\ , \\| x \\| = 1 , \\| y \\| = \\epsilon \\right \\} , \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\left \\| p / q \\right \\| & = \\sum _ { i } m _ { i } \\left \\| p / q \\right \\| _ { i } = 2 \\sum _ { i } m _ { i } \\left ( \\sum _ { j = 1 } ^ { m } a ^ i _ j \\Delta \\left ( p / q , b _ j / c _ j \\right ) \\right ) \\\\ & = \\sum _ { j = 1 } ^ { m } \\left ( 2 \\sum _ { i } a ^ { i } _ { j } m _ { i } \\right ) \\Delta \\left ( p / q , b _ j / c _ j \\right ) . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\Z _ p = \\varprojlim _ k \\Z / p ^ k \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} L ( a ( 1 + q ) , b ( 1 + q ) ^ 2 ) & = \\Delta \\frac { \\frac 1 \\Delta - 1 + q ( \\alpha - \\beta ) } { 2 } + \\Delta \\frac { 1 - q ( \\alpha - \\beta ) - \\sqrt { 1 - 2 q ( \\alpha + \\beta ) + q ^ 2 ( \\alpha - \\beta ) } \\ , } { 2 } \\\\ & = \\Delta \\frac { \\frac 1 \\Delta - 1 + q ( \\alpha - \\beta ) } { 2 } + \\Delta T ( \\alpha q , \\beta q ) . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\left \\vert \\sum _ { j = 1 } ^ { \\infty } a _ { j } r _ { j } \\left ( t \\right ) \\right \\vert ^ { 2 } d t = \\sum _ { j = 1 } ^ { \\infty } \\left \\vert a _ { j } \\right \\vert ^ { 2 } \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\| u ^ { * } ( \\varphi ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) } ^ m ) u \\\\ & - \\psi ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\rho ( p \\otimes a ) } ^ m \\| < \\varepsilon \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} T _ 0 = \\left \\{ \\sum t _ j B p _ j , \\ , \\sum t _ j = 1 , \\ , t _ j \\geq 0 \\forall j \\right \\} . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\overline { Q } \\frac { \\partial ^ 2 z _ 0 ( \\mathbf { x } , \\tau ) } { \\partial \\tau ^ 2 } = - \\mathbf { D } ^ * g ^ 0 \\mathbf { D } z _ 0 ( \\mathbf { x } , \\tau ) , \\\\ & z _ 0 ( \\mathbf { x } , 0 ) = \\phi ( \\mathbf { x } ) , \\overline { Q } \\frac { \\partial z _ 0 } { \\partial \\tau } ( \\mathbf { x } , 0 ) = \\psi ( \\mathbf { x } ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} u = u _ 1 + \\ldots + u _ s + u _ { s + 1 } , \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n ( \\xi _ i ^ N + N ) I _ { A _ i } ] = \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { t _ i } [ \\xi ^ N _ i + N ] I _ { A _ i } ] . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} h _ { \\mathrm { F a l } } ( X ) = - \\frac { 1 } { 2 } \\frac { L ^ { \\prime } ( \\chi _ { E / F } , 0 ) } { L ( \\chi _ { E / F } , 0 ) } - \\frac { 1 } { 4 } \\log \\left ( \\frac { | d _ E | } { d _ F } \\right ) - \\frac { d } { 2 } \\log ( 2 \\pi ) , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\bar Y _ t & = \\bar Y _ n + \\int _ { t \\wedge \\tau } ^ { n \\wedge \\tau } \\bigl ( f \\bigl ( s , Y ^ m _ s , Z ^ m _ s , V ^ m _ s \\bigr ) - f \\bigl ( s , Y ^ n _ s , Z ^ n _ s , V ^ n _ s \\bigr ) \\bigr ) d s - \\int _ { t \\wedge \\tau } ^ { n \\wedge \\tau } \\bar Z _ s d W _ s \\\\ & - \\int _ { t \\wedge \\tau } ^ { n \\wedge \\tau } \\int _ U \\widehat V _ s ( e ) \\widehat \\pi ( d e , d s ) - \\bar M _ { n \\wedge \\tau } + \\bar M _ { t \\wedge \\tau } . \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} D ^ { \\boldsymbol { r } } f = \\sum _ { \\boldsymbol { m } } \\vartheta _ { \\boldsymbol { m } } D ^ { \\boldsymbol { r } } \\varphi _ { \\boldsymbol { N } , \\boldsymbol { m } } + \\sum _ { \\boldsymbol { j } , \\boldsymbol { k } } \\theta _ { \\boldsymbol { j } , \\boldsymbol { k } } D ^ { \\boldsymbol { r } } \\psi _ { \\boldsymbol { j } , \\boldsymbol { k } } , \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} \\varphi _ { j 2 } x _ 2 + \\cdots + \\varphi _ { j d } x _ d - x _ j = \\Phi _ j x _ 1 \\quad , j = d + 1 , \\dots , n \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} g _ { n , v } ( x ) = \\theta _ { n , v } ( x ) + \\sum _ { i = 1 } ^ r a _ { n i } h _ { i , v } ( x ) \\geqslant 0 , \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} \\mathcal { U } \\mathcal { A } \\mathcal { U } ^ { - 1 } = \\int _ { \\widetilde \\Omega } \\oplus \\mathcal { A } ( \\mathbf { k } ) \\ , d \\mathbf { k } . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\rho ^ { \\gamma _ n } = \\rho ^ { \\widehat { \\gamma } } , \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , V ) & = - g _ { 1 } ( V , \\mathcal { A } _ { X } \\phi \\mathcal { B } Y + \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\phi \\mathcal { B } Y ) - g _ { 1 } ( V , \\mathcal { A } _ { X } \\omega \\mathcal { B } Y + \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { X } \\omega \\mathcal { B } Y ) \\\\ & + g _ { 2 } ( \\pi _ * ( C Y ) , \\pi _ * ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) ) . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} | | u _ j \\cdot \\nabla u _ j | | _ { \\tilde { L } ^ { q } } = \\max \\{ | | u _ j \\cdot \\nabla u _ j | | _ { L ^ { q } } , | | u _ j \\cdot \\nabla u _ j | | _ { L ^ { 2 } } \\} \\leq \\frac { C K _ j ^ { 2 } } { s ^ { \\frac { 3 } { 2 } ( 1 - \\frac { 1 } { q } ) } } . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\| ( \\widetilde { w } _ { 2 n } - \\widetilde { w } _ { 2 n - 2 } ) x _ i \\| & = \\| w _ { 2 n } v _ { 2 n } \\widetilde { w } _ { 2 n - 2 } v _ { 2 n } ^ * x _ i - \\widetilde { w } _ { 2 n - 2 } v _ { 2 n } v _ { 2 n } ^ * x _ i \\| \\\\ & \\leq \\| w _ { 2 n } [ v _ { 2 n } , \\widetilde { w } _ { 2 n - 2 } ] v _ { 2 n } ^ * x _ i \\| + \\| ( w _ { 2 n } - 1 ) \\widetilde { w } _ { 2 n - 2 } x _ { i } \\| \\\\ & < \\frac { 1 } { 2 ^ { 2 n - 1 } } \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} s _ \\lambda = \\det \\left ( s _ { ( \\alpha _ i | \\beta _ j ) } \\right ) _ { 1 \\le i , j \\le p } . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} \\zeta _ K ( O _ 1 ) - \\zeta _ K ( O _ 2 ) = { \\frac { \\alpha ( O _ 2 K ) - \\alpha ( O _ 1 K ) } { \\alpha ( O _ 1 K ) \\alpha ( O _ 2 K ) } } \\leq { 2 R ^ 2 ( K ^ { \\circ } ) } { R ^ 2 ( K ) } \\| O _ 1 - O _ 2 \\| _ 2 . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} K _ r ( x , \\{ z \\} ) = \\frac { \\delta _ { r , \\pi ( x , z ) } } { \\omega _ D ( \\{ r \\} ) } \\quad \\quad ( r \\in D , \\ > x , z \\in X ) . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} P ( x ) = \\frac { 1 } { n ( n - 2 ) | B _ 1 | } | x | ^ { 2 - n } \\mbox { r e s p . } P ( x ) = - \\frac { 1 } { 2 \\pi } \\log | x | . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\phi _ 1 ( a , h ) = \\frac { e ^ { a h } - 1 } a , \\phi _ 2 ( c , h ) = \\frac { 1 - e ^ { - c h } } c , \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} b _ k = - \\frac { \\det { \\overline C ^ { k } _ { k + 1 } } } { \\det { \\overline C ^ { k } } } + \\frac { \\det { \\overline C ^ { k - 1 } _ k } } { \\det { \\overline C ^ { k - 1 } } } \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k \\binom { m } { j } \\binom { n - m } { k - j } = \\binom { n } { k } \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} a _ 0 = 1 \\textrm { a n d } a _ n = - \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } a _ k \\textrm { t r } ^ { \\flat } \\ ( A ^ { n - k } \\ ) \\textrm { f o r } n \\geqslant 1 . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} d \\rho ^ { ( a , b ) } ( x ) = w ^ { ( a , b ) } ( x ) d x \\Bigr | _ { [ - 1 , 1 ] } + \\frac { b - a } { b } d \\delta _ { s _ 0 } \\quad { \\rm f o r } b > a \\ge 2 \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} c _ j = \\frac { 1 } { P _ { r } ( \\vec { 1 } ) } \\left [ \\frac { \\partial P _ r } { \\partial z _ j } ( \\vec { 1 } ) \\left ( 1 - \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) + \\frac { \\partial P _ { r + 1 } } { \\partial z _ j } ( \\vec { 1 } ) \\right ] \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} \\tilde { g } ^ - _ j \\tilde { e } ^ { ( - 1 ) } _ { j i } = \\sum _ { s \\geq 1 } \\sum _ { j _ 1 < \\ldots < j _ s = j } ( - 1 ) ^ { s - 1 } \\tilde { f } ^ { ( 0 ) } _ { j _ s j _ { s - 1 } } \\cdots \\tilde { f } ^ { ( 0 ) } _ { j _ 2 j _ 1 } \\cdot \\left ( [ z ] T ^ - _ { j _ 1 i } ( z ) \\right ) . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} x = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } u x _ i = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } ( u V ( f _ i ) + \\alpha _ i u e ^ { \\abs { x } } \\{ \\mu ( \\infty , x ) \\} ) , \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { l = 1 } ^ { p - q } z _ { i l } \\overline { a _ { k u } ^ { i l } \\left ( Z \\right ) } + \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\overline { z } _ { j l } a _ { u k } ^ { j l } \\left ( Z \\right ) = 0 , \\mbox { f o r a l l $ i , j , k , u = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\tilde \\omega _ { \\bullet } ( t ) = \\sum _ { i = 1 } ^ { \\infty } \\kappa ^ { i \\beta } \\left ( \\omega _ { \\bullet } ( \\kappa ^ { - i } t ) \\ , [ \\kappa ^ { - i } t \\le 1 ] + \\omega _ { \\bullet } ( 1 ) \\ , [ \\kappa ^ { - i } t > 1 ] \\right ) . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\mathcal { F } _ q ( H , q ^ { - 2 m } X ^ { - 1 } ) = \\underline { \\chi } _ { \\boldsymbol { K } , q } ( \\gamma ( H ) ) ^ { m - 1 } ( q ^ m X ) ^ { - { \\rm o r d } _ q ( \\gamma ( H ) ) } \\mathcal { F } _ q ( H , X ) . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\ell ^ \\sigma _ y W ^ \\rho _ u ( \\lambda ) ( x ) & = W ^ \\rho _ u ( \\rho ^ * ( y ) \\lambda ) ( x ) \\\\ r ^ \\sigma _ y W ^ \\rho _ u ( \\lambda ) ( x ) & = W ^ \\rho _ { \\rho ( y ) u } ( \\lambda ) ( x ) . \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { \\tau , t } } ( u ) \\right ) = 0 . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\mathcal { S } & : = \\{ ( i + 1 , j ) \\geq ( i , j ) > ( i + 1 , j + 1 ) \\ | \\ 1 \\leq j \\leq i \\leq n - 1 \\} . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} u _ p \\ , : = \\ , u _ { b _ n } \\circ \\cdots \\circ u _ { b _ 1 } \\in N \\ \\ , \\ \\ p : a \\to a \\ , , \\ , p = b _ n * \\ldots * b _ 1 \\ . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} f ^ { 2 , 0 } ( Y , w ) = f ^ { 0 , 2 } ( Y , w ) = f ^ { 1 , 3 } ( Y , w ) = f ^ { 3 , 1 } ( Y , w ) = 0 . \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\big [ \\mu _ { i _ 1 } , \\ldots \\widehat \\mu _ { i _ k } , \\ldots , \\mu _ { i _ p } , [ \\mu _ { j _ 1 } , \\ldots , \\mu _ { j _ q } ] \\big ] , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\tilde { R } ( t ) = \\begin{cases} C t ^ { 1 / 2 } & \\gamma = 1 \\\\ C t ^ { ( 2 - \\gamma ) / 2 } \\ln \\frac { 1 } { t } & \\gamma > 1 , \\end{cases} \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} w [ \\mathbf { u } , \\mathbf { u } ] = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ d } \\left \\langle \\sigma _ * ( \\mathbf { u } ) , e _ * ( \\mathbf { u } ) \\right \\rangle _ { \\mathbb { C } ^ m } d \\mathbf { x } = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ d } \\left \\langle g ( \\mathbf { x } ) b ( \\mathbf { D } ) \\mathbf { u } , b ( \\mathbf { D } ) \\mathbf { u } \\right \\rangle _ { \\mathbb { C } ^ m } d \\mathbf { x } , \\mathbf { u } \\in H ^ 1 ( \\mathbb { R } ^ d ; \\mathbb { C } ^ d ) . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ { m - 1 } S _ { j _ k } \\cdot \\Big ( D _ { i _ 1 } \\cdots D _ { i _ { n + 1 } } S _ j + ( \\sum \\limits _ { k = 1 } ^ { n + 1 } D _ { i _ 1 } \\cdots S _ { i _ k } \\cdots D _ { i _ { n + 1 } } ) D _ j \\Big ) , \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 1 & x _ 2 & \\cdots & x _ { n - 1 } & 1 \\\\ y _ 1 & y _ 2 & \\cdots & y _ { n - 1 } & y _ n \\\\ 1 & 0 & \\cdots & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} D _ { 0 ^ + } ^ { \\alpha } f ( t ) = \\frac { 1 } { \\Gamma ( n - \\alpha ) } \\frac { d ^ { n } } { d t ^ { n } } \\int _ { 0 } ^ { t } \\frac { f ( s ) d s } { ( t - s ) ^ { \\alpha - n + 1 } } , \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\int _ { T _ 0 ( F _ x ) \\times T _ 3 ( F _ x ) } f _ x ( t _ 0 ^ { - 1 } \\gamma t _ 3 ) \\ , | t _ 0 | ^ { 2 s } \\eta _ x ( t _ 3 ) \\ , d t _ 0 \\ , d t _ 3 = | \\epsilon | _ x ^ { - 2 s } \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { k - 1 } } e ^ { - n / X } = C X ^ { \\frac { 3 } { 2 } } + O ( X ^ { \\frac { 1 } { 2 } + \\epsilon } ) , \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} A _ n ^ - ( n - 1 ) & = 2 ^ { n - 3 } + \\sum _ { j = 2 } ^ { n - 3 } a ( n ; n - 1 , j ) + a ( n - 3 ) \\\\ & = 2 ^ { n - 3 } + a ( n - 3 ) + \\sum _ { j = 2 } ^ { n - 3 } a ( n - 1 ; n - 2 ; j ) + ( 2 ^ 0 + 2 ^ 1 + \\cdots + 2 ^ { n - 5 } ) \\\\ & = 2 ^ { n - 3 } + a ( n - 3 ) + A _ { n - 1 } ^ - ( n - 2 ) - 2 ^ { n - 4 } + 2 ^ { n - 4 } - 1 \\\\ & = A _ { n - 1 } ^ - ( n - 2 ) + a ( n - 3 ) + 2 ^ { n - 3 } - 1 , \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} Q ( z ) = ( - 1 ) ^ k T _ { 1 , \\ldots , k } V ( z , z _ 1 , \\ldots , z _ k ) \\bigg [ \\prod ^ k _ { i = 1 } \\Psi _ p ( z _ i ) \\bigg ] ^ { - 1 } \\bigg [ 1 + O \\bigg ( \\frac { \\Psi _ p ( z _ k ) } { \\widetilde { \\Psi } _ { p , k } } \\bigg ) \\bigg ] , \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} d X ^ i _ t = \\sigma ^ i _ { \\alpha } ( X _ t ) d Z ^ { \\alpha } _ t , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} M _ 1 & = \\dfrac { H _ 2 ( \\widetilde { X } ) } { H _ 2 ( E _ 1 ) \\oplus H _ 2 ( E _ 2 ) } , \\\\ M _ 2 & = H _ 2 ^ { ( 0 ) } ( \\widetilde { X } , E _ 1 \\cup E _ 2 ) , \\\\ M _ 3 & = H _ 1 ( E _ 1 ) \\oplus H _ 1 ( E _ 2 ) , \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} C _ 2 = \\left ( \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right | ^ 2 - 2 \\Re \\left ( \\frac { P _ { r + 2 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) \\right ) . \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\begin{array} { c c c } F _ 2 = \\left [ \\begin{matrix} I _ n & - I _ n & 0 _ n \\\\ I _ n & I _ n & - 2 I _ n \\end{matrix} \\right ] , & \\ \\ \\ \\ \\ & \\tilde { R } = \\left ( R , 3 R \\right ) , \\\\ \\end{array} \\\\ \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } \\phi ( x ) | ^ { p } d x = \\frac { Q ( q - p ) - a _ { 2 } p q } { a _ { 1 } p q - Q ( q - p ) } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} O \\left ( n ^ { 2 } \\left ( 1 - p \\right ) ^ { 2 n } \\left ( p + n p ^ { 2 } + n ^ { 2 } p ^ { 3 } + n ^ { 3 } p ^ { 4 } \\right ) \\right ) = o \\left ( \\log ^ { 2 } n \\right ) . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} L ( \\tfrac { 1 } { 2 } , f \\times \\overline { f } \\times \\mu _ j ) = L ( \\tfrac { 1 } { 2 } , ^ 2 f \\times \\mu _ j ) L ( \\tfrac { 1 } { 2 } , \\mu _ j ) . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } f _ L v _ i d v _ 2 d v _ 3 = \\int _ { \\mathbb { R } ^ 2 } f _ R v _ i d v _ 2 d v _ 3 = 0 \\quad ( i = 2 , 3 ) \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( \\phi _ m ( X ^ x _ { \\tau + t ^ m _ 0 } , X ^ x _ { \\tau + t ^ m _ 1 } , X ^ x _ { \\tau + t ^ m _ 2 } , \\cdots , X ^ x _ { \\tau + t _ m ^ m } ) ) ] = \\hat { \\mathbb { E } } [ \\varphi ( \\phi _ m ( X ^ y _ { t ^ m _ 0 } , X ^ y _ { t ^ m _ 1 } , X ^ y _ { t ^ m _ 2 } , \\cdots , X ^ y _ { t _ m ^ m } ) ) ] _ { y = X ^ x _ { \\tau } } . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\beta ^ N = 0 . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} J _ { \\nu } ( x ) = \\sum \\limits _ { m = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ m } { m ! \\varGamma ( m + k + 1 ) } \\biggl ( \\frac { x } { 2 } \\biggr ) ^ { 2 m + \\nu } . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} a f _ 0 & = v _ 0 + a \\sum _ { i = 1 } ^ d \\lambda _ i v _ i \\\\ & = v _ 0 + a \\sum _ { i = 1 } ^ d \\sum _ { j = i } ^ d \\lambda _ i \\alpha _ { i , j } f _ j . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\| j ( x ) \\| & = \\sup _ { x ^ * \\in X ^ * , \\| k ( x ^ * ) \\| = 1 } \\langle k ( x ^ * ) , j ( x ) \\rangle = \\sup _ { x ^ * \\in X ^ * , \\| k ( x ^ * ) \\| = 1 } \\langle x ^ * , x \\rangle \\\\ & \\geq \\sup _ { x ^ * \\in X ^ * , \\| x ^ * \\| = \\frac 1 { \\| k \\| } } \\langle x ^ * , x \\rangle = \\frac 1 { \\| k \\| } \\| x \\| , \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\sum _ { k = 1 } ^ K R _ k ( t ) \\zeta _ k . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} B ^ { ( 1 + s ) } & \\stackrel { d e f } { = } \\widetilde { \\Omega } _ 2 \\ , A ^ { ( 1 + s ) } = \\widetilde { B } _ 2 - \\left ( \\widetilde { \\Omega } _ 1 + \\widetilde { \\Omega } _ 2 \\ , L _ { 2 1 } \\right ) A _ { 1 1 } \\ , L _ { 2 1 } ^ T \\\\ & = \\widetilde { B } _ 2 - \\widetilde { B } _ 1 \\ , L _ { 2 1 } ^ T . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} s ^ + _ \\lambda ( { v } ) = \\left ( \\frac { { H } _ { \\lambda } ( { v } ) } { F ( v ) } \\right ) ^ { 1 / ( \\gamma - p ) } . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} u _ x & = - \\beta y z \\\\ u _ y & = \\beta x z \\\\ u _ z & = \\beta \\left [ x y + \\sum \\limits _ { n = 1 } ^ \\infty \\frac { 3 2 a ^ 2 ( - 1 ) ^ n } { \\pi ^ 3 ( 2 n - 1 ) ^ 3 } \\sin \\left ( ( 2 n - 1 ) \\frac { \\pi x } { 2 a } \\right ) \\frac { \\sinh ( ( 2 n - 1 ) \\frac { \\pi y } { 2 a } ) } { \\cosh ( ( 2 n - 1 ) \\frac { \\pi y } { 2 a } ) } \\right ] \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\begin{aligned} p ( n ) & = p ( n + 1 ) + p ( n , 1 ) , \\\\ p ( n _ 1 , n _ 2 ) & = p ( n _ 1 + 1 , n _ 2 ) + p ( n _ 1 , n _ 2 + 1 ) + p ( n _ 1 , n _ 2 , 1 ) \\end{aligned} \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} u ( t ) = \\exp ^ { - t A ^ { \\frac { 1 } { 2 } } } u ( 0 ) \\quad ( t \\geq 0 ) , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\frac { E ' \\otimes W ' - \\left ( \\overline { E ' \\otimes W ' } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } = \\left ( V ' \\otimes Z ' \\right ) \\left ( \\overline { V ' \\otimes Z ' } \\right ) ^ { t } , \\mbox { f o r a n i n v e r t i b l e m a t r i x $ V ' \\in \\mathcal { M } _ { q N \\times q N } \\left ( \\mathbb { C } \\right ) $ . } \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log D ( f ^ n ( x ) ) = \\frac { 1 } { n } \\log D ( x ) - \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\tilde D ( f ^ j ( x ) ) , \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\sup _ { \\lambda = \\pm 1 , \\pm i } \\norm { x + \\lambda y } \\geq 1 + \\delta ( x , \\epsilon ) \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} H & = \\frac 1 n \\log \\binom { n } { a _ 1 , \\dots , a _ n } \\\\ & \\geq \\frac 1 n \\log \\ ( \\frac { n ! } { ( n - k + 1 ) ! 1 ^ { k - 1 } } \\ ) \\\\ & = \\frac { n - k + 1 } n \\log \\frac { n } { n - k + 1 } + \\frac { k - 1 } n \\log n + O \\ ( \\frac { k } n + \\frac { \\log n } n \\ ) \\\\ & = \\frac { k } n \\log n + O \\ ( \\frac { k } n + \\frac { \\log n } n \\ ) . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left \\langle A _ { \\varepsilon } \\nabla u _ { \\varepsilon } , \\nabla v \\right \\rangle = F ( v ) \\qquad \\forall v \\in H _ { 0 } ^ { 1 } \\left ( \\Omega \\right ) . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} D _ M ( A B ) = D _ M ( A ) B + ( - 1 ) ^ { | M | | A | } A D _ M ( B ) . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} E ( z _ { n \\ell } ^ 4 ) = E ( E ( z _ { n \\ell } ^ 4 | w _ { \\ell } ) ) = E ( E ( ( \\sum ^ { \\ell - 1 } _ { i = 1 } q _ { \\ell i } ) ^ 4 | w _ { \\ell } ) ) = ( \\ell - 1 ) d _ 4 + 6 ( \\ell - 1 ) ( \\ell - 2 ) d _ 2 ^ 2 \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} F = \\begin{pmatrix} 2 \\overline { M ' } ( M + \\overline { M ' } ) ^ { - 1 } M & - i ( M - \\overline { M ' } ) ( M + \\overline { M ' } ) ^ { - 1 } \\\\ - i ( M + \\overline { M ' } ) ^ { - 1 } ( M - \\overline { M ' } ) & 2 ( M + \\overline { M ' } ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\frac { f ^ \\prime _ \\alpha ( z ) } { f _ \\alpha ( z ) } = \\int _ 0 ^ 1 \\frac { t ^ { x - 1 } } { t - 1 } g _ \\alpha ( t ) d t , \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} - T _ 0 ^ 3 T _ 2 - 2 T _ 0 ^ 2 T _ 1 T _ 2 + 1 4 T _ 0 ^ 2 T _ 2 ^ 2 - 2 T _ 0 T _ 1 ^ 3 + 9 T _ 0 T _ 1 ^ 2 T _ 2 + 4 T _ 0 T _ 1 T _ 2 ^ 2 - 7 T _ 0 T _ 2 ^ 3 + T _ 1 ^ 4 - 2 T _ 1 ^ 3 T _ 2 \\\\ { } - 7 T _ 1 ^ 2 T _ 2 ^ 2 + 3 T _ 2 ^ 4 = 0 \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla U _ i \\cdot \\nabla U _ j = \\int _ { \\Omega } ( - \\Delta U _ i ) U _ j = \\int _ { \\Omega } ( \\lambda U _ i + w _ i ^ 5 ) U _ j . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( \\mathcal C _ \\Omega \\mathcal E ^ C P _ \\alpha ^ u \\langle \\rangle ) & = \\langle \\rangle , \\\\ o _ { \\langle \\rangle } ( \\mathcal C _ \\Omega \\mathcal E ^ C P _ \\alpha ^ u \\langle \\rangle ) & = \\vartheta ( \\Omega + \\Omega ^ { \\varepsilon _ { \\langle \\rangle } } ) . \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} \\left ( A ^ { \\alpha } g \\right ) ( x ) = \\lim \\limits _ { \\varepsilon \\rightarrow 0 + } \\left ( \\left ( J ^ { \\alpha } _ { \\left ( A + \\varepsilon \\right ) ^ { - 1 } } \\right ) ^ { - 1 } g \\right ) ( x ) \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} & \\{ ( k , i ) > ( k + 1 , s ) \\geq ( k , j ) , \\ ( k , i ) \\geq ( k - 1 , t ) > ( k , j ) \\} \\subseteq \\mathcal { C } , \\\\ & \\{ ( k , i ) > ( k + 1 , s ) , ( k + 1 , t ) \\geq ( k , j ) \\} \\subseteq \\mathcal { C } . \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} w _ i : = w _ { \\mu _ i , \\zeta _ i } . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} A _ 1 = \\left ( \\iint _ { Q _ 0 } v ^ { 2 p } \\ , d v _ 0 \\ , d t \\right ) ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\widehat { \\mathcal { A } } = \\mathbf { D } ^ * g ( \\mathbf { x } ) \\mathbf { D } = - \\operatorname { d i v } g ( \\mathbf { x } ) \\nabla . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\sum _ { k _ 1 \\leq M } \\frac { \\mu ^ 2 ( k _ 1 ) \\chi ( k _ 1 ) } { k _ 1 \\tau ( k _ 1 ) } \\prod _ { p \\mid k _ 1 } \\left ( 1 + \\frac { 1 } { p } \\right ) ^ { - 1 } & = \\sum _ { k _ 1 \\leq M } \\frac { \\mu ^ 2 ( k _ 1 ) \\chi ( k _ 1 ) } { k _ 1 \\tau ( k _ 1 ) } \\sum _ { d \\mid k } \\mu ( d ) f ( d ) , \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} Z ( X _ 0 ( 8 6 ) _ 3 , t ) = & ( 1 + 2 t + 3 t ^ 2 ) ^ 2 ( 1 + 4 t ^ 2 + 9 t ^ 4 ) ^ 2 ( 1 - t + 5 t ^ 2 - 3 t ^ 3 + 9 t ^ 4 ) \\\\ & ( 1 + t + t ^ 2 + 3 t ^ 3 + 9 t ^ 4 ) / ( 1 - t ) ( 1 - 3 t ) . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\norm { v } _ 3 + \\norm { \\partial _ t v } _ { 2 . 5 } + \\norm { \\partial ^ 2 _ t v } _ { 1 . 5 } + \\norm { \\partial ^ 3 _ t v } _ 0 + \\norm { q } _ 3 + \\norm { \\partial _ t q } _ { 2 } + \\norm { \\partial ^ 2 _ t q } _ 1 \\leq C _ 0 . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} & = \\frac { ( - g ) ^ { n } ( g h ^ { 2 } - k ^ { 2 } + f h k ) } { \\left ( f ^ { 2 } + 4 g \\right ) } \\left | B ^ { - n } \\right | \\left | M B ^ n + ( - g ) ^ n B ^ { - n } \\right | \\\\ & = \\frac { g h ^ { 2 } - k ^ { 2 } + f h k } { \\left ( f ^ { 2 } + 4 g \\right ) } \\left | B ^ { - n } \\right | \\left | M B ^ n + ( - g ) ^ n B ^ { - n } \\right | . \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} a ( i + 1 ) - a ( i ) \\geq m , \\ : i = 1 , 2 , \\ldots , p - 1 \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\omega | ^ { 2 } - | \\bigtriangledown \\psi | ^ { 2 } ) d x d y = - 2 \\nu \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\bigtriangledown \\omega | ^ { 2 } - | \\omega | ^ { 2 } ) d x d y , \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} \\int _ M f _ x ! ( h _ x ; \\mathcal S _ x ) \\wedge \\rho = ( - 1 ) ^ { | \\rho | | \\omega | } \\int _ { ( s _ x ) ^ { - 1 } ( 0 ) } \\pi _ 1 ^ * \\widetilde h _ x \\wedge \\pi _ 1 ^ * f _ x ^ * \\rho \\wedge \\pi _ 2 ^ * \\omega _ x \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} V ( r , z ) = \\begin{cases} \\gamma [ ( 1 + | r | ^ { n - 1 } ) ^ { \\frac { 1 } { n - 1 } } - 1 ] \\arctan ( z ^ 2 ) & z \\ge 0 \\ , , \\\\ 0 & \\ , . \\end{cases} \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} v ( 0 ) = v _ 0 \\ , , v ' ( 0 ) = v _ 1 \\ , . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} p _ a \\ , = \\ , \\{ p _ { o a } : a \\to o \\} _ o \\ , \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} \\| w \\| _ { H ^ 1 ( y ^ { \\alpha } , \\C ) } : = \\left ( \\| w \\| _ { L ^ 2 ( y ^ { \\alpha } , \\C ) } + \\| \\nabla w \\| _ { L ^ 2 ( y ^ { \\alpha } , \\C ) } \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} 2 ^ { J _ { n , l } ( \\boldsymbol { \\alpha } ) } = \\left ( \\frac { R } { M } \\right ) ^ { \\left [ \\sum _ { l = 1 } ^ d \\alpha _ l \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha { * } } - \\frac { 1 } { 2 \\alpha _ l } \\right ) \\right ] ^ { - 1 } } \\left ( \\frac { n } { \\log { n } } \\right ) ^ { \\frac { \\alpha ^ { * } } { \\alpha _ l ( 2 \\alpha ^ { * } + d ) } } . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\mathcal { K } _ m ' ( \\varphi ) v ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\frac { \\partial \\kappa } { \\partial u } ( s , \\zeta _ i ^ j , \\varphi ( \\zeta _ i ^ j ) ) v ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} R _ \\mu ( \\delta | P _ { X Y } ) : = \\min \\{ r _ 0 + \\mu r _ 2 : ( r _ 0 , r _ 2 ) \\in { \\cal R } _ { \\mathtt { W A K } } ^ * ( \\delta | P _ { X Y } ) \\} \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} \\varphi = \\left ( \\frac { z _ 0 A ( z _ 1 , z _ 2 ) + B ( z _ 1 , z _ 2 ) } { z _ 0 C ( z _ 1 , z _ 2 ) + D ( z _ 1 , z _ 2 ) } , \\psi ( z _ 1 , z _ 2 ) \\right ) \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} D _ { \\zeta _ i ' } \\phi = \\partial _ { \\zeta _ i ' } A ( \\phi ; \\mu ' , \\zeta ' ) + \\partial _ \\phi A ( \\phi ; \\mu ' , \\zeta ' ) [ D _ { \\zeta _ i ' } \\phi ] . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} g ' : = \\lim _ { n \\to \\infty } { | E _ n ' | } ^ { 1 / n } . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} { \\rm d i m } \\ , { \\rm K e r } \\ , L ( t , x , \\tau , \\xi ) = \\mbox { m u l t i p l i c i t y o f t h e e i g e n v a l u e $ \\tau $ } . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} L u = \\Delta u - \\sum _ { i = 1 } ^ n A ( x - a _ i ) \\cdot \\nabla u , \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} \\beta ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) : = 1 - \\frac { 1 - \\alpha } { 1 + d \\delta ' ( 1 - \\alpha ) } . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} 0 = \\tilde { H } ( t , x , Q ^ \\ast , p , g ) - H ( t , x , a ^ \\ast , p , g ) = \\int _ A \\left [ H ( t , x , a , p , g ) - H ( t , x , a ^ \\ast , p , g ) \\right ] Q ^ \\ast ( d a ) , \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} M _ { n } \\left ( { u , \\xi } \\right ) = { \\sup _ { w \\in \\mathcal { L } \\left ( \\xi \\right ) } } \\left \\vert { \\varepsilon _ { n } \\left ( { u , w } \\right ) } \\right \\vert , \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} F _ \\pm ^ 3 + F _ \\pm = 0 , \\ ; \\gamma ( F _ \\pm X , Y ) + \\gamma ( X , F _ \\pm Y ) = 0 \\ ; ( X , Y \\in T M ) \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} W ( \\chi _ c ) = \\sum _ { a \\in \\mathcal { O } _ { K } / ( c ) } \\chi _ c ( a ) e \\Big ( \\Big ( \\frac { a } { \\delta c } \\Big ) \\Big ) , \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\xi = \\ln \\left ( { { \\tfrac { 1 } { 2 } } z } \\right ) + 1 + \\mathcal { O } \\left ( { z ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} | \\zeta - \\zeta _ 0 | ^ 2 + \\tau ^ 2 & \\leqslant | \\zeta - \\zeta _ 0 | ^ 2 + \\frac 4 9 | \\zeta - \\zeta _ 0 | ^ 2 = \\frac { 1 3 } { 9 } | \\zeta - \\zeta _ 0 | ^ 2 , \\\\ | ( \\zeta - \\zeta _ 0 ) ^ 2 + \\tau ^ 2 | & \\geqslant | \\zeta - \\zeta _ 0 | ^ 2 - \\tau ^ 2 \\geqslant \\frac 5 9 | \\zeta - \\zeta _ 0 | ^ 2 , \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} T ( z , t ) = \\frac { 1 - ( z - t ) - \\sqrt { 1 - 2 ( z + t ) + ( z - t ) ^ { 2 } } \\ , } { 2 } . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\left \\{ x _ j \\ , x _ { n + j } \\ , , \\ , x _ 1 + x _ { n + 1 } \\ , , \\ , x _ j + x _ { n + j } + \\sum _ { i = 1 } ^ { j - 1 } c _ { i , j } \\ , x _ { n + i } \\ , \\forall \\ 2 \\leq j \\leq n \\right \\} \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\eqref { v a 1 } & = T ( x ) \\psi ( x ) \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ + e ^ { - \\frac { i t } { 2 } ( \\xi - x ) ^ 2 } \\ , d \\xi \\\\ & + \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ + e ^ { - \\frac { i t } { 2 } ( \\xi - x ) ^ 2 } [ T ( \\xi ) \\psi ( \\xi ) - T ( x ) \\psi ( x ) ] \\ , d \\xi . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} \\pi _ 1 ( X ) = \\langle \\gamma _ 0 , \\ldots , \\gamma _ n \\mid \\gamma _ j ^ { p _ j } = 1 , ~ \\gamma _ 0 \\gamma _ 1 \\cdots \\gamma _ n = 1 \\rangle , \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\int _ 0 ^ R \\frac { r ^ 3 ( 1 - r ^ 2 ) } { ( 1 + r ^ 2 ) ^ 3 } \\mathrm { d } r = ~ & \\frac { 2 R ^ 4 + R ^ 2 } { 2 ( R ^ 2 + 1 ) ^ 2 } - \\frac { 1 } { 2 } \\log ( R ^ 2 + 1 ) = - \\log R + 1 + O ( R ^ { - 2 } ) . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} - \\kappa ^ { \\varepsilon } \\nabla \\theta ^ { \\varepsilon } \\cdot \\mbox { n } = 0 \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} T _ t f = \\lim _ { n \\rightarrow \\infty } T _ t f _ n = \\lim _ { n \\rightarrow \\infty } P _ t f _ n , \\ ; m \\R ^ d . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s \\left ( \\varphi _ j ( x _ i ) - \\varphi _ j ( y _ i ) \\right ) = 0 ( 1 \\le j \\le k ) , \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} u ^ \\ast ( t ) = \\left \\{ \\begin{aligned} \\begin{pmatrix} k _ M \\\\ 0 \\end{pmatrix} & \\quad \\textnormal { f o r } t < \\tau _ s \\\\ 0 & \\quad \\textnormal { f o r } t > \\tau _ s \\end{aligned} \\right . \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} Q _ p ( x _ 0 , v _ 0 , t ) = e ^ { - p t } Q _ { - p } ( - x _ 0 , - v _ 0 , t ) \\ , , \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} M _ t ^ { \\lambda , \\phi } = \\phi ( x , y , 0 ) + \\int _ 0 ^ t e ^ { - \\lambda \\tau } \\frac { \\partial \\phi } { \\partial y } ( X _ \\tau , Y _ \\tau , \\tau ) \\d W _ \\tau \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\kappa _ - & = \\inf _ { x \\in \\Sigma } \\inf _ { i = 1 , \\dots , n } \\kappa _ i ( x ) , \\\\ \\kappa _ + & = \\sup _ { x \\in \\Sigma } \\sup _ { i = 1 , \\dots , n } \\kappa _ i ( x ) , \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} \\mathcal { N } = \\{ ( L , v ) \\in \\beta ^ { * } ( T M ) ; v \\perp L \\} . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\dim { \\mathcal M } _ { k + 1 } ( \\beta ) = \\mu ( \\beta ) + \\dim L + k - 2 . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} x = y _ { 0 } + \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ( y _ { i } - y _ { 0 } ) , | \\alpha _ { i } | \\leq C ( m , \\rho ) | x - y _ { 0 } | . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus B ( x _ 0 , 8 r ) } \\abs { D u } \\lesssim \\sum _ { k = 1 } ^ N \\left ( 2 ^ { - \\beta k } + \\{ \\ln ( 4 / r ) \\} ^ { - 1 } \\right ) \\ \\norm { \\vec b } _ { L ^ 1 ( B ( x _ 0 , r ) ) } \\sim \\int _ { B ( x _ 0 , r ) } \\abs { \\vec b } , \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} C F ( \\mathcal C ) ^ 0 = \\bigoplus _ { \\alpha \\in \\frak A } \\Omega ( R _ { \\alpha } ; o _ { R _ { \\alpha } } ) , \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} { \\lim } _ { t \\to \\nu ^ { - } } x ( t _ n ) = x ^ { * } . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} ( K _ r \\circ K _ { \\bar r } ) ( x , \\{ x \\} ) = \\sum _ { z \\in X : \\ > \\pi ( x , z ) = r } \\frac { \\omega _ X ( \\{ z \\} ) } { \\omega _ D ( \\{ r \\} ) } \\cdot \\frac { \\omega _ X ( \\{ x \\} ) } { \\omega _ D ( \\{ \\bar r \\} ) } \\ > \\delta _ { \\bar r , \\pi ( z , x ) } = \\frac { \\omega _ X ( \\{ x \\} ) } { \\omega _ D ( \\{ \\bar r \\} ) } . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 - t } { p _ { 0 } } + \\frac { t } { p _ { 1 } } . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\eta _ x : = \\inf \\{ n \\geq 0 : \\ : S _ n \\geq x \\} \\ , . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} t _ { n p } = \\sum _ { 1 \\le j < i \\le p } r _ { i j } ^ 2 . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} X = \\left \\{ \\omega = \\sum _ { k \\in \\mathbf { Z } , \\ k \\neq 0 } e ^ { i k \\alpha x } \\omega _ { k } \\left ( y \\right ) , \\ \\Vert \\omega \\Vert _ { X } ^ { 2 } = \\Vert \\frac { 1 } { \\sqrt { K _ { 2 } \\left ( y \\right ) } } \\omega \\Vert _ { L ^ { 2 } } ^ { 2 } < \\infty \\right \\} . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\dot { y } ( t , p ) = L ( p ) ^ \\top \\tilde { f } ( L ( p ) ^ { - \\top } y ( t , p ) , p ) , \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\psi ( u ) & = & s g n ( g ( t _ 1 ( u ) ) ) \\frac { f ( t _ 1 ( u ) ) } { g ( t _ 1 ( u ) ) } + s g n ( g ( t _ 2 ( u ) ) ) \\frac { f ( t _ 2 ( u ) ) } { g ( t _ 2 ( u ) ) } \\\\ & = & \\frac { f ( t _ 1 ( u ) ) } { g ( t _ 1 ( u ) ) } - \\frac { f ( t _ 2 ( u ) ) } { g ( t _ 2 ( u ) ) } \\\\ & = & \\frac { 1 } { ( 1 + n u ) ^ { \\frac { n - 1 } { n } } } [ f ( ( 1 + n u ) ^ { \\frac { 1 } { n } } ) + f ( - ( 1 + n u ) ^ { \\frac { 1 } { n } } ) ] . \\end{array} \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} g _ 1 ( x ) : = \\sup _ { y \\in B _ r ( x ) } f ( y ) , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} f _ + = \\sum _ { \\substack { x \\in \\{ \\pm 1 \\} ^ d \\\\ \\eta _ d ( x ) = 1 } } e _ x , f _ - = \\sum _ { \\substack { x \\in \\{ \\pm 1 \\} ^ d \\\\ \\eta _ d ( x ) = - 1 } } e _ x \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} R _ k ^ { ( \\alpha , \\alpha ) } ( x ) = \\ > _ 2 F _ 1 ( 2 \\alpha + k + 1 , - k , \\alpha + 1 ; ( 1 - x ) / 2 ) \\quad ( k \\in \\mathbb N _ 0 ) \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\omega ( \\phi ( q ) ) = \\omega ( q - 1 ) \\ge \\omega ( m ) = \\pi ( U ) & = \\frac U { \\log U } + O \\bigg ( \\frac U { \\log ^ 2 U } \\bigg ) \\\\ & = \\frac { \\log x } { 5 \\log \\log x } + O \\bigg ( \\frac { \\log x } { ( \\log \\log x ) ^ 2 } \\bigg ) . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\begin{cases} D _ { 0 ^ + } ^ { \\alpha , \\beta } x ( t ) & = f ( t , x ) , t \\in ( 0 , + \\infty ) , \\\\ I _ { 0 ^ + } ^ { 1 - \\gamma } x ( 0 ^ + ) & = x _ 0 , \\gamma = \\alpha + \\beta - \\alpha \\beta , \\end{cases} \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} x ^ { - 1 } y = h ^ { - 1 } ( h x ^ { - 1 } u ^ { - 1 } g ^ n u x h ^ { - 1 } ) \\in \\langle S \\rangle N , \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} ( 1 \\pm 2 \\gamma ) \\alpha _ S ^ { \\binom { r } { 2 } - 1 } ( n / t ) ^ { r - 2 } . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} f ^ * ( x ^ * ) : = \\sup \\{ \\langle x ^ * , x \\rangle - f ( x ) : x \\in X \\} , x ^ * \\in X , \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} G _ K : = G ^ 0 _ K \\cup \\left ( \\tau ' G ^ 0 _ K \\right ) . \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} H ( x , t ) = ( f _ 1 ( x ) , t ) = ( f _ 2 ( x ) , t ) \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { r _ n ( x , 2 ) } { \\log _ 2 n } = 1 . \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} b _ { 2 } = K ^ { 2 } \\kappa _ { 1 } ^ { 2 } \\mathcal { \\alpha } _ { 1 } ^ { 2 } , \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} U ( z ) : = \\sum _ { k = 0 } ^ { \\infty } u _ k z ^ k \\stackrel { 1 } { = } \\frac { 2 } { ( 1 + z ) ( 2 + z ) } \\Bigg [ 1 + \\Bigg ( 2 - \\gamma - \\Psi ( 1 - z ) \\Bigg ) z \\Bigg ] , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} v '' ( t ) + A v ( t ) - \\beta \\int _ 0 ^ t \\ e ^ { - \\eta ( t - s ) } A v ( s ) d s = 0 \\ , , t \\ge 0 \\ , , \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} & F _ T ( x ) - 1 - x F _ T ( x ) - G _ 2 ( x ) - G _ 3 ( x ) \\\\ & = \\frac { x } { 1 - x } \\big ( F _ T ( x ) - 1 - x F _ T ( x ) - G _ 2 ( x ) \\big ) + \\frac { x ^ 3 } { ( 1 - x ) ^ 3 } \\big ( F _ T ( x ) - 1 - x F _ T ( x ) \\big ) . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} w ( \\infty ) = \\frac 1 2 \\ddot c ^ 0 = \\mbox { c o n s t a n t } . \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\psi ( a ) - \\psi ( 2 a ) = - \\int _ a ^ { 2 a } \\varphi ' ( t ) t ^ { m - 2 } \\geq a ^ { m - 1 } \\left ( \\frac { 2 ^ { m - 1 } - 1 } { m - 1 } - a \\frac { 2 ^ m - 1 } { m } \\right ) \\ge C _ m a ^ { m - 1 } . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} s x = y t , s ^ { - 1 } y = x t ^ { - 1 } , x ^ { - 1 } t = s y ^ { - 1 } , y ^ { - 1 } t ^ { - 1 } = s ^ { - 1 } x ^ { - 1 } . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} \\hat { 1 _ { S _ m } } ( \\chi _ 1 , \\dots , \\chi _ m ) = \\frac { n ^ { n - m } } { ( n - m ) ! } \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\sum _ { \\overrightarrow { \\omega } \\in I } \\exp \\ ( \\sup _ { V _ { \\overrightarrow { \\omega } , t ' } } g _ { m , t ' } \\ ) \\leqslant 2 \\inf _ { J \\subseteq \\Omega ^ m , S ^ 1 = \\bigcup _ { \\overrightarrow { \\omega } \\in J } V _ { \\overrightarrow { \\omega } , t ' } } \\sum _ { \\overrightarrow { \\omega } \\in J } \\exp \\ ( \\sup _ { V _ { \\overrightarrow { \\omega } , t ' } } g _ { m , t ' } \\ ) ; \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} T n = 0 \\mbox { o n } \\partial \\Omega \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\prod _ { q \\mid D ( T ) } F _ q ( T , q ^ { k - n - 1 } ) & = \\prod _ { q \\mid D ( T ) } F _ q ( T , q ^ { - \\tfrac { n + 1 } { 2 } + ( p - 1 ) \\cdot t } ) \\\\ & \\equiv \\prod _ { q \\mid D ( T ) } F _ q ( T , q ^ { - \\tfrac { n + 1 } { 2 } } ) = 0 \\pmod { p } . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} N _ 1 \\leq N \\prod _ { i = 1 } ^ s \\left ( 1 - \\frac 1 { p _ i } \\right ) + 2 ^ s < \\frac { c _ 1 } { 2 } N , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} T _ f g ( x ) : = \\int _ D \\int _ X g ( y ) \\ > K _ h ( x , d y ) \\ > f ( h ) \\ > d \\omega _ D ( h ) \\quad ( x \\in X , \\ > g \\in C _ c ( X ) ) \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\| v \\| _ { C ^ { k , \\alpha } ( \\Omega ^ 0 ) } \\le C ( \\boldsymbol { \\mathcal C ^ 0 } ) \\big ( \\| \\Delta \\dot h ^ 0 \\| _ { C ^ { k - 2 , \\alpha } ( \\Omega ^ 0 ) } + | \\dot c ^ 0 | \\big ) \\le C ( \\boldsymbol { \\mathcal C ^ 0 } ) \\big ( \\| \\dot h ^ 0 \\| _ { C ^ { k , \\alpha } ( \\Omega ^ 0 ) } + | \\dot c ^ 0 | \\big ) . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} C [ u , u , u _ x ] = R _ 1 [ u , u _ x , u _ x ] + R _ 2 [ u , u _ x , u _ x ] , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\Phi ( \\mathtt p ) : = { \\cal H } \\mathtt f \\big ( ( \\mathtt h + c ) | D | \\big ) [ \\eta ( \\cdot + { \\mathtt p } ( \\cdot ) ) ] \\ , , \\mathtt f ( \\xi ) : = \\frac { 1 } { \\tanh ( \\xi ) } , \\xi \\neq 0 \\ , , \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\textrm { t r } ^ { \\flat } A _ { \\omega , \\omega , t } = \\frac { \\chi _ { \\overrightarrow { \\omega } , t } \\ ( y \\ ) e ^ { g _ { m , t } \\ ( y \\ ) } \\theta _ \\omega \\ ( y \\ ) } { 1 - \\ ( T _ t ^ m \\ ) ' \\ ( y \\ ) ^ { - 1 } } . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} S ( x , x ) = \\frac { 1 } { B ( L / 2 , 1 / 2 ) } \\frac { I _ { 1 - x ^ 2 } ( L + 1 , N - 1 ) } { 1 - x ^ 2 } + \\frac { ( 1 - x ^ 2 ) ^ { ( L - 2 ) / 2 } | x | ^ { N - 1 } } { B ( N / 2 , L / 2 ) } I _ { x ^ 2 } \\left ( \\frac { N - 1 } 2 , \\frac { L + 2 } 2 \\right ) , \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\left ( \\sum _ { n \\leq X } \\lvert S _ f ( n ) \\rvert \\right ) ^ 2 & = \\left ( \\sum _ { n \\leq X } \\lvert S _ f ( n ) \\rvert \\cdot 1 \\right ) ^ 2 \\\\ & \\leq \\sum _ { n \\leq X } \\lvert S _ f ( n ) \\rvert ^ 2 \\sum _ { m \\leq X } 1 = X \\sum _ { n \\leq X } \\lvert S _ f ( n ) \\rvert ^ 2 \\\\ & \\ll X ^ { k - 1 + \\frac { 5 } { 2 } } . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\operatorname { w t } ( U ; q , t ) = ( - ( 1 - q ) ( 1 - t ) ) ^ { \\operatorname { p o s } ( U ) - n } \\prod _ { u _ { i , j } > 0 } [ u _ { i , j } ] _ { q , t } \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} \\Delta _ s ( L ^ 1 ( D , \\omega _ D ) ) : = \\{ \\phi \\in L ^ 1 ( D , \\omega _ D ) ^ * : \\ > \\phi \\ > \\ > \\ > \\ > \\ > \\phi ( f ^ * ) = \\overline { \\phi ( f ) } \\ > \\ > \\ > \\ > \\ > f \\in L ^ 1 ( D ) \\} \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} d \\upsilon ( l , m ) \\ , : = \\ , \\upsilon _ l \\upsilon _ m \\upsilon _ { l m } ^ * \\ , = \\ , \\rho ( \\gamma ( l , m ) ) \\ , \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\alpha _ i = \\frac { \\partial F } { \\partial q ^ i } = - \\frac { \\partial L } { \\partial q ^ i } , \\beta ^ i = \\frac { \\partial F } { \\partial p _ i } = \\dot { q } ^ i , \\frac { \\partial F } { \\partial \\dot { q } ^ i } = p _ i - \\frac { \\partial L } { \\partial \\dot { q } ^ i } = 0 \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} D ^ { i j } _ { k u k ' u ' } = 0 , \\quad \\quad \\quad \\quad \\quad \\quad \\mbox { f o r a l l $ i , j , k , u , k ' , u ' = 1 , \\dots , q $ w i t h $ k \\neq u $ , $ k ' \\neq u ' $ , } \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\mathcal { U } = & \\left \\{ \\alpha _ { k _ { 0 } } \\times \\alpha _ { k _ { 1 } } \\times \\ldots \\times \\alpha _ { k _ { ( M + 1 ) N } } \\times I ^ { \\infty } : \\{ k _ { j } \\} _ { j = 0 } ^ { ( M + 1 ) N } \\right . \\\\ & \\{ 1 , \\ldots , n \\} ( M + 1 ) N + 1 \\Big \\} . \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} \\log \\ ( 2 ^ m n ^ { \\delta m + 1 } \\binom { n } { m } ^ { - 1 / 2 } \\ ) & \\leq \\log \\ ( 2 ^ m n ^ { \\delta m + 1 } ( n / m ) ^ { - m / 2 } \\ ) \\\\ & = O ( m + \\log n ) + \\delta m \\log n - ( m / 2 ) \\log ( n / m ) . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} [ \\sigma ] _ { ( A _ { p ' } ) ^ { \\frac { 1 } { p ' } } ( A ^ * _ { \\infty } ) ^ { \\frac { 1 } { p } } } : = \\sup \\limits _ { i \\in \\mathbb Z , Q \\in \\mathcal { F } ^ 0 _ { i } } \\Bigg ( \\mathop { \\hbox { e s s s u p } } \\limits _ { Q } ( \\mathbb E ( \\omega | \\mathcal { F } _ i ) \\mathbb E ( \\sigma | \\mathcal { F } _ i ) ^ { p - 1 } ) \\frac { \\int _ { Q } { ^ * M } _ i ( \\sigma \\chi _ { Q } ) d \\mu } { | Q | } \\Bigg ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} h ^ { 2 , 1 } ( Z _ \\Delta ) & = \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} z ^ { T } ( F + S ) z & = v ^ { T } ( F + S ) v - 2 u ^ { T } S v + u ^ { T } S ( F + S ) ^ { - 1 } S u . \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} & x _ 1 ^ { p ^ 2 } = x _ 2 ^ { p ^ 2 } = x _ 3 ^ { p ^ 2 } = x _ 4 ^ { p ^ 2 } = 1 , \\\\ & [ x _ 1 , x _ 2 ] = 1 , \\ , [ x _ 1 , x _ 3 ] = x _ 4 ^ p , \\ , [ x _ 1 , x _ 4 ] = x _ 4 ^ p , \\\\ & [ x _ 2 , x _ 3 ] = x _ 1 ^ p , \\ , [ x _ 2 , x _ 4 ] = x _ 2 ^ p , \\ , [ x _ 3 , x _ 4 ] = x _ 4 ^ p . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{gather*} \\left [ \\sum _ { \\lambda = ( \\lambda _ 1 \\geq \\cdots \\geq \\lambda _ N ) \\in \\Z ^ N } { a _ { \\lambda } x _ 1 ^ { \\lambda _ 1 } \\cdots x _ N ^ { \\lambda _ N } } \\right ] _ 0 = a _ { ( 0 , \\dots , 0 ) } . \\end{gather*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\mathcal { L } f ( x ) : = \\frac { 1 } { 2 \\ell ( x ) } \\nabla \\big ( \\ell ( x ) \\ , \\alpha ( x ) \\ , \\nabla f ( x ) \\big ) , x \\in \\mathring { \\Gamma } , \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} \\bar a _ { i j } D _ { i j } u = 0 \\ ; \\ ; B ^ + ( 0 , 2 ) , u = \\ ; \\ ; T ( 0 , 2 ) . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} g ( \\xi ) = \\left \\{ \\begin{aligned} - a _ n , \\quad & \\ , A _ { \\xi _ i \\xi _ { i + 1 } } = 1 i < n A _ { \\xi _ n \\xi _ { n + 1 } } = 0 \\\\ - h ( Y ) , & \\ , A _ { \\xi _ i \\xi _ { i + 1 } } = 1 i \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\epsilon \\simeq \\gamma ^ { n } \\ ; , \\alpha = \\gamma ^ { 2 n - 2 } \\ ; , \\delta = \\gamma \\ ; . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} | E _ { n + 2 } '' | + | E _ { n + 1 } '' | = | E _ n ' | . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} b = \\lambda b _ 1 , \\ , \\ , \\ , c = \\lambda c _ 1 . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} a ( H ; G _ { k } ^ { ( m ) } ) & = ( \\ , p \\ , C ) \\times \\prod _ { q \\mid \\gamma ( H ) } \\mathcal { F } _ q ( H , q ^ { k - 2 m } ) \\\\ & \\equiv C \\times 0 = 0 \\pmod { p } . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{gather*} \\left | \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i + 1 ) - z ) } \\right | \\leq c _ 1 \\cdot q ^ { c _ 2 \\Re z } , i = 1 , 2 , \\dots , N , \\end{gather*}"}
-{"id": "9872.png", "formula": "\\begin{align*} W ( 1 ) = \\max \\{ 8 , | 2 - n | \\} + \\frac { 1 } { 4 } \\sum _ { \\gamma \\in \\Gamma ^ S ( \\chi _ 0 ) } \\frac { 1 } { \\gamma ^ 3 } \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} ( S _ { q , c } ^ p ) ^ * = S _ { q ' , r } ^ { p ' } , ( S _ { q , c } ^ p ) ^ * = S _ { q ' , r } ^ { p ' } \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} | \\Lambda _ \\iota | \\leq C ( c \\Im m _ { \\mu _ A \\boxplus \\mu _ B } + \\hat { \\Lambda } ) , \\iota = A , B , \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} W ( s ; f , \\overline { g } ) = \\frac { L ( s , f \\times \\overline { g } ) } { \\zeta ( 2 s ) } + Z ( s , 0 , f \\times \\overline { g } ) . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} \\int _ { B _ { \\sigma } ( x ) } | D _ { e } u | ^ { 2 } & \\leq \\rho ^ { - 2 } \\int _ { B _ { \\sigma } ( x ) } | D u ( z ) \\cdot v ( x ) | ^ { 2 } \\\\ & \\leq 2 \\rho ^ { - 2 } \\left ( \\int _ { B _ { \\sigma } ( x ) } | D u ( z ) \\cdot v ( z ) | ^ { 2 } + \\int _ { B _ { \\sigma } ( x ) } | D u ( z ) \\cdot ( v ( z ) - v ( x ) ) | ^ { 2 } \\right ) \\\\ & \\leq C \\int _ { B _ { 1 } ( 0 ) } | D _ { v } u | ^ { 2 } + C \\sigma ^ { 2 } \\int _ { B _ { \\sigma } ( x ) } | D u | ^ { 2 } \\\\ & \\leq C \\delta + C \\Lambda \\sigma ^ { m } , \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\Lambda _ { t , 0 } = \\exp \\left ( \\sum _ { q = 1 } ^ { \\infty } \\ln \\left ( \\lambda _ q \\right ) \\right ) = 0 , \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} \\frac { p \\sqrt [ p ] { \\lambda _ { 1 , p } ( M , F ) } } { q \\sqrt [ q ] { \\lambda _ { 1 , q } ( M , F ) } } \\leq \\kappa _ F ^ 2 = \\sigma _ F . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\frac { \\partial S } { \\partial t } + H \\left ( q ^ i , \\frac { \\partial S } { \\partial q ^ i } \\right ) = 0 . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} K ^ { r r } ( x , y ) = \\begin{bmatrix} D ( x , y ) & S ( x , y ) \\\\ - S ( y , x ) & \\tilde { I } ( x , y ) \\end{bmatrix} . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\sup _ { y \\in U _ \\epsilon } & | H ( y ) - H ( e ) | \\\\ & \\leq D _ \\epsilon \\sum _ { \\stackrel { 1 \\leq | \\alpha | \\leq n } { | \\delta | = | \\alpha | } } \\int _ { [ - \\epsilon , \\epsilon ] ^ { | \\delta | } } | \\langle \\lambda , \\rho ( \\tau _ \\delta ( \\mathbf { t } ) ) \\rho ( X _ n ) ^ { \\delta _ n } \\cdots \\rho ( X _ 1 ) ^ { \\delta _ 1 } u ) | ( d t _ 1 ) ^ { \\delta _ 1 } \\dots ( d t _ n ) ^ { \\delta _ n } , \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} A _ n ^ - ( i ) & = \\sum _ { j = 1 } ^ { i - 1 } a ( n ; i , j ) , & & A _ n ^ - = \\sum _ { i = 2 } ^ n A _ n ^ - ( i ) , & & A ^ - ( x ) = \\sum _ { n \\geq 2 } A _ n ^ - x ^ n . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} H ( X \\cup \\{ z \\} ) - H ( Y \\cup \\{ z \\} ) & = - H ( Y \\setminus X \\mid X \\cup \\{ z \\} ) \\\\ & \\ge - H ( Y \\setminus X \\mid X ) \\\\ & = - H ( Y ) + H ( X ) , \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\gamma ( q ) = ( q ^ 1 , q ^ 2 ; q ^ 1 , 0 ) . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} \\ln \\det \\mathcal { T } _ { \\nu , \\theta } & \\geq \\sum _ { 1 \\leq i \\leq 3 } \\big \\{ ( 1 - \\theta ) ( 1 - \\nu ) \\ln T _ { t r } + ( 1 - \\theta ) \\nu \\ln \\Theta _ i + \\theta \\ln T _ { \\delta } \\big \\} \\cr & = 3 ( 1 - \\theta ) ( 1 - \\nu ) \\ln T _ { t r } + ( 1 - \\theta ) \\nu \\ln \\Theta _ 1 \\Theta _ 2 \\Theta _ 3 + 3 \\theta \\ln T _ { \\delta } \\cr & = 3 ( 1 - \\theta ) ( 1 - \\nu ) \\ln T _ { t r } + ( 1 - \\theta ) \\nu \\ln \\det \\Theta + 3 \\theta \\ln T _ { \\delta } . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\langle a , \\chi _ j \\rangle = a _ j . \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} \\min \\limits _ { \\sum _ { i = 1 } ^ 2 \\alpha _ k = 1 } \\sum _ { i = 1 } ^ 2 \\frac { ( 2 ^ { \\frac { \\bar R _ i } { \\alpha _ i } } - 1 ) \\alpha _ i } { g _ i } \\ge \\frac { 2 ^ { \\bar R _ 2 } - 1 } { g _ 2 } + \\frac { 2 ^ { \\bar R _ 2 } ( 2 ^ { \\bar R _ 1 } - 1 ) } { g _ 1 } , \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\rho ( z ) = \\sqrt { \\det \\Sigma ^ { - 1 } } \\ , e ^ { - \\pi \\ , \\Sigma ^ { - 1 } z ^ { 2 } } \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , U ) & = - g _ { 1 } ( \\phi W , \\hat { \\nabla } _ { Z } \\varphi U ) - g _ { 2 } ( \\omega W , \\pi _ { \\ast } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { Z } \\varphi U ) ) . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} f ( x ) = \\inf _ { y \\in X } f ^ c ( y ) + D ( x , y ) ^ p \\forall x \\in X \\ , . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} E = V ^ 5 - \\sum _ { i = 1 } ^ k w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 5 , \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\nabla ( | f | ^ { \\alpha } ) = \\alpha | f | ^ { \\alpha - 2 } \\Re ( f \\nabla \\overline { f } ) , \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} L ^ \\alpha [ u ^ + ] = f ( x , u ^ + ) . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} W _ { q , k } ( \\pi _ { k } ^ { \\mu } , \\pi _ { k } ^ { \\nu } ) = W _ { q } ( \\pi _ { k , \\infty } ^ { \\mu } , \\pi _ { k , \\infty } ^ { \\nu } ) , \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} \\| M + N \\| _ { \\widetilde { S } ^ { p } _ q } & = \\lim _ { n \\to \\infty } \\| M ^ n + N ^ n \\| _ { \\widetilde { S } ^ { p } _ q } \\\\ & = \\lim _ { n \\to \\infty } \\| M ^ n + N ^ n \\| _ { \\widetilde { S } ^ { p , \\mathcal { T } } _ q } \\\\ & \\leq \\lim _ { n \\to \\infty } \\| M ^ n \\| _ { \\widetilde { S } ^ { p , \\mathcal { T } } _ q } + \\| N ^ n \\| _ { \\widetilde { S } ^ { p , \\mathcal { T } } _ q } = \\| M \\| _ { \\widetilde { S } ^ { p } _ q } + \\| N \\| _ { \\widetilde { S } ^ { p } _ q } . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} 2 + D x ^ 2 = y ^ n , n > 2 , \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\frac { d ( T _ \\Delta x , T _ \\Delta y ) } { d ( x , y ) } = \\frac { f ( \\Delta ) } { f ( 0 ) } \\ge \\frac { R - \\Delta } { R } . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} D _ { \\zeta ' } I _ \\lambda ( \\zeta ' , \\mu ' ) = 0 , D _ { \\mu ' } I _ \\lambda ( \\zeta ' , \\mu ' ) = 0 . \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\abs { D u ( x ) - \\vec q _ { x , r } } \\le \\sum _ { i = 0 } ^ { \\infty } \\ , \\abs { \\vec q _ { x , \\kappa ^ i r } - \\vec q _ { x , \\kappa ^ { i + 1 } r } } \\lesssim \\sum _ { i = 0 } ^ \\infty \\phi ( x , \\kappa ^ i r ) . \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} P _ i ( u ) = P _ i ( - u + n - i + 2 ) \\ ; \\ ; i \\geq 2 \\ ; \\ ; P _ 1 ( u ) = P _ 1 ( - u + \\tfrac { N } { 2 } ) , \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} r * ( a , b , c , d ) & = ( d , a , b , c ) \\\\ s * ( a , b , c , d ) & = ( a , c , b , d ) \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} r ( { \\boldsymbol { x } } ) \\doteq ( x _ { 0 } ) ^ { + } + \\sum _ { k = 1 } ^ { \\infty } k x _ { k } , r _ { 0 } ( { \\boldsymbol { x } } ) \\doteq \\frac { ( x _ { 0 } ) ^ { + } } { r ( { \\boldsymbol { x } } ) } { { 1 } } _ { \\{ r ( { \\boldsymbol { x } } ) > 0 \\} } , r _ { k } ( { \\boldsymbol { x } } ) \\doteq \\frac { k x _ { k } } { r ( { \\boldsymbol { x } } ) } { { 1 } } _ { \\{ r ( { \\boldsymbol { x } } ) > 0 \\} } , k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} \\begin{cases} c _ { 1 , \\infty } c _ { 2 , \\infty } = c _ { 3 , \\infty } = c _ { 1 , \\infty } c _ { 4 , \\infty } , \\\\ c _ { 1 , \\infty } + c _ { 3 , \\infty } = M _ { 1 3 } \\\\ c _ { 2 , \\infty } + c _ { 3 , \\infty } + c _ { 4 , \\infty } = M _ { 2 3 4 } . \\end{cases} \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} ( \\mathcal { H } _ 0 + I ) ^ { - 1 } = \\varepsilon ^ 2 T _ \\varepsilon ^ * ( \\mathcal { H } _ 0 + \\varepsilon ^ 2 I ) ^ { - 1 } T _ \\varepsilon = T _ \\varepsilon ^ * \\mathcal { R } ( \\varepsilon ) T _ \\varepsilon , \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} A ( X , Y ) = g ( A ( X , Y ) , \\xi ) \\xi , \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} S _ k ( \\alpha ) : = \\left \\{ s \\in \\R \\colon \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ s w _ k ( n ) } = \\sum _ { n = 1 } ^ \\infty \\frac { e ^ { \\alpha _ n / k } } { n ^ s } < \\infty \\right \\} , k \\in \\N . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} A ( t : s ) = A ( t ) A ( t - 1 ) \\cdots A ( s ) , \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{gather*} \\Phi ( [ \\gamma ] ) = \\int _ I \\gamma ^ * \\omega ^ 0 . \\end{gather*}"}
-{"id": "7684.png", "formula": "\\begin{align*} d \\Omega ( Z , X , J Y ) + d \\Omega ( Z , J X , Y ) = 2 [ \\gamma ( b ( J X , Y ) + b ( X , J Y ) , J Z ) ] . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } \\partial _ { i } ( \\bar q ( R ) ) = \\bar q ' ( R ) \\partial _ { t } ^ { 2 } \\partial _ { i } R + 2 \\bar q '' ( R ) \\partial _ { t } \\partial _ { i } R \\partial _ { t } R + \\bar q '' ( R ) \\partial _ { i } R \\partial _ { t } ^ 2 R + \\bar q ''' ( R ) ( \\partial _ { t } R ) ^ 2 \\partial _ { i } R \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} H ( \\Delta Y ^ { ( n ) } _ { 2 } ( \\Delta _ n ) | M _ 2 ) & = \\sum _ { i = 1 } ^ n h ( Y ^ { ( n ) } _ { 2 } ( t _ { n , i } ) - Y ^ { ( n ) } _ { 2 } ( t _ { n , i - 1 } ) | Y _ { 2 , t _ { n , 0 } } ^ { ( n ) , t _ { n , i - 1 } } , M _ 2 ) \\\\ & \\geq \\sum _ { i = 1 } ^ n h ( \\sqrt { N _ 2 } B _ { 2 } ( t _ { n , i } ) - \\sqrt { N _ 2 } B _ { 2 } ( t _ { n , i - 1 } ) ) \\\\ & = \\sum _ { i = 1 } ^ n \\log ( 2 \\pi e N _ 2 \\delta _ n ) , \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{gather*} \\sum _ { r = 1 } ^ { \\infty } \\sum _ { s = 0 } ^ { \\theta - 1 } \\left \\{ x ^ { \\nu _ r + \\theta ( r - 1 ) + s } \\prod _ { \\substack { i \\geq 1 \\\\ i \\neq r } } { \\frac { \\big ( q ^ { - \\nu _ r + \\nu _ i - s } t ^ { i - r + 1 } ; q \\big ) _ { \\infty } } { \\big ( q ^ { - \\nu _ r + \\nu _ i - s } t ^ { i - r } ; q \\big ) _ { \\infty } } } \\times \\prod _ { \\substack { \\theta > j \\geq 0 \\\\ j \\neq s } } { \\frac { 1 } { ( 1 - q ^ { j - s } ) } } \\right \\} . \\end{gather*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\| ( R ( v _ 2 - v _ 1 ) ) ^ { ( \\beta ) } \\| _ \\infty \\leq \\frac { C _ { 1 1 } ( b - a ) } { 2 } \\left ( \\sum _ { i = 1 } ^ \\rho | w _ i | \\right ) \\| v _ 2 - v _ 1 \\| _ \\infty ^ 2 \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\frac { C _ { 1 , n } } { ( k - 3 ) ! } = - [ 2 ( n - 1 ) a _ k + b _ { k - 1 } ] e _ 1 - \\frac { 2 n ( n - 1 ) } { k } a _ { k - 1 } - \\frac { n } { k - 1 } b _ { k - 2 } , \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\pi ^ \\varrho _ o : { } ^ 2 G _ \\bullet \\to { } ^ 1 U ( H ^ \\varrho _ o ) \\ \\ \\ , \\ \\ \\ \\pi ^ \\varrho _ o ( v ) : = \\{ \\bar { v } _ { o a } \\cdot v \\cdot \\bar { v } _ { a o } \\} ^ { \\varrho _ o } \\ , \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} | U | = \\frac { | \\rho U | } { \\rho } = \\frac { \\Big | \\int _ { \\mathbb { R } ^ 3 } f v d v \\Big | } { \\int _ { \\mathbb { R } ^ 3 } f d v } \\leq \\frac { a _ u + c _ u } { 2 a _ { \\ell } } . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ { \\theta N - 1 } { \\big ( z _ r q ^ { \\theta r - \\theta + \\tau _ r ^ + - \\tau _ r ^ - } - q ^ { \\theta ( N - 1 ) - j } \\big ) } = T _ { q , z _ r } ^ { \\theta r - \\theta + \\tau _ r ^ + - \\tau _ r ^ - } \\prod _ { j = 1 } ^ { \\theta N - 1 } { \\big ( z _ r - q ^ { \\theta ( N - 1 ) - j } \\big ) } . \\end{gather*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\left ( \\mathcal { W } \\circ \\varphi _ { A } \\left ( Z ' , Z '' \\right ) \\right ) ^ { t } = \\left ( \\frac { \\sqrt { 1 - A } { Z ' } ^ { t } } { I _ { q } - A { Z '' } ^ { t } } , \\frac { \\sqrt { 1 - A } \\left ( A - Z ''' \\right ) { Z ' } ^ { t } } { \\left ( I _ { q } - A Z ''' \\right ) \\left ( I _ { q } - A { Z '' } ^ { t } \\right ) } , \\frac { \\left ( A - Z ''' \\right ) \\left ( A - { Z '' } ^ { t } \\right ) } { \\left ( I _ { q } - A Z ''' \\right ) \\left ( I _ { q } - A { Z '' } ^ { t } \\right ) } \\right ) . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ \\frac { 4 \\sqrt 2 } { ( 2 \\pi ) ^ { d + 1 } } \\int _ r ^ \\infty \\frac { \\sinh s } { \\sqrt { \\cosh 2 s - \\cosh 2 r } } \\cdot \\left ( \\frac { - 1 } { \\sinh s } \\partial _ s \\right ) ^ { d } \\frac { \\sin ( s \\cdot \\sqrt { \\lambda - d ^ 2 } ) } { s } \\ , d s \\ , . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\Delta = t ^ 2 ( t - p ) ^ 3 ( t + p ) ^ 3 P ^ 2 Q ^ 2 R ^ 2 S , \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} u ^ { L } _ { i + \\frac { 1 } { 2 } } = q _ i ( x _ { i + \\frac { 1 } { 2 } } ) \\ \\ { \\rm a n d } \\ \\ u ^ { R } _ { i - \\frac { 1 } { 2 } } = q _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} q = z ^ * \\cdot \\prod _ { F \\in \\ + F ^ * } \\frac { 1 } { { z _ F } ^ { x _ F } } \\cdot \\prod _ { F \\in \\ + F \\setminus \\ + F ^ * } \\left ( \\frac { | \\pi _ F [ R ] | } { z _ F } \\right ) ^ { x _ F } . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} E ( y ) & = 5 w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 4 \\Bigl ( \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ i ( \\varepsilon \\ , y ) + \\sum _ { j \\not = i } w _ { \\mu _ j ^ { \\prime } , \\zeta _ j ^ { \\prime } } ( y ) + \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ j ( \\varepsilon \\ , y ) \\Bigr ) \\\\ & + O ( w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 3 \\varepsilon ^ 2 ) + O ( \\varepsilon ^ 5 ) , y \\in B _ { \\delta / \\varepsilon } ( \\zeta _ i ' ) . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} | f _ i ( u ) | \\leq K ( | u | ^ { \\mu } + 1 ) i = 1 , \\ldots , N u \\in \\mathbb R ^ N . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\mathrm { d } X _ { t } = b ( X _ { t } ) \\mathrm { d } t + \\sigma ( X _ { t } ) \\mathrm { d } B _ { t } , \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} w ( x , \\xi ) = \\int _ 0 ^ { l ( x , \\xi ) } h _ { i _ 1 \\dots i _ m } ( \\gamma _ { x , \\xi } ( t ) ) \\dot { \\gamma } _ { x , \\xi } ^ { i _ 1 } ( t ) \\cdots \\dot { \\gamma } _ { x , \\xi } ^ { i _ m } ( t ) d t . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\sum _ { A , B } \\left ( h _ { B A } , { H _ { A B } } \\right ) _ { \\varphi } = \\sum _ { A , B } \\left ( h _ { A B } , { H _ { A B } } \\right ) _ { \\varphi } - 2 \\sum _ { A , B } \\left ( h _ { [ A B ] } , { H _ { [ A B ] } } \\right ) _ { \\varphi } . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\| R ( v _ 2 - v _ 1 ) \\| _ \\infty \\leq \\frac { C _ { 6 } ( b - a ) } { 2 } \\left ( \\sum _ { i = 1 } ^ \\rho | w _ i | \\right ) \\| v _ 2 - v _ 1 \\| _ \\infty ^ 2 . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} I _ { \\nu } \\left ( { \\nu z } \\right ) = \\frac { \\nu ^ { \\nu } } { e ^ { \\nu } \\Gamma \\left ( { \\nu + 1 } \\right ) \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 4 } } \\exp \\left \\{ { \\nu \\xi + \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { \\tilde { { E } } _ { s } \\left ( p \\right ) - k _ { s } } { \\nu ^ { s } } } } \\right \\} \\left \\{ { 1 + \\eta _ { n , 1 } \\left ( { \\nu , z } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} & \\nabla u ^ \\# ( x ) = u ^ { * ' } ( \\kappa _ n ( F ^ o ( x ) ) ^ n ) n \\kappa _ n ( F ^ o ( x ) ) ^ { n - 1 } \\nabla F ^ o ( x ) ; \\\\ & F ( \\nabla u ^ \\# ( x ) ) = - u ^ { * ' } ( \\kappa _ n ( F ^ o ( x ) ) ^ n ) n \\kappa _ n ( F ^ o ( x ) ) ^ { n - 1 } ; \\\\ & \\nabla F ( \\nabla u ^ \\# ( x ) ) = \\frac { x } { F ^ o ( x ) } . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\{ t : \\Delta N ^ { k , i } _ t \\neq 0 , i = 1 , 2 , 3 \\} \\subset \\{ \\tau _ { n _ { k - 1 } + 1 } , \\ldots , \\tau _ { n _ { k } } \\} , \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} a & = b = 0 \\\\ a ' & = b ' = 1 . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\partial _ { t } \\theta ^ { \\varepsilon } + \\mathcal { A } _ { \\kappa } ^ { \\varepsilon } \\theta ^ { \\varepsilon } = \\tau ^ { \\varepsilon } \\sum _ { i = 1 } ^ { N } \\nabla ^ { \\delta } u _ { i } ^ { \\varepsilon } \\cdot \\nabla \\theta ^ { \\varepsilon } \\quad \\mbox { i n } \\ ; Q _ { T } ^ { \\varepsilon } , \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} v ( x , t ) = \\min ( 0 , | \\lambda _ 1 | t - ( d + \\alpha ) \\log ( | x | ) ) . \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} G _ \\lambda ( \\zeta _ i + \\mu _ i z , \\zeta _ j ) = G _ \\lambda ( \\zeta _ i , \\zeta _ j ) + \\mu _ i \\ , { \\bf c } \\cdot z + \\theta _ 2 ( \\zeta _ i + \\mu _ i z , \\zeta _ j ) , \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} W ^ \\rho _ u ( \\rho ^ * ( y ) \\phi ) ( x ) = & \\overline { w } z W ^ { \\rho _ \\sigma } _ u ( \\rho _ \\sigma ^ * ( y , w ) \\phi ) ( x , z ) = \\overline { w } z \\ell _ { ( y , w ) } W ^ { \\rho _ \\sigma } _ u ( \\phi ) ( x , z ) \\\\ = & \\overline { \\sigma ( y , y ^ { - 1 } ) } \\sigma ( y ^ { - 1 } , x ) \\ell _ y W ^ \\rho _ u ( \\phi ) ( x ) \\\\ = & \\ell ^ \\sigma _ y W ^ \\rho _ u ( \\phi ) ( x ) . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} R ( 0 ) \\xrightarrow { ( g r ) ^ { \\otimes m } } ( B ^ { 0 } ) ^ { \\otimes m } \\hookrightarrow ( B ^ { \\otimes m } ) ^ { 0 } \\\\ \\intertext { w h i c h f a c t o r s a s } \\\\ R ( 0 ) \\xrightarrow { r ^ { \\circ m } = 0 } R ( 0 ) \\xrightarrow { g ^ { \\otimes m } } ( B ^ { 0 } ) ^ { \\otimes m } \\hookrightarrow ( B ^ { \\otimes m } ) ^ { 0 } . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\partial _ { t } \\omega _ { s } ^ { \\nu } & = \\nu \\partial _ { y y } \\omega _ { s } ^ { \\nu } + P _ { 0 } \\left ( U _ { n } ^ { \\nu } \\cdot \\nabla \\omega _ { n } ^ { \\nu } \\right ) \\\\ & = \\nu \\partial _ { y y } \\omega _ { s } ^ { \\nu } + P _ { 0 } \\left ( U _ { n 1 } ^ { \\nu } \\cdot \\nabla \\left ( \\omega _ { n 2 } ^ { \\nu } - \\psi _ { n 2 } ^ { \\nu } \\right ) + U _ { n 2 } ^ { \\nu } \\cdot \\nabla \\omega _ { n 2 } ^ { \\nu } \\right ) . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\varphi _ 4 ' & = \\int _ { \\gamma _ 4 } u ^ { - \\frac { 2 } { 3 } } ( 1 - u ) ^ { - \\frac { 1 } { 3 } } ( 1 - t u ) ^ { - \\frac { 1 } { 3 } } d u , \\\\ \\gamma _ 4 & = \\{ ( w , u ) \\in C \\mid u \\in ( 0 , { 1 - x _ 2 } ) , w \\in \\bold e ( \\dfrac { 1 } { 3 } ) \\} . \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} I _ { m - 1 } ( t ) & = \\frac { \\alpha } { m - \\alpha } t ^ { m - \\alpha } \\int _ 0 ^ \\infty L _ 1 ( t / w ) \\left ( 1 - e ^ { - w } \\right ) ^ { m - 1 } w ^ { \\alpha - m } d w \\\\ & = \\frac { \\alpha } { m - \\alpha } t ^ { m - \\alpha } \\int _ 0 ^ \\infty L _ 1 ( t z ) \\left ( 1 - e ^ { - \\frac { 1 } { z } } \\right ) ^ { m - 1 } z ^ { m - \\alpha - 2 } d z . \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} 1 = | x _ r ^ * ( u _ r ) | \\stackrel { \\mathrm { ( I I ) } } { = } | y _ 1 ^ * ( x _ r ^ * ( u _ r ) F e ) | \\stackrel { \\mathrm { ( I I I ) } } { = } | y _ 1 ^ * ( S ( u _ r ) ) | \\leq \\norm { S ( u _ r ) } \\leq 1 . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } ( w _ { 0 } ( x , t , y , s ) - \\nabla u ( x , t ) ) \\cdot v ( x , y ) c _ { 1 } ( t ) c _ { 2 } ( s ) d y d s d x d t = 0 \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} q ( w _ 1 , w _ 2 ) = p ( w _ 1 ( \\tilde { y } + x ) + w _ 2 ( \\tilde { y } - x ) ) \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} ( 3 u - 1 ) ( 3 u + 1 ) = 2 ^ { 2 b + 1 } 9 u ^ 2 + 1 = 2 ^ { 2 b + 1 } . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} \\tilde \\kappa = \\kappa - M \\gamma ^ { 1 / 3 } ( 1 + \\gamma ) ^ { - 4 / 3 } > 0 \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { K ^ i _ { m _ 0 } + 1 } | \\zeta ^ i _ n | + R ^ i _ n < r \\ , . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\mathcal { B } _ { u u } ^ { i } = U _ { u u } ^ { i } \\begin{pmatrix} \\alpha _ { u u } ^ { 1 i } & 0 & \\dots & 0 \\\\ 0 & \\alpha _ { u u } ^ { 2 i } & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & \\alpha _ { u u } ^ { N i } \\end{pmatrix} \\left ( U _ { u u } ^ { i } \\right ) ^ { - 1 } , \\quad \\mbox { f o r a l l $ u , i = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\frac { N _ { 2 d } ^ { \\mathrm { W e y l } } ( X ) } { N _ { 2 d } ^ { \\mathrm { c m } } ( X ) } = 1 + O _ { d , \\epsilon } ( X ^ { - C _ 3 ( \\delta _ d , \\delta ^ { \\prime } ) + \\epsilon } ) , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ p } { \\rm I m } \\left ( \\varphi _ d \\right ) = n - 1 , \\ \\mbox { t h a t i s } , \\ \\# { \\rm I m } \\left ( \\varphi _ d \\right ) = p ^ { n - 1 } . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} [ M ] _ t = \\sum _ { n \\geq 1 } [ \\langle M , h _ n \\rangle ] _ t . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\dim D ^ { ( p - 1 , p - 2 , 2 ) } & = \\dim ( p - 1 , p - 2 , 2 ) - \\dim D ^ { ( p , p - 2 , 1 ) } \\cr & = \\dim ( p - 1 , p - 2 , 2 ) - \\dim ( p , p - 2 , 1 ) + \\dim ( 2 p - 3 , 1 , 1 ) . \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\beta \\le B _ h \\\\ k _ { h \\beta } = h } } r _ { h \\beta } \\prod _ { i = 1 } ^ h x _ { v ( h , \\beta , i ) } \\prod _ { j = 1 } ^ { \\tilde k _ { h \\beta } } y _ { w ( h , \\beta , j ) } = \\bigg ( \\sum _ { i = 0 } ^ { \\rho ( X ) } x _ i Q _ i ( y _ 0 , \\ldots , y _ { \\rho ( X ) } ) \\bigg ) ^ h . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} s _ { i j } ^ { \\circ m } ( u ) = s _ { i j } ( u + \\tfrac { m } { 2 } ) + \\tfrac { \\delta _ { i j } } { 2 u } \\sum _ { a = n - m + 1 } ^ n s _ { a a } ( u + \\tfrac { m } { 2 } ) . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\omega _ { t } = U ^ { \\prime \\prime } \\left ( y \\right ) \\partial _ { x } \\left ( \\frac { \\omega } { K _ { 2 } \\left ( y \\right ) } - \\psi \\right ) = J L \\omega , \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\lvert \\mathrm { I m } \\log \\Phi ' ( z ) \\rvert & \\leqslant \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ { + \\infty } \\frac { y \\lvert \\mathrm { I m } \\log \\Phi ' ( t ) \\rvert \\ , \\mathrm { d } t } { ( t - x ) ^ 2 + y ^ 2 } \\\\ & \\leqslant \\frac { \\theta _ 0 } { \\pi } \\int _ { - \\infty } ^ { + \\infty } \\frac { y \\ , \\mathrm { d } t } { ( t - x ) ^ 2 + y ^ 2 } = \\theta _ 0 , \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\Gamma _ M = M - \\Gamma . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( T _ { \\alpha } \\right ) & : = \\left \\{ x \\in D \\ , \\big | \\lim \\limits _ { t \\rightarrow 0 + } - t ^ { 1 - 2 \\alpha } U ' ( t ) x \\right \\} , \\\\ T _ { \\alpha } x & : = \\lim \\limits _ { t \\rightarrow 0 + } - t ^ { 1 - 2 \\alpha } U ' ( t ) x . \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} A ( p ) : = \\textstyle { \\frac { 1 } { 1 0 0 } } { \\footnotesize \\begin{pmatrix} 1 2 8 p ^ 2 - 7 2 p - 3 2 & 2 9 5 p ^ 2 - 1 9 9 p + 4 & 1 6 5 p ^ 2 - 2 3 4 p + 4 6 \\\\ - 8 2 p ^ 2 - 5 9 p + 2 7 0 & - 2 6 6 p ^ 2 + 1 4 4 p - 7 3 & - 1 4 7 p ^ 2 - 2 1 0 p + 2 8 6 \\\\ 7 0 p ^ 2 + 2 9 6 p - 8 0 & 4 3 p ^ 2 + 9 6 p + 8 & 1 5 p ^ 2 + 1 4 6 p - 2 5 1 \\\\ \\end{pmatrix} } \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} I _ 1 ( t ) = | t | S ^ { 1 / 2 } P , I _ 2 ( t ) = t | t | Z S ^ { 1 / 2 } P , I _ 3 ( t ) = t | t | S ^ { - 1 / 2 } P K . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} J \\Delta J = \\Delta ^ { - 1 } . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } \\mathcal { C } X ) & = g _ { 1 } ( V , \\hat { \\nabla } _ { U } \\phi \\mathcal { B } X + \\mathcal { T } _ { U } \\omega \\mathcal { B } X ) + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) , \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} z _ { n \\ell } = \\sum ^ { \\ell - 1 } _ { i = 1 } q _ { \\ell i } . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\# T & \\geq \\left ( 1 - \\frac { 1 } { q } \\right ) \\cdot ( q - 1 ) q ^ d \\cdot ( q - 1 ) q ^ { d - 1 } \\\\ & \\geq \\frac { ( q - 1 ) ^ 3 } { q ^ 2 } q ^ { 4 \\log _ q n } = \\frac { ( q - 1 ) ^ 3 } { q ^ 2 } n ^ 4 . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} \\hat u ( x _ 1 , x ' ) = u ( | x _ 1 | , x ' ) ~ \\ ! . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\sum _ { n \\leq R } d ( n ) = \\sum _ { n \\leq R } \\sum _ { d \\mid n } 1 = \\sum _ { d \\leq R } \\Big \\lfloor \\frac { R } { d } \\Big \\rfloor , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\mu _ 0 = 1 \\mbox { a n d } \\mu _ i = \\frac { \\mathbb { P } ( X _ 1 > i ) } { \\prod _ { j = 1 } ^ { i } [ 1 - \\mathbb { P } ( X _ 1 > j ) ] } \\mbox { f o r } i \\geq 1 . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { n = 1 } ^ N G ( S ^ n y ) \\prod _ { i = 1 } ^ k R ^ { i n } \\tilde { f } _ i ( x ) = 0 . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\sqrt { g } \\Delta _ { g } \\eta ^ { \\alpha } = \\sqrt { g } g ^ { i j } \\Pi _ { \\mu } ^ { \\alpha } \\partial _ { i j } ^ 2 \\eta ^ { \\mu } . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} \\varepsilon \\int _ { \\Gamma ^ { \\varepsilon } } \\partial _ { t } v _ { i } ^ { 0 } \\psi _ { i } d S _ { \\varepsilon } = \\varepsilon \\int _ { \\Gamma ^ { \\varepsilon } } \\left ( A _ { i } u _ { i } ^ { 0 } - B _ { i } v _ { i } ^ { 0 } \\right ) \\psi _ { i } d S _ { \\varepsilon } . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\hat { R } ( x ) = R _ { 1 2 } ( x ) - \\frac { [ R _ { 1 1 } R _ { 1 2 } ] } { [ R _ { 1 1 } ^ 2 ] } R _ { 1 1 } ( x ) . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} T b _ { \\ell m } ( x ) = \\int _ { K _ { \\ell } } \\left ( \\partial _ { i } \\partial _ { j } G ( x - y ) - \\partial _ { i } \\partial _ { j } G ( x - \\bar { y } _ { \\ell } ) \\right ) b _ { \\ell m } ( y ) d y \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} 1 + ( b _ i - z + \\omega _ B ) m _ { \\mu _ A \\boxplus \\mu _ B } = m _ { \\mu _ A \\boxplus \\mu _ B } \\Big ( \\frac { 1 } { m _ { \\mu _ A \\boxplus \\mu _ B } } + b _ i - z + \\omega _ B \\Big ) = m _ { \\mu _ A \\boxplus \\mu _ B } ( b _ i - \\omega _ A ) \\ , . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\norm { T _ i f \\cdot g _ \\frac { 1 } { 2 } } _ { \\ell ^ 1 ( \\N ^ * ) } \\leq \\begin{cases} \\frac { q } { ( q - 1 ) ^ 2 } q ^ { - ( \\abs { i } - 1 ) } & i < 0 , \\\\ \\frac { q ( q + 1 ) } { ( q - 1 ) ^ 2 } & 0 \\leq i < d , \\\\ \\frac { q } { ( q - 1 ) ^ 2 } q ^ { - ( i - d ) } & d \\leq i , \\end{cases} \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} m _ k = \\int _ { - 1 } ^ 1 f ( t ) ( G ( t ) ) ^ k d t = 0 , \\ , \\ , k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} \\vert u ( t _ { 1 } , x _ { 1 } ) - u ( t _ { 2 } , x _ { 2 } ) \\vert & \\leq \\underset { Q ( ( t _ { 2 } , x _ { 2 } ) , ( 1 - \\delta ) ^ { K } R ) } { O s c } u \\leq \\theta ^ { K - 1 } \\underset { Q ( ( t _ { 2 } , x _ { 2 } ) , ( 1 - \\delta ) R ) } { O s c } u \\\\ & \\leq \\theta ^ { K - 1 } \\underset { Q ( ( t _ { 0 } , x _ { 0 } ) , R ) } { O s c } u = \\theta ^ { - 2 } ( \\theta ^ { K + 1 } ) \\underset { Q ( ( t _ { 0 } , x _ { 0 } ) , R ) } { O s c } u . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\varepsilon } X _ { n } \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } + \\int _ { \\Omega _ \\varepsilon } \\phi \\ , \\partial _ { ( \\zeta ^ \\prime ) _ n } \\bigl [ w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\bigr ] = 0 \\quad j = 1 , 2 , 3 , 4 , \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} S _ { \\lambda f , E } ( x ) \\cdot \\mathcal { U } ( x ) = e ^ { 2 \\pi i \\theta } \\mathcal { U } ( x + \\alpha ) . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\epsilon ^ { \\frac { p ^ m - 1 } { 2 } } = 1 . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} | \\mathcal { I } _ { \\epsilon , \\alpha } | = \\left | \\int _ { | y | < r } G _ \\epsilon ( \\bar { x } , y ) ( 1 - h ( y ) ) K d y \\right | \\lesssim \\left ( \\sup _ { | z | < r } h ( z ) \\right ) \\int _ { | y | < r } O ( | y | ) K d y \\lesssim \\left ( \\sup _ { | z | < r } h ( z ) \\right ) \\frac { | \\bar { x } | ^ { 1 - 1 / \\epsilon } } { \\epsilon } r ^ { 1 - \\alpha } \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} I _ j & = \\left ( \\frac { 1 } { \\lambda _ 1 \\lambda _ 2 } \\right ) \\left ( \\frac { \\lambda _ j ^ { - 1 } v _ j } { ( T ) } \\right ) \\\\ & = \\left ( \\frac { 1 } { \\lambda _ 1 \\lambda _ 2 } \\right ) \\left ( \\frac { \\lambda _ j ^ { - 1 } } { ( \\lambda _ 1 ^ { - 1 } \\lambda _ 2 ^ { - 1 } / 2 ) } \\right ) \\\\ & = \\frac { 2 } { \\lambda _ j } . \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} K i G ( f g ) \\alpha _ m & = K i G ( f ) K i G ( g ) \\alpha _ m \\\\ & = K i G ( f ) \\alpha _ k \\alpha _ k ^ \\dagger K i G ( g ) \\alpha _ m \\\\ & = \\alpha _ n \\alpha _ n ^ \\dagger K i G ( f ) \\alpha _ k \\alpha _ k ^ \\dagger K i G ( g ) \\alpha _ m \\\\ & = \\alpha _ n \\alpha _ n ^ \\dagger K i G ( f ) K i G ( g ) \\alpha _ m \\\\ & = \\alpha _ n \\alpha _ n ^ \\dagger K i G ( f g ) \\alpha _ m . \\\\ \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} x ( t , p ) = \\sum _ { i = 1 } ^ { \\infty } v _ i ( t ) \\Phi _ i ( p ) . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\omega _ { t } & = - e ^ { - \\nu t } \\sin y \\partial _ { x } ( \\omega - \\psi ) - ( e ^ { - \\nu t } - 1 ) \\sin y \\partial _ { x } ( \\omega ^ { 0 } - \\psi ^ { 0 } ) + \\nu \\bigtriangleup \\omega _ { n } ^ { \\nu } \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + u _ { s } ^ { \\nu } \\partial _ { x } \\omega _ { n } ^ { \\nu } + v _ { n } ^ { \\nu } \\partial _ { y } \\omega _ { s } ^ { \\nu } + P _ { \\neq 0 } \\left ( U _ { n } ^ { \\nu } \\cdot \\nabla \\omega _ { n } ^ { \\nu } \\right ) . \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} p _ 1 = \\frac { \\log n - \\log \\left ( f + 1 \\right ) - h } { n } = \\frac { \\log n / 2 + \\log \\log n - \\left ( 1 + o \\left ( 1 \\right ) \\right ) \\log g } { n } , \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\mu _ { \\alpha } \\boxplus \\mu _ \\beta \\big ( ( - \\infty , \\gamma _ j ] \\big ) = \\frac { j } { N } . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} & 2 m _ { 2 } \\partial ( a _ { 2 } e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } ) \\wedge \\bar \\partial ( \\varphi - \\eta ) \\\\ & = 2 m _ { 2 } ( e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\partial a _ { 2 } \\wedge \\bar \\partial ( \\varphi - \\eta ) + 2 m _ { 2 } a _ { 2 } e ^ { 2 m _ { 2 } ( \\varphi - \\eta ) } \\partial ( \\varphi - \\eta ) \\wedge \\bar \\partial ( \\varphi - \\eta ) ) . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} x _ 0 : = \\frac { 1 } { 1 - \\tau } ( E ^ { - 1 } w - E ^ { - 1 } c ) . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} C = \\nu ( g ) \\mbox { a n d } C ' = \\nu \\left ( \\frac { \\partial U } { \\partial y } \\frac { \\partial V } { \\partial y } \\right ) . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} J _ \\lambda ( u ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\vert \\nabla u \\vert ^ 2 - \\frac { \\lambda } { 2 } \\int _ { \\Omega } u ^ 2 - \\frac { 1 } { p + 1 } \\int _ { \\Omega } | u | ^ { p + 1 } . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} T _ { g _ 1 ^ H } \\phi ( g _ 2 ) & = \\int _ G \\phi ( y ) \\ > K _ { g _ 1 ^ H } ( g _ 2 , d y ) = \\int _ H \\phi ( g _ 2 \\cdot h ( g _ 1 ) ) \\ > d \\omega _ H ( h ) \\\\ & = \\int _ H \\phi ( g _ 2 ) \\cdot \\phi ( h ( g _ 1 ) ) \\ > d \\omega _ H ( h ) = \\phi ( g _ 2 ) \\cdot \\alpha ( g _ 1 ^ H ) , \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} u _ i ( T , x ) = u _ { i } ^ { 0 } ( x ) \\ , , u _ { i t } ( T , x ) = u _ { i } ^ { 1 } ( x ) \\ , , x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} V = \\left ( \\hat { v } ( t _ 0 ) , \\hat { v } ( t _ 1 ) , \\ldots , \\hat { v } ( t _ { \\ell - 1 } ) \\right ) \\in \\real ^ { m n \\times \\ell } . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} z _ i ( T , \\cdot ) = z _ { i } ^ { 0 } \\ , , z _ { i t } ( T , \\cdot ) = z _ { i } ^ { 1 } \\ , , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} \\sum _ { p + q = i } h ^ p ( Z , \\Omega ^ q ( \\log D _ \\infty , \\mathrm { r e l } \\ , f ^ { - 1 } ( \\epsilon _ 0 ) ) ) & = \\dim H ^ { i } ( Y , f ^ { - 1 } ( \\epsilon _ 0 ) ) \\\\ & = \\dim \\mathbb { H } ^ { i } ( Z , \\Omega ^ \\bullet _ Z ( \\log D _ \\infty , f ) ) \\\\ & = \\sum _ { p + q = i } \\dim H ^ p ( Z , \\Omega ^ q ( \\log D _ \\infty , f ) ) . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} \\mathbf { x } ^ \\ast A \\mathbf { x } & = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n A ( Y _ i , Y _ j ) \\overline { x _ i } x _ j \\\\ & = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\sum _ { y \\in Y } I _ y ( Y _ i ) I _ y ( Y _ j ) \\overline { x _ i } x _ j \\\\ & = \\sum _ { y \\in Y } \\left | \\sum _ i I _ y ( Y _ i ) x _ i \\right | ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\langle Q x , x \\rangle + \\langle q , x \\rangle + \\theta \\le f ( x ) , \\forall x \\in X . \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} P _ n ( x ) = \\log 2 \\cdot \\omega ( \\phi ( n ) ) + \\frac { 1 } { 4 } \\sum _ { q \\leq X } \\omega _ q ( n ) ^ 2 \\Lambda ( q ) , \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\bar \\Lambda = \\bar \\Lambda ^ 0 + \\widehat { \\Lambda } , | \\widehat { \\Lambda } | \\leq \\varepsilon ^ { 1 - \\sigma } . \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} g _ { f , p } '' ( y ) = g _ { f , p } '' ( \\tau ^ { - 1 } y - \\tau ^ { - 1 } x _ 1 ) = g _ { f , p } ( \\alpha y + y _ 1 ) \\forall y \\in X , \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} & F ^ 2 H _ { d R } ^ 2 ( \\widetilde { X } ) ( \\chi ) = \\lambda ^ * ( H ^ { 1 0 } ( E ) ( \\chi ) \\otimes H ^ { 1 0 } ( C ) ( \\chi ) ) , \\\\ & F ^ 1 H _ { d R } ^ 2 ( \\widetilde { X } ) ( \\overline { \\chi } ) = \\lambda ^ * ( H ^ { 0 1 } ( E ) ( \\overline { \\chi } ) \\otimes H ^ { 1 0 } ( C ) ( \\overline { \\chi } ) ) . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} A _ k = U _ k \\Sigma _ k V _ k ^ T \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} E \\left ( r _ { i j } ^ 2 - \\frac { 1 } { n - 1 } \\right ) \\left ( r _ { s t } ^ 2 - \\frac { 1 } { n - 1 } \\right ) = \\left \\{ \\begin{array} { l l } 0 , & \\hbox { i f } ( i , j ) \\ne ( s , t ) , \\\\ \\frac { 3 } { ( n - 1 ) ( n + 1 ) } - \\frac { 1 } { ( n - 1 ) ^ 2 } = \\frac { 2 n - 4 } { ( n - 1 ) ( n + 1 ) } , & \\hbox { i f } ( i , j ) \\ne ( s , t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} < X _ { \\varepsilon } ( p ) > + T _ p D _ { \\alpha } ' = T _ p P \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\Delta ^ 4 + a \\Delta ^ 3 + \\Delta ^ 2 + a \\Delta + \\frac { 1 } { a } = 0 . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\langle \\Delta ^ { ( n , \\sigma ) } _ { 2 } ( P ) ^ { - 1 } \\xi , \\xi \\rangle & = \\left \\langle \\left ( \\sum _ { i = 1 } ^ { n } A _ { i } ^ { - 1 } \\otimes E _ { i i } \\right ) \\left ( \\sum _ { j = 1 } ^ { n } z _ { j } \\otimes e _ { j } \\right ) , \\sum _ { j = 1 } ^ { n } z _ { j } \\otimes e _ { j } \\right \\rangle \\\\ & = \\sum _ { i = 1 } ^ { n } \\langle A _ { i } ^ { - 1 } z _ i , z _ i \\rangle = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a } \\\\ & = \\frac { n } { a } \\leq 1 . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} f ( n , k , c ) : = \\binom { c - k + 1 } { 2 } + k \\cdot ( n - c + k - 1 ) . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\| ( \\tilde W - W ) \\| _ { C ^ { m + 2 , \\alpha } ( \\overline { B _ { 2 R } } \\setminus U ) } + \\| ( \\tilde W - W ) \\| _ { C ^ { m + 2 , \\alpha } ( \\overline U ) } & \\le C \\| \\partial _ { \\nu , \\ , { \\rm o u t } } \\tilde W \\| _ { C ^ { m + 1 , \\alpha } ( \\partial U ) } \\\\ & \\le C \\| \\tilde W \\| _ { C ^ { m + 2 , \\alpha } ( \\overline { B _ { 2 R } } \\setminus U ) } \\le C \\| f \\| _ { C ^ { m , \\alpha } ( \\overline { B _ 4 } ) } . \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ N { \\frac { ( t x _ i y ; q ) _ { \\infty } } { ( x _ i y ; q ) _ { \\infty } } } = \\sum _ { r \\geq 0 } g _ r ( x _ 1 , \\dots , x _ N ; q , t ) y ^ r . \\end{gather*}"}
-{"id": "1378.png", "formula": "\\begin{align*} | \\log \\gamma _ 2 | & < \\frac { B _ 2 + | \\log \\gamma _ 1 | } { 2 h } < \\frac { G ( B _ 2 ) \\log \\alpha _ 2 + G ( | \\log \\gamma _ 1 | ) \\cdot \\log \\alpha _ 2 \\log c } { 2 h } \\\\ & < \\frac { \\left ( \\frac { G ( B _ 2 ) } { \\log 1 0 ^ 5 } + G ( | \\log \\gamma _ 1 | ) \\right ) \\log \\alpha _ 2 \\log c } { 2 G ( h ) \\cdot \\log \\alpha _ 2 \\log c } = \\frac { \\frac { G ( B _ 2 ) } { \\log 1 0 ^ 5 } + G ( | \\log \\gamma _ 1 | ) } { 2 G ( h ) } = : G ( | \\log \\gamma _ 2 | ) . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} { \\mathcal M } _ { k _ 1 + 1 } ( \\beta _ 1 ) \\ , \\ , { } _ { { \\rm e v } _ i } \\times _ { { \\rm e v } _ 0 } { \\mathcal M } _ { k _ 2 + 1 } ( \\beta _ 2 ) = ( - 1 ) ^ { \\delta } { \\mathcal M } _ { k _ 1 + 1 } ( \\beta _ 1 ) \\ , \\ , { } _ { { \\rm e v } _ 1 } \\times _ { { \\rm e v } _ 0 } { \\mathcal M } _ { k _ 2 + 1 } ( \\beta _ 2 ) \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} { \\overline { \\zeta _ n } ( z ) = \\int _ M \\zeta _ n \\mathrm { d } \\mu + O ( 1 ) \\Big ( \\sum _ { { l } = 1 } ^ m ( | A _ { n , { l } } | + e ^ { - \\frac { \\lambda _ { n , j } ^ { ( 1 ) } } { 2 } } ) ( \\frac { e ^ { \\frac { \\lambda _ { n , j } ^ { ( 1 ) } } { 2 } } } { | z | } + 1 ) \\Big ) , } \\ \\ \\textrm { a n d t h u s } \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} f \\in D ( \\Lambda ^ { s } ) : = \\left \\{ f \\in L ^ 2 ( \\Omega ) : \\L ^ s f \\in L ^ 2 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} T ( a ( 1 + q ) , b ( 1 + q ) ) & = \\frac { 1 - ( a - b ) ( 1 + q ) - \\sqrt { 1 - 2 ( 1 + q ) ( a + b ) + ( 1 + q ) ^ 2 ( a - b ) ^ 2 } \\ , } 2 \\\\ & = \\frac { 1 - ( a - b ) - ( a - b ) q } 2 \\\\ & \\qquad - \\frac { \\sqrt { 1 - 2 ( a + b ) + ( a - b ) ^ 2 - 2 q ( a + b - ( a - b ) ^ 2 ) + q ^ 2 ( a - b ) ^ 2 } \\ , } 2 . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\frac { \\| \\bar { x } _ { L , k _ 0 } - x _ { t r u e } \\| } { \\| x _ { t r u e } \\| } = \\frac { \\| x _ { L , k _ 0 } - x _ { t r u e } \\| } { \\| x _ { t r u e } \\| } \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\norm { u _ { s + 1 } } _ { x , \\delta , s + 1 } = \\sum _ { n = 0 } ^ { + \\infty } \\norm { A ^ { n } ( x ) \\tilde { u } } e ^ { - \\tilde { \\lambda } n } . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} | { \\bar { B } } ^ n _ k ( t ) - { \\bar { B } } ^ n _ k ( t - ) | \\le \\frac { 1 } { n } , \\ ; | { \\bar { \\eta } } ^ n ( t ) - { \\bar { \\eta } } ^ n ( t - ) | \\le \\frac { 2 } { n } , \\ ; | { \\bar { Y } } ^ n _ k ( t ) - { \\bar { Y } } ^ n _ k ( t - ) | \\le \\frac { K } { n } + \\sum _ { j = K + 1 } ^ \\infty \\frac { j n _ j } { n } \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} a _ { k , k } = \\left ( \\prod _ { j = 0 } ^ { k - 1 } a _ j \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} \\sum _ { q = 0 } ^ { \\infty } \\frac { 1 } { n ^ { \\ell _ q } } = \\infty , \\quad \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\theta _ a ( t ) : = \\begin{cases} a t & t \\in [ 0 , 1 ] , \\\\ a & t \\in [ 1 , \\infty ] . \\end{cases} \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} D ( x ) = \\log 2 \\cdot \\mu ( \\omega \\circ \\phi ) + \\frac { 1 } { 4 } \\sum _ { q \\leq X } \\mu ( \\omega _ q ) ^ 2 \\Lambda ( q ) \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { ( 1 - 2 x ) \\big ( ( 1 - 5 x + 9 x ^ 2 - 6 x ^ 3 ) \\sqrt { 1 - 4 x } - ( 1 - 9 x + 2 9 x ^ 2 - 3 8 x ^ 3 + 1 8 x ^ 4 ) \\big ) } { 2 ( 1 - x ) ^ 2 x ( 1 - 7 x + 1 4 x ^ 2 - 9 x ^ 3 ) } \\ , . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\frac 1 i \\partial _ x m _ 1 ( \\lambda + ) = & ~ i \\lambda e ^ { i \\lambda x } \\int _ { - \\infty } ^ x e ^ { - i \\lambda y } u ( y ) m _ 1 ( y , \\lambda + ) ~ d y + u m _ 1 ( \\lambda + ) \\\\ & - \\frac 1 i \\partial _ x \\widetilde { G } _ { \\lambda } * ( u m _ 1 ( \\lambda + ) ) . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} { \\rm d i v } ( b ^ 0 u _ 0 ( t ) ) = \\int _ 0 ^ t \\int _ 0 ^ s { \\rm d i v } f ( r ) d r d s + t { \\rm d i v } ( b ^ 0 g _ 1 ) + { \\rm d i v } ( b ^ 0 g _ 0 ) \\in L ^ \\infty ( 0 , T ; L ^ 2 ( D ) ) . \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\eta ( x ) : = \\begin{cases} \\dfrac { 1 } { 2 } d ( x ) ^ 2 , & K _ M \\leq 0 , \\\\ 1 - \\cos \\sqrt { a } d ( x ) , & 0 < K _ M \\leq a . \\end{cases} \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\phi ( x ) \\cdot \\alpha ( h _ 1 * h _ 2 ) & = \\phi ( x ) \\cdot \\int _ D \\alpha ( h ) \\ > d ( \\delta _ { h _ 1 } * \\delta _ { h _ 1 } ) ( h ) = \\int _ D T _ h \\phi ( x ) \\ > d ( \\delta _ { h _ 1 } * \\delta _ { h _ 1 } ) ( h ) \\\\ & = T _ { h _ 1 } \\circ T _ { h _ 1 } \\phi ( x ) = \\phi ( x ) \\cdot \\alpha ( h _ 1 ) \\alpha ( h _ 2 ) \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} A _ { \\omega , \\omega ' , t } \\phi = h _ \\omega \\ ( \\chi _ { \\overrightarrow { \\omega } , t } e ^ { g _ { m , t } } \\theta _ { \\omega ' } \\ ) \\circ \\ ( \\left . T _ t ^ m \\right | _ { W _ { \\overrightarrow { \\omega } , t } } \\ ) ^ { - 1 } \\circ \\pi . \\ ( \\phi \\circ \\kappa _ { \\omega ' } \\circ \\ ( \\left . T _ t ^ m \\right | _ { W _ { \\overrightarrow { \\omega } , t } } \\ ) ^ { - 1 } \\circ \\pi \\ ) . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} N _ a = \\frac { a _ 1 } { ( T _ 1 ) } , \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} C _ { x y } = \\frac { E [ ( x - \\mu _ x ) ( y - \\mu _ y ) ] } { \\sigma _ x \\sigma _ y } , \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\| w f \\| _ { L ^ 2 } ^ 2 + \\| f _ x \\| _ { L ^ 2 } ^ 2 = \\| L _ z ^ - f \\| _ { L ^ 2 } ^ 2 + 2 \\Im \\int w f \\cdot \\bar f _ x . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} C _ { i ( u ) u } ^ { k } \\overline { \\left ( C _ { i \\left ( u ' \\right ) u ' } ^ { k ' } \\right ) ^ { t } } = \\beta ^ { 1 i } _ { u u } \\overline { \\beta ^ { 1 i ' } _ { u ' u ' } } , \\quad \\mbox { f o r a l l $ k , k ' , u , u ' = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} & \\left | f l \\left ( B ^ { ( k + 1 ) } \\right ) - B ^ { ( k + 1 ) } \\right | \\le \\left | f l \\left ( B ^ { ( 1 ) } \\right ) - B ^ { ( 1 ) } \\right | \\begin{pmatrix} 0 \\\\ I \\end{pmatrix} ( 1 + u ) ^ { k } \\\\ & + u \\sum _ { i = 1 } ^ { k } \\left | B ^ { ( i ) } \\right | \\begin{pmatrix} 0 \\\\ I \\end{pmatrix} ( 1 + u ) ^ { k - i } + 3 u \\sum _ { i = 1 } ^ { k } \\left | B ^ { ( i ) } \\right | \\begin{pmatrix} I \\\\ 0 \\end{pmatrix} \\left | L _ i ^ T \\right | ( 1 + u ) ^ { k - i } + O ( u ^ 2 ) , \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} \\psi _ { s , F } ( \\pm v ) = F ^ { - 1 / 6 } { \\rm A i } \\left ( \\pm F ^ { 1 / 3 } v + F ^ { - 2 / 3 } s \\right ) , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\ell ( \\Delta ) - 4 = 2 4 - \\ell ( \\Delta ^ \\circ ) + \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) - \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | \\varphi | ^ 2 ~ d x - \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { \\infty } \\left ( \\widehat { u \\varphi } \\bar { \\hat \\varphi } ( 1 - \\chi ) \\right ) ' ~ d \\xi = 0 , \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} J _ { l } = U ^ { \\prime \\prime } \\left ( y \\right ) i \\alpha l , \\ \\ \\ \\ L _ { l } = \\frac { 1 } { K _ { 2 } \\left ( y \\right ) } - \\left ( - \\frac { d ^ { 2 } } { d y ^ { 2 } } + \\alpha ^ { 2 } l ^ { 2 } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} r \\mathtt { I } ' _ \\nu ( r ) - \\beta \\nu \\mathtt { I } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} X ^ * ( t ) = \\int _ 0 ^ t X ( s ) d s . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} | p ( z ) | \\leq \\exp ( \\Re ( \\sum _ { j = 1 } ^ { 2 } z _ j p _ j ( 0 ) ) + \\frac { 1 } { 2 } \\| z \\| _ { \\infty } ^ 2 ( | \\sum _ { j = 1 } ^ { 2 } p _ j ( 0 ) | ^ 2 - \\Re ( \\sum _ { j , k = 1 } ^ { 2 } p _ { j , k } ( 0 ) ) ) ) . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\sigma ( T _ 1 ) = \\sigma ( T _ 3 ) = \\sigma ( T _ 5 ) = + ; \\sigma ( T _ 2 ) = \\sigma ( T _ 4 ) = \\sigma ( T _ 6 ) = - , \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\left \\{ \\varphi = \\big ( \\varphi _ 0 ( z _ 0 , z _ 1 , z _ 2 ) , \\psi ( z _ 1 , z _ 2 ) \\big ) \\ , \\big \\vert \\ , \\varphi _ 0 \\in \\mathrm { P G L } ( 2 , \\mathbb { C } [ z _ 1 , z _ 2 ] ) , \\ , \\psi \\in \\mathrm { B i r } ( \\mathbb { P } ^ 2 _ \\mathbb { C } ) \\right \\} . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} u _ j ( A _ j ( 0 , 1 ) ) = \\frac { r _ j ^ { n - 2 } G ( X _ 0 , q _ j + r _ j A _ j ( 0 , 1 ) ) } { \\omega ( B ( q _ j , r _ j ) ) } = \\frac { r _ j ^ { n - 2 } G ( X _ 0 , A ( q _ j , r _ j ) ) } { \\omega ( B ( q _ j , r _ j ) ) } \\sim 1 . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k ( - \\lambda _ i ) ^ 3 \\geq ( k + 3 l ) , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} & E A = c i r c s h i f t ( E A , [ y _ 1 + y _ 2 ~ ~ y _ 2 + y _ 3 ~ ~ y _ 1 + y _ 3 ] ) , \\\\ & E A = c i r c s h i f t ( E A , [ \\lfloor \\frac { y _ 1 } { 5 0 } \\rfloor ~ ~ \\lfloor \\frac { y _ 2 } { 5 0 } \\rfloor ~ ~ - \\lfloor \\frac { y _ 3 } { 5 0 } \\rfloor ] ) . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} a = \\left \\{ \\begin{array} { l l } 3 - 2 \\alpha / \\beta & \\mbox { f o r e q u a t i o n \\eqref { H A M } } \\\\ 1 - \\alpha / \\beta & \\mbox { f o r e q u a t i o n \\eqref { P A M } } \\end{array} \\right . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} \\sum _ { q \\geq 2 } \\frac { \\Lambda ( q ) } { \\phi ( q ) ^ 2 } = \\sum _ { p } \\sum _ { j = 1 } ^ \\infty \\frac { \\Lambda ( p ^ j ) } { \\phi ( p ^ j ) ^ 2 } = \\sum _ { p } \\frac { \\log p } { ( p - 1 ) ^ 2 } \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { ( p ^ { j - 1 } ) ^ 2 } = \\sum _ { p } \\frac { \\log p } { ( p - 1 ) ^ 2 } \\frac { p ^ 2 } { p ^ 2 - 1 } = 4 A _ 0 , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} y = y - \\gamma 0 = \\sum _ { i \\in J } \\alpha _ { i } x _ { i } - \\gamma \\sum _ { i \\in J } \\beta _ { i } x _ { i } = \\sum _ { i \\in J \\smallsetminus \\{ j \\} } \\delta _ { i } x _ { i } . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\begin{aligned} & [ L ^ 2 , H ^ 1 _ 0 ] _ \\gamma = D ( \\L ^ \\gamma ) , \\\\ & [ H ^ { - 1 } , L ^ 2 ] _ \\gamma = \\big ( [ H ^ 1 , L ^ 2 ] _ \\gamma \\big ) ^ * = \\big ( [ L ^ 2 , H ^ 1 ] _ { 1 - \\gamma } \\big ) ^ * = D ( \\L ^ { 1 - \\gamma } ) ^ * = D ( \\L ^ { \\gamma - 1 } ) . \\end{aligned} \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} \\| T _ a \\chi _ { k _ 0 } - T _ a \\chi _ { k _ 1 } \\| _ 1 \\geq | \\widehat { T _ a \\chi _ { k _ 0 } } ( 0 ) - \\widehat { T _ a \\chi _ { k _ 1 } } ( 0 ) | = | \\hat a ( - k _ 0 ) - \\hat a ( - k _ 1 ) | \\geq ( 1 - \\epsilon ) c . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} W _ { 2 } ( P _ { t } ( x , \\cdot ) , P _ { t } ( x ^ { \\prime } , \\cdot ) ) = | m _ { t } ( x ) - m _ { t } ( x ^ { \\prime } ) | = \\exp ( - \\lambda _ { \\mathrm { s i g } } t ) | x - x ^ { \\prime } | . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\gamma ' ( ( \\nabla ^ { \\gamma ' } _ X F _ \\pm ) Y , F _ \\pm Z ) = \\mp \\frac { 1 } { 2 } [ d \\psi ( X , F _ \\pm Y , F _ \\pm Z ) - d \\psi ( X , Y , Z ) ] , \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} ( \\underline X , \\underline A ) ^ I = \\{ ( x _ 1 , \\ldots , x _ m ) \\in X _ 1 \\times \\cdots \\times X _ m \\ ; | \\ ; x _ j \\in A _ j \\} . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} M = \\Gamma ( x _ 1 , t _ 1 ) > 0 . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } d ( K _ N ) \\left | \\frac { 7 a _ 2 ^ { 2 } ( K _ N ) - a _ 2 ( K _ N ) - 1 0 a _ 4 ( K _ N ) } { 3 2 a _ { 2 } ( K _ N ) v _ 3 ( K _ N ) } \\right | = 0 \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} R _ { 1 , \\epsilon } ( x ) = e ^ { \\epsilon \\mathfrak { Y } ( x ) } , \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} 0 < u _ n \\le u _ 0 = 1 \\mbox { a n d } u _ n ^ 2 \\le u _ { n - 1 } u _ { n + 1 } \\mbox { f o r a l l } n \\ge 1 , \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} \\eta \\nabla u ^ { p / 2 } = \\nabla ( \\eta u ^ { p / 2 } ) - u ^ { p / 2 } \\nabla \\eta , \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} ( n , 1 ) ^ { \\dagger } = n ! \\left ( 1 - \\frac { 2 } { n } + O \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} I _ 0 = 2 n ^ { 1 / 2 } \\int _ { T _ 0 } | v | ^ 2 d y . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{gather*} a _ n ^ { ( \\theta ) } = \\frac { 1 } { x _ 1 - x _ 2 } \\big ( T _ { q , x _ 2 } a _ n ^ { ( \\theta - 1 ) } - T _ { q , x _ 1 } a _ { n - 1 } ^ { ( \\theta - 1 ) } \\big ) , n = 1 , 2 , \\dots , \\theta - 1 , \\\\ a _ 0 ^ { ( \\theta ) } = \\frac { 1 } { \\prod \\limits _ { i = 0 } ^ { \\theta - 1 } { ( x _ 1 - q ^ i x _ 2 ) } } , a _ { \\theta } ^ { ( \\theta ) } = \\frac { 1 } { \\prod \\limits _ { i = 0 } ^ { \\theta - 1 } { ( x _ 2 - q ^ i x _ 1 ) } } . \\end{gather*}"}
-{"id": "7010.png", "formula": "\\begin{align*} | T _ { \\mu _ t } g ( x ) - g ( x ) | & = \\Bigl | \\int _ D \\int _ X ( g ( x ) - g ( y ) ) \\ > K _ h ( x , d y ) \\ > d \\mu _ t ( h ) \\Bigr | \\\\ & = \\epsilon + 2 \\| g \\| _ \\infty \\ > \\mu _ t ( D \\setminus U ) \\le 2 \\epsilon \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} h _ { t } ( \\exp { H } ) & \\asymp t ^ { - n / 2 } \\left ( \\underset { \\alpha \\in \\Sigma _ { 0 } ^ { + } } { \\prod } ( 1 + \\langle \\alpha , H \\rangle ) ( 1 + t + \\langle \\alpha , H \\rangle ) ^ { \\frac { m _ { \\alpha } + m _ { 2 \\alpha } } { 2 } - 1 } \\right ) \\\\ & \\times e ^ { - \\left \\Vert \\rho \\right \\Vert ^ { 2 } t - \\langle \\rho , H \\rangle - \\left \\Vert H \\right \\Vert ^ { 2 } / ( 4 t ) } , \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} [ w : x : y : z ] \\ = \\ [ { \\textstyle s + t + \\frac { 1 } { s } + \\frac { 1 } { t } \\ , : \\ s - \\frac { 1 } { s } \\ , : \\ t - \\frac { 1 } { t } \\ , : \\ 1 } ] \\ , . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\sigma : \\begin{cases} y _ 1 & = y _ 1 \\\\ u & = \\frac { y _ 2 } { y _ 1 } \\\\ y _ 3 & = y _ 3 \\end{cases} \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 { \\sqrt N } \\rho _ { ( 1 ) } ^ r ( x ) = \\sqrt { \\frac { ( 1 - \\alpha ) } { \\pi \\alpha } } \\frac { 1 } { 1 - x ^ 2 } \\ , \\chi _ { - \\sqrt { \\alpha } < x < \\sqrt { \\alpha } } , \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} Q & = 2 ^ { r - 5 } [ ( r - 2 ) ( r - 3 ) - ( ( r - 2 ) ( r - 3 ) + 8 ( r - 2 ) + 8 ) ] \\\\ & = - 2 ^ { r - 2 } ( r - 1 ) . \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} M ( \\alpha , \\beta , \\gamma , a _ { i , j } ) = \\left ( \\begin{array} { c | c c c c c } \\alpha & \\beta & \\gamma & 0 & \\ldots & 0 \\\\ \\hline \\\\ 0 & & & a _ { i , j } & & \\\\ \\vdots & & & & & \\\\ 0 & & & & & \\end{array} \\right ) \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} { 1 \\over \\Delta t } ( u _ { 0 , m + 1 } - u _ { 0 , m } ) - { \\partial u _ 0 \\over \\partial t } ( t _ m ) = { \\partial u _ 0 \\over \\partial t } ( \\tau ) - { \\partial u _ 0 \\over \\partial t } ( t _ m ) = \\int _ { t _ m } ^ \\tau { \\partial ^ 2 u _ 0 \\over \\partial t ^ 2 } ( \\sigma ) d \\sigma , \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} a ' ( n ; i + 1 ) = a ' ( n ; i ) + a ( n - 1 ; i ) - \\sum _ { i ' = i - 2 } ^ { n - 5 } w _ { i ' , i - 2 } + \\sum _ { j = 0 } ^ { i - 3 } w _ { i - 3 , j } \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} p _ { N , k } ^ { P _ m } = \\prod _ { l = 1 } ^ k \\int _ { - 1 } ^ { + 1 } \\mathrm { d } \\lambda _ l \\prod _ { j = 1 } ^ { ( N - k ) / 2 } \\int _ { D _ { + } } \\mathrm { d } x _ j \\mathrm { d } y _ j Q _ { N , k } \\left ( \\{ \\lambda _ l \\} _ { l = 1 } ^ k , \\{ x _ j \\pm \\mathrm { i } y _ j \\} _ { j = 1 } ^ { ( N - k ) / 2 } \\right ) , \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} 2 u _ { k } + 3 u _ { k - 1 } + u _ { k - 2 } \\stackrel { 1 } { = } 2 \\zeta ( k ) \\mbox { w i t h } u _ 0 = 1 , \\ , u _ 1 = 1 / 2 , \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\eta \\circ \\nu _ l \\ , = \\ , \\nu ' _ l \\circ \\eta \\ \\ \\ , \\ \\ \\ \\eta \\circ \\alpha _ g \\ , = \\ , \\alpha ' _ { \\phi ( g ) } \\circ \\eta \\ , \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} f ( T ) & = \\prod _ { j = 1 } ^ g ( T - \\sqrt { q } e ^ { i \\theta _ j } ) ( T - \\sqrt { q } e ^ { - i \\theta _ j } ) \\\\ & = T ^ { 2 g } + c _ 1 T ^ { 2 g - 1 } + \\dots + c _ g T ^ g + c _ { g - 1 } q T ^ { g - 1 } + \\dots + c _ 1 q ^ { g - 1 } T + q ^ g , \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} N _ { d , h } ( X ) = c X ^ { \\frac { 1 } { 2 } } \\log X + O \\big ( X ^ { \\frac { 1 } { 2 } } ( \\log X ) ^ { \\frac { 3 } { 4 } } \\big ) \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} S _ { 1 } - S _ { 2 } + S _ { 3 } + S _ { 4 } + S _ { 5 } + S _ { 6 } - S _ { 7 } = \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\left ( \\varphi _ { i l } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } \\overline { \\left ( \\varphi _ { j l } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } } + 2 \\mbox { R e } \\left \\{ \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\overline { z } _ { i l } b _ { j l } \\left ( Z \\right ) \\right \\} , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\widehat { R } _ { \\epsilon } ^ { - 1 } ( x + \\alpha ) A ^ { E + \\epsilon } ( x ) \\widehat { R } _ { \\epsilon } ( x ) = e ^ { \\mathfrak { P } + \\epsilon \\mathfrak { P } _ 1 + \\epsilon ^ 2 \\mathfrak { P } _ 2 + \\epsilon ^ 3 \\mathfrak { R } _ \\epsilon ( x ) } , \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\sum _ { s = 2 } ^ { d + 1 } \\frac { x ^ { d + 3 } } { ( 1 - x ) ^ { d + 3 - s } } \\big ( K ( x ) - 1 \\big ) \\ , . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} | | \\delta ^ { \\ell } ( a ) | | _ { \\infty , 0 } & \\le C | | \\delta ^ { \\ell } ( a ) | | _ { s - | \\ell | } \\\\ & = C | | \\sum _ m m ^ { \\ell } a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } | | _ { s - | \\ell | } \\\\ & < C | | \\sum _ m ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { | \\ell | } a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } | | _ { s - | \\ell | } \\\\ & = C | | P _ { \\lambda ^ { | \\ell | } } ( a ) | | _ { s - | \\ell | } \\\\ & = C | | a | | _ s \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{gather*} M ' ( S _ { \\phi } ) = \\int _ { M \\in \\Omega _ { q , t } } { M ( S _ { \\phi } ) \\pi ( { \\rm d } M ) } , \\end{gather*}"}
-{"id": "1784.png", "formula": "\\begin{align*} N _ { B ^ u _ \\sigma ( \\Lambda ) } = \\beta ^ * ( E ^ s _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } \\oplus ( \\beta ^ * ( E ^ u _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } \\cap N _ { B ^ u _ \\sigma ( \\Lambda ) } ) \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} E ( u , - T ) = - \\int _ { D ^ 2 } w _ { T } ^ * \\omega - \\int _ { t \\in S ^ 1 } H _ { \\tau , t } ( \\gamma _ T ( t ) ) \\in \\R . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} h _ R ( x _ i ^ { 2 k + 1 } ) & = h _ R ( x _ i ^ { 2 k } ) x _ i + ( - 1 ) ^ { | x _ i ^ { 2 k } | | x _ i | } h _ R ( x _ i ) x _ i ^ { 2 k } \\\\ & = \\bigtriangleup ( 2 k ) y _ i x _ i ^ { 2 k - 1 } x _ i + ( - 1 ) ^ { | x _ i ^ { 2 k } | | x _ i | } y _ i x _ i ^ { 2 k } \\\\ & = ( k + 1 + k ( - 1 ) ^ { ( 2 k - 1 ) | x _ i | ^ 2 } ) y _ i x _ i ^ { 2 k } \\\\ & = \\bigtriangleup ( n + 1 ) y _ i x _ i ^ { n } . \\\\ \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} N _ { d , h } ( R ) = \\# \\big ( \\mathbb { Z } ^ d \\cap \\mathcal { H } _ { d , h } \\cap B ( \\sqrt { R } ) \\big ) . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} E ( y _ { n \\ell } ^ 4 ) = \\sigma _ { n p _ n } ^ { - 4 } \\sum _ { 1 \\le j _ 1 , j _ 2 , j _ 3 , j _ 4 \\le \\ell - 1 } E ( \\hat { r } _ { \\ell j _ 1 } \\hat { r } _ { \\ell j _ 2 } \\hat { r } _ { \\ell j _ 3 } \\hat { r } _ { \\ell j _ 4 } ) = \\sigma _ { n p _ n } ^ { - 4 } O \\left ( \\frac { \\ell ^ 2 } { n ^ 4 } \\right ) = O \\left ( \\frac { \\ell ^ 2 } { p _ n ^ 4 } \\right ) \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { v _ { K , N } ( r ) } { \\omega _ N r ^ N } = 1 , \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\sin \\| \\Theta \\| = \\| P - Q \\| \\leq \\sin \\left ( \\frac { 1 } { 2 } \\arctan \\gamma \\right ) , \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} \\| g \\| _ { F ^ \\times } = \\sup \\left \\{ \\int _ I f g : \\ \\| g \\| _ F \\le 1 \\right \\} , \\ \\ \\ g \\in F ^ \\times , \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( | u _ 1 ( t ) | ^ 2 + | u _ 2 ( t ) | ^ 2 \\big ) \\ d t \\asymp \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) + | \\mathcal { E } | ^ 2 \\ , . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = K + ( l _ 2 ) , \\\\ ( 0 ) : l _ 2 & = K + ( l _ 1 ) , \\\\ K \\cap ( l _ 1 ) & = K \\cap ( l _ 2 ) = ( 0 ) . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\nabla u ( x ) = - \\dfrac { 1 } { m } x + \\alpha \\dfrac { x } { \\abs { x } } \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} o ( 0 ) & : = 0 , \\\\ o ( \\varepsilon _ s ) & : = \\varepsilon _ { \\eta + 1 + c ( s ) } , \\\\ o ( \\Omega ^ { s _ 0 } \\cdot \\beta _ 0 + \\dots + \\Omega ^ { s _ n } \\cdot \\beta _ n ) & : = ( \\omega ^ { 1 + \\alpha } ) ^ { o ( s _ 0 ) } \\cdot \\beta _ 0 + \\dots + ( \\omega ^ { 1 + \\alpha } ) ^ { o ( s _ n ) } \\cdot \\beta _ n . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} D \\left ( \\ln \\omega \\right ) _ { , t } + \\frac { 2 D _ { , t } } { T } & = 0 \\\\ D _ { , t t } & = 0 \\\\ \\frac { C } { T } \\left ( \\ln \\omega \\right ) _ { , t } + 2 \\frac { C _ { , t } } { T } + \\frac { \\lambda } { \\omega T } T _ { , t t } & = \\lambda _ { 1 } \\\\ T _ { , t } & = a _ { 7 } \\omega C \\\\ C _ { , t } & = a _ { 0 } T \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} 1 = ( - 1 ) ^ { b _ 0 } \\epsilon ^ { b _ 1 } \\sqrt { 5 } ^ { b _ 2 } w ^ b 2 ^ { - 3 } - ( - 1 ) ^ { b _ 0 } \\bar \\epsilon ^ { b _ 1 } ( - \\sqrt { 5 } ) ^ { b _ 2 } \\bar w ^ b 2 ^ { - 3 } . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} a s ^ { - 1 } v s a ^ { - 1 } = \\gamma { r } , ~ v s = r v , ~ v = \\tau { s } r \\tau { s ^ { - 1 } } \\tau ^ { - 1 } = \\beta ^ { - 1 } s ^ { - 1 } \\tau { s ^ 2 } r \\tau { s ^ { - 1 } } \\rangle . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\begin{aligned} u ^ \\ast ( t ) & = \\arg \\max _ { u \\in \\Omega } x _ E ^ \\ast ( t ) \\lambda S u \\\\ & = \\arg \\max _ { u \\in \\Omega } ( \\lambda _ 2 - a _ M b _ M \\lambda _ 1 ) u _ M + ( \\lambda _ 3 - a _ E b _ E \\lambda _ 1 ) u _ E \\end{aligned} \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} P ( x ) = P _ { e x } - x _ 3 . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} f ^ { T } ( q ^ i , \\dot { q } ^ i ) = \\dot { q } ^ i \\frac { \\partial f } { \\partial q ^ i } . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } E [ X _ { 3 } ^ \\theta ( t ) ] \\big | _ { \\theta = 1 5 } \\approx \\frac 1 h E \\big [ X _ { 3 } ^ { 1 5 + h / 2 } ( t ) - X _ { 3 } ^ { 1 5 - h / 2 } ( t ) \\big ] \\quad \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} P \\left ( \\N _ { \\rho ^ n } \\notin K _ \\epsilon \\right ) = P \\left ( \\N _ { \\rho ^ n } ( T , U , A ) > \\frac { T \\nu ( U ) } { \\epsilon } \\right ) \\leq E [ \\N _ { \\rho ^ n } ( T , U , A ) ] \\cdot \\frac { \\epsilon } { T \\nu ( U ) } = \\epsilon \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} F _ { ( A _ 1 ' \\ldots A _ k ' ) } = \\frac 1 k ( F _ { A _ 1 ' A _ 2 ' \\ldots A _ k ' } + \\cdots + F _ { A _ s ' A _ 2 ' \\ldots A _ 1 ' \\ldots A _ k ' } + \\cdots + F _ { A _ k ' A _ 2 ' \\ldots A _ 1 ' } ) , \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} v ^ * = \\max \\left ( v _ 1 ^ * , \\ldots , v _ h ^ * \\right ) = \\max \\left ( \\bigcup _ { i = 1 } ^ h V _ i \\right ) . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} \\begin{cases} \\psi _ 0 & : \\pi ^ { - 1 } ( U _ 0 ) \\rightarrow U _ 0 \\times S ^ 1 \\\\ \\psi _ 1 & : \\pi ^ { - 1 } ( U _ 1 ) \\rightarrow U _ 1 \\times S ^ 1 \\end{cases} \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\ddot x = - \\sum _ { i = 1 } ^ N \\frac { m _ i ( x - c _ i ) } { \\vert x - c _ i \\vert ^ { \\alpha + 2 } } , x \\in \\mathbb { R } ^ 2 \\setminus \\{ c _ 1 , \\ldots , c _ N \\} , \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} \\eta ^ { 1 2 } b _ 1 = & ( x , y ) + ( x a + y c ) e _ 1 + ( x b + y d ) e _ 2 \\\\ = & ( x , y ) - u ( x a + y c ) ( E _ 6 , 6 E _ 4 ) + u ( x b + y d ) ( E _ 4 ^ 2 , 6 E _ 6 ) \\\\ & + ( x a + y c ) ( \\eta ^ { 1 2 } , 0 ) + ( x b + y d ) ( 0 , 6 \\eta ^ { 1 2 } ) \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\nabla _ { y } \\cdot \\left ( - d _ { i } \\left ( y \\right ) \\nabla _ { y } \\bar { u } _ { i } ^ { j } ( x , y ) \\right ) = \\nabla _ { y } \\cdot ( d _ i n _ j ) \\quad \\mbox { i n } \\ ; Y _ { 1 } , \\\\ \\\\ - d _ { i } \\left ( y \\right ) \\nabla _ { y } \\bar { u } _ { i } ^ { j } \\cdot \\mbox { n } = d _ { i } n _ { j } \\quad \\mbox { o n } \\ ; \\partial Y _ { 0 } , \\\\ \\\\ \\bar { u } _ { i } ^ { j } \\ ; \\mbox { i s } \\ ; Y \\mbox { - p e r i o d i c } , \\end{array} \\right . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} [ \\delta ( a ) , \\delta ( b ) ] ( c ) & = \\delta ( a ) \\delta ( b ) ( c ) - ( - 1 ) ^ { | a | | b | } \\delta ( b ) \\delta ( a ) ( c ) \\\\ & = \\{ a , \\{ b , c \\} \\} - ( - 1 ) ^ { | a | | b | } \\{ b , \\{ a , c \\} \\} , \\\\ \\delta ( \\{ a , b \\} ) ( c ) & = \\{ \\{ a , b \\} , c \\} , \\\\ ( \\delta d ) ( a ) ( b ) & = \\delta ( d ( a ) ) ( b ) = \\{ d ( a ) , b \\} \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} i ( c ^ { m _ { l _ 2 } } ) = 2 n \\bar { q } l _ { 2 } + 2 [ Q _ { 0 } ] - 2 i '' . \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} B = \\frac { 1 } { 2 } \\left ( ( 2 m ) ^ 2 \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right | ^ 2 - 2 ( 2 m ) ^ 2 \\Re \\left ( \\frac { P _ { r + 2 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) + r \\right ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} C _ k : = ( [ 1 , k - 1 ] , [ k + 1 , n ] ) . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\begin{cases} & s _ 0 \\triangleq \\inf \\{ t > \\tau _ 0 , Y _ t = 0 \\mbox { a n d } Z _ t = - \\delta P _ Y \\} , \\\\ & \\tau _ 1 \\triangleq \\inf \\{ t > s _ 0 , Y _ t = 0 \\mbox { a n d } Z _ t = \\delta P _ Y \\} , \\end{cases} \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} f ( \\gamma z ) = ( c z + d ) ^ k f ( z ) \\gamma = \\left ( \\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix} \\right ) \\in \\Gamma , \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} ( e ^ { A t } \\xi ) ^ { ( i ) } = e ^ { \\lambda t } \\sum _ { j = 0 } ^ { d - i } \\frac { t ^ j } { j ! } \\xi ^ { ( i + j ) } , ( e ^ { - A s } \\xi ) ^ { ( i ) } = e ^ { - \\lambda s } \\sum _ { j = 0 } ^ { d - i } \\frac { ( - s ) ^ j } { j ! } \\xi ^ { ( i + j ) } . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} f ( x ) = a ^ n , \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} D _ h ^ k ( s ) = \\frac { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\langle P _ h ^ { \\frac { 1 } { 2 } } ( \\cdot , s ) , V \\rangle . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\mbox { r a n k } \\left ( \\mathcal { B } _ { k u } ^ { i } \\right ) = \\mbox { r a n k } \\left ( \\mathcal { B } _ { u k } ^ { j } \\right ) , \\quad \\mbox { f o r a l l $ k , u , i , j = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { A } } : = ( S ^ d ) ^ k \\rightarrow \\oplus _ { i = 1 } ^ k V _ { \\mathbf { e } _ i } ^ { a _ i } \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} v _ { p , n } ( k ) & = \\# \\{ ( a , b , c , d ) \\in \\Z _ { > 0 } : a d + b c = n , \\ ; c \\equiv d k \\bmod p \\} \\\\ v ' _ { p , n } ( k ) & = \\# \\{ ( a , b , c , d ) \\in \\Z _ { > 0 } : a d + b c = n , \\ ; \\gcd ( c , d ) = 1 , \\ ; c \\equiv d k \\bmod p \\} \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} \\beta ( a ^ r c ^ s ) & = ( Q \\otimes i d ) \\big ( \\Delta _ \\mathcal { C } ( a ^ r c ^ s ) \\big ) = ( Q \\otimes i d ) \\big ( \\Delta _ \\mathcal { C } ( a ) ^ r \\Delta _ \\mathcal { C } ( c ) ^ s \\big ) \\\\ & = ( Q \\otimes i d ) \\Big ( ( a \\otimes a - q \\ , c ^ * \\otimes c ) ^ r ( c \\otimes a + a ^ * \\otimes c ) ^ s \\Big ) \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} P _ 0 = ( 0 , 0 ) , P _ 1 = ( 1 , 0 ) , P _ t = ( 1 / t , \\infty ) , P _ \\infty = ( \\infty , \\infty ) . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} Q ' ( s ) = \\frac { Q ' ( 0 ) e ^ { f ( s ) } } { b ( s ) ^ { 2 n + 1 } } + ( 2 n + 2 ) \\frac { e ^ { f ( s ) } } { b ( s ) ^ { 2 n + 1 } } \\int _ { 0 } ^ s b ( t ) ^ { 2 n - 1 } e ^ { - f ( t ) } \\left [ Q ( t ) ^ 3 - Q ( t ) \\right ] \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\begin{aligned} & { \\cal K } = p ^ 2 - V ( x , E ) + h ^ 2 V _ 2 ( x , p , E ) + \\cdots = \\cr & F ^ * \\sharp ( L _ 0 ( x , p ) - E + h ^ 2 L _ 2 ( x , p ) + \\cdots ) \\sharp F \\cr \\end{aligned} \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s ( \\varphi _ j ( x _ i ) - \\varphi _ j ( y _ i ) ) = 0 ( 1 \\le j \\le k ) , \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} 2 d \\geq 2 g _ Y - 1 = 2 ( 2 g _ 3 - 1 ) - 1 . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\ell = \\sqrt { b c _ r - 1 } > \\sqrt { b r } \\sqrt { \\frac { 1 } { 2 } - \\frac { 1 } { b } } \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\l ( H _ i ) = \\O ( 3 ^ i ) + \\l ( { \\rm L } _ { 2 } ( 3 ) ) = i + 2 . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} G = L \\left ( \\begin{array} { c c } \\lambda _ 1 & 0 \\\\ 0 & \\lambda _ 2 \\end{array} \\right ) , \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} \\rho = \\frac { \\| T \\| _ { 1 , \\infty } } { \\| A \\| _ { 1 , \\infty } } , \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\frac { H ( y ) } { H ( x ) } \\det [ q _ t ( x _ i , y _ j ) ] , x , y \\in \\mathbb { R ^ + } _ { < } ^ { n } , \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\pi _ 1 + \\cdots + \\pi _ d = f \\qquad ( \\pi _ 1 , \\dots , \\pi _ d \\in S ) \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} T _ { , t t } = - \\omega ( t ) T . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( e ^ { i ( k , B ^ { \\beta , \\alpha } ( t ) ) } \\big ) = E _ { \\beta } \\left ( - \\frac { | k | ^ { 2 } } { 2 } t ^ { \\alpha } \\right ) , \\ ; k \\in \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} E ( a , x + y ) \\sqcup & E ( x + y - a , y ) \\sqcup E ( x + y - b , x ) \\hookrightarrow E ( a , x + y ) \\sqcup \\sqrt { \\alpha } E ( x + y - a , x + y ) \\\\ & \\subset \\sqrt \\alpha \\left ( E ( a , x + y ) \\sqcup E ( x + y - a , x + y ) \\right ) \\hookrightarrow \\alpha E ( x + y , x + y ) = \\alpha B ^ 4 ( x + y ) . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\widetilde { u } \\mid _ { x \\in \\partial G } = 0 , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} \\partial _ { 0 } M _ { 0 } U + M _ { 1 } \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) U + A U = F \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} A = u ( x , y , 0 ) \\mbox { a n d } A ' = ( u v + w ) ( x , y , 0 ) . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ { \\max } ( D , g ) = \\lambda _ { \\max } ( D + ( s ) , g - \\log | s | ) , \\\\ \\lambda _ { \\max } ^ { \\mathrm { a s y } } ( D , g ) = \\lambda _ { \\max } ^ { \\mathrm { a s y } } ( D + ( s ) , g - \\log | s | ) , \\\\ \\hat { \\mu } _ { \\mathrm { e s s } } ( D , g ) = \\hat { \\mu } _ { \\mathrm { e s s } } ( D + ( s ) , g - \\log | s | ) . \\end{cases} \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} D _ { f , g } ( s ; h ) : = \\sum _ { n \\geq 1 } \\frac { a ( n ) \\overline { b ( n - h ) } + a ( n - h ) \\overline { b ( n ) } } { n ^ { s + k - 1 } } , \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} I \\cap A = \\bigcup _ { j = 1 } ^ q ( I _ j \\cap A ) \\cup \\left ( [ a + q d + 1 , n ] \\cap A \\right ) \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} S : = \\mathcal { P } ( \\Sigma ) = \\{ p \\in \\mathbb { R } ^ d : p _ j \\geq 0 , j = 1 , \\ldots , d ; p _ 1 + \\ldots + p _ d = 1 \\} \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\ell _ q = k _ { q n } - k _ { ( q - 1 ) n } , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} g = \\sum _ { m , n } \\mathbf 1 _ { A _ n } \\Big ( \\sum _ m \\nu ( B _ { n m } ) \\| x _ { n m } ^ * \\| _ { X ^ * } ^ { q ' } \\Big ) ^ { \\frac { p ' } { q ' } - 1 } \\ \\mathbf 1 _ { B _ { n m } } \\ x _ { n m } \\| x _ { n m } ^ * \\| _ { X ^ * } ^ { q ' - 1 } , \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\tilde { A } \\left ( W ' , \\overline { W } ' \\right ) \\otimes \\left ( W ' - \\overline { W ' } ^ { t } \\right ) = 2 \\sqrt { - 1 } \\left ( V \\otimes Z ' \\right ) \\overline { \\left ( V \\otimes Z ' \\right ) } ^ { t } , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { \\infty } \\left ( \\sum _ { i = 1 } ^ { \\infty } | T ( e _ { i } , e _ { j } ) | ^ { \\frac { p } { p - 1 } } \\right ) ^ { 2 \\frac { p - 1 } { p } } \\right ) ^ { \\frac { 1 } { 2 } } & \\leq \\left ( \\sum _ { i = 1 } ^ { \\infty } \\left ( \\sum _ { j = 1 } ^ { \\infty } | T ( e _ { i } , e _ { j } ) | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } \\frac { p } { p - 1 } } \\right ) ^ { \\frac { p - 1 } { p } } \\\\ & \\leq C _ { p , \\infty } \\Vert T \\Vert . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} P _ 1 ^ { ( a , b ) } P _ n ^ { ( a , b ) } = \\frac { 1 } { a ( b - 1 ) } P _ { n - 1 } ^ { ( a , b ) } + \\frac { b - 2 } { a ( b - 1 ) } P _ n ^ { ( a , b ) } + \\frac { a - 1 } { a } P _ { n + 1 } ^ { ( a , b ) } \\quad \\quad ( n \\ge 1 ) . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\left ( { \\tau _ - } _ { \\gamma \\cdot ( x , \\xi ) } \\right ) ^ { - 1 } \\left ( { U _ { \\gamma * } } _ { | B _ - ( x , \\xi ) } ( \\zeta ) \\right ) = \\frac { 1 } { T _ \\gamma ( B _ - ( x , \\xi ) ) } \\ , \\gamma \\cdot \\left ( \\left ( { \\tau _ - } _ { ( x , \\xi ) } \\right ) ^ { - 1 } ( \\zeta ) \\right ) \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} p ( z ) = \\det ( I + \\sum _ { j = 1 } ^ { n } z _ j X _ j ) \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\| u \\| _ { X _ h } ^ 2 = h ^ { - \\frac 1 2 } \\| u \\| _ { L ^ 2 } ^ 2 + h ^ { \\frac { 1 5 } 2 } \\| \\partial _ x ^ 4 u \\| _ { L ^ 2 } ^ 2 + h ^ { - \\frac 9 2 } \\| L _ y ^ 2 \\partial _ x u \\| _ { L ^ 2 } ^ 2 + h ^ { - \\frac 1 2 } \\| J _ h u \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\left ( W ^ T f ^ T \\right ) _ k = y _ k , k = 1 , \\ldots , T . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} G _ 2 ( z _ 1 , z _ 2 ) ^ 2 = & \\left ( \\sqrt { 1 - \\frac { | z _ 1 | ^ 2 } { 4 } } \\operatorname { R e } ( z _ 1 ) + \\sqrt { 1 - \\frac { | z _ 2 | ^ 2 } { 4 } } \\operatorname { R e } ( z _ 2 ) \\right ) ^ 2 \\\\ & \\ , + \\left ( \\sqrt { 1 - \\frac { | z _ 1 | ^ 2 } { 4 } } \\operatorname { I m } ( z _ 1 ) + \\sqrt { 1 - \\frac { | z _ 2 | ^ 2 } { 4 } } \\operatorname { I m } ( z _ 2 ) \\right ) ^ 2 + \\left ( 2 - \\frac { | z _ 1 | ^ 2 + | z _ 2 | ^ 2 } { 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} B ^ 2 & = A C \\\\ B ^ 3 & = \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 \\\\ B & = \\lambda _ 2 = ( \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ) ^ { 1 / 3 } . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} f ( x + h ) = f ( x ) + \\langle f ' ( x ) , h \\rangle + \\frac { 1 } { 2 } \\langle f '' ( x ) h , h \\rangle + o ( \\| h \\| ^ 2 ) , h \\in X . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\partial _ { t } v _ { i } ^ { \\varepsilon } = a _ { i } ^ { \\varepsilon } u _ { i } ^ { \\varepsilon } - b _ { i } ^ { \\varepsilon } v _ { i } ^ { \\varepsilon } \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\Gamma ^ { \\varepsilon } , \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} f ( t ) = \\max _ { i \\in [ n ] } \\frac { \\sum _ { j = 1 } ^ { n } | [ W ( t : 0 ) ] _ { i j } - \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } [ W ( t : 0 ) ] _ { i k } | } { y _ i ( t ) } . \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} { h _ 2 ^ \\pm } ^ { ( k ) } = \\sum _ { i } \\frac { \\partial } { \\partial x ^ { ( i ) } } \\pi ^ { ( k ) } ( g ( \\pm R e _ 1 ) ) { u ^ \\pm _ 2 } ^ { ( i ) } + \\frac { 1 } { 2 } \\sum _ { i , j } \\frac { \\partial ^ 2 } { \\partial x ^ { ( i ) } \\partial x ^ { ( j ) } } \\pi ^ { ( k ) } ( g ( \\pm R e _ 1 ) ) { u ^ \\pm _ 1 } ^ { ( i ) } { u ^ \\pm _ 1 } ^ { ( j ) } . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\| X ^ { \\beta } ( \\varphi ) \\| _ { L ^ { 2 } ( \\mu _ { \\beta } ) } ^ { 2 } = \\frac { 1 } { \\Gamma ( \\beta + 1 ) } | \\varphi | _ { 0 } ^ { 2 } . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} g _ 1 ( \\phi U _ 1 , V _ 1 ) = - g _ 1 ( U _ 1 , \\phi V _ 1 ) \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} \\breve { H } _ \\lambda ( t ) \\ ; = \\ ; \\begin{pmatrix} \\widehat { H } ( t ) & \\lambda \\ , k ( t ) \\\\ \\lambda \\ , k ( t ) ^ * & \\widehat { H } ' ( t ) \\end{pmatrix} \\ ; = \\ ; \\begin{pmatrix} \\widehat { H } ( t ) & 0 \\\\ 0 & \\widehat { H } ' ( t ) \\end{pmatrix} \\ ; + \\ ; \\lambda \\ , K ( t ) \\ ; , \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\| \\varphi - z _ n ^ M \\| _ \\infty = O \\left ( \\max \\left \\{ \\tilde { h } ^ { d } , h ^ { 3 r } \\right \\} \\right ) . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} ( 1 \\pm \\gamma ) \\left ( \\prod _ { \\substack { \\{ k , \\ell \\} \\in \\binom { [ r ] } { 2 } \\\\ \\{ k , \\ell \\} \\neq \\{ i , j \\} } } d _ { k \\ell } \\right ) \\prod _ { s \\in [ r ] , s \\neq i , j } | V _ s | \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} h ( ( C _ { A _ i } ) _ { i = 1 } ^ t , ( b _ i ) _ { i = 1 } ^ t , C _ B ) \\triangleq h ( C _ { A _ t } , b _ t , h ( ( C _ { A _ i } ) _ { i = 1 } ^ { t - 1 } , ( b _ i ) _ { i = 1 } ^ { t - 1 } , C _ B ) ) , \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} 2 \\gamma ( \\nabla ^ \\gamma _ X J ( Y ) , Z ) = d \\Omega ( X , Y , Z ) - d \\Omega ( X , J Y , J Z ) , \\ , \\forall X , Y , Z \\in T M \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } 2 a _ 0 + \\frac { 2 } { 3 } a _ 2 + \\frac { 2 } { 5 } a _ 4 & = & 0 \\\\ - \\frac { 2 n } { 1 + n } a _ 0 - \\frac { 2 n } { 3 ( 3 + n ) } a _ 2 - \\frac { 2 n } { 5 ( 5 + n ) } a _ 4 & = & 0 \\\\ \\frac { 4 n ^ 2 } { ( 1 + 2 n ) ( 1 + n ) } a _ 0 + \\frac { 4 n ^ 2 } { 3 ( 3 + 2 n ) ( 3 + n ) } a _ 2 + \\frac { 4 n ^ 2 } { 5 ( 5 + 2 n ) ( 5 + n ) } a _ 4 & = & 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 F \\bigl ( d _ W ( x ) \\bigr ) \\mathrm { d } x = F ( \\gamma ) \\le \\bigl ( 1 - \\sqrt { \\gamma } \\bigr ) F ( 0 ) + \\sqrt { \\gamma } F ( \\sqrt { \\gamma } ) \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } ( x ) \\cdot P _ n ^ { ( a , b ) } ( y ) = \\int _ { - s _ 1 ^ { ( a , b ) } } ^ { s _ 1 ^ { ( a , b ) } } P _ n ^ { ( a , b ) } ( z ) \\ > d \\mu _ { x , y } ( z ) { f o r \\ > \\ > a l l } n \\in \\mathbb N _ 0 . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\zeta ' _ k ( t ) & = - \\frac { k \\zeta _ k ( t ) } { \\mu - 2 t } = - r _ k ( \\boldsymbol { \\zeta } ( t ) ) . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} f _ ! ( h ; \\widehat { \\frak S } ^ { \\epsilon } ) = f _ ! ( h ; \\widehat { \\frak S ^ { \\prime \\epsilon } } ) . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , X ) & = - g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { Z } \\varphi W , X ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ M ( \\tilde { u } _ n ^ { ( 1 ) } - \\tilde { u } _ n ^ { ( 2 ) } ) \\mathrm { d } \\mu = - ( \\lambda _ { n , j } ^ { ( 1 ) } - \\lambda _ { n , j } ^ { ( 2 ) } ) + O \\left ( \\sum _ { i = 1 } ^ 2 \\lambda _ { n , j } ^ { ( i ) } e ^ { - \\lambda _ { n , j } ^ { ( i ) } } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} T _ { 1 } \\left ( W ' , Z ' \\right ) = \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\otimes W ' , V \\otimes Z ' \\right ) , \\quad \\mbox { f o r s u i t a b l e $ V = V \\left ( W ' , \\overline { W ' } \\right ) \\in \\mathcal { M } _ { q ' N ' \\times q ' N ' } \\left ( \\mathbb { C } \\right ) $ . } \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\operatorname { V a r } _ { \\nabla _ t } ( \\alpha ) \\colon ( X _ 1 , \\dots , X _ n ) \\mapsto \\sum _ { i = 1 } ^ n \\alpha ( \\nabla _ 0 X _ 1 , \\dots , \\nabla _ 0 X _ { i - 1 } , \\dot { \\nabla } _ 0 X _ i , \\nabla _ 0 X _ { i + 1 } , \\dots \\nabla _ 0 X _ n ) . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} ( k ^ j \\lambda ^ k _ i ) _ { k = - n + 1 } ^ \\infty , i = 1 , \\dots , \\nu , j = 0 , \\dots , \\mu _ i - 1 , \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { l } ( - \\lambda _ i ) = \\lambda + ( n - k - 1 ) + \\sum _ { i = l + 1 } ^ { k } \\lambda _ i \\geq n + \\lambda - d - 2 \\geq n - d - 1 , \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\Lambda ( y ^ 0 , r , n ) : = ( x _ 0 , x _ 1 , \\ldots , x _ { n - 1 } ) , \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , W ) & = g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( X , Y ) , \\pi _ { \\ast } \\omega \\phi W ) + g _ { 1 } ( \\mathcal { A } _ { X } \\mathcal { B } Y , \\omega W ) - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( X , \\mathcal { C } Y ) , \\pi _ { \\ast } \\omega W ) \\\\ & - \\eta ( Y ) g _ { 1 } ( X , \\omega W ) \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} & R _ k ^ { \\prime \\rm N O M A } ( \\nu ) = \\\\ & \\left \\{ \\begin{array} { l } \\log _ 2 \\left ( 1 + p _ s g _ k ( \\nu ) \\right ) , \\ { \\rm i f } \\ g _ k ( \\nu ) > g _ { \\bar k } ( \\nu ) , \\\\ \\log _ 2 \\left ( 1 + \\frac { p _ w g _ k ( \\nu ) } { p _ s g _ k ( \\nu ) + 1 } \\right ) , \\ { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} g ^ { ( i ) } ( 0 ) = 0 , \\ : ( g ^ { ( i ) } ) ' ( t ) = r ( \\boldsymbol { \\zeta } ^ { ( i ) } ( g ^ { ( i ) } ( t ) ) ) { { 1 } } _ { \\{ g ^ { ( i ) } ( t ) < \\tau _ { \\boldsymbol { \\zeta } ^ { ( i ) } } \\} } + { { 1 } } _ { \\{ g ^ { ( i ) } ( t ) \\ge \\tau _ { \\boldsymbol { \\zeta } ^ { ( i ) } } \\} } , \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\Im ( d \\log \\wedge _ \\Z \\log ) ( f \\wedge _ \\Z g ) = \\Im \\big ( \\log \\vert g \\vert \\cdot d ( \\log f ) - \\log \\vert f \\vert \\cdot d ( \\log g ) \\big ) \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} N _ { d , h } ( R ) = \\sum _ { 2 X _ { 2 k + 2 } ^ 2 + h \\leq R } r _ { 2 k + 1 } ( X _ { 2 k + 2 } ^ 2 + h ) = \\sum _ { 2 m ^ 2 + h \\leq R } r _ { 2 k + 1 } ( m ^ 2 + h ) . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} M ( t ) = \\int _ 0 ^ t e ^ { - A s } \\sigma ( 0 ) d W ( s ) , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } k ^ * ( t ) e ^ { i u t } d t = \\cos ( u T ) K ^ * ( u ) \\ , , \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\chi \\left ( x ( \\nu ( \\Gamma _ d ) ) ^ { 1 / \\alpha } \\Gamma _ d \\right ) - x ^ { - \\alpha } \\chi \\left ( ( \\nu ( \\Gamma _ d ) ) ^ { 1 / \\alpha } \\Gamma _ d \\right ) = c _ { a } x ^ { - \\alpha } \\frac { x ^ { \\rho } - 1 } { \\rho } \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} { C _ { d , \\alpha } = \\pi ^ { - d / 2 } 2 ^ { - \\alpha } \\ , \\frac { \\Gamma ( \\frac { d - \\alpha } { 2 } ) } { \\Gamma ( \\frac { \\alpha } { 2 } ) } , } \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} & 4 C _ 2 \\underline { \\gamma } ^ 2 \\log { n } + \\sigma _ 0 ^ 2 ( 2 \\log { 2 ^ { \\sum _ { l = 1 } ^ d j _ l } } + \\underline { \\gamma } \\log { n } ) \\\\ & \\qquad + 4 \\sigma _ 0 \\underline { \\gamma } \\sqrt { C _ 2 \\log { n } ( 2 \\log { 2 ^ { \\sum _ { l = 1 } ^ d j _ l } } + \\underline { \\gamma } \\log { n } ) } \\leq \\kappa ( \\underline { \\gamma } ) \\log { n } + 2 \\sigma _ 0 ^ 2 \\log { 2 ^ { \\sum _ { l = 1 } ^ d j _ l } } , \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} 0 \\to { H _ 1 ( M ; R ) } \\to { R } \\otimes _ { \\Lambda _ \\beta } { A ( \\pi ) } \\cong { R ^ \\beta } \\to { R } \\to { R } \\otimes _ { \\Lambda _ \\beta } \\mathbb { Z } = \\mathbb { Z } \\to 0 , \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} a _ { m + k } & = \\log ( N ( \\star _ { i = 1 } ^ { m + k } H , \\alpha ^ { m N + k N + 1 } ) ) \\le \\log ( N ( \\star _ { i = 1 } ^ { m } H , \\star _ { i = 1 } ^ { m } \\beta ) N ( \\star _ { i = 1 } ^ { k } H , \\star _ { i = 1 } ^ { k } \\beta ) ) \\\\ & = \\log ( N ( \\star _ { i = 1 } ^ { m } H , \\star _ { i = 1 } ^ { m } \\beta ) + \\log ( N ( \\star _ { i = 1 } ^ { k } H , \\star _ { i = 1 } ^ { k } \\beta ) ) = a _ { m } + a _ { k } . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d s ^ { 2 } } + \\Gamma _ { j k } ^ { i } \\frac { d x ^ { j } } { d s } \\frac { d x ^ { k } } { d s } + \\omega \\left ( s \\right ) V ^ { , i } = 0 , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} f _ { 1 } ( w _ { n } ) w _ { n } - f _ { 1 } ( t _ { n } w _ { n } ) t _ { n } w _ { n } = ( f _ { 1 } ' ( \\sigma ^ { i } _ { n } ) \\sigma ^ { i } _ { n } + f _ { 1 } ( \\sigma ^ { i } _ { n } ) ) w _ { n } ( 1 - t _ { n } ) , \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} \\rho ( x ^ { - 1 } ) = \\overline { \\sigma ( x , x ^ { - 1 } ) } \\rho ( x ) ^ { - 1 } = \\overline { \\sigma ( x ^ { - 1 } , x ) } \\rho ( x ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\int _ { \\O ^ \\# } \\left [ u ^ \\# ( x , y ) - v ( x , y ) \\right ] & \\left ( - \\Delta W ( x , y ) \\right ) d x d y \\\\ & = \\int _ { \\O ^ \\# } \\left [ u ^ \\# ( x , y ) - v ( x , y ) \\right ] h ( x , y ) d x d y \\le 0 \\ , . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} & \\ell _ 1 = p , \\ell _ 2 = q , \\ell _ 3 = 1 - p , \\ell _ 4 = 1 - q , \\\\ & \\ell _ 5 = 1 - x _ 1 p - x _ 2 q . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = - e _ 3 , [ e _ 1 , e _ 3 ] = e _ 2 \\textrm { a n d } [ e _ 2 , e _ 3 ] = - e _ 1 \\ , . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { M _ { \\mathbf { m } } } } \\oplus \\phi _ { \\mathcal { A } } \\oplus \\phi _ { \\mathcal { O } } : ( S ^ d ) ^ k \\rightarrow \\oplus _ { i = 1 } ^ k U _ { k , i } ^ { \\oplus m _ i } \\oplus _ { i = 1 } ^ { a _ i } V _ { \\mathbf { e } _ i } ^ { a _ i } \\oplus V _ { A ( \\mathcal { O } ) } . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 0 } a ( n ) e ( n z ) , \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{gather*} v ( s , x ' _ 0 , { \\bf x ' } ) = s ^ \\sigma \\varpi ( x ' _ 0 , { \\bf x ' } ) , \\end{gather*}"}
-{"id": "677.png", "formula": "\\begin{gather*} p ^ { k } \\left ( x , t , y ^ { j } , s ^ { m } \\right ) \\\\ = p ^ { k , 0 } \\left ( x , t \\right ) + \\sum _ { j = 1 } ^ { n } p ^ { k , j } \\left ( x , t , y ^ { j } , s ^ { m - d _ { j } } \\right ) + \\delta c \\left ( x , t , y ^ { n } , s ^ { m } \\right ) \\end{gather*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\sum _ { a \\mod { r } } ' \\chi _ * ( a ) e \\left ( n \\bar { s } \\tfrac { a } { r } \\right ) = \\chi _ * ( s ) \\sum _ { a \\mod { r _ 0 } } ' \\chi _ * ( a ) e \\left ( n \\tfrac { a } { r } \\right ) \\sum _ { j \\mod { r / r _ 0 } } e \\left ( n \\tfrac { j } { r / r _ 0 } \\right ) = \\delta _ { \\frac { r } { r _ 0 } \\mid n } \\frac { r } { r _ 0 } \\chi _ * ( s ) c _ { \\chi _ * } ( n ) . \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} Y _ m = & \\iint _ { Q _ m } ( v - k _ m ) _ + ^ p \\ , d v _ 0 \\ , d t \\ge ( k _ { m + 1 } - k _ m ) ^ p | E _ m | = \\frac { k ^ p } { 2 ^ { ( m + 1 ) p } } | E _ m | \\\\ \\Rightarrow \\quad | E _ m | \\le & \\frac { 2 ^ { ( m + 1 ) p } } { k ^ p } Y _ m . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} U ( f ( O ) ) = U ( O ^ T A O y ) = ( O U ^ T ) ^ T A ( O U ^ T ) ( U y ) = f ( O U ^ T ) . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} d _ 0 \\circ \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } + \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha _ - } \\circ d _ 0 + \\sum _ { \\alpha ; E ( \\alpha _ - ) < E ( \\alpha ) < E ( \\alpha _ + ) } \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha } \\circ \\frak m ^ { \\epsilon } _ { 1 ; \\alpha , \\alpha _ - } = 0 . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} - \\frac { \\lambda + i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } \\widehat 1 = \\widehat 1 . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\| { \\mathcal E } ( t , \\tau ) \\| \\le { C } _ { 1 5 } + C _ { 1 6 } | \\tau | | t | , 0 < | t | \\le t ^ 0 ; { C } _ { 1 5 } = C _ { 1 4 } + 2 c _ * ^ { - 1 / 2 } C _ 1 , \\ C _ { 1 6 } = c _ * ^ { - 1 / 2 } C _ 5 . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{gather*} H _ { t , \\tau } ( u ) = - \\mathrm { d i v } [ a _ { t , \\tau } ( x , u , \\nabla u ) ] + b _ { t , \\tau } ( x , u , \\nabla u ) \\ , , \\\\ K _ { t , \\tau } = K _ t \\ , . \\end{gather*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\gamma z = \\frac { a z + b } { c z + d } . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} & \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\prec \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { N \\eta } + \\frac { 1 } { ( N \\eta ) ^ 2 } , \\iota = A , B \\ , . \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\frac { e ^ { \\tilde { u } _ n ^ { ( 1 ) } } - e ^ { \\tilde { u } _ n ^ { ( 2 ) } } } { \\| \\tilde { u } _ n ^ { ( 1 ) } - \\tilde { u } _ n ^ { ( 2 ) } \\| _ { L ^ \\infty ( M ) } } = e ^ { \\tilde { u } _ n ^ { ( 1 ) } } \\zeta _ n \\left ( 1 + O \\left ( \\frac { 1 } { \\lambda _ { n , j } ^ { ( 1 ) } } \\right ) \\right ) . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { x } _ 1 & = & - p _ 1 x _ 1 + p _ 2 x _ 2 - p _ 5 x _ 1 x _ 3 \\\\ \\dot { x } _ 2 & = & p _ 1 x _ 1 - p _ 2 x _ 2 - 2 p _ 3 x _ 2 ^ 2 + 2 p _ 4 x _ 3 + p _ 5 x _ 1 x _ 3 \\\\ \\dot { x } _ 3 & = & p _ 3 x _ 2 ^ 2 - p _ 4 x _ 3 \\\\ \\end{array} \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\textsc { K } ^ 2 ( \\gamma + 1 ) - \\textsc { K } ( \\gamma ) \\textsc { K } ( \\gamma + 2 ) = \\sum _ { k = 0 } ^ \\infty \\sum _ { j = 0 } ^ k \\delta _ { j , k } T _ { j , k } z ^ k , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} e ^ { - x } = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } x ^ { - s } \\Gamma ( s ) d s . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\nabla _ { P ( p , 1 , 1 ) } ( z ) & = 1 + z + \\frac { p - 1 } { 2 } z = \\frac { ( p + 1 ) } { 2 } z , \\\\ \\nabla _ { P ( p , 1 , r ) } ( z ) & = \\nabla _ { P ( 1 , 1 , r ) } ( z ) + \\frac { ( p - 1 ) ( r + 1 ) } { 4 } z = 1 + \\frac { ( p + 1 ) ( r + 1 ) } { 4 } z , \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} z \\mathtt { J } ' _ { \\nu } ( z ) = z \\mathtt { J } _ { \\nu - 1 } ( z ) - \\nu \\mathtt { J } _ { \\nu } ( z ) , \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\Psi _ j ( p ) : = \\sum _ { i = 1 } ^ m \\bar { c } _ { i j } \\Phi _ i ( p ) \\mbox { f o r } \\ ; \\ ; j = 1 , \\ldots , r \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} \\tilde { l } _ { \\gamma } ^ { G E L } ( \\theta ) = - 2 \\sum _ { i = 1 } ^ n \\log ( n w ^ { \\gamma } _ i ( \\theta ) ) \\ , . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} P ( F + t G ) - P ( F ) & \\ge \\big ( h _ { \\mu _ { 0 } } ( g ) + \\int ( F + t G ) d \\mu _ { 0 } \\big ) - \\big ( h _ { \\mu _ { 0 } } ( g ) + \\int F d \\mu _ { 0 } \\big ) \\\\ & = t \\int G d \\mu _ 0 . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\Upsilon _ \\alpha ( \\beta ) \\ ; = \\ ; \\inf _ { m } \\frac { \\pi } { 4 } \\int _ 0 ^ { 1 / 2 } \\frac { m ' ( r ) ^ 2 } { \\sigma ( m ( r ) ) } \\ , d r \\ ; , \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\beta _ 1 & = \\rho ( 1 - \\rho ^ 2 ) \\gamma _ 2 , \\beta _ 2 = ( 1 - \\rho ^ 2 ) \\gamma _ 1 + ( 1 - \\rho ^ 2 ) \\gamma _ 2 , \\\\ \\beta _ 3 & = ( 1 - \\rho ) \\gamma _ 3 , \\beta _ 4 = ( 1 - \\rho ^ 2 ) \\gamma _ 4 - \\rho ^ 2 ( 1 - \\rho ^ 2 ) \\gamma _ 2 , \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\mathcal { J } _ 2 \\leqslant \\frac { \\sum _ { n = K + 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { n } \\overline { F } _ { \\xi _ k } ( \\mathit { x y } ) \\mathbb { P } ( \\eta = n ) } { \\overline { F } _ { \\xi _ \\varkappa } ( x ) \\mathbb { P } ( \\eta = \\varkappa ) } , \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} D ( s , S _ f \\times \\overline { S _ g } ) = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { s + k - 1 } } = \\sum _ { n \\geq 1 } \\frac { 1 } { n ^ { s + k - 1 } } \\sum _ { m = 1 } ^ n a ( m ) \\sum _ { h = 1 } ^ n \\overline { b ( h ) } . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} 2 [ \\sigma ] + 2 [ \\tau ] = ( a - 1 ) [ \\gamma ] . \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\begin{array} { r l l l } \\Delta \\mathcal { D } _ 0 & = & - \\lambda \\ , \\alpha _ 3 \\ , \\Bigl ( \\frac { 1 } { ( 1 + \\vert z \\vert ^ 2 ) ^ { 1 / 2 } } - \\frac { 1 } { \\vert z \\vert } \\Bigr ) & \\mathbb { R } ^ 3 , \\\\ \\mathcal { D } _ 0 & \\rightarrow & 0 & \\vert z \\vert \\rightarrow \\infty . \\end{array} \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\sum _ { k = 1 } ^ K Y _ k \\big ( \\int _ 0 ^ t \\lambda _ k ( X ( s ) ) d s \\big ) \\zeta _ k . \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} d i v _ { \\mathfrak { m } } V : = \\sum _ { i = 1 } ^ n \\left ( \\frac { \\partial V ^ i } { \\partial x ^ i } + V ^ i \\frac { \\partial \\Phi } { \\partial x ^ i } \\right ) . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} | k | x \\left | \\int _ { 2 | k | x } ^ 2 \\frac { e ^ { i \\xi } - 1 } { \\xi } \\frac { 1 } { \\xi - k x } ~ d \\xi \\right | & \\le C | k | x \\int _ { 2 | k | x } ^ 2 \\frac { 1 } { \\xi - | k | x } ~ d \\xi \\\\ & = C | k | x \\log \\left ( \\frac { 2 } { | k | x } - 1 \\right ) \\\\ & \\le C | k | ^ { \\epsilon } | x | ^ { \\epsilon } . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} y '' ( z ) + \\frac { \\tau ( z ) } { \\sigma ( z ) } y ' ( z ) + \\frac { \\bar { \\sigma } ( z ) } { \\sigma ^ 2 ( z ) } y ( z ) = 0 . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} v _ { i _ 1 \\dots i _ { m - 1 } } ( x ) = \\frac { 1 } { ( m - 1 ) ! } \\left . \\frac { \\partial ^ { m - 1 } } { \\partial \\xi ^ { i _ 1 } \\cdots \\partial \\xi ^ { i _ { m - 1 } } } u ( x , \\xi ) \\right | _ { \\xi = e _ n } . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\lambda _ 1 ( H _ { \\Lambda } ^ D ) & \\leq ~ ~ < \\varphi _ 0 , H _ { \\Lambda } ^ D \\varphi _ 0 > \\\\ ~ & = \\lambda _ 0 + \\int _ { \\Lambda } V \\varphi _ 0 ^ 2 ~ d x \\\\ & \\leq ~ \\lambda _ 0 + \\frac { C } { | \\Lambda ^ { ' } | } \\int _ { \\Lambda } V ( x ) ~ d x , \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} f _ { i } ' ( s ) s ^ { 2 } - f _ { i } ( s ) s > 0 , \\\\ f _ { i } ' ( s ) > 0 , \\\\ \\phi _ { i } ( s ) = f _ { i } ( s ) s - 2 F _ { i } ( s ) > 0 , \\\\ \\phi _ { i } ( s ) > \\phi _ { i } ( t s ) , \\ \\mbox { f o r a l l } \\ t \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} ( a \\star b , c ) = ( a \\otimes b , \\Delta ( c ) ) ( c , a \\star b ) = ( \\Delta ( c ) , a \\otimes b ) . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} & \\sigma ^ 2 ( y _ 0 ) = ( x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 0 , x _ 1 , \\ldots , x _ 5 , x _ 0 , \\ldots ) : = y _ 1 \\\\ & \\sigma ^ 2 ( y _ 1 ) = ( x _ 4 , x _ 5 , x _ 0 , x _ 1 , \\ldots , x _ 5 , x _ 0 , \\ldots ) : = y _ 2 \\\\ & \\sigma ^ 2 ( y _ 2 ) = ( x _ 0 , x _ 1 , \\ldots , x _ 5 , x _ 0 , \\ldots ) = y _ 0 \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} m _ 1 = 2 ^ { q + 1 } - t - a _ 1 \\ , \\ , \\ , a n d \\ , \\ , \\ , m _ i = 2 ^ q \\cdot [ 2 ^ { i - 2 } - 1 ] + t + i - 3 + 2 a _ { i - 1 } - a _ i \\ , \\ , \\ , f o r \\ , \\ , a l l \\ , \\ , 2 \\leq i \\leq k . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} L ( x , u , \\lambda _ 0 , \\lambda , \\mu , \\nu ) = H ( x , u , \\lambda _ 0 , \\lambda ) + \\mu g ( u ) + \\nu h ( x ) . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} | A \\cap I _ { u , n } | & = \\sum _ { j = 1 } ^ { q + 1 } \\left | A \\cap I _ { u , n } ^ { ( j ) } \\right | \\leq \\sum _ { j = 1 } ^ q \\left ( \\left | I _ { u , n } ^ { ( j ) } \\right | - 1 \\right ) + \\left | I _ { u , n } ^ { ( q + 1 ) } \\right | \\\\ & = \\sum _ { j = 1 } ^ { q + 1 } \\left | I _ { u , n } ^ { ( j ) } \\right | - q = | I _ { u , n } | - q = n - q \\\\ & < n - \\frac { n } { d } + 1 = \\left ( 1 - \\frac { 1 } { d } \\right ) n + 1 . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\| v _ i ( t _ 0 ) \\| _ 2 \\leq 2 \\sum _ { i = 1 } ^ { N } \\| v _ { i , 0 } \\| _ 2 . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} Y _ i ( r ) = B \\sqrt { n } G _ i ( Y ( r ) , r ) , \\ ; i = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\mu ^ { q / p } \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) \\triangleleft \\mu \\left ( \\sum _ { i = 1 } ^ n \\mu ^ { q / p } ( | x _ i | ^ p ) \\right ) = \\mu \\left ( \\sum _ { i = 1 } ^ n \\mu ^ q ( x _ i ) \\right ) . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\partial _ { \\boldsymbol e _ i } \\eta ^ { t , \\bar t } = \\left ( \\frac { \\partial _ N \\partial _ { \\boldsymbol e _ i } ( u ^ { t , \\bar t } - h ^ { t , \\bar t } ) } { ( N \\cdot \\nu ^ { t , \\bar t } ) ^ 2 \\ , \\Delta h ^ { t , \\bar t } } \\right ) ( z , \\eta ^ { t , \\bar t } ( x ) ) . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} F ( z ) = \\sum ^ \\mu _ { s = 1 } \\frac { v _ s } { z - z _ s } + u ( z ) , \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\phi _ { A B } \\circ \\phi _ { B C } = \\phi _ { A C } . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\varphi ( x ) = T _ { \\lambda + 0 i } \\varphi ( x ) & = i \\int _ { - \\infty } ^ x e ^ { i \\lambda ( x - y ) } u \\varphi ( y ) ~ d y - \\widetilde { G } _ { \\lambda } * ( u \\varphi ) ( x ) \\\\ & = - i \\int _ x ^ { \\infty } e ^ { i \\lambda ( x - y ) } u \\varphi ( y ) ~ d y - \\widetilde { G } _ { \\lambda } * ( u \\varphi ) ( x ) \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} y ^ 3 - 3 y - 2 q = 0 \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\frac { 5 9 . 4 8 8 A ' B ( B - A ) ^ 2 } { A g ^ 4 } & = 2 3 7 . 9 5 2 \\frac { ( b - a ) ^ 3 b } { a g ^ 4 } < 2 3 7 . 9 5 2 \\frac { ( b - a ) ^ 3 b } { a } \\\\ & \\leq 2 3 7 . 9 5 2 \\left ( \\frac { k - 1 } { k } \\right ) ^ 3 k b ^ 3 , \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} R ^ { - 1 } ( x + \\alpha ) A ^ { E } ( x ) R ( x ) = \\left [ \\begin{array} { c c } \\pm 1 & \\mu _ m \\\\ 0 & \\pm 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} Z ^ 2 + 1 = p _ 1 ^ { k _ 1 } \\cdots p _ r ^ { k _ r } . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\int e ^ { i \\lambda \\phi ( x , \\xi ) } \\tilde { a _ { N } } ( x , \\xi ) f _ { i _ { 1 } \\dots i _ { m } } ( x ) { \\tilde { b } } ^ { i _ { 1 } } ( x , \\xi ) \\cdots { \\tilde { b } } ^ { i _ { m } } ( x , \\xi ) d x = 0 . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\zeta \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { P f } \\begin{bmatrix} { [ } \\zeta ^ 2 \\alpha _ { j , l } + \\beta _ { j , l } ] & [ \\mu _ j ] \\\\ { [ } - \\mu _ l ] & 0 \\end{bmatrix} _ { j , l = 1 , \\ldots , N } . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\left \\Vert T \\right \\Vert & = \\sup \\left \\{ \\left \\vert T \\left ( x , y \\right ) \\right \\vert : \\left \\Vert x \\right \\Vert _ { X _ { p } } \\leq 1 , ~ \\left \\Vert y \\right \\Vert _ { X _ { q } } \\leq 1 \\right \\} \\\\ & = \\sup \\left \\{ \\left \\vert \\sum _ { i , j } a _ { i j } x _ { i } y _ { j } \\right \\vert : \\left \\Vert x \\right \\Vert _ { X _ { p } } \\leq 1 , ~ \\left \\Vert y \\right \\Vert _ { X _ { q } } \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\chi ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) \\\\ & = 1 8 - 6 c _ 2 ( E _ { C , A } ) + 2 { c _ 1 ( E _ { C , A } ) } ^ 2 \\\\ & = 1 8 - 6 ( d + 2 ) + 2 ( 2 g - 2 ) \\\\ & = 2 - 2 \\rho ( g , 2 , d + 2 ) . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\mathcal { R } _ { T } ^ c ( \\alpha ) = \\left \\{ T _ { n p } \\ge \\frac { p ( p - 1 ) } { 2 ( n - 4 ) } \\left ( \\sqrt { \\frac { n - 3 } { n - 6 } } - 1 \\right ) + \\chi ^ 2 _ { \\alpha } ( p ( p - 1 ) / 2 ) \\sqrt { \\frac { ( n - 3 ) } { ( n - 4 ) ^ 2 ( n - 6 ) } } \\right \\} . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} \\mu ( t , x ) = \\inf \\left \\{ s \\geq 0 : d ( s , x ) \\leq t \\right \\} , t \\geq 0 , \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} w = \\int _ { 0 } ^ { t } e ^ { ( t - s ) B } f ( s ) \\dd s \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} J ( 0 , a ) & : = a , \\\\ J ( m + 1 , a ) & : = \\{ s \\in S _ { \\omega ^ \\alpha } ^ u \\ , | \\ , \\forall _ r ( \\forall _ { t < r } \\ , t \\in J ( m , a ) \\rightarrow \\forall _ { t < r + \\Omega ^ s } \\ , t \\in J ( m , a ) ) \\} . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} \\sum _ { \\substack { h _ 0 , h _ 1 , h _ 2 \\ge 0 \\\\ h _ 0 + h _ 1 + h _ 2 = h \\\\ h _ 0 + 2 h _ 1 + h _ 2 = m } } \\binom h { h _ 0 , h _ 1 , h _ 2 } \\sum _ { n \\leq x } \\big ( \\log 2 \\cdot F _ { \\omega _ 0 } ( n ) \\big ) ^ { h _ 0 } S _ 1 ^ { h _ 1 } S _ 2 ^ { h _ 2 } = \\sum _ { \\substack { \\beta \\le B _ h \\\\ k _ { h \\beta } = m } } r _ { h \\beta } \\prod _ { j = 1 } ^ { \\tilde k _ { h \\beta } } \\mu ( \\omega _ { q _ { w ( h , \\beta , j ) } } ) \\sum _ { n \\leq x } \\prod _ { i = 1 } ^ { k _ { h \\beta } } F _ { \\omega _ { q _ { v ( h , \\beta , i ) } } } ( n ) . \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} \\varphi = ( z _ 1 , P ( z _ 1 ) - \\delta z _ 0 ) \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} | B ^ T e _ 2 | ^ 2 = ( 1 - \\epsilon ^ 2 ) + \\epsilon ^ 2 = 1 , \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} \\nu _ { x } ( z ) = \\nu _ S ( y ) z = y - t \\nu _ S ( x ) \\ , . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} Z + i = \\pi _ 1 ^ { k _ 1 } \\cdots \\pi _ r ^ { k _ r } . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\frak h _ { c a } ^ { i } = \\psi _ { c b } ^ { i + 1 } \\circ \\frak h _ { b a } ^ { i } + \\frak h _ { c b } ^ { i } \\circ \\psi _ { b a } ^ { i } . \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} a _ k ( x ) & = x \\sum _ { j \\geq k - 1 } a _ j ( x ) \\ , , \\\\ b _ k ( x ) & = x \\big ( a _ { k - 1 } ( x ) + b _ { k - 1 } ( x ) + c _ { k - 1 } ( x ) \\big ) \\ , , \\\\ c _ k ( x ) & = x \\sum _ { j \\geq k } \\big ( b _ j ( x ) + c _ j ( x ) \\big ) \\ , , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\mathcal { B } u ( t ) = g ( t ) t \\in ( 0 , T ) \\ , . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} d - 2 \\le M . N - 2 h ^ 1 ( M ) - 2 = d + 3 - | Z | - c _ 2 ( F ^ { * * } ) - 2 h ^ 1 ( M ) - 2 \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\biggr | \\frac { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 2 } } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } } - \\frac { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k } } \\biggr | < \\epsilon . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} \\{ h ( x , \\xi ) , X _ { j } ( x , \\xi ) \\} = \\sum _ { \\ell = 1 } ^ { N } \\alpha _ { j \\ell } ( x , \\xi ) X _ { \\ell } ( x , \\xi ) , \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\partial _ { e e } \\tilde u ^ 0 = ( e \\cdot \\nu ) ^ 2 \\partial _ { \\nu \\nu } \\tilde u ^ 0 = ( e \\cdot \\nu ) ^ 2 \\Delta \\tilde u ^ 0 = - ( e \\cdot \\nu ) ^ 2 \\Delta h ^ 0 \\mbox { o n } \\Gamma ^ 0 \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\rho ( \\tilde { \\theta } _ { 1 } ^ \\alpha , x ) < \\alpha | x \\right ) = \\alpha + \\phi ( \\Phi ^ { - 1 } ( \\alpha ) ) E _ { F } \\left ( R ^ { * } ( \\alpha , x ) \\right ) + O ( n ^ { - 1 } ) , \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} h = \\left ( \\begin{matrix} i / 2 & 0 \\\\ 0 & - i / 2 \\end{matrix} \\right ) , e = \\left ( \\begin{matrix} 0 & 1 / 2 \\\\ - 1 / 2 & 0 \\end{matrix} \\right ) , f = \\left ( \\begin{matrix} 0 & i / 2 \\\\ i / 2 & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\begin{cases} a _ 1 + \\cdots + a _ n & = n , \\\\ \\frac 1 n \\log \\binom { n } { a _ 1 , \\dots , a _ n } & \\leq H . \\end{cases} \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} p _ { t } ^ { M } ( \\tilde { x } , \\tilde { y } ) = \\frac { t } { 2 \\sqrt { \\pi } } \\int _ { 0 } ^ { \\infty } u ^ { - 3 / 2 } e ^ { - t ^ { 2 } / 4 u } h _ { u } ^ { M } ( \\tilde { x } , \\tilde { y } ) d u , \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\varphi _ { \\underline { d } , \\underline { d } '' } ( \\varphi _ { \\underline { d } ' , \\underline { d } } ( V _ { \\underline { d } ' } ) ) = \\varphi _ { \\underline { d } ' , \\underline { d } '' } ( V _ { \\underline { d } ' } ) = V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\Pi _ { \\epsilon } \\left [ d ( b ( \\theta ) , b _ { 0 } ) \\geq \\epsilon ^ * + \\delta _ { n } | \\eta ( \\mathbf { y } ) \\right ] = o _ { P _ 0 } ( 1 ) , \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} g _ v ( x ) & = \\sum _ { i = 1 } ^ r a _ { i } h _ { i , v } ( x ) \\\\ & = \\lim _ { n \\to \\infty } \\theta _ { n , v } ( x ) + \\sum _ { i = 1 } ^ r \\left ( \\lim _ { n \\to \\infty } a _ { n i } \\right ) h _ { i , v } ( x ) \\\\ & = \\lim _ { n \\to \\infty } \\left ( \\theta _ { n , v } ( x ) + \\sum _ { i = 1 } ^ r a _ { n i } h _ { i , v } ( x ) \\right ) \\geqslant 0 . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} C _ 1 [ f , g , h ] = \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\partial _ x ^ { - 1 } P _ { N _ 1 } ( \\partial _ x f _ { N _ 2 } g _ { \\leq N _ 2 } h _ { \\leq N _ 2 } ) , C _ 2 [ f , g , h ] = \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\partial _ x ^ { - 1 } P _ { N _ 1 } ( f _ { N _ 2 } g _ { \\leq N _ 2 } \\partial _ x h _ { \\leq N _ 2 } ) , \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} \\bar { R } _ o ( X , Y ) Z = \\tilde { R } ( X , Y ) Z - [ [ X , Y ] _ { J } , Z ] , \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\hat { \\alpha } _ { j l } = ( p _ { j } ) \\alpha + \\xi _ { j l } , \\ \\forall \\ 1 \\leq j \\leq k \\ \\ 0 \\leq l \\leq | p _ { j } | - 1 , \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} \\cos ( \\varepsilon ^ { - 1 } \\tau A ( t ) ^ { 1 / 2 } ) F ( t ) = \\sum _ { l = 1 } ^ { n } \\cos ( \\varepsilon ^ { - 1 } \\tau \\sqrt { \\lambda _ l ( t ) } ) ( \\cdot , \\varphi _ l ( t ) ) \\varphi _ l ( t ) . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} f ( M ) = I \\left ( M ; \\overline { M } \\right ) = \\log _ { 2 } | \\Sigma _ { M } | + \\log _ { 2 } | \\Sigma _ { \\overline { M } } | - \\log _ { 2 } | \\Sigma _ X | , n o \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} \\partial ( y ) - \\phi _ 0 y = y _ 0 ^ { - 1 } \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} \\phi _ { n , j } ( y ) = \\frac { \\rho _ n } { m } ( R ( y , x _ { n , j } ^ { ( 1 ) } ) - R ( x _ { n , j } ^ { ( 1 ) } , x _ { n , j } ^ { ( 1 ) } ) ) + \\frac { \\rho _ n } { m } \\sum _ { l \\neq j } ( G ( y , x _ { n , l } ^ { ( 1 ) } ) - G ( x _ { n , j } ^ { ( 1 ) } , x _ { n , l } ^ { ( 1 ) } ) ) , \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| f - f * K _ n \\| _ X = 0 ; \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\omega = \\left ( \\frac { \\int _ { - 1 } ^ 1 \\vert \\dot u ( t ) \\vert ^ 2 \\ , d t } { 2 \\int _ { - 1 } ^ 1 U ( u ( t ) ) \\ , d t } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\phi ( \\rho ) = \\frac { ( 1 + \\rho ^ { n - 1 } ) ^ \\frac { 1 } { n - 1 } - 1 } { \\rho ^ { n - 2 } } \\ , . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} f _ A ( n ) = \\max _ { u \\in \\N _ 0 } \\left | A \\cap [ u , u + n - 1 ] \\right | \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} E \\bigl ( Z _ t ^ H ( a , b ) - Z _ s ^ H ( a , b ) \\bigr ) ^ 2 & = \\bigl ( a ^ 2 + b ^ 2 \\bigr ) | t - s | ^ { 2 H } \\\\ & - 2 ^ { 2 H } a b \\bigl ( | t | ^ { 2 H } + | s | ^ { 2 H } \\bigr ) + 2 a b | t + s | ^ { 2 H } ; \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} \\phi \\left ( \\xi \\right ) = f _ { 1 } \\left ( z \\right ) / f _ { 0 } \\left ( z \\right ) , \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\psi ^ { P , \\epsilon } _ { \\alpha _ 2 , \\alpha _ 1 } ( h ) = { \\rm e v } _ { + } ! ( { \\rm e v } _ { - } ^ * h ; \\widehat { \\frak S ^ { + \\epsilon } } ( \\alpha _ 1 , \\alpha _ 2 ; P ) ) . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\Sigma = \\{ \\lambda \\in \\mathbb R : \\} . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} a _ 0 & a _ { k - 1 } & a _ { k - 2 } & \\ldots & a _ { 2 } & a _ { 1 } \\\\ a _ 1 & a _ 0 & a _ { k - 1 } & \\ldots & a _ { 3 } & a _ { 2 } \\\\ a _ 2 & a _ 1 & a _ 0 & \\ldots & a _ { 4 } & a _ { 3 } \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ a _ { k - 1 } & a _ { k - 2 } & a _ { k - 3 } & \\ldots & a _ { 1 } & a _ { 0 } \\end{matrix} \\right ] \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} m ( t , x ) \\leq \\frac { C _ { 1 } \\vert x \\vert ^ { 2 } } { 1 6 C _ { 2 } ^ { 2 } t } + \\frac { C _ { 1 } \\Lambda ^ { \\mu } \\vert x \\vert ^ { \\mu } } { 4 ^ { \\mu } C _ { 2 } ^ { \\mu } t ^ { \\mu - \\nu } } - \\frac { \\vert x \\vert ^ { 2 } } { 4 C _ { 2 } t } = \\frac { C _ { 1 } \\vert x \\vert ^ { 2 } } { 1 6 C _ { 2 } ^ { 2 } t } + \\frac { C _ { 1 } \\Lambda ^ { \\mu } \\vert x \\vert ^ { 2 } } { 4 ^ { \\mu } C _ { 2 } ^ { \\mu } t ^ { 1 } } \\cdot \\frac { \\vert x \\vert ^ { \\mu - 2 } } { t ^ { \\mu - \\nu - 1 } } - \\frac { \\vert x \\vert ^ { 2 } } { 4 C _ { 2 } t } \\leq - \\frac { \\vert x \\vert ^ { 2 } } { 8 C _ { 2 } t } \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} \\Gamma = \\langle e _ 0 , \\ldots , e _ n , \\sigma _ 1 \\ldots , \\sigma _ m \\mid e _ j ^ { p _ j } = 1 \\textrm { f o r a l l } j , ~ e _ 0 \\cdots e _ n \\sigma _ 1 \\cdots \\sigma _ m = 1 \\rangle . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert \\phi _ n \\Vert _ { \\ast } = 0 . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } | \\phi ( x ) | ^ { q } d x = \\frac { ( a _ { 1 } - a _ { 2 } ) p q } { a _ { 1 } p q - Q ( q - p ) } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\underbrace { \\sum _ { r = 1 } ^ n c _ { j , r } \\ , c _ { k , r } } _ { P _ { j k } } + \\underbrace { c _ { j , k } \\cdot \\sum _ { \\substack { r , s = 1 \\\\ r < s } } ^ n c _ { k , r } \\ , c _ { k , s } } _ { Q _ { j k } } \\equiv 0 \\bmod { 2 } \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} m ( G _ { 1 } , k ) = & m ( G _ { 1 } - v , k ) + \\sum \\limits _ { w ' \\in N ( v ) } m ( G _ { 1 } - v - w ' , k - 1 ) . \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} 0 \\leqslant g _ n - \\log | \\varphi _ n | = g + ( h _ n - h ) - \\log | \\varphi _ n | \\leqslant g + \\epsilon - \\log | \\varphi _ n | , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\int _ 1 ^ X \\Big \\lvert \\sum _ { n \\leq r } d ( n ) - c r \\log r - c ' r \\Big \\rvert ^ 2 d r = c '' X ^ { \\frac { 3 } { 2 } } + O ( X ^ { \\frac { 5 } { 4 } + \\epsilon } ) , \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\langle E ^ { ( 1 ) } ( \\phi ) \\rangle _ t \\leq & \\ , \\frac { A ^ n _ { e , e } } { ( M L ) ^ 2 } \\sum _ z \\sum _ { w \\sim z } \\int _ 0 ^ t ( \\phi _ s ( z ) - \\phi _ s ( w ) ) ^ 2 \\ , d s \\\\ = & \\ , \\frac { 2 A ^ n _ { e , e } } { L ^ 2 } \\int _ 0 ^ t \\ < 1 , \\ , | \\nabla _ L \\phi _ s | ^ 2 \\ > _ e \\ , d s \\to 0 \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} \\Big | \\int _ \\R h ( x ) ( \\dd \\mu _ A ( x ) - \\dd \\mu _ \\alpha ( x ) ) \\Big | & = \\Big | \\int _ \\R \\big ( h ' ( x ) - h ' ( x + s ) \\big ) \\mathcal { F } _ { \\mu _ A } ( x ) \\dd x + \\int _ \\R h ' ( x + s ) \\big ( \\mathcal { F } _ { \\mu _ A } ( x ) - \\mathcal { F } _ { \\mu _ \\alpha } ( x + s ) \\big ) \\dd x \\Big | \\\\ & \\le C s \\Big ( \\sup _ { x \\in \\R } | h '' ( x ) | + \\int _ \\R | h ' ( x ) | \\dd x \\Big ) \\ , , \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} T _ x M = \\langle X \\rangle \\oplus E _ { 1 } ( x ) \\oplus \\cdots \\oplus E _ { l ( x ) } ( x ) \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} V _ { - P } = \\begin{pmatrix} I & 0 \\\\ P & I \\end{pmatrix} M _ { L , m } = \\begin{pmatrix} L ^ { - 1 } & 0 \\\\ 0 & L ^ { T } \\end{pmatrix} . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\widehat { \\underline \\mu } _ 1 ^ S ( t ) = \\prod _ { \\substack { \\chi \\mod q \\\\ \\chi \\ne \\chi _ 0 \\\\ \\chi } } \\prod _ { \\gamma \\in \\Gamma ^ S ( \\chi ) } J _ 0 \\Bigg ( \\frac { 2 \\big | \\big ( \\chi ( a ) - \\chi ( b ) \\big ) t \\big | } { \\sqrt { \\frac { 1 } { 4 } + \\gamma ^ 2 } } \\Bigg ) . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\mathcal { S } [ X ] = 1 + S _ 1 X + S _ 2 X ^ 2 + S _ 3 X ^ 3 + \\ldots + S _ { m d _ c } X ^ { m d _ c } , \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} - \\dot { \\tilde { f } } ( 0 ) \\le \\frac { 1 } { | Y | } \\frac { 1 } { r } Y ^ 2 = \\frac { \\tilde { f } ( 0 ) } { r } . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\begin{aligned} & - h ^ 2 \\Delta _ { x } \\Phi - { \\partial ^ 2 \\Phi \\over \\partial z ^ 2 } = f , - D ( x ) < z < 0 \\cr & \\Phi = \\varphi ^ + ( x ) \\ @ \\ z = 0 \\cr & \\partial _ { \\vec n _ h } \\Phi = \\psi ^ - ( x ) \\ @ \\ z = - D ( x ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\frac { \\partial L } { \\partial \\dot { q } ^ i } \\right ) = \\frac { \\partial L } { \\partial q ^ i } . \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} h ^ { 1 , 1 } ( Z _ \\Delta ) = 2 \\ell ( \\Delta ^ \\circ ) - 5 - \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) + \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\binom { n + k - 1 } { k - 1 } \\leq \\exp \\ ( O \\ ( \\frac { H n \\log \\log n } { \\log n } \\ ) \\ ) = \\exp ( o ( H n ) ) . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\log \\left ( { \\Gamma \\Big ( { r ( n + \\nu - 2 ) + n + \\nu \\over 2 } + { ( r + 1 ) t \\over 2 } \\Big ) \\over \\Gamma \\Big ( { r ( n + \\nu - 2 ) + n + \\nu \\over 2 } + { r t \\over 2 } \\Big ) } \\right ) & = \\frac t 2 \\log \\left ( { r ( n + \\nu - 2 ) + n + \\nu \\over 2 } + { r t \\over 2 } \\right ) + o ( 1 ) \\\\ & = \\frac t 2 \\log \\left ( \\frac { ( r + 1 ) n } { 2 } \\right ) + o ( 1 ) . \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} f _ s & = f _ 2 \\bigl ( Y ^ 1 _ s , Z ^ 1 _ s , \\psi ^ 1 _ s \\bigr ) - f _ 1 \\bigl ( Y ^ 1 _ s , Z ^ 1 _ s , \\psi ^ 1 _ s \\bigr ) , \\\\ \\alpha _ s & = \\frac { f _ 2 ( Y ^ 2 _ s , Z ^ 1 _ s , \\psi ^ 1 _ s ) - f _ 2 ( Y ^ 1 _ s , Z ^ 1 _ s , \\psi ^ 1 _ s ) } { \\overline Y _ s } \\ 1 _ { \\overline Y _ s \\neq 0 } , \\\\ \\beta _ s & = \\frac { f _ 2 ( Y ^ 2 _ s , Z ^ 2 _ s , \\psi ^ 1 _ s ) - f _ 2 ( Y ^ 2 _ s , Z ^ 1 _ s , \\psi ^ 1 _ s ) } { \\overline Z _ s } \\ 1 _ { \\overline Z _ s \\neq 0 } . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\frac { \\omega ( V ) } { \\omega ( \\Delta ) } \\leq \\sum _ { j \\in J } \\frac { \\omega \\left ( 5 \\Delta _ { j } \\right ) } { \\omega ( \\Delta ) } \\le C \\sum _ { j \\in J } \\frac { \\omega \\left ( \\Delta _ { j } \\right ) } { \\omega ( \\Delta ) } \\le C \\sum _ { j \\in J } \\omega ^ { A _ { \\Delta } } ( \\Delta _ { j } ) = C \\omega ^ { A _ { \\Delta } } \\left ( \\bigcup _ { j \\in J } \\Delta _ { j } \\right ) \\leq C \\omega ^ { A _ { \\Delta } } ( V ) , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\langle \\rho ^ * ( x y ) \\lambda , v \\rangle = & \\langle \\lambda , \\rho ( x y ) ^ { - 1 } v \\rangle \\\\ = & \\langle \\lambda , ( \\sigma ( x , y ) \\rho ( x ) \\rho ( y ) ) ^ { - 1 } v \\rangle \\\\ = & \\langle \\lambda , \\overline { \\sigma ( x , y ) } \\rho ( y ) ^ { - 1 } \\rho ( x ) ^ { - 1 } v \\rangle \\\\ = & \\langle \\sigma ( x , y ) \\lambda , \\rho ( y ) ^ { - 1 } \\rho ( x ) ^ { - 1 } v \\rangle \\\\ = & \\langle \\sigma ( x , y ) \\rho ^ * ( x ) \\rho ^ * ( y ) \\lambda , v \\rangle \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{gather*} \\int _ { S _ { m - d _ { i } + 1 } } \\cdots \\int _ { S _ { m } } \\int _ { Y _ { i } } \\cdots \\int _ { Y _ { n } } a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) \\\\ \\cdot \\nabla _ { y _ { i } } v _ { i + 1 } \\left ( y _ { i } \\right ) d y _ { n } \\cdots d y _ { i } d s _ { m } \\cdots d s _ { m - d _ { i } + 1 } = 0 \\end{gather*}"}
-{"id": "7454.png", "formula": "\\begin{align*} D _ { \\zeta _ { n } ' } I _ \\lambda ( \\zeta ' , \\mu ' ) & = - \\sum _ { i , j } c _ { i j } \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , ( D _ { \\xi _ { n } ' } V + D _ { \\xi _ { n } ' } \\phi ) . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\partial _ t \\Gamma + b \\cdot \\nabla \\Gamma - \\Delta \\Gamma + \\frac { 2 } { r } \\partial _ r \\Gamma & \\leq 0 \\textrm { i n } \\ \\Pi \\times ( 0 , T ] , \\\\ \\partial _ n \\Gamma + 2 \\Gamma & \\leq 0 \\textrm { o n } \\ \\partial \\Pi \\times ( 0 , T ] , \\\\ \\Gamma & \\leq 0 \\textrm { o n } \\ \\Pi \\times \\{ t = 0 \\} . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} ( \\alpha _ { 2 \\underline { d } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } ) ( s ) = ( \\alpha _ { 2 \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } ) ( s ) = ( \\alpha _ { 2 \\widetilde { \\underline { d } } } \\circ 0 ) ( s ) = 0 \\in V _ { X _ 2 } ( - i A - l B ) , \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\left ( - \\epsilon _ { 1 } ^ { 2 } \\Delta U _ { \\epsilon , y _ { i } } ^ { ( i ) } + V ( x ) U _ { \\epsilon , y _ { i } } ^ { ( i ) } \\right ) + ( - \\epsilon _ { 1 } ^ { 2 } \\Delta \\varphi _ { \\epsilon } + V ( x ) \\varphi _ { \\epsilon } ) = \\Big ( \\sum _ { i = 1 } ^ { k } U _ { \\epsilon , y _ { i } } ^ { ( i ) } + \\varphi _ { \\epsilon } \\Big ) ^ { p } . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} F _ j : = \\{ \\mathbb E ^ { \\omega } _ j ( g ) ^ { q ' } \\leq \\mathbb E ^ { \\omega } _ j ( f ) ^ { p } \\} \\hbox { a n d } G _ j : = \\Omega \\setminus F _ j . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} E _ { ( B , \\beta ) _ 1 } ( z ) = E _ { B , \\beta } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { \\Gamma ( \\beta + k B ) } , \\ ; z \\in \\mathbb { C } , \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} { \\frak M } = \\bigcup _ { q \\le L ^ { B ^ 2 } } \\sideset { } { ^ { * } } \\bigcup _ { 1 \\le a \\le q } J _ { q , a } \\mbox { a n d } \\quad { \\frak m } = \\left ( L ^ { B ^ 2 } / { X } , 1 + L ^ { B ^ 2 } / { X } \\right ] \\setminus { \\frak M } , \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} ( m ( \\Phi ' ) - v ) \\Phi '' = 0 , \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} p ^ H ( x ; s , t ) = 1 / \\sqrt { 2 \\pi { \\Bbb E } \\bigl ( Z _ t ^ H - Z _ s ^ H \\bigr ) ^ 2 } \\exp \\bigl ( - x ^ 2 / 2 { \\Bbb E } \\bigl ( Z _ t ^ H - Z _ s ^ H \\bigr ) ^ 2 \\bigr ) . \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( X ^ x _ { \\tau + \\cdot } ) ] = \\hat { \\mathbb { E } } [ \\varphi ( X ^ y _ \\cdot ) ] _ { y = X ^ x _ { \\tau } } . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} B : = \\sum _ { i = 1 } ^ { n } c _ i E _ { n , i } \\ , c _ i \\in K . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\frac { \\eta ( 0 , t _ { k } ] } { \\eta ( 0 , t _ { k + 1 } ] } \\sim \\mathrm { B e t a } ( \\gamma t _ k , \\gamma ( t _ { k + 1 } - t _ k ) ) , 1 \\le k \\le n , t _ { n + 1 } : = t . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} ( 1 + x ) ^ { 1 / 3 } = 1 + \\frac 1 3 x - \\frac 1 9 x ^ 2 + o ( x ^ 2 ) \\ , , x \\to 0 \\ , , \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} V _ { ( + , m ) } = \\{ v \\in V \\ , : \\ , s _ { i j } ( u ) v = 0 \\ ; \\ ; i < j n - m + 1 \\leq j \\leq n \\} . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} t Y ( t ) = \\left ( \\frac { t } { 1 + X ( t ) } \\right ) ^ { \\langle - 1 \\rangle } , \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} & m ( \\{ f g , a \\} ) \\\\ = & m ( f \\{ g , a \\} + ( - 1 ) ^ { | f | | g | } g \\{ f , a \\} ) \\\\ = & m ( f ) [ h ( g ) m ( a ) - ( - 1 ) ^ { | g | | a | } m ( a ) h ( g ) ] + ( - 1 ) ^ { | f | | g | } m ( g ) [ h ( f ) m ( a ) - ( - 1 ) ^ { | f | | a | } m ( a ) h ( f ) ] \\\\ = & [ m ( f ) h ( g ) + ( - 1 ) ^ { | f | | g | } m ( g ) h ( f ) ] m ( a ) - ( - 1 ) ^ { | f | | a | + | g | | a | } m ( a ) [ m ( f ) h ( g ) + ( - 1 ) ^ { | f | | g | } m ( g ) h ( f ) ] \\\\ = & h ( f g ) m ( a ) - ( - 1 ) ^ { | f g | | a | } m ( a ) h ( f g ) . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ { \\bar { k } } \\xi _ { i } ^ { \\prime } \\right \\} \\in ( 0 , 1 ) \\backslash \\{ 1 / 2 \\} . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\omega \\left ( \\xi , _ { k } \\delta _ { j } ^ { i } + 2 \\xi , _ { j } \\delta _ { k } ^ { i } \\right ) V ^ { , k } + 2 \\eta ^ { i } , _ { t | j } - \\xi , _ { t t } \\delta _ { j } ^ { i } = 0 \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} 5 ( 3 u ^ 2 - 2 \\cdot 5 ^ { 2 b } ) ^ 2 - 1 6 \\cdot 5 ^ { 4 b } = \\pm 1 \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} o ( \\alpha _ 1 , \\alpha ' _ 1 ) = ( \\widehat d ' _ { 1 } \\circ \\widehat \\psi _ { 1 } - \\widehat \\psi _ { 1 } \\circ \\widehat d _ { 1 } ) _ { \\alpha ' _ 1 \\alpha _ 1 } . \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} m ^ { - 1 } ( t ^ { - 1 } z ) - \\xi = t ^ { - 1 } ( z - t m ( \\xi ) ) p ( t ^ { - 1 } z , \\xi ) , \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} P _ n ( x ) - D ( x ) = \\log 2 \\cdot F _ { \\omega _ 0 } ( n ) + \\frac 1 2 \\sum _ { 2 \\le q \\leq X } \\mu ( \\omega _ q ) F _ { \\omega _ q } ( n ) + \\frac 1 4 \\sum _ { 2 \\le q \\leq X } F _ { \\omega _ q } ( n ) ^ 2 . \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m - 1 } \\frac { 1 } { j } \\le 1 + \\ln ( m - 1 ) , \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} ( T ' ) ^ m ( f , g ( x ) : = \\sum _ { j > N } \\int \\left | \\psi _ { a j + m } f * \\Phi _ { a j + m } \\left ( x - \\frac { t ^ a + \\epsilon _ P ( t ) } { 2 ^ { a j } } \\right ) \\right | \\\\ \\left | \\psi _ { b j + m } g * \\Phi _ { b j + m } \\left ( x - \\frac { t ^ b + \\epsilon _ Q ( t ) } { 2 ^ { b j } } \\right ) \\rho ( t ) \\right | \\ , d t . \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\| q \\| _ { H ^ { - 1 } ( \\R ^ { n } ) } \\leq C \\| N _ { A _ { 2 } , q _ { 2 } } - N _ { A _ { 1 } , q _ { 1 } } \\| ^ { \\mu _ { 2 } } , \\mu _ { 2 } = 1 / ( k + 1 ) \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } U _ { \\gamma _ 0 ( r ) } U _ { \\gamma _ 1 ( r ) } ^ { - 1 } = U _ { ( - 1 , 1 ) } \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & \\delta \\end{pmatrix} \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} D = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & - \\frac { 1 } { 2 } & \\delta \\\\ 0 & - \\delta & - \\frac { 1 } { 2 } \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\int _ { [ 0 , T ] \\times [ 0 , 1 ] } \\ell ( \\varphi _ k ^ n ( s , y ) ) \\ , d s \\ , d y \\le I _ T ( \\boldsymbol { \\zeta } ^ n , \\psi ^ n ) + \\frac { 1 } { n } \\le N + \\frac { 1 } { n } . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} f ( x + m _ 1 p ) = f ( x + m _ 2 p ) . \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\sum _ { L ^ { 2 B } < u \\le y } \\left | { \\frak P } _ { \\ell } ( u , L ^ B ) - { \\frak S } _ { \\ell } ( u ) \\right | ^ 2 & = \\sum _ { L ^ { 2 B } < u \\le y } \\left | { \\frak P } _ { \\ell } ^ { \\prime } ( u , L ^ B ) f _ { \\ell } ( u , L ^ B ) - { \\frak S } _ { \\ell } ^ { \\prime } ( u ) f _ { \\ell } ( u ) \\right | ^ 2 \\\\ & \\ll L ^ 2 \\sum _ { L ^ { 2 B } < u \\le y } \\left | { \\frak P } _ { \\ell } ^ { \\prime } ( u , L ^ B ) - { \\frak S } _ { \\ell } ^ { \\prime } ( u ) \\right | ^ 2 + L ^ { 1 0 \\ell } R _ { f } . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} M _ { \\lambda , \\theta } : = M ( \\lambda , \\theta ) = L \\Biggl ( \\dfrac { \\lambda } { \\theta } \\Biggl ) ^ { \\frac { p ^ { - } - 1 } { ( \\eta ^ { + } + \\gamma ^ { + } ) - ( \\beta ^ { + } + \\alpha ^ { + } ) } } \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} \\xi ( \\theta ) = \\xi ( m _ 0 ) + \\frac { 1 } { \\sqrt { n } } \\sum _ { r , s } \\delta _ r ( \\hat { \\theta } ^ M - m _ 0 ) ^ s \\hat { \\xi } ^ { '' } _ { r s } ( m _ 0 ) + \\frac { 1 } { 2 n } \\sum _ { r , s } \\delta _ r \\delta _ s \\hat { \\xi } ^ { '' } _ { r s } ( m _ 0 ) + O _ p \\left ( n ^ { - 1 } \\right ) \\ , . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{gather*} \\lambda _ l ( t , \\boldsymbol { \\theta } ) = \\gamma _ l ( \\boldsymbol { \\theta } ) t ^ 2 + \\mu _ l ( \\boldsymbol { \\theta } ) t ^ 3 + \\ldots , l = 1 , \\ldots , n , \\\\ \\varphi _ l ( t , \\boldsymbol { \\theta } ) = \\omega _ l ( \\boldsymbol { \\theta } ) + t \\psi ^ { ( 1 ) } _ l ( \\boldsymbol { \\theta } ) + \\ldots , l = 1 , \\ldots , n . \\end{gather*}"}
-{"id": "3211.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ r t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } = \\frac { ( q ; q ) _ r } { \\big ( t ^ N ; q \\big ) _ r } g _ r \\big ( q ^ { \\lambda _ 1 } t ^ { N - 1 } , \\dots , q ^ { \\lambda _ N } ; q , t \\big ) . \\end{gather*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left \\langle \\nabla \\psi , \\nabla v \\right \\rangle = F \\left ( v \\right ) \\qquad \\forall v \\in W _ { 0 } ^ { 1 , p ^ { \\prime } } \\left ( \\Omega \\right ) . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} G \\left ( Z , W \\right ) = \\begin{pmatrix} W & 0 \\\\ 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\mathbb E \\| F \\star \\bar { \\mu } \\| \\leq \\mathbb E \\| F \\star \\mu \\| + \\mathbb E \\| F \\star \\nu \\| \\leq \\mathbb E \\| F \\| \\star \\mu + \\mathbb E \\| F \\| \\star \\nu = 2 \\mathbb E \\| F \\| \\star \\nu , \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\widehat { \\mathcal { A } } _ \\varepsilon & : = b ( \\mathbf { D } ) ^ * g ^ { \\varepsilon } ( \\mathbf { x } ) b ( \\mathbf { D } ) , \\\\ \\mathcal { A } _ \\varepsilon & : = ( f ^ { \\varepsilon } ( \\mathbf { x } ) ) ^ * b ( \\mathbf { D } ) ^ * g ^ { \\varepsilon } ( \\mathbf { x } ) b ( \\mathbf { D } ) f ^ { \\varepsilon } ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\{ \\frak m ^ { [ 0 , 1 ] , 1 } _ k \\mid i = 1 , 2 , \\dots \\} \\longrightarrow \\{ \\frak m ^ { j 1 } _ k \\mid i = 1 , 2 \\dots \\} , j = 0 , 1 , \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\sum _ { y \\in B ( x , r ) \\cap \\Gamma ^ n } | f ( y ) - \\bar { f } _ { B } | ^ 2 \\ , m _ n ( y ) \\leq \\ , \\kappa _ 2 \\ , r ^ 2 \\ , \\mathcal { E } _ n ( f \\ , 1 _ { B ( x , 2 r ) } ) \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} f _ 1 ( k ) = 3 \\log _ 2 k + 2 k / \\log _ 2 ( 2 k ) + 3 5 . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} A ^ + _ { n ; i } ( v ) = b ' ( n ; i ) v ^ { n - 1 - i } - b ' ( n - 1 ; i ) v ^ { n - 1 - i } + ( 1 + v ) A ^ + _ { n - 1 ; i } ( v ) + \\frac { 1 - v ^ { n - 1 - i } } { 1 - v } b ( n ; i - 1 ) , \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\aligned & \\widehat { \\mathcal U ^ + } ( { \\rm m o r } ; 3 , 1 ; \\alpha _ 1 , \\alpha _ 3 ) \\\\ = & \\bigcup _ { \\alpha _ 2 \\in \\frak A _ 2 } \\widehat { \\mathcal U ^ + } ( { \\rm m o r } ; 2 , 1 ; \\alpha _ 1 , \\alpha _ 2 ) \\times _ { R _ { \\alpha _ 2 } } \\widehat { \\mathcal U ^ + } ( { \\rm m o r } ; 3 , 2 ; \\alpha _ 2 , \\alpha _ 3 ) \\endaligned \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } E [ X _ { 1 } ^ \\theta ( t ) ] \\big | _ { \\theta = 1 } \\frac { d } { d \\theta } E [ X _ { 3 } ^ \\theta ( t ) ] \\big | _ { \\theta = 1 } . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} 3 x = 3 y & = 2 z = 2 w \\\\ 3 \\bigg ( \\frac { A } { B E } \\bigg ) = 3 \\bigg ( \\frac { A } { C D } \\bigg ) & = 2 \\bigg ( \\frac { B } { C E } \\bigg ) = 2 \\bigg ( \\frac { C } { D E } \\bigg ) . \\\\ \\intertext { P l u g g i n g i n $ t = 0 $ g i v e s t h e i n i t i a l r e q u i r e m e n t s f o r t h i s s e t o f s o l u t i o n s : } \\lambda _ 2 \\lambda _ 5 = \\lambda _ 3 \\lambda _ 4 , \\lambda _ 2 \\lambda _ 4 & = \\lambda _ 3 ^ 2 , 3 \\lambda _ 1 \\lambda _ 5 = 2 \\lambda _ 3 ^ 2 . \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ { c c } } ( \\frac { n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , c } } } { P _ { c } G _ 0 } ) = \\exp \\bigg ( - 2 \\pi \\lambda _ B \\bigg ( \\sum _ { j \\in \\{ L , N \\} } \\sum _ { i = 1 } ^ 3 p _ { G _ i } \\times \\\\ & \\bigg ( \\int _ 0 ^ { \\infty } \\bigg ( 1 - 1 / \\bigg ( 1 + \\frac { n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , c } } G _ i } { G _ 0 N _ j t ^ { \\alpha _ { j , c } } } \\bigg ) ^ { N _ j } \\bigg ) p _ { j , c } ( t ) t d t \\bigg ) \\bigg ) \\bigg ) \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} W _ { \\ell } ( y , H ) : = \\int _ { y } ^ { 2 y } { \\rm d } t \\left | \\sum _ { t < m ^ { \\ell } \\le t + H } e ( m ^ { \\ell } a / q ) \\right | ^ 2 \\ll _ { \\ell , A } H ^ 2 y ^ { \\frac { 2 } { \\ell } - 1 } L ^ { - A - 2 } \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} Z = [ Y , Y ] . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} X ^ { ( p ) } = \\{ f \\in L ^ 0 \\colon | f | ^ p \\in X \\} \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} f _ 0 \\widehat { S } ( \\mathbf { k } ) f _ 0 \\widehat { P } = \\mathcal { A } ^ 0 ( \\mathbf { k } ) \\widehat { P } . \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} \\left | 5 \\Bigl ( \\sum _ { i = 1 } ^ k w _ { \\mu _ i ' , \\zeta _ i ' } + \\varphi _ i \\Bigr ) ^ 4 \\ , D _ { \\zeta _ 1 ' } \\varphi _ 1 \\right | & \\leq C \\left ( w _ { \\mu _ 1 ' , \\zeta _ 1 ' } ^ 4 + \\varepsilon ^ 4 \\right ) \\varepsilon ^ 2 \\\\ & \\leq C \\ , \\varepsilon ^ 2 \\ , w _ { \\mu _ 1 ' , \\zeta _ 1 ' } ^ 4 . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\prod _ { \\ell = 1 } ^ n F _ { \\ell } ( X , u ( X ) ) ^ { k _ { \\ell } } = 1 , \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\| \\phi \\| _ { \\mathcal D _ { q ' } ^ { p ' } ( X ^ * ) } = \\| \\phi \\| _ { ( \\mathcal D _ { q } ^ p ( X ) ) ^ * } , \\ ; \\ ; \\ ; \\phi \\in \\mathcal D _ { q ' } ^ { p ' } ( X ^ * ) . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} \\sum _ { 1 \\leqslant k _ 1 \\leqslant k _ 2 \\leqslant \\cdots \\leqslant k _ m \\leqslant n } \\prod _ { i = 0 } ^ { m } k _ { i } ( n + m - i ) ^ { k _ { i + 1 } - k _ { i } } = { n + m - 1 \\choose m } n ^ { n + m - 1 } , \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} P _ { k , i } : = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k - i + 1 } } u _ { \\sigma ( i ) } ^ { 2 ^ { k - i } } u _ { \\sigma ( 2 ) } ^ { 2 ^ { k - i - 1 } } \\cdots u ^ 1 _ { \\sigma ( k ) } , \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\psi ( ( \\tilde { a } * h ^ * ) * h ) & = \\int _ { s ( V ) } \\psi _ y \\big ( \\big ( \\tilde { a } ( U _ y ) ( h ^ * * h ) ( y ) \\big ) \\cdot \\delta _ { U _ y } \\big ) \\ , d \\mu ( y ) \\\\ & = \\int _ { s ( V ) } | q ( y ) | ^ 2 \\psi _ y \\big ( \\tilde { a } ( U _ y ) \\cdot \\delta _ { U _ y } \\big ) \\ , d \\mu ( y ) . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} 2 \\Biggl \\{ \\sum _ { i = 0 } ^ { k - 1 } a ^ { i } - k a ^ { k - 1 } \\Biggr \\} < ( b - 1 ) \\Biggl \\{ ( k - 1 ) k + \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) a ^ { i } \\Biggr \\} \\ , . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{gather*} \\Phi ^ { A _ k \\nu } ( x _ 1 , \\dots , x _ m ; q , t ) = t ^ { k { m \\choose 2 } } ( x _ 1 \\cdots x _ m ) ^ k \\Phi ^ { \\nu } ( x _ 1 , \\dots , x _ m ; q , t ) , \\\\ \\forall \\ , ( x _ 1 , \\dots , x _ m ) \\in ( \\C \\setminus \\{ 0 \\} ) ^ m . \\end{gather*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\Delta v + e ^ { v } = 0 \\quad \\textrm { i n } \\ \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} 0 = m p d _ 1 + ( m p ) ^ 2 d _ 2 + \\cdots + ( m p ) ^ { k - 1 } d _ { k - 1 } \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\| \\ + F \\| \\geq \\sum _ { F \\in \\ + F } x _ F = 1 + \\binom { n - k + 1 } { k } \\frac { n - k + 1 } { k \\binom { n - k + 1 } { k } } = \\frac { n + 1 } { k } = 1 / p . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} T _ { n } \\left ( t \\right ) = \\mathcal { \\alpha } _ { n } s i n \\left ( \\omega _ { n } t \\right ) + \\beta _ { n } c o s \\left ( \\omega _ { n } t \\right ) , \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\iota _ 1 ( a ) = \\nu ( b ) , \\ , \\ , \\ , \\iota _ 1 ( b ) = \\nu ( a ) , \\ , \\ , \\ , \\iota _ 1 ( \\nu ( a ) ) = b ; \\\\ \\iota _ 2 ( \\nu ( a ) ) = \\nu ( b ) , \\ , \\ , \\ , \\iota _ 2 ( \\nu ( b ) ) = \\nu ( a ) , \\ , \\ , \\ , \\iota _ 2 ( \\iota _ 1 ( c ) ) = \\nu ( c ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} & \\mathbb { E } [ D _ { c t } ( t ) ] = \\bar D _ { c t } = \\tau K , \\\\ & \\mathbb { E } [ D _ { l t } ( t ) ] = \\tau \\sum \\limits _ { n = M + 1 } ^ { N } \\sum \\limits _ { k } 1 - ( 1 - p _ n ) ^ k \\Pr ( K _ t = k ) . \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} F = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k T ^ { i n } f _ i \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\kappa ( r ) = 1 + r ^ q , q \\in [ 2 , \\infty ) , r \\ge 0 . \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} \\Lambda ( z ) & = \\Im ( \\log \\Gamma ( 1 + \\frac { z } { 2 \\pi \\sqrt { - 1 } } ) ) \\\\ & = \\frac { \\gamma z } { 2 \\pi } + \\sum _ { j \\ge 1 } ( - 1 ) ^ j \\frac { \\zeta ( 2 j + 1 ) } { 2 j + 1 } \\left ( \\frac { z } { 2 \\pi } \\right ) ^ { 2 j } \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} 4 \\pi a [ u ] ( x , t ) & = \\int _ { B _ r ( x ) } \\frac { u ( y ) } { | x - y | } \\ ; d y + \\int _ { B _ r ^ c ( x ) } \\frac { u ( y ) } { | x - y | } \\ ; d y \\\\ & \\le \\frac { 1 } { r } \\| u \\| _ { L ^ \\infty ( L ^ 1 ) } + 4 \\pi \\| u \\| _ { L ^ \\infty ( L ^ p ) } r ^ { 2 - 3 / p } , \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\Phi _ { k } ( u ) : = \\frac { \\vartheta ( u + \\hbar / 2 ) } { \\vartheta ( u - \\hbar / 2 ) } \\sum _ { i = 1 } ^ { \\infty } H _ { k } \\otimes \\wp ^ { ( i ) } ( z ) u ^ i \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{gather*} P _ { \\widetilde { \\lambda } } \\big ( x _ 1 , \\dots , x _ m ; N , q , q ^ { \\theta } \\big ) = P _ { \\lambda } \\big ( x _ 1 , \\dots , x _ m ; N , q , q ^ { \\theta } \\big ) ( x _ 1 \\cdots x _ m ) ^ p q ^ { - \\theta p N m } q ^ { \\theta p m ( m + 1 ) / 2 } \\end{gather*}"}
-{"id": "876.png", "formula": "\\begin{align*} ( N _ r V ) _ s & = N _ r ( s , r - 1 ) - ( r - 2 ) N _ r ( s , r ) \\\\ & = \\binom { s } { 2 } \\binom { r - 2 } { 1 } - ( r - 2 ) \\binom { s } { 2 } \\binom { r - 2 } { 0 } \\\\ & = ( r - 2 ) \\binom { s } { 2 } ( 1 - 1 ) = 0 . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} p ( z ) = \\sum _ { j = r } ^ { d } P _ j ( z ) \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\neq 0 } A _ f ( n ) \\sqrt { y } K _ { i t _ j } ( 2 \\pi \\lvert y \\rvert n ) e ^ { 2 \\pi i n x } \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\Phi ( x ) = \\Phi ( \\sqrt { x } \\sqrt { x } ) = \\Phi ( \\sqrt { x } ) \\Phi ( \\sqrt { x } ) = ( \\Phi ( \\sqrt { x } ) ) ^ * \\Phi ( \\sqrt { x } ) = \\abs { \\Phi ( \\sqrt { x } ) } ^ 2 \\ge 0 . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\operatorname { v o l } ( \\rho _ u ) = \\operatorname { v o l } ( \\rho _ 0 ) + \\frac { 1 } { 4 } \\Im ( u \\overline { v } ) + O ( | u | ^ 4 ) = \\operatorname { v o l } ( \\rho _ 0 ) + \\frac { 1 } { 4 } \\Im ( \\tau ) | u | ^ 2 + O ( | u | ^ 4 ) . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\dim ( \\overline { \\Lambda , \\Gamma } ) + \\dim ( \\Lambda \\cap \\Gamma ) = \\dim ( \\Lambda ) + \\dim ( \\Gamma ) . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\langle \\tilde { M } _ { g ^ { * } } \\phi , \\psi \\rangle _ { \\mathcal { P } } = \\langle \\phi , \\tilde { M } _ { g } \\ , \\psi \\rangle _ { \\mathcal { P } } \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\chi ( X ) - \\chi ( X \\cap Q ) - \\chi ( X \\cap H ) + \\chi ( X \\cap Q \\cap H ) = \\begin{cases} ( n - 1 ) - 2 ( n - 1 ) + ( n - 3 ) = - 2 & \\\\ n - 2 ( n - 2 ) + ( n - 2 ) = 2 & \\end{cases} \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\lVert A _ i ( x ) ^ { - 1 } \\rVert ' = \\sup _ { \\lVert v \\rVert _ { E _ i ( x ) } \\le 1 } \\lVert A _ i ( x ) ^ { - 1 } v \\rVert _ { E _ i ( x ) } . \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\sum _ { i , j } c _ { i j } ^ n \\ , \\int _ { \\Omega _ { \\varepsilon _ n } } w _ { \\mu _ { i , n } ^ { \\prime } , \\zeta _ { i , n } ^ { \\prime } } ^ 4 \\ , z _ { i j } ^ n \\ , { \\bf z } _ { k l } ^ n = \\int _ { \\Omega _ { \\varepsilon _ n } } L ( { \\bf z } _ { k l } ^ n ) \\ , \\phi _ n - \\int _ { \\Omega _ { \\varepsilon _ n } } h _ n \\ , { \\bf z } _ { k l } ^ n . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t Z _ t & = \\Delta Z _ t + \\sqrt { 2 } d W _ t \\\\ Z _ 0 & \\equiv 0 \\end{cases} \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial u } { \\partial t } \\left ( x , r , t \\right ) d r + 2 \\pi a \\left ( x , t \\right ) \\frac { \\partial a } { \\partial t } \\left ( x , t \\right ) u \\left ( x , a \\left ( x , t \\right ) , t \\right ) . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 ' & = \\mathcal { R } ' \\cap \\mathcal { G } _ { l _ 1 } = \\mathcal { R } _ 1 \\sqcup \\Delta \\mathcal { R } _ 1 \\\\ \\mathcal { R } _ 2 ' & = \\mathcal { R } ' \\cap \\mathcal { G } _ { l _ 2 } = \\mathcal { R } _ 2 \\setminus \\Delta \\mathcal { R } _ 2 \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\varepsilon _ { n } } { d \\xi ^ { 2 } } - u ^ { 2 } \\varepsilon _ { n } = \\psi \\varepsilon _ { n } - \\frac { G _ { n } } { u ^ { 2 n } } . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} L \\Psi = \\chi _ 1 ( \\tau ) \\Psi \\ . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\partial _ x ( e ^ { - i \\phi } f ) = - i e ^ { - i \\phi } L _ z ^ + f . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\{ x _ i x _ { n + i } ~ , ~ x _ i - x _ { n + i } + \\sum _ { j = 1 } ^ { i - 1 } c _ { j , i } \\cdot x _ { n + j } ~ ~ \\mbox { f o r } ~ ~ 1 \\leq i \\leq n - 1 \\} . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\| f \\| _ { M ( X , Y ) } = \\sup _ { \\| g \\| _ X \\le 1 } \\| f g \\| _ Y . \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} p ^ { k - \\ell } \\le \\frac { p ^ { k - \\ell + j } - 1 } { p ^ j - 1 } \\le \\frac { p ^ { k - \\ell + j } } { p ^ j - 1 } = \\frac { p ^ { k - \\ell } } { 1 - p ^ { - j } } . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} u ( t , x ) = \\big ( U ( t ) g \\big ) ( x ) = & \\ ; \\frac { 1 } { \\Gamma ( \\alpha ) } \\left ( \\frac { t } { 2 } \\right ) ^ { 2 \\alpha } \\int \\limits _ 0 ^ \\infty r ^ { - \\alpha } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } e ^ { - r f ( x ) } g ( x ) \\ , \\frac { d r } { r } \\\\ = & \\ ; \\frac { 2 g ( x ) } { \\Gamma ( \\alpha ) } \\left ( \\frac { t f ( x ) ^ { \\frac { 1 } { 2 } } } { 2 } \\right ) ^ { \\alpha } K _ { \\alpha } \\left ( t f ( x ) ^ { \\frac { 1 } { 2 } } \\right ) \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) = 1 + \\int _ \\R \\frac { \\dd \\widehat \\mu _ \\beta ( x ) } { ( x - \\omega _ \\alpha ( E _ - ) ) ^ 2 } > 1 \\ , , F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) = 1 + \\int _ \\R \\frac { \\dd \\widehat \\mu _ \\alpha ( x ) } { ( x - \\omega _ \\beta ( E _ - ) ) ^ 2 } > 1 \\ , , \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\vartheta = 6 v \\frac { \\partial } { \\partial u } + \\frac { u ^ 2 } { 3 } \\frac { \\partial } { \\partial v } . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\frac { d \\| \\boldsymbol { w } ( t ) \\| _ p ^ p } { d t } = \\left ( \\frac { \\partial \\| \\boldsymbol { w } ( t ) \\| _ p ^ p } { \\partial \\boldsymbol { w } ( t ) ^ \\ast } \\right ) ^ { \\ ! H } \\ , \\frac { d \\boldsymbol { w } ( t ) } { d t } = : \\boldsymbol { b } ( t ) ^ H \\frac { d \\boldsymbol { w } ( t ) } { d t } = 0 \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} F ( z ) - G _ { m , n } ( z ) = \\psi _ { m , n } ( z ) \\sum ^ \\mu _ { s = 1 } \\frac { v _ s } { z - z _ s } \\ , \\frac { 1 } { \\psi _ { m , n } ( z _ s ) } . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\nu _ 1 ^ T \\Gamma \\nu _ 1 & = ( - 1 , 0 ) \\Gamma \\left ( \\begin{array} { c } - 1 \\\\ 0 \\end{array} \\right ) \\\\ & = \\Gamma _ { 1 1 } , \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\begin{array} { c c l l l l l l } \\theta ( q _ a , z _ a ) & = & - & 1 . 6 \\ldots \\times 1 0 ^ { - 1 5 } & + & i \\ , 2 . 8 \\ldots \\times 1 0 ^ { - 1 6 } & = : & \\chi _ 0 ~ , \\\\ \\theta _ z ( q _ a , z _ a ) & = & - & 8 . 0 \\ldots \\times 1 0 ^ { - 1 6 } & + & i \\ , 0 . 0 \\ldots \\times 1 0 ^ { - 1 5 } & = : & \\lambda ^ * ~ . \\end{array} \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\widetilde M ^ i _ t = \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Delta M ^ i _ s , \\ ; \\ ; \\ ; t \\geq 0 , \\ ; i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\sigma = \\int \\sigma _ \\omega \\ , d P ( \\omega ) \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} ( X \\cap Y ) ^ * = X ^ * + Y ^ * , ( X + Y ) ^ * = X ^ * \\cap Y ^ * \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\delta _ { n , 1 } = \\langle s _ { ( n ) } , s _ { ( 1 ^ n ) } \\rangle = \\langle h _ n , e _ n \\rangle = \\sum _ { \\lambda \\vdash n } \\langle \\frac { p _ \\lambda } { z _ \\lambda } , \\frac { \\varepsilon _ { \\lambda } p _ \\lambda } { z _ \\lambda } \\rangle = \\sum _ { \\lambda \\vdash n } \\frac { \\varepsilon _ { \\lambda } } { z _ { \\lambda } } \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\P ( T _ 1 ' \\le t _ 1 , \\ldots , T _ n ' \\le t _ n \\mid N ( t ) = n ) = \\prod _ { k = 1 } ^ n \\left ( \\frac { \\gamma t _ k + k - 1 } { \\gamma t _ { k + 1 } + k - 1 } \\right ) . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { t } } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} - \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = \\mathcal { C } \\left ( x , t \\right ) \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} t \\mapsto \\{ x _ 0 = p x _ 1 + t ( x _ 2 - q x _ 3 ) \\} . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} 1 = c _ s \\int _ 0 ^ \\infty t ^ { - 1 - \\frac s 2 } ( 1 - e ^ { - t } ) d t \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\sum _ { t \\in \\Z / p \\Z } ( 1 - ( \\frac { 1 - t } { p } ) ) ( 1 - ( \\frac { t ( t - 1 ) } { p } ) ) & = \\sum _ { t \\in \\Z / p \\Z } 1 - ( \\frac { 1 - t } { p } ) - ( \\frac { t ( t - 1 ) } { p } ) + ( \\frac { - t ( t - 1 ) ^ 2 } { p } ) \\\\ & \\ge p - 4 \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} \\mathcal { T } = \\left \\{ t _ { a , b } : = \\left ( \\begin{array} { c c c c c } a & 0 & 0 & \\dots & 0 \\\\ 0 & b & 0 & \\dots & 0 \\\\ 0 & 0 & a b & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\dots & a ^ { n - 2 } b \\end{array} \\right ) : a , b \\in \\mathbb { C } ^ { * } \\right \\} \\cong \\mathbb { C } ^ { * } \\times \\mathbb { C } ^ { * } . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} r ^ 2 + q _ { j } r - { p _ { j } ^ 3 \\over 2 7 } = 0 \\ , . \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} \\overline { R ( Q _ t ) } \\supseteq \\overline { R ( Q _ s ) } \\supseteq \\overline { R ( B ) } = \\overline { R ( B B ^ * ) } , \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\nabla _ { \\phi p } R _ 0 { } ^ { \\beta } { } _ { l \\bar { q } m } = \\nabla _ { 0 p } R _ 0 { } ^ { \\beta } { } _ { l \\bar { q } m } + U _ { p s } ^ { \\beta } R _ 0 { } ^ s { } _ { l \\bar { q } m } - U _ { p l } ^ s R _ 0 { } ^ { \\beta } { } _ { s \\bar { q } m } - U _ { p m } ^ s R _ 0 { } ^ { \\beta } { } _ { l \\bar { q } s } . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} L \\Psi = \\lambda \\Psi , \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\phi _ n ( \\lambda _ 1 , x ) \\cdot \\phi _ n ( \\lambda _ 2 , x ) = \\int _ { P ( D ) } \\phi _ n ( \\lambda , x ) \\ > d \\mu _ { n , \\lambda _ 1 , \\lambda _ 2 } ( \\lambda ) \\quad \\quad x \\in D . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\rho \\frac { \\partial v } { \\partial t } + \\frac { \\partial P } { \\partial r } = \\mu \\left ( \\frac { \\partial ^ { 2 } v } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial v } { \\partial r } - \\frac { v } { r ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l w _ { n , s + 1 } + w _ { n , s - 1 } - a _ n w _ { n + 1 , s } - a _ { n - 1 } w _ { n - 1 , s } - b _ n w _ { n , s } = - \\delta _ { s , n } ( 1 - a _ n ^ 2 ) \\prod _ { k = 0 } ^ { n - 1 } a _ k , \\ , n , s \\in \\mathbb { N } , \\ , \\ , s > n , \\\\ w _ { n , n } - b _ n \\prod _ { k = 0 } ^ { n - 1 } a _ k - a _ { n - 1 } w _ { n - 1 , n - 1 } = 0 , n \\in \\mathbb { N } , \\\\ w _ { 0 , t } = 0 , t \\in \\mathbb { N } _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} \\bar W _ i ^ l = V ^ l \\otimes \\underbrace { V ^ l _ \\# \\otimes \\ldots \\otimes V ^ l _ \\# } _ { i - 1 \\mbox { t i m e s } } \\otimes W ^ l _ \\# \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } + \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi \\frac { \\partial \\left ( r v \\right ) } { \\partial r } d r = 0 . \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} r _ k = - { ( - 1 ) ^ k } / { k ! } \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } R _ \\mu ( \\delta | P _ { X Y } ) = R _ \\mu ( P _ { X Y } ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} { \\bf \\Phi } : = \\begin{pmatrix} \\Phi & 0 \\\\ 0 & \\overline \\Phi \\end{pmatrix} \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\dot { q } ^ i = \\lambda ^ a X _ a ^ i ( q ) , \\dot { p } _ j = - p _ i \\lambda ^ a \\frac { \\partial X _ a ^ i } { \\partial q ^ j } , p _ i X _ a ^ i ( q ) = 0 . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} s _ i ^ 2 = e , \\ \\ s _ i s _ j = s _ j s _ i \\ \\mathrm { i f } \\ | i - j | > 1 , \\ \\ s _ i s _ { i + 1 } s _ i = s _ { i + 1 } s _ i s _ { i + 1 } , \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} | e ^ x - e ^ y | = | e ^ y ( e ^ { x - y } - 1 ) | \\leq e ^ y ( | x - y | e ^ { x - y } + | x - y | e ^ { y - x } ) = | x - y | ( e ^ x + e ^ { 2 y - x } ) , \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\partial _ t u + ( - \\Delta ) ^ s u = 0 , 0 < s < 1 \\ , , \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} \\varphi _ U ( p _ n ) = \\sum _ { d | n } d T _ d ( U ^ { \\frac { n } { d } } ) \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} I _ { 1 , a } \\ge \\int _ { 0 } ^ { 1 / 4 c _ r } \\frac { 2 \\sqrt { 2 } c _ r ( 1 - \\delta ) v } { \\pi v } ( 1 - v ) ^ { \\ell ^ 2 } \\ , d v = \\frac { 2 \\sqrt { 2 } c _ r ( 1 - \\delta ) } { \\pi ( \\ell ^ 2 + 1 ) } \\left ( 1 - \\left ( 1 - \\frac { 1 } { 4 c _ r } \\right ) ^ { \\ell ^ 2 + 1 } \\right ) . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\tau _ y : = \\inf \\{ n \\geq 0 : \\ : X ^ 1 _ n = y X ^ 2 _ n = y \\} \\ , . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} & ~ \\epsilon \\chi _ { \\mathbb { R } ^ + } ( x ) e ^ { i ( \\lambda + i \\epsilon ) x } * ( m _ 1 ( \\lambda + ) - 1 ) \\\\ = & ~ \\epsilon \\int _ 0 ^ { \\infty } e ^ { i ( \\lambda + i \\epsilon ) y } ( m _ 1 ( \\lambda + ) - 1 ) ( x - y ) ~ d y \\\\ = & ~ \\int _ 0 ^ { \\infty } e ^ { i \\lambda y / \\epsilon } e ^ { - y } ( m _ 1 ( \\lambda + ) - 1 ) \\left ( x - \\frac { y } { \\epsilon } \\right ) ~ d y \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{gather*} n \\circ \\eta = \\frac { a ^ T N } { | a ^ T N | } . \\end{gather*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\mathfrak A _ { P / X } \\subset \\bigcup _ { i = 1 } ^ N \\{ X _ i = 0 \\} \\quad \\quad \\mathfrak B _ { P / X } \\subset D _ 0 . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} \\Phi _ h \\bigg ( s _ h \\bigg ( \\sum _ { j = 0 } ^ { \\rho ( X ) } x _ j Q _ j ( y _ 1 , \\dots , y _ { \\rho ( X ) } ) \\bigg ) ^ h \\bigg ) & = s _ h \\Phi _ h \\bigg ( \\bigg ( \\sum _ { j = 0 } ^ { \\rho ( X ) } Q _ j ( y _ 1 , \\dots , y _ { \\rho ( X ) } ) x _ j \\bigg ) ^ h \\bigg ) \\\\ & = s _ h \\bigg ( \\sum _ { i = 0 } ^ { \\rho ( X ) } \\sum _ { j = 0 } ^ { \\rho ( X ) } Q _ i ( y _ 0 , \\dots , x _ { \\rho ( X ) } ) Q _ j ( y _ 0 , \\dots , y _ { \\rho ( X ) } ) z _ { i j } \\bigg ) ^ { h / 2 } \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} & X _ k ^ { \\rm N O M A } ( \\nu ) = \\\\ & \\left \\{ \\begin{array} { l l } 1 , & { \\rm i f } \\ R _ { k \\rightarrow \\bar { k } } ^ { \\rm N O M A } ( \\nu ) < \\bar R _ { \\bar k } , \\ R _ { k \\rightarrow k } ^ { \\rm N O M A } ( \\nu ) < \\bar R _ k , \\\\ 1 , & { \\rm i f } \\ R _ { k \\rightarrow \\bar { k } } ^ { \\rm N O M A } ( \\nu ) \\ge \\bar R _ { \\bar k } , \\ R _ k ^ { \\rm N O M A } ( \\nu ) < \\bar R _ k , \\\\ 0 , & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} { F _ { s + 1 } ^ { \\pm } } \\left ( \\xi \\right ) = - { \\dfrac { 1 } { 2 } } \\frac { d { F _ { s } ^ { \\pm } } \\left ( \\xi \\right ) } { d \\xi } - { \\dfrac { 1 } { 2 } } \\sum \\limits _ { j = 0 } ^ { s } { F _ { j } ^ { \\pm } \\left ( \\xi \\right ) F _ { s - j } ^ { \\pm } \\left ( \\xi \\right ) } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { v } \\\\ \\dot { q } \\end{bmatrix} = \\begin{bmatrix} - \\frac { d } { m } & - \\frac { 1 } { m } \\\\ 1 & - s \\end{bmatrix} \\begin{bmatrix} v \\\\ q \\end{bmatrix} + \\begin{bmatrix} \\frac { e _ 1 } { m } \\\\ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ d \\tau _ i D _ i \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) | = O _ m \\ ( \\frac 1 { n ^ { m - k } } \\frac { n ! } { n ^ n } \\ ) . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} a \\cdot x = b \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} A _ 0 = C _ 0 , A _ 1 = t ^ { a _ 1 } C _ 1 , A _ 2 = t ^ { a _ 2 } C _ 2 , A _ 3 = t ^ { a _ 3 } C _ 3 \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( e ^ { i ( k , X ^ { \\beta } ( \\varphi ) ) } \\big ) = E _ { \\beta } \\left ( - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { d } k _ { j } ^ { 2 } | \\varphi _ { j } | _ { L ^ { 2 } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\mathcal { A } ^ 0 : = f _ 0 \\widehat { \\mathcal { A } } ^ 0 f _ 0 = f _ 0 b ( \\mathbf { D } ) ^ * g ^ 0 b ( \\mathbf { D } ) f _ 0 . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} \\tilde a ^ { i j } D _ { i j } \\tilde u = \\tilde g ' - \\tilde b ^ i D _ i \\tilde u = : \\tilde h , \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\min \\left \\| \\begin{bmatrix} E & r \\\\ \\end{bmatrix} \\right \\| _ F , \\textrm { s u b j e c t t o } b - r \\in \\mathcal { R } ( A + E ) . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} L ( f , s ) = \\prod _ p ( 1 - a _ p p ^ { - s } + p ^ { 1 - 2 s } ) ^ { - 1 } . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\pi = y _ { 0 } + \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ( y _ { i } - y _ { 0 } ) , | \\alpha _ { i } | \\leq C ( m , \\rho ) | \\pi - y _ 0 | \\leq C ( m , \\rho ) r , \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} \\phi ( n ) = \\prod _ { p ^ { k _ p } \\parallel \\phi ( n ) } p ^ { k _ p } ; \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n - 1 \\} , \\ , \\forall j \\in \\{ 1 , \\ldots , n - 1 \\} , \\ , \\forall g \\in \\mathcal { E } _ f , \\ \\kappa ( s _ i , s _ j ( g ) ) = \\kappa ( s _ i , g ) . \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} w ( Y _ n ) = w ( Y _ { n - 1 } ) \\cdot ( 1 + x _ n ) ( 1 + x _ { 2 n } ) , \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} { } _ a \\mathtt { B } _ { b , p , c } ( z ) : = \\sum _ { k = 0 } ^ \\infty \\frac { ( - c ) ^ k } { k ! \\ ; \\mathrm { \\Gamma } { \\left ( a k + p + \\frac { b + 1 } { 2 } \\right ) } } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 k + p } . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} Y _ i ( t ) = \\sqrt { s n r _ i } \\int _ 0 ^ t X ( s , M _ 1 , M _ 2 , \\ldots , M _ m ) d s + B _ i ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\phi ( \\pm 1 , y ) = \\phi ( \\pm 1 , - e y ) \\ : \\mbox { i n } \\ : \\pm y > 0 \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\omega _ h : = \\{ x \\in \\Omega : d _ 0 ( x ) < h \\} . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\int \\ , P ( D ^ { ( k ) } + T ) ( \\mathrm { d } T ) \\int \\prod _ { s = 1 } ^ { ( N - k ) / 2 } \\delta \\left ( G _ s - \\mathrm { d i a g } ( x _ s \\pm \\mathrm { i } y _ s ) \\right ) ( \\mathrm { d } G _ s ) = \\prod _ { l = 1 } ^ k w _ r ( \\lambda _ l ) \\prod _ { j = 1 } ^ { ( N - k ) / 2 } w _ c ( x _ j , y _ j ) \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} \\Delta ^ 2 + u \\Delta + v = 0 , \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} A _ j ( p , r ) = \\frac { A ( \\widetilde p , r r _ j ) - q _ j } { r _ j } \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\| v _ i ( t ) \\| _ { p _ 0 + \\delta } \\leq \\widehat C ( t ) t > 0 , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} \\gamma ( \\nabla ^ { \\gamma } _ X Y , Z ) = \\pm \\frac { 1 } { 2 } d \\psi ( X , Y , Z ) . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\int K a _ 0 \\equiv \\int _ 0 ^ \\infty d G \\thinspace K ( F , G ) \\thinspace a _ 0 ( G ) = a _ 1 ( F ) + a _ 2 ( F ) + a _ 3 ( F ) + { \\rm O } \\left ( p ^ 5 \\right ) , \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\int _ { Q } q ( x ) ( \\phi _ { 2 } \\overline { \\phi } _ { 1 } ) ( x , t ) ( b _ { 2 , \\lambda } ^ { \\sharp } \\overline { b _ { 1 } ^ { \\sharp } } _ { \\lambda } ) ( x , t ) \\ , d x \\ , d t = \\displaystyle \\int _ { \\Sigma } ( N _ { A ' _ { 1 } , q _ { 1 } } - N _ { A ' _ { 2 } , q _ { 2 } } ) ( f ) \\overline { v } ( x , t ) \\ , d \\sigma \\ , d t + I _ { \\lambda } , \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\frac { d W } { d t } & \\le \\left | \\frac { d W } { d t } \\right | = | \\nabla w _ \\Omega ( x _ 0 + t n ( x _ 0 ) ) \\cdot n ( x _ 0 ) | \\\\ & \\le | \\nabla w _ \\Omega ( x _ 0 + t n ( x _ 0 ) ) | \\le G ( w _ \\Omega ( x _ 0 + t n ( x _ 0 ) ) ) \\\\ & = G ( W ( t ) ) . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} N _ g ( \\xi ) = g \\xi = \\left ( \\sum _ { n = 1 } ^ \\infty g _ n \\right ) \\xi = \\left ( \\sum _ { n = 1 } ^ \\infty x _ n \\right ) ( \\xi ) = \\abs { x } ( \\xi ) , \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\phi _ + + \\phi _ - = z \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\frac { \\sin ( s \\cdot \\sqrt { \\lambda - b _ d } ) } { s } = \\int _ 0 ^ { \\sqrt { \\lambda - b _ d } } \\cos ( s \\cdot a ) \\ , d a \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ^ { \\gamma , \\sigma } ( z ) = \\sum _ { n = 0 } ^ \\infty \\frac { ( \\gamma ) _ n } { ( \\sigma ) _ n \\Gamma ( \\alpha n + \\beta ) } \\frac { z ^ n } { n ! } , \\ ; \\alpha \\in \\mathbb { R } , \\beta , \\gamma , \\sigma , z \\in \\mathbb { C } , \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\widehat { N } _ { 0 , Q } = \\widehat { P } ( M ^ * ) ^ { - 1 } N _ 0 M ^ { - 1 } \\widehat { P } , \\widehat { N } _ { * , Q } = \\widehat { P } ( M ^ * ) ^ { - 1 } N _ * M ^ { - 1 } \\widehat { P } . \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\alpha _ X + \\beta _ { X ' } = 1 , \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\bigcup _ { G \\leq S _ d } \\{ \\textrm { $ G $ - W e y l C M f i e l d s o f d e g r e e $ 2 d $ } \\} \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} X = Q ( D ^ { ( k ) } + T ) Q ^ T . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\dot \\lambda _ 1 & = 0 \\\\ \\dot \\lambda _ 2 & = - \\lambda _ 0 b _ M \\\\ \\dot \\lambda _ 3 & = - \\lambda _ 0 b _ E + a _ M b _ M k _ M \\lambda _ 1 - k _ M \\lambda _ 2 , \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { n } < \\frac { 2 } { n } < \\cdots < \\frac { n } { n } = 1 . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} Q _ i \\big ( \\mu ( \\omega _ { q _ 0 } ) , \\dots , \\mu ( \\omega _ { q _ { \\rho ( X ) } } ) \\big ) = \\frac { 1 } { 2 } \\Lambda ( q _ i ) \\mu ( \\omega _ { q _ i } ) = \\frac { 1 } { 2 } \\frac { \\Lambda ( q _ i ) } { \\phi ( q _ i ) } \\log \\log x + O \\bigg ( \\frac { \\Lambda ( q _ i ) \\log q _ i } { \\phi ( q _ i ) } \\bigg ) . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\mathcal { X } _ n = \\left \\{ g \\in L ^ \\infty [ a , b ] : g | _ { [ t _ { k - 1 } , t _ k ] } \\ ; \\mbox { i s a p o l y n o m i a l o f d e g r e e } \\ ; \\leq r - 1 , \\ ; k = 1 , \\ldots , n \\right \\} . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\widehat { \\phi } _ k = \\pm \\frac { \\widehat { \\nu } _ k } { e ^ { 2 \\pi i k \\alpha } - 1 } \\ ( k \\neq 0 ) , \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} y = x + \\alpha ( x ) { \\rm w i t h \\ i n v e r s e } x = y + \\breve \\alpha ( y ) \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\mathbb E | N _ { \\infty } - N ^ n _ { \\infty } | ^ p & \\eqsim _ { p } \\mathbb E [ N - N ^ n ] _ { \\infty } ^ { \\frac p 2 } \\\\ & = \\mathbb E \\bigl ( [ N ] ^ c _ { \\infty } + [ N ] ^ q _ { \\infty } + [ N - N ^ n ] ^ a _ { \\infty } \\bigr ) ^ { \\frac { p } { 2 } } \\geq \\mathbb E \\bigl ( [ N ] ^ c _ { \\infty } + [ N ] ^ q _ { \\infty } \\bigr ) ^ { \\frac { p } { 2 } } . \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} E ' ( u , - T ) = \\begin{cases} E ( u , - T ) & \\\\ \\frac { T _ 1 - T } { T _ 1 } ( E ( \\alpha _ + ) + c - E ( u , - T _ 1 ) ) + E ( u , - T _ 1 ) & \\end{cases} \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} v ^ { ( k ) } : = ( \\Phi _ - ) ^ { - \\lambda _ 0 } \\sum _ { \\ell = 1 } ^ k \\frac { ( - \\log \\Phi _ - ) ^ { k - \\ell } } { ( k - \\ell ) ! } \\tilde u ^ { ( \\ell ) } . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} D _ i = \\sum _ { j = 1 } ^ { M } P _ { i , j } D _ { i , j } , \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\widetilde { X } ^ N _ 1 ( t ) = \\xi _ 1 ^ N + \\int _ 0 ^ t \\int _ U f ( s , \\widetilde { X } ^ N _ 1 ( s ^ - ) , u , \\beta ( s ) , \\widetilde { \\mu } _ N ( s ^ - ) ) \\N _ 1 ^ N ( d s , d u ) \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { 1 6 } a _ i ^ 2 = 1 6 \\end{gather*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\mu \\left ( t , \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) ^ { \\frac 1 p } \\right ) = \\left ( \\mu \\left ( t , \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) \\right ) ^ { \\frac 1 p } , \\ \\ \\mu \\left ( t , \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) \\leq \\sum _ { i = 1 } ^ n \\mu \\left ( \\frac { t } { n } , \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) , \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} & a _ { t , \\tau } ( x , s , \\xi ) = \\psi ( \\tau | s | ) \\ , a _ t ( x , s , \\xi ) + [ 1 - \\psi ( \\tau | s | ) ] \\ , | \\xi | ^ { p - 2 } \\xi \\ , , \\\\ & b _ { t , \\tau } ( x , s , \\xi ) = \\Psi _ { \\tau } ( b _ t ( x , s , \\xi ) ) \\ , , \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\frac { f ^ \\prime _ \\alpha ( z ) } { f _ \\alpha ( z ) } = \\psi ( z ) + \\psi ( z + 2 \\alpha ) - 2 \\psi ( z + \\alpha ) , \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} h ^ { \\mathrm { a n } } _ { ( D , g ) } ( f ^ { \\mathrm { a n } } ( \\xi ) ) = h ^ { \\mathrm { a n } } _ { f ^ * ( D , g ) } ( \\xi ) = h ^ { \\mathrm { a n } } _ { d ( D , g ) + \\widehat { ( \\varphi ) } } ( \\xi ) = d h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\langle { \\alpha , \\beta , \\gamma , \\delta , x , y } \\mid { [ x , y ] = x y \\gamma ( x y ) ^ { - 1 } } \\ ! . x \\delta { x ^ { - 1 } } , ~ \\beta . { y } \\beta { y ^ { - 1 } } = \\delta . { x } \\beta { x } ^ { - 1 } \\ ! . x y \\beta ^ { - 1 } ( x y ) ^ { - 1 } \\ ! . [ x , y ] , \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} d N _ s & = \\varGamma _ { t , s - } \\int _ U \\bigl ( V _ s ( e ) + Y _ { s - } \\gamma _ s ( e ) + V _ s ( e ) \\gamma _ s ( e ) \\bigr ) \\widehat { \\pi } ( d e , d s ) \\\\ & + \\varGamma _ { t , s - } ( Z _ s + Y _ s \\beta _ s ) d W _ s + \\varGamma _ { t , s - } d M _ s . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\langle { u , w , x } \\mid { x u ^ 2 = ( x ^ 2 w ) ^ 4 u ^ { - 2 } x ( x ^ 2 w ) ^ 4 } , ~ u = x ^ 2 w u x ^ 2 w , ~ x ^ 2 ( x ^ 2 w ) ^ 9 = u ^ 4 \\rangle . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} | G | \\leq \\sum _ { j = 1 } ^ { \\infty } | q | ^ { j ( j - 1 ) / 2 } / 2 ^ j \\leq \\sum _ { j = 1 } ^ { \\infty } 1 / 2 ^ j = 1 ~ . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log | { \\cal M } _ i ^ { ( n ) } | \\le r _ i , ~ ~ ~ i = 0 , 1 , 2 \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} ( { \\rm i } _ { Q } ) \\iff ( { \\rm i i } _ { Q } ) \\iff ( { \\rm i i i } _ { Q } ) \\quad & { \\rm i f } Q = P . \\\\ \\intertext { F u r t h e r , i t i s o b v i o u s f r o m t h e F r o b e n i u s r e c i p r o c i t y t h a t t h e c o n d i t i o n ( i $ _ { Q } $ ) a u t o m a t i c a l l y h o l d s i f $ Q = G $ ; ( i i $ _ { Q } $ ) a n d ( i i i $ _ { Q } $ ) o b v i o u s l y h o l d . H e n c e } ( { \\rm i } _ { Q } ) \\iff ( { \\rm i i } _ { Q } ) \\iff ( { \\rm i i i } _ { Q } ) \\quad & { \\rm i f } Q = G . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} ( W ^ r _ d ( C ) ) = \\{ L \\in ^ d ( C ) : h ^ 0 ( C , L ) \\ge r + 1 \\} . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { n } \\psi \\left ( \\frac { j } { 2 } \\right ) \\sim { n \\over 2 } \\log n , \\frac { 1 } { 4 } \\sum _ { j = 1 } ^ { n } \\psi ^ { ( 1 ) } \\left ( \\frac { j } { 2 } \\right ) = { 1 \\over 2 } \\log { n } + c _ 1 + o ( 1 ) , \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} P \\left ( x , \\nabla V ^ { \\left ( 1 \\right ) } \\right ) - P \\left ( x , \\nabla V ^ { \\left ( 2 \\right ) } \\right ) = \\widehat { P } \\left ( x , \\nabla V ^ { \\left ( 1 \\right ) } , \\nabla V ^ { \\left ( 2 \\right ) } \\right ) \\nabla \\widetilde { V } \\left ( x \\right ) , \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} D _ { n n n } v = D _ { n n } D _ n v = - \\frac { 1 } { \\bar a ^ { n n } } \\sum _ { ( i , j ) \\neq ( n , n ) } \\bar a ^ { i j } D _ { i j n } v , \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} \\C \\setminus ( - \\infty , 0 ] \\ni z \\mapsto z ^ { \\alpha } : = \\exp ^ { \\alpha \\ln \\abs { z } + i \\alpha \\arg z } . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\phi ^ { 2 } V _ { 2 } , Z ) + g _ { 1 } ( \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\omega \\phi V _ { 2 } ) , \\pi _ * Z ) - g _ { 1 } ( \\mathcal { T } _ { U _ { 2 } } \\omega V _ { 2 } , \\mathcal { B } Z ) \\\\ & - g _ { 2 } ( \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U _ { 2 } } \\omega V _ { 2 } ) , \\pi _ * ( \\mathcal { C } Z ) ) - g _ { 1 } ( V _ { 2 } , \\phi U _ { 2 } ) \\eta ( Z ) . \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\bar V _ i ^ l = V ^ l \\otimes \\underbrace { V ^ l _ \\# \\otimes \\ldots \\otimes V ^ l _ \\# } _ { i \\mbox { t i m e s } } \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{gather*} \\partial ^ 2 _ t Q ( \\overline { \\partial } \\eta ) = Q ( \\overline { \\partial } \\eta ) ( \\partial _ t \\overline { \\partial } \\eta ) ^ 2 + Q ( \\overline { \\partial } \\eta ) \\partial ^ 2 _ t \\overline { \\partial } \\eta = Q ( \\overline { \\partial } \\eta ) ( \\overline { \\partial } v ) ^ 2 + Q ( \\overline { \\partial } \\eta ) \\partial _ t \\overline { \\partial } v , \\end{gather*}"}
-{"id": "550.png", "formula": "\\begin{align*} & \\sum _ { p = 0 } ^ { r - 3 } ( - 1 ) ^ p \\left [ a _ { k - r + 2 + p } ( n - p ) ( n - p - 1 ) + b _ { k - r + 1 + p } ( n - p ) + c _ { k - r + p } \\right ] e _ p \\\\ & + ( - 1 ) ^ { r - 2 } [ a _ k ( n - r + 2 ) ( n - r + 1 ) + b _ { k - 1 } ( n - r + 2 ) + c _ { k - 2 } ] e _ { r - 2 } = 0 , \\\\ & r = 3 , 4 , \\ldots , k . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} E \\left [ g ( X ( t ) ) \\right ] = E \\left [ g ( \\xi ) \\right ] + E \\left [ \\int _ 0 ^ t \\int _ A \\Lambda _ s ^ a g ( X ( s ) ) [ \\widehat { \\gamma } ( s , X ( s ) ) ] ( d a ) d s \\right ] , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} u _ n & : = \\mathbb { P } ( \\Pi \\mbox { r e g e n e r a t e s a t } n ) , \\\\ f _ n & : = \\mathbb { P } ( \\Pi \\mbox { r e g e n e r a t e s f o r t h e f i r s t t i m e a t } n ) . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\Lambda ( s , \\chi ) = W ( \\chi ) ( N ( m ) ) ^ { - 1 / 2 } \\Lambda ( 1 - s , \\overline { \\chi } ) , \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{gather*} \\big [ m ^ { 2 } , j ^ { 2 } \\big ] = 0 . \\end{gather*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\left . \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right \\vert _ { W = O _ { q \\times q } } = I _ { { q ' } ^ { 2 } } . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{gather*} \\overline E ^ 0 = \\big \\{ v _ n \\mid v \\in E ^ 0 n \\in \\Z \\big \\} , \\overline E ^ 1 = \\big \\{ e _ n \\mid e \\in E ^ 1 n \\in \\Z \\big \\} , \\\\ s ( e _ n ) = s ( e ) _ n , \\qquad r ( e _ n ) = r ( e ) _ { n - 1 } . \\end{gather*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\partial _ t f = - \\frac 1 { 2 t } ( z - t v ) \\partial _ z \\chi - \\frac 1 { 2 t } ( y - t v _ y ) \\partial _ y \\chi - v \\partial _ z \\chi - v _ y \\partial _ y \\chi , \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} m s \\le ( 2 s - 1 ) + ( m - 2 ) s + \\left | \\bigcap _ { i = 1 } ^ m { V _ i } \\right | \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} T _ e & = \\begin{cases} e & \\ e \\in s ^ { - 1 } _ E ( H ) , \\\\ \\alpha & \\ e = \\overline { \\alpha } \\in \\overline { F } ( H ) . \\end{cases} \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} p ( z ) = c _ 0 \\det ( J + \\sum _ { j = 1 } ^ { d } z _ j X _ j ) \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} \\sum _ { k , l \\in \\Z } e ^ { - r \\left ( \\alpha ^ 2 k ^ 2 + 2 \\alpha \\beta \\gamma \\ , k l + \\beta ^ 2 ( 1 + \\gamma ) ^ 2 l ^ 2 \\right ) } \\leq \\sum _ { k , l \\in \\Z } e ^ { - r \\left ( \\tfrac { 1 } { \\alpha ^ 2 } k ^ 2 + \\tfrac { 1 } { \\beta ^ 2 } l ^ 2 \\right ) } . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} U _ { \\varepsilon _ i - \\varepsilon _ j } & = \\{ E + \\xi ( E ( i , j ) - E ( - j , - i ) ) \\mid \\xi \\in \\Bbbk \\} , \\\\ U _ { - \\varepsilon _ i + \\varepsilon _ j } & = \\{ E + \\xi ( E ( j , i ) - E ( - i , - j ) ) \\mid \\xi \\in \\Bbbk \\} , \\\\ U _ { \\varepsilon _ i + \\varepsilon _ j } & = \\{ E + \\xi ( E ( i , - j ) - E ( j , - i ) ) \\mid \\xi \\in \\Bbbk \\} , \\\\ U _ { - \\varepsilon _ i - \\varepsilon _ j } & = \\{ E + a ( E ( - j , i ) - E ( - i , j ) ) \\mid a \\in \\Bbbk \\} , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} E & = \\frac { \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 4 ^ 2 \\lambda _ 5 } { A ^ 2 B D ^ 2 } \\\\ & \\approx \\frac { \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 4 ^ 2 \\lambda _ 5 ( M t ) ^ { 6 / 7 } } { m ^ 2 ( \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ) ^ { 1 / 3 } ( M t ) ^ { 2 / 7 } } \\\\ & = M _ E ( M t ) ^ { 4 / 7 } . \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} I _ { \\delta , \\theta } & \\geq ( 1 - \\theta ) \\frac { \\rho } { 2 } \\big \\{ ( 1 - \\nu ) \\ln \\det \\Theta + \\nu \\ln \\det \\Theta - 3 \\ln T _ { \\delta } + \\delta \\big ( \\ln T _ { i n t } - \\ln T _ { \\delta } \\big ) \\big \\} \\cr & = ( 1 - \\theta ) \\frac { \\rho } { 2 } \\big \\{ \\ln \\det \\Theta + \\delta \\ln T _ { i n t } - ( 3 + \\delta ) \\ln T _ { \\delta } \\big \\} . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} \\Delta _ { h _ k e _ n } e _ n \\mid _ { q = 0 } = \\left [ \\sum _ { \\lambda \\vdash n } h _ k [ B _ { \\lambda } ] e _ n [ B _ { \\lambda } ] \\frac { M B _ { \\lambda } \\Pi _ { \\lambda } \\widetilde { H } _ { \\lambda } } { w _ { \\lambda } } \\right ] _ { q = 0 } . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} 0 & = \\frac { 1 } { a } \\left ( \\frac { a ' } { p } \\right ) ' + 2 n \\frac { 1 } { b } \\left ( \\frac { b ' } { p } \\right ) ' \\\\ \\frac { 1 } { p } \\left ( \\frac { a ' } { p } \\right ) ' & = 2 n \\left ( \\frac { a ^ 3 } { b ^ 4 } - \\frac { a ' b ' } { b p ^ 2 } \\right ) \\\\ \\frac { 1 } { p } \\left ( \\frac { b ' } { p } \\right ) ' & = \\frac { 2 n + 2 } { b } - 2 \\frac { a ^ 2 } { b ^ 3 } - \\frac { a ' b ' } { a p ^ 2 } - ( 2 n - 1 ) \\frac { 1 } { b } \\left ( \\frac { b ' } { p } \\right ) ^ 2 . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} u _ { k } \\stackrel { 1 } { = } \\sum _ { j = 1 } ^ { \\infty } \\frac { 2 } { j ^ k ( j + 1 ) ( j + 2 ) } . \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} ( g _ { i j } ( v ) ) _ { 1 \\leq i , j \\leq n } : = ( \\frac { 1 } { 2 } \\frac { \\partial ^ 2 ( F ^ 2 ) } { \\partial v ^ i \\partial v ^ j } ( v ) ) _ { 1 \\leq i , j \\leq n } \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { D _ { 1 - x / t } - K _ { d } } { A _ { d } ( b _ { d } ( t ) ) } = \\lim _ { t \\to \\infty } \\left [ \\frac { { \\bf { I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } - \\frac { { \\bf { I I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } \\right ] = \\frac { c _ { d } K _ { d } } { \\alpha \\rho } ( x ^ { - \\rho / \\alpha } - 1 ) . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\mathtt { h } _ { a , \\nu } ( z ) & = z - \\frac { \\mathrm { \\Gamma } { ( a \\nu + 1 ) } } { 1 ! 2 ^ 2 \\mathrm { \\Gamma } { ( a + a \\nu + 1 ) } } a ^ a z ^ 2 + \\frac { \\mathrm { \\Gamma } { ( a \\nu + 1 ) } } { 2 ! 2 ^ 4 \\mathrm { \\Gamma } { ( 2 a + a \\nu + 1 ) } } a ^ { 2 a } z ^ 3 + \\cdots \\\\ & + ( - 1 ) ^ k \\frac { \\mathrm { \\Gamma } { ( a \\nu + 1 ) } } { k ! 2 ^ { 2 k } \\mathrm { \\Gamma } { ( a k + a \\nu + 1 ) } } a ^ { a k } z ^ { k + 1 } + \\cdots \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} 2 D ' = \\begin{cases} ( 1 + x ) & , \\\\ ( 1 ) & , \\end{cases} \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\frac { d } { d t } r ( \\boldsymbol { \\zeta } ^ { ( i ) } ( t ) ) & = ( \\zeta _ 0 ^ { ( i ) } ) ' ( t ) + \\sum _ { k = 1 } ^ \\infty k ( \\zeta _ k ^ { ( i ) } ) ' ( t ) \\ge \\sum _ { k = 0 } ^ \\infty ( k - 2 ) r _ k ( \\boldsymbol { \\zeta } ^ { ( i ) } ( t ) ) - \\sum _ { k = 1 } ^ \\infty k r _ k ( \\boldsymbol { \\zeta } ^ { ( i ) } ( t ) ) \\\\ & = - 2 \\cdot { { 1 } } _ { \\{ r ( \\boldsymbol { \\zeta } ^ { ( i ) } ( t ) ) > 0 \\} } \\ge - 2 . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} ( S ^ d ) ^ k _ 1 = \\{ x \\in ( S ^ d ) ^ k \\mid \\mathfrak { S } _ k ^ \\pm ( x ) \\neq \\{ e \\} \\} \\ , \\ , \\ , \\ , ( S ^ d ) ^ { \\star k } _ 1 = \\{ \\lambda \\ , x \\in ( S ^ d ) ^ { \\star k } \\mid \\mathfrak { S } _ k ^ \\pm ( \\lambda \\ , x ) \\neq \\{ e \\} \\} \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\imath _ b \\circ g \\ , = \\ , \\nu _ b ( g ) \\circ \\imath _ b \\ \\ \\ , \\ \\ \\ \\imath _ b | A _ \\bullet \\ , = \\ , \\jmath _ b \\ \\ \\ , \\ \\ \\ \\forall b \\in \\Sigma _ 1 ( \\Delta ) \\ , , \\ , g \\in G \\ , , \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\| u _ { x _ 1 , r } - Q _ { x _ 1 } \\| _ { L ^ 1 ( \\partial B _ 1 ) } + \\| u _ { x _ 2 , r } - Q _ { x _ 2 } \\| _ { L ^ 1 ( \\partial B _ 1 ) } & \\leq C ( - \\log ( r ) ) ^ { - \\frac { 1 - \\gamma } { 2 \\gamma } } \\\\ & = C ( - \\log | x _ 1 - x _ 2 | - \\frac { 1 - \\gamma } { 2 \\gamma } \\log ( \\log | x _ 1 - x _ 2 | ) ) ^ { - \\frac { 1 - \\gamma } { 2 \\gamma } } \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { ( \\tau \\wedge T ) + } [ \\varphi ( X _ { \\tau \\wedge T + t _ 1 } ^ x , \\cdots , X _ { \\tau \\wedge T + t _ m } ^ x ) ] = \\hat { \\mathbb { E } } [ \\varphi ( X _ { t _ 1 } ^ y , \\cdots , X _ { t _ m } ^ y ) ] _ { y = X ^ x _ { \\tau \\wedge T } } . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\partial _ { \\zeta _ i ' } A ( \\phi ; \\mu ' , \\zeta ' ) & = - \\partial _ { \\zeta _ i ' } T ( N ( \\phi ; \\mu ' , \\zeta ' ) + E ; \\mu ' , \\zeta ' ) - T ( \\partial _ { \\zeta _ i ' } N ( \\phi ; \\mu ' , \\zeta ' ) ; \\mu ' , \\zeta ' ) \\\\ & - T ( D _ { \\zeta _ i ' } E ; \\mu ' , \\zeta ' ) . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} \\langle v , w \\rangle : = \\int _ 0 ^ \\infty v ( t ) ^ T w ( t ) \\ , d t \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} 0 \\ , = \\big ( \\nabla _ { o u t } f \\cdot [ \\ell \\alpha ] \\big ) ( v ) , v \\in V , \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} J = \\left \\{ \\begin{aligned} a \\varphi , & \\mathfrak { b } _ + \\\\ \\frac { 1 } { a } \\varphi , & \\quad \\mathfrak { b } _ - \\end{aligned} \\right . , \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\Gamma ^ t \\ = \\ \\{ \\sigma = \\lambda ^ t ( z ) \\} \\mbox { f o r } t \\in ( 0 , t _ \\circ ) . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} a = \\left \\{ \\begin{array} { l l } 3 - \\alpha & \\mbox { f o r e q u a t i o n \\eqref { H A M } } \\\\ 1 - \\alpha / 2 & \\mbox { f o r e q u a t i o n \\eqref { P A M } } \\end{array} \\right . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\dot { q } = - s q + u _ 2 , \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\mathcal { C } : = \\bigcap _ { \\{ | \\theta _ { \\boldsymbol { j } , \\boldsymbol { k } } ^ 0 | > \\overline { \\gamma } \\sqrt { \\log { n } / n } \\} } [ \\theta _ { \\boldsymbol { j } , \\boldsymbol { k } } \\neq 0 ] , \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} S ( t , s ) = t ^ { 2 H } + s ^ { 2 H } - \\frac { 1 } { 2 } \\bigl ( ( t + s ) ^ { 2 H } + | t - s | ^ { 2 H } \\bigr ) , \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\log ( e ^ { Q _ 1 } e ^ { Q _ 2 } ) = \\left ( \\begin{matrix} - 0 . 0 5 7 1 7 5 2 & 0 . 0 7 1 8 2 4 8 & 0 . 0 4 9 8 3 4 8 & 0 . 0 4 9 8 3 4 8 \\\\ 0 . 0 2 9 1 0 5 1 & - 0 . 0 9 9 8 9 4 9 & 0 . 0 2 0 0 9 5 1 & 0 . 0 2 0 0 9 5 1 \\\\ 0 . 0 1 0 9 9 6 7 & 0 . 0 1 0 9 9 6 7 & - 0 . 0 9 4 7 0 4 7 & 0 . 0 1 5 8 9 5 3 \\\\ 0 . 0 1 7 0 7 3 4 & 0 . 0 1 7 0 7 3 4 & 0 . 0 2 4 7 7 4 8 & - 0 . 0 8 5 8 2 5 2 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\chi ^ i = ( \\chi _ 1 , \\dots , \\chi _ i + \\chi _ m , \\dots , \\chi _ { m - 1 } , 0 ^ { n - m + 1 } ) . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} 2 ^ { r } \\left ( \\prod _ { i = 1 } ^ { r } L ( i - k , \\chi _ { \\boldsymbol { K } } ^ { i - 1 } ) ^ { - 1 } \\right ) \\left ( \\prod _ { q : { \\rm p r i m e } } \\mathcal { F } _ q ( H ; q ^ { k - 2 r } ) \\right ) . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\tilde { \\rho } ( \\theta _ { 0 1 } , x ) < \\alpha \\right ) = \\alpha + O ( n ^ { - 1 / 2 } ) \\ , . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\widetilde { A } _ k = \\widetilde { U } _ k \\widetilde { \\Sigma } _ k \\widetilde { V } _ k ^ T , \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n = 1 } ^ \\infty f _ n n ^ { - s - \\frac { k - 1 } 2 } \\quad \\Lambda _ g ( s ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n = 1 } ^ \\infty g _ n n ^ { - s - \\frac { k - 1 } 2 } \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} X _ 0 : = X \\setminus X ^ { \\boxminus \\tau } . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} C _ { 6 } = \\max _ { \\stackrel { s , t \\in [ a , b ] } { | u | \\leq \\| \\varphi \\| _ \\infty + \\delta } } \\left | \\frac { \\partial ^ { 2 } \\kappa } { \\partial u ^ 2 } ( s , t , u ) \\right | . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} | \\Delta _ { x , t } | = \\max _ { \\lambda \\in \\Gamma ( T ) } \\min _ { m \\in \\Xi _ { x , t } } \\bigg \\langle \\eta ( \\lambda ) \\otimes \\frac { 1 } { \\Vert \\lambda \\Vert } , m \\bigg \\rangle . \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} D ^ { \\frac { 1 } { 2 } } _ h ( s ) = \\frac { ( 2 \\pi ) ^ { s - \\frac { 1 } { 4 } } } { \\Gamma ( s - \\frac { 1 } { 4 } ) } \\langle P _ h ^ { \\frac { 1 } { 2 } } ( \\cdot , s ) , V \\rangle = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 3 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s - \\frac { 1 } { 4 } } } - \\mathfrak { E } _ h ^ { \\frac { 1 } { 2 } } ( s ) . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} 2 \\ , h \\ , \\Re \\ , G ( 1 / 2 + i \\ , t _ 1 ) = - \\ , \\frac { ( 2 \\ , h ) ^ 3 } { 2 4 } \\ , ( 2 \\ , \\Re \\ , G ( \\xi _ T + i \\ , t _ 1 ) + \\Re \\ , G ( \\xi _ M + i \\ , t _ 1 ) \\ , . \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\Big \\| \\Big ( \\sum _ { i = 1 } ^ k a _ i x _ i ^ 2 - \\sum _ { i = k + 1 } ^ d a _ i x _ i ^ 2 \\Big ) _ + \\Big \\| & \\ge \\Big \\| \\Big ( a _ 1 x _ 1 ^ 2 - \\sum _ { i = k + 1 } ^ d a _ i x _ i ^ 2 \\Big ) _ + \\Big \\| \\ge \\left \\| \\left ( a _ 1 x _ 1 ^ 2 - | X '' | ^ 2 \\right ) _ + \\right \\| \\\\ & = \\frac 1 k \\sum _ { j = 1 } ^ k \\left \\| \\left ( a _ 1 x _ j ^ 2 - | X '' | ^ 2 \\right ) _ + \\right \\| \\ge \\frac 1 d \\left \\| \\left ( a _ 1 | X ' | ^ 2 - | X '' | ^ 2 \\right ) _ + \\right \\| . \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\kappa _ { X ^ + } ( \\zeta , t ) = \\int _ { 0 } ^ \\infty \\left ( \\sum _ { k = 1 } ^ { \\lfloor t \\rfloor } \\bigg ( \\kappa _ X \\left ( e ^ { \\xi } \\eta \\left ( k , \\xi \\right ) \\zeta \\right ) - \\kappa _ X \\big ( \\eta \\left ( k , \\xi \\right ) \\zeta \\big ) \\bigg ) + \\kappa _ X \\big ( \\eta \\left ( \\lfloor t \\rfloor , \\xi \\right ) \\zeta \\big ) \\right ) \\pi ( d \\xi ) , \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\mathtt { f } _ { \\nu } ( z ) & = \\left ( 2 ^ { \\nu } \\Gamma ( \\nu + 1 ) \\mathtt { J } _ { \\nu } ( z ) \\right ) ^ { \\tfrac { 1 } { \\nu } } , \\\\ \\mathtt { g } _ { \\nu } ( z ) & = 2 ^ { \\nu } \\Gamma ( \\nu + 1 ) z ^ { 1 - \\nu } \\mathtt { J } _ { \\nu } ( z ) , \\\\ \\mathtt { h } _ { \\nu } ( z ) & = 2 ^ { \\nu } \\Gamma ( \\nu + 1 ) z ^ { 1 - \\frac { \\nu } { 2 } } \\mathtt { J } _ { \\nu } ( \\sqrt { z } ) . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} L ( s , f \\times \\overline { g } ) : = \\zeta ( 2 s ) \\sum _ { n \\geq 1 } \\frac { a ( n ) \\overline { b ( n ) } } { n ^ { s + k - 1 } } , \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\delta _ { n , j } \\left ( { u , \\xi } \\right ) = \\ln \\left [ { 1 + \\exp } \\left \\{ \\left ( - 1 \\right ) ^ { j + 1 } u \\xi \\right \\} { \\varepsilon _ { n , j } \\left ( { u , \\xi } \\right ) } \\right ] . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} F ( x ) = \\Omega ( G ( x ) ) \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\left \\| \\partial _ { N } w ^ t _ { 2 1 } - \\frac 1 2 f \\right \\| _ { C ^ { k - 2 , \\alpha } ( \\Gamma ^ 0 ) } & \\le C \\epsilon \\left \\| w ^ t _ { 2 1 } \\right \\| _ { C ^ { k - 1 , \\alpha } ( \\overline { \\Omega ^ 0 } ) } + \\| \\partial _ { \\nu ^ 0 , 0 } w ^ t _ { 2 1 } \\| _ { C ^ { k - 2 , \\alpha } ( \\Gamma ^ 0 ) } \\\\ & \\le C \\epsilon \\left \\| f \\right \\| _ { C ^ { k - 2 , \\alpha } ( \\Gamma ^ 0 ) } + C \\| f \\| _ { C ^ { k - 3 , \\alpha } ( \\Gamma ^ 0 ) } \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\big | y ( x ) - \\hat { y } ( x ) \\big | & \\leq | { \\epsilon } | ( x - a ) ^ { \\gamma - 1 } \\sum _ { j = 0 } ^ { \\infty } \\frac { A ^ j ( x - a ) ^ { \\alpha { j } } } { \\Gamma ( \\alpha { j } + \\gamma ) } \\\\ & = | { \\epsilon } | ( x - a ) ^ { \\gamma - 1 } E _ { \\alpha , \\gamma } ( A ( x - a ) ^ { \\alpha } ) , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\int _ { [ 0 , T ] \\times [ 0 , 1 ] } \\ell ( \\varphi _ k ^ \\varepsilon ( s , y ) ) \\ , d s \\ , d y \\le \\sum _ { k = 0 } ^ \\infty \\int _ { [ 0 , T ] \\times [ 0 , 1 ] } \\ell ( \\varphi _ k ( s , y ) ) \\ , d s \\ , d y + \\frac { \\sigma } { 2 } \\le I _ T ( \\boldsymbol { \\zeta } , \\psi ) + \\sigma . \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} M _ h ( x ) = \\sum _ { n \\leq x } ( P _ n ( x ) - D ( x ) ) ^ h \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} X _ k ^ { \\rm O M A I I } ( \\nu ) = \\left \\{ \\begin{array} { l l } 1 , & { \\rm i f } \\ R _ k ^ { \\rm O M A I I } ( \\nu ) < \\bar R _ k , \\\\ 0 , & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\| \\Phi ( t ) \\| \\le C _ { 8 } | t | ^ 3 , | t | \\le t ^ 0 ; C _ { 8 } = \\beta _ { 8 } \\delta ^ { - 1 } \\bigl ( \\| X _ 1 \\| ^ 4 c _ * ^ { - 1 / 2 } + \\| X _ 1 \\| ^ 6 c _ * ^ { - 3 / 2 } + \\| X _ 1 \\| ^ 8 c _ * ^ { - 5 / 2 } \\bigr ) . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} \\theta _ { k } \\circ \\pi ^ * _ k = \\deg \\theta \\cdot { \\rm { i d } } _ { H ^ k ( Y ) } . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} y _ i = h ^ j _ { i } x _ i + \\sum _ { k \\in \\bold { \\Psi } _ { j } ( t ) , k \\neq i } h ^ j _ { k , i } x _ k + n ^ j _ i \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\left \\Vert \\omega \\left ( t \\right ) \\right \\Vert _ { L ^ { 2 } } = \\left \\Vert e ^ { t J L } \\omega \\left ( 0 \\right ) \\right \\Vert _ { L ^ { 2 } } \\leq C \\left \\Vert \\omega \\left ( 0 \\right ) \\right \\Vert _ { L ^ { 2 } } , \\ \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n D _ i ( a ^ { i j } ( x ) D _ j u ) = 0 \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { n = 1 } ^ N G ( S ^ n y ) \\prod _ { i = 1 } ^ k T ^ { i n } f _ i = 0 . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} J ( \\rho _ 3 , m ) & = J ( \\rho _ 1 , m ) P ( \\zeta = 1 ) + J ( \\rho _ 2 , m ) P ( \\zeta = 0 ) \\\\ & \\leq \\theta J ( \\sigma , m ) + ( 1 - \\theta ) J ( \\sigma , m ) = J ( \\sigma , m ) \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\begin{cases} 2 r [ c ] - ( 1 - r + 2 r m ) [ h ] = p [ \\mu ] + q [ \\lambda ] \\\\ 2 r [ c ] - ( 1 - 3 r - 2 r m ) [ h ] = p [ \\mu ] + q ' [ \\lambda ] , \\end{cases} \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathfrak { m } ^ { ( p , p ) } \\big ] = & \\mathbb { E } \\big [ ( O _ \\prec ( \\hat { \\Pi } ^ 2 ) + O _ \\prec ( \\Psi \\hat { \\Upsilon } ) ) \\mathfrak { m } ^ { ( p - 1 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 2 , p ) } \\big ] \\\\ & \\qquad + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 1 , p - 1 ) } \\big ] \\ , . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla ( \\eta u _ k ^ { p / 2 } ) ) & = \\frac { p ^ 2 } { 4 } u _ k ^ { p - 2 } \\eta ^ 2 ( a \\nabla u _ k , \\nabla u _ k ) + p \\eta u _ k ^ { p - 1 } ( a \\nabla u _ k , \\nabla \\eta ) + u _ k ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) , \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} L _ m ( x ) & = \\sum _ { s = 0 } ^ { m - 2 } \\binom { m - 2 } { s } x ^ m F _ T ( x ) C ( x ) \\big ( C ( x ) - 1 \\big ) ^ { s } \\\\ & + \\sum _ { s = 1 } ^ { m - 1 } \\binom { m - 1 } { s } x ^ m \\big ( F _ T ( x ) - 1 \\big ) C ^ 2 ( x ) \\big ( C ( x ) - 1 \\big ) ^ { s - 1 } \\\\ & = x ^ m C ^ { m - 1 } ( x ) F _ T ( x ) + \\frac { x ^ m C ( x ) ^ { m + 1 } \\big ( F _ T ( x ) - 1 \\big ) } { C ( x ) - 1 } - x ^ m \\big ( F _ T ( x ) - 1 \\big ) C ( x ) ^ 2 \\ , , \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} ( 0 ) : ( x ) & = ( z , u ) , \\\\ ( 0 ) : ( z ) & = ( x ) , \\\\ ( z ) : ( x , z ) & = ( x , z , u ) , \\\\ ( x ) : ( x , y ) & = ( x , y , z , u ) , \\\\ ( z ) : ( z , u ) & = ( x , z ) , \\\\ ( z , u ) : ( x , z , u ) & = ( x , y , z , u ) , \\\\ ( x , z , u ) : ( x , y , z , u ) & = ( x , y , z , u ) . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\Delta _ 3 & = \\frac { M _ 3 - S _ 3 ^ f } { \\sigma _ 3 } \\\\ R _ 3 & = \\frac { S _ 3 ^ f } { M _ 3 } \\times 1 0 0 \\ : \\ : ( \\textrm { i n } \\ : \\ \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\mathcal { A } y y = \\lambda x , ~ x \\mathcal { A } y = \\lambda y , ~ x ^ T x = 1 ~ a n d ~ y ^ T y = 1 , \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} G : = \\mu ^ 2 \\frac { | { \\rm R m } _ { \\phi } | _ { \\omega _ { \\phi } } ^ 2 } { L - S } + B S , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} \\tilde { g } ^ { ( - 1 ) } _ i = [ z ] T ^ - _ { i i } ( z ) - \\sum _ { j < i } A ^ - _ { i j } \\cdot \\left ( [ z ] T ^ - _ { j i } ( z ) \\right ) , \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} \\bar { \\sigma } ( z ) = \\tilde { \\sigma } ( z ) + \\pi ^ 2 ( z ) + \\pi ( z ) [ \\tilde { \\tau } ( z ) - \\sigma ' ( z ) ] + \\pi ' ( z ) \\sigma ( z ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\frac { z ' } { z ^ { \\frac { \\alpha - 1 } { \\alpha + 1 } } } = \\frac { \\alpha + 1 } { 2 } \\gamma ( s ) \\left ( 1 + \\epsilon ( s ) \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} C o v \\bigl ( B _ t ^ H , B _ s ^ H \\bigr ) = \\frac { 1 } { 2 } \\bigl ( | s | ^ { 2 H } + | t | ^ { 2 H } - | t - s | ^ { 2 H } \\bigr ) . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} v : \\R ^ { n } \\times \\R ^ { 2 - 2 \\alpha } \\rightarrow \\R , v ( x , y ) = u \\left ( \\norm { y } , x \\right ) , \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\sum _ { r = j + 1 } ^ n \\sum _ { \\substack { s = k + 1 \\\\ s \\neq r } } ^ n c _ { j , r } \\ , c _ { k , s } & = \\sum _ { r = j + 1 } ^ n c _ { j , r } \\left ( \\sum _ { s = k + 1 } ^ n c _ { k , s } - c _ { k , r } \\right ) \\\\ ~ & = \\sum _ { r = j + 1 } ^ n c _ { j , r } \\cdot \\sum _ { s = k + 1 } ^ n c _ { k , s } - \\sum _ { r = j + 1 } ^ n c _ { j , r } \\ , c _ { k , r } \\\\ ~ & = \\sum _ { r = 1 } ^ n c _ { j , r } \\ , c _ { k , r } = : P _ { j k } \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\deg P & = \\deg H ( \\deg u - l ) + ( \\deg G ) s + \\deg F ( l - s ) , \\\\ \\deg Q & = \\deg H ( \\deg u - m ) + ( \\deg G ) t + \\deg F ( m - t ) . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} J ( t ) = \\# \\ , \\textrm { p a r t i c l e s w h i c h j u m p e d f r o m s i t e } 0 \\textrm { t o s i t e } 1 \\textrm { d u r i n g t i m e } [ 0 , t ] . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} F ( x , u ; v ) & = 1 + x u K + x u \\big ( F ( x , u ; 0 ) - 1 + v F ( x , u ; v ) - v \\big ) \\\\ & + \\frac { x } { 1 - v } \\big ( F ( x , u ; 1 ) - v F ( x , u ; v ) \\big ) - x - u x ^ 2 K \\\\ & + \\frac { u x ^ 3 } { ( 1 - x ) ^ 2 } \\left ( \\frac { F ( x , u ; 0 ) + F ( x , u ; 1 ) - 1 } { ( 1 - u x ) ( 1 - u v x ) } - 1 \\right ) \\\\ & + \\frac { u x ^ 3 } { ( 1 - x ) ^ 2 ( 1 - u x ) ( 1 - v u x ) } \\left ( \\frac { 1 } { 1 - x } F \\Big ( x , u ; \\frac { 1 } { 1 - x } \\Big ) - \\frac { 1 } { 1 - x } - F ( x , u ; 1 ) + 1 \\right ) , \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} \\delta ( a ) \\gamma ( b ) ( c ) - ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\delta ( a ) ( c ) & = \\delta ( a ) ( b c ) - ( - 1 ) ^ { | a | | b | } \\gamma ( b ) \\{ a , c \\} \\\\ & = \\{ a , b c \\} - ( - 1 ) ^ { | a | | b | } b \\{ a , c \\} \\\\ & = \\gamma ( \\{ a , b \\} ) ( c ) \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} f _ c ( T _ 0 : T _ 1 ) = ( T _ 0 ^ 2 : T _ 1 ^ 2 + c T _ 0 ^ 2 ) ( c \\in K ) , \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} Q ( r ) \\le \\sum _ { s = 2 } ^ { r } \\exp ( - c \\sqrt { s } ) + \\frac 1 2 . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} r _ { 0 , n } & : = \\frac { 1 } { n } \\log | { \\cal M } _ 0 ^ { ( n ) } | + \\frac { | { \\cal X } | | { \\cal Y } | \\log ( n + 1 ) } { n } + ( \\alpha _ n + \\beta _ n ) , \\\\ r _ { 1 , n } & : = \\frac { 1 } { n } \\log | { \\cal M } _ 1 ^ { ( n ) } | + \\frac { 1 } { n } + 2 ^ { - n \\beta _ n } \\log | { \\cal X } | , \\\\ r _ { 2 , n } & : = \\frac { 1 } { n } \\log | { \\cal M } _ 2 ^ { ( n ) } | + \\frac { 1 } { n } + 2 ^ { - n \\beta _ n } \\log | { \\cal Y } | \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\| u \\| _ { 1 / r } ^ { 2 } + \\sum _ { j = 1 } ^ { N } \\| X _ { j } u \\| _ { 0 } ^ { 2 } \\leq C \\left ( | \\langle P u , u \\rangle | + \\| u \\| ^ { 2 } _ { 0 } \\right ) , \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} | p ( z ) | \\leq | P _ r ( \\vec { 1 } ) | e ^ { - r / 2 } \\exp \\left [ \\Re ( \\sum _ { j = 1 } ^ { 2 } c _ j z _ j ) + B \\| z \\| _ { \\infty } ^ 2 \\right ] \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} ( \\alpha _ { 3 \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } ) ( s ) = \\alpha _ { 3 \\underline { d } } ( s ) \\in V _ { X _ 3 } ( - ( d - l ) B ) . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } U _ i ^ 5 \\ , U _ j & = \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 5 \\ , U _ j + 5 \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 4 \\ , \\pi _ i \\ , U _ j + \\mathcal { R } _ { i , j } ^ 1 , \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\lambda ( t , x , y , a , p ) = a _ y + \\zeta ( p ) , \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t \\eta + \\langle \\nabla _ x \\Phi , \\nabla _ x \\eta \\rangle - \\partial _ z \\Phi = 0 \\ @ \\ z = \\eta ( x , t ) \\cr & \\partial _ t \\Phi + { 1 \\over 2 } | \\nabla \\Phi | ^ 2 + g \\eta + { p \\over \\rho } = 0 \\ @ \\ z = \\eta ( x , t ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\varphi _ 4 = & B ( \\dfrac { 1 } { 3 } , \\dfrac { 1 } { 3 } ) ( 1 - x _ 1 ) ^ { - \\frac { 1 } { 3 } } ( 1 - x _ 2 ) ^ { - \\frac { 1 } { 3 } } \\varphi _ 4 ' , \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\Lambda _ B = - \\Phi _ 2 ^ c + ( F ' _ B ( \\omega _ A ) - 1 ) \\Lambda _ A + \\frac { 1 } { 2 } F '' _ B ( \\omega _ A ) \\Lambda _ A ^ 2 + O ( \\Lambda _ A ^ 3 ) \\ , . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{gather*} f _ { i _ 1 , i _ 2 , i _ 3 } ( x _ 1 , x _ 2 , x _ 3 ; q , 2 ) = 0 , \\textrm { f o r a l l } i _ 1 + i _ 2 + i _ 3 \\leq 2 . \\end{gather*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\left ( \\frac { I _ T ( s n r ) } { s n r } \\right ) ' & = \\frac { 1 } { s n r } \\left ( I ' _ T ( s n r ) - \\frac { I _ T ( s n r ) } { s n r } \\right ) \\\\ & = \\frac { 1 } { 2 s n r } \\left ( \\int _ 0 ^ T E [ ( X ( s ) - E [ X ( s ) | Y _ 0 ^ T ] ) ^ 2 ] d s - \\int _ 0 ^ T E [ ( X ( s ) - E [ X ( s ) | Y _ 0 ^ s ] ) ^ 2 ] d s \\right ) \\leq 0 , \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} U _ { i l } ^ k : = ( \\nabla _ { \\phi i } h \\cdot h ^ { - 1 } ) ^ k { } _ l , \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} ( \\mathfrak { X } _ k ^ { + } ( u , \\lambda ) , \\Phi _ l ( x ) ) = 0 , \\ , \\ ( 1 , \\mathfrak { X } _ k ^ { + } ( u , \\lambda ) ) = 0 . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} L ( f _ s J , \\omega ) f _ s ^ \\ast S _ s = f _ s ^ \\ast ( L ( J , f _ { - s } ^ \\ast \\omega ) S _ s ) = 0 . \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} m _ t = \\frac { l \\circ \\Pi _ t } { l \\ ( \\rho _ t \\ ) } \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} | I _ { t _ 1 } ( y _ 1 ) - I _ { t _ 2 } ( y _ 2 ) | & = \\Big | \\int _ { t _ 2 } ^ { t _ 1 } v ^ n _ { s - } ( x ) p ^ n _ { t _ 1 - s } ( x , y _ 1 ) d s + \\int _ 0 ^ { t _ 2 } v ^ n _ { s - } ( x ) [ p ^ n _ { t _ 1 - s } ( x , y _ 1 ) - p ^ n _ { t _ 2 - s } ( x , y _ 2 ) ] d s \\Big | \\\\ & \\leq \\int _ { t _ 2 } ^ { t _ 1 } p ^ n _ { t _ 1 - s } ( x , y _ 1 ) \\ , d s + \\int _ 0 ^ { t _ 2 } \\big | p ^ n _ { t _ 1 - s } ( x , y _ 1 ) - p ^ n _ { t _ 2 - s } ( x , y _ 2 ) \\big | \\ , d s . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} & f _ 1 ( 0 ) = g ( 0 ) = f _ 2 ( 0 ) = 0 , \\\\ & f _ 1 ( 1 / \\theta ) \\leq g ( 1 / \\theta ) \\leq f _ 2 ( 1 / \\theta ) , \\\\ & f _ 1 ( - 1 / \\theta ) \\leq g ( - 1 / \\theta ) \\leq f _ 2 ( - 1 / \\theta ) , \\\\ \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} B V ( q ) _ { i + \\frac { 1 } { 2 } } = | q ^ { \\xi } _ i ( x _ { i + \\frac { 1 } { 2 } } ) - q ^ { \\eta } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) | , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\left \\langle \\partial _ { t } u \\left ( t \\right ) , u \\left ( t \\right ) \\right \\rangle _ { H ^ { - 1 } ( \\Omega ) , H _ { 0 } ^ { 1 } ( \\Omega ) } d t \\\\ + \\int _ { \\Omega _ { T } } \\left ( \\int _ { \\mathcal { Y } _ { n , m } } a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) d y ^ { n } d s ^ { m } \\right ) \\cdot \\nabla u \\left ( x , t \\right ) d x d t \\\\ = \\int _ { \\Omega _ { T } } f \\left ( x , t \\right ) u \\left ( x , t \\right ) d x d t \\end{gather*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\beta ( x , y , z ) = \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} M ^ { \\otimes _ A m } = ( M ^ { \\otimes _ A m } ) _ D \\oplus ( M ^ { \\otimes _ A m } ) _ C . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , \\partial _ { \\mu _ n ' } V = \\delta _ { j 4 } \\ , \\delta _ { i k } \\int _ { \\R ^ 3 } w _ { \\mu ' , 0 } ^ 4 ( \\partial _ \\mu w _ { \\mu ' , 0 } ) ^ 2 + o ( 1 ) \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\| N _ m \\| _ { 2 \\to 2 } = 1 + O ( m ^ { 1 / 2 } / n ^ { 1 / 2 } + 1 / m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) = - g _ { 1 } ( \\mathcal { T } _ { U } w \\phi Z , V ) + g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } ( w Z ) ) , \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} M ( u ) = \\sum _ { j = 1 } ^ { N ( u ) } L _ { j } ( u - T _ { j } ) \\ , , u \\ge 0 \\ , , \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\gamma _ { i } = C _ { 1 } \\left ( 1 - \\sqrt { 1 - \\lambda { } _ { \\epsilon } } \\right ) ^ { i } + C _ { 2 } \\left ( 1 + \\sqrt { 1 - \\lambda { } _ { \\epsilon } } \\right ) ^ { i } , \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } + \\frac { \\partial A } { \\partial t } = 0 . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} K ^ 2 _ S = 8 \\chi ( { \\mathcal O } _ S ) , \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} ( B * \\omega ) ( A ) : = \\frac { \\omega ( B ^ * A B ) } { \\omega ( B ^ * B ) } , \\omega ( B ^ * B ) > 0 \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} L \\circ \\Gamma _ \\iota = \\Gamma _ \\iota \\circ L \\ \\ \\ \\hbox { i n } \\ A ^ 2 ( X \\times X ) \\ . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} a _ { m k } = \\left [ \\psi _ { k } ^ { \\prime } , \\psi _ { m } \\right ] = \\left \\{ \\begin{array} { c } 1 k = m , \\\\ 0 k < m . \\end{array} \\right . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\overset { \\circ \\circ } S _ k ( X ^ { \\frak C \\boxplus \\tau } ) = S _ k ( X ^ { \\frak C \\boxplus \\tau } , \\widehat { \\mathcal U } ^ { \\frak C \\boxplus \\tau } ) \\cap ( \\mathcal R ^ { \\frak C } ) ^ { - 1 } ( \\overset { \\circ } S _ k ( X , \\widehat { \\mathcal U } ) ) . \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} \\frac { ( \\omega _ 0 + \\sqrt { - 1 } \\partial \\bar { \\partial } \\phi _ { \\epsilon } ) ^ n } { \\omega _ 0 ^ n } = \\frac { \\exp ( \\dot { \\phi _ { \\epsilon } } - F _ 0 - \\gamma \\phi _ { \\epsilon } - \\theta _ X - X ( \\phi _ { \\epsilon } ) ) } { \\prod _ { i = 1 } ^ d ( \\epsilon ^ 2 + | s _ i | _ { H _ i } ^ 2 ) ^ { ( 1 - \\beta ) \\tau _ i } } , \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} C = ( \\sum _ { j = 1 } ^ { 2 } | a ( e _ j ) | ) ^ 2 + \\sum _ { j , k = 1 } ^ { 2 } | \\Re [ a ( e _ j + e _ k ) ] | . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { R } ^ { n } } \\| A x - b \\| { \\rm o r } A x = b , A \\in \\mathbb { R } ^ { m \\times n } , b \\in \\mathbb { R } ^ { m } , \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{gather*} \\| F ( \\tau ) - F ( t ) \\| = { \\sup } _ { x \\in X } d ( f ( \\tau , x ) , f ( t , x ) ) < \\varepsilon \\forall t \\in U . \\end{gather*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\partial _ { t } \\omega ^ { \\nu } & = \\nu \\Delta \\omega ^ { \\nu } - e ^ { - \\nu t } \\left [ \\sin y \\partial _ { x } \\left ( 1 + \\Delta ^ { - 1 } \\right ) \\right ] \\omega ^ { \\nu } + U ^ { \\nu } \\cdot \\nabla \\omega ^ { \\nu } \\\\ & = L \\left ( t \\right ) \\omega ^ { \\nu } + U ^ { \\nu } \\cdot \\nabla \\omega ^ { \\nu } , \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c } 1 & \\frac { 1 } { 3 } & \\frac { 1 } { 5 } \\\\ \\\\ \\frac { 1 } { 1 + n } & \\frac { 1 } { 3 ( 3 + n ) } & \\frac { 1 } { 5 ( 5 + n ) } \\\\ \\\\ \\frac { 1 } { ( 1 + 2 n ) ( 1 + n ) } & \\frac { 1 } { 3 ( 3 + 2 n ) ( 3 + n ) } & \\frac { 1 } { 5 ( 5 + 2 n ) ( 5 + n ) } \\\\ \\end{array} \\right ] \\left [ \\begin{array} { c } a _ 0 \\\\ a _ 2 \\\\ a _ 4 \\end{array} \\right ] = \\left [ \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { n = 1 } ^ N G ( S ^ n y ) S ^ n F ( x , \\ldots , x ) = 0 . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\eta \\circ \\alpha _ g \\ , = \\ , \\alpha ' _ g \\circ \\eta \\ \\ \\ , \\ \\ \\ \\alpha ' _ { \\kappa ( l ) } \\circ \\eta \\circ \\nu _ l \\ = \\ \\nu ' _ l \\circ \\eta \\ . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} i \\partial _ t w = \\lambda t ^ { - 1 } | w | ^ 2 w + \\mathcal { O } ( t ^ { - 1 - } ) \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in \\hat { G } ^ n } | \\hat { 1 _ S } ( \\chi ) | ^ 3 = ( e ^ { 1 / 2 } + o ( 1 ) ) n \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 , \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\omega ^ T _ Q = d \\dot { p } _ i \\wedge d q ^ i + d p _ i \\wedge d \\dot { q } ^ i \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\frac { \\Im F _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) - \\Im \\omega _ \\beta ( z ) } { \\Im \\omega _ \\beta ( z ) } = \\int _ \\R \\frac { 1 } { | x - \\omega _ \\beta ( z ) | ^ 2 } \\ , \\dd \\widehat { \\mu } _ { \\alpha } ( x ) \\ , . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} g _ { k } ( x , y ) = \\exp \\left \\{ \\sum _ { i = 1 } ^ { n } \\sum _ { j > i } y ^ { ( i ) } y ^ { ( j ) } x ^ { ( i , j ) } + \\sum _ { i = 1 } ^ { n } y ^ { ( i ) } x ^ { ( i ) } - \\psi ( x ) \\right \\} , \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\L ^ { - s } [ \\L ^ s , a ] f = [ a , \\L ^ { - s } ] \\L ^ s f , \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} e _ i e _ j = a _ { i j } e _ 1 . \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} f _ 6 = \\big ( z _ 0 : z _ 1 : z _ 3 : - z _ 2 \\big ) \\circ \\big ( z _ 0 z _ 3 : z _ 1 z _ 3 : z _ 1 ^ 2 - z _ 2 z _ 3 : z _ 3 ^ 2 \\big ) \\circ \\big ( z _ 0 : z _ 2 : z _ 1 : z _ 3 \\big ) . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\| u \\| _ { \\Phi } ^ { 2 } = \\int _ { \\Omega } e ^ { - 2 \\lambda \\Phi ( x ) } | u ( x ) | ^ { 2 } L ( d x ) . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} P ( t ) = \\binom { r + t } { r } - \\binom { r + t - d } { r } . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} \\tilde h _ 2 ( x , u , \\varphi ) = \\frac { u + x g _ 2 ( u , \\varphi ) + \\cdots + x ^ { k - 1 } g _ k ( u , \\varphi ) + \\frac { G _ k ( x , x u , \\varphi ) } { x } } { 1 + x f _ 2 ( u , \\varphi ) + \\cdots + x ^ { k - 1 } f _ k ( u , \\varphi ) + \\frac { F _ k ( x , x u , \\varphi ) } { x } } \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} D ( s , S _ f \\times \\overline { S _ g } ) = W ( s ; f , \\overline { g } ) + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } W ( s - z ; f , \\overline { g } ) \\zeta ( z ) \\frac { \\Gamma ( z ) \\Gamma ( s - z + k - 1 ) } { \\Gamma ( s + k - 1 ) } \\ ; d z , \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} c _ { U , V } ( u \\otimes v ) = x . v \\otimes u , u \\in U _ x , v \\in V \\ . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\overline { K ^ * ( u ) } = K ^ * ( \\overline { u } ) \\ , , \\big | K ^ * ( u ) \\big | = \\big | K ^ * ( \\overline { u } ) \\big | . \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} g ' : = \\lim _ { n \\to \\infty } { | E _ n ' | } ^ { 1 / n } \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log | \\tilde { { \\cal M } } _ i ^ { ( n ) } | \\le r _ i + \\nu , ~ ~ ~ i = 0 , 2 \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} \\mu ( f ) = \\mu ( f ; x ) = \\sum _ { p \\leq x } \\frac { f ( p ) } { p } , \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} f ( \\lambda , 0 ) = 0 \\textrm { a n d } \\nabla f _ \\lambda ( 0 ) = 0 , \\forall \\lambda \\in \\R . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\mu _ 3 ^ 2 \\geq \\nu _ 2 \\mu _ 4 ^ 2 \\nu _ 2 : = \\left ( \\sqrt { 1 + \\frac { c _ { 1 , \\infty } } { 2 c _ { 3 , \\infty } } } - 1 \\right ) ^ 2 \\left ( - 1 + \\sqrt { 1 + \\frac { c _ { 3 , \\infty } + c _ { 2 , \\infty } } { c _ { 4 , \\infty } } } \\right ) ^ { - 2 } . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 F \\bigl ( d _ W ( x ) \\bigr ) \\mathrm { d } x = y F ( y + z ) + z F ( y ) \\le \\tfrac 3 5 \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} X : = \\mathrm { P r o j } \\left ( \\mathbb C [ T ^ N , Z _ 0 T ^ N , Z _ 1 T ^ N , Z _ 2 T ^ N , Z _ 3 T ^ N ] \\right ) . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\varphi ^ { A + \\boxplus \\tau } _ { \\frak r \\frak o } ( y ) = ( ( \\varphi ^ { A + } _ { \\frak r \\frak o } ) ^ { \\boxplus \\tau } ( y ' ) , ( t _ i ) _ { i \\in A } ) \\in V _ { \\frak r } ^ + ( p , A ) . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} w _ k = \\int _ { \\R _ + } ( e ^ { - ( u _ k - i v _ k ) x } - 1 ) \\nu ( d x ) = : f ( u _ k , v _ k ) , \\ , k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} N _ { A _ { 1 } , q _ { 1 } } = N _ { A _ { 1 } + \\frac { 1 } { 2 } \\nabla \\psi , q _ { 1 } } , N _ { A _ { 2 } , q _ { 2 } } = N _ { A _ { 2 } - \\frac { 1 } { 2 } \\nabla \\psi , q _ { 2 } } . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} 3 x ^ 2 + 2 d ^ 2 = y ^ n . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} G _ 2 ( x ) & = \\sum _ { k \\geq 0 } \\left ( \\frac { x ^ { k + 2 } } { ( 1 - x ) ^ k } K ( x ) F _ T ( x ) + \\sum _ { j = 0 } ^ { k - 1 } \\frac { x ^ { k + 2 } } { ( 1 - x ) ^ j } \\left ( K ( x ) - \\frac { 1 } { 1 - x } \\right ) F _ T ( x ) \\right ) \\\\ & = \\frac { x ^ 2 ( 1 - 4 x + 5 x ^ 2 - x ^ 3 ) } { ( 1 - 3 x + x ^ 2 ) ( 1 - 2 x ) ( 1 - x ) } F _ T ( x ) \\ , . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} f ( z + e _ i ) = - f ( z ) , \\ , \\ f ( z + \\tau e _ i ) = - e ^ { - \\pi i \\tau \\xi _ i } e ^ { - 2 \\pi i z _ i } f ( z ) . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\tau ^ 1 _ { [ x , y ] } = \\infty \\ , , \\ , \\ , \\tau ^ 2 _ { [ x , y ] } = \\infty \\ , , \\ , \\ , \\sigma = \\infty \\ , , \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} \\langle [ v = 0 ] , g \\rangle = \\frac { i } { 2 \\pi } \\int \\tilde \\partial \\bar { \\tilde \\partial } \\log | v | ^ { 2 } \\wedge g , \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} f _ { j i } = \\prod _ { v \\in W _ { j , i - 1 } ^ { * } } ( x _ { j i } + v ) \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} - a _ 1 ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 + \\mathcal P o l y _ 4 ( 0 , \\zeta ^ 0 , \\bar \\Lambda ^ 0 ) = - \\frac { 1 } { 2 } a _ 1 ( \\bar { \\Lambda } _ 1 ^ 0 ) ^ 2 . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\mbox { I m } \\left ( G \\left ( Z , W \\right ) \\right ) = F \\left ( Z , W \\right ) \\overline { F \\left ( Z , W \\right ) } ^ { t } . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} C ' ( x ) = \\left ( u ( x ) c ' ( x ) + c _ 0 \\sum _ { i = 0 } ^ { n - 2 } x ^ i \\right ) \\bmod { ( x ^ { n - 1 } - 1 ) } , \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} C _ 1 ( \\delta _ 5 , M ( G ) ) \\geq \\frac { 1 } { 4 } , \\ \\alpha ( \\delta _ 5 , M ( G ) , \\delta ' ) \\leq \\frac { 1 9 } { 2 5 } , \\ \\beta ( \\delta _ 5 , \\delta ' ) = \\frac { 1 7 } { 2 0 } , \\ C _ 2 ( \\delta _ d , M ( G ) , \\delta ' ) = \\frac { 3 } { 2 0 } , \\ C _ 3 ( \\delta _ d , \\delta ' ) = \\frac { 3 } { 2 0 } , \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} { \\bar { X } } _ { k } ( t ) & = p _ { k } - { \\bar { B } } _ { k } ( t ) \\geq 0 , k \\in \\mathbb { N } , \\\\ { \\bar { Y } } ( t ) & = \\sum _ { k = 0 } ^ { \\infty } ( k - 2 ) { \\bar { B } } _ { k } ( t ) , \\\\ { \\bar { X } } _ { 0 } ( t ) & = { \\bar { Y } } ( t ) + { \\bar { \\eta } } ( t ) \\geq 0 . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} \\left [ \\ ; . . . \\ ; \\right ] = \\textstyle { 3 \\over 2 } \\displaystyle x ^ { - 1 } \\left ( h { \\partial \\over \\partial h } - g { \\partial \\over \\partial g } \\right ) { \\rm A i } \\left ( x ( 1 + h ) ^ { 2 / 3 } \\right ) { \\rm A i } \\left ( x h ^ { 2 / 3 } ( 1 + g ) ^ { 2 / 3 } \\right ) \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} m ( B _ { n , x , y , c } ^ { ( t ) } , k ) = & m ( B _ { n , x , y , c } ^ { ( t ) } - v _ { 1 } , k ) + m ( B _ { n , x , y , c } ^ { ( t ) } - v _ { 1 } - v , k - 1 ) \\\\ = & m ( B _ { n - 1 , x , y , c } ^ { ( t - 1 ) } , k ) + m ( T , k - 1 ) \\\\ = & m ( B _ { n - 2 , x , y , c } ^ { ( t - 2 ) } , k ) + 2 m ( T , k - 1 ) \\\\ = & \\cdots \\\\ = & m ( B _ { n - t , x , y , c } ^ { ( 0 ) } , k ) + t m ( T , k - 1 ) \\\\ \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\bar \\Lambda | _ { K } = 1 , \\bar Z | _ { K } > 0 , \\bar \\Lambda | _ { K } = - 1 , \\bar Z | _ { K } < 0 , \\bar \\Lambda | _ { K } \\in [ - 1 , 1 ] , \\bar Z | _ { K } = 0 , \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} F _ { r + 1 } ( z ) ^ { 2 } - F _ { r } ( z ) F _ { r + 2 } ( z ) = ( - z ) ^ { r } , \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} e ( G _ c ) & = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { c } f _ i = \\frac { 1 } { 2 } ( \\sum _ { v \\in A ' } d ( v ) + \\sum _ { v \\in A \\backslash A ' } d ( v ) + \\sum _ { v \\in B ' } d ( v ) + \\sum _ { v \\in B \\backslash B ' } d ( v ) ) \\\\ & \\leq \\frac { 1 } { 2 } \\left ( ( i - 1 ) i + ( | A | - i + 1 ) \\alpha + | B ' | ( c - i ) + ( | B | - | B ' | ) ( c - 1 ) \\right ) \\\\ & \\leq i ^ 2 - i ( | B | + \\alpha + 2 ) / 2 + \\alpha ( c + 1 ) / 2 + | B | ( c - \\alpha ) / 2 , \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} \\| \\partial _ t q ^ L _ { 0 , m + 1 / 2 } \\| _ { H } = \\left \\| { q ^ L _ { 0 , m + 1 } - q ^ L _ { 0 , m } \\over \\Delta t } \\right \\| _ { H } \\le \\sup _ { t \\in ( 0 , T ) } \\left \\| { \\partial q ^ L _ 0 \\over \\partial t } \\right \\| _ { H } \\le c h _ L ^ s \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} | N ( x _ i ) \\cap N ( y _ 1 ) | = d ( x _ i ) - | N ( x _ i ) \\setminus N ( y _ 1 ) | \\le ( 2 r - 4 ) - 3 = 2 r - 7 . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} & M \\cos ( \\tau ( t ^ 2 S ) ^ { 1 / 2 } P ) P M ^ * = M _ 0 \\cos ( \\tau ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { 1 / 2 } ) M _ 0 \\widehat { P } , \\\\ & M ( t ^ 2 S ) ^ { - 1 / 2 } \\sin ( \\tau ( t ^ 2 S ) ^ { 1 / 2 } P ) P M ^ * = M _ 0 ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { - 1 / 2 } \\sin ( \\tau ( t ^ 2 M _ 0 \\widehat { S } M _ 0 ) ^ { 1 / 2 } ) M _ 0 \\widehat { P } . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} u _ n = u + v ^ * \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} R _ d ( F ) = \\textrm { R e s } _ { s = 1 } A ( s ) = S \\cdot \\textrm { R e s } _ { s = 1 } \\zeta _ F ( s ) , \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { T - 1 } b _ k = - \\frac { \\det { \\overline C ^ { T - 1 } _ T } } { \\det { \\overline C ^ { T - 1 } } } , \\sum _ { k = 1 } ^ { T } b _ k = - \\frac { \\det { \\overline C ^ { T } _ { T + 1 } } } { \\det { \\overline C ^ { T } } } , \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} b X _ 0 + r Y _ 0 & < \\frac { 4 b ^ 2 } { b X _ 0 - r Y _ 0 } = \\frac { 4 b ^ 2 } { 2 b - 4 } \\\\ & = 2 b + \\frac { 8 b } { 2 b - 4 } = 2 b + 4 + \\frac { 1 6 } { 2 b - 4 } < 2 b + 4 . 1 . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} i \\partial _ t f _ j = \\lambda t ^ { - 1 } | f _ j | ^ 2 f _ j + \\mathcal { O } ( t ^ { - \\frac 5 4 + \\beta } ) , j \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} A '' _ n ( v ) & = A '' _ { n - 1 } ( v ) + B _ n ( v ) + B ' _ n ( v ) - B ' _ { n - 1 } ( v ) , \\\\ B ' _ n ( v ) & = A _ { n - 2 } ( 1 ) v ^ { n - 2 } + A _ { n - 2 } ( 1 ) v ^ { n - 3 } + \\frac { 1 } { 1 - v } ( B ' _ { n - 1 } ( v ) - v ^ { n - 3 } B ' _ { n - 1 } ( 1 ) ) , \\\\ B _ n ( v ) & = B ' _ { n - 1 } ( v ) + \\frac { ( 1 + v ) ^ { n - 3 } - 1 } { v } . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} F ( w ) & = \\frac 1 { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma } F ( \\zeta ) \\Big ( \\frac 1 { \\zeta - ( \\zeta _ 0 + \\mathrm { i } \\tau ) } - \\frac 1 { \\zeta - ( \\zeta _ 0 - \\mathrm { i } \\tau ) } \\Big ) \\mathrm { d } \\zeta \\\\ & = \\int _ \\Gamma F ( \\zeta ) K _ { \\mathrm { i } \\tau } ( \\zeta , \\zeta _ 0 ) \\ , \\mathrm { d } \\zeta . \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\big | { y } _ { 0 } ( x ) - \\hat { y } _ { 0 } ( x ) \\big | & = \\bigg | \\frac { y _ a } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } - \\frac { ( y _ a + { \\epsilon } ) } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } \\bigg | \\\\ \\big | { y } _ { 0 } ( x ) - \\hat { y } _ { 0 } ( x ) \\big | & \\leq \\big | { \\epsilon } \\big | \\frac { ( x - a ) ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) } . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} & \\phi ' m ' ( r + I ) = \\phi ' ( m _ R ( r ) + Q ) = \\phi ( m _ R ( r ) ) = f \\pi ' ( r ) = f ( r + I ) , \\\\ & \\phi ' h ' ( r + I ) = \\phi ' ( h _ R ( r ) + Q ) = \\phi ( h _ R ( r ) ) = g \\pi ' ( r ) = g ( r + I ) . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} \\min _ { Q \\in \\mathcal { P } ( A ) } \\int _ A H ( t , x , a , p , g ) Q ( d a ) \\geq \\min _ { Q \\in \\mathcal { P } ( A ) } \\int _ A H ( t , x , a ^ \\ast , p , g ) Q ( d a ) = H ( t , x , a ^ \\ast , p , g ) , \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { ( 1 - x ) ^ 2 ( 1 - 4 x + 6 x ^ 2 - 5 x ^ 3 + x ^ 4 ) \\ , C ( x ) - 1 + 6 x - 1 4 x ^ 2 + 1 5 x ^ 3 - 8 x ^ 4 + x ^ 5 } { x ( 1 - 3 x + x ^ 2 ) ( 1 - x + x ^ 3 ) } \\ , . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\widetilde { Y } _ t & = \\widetilde { \\xi } + \\int ^ T _ t \\widetilde { f } ( s , \\widetilde { Y } _ s , \\widetilde { Z } _ s , \\widetilde { V } _ s ) d s + \\int _ t ^ T d \\widetilde { R } _ s - \\int ^ T _ t \\widetilde { Z } _ s d W _ s \\\\ [ - 1 p t ] & - \\int _ { t } ^ { T } \\int _ U \\widetilde { V } _ { s } ( e ) \\widehat \\pi ( d e , d s ) - \\int _ t ^ T d \\widetilde { M } _ s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} ( X \\cap U ) ^ { \\boxplus \\tau } = \\bigcup _ { x \\in \\psi ( s ^ { - 1 } ( 0 ) ) } \\left ( ( s _ x ^ { \\boxplus \\tau } ) ^ { - 1 } ( 0 ) / \\Gamma _ x \\right ) / \\sim . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\mathbb E ( \\| F \\star \\bar { \\mu } \\| ^ p \\cdot \\mathbf 1 _ { A } ) = \\mathbb E \\| ( F \\cdot \\mathbf 1 _ { A } ) \\star \\bar { \\mu } \\| ^ p \\eqsim _ p \\mathbb E \\| F \\cdot \\mathbf 1 _ { A } \\| ^ p \\star \\nu = \\mathbb E ( \\| F \\| ^ p \\star \\nu \\cdot \\mathbf 1 _ { A } ) . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\Phi = 0 & \\{ - h < y < \\eta ( t , x ) \\} \\\\ \\partial _ y \\Phi = 0 & y = - h \\\\ \\Phi = \\psi & \\{ y = \\eta ( t , x ) \\} \\ , . \\end{cases} \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} w _ { \\rm i m p . } ( x ) : = \\int _ { \\Gamma _ 0 } d \\mathcal { H } ^ { n - 1 } ( y ) \\frac { \\Theta } { ( N \\cdot \\nu ) } ( y ) \\ , P ( x - y ) , \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\begin{aligned} Y ( t , \\zeta ) & : = \\Xi ( t , \\zeta ) \\Psi ( t ) + t ( Z \\Xi ( t , \\zeta ) + \\Xi ( t , \\zeta ) Z ^ * ) ( t ^ 3 K + \\Psi ( t ) ) \\\\ & - t ^ 3 \\Xi ( t , \\zeta ) N \\Xi ( t , \\zeta ) ( t ^ 3 K + \\Psi ( t ) ) + \\mathcal { J } ( t , \\zeta ) \\left ( t ^ 2 S P + t ^ 3 K + \\Psi ( t ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} g _ { k } ( x , y _ { k } ) = \\exp \\left [ \\sum _ { i = 1 } ^ { n } \\left \\{ \\sum _ { j = 1 } ^ { p } y _ { k } ^ { ( i ) } z _ { k } ^ { ( i , j ) } x ^ { ( j ) } - \\log \\left ( 1 + e ^ { \\sum _ { j = 1 } ^ { p } z _ { k } ^ { ( i , j ) } x ^ { ( j ) } } \\right ) \\right \\} \\right ] . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\langle \\Psi ( z _ 0 , z _ 1 ) , ( \\xi _ 0 , \\xi _ 1 ) \\rangle _ { X \\times X } = \\int _ 0 ^ T \\langle \\mathcal { B } \\phi ( t ) , D _ \\nu \\xi ( t ) - \\int _ t ^ T \\ H ( s - t ) D _ \\nu \\xi ( s ) d s \\rangle _ Y \\ d t \\ , . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ 2 ^ { 7 / 2 } \\int _ r ^ \\infty \\frac { \\sinh s } { \\sqrt { \\cosh 2 s - \\cosh 2 r } } & \\cdot \\frac { - 1 } { 2 \\pi \\sinh s } \\partial _ s \\left ( \\frac { - 1 } { 2 \\pi \\sinh 2 s } \\partial _ s \\right ) ^ 3 \\cdot \\\\ & \\cdot \\left ( \\frac { - 1 } { 2 \\pi \\sinh s } \\partial _ s \\right ) ^ { 4 } \\frac { \\sin ( s \\cdot \\sqrt { \\lambda - 1 1 ^ 2 } ) } { \\pi s } \\ , d s \\ , . \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\lambda ( \\beta ) : = \\limsup _ { t \\to \\infty } \\frac { 1 } { t } \\log E | u ^ { h , \\beta } ( t , x ) | ^ 2 < \\infty . \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | p _ n ( i y ) | = \\exp ( \\gamma y ^ 2 ) . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} \\Lambda _ i : = \\mu _ i ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} c _ { j , k } \\cdot \\sum _ { \\substack { r , s = 1 \\\\ r < s } } ^ n c _ { k , r } \\ , c _ { k , s } = : Q _ { j k } . \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} - \\Delta _ g u _ k + c ( n ) R ( g ) u _ { k } = c ( n ) R ( g _ { k } ) u _ k ^ \\frac { n + 2 } { n - 2 } , \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} e r r _ 2 ( t ) & = \\ , - \\sum _ { e \\in E ( v ) } \\frac { 2 C ^ n _ { e e } } { L ^ e } \\int _ 0 ^ t \\big ( u ^ n _ { s - } ( x ^ e _ 2 ) - u ^ n _ { s - } ( x ^ e _ 1 ) \\big ) \\ , \\nabla _ L \\phi _ s ( x ^ e _ 1 ) \\ , d s . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\quad \\left ( H ( 0 ) \\right ) ^ { t } \\equiv \\begin{pmatrix} 0 & 0 & \\dots & 0 & b _ { 1 } & 0 & \\dots & 0 \\\\ 0 & 0 & \\dots & 0 & 0 & b _ { 2 } & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & b _ { q } \\end{pmatrix} , \\quad \\mbox { w h e r e $ b _ { 1 } , b _ { 2 } , \\dots , b _ { q } \\in [ 0 , 1 ) $ . } \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq t \\leq T } Z _ 2 ( t ) \\leq d | | \\N _ { \\rho ^ { \\widehat { \\gamma } , X } } - \\N _ { \\rho ^ { \\widehat { \\gamma } , Y } } | | _ { T V } = d \\sup _ { E \\subset [ 0 , T ] \\times U \\times A } \\left | \\N _ { \\rho ^ { \\widehat { \\gamma } , X } } ( E ) - \\N _ { \\rho ^ { \\widehat { \\gamma } , Y } } ( E ) \\right | . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} g _ 0 ( s ) = \\int _ { 0 } ^ { \\infty } \\ ! \\ ! \\ ! \\ ! \\int _ { 0 } ^ { \\infty } \\ ! \\ ! \\ ! \\ ! \\int _ { 0 } ^ { \\infty } \\ ! \\ ! \\ ! \\ ! \\int _ { 0 } ^ { \\infty } e ^ { - \\sum _ { i = 0 } ^ 3 \\tau _ i } \\overset { \\scriptstyle 3 } { \\underset { \\scriptstyle i = 1 } \\Pi } \\tau _ i ^ { \\frac { a _ i } 4 - 1 } \\left ( \\int _ { c - \\i \\infty } ^ { c + \\i \\infty } e ^ { - x t } d t \\right ) \\ \\frac { d \\tau _ 0 } { \\tau _ 0 } d \\tau _ 1 d \\tau _ 2 d \\tau _ 3 . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} H ^ { o p t } = \\arg \\min _ { H \\in { ( q ) ^ * } ^ { d _ c } } \\{ S _ 4 ( H ) / ( S _ 3 ( H ) = 0 , S _ 2 ( H ) = 0 ) \\} . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} X ( t ) = \\int _ { \\R _ + } e ^ { - \\xi t } \\int _ { - \\infty } ^ { \\xi t } e ^ { s } \\Lambda ( d \\xi , d s ) = \\int _ { \\R _ + } \\int _ { \\R } e ^ { - \\xi t + s } \\mathbf { 1 } _ { [ 0 , \\infty ) } ( \\xi t - s ) \\Lambda ( d \\xi , d s ) . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\sum _ { i \\in \\Z } \\Big ( \\sup _ { n \\ge 0 } \\frac 1 { A _ { n } } | \\sum _ { j = 0 } ^ { n } a _ j v _ { i + j } | \\Big ) ^ p \\le C ^ p \\sum _ { i \\in \\Z } | v _ j | ^ p \\ , . \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\widetilde f ( s ) = \\sum _ { n \\geq 0 } \\mathbb E \\bigl ( f ( s ) \\big | \\mathcal F _ { \\frac { n } { 2 ^ m } } \\bigr ) \\mathbf 1 _ { s \\in ( \\frac { n } { 2 ^ m } , \\frac { n + 1 } { 2 ^ m } ] } , \\ ; \\ ; \\ ; s \\geq 0 . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} A ( \\psi , \\phi ) ( x , p ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\R ^ n } e ^ { - \\tfrac { i } { \\hbar } p y } \\psi ( y + \\tfrac { 1 } { 2 } x ) \\overline { \\phi ( y - \\tfrac { 1 } { 2 } x ) } \\ , d ^ { n } y . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} h _ R ( x _ i ^ { 2 k + 2 } ) & = h _ R ( x _ i ^ { 2 k + 1 } ) x _ i + ( - 1 ) ^ { | x _ i ^ { 2 k + 1 } | | x _ i | } h _ R ( x _ i ) x _ i ^ { 2 k + 1 } \\\\ & = \\bigtriangleup ( 2 k + 1 ) y _ i x _ i ^ { 2 k } x _ i + ( - 1 ) ^ { | x _ i ^ { 2 k + 1 } | | x _ i | } y _ i x _ i ^ { 2 k + 1 } \\\\ & = ( k + 1 + ( k + 1 ) ( - 1 ) ^ { ( 2 k + 1 ) | x _ i | ^ 2 } ) y _ i x _ i ^ { 2 k + 1 } \\\\ & = \\bigtriangleup ( n + 1 ) y _ i x _ i ^ { n } , \\\\ \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} 1 = \\sum \\limits _ { N \\in 2 ^ { \\delta \\Z } } P _ N . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} g ^ * _ F ( \\alpha , \\beta ) : = \\frac { n + 2 } { \\lambda ( B _ x M ) } \\int _ { B _ x M } \\alpha ( v ) . \\beta ( v ) \\ d \\lambda ( v ) , \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} p = \\frac { \\log n - \\log \\left ( f + 1 \\right ) - h } { n } = \\frac { \\log n / 2 + \\log \\log n + \\left ( 1 + o \\left ( 1 \\right ) \\right ) \\log g } { n } . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} 3 x + d \\sqrt { - 6 } = \\frac { \\gamma ^ n } { 3 ^ { ( n - 1 ) / 2 } } . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} C = \\{ ( q ^ 1 , q ^ 2 , p _ 1 , p _ 2 ) \\in T ^ * \\mathbb { R } ^ 2 : p _ 2 = 0 \\} . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\begin{aligned} & { q } ^ { T } _ { i } ( x _ { i + 1 / 2 } ) = \\bar { q } _ { m i n } + \\dfrac { \\bar { q } _ { m a x } } { 2 } \\left ( 1 + \\theta \\dfrac { \\tanh ( \\beta ) + A } { 1 + A ~ \\tanh ( \\beta ) } \\right ) \\\\ & { q } ^ { T } _ { i } ( x _ { i - 1 / 2 } ) = \\bar { q } _ { m i n } + \\dfrac { \\bar { q } _ { m a x } } { 2 } \\left ( 1 + \\theta ~ A \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} \\mathbb { E } [ U ( t ) | Q ( t ) ] = \\lambda + \\theta , \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{gather*} \\lim _ { q \\rightarrow 1 } { \\frac { ( x q ^ a ; q ) _ { \\infty } } { ( x q ^ b ; q ) _ { \\infty } } } = ( 1 - x ) ^ { b - a } \\end{gather*}"}
-{"id": "6560.png", "formula": "\\begin{align*} M ( X ) = M _ 0 ( X ) \\oplus \\cdots \\oplus M _ { 2 n } ( X ) \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} \\psi = \\frac { 1 } { 1 2 } \\frac { d \\Delta } { \\Delta } , \\omega = \\frac { 3 } { 2 } \\Big ( \\frac { 2 u \\ , d v - 3 v \\ , d u } { \\Delta } \\Big ) \\in \\Omega ^ 1 ( X ) \\ . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} w ^ { t _ m } ( \\infty ) = \\frac 1 2 \\lim _ { x \\to \\infty } \\delta ^ 2 _ { t _ m } \\tilde u ^ 0 ( \\infty ) = \\frac 1 2 \\delta ^ 2 _ { t _ m } c ^ 0 \\rightarrow \\frac 1 2 \\ddot c ^ 0 \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\int _ { \\Gamma } \\frac { F ( \\zeta ) } { \\zeta - \\alpha } \\ , \\mathrm { d } \\zeta & = \\int _ { \\Gamma } \\frac { f \\big ( \\Psi ( \\zeta ) \\big ) \\big ( \\Psi ' ( \\zeta ) \\big ) ^ { \\frac 1 p } } { \\zeta - \\alpha } \\ , \\mathrm { d } \\zeta \\\\ & = \\int _ { \\mathbb { R } } \\frac { f ( t ) \\big ( \\Phi ' ( t ) \\big ) ^ { \\frac 1 q } } { \\Phi ( t ) - \\alpha } \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{gather*} J '' + \\overset { \\mathrm { s f } } W J = 0 . \\end{gather*}"}
-{"id": "4390.png", "formula": "\\begin{align*} ( P ^ { \\pi } ( R + P ) ) ^ d = P ^ { \\pi } ( R + P ) ^ d \\quad \\quad ( P ^ { \\pi } ( R + P ) ) ^ { \\pi } = 1 - P ^ { \\pi } ( R + P ) ( R + P ) ^ { d } . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( \\mathcal C _ t P ) & = l _ { \\langle \\rangle } ( P ) , & o _ { \\langle \\rangle } ( \\mathcal C _ t P ) & = \\vartheta ( t + \\Omega ^ { o _ { \\langle \\rangle } ( P ) } ) , \\\\ d ( \\mathcal C _ t P ) & = 1 , & \\langle h _ 0 ( \\mathcal C _ t P ) , h _ 1 ( \\mathcal C _ t P ) \\rangle & = \\langle t + \\Omega ^ { o _ { \\langle \\rangle } ( P ) } , 0 \\rangle \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} y ' ( r ) = - Q _ { \\infty } A ^ * Q _ { \\infty } ^ { - 1 } y ( r ) , r \\in \\ , ] - \\infty , 0 ] . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\hat { y } _ { m } ( x ) = \\hat { y } _ { 0 } ( x ) + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } ( x - t ) ^ { \\alpha - 1 } f ( t , \\hat { y } _ { m - 1 } ( t ) ) d t . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} r a n k \\left ( \\left [ \\begin{array} { c c c c } \\otimes & 0 & \\times & \\times \\\\ \\times & \\otimes & 0 & 0 \\\\ \\times & \\times & \\otimes & 0 \\end{array} \\right ] \\right ) = 3 \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} I _ { m - 1 } ( t ) & = \\int _ 0 ^ \\infty \\int _ 0 ^ { \\xi t } \\left ( 1 - e ^ { - w } \\right ) ^ { m - 1 } d w \\xi ^ { - m } \\pi ( d \\xi ) \\\\ & = \\int _ 0 ^ \\infty \\left ( 1 - e ^ { - w } \\right ) ^ { m - 1 } \\int _ { w / t } ^ { \\infty } \\xi ^ { - m } \\pi ( d \\xi ) d w . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} { \\mathcal U '' _ p } = { \\mathcal U ^ { \\boxplus \\tau } _ { \\mathcal R ( p ) } } \\vert _ { U ^ { \\boxplus \\tau _ 1 } _ { p ' } \\cap U _ p ( \\tau _ 2 ) } . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\sum _ { h \\geq 1 } \\frac { \\rho _ j ( h ) } { h ^ { s + w - \\frac { 1 } { 2 } } } & = L ( s + w - \\tfrac { 1 } { 2 } , \\mu _ j ) \\\\ \\sum \\frac { \\sigma _ { 1 - 2 w } ( h ) } { h ^ { s + \\frac { 1 } { 2 } - w } } & = L ( s , E ( \\cdot , w ) ) = \\zeta ( s + w - \\tfrac { 1 } { 2 } ) \\zeta ( s - w + \\tfrac { 1 } { 2 } ) , \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} 2 A y = - B B ^ * Q _ \\infty ^ { - 1 } y \\forall y \\in R ( Q _ \\infty ) \\subseteq D ( A ) . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\sum _ { T \\leq \\lvert t _ j \\rvert \\leq 2 T } \\rho _ j ( h ) \\langle \\mu _ j , V \\rangle = \\sum _ { T \\leq \\lvert t _ j \\rvert \\leq 2 T } \\rho _ j ( h ) \\langle \\mu _ j , \\theta ^ { 2 k + 1 } \\overline { \\theta } y ^ { \\frac { k + 1 } { 2 } } \\rangle \\ll T ^ { 3 k + 8 + \\epsilon } . \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\mathfrak o _ { \\xi } : = \\{ \\alpha \\in \\kappa ( \\xi ) \\mid v _ { \\xi } ( \\alpha ) \\leqslant 1 \\} \\quad \\mathfrak m _ { \\xi } : = \\{ \\alpha \\in \\kappa ( \\xi ) \\mid v _ { \\xi } ( \\alpha ) < 1 \\} . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} P _ n ( x ) & = \\log 2 \\cdot \\omega _ 0 ( n ) + \\frac 1 4 \\sum _ { 2 \\le q \\le X } \\omega _ { q } ( n ) ^ 2 \\Lambda ( q ) \\\\ & = \\log 2 \\cdot ( \\mu ( \\omega _ 0 ) + F _ { \\omega _ 0 } ( n ) ) + \\frac 1 4 \\sum _ { 2 \\le q \\le X } ( \\mu ( \\omega _ q ) + F _ { \\omega _ q } ( n ) ) ^ 2 \\Lambda ( q ) . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} - \\Delta \\ , \\psi + \\lambda \\ , \\psi = 0 \\quad \\Omega \\setminus \\{ \\zeta _ 1 , \\ldots , \\zeta _ k \\} , \\psi = 0 \\quad \\partial \\Omega , \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} X ( t ) = e ^ { - \\lambda t } \\int _ { - \\infty } ^ t e ^ { \\lambda s } d L ( \\lambda s ) = \\int _ { \\R } e ^ { - \\lambda t + s } \\mathbf { 1 } _ { [ 0 , \\infty ) } ( \\lambda t - s ) d L ( s ) , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} P _ k ( \\mathcal { O } ) : = \\Pi _ { ( r , s ) \\in \\mathcal { O } } ( u _ r + u _ s ) . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\overline { Z } ^ i _ { j , t } = \\int _ 0 ^ t { ( Z ^ { - 1 } _ { ( 1 ) } ) ^ k _ { j , s _ - } { d Z ^ i _ { k , ( 1 ) , s } } } , \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} C ^ 3 & = \\lambda _ 3 ^ 3 + \\frac { 3 \\lambda _ 1 \\lambda _ 3 ^ 2 } { \\lambda _ 2 } t \\\\ C & = \\lambda _ 3 \\Bigg ( 1 + \\frac { 3 \\lambda _ 1 } { \\lambda _ 2 \\lambda _ 3 } t \\Bigg ) ^ { 1 / 3 } . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\frac { T _ { n p } ^ c - \\frac { p ( p - 1 ) } { 2 } } { \\sqrt { p ( p - 1 ) } } = T _ { n p } ^ * \\overset { d } \\to N ( 0 , 1 ) , \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\widetilde { T } ( z _ { 0 } ) ^ { \\ast } = \\widetilde { T } ( z _ { 0 } ) ^ { - 1 } = \\widetilde { T } ( - z _ { 0 } ) , \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\partial \\bar \\partial \\log F = \\frac { \\partial \\bar \\partial F } { F } - \\frac { \\partial F \\wedge \\bar \\partial F } { F ^ { 2 } } . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\sigma \\circ \\tilde { h _ t } = h _ t \\circ \\sigma . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} ( \\psi | \\phi ) = \\int _ { \\R ^ n } \\psi ( x ) \\overline { \\phi ( x ) } d ^ { n } x \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\sigma ( ( a c , b d ) ) = + , \\sigma ( ( a , 1 b ) ) = - , \\sigma ( ( a d , 1 c ) ) = + , \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} ( \\theta - 1 ) ( \\theta x + q - 1 ) \\left ( ( \\theta + 1 ) x + \\theta + 2 q - 3 \\right ) = ( x + q - 1 ) ( x + \\theta + q - 2 ) ^ 2 . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\chi ( x _ 1 , \\ldots , x _ { 2 d } ) = \\eta _ { d _ 1 } ( x _ 1 , \\ldots , x _ { d _ 1 } ) . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ \\infty \\sum _ { s = 1 } ^ \\infty x ^ r x ^ s x ^ { \\max \\{ r , s \\} } & = 2 \\sum _ { r = 1 } ^ \\infty \\sum _ { s = r } ^ \\infty x ^ r x ^ s x ^ s - \\sum _ { r = 1 } ^ \\infty x ^ r x ^ r x ^ r \\\\ & = 2 \\sum _ { r = 1 } ^ \\infty \\frac { x ^ { 3 r } } { 1 - x ^ 2 } - \\sum _ { r = 1 } ^ \\infty x ^ { 3 r } = \\frac { 1 + x ^ 2 } { 1 - x ^ 2 } \\frac { x ^ 3 } { 1 - x ^ 3 } , \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\pi _ 1 + \\cdots + \\pi _ d = \\pi _ { d + 1 } \\qquad ( \\pi _ 1 , \\dots , \\pi _ { d + 1 } \\in S ) . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A g = g g ^ { k l } \\overline { \\partial } _ A g _ { k l } , \\end{gather*}"}
-{"id": "9238.png", "formula": "\\begin{align*} F u t _ f ( X ) = \\int _ M ( S ( J , f ) - c _ { J , f } ) \\ , u _ X \\ , \\frac { \\omega _ g ^ m } { f ^ { 2 m + 1 } } . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} E \\dot x \\left ( t \\right ) & = A x \\left ( t \\right ) + B u \\left ( t \\right ) , \\\\ y \\left ( t \\right ) & = C x \\left ( t \\right ) , \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ^ { 1 , \\sigma } ( z ) = \\Gamma ( \\sigma ) E _ { \\alpha , \\beta ; 1 , \\sigma } ( z ) , \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\prod _ { j = 1 } ^ k F _ { g _ j } ( n ) = x \\sum _ { p _ 1 , \\dots , p _ k \\leq z } H ( p _ 1 \\cdots p _ k ) g _ 1 ( p _ 1 ) \\cdots g _ k ( p _ k ) + O ( \\sqrt x ) . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} f ( \\{ a , b \\} ) & = g ( a ) f ( b ) - ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\\\ g ( a b ) & = f ( a ) g ( b ) + ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } + \\frac { \\partial v } { \\partial r } + \\frac { v } { r } = 0 . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} ( ( \\mathcal U _ p ) ^ { \\boxplus \\tau _ 1 } ) ^ { \\boxplus \\tau _ 2 } = \\mathcal U _ p ^ { \\boxplus ( \\tau _ 1 + \\tau _ 2 ) } \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} i ^ { 3 , 3 } ( H ^ 3 ( Y , V & ; \\mathbb { Q } ) ) = 1 , \\ , i ^ { 2 , 2 } ( H ^ 3 ( Y , V ; \\mathbb { Q } ) ) = p h ( Y , w ) - 3 , \\ , i ^ { 1 , 1 } ( H ^ 3 ( Y , V ; \\mathbb { Q } ) ) = p h - 2 , \\\\ i ^ { 2 , 1 } ( H ^ 3 ( Y , V ; \\mathbb { Q } ) ) = & \\ , i ^ { 1 , 2 } ( H ^ 3 ( Y , V ; \\mathbb { Q } ) ) = h ^ { 1 , 2 } ( Z ) , \\ , i ^ { 2 , 0 } ( H ^ 3 ( Y , V ; \\mathbb { Q } ) ) = i ^ { 0 , 2 } ( H ^ 3 ( Y , V ) ) = 1 . \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\Omega ( M , \\beta ) : = \\prod ^ \\infty _ { j = 1 } S A _ { b _ j } ( M ) \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } \\partial _ t u ( x , t ) = L + V ( x ) u ( x , t ) , x \\in \\R ^ N , t > 0 , \\\\ u ( \\cdot , t ) = u _ 0 \\in L ^ 2 _ \\mu , \\end{array} \\right . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A g = g g ^ { k l } \\overline { \\partial } _ A g _ { k l } , \\end{gather*}"}
-{"id": "4997.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ 1 } F ^ p \\left ( \\frac { \\partial v } { \\partial x _ 1 } , k \\nabla _ { x ' } v \\right ) \\ d x = k ^ { n - 1 } \\int _ { \\Omega _ k } F ^ p ( \\nabla u ) \\ d x , \\\\ \\int _ { \\Omega _ 1 } | v | ^ p = k ^ { n - 1 } \\int _ { \\Omega _ k } | u | ^ p \\ d x , \\\\ \\int _ { \\Omega _ 1 } | v | ^ { p - 2 } v \\ d x = k ^ { n - 1 } \\int _ { \\Omega _ k } | u | ^ { p - 2 } u \\ d x = 0 . \\end{gather*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\partial _ { i } \\hat n _ { \\mu } = - g ^ { k l } \\partial _ { i k } \\eta ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\chi _ { _ { ] - \\infty , a [ } } \\left ( m _ { 0 } \\right ) \\ ; \\overline { \\partial _ { 0 } M \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) + A } ^ { - 1 } \\ : \\chi _ { _ { [ a , \\infty [ } } \\left ( m _ { 0 } \\right ) = 0 \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\exp ( \\hat { h } ^ { \\alpha } ( z ) Y _ { \\alpha } ) = z . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} R ( x ) = R ^ { ( 1 ) } ( x ) \\left [ \\begin{array} { c c } 1 & \\phi ( x ) \\\\ 0 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} T ' _ { \\tilde { k } _ 1 , k _ 1 , k _ 2 , k _ 3 } = \\delta _ { k _ 1 = k _ 2 = k _ 3 = 1 } \\frac { u ( 1 , 1 , 1 ) M } { 4 \\tilde { k } _ 1 \\pi ^ 2 } & \\prod _ { p \\mid \\tilde { k } _ 1 } \\ ! \\left ( 1 + \\frac { 1 } { p } \\right ) ^ { - 1 } \\ ! \\left \\{ 1 + O \\left ( \\tau ( \\tilde { k } _ 1 ) \\sqrt { \\frac { \\tilde { k } _ 1 } { M } } \\right ) \\ ! \\ ! \\right \\} \\ ! \\\\ & + O _ C ( \\sqrt { M } V ^ 2 \\log V ( \\log M ) ^ { - C } ) . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} u ( r ) = \\dfrac { 1 } { 2 m } ( b ^ 2 - r ^ 2 ) , \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\int _ { B _ { R ( t ) } } u ( x , t ) d x = \\| u _ 0 \\| _ { L ^ 1 } - \\int _ { \\R ^ 3 \\backslash B _ { R ( t ) } } u ( x , t ) d x \\geq \\frac { 1 } { 2 } \\| u _ 0 \\| _ { L ^ 1 } . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} f ( x ) \\approx \\sum _ { k = 0 } ^ \\infty { x \\choose k } \\Delta ^ k f ( 0 ) , ( \\textnormal { s e e } \\ , \\ , [ 3 , 7 ] ) . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} ~ g ( \\mathbf { x } _ { n + 1 } ) = e ( \\mathbf { x } _ { n + 1 } ) _ { i = 1 } ^ n \\| \\mathbf { x } _ { n + 1 } - \\mathbf { x } _ i \\| . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 0 } ^ n u _ k \\right \\| _ { H ^ s \\to H ^ s } \\leqslant M . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\sum _ { \\boldsymbol { a } \\neq \\boldsymbol { j } } \\| ( \\boldsymbol { \\Psi } _ { \\boldsymbol { j } } ^ T \\boldsymbol { \\Psi } _ { \\boldsymbol { a } } ) _ { \\boldsymbol { k } , \\cdot } \\| & \\lesssim \\sum _ { \\boldsymbol { a } \\neq \\boldsymbol { j } } \\prod _ { l = 1 } ^ d 2 ^ { a _ l + j _ l / 2 } . \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\biggr ( \\frac { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k } } \\biggr ) ^ \\infty _ { k = k _ 0 + 1 } \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\textrm { i n d } ( g ) : = d - \\# ( \\{ 1 , \\ldots , d \\} / \\langle g \\rangle ) . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\frac { \\vartheta ( v + z _ j + \\lambda _ j ) \\vartheta ( u + z _ j + a ) \\vartheta ( \\lambda _ j + 2 a ) } { \\vartheta ( v + z _ j ) \\vartheta ( u + z _ j - a ) } & - \\frac { \\vartheta ( u - v + a ) \\vartheta ( v + z _ j + \\lambda _ j + 2 a ) \\vartheta ( \\lambda _ j ) } { \\vartheta ( u - v - a ) \\vartheta ( v + z _ j ) } \\\\ & - \\frac { \\vartheta ( 2 a ) \\vartheta ( u - v - a - \\lambda _ j ) \\vartheta ( u + z _ j + \\lambda _ j + a ) } { \\vartheta ( u - v - a ) \\vartheta ( u + z _ j - a ) } = 0 . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\omega ( \\nabla _ t X _ 1 , \\dots , \\nabla _ t X _ n ) = \\omega ( \\widetilde { \\nabla } _ t X _ 1 , \\dots , \\widetilde { \\nabla } _ t X _ n ) , \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} - \\frac { \\partial P } { \\partial x } \\left ( x , t \\right ) = L \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) + R Q \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\Phi & ( w , z ) = \\bigl ( \\Gamma ( w , z ) , \\Gamma ( - w , z ) \\bigr ) \\quad \\mbox { w h e r e } \\\\ \\Gamma ( w , z ) & = \\left ( \\frac { \\sqrt { 8 - | w | ^ 2 } \\left ( ( 8 - 2 | w | ^ 2 - | z | ^ 2 ) w + \\bar { w } z ^ 2 \\right ) } { 8 ( 4 - | w | ^ 2 ) } + \\frac { i z } { 4 } \\sqrt { 8 - 2 | w | ^ 2 - | z | ^ 2 } , \\right . \\\\ & \\left . 1 - \\frac { | w | ^ 2 + | z | ^ 2 } { 4 } - \\frac { \\sqrt { ( 8 - | w | ^ 2 ) ( 8 - 2 | w | ^ 2 - | z | ^ 2 ) } } { 4 ( 4 - | w | ^ 2 ) } I m ( w \\bar { z } ) \\right ) . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} D _ { m , n } = F [ \\xi _ m , \\xi _ { m + 1 } , \\ldots , \\xi _ n ] , n \\geq m . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\beta \\int _ 0 ^ t \\ e ^ { - \\eta ( t - s ) } u _ { 1 x x } ( s , x ) d s + a u _ 2 ( t , x ) = 0 \\ , , \\\\ \\phantom { u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\int _ 0 ^ t \\ k ( t - s ) u _ { 1 x x } ( s , x ) d s + } t \\in ( 0 , T ) \\ , , \\ , \\ , \\ , x \\in ( 0 , \\pi ) \\\\ \\displaystyle u _ { 2 t t } ( t , x ) - u _ { 2 x x } ( t , x ) + b u _ 1 ( t , x ) = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) = \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial u } { \\partial t } \\left ( x , r , t \\right ) d r . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} Y _ { ( k ) } ( t ) = \\rho \\int _ 0 ^ t g _ { ( k ) } ( s , W _ 0 ^ s , Y _ { ( k ) , 0 } ^ s ) d s + B ( t ) , t \\in [ 0 , \\tau _ k \\wedge T ] . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} f ( \\lambda , 0 ) = 0 \\textrm { a n d } \\nabla f _ \\lambda ( 0 ) = 0 , \\forall \\lambda \\in \\R . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\pi _ { k , \\infty } ^ { \\mu } ( A ) = \\pi _ { 0 , \\infty } ^ { \\mu } R _ { 1 , \\infty } \\cdots R _ { k , \\infty } ( A ) , \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} S ( \\alpha , \\beta ) = \\begin{cases} \\{ \\beta , \\beta + 1 , \\ldots , \\alpha - 1 \\} & \\alpha - \\beta \\in \\Z _ { > 0 } , \\\\ \\emptyset & \\end{cases} \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} B = ( E + B ^ * ) ^ { - 1 } ( E - B ^ * ) ~ . \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} b \\leq - 1 + \\sqrt { 1 + \\frac { ( \\alpha + 1 ) c _ { 1 , \\infty } + c _ { 3 , \\infty } } { c _ { 2 , \\infty } } } = : b _ { m a x } . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\langle x _ 1 \\boxtimes y _ 1 , x _ 2 \\boxtimes y _ 2 \\rangle _ { K _ \\lambda \\boxtimes _ N K _ { \\lambda ^ { - 1 } } } & = \\langle \\langle x _ 2 | x _ 1 \\rangle _ N y _ 1 , y _ 2 \\rangle _ { K _ { \\lambda ^ { - 1 } } } = \\langle ( x _ 2 ^ * x _ 1 ) y _ 1 , y _ 2 \\rangle _ { K _ { \\lambda ^ { - 1 } } } \\\\ & = \\phi ( y _ 2 ^ * x _ 2 ^ * x _ 1 y _ 1 ) = \\tau ( y _ 2 ^ * x _ 2 ^ * x _ 1 y _ 1 ) = \\langle x _ 1 y _ 1 , x _ 2 y _ 2 \\rangle _ { L ^ 2 ( N ) } . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} W ( s ; f , \\overline { g } ) : = \\frac { L ( s , f \\times \\overline { g } ) } { \\zeta ( 2 s ) } + Z ( s , 0 , f \\times \\overline { g } ) , \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\omega _ \\beta ( z ) - \\omega _ \\beta ( E _ - ) = \\sqrt { \\frac { - 2 } { \\widetilde z '' ( \\omega _ \\beta ( E _ - ) ) } } \\sqrt { E _ - - z } + O ( | z - E _ - | ) \\ , , \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\xi \\left ( x , t \\right ) = \\left [ x _ { n } + \\left ( x _ { 1 } - 1 / 2 \\right ) ^ { 2 } / \\omega ^ { 2 } + \\displaystyle \\sum \\limits \\limits _ { k = 2 } ^ { n - 1 } x _ { k } ^ { 2 } + \\left ( t - T / 2 \\right ) ^ { 2 } + 1 / 4 \\right ] ^ { - \\nu } . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\lambda _ j = \\inf _ { \\substack { V \\subset H ^ 1 ( \\Sigma ) , \\\\ d i m V = j + 1 } } \\sup _ { \\substack { 0 \\neq u \\in V , \\\\ \\int _ \\Sigma u ^ 2 d \\sigma = 1 } } \\int _ \\Sigma | \\nabla _ \\Sigma u | ^ 2 d \\sigma , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\varphi _ { \\underline { d } , \\underline { d } '' } ( \\left \\langle v _ 1 , \\ldots , v _ { \\beta } \\right \\rangle ) = \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle . \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\tilde { Q } _ p ( x _ 0 , v _ 0 , s ) = \\tilde { Q } _ { - p } ( - x _ 0 , - v _ 0 , s + p ) \\ , . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ H = \\langle Q _ \\infty ^ { - 1 / 2 } x , Q _ \\infty ^ { - 1 / 2 } y \\rangle _ X \\forall x , y \\in H . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} G _ j : = \\sum _ { k = 0 } ^ \\infty 1 ( \\widehat { Q } _ k = j ) \\mbox { f o r } ~ j \\ge 1 , \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} T _ { j , m , n } ( f , g ) ( x ) : = \\iint \\hat { f } ( \\xi ) \\hat { g } ( \\eta ) e ^ { 2 \\pi i ( \\xi + \\eta ) x } m _ j ( \\xi , \\eta ) \\hat { \\Phi } \\left ( \\frac { \\xi } { 2 ^ { a j + m } } \\right ) \\hat { \\Phi } \\left ( \\frac { \\eta } { 2 ^ { b j + n } } \\right ) \\ , d \\xi d \\eta , \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} 1 - \\rho _ k & = \\frac { f ( z _ k ) - f ( x _ k ) - m _ k ( z _ k ) } { - m _ k ( z _ k ) } \\\\ & \\leq \\frac { \\alpha _ 0 \\bar \\varpi } { 2 \\rho \\lambda \\delta } \\left [ \\frac { \\| ( \\nabla ^ 2 f ( x _ k ) - H _ k ) [ y _ k ] \\| } { \\| y _ k \\| } + \\frac { L _ H } { 3 } \\| y _ k \\| \\right ] \\frac { \\| y _ k \\| ^ 2 } { g _ k } \\max \\left \\{ \\frac { 1 } { \\chi } , \\frac { A _ 4 } { g _ k } , \\frac { A _ 5 \\sigma _ k } { g _ k } \\right \\} , \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ { g + 1 } } \\sum _ { c _ 2 , \\ldots , c _ n } \\frac { \\psi _ 1 ^ { g + 1 - \\sum _ j c _ j } } { ( g + 1 - \\sum _ j c _ j ) ! \\prod _ { j = 2 } ^ n c _ j ! } \\cdot \\frac { ( 2 g + 2 - 2 \\sum _ j c _ j ) ! } { \\prod _ j ( b _ j - 2 c _ j ) ! } \\prod _ { j = 2 } ^ n \\psi _ j ^ { c _ j } , \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} U ( x ) = ( u _ 1 ( x ) , u _ 0 ( x ) ) \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} [ e ^ { - 2 \\pi \\imath \\ , G _ { \\theta } ( \\widehat { U } ) } ] _ 1 \\ ; = \\ ; [ e ^ { - 2 \\pi \\imath \\ , G _ { \\theta } ( \\widehat { U } e ^ { 2 \\pi \\imath \\hat { h } _ \\theta } ) } ] _ 1 \\ ; . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\frak E _ { \\le E _ 0 } = \\{ E ( \\alpha _ + ) - E ( \\alpha _ - ) \\mid \\mathcal M ( \\alpha _ - , \\alpha _ + ) \\ne \\emptyset , ~ E ( \\alpha _ + ) - E ( \\alpha _ - ) \\le E _ 0 \\} . \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\begin{pmatrix} \\hat { u } \\\\ \\hat { v } \\\\ \\hat { w } \\end{pmatrix} = \\begin{pmatrix} 0 . 1 + 0 . 1 x + 0 . 2 y + 0 . 2 z \\\\ 0 . 0 5 + 0 . 1 5 x + 0 . 1 y + 0 . 2 z \\\\ 0 . 0 5 + 0 . 1 x + 0 . 2 y + 0 . 2 z \\end{pmatrix} . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\left | \\log \\Phi _ s ^ * ( e ^ { i \\theta _ j } ) \\right | \\leq \\left | \\sum _ { m = G ( k ) } ^ s \\alpha _ m e ^ { i \\gamma _ m ( \\theta _ j ) } \\right | + O \\left ( \\sum _ { m = G ( k ) } ^ s | \\alpha _ m | ^ 2 \\right ) + \\log | \\Phi _ { G ( k ) } ^ * | . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\frac { x ^ { \\nu + 1 } } { ( x ^ 2 + a ^ 2 ) ^ { \\nu + 3 / 2 } } J _ { \\nu } ( b x ) { \\rm d } x = \\frac { b ^ { \\nu } \\sqrt { \\pi } } { 2 ^ { \\nu } a \\varGamma ( \\nu + 3 / 2 ) } e ^ { - a b } . \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} & T ^ m ( f , g ) ( x ) : = \\\\ & \\sum _ { j > N } \\int \\left | f * \\Phi _ { a j + m } \\left ( x - \\frac { t ^ a + \\epsilon _ P ( t ) } { 2 ^ { a j } } \\right ) g * \\Phi _ { b j + m } \\left ( x - \\frac { t ^ b + \\epsilon _ Q ( t ) } { 2 ^ { b j } } \\right ) \\rho ( t ) \\right | \\ , d t . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\begin{array} { c l } \\langle x _ t , p ( x _ t ) \\rangle & = \\displaystyle \\frac { 1 } { \\Xi } \\int _ { - h } ^ 0 \\int _ { \\theta } ^ 0 x _ t ^ { \\top } ( s ) R \\int _ { - h } ^ 0 \\int _ { \\theta _ 1 } ^ 0 x _ t ( s _ 1 ) d s _ 1 d \\theta _ 1 d s d \\theta \\\\ & = \\displaystyle \\frac { 1 } { \\Xi } \\int _ { - h } ^ 0 \\int _ { \\theta } ^ 0 x _ t ^ { \\top } ( s ) d s d \\theta R \\int _ { - h } ^ 0 \\int _ { \\theta } ^ 0 x _ t ( s ) d s d \\theta , \\end{array} \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} \\biggr ( \\frac { \\sum _ { l = 1 } ^ \\eta \\sum _ { m = l + 1 } ^ \\eta c ^ { \\bf a } _ l c ^ { \\bf a } _ m \\gamma ^ k _ l \\gamma ^ k _ m ( \\gamma _ l - \\gamma _ m ) ^ 2 } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ k } \\biggr ) ^ \\infty _ { k = k _ 0 + 1 } \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} K ( t , x , y ) = \\frac { 1 } { 4 \\pi t } \\exp { \\left ( - \\frac { x ^ 2 + y ^ 2 } { 4 \\kappa t } \\right ) } . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\norm { f ^ * c } _ \\frac { n } { k } & = \\sup _ { \\theta \\in [ 0 , 2 \\pi ] } \\norm { \\cos ( \\theta ) f ^ * a + \\sin ( \\theta ) f ^ * b } = \\sup _ { \\theta \\in [ 0 , 2 \\pi ] } \\norm { f ^ * ( \\cos ( \\theta ) a + \\sin ( \\theta ) b ) } \\\\ & \\leq C \\left ( \\deg f \\right ) ^ { \\frac { k } { n } } \\sup _ { \\theta \\in [ 0 , 2 \\pi ] } \\norm { \\cos ( \\theta ) a + \\sin ( \\theta ) b } = C \\left ( \\deg f \\right ) ^ { \\frac { k } { n } } \\norm { c } _ \\frac { n } { k } , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ { \\theta ( N - r + 1 ) - 1 } { \\big ( z _ r - q ^ { \\theta ( N - r ) - j } \\big ) } = \\prod _ { j = 1 } ^ { \\theta ( N - r + 1 ) - 1 } { \\big ( z _ r - q ^ { j - \\theta } \\big ) } , \\end{gather*}"}
-{"id": "5962.png", "formula": "\\begin{align*} U u & = U e ^ { t A } u _ 0 - \\int _ { 0 } ^ { t } U e ^ { ( t - s ) A } \\mathbb { P } ( u \\cdot \\nabla u ) ( s ) \\dd s \\\\ & = e ^ { t A } U u _ 0 - \\int _ { 0 } ^ { t } e ^ { ( t - s ) A } \\mathbb { P } ( U u \\cdot \\nabla U u ) ( s ) \\dd s . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } k ( t ) \\Big | \\sum _ { n = n _ 0 } ^ { \\infty } R _ { n } e ^ { r _ n t } \\Big | ^ 2 \\ d t \\le \\varepsilon \\ \\pi T \\sum _ { n = n _ 0 } ^ \\infty \\frac { | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 } { \\pi ^ 2 + T ^ 2 r _ { n } ^ 2 } \\ , . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} P \\bigg ( | V _ \\chi ^ S | \\ge \\frac { \\rho } { 4 \\sqrt { r } \\phi ( q ) } \\bigg ) \\le \\bigg ( \\frac { 1 6 r \\phi ( q ) ^ 2 } { \\rho ^ 2 } \\bigg ) \\sigma ^ 2 ( V _ \\chi ^ S ) = \\bigg ( \\frac { 1 6 r \\phi ( q ) ^ 2 } { \\rho ^ 2 } \\bigg ) \\cdot 4 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma > T } } \\frac 1 { \\frac 1 4 + \\gamma ^ 2 } < 1 , \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} \\int _ { \\Pi } | \\nabla v | ^ { 2 } \\dd x & = - \\int _ { \\Pi } \\Delta v \\cdot v \\dd x + \\int _ { \\partial \\Pi } \\frac { \\partial v } { \\partial n } \\cdot v \\dd { \\mathcal { H } } \\\\ & = \\int _ { \\Pi } \\textrm { c u r l } \\ ( \\omega ^ { \\theta } e _ { \\theta } ) \\cdot v \\dd x - \\int _ { \\partial \\Pi } ( \\partial _ r u ^ { r } u ^ { r } + \\partial _ { r } u ^ { z } u ^ { z } ) \\dd { \\mathcal { H } } \\\\ & = \\int _ { \\Pi } | \\omega ^ { \\theta } | ^ { 2 } \\dd x . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} F ( d t _ { J } \\wedge h ) \\vert _ { \\{ { \\bf t } \\} \\times M _ t } = \\sum _ { I } d t _ I \\wedge d t _ { J } \\wedge F ^ { \\bf t } _ { I ; J } ( h \\vert _ { { \\{ { \\bf t } \\} } \\times M _ t } ) . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\xi '' ( t ) + A \\xi ( t ) - \\int _ t ^ T \\ H ( s - t ) A \\xi ( s ) d s = 0 \\ , , t \\in [ 0 , T ] \\ , , \\\\ \\\\ \\xi ( T ) = \\xi _ 0 \\ , , \\xi ' ( T ) = \\xi _ 1 \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus N _ \\Omega } \\nabla v _ m \\nabla \\varphi + \\int _ { \\Omega \\setminus N _ \\Omega } \\beta ' _ m ( w _ m ) v _ m \\varphi = 0 . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} & \\frac { C _ { 2 , n } } { ( k - 4 ) ! } - \\frac { C _ { 1 , n } } { ( k - 3 ) ! } e _ 1 \\\\ & = 2 [ ( 2 n - 3 ) a _ k + b _ { k - 1 } ] e _ 2 + \\Bigl [ \\frac { 2 } { k } ( n - 1 ) ( n - k ) a _ { k - 1 } + \\frac { 1 } { k - 1 } ( n - k + 1 ) b _ { k - 2 } \\Bigr ] e _ 1 \\\\ & - \\frac { 2 n ( n - 1 ) } { k ( k - 1 ) } ( 2 k - 3 ) a _ { k - 2 } - \\frac { 2 n } { k - 1 } b _ { k - 3 } . \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ 1 = 1 , \\Pi _ 2 = 2 ) & = \\mathbb { P } ( Y _ 1 = 1 , \\Pi _ 1 = 1 , \\Pi _ 2 = 2 ) + \\mathbb { P } ( Y _ 1 \\ge 2 , \\Pi _ 1 = 1 , \\Pi _ 2 = 2 ) \\\\ & = ( 1 - q ) \\cdot \\frac { 1 - q } { q } \\lambda _ 1 ( q ) + \\sum _ { y = 2 } ^ \\infty q ^ { y - 1 } ( 1 - q ) \\frac { 1 } { y ( y - 1 ) } \\\\ & = \\frac { ( 1 - q ) ^ 2 } { q } \\lambda _ 1 ( q ) + \\frac { 1 - q } { q } \\lambda _ 2 ( q ) , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} T & = \\frac { \\Big ( \\int _ { \\mathbb { R } ^ 3 } f d v \\Big ) \\Big ( \\int _ { \\mathbb { R } ^ 3 } f | v | ^ 2 d v \\Big ) - \\Big | \\int _ { \\mathbb { R } ^ 3 } f v d v \\Big | ^ 2 } { 3 \\Big ( \\int _ { \\mathbb { R } ^ 3 } f d v \\Big ) ^ 2 } \\geq \\frac { \\gamma _ { \\ell } } { 3 a _ { u } ^ 2 } . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\tau ( G ) : = ( H _ i / H _ i ^ \\prime ) _ { 1 \\le i \\le p + 1 } \\varkappa ( G ) : = ( \\ker ( T _ { G , H _ i } ) ) _ { 1 \\le i \\le p + 1 } \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} q _ \\pm ( e _ 1 ) = 0 , \\ q _ \\pm ( e _ 2 ) = \\pm \\frac { 1 } { 4 } , \\ q _ \\pm ( e _ 1 + e _ 2 ) = \\mp \\frac { 1 } { 4 } \\ . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} f : S _ { A \\cup B } & \\to \\{ 0 , 1 \\} ^ { | A | + | B | } \\\\ f ( \\pi ) _ i & = \\begin{cases} 1 & \\pi _ i \\in A \\\\ 0 & \\pi _ i \\in B \\end{cases} . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\rho \\frac { \\partial Q } { \\partial t } + \\pi a ^ { 2 } \\frac { \\partial P } { \\partial x } = 2 \\pi \\mu \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } r \\left ( \\frac { \\partial ^ { 2 } u } { \\partial r ^ { 2 } } + \\frac { 1 } { r } \\frac { \\partial u } { \\partial r } \\right ) d r . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\varphi _ { \\underline { d } , \\underline { d } '' } ( V _ { \\underline { d } } ) = ( V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } + V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } ) + \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle = V _ { \\underline { d } '' } ^ { X _ { 1 } , 0 } . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} \\sup _ { \\boldsymbol { x } \\in [ 0 , 1 ] ^ d } \\| \\boldsymbol { \\psi } _ { \\boldsymbol { J } _ n } ( \\boldsymbol { x } ) \\| _ 1 \\| ( \\boldsymbol { \\Psi } ^ T \\boldsymbol { \\Psi } ) ^ { - 1 } \\| _ { ( \\infty , \\infty ) } \\mathrm { E } \\| \\boldsymbol { \\eta } \\| _ \\infty \\lesssim n ^ { - 1 } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } / 2 } \\mathrm { E } \\| \\boldsymbol { \\eta } \\| _ \\infty , \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s ( x _ i ^ j - y _ i ^ j ) = 0 ( 1 \\le j \\le 3 ) , \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\mathcal { R } ( \\varepsilon ) : = \\varepsilon ^ 2 ( \\mathcal { H } _ 0 + \\varepsilon ^ 2 I ) ^ { - 1 } . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\| v _ i ( t ) \\| _ { p _ 0 } \\leq \\varepsilon t > 0 , \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} g _ 0 ( x ) - G _ 0 ( x , - x ) & = \\frac { 1 } { \\omega _ { 2 } } \\left [ - \\frac { 1 } { 2 \\vert x \\vert } + \\sum _ { m = 0 } ^ \\infty [ 1 + ( - 1 ) ^ m ] \\ , P _ m ( x ) \\right ] \\\\ & \\geq \\frac { 1 } { \\omega _ { 2 } } \\left [ - \\frac { 1 } { 2 \\vert x \\vert } + 2 P _ 0 ( x ) \\right ] \\forall x \\in \\Omega _ a . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} ( d _ n ) _ { n \\geq 0 } : = \\left ( ( F _ n \\star \\bar { \\mu } ) _ { \\infty } \\right ) _ { n \\geq 0 } \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\left | e \\left ( G _ 1 \\right ) - \\left ( \\log n - \\log \\left ( f + 1 \\right ) - h \\right ) \\frac { n } { 2 } \\right | & = o \\left ( n \\right ) , \\\\ \\left | e \\left ( G _ 2 \\setminus G _ 1 \\right ) - 2 h n \\right | & = o \\left ( n \\right ) , \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ b & c \\end{pmatrix} = - d ^ 2 . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\langle \\theta ( A ) , x \\rangle : = F ( \\mathbf 1 _ { A } \\cdot x ) ( \\mathcal B ( \\mathbb R _ + ) \\otimes A \\in \\mathcal F \\otimes \\mathcal J , x \\in X ) . \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{gather*} c ( N , m , \\theta ) = \\theta ^ 2 \\dbinom { m + 1 } { 3 } - \\left \\{ N \\theta ^ 2 - \\dbinom { \\theta + 1 } { 2 } \\right \\} \\dbinom { m } { 2 } \\end{gather*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\mathbb E \\Bigl ( \\sum _ { t \\geq 0 } \\| \\Delta ( \\Phi \\cdot M ) _ t \\| ^ 2 \\Bigr ) ^ { \\frac p 2 } = \\mathbb E [ \\Phi \\cdot M ] _ { \\infty } ^ { \\frac p 2 } = \\infty . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} f _ i ( u ) = \\sum _ { r = 1 } ^ { R } k _ r ( y _ { r , i } ' - y _ { r , i } ) u ^ { y _ r } u ^ { y _ r } = \\prod _ { i = 1 } ^ { N } u _ i ^ { y _ { r , i } } . \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\Sigma _ { n , j } : = \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ j \\leq n } \\mathbb { P } \\left ( \\bigcap _ { k = 1 } ^ j A _ { n , i _ k } \\right ) . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} \\left ( [ \\bar E ] , [ \\bar A ] \\right ) ^ { \\star } = \\{ ( E , A ) \\in [ \\bar E ] \\times [ \\bar A ] : \\det ( A - \\lambda E ) \\not \\equiv 0 \\} . \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\theta ( y _ { i } , 4 r ) - \\theta ( y _ i , 2 r ) < \\delta \\mbox { f o r e v e r y } i = 0 , \\dots , m - 3 , \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} R \\pi _ * \\mathbb Q _ X = \\mathcal J ^ { \\bullet } = \\mathcal K ^ { \\bullet } \\oplus \\mathcal H ^ { \\bullet } . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) } ^ { c } ( z ) = S ( z , z ) . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} ( o s c _ { r + 0 } f ) ( x ) = \\lim _ { R \\searrow r } ( o s c _ R f ) ( x ) \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} { X _ { \\varepsilon } } : \\left ( \\begin{array} { c } \\dot x \\\\ \\dot y \\\\ \\dot \\varphi \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & E \\end{array} \\right ) \\left ( \\begin{array} { c } x \\\\ y \\\\ \\varphi \\end{array} \\right ) + \\underline G ( x , y , \\varphi , \\varepsilon ) \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) , \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\tilde \\theta _ { j _ 0 } & = ( C \\ , 2 ^ { - \\frac { n + 1 - m } { m } t } , \\eta C 2 ^ { t } , \\underbrace { C \\ , 2 ^ t , \\dots , C \\ , 2 ^ t } _ { n - m } ) \\in \\R ^ { n + 2 - m } \\ , , \\\\ [ 1 e x ] \\tilde \\theta _ j & = C \\ , 2 ^ { - \\frac { n + 1 - m } { m } t } \\in \\R ( 1 \\le j \\le m , ~ j \\not = { j _ 0 } ) \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\Vert \\L [ \\nabla \\phi , \\L ^ { - \\alpha } ] \\L ^ \\alpha \\psi \\Vert _ { L ^ 2 } \\le C \\Vert \\nabla \\phi \\Vert _ { W ^ { 1 , \\infty } } \\Vert \\L ^ \\alpha \\psi \\Vert _ { L ^ 2 } = C \\Vert \\nabla \\phi \\Vert _ { W ^ { 1 , \\infty } } \\Vert \\psi \\Vert _ { D ( \\L ^ \\alpha ) } . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} \\alpha = \\alpha ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) : = \\max \\left \\{ \\frac { \\delta _ d + \\delta ^ { \\prime } + M ( G ) } { \\delta ^ { \\prime } + 2 } , \\frac { M ( G ) } { 2 } \\right \\} < 1 , \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\omega _ { a , b } = \\sum _ { i = 1 } ^ m \\nu _ { _ { S _ i ; { \\widehat \\omega } _ { a , b } ( \\mu _ i ) } } = \\sum _ { i = 1 } ^ m F _ i ( a , b ) \\kappa _ { S _ i } . \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\frak x = \\sum _ { i = 1 } ^ { \\infty } P _ i ( e ) T ^ { \\lambda _ i } \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ { 1 , 0 , 2 , 0 } & 0 & a _ { 0 , 1 , 2 , 0 } & 0 \\\\ a _ { 1 , 0 , 1 , 1 } & a _ { 1 , 0 , 2 , 0 } & a _ { 0 , 1 , 1 , 1 } & a _ { 0 , 1 , 2 , 0 } \\\\ a _ { 1 , 0 , 0 , 2 } & a _ { 1 , 0 , 1 , 1 } & a _ { 0 , 1 , 0 , 2 } & a _ { 0 , 1 , 1 , 1 } \\\\ 0 & a _ { 1 , 0 , 0 , 2 } & 0 & a _ { 0 , 1 , 0 , 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\widehat L = C _ + \\cup C _ - \\bigcup _ { | v | \\geq N ^ { \\nu / 3 } } { \\widehat A ( v ) } \\bigcup _ { k = 1 } ^ { N ^ { \\nu / 3 } } { \\cal D } _ k \\bigcup _ { \\ell = 1 } ^ { N ^ { \\nu / 3 } } \\widetilde { \\cal D } _ { \\ell } , \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} { z } _ n ^ S = \\mathcal { K } _ m ( { z } _ n ^ C ) + f . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} { \\rm V a r } ( X ) = \\frac { C _ 1 ^ 2 } { 2 } , { \\rm V a r } ( Y ) = \\frac { C _ 2 ^ 2 } { 2 } , \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} \\| \\mathcal { K } _ m ' ( \\varphi _ m ) ( I - Q _ n ) \\mathcal { K } _ m ' ( \\varphi _ m ) \\| = O ( h ^ { 2 r } ) . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} { \\mathcal E } ( t , \\tau ) : = A ( t ) ^ { - 1 / 2 } e ^ { - i \\tau A ( t ) ^ { 1 / 2 } } P - ( t ^ 2 S ) ^ { - 1 / 2 } e ^ { - i \\tau ( t ^ 2 S ) ^ { 1 / 2 } P } P . \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} v = \\sum _ { \\lambda \\in v \\Lambda ^ n } \\lambda \\lambda ^ * . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\| u ^ { * } ( \\varphi _ n ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ n f ( z _ k ) \\sigma _ n ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ n f ( z _ k ) \\sigma _ n ( a ) } ^ n ) u \\\\ & - \\psi _ n ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ n f ( z _ k ) \\sigma _ n ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ n f ( z _ k ) \\sigma _ n ( a ) } ^ n \\| < \\varepsilon \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} \\theta _ n & = \\sum _ { i = 1 } ^ r b _ { n i } h _ i = \\sum _ { i = 1 } ^ r b _ { n i } \\left ( \\nu _ i + \\sum _ { j = 1 } ^ s \\alpha _ { i j } e _ j \\right ) \\\\ & = \\sum _ { i = 1 } ^ r b _ { n i } \\nu _ i + \\sum _ { j = 1 } ^ s \\left ( \\sum _ { i = 1 } ^ r b _ { n i } \\alpha _ { i j } \\right ) e _ j = \\sum _ { i = 1 } ^ r b _ { n i } \\nu _ i . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{gather*} b _ n ^ { ( \\theta ) } = \\frac { 1 } { x _ 1 - x _ 2 } \\big ( T _ { q , x _ 2 } b _ n ^ { ( \\theta - 1 ) } - T _ { q , x _ 1 } b _ { n - 1 } ^ { ( \\theta - 1 ) } \\big ) , n = 1 , 2 , \\dots , \\theta - 1 , \\\\ b _ 0 ^ { ( \\theta ) } = \\frac { 1 } { x _ 1 - x _ 2 } T _ { q , x _ 2 } b _ 0 ^ { ( \\theta - 1 ) } , b _ { \\theta } ^ { ( \\theta ) } = \\frac { 1 } { x _ 2 - x _ 1 } T _ { q , x _ 1 } b _ { \\theta - 1 } ^ { ( \\theta - 1 ) } . \\end{gather*}"}
-{"id": "5696.png", "formula": "\\begin{align*} u ^ 3 ( t ) = \\\\ \\begin{cases} ( 0 , 0 , 0 , 0 ) \\\\ ( 0 , 0 , 1 , 0 ) \\\\ \\end{cases} ~ Q ( t ) \\ge 1 . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\| V ^ * ( \\Phi ^ { \\prime } ( f \\otimes a ) + & \\sum _ { j = 1 } ^ m \\sum _ { k = 1 } ^ m f ( z _ k ) \\rho ( p _ { j , k } \\otimes a ) + \\sigma ^ { \\prime \\prime } ( f ) ) V \\\\ & - ( \\Psi ^ { \\prime } ( f \\otimes a ) + \\sum _ { j = 1 } ^ m \\sum _ { k = 1 } ^ m f ( z _ k ) \\rho ( p _ { j , k } \\otimes a ) + \\sigma ^ { \\prime \\prime } ( f ) ) \\| < \\frac { \\varepsilon } { 7 } . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\{ a ' , b ' \\} & = \\sum _ { \\alpha \\in \\Lambda } \\psi _ { \\alpha } ( a ' ) \\{ x _ { \\alpha } b ' \\} , \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} D ^ { \\frac { 1 } { 2 } } _ h ( s ) = \\frac { ( 2 \\pi ) ^ { s - \\frac { 1 } { 4 } } } { \\Gamma ( s - \\frac { 1 } { 4 } ) } \\langle P _ h ^ { \\frac { 1 } { 2 } } ( \\cdot , s ) , V \\rangle = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 3 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s - \\frac { 1 } { 4 } } } - \\mathfrak { E } _ h ^ { \\frac { 1 } { 2 } } ( s ) . \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} \\mathbf { { h } } ( x ) = \\sum _ { y \\in S ( x ) } \\mathbf { F } ( \\mathbf { h } ( y ) ) , \\forall \\ x \\in V \\setminus \\{ x ^ 0 \\} , \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} L U = \\nu ( g ) - g \\mbox { i n } \\mathbb { R } ^ 2 , L V = \\nu ( \\psi ) - \\psi \\mbox { i n } \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { M } _ \\mathbf { m } } = \\phi _ { \\mathcal { M } _ 1 } \\oplus \\cdots \\oplus \\phi _ { \\mathcal { M } _ k } : ( S ^ d ) ^ k \\rightarrow U _ { k , 1 } ^ { \\oplus m _ 1 } \\oplus \\cdots \\oplus U _ { k , k } ^ { \\oplus m _ k } \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} x ( \\mathbf { e } \\backslash \\mathbf { d } ) y & = \\sup _ { z \\in Z } ( z \\mathbf { d } y - z \\mathbf { e } x ) _ + . \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} A \\chi _ j = T _ a \\chi _ j \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\varepsilon ( 1 ) & = 1 , \\\\ \\varepsilon ( a ) & = 0 , ~ ~ ~ ~ ~ \\ \\ \\ \\ \\forall a \\in L , \\\\ \\varepsilon ( u v ) & = \\varepsilon ( u ) \\varepsilon ( v ) , ~ ~ ~ ~ ~ \\ \\forall u , v \\in S ( L ) , \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\log G ( z + 1 ) = \\frac 1 2 z ^ 2 \\log z - \\frac 3 4 z ^ 2 + \\frac z 2 \\log ( 2 \\pi ) - \\frac 1 { 1 2 } \\log z + \\zeta ' ( - 1 ) + O ( 1 / z ) , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\begin{cases} g _ n : = g _ 0 + h _ n & ( n \\geqslant 1 ) \\\\ { \\displaystyle \\varphi _ 0 = 1 , \\varphi _ n = \\prod _ { i = 0 } ^ { n - 1 } ( f ^ i ) ^ * ( \\varphi ) ^ { 1 / d ^ { i + 1 } } } & ( n \\geqslant 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\sigma _ d ( N ) & = \\prod _ { p \\vert N } \\left ( \\begin{array} { c c } e _ p + d - 1 \\\\ d - 1 \\end{array} \\right ) _ p \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ q b _ { k + m - 1 - q } m _ { ( m , \\dot { 0 } ) } = \\sum _ { t = 0 } ^ { q - 1 } b _ { k - t - 1 } m _ { ( q - t , \\dot { 0 } ) } , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( E _ - ) ) - 1 ) ( F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( E _ - ) ) - 1 ) = 1 \\ , . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\mu | _ T = \\lambda | _ T - \\sum _ { r = 1 } ^ { n - 2 } { ( c _ r + c _ { 2 n - r } ) \\beta _ r - ( c _ { n - 1 } - c _ n + c _ { n + 1 } ) \\beta _ { n - 1 } - c _ n \\beta _ n } . \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} [ \\eta \\cdot \\nu ] = - [ \\xi \\cdot e _ 2 ] ( 0 ) - e _ 2 \\cdot e _ 2 = \\| \\xi \\| _ \\infty - 1 = 0 \\ , , \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} { \\bf E } X _ 1 ^ 2 = \\bigl ( { \\bf E } R _ 1 ^ 2 + 2 \\varDelta _ 1 { \\bf E } R _ 1 + \\varDelta _ 1 ^ 2 \\bigr ) { \\bf E } \\bigl [ ( \\cos \\theta ) ^ 2 \\bigr ] \\approx n ^ { - 1 } \\biggl ( \\frac { t ^ 2 } { \\mu ^ 2 } + \\frac { C _ 1 t } { \\mu } + \\frac { C _ 1 ^ 2 } { 2 } \\biggr ) , \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} D _ { n n } v = - \\frac { 1 } { \\bar a ^ { n n } } \\sum _ { ( i , j ) \\neq ( n , n ) } \\bar a ^ { i j } D _ { i j } v , \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { Y } } _ { i j } ( 0 ) = 0 \\ \\mbox { ( f o r a n y $ 1 \\leq i , j \\leq 2 $ ) } , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\big | K ( u ) \\big | = \\big | K ( \\overline { u } ) \\big | \\ , , \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} | z ^ n | + ( 1 - \\varepsilon ) | 1 - z ^ n | & = | z ^ n | + ( 1 - \\varepsilon ) | 1 - z | | 1 + z + \\ldots + z ^ { n - 1 } | \\\\ & \\leq | z | ^ n + ( 1 - | z | ) | 1 + z + \\ldots + z ^ { n - 1 } | \\\\ & \\leq | z | ^ n + ( 1 - | z | ) ( 1 + | z | + \\ldots + | z | ^ { n - 1 } ) \\\\ & = | z | ^ n + ( 1 - | z | ^ n ) = 1 , \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho } x ( t , \\rho ) = 1 + \\frac { \\partial } { \\partial \\rho } ( H ( t , \\rho ) \\rho ^ 2 ) + \\frac { \\partial } { \\partial \\rho } ( H ^ 2 ( t , \\rho ) \\rho ^ 3 ) + \\dots , \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} d _ p ^ { 2 k } \\circ u = \\sum _ { \\ell = 1 } ^ Q d _ p ^ { 2 k } ( u _ \\ell ) \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} d ^ { 2 } W / d \\xi ^ { 2 } = \\left \\{ { u ^ { 2 } + \\psi \\left ( \\xi \\right ) } \\right \\} W , \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{gather*} \\Lambda ^ { N + 1 } _ N ( \\lambda , \\mu ) = \\psi _ { \\lambda / \\mu } ( q , t ) \\frac { P _ { \\mu } \\big ( t ^ N , \\dots , t ^ 2 , t ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ N , \\dots , t , 1 ; q , t \\big ) } , \\end{gather*}"}
-{"id": "9459.png", "formula": "\\begin{align*} L _ z = z - 2 t | D _ x | . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} r ^ { - 1 } \\sigma { r } . r \\tau \\sigma { r ^ { - 1 } } = \\tau \\sigma . { r ^ 2 } \\tau { r ^ { - 2 } } , ~ a r ^ { - 1 } \\sigma ^ { - 1 } \\xi { r a ^ { - 1 } } . a \\sigma { a ^ { - 1 } } = \\gamma . r \\gamma ^ { - 1 } r ^ { - 1 } , ~ \\xi = { r } \\xi \\sigma ^ { - 1 } r ^ { - 1 } , \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} T B V _ { i } ^ { T } = | q ^ { T } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q ^ { T } _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) | + | q ^ { T } _ { i } ( x _ { i + \\frac { 1 } { 2 } } ) - q ^ { T } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) | . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } I ( W _ 0 ^ T ; Y ^ { ( n ) } ( \\Delta _ n ) ) = I ( W _ 0 ^ T ; Y _ 0 ^ T ) . \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\psi ( W _ p ) = \\psi ( W _ { q _ i } ^ * W _ p W _ { q _ i } ) & = \\omega ( p , q _ i ) \\omega ( { q _ i } ^ { - 1 } , p q _ i ) \\psi ( W _ p ) \\\\ & = \\omega ( p , q _ i ) \\omega ( q _ i ^ { - 1 } , p ) \\omega ( q _ i ^ { - 1 } , q _ i ) \\psi ( W _ p ) \\\\ & = \\omega ( p , q _ i ) \\overline { \\omega ( q _ i , p ) } \\omega ( q _ i ^ { - 1 } , q _ i ) \\psi ( W _ p ) \\\\ & = ( \\omega \\omega ^ * ) ( q _ i , p ) \\omega ( q _ i ^ { - 1 } , q _ i ) \\psi ( W _ p ) . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\zeta _ n ( x ) - \\int _ M \\zeta _ n \\mathrm { d } \\mu - \\sum _ { j = 1 } ^ m A _ { n , j } G ( x _ { n , j } ^ { ( 1 ) } , x ) \\right | \\le { C } \\sum _ { j = 1 } ^ m e ^ { - \\frac { \\lambda _ { n , j } ^ { ( 1 ) } } { 2 } } \\left ( \\frac { 1 _ { U _ { 2 r _ 0 } ( x _ { n , j } ^ { ( 1 ) } ) } ( x ) } { ( | T _ j ( x ) - T _ j ( x _ { n , j } ^ { ( 1 ) } ) | ) } + 1 _ { M \\setminus U _ { 2 r _ 0 } ( x _ { n , j } ^ { ( 1 ) } ) } ( x ) \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\tau _ 2 = \\left [ \\begin{array} { c c c c } & & 1 & \\\\ & & & 1 \\\\ 1 & & & \\\\ & 1 & & \\end{array} \\right ] . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} K _ r ( x , \\{ z \\} ) = \\frac { \\omega _ X ( \\{ z \\} ) } { \\omega _ D ( \\{ r \\} ) } \\ > \\delta _ { r , \\pi ( x , z ) } \\quad \\quad ( x , z \\in X , \\ > r \\in D ) . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( i + 1 ) ( 2 k - 4 - 3 i ) a ^ { i } \\ge 0 \\ , , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} h ( \\sigma ^ { N } , \\alpha ^ { M N + 1 } \\times I ^ { \\infty } ) & = \\lim _ { l \\to \\infty } \\frac { \\log ( N ( \\mathbf { H } , ( \\vee _ { i = 0 } ^ { l } \\sigma ^ { - i N } ( \\mathcal { V } ) ^ { * } ) ) } { l } \\\\ & = \\lim _ { l \\to \\infty } \\frac { \\log ( N ( \\star _ { i = 1 } ^ { ( M + l ) N } H , \\alpha ^ { ( M + l ) N + 1 } ) ) } { l } , \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} h ' ( \\gamma ) = \\mu h ( \\gamma ) \\mu ^ { - 1 } , \\tilde \\varphi ' ( x ) = \\mu \\tilde \\varphi ( x ) . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\mathcal Z \\colon x _ 0 x _ 3 ^ 3 + x _ 1 x _ 2 ^ 3 + x _ 2 x _ 3 q ( x _ 0 , x _ 1 ) + g ( x _ 0 , x _ 1 ) = 0 , \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} J _ { \\gamma } ( m ( r ) \\ , d r ) \\ ; = \\ ; - \\ , \\pi \\int _ 0 ^ { \\infty } \\Gamma '' ( r ) \\ , m ( r ) \\ , d r \\ ; - \\ ; \\pi \\int _ 0 ^ { \\infty } \\sigma ( m ( r ) ) \\ , \\Gamma ' ( r ) ^ 2 \\ , d r \\ ; . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} r ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ n \\left \\{ d ( \\mathbf { x } _ i ) ^ { - 2 } d ( \\mathbf { x } ) ^ { - 2 } \\| \\mathbf { x } _ i - \\mathbf { x } \\| ^ { - 4 p } \\right \\} . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} \\left | f ( x ) - f ( y ) \\right | _ p = p ^ { \\tau _ j } | x - y | _ p . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} E = \\{ & ( m ( x _ 1 , m ( x _ 2 , x _ 3 ) ) , m ( m ( x _ 1 , x _ 2 ) , x _ 3 ) ) , \\\\ & ( m ( e , x _ 1 ) , x _ 1 ) , \\\\ & ( m ( x _ 1 , e ) , x _ 1 ) , \\\\ & ( m ( x _ 1 , i ( x _ 1 ) ) , e ) \\} . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\mathcal { R } ' & \\triangleq \\mathcal { R } \\sqcup \\Delta \\mathcal { R } _ 1 \\setminus \\Delta \\mathcal { R } _ 2 \\\\ & = \\mathcal { R } _ 0 \\sqcup \\mathcal { R } _ 1 ' \\sqcup \\mathcal { R } _ 2 ' \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\varphi = \\left ( \\psi \\circ \\left ( z _ 0 , \\frac { \\alpha z _ 1 + \\beta } { \\gamma z _ 1 + \\delta } \\right ) \\circ \\psi ^ { - 1 } \\right ) \\circ \\left ( \\psi \\circ \\left ( \\frac { a ( z _ 1 ) z _ 0 + b ( z _ 1 ) } { c ( z _ 1 ) z _ 0 + d ( z _ 1 ) } , z _ 1 \\right ) \\circ \\psi ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} 2 ( - 1 ) ^ { | a | | b | } f ( \\{ a , b \\} ) & = ( - 1 ) ^ { | a | | b | } g ( a ) f ( b ) + ( - 1 ) ^ { | a | | b | } f ( a ) g ( b ) - g ( b ) f ( a ) - f ( b ) g ( a ) \\\\ & = ( - 1 ) ^ { | a | | b | } f ( a ) g ( b ) - g ( b ) f ( a ) + ( - 1 ) ^ { | a | | b | } ( - 1 ) ^ { | a | | b | } f ( \\{ a , b \\} ) \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\forall \\sigma \\in S _ n , \\ , \\forall \\tau \\in S _ n , \\ , \\forall g \\in \\mathcal { E } _ f , \\ \\kappa ( \\sigma , \\tau ( g ) ) = \\kappa ( \\sigma , g ) . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\mathcal { G } _ { f , \\mathrm { i d } } ( s ) = s f \\bigl ( \\mathcal { G } _ { f , \\mathrm { i d } } ( s ) \\bigr ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} h _ 1 ( x ) : = \\sup _ { z \\in B _ { r + \\delta } ( x ) } f ( z ) , h _ 2 ( x ) : = \\sup _ { z \\in B _ { r - \\delta } ( x ) } f ( z ) . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} \\Lambda ( s , f ) : = ( 2 \\pi ) ^ { - ( s + \\tfrac { k - 1 } { 2 } ) } \\Gamma ( s + \\tfrac { k - 1 } { 2 } ) L ( s , f ) = \\varepsilon \\Lambda ( 1 - s , f ) , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} & A = \\begin{pmatrix} x & y \\\\ 0 & 1 / x \\end{pmatrix} & & B = \\begin{pmatrix} z & w \\\\ 0 & 1 / z \\end{pmatrix} & & C = \\begin{pmatrix} 1 & \\frac { - 1 } { 2 } \\left ( x y ( 1 - z ^ 2 ) - z w ( 1 - x ^ 2 ) \\right ) \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} V _ { , k } Y ^ { k } + d _ { 2 } V + c _ { 2 } = 0 . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} ( S ^ d ) ^ { \\star k } = \\{ \\lambda \\ , x : = \\sum _ { i = 1 } ^ k \\lambda _ i x _ i \\mid x \\in ( S ^ d ) ^ k , \\ , 0 \\leq \\lambda _ i \\leq 1 , 1 \\leq i \\leq k , \\ , \\ , \\sum _ { i = 1 } ^ k \\lambda _ i = 1 \\} \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} S ( [ 0 , \\alpha ) , m ) = \\{ f \\in L ^ 0 [ 0 , \\alpha ) : \\ , \\exists A , m ( A ^ c ) < \\infty , f \\chi _ A \\in L _ { \\infty } [ 0 , \\alpha ) \\} . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} e ^ { t \\Delta } f ( x ) = \\int _ \\Omega H ( x , y , t ) f ( y ) d y \\quad \\forall x \\in \\Omega . \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} C \\left \\{ \\zeta \\ddagger \\int _ { A } f d \\Lambda \\right \\} = \\int _ { A } \\kappa _ { L } ( \\zeta f ( w ) ) M ( d w ) \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} T _ { n } \\left ( { u , \\xi } \\right ) = u \\frac { \\partial S _ { n } \\left ( { u , \\xi } \\right ) } { \\partial \\xi } = \\sum \\limits _ { s = 0 } ^ { n - 2 } { \\frac { F _ { s + 1 } \\left ( \\xi \\right ) } { u ^ { s } } , } \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\det X ( z ) = \\det X _ s ( z ) + . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} M = W ^ { r } M \\rightarrow \\cdots \\rightarrow W ^ { - 1 } M = 0 \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} I _ 1 = \\int _ 0 ^ { \\infty } \\frac { J _ 1 ( r \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } ) } { r ^ 2 + a ^ 2 } { \\rm d } r = \\frac { 1 } { a ^ 2 ( \\sqrt { \\alpha ^ 2 + \\beta ^ 2 ) } } \\bigl ( 1 - a \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } K _ 1 \\bigl ( a \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } \\bigr ) \\bigr ) . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} ( 1 \\pm 2 \\gamma ) \\alpha _ S ^ { \\binom { r } { 2 } - 1 } ( n / t ) ^ { r - 2 } \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} \\int _ D \\int _ D g _ x ( h ) \\ > d ( \\delta _ l * \\delta _ w ) ( h ) \\ > \\alpha ( w ) \\ > d \\omega _ D ( w ) & = \\int _ D g _ x ( l * w ) \\alpha ( w ) \\ > d \\omega _ D ( w ) \\\\ & = \\int _ D g _ x ( w ) \\alpha ( \\bar l * w ) \\ > d \\omega _ D ( w ) \\\\ & = \\alpha ( \\bar l ) \\int _ D g _ x ( w ) \\alpha ( w ) \\ > d \\omega _ D ( w ) = \\alpha ( \\bar l ) T _ \\alpha g ( x ) . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\ddot { \\eta } ^ i ( t ) + \\sum _ { j , k = 1 } ^ n \\gamma ^ i _ { j k } \\big ( \\dot { \\eta } ( t ) \\big ) \\dot { \\eta } ^ j ( t ) \\dot { \\eta } ^ k ( t ) = 0 \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} d X ^ i _ t = \\sigma ^ { i } _ { \\alpha } ( X _ { t _ - } ) ( B ^ { - 1 } ) ^ { \\alpha } _ { \\beta , t } d \\tilde { Z } ^ { \\beta } _ t , \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\varphi _ { \\epsilon } } { \\partial t } = \\log \\frac { \\omega _ { \\varphi _ { \\epsilon } } ^ n } { \\omega _ 0 ^ n } + F _ 0 + \\gamma ( k \\chi + \\varphi _ { \\epsilon } ) + \\log ( \\prod _ { i = 1 } ^ d ( \\epsilon ^ 2 + | s _ i | _ { H _ i } ^ 2 ) ) ^ { ( 1 - \\beta ) \\tau _ i } + \\theta _ X ( \\omega _ { \\varphi _ { \\epsilon } } ) \\\\ \\varphi _ { \\epsilon } | _ { t = 0 } = c _ { \\epsilon 0 } . \\end{cases} \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\Gamma _ { j k } ^ { i } \\dot { x } ^ { j } \\dot { x } ^ { k } + \\omega \\left ( t \\right ) V ^ { , i } = 0 ~ ~ ~ , ~ \\omega _ { , t } \\neq 0 , \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\# = 2 7 , \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\chi = \\rho ( 1 - \\rho ) . \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{gather*} \\big \\{ m - n \\colon a _ { m , n } ^ { q - 2 } ( f ) > 0 , \\ , m , n \\geq 0 \\big \\} = \\left ( \\bigcup _ { j = 2 } ^ 5 5 \\mathbb Z _ + + j \\right ) \\cup ( - 5 \\mathbb Z _ + - 4 ) , \\end{gather*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\| \\ + F \\| \\geq \\sum _ { F \\in \\ + F } x _ F = \\frac { n - k } { k - 1 } + \\binom { k - 1 } { 2 k - 1 - n } \\frac { 1 } { \\binom { k - 1 } { 2 k - 1 - n } } = \\frac { n - 1 } { k - 1 } = 1 / p . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} C F ( h _ * ) = \\Omega \\left ( \\coprod _ { p \\in R ( h ) } [ 0 , 1 ] _ p \\right ) \\widehat \\otimes \\Lambda _ 0 . \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} T '' _ { k _ 2 , \\tilde { k } _ 2 , k _ 3 , \\tilde { k } _ 3 } = \\delta _ { k _ 2 = \\tilde { k } _ 2 = k _ 3 = \\tilde { k } _ 3 = \\delta _ 2 \\delta _ 3 2 ^ { \\alpha + \\beta } = 1 } \\frac { M } { \\pi ^ 2 } & \\left \\{ 1 + O \\left ( \\frac { 1 } { \\sqrt { M } } \\right ) \\ ! \\ ! \\right \\} \\\\ & + O _ C ( \\sqrt { M } V ^ 2 ( \\log V ) ^ 2 ( \\log M ) ^ { - C } ) . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} & ( 1 + E ( t ) ) ^ { 1 - 3 / 2 p } - ( 1 + E _ 0 ) ^ { 1 - 3 / 2 p } \\leq C _ p \\int _ 0 ^ t \\left ( 1 - \\frac { d H [ u ] } { d t } \\right ) ^ { 1 / p } d t ' \\\\ & \\leq C _ p t ^ { 1 - 1 / p } \\left ( \\int _ 0 ^ t \\left ( 1 - \\frac { d H [ u ] } { d t } \\right ) d t ' \\right ) ^ { 1 / p } = C _ p t ^ { 1 - 1 / p } ( t + H [ u _ 0 ] - H [ u ( t ) ] ) ^ { 1 / p } . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} { \\rm e v } _ j ( { \\bf x } ) = \\begin{cases} { \\rm e v } _ j ( { \\bf x } _ 1 ) & \\\\ { \\rm e v } _ { j - i + 1 } ( { \\bf x } _ 2 ) & \\\\ { \\rm e v } _ { j - k _ 2 + 1 } ( { \\bf x } _ 1 ) & \\end{cases} \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} G _ 2 ( x ) = x ^ 2 F _ T ( x ) C ( x ) + \\frac { x ^ 3 C ( x ) ^ 2 \\big ( F _ T ( x ) - 1 \\big ) } { 1 - 2 x } + \\frac { x ^ 3 C ( x ) ^ 2 } { 1 - 2 x } - \\frac { x ^ 3 C ( x ) ^ 2 } { ( 1 - x ) ^ 2 } + \\frac { x ^ 3 C ( x ) } { ( 1 - x ) \\big ( 1 - x - x C ( x ) \\big ) } \\ , . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\sigma \\circ \\tilde { h } = h \\circ \\sigma . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} 0 = - \\ , \\frac { ( 2 \\ , h ) ^ 2 } { 1 2 } \\ , \\Re \\ , G ^ { ( 2 \\ , n + 2 ) } ( 1 / 2 ) \\ , . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} G _ k ^ 0 ( x ) & = G _ { k } ( x ) - l ( k ) = \\frac 1 { 2 \\pi } \\int _ 0 ^ { \\infty } \\frac { e ^ { i x \\xi } - \\chi ( \\xi ) } { \\xi - k } ~ d \\xi , \\\\ T _ k ^ 0 ( \\varphi ) & = G _ k ^ 0 * ( u \\varphi ) = T _ k ( \\varphi ) - l ( k ) \\langle \\varphi , u \\rangle . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\mathcal { Z } _ 1 = \\mathcal { S } \\Lambda _ A + \\mathcal { T } _ A \\Lambda _ A ^ 2 + O ( ( \\Phi _ 2 ^ c ) ^ 2 ) + O ( \\Phi _ 2 ^ c \\Lambda _ A ) + O ( \\Lambda _ A ^ 3 ) \\ , . \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} T _ f g = T ^ { f \\circ \\pi } g = \\tilde T _ f g \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} x ' ( t ) = g ( t ) x ^ 2 ( t ) + \\epsilon f ( t ) x ^ 3 ( t ) , \\ , \\ , t \\in [ a , b ] , \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} p ^ \\star ( x ) - q ( x ) = \\frac { ( n - 1 ) p ( x , x ) } { n - s p ( x , x ) } - q ( x ) \\geq k > 0 , \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} C ^ T f ^ T = a _ 0 \\left [ \\beta \\varkappa ^ T - \\alpha \\left ( R ^ T \\right ) ^ * \\varkappa ^ T \\right ] . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} ( e _ g \\ , e _ h ) ( \\phi ) & = ( e _ g \\otimes e _ h ) ( \\phi ^ { ( 1 ) } \\otimes \\phi ^ { ( 2 ) } ) \\\\ & = e _ g ( \\phi ^ { ( 1 ) } ) e _ h ( \\phi ^ { ( 2 ) } ) \\\\ & = \\langle g , \\phi ^ { ( 1 ) } \\rangle _ { \\mathcal { C } } \\ , \\langle h , \\phi ^ { ( 2 ) } \\rangle _ { \\mathcal { C } } . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\zeta _ k ^ { ( i ) } ( t ) & = p _ k - \\int _ 0 ^ t r _ k ( \\boldsymbol { \\zeta } ^ { ( i ) } ( s ) ) \\ , d s , k \\in \\mathbb { N } , \\\\ \\psi ^ { ( i ) } ( t ) & = \\sum _ { k = 0 } ^ \\infty ( k - 2 ) \\int _ 0 ^ t r _ k ( \\boldsymbol { \\zeta } ^ { ( i ) } ( s ) ) \\ , d s . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\frac { 1 } { z _ 0 } = \\frac { \\int _ { i + 2 } ^ { i } \\eta ^ 4 d \\tau } { \\int _ { i + 2 } ^ { i + 1 } \\eta ^ 4 d \\tau } = 1 + \\frac { \\int _ { i + 1 } ^ { i } \\eta ^ 4 d \\tau } { \\int _ { i + 2 } ^ { i + 1 } \\eta ^ 4 d \\tau } = 1 + \\chi ( T ^ { - 1 } ) = 1 + e ^ { - 2 \\pi i / 6 } , \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} x = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } u V ( f _ i ) , \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} I _ D ( X _ 0 ^ T \\to Y _ 0 ^ T ) = \\inf _ { \\Delta _ n } \\sum _ { i = 1 } ^ n I ( X _ { t _ { n , 0 } } ^ { t _ { n , i } } ; Y _ { t _ { n , i - 1 } } ^ { t _ { n , i } } | Y _ { t _ { n , 0 } } ^ { t _ { n , i - 1 } } ) . \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B _ { 2 \\Lambda _ { n , j , d } ^ { + } } ( 0 ) } \\rho _ n \\overline { h } _ j ( z ) e ^ { \\overline { \\tilde { u } _ n ^ { ( 1 ) } } ( z ) } e ^ { - \\lambda _ { n , j } ^ { ( 1 ) } } \\Big ( \\frac { 1 - e ^ { \\overline { \\tilde { u } _ n ^ { ( 2 ) } } - \\overline { \\tilde { u } _ n ^ { ( 1 ) } } } } { \\| \\tilde { u } _ n ^ { ( 1 ) } - \\tilde { u } _ n ^ { ( 2 ) } \\| _ { L ^ \\infty ( M ) } } \\Big ) \\mathrm { d } z = o \\Big ( \\frac { 1 } { \\lambda _ { n , j } ^ { ( 1 ) } } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\tilde { h } _ k ^ { ( 1 ) } \\big ( 2 \\ln ( k ) \\big ) & = \\gamma _ { 2 \\ln ( k ) } k ^ { \\frac { 1 } { 2 \\ln ( k ) } } q _ p \\\\ & = \\Bigg [ \\frac { \\Gamma \\big ( \\ln ( k ) + \\frac { 1 } { 2 } \\big ) k } { - \\big ( 2 \\ln ( k ) \\big ) ^ { 1 - \\ln ( k ) } \\sqrt { \\pi } \\Gamma \\big ( \\ln ( k ) \\big ) \\ln ( p ) } \\Bigg ] ^ { \\frac { 1 } { 2 \\ln ( k ) } } \\\\ & \\sim \\sqrt { 2 e \\ln ( k ) } . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L _ a U = 0 \\mbox { i n } D _ { \\delta } , \\\\ U ( 0 , x , y ) = 0 \\mbox { o n } ( x , y ) \\in D _ { \\delta } \\cap \\{ t = 0 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\theta ( R / I ) = I ^ \\perp \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} \\Gamma _ { t } ^ { x _ { 0 } , R } f ( x ) = \\int _ { B ( x _ { 0 } , R ) } \\Gamma ^ { x _ { 0 } , R } ( t , x ; 0 , \\xi ) f ( \\xi ) d \\xi \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} x ( t , p ) = \\sum _ { i = 1 } ^ \\infty v _ i ( t ) \\Phi _ i ( p ) \\mbox { a n d } y ( t , p ) = \\sum _ { i = 1 } ^ \\infty w _ i ( t ) \\Phi _ i ( p ) , \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} P ( ( x _ 1 , x _ 2 ) , \\mu ) = & \\begin{pmatrix} y _ 1 & y _ 1 \\\\ y _ 2 & - y _ 2 \\\\ y _ 3 & y _ 4 \\end{pmatrix} \\begin{pmatrix} c _ E & 0 \\\\ 0 & \\overline { c _ E } \\\\ \\end{pmatrix} , y _ i = \\int _ { \\beta _ i } \\psi _ 1 \\quad ( i = 1 , 2 , 3 , 4 ) . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\partial _ t u \\ , = \\alpha \\ , \\Delta u + \\beta \\ , u ( 1 - u ) + \\sqrt { \\gamma \\ , u ( 1 - u ) } \\dot { W } , \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log n } { \\log \\beta _ n } \\not = 0 . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\max _ { 1 \\le i \\le p } \\left | \\frac { z _ i ' z _ i } { n - 1 } - 1 \\right | = o _ p ( 1 ) ~ ~ ~ \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} [ \\phi , \\psi ] = \\int _ { \\Omega _ \\varepsilon } \\Bigl [ 5 ( V _ 1 + V _ 2 ) ^ 4 \\phi - h - \\Bigr ] \\ , \\psi \\quad \\psi \\in H . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} f _ { k l } ^ { \\left ( \\alpha \\right ) } \\left ( Z , W \\right ) = 0 , \\varphi _ { k l } ^ { \\left ( \\left [ \\frac { \\alpha + 1 } { 2 } \\right ] \\right ) } \\left ( Z , W \\right ) = 0 , \\quad \\mbox { f o r a l l $ k = 1 , \\dots , q $ a n d $ l = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k & \\int _ { [ 0 , t ] \\times [ 0 , \\infty ) } 1 _ { \\left [ \\overline q _ { k - 1 } , \\ \\overline q _ { k - 1 } + \\lambda _ k ^ X ( s - , X ( s - ) \\right ) } ( x ) N ( d s \\times d x ) \\\\ Z ( t ) = Z ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k & \\int _ { [ 0 , t ] \\times [ 0 , \\infty ) } 1 _ { \\left [ \\overline q _ { k - 1 } , \\ \\overline q _ { k - 1 } + \\lambda _ k ^ Z ( s - , Z ( s - ) \\right ) } ( x ) N ( d s \\times d x ) , \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } - u _ { 1 } ( x , t , y , s ) v _ { 1 } ( x ) v _ { 2 } ( y ) c _ { 1 } ( t ) \\partial _ { s } c _ { 2 } ( s ) \\\\ + A ( y , s ) ( \\nabla u ( x , t ) + \\nabla _ { y } u _ { 1 } ( x , t , y , s ) ) \\\\ \\cdot v _ { 1 } ( x ) \\nabla _ { y } v _ { 2 } \\left ( y \\right ) c _ { 1 } ( t ) c _ { 2 } \\left ( s \\right ) d y d s d x d t = 0 \\end{gather*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\Lambda _ f ( s , c _ \\chi ) = i ^ k \\xi ( q ) \\overline { \\chi ( - N ) } ( N q ^ 2 ) ^ { \\frac 1 2 - s } \\Lambda _ g ( 1 - s , c _ { \\overline { \\chi } } ) . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\frac { C _ { q , n } } { ( k - 2 - q ) ! } & = - \\sum _ { t = 0 } ^ { q - 1 } [ 2 ( n - 1 ) a _ { k - t } + b _ { k - t - 1 } ] m _ { ( q - t , \\dot { 0 } ) } - \\sum _ { s = 1 } ^ { [ q / 2 ] } \\sum _ { t = 0 } ^ { q - 2 s } 2 a _ { k - t } \\ , m _ { ( q - t - s , s , \\dot { 0 } ) } \\\\ & - \\frac { n ( n - 1 ) } { k ( k - 1 ) } q ( 2 k - q - 1 ) a _ { k - q } - \\frac { n } { k - 1 } q b _ { k - q - 1 } , \\\\ & q = 1 , 2 , \\ldots , k - 2 , \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\varphi _ i = \\ < \\Gamma _ i , \\Omega _ X \\ > = \\int _ { \\Gamma _ i } \\Omega _ X , ( i = 1 , 2 , 3 , 4 ) . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\rho ^ * ( \\psi _ 1 ) = \\chi ( \\rho ) \\psi _ 1 , \\rho ^ * ( \\psi _ 2 ) = \\overline { \\chi } ( \\rho ) \\psi _ 2 , \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\frac { \\| \\bar { x } _ { L , k } - x _ { t r u e } \\| } { \\| x _ { t r u e } \\| } = \\frac { \\| x _ { L , k } - x _ { t r u e } \\| } { \\| x _ { t r u e } \\| } \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} f = \\sum _ { U \\in F } a _ { U } 1 _ { U } , \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} ( y _ 1 , y _ 2 ) ^ { \\tau _ 1 } = ( y _ 1 , y _ 2 ^ { \\sigma _ 2 } ) , ( y _ 1 , y _ 2 ) ^ { \\tau _ 2 } = ( y _ 1 ^ { \\sigma _ 1 } , y _ 2 ) , ( y _ 1 , y _ 2 ) ^ { \\tau _ 3 } = ( y _ 1 ^ { \\sigma _ 1 } , y _ 2 ^ { \\sigma _ 2 } ) . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\kappa ( x , g ) = \\kappa ( t _ { i _ 1 } , t _ { i _ 2 } \\dots t _ { i _ m } ( g ) ) \\ , \\kappa ( t _ { i _ 2 } , t _ { i _ 3 } \\dots t _ { i _ m } ( g ) ) \\ldots \\kappa ( t _ { i _ m } , g ) . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} q ( w _ 1 , w _ 2 ) = p ( w _ 1 ( \\tilde { y } + x ) + w _ 2 ( \\tilde { y } - x ) ) \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} Q _ { 2 t } x = Q _ t x + Q _ { t } e ^ { 2 t A } x = Q _ t ( x + e ^ { 2 t A } x ) , x \\in H . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} \\log G ( n ) = \\sum _ { p \\mid \\phi ( n ) } \\log G _ p ( n ) \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\frac { L ^ \\alpha [ e ^ a ] } { e ^ a } & = \\int \\left ( 1 - \\frac { \\exp ( a ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( a / \\epsilon ) } \\right ) K d y \\\\ & = \\frac { L ^ \\alpha [ e ^ g ] } { e ^ g } + \\int \\left ( \\frac { \\exp ( g ( \\hat { x } | x | ^ { 1 / \\epsilon } - y ) ) } { \\exp ( g ( \\hat { x } | x | ^ { 1 / \\epsilon } ) ) } - \\frac { \\exp ( a ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( a / \\epsilon ) } \\right ) K d y , \\\\ \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} E _ f : = \\overline { S _ f \\setminus Z } \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} S \\omega _ l = \\gamma _ l \\omega _ l , l = 1 , \\ldots , n , \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} z _ 0 = \\frac { a ( i + 3 ) } { a ( i ) } = 1 . 0 9 1 0 8 4 9 0 8 9 \\ldots + 0 . 4 9 4 2 8 1 8 1 8 6 \\ldots i . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\vartheta ( \\hbar ) [ \\mathfrak { X } ^ + ( u , \\lambda _ 1 ) , \\mathfrak { X } ^ { - } ( v , \\lambda _ 2 ) ] = \\frac { \\vartheta ( u - v + \\lambda _ { 1 } ) } { \\vartheta ( u - v ) \\vartheta ( \\lambda _ { 1 } ) } \\Phi ( v ) + \\frac { \\vartheta ( u - v - \\lambda _ { 2 } ) } { \\vartheta ( u - v ) \\vartheta ( \\lambda _ { 2 } ) } \\Phi ( u ) . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\mu _ { 1 } \\left \\vert \\eta \\right \\vert ^ { 2 } \\leq \\displaystyle \\sum \\limits _ { i , j = 1 } ^ { n } a _ { i , j } \\left ( x \\right ) \\eta _ { i } \\eta _ { j } \\leq \\mu _ { 2 } \\left \\vert \\eta \\right \\vert ^ { 2 } , \\forall x \\in \\mathbb { R } ^ { n } , \\forall \\eta = \\left ( \\eta _ { 1 } , . . . \\eta _ { n } \\right ) \\in \\mathbb { R } ^ { n } . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\otimes W ' - \\left ( \\overline { \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\otimes W ' } \\right ) ^ { t } = 2 \\sqrt { - 1 } \\left ( V \\otimes Z ' \\right ) \\overline { \\left ( V \\otimes Z ' \\right ) } ^ { t } , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} R ( X , Y ) Z = - \\nabla _ X \\nabla _ Y Z + \\nabla _ Y \\nabla _ X Z + \\nabla _ { [ X , Y ] } Z \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\langle M _ { g ^ { * } } \\phi , \\psi \\rangle _ { \\mathcal { P } } = \\langle \\phi , M _ { g } \\psi \\rangle _ { \\mathcal { P } } \\mathrm { o r } \\langle \\phi g ^ { * } , \\psi \\rangle _ { \\mathcal { A } } = \\langle \\phi , \\psi g \\rangle _ { \\mathcal { A } } \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\mathbb { P } ( K _ n = k ) = \\frac { ( n , k ) ^ { \\dagger } } { n ! } \\mbox { f o r } ~ 1 \\le k \\le n , \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} K = \\{ ( \\lambda , c ) \\in I \\times [ r , s ] : \\ ; \\exists x \\in X , f _ \\lambda ( x ) = c , f _ \\lambda ' ( x ) = 0 \\} . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} F ( f ) = \\mathbb E \\sum _ { n \\geq 0 } \\langle f _ n , g _ n \\rangle , \\ ; \\ ; \\ ; & f = ( f _ n ) _ { n \\geq 0 } \\in Q _ q ^ p , \\\\ \\| ( g _ n ) _ { n \\geq 0 } \\| _ { Q _ { p ' } ^ { q ' } } & \\lesssim _ { p , q } \\| F \\| _ { ( Q _ q ^ p ) ^ * } . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} S _ 2 ( R ) = \\sum _ { n \\leq R } r _ 2 ( n ) , \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} | | \\delta ^ { \\ell } ( \\vec { a } ) | | _ { \\infty , 0 } & \\le C | | \\delta ^ { \\ell } ( \\vec { a } ) | | _ { s - | \\ell | } \\\\ & = C | | \\sum _ { j , m } m ^ { \\ell } a _ { j , m } \\prod _ { g = 1 } ^ n U _ g ^ { m _ g } f _ j | | _ { s - | \\ell | } \\\\ & < C | | \\sum _ { j , m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { | \\ell | } a _ { j , m } \\prod _ { g = 1 } ^ n U _ g ^ { m _ g } f _ j | | _ { s - | \\ell | } \\\\ & = C | | P _ { \\lambda ^ { | \\ell | } } ( \\vec { a } ) | | _ { s - | \\ell | } \\\\ & = C | | \\vec { a } | | _ s \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\partial _ { t } \\Pi _ { \\mu } ^ { \\alpha } = \\partial _ { t } \\hat n ^ { \\alpha } \\hat n _ { \\mu } + \\hat n ^ { \\alpha } \\partial _ { t } \\hat n _ { \\mu } . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\theta = ( \\theta _ 1 \\cdots \\theta _ k ) ^ { \\frac 1 k } . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} D _ { i j } ( a ^ { i j } w ) = 0 B _ 2 \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} \\det \\tilde { X } = ( - 1 ) ^ q \\det X \\begin{pmatrix} I \\sqcup K \\\\ I ' \\sqcup K ' \\end{pmatrix} \\cdot \\left ( \\det X \\begin{pmatrix} J \\sqcup K \\\\ J ' \\sqcup K ' \\end{pmatrix} \\right ) ^ { p + q - 1 } . \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} V \\left ( Z ' , Z '' \\right ) = \\left ( Z ' , Z _ { 2 } '' \\odot \\varphi _ { B } \\left ( Z ' , Z '' \\right ) \\right ) . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\| h _ n - h _ m \\| _ { \\sup } & \\leqslant \\sum _ { i = m } ^ { n - 1 } \\frac { 1 } { d ^ { i + 1 } } \\| ( ( f ^ { \\mathrm { a n } } ) ^ i ) ^ * ( \\lambda ) \\| _ { \\sup } = \\frac { \\| \\lambda \\| _ { \\sup } } { d ^ { m + 1 } } \\sum _ { i = 0 } ^ { n - m - 1 } \\frac { 1 } { d ^ { i } } \\\\ & \\leqslant \\frac { \\| \\lambda \\| _ { \\sup } } { d ^ { m + 1 } } \\sum _ { i = 0 } ^ { \\infty } \\frac { 1 } { d ^ { i } } = \\frac { \\| \\lambda \\| _ { \\sup } } { d ^ m ( d - 1 ) } . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{gather*} \\Upsilon ^ { a b } = \\eta ^ a \\circ \\zeta ^ b - \\eta ^ b \\circ \\zeta ^ a . \\end{gather*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 g ( t ) d t = 0 \\mbox { a n d } \\int _ { - 1 } ^ 1 f ( t ) r _ k ( t ) d t = 0 , \\ , \\ , k \\geq 0 \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\dim _ { \\mathbb Q } H ^ 0 ( X \\stackrel { \\pi } \\to Y ) = 1 . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{gather*} C _ n ^ { ( q , q ^ { \\theta } ) } \\big ( x _ 1 / \\big ( q ^ { \\theta } x _ 2 \\big ) \\big ) = \\frac { q ^ { \\theta } x _ 2 - q ^ { 2 n } x _ 1 } { q ^ { \\theta } x _ 2 - x _ 1 } \\frac { 1 } { t ^ n } \\prod _ { i = 1 } ^ n { \\left \\{ \\frac { q ^ { \\theta } - q ^ { i - 1 } } { 1 - q ^ i } \\frac { q ^ { \\theta } x _ 2 - q ^ { i - 1 } x _ 1 } { x _ 2 - q ^ i x _ 1 } \\right \\} } . \\end{gather*}"}
-{"id": "3273.png", "formula": "\\begin{gather*} \\Lambda ^ M _ N \\delta _ { \\lambda } ( \\mu ) = \\Lambda ^ M _ N ( \\lambda , \\mu ) . \\end{gather*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\int \\ , f \\ , L _ \\eta g \\ , \\dd \\mu = - \\ , \\int \\ , ( H ^ { - 2 \\eta } \\ , \\nabla _ x f . \\nabla _ x g + \\nabla _ y f . \\nabla _ y g ) \\ , \\dd \\mu \\ , . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} - \\Delta _ y G _ { B _ 1 } ( y , x ) = \\delta _ x B _ 1 , G _ { B _ 1 } ( y , x ) = 0 y \\in \\partial B _ 1 . \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} c _ i ( \\lambda ) = \\frac { \\partial P _ i } { \\partial r } ( \\vec { 0 } , 0 , \\lambda ) \\\\ c _ { i j } ( \\lambda ) = \\frac { \\partial P _ i } { \\partial u _ j } ( \\vec { 0 } , 0 , \\lambda ) \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} B _ 1 ^ * & = \\rho ( 1 - \\rho ^ 2 ) \\Gamma _ 2 , \\ B _ 2 ^ * = ( 1 - \\rho ^ 2 ) \\Gamma _ 1 + ( 1 - \\rho ^ 2 ) \\Gamma _ 2 , \\ B _ 3 ^ * = ( 1 - \\rho ) \\Gamma _ 3 . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} T _ { l , \\alpha ^ { ( l ) } } ( W ) = T _ { l } ( W _ { \\alpha ^ { ( l ) } , 1 } ) T _ { l } ( W _ { \\alpha ^ { ( l ) } , 2 } ) \\cdots T _ { l } ( W _ { \\alpha ^ { ( l ) } , l ( \\alpha ^ { ( l ) } ) } ) \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta W = h & \\mbox { i n } \\O ^ \\# \\\\ W = 0 & \\mbox { o n } \\partial \\O ^ \\# \\ , . \\end{cases} \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} X _ K : = ( \\P ( W _ 1 ) \\times \\P ( W _ 2 ) ) \\cap \\P ( K ^ \\perp ) \\subset \\P ( W _ 1 \\otimes W _ 2 ) , \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\Lambda _ p ^ p ( \\Omega _ k ) = \\lambda _ p ^ p ( W _ \\frac d 2 ) \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} E : = ( R ^ { \\ast } ) ^ { - 1 } \\begin{pmatrix} B _ { X } \\\\ B _ { Y } \\end{pmatrix} , \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} H \\left ( q , \\frac { \\partial W } { \\partial q } ( \\bar { q } , q ) \\right ) = , \\frac { \\partial W } { \\partial \\bar { q } } = 0 . \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\mathcal { B S D } : \\quad \\mbox { I m } W : = \\frac { 1 } { 2 \\sqrt { - 1 } } \\left ( W - \\overline { W } ^ { t } \\right ) = Z \\overline { Z } ^ { t } . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ \\begin{cases} \\displaystyle \\frac { 2 } { ( 2 \\pi ) ^ { \\frac { d + 1 } { 2 } } } \\left ( \\frac { - 1 } { \\sinh ( r ) } \\partial _ r \\right ) ^ { \\frac { d - 1 } { 2 } } \\frac { \\sin ( r \\cdot \\sqrt { \\lambda - b _ d } ) } { r } & \\lambda \\ge b _ d , \\\\ [ 3 e x ] 0 & . \\end{cases} \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\tilde { { \\cal M } } _ 0 ( 0 ) = \\bigg \\{ m : | \\tilde { \\varphi } _ 0 ^ { - 1 } ( m ) \\cap { \\cal T } _ { \\bar { X } } ^ n | \\le \\frac { | { \\cal T } _ { \\bar { X } } ^ n | } { | \\tilde { { \\cal M } } _ 0 ^ { ( n ) } | } \\bigg \\} . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\Pi _ { \\mu } ^ { \\alpha } = \\hat n ^ { \\alpha } \\hat n _ { \\mu } , \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} d t + e = 2 \\left ( \\dfrac { 2 m - 1 } { m - 1 } \\right ) t ^ { m + 1 } \\Delta _ g ( t ^ { 1 - m } ) + ( c - \\Psi '' ( t ) ) t ^ 2 . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } D ( s , S _ f \\times \\overline { S _ g } ) X ^ s \\Gamma ( s ) d s = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { k - 1 } } e ^ { - n / X } , \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{gather*} M ^ { A _ k \\nu } ( S _ { A _ k \\phi } ) = M ^ { \\nu } ( S _ { \\phi } ) , \\textrm { f o r a l l f i n i t e p a t h s } \\phi = \\big ( \\phi ^ { ( 0 ) } \\prec \\phi ^ { ( 1 ) } \\prec \\cdots \\prec \\phi ^ { ( n ) } \\big ) . \\end{gather*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } A = \\left [ \\begin{matrix} 0 . 2 & \\ & 0 \\\\ 0 . 2 & & 0 . 1 \\end{matrix} \\right ] & , & A _ D = \\left [ \\begin{matrix} - 1 & \\ & 0 \\\\ - 1 & & - 1 \\end{matrix} \\right ] & & A _ d = 0 _ 2 \\end{array} . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} a _ 2 s _ 2 b _ 2 - a _ 1 s _ 1 b _ 1 & = a _ 1 u _ 1 s _ 2 b _ 2 - a _ 1 s _ 1 u _ 3 b _ 2 \\\\ & = - a _ 1 ( s _ 1 u _ 3 - u _ 1 s _ 2 ) b _ 2 = a _ 1 ( s _ 1 , s _ 2 ) _ { u _ 1 u _ 2 u _ 3 } b _ 2 . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\overline { K ( u ) } = K ( \\overline { u } ) \\ , , \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 3 / 2 } \\psi _ { s , F } ( - v ) = \\left \\{ \\begin{array} { l } ( 2 s ) ^ { - 1 } \\left ( 2 - e ^ { - \\sqrt { 3 s } \\thinspace | v | } \\right ) \\ , , \\\\ ( 2 s ) ^ { - 1 } e ^ { - \\sqrt { 3 s } \\thinspace | v | } \\ , , \\end{array} \\right . \\begin{array} { l } v > 0 \\ , , \\\\ v < 0 \\ , . \\end{array} \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} A _ n ^ + ( i ) & = \\sum _ { j = i + 1 } ^ n a ( n ; i , j ) , & & A _ n ^ + = \\sum _ { i = 1 } ^ { n - 1 } A _ n ^ + ( i ) , & & A ^ + ( x ) = \\sum _ { n \\geq 2 } A _ n ^ + x ^ n . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\| \\Phi \\| _ { - p , - q , \\mu _ { \\beta } } ^ { 2 } : = \\sum _ { n = 0 } ^ { \\infty } 2 ^ { - n q } | \\Phi ^ { ( n ) } | _ { - p } ^ { 2 } < \\infty , p , q \\in \\mathbb { N } _ { 0 } . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } f _ n c _ q ( n ) e ^ { 2 \\pi i n z } = ( - 1 ) ^ k \\xi ( q ) ( N q ^ 2 ) ^ { - \\frac { k } { 2 } } z ^ { - k } \\sum _ { n = 1 } ^ { \\infty } g _ n c _ q ( n ) e ^ { 2 \\pi i \\frac { - n } { N q ^ 2 z } } . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} p _ i = c _ i z ^ { k _ i } \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } I _ c ^ { x x } \\\\ I _ c ^ { x y } \\\\ I _ c ^ { y y } \\end{array} \\right \\} = \\int \\limits _ \\Omega \\left \\{ \\begin{array} { c } ( x - x _ c ) ^ 2 \\\\ ( x - x _ c ) ( y - y _ c ) \\\\ ( y - y _ c ) ^ 2 \\end{array} \\right \\} ~ \\mathrm { d } \\Omega \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} [ A ^ { \\epsilon } _ i ( z ) , B ^ { \\epsilon ' } _ j ( w ) ] = [ A ^ { \\epsilon } _ i ( z ) , C ^ { \\epsilon ' } _ j ( w ) ] = [ B ^ { \\epsilon } _ i ( z ) , C ^ { \\epsilon ' } _ j ( w ) ] = 0 \\ \\mathrm { f o r } \\ i \\ne j , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\beta t } E | u ^ { w } ( t , x ) | ^ 2 d t = \\frac { \\beta } { 4 } \\int _ 0 ^ { \\infty } e ^ { - t \\beta ^ 2 / 4 } E | u ^ { h , \\beta } ( t , x ) | ^ 2 d t . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\tilde { f } _ D ( t _ 0 ^ { - 1 } \\gamma t _ 3 ) = \\begin{cases} 1 & \\mbox { i f } ( E _ 1 , E _ 2 , E _ 3 ) \\in \\mathfrak { N } _ { D , \\gamma } \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} u ( t ) = M ( t ) D ( t ) V ( t ) w ( t ) \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} \\chi _ { _ { ] - \\infty , a [ } } \\left ( m _ { 0 } \\right ) \\ ; \\mathcal { S } \\ : \\chi _ { _ { [ a , \\infty [ } } \\left ( m _ { 0 } \\right ) = 0 \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\begin{pmatrix} e ^ { ( l + i \\theta ) / 2 } & 0 \\\\ 0 & e ^ { - ( l + i \\theta ) / 2 } \\end{pmatrix} = \\exp \\begin{pmatrix} ( l + i \\theta ) / 2 & 0 \\\\ 0 & { - ( l + i \\theta ) / 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} \\gamma \\geq 0 \\textnormal { a n d } \\gamma h ^ \\ast [ t _ f ] = 0 . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} & \\iota _ + \\ ; = \\ ; \\iota _ + ( \\varepsilon , r _ 0 , T ) \\ ; = \\ ; \\frac 1 { \\log T } \\log \\frac { T ^ { r _ 0 } + T ^ { \\varepsilon } } { T ^ { r _ 0 } } \\ ; , \\\\ & \\iota _ - \\ ; = \\ ; \\iota _ - ( \\varepsilon , r _ 0 , T ) \\ ; = \\ ; \\frac 1 { \\log T } \\log \\frac { T ^ { r _ 0 } } { T ^ { r _ 0 } - T ^ { \\varepsilon } } \\ ; \\cdot \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\widehat { G } _ t ( X ) : = \\xi + \\int _ 0 ^ t \\int _ U \\int _ A f ( s , X ( s ^ - ) , u , a ) \\N _ { \\rho ^ { \\widehat { \\gamma } , X } } ( d s , d u , d a ) \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} F _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z _ 0 ) ) + F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z _ 0 ) ) \\big ( \\omega _ B ( z _ 0 ) - \\omega _ \\beta ( z _ 0 ) \\big ) - \\omega _ A ( z _ 0 ) - \\omega _ B ( z _ 0 ) + z _ 0 & = r _ 1 ( z _ 0 ) + O ( | \\omega _ B ( z _ 0 ) - \\omega _ \\beta ( z _ 0 ) | ^ 2 ) \\ , , \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} S _ 0 & = S , & S _ 1 & = T ^ 3 S T ^ { - 3 } , & R _ 2 & = T ^ 2 R T ^ { - 2 } = T ^ 2 S T ^ { - 1 } . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} C ^ T _ { i , j } = \\int _ { - \\infty } ^ \\infty T _ { T - i + 1 } ( \\lambda ) T _ { T - j + 1 } ( \\lambda ) \\ , d \\rho ( \\lambda ) , i , j = 1 , \\ldots , T . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} E ' \\otimes W ' = \\begin{pmatrix} W ' _ { 1 1 } & W ' _ { 1 2 } - \\tilde { B } \\otimes W ' _ { 1 1 } \\\\ W ' _ { 2 1 } - \\tilde { C } \\otimes W ' _ { 1 1 } & W ' _ { 2 2 } - \\tilde { D } \\otimes W ' _ { 1 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} D _ { \\pi ^ { - s } } = \\sum _ { r = 0 } ^ { n - 2 } \\beta ^ { \\pi ^ { - s + r } } \\sum _ { t = 0 } ^ { n - 2 } \\beta ^ { \\pi ^ { - r + t } } C _ { \\pi ^ t } . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} L _ N ^ { \\rm r e s c } ( v ) : = \\frac { L _ { A ( v ) \\to E _ N ( w ) } - N / \\chi } { \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} P _ { i , j } = \\frac { f _ { i , j } ^ { - \\beta } } { \\sum _ { k = 1 } ^ { M } k ^ { - \\beta } } , \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} \\left ( s M _ s \\right ) _ { v \\sigma } & = \\sum _ { \\rho \\in \\Sigma } s ( \\rho v ) \\cdot ( M _ s ) _ { \\rho v , v \\sigma } \\\\ & = \\sum _ { \\rho \\in \\Sigma } \\frac { s ( \\rho v ) \\cdot s ( v \\sigma ) } { \\sum _ { \\tau \\in \\Sigma } s ( v \\tau ) } \\\\ & = s ( v \\sigma ) \\cdot \\frac { \\Sigma _ { \\rho \\in \\Sigma } s ( \\rho v ) } { \\Sigma _ { \\tau \\in \\Sigma } s ( v \\tau ) } = s ( v \\sigma ) , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\begin{array} { l l l } A & = & \\tau E ^ * A ^ { - 1 } E , \\\\ ( \\tau E ^ * A ^ { - 1 } + I ) b & = & w + \\tau E ^ { * } A ^ { - 1 } c , \\\\ \\gamma & = & \\displaystyle { \\frac { \\beta + \\langle \\tau ( c - b ) , \\frac { 1 } { 2 } A ^ { - 1 } ( c - b ) \\rangle } { \\tau + 1 } } . \\end{array} \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , V ) & = - g _ { 1 } ( V , \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\phi \\mathcal { B } Y + \\mathcal { A } _ { X } \\omega \\mathcal { B } Y ) + g _ { 2 } ( \\pi _ * ( C Y ) , ( \\nabla \\pi _ * ) ( X , \\varphi V ) ) , \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} J V _ g ( v , \\lambda ) & = J ( U _ g v , U ^ * _ { \\tau ( g ) } \\lambda ) = ( \\Phi ^ { - 1 } U ^ * _ { \\tau ( g ) } \\lambda , \\Phi U _ { r ^ 2 } U _ g v ) = ( U _ { \\tau ( g ) } \\Phi ^ { - 1 } \\lambda , \\Phi U _ { \\tau ^ 2 ( g ) } U _ { r ^ 2 } v ) \\\\ & = ( U _ { \\tau ( g ) } \\Phi ^ { - 1 } \\lambda , U ^ * _ { \\tau ^ 2 ( g ) } \\Phi U _ { r ^ 2 } v ) = V _ { \\tau ( g ) } ( \\Phi ^ { - 1 } \\lambda , \\Phi U _ { r ^ 2 } v ) = V _ { \\tau ( g ) } J ( v , \\lambda ) . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} [ T _ { n } ( U _ 1 ) , T _ { n } ( U _ 2 ) ] = \\sum _ { U _ 3 } ( N _ { U _ 1 , U _ 2 } ^ { U _ 3 } - N _ { U _ 2 , U _ 1 } ^ { U _ 3 } ) T _ { n } ( U _ 3 ) \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\int _ \\tau ^ t { { 1 } } _ { \\{ \\zeta _ 0 ( s ) > { \\tilde { \\zeta } } _ 0 ( s ) \\} } ( \\zeta _ 0 ( s ) - { \\tilde { \\zeta } } _ 0 ( s ) ) ( \\eta _ 0 ' ( s ) - { \\tilde { \\eta } } _ 0 ' ( s ) ) \\ , d s & \\le \\int _ \\tau ^ t { { 1 } } _ { \\{ \\zeta _ 0 ( s ) > { \\tilde { \\zeta } } _ 0 ( s ) \\} } ( \\zeta _ 0 ( s ) - { \\tilde { \\zeta } } _ 0 ( s ) ) \\eta _ 0 ' ( s ) \\ , d s \\\\ & \\le \\int _ \\tau ^ t { { 1 } } _ { \\{ \\zeta _ 0 ( s ) > 0 \\} } \\zeta _ 0 ( s ) \\ , \\eta _ 0 ( d s ) \\\\ & = 0 , \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) ( 1 - 2 \\tilde { \\alpha } _ i ( x ) ^ 2 ) & \\geq - 1 ; \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} & | ( \\zeta - \\zeta _ 0 ) ^ 2 + \\tau ^ 2 | \\\\ = { } & \\bigl | ( u - u _ 0 ) ^ 2 - \\big ( a ( u ) - a ( u _ 0 ) \\big ) ^ 2 + \\tau ^ 2 + 2 \\mathrm { i } ( u - u _ 0 ) \\big ( a ( u ) - a ( u _ 0 ) \\big ) \\bigr | , \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\widehat { N } _ Q = \\widehat { P } ( M ^ * ) ^ { - 1 } N M ^ { - 1 } \\widehat { P } . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} \\alpha = [ p _ 1 , \\ldots , p _ { t - 1 } ] , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} u _ 1 ( t , 0 ) = u _ 2 ( t , 0 ) = 0 \\ , , u _ 1 ( t , \\pi ) = g _ 1 ( t ) \\ , , u _ 2 ( t , \\pi ) = g _ 2 ( t ) t \\in ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} r \\cdot \\nabla _ r V = | r | ^ { n - 1 } h \\ , , \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} K i G ( f ) \\alpha _ a & = \\alpha _ b \\alpha _ b ^ \\dagger K i G ( f ) \\alpha _ a \\\\ K i G ( g ) \\alpha _ c & = \\alpha _ d \\alpha _ d ^ \\dagger K i G ( g ) \\alpha _ c \\\\ \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} a = \\lambda a _ 1 \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} j ^ \\ast ( \\rho ^ \\ast ( \\operatorname { v o l } _ { \\mathcal { H } , \\partial } ) ) = \\operatorname { C h a r } _ { \\nabla _ \\rho , { \\nabla \\vert _ { \\partial M _ i } } } ( \\varpi , \\{ \\beta _ i \\} ) . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} E _ { ( B , \\beta ) _ 2 } ( z ) = E _ { B _ 1 , \\beta _ 1 ; B _ 2 \\beta _ 2 } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { \\Gamma ( \\beta _ 1 + k B _ 1 ) \\Gamma ( \\beta _ 2 + k B _ 2 ) } , \\ ; z \\in \\mathbb { C } , \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} V ^ l = \\bigoplus _ { 0 \\le i \\le l } { \\cal V } ^ i \\mbox { \\ \\ a n d \\ \\ } V ^ l _ \\# = \\bigoplus _ { 0 \\le i \\le l } { \\cal V } ^ i _ \\# . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} \\tilde e ( \\mathbf { x } ) = \\left [ \\left \\{ \\sum _ { i = 1 } ^ { n _ 1 } \\left ( \\hat f _ { - i } ( \\mathbf { x } ) - \\hat f ( \\mathbf { x } ) \\right ) ^ 2 \\right \\} / n _ 1 \\right ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} r = \\{ h ^ 0 ( A ) - 1 | A C \\} . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\frac { \\sigma } { 3 } \\leq \\bar { m } e ^ { v ^ \\epsilon ( \\bar { x } , \\bar { t } ) / \\epsilon } = \\bar { m } e ^ { \\psi ^ \\epsilon ( \\bar { x } , \\bar { t } ) / \\epsilon } . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\textrm { H i l b } ^ { P } ( \\mathbb { P } ^ { r } _ { k } ) ^ { \\textrm { u s } } _ { d + t } = \\coprod _ { [ \\lambda ] , \\delta > 0 } { E _ { d + t , [ \\lambda ] , \\delta } } \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} V _ \\omega ( x ) = U _ \\omega ( x ) ^ 2 = \\Big ( \\sum q _ i f ( x - i ) \\Big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\kappa ^ { \\chi _ n } _ { 1 } ( \\gamma ) \\ n ^ { - \\frac { 1 } { 2 } } = \\frac { \\gamma } { \\sqrt { n } } \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} I ^ q f ( x , \\xi ) = \\int _ { 0 } ^ { l ( \\gamma _ { x , \\xi } ) } t ^ q \\langle f ( \\gamma _ { x , \\xi } ( t ) ) , \\dot { \\gamma } _ { x , \\xi } ^ m ( t ) \\rangle d t = \\int _ { 0 } ^ { l ( \\gamma _ { x , \\xi } ) } t ^ q f _ { i _ 1 \\dots i _ m } ( \\gamma _ { x , \\xi } ( t ) ) \\dot { \\gamma } _ { x , \\xi } ^ { i _ 1 } ( t ) \\cdots \\dot { \\gamma } _ { x , \\xi } ^ { i _ m } ( t ) d t . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} & \\limsup _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { \\tau \\in \\mathcal { T } ^ \\delta } { { E } } \\left [ \\sum _ { k = 0 } ^ \\infty ( k + 2 ) \\left | { \\bar { B } } ^ n _ k ( \\tau + \\delta ) - { \\bar { B } } ^ n _ k ( \\tau ) \\right | + \\left | { \\bar { \\eta } } ^ n ( \\tau + \\delta ) - { \\bar { \\eta } } ^ n ( \\tau ) \\right | \\right ] \\\\ & \\le ( K + 3 ) \\gamma ( M ) M _ 0 + 5 { \\bar { U } } _ K . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\frac { d } { d t } ( A B C ) = \\frac { d } { d t } ( A B E ) = \\frac { d } { d t } ( A C D ) = \\frac { d } { d t } ( A D E ) = 0 . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\mu ^ { \\lambda } = \\sup \\{ \\mu \\in ( \\mu _ 0 , \\lambda ^ * ) : \\ \\hat { J } _ \\lambda ^ + ( \\mu ) = \\hat { J } _ \\lambda ^ + ( \\mu _ 0 ) \\} , \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\mathcal { D } ( K - 1 ) \\log \\widetilde { b } & < 4 M \\Omega L \\log \\widetilde { b } \\\\ & = 3 2 M L ( \\rho + 3 ) ^ 2 \\log \\widetilde { b } \\log \\alpha _ 1 \\log \\alpha _ 2 \\log c \\\\ & + 3 2 M L ( \\rho + 3 ) ^ 2 \\log \\widetilde { b } \\cdot 0 . 0 8 6 7 5 ( \\rho - 1 ) \\log \\alpha _ 1 \\log \\alpha _ 2 . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} V _ \\# ^ l = \\{ v \\in H ^ 1 _ \\# ( Y ) , \\ v | _ T \\in { \\cal P } _ 1 ( T ) \\ \\forall \\ , T \\in { \\cal T } ^ l _ \\# \\} . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\mathfrak { I } _ { 1 } ( u ) : = \\int \\limits _ { \\mathbb { G } } \\left ( \\sum _ { j = 1 } ^ { \\ell } | \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } u ( x ) | ^ { p } - | u ( x ) | ^ { q } \\right ) d x . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} { \\cal E } : = { \\cal E } _ 0 + { \\cal E } _ 1 + D G _ \\delta ( { \\mathtt u } _ \\delta ) R _ Z D { \\widetilde G } _ \\delta ( { \\mathtt u } _ \\delta ) ^ { - 1 } \\ , , { \\cal E } _ \\omega : = D G _ \\delta ( { \\mathtt u } _ \\delta ) { \\mathbb R } _ \\omega D { \\widetilde G } _ \\delta ( { \\mathtt u } _ \\delta ) ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\frac { \\sin ( A ) } { 1 } = \\frac { \\sin ( B ) } { y _ 1 } = \\frac { \\sin ( C ) } { y _ 1 ^ 2 - 1 } \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} a ( x , v ) = \\ln \\left ( \\frac { f _ - ( x - \\tau _ - ( x , v ) v , v ) } { \\phi _ 1 ( x , v ) } \\right ) \\ , . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} q _ 2 ( w ) = \\sum _ j p _ j ( w _ 1 ( \\tilde { y } + x ) + w _ 2 ( \\tilde { y } - x ) ) ( \\tilde { y } _ j - x _ j ) \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} ( 1 - 2 a ) ^ 2 - 4 b = \\Delta ^ 2 \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} T _ { n } = \\frac { T } { 4 } \\left ( 1 - \\frac { 1 } { 2 ^ { n - 1 } } \\right ) . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\int _ { \\Omega } | ( \\mathbf { D } + \\mathbf { k } ) \\mathbf { u } | ^ 2 d \\mathbf { x } \\ge \\sum _ { \\mathbf { b } \\in \\widetilde { \\Gamma } } | \\mathbf { k } | ^ 2 | \\hat { \\mathbf { u } } _ { \\mathbf { b } } | ^ 2 = | \\mathbf { k } | ^ 2 \\int _ { \\Omega } | \\mathbf { u } | ^ 2 d \\mathbf { x } , \\mathbf { u } \\in \\widetilde { H } ^ 1 ( \\Omega ; \\mathbb { C } ^ n ) , \\ ; \\mathbf { k } \\in \\widetilde { \\Omega } . \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} a ^ { a / 2 } r \\mathtt { J } _ \\nu ' ( r ) + \\left ( ( a ^ { a / 2 } - 1 ) ( 1 - a + a \\nu ) - a ^ { a / 2 } \\nu + 2 ( 1 - \\beta ) \\right ) \\mathtt { J } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} C _ { 1 - \\gamma } ^ { \\alpha , \\beta } [ a , b ] = \\big \\{ f \\in { C _ { 1 - \\gamma } [ a , b ] } : D _ { a ^ + } ^ { \\alpha , \\beta } f \\in { C _ { 1 - \\gamma } [ a , b ] } \\big \\} , \\quad \\gamma = \\alpha + \\beta ( 1 - \\alpha ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} J ( u ) : = \\left ( \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } d x \\right ) ^ { \\frac { Q ( q - p ) - a _ { 2 } p q } { ( a _ { 1 } - a _ { 2 } ) p ^ { 2 } } } \\left ( \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } d x \\right ) ^ { \\frac { a _ { 1 } p q - Q ( q - p ) } { ( a _ { 1 } - a _ { 2 } ) p ^ { 2 } } } \\left ( \\int _ { \\mathbb { G } } | u ( x ) | ^ { q } d x \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} v ^ \\delta _ { 1 , t } = \\sum _ { k = 1 } ^ N \\frac { a _ 0 } { \\rho _ k } T _ t ( \\lambda _ k ) . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} h ^ { 1 , 2 } ( Z _ \\Delta ) = h ^ { 2 , 1 } ( Z _ { \\Delta } ) & = \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) \\\\ h ^ { 1 , 1 } ( Z _ \\Delta ) = h ^ { 2 , 2 } ( Z _ \\Delta ) & = 2 \\ell ( \\Delta ^ \\circ ) - 5 - \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) + \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} 2 ( 1 - a ) \\left ( 1 - a ^ k - k ( 1 - a ) a ^ { k - 1 } - \\binom { k } { 2 } ( 1 - a ) ^ 2 a ^ { k - 2 } \\right ) \\\\ < ( b - 1 ) \\left ( a ^ k + k ( 1 - a ) + k ( k - 1 ) ( 1 - a ) ^ 2 - 1 \\right ) \\ , , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} u ( x ) = u ^ { r } ( r , z ) e _ { r } ( \\theta ) + u ^ { \\theta } ( r , z ) e _ { \\theta } ( \\theta ) + u ^ { z } ( r , z ) e _ { z } . \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{gather*} \\widehat { \\lambda } _ l ( t , \\boldsymbol { \\theta } ) = \\widehat { \\gamma } _ l ( \\boldsymbol { \\theta } ) t ^ 2 + \\widehat { \\mu } _ l ( \\boldsymbol { \\theta } ) t ^ 3 + \\ldots , l = 1 , \\ldots , n , \\\\ \\widehat { \\varphi } _ l ( t , \\boldsymbol { \\theta } ) = \\widehat { \\omega } _ l ( \\boldsymbol { \\theta } ) + t \\widehat { \\psi } ^ { ( 1 ) } _ l ( \\boldsymbol { \\theta } ) + \\ldots , l = 1 , \\ldots , n . \\end{gather*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\hbox { s u p p } ( p _ t ( x , d y ) ) = E , x \\in \\mathring { E } , t > 0 . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} y _ { g , i } : = \\frac { g ( \\sqrt { - \\alpha _ i } ) } { \\sqrt { - \\alpha _ i } } . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} q _ { i } ( x ) = \\left \\{ \\begin{array} { l } q ^ { W } _ { i } ( x ) , \\ { \\rm i f } \\ \\left ( q ^ { \\xi } _ i ( x _ { i + \\frac { 1 } { 2 } } ) - q ^ { \\eta } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) \\right ) \\left ( q ^ { \\xi ' } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q ^ { \\eta ' } _ { i } ( x _ { i - \\frac { 1 } { 2 } } \\right ) < 0 , \\\\ q ^ { T } _ { i } ( x ) , \\ { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} ( x ^ 2 - x + 1 ) [ \\gamma ] = 0 , \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } a ( n ) e ( n z ) . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\varphi _ n ^ M - \\mathcal { K } _ n ^ M ( \\phi _ n ^ M ) = f , \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } 2 ^ { \\sum _ { l = j } ^ K \\bar R _ l } - 2 ^ { \\sum _ { l = j + 1 } ^ K \\bar R _ l } , & i = j \\\\ 2 ^ { \\sum _ { l = j + 1 } ^ K \\bar R _ l } - 2 ^ { \\sum _ { l = j + 2 } ^ K \\bar R _ l } , & i = j + 1 \\\\ \\vdots & i = j + 2 , \\ldots , K - 1 \\\\ 2 ^ { \\bar R _ K } - 1 . & i = K \\end{array} \\right . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} g * \\tau _ { i \\beta } ( f ) ( \\gamma ) & = \\begin{cases} e ^ { - \\beta D ( U _ x ) } g ( \\gamma ( U _ x ) ^ { - 1 } ) f ( U _ x ) & x = s ( \\gamma ) \\in s ( U ) \\\\ 0 & s ( \\gamma ) \\notin s ( U ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\varphi = \\big ( z _ 0 , \\psi ( z _ 1 , z _ 2 ) \\big ) \\circ \\left ( \\frac { z _ 0 A ( z _ 1 , z _ 2 ) + B ( z _ 1 , z _ 2 ) } { z _ 0 C ( z _ 1 , z _ 2 ) + D ( z _ 1 , z _ 2 ) } , z _ 1 , z _ 2 \\right ) . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} 2 p ( N ^ + [ v ] ) & = 2 p ( N ^ + ( v ) ) + p ( v ) \\\\ & \\geq p ( N ^ + ( v ) ) + p ( N ^ - ( v ) ) + p ( v ) \\\\ & = p ( V ) , \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , W ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , \\phi ^ { 2 } W ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , \\omega \\phi W ) + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { B } Y , \\omega W ) \\\\ & + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { C } Y , \\omega W ) - \\eta ( Y ) g _ { 1 } ( X , \\omega W ) . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} & \\frac { \\partial \\tilde u } { \\partial \\bar z _ { j } } + m \\frac { \\partial \\varphi } { \\partial \\bar z _ { j } } \\tilde u \\\\ & = \\frac { \\partial } { \\partial \\bar z _ { j } } ( \\tilde u e ^ { m \\varphi } ) = 0 , \\ \\forall j . \\\\ \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} x _ 1 ^ 2 + \\cdots + x _ { n - 1 } ^ 2 = c x _ n ^ 2 \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{gather*} \\frac { ( - 1 ) ^ { \\theta m ( m - 1 ) / 2 } q ^ { c ( N , m , \\theta ) } } { \\prod \\limits _ { r = 1 } ^ m { \\prod \\limits _ { j = 1 } ^ { \\theta ( N - r + 1 ) - 1 } { \\big ( z _ r - q ^ { j - \\theta } \\big ) } } } \\cdot \\prod _ { r = 1 } ^ m { T _ { q , z _ r } ^ { \\theta r - \\theta + \\tau _ r ^ + - \\tau _ r ^ - } } \\left ( \\prod _ { j = 1 } ^ { \\theta N - 1 } { \\big ( z _ r - q ^ { \\theta ( N - 1 ) - j } \\big ) } \\right ) . \\end{gather*}"}
-{"id": "5435.png", "formula": "\\begin{align*} f ( S ^ t x ) = e ^ { A t } f ( x ) \\qquad \\textrm { o r } D f ( x ) b ( x ) = A f ( x ) . \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} w ( e ) & = \\begin{cases} l ( \\tau ( r ( e ) ) ) - l ( \\tau ( s ( e ) ) ) + 1 & \\ e \\not \\in T ^ 1 ; s ( e ) , r ( e ) \\in T ^ 0 , \\\\ 1 & \\ o t h e r w i s e . \\end{cases} \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\bar { \\jmath } ^ { \\ , ' \\ , a } _ l \\circ \\eta _ * \\ , = \\ , \\alpha ' _ { \\kappa ( l ) } \\circ \\eta _ * \\circ \\bar { \\jmath } ^ { \\ , a } _ l \\ \\ \\ , \\ \\ \\ \\forall l \\in \\Pi \\ , \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} A _ { 0 } \\left ( x , v \\right ) + F _ { 1 } \\left ( x , \\nabla v \\right ) = - a _ { 0 } \\left ( x \\right ) , x \\in \\Omega , x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} V = U \\begin{pmatrix} \\Sigma \\\\ 0 \\\\ \\end{pmatrix} Q ^ \\top \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} u ^ f _ { n , t } = v ^ f _ { n , t } , n \\leqslant t \\leqslant N , W ^ N = W ^ N _ { N , h } . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} { \\rm e v } _ i ( { \\bf x } ) = { \\rm e v } _ j ( { \\bf x } _ { \\rm v } ) . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} [ \\psi ^ + _ { i , 0 } , \\psi ^ \\pm _ { j , \\pm s ^ \\pm _ j } ] = 0 , \\ [ \\psi ^ - _ { i , b _ i } , \\psi ^ \\pm _ { j , \\pm s ^ \\pm _ j } ] = 0 \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | \\leq \\frac { 1 } { n - m + 1 } \\sum _ { i = 1 } ^ { m - 1 } | \\hat { 1 _ S } ( \\chi ^ i ) | . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} R ( T ) = \\left \\{ v : \\ \\ v = \\left ( \\begin{array} { c } \\alpha _ 1 \\\\ \\alpha _ 2 \\end{array} \\right ) + \\beta \\left ( \\begin{array} { c } x _ 2 \\\\ - x _ 1 \\end{array} \\right ) \\right \\} \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} \\hat u _ { t , x } ( r ) = B ^ * e ^ { - r A ^ * } { Q } ^ { - 1 } _ { t } x \\forall r \\in [ - t , 0 ] , \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\nu ( k A ) = k ^ { - \\alpha } \\nu ( A ) , k > 0 \\ , . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\int _ a ^ b f ( t ) ( G ( t ) ) ^ k d t = 0 , \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} h = \\left \\{ \\begin{array} { l } \\textrm { s i g n } ( w ) \\left ( \\sqrt { \\dfrac { t } { 2 } } + \\sqrt { \\dfrac { | w | } { \\sqrt { 8 t } } - \\dfrac { t } { 2 } } \\right ) ^ 3 \\\\ \\quad \\quad \\quad \\quad \\textrm { f o r } ~ | w | > \\frac { 2 } { 3 } \\sqrt [ 4 ] { 3 \\lambda ^ 3 } \\\\ \\\\ 0 , ~ ~ \\textrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\widehat \\Theta \\le \\frac { 1 } { \\theta ^ { n + 1 } } \\max _ { 1 \\le r \\le n } \\ \\min \\limits _ { \\substack { r _ 1 + \\dots + r _ m = r \\\\ [ 0 . 5 e x ] 0 \\le r _ j \\le \\ell _ j \\ ( 1 \\le j \\le m ) } } ~ \\prod _ { j = 1 } ^ m \\prod _ { i = 0 } ^ { r _ j - 1 } \\theta _ { j , i } \\ , . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\mathcal { P } } s _ \\lambda X _ \\lambda = \\left ( \\sum _ { i \\geq 0 } ( - 1 ) ^ i e _ i \\right ) \\prod _ { l \\geq 1 } \\left ( 1 + p _ l \\right ) ^ { T _ l } \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } I _ c ^ { x x } \\\\ I _ c ^ { x y } \\\\ I _ c ^ { y y } \\end{array} \\right \\} = \\int \\limits _ { \\Omega _ c } \\left \\{ \\begin{array} { c } ( x - x _ c ) ^ 2 \\\\ ( x - x _ c ) ( y - y _ c ) \\\\ ( y - y _ c ) ^ 2 \\end{array} \\right \\} ~ \\mathrm { d } \\Omega \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V , \\varphi X ) \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} u \\left ( x , t , x _ { 0 } \\right ) = g _ { 0 } \\left ( x , t , x _ { 0 } \\right ) , \\partial _ { n } u \\left ( x , t , x _ { 0 } \\right ) = g _ { 1 } \\left ( x , t , x _ { 0 } \\right ) , \\forall \\left ( x , t , x _ { 0 } \\right ) \\in \\Gamma \\times \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} - d _ { i } ^ { \\varepsilon } \\nabla u _ { i } ^ { \\varepsilon } \\cdot \\mbox { n } = 0 \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} \\l ( p ^ k . ( ( p ^ k - 1 ) / 2 ) ) = \\l ( D _ { p ^ k - 1 } ) = \\O ( p ^ k - 1 ) , \\ ; \\ ; \\l ( D _ { p ^ k + 1 } ) = \\O ( p ^ k + 1 ) . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{gather*} \\Phi _ 0 = 0 . \\end{gather*}"}
-{"id": "2607.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ { \\infty } \\exp \\left ( T _ l \\left ( \\log \\left ( 1 + \\sum _ { U } p _ l ^ { ( U ) } [ U ] \\right ) \\right ) \\right ) \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\begin{pmatrix} O _ s & \\frac { 1 } { 2 i } B \\\\ - \\frac { 1 } { 2 i } B ^ * & \\Im C \\end{pmatrix} \\geq 0 . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\xi _ k = - 1 / q ^ k + ( - 1 ) ^ k q ^ { k ( k - 1 ) / 2 } ( 1 + \\Phi _ k ( q ) ) ~ , \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} V ( \\mu ( x ) ) = U ^ { - 1 } ( \\mu ( x ) \\circ \\sigma ) = U ^ { - 1 } ( \\mu ( f ) \\circ \\sigma ) = U ^ { - 1 } ( f ) = x . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\inf _ { n \\geq 1 } \\inf _ { ( x , y ) : x \\sim y } \\frac { C ^ n _ { x , y } } { C ^ n _ x } > 0 \\quad C ^ n _ x : = \\sum _ { z : z \\sim x } C ^ n _ { x , z } , \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\lambda _ 1 ( t , x ) = 6 0 + 1 5 \\sin \\left ( \\tfrac { 2 \\pi t } { 2 4 } \\right ) . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} X _ q : = \\{ x \\in X : e ^ { i \\theta } \\circ x \\neq x , \\forall \\theta \\in ( 0 , \\frac { 2 \\pi } { q } ) , \\ e ^ { i \\frac { 2 \\pi } { q } } \\circ x = x \\} . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} P _ h ( z , s ) & = \\sum _ j \\langle P _ h ( \\cdot , s ) , \\mu _ j \\rangle \\mu _ j ( z ) \\\\ & + \\frac { 1 } { 4 \\pi } \\int _ { - \\infty } ^ \\infty \\langle P _ h ( \\cdot , s ) , E ( \\cdot , \\tfrac { 1 } { 2 } + i t ) \\rangle E ( z , \\tfrac { 1 } { 2 } + i t ) \\ , d t . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} ( g ^ { - 1 } ) ^ \\cdot = - g ^ { - 1 } \\dot { g } g ^ { - 1 } = - J ^ { - 1 } \\dot { J } g ^ { - 1 } . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} q _ { 1 1 } ( 0 ) = \\sum _ { j , k } p _ { j k } ( 0 ) ( \\tilde { y } _ j + x _ j ) ( \\tilde { y } _ k + x _ k ) \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} \\partial _ { N } v = ( N \\cdot \\nu ^ 0 ) \\partial _ \\nu v \\quad \\mbox { o n } \\Gamma ^ 0 _ { \\rm o u t } \\mbox { w h i l e } \\partial _ { \\nu } v = 0 \\mbox { o n } \\Gamma ^ 0 _ { \\rm i n } . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\lambda _ 1 ( t ) = 6 0 + 1 5 \\sin ( \\tfrac { 2 \\pi t } { 2 4 } ) . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\exp ( - Y ) \\exp ( X ) = b a ^ { - 1 } . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} g \\mapsto F _ g , \\ ; \\ ; \\ ; F _ g ( f ) = \\mathbb E \\Bigl ( \\sum _ { k = 0 } ^ { \\infty } \\langle f _ k , g _ k \\rangle \\Bigr ) \\ ; \\ ; \\ ; \\bigl ( f \\in H ^ { s _ q } _ p ( X ) , g \\in H ^ { s _ { q ' } } _ { p ' } ( X ^ * ) \\bigr ) , \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\tilde a ^ { i j } ( y ) = D _ l \\Phi ^ i D _ k \\Phi ^ j a ^ { k l } ( x ) , \\tilde b ^ i ( y ) = D _ { k l } \\Phi ^ i a ^ { k l } ( x ) , \\tilde g ' ( y ) = g ' ( x ) . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} h ^ { 1 , 1 } ( \\hat { X } _ \\Delta ) = h ^ { 2 , 2 } ( \\hat { X } _ \\Delta ) = \\ell ( \\Delta ) - 4 , \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } + 1 } \\right ) ^ r ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty \\mathcal { E } _ { n , \\lambda } ^ { ( r ) } ( x ) \\frac { t ^ n } { n ! } , \\ , \\ , ( r \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} | f | _ { 0 } ^ { 2 } : = \\sum _ { k = 1 } ^ { d } | f _ { k } | _ { L ^ { 2 } } ^ { 2 } = \\sum _ { k = 1 } ^ { d } \\int _ { \\mathbb { R } } f _ { k } ^ { 2 } ( x ) \\ , d x . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} U ^ * A U = \\begin{pmatrix} O _ { s } & 0 \\\\ 0 & C \\end{pmatrix} \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} I & = \\int _ { \\mathbb { R } } \\lvert h ( t + \\mathrm { i } y ) \\rvert ^ q \\ , \\mathrm { d } t \\\\ & = \\int _ { \\{ t \\colon \\lvert t - x _ 0 \\rvert > \\delta _ 0 \\} } \\lvert h ( t + \\mathrm { i } y ) \\rvert ^ q \\ , \\mathrm { d } t + \\int _ { \\{ t \\colon \\lvert t - x _ 0 \\rvert \\leqslant \\delta _ 0 \\} } \\lvert h ( t + \\mathrm { i } y ) \\rvert ^ q \\ , \\mathrm { d } t \\\\ & = I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k \\P \\big ( N ( 0 , t _ j - t _ { j - 1 } ] = \\ell _ j \\big ) \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} N _ 1 ( R ) & : = \\# \\{ X ^ 2 + Y ^ 2 = Z ^ 2 + 1 \\ ; \\lvert Z \\rvert \\leq R \\} \\\\ N _ 2 ( R ) & : = \\# \\{ X ^ 2 + Y ^ 2 = ( 2 Z ) ^ 2 + 1 \\ ; \\lvert Z \\rvert \\leq R \\} , \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\phi _ 0 ( t ) & = x _ 0 ( t - a ) ^ { \\gamma - 1 } , t \\in ( a , a + l ] , \\\\ \\phi _ n ( t ) = \\phi _ 0 ( t ) + & \\int _ { a } ^ { t } \\frac { ( t - s ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } f ( s , \\phi _ { n - 1 } ( s ) ) d s , t \\in ( a , a + l ] , n = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\left ( \\xi , _ { k } \\delta _ { j } ^ { i } + 2 \\xi , _ { j } \\delta _ { k } ^ { i } \\right ) P ^ { k } + 2 \\eta ^ { i } , _ { t | j } - \\xi , _ { t t } \\delta _ { j } ^ { i } = 0 \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} P _ m ( x ) : = \\frac { a ^ { 2 m + 1 } - 2 a ^ { 2 m + 1 } | x | ^ { 2 m + 1 } + | x | ^ { 2 ( 2 m + 1 ) } } { ( 2 m + 1 ) | x | ^ { 2 ( m + 1 ) } ( 1 - a ^ { 2 m + 1 } ) } . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} f _ { [ - N , N ] } ^ { - 1 } ( [ - N , N ] ) : = \\{ ( \\l , x ) \\in [ - N , N ] \\times X : f ( \\l , x ) \\in [ - N , N ] \\} , \\forall N \\in \\N . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\Delta _ \\mathcal { C } ( a ) & = a \\otimes a - q \\ , c ^ * \\otimes c , \\\\ \\Delta _ \\mathcal { C } ( c ) & = c \\otimes a + a ^ * \\otimes c . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} h \\circ f _ 1 = f _ 2 \\circ h ; \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\min _ { Q \\in \\mathcal { P } ( A ) } \\widetilde { H } ( t , x , Q , p , g ) \\leq \\min _ { Q = \\delta _ a , a \\in A } \\int _ A H ( t , x , a , p , g ) Q ( d a ) = \\min _ { a \\in A } H ( t , x , a , p , g ) \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\to 0 } C _ \\gamma ( \\delta , \\varrho ) \\ ; = \\ ; - \\infty \\quad \\limsup _ { \\varepsilon \\to 0 } C ( \\varepsilon , q , \\ell ) \\ ; \\le \\ ; - ( \\ell - 1 ) \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} m \\geq \\frac { 1 } { 2 \\log \\Big ( \\frac { \\sum _ { i = 1 } ^ g \\sqrt { 2 \\bar { k } _ i ( 1 + \\delta _ { \\bar { k } _ i } ) } + \\max _ i ( \\sqrt { \\bar { k } _ i ( 1 - \\delta _ { \\bar { k } _ i } ) / 8 } ) } { \\min _ i ( \\sqrt { \\bar { k } _ i ( 1 - \\delta _ { \\bar { k } _ i } ) / 8 } ) } \\Big ) } \\bar { k } \\log \\Big ( \\frac { n } { \\bar { k } } \\Big ) \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} { \\bf z } _ { k l } ^ n ( y ) : = \\eta ( 2 \\ , \\varepsilon _ n \\ , \\vert y - \\zeta _ { k , n } ^ { \\prime } \\vert ) \\ , z _ { k l } ^ n ( y ) . \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} r ^ { - m } \\int _ { B _ { r } ( 0 ) } \\left ( r ^ { 2 } | D _ { \\hat { L } } u ( z ) | ^ { 2 } + | D _ { v } u ( z ) | ^ { 2 } \\right ) \\ , \\mathrm { d } z \\leq C ( m , \\rho ) \\sum _ { i = 0 } ^ { k } \\left ( \\theta ( y _ { i } , 4 r ) - \\theta ( y _ { i } , 2 r ) \\right ) , \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} & M _ { 3 2 } ^ { \\tau } = M _ { 2 2 } , U _ n M _ { 2 2 } U = M _ { 2 2 } . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} V \\left ( x \\right ) = \\left ( v _ { 0 } \\left ( x \\right ) , . . . , v _ { N - 1 } \\left ( x \\right ) \\right ) ^ { T } , A _ { 0 } \\left ( x , V \\right ) = \\left ( A _ { 0 } \\left ( x , v _ { 0 } \\right ) , . . . , A _ { 0 } \\left ( x , v _ { N - 1 } \\right ) \\right ) ^ { T } . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} 1 - \\frac { | V ( H ) | } { | V ( G ) | } \\le \\frac { q + 1 - ( q + 1 - t ) } { q + 1 - \\lambda } = \\frac { t } { q + 1 - \\lambda } . \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\begin{aligned} \\limsup _ { h \\to 0 } h ^ { n - 1 } \\| \\chi _ j u _ \\Sigma \\| _ { L ^ \\infty ( B ( x _ 0 , r ( h ) ) ) } ^ 2 & \\leq C _ { n , l } { 4 } \\delta _ 0 ^ { - 1 } \\alpha ^ { n - 1 } \\int _ { \\Lambda _ { x _ 0 , 3 \\delta } } \\chi _ j ^ 2 d \\mu \\\\ & + C _ { n , l } \\alpha ^ { n - 2 l - 1 } \\sum _ { i = 2 } ^ { n } \\int _ { \\Lambda _ { x _ 0 , 3 \\delta } } \\chi _ j ^ 2 ( { 4 } \\delta _ 0 ^ { - 1 } q _ { j , i } ^ 2 + { C _ 0 ^ 2 } \\delta _ 0 | H _ p q _ { i , j } | ^ 2 ) d \\mu \\end{aligned} \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ | \\hat { \\mathbb { E } } _ { \\tau + } [ X ] - \\hat { \\mathbb { E } } _ { ( \\tau \\wedge T ) + } [ X I _ { \\{ \\tau \\leq T \\} } ] | ] & = \\hat { \\mathbb { E } } [ | \\hat { \\mathbb { E } } _ { \\tau + } [ X ] - \\hat { \\mathbb { E } } _ { ( \\tau \\wedge T ) + } [ X I _ { \\{ \\tau \\leq T \\} } ] | I _ { \\{ \\tau > T \\} } ] \\\\ & \\leq C _ X c ( { \\{ \\tau > T \\} } ) \\rightarrow 0 , \\ \\ \\ \\ \\ T \\rightarrow \\infty . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( A ) = \\lim _ { \\delta \\searrow 0 } \\mathcal { H } ^ s _ \\delta ( A ) \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} x ( t ) & = ( x _ 1 ( t ) , \\ldots , x _ n ( t ) ) ' , \\cr y ( t ) & = ( y _ 1 ( t ) , \\ldots , y _ n ( t ) ) ' . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} z ^ 2 + t = \\pm \\bigg ( \\sqrt { 2 t } z + \\frac { | w | } { \\sqrt { 8 t } } \\bigg ) \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} z ^ 3 \\textrm { s i g n } ( h ) - w z + \\frac { \\lambda } { 4 } \\textrm { s i g n } ( h ) = 0 \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\frac { d + \\varepsilon \\sqrt { b d } } { \\sqrt { b d } } \\left ( \\frac { y + \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } - \\frac { d + \\sqrt { b d } } { \\sqrt { b d } } & < W _ { 2 m } ^ { ( b , d ) } = W _ { 2 m } ^ { ( a , d ) } \\\\ & < \\frac { d + \\varepsilon \\sqrt { a d } } { \\sqrt { a d } } \\left ( \\frac { x + \\sqrt { a d } } { 2 } \\right ) ^ { 2 m } \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} & C = \\frac { \\mu } { \\lambda '^ 3 \\sigma } \\left ( \\frac { \\omega } { 6 } + \\frac { 1 } { 2 } \\sqrt { \\frac { \\omega ^ 2 } { 9 } + \\frac { 8 \\lambda ' \\omega ^ { 5 / 4 } \\theta ^ { 1 / 4 } } { 3 \\sqrt { a _ 1 ' a _ 2 ' } H ^ { 1 / 2 } } + \\frac { 4 } { 3 } \\left ( \\frac { 1 } { a _ 1 ' } + \\frac { 1 } { a _ 2 ' } \\right ) \\frac { \\lambda ' \\omega } { H } } \\right ) ^ 2 , \\\\ & C ' = \\sqrt { \\frac { C \\sigma \\omega \\theta } { \\lambda '^ 3 \\mu } } . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\left ( R ^ T _ { N , h } f \\right ) _ t = v ^ f _ { 1 , t } , t = 1 , \\ldots , T . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\nu _ { x _ n - x , z } ( t ) & = 1 \\Rightarrow \\displaystyle \\lim _ { n \\rightarrow \\infty } \\frac { t } { t + \\| x _ n - x , z \\| } = 1 \\\\ & \\Rightarrow \\displaystyle \\lim _ { n \\rightarrow \\infty } \\| x _ n - x , z \\| = 0 \\Rightarrow x _ n \\rightarrow x \\textrm \\quad { ~ ~ f o r ~ ~ t h e ~ ~ 2 - n o r m ~ ~ } . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\big ( \\theta _ x ( f ) ( h _ x \\otimes \\zeta _ x ) \\ , \\big | \\ , ( h _ x \\otimes \\zeta _ x ) \\big ) & = \\big ( f * h _ x \\otimes \\zeta _ x \\ , \\big | \\ , h _ x \\otimes \\zeta _ x \\big ) \\\\ & = \\big ( \\pi _ x ( \\langle h _ x , f * h _ x \\rangle ) \\zeta _ x \\ , \\big | \\ , \\zeta _ x \\big ) \\\\ & = \\psi _ x \\big ( \\langle h _ x , f * h _ x \\rangle \\big ) . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\sum _ { \\{ r : \\ ; y _ r = y \\} } k _ r u _ { \\infty } ^ { y _ r } = \\sum _ { \\{ r : \\ ; y _ r ' = y \\} } k _ r u _ { \\infty } ^ { y _ r } . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} D ( s , & S _ f \\times \\overline { S _ g } ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { s + k - 1 } } \\\\ & = W ( s ; f , \\overline { g } ) + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } W ( s - z ; f , \\overline { g } ) \\zeta ( z ) \\frac { \\Gamma ( z ) \\Gamma ( s - z + k - 1 ) } { \\Gamma ( s + k - 1 ) } \\ ; d z , \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} B _ n ( v ) & = \\sum _ { i = 1 } ^ { n - 2 } b ( n ; i ) v ^ { i - 1 } , & & B ' _ n ( v ) = \\sum _ { i = 1 } ^ { n - 1 } b ' ( n ; i ) v ^ { i - 1 } , \\\\ A ' _ n ( v ) & = \\sum _ { i = 1 } ^ n a ' ( n ; i ) v ^ { i - 1 } , & & A '' _ n ( v ) = \\sum _ { i = 1 } ^ n a '' ( n ; i ) v ^ { i - 1 } , \\\\ A _ n ( v ) & = \\sum _ { i = 1 } ^ n a ( n ; i ) v ^ { i - 1 } . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} c ^ { s ( \\tau ) } = \\prod _ { \\sigma \\in S } { c _ \\sigma } ^ { s ( \\tau ) ( \\sigma ) } . \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u & \\ , = \\alpha _ e \\ , \\Delta u + \\beta _ e \\ , u ( 1 - u ) + \\sqrt { \\gamma _ e \\ , u ( 1 - u ) } \\dot { W } & & \\quad \\overset { \\circ } { e } \\\\ \\nabla _ { o u t } u \\cdot [ \\alpha ] & \\ , = - \\hat { \\beta } \\ , u ( 1 - u ) & & V . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} a ^ { ( j ) } _ { i + 1 } \\beta ^ { ( j ) } _ { i } : = a ^ { ( j ) } _ { i + 1 } \\cdot ( \\beta , - E _ i ^ { ( j ) * } ) = a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\beta ^ { ( j ) } _ { i + 1 } + \\Big ( \\frac { a ^ { ( j ) } _ { i + 1 } D ^ { ( j ) } _ { i } } { p ^ { ( j ) } _ { i + 1 } D ^ { ( j ) } _ { i + 1 } } - a ^ { ( j ) } _ { i } p ^ { ( j ) } _ { i } \\Big ) \\cdot \\beta ^ { \\Gamma ^ { ( j ) } _ i } _ { i + 1 } . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\ , \\varphi _ { \\underline { d } , { \\underline { d } } '' } ( K _ { i l } ) = & \\ , V _ { X _ 1 } ( - ( d - i ) A ) + \\ , V _ { X _ 2 } ( - i A - l B ) + \\ , V _ { X _ 3 } ( - ( d - l ) B ) \\\\ & - \\ , ( e v ^ { i l } ) - \\ , V _ { X _ 1 } ( - ( d - i + 1 ) A ) . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\ln \\left ( \\left | z _ i ( t + 1 ) - \\bar { x } \\right | \\right ) \\leq & \\ln \\left ( \\frac { 2 \\| x ( 0 ) \\| _ 1 } { { y _ { i } ( t + 1 ) } } \\Lambda _ { t , 0 } \\right ) \\cr = & \\ln \\left ( 2 \\| x ( 0 ) \\| _ 1 \\right ) + \\ln \\left ( \\frac { 1 } { { y _ { i } ( t + 1 ) } } \\right ) \\cr & + \\ln \\left ( \\Lambda _ { t , 0 } \\right ) . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} p ^ { \\dagger } _ j = \\frac { \\mu ^ { \\dagger } } { \\mu } ( j , 1 ) ^ { \\dagger } \\sum _ { \\ell = 1 } ^ \\infty \\frac { p _ { \\ell } } { \\ell ! } \\sum _ { k = 1 } ^ { \\ell } k \\ , ( l - j , k - 1 ) ^ { \\dagger } \\mbox { f o r } j \\in \\mathbb { N } _ { + } . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} 2 \\frac { \\phi ' ( z ) } { \\phi ( z ) } + \\frac { \\tilde { \\tau } ( z ) } { \\sigma ( z ) } = \\frac { \\tau ( z ) } { \\sigma ( z ) } , \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\chi ( P _ k ) = 1 + ( - 1 ) ^ { k } \\beta _ k ( P _ k ) . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} P \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) = \\left \\Vert \\mathbf { Q } \\left ( \\omega _ { 1 } , \\omega _ { 1 } , a _ { 1 , i } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) , b _ { 1 , i } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) , b _ { 2 , i } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) ; \\mathbf { t } \\right ) + \\bar { p } _ { , i } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\mathbf { 1 } - \\mathbf { f } \\right \\Vert _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} T _ { \\alpha } = c _ { \\alpha } A ^ { \\alpha } \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\operatorname { R e s } ( P _ 1 ( 0 , 0 , x _ { 3 } , x _ { 4 } ) , P _ 2 ( 0 , 0 , x _ { 3 } , x _ { 4 } ) ) = & a _ 1 ^ 2 b _ 0 b _ 2 - a _ 2 a _ 1 b _ 0 b _ 1 - a _ 0 a _ 1 b _ 1 b _ 2 + a _ 2 ^ 2 b _ 0 ^ 2 + a _ 0 a _ 2 b _ 1 ^ 2 \\\\ & + a _ 0 ^ 2 b _ 2 ^ 2 - 2 a _ 0 a _ 2 b _ 0 b _ 2 \\\\ = & 1 / 1 6 \\cdot \\operatorname { D i s c } _ { a , b } ( \\operatorname { D i s c } _ { x _ { 3 } , x _ { 4 } } ( a \\cdot P _ { 1 } ( 0 , 0 , x _ { 3 } , x _ { 4 } ) + b \\cdot P _ { 2 } ( 0 , 0 , x _ { 3 } , x _ { 4 } ) ) ) . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} D ^ \\perp = \\left \\{ \\left . \\left ( e , \\varepsilon \\right ) \\in E \\oplus \\widehat { E } \\ \\right | \\sigma _ \\mathrm { s t d } ( ( e , \\varepsilon ) , ( 0 , \\widehat { \\pi } ( d ) ) = 0 \\ \\forall d \\in D \\right \\} \\ = \\ B \\oplus \\widehat { E } \\ . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\mathcal { R } _ { i , j } ^ 8 : = & \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 3 \\ , U _ j ^ 3 + 3 \\int _ 0 ^ 1 d s \\ , \\int _ { B _ \\rho ( \\zeta _ i ) } ( w _ i + s \\pi _ i ) ^ 2 \\ , \\pi _ i \\ , U _ j ^ 3 . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} V _ { ( d _ 1 , d _ 2 ) } = \\mathrm { S p a n } \\left \\{ \\Psi _ \\chi : \\begin{array} { c } \\mathrm { p o s } ( \\chi ) = d _ 1 \\\\ \\mathrm { n e g } ( \\chi ) = d _ 2 \\end{array} \\right \\} \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\lim _ { t \\ , \\rightarrow \\ , 0 } \\ , t ^ { \\ : \\ ! m / 2 } \\ , \\| \\ , D ^ { m } \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { 2 } ( \\mathbb { R } ^ { n } ) } } \\ ! = \\ , 0 \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\sum _ { \\lvert t _ j \\rvert \\sim T } \\langle P _ h ^ k ( \\cdot , s ) , \\mu _ j \\rangle \\langle \\mu _ j , V \\rangle = \\sum _ { \\lvert t _ j \\rvert \\sim T } \\frac { \\Gamma ( s - \\frac { 1 } { 2 } + i t _ j ) \\Gamma ( s - \\frac { 1 } { 2 } - i t _ j ) } { ( 4 \\pi h ) ^ { s - 1 } \\Gamma ( s - \\frac { k } { 2 } ) } \\rho _ j ( h ) \\langle \\mu _ j , V \\rangle . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\ ( \\frac { n ^ n } { n ! } \\ ) ^ 4 \\sum _ { \\substack { \\chi \\in \\hat { G } ^ m \\\\ ( \\chi , 0 ^ { n - m } ) \\in X , \\chi \\neq 0 } } | \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) | ^ 4 = O ( m ^ 2 / n ^ 3 ) , \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\iint _ { \\Gamma \\backslash \\mathcal { H } } f ( z ) \\overline { g ( z ) } \\frac { d x d y } { y ^ 2 } . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} f _ { m , p ; i } ( x ) & = x f _ { m , p + i - 1 } + \\frac { x ^ { p + 1 } } { ( 1 - x ) ^ 2 } \\sum _ { j = - 1 } ^ { i - 2 } x ^ { i - j } f _ { m - i + j + 1 - p ; j } ( x ) \\\\ & + \\frac { x ^ { 3 + p } } { ( 1 - x ) ^ 3 } \\sum _ { k = 0 } ^ { i - 1 } \\sum _ { j = k } ^ { m - i - 1 + k - p } \\frac { x ^ { i - k } } { ( 1 - x ) ^ { j - k } } f _ { m - i + k - p , j } ( x ) \\ , . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\mathcal { W } \\left ( \\partial _ { 0 } M _ { 0 } + M _ { 1 } \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) + A \\right ) \\mathcal { V } \\left ( \\mathcal { V } ^ { - 1 } U \\right ) = \\mathcal { W } F \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} 2 g ( A ( X , Y ) , Z ) = - g ( \\tilde { T } ( X , Y ) , Z ) + g ( \\tilde { T } ( Y , Z ) , X ) - g ( \\tilde { T } ( Z , X ) , Y ) . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} T [ E , c , w , \\tau , \\beta ] ( f ) ( x ) : = \\tau f ^ * ( E x + c ) + \\langle w , x \\rangle + \\beta , x \\in X \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} P = \\sum _ { j = 1 } ^ { p } P _ j , P _ j P _ l = 0 \\ j \\ne l . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} g _ { 1 } ( [ X , Y ] , V ) & = - g _ { 1 } ( \\mathcal { B } Y , \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) - g _ { 1 } ( \\mathcal { C } Y , \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) + g _ { 1 } ( \\mathcal { B } X , \\nabla ^ { ^ { M _ 1 } } _ { Y } \\varphi V ) + g _ { 1 } ( \\mathcal { C } X , \\nabla ^ { ^ { M _ 1 } } _ { Y } \\varphi V ) . \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} u _ n < b _ { n + 1 , 1 } \\leq \\sum _ { i \\in I } b _ { i , 1 } + b _ { n + 1 , 1 } \\leq \\sum _ { i = 1 } ^ n b _ { i , 1 } + b _ { n + 1 , 1 } = u _ n + 2 \\sum _ { i = 1 } ^ n b _ { i , 1 } \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot V & = ( - 1 ) ^ { r - | S | + a - b } | T _ 1 | | T _ 2 | \\\\ & = ( - 1 ) ^ { r - | S | + a - b } \\binom { | S \\backslash U | } { a - | S \\cap U | } \\binom { | \\bar { S } \\backslash U | } { b } \\\\ & = ( - 1 ) ^ { r - | S | + a - b } \\binom { | S | - | S \\cap U | } { a - | S \\cap U | } \\binom { r - | S | - | U | + | S \\cap U | } { b } \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} \\int _ X f \\cdot T _ h g \\ > d \\omega _ X & = \\int _ G \\int _ G f ( x H ) g ( x y H ) \\ > d ( \\omega _ H * \\delta _ h * \\omega _ H ) ( y ) \\ > d \\omega _ G ( x ) \\\\ & = \\int _ G \\int _ G f ( x y H ) g ( x H ) \\ > d ( \\omega _ H * \\delta _ { h ^ { - 1 } } * \\omega _ H ) ( y ) \\ > d \\omega _ G ( x ) \\\\ & = \\int _ X T _ { \\bar h } f \\cdot g \\ > d \\omega _ X . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} [ \\alpha ^ { N , - i } ; \\beta ] _ j = \\begin{cases} \\alpha ^ N _ j & j \\neq i \\\\ \\beta & j = i . \\end{cases} \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\phi '' ( t ) + \\mathcal { A } \\phi ( t ) - \\int _ 0 ^ t \\ H ( t - s ) \\mathcal { A } \\phi ( s ) d s = 0 \\ , , t \\in [ 0 , T ] \\ , , \\\\ \\\\ \\displaystyle \\mathcal { B } \\phi ( t ) = D _ \\nu z ( t ) - \\int _ t ^ T \\ H ( s - t ) D _ \\nu z ( s ) d s , t \\in [ 0 , T ] \\ , , \\\\ \\\\ \\phi ( 0 ) = \\phi ' ( 0 ) = 0 \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} v _ i ( \\varepsilon , \\zeta ) = \\bar v _ i - ( \\bar v _ i \\cdot v _ 1 ( \\varepsilon , \\zeta ) ) v _ 1 ( \\varepsilon , \\zeta ) , 2 \\leq i \\leq k , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} P _ 0 = 2 \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\Pi ^ T \\Omega ^ { - 1 } \\psi _ i - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\bar { \\psi ^ T } \\Omega ^ { - 1 } \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\ , . \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\Gamma ^ { \\psi } ( t , x ; \\tau , \\xi ) = \\exp ( - \\psi ( x ) ) \\Gamma ( t , x ; \\tau , \\xi ) \\exp ( \\psi ( \\xi ) ) . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} D ^ { \\otimes m ( l - 1 ) } \\otimes L ( \\lambda ^ * ) & = L ( m ( l - 1 ) - \\lambda _ n , \\ldots , m ( l - 1 ) - \\lambda _ 2 , m ( l - 1 ) - \\lambda _ 1 ) \\cr & = L ( \\lambda ^ \\dagger ) \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } U _ i ^ 4 \\ , U _ j \\ , U _ m & = \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 4 \\ , U _ j \\ , U _ m + \\mathcal { R } _ { i , j , m } ^ 5 , \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } A = \\left [ \\begin{matrix} - 3 & \\ \\ & - 2 \\\\ 1 & & 0 \\end{matrix} \\right ] & , & A _ d = \\left [ \\begin{matrix} - 0 . 5 & \\ \\ & 0 . 1 \\\\ 0 . 3 & & 0 \\end{matrix} \\right ] & & A _ D = 0 _ 2 , \\end{array} . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} P _ j ( i ) = \\frac { - 2 q ^ { - i } } { q - 1 } g _ \\frac { 1 } { 2 } ( j ) ( 1 - q ^ { - f ( i ) } ) , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | \\psi _ { j _ l , k _ l } ( x _ l ) | d x _ l = \\int _ { 2 ^ { - j _ l } ( - N _ l + 1 + k _ l ) } ^ { 2 ^ { - j _ l } ( N _ l + k _ l ) } 2 ^ { j _ l / 2 } | \\psi _ { k _ l } ( 2 ^ { j _ l } x _ l ) | d x _ l \\lesssim 2 ^ { - j _ l / 2 } , \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} | p ( z ) | \\leq \\exp ( \\Re ( \\sum _ { j = 1 } ^ { n } z _ j p _ j ( 0 ) ) + \\frac { 1 } { 2 } \\| z \\| _ { \\infty } ^ 2 ( | \\sum _ { j = 1 } ^ { n } p _ j ( 0 ) | ^ 2 - \\Re ( \\sum _ { j , k = 1 } ^ { n } p _ { j , k } ( 0 ) ) ) ) . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\mu _ k = \\int _ { - 1 } ^ { 1 } w _ r ^ { ( m ) } ( x ) p _ { k - 1 } ( x ) \\mathrm { d } x . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} Y ^ { n } ( t ) & \\doteq X _ { 0 } ^ { n } ( 0 ) + \\sum _ { k = 0 } ^ { \\infty } \\frac { k - 2 } { n } \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { [ 0 , r _ { k } ( \\boldsymbol { X } ^ { n } ( s - ) ) ) } ( y ) \\ , N _ { k } ^ { n } ( d s \\ , d y ) , \\\\ \\eta ^ { n } ( t ) & \\doteq \\sum _ { k = 0 } ^ { \\infty } \\frac { 2 } { n } \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { \\{ X _ { 0 } ^ { n } ( s - ) < 0 \\} } { { 1 } } _ { [ 0 , r _ { k } ( \\boldsymbol { X } ^ { n } ( s - ) ) ) } ( y ) \\ , N _ { k } ^ { n } ( d s \\ , d y ) . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\big ( \\nabla _ { o u t } \\ , \\phi _ t \\ , \\cdot [ \\alpha ] \\big ) ( v ) = 0 \\quad t \\geq 0 , \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\mathcal { R } = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\rho , \\xi , \\N ) \\right \\} \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} v ^ t : = \\delta _ t \\tilde u ^ 0 = \\frac 1 t ( \\tilde u ^ t - \\tilde u ^ 0 ) . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} w ( z ; L ) = \\left \\{ \\begin{array} { l l } \\left ( \\frac { L ( L - 1 ) } { 2 \\pi } | 1 - z ^ 2 | ^ { L - 2 } \\int _ { 2 | \\mathrm { I m } \\ , z | / | 1 - z ^ 2 | } ^ 1 ( 1 - t ^ 2 ) ^ { ( L - 3 ) / 2 } \\mathrm { d } t \\right ) ^ { 1 / 2 } , & L > 1 \\\\ \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { 1 / 2 } | 1 - z ^ 2 | ^ { - 1 / 2 } , & L = 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} H ^ { o p t } = \\arg \\min _ { H \\in { ( q ) ^ * } ^ { d _ c } } \\{ S _ 3 ( H ) / S _ 2 ( H ) = 0 \\} , \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} \\mathcal { B } \\mathcal { T } _ { U } V + \\phi \\hat { \\nabla } _ { U } V & = \\hat { \\nabla } _ { U } \\phi V + \\mathcal { T } _ { U } \\omega V , \\\\ g _ 1 ( U , V ) \\xi + \\mathcal { C } \\mathcal { T } _ { U } V + \\omega \\hat { \\nabla } _ { U } V & = \\mathcal { T } _ { U } \\phi V + \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\omega V , \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} ( U _ { ( t , e ^ s ) } f ) ( x ) = e ^ { i t x } e ^ { s / 2 } f ( e ^ s x ) . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} \\mathcal E _ s ( u - P _ { \\ ! s } u ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) = \\mathcal E _ s ( u ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) - \\mathcal E _ s ( P _ { \\ ! s } u ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) ~ \\ ! . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} ( A u ) ( \\varphi , x ) : = ( A ( \\varphi ) u ( \\varphi , \\cdot ) ) ( x ) \\ , . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} z _ 1 = \\sqrt { \\frac { | w | } { 3 } } \\frac { \\sin ( A ) } { \\sin ( B ) } = \\sqrt { \\frac { 4 | w | } { 3 } } \\cos ( A ) = \\sqrt { \\frac { 4 | w | } { 3 } } \\cos \\ ! \\bigg ( \\frac { \\pi } { 3 } - \\frac { C } { 3 } \\bigg ) \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} g _ { 1 } ( V , \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\phi \\mathcal { B } Y + \\mathcal { A } _ { X } \\omega \\mathcal { B } Y ) & = g _ { 2 } ( \\pi _ * ( \\mathcal { C } Y ) , ( \\nabla \\pi _ * ) ( X , \\varphi V ) ) \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} [ Y _ 1 , Y _ 2 ] & = 0 & [ Y _ 1 , Y _ 3 ] & = 0 & [ Y _ 1 , Y _ 4 ] & = 0 & [ Y _ 1 , Y _ 5 ] & = 0 \\\\ [ Y _ 2 , Y _ 3 ] & = 0 & [ Y _ 2 , Y _ 4 ] & = 0 & [ Y _ 2 , Y _ 5 ] & = Y _ 1 & [ Y _ 3 , Y _ 4 ] & = Y _ 1 \\\\ [ Y _ 3 , Y _ 5 ] & = \\alpha Y _ 1 + Y _ 2 - Y _ 3 & [ Y _ 4 , Y _ 5 ] & = \\beta Y _ 1 + \\gamma Y _ 2 + Y _ 3 \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} & E [ g ( X ( t ) ) ] - E [ g ( X ( s ) ) ] \\\\ & = \\int _ s ^ t \\int _ U \\int _ A E [ g ( X ( r ) + f ( r , X ( r ) , u , a , m ( r ) ) ) - g ( X ( r ) ) ] \\rho _ r ( d a ) \\nu ( d u ) d r . \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} A J _ X ^ d ( a _ 1 i _ { y _ 1 } ^ * Z + \\cdots + a _ s i _ { y _ s } ^ * Z ) = a _ 1 A J _ X ^ d ( i _ { y _ 1 } ^ * Z ) + \\cdots + a _ s A J _ X ^ d ( i _ { y _ s } ^ * Z ) . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} J \\psi _ { 1 } = a _ { 1 } \\cos x + a _ { 2 } \\cos y + b _ { 1 } \\sin x + b _ { 2 } \\sin y . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\displaystyle u _ { \\bar { \\zeta } } + A ( u ) \\overline { u _ { \\zeta } } = 0 , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} U ^ { a } \\left ( \\bar { q } , q \\right ) = 0 , a = 1 , . . . , l . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\partial _ { x _ { i } } b _ { i } ( t , x ) = 0 \\tag * { \\textbf { S } } \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} X : = E ^ { \\infty } \\cup \\{ \\mu \\in E ^ { * } \\mid r ( \\mu ) \\} . \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} ( x , y ) \\mapsto A _ x ( y ) ~ : = ~ \\int _ 0 ^ \\infty a ( \\lambda ) \\ , d e _ \\lambda ( x , y ) \\ , . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} \\left \\Vert W ^ { \\ast } - W _ { n } \\right \\Vert _ { H ^ { 1 } \\left ( \\Omega _ { d + c } \\right ) } \\leq C _ { 4 } \\sigma ^ { \\theta } + q ^ { n } \\left \\Vert W _ { 0 } - W _ { \\min } \\right \\Vert _ { H ^ { s } \\left ( \\Omega \\right ) } , n = 1 , 2 , . . . , \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} 0 \\to \\sharp _ { i = 1 } ^ m M _ i ^ \\ast \\to F ^ \\ast \\sharp ( \\sharp _ { i > 1 } M _ i ) \\to G ^ \\ast \\sharp ( \\sharp _ { i > 1 } M _ i ) . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = i ^ k N ^ { \\frac 1 2 - s } \\Lambda _ g ( 1 - s ) . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\chi _ { ( 0 , \\infty ) } ( \\xi ) = \\sum _ n \\psi _ n ^ 2 ( \\xi ) . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\widehat \\alpha : = - \\langle M _ 1 \\rangle ^ { - 1 } ( \\langle g _ 1 \\rangle + \\langle M _ 2 g _ 2 \\rangle + \\langle M _ 3 g _ 3 \\rangle ) \\ , . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} P _ 2 = \\sum _ a \\binom { a } { 2 } \\binom { | S | - 2 } { a } \\sum _ b \\binom { r - | S | - 2 } { b - 2 } + \\sum _ a \\binom { | S | - 2 } { a } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 2 } { b - 2 } \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} K ^ i _ m : = \\sup \\ , \\{ k \\geq 0 : \\ : T ^ i _ k \\leq m \\} \\ , . \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\begin{aligned} p _ j ( z ) & = ( X _ j ( X ( z ) ) ^ { - 1 } ) p ( z ) \\\\ p _ { j k } ( z ) & = - ( X _ j ( X ( z ) ) ^ { - 1 } X _ k ( X ( z ) ) ^ { - 1 } ) p ( z ) + ( X _ j ( X ( z ) ) ^ { - 1 } ) ( X _ k ( X ( z ) ) ^ { - 1 } ) p ( z ) \\end{aligned} \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} x ^ * ( p ) = \\begin{pmatrix} \\sin ( p ) \\\\ \\cos ( p ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\Gamma ' _ r = \\left \\{ \\gamma _ S ( 1 , j ) ( a , r ) \\ , \\Big | \\ , j \\in \\{ 2 , \\ldots , r - 1 \\} , a = \\min ( \\{ 2 , \\ldots , r - 1 \\} \\setminus \\{ j \\} ) \\right \\} . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\delta _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } + \\frac { 1 } { 2 } r ^ { - 2 } . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\sum _ { p + q = i } h ^ { p , q } ( Y , V ) = \\dim H ^ { i } ( Y , V ) = \\sum _ { p + q = i } f ^ { p , q } ( Y , w ) . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} J = \\begin{pmatrix} 0 & - 1 & 1 & 0 \\\\ 1 & 0 & 0 & - 1 \\\\ 0 & 0 & 0 & - 1 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\big ( 1 - ( 1 + v ) x \\big ) A ^ + ( x ; v ) & = \\big ( 1 - ( 1 + v ) x \\big ) G ( x ; v ) + \\frac { 1 - v x } { v ^ 2 } B ' ( v x ; 1 / v ) - x ^ 3 v - \\frac { v ^ 2 x ^ 4 } { 1 - 2 v x } \\\\ & + \\frac { 1 } { 1 - v } ( B ( x ; 1 ) - \\frac { 1 } { v ^ 3 } B ( v x ; 1 / v ) ) + \\frac { x ^ 2 } { v } C ( x ) . \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} \\partial _ t X = \\Delta X + a _ 1 X + a _ 2 : X ^ { 2 } : + \\dots + a _ { 2 n - 1 } : X ^ { 2 n - 1 } : + \\xi \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\liminf _ \\lambda r _ \\lambda & = \\lim _ \\gamma \\inf _ { \\gamma \\prec \\lambda } r _ \\lambda . \\\\ \\limsup _ \\lambda r _ \\lambda & = \\lim _ \\gamma \\sup _ { \\gamma \\prec \\lambda } r _ \\lambda . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\dot \\lambda ( t ) = - \\frac { \\partial L ^ \\ast } { \\partial x } [ t ] \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} [ E _ { \\Phi } : \\Q ] = 2 ^ d [ G : S ] \\geq 2 ^ d . \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} \\mathcal { W } \\left [ { \\exp \\left \\{ { u \\xi + E _ { 0 } ^ { + } \\left ( \\xi \\right ) } \\right \\} , \\exp \\left \\{ { - u \\xi + E _ { 0 } ^ { - } \\left ( \\xi \\right ) } \\right \\} } \\right ] = - \\left \\{ { 2 u + \\phi \\left ( \\xi \\right ) } \\right \\} ^ { - 1 } , \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} e _ i = ( 1 , \\ldots , 1 , \\underbrace { - 1 } _ { i ^ \\mathrm { t h } \\mathrm { \\ , p l a c e } } , 1 , \\ldots , 1 ) \\in \\{ \\pm 1 \\} ^ { 2 d } , \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} & = \\frac { r } { m } \\Delta ^ k \\left ( \\frac { r } { m } \\right ) ^ j + k \\sum _ { l = 0 } ^ k { k \\choose l } ( - 1 ) ^ { k - l } \\Big ( l + \\frac { r } { m } \\Big ) ^ j + k \\sum _ { l = 0 } ^ { k - 1 } { k - 1 \\choose l } ( - 1 ) ^ { k - 1 - l } \\Big ( l + \\frac { r } { m } \\Big ) ^ j \\\\ & = \\frac { r } { m } \\Delta ^ k \\left ( \\frac { r } { m } \\right ) ^ j + k \\Delta ^ k \\left ( \\frac { r } { m } \\right ) ^ j + k \\Delta ^ { k - 1 } \\left ( \\frac { r } { m } \\right ) ^ j . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} k _ { 1 } = - { \\tfrac { 1 } { { 1 2 } } } , \\ k _ { 3 } = { \\tfrac { 1 } { { 3 6 0 } } } , \\ k _ { 5 } = - { \\tfrac { 1 } { { 1 2 6 0 } } . } \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\sup _ { \\sigma \\in \\mathcal { V } _ n } \\sum _ { j _ 1 = N _ 1 } ^ { J _ { n , 1 } - 1 } \\cdots \\sum _ { j _ d = N _ d } ^ { J _ { n , d } - 1 } \\sum _ { k _ 1 = 0 } ^ { 2 ^ { j _ 1 } - 1 } \\cdots \\sum _ { k _ d = 0 } ^ { 2 ^ { j _ d } - 1 } n ^ { - \\frac { C _ 1 \\overline { \\gamma } ^ 2 } { 1 6 \\sigma ^ 2 } } \\sigma \\lesssim ( \\sigma _ 0 + o ( 1 ) ) n ^ { - \\left [ \\frac { C _ 1 } { 1 6 ( \\sigma _ 0 ^ 2 + o ( 1 ) ) } \\overline { \\gamma } ^ 2 - 1 \\right ] } , \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} S ( T ( y , s ) ) = T ( y , - s ) , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} ( \\frac { d } { d s } | _ { s = 0 } L ) S + L ( - \\frac 1 2 ( J X _ u ) S + S ^ \\alpha u _ \\alpha ) = 0 . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\| f _ y \\| _ { L ^ 4 } ^ 4 & = - \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\int f _ { N _ 1 } f _ { N _ 2 , y y } ( f _ { \\leq N _ 2 , y } ) ^ 2 \\ , d x d y - 2 \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\int f _ { N _ 1 } f _ { N _ 2 , y } f _ { \\leq N _ 2 , y } f _ { \\leq N _ 2 , y y } \\ , d x d y \\\\ & \\lesssim \\| f _ x \\| _ { L ^ \\infty } \\left ( \\| C _ 1 [ \\partial _ x ^ { - 1 } f _ { y y } , f _ y , f _ y ] \\| _ { L ^ 1 } + \\| C _ 2 [ f _ y , f _ y , \\partial _ x ^ { - 1 } f _ { y y } ] \\| _ { L ^ 1 } \\right ) , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} C L \\frac { \\partial ^ { 2 } P } { \\partial t ^ { 2 } } \\left ( x , t \\right ) + C R \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) = \\frac { \\partial ^ { 2 } P } { \\partial x ^ { 2 } } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { N } } = \\left \\lbrace \\mathbf { u } \\in L _ 2 ( \\Omega ; \\mathbb { C } ^ n ) \\colon \\mathbf { u } = \\mathbf { c } \\in \\mathbb { C } ^ n \\right \\rbrace , \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} ( h ^ 0 _ j = \\alpha ^ { a _ j } / h ^ 0 _ { j - 1 } = \\alpha ^ { a _ { j - 1 } } ) = \\\\ \\frac { \\gamma _ m ( d _ c - j , 2 ^ m - 2 m - { a _ j } ) } { \\gamma _ m ( d _ c - j + 1 , 2 ^ m - 2 m - a _ { j - 1 } ) } . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\Gamma = \\left \\{ \\left ( x , t \\right ) : \\left \\vert x \\right \\vert = 1 , t \\in \\left ( T / 4 , T \\right ) \\right \\} , \\Gamma _ { d } = \\left \\{ \\left ( x , t \\right ) : \\left \\vert x \\right \\vert = 1 , \\xi \\left ( x , t \\right ) > d \\right \\} . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { 1 - 4 x + 2 x ^ 2 - ( 1 - 6 x + 9 x ^ 2 ) C ( x ) } { x ( 1 - 4 x ) } \\ , . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} R _ { J _ r x } L _ x = \\varepsilon . \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\exp \\left ( i \\sum _ { j = 1 } ^ { n } k _ { j } \\big ( B ^ { \\beta , \\alpha } ( t _ { j } + h ) - B ^ { \\beta , \\alpha } ( h ) \\big ) \\right ) \\right ) = \\mathbb { E } \\left ( \\exp \\left ( i \\sum _ { j = 1 } ^ { n } k _ { j } B ^ { \\beta , \\alpha } ( t _ { j } ) \\right ) \\right ) . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} 3 x + p ^ b \\sqrt { - 6 } = \\frac { \\gamma ^ n } { 3 ^ { ( n - 1 ) / 2 } } . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\widehat { T } ( z _ { 0 } ) \\phi ( x ) = e ^ { \\tfrac { i } { \\hbar } ( p _ { 0 } x - \\frac { 1 } { 2 } p _ { 0 } x _ { 0 } ) } \\phi ( x - x _ { 0 } ) , \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} z = \\sum _ { 1 \\leq i \\leq n } z _ { i , 1 } \\cdots z _ { i , l _ i } \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} \\div \\frac { \\nabla u } { \\sqrt { 1 + | \\nabla u | ^ 2 } } = H \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} I = ( y ^ 4 , y z , z ^ 4 ) \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} x _ { n + i } ^ 2 = \\sum _ { j = 1 } ^ { i - 1 } c _ { j , i } \\cdot x _ { n + i } \\cdot x _ { n + j } \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\{ A ^ r _ { i } , A ^ g _ { i } , A ^ b _ { i } \\} _ { i = 1 } ^ { 4 } = D i v ( A ) . \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\phi ( t , x , y ) = t \\Phi ( t ^ { - 1 } z ) , \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} & t ( y ^ { - 1 } x ) ^ { \\pm n } = ( y ^ { - 1 } x ) ^ { \\pm n } t \\ \\mbox { f o r e v e r y $ t \\in S ^ { - } $ } , \\\\ & t ( y ^ { - 1 } x ) ^ { \\pm n } = ( x ^ { - 1 } y ) ^ { \\pm n } t \\ \\mbox { f o r e v e r y $ t \\not \\in S ^ { - } $ } . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\gamma ( F _ \\pm ( \\nabla ^ \\gamma _ X F _ \\pm ) Y , Z ) = \\pm \\frac { 1 } { 2 } [ d \\psi ( X , Y , F ^ 2 _ \\pm Z ) + d \\psi ( X , F _ \\pm Y , F _ \\pm Z ) ] . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus B ( x _ l , 8 r _ l ) } \\abs { D ^ 2 w _ l } \\lesssim \\sum _ { k = 1 } ^ { N } 2 ^ { - \\alpha k } \\norm { b _ l } _ { L ^ 1 ( Q _ l ) } \\simeq \\int _ { Q _ l } \\abs { b _ l } \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} d _ 3 = c _ 3 - 3 c _ 2 \\frac { 1 } { n - 1 } + 3 c _ 1 \\frac { 1 } { ( n - 1 ) ^ 2 } - \\frac { 1 } { ( n - 1 ) ^ 3 } = O \\left ( \\frac { 1 } { n ^ 3 } \\right ) , \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} \\frac { \\partial A } { \\partial t } = \\frac { \\partial A } { \\partial P } \\frac { \\partial P } { \\partial t } . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { d e t } [ b _ { j , k } ( \\zeta ) ] _ { j , k = 1 , \\ldots , N / 2 } \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{gather*} \\| \\mathrm { E } _ f \\widehat { f } _ n - f \\| _ \\infty \\leq C \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l J _ { n , l } } , \\\\ P _ f \\left ( \\| \\widehat { f } _ n - \\mathrm { E } _ f \\widehat { f } _ n \\| _ \\infty \\geq Q _ 1 \\sqrt { \\frac { 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } } \\log { n } } { n } } + \\sqrt { 2 Q _ 2 \\frac { 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } } } { n } x } \\right ) \\leq e ^ { - x } . \\end{gather*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\phi ^ { 2 } W = - \\cos ^ 2 { \\theta } W , \\ , \\ , \\ W \\in \\Gamma ( D _ 2 ) , \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} [ R _ { 1 1 } ^ 2 ] = [ R _ { 2 1 } ^ 2 ] \\geq \\frac { 1 } { 2 | | R | | _ 0 } ; \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\Phi ( L , z _ 1 , \\ldots , z _ m ) = \\left \\{ \\begin{array} { c c } 1 , & T ( L + z _ 1 + \\ldots + z _ t ) \\in \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) t \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\{ h ( x , \\xi ) , X _ { j } ( x , \\xi ) \\} = \\alpha ( x , \\xi ) X ( x , \\xi ) , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} f _ + ( x , \\xi ) = f ( x , \\xi ) \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ p \\ ; \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\left ( \\frac { \\nu - 1 + \\mu _ i } { p } \\right ) ^ j \\ ; \\Psi \\left ( \\frac { \\nu - 1 + \\mu _ i } { p } \\right ) = 0 , \\ ; \\ ; \\ ; 0 \\leq j \\leq r - 1 . \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} f ( z , u ) = \\frac { 1 } { 2 } \\left ( z ^ 2 + \\frac { 1 } { z ^ 2 } \\right ) , \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { \\tau , t } } ( u ) \\right ) = 0 . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\rho ( \\xi - y ) & = \\sum _ { | \\ell | < N _ 1 } \\frac { ( - y ) ^ { \\ell } } { \\ell ! } ( \\partial ^ { \\ell } \\rho ) ( \\xi ) + R _ { N _ 1 } ( \\xi , y ) \\\\ \\alpha _ x ( \\rho ( \\xi - y ) ) & = \\sum _ { | \\ell | < N _ 1 } \\sum _ m \\frac { ( - y ) ^ { \\ell } } { \\ell ! } ( \\partial ^ { \\ell } \\rho _ m ) ( \\xi ) e ^ { i x \\cdot m } \\left ( \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\right ) + \\alpha _ x ( R _ { N _ 1 } ( \\xi , y ) ) \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\alpha ( O K ) \\leq 9 \\Biggl ( \\sup _ { w \\in K ^ { \\circ } \\cap L } \\langle J ( O ) v , w \\rangle + \\sup _ { u , w \\in K ^ { \\circ } \\cap L } \\langle J ( O ) u , w \\rangle \\Biggr ) . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} w ( a , b ) = c _ 1 ^ { n _ 1 } c _ 2 ^ { n _ 2 } c _ 3 ^ { n _ 3 } . . . c _ k ^ { n _ k } \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} i ^ { 0 , 0 } ( H ^ 3 _ c ( Y ; \\mathbb { Q } ) ) & = 1 , i ^ { 1 , 1 } ( H ^ 3 _ c ( Y ; \\mathbb { Q } ) ) = p h ( Y , w ) - 3 , \\\\ i ^ { 2 , 1 } ( H ^ 3 _ c ( Y ; \\mathbb { Q } ) ) & = i ^ { 1 , 2 } ( H ^ 3 _ c ( Y ; \\mathbb { Q } ) ) = h ^ { 1 , 2 } ( Z ) . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\int _ a ^ b k ( s , t ) x ( t ) d t = f ( s ) , c \\leq s \\leq d . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} d x ( t ) = b ( x ( t ) ) d t + \\sigma ( x ( t ) ) u ( t ) d t , \\ ; x ( 0 ) = x _ { 0 } , \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\bar A ^ { \\lambda } = \\left [ \\begin{array} { c c c } \\otimes & 0 & \\times \\\\ \\times & \\otimes & 0 \\\\ \\times & \\times & \\otimes \\end{array} \\right ] , \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} & E \\left [ \\int _ 0 ^ T h ( t , X ( t ) ) \\int _ A \\varphi ( t , X ( t ) , a ) [ \\widehat { \\gamma } ( t , X ( t ) ) ] ( d a ) d t \\right ] \\\\ & = T \\int _ { [ 0 , T ] \\times \\Sigma } h ( t , x ) \\int _ A \\varphi ( t , x , a ) [ \\widehat { \\gamma } ( t , x ) ] ( d a ) \\Theta _ 1 ( d t , d x ) \\\\ & = T \\int _ { [ 0 , T ] \\times \\Sigma \\times A } h ( t , x ) \\varphi ( t , x , a ) \\Theta ( d t , d x , d a ) \\\\ & = E \\left [ \\int _ 0 ^ T h ( t , X ( t ) ) \\int _ A \\varphi ( t , X ( t ) , a ) \\rho _ t ( d a ) d t \\right ] , \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} | | \\Gamma | | _ { \\infty } & = - \\inf _ { x \\in \\Pi } \\Gamma ( x , t ) \\\\ & \\leq - m = | | \\Gamma _ 0 | | _ { \\infty } . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} ~ e ( \\mathbf { x } _ { n + 1 } ) = \\left [ \\left \\{ \\sum _ { i = 1 } ^ n \\left ( \\hat f _ { - i } ( \\mathbf { x } _ { n + 1 } ) - \\hat f ( \\mathbf { x } _ { n + 1 } ) \\right ) ^ 2 \\right \\} / n \\right ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} \\mathfrak { g } _ i = \\mathfrak { s o } ( n ) \\oplus \\mathfrak { m } _ i , \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\bigg | \\prod ^ { p + k } _ { i = 1 } \\big ( z - \\xi ^ { ( p ) } _ i \\big ) \\bigg | ^ { 1 / p } = \\kappa \\Phi ( z ) ; \\kappa = \\ , ( E ) , \\Phi ( z ) = \\exp [ g ( z ) ] , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} E | X ^ * ( t ) | ^ m = E ( X ^ * ( t ) ) ^ m = \\sum _ { k = 1 } ^ m B _ { m , k } \\left ( \\kappa _ { X ^ * } ^ { ( 1 ) } ( t ) , \\dots , \\kappa _ { X ^ * } ^ { ( m - k + 1 ) } ( t ) \\right ) , \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} \\mathcal { J } _ n ( \\underline { \\gamma } ) \\subset \\mathcal { I } _ n ( \\boldsymbol { \\alpha } ) : = \\{ ( \\boldsymbol { j } , \\boldsymbol { k } ) : j _ l < J _ { n , l } ( \\boldsymbol { \\alpha } ) , k _ l < 2 ^ { j _ l } , l = 1 , \\dotsc , d \\} , \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} Q ( z ) = ( - 1 ) ^ k \\sum _ { 1 \\leq s _ 1 < s _ 2 < \\cdots < s _ k \\leq \\mu } T _ { s _ 1 , \\ldots , s _ k } V ( z , z _ { s _ 1 } , z _ { s _ 2 } , \\ldots , z _ { s _ k } ) \\bigg [ \\prod ^ k _ { i = 1 } \\Psi _ p ( z _ { s _ i } ) \\bigg ] ^ { - 1 } . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\frac { \\partial ^ { \\beta + 1 } \\kappa } { \\partial s ^ \\beta \\partial u } ( s , t , \\varphi _ m ( t ) ) = D ^ { ( \\beta , 0 ) } \\ell _ m ( s , t ) \\in C ^ { 3 r - \\beta } ( [ a , b ] \\times [ a , b ] ) . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} ( a g + a ' g ' ) + \\log | f ^ a { f ' } ^ { a ' } | = a ( g + \\log | f | ) + a ' ( g ' + \\log | f ' | ) \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} S ( \\mathfrak { C } ) = ( \\gamma \\mid _ { \\mathfrak { C } } ) ^ { - 1 } ( \\mathbb { G } ^ { \\vee } ) \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} 0 = \\int _ { \\R ^ d } 1 _ { A \\cap B _ { r } ( x _ 0 ) } P _ t 1 _ { A \\cap B _ { r } ( x _ 0 ) } d m \\underset { t \\rightarrow 0 } { \\ ; \\longrightarrow } m ( B _ r ( x _ 0 ) \\cap A ) > 0 , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} D ' ( 1 / x ) & = g ^ 2 ( x ) C ' ( x ) \\pmod { x ^ { n - 1 } - 1 } \\\\ & = \\left ( \\sum _ { r = 0 } ^ { n - 2 } \\beta ^ { \\pi ^ { - r } } x ^ r \\right ) ^ 2 \\cdot \\left ( \\sum _ { w = 0 } ^ { m - 1 } C ' _ { 2 u + h w } x ^ { 2 u + h w } \\right ) \\pmod { x ^ { n - 1 } - 1 } \\\\ & = x ^ { 2 u } \\cdot \\left ( \\sum _ { r = 0 } ^ { n - 2 } \\beta ^ { 2 \\pi ^ { - r } } x ^ { 2 r } \\right ) \\cdot \\left ( \\sum _ { w = 0 } ^ { m - 1 } C _ l ^ { 2 ^ w } x ^ { h w } \\right ) \\pmod { x ^ { n - 1 } - 1 } \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\rho ( \\mu , \\nu ) & \\doteq \\inf \\left \\{ \\int _ { \\mathbb { R } ^ d \\times \\mathbb { R } ^ d } \\| x - y \\| \\pi ( d x , d y ) ; \\ ; \\pi \\mu \\nu \\right \\} \\\\ & = \\sup \\left \\{ \\langle g , \\mu \\rangle - \\langle g , \\nu \\rangle : g \\ ! : \\R ^ { d } \\to \\R , \\ ; \\| g ( x ) - g ( y ) \\| \\leq \\| x - y \\| \\right \\} . \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\left | \\frac { Q _ { j , k } f ( x ) } { \\prod _ { i = j } ^ { k - 1 } \\varsigma _ { i } } - \\phi { } _ { j , k - 1 } ( x ) \\eta _ { k } ^ { \\mu _ { 0 } } f \\right | \\leq \\rho ^ { k - j } \\| f \\| _ { e ^ { V } } c _ { \\mu _ { 0 } } e ^ { V ( x ) } \\mu _ { 0 } ( e ^ { V } ) , \\quad \\forall x \\in \\mathbb { R } ^ { p } , \\ , 0 \\leq j < k \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} { n - 3 } = r t \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} \\mathcal { A } = f ( \\mathbf { x } ) ^ * b ( \\mathbf { D } ) ^ * g ( \\mathbf { x } ) b ( \\mathbf { D } ) f ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} N = W ^ { 2 n } N \\rightarrow \\cdots \\rightarrow W ^ { d } N = 0 \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\| a \\| _ { M ( X , Y ) } = \\| B \\| _ { X \\to Y } \\leq \\| A \\| _ { H [ X ] \\to H [ Y ] } = \\| T _ a \\| _ { H [ X ] \\to H [ Y ] } , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\eqref { v a 5 } & = - \\sqrt { \\tfrac { i t } { 2 \\pi } } e ^ { - \\frac { i t x ^ 2 } { 2 } } \\tfrac { x } { 1 - i t x ^ 2 } [ T ( x ) \\psi ( x ) - T ( 0 ) \\psi ( 0 ) ] \\\\ & - \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ + e ^ { - \\frac { i t } { 2 } ( \\xi - x ) ^ 2 } \\tfrac { \\xi - x } { 1 - i t ( \\xi - x ) ^ 2 } \\partial _ \\xi [ T ( \\xi ) \\psi ( \\xi ) ] \\ , d \\xi \\\\ & - \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ + e ^ { - \\frac { i t } { 2 } ( \\xi - x ) ^ 2 } \\tfrac { 2 i t ( \\xi - x ) } { [ 1 - i t ( \\xi - x ) ^ 2 ] ^ 2 } [ T ( \\xi ) \\psi ( \\xi ) - T ( x ) \\psi ( x ) ] \\ , d \\xi . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} | D ^ j U | _ { \\omega _ { \\phi } } ^ 2 \\leq C _ { 4 6 } \\ ; \\ ; . \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\sum _ { i } m _ i ^ 2 m ^ { r - 6 } \\le \\tfrac { r ( m + 1 ) } { 2 } m ^ { r - 6 } \\sum _ i m _ i \\le \\tfrac { r ( m + 1 ) } { 2 } m ^ { r - 6 } \\binom { m ( r + 1 ) } { 2 } = o ( m ^ { r - 2 } ) . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\begin{array} { r l l l } - \\Delta _ y G _ \\lambda - \\lambda G _ \\lambda & = & \\delta _ x & y \\in \\Omega , \\\\ G _ \\lambda ( x , y ) & = & 0 & y \\in \\partial \\Omega , \\end{array} \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} A = \\langle e _ 1 \\rangle _ { \\overline { 0 } } \\oplus \\langle e _ 2 , e _ 3 , . . . , e _ n \\rangle _ { \\overline { 1 } } \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\langle L ^ \\alpha [ u ] , u \\rangle = L _ 1 + L _ 2 , \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u _ i - d _ i \\Delta u _ i & = f _ i ( u ) , & & x \\in \\Omega , t > 0 , \\\\ \\nabla u _ i \\cdot \\nu & = 0 , & & x \\in \\partial \\Omega , t > 0 , \\\\ u _ i ( x , 0 ) & = u _ { i , 0 } ( x ) , & & x \\in \\Omega \\end{aligned} \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} x . ( u \\otimes v ) = \\alpha ( x | y , z ) ( x . u \\otimes x . v ) , u \\in U _ x , \\ v \\in V _ y \\ . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} D ( x ) = A ( \\log \\log x ) ^ 2 + O \\bigg ( \\frac { ( \\log \\log x ) ^ { 3 / 2 } } { ( \\log \\log \\log x ) ^ 2 } \\bigg ) , \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} d _ 0 \\left ( \\sum _ { i = 1 } ^ { \\infty } h _ i \\right ) = \\sum _ { i = 1 } ^ { \\infty } ( - 1 ) ^ { \\dim R _ { \\alpha _ i } + \\mu ( \\alpha _ i ) + 1 + \\deg h _ i } d _ { d R } h _ i \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\Sigma ^ G _ P \\cap \\Sigma ^ G _ Q = \\{ \\check { \\alpha } \\} \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } H ( t , \\rho ) = 2 \\int \\limits _ { - 1 } ^ t f ( s ) H ( s , \\rho ) d s + \\int \\limits _ { - 1 } ^ t R _ 1 ( s , \\rho ) d s \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} g ( c ) = N ( c ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} m _ 1 ( \\lambda + ) = F ^ { - 1 } \\left ( - \\frac { i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } F ( m _ 1 ( \\lambda + ) - 1 ) \\right ) + 1 + G _ { \\lambda + i \\epsilon } * ( u m _ 1 ( \\lambda + ) ) . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\zeta ^ n _ 0 ( t ) = \\Gamma ( \\psi ^ n ) = \\psi ^ n ( t ) + \\eta ^ n ( t ) = \\sum _ { k = 0 } ^ \\infty ( k - 2 ) B _ k ^ n ( t ) + \\eta ^ n ( t ) , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla ( \\eta u ^ { p / 2 } ) ) & = \\frac { p ^ 2 } { 4 } u ^ { p - 2 } \\eta ^ 2 ( a \\nabla u , \\nabla u ) + p \\eta u ^ { p - 1 } ( a \\nabla u , \\nabla \\eta ) + u ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) , \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} ( b X _ 0 - r Y _ 0 ) ( b X _ 0 + r Y _ 0 ) & = b ^ 2 X _ 0 ^ 2 - r ^ 2 Y _ 0 ^ 2 = b ( a Y _ 0 ^ 2 + 4 ( b - a ) ) - a b Y _ 0 ^ 2 - 4 Y _ 0 ^ 2 \\\\ & = 4 b ( b - a ) - 4 Y _ 0 ^ 2 . \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} V ( t , x ) = \\inf _ { u \\in { \\cal U } _ { [ 0 , t ] } ( 0 , x ) } J _ { [ 0 , t ] } ( u ) , \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\forall k \\geq 1 : \\ : \\zeta ^ i _ k : = d ^ i \\ , , \\ , \\ , \\nu ^ i _ k = 1 \\ , , \\ , \\ , R ^ i _ k = r ^ i \\quad ; \\forall n \\geq 0 : \\ : S ^ i _ n : = s _ 0 ^ i + d ^ i n \\ , , \\ , \\ , T ^ i _ n = n \\ , . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\frac { d \\widetilde { y } } { d w } = \\widetilde { f } ( w ) \\widetilde { y } ^ 3 + \\widetilde { g } ( w ) \\widetilde { y } ^ 2 \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} B = \\frac { 1 } { 2 } \\left ( \\left | \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ { r } ( \\vec { 1 } ) } \\right | ^ 2 - 2 \\Re \\left ( \\frac { P _ { r + 2 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) + r \\right ) \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\frac { \\alpha _ 2 ( \\lambda ) } { \\alpha _ 1 ( \\lambda ) } = \\frac { \\lambda } { 2 } . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} r ( d ^ { - 1 } ( U ) ) = U = d ( r ^ { - 1 } ( U ) ) . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} f ( t ) = \\hat { f } ( \\sigma ( t ) ) \\sigma ' ( t ) , \\ , \\ , g ( t ) = \\hat { g } ( \\sigma ( t ) ) \\sigma ' ( t ) \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} e ^ { \\lambda _ { n , j } } h ^ 2 ( x _ { n , j } ) e ^ { G _ j ^ * ( x _ { n , j } ) } = e ^ { \\lambda _ { n , 1 } } h ^ 2 ( x _ { n , 1 } ) e ^ { G _ 1 ^ * ( x _ { n , 1 } ) } ( 1 + O ( e ^ { - \\frac { \\lambda _ { n , 1 } } { 2 } } ) ) , \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\varphi _ i = \\left ( \\frac { z _ i A _ i ( z _ { i + 1 } , z _ { i + 2 } , \\ldots , z _ { n - 1 } ) + B _ i ( z _ { i + 1 } , z _ { i + 2 } , \\ldots , z _ { n - 1 } ) } { z _ i C _ i ( z _ { i + 1 } , z _ { i + 2 } , \\ldots , z _ { n - 1 } ) + D _ i ( z _ { i + 1 } , z _ { i + 2 } , \\ldots , z _ { n - 1 } ) } \\right ) \\in \\mathrm { P G L } ( 2 , \\mathbb { C } ( z _ { i + 1 } , z _ { i + 2 } , \\ldots , z _ { n - 1 } ) ) \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} H = \\frac { x ^ 2 ( C ( x ) - 1 ) + x ^ 5 C ( x ) ^ 5 } { 1 - x - x C ( x ) } = x ( C ( x ) - 1 ) ^ 2 + x ^ 4 C ( x ) ^ 5 ( C ( x ) - 1 ) \\ , . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} R ^ { 2 M } = R ^ { 2 M } _ { M , h } . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} N = \\big ( z _ 0 + 2 z _ 1 ( z _ 0 z _ 2 - z _ 1 ^ 2 ) + z _ 2 ( z _ 0 z _ 2 - z _ 1 ^ 2 ) ^ 2 , z _ 1 + z _ 2 ( z _ 0 z _ 2 - z _ 1 ^ 2 ) , z _ 2 \\big ) \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} L ( s , f \\times \\overline { g } ) : = \\frac { ( 4 \\pi ) ^ { s + k - 1 } \\zeta ( 2 s ) } { \\Gamma ( s + k - 1 ) } \\langle \\Im ( \\cdot ) ^ k f \\overline { g } , E ( \\cdot , \\overline { s } ) \\rangle , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} G _ { r } ( z , v _ { I } , v _ { L } ) = \\Phi ^ { r } ( T ( z v _ { I } , t v _ { L } ) ) | _ { t = z } = \\frac { 1 - u ^ { r + 2 } } { ( 1 - u ^ { r + 1 } ) ( 1 + u ) } T \\Big ( \\frac { u ( 1 - u ^ { r + 1 } ) ^ { 2 } } { ( 1 - u ^ { r + 2 } ) ^ { 2 } } v _ { I } , \\frac { u ^ { r + 1 } ( 1 - u ) ^ { 2 } } { ( 1 - u ^ { r + 2 } ) ^ { 2 } } v _ { L } \\Big ) . \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\phi _ 1 = \\frac { \\mu + \\frac { 1 } { 2 } \\partial ( \\varphi ) } { \\varphi } = \\frac { ( z ^ 2 + 1 ) ^ 3 \\mu + z ^ 4 - z ^ 2 } { ( z ^ 2 + 1 ) ( ( z ^ 2 + 1 ) ^ 2 \\lambda + z ^ 4 + z ^ 2 + 1 ) } , \\mbox { w h e r e } \\varphi = \\lambda + 1 - \\frac { 1 } { \\eta ^ 2 } , \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} G _ { \\emptyset } ( \\mathbb Q ^ { \\mathrm { c y c } } ( \\sqrt { - \\ell _ 1 } ) ) ^ { \\mathrm { a b } } \\simeq \\varLambda / T \\simeq \\mathbb Z _ 2 \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} m _ { \\mu _ \\alpha \\boxplus \\mu _ \\beta } ( z ) - m _ { \\mu _ \\alpha \\boxplus \\mu _ \\beta } ( E _ - ) = m _ { \\mu _ \\alpha } ' ( \\omega _ \\beta ( E _ - ) ) \\sqrt { \\frac { - 2 ( \\omega _ \\beta ( E _ - ) ) } { \\widetilde z '' ( \\omega _ \\beta ( E _ - ) ) } } \\sqrt { E _ - - z } + O ( | z - E _ - | ) \\ , , \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} Z ( s , w , f \\times \\overline { g } ) : = \\sum _ { n , h \\geq 1 } \\frac { D _ { f , \\overline { g } } ( s ; h ) } { h ^ w } = \\frac { ( 4 \\pi ) ^ { s + k - 1 } } { \\Gamma ( s + k - 1 ) } \\sum _ { h \\geq 1 } \\frac { \\langle \\mathcal { V } _ { f , \\overline { g } } , P _ h \\rangle } { h ^ w } , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , \\chi _ c \\right ) = 2 \\sum _ { 0 \\neq \\mathcal { A } \\subset \\mathcal { O } _ K } \\frac { \\chi _ c ( \\mathcal { A } ) } { N ( \\mathcal { A } ) ^ { 1 / 2 } } V \\left ( \\frac { 2 \\pi N ( \\mathcal { A } ) } { ( 3 N ( c ) ) ^ { 1 / 2 } } \\right ) . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} & \\delta _ { [ k = \\frac { 1 } { 2 } ] } \\delta _ { [ h = a ^ 2 ] } \\bigg ( 2 R ' _ h X ^ { \\frac { 1 } { 2 } } \\log X - 4 R ' _ h X ^ { \\frac { 1 } { 2 } } \\bigg ) + \\tfrac { 1 } { k } R _ { k , h } ^ k X ^ { k } + O \\big ( \\frac { X ^ { \\frac { 1 } { 2 } } \\log X } { Y } \\big ) \\\\ & + O \\big ( \\frac { X ^ { k } } { Y } \\big ) + O ( X ^ { \\frac { k - 1 } { 2 } } ) + O ( X ^ { \\frac { k } { 2 } + \\epsilon } Y ^ { 3 k + \\frac { 1 7 } { 2 } + 2 \\epsilon } ) . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} ( \\frak N _ { 4 3 } \\circ \\frak N _ { 3 2 } ) \\circ \\frak N _ { 2 1 } = \\frak N _ { 4 3 } \\circ ( \\frak N _ { 3 2 } \\circ \\frak N _ { 2 1 } ) . \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{gather*} ( I f ) ( x ) = \\int _ { - 1 } ^ x f ( u ) { \\rm d } u , x \\in [ - 1 , 1 ] , \\end{gather*}"}
-{"id": "6699.png", "formula": "\\begin{align*} \\deg u \\circ F = ( \\deg u ) ( \\deg F ) . \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( B _ { \\tau + \\cdot } ) ] = \\hat { \\mathbb { E } } [ \\varphi ( B _ \\cdot ^ y ) ] _ { y = B _ { \\tau } } , \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\mathcal { E } : = \\operatorname { S y m } ^ { 3 } ( \\mathcal { S } ^ { \\vee } ) . \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\frac { w } { h } = \\alpha . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} 0 < \\underline { \\alpha } \\leq \\alpha _ { \\varepsilon } : = \\left \\vert \\left \\vert \\left \\vert A _ { \\varepsilon } ^ { - 1 } \\right \\vert \\right \\vert \\right \\vert _ { \\infty , \\Omega } ^ { - 1 } \\leq \\left \\vert \\left \\vert \\left \\vert A _ { \\varepsilon } \\right \\vert \\right \\vert \\right \\vert _ { \\infty , \\Omega } = : \\beta _ { \\varepsilon } \\leq \\overline { \\beta } < \\infty \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} X _ n = \\sum _ { k = 0 } ^ { n - 1 } X \\circ T _ 0 ^ k \\ ( T _ 0 ^ { n - 1 - k } \\ ) ' \\circ T _ 0 ^ k , \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} [ \\mathfrak { X } _ i ^ { + } ( u , \\lambda _ 1 ) , \\mathfrak { X } _ j ^ { - } ( v , \\lambda _ 2 ) ] = 0 . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} P _ { \\frac 1 2 m ^ { - 1 } ( v ) \\leq \\cdot < \\frac 3 2 m ^ { - 1 } ( v ) } ( u _ { N _ 1 } u _ { N _ 2 } ) = 0 , \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} A = \\bigoplus \\limits _ { g \\in G } A _ g . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} ( x \\otimes y ) ^ { \\tau _ 1 } = x \\otimes y ^ { \\sigma _ 2 } , ( x \\otimes y ) ^ { \\tau _ 2 } = x ^ { \\sigma _ 1 } \\otimes y , ( x \\otimes y ) ^ { \\tau _ 3 } = x ^ { \\sigma _ 1 } \\otimes y ^ { \\sigma _ 2 } . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} M ( x ) = \\sum _ { d \\ge 0 } M _ d ( x ) = x ^ 3 \\left ( \\frac { 1 - 2 x } { 1 - 3 x + x ^ 2 } \\right ) ^ 2 \\ , . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} N ^ + _ { D _ C } ( u _ i ) : = \\{ \\ , u _ j \\ , | \\ , c _ { i , j } = 1 \\ , \\} N _ i : = | N ^ + _ { D _ C } ( u _ i ) | u _ i \\\\ N ^ - _ { D _ C } ( u _ i ) : = \\{ \\ , u _ j \\ , | \\ , c _ { j , i } = 1 \\ , \\} I _ i : = | N ^ - _ { D _ C } ( u _ i ) | u _ i \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\bar { \\psi } + \\frac { 2 } { 3 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\psi _ i \\right ) ^ 3 + \\frac { 3 } { 4 } \\Pi ^ T \\Omega ^ { - 1 } \\Pi - \\frac { 1 } { 4 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\psi _ i \\right ) ^ 4 + O _ p ( n ^ { - 5 / 2 } ) \\ , . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{gather*} \\beta ( \\beta - \\beta \\alpha ) \\beta = \\beta ^ 3 - \\beta ^ 2 \\alpha \\beta = \\beta , \\\\ ( \\beta - \\beta \\alpha ) \\beta ( \\beta - \\beta \\alpha ) = \\beta ^ 3 - \\beta \\alpha \\beta ^ 2 - \\beta ^ 3 \\alpha + \\beta \\alpha \\beta ^ 2 \\alpha = \\beta - \\beta \\alpha , \\end{gather*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\partial ^ { j } } { \\partial \\rho ^ { j } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = j ! \\int \\limits _ { - 1 } ^ 1 f ( t ) r _ j ( t ) d t \\mbox { f o r e a c h } j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\sum _ { 1 \\leqslant k _ 1 \\leqslant k _ 2 \\leqslant k _ 3 \\leqslant n } k _ 1 k _ 2 k _ 3 ( n + 3 ) ^ { k _ 1 - 1 } ( n + 2 ) ^ { k _ 2 - k _ 1 } ( n + 1 ) ^ { k _ 3 - k _ 2 } n ^ { n - k _ 3 } = { n + 2 \\choose 3 } n ^ { n + 2 } . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } \\phi ( x ) | ^ { p } d x + \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x = \\int _ { \\mathbb { G } } | \\phi ( x ) | ^ { q } d x . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} ( \\Delta ^ e \\otimes I ) \\Delta ^ e m ( a ) & = ( \\Delta ^ e \\otimes I ) ( m \\otimes m ) \\Delta ( a ) = \\Delta ^ e m ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) \\\\ & = ( m \\otimes m ) \\Delta ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) = m ( a _ { ( 1 1 ) } ) \\otimes m ( a _ { ( 1 2 ) } ) \\otimes m ( a _ { ( 2 ) } ) , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} \\mathbf { G } = \\beta ^ { \\rho , A } + \\mathbf { B } . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\Vert \\L [ \\nabla \\phi , \\L ^ { - \\alpha + \\delta } ] \\L ^ \\alpha \\psi \\Vert _ { L ^ 2 } = \\Vert [ \\nabla \\phi , \\L ^ { - \\alpha + \\delta } ] \\L ^ \\alpha \\psi \\Vert _ { H ^ 1 } \\le C \\Vert \\nabla \\phi \\Vert _ { W ^ { 1 , \\infty } } \\Vert \\psi \\Vert _ { D ( \\L ^ \\alpha ) } \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} L _ X J = 2 \\sqrt { - 1 } \\nabla ^ { \\prime \\prime } _ J X ^ \\prime - 2 \\sqrt { - 1 } \\nabla ^ \\prime _ J X ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\{ \\Upsilon _ 1 ( j ) , \\Upsilon _ 2 ( j ) \\} = \\{ \\sigma ( 2 j - 1 ) , \\sigma ( 2 j ) \\} 1 \\le j \\le \\tfrac k 2 . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} & O _ P \\left ( \\sqrt { n } 2 ^ { \\sum _ { l = 1 } ^ d ( j _ l / 2 + J _ { n , l } ) } ( \\log { n } / n ) ^ { \\alpha ^ { * } / ( 2 \\alpha ^ { * } + d ) } + n \\prod _ { l = 1 } ^ d 2 ^ { - j _ l / 2 } \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l J _ { n , l } } \\right ) \\\\ & = O _ P \\left ( \\sqrt { n } n ^ { 1 / 4 } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } / 2 } \\epsilon _ n ^ { 1 / 2 } \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l J _ { n , l } / 2 } \\right ) , \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\xi _ { d } = \\left \\{ x \\in \\Omega : \\xi \\left ( x \\right ) = d \\right \\} , \\Omega _ { d } = \\left \\{ x \\in \\Omega : \\xi \\left ( x \\right ) > d \\right \\} . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\| T _ h f \\| _ p ^ p & = \\int _ X \\Bigl | \\int _ X f ( y ) \\ > K _ h ( x , d y ) \\Bigr | ^ p \\ > d \\omega _ X ( x ) \\\\ & \\le \\int _ X \\Bigl ( \\int _ X | f ( y ) | ^ p \\ > K _ h ( x , d y ) \\Bigr ) \\Bigl ( \\int _ X 1 \\ > K _ h ( x , d y ) \\Bigr ) ^ { p / q } d \\omega _ X ( x ) = \\| f \\| _ p ^ p . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} \\begin{cases} \\lim _ { | x | \\to \\infty } h ^ t ( x ) < c ^ t & n \\ge 3 \\\\ \\lim _ { | x | \\to \\infty } \\frac { h ^ t ( x ) } { - \\log | x | } < c ^ t & n = 2 , \\end{cases} \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} & ( M _ r { V } ) _ S = \\\\ & \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\left [ \\binom { | S | - 2 } { a - 2 } \\binom { r | S | - 2 } { b } + \\binom { | S | - 2 } { a } \\binom { r - | S | - 2 } { b - 2 } \\right . \\\\ & \\left . - \\binom { | S | - 2 } { a - 1 } \\binom { r - | S | - 2 } { b - 1 } - \\binom { | S | - 2 } { a - 1 } \\binom { r - | S | - 2 } { b - 1 } \\right ] \\\\ & = P _ 1 + P _ 2 - 2 P _ 3 \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\left | Z _ T \\right | = P _ Y \\right ) \\ : \\mbox { f o r } \\ : T \\geq 0 , \\mbox { a n d } \\lim _ { T \\to \\infty } \\mathbb { P } \\left ( \\left | Z _ T \\right | = P _ Y \\right ) , \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} t _ i ( M ) = t _ { i - 1 } ( \\Omega ^ R _ 1 ( M ) ) & \\le \\max \\{ t _ { i - 1 } ( M ) + 1 , t _ { i - 1 } ( K ) , t _ { i - 2 } ( L ) \\} \\\\ & = \\max \\{ t _ { i - 1 } ( M ) + 1 , i + 2 \\} = i + 2 . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} L ^ { \\rho _ + } ( \\xi _ 0 n ^ { 2 / 3 } , 0 ) \\simeq \\xi _ 0 n ^ { 2 / 3 } / ( 1 - \\rho _ + ) = \\tfrac { 1 + \\gamma } { \\gamma } \\xi _ 0 n ^ { 2 / 3 } + \\tfrac { ( 1 + \\gamma ) ^ 2 } { \\gamma ^ 2 } \\xi _ 0 \\kappa n ^ { 1 / 3 } + O ( 1 ) \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\mathbf { h } ( M ) : = \\inf _ { \\Gamma } \\frac { \\min \\{ A _ { \\pm } ( \\Gamma ) \\} } { \\min \\{ \\mathfrak { m } ( D _ 1 ) , \\mathfrak { m } ( D _ 2 ) \\} } , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ M \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) & = \\sum _ { \\substack { m \\leq M \\\\ m ~ } } \\frac 1 { ( m / 2 ) ! } \\ ( - \\frac 1 { 2 n ^ 2 } \\sum _ { x \\in G } | f ^ { - 1 } ( x ) | ^ 2 \\ ) ^ { m / 2 } \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\\\ & \\qquad + O _ M \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} \\frac { t ^ { b } } { \\prod _ i ( 1 - t ^ { a _ i } ) } = - \\frac { t ^ { b - a _ { i _ 0 } } } { \\prod _ { i \\neq i _ 0 } ( 1 - t ^ { a _ i } ) } + \\frac { t ^ { b - a _ { i _ 0 } } } { \\prod _ { i } ( 1 - t ^ { a _ i } ) } = \\sum _ { \\substack { x _ i \\geq 1 \\\\ \\sum _ i x _ i a _ i \\leq b } } p _ { ( x _ i ) } \\cdot t ^ { b - \\sum _ i x _ i a _ i } + \\substack { \\textnormal { n e g a t i v e d e g r e e } \\\\ \\\\ \\textnormal { r a t i o n a l f u n c t i o n } } , \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} A ^ + _ n ( v ) & = A ^ + _ { n ; 1 } ( v ) + \\left ( B ' _ n ( 1 / v ) - 2 ^ { n - 3 } \\right ) v ^ { n - 2 } - \\left ( B ' _ { n - 1 } ( 1 / v ) - 2 ^ { n - 4 } \\right ) v ^ { n - 2 } \\\\ & + ( 1 + v ) \\left ( A ^ + _ { n - 1 } ( v ) - A ^ + _ { n - 1 ; 1 } ( v ) \\right ) + \\frac { 1 } { 1 - v } \\left ( B _ n ( 1 ) - v ^ { n - 3 } B _ n ( 1 / v ) \\right ) + C _ { n - 2 } / v , \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\sigma _ F : = \\sup _ { x \\in M } \\sup _ { v , w \\in T _ x M \\backslash \\{ 0 \\} } \\frac { g _ v ( w , w ) } { F ( w ) ^ 2 } = \\sup _ { x \\in M } \\sup _ { \\alpha , \\beta \\in T ^ * _ x M \\backslash \\{ 0 \\} } \\frac { F ^ * ( \\beta ) } { g ^ * _ \\alpha ( \\beta , \\beta ) } \\in [ 1 , \\infty ] . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} \\frac { \\mathtt { h } _ { a , \\nu } ' ( z ) } { \\mathtt { h } _ { a , \\nu } ( z ) } = \\frac { 1 + a - a \\nu } { 2 z } + \\frac { a ^ { a / 2 } } { 2 \\sqrt { z } } \\frac { { } _ a \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ' ( a ^ { a / 2 } \\sqrt { z } ) } { { } _ a \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } \\sqrt { z } ) } , \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} m _ { \\mu _ A } ( \\omega _ B ( z ) ) & = m _ { \\mu _ B } ( \\omega _ A ( z ) ) = m _ { \\mu _ A \\boxplus \\mu _ B } ( z ) , \\\\ \\omega _ A ( z ) + \\omega _ B ( z ) - z & = - \\frac { 1 } { m _ { \\mu _ A \\boxplus \\mu _ B } ( z ) } . \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} p _ t ( x , d y ) = p _ t ( x , y ) d y , x , y \\in \\mathring { S } . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } A _ f ( n ) n ^ { \\frac { k - 1 } { 2 } } e ^ { 2 \\pi i n z } \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} ( r ^ 2 - r + 1 ) [ \\sigma ] = r ( r ^ 2 - r + 1 ) [ \\tau ] = 0 , \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi ) Y = g _ { M } ( X , Y ) \\xi - \\eta ( Y ) X \\ \\ \\ \\ \\nabla ^ { ^ { M _ 1 } } _ { X } \\xi = - \\varphi X , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\psi \\left ( \\xi \\right ) = \\frac { 4 f _ { 0 } \\left ( z \\right ) { f } _ { 0 } ^ { \\prime \\prime } \\left ( z \\right ) - 5 { f } _ { 0 } ^ { \\prime } { } ^ { 2 } \\left ( z \\right ) } { 1 6 f _ { 0 } ^ { 3 } \\left ( z \\right ) } + \\frac { g \\left ( z \\right ) } { f _ { 0 } \\left ( z \\right ) } . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\rho ( [ 0 , t [ \\times A ) = \\rho ( [ 0 , t ] \\times A ) = t \\forall t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d A } { d D } + \\frac { A } { D } = - \\ell A ^ 2 D ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\frac { d x } { d t } = a x , \\frac { d y } { d t } = - c y . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X _ 1 ( s ) d s + \\int _ 0 ^ t X _ 2 ( s ) d s + B ( t ) , t \\geq 0 . \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} T _ r ( U ) = \\frac { 1 } { r } \\sum _ { d | r } \\varphi _ { ( U ^ { \\frac { r } { d } } ) } ( p _ d ) \\mu ( r / d ) \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\tilde { X _ 0 } _ { \\vert \\mathcal { E } } : \\begin{cases} \\dot u & = 1 + u ^ 2 \\\\ \\dot y _ 3 & = \\frac { 1 } { 2 } ( 1 + y _ 3 ^ 2 ) \\end{cases} \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\alpha _ { i } = \\alpha _ { \\sigma ^ { k } ( 1 ) } = \\frac { \\alpha _ { 1 } d ^ { k } } { c _ { \\sigma ^ { 0 } ( 1 ) } c _ { \\sigma ( 1 ) } \\cdots c _ { \\sigma ^ { k - 1 } ( 1 ) } } , \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} F \\left ( q , \\frac { \\partial W } { \\partial q } , \\lambda \\right ) = c s t , \\frac { \\partial W } { \\partial \\bar { q } } ( \\bar { q } , q ) = 0 , \\frac { \\partial F } { \\partial \\lambda } \\left ( q , \\frac { \\partial W } { \\partial q } , \\lambda \\right ) = 0 . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\hat { \\Lambda } ( z ) = \\frac { N ^ { 3 \\varepsilon } } { ( N \\eta ) ^ { \\frac 1 3 } } . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} L f ( \\eta ) = \\sum _ { j \\in \\Z } \\eta _ j ( 1 - \\eta _ { j + 1 } ) \\big ( f ( \\eta ^ { j , j + 1 } ) - f ( \\eta ) \\big ) , \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} X _ i ^ { p _ i } + X _ 1 ^ { p _ 1 } - \\lambda _ i X _ 0 ^ { p _ 0 } , i = 2 , \\ldots , n , \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\phi } } \\right ) U _ { m l } ^ { \\beta } = \\nabla _ { \\phi m } ( \\widetilde { \\eta } ^ { \\beta } { } _ l + \\nabla _ { \\phi l } X ^ { \\beta } ) - \\nabla _ { \\phi } ^ { \\bar { q } } R _ 0 { } ^ { \\beta } { } _ { l \\bar { q } m } . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} B = 2 ^ { n - 1 } \\frac { \\sqrt { 2 e ^ 2 - e } } { e - 1 } = 2 ^ { n - 1 } \\cdot 2 . 0 2 1 0 \\dots , \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} \\bar F _ { \\lambda } ( \\zeta , \\bar { \\Lambda } ) & = k a _ 0 - a _ 1 \\sigma _ 1 ( \\varepsilon , \\zeta ) ^ 2 \\bar \\Lambda _ 1 ^ 2 - a _ 1 \\sigma _ 1 ( \\varepsilon , \\zeta ) ( \\bar \\Lambda ' ) ^ T Q ( \\varepsilon , \\zeta ) \\bar \\Lambda ' \\\\ & + \\sigma _ 1 ( \\varepsilon , \\zeta ) ^ 2 \\ , \\mathcal P o l y _ 4 ( \\varepsilon , \\zeta , \\bar \\Lambda ) + \\theta _ \\lambda ( \\zeta , \\bar \\Lambda ) , \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} \\begin{dcases} & L u _ \\lambda - \\lambda u _ \\lambda = - g \\mbox { i n } \\mathbb { R } ^ 2 , \\\\ & L v _ \\mu - \\mu v _ \\mu = - \\psi \\mbox { i n } \\mathbb { R } ^ 2 , \\\\ & L w _ { \\mu + \\lambda } - ( \\mu + \\lambda ) w _ { \\lambda + \\mu } = - \\frac { \\partial u _ \\lambda } { \\partial y } \\frac { \\partial v _ \\mu } { \\partial y } \\mbox { i n } \\mathbb { R } ^ 2 . \\end{dcases} \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\dd u _ t \\ , = \\ , ( - u _ t + g X u _ t ) \\dd t + \\dd B _ t , \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} E ( x ) = \\big ( E ( x ; q , a _ 1 ) , \\ldots , E ( x ; q , a _ r ) \\big ) . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\downarrow 0 } \\limsup _ { n \\to \\infty } \\P \\left ( W _ { \\Gamma ^ 1 _ n } ( \\delta ) > \\eta \\right ) = 0 \\ , , \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} [ c - b ] ^ 2 = c ^ 2 - 2 c b + b ^ 2 \\geq c ^ 2 + b ^ 2 . \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\Gamma _ { j k } ^ { i } \\dot { x } ^ { j } \\dot { x } ^ { k } + \\omega \\left ( t \\right ) V ^ { , i } = 0 ~ ~ , \\omega _ { , t } V ^ { , i } \\neq 0 , V ( x ^ { i } ) . \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} f = \\sum _ { m , l } b _ { m l } \\ , L ( { \\bf z } _ { m l } ) - 5 \\ , \\phi \\ , \\partial _ { ( \\zeta ^ \\prime ) _ n } V ^ 4 + \\sum _ { i , j } c _ { i j } \\ , \\partial _ { ( \\zeta ^ \\prime ) _ n } \\bigl [ w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\bigr ] . \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} v ' ( c ' ) + { i ' } _ * ( I ' ) - { o ' } _ * ( O ' ) = 0 . \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} g ( c ) = \\prod ^ k _ { i = 1 } g ( \\pi _ i ) . \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( M , g ) : = \\inf _ { f \\in \\mathcal { H } _ 0 ^ { p } } \\frac { \\int _ M | d f | ^ p \\ d V _ g } { \\int _ M | f | ^ p \\ d V _ g } . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} { \\mathcal P } ( t , x , \\tau , \\xi ) = h ( t , x , \\tau , \\xi ) I + P _ { m - 1 } + \\cdots + P _ 0 \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} w u = t w , ~ x s = t x \\rangle \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} u _ { i , i } + u _ { i , i + 1 } + \\ldots + u _ { i , n } - ( u _ { 1 , i } + u _ { 2 , i } + \\ldots + u _ { i - 1 , i } ) = \\alpha _ i . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} X = \\begin{pmatrix} 1 & x _ 1 & \\dots & x _ { n - 1 } \\\\ x _ 1 & 1 & x _ n & \\dots \\\\ \\dots & \\dots & \\dots & \\dots \\end{pmatrix} . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} & F ( \\mu _ 1 , \\mu _ 2 ) : = \\ 1 ^ * ( A _ 1 ^ * - \\mu _ 1 I ) ^ { - 1 } ( A _ 2 ^ * - \\mu _ 2 I ) ^ { - 1 } , \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\kappa ( L ( I _ n - \\widetilde { V } _ k \\widetilde { V } _ k ^ T ) ) \\geq \\kappa ( L ( I _ n - \\widetilde { V } _ { k - 1 } \\widetilde { V } _ { k - 1 } ^ T ) ) , k = 1 , 2 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega W ) , \\pi _ { \\ast } ( X ) ) - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Z , \\omega \\phi W ) , \\pi _ { \\ast } ( X ) ) & = g _ { 1 } ( \\mathcal { T } _ { Z } \\omega W , \\mathcal { B } X ) + g _ { 1 } ( W , \\varphi Z ) \\eta ( X ) \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\mathcal { J } _ o = \\left \\{ \\begin{aligned} a \\varphi , & \\mathfrak { b } _ + \\\\ \\frac { 1 } { a } \\varphi , & \\mathfrak { b } _ - \\end{aligned} \\right . , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\left ( W , Z \\right ) : = \\left ( E '^ { - 1 } \\otimes W ' , V '^ { - 1 } \\otimes Z ' \\right ) , \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{gather*} \\rho _ { j } \\partial _ { s _ { m - d _ { j } } } u _ { j } \\left ( x , t , y ^ { j } , s ^ { m - d _ { j } } \\right ) \\\\ = \\left ( \\ ! \\int _ { S _ { m - d _ { j } + 1 } } \\ ! \\ ! \\ldots \\ ! \\int _ { S _ { m } } \\ ! \\int _ { Y _ { j + 1 } } \\ ! \\ ! \\ldots \\ ! \\int _ { Y _ { n } } \\ ! \\ ! \\ ! a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) d y _ { n } \\cdots d y _ { j + 1 } d s _ { m } \\cdots d s _ { m - d _ { j } + 1 } \\ ! \\right ) \\ ! \\nabla _ { y _ { j } } \\end{gather*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } & = \\left ( \\frac { e ^ t - 1 } { 2 } + 1 \\right ) ^ { - 1 } = \\sum _ { l = 0 } ^ \\infty \\left ( \\frac { e ^ t - 1 } { 2 } \\right ) ^ l ( - 1 ) ^ l \\\\ & = \\sum _ { l = 0 } ^ \\infty ( - 1 ) ^ l 2 ^ { - l } l ! \\sum _ { n = l } ^ \\infty S _ 2 ( n , l ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { l = 0 } ^ n S _ 2 ( n , l ) ( - 1 ) ^ l 2 ^ { - l } l ! \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} K _ d ( x ) = \\frac { 1 } { 1 - x } K _ { d + 1 } ( x ) + \\frac { 1 } { 1 - x } H _ d ( x ) - \\frac { 1 } { 1 - x } H _ { d + 1 } ( x ) \\ , . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} \\frac { \\tilde { A } \\left ( W ' , \\overline { W } ' \\right ) \\otimes W ' - \\left ( \\overline { \\tilde { A } \\left ( W ' , \\overline { W } ' \\right ) \\otimes W ' } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } , \\quad \\quad \\mbox { f o r a l l $ i , j = 1 , 2 $ , } \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\begin{pmatrix} 0 & 1 & 0 & 1 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} \\frac { d + 2 } { d - 2 } q _ d ' = q _ d , \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} p _ { 2 n } ( z ) = z ^ { 2 n } , p _ { 2 n + 1 } ( z ) = z ^ { 2 n + 1 } - \\left ( \\prod _ { i = 1 } ^ m \\frac { 2 n } { L _ i + 2 n } \\right ) z ^ { 2 n - 1 } . \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\# - \\# = 3 . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\sum _ { i , j } c _ { i j } \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , ( \\partial _ { \\mu _ n ' } V + \\partial _ { \\mu _ n ' } \\phi ) & = 0 \\\\ \\sum _ { i , j } c _ { i j } \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , ( D _ { \\zeta _ { n } ' } V \\cdot e _ l + D _ { \\zeta _ { n } ' } \\phi \\cdot e _ l ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} u ( t ) \\leq u ( T ) + \\int _ { t } ^ { T } \\alpha ( s ) d s < - C _ { 1 } + C _ { 2 } = : - C _ { 3 } , \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\Delta ^ k 0 ^ n = \\begin{cases} S _ 2 ( n , k ) & \\ , \\ , n \\geq k \\\\ 0 & \\ , \\ , n < k . \\end{cases} \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} S _ i = \\frac { 1 } { t _ i } , \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\aligned & \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak H _ { a b } ^ { i + 1 i } - \\frak H _ { a b } ^ { i + 1 i } \\circ \\hat d _ { 1 } ^ { i } \\\\ & = \\frak h _ b ^ { i + 1 i } - \\frak h _ a ^ { i + 1 i } + \\frak h _ { a b } ^ { i + 1 } \\circ \\psi _ 1 ^ { i + 1 i } - \\psi _ 2 ^ { i + 1 i } \\circ \\frak h _ { a b } ^ { i } \\endaligned \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} ( u \\star _ i v ) \\star _ j ( u ' \\star _ i v ' ) = ( u \\star _ j u ' ) \\star _ i ( v \\star _ j v ' ) . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\mathfrak { E } _ h ^ { \\frac { 1 } { 2 } } ( s ) : = \\mathfrak { E } _ { h , \\infty } ^ { \\frac { 1 } { 2 } } ( s ) + \\mathfrak { E } _ { h , 0 } ^ { \\frac { 1 } { 2 } } ( s ) . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} x y \\cdot J _ r x = y , \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } \\big ( ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 \\big ) ^ k = \\sum _ { n = k } ^ \\infty S _ { 2 , \\lambda } ( n , k | x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\aligned & \\widehat { \\mathcal U } ( { \\rm m o r } ; 3 , 1 ; \\alpha _ 1 , \\alpha _ 3 ) \\\\ & = \\bigcup _ { \\alpha _ 2 \\in \\frak A _ 2 } \\widehat { \\mathcal U } ( { \\rm m o r } ; 2 , 1 ; \\alpha _ 1 , \\alpha _ 2 ) \\times _ { R _ { \\alpha _ 2 } } ^ { \\boxplus \\tau } \\widehat { \\mathcal U } ( { \\rm m o r } ; 3 , 2 ; \\alpha _ 2 , \\alpha _ 3 ) . \\endaligned \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ n \\left | \\| \\mathbf v _ 1 - \\mathbf v _ j \\| ^ 2 - 1 \\right | \\geq \\lambda \\geq 3 d ^ { \\beta - 2 / 3 } . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} x _ 1 ^ 2 + 2 x _ 2 ^ 2 + 2 x _ 3 ^ 2 + x _ 4 ^ 2 + 2 x _ 5 ^ 2 + x _ 6 ^ 2 = 0 \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} Y _ { m + 1 } = & \\iint _ { Q _ { m + 1 } } ( v - k _ { m + 1 } ) _ + ^ p \\ , d v _ 0 \\ , d t \\\\ \\le & \\iint _ { Q _ m } ( v - k _ { m + 1 } ) _ + ^ p \\phi _ m ^ 2 \\ , d v _ 0 \\ , d t \\\\ \\le & \\left ( \\iint _ { Q _ m } [ ( v - k _ m ) _ + ^ p \\phi _ m ^ 2 ] ^ { \\frac { n + 2 } { n } } \\ , d v _ 0 \\ , d t \\right ) ^ { \\frac { n } { n + 2 } } | E _ m | ^ { \\frac { 2 } { n + 2 } } \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} p _ i ( z ) = c _ i z ^ { k _ i } \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{gather*} \\Delta _ g ( \\cdot ) = \\frac { 1 } { \\sqrt { g } } \\partial _ i ( \\sqrt { g } g ^ { i j } \\partial _ j ( \\cdot ) ) , \\end{gather*}"}
-{"id": "622.png", "formula": "\\begin{align*} ( g - \\log | s | ) ( \\xi ) = h ^ { \\mathrm { a n } } _ { ( D , g ) + \\widehat { ( s ) } } ( \\xi ) = h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) \\leqslant 0 , \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\phi _ n ( g ) \\phi _ n ( h ) = \\int _ { H _ n } \\phi _ n ( g k h ) \\ > d \\omega _ { H _ n } ( k ) \\quad ( g , h \\in G _ n ) \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} A ^ + ( x ; 1 ) & = x H ( x ) + \\frac { 1 - x } { 1 - 2 x } B ' ( x ; 1 ) - \\frac { x ( 1 - x ) ^ 2 } { 1 - 2 x } - \\frac { x ^ 4 } { ( 1 - 2 x ) ^ 2 } \\\\ & + \\frac { 1 - 3 x + x ^ 2 } { 1 - 2 x } C ( x ) - 1 . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\tilde { \\chi } _ { \\overrightarrow { \\omega } , t } : x \\in S ^ 1 \\mapsto \\prod _ { i = 0 } ^ { m - 1 } \\tilde { \\chi } _ { \\omega _ i } \\ ( T _ t ^ i \\ ( x \\ ) \\ ) \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } Q & = 1 \\\\ \\lim _ { s \\rightarrow 0 } Q ' & = 0 \\\\ \\lim _ { s \\rightarrow 0 } Q '' & = a ''' ( 0 ) - b ''' ( 0 ) \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ ( 2 \\pi r / \\sqrt \\lambda ) ^ { - d / 2 } J _ { d / 2 } ( r \\sqrt \\lambda ) \\ , , \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\mathrm { H o m } ( \\mathrm { C o k e r } \\ , \\phi , \\mathbb F _ p ) & \\simeq \\mathrm { K e r } \\ , \\phi ^ { \\vee } \\simeq \\mathrm { K e r } \\ , \\xi ' \\simeq \\mathrm { K e r } \\ , \\xi \\\\ & \\simeq ( \\widetilde { \\mathcal { U } } _ S \\cap \\overline { k ^ { \\times } } \\mathcal { J } _ k ^ p ) \\mathcal { J } _ k ^ p / \\mathcal { J } _ k ^ p = \\widetilde { V } _ S \\mathcal { J } _ k ^ p / \\mathcal { J } _ k ^ p \\simeq \\widetilde { V } _ S / ( \\overline { k ^ { \\times } } ) ^ p . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} F _ \\lambda ( \\mu , r ) = k a _ 0 + 2 a _ 1 \\mu f _ \\lambda ( r ) + k a _ 2 \\lambda \\mu ^ 2 - a _ 3 \\mu ^ 2 f _ \\lambda ( r ) ^ 2 \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } X ( t _ { n , i } , M , Y _ 0 ^ { ( n ) , t _ { n , i } } ) d s + B ( t _ { n , i + 1 } ) - B ( t _ { n , i } ) . \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\hat { 1 _ S } ( \\chi ' ) ^ 3 \\chi ' ( f ) = \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) \\psi ( 3 \\Sigma G + \\Sigma f ) = \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} P ( F + t G ) - P ( F ) & \\le \\big ( h _ { \\mu _ { t } } ( g ) + \\int ( F + t G ) d \\mu _ { t } + t ^ 2 \\big ) - \\big ( h _ { \\mu _ { t } } ( g ) + \\int F d \\mu _ { t } \\big ) \\\\ & = t \\int G d \\mu _ { t } + t ^ 2 . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} E _ P [ \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ { t _ i } [ \\xi _ i ] I _ { A _ i } ] \\leq \\hat { \\mathbb { E } } [ \\sum _ { i = 1 } ^ n \\xi _ i I _ { A _ i } ] . \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\Sigma } \\frac { s ( v \\sigma ) } { \\sum _ { \\rho \\in \\Sigma } s ( v \\rho ) } = \\frac { \\sum _ { \\sigma \\in \\Sigma } s ( v \\sigma ) } { \\sum _ { \\rho \\in \\Sigma } s ( v \\rho ) } = 1 , \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} S _ 2 ( H ) = \\sum _ { i = 1 } ^ { d _ c - 1 } \\sum _ { j = i + 1 } ^ { d _ c } S _ 2 ( \\{ h _ i , h _ j \\} ) , \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} C _ d ( x ) = P _ d \\left ( x C _ d ( x ) ^ d \\right ) \\ , \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} w _ { 1 1 } = \\Re w _ { 1 1 } + \\sqrt { - 1 } \\left < Z _ { 1 } , Z _ { 1 } \\right > , w _ { 2 2 } = \\Re w _ { 1 1 } + \\sqrt { - 1 } \\left < Z _ { 2 } , Z _ { 2 } \\right > , w _ { 1 2 } = w _ { 2 1 } - \\sqrt { - 1 } \\Re w _ { 1 1 } + \\sqrt { - 1 } \\left < Z _ { 1 } , Z _ { 2 } \\right > . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\kappa ( s _ i , s _ i ^ { - 1 } ( g ) ) \\ , \\kappa ( s _ i ^ { - 1 } , g ) = 1 = \\kappa ( s _ i ^ { - 1 } , s _ i ( g ) ) \\ , \\kappa ( s _ i , g ) . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{gather*} \\lim _ { R \\rightarrow \\infty } \\left ( \\frac { x \\cdot \\xi } { \\mu R } \\right ) ^ { \\sigma - 2 } = 0 . \\end{gather*}"}
-{"id": "3487.png", "formula": "\\begin{align*} h _ { m + 1 } ( x _ i , x _ { i + 1 } , \\dots , x _ n ) - x _ i \\cdot h _ m ( x _ i , x _ { i + 1 } , \\dots , x _ n ) = h _ { m + 1 } ( x _ { i + 1 } , \\dots , x _ n ) \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} \\sum _ { \\left ( \\Pi , \\Delta \\right ) } \\mathbb { P } \\left ( \\Pi , \\Delta \\right ) = 1 . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} e _ l & \\equiv e _ l ( z _ 1 , z _ 2 , \\ldots , z _ n ) = \\sum _ { 1 \\le i _ 1 < i _ 2 < \\cdots < i _ l \\le n } z _ { i _ 1 } z _ { i _ 2 } \\ldots z _ { i _ l } , l = 1 , 2 , \\ldots , n , \\\\ e _ 0 & \\equiv 1 . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} E _ { m - 1 } ^ { ( \\textrm { r } ) } \\cdots E _ 1 ^ { ( \\textrm { r } ) } E _ a A E _ 1 ^ { ( \\textrm { c } ) } \\cdots E _ { m - 1 } ^ { ( \\textrm { c } ) } = \\left [ \\begin{array} { c | c } 1 & 0 \\\\ \\hline 0 & A _ { m - 1 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} \\hat d ^ j = d _ 0 \\oplus \\bigoplus _ { \\alpha _ 1 , \\alpha _ 2 \\atop E ( { \\alpha _ 2 } ) - E ( { \\alpha _ 1 } ) \\le E _ { k _ 1 } } \\frak m ^ { j } _ { 1 ; \\alpha _ 2 , \\alpha _ 1 } , \\widehat \\psi = \\bigoplus _ { \\alpha _ 1 , \\alpha _ 2 \\atop E ( { \\alpha _ 2 } ) - E ( { \\alpha _ 1 } ) \\le E _ { k _ 1 } } \\psi ^ { } _ { \\alpha _ 2 , \\alpha _ 1 } . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} P _ { j k } = M _ { j k } \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} g _ { 1 } ( \\omega V , \\mathcal { T } _ { U } \\mathcal { B } X ) + g _ { 1 } ( V , \\phi U ) \\eta ( X ) & = g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\mathcal { C } X ) , \\pi _ { \\ast } \\omega V ) - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , X ) , \\pi _ { \\ast } \\omega \\phi V ) \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} a _ { 1 } = K ^ { 1 } \\kappa _ { 1 } ^ { 1 } \\beta _ { 1 } ^ { 1 } , \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\nu ( E ) : = \\sum _ { y = 1 } ^ d \\ell ( E \\cap U _ y ) , E \\in \\mathcal { B } ( U ) , \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} x _ { g h , \\sigma _ { g h } ( i ) } : = \\frac { ( g h ) ( \\sqrt { \\alpha _ i } ) } { \\sqrt { \\alpha _ { \\sigma _ { g h } ( i ) } } } = x _ { h , \\sigma _ h ( i ) } x _ { g , \\sigma _ { g h } ( i ) } . \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} A ^ - ( x ) - x ^ 2 = \\ & x A ^ - ( x ) - 2 x ^ 2 \\big ( A ( x ) - 1 \\big ) + \\frac { x } { 1 - x } \\big ( A ( x ) - 1 - 2 x \\big ) + x \\big ( C ( x ) - 1 - x \\big ) \\\\ & - x ^ 2 \\big ( C ( x ) - 1 \\big ) - \\frac { x ^ 3 ( 1 - 3 x ) } { ( 1 - x ) ^ 2 ( 1 - 2 x ) } \\ , , \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} N _ { 2 d } ^ { } ( X , G ) \\leq \\sum _ { n = 1 } ^ { \\infty } a ( n ) \\phi _ Y \\left ( \\frac { n } { X } \\right ) = r _ d ( G ) X + O ( X Y ^ { - 1 } ) + O ( X ^ { \\alpha ' } Y ^ { d \\delta ^ { \\prime } ( 1 - \\alpha ' ) + \\epsilon } ) . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} E ( t ) = \\lambda _ 1 ( H _ { \\Lambda } ^ { N } ( t ) ) \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\theta ^ { 0 } \\left ( t = 0 \\right ) = \\theta ^ { 0 , 0 } \\quad \\mbox { i n } \\ ; \\overline { \\Omega } , \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} \\begin{cases} u \\in K _ t \\ , , \\\\ \\noalign { \\medskip } \\langle H _ t ( u ) , v - u \\rangle \\geq \\langle \\varphi , v - u \\rangle \\qquad \\forall v \\in K _ t \\ , , \\end{cases} \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} D _ { i j } ( a ^ { i j } \\tilde v W ) = 0 \\ ; \\ ; \\Omega , \\tilde v = \\frac { \\psi } { W } \\ ; \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} G & = \\{ \\widetilde { g } = \\begin{pmatrix} g & 0 \\\\ b & 1 \\end{pmatrix} \\in G L ( 3 , \\bold Z [ \\rho ] ) \\mid \\ g \\in \\Gamma \\} , \\\\ G ( 1 - \\rho ) & = \\{ \\widetilde { g } \\in G \\mid \\ \\widetilde { g } \\equiv i d ( 1 - \\rho ) \\} . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | \\leq \\binom { n + k - 1 } { k - 1 } ^ { 1 / 2 } e ^ { - H n / 2 } \\frac { n ! } { n ^ n } , \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\Lambda _ p ^ p ( \\Omega ) = \\min \\left \\{ \\frac { \\int _ \\Omega F ^ p ( \\nabla u ) \\ d x } { \\int _ \\Omega | u | ^ p \\ d x } : \\ u \\in W ^ { 1 , p } ( \\Omega ) , \\int _ \\Omega u | u | ^ { p - 2 } \\ d x = 0 \\right \\} . \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\delta _ * ( \\L ) = \\liminf _ { r \\to \\infty } \\frac { 1 } { r ^ 2 } \\min _ { ( x , p ) \\in \\R ^ 2 } \\# \\{ z \\in \\L \\ , \\colon z \\in ( x , p ) + [ 0 , r ] ^ 2 \\} > ( 2 \\pi \\hbar ) ^ { - 1 } . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ h \\lambda _ i ( n ) \\left ( \\sum _ { \\ell = 1 } ^ { \\kappa _ i } e \\ ! \\left ( n \\tfrac { a _ { i \\ell } } { q m } \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\rho _ { \\frac { 3 } { 2 } - m } ( s ) = & \\frac { ( - 1 ) ^ { m - 1 } ( 4 \\pi ) ^ k \\zeta ( 1 - m ) \\zeta ( 2 s + m - 2 ) \\Gamma ( 2 s + m - 2 ) } { \\Gamma ( m ) \\Gamma ( s ) \\Gamma ( s + k - 1 ) \\zeta ^ * ( 4 - 2 s - 2 m ) } \\times \\\\ & \\quad \\langle \\mathcal { V } _ { f , \\overline { g } } , \\overline { E ( \\cdot , s + m - 1 ) } \\rangle . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\Phi _ { t } ( x ) = \\Phi _ { 0 } ( x ) + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } h \\Bigg ( x , \\frac { 2 } { i } \\partial _ { x } \\Phi _ { s } ( x ) \\Bigg ) d s , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ { c d } } ( \\frac { \\beta n \\eta _ L \\Gamma r _ 0 ^ { \\alpha _ { L , d } } } { P _ { d } G _ 0 } ) = \\exp \\bigg ( - 2 \\pi \\lambda _ B \\bigg ( \\sum _ { j \\in \\{ L , N \\} } \\sum _ { i = 1 } ^ 3 p _ { G _ i } \\times \\\\ & \\bigg ( \\int _ 0 ^ { \\infty } \\bigg ( 1 - 1 / \\bigg ( 1 + \\frac { \\beta n \\eta _ s \\Gamma r _ 0 ^ { \\alpha _ { s , d } } P _ d G _ i } { P _ d G _ 0 N _ j t ^ { \\alpha _ { j , d } } } \\bigg ) ^ { N _ j } \\bigg ) p _ { j , d } ( t ) t d t \\bigg ) \\bigg ) \\bigg ) \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} a [ u ] ( x , t ) = \\frac { 1 } { 4 \\pi } \\int _ { \\R ^ 3 } \\frac { u ( y , t ) } { | x - y | } d y x \\in \\R ^ 3 , ~ ~ t > 0 . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} E = \\left \\{ r = p , s = 0 , \\dot { r } = - \\frac { \\partial H } { \\partial x } ( x , p ) , \\dot { s } = \\dot { x } - \\frac { \\partial H } { \\partial p } ( x , p ) \\right \\} \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\tilde \\rho ( t ) = \\exp ( t D ) = \\begin{pmatrix} \\cos \\frac { t } { \\sigma } & \\sin \\frac { t } { \\sigma } & 0 \\\\ - \\sin \\frac { t } { \\sigma } & \\cos \\frac { t } { \\sigma } & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi \\notin \\mathfrak { M } _ m } | \\hat { 1 _ S } ( \\chi ) | ^ 3 = O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ n } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} \\kappa _ { 0 } \\left ( u , \\xi \\right ) = \\sup _ { t \\in \\mathcal { L } _ { j } } \\left \\{ \\frac { 1 } { \\left \\vert { 1 + \\phi \\left ( t \\right ) / \\left ( { 2 u } \\right ) } \\right \\vert } \\right \\} , \\ \\kappa _ { 2 } \\left ( \\xi \\right ) = \\sup _ { t \\in \\mathcal { L } _ { j } } \\left \\vert { \\phi \\left ( t \\right ) } \\right \\vert . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} Z _ 2 ( t ) : = \\int _ 0 ^ t \\int _ U \\int _ A | f ( s , X ( s ^ - ) , u , a ) | \\left | \\N _ { \\rho ^ { \\widehat { \\gamma } , X } } - \\N _ { \\rho ^ { \\widehat { \\gamma } , Y } } \\right | ( d s , d u , d a ) , \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} M ( y ^ { n } ) = \\{ m \\in \\mathcal { M } : W ^ { n } ( y ^ { n } | c _ { o u } ^ { ( n ) } ( m ) ) > 0 \\} . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\varphi _ { k l } ^ { ( 2 ) } \\left ( Z , W \\right ) = 0 , \\mbox { f o r a l l $ k = 1 , \\dots , q $ a n d $ l = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} v _ t + ( - \\Delta ) ^ { s } _ x v = \\frac 1 { 2 s } \\nabla _ x \\cdot ( x \\ , v ) \\ , . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} [ \\prod _ { i = 3 } ^ k r _ q ^ { ( i ) } A ^ { - 1 } , a ] \\ \\cdot \\ [ \\prod _ { i = 3 } ^ k s _ q ^ { ( i ) } B ^ { - 1 } , b ] \\in \\gamma _ { k + 3 } ( F ) . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} c ( q , 8 ) \\le \\begin{cases} 2 ( q ^ 3 - 2 q ) , & q \\\\ 2 ( q ^ 3 - 3 q - 2 ) , & q . \\end{cases} \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} K _ { 2 , r } & \\geq \\frac { \\left ( \\sum _ { i , j = 1 } ^ { N } \\left \\vert \\left ( y _ { i j } \\right ) \\right \\vert ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } } { \\left ( { \\displaystyle \\int \\limits _ { I ^ { 2 } } } \\left \\vert \\sum _ { i , j = 1 } ^ { N } r _ { i } \\left ( t \\right ) r _ { j } \\left ( s \\right ) y _ { i j } \\right \\vert ^ { r } d t d s \\right ) ^ { \\frac { 1 } { r } } } \\\\ & = \\frac { 2 } { 2 ^ { ^ { \\frac { 2 r - 2 } { r } } } } = 2 ^ { \\frac { 2 - r } { r } } . \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\lambda ^ * \\Omega _ X = & ( 1 - x _ 1 ) ^ { - \\frac { 1 } { 3 } } ( 1 - x _ 2 ) ^ { - \\frac { 1 } { 3 } } w ' d r \\wedge \\psi _ 1 \\\\ = & ( 1 - x _ 1 ) ^ { - \\frac { 1 } { 3 } } ( 1 - x _ 2 ) ^ { - \\frac { 1 } { 3 } } r ^ { - \\frac { 2 } { 3 } } ( 1 - r ) ^ { - \\frac { 2 } { 3 } } d r \\\\ & \\wedge u ^ { - \\frac { 2 } { 3 } } ( 1 - u ) ^ { - \\frac { 1 } { 3 } } ( 1 - t u ) ^ { - \\frac { 1 } { 3 } } d u . \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} \\prod _ { \\substack { p \\\\ p \\neq 2 \\\\ p \\nmid h } } \\Big ( 1 + \\frac { \\big ( \\frac { h ( - 1 ) ^ { k - \\frac { 1 } { 2 } } } { p } \\big ) } { p ^ { 2 w - \\frac { 1 } { 2 } } } \\Big ) = \\frac { L ^ { ( 2 ) } ( 2 w - \\frac { 1 } { 2 } , \\chi _ { k , h } ) } { \\zeta ^ { ( 2 h ) } ( 4 w - 1 ) } . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( x ) = \\Phi ( x ) + \\frac { 1 } { C } d ^ { 2 } ( x ) , \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\geq 1 / \\| P \\| _ { Z \\to Z } \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| P f _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in Z , n \\in \\N \\} \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\Lambda _ 0 : = \\sup _ { B _ { g ( 0 ) } \\left ( x _ 0 , \\frac { 2 \\rho } { \\sqrt { K } } \\right ) } | R m | ( x , 0 ) < \\infty . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\theta ^ * = 1 + 3 q ^ 3 z + 6 q ^ 7 z ^ 2 + 1 0 q ^ { 1 2 } z ^ 3 + 1 5 q ^ { 1 8 } z ^ 4 + 2 1 q ^ { 2 5 } z ^ 5 + 2 8 q ^ { 3 3 } z ^ 6 + \\cdots ~ . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} \\eta _ { n , 1 } \\left ( { \\nu , z } \\right ) = e ^ { - \\nu \\xi } \\varepsilon _ { n , 1 } \\left ( { \\nu , \\xi } \\right ) . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} H _ 1 ^ 1 & = Y ^ 3 & H _ 1 ^ 2 & = Z ^ 3 \\\\ H _ 2 ^ 1 & = X H _ 1 ^ 1 & H _ 2 ^ 2 & = X H _ 1 ^ 2 \\\\ H _ 3 ^ 1 & = X ^ 2 H _ 1 ^ 1 & H _ 3 ^ 2 & = X ^ 2 H _ 1 ^ 2 \\\\ H _ 4 ^ 1 & = X ^ 3 H _ 1 ^ 1 & H _ 4 ^ 2 & = X ^ 3 H _ 1 ^ 2 \\\\ H _ 5 ^ 1 & = X ^ 4 H _ 1 ^ 1 & H _ 5 ^ 2 & = X ^ 4 H _ 1 ^ 2 . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} { \\cal L } _ \\omega = { \\cal V } _ n { \\cal L } _ n { \\cal V } _ n ^ { - 1 } , { \\cal V } _ n : = { \\cal P } _ \\bot { \\cal U } _ n . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\big \\langle x _ 1 , y _ 1 , \\dots , x _ \\ell , y _ \\ell ; r \\big \\rangle , r = \\left [ x _ 1 , y _ 1 \\right ] \\cdot \\ldots \\cdot \\left [ x _ \\ell , y _ \\ell \\right ] \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} \\left ( \\omega _ { 1 } ^ { * } , \\omega _ { 2 } ^ { * } \\right ) = \\underset { l , m } { a r g m i n } \\left ( P \\left ( \\omega _ { 1 } ^ { l } , \\omega _ { 2 } ^ { m } \\right ) \\right ) . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} E ( \\max _ { 2 \\le \\ell \\le p _ n } ( y _ { n \\ell } ) ^ 2 ) \\le \\sqrt { E ( \\max _ { 2 \\le \\ell \\le p _ n } y _ { n \\ell } ^ 4 ) } \\le \\sqrt { \\sum ^ { p _ n } _ { \\ell = 2 } E ( y _ { n \\ell } ^ 4 ) } \\to 0 . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} F ^ * ( \\alpha ) : = \\sup _ { v \\in T _ x M , F ( v ) \\leq 1 } \\alpha ( v ) = \\sup _ { v \\in T _ x M , F ( v ) = 1 } \\alpha ( v ) . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} N _ d ( x ) = \\frac { x ^ { d + 4 } ( 1 - x ) ( 1 - 2 x ) } { ( 1 - 3 x + x ^ 2 ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { L } } _ 1 = \\mathcal { L } _ 1 | _ { Y _ S } , \\widetilde { \\mathcal { L } } _ 2 = \\mathcal { L } _ 2 | _ { Y _ S } \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} f ^ * ( D , g _ 0 ) = d ( D , g _ 0 ) + \\widehat { ( \\varphi ) } + ( 0 , \\lambda ) . \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\tau _ { i j } ^ { \\Delta r } ( x ) : = \\int _ { i \\Delta } ^ { ( i + 1 ) \\Delta } [ \\widehat { \\gamma } ( s , x ) ] ( C _ j ^ r ) d s . \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | ^ 2 \\leq O ( m ^ { 1 / 4 } ) e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } ) } \\binom { n } { m } ^ { 1 / 2 } \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 2 . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{gather*} 1 = \\sum _ { \\mu \\colon \\mu \\prec \\lambda } { \\Lambda ^ { N + 1 } _ N ( \\lambda , \\mu ) } . \\end{gather*}"}
-{"id": "789.png", "formula": "\\begin{align*} ( A ( y , s ) \\left [ \\nabla u ( x , t ) + \\nabla _ { y } u _ { 1 } ( x , t , y , s ) \\right ] ) \\cdot n = 0 ( \\partial Y ^ { \\ast } - \\partial Y ) \\times ( 0 , 1 ) \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\phi _ { 1 } ( t _ { n } u _ { n } - t _ { 0 } u _ { 0 } ) - \\phi _ { 1 } ( t _ { n } w _ { n } ) = \\phi _ { 1 } ' ( \\zeta _ { n } ) \\hat { t } _ { n } u _ { 0 } , \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} T _ e & = \\begin{cases} e & \\ e \\neq e _ 0 , \\\\ e _ { n + 1 } e _ { n } \\cdots e _ 1 & \\ e = e _ 0 . \\end{cases} \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} & x _ { n + 1 } = G _ r ( x _ n ) : = \\\\ & \\left \\{ \\begin{array} { l l } \\omega _ 1 f _ 1 \\circ F ( r , x _ n ) + \\alpha _ 1 g _ 1 ( r x _ n ) + \\xi _ 1 \\frac { ( \\beta _ 1 - r ) x _ n } { 2 } ~ m o d ~ 1 , & w h e n ~ x _ n < 0 . 5 , \\\\ \\\\ \\omega _ 2 f _ 2 \\circ F ( r , x _ n ) + \\alpha _ 2 g _ 2 ( r x _ n ) + \\xi _ 2 \\frac { ( \\beta _ 2 - r ) ( 1 - x _ n ) } { 2 } ~ m o d ~ 1 , & w h e n ~ x _ n \\geq 0 . 5 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} 0 & = \\int _ \\Omega \\sum _ { i = 1 } ^ { m } \\sum _ { \\ell = 1 } ^ Q \\left ( \\langle D _ i u _ \\ell , \\nabla \\hat { f } ( u _ \\ell ) \\rangle D _ i \\varphi + \\langle D _ i u _ \\ell , D \\nabla \\hat { f } \\cdot D _ i u _ \\ell \\rangle \\varphi \\right ) \\\\ & = \\int _ \\Omega \\sum _ { i = 1 } ^ { m } \\left ( D _ i ( f \\circ u ) D _ i \\varphi + \\sum _ { \\ell = 1 } ^ Q \\nabla ^ 2 f ( u _ \\ell ) ( D _ i u _ \\ell , D _ i u _ \\ell ) \\varphi \\right ) . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\widehat A ( v ) = A ^ { \\rm f l a t } ( v ) - \\delta v ^ 2 N ^ { 1 / 3 } \\mathbf { e } _ \\rho , \\mathbf { e } _ \\rho = ( ( 1 - \\rho ) ^ 2 , \\rho ^ 2 ) , \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\epsilon ^ { * } = \\inf _ { { \\theta \\in \\Theta } } d ( b _ { 0 } , b ( \\theta ) ) > 0 \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\delta ( A ) = \\lim _ { d \\rightarrow \\infty } \\max _ { u \\in \\N _ 0 } \\left ( \\frac { | A \\cap [ u , u + d - 1 ] | } { d } \\right ) = 1 . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{gather*} \\psi ( \\beta , \\varphi _ 1 , \\ldots , \\varphi _ { n - 2 } , \\phi ) = V ( \\beta ) Y ^ { m } _ { l _ 1 , \\ldots , l _ { n - 2 } } ( \\varphi _ 1 , \\ldots , \\varphi _ { n - 2 } , \\phi ) , \\end{gather*}"}
-{"id": "4872.png", "formula": "\\begin{align*} \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } = \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\left ( \\frac { z \\mathtt { J } ' _ { \\nu } ( z ) } { \\mathtt { J } _ { \\nu } ( z ) } - ( \\nu - 1 ) ( 1 - a ) \\right ) . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\langle P _ h ^ k ( \\cdot , s ) , \\mu _ { j , \\ell } \\rangle = \\frac { \\overline { \\rho _ { j , \\ell } ( h ) } } { ( 4 \\pi h ) } ^ { s - 1 } \\frac { \\Gamma ( s + \\frac { \\ell } { 2 } - 1 ) \\Gamma ( s - \\frac { \\ell } { 2 } ) } { \\Gamma ( s - \\frac { k } { 2 } ) } . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} A \\left ( \\widetilde { u } \\right ) = f \\left ( x _ { 1 } - x _ { 0 } \\right ) \\chi \\left ( \\overline { x } - \\overline { x } ^ { 0 } \\right ) , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\left ( \\overline W ^ T \\right ) ^ { - 1 } = \\begin{pmatrix} a _ { 1 , 1 } & a _ { 1 , 2 } & a _ { 1 , 3 } & \\ldots & a _ { 1 , T } \\\\ 0 & a _ { 2 , 2 } & a _ { 2 , 3 } & \\ldots & . . \\\\ \\cdot & \\cdot & \\cdot & a _ { T - 1 , T - 1 } & a _ { T - 1 , T } \\\\ 0 & \\ldots & \\ldots & 0 & a _ { T , T } \\end{pmatrix} , \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} f _ { n , q } = ( 1 - q ) ^ n \\ , Z _ { n , q } ^ \\dagger , \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} X ( 0 ) = ( 0 , 1 0 0 0 , 0 ) , \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\Omega ( { \\mathbf Z } , x ) = \\overline { \\big \\{ S ^ n ( x , x , \\ldots , x ) : n \\in \\Z \\big \\} } \\subset ( G / \\Gamma ) ^ { k } . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\norm { p } = \\norm * { \\frac { \\rho } { \\norm { P _ { K } x } } P _ { K } x } = \\rho , \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\frac { \\tilde { A } \\left ( W ' , Z ' , \\overline { W } ' , \\overline { Z } ' \\right ) W ' - \\left ( \\overline { \\tilde { A } \\left ( W ' , Z ' , \\overline { W } ' , \\overline { Z } ' \\right ) \\otimes W ' } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } , \\quad \\quad \\mbox { f o r a l l $ i , j = 1 , 2 $ , } \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\| A ' \\| _ { W ^ { 1 , p _ { 0 } } ( \\Omega ) } \\leq C \\| d \\alpha _ { A _ { 1 } } - d \\alpha _ { A _ { 2 } } \\| _ { L ^ { p _ { 0 } } ( \\Omega ) } = C \\| d \\alpha _ { A ' _ { 1 } } - d \\alpha _ { A ' _ { 2 } } \\| _ { L ^ { p _ { 0 } } ( \\Omega ) } . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\limsup _ { s \\rightarrow \\pm \\infty } \\frac { | f _ { i } ( s ) | } { e ^ { \\alpha s ^ { 2 } } - 1 } = \\left \\{ \\begin{array} { c l l } 0 & \\mbox { i f } & \\alpha > \\alpha ^ { i } _ { 0 } , \\\\ + \\infty & \\mbox { i f } & \\alpha < \\alpha ^ { i } _ { 0 } . \\end{array} \\right . \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\mathrm { d i v } \\left [ A ( x , u ) \\nabla u \\right ] + B ( x , u ) | \\nabla u | ^ 2 = g ( x , u ) & \\qquad \\ , , \\\\ \\noalign { \\medskip } u = 0 & \\qquad \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\left ( \\frac { b - a } { a + b \\omega } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) \\left ( \\frac { a - b } { a + b \\omega } \\right ) = \\left ( \\frac { - 1 } { a + b \\omega } \\right ) ( - 1 ) ^ { ( ( N ( a - b ) - 1 ) / 2 ) ( ( N ( a + b \\omega ) - 1 ) / 2 ) } \\left ( \\frac { a + b \\omega } { a - b } \\right ) . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\Lambda _ j ( s ) = \\pi ^ { - s } \\Gamma \\left ( \\tfrac { s + i t _ j } { 2 } \\right ) \\Gamma \\left ( \\tfrac { s - i t _ j } { 2 } \\right ) L ( s , \\mu _ j ) = \\Lambda _ j ( 1 - s ) . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} & A ^ { i j } = \\overline { A ^ { j i } } ^ { t } , \\quad \\mbox { f o r a l l $ i , j = 1 \\dots , q $ , } \\\\ & B ^ { i j } = \\overline { C ^ { j i } } ^ { t } , \\quad \\mbox { f o r a l l c o r r e s p o n d i n g $ i , j $ , } \\\\ & D ^ { i j } = \\overline { D ^ { j i } } ^ { t } , \\quad \\mbox { f o r a l l $ i , j = 1 \\dots , q ' - q $ . } \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\Lambda _ S ( f , g ) : = \\sum _ { j > N } \\iint & \\left | \\sum _ { n \\in S _ j } f _ { n , m , j } \\left ( x - \\frac { t ^ a + \\epsilon _ P ( t ) } { 2 ^ { a j } } \\right ) \\right | \\\\ & \\left | \\sum _ { n \\in S _ j } g _ { n , m , j } \\left ( x - \\frac { t ^ b + \\epsilon _ Q ( t ) } { 2 ^ { b j } } \\right ) \\right | | \\rho ( t ) | \\ , d t d x \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} P = \\left [ \\begin{array} { c c } 1 & \\mu _ m \\\\ 0 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ | \\varphi _ m ( X ^ x _ { \\tau + \\cdot } ) - \\varphi ( X ^ x _ { \\tau + \\cdot } ) | ] & \\leq \\hat { \\mathbb { E } } [ \\sup _ { 0 \\leq t \\leq T } | \\varphi _ m ( X ^ x _ { t + \\cdot } ) - \\varphi ( X ^ x _ { t + \\cdot } ) | ] \\\\ & = \\hat { \\mathbb { E } } ^ x _ 2 [ \\sup _ { 0 \\leq t \\leq T } | \\varphi _ m ( B ' _ { t + \\cdot } ) - \\varphi ( B ' _ { t + \\cdot } ) | ] . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} E \\bigl ( Z _ t ^ H ( a , b ) ^ 2 \\bigr ) = \\bigl ( a ^ 2 + b ^ 2 - \\bigl ( 2 ^ { 2 H } - 2 \\bigr ) a b \\bigr ) t ^ { 2 H } . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} f \\left ( z \\right ) = z ^ { m } \\sum \\limits _ { s = 0 } ^ { \\infty } { f } _ { s } { z } ^ { - s } , \\ g \\left ( z \\right ) = z ^ { p } \\sum \\limits _ { s = 0 } ^ { \\infty } { g } _ { s } { z } ^ { - s } , \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} S ( x , \\chi ) = \\sum _ { \\substack { N ( \\varpi ) \\leq x \\\\ \\varpi \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\chi ( \\varpi ) \\log N ( \\varpi ) \\ll \\min \\left \\{ x , \\sqrt { x } \\log ^ { 3 } x \\log N ( n ) \\right \\} . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} X ^ N _ i ( t ) = \\xi ^ N _ i + \\int _ 0 ^ t \\int _ U f ( s , X ^ N _ i ( s ^ - ) , u , \\alpha ^ N _ i ( s ) , \\mu ^ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\frac { 1 + z ^ { 2 } } { z ^ { 2 } } , \\ g \\left ( z \\right ) = - \\frac { 1 } { 4 z ^ { 2 } } . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\textrm { t r } ^ { \\flat } A _ { \\omega , \\omega , t } = \\sum _ { T _ t ^ m \\ ( x \\ ) = x } \\frac { \\chi _ { \\overrightarrow { \\omega } , t } \\ ( x \\ ) e ^ { g _ { m , t } \\ ( x \\ ) } \\theta _ \\omega \\ ( x \\ ) } { 1 - \\ ( T _ t ^ m \\ ) ' \\ ( x \\ ) ^ { - 1 } } . \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} \\sum _ { 2 \\le q _ i \\leq X } \\Lambda ( q _ i ) \\ll X = ( \\log \\log x ) ^ { 1 / 2 } ( \\log \\log \\log x ) ^ 2 \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} ( h ^ 0 _ 2 = \\alpha ^ { a _ 2 } ) = \\frac { \\gamma _ m ( d _ c - 2 , 2 ^ m - a _ 2 - 2 m ) } { \\gamma _ m ( d _ c - 1 , 2 ^ m - 2 m ) } . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} a _ { 2 } = K ^ { 2 } \\kappa _ { 1 } ^ { 2 } \\beta _ { 1 } ^ { 2 } , \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\S ( f ) = \\exp \\ ( - \\frac 1 2 e ^ { - H _ 2 ( f ) } \\ ) . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} f \\biggr ( \\frac { F _ { k _ 0 + 2 } } { F _ { k _ 0 + 1 } } , \\dots , \\frac { F _ { k _ 0 + n + 1 } } { F _ { k _ 0 + n } } \\biggr ) = \\biggr ( \\frac { F _ { k _ 0 + 3 } } { F _ { k _ 0 + 2 } } , \\dots , \\frac { F _ { k _ 0 + n + 2 } } { F _ { k _ 0 + n + 1 } } \\biggr ) . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} \\epsilon = 0 . 0 0 0 \\thinspace 8 7 2 \\thinspace 0 7 3 \\thinspace 2 , \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} D _ { \\zeta ' } E & = O ( \\varepsilon ^ { 2 - \\sigma } ) w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 4 + O ( w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 3 \\varepsilon ^ 2 ) + O ( \\varepsilon ^ 5 ) , \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{gather*} C H _ { a l g } ^ d ( X ) : = \\{ Z \\in C H ^ d ( X ) : Z \\sim _ { a l g } 0 \\} , \\\\ C H _ { h o m } ^ d ( X ) : = \\{ Z \\in C H ^ d ( X ) : Z \\sim _ { h o m } 0 \\} , \\\\ { \\rm N S } _ { a l g } ^ d ( X ) : = C H ^ d ( X ) / C H _ { a l g } ^ d ( X ) , \\ ; { \\rm N S } _ { h o m } ^ d ( X ) : = C H ^ d ( X ) / C H _ { h o m } ^ d ( X ) , \\\\ { \\rm G r i f f } ^ d ( X ) : = C H _ { h o m } ^ d ( X ) / C H _ { a l g } ^ d ( X ) . \\end{gather*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\| K _ { \\boldsymbol { W } } ( g ) - g \\| _ p \\leq C \\| g \\| _ { \\mathcal { B } ^ { \\boldsymbol { \\alpha } } _ { p , q } } \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l W _ l } . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\lim _ { k _ 0 \\to \\infty } \\frac { F _ { k _ 0 + i + 1 } } { F _ { k _ 0 + n + 1 } } = \\Psi ^ { - n + i } \\ ! , i = 1 , \\dots , n , \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} A ( x ) - 1 - x = & \\ \\frac { - x } { 1 - x } + \\frac { x } { ( 1 - x ) ^ 2 } A ( x ) - \\frac { 2 x ^ 2 } { 1 - x } A ( x ) + x \\big ( C ( x ) - 1 \\big ) - \\frac { x ^ 3 ( 1 - 3 x ) } { ( 1 - x ) ^ 3 ( 1 - 2 x ) } \\\\ & + x A ( x ) + ( 1 - 2 x ) C ( x ) - 1 . \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\displaystyle { \\begin{array} { c c c } \\dot { x } & = & - y + P ( x , y ) \\\\ \\dot { y } & = & x + Q ( x , y ) , \\\\ \\end{array} } \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} n \\left ( 1 - p \\right ) ^ { n - 1 } + n \\left ( n - 1 \\right ) p \\left ( 1 - p \\right ) ^ { n - 2 } = o \\left ( \\sqrt { n } \\right ) , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} \\begin{cases} ( e ^ { - 1 / 2 } + o ( 1 ) ) n ! ^ 2 / n ^ { n - 1 } & ~ d = 2 ~ ~ \\Sigma G = 0 , \\\\ ( 1 + O _ d ( n ^ { 2 - d } ) ) n ! ^ d / n ^ { n - 1 } & ~ d \\geq 3 ~ ~ ( d + 1 ) \\Sigma G = 0 , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} G w ( x , \\xi ) = h _ { i _ 1 \\dots i _ m } ( x ) \\xi ^ { i _ 1 } \\cdots \\xi ^ { i _ m } \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\nu _ { \\tau } ( ( s , t ] \\times J ) & = \\nu ( ( \\tau _ s , \\tau _ t ] \\times J ) = \\nu ( ( 0 , \\tau _ t ] \\times J ) - \\nu ( ( 0 , \\tau _ s ] \\times J ) \\\\ & \\leq \\nu ( ( 0 , \\tau _ t ] \\times J ) - \\nu ( ( 0 , \\tau _ s ] \\times J ) + ( \\tau _ t - \\tau _ s ) \\\\ & = ( \\nu ( ( 0 , \\tau _ t ] \\times J ) + \\tau _ t ) - ( \\nu ( ( 0 , \\tau _ s ] \\times J ) + \\tau _ s ) = t - s . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\widetilde { z } ( \\omega ) = E _ - + \\widetilde { z } ' ( \\omega _ \\beta ( E _ - ) ) ( \\omega - \\omega _ \\beta ( E _ - ) ) + \\frac { 1 } { 2 } \\widetilde { z } '' ( \\omega _ \\beta ( E _ - ) ) ( \\omega - \\omega _ \\beta ( E _ - ) ) ^ 2 + O \\left ( ( \\omega - \\omega _ \\beta ( E _ - ) ) ^ 3 \\right ) \\ , . \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} \\left ( l _ { j , 1 } , \\ ; \\ ; l _ { j , 2 } \\right ) & = \\left ( a _ { i , 1 } , \\ ; \\ ; a _ { i , r } \\right ) \\left ( \\begin{array} { c c } a _ { 1 , 1 } & a _ { r , 1 } \\\\ a _ { r , 1 } & a _ { r , r } \\end{array} \\right ) ^ { - 1 } \\\\ & = \\frac { 1 } { a _ { 1 , 1 } \\ , a _ { r , r } - a _ { r , 1 } ^ 2 } \\left ( a _ { i , 1 } \\ , a _ { r , r } - a _ { r , 1 } \\ , a _ { i , r } , \\ ; \\ ; a _ { 1 , 1 } \\ , a _ { i , r } - a _ { r , 1 } \\ , a _ { i , 1 } \\right ) . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{gather*} \\lim _ { q \\rightarrow 1 } { \\Gamma _ q ( z ) } = \\Gamma ( z ) . \\end{gather*}"}
-{"id": "683.png", "formula": "\\begin{gather*} 4 \\partial _ { s _ { 1 } } u _ { 1 } \\left ( x , t , y _ { 1 } , s _ { 1 } \\right ) - \\nabla _ { y _ { 1 } } \\cdot \\int _ { S _ { 2 } } \\int _ { S _ { 3 } } \\int _ { Y _ { 2 } } a \\left ( y ^ { 2 } , s ^ { 3 } , \\nabla u \\left ( x , t \\right ) \\right . \\\\ + \\left . \\nabla _ { y _ { 1 } } u _ { 1 } \\left ( x , t , y _ { 1 } , s _ { 1 } \\right ) + \\nabla _ { y _ { 2 } } u _ { 2 } \\left ( x , t , y ^ { 2 } , s ^ { 3 } \\right ) \\right ) d y _ { 2 } d s _ { 3 } d s _ { 2 } = 0 \\end{gather*}"}
-{"id": "3949.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty G ( r ) \\ , \\mu _ \\delta ( d r ) \\ ; = \\ ; \\int _ 0 ^ \\infty G ( r ) \\ , \\mu \\big ( \\psi _ { r , \\delta } \\big ) \\ , d r \\ ; . \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\widetilde A ( v ) = ( - \\alpha _ 1 v N ^ { 2 / 3 } , 0 ) , \\alpha _ 1 = 2 \\frac { ( 1 - \\rho ) ^ { 2 / 3 } } { \\rho ^ { 4 / 3 } } , \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} P & = 2 ^ { r - 5 } [ ( r - 2 ) ( r - 3 ) + 8 ( r - 2 ) + 8 + ( r - 2 ) ( r - 3 ) ] \\\\ & = 2 ^ { r - 4 } ( r ^ 2 - r + 2 ) \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\omega _ i = \\frac { 1 / \\bar { k } _ i } { \\sum _ { j = 1 } ^ g 1 / \\bar { k } _ j } ~ ~ ~ ~ \\forall ~ i \\in \\{ 1 , . . . , g \\} \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\| s ^ { n ' } { s ' } ^ { n } \\| _ { ( h \\otimes h ) ^ { n n ' } , \\hat { \\kappa } ( \\xi ) } & \\leqslant \\left ( \\| s \\| _ { h ^ { n } , \\hat { \\kappa } ( \\xi ) } \\right ) ^ { n ' } \\left ( \\| s ' \\| _ { h ^ { n ' } , \\hat { \\kappa } ( \\xi ) } \\right ) ^ { n } \\\\ & \\leqslant e ^ { n n ' \\epsilon } ( | s | _ { h ^ { n } } ( \\xi ) ) ^ { n ' } ( | s ' | _ { h ^ { n ' } } ( \\xi ) ) ^ n \\\\ & = e ^ { n n ' \\epsilon } | s ^ { n ' } { s ' } ^ n | _ { ( h \\otimes h ' ) ^ { n n ' } } ( \\xi ) . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} e _ { \\rm n u m } = { \\cal W } ^ { - p } \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} f = \\sum _ { n = 1 } ^ { \\infty } g _ n h _ n , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} d \\Pi ( \\boldsymbol { \\theta } _ { - ( \\boldsymbol { j } , \\boldsymbol { k } ) } ) = \\prod _ { ( \\boldsymbol { x , y } ) \\neq ( \\boldsymbol { j } , \\boldsymbol { k } ) } [ ( 1 - \\omega _ { \\boldsymbol { x } , n } ) d \\delta _ 0 ( \\theta _ { \\boldsymbol { x , y } } ) + \\omega _ { \\boldsymbol { x } , n } p ( \\theta _ { \\boldsymbol { x , y } } ) d \\theta _ { \\boldsymbol { x , y } } ] . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} & u _ { 1 , j } = c _ j u _ { 1 , j _ e } + ( 1 - c _ j ) u _ { 1 , j _ e + 1 } \\mbox { a n d } v _ { 1 , j } = c _ j v _ { 1 , j _ e } + ( 1 - c _ j ) v _ { 1 , j _ e + 1 } \\quad 1 \\le j \\le \\tilde J , \\\\ & u _ { I , j } = c _ j u _ { I , j _ e } + ( 1 - c _ j ) u _ { I , j _ e + 1 } \\mbox { a n d } v _ { I , j } = c _ j v _ { I , j _ e } + ( 1 - c _ j ) v _ { I , j _ e + 1 } \\tilde J + 2 \\le j \\le J , \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , & \\forall \\ 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k , l } = ( p _ { k } ) \\theta + { l \\over | p _ { k } | } , & \\forall \\ 0 \\leq l \\leq | p _ { k } | - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} g _ { 1 } ( [ X , Y ] , W ) & = g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { X } Y , \\phi W ) + g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { X } Y , \\omega W ) + \\eta ( Y ) g _ { 1 } ( X , \\omega W ) \\\\ & - g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { Y } X , \\phi W ) - g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { Y } X , \\omega W ) - \\eta ( X ) g _ { 1 } ( Y , \\omega W ) \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} \\beta _ 2 ( k , r ) \\le \\beta _ 2 ( k - 1 , r - 1 ) + 3 \\le ( 4 ( r - 1 ) - ( k - 1 ) - 2 ) + 3 = 4 r - k - 2 , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} d ^ { 2 } \\varepsilon / d \\xi ^ { 2 } - \\left \\{ { u ^ { 2 } + u \\phi \\ } \\right \\} \\varepsilon = h , \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\langle \\Psi ( z _ 0 , z _ 1 ) , ( z _ 0 , z _ 1 ) \\rangle _ { X \\times X } = \\int _ 0 ^ T \\Big \\| D _ \\nu z ( t ) - \\int _ t ^ T \\ H ( s - t ) D _ \\nu z ( s ) d s \\Big \\| _ Y ^ 2 \\ d t \\ , . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} a _ 1 & = ( \\rho + 3 ) \\log \\alpha _ 1 \\\\ a _ 2 & = ( \\rho + 3 ) \\log \\alpha _ 2 \\\\ a _ 3 & = 8 ( \\log c + 0 . 0 8 6 7 5 ( \\rho - 1 ) ) . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} \\nabla ^ \\gamma _ X Y = \\nabla ^ { \\gamma ' } _ X Y + b ( X , Y ) , \\ ; \\nabla ^ \\gamma _ X U = - W _ U X + \\nabla ^ \\nu _ X U , \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} P ( ( x _ 1 , x _ 2 ) , \\mu ) = \\begin{pmatrix} \\ < \\mu ( B _ 1 ) , \\xi \\ > & \\ < \\mu ( B _ 1 ) , \\xi ' \\ > \\\\ \\ < \\mu ( B _ 2 ) , \\xi \\ > & \\ < \\mu ( B _ 2 ) , \\xi ' \\ > \\\\ \\ < \\mu ( B _ 3 ) , \\xi \\ > & \\ < \\mu ( B _ 3 ) , \\xi ' \\ > \\end{pmatrix} . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\log \\left ( { \\Gamma \\Big ( { n ( n + \\nu - 1 ) + n t + t + \\nu \\over 2 } \\Big ) \\over \\Gamma \\Big ( { n ( n + \\nu - 1 ) + n t + \\nu \\over 2 } \\Big ) } \\right ) = t \\log n - { t \\over 2 } \\log 2 + o ( 1 ) . \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{gather*} R _ { g } ( \\vartheta _ p ) = g ^ { - 1 } \\cdot \\vartheta _ p , \\end{gather*}"}
-{"id": "887.png", "formula": "\\begin{align*} ( M _ r V ) _ S & = \\sum _ T M _ r ( S , T ) V _ T \\\\ & = \\sum _ T \\left [ \\binom { | S \\cap T | } { 2 } + \\binom { | \\bar { S } \\cap \\bar { T } | } { 2 } \\right ] V _ T \\\\ & = \\sum _ { \\substack { a , b \\\\ a = | S \\cap T | \\\\ b = | \\bar { S } \\cap \\bar { T } | } } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] V _ T \\\\ & = \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\chi ( S ; a , b ) \\cdot V , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d t ^ { 2 } } + \\Gamma _ { j k } ^ { i } \\frac { d x ^ { j } } { d t } \\frac { d x ^ { k } } { d t } + \\phi \\left ( t \\right ) \\frac { d x ^ { i } } { d t } + V ^ { , i } = 0 . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} & ( i ) \\ \\ \\phi ^ 2 + \\mathcal { B } \\omega = - i d , \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( i i ) \\ \\ \\mathcal { C } ^ 2 + \\omega \\mathcal { B } = - i d , \\\\ & ( i i i ) \\ \\ \\omega \\phi + \\mathcal { C } \\omega = 0 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( i v ) \\ \\ \\mathcal { B } \\mathcal { C } + \\phi \\mathcal { B } = 0 , \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} \\Gamma ^ { ( j ) } _ i \\ \\ \\mbox { f o r t h e s u b g r a p h s p a n n e d b y t h e n o d e s $ \\{ v ^ { ( j ) } _ { i ' } \\} _ { i ' = 1 } ^ { i } $ a n d t h e i r c o r r e s p o n d i n g e n d - v e r t i c e s , } \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\Gamma _ { I , J } ( L ) _ u = \\{ y \\in L _ u \\mid I ^ n y \\subseteq J y \\mbox { f o r s o m e } n \\geq 1 \\} \\mbox { f o r a l l } u \\in \\Z . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} \\rho _ \\pm = \\rho _ 0 \\pm \\kappa n ^ { - 1 / 3 } \\textrm { w i t h } \\rho _ 0 = \\frac { 1 } { \\gamma + 1 } . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} a ( \\tau ) = \\int _ { \\zeta + 2 } ^ \\tau f ( z ) d z . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} D _ \\infty ^ k ( h , w ) = \\sum _ { c \\mid h } \\frac { c } { ( 4 c ) ^ { 2 w } } \\Big ( e ^ { \\frac { \\pi i h } { 2 c } } + ( - 1 ) ^ k e ^ { \\frac { 3 \\pi i h } { 2 c } } \\Big ) \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} P & = \\sum _ t \\left [ \\binom { t } { 2 } \\binom { r - 2 } { t - s } - \\binom { r - t } { 2 } \\binom { r - 2 } { r - t } \\right ] \\\\ Q & = \\sum _ t \\left [ \\binom { t } { 2 } \\binom { r - 2 } { t } - \\binom { r - t } { 2 } \\binom { r - 2 } { r - t - 2 } \\right ] . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\omega _ { i , j } = \\begin{cases} 0 & i = 0 , j = 0 , \\\\ { \\rm E x p } ( 1 - \\rho ) & i \\geq 1 , j = 0 , \\\\ { \\rm E x p } ( \\rho ) & i = 0 , j \\geq 1 , \\\\ { \\rm E x p } ( 1 ) & i \\geq 1 , j \\geq 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\psi _ \\lambda ( \\zeta ) = \\det M _ \\lambda ( \\zeta ) , \\zeta \\in \\Omega _ k ^ * . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} I = ( x ^ 3 - w , x ^ 4 - y z , x z - y ^ 2 , x ^ 3 y - z ^ 2 ) . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\eta ^ { 1 2 } b _ 1 = & ( 1 + u ( b E _ 4 ^ 2 - a E _ 6 ) ) ( \\eta ^ { 1 2 } , 0 ) + u ( b E _ 4 ^ 2 - a E _ 6 ) ( - u E _ 6 , - 6 u E _ 4 ) + \\\\ & u ( b E _ 6 - a E _ 4 ) ( 0 , 6 \\eta ^ { 1 2 } ) + u ( b E _ 6 - a E _ 4 ) ( u E _ 4 ^ 2 , 6 u E _ 6 ) \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\Gamma _ { 2 d } = \\{ \\pm 1 \\} ^ { 2 d } \\rtimes S _ { 2 d } . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\mathtt { D C } ( \\gamma , \\tau ) : = \\Big \\{ \\omega \\in \\mathtt \\R ^ \\nu : | \\omega \\cdot \\ell | \\geq \\frac { \\gamma } { | \\ell | ^ { \\tau } } \\ \\ \\forall \\ell \\in \\Z ^ \\nu \\setminus \\{ 0 \\} \\Big \\} \\ , , \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 3 } } { \\partial \\rho ^ { 3 } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ { 2 } ( t ) d t + 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) F ( t ) d t , \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\mathcal { R } _ { i j } ^ 1 : = 2 0 \\int _ 0 ^ 1 d \\tau \\ , ( 1 - \\tau ) \\int _ { B _ \\rho ( \\zeta _ i ) } ( w _ i + \\tau \\pi _ i ) ^ 3 \\ , \\pi _ i ^ 2 \\ , U _ j . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} h ( x , y ) = x U ^ t \\overline { y } , \\ < x , y \\ > = \\dfrac { 2 } { 3 } \\Re ( h ( x , y ) ) , \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\frac { A \\otimes W - \\left ( \\overline { A \\otimes W } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } = \\left ( V \\otimes Z \\right ) \\left ( \\overline { V \\otimes Z } \\right ) ^ { t } . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} A _ 1 ( x ) = L ( f _ 1 ( x ) ) ^ { - 1 } A _ 2 ( h ( x ) ) L ( x ) , \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\epsilon = ( k _ 1 - 1 ) ( k _ 2 - 1 ) + \\dim L + k _ 1 + ( i - 1 ) \\Big ( 1 + ( \\mu ( \\beta _ 2 ) + k _ 2 ) \\dim L \\Big ) . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} \\begin{array} { l l l } A & = & \\sqrt { \\tau } E , \\\\ b & = & \\displaystyle { \\frac { w + \\sqrt { \\tau } c } { 1 + \\sqrt { \\tau } } } , \\\\ \\gamma & = & \\displaystyle { \\frac { \\beta ( 1 + \\sqrt { \\tau } ) ^ 2 + \\frac { 1 } { 2 } \\sqrt { \\tau } \\langle c - w , E ^ { - 1 } ( c - w ) \\rangle } { ( 1 + \\sqrt { \\tau } ) ^ 2 ( \\tau + 1 ) } } . \\end{array} \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} D _ 4 = \\langle ~ r , s ~ | ~ r ^ 4 , s ^ 2 , r ^ k s = s r ^ { - k } ~ \\rangle \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{gather*} n \\circ \\eta = \\frac { \\partial _ 1 r \\times \\partial _ 2 r } { | \\partial _ 1 r \\times \\partial _ 2 r | } . \\end{gather*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\aligned \\sum _ { k _ 1 + k _ 2 = k + 1 } & \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\sum _ { i = 1 } ^ { k - k _ 2 + 1 } \\\\ & ( - 1 ) ^ * { \\frak m } ^ P _ { k _ 1 , \\beta _ 1 } ( h _ 1 , \\ldots , { \\frak m } ^ P _ { k _ 2 , \\beta _ 2 } ( h _ i , \\ldots , h _ { i + k _ 2 - 1 } ) , \\ldots , h _ { k } ) = 0 , \\endaligned \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} H ( x _ { 1 } , x _ { 2 } ) & = \\chi ( [ ( 0 , 0 ) , ( x _ { 1 } , x _ { 2 } ) ] ^ { c } ) \\\\ & = \\chi _ { + } ( [ ( 0 , 0 ) , ( x _ { 1 } , x _ { 2 } ) ] ^ { c } ) - \\chi _ { - } ( [ ( 0 , 0 ) , ( x _ { 1 } , x _ { 2 } ) ] ^ { c } ) \\\\ & = : \\int \\limits _ { [ ( 0 , 0 ) , ( x _ { 1 } , x _ { 2 } ) ] ^ { c } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! h _ { + } ( x _ { 1 } , x _ { 2 } ) \\ ; \\mathrm d x _ { 1 } \\mathrm d x _ { 2 } - \\int \\limits _ { [ ( 0 , 0 ) , ( x _ { 1 } , x _ { 2 } ) ] ^ { c } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! h _ { - } ( x _ { 1 } , x _ { 2 } ) \\ ; \\mathrm d x _ { 1 } \\mathrm d x _ { 2 } . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } \\prod _ { i = 1 } ^ k T _ { [ 1 , i ] } ^ g f _ i \\prod _ { j = 1 } ^ \\ell S _ { [ 1 , j ] } ^ g h _ j = \\left ( \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } \\prod _ { i = 1 } ^ k T _ { [ 1 , i ] } ^ g f _ i \\right ) \\ ! \\ ! \\ ! \\left ( \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } \\prod _ { j = 1 } ^ \\ell S _ { [ 1 , j ] } ^ g h _ j \\right ) \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} V _ { \\frak r } ( p ; A ) = ( S _ A ( V _ { \\frak r } ) ) ^ { \\boxplus \\tau } \\times [ - \\tau , 0 ) ^ { A } . \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} \\Phi ( f ( z , w ) ) = f ( z + P , z P + P ^ { 2 } ) . \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} \\triangle _ g u - \\partial _ t u - \\tilde { V } ( x , t ) u = 0 \\mbox { o n } \\ \\mathcal { M } ^ 1 , \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\big ( a \\cdot \\hat { \\gamma } \\big ) ( \\chi ) = \\chi ( \\gamma ) a . \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\Sigma = ( L _ 1 \\mathbb { T } ) \\times ( L _ 2 \\mathbb { T } ) , \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} V = V _ { 1 } - V _ { 2 } = - 2 i A ' + \\nabla ( 2 i \\psi ) = V ' + \\nabla \\varphi . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha , \\beta } f ( x ) = I _ { a ^ + } ^ { \\beta ( n - \\alpha ) } D ^ { n } I _ { a ^ + } ^ { ( 1 - \\beta ) ( n - \\alpha ) } f ( x ) , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigg ( \\frac { A C } { B ^ 2 } \\Bigg ) & = \\frac { 3 A } { B ^ 2 E } ( B ^ 2 - A C ) \\\\ & = 3 \\Bigg ( \\frac { A C } { B ^ 2 } \\Bigg ) \\Bigg ( \\frac { B ^ 2 - A C } { B C E } \\Bigg ) . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\frac { 1 } { 2 | \\Omega | } \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } \\left ( \\sum _ { i = 1 } ^ { N } ( y _ { r , i } ' - y _ { r , i } ) \\frac { \\overline v _ i } { u _ { i , \\infty } } \\right ) ^ 2 \\geq \\beta \\sum _ { i = 1 } ^ { N } \\frac { \\overline v _ i ^ 2 } { u _ { i , \\infty } } . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\| U ^ { * } ( \\Phi ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\Sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\Sigma ( a ) } ^ m ) U \\\\ & - \\Psi ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\Sigma ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\Sigma ( a ) } ^ m \\| < \\varepsilon \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} A _ { 1 , 2 } = \\left ( \\begin{matrix} 2 & 1 \\end{matrix} \\right ) , A _ { 2 , 2 } = \\left ( \\begin{matrix} 2 & 2 & 1 & 0 \\\\ 2 & 1 & 1 & 1 \\end{matrix} \\right ) , A _ { 3 , 2 } = \\left ( \\begin{matrix} 2 & 2 & 2 & 1 & 0 & 0 \\\\ 2 & 2 & 1 & 1 & 1 & 0 \\\\ 2 & 1 & 1 & 1 & 1 & 1 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{gather*} F ^ a = 0 . \\end{gather*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\Delta ( a ) & = a \\otimes 1 + 1 \\otimes a , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ \\forall a \\in L , \\\\ \\Delta ( u v ) & = \\Delta ( u ) \\Delta ( v ) , ~ ~ ~ ~ ~ ~ ~ ~ \\forall u , v \\in S ( L ) \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} { \\rm C o r r } _ { ( \\frak X _ { 2 3 } , \\widehat { \\frak S } ^ { \\epsilon } _ { 2 3 } ) , P _ 1 } \\circ { \\rm C o r r } _ { ( \\frak X _ { 1 2 } , \\widehat { \\frak S } ^ { \\epsilon } _ { 1 2 } ) , P _ 2 } = { \\rm C o r r } _ { ( \\frak X _ { 1 3 } , \\widehat { \\frak S } ^ { \\epsilon } _ { 1 3 } ) } \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} A _ { 2 s + 1 } = \\sum _ { i = 0 } ^ s q _ { 2 i + 1 } P _ { 2 i + 1 } ^ s + \\sum _ { i = 0 } ^ s q _ { 2 i } L _ s ^ i \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\aligned & ( \\hat d _ { 2 } ^ { i + 1 } \\circ A - ( - 1 ) ^ { \\deg A } A \\circ \\hat d _ { 1 } ^ { i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } \\\\ & \\quad - ( \\hat d _ { 2 } ^ { i + 1 } \\circ A \\vert _ { E ^ i } - ( - 1 ) ^ { \\deg A } A \\vert _ { E ^ i } \\circ \\hat d _ { 1 } ^ { i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } \\\\ & = d _ 0 \\circ A _ { \\alpha ' _ 2 \\alpha _ 1 } - ( - 1 ) ^ { \\deg A } A _ { \\alpha ' _ 2 \\alpha _ 1 } \\circ d _ 0 . \\endaligned \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l u _ { n , t + 1 } + u _ { n , t - 1 } - a _ { n } u _ { n + 1 , t } - a _ { n - 1 } u _ { n - 1 , t } - b _ n u _ { n , t } = 0 , n , t \\in \\mathbb { N } , \\\\ u _ { n , - 1 } = u _ { n , 0 } = 0 , n \\in \\mathbb { N } , \\\\ u _ { 0 , t } = f _ t , t \\in \\mathbb { N } \\cup \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} H : = f ^ { - 1 } ( I ) \\subset D . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\left < \\left ( d ^ { i j } _ { k l } \\right ) _ { 1 \\leq k , l \\leq q } , \\left ( \\overline { A } ^ { - 1 } \\right ) ^ { t } \\right > , \\quad \\mbox { f o r a l l $ i , j = 1 \\dots , q ' - q $ . } \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\Big [ \\mu _ { j _ 3 } , \\big [ \\mu _ { j _ 4 } , \\big [ \\ldots \\big [ \\mu _ { j _ { | J | } } , [ \\mu _ { j _ 1 } , \\mu _ { j _ 2 } ] \\big ] \\ldots \\Big ] , J = \\{ j _ 1 , \\ldots , j _ { | J | } \\} . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{gather*} A = \\mathfrak { G } ( X , x _ 0 ) = \\{ \\alpha \\in G : \\alpha x _ 0 = x _ 0 \\} , \\ \\ \\\\ F = \\mathfrak { G } ( Z , z _ 0 ) = \\mathfrak { G } ( Z ^ * , z ^ * _ 0 ) = \\{ \\alpha \\in G : \\alpha z _ 0 = z _ 0 \\} = \\{ \\alpha \\in G : \\alpha z ^ * _ 0 = z ^ * _ 0 \\} . \\end{gather*}"}
-{"id": "4516.png", "formula": "\\begin{align*} [ a , b , b , a ] = [ a , b , a , b ] . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = ( y u ) + ( l _ 2 ) , \\\\ ( 0 ) : l _ 2 & = ( y u ) + ( l _ 1 ) , \\\\ ( y u ) \\cap ( l _ 1 ) & = ( y u ) \\cap ( l _ 2 ) = ( 0 ) . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} I ( f ) \\lesssim \\int _ { \\mathcal { I } _ { \\boldsymbol { j } } } \\left | \\prod _ { l = 1 } ^ d 2 ^ { j _ l / 2 + j _ l } \\psi ^ { ' } _ { k _ l } ( 2 ^ { j _ l } x _ l ) \\mathrm { s g n } [ \\psi _ { k _ l } ( 2 ^ { j _ l } x _ l ) ] \\right | d \\boldsymbol { x } \\lesssim \\prod _ { l = 1 } ^ d 2 ^ { j _ l / 2 } , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} \\| \\mathcal { A } x - \\mathcal { A } y \\| _ { \\ell _ 2 } \\leq \\sum _ { i = 1 } ^ g \\| \\mathcal { A } _ i x _ i - \\mathcal { A } _ i y _ i \\| _ { \\ell _ 2 } \\leq \\sum _ { i = 1 } ^ g \\sqrt { 2 \\bar { k } _ i ( 1 + \\delta _ { \\bar { k } _ i } ) } \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} i \\partial _ t w = \\lambda t ^ { - 1 } V ( t ) ^ { - 1 } \\bigl [ | V ( t ) w | ^ 2 V ( t ) w \\bigr ] . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} & \\sum _ { e \\in E ( v ) } ( M ^ e L ^ e ) ^ { - 1 } \\ , \\frac { 2 C ^ n _ { e e } L ^ e } { M ^ e } \\ , \\sum _ { z \\in x ^ e _ 2 } \\sum _ { w \\in x ^ e _ 1 } \\int _ 0 ^ t ( \\xi _ { s - } ( w ) - \\xi _ { s - } ( z ) ) \\phi _ s ( z ) + ( \\xi _ { s - } ( z ) - \\xi _ { s - } ( w ) ) \\phi _ s ( w ) \\ , d s \\\\ = & \\sum _ { e \\in E ( v ) } 2 C ^ n _ { e e } \\int _ 0 ^ t \\big ( u ^ n _ { s - } ( x ^ e _ 2 ) - u ^ n _ { s - } ( x ^ e _ 1 ) \\big ) \\ , \\big ( \\phi _ s ( x ^ e _ 1 ) - \\phi _ s ( x ^ e _ 2 ) \\big ) \\ , d s \\\\ = & \\ , e r r _ 2 ( t ) \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\vert \\zeta _ i ^ { \\prime } - \\zeta _ j ^ { \\prime } \\vert > \\frac { \\delta } { \\varepsilon } , \\ i \\not = j ; d i s t ( \\zeta _ i ^ { \\prime } , \\partial \\Omega _ { \\varepsilon } ) > \\frac { \\delta } { \\varepsilon } \\delta < \\mu _ i ^ { \\prime } < \\delta ^ { - 1 } , \\ i = 1 , \\ldots , k , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\theta ( E ) ( x ) : = \\lim _ { r \\to 0 ^ + } \\frac { | E \\cap B _ r ( x ) | } { | B _ r ( x ) | } \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\ , K _ { b _ { j } , l _ { j } - 1 } ^ { X _ { 2 } ^ { c } , 0 } - \\ , K _ { b _ { j } , l _ { j } } ^ { X _ { 2 } ^ { c } , 0 } = 1 - 0 = 1 \\ , \\ , \\ , \\ , l _ { j } > 0 . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} \\tilde { \\psi } ( X ) = \\tilde { \\psi } ( X ^ { t } ) , \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} x a + y c & = 1 + u ( b E _ 4 ^ 2 - a E _ 6 ) , \\\\ x b + y d & = u ( b E _ 6 - a E _ 4 ) , \\\\ x - u ( x a + y c ) E _ 6 + u ( x b + y d ) E _ 4 ^ 2 & = - u ^ 2 ( b E _ 4 ^ 2 - a E _ 6 ) E _ 6 + u ^ 2 ( b E _ 6 - a E _ 4 ) E _ 4 ^ 2 , \\\\ y - 6 u ( x a + y c ) E _ 4 + 6 u ( x b + y d ) E _ 6 & = - 6 u ^ 2 ( b E _ 4 ^ 2 - a E _ 6 ) E _ 4 + 6 u ^ 2 ( b E _ 6 - a E _ 4 ) E _ 6 . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\exp \\mathcal H = \\varinjlim _ { n \\to \\infty } \\sum _ { k = 0 } ^ n \\frac { 1 } { k ! } \\mathcal H ^ { \\otimes k } \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} S _ m = \\sum _ { i = 1 } ^ n \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\frac { z _ i ^ m } { z _ i - z _ j } , T _ m = \\sum _ { i = 1 } ^ n z _ i ^ m , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi i } \\int _ { ( 4 ) } \\bigg ( \\frac { 1 } { 2 } W ( s ; f , \\overline { g } ) + \\frac { W ( s - 1 ; f , \\overline { g } ) } { s + k - 2 } \\bigg ) X ^ s \\Gamma ( s ) d s \\\\ & \\quad + \\frac { 1 } { ( 2 \\pi i ) ^ 2 } \\int _ { ( 4 ) } \\int _ { ( - 1 + \\epsilon ) } W ( s - z ; f , \\overline { g } ) \\zeta ( z ) \\frac { \\Gamma ( z ) \\Gamma ( s - z + k - 1 ) } { \\Gamma ( s + k - 1 ) } d z X ^ s \\Gamma ( s ) d s . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle { \\frac { d A ( t ) } { d t } } & = & - r ( \\alpha ) A ( t ) B ( t ) + I _ 1 ( t ) \\\\ \\\\ \\displaystyle { \\frac { d B ( t ) } { d t } } & = & - r ( \\alpha ) A ( t ) B ( t ) + I _ 2 ( t ) + I _ 3 ( t ) \\\\ \\\\ \\displaystyle { \\frac { d C ( t ) } { d t } } & = & 2 r ( \\alpha ) A ( t ) B ( t ) - O _ 4 ( t ) . \\end{array} \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } + 2 \\pi a \\left ( x , t \\right ) v \\left ( a \\left ( x , t \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { C _ { n , 1 , q } } { n } = \\alpha ( q ) a . s . \\mbox { w i t h } \\lim _ { q \\downarrow 0 } \\alpha ( q ) = 1 . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\partial _ a ( \\tilde { \\Psi } ( a , x , z ) ) = \\mathcal { L } ( \\tilde { \\Psi } ( a , x , z ) ) + F ( \\tilde { \\Psi } ( a , x , z ) , x , z ) , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\frac { 1 } { \\omega _ \\alpha ( z ) + \\omega _ \\beta ( z ) - z } = - \\int _ \\R \\frac { \\dd \\mu _ \\alpha ( x ) } { x - \\omega _ \\beta ( z ) } & = \\frac { 1 } { \\omega _ \\beta ( z ) } + O ( ( \\omega _ \\beta ( z ) ) ^ { - 2 } ) \\ , , \\\\ \\frac { 1 } { \\omega _ \\alpha ( z ) + \\omega _ \\beta ( z ) - z } = - \\int _ \\R \\frac { \\dd \\mu _ \\beta ( x ) } { x - \\omega _ \\alpha ( z ) } & = \\frac { 1 } { \\omega _ \\alpha ( z ) } + O ( ( \\omega _ \\alpha ( z ) ) ^ { - 2 } ) \\ , , \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\rho _ t ^ { \\widehat { \\gamma } } ( d a ) : = [ \\widehat { \\gamma } ( t , X _ { \\widehat { \\gamma } , m } ( t ^ - ) ) ] ( d a ) . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\begin{pmatrix} \\hat { u } \\\\ \\hat { v } \\end{pmatrix} = \\begin{pmatrix} 0 . 1 + 0 . 1 x + 0 . 2 y \\\\ 0 . 0 5 + 0 . 1 5 x + 0 . 1 y \\end{pmatrix} \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} d e ^ { i \\theta _ 0 } : T _ x ^ { 1 , 0 } X \\rightarrow T ^ { 1 , 0 } _ { e ^ { i \\theta _ 0 } x } X , \\\\ d e ^ { i \\theta _ 0 } : T _ x ^ { 0 , 1 } X \\rightarrow T ^ { 0 , 1 } _ { e ^ { i \\theta _ 0 } x } X , \\\\ d e ^ { i \\theta _ 0 } ( T ( x ) ) = T ( e ^ { i \\theta _ 0 } x ) . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} x _ n \\cdot x _ { 2 n } = 0 , x _ n = x _ { 2 n } - \\sum _ { i = 1 } ^ { n - 1 } c _ { i , n } x _ { n + i } . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } U _ n = 0 , \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\pi _ 1 \\pi ^ { - 1 } \\pi _ { d + 1 } + \\cdots + \\pi _ d \\pi ^ { - 1 } \\pi _ { d + 1 } = \\pi _ { d + 1 } . \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\begin{aligned} A ( t ) & \\sim k _ A t ^ { - 1 / 3 } \\\\ C ( t ) = B ( t ) & \\sim k _ B t ^ { 1 / 3 } \\\\ D ( t ) & \\rightarrow K _ 1 \\\\ E ( t ) & \\sim k _ E t ^ { - 1 / 3 } . \\end{aligned} \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} \\Gamma = \\left \\{ x _ { n } = 0 , \\left ( x _ { 1 } - 1 / 2 \\right ) ^ { 2 } / \\omega ^ { 2 } + \\displaystyle \\sum \\limits _ { k = 2 } ^ { n - 1 } x _ { k } ^ { 2 } < \\frac { 1 } { 4 } \\right \\} \\subset \\partial \\Omega . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} y _ i = \\int _ { \\beta _ i } \\psi _ 1 ( i = 1 , \\dots , 4 ) . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j } s ' _ i = \\begin{cases} j - 1 & 1 \\le j < m \\\\ \\sum _ { i = 1 } ^ { j } s _ i & m \\le j \\end{cases} . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} g ( \\varpi ) = \\begin{cases} N ( \\varpi ) ^ { 1 / 2 } & N ( \\varpi ) \\equiv 1 \\pmod 4 , \\\\ - i N ( \\varpi ) ^ { 1 / 2 } & N ( \\varpi ) \\equiv - 1 \\pmod 4 . \\end{cases} \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} E ^ { c } = \\left \\{ \\omega \\in X \\ | \\ \\left \\langle L \\omega , \\omega _ { 1 } \\right \\rangle = 0 , \\ \\forall \\omega _ { 1 } \\in E ^ { s } \\oplus E ^ { u } \\right \\} . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} R _ \\alpha ( \\omega _ \\beta ( z ) ) = \\frac { \\left | \\int _ \\R \\frac { \\dd \\mu _ \\alpha ( x ) } { x - \\omega _ \\beta ( z ) } \\right | ^ 2 } { \\int _ \\R \\frac { \\dd \\mu _ \\alpha ( x ) } { | x - \\omega _ \\beta ( z ) | ^ 2 } } \\le C k ^ { 1 - | t _ - ^ \\alpha | } \\ , , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} a [ u ] ( x , t ) = \\int _ { | x - y | < 1 } \\frac { u ( y , t ) } { | x - y | } d y + \\int _ { | x - y | \\geq 1 } \\frac { u ( y , t ) } { | x - y | } d y \\equiv I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} f _ { \\Lambda } ( u ) = \\frac { \\omega _ \\Lambda ( u ) - | u | } { 2 } . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\norm { F e } = \\norm { F _ X e _ X \\otimes F _ Y e _ Y } = \\norm { F _ X e _ X } \\norm { F _ Y e _ Y } = 1 . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) & = \\frac 1 { ( m / 2 ) ! } \\ ( - \\frac 1 { 2 n ^ 2 } \\sum _ { x \\in G } | f ^ { - 1 } ( x ) | ^ 2 \\ ) ^ { m / 2 } \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\\\ & \\qquad + O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\ ) , \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\| V ^ { - 1 } \\bigl ( | V w | ^ 2 V w \\bigr ) \\| _ { \\dot H ^ 1 } & \\lesssim \\| | V w | ^ 2 V w \\| _ { H ^ 1 } + t ^ { \\frac 1 2 } \\bigl | ( V w ) | _ { x = 0 } \\bigr | ^ 3 \\\\ & \\lesssim \\| V w \\| _ { L ^ \\infty } ^ 2 \\| V w \\| _ { H ^ 1 } + t ^ { - \\frac 1 4 } \\| w \\| _ { H ^ 1 } ^ 3 \\\\ & \\lesssim \\bigl [ \\| w \\| _ { L ^ \\infty } + t ^ { - \\frac 1 4 } \\| w \\| _ { H ^ 1 } \\bigr ] ^ 2 \\| w \\| _ { H ^ 1 } + t ^ { - \\frac 1 4 } \\| w \\| _ { H ^ 1 } ^ 3 , \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { n p _ n } ^ 2 } \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 2 I ( | z _ { n \\ell } | \\ge \\varepsilon \\tau _ { n p _ n } ) | \\mathcal { F } _ { n ( \\ell - 1 ) } ) \\to 0 ~ ~ ~ \\mbox { i n p r o b a b i l i t y } \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\langle B \\chi _ j , \\chi _ k \\rangle = \\langle a , \\chi _ { k - j } \\rangle = a _ { k - j } . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} P = M ^ { - 1 } ( Q _ { \\widehat { \\mathfrak { N } } } ) ^ { - 1 } \\widehat { P } ( M ^ * ) ^ { - 1 } . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} \\varphi ( E _ { i j } ) = & y _ { i j } E _ { i j } + y _ { i , \\sigma ^ { - 1 } ( j ) } E _ { i , \\sigma ^ { - 1 } ( j ) } + y _ { \\sigma ^ { - 1 } ( i ) , j } E _ { \\sigma ^ { - 1 } ( i ) , j } \\\\ & + y _ { \\sigma ^ { - 1 } ( i ) , \\sigma ^ { - 1 } ( j ) } E _ { \\sigma ^ { - 1 } ( i ) , \\sigma ^ { - 1 } ( j ) } \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} Q _ { K _ 3 , L ' } ( y ) & \\leq \\Sigma _ { \\emptyset } - \\Sigma _ { \\{ 1 \\} } - \\Sigma _ { \\{ 2 \\} } - \\Sigma _ { \\{ 3 \\} } + \\Sigma _ { \\{ 1 , 2 \\} } + \\Sigma _ { \\{ 1 , 3 \\} } + \\Sigma _ { \\{ 2 , 3 \\} } - \\Sigma _ { \\{ 1 , 2 , 3 \\} } \\\\ & = y ( y ^ 2 - 1 ) - 3 y ( y - 1 ) + 3 ( y - 1 ) - 1 = y ^ 3 - 3 y ^ 2 + 5 y - 4 . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} T _ n & = \\frac { 1 } { 4 } \\left ( 2 - \\frac { 1 } { 2 ^ n } \\right ) T , R _ n = \\frac { 1 } { 2 } \\left ( 1 + \\frac { 1 } { 2 ^ n } \\right ) R , \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} & U = U _ { m n } = \\{ \\delta _ { m n - i - k + 1 } \\} _ { i , k = 1 } ^ { m n } , U _ m = \\{ \\delta _ { m - i - k + 1 } \\} _ { i , k = 1 } ^ { m } . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} S _ s = S + s S ^ \\alpha \\ , u _ \\alpha + O ( s ^ 2 ) . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} H _ j ( x , y ) = \\frac { \\eta ( t _ { j - 1 } , y ) } { \\eta ( t _ { j - 1 } , t _ j ] } + x \\frac { \\eta ( \\{ y \\} ) } { \\eta ( t _ { j - 1 } , t _ j ] } , t _ { j - 1 } < y \\le t _ j \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} N = \\left [ \\begin{array} { c c c c } A & 0 _ n & - I _ n & 0 _ n \\end{array} \\right ] , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} \\ell _ 2 ( z ) = \\dot \\lambda ^ 0 ( z ) : = \\partial _ N v ( z , \\sigma = 0 ) \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} J ( \\rho , m ) & = E \\left [ \\int _ 0 ^ T \\int _ A c ( t , Y ( t ) , a , m ( t ) ) [ \\widehat { \\gamma } ( s , Y ( s ) ) ] ( d a ) d t + \\psi ( Y ( T ) , m ( T ) ) \\right ] \\\\ & = J ( \\widehat { \\gamma } , m ) . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\Phi _ s : = \\sum _ { k = 1 } ^ j \\Phi ^ { ( k ) } ( s - s _ 0 ) ^ { k - 1 } , \\Phi ^ { ( k ) } : = \\rho ^ { - s _ 0 + m } \\sum _ { \\ell = 0 } ^ { k - 1 } \\tfrac { ( - \\log \\rho ) ^ \\ell } { \\ell ! } \\varphi ^ { ( k - \\ell ) } . \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} T _ { s _ 1 , \\ldots , s _ k } = \\begin{vmatrix} \\alpha _ { 1 , s _ 1 } & \\alpha _ { 1 , s _ 2 } & \\cdots & \\alpha _ { 1 , s _ k } \\\\ \\alpha _ { 2 , s _ 1 } & \\alpha _ { 2 , s _ 2 } & \\cdots & \\alpha _ { 2 , s _ k } \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\alpha _ { k , s _ 1 } & \\alpha _ { k , s _ 2 } & \\cdots & \\alpha _ { k , s _ k } \\end{vmatrix} . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) & = \\cos ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) - g _ { 1 } ( \\mathcal { T } _ { U } V , w \\phi Z ) + g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } ( w Z ) ) \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} V _ n ^ c ( M , [ g ] ) : = \\inf _ { \\phi \\in I _ n ( M , [ g ] ) } \\sup _ { \\gamma \\in G _ n } V o l ( M , ( \\gamma \\circ \\phi ) ^ * c a n ) , \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} T ^ { d } = & P ^ \\pi \\sum _ { j = 0 } ^ { n - 1 } \\sum _ { k = 0 } ^ { j } P ^ k R ^ { j - k } \\sum _ { h = 0 } ^ { l - 1 } ( Q ^ d ) ^ { h + j + 1 } S ^ { h } . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} M ( y _ i | c _ i ) & = \\begin{cases} \\log _ 2 2 p _ i - R & \\\\ \\log _ 2 2 ( 1 - p _ i ) - R & \\end{cases} \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} 0 & = \\int _ { \\mathbb R ^ n } \\left ( - \\frac { u _ \\tau - u _ 0 } { \\tau } \\Delta \\varphi + \\frac { g ( u _ \\tau ) - g ( u _ 0 ) } { \\tau } \\varphi \\right ) \\\\ & = \\int _ { \\mathbb R ^ n } \\frac { u _ \\tau - u _ 0 } { \\tau } \\left ( - \\Delta \\varphi + \\frac { g ( u _ \\tau ) - g ( u _ 0 ) } { u _ \\tau - u _ 0 } \\varphi \\right ) . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} ( d f _ \\tau ) ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n + 1 } ) & = x _ 1 f _ \\tau ( x _ 2 \\otimes x _ 3 \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i f _ \\tau ( x _ 1 \\otimes \\dots \\otimes x _ i x _ { i + 1 } \\otimes \\dots \\otimes x _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f _ \\tau ( x _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n } ) x _ { n + 1 } . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) & = \\cos ^ { 2 } \\theta g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 2 } , V _ { 2 } ) , \\pi _ * Z ) + g _ { 2 } ( \\nabla \\pi _ * ( U _ { 2 } , \\omega \\phi V _ { 2 } ) , \\pi _ * Z ) - g _ { 1 } ( \\mathcal { T } _ { U _ { 2 } } \\omega V _ { 2 } , \\mathcal { B } Z ) \\\\ & + g _ { 2 } ( \\nabla \\pi _ * ( U _ { 2 } , \\omega V _ { 2 } ) , \\pi _ * ( \\mathcal { C } Z ) ) - g _ { 1 } ( V _ { 2 } , \\phi U _ { 2 } ) \\eta ( Z ) , \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} d _ \\ell ( p ^ m ) = \\frac { \\Gamma ( m + \\ell ) } { \\Gamma ( \\ell ) m ! } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\mathbb { E } ( N _ j \\ , | \\ , N > 0 ) = \\frac { \\mathbb { E } N _ j } { \\mathbb { P } ( N > 0 ) } = \\frac { q ^ { j - 1 } } { j } \\mbox { f o r } j \\ge 1 . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} \\partial { \\mathcal M } _ { k + 1 } ( \\beta ) = \\bigcup _ { k _ 1 + k _ 2 = k } \\bigcup _ { i = 1 , \\dots , k _ 2 } \\bigcup _ { \\beta _ 1 + \\beta _ 2 = \\beta } { \\mathcal M } _ { k _ 1 + 1 } ( \\beta _ 1 ) \\ , \\ , { } _ { { \\rm e v } _ 0 } \\times _ { { \\rm e v } _ i } { \\mathcal M } _ { k _ 2 + 1 } ( \\beta _ 2 ) , \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} X ( t ) = \\xi + \\int _ 0 ^ t \\int _ U f ( s , X ( s ^ - ) , u , \\alpha ( s ) , m ( s ) ) \\N ( d s , d u ) , \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} a _ 2 ( K _ N ) & = a _ 2 ( K ) + l k ( K ' , K '' ) N \\\\ v _ 3 ( K _ N ) & = v _ 3 ( K ) - \\frac { 1 } { 4 } ( a _ 2 ( K ' ) + a _ 2 ( K '' ) ) N + \\frac { 1 } { 8 } \\left ( 2 a _ 2 ( K ) + l k ( K ' , K '' ) ^ { 2 } \\right ) N \\\\ & \\quad + \\frac { 1 } { 8 } l k ( K ' , K '' ) ^ { 2 } N ^ { 2 } \\\\ a _ 4 ( K _ N ) & = a _ 4 ( K ) + a _ 3 ( L ) N \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} T _ 1 f & = \\sum _ i f ( x _ i ) \\sigma ( x , x ^ { - 1 } x _ i ) \\psi _ i \\# W ^ \\rho _ u ( u ) \\\\ T _ 2 f & = \\sum _ i \\lambda _ i ( f ) \\ell _ { x _ i } ^ \\sigma W ^ \\rho _ u ( u ) \\\\ T _ 3 f & = \\sum _ i c _ i f ( x _ i ) \\ell _ { x _ i } ^ \\sigma W ^ \\rho _ u ( u ) \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} & E | \\mu ^ N ( t ) - m ( t ) | + \\frac C N \\sum _ { i = 1 } ^ N E | X ^ N _ i ( t ) - Y ^ N _ i ( t ) | \\\\ & \\leq E | \\overline { \\mu } ^ N ( t ) - m ( t ) | + \\frac { 2 C } { N } \\sum _ { i = 1 } ^ N E | X ^ N _ i ( t ) - Y ^ N _ i ( t ) | \\\\ & \\leq \\frac { C } { \\sqrt { N } } + 2 K _ 1 \\frac 1 N \\sum _ { i = 1 } ^ N \\int _ 0 ^ t \\left [ E | X ^ N _ i ( s ) - Y ^ N _ i ( s ) | + E | \\mu ^ N ( s ) - m ( s ) | \\right ] d s . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} u \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = g _ { 0 } \\left ( x , x _ { 0 } \\right ) , \\partial _ { n } u \\left ( x , x _ { 0 } \\right ) \\mid _ { x \\in \\Gamma , x _ { 0 } \\in \\left [ 0 , 1 \\right ] } = g _ { 1 } \\left ( x , x _ { 0 } \\right ) , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} P \\phi \\ | _ { \\gamma _ { i } \\left ( c \\right ) } = \\frac { \\oint _ { \\gamma _ { i } \\left ( c \\right ) } \\frac { \\phi \\left ( x , y \\right ) } { \\left \\vert \\nabla \\psi _ { 0 } \\right \\vert } d l } { \\oint _ { \\gamma _ { i } \\left ( c \\right ) } \\frac { 1 } { \\left \\vert \\nabla \\psi _ { 0 } \\right \\vert } d l } , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\frac { 2 h } { ( 1 + h ) ( 1 + 2 h ) } = \\sum _ { j \\ge 0 } ( - 1 ) ^ { j + 1 } ( 2 ^ { j + 1 } - 2 ) h ^ j \\quad . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v ^ t = - \\Delta \\delta _ t h ^ 0 & \\mbox { i n } \\Omega ^ t \\\\ v ^ t = - \\frac 1 t \\tilde u ^ 0 & \\mbox { o n } \\Gamma ^ t \\\\ \\lim _ { x \\to \\infty } v ^ t = \\delta _ t c ^ 0 , \\lim _ { x \\to \\infty } \\frac { v ^ t ( x ) } { - \\log | x | } = \\delta _ t c ^ 0 . \\end{cases} \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} a ( n ; i , j ) & = a ( n - 1 ; i , j ) , \\qquad \\mbox { i f $ 1 \\leq i < j \\leq n - 2 $ } , \\\\ a ( n ; n - 1 , i ) & = a ( n - 1 ; i ) , \\qquad \\mbox { i f $ 1 \\leq i \\leq n - 2 $ } , \\\\ a ( n ; i , i - 1 ) & = a ( n - 1 , i - 1 ) , \\qquad \\mbox { $ 2 \\leq i \\leq n $ } , \\\\ a ( n ; i , i + 1 ) & = a ( i - 1 ) , \\qquad \\mbox { $ 1 \\leq i \\leq n - 1 $ } , \\\\ a ( n ; i , j ) + w _ { i - 3 , j - 1 } & = a ( n ; i + 1 , j ) , \\qquad \\mbox { $ 1 \\leq j < i - 1 < n - 1 $ } , \\\\ a ( n ; n - 1 , n ) & = a ( n - 2 ) , \\\\ a ( n ; n ; i ) & = a ( n - 1 , i ) , \\qquad \\mbox { i f $ 1 \\leq i \\leq n - 1 $ } . \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} s _ h = \\frac { h ! } { 2 ^ { h / 2 } ( h / 2 ) ! } . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} & \\left ( 1 - \\frac { x } { 1 - v } \\right ) A ( \\frac { x } { v } , v ) = 1 + \\frac { x } { v } - \\frac { x } { 1 - v } A ( x , 1 ) + \\frac { ( x ^ 2 + v x - v ) x ^ 5 } { v ( 1 - 2 x ) ( v - ( 1 + v ) x ) ( 1 - x ) ^ 2 ( v - x ) ^ 2 } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac { v } { v - x } B ' ( x / v , v ) + \\frac { x ^ 4 } { v ( v - ( 1 + v ) x ) ( v - x ) ^ 2 } , \\\\ & \\left ( 1 - \\frac { x } { v ( 1 - v ) } \\right ) B ' ( \\frac { x } { v } , v ) = \\frac { x ^ 2 } { v ^ 2 } A ( x , 1 ) + \\frac { x ^ 2 } { v ^ 3 } ( A ( x , 1 ) - 1 ) - \\frac { x } { v ^ 3 ( 1 - v ) } B ' ( x , 1 ) . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} I _ 0 ( t ) = - \\frac { 1 } { 2 } t | t | N _ 0 S ^ { - 1 / 2 } P . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} a _ 0 * \\ldots * a _ { k } = \\left ( \\frac { \\det { \\overline C ^ { k + 1 } } } { \\det { \\overline C ^ { k } } } \\right ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} h ^ { t } - h ^ 0 = \\xi _ + ^ { t } + \\xi _ - ^ { t } \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} J ^ N _ i ( \\alpha ^ N ) : = E \\left [ \\int _ 0 ^ T c ( t , X _ i ^ N ( t ) , \\alpha _ i ^ N ( t ) , \\mu ^ N ( t ) ) d t + \\Psi ( X _ i ^ N ( T ) , \\mu ^ N ( T ) ) \\right ] . \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} V _ n \\subset \\overline { V } _ n \\subset V _ { n + 1 } \\ ; n \\in \\N \\ ; \\cup _ { n = 1 } ^ { \\infty } V _ n = U . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} E ( \\gamma a , \\gamma b ) \\hookrightarrow \\triangle ( 1 , 1 ) \\times _ L \\square ( a , b ) = \\{ ( x _ 1 + i y _ 1 , x _ 2 + i y _ 2 ) | x _ 1 , x _ 2 , y _ 1 , y _ 2 > 0 , \\ , x _ 1 + x _ 2 < 1 , \\ , y _ 1 < a , y _ 2 < b \\} , \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} d \\operatorname { V a r } _ { \\widetilde { \\nabla } _ t } ( \\beta ^ A _ t ) = j ^ * _ A ( \\operatorname { V a r } _ { \\nabla _ t } ( \\omega ) ) = j ^ * _ A ( \\operatorname { V a r } _ { \\widetilde { \\nabla } _ t } ( \\omega ) ) , \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} \\mathcal { X } = L ^ { 2 } \\left ( \\Omega , \\mathbb { C } ^ { 3 } \\right ) \\oplus \\left ( L ^ { 2 } \\left ( \\Omega , \\mathrm { s y m } \\left [ \\mathbb { C } ^ { 3 \\times 3 } \\right ] \\right ) \\oplus \\tilde { X } \\right ) \\oplus L ^ { 2 } \\left ( \\Omega , \\mathbb { C } ^ { 3 } \\right ) \\oplus \\left ( L ^ { 2 } \\left ( \\Omega , \\left [ \\mathbb { C } ^ { 3 } \\right ] \\right ) \\oplus X \\right ) . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\theta = \\frac { 4 N - 8 } { 4 N - 7 } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ( - z ) = u _ 0 ( z ) - u _ 1 ( z ) + \\sum _ { k = 2 } ^ \\infty ( - 1 ) ^ k u _ k ( z ) . \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\tilde \\rho _ { ( 1 ) } ^ r ( x ) = \\frac { x ^ { L / 2 - 1 } e ^ { - x } } { 2 \\Gamma ( L / 2 ) } \\left ( 1 - x ^ { L / 2 + 1 } \\overline { \\gamma } ( L / 2 + 1 , x ) + \\frac { ( 2 x ) ^ L } { B ( L / 2 , 1 / 2 ) } \\overline { \\gamma } ( L + 1 , 2 x ) \\right ) , \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} 1 5 \\ge p _ a ( \\bar { C } ) = \\frac { 1 } { 2 } \\left ( 6 ^ 2 - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { 1 6 } a _ i ^ 2 \\right ) + 1 . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\dfrac { d ^ 2 y } { d x ^ 2 } = ( \\varphi ( x ) + h ) y ( I ) \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} v ^ t _ 1 ( x ) : = - \\int _ { \\R ^ n } d y \\left ( \\frac { \\Delta h ^ 0 } { t } \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } \\right ) ( y ) P ( x - y ) \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\frac { N _ { 2 d } ^ { \\textrm { W e y l } } ( X ) } { N _ { 2 d } ^ { \\textrm { c m } } ( X ) } = 1 + O ( X ^ { - \\beta ( d ) } ) \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{gather*} P _ { ( r ) } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) = \\prod _ { j = 2 } ^ N { \\frac { \\big ( q ^ { r } t ^ { j - 1 } ; q \\big ) _ { \\infty } ( t ^ j ; q ) _ { \\infty } } { ( q ^ r t ^ j ; q ) _ { \\infty } ( t ^ { j - 1 } ; q ) _ { \\infty } } } = { \\frac { ( q ^ r t ; q ) _ { \\infty } ( t ^ N ; q ) _ { \\infty } } { ( q ^ r t ^ N ; q ) _ { \\infty } ( t ; q ) _ { \\infty } } } = \\frac { \\big ( t ^ N ; q \\big ) _ r } { ( t ; q ) _ r } . \\end{gather*}"}
-{"id": "776.png", "formula": "\\begin{align*} a ^ { i j } _ k D _ { i j } ( u - u _ k ) = f - f _ k + ( a ^ { i j } _ k - a ^ { i j } ) D _ { i j } u \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} x _ { n + 1 } ^ 2 = 0 ; \\ , x _ { n + 1 } \\cdot x _ { n + 2 } ^ 2 = 0 ; \\ , x _ { n + 1 } \\cdot x _ { n + 2 } \\cdot x _ { n + 3 } ^ 2 = 0 ; \\ , \\cdots ; \\ , x _ { n + 1 } \\cdot x _ { n + 2 } \\cdots x ^ 2 _ { 2 n - 2 } = 0 . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} F ( M ' ) \\circ F ( D ) \\circ F ( M ) \\circ F ( D ' ) & = ( + \\oplus \\Delta ^ { \\dagger } ) \\circ ( \\Delta \\oplus + ^ { \\dagger } ) \\circ ( \\Delta ^ { \\dagger } \\oplus + ) \\circ ( + ^ { \\dagger } \\oplus \\Delta ) \\\\ & = \\mathrm { i d } _ 2 \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} F ( z ) - R _ { p , k } ( z ) = O \\bigg ( \\frac { \\Psi _ { p } ( z ) } { \\widetilde { \\Psi } _ { p , k } } \\bigg ) \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} \\mathcal { M } _ { Y } ( t ) = \\prod _ { i = 1 } ^ n ( e ^ t p _ i + 1 - p _ i ) . \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\| u \\| _ { Z ^ k } ^ 2 = \\| u \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ x ^ k u \\| _ { L ^ 2 } ^ 2 + \\| L _ y ^ 2 \\partial _ x u \\| _ { L ^ 2 } ^ 2 , \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\mathbf x = ( \\mathbf x _ 1 + \\cdots + \\mathbf x _ n ) / n , \\ \\mathbf y = ( \\mathbf y _ 1 + \\cdots + \\mathbf y _ n ) / n . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\Phi _ H ( t , \\omega , h ) : = \\Phi ( t , \\omega ) h , \\ ; \\ ; \\ ; t \\geq 0 , \\omega \\in \\Omega , h \\in H . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\Gamma : = ( c \\ , \\Im m _ { \\mu _ A \\boxplus \\mu _ B } + \\hat { \\Lambda } ) ^ { - 2 } \\big ( | \\Lambda _ A | ^ 2 + | \\Lambda _ B | ^ 2 \\big ) + \\Big ( \\frac { N ^ { 5 \\varepsilon } } { ( N \\eta ) ^ { \\frac 1 3 } } \\Big ) ^ { - 2 } | \\Upsilon | ^ 2 + \\Big ( \\frac { N ^ { 5 \\varepsilon } } { \\sqrt { N \\eta } } \\Big ) ^ { - 1 } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( | T _ { i } | ^ 2 + N ^ { - 1 } ) ^ { \\frac 1 2 } , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\ , V _ { \\underline { d } '' } ^ { X _ 1 , 0 } - \\ , V _ { \\underline { d } ''' } ^ { X _ { 3 } ^ { c } , 0 } = \\ , V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} \\partial { \\mathcal M } ( H ; \\alpha _ - , \\alpha _ + ) = \\bigcup _ { \\alpha } { \\mathcal M } ( H ; \\alpha _ - , \\alpha ) \\ , \\ , { } _ { { \\rm e v } _ { + } } \\times _ { { \\rm e v } _ { - } } { \\mathcal M } ( H ; \\alpha , \\alpha _ + ) . \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} a _ i = \\lim _ { n \\to \\infty } a _ { n i } ( i = 1 , \\ldots , r ) \\quad D = a _ { n 1 } A _ 1 + \\cdots + a _ { n r } A _ r . \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} | f ( a _ n ) | \\ge | p _ n ( a _ n ) | - \\sum _ { k = 1 } ^ { n - 1 } | p _ k ( a _ n ) | - \\sum _ { k = n + 1 } ^ \\infty | p _ k ( a _ n ) | > n . \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} \\theta _ { v } ( 0 , 2 ) - \\theta _ { v } ( 0 , 1 ) & = 0 \\ , , \\\\ \\int _ { B _ 1 } \\abs { D _ L v } ^ { 2 } & = 0 \\ , . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} N _ { | f | \\chi _ B } = N _ { | f | } N _ { \\chi _ B } = | x | e ^ { \\abs { x } } ( s , \\infty ) = \\int _ { ( s , \\infty ) } \\lambda d e ^ { | x | } ( \\lambda ) \\geq s e ^ { \\abs { x } } ( s , \\infty ) = N _ { s \\chi _ B } , \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Bigl ( \\frac { r } { 1 + r ^ 2 } \\Bigr ) ^ q \\frac { d r } { r ^ { \\alpha + 1 } } = \\frac { \\Gamma \\bigl ( \\frac { q - \\alpha } { 2 } \\bigr ) \\ , \\Gamma \\bigl ( \\frac { q + \\alpha } { 2 } \\bigr ) } { 2 \\ , \\Gamma ( q ) } \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} T ( z , t ) = t + \\Phi ( T ( z , t ) ) . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\alpha = 0 \\ , . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } H ( t , \\rho ) | _ { \\rho = 0 } & = & 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) H ^ 2 ( s , 0 ) d s + 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) \\frac { \\partial } { \\partial \\rho } H ( s , 0 ) d s \\\\ & = & 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) G ^ 2 ( s ) d s + 3 ! \\int \\limits _ { - 1 } ^ t f ( s ) F ( s ) d s . \\end{array} \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\eta ^ \\epsilon ( x , y ) : = \\hat { ( \\hat { x } | x | ^ { 1 / \\epsilon } - y ) } | \\hat { x } | x | ^ { 1 / \\epsilon } - y | ^ { \\epsilon } \\ \\ K : = K ( \\hat { x } | x | ^ { 1 / \\epsilon } , y ) . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\delta ( B ^ { \\beta , \\alpha } ( s ) - B ^ { \\beta , \\alpha } ( u ) ) = \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { d / 2 } \\int _ { \\mathbb { R } ^ { d } } e ^ { i ( \\lambda , B ^ { \\beta , \\alpha } ( s ) - B ^ { \\beta , \\alpha } ( u ) ) } \\ , d \\lambda \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} \\sum _ { \\substack { q _ m \\le z \\\\ \\omega _ 0 ( q _ m ) > 4 k \\log \\log z } } \\frac 1 { q _ m } = \\sum _ { \\substack { q _ m \\le z \\\\ \\omega ( q _ m - 1 ) > 4 k \\log \\log z } } \\frac 1 { q _ m } \\le \\sum _ { \\substack { n \\le z \\\\ \\omega ( n ) > 4 k \\log \\log z } } \\frac 1 n . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\gamma _ m ( 2 , n ) = \\gamma _ m ( 2 , n - 1 ) + \\gamma _ m ( 1 , n - m ) . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\tilde { F } = ( \\tilde { F } _ 1 ^ \\top , \\ldots , \\tilde { F } _ m ^ \\top ) ^ \\top , \\tilde { F } _ i ( \\tilde { v } ) : = \\mathbb { E } \\left [ \\tilde { f } \\left ( \\sum _ { j = 1 } ^ m \\tilde { v } _ j \\Phi _ j ( \\cdot ) , \\cdot \\right ) \\Phi _ i ( \\cdot ) \\right ] \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} R _ Z ( p , p + n ) = { \\mathbb E } \\bigl ( \\bigl ( Z _ { p + 1 } ^ H - Z _ p ^ H \\bigr ) \\bigl ( Z _ { p + n + 1 } ^ H - Z _ { p + n } ^ H \\bigr ) \\bigr ) , p \\ge 1 . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\widetilde { V } ( z ) = \\theta ^ { 2 k + 1 } ( z ) \\overline { \\theta ( z ) } y ^ { \\frac { k + 1 } { 2 } } , \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\log | Q _ N | = k N \\log N + O ( N ) . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} K _ h ( t , x , y ) = \\frac 1 { \\sqrt { 2 h t } } \\ k \\left ( h ^ { - 2 } t , h ^ { - 1 } ( x + \\frac 1 { 4 t } y ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} \\chi _ { n } \\left ( { u , \\xi } \\right ) = 2 F _ { n } \\left ( \\xi \\right ) - \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { G _ { n , s } \\left ( \\xi \\right ) } { u ^ { s } } , } \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} ( D ( u ) n ) _ { \\textrm { t a n } } = 0 , \\ u \\cdot n = 0 \\textrm { o n } \\partial \\Pi . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\| \\xi - \\xi \\chi _ { A _ n } \\| _ { L _ 2 } & \\leq \\| \\xi \\| _ { L _ 2 } \\| \\| \\chi _ { [ 0 , \\alpha ) } - \\chi _ { A _ n } \\| _ { L _ 2 } = \\| \\xi \\| _ { L _ 2 } m ( A _ n ^ c ) \\\\ & \\leq \\frac 1 n \\| \\xi \\| _ { L _ 2 } \\to 0 n \\to \\infty . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} ( m \\Omega ) \\vartriangleleft n = ( J _ \\phi n ^ * J _ \\phi ) m \\Omega = ( J _ \\phi \\sigma _ { i / 2 } ^ \\phi ( n ) ^ * J _ \\phi ) m \\Omega = m n \\Omega = ( m \\Omega ) \\cdot n . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} f '' + \\left ( \\frac { a ' } { a } + 2 n \\frac { b ' } { b } \\right ) f ' - ( f ' ) ^ 2 = 2 f '' ( 0 ) \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} | I _ { u , n } | = n = q d + r \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} X _ 1 \\times _ Y X _ 2 = ( - 1 ) ^ { ( \\dim X _ 1 - \\dim Y ) ( \\dim X _ 2 - \\dim Y ) } X _ 2 \\times _ Y X _ 1 , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) = \\sum _ { \\ell \\in \\N ^ n : | j + \\ell | \\leq k } \\frac { 1 } { \\ell ! } \\ , f ^ { ( j + \\ell ) } ( y ) \\ , ( x - y ) ^ \\ell + R _ j ( x , y ) , x , y \\in F , \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon } ( x , t ) = \\Gamma ( x , t ) - \\varepsilon ( A t + | x | ^ { 2 } ) , \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\| N ^ { k , 1 } \\| _ { \\widetilde S _ q ^ p } + \\| N ^ { k , 2 } \\| _ { \\widetilde D _ { q , q } ^ p } + \\| N ^ { k , 3 } \\| _ { \\widetilde D _ { p , q } ^ p } & < \\frac { 1 } { 2 ^ k } , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; k \\geq 1 , \\\\ \\| N ^ { 0 , 1 } \\| _ { \\widetilde S _ q ^ p } + \\| N ^ { 0 , 2 } \\| _ { \\widetilde D _ { q , q } ^ p } + \\| N ^ { 0 , 3 } \\| _ { \\widetilde D _ { p , q } ^ p } & \\leq 2 \\| M ^ { n _ 0 } \\| _ { \\mathcal A _ { p , q } } . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\Delta ^ e ( x _ i ) & = x _ i \\otimes 1 + 1 \\otimes x _ i & \\Delta ^ e ( y _ i ) & = y _ i \\otimes 1 + 1 \\otimes y _ i \\\\ \\varepsilon ^ e ( x _ i ) & = 0 & \\varepsilon ^ e ( y _ i ) & = 0 \\\\ S ^ e ( x _ i ) & = - x _ i & S ^ e ( y _ i ) & = - y _ i , \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n K ( p _ { f _ 0 , i } , p _ { f , i } ) & = \\frac { 1 } { 2 } \\log { \\left ( \\frac { \\sigma ^ 2 } { \\sigma _ 0 ^ 2 } \\right ) } - \\frac { 1 } { 2 } \\left ( 1 - \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } \\right ) + \\frac { \\| f - f _ 0 \\| _ n ^ 2 } { 2 \\sigma ^ 2 } , \\\\ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n V ( p _ { f _ 0 , i } , p _ { f , i } ) & = \\frac { 1 } { 2 } \\left ( 1 - \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } \\right ) ^ 2 + \\frac { \\sigma _ 0 ^ 2 \\| f - f _ 0 \\| _ n ^ 2 } { \\sigma ^ 4 } . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\| f \\| _ { H [ Z ] ^ { \\wedge } } = \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| f _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in H [ Z ] , n \\in \\N \\} . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} Q ( x _ 0 + y _ 0 , \\dots , x _ { \\rho ( X ) } + y _ { \\rho ( X ) } ) - Q ( y _ 0 , \\dots , y _ { \\rho ( X ) } ) & = \\log 2 \\cdot x _ 0 + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\rho ( X ) } \\Lambda ( q _ i ) x _ i y _ i + \\sum _ { i = 1 } ^ { \\rho ( X ) } \\Lambda ( q _ i ) x _ i ^ 2 \\\\ & = \\sum _ { i = 0 } ^ { \\rho ( X ) } x _ i Q _ i ( y _ 0 , \\ldots , y _ { \\rho ( X ) } ) + \\sum _ { i = 1 } ^ { \\rho ( X ) } \\Lambda ( q _ i ) x _ i ^ 2 . \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} \\Omega _ { s + 1 } - \\Omega _ s > \\delta \\cdot ( \\beta _ { s + 1 } - \\beta _ s ) = 2 ^ { - m _ i } , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { - 2 \\pi i h x } } { \\lvert z \\rvert ^ { 2 w - k } z ^ k } d x = \\frac { y ^ { - w } i ^ { - k } \\pi ^ w \\lvert h \\rvert ^ { w - 1 } } { \\Gamma ( w + \\frac { \\lvert h \\rvert } { h } \\frac { k } { 2 } ) } W _ { \\frac { \\lvert h \\rvert } { h } \\frac { k } { 2 } , \\frac { 1 } { 2 } - w } ( 4 \\pi \\lvert h \\rvert y ) . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\left | { \\dd ^ m \\over \\dd z ^ m } S _ { n , r } ( z ) \\Big | _ { z = 0 } \\right | \\le 2 ( m - 1 ) ! \\ , . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} ( \\mathbb E \\| F \\| ^ p \\star \\nu ) ^ { \\frac 1 p } & = \\sup _ { G \\in Y ^ * : \\| G \\| \\leq 1 } \\mathbb E \\langle F , G \\rangle \\star \\nu = \\sup _ { G \\in Y ^ * : \\| G \\| \\leq 1 } \\mathbb E \\langle F \\star \\bar { \\mu } , G \\star \\bar { \\mu } \\rangle \\\\ & \\lesssim _ { p } \\sup _ { \\xi \\in L ^ { p ' } ( \\Omega ; X ^ * ) : \\| \\xi \\| \\leq 1 } \\mathbb E \\langle F \\star \\bar { \\mu } , \\xi \\rangle = ( \\mathbb E \\| F \\star \\bar { \\mu } \\| ^ p ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} { \\hat L } ( { \\tilde t } - \\epsilon | { \\tilde x } | ^ 2 , { \\tilde x } , { \\tilde \\tau } , { \\tilde \\xi } + 2 \\epsilon { \\tilde \\tau } { \\tilde x } ) = C \\big ( { \\tilde \\tau } - C ^ { - 1 } \\sum _ { j = 1 } ^ d A _ j ^ * ( t , x ) { \\tilde \\xi } _ j \\big ) \\\\ = C \\big ( { \\tilde \\tau } - C ^ { - 1 } A ^ * ( t , x , { \\tilde \\xi } ) \\big ) \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\langle { u , v , x , y } \\mid { u v u ^ { - 1 } = v ^ { - 1 } , ~ x y x ^ { - 1 } = y ^ { - 1 } } , ~ x ^ 2 = u ^ 4 v ^ { - 9 } , ~ y = u ^ 2 v ^ { - 4 } \\rangle . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} | \\Lambda _ \\iota ( z ) | = | \\omega _ \\iota ^ c ( z ) - \\omega _ \\iota ( z ) | \\prec \\frac { 1 } { \\sqrt { N } } \\ , , \\iota = A , B \\ , , z = E + \\mathrm { i } \\eta _ \\mathrm { M } \\ , , \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\frac { C _ { 3 , n } } { ( k - 5 ) ! } & = - [ 2 ( n - 1 ) a _ k + b _ { k - 1 } ] m _ { ( 3 , \\dot { 0 } ) } - [ 2 ( n - 1 ) a _ { k - 1 } + b _ { k - 2 } ] m _ { ( 2 , \\dot { 0 } ) } \\\\ & - [ 2 ( n - 1 ) a _ { k - 2 } + b _ { k - 3 } ] m _ { ( 1 , \\dot { 0 } ) } - 2 a _ k m _ { ( 2 , 1 , \\dot { 0 } ) } - 2 a _ { k - 1 } m _ { ( 1 ^ 2 , \\dot { 0 } ) } \\\\ & - \\frac { 6 n ( n - 1 ) } { k ( k - 1 ) } ( k - 2 ) a _ { k - 3 } - \\frac { 3 n } { k - 1 } b _ { k - 4 } , \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} p \\mid _ { \\Gamma } = p _ { 0 } \\left ( x \\right ) , \\partial _ { n } p \\mid _ { \\Gamma } = p _ { 1 } \\left ( x \\right ) , \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } ^ x = \\sum _ { \\substack { T \\\\ S \\thicksim T \\\\ x \\in T } } ( r - 2 | T | ) . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{gather*} \\theta ' = - ( n - 1 ) p _ { n n } - \\operatorname { t r } \\big ( \\omega ^ 2 \\big ) - \\operatorname { t r } \\big ( \\sigma ^ 2 \\big ) - \\frac { \\theta ^ 2 } { n - 1 } , \\end{gather*}"}
-{"id": "6679.png", "formula": "\\begin{align*} P _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) = \\sum _ { \\overline { n n ' } \\ e d g e \\ o f \\ \\Gamma ^ { o r b } } P ^ { n , n ' } _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) - \\sum _ { n \\in \\mathcal { N } } ( \\delta _ { n , \\mathcal { N } } - 1 ) P _ { h } ^ { n } ( \\mathbf { t } _ { \\mathcal { N } } ) , \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} ( 1 + 2 / k ) b \\le b ^ \\prime \\le k ^ 2 b , b ^ \\prime = \\left \\lceil \\frac { r ^ \\prime + 1 } { k - r ^ \\prime } a ^ \\prime \\right \\rceil , \\rho b ' \\ge b . \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\frac { b ^ k - y ^ k } { b - y } - \\frac { y ^ k - a ^ k } { y - a } = k ( y ^ { k - 1 } - a ^ { k - 1 } ) \\ , , \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} X ( t ^ * ) = X ( t ^ * - ) + \\zeta _ k , \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} B [ u , u u _ x ] - u B [ u , u _ x ] = C [ u , u , u _ x ] , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} W _ { \\alpha , \\nu } ( y ) = \\frac { y ^ { \\nu + \\frac { 1 } { 2 } } e ^ { - \\frac { y } { 2 } } } { \\Gamma ( \\nu - \\alpha + \\frac { 1 } { 2 } ) } \\int _ 0 ^ \\infty e ^ { - y t } t ^ { \\nu - \\alpha - \\frac { 1 } { 2 } } ( 1 + t ) ^ { \\nu + \\alpha - \\frac { 1 } { 2 } } d t , \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} m _ e ( \\lambda \\pm ) & = m _ e ^ 0 ( \\lambda \\pm ) + l ( \\lambda \\pm ) \\langle m _ e ^ 0 ( \\lambda \\pm ) , u \\rangle m _ 1 ( \\lambda \\pm ) \\\\ & = \\frac { m _ e ^ 0 ( \\lambda \\pm ) + l ( \\lambda \\pm ) \\big ( \\langle m _ e ^ 0 ( \\lambda \\pm ) , u \\rangle m _ 1 ^ 0 ( \\lambda \\pm ) - \\langle m _ 1 ^ 0 ( \\lambda \\pm ) , u \\rangle m _ e ^ 0 ( \\lambda \\pm ) \\big ) } { 1 - l ( \\lambda \\pm ) \\langle m _ 1 ^ 0 ( \\lambda \\pm ) , u \\rangle } . \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} w [ \\mathbf { u } , \\mathbf { u } ] = \\int _ { \\mathbb { R } ^ d } \\left \\langle g _ \\wedge ( \\mathbf { x } ) b _ \\wedge ( \\mathbf { D } ) \\mathbf { u } , b _ \\wedge ( \\mathbf { D } ) \\mathbf { u } \\right \\rangle _ { \\mathbb { C } ^ { m _ \\wedge } } d \\mathbf { x } . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta _ p u & = \\lambda | u | ^ { p - 2 } u + f | u | ^ { \\gamma - 2 } u & & \\mbox { i n } \\ \\ \\Omega \\\\ u & = 0 & & \\mbox { o n } \\ \\ \\partial \\Omega \\end{aligned} \\right . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} X _ \\pi ^ R = 2 \\Re \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi _ 0 ) } } \\frac { Z _ \\gamma } { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{gather*} \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { 0 } ^ { T } \\int _ { \\Omega } u _ { \\varepsilon } ( x , t ) v \\left ( x , t , \\frac { x } { \\varepsilon _ { 1 } } , \\cdots , \\frac { x } { \\varepsilon _ { n } } , \\frac { t } { \\varepsilon _ { 1 } ^ { \\prime } } , \\cdots , \\frac { t } { \\varepsilon _ { m } ^ { \\prime } } \\right ) d x d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { \\mathcal { Y } _ { n , m } } u _ { 0 } ( x , t , y ^ { n } , s ^ { m } ) v ( x , t , y ^ { n } , s ^ { m } ) d y ^ { n } d s ^ { m } d x d t \\end{gather*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\max _ { u \\in \\N _ 0 } \\left ( \\frac { | A \\cap [ u , u + d - 1 ] | } { d } \\right ) = 1 \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( M , g _ F ) = \\alpha ^ { \\frac { p } { 2 } } \\lambda _ { 1 , p } ( M , \\tilde { g } ) . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} ( x _ 1 , \\dotsc , x _ k \\ , , \\ , y _ 1 , \\dotsc , y _ m ) \\ \\longmapsto \\ ( x _ i y _ j \\mid i = 1 , \\dotsc , k \\mbox { a n d } j = 1 , \\dotsc , m ) \\ , . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\partial _ { t } ( n _ \\mu \\circ \\eta ) = - g ^ { k l } \\partial _ { k } v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\textrm { t r } ^ { \\flat } \\ ( \\mathcal { K } _ t ^ m \\ ) = \\sum _ { T _ t ^ m x = x } \\frac { \\exp \\ ( g _ { m , t } \\ ( x \\ ) \\ ) } { 1 - \\ ( \\ ( T _ t ^ m \\ ) ' \\ ( x \\ ) \\ ) ^ { - 1 } } . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} d E _ f ( \\dot c ) = - 4 R e ( \\langle \\Phi _ f , \\dot c \\rangle ) ~ . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} \\Delta ( i ) & = \\norm { \\tilde { T } _ i f \\cdot h } _ { \\ell ^ 1 ( \\N ^ * ) } = \\sum _ { k = 1 } ^ { d - i } f ( i ) q ^ { - k + 1 } - \\sum _ { k = i } ^ { d - i } ( k - i ) q ^ { - k + 1 } \\\\ & = f ( i ) \\sum _ { k = 1 } ^ { d - i } q ^ { - k + 1 } - q ^ { - i } \\sum _ { k = 0 } ^ { f ( i ) } k q ^ { - k + 1 } . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\det ( N M N ^ t ) = f ^ 2 g ^ 2 + f ^ 2 \\delta + g ^ 2 + \\delta - f ^ 2 g ^ 2 = ( f ^ 2 + 1 ) \\delta + g ^ 2 \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} [ E _ { \\Phi } : \\Q ] = 2 ^ v [ G : S ] . \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} N ( H _ { \\Lambda } ^ N , E ) ~ : = & ~ \\# \\{ \\lambda _ k ( H _ { \\Lambda } ^ N ) \\leq E \\} \\ , , \\\\ N ( H _ { \\Lambda } ^ D , E ) ~ : = & ~ \\# \\{ \\lambda _ k ( H _ { \\Lambda } ^ D ) \\leq E \\} \\ , . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} J _ { \\mathcal { G } _ { f , g } , g } ( x ) & = \\inf \\biggl \\{ I _ { \\mathcal { G } _ { f , g } , g } \\biggl ( \\frac { z } { 1 - x } , z \\biggr ) : z > 0 \\biggr \\} \\\\ & = \\inf \\biggl \\{ \\frac { z } { 1 - x } I _ f \\biggl ( \\frac { \\frac { z } { 1 - x } - z } { \\frac { z } { 1 - x } } \\biggr ) + I _ g ( z ) : z > 0 \\biggr \\} \\\\ & = \\inf \\biggl \\{ \\frac { z } { 1 - x } I _ f ( x ) + I _ g ( z ) : z > 0 \\biggr \\} \\\\ & = - \\sup \\biggl \\{ - z \\frac { I _ f ( x ) } { 1 - x } - I _ g ( z ) : z > 0 \\biggr \\} ; \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\theta _ { u } ( y _ j , 8 ) - \\theta _ { u } ( y _ j , \\rho _ 0 ) = 0 , \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} a _ c ( u , v ) : = \\int _ { \\R ^ N } \\nabla u \\cdot \\nabla v \\ , d \\mu - c \\sum _ { i = 1 } ^ n \\int _ { \\R ^ N } \\frac { u v } { | x - a _ i | ^ 2 } \\ , d \\mu \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\begin{aligned} & S ^ N ( m ) = \\{ R \\in C ^ \\infty ( T ^ * { \\bf R } ^ d ) : \\exists M , \\forall \\alpha , \\beta \\in { \\bf N } ^ { 2 } \\cr & \\exists C _ { \\alpha \\beta } > 0 , \\ | \\partial ^ \\alpha _ { x } \\partial ^ \\beta _ p H ( x , p ; h ) | \\leq C _ \\alpha h ^ N m ^ M \\} \\cr \\end{aligned} \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} { \\rm A i } ( z ) = { 1 \\over 2 \\pi } \\int _ { - \\infty } ^ \\infty d q \\thinspace e ^ { i \\left ( { 1 \\over 3 } q ^ 3 + q z \\right ) } \\ , , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} I _ { a ^ + } ^ { 1 - \\gamma - \\delta ( \\beta - 1 ) } \\hat { y } ( x ) { \\big | } _ { x = a } = { \\hat { y } } _ a , \\gamma = \\alpha + \\beta ( 1 - \\alpha ) . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} \\{ b _ i , b _ { i + 1 } \\} \\neq 0 , \\ ; \\ ; i = 0 , \\ldots , \\ell - 1 , \\{ b _ i , b _ j \\} = 0 , \\ ; \\ ; j = i + 2 , \\ldots , k \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} n - 3 - \\d = p _ 1 + p _ 2 + p _ 3 , \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\tilde { F } ^ { G E L } _ \\gamma ( x ) = \\sum _ { i = 1 } ^ n w ^ { \\gamma } _ i ( \\theta ) I _ { \\{ x _ i \\leq x \\} } \\ , . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} \\aligned \\sum _ { k _ 1 + k _ 2 = k + 1 } \\sum _ { i = 1 } ^ { k - k _ 2 + 1 } ( - 1 ) ^ * { \\frak m } _ { k _ 1 } ( x _ 1 , \\ldots , { \\frak m } _ { k _ 2 } ( x _ i , \\ldots , x _ { i + k _ 2 - 1 } ) , \\ldots , x _ { k } ) = 0 . \\endaligned \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} z & = [ n ' , n '' ] = [ z ' + y ' , z '' + y '' ] \\\\ & = [ z ' , z '' ] + [ z ' , y '' ] + [ y ' , z '' ] + [ y ' , y '' ] = [ y ' , y '' ] , \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} ( D _ s f ) ( e ^ { i \\theta } ) = \\left \\{ \\begin{array} { l l } f ( e ^ { i \\theta s } ) , & \\theta s \\in [ 0 , 2 \\pi ) , \\\\ 0 , & \\theta s \\not \\in [ 0 , 2 \\pi ) , \\end{array} \\right . \\theta \\in [ 0 , 2 \\pi ) . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle { \\frac { d D ( t ) } { d t } } & = & - r ( \\beta ) D ( t ) + I _ 4 ( t ) \\\\ \\\\ \\displaystyle { \\frac { d E ( t ) } { d t } } & = & r ( \\beta ) D ( t ) - O _ 5 ( t ) \\\\ \\\\ \\displaystyle { \\frac { d F ( t ) } { d t } } & = & r ( \\beta ) D ( t ) - O _ 6 ( t ) . \\end{array} \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] & = 0 & [ X _ 1 , X _ 3 ] & = 0 & [ X _ 1 , X _ 4 ] & = 0 & [ X _ 1 , X _ 5 ] & = 0 & [ X _ 2 , X _ 3 ] & = X _ 1 \\\\ [ X _ 2 , X _ 4 ] & = 0 & [ X _ 2 , X _ 5 ] & = X _ 3 & [ X _ 3 , X _ 4 ] & = 0 & [ X _ 3 , X _ 5 ] & = - X _ 2 & [ X _ 4 , X _ 5 ] & = \\epsilon X _ 1 \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} a _ \\gamma = \\int _ { \\widehat { \\Gamma } } \\overline { \\chi ( \\gamma ) } \\alpha ^ c _ \\chi ( a ) \\ , d \\mu ( \\chi ) . \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\int _ { G } f \\left ( g \\right ) d g = c \\int _ { K } d k _ { 1 } \\int _ { \\mathfrak { a } ^ { + } } \\delta \\left ( H \\right ) d H \\int _ { K } f \\left ( k _ { 1 } \\left ( \\exp H \\right ) k _ { 2 } \\right ) d k _ { 2 } , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} w ( a , b ) = a ^ { n _ j } b ^ { n _ { j - 1 } } . . . b ^ { n _ 2 } a ^ { n _ 1 } \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} - \\sum _ { j \\ne i } | a _ i - a _ j | ^ 2 + \\frac { n + 1 } { 2 } | x - a _ i | ^ 2 \\le & \\sum _ { i = 1 } ^ n | x - a _ i | ^ 2 \\\\ & \\le ( 2 n - 1 ) | x - a _ i | ^ 2 + 2 \\sum _ { j \\ne i } | a _ i - a _ j | ^ 2 \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ n \\left ( 1 + x _ j + x _ { n + j } + x _ j \\cdot x _ { n + j } \\right ) . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( \\mathcal I _ { \\forall _ x \\psi , b } P ) & = ( l _ { \\langle \\rangle } ( P ) \\backslash \\{ \\forall _ x \\psi \\} ) \\cup \\{ \\psi ( b ) \\} , \\\\ o _ { \\langle \\rangle } ( \\mathcal I _ { \\forall _ x \\psi , b } P ) & = o _ { \\langle \\rangle } ( P ) \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} w = \\pm \\sqrt { \\frac { c } \\lambda } q \\ , . \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} 0 > \\alpha ( h ^ { \\ast } ) = \\sum _ { i \\ne L } a _ i { \\alpha } _ i ( h ^ { \\ast } ) + a _ L { \\alpha } _ L ( h ^ { \\ast } ) = a _ L { \\alpha } _ L ( h ^ { \\ast } ) , \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} P _ h ( z , s ) : = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma _ 0 ( N ) } \\Im ( \\gamma z ) ^ s e ( h \\gamma \\cdot z ) . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} { q } ^ { T } _ { i } ( x ) = \\bar { q } _ { m i n } + \\dfrac { \\bar { q } _ { m a x } } { 2 } \\left ( 1 + \\theta ~ \\tanh \\left ( \\beta \\left ( \\dfrac { x - x _ { i - 1 / 2 } } { x _ { i + 1 / 2 } - x _ { i - 1 / 2 } } - \\tilde { x } _ { i } \\right ) \\right ) \\right ) , \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\frac { f ( n ) } { n } = \\frac { f ( q d + r ) } { n } \\leq \\frac { q f ( d ) } { q d } + \\frac { f ( r ) } { n } = \\frac { f ( d ) } { d } + \\frac { f ( r ) } { n } . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} g _ { 1 } ( [ U , V ] , Z ) & = \\cos ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , Z ) - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) + \\nabla _ { U } ^ { \\pi } \\pi _ { \\ast } \\varphi V , \\pi _ { \\ast } w Z ) \\\\ & - \\cos ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { V } U , Z ) + g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( V , \\varphi U ) + \\nabla _ { V } ^ { \\pi } \\pi _ { \\ast } \\varphi U , \\pi _ { \\ast } w Z ) . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { M i n i m i z e } & f ( x ) , \\\\ \\mbox { s u b j e c t t o } & h ( x ) = 0 , \\\\ & g ( x ) \\leq 0 , \\end{array} \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} r _ 1 + \\cdots + r _ { m - k + 1 } = k \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 ^ + } \\sup _ { t \\in [ t _ 0 - \\epsilon , t _ 0 + \\epsilon ] } \\bigg | \\frac { Z ^ H ( t ) - Z ^ H ( t _ 0 ) } { t - t _ 0 } \\bigg | = + \\infty , \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\alpha _ 2 = \\{ ( V , ' I ' , V '' , I '' , \\phi _ 1 , \\ldots , I _ 4 ) | V ' = \\phi _ 2 - \\phi _ 1 , I ' = I _ 1 = - I _ 2 \\phantom { h h h h h } \\\\ [ - 3 e m ] \\begin{aligned} \\\\ \\\\ \\phantom { h h h } V '' = \\phi _ 4 - \\phi _ 3 , \\\\ \\end{aligned} \\begin{aligned} \\\\ \\\\ I '' = I _ 3 = - I _ 4 \\} \\phantom { h h h } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\mathbb { E } \\phi ( X _ t , Y _ t ) - \\phi ( x , y ) } { t } = \\dfrac { 1 } { 2 } \\dfrac { \\partial ^ 2 \\phi } { \\partial y ^ 2 } - ( c _ 0 y + k x ) \\dfrac { \\partial \\phi } { \\partial y } + y \\dfrac { \\partial \\phi } { \\partial x } , \\mbox { i f } | x | < 1 . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k \\frac { \\eta _ j ( 0 , t _ j - t _ { j - 1 } ] ^ { \\ell _ j } } { \\ell _ j ! } e ^ { - \\eta _ j ( 0 , t _ j - t _ { j - 1 } ] } = \\prod _ { j = 1 } ^ k \\P \\big ( N _ j ( 0 , t _ j - t _ { j - 1 } ] = \\ell _ j \\mid \\eta _ j \\big ) , \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\int _ { t _ \\ell } ^ { t _ \\ell + \\Delta } \\overline \\lambda _ 0 \\ , d s = \\mathcal E _ \\ell \\iff \\Delta = \\frac { \\mathcal E _ \\ell } { \\overline \\lambda _ 0 } . \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\mathfrak { X } _ i ^ + ( u , \\lambda ) * \\mathfrak { X } _ j ^ + ( v , \\zeta ) = - g _ { \\lambda _ i } ( u + z _ i ) g _ { \\zeta _ j } ( v + z _ j ) \\prod _ { m \\in S } \\vartheta ( z _ { j } - z _ i + m \\frac { \\hbar } { 2 } ) . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} P _ 2 = 2 ^ { r - 7 } \\left [ ( | S | - 2 ) ( | S | - 3 ) + ( r - | S | - 2 ) ( r - | S | - 3 ) + 8 ( r - | S | - 2 ) + 8 \\right ] , \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} q \\left ( \\frac { i + 1 } { 2 } , \\frac { i - 1 } { 2 } \\right ) = p ( x + i \\tilde { y } ) \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} & T ^ { 1 , 0 } X \\oplus \\{ \\mathbb { C } ( T - i \\frac { \\partial } { \\partial \\eta } ) \\} , \\\\ & J T = \\frac { \\partial } { \\partial \\eta } , \\ J u = i u \\ \\ u \\in T ^ { 1 , 0 } X . \\\\ \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\mathcal { A } ^ N : = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\alpha ^ N , \\xi ^ N , \\N ^ N ) \\right \\} \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\lambda ( t ) = 1 - c + c \\tanh \\bigl ( 4 \\cos ( 2 \\pi \\nu t ) \\bigr ) , \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} H _ d ( x ) = x ^ { d + 2 } \\left ( F _ T ( x ) - \\frac { 1 } { 1 - x } \\right ) + x ^ { d + 2 } \\left ( \\frac { 1 } { 1 - x } + d \\frac { x } { ( 1 - x ) ^ 2 } \\right ) \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} \\omega & = | V ( W ) \\backslash ( A \\cup B ) | + | V ( W ) \\cap ( A \\cup B ) | \\leq | V ( C ) \\backslash ( A \\cup B ) | + | V ( W ) \\cap ( A \\cup B ) | \\\\ & \\leq | V ( C ) | - | A \\cup B | + | V ( W ) \\cap A | + | V ( W ) \\cap B | \\leq c - ( d - 1 ) ( t - 1 ) . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} F = ( F _ 1 ^ \\top , \\ldots , F _ m ^ \\top ) ^ \\top , F _ i ( \\hat { v } ) : = \\mathbb { E } \\left [ f \\left ( \\sum _ { j = 1 } ^ m \\hat { v } _ j \\Phi _ j ( \\cdot ) , \\cdot \\right ) \\Phi _ i ( \\cdot ) \\right ] \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} F _ T ( x ) - 1 - x & = x C ( x ) \\big ( F _ T ( x ) - 1 \\big ) \\\\ & + \\frac { ( x ^ 4 - 2 x ^ 3 + 5 x ^ 2 - 4 x + 1 ) C ( x ) - x ^ 3 - 2 x ^ 2 + 3 x - 1 } { ( 1 - 2 x ) ^ 2 } . \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{gather*} \\Gamma _ q ( \\theta N ) \\prod _ { i = 1 } ^ N { \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i + 1 ) - z ) } } = ( 1 - q ) \\frac { ( q ; q ) _ { \\infty } } { ( q ^ { \\theta N } ; q ) _ { \\infty } } \\prod _ { i = 1 } ^ N { \\frac { \\big ( q ^ { \\lambda _ i + \\theta ( N - i + 1 ) - z } ; q \\big ) _ { \\infty } } { \\big ( q ^ { \\lambda _ i + \\theta ( N - i ) - z } ; q \\big ) _ { \\infty } } } \\end{gather*}"}
-{"id": "7808.png", "formula": "\\begin{align*} d _ { i , \\alpha } { \\cal F } ( i _ 0 ) \\circ { \\bf T } _ 0 - { \\rm I d } = { \\cal P } ( i _ 0 ) + { \\cal P } _ \\omega ( i _ 0 ) + { \\cal P } _ \\omega ^ \\bot ( i _ 0 ) \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\psi _ 1 ^ k \\prod _ { j = 2 } ^ n \\psi _ i ^ { l _ j } + \\Psi _ { \\vec { l } } , \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\mathbf { T } _ N M = \\mathbf { T } N \\oplus \\mathbf { T } ^ { \\perp _ g } N . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\frak H _ { c a } ^ i = ( \\frak N _ { c b } ^ { i + 1 } \\circ \\frak H _ { b a } ^ i ) \\cup ( \\frak H _ { c b } ^ i \\circ \\mathcal { I D } ) . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} \\prod _ { i \\in I } \\prod _ { j = 1 } ^ { v ^ i } \\frac { \\vartheta ( z ^ { ( i ) } _ j + \\lambda _ i + c _ { k i } \\hbar ) } { \\vartheta ( z ^ { ( i ) } _ j ) \\vartheta ( \\lambda _ i + c _ { k i } \\hbar ) } , \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} D ( w _ { d + 1 } , 1 , c , 0 ) & = D ( \\epsilon , L , K , K ) , \\\\ D ( w _ { d + 1 } , 1 , c , K ) & = \\mathcal { A } ( [ w _ 1 , . . . , w _ K ] , [ w _ { K + 1 } , . . . , w _ { K + L } ] ) , \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} K ( w ) : = \\frac { T \\pi } { \\pi ^ 2 - T ^ 2 w ^ 2 } \\ , , w \\in \\C \\ , , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} u ( t ) = \\log ( t ) + \\phi ( t ) ; g _ 1 ( t ) = 1 + t \\phi ' ( t ) g _ 2 ( t ) = \\frac { 1 } { g _ 1 ( t ) ^ 2 } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\psi ( t ) - { \\tilde { \\psi } } ( t ) = - 2 \\int _ { [ 0 , t ] \\times [ 0 , 1 ] } \\left ( { { 1 } } _ { [ 0 , r _ 0 ( \\boldsymbol { \\zeta } ( s ) ) ) } ( y ) - { { 1 } } _ { [ 0 , r _ 0 ( { \\tilde { \\boldsymbol { \\zeta } } } ( s ) ) ) } ( y ) \\right ) \\varphi _ 0 ^ \\varepsilon ( s , y ) \\ , d s \\ , d y . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} ( \\alpha \\beta ) ^ n = 3 x ^ 2 + 2 d ^ 2 = y ^ n , \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\max _ { x \\in M } \\log \\lVert A ^ n ( x ) \\rVert = \\sup _ { ( x , p ) \\in M \\times \\mathbb N : f ^ p ( x ) = x } \\log \\rho ( A ^ p ( x ) ) ^ { 1 / p } . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\beta = \\norm { u } = \\norm * { \\sum _ { i \\in I } \\alpha _ { i } x _ { i } } \\leq \\sum _ { i \\in I } \\alpha _ { i } \\norm { x _ { i } } = \\beta \\sum _ { i \\in I } \\alpha _ { i } . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} M _ { i , l } = \\begin{cases} \\frac { 1 } { n } & l \\in t _ i \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\sum _ { \\alpha ; E ( \\alpha _ - ) < E ( \\alpha ) < E ( \\alpha _ + ) } \\frak m ^ { \\epsilon } _ { 1 ; \\alpha _ + , \\alpha } \\circ \\frak m ^ { \\epsilon } _ { 1 ; \\alpha , \\alpha _ - } = 0 . \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} 1 = ( - 1 ) ^ { b _ 0 } \\epsilon ^ { b _ 1 } w ^ { b } ( \\sqrt { 2 } ) ^ { - 3 } + ( - 1 ) ^ { b _ 0 } \\bar \\epsilon ^ { b _ 1 } \\bar w ^ { b } ( - \\sqrt { 2 } ) ^ { - 3 } , \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\mathrm { s u p p } \\ , \\mu _ \\alpha \\boxplus \\mu _ \\beta = [ E _ - , E _ + ] \\ , , \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\partial ^ { j } } { \\partial \\rho ^ { j } } \\left ( H ( t , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = j ! \\int \\limits _ { - 1 } ^ t f ( s ) r _ j ( s ) d s \\mbox { f o r e a c h } j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\lambda _ 1 ^ \\theta ( t ) = 6 0 + \\theta \\sin ( \\tfrac { 2 \\pi t } { 2 4 } ) \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} V + V ' = V '' & & & I = I ' = I '' & & & \\\\ V '' = \\phi _ 8 - \\phi _ 7 & & & I '' = I _ 7 = - I _ 8 \\\\ \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} \\mathcal { G } _ 0 ( U ) : = \\bigg \\{ & f \\in \\mathcal { O } _ { X \\times \\mathbb { B } ^ 1 } ( U \\times \\mathbb { B } ^ 1 ) = \\mathcal { O } _ X ( U ) \\langle T \\rangle \\ , \\bigg | \\ , f = 1 + \\sum _ { i = 1 } ^ \\infty f _ i T ^ i \\\\ & \\vert f _ i ( u ) \\vert < 1 \\ \\forall i \\geq 1 \\ \\forall u \\in U \\bigg \\} \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\quad \\lim _ { k _ 0 < k \\to \\infty } \\frac { F ^ { \\bf a } _ { k + 1 } } { F ^ { \\bf a } _ k } = \\lambda _ 0 , \\ , \\ , \\ , F ^ { \\bf a } _ k \\ne 0 \\ , \\ , \\ , \\ , \\ , \\ , k > k _ 0 , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} E _ { ( B , \\beta ) _ n } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { \\prod _ { j = 1 } ^ n \\Gamma ( \\beta _ j + k B _ j ) } , \\ ; z \\in \\mathbb { C } . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } \\right ) \\geq a ^ { a / 2 } - \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\sum _ { n = 1 } ^ \\infty \\frac { 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - 1 } = \\frac { \\mathtt { f } ' _ { a , \\nu } ( 1 ) } { \\mathtt { f } _ { a , \\nu } ( 1 ) } . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} B ' = A ' \\times _ A B , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} ( \\mathcal { O } ( 2 \\Delta ) { \\mid _ C } ) = 4 + 2 \\Delta . c _ 1 ( M ) \\le M . N + c _ 2 ( F ^ { * * } ) + | Z | = d + 2 \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} - \\kappa ^ { \\varepsilon } \\nabla \\theta ^ { \\varepsilon } \\cdot \\mbox { n } = \\varepsilon g _ 0 ^ { \\varepsilon } \\theta ^ { \\varepsilon } \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\Gamma _ { R } ^ { \\varepsilon } , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k \\int _ 0 ^ t 1 _ { \\left [ \\frac { q _ { k - 1 } ( s - ) } { \\lambda _ 0 ( X ( s - ) ) } , \\frac { q _ { k } ( s - ) } { \\lambda _ 0 ( X ( s - ) ) } \\right ) } ( \\xi _ { R _ 0 ( s - ) } ) d R _ 0 ( s ) \\\\ R _ 0 ( s ) & = Y \\big ( \\int _ 0 ^ t \\lambda _ 0 ( X ( s ) ) d s \\big ) , \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\pi _ { k } ( x , y _ { 0 : k } , A ) = R _ { 1 , k } R _ { 2 , k } \\cdots R _ { k , k } ( x , A ) . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\epsilon = ( k _ 1 - 1 ) ( k _ 2 - 1 ) + \\dim L + k _ 1 + ( i - 1 ) \\Big ( 1 + ( \\mu ( \\beta _ 2 ) + k _ 2 + \\dim P ) \\dim L \\Big ) \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} S = \\dfrac { 1 - T B V _ { i } ^ { ( W ) } } { \\max ( T B V _ { i } ^ { ( W ) } , \\epsilon ) } , \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} H ^ k ( X ) = \\pi ^ * _ k H ^ k ( Y ) \\oplus \\ker \\theta _ k . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{gather*} E _ * ( t , \\tau ) = - i e ^ { - i \\tau A ( t ) ^ { 1 / 2 } } \\sum _ { { 1 \\le j , l \\le p : \\ ; j \\ne l } } J _ { j l } ( t , \\tau ) , \\\\ J _ { j l } ( t , \\tau ) : = t | t | \\int _ 0 ^ { \\tau } e ^ { i \\tilde { \\tau } A ( t ) ^ { 1 / 2 } } F ( t ) P _ j G _ * P _ l e ^ { - i \\tilde { \\tau } ( t ^ 2 S ) ^ { 1 / 2 } P } P \\ , d \\tilde { \\tau } . \\end{gather*}"}
-{"id": "8026.png", "formula": "\\begin{align*} Q _ p ( x _ 0 , v _ 0 , t ) = \\left \\langle \\exp \\left [ - p \\int _ 0 ^ t d t ' \\thinspace \\theta \\left ( x ( t ' ) \\right ) \\right ] \\right \\rangle , \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { \\tau - T , t } } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} \\rho _ { \\frac { 1 } { 2 } } ( s ) = \\frac { ( 4 \\pi ) ^ k \\zeta ( 0 ) } { \\Gamma ( s + k - 1 ) } \\frac { \\zeta ( 2 s - 1 ) } { \\zeta ^ * ( 2 - 2 s ) } \\frac { \\Gamma ( 2 s - 1 ) } { \\Gamma ( s ) } \\langle \\mathcal { V } _ { f , \\overline { g } } , E ( \\cdot , \\overline { s } ) \\rangle . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} E \\left [ h ( X ( t _ i ) ; i \\leq j ) ( M _ g ^ X ( t + s ) - M _ g ^ X ( t ) ) \\right ] = 0 \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} ( A ( t ) + \\zeta I ) ^ { - 1 } A ( t ) F ( t ) = t ^ 2 \\Xi ( t , \\zeta ) S P + t ^ 3 Z \\Xi ( t , \\zeta ) S P + t ^ 3 \\Xi ( t , \\zeta ) K - t ^ 5 \\Xi ( t , \\zeta ) N \\Xi ( t , \\zeta ) S P + Y ( t , \\zeta ) , \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} A ( v ) = A ^ { \\rm f l a t } ( v ) + \\lambda ( v ) \\mathbf { e } _ \\rho , \\textrm { w i t h } \\lambda ( v ) \\simeq \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } { \\cal R } ( v ) \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} m _ H ( z ) = \\int _ \\R \\frac { 1 } { x - z } \\dd \\mu _ H ( x ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\frac { 1 } { \\lambda _ i - z } , z \\in \\mathbb { C } ^ + . \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} T _ { h } f ( x _ 1 , x _ 2 ) = \\left \\{ \\begin{array} { r @ { \\quad \\quad } l } f _ 1 ( x _ 1 ) \\cdot T _ h ^ 2 f _ 2 ( x _ 2 ) & h \\in D _ 2 \\\\ T _ h ^ 1 f _ 1 ( x _ 1 ) \\cdot \\int _ { X _ 2 } f _ 2 \\ > d \\omega _ { X _ 2 } & h \\in D _ 1 \\setminus \\{ e _ 1 \\} \\end{array} \\right . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\Delta ^ \\perp ( 2 ^ { q + 1 } - t , t - 1 , 2 ^ q + t , 3 \\cdot 2 ^ q + t , \\cdots , 2 ^ q \\cdot [ 2 ^ { k - 2 } - 1 ] + t + k - 3 ) ; k ) = U ( 2 ^ { q + 1 } - t ; k ) . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} b _ { j , k } ( \\zeta ) : = ( \\zeta ^ 2 - 1 ) \\left ( a _ { 2 j - 1 , 2 k } - \\left ( \\prod _ { i = 1 } ^ m \\frac { 2 k - 2 } { L _ i + 2 k - 2 } \\right ) a _ { 2 j - 1 , 2 k - 2 } \\right ) + h _ { j - 1 } \\delta _ { j , k } , \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} ( \\partial - \\tilde { \\phi } _ s ( x , \\tau ) ) \\Upsilon = 0 \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} K ( x , \\xi ) = 2 P _ e \\bigl [ \\ 1 ( \\xi ) T ( \\xi ) f ( x , \\xi ) \\bigr ] , \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} W _ i ( \\R ) = W _ { i , \\mathrm { c u s p } } \\oplus W _ { i , \\mathrm { E i s } } = \\left ( \\bigoplus _ \\pi W _ { i , \\pi } \\right ) \\oplus W _ { i , \\mathrm { E i s } } \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} U ( \\mathcal { N } ) = \\{ N _ { f } : \\ , f \\in L _ \\infty [ 0 , \\alpha ) , \\abs { f } = \\chi _ { [ 0 , \\alpha ) } \\} . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} S _ f ( n ) : = \\sum _ { n \\leq X } a ( n ) \\ll X ^ { \\frac { k - 1 } { 2 } + \\frac { 1 } { 4 } + \\epsilon } . \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} L ( \\boldsymbol { w } _ k , \\lambda _ k ) = ( R - \\boldsymbol { w } _ k ^ H \\boldsymbol { x } _ k \\boldsymbol { x } _ k ^ H \\boldsymbol { w } _ k ) ^ 2 - \\lambda _ k ( \\| \\boldsymbol { w } _ k \\| _ p ^ p - c ) , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } P _ { \\phi _ k } ( a ) & = \\frac { 1 } { ( 2 \\pi ) ^ n } \\int _ { \\mathbb R ^ n } \\ ! \\int _ { \\mathbb R ^ n } \\ ! e ^ { - i s \\cdot \\xi } \\sum _ { k = 0 } ^ { \\infty } \\phi _ k ( \\xi ) \\alpha _ s ( a ) \\ , \\mathrm d s \\ , \\mathrm d \\xi \\\\ & = \\frac { 1 } { ( 2 \\pi ) ^ n } \\int _ { \\mathbb R ^ n } \\ ! \\int _ { \\mathbb R ^ n } \\ ! e ^ { - i s \\cdot \\xi } \\phi ( \\xi ) \\alpha _ s ( a ) \\ , \\mathrm d s \\ , \\mathrm d \\xi , \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} ( 3 x ) ^ 2 + 6 d ^ 2 = 3 y ^ n . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} Q _ N : = \\prod _ { n = 1 } ^ N f ( n ) , \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 2 } ^ { k } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} & = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} \\\\ & = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ p _ { j } \\{ m _ { l } \\theta \\} + \\xi _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ p _ { j } \\{ m _ { l } \\theta \\} \\right \\} , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} P _ i : = ( 1 - W _ 1 ) \\cdots ( 1 - W _ { i - 1 } ) W _ i , \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} F ( z ) - R _ { p , k } ( z ) = \\frac { \\Delta ( z ) } { Q ( z ) } , \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} [ V ( t ) \\psi ] ( x ) = \\eqref { v a 1 } + \\eqref { v a 2 } + \\eqref { v a 3 } . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } \\exp \\left ( \\frac { \\pi s ^ 2 } { Y ^ 2 } \\right ) \\frac { X ^ s } { Y } d s = \\frac { 1 } { 2 \\pi } \\exp \\left ( - \\frac { Y ^ 2 \\log ^ 2 X } { 4 \\pi } \\right ) . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} \\delta ( A ) = \\lim _ { n \\rightarrow \\infty } \\max _ { u \\in \\N _ 0 } \\frac { | A \\cap I _ { u , n } | } { n } \\leq 1 - \\frac { 1 } { d } < 1 \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{gather*} \\sigma = - \\frac { n - 1 } { 2 } + \\mu '' \\textrm { a n d } \\mu '' = \\pm \\sqrt { ( n - 1 ) ^ { 2 } / 4 - \\mu ^ { 2 } R ^ { 2 } } . \\end{gather*}"}
-{"id": "9804.png", "formula": "\\begin{align*} \\log I ( n ) = \\sum _ { p \\mid \\phi ( n ) } \\log I _ p ( n ) . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} M ( \\alpha , \\beta , \\gamma , \\delta , a _ { i , j } ) = \\left ( \\begin{array} { c | c c c c c } \\alpha & \\beta & \\gamma & 0 & \\ldots & 0 \\\\ \\hline \\\\ \\delta & & & & & \\\\ 0 & & & a _ { i , j } & & \\\\ \\vdots & & & & & \\\\ 0 & & & & & \\end{array} \\right ) \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d s ^ { 2 } } + \\Gamma _ { j k } ^ { i } \\frac { d x ^ { j } } { d s } \\frac { d x ^ { k } } { d s } + \\frac { 1 } { \\left ( \\frac { d S \\left ( t \\right ) } { d t } \\right ) ^ { 2 } } \\left ( \\left ( \\frac { d ^ { 2 } S \\left ( t \\right ) } { d t ^ { 2 } } \\right ) + \\phi \\left ( t \\right ) \\frac { d S \\left ( t \\right ) } { d t } \\right ) \\frac { d x ^ { i } } { d s } + \\frac { 1 } { \\left ( \\frac { d S \\left ( t \\right ) } { d t } \\right ) ^ { 2 } } V ^ { , i } = 0 . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} Q _ { s } \\left ( x _ { 0 } \\right ) = \\displaystyle \\sum \\limits _ { j = 0 } ^ { s } b _ { j } \\left ( Q _ { s } \\right ) P _ { j } \\left ( x _ { 0 } \\right ) . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} F = \\begin{pmatrix} 2 ( M + I ) ^ { - 1 } M & - i ( M - I ) ( M + I ) ^ { - 1 } \\\\ - i ( M + I ) ^ { - 1 } ( M - I ) ^ { - 1 } & 2 ( M + I ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\Lambda _ 2 \\left [ \\left [ \\partial ( p _ N ) \\Lambda _ 1 - p _ N \\partial ( \\Lambda _ 1 ) \\right ] \\varphi - n p _ N \\Lambda _ 1 ( \\mu + \\alpha ) \\right ] = n p _ N \\Lambda _ 1 \\varphi N f _ s . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} \\frac { \\log T } { T } \\log \\Psi _ { \\rm d y n } \\ ; = \\ ; - \\ , \\pi \\ , \\int \\Gamma '' ( r ) \\ , \\bar \\mu ^ T ( d r ) \\ ; - \\ ; \\int _ 0 ^ 1 W _ \\gamma ( \\eta _ s ) \\ , d s \\ ; + \\ ; o _ T ( 1 ) \\ ; , \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{gather*} y ^ 0 = F \\big ( x ^ 0 , \\ldots , x ^ n ; y ^ 1 , \\ldots , y ^ { n - 1 } \\big ) . \\end{gather*}"}
-{"id": "1502.png", "formula": "\\begin{align*} L _ { M } = \\max _ { | v | \\leq M } L _ { A ( v ) \\to E _ N ( w ) } \\quad \\textrm { a n d } L _ { M ^ c } = \\max _ { | v | > M } L _ { A ( v ) \\to E _ N ( w ) } . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} u _ { n , q } : = \\mathbb { P } _ q ( [ n ] \\mbox { i s a b l o c k o f } \\Pi ) = ( 1 - q ) ^ n Z _ { n , q } , \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\sup _ { n \\geq 1 } \\sum _ { e \\in E ( v ) } \\sum _ { \\tilde { e } \\in E ( v ) } C ^ n _ { \\tilde { e } , e } \\Big ( \\frac { 1 } { L ^ e } + \\frac { 1 } { L ^ { \\tilde { e } } } \\Big ) < \\infty . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\ell } c _ j H _ { b _ j } \\in T _ { \\rho } \\Sigma \\cap ( T _ { \\rho } \\Sigma ) ^ { \\sigma } , d t \\big ( \\sum _ { j = 0 } ^ { \\ell } c _ j H _ { b _ j } \\big ) \\neq 0 \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} H \\left ( q ^ { i } , \\gamma _ { j } \\left ( q \\right ) \\right ) = . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} _ 2 ( F _ 4 ) & = | ( - 1 , 1 , 0 ) \\times ( - 1 , 0 , 1 ) | / 2 \\\\ & = \\sqrt { 3 } / 2 . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} q q ' = q + q ' \\frac { p p ' } { q ' } - p + \\frac { p } { q } = \\frac { p ' } { q ' } . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} I : = \\int _ \\Omega \\nabla ^ \\perp \\psi \\cdot [ \\Lambda ^ { 2 - \\beta } , \\nabla \\phi ] \\psi d x = \\int _ \\Omega \\L ^ { - 1 + \\alpha - \\delta } \\nabla ^ \\perp \\psi \\cdot \\L ^ { 1 - \\alpha + \\delta } [ \\Lambda ^ \\alpha , \\nabla \\phi ] \\psi d x . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} I = \\left \\{ \\begin{aligned} a \\varphi , & \\quad \\mathfrak { b } _ + \\\\ - \\frac { 1 } { a } \\varphi , & \\mathfrak { b } _ - \\end{aligned} \\right . , \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} c _ 1 = \\int _ 0 ^ 1 \\sqrt { 2 \\hat { \\psi } ( r ) } d r . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\tilde { \\psi } ( ( x _ { i j } ) ) & = \\frac { 1 } { 2 } \\left ( \\psi ( ( x _ { i j } ) ) + \\overline { \\psi ( ( x _ { i j } ) ) } \\right ) = \\frac { 1 } { 2 } \\left ( \\sum _ { i , j = 1 } ^ { n } x _ { i j } \\psi ( E _ { i j } ) + \\sum _ { i , j = 1 } ^ { n } x _ { i j } \\overline { \\psi ( E _ { i j } ) } \\right ) \\\\ & = \\frac { 1 } { 2 } \\left ( \\sum _ { i = 1 } ^ { n } x _ { i i } ( \\psi ( E _ { i i } ) + \\psi ( E _ { i i } ) ^ { t } ) + \\sum _ { 1 \\leq i \\neq j \\leq n } x _ { i j } ( \\psi ( E _ { i j } ) + \\psi ( E _ { j i } ) ) \\right ) \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} \\int _ X f ( \\pi ( x , y ) ) \\ > d \\omega _ X ( y ) & = T ^ { f \\circ \\pi } g ( x ) = T _ f g ( x ) \\\\ & = \\int _ D \\int _ X 1 \\ > K _ h ( x , d y ) \\ > f ( h ) \\ > d \\omega _ D ( h ) = \\int _ D f \\ > d \\omega _ D ( h ) \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi \\notin \\mathfrak { M } _ m } | \\hat { 1 _ S } ( \\chi ) | ^ 3 = \\binom { n } { m } \\sum _ { \\substack { \\chi _ 1 , \\dots , \\chi _ m \\neq 0 \\\\ ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) \\notin \\mathfrak { M } _ m } } | \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 , \\dots , 0 ) | ^ 3 . \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} | | P _ { \\lambda ^ t } ( \\vec { a } ) | | _ { s - t } ^ 2 & = \\sum _ { j , m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { s - t } \\lambda ^ { 2 t } ( m ) | a _ { j , m } | ^ 2 \\\\ & = \\sum _ { j , m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { s - t } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ t | a _ { j , m } | ^ 2 \\\\ & = \\sum _ { j , m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ s | a _ { j , m } | ^ 2 \\\\ & = | | \\vec { a } | | _ s \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } V \\left ( \\sum _ { i = 1 } ^ { n + 1 } ( J _ i \\varphi ) ^ 2 \\right ) \\ , d \\mu = \\int _ { \\R ^ N } V \\left ( \\sum _ { i = 1 } ^ { n + 1 } J _ i ^ 2 \\right ) \\varphi ^ 2 \\ , d \\mu = \\int _ { \\R ^ N } V \\varphi ^ 2 \\ , d \\mu . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\xi _ t ( z ) = \\xi _ 0 ( z ) & + \\sum _ { w \\sim z } \\int _ 0 ^ t ( \\xi _ { s - } ( w ) - \\xi _ { s - } ( z ) ) \\ , d P ^ { z , w } _ s \\\\ & + \\sum _ { w \\sim z } \\int _ 0 ^ t \\xi _ { s - } ( w ) ( 1 - \\xi _ { s - } ( z ) ) \\ , d \\tilde P ^ { z , w } _ s . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\frak m _ k = \\sum _ { \\beta } T ^ { E ( \\beta ) } \\frak m _ { k , \\beta } . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\textstyle P _ m ^ { ( a , b ) } P _ n ^ { ( a , b ) } = \\sum _ { k = | m - n | } ^ { m + n } g _ { m , n , k } P _ k ^ { ( a , b ) } ( m , n \\ge 0 ) . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\forall x \\in S ^ 1 : \\sum _ { \\ ( \\omega _ 0 , \\dots , \\omega _ { m - 1 } \\ ) \\in I } \\prod _ { i = 0 } ^ { m - 1 } d \\ ( T _ t ^ i \\ ( x \\ ) , S ^ 1 \\setminus W _ { \\omega _ i } \\ ) > 0 . \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\left ( Z _ { A } ^ { A ' } \\right ) _ \\varphi ^ * = Z ^ { A } _ { A ' } - 2 Z ^ { A } _ { A ' } \\varphi . \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} X = \\xi ( t ) \\partial _ { t } + T ( t ) S _ { J } ^ { i } \\partial _ { i } ~ , ~ f = T _ { , t } S , \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 t ^ j \\ ; \\Psi ( t ) d t = \\int _ 0 ^ 1 t ^ j ( t - q _ 1 ) \\cdots ( t - q _ r ) = 0 \\ ; \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; \\ ; 0 \\leq j \\leq r - 1 . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} I _ { \\nu } \\left ( { \\nu z } \\right ) = \\frac { c _ { 1 } \\left ( \\nu \\right ) } { \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 4 } } \\exp \\left \\{ { \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { \\tilde { { E } } _ { s } \\left ( p \\right ) - k _ { s } } { \\nu ^ { s } } } } \\right \\} \\left \\{ { e ^ { \\nu \\xi } + \\varepsilon _ { n , 1 } \\left ( { \\nu , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\begin{aligned} A B C & = \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 \\\\ A ^ 2 B D ^ 2 E & = \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 4 ^ 2 \\lambda _ 5 . \\end{aligned} \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{gather*} \\prod \\limits _ { s = 1 } ^ m { T _ { q , x _ s } ^ { \\theta ( s - 1 ) + \\tau _ s ^ + - \\tau _ s ^ - } } \\left ( \\prod _ { s = 1 } ^ m { P _ { \\lambda } \\big ( z _ s ; N , q , q ^ { \\theta } \\big ) } \\right ) . \\end{gather*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } k ( t ) \\Big | \\sum _ { n = n _ 0 } ^ { \\infty } R _ { n } e ^ { r _ n t } \\Big | ^ 2 \\ d t \\le \\pi T \\varepsilon ' \\sum _ { n = n _ 0 } ^ \\infty \\frac { | d _ n D _ { n } | ^ 2 } { \\pi ^ 2 + T ^ 2 r _ { n } ^ 2 } \\le \\pi T \\varepsilon ' \\sum _ { n = n _ 0 } ^ \\infty \\frac { | d _ n D _ { n } | ^ 2 } { \\pi ^ 2 + 4 T ^ 2 ( \\Im \\zeta _ { n } ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} P _ E \\min c ^ \\top x , \\ ; A x - s = b , \\ ; E s \\geq 0 \\ , . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\chi ^ i = ( \\chi _ 1 , \\dots , \\chi _ i + \\chi _ m , \\dots , \\chi _ { m - 1 } , 0 ^ { n - m + 1 } ) . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} y ^ { m + 1 } _ n \\mathbf { d } y ^ { m + 1 } _ { n + 1 } & \\leq y ^ { m + 1 } _ n \\mathbf { d } y ^ m _ { n + 1 } + y ^ m _ { n + 1 } \\underline { \\mathbf { d } } y ^ { m + 1 } _ { n + 1 } = y ^ m _ { n + 1 } \\underline { \\mathbf { d } } y ^ { m + 1 } _ { n + 1 } \\\\ & \\leq \\tfrac { \\underline { \\mathbf { d } } } { \\mathbf { e } } ( y ^ m _ { n + 1 } \\mathbf { e } y ^ { m + 1 } _ { n + 1 } ) \\leq \\tfrac { \\underline { \\mathbf { d } } } { \\mathbf { e } } ( s ^ { m + 1 } _ { n + 1 } ) < t ^ { m + 1 } _ n \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } H ( 1 , \\rho ) | _ { \\rho = 0 } = 2 \\int \\limits _ { - 1 } ^ 1 f ( t ) G ( t ) d t . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} D = U \\begin{pmatrix} I & 0 \\\\ 0 & K \\end{pmatrix} U ^ * \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} G \\left ( Z , W \\right ) = \\left ( g _ { k l } \\left ( Z , W \\right ) \\right ) _ { 1 \\leq k , l \\leq q ' } , \\quad F \\left ( Z , W \\right ) = \\left ( f _ { k l } \\left ( Z , W \\right ) \\right ) _ { 1 \\leq l \\leq p ' - q ' \\atop 1 \\leq k \\leq q ' } . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} P \\colon \\sum _ { n = - \\infty } ^ { \\infty } \\widehat f ( n ) t ^ n \\mapsto \\sum _ { n = 0 } ^ { \\infty } \\widehat f ( n ) t ^ n , \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} \\frac 1 { \\sqrt x } \\big ( \\phi ( q ) \\theta ( x ; q , a ) - \\theta ( x ) \\big ) - \\frac 1 { \\sqrt x } \\big ( \\phi ( q ) \\psi ( x ; q , a ) - \\psi ( x ) \\big ) = - c ( q , a ) + O \\bigg ( \\frac 1 { \\log x } \\bigg ) . \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} r _ { \\langle \\rangle } ( \\mathcal B _ { \\exists , \\varphi } ^ \\beta P ) & = ( \\exists _ x , b , \\theta ^ \\beta ) , \\\\ n ( \\mathcal B _ { \\exists , \\varphi } ^ \\beta P , a ) & = \\mathcal B _ { \\exists , \\theta ( b ) } ^ \\beta \\mathcal B _ { \\exists , \\varphi } ^ \\beta n ( P , 0 ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} S : = | \\nabla _ 0 g _ { \\phi } | _ { \\omega _ { \\phi } } ^ 2 = g _ { \\phi } ^ { i \\bar { j } } g _ { \\phi } ^ { k \\bar { l } } g _ { \\phi } ^ { p \\bar { q } } \\nabla _ { 0 i } g _ { \\phi k \\bar { q } } \\nabla _ { 0 \\bar { j } } g _ { \\phi p \\bar { l } } . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{gather*} \\sum _ { n \\in \\N } \\Big ( \\sum _ { k \\ge 0 } | a _ k | u _ { n + k } \\Big ) ^ 2 \\le \\sum _ { n \\in \\N } \\Big ( \\sum _ { k \\ge n } | a _ { k - n } | u _ { k } \\Big ) ^ 2 = \\| N _ { \\bf | a | } ^ * { \\bf v } \\| _ { \\ell ^ 2 ( \\N ) } ^ 2 \\\\ \\le \\| N _ { \\bf | a | } ^ * \\| ^ 2 \\ , \\| { \\bf v } \\| _ { \\ell ^ 2 ( \\N ) } ^ 2 = \\| N _ { \\bf | a | } \\| ^ 2 \\ , \\sum _ { n \\in \\N } A _ n ^ 2 u _ n ^ 2 \\ , . \\end{gather*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\lambda _ 2 ( q ) : = \\sum _ { h = 2 } ^ \\infty \\frac { q ^ { h } } { h ( h - 1 ) } = q - ( 1 - q ) \\lambda _ 1 ( q ) . \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\theta _ { q z } = 1 + 6 q ^ 2 z + 1 8 q ^ 5 z ^ 2 + 4 0 q ^ 9 z ^ 3 + 7 5 q ^ { 1 4 } z ^ 4 + 1 2 6 q ^ { 2 0 } z ^ 5 + 1 9 6 q ^ { 2 7 } z ^ 6 + \\cdots ~ . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\theta _ { 1 } = i _ { T } \\omega _ { Q } = \\dot { p } _ i d q ^ i - \\dot { q } ^ i d p _ i , \\theta _ { 2 } = d _ { T } \\theta _ { Q } = \\dot { p } _ i d q ^ i + p _ i d \\dot { q } ^ i . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} ( \\alpha _ t f ) ( x ) = \\beta _ t ( x ) ( f ( x - t ) ) , \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} T _ k ^ i & : = \\inf \\{ n \\geq T _ { k - 1 } ^ i : Q _ n ^ i = q ^ i _ 0 \\} i = 1 , 2 \\ , , \\\\ T _ k ^ \\Delta & : = \\inf \\{ n \\geq T _ { k - 1 } ^ \\Delta : Q _ n ^ 1 = q ^ 1 _ 0 , Q _ n ^ 2 = q ^ 2 _ 0 \\} \\ , . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\omega _ A ' ( z ) \\\\ \\omega _ B ' ( z ) \\end{array} \\right ) = - \\mathcal { S } ^ { - 1 } ( z ) \\left ( \\begin{array} { c c } F ' _ A ( \\omega _ B ( z ) ) \\\\ F ' _ B ( \\omega _ A ( z ) ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 + \\epsilon ) } D _ G ^ { \\mathrm { c m } } ( s ) \\widehat { \\phi _ { Y } } ( s ) X ^ s d s = \\widehat { \\phi _ { Y } } ( 1 ) r _ d ( G ) X + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\alpha ' ) } D _ G ^ { \\mathrm { c m } } ( s ) \\widehat { \\phi _ { Y } } ( s ) X ^ s d s . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} & \\P \\big ( \\cap ^ k _ { j = 1 } \\{ T _ { s _ { j - 1 } + 1 } \\le t _ { j 1 } , \\ldots , T _ { s _ j } \\le t _ { j \\ell _ j } , \\ , N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\} \\big ) \\\\ & = \\prod _ { j = 1 } ^ k \\P \\big ( T _ { s _ { j - 1 } + 1 } \\le t _ { j 1 } , \\ldots , T _ { s _ j } \\le t _ { j \\ell _ j } , \\ , N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\big ) . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} \\varphi _ d ( x ) \\coloneqq \\tilde { D } _ 1 f _ d ( x ) = x + x ^ { p ^ { i _ 2 } } + \\cdots + x ^ { p ^ { i _ { p } } } \\ \\ ( x \\in F ) , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} T _ { l } ( W _ { \\alpha ^ { ( l ) } , r } ) = \\sum _ { U \\in I ( \\mathcal { C } ) } M _ { W , \\alpha ^ { ( l ) } , r } ^ { ( U ) } T _ l ( U ) \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} & R _ k ^ { \\prime \\rm O M A I I } ( \\nu ) = \\\\ & \\left \\{ \\begin{array} { l } \\alpha _ k \\log _ 2 \\left ( 1 + \\frac { p _ s g _ k ( \\nu ) } { \\alpha _ k } \\right ) , \\ { \\rm i f } \\ g _ k ( \\nu ) > g _ { \\bar k } ( \\nu ) , \\\\ \\alpha _ k \\log _ 2 \\left ( 1 + \\frac { p _ w g _ k ( \\nu ) } { \\alpha _ k } \\right ) , \\ { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\sigma _ 0 ( u _ 1 , u _ 2 , e _ 1 , e _ 2 ) & : = \\left \\langle \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\\\ e _ 1 \\\\ e _ 2 \\end{bmatrix} , \\begin{bmatrix} I & 0 & 0 & 0 \\\\ 0 & I & 0 & 0 \\\\ 0 & 0 & - \\gamma ^ 2 I & 0 \\\\ 0 & 0 & 0 & - \\gamma ^ 2 I \\end{bmatrix} \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\\\ e _ 1 \\\\ e _ 2 \\end{bmatrix} \\right \\rangle \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} \\int \\limits _ { { \\mathbb R } ^ { 2 n } } h ( z ) d \\nu _ y ^ A ( z ) = \\int \\limits _ { { \\rm S O } ( 2 n ) } h ( f ( O ) ) d \\mu _ { 2 n } ( O ) = \\int \\limits _ { S ^ { 2 n - 1 } } \\Bigl ( \\int \\limits _ { G _ y } h ( f ( O _ v U ) ) d \\mu _ { 2 n - 1 } ( U ) \\Bigr ) d \\sigma _ { 2 n - 1 } ( v ) . \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} a ( n ; i , n ) & = a ( n ; i , n - 1 ) , \\qquad \\mbox { i f $ 1 \\leq i \\leq n - 2 $ } , \\\\ a ( n ; i , j ) & = a ( n - 1 ; i , j ) + b ( n ; i - 1 ) , \\qquad \\mbox { i f $ 2 \\leq i < j \\leq n - 1 $ } , \\\\ a ( n ; i , j ) & = \\sum _ { k = 1 } ^ j a ( n - 1 ; i - 1 , k ) , \\qquad \\mbox { i f $ 1 \\leq j < i - 1 \\leq n - 2 $ } , \\\\ a ( n ; i , i - 1 ) & a ( n - 1 , i - 1 ) , \\qquad \\mbox { i f $ 2 \\leq i \\leq n $ } , \\\\ b ( n ; i ) & = \\sum _ { k = 1 } ^ i b ( n - 1 ; k ) , \\qquad \\mbox { $ 1 \\leq i \\leq n - 1 $ } , \\\\ a ( n ; n - 1 ) & = a ( n - 1 ) , \\\\ a ( n ; n ) & = a ( n - 1 ) , \\\\ b ( n ; n ) & = 0 . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} = \\frac { \\alpha { \\left ( s ( x ) , s ( y ) , s ( z ) \\right ) } \\alpha { \\left ( \\gamma ( x , y ) , s ( x + y ) , s ( z ) \\right ) } c { \\left ( s ( x + y ) + s ( z ) , \\gamma ( x , y ) \\right ) } \\eta { \\left ( \\gamma ( x , y + z ) , \\gamma ( y , z ) \\right ) } } { \\alpha { \\left ( s ( x ) , s ( y + z ) , \\gamma ( y , z ) \\right ) } c { \\left ( s ( x + y ) , \\gamma ( x , y ) \\right ) } \\eta { \\left ( \\gamma ( x + y , z ) , \\gamma ( x , y ) \\right ) } } \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} 1 - \\left ( 1 - \\frac { \\left | P _ { G ( \\tau ) } \\right | } { \\binom { n } { 2 } } \\right ) ^ { h n / 2 } \\ge 1 - e ^ { - \\Omega \\left ( \\left | P _ { G ( \\tau ) } \\right | h / n \\right ) } , \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} F _ p ( p , p , b ) & = \\dim ( p - 1 , b ) + \\dim ( p + b - 2 , 1 ) + \\dim ( p , b - 1 ) \\cr & \\ \\ \\ \\ \\ + \\dim ( p + b - 3 , 1 , 1 ) \\cr & = \\dim ( p , b ) + \\dim ( p + b - 2 , 1 , 1 ) \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} | F _ v ( \\ell ) | \\leq \\gamma ( G ) - t + k + 1 = 4 . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\bar { v } _ t ^ \\epsilon ( \\bar { x } , \\bar { t } ) + \\int \\left ( 1 - \\frac { \\exp ( \\bar { v } ^ \\epsilon ( \\eta ^ \\epsilon ( \\bar { x } , y ) , \\bar { t } ) / \\epsilon ) } { \\exp ( \\bar { v } ^ \\epsilon ( \\bar { x } , \\bar { t } ) / \\epsilon ) } \\right ) K d y + \\mathcal { J } _ \\epsilon = \\mu ( \\hat { \\bar { x } } | \\bar { x } | ^ { 1 / \\epsilon } ) - E e ^ { - \\bar { v } ^ \\epsilon ( \\bar { x } , \\bar { t } ) / \\epsilon } , \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} N _ r ( s , t ) & = \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\left ( \\binom { a } { 2 } + \\binom { b } { 2 } \\right ) \\binom { s } { a } \\binom { r - s } { b } \\\\ & = \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\binom { a } { 2 } \\binom { s } { a } \\binom { r - s } { b } + \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\binom { b } { 2 } \\binom { s } { a } \\binom { r - s } { b } . \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} N _ 0 = \\sum _ { j = 1 } ^ { p } P _ j N P _ j , N _ * = \\sum _ { { 1 \\le j , l \\le p : \\ ; j \\ne l } } P _ j N P _ l . \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\sum ^ \\infty _ { j = 0 } \\abs { ( \\ , g _ j \\ , | \\ , \\chi _ \\varepsilon \\ , ) _ M } ^ 2 = ( \\ , B \\chi _ \\varepsilon \\ , | \\ , \\chi _ \\varepsilon \\ , ) _ M . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\limsup _ { p \\to \\infty } \\big \\| F ( z ) - R _ { p , k } ( z ) \\big \\| ^ { 1 / p } \\leq \\frac { \\Phi ( z ) } { \\rho } , z \\in \\widetilde { K } = E _ \\rho \\setminus \\{ z _ 1 , \\ldots , z _ \\mu \\} , \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{gather*} J '' + \\overset { \\mathrm { s f } } R J = 0 , \\end{gather*}"}
-{"id": "1066.png", "formula": "\\begin{align*} a c - b ^ 2 = - d ^ 2 \\iff a c = b ^ 2 - d ^ 2 = ( b + d ) ( b - d ) . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} & \\sup _ { { \\bar { \\tau } } < t \\le T } | { \\bar { B } } ^ n _ 0 ( t ) - { \\bar { B } } ^ n _ 0 ( { \\bar { \\tau } } ) | \\\\ & \\le \\sup _ { { \\bar { \\tau } } < t \\le T } | { \\bar { X } } ^ n _ 0 ( t ) - { \\bar { X } } ^ n _ 0 ( { \\bar { \\tau } } ) | + \\sup _ { { \\bar { \\tau } } < t \\le T } | { \\bar { \\eta } } ^ n ( t ) - { \\bar { \\eta } } ^ n ( { \\bar { \\tau } } ) | + \\sum _ { k = 1 } ^ \\infty | k - 2 | \\sup _ { { \\bar { \\tau } } < t \\le T } | { \\bar { B } } ^ n _ k ( t ) - { \\bar { B } } ^ n _ k ( { \\bar { \\tau } } ) | . \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} \\langle M ^ { u _ i } , M ^ { u _ j } \\rangle _ t = \\int _ 0 ^ t a _ { i j } ( X _ s ) \\ , d s , 1 \\le i , j \\le d , \\ t \\ge 0 . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} { \\rm H o m } ( M ' , N '' ) = 0 , \\ ; { \\rm H o m } ( M ' , N '' [ - 1 ] ) = 0 . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\psi _ t ( x _ 0 , t _ 0 ) - | \\lambda _ 1 | + A _ { k + 1 } = - \\sigma . \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} \\emph { e r r } \\stackrel { d e f } { = } \\frac { \\| A \\hat { x } - b \\| _ \\infty } { \\| A \\| _ \\infty \\| \\hat { x } \\| _ \\infty } , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} \\mathbf { S } \\left ( \\omega _ { 1 } , \\omega _ { 1 } , a _ { 1 } , b _ { 1 } , b _ { 2 } , \\bar { p } ; \\mathbf { t } \\right ) = \\mathbf { Q } \\left ( \\omega _ { 1 } , \\omega _ { 1 } , a _ { 1 } , b _ { 1 } , b _ { 2 } ; \\mathbf { t } \\right ) + \\bar { p } \\mathbf { 1 } , \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} E _ k \\stackrel { d e f } { = } \\bigcap _ { k \\le j \\le n } E _ { k , j } \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\omega _ { \\varphi _ { \\epsilon } } } { \\partial t } = - { \\rm R i c } ( \\omega _ { \\varphi _ { \\epsilon } } ) + \\gamma \\omega _ { \\varphi _ { \\epsilon } } + ( 1 - \\beta ) \\eta _ { \\epsilon } + L _ X \\omega _ { \\varphi _ { \\epsilon } } \\\\ \\omega _ { \\varphi _ { \\epsilon } } | _ { t = 0 } = \\omega _ { \\epsilon } , \\end{cases} \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\rho _ { \\frac { 3 } { 2 } } ( s ) = \\frac { ( 4 \\pi ) ^ k \\zeta ( 2 s - 2 ) \\Gamma ( 2 s - 2 ) } { \\Gamma ( s ) \\Gamma ( s + k - 1 ) \\zeta ^ * ( 2 s - 2 ) } \\langle \\mathcal { V } _ { f , \\overline { g } } , \\overline { E ( \\cdot , 2 - s ) } \\rangle . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} z = \\sqrt { \\frac { t } { 2 } } + \\sqrt { \\frac { | w | } { \\sqrt { 8 t } } - \\frac { t } { 2 } } \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} 1 = \\beta ^ n = \\beta ^ { 2 t + 1 } = \\beta \\left ( - 1 - \\beta \\right ) ^ 2 = \\beta - \\beta ^ 2 + \\beta ^ 3 , \\ \\ \\mbox { t h a t , i s } , \\ \\ \\beta ^ 2 + 1 = \\beta ( \\beta ^ 2 + 1 ) . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} L _ 0 ( x ) = \\biggl ( \\frac { x } { 2 } \\biggr ) \\sum \\limits _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( \\varGamma ( 3 / 2 + k ) ) ^ 2 } \\biggl ( \\frac { x } { 2 } \\biggr ) ^ { 2 k } , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\lambda ( t ) & = \\tilde \\lambda _ - ( c ( t ) ) = \\tilde \\lambda _ + ( d ( t ) ) , \\\\ a ( t ) & = y _ - ( c ( t ) ) , \\\\ b ( t ) & = y _ + ( d ( t ) ) . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\mu _ { t 1 } ^ { \\pm } ( x , \\infty ) & : = \\left ( \\frac { t \\ , \\P \\left [ \\frac { X _ { 1 } } { b ( t ) } \\in ( x , \\infty ) \\right ] - \\nu _ { 2 } ( x , \\infty ) } { A ( b ( t ) ) } \\right ) ^ { \\pm } \\to \\left ( x ^ { - 2 } ( x ^ { - 1 } - 1 ) \\right ) ^ { \\pm } = : \\chi _ { 1 } ^ { \\pm } ( x , \\infty ) , \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} a _ { 1 } = \\frac { b _ { 1 } \\sin \\omega _ { 1 } T _ { 0 } \\cos \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) + b _ { 2 } \\sin \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) } { 1 - \\cos \\omega _ { 1 } T _ { 0 } \\cos \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) } , \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} r \\dot { F } & = - F ^ 2 r + 4 n r \\frac { F B } { b } - 2 n \\frac { B } { b } - 2 n ( 2 n - 1 ) r \\frac { B ^ 2 } { b ^ 2 } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac { n ( n + 1 ) } { h b ^ 2 } - \\frac { n } { 2 h b ^ 4 } \\left ( r + 2 F r ^ 2 \\right ) \\\\ r \\dot { h } & = \\frac { 2 n r ^ 2 } { b ^ 4 } - 4 n h r \\frac { B } { b } + 2 h r F \\\\ r \\dot { b } & = B r \\\\ r \\dot { B } & = \\frac { n + 1 } { 2 h b } - \\frac { r } { 2 h b ^ 3 } - B - \\frac { B r ^ 2 n } { h b ^ 4 } + \\frac { r B ^ 2 } { b } \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} | k | x \\left | \\int _ 2 ^ x \\frac { e ^ { i \\xi } - 1 } { \\xi } \\frac { 1 } { \\xi - k x } ~ d \\xi \\right | & \\le C | k | x \\left | \\int _ 2 ^ x \\frac { 1 } { \\xi ( \\xi - | k | x ) } ~ d \\xi \\right | \\\\ & = C \\left | \\log \\left ( \\frac { 2 ( 1 - | k | ) } { 2 - | k | x } \\right ) \\right | \\\\ & \\le C \\log [ ( 1 + | k | ) ( 1 + | k | x ) ] \\\\ & \\le C | k | ^ { \\epsilon } ( 1 + | x | ) ^ { \\epsilon } . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\P ( W > w ) & = \\frac 1 2 w ^ { - 1 } + \\frac 1 2 w ^ { - 2 } , w > 1 , \\\\ \\P ( Z > z ) & = 2 z ^ { - 1 } - z ^ { - 2 } , z > 1 . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} V _ { \\underline { d } '' } ^ { X _ { 1 } , 0 } = ( V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } \\oplus V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } ) \\oplus \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle = V _ { \\underline { d } '' } ^ { X _ { 2 } ^ { c } , 0 } \\oplus \\left \\langle u _ 1 , \\ldots , u _ { \\beta } \\right \\rangle . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} - \\mathbb { D } ^ { i } \\nabla u _ { i } ^ { 0 } \\cdot \\mbox { n } = 0 \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | \\leq \\binom { n + k - 1 } { k - 1 } ^ { 1 / 2 } \\binom { n } { a _ 1 , \\dots , a _ k } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\sum _ { I \\subseteq [ 5 ] } ( - 1 ) ^ { | I | } \\Sigma _ I & = ( y ^ 5 - y ^ 4 + y ^ 3 + 2 y ^ 2 - y ) - 5 ( y ( y - 1 ) ^ 3 + y ^ 3 ) + y ^ 3 + 6 ( y ^ 2 + ( y - 1 ) ^ 3 ) \\\\ & \\qquad + 3 y ( y ^ 2 - y + 1 ) - 3 y ^ 2 - 6 ( y ^ 2 - y + 1 ) - y ^ 2 + 5 y - 1 \\\\ & = y ^ 5 - y ^ 4 - 5 y ^ 2 + 1 3 y - 7 - 5 y ( y ^ 3 - 3 y ^ 2 + 3 y - 1 ) + 6 ( y ^ 3 - 3 y ^ 2 + 3 y - 1 ) \\\\ & = y ^ 5 - 6 y ^ 4 + 2 1 y ^ 3 - 3 8 y ^ 2 + 3 6 y - 1 3 . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\mathcal { L } : = \\left \\{ m : [ 0 , T ] \\longrightarrow S : | m ( t ) - m ( s ) | \\leq K | t - s | , m ( 0 ) = m _ 0 \\right \\} \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} B _ \\mu ^ t ( v _ 0 ) : = M _ t [ v _ 0 e ^ { - \\mu i } ] ( 0 ) = v ( t , v _ 0 ) , \\ , \\forall \\ , \\ , v ( 0 ) = v _ 0 \\in [ 0 , \\infty ) , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\sum _ { \\substack { m = n _ 0 \\\\ m \\not = n } } ^ \\infty \\ \\frac { 1 } { 4 ( m - n ) ^ 2 - 1 } \\le 1 \\ , . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} \\omega ( q _ i , q _ j ) = \\begin{cases} \\sigma ( q _ i , q _ j ) \\overline { \\sigma ( q _ j , q _ i ) } & \\\\ 1 & \\\\ \\end{cases} \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} J _ { \\mu _ g } ( x ) = \\left \\{ \\begin{array} { @ { } l l } I _ g ( \\frac { \\mu _ g } { 1 - x } ) & \\ \\mbox { i f } \\ 1 - \\frac { \\mu _ g } { r _ { \\mathrm { m i n } } } \\leq x < 1 , \\\\ \\infty & \\ \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } + L u - \\lambda u = - g , u ( T ) = f \\mbox { i n } \\mathbb { R } ^ 2 , \\frac { \\partial v } { \\partial t } + L v - \\mu v = - \\psi , v ( T + h ) = \\varphi \\mbox { i n } \\mathbb { R } ^ 2 , \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} E _ { p , q } ^ { 1 } = H _ { p + q } \\left ( \\frac { F _ p C _ * } { F _ { p - 1 } C _ * } \\right ) . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{gather*} \\mu ^ { 2 } R ^ { 2 } = - \\sigma ( n - 1 + \\sigma ) , \\end{gather*}"}
-{"id": "5452.png", "formula": "\\begin{align*} m [ b _ 1 + . . . + b _ { n - 1 } ] + k b _ m & = m [ b _ 1 + . . . + b _ { n - 1 } + t b _ m ] + r b _ m \\\\ & = m [ b _ 1 + . . . + b _ { m - 1 } ] + m [ b _ m + . . . + b _ { n - 1 } + t b _ m ] + r b _ m \\\\ & \\ge m b _ { 1 + . . . + ( m - 1 ) + r } + m [ b _ m + . . . + b _ { n - 1 } + t b _ m ] \\\\ & \\ge m b _ { 1 + . . . + ( n - 1 ) + t m + r } \\\\ & = m b _ s \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} | g ( n ) | & = \\begin{cases} \\sqrt { N ( n ) } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} f _ { \\theta , q , k } ( x ) = \\left ( \\frac { \\theta x + q - 1 } { x + \\theta + q - 2 } \\right ) ^ k \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\frac { d \\ell } { d \\tau } ( \\tau ) = { \\rm g r a d } \\ , h _ { \\chi ( \\tau ) } \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} E [ f ( X ) ] = E [ f ( X ) - f ( Z _ { L } ) ] + \\sum _ { \\ell = \\ell _ 0 + 1 } ^ L E [ f ( Z _ { \\ell } ) - f ( Z _ { \\ell - 1 } ) ] + E [ f ( Z _ { \\ell _ 0 } ) ] , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} T _ h \\phi ( v ) = \\frac { 1 } { | \\{ w \\in \\Gamma : \\ > d ( v , w ) = h \\} | } \\sum _ { w \\in \\Gamma : \\ > d ( v , w ) = h } \\phi ( w ) = P _ h ^ { ( a , b ) } ( x _ c ) \\cdot \\phi ( v ) \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\frac { H ( y ) } { H ( x ) } \\det [ q _ t ( x _ i , y _ j ) ] , x , y \\in \\mathbb { R } _ { < } ^ n , \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} F ^ * x ^ * = f _ 0 ( t ) \\left ( \\delta _ t \\otimes F _ Z ^ * z ^ * \\right ) . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\P ( \\Pi _ n = \\{ C _ 1 , \\ldots , C _ k \\} ) = p ( \\# C _ 1 , , \\ldots , \\# C _ k ) \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} ( \\mathcal { K } x ) ( s ) = \\int _ a ^ b \\kappa ( s , t ) x ( t ) d t , \\ ; \\ ; \\ ; s \\in [ a , b ] \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} F _ { v _ { 1 , 1 } , g _ 1 } = F _ { v _ { 2 , n } , g _ n } & = F _ { v _ { 2 , i } , g _ i } = F _ { v _ { i + 1 , 1 } , g _ i } = f _ 3 , F _ { v _ { 1 , j } , f _ { 1 , j } } = N _ { Z / Y _ i } \\otimes ^ \\mathbb { L } f _ 2 , \\\\ & F _ { v _ { 2 , j } , f _ { 2 , j } } = F _ { v _ { 1 , j } , f _ { 2 , j } } = f _ 1 , F _ { v _ { 2 , j } , f _ { 1 , j } } = f _ 2 . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\Gamma _ { 2 2 } = \\frac { 2 } { J _ 2 } , \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} U _ { \\ ! s } ( x ) = \\left ( 1 + | x | ^ 2 \\right ) ^ { \\frac { 2 s - n } { 2 } } . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\theta ( r ) = \\sup \\limits _ { z \\in \\partial A _ r } \\max \\limits _ { i , j = 1 , . . . , 2 d } | \\frac { \\partial ^ 2 H } { \\partial z _ i \\partial z _ j } | \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\hat { w } _ i ( t ) = \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } \\Phi _ i ( p ^ { ( \\ell ) } ) \\ , C ( p ^ { ( \\ell ) } ) x ( t , p ^ { ( \\ell ) } ) \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} \\Delta ^ e d ^ e h ( a ) = & ( m \\otimes h + h \\otimes m ) \\Delta d ( a ) \\\\ = & ( m \\otimes h + h \\otimes m ) ( d ( a _ { ( 1 ) } ) \\otimes a _ { ( 2 ) } + ( - 1 ) ^ { | a _ { ( 1 ) } | } a _ { ( 1 ) } \\otimes d ( a _ { ( 2 ) } ) ) \\\\ = & m d ( a _ { ( 1 ) } ) \\otimes h ( a _ { ( 2 ) } ) + h d ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) \\\\ & + ( - 1 ) ^ { | a _ { ( 1 ) } | } m ( a _ { ( 1 ) } ) \\otimes h d ( a _ { ( 2 ) } ) + ( - 1 ) ^ { | a _ { ( 1 ) } | } h ( a _ { ( 1 ) } ) \\otimes m d ( a _ { ( 2 ) } ) \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { 2 t + 1 } \\left | \\left ( \\| \\mathbf v _ 1 - \\mathbf v _ j \\| ^ 2 - 1 \\right ) \\right | \\geq 3 d ^ { \\beta - 2 / 3 } \\cdot \\frac { 2 d ^ { 4 / 9 } } { d + 1 } \\geq 4 d ^ { 2 / 9 } > 3 \\sqrt { t } . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\langle M ^ { v _ k } , M ^ { v _ l } \\rangle _ t = \\int _ 0 ^ t \\Phi _ { k l } ( X _ s ) \\ , d s , 1 \\le k , l \\le d , \\ t \\ge 0 , \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\left [ | R _ { 1 1 } ( \\cdot + \\alpha ) \\hat { R } ( \\cdot ) - R _ { 1 1 } ( \\cdot ) \\hat { R } ( \\cdot + \\alpha ) | \\right ] \\leq 2 | | R | | _ 0 \\sqrt { [ \\hat { R } ^ 2 ] } . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) & : = \\int _ { \\Gamma } \\alpha ( x ) \\ , \\nabla f ( x ) \\cdot \\nabla g ( x ) \\ , \\nu ( d x ) \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\delta _ h * \\delta _ l : = & ( \\delta _ h * _ n \\delta _ l ) | _ { D _ n \\setminus \\{ e _ n \\} } + \\\\ + & ( \\delta _ h * _ n \\delta _ l ) ( \\{ e _ n \\} ) \\Bigl [ \\sum _ { k = 2 } ^ { n - 1 } \\Bigl ( \\prod _ { m = k + 1 } ^ { n - 1 } \\omega _ { D _ m } ( \\{ e _ m \\} ) \\Bigr ) \\cdot \\omega _ { D _ k } | _ { D _ k \\setminus \\{ e _ k \\} } + \\Bigl ( \\prod _ { m = 2 } ^ { n - 1 } \\omega _ { D _ m } ( \\{ e _ m \\} ) \\Bigr ) \\cdot \\omega _ { D _ 1 } \\Bigr ] . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} v _ A = \\tilde a _ M b _ M v _ M + \\tilde a _ E b _ E v _ E . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\tilde { \\rho } ( \\theta _ { 0 1 } , x ) = \\Phi \\left ( Z _ n \\right ) + O _ p ( n ^ { - 1 / 2 } ) \\ , , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} E _ { \\Delta } = E _ { \\Delta ^ \\circ } , V _ { \\Delta } = F _ { \\Delta ^ \\circ } , F _ { \\Delta } = V _ { \\Delta ^ \\circ } . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} d \\vartheta = 0 , d \\Omega - \\vartheta \\wedge \\Omega = 0 . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} ( I - P + Q ) h _ { n } = ( I - P + Q ) ( R _ { X } x _ { n } + P R _ { Y } y ) = R _ { X } x _ { n } + P R _ { Y } y = h _ { n } \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n - 2 \\} , \\ , \\forall g \\in \\mathcal { E } _ f , \\ ( - 1 ) ^ { | g _ i | | g _ { i + 2 } | } = ( - 1 ) ^ { | g _ i | | g _ { i + 1 } | } = ( - 1 ) ^ { | g _ { i + 1 } | | g _ { i + 2 } | } , \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} | L a w ( X ( t ) ) - L a w ( X ( s ) ) | & = \\sqrt { \\sum _ { x = 1 } ^ d \\left [ e _ x \\cdot ( L a w ( X ( t ) ) - L a w ( X ( s ) ) ) \\right ] ^ 2 } \\\\ & \\leq \\sqrt { \\sum _ { x = 1 } ^ d | t - s | ^ 2 4 | e _ x | _ { \\infty } ^ 2 \\nu ( U ) ^ 2 } = 2 \\nu ( U ) \\sqrt { d } | t - s | , \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} W _ { 2 l + 2 } & = x W _ { 2 l + 1 } - W _ { 2 l } = x ^ 2 W _ { 2 l } - ( x W _ { 2 l - 1 } - W _ { 2 l - 2 } ) - W _ { 2 l - 2 } - W _ { 2 l } = \\\\ & = x ^ 2 W _ { 2 l } - W _ { 2 l } - W _ { 2 l - 2 } - W _ { 2 l } = ( x ^ 2 - 2 ) W _ { 2 l } - W _ { 2 l - 2 } . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} d & \\leq n - \\lvert \\mathcal { T } ' \\rvert = n - \\lvert \\mathcal { T } '' \\rvert - \\gamma \\\\ & \\leq n - k + 1 - \\gamma \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\frac { d B } { d t } = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 } { \\lambda _ 4 B ^ 2 C } & = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 } { \\lambda _ 4 B ^ 2 } \\left ( \\frac { 1 } { \\omega } ( B ^ 2 ) \\right ) ^ { - 1 / 2 } \\\\ B ^ 3 \\frac { d B } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 \\omega ^ { 1 / 2 } } { \\lambda _ 4 } \\\\ B ( t ) & = \\lambda _ 2 \\left ( \\frac { 4 \\lambda _ 1 \\lambda _ 3 \\omega ^ { 1 / 2 } } { \\lambda _ 2 ^ 2 \\lambda _ 4 } t + 1 \\right ) ^ { 1 / 4 } . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\| M _ m \\| _ { 2 \\to 2 } = 1 + O ( m / n ) . \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} V + V ' = \\phi _ 6 - \\phi _ 5 & & & I = I ' = I _ 5 = - I _ 6 . \\\\ \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} L = L _ \\R L _ + \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} Z _ s = \\Gamma _ s \\cap ( C \\times \\{ 0 \\} ) = \\{ ( \\lambda , 0 ) \\mid R _ { 2 s + 1 } ( \\lambda ) = 0 \\} . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} { \\cal R } \\begin{pmatrix} \\eta \\\\ \\psi \\end{pmatrix} = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\begin{pmatrix} \\eta \\\\ \\psi \\end{pmatrix} \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} \\mathcal { G } _ 0 ( U ) : = \\bigg \\{ & f \\in \\mathcal { O } _ { X \\times \\mathbb { B } ^ 1 } ( U \\times \\mathbb { B } ^ 1 ) = \\mathcal { O } _ X ( U ) \\langle T \\rangle \\ , \\bigg | \\\\ & f = 1 + \\sum _ { i = 1 } ^ \\infty f _ i T ^ i \\vert f _ i ( u ) \\vert < 1 \\ \\forall i \\geq 1 \\ \\forall u \\in U \\bigg \\} . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} b _ { n + 1 } \\leq a & \\leq \\left ( b _ { n + 1 } + \\ell _ { n + 1 } - 1 \\right ) + \\sum _ { j \\in J } \\left ( b _ { j } + \\ell _ j - 1 \\right ) \\\\ & \\leq \\sum _ { j = 1 } ^ { n + 1 } ( b _ j + \\ell _ j ) - 1 \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} U _ i = \\sum _ { j = 1 } ^ { \\mu _ i } C _ { j , i } , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\limsup _ { T \\to \\infty } \\frac 1 { M _ T } \\log ( a _ T + b _ T ) = \\max \\Big \\{ \\limsup _ { T \\to \\infty } \\frac 1 { M _ T } \\log a _ T \\ , , \\ , \\limsup _ { T \\to \\infty } \\frac 1 { M _ T } \\log b _ T \\Big \\} \\ ; . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } \\left ( \\sum _ { j = 1 } ^ { \\ell } | \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } u ( x ) | ^ { p - 2 } \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } u ( x ) \\overline { \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } \\psi ( x ) } - | u ( x ) | ^ { q - 2 } u ( x ) \\overline { \\psi ( x ) } \\right ) d x = 0 \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} u ( x , \\xi ) = \\int _ 0 ^ { l ( x , \\xi ) } f _ { i _ 1 \\dots i _ m } ( \\gamma _ { x , \\xi } ( t ) ) \\dot { \\gamma } _ { x , \\xi } ^ { i _ 1 } ( t ) \\cdots \\dot { \\gamma } _ { x , \\xi } ^ { i _ m } ( t ) d t . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} f ( x ) = f ^ * ( E x + c ) + \\langle c , x \\rangle + \\beta , x \\in X . \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} \\mathcal { K } _ G ( X ^ { 1 / 2 } ) : = \\{ K / \\Q : ~ \\textrm { G a l } ( K ^ c / \\Q ) \\cong G , ~ | d _ K | \\leq X ^ { 1 / 2 } \\} . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} & \\mathrm { E } _ f ( 1 - \\Phi _ n ) = P _ f \\left ( \\| \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - f _ 0 \\| _ \\infty \\leq M _ 0 \\rho _ n \\epsilon _ n \\right ) \\\\ & \\leq P _ f \\left ( \\| \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - \\mathrm { E } _ f \\widehat { f } _ { n , \\boldsymbol { \\alpha } } \\| _ \\infty \\geq \\| f - f _ 0 \\| _ \\infty - M _ 0 \\rho _ n \\epsilon _ n - \\| \\mathrm { E } _ f \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - f \\| _ \\infty \\right ) . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\left \\{ \\begin{pmatrix} 0 & 1 / 2 \\\\ 1 / 2 & 0 \\end{pmatrix} , \\begin{pmatrix} 0 & i / 2 \\\\ - i / 2 & 0 \\end{pmatrix} , \\begin{pmatrix} 1 / 2 & 0 \\\\ 0 & 1 / 2 \\end{pmatrix} \\right \\} \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\sum _ { \\lvert 2 m ^ 2 + h - X \\rvert \\leq X } r _ { 2 k + 1 } ( m ^ 2 + h ) = \\tfrac { 1 } { k } R _ { k , h } ^ k X ^ { k } + O ( X ^ { k + \\epsilon - \\lambda ( k ) } ) . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\{ a , b \\} & = \\sum _ { \\alpha \\in \\Lambda } \\psi _ { \\alpha } ( a ) \\{ x _ { \\alpha } , b \\} \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} \\det \\mathcal { T } _ { \\nu , \\theta } & = \\det \\Big \\{ P ^ { \\top } \\mathcal { T } _ { \\nu , \\theta } P \\Big \\} \\cr & = \\det \\Big \\{ ( 1 - \\theta ) \\big \\{ ( 1 - \\nu ) T _ { t r } I d + \\nu P ^ { \\top } \\Theta P \\big \\} + \\theta T _ { \\delta } I d \\Big \\} \\cr & = \\prod _ { 1 \\leq i \\leq 3 } \\big \\{ ( 1 - \\theta ) \\big ( ( 1 - \\nu ) T _ { t r } + \\nu \\Theta _ i \\big ) + \\theta T _ { \\delta } \\big \\} . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} h ^ { \\pm } ( c _ r ) : = c _ r \\mp \\frac { \\Re \\left ( \\displaystyle \\sum _ { k n \\le X } \\frac { a ^ \\pm ( n ) \\overline { a ^ \\pm ( k n ) } g _ { c _ r } ( k ) \\Lambda ( k ) } { k n } \\right ) } { \\displaystyle \\sum _ { n \\le X } \\frac { | a ^ \\pm ( n ) | ^ 2 } { n } } \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} X & = G ( x _ 1 ) ^ { o _ 1 } \\otimes \\cdots \\otimes G ( x _ m ) ^ { o _ m } \\\\ Y & = G ( y _ 1 ) ^ { o ' _ 1 } \\otimes \\cdots \\otimes G ( y _ n ) ^ { o ' _ n } \\\\ X ' & = G ( x ' _ 1 ) ^ { o '' _ 1 } \\otimes \\cdots \\otimes G ( x ' _ { m ' } ) ^ { o '' _ { m ' } } \\\\ Y ' & = G ( y ' _ 1 ) ^ { o ''' _ 1 } \\otimes \\cdots \\otimes G ( y ' _ { n ' } ) ^ { o ''' _ { n ' } } \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} f _ x ( ( 1 , \\varpi ^ { - k } ) \\cdot \\gamma ) = \\begin{cases} 1 & \\mbox { i f } | \\varpi | ^ k = | c | \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} I ( u _ { n } , v _ { n } ) = c _ { \\tilde { \\mathcal { N } } } + o _ { n } ( 1 ) \\mbox { a n d } \\langle I ' ( u _ { n } , v _ { n } ) , ( u _ { n } , v _ { n } ) \\rangle = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} d _ r & = 3 r , 1 \\le r \\le [ \\frac { g - 1 } { 3 } ] , \\\\ & = r + g - 1 - [ \\frac { g - r - 1 } { 2 } ] , [ \\frac { g - 1 } { 3 } ] < r \\le g - 1 \\\\ & = r + g , r \\ge g \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ i = \\pi _ i , 1 \\le i \\le n ) = \\prod _ { j = 1 } ^ n p \\left ( \\pi _ j - \\sum _ { 1 \\le i < j } 1 ( \\pi _ i < \\pi _ j ) \\right ) , \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) - 1 ) ( F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) - 1 ) = 1 \\ , , z \\in \\R \\ , . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} & \\Lambda _ { m } ( \\mathcal { O } _ { \\boldsymbol { K } , q } ) \\\\ & = \\left \\{ H = ( h _ { j l } ) \\in M _ { m } ( \\boldsymbol { K } \\otimes _ { \\mathbb { Q } } \\mathbb { Q } _ { q } ) \\bigm | H ^ { * } = H , \\ h _ { j j } \\in \\mathbb { Z } _ { q } , \\ \\sqrt { - D _ { \\boldsymbol { K } } } \\ , h _ { j l } \\in \\mathcal { O } _ { \\boldsymbol { K } , q } \\right \\} . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\chi _ n ( \\pi ) : = \\chi _ n ( \\pi , 1 ^ { n - | \\pi | } ) | \\pi | \\leq n , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} & T ^ { 1 , 0 } X \\oplus \\{ \\mathbb { C } ( T - i \\frac { \\partial } { \\partial \\eta } ) \\} , \\\\ & J T = \\frac { \\partial } { \\partial \\eta } , \\ J u = i u \\ \\ u \\in T ^ { 1 , 0 } X . \\\\ \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} A = u ( x , y , 0 ) \\mbox { a n d } A ' = ( u v + w ) ( x , y , 0 ) . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} L _ u \\varphi & = \\frac { 1 } { i } \\varphi _ x - C _ + ( u C _ + \\varphi ) , \\\\ B _ u \\varphi & = \\frac { 1 } { i } \\varphi _ { x x } + 2 [ ( C _ + u _ x ) ( C _ + \\varphi ) - C _ + ( ( u C _ + \\varphi ) _ x ) ] . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla v _ m \\nabla ( v _ m + g _ m ) + \\int _ \\Omega \\beta _ m ' ( w _ m ) v _ m ( v _ m + g _ m ) = 0 . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\max _ { m ~ \\chi } | \\hat { 1 _ S } ( \\chi ) | \\leq e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) } 2 ^ { - m / 2 } \\binom { n } { m } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} \\Gamma ^ t _ \\theta : = \\partial \\Omega ^ t _ \\theta = \\{ \\sigma = \\theta \\lambda ^ t ( z ) \\} \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\beta _ { X \\odot Y ' } \\leq \\beta _ X + \\beta _ { Y ' } = 1 + \\beta _ X - \\alpha _ { Y } < 1 . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\mathbf { h } _ i ^ * G \\mathbf { h } _ i = \\mathbf { u } _ i ^ * G \\mathbf { u } _ i = \\mathbf { e } _ i ^ * U ^ * G U \\mathbf { e } _ i = \\mathcal { G } _ { i i } \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} L _ { \\eta } g _ { i j } & = 2 \\left ( \\frac { 1 } { 2 } \\xi _ { , t } \\right ) g _ { i j } \\\\ V _ { , k } \\eta ^ { k } + \\left ( \\xi _ { , t } + \\left ( \\ln \\omega \\right ) _ { , t } \\xi \\right ) V & = - \\frac { f _ { , t } } { \\omega } \\\\ \\eta _ { i , t } & = f _ { , i } . \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\sup _ { 0 < r \\leq 1 } \\frac { 1 } { | B | } \\int _ { B } | f ( x ) - f _ { B } | d x = 0 . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} ( m / n ) ^ { 1 / 2 } = o \\ ( \\frac { \\log ( n / m ) } { \\log n } \\ ) : \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ M \\| \\partial ^ 2 _ t q ^ L _ { 0 , m } \\| ^ 2 _ { H } \\Delta t \\le \\frac 4 3 \\left \\| { \\partial ^ 2 q ^ L _ { 0 } \\over \\partial t ^ 2 } \\right \\| ^ 2 _ { L ^ 2 ( 0 , T ; H ) } , \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} U \\cap \\big \\{ | y ' | < r , | y _ n | < r \\big \\} = \\{ y _ n < F _ { x _ o } ( y ' ) \\} \\cap \\big \\{ | y ' | < r , | y _ n | < r \\big \\} , \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} v _ 1 ^ t ( x ) = \\int _ 0 ^ 1 V ^ \\theta ( x ) \\ , d \\theta \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} D \\bar J _ \\lambda ( V + \\phi ) [ \\phi ] & = - \\sum _ { i , j } c _ { i j } \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\phi = 0 , \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{gather*} K = \\sum _ { i = - \\mu } ^ \\mu K _ p , \\end{gather*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } & = \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\frac { z \\ ; { } _ { a } \\mathtt { B } ' _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } \\\\ & = \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\left ( \\frac { z \\mathtt { J } _ { \\nu - 1 } \\left ( z \\right ) } { \\mathtt { J } _ { \\nu } \\left ( z \\right ) } - \\nu ( 2 - a ) + 1 - a \\right ) . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\gamma _ 1 & = \\{ ( w , u ) \\in C \\mid u \\in ( - \\infty , 0 ) , w \\in \\bold e ( \\dfrac { 1 } { 2 } ) \\} , \\\\ \\gamma _ 2 & = \\{ ( w , u ) \\in C \\mid u \\in ( \\dfrac { 1 } { t } , \\infty ) , w \\in \\bold e ( \\dfrac { 1 } { 6 } ) \\} , \\\\ \\gamma _ 3 & = \\{ ( w , u ) \\in C \\mid u \\in ( 0 , 1 - x _ 1 ) , w \\in \\bold e ( \\dfrac { 1 } { 3 } ) \\} . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} \\varphi _ { \\varepsilon } ( p ) = \\phi _ 0 ^ { \\tau ( \\varepsilon , p ) } ( p ( \\varepsilon ) ) . \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { d e t } \\left [ \\zeta ^ 2 \\alpha _ { 2 j - 1 , 2 l } + \\beta _ { 2 j - 1 , 2 l } \\right ] _ { j , l = 1 , \\ldots , N / 2 } \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} v _ i = u _ i x _ E , \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\partial _ { \\mu _ i ^ \\prime } w _ { \\mu _ i ' , \\zeta _ i ' } & = \\frac { \\alpha _ 3 \\ , \\bigl ( | y - \\zeta _ i ' | ^ 2 - \\mu _ i '^ 2 \\bigr ) } { 2 \\ , \\sqrt { \\mu _ i ' } \\ , \\bigl ( | y - \\zeta _ i ' | ^ 2 + \\mu _ i '^ 2 \\bigr ) ^ { \\frac { 3 } { 2 } } } , D _ { \\zeta _ i ^ \\prime } w _ { \\mu _ i ' , \\zeta _ i ' } = \\frac { \\alpha _ 3 \\ , \\sqrt { \\mu _ i ' } \\ , ( y - \\zeta _ i ' ) } { \\bigl ( | y - \\zeta _ i ' | ^ 2 + \\mu _ i '^ 2 \\bigr ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\theta _ Q \\ominus \\theta _ { \\bar { Q } } = p _ { i } d q ^ { i } - \\bar { p } _ { i } d \\bar { q } ^ { i } = d \\left ( W ' \\left ( q , \\bar { q } \\right ) + \\nu _ { a } U ^ { a } \\left ( q , \\bar { q } \\right ) \\right ) , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} ( \\hat d ^ 1 \\circ \\hat d ^ 1 ) _ { \\alpha _ + , \\alpha _ - } = o ( \\alpha _ + , \\alpha _ - ) , \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\Delta ( m ; k ) \\leq U ( m ; k ) : = 2 ^ { q + k - 1 } + r \\ , \\ , \\ , \\ , \\ , \\ , m = 2 ^ q + r , \\ , \\ , \\ , 0 \\leq r < 2 ^ q , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} u ^ 4 ( t ) = \\\\ \\begin{cases} ( 0 , 0 , 0 , 0 ) \\\\ ( 1 , 0 , 0 , 0 ) \\\\ \\end{cases} ~ Q ( t ) \\ge 1 . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\frac { \\alpha ^ n - \\beta ^ n } { \\alpha - \\beta } = d / v , \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} \\beta = k _ + E _ + ^ * + \\sum _ j ( \\sum _ i k ^ { ( j ) } _ i E _ i ^ { ( j ) * } - \\sum _ { v \\in \\mathcal { E } ^ { ( j ) } } x ^ { ( j ) } _ v E _ v ^ { ( j ) * } ) - x _ + E ^ * _ { + s } \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\begin{array} { c c c } Y ( x ) & : = & \\partial _ a ( \\Phi _ a ( x ) ) | _ { a = 0 } \\\\ C ( x ) & : = & \\partial _ a ( B _ a ( x ) ) | _ { a = 0 } \\\\ \\tau ( x ) & : = & \\partial _ a ( \\eta _ a ( x ) ) | _ { a = 0 } . \\end{array} \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} Q _ n ^ * ( k ) : = \\sum _ { i = 1 } ^ n 1 ( X _ i > k ) \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\Psi ( x , z _ { ( 1 ) } , z _ { ( 2 ) } ) = z _ { ( 1 ) } \\cdot x + z _ { ( 2 ) } . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} K ( \\Gamma _ s ) \\langle \\Psi _ + , \\Psi _ { - } \\rangle = K ( \\Gamma _ s ) \\langle \\Psi _ + \\rangle = K ( \\Gamma _ s ) \\langle \\Psi _ - \\rangle . \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} f _ { k i } ( L ) = \\left \\{ \\begin{array} { c c } 0 , & T ( L ) \\notin \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\\\ \\frac { \\prod _ { j = 1 } ^ { k - 1 } [ l _ { k i } - l _ { k - 1 , j } ] _ q } { \\prod _ { j \\neq i } ^ { k } [ l _ { k i } - l _ { k j } ] _ q } , & T ( L ) \\in \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} f _ 1 = & p _ 1 , \\\\ f _ 2 = & p _ 1 p _ 2 , \\\\ f _ 3 = & p _ 1 p _ 2 ^ 2 + p _ 1 ^ 2 p _ 3 + p _ 1 p _ 2 p _ 3 , \\\\ f _ 4 = & p _ 1 p _ 2 ^ 3 + 2 p _ 1 ^ 2 p _ 2 p _ 3 + 2 p _ 1 p _ 2 ^ 2 p _ 3 + p _ 1 ^ 2 p _ 3 ^ 2 + p _ 1 p _ 2 p _ 3 ^ 2 \\\\ & + p _ 1 ^ 3 p _ 4 + 2 p _ 1 ^ 2 p _ 2 p _ 4 + p _ 1 p _ 2 ^ 2 p _ 4 + p _ 1 ^ 2 p _ 3 p _ 4 + p _ 1 p _ 2 p _ 3 p _ 4 . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } W _ f ( \\sqrt { p } \\ , \\beta , p \\Theta ) = 1 - \\frac 4 \\pi \\sum _ { k = 0 } ^ { + \\infty } \\frac { ( - 1 ) ^ k } { 2 k + 1 } \\exp \\left ( - \\frac { ( 2 k + 1 ) ^ 2 \\pi ^ 2 \\Theta } { 8 \\beta ^ 2 } \\gamma _ f ^ 2 \\right ) , \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} p e r ( ( x _ 1 , x _ 2 ) , \\mu ) ) = ( \\dfrac { y _ 1 } { y _ 2 } , \\dfrac { y _ 3 } { y _ 2 } , \\dfrac { y _ 4 } { y _ 2 } ) . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} e ( G ) & \\leq e ( G ^ { * } ) \\leq e ( G ^ { * } [ C ] ) + \\sum _ { i } \\left ( \\frac { ( d _ i - 2 ) } { 2 } | V ( R _ i ) | + p _ i \\cdot | V ( R _ i ) | \\right ) \\\\ & \\leq \\binom { c - s + 1 } { 2 } + ( s - 1 ) s + \\frac { d + 2 p - 2 } { 2 } ( n - c ) . \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\begin{aligned} A ^ 2 E ^ 2 ( B ^ 2 - C ^ 2 ) & = \\lambda _ 1 ^ 2 \\lambda _ 5 ^ 2 ( \\lambda _ 2 ^ 2 - \\lambda _ 3 ^ 2 ) \\\\ A ^ 2 B C D ^ 2 & = \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 3 \\lambda _ 4 ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\zeta \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { d e t } \\left [ \\left [ \\zeta ^ 2 \\alpha _ { 2 j - 1 , 2 l } + \\beta _ { 2 j - 1 , 2 l } \\right ] \\left [ \\mu _ { 2 j - 1 } \\right ] \\right ] _ { \\substack { j = 1 , \\ldots , ( N + 1 ) / 2 , \\\\ l = 1 , \\ldots , ( N - 1 ) / 2 } } . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\sum _ { \\substack { c \\geq 1 \\\\ \\gcd ( c , 2 ) = 1 } } \\frac { 1 } { c ^ { 2 w } } \\chi _ k ( c ) H _ h ( c ) = \\prod _ { \\substack { p \\\\ p \\neq 2 } } \\Big ( 1 + \\frac { \\chi _ k ( p ) H _ h ( p ) } { p ^ { 2 w } } + \\frac { \\chi _ k ( p ^ 2 ) H _ h ( p ^ 2 ) } { p ^ { 4 w } } + \\cdots \\Big ) . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ p ^ p + C \\sum _ { i = 1 } ^ { N } \\| \\nabla ( v _ i ^ { p / 2 } ) \\| _ 2 ^ 2 \\leq C \\sum _ { i = 1 } ^ { N } \\left ( \\| v _ i \\| _ { \\mu + p - 1 } ^ { \\mu + p - 1 } + \\| v _ i \\| _ { p + { \\delta } } ^ { p + { \\delta } } + \\| v _ i \\| _ p ^ p \\right ) . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} U \\otimes H ( 0 ) \\equiv \\begin{pmatrix} 0 & 0 & \\dots & 0 & b _ { 1 } & 0 & \\dots & 0 \\\\ 0 & 0 & \\dots & 0 & 0 & b _ { 2 } & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & b _ { q } \\end{pmatrix} , \\quad \\mbox { w h e r e $ b _ { 1 } , b _ { 2 } , \\dots , b _ { q } \\in [ 0 , 1 ) $ . } \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\frac { \\phi ' ( z ) } { \\phi ( z ) } = \\frac { \\pi ( z ) } { \\sigma ( z ) } . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} u ( x , y ) & = \\frac { P y } { 6 E I } \\left [ ( 9 L - 3 x ) x + ( 2 + \\nu ) \\left ( y ^ 2 - \\frac { D ^ 2 } { 4 } \\right ) \\right ] , \\\\ v ( x , y ) & = - \\frac { P } { 6 E I } \\left [ 3 \\nu y ^ 2 ( L - x ) + ( 4 + 5 \\nu ) \\frac { D ^ 2 x } { 4 } + ( 3 L - x ) x ^ 2 \\right ] . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\omega _ { a , b } ( x , y ) = - ( - 1 ) ^ { | x | | y | } { \\omega } _ { a , b } ( y , x ) = - ( - 1 ) ^ { | x | | y | + | b | | y | } \\omega ( a y , b x ) = & ( - 1 ) ^ { | a | | b | + | a | | x | } \\omega ( b x , a y ) \\\\ = & ( - 1 ) ^ { | a | | b | } \\omega _ { b , a } ( x , y ) . \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | \\Phi _ N | } \\sum _ { g \\in \\Phi _ N } \\big | \\phi _ k ( S ^ g y ) - \\phi ( S ^ g y ) \\big | = \\norm { \\phi _ k - \\phi } { 1 } \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} I _ { \\Delta } = ( x _ 1 x _ 3 , x _ 2 x _ 4 , x _ 1 x _ 5 , x _ 3 x _ 5 ) . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\mu } ( F _ \\lambda ( \\mu , r ) + R _ \\lambda ( \\mu , r ) ) = 0 \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\begin{aligned} t ^ 0 = \\frac { r _ 0 } { 2 } \\alpha _ 0 ^ { 1 / 2 } \\alpha _ 1 ^ { - 1 / 2 } \\left ( \\| h \\| _ { L _ { \\infty } } \\| h ^ { - 1 } \\| _ { L _ { \\infty } } \\| f \\| _ { L _ \\infty } \\| f ^ { - 1 } \\| _ { L _ \\infty } \\right ) ^ { - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\left | { \\dd ^ m \\over \\dd z ^ m } S _ { n , r } ( z ) \\Big | _ { z = 0 } \\right | & = \\frac { 1 } { 2 ^ m } \\sum _ { j = 1 } ^ { r } \\left | \\psi ^ { ( m - 1 ) } \\left ( \\frac { n - r + j } { 2 } \\right ) \\right | \\leq \\frac { 1 } { 2 ^ m } \\sum _ { j = 1 } ^ { n } \\left | \\psi ^ { ( m - 1 ) } \\left ( \\frac { j } { 2 } \\right ) \\right | . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} F ( w ) & = \\frac 1 { 2 \\pi \\mathrm { i } } \\int _ { \\Gamma } F ( \\zeta ) \\Big ( \\frac 1 { \\zeta - ( \\zeta _ 0 + z ) } - \\frac 1 { \\zeta - ( \\zeta _ 0 - z ) } \\Big ) \\mathrm { d } \\zeta \\\\ & = \\int _ \\Gamma F ( \\zeta ) K _ z ( \\zeta , \\zeta _ 0 ) \\ , \\mathrm { d } \\zeta . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} k ^ { n / 2 } F ( t ) = t ^ n F ( k / t ) , \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\| F ^ { ( 2 ) } _ { j l } ( t ) P _ j \\| = \\| ( F ^ { ( 2 ) } _ { j l } ( t ) - ( P _ { i _ 0 + 1 } + \\ldots + P _ p ) ) P _ j \\| \\le C _ { 4 , j l } | t | , | t | \\le t ^ { 0 0 } _ { j l } . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\Pr \\left \\{ X < \\varepsilon _ { k , 1 } , X \\ge \\varepsilon _ { k , 4 } , X > Y \\right \\} = 0 . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} \\mathcal { R } _ { i , j , m } ^ 5 : = 4 \\int _ 0 ^ 1 d \\tau \\ , \\int _ { B _ \\rho ( \\zeta _ i ) } ( w _ i + \\tau \\pi _ i ) ^ 3 \\ , \\pi _ i \\ , U _ j \\ , U _ m . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} ( \\alpha + 1 ) \\ , \\overline { c _ 1 } ( t ) + \\overline { c _ 2 } ( t ) + \\overline { c _ 3 } ( t ) = ( \\alpha + 1 ) \\ , \\overline { c _ { 1 , 0 } } + \\overline { c _ { 2 , 0 } } + \\overline { c _ { 3 , 0 } } = : M , t > 0 . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\mathfrak { N } = \\left \\lbrace \\mathbf { u } \\in L _ 2 ( \\Omega ; \\mathbb { C } ^ n ) \\colon f \\mathbf { u } = \\mathbf { c } \\in \\mathbb { C } ^ n \\right \\rbrace , \\dim \\mathfrak { N } = n . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} D ( \\Lambda ^ s ) = [ L ^ 2 ( \\Omega ) , D ( - \\Delta ) ] _ { \\frac s 2 } \\quad \\forall s \\in [ 0 , 2 ] . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\bar W _ i ^ L = \\left ( \\bigoplus _ { 0 \\le l _ 0 , \\ldots , l _ { i - 1 } \\le L } { \\cal V } ^ { l _ 0 } \\otimes { \\cal V } ^ { l _ 1 } _ \\# \\otimes \\ldots \\otimes { \\cal V } ^ { l _ { i - 1 } } _ \\# \\right ) \\otimes W ^ L _ \\# , \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} | \\hat F _ 1 ( x _ 1 ) | & \\ll \\sum _ { i = 0 } ^ { N _ 1 } | \\hat F _ 1 ^ { ( i ) } ( y _ 1 ) | \\cdot | x _ 1 - y _ 1 | ^ { i } \\\\ [ 0 e x ] & \\stackrel { \\eqref { e q 8 7 } } { \\ll } ~ \\sum _ { i = 1 } ^ { N _ 1 } H ^ { \\ell _ { 1 , i } \\delta } \\cdot | \\sigma _ 1 ( F , t _ 1 ) | ^ { i } \\\\ [ 0 e x ] & \\stackrel { \\eqref { h j } } { \\ll } ~ \\sum _ { i = 1 } ^ { N _ 1 } H ^ { \\ell _ { 1 , i } \\delta } \\cdot H ^ { - ( \\ell _ { 0 } + \\ell _ { 1 , i } - 1 ) \\delta } \\\\ [ 0 e x ] & \\ll ~ H ^ { - ( \\ell _ { 0 } - 1 ) \\delta } \\ , . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\phi _ { F _ 2 } ^ { t } \\left ( v , w \\right ) = \\left ( \\frac { v + w } { 2 } + ( \\cos t ) \\frac { v - w } { 2 } + ( \\sin t ) \\frac { w \\times v } { \\| v + w \\| } , \\frac { v + w } { 2 } + ( \\cos t ) \\frac { w - v } { 2 } + ( \\sin t ) \\frac { v \\times w } { \\| v + w \\| } \\right ) . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k ( - \\lambda _ i ) = \\sum _ { i = 1 } ^ k ( - \\lambda _ i ) ^ 3 = ( k + l ) . \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{gather*} \\xi _ a = k _ { a b } \\ , \\theta ^ b , \\end{gather*}"}
-{"id": "3209.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ r t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } = \\frac { P _ { ( r ) } \\big ( q ^ { \\lambda _ 1 } t ^ { N - 1 } , \\dots , q ^ { \\lambda _ { N - 1 } } t , q ^ { \\lambda _ N } ; q , t \\big ) } { P _ { ( r ) } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } . \\end{gather*}"}
-{"id": "7959.png", "formula": "\\begin{align*} \\mathcal D ^ t _ 2 : = \\frac 1 t \\bigg ( \\frac { \\partial _ N v } { N \\cdot \\nu ^ 0 } \\ , \\mathcal { H } ^ { n - 1 } \\restriction _ { \\Gamma ^ 0 } - \\frac 1 t \\Delta h ^ 0 \\ , \\chi _ { \\Omega ^ 0 \\setminus \\Omega ^ t } \\bigg ) . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} w _ \\Omega ( x ) \\le \\Psi ^ { - 1 } ( d ( x , { N _ { \\Omega } } ) ) , \\textrm { w h e r e } \\Psi ( s ) = \\int _ { 0 } ^ s \\frac { d t } { G ( t ) } , \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} ( 1 - r ) [ \\gamma ] = 0 , \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\vert \\alpha \\vert ^ { \\frac { 2 } { 2 - \\gamma } } \\Lambda ^ { \\frac { 2 } { 2 - \\gamma } } T = ( \\vert \\alpha \\vert ^ { \\frac { 2 } { 1 - \\theta } } \\Lambda ^ { \\mu } T ^ { \\nu } ) ^ { \\frac { 2 - \\gamma + \\frac { 2 } { l } } { 2 - \\gamma } } , \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} W _ { m , r } ( n + 1 , k ) = ( r + m k ) W _ { m , r } ( n , k ) + W _ { m , r } ( n , k - 1 ) . \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} F _ 0 & = 1 \\\\ F _ 1 & = t ^ { - a _ 1 } \\hat c _ 0 ^ { - 1 } \\hat c _ 1 a _ 1 \\\\ F _ 2 & = t ^ { - a _ 2 } \\hat c _ 0 ^ { - 1 } \\hat c _ 2 a _ 2 ( a _ 2 - a _ 1 ) \\\\ F _ 3 & = t ^ { - a _ 3 } \\hat c _ 0 ^ { - 1 } \\hat c _ 3 a _ 3 ( a _ 3 - a _ 1 ) ( a _ 3 - a _ 2 ) \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} X _ 0 : \\begin{cases} \\dot y _ 1 & = - y _ 2 + y _ 1 y _ 3 \\\\ \\dot y _ 2 & = y _ 1 + y _ 2 y _ 3 \\\\ \\dot y _ 3 & = \\frac { 1 } { 2 } ( 1 + y _ 3 ^ 2 - ( y _ 1 ^ 2 + y _ 2 ^ 2 ) ) . \\end{cases} \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} s ( z ) = \\sigma + s ' ( z ) = \\sigma + \\omega \\otimes \\lambda \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\mu = \\max _ { y \\in \\mathcal C } | y | | y | = \\sum _ { i = 1 } ^ { N } | y _ i | y \\in \\mathbb R ^ N . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} S _ n : = s _ 0 + \\sum _ { k = 1 } ^ n \\zeta _ k \\ , \\ \\ \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} \\limsup _ { r \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( L [ \\pm n ^ { 2 / 3 } ] _ n - 4 n \\leq - \\frac { r ^ 2 n ^ { 1 / 3 } } { 1 2 8 } \\right ) = 0 \\ , . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A g ^ { i j } = - g ^ { l j } g ^ { i k } \\overline { \\partial } _ A g _ { k l } \\end{gather*}"}
-{"id": "8549.png", "formula": "\\begin{align*} D _ { i _ 1 } ^ 2 \\times \\cdots \\times D _ { i _ { p - 1 } } ^ 2 \\times \\left ( \\bigcup \\limits _ { k = 1 } ^ q D _ { j _ 1 } ^ 2 \\times \\cdots \\times S _ { j _ k } ^ 1 \\times \\cdots \\times D _ { j _ q } ^ 2 \\right ) \\hookrightarrow \\mathcal { Z _ K } , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\sigma ( z ) y '' ( z ) + \\tau ( z ) y ' ( z ) + h ( z ) y ( z ) = 0 . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} ( T _ k 1 ) ( x ) & = - \\frac { 1 } { 2 \\pi k } \\int _ 0 ^ \\infty e ^ { i x \\xi } \\hat u ( \\xi ) ~ d \\xi + \\frac 1 { 2 \\pi k } \\int _ 0 ^ \\infty \\frac { \\xi e ^ { i x \\xi } } { \\xi - k } \\hat u ( \\xi ) ~ d \\xi \\\\ & = - \\frac { C _ + u ( x ) } { k } + \\frac 1 { 2 \\pi k } \\int _ 0 ^ { \\infty } \\frac { \\xi ^ { 1 - \\epsilon } e ^ { i x \\xi } } { \\xi - k } [ \\xi ^ { \\epsilon } \\hat u ( \\xi ) ] ~ d \\xi . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\omega ^ { \\ast } : = \\omega _ 0 + k \\sum _ { i = 1 } ^ d \\sqrt { - 1 } \\partial \\bar { \\partial } | s _ i | _ { H _ i } ^ { 2 ( 1 - ( 1 - \\beta ) \\tau _ i ) } \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} c _ i = . \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\| E ( t , \\tau ) \\| \\le 2 C _ 1 | t | + C _ { 9 } | \\tau | | t | ^ 3 , | t | \\le t ^ 0 , \\ N = 0 . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} \\sum _ { p \\mid m _ 0 } \\frac { 1 } { \\log p } = \\sum _ { p \\le y } \\frac { 1 } { \\log p } = \\frac { y } { \\log ^ 2 y } < \\frac { \\log x } { ( \\log \\log x ) ^ 2 } + O \\bigg ( \\frac { \\log x } { ( \\log \\log x ) ^ 3 } \\bigg ) . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { \\Pi } | \\omega ^ { \\theta } | ^ { 2 } \\dd x + 2 \\int _ { \\Pi } \\Big ( | \\nabla \\omega ^ { \\theta } | ^ { 2 } + \\Big | \\frac { \\omega ^ { \\theta } } { r } \\Big | ^ { 2 } \\Big ) \\dd x & = 2 \\int _ { \\Pi } \\frac { u ^ { r } } { r } \\omega ^ { \\theta } \\omega ^ { \\theta } \\dd x + 2 \\int _ { \\Pi } \\frac { \\partial _ z | u ^ { \\theta } | ^ { 2 } } { r } \\omega ^ { \\theta } \\dd x \\\\ & = : I + I I . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} 2 4 = \\sum _ { F \\in \\Delta [ 1 ] } ( \\ell ^ * ( F ) \\ell ^ * ( F ) + \\ell ^ * ( F ) + \\ell ^ * ( F ^ \\circ ) + 1 ) . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} R _ 0 u : = S _ 1 u , R _ j u : = ( S _ { j + 1 } - S _ j ) u , j \\geq 1 . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} d ^ { 2 } w / d z ^ { 2 } - \\left \\{ { u ^ { 2 } f \\left ( z \\right ) + g \\left ( z \\right ) } \\right \\} w = p \\left ( z \\right ) . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} & \\sum _ { l = 0 } ^ k \\binom { n + k - 2 } { k - l } \\sigma ^ { ( k - l ) } v _ n ^ { ( l ) } + \\sum _ { l = 0 } ^ { k - 1 } \\binom { n + k - 2 } { k - l - 1 } \\tau ^ { ( k - l - 1 ) } v _ n ^ { ( l ) } \\\\ & + \\sum _ { l = 0 } ^ { k - 2 } \\binom { n + k - 2 } { k - l - 2 } h ^ { ( k - l - 2 ) } v _ n ^ { ( l ) } = 0 . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 a _ 1 ^ 2 & a _ 2 ^ 2 & a _ 3 ^ 2 \\\\ a _ 2 ^ 2 & 2 a _ 4 ^ 2 & a _ 5 ^ 2 \\\\ a _ 3 ^ 2 & a _ 5 ^ 2 & 2 a _ 6 ^ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} \\rho \\frac { \\partial Q } { \\partial t } + \\pi a ^ { 2 } \\frac { \\partial P } { \\partial x } = 2 \\pi \\mu a \\left ( x , t \\right ) \\bar { u } \\left ( x , t \\right ) \\left ( \\frac { d U } { d r } \\mid _ { a \\left ( x , t \\right ) } \\right ) . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\theta _ { \\phi } ( 0 , \\rho ) = \\lim _ { j \\to \\infty } \\theta _ { u _ { j } } ( 0 , \\rho ) \\forall \\ , \\rho > 0 . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\chi _ V ( t ) = \\sum _ { j \\ge 0 } ( - 1 ) ^ j \\chi ( V \\cap H _ 1 \\cap \\cdots \\cap H _ j ) \\ , t ^ j \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} d = 3 \\colon & 1 9 \\le p \\le 2 2 8 1 ; \\\\ d = 4 \\colon & 1 9 \\le p \\le 2 2 8 1 , p \\neq 2 9 ; \\\\ d = 5 \\colon & 2 3 \\le p \\le 2 2 8 1 , p \\neq 2 9 ; \\\\ d = 6 \\colon & 2 3 \\le p \\le 2 2 8 1 , p \\neq 2 9 ; \\\\ d = 7 \\colon & 3 7 \\le p \\le 2 2 8 1 . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} Y ( x ) = \\frac { 1 } { 4 } \\sum _ { \\omega \\in \\{ - N , \\dots , N \\} ^ 2 } \\hat { Y } ( \\omega ) e ^ { \\pi i x \\cdot \\omega } \\ ; . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\ , K _ { i l } = \\ , K _ { b _ { j } , l _ { j } } = r + 1 . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} X _ t = x + \\int _ 0 ^ t Y _ s \\d s , Y _ t = y - \\int _ 0 ^ t ( c _ 0 Y _ s + k X _ s ) \\d s + W _ t - ( 1 + e ) \\sum _ { 0 \\leq s \\leq t } \\dot { Y } _ { s - } \\mathbf { 1 } _ { \\{ | X _ s | = P _ O \\} } \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} u = 0 \\mbox { o n } \\partial \\Omega \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} d ( ( 0 , \\Phi ^ { - 1 } ( u ) , 0 ) , 0 ) & = ( ( 0 , \\Phi ^ { - 1 } ( s ) - r , 0 ) , - u ) \\\\ & = ( ( s , \\Phi ^ { - 1 } ( s ) , t ) , 0 ) - ( ( s , r , t ) , u ) . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} 2 \\sum _ { i = 0 } ^ { k - 1 } ( k - 1 - i ) b ^ i & \\le ( k - 1 ) k + \\frac { 2 ( k - 2 ) } 3 \\Biggl \\{ \\sum _ { i = 1 } ^ { k - 1 } b ^ i - ( k - 1 ) \\Biggr \\} \\\\ & = ( k - 1 ) k + \\frac { 2 ( k - 2 ) } 3 \\Biggl \\{ \\sum _ { i = 0 } ^ { k - 1 } a ^ i - k a ^ { k - 1 } \\Biggr \\} \\\\ & < ( k - 1 ) k + \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) a ^ { i } \\ , . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} { \\frak S } _ { \\ell } ( u ) = { \\frak S } _ { \\ell } ^ { \\prime } ( u ) f _ { \\ell } ( u ) , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\ , \\ , \\ , F ^ { \\bf a } _ k = a _ k \\ , \\ , \\ , \\ , \\ , - n + 1 \\le k \\le 0 \\ , \\ , \\ , \\ , \\ , \\ , F ^ { \\bf a } _ k = b _ 1 F ^ { \\bf a } _ { k - 1 } + \\cdots + b _ n F ^ { \\bf a } _ { k - n } \\ , \\ , \\ , \\ , \\ , k > 0 . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z \\mathtt { g } _ { a , \\nu } ' ( z ) } { \\mathtt { g } _ { a , \\nu } ( z ) } \\right ) \\geq \\frac { r \\mathtt { g } _ { a , \\nu } ' ( r ) } { \\mathtt { g } _ { a , \\nu } ( r ) } = a ( 1 - \\nu ) + a ^ { a / 2 } \\left ( a \\nu - a + 1 - \\sum _ { n = 1 } ^ \\infty \\frac { 2 r ^ 2 } { \\mathtt { j } _ { \\nu , n } ^ 2 - r ^ 2 } \\right ) \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} \\min \\| x \\| \\quad { \\rm s u b j e c t \\ \\ t o } x \\in \\mathcal { S } _ k = \\left \\{ x \\mid \\| A _ k x - b \\| = \\min \\right \\} \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} x ' ( t ) = f ( t ) x ^ 3 ( t ) + g ( t ) x ^ 2 ( t ) , \\ , \\ , t \\in [ a , b ] \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} v ( \\alpha ) : = \\inf \\{ \\alpha _ { n + 1 } - \\alpha _ { n } \\colon n \\in \\N \\} . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\abs { x } & = \\abs { x } q + \\abs { x } e ^ { \\abs { x } } \\{ \\mu ( \\infty , x ) \\} = \\abs { x } q + \\mu ( \\infty , x ) e ^ { \\abs { x } } \\{ \\mu ( \\infty , x ) \\} \\\\ & = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } V ( f _ i ) + \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } \\alpha _ i e ^ { \\abs { x } } \\{ \\mu ( \\infty , x ) \\} = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } x _ i . \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} h _ n ( z ) = - \\frac { n ( n - 1 ) } { k ( k - 1 ) } \\sigma '' ( z ) - \\frac { n } { k - 1 } \\tau ' ( z ) + \\sum _ { l = 0 } ^ { k - 3 } C _ { k - l - 2 , n } \\frac { z ^ l } { l ! } , \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} ( ( j i ) ^ * ( b ) , I , { ( j ' o ' ) } ^ * ( b ) , O ' ) = ( i ^ * ( c ) , I , { o ' } ^ * ( c ' ) , O ' ) \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} ( \\partial _ t L _ u ) \\varphi = - C _ + ( u _ t \\varphi ) = 2 C _ + ( u u _ x \\varphi ) + \\frac { 1 } { i } C _ + ( [ ( u _ { x x } - 2 ( C _ + u _ { x x } ) ] \\varphi ) \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} \\displaystyle \\tilde { l } ^ { E L } ( \\theta ) = 2 \\sum _ { i = 1 } ^ n \\log ( 1 + \\lambda ^ T _ { E L } \\psi ( x _ i , \\theta ) ) \\ , . \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb { R } } u \\varphi ~ d x \\right | ^ 2 = - 2 \\pi k \\int _ { \\mathbb { R } } | \\varphi | ^ 2 ~ d x . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\psi '' ( z ) + \\frac { \\tilde { \\tau } ( z ) } { \\sigma ( z ) } \\psi ' ( z ) + \\frac { \\tilde { \\sigma } ( z ) } { \\sigma ^ 2 ( z ) } \\psi ( z ) = 0 , \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} H [ U _ { q } ] = \\begin{pmatrix} H _ { q } ' & 0 \\\\ 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} \\{ C ^ T \\} _ { l + 1 , m + 1 } = \\int _ { - \\infty } ^ \\infty T _ { T - l } ( \\lambda ) T _ { T - m } ( \\lambda ) \\ , d \\rho ( \\lambda ) , l , m = 0 , \\ldots , T - 1 , \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\pm p ^ b = f _ n ( u , \\pm 1 ) . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\frac 1 n \\log \\binom { n } { a _ 1 , \\dots , a _ n } = \\sum _ { a _ i > 0 } \\ ( \\frac { a _ i } { n } \\log \\frac { n } { a _ i } + O \\ ( \\frac { \\log ( a _ i + 2 ) } n \\ ) \\ ) + O \\ ( \\frac { \\log n } { n } \\ ) . \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} T ( a ( 1 + q ) , b ( 1 + q ) ) & = \\frac { \\Delta } { 2 } \\Bigl ( \\frac 1 { \\Delta } - ( \\alpha - \\beta ) - ( \\alpha - \\beta ) q - \\sqrt { 1 - 2 q ( \\alpha + \\beta ) + q ^ 2 ( \\alpha - \\beta ) ^ 2 } \\ , \\Bigr ) \\\\ & = \\frac { \\Delta ( \\frac 1 \\Delta - 1 - ( \\alpha - \\beta ) ) } 2 + \\Delta T ( \\alpha q , \\beta q ) . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\sum _ { s \\in \\mathcal S } \\chi _ s { \\rm { t r } } ( P _ { U _ s } P _ M ) = \\chi _ 0 \\dim M . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} f ( \\{ a , b \\} ) & = g ( a ) f ( b ) - ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\\\ g ( a b ) & = f ( a ) g ( b ) + ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} ( I - T _ k ) ^ { - 1 } & = ( I - S _ k + \\widetilde { T } _ k ) ^ { - 1 } = ( I + ( I - S _ k ) ^ { - 1 } \\widetilde { T } _ k ) ^ { - 1 } ( I - S _ k ) ^ { - 1 } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } ( - ( I - S _ k ) ^ { - 1 } \\widetilde { T } _ k ) ^ { n } ( I - S _ k ) ^ { - 1 } . \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{gather*} ( x _ 0 ) ^ { 2 } + R ^ { 2 } = \\left ( \\frac { x _ 0 } { \\xi _ 0 } \\right ) ^ { 2 } \\sum _ { i = 1 } ^ { n } ( \\xi _ i ) ^ { 2 } = ( x _ 0 ) ^ { 2 } \\therefore R = 0 , \\end{gather*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } & \\left | z _ i ( t + 1 ) - \\bar { x } \\right | = 0 , \\quad . \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\lambda = \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\right ) ^ { - 1 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i + \\epsilon _ 1 \\ , . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} \\langle \\varepsilon _ { k l m } , \\varepsilon _ { r s t } \\rangle _ { \\mathcal { C } } = w ( k , l - m ) \\ , \\delta _ { k , r } \\ , \\delta _ { l - m , s - t } \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\frac { h _ K } { h _ { K ^ + } } & = \\sqrt { \\abs { \\frac { \\Delta _ K } { \\Delta _ { K ^ + } } } } \\frac { \\omega _ K R _ { K ^ + } } { 2 \\pi ^ g R _ K } \\prod _ { \\ell } \\nu _ { \\ell } ( K ) \\\\ & = \\sqrt { \\abs { \\frac { \\Delta _ K } { \\Delta _ { K ^ + } } } } \\frac { \\omega _ K } { ( 2 \\pi ) ^ g } \\prod _ { \\ell } \\nu _ { \\ell } ( K ) \\\\ & = { \\omega _ K } \\nu _ \\infty ( f ) \\prod _ { \\ell } \\nu _ { \\ell } ( K ) . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( t ) d t \\approx \\sum _ { i = 1 } ^ \\rho w _ i f ( \\mu _ i ) \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} | \\ 4 { R m } | ( x , y , t ) = | R m | ( x , t ) , | \\ 4 { R i c } | ( x , y , t ) = | R i c | ( x , t ) , \\ 4 { R } ( x , y , t ) = R ( x , t ) \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\eta ( r , z ) = \\begin{cases} \\gamma | r | ^ { 1 - n } \\left ( - 2 [ ( 1 + | r | ^ { n - 1 } ) ^ \\frac { 1 } { n - 1 } - 1 ] \\frac { z } { 1 + z ^ 4 } \\ , r , \\ \\arctan ( z ^ 2 ) ( 1 + | r | ^ { n - 1 } ) ^ { \\frac { 2 - n } { n - 1 } } | r | ^ { n - 1 } \\right ) & z > 0 , \\\\ 0 & z \\le 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\left . \\partial _ { w _ { a a } } \\left ( G \\left ( W , Z \\right ) \\right ) \\right \\vert _ { Z = O _ { q \\times N } \\atop { W = O _ { q \\times q } } } = \\begin{pmatrix} 0 & \\dots & 0 & \\dots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\dots & 0 & \\dots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\dots & 0 & \\dots & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } v _ { \\alpha } & = ( \\delta ^ { \\mu \\alpha } - a ^ { \\mu \\alpha } ) \\partial _ { \\mu } v _ { \\alpha } - \\frac { 1 } { R } \\partial _ { t } R \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} ( \\partial _ n + 2 ) \\Gamma _ m & = 2 m - ( \\partial _ n + 2 ) \\Gamma \\\\ & = 2 m \\leq 0 . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} M _ \\lambda ( \\zeta ) : = \\begin{pmatrix} g _ { \\lambda } ( \\zeta _ 1 ) & - G _ { \\lambda } ( \\zeta _ 1 , \\zeta _ 2 ) & \\ldots & - G _ { \\lambda } ( \\zeta _ 1 , \\zeta _ k ) \\\\ - G _ { \\lambda } ( \\zeta _ 1 , \\zeta _ 2 ) & g _ { \\lambda } ( \\zeta _ 2 ) & \\ldots & - G _ \\lambda ( \\zeta _ 2 , \\zeta _ k ) \\\\ \\vdots & & & \\vdots \\\\ - G _ \\lambda ( \\zeta _ 1 , \\zeta _ k ) & - G _ \\lambda ( \\zeta _ 2 , \\zeta _ k ) & \\ldots & g _ \\lambda ( \\zeta _ k ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} { \\bf c ^ { \\bf a } } = C ^ { - 1 } { \\bf a } , \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u & \\ , = \\mathcal { L } \\ , u + \\beta \\ , u ( 1 - u ) + \\sqrt { \\gamma \\ , u ( 1 - u ) } \\dot { W } & & \\quad \\overset { \\circ } { \\Gamma } \\\\ \\nabla _ { o u t } u \\cdot [ \\alpha ] & \\ , = - \\hat { \\beta } \\ , u ( 1 - u ) & & V , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} W ^ { \\rho _ \\sigma } _ { u _ i } ( \\lambda ) * W ^ { \\rho _ \\sigma } _ { u _ j } ( u _ i ) ( x , t ) = & \\overline { t } W ^ \\rho _ { u _ i } ( \\lambda ) \\# W ^ \\rho _ { u _ j } ( u _ i ) ( x ) \\\\ = & \\overline { t } C _ { i , j } W ^ \\rho _ { u _ j } ( \\lambda ) ( x ) \\\\ = & C _ { i , j } W ^ { \\rho _ \\sigma } _ { u _ j } ( \\lambda ) ( x , t ) \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\wedge ^ 3 E \\otimes L = ( 4 { \\cal O } ( 3 ) \\oplus 1 2 { \\cal O } ( 2 ) \\oplus 4 { \\cal O } ( 1 ) ) \\otimes { \\cal O } ( - 1 ) = 4 { \\cal O } ( 2 ) \\oplus 1 2 { \\cal O } ( 1 ) \\oplus 4 { \\cal O } _ X \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} f _ p ( a ) = \\begin{cases} 1 - 1 / p , & p \\mid a , \\\\ - 1 / p , & p \\nmid a . \\end{cases} \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} L _ { i p } ( \\Omega ) : = \\bigcup _ { m = 1 } ^ \\infty L _ { i p } ( \\Omega _ m ) . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\sum _ { m = 2 } ^ { q + 1 } a _ { k + m - 1 - q } m _ { ( m - 1 , \\dot { 0 } ) } = \\sum _ { t = 0 } ^ { q - 1 } a _ { k - t } m _ { ( q - t , \\dot { 0 } ) } \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\max _ { x \\in M } \\log \\lVert A ^ n ( x ) \\rVert = \\sup _ { \\mu \\in \\mathcal E ( f ) } \\lambda _ 1 ( \\mu ) = \\sup _ { \\mu \\in \\mathcal E _ { p e r } ( f ) } \\lambda _ 1 ( \\mu ) . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} ( N + 2 ) \\Gamma _ { \\varepsilon } = ( N + 2 ) \\Gamma - \\varepsilon ( \\partial _ n + 2 ) ( A t + | x | ^ { 2 } ) \\leq 0 . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} R _ { p , k } ( z ) = \\frac { \\sum ^ { k } _ { j = 0 } M _ j \\ , \\psi _ { 1 , j } ( z ) \\ , G _ { j + 1 , p } ( z ) } { \\sum ^ { k } _ { j = 0 } M _ j \\ , \\psi _ { 1 , j } ( z ) } . \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} V ( z ) : = \\widetilde { V } ( z ) - E _ \\infty ^ k ( z , \\tfrac { k + 1 } { 2 } ) - E _ 0 ^ k ( z , \\tfrac { k + 1 } { 2 } ) . \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { j = 1 } ^ { k } \\sum _ { l = 0 } ^ { | p _ { j } | - 1 } \\xi _ { j l } \\right \\} \\in ( 0 , 1 ) \\backslash \\{ { 1 / 2 } \\} . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\varphi _ n ^ C - Q _ n \\mathcal { K } ( \\phi _ n ^ C ) = Q _ n f . \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\rho _ i \\le \\min \\{ \\epsilon _ { k , i + 1 } / \\epsilon _ { k , i } \\mid k = 0 , 1 , 2 , \\dots \\} \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} T ^ { f \\circ \\pi } g ( x ) & = \\int _ X g ( y ) \\cdot f ( \\pi ( x , y ) ) \\ > d \\omega _ X ( y ) = \\int _ D \\int _ X g ( y ) \\cdot f ( \\pi ( x , y ) ) \\ > K _ h ( x , d y ) \\ > d \\omega _ D ( h ) \\\\ & = \\int _ D \\int _ X g ( y ) \\cdot f ( h ) \\ > K _ h ( x , d y ) \\ > d \\omega _ D ( h ) = T _ f g ( x ) . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\langle \\mathcal { A } \\varphi , \\xi \\rangle = \\langle \\varphi , A \\xi \\rangle - \\langle \\mathcal { B } \\varphi , D _ \\nu \\xi \\rangle _ Y \\ , , \\quad \\forall \\varphi \\in D ( \\mathcal { A } ) \\ , , \\xi \\in D ( A ) \\ , , \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} \\varepsilon _ 3 \\dot { x } ( t ) = - \\varepsilon _ 1 x ( t ) - \\varepsilon _ 2 x ( t - h ) - \\varepsilon _ 4 \\frac { 1 } { h } \\int _ { - h } ^ 0 x ( t + s ) d x \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\| S ^ { 1 / 2 } U ( t ) \\| + \\gamma \\int _ 0 ^ t e ^ { \\gamma ( t - s ) } \\| S ^ { 1 / 2 } U ( s ) \\| d s \\\\ \\leq e ^ { \\gamma t } \\ , \\| S ^ { 1 / 2 } U ( 0 ) \\| + \\int _ 0 ^ t e ^ { \\gamma ( t - s ) } \\| S ^ { 1 / 2 } { \\tilde L } _ a U ( s ) \\| d s . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} p _ { j 1 } + \\sum _ { l = 2 } ^ { r } s _ { l } p _ { j l } \\in \\Z \\backslash \\{ 0 \\} , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} F _ + = \\int _ { \\mathbb { R } ^ 2 } \\Phi _ + ( f ) v _ 2 d v _ 2 d v _ 3 , ~ G = \\int _ { \\mathbb { R } ^ 2 } \\mathcal { M } _ { \\nu } ( f ) v _ 2 d v _ 2 d v _ 3 , ~ F _ { L , + } ( v _ 1 ) = \\int _ { \\mathbb { R } ^ 2 } f _ L ( v ) v _ 2 d v _ 2 d v _ 3 . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} & E A = c i r c s h i f t ( E A , [ y _ 1 ~ ~ - 2 y _ 1 ) , \\\\ & E A = c i r c s h i f t ( E A , [ \\lfloor \\frac { y _ 1 } { 5 0 } \\rfloor ~ ~ - \\lfloor \\frac { y _ 1 } { 3 5 } \\rfloor ] ) . \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} c _ { 1 , \\infty } = \\frac { k _ 3 } { k _ 1 } c _ { 2 , \\infty } ^ { \\alpha + 1 } , \\ c _ { 3 , \\infty } = \\frac { k _ 3 } { k _ 2 } c _ { 2 , \\infty } ^ { \\alpha } . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} P \\left [ X ( t + h ) = y | X ( t ) = x \\right ] = E _ { t , x } [ \\lambda ( t , x , y , \\alpha ( t ) , m ( t ) ) ] \\cdot h + o ( h ) , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} | d f | _ x : = \\sqrt { g ^ { i j } ( x ) \\frac { \\partial f } { \\partial x ^ i } ( x ) \\frac { \\partial f } { \\partial x ^ j } ( x ) } , \\ a n d \\ \\langle \\beta , d f \\rangle _ x : = g ^ { i j } ( x ) b _ i ( x ) \\frac { \\partial f } { \\partial x ^ j } ( x ) . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} U _ { \\varepsilon _ k } & = \\{ E + \\xi ( 2 E ( k , 0 ) - E ( 0 , - k ) ) - \\xi ^ 2 E ( k , - k ) \\mid \\xi \\in \\Bbbk \\} , \\\\ U _ { - \\varepsilon _ k } & = \\{ E + \\xi ( E ( 0 , k ) - 2 E ( - k , 0 ) ) - \\xi ^ 2 E ( - k , k ) \\mid \\xi \\in \\Bbbk \\} . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} f '' & = \\frac { a '' } { a } + 2 n \\frac { b '' } { b } \\\\ & = - 2 n \\frac { a ^ 2 } { b ^ 4 } + \\frac { 4 n ( n + 1 ) } { b ^ 2 } - 4 n \\frac { a ' b ' } { a b } - 2 n ( 2 n - 1 ) \\left ( \\frac { b ' } { b } \\right ) ^ 2 + \\left ( \\frac { a ' } { a } + 2 n \\frac { b ' } { b } \\right ) f ' . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\langle A x _ { 1 } , x _ { 2 } \\rangle _ { X } = \\overline { \\langle A ' x _ { 2 } , x _ { 1 } \\rangle } _ { X } , x _ { 1 } , x _ { 2 } \\in X . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} \\chi ( P _ k ) = \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { i } f _ i ( P _ k ) . \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\frac 1 n \\sum _ { i = 1 } ^ n \\frac 1 h ( f ( X ^ { \\theta + h / 2 } _ { [ i ] } ) - f ( X ^ { \\theta - h / 2 } _ { [ i ] } ) ) , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} c ^ { \\left ( 0 , J \\right ) } _ { i j } W ^ { J } = \\left < d ^ { \\left ( I ' , J ' \\right ) } _ { i , j } Z _ { i } , \\Xi _ { j } \\right > W ^ { J ' } + \\dots , \\quad \\mbox { f o r s u i t a b l e m u l t i - i n d e x e s $ J , J ' \\in \\mathbb { N } ^ { q ^ { 2 } } $ a n d $ I ' \\in \\mathbb { N } ^ { q \\left ( p - q \\right ) } $ , } \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} R ( B ) = \\bigoplus _ { b \\in B } { I ( b ) } \\ . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} u ^ n _ t ( x ) : = \\frac { 1 } { M ^ e } \\sum _ { i = 1 } ^ { M ^ e } \\xi _ t ( x , i ) . \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) ( 3 i + 2 - k ) b ^ i \\ge 0 \\ , . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\lvert \\mathcal { T } \\rvert = \\sum _ { j = 1 } ^ { s ^ * - 1 } n _ j + P ( r _ { s ^ * } + \\delta _ { s ^ * } - 1 ) + Q + 1 \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} h ( x ) = x \\geq 0 . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} R ( u - v ) \\ , K _ 1 ( u ; a ) \\ , R ( u + v ) \\ , K _ 2 ( v ; a ) = K _ 2 ( v ; a ) \\ , R ( u + v ) \\ , K _ 1 ( u ; a ) \\ , R ( u - v ) . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} { \\cal E } _ \\omega ^ \\bot : = D G _ \\delta ( { \\mathtt u } _ \\delta ) { \\mathbb R } _ \\omega ^ \\bot D { \\widetilde G } _ \\delta ( { \\mathtt u } _ \\delta ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\frac { d D } { d E } & = \\frac { E } { D } \\Bigg ( \\frac { \\lambda _ 3 \\lambda _ 4 } { \\lambda _ 2 \\lambda _ 5 } \\Bigg ) \\\\ D ^ 2 & = \\epsilon E ^ 2 + \\ell \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} X _ { \\sigma } ^ { \\gamma } = T _ { \\pi } \\circ X _ { \\sigma } \\circ \\gamma . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\alpha ( K ) = \\mathrm { S p a n } _ F \\{ \\alpha _ 1 ( x ) \\alpha _ 2 ( y ) : x \\in K _ 1 , y \\in K _ 2 \\} , \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\int _ \\Omega \\sum _ { j = 1 } ^ n \\eta ^ 2 \\langle D ^ 2 L ( \\nabla u ) \\nabla u _ { x _ j } , \\nabla \\gamma ^ j \\rangle & = - \\int _ \\Omega \\sum _ { j = 1 } ^ n \\gamma ^ j \\langle D ^ 2 L ( \\nabla u ) \\nabla u _ { x _ j } , \\nabla \\eta ^ 2 \\rangle \\\\ & \\phantom { = } - \\int _ \\Omega \\sum _ { j = 1 } ^ n \\eta ^ 2 \\gamma ^ j f _ { x _ j } . \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\nabla [ \\L ^ { - s } , a ] f ( x ) & = c _ s \\int _ 0 ^ \\infty t ^ { - 1 + \\frac s 2 } \\int _ \\Omega \\nabla _ x H ( x , y , t ) [ a ( y ) - a ( x ) ] f ( y ) d t \\\\ & - c _ s \\int _ 0 ^ \\infty t ^ { - 1 + \\frac s 2 } \\int _ \\Omega H ( x , y , t ) \\nabla a ( x ) f ( y ) d t \\\\ & = : I + I I . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} _ { B ( r ( \\gamma ) ) } \\langle a , b \\rangle = a b ^ * \\langle a , b \\rangle _ { B ( s ( \\gamma ) ) } = a ^ * b . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} x _ { g , j } : = \\frac { g ( \\sqrt { \\alpha _ i } ) } { \\sqrt { \\alpha _ j } } = \\frac { g \\big ( \\sqrt { \\alpha _ { \\sigma _ g ^ { - 1 } ( j ) } } \\big ) } { \\sqrt { \\alpha _ j } } . \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\boldsymbol { b } _ k ^ H \\left ( \\boldsymbol { g } _ k - \\lambda _ k \\boldsymbol { b } _ k \\right ) = 0 ~ ~ ~ \\Rightarrow ~ ~ ~ \\lambda _ k = \\dfrac { \\boldsymbol { b } _ k ^ H \\boldsymbol { g } _ k } { \\| \\boldsymbol { b } _ k \\| ^ 2 } \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\left < Z ^ { \\star } _ { i } U _ { a _ { i } } , Z ^ { \\star } _ { j } U _ { a _ { j } } \\right > = \\delta _ { i } ^ { j } , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n | \\psi _ { \\boldsymbol { j } , \\boldsymbol { k } } ( \\boldsymbol { X } _ i ) | \\lesssim n \\prod _ { l = 1 } ^ d 2 ^ { - j _ l / 2 } . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} \\| D _ { \\zeta ' } \\phi \\| _ * \\leq \\varepsilon ^ { 2 - \\sigma } , \\| D _ { \\bar \\Lambda _ 1 } \\phi \\| _ * \\leq \\varepsilon ^ { 2 - \\sigma } , \\| D _ { \\bar \\Lambda _ l } \\phi \\| _ * \\leq \\varepsilon , l = 2 , \\ldots , k . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} x ( t ) = x ( 0 ) + \\int _ { [ 0 , t ] \\times S } 1 _ { [ 0 , g ( x ( s ) ) ] } ( y ) \\varphi ( s , y ) d s \\ , m ( d y ) , \\ ; 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} z ^ { \\alpha } = \\frac { \\sin \\left ( \\alpha \\pi \\right ) } { \\pi } \\int \\limits _ { 0 } ^ { \\infty } t ^ { \\alpha - 1 } \\frac { z } { t + z } \\ , d t . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\overline { U } \\times [ \\tau _ 1 , \\tau _ 2 ] \\subset \\bigcup _ { i = 1 } ^ { N } R _ { x _ i } ( r ) \\times ( t _ i - r ^ 2 , t _ i ) = : Q . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} g = d s ^ 2 + g _ { a ( s ) , b ( s ) } = d s ^ 2 + a ( s ) ^ 2 \\sigma \\otimes \\sigma + b ( s ) ^ 2 \\pi ^ { \\ast } g _ { \\mathbb { C } P ^ n } . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\mu ( G _ \\tau ) = 2 ^ { - | \\tau | } | B ( \\tau , 2 \\delta ) | \\le 2 ^ { - | \\tau | } 2 ^ { ( 1 - p ) | \\tau | } = 2 ^ { - p | \\tau | } . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\mathbb E ( f | \\mathcal P _ m ) ( s ) = \\sum _ { n \\geq 0 } \\mathbb E \\bigl ( f ( s ) \\big | \\mathcal F _ { \\frac { n } { 2 ^ m } } \\bigr ) , \\ ; \\ ; \\ ; s \\in \\Bigl ( \\frac { n } { 2 ^ m } , \\frac { n + 1 } { 2 ^ m } \\Bigr ] , n \\geq 0 . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} { \\rm N S } _ { a l g , X } ^ d ( { \\rm S p e c } \\ , { \\bf C } ) = { \\rm N S } _ { a l g } ^ d ( X ) . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} ( A _ { d } ( \\delta _ { n } ) ) & = \\int _ { \\{ d ( b ( \\theta ) , b _ { 0 } ) \\geq \\epsilon ^ * + \\delta _ { n } \\} } P _ { \\theta } \\left [ d ( \\eta ( \\mathbf { z } ) , \\eta ( \\mathbf { y } ) ) \\leq \\epsilon _ n \\right ] d \\Pi ( \\theta ) \\\\ & \\leq \\int _ \\Theta P _ \\theta \\left [ d ( b ( \\theta ) , \\eta ( \\mathbf { z } ) ) \\geq \\delta _ { n } / 4 \\right ] d \\Pi ( \\theta ) . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} f ^ * ( D ' ) = d D ' + ( f ^ * ( \\theta ) \\theta ^ { - d } \\varphi ) . \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} x P _ n ( u \\varphi ) ( x ) & = x \\int _ { \\real } \\check { \\psi _ n } ( x - y ) u ( y ) \\varphi ( y ) ~ d y \\\\ & = \\int _ { \\real } ( x - y ) \\check { \\psi _ n } ( x - y ) u ( y ) \\varphi ( y ) ~ d y + \\int _ { \\real } \\check { \\psi _ n } ( x - y ) y u ( y ) \\varphi ( y ) ~ d y \\\\ & = ( x \\check { \\psi _ n } ) * ( u \\varphi ) + \\check { \\psi _ n } * ( x u \\varphi ) . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} P ( z _ 1 , z _ 2 ) = p ( z _ 1 x _ + + z _ 2 x _ { - } ) \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} & E A ( : , : , 1 ) = c i r c s h i f t ( E A ( : , : , 1 ) , [ - y _ 1 + t _ 0 ~ ~ t _ 0 ] ) , \\\\ & E A ( : , : , 2 ) = c i r c s h i f t ( E A ( : , : , 2 ) , [ t _ 0 ~ ~ - y _ 2 + t _ 0 ] ) , \\\\ & E A ( : , : , 3 ) = c i r c s h i f t ( E A ( : , : , 3 ) , [ t _ 0 ~ ~ - y _ 3 + t _ 0 ] ) . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\gamma ^ * ( G ) & = \\sum _ { i } g ( v _ i ) , \\\\ & = \\sum _ { 1 \\le i \\le r } g ( v _ i ) + \\sum _ { r \\le i \\le 2 r - 1 } g ( v _ i ) - g ( v _ r ) \\\\ & = \\sum _ { v _ i \\in N ^ - [ v _ r ] } g ( v _ i ) + \\sum _ { v _ i \\in N ^ - [ v _ { 2 r - 1 } ] } g ( v _ i ) - g ( v _ r ) \\\\ & > 2 - \\varepsilon . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\widetilde { T } ( z _ { 0 } ) W ( \\psi , \\phi ) = W ( \\widehat { T } ( z _ { 0 } ) \\psi , \\phi ) . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} M ^ n _ t = \\sum _ { k = 0 } ^ n \\Delta M _ { \\tau _ k } \\mathbf 1 _ { [ 0 , t ] } ( \\tau _ k ) \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} T \\overline \\partial _ b = \\overline \\partial _ b T ~ ~ \\Omega ^ { 0 , q } ( X ) . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} M ( x , \\lambda ) & = m _ 1 ( x , \\lambda + 0 i ) , ~ \\overline { M } ( x , \\lambda ) = m _ e ( x , \\lambda + 0 i ) , \\\\ N ( x , \\lambda ) & = m _ e ( x , \\lambda - 0 i ) , ~ \\overline { N } ( x , \\lambda ) = m _ 1 ( x , \\lambda - 0 i ) . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} V _ m = ( \\alpha _ 1 \\alpha _ 2 \\dots \\alpha _ m ) ^ { - 1 / 2 } U _ m . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} V _ { g \\circ f } = V _ g \\cap g ( V _ f ) & & U _ { g \\circ f , ( i , j ) } = U _ { f , i } \\cap f ^ { - 1 } U _ { g , j } . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} A ( t ) F ( t ) & = t ^ 2 S P + t ^ 3 K + \\Psi ( t ) , \\\\ \\| \\Psi ( t ) \\| & \\le C _ 3 t ^ 4 ; C _ 3 = \\beta _ 3 \\delta ^ { - 1 } \\| X _ 1 \\| ^ 4 . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} & \\Big [ \\mu _ { k _ 1 } , \\big [ \\mu _ { k _ 2 } , \\big [ \\ldots \\big [ \\mu _ { k _ { r - 1 } } , [ \\mu _ { k _ r } , w _ { \\alpha } ( J _ 1 , p ) ] \\big ] \\ldots \\Big ] & & \\Big [ \\mu _ { l _ 1 } , \\big [ \\mu _ { l _ 2 } , \\big [ \\ldots \\big [ \\mu _ { l _ { s - 1 } } , [ \\mu _ { l _ s } , w _ { \\beta } ( J _ 2 , p ) ] \\big ] \\ldots \\Big ] \\\\ & J \\setminus J _ 2 = \\{ k _ 1 , \\ldots , k _ r \\} & & J \\setminus J _ 1 = \\{ l _ 1 , \\ldots , l _ s \\} \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\mathbb { E } g ( D ^ * _ z , D ^ * _ { z + 1 } , \\ldots ) = \\frac { 1 } { \\mu } \\mathbb { E } \\left ( \\sum _ { k = 1 } ^ \\infty g ( D _ k , D _ { k + 1 } , \\ldots ) 1 ( Y _ 1 \\ge k ) \\right ) , \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} & \\int _ { D _ 4 } \\big ( \\triangle ( u \\xi ) - \\partial _ t ( u \\xi ) - \\tilde { V } ( r , \\theta , t ) u \\xi \\big ) ^ 2 e ^ { - 2 \\tau g ( r ) } r ^ { 4 - n } d v _ g d t \\\\ & = \\int _ { D _ 4 } \\big ( \\triangle \\xi u + 2 \\nabla \\xi \\cdot \\nabla u + \\xi \\triangle u - \\xi _ t u - \\xi u _ t - \\tilde { V } ( r , \\theta , t ) u \\xi \\big ) ^ 2 e ^ { - 2 \\tau g ( r ) } r ^ { 4 - n } d v _ g d t \\\\ & = 0 , \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\| y \\xi \\| ^ 2 & = \\langle y \\xi , y \\xi \\rangle = \\langle y ( \\sum _ { n } \\alpha _ n e _ n + \\phi ) , y ( \\sum _ { k } \\alpha _ k e _ k + \\phi ) \\rangle = \\langle y ( \\sum _ { n } \\alpha _ n e _ n ) , y ( \\sum _ { k } \\alpha _ k e _ k ) \\rangle = \\sum _ n \\alpha _ n ^ 2 \\\\ & \\leq \\sum _ n \\alpha _ n ^ 2 + \\langle \\phi , \\phi \\rangle = \\langle \\sum _ { n } \\alpha _ n e _ n + \\phi , \\sum _ { k } \\alpha _ k e _ k + \\phi \\rangle = \\| \\xi \\| ^ 2 . \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } V ( t , X _ { \\sigma } ( t ) ) + \\tilde { H } ( t , X _ { \\sigma } ( t ) , \\sigma _ t , V ( t , \\cdot ) ) = 0 \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { x } { v ( 1 - v ) } \\right ) A ' \\Big ( \\frac { x } { v } , v \\Big ) & = - \\frac { x } { v ^ 2 ( 1 - v ) } A ' ( x , 1 ) + \\frac { x ( 1 + v ) } { v ^ 2 } \\big ( A ( x , 1 ) - 1 \\big ) - \\frac { x ^ 2 } { v ^ 2 } A ( v , 1 ) . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} \\langle \\partial f , g \\rangle = \\langle f , p _ 1 g \\rangle \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} N _ { 3 , h } ( R ) = c R ^ { \\frac { 1 } { 2 } } \\log R + O \\big ( R ^ { \\frac { 1 } { 2 } } ( \\log R ) ^ { \\frac { 3 } { 4 } } \\big ) \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} L _ \\rho ( \\dot \\rho ) = \\varpi _ \\rho ( L ' ( \\rho ) \\cdot \\dot \\rho ) \\ , \\imath _ \\rho \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} X = \\sum _ { - \\infty } ^ { + \\infty } c _ n z ^ n , c _ n \\in \\mathfrak g \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\alpha _ 1 ^ \\ast = \\left \\{ \\begin{array} { l l } \\forall \\in [ 0 , \\min \\{ \\frac { \\hat P - c _ 2 } { c _ 1 - c _ 2 } , 1 \\} ] , & { \\rm i f } \\ c _ 1 > c _ 2 , \\\\ \\forall \\in [ ( \\frac { \\hat P - c _ 2 } { c _ 1 - c _ 2 } ) ^ + , 1 ] , & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} I ( M _ 2 ; Y _ { 2 , 0 } ^ T ) & = \\lim _ { n \\to \\infty } I ( M _ 2 ; Y ^ { ( n ) } _ { 2 } ( \\Delta _ n ) ) \\\\ & = \\lim _ { n \\to \\infty } I ( M _ 2 ; \\Delta Y ^ { ( n ) } _ { 2 } ( \\Delta _ n ) ) \\\\ & = \\lim _ { n \\to \\infty } h ( \\Delta Y ^ { ( n ) } _ { 2 } ( \\Delta _ n ) ) - h ( \\Delta Y ^ { ( n ) } _ { 2 } ( \\Delta _ n ) | M _ 2 ) , \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} \\sigma _ { n p } ^ 2 = \\frac { p ( p - 1 ) ( n - 3 ) } { ( n - 4 ) ^ 2 ( n - 6 ) } . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} u ^ n _ t ( y ) = \\ , & P ^ n _ { t } u ^ n _ 0 ( y ) + \\sum _ e \\Big ( Y ^ e _ t ( \\phi ) + Z ^ e _ t ( \\phi ) + E ^ { ( 1 , e ) } _ t ( \\phi ) + E ^ { ( 2 , e ) } _ t ( \\phi ) \\Big ) \\\\ & + \\sum _ e \\Big ( E ^ { ( 3 , e ) } _ t ( \\phi ) + E ^ { ( 4 , e ) } _ t ( \\phi ) \\Big ) + o ( 1 ) . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} F ( x ) = o ( G ( x ) ) \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} [ [ a , _ { 2 n - i } b ] , [ a , _ { i } b ] ] + [ [ a , _ { 2 n - 1 - i } , b ] , [ a , _ { i + 1 } b ] ] = [ [ a , _ { 2 n - 1 - i } b ] , [ a , _ { i } b ] , b ] \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} L _ { \\eta } P ^ { i } + \\xi P _ { ~ , t } ^ { i } + 2 \\xi , _ { t } P ^ { i } + \\eta ^ { i } , _ { t t } = 0 \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\begin{aligned} f ( b ) & = k _ { 0 } ^ { + } + \\sum _ { j = 2 } ^ { k } p _ { j } ( b - 1 ) + \\sum _ { j = 2 } ^ { k _ 1 } \\xi _ { j } . \\end{aligned} \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\frac { 1 } { \\| k \\| } \\| x \\| & \\leq \\| j ( x ) \\| \\leq \\| j \\| \\| x \\| , \\\\ \\frac { 1 } { \\| j \\| } \\| x ^ * \\| & \\leq \\| k ( x ^ * ) \\| \\leq \\| k \\| \\| x ^ * \\| . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} p ( z ) = \\sum _ { j = r } ^ { d } P _ j ( z ) \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\alpha _ t > \\frac { p \\left ( \\frac { t } { B } - 2 \\right ) - \\frac { p t } { 2 B } } { \\frac { t } { B } } = \\frac { p } { 2 } - \\frac { 2 p B } { t } . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} u _ { i , 0 } ^ { \\varepsilon } \\left ( t , x \\right ) : = u _ { i } ^ { 0 } \\left ( t , x \\right ) , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} \\pi _ 1 + \\cdots + \\pi _ d = \\pi \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} x = \\begin{bmatrix} - 1 & 0 \\\\ 0 & 0 \\end{bmatrix} \\quad y = \\frac 1 2 \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} w = \\sum _ { \\gamma \\in H _ \\varepsilon } y _ \\gamma \\leq \\sum _ { \\gamma \\in H _ \\varepsilon } x \\wedge n _ \\varepsilon e _ \\gamma = x _ { n _ \\varepsilon , H _ \\varepsilon } . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\varphi ^ \\hbar _ { M } ( x ) = \\left ( \\tfrac { 1 } { \\pi \\hbar } \\right ) ^ { n / 4 } ( \\det X ) ^ { 1 / 4 } e ^ { - \\tfrac { 1 } { 2 \\hbar } M x ^ { 2 } } , \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{gather*} J = ( z _ { 1 1 } , z _ { 1 2 } - z _ { 2 1 } , z _ { 1 3 } - z _ { 2 2 } - z _ { 3 1 } , z _ { 1 4 } - z _ { 2 3 } - z _ { 3 2 } , z _ { 1 5 } - z _ { 2 4 } - z _ { 3 3 } , \\\\ z _ { 1 6 } - z _ { 2 5 } - z _ { 3 4 } , z _ { 2 6 } - z _ { 3 5 } , z _ { 3 6 } ) \\end{gather*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\frac { d } { d z } W _ { \\alpha , \\beta } ^ { \\gamma , \\sigma } ( z ) = \\frac { \\gamma } { \\sigma } W _ { \\alpha , \\beta + \\alpha } ^ { \\gamma + 1 , \\sigma + 1 } ( z ) , \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} H _ 1 ^ 1 & = Y ^ 3 - Z ^ 3 & H _ 1 ^ 2 & = Y ^ 2 Z \\\\ H _ 2 ^ 1 & = X H _ 1 ^ 1 + Y Z ^ 3 & H _ 2 ^ 2 & = X H _ 1 ^ 2 \\\\ H _ 3 ^ 1 & = X H _ 2 ^ 1 - Y ^ 2 Z ^ 3 & H _ 3 ^ 2 & = X H _ 2 ^ 2 \\\\ H _ 4 ^ 1 & = X H _ 3 ^ 1 + Y ^ 3 Z ^ 3 - 4 Z ^ 6 & H _ 4 ^ 2 & = X H _ 3 ^ 2 \\\\ H _ 5 ^ 1 & = X H _ 4 ^ 1 + Y ^ 7 - Y ^ 4 Z ^ 3 + 4 Y Z ^ 6 & H _ 5 ^ 2 & = X H _ 4 ^ 2 . \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} a \\cdot b J _ r a = b . \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\Pr ( E _ e ) = \\frac { 2 | B | } { 2 ( n - 2 ) } \\cdot \\frac { | B | - 1 } { ( n - 3 ) + 1 + ( | A | - 2 ) } = 1 - O \\left ( \\frac k n \\right ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} ( K _ { i l } ^ { X _ { 3 } ^ { c } , 0 } + K _ { i l } ^ { X _ { 2 } ^ { c } , 0 } ) & = \\ , V _ { X _ 2 } ( - ( i + 1 ) A - ( l + 1 ) B ) + \\ , V _ { X _ 3 } ( - ( d - l + 1 ) B ) \\\\ & = \\ , V _ { X _ 2 } ( - ( i + 1 ) A - l B ) - 1 + \\ , V _ { X _ 3 } ( - ( d - l + 1 ) B ) \\\\ & = \\ , K _ { i l } ^ { X _ 1 , 0 } - 1 . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} I _ n = [ u + v ^ * , u + v ^ * + n - 1 ] = [ u _ n , u _ n + n - 1 ] . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\mathrm { v o l } \\left ( O ( p ) \\right ) = \\frac { 2 ^ p \\pi ^ { p ( p + 1 ) / 4 } } { \\prod _ { j = 1 } ^ p \\Gamma ( j / 2 ) } \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} P _ 1 = \\sum _ a \\binom { a } { 2 } \\binom { | S | - 2 } { a - 2 } \\sum _ b \\binom { r - | S | - 2 } { b } + \\sum _ a \\binom { | S | - 2 } { a - 2 } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 2 } { b } \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} \\theta _ { p } ^ { i _ 1 } \\dots \\theta _ { p } ^ { i _ m } f _ { i _ 1 \\dots i _ m } & = 0 , p = \\{ 1 , \\dots , N \\} \\\\ \\xi _ { 0 } ^ { i _ m } f _ { i _ 1 \\dots i _ m } & = 0 , 1 \\leq i _ 1 \\leq \\dots \\leq i _ { m - 1 } \\leq n . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} E ( t , \\tau ) : = e ^ { - i \\tau A ( t ) ^ { 1 / 2 } } P - e ^ { - i \\tau ( t ^ 2 S ) ^ { 1 / 2 } P } P . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } i \\displaystyle \\frac { \\partial \\psi } { \\partial t } = ( - \\Delta ) ^ { 1 / 2 } \\psi + V _ { 1 } ( x ) \\psi - f _ { 1 } ( \\psi ) - \\lambda ( x ) \\phi , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { N } , \\\\ i \\displaystyle \\frac { \\partial \\phi } { \\partial t } = ( - \\Delta ) ^ { 1 / 2 } \\phi + V _ { 2 } ( x ) \\phi - f _ { 2 } ( \\phi ) - \\lambda ( x ) \\psi , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\frac 1 t \\log E | u ^ { w } ( t , x ) | ^ 2 = \\beta _ 0 , \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\bold B = \\{ \\eta \\in \\bold C \\mid \\Re ( \\eta ) > 0 \\} . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\mathcal { H } ^ s _ \\delta ( A ) = c _ s \\inf \\left \\{ \\sum _ { k = 1 } ^ \\infty d i a m ( V _ k ) ^ s \\ , | \\ , d i a m ( V _ k ) \\le \\delta , A \\subset \\bigcup _ { k = 1 } ^ \\infty V _ k \\right \\} \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} x ( \\mathbf { d } / \\mathbf { e } ) y & = \\sup _ { z \\in Z } ( x \\mathbf { d } z - y \\mathbf { e } z ) _ + , \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s \\left ( \\varphi _ j ( u _ i ) - \\varphi _ j ( v _ i ) \\right ) = 0 ( 1 \\le j \\le k ) , \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} f = f ^ * , \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} | \\zeta _ n ( x _ n ^ * ) | = 1 . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} A ^ { E } ( x ) \\widehat { \\mathcal { U } } ( x ) = \\pm \\widehat { \\mathcal { U } } ( x + \\alpha ) . \\\\ \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} c _ \\chi ( n ) = \\sum _ { a \\mod { r } } ' \\sum _ { b \\mod { s } } ' \\chi ( a s \\bar { s } + b r \\bar { r } ) e \\left ( n \\tfrac { a s \\bar { s } + b r \\bar { r } } { r s } \\right ) = c _ s ( n ) \\sum _ { a \\mod { r } } ' \\chi _ * ( a ) e \\left ( n \\bar { s } \\tfrac { a } { r } \\right ) . \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\Lambda _ f ( s , \\chi ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n = 1 } ^ \\infty f _ n \\chi ( n ) n ^ { - s - \\frac { k - 1 } 2 } \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} & { [ \\ell _ a , \\ell _ b , \\ell _ a ] } = { \\textstyle \\big ( \\frac { \\ell _ b } { \\ell _ a } \\big ) _ 4 } \\hbox { i f $ \\ell _ a \\not \\equiv 3 \\pmod { 4 } $ a n d $ \\ell _ b \\not \\equiv 3 \\pmod { 4 } $ , } \\\\ & { [ \\ell _ a , \\ell _ b , \\ell _ a ] } = { [ \\ell _ a , \\ell _ b , \\ell _ b ] } = { \\textstyle \\big ( \\frac { - \\ell _ b } { \\ell _ a } \\big ) _ 4 } \\hbox { i f $ \\ell _ a \\not \\equiv 3 \\pmod { 4 } $ a n d $ \\ell _ b \\equiv 3 \\pmod { 4 } $ . } \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} x _ n - \\tilde { \\mathcal { K } } _ n ^ M ( x _ n ) = f \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} u _ \\infty \\stackrel { \\theta } { = } \\frac { 1 } { 1 + \\theta } \\prod _ { j = 2 } ^ { \\infty } \\frac { j ( j + 2 \\theta ) } { ( j + \\theta ) ^ 2 } = \\frac { \\Gamma ( \\theta + 2 ) \\Gamma ( \\theta + 1 ) } { \\Gamma ( 2 \\theta + 2 ) } . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\theta } \\| ^ 2 \\left ( n - \\prod _ { l = 1 } ^ d 2 ^ { J _ { n , l } } \\right ) \\lesssim n \\| K _ { \\boldsymbol { J } _ n } ( f ) \\| _ n ^ 2 = \\boldsymbol { \\theta } ^ T \\boldsymbol { \\Psi } ^ T \\boldsymbol { \\Psi } \\boldsymbol { \\theta } \\lesssim \\| \\boldsymbol { \\theta } \\| ^ 2 \\left ( n + \\prod _ { l = 1 } ^ d 2 ^ { J _ { n , l } } \\right ) . \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } u _ n = 1 / \\mu > 0 . \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { p ^ j } \\frac { \\log p } { ( p ^ { j - 1 } ) ^ 2 ( p - 1 ) ^ 2 } \\bigg ) ^ 2 = ( 4 A _ 0 ) ^ 2 . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} \\| f \\| _ X = \\sup \\{ | \\langle f , g \\rangle | \\colon g { \\rm \\ i s \\ s i m p l e \\ f u n c t i o n \\ a n d \\ } \\| g \\| _ { X ' } \\le 1 \\} . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\varpi = - { \\sum _ { j < k } } ( i h _ { j k } ) ^ \\vee \\wedge ( i e _ { j k } ) ^ \\vee \\wedge ( i f _ { j k } ) ^ \\vee . \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} V _ { , k } Y ^ { k } + ( 2 \\psi _ { Y } + d _ { 2 } ) V + k = 0 . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} ( f ^ { \\mathrm { a n } } ) ^ * ( h _ n ) = \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { d ^ { i + 1 } } ( ( f ^ { \\mathrm { a n } } ) ^ { i + 1 } ) ^ * ( \\lambda ) = d h _ { n + 1 } - \\lambda , \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\frac { 1 } { p } H _ \\lambda ( u _ n ) - \\frac { 1 } { \\gamma } F ( u _ n ) = \\frac { \\gamma - p } { p \\gamma } H _ \\lambda ( u _ n ) + o ( 1 ) \\to c < 0 ~ ~ \\mbox { a s } ~ ~ n \\to \\infty . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\langle P _ h ^ k ( \\cdot , s ) , V \\rangle & = \\int \\int _ { \\Gamma _ 0 ( 4 ) \\backslash \\mathcal { H } } P _ h ( z , s ) \\overline { V ( z ) } \\frac { d x d y } { y ^ 2 } \\\\ & = \\int _ 0 ^ \\infty \\int _ 0 ^ 1 y ^ { s - 1 } e ^ { 2 \\pi i h z } \\overline { V ( z ) } d x \\frac { d y } { y } . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert \\phi _ n \\Vert _ { \\infty } = 0 . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\sigma ^ 1 : = \\inf \\{ n \\geq 0 : \\ : Y ^ 1 _ n - d \\nu ^ 1 _ { n + 1 } - R ^ 1 _ { n + 1 } \\leq z \\} \\ , \\ \\ \\sigma ^ 2 : = \\inf \\{ n \\geq 0 : \\ : Y ^ 2 _ n + R ^ 2 _ { n + 1 } \\geq z \\} \\ , . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\mu _ { \\mathrm { t o t } } ( g ) : = \\sum _ { x \\in X ^ { ( 1 ) } } \\mu _ x ( g ) [ \\kappa ( x ) : K ] \\in [ - \\infty , + \\infty [ \\ , , \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) \\cdot \\overline { \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) } ^ { t } = \\overline { \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) } ^ { t } \\cdot \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} \\langle \\mathcal { V } _ { f , \\overline { g } } , P _ h ( \\cdot , \\overline { s } ) \\rangle = \\frac { \\Gamma \\left ( { s } + k - 1 \\right ) } { \\left ( 4 \\pi \\right ) ^ { { s } + k - 1 } } D _ { f , \\overline { g } } ( { s } ; h ) , \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\begin{aligned} A B C & = \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 , & A B E & = \\lambda _ 1 \\lambda _ 2 \\lambda _ 5 , \\\\ A C D & = \\lambda _ 1 \\lambda _ 3 \\lambda _ 4 , & A D E & = \\lambda _ 1 \\lambda _ 4 \\lambda _ 5 . \\end{aligned} \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} Y ^ { t } ( x , \\xi ) = b _ { t } ( x , \\xi ) X ( x , \\xi ) , \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} c _ \\chi ( n ) = \\sum _ { \\substack { a \\pmod * { q } \\\\ ( a , q ) = 1 } } \\chi ( a ) e \\ ! \\left ( \\frac { a n } q \\right ) , \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} ( R ^ { ( 1 ) } ) ^ { - 1 } ( x + \\alpha ) A ^ { E } ( x ) R ^ { ( 1 ) } ( x ) = \\left [ \\begin{array} { c c } \\pm 1 & \\nu ( x ) \\\\ 0 & \\pm 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} K _ X = ( \\det E ) ^ { - 5 } \\otimes L \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\overline C _ { i j } = C _ { T + 1 - j , T + 1 - i } , \\overline C ^ T = a _ 0 \\begin{pmatrix} r _ 0 & r _ 1 & r _ 2 & \\ldots & r _ { T - 1 } \\\\ r _ 1 & r _ 0 + r _ 2 & r _ 1 + r _ 3 & \\ldots & . . \\\\ r _ 2 & r _ 1 + r _ 3 & r _ 0 + r _ 2 + r _ 4 & \\ldots & . . \\\\ \\cdot & \\cdot & \\cdot & \\cdot & \\cdot \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\lambda _ { E L } : \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { 1 } { 1 + \\lambda ^ T _ { E L } \\psi ( x _ i , \\theta ) } \\psi ( x _ i , \\theta ) ) = 0 \\ , , \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\mathbb { P } ( N _ j = k \\ , | \\ , N > 0 ) = \\frac { 1 } { k ! \\ , q } \\left ( \\frac { q ^ j } { j } \\right ) ^ k \\exp \\left ( - \\frac { q ^ j } { j } \\right ) \\mbox { f o r } j \\ge 1 , \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\widetilde { V } \\big | _ { \\sigma _ 0 } ( z ) = \\theta ^ { 2 k + 1 } \\Big ( \\frac { - 1 } { 4 z } \\Big ) \\overline { \\theta \\Big ( \\frac { - 1 } { 4 z } \\Big ) } \\Im ^ { \\frac { k + 1 } { 2 } } \\Big ( \\frac { - 1 } { 4 z } \\Big ) \\frac { \\lvert - 2 i z \\rvert ^ k } { ( - 2 i z ) ^ k } \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} \\mathcal { F } _ C = \\{ p \\in \\C [ z ] : p , p ( 0 ) = 1 , | p ' ( 0 ) | , | p '' ( 0 ) | \\leq C \\} \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} n ^ { \\frac { 1 } { 2 \\sigma ^ 2 } \\kappa ( \\underline { \\gamma } ) - 1 / 2 } \\sigma \\prod _ { l = 1 } ^ d 2 ^ { J _ { n , l } ( \\sigma _ 0 ^ 2 / \\sigma ^ 2 - \\mu _ l ) } \\lesssim n ^ { - \\mu _ { \\mathrm { m i n } } - 1 / 2 + \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } + \\frac { 1 } { 2 \\sigma ^ 2 } \\kappa ( \\underline { \\gamma } ) } \\sigma . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ i ( \\varepsilon \\ , y ) = O ( \\varepsilon ) , w _ { \\mu _ j ^ { \\prime } , \\zeta _ j ^ { \\prime } } ( y ) + \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ j ( \\varepsilon \\ , y ) = O ( \\varepsilon ) . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\theta _ \\lambda ^ { ( 2 ) } ( \\zeta ' , \\mu ' ) & = - \\int _ 0 ^ 1 s D ^ 2 \\bar J _ \\lambda ( V + s \\phi ) [ \\phi ^ 2 ] \\ , d s . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} \\langle T _ a \\chi _ j , \\chi _ k \\rangle = a _ { k - j } = \\langle A \\chi _ j , \\chi _ k \\rangle \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\sum ^ \\infty _ { j = 1 } \\abs { g _ j ( x _ 0 ) } ^ 2 \\geq B ( x _ 0 , x _ 0 ) m ( x _ 0 ) . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} E ( u ) = \\mathcal A _ { H } ( [ \\gamma _ 0 , w _ 0 ] ) . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\aligned B _ { \\tau } ( \\overset { \\circ \\circ } S _ 0 ( X ' , \\widehat { \\mathcal U } ) ) & = \\overset { \\circ \\circ } S _ 0 ( X ' , \\widehat { \\mathcal U } ) = \\overset { \\circ } S _ 0 ( X , \\widehat { \\mathcal U } ) , \\\\ B _ { \\tau } ( \\overset { \\circ \\circ } S _ k ( X ' , \\widehat { \\mathcal U } ) ) \\cap X & = \\overset { \\circ } S _ k ( X , \\widehat { \\mathcal U } ) . \\endaligned \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\tilde { { E } } _ { 1 } \\left ( p \\right ) = { \\tfrac { 1 } { { 2 4 } } } p \\ , \\left ( { 3 - 5 \\ , p ^ { 2 } } \\right ) , \\ \\tilde { { E } } _ { 2 } \\left ( p \\right ) = { \\tfrac { 1 } { { 1 6 } } } \\ , p ^ { 2 } \\left ( { 1 - p ^ { 2 } } \\right ) \\left ( { 1 - 5 \\ , p ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\frac { t } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty \\beta _ { n , \\lambda } ( x ) \\frac { t ^ n } { n ! } , \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} p _ t ( x , y ) ~ = ~ \\frac { \\sqrt { \\pi } } { ( 2 \\pi ) ^ { \\frac { d + 1 } { 2 } } } \\cdot \\frac { \\exp \\left ( - b _ d \\cdot t \\right ) } { \\sqrt { t } } \\cdot \\left ( \\frac { - 1 } { \\sinh ( r ) } \\partial _ r \\right ) ^ { \\frac { d - 1 } { 2 } } \\exp \\left ( \\frac { - r ^ 2 } { 4 t } \\right ) \\ , , \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} I _ { \\rho , m } : = \\int \\rho ( t ) e ^ { - 2 \\pi i 2 ^ m ( \\xi ( t ^ a + \\epsilon _ P ( t ) ) + \\eta ( t ^ b + \\epsilon _ Q ( t ) ) ) } \\ , d t \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{gather*} \\sum _ { n \\ge 1 } \\frac { \\max _ { 1 \\le i \\le n } V _ n ^ q } { n ^ p } \\le \\sum _ { r \\ge 0 } \\frac 1 { 2 ^ { ( p - 1 ) r } } ( \\sum _ { j = 0 } ^ r 2 ^ { k \\varepsilon q ' q } ) ^ { q / q ' } ( \\sum _ { k = 0 } ^ r 2 ^ { - k \\varepsilon } V _ { 2 ^ k } ^ q ) \\le C \\sum _ { n \\ge 0 } \\frac { V _ { 2 ^ n } ^ q } { 2 ^ { n p } } \\ , . \\end{gather*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\textstyle P _ m ^ { ( a , b ) } P _ n ^ { ( a , b ) } = \\sum _ { k = | m - n | } ^ { m + n } g _ { m , n , k } P _ k ^ { ( a , b ) } ( m , n \\ge 0 ) . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\frac { N _ { 1 0 } ^ { \\mathrm { W e y l } } ( X ) } { N _ { 1 0 } ^ { \\mathrm { c m } } ( X ) } = 1 + O _ { \\epsilon } ( X ^ { - \\frac { 1 } { 2 } + \\epsilon } ) . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} p ! ( p ^ { k - 1 } ) ^ p [ p ^ { k - 2 } ( p - 1 ) ] ^ p [ p ^ { k - 2 } p ^ { k - 3 } \\cdots p ] ^ p = p ! [ ( p - 1 ) p ^ { ( k ^ 2 + k - 4 ) / 2 } ] ^ p \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} \\ker p = \\left \\{ \\left ( u ( b E _ 4 ^ 2 - a E _ 6 ) + a \\eta ^ { 1 2 } , 6 u ( b E _ 6 - a E _ 4 ) + 6 b \\eta ^ { 1 2 } \\right ) \\mid a , b \\in R ( 1 ) \\right \\} . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} K _ j \\leq 2 K _ 1 \\textrm { f o r } \\ j = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} U ^ 0 = U _ 1 + \\ldots + U _ k , \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} { \\cal L } _ n = { \\cal U } _ n ^ { - 1 } { \\cal L } _ 0 { \\cal U } _ n \\ , . \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\gamma ( a ) = b , \\ \\gamma ( b ) = a , \\ \\gamma ( x ) = x ^ { - 1 } , \\ \\gamma ( y ) = y ^ { - 1 } . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\sum _ { H ( \\chi ) > \\epsilon } | \\hat { 1 _ S } ( \\chi ) | ^ 3 = O _ \\epsilon \\ ( \\exp ( - c \\epsilon n ) \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\ ) , \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} T _ { 2 } \\left ( W ' , Z ' \\right ) = \\left ( \\frac { 1 } { I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) } \\otimes W ' , V \\otimes \\left ( Z ' - A \\otimes W \\right ) \\right ) , \\quad \\mbox { f o r a s u i t a b l e m a t r i x $ V = V \\left ( W ' , Z ' \\right ) \\in \\mathcal { M } _ { q ' N ' \\times q ' N ' } , $ } \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\log ( e ^ A e ^ B ) = { A + B + \\frac { 1 } { 2 } \\left [ A , B \\right ] + \\frac { 1 } { 1 2 } \\left ( \\left [ A , \\left [ A , B \\right ] \\right ] + \\left [ B , \\left [ B , A \\right ] \\right ] \\right ) + \\ldots } \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\sigma _ i ( f g ) = f \\sigma _ i ( g ) . \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} A u = \\delta \\left ( x _ { 1 } - x _ { 0 } , \\overline { x } - \\overline { x } ^ { 0 } , t \\right ) , \\left ( x , t \\right ) \\in D _ { T } ^ { n + 1 } , \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\widehat f ! ( \\widehat h ; \\widehat { \\frak S } ) = \\widehat { f ^ { \\boxplus \\tau } } ! ( \\widehat { h ^ { \\boxplus \\tau } } ; \\widehat { \\frak S ^ { \\boxplus \\tau } } ) . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} q ( M / R , \\alpha , \\varphi ^ * _ { 0 } ( \\beta ^ * ) ) = q ( N / R , \\varphi _ { 0 } ( \\alpha ) , \\beta ^ * ) \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\Phi - v \\Phi ' + ( \\Phi ' ) ^ 2 \\coth ( \\Phi ' ) - \\Phi ' = 0 . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ a p ^ { ( a - b ) b } & = 2 \\bigg ( p ^ { c ( c + 1 ) } + \\sum _ { b = 0 } ^ { c - 1 } p ^ { ( 2 c + 1 - b ) b } \\bigg ) \\\\ & = 2 \\bigg ( p ^ { c ( c + 1 ) } + O \\bigg ( \\sum _ { b = 0 } ^ { c - 1 } p ^ { ( c + \\frac 1 2 ) ^ 2 - ( c + \\frac 1 2 - b ) } \\bigg ) \\bigg ) = 2 p ^ { c ( c + 1 ) } + O \\big ( p ^ { ( c + \\frac 1 2 ) ^ 2 - \\frac 1 2 } \\big ) . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A ( \\sqrt { g } g ^ { i j } ) = \\sqrt { g } \\left ( \\frac { 1 } { 2 } g ^ { i j } g ^ { k l } - g ^ { l j } g ^ { i k } \\right ) \\overline { \\partial } _ A g _ { k l } , \\end{gather*}"}
-{"id": "7327.png", "formula": "\\begin{align*} f _ { n , m , j } ( x ) : = 1 ^ * _ { n , a j } \\psi _ { a j + m } f * \\Phi _ { a j + m } ( x ) ; \\\\ g _ { n , m , j } ( x ) : = 1 ^ * _ { n , a j } \\psi _ { b j + m } g * \\Phi _ { b j + m } ( x ) . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} L ( Q ( t ) ) = \\frac { 1 } { 2 } Q ^ 2 ( t ) . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\widehat { m _ 1 ( \\lambda + ) } = & ~ - \\frac { i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } F ( m _ 1 ( \\lambda + ) - 1 ) - \\frac { \\lambda + i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } \\widehat 1 \\\\ & + \\frac { 1 } { \\xi - ( \\lambda + i \\epsilon ) } \\chi _ { \\mathbb { R } ^ + } F ( u m _ 1 ( \\lambda + ) ) . \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} H _ 1 ^ 1 & = Y W & H _ 1 ^ 2 & = Z W \\\\ H _ 2 ^ 1 & = X H _ 1 ^ 1 & H _ 2 ^ 2 & = X H _ 1 ^ 2 + Y ^ 2 W \\\\ H _ 3 ^ 1 & = X ^ 2 H _ 1 ^ 1 + Z ^ 2 W & H _ 3 ^ 2 & = X ^ 2 H _ 1 ^ 2 + X Y ^ 2 W = X H _ 2 ^ 2 \\\\ H _ 4 ^ 1 & = X ^ 3 H _ 1 ^ 1 + X Z ^ 2 W + Y ^ 2 Z W + W ^ 3 & H _ 4 ^ 2 & = X ^ 3 H _ 1 ^ 2 + X ^ 2 Y ^ 2 W + Y Z ^ 2 W \\\\ & = X H _ 3 ^ 1 + Y ^ 2 Z W + W ^ 3 & & = X H _ 3 ^ 2 + Y Z ^ 2 W . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} y ^ 3 - ( 1 + \\theta + \\theta ^ 2 ) y ^ 2 - ( 2 \\theta + 1 ) ( 1 - \\theta - q ) y - ( 1 - \\theta - q ) ^ 2 = 0 . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } e ^ { - i \\lambda x } u \\varphi ( x ) ~ d x = 0 . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\begin{array} { r c l } I ( f ) - T ( f ) & : = & - \\ , \\frac { ( 2 \\ , h ) ^ 3 } { 1 2 } \\ , f '' ( \\xi _ T ) \\\\ & & \\\\ I ( f ) - M ( f ) & : = & \\frac { ( 2 \\ , h ) ^ 3 } { 2 4 } \\ , f '' ( \\xi _ M ) \\ , . \\end{array} \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\alpha = 0 \\ , . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\ker T ^ { \\ast } = \\ker R ^ { \\ast } \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} u ( x ) = \\int _ { 0 } ^ { + \\infty } e ^ { i x _ { 2 } \\rho } \\phi _ { \\lambda } ( x _ { 1 } \\rho ^ { 1 / k } ) ( 1 + \\rho ^ { 4 } ) ^ { - 1 } d \\rho . \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\tilde { X _ 0 } : \\begin{cases} \\dot x & = E x u \\\\ \\dot u & = - E ( 1 + u ^ 2 ) \\\\ \\dot \\varphi & = 1 . \\end{cases} \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\phi ( t ) - \\phi ( p _ 2 ) & = \\sum _ { i = 2 } ^ { n - 1 } \\int _ { p _ i } ^ { p _ { i + 1 } } \\phi ' ( s ) d s + \\int _ { p _ { n } } ^ t \\phi ' ( s ) d s \\\\ & \\ge \\sum _ { i = 2 } ^ { n - 1 } \\int _ { p _ i } ^ { p _ { i + 1 } } - \\dfrac { b _ i } { s } d s + \\int _ { p _ { n } } ^ t - \\dfrac { b _ n } { t } d s \\\\ & = - \\sum _ { i = 2 } ^ { n - 1 } b _ i \\log ( p _ { i + 1 } / p _ i ) - b _ n \\log ( t ) + b _ n \\log ( p _ n ) \\\\ & \\ge - b _ n \\log ( p _ { n + 1 } ) - b _ n \\log ( t ) \\\\ & \\ge - 2 b _ n \\log ( t ) . \\\\ \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} t \\ , \\P ( X _ { 1 } > x b ( t ) ) & = t \\frac 1 2 x ^ { { - 2 } } \\frac { 2 } { t } \\left ( 1 + \\frac 1 { \\sqrt { 2 t } } \\right ) ^ { - 2 } \\left ( 1 + x ^ { - 1 } { \\sqrt \\frac 2 t } \\left ( 1 + \\frac { 1 } { \\sqrt { 2 t } } \\right ) ^ { - 1 } \\right ) \\\\ & = x ^ { - 2 } + \\sqrt { 2 } x ^ { - 2 } ( x ^ { - 1 } - 1 ) \\frac 1 { \\sqrt t } + o \\left ( \\frac 1 { \\sqrt t } \\right ) , \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} T _ g : = P \\ , \\alpha \\ , ( \\cdot \\otimes g ) \\in \\mathcal { L } ( \\mathcal { P } ) . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} C _ n = R _ n = 0 \\qquad \\mbox { f o r a l l } n \\in N \\ , , \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } Q [ J _ i \\varphi ] \\ge - \\frac { 1 } { 2 } { \\rm T r } A \\sum _ { i = 1 } ^ { n } \\int _ { \\Omega _ i } \\varphi ^ 2 \\ , d \\mu - \\frac { ( n - 1 ) c } { r _ 0 ^ 2 } \\sum _ { i = 1 } ^ { n } \\int _ { \\Omega _ i } J _ i ^ 2 \\varphi ^ 2 \\ , d \\mu \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} h ^ - ( c _ r ) = c _ r + 2 \\ell \\int _ { 0 } ^ { 1 } \\frac { \\sin ( \\pi c _ r v ( 1 - \\delta ) ) } { \\pi v } ( 1 - v ) ^ { \\ell ^ 2 } \\ , d v + O \\left ( 1 / \\log T \\right ) . \\\\ \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\| c _ 1 \\| _ { L ^ r ( Q _ T ) } \\leq C _ T 1 \\leq r < \\infty N = 1 , 2 . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} & \\left ( 1 - \\sqrt { b / d } \\right ) \\left ( \\frac { y + \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } < \\frac { - b + \\sqrt { b d } } { \\sqrt { b d } } \\left ( \\frac { y + \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } < Y _ { 2 m } ^ { ( b , d ) } = Y _ { 2 h } ^ { ( a , b ) } < \\\\ & < \\frac { b + \\sqrt { a b } } { \\sqrt { a b } } \\left ( \\frac { r + \\sqrt { a b } } { 2 } \\right ) ^ { 2 h } \\leq \\left ( 1 + \\sqrt { b / a } \\right ) \\left ( \\frac { r + \\sqrt { a b } } { 2 } \\right ) ^ { 2 h } . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} L ( I _ n - \\widetilde { V } _ k \\widetilde { V } _ k ^ T ) = L \\widetilde { V } _ { n - k } \\widetilde { V } _ { n - k } ^ T . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\nu _ l = \\sum _ { j = 0 } ^ { k - 1 } a _ j e ^ { \\frac { 2 \\pi i } { k } j l } , l = 0 , \\ldots , k - 1 . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} ( X , \\widehat { \\mathcal U ''' } ) ^ { \\boxplus ( \\tau - \\tau ' ) } \\vert _ { \\overline { X \\setminus X ^ { \\boxplus ( \\tau - \\tau ' ) } } } = ( X ' , \\widehat { \\mathcal U ^ + } ) \\vert _ { \\overline { X \\setminus X ^ { \\boxplus ( \\tau - \\tau ' ) } } } . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\Phi _ { 3 1 } \\vert _ { U _ { 3 2 1 } } = \\Phi _ { 3 2 } \\circ \\Phi _ { 2 1 } \\vert _ { U _ { 3 2 1 } } . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } ( \\zeta _ i ) } E _ i & = 6 \\ , \\int _ { B _ { \\rho } ( \\zeta _ i ) } U _ i ^ 5 \\ , Q _ i + 1 5 \\int _ { B _ { \\rho } ( \\zeta _ i ) } U _ i ^ 4 \\ , Q _ i ^ 2 + \\mathcal { R } \\\\ & = 6 \\ , \\sum _ { j \\neq i } \\int _ { B _ { \\rho } ( \\zeta _ i ) } U _ i ^ 5 \\ , U _ j + 1 5 \\sum _ { j \\neq i } \\sum _ { m \\neq i } \\int _ { B _ { \\rho } ( \\zeta _ i ) } U _ i ^ 4 \\ , U _ j \\ , U _ m + \\mathcal { R } . \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} p _ { j } ^ { * } H ^ { 0 } ( E _ { j , x } , \\Omega ^ { 1 } _ { E _ { j , x } } ) \\cap p _ { k } ^ { * } H ^ { 0 } ( E _ { k , x } , \\Omega ^ { 1 } _ { E _ { k , x } } ) = 0 \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} D ( s , S _ f ) & = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) } { n ^ { s + \\frac { k - 1 } { 2 } } } \\\\ D ( s , S _ f \\times S _ g ) & = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) S _ g ( n ) } { n ^ { s + k - 1 } } \\\\ D ( s , S _ f \\times \\overline { S _ g } ) & = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { s + k - 1 } } . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{gather*} F ( z ) - R _ { p , k } ( z ) \\sim B _ p ( z ) \\frac { \\psi _ { 1 , p } ( z ) } { \\Psi _ p ( z _ { k + 1 } ) } \\quad \\\\ B _ p ( z ) = ( - 1 ) ^ { k } \\frac { \\widehat { T } ^ { ( p ) } _ { 1 , \\ldots , k + 1 } ( z ) } { T _ { 1 , \\ldots , k } } \\prod ^ k _ { i = 1 } \\frac { z _ { k + 1 } - z _ i } { z - z _ i } , \\end{gather*}"}
-{"id": "1010.png", "formula": "\\begin{align*} x ( P _ n \\varphi ) _ x \\overline { P _ n \\varphi } + x P _ n ( u \\varphi ) \\overline { P _ n \\varphi } = 0 , \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} c ^ * _ + = \\inf _ { \\mu > 0 } \\Phi ^ + ( \\mu ) = \\inf _ { \\mu > 0 } \\frac { \\ln \\lambda ( \\mu ) } { \\mu } = \\inf _ { \\mu > 0 } \\frac { a - 1 + \\alpha e ^ { \\mu } + \\beta e ^ { - \\mu } } { \\mu } , \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} \\Pi = \\Pi \\cap \\Gamma = ( \\cup _ { i } \\Pi _ { i } ) \\cap ( \\cup _ { j } \\Gamma _ { j } ) = \\cup _ { i , j } ( \\Pi _ { i } \\cap \\Gamma _ { j } ) \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { q } \\left ( 1 - \\frac { 1 } { n ^ { l _ i } } \\right ) \\leq \\left ( 1 - \\frac { 1 } { n ^ { \\frac { t } { q } } } \\right ) ^ { q } , \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} M ( L ^ { \\varphi _ 1 } , L ^ { \\varphi } ) = L ^ { \\varphi \\ominus \\varphi _ 1 } , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} f ( x + m p ) = f ( x ) + m p f ' ( x ) + ( m p ) ^ 2 f '' ( x ) + \\cdots + ( m p ) ^ { k - 1 } f ^ { ( k - 1 ) } ( x ) \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\chi = \\tilde { \\xi } _ 0 + N , \\quad \\eta = - \\lambda \\frac { \\chi ^ { ( d - 1 ) } } { \\chi ^ { ( d ) } } + \\log \\frac { | \\chi ^ { ( d ) } | } { R ( d - 1 ) ! \\lambda ^ { d - 1 } } ; \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} h ^ 0 ( N _ C ( - 1 ) ( - x - y ) [ p _ 1 \\to p _ 1 ' ] [ p _ 2 \\to p _ 2 ' ] [ p _ 3 \\to p _ 3 ' ] ) = 0 . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\nabla _ k R ^ k _ { \\ ; j } & = - \\nabla _ k \\nabla ^ k \\nabla _ j f \\\\ & = ( \\nabla _ j \\nabla _ k - \\nabla _ k \\nabla _ j ) \\nabla ^ k f - \\nabla _ j \\nabla _ k \\nabla ^ k f \\\\ & = R ^ k _ { \\ ; a j k } \\nabla ^ a f + \\nabla _ j R \\\\ & = - R _ { a j } \\nabla ^ a f + 2 \\nabla _ k R ^ k _ { \\ ; j } \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ t [ \\eta _ n X _ n + Y _ n ] = \\eta _ n ^ + \\hat { \\mathbb { E } } _ t [ X _ n ] + \\eta _ n ^ - \\hat { \\mathbb { E } } _ t [ - X _ n ] + Y _ n . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} E _ 1 \\stackrel { d e f } { = } \\bigcap _ { 1 \\le j \\le n } E _ { 1 , j } \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } [ \\varphi ( X _ { t _ 1 } ^ { y } , X _ { t _ 2 } ^ { y } , \\cdots , X _ { t _ m } ^ { y } ) ] = \\hat { \\mathbb { E } } [ \\varphi _ { 1 } ( X _ { t _ 1 } ^ { y } ) ] , \\ \\ \\ \\ y \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ h \\left ( \\sum _ { \\ell = 1 } ^ { \\kappa _ { i , j } } e \\ ! \\left ( n \\tfrac { a ^ { ( j ) } _ { i \\ell } } { q _ j m } \\right ) \\right ) \\lambda _ i ( n ) = 0 \\quad j \\in \\{ 1 , \\ldots , h \\} , \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} \\kappa ^ 2 = - e s ^ 2 / 2 n = - 2 ( n a ^ 2 + b ^ 2 + m c ^ 2 ) \\quad \\textnormal { a n d } \\kappa _ * = ( 2 n a / s , 2 b / s ) \\in \\Z / 2 n \\Z \\times \\Z / 2 \\Z . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\norm { v ^ { j + 1 } _ V } _ { j + 1 } = \\norm { A ( f ^ j ( x ) ) u ^ j _ V } _ { j + 1 } \\leq e ^ { \\lambda _ 3 + \\delta } \\norm { u ^ j _ V } _ j . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\beta _ i ( U ) = \\beta _ i ( ( y ) ) + \\beta _ i ( U / ( y ) ) . \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( Y , \\phi V ) , \\pi _ { \\ast } ( X ) ) + g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( X , \\phi V ) , \\pi _ { \\ast } ( X ) ) & = g _ { 1 } ( \\phi V , \\mathcal { V } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { B } Y + \\nabla ^ { ^ { M _ 1 } } _ { Y } \\mathcal { B } X ) ) \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} I ( \\tilde { X } ^ { ( n ) } ( \\Delta _ n ) \\to Y ^ { ( n ) } ( \\Delta _ n ) ) = I ( M ; Y ^ { ( n ) } ( \\Delta _ n ) ) . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} X = \\left ( d _ { 1 } t + d _ { 2 } \\right ) \\partial _ { t } + a _ { 1 } Y ^ { i } , \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\left ( \\forall y \\in C \\right ) \\norm { y } = \\rho \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} ( - \\partial ^ 2 + u _ s - \\chi _ 1 ( \\tau ) ) \\Upsilon = 0 \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} 1 + ( p - q ) - ( p - 2 ) ( q - 2 ) = ( p - 1 ) ( 3 - q ) > 0 . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\tilde \\varphi ^ { ( k ) } & : = \\pi _ { 0 * } \\tilde u ^ { ( k ) } , \\\\ \\varphi ^ { ( k ) } & : = \\pi _ { 0 * } u ^ { ( k ) } . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} Y _ 1 ^ { ( n ) } ( t ) = Y _ 1 ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t } X ^ { ( n ) } ( t _ { n , i } ) d s + B _ 1 ( t ) - B _ 1 ( t _ { n , i } ) , \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} g _ { 1 } ( [ U , V ] , Z ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , \\varphi \\phi Z ) + g _ { 1 } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V , w Z ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { V } U , \\varphi \\phi Z ) - g _ { 1 } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { V } \\varphi U , w Z ) . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} [ \\L ^ s , a ] f ( x ) = c _ s \\int _ 0 ^ \\infty t ^ { - 1 - \\frac s 2 } \\int _ \\Omega H ( x , y , t ) d t [ a ( x ) - a ( y ) ] f ( y ) d y . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\bigoplus _ { | I | = r } M ( Z _ I ) \\rightarrow \\cdots \\rightarrow \\bigoplus _ { | I | = 1 } M ( Z _ I ) \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} \\theta ^ { \\varepsilon } \\left ( 0 , x \\right ) = \\theta ^ { \\varepsilon , 0 } \\left ( x \\right ) \\quad \\mbox { f o r } \\ ; x \\in \\Omega ^ { \\varepsilon } . \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} P _ 3 = \\sum _ a \\binom { a } { 2 } \\binom { | S | - 2 } { a - 1 } \\sum _ b \\binom { r - | S | - 2 } { b - 1 } + \\sum _ a \\binom { | S | - 2 } { a - 1 } \\sum _ b \\binom { b } { 2 } \\binom { r - | S | - 2 } { b - 1 } . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} E = \\left \\{ \\left ( q ^ i , p _ i ; \\frac { \\partial F } { \\partial p _ i } , - \\frac { \\partial F } { \\partial q ^ i } \\right ) \\in T T ^ * Q : \\frac { \\partial F } { \\partial \\lambda ^ a } = 0 \\right \\} \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} & M E ( B _ { n , 3 , 3 , 3 } ^ { ( n - 5 ) } ) = 2 \\sqrt { \\frac { n + 1 + \\sqrt { ( n + 1 ) ^ { 2 } - 4 ( 3 n - 9 ) } } { 2 } } + 2 \\sqrt { \\frac { n + 1 - \\sqrt { ( n + 1 ) ^ { 2 } - 4 ( 3 n - 9 ) } } { 2 } } ; \\\\ & M E ( B _ { n , 3 , 3 } ^ { ( n - 5 ) } ) = 2 + 2 \\sqrt { \\frac { n + \\sqrt { n ^ { 2 } - 4 ( n - 5 ) } } { 2 } } + 2 \\sqrt { \\frac { n - \\sqrt { n ^ { 2 } - 4 ( n - 5 ) } } { 2 } } . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} - \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) = C \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} D _ { i _ { 0 1 } } ^ 2 \\times \\cdots \\times D _ { i _ { 0 p _ 0 } } ^ 2 \\times \\partial D ^ { d ( w ) } \\cup \\left ( \\left ( \\bigcup \\limits _ { k = 1 } ^ p D _ { i _ { 0 1 } } ^ 2 \\times \\cdots \\times D _ { i _ { 0 ( k - 1 ) } } ^ 2 \\times S _ { i _ { 0 k } } ^ 1 \\times D _ { i _ { 0 ( k + 1 ) } } ^ 2 \\times \\cdots \\times D _ { i _ { 0 p _ 0 } } ^ 2 \\right ) \\times D _ { w } ^ { d ( w ) } \\right ) . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\lambda _ l ( t , \\boldsymbol { \\theta } ) = \\gamma _ l ( \\boldsymbol { \\theta } ) t ^ 2 + \\mu _ l ( \\boldsymbol { \\theta } ) t ^ 3 + \\ldots , \\ ; l = 1 , \\ldots , n , \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\Omega _ a = \\{ x \\in \\R ^ 3 \\ : \\ a < | x | < 1 \\} , \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\phi _ { i } \\left ( t , x \\right ) : = u _ { i } ^ { \\varepsilon } \\left ( t , x \\right ) - \\left ( u _ { i , 0 } ^ { \\varepsilon } \\left ( t , x \\right ) + \\varepsilon m ^ { \\varepsilon } \\left ( x \\right ) \\bar { u } _ { i } \\left ( x , \\frac { x } { \\varepsilon } \\right ) \\cdot \\nabla _ { x } u _ { i } ^ { 0 } \\left ( t , x \\right ) \\right ) , \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} P _ j N _ * P _ j = 0 , j = 1 , \\ldots , p ; P _ l N _ 0 P _ j = 0 \\ l \\ne j . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} X ^ { \\beta } ( \\varphi ) : S ' _ { d } \\longrightarrow \\mathbb { R } ^ { d } , \\ ; w \\mapsto X ^ { \\beta } ( \\varphi , w ) : = \\big ( \\langle w _ { 1 } , \\varphi _ { 1 } \\rangle , \\ldots , \\langle w _ { d } , \\varphi _ { d } \\rangle \\big ) . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} J ^ n ( X ) = J _ a ^ n ( X ) = { \\rm A l b } ( X ) . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\widetilde { T } ( z _ { 0 } ) \\widetilde { T } ( z _ { 1 } ) & = e ^ { \\tfrac { i } { \\hslash } \\sigma ( z _ { 0 } , z _ { 1 } ) } \\widetilde { T } ( z _ { 1 } ) \\widetilde { T } ( z _ { 0 } ) \\\\ \\widetilde { T } ( z _ { 0 } + z _ { 1 } ) & = e ^ { - \\tfrac { i } { 2 \\hslash } \\sigma ( z _ { 0 } , z _ { 1 } ) } \\widetilde { T } ( z _ { 0 } ) \\widetilde { T } ( z _ { 1 } ) . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( | u _ 1 ( t ) | ^ 2 + | u _ 2 ( t ) | ^ 2 \\big ) \\ d t \\asymp \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\ , . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} | | \\Gamma | | _ { \\infty } & = \\max \\Big \\{ - \\inf _ { x \\in \\Pi } \\Gamma ( x , t ) , \\ \\sup _ { x \\in \\Pi } \\Gamma ( x , t ) \\Big \\} \\\\ & \\leq \\max \\{ - m , \\ M \\} = | | \\Gamma _ 0 | | _ { \\infty } . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\left | \\det \\left ( A ^ { ( k ) } ( k : m , k : m ) \\right ) \\right | = \\prod _ { j = k } ^ m p _ j . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\frac { d + \\varepsilon \\sqrt { a d } } { \\sqrt { a d } } \\left ( \\frac { x + \\sqrt { a d } } { 2 } \\right ) ^ { 2 l } - \\frac { d + \\sqrt { a d } } { \\sqrt { a d } } & < W _ { 2 l } ^ { ( a , d ) } = W _ { 2 m } ^ { ( b , d ) } < \\\\ & < \\frac { d + \\varepsilon \\sqrt { b d } } { \\sqrt { b d } } \\left ( \\frac { y + \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\left ( \\frac { \\alpha - \\beta } { 2 \\sqrt { - 2 } } \\right ) \\left ( \\frac { \\alpha ^ n - \\beta ^ n } { \\alpha - \\beta } \\right ) = d . \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} ( G ( M ' ) \\circ G ( D ) \\circ G ( M ) \\circ G ( D ' ) ) ^ 2 & = ( \\mu _ 2 \\circ d _ 2 \\circ m _ 2 \\circ \\delta _ 2 ) \\circ ( \\mu _ 2 \\circ d _ 2 \\circ m _ 2 \\circ \\delta _ 2 ) \\\\ & = ( \\mu _ 2 \\circ d _ 2 \\circ \\mu _ 2 \\circ d _ 2 ) \\circ ( m _ 2 \\circ \\delta _ 2 \\circ m _ 2 \\circ \\delta _ 2 ) \\\\ & = ( \\mu _ 2 \\circ d _ 2 \\circ m _ 2 \\circ \\delta _ 2 ) \\\\ & = G ( M ' ) \\circ G ( D ) \\circ G ( M ) \\circ G ( D ' ) \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} s ' = f ^ * _ { \\hat { \\kappa } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) } ( s ) \\otimes _ { \\hat { \\kappa } ( f ^ { \\mathrm { a n } } ( \\zeta ) ) } 1 _ { \\hat { \\kappa } ( \\zeta ) } \\in H ^ 0 ( Y , f ^ * ( L ) ) _ { \\hat { \\kappa } ( \\zeta ) } \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} P _ f \\left ( \\| \\widehat { f } _ n - \\mathrm { E } _ f \\widehat { f } _ n \\| _ \\infty \\geq \\mathrm { E } _ f \\| \\widehat { f } _ n - \\mathrm { E } _ f \\widehat { f } _ n \\| _ \\infty + \\sqrt { 2 Q _ 2 n ^ { - 1 } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } } x } \\right ) \\leq e ^ { - x } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\displaystyle { [ A ] ^ 2 = - 1 , K ^ { * } ( [ A ] ) = 1 . } \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} & \\int _ { | u - \\alpha _ 1 | > \\frac { d } { M } } \\frac { \\ , \\mathrm { d } u } { \\big ( ( u - \\alpha _ 1 ) ^ 2 + ( v - \\alpha _ 2 ) ^ 2 \\big ) ^ { \\frac { p } { 2 } } } \\\\ \\leqslant { } & \\int _ { | u - \\alpha _ 1 | > \\frac { d } { M } } \\frac { \\ , \\mathrm { d } u } { | u - \\alpha _ 1 | ^ p } \\\\ = { } & \\frac 2 { p - 1 } d ^ { 1 - p } M ^ { p - 1 } , \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { k - 1 } } e ^ { - n / X } = C X ^ { \\frac { 3 } { 2 } } + O _ { f , g , \\epsilon } ( X ^ { \\frac { - 1 } { 2 } + \\theta + \\epsilon } ) \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} L _ s \\Psi = \\lambda _ 0 \\Psi , \\ , \\ , \\ , A _ { 2 s + 1 } \\Psi = \\mu _ 0 \\Psi . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ r { \\Gamma \\Big ( { ( t + 1 ) n - r + j \\over 2 } \\Big ) \\over \\Gamma \\Big ( { n - r + j \\over 2 } \\Big ) } & \\sim \\prod _ { j = 1 } ^ r e ^ { - { t n \\over 2 } } \\Big ( { n \\over 2 } \\Big ) ^ { t n \\over 2 } ( t + 1 ) ^ { { ( t + 1 ) n \\over 2 } + { j - r - 1 \\over 2 } } \\\\ & = e ^ { - { t n r \\over 2 } } \\Big ( { n \\over 2 } \\Big ) ^ { r t n \\over 2 } ( t + 1 ) ^ { \\big ( { ( t + 1 ) n \\over 2 } - { 1 \\over 2 } \\big ) r - { r ( r - 1 ) \\over 4 } } . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} V _ m ( t , x ) = V _ m ( T , x ) + \\int _ t ^ T H ( s , x , a ^ \\ast ( s , x , m ( s ) , V _ m ( s , \\cdot ) ) , m ( s ) , V _ m ( s , \\cdot ) ) d s . \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) ( 3 i + 2 - k ) a ^ { k - 2 - i } \\ge 0 \\ , . \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} k ^ * ( t ) : = \\left \\{ \\begin{array} { l } \\cos \\frac { \\pi t } { 2 T } \\ , \\qquad \\mbox { i f } \\ | t | \\le T \\ , , \\\\ \\\\ 0 \\ , \\qquad \\qquad \\quad \\ \\ \\ \\ \\mbox { i f } \\ | t | > T \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\mathbb { E } [ \\Phi _ { j } ^ * ( z ) | \\mathcal { F } _ n ] = \\Phi _ n ^ * ( z ) j \\geq n \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial t } + L v - \\mu v = - \\psi \\ : \\mbox { i n } \\ : D _ { T + h } , v ( \\pm 1 , y ) = v ( \\pm 1 , - e y ) \\ : \\mbox { i n } \\ : D _ { T + h } ^ \\pm , v ( T + h ) = \\varphi \\ : \\mbox { i n } \\ : D . \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\nabla _ L f ( x ) & : = L \\ , \\Big ( f ( x + L ^ { - 1 } ) - f ( x ) \\Big ) \\qquad \\qquad x \\in e ^ n , \\\\ \\Delta _ L f ( x ) & : = L ^ 2 \\ , \\Big ( f ( x + L ^ { - 1 } ) + f ( x - L ^ { - 1 } ) - 2 f ( x ) \\Big ) x \\in e ^ n \\setminus \\{ x ^ e _ 1 \\} . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\frac { k _ { s ^ * } } { r _ { s ^ * } } \\geq \\frac { k - \\sum _ { j = 1 } ^ { s ^ * - 1 } k _ j } { r _ { s ^ * } } > \\left \\lceil \\frac { k - \\sum _ { j = 1 } ^ { s ^ * - 1 } k _ j } { r _ { s ^ * } } \\right \\rceil - 1 \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\mathcal { I } _ { 1 } \\leqslant \\frac { \\sum _ { \\substack { n = 1 } } ^ { K } \\mathbb { P } ( S _ { ( n ) } > \\mathit { x y } ) \\mathbb { P } ( \\eta = n ) } { \\sum _ { \\substack { n = 1 } } ^ { K } \\mathbb { P } ( S _ { ( n ) } > x ) \\mathbb { P } ( \\eta = n ) } \\leqslant \\max \\limits _ { \\substack { 1 \\leqslant n \\leqslant K \\\\ n \\in { \\rm s u p p } ( \\eta ) } } \\frac { \\mathbb { P } ( S _ { ( n ) } > \\mathit { x y } ) } { \\mathbb { P } ( S _ { ( n ) } > x ) } , \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\xi \\varGamma _ { t , T } - Y _ t = - \\int _ t ^ T \\varGamma _ { t , s } f _ s d s - \\int _ t ^ T \\varGamma _ { t , s - } d R _ s + \\int _ t ^ T d N _ s \\quad \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) v \\rVert = \\lambda _ j ( \\mu ) . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\| N _ { A _ { 1 } , q _ { 1 } } - N _ { A _ { 2 } , q _ { 2 } } \\| = \\| \\Lambda _ { V _ { 1 } } - \\Lambda _ { V _ { 2 } } \\| , \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} f _ 1 & = \\left ( \\frac { \\eta ( q ^ 2 ) ^ 5 } { \\eta ( q ) \\eta ( q ^ 4 ) ^ 2 } \\right ) ^ 2 , & f _ 2 & = \\eta ( q ^ 2 ) ^ 4 , & f _ 3 & = \\left ( \\frac { \\eta ( q ) \\eta ( q ^ 4 ) ^ 2 } { \\eta ( q ^ 2 ) } \\right ) ^ 2 . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\widehat { P } \\mathbf { u } = | \\Omega | ^ { - 1 } \\int _ { \\Omega } \\mathbf { u } ( \\mathbf { x } ) \\ , d \\mathbf { x } . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} { \\rm S p } _ 2 ( p ) \\wr S _ n = M _ 0 > M _ 1 > \\cdots > M _ s = C _ 6 \\wr S _ n \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\left \\| \\frac { \\partial ^ d } { \\partial x _ 1 \\cdots \\partial x _ d } K _ { \\boldsymbol { J } _ n } ( f ) \\right \\| _ 2 \\lesssim \\sqrt { \\sum _ { j _ 1 = N _ 1 - 1 } ^ { J _ { n , 1 } - 1 } \\cdots \\sum _ { j _ d = N _ d - 1 } ^ { J _ { n , d } - 1 } \\sum _ { \\boldsymbol { k } } 2 ^ { 2 \\sum _ { l = 1 } ^ d j _ l } \\theta _ { \\boldsymbol { j } , \\boldsymbol { k } } ^ 2 } \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} p _ s ( x ^ N | \\theta ) & = c ( \\emptyset ) p ( x _ 1 | \\theta ) c ( x _ 1 ) p ( x _ 2 | x ^ 1 , \\theta ) \\ldots c ( x ^ { N - 1 } ) p ( x _ N | x ^ { N - 1 } , \\theta ) ( 1 - c ( x ^ N ) ) \\\\ & \\propto \\Pi _ { i = 1 } ^ N p ( x _ i | x ^ { i - 1 } , \\theta ) \\\\ & = p ( x ^ N | \\theta ) \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} P = u _ 1 ^ d \\cdots u _ i ^ d \\cdot P _ { k , i + 1 } ^ { 2 ^ { q + i } + 2 ^ q - t - i - a _ { i + 1 } } \\cdot u _ { i + 1 } ^ { a _ { i + 1 } } \\cdot P _ { k , i + 1 } ( \\mathcal { O } ) \\cdot h _ { i + 1 } \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} L ( z , w ) & = \\frac 1 2 \\biggl ( 1 - ( 1 - 2 z ) \\sqrt { 1 - \\frac { 4 w } { ( 1 - 2 z ) ^ 2 } } \\ , \\biggr ) = \\frac 1 2 \\biggl ( 1 - \\sum _ { k \\ge 0 } \\binom { 1 / 2 } { k } \\frac { ( - 1 ) ^ k 4 ^ k w ^ k } { ( 1 - 2 z ) ^ { 2 k - 1 } } \\biggr ) \\\\ & = z + \\sum _ { k \\ge 1 } C _ { k - 1 } \\frac { w ^ k } { ( 1 - 2 z ) ^ { 2 k - 1 } } = z + \\sum _ { \\substack { k \\ge 1 \\\\ n \\ge 0 } } C _ { k - 1 } \\binom { n + 2 k - 2 } { n } 2 ^ n w ^ k z ^ { n } . \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} 2 \\sum _ { 2 \\leq q _ 1 \\leq X } \\sum _ { q _ 2 > X } \\frac { \\Lambda ( q _ 1 ) \\Lambda ( q _ 2 ) } { \\phi ( q _ 1 ) \\phi ( q _ 2 ) ^ 2 } = 2 \\bigg ( \\sum _ { 2 \\leq q _ 1 \\leq X } \\frac { \\Lambda ( q _ 1 ) } { \\phi ( q _ 1 ) } \\bigg ) \\bigg ( \\sum _ { q _ 2 > X } \\frac { \\Lambda ( q _ 2 ) } { \\phi ( q _ 2 ) ^ 2 } \\bigg ) \\ll { \\log X } \\cdot \\frac 1 { X } . \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} r \\frac { d u ( r , \\lambda ) } { d r } & = P ( u ( r , \\lambda ) , r , \\lambda ) \\\\ u ( 0 , \\lambda ) & = 0 \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} F _ T ( x ) & = \\frac { ( 1 - 7 x + 1 8 x ^ 2 - 1 9 x ^ 3 + 6 x ^ 4 ) C ( x ) - ( 1 - 6 x + 1 2 x ^ 2 - 8 x ^ 3 + x ^ 4 ) } { x ^ 2 ( 1 - x ) ( 1 - 2 x ) } \\ , . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\| { \\mathcal E } ( t , \\tau ) \\| \\le { C } _ { 1 5 } + C _ { 1 7 } | \\tau | t ^ 2 , 0 < | t | \\le t ^ 0 , \\ N = 0 ; C _ { 1 7 } = c _ * ^ { - 1 / 2 } C _ { 9 } . \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} c _ { n - 1 , n } = 0 . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} { \\rm C o r r } _ { ( \\frak X , \\widehat { \\frak S } ^ { \\epsilon } ) } ( h ) = \\widehat f _ t ! ( \\widehat f _ s ^ * h ; \\widehat { \\frak S } ^ { \\epsilon } ) \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\tau _ { \\tau _ t } = \\inf \\{ { s > \\tau _ t } : B _ s = a \\} = { \\tau _ t } \\ \\ \\ \\ \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} A = 1 8 \\cdot 2 6 \\cdot 2 3 \\cdot 2 7 \\cdot 1 5 \\cdot 2 \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\mathrm { h \\ddot { o } l } _ \\beta ( f , \\overline { V } ) : = \\sup \\left \\{ \\frac { | f ( x ) - f ( y ) | } { \\| x - y \\| ^ \\beta } : x , y \\in \\overline { V } , x \\not = y \\right \\} \\in [ 0 , \\infty ] , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} | I | = 2 , \\ : 6 , \\ : 1 0 , \\ : 1 4 . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\min _ { \\overline { \\Omega _ { d } } } \\varphi _ { \\lambda } \\left ( x \\right ) = \\varphi _ { \\lambda } \\left ( x \\right ) \\mid _ { \\xi _ { d } } = e ^ { \\lambda d } , \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} f \\ = \\ \\sum _ a c _ a x ^ a \\ , , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\Vert f _ { t } \\Vert _ { L _ { ( \\frac { T } { 2 } , T ) \\times \\mathbb { R } ^ { n } } ^ { \\infty } } \\leq \\left ( \\prod _ { k = 1 } ^ { \\infty } C ^ { \\frac { 1 } { 2 p _ { k - 1 } } } ( \\vert \\alpha \\vert ^ { 2 } p _ { k - 1 } ^ { 2 } + p _ { k - 1 } ^ { \\frac { 2 } { 2 - \\gamma } } \\Vert b \\Vert _ { L _ { t } ^ { l } L _ { x } ^ { q } } ^ { \\frac { 2 } { 2 - \\gamma } } \\vert \\alpha \\vert ^ { \\frac { 2 } { 2 - \\gamma } } + \\sigma \\frac { 4 ^ { k } } { T } ) ^ { \\frac { 1 } { 2 p _ { k - 1 } } } \\right ) \\Vert f _ { t } \\Vert _ { L _ { I _ { 0 } \\times \\mathbb { R } ^ { n } } ^ { 2 } } . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} d ^ { 2 } W / d \\xi ^ { 2 } - \\left \\{ { u ^ { 2 } + \\psi \\left ( \\xi \\right ) } \\right \\} W = \\varpi \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\tilde { { F } } _ { 2 } \\left ( p \\right ) = { \\tfrac { 1 } { 8 } } \\ , p ^ { 3 } \\left ( { 1 - p ^ { 2 } } \\right ) \\left ( { 1 2 \\ , p ^ { 2 } - 1 5 \\ , p ^ { 4 } - 1 } \\right ) , \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} F _ p ( p , p - 1 , p - 1 ) & = 1 + \\dim ( p - 1 , p - 2 ) + \\dim ( 2 p - 4 , 1 ) \\cr & = \\dim ( p - 1 , p - 2 ) + \\dim ( 2 p - 4 , 1 ) + \\dim ( 2 p - 3 ) \\cr & = \\dim ( p - 1 , p - 1 ) + \\dim ( 2 p - 3 , 1 ) \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} I _ 1 & = \\int _ { | v _ 1 | < \\tau } \\left \\{ \\int _ { 0 } ^ x \\frac { 1 } { \\tau | v _ 1 | } e ^ { - \\frac { a _ { \\ell } ( x - y ) } { \\tau | v _ 1 | } } d y \\right \\} \\mathcal { M } _ 1 ( v _ 1 ) d v _ 1 \\cr & = \\frac { 1 } { a _ { \\ell } } \\int _ { | v _ 1 | < \\tau } \\left \\{ 1 - e ^ { - \\frac { a _ { \\ell } x } { \\tau | v _ 1 | } } \\right \\} \\mathcal { M } _ 1 ( v _ 1 ) d v _ 1 \\cr & \\leq \\frac { 1 } { a _ { \\ell } } \\int _ { | v _ 1 | < \\tau } \\left \\{ 1 - e ^ { - \\frac { a _ { \\ell } } { \\tau | v _ 1 | } } \\right \\} \\mathcal { M } _ 1 ( v _ 1 ) d v _ 1 , \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} L _ 0 \\gamma = \\Delta \\gamma - \\frac { 1 } { r ^ { 2 } } \\gamma , \\\\ L _ 1 \\Gamma = \\Delta \\Gamma - \\frac { 2 } { r } \\partial _ r \\Gamma , \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} T _ { n p } ^ * : = \\frac { T _ { n p } - \\frac { p ( p - 1 ) } { 2 ( n - 4 ) } } { \\sigma _ { n p } } \\overset { d } \\to N ( 0 , 1 ) \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} \\sum _ { n = k } ^ \\infty ( n , k ) ^ { \\dagger } z ^ n = \\left ( 1 - \\frac { 1 } { G ( z ) } \\right ) ^ k \\mbox { f o r } ~ 1 \\leq k \\leq n . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} B = u _ \\lambda ( x , y ) \\mbox { a n d } B ' = ( u _ \\lambda v _ \\mu + w _ { \\lambda + \\mu } ) ( x , y ) . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} I _ n + \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ h V _ i & \\subseteq [ u + v ^ * , u + v ^ * + n - 1 ] + [ 0 , ( h - 1 ) v ^ * ] \\\\ & = [ u + v ^ * , u + h v ^ * + n - 1 ] \\\\ & \\subseteq I \\subseteq A . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} S _ f ( X ) = & \\sum _ { n \\leq X } a ( n ) = Q ( X ) + O ( X ^ { \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } + 2 A ( w - \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } ) \\eta + \\epsilon } ) \\\\ & + O ( X ^ { q - \\frac { 1 } { 2 A } - \\eta } \\log ( X ) ^ { r - 1 } ) + O \\bigg ( \\sum _ { X \\leq n \\leq X ' } | a ( n ) | \\bigg ) \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} a ( \\xi ) = \\sqrt { m ( \\xi ) } , \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} ( M _ 0 / 2 ) \\rho _ n \\epsilon _ n \\geq Q _ 1 \\sqrt { \\frac { 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } ( \\boldsymbol { \\alpha } ) } \\log { n } } { n } } + \\sqrt { 2 Q _ 2 C _ { I } } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } ( \\boldsymbol { \\alpha } ) / 2 } \\epsilon _ n , \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\frac { \\zeta _ i ^ { ( k - 1 ) p + \\nu } - t _ { k - 1 } } { h } = \\frac { ( \\nu - 1 + \\mu _ i ) \\ ; \\tilde h } { p \\ ; \\tilde h } = \\frac { \\nu - 1 + \\mu _ i } { p } . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} \\inf \\{ k \\geq 0 : \\| Y _ k - x \\| \\leq R _ { k + 1 } \\} \\ , \\leq \\ , \\inf \\{ n \\geq 0 : X ^ { 1 } _ { n } = x \\ , X ^ { 2 } _ { n } = x \\} \\ , . \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} - T h ( 1 , T ) + 2 T ^ 3 g ( 1 , T ) - g ^ 2 ( 1 , T ) - ( T ^ 2 - T ) T ^ 4 = - T h ( 1 , T ) + T ^ 5 - [ T ^ 3 - g ( 1 , T ) ] ^ 2 \\ , . \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{gather*} Y \\mathbf { d } = \\sup _ { y \\in Y } y \\mathbf { d } \\leq \\sup _ { y \\in Y } \\liminf _ \\lambda ( y \\mathbf { d } x _ \\lambda + x _ \\lambda \\underline { \\mathbf { d } } ) \\leq ( x _ \\lambda ) \\underline { \\mathbf { d } } . \\\\ \\mathbf { d } ( x _ \\lambda ) = \\liminf _ \\lambda \\mathbf { d } x _ \\lambda \\leq \\inf _ { y \\in Y } \\liminf _ \\lambda ( \\overline { \\mathbf { d } } y + y \\mathbf { d } x _ \\lambda ) = \\overline { \\mathbf { d } } Y . \\end{gather*}"}
-{"id": "7206.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { \\infty } \\frac { 1 } { n ^ { \\ell _ q } } = \\infty , \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} u ^ { L } _ { i + \\frac { 1 } { 2 } } = q _ i ( x _ { i + \\frac { 1 } { 2 } } ) \\ \\ { \\rm a n d } \\ \\ u ^ { R } _ { i - \\frac { 1 } { 2 } } = q _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\max \\{ j , j ' \\} = \\min \\{ j , j ' \\} + | j - j ' | \\leq C _ 0 N _ n ^ { 2 ( \\tau + 1 ) } \\gamma ^ { - 2 } + N _ n \\leq 2 C _ 0 N _ n ^ { 2 ( \\tau + 1 ) } \\gamma ^ { - 2 } \\ , . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} D ' \\left ( \\frac { 1 } { x } \\right ) = u ( x ) ^ 2 C ' ( x ) \\pmod { x ^ { n - 1 } - 1 } . \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} m _ 1 = 2 ^ { q + 1 } - t - a _ 1 \\ , \\ , \\ , a n d \\ , \\ , \\ , m _ i = 2 ^ q \\cdot [ 2 ^ { i - 2 } - 1 ] + t + 2 a _ { i - 1 } - a _ i \\ , \\ , \\ , f o r \\ , \\ , a l l \\ , \\ , 2 \\leq i \\leq k . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} E \\left [ \\int _ { 0 } ^ { T } \\frac { \\partial } { \\partial t } V ( t , X _ { \\sigma } ( t ) ) + \\tilde { H } ( t , X _ { \\sigma } ( t ) , \\sigma _ t , V ( t , \\cdot ) ) d t \\right ] = 0 , \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} & \\sum _ { \\lvert 2 m ^ 2 + h \\rvert \\leq X } r _ { 2 k + 1 } ( m ^ 2 + h ) \\\\ & = \\delta _ { [ k = \\frac { 1 } { 2 } ] } \\delta _ { [ h = a ^ 2 ] } \\bigg ( 2 R ' _ h X ^ { \\frac { 1 } { 2 } } \\log X - 4 R ' _ h X ^ { \\frac { 1 } { 2 } } \\bigg ) + \\tfrac { 1 } { k } R _ { k , h } ^ k X ^ { k } + O ( X ^ { k + \\epsilon - \\lambda ( k ) } ) \\\\ & + O \\bigg ( \\sum _ { \\lvert 2 m ^ 2 + h - X \\rvert \\leq X ^ { 1 + \\epsilon - \\lambda ( k ) } } r _ { 2 k + 1 } ( m ^ 2 + h ) \\bigg ) . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\kappa ( s _ i , g ) = \\kappa ( s _ i ^ { - 1 } , g ) = ( - 1 ) ^ { | g _ i | | g _ { i + 1 } | } . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} p _ 1 ^ \\ast = c _ 1 \\alpha _ 1 ^ \\ast , \\ p _ 2 ^ \\ast = c _ 2 ( 1 - \\alpha _ 1 ^ \\ast ) , \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\nabla _ { y } \\cdot \\left ( - \\kappa \\left ( y \\right ) \\nabla _ { y } \\bar { \\theta } ^ { j } ( x , y ) \\right ) = \\nabla _ { y } \\cdot ( \\kappa n _ j ) \\quad \\mbox { i n } \\ ; Y _ { 1 } , \\\\ \\\\ - \\kappa \\left ( y \\right ) \\nabla _ { y } \\bar { \\theta } ^ { j } \\cdot \\mbox { n } = \\kappa n _ { j } \\quad \\mbox { o n } \\ ; \\partial Y _ { 0 } , \\\\ \\\\ \\bar { \\theta } ^ { j } \\ ; \\mbox { i s } \\ ; Y \\mbox { - p e r i o d i c } , \\end{array} \\right . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} ( x ^ { \\frac { 1 } { n } } + a ) ^ { n - 1 } ( x ^ { \\frac { 1 } { n } } + a - n ) = \\sum _ { m = 0 } ^ { n } a ^ { m - 1 } ( a - m ) \\frac { n ! } { ( n - m ) ! m ! } x ^ { \\frac { n - m } { n } } . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} f _ { 0 } ( t ) = f ( t ) f _ { i } ( t ) = \\int _ { 0 } ^ { t } f _ { i - 1 } ( s ) d s , \\ ; i \\geq 1 . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{gather*} \\lambda ^ 3 = \\sum _ { i = 1 } ^ l ( - \\lambda _ i ) ^ 3 + \\sum _ { i = l + 1 } ^ k ( - \\lambda _ i ) ^ 3 - ( n - k - 1 ) \\geq \\frac { ( - \\lambda _ 1 - \\dots - \\lambda _ l ) ^ 3 } { l ^ 2 } - ( k - l ) - ( n - k - 1 ) \\\\ \\geq \\frac { ( n - d - 1 ) ^ 3 } { ( d + 1 ) ^ 2 } - n . \\end{gather*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { l ' = 1 } ^ { p - q } \\left ( \\varphi _ { i l ' } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } \\overline { \\left ( \\varphi _ { i l ' } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } } \\equiv 0 , \\quad \\mbox { f o r a l l $ i = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} \\rho = \\sum _ { j } \\lambda _ { j } W \\psi _ { j } , \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } ( t , p ) = \\tilde { f } ( \\tilde { x } ( t , p ) , p ) \\mbox { w i t h } \\tilde { f } ( x , p ) : = f ( x + x ^ * ( p ) , p ) \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{gather*} \\mathbb { P } - \\lim _ { N \\rightarrow \\infty } { M ^ { \\lambda ( N ) } _ m } = M _ m \\forall \\ , m = 0 , 1 , 2 , \\dots . \\end{gather*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\partial _ x \\bigl [ \\eqref { v a 1 } \\bigr ] & = \\sqrt { \\tfrac { i t } { 2 \\pi } } e ^ { - \\frac { i t } { 2 } x ^ 2 } T ( 0 ) \\psi ( 0 ) \\\\ & + \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ + e ^ { - \\frac { i t } { 2 } ( \\xi - x ) ^ 2 } \\partial _ \\xi \\bigl [ T ( \\xi ) \\psi ( \\xi ) \\bigr ] \\ , d \\xi . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} q _ i = \\binom { k + i - 2 } { i } - \\binom { k + 1 } { i } + ( - 1 ) ^ i \\binom { k - 1 } { i - 2 } \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\langle \\mathbf { G } - \\beta ^ { \\rho , A } , \\nabla \\varphi \\rangle \\rho \\ , d x = 0 , \\forall \\varphi \\in C _ 0 ^ { \\infty } ( \\R ^ d ) , \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\mu ( I _ { n _ { k + 1 } } ( u ) ) = \\frac { \\mu ( I _ { n _ k } ( u _ 1 , \\ldots , u _ k ) ) } { \\sharp G _ k } . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} & = - \\int _ { \\mathbb { T } _ { \\alpha } } ( e ^ { - \\nu t } - 1 ) \\sin y \\partial _ { x } ( \\omega ^ { 0 } - \\psi ^ { 0 } ) \\left ( \\omega - \\psi \\right ) d x d y + \\nu \\int _ { \\mathbb { T } _ { \\alpha } } \\bigtriangleup \\omega ^ { \\nu } ( \\omega ^ { \\nu } - \\psi ^ { \\nu } ) d x d y \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ - \\nu \\int _ { \\mathbb { T } _ { \\alpha } } \\bigtriangleup \\omega ^ { \\nu } ( \\omega ^ { 0 } - \\psi ^ { 0 } ) d x d y \\\\ & = I + I I + I I I . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} a _ { \\sigma } ( z ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\mathbb { R } ^ { 2 n } } e ^ { - \\frac { i } { \\hbar } \\sigma ( z , z ^ { \\prime } ) } a ( z ^ { \\prime } ) d ^ { 2 n } z ^ { \\prime } . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} I + I I = I I I \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\dot { y } ( t , p ) = B ( p ) y ( t , p ) \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} t _ n < t _ { n + 1 } & , \\ , \\ , n \\in \\N , \\\\ { \\lim } _ { n \\to \\infty } t _ n & = \\nu , \\ , \\ , | M ( t _ n ) | \\leq L , \\\\ i . e . \\ , \\ , { t _ n } ^ 2 + x ^ 2 ( t _ n ) & \\leq { L } ^ { 2 } . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} F _ { \\tau - } : = \\sigma ( \\mathcal F _ { \\tau _ n } ) _ { n \\geq 1 } . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} { \\bf A } = \\sum _ { i = 1 } ^ { n } { \\bf X } _ { i } { \\bf X } ^ { ' } _ { i } \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} [ V ( t ) \\psi ] ( x ) & = \\int _ \\R \\sqrt { \\tfrac { i t } { 2 \\pi } } e ^ { - \\frac { i t } { 2 } ( x ^ 2 + \\xi ^ 2 ) } K ( t x , \\xi ) \\psi ( \\xi ) \\ , d \\xi , \\\\ [ V ^ { - 1 } ( t ) \\phi ] ( \\xi ) & = \\int _ \\R \\overline { \\sqrt { \\tfrac { i t } { 2 \\pi } } e ^ { - \\frac { i t } { 2 } ( x ^ 2 + \\xi ^ 2 ) } K ( t x , \\xi ) } \\phi ( x ) \\ , d x , \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} s : = \\sum _ { i = 1 } ^ k \\left ( \\| \\mathbf w _ 0 - \\mathbf w _ i \\| ^ 2 - 1 \\right ) | s | \\geq \\sqrt { k } . \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\Gamma ' ( s ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { B } _ 1 ( s ) } \\frac { \\Gamma ( z ) } { ( z - s ) ^ 2 } d z \\ll \\max _ { 0 \\leq \\theta \\leq 2 \\pi } \\lvert \\Gamma ( s + e ^ { i \\theta } ) \\rvert , \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} f ^ { ( n ) } ( X ) = f ^ { ( e ) } \\left ( f ^ { ( n - e ) } ( X ) \\right ) = \\left ( f ^ { ( n - e ) } ( X ) \\right ) ^ S \\phi \\left ( f ^ { ( n - e ) } ( X ) \\right ) . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\lim \\limits _ { i \\rightarrow \\infty } \\frac { n _ i } { \\varphi ( n _ i ) } = + \\infty \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} \\kappa _ { X ^ * } ( \\zeta , t ) = \\zeta \\int _ 0 ^ \\infty \\int _ 0 ^ t \\kappa _ X ' \\left ( \\varepsilon ( s , \\xi ) \\zeta \\right ) d s \\ , \\pi ( d \\xi ) , \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} ( x , 0 , z ) ^ p & = ( 0 , 0 , x ^ p - x ) , \\\\ ( 0 , y , z ) ^ p & = ( 0 , 0 , y + y ^ p + \\cdots + y ^ { p ^ { n - 1 } } ) . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} L ^ e ( M ^ e ) ^ 2 = L ^ { \\tilde { e } } ( M ^ { \\tilde { e } } ) ^ 2 \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} J _ m = x J _ m + ( 2 x - x ^ 2 ) J _ { m - 1 } - x ^ 2 J _ { m - 2 } + \\frac { x ^ { m + 2 } } { ( 1 - x ) ^ { m - 1 } ( 1 - 2 x ) } + \\frac { x ^ { m + 3 } } { ( 1 - x ) ^ 2 ( 1 - 2 x ) } \\ , . \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} V _ { p , k } ( z ) - S ( z ) = O \\bigg ( \\frac { \\Psi _ p ( z _ k ) } { \\widetilde { \\Psi } _ { p , k } } \\bigg ) , \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X ( s , M , Y _ 0 ^ { s - D } ) d s + B ( t ) , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} \\kappa _ F : = \\sup _ { x \\in M } \\sup _ { v \\in T _ x M \\backslash \\{ 0 \\} } \\frac { F ( v ) } { F ( - v ) } \\in [ 1 , \\infty ] . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\left \\| { \\partial p _ 0 ^ L \\over \\partial t } ( 0 ) + \\sum _ { i = 1 } ^ n \\nabla _ { y _ i } { \\partial p _ i ^ L \\over \\partial t } ( 0 ) \\right \\| _ { H _ n } \\le c h _ L ^ s , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\varphi ( s ) - \\int _ 0 ^ 1 \\frac { d s } { s + t + \\varphi ( t ) } = f ( s ) , \\ ; \\ ; \\ ; 0 \\leq s \\leq 1 , \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } { 2 } \\ , . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\tilde { G } \\left ( 0 , W \\right ) = \\begin{pmatrix} W & 0 \\\\ 0 & 0 \\end{pmatrix} , \\tilde { F } \\left ( 0 , W \\right ) = 0 , \\quad \\dots , \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\mbox { s u m o f c o l u m n ~ 0 } & = y _ 0 - ( q - 2 ) \\varepsilon + x _ { 0 , 0 } + \\sum _ { i = 1 } ^ { q - 2 } ( x _ { i , 0 } + \\varepsilon ) = y _ 0 + \\sum _ { i = 0 } ^ { q - 2 } x _ { 0 , i } = \\mbox { s u m o f r o w ~ 0 } . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} r _ + = \\inf _ { v \\in C _ + } \\frac { F ( v , v ) } { \\langle J v , v \\rangle } \\ge \\sup _ { u \\in C _ - } \\frac { F ( u , u ) } { \\langle J u , u \\rangle } = r _ - \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} \\begin{aligned} f _ { m , p } ( x ) & = x f _ { m - 1 , p - 1 } ( x ) + x \\sum _ { i = 1 } ^ { m - p } f _ { m , p + i - 1 } + \\frac { x ^ { p + 1 } } { ( 1 - x ) ^ 2 } \\sum _ { i = 1 } ^ { m - p } \\sum _ { j = - 1 } ^ { i - 2 } x ^ { i - j } f _ { m - i + j + 1 - p ; j } ( x ) \\\\ & + \\frac { x ^ { 3 + p } } { ( 1 - x ) ^ 3 } \\sum _ { i = 1 } ^ { m - p } \\sum _ { k = 0 } ^ { i - 1 } \\sum _ { j = k } ^ { m - i - 1 + k - p } \\frac { x ^ { i - k } } { ( 1 - x ) ^ { j - k } } f _ { m - i + k - p , j } ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\begin{vmatrix} u _ { 1 , 0 } & u _ { 1 , 1 } & \\cdots & u _ { 1 , k - 1 } \\\\ u _ { 2 , 0 } & u _ { 2 , 1 } & \\cdots & u _ { 2 , k - 1 } \\\\ \\vdots & \\vdots & & \\vdots \\\\ u _ { k , 0 } & u _ { k , 1 } & \\cdots & u _ { k , k - 1 } \\end{vmatrix} \\neq 0 . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathfrak { m } ^ { ( p , p ) } \\big ] = \\mathbb { E } \\big [ O _ \\prec ( \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 1 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 2 , p ) } \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\Psi ^ 2 \\hat { \\Pi } ^ 2 ) \\mathfrak { m } ^ { ( p - 1 , p - 1 ) } \\big ] . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} F ^ { - 1 } \\left ( - \\frac { i \\epsilon } { \\xi - ( \\lambda + i \\epsilon ) } F ( m _ 1 ( \\lambda + ) - 1 ) \\right ) = \\epsilon \\chi _ { \\mathbb { R } ^ + } ( x ) e ^ { i ( \\lambda + i \\epsilon ) x } * ( m _ 1 ( \\lambda + ) - 1 ) . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} \\mathcal { V } _ { i } ^ { ( n ) } = \\{ e _ { i } \\otimes e _ { j } : j \\neq i , \\ ; \\ ; 1 \\leq j \\leq n \\} . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} & C : w ^ 3 = u ( 1 - u ) ^ 2 ( 1 - t u ) ^ { - 1 } , \\\\ & E : { w ' } ^ 3 = r ^ { - 2 } ( 1 - r ) ^ { - 2 } . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} { \\bar { B } } _ { k } ( t ) = \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { [ 0 , r _ { k } ( \\bar { \\boldsymbol { X } } ( s ) ) ) } ( y ) \\ , \\varphi _ { k } ( s , y ) d s \\ , d y . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} w ( y _ 1 , x ' ) = \\int \\psi ( t ) e ^ { - \\frac { i } { h } A ( t , y _ 1 , x ' , h D _ { x ' } ) } w | _ { x _ 1 = t } d t - \\frac { i } { h } \\int \\psi ( t ) \\int _ { y _ 1 } ^ { t } e ^ { - \\frac { i } { h } A ( s , y _ 1 , x ' , h D _ { x ' } ) } f ( s , x ' ) d s d t \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} ( x + 1 ) ( y - 1 ) ( x - 1 ) [ \\gamma ] = x y [ \\gamma ] - x [ \\delta ] , \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\Phi \\bigl ( 2 u + \\tfrac { n + 1 } { 2 } v + \\ell \\bigr ) = 2 u + \\tfrac { n + 1 } { 2 } v + \\ell \\ , \\ \\ \\Phi ( v + \\ell ) = - v - \\ell \\ , \\ \\ \\Phi \\bigl ( u - \\tfrac { n + 1 } { 4 } v \\bigr ) = - u + \\tfrac { n + 1 } { 4 } v , \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\kappa ( \\mu ) = \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) \\rVert _ { i c } \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\rho ( \\tilde { \\theta } _ { 1 } ^ \\alpha , x ) < \\alpha | x \\right ) = \\alpha + O ( n ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\xi _ m ( d _ c ) = \\gamma _ m ( d _ c - 1 , 2 ^ m - 2 m ) . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) , \\pi _ { \\ast } \\omega Z ) & = g _ { 1 } ( \\mathcal { T } _ { U } \\omega \\phi Z , V ) , \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} H = - \\tfrac 1 2 \\partial _ x ^ 2 + q \\delta ( x ) , \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} n _ j & \\geq p _ j r _ j + 1 + ( p _ j + 1 ) ( \\delta _ j - 1 ) \\\\ & = p _ j ( r _ j + \\delta _ j - 1 ) + \\delta _ j \\\\ & > p _ j ( r _ j + \\delta _ j - 1 ) + q _ j \\\\ & = n _ j \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} | ( \\nabla F _ x ) ( y ) | ^ 2 + \\lambda F _ x ( y ) ^ 2 ~ = ~ | ( \\nabla F _ z ) ( x ) | ^ 2 + \\lambda F _ z ( x ) ^ 2 ~ = ~ | \\nabla h ( x ) | ^ 2 + \\lambda a ^ 2 F _ x ( x ) ^ 2 \\ , . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} C ( d ) : = 2 ^ { d + 2 } + 2 ^ { d + 1 } d \\left ( d + 5 \\right ) + \\frac { 2 \\pi ^ { d / 2 } } { \\Gamma ( d / 2 ) } d ^ { d / 2 - 1 } \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} a ^ { i j } D _ { i j } w _ l = b _ l \\ ; \\mbox { i n } \\ ; \\Omega , w _ l = 0 \\ ; \\mbox { o n } \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} R ^ * : = \\sup _ { t \\in [ 0 , T ^ * ] } \\| v ( t ) \\| _ { H ^ 1 } \\leq C ( T ^ * , A ^ * , M ^ * ) < \\infty . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} d = p ^ { i _ 1 } + p ^ { i _ 2 } + \\cdots + p ^ { i _ p } \\ \\ \\mbox { w i t h } \\ \\ i _ 1 \\leq i _ 2 \\leq \\cdots \\leq i _ p . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} L ( X ) = f + \\sum _ { i , j } c _ { i j } ^ n \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\quad \\Omega _ \\varepsilon , \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\{ ( x , y ) \\in \\mathbb { R } ^ 2 : x \\ ! - \\ ! y = 0 , x , y \\geq 0 \\} . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\widetilde { e } \\left ( \\frac { k b } { \\varpi ^ l } \\right ) \\sum _ { c \\bmod { \\varpi ^ { l - 1 } } } \\widetilde { e } \\left ( \\frac { k \\varpi ^ { - h } c } { \\varpi ^ { l - h - 1 } } \\right ) = \\widetilde { e } \\left ( \\frac { k b } { \\varpi ^ l } \\right ) \\sum _ { c \\bmod { \\varpi ^ { l - 1 } } } \\widetilde { e } \\left ( \\frac { c } { \\varpi ^ { l - h - 1 } } \\right ) . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\widetilde { U } = ( \\widetilde { u } _ 1 , \\widetilde { u } _ 2 , \\ldots , \\widetilde { u } _ l ) \\in \\mathbb { R } ^ { m \\times l } , \\widetilde { V } = ( \\widetilde { v } _ 1 , \\widetilde { v } _ 2 , \\ldots , \\widetilde { v } _ l ) \\in \\mathbb { R } ^ { n \\times l } \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\xi _ L = \\dot { q } ^ { i } \\frac { \\partial } { \\partial q ^ i } + \\xi ^ i ( q , \\dot { q } ) \\frac { \\partial } { \\partial { \\dot q } ^ i } . \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\Phi _ j = \\varphi _ { j 2 } s _ 2 + \\cdots + \\varphi _ { j d } s _ d - \\varphi _ j \\quad . \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} z _ n ^ C - Q _ n \\mathcal { K } _ m ( z _ n ^ C ) = Q _ n f . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\sup _ { \\boldsymbol { x } \\in [ 0 , 1 ] ^ d } | G _ n ( \\boldsymbol { x } ) - U ( \\boldsymbol { x } ) | = O \\left ( \\frac { 1 } { n } \\right ) , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} 0 & = a F + b \\eta ^ { 1 2 } F + c D F + d \\eta ^ { 1 2 } D F \\\\ & = a F + b u ( E _ 6 F + 6 E _ 4 D F ) + c D F - \\frac { 1 } { 6 } d u \\left ( E _ 4 ^ 2 F + 6 E _ 6 D F \\right ) \\\\ & = \\left ( a + b u E _ 6 - \\frac { 1 } { 6 } d u E _ 4 ^ 2 \\right ) F + \\left ( c + 6 b u E _ 4 - d u E _ 6 \\right ) D F \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\mathbb Q \\ , \\overline { u } ( t ) = \\mathbb Q \\ , \\overline { u } _ 0 = : M \\in \\mathbb R ^ m t > 0 \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} D ( \\beta ) = 1 + \\beta + \\beta ^ t \\ne 0 \\ \\ \\mbox { f o r a n y $ \\beta \\in \\overline { \\mathbb { F } _ p } \\setminus \\{ 1 \\} $ w i t h $ \\beta ^ n = 1 $ } . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty d v \\thinspace \\left [ { \\partial ^ 2 \\over \\partial v ^ 2 } \\thinspace \\psi _ { s , F } ( - v ) \\right ] \\psi _ { s + p , G } ( v ) = \\int _ { - \\infty } ^ \\infty d v \\thinspace \\psi _ { s , F } ( - v ) { \\partial ^ 2 \\over \\partial v ^ 2 } \\thinspace \\psi _ { s + p , G } ( v ) , \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\| f \\| _ X = \\sup \\{ | \\langle f , g \\rangle | \\colon g \\in X ' , \\ \\| g \\| _ { X ' } \\le 1 \\} . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} Z \\mathbf { d } x & = \\sup _ { z \\in Z } z \\mathbf { d } x . \\\\ x \\mathbf { d } Z & = \\inf _ { z \\in Z } x \\mathbf { d } z . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} \\beta _ { 0 } ^ { i } = 0 , \\gamma _ { 0 } ^ { i } = 0 , i \\geq 1 , \\ ; \\beta _ { \\ell } ^ { 0 } = 1 , \\gamma _ { \\ell } ^ { 0 } = 1 , \\ell \\geq 0 . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\mathbf { t } = \\left ( \\mathbf { t } _ { 1 } ^ { \\prime } , \\mathbf { t } _ { 2 } ^ { \\prime } \\right ) ^ { \\prime } = \\left ( t _ { 1 } ^ { 1 } , t _ { 1 } ^ { 2 } , \\ldots , t _ { 1 } ^ { n } , t _ { 2 } ^ { 1 } , t _ { 2 } ^ { 2 } , \\ldots , t _ { 2 } ^ { m } \\right ) ^ { \\prime } \\in \\mathbb { R } ^ { \\left ( n + m \\right ) \\times 1 } \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} \\left [ A _ { 1 } w _ { 1 } , w _ { 2 } \\right ] & = \\left \\langle L \\left ( I - P _ { 1 } \\right ) J L w _ { 1 } , w _ { 2 } \\right \\rangle = \\left \\langle L J L w _ { 1 } , w _ { 2 } \\right \\rangle = - \\left \\langle L w _ { 1 } , J L w _ { 2 } \\right \\rangle \\\\ & = - \\left \\langle L w _ { 1 } , \\left ( I - P _ { 1 } \\right ) J L w _ { 2 } \\right \\rangle = - \\left [ w _ { 1 } , A _ { 1 } w _ { 2 } \\right ] . \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { X _ \\infty ( 0 ) } = \\| u _ 0 \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ x ^ 4 u _ 0 \\| _ { L ^ 2 } ^ 2 + \\| y ^ 2 \\partial _ x u _ 0 \\| _ { L ^ 2 } ^ 2 + \\| ( x \\partial _ x + y \\partial _ y ) u _ 0 \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\Psi = A t ^ { 2 m } + B t ^ { 2 m - 1 } + \\dfrac { c } { 2 ( m - 1 ) ( 2 m - 3 ) } t ^ 2 - \\dfrac { d } { 2 ( m - 1 ) ( 2 m - 1 ) } t - \\dfrac { e } { 2 m ( 2 m - 1 ) } \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} ( \\mathrm { d } X ) = \\prod _ { j < p } | \\lambda ( D ^ { ( k ) } _ { p p } ) - \\lambda ( D ^ { ( k ) } _ { j j } ) | \\ , ( \\mathrm { d } T ) ( Q ^ T \\mathrm { d } Q ) \\prod _ { j = 1 } ^ k \\mathrm { d } \\lambda _ j \\prod _ { s = 1 } ^ { ( N - k ) / 2 } \\mathrm { d } G _ s , \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} ( x _ \\lambda ) \\mathbf { d } & = \\limsup _ \\lambda x _ \\lambda \\mathbf { d } . \\\\ \\mathbf { d } ( x _ \\lambda ) & = \\liminf _ \\gamma \\mathbf { d } x _ \\lambda . \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} \\mathbb { P } ( E _ k ) & \\ge \\left ( 1 - 2 ( n - k + 1 ) \\exp \\left ( - \\frac { ( \\epsilon ^ 2 - \\epsilon ^ 3 ) p } { 4 } \\right ) \\right ) \\sum _ { \\left ( \\Pi , \\Delta \\right ) } \\mathbb { P } \\left ( \\Pi , \\Delta \\right ) \\\\ & = 1 - 2 ( n - k + 1 ) \\exp \\left ( - \\frac { ( \\epsilon ^ 2 - \\epsilon ^ 3 ) p } { 4 } \\right ) . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} E ^ { \\pm } ( x ) = \\left ( \\begin{array} { c } u ^ { \\pm } ( x ) \\\\ v ^ { \\pm } ( x ) \\\\ w ^ { \\pm } ( x ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\Lambda : = \\frac { | A - A ' | } { \\min \\{ \\delta ( A ) , \\delta ( A ' ) \\} } > 1 , \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\int _ { | x - y | \\leq \\delta / C _ { 0 } } e ^ { i \\lambda \\psi ( x , \\beta ) } f _ { i _ { 1 } \\dots i _ { m } } ( x ) B ^ { i _ { 1 } \\dots i _ { m } } ( x , \\beta ; \\lambda ) d x = \\mathcal { O } ( e ^ { - \\lambda / C } ) \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} f ( X ) & = v ( X ^ r ) , \\\\ f ^ { ( 2 ) } ( X ) & = v ( v ( X ^ r ) ^ r ) ) = v \\left ( ( w _ 1 ( X ) ) ^ { r ^ 2 } \\right ) = ( w _ 2 \\circ w _ 1 ( X ) ) ^ { r ^ 2 } . \\\\ & \\vdots \\\\ f ^ { ( k ) } ( X ) & = ( w _ k \\circ w _ { k - 1 } \\circ \\ldots \\circ w _ 1 ( X ) ) ^ { r ^ k } , \\ : \\ : k \\geq 1 . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} & \\div ( \\Delta ) = \\sum \\limits _ { \\substack { d | N \\\\ 1 \\leq d \\leq N } } \\frac { N } { d } \\frac { 1 } { \\gcd ( d , N / d ) } \\mathfrak { a } _ { c / d } \\\\ & \\div ( \\Delta ( N \\cdot ) ) = \\sum \\limits _ { \\substack { d | N \\\\ 1 \\leq d \\leq N } } \\frac { d } { \\gcd ( d , N / d ) } \\mathfrak { a } _ { c / d } . \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} f _ { i + 1 / 2 } ^ { } ( q _ { i + 1 / 2 } ^ { L } , q _ { i + 1 / 2 } ^ { R } ) = \\frac { 1 } { 2 } \\left ( f ( q _ { i + 1 / 2 } ^ { L } ) + f ( q _ { i + 1 / 2 } ^ { R } ) \\right ) - \\frac { | a _ { i + 1 / 2 } | } { 2 } \\left ( q _ { i + 1 / 2 } ^ { R } - q _ { i + 1 / 2 } ^ { L } ) \\right ) , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f \\right ] : = \\int _ { \\Omega } f ( p ( \\omega ) ) \\ ; \\mbox { d } \\mu ( \\omega ) = \\int _ { \\Pi } f ( p ) \\rho ( p ) \\ ; \\mbox { d } p \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\lim _ { T \\to T ^ * } \\| v \\| _ { L ^ { q _ d } _ t ( [ 0 , T ) ; W _ x ^ { 1 , r _ d } ) } = \\infty . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} g _ { 1 } ( \\mathcal { A } _ { X } \\omega W , \\mathcal { B } Y ) + \\eta ( Y ) g _ { 1 } ( X , \\omega W ) & = g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , Y ) , \\pi _ * \\omega \\phi W ) \\\\ & - g _ { 2 } ( ( \\nabla \\pi _ * ) ( X , \\mathcal { C } Y ) , \\pi _ * \\omega W ) , \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\hat { Y _ 1 } = \\frac { Y _ 1 } { \\sqrt { A } } , \\ ; \\ ; \\hat { Y _ 2 } = \\frac { Y _ 2 } { \\sqrt { B } } , \\ ; \\ ; \\hat { Y _ 3 } = \\frac { Y _ 3 } { \\sqrt { C } } , \\ ; \\ ; \\hat { Y _ 4 } = \\frac { Y _ 4 } { \\sqrt { D } } , \\hat { Y _ 5 } = \\frac { Y _ 5 } { \\sqrt { E } } . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\sigma _ j ( F , t _ j ) ) \\ll \\frac { \\Psi ( H ) } { | F _ j ' ( \\kappa _ j ) | } \\le \\frac { \\Psi ( H ) } { H ^ { ( \\ell _ { j , 1 } - 1 ) \\delta } } \\ , , \\qquad \\kappa _ j = \\kappa _ j ( F , t _ j ) . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} p _ j ( 0 ) = X _ j p _ { j k } ( 0 ) = - ( X _ j X _ k ) + ( X _ j ) ( X _ k ) . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} A ( \\pi ) = \\{ 1 = L _ 1 < L _ 2 < \\cdots < L _ { k - 1 } \\le n - 1 \\} \\cup \\{ n + 1 \\} \\ , . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\Phi _ a ( u , v ) = \\varphi _ a ( u ) + v , a \\in A , \\ u \\in E , \\ v \\in F , \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\left ( \\mathcal { K } _ m ' ( \\varphi _ m ) v \\right ) ^ { ( \\beta ) } ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\frac { \\partial ^ { \\beta + 1 } \\kappa } { \\partial s ^ \\beta \\partial u } ( s , \\zeta _ i ^ j , \\varphi _ m ( \\zeta _ i ^ j ) ) v ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} P ( \\psi ( X _ { t _ { n + 1 } } ) \\in A | \\ > \\sigma ( \\psi ( X _ { t _ n } ) ) ) & = K _ { \\psi ( X _ { t _ n } ) } \\circ K ^ X _ { \\mu _ { t _ { n + 1 } - t _ n } } ( x _ 0 , \\psi ^ { - 1 } ( A ) ) \\\\ & = P ( \\psi ( X _ { t _ { n + 1 } } ) \\in A | \\ > \\sigma ( \\psi ( X _ { t _ 0 } ) , \\ldots \\psi ( X _ { t _ n } ) ) ) \\quad \\quad \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\sum _ { k = p } ^ q | a _ k | u _ { n + k } \\underset { p , q \\to + \\infty } \\longrightarrow 0 \\ , , \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} X = D \\left ( t \\right ) \\partial _ { t } + T ( t ) Y ^ { i } \\partial _ { i } \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ t [ \\varphi ( X _ { t + t _ 1 } ^ { x } , X _ { t + t _ 2 } ^ { x } , \\cdots , X _ { t + t _ { m } } ^ { x } ) ] & = \\hat { \\mathbb { E } } _ t [ \\varphi _ { m - 1 } ( X _ { t + t _ 1 } ^ { x } , X _ { t + t _ 2 } ^ { x } , \\cdots , X _ { t + t _ { m - 1 } } ^ { x } ) ] \\\\ & \\ \\vdots \\\\ & = \\hat { \\mathbb { E } } _ t [ \\varphi _ { 1 } ( X _ { t + t _ 1 } ^ { x } ) ] \\\\ & = \\hat { \\mathbb { E } } [ \\varphi _ { 1 } ( X _ { t _ 1 } ^ { y } ) ] _ { y = X _ { t } ^ x } , \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} \\frac { d x } { d \\theta } = ( \\sin \\theta - \\sin 2 \\theta + \\sin 3 \\theta ) x ^ 3 + ( \\cos \\theta + 2 \\cos 2 \\theta ) x ^ 2 , ~ \\theta \\in [ 0 , 2 \\pi ] , \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\int _ { \\mathcal { L } _ { j } } { \\left \\vert { T _ { n } \\left ( \\pm { u , t } \\right ) d t } \\right \\vert } \\leq \\sum \\limits _ { s = 0 } ^ { n - 2 } { \\frac { 1 } { \\left \\vert u \\right \\vert ^ { s } } \\int _ { \\mathcal { L } _ { j } } { \\left \\vert { F _ { s + 1 } \\left ( t \\right ) d t } \\right \\vert . } } \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} h ^ { t , \\bar t } : = h ^ 0 + \\xi _ + ^ { t } + \\xi _ - ^ { \\bar t } \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} h = \\left \\{ \\begin{array} { l } \\frac { 2 } { 3 } w \\bigg ( 1 + \\cos \\ ! \\bigg ( \\frac { 2 } { 3 } \\pi - \\frac { 2 } { 3 } C \\bigg ) \\bigg ) \\\\ \\quad \\quad \\quad \\quad \\textrm { f o r } ~ | w | > \\frac { \\sqrt [ 3 ] { 5 4 } } { 4 } \\lambda ^ { 2 / 3 } \\\\ \\\\ 0 , \\quad \\quad \\quad \\textrm { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} [ \\gamma ^ { N , - i } ; \\beta ] _ j ( t ) = \\begin{cases} \\gamma ^ N _ j ( t , \\widetilde { X } ^ N _ j ( t ^ - ) ) & j \\neq i \\\\ \\beta ( t ) & j = i . \\end{cases} \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\varepsilon _ { i 2 } = O _ \\prec \\big ( \\frac { 1 } { \\sqrt { N } } \\big ) \\ , . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} q _ k ^ X ( t + \\Delta ) = \\sum _ { \\ell = 1 } ^ k \\lambda _ \\ell ^ X ( t + \\Delta , X ) q _ k ^ Z ( t + \\Delta ) = \\sum _ { \\ell = 1 } ^ k \\lambda _ \\ell ^ Z ( t + \\Delta , Z ) , \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} E = \\left \\{ ( q ^ 1 , q ^ 2 , p _ 1 , p _ 2 ; \\dot { q } ^ 1 , \\dot { q } ^ 2 , 2 q ^ 2 q ^ 1 , ( q ^ 1 ) ^ 2 ) \\in T T ^ * \\mathbb { R } ^ 2 : p _ 1 = \\dot { q } ^ 1 , p _ 2 = 0 \\right \\} . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\widehat { S } ( \\mathbf { k } ) \\widehat { P } = \\widehat { \\mathcal { A } } ^ 0 ( \\mathbf { k } ) \\widehat { P } . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} T ' = T + 1 \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\{ ( t _ 1 , \\ldots , t _ n ) \\in \\overline { K } ^ { n } \\mid F _ 1 ( 0 , t _ 1 , \\ldots , t _ n ) = \\cdots = F _ n ( 0 , t _ 1 , \\ldots , t _ n ) = 0 \\} = \\{ ( 0 , \\ldots , 0 ) \\} . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\int _ { \\Gamma ^ t _ { \\theta } } \\frac { ( N \\cdot \\nu ^ t _ { \\theta } ) } { \\frac { \\partial \\sigma } { \\partial s } } ( x ) f ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) = \\int _ { Z } f ( z , \\theta \\lambda ^ t ( z ) ) \\bar J ( z , \\theta \\lambda ^ t ( z ) ) d z . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} \\tilde { L } _ { \\gamma } ^ { G E L } ( \\theta | x ) = \\prod _ { i = 1 } ^ n w ^ { \\gamma } ( x _ i , \\theta ) \\ , , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} F _ { 1 } ^ { \\pm } \\left ( \\xi \\right ) = { \\tfrac { 1 } { 2 } } \\psi _ { 0 } \\left ( \\xi \\right ) - { \\tfrac { 1 } { 8 } } \\phi ^ { 2 } \\left ( \\xi \\right ) \\mp { \\tfrac { 1 } { 4 } } { \\phi } ^ { \\prime } \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } F ( s ) x ^ { - s } d s . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\Big ( \\Gamma ( a n + 1 ) J _ n ( 1 ) \\Big ) = \\gamma . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} ( L + 1 ) \\Gamma & \\leq 0 \\textrm { i n } \\ \\Pi \\times ( 0 , T ] , \\\\ ( N + 2 ) \\Gamma & \\leq 0 \\textrm { o n } \\ \\partial \\Pi \\times ( 0 , T ] , \\\\ \\Gamma & \\leq 0 \\textrm { o n } \\ \\Pi \\times \\{ t = 0 \\} . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\sum _ { i = i _ 0 + 1 } ^ \\infty \\phi ( x , \\kappa ^ i r ) \\lesssim \\phi ( x , \\kappa ^ { i _ 0 + 1 } r ) + \\norm { D u } _ { L ^ \\infty ( B ( x , r ) \\cap B ^ + _ 4 ) } \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\mathbf { A } } ( t ) } t \\ , d t + \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\vec g } ( t ) } t \\ , d t . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} \\nu _ 0 < \\nu _ j j = 2 , \\ldots , k - 1 . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\langle b _ { n _ k } , \\chi _ j \\rangle = \\langle A \\chi _ { n _ k } , \\chi _ { n _ k + j } \\rangle = a _ j , \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\S ( f ) = \\exp \\ ( - \\frac 1 { 2 n ^ 2 } \\sum _ { x \\in G } | f ^ { - 1 } ( x ) | ^ 2 \\ ) . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} & \\sum _ { r = 2 } ^ k \\left \\{ \\sum _ { p = 0 } ^ { r - 2 } ( - 1 ) ^ p \\left [ a _ { k - r + 2 + p } ( n - p ) ( n - p - 1 ) + b _ { k - r + 1 + p } ( n - p ) + c _ { k - r + p } \\right ] e _ p \\right \\} z ^ { k + n - r } \\\\ & + = 0 . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} u _ h & = \\rho ( h ^ { - 1 } [ { \\sqrt { - h ^ 2 \\Delta _ g } } - 1 ] ) u _ h = \\int _ { \\R } \\hat { \\rho } ( t ) e ^ { i t [ { \\sqrt { - h ^ 2 \\Delta _ g } } - 1 ] / h } \\chi ( x , h D ) u _ h \\ , d t + O _ { \\gamma } ( h ^ { \\infty } ) . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\textrm { P r } ( X = k ) = \\displaystyle { \\sum _ { \\Lambda \\in \\mathcal { S } _ k } } \\Big [ \\displaystyle { \\prod _ { i \\in \\Lambda } } ~ p _ i \\Big ] \\Big [ \\displaystyle { \\prod _ { j \\in \\Lambda ^ c } } ( 1 - p _ j ) \\Big ] \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} a _ { k u } ^ { i l } \\left ( Z \\right ) = a _ { k u } ^ { i l } \\left ( Z _ { i } \\right ) , \\quad \\quad \\quad \\quad \\mbox { f o r a l l $ i , k , u = 1 , \\dots , q $ a n d $ l = 1 , \\dots , p - q $ , } \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} V \\mid _ { \\Gamma } = p _ { 0 } \\left ( x \\right ) , \\partial _ { n } V \\mid _ { \\Gamma } = p _ { 1 } \\left ( x \\right ) , \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} K _ 0 = F \\oplus F . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { l ' = 1 } ^ { p - q } \\left ( \\varphi _ { i _ { 0 } l ' } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } \\overline { \\left ( \\varphi _ { i _ { 0 } l ' } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } } \\not \\equiv 0 . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} \\hat V _ i ^ L = \\bigoplus _ { l _ 0 + \\ldots + l _ { i - 1 } \\le L } { \\cal V } ^ { l _ 0 } \\otimes { \\cal V } ^ { l _ 1 } _ \\# \\otimes \\ldots \\otimes { \\cal V } ^ { l _ { i - 1 } } _ \\# \\otimes V ^ { L - ( l _ 0 + \\ldots + l _ { i - 1 } ) } _ \\# , \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} \\frac { C _ { q , n } } { ( k - 2 - q ) ! } & = - 2 \\sum _ { m = 2 } ^ { q + 1 } a _ { k + m - 1 - q } \\biggl [ ( n - 1 ) m _ { ( m - 1 , \\dot { 0 } } + \\sum _ { p = 1 } ^ { [ ( m - 1 ) / 2 ] } m _ { ( m - 1 - p , p , \\dot { 0 } ) } \\biggr ] \\\\ & - \\sum _ { m = 1 } ^ { q } b _ { k + m - 1 - q } m _ { ( m , \\dot { 0 } ) } - \\frac { n ( n - 1 ) } { k ( k - 1 ) } q ( 2 k - q - 1 ) a _ { k - q } \\\\ & - \\frac { n } { k - 1 } q b _ { k - q - 1 } , l = 0 , 1 , \\ldots , k - 3 , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\| \\widetilde { J } _ 2 ( t , \\tau ) - { J } _ 2 ( t , \\tau ) \\| \\le \\| M \\| ^ 2 ( C _ 1 c _ * ^ { - 1 / 2 } + \\delta ^ { - 1 / 2 } ) = : \\widetilde { C } , \\tau \\in { \\mathbb R } , \\ | t | \\le t ^ 0 . \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot H ^ { s _ 1 , s _ 2 } } = \\| | D _ x | ^ { s _ 1 } | D _ y | ^ { s _ 2 } u \\| _ { L ^ 2 _ { x , y } } , \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} L _ { \\eta } \\Gamma _ { ( j k ) } ^ { i } = 2 \\xi , _ { t ( j } \\delta _ { k ) } ^ { i } \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} H _ \\omega = - \\Delta + U _ \\omega \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} a ^ { i j } _ k D _ { i j } u _ k = f _ k \\ ; \\mbox { i n } \\ ; \\Omega , u _ k = 0 \\ ; \\mbox { o n } \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} 3 u - \\sqrt { 2 } = ( - 1 ) ^ { b _ 0 } \\epsilon ^ { b _ 1 } w ^ { b } , \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\chi ( k \\Gamma _ { 2 } ) = H ( k , k ) = 2 k ^ { - 2 } ( k ^ { - 1 } - 1 ) . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\nu ^ A _ y = \\int \\limits _ { v \\in S ^ { 2 n - 1 } } \\nu ^ A _ { y , v } d \\sigma _ { 2 n - 1 } ( v ) . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\min \\left \\| \\begin{bmatrix} E & r \\\\ \\end{bmatrix} \\right \\| _ F , \\textrm { s u b j e c t t o } \\lambda b - r \\in \\mathcal { R } ( A + E ) , \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} p ( N ^ + [ S ] ) & \\ge p ( N ^ + [ v ] ) + p \\big ( N ^ + [ S ' ] \\cap N ^ o ( v ) \\big ) \\\\ & = p ( v ) + p ( N ^ + ( v ) ) + p ( N _ { G [ N ^ o ( v ) ] } ^ + [ S ' ] ) \\\\ & \\geq p ( v ) + \\big ( p ( N ^ + ( v ) ) + p ( N ^ - ( v ) \\big ) / 2 + p ( N ^ o ( v ) ) / 2 \\\\ & = p ( V ) / 2 . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| v _ i \\| _ p ^ p + d _ i p ( p - 1 ) \\int _ { \\Omega } | v _ i | ^ { p - 2 } | \\nabla v _ i | ^ 2 d x = p \\int _ { \\Omega } \\left ( L _ i v + g _ i ( v ) \\right ) | v _ i | ^ { p - 2 } v _ i d x . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\lambda _ { n } = \\max _ { M } \\tilde { u } _ n , \\ \\ \\textrm { a n d } \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} e _ { k i } ( L ) = \\left \\{ \\begin{array} { c c } 0 , & T ( L ) \\notin \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\\\ - \\frac { \\prod _ { j = 1 } ^ { k + 1 } [ l _ { k i } - l _ { k + 1 , j } ] _ q } { \\prod _ { j \\neq i } ^ { k } [ l _ { k i } - l _ { k j } ] _ q } , & T ( L ) \\in \\mathcal { B } _ { \\mathcal { C } } ( T ( L ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\overline \\lambda _ 0 \\ge \\lambda _ 0 ( s , X ( s ) ) = \\sum _ { k = 1 } ^ K \\lambda _ k ( s , X ( s ) ) , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\int _ { \\mathcal { L } _ { j } } { \\left \\vert { \\chi _ { n } \\left ( \\pm { u , t } \\right ) d t } \\right \\vert } \\leq 2 \\int _ { \\mathcal { L } _ { j } } { \\left \\vert { F _ { n } \\left ( t \\right ) d t } \\right \\vert } + \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { 1 } { \\left \\vert u \\right \\vert ^ { s } } \\int _ { \\mathcal { L } _ { j } } { \\left \\vert { G _ { n , s } \\left ( t \\right ) d t } \\right \\vert , } } \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} Q _ v & = \\begin{cases} v & \\ v \\in H , \\\\ \\alpha \\alpha ^ * & \\ v = \\alpha \\in F ( H ) \\end{cases} \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} { \\tilde \\tau } = { \\tilde \\mu } _ j ( { \\tilde t } , { \\tilde x } , { \\tilde \\xi } ) , { \\tilde b } _ { j i } ( { \\tilde t } , { \\tilde x } , { \\tilde \\xi } ) = b _ { j i } ( { \\tilde t } - \\epsilon | { \\tilde x } | ^ 2 , { \\tilde x } , { \\tilde \\xi } + 2 \\epsilon { \\tilde \\mu } _ j ( { \\tilde t } , { \\tilde x } , { \\tilde \\xi } ) { \\tilde x } ) = 0 . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} S _ { \\square , M } & = - \\widetilde { \\Phi } ( 0 ) \\frac { X } { 4 } \\sum _ { \\substack { N ( l ) \\leq Z \\\\ l \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\frac { \\mu _ { [ i ] } ( l ) } { N ( l ^ 2 ) } \\sum _ { \\substack { \\varpi \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } \\\\ N ( \\varpi ) \\geq X / N ( l ^ 2 ) } } \\frac { \\log N ( \\varpi ) } { N ( \\varpi ) } \\hat { \\phi } \\left ( \\frac { \\log N ( \\varpi ) } { \\log X } \\right ) + O ( X ) . \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} e _ \\lambda ( x , y ) ~ = ~ 4 \\int _ r ^ \\infty \\frac { \\sinh s } { \\sqrt { \\cosh 2 s - \\cosh 2 r } } & \\cdot \\frac { - 1 } { 2 \\pi \\sinh s } \\partial _ s \\frac { - 1 } { 2 \\pi \\sinh 2 s } \\partial _ s \\cdot \\\\ & \\cdot \\left ( \\frac { - 1 } { 2 \\pi \\sinh s } \\partial _ s \\right ) ^ { 2 d - 2 } \\frac { \\sin ( s \\cdot \\sqrt { \\lambda - ( 2 d + 1 ) ^ 2 } ) } { \\pi s } \\ , d s \\ , . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} a ^ { i j } D _ { i j } v = g \\ ; \\mbox { i n } \\ ; \\Omega , v = 0 \\ ; \\mbox { o n } \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} \\sum _ { j _ 1 \\in \\mathcal { Q } _ 1 } \\cdots \\sum _ { j _ d \\in \\mathcal { Q } _ d } 2 ^ { \\sum _ { l = 1 } ^ d j _ l } \\omega _ { \\boldsymbol { j } , n } \\leq \\sum _ { j _ 1 \\in \\mathcal { Q } _ 1 } \\cdots \\sum _ { j _ d \\in \\mathcal { Q } _ d } 2 ^ { - \\sum _ { l = 1 } ^ d j _ l \\mu _ l } . \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} { \\mathcal U } _ { 1 } ^ C ( t ) = \\sum _ { n = 1 } ^ { \\infty } \\big ( C _ { n } e ^ { i \\omega _ { n } t } + \\overline { C _ { n } } e ^ { - i \\overline { \\omega _ { n } } t } \\big ) , { \\mathcal U } _ { 1 } ^ D ( t ) = \\sum _ { n = 1 } ^ { \\infty } \\big ( D _ { n } e ^ { i \\zeta _ { n } t } + \\overline { D _ { n } } e ^ { - i \\overline { \\zeta _ { n } } t } \\big ) , { \\mathcal U } _ { 1 } ^ R ( t ) = \\sum _ { n = 1 } ^ { \\infty } R _ { n } e ^ { r _ { n } t } , \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\tilde { { E } } _ { s } \\left ( 1 \\right ) = k _ { s } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\mu _ 3 ^ 2 \\geq \\nu _ 1 \\mu _ 2 ^ 2 \\nu _ 1 : = \\left ( \\sqrt { 1 + \\frac { c _ { 1 , \\infty } } { 2 c _ { 3 , \\infty } } } - 1 \\right ) ^ 2 \\left ( - 1 + \\sqrt { 1 + \\frac { c _ { 3 , \\infty } + c _ { 4 , \\infty } } { c _ { 2 , \\infty } } } \\right ) ^ { - 2 } \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} X ^ N _ i ( t ) = \\xi ^ N _ i + \\int _ 0 ^ t \\int _ U f ( s , X ^ N _ i ( s ^ - ) , u , \\alpha ^ N _ i ( s ) , \\mu ^ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) , i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} K _ { 2 } \\left ( y \\right ) = - \\frac { U ^ { \\prime \\prime } } { U - U _ { s } } > 0 \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\Phi ( f ( z , w ) ) = f ( z + w , ( 2 z + w ) w ) . \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\widetilde { \\Sigma } _ k = { \\rm d i a g } ( \\widetilde { \\sigma } _ 1 , \\widetilde { \\sigma } _ 2 , \\ldots , \\widetilde { \\sigma } _ k ) \\in \\mathbb { R } ^ { k \\times k } , \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} U _ { i l } ^ k = \\Gamma _ { \\phi i l } ^ k - \\Gamma _ { 0 i l } ^ k , \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } f ( \\theta ) ( G ( \\theta ) ) ^ k d \\theta = 0 , \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { R } _ o ( X , Y ) Z = & R ( X , Y ) Z - g ( A ( Y , Z ) , J ) A ( X , J ) + g ( A ( X , Z ) , J ) A ( Y , J ) \\\\ & + 2 g ( X , \\varphi Y ) A ( J , Z ) + [ \\tilde { T } ( X , Y ) , Z ] \\\\ = & R ( X , Y ) Z - g ( A ( Y , Z ) , J ) ( \\varphi X + \\varphi h X ) \\\\ & + g ( A ( X , Z ) , J ) ( \\varphi Y + \\varphi h Y ) + 2 g ( X , \\varphi Y ) A ( J , Z ) \\\\ & + 2 g ( X , \\varphi Y ) [ J , Z ] , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\hat u _ { \\infty , x } ( r ) = B ^ * Q _ { \\infty } ^ { - 1 } \\hat y _ { \\infty , x } ( r ) , r \\in \\ , ] - \\infty , 0 ] . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} w _ n ( x ) = \\tilde { u } _ n ( x ) - \\sum _ { j = 1 } ^ m \\rho _ { n , j } G ( x , x _ { n , j } ) - \\int _ M \\tilde { u } _ n \\mathrm { d } \\mu . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} \\big | { y } _ { m } ( x ) - \\hat { y } _ { m } ( x ) \\big | \\leq | { \\epsilon } | ( x - a ) ^ { \\gamma - 1 } \\sum _ { j = 0 } ^ { m } \\frac { A ^ j ( x - a ) ^ { \\alpha { j } } } { \\Gamma ( \\alpha { j } + \\gamma ) } . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} Z ( s , w , f \\times \\overline { g } ) : = \\sum _ { n , h \\geq 1 } \\frac { a ( n ) \\overline { b ( n - h ) } + a ( n - h ) \\overline { b ( n ) } } { n ^ { s + k - 1 } h ^ w } . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} C '' = C '' ( N , \\gamma , \\omega _ 0 , X , \\| \\phi ( \\cdot , 0 ) \\| _ { C ^ 4 ( B _ r ( p ) ) } , \\| \\widetilde { \\eta } \\| _ { C ^ 2 ( B _ r ( p ) ) } ) \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} k e r \\pi _ { * } = D _ { 1 } \\oplus D _ { 2 } , \\ , \\ , \\ , \\varphi ( D _ { 1 } ) = D _ { 1 } , \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} V o l ( M , \\tilde { g } ) = \\alpha ^ { \\frac { n } { 2 } } V o l ( M , g _ F ) = 1 \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} \\{ y \\in T _ \\Delta A \\ , | \\ , d ( y , H ) = t \\} = T _ \\Delta ( \\{ x \\in A \\ , | \\ , d ( x , H ) = t + \\Delta \\} ) , \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} E = \\max \\left ( \\max _ { 1 \\leq i , j \\leq n } \\left \\{ \\frac { | b _ i | } { A _ j } + \\frac { | b _ j | } { A _ i } \\right \\} , 3 \\right ) . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} f ( x ) = f ( x + w ) + \\langle w , x \\rangle , x \\in X . \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\mathcal { B } ^ { \\boldsymbol { \\alpha } } _ { p , q } \\equiv \\begin{cases} \\{ f \\in L _ p ( [ 0 , 1 ] ^ d ) : \\| f \\| _ { \\mathcal { B } ^ { \\boldsymbol { \\alpha } } _ { p , q } } < \\infty \\} , & 1 \\leq p < \\infty , \\\\ \\{ f \\in C _ u ( [ 0 , 1 ] ^ d ) : \\| f \\| _ { \\mathcal { B } ^ { \\boldsymbol { \\alpha } } _ { p , q } } < \\infty \\} , & p = \\infty \\end{cases} \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} e _ g : = \\langle g , \\cdot \\rangle _ { \\mathcal { C } } \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\lambda ^ { ( n ) } = \\max \\limits _ { m \\in \\mathcal { M } } \\lambda _ m ^ { ( n ) } , \\ ; \\ ; \\ ; \\ ; \\ ; P _ { e } ^ { ( n ) } = \\frac { 1 } { | \\mathcal { M } | } \\sum \\limits _ { m \\in \\mathcal { M } } \\lambda _ m ^ { ( n ) } . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} ( o s c _ r f ) ( x ) : = \\sup _ { y \\in B _ r ( x ) \\cap D } f ( y ) - \\inf _ { y \\in B _ r ( x ) \\cap D } f ( y ) . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\tilde { T } ( X , Y ) = g ( \\tilde { T } ( X , Y ) , \\xi ) \\xi = - g ( [ X , Y ] , \\xi ) \\xi = 2 g ( X , \\varphi Y ) \\xi . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} D _ { i j } ( a ^ { i j } v ) = 0 \\ ; \\ ; \\Omega , v = \\psi \\ ; \\ ; \\partial \\Omega \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} \\det \\left ( \\det X \\begin{pmatrix} ( i _ \\alpha ) \\sqcup K \\\\ ( i ' _ \\beta ) \\sqcup K ' \\end{pmatrix} \\right ) _ { 1 \\le \\alpha , \\ , \\beta \\le p } = \\det X \\begin{pmatrix} I \\sqcup K \\\\ I ' \\sqcup K ' \\end{pmatrix} \\cdot \\left ( \\det X \\begin{pmatrix} K \\\\ K ' \\end{pmatrix} \\right ) ^ { p - 1 } . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\sum _ { i _ 1 , i _ 2 = 0 } ^ p m _ { i _ 1 + i _ 2 } ^ { \\gamma } \\bar { z } _ { i _ 1 } z _ { i _ 2 } \\geq 0 \\ ; . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} v : = v ^ i \\left . \\frac { \\partial } { \\partial x ^ i } \\right | _ x . \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\sum _ { n \\geqslant 1 } \\frac { 1 } { n } \\sum _ { T _ t ^ n x = x } \\frac { \\exp \\ ( g _ { n , t } \\ ( x \\ ) \\ ) } { 1 - \\ ( \\ ( T _ t ^ n \\ ) ' \\ ( x \\ ) \\ ) ^ { - 1 } } z ^ n , \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} z ' & = w \\\\ w ' & = c _ 2 ( s ) \\left ( f ( z , s ) - w \\right ) . \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\gamma _ m ( t , x ) : = a ^ \\ast ( t , x , m ( t ) , V _ m ( t , \\cdot ) ) \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} C = \\lim _ { \\omega \\to \\infty } C ^ { ( \\omega ) } = P / 2 . \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } x ( t , \\rho ) & = & \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } ( H ( t , \\rho ) \\rho ^ 2 ) + \\frac { \\partial ^ 3 } { \\partial \\rho ^ 3 } ( H ^ 2 ( t , \\rho ) \\rho ^ 3 ) + \\dots \\\\ & = & 3 ! H ^ 2 ( t , \\rho ) + 3 ! \\frac { \\partial } { \\partial \\rho } H ( t , \\rho ) + R _ 2 ( t , \\rho ) , \\end{array} \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\ell ( \\phi _ t , \\phi _ x , \\phi _ y ) = 0 , \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} S _ \\gamma = \\{ ( t _ 0 , t _ 3 ) : t _ 0 ^ { - 1 } \\gamma t _ 3 = \\gamma \\} \\subset T _ { 0 F } \\times T _ { 3 F } , \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} a _ { 1 , 1 } = \\left ( \\det { \\overline C ^ 1 } \\right ) ^ { - \\frac { 1 } { 2 } } , a _ { 2 , 2 } = \\left ( \\frac { \\det { \\overline C ^ 2 } } { \\det { \\overline C ^ 1 } } \\right ) ^ { - \\frac { 1 } { 2 } } , a _ { k , k } = \\left ( \\frac { \\det { \\overline C ^ k } } { \\det { \\overline C ^ { k - 1 } } } \\right ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} d _ { 1 } : = \\inf \\{ \\L _ { 1 } ( u ) : u \\in \\mathcal { N } _ { 1 } \\} , \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\aligned \\partial ( \\frak H _ { ( 3 2 ) } ^ i \\circ \\frak H _ { ( a b ) } ^ { i } ) = & \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\frak N ^ { i } _ { 2 } \\circ \\frak H _ { ( a b ) } ^ { i } \\cup \\frak H ^ { i } _ { 3 2 } \\circ \\frak N ^ { i } _ { b , 2 1 } \\\\ & \\cup \\frak N ^ { i } _ { 3 } \\circ \\frak N ^ { i } _ { 3 2 } \\circ \\frak H _ { ( a b ) } ^ { i } \\cup \\frak H ^ { i } _ { 3 2 } \\circ \\frak N ^ { i } _ { a , 2 1 } . \\endaligned \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} I ( W \\wedge X , Y ) + \\mu H ( Y | W ) = R _ \\mu ( \\delta | P _ { X Y } ) \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} S = A ^ { - 1 } \\left ( d F \\right ) = \\left \\{ y \\in Y : \\left \\langle y , u \\right \\rangle = \\left \\langle d E , T \\tau ( u ) \\right \\rangle , \\forall u \\in T _ { y } Y \\right \\} \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\psi ^ { 3 1 ; \\epsilon } _ { \\alpha _ { 3 } , \\alpha _ 1 } = \\sum _ { \\alpha _ 2 \\in \\frak A _ 2 } \\psi ^ { 3 2 ; \\epsilon } _ { \\alpha _ { 3 } , \\alpha _ 2 } \\circ \\psi ^ { 2 1 ; \\epsilon } _ { \\alpha _ { 2 } , \\alpha _ 1 } \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} c _ 2 ( E _ { C , A } ) = d + 2 = M . N + | Z | + c _ 2 ( F ^ { * * } ) . \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} H ^ { ( r ) } : = \\{ z \\in D \\ , : \\ , d i s t ( z , H ) < r \\} . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} Q _ \\tau = Q _ t + e ^ { t A } Q _ { \\tau - t } e ^ { t A ^ * } . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} _ { \\mathbb { R } } ( \\mathcal { R ^ + } ) = \\left \\{ \\alpha _ 1 Q _ 1 + \\alpha _ 2 Q _ 2 + \\ldots + \\alpha _ k Q _ k : Q _ i \\in \\mathcal { R } ^ + , \\alpha _ i \\in \\mathbb { R } \\right \\} \\subseteq \\mathcal { R } , \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} \\int _ { \\varepsilon Y _ { 1 } } p \\left ( \\frac { x } { \\varepsilon } \\right ) \\phi \\left ( x \\right ) d x & = \\varepsilon ^ { d } \\int _ { Y _ { 1 } } p \\left ( y \\right ) \\phi \\left ( \\varepsilon y \\right ) d y , \\\\ \\int _ { \\varepsilon Y _ { 1 } } \\int _ { Y _ { 1 } } p \\left ( y \\right ) \\phi \\left ( x \\right ) d y d x & = \\varepsilon ^ { d } \\int _ { Y _ { 1 } } \\int _ { Y _ { 1 } } p \\left ( y \\right ) \\phi \\left ( \\varepsilon z \\right ) d y d z , \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{gather*} \\frac { \\ln { q } } { 1 - q } \\frac { \\big ( x t ^ N ; q \\big ) _ { \\infty } } { ( x q ; q ) _ { \\infty } } \\times \\Gamma _ q ( \\theta N ) \\times \\sum _ { n = 0 } ^ { \\infty } { x ^ n \\frac { \\operatorname { R e s } _ { z = n } { \\Gamma _ q ( - z ) } } { \\Gamma _ q ( \\theta N - n ) } } . \\end{gather*}"}
-{"id": "4254.png", "formula": "\\begin{align*} F ( f ) = F _ g ( f ) = \\mathbb E \\sum _ { k = 0 } ^ n \\langle f _ k , g _ k \\rangle . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} I ( f ) = \\int _ { [ 0 , 1 ] ^ d } \\left | \\psi _ { \\boldsymbol { a } , \\boldsymbol { c } } ( \\boldsymbol { x } ) \\frac { \\partial ^ d } { \\partial x _ 1 \\cdots \\partial x _ d } \\psi _ { \\boldsymbol { b } , \\boldsymbol { e } } ( \\boldsymbol { x } ) + \\psi _ { \\boldsymbol { b } , \\boldsymbol { e } } ( \\boldsymbol { x } ) \\frac { \\partial ^ d } { \\partial x _ 1 \\cdots \\partial x _ d } \\psi _ { \\boldsymbol { a } , \\boldsymbol { c } } ( \\boldsymbol { x } ) \\right | d \\boldsymbol { x } . \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} \\omega _ q ( n ) = \\mu ( \\omega _ q ) + F _ { \\omega _ q } ( n ) . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\widetilde { X } ^ N _ i ( t ) = \\xi _ i ^ N + \\int _ 0 ^ t \\int _ U f ( s , \\widetilde { X } ^ N _ i ( s ^ - ) , u , \\gamma ( s , \\widetilde { X } ^ N _ i ( s ^ - ) ) , \\widetilde { \\mu } _ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{gather*} \\Gamma _ q ( z + 1 ) = [ z ] _ q \\Gamma _ q ( z ) , z \\notin \\{ \\dots , - 2 , - 1 , 0 \\} , \\end{gather*}"}
-{"id": "267.png", "formula": "\\begin{align*} ( \\Delta \\otimes I ) \\Delta ( c ) & = ( I \\otimes \\Delta ) \\Delta ( c ) = c _ { ( 1 ) } \\otimes c _ { ( 2 ) } \\otimes c _ { ( 3 ) } , \\\\ c & = \\varepsilon ( c _ { ( 1 ) } ) c _ { ( 2 ) } = c _ { ( 1 ) } \\varepsilon ( c _ { ( 2 ) } ) , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\mathcal O _ { S , x } \\subset O _ { \\nu } \\quad \\quad \\mathfrak { m } _ { x } = \\mathfrak { m } _ { \\nu } \\cap \\mathcal O _ { S , x } , \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} X _ i = Q _ i ( D _ i ^ { ( k ) } + T _ i ) Q _ { i + 1 } ^ { - 1 } , \\quad ( i = 1 , \\ldots , m ) , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} ( \\mathrm { d } X ) = \\prod _ { 1 \\leq i < j \\leq N } | \\lambda _ j - \\lambda _ i | ( Q ^ T \\mathrm { d } Q ) , \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\frac { 2 ( k - 2 ) } { 3 } \\Biggl \\{ \\sum _ { i = 0 } ^ { k - 1 } a ^ i - k a ^ { k - 1 } \\Biggr \\} < \\sum _ { i = 0 } ^ { k - 2 } ( k - 1 - i ) a ^ { i } \\ , . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d w } { d t } & = - i E w \\\\ \\frac { d z _ 0 } { d t } & = i z _ 0 \\end{cases} \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\mathrm { E } N ^ { ( i ) } N ^ { ( j ) } = \\sum _ { p = 0 } ^ { d - i } \\sum _ { q = 0 } ^ { d - j } \\binom { p + q } { q } \\frac { ( - 1 ) ^ { p + q } } { ( 2 \\lambda ) ^ { p + q + 1 } } a _ { p + i , q + j } ( 0 ) ; \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} g ( \\sqrt { \\alpha _ i } ) = x _ { g , \\sigma _ g ( i ) } \\sqrt { \\alpha _ { \\sigma _ g ( i ) } } \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\frac { N _ { 1 0 } ^ { \\mathrm { W e y l } } ( X , S _ 5 ) } { N _ { 1 0 } ^ { \\mathrm { c m } } ( X , S _ 5 ) } = 1 + O _ { \\epsilon } ( X ^ { - \\frac { 3 } { 1 0 } + \\epsilon } ) . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} P q ( x , h D ) u & = q ( x , h D ) P u + [ P , q ( x , h D ) ] u \\\\ & = q ( x , h D ) P u + \\frac { h } { i } \\{ p , q \\} ( x , h D ) u + O _ { L ^ 2 } ( h ^ 2 ) . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\lambda = \\frac { ( T ) } { ( T _ 1 ) } . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} \\dim T ( \\varpi ) = \\dim V _ G ( \\varpi ) + 2 ^ n > \\dim V _ G ( \\varpi ) , \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} \\begin{aligned} f _ n ^ * { ( y ) } & = \\rho _ n h e ^ { \\tilde { u } _ n ^ { ( 1 ) } } ( \\zeta _ n { ( y ) } + O ( \\| \\tilde { u } _ n ^ { ( 1 ) } - \\tilde { u } _ n ^ { ( 2 ) } \\| _ { L ^ { \\infty } ( M ) } ) ) = \\rho _ n h e ^ { \\tilde { u } _ n ^ { ( 1 ) } { ( y ) } } ( - b _ 0 + o ( 1 ) ) , \\end{aligned} \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} D _ h ^ { \\frac { 1 } { 2 } } ( s ) : = \\frac { ( 2 \\pi ) ^ { s - \\frac { 1 } { 4 } } } { \\Gamma ( s - \\frac { 1 } { 4 } ) } \\langle P _ h ^ { \\frac { 1 } { 2 } } ( \\cdot , s ) , V \\rangle . \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} u _ k = \\mathbb { P } ( \\widehat { Q } _ 0 < \\widehat { Q } _ 1 < \\cdots < \\widehat { Q } _ k \\ , | \\ , \\widehat { Q } _ 0 = 1 ) . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } e ^ { x t } \\big ( e ^ t - 1 \\big ) ^ k = \\sum _ { n = k } ^ \\infty S _ 2 ( n , k | x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} f _ - ( x , v ) = \\delta ( x - x ' ) \\delta ( v - v ' ) , ( x ' , v ' ) \\in \\Gamma _ - \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} A ^ \\top M + M A + Q = 0 \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\big \\| \\ddot \\Psi ^ { t + s } ( x ) - \\ddot \\Psi ^ t ( x ) \\big \\| _ { C ^ { 0 } ( \\R ^ n ; \\R ^ n ) } = o ( 1 ) \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\mathbb E ( d _ { 2 n + 1 } | { \\mathcal G _ { 2 n } } ) = \\mathbb E ( \\Delta M _ { \\tau _ n ' } | \\mathcal F _ { \\tau _ n ' - } ) = 0 . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} \\frac { 1 } { r ( x ) } = \\frac { 1 } { p ( x ) } + \\frac { 1 } { q ( x ) } \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} d ( s _ n , s _ m ) = d ( s _ n - s _ m , 0 ) = d \\Big ( \\sum _ { k = m + 1 } ^ n k y _ { \\beta _ k } , 0 \\Big ) & \\leq \\sum _ { k = m + 1 } ^ n d \\big ( k y _ { \\beta _ k } , 0 \\big ) \\\\ & \\leq \\sum _ { k = m + 1 } ^ n \\frac { 1 } { 2 ^ k } \\to 0 , n , m \\to \\infty . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\overline { D } + \\widehat { ( s ^ { t } { s ' } ^ { ( 1 - t ) } ) } & = \\overline { D } + t \\widehat { ( s ) } + ( 1 - t ) \\widehat { ( s ' ) } \\\\ & = t ( \\overline { D } + \\widehat { ( s ) } ) + ( 1 - t ) ( \\overline { D } + \\widehat { ( s ' ) } ) \\geqslant 0 , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\delta _ { h _ 1 } \\tilde * \\delta _ { h _ 2 } : = \\frac { \\alpha _ 0 } { \\alpha _ 0 ( h _ 1 ) \\cdot \\alpha _ 0 ( h _ 2 ) } \\cdot ( \\delta _ { h _ 1 } * \\delta _ { h _ 2 } ) \\in M ^ 1 ( D ) \\quad ( h _ 1 , h _ 2 \\in D ) \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\gamma ( \\chi , s ) : = \\pi ^ { - d s / 2 } \\Gamma \\left ( \\frac { s } { 2 } \\right ) ^ { d - | \\frak { c } _ { \\infty } | } \\Gamma \\left ( \\frac { s + 1 } { 2 } \\right ) ^ { | \\frak { c } _ { \\infty } | } . \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\mathfrak { s w } ^ { n o r m } _ h ( M ) : = - \\frac { ( K + 2 r _ h ) ^ 2 + | \\mathcal { V } | } { 8 } - \\mathfrak { s w } _ { - h * \\sigma _ { c a n } } ( M ) \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { { \\bf { I I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } = 0 . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} W _ { n , 2 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ { S _ { n } \\left ( { - u , \\xi } \\right ) } \\right \\} \\left \\{ { e ^ { - u \\xi } + \\varepsilon _ { n , 2 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} A = \\langle e _ 1 \\rangle _ { \\overline { 0 } } \\oplus \\langle e _ 2 , e _ 3 \\rangle _ { \\overline { 1 } } . \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - \\nu \\cdot t } \\ , d _ { \\nu } \\left ( \\frac { \\sin ( r \\sqrt { \\nu } ) } { r } \\right ) ~ = ~ \\frac { \\sqrt \\pi } { 2 \\sqrt t } \\cdot \\exp \\left ( \\frac { - r ^ 2 } { 4 t } \\right ) \\ , , \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} n ^ { 2 n } 1 _ S * 1 _ S * 1 _ S ( f ) = n ^ { 2 n } \\sum _ { \\chi \\in \\hat { G } ^ n } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\delta _ t ^ 2 f ^ s : = 2 \\frac { \\delta _ t f ^ s - \\dot f ^ s } { t } \\mbox { a n d } \\ddot f ^ s : = \\lim _ { t \\downarrow 0 } \\delta _ t ^ 2 f ^ s = \\partial _ { t t } f ( y , 0 ) . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} G _ { \\lambda \\pm 0 i } ( x ) = \\lim _ { \\epsilon \\searrow 0 } G _ { \\lambda \\pm i \\epsilon } ( x ) = \\pm i e ^ { i \\lambda x } \\chi _ { \\real ^ { \\pm } } ( x ) - \\widetilde { G } _ { \\lambda } ( x ) , \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\{ \\Phi ^ A , \\Phi ^ B \\} = 0 . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\sum _ { k = 1 } ^ { K + 1 } \\zeta _ k \\int _ 0 ^ t 1 _ { \\left [ \\frac { q _ { k - 1 } ( s - ) } { \\overline \\lambda _ 0 } , \\frac { q _ { k } ( s - ) } { \\overline \\lambda _ 0 } \\right ) } ( \\xi _ { R _ 0 ( s - ) } ) d R _ 0 ( s ) \\\\ R _ 0 ( s ) & = Y \\big ( \\int _ 0 ^ t \\overline \\lambda _ 0 d s \\big ) , \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} f _ { \\theta , q , 3 } ( x ) = \\left ( \\frac { \\theta x + q - 1 } { x + \\theta + q - 2 } \\right ) ^ 3 \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} C D C D ^ { - 1 } = \\begin{pmatrix} x ^ 2 & x y ( z ^ 2 + 1 / x ^ 2 ) + z w ( 1 - x ^ 2 ) \\\\ 0 & 1 / x ^ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} D _ { G } ^ { \\textrm { c m } } ( s ) = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { 1 } { d _ F ^ { 2 s } } \\sum _ { [ E : F ] = 2 } \\frac { 1 } { \\mathcal { N } _ { F / \\Q } ( \\frak { D } _ { E / F } ) ^ s } = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { D _ { F , C _ 2 } ^ { - } ( s ) } { d _ F ^ { 2 s } } . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} F \\left ( \\frac 1 i \\partial _ x \\widetilde { G } _ { \\lambda } * ( u m _ 1 ( \\lambda + ) ) \\right ) & = \\xi \\frac { \\chi _ { \\real ^ - } } { \\xi - \\lambda } F ( u m _ 1 ( \\lambda + ) ) \\\\ & = \\chi _ { \\real ^ - } F ( u m _ 1 ( \\lambda + ) ) + \\lambda \\frac { \\chi _ { \\real ^ - } } { \\xi - \\lambda } F ( u m _ 1 ( \\lambda + ) ) \\\\ & = F [ C _ - ( u m _ 1 ( \\lambda + ) ) + \\lambda \\widetilde { G } _ { \\lambda } * ( u m _ 1 ( \\lambda + ) ) ] . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\lvert f ( x ) \\rvert ^ p \\ , \\mathrm { d } x & = \\int _ { \\mathbb { R } } \\big \\lvert F \\big ( \\Phi ( x ) \\big ) \\big \\rvert ^ p \\cdot \\lvert \\Phi ' ( x ) \\rvert \\ , \\mathrm { d } x \\\\ & = \\int _ { \\Gamma } \\lvert F ( \\zeta ) \\rvert ^ p \\lvert \\mathrm { d } \\zeta \\rvert \\leqslant \\lVert F \\rVert _ { H ^ p ( \\Omega _ + ) } ^ p < + \\infty , \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ p ^ p + C \\sum _ { i = 1 } ^ { N } \\| v _ i ^ { p / 2 } \\| _ { H ^ 1 ( \\Omega ) } ^ 2 \\leq C \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ { \\mu + p - 1 } ^ { \\mu + p - 1 } + C . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} ( \\mathbf { x } _ { n + 1 } ) = \\left ( \\left \\{ \\min _ { i = 1 , \\ldots , n } f ( \\mathbf { x } _ i ) - f ( \\mathbf { x } _ { n + 1 } ) \\right \\} ^ + \\mid \\mathbf { x } _ 1 , \\ldots , \\mathbf { x } _ n , f ( \\mathbf { x } _ 1 ) , \\ldots , f ( \\mathbf { x } _ n ) \\right ) , \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = \\Gamma _ \\C ( s + \\tfrac { k - 1 } 2 ) \\sum _ { n \\mid N ^ \\infty } \\lambda _ n n ^ { - s } \\prod _ { p \\nmid N } \\bigl ( 1 - \\lambda _ p p ^ { - s } + \\xi ( p ) p ^ { - 2 s } \\bigr ) ^ { - 1 } , \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} K ( u ( t ) ) & = - \\dfrac { 2 t u ' ( t ) + t ^ 2 u '' ( t ) } { ( u ' t ) ^ 3 } , \\\\ & = - \\dfrac { 1 + 2 t \\phi ' ( t ) + t ^ 2 \\phi '' ( t ) } { ( 1 + t \\phi ' ( t ) ) ^ 3 } , \\\\ & = - \\dfrac { g _ 1 ( t ) + t g _ 1 ' ( t ) } { g _ 1 ( t ) ^ 3 } , \\\\ & = - ( g _ 2 ( t ) - \\frac { t } { 2 } g _ 2 ' ( t ) ) , \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\bigl \\| ( g _ { k , m } ) _ { k = 1 , m = 0 } ^ { k = K , m = M } \\bigr \\| _ { Q ^ { p } _ { q } } = 1 \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} M _ t = \\sum _ { n = 0 } ^ N d _ { 2 n + 1 } \\mathbf 1 _ { [ 0 , t ] } ( \\tau _ n ' ) , \\ ; \\ ; \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\phi } } \\right ) \\nabla _ { \\phi } U = \\nabla _ { \\phi } \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\phi } } \\right ) U + U \\ast \\nabla _ { \\phi } ( { \\rm R m } _ { \\phi } + \\widetilde { \\eta } + \\nabla _ { \\phi } X ) + { \\rm R m } _ { \\phi } \\ast \\nabla _ { \\phi } U , \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} v ^ k _ { r s t } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\partial ^ 3 } { \\partial \\theta _ r \\partial \\theta _ s \\partial \\theta _ t } \\psi ^ k ( x _ i , \\theta ) \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} I : = \\varrho ^ 2 | \\nabla _ { \\phi } U | _ { \\omega _ { \\phi } } ^ 2 + E S + 2 | { \\rm R m } _ { \\phi } | _ { \\omega _ { \\phi } } ^ 2 , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\Pi = \\{ x = ( x _ { 1 } , x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ { 3 } \\ | \\ | x _ { h } | > 1 , \\ x _ { h } = ( x _ 1 , x _ 2 ) \\} , \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} - c _ 0 = \\left ( \\sum _ { l = 0 } ^ k a _ l z ^ l \\right ) \\sum _ { i = 1 } ^ n \\frac { 1 } { z - z _ i } \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\frac { 2 } { z _ i - z _ j } + \\left ( \\sum _ { l = 0 } ^ { k - 1 } b _ l z ^ l \\right ) \\sum _ { i = 1 } ^ n \\frac { 1 } { z - z _ i } + \\sum _ { l = 1 } ^ { k - 2 } c _ l z ^ l . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\frac { 1 } { \\lvert z \\rvert ^ { 2 w - k } z ^ k } = y ^ { 1 - 2 w } \\pi 4 ^ { 1 - w } \\frac { \\Gamma ( 2 w - 1 ) } { \\Gamma ( w + \\frac { k } { 2 } ) \\Gamma ( w - \\frac { k } { 2 } ) } . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} f ( t ) = a _ 0 + a _ 1 t + a _ 2 t ^ 2 + a _ 3 t ^ 3 + a _ 4 t ^ 4 + a _ 5 t ^ 5 . \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} \\mathbb { \\hat { E } } _ { 1 } [ \\varphi ( X _ { 1 } ) ] = \\mathbb { \\hat { E } } _ { 2 } [ \\varphi ( X _ { 2 } ) ] , \\ \\ \\ \\ \\ \\varphi \\in C _ { b . L i p } ( \\mathbb { R } ^ { d } ) . \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} v _ 3 = 0 , \\ , \\partial _ i v _ 3 = 0 \\ , \\ , \\partial ^ 3 _ t v _ 3 = 0 , \\\\ \\ , \\partial _ i \\eta _ 3 = 0 , \\ , \\ , a _ { 3 1 } = a _ { 3 2 } = 0 , \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} b _ l = \\chi _ { Q _ l } \\left ( f - m _ l ( f ) \\right ) . \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} R T ^ { d } = & \\sum _ { j = 0 } ^ { n - 1 } R ^ { j + 1 } \\sum _ { h = 0 } ^ { l - 1 } ( Q ^ d ) ^ { h + j + 1 } S ^ { h } . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{gather*} \\cos \\theta = \\frac { \\| \\mathbf z - \\mathbf w _ 0 \\| ^ 2 + \\| \\mathbf w _ 0 - \\mathbf o \\| ^ 2 - \\| \\mathbf z - \\mathbf o \\| ^ 2 } { 2 \\| \\mathbf z - \\mathbf w _ 0 \\| \\| \\mathbf w _ 0 - \\mathbf o \\| } \\\\ = \\frac { 1 + x ^ 2 - ( k + 1 ) / 2 k } { 2 x } = \\frac { k - 1 } { 4 k x } + \\frac { x } { 2 } = : g ( x ) . \\end{gather*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\big | \\mathcal { Z } _ \\iota \\big | \\leq N ^ { \\frac { \\varepsilon } { 3 } } \\hat { \\Pi } \\leq N ^ { \\frac { \\varepsilon } { 2 } } \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { ( N \\eta ) ^ { \\frac 1 3 } } , \\iota = 1 , 2 \\Theta \\Big ( z , { \\frac { N ^ { 3 \\varepsilon } } { ( N \\eta ) ^ { \\frac 1 3 } } } , { \\frac { N ^ { 3 \\varepsilon } } { \\sqrt { N \\eta } } } \\Big ) \\cap \\Omega _ 3 ( z ) , \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\omega ( x ) : = \\big ( N - ( N \\cdot \\nu ) \\nu \\big ) ( x ) \\mbox { f o r $ x $ o n $ \\Gamma ^ 0 $ } \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\tau ( n ) q ^ n = q \\prod _ { n \\geq 1 } ( 1 - q ^ n ) ^ { 2 4 } . \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} ( \\overline { H [ X \\odot Y ' ] } ) ^ * = ( \\overline { H [ X \\odot Y ' ] } ^ { \\wedge } ) ^ * = ( \\overline { H [ ( X \\odot Y ' ) ^ { \\wedge } ] } ) ^ * . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } u ^ { m } \\widetilde { \\pi } ( d u ) = \\frac { \\alpha } { m - \\alpha } L _ 1 ( t ) t ^ { m - \\alpha } , \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\frac { d u _ \\tau } { d \\tau } + g ^ { \\prime } ( u _ \\Omega ) \\frac { d u _ \\tau } { d \\tau } = 0 , & \\Omega , \\\\ \\frac { d u _ \\tau } { d \\tau } = - \\nabla u _ \\Omega \\cdot \\theta , & \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} u _ { i , j } = - \\sum ^ \\mu _ { s = 1 } \\alpha _ { i , s } \\frac { \\psi _ { 1 , j } ( z _ s ) } { \\Psi _ p ( z _ s ) } + \\big ( q , \\Theta [ \\xi _ { j + 1 } , \\ldots , \\xi _ { p + i } ] \\big ) , \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} c _ 4 = \\frac { 1 0 5 } { ( n - 1 ) ( n + 1 ) ( n + 3 ) ( n + 5 ) } . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} \\mu _ { \\mathrm { t o t } } ( g ' ) = 1 - \\sum _ { n = 1 } ^ { \\infty } ( 1 / 2 ) ^ n = 0 . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\gamma ( l , 1 ) \\ , = \\ , \\gamma ( 1 , l ) \\ , = \\ , 1 \\ \\ \\ , \\ \\ \\ \\forall l \\in \\Pi \\ . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} F [ F X , Z ] - F ^ 2 [ X , Z ] = 0 , \\ ; \\forall X \\in P , Z \\in Q . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} f _ r ( a ) = \\prod _ { p ^ \\alpha \\parallel r } f _ p ( a ) ^ \\alpha . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} u ^ { p - 1 } \\eta ( a \\nabla u , \\nabla \\eta ) = \\frac { 2 } { p } u ^ { p / 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla \\eta ) - \\frac { 2 } { p } u ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\psi ( u ) = \\sum _ { i = 1 } ^ m s g n ( g ( t _ i ( u ) ) ) \\frac { f ( t _ i ( u ) ) } { g ( t _ i ( u ) ) } \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} A ^ 2 & = \\frac { \\lambda _ 1 ^ 2 \\lambda _ 2 ^ 2 \\lambda _ 4 ^ 2 } { B ^ 2 D ^ 2 } \\\\ & > \\frac { \\lambda _ 1 ^ 2 \\lambda _ 2 ^ 2 \\lambda _ 4 ^ 2 } { \\lambda _ 2 ^ 2 K _ 1 ^ 2 } \\Bigg ( 1 + \\frac { 3 \\lambda _ 1 } { \\lambda _ 2 ^ 2 } t \\Bigg ) ^ { - 2 / 3 } \\\\ A & > \\frac { \\lambda _ 1 \\lambda _ 4 } { K _ 1 } \\Bigg ( 1 + \\frac { 3 \\lambda _ 1 } { \\lambda _ 2 ^ 2 } t \\Bigg ) ^ { - 1 / 3 } . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\varepsilon _ { k l m } & = a ^ k \\ , c ^ l \\ , ( c ^ * ) ^ m \\mathrm { i f ~ } k \\ge 0 , \\\\ \\varepsilon _ { k l m } & = ( a ^ * ) ^ { - k } \\ , c ^ l \\ , ( c ^ * ) ^ m \\mathrm { ~ i f ~ } k < 0 . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\sum ^ \\infty _ { j = 1 } \\abs { g _ j ( x _ 0 ) } ^ 2 \\leq B ( x _ 0 , x _ 0 ) m ( x _ 0 ) . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\textstyle \\sum _ { k \\in D } \\frac { \\omega _ k } { \\omega _ i \\omega _ j } p _ { i , j } ^ k = 1 \\quad i , j \\in D . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\Psi ( n ) = \\min _ { \\substack { k _ 1 , \\ldots , k _ n \\in \\mathbb { Z } \\\\ k _ n \\neq 0 } } \\left ( \\deg \\left ( \\left ( f ^ { ( 1 ) } \\right ) ^ { k _ 1 } \\ldots \\left ( f ^ { ( n ) } \\right ) ^ { k _ n } \\right ) \\right ) . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} { q } _ { i } ( x ) = \\left \\{ \\begin{array} { l } { q } ^ { T } _ { i } ~ ~ ~ { \\rm i f } \\ \\ T B V _ { i } ^ { T } < T B V _ { i } ^ { W } , \\\\ { q } ^ { W } _ { i } ~ ~ ~ ~ \\mathrm { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} \\sum _ { p ^ { k _ p } \\parallel \\phi ( n ) } k _ p \\log p = \\log \\phi ( n ) \\le \\log x . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} R _ \\mu ( P _ { X Y } ) & \\le I ( \\tilde { W } \\wedge \\tilde { X } ) + \\mu H ( \\tilde { Y } | \\tilde { W } ) \\\\ & = I ( \\tilde { W } \\wedge \\tilde { X } , \\tilde { Y } ) + \\mu H ( \\tilde { Y } | \\tilde { W } ) \\\\ & \\le I ( W \\wedge X , Y ) + \\mu H ( Y | W ) + \\delta ^ \\prime \\\\ & = R _ \\mu ( \\delta | P _ { X Y } ) + \\delta ^ \\prime , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} | \\mathcal { E } | ^ 2 \\le M \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) , ( M > 0 ) \\ , . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a ^ { i j } ( x ) D _ { i j } u = f \\ ; \\mbox { i n } \\ ; B ( 0 , 1 ) , u = 0 \\ ; \\mbox { o n } \\ ; \\partial B ( 0 , 1 ) , \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} x - \\mathcal { K } ( x ) = f \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} v _ { 2 } ^ { { } } ( y ) = \\bigtriangleup _ { y } \\rho ( y ) = \\nabla _ { y } \\cdot \\left ( \\nabla _ { y } \\rho ( y ) \\right ) \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { 1 } { n ^ { s + k - 1 } } \\Bigg ( \\sum _ { h = m > 0 } + \\sum _ { h > m \\geq 0 } + \\sum _ { m > h \\geq 0 } \\Bigg ) a ( n - m ) \\overline { b ( n - h ) } . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\begin{aligned} \\{ { \\eta } ( { \\xi } _ { 1 } ) + { \\eta } ( { \\xi } _ { 2 } ) + \\cdots + { \\eta } ( { \\xi } _ { k } ) \\} & = \\{ \\{ \\xi _ { 1 } - p _ { 1 } \\eta \\} + \\{ \\xi _ { 2 } - p _ { 2 } \\eta \\} + \\cdots + \\{ \\xi _ { k } - p _ { k } \\eta \\} \\} \\\\ & = \\{ \\xi _ { 1 } + \\xi _ { 2 } + \\cdots + \\xi _ { k } - ( p _ { 1 } + p _ { 2 } + \\dots + p _ { k } ) \\eta \\} \\\\ & = \\{ \\xi _ { 1 } + \\xi _ { 2 } + \\cdots + \\xi _ { k } \\} , \\end{aligned} \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} u _ n : = \\rho P _ { \\cdot } f _ n \\rightarrow u \\ ; C ^ { \\gamma ; \\frac { \\gamma } { 2 } } ( \\overline { U } \\times [ \\tau _ 1 , \\tau _ 2 ] ) . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} \\P _ x \\Big ( \\lim _ { n \\rightarrow \\infty } \\sigma _ { n } \\ge \\zeta \\Big ) = 1 . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\sup _ { r > 1 } \\frac { 1 } { | B | } \\int _ { B } | f ( x ) - f _ { B } | d x \\leq \\sup _ { r > 1 } C 2 ^ { - k _ { 0 } } \\sum ^ { k _ { 0 } + 1 } _ { k = 0 } \\int _ { A _ { k } } 2 ^ { k } d x \\leq C . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\delta _ m * \\delta _ n = \\sum _ { k = | m - n | } ^ { m + n } g _ { m , n , k } \\delta _ k \\in M ^ 1 ( \\mathbb N _ 0 ) \\quad ( m , n \\in \\mathbb N _ 0 ) \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} L _ { 0 } = - \\frac { d ^ { 2 } } { d y ^ { 2 } } - K _ { 2 } \\left ( y \\right ) : H ^ { 2 } \\left ( y _ { 1 } , y _ { 2 } \\right ) \\rightarrow L ^ { 2 } \\left ( y _ { 1 } , y _ { 2 } \\right ) , \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\bar { I } & = \\int ( h ( x , t ) - h ( x + y , t ) ) w ( x + y , t ) \\overline { K } ( x , y ) d y \\\\ & = e ^ { - | \\lambda _ 1 | t } \\int ( | x | ^ { \\xi ( d + \\alpha ) } - | x + y | ^ { \\xi ( d + \\alpha ) } ) w ( x + y , t ) \\overline { K } ( x , y ) d y . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\omega \\left ( t \\right ) x = 0 ~ , ~ \\omega \\left ( t \\right ) \\neq \\frac { 1 } { 4 t ^ { 2 } } . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} \\gamma = e ^ { t L _ 0 } u _ 0 ^ { \\theta } - \\int _ { 0 } ^ { t } e ^ { ( t - s ) L _ 0 } \\Big ( v \\cdot \\nabla \\gamma + \\frac { u ^ { r } } { r } \\gamma \\Big ) \\dd s . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\sum _ { A _ 1 ' , \\ldots , A _ k ' } \\left ( g _ { A _ 1 ' \\ldots A _ k ' } , G _ { A _ 1 ' \\ldots A _ k ' } \\right ) _ { \\varphi } = \\sum _ { A _ 1 ' , \\ldots , A _ k ' } \\left ( g _ { A _ 1 ' \\ldots A _ k ' } , G _ { ( A _ 1 ' \\ldots A _ k ' ) } \\right ) _ { \\varphi } \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} p ( x , D ) u = \\frac 1 { ( 2 \\pi ) ^ { \\frac d 2 } } \\int _ { \\R ^ d } p ( x , \\xi ) \\hat f ( \\xi ) e ^ { i x \\cdot \\xi } \\ , d \\xi , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} d \\widehat f ! ( h ; \\widehat { \\frak S } ^ { \\epsilon } ) = \\widehat f ! ( d h ; \\widehat { \\frak S } ^ { \\epsilon } ) + \\widehat f ! ( h \\vert _ { \\partial _ { \\frak C ^ h } ( X , \\widehat { \\mathcal U } ) } ; \\widehat { \\frak S } ^ { \\epsilon } \\vert _ { \\partial _ { \\frak C ^ h } ( X , \\widehat { \\mathcal U } ) } ) . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} M = H ^ 0 \\bigsqcup 2 g H ^ 1 \\bigsqcup H ^ 2 \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{gather*} \\frac { \\Gamma _ q ( \\lambda _ N - z ) } { \\Gamma _ q ( \\lambda _ 1 + \\theta N - z ) } = ( 1 - q ) ^ { \\lambda _ 1 + \\theta N - \\lambda _ N } \\frac { \\big ( q ^ { \\lambda _ 1 + \\theta N - z } ; q \\big ) _ { \\infty } } { \\big ( q ^ { \\lambda _ N - z } ; q \\big ) _ { \\infty } } . \\end{gather*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi _ 1 = 1 , \\Pi _ 2 = 2 ) = 1 - q . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\mathcal { I } _ o = \\left \\{ \\begin{aligned} a \\varphi , & \\mathfrak { b } _ + \\\\ - \\frac { 1 } { a } \\varphi , & \\mathfrak { b } _ - \\end{aligned} \\right . , \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} G \\left ( { u , \\xi } \\right ) \\sim \\frac { 1 } { u ^ { 2 } } \\sum \\limits _ { s = 0 } ^ { \\infty } { \\frac { G _ { s } \\left ( \\xi \\right ) } { u ^ { 2 s } } , } \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} D _ { \\tau } f ( t ) = \\tau ^ { - 1 } ( f ( t ) - f ( t - \\tau ) ) , \\quad \\tau \\leq t \\leq T , \\quad \\mbox { f o r a n y } f : [ 0 , T ] \\to \\R . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} P _ n : = \\{ \\zeta \\in P : u _ n ( \\zeta ) \\le n \\} . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} & F _ T ( x ) - 1 - x F _ T ( x ) \\\\ & = x ^ 2 F _ T ( x ) C ( x ) ^ 2 + \\frac { x ^ 3 C ( x ) ^ 3 ( F _ T ( x ) - 1 ) } { 1 - 2 x } + \\frac { x ^ 3 C ( x ) ^ 3 } { 1 - 2 x } - \\frac { x ^ 3 C ( x ) ^ 3 } { ( 1 - x ) ^ 2 } + \\frac { x ^ 3 C ( x ) ^ 4 } { 1 - x } \\ , . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\Gamma _ { d } = \\left \\{ x \\in \\Gamma : \\xi \\left ( x \\right ) > d \\right \\} . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} T _ K f ( x H , H h H ) & = \\int _ G f ( y H ) \\ > d ( \\delta _ x * \\omega _ H * \\delta _ h * \\omega _ H ) ( y ) \\\\ & = \\int _ G \\int _ G f ( x z _ 1 h z _ 2 H ) \\ > d \\omega _ H ( z _ 1 ) \\ > d \\omega _ H ( z _ 2 ) \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\frac { 1 } { I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) } \\otimes W ' - \\overline { \\frac { 1 } { I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) } \\otimes W ' } ^ { t } = 2 \\sqrt { - 1 } \\left ( V \\otimes \\left ( Z ' - A \\otimes W \\right ) \\right ) \\overline { V \\otimes \\left ( Z ' - A \\otimes W \\right ) } ^ { t } , \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\Vert \\Phi _ T \\vert _ { \\{ L \\} \\times E ^ s _ \\sigma } \\Vert = \\Vert \\Phi ^ * _ T \\vert _ { \\{ L \\} \\times E ^ s _ \\sigma } \\Vert = \\Vert X _ T \\vert _ { E ^ s _ \\sigma } \\Vert . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} X = D \\left ( t \\right ) \\partial _ { t } + T \\left ( t \\right ) V ^ { , i } \\partial _ { i } , \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\mathbf { 1 } \\Big ( \\Lambda \\leq \\frac { | \\mathcal { S } | } { K _ 0 } \\Big ) | \\Lambda _ \\iota | \\leq N ^ { - \\varepsilon } | \\mathcal { S } | , \\iota = A , B \\ , , \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} J _ { q , a } = \\left ( { a } / { q } - { L ^ { B ^ 2 } } / { X } , { a } / { q } + { L ^ { B ^ 2 } } / { X } \\right ] , \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} r _ { i j } ^ 2 = s _ { i , j } ^ 2 / ( ( z _ i ' z _ i ) ( z _ j ' z _ j ) ) = \\frac { s _ { i , j } ^ 2 } { ( n - 1 ) ^ 2 } ( 1 + o _ p ( 1 ) ) , \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\{ f \\times f ' \\} ( l ) \\ , : = \\ , \\int _ \\Pi f ( m ) \\ , \\bar \\nu _ m ( f ' ( m ^ { - 1 } l ) ) \\ , \\bar \\gamma ( m , m ^ { - 1 } l ) \\ , d m \\ , \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} L ( a ( 1 + q ) , b ( 1 + q ) ^ 2 ) & = \\frac { 1 - \\sqrt { ( 1 - 2 a - 2 a q ) ^ 2 - 4 b ( 1 + q ) ^ 2 } \\ , } { 2 } \\\\ & = \\frac { 1 - \\sqrt { ( 1 - 2 a ) ^ 2 - 4 b - 2 q ( 2 a ( 1 - 2 a ) + 4 b ) + q ^ 2 ( 4 a ^ 2 - 4 b ) } \\ , } 2 . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\aligned & \\widehat { \\frak S ^ + } ( { \\rm m o r } ; 3 , 1 ; \\alpha _ 1 , \\alpha _ 3 ) \\\\ = & \\bigcup _ { \\alpha _ 2 \\in \\frak A _ 2 } \\widehat { \\frak S ^ + } ( { \\rm m o r } ; 2 , 1 ; \\alpha _ 1 , \\alpha _ 2 ) \\times _ { R _ { \\alpha _ 2 } } \\widehat { \\frak S ^ + } ( { \\rm m o r } ; 3 , 2 ; \\alpha _ 2 , \\alpha _ 3 ) . \\endaligned \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} Q \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) = a . \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\frac { z \\mathtt { g } _ { a , \\nu } ' ( z ) } { \\mathtt { g } _ { a , \\nu } ( z ) } & = a ( 1 - \\nu ) + a ^ { a / 2 } \\left ( \\frac { z \\mathtt { J } ' _ { \\nu } \\left ( z \\right ) } { \\mathtt { J } _ { \\nu } \\left ( z \\right ) } - ( \\nu - 1 ) ( 1 - a ) \\right ) \\\\ & = a ( 1 - \\nu ) + a ^ { a / 2 } \\left ( a \\nu - a + 1 - \\sum _ { n = 1 } ^ \\infty \\frac { 2 z ^ 2 } { \\mathtt { j } _ { \\nu , n } ^ 2 - z ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\bar { S } = \\left \\{ \\left ( \\bar { q } ^ i , \\frac { \\partial W } { \\partial \\bar { q } ^ i } ( \\bar { q } , q ) \\right ) \\in T ^ * \\bar { Q } : \\frac { \\partial W } { \\partial { q } ^ j } ( \\bar { q } , q ) = 0 \\right \\} . \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\beta ^ * ( E ^ s _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } = N ^ { s } _ { B ^ u _ \\sigma ( \\Lambda ) } \\subset \\oplus ( N ^ { u } _ { B ^ u _ \\sigma ( \\Lambda ) } \\cap \\beta ^ * ( E ^ s _ \\sigma ) \\vert _ { B ^ u _ \\sigma ( \\Lambda ) } ) \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} Z ( G _ n ) = \\langle x _ 1 ^ { p ^ 2 } , x _ 2 ^ { p ^ 2 } , x _ 3 ^ { p ^ 2 } \\rangle , \\ ; \\Phi ( G _ n ) = \\langle x _ 1 ^ p , x _ 2 ^ p , x _ 3 ^ p , x _ 4 ^ p \\rangle , \\ ; G _ n ' = \\langle x _ 1 ^ { p ^ { n - 2 } } , x _ 2 ^ { p ^ 2 } , x _ 3 ^ { p ^ 2 } \\rangle . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} P ( \\ | \\ \\mathcal { F } _ t ) & = \\lambda _ k ( X ( t ) ) h + o ( h ) \\\\ P ( \\ | \\ \\mathcal { F } _ t ) & = o ( h ) \\\\ P ( \\ | \\ \\mathcal { F } _ t ) & = 1 - \\sum _ { k } \\lambda _ k ( X ( t ) ) h + o ( h ) , \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} L _ 1 u ^ 2 = \\langle A \\nabla u , \\nabla u \\rangle + 2 u L _ 1 u . \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} \\sum _ { p \\leq t } \\omega _ q ( p ) \\omega _ 0 ( p ) = \\sum _ { \\substack { p \\leq t \\\\ p \\equiv 1 \\mod q } } \\omega ( p - 1 ) = \\frac { t \\log \\log t } { \\phi ( q ) \\log t } + O \\bigg ( \\frac { t } { \\log t } \\bigg ) \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\delta _ m * \\delta _ n = \\sum _ { k = | m - n | } ^ { m + n } g _ { m , n , k } \\delta _ k \\in M ^ 1 ( \\mathbb N _ 0 ) \\quad ( m , n \\in \\mathbb N _ 0 ) \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\nabla ^ 2 u ( F , \\nabla u ) & = u _ { i j } F _ i u _ j = ( u _ j F _ i u _ j ) _ i - u _ j F _ { i , i } u _ j - u _ j F _ i u _ { j i } \\\\ & = d i v ( | \\nabla u | ^ 2 F ) - | \\nabla u | ^ 2 \\cdot d i v F - \\nabla ^ 2 u ( F , \\nabla u ) , \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} c = \\frac { e ^ 2 - 2 \\ , \\zeta - 1 } { 3 \\ , e } , d = \\frac 1 9 ( e ^ 2 - 2 0 \\ , \\zeta - 1 0 ) . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\frac { F ( b ) - F ( y ) } { b - y } - \\frac { F ( y ) - F ( a ) } { y - a } = F ' ( b ) - F ' ( y ) \\ , , \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} A _ { \\lambda } ( x ) = e ^ { - \\lambda } A ( x ) , x \\in M . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\partial _ N v ^ t = - \\frac { \\partial _ N \\tilde u ^ 0 } { t } = - \\frac { \\sigma } { t } = - \\frac { \\lambda ^ t } { t } \\circ \\bar \\pi _ 1 \\mbox { o n } \\Gamma ^ t \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\kappa ( L \\widetilde { V } _ { n - k } ) \\geq \\kappa ( L \\widetilde { V } _ { n - ( k + 1 ) } ) , k = 1 , 2 , \\ldots , n - 1 , \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} \\begin{array} { r c l } M _ { 0 } & = & \\left ( \\begin{array} { c c c c } \\rho _ { * } & 0 & 0 & \\quad 0 \\\\ 0 & C ^ { - 1 } & C ^ { - 1 } e & \\quad 0 \\\\ 0 & e ^ { * } C ^ { - 1 } & \\epsilon + e ^ { * } C ^ { - 1 } e & \\quad 0 \\\\ 0 & 0 & 0 & \\quad \\mu \\end{array} \\right ) , \\\\ M _ { 1 } & = & \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & \\sigma & 0 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right ) . \\end{array} \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\nu ^ { f , g } : = \\sum _ { k \\geq 0 } k \\pi _ k ^ { f , g } , \\mbox { a n d w e h a v e } \\nu ^ { f , g } = \\frac { \\mu _ g } { 1 - \\mu _ f } ; \\vadjust { \\eject } \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\left \\{ e _ { \\mu _ { \\beta } } ( \\cdot , \\varphi ) : = \\frac { e ^ { \\langle \\cdot , \\varphi \\rangle } } { \\mathbb { E } \\big ( e ^ { \\langle \\cdot , \\varphi \\rangle } \\big ) } , \\ ; \\varphi \\in S _ { d , \\mathbb { C } } , \\ ; | \\varphi | _ { p } < 2 ^ { - q } \\right \\} \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} x ^ 2 \\beta { x ^ { - 2 } } \\ ! . x ^ 2 y ^ { - 1 } \\beta ^ { - 1 } y x ^ { - 2 } = \\gamma \\delta . x \\alpha { x ^ { - 1 } } . x y ^ { - 1 } [ x , y ] ^ { - 1 } y x ^ { - 1 } ~ \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} L _ 3 = \\left ( \\begin{array} { c } L _ 1 \\\\ L _ 2 \\\\ \\end{array} \\right ) \\in \\mathbb { R } ^ { ( 2 n - 3 ) \\times n } , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} & D ( i , j ) : = \\left \\{ \\begin{array} { l l } 1 , & w h e n ~ C _ 1 ( i , j ) \\neq C _ 2 ( i , j ) , \\\\ \\\\ 0 , & w h e n ~ C _ 1 ( i , j ) = C _ 2 ( i , j ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} f ( \\{ a , b \\} ) & = g ( a ) f ( b ) - ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\\\ g ( a b ) & = f ( a ) g ( b ) + ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} & \\sum _ { n _ 1 + n _ 2 = 2 g + 1 - D } { 2 g + 1 - D \\choose n _ 1 } \\frac { ( 2 g - 3 + n + n _ 1 ) ! } { ( 2 g - 3 + n ) ! } \\cdot n _ 2 ! \\cdot { n _ 2 \\choose 2 c _ i + 2 - e _ i - e _ { n + 1 } } \\\\ & = \\left ( 2 g + 2 - \\sum b _ j - 1 \\right ) ! \\cdot { 4 g - 1 + n - \\sum b _ j \\choose 2 g - 1 - \\sum b _ j - 2 c _ i + e _ i + e _ { n + 1 } } , \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} \\partial _ N v ^ t _ 1 ( z , \\lambda ^ t ( z ) ) = \\int _ 0 ^ 1 \\partial _ N V ^ \\theta ( z , \\lambda ^ t ( z ) ) d \\theta . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} 2 ^ { - n p / 2 } \\prod _ { i = 1 } ^ { p } \\frac { \\Gamma ( 3 / 2 ) } { \\Gamma ( ( 1 + i / 2 ) \\Gamma ( ( n - p + i ) / 2 ) } l ^ { \\frac { n - p + i } { 2 } - 1 } _ { i } \\mbox { e } ^ { - { l _ { i } } / 2 } \\prod _ { i < j } | l _ i - l _ j | . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} b _ n \\ ( u , \\tau \\ ) = \\sum _ { T _ \\tau ^ n x = x } \\frac { \\exp \\ ( - u \\sum _ { k = 0 } ^ { n - 1 } g \\ ( T _ \\tau ^ k x \\ ) \\ ) } { \\left | \\ ( T _ \\tau ^ n \\ ) ' \\ ( x \\ ) - 1 \\right | } = \\sum _ { T _ \\tau ^ n x = x } \\frac { \\exp \\ ( - u \\sum _ { k = 0 } ^ { n - 1 } g \\ ( T _ \\tau ^ k x \\ ) \\ ) } { \\ ( T _ \\tau ^ n \\ ) ' \\ ( x \\ ) - 1 } , \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d v } { d D } & + ( - D ^ { - 1 } ) v = \\ell D ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} d = p ^ t + p + 1 , \\ \\ \\ t = \\left \\{ \\begin{array} { c l } \\frac { n - 1 } { 2 } & ( \\mbox { $ n $ i s o d d } ) , \\\\ \\frac { n } { 2 } & ( \\mbox { $ n $ i s e v e n } ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\begin{array} { c c c } A = \\left [ \\begin{matrix} 0 . 2 & \\ \\ & 0 \\\\ 0 . 2 & & 0 . 1 \\end{matrix} \\right ] & & B = \\left [ \\begin{matrix} - 1 & \\ \\ & 0 \\\\ - 1 & & - 1 \\end{matrix} \\right ] \\end{array} . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} f ( 1 : x _ 1 : \\cdots : x _ n ) = ( 1 : f _ 1 ( x _ 1 , \\ldots , x _ n ) : \\cdots : f _ n ( x _ 1 , \\ldots , x _ n ) ) . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\langle A x , x \\rangle = \\| R _ { A } x \\| ^ { 2 } = \\| W ^ { \\ast } R _ { D } x \\| ^ { 2 } \\leq \\alpha ^ { 2 } \\| R _ { D } x \\| ^ { 2 } = \\alpha ^ { 2 } \\langle D x , x \\rangle , x \\in X , \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\langle W ( \\ell _ 1 ) ^ { 1 / 2 } \\ell _ { 1 } | \\Delta _ n W ( \\ell _ 2 ) ^ { 1 / 2 } \\ell _ { 2 } \\rangle _ { n } & = W ( \\ell _ 1 ) ^ { 1 / 2 } W ( \\ell _ 2 ) ^ { 1 / 2 } { } _ n \\langle \\ell _ { 2 } , \\ell _ { 1 } \\rangle \\\\ & = \\delta _ { \\ell _ 1 = \\ell _ 2 } W ( \\ell _ 1 ) ^ { 1 / 2 } W ( \\ell _ 2 ) ^ { 1 / 2 } W ( \\ell _ 1 ) ^ { 1 / 2 } W ( \\overline { \\ell _ { 2 } } ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\delta _ { \\ell _ 1 = \\ell _ 2 } W ( \\ell _ 1 ) . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} L _ { ( d _ 1 , d _ 2 ) } = j ^ * ( L _ { d _ 1 } \\boxtimes \\Q _ \\ell ) \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\sigma & = \\frac { 1 + 2 \\mu - \\mu ^ 2 } { 2 } , \\lambda ' = \\sigma \\log \\varrho , H = \\frac { h ' } { \\lambda ' } + \\frac { 1 } { \\sigma } , \\\\ \\omega & = 2 \\left ( 1 + \\sqrt { 1 + \\frac { 1 } { 4 H ^ 2 } } \\right ) , \\theta = \\sqrt { 1 + \\frac { 1 } { 4 H ^ 2 } } + \\frac { 1 } { 2 H } . \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} F _ x ( t , w ) : = \\min _ { a \\in A } \\left \\{ \\int _ U [ w _ { x + f ( t , x , u , a , m ( t ) ) ) } - w _ x ] \\nu ( d u ) + c ( t , x , a , m ( t ) ) \\right \\} , \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} R _ { 2 } ( t ) = C t ^ { \\nu / \\mu } ( 1 + ( \\frac { 1 } { 2 } - \\frac { \\nu } { \\mu } ) \\ln \\frac { 1 } { t } ) = C t ^ { ( 2 - \\gamma ) / 2 } ( 1 + ( \\frac { 1 } { 2 } - \\frac { 2 - \\gamma } { 2 } ) \\ln \\frac { 1 } { t } ) \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\mathcal { A } = f ( \\mathbf { x } ) ^ * b ( \\mathbf { D } ) ^ * g ( \\mathbf { x } ) b ( \\mathbf { D } ) f ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 1 & x _ 2 & \\cdots & x _ { n - 1 } & x _ n \\\\ y _ 1 & y _ 2 & \\cdots & y _ { n - 1 } & y _ n \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} R _ 1 & = \\lfloor c _ 1 a _ 2 a _ 3 \\rfloor , S _ 1 = \\lfloor c _ 1 a _ 1 a _ 3 \\rfloor , T _ 1 = \\lfloor c _ 1 a _ 1 a _ 2 \\rfloor , \\\\ R _ 2 & = \\lfloor c _ 2 a _ 2 a _ 3 \\rfloor , S _ 2 = \\lfloor c _ 2 a _ 1 a _ 3 \\rfloor , T _ 2 = \\lfloor c _ 2 a _ 1 a _ 2 \\rfloor , \\\\ R _ 3 & = \\lfloor c _ 3 a _ 2 a _ 3 \\rfloor , S _ 3 = \\lfloor c _ 3 a _ 1 a _ 3 \\rfloor , T _ 3 = \\lfloor c _ 3 a _ 1 a _ 2 \\rfloor . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} ( - \\Delta _ H ) ^ { m _ { 1 } } u + ( - \\Delta _ H ) ^ { m _ { 2 } } u = | u | ^ { q - 2 } u . \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\frac { E ( \\Delta _ n ^ 2 ) } { \\tau _ { n p _ n } ^ 4 } = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\to 0 ~ \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} a _ 1 b _ 1 - a _ 2 - b _ 2 & = a _ 1 c _ 1 - a _ 2 - c _ 2 \\\\ a _ 1 ( b _ 1 - c _ 1 ) & = b _ 2 - c _ 2 . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\delta ( s ^ { - 1 } , y ) = ( x , t ^ { - 1 } ) \\mbox { i f $ \\lambda ( s , x ) = ( y , t ) $ } , \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} [ \\psi _ i ^ \\epsilon ( z ) , \\psi _ j ^ { \\epsilon ' } ( w ) ] = 0 , \\ \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } \\cdot ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } = ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } \\cdot \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } = 1 , \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } \\bigl ( \\frac { z + z ^ { - 1 } } { 2 } \\bigr ) = \\frac { c ( z ) z ^ n + c ( z ^ { - 1 } ) z ^ { - n } } { ( ( a - 1 ) ( b - 1 ) ) ^ { n / 2 } } \\quad \\quad \\ > \\ > z \\in \\mathbb C \\setminus \\{ 0 , \\pm 1 \\} \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\delta ( T ) _ i : = \\begin{cases} i + k - 1 & i \\in T \\\\ j - 1 & i \\notin T i = s _ j , \\end{cases} \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\mathfrak { c } _ { \\epsilon } = \\frac { 1 } { 4 } \\sum _ { 0 < | \\omega | \\leq \\epsilon ^ { - 1 } } \\frac { 1 } { \\pi ^ 2 | \\omega | ^ 2 } \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\ , \\P \\ , \\bigg ( \\sup _ { \\substack { t _ 1 - t _ 2 < \\delta \\\\ 0 \\leq t _ 2 \\leq t _ 1 \\leq T } } \\big \\| u ^ n _ { t _ 1 } - u ^ n _ { t _ 2 } \\big \\| \\ , > \\epsilon \\bigg ) = 0 . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} & T ^ * _ { G } X ^ { s } = G \\times _ { P } ( \\mu _ { v _ 1 , w } ^ { - 1 } ( 0 ) ^ { s s } \\times \\mu ^ { - 1 } _ { v _ 2 } ( 0 ) ) , \\ , \\ \\ , \\ Z _ { G } ^ s = G \\times _ P \\{ ( x , x ^ * , i , j ) \\in \\mu ^ { - 1 } _ { v , w } ( 0 ) ^ { s s } \\mid ( x , x ^ * ) ( V _ 1 ) \\subset V _ 1 , ( i ) \\subset V _ 1 \\} . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\mathfrak { A } _ \\theta ( u , u ) = \\mathfrak { A } _ \\theta ( v , v ) u , v \\in U _ \\omega . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} \\left \\Vert V ^ { \\left ( 1 \\right ) } - V ^ { \\left ( 2 \\right ) } \\right \\Vert _ { H ^ { 1 } \\left ( \\Omega _ { d + c } \\right ) } \\leq C _ { 2 } \\left ( 1 + \\left \\Vert V ^ { \\left ( 1 \\right ) } - V ^ { \\left ( 2 \\right ) } \\right \\Vert _ { H ^ { 2 } \\left ( \\Omega \\right ) } \\right ) \\sigma ^ { \\rho } , \\rho = c / \\left ( m + c \\right ) . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} A _ { 0 } \\left ( x , V \\right ) - P \\left ( x , \\nabla V \\right ) = 0 , \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} [ Q _ \\infty ^ { 1 / 2 } Q _ t ^ { - 1 / 2 } ] ^ { * H } y = Q _ \\infty [ Q _ \\infty ^ { 1 / 2 } Q _ t ^ { - 1 / 2 } ] ^ * Q _ \\infty ^ { - 1 } y = Q _ \\infty Q _ t ^ { - 1 / 2 } Q _ \\infty ^ { - 1 / 2 } y \\forall y \\in R ( Q _ \\infty ) . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\int _ { M } u ( x ) \\cdot \\partial ^ i h ( x ) \\ , d \\mu ( x ) ~ = \\int _ { M } u ( x ) \\int _ M h ( y ) \\cdot \\partial ^ i _ x F _ x ( y ) \\ , d \\mu ( y ) \\ , d \\mu ( x ) \\ , . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\left ( \\tilde { G } , \\tilde { F } \\right ) = T _ { 2 } \\circ \\left ( G , F \\right ) , \\mbox { f o r $ T _ { 2 } = T _ { 2 } \\left ( P \\right ) $ . } \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} I _ K ( x ) = c x + O ( x ^ { 1 - 3 / ( n + 2 ) } ) . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} [ M , N ] _ t & = \\mathbb P - \\lim \\sum _ { i = 1 } ^ k \\langle M _ { t _ i } - M _ { t _ { i - 1 } } , N _ { t _ i } - N _ { t _ { i - 1 } } \\rangle , \\ ; \\ ; \\ ; t \\geq 0 , \\\\ [ M ] _ t & = [ M , M ] _ t \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { ( 1 - 2 x ) ( 1 - 6 x + 1 2 x ^ 2 - 1 0 x ^ 3 + 2 x ^ 4 ) - x ^ 2 ( 1 - 2 x + 2 x ^ 2 ) ^ 2 C ( x ) } { 1 - 9 x + 3 0 x ^ 2 - 4 9 x ^ 3 + 3 8 x ^ 4 - 8 x ^ 5 - 4 x ^ 6 } \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { u } _ n ^ { ( 1 ) } ( x ) - \\tilde { u } _ n ^ { ( 2 ) } ( x ) = - ( \\lambda _ { n , j } ^ { ( 1 ) } - \\lambda _ { n , j } ^ { ( 2 ) } ) + O \\left ( \\sum _ { i = 1 } ^ 2 e ^ { - \\frac { \\lambda _ { n , j } ^ { ( i ) } } { 2 } } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} 0 \\le f \\le M : = \\sup _ D f - \\inf _ D f . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} 0 = \\int _ 0 ^ T ( L ^ * _ a { W } ^ { \\pm } _ { \\eta } , U ) d t = \\int _ 0 ^ T ( { W } ^ { \\pm } _ { \\eta } , L _ a U ) d t \\\\ + ( W ^ { \\pm } _ { \\eta } ( T ) , U ( T ) ) - ( W ^ { \\pm } _ { \\eta } ( 0 ) , U ( 0 ) ) \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\varphi \\xi = 0 , \\ \\ \\eta o \\varphi = 0 \\ \\ \\ \\ \\ g _ { M } ( \\varphi X , \\varphi Y ) = g _ { M } ( X , Y ) - \\eta ( X ) \\eta ( Y ) , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} & Z ^ 1 _ i : = \\Omega ( K _ i , r ^ i ) \\times 2 5 5 , \\\\ & Z ^ 2 _ i : = c i r c s h i f t ( Z ^ 1 _ i , t _ i ) . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\widehat { \\phi _ { Y } } ( s ) : = \\int _ { 0 } ^ { \\infty } \\phi _ Y ( t ) t ^ { s - 1 } d t , \\textrm { R e } ( s ) > 0 \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} W = f ^ { 1 / 4 } \\left ( z \\right ) w \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} Z ( \\mu , \\nu ) = \\{ ( \\mu x , | \\mu | - | \\nu | , \\nu x ) \\mid x \\in X , r ( \\mu ) = s ( x ) \\} , \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\frac { f } { g } = \\frac { f h ^ n } { g h ^ n } \\{ f h ^ n , g h ^ n \\} \\subset V _ { m + \\ell n } \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} { B } ' \\odot H _ t = B \\odot R + { B } ' \\odot \\Phi ( H ) , \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\tilde { h } ( 0 , \\tilde { x } , \\tilde { y } ) = 1 - 0 . 8 e ^ { - 6 ( \\tilde { y } + 0 . 2 ) ^ 2 - 4 ( \\tilde { x } - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} d e ^ i & = - \\theta ^ i _ j \\wedge e ^ j \\\\ \\theta ^ i _ j & = - \\theta ^ j _ i \\\\ \\Omega ^ i _ j & = d \\theta ^ i _ j + \\theta ^ i _ k \\wedge \\theta ^ k _ j . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} H : = \\tau ^ { - 1 } ( E ^ { - 1 } ) ^ * , v : = - \\tau ^ { - 1 } ( E ^ { - 1 } ) ^ * w , z : = - \\tau ^ { - 1 } E ^ { - 1 } c , \\rho : = \\tau ^ { - 1 } ( \\langle w , E ^ { - 1 } c \\rangle - \\beta ) . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} z _ { n \\ell } = \\sum ^ { \\ell - 1 } _ { i = 1 } r _ { \\ell i } ^ 2 - \\frac { \\ell - 1 } { n - 1 } . \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} X _ N & = \\frac { Z _ 0 } { X _ 1 \\cdots X _ { N - 1 } } , \\\\ Y _ { N - r + 1 } & = X _ { N - r + 1 } M _ 1 \\left ( \\frac { Y _ 1 } { X _ 1 } , \\dots , \\frac { Y _ { N - r } } { X _ { N - r } } , \\frac { Z _ 1 } { Z _ 0 } , \\dots , \\frac { Z _ r } { Z _ 0 } \\right ) , \\\\ & \\vdots \\\\ Y _ N & = \\frac { Z _ 0 } { X _ 1 \\cdots X _ { N - 1 } } M _ r \\left ( \\frac { Y _ 1 } { X _ 1 } , \\dots , \\frac { Y _ { N - r } } { X _ { N - r } } , \\frac { Z _ 1 } { Z _ 0 } , \\dots , \\frac { Z _ r } { Z _ 0 } \\right ) , \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 = A ^ 2 = \\underset { n \\to + \\infty } { \\lim } \\int _ { \\mathbb { R } ^ N } | \\nabla u _ n | ^ 2 , \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} V ( t , x ) : = V _ 1 ( 0 , t ; 0 , x ) = \\inf _ { u \\in { \\cal U } _ { [ 0 , t ] } ( 0 , x ) } J _ { [ 0 , t ] } ( u ) \\forall t \\in \\ , ] 0 , + \\infty [ \\ , , \\forall x \\in X . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\zeta _ k ^ { \\rm N O M A } = \\Pr \\left \\{ R _ k ^ { \\rm N O M A } ( \\nu ) < \\bar R _ k \\right \\} . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\int _ { E } g ( x ) P _ t f ( x ) m ( d x ) = \\int _ { E } f ( x ) \\hat { P } _ t g ( x ) m ( d x ) . \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} \\lambda _ 1 ( t ) & = \\lambda _ 1 ( \\tau _ 1 ) \\\\ \\lambda _ 2 ( t ) & = \\lambda _ 2 ( \\tau _ 1 ) + \\lambda _ 0 b _ M ( \\tau _ 1 - t ) \\\\ \\lambda _ 3 ( t ) & = \\lambda _ 3 ( \\tau _ 1 ) + \\frac { 1 } { 2 } \\lambda _ 0 b _ M k _ M ( \\tau _ 1 - t ) ^ 2 \\\\ & + ( \\lambda _ 0 b _ E - a _ M b _ M k _ M \\lambda _ 1 ( \\tau _ 1 ) + k _ M \\lambda _ 2 ( \\tau _ 1 ) ) ( \\tau _ 1 - t ) . \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( V , \\mathcal { T } _ { U } \\phi \\mathcal { B } X + \\hat { \\nabla } _ { U } \\phi \\mathcal { B } X ) + g _ { 1 } ( V , \\mathcal { T } _ { U } \\omega \\mathcal { B } X + \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\omega \\mathcal { B } X ) \\\\ & - g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { C } X ) , \\pi _ { \\ast } ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V ) ) + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) . \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} S ^ t _ 3 ( H ) = & \\sum _ { 1 \\leq i < j < k \\leq d _ c } | \\mathcal { C } ^ { i , j , k } | \\\\ & + S _ 3 ( \\{ h _ i , h _ j \\} ) + S _ 3 ( \\{ h _ i , h _ k \\} ) + S _ 3 ( \\{ h _ i , h _ k \\} ) . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} T _ i = e ^ { 1 / t _ i } , ( \\ , \\ , t _ i = \\frac { 1 } { \\log T _ i } ) . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\norm { x - p } = \\sqrt { d _ { K } ^ { 2 } { \\left ( x \\right ) } + \\left ( \\norm { P _ { K } x } - \\rho \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\mathrm { R i c } _ { i j } + \\nabla _ i \\nabla _ j f = 0 , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} a = 0 & & a ' & = ( n + 1 ) \\frac { k } { p } \\\\ b = 1 & & b ' & = 0 \\\\ f = 0 & & f ' & = 0 . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{gather*} A ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) \\le 0 \\ u ( x _ 0 ) > 0 \\\\ B ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) \\le 0 \\ u ( x _ 0 ) < 0 \\\\ - \\mathcal { Q } _ \\infty \\phi ( x _ 0 ) \\le 0 \\ u ( x _ 0 ) = 0 \\end{gather*}"}
-{"id": "2434.png", "formula": "\\begin{align*} w ^ * ( t ) = ( \\bar { C } ^ \\top \\bar { C } ) ^ { - 1 } \\bar { C } ^ \\top \\hat { w } ( t ) \\mbox { f o r a l l } \\ ; \\ ; t \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\epsilon ( \\| y _ 1 \\| _ 2 ^ 2 + \\| u _ 1 \\| _ 2 ^ 2 ) & \\leq \\langle u _ 1 , y _ 1 \\rangle + \\langle u _ 2 , y _ 2 \\rangle \\\\ & = \\langle e _ 1 - y _ 2 , y _ 1 \\rangle + \\langle e _ 2 + y _ 1 , y _ 2 \\rangle \\\\ & = \\langle e _ 1 , y _ 1 \\rangle + \\langle e _ 2 , y _ 2 \\rangle . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} c _ { i 0 } & = \\mathrm { l k } ( \\ell _ i , \\ell _ 0 ) , \\\\ c _ { i j } & \\equiv \\mathrm { l k } ( \\ell _ i , \\ell _ j ) \\mod { 2 } \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\texttt { C } _ { \\Omega } = \\left \\{ \\begin{array} { l l } \\frac { \\texttt { C } _ { P , \\Omega } } { \\sqrt { \\varepsilon } } , & \\textrm { i f } ~ \\kappa = 0 , \\\\ \\min \\left \\{ \\frac { \\texttt { C } _ { P , \\Omega } } { \\sqrt { \\varepsilon } } , \\frac { 1 } { \\sqrt { \\kappa } } \\right \\} , & \\textrm { i f } ~ \\kappa \\neq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\left ( V \\left ( Z ' , Z '' \\right ) \\right ) ^ { t } = \\left ( { Z ' } ^ { t } , Z ''' \\left ( \\varphi _ { B } \\left ( Z ' , Z '' \\right ) \\right ) ^ { t } \\right ) . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\sum _ { d _ 1 n _ 1 + \\dots + d _ r n _ r = n } \\alpha _ { ( n _ 1 , \\dots , n _ r ) } f _ 1 ^ { n _ 1 } \\cdots f _ r ^ { n _ r } \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\frac 1 p + \\frac { d - s } { d } = 1 + \\frac 1 r . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} & \\left < B _ { 1 \\beta } ^ { k } , B _ { 1 \\beta } ^ { k ' } \\right > > 0 , \\quad \\quad \\mbox { f o r a l l $ k , k ' , \\beta = 1 , \\dots , q $ . } \\\\ & \\left < B _ { 1 \\beta } ^ { k } , B _ { 1 \\alpha } ^ { k ' } \\right > = 0 , \\quad \\quad \\mbox { f o r a l l $ k , k ' , \\alpha , \\beta = 1 , \\dots , q $ w i t h $ \\alpha \\neq \\beta $ , } \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} w _ k ( Y _ n ) = \\sum _ { t = 1 } ^ { n - 1 } w _ { k - 1 } ( Y _ t ) \\ , A _ { t + 1 } \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\xi = \\int { f ^ { 1 / 2 } \\left ( z \\right ) d z . } \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} \\int F _ { h _ 1 , \\ldots , h _ { m + 1 } , n } \\ , d \\nu = 0 \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} e ( X , X ^ c ) & \\ge e ( X ' , H \\setminus X ' ) + d ( x _ 1 ) + e ( X , y ) + e ( X ' , y _ 2 ) + e ( X ' , y _ 3 ) \\\\ & \\ge 1 8 + 6 + 4 + 3 + 3 = 3 4 , \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} G ( 1 , x ) = \\int _ { \\mathbb { R } ^ { n } } \\ln ( \\varGamma ( 1 , x ; 0 , \\xi ) ) \\ ; \\mu ( d \\xi ) \\geq - C , \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\sum _ { z < m ^ { \\ell } \\le 2 ^ { \\ell } z } \\Lambda ( m ^ { \\ell } + u ) \\Lambda ( m ) = \\left \\{ \\int _ { { \\frak M } } + \\int _ { { \\frak m } } \\right \\} J ( \\alpha , z ) J _ { \\ell } ( \\alpha , z ) e \\left ( u \\alpha \\right ) { \\rm d } \\alpha . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\int _ M | \\nabla _ x F _ x ( y ) | ^ 2 \\ , d \\mu ( y ) ~ = ~ \\int _ M F _ x ( y ) \\cdot \\Delta _ x F _ x ( y ) \\ , d \\mu ( y ) \\ , . \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} a \\cdot \\delta _ u ( v ) = \\begin{cases} a & \\\\ 0 & \\\\ \\end{cases} \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} f ^ { ( k ) } ( X ) & = f ( f ^ { ( k - 1 ) } ( X ) ) \\\\ & = a _ 1 ( b _ 1 X ^ { e _ 1 } + \\ldots + b _ t X ^ { e _ t } ) ^ { d _ 1 } + \\ldots + a _ s \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} E \\times n = 0 \\mbox { o n } \\partial \\Omega , \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} W _ { n , 1 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ S _ { n } ^ { + } \\left ( { u , \\xi } \\right ) \\right \\} \\left [ \\exp \\left \\{ u \\xi + { E _ { 0 } ^ { + } \\left ( \\xi \\right ) } \\right \\} { + \\varepsilon _ { n , 1 } \\left ( { u , \\xi } \\right ) } \\right ] , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\Phi _ j ( x , \\mathbf { q } ) = \\sum _ { l = 1 } ^ m 8 \\pi G ( x , q _ l ) - G _ j ^ * ( q _ j ) + \\log h ( x ) - \\log h ( q _ j ) . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\chi _ M u _ { < t ^ { - \\frac 1 3 } } = \\chi _ M u _ { < ( t M ) ^ { - 1 } } + \\sum \\limits _ { ( t M ) ^ { - 1 } \\leq N < t ^ { - \\frac 1 3 } } \\chi _ M u _ N . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} | x _ 1 + \\dots + x _ n | ^ 2 & = ( x _ 1 + \\dots + x _ n ) ^ * ( x _ 1 + \\dots + x _ n ) = x _ 1 ^ * x _ 1 + x _ 2 ^ * x _ 2 + \\dots + x _ n ^ * x _ n \\\\ & = | x _ 1 | ^ 2 + | x _ 2 | ^ 2 + \\dots + | x _ n | ^ 2 . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( \\phi ^ { 2 } V , \\nabla ^ { ^ { M _ 1 } } _ { U } X ) + g _ { 1 } ( \\omega \\phi V , \\nabla ^ { ^ { M _ 1 } } _ { U } X ) - g _ { 1 } ( \\omega V , \\mathcal { T } _ { U } \\mathcal { B } X ) \\\\ & - g _ { 1 } ( \\omega V , \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\mathcal { C } X ) - g _ { 1 } ( V , \\phi U ) \\eta ( X ) . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} L ( A t + | x | ^ { 2 } ) & = \\Big ( \\partial _ t + b \\cdot \\nabla - \\Delta + \\frac { 2 } { r } \\partial _ r \\Big ) ( A t + | x | ^ { 2 } ) \\\\ & = A + 2 b \\cdot x - 2 , \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} x ^ \\ast ( t ) = \\begin{pmatrix} x _ N ( \\tau _ 0 ) - a _ E b _ E x _ E ( \\tau _ 0 ) \\bigl ( e ^ { k _ E ( t - \\tau _ 0 ) } - 1 \\bigr ) \\\\ x _ M ( \\tau _ 0 ) \\\\ x _ E ( \\tau _ 0 ) e ^ { k _ E ( t - \\tau _ 0 ) } \\end{pmatrix} . \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} F _ j : = \\{ \\mathbb E ^ { \\omega } _ j ( g ) ^ { q ' } \\omega \\leq \\mathbb E ^ { \\sigma } _ j ( f ) ^ { p } \\sigma \\} \\hbox { a n d } G _ j : = \\Omega \\setminus F _ j . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} h ^ 0 ( N _ { C } ( - 1 ) ( - q _ 1 - \\cdots - q _ 5 ) ) = 0 . \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( P _ { \\pi ^ * } ( \\theta < U | x ) < \\alpha \\right ) = \\alpha + O ( n ^ { - 1 / 2 } ) \\ , . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} U = \\{ ( a _ 1 , \\ldots , a _ d ) \\ , : \\ , \\sum _ { i } a _ i = 0 \\} , \\ ; \\ ; W = \\{ ( a , \\ldots , a ) \\ , : \\ , a \\in \\mathbb { F } _ { p } \\} \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} & \\sum ^ \\infty _ { j = 0 } \\abs { ( \\ , g _ j \\ , | \\ , \\chi _ { \\varepsilon _ 1 } \\ , ) _ M - ( \\ , g _ j \\ , | \\ , \\chi _ { \\varepsilon _ 2 } \\ , ) _ M } ^ 2 \\\\ & = ( \\ , B \\chi _ { \\varepsilon _ 1 } \\ , | \\ , \\chi _ { \\varepsilon _ 1 } \\ , ) _ M - ( \\ , B \\chi _ { \\varepsilon _ 1 } \\ , | \\ , \\chi _ { \\varepsilon _ 2 } \\ , ) _ M - ( \\ , B \\chi _ { \\varepsilon _ 2 } \\ , | \\ , \\chi _ { \\varepsilon _ 1 } \\ , ) _ M + ( \\ , B \\chi _ { \\varepsilon _ 2 } \\ , | \\ , \\chi _ { \\varepsilon _ 2 } \\ , ) _ M . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} B _ \\pm ( \\gamma \\cdot ( x , \\xi ) ) = U _ \\gamma ( B _ \\pm ( x , \\xi ) ) , \\Phi _ \\pm ( \\gamma \\cdot ( x , \\xi ) ) = T _ \\gamma ( B _ \\pm ( x , \\xi ) ) \\Phi _ \\pm ( x , \\xi ) . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\mu ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n p } \\log \\lVert A ^ { n p } ( x ) \\rVert & = \\limsup _ { n \\to \\infty } \\frac { 1 } { n p } \\log \\rho ( A ^ { n p } ( x ) ) \\\\ & = \\limsup _ { n \\to \\infty } \\frac { 1 } { n p } \\log \\rho ( ( A ^ { p } ( x ) ) ^ n ) \\\\ & = \\limsup _ { n \\to \\infty } \\frac { 1 } { n p } \\log ( \\rho ( A ^ p ( x ) ) ) ^ n \\\\ & = \\frac 1 p \\log \\rho ( A ^ p ( x ) ) . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} ( n ) _ \\lambda ! & = n ( n - \\lambda ) ( n - 2 \\lambda ) \\cdots ( n - ( n - 1 ) \\lambda ) \\\\ & = ( n ) _ { n , \\lambda } , \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} g _ { \\lambda _ 0 } ( \\zeta _ 0 ) = 0 , \\nabla g _ { \\lambda _ 0 } ( \\zeta _ 0 ) = 0 , \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} I _ { \\nu } \\left ( { \\nu z } \\right ) = \\frac { \\left ( { { \\frac { 1 } { 2 } } \\nu z } \\right ) ^ { \\nu } } { \\Gamma \\left ( { \\nu + 1 } \\right ) } \\left \\{ { 1 + \\mathcal { O } \\left ( z \\right ) } \\right \\} . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\lambda _ { N - i + 1 } ( N ) } = \\nu _ i , \\forall \\ , i = 1 , 2 , \\dots . \\end{gather*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\lambda _ l ( t ) & = \\gamma _ l t ^ 2 + \\mu _ l t ^ 3 + \\dots , \\gamma _ l \\ge 0 , \\ ; \\mu _ l \\in \\mathbb { R } , l = 1 , \\dots , n , \\\\ \\varphi _ l ( t ) & = \\omega _ l + t \\psi _ l ^ { ( 1 ) } + \\dots , l = 1 , \\dots , n . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 3 } } { \\partial \\rho ^ { 3 } } \\left ( H ( 1 , \\rho ) \\right ) { \\big | _ { \\rho = 0 } } = 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) G ^ { 2 } ( t ) d t + 3 ! \\int \\limits _ { - 1 } ^ 1 f ( t ) F ( t ) d t = 3 ! \\int _ { - 1 } ^ 1 f ( t ) r _ 3 ( t ) d t = 0 . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} A = \\langle e _ 1 \\rangle _ { 1 } \\oplus \\langle e _ 2 \\rangle _ { 2 } \\oplus \\langle e _ 2 \\rangle _ { 3 } . \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} f = ( f _ { 1 } \\otimes e _ { 1 } , \\ldots , f _ { d } \\otimes e _ { d } ) , \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | p _ n ( i y ) | = \\exp { \\gamma y ^ 2 } \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} k ( Y , w ) : = \\sum _ { s \\in \\Sigma } ( \\rho _ s - 1 ) . \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} B [ u , v ] = Q _ 1 [ u , v _ x ] + Q _ 1 [ v , u _ x ] + \\partial _ x Q _ 2 [ u , v ] , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} f * g ( \\gamma ) = \\sum _ { \\alpha \\beta = \\gamma } f ( \\alpha ) g ( \\beta ) = \\begin{cases} f ( U ^ x ) g ( ( U ^ x ) ^ { - 1 } \\gamma ) & x = r ( \\gamma ) \\in r ( U ) \\\\ 0 & r ( \\gamma ) \\notin r ( U ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} H _ k ( x ) = \\left ( x + \\frac { x ^ 2 } { 1 - x } \\big ( K ( x ) - 1 \\big ) \\right ) \\sum _ { j \\geq k - 1 } H _ j ( x ) . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} g ^ { - 1 } \\dot { g } = J ^ { - 1 } \\omega ^ { - 1 } \\omega \\dot { J } = J ^ { - 1 } \\dot { J } , \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} F l o w ( X _ { \\rho _ 3 , m } ) = \\theta F l o w ( X _ { \\rho _ 1 , m } ) + ( 1 - \\theta ) F l o w ( X _ { \\rho _ 2 , m } ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} \\bar \\Sigma _ n & = \\frac { \\nu + n - 1 } { \\nu + n } \\bar \\Sigma _ { n - 1 } + \\frac { \\rho ^ 2 \\nu + n - 1 } { ( \\nu + n ) ( \\rho ^ 2 \\nu + n ) } ( x _ n - \\bar \\mu _ { n - 1 } ) ( x _ n - \\bar \\mu _ { n - 1 } ) ^ \\top . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\overline F ( x ) & = \\frac { 1 } { 2 } x ^ { - \\alpha } ( 1 + x ^ { { \\rho } } ) , x \\ge 1 , \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\| \\mu ( | x _ i | ^ q ) \\| _ { E ^ { ( 1 / q ) } } & = \\sum _ { i = 1 } ^ n \\| \\mu ^ q ( x _ i ) \\| _ { E ^ { ( 1 / q ) } } \\le \\left \\| \\sum _ { i = 1 } ^ n \\mu ^ q ( x _ i ) \\right \\| _ { E ^ { ( 1 / q ) } } \\\\ & \\le \\left \\| \\mu ^ { q / p } \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) \\right \\| _ { E ^ { ( 1 / q ) } } = \\left \\| \\mu \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) ^ { q / p } \\right \\| _ { E ^ { ( 1 / q ) } } . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} F _ T ( x ) = \\frac { 2 - 1 1 x + 1 3 x ^ 2 - 6 x ^ 3 + ( 1 - x ) x ( 1 - 6 x + 4 x ^ 2 ) ( 1 - 4 x ) ^ { - 1 / 2 } } { 2 ( 1 - 6 x + 8 x ^ 2 - 4 x ^ 3 ) } \\ , . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{gather*} \\big \\{ \\omega ^ i , \\theta ^ a , \\phi _ 0 , \\phi ^ a { } _ b , \\phi _ n , \\gamma _ a , \\pi _ i \\big \\} _ { i = 0 , \\ , a = 1 } ^ { n , \\ n - 1 } \\end{gather*}"}
-{"id": "5809.png", "formula": "\\begin{align*} H ( f ) - H ( \\mathcal { M } _ { \\nu , \\theta } ) & \\geq H ( f | \\mathcal { M } _ { 0 , 1 } ) + ( 1 - \\theta ) \\{ H ( \\mathcal { M } _ { 0 , 1 } ) - H ( \\mathcal { M } _ { \\Theta } ) \\} \\cr & \\geq H ( f | \\mathcal { M } _ { 0 , 1 } ) + ( 1 - \\theta ) \\{ H ( \\mathcal { M } _ { 0 , 1 } ) - H ( f ) \\} \\cr & = H ( f | \\mathcal { M } _ { 0 , 1 } ) + ( \\theta - 1 ) H ( f | \\mathcal { M } _ { 0 , 1 } ) \\cr & = \\theta H ( f | \\mathcal { M } _ { 0 , 1 } ) . \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l a _ n \\phi _ { n + 1 } + a _ { n - 1 } \\phi _ { n - 1 } + b _ n \\phi _ n = \\lambda \\phi _ n , \\\\ \\phi _ 0 = 0 , \\ , \\ , \\phi _ 1 = 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} \\alpha ( f ) = \\begin{cases} 0 & , \\\\ \\frac { 3 } { 1 6 } & , \\\\ \\frac { 7 } { 6 4 } & . \\end{cases} \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\varepsilon _ { n , 1 } \\left ( { u , \\xi } \\right ) = \\frac { 1 } { u } \\int _ { \\alpha _ { 1 } } ^ { \\xi } \\mathsf { K } { \\left ( { \\xi , t } \\right ) h } _ { n } { \\left ( { u , t } \\right ) d t , } \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\rho _ F ( ( s , t ] ) = F ( t ) - F ( s ) , \\ ; \\ ; \\ ; 0 \\leq s < t < \\infty . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} M & = \\sup _ { x \\in \\Pi } \\Gamma _ 0 ( x ) , \\\\ m & = \\inf _ { x \\in \\Pi } \\Gamma _ 0 ( x ) . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\Lambda _ { t , 0 } = \\lim _ { t \\rightarrow \\infty } \\prod _ { q \\in \\Q _ { t , 0 } } \\lambda _ q = \\lim _ { t \\rightarrow \\infty } \\exp \\left ( \\sum _ { q \\in \\Q _ { t , 0 } } \\ln \\left ( \\lambda _ q \\right ) \\right ) . \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\xi = \\int { f _ { 0 } ^ { 1 / 2 } \\left ( z \\right ) d z } , \\ W = f _ { 0 } ^ { 1 / 4 } \\left ( z \\right ) w . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} \\mathrm { J } _ n = \\mathrm { P G L } ( 2 , \\mathbb { C } ( z _ 1 , z _ 2 , \\ldots , z _ { n - 1 } ) ) \\times \\mathrm { P G L } ( 2 , \\mathbb { C } ( z _ 2 , z _ 3 , \\ldots , z _ { n - 1 } ) ) \\times \\ldots \\times \\mathrm { P G L } ( 2 , \\mathbb { C } ( z _ { n - 1 } ) ) \\times \\mathrm { P G L } ( 2 , \\mathbb { C } ) \\subset \\mathrm { B i r } ( \\mathbb { P } ^ n _ \\mathbb { C } ) . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\omega ^ { ( k ) } _ i = \\left \\{ \\begin{aligned} ( 1 , 0 ^ { n _ { k + i } - n _ { k + i - 1 } - 1 } ) & , & { \\rm w h e n } \\ \\ i { \\rm \\ i s \\ o d d } ; \\\\ \\left ( ( 1 , 0 ^ { h - 1 } ) ^ { \\lfloor \\frac { n _ { k + i } - n _ { k + i - 1 } } { h } \\rfloor h } , 0 ^ { n _ { k + i } - n _ { k + i - 1 } - \\lfloor \\frac { n _ { k + i } - n _ { k + i - 1 } } { h } \\rfloor h } \\right ) & , & { \\rm w h e n } \\ \\ i { \\rm \\ i s \\ e v e n } . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} x _ k = - \\cos \\left ( \\frac { ( k - 1 ) \\pi } { N - 1 } \\right ) , k = 1 , \\ldots , N . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} & \\varepsilon _ { ( a b ) , 2 } ( [ x _ a , x _ b ] ) = 1 , \\\\ & \\varepsilon _ { ( a b ) , 2 } ( y _ i ) = \\varepsilon _ { ( a b ) , 2 } ( [ x _ a , x _ b ] ^ { c _ { i a b } } ) + \\varepsilon _ { ( a ) , 2 } ( [ x _ a , x _ b ] ^ { c _ { i a b } } ) \\varepsilon _ { ( b ) , 2 } ( h _ i ) + \\varepsilon _ { ( a b ) , 2 } ( h _ i ) , \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\dim { \\mathcal M } ( { \\alpha _ - } , { \\alpha _ + } ) = \\mu ( \\alpha _ + ) - \\mu ( \\alpha _ - ) - 1 + \\dim R _ { \\alpha _ + } . \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } ( | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u | ^ { p - 2 } \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ) + \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } ( | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u | ^ { p - 2 } \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ) = | u | ^ { q - 2 } u . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} b _ q ( x , m , w ) = \\sup \\left \\{ \\alpha m + \\beta \\cdot w \\ ; : \\ ; ( \\alpha , \\beta ) \\in A _ { q ' } ( x ) \\right \\} , \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} [ x + \\gamma _ j , B _ u ] \\phi _ j & = - \\frac 2 i \\partial _ x \\phi _ j + 2 C _ + ( u \\phi _ j ) - 2 [ x C _ + ( u \\phi _ j ) _ x - C _ + ( x ( u \\phi _ j ) _ x ) ] \\\\ & = - \\frac 2 i \\partial _ x \\phi _ j + 2 C _ + ( u \\phi _ j ) \\\\ & = - 2 L _ u \\phi _ j = - 2 \\lambda _ j \\phi _ j . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } a ( n ) e ( n z ) . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { X _ h ( 0 ) } = h ^ { - \\frac 1 2 } \\| u _ 0 \\| _ { L ^ 2 } ^ 2 + h ^ { \\frac { 1 5 } 2 } \\| \\partial _ x ^ 4 u _ 0 \\| _ { L ^ 2 } ^ 2 + h ^ { - \\frac 9 2 } \\| y ^ 2 \\partial _ x u _ 0 \\| _ { L ^ 2 } ^ 2 + h ^ { - \\frac 1 2 } \\| ( x \\partial _ x + y \\partial _ y ) u _ 0 \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} h _ i ( \\underline { \\rm H o m } ( { \\bf L } ^ { n - d } , M ( X ) ) ) \\cong \\left \\{ \\begin{array} { l l } 0 & i < 0 , \\\\ C H _ { X } ^ d & i = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} p ^ c ( t ) = \\overline { p ( 1 / t ) } \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} & \\tilde \\sigma _ j ( 0 , r ) > 0 , \\forall r \\in ( a , 1 ) , \\ j = 1 , \\ldots , k , \\\\ & g _ 0 ( \\zeta _ 1 ( r ) ) ^ 2 - G _ 0 ( \\zeta _ 1 ( r ) , \\zeta _ j ( r ) ) ^ 2 > 0 \\forall r \\in ( a , 1 ) , \\ j = 2 , \\ldots , k . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} H ( X ) : = - \\sum _ { x \\in X } P ( x ) \\log P ( x ) . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} u ^ { o p } ( a \\otimes b ) = ( - 1 ) ^ { | a | | b | } b \\cdot a = a \\cdot b = u ( a \\otimes b ) , & \\\\ \\Delta ^ { o p } = T \\Delta , & \\\\ \\{ a , b \\} ^ { o p } = ( - 1 ) ^ { | a | | b | } \\{ b , a \\} = - \\{ a , b \\} , & \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\frac { 1 } { m ^ k k ! } e ^ { r t } ( e ^ { m t } - 1 ) ^ k & = \\frac { 1 } { m ^ k } \\left ( \\sum _ { l = 0 } ^ \\infty r ^ l \\frac { t ^ l } { l ! } \\right ) \\left ( \\sum _ { i = k } ^ \\infty S _ 2 ( i , k ) \\frac { m ^ i t ^ i } { i ! } \\right ) \\\\ & = \\frac { 1 } { m ^ k } \\sum _ { n = k } ^ \\infty \\left ( \\sum _ { i = k } ^ n S _ 2 ( i , k ) { n \\choose i } r ^ { n - i } m ^ i \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} Q \\left ( x , \\nabla p , \\nabla W _ { 1 } + \\nabla h \\right ) = \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\bar { \\Lambda } ^ 0 _ 1 : = \\sqrt { \\frac { a _ 1 } { 2 a _ 2 \\lambda _ 0 \\sum _ { i = 1 } ^ k P _ { i 1 } ( 0 , \\zeta ^ 0 ) ^ 4 } } . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} ( \\Delta - s _ 0 ( n - s _ 0 ) - m ) \\varphi ^ { ( k ) } = - ( 2 s _ 0 - n ) \\varphi ^ { ( k - 1 ) } - \\varphi ^ { ( k - 2 ) } . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\mathbf { 1 } \\Big ( \\Lambda \\leq \\frac { | \\mathcal { S } | } { K _ 0 } \\Big ) | \\Lambda _ \\iota | \\leq C N ^ { - \\varepsilon } | \\mathcal { S } | \\ , , \\iota = A , B . \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} \\sum _ { N ( a ) \\leq x } 1 = \\pi x + O ( x ^ { \\theta } ) \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\phi } = K _ { 2 0 } ( \\widetilde \\omega t ) [ y ] + K _ { 1 1 } ^ T ( \\widetilde \\omega t ) [ w ] \\\\ \\dot { y } = 0 \\\\ \\dot { w } = J K _ { 0 2 } ( \\widetilde \\omega t ) [ w ] + J K _ { 1 1 } ( \\widetilde \\omega t ) [ y ] \\ , . \\end{cases} \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N ( X _ j ^ 8 + Y _ j ^ 8 ) u + \\triangle _ { H } ^ 2 u - \\triangle _ H u + u = | u | ^ { q - 2 } u , \\ ; \\ ; u \\in H ^ 4 ( \\mathbb { H } ^ { N } ) , \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} ( A ( t ) + \\zeta I ) ^ { - 1 } F ( t ) = \\Xi ( t , \\zeta ) + t ( Z \\Xi ( t , \\zeta ) + \\Xi ( t , \\zeta ) Z ^ * ) - t ^ 3 \\Xi ( t , \\zeta ) N \\Xi ( t , \\zeta ) + \\mathcal { J } ( t , \\zeta ) , \\ | t | \\le t ^ 0 , \\ ; \\zeta > 0 . \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} r _ { j } & : \\left [ 0 , 1 \\right ] \\longrightarrow \\mathbb { R } j \\in \\mathbb { N } \\\\ r _ { j } \\left ( t \\right ) & : = s i g n \\left ( \\sin 2 ^ { j } \\pi t \\right ) \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\ + U _ \\sigma = \\left \\{ X \\ : \\ \\Upsilon ^ X \\succeq \\hat { \\sigma } \\right \\} . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ { n , j } ^ * ( r ) = \\zeta _ { n , j } ^ * ( \\Lambda _ { n , j , R } ^ { - } ) + o ( 1 ) + o \\Big ( \\frac { 1 } { \\lambda _ { n , j } ^ { ( 1 ) } } \\Big ) R + O ( { R ^ { - 2 } } ) \\ \\ \\textrm { f o r a l l } \\ \\ r \\in ( \\Lambda _ { n , j , R } ^ { - } , { \\delta } ) . \\end{aligned} \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} V + V ' = V '' & & & I = I ' = I '' . & & & \\\\ \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} A ( t ) ^ { 1 / 2 } F ( t ) = | t | S ^ { 1 / 2 } P + t | t | \\bigl ( Z S ^ { 1 / 2 } P + S ^ { 1 / 2 } P Z ^ * \\bigr ) + \\frac { 1 } { 2 } t | t | N _ 0 S ^ { - 1 / 2 } P + t | t | S ^ { - 1 / 2 } P N _ * + I _ * ( t ) + \\Phi ( t ) . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\frac { d f } { d t } = f _ { t } + \\dot { x } ^ { k } f _ { , k } . \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} & [ a _ 1 , a _ 2 ] = a _ 1 ^ p , \\ , [ a _ 1 , a _ 3 ] = a _ 3 ^ p , \\ , [ a _ 1 , a _ 4 ] = a _ 4 ^ p , \\\\ & [ a _ 2 , a _ 3 ] = a _ 2 ^ p , \\ , [ a _ 2 , a _ 4 ] = 1 , \\ , [ a _ 3 , a _ 4 ] = a _ 3 ^ p . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} e = r ^ { - 1 } \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ( y _ { i } - y _ { 0 } ) , | \\alpha _ { i } | \\leq C ( m , \\rho ) . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} M _ { N } A _ { 0 } \\left ( x , V \\right ) = F \\left ( x , \\nabla V \\right ) , \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\int _ M f _ x ^ { \\boxplus \\tau } ! ( h _ x ^ { \\boxplus \\tau } ; \\mathcal S _ x ^ { \\boxplus \\tau } ) \\wedge \\rho = ( - 1 ) ^ { | \\rho | | \\omega | } \\int _ { ( s _ x ^ { \\boxplus \\tau } ) ^ { - 1 } ( 0 ) } \\pi _ 1 ^ * h ^ { \\boxplus \\tau } _ x \\wedge \\pi _ 1 ^ * ( f ^ { \\boxplus \\tau } _ x ) ^ * \\rho \\wedge \\pi _ 2 ^ * \\omega _ x \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - \\eta t } h ( t ) d t = \\infty \\ f o r \\ a l l \\ 0 < \\eta < A . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} ( 0 ) : l _ 1 & = K + ( l _ 2 ) , \\\\ ( 0 ) : l _ 2 & = K + ( l _ 1 ) , \\\\ K \\cap ( l _ 1 ) & = K \\cap ( l _ 2 ) = ( a _ 1 a _ { 1 2 } a _ { 1 9 } ) . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} T _ j ( p ) f ( z ) = \\sum _ { n \\ge 1 } \\left ( a ( p ^ { j } n ) + p ^ { j ( k - 1 ) } \\chi ^ { j } ( p ) a \\left ( \\frac { n } { p ^ { j } } \\right ) \\right ) e ( n z ) , \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\frac { \\partial R _ i } { \\partial g _ { i k } } = - c _ i \\mathbf { e } _ k ( \\mathbf { e } _ i + \\mathbf { h } _ i ) ^ * + \\Delta _ R ( i , k ) , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} u \\left ( x , 0 , x _ { 0 } \\right ) = 0 , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} h \\circ I = 1 / h \\hbox { a n d } v \\circ I = 1 / v . \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} R ( \\alpha , x ) = \\frac { \\sqrt { n } \\left ( \\hat { \\theta } ^ M _ 1 - \\hat { \\theta } ^ { M L } _ 1 \\right ) } { \\sqrt { \\hat { L } ^ { 1 1 } } } + \\left ( \\sqrt { \\frac { \\hat { \\nu } ^ { 1 1 } } { \\hat { L } ^ { 1 1 } } } - 1 \\right ) \\Phi ^ { - 1 } ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} C _ 0 = r ( \\log ( 2 ) - 1 / 4 ) + \\frac { \\| \\nabla P _ r ( \\vec { 1 } ) \\| _ 1 } { \\sqrt { 2 } | P _ r ( \\vec { 1 } ) | } \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} y _ n ( z ) = \\prod _ { i = 1 } ^ n ( z - z _ i ) , \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} M _ n = A _ n \\oplus \\operatorname { d i a g } ( \\lambda _ { n + 1 } , \\lambda _ { n + 2 } , \\dots ) \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\big | \\Psi _ p ( z ) \\big | ^ { 1 / p } = \\kappa \\Phi ( z ) , \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\ , \\varphi _ { \\underline { d } ' , \\underline { d } } ( V _ { \\underline { d } ' } ) = & r + 1 - \\ , V _ { \\underline { d } ' } ^ { X _ { 2 } ^ { c } , 0 } \\\\ = & r + 1 - ( \\ , V _ { \\underline { d } '' } ^ { X _ 1 , 0 } - \\ , V _ { \\underline { d } '' } ^ { X _ { 3 } ^ { c } , 0 } ) . \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\zeta _ { n , j } ^ * ( r ) = - 2 \\pi b _ { j , 0 } + o _ R ( 1 ) + o _ n ( 1 ) ( 1 + O ( R ) ) , \\ \\ \\textrm { f o r a l l } \\ \\ r \\in ( \\Lambda _ { n , j , R } ^ { - } , { \\delta } ) , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s \\tau } M _ { \\beta } ( \\tau ) \\ , d \\tau = E _ { \\beta } ( - s ) . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} f ' : ( u , r ) \\mapsto ( p , r ) = ( \\dfrac { - u } { 1 - x _ 1 - u } , r ) . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\Gamma _ { 1 1 } = \\frac { 2 } { J _ 1 } , \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} B ( X ) = \\begin{cases} O ( X ^ k \\log ^ 2 X ) \\\\ \\Omega ( X ^ { k - \\frac { 1 } { 4 } } \\frac { ( \\log \\log \\log X ) ^ 2 } { \\log X } ) , \\end{cases} \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} \\tilde \\psi _ \\lambda ( r ) = \\tilde \\psi _ \\lambda ( \\zeta ( r ) ) , \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} C H _ { J a c , { \\bf Q } } ^ d ( X ) : = { \\rm k e r } \\ , A J _ { X , { \\bf Q } } ^ d . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} f ( x + m p ) = f _ 0 ( x ) + m p f _ 1 ( x ) + \\cdots + ( m p ) ^ { k - 1 } f _ { k - 1 } ( x ) \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} B = R ^ { \\ast } _ { X } R _ { Y } = R ^ { \\ast } _ { A } U _ { A } V ^ { \\ast } _ { A } V _ { D } U ^ { \\ast } _ { D } R _ { D } = R ^ { \\ast } _ { A } K R _ { D } , \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{gather*} P _ { \\lambda } ( q , t ) = \\sum _ { \\mu } { c _ { \\lambda , \\mu } m _ { \\mu } } \\end{gather*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\langle w , \\alpha _ Q ( v ) \\rangle = \\frac { d } { d t } \\langle \\gamma , \\xi \\rangle | _ { t = 0 } \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\big ( F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z _ 0 ) ) - 1 \\big ) \\Omega _ A ( z _ 0 ) - \\Omega _ B ( z _ 0 ) & = r _ 2 ( z _ 0 ) + O \\left ( | \\Omega _ A ( z _ 0 ) | ^ 2 \\right ) \\ , . \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} N ( \\phi ) = ( V + \\phi ) ^ 5 - V ^ 5 - 5 V ^ 4 \\ , \\phi , \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} C _ 0 = \\frac { 2 ( 1 + 1 / \\sqrt { a } ) } { \\sqrt { 1 - \\delta _ { ( a + 1 ) k } } - \\sqrt { 1 + \\delta _ { a k } } / \\sqrt { a } } \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} & \\sum _ { m = 2 } ^ { q + 1 } a _ { k + m - 1 - q } \\sum _ { p = 1 } ^ { [ ( m - 1 ) / 2 ] } m _ { ( m - 1 - p , p , \\dot { 0 } ) } \\\\ & = \\sum _ { t = 0 } ^ { q - 1 } a _ { k - t } \\sum _ { s = 1 } ^ { [ ( q - t ) / 2 ] } m _ { ( q - t - s , s , \\dot { 0 } ) } \\\\ & = \\sum _ { s = 1 } ^ { [ q / 2 ] } \\sum _ { t = 0 } ^ { q - 2 s } a _ { k - t } m _ { ( q - t - s , s , \\dot { 0 } ) } . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\varphi ( t ) = \\left \\{ \\begin{array} { l l l } \\varphi _ 1 ( t ) t \\in ( - T _ 1 , \\ - T _ 2 ) , \\medskip \\\\ \\varphi _ 2 ( t ) t \\in ( T _ 2 , \\ T _ 1 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} ( n + 3 ) \\tilde { N } _ { n + 2 } ( t ) - ( 2 n + 3 ) ( t + 1 ) \\tilde { N } _ { n + 1 } ( t ) + n ( t - 1 ) ^ { 2 } \\tilde { N } _ { n } ( t ) = 0 \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\sum _ { s = 2 } ^ { d + 1 } \\frac { ( s - 1 ) x ^ { d + 4 } } { ( 1 - x ) ^ { d + 5 - s } } \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} \\gamma ( ( \\nabla ^ \\gamma _ Z J ) X , J Y ) - \\gamma ( \\nabla ^ \\gamma _ { J X } Z , J Y ) + \\gamma ( \\nabla ^ \\gamma _ X Z , Y ) = 0 . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} { \\rm d e t } \\ , L _ { \\rho } ( t , x , \\tau , \\xi ) = h _ { \\rho } ( t , x , \\tau , \\xi ) \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } U _ i ^ 4 \\ , U _ j \\ , U _ m & = \\frac { 2 } { 5 } \\ , a _ 3 \\ , \\mu _ i \\ , \\mu _ j \\ , \\mu _ m \\ , G _ \\lambda ( \\zeta _ i , \\zeta _ j ) \\ , G _ \\lambda ( \\zeta _ i , \\zeta _ m ) + \\mathcal { R } _ { i , j , m } ^ 5 + \\mathcal { R } _ { i , j , m } ^ 6 . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} - \\kappa ^ { \\varepsilon } \\nabla \\theta ^ { \\varepsilon } \\cdot \\mbox { n } = 0 \\quad \\mbox { o n } \\ ; \\left ( 0 , T \\right ) \\times \\Gamma _ { N } ^ { \\varepsilon } , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} a _ { j , k } = \\int _ { - 1 } ^ 1 \\ , \\mathrm { d } x \\int _ { - 1 } ^ 1 \\ , \\mathrm { d } y \\ , w _ r ^ { ( m ) } ( x ) w _ r ^ { ( m ) } ( y ) x ^ { j - 1 } y ^ { k - 1 } \\mathrm { s g n } ( y - x ) . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} K = \\det ( l _ { i j } ) _ { i , j = 0 , \\dots , 3 } . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - M P _ t \\| _ \\infty = 0 \\ , , \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\lim _ { t \\ , \\rightarrow \\ , \\infty } \\ , \\| \\ , D \\ : \\ ! \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { 2 } ( \\mathbb { R } ^ { n } ) } } = \\ ; 0 , \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{gather*} z ^ 1 _ { \\mu } z ^ 2 _ { \\nu } - z ^ 2 _ { \\mu } z ^ 1 _ { \\nu } = 0 , \\end{gather*}"}
-{"id": "9535.png", "formula": "\\begin{align*} p q ' = \\frac { 5 } { 3 } \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} T \\Xi _ g ( Y _ { \\alpha } ) = \\Upsilon ^ { \\beta } _ { g , \\alpha } Y _ { \\beta } . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n k ^ 2 \\binom { n } { k } = ( n + n ^ 2 ) 2 ^ { n - 2 } \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} P ( Q ^ T X Q ) = P ( X ) \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} w _ i ^ { E T } ( \\theta ) = \\frac { e ^ { \\lambda ^ T _ { E T } \\psi ( x _ i , \\theta ) } } { \\sum _ { i = 1 } ^ n e ^ { \\lambda _ { E T } ^ T \\psi ( x _ i , \\theta ) } } \\ , , \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} \\gamma = \\left \\{ \\begin{array} { l l } \\log ( 2 ^ { 1 - \\alpha } \\rho ) & \\mbox { f o r e q u a t i o n \\eqref { H A M } } \\\\ \\log \\rho & \\mbox { f o r e q u a t i o n \\eqref { P A M } } \\end{array} \\right . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} X = T \\left ( t \\right ) S _ { J } ^ { , i } \\partial _ { i } . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } g ( s , W _ 0 ^ { t _ { n , i } } , Y _ 0 ^ { ( n ) , t _ { n , i } } ) d s + B ( t _ { n , i + 1 } ) - B ( t _ { n , i } ) , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} | F _ { v ^ * } | \\leq \\gamma ( G ) - t + 2 \\leq ( t + m + 1 ) - t + 2 = m + 3 \\leq t + 1 . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} X ^ { 1 - \\theta } Y ^ { \\theta } = \\{ f \\in L ^ 0 : | f | = g ^ { 1 - \\theta } h ^ { \\theta } , \\ g \\in X , \\ h \\in Y \\} , \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\mathcal { P } f ( x ) = \\sum _ { n \\in ( L _ 1 ^ { - 1 } \\mathbb { Z } ) \\times ( L _ 2 ^ { - 1 } \\mathbb { Z } ) } e ^ { 2 \\pi i n \\cdot x ' } e ^ { 2 \\pi \\abs { n } x _ 3 } \\hat { f } ( n ) , \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} F ^ \\times = \\left \\{ f \\in L ^ 0 ( I ) : \\int _ I f g < \\infty \\ \\ \\ g \\in F \\right \\} . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} [ S ^ p E \\otimes S ^ { r - p } E : L ( \\lambda ) ] & = [ f ( S ^ p E \\otimes S ^ { r - p } E ) : f ( L ( \\lambda ) ) ] \\cr & = [ M ( p , r - p ) : f ( L ( \\lambda ) ) ] \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} s \\xrightarrow { x | y } t \\ \\ \\mbox { w h e n e v e r } \\ \\ \\lambda ( s , x ) = ( y , t ) . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\phi \\ , d \\mu = \\lim _ k \\int _ { \\R ^ n } \\phi \\ , d \\mu _ k = \\lim _ k \\int _ { \\R ^ n } \\nabla \\phi \\cdot z _ k = 0 \\ , , \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} s _ { i } ^ { ( j ) } = ( - 1 ) ^ { h _ { i } \\oplus r _ { i } ^ { ( j ) } } \\cdot ( \\log ( 1 - p _ { e _ { i } } ) - \\log ( p _ { e _ { i } } ) ) \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} n ! = \\sum _ { k = 1 } ^ n ( k , 1 ) ^ { \\dagger } ( n - k ) ! , \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} m ( T - r _ { 2 } , k - 1 ) = & m ( P _ { x - 2 } \\cup P _ { y - 2 } \\cup P _ { c - 2 } , k - 1 ) + m ( P _ { x - 3 } \\cup P _ { y - 2 } \\cup P _ { c - 3 } , k - 2 ) \\\\ & + m ( P _ { x - 2 } \\cup P _ { y - 3 } \\cup P _ { c - 3 } , k - 2 ) . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\tilde { \\rho } ( \\tilde { \\theta } _ 1 ^ \\alpha , x ) = \\Phi \\left ( \\tilde { Z } ( \\eta _ 1 ^ \\alpha ) \\right ) + O _ p ( n ^ { - 1 } ) \\ , , \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} R ^ { - 1 } ( x + \\alpha ) A ^ { E + \\epsilon } ( x ) R ( x ) = P + \\epsilon \\widetilde { P } ( x ) , \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} | f _ j ( x ) | = \\left | \\frac 1 { | B ( x , 1 ) | } \\int _ { B ( x , 1 ) } f _ j ( y ) d V ( y ) \\right | \\leq \\frac 1 { | B ( x , 1 ) | } \\| f \\| _ \\varphi \\left ( \\int _ { B ( x , 1 ) } e ^ { 2 \\varphi ( y ) } d V ( y ) \\right ) ^ { \\frac 1 2 } \\leq C _ x \\| f \\| _ \\varphi , \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} F u t _ { K , a } ( X ) = \\int _ M ( S ( \\omega _ g , f _ { K , g , a } ) - c _ { \\Omega , K , a } ) \\ , u _ X \\ , \\frac { \\omega _ g ^ m } { f _ { K , g , a } ^ { 2 m + 1 } } \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} \\left ( C ^ T f , g \\right ) = \\sum _ { k , \\tau \\in K ( T , T - 1 ) } h ( k , \\tau ) . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\omega _ { t } + \\left ( U \\left ( y \\right ) - U _ { s } \\right ) \\partial _ { x } \\omega + U ^ { \\prime \\prime } \\left ( y \\right ) \\partial _ { x } \\psi = 0 , \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} p ^ n - 2 = ( p - 1 ) \\left ( 1 + p + \\cdots + p ^ { n - 1 } \\right ) - 1 = ( p - 2 ) + ( p - 1 ) \\left ( p + \\cdots + p ^ { n - 1 } \\right ) . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\pi _ 1 \\Sigma _ g \\cong \\langle a _ 1 , b _ 1 , \\hdots , a _ g , b _ g \\ , | \\ , [ a _ 1 , b _ 1 ] \\cdots [ a _ g , b _ g ] = 1 \\rangle \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\begin{cases} \\Delta V ^ t = - \\Delta \\dot h ^ t & \\mbox { i n } \\Omega ^ t \\\\ V ^ t = 0 & \\mbox { o n } \\Gamma ^ t \\\\ \\lim _ { x \\to \\infty } V ^ t ( x ) = \\dot c ^ t , & \\lim _ { x \\to \\infty } \\frac { V ^ t ( x ) } { - \\log | x | } = \\dot c ^ t ( \\ n = 2 ) . \\end{cases} \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\int _ t ^ { t + \\Delta } g ( s ) d s = \\mathcal { E } , \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} t _ 4 ( \\mbox { s o r t } ( y ) , \\mbox { s o r t } ( 5 \\cdot \\chi ' ) ) = ( 5 , 1 0 , 1 1 , 1 2 , 1 3 , 1 5 , 2 0 , 2 1 , 2 5 , 3 0 , 3 5 , 4 0 , 4 5 ) , \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\frac { n _ k } { \\varphi ( n _ k ) } = + \\infty \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\sigma _ j ( E ) = \\frac { \\sigma ( q _ j + r _ j E ) } { r _ j ^ { n - 1 } } \\qquad \\hbox { a n d } \\omega _ j ( E ) = \\frac { \\omega ( q _ j + r _ j E ) } { \\omega ( B ( q _ j , r _ j ) ) } . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} F ( z ) = \\sum ^ \\mu _ { s = 1 } \\frac { v _ s } { z - z _ s } + \\Theta ( z ) , \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} X _ 1 ( t ) = \\left \\{ \\begin{array} { l l } Z ^ 1 ( t ) & t \\ , \\in \\ , [ 0 , \\sigma _ 1 ) , \\\\ Z ^ 1 ( \\sigma _ 1 ^ - ) + \\psi ( Z ^ 1 ( \\sigma _ 1 ^ - ) , \\alpha ( \\sigma _ 1 ^ - ) , \\pi _ 1 \\circ p ( \\sigma _ 1 ) ) & t = \\sigma _ 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\alpha _ { k , c _ 1 ^ 1 } = 0 \\\\ \\alpha _ { k , c _ 2 ^ 1 } = \\max \\left \\{ \\frac { \\tau _ k ( \\tau _ { \\bar k } + 1 ) } { \\tau _ { \\bar k } ( \\tau _ k + 1 ) } , \\tau _ k \\right \\} \\ , \\mbox { o r } \\ , \\max \\left \\{ \\tau _ k , \\frac { 1 } { \\tau _ { \\bar k } } \\right \\} \\\\ \\alpha _ { k , c _ 3 ^ 1 } = \\max \\left \\{ \\frac { \\tau _ k } { \\tau _ { \\bar k } ( \\tau _ k + 1 ) } , \\tau _ k \\right \\} \\ , \\mbox { o r } \\ , \\frac { \\tau _ k } { \\tau _ { \\bar k } ( \\tau _ k + 1 ) } \\end{array} \\right . \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\bigcup _ { r = 1 } ^ { \\infty } \\bigcup _ { s = 0 } ^ { \\theta - 1 } { \\{ \\nu _ r + \\theta ( r - 1 ) + s \\} } \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} L a w \\left ( ( \\alpha ^ N _ 1 , \\xi ^ N _ 1 , \\N _ 1 ^ N ) , \\ldots , ( \\alpha ^ N _ N , \\xi ^ N _ N , \\N _ N ^ N ) \\right ) = ( L a w ( \\alpha , \\xi , \\N ) ) ^ { \\otimes N } . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = \\sum _ { l = 1 } ^ { r } p _ { j l } \\theta _ { k _ l } + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq k . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} \\psi ( x , \\bar { x } ) = \\Phi ( x ) . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\operatorname { C o n e } ( f ) ^ n = L ^ { n - 1 } \\oplus K ^ { n } \\textrm { a n d } d _ { \\operatorname { C o n e } ( f ) } = \\left ( \\begin{matrix} - d _ L & f \\\\ 0 & d _ K \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} f _ \\lambda ( r ) & = k g _ \\lambda ( \\zeta _ 1 ( r ) ) - k \\sum _ { j = 2 } ^ k G _ \\lambda ( \\zeta _ 1 ( r ) , \\zeta _ j ( r ) ) \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ^ { \\gamma , \\sigma } ( z ) = c _ { \\alpha , \\beta } \\int _ 0 ^ 1 ( 1 - t ^ { 1 / \\alpha } ) ^ { \\beta - \\alpha - 1 } W _ { \\alpha , \\alpha } ^ { \\gamma , \\sigma } ( z t ) d t , \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} \\sigma ( x , t ) & = \\int _ { | x - y | < \\rho } \\xi ( c _ 1 u ( y , t ) ) d y \\geq C \\int _ { | x - y | < \\rho } u ( y , t ) d y = C \\left ( m - \\int _ { | x - y | > \\rho } u ( y , t ) d y \\right ) \\\\ & \\geq C \\left ( m - \\frac { 1 } { \\rho ^ 2 } \\int _ { | x - y | > \\rho } | x - y | ^ 2 u ( y , t ) d y \\right ) \\geq C \\left ( m - \\frac { 2 } { \\rho ^ 2 } \\int _ { \\R ^ 3 } ( | x | ^ 2 + | y | ^ 2 ) u ( y , t ) d y \\right ) \\\\ & = C \\left ( m - \\frac { 2 ( m | x | ^ 2 + E ( t ) ) } { \\rho ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\mathbb { P } ( \\Pi \\mbox { s p l i t s a t } n ) = 1 + \\sum _ { j = 1 } ^ n ( - 1 ) ^ j \\Sigma _ { n , j } , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} R ( \\hbar ) : = \\hbar ^ { n / 2 } R _ { 0 } + \\hbar ^ { ( n - 1 ) / 2 } R _ { 1 } + \\cdots + \\hbar ^ { 1 / 2 } R _ { n - 1 } + R _ { n } , \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\big \\{ \\sigma ^ 1 > n \\ , , \\ , \\sigma ^ 2 > n \\ , , \\ , N ^ 2 _ { [ x , y ] } > n \\big \\} \\subseteq \\big \\{ \\sigma > n \\ , , \\ , \\tau ^ 1 _ { [ x , y ] } > n \\ , , \\ , \\tau ^ 2 _ { [ x , y ] } > n \\big \\} \\\\ = \\big \\{ \\min \\{ \\sigma , \\tau ^ 1 _ { [ x , y ] } , \\tau ^ 2 _ { [ x , y ] } \\} > n \\big \\} \\ , . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\| w \\| _ { X _ T } : = \\sup _ { t \\in [ 1 , T ] } \\bigl \\{ \\| w ( t ) \\| _ { L ^ \\infty } + t ^ { - \\beta } \\| w ( t ) \\| _ { H ^ 1 } \\bigr \\} . \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} & Q ( w ^ \\prime \\ , c ^ l \\ , ( c ^ * ) ^ m ) = Q ( u \\ , ( a a ^ * ) \\ , v \\ , c ^ l \\ , ( c ^ * ) ^ m ) = Q ( u \\ , ( 1 - q ^ 2 c c ^ * ) \\ , v \\ , c ^ l \\ , ( c ^ * ) ^ m ) \\\\ & = Q ( u \\ , v \\ , c ^ l \\ , ( c ^ * ) ^ m ) - q ^ 2 Q ( u \\ , ( c c ^ * ) \\ , v \\ , c ^ l \\ , ( c ^ * ) ^ m ) \\\\ & = Q ( u \\ , v \\ , c ^ l \\ , ( c ^ * ) ^ m ) - q ^ r Q ( u \\ , v \\ , c ^ { l + 1 } \\ , ( c ^ * ) ^ { m + 1 } ) = 0 - 0 = 0 . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} - \\partial _ { \\nu ^ t _ \\theta , { \\rm o u t } } V ^ \\theta = \\frac { 1 } { 2 } ( N \\cdot \\nu ^ t _ \\theta ) \\frac { - \\Delta h ^ 0 } { \\partial _ { s s } u ^ 0 } \\ , \\frac { \\lambda ^ t } { t } \\circ \\bar \\pi _ 1 + \\partial _ { \\nu ^ t _ \\theta , 0 } V ^ \\theta \\mbox { o n } \\Gamma ^ t _ \\theta , \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t ^ 2 u ( t , x ) + \\frac { 1 - 2 \\alpha } { t } \\partial _ t u ( t , x ) & = - \\Delta _ x u ( t , x ) & \\big ( ( t , x ) \\in ( 0 , \\infty ) \\times \\R ^ n \\big ) , \\\\ u ( 0 , x ) & = f ( x ) & ( x \\in \\R ^ n ) , \\end{aligned} \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} f ( h \\tilde * \\pi _ z ( y ) ) = ( \\tilde T _ h ( f \\circ \\pi _ z ) ) ( y ) \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} h _ 0 = p _ 0 ^ { - 3 / 8 } p _ 1 ^ { 2 / 8 } p _ 2 ^ { 1 / 8 } = h _ 3 ^ { - 1 } , h _ 1 = p _ 0 ^ { - 1 / 8 } p _ 1 ^ { - 2 / 8 } p _ 2 ^ { 3 / 8 } = h _ 2 ^ { - 1 } . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( e ^ { i ( k , X ^ { \\beta } ( \\varphi ) - X ^ { \\beta } ( \\psi ) ) } \\big ) = E _ { \\beta } \\left ( - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { d } k _ { j } ^ { 2 } | \\varphi _ { j } - \\psi _ { j } | _ { L ^ { 2 } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} u ^ { T } \\beta u = \\frac { 1 } { 2 } u ^ { T } ( \\beta + \\beta ^ { T } ) u \\leq - \\lambda _ { \\mathrm { s i g } } \\| u \\| ^ { 2 } , \\qquad \\forall u \\in \\mathbb { R } ^ { p } . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\begin{array} { l c l } v ( c ) + i _ * ( I ) - { o } _ * ( O ) & = & 0 \\\\ \\\\ v ' ( c ' ) + { i ' } _ * ( I ' ) - { o ' } _ * ( O ' ) & = & 0 . \\end{array} \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\lim _ { d \\rightarrow \\infty } \\frac { p _ \\alpha ( d ) } { p ( d ) } e ^ { d ^ { 2 + \\epsilon } } & = \\infty \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} F _ { 1 } ^ { \\star \\star } \\left ( Z , W \\right ) = \\left ( f ^ { \\star \\star } , \\varphi ^ { \\star \\star } \\right ) \\left ( Z , W \\right ) . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} P ( \\psi ( X _ { t _ { n + 1 } } ) \\in A | \\ > \\sigma ( \\psi ( X _ { t _ n } ) ) ) = P ( \\psi ( X _ { t _ { n + 1 } } ) \\in A | \\ > \\sigma ( \\psi ( X _ t ) ; t \\in [ 0 , t _ n ] ) ) \\quad \\quad . \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\mathcal { H } _ { h , d } = X _ 1 ^ 2 + \\cdots + X _ { d - 1 } ^ 2 = X _ d ^ 2 + h , \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\frac { ( M _ r { V } ^ x ) _ S } { 2 ^ { r - 4 } } & = \\left [ 6 ( S ( x ) ) ^ 2 + 2 S ( x ) r + r ^ 2 - 4 S ( x ) | S | - 2 | S | ^ 2 - 6 S ( x ) - 3 r + 6 | S | + 2 \\right ] \\\\ & = \\left [ 2 S ( x ) r + r ^ 2 - 4 S ( x ) | S | - 2 | S | ^ 2 - 3 r + 6 | S | + 2 \\right ] , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\widehat { K } = M ^ { - 1 } \\begin{bmatrix} A ^ T & \\| x _ S \\| _ 2 A ^ T \\left ( I _ m - \\frac { 1 } { \\| r \\| _ 2 ^ 2 } r r ^ T \\right ) & \\| r \\| _ 2 \\left ( I _ n - \\frac { 1 } { \\| r \\| _ 2 ^ 2 } A ^ T r x _ S ^ T \\right ) \\end{bmatrix} . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} \\nabla ^ { ^ M } _ { V } W = \\mathcal { T } _ { V } W + \\hat { \\nabla } _ { V } W ; \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ N \\sum _ { j = 0 } ^ { \\theta - 1 } { \\big | q ^ { - z + \\lambda _ { N - i + 1 } ( N ) + \\theta ( i - 1 ) + j } \\big | } \\\\ \\qquad { } \\leq \\sum _ { i = 1 } ^ N \\sum _ { j = 0 } ^ { \\theta - 1 } { q ^ { - \\Re z + \\lambda _ N ( N ) + \\theta ( i - 1 ) + j } } \\leq c \\sum _ { i = 1 } ^ N \\sum _ { j = 0 } ^ { \\theta - 1 } { q ^ { \\theta ( i - 1 ) + j } } < \\frac { c } { 1 - q } , \\end{gather*}"}
-{"id": "9278.png", "formula": "\\begin{align*} [ v , \\omega ] _ { S ^ * _ p } : = \\sup \\limits _ { i \\in \\mathbb Z , E \\in \\mathcal { F } ^ 0 _ i } \\Bigg ( \\frac { \\int _ E { ^ * M _ i } ( \\sigma \\chi _ { E } ) ^ p v d \\mu } { \\sigma ( { E } ) } \\Bigg ) ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\mathcal { K } _ m ( x ) ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\kappa \\left ( s , \\zeta _ i ^ j , x \\left ( \\zeta _ i ^ j \\right ) \\right ) . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} f ^ { ( k ) } ( 0 ) = f ^ { ( \\epsilon ) } \\left ( f ^ { ( ( j - 1 ) e ) } \\left ( f ^ { ( \\ell ) } ( \\beta ) \\right ) \\right ) = f ^ { ( \\epsilon ) } \\left ( f ^ { ( ( j - 1 ) e ) } ( 0 ) \\right ) = f ^ { ( \\epsilon ) } ( 0 ) . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\vartheta } _ 0 \\| _ \\infty \\leq R , \\qquad \\| \\boldsymbol { \\theta } _ { \\boldsymbol { j } } ^ 0 \\| _ \\infty \\leq R 2 ^ { - \\sum _ { l = 1 } ^ d \\alpha _ l j _ l \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha ^ { * } } \\right ) } , \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} \\langle j x , x ^ { \\sim } \\rangle _ { X ' } : = \\overline { \\langle x ^ { \\sim } , x \\rangle } _ { X } , x ^ { \\sim } \\in X ^ { \\sim } , \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\begin{array} { c c l } | U | & \\geq & \\prod _ { m = 1 } ^ { n } ( | q | ^ { - n - 1 / 2 + m } - 1 ) \\prod _ { m = 1 } ^ { \\infty } ( 1 - | q | ^ { m - 1 / 2 } ) \\\\ \\\\ & = & | q | ^ { - n ^ 2 / 2 } \\prod _ { m = 1 } ^ { n } ( 1 - | q | ^ { m - 1 / 2 } ) \\prod _ { m = 1 } ^ { \\infty } ( 1 - | q | ^ { m - 1 / 2 } ) \\\\ \\\\ & \\geq & | q | ^ { - n ^ 2 / 2 } ( \\prod _ { m = 1 } ^ { \\infty } ( 1 - | q | ^ { m - 1 / 2 } ) ) ^ 2 ~ . \\end{array} \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\omega ^ { \\sharp } _ { \\bullet } ( t ) \\lesssim \\sum _ { j = 0 } ^ { \\infty } \\frac { 1 } { 2 ^ { j \\beta } } \\left ( \\tilde \\omega _ { \\bullet } ( 2 ^ j t ) \\ , [ 2 ^ j t \\le 1 ] + \\tilde \\omega _ { \\bullet } ( 1 ) \\ , [ 2 ^ j t > 1 ] \\right ) . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\frac { f _ i } { g _ i } = \\frac { f _ i h ^ n } { g _ i h ^ n } \\{ f _ i h ^ n , g _ i h ^ n \\} \\subset V _ { p + q _ 0 n } . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} \\left ( - \\partial ^ 2 + \\frac { - 2 } { \\cosh ^ 2 ( x ) } \\right ) \\Psi = - \\tau ^ 2 \\Psi \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} g = d s ^ 2 + a ( s ) ^ 2 \\left ( d \\tau - 2 A \\right ) ^ 2 + b ( s ) ^ 2 \\hat { g } , \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} - \\frac { 1 } { n } \\log { { E } } e ^ { - n h ( \\boldsymbol { X } ^ { n } , Y ^ { n } ) } = \\inf _ { \\boldsymbol { \\varphi } ^ { n } = ( \\varphi _ { k } ^ { n } ) _ { k \\in \\mathbb { N } _ { 0 } } \\in \\bar { \\mathcal { A } } _ { b } } { { E } } \\left \\{ \\sum _ { k = 0 } ^ { \\infty } \\int _ { [ 0 , T ] \\times \\lbrack 0 , 1 ] } \\ell ( \\varphi _ { k } ^ { n } ( s , y ) ) \\ , d s \\ , d y + h ( \\bar { \\boldsymbol { X } } ^ { n } , { \\bar { Y } } ^ { n } ) \\right \\} . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} p _ 0 = ( 0 , 0 ) , p _ 1 = ( 1 , 0 ) , p _ t = ( 0 , 2 ) , p _ \\infty = ( 1 , 2 ) . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} S _ 1 = \\sum _ { 2 \\le q \\leq X } \\Lambda ( q ) F _ { \\omega _ { q } } ( n ) ^ 2 \\quad S _ 2 = \\sum _ { 2 \\le q \\leq X } 2 \\Lambda ( q ) \\mu ( \\omega _ { q } ) F _ { \\omega _ { q } } ( n ) \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} h _ R ( x _ 1 ^ { i _ 1 } x _ 2 ^ { i _ 2 } \\cdots x _ { n - 1 } ^ { i _ { n - 1 } } ) = \\sum _ { k = 1 } ^ { n - 1 } ( - 1 ) ^ { | x _ 1 ^ { i _ 1 } \\cdots x _ { k - 1 } ^ { i _ { k - 1 } } | | x _ k | } \\bigtriangleup ( i _ k ) y _ k x _ 1 ^ { i _ 1 } \\cdots x _ { k - 1 } ^ { i _ { k - 1 } } x _ k ^ { i _ k - 1 } x _ { k + 1 } ^ { i _ { k + 1 } } \\cdots x _ { n - 1 } ^ { i _ { n - 1 } } . \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\hat { f } ( n ) = \\int _ \\Sigma f ( x ' ) \\frac { e ^ { - 2 \\pi i n \\cdot x ' } } { L _ 1 L _ 2 } d x ' . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} { \\mathbb E } _ { \\mu } \\left ( \\sup _ { w \\in K ^ { \\circ } \\cap L } \\langle J ( O ) v , w \\rangle \\right ) = R ( K ^ { \\circ } ) \\int \\limits _ { x \\in S ^ { 2 n - 2 } } \\sup _ { w \\in K ^ { \\circ } \\cap L } \\langle x , w \\rangle d \\sigma _ { 2 n - 2 } = R ( K ^ { \\circ } ) M ^ * ( K ^ { \\circ } \\cap L ) . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\Re \\partial _ { 0 } = \\nu > 0 . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} w ( \\varphi _ { x } ) + \\sum _ { y \\neq x } w ( \\varphi _ { \\{ x , y \\} } ) = \\sum _ { i = 1 } ^ { k } \\sum _ { \\substack { ( y , z ) \\in S _ { i } \\\\ x \\not \\in \\{ y , z \\} } } \\frac { 1 + b _ { k } - b _ { i } } { 4 } + \\sum _ { i = 1 } ^ { k } \\sum _ { \\substack { ( y , z ) \\in S _ { i } \\\\ x \\in \\{ y , z \\} } } \\frac { 1 + b _ { k } - b _ { i } } { 4 } = \\sum _ { i = 1 } ^ { k } \\sum _ { ( y , z ) \\in S _ { i } } \\frac { 1 + b _ { k } - b _ { i } } { 4 } \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} & u _ { i } ^ { 0 } = \\sum _ { n = 1 } ^ { \\infty } \\alpha _ { i n } w _ { n } \\ , , \\qquad \\quad \\alpha _ { i n } = \\langle u _ { i } ^ { 0 } , w _ n \\rangle \\ , , \\| u _ i ^ 0 \\| ^ 2 _ { D ( \\sqrt { L } ) } : = \\sum _ { n = 1 } ^ { \\infty } \\alpha _ { i n } ^ 2 \\lambda _ n \\ , , \\\\ & u _ { i } ^ { 1 } = \\sum _ { n = 1 } ^ { \\infty } \\rho _ { i n } w _ { n } \\ , , \\qquad \\quad \\rho _ { i n } = \\langle u _ { i } ^ { 1 } , w _ n \\rangle \\ , , \\| u _ i ^ 1 \\| ^ 2 _ { H } : = \\sum _ { n = 1 } ^ { \\infty } \\rho _ { i n } ^ 2 \\ , . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\tilde { Z } _ t & = & \\int _ 0 ^ t { \\partial _ { z ^ { \\alpha } } ( \\Xi _ { G _ s } ) ( 0 ) d Z ^ { \\alpha } _ s } + \\frac { 1 } { 2 } \\int _ 0 ^ t { \\partial _ { z ^ { \\alpha } z ^ { \\beta } } ( \\Xi _ { G _ s } ) ( 0 ) d [ Z ^ { \\alpha } , Z ^ { \\beta } ] _ s ^ c } + \\\\ & & + \\sum _ { 0 \\leq s \\leq t } ( \\Xi _ { G _ s } ( \\Delta Z _ s ) - \\partial _ { z ^ { \\alpha } } ( \\Xi _ { G _ s } ) ( 0 ) \\Delta Z ^ { \\alpha } _ s ) . \\end{array} \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} \\mathfrak { Y } ( x + \\alpha ) P - P \\mathfrak { Y } ( x ) = \\widetilde { P } - [ \\widetilde { P } ] . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} D _ { \\dot { \\eta } } ^ { \\dot { \\eta } } D _ { \\dot { \\eta } } ^ { \\dot { \\eta } } J + R _ { \\dot { \\eta } } ( J ) = 0 \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} B ' ( x , 1 ) = x ^ 2 A ( x , 1 ) ( 1 + C ( x ) ) - x ^ 2 C ( x ) . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\sum _ { B \\subset A \\subset [ n ] } ( - 1 ) ^ { n - | A | } \\ = \\ ( 1 - 1 ) ^ { n - | B | } \\ = \\ \\begin{cases} 0 & \\mbox { i f } B \\neq [ n ] \\\\ 1 & \\mbox { i f } B = [ n ] \\end{cases} \\ . \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} & \\frac { \\dd ^ m } { \\dd z ^ m } S _ { n , r } ( z ) + \\Big ( { r + 1 \\over 2 } \\Big ) ^ { m } \\psi ^ { ( m - 1 ) } \\Big ( { r ( n + \\nu - 2 ) + ( n + \\nu ) \\over 2 } + \\frac { ( r + 1 ) z } { 2 } \\Big ) \\\\ & - \\Big ( { r \\over 2 } \\Big ) ^ { m } \\psi ^ { ( m - 1 ) } \\Big ( { r ( n + \\nu - 2 ) + ( n + \\nu ) \\over 2 } + \\frac { r z } { 2 } \\Big ) - { r + 1 \\over 2 ^ { m } } \\psi ^ { ( m - 1 ) } \\left ( \\frac { n + \\nu } { 2 } + \\frac { z } { 2 } \\right ) . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\varepsilon } \\Bigl ( X _ { n } - \\sum _ { m , l } b _ { m l } \\ , { \\bf z } _ { m l } \\Bigr ) \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } = 0 , \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} L _ { d } ^ { 2 } : = L ^ { 2 } ( \\mathbb { R } ) \\otimes \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} Q '' = \\left ( f ' - ( 2 n + 1 ) \\frac { b ' } { b } \\right ) Q ' + \\frac { 2 n + 2 } { b ^ 2 } \\left ( Q ^ 3 - Q \\right ) , \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} ( M _ 1 ) _ { \\rho } = H _ 2 ( \\widetilde { X } ) _ { \\rho } , ( M _ 2 ) _ { \\rho } = H _ 2 ^ { ( 0 ) } ( \\widetilde { X } , E _ 1 \\cup E _ 2 ) _ { \\rho } . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } d _ i \\frac { \\| \\nabla v _ i \\| _ 2 ^ 2 } { u _ { i , \\infty } } \\geq P ( \\Omega ) \\min _ { i = 1 , \\ldots , N } \\{ d _ i \\} \\sum _ { i = 1 } ^ { N } \\frac { \\| v _ i - \\overline { v } _ i \\| _ 2 ^ 2 } { u _ { i , \\infty } } . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} k & : = m + ( p - 1 ) \\cdot t \\\\ \\beta _ p ( m , k ) & : = _ p \\left ( \\prod _ { i = 1 } ^ m L ( i - k , \\chi _ { \\boldsymbol { K } } ^ { i - 1 } ) ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} A ( x ) = D f _ 1 ( x ) , B ( x ) = D f _ 2 ( x ) L ( x ) = D h ( x ) . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\limsup _ { p \\to \\infty } \\big | z ^ { ( p ) } _ m - z _ m \\big | ^ { 1 / p } \\leq \\frac { \\Phi ( z _ m ) } { \\rho } , m = 1 , \\ldots , k . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\omega \\bigl ( \\omega ( A , B ) , B ^ { \\sim } \\bigr ) = \\bigl [ T ^ { \\ast [ - 1 ] } B ^ { \\sim } \\bigr ] ^ { \\ast } T ^ { \\ast [ - 1 ] } B ^ { \\sim } \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} [ \\psi ^ { \\varepsilon } ] = T _ \\varepsilon ^ * [ \\psi ] T _ \\varepsilon . \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} u \\left ( x , k _ { 0 } \\right ) \\mid _ { \\Gamma _ { t r } = } g _ { 0 } \\left ( x _ { 1 } , x _ { 2 } , x _ { 0 } \\right ) , \\partial _ { n } u \\left ( x , k _ { 0 } \\right ) \\mid _ { \\Gamma _ { t r } = } g _ { 1 } \\left ( x _ { 1 } , x _ { 2 } , x _ { 0 } \\right ) , \\forall x _ { 0 } \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} L ( s , \\psi \\chi _ a ) = \\sum _ { \\mathcal { A } \\neq 0 } \\frac { \\psi ( \\mathcal { A } ) \\chi _ a ( \\mathcal { A } ) } { N ( \\mathcal { A } ) ^ s } . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} G = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N \\prod _ { j = 1 } ^ \\ell S ^ { j n } g _ j \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } | C _ n | ^ 2 < + \\infty \\ , , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\frac { z \\ ; { } _ { a } \\mathtt { B } ' _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } & = \\frac { a ^ { a / 2 } z } { a } \\frac { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a , 1 } ( a ^ { a / 2 } z ) } { { } _ { a } \\mathtt { B } _ { 2 a - 1 , a \\nu - a + 1 , 1 } ( a ^ { a / 2 } z ) } - ( 2 - a ) \\nu + 1 - a \\\\ & = \\frac { z \\mathtt { J } _ { \\nu - 1 } \\left ( z \\right ) } { \\mathtt { J } _ { \\nu } \\left ( z \\right ) } - ( 2 - a ) \\nu + 1 - a , \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} Y ( t ) = X _ 1 ( t ) + X _ 2 ( t ) + Z ( t ) , t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\mu ^ { ( n ) } = \\frac { \\mu } { t } n ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( X , \\mathcal { C } Y ) - ( \\nabla \\pi _ { \\ast } ) ( Y , \\mathcal { C } X ) , \\pi _ { \\ast } \\omega W ) & = g _ { 1 } ( \\mathcal { A } _ { X } \\mathcal { B } Y + \\mathcal { A } _ { Y } \\mathcal { B } X , \\omega W ) \\\\ & + \\eta ( Y ) g _ 1 ( X , \\omega W ) - \\eta ( X ) g _ 1 ( Y , \\omega W ) \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\rho _ \\pm = \\rho _ 0 \\pm \\kappa n ^ { - 1 / 3 } \\textrm { w i t h } \\rho _ 0 = \\frac { 1 } { \\gamma + 1 } . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} ( \\delta _ { H x H } * \\delta _ { H y H } ) ( \\{ H g H \\} ) = \\frac { | \\{ ( k , l ) : H \\tilde x _ k ^ { - 1 } y _ l H = H g H \\} | } { \\omega _ { H x ^ { - 1 } H } \\cdot \\omega _ { H y H } } . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\nu _ l & = g _ \\lambda ( \\zeta _ 1 ( r ) ) - \\sum _ { j = 1 } ^ { k - 1 } R e \\left [ G _ \\lambda ( \\zeta _ 1 ( r ) , \\zeta _ { j + 1 } ( r ) ) e ^ { \\frac { 2 \\pi i } { k } j l } \\right ] \\\\ & > g _ \\lambda ( \\zeta _ 1 ( r ) ) - \\sum _ { j = 1 } ^ { k - 1 } G _ \\lambda ( \\zeta _ 1 ( r ) , \\zeta _ { j + 1 } ( r ) ) = \\nu _ 0 , \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} 0 = \\sum _ { n = 1 } ^ N \\left ( v _ { n , t + 1 } \\phi ^ l _ n + v _ { n , t - 1 } \\phi ^ l _ n - v _ { n , t } \\left ( a _ { n - 1 } \\phi ^ l _ { n - 1 } + a _ n \\phi ^ l _ { n + 1 } + b _ n \\phi ^ l _ n \\right ) \\right ) - a _ 0 f _ t = 0 \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} X ^ 2 + Y ^ 2 = Z ^ 2 + 1 \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} \\rho \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) + \\pi a ^ { 2 } \\left ( x , t \\right ) \\frac { \\partial P } { \\partial x } \\left ( x , t \\right ) = \\mu a \\left ( x , t \\right ) \\left ( \\frac { d U } { d r } \\mid _ { a \\left ( x , t \\right ) } \\right ) \\left ( \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } r U \\left ( r \\right ) d r \\right ) ^ { - 1 } Q \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\tilde { z } _ n ^ M = \\mathcal { K } _ m z _ n ^ M + f . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\partial _ { \\nu , \\ , { \\rm o u t } } w ( x ) = \\partial _ { \\nu , 0 } w ( x ) - \\frac { 1 } { 2 } f ( x ) \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\psi ( x _ 0 , t _ 0 ) = v _ * ( x _ 0 , t _ 0 ) < - A _ { k + 1 } t _ 0 + \\min ( 0 , | \\lambda _ 1 | t _ 0 - ( d + \\alpha ) \\log ( | x _ 0 | ) ) = W _ { k + 1 } ( x _ 0 , t _ 0 ) , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\int _ 0 ^ t \\ k ( t - s ) u _ { 1 x x } ( s , x ) d s + a u _ 2 ( t , x ) = 0 \\ , , \\\\ \\phantom { u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\int _ 0 ^ t \\ k ( t - s ) u _ { 1 x x } ( s , x ) d s + a u _ 2 ( t , x ) = 0 \\ , , \\qquad } t \\in ( 0 , T ) \\ , , x \\in ( 0 , \\pi ) \\\\ \\displaystyle u _ { 2 t t } ( t , x ) - u _ { 2 x x } ( t , x ) + b u _ 1 ( t , x ) = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} k _ 1 = \\frac { 2 } { 1 1 } , k _ 2 = \\frac { 2 } { 1 1 } , k _ 3 = \\frac { 3 } { 1 1 } , k _ 4 = \\frac { 3 } { 1 1 } . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} S _ m = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\frac { z _ i ^ m - z _ j ^ m } { z _ i - z _ j } = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\sum _ { p = 0 } ^ { m - 1 } z _ i ^ { m - 1 - p } z _ j ^ p , \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{gather*} \\max \\{ A ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\ge 0 \\ u ( x _ 0 ) > 0 \\\\ \\max \\{ B ( \\phi ( x _ 0 ) , \\nabla \\phi ( x _ 0 ) , \\nabla ^ 2 \\phi ( x _ 0 ) ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\ge 0 \\ u ( x _ 0 ) < 0 \\\\ \\max \\{ - \\mathcal { Q } _ \\infty \\phi ( x _ 0 ) , \\nabla F ( \\nabla \\phi ( x _ 0 ) ) \\cdot \\nu \\} \\ge 0 \\ u ( x _ 0 ) = 0 \\end{gather*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ p \\frac { \\hat z _ i ^ { m _ i } - z _ i ^ { m _ i } } { \\hat z _ i - z _ i } \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k a _ j ^ 2 \\le C _ d \\| \\nabla _ \\theta \\varphi \\| _ { L ^ 2 ( \\partial B _ 1 ) } ^ { 2 ( 1 - \\gamma _ k ) } , \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\begin{cases} v \\partial _ y f ^ L = \\langle f ^ L \\rangle - f ^ L \\ , , \\\\ f ^ L ( y = 0 , v > 0 ) = \\phi ( v ) - \\eta \\ , , \\\\ f ^ L ( y = \\infty ) = 0 \\ , ; \\end{cases} \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} f _ h ( \\mathbf { t } _ { \\mathcal { N } } ) = P _ h ( \\mathbf { t } _ { \\mathcal { N } } ) + f ^ { - } _ h ( \\mathbf { t } _ { \\mathcal { N } } ) \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 F \\bigl ( d _ { [ \\lambda , W ] } ( x ) \\bigr ) \\ ; \\mathrm { d } x & = \\lambda F ( 0 ) + \\int _ { \\lambda } ^ 1 F \\left ( ( 1 - \\lambda ) d _ { W } \\left ( \\frac { x - \\lambda } { 1 - \\lambda } \\right ) \\right ) \\ ; \\mathrm { d } x \\\\ & = \\lambda F ( 0 ) + ( 1 - \\lambda ) \\int _ 0 ^ 1 H \\bigl ( d _ W ( x ) \\bigr ) \\ ; \\mathrm { d } x \\ , , \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} v ^ * ( x _ 0 , t _ 0 ) > V _ { k + 1 } ( x _ 0 , t _ 0 ) = \\min ( 0 , B _ { k + 1 } t _ 0 - ( d + \\alpha ) \\log ( | x _ 0 | ) ) , \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} \\tilde { { \\cal M } } _ 0 = \\bigcup _ { i = 0 } ^ { L _ n } \\tilde { { \\cal M } } _ 0 ( i ) \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\Omega _ j = \\frac { 1 } { r _ j } ( \\Omega - q _ j ) \\qquad \\hbox { a n d } \\qquad \\partial \\Omega _ j = \\frac { 1 } { r _ j } ( \\partial \\Omega - q _ j ) . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} e _ x = e _ { x _ 1 } \\otimes \\cdots \\otimes e _ { x _ d } \\in E _ d . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} K ( V , W ) : = \\frac { g _ V ( R ^ V ( V , W ) W , V ) } { g _ V ( V , V ) g _ V ( W , W ) - g _ V ( V , W ) ^ 2 } , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\varphi ( \\bar x , 2 \\kappa ^ i r ) \\lesssim \\varphi ( \\bar x , 2 r ) + \\norm { D u } _ { L ^ \\infty ( B ( x , 1 ) \\cap B ^ + _ 4 ) } \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\mathbf { A } } ( 4 t ) } t \\ , d t + \\int _ 0 ^ r \\frac { \\tilde \\omega _ { \\vec g } ( 4 t ) } t \\ , d t . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} d _ 1 = 0 , ~ ~ ~ ~ d _ 2 = c _ 2 - \\left ( \\frac { 1 } { n - 1 } \\right ) ^ 2 = \\frac { 2 ( n - 2 ) } { ( n - 1 ) ^ 2 ( n + 1 ) } , \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\rho ^ 2 } H ( t , \\rho ) | _ { \\rho = 0 } = 2 \\int \\limits _ { - 1 } ^ t f ( s ) H ( s , 0 ) d s = 2 \\int \\limits _ { - 1 } ^ t f ( s ) G ( s ) d s . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} \\ker ( T _ 4 ) = \\begin{cases} H _ 4 / H _ 4 ^ \\prime \\simeq \\langle x y ^ 2 , s _ 2 \\rangle \\simeq C _ 3 \\times C _ 3 a = w = z = 0 a = 1 , \\ w = - 1 , \\\\ G ^ \\prime / H _ 4 ^ \\prime \\simeq \\langle s _ 2 \\rangle \\simeq C _ 3 . \\end{cases} \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} w ( x ) : = \\int _ { \\partial U } d \\mathcal { H } ^ { n - 1 } ( y ) \\ , f ( y ) P ( x - y ) \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha f ( x _ { i - 1 } ( t ) ) + a f ( x _ { i } ( t ) ) + \\beta f ( x _ { i + 1 } ( t ) ) , \\ , \\ , i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} & \\mathrm { E } _ f ( 1 - \\Phi _ n ) = P _ f \\left ( \\| \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - f _ 0 \\| _ \\infty \\leq M _ 0 \\rho _ n \\epsilon _ n \\right ) \\\\ & \\leq P _ f \\left ( \\| \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - \\mathrm { E } _ f \\widehat { f } _ { n , \\boldsymbol { \\alpha } } \\| _ \\infty \\geq \\| f - f _ 0 \\| _ \\infty - M _ 0 \\rho _ n \\epsilon _ n - \\| \\mathrm { E } _ f \\widehat { f } _ { n , \\boldsymbol { \\alpha } } - f \\| _ \\infty \\right ) . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} r _ { t - 1 } ^ { N , h } = \\left ( R ^ T _ { N , h } \\delta \\right ) _ t = v ^ \\delta _ { 1 , t } , t = 1 , \\ldots , T . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\mu ( w , y ) = \\binom { | S | - f } { a - S ( w ) - S ( y ) } \\binom { r - | S | + f - 4 } { b - f + S ( w ) + S ( y ) } \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} F _ x ( x ) ~ = ~ \\norm { F _ x } ^ 2 , \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} \\aligned \\lim _ { \\tau \\to - \\infty } u ( \\tau , t ) & = \\gamma _ { - \\infty } ( t ) , \\\\ \\lim _ { \\tau \\to + \\infty } u ( \\tau , t ) & = \\gamma _ { + \\infty } ( t ) . \\endaligned \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} ( 1 - \\omega _ { \\boldsymbol { j } , n } ) \\exp { \\left ( - \\kappa ( \\underline { \\gamma } ) \\frac { \\log { n } } { 2 \\sigma ^ 2 } - \\frac { \\sigma _ 0 ^ 2 } { \\sigma ^ 2 } \\log { 2 ^ { \\sum _ { l = 1 } ^ d j _ l } } \\right ) } \\int _ { \\mathcal { W } _ n } K _ n ( \\widetilde { \\Theta } ) d \\Pi ( \\widetilde { \\Theta } ) . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} a _ 1 = 8 ( \\alpha _ 3 \\pi ) ^ 2 , a _ 3 = 1 2 0 \\ , ( \\alpha _ 3 \\pi ^ 2 ) ^ 2 . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} \\underset { n \\to + \\infty } { \\lim } \\int _ { \\mathbb { R } ^ N } f ( u _ n ) z = \\int _ { \\mathbb { R } ^ N } f ( u ) z . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} A ( \\psi , \\phi ) ( z ) = 2 ^ { - n } W ( \\psi , \\phi ^ { \\vee } ) ( \\tfrac { 1 } { 2 } z ) , \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { C _ { n , 1 , \\theta } } { n } = \\beta ( \\theta ) a . s . \\mbox { w i t h } \\lim _ { \\theta \\downarrow 0 } \\beta ( \\theta ) = 1 . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} W ( s ; f , \\overline { g } ) = \\frac { L ( s , f \\times \\overline { g } ) } { \\zeta ( 2 s ) } + Z ( s , 0 , f \\times \\overline { g } ) . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} ( 1 + x ) ^ p = 1 + p x + \\mathcal { O } ( x ^ 2 ) , \\log ( 1 + x ) = x + \\mathcal { O } ( x ^ 2 ) , x \\to 0 . \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} y = \\vec \\Phi ( x ) = ( \\Phi ^ 1 ( x ) , \\ldots , \\Phi ^ n ( x ) ) , \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} x _ { 0 } ^ { 3 } x _ { 2 } + x _ { 1 } x _ { 2 } ^ { 3 } + x _ { 1 } ^ { 3 } x _ { 3 } + x _ { 0 } x _ { 3 } ^ { 3 } = 0 . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} ( N _ 2 \\boldsymbol { u } ) _ { i , j } = 0 , ( N _ 2 \\boldsymbol { v } ) _ { i , j } = 0 , ( N _ 2 \\boldsymbol { w } ) _ { i , j } = 0 , \\mbox { w h e r e } ( N _ 2 \\boldsymbol { u } ) _ { i , j } \\triangleq \\left \\{ \\begin{matrix} \\dfrac { u _ { i , j + 1 } - u _ { i , j } } { \\delta y } j = 1 , \\\\ \\\\ \\dfrac { u _ { i , j } - u _ { i , j - 1 } } { \\delta y } \\quad j = J . \\end{matrix} \\right . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { x \\in M } \\log \\lVert A ^ n ( x ) \\rVert ^ { 1 / n } = \\sup _ { ( x , p ) \\in M \\times \\mathbb N : f ^ p ( x ) = x } \\log \\rho ( A ^ p ( x ) ) ^ { 1 / p } , \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} w = P _ { t ^ { - \\frac 1 6 } < \\cdot \\leq t ^ { \\frac 1 { 1 2 } } } u , \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\phi ( g h ) = \\phi ( g ) \\phi ( h ) \\Longleftrightarrow ( x _ { g h } , \\sigma _ { g h } ) = ( x _ { g } ( \\sigma _ { g } \\circ x _ h ) , \\sigma _ { g } \\sigma _ { h } ) \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\partial _ t [ ( \\alpha + 1 ) c _ 1 ^ { \\varepsilon } + c _ 2 ^ { \\varepsilon } + c _ 3 ^ { \\varepsilon } ] - \\Delta [ ( \\alpha + 1 ) d _ 1 c _ 1 ^ { \\varepsilon } + d _ 2 c _ 2 ^ { \\varepsilon } + d _ 2 c _ 3 ^ { \\varepsilon } ] = 0 . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} J _ \\nu ( z ) : = \\sum _ { n \\geq 0 } \\frac { ( - 1 ) ^ n } { \\Gamma ( n + 1 ) \\Gamma ( \\nu + n + 1 ) } ( z / 2 ) ^ { \\nu + 2 n } . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} \\hat u _ { t , x } ( r ) = B ^ * Q _ { t + r } ^ { - 1 } \\hat y _ { t , x } ( r ) , r \\in \\ , ] - t , 0 ] , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} w ( Y _ n ) = \\prod _ { i = 1 } ^ { 2 n } ( 1 + x _ i ) \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ f : f \\in L ^ 1 ( \\Omega \\times V ) , v \\cdot \\nabla _ x f \\in L ^ 1 ( \\Omega \\times V ) \\} \\ , . \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} ( ( I ( X _ { \\gamma _ { t } } ) , t \\geq 0 ) , \\mathbb { P } _ x ) \\stackrel { ( d ) } { = } ( ( \\hat { X } ^ { \\hat h } _ t , t \\geq 0 ) , \\mathbb { P } _ { I ( x ) } ) , \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} C = 6 e ^ 2 \\left ( \\sum _ { i = 1 } ^ { n } | a ( e _ i ) | \\right ) ^ 2 + 4 e ^ 2 \\sum _ { i , j = 1 } ^ { n } | a ( e _ i + e _ j ) | . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} n _ { s ^ * } - k _ { s ^ * } & = n _ { s ^ * } - \\lfloor m _ { s ^ * } \\rfloor r _ { s ^ * } \\geq n _ { s ^ * } - m _ { s ^ * } r _ { s ^ * } \\\\ & = m _ { s ^ * } ( \\delta _ { s ^ * } - 1 ) \\geq \\lfloor m _ { s ^ * } \\rfloor ( \\delta _ { s ^ * } - 1 ) \\\\ & = \\frac { k _ { s ^ * } } { r _ { s ^ * } } ( \\delta _ { s ^ * } - 1 ) \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} C _ k ^ i = C _ k ^ i ( N , \\gamma , \\omega _ 0 , X , \\| \\phi ( \\cdot , 0 ) \\| _ { C ^ { k + 4 } ( B _ r ( p ) ) } , \\| \\phi \\| _ { C ^ 0 ( B _ r ( p ) \\times [ 0 , T ] ) } , \\| \\widetilde { \\eta } \\| _ { C ^ { k + 2 } ( B _ r ( p ) ) } , \\| F \\| _ { C ^ 0 ( B _ r ( p ) ) } ) \\ ; \\ ; ( i = 1 , 2 , 3 ) \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\| Q _ \\infty ^ { - 1 / 2 } x \\| _ X = \\| x \\| _ H \\forall x \\in H . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} w _ { \\eta } ( x ) & = { } ^ { t } R w ( R x ) \\\\ & = w ^ { r } ( r , \\theta + \\eta , z ) e _ { r } ( \\theta ) + w ^ { \\theta } ( r , \\theta + \\eta , z ) e _ { \\theta } ( \\theta ) + w ^ { z } ( r , \\theta + \\eta , z ) e _ { z } . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\norm { T _ { \\lambda , k } ^ z } _ 2 & \\lesssim \\abs { z ( z - 1 ) } 2 ^ { - ( d - 2 ) k \\sigma } \\abs { \\lambda 2 ^ { - ( d - 2 ) k } } ^ { - 1 / 2 } 2 ^ { - ( 2 n - 2 ) k / 2 } \\\\ & = C _ z 2 ^ { - ( d - 2 ) k \\sigma } 2 ^ { ( d - 2 n ) k / 2 } \\abs { \\lambda } ^ { - 1 / 2 } \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\int _ { S ' _ { d } } \\langle w , \\varphi \\rangle _ { 0 } ^ { 2 n + 1 } \\ , d \\mu _ { \\beta } ( w ) & = 0 , \\\\ \\int _ { S ' _ { d } } \\langle w , \\varphi \\rangle _ { 0 } ^ { 2 n } \\ , d \\mu _ { \\beta } ( w ) & = \\frac { ( 2 n ) ! } { 2 ^ { n } \\Gamma ( \\beta n + 1 ) } | \\varphi | _ { 0 } ^ { 2 n } . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\hat { \\theta } _ { i } ^ { \\prime } = p _ { i } ^ { \\prime } \\alpha + \\xi _ { i } ^ { \\prime } , \\ \\forall \\ 1 \\leq i \\leq \\bar { k } , \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} \\rho ^ { N , h } ( \\lambda ) = \\sum _ { \\{ k \\ , | \\ , \\lambda _ k < \\lambda \\} } \\frac { a _ 0 } { \\rho _ k } , \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} \\tilde \\sigma _ 1 ( \\lambda , r ) = g _ \\lambda ( \\zeta _ 1 ( r ) ) - \\sum _ { j = 1 } ^ { k - 1 } G _ \\lambda ( \\zeta _ 1 ( r ) , \\zeta _ { j + 1 } ( r ) ) , \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} T u = \\frac { \\nabla u } { \\sqrt { 1 + | \\nabla u | ^ 2 } } \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\mu = \\int \\mu _ \\omega \\ , d P ( \\omega ) . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} A ( p , u ) & = \\begin{cases} \\frac { p } { p - 1 } ( \\varrho _ { \\ell } ( p , u ) - 1 ) - 1 & u \\equiv 0 ( \\bmod p ) \\\\ \\frac { p } { p - 1 } ( \\varrho _ { \\ell } ( p , u ) - 1 ) + \\frac { 1 } { p - 1 } & u \\not \\equiv 0 ( \\bmod p ) \\end{cases} \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\overline { D } ' = ( D , g ) + \\left ( 0 , \\sum _ { v \\in M _ K \\setminus ( U \\cap M _ K ^ { \\mathrm { f i n } } ) } c [ v ] \\right ) , \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} \\P _ x \\left ( \\left \\{ \\int _ 0 ^ t | f | ( X _ s ) d s < \\infty \\right \\} \\cap \\{ t < \\zeta \\} \\right ) = \\P _ x \\left ( \\{ t < \\zeta \\} \\right ) , \\ \\ f \\in L ^ q _ { l o c } ( \\R ^ d , m ) . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} C _ { \\pi ^ { - s } } = c _ 0 + \\sum _ { r = 0 } ^ { n - 2 } \\beta ^ { \\pi ^ { r - s } } c _ { \\pi ^ r } , \\ \\ \\ s = 0 , 1 , \\cdots , n - 2 . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\hat { \\rho } _ j ( x _ { \\alpha _ \\varepsilon } - n e \\wedge x _ { \\alpha _ \\varepsilon } ) = \\rho _ j ( x _ { \\alpha _ \\varepsilon } - n e \\wedge x _ { \\alpha _ \\varepsilon } ) < \\varepsilon ~ ~ ~ ~ ( \\forall n \\ge n _ \\varepsilon ) . \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d + 3 } ( v _ i , \\ , e _ j ) \\widetilde { D } _ i = 0 j = 1 , 2 , 3 , \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\partial _ t X = \\Delta X - \\frac { m } { m + 2 } : | X | ^ { 2 } X : + \\frac { 1 } { \\sqrt { m } } \\xi \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} y ' ( r ) = ( A + B B ^ * Q _ { t + r } ^ { - 1 } ) y ( r ) , r \\in \\ , ] - t , 0 ] \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\ ( \\frac { n ^ m ( n - m ) ! } { n ! } \\ ) ^ 4 \\sum _ { \\substack { \\chi \\in \\hat { G } ^ m \\\\ \\chi \\neq 0 } } | \\hat { 1 _ { S _ m } } ( \\chi ) | ^ 4 = \\ ( \\frac { n ^ n } { n ! } \\ ) ^ 4 \\sum _ { \\substack { \\chi \\in \\hat { G } ^ m \\\\ \\chi \\neq 0 } } | \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) | ^ 4 . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\left ( C ^ T _ { N , h } f , g \\right ) _ { \\mathcal { F } ^ T } = \\left ( v ^ f _ { \\cdot , T } , v ^ g _ { \\cdot , T } \\right ) _ { \\mathcal { H } ^ N } = \\left ( W ^ T _ { N , h } f , W ^ T _ { N , h } g \\right ) _ { \\mathcal { H } ^ N } . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\alpha _ { Z ^ { ( 1 / 2 ) } } = 2 \\alpha _ { Z } { \\rm \\ a n d \\ } \\beta _ { Z ^ { ( 1 / 2 ) } } = 2 \\beta _ { Z } , \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\omega _ { \\Lambda _ n } = \\omega _ { \\Lambda _ \\infty } , \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } - \\Delta u + \\mu u = | u | ^ { p - 1 } u + \\lambda v , & x \\in \\mathbb { R } ^ { N } , \\\\ - \\Delta v + \\nu v = | v | ^ { 2 ^ { * } - 1 } v + \\lambda u , & x \\in \\mathbb { R } ^ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} ( L _ a ) _ { \\rho } ( \\dot { x } , { \\dot \\tau } , { \\dot \\xi } ) = { \\dot \\tau } I + A _ 1 { \\dot \\xi } + { \\bar \\eta } A _ 2 ( a ) { \\dot x } \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n c _ { i , j } \\equiv 0 \\bmod { 2 } \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} y F ( y + z ) + z F ( y ) = y \\cdot ( 5 y + 5 z - 2 ) + z \\cdot \\tfrac { 5 y - 1 } { 2 } \\le \\tfrac 3 5 \\ , . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\chi ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) = \\frac { { c _ 1 ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) } ^ 2 } { 2 } - c _ 2 ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) + ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) \\chi ( \\mathcal { O } _ X ) . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\sigma ( z , z ' ) = p x ' - p ' x , \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} m ( t , x ) = \\min _ { \\alpha \\in \\mathbb { R } ^ { n } } ( C ( \\vert \\alpha \\vert ^ { 2 } t + \\vert \\alpha \\vert ^ { \\mu } \\Lambda ^ { \\mu } t ^ { \\nu } ) + \\alpha \\cdot x ) \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} U ( t ) x : = \\begin{cases} \\frac { 1 } { \\Gamma ( \\alpha ) } \\left ( \\frac { t } { 2 } \\right ) ^ { 2 \\alpha } \\int \\limits _ 0 ^ \\infty r ^ { - \\alpha } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } T ( r ) x \\ , \\frac { d r } { r } & t > 0 , \\\\ x & t = 0 . \\end{cases} \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} V ( x ) = \\sum _ { i = 1 } ^ N \\frac { m _ i } { \\alpha \\vert x - c _ i \\vert ^ { \\alpha } } , \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\phi } } \\right ) \\nabla _ { \\phi } { \\rm R m } _ { \\phi } = \\nabla _ { \\phi } \\left ( \\frac { d } { d t } - \\Delta _ { \\omega _ { \\phi } } \\right ) { \\rm R m } _ { \\phi } + { \\rm R m } _ { \\phi } \\ast \\nabla _ { \\phi } ( { \\rm R m } _ { \\phi } + \\widetilde { \\eta } + \\nabla _ { \\phi } X ) , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} a : = \\min \\left \\{ \\frac { \\nu ( f _ 1 ) } { d _ 1 } , \\dots , \\frac { \\nu ( f _ r ) } { d _ r } \\right \\} . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} U _ k : = \\left \\{ \\sum _ { h \\in \\mathbb { Z } _ 2 ^ { \\oplus k } } r _ h h \\mid r _ h \\in \\mathbb { R } \\ , \\ , \\ , \\ , \\ , \\sum _ h r _ h = 0 \\right \\} , \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} A \\otimes W = \\left ( \\displaystyle \\sum _ { k , l = 1 } ^ { q ' } a _ { k l } ^ { i j } w _ { k l } \\right ) _ { 1 \\leq i \\leq q ' \\atop 1 \\leq j \\leq N ' } . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} X = \\left ( \\begin{matrix} 1 & 0 & 2 & 0 & 3 \\\\ 0 & 1 & 0 & 2 & 0 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} P & = \\sum _ a ( - 1 ) ^ a \\binom { a } { 2 } \\binom { | S | - | U ' | } { a - | U ' | } \\\\ Q & = \\sum _ b ( - 1 ) ^ b \\binom { r - | S | - | U | + | U ' | } { b } \\\\ P ' & = \\sum _ a ( - 1 ) ^ a \\binom { | S | - | U ' | } { a - | U ' | } \\\\ Q ' & = \\sum _ b ( - 1 ) ^ b \\binom { b } { 2 } \\binom { r - | S | - | U | + | U ' | } { b } . \\\\ \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} f '' & = ( 2 n + 1 ) \\frac { a '' } { a } \\\\ a '' & = \\frac { 2 n } { a } \\left ( 1 - ( a ' ) ^ 2 \\right ) + a ' f ' . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\Delta ^ { 1 / 4 } S \\Delta ^ { - 1 / 4 } = \\Delta ^ { 1 / 4 } J \\Delta ^ { 1 / 4 } = \\Delta ^ { 1 / 4 } \\Delta ^ { - 1 / 4 } J = J \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} F _ p ( p , p , 2 ) & = \\dim ( p - 1 , 2 ) + \\dim ( p , 1 ) + \\dim ( p , 1 , 1 ) \\cr & = \\dim ( p , 2 ) + \\dim ( p , 1 , 1 ) \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\beta = ( Q \\otimes i d _ { \\mathcal { C } } ) \\ , \\Delta _ { \\mathcal { C } } . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} F ( f ) = \\mathbb E \\sum _ { n = 1 } ^ { \\infty } \\langle f _ n , \\tilde g _ n \\rangle = \\mathbb E \\sum _ { n = 1 } ^ { \\infty } \\mathbb E _ { n } \\langle f _ n , \\tilde g _ n \\rangle = \\mathbb E \\sum _ { n = 1 } ^ { \\infty } \\langle f _ n , \\mathbb E _ { n } \\tilde g _ n \\rangle = \\mathbb E \\sum _ { n = 1 } ^ { \\infty } \\langle f _ n , g _ n \\rangle . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} u ^ 2 ( t ) = \\\\ \\begin{cases} ( 0 , 0 , 0 , 0 ) \\\\ ( 0 , 0 , 1 , 0 ) \\\\ ( 1 , 0 , 0 , 0 ) \\end{cases} ~ Q ( t ) = 1 . \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} C \\| G X \\mathbf { e } _ i \\| \\| G \\mathbf { e } _ i \\| = \\frac { C } { \\eta } \\sqrt { \\Im ( X G X ) _ { i i } } \\sqrt { \\Im G _ { i i } } \\leq C ' \\frac { \\Im G _ { i i } } { \\eta } . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\overline { \\rho _ { 0 , 4 } } = & \\underset { j = 1 } { \\stackrel { d } { \\textstyle { \\prod } } } ( w _ { 0 1 } w _ j w _ { 0 2 } ^ { - 1 } w _ j ^ { - 1 } ) ^ { c _ { 0 j } } { [ w _ j w _ { 0 2 } w _ j ^ { - 1 } , w _ { 0 1 } ] ^ { c _ { 0 0 j } } } \\\\ & \\cdot \\underset { 0 < a < b } { \\textstyle { \\prod } } ( w _ { 0 2 } ^ { - 1 } w _ a ^ { - 1 } w _ { 0 1 } w _ a ) ^ { 2 c _ { 0 a } c _ { 0 b } } { [ ( w _ a w _ b ^ { - 1 } ) ^ 2 , w _ { 0 1 } ] ^ { c _ { 0 a b } } } \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\bar { \\psi } + \\frac { 2 } { 3 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\psi _ i \\right ) ^ 3 + \\Pi ^ T \\Omega ^ { - 1 } \\Pi - \\frac { 1 } { 2 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\psi _ i \\right ) ^ 4 + O _ p ( n ^ { - 5 / 2 } ) \\ , . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\dd u _ t \\ , = \\ , ( - u _ t + g X u _ t ) \\dd t + \\dd B _ t \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} \\frac { d } { d t } \\log ( E ( t ) ) = P ( - t ) \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\sigma _ Y ( m ) = \\lim _ { t \\to \\infty } \\frac { \\log \\left | \\kappa _ Y ^ { ( m ) } ( t ) \\right | } { \\log t } , m \\in \\N , \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\| A ( \\chi _ n f ) \\| _ Y \\leq \\| A \\| _ { H [ X ] \\to H [ Y ] } \\| \\chi _ n f \\| _ { H [ X ] } = \\| A \\| _ { H [ X ] \\to H [ Y ] } \\| f \\| _ { X } . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} \\Delta ^ { ( n , \\sigma ) } _ { 2 } ( P ) = \\left ( \\begin{array} { c c } \\Delta ^ { ( n , \\sigma ) } ( x x ^ { * } ) & \\Delta ^ { ( n , \\sigma ) } ( x y ^ { * } ) \\\\ \\Delta ^ { ( n , \\sigma ) } ( y x ^ { * } ) & \\Delta ^ { ( n , \\sigma ) } ( y y ^ { * } ) \\\\ \\end{array} \\right ) = \\sum _ { i = 1 } ^ { n } A _ { i } \\otimes E _ { i i } , \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\sup _ { P _ n } | p _ n | < 2 ^ { - n } \\quad | p _ n ( a _ n ) | > n + \\sum _ { k = 1 } ^ { n - 1 } | p _ k ( a _ n ) | . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\forall x \\in \\R ^ n , \\alpha _ x ( A - C ) x + \\beta _ x ( B - C ) x = 0 , \\mbox { f o r s o m e } ( \\alpha _ x , \\beta _ x ) \\neq ( 0 , 0 ) . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 0 ^ { m - k } , \\chi _ 1 , \\dots , \\chi _ k , 0 ^ { n - m } ) \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} w _ r ^ { ( 1 ) } ( \\lambda ) = \\frac { ( 1 - \\lambda ^ 2 ) ^ { L / 2 - 1 } } { \\sqrt { 2 \\pi } } \\left ( L \\frac { \\Gamma ( 1 / 2 ) \\Gamma \\left ( ( L + 1 ) / 2 \\right ) } { \\Gamma ( L / 2 ) } \\right ) ^ { 1 / 2 } , | \\lambda | < 1 \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} X _ b : = ( \\P ( V _ 1 ) \\times \\P ( V _ 2 ) \\times \\P ( V _ 3 ) ) \\cap \\P ( b ^ \\perp ) \\subset \\P ( V _ 1 \\otimes V _ 2 \\otimes V _ 3 ) \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} E ( t ) = \\sum _ { r \\geq 0 } e _ r ( \\mathbf { x } ) t ^ r = \\prod _ { i } ( 1 + x _ i t ) \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\Xi \\wedge \\Xi ' & = ( z - \\bar { z } ) \\Psi \\wedge e _ { h + 1 } \\wedge f _ { h + 1 } + \\Phi \\wedge \\Xi ' \\\\ & = ( z - \\bar { z } ) M _ 1 e _ 0 \\wedge f _ 0 \\wedge \\cdots \\wedge e _ { h + 1 } \\wedge f _ { h + 1 } + \\Phi \\wedge \\Xi ' , \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{gather*} \\sup _ { i \\geq 1 } \\sup _ { x \\in K } \\left | x ^ z \\cdot \\prod _ { j = 1 } ^ { \\infty } { \\frac { \\big ( q ^ { - z + \\nu _ j ^ { ( i ) } } t ^ j ; q \\big ) _ { \\infty } } { \\big ( q ^ { - z + \\nu _ j ^ { ( i ) } } t ^ { j - 1 } ; q \\big ) _ { \\infty } } } \\right | \\leq c \\cdot a ^ r , \\textrm { i f } z = r \\pm \\frac { \\pi \\sqrt { - 1 } } { \\ln { q } } . \\end{gather*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\L = \\begin{pmatrix} \\alpha & 0 \\\\ \\beta \\gamma & \\beta \\end{pmatrix} \\Z ^ 2 = \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\beta \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ \\gamma & 1 \\end{pmatrix} \\Z ^ 2 \\ , . \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\jmath ^ { \\ , ' } _ p \\circ \\eta _ a \\ , = \\ , \\beta _ p \\circ \\eta _ a \\circ \\jmath _ p \\ , \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} P _ n ( x ) & = \\log 2 \\cdot \\omega _ 0 ( n ) + \\frac 1 4 \\sum _ { 2 \\le q \\le X } \\omega _ { q } ( n ) ^ 2 \\Lambda ( q ) \\\\ D ( x ) & = \\log 2 \\cdot \\mu ( \\omega _ 0 ) + \\frac { 1 } { 4 } \\sum _ { 2 \\le q \\leq X } \\mu ( \\omega _ q ) ^ 2 \\Lambda ( q ) , \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} \\tau _ 0 \\triangleq \\inf \\{ t > 0 , Y _ t = 0 \\mbox { a n d } \\vert Z _ t \\vert = P _ Y \\} , \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq t \\leq T } & | L a w ( X ( t ) ) - L a w ( Y ( t ) ) | = : | | F l o w ( X ) - F l o w ( Y ) | | _ { \\infty } \\\\ & \\leq 2 T ^ \\ast \\sqrt { d } e ^ { T ^ \\ast C _ 1 } \\left [ C _ 2 + C _ 3 ( K _ 2 + T ^ \\ast C _ 4 ) e ^ { T ^ \\ast C _ 5 } \\right ] | | m - n | | _ { \\infty } , \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\nu _ \\infty ( f ) \\prod _ { \\ell } \\nu _ \\ell ( f ) = \\frac { 1 } { \\omega _ K } \\frac { h _ { K } } { h _ { K ^ + } } . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} f _ { V , W } ( x _ 1 , x _ 2 ) = \\frac { 1 } { 2 \\pi } \\frac { b } { ( b ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 ) ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\sum ^ \\infty _ { j = 0 } \\abs { ( \\ , g _ j \\ , | \\ , \\chi _ { \\varepsilon _ 1 } \\ , ) _ M - ( \\ , g _ j \\ , | \\ , \\chi _ { \\varepsilon _ 2 } \\ , ) _ M } ^ 2 < \\delta . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus N _ \\Omega } \\nabla v \\nabla \\varphi + \\int _ { \\Omega \\setminus N _ \\Omega } \\beta ' ( w ) v \\varphi = 0 \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} Q _ t ^ { 1 / 2 } Q _ t ^ { - 1 / 2 } z = z \\forall z \\in R ( Q _ t ^ { 1 / 2 } ) , Q _ t ^ { - 1 / 2 } Q _ t ^ { 1 / 2 } x = P _ { [ \\ker Q _ t ] ^ \\perp } \\ , x \\forall x \\in X . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} \\int _ { B ( 0 , R ) } | \\nabla u _ j | ^ 2 \\ , d Z & = \\int _ { B ( q _ j , R r _ j ) } \\frac { r _ j ^ { n - 2 } } { ( \\omega ( B ( q _ j , r _ j ) ) ) ^ 2 } | \\nabla G ( X _ 0 , Y ) | ^ 2 \\ , d Y \\\\ & \\leq C \\frac { r _ j ^ { n - 4 } } { ( \\omega ( B ( q _ j , r _ j ) ) ) ^ 2 } \\int _ { B ( q _ j , 2 R r _ j ) } G ( X _ 0 , Y ) ^ 2 d Y . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} L ( a ( 1 + q ) , b ) & = \\frac { 1 - \\sqrt { ( 1 - 2 a - 2 a q ) ^ 2 - 4 b } \\ , } { 2 } \\\\ & = \\frac { 1 - \\sqrt { ( 1 - 2 a ) ^ 2 - 4 b - 2 q ( 2 a - 4 a ^ 2 ) + q ^ 2 4 a ^ 2 } \\ , } 2 . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\underline { \\lim } _ n F ( x + y _ n ) = \\underline { \\lim } _ n ( 2 F ( x ) - F ( x - y _ n ) ) = 2 - \\overline { \\lim } _ n F ( x - y _ n ) \\geq 1 . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} f ( r ) = \\frac { 2 } { \\pi } \\frac { a } { r ^ 2 + a ^ 2 } , a > 0 , r > 0 . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} H ( t ) \\ ; = \\ ; \\sum _ { n = 1 } ^ N \\ ; N \\ ; H ( n ) \\ ; \\chi \\big ( t \\in [ 2 \\pi \\tfrac { n - 1 } { N } , 2 \\pi \\tfrac { n } { N } ) \\big ) \\ ; . t \\in [ 0 , 2 \\pi ) \\ ; , \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} C _ 1 = \\frac { 2 \\sqrt { 1 - \\delta _ { ( a + 1 ) k } } + \\sqrt { 1 + \\delta _ { a k } } / \\sqrt { a } } { \\sqrt { a } \\sqrt { 1 - \\delta _ { ( a + 1 ) k } } - \\sqrt { 1 + \\delta _ { a k } } } \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ \\{ a _ n \\} _ { n = 1 } ^ \\infty \\in \\ell ^ 2 \\mid \\{ \\lambda _ n a _ n \\} _ { n = 1 } ^ \\infty \\in \\ell ^ 2 \\} \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} \\Lambda ( s , f _ \\chi ) = \\int _ 0 ^ \\infty f _ \\chi ( i y ) y ^ { s + \\frac { k - 1 } { 2 } } \\ ; \\frac { d y } { y } = \\frac { \\Gamma \\left ( s + \\frac { k - 1 } { 2 } \\right ) } { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } L ( s , f _ \\chi ) . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} M ( X ) = M _ 0 ( X ) \\oplus \\cdots \\oplus M _ { 2 n } ( X ) \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} \\Vert T ( h ) \\Vert _ { \\ast } \\leq C \\ , \\Vert h \\Vert _ { \\ast \\ast } \\quad \\vert c _ { i j } \\vert \\leq C \\ , \\Vert h \\Vert _ { \\ast \\ast } , \\ , \\ , i = 1 , \\ldots , k , \\ , \\ , j = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} 2 4 = \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) + \\sum _ { F ^ \\circ \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ^ \\circ ) + E _ \\Delta . \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\lim _ { r \\to + \\infty } \\int _ { - r } ^ r | \\eta _ 2 ( x _ 1 , t ) | \\ , d x _ 1 = \\lim _ { r \\to + \\infty } \\int _ 0 ^ t \\eta _ 1 ( - r , x _ 2 ) - \\eta _ 1 ( r , x _ 2 ) \\ , d x _ 2 = \\int _ 0 ^ t \\lim _ { r \\to + \\infty } ( \\eta _ 1 ( - r , x _ 2 ) - \\eta _ 1 ( r , x _ 2 ) ) \\ , d x _ 2 = 0 \\ , . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\begin{cases} P _ { i m a x } \\bar { z } _ { i i } / ( P _ { j m a x } \\bar { z } _ { j i } ) \\geq \\gamma _ i \\\\ P _ { j m a x } \\bar { z } _ { j j } / ( P _ { i m a x } \\bar { z } _ { i j } ) \\geq \\gamma _ j \\end{cases} \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , \\chi _ c \\right ) = 2 \\sum _ { 0 \\neq \\mathcal { A } \\subset \\mathcal { O } _ K } \\frac { \\chi _ c ( \\mathcal { A } ) } { N ( \\mathcal { A } ) ^ { 1 / 2 } } V \\left ( \\frac { \\pi N ( \\mathcal { A } ) } { N ( c ) ^ { 1 / 2 } } \\right ) . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} ( \\varepsilon _ { A ' B ' } ) = \\left ( \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ) , ( \\varepsilon ^ { A ' B ' } ) = \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } a ( n ) + \\sum _ { X < n \\leq X + X / Y } a ( n ) \\phi _ Y \\Big ( \\frac { n } { X } \\Big ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } D ( s ) \\Phi _ Y ( s ) X ^ s d s . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} U ( z ) : = \\frac { \\alpha _ 3 } { ( 1 + \\vert z \\vert ^ 2 ) ^ { 1 / 2 } } , \\alpha _ 3 = 3 ^ { 1 / 4 } , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} N \\in C ^ \\infty ( U _ \\circ ; \\R ^ n ) , | N | = 1 \\mbox { a n d } N \\cdot \\nu ^ t \\ge ( 1 - \\varepsilon _ o ) \\mbox { f o r } t \\in ( - t _ \\circ , t _ \\circ ) , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} A _ g = \\langle e _ 1 \\rangle \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} y _ 0 = x _ 0 , a n d y _ 1 ( r ) = a _ 1 ^ { - 1 } x _ 1 ( - d - r ) , r \\in \\ , ] - d , 0 ] . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\alpha ( k , 5 ) & = 6 k \\quad k = 7 k \\ge 9 , \\\\ & 3 0 \\le \\alpha ( 5 , 5 ) \\le 3 6 , \\\\ & 3 6 \\le \\alpha ( 6 , 5 ) \\le 4 0 , \\\\ & 4 8 \\le \\alpha ( 8 , 5 ) \\le 4 9 . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} b _ l = \\chi _ { Q _ l } \\left ( f - \\tilde m _ l ( f ) \\right ) . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\lambda _ \\infty ( \\Omega ) : = \\frac { 1 } { i _ F ( \\Omega ) } , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } ^ x = \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - 1 + S ( x ) } { b } \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\phi _ { \\epsilon } : = \\varphi _ { \\epsilon } + k \\chi . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} i \\partial _ t \\vec w = \\lambda t ^ { - 1 } A ( w , x ) \\vec w + \\mathcal { O } ( t ^ { - 1 - } ) , \\vec w ( t , x ) = ( w ( t , x ) , w ( t , - x ) ) . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\alpha ( \\{ a , b \\} ) & = \\beta ( a ) \\alpha ( b ) - ( - 1 ) ^ { | a | | b | } \\alpha ( b ) \\beta ( a ) , \\\\ \\beta ( a b ) & = \\alpha ( a ) \\beta ( b ) + ( - 1 ) ^ { | a | | b | } \\alpha ( b ) \\beta ( a ) , \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} S = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 / ( 1 - \\mu ^ 2 ) \\end{array} \\right ) \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} S ( w , z ) = 2 \\mathrm { i } \\left ( w _ c ^ { ( m ) } \\left ( ( u , v ) \\right ) w _ c ^ { ( m ) } \\left ( ( x , y ) \\right ) \\right ) ^ { 1 / 2 } \\sum _ { j = 0 } ^ { N - 2 } \\bigg ( \\prod _ { i = 1 } ^ m \\frac { ( L _ i + j ) ! } { L _ i ! j ! } \\bigg ) ( \\bar { z } - w ) ( w \\bar { z } ) ^ j , \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\left ( m | \\nabla u | ^ { 2 } + 2 u \\right ) = m c ^ { 2 } \\partial \\Omega , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\real \\frac { z \\mathtt { h } _ { a , \\nu } ' ( z ) } { \\mathtt { h } _ { a , \\nu } ( z ) } \\geq 1 - \\frac { ( a \\nu + 1 - a ) ( 1 - a ^ { a / 2 } ) } { 2 } - a ^ { a / 2 } \\sum _ { n = 1 } ^ \\infty \\frac { | z | } { \\mathtt { j } _ { \\nu , n } ^ 2 - | z | } = \\frac { | z | \\mathtt { h } _ { a , \\nu } ' ( | z | ) } { \\mathtt { h } _ { a , \\nu } ( | z | ) } = \\beta \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\Omega _ k ^ * = \\{ \\zeta = ( \\zeta _ 1 , \\dots , \\zeta _ k ) \\in \\Omega ^ k : \\zeta _ i \\not = \\zeta _ j i \\not = j \\} . \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} R _ \\alpha ( \\omega _ \\beta ( z ) ) = \\frac { \\left | \\int _ \\R \\frac { \\dd \\mu _ \\alpha ( x ) } { x - \\omega _ \\beta ( z ) } \\right | ^ 2 } { \\int _ \\R \\frac { \\dd \\mu _ \\alpha ( x ) } { | x - \\omega _ \\beta ( z ) | ^ 2 } } \\le 1 - C _ S ( k , \\varrho ) \\ , , \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} \\tilde { f } _ { k l } \\left ( Z , W \\right ) \\equiv 0 , \\quad \\mbox { f o r a l l $ l = 2 , \\dots , N $ a n d $ k = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} S _ { n } \\left ( { u , \\xi } \\right ) = \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { E _ { s } \\left ( \\xi \\right ) } { u ^ { s } } . } \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} \\mathcal { Q } ( K _ 1 , K _ 2 ) = \\{ \\mbox { i s o m o r p h i s m c l a s s e s o f q u a t e r n i o n e m b e d d i n g s } ( B , \\alpha _ 1 , \\alpha _ 2 ) \\} , \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} \\theta ( V , p , 1 ) & = \\frac V { p - 1 } + O \\bigg ( \\frac { V } { ( \\log V ) ^ 3 } \\frac { \\log Q } Q \\bigg ) \\\\ & \\le \\frac V Q + O \\bigg ( \\frac { V } { ( \\log V ) ^ 3 } \\frac { \\log Q } Q \\bigg ) = \\log x - \\frac { \\log x } { \\log \\log x } + O \\bigg ( \\frac { \\log x } { ( \\log \\log x ) ^ 2 } \\bigg ) . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} \\frac { \\partial Q } { \\partial x } \\left ( x , t \\right ) + \\frac { \\pi a ^ { 2 } \\left ( x , t \\right ) } { \\rho c _ { 0 } ^ { 2 } } \\frac { \\partial P } { \\partial t } \\left ( x , t \\right ) = 0 . \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} \\gamma _ k ^ { ( k + 1 ) / 2 } = \\eta _ k + ( 1 - \\eta _ k ) \\eta _ k ^ k \\ , . \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\left ( Q _ { n ( k , m ) } ^ * ( k - j ) , 0 \\le j \\le k \\right ) \\stackrel { ( d ) } { = } ( \\widehat { Q } _ j , 0 \\le j \\le k \\ , | \\ , \\widehat { Q } _ 0 = m ) \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} D ^ { i j } _ { k k k ' u ' } = \\overline { D ^ { i j } _ { k k u ' k ' } } , \\mbox { f o r a l l $ i , j , k , k ' , u ' = 1 , \\dots , q $ w i t h $ k ' \\neq u ' $ , } \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\max _ { \\boldsymbol { j } } 2 ^ { \\sum _ { l = 1 } ^ d j _ l / 2 } \\max _ { \\boldsymbol { k } } | \\theta ^ { * } _ { \\boldsymbol { j } , \\boldsymbol { k } } - \\theta ^ 0 _ { \\boldsymbol { j } , \\boldsymbol { k } } | = R 2 ^ { - \\sum _ { l = 1 } ^ d \\alpha _ l J _ { n , l } ( \\boldsymbol { \\alpha } ) \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha ^ { * } } - \\frac { 1 } { 2 \\alpha _ l } \\right ) } = M \\epsilon _ n \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , D _ { \\zeta _ { n } ' } V \\cdot e _ l = \\delta _ { i k } \\ , \\delta _ { j l } \\int _ { \\R ^ 3 } w _ { \\mu ' , 0 } ^ 4 ( \\nabla w _ { \\mu ' , 0 } \\cdot e _ 1 ) ^ 2 + o ( 1 ) \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} v ( y ) = \\varepsilon ^ { 1 / 2 } u ( \\varepsilon y ) . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} H \\left ( q ^ i , \\frac { \\partial W } { \\partial q ^ i } \\right ) = E . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\pi _ 1 ( \\Upsilon _ n ) & = \\langle \\ , c _ 1 , \\hdots , c _ n \\ , | \\ , c _ 1 ^ 2 c _ 2 ^ 2 \\cdots c _ n ^ 2 = 1 \\ \\rangle \\\\ \\pi _ 1 ( \\Upsilon _ { 2 g + 1 } ) & \\cong \\langle \\ , a _ 1 , b _ 1 , \\hdots , a _ g , b _ g , c \\ , | \\ , [ a _ 1 , b _ 1 ] \\cdots [ a _ g , b _ g ] c ^ 2 = 1 \\ \\rangle \\\\ \\pi _ 1 ( \\Upsilon _ { 2 g + 2 } ) & \\cong \\langle \\ , a _ 1 , b _ 1 , \\hdots , a _ { g } , b _ { g } , c , d \\ , | \\ , [ a _ 1 , b _ 1 ] \\cdots [ a _ { g } , b _ { g } ] c d c d ^ { - 1 } = 1 \\ \\rangle \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\bigg | \\nu _ i ( \\xi ) \\gamma ' _ j ( t ) \\frac { | z | ^ 2 \\delta _ { i j } - n z _ i z _ j } { | z | ^ { n + 2 } } \\bigg | = C \\frac { | z _ 1 z _ 2 | } { | z | ^ { n + 2 } } \\le C \\frac { | z | ^ 2 | z | } { | z | ^ { n + 2 } } \\le C | z | ^ { 1 - n } = C | \\xi - y | ^ { 1 - n } \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\widetilde { P } ( x ) = \\left [ \\begin{array} { c c } R _ { 1 1 } ( x ) R _ { 1 2 } ( x ) - \\mu _ m R _ { 1 1 } ^ 2 ( x ) & R _ { 1 2 } ^ 2 ( x ) - \\mu _ m R _ { 1 1 } ( x ) R _ { 1 2 } ( x ) \\\\ - R _ { 1 1 } ^ 2 ( x ) & - R _ { 1 1 } ( x ) R _ { 1 2 } ( x ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} h _ R ( x _ i ^ { n } ) = \\begin{cases} ( k + k ( - 1 ) ^ { ( 2 k - 1 ) | x _ i | ^ 2 } ) y _ i x _ i ^ { 2 k - 1 } , & i f ~ n = 2 k , \\\\ ( k + 1 + k ( - 1 ) ^ { ( 2 k - 1 ) | x _ i | ^ 2 } ) y _ i x _ i ^ { 2 k } , & i f ~ n = 2 k + 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\rho _ t ( E ) : = \\lim _ { \\delta \\rightarrow 0 } \\frac { \\rho ( [ t - \\delta , t ] \\times E ) } { \\delta } , E \\in \\mathcal { B } ( A ) . \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\pi _ g ( \\phi \\otimes f ) : = \\langle g , \\ , f \\rangle _ \\mathcal { C } \\ , \\phi \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} u ( \\eta + \\delta ) = e ^ { \\eta A } u ( \\delta ) + \\int _ { 0 } ^ { \\eta } e ^ { ( \\eta - s ) A } f ( \\delta + s ) \\dd s , \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\begin{pmatrix} { X } _ { n + 1 } \\\\ { Y } _ { n + 1 } \\end{pmatrix} \\triangleq M \\left ( \\delta t , \\begin{pmatrix} X _ n \\\\ Y _ n \\end{pmatrix} \\right ) + \\Sigma \\begin{pmatrix} G _ { n , 1 } \\\\ G _ { n , 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} [ Y _ 1 , Y _ 2 ] & = 0 & [ Y _ 1 , Y _ 3 ] & = 0 & [ Y _ 1 , Y _ 4 ] & = 0 & [ Y _ 1 , Y _ 5 ] & = 0 \\\\ [ Y _ 2 , Y _ 3 ] & = 0 & [ Y _ 2 , Y _ 4 ] & = Y _ 1 & [ Y _ 2 , Y _ 5 ] & = \\alpha Y _ 1 & [ Y _ 3 , Y _ 4 ] & = \\beta Y _ 1 \\\\ [ Y _ 3 , Y _ 5 ] & = ( \\alpha \\beta + 1 ) Y _ 1 & [ Y _ 4 , Y _ 5 ] & = \\gamma Y _ 1 \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} A \\cap K ' = \\bigcup _ { n \\in \\mathbb N } ( B : I ^ n ) , \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } - u ^ { \\varepsilon } \\left ( x , t \\right ) v \\left ( x \\right ) \\partial _ { t } c \\left ( t \\right ) + a \\left ( \\frac { x } { \\hat { \\varepsilon } ^ { n } } , \\frac { t } { \\check { \\varepsilon } ^ { m } } , \\nabla u ^ { \\varepsilon } \\left ( x , t \\right ) \\right ) \\cdot \\nabla v \\left ( x \\right ) c \\left ( t \\right ) d x d t \\\\ = \\int _ { \\Omega _ { T } } f \\left ( x , t \\right ) v \\left ( x \\right ) c \\left ( t \\right ) d x d t \\end{gather*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\lambda ^ { p } \\mu ^ { a _ { 2 } p - Q } \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } d x = \\int _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi ( x ) | ^ { p } d x \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\Pi _ t = m _ t \\otimes \\rho _ t \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J _ { e , e } & \\ , = \\Big ( 1 + \\sum _ { \\tilde { e } \\in E ( v ) : \\tilde { e } \\neq e } q _ { e , \\tilde { e } } \\Big ) ^ { - 1 } & & \\qquad e E ( v ) \\\\ J _ { e , \\tilde { e } } & \\ , = \\ , J _ { e , e } \\ , q _ { e , \\tilde { e } } & & e \\neq \\tilde { e } \\in E ( v ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} \\abs { x } q = V \\mu ( x ) = \\frac 1 { k + 1 } \\sum _ { i = 1 } ^ { k + 1 } V ( f _ i ) . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\binom { n } { a _ 1 , \\dots , a _ k } . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} ( \\partial _ n + 2 ) ( A t + | x | ^ { 2 } ) & = ( - \\partial _ r + 2 ) ( A t + r ^ { 2 } + | z | ^ { 2 } ) \\\\ & = 2 ( A t + | z | ^ { 2 } ) + 2 r ( r - 1 ) \\\\ & \\geq 0 , \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} J ^ { ( 1 ) } _ { j l } ( t , \\tau ) = t | t | \\sum _ { r = 1 } ^ { i _ 0 } \\sum _ { q = 1 } ^ { k _ r } \\left ( \\int _ 0 ^ { \\tau } e ^ { i \\tilde { \\tau } \\bigl ( \\sqrt { \\mathstrut \\lambda ^ { ( r ) } _ q ( t ) } - \\sqrt { \\mathstrut \\gamma ^ { \\circ } _ l } | t | \\bigr ) } d \\tilde { \\tau } \\right ) ( P _ j G _ * P _ l \\cdot , \\varphi ^ { ( r ) } _ { q } ( t ) ) \\varphi ^ { ( r ) } _ { q } ( t ) . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\ , K _ { i l } = \\ , K _ { i - 1 , l } = \\ , K _ { b _ { 0 } , l _ { 0 } } = r + 1 . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\| \\mathcal { Q } _ \\tau ( v ) \\| ^ 2 = \\| \\mathcal { Q } ^ 1 _ \\tau ( v ) \\| ^ 2 + \\| \\mathcal { Q } ^ 2 _ \\tau ( v ) \\| ^ 2 + 2 < \\mathcal { Q } ^ 1 _ \\tau ( v ) , \\mathcal { Q } ^ 2 _ \\tau ( v ) > . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} B ^ u _ \\sigma ( \\Lambda ) = \\{ L \\in B _ \\sigma ( \\Lambda ) : L \\in E ^ u _ \\sigma \\} . \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\langle \\Delta ^ k \\psi , u \\rangle ~ = ~ \\langle \\psi , v \\rangle \\ , . \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{gather*} F \\big ( x ^ i ; y ^ a \\big ) = \\textstyle \\frac { 1 } { 3 } c _ { 1 1 1 } \\big ( y ^ 1 \\big ) ^ 3 + c _ { 1 2 } y ^ 1 y ^ 2 + c _ 1 y ^ 1 + c _ 2 y ^ 2 + c _ 0 , \\end{gather*}"}
-{"id": "2138.png", "formula": "\\begin{align*} P _ V ( t ) = [ Q _ \\infty ^ { 1 / 2 } Q _ t ^ { - 1 / 2 } ] ^ { * H } Q _ \\infty ^ { 1 / 2 } Q _ t ^ { - 1 / 2 } \\forall t > 0 . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\alpha _ { i } = 1 . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} B \\ , \\widetilde { \\Pi } = \\left ( \\Omega \\ , \\widetilde { \\Pi } \\right ) \\left ( \\widetilde { \\Pi } ^ T \\ , A \\ , \\widetilde { \\Pi } \\right ) . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\alpha _ t ( u ) = e ^ { i \\lambda t } u \\quad \\mbox { f o r a l l } t \\in \\R \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} I = [ u , u + h v ^ * + n - 1 ] . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} h ( r , s ; \\chi ) = \\sum _ { \\substack { ( n , r ) = 1 \\\\ n \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\frac { \\chi ( n ) g _ 4 ( r , n ) } { N ( n ) ^ s } . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} & \\{ \\Delta ( x _ i ) , \\Delta ( x _ j ) \\} \\\\ = & \\{ x _ i \\otimes 1 + 1 \\otimes x _ i , x _ j \\otimes 1 + 1 \\otimes x _ j \\} \\\\ = & \\{ x _ i , x _ j \\} \\otimes 1 + 1 \\otimes \\{ x _ i , x _ j \\} \\\\ = & x _ i x _ j \\otimes 1 - ( - 1 ) ^ { | x _ i | | x _ j | } ( x _ j x _ i \\otimes 1 ) + 1 \\otimes x _ i x _ j - ( - 1 ) ^ { | x _ i | | x _ j | } ( 1 \\otimes x _ j x _ i ) , \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\langle A g _ 1 , g _ 2 \\rangle _ { X } : = \\int _ X A g _ 1 ( x ) \\cdot \\overline { g _ 2 ( x ) } \\ > d \\omega _ X ( x ) \\quad \\quad g _ 1 , g _ 2 \\in C _ c ( X ) . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ { ( \\eta ) } > x ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } \\mathbb { P } ( S _ { ( n ) } > x ) \\mathbb { P } ( \\eta = n ) \\\\ & \\geqslant \\mathbb { P } \\bigl ( \\max \\{ S _ 1 , \\dots , S _ a \\} > x \\bigr ) \\mathbb { P } ( \\eta = a ) \\\\ & \\geqslant \\mathbb { P } ( S _ a > x ) \\mathbb { P } ( \\eta = a ) . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} B ( z , w ) = \\int ^ \\infty _ 0 \\ ! \\ ! e ^ { i \\phi ( z , w ) t } b ( z , w , t ) d t + H ( z , w ) , \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} \\phi ( t ) = \\phi ( t + 1 ) + \\delta t + \\rho , t \\in \\R \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} a : k [ x _ 1 , \\ldots , x _ r ] _ q / \\langle g _ 1 , \\ldots , g _ r \\rangle ^ e \\to k [ x _ 1 , \\ldots , x _ r ] _ q / ( \\phi ^ * ) ^ { - 1 } ( \\langle f _ 1 , \\ldots , f _ r \\rangle ^ e ) \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} { \\bf A } _ { ( k + 1 ) } = { \\bf A } _ { ( k ) 2 2 } - { \\bf A } _ { ( k ) 2 1 } { \\bf A } _ { ( k ) 1 2 } / a _ { ( k ) 1 1 } . \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { D _ { 1 - x / t } - K _ { d } } { A _ { d } ( b _ { d } ( t ) ) } = \\lim _ { t \\to \\infty } \\left [ \\frac { { \\bf { I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } - \\frac { { \\bf { I I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } \\right ] = \\frac { ( c _ { d } - \\kappa c _ { 1 } ) K _ { d } } { \\alpha \\rho } ( x ^ { - \\rho / \\alpha } - 1 ) . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} f _ L ( v ) = C _ L 1 _ { r _ 1 \\leq v _ 1 \\leq r _ 2 } e ^ { - \\frac { | v _ 2 | ^ 2 } { 2 } } e ^ { - \\frac { | v _ 3 | ^ 2 } { 2 } } , f _ R ( v ) = C _ R 1 _ { - r _ 2 \\leq v _ 1 \\leq - r _ 1 } e ^ { - \\frac { | v _ 2 | ^ 2 } { 2 } } e ^ { - \\frac { | v _ 3 | ^ 2 } { 2 } } , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\sum _ \\chi \\chi ( f ) = n ^ { m / 2 } 1 _ { f ( 1 ) = f ( 2 ) , \\dots , f ( m - 1 ) = f ( m ) } + O _ m ( n ^ { m / 2 - 1 } ) . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} c _ q = - 2 \\sum _ { m = 1 } ^ { k - 1 - q } a _ { q + m + 1 } S _ m - \\sum _ { m = 0 } ^ { k - 2 - q } b _ { q + m + 1 } T _ m , q = 1 , 2 , \\ldots , k - 2 , \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} S _ f = \\{ ( \\lambda , y ) \\in \\R ^ 2 / f _ \\lambda ( x ) = y , \\ f ' _ \\lambda ( x ) = 0 \\hbox { f o r s o m e } x \\in X \\} \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\nabla _ { J ( \\ell , m ) } ( z ) = \\nabla _ { J ( 0 , m ) } ( z ) - \\frac { \\ell } { 2 } z \\nabla _ { ( T ( 2 , - m ) ) } ( z ) = 1 + \\frac { \\ell m } { 4 } z ^ { 2 } . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Delta _ { \\gamma , n } ( \\cdot , \\cdot ) \\stackrel { d i s t . } { = } B _ 1 ( \\cdot ) + B _ 2 ( \\cdot ) \\ , , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ X ] : = \\sum _ { i = 1 } ^ { \\infty } \\hat { \\mathbb { E } } _ { t _ i } [ X ] I _ { \\{ \\tau = t _ i \\} } , \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} & \\Sigma _ { \\{ 6 \\} } = y ^ 4 - y ^ 3 + y ^ 2 , & & \\Sigma _ { \\{ i , 6 \\} } = ( y - 1 ) ( y ^ 2 - y + 1 ) i = 2 , 4 , \\\\ & \\Sigma _ { \\{ 3 , 6 \\} } = ( y - 1 ) ^ 3 + y ^ 2 , & & \\Sigma _ { \\{ i , 3 , 6 \\} } = y ^ 2 - 2 y + 2 i = 2 , 4 , \\\\ & \\Sigma _ { \\{ 2 , 4 , 6 \\} } = y ( y - 1 ) , & & \\Sigma _ { \\{ 2 , 3 , 4 , 6 \\} } = y - 1 . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\left < Z _ { k } , \\mathcal { B } _ { k k } ^ { k } \\left ( Z _ { k } \\right ) \\right > \\left < Z _ { k } , Z _ { k } \\right > = \\displaystyle \\sum _ { l ' = 1 } ^ { p - q } \\varphi _ { k l ' } ^ { k k } \\left ( Z _ { k } , Z _ { k } \\right ) \\overline { \\varphi _ { k l ' } ^ { k k } \\left ( Z _ { k } , Z _ { k } \\right ) } . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\int _ { | x - y | \\leq \\delta / C _ { 0 } } e ^ { i \\lambda \\psi _ { ( p ) } ( x , \\beta ) } f _ { i _ { 1 } \\dots i _ { m } } ( x ) B _ { ( p ) } ^ { i _ { 1 } \\dots i _ { m } } ( x , \\beta ; \\lambda ) d x = \\mathcal { O } ( e ^ { - \\lambda / C } ) \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} C F _ { j } ^ { i } : = \\left ( C F ( \\mathcal F _ { j } ^ { i } ) , \\{ \\frak m ^ { j i ; \\epsilon } _ { 1 ; \\alpha _ + \\alpha _ - } \\} \\right ) . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} N \\ge q ^ { n - 1 } - q ^ { ( n - 2 ) / 2 } ( q - 1 ) \\prod _ { j = 1 } ^ n ( d _ j ^ + + d _ j ^ - ) . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} M _ t = \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Delta M _ s & = \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Bigl ( \\Delta M ^ 1 _ s + \\Delta M ^ 2 _ s + \\Delta M ^ 3 _ s \\Bigr ) \\\\ & = \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Delta M ^ 1 _ s + \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Delta M ^ 2 _ s + \\sum _ { s \\in \\mathcal T \\cap [ 0 , t ] } \\Delta M ^ 3 _ s \\\\ & = \\widetilde M ^ 1 _ t + \\widetilde M ^ 2 _ t + \\widetilde M ^ 3 _ t . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\lvert a _ j ( x ) - a ( x ) \\rvert & \\leqslant a ( k \\cdot 2 ^ { - j } ) + 2 M \\cdot 2 ^ { - j } + M ( x - k \\cdot 2 ^ { - j } ) \\\\ & \\phantom { \\leqslant { } } - \\big ( a ( k \\cdot 2 ^ { - j } ) - M ( x - k \\cdot 2 ^ { - j } ) \\big ) \\\\ & = 2 M \\cdot 2 ^ { - j } + 2 M ( x - k \\cdot 2 ^ { - j } ) \\\\ & \\leqslant 4 M \\cdot 2 ^ { - j } . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} g _ n : = a _ { n 1 } h _ 1 + \\cdots + a _ { n r } h _ r + \\theta _ { n } \\quad g : = a _ { 1 } h _ 1 + \\cdots + a _ { r } h _ r . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} \\ , K _ { i \\tilde { l } } = \\ , K _ { i - 1 , \\tilde { l } } = \\ , K _ { b _ { j } , \\tilde { l } } = r + 1 \\ , \\ , \\ , l _ { j } < \\tilde { l } \\leq b ' _ { r - j } \\ , \\ , \\ , \\ , i + \\tilde { l } \\leq d . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\| \\partial U \\| _ { C ^ { k , \\alpha } _ r } : = \\sup _ { x _ o \\in \\partial U } \\| F _ { x _ o } \\| _ { C ^ { k , \\alpha } ( \\overline { B _ { r } ' } ) } < \\infty , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} P \\left ( \\mathrm { d i a g } ( \\lambda _ 1 , \\ldots , \\lambda _ N ) \\right ) = \\prod _ { l = 1 } ^ N w ( \\lambda _ l ) . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} \\| f \\xi \\chi _ { A _ n } \\| _ { L _ 2 } ^ 2 = \\int _ 0 ^ { \\alpha } \\abs { f \\xi \\chi _ { A _ n } } ^ 2 = \\int _ { A _ n } \\abs { f } ^ 2 \\abs { \\xi } ^ 2 \\leq n ^ 2 \\| \\xi \\| _ { L _ 2 } ^ 2 . \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\Sigma \\Sigma ^ { T } = \\begin{pmatrix} \\sigma _ x ^ 2 ( \\delta t ) & \\sigma _ { x , y } ( \\delta t ) \\\\ \\sigma _ { x , y } ( \\delta t ) & \\sigma _ { y } ^ 2 ( \\delta t ) \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} 0 = 1 + \\beta + \\beta ^ t = \\left \\{ \\begin{array} { c l } \\beta - 1 & ( t \\equiv 0 \\mod 4 ) , \\\\ - \\beta + 1 & ( t \\equiv 1 \\mod 4 ) , \\\\ \\beta & ( t \\equiv 2 \\mod 4 ) , \\\\ 1 & ( t \\equiv 3 \\mod 4 ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho } H ( t , \\rho ) = \\int \\limits _ { - 1 } ^ t f ( s ) \\frac { \\partial } { \\partial \\rho } x ( s , \\rho ) d s \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} \\Gamma = e ^ { t L _ 1 } \\Gamma _ 0 - \\int _ { 0 } ^ { t } e ^ { ( t - s ) L _ 1 } ( b \\cdot \\nabla \\Gamma ) ( s ) \\dd s . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} V = \\left ( v _ { \\alpha } ^ { \\beta } \\right ) _ { 1 \\leq \\alpha \\leq q N } ^ { 1 \\leq \\beta \\leq q N } \\in \\mathcal { M } _ { q N \\times q N } \\left ( \\mathbb { C } \\right ) , \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} x _ n - \\tilde { \\mathcal { K } } _ n ^ M x _ n = f \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} 0 \\leq \\rho _ i ( x ) \\leq 1 , \\ : 1 \\leq i \\leq n ; \\ : \\rho _ { n + 1 } ( x ) = 1 . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\psi ( z ) = \\log z + O ( 1 / z ) \\quad \\psi ^ { ( m ) } ( z ) = ( - 1 ) ^ { m - 1 } \\ , { ( m - 1 ) ! \\over z ^ m } + O ( 1 / z ^ { m + 1 } ) \\ , . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} \\frac { \\gamma ^ n } { 3 ^ { n / 2 } } - \\frac { \\bar { \\gamma } ^ n } { 3 ^ { n / 2 } } = 2 d \\sqrt { - 2 } , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} & H _ 2 ^ { ( 0 ) } ( \\widetilde { X } , E _ 1 \\cup E _ 2 ) = \\partial ^ { - 1 } ( H _ 1 ( E _ 1 ) \\oplus H _ 1 ( E _ 2 ) ) , \\\\ & H _ 2 ^ { ( i ) } ( \\widetilde { X } , E _ 1 \\cup E _ 2 ) = \\partial ^ { - 1 } ( H _ 1 ( E _ i ) ) ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} ( \\partial - \\tilde { \\phi } _ s ( x , \\tau ) ) \\Upsilon = 0 \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\rho ( d t , d a ) = \\rho _ t ( d a ) d t . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} p : = \\sqrt { \\frac { a _ 5 } { a _ 6 } } \\ , , a _ 6 p = p ^ { - 1 } a _ 5 = \\sqrt { a _ 5 a _ 6 } \\ , = : a _ 7 \\ , . \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\tilde { \\sigma } ( y ) = D f \\left ( g ( y ) \\right ) \\sigma ( g ( y ) ) , L _ i ( y ) = \\sum _ { j , l = 1 } ^ d \\partial _ j \\partial _ l f _ i ( g ( y ) ) a _ { j l } \\left ( g ( y ) \\right ) , i = 1 , \\dots , d , \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} x ( t , \\rho ) = \\frac { \\rho } { \\displaystyle 1 - \\rho \\int _ { - 1 } ^ t ( f ( s ) x ( s ) + g ( s ) ) d s } . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} d \\psi ( X , Y , Z ) = \\mp d \\Omega _ \\pm ( J _ \\pm X , J _ \\pm Y , J _ \\pm Z ) , \\ ; \\forall X , Y , Z \\in T M . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} P _ k ( n ) : = S _ { \\theta ^ k } ( n ) - B _ k ( \\sqrt n ) . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} y \\cdot a = b \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\mathcal { R } \\left ( x , t \\right ) = - \\mu \\left ( \\frac { d U } { d r } \\mid _ { a \\left ( x , t \\right ) } \\right ) \\left ( \\pi a \\left ( x , t \\right ) \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } r U \\left ( r \\right ) d r \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\widehat { \\mathcal { U } } ( x ) = e ^ { i \\pi \\widetilde { n } x } \\mathcal { U } ( x ) . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\| \\varphi - \\tilde { z } _ n ^ M \\| _ \\infty = O \\left ( \\max \\left \\{ \\tilde { h } ^ { d } , h ^ { 4 r } \\right \\} \\right ) . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} \\widetilde U ' = U ' { } _ { \\Phi } \\ ! \\times _ U \\tilde U \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} ( A ) = ( A ) - 2 r ( A ) , \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} m _ k = \\int \\limits _ { - 1 } ^ 1 f ( t ) ( G ( t ) ) ^ k d t = 0 , ~ ~ k = 0 , 1 , 2 , \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\phi _ 0 ( t ) & = x _ 0 ( t - a ) ^ { \\gamma - 1 } , t \\in ( a , a + l ] , \\\\ \\phi _ n ( t ) = \\phi _ 0 ( t ) & + \\int _ { a } ^ { t } \\frac { ( t - s ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } f ( s , \\phi _ { n - 1 } ( s ) ) d s , t \\in ( a , a + l ] , \\ , n = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\blacksquare ( F ' F ) = \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } g _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - \\omega \\left ( t \\right ) V \\left ( x ^ { k } \\right ) ~ , ~ \\omega _ { , t } \\neq 0 , \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\ell } s _ { j } = 1 \\ ; \\textrm { a n d } \\ ; \\sum _ { j = 1 } ^ { \\ell } \\frac { s _ { j } } { p _ { j } } = \\frac 1 q . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\rho _ { ( k ) } ^ { \\mathrm { r e a l } } ( x _ 1 , \\ldots , x _ k ) = \\mathrm { P f } \\left [ K ^ { r r } ( x _ j , x _ l ) \\right ] _ { j , l = 1 , \\ldots , k } \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\Psi ^ t ( \\Omega ^ 0 ) = \\Omega ^ t , \\Psi ^ t ( \\Gamma ^ 0 ) = \\Gamma ^ t \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\Gamma _ { 1 1 } = \\frac { 2 } { J _ 1 } \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\rho ^ { \\widehat { \\gamma } , X } ( d t , d a ) : = [ \\widehat { \\gamma } ( t , X ( t ^ - ) ) ] ( d a ) d t . \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\| x ^ * & \\otimes y ^ * - F _ X ^ * x ^ * \\otimes F _ Y ^ * y ^ * \\| = \\\\ & = \\norm { x ^ * \\otimes ( y ^ * - F _ Y ^ * y ^ * ) - ( x ^ * - F _ X ^ * x ^ * ) \\otimes F _ Y ^ * y ^ * } \\\\ & \\leq \\norm { y ^ * - F _ Y ^ * y ^ * } + \\norm { F _ Y ^ * y ^ * } \\norm { ( x ^ * - F _ X ^ * x ^ * ) } . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} P = h ( t , x , D _ t , D _ x ) + p _ { m - 1 } ( t , x , D _ t , D _ x ) + p _ { m - 2 } ( t , x , D _ t , D _ x ) + \\cdots \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} \\P _ m f = \\sum _ { j = 1 } ^ m f _ j w _ j \\quad ~ f = \\sum _ { j = 1 } ^ \\infty f _ j w _ j . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} g _ F ( s ) : = D _ { F , C _ 2 } ^ - ( s ) - \\frac { R _ d ( F ) } { s - 1 } \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} M E ( G ) = \\frac { 2 } { \\pi } \\int ^ { \\infty } _ { 0 } \\frac { 1 } { x ^ { 2 } } \\ln \\left [ \\sum \\limits _ { k \\geq 0 } m ( G , k ) x ^ { 2 k } \\right ] d x . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} C _ 1 ( \\delta _ d , M ( G ) ) : = \\begin{cases} 1 / 2 , & \\textrm { i f $ \\delta _ d + M ( G ) \\leq 1 $ } \\\\ \\displaystyle 1 - \\frac { \\delta _ d + M ( G ) } { 2 } , & \\textrm { i f $ 1 < \\delta _ d + M ( G ) < 2 $ } . \\end{cases} \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} W _ { 2 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ { S _ { n } \\left ( - { u , \\xi } \\right ) - S _ { n } \\left ( - { u , } \\alpha _ { 2 } \\right ) } \\right \\} \\left \\{ { e ^ { - u \\xi } + \\varepsilon _ { n , 2 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} D ( x ) : = \\sup _ { n \\ge 0 } \\{ \\lVert A _ i ^ n ( x ) \\rVert \\cdot e ^ { - ( \\lambda _ i + \\epsilon ) n } \\} < \\infty , \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} h ^ { 2 , 2 } ( \\widetilde { X } ) = h ^ { 1 , 1 } ( \\widetilde { X } ) & = h ^ { 1 , 1 } ( X ) + 1 \\\\ h ^ { 2 , 1 } ( \\widetilde { X } ) = h ^ { 1 , 2 } ( \\widetilde { X } ) & = h ^ { 1 , 2 } ( X ) + g \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\biggl | 2 a _ 1 \\sum _ { j = 2 } ^ k ( Q _ { j - 1 , l - 1 } ( \\varepsilon , \\zeta ) - \\delta _ { j l } ) \\widehat \\Lambda _ j \\biggr | \\leq C \\varepsilon ^ { 2 - 2 \\sigma } , \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} A ^ - ( x ) & = \\frac { - x } { 1 - x } + \\frac { x } { ( 1 - x ) ^ 2 } A ( x ) - \\frac { 2 x ^ 2 } { 1 - x } A ( x ) + x \\big ( C ( x ) - 1 \\big ) - \\frac { x ^ 3 ( 1 - 3 x ) } { ( 1 - x ) ^ 3 ( 1 - 2 x ) } . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} \\bar g \\in \\bar G \\quad g \\in G \\quad \\quad \\bar g \\bar H = g \\bar H . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\| v _ i ^ 2 \\| _ 4 ^ 4 \\leq C \\| v _ i ^ 2 \\| _ { H ^ 1 ( \\Omega ) } ^ 2 \\| v _ i ^ 2 \\| _ { 1 } ^ 2 = C \\| v _ i ^ 2 \\| _ { H ^ 1 ( \\Omega ) } ^ 2 \\| v _ i \\| _ 2 ^ 4 . \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} p _ n = p \\ge 2 \\mbox { i s a f i x e d i n t e g e r f o r a l l l a r g e } n ; \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} i , j = 1 , \\ldots , n , \\ > u , v \\in A _ i , \\ > x , y \\in A _ j : | \\beta ( \\pi ( u , x ) ) - \\beta ( \\pi ( v , y ) ) | \\le \\epsilon . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} D = \\begin{pmatrix} 0 & \\frac { 1 } { \\sigma } & 0 \\\\ - \\frac { 1 } { \\sigma } & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\beta ( \\delta _ d , M ( G ) , \\delta ^ { \\prime } ) : = 1 - \\frac { 1 - \\alpha } { 1 + d \\delta ' ( 1 - \\alpha ) } \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\sigma ( \\eta ^ { - 1 } , \\gamma ^ { - 1 } ) \\sigma ( \\gamma \\eta , \\eta ^ { - 1 } \\gamma ^ { - 1 } ) \\sigma ( \\gamma , \\eta ) & = \\sigma ( \\gamma \\eta , \\eta ^ { - 1 } ) \\sigma ( \\gamma , \\gamma ^ { - 1 } ) \\sigma ( \\gamma , \\eta ) \\\\ & = \\sigma ( \\eta , \\eta ^ { - 1 } ) \\sigma ( \\eta , r ( \\eta ) ) \\sigma ( \\gamma , \\gamma ^ { - 1 } ) \\\\ & = \\sigma ( \\eta , \\eta ^ { - 1 } ) \\sigma ( \\gamma , \\gamma ^ { - 1 } ) . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} v ( \\mathcal F ) \\circ v ( \\mathcal E ) & = v ( \\mathcal O _ { \\Delta _ X } ) = \\Delta _ X \\in \\operatorname { C H } ( X \\times _ K X ) \\otimes \\mathbb Q , \\\\ v ( \\mathcal E ) \\circ v ( \\mathcal F ) & = v ( \\mathcal O _ { \\Delta _ Y } ) = \\Delta _ Y \\in \\operatorname { C H } ( Y \\times _ K Y ) \\otimes \\mathbb Q . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} & g ( \\lambda , \\delta , \\mu ) = \\max \\mathcal { L } _ 1 ^ { \\rm N O M A } ( \\{ p _ k ( \\nu ) \\} , \\{ p _ { \\bar k } ( \\nu ) \\} , \\lambda , \\delta , \\mu ) , \\\\ & \\mathtt { s . t . } ~ ~ p _ k ( \\nu ) \\ge 0 , p _ { \\bar k } ( \\nu ) \\ge 0 , p _ k ( \\nu ) + p _ { \\bar k } ( \\nu ) \\le \\hat P , \\ ; \\forall \\nu . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} m = \\sum _ { i = 1 } ^ { n } a _ { i } x _ { i } = \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { t _ { i } } a _ { i } b _ { j } x _ { i j } \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} l ( H _ i ) < 1 + 3 ^ i ( 1 + \\log _ 7 { 3 } ) = 1 + 3 ^ i \\log _ 7 { 2 1 } < 3 ^ { i + 1 } \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ g ( n ) } } { n ^ { k - 1 } } e ^ { - n / X } & = C X ^ { \\frac { 3 } { 2 } } + O _ { f , g , \\epsilon } ( X ^ { \\frac { 1 } { 2 } + \\epsilon } ) \\\\ \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) S _ g ( n ) } { n ^ { k - 1 } } e ^ { - n / X } & = C ' X ^ { \\frac { 3 } { 2 } } + O _ { f , g , \\epsilon } ( X ^ { \\frac { 1 } { 2 } + \\epsilon } ) \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} L ^ { 2 } = \\ker \\left ( J L \\right ) + R \\left ( J \\right ) . \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\int _ \\Omega \\tt u \\cdot \\nabla \\phi d x = \\int _ { \\Omega } \\Lambda ^ { \\alpha } \\psi \\nabla ^ \\perp \\psi \\cdot \\nabla \\phi d x = - \\int _ { \\Omega } \\psi \\nabla ^ \\perp \\L ^ { \\alpha } \\psi \\cdot \\nabla \\phi d x , \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} M _ d ( x ) = \\frac { 1 - 2 x } { ( 1 - 3 x + x ^ 2 ) ^ 2 } x ^ { d + 3 } \\big ( 1 + ( d - 3 ) x + ( 1 - d ) x ^ 2 \\big ) \\ , , \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\int _ { t _ \\ell } ^ { t _ \\ell + \\Delta } \\lambda _ 0 ( s , X ( s ) ) d s = \\mathcal E _ { \\ell } , \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\phi ^ { \\omega } : = \\sum _ { n \\in \\mathbb { Z } ^ d } g _ n ( \\omega ) \\psi ( D - n ) \\phi , \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} P _ 1 ( x ) = & 2 ^ { r - 4 } ( 2 | S | - r ) ( ( | S | - S ( x ) ) ( | S | - S ( x ) - 1 + 4 S ( x ) ( | S | - S ( x ) ) \\\\ P _ 2 ( x ) = & 2 ^ { r - 4 } ( 2 | S | - r ) ( r - | S | + S ( x ) - 1 ) ( r - | S | + S ( x ) - 2 ) \\\\ P _ 4 ( x ) = & - 2 ^ { r - 4 } ( r - | S | + S ( x ) - 1 ) ( r - | S | + S ( x ) - 2 ) ( | S | + S ( x ) ) \\\\ P _ 5 ( x ) = & 2 ^ { r - 4 } \\left [ ( | S | - S ( x ) ) ( | S | - S ( x ) - 1 ) \\right . \\\\ & \\left . + 4 ( S ( x ) ) ( | S | - S ( x ) ) \\right ] ( r - | S | + S ( x ) - 1 ) \\\\ P _ 6 ( x ) = & 2 ^ { r - 4 } ( r - | S | + S ( x ) - 1 ) ( r - | S | + S ( x ) - 2 ) ( r - | S | + S ( x ) + 1 ) . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\partial _ t \\Gamma ( x _ 1 , t _ 1 ) & \\geq 0 , \\\\ \\nabla \\Gamma ( x _ 1 , t _ 1 ) & = 0 , \\\\ \\Delta \\Gamma ( x _ 1 , t _ 1 ) & \\leq 0 . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( E \\right ) & = \\mathbb { P } \\left ( \\bigcap _ { k = 1 } ^ n E _ k \\right ) \\ge \\sum _ { k = 1 } ^ n \\mathbb { P } ( E _ k ) - ( n - 1 ) \\\\ & \\ge \\sum _ { k = 1 } ^ n \\left ( 1 - 2 ( n - k + 1 ) \\exp \\left ( - \\frac { ( \\epsilon ^ 2 - \\epsilon ^ 3 ) p } { 4 } \\right ) \\right ) - ( n - 1 ) \\\\ & \\ge 1 - n ( n + 1 ) \\exp \\left ( - \\frac { ( \\epsilon ^ 2 - \\epsilon ^ 3 ) p } { 4 } \\right ) , \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\limsup _ { p \\to \\infty } \\big | z ^ { ( p ) } _ m - z _ m \\big | ^ { 1 / p } \\leq \\frac { \\Phi ( z _ m ) } { \\Phi ( z _ { k + 1 } ) } , m = 1 , \\ldots , k . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} f ^ \\R _ 2 = \\tfrac { F _ 3 } { F _ 0 } - \\tfrac { ( F _ 1 ) ^ 2 } { ( F _ 0 ) ^ 2 } \\tfrac { \\tilde A _ 2 ^ \\flat } { \\tilde A _ 3 } + 2 \\tfrac { F _ 1 } { F _ 0 } \\tfrac { ( \\tilde A _ 2 ^ \\flat ) ^ 2 } { ( \\tilde A _ 3 ) ^ 2 } - 2 \\tfrac { ( \\tilde A _ 2 ^ \\flat ) ^ 3 } { ( \\tilde A _ 3 ) ^ 3 } + 2 \\tfrac { \\tilde A _ 1 ^ \\flat \\tilde A _ 2 ^ \\flat } { ( \\tilde A _ 3 ) ^ 2 } - 2 \\tfrac { \\tilde A _ 0 ^ \\flat } { \\tilde A _ 3 } . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\frac { P _ { \\lambda ( N ) } \\big ( x , t ^ { - 1 } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , t ^ { - 2 } , \\dots , t ^ { 1 - N } \\big ) } } = \\Phi ^ { \\nu } ( x ; q , t ) , \\end{gather*}"}
-{"id": "2508.png", "formula": "\\begin{align*} f _ { 2 n } ( t ) = \\Upsilon _ n \\big ( R _ { n } e ^ { r _ { n } t } + C _ { n } e ^ { i \\omega _ { n } t } + \\overline { C _ { n } } e ^ { - i \\overline { \\omega _ { n } } t } + D _ { n } e ^ { i \\zeta _ { n } t } + \\overline { D _ { n } } e ^ { - i \\overline { \\zeta _ { n } } t } \\big ) \\ , . \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} F ( t ) : = - 3 2 t ^ 3 + 1 0 8 t ^ 2 - \\frac { 2 4 3 } { 2 } t + \\frac { 7 2 9 } { 1 6 } \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{gather*} \\alpha _ k : = \\int _ 0 ^ \\infty \\frac { d x } { x ^ { \\alpha } ( x + k ) ^ { 3 / 2 - \\alpha } ( \\log ( x + k ) ( \\log \\log ( x + k + 2 ) ) ^ 3 ) ^ { 1 / 2 } } \\\\ = k ^ { - 1 / 2 } \\int _ 0 ^ \\infty \\frac { d u } { u ^ { \\alpha } ( u + 1 ) ^ { 3 / 2 - \\alpha } ( \\log ( k u + k ) ( \\log \\log ( k u + k + 2 ) ) ^ 3 ) ^ { 1 / 2 } } \\ , . \\end{gather*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\widehat C K ^ h _ { ( \\Lambda , d \\mu ) } A = K ^ h _ { ( \\Lambda , d \\mu ) } \\bigl ( C _ 0 ( x , p ; E ) | _ { \\Lambda } A \\bigr ) ( 1 + { \\cal O } ( h ) ) \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\left \\langle P _ h ( \\cdot , s ) , E ( \\cdot , w ) \\right \\rangle = \\frac { 2 \\pi ^ { \\overline { w } + \\frac { 1 } { 2 } } h ^ { \\overline { w } - \\frac { 1 } { 2 } } \\sigma _ { 1 - 2 \\overline { w } } ( h ) } { \\zeta ( 2 \\overline { w } ) ( 4 \\pi h ) ^ { s - \\frac { 1 } { 2 } } } \\frac { \\Gamma ( s + \\overline { w } - 1 ) \\Gamma ( s - \\overline { w } ) } { \\Gamma ( \\overline { w } ) \\Gamma ( s ) } , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} E _ i : = & \\Bigl [ ( U _ i + Q _ i ) ^ 6 - U _ i ^ 6 \\Bigr ] - \\sum _ { j \\neq i } U _ j ^ 6 \\\\ = & 6 \\ , ( U _ i ^ 5 \\ , Q _ i + U _ i \\ , Q _ i ^ 5 ) + 1 5 \\ , ( U _ i ^ 4 \\ , Q _ i ^ 2 + Q _ i ^ 2 \\ , U _ i ^ 4 ) + 2 0 \\ , U _ i ^ 3 \\ , Q _ i ^ 3 - \\sum _ { j \\neq i } U _ j ^ 6 . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} & t _ 0 : = \\lfloor \\frac { \\sum ^ { 4 } _ { i = 1 } t _ i } { 4 } \\rfloor , \\\\ & K _ 0 : = \\Lambda ( y ^ 0 _ 0 , r _ 0 , m \\times n ) . \\\\ \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} p ( z ) = \\sum _ { j = r } ^ { d } P _ j ( z ) \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} \\widetilde { V } = \\left ( \\begin{array} { c c } \\widetilde { V } _ k & \\widetilde { V } _ { n - k } \\\\ \\end{array} \\right ) \\in \\mathbb { R } ^ { n \\times n } \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\alpha _ t = \\frac { p \\left ( \\left \\lfloor \\frac { t } { B } \\right \\rfloor - 1 \\right ) - \\frac { p t } { 2 B } } { \\left \\lfloor \\frac { t } { B } \\right \\rfloor - 1 } . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\mathcal { D } _ { f r } = \\left \\{ \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\left | 0 < \\omega _ { 1 } \\leq C , \\ , 0 < \\omega _ { 2 } \\leq C \\right . \\right \\} \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\left [ b _ { n + 1 } , \\sum _ { j = 1 } ^ { n + 1 } ( b _ j + \\ell _ j ) - 1 \\right ] \\subseteq A . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq t < \\infty } B _ t = + \\infty \\ \\ \\ \\ \\inf _ { 0 \\leq t < \\infty } B _ t = - \\infty , \\ \\ \\ \\ \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} e ( A ) = \\dim _ K \\langle H ^ j _ { \\bf 1 _ d } : j = 1 , \\dots , \\tau \\rangle \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 ( \\mu ) } : = \\sqrt { \\int _ { \\Gamma } | f ( x ) | ^ 2 \\ , \\mu ( d x ) } = \\sqrt { \\sum _ { e } \\int _ { e } | f ( x ) | ^ 2 \\ , \\mu ( d x ) } \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} S _ { \\frak M } ( y , z ) & = \\sum _ { u \\le y } \\left | \\mathcal { I } _ 1 ( u , z ) + \\mathcal { I } _ 2 ( u , z ) + \\mathcal { I } _ 3 ( u , z ) \\right | ^ 2 \\\\ & \\ll \\sum _ { u \\le y } \\left ( \\left | \\mathcal { I } _ 1 ( u , z ) \\right | ^ 2 + \\left | \\mathcal { I } _ 2 ( u , z ) \\right | ^ 2 + \\left | \\mathcal { I } _ 3 ( u , z ) \\right | ^ 2 \\right ) \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\phi ( \\gamma ( l , m ) ) \\ = \\ \\gamma ' ( l , m ) \\ \\ \\ , \\ \\ \\ l , m \\in \\Pi \\ , \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} - \\partial _ { w _ { a a } } \\left ( R \\left ( W ' \\right ) \\right ) = \\partial _ { w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) , \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\Delta f - | \\nabla f | ^ 2 = - R ( 0 ) = 2 f '' ( 0 ) \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\widehat { S } \\zeta _ l = \\gamma _ l Q _ { \\widehat { \\mathfrak { N } } } \\zeta _ l , l = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\alpha _ i \\geq 0 , \\ ; \\ ; \\forall \\ ; i = 1 , . . . , N , \\ ; \\ ; \\exists \\ ; \\bar { i } \\in \\{ 1 , . . . , N \\} \\ ; \\mbox { s u c h t h a t $ \\alpha _ { \\bar { i } } > 0 $ } \\ ; \\ ; \\mbox { a n d } \\sum _ { i = 1 } ^ { N } \\alpha _ i < \\| \\kappa \\| _ { 1 } . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\dim { \\mathcal M } _ { k + 1 } ( \\beta ; P ) = \\mu ( \\beta ) + \\dim L + k - 2 + \\dim P . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} d \\tilde { Z _ t } = \\Xi _ { G _ t } ( d Z _ t ) \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} T _ n ^ f ( U _ 1 ) T _ n ^ f ( U _ 2 ) = T _ n ^ f ( U _ 1 \\otimes U _ 2 ) \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} W \\phi = W ( \\phi , \\phi ) \\ , \\ , A \\phi = A ( \\phi , \\phi ) , \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} { \\Delta _ M } \\zeta _ 0 = 0 \\ \\ \\textrm { i n } \\ \\ \\ M \\setminus \\{ q _ 1 \\cdots q _ m \\} . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} \\langle j x , x ^ { \\sim } \\rangle _ { X ' } = : \\langle x , x ^ { \\sim } \\rangle _ { X ' } , x \\in X , x ^ { \\sim } \\in X ^ { \\sim } . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\kappa _ { L } ( \\zeta ) = i \\zeta a - \\frac { \\zeta ^ { 2 } } { 2 } b + \\int _ { \\R } \\left ( e ^ { i \\zeta x } - 1 - i \\zeta \\mathbf { 1 } _ { [ - 1 , 1 ] } ( x ) \\right ) \\mu _ L ( d x ) . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} r _ i ( t ) = T B \\log _ 2 \\left ( 1 + \\frac { P _ i z ^ j _ i } { B \\sigma ^ 2 + \\sum _ { k \\in \\bold { \\Psi } _ { j } ( t ) , k \\neq i } P _ k z ^ j _ { k , i } } \\right ) \\ : \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} ( \\nabla \\pi _ { * } ) ( X , Z ) & = \\nabla ^ { \\pi } _ { X } \\pi _ { * } Z + \\pi _ { * } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Z - \\varphi ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi ) Z - \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { X } Z ) \\xi ) \\\\ & = \\nabla ^ { \\pi } _ { X } \\pi _ { * } Z + \\pi _ { * } ( \\varphi ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Z _ 1 + \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Z _ 2 ) - \\eta ( Z ) \\varphi X - \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { X } Z ) \\xi ) . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} \\eta ( r , z ) = \\big ( r f ( r , z ) , \\ h ( r , z ) \\big ) \\ , , \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} w _ { \\mu , \\zeta } ( x ) : & = \\mu ^ { - 1 / 2 } \\ , U \\Bigl ( \\frac { x - \\zeta } { \\mu } \\Bigr ) = \\frac { \\alpha _ 3 \\ , \\mu ^ { 1 / 2 } } { \\Bigl ( \\mu ^ 2 + \\vert x - \\zeta \\vert ^ 2 \\Bigr ) ^ { 1 / 2 } } , \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} k ( s + k ) = \\frac { k ( k - 1 ) } { 2 } + k ^ 2 x ^ 2 , \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} H _ a ( u ) & : = \\int _ { 0 } ^ \\infty \\big [ ( 1 + z u ^ { - 1 } ) ^ { 1 - a } - 1 - ( 1 - a ) z u ^ { - 1 } \\big ] \\pi ( \\dd z ) \\\\ & = a ( a - 1 ) u ^ { - 2 } \\int _ 0 ^ \\infty z ^ 2 \\pi ( \\dd z ) \\int _ 0 ^ 1 ( 1 + z u ^ { - 1 } v ) ^ { - 1 - a } ( 1 - v ) \\dd v , \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} N _ { | f | \\chi _ { B ^ c } } = N _ { | f | } N _ { \\chi _ { B ^ c } } = | x | e ^ { \\abs { x } } [ 0 , s ] = \\int _ { [ 0 , s ] } \\lambda d e ^ { | x | } ( \\lambda ) \\leq s e ^ { \\abs { x } } [ 0 , s ] = N _ { s \\chi _ { B ^ c } } . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\frac { \\lambda ^ t } { t } ( z ) = - \\partial _ N v ^ t ( z , \\lambda ^ t ( z ) ) . \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} | F _ g ( f ) | & \\leq \\mathbb E \\Bigl ( \\sum _ { k = 0 } ^ { n } \\mathbb E _ { k - 1 } ( \\| f _ k \\| \\| g _ k \\| ) \\Bigr ) \\\\ & \\leq \\mathbb E \\Bigl ( \\sum _ { k = 0 } ^ { n } ( \\mathbb E _ { k - 1 } \\| f _ k \\| ^ q ) ^ { 1 / q } ( \\mathbb E _ { k - 1 } \\| g _ k \\| ^ { q ' } ) ^ { 1 / { q ' } } \\Bigr ) \\\\ & \\leq \\| f \\| _ { H ^ { s _ q ^ n } _ p ( X ) } \\| g \\| _ { H ^ { s _ { q ' } ^ n } _ { p ' } ( X ^ * ) } . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} V = V _ { 1 } \\otimes \\dots \\otimes V _ { n } , \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} 2 D = \\begin{cases} ( x ) & , \\\\ ( 1 ) & , \\end{cases} \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} a _ 0 & = g _ \\lambda ( \\zeta _ 1 ( r ) ) , \\\\ a _ j & = - G _ \\lambda ( \\zeta _ 1 ( r ) , \\zeta _ { j + 1 } ( r ) ) , j = 1 , \\ldots , k - 1 , \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} 5 ( 3 u ^ 2 - 2 \\cdot 5 ^ { 2 ( b - 1 ) } ) ^ 2 - 1 6 \\cdot 5 ^ { 4 ( b - 1 ) } = \\pm 5 . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} d U _ x ( t ) = & \\left [ \\mathcal { L } _ n U _ x + L ^ e \\ , \\frac { \\hat { \\beta } ( v ) } { d e g ( v ) } \\ , U _ x ( 1 - U _ x ) \\right ] \\ , d t \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ { ( \\eta ) } > x ) \\geqslant \\frac { 1 } { 2 } \\overline { F } _ { \\xi _ 1 } ( x ) \\mathbb { P } ( \\eta = a ) , \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} T B V _ { i } ^ { W } = | q ^ { W } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q ^ { W } _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) | + | q ^ { W } _ { i } ( x _ { i + \\frac { 1 } { 2 } } ) - q ^ { W } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) | \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} & \\delta _ { [ k = \\frac { 1 } { 2 } ] } \\delta _ { [ h = a ^ 2 ] } \\bigg ( R ' _ h X ^ { \\frac { 1 } { 2 } } \\log X + R ' _ h X ^ { \\frac { 1 } { 2 } } \\frac { \\pi } { Y } \\bigg ) \\frac { \\exp ( \\frac { \\pi } { 4 Y } ) } { Y } + R _ k ^ k X ^ k \\frac { \\exp ( \\frac { \\pi k ^ 2 } { Y } ) } { Y } \\\\ & + O ( \\frac { X ^ { k - \\frac { 1 } { 2 } } } { Y } ) + O ( X ^ { \\frac { k } { 2 } + \\epsilon } Y ^ { 3 k + \\frac { 1 7 } { 2 } + \\epsilon } ) . \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} J _ { \\lambda _ 0 } ^ + ( v ) \\leq \\liminf _ { n \\to \\infty } J _ { \\lambda _ n } ^ + ( v _ n ) = : \\tilde { J } < + \\infty . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} Q _ \\tau = Q _ t + e ^ { 2 t A } Q _ { \\tau - t } = Q _ t + Q _ { \\tau - t } e ^ { 2 t A } ; \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\langle \\eta | \\xi \\rangle _ N = { } _ N \\langle \\overline { \\eta } , \\overline { \\xi } \\rangle \\qquad \\eta , \\xi \\in H ^ \\circ . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\Phi _ h ( M ) = \\frac 1 { h ! } \\sum _ { \\sigma \\in \\Sigma _ h } z _ { m _ { \\sigma 1 } m _ { \\sigma 2 } } \\cdots z _ { m _ { \\sigma ( h - 1 ) } m _ { \\sigma h } } . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\partial ( \\frak N _ { 3 2 } ^ { P _ 2 } \\circ \\frak N _ { 2 1 } ^ { P _ 1 } ) = ( \\frak N _ { 3 2 } ^ { \\partial P _ 2 } \\circ \\frak N _ { 2 1 } ^ { P _ 1 } ) \\cup ( \\frak N _ { 3 2 } ^ { P _ 2 } \\circ \\frak N _ { 2 1 } ^ { \\partial P _ 1 } ) \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} A \\left \\| z \\right \\| ^ { 2 } \\leq \\sum ^ { M } _ { i = 1 } \\left | \\left \\langle z , \\phi _ { i } \\right \\rangle \\right | ^ { 2 } \\leq B \\left \\| z \\right \\| ^ { 2 } , \\forall z \\in \\mathbb { C } ^ { m } \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} J - J _ 0 - J _ 1 = ( 3 x - 2 x ^ 2 ) J - x J _ 1 - ( 3 x - x ^ 2 ) J _ 0 + \\frac { x ^ 4 } { ( 1 - 2 x ) ^ 2 } + \\frac { x ^ 5 } { ( 1 - x ) ^ 3 ( 1 - 2 x ) } \\ , . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{gather*} \\frac { \\big ( q ^ { \\theta N - z } ; q \\big ) _ { \\infty } } { ( q ^ { - z } ; q ) _ { \\infty } } = \\frac { \\big ( 1 - q ^ { \\theta N - z } \\big ) \\cdots \\big ( 1 - q ^ { \\theta N - z + M } \\big ) } { ( 1 - q ^ { - z } ) \\cdots \\big ( 1 - q ^ { - z + M } \\big ) } \\cdot \\frac { \\big ( q ^ { \\theta N - z + M + 1 } ; q \\big ) _ { \\infty } } { \\big ( q ^ { - z + M + 1 } ; q \\big ) _ { \\infty } } , \\end{gather*}"}
-{"id": "7000.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } ( x ) \\cdot P _ n ^ { ( a , b ) } ( y ) = \\int _ { - s _ 1 } ^ { s _ 1 } P _ n ^ { ( a , b ) } ( z ) \\ > d \\mu _ { x , y } ( z ) { f o r \\ > \\ > a l l } n \\in \\mathbb N _ 0 . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} Y _ { l } & = \\ X ^ { l } \\oplus X ^ { - l } \\\\ & = \\left \\{ \\cos \\left ( \\alpha l x \\right ) \\omega _ { 1 } \\left ( y \\right ) + \\sin \\left ( \\alpha l x \\right ) \\omega _ { 2 } \\left ( y \\right ) , \\ \\omega _ { 1 } , \\omega _ { 2 } \\in L _ { \\frac { 1 } { K _ { 2 } \\left ( y \\right ) } } ^ { 2 } \\left ( y _ { 1 } , y _ { 2 } \\right ) \\right \\} , \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\frac { \\vartheta ( x _ 1 - y _ 1 ) \\vartheta ( x _ 1 - y _ 2 ) \\vartheta ( x _ 1 - y _ 3 ) } { \\vartheta ( x _ 1 - x _ 2 ) \\vartheta ( x _ 1 - x _ 3 ) } + \\frac { \\vartheta ( x _ 2 - y _ 1 ) \\vartheta ( x _ 2 - y _ 2 ) \\vartheta ( x _ 2 - y _ 3 ) } { \\vartheta ( x _ 2 - x _ 1 ) \\vartheta ( x _ 2 - x _ 3 ) } + \\frac { \\vartheta ( x _ 3 - y _ 1 ) \\vartheta ( x _ 3 - y _ 2 ) \\vartheta ( x _ 3 - y _ 3 ) } { \\vartheta ( x _ 3 - x _ 1 ) \\vartheta ( x _ 3 - x _ 2 ) } = 0 . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} \\left \\{ B _ { 1 1 } ^ { k } , B _ { 1 2 } ^ { k } , \\dots , B _ { 1 N } ^ { k } \\right \\} \\subset \\mathbb { C } ^ { N } , \\quad \\mbox { f o r a l l $ k = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { \\scriptstyle L ^ { q ( \\cdot ) } ( \\partial \\Omega ) } \\leq \\sum _ { i = 0 } ^ { N } \\Vert f \\Vert _ { \\scriptstyle L ^ { q ( \\cdot ) } ( \\overline { B _ { i } } \\cap \\partial \\Omega ) } , \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} W _ t & = - | \\lambda _ 1 | e ^ { - | \\lambda _ 1 | t } | x | ^ { \\xi ( d + \\alpha ) } w + h \\left ( w N ( x , w ) + \\int ( w ( x + y , t ) - w ( x , t ) ) \\overline { K } ( x , y ) d y \\right ) \\\\ & = W \\left ( N ( x , w ) - \\frac { | \\lambda _ 1 | e ^ { - | \\lambda _ 1 | t } | x | ^ { \\xi ( d + \\alpha ) } } { M + e ^ { - | \\lambda _ 1 | t } | x | ^ { \\xi ( d + \\alpha ) } } \\right ) + \\int ( W ( x + y , t ) - W ( x , t ) ) \\overline { K } ( x , y ) d y + \\bar { I } , \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\widehat { T } ^ { ( p ) } _ { s _ 0 , s _ 1 , \\ldots , s _ k } ( z ) = \\begin{vmatrix} \\widehat { e } ^ { ( p ) } _ { s _ 0 } ( z ) & \\widehat { e } ^ { ( p ) } _ { s _ 1 } ( z ) & \\cdots & \\widehat { e } ^ { ( p ) } _ { s _ k } ( z ) \\\\ \\alpha _ { 1 , s _ 0 } & \\alpha _ { 1 , s _ 1 } & \\cdots & \\alpha _ { 1 , s _ k } \\\\ \\alpha _ { 2 , s _ 0 } & \\alpha _ { 2 , s _ 1 } & \\cdots & \\alpha _ { 2 , s _ k } \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\alpha _ { k , s _ 0 } & \\alpha _ { k , s _ 1 } & \\cdots & \\alpha _ { k , s _ k } \\end{vmatrix} . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } + L u - \\lambda u = - g \\ : \\mbox { i n } \\ : D _ T , \\frac { \\partial u } { \\partial t } + L _ \\pm u - \\lambda u = - g \\ : \\mbox { i n } \\ : D _ T ^ \\pm , u ( T ) = f \\ : \\mbox { i n } \\ : D . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\begin{cases} D _ { 0 ^ + } ^ { \\alpha , \\beta } x _ { 1 } ( t ) & = f _ { 1 } ( t , x _ { 1 } , x _ { 2 } , . . , x _ { n } ) , \\\\ D _ { 0 ^ + } ^ { \\alpha , \\beta } x _ { 2 } ( t ) & = f _ { 2 } ( t , x _ { 1 } , x _ { 2 } , . . , x _ { n } ) , \\\\ & \\cdots \\\\ D _ { 0 ^ + } ^ { \\alpha , \\beta } x _ { n } ( t ) & = f _ { n } ( t , x _ { 1 } , x _ { 2 } , . . , x _ { n } ) , \\\\ I _ { 0 ^ + } ^ { 1 - \\gamma } x _ { i } ( 0 ^ + ) & = x _ 0 , \\gamma = \\alpha + \\beta - \\alpha \\beta , i = 1 , 2 , . . , n , \\end{cases} \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} k _ { r } + 2 k _ { c } + 2 k _ { i } ^ { \\leq 0 } + k _ { 0 } ^ { \\leq 0 } = n ^ { - } \\left ( L \\right ) . \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} ( 3 x + d \\sqrt { - 6 } ) ( 3 x - d \\sqrt { - 6 } ) = 3 y ^ n . \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} r _ d ( G ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { \\textrm { R e s } _ { s = 1 } \\zeta _ F ( s ) } { 2 ^ d d _ F ^ 2 \\zeta _ F ( 2 ) } > 0 , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\phi _ { 0 + } = \\phi _ 0 = \\frac { \\mu _ 0 + \\alpha ( \\lambda _ 0 ) } { \\varphi ( \\lambda _ 0 ) } \\mbox { a n d } \\phi _ { 0 - } = \\frac { - \\mu _ 0 + \\alpha ( \\lambda _ 0 ) } { \\varphi ( \\lambda _ 0 ) } . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\varphi ( 0 , w ) & = \\sqrt \\pi \\frac { \\Gamma ( w - \\frac { 1 } { 2 } ) } { \\Gamma ( w ) } \\sum _ c c ^ { - 2 w } S _ \\mathfrak { a } ( 0 , 0 ; c ) \\\\ \\varphi ( m , w ) & = \\frac { \\pi ^ w } { \\Gamma ( w ) } \\lvert m \\rvert ^ { w - 1 } \\sum _ c c ^ { - 2 w } S _ \\mathfrak { a } ( 0 , m ; c ) \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} K ^ * ( u ) = 2 K _ { 2 T } ( u ) \\ , . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\sum _ { \\substack { N ( a ) \\leq x \\\\ a \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\frac { 1 } { N ( a ) } = \\frac { \\pi } { 8 } \\log x + C _ 0 + O ( x ^ { \\theta - 1 } ) , \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\frac { d } { d z } W _ { \\alpha , \\beta } ( z ) = W _ { \\alpha , \\beta + \\alpha } ( z ) , \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { n } | a _ { j } | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } \\leq \\mathrm { A } _ { p } \\left ( \\int _ { 0 } ^ { 1 } \\left \\vert \\sum _ { j = 1 } ^ { n } a _ { j } r _ { j } ( t ) \\right \\vert ^ { p } d t \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\frak V _ p = ( \\overline V _ { \\overline p } \\times [ - \\tau , 0 ) ^ k , \\Gamma _ { \\overline p } , \\phi _ p ) . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} P ( x , y ' ) = \\frac { \\rho } { \\rho ^ 2 + r ^ 2 } ( 1 + | y | ^ 2 ) \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in \\hat { G } ^ n } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) = \\sum _ { H ( \\chi ) \\leq \\epsilon } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) + \\sum _ { H ( \\chi ) > \\epsilon } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} & x = \\frac { k _ 1 } { t + c } , y = \\frac { k _ 2 } { t + c } , z = \\frac { k _ 3 } { t + c } , w = \\frac { k _ 4 } { t + c } \\intertext { w h i c h , i n t u r n , i m p l y } & x ' = \\frac { - k _ 1 } { ( t + c ) ^ 2 } , y ' = \\frac { - k _ 2 } { ( t + c ) ^ 2 } , z ' = \\frac { - k _ 3 } { ( t + c ) ^ 2 } , w ' = \\frac { - k _ 4 } { ( t + c ) ^ 2 } \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} w _ R '' = \\frac { w _ R } { 2 } U _ 1 ( c _ 1 + w _ R ^ 2 ) + \\frac { \\overline { w } _ R } { 2 } \\vert w _ R \\vert ^ 2 \\nabla U _ 1 ( c _ 1 + w _ R ^ 2 ) ; \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\sum _ { a _ 1 \\in \\mathcal { T } _ 1 } \\cdots \\sum _ { a _ d \\in \\mathcal { T } _ d } \\prod _ { l = 1 } ^ d 2 ^ { a _ l + j _ l / 2 } = \\prod _ { l = 1 } ^ d \\sum _ { a _ l \\in \\mathcal { T } _ l } 2 ^ { a _ l + j _ l / 2 } . \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} \\frac { S ( z ) } { z } = \\frac { \\log \\log z } { \\phi ( q ) \\log z } + O \\bigg ( \\frac { 1 } { \\log z } \\bigg ) \\ll 1 \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\binom { a } { 2 } \\binom { s } { a } \\binom { r - s } { b } & = \\binom { s } { 2 } \\sum _ { \\substack { a , b \\\\ r = s + t - a + b } } \\binom { s - 2 } { a - 2 } \\binom { r - s } { b } \\\\ & = \\binom { s } { 2 } \\sum _ b \\binom { s - 2 } { r - t - b } \\binom { r - s } { b } , \\\\ & = \\binom { s } { 2 } \\binom { r - 2 } { r - t } . \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\varphi _ U ( p _ n ) = \\sum _ { d | n } d T _ d ( U ^ { \\frac { n } { d } } ) \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} G _ 2 ( x ; d ) & = \\frac { x ^ { 2 + d } ( F _ T ( x ) - 1 ) C ( x ) } { ( 1 - x ) ^ d \\big ( 1 - x C ( x ) \\big ) } + \\left ( \\sum _ { m = 2 } ^ d \\frac { x ^ { 3 + d } C ( x ) } { ( 1 - x ) ^ m \\big ( 1 - x C ( x ) \\big ) } \\right ) + \\frac { x ^ { 2 + d } C ( x ) } { ( 1 - x ) \\big ( 1 - x C ( x ) \\big ) ^ d } \\\\ & = \\frac { x ^ { 2 + d } ( F _ T ( x ) - 1 ) C ( x ) ^ 2 } { ( 1 - x ) ^ d } + \\left ( \\sum _ { m = 2 } ^ d \\frac { x ^ { 3 + d } C ( x ) ^ 2 } { ( 1 - x ) ^ m ) } \\right ) + \\frac { x ^ { 2 + d } C ( x ) } { ( 1 - x ) \\big ( 1 - x C ( x ) \\big ) ^ d } \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} V _ { , k } Y ^ { k } + 2 \\psi _ { Y } V + d _ { 1 } V + m S + k = 0 . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} L ^ { \\rm r e s c , h } _ n ( u ) : = \\frac { L _ { ( 0 , 0 ) \\to ( \\gamma ^ 2 n + \\beta _ 1 u n ^ { 2 / 3 } , n ) } - n ( 1 + \\sqrt { \\gamma ^ 2 + \\beta _ 1 u n ^ { - 1 / 3 } } ) ^ 2 } { \\beta _ 2 n ^ { 1 / 3 } } , \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} M _ { 2 n - 2 } ( X ) : = M _ 2 ( X ) \\otimes { \\bf L } ^ { n - 2 } . \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} E ( \\prod ^ 4 _ { i = 1 } \\hat { r } _ { \\ell j _ i } ) = \\left \\{ \\begin{array} { l l } \\frac { 1 2 ( n - 3 ) ( 5 n ^ 2 - 2 7 n + 4 0 ) } { ( n - 4 ) ^ 2 ( n - 6 ) ( n - 8 ) ( n - 1 0 ) } , & \\hbox { i f $ j _ 1 = j _ 2 = j _ 3 = j _ 4 $ ; } \\\\ \\frac { 4 ( n - 3 ) ^ 2 } { ( n - 4 ) ^ 2 ( n - 6 ) ^ 2 } , & \\hbox { i f $ \\{ j _ 1 , j _ 2 , j _ 3 , j _ 4 \\} $ f o r m s t w o d i s t i n c t p a i r s ; } \\\\ 0 , & \\hbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} & \\frac { ( 2 g + 1 - \\sum _ j b _ j ) ! } { b _ i ! } \\sum _ { c _ 2 , \\ldots , c _ n } ( 2 k - 1 ) ! ! ( 2 c _ i + 1 ) ! ! \\psi _ 1 ^ k ( \\psi ' ) ^ { c _ i } \\prod _ { j \\neq i } \\frac { \\psi _ j ^ { c _ j } } { 2 ^ { c _ j } c _ j ! ( b _ j - 2 c _ j ) ! } \\\\ & \\left ( - { 4 g + n - \\sum _ { j \\neq i } b _ j \\choose 2 g - 2 c _ i - \\sum _ { j \\neq i } b _ j } + \\sum _ { d = 0 } ^ { b _ i - 2 c _ i + 2 } { 4 g - 1 + n - \\sum _ j b _ j \\choose 2 g - 2 c _ i - \\sum _ { j \\neq i } b _ j - d } { b _ i + 1 \\choose d } \\right ) , \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} L ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { a ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\underline { \\rm H o m } ( { \\bf L } ^ { n } , M ( X ) ) \\otimes { \\bf L } ^ n \\rightarrow & \\underline { \\rm H o m } ( { \\bf L } ^ { n - 1 } , M ( X ) ) \\otimes { \\bf L } ^ { n - 1 } \\\\ \\rightarrow & \\cdots \\rightarrow \\underline { \\rm H o m } ( { \\bf 1 } , M ( X ) ) = M ( X ) . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} L ( s , f \\times g ) = \\zeta ( 2 s ) \\sum _ { n \\geq 1 } \\frac { a ( n ) b ( n ) } { n ^ { s + k - 1 } } . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} - \\Delta _ \\Sigma f = \\lambda f , \\Sigma , \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} K _ 3 = \\min \\left \\{ 2 \\min _ { j } \\{ d _ j \\} , \\min \\left \\{ \\frac 1 2 \\min \\{ 1 , \\beta _ 2 ^ { - 1 } \\beta _ 3 \\} \\beta _ 1 , \\beta _ 4 \\right \\} \\right \\} \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} ( \\wp _ \\lambda ) \\ , \\left \\{ \\begin{aligned} & \\Delta u + \\lambda u + u ^ p = 0 \\quad \\Omega , \\\\ & u > 0 \\quad \\Omega , \\\\ & u = 0 \\quad \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\mathcal S = \\inf _ { u \\in { \\mathcal D } ^ 1 ( \\R ^ n ) \\atop \\scriptstyle u \\ne 0 } \\frac { \\langle - \\Delta u , u \\rangle } { \\| u \\| ^ 2 _ { L ^ { 2 ^ * } ( \\R ^ n ) } } \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} S _ 1 ^ { h _ 1 } & = \\bigg ( \\sum _ { 2 \\le q \\leq X } \\Lambda ( q ) F _ { \\omega _ { q } } ( n ) ^ 2 \\bigg ) ^ { h _ 1 } = \\sum _ { 2 \\le q _ 1 , \\ldots , q _ { h _ 1 } \\leq X } \\prod _ { i = 1 } ^ { h _ 1 } \\Lambda ( q _ i ) F _ { \\omega _ { q _ i } } ( n ) ^ 2 \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\binom { n } { m } 2 ^ m n ^ { \\delta m + 1 } \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} 0 & = \\frac { \\mu ( G _ 1 ( \\rho ) - \\rho ) } { \\rho - 1 } = \\frac { 1 } { \\rho - 1 } \\sum _ { k = 1 } ^ \\infty k p _ k ( \\rho ^ { k - 1 } - \\rho ) = - p _ 1 + \\rho \\sum _ { k = 3 } ^ \\infty k p _ k \\frac { \\rho ^ { k - 2 } - 1 } { \\rho - 1 } \\\\ & = - p _ 1 + \\rho \\sum _ { k = 3 } ^ \\infty k p _ k ( \\rho ^ { k - 3 } + \\rho ^ { k - 4 } + \\dotsb + 1 ) \\\\ & \\ge - p _ 1 + \\rho \\sum _ { k = 3 } ^ \\infty k p _ k ( k - 2 ) \\rho ^ { k - 3 } \\\\ & \\ge \\sum _ { k = 1 } ^ \\infty k ( k - 2 ) p _ k \\rho ^ { k - 1 } \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} g ^ { i j } \\Pi ^ { \\alpha } _ { \\mu } \\partial _ { i j } ^ 2 \\eta ^ { \\mu } = - \\frac { 1 } { \\sigma } \\frac { J } { \\sqrt { g } } a ^ { \\mu \\alpha } N _ { \\mu } q , \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\xi _ { , t } & = 2 \\psi _ { Y } T \\\\ \\lambda \\frac { 1 } { T } \\frac { T _ { , t t } } { \\omega } + \\frac { \\left ( \\ln \\omega \\right ) _ { , t } \\xi } { T } & = d _ { 2 } \\\\ \\frac { 1 } { T } \\frac { K _ { , t } } { \\omega } & = k \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{gather*} H ^ { \\nu } ( x ) = \\frac { \\prod \\limits _ { i = 0 } ^ { \\infty } { ( 1 - q ^ i t ) } } { \\prod \\limits _ { j = 1 } ^ { \\infty } { ( 1 - q ^ { \\nu _ j + j - 1 } t ) } } . \\end{gather*}"}
-{"id": "3266.png", "formula": "\\begin{gather*} C _ { \\tau } ^ { ( q , q ^ { \\theta } ) } \\big ( t ^ { N - 1 } x _ 1 , \\dots , t ^ { N - 1 } x _ m \\big ) = C _ { \\tau } ^ { ( q , q ^ { \\theta } ) } ( x _ 1 , \\dots , x _ m ) . \\end{gather*}"}
-{"id": "8773.png", "formula": "\\begin{align*} A ^ - _ { i j } = \\tilde { f } ^ { ( 0 ) } _ { i j } - \\sum _ { j < j ' < i } A ^ - _ { i j ' } \\tilde { f } ^ { ( 0 ) } _ { j ' j } . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\mathcal { H } ^ { d - 1 } ( \\{ y \\in T _ \\Delta A \\ , | \\ , d ( y , H ) = t \\} ) \\ge \\left ( \\frac { R - \\Delta } { R } \\right ) ^ { d - 1 } \\mathcal { H } ^ { d - 1 } ( \\{ x \\in A \\ , | \\ , d ( x , H ) = t + \\Delta \\} ) . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ { n _ 1 } t ^ { N - 1 } , \\dots , q ^ { n _ m } t ^ { N - m } , t ^ { N - m - 1 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) } \\\\ { } = \\frac { Q _ { ( n _ 1 , \\dots , n _ m ) } \\big ( q ^ { \\lambda _ 1 } t ^ { N - 1 } , q ^ { \\lambda _ 2 } t ^ { N - 2 } , \\dots , q ^ { \\lambda _ N } ; q , t \\big ) } { Q _ { ( n _ 1 , \\dots , n _ m ) } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) } . \\end{gather*}"}
-{"id": "8821.png", "formula": "\\begin{align*} S _ 3 ( H ) = S ^ t _ 3 ( H ) - ( d _ c - 3 ) S ^ c _ 3 ( H ) \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} W _ { n , 1 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ { S _ { n } \\left ( { u , \\xi } \\right ) } \\right \\} \\left \\{ { e ^ { u \\xi } + \\varepsilon _ { n , 1 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} ( s : t ) \\mapsto \\left ( \\sqrt { \\binom { n - 1 } j } s ^ { n - 1 - j } t ^ j \\right ) _ { j = 0 , \\dots , n - 1 } \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} U _ 0 : = & \\left ( \\frac { 1 } { T } + C ( \\varepsilon , p ) \\right ) \\int _ { T _ 0 } ^ { T } \\int \\eta _ 0 ^ 2 u ^ p + a \\eta _ 0 ^ 2 u ^ p \\ ; d x d t , T _ 0 = T / 4 , \\\\ \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} K _ \\epsilon : = \\left \\{ \\Theta \\in \\mathcal { M } : \\Theta ( [ 0 , T ] \\times U \\times A ) \\leq \\frac { T \\nu ( U ) } { \\epsilon } \\right \\} \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} M _ h ( x ) = \\sum _ { n \\leq x } \\big ( P _ n ( x ) - D ( x ) \\big ) ^ h & = \\sum _ { n \\leq x } \\big ( \\log 2 \\cdot F _ { \\omega _ 0 } ( n ) + S _ 1 + S _ 2 \\big ) ^ h \\\\ & = \\sum _ { \\substack { h _ 0 , h _ 1 , h _ 2 \\ge 0 \\\\ h _ 0 + h _ 1 + h _ 2 = h } } \\binom h { h _ 0 , h _ 1 , h _ 2 } \\sum _ { n \\leq x } \\big ( \\log 2 \\cdot F _ { \\omega _ 0 } ( n ) \\big ) ^ { h _ 0 } S _ 1 ^ { h _ 1 } S _ 2 ^ { h _ 2 } , \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} I ^ { \\Omega } u ( x , \\lambda ) = \\left ( \\frac { \\lambda } { \\pi } \\right ) ^ { n } \\int _ { \\Omega } e ^ { 2 \\lambda \\psi ( x , \\bar { y } ) } i ( x , \\bar { y } , \\lambda ) e ^ { - 2 \\lambda \\Phi ( y ) } u ( y , \\lambda ) L ( d y ) , \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} ( \\sin ^ { 2 } \\theta ) g _ { 1 } ( [ U , V ] , Z ) & = - g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , \\varphi V ) - ( \\nabla \\pi _ { \\ast } ) ( V , \\varphi U ) , \\pi _ { \\ast } w Z ) , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{gather*} \\left ( 2 H - \\sum _ { i = 1 1 } ^ { 1 6 } s _ i E _ i - \\sum _ { j = 1 } ^ { 1 6 } r _ j E _ j \\right ) - \\frac { 1 } { 2 } \\left ( 3 H - \\sum _ { i = 1 1 } ^ { 1 6 } E _ i \\right ) \\\\ \\equiv \\frac { 1 } { 2 } \\left ( H - \\sum _ { i = 1 1 } ^ { 1 6 } ( 2 s _ i - 1 ) E _ i - \\sum _ { j = 1 } ^ { 1 6 } 2 r _ j E _ j \\right ) \\end{gather*}"}
-{"id": "7432.png", "formula": "\\begin{align*} E ( y ) = \\Bigl ( \\sum _ { i = 1 } ^ k \\bigl [ w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) + \\varepsilon ^ { \\frac { 1 } { 2 } } \\pi _ i ( \\varepsilon \\ , y ) \\bigr ] \\Bigr ) ^ 5 - \\sum _ { i = 1 } ^ k w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 5 ( y ) , y \\in \\Omega _ { \\varepsilon } . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{gather*} \\prod _ { 1 \\leq i < j \\leq m } { ( T _ { q , x _ j } - T _ { q , x _ i } ) } = \\sum _ { \\tau \\in M ^ { ( m ) } _ 1 } { ( - 1 ) ^ { | \\tau | } \\prod _ { k = 1 } ^ m { T _ { q , x _ k } ^ { k - 1 + \\tau _ k ^ + - \\tau _ k ^ - } } } , \\end{gather*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ n ^ M = \\mathcal { K } ( \\phi _ n ^ M ) + f . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\frac { 1 } { | m _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) | } = | \\omega _ \\alpha ( z ) + \\omega _ \\beta ( z ) - z | \\ge \\frac { L M } { 2 } \\ , , \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} v _ 0 = \\sum _ { j = 1 } ^ { \\infty } \\alpha _ { j } w _ { j } \\ , , \\qquad \\quad \\alpha _ { j } = \\langle v _ 0 , w _ j \\rangle \\ , , \\quad \\sum _ { j = 1 } ^ { \\infty } \\alpha _ { j } ^ 2 \\lambda _ j < \\infty \\ , , \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\delta Y ^ { n } _ t & = \\int ^ T _ t \\bigl [ f \\bigl ( s , Y ^ { n + 1 } _ s , Z ^ n _ s , V ^ n _ s \\bigr ) - f \\bigl ( s , Y ^ { n } _ s , Z ^ { n - 1 } _ s , V ^ { n - 1 } _ s \\bigr ) \\bigr ] d s - \\int ^ T _ t \\delta Z ^ { n } _ s d W _ s \\\\ & - \\int _ { t } ^ { T } \\int _ U \\delta V ^ { n } _ { s } ( e ) \\widehat \\pi ( d e , d s ) - \\int _ t ^ T d \\delta M ^ { n } _ s , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{gather*} J = ( z _ { 1 1 } , z _ { 1 2 } - z _ { 2 1 } , z _ { 1 3 } - z _ { 2 2 } - z _ { 3 1 } , z _ { 1 4 } - z _ { 2 3 } - z _ { 3 2 } - z _ { 4 1 } , z _ { 1 5 } - z _ { 2 4 } - z _ { 3 3 } - z _ { 4 2 } , \\\\ z _ { 2 5 } - z _ { 3 4 } - z _ { 4 3 } , z _ { 3 5 } - z _ { 4 4 } , z _ { 4 5 } ) \\end{gather*}"}
-{"id": "1606.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H ^ { i j } _ { \\tau , t } } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} P _ t u ( x ) - u ( x ) = \\int _ 0 ^ t P _ s ( L _ r u ) ( x ) \\ , d s . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} | \\chi _ { \\mathbb { R } ^ + } ( x ) e ^ { i ( \\lambda + i \\epsilon ) x } * g _ n | & = \\left | \\int _ { - \\infty } ^ x e ^ { - \\epsilon ( x - y ) } e ^ { i \\lambda ( x - y ) } g _ n ( y ) ~ d y \\right | \\\\ & \\le e ^ { - \\epsilon x } ( \\sup _ { y \\le x } e ^ { \\epsilon y } [ w ( y ) ] ^ { 2 + s - s _ 1 } ) \\| w ^ { - 2 - ( s - s _ 1 ) } g _ n \\| _ { L ^ 1 } \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\Omega _ X = & z d p \\wedge d q \\ \\big ( = p ^ { - \\frac { 2 } { 3 } } q ^ { - \\frac { 2 } { 3 } } ( 1 - p ) ^ { - \\frac { 2 } { 3 } } ( 1 - q ) ^ { - \\frac { 2 } { 3 } } ( 1 - x _ 1 p - x _ 2 q ) ^ { - \\frac { 2 } { 3 } } d p \\wedge d q \\big ) \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} u _ { n } : = \\mathbb { P } ( [ n ] \\mbox { i s a b l o c k o f } \\Pi ) = \\prod _ { j = 1 } ^ n \\sum _ { i = 1 } ^ j p _ i . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\nu _ { x _ n - x , z } ( t ) & = \\displaystyle \\lim _ { n \\rightarrow \\infty } \\frac { t } { t + \\| x _ n - x , z \\| } \\\\ & = \\frac { t } { t + \\displaystyle \\lim _ { n \\rightarrow \\infty } \\| x _ n - x , z \\| } = \\frac { t } { t + 0 } = 1 , i . e . x _ n \\rightarrow x \\textrm { f o r t h e 2 - P n o r m } . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} B = X _ n ^ { - 1 } \\ , D _ { \\hat { y } } \\hat { X } - X _ n ^ { - 2 } \\ , \\hat { X } \\otimes D _ { \\hat { y } } X _ n \\ , . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} T _ l ( T _ { \\alpha } g ) ( x ) & = \\int _ D \\int _ X \\int _ X g ( y ) \\ > K _ h ( z , d y ) \\ > K _ l ( x , d z ) \\ > \\alpha ( h ) \\ > d \\omega _ D ( h ) \\\\ & = \\int _ D \\int _ D \\int _ X g ( y ) \\ > K _ h ( x , d y ) \\ > d ( \\delta _ l * \\delta _ w ) ( h ) \\ > \\alpha ( w ) \\ > d \\omega _ D ( w ) \\\\ & = \\int _ D \\int _ D g _ x ( h ) \\ > d ( \\delta _ l * \\delta _ w ) ( h ) \\ > \\alpha ( w ) \\ > d \\omega _ D ( w ) = \\alpha ( \\bar l ) T _ \\alpha g ( x ) \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\overline { m } u ^ 2 \\leq E ( x , u ) = : \\partial _ u f ( x , 0 ) u - f ( x , u ) \\leq \\overline { M } u ^ 2 . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\sum _ { n _ 1 + n _ 2 = 2 g + 1 - D } { 2 g + 1 - D \\choose n _ 1 } \\cdot \\frac { ( 2 g - 3 + n + n _ 1 ) ! } { ( 2 g - 3 + n ) ! } \\cdot \\frac { n _ 2 ! } { 0 ! } \\cdot ( \\widetilde { \\Gamma } _ i ^ { n _ 1 , n _ 2 } ) , \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\int _ M f F ( \\nabla \\phi ) \\ d \\mathfrak { m } = \\int _ { - \\infty } ^ { \\infty } \\left ( \\int _ { \\phi ^ { - 1 } ( t ) } f \\ d A _ \\mathbf { n } \\right ) \\ d t , \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\Phi ( f ( z , t ) ) = ( 1 - t ) f \\Big ( \\frac { z } { ( 1 - t ) ^ { 2 } } , \\frac { z t } { ( 1 - t ) ^ { 2 } } \\Big ) . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} u _ i ( r ) = \\sum _ { j = 1 } ^ { \\infty } a _ { i j } r ^ j \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\xi _ { ( , i | j } \\delta _ { r ) } ^ { k } = 0 , \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\psi ( f * g ) = \\psi ( g * \\tau _ { i \\beta } ( f ) ) \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} h ( l ) = \\max \\left \\{ \\log _ 2 ( l - 2 ) + \\frac { l - 2 } { \\log _ 2 ( l - 2 ) } + 1 , \\ ; 3 \\log _ 2 ( ( l - 4 ) / 3 ) + \\frac { 2 ( l - 4 ) } { 3 \\log _ 2 ( 2 ( l - 4 ) / 3 ) } + 3 5 \\right \\} . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} L _ z ^ \\pm = m ^ { - 1 } ( t ^ { - 1 } z ) \\pm i \\partial _ x , \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s \\left ( \\psi _ { l j } ( x _ { i 1 } , \\ldots , x _ { i n } ) - \\psi _ { l j } ( y _ { i 1 } , \\ldots , y _ { i n } ) \\right ) = 0 ( 1 \\le l \\le n , 1 \\le j \\le k ) , \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho } H ( 1 , \\rho ) | _ { \\rho = 0 } = \\int \\limits _ { - 1 } ^ 1 f ( t ) d t . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\big ( \\nabla _ { o u t } f \\cdot [ g ] \\big ) ( v ) \\ , & : = \\sum _ { e \\in E ^ + ( v ) } \\nabla f _ { e - } ( v ) \\ , g _ { e + } \\ , + \\ , \\sum _ { e \\in E ^ - ( v ) } \\nabla f _ { e + } ( v ) \\ , g _ { e - } . \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\partial _ { n } ( \\tilde { v } ) - A ( x ^ { \\prime } , x ^ { n } ) \\tilde { v } & = w , \\\\ \\tilde { v } | _ { x ^ n < < 0 } & = 0 \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} E \\frac { u ( t + h ) - u ( t ) } { h } u ( t ) = E \\frac { P _ h u ( t ) - u ( t ) } { h } u ( t ) + R ( h ) , \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{gather*} \\prod _ { \\substack { 1 \\leq i < j \\leq m \\\\ 0 \\leq k < \\theta } } { \\big ( t ^ { N - 1 } x _ i - t ^ { N - 1 } x _ j q ^ k \\big ) ^ { - 1 } } \\times ( q - 1 ) ^ { - \\theta { m \\choose 2 } } \\times \\prod _ { i = 1 } ^ m { \\frac { [ \\theta ( N - i + 1 ) - 1 ] _ q ! } { [ \\theta N - 1 ] _ q ! } } \\end{gather*}"}
-{"id": "1018.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | \\varphi | ^ 2 ~ d x + \\frac { 1 } { 4 \\pi } \\widehat { u \\varphi } ( 0 ) \\bar { \\hat \\varphi } ( 0 + ) = 0 . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} W ( z ) - \\Theta & = - \\frac { ( 1 - b - c _ 0 ) ^ 2 + ( 1 - b ) ^ 2 } 2 \\Theta + \\frac { 1 } { d + 2 } \\int _ { \\partial B _ 1 } \\left ( | \\nabla _ \\theta \\phi | ^ 2 - 2 d \\phi ^ 2 \\right ) - \\frac { c _ 0 } { d + 2 } \\int _ { \\partial B _ 1 ^ + } \\phi \\\\ & \\le - \\frac { c _ 0 ^ 2 } 4 \\Theta + \\frac { 1 } { d + 2 } \\int _ { \\partial B _ 1 } \\left ( | \\nabla _ \\theta \\phi | ^ 2 - 2 d \\phi ^ 2 \\right ) - \\frac { c _ 0 } { d + 2 } \\int _ { \\partial B _ 1 ^ + } \\phi , \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} \\zeta _ i ^ j = s _ { j - 1 } + \\mu _ i \\ ; \\tilde h , \\ ; \\ ; \\ ; i = 1 , \\ldots , \\rho , \\ ; \\ ; \\ ; j = 1 , \\ldots , m . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\langle T ^ n _ + \\rangle ( x _ 0 , v _ 0 , t ) = \\left . ( - 1 ) ^ n \\frac { \\partial ^ n } { \\partial p ^ n } Q _ p ( x _ 0 , v _ 0 , t ) \\right \\vert _ { p = 0 } \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ { \\mu _ 1 } t ^ { N - 1 } , q ^ { \\mu _ 2 } t ^ { N - 2 } , \\dots , q ^ { \\mu _ N } ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) } = \\frac { P _ { \\mu } \\big ( q ^ { \\lambda _ 1 } t ^ { N - 1 } , q ^ { \\lambda _ 2 } t ^ { N - 2 } , \\dots , q ^ { \\lambda _ N } ; q , t \\big ) } { P _ { \\mu } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) } . \\end{gather*}"}
-{"id": "7393.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 z _ { i , j } \\ , z _ { i , l } = 0 \\quad j \\neq l . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} \\sum _ { n \\leq R } d ( n ^ 2 + 1 ) = \\frac { N _ 1 ( R ) } { 2 } - \\frac { N _ 2 ( R / 2 ) } { 4 } . \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} { \\cal V } ^ l = ( P ^ { l 0 } - P ^ { ( l - 1 ) 0 } ) V ^ { l } , \\ \\ { \\cal V } ^ l _ \\# = ( P ^ { l 0 } _ \\# - P ^ { ( l - 1 ) 0 } _ \\# ) V ^ l . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} X _ { \\sigma } = \\sigma ^ i ( q , \\gamma ( q ) ) \\frac { \\partial } { \\partial q ^ i } + \\sigma _ i ( q , \\gamma ( q ) ) \\frac { \\partial } { \\partial p _ i } , X _ { \\sigma } ^ { \\gamma } = \\sigma ^ i ( q , \\gamma ( q ) ) \\frac { \\partial } { \\partial q ^ i } , \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , V ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Y , \\varphi V ) . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} S & = \\sum \\limits _ { i = 1 } ^ { n } \\left ( k ^ { 2 } - g h ^ { 2 } - f h k \\right ) ( - g ) ^ { i - 1 } \\\\ & = \\left \\{ \\begin{array} [ c ] { c } \\left ( k ^ { 2 } - g h ^ { 2 } - f h k \\right ) \\dfrac { 1 - ( - g ) ^ { n } } { 1 + g } g \\neq - 1 \\\\ n \\left ( k ^ { 2 } - g h ^ { 2 } - k h t \\right ) g = - 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} \\underset { E \\searrow 0 } { { \\lim } } ~ ~ \\frac { \\ln | \\ln N ( E ) | } { \\ln E } = - \\frac { d } { 2 ( \\alpha - d ) } \\ , . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} A _ t = A _ 0 + \\sum _ { 0 \\leq s \\leq t } \\Delta A _ s . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\dot { q } ^ i = g ^ i ( q , \\lambda ) , f ^ a ( q , \\lambda ) = 0 \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\overline { M } ^ { ( n ) } \\overline { M } _ 2 / \\overline { M } _ 2 = ( 2 , T ) ^ { n - 1 } ( \\overline { M } / \\overline { M } _ 2 ) \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} \\Xi ( Y ) \\equiv _ { \\mathrm { t t } } \\bigoplus _ { i = 0 } ^ { j } \\Phi _ { A _ i } ( Y _ i ) \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\operatorname { v o l } ( \\rho _ u ) - \\operatorname { v o l } ( \\rho _ 0 ) = - \\Im ( \\tau ) | z | + O ( | z | ^ 2 ) . \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\widetilde { f } ( X ) = \\left ( a _ 3 ^ 2 X ^ 2 + 3 a _ 1 a _ 3 - a _ 2 ^ 2 \\right ) ^ 2 X ^ 2 - \\Delta _ f , \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\mathfrak { I } ( u ) : = \\int \\limits _ { \\mathbb { G } } ( | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } + | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } - | u ( x ) | ^ { q } ) d x . \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal E _ s ( \\hat u ; \\R ^ n \\ ! \\ ! \\times \\ ! \\R ^ n ) = 2 \\mathcal E _ s ( u ; \\R ^ n _ + \\ ! \\times \\ ! \\R _ + ^ n ) + 2 \\mathcal E _ s ( \\hat u ; \\R ^ n _ + \\times \\R _ - ^ n ) \\\\ \\mathcal E _ s ( \\hat u ; \\R ^ n _ + \\times \\R _ - ^ n ) \\le \\mathcal E _ s ( u ; \\R ^ n _ + \\ ! \\times \\ ! \\R _ + ^ n ) ~ \\ ! . \\end{cases} \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} a _ 1 x _ 1 ^ k + \\ldots + a _ s x _ s ^ k = 0 \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} c _ j = \\frac { 1 } { P _ r ( \\vec { 1 } ) } \\left [ \\left ( 1 - \\frac { P _ { r + 1 } ( \\vec { 1 } ) } { P _ r ( \\vec { 1 } ) } \\right ) \\frac { \\partial P _ r } { \\partial z _ j } ( \\vec { 1 } ) + \\frac { \\partial P _ { r + 1 } } { \\partial z _ j } ( \\vec { 1 } ) \\right ] \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} P ( x , t ; s ) = t ^ { - n / 2 s } F _ s ( | x | t ^ { - 1 / 2 s } ) \\ , . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} S _ { n , m } ( \\Psi ) = \\sum _ { h = 1 } ^ { \\infty } h ^ { n - m } \\Psi ^ m ( h ) \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} \\alpha _ n ^ \\dagger \\circ K i G ( f g ) \\circ \\alpha _ m & = \\alpha _ n ^ \\dagger \\circ K i G ( f ) K i G ( g ) \\circ \\alpha _ m \\\\ & = \\alpha _ n ^ \\dagger \\circ K i G ( f ) \\alpha _ k \\alpha _ k ^ \\dagger K i G ( g ) \\circ \\alpha _ m \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\tilde D ( x ) = \\log D ( x ) - \\log D ( f ( x ) ) . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\Omega \\tt ( x , t ) \\partial _ t \\phi ( x , t ) d x d t + \\int _ 0 ^ T \\int _ \\Omega u ( x , t ) \\tt ( x , t ) \\cdot \\nabla \\phi ( x , t ) d x d t = 0 \\quad \\forall \\phi \\in C ^ \\infty _ 0 ( \\Omega \\times ( 0 , T ) ) . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} \\gamma ^ N _ i ( t , x ^ N ) : = \\gamma ( t , x ^ N _ i ) \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} u ^ \\ast ( t ) = \\left \\{ \\begin{aligned} \\begin{pmatrix} 0 \\\\ k _ E \\end{pmatrix} & \\quad \\textnormal { f o r } t < \\tau _ s \\\\ 0 & \\quad \\textnormal { f o r } t > \\tau _ s . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\frac { P _ n ( x ) - D ( x ) } { \\sqrt C ( \\log \\log x ) ^ { 3 / 2 } } = \\frac { \\log G ( n ) - A ( \\log \\log x ) ^ 2 } { \\sqrt C ( \\log \\log x ) ^ { 3 / 2 } } + O \\bigg ( \\frac 1 { \\log \\log \\log x } \\bigg ) \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} X ( t ) = \\xi + \\int _ 0 ^ t \\int _ U f ( s , X ( s ^ - ) , u , \\alpha ( s ) , m ( s ) ) \\N ( d s , d u ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\tau = 0 , { \\tilde b } _ 1 ( t , x , \\xi ) = \\cdots = { \\tilde b } _ k ( t , x , \\xi ) = 0 , \\ ; \\ ; { \\tilde b } _ i ( t , x , \\xi ) = b _ i ( t , \\chi _ t ( x , \\xi ) ) \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} \\lambda \\pi ^ - + A _ 2 \\pi ^ - = 0 \\hbox { i n } \\ : D _ 2 , \\lambda \\pi ^ - + B _ { 2 , + } \\pi ^ - = 0 \\hbox { i n } \\ : D _ 2 ^ + , \\lambda \\pi ^ - + B _ { 2 , - } \\pi ^ - = 0 \\hbox { i n } \\ : D _ 2 ^ - , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\psi _ k ( x _ i - \\theta ) = ( x _ i - \\theta ) \\left ( 1 - \\left ( \\frac { x _ i - \\theta } { k } \\right ) ^ 2 \\right ) ^ 2 I _ { \\{ | x _ i - \\theta | \\leq k \\} } \\ , , \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 4 ) = O \\left ( \\frac { p _ n ^ 3 } { n ^ 4 } \\right ) . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} b ( \\mathbf { D } ) ^ * g ( \\mathbf { x } ) \\left ( b ( \\mathbf { D } ) \\Lambda ( \\mathbf { x } ) + \\mathbf { 1 } _ m \\right ) = 0 , \\int _ { \\Omega } \\Lambda ( \\mathbf { x } ) \\ , d \\mathbf { x } = 0 . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} S H _ 2 = W ^ 1 ( - 1 ) W ^ 2 ( 1 ) H _ 2 = W ^ 1 ( - 1 ) ( H _ 1 \\cap H _ 2 ) = H _ 1 \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{gather*} Q = M A N \\subset L . \\end{gather*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\sum _ { j \\ge 0 } | \\sum _ { i \\ge j } a _ { i - j } v _ i | ^ q = \\sum _ { j \\ge 0 } | \\sum _ { i \\ge 0 } a _ i v _ { i + j } | ^ q \\le C _ p ^ q \\sum _ { j \\ge 0 } | A _ j v _ j | ^ q \\ , . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} | V _ m | & \\leq \\prod _ { r = 2 } ^ m \\ ( 1 + O ( r ^ { 1 / 2 } / n ^ { 1 / 2 } + 1 / r ^ { 1 / 2 } ) \\ ) \\frac { n ! } { n ^ n } \\\\ & = e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\sup u _ \\infty = - \\inf u _ \\infty u _ \\infty . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} H _ 1 ^ 1 & = Y & H _ 1 ^ 2 & = Z \\\\ H _ 2 ^ 1 & = X H _ 1 ^ 1 & H _ 2 ^ 2 & = X H _ 1 ^ 2 + Y ^ 2 \\\\ H _ 3 ^ 1 & = X ^ 2 H _ 1 ^ 1 & H _ 3 ^ 2 & = X ^ 2 H _ 1 ^ 2 + X Y ^ 2 = X H _ 2 ^ 2 \\\\ H _ 4 ^ 1 & = X ^ 3 H _ 1 ^ 1 + Y W + Z ^ 2 & H _ 4 ^ 2 & = X ^ 3 H _ 1 ^ 2 + X ^ 2 Y ^ 2 + Z W = X H _ 3 ^ 2 + Z W \\\\ H _ 5 ^ 1 & = X ^ 4 H _ 1 ^ 1 + X Y W + X Z ^ 2 + Y ^ 2 Z & H _ 5 ^ 2 & = X ^ 4 H _ 1 ^ 2 + X ^ 3 Y ^ 2 + X Z W + Y Z ^ 2 + Y ^ 2 W \\\\ & = X H _ 4 ^ 1 + Y ^ 2 Z & & = X H _ 4 ^ 2 + Y Z ^ 2 + Y ^ 2 W \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{gather*} Q = Q ( \\overline { \\partial } \\eta ) . \\end{gather*}"}
-{"id": "5302.png", "formula": "\\begin{align*} d \\xi / d z = f ^ { 1 / 2 } \\left ( z \\right ) , \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} = & m ( P _ { a - 1 } \\cup P _ { b - 1 } , k ) + 2 m ( P _ { a - 2 } \\cup P _ { b - 1 } , k - 1 ) \\\\ & + 2 m ( P _ { a - 1 } \\cup P _ { b - 2 } , k - 1 ) + t m ( Q , k - 1 ) , \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} D ( e _ 1 ) = \\frac { 1 } { 2 } e _ 1 , D ( e _ 2 ) = 0 \\textrm { a n d } D ( e _ 3 ) = \\frac { 1 } { 2 } e _ 3 . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} D _ { p , q } = \\left \\{ Z \\in \\mathcal { M } _ { p , q } \\left ( \\mathbb { C } \\right ) ; I _ { q } - \\overline { Z } ^ { t } Z > 0 \\right \\} , S _ { p , q } = \\left \\{ Z \\in \\mathcal { M } _ { p , q } \\left ( \\mathbb { C } \\right ) ; I _ { q } - \\overline { Z } ^ { t } Z = 0 \\right \\} , p > q . \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } | P _ n \\varphi | ^ 2 ~ d x + \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { \\infty } \\left ( \\left ( \\frac { \\psi _ n ^ 2 } { 2 } \\right ) ' \\widehat { u \\varphi } \\bar { \\hat \\varphi } + \\psi _ n ^ 2 \\widehat { u \\varphi } ' \\bar { \\hat \\varphi } \\right ) ~ d \\xi = 0 . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\| F ( t ) - P \\| & \\le C _ 1 | t | ; C _ 1 = \\beta _ 1 \\delta ^ { - 1 / 2 } \\| X _ 1 \\| , \\\\ \\| A ( t ) F ( t ) - t ^ 2 S P \\| & \\le C _ 2 | t | ^ 3 ; C _ 2 = \\beta _ 2 \\delta ^ { - 1 / 2 } \\| X _ 1 \\| ^ 3 . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\langle p , g \\rangle = \\sum _ { k = 1 } ^ n p _ k \\langle \\chi _ k , g \\rangle = 0 \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} & A _ m ( x ) = S _ { m } ( x ) + S _ { k _ { 1 } m } ( x ) + \\cdot \\cdot \\cdot + S _ { k _ { t - 1 - j _ 0 } m } ( x ) \\\\ & = \\Bigr ( S _ m ( x _ 2 ) + S _ { k _ { 1 } m } ( x _ 2 ) + \\cdot \\cdot \\cdot + S _ { k _ { t - 1 - j _ 0 } m } ( x _ 2 ) \\Bigr ) \\\\ & \\quad \\quad + \\Bigr ( R _ m ( x , x _ 2 ) + R _ { k _ { 1 } m } ( x , x _ 2 ) + \\cdot \\cdot \\cdot + R _ { k _ { t - 1 - j _ 0 } m } ( x , x _ 2 ) \\Bigr ) \\\\ & = A _ { m } ( x _ { 2 } ) ( 1 + v _ { m } ( x , x _ 2 ) ) , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} f ^ { ( k ) } ( X ) & = f ^ { ( k - 1 ) } ( f ( X ) ) \\\\ & = b _ 1 ( a _ 1 X ^ { d _ 1 } + a _ 2 X ^ { d _ 2 } + \\ldots + a _ s ) ^ { e _ 1 } + b _ 2 . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} H \\cdot \\Psi _ \\chi = \\big ( \\mathrm { p o s } ( \\chi ) - \\mathrm { n e g } ( \\chi ) \\big ) \\cdot \\Psi _ \\chi , \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} h _ { V _ { n , c } } ( t ) = \\sum _ { i = 0 } ^ { n - 1 } h _ i t ^ i , ~ h _ i = \\sum _ { j = 0 } ^ i ( - 1 ) ^ { i - j } \\binom { n - 1 + j c } { n - 1 } \\binom { n } { i - j } . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ { \\infty } { \\frac { ( a ; q ) _ n } { ( q ; q ) _ n } z ^ n } = \\frac { ( a z ; q ) _ { \\infty } } { ( z ; q ) _ { \\infty } } \\end{gather*}"}
-{"id": "9871.png", "formula": "\\begin{align*} 0 = y _ 1 z _ 1 + y _ 2 z _ 2 + \\cdots + y _ r z _ r & \\ge ( y _ 1 + \\cdots + y _ k ) z _ k + ( y _ \\ell + \\cdots + y _ r ) z _ \\ell \\\\ & = ( z _ \\ell - z _ k ) ( y _ \\ell + \\cdots + y _ r ) > 0 , \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} h = \\sum _ { I \\subset \\{ 1 , \\dots , d \\} } d t _ I \\wedge h _ I . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\P \\big ( \\cap _ { j = 1 } ^ k N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\mid \\eta \\big ) = \\prod _ { j = 1 } ^ k \\frac { \\eta ( t _ { j - 1 } , t _ j ] ^ { \\ell _ j } } { \\ell _ j ! } e ^ { - \\eta ( t _ { j - 1 } , t _ j ] } . \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} ( M _ r V ) _ S = \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] ( - 1 ) ^ { r - | S | + a - b } \\binom { s - | U ' | } { a - | U ' | } \\binom { r - s - | U | + | U ' | } { b } . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} E _ \\alpha : = \\{ \\phi \\in C ( X ) : \\ > ( \\alpha , \\phi ) \\quad \\} \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} f _ 2 ( h , k ) & : = \\Big [ \\int _ { - \\infty } ^ { \\infty } G _ \\nu ( t + h ) g _ \\nu ( t ) d t \\Big ] ^ k \\\\ & = \\mathbb { P } \\big ( \\max _ { i = 1 , \\ldots , k } \\{ T _ i ^ { ( 1 ) } - T _ i ^ { ( 2 ) } \\} \\leq h \\big ) \\\\ & = \\mathbb { P } \\big ( \\max _ { i = 1 , \\ldots , k } \\{ T _ i ^ { ( 1 ) } + T _ i ^ { ( 2 ) } \\} \\leq h \\big ) . \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{gather*} \\big \\{ m - n \\colon a _ { m , n } ^ { q - 1 } ( \\mathcal D _ z f ) > 0 , \\ , m , n \\geq 0 \\big \\} = \\mathbb Z _ + \\setminus 5 \\mathbb { Z } \\end{gather*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\sum _ { n \\geq 1 } a ( n ) \\exp \\bigg ( - \\frac { Y ^ 2 \\log ^ 2 ( X / n ) } { 4 \\pi } \\bigg ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } \\exp \\big ( \\frac { \\pi s ^ 2 } { Y ^ s } \\big ) \\frac { X ^ s } { Y } d s . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\frac { 1 + i ^ { 2 k } } { 2 } \\sum _ { c > 0 } \\frac { 1 } { ( 4 c ^ 2 ) ^ { 2 w } } \\varphi ( 4 c ^ 2 ) = \\frac { 1 + i ^ { 2 k } } { 2 } \\frac { 1 } { 2 ^ { 4 w - 1 } - 1 } \\frac { \\zeta ( 4 w - 2 ) } { \\zeta ( 4 w - 1 ) } . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S & = 2 ^ { r - | S | - 1 } \\sum _ a \\binom { | S | } { 2 } \\binom { | S | - 2 } { a - 2 } + 2 ^ { | S | } \\sum _ b \\binom { r - | S | - 1 } { 2 } \\binom { r - | S | - 3 } { b - 2 } \\\\ & = 2 ^ { r - 4 } [ | S | ^ 2 - | S | + ( r - | S | - 1 ) ( r - | S | - 2 ) ] \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\hat { f } _ s ( r _ c ) = 2 \\pi \\lambda _ B r _ c p _ { s , c } ( r _ c ) e ^ { - 2 \\pi \\lambda _ B \\big ( \\psi _ s ( r _ c ) + \\psi _ { s ' } ( r _ c ^ { \\alpha _ { s , c } / \\alpha _ { s ' , c } } ) \\big ) } / \\mathcal { A } _ { s , c } \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\frac { d A } { d D } & = \\left ( \\frac { - A ^ 4 D ^ 2 } { \\lambda _ 1 ^ 2 \\lambda _ 2 \\lambda _ 4 ^ 2 \\lambda _ 5 } + \\frac { - A ^ 3 B } { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 D } \\right ) \\left ( \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 } { A ^ 2 B } \\right ) \\\\ & = \\frac { - \\lambda _ 3 A ^ 2 D ^ 2 } { \\lambda _ 1 \\lambda _ 4 ^ 2 \\lambda _ 5 B } + \\frac { - A } { D } \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} J ^ { \\alpha } _ A x : = \\frac { \\sin \\left ( \\alpha \\pi \\right ) } { \\pi } \\int \\limits _ { 0 } ^ { \\infty } t ^ { \\alpha - 1 } \\left ( t + A \\right ) ^ { - 1 } A x \\ , d t . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} S _ f ^ \\nu ( n ) = \\sum _ { m \\leq n } \\frac { a _ f ( m ) } { m ^ \\nu } \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} h ( y ) = \\{ h _ r ( y ) \\} _ { r = 1 } ^ { m n } , & h _ r ( y ) : = y _ 1 ^ { i - 1 } y _ 2 ^ { j - 1 } { \\mathrm { f o r } } \\ , \\ , r = m ( i - 1 ) + j , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} I _ 0 = 2 ( \\lambda _ 1 ^ { - 2 } + \\lambda _ 2 ^ { - 2 } ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial t } + L w - ( \\lambda + \\mu ) w = - \\frac { \\partial u } { \\partial y } \\frac { \\partial v } { \\partial y } \\ : \\mbox { i n } \\ : D _ T , \\frac { \\partial w } { \\partial t } + L _ \\pm w - ( \\lambda + \\mu ) w = - \\frac { \\partial u } { \\partial y } \\frac { \\partial v } { \\partial y } \\ : \\mbox { i n } \\ : D _ T ^ \\pm , w ( T ) = 0 \\ : \\mbox { i n } \\ : D \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} d _ 0 ( h ) = ( - 1 ) ^ { \\dim R _ { \\alpha } + \\mu ( \\alpha ) + 1 + \\deg h } d _ { d R } ( h ) \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\ell = \\sqrt { b c _ r - 1 } \\ge \\sqrt { b r } \\sqrt { 1 - \\frac { 1 } { b } } \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} u _ 1 < b _ { 2 , 1 } < b _ { 1 , 1 } + b _ { 2 , 1 } = u _ 1 + 2 b _ { 1 , 1 } \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} a _ 0 \\lambda _ 0 ^ { n - 1 } + \\sum _ { i = 1 } ^ { n - 1 } a _ { - i } \\sum ^ { n - i } _ { j = 1 } b _ { i + j } \\lambda _ 0 ^ { n - j - 1 } \\ne 0 , \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} L y = x ^ 2 \\frac { d ^ 2 y } { d x ^ 2 } + x \\frac { d y } { d x } - \\bigl ( x ^ 2 + \\nu ^ 2 \\bigr ) y = 0 \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} U _ f = \\int _ \\R f ( t ) U _ t \\ , d t = \\int _ \\R \\int _ \\R e ^ { - i t p } f ( t ) \\ , d t \\ , d E ( p ) = \\int _ \\R \\hat f ( p ) \\ , d E ( p ) = \\hat f ( H ) . \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\nabla _ { P ( p , q , r ) } ( z ) & = \\nabla _ { P ( 1 , q , r ) } ( z ) + \\frac { ( p - 1 ) ( q + r ) } { 4 } z \\\\ & = 1 + \\frac { ( q + 1 ) ( r + 1 ) } { 4 } z + \\frac { ( p - 1 ) ( q + r ) } { 4 } z = 1 + \\frac { p q + q r + r p + 1 } { 4 } z . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\int ( a \\nabla u , \\nabla ( \\eta ^ 2 u _ k ^ { p - 1 } ) ) \\ ; d x = \\int ( p - 1 ) \\eta ^ 2 u _ k ^ { p - 2 } ( a \\nabla u _ k , \\nabla u _ k ) + 2 u _ k ^ { p - 1 } \\eta ( a \\nabla u _ k , \\nabla \\eta ) \\ ; d x . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\norm { u } _ { x , i } & = \\sum _ { n \\in \\mathbb { Z } } \\norm { A ^ n _ i ( x ) u } e ^ { - \\lambda _ i n - \\delta \\mid n \\mid } \\\\ & \\leq \\sum _ { n \\in \\mathbb { Z } } \\left ( C ( x ) e ^ { \\lambda _ i n + \\varepsilon \\mid n \\mid } \\norm { u } \\right ) e ^ { - \\lambda _ i n - \\delta \\mid n \\mid } \\\\ & = C ( x ) \\sum _ { n \\in \\mathbb { Z } } e ^ { ( \\varepsilon - \\delta ) \\mid n \\mid } \\norm { u } . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} X _ 1 ^ 2 + \\cdots + X _ { d - 1 } ^ 2 = X _ d ^ 2 + h \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} | \\mathbf { S } \\cdot \\vec w | ^ 2 = | ( S _ 1 ^ 2 - S _ 2 ^ 2 ) f _ 1 + ( S _ 1 \\bar S _ 2 + \\bar S _ 1 S _ 2 ) f _ 2 | ^ 2 = | f _ 1 | ^ 2 , \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\Psi \\bmod e _ { h + 1 } = \\Psi ' \\bmod e _ { h + 1 } & = \\pm | \\det G | ^ 2 \\sigma _ 0 \\wedge \\bar { \\sigma } _ 0 \\wedge \\cdots \\wedge \\sigma _ h \\wedge \\bar { \\sigma } _ h \\\\ & = \\pm | \\det G | ^ 2 A e _ 0 \\wedge f _ 0 \\wedge \\cdots \\wedge e _ h \\wedge f _ h . \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { 1 } { x } \\# \\bigg \\{ n \\leq x \\colon \\frac { \\log G ( n ) - A ( \\log \\log n ) ^ 2 } { \\sqrt C ( \\log \\log n ) ^ { 3 / 2 } } < u \\bigg \\} = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ u e ^ { - t ^ 2 / 2 } \\ , d t \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} H = \\ ( 1 + O \\ ( \\frac { \\log \\log n } { \\log n } \\ ) \\ ) \\sum _ { i = 1 } ^ n \\frac { a _ i } { n } \\log \\frac { n } { a _ i } . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\lambda _ 0 = d \\left ( \\frac { \\pi } { L } \\right ) ^ 2 \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} x _ n \\cdot x _ { 2 n } , ~ ~ x _ { n } - x _ { 2 n } + \\sum _ { i = 1 } ^ { n - 1 } c _ { i , n } x _ { n + i } . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} h ^ i ( N _ { X / H } ( - 1 ) ( - \\Gamma - F ) ) = 0 , h ^ i ( \\O _ X ( \\Gamma - F ) ) = 0 , h ^ i ( N _ Y ( - 1 ) ( - E ) ) = 0 . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\begin{cases} u \\in K \\ , , \\\\ \\noalign { \\medskip } \\displaystyle { \\int _ { \\Omega } \\bigl [ a ( x , u , \\nabla u ) \\cdot \\nabla ( v - u ) + b ( x , u , \\nabla u ) \\ , ( v - u ) \\bigr ] \\ , d x \\geq 0 } \\\\ \\noalign { \\medskip } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\ , . \\end{cases} \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\Big ( \\sum \\limits _ { k = 1 } ^ { n + 1 } D _ { i _ 1 } \\cdots S _ { i _ k } \\cdots D _ { i _ { n + 1 } } \\Big ) h ( w ) , \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { \\infty } \\mathbb { E } \\exp \\left ( - \\frac { x W _ i } { T _ { i } } \\right ) < \\infty \\mbox { f o r s o m e } x > 0 . \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\sum _ { c \\bmod { \\varpi ^ { l - 1 } } } \\widetilde { e } \\left ( \\frac { c } { \\varpi ^ { l - h - 1 } } \\right ) = N ( \\varpi ^ h ) \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - h - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 } { \\varpi ^ { l - h - 1 } } \\right ) . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} E | u ( t , x ) | ^ 2 = 1 + \\sum _ { n \\geq 1 } E | I _ n ( f _ n ( \\cdot , t , x ) ) | ^ 2 = 1 + \\sum _ { n \\geq 1 } J _ n ( t ) , \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} V _ m ( t , x ) : = \\inf _ { \\alpha \\in \\mathcal { A } } J ( t , x , \\alpha , m ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} B ( g , e ^ { 2 u + 2 v } g ) = B ( g , e ^ { 2 u } g ) + B ( e ^ { 2 u } g , e ^ { 2 u + 2 v } g ) ~ . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} \\int _ 0 ^ \\vartheta \\frac { x ^ t \\ , \\dd x } { ( x - \\lambda ) ^ 2 + \\nu ^ 2 } \\sim \\begin{cases} \\frac { \\lambda ^ t } { \\nu } \\ , , & \\textrm { i f } \\lambda > \\nu \\ , , \\\\ | \\omega | ^ { t - 1 } \\sim \\lambda ^ { t - 1 } \\ , , & \\textrm { i f } \\lambda < - \\nu \\ , , \\\\ \\nu ^ { t - 1 } \\ , , & \\textrm { i f } \\nu > | \\lambda | \\ , . \\end{cases} \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\begin{aligned} p ^ { ( 1 ) } ( n ) & = p ( 1 , n ) + p ( n + 1 ) , \\\\ p ^ { ( 1 ) } ( n _ 1 , n _ 2 ) & = p ( 1 , n _ 1 , n _ 2 ) + p ( n _ 1 + 1 , n _ 2 ) + p ( n _ 1 , n _ 2 + 1 ) \\end{aligned} \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} E \\left [ g ( X ( t ) ) \\right ] & = E \\left [ g ( \\xi ) \\right ] + E \\left [ \\int _ 0 ^ t \\int _ A \\Lambda _ s ^ a g ( X ( s ) ) \\rho _ s ( d a ) d s \\right ] \\\\ & = E \\left [ g ( \\xi ) \\right ] + E \\left [ E \\left [ \\left . \\int _ 0 ^ t \\int _ A \\Lambda _ s ^ a g ( X ( s ) ) \\rho _ s ( d a ) \\right | X ( s ) \\right ] d s \\right ] \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\dim ( p - 2 , 2 ) + \\dim ( p - 1 , 1 , 1 ) + 2 = & \\dim ( p - 2 , 2 ) + \\dim ( p - 2 , 1 , 1 ) \\cr & + \\dim ( p - 1 , 1 ) + 2 \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} v \\left ( z \\right ) = v _ { \\mu , a } ( z ) = \\log \\frac { 8 e ^ { \\mu } } { ( 1 + e ^ { \\mu } \\vert z + a \\vert ^ { 2 } ) ^ { 2 } } , \\mu \\in \\mathbb { R } , \\ ; a = ( a _ 1 , a _ 2 ) \\in \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} ~ u _ i ( ( L _ i / R ) ^ 3 x ) \\cap ~ \\eta ( R , s , t ; x ) = \\emptyset . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} k - 1 - \\sum _ { j = 1 } ^ { s ^ * - 1 } m _ j r _ j = P r _ { s ^ * } + Q \\geq 0 \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\big ( \\pi ( a ) \\xi ( s ( u ) ) \\ , \\big | \\ , \\xi ( x ) \\big ) = 0 . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\eta \\nabla u _ k ^ { p / 2 } = \\nabla ( \\eta u _ k ^ { p / 2 } ) - u _ k ^ { p / 2 } \\nabla \\eta , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} ( P _ k ^ { a c } \\circ \\Q [ \\eth ] ) ^ ! = \\Q [ \\eth ] ^ ! \\circ ( P _ k ^ { a c } ) ^ ! = \\Q [ \\hbar ^ 2 ] \\circ s ^ { k - 1 } \\hbar ^ { - 1 } P _ k ^ { a c } . \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { t \\in [ 0 , T ] } | L a w ( X _ k ( t ) ) - L a w ( X ( t ) ) | = 0 , \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} P = P _ { k , 1 } ^ { m _ 1 } \\cdot u _ 1 ^ { a _ 1 } \\cdot P _ { k , 1 } ( \\mathcal { O } ) \\cdot P _ { k , 2 } ^ { m _ 2 } \\cdot u _ 2 ^ { a _ 2 } \\cdots P _ k ^ { m _ k } \\cdot u _ k ^ { a _ k } . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 f ( t ) x ( t , \\rho ) d t = 0 , \\ , \\ , | \\rho | < \\rho _ 0 \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} f ' _ { \\theta , q , 3 } ( \\mathbf { x } ^ { ( i ) } ) = \\frac { 3 ( \\theta - 1 + q ) | 1 - \\theta - q | _ p ^ 2 \\mathbf { x } ^ { ( i ) } } { ( \\theta - 1 ) \\left ( \\mathbf { y } ^ { ( i ) } \\right ) ^ * \\left ( \\theta \\left ( \\mathbf { y } ^ { ( i ) } \\right ) ^ { * } + \\left ( 1 - \\theta - q \\right ) ^ { * } \\right ) } . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} ( \\Delta - s _ 0 ( n - s _ 0 ) - m ) \\tilde \\varphi ^ { ( k ) } = - ( 2 s _ 0 - n ) \\tilde \\varphi ^ { ( k - 1 ) } - \\tilde \\varphi ^ { ( k - 2 ) } \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} q ^ 2 - q _ 1 q _ 2 = 0 \\ , . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega _ { \\varepsilon } } \\nabla u _ { \\varepsilon } ( x , t ) \\cdot v \\left ( x , \\frac { x } { \\varepsilon } \\right ) c _ { 1 } ( t ) c _ { 2 } \\left ( \\frac { t } { \\varepsilon ^ { r } } \\right ) d x d t \\\\ = - \\int _ { 0 } ^ { T } \\int _ { \\Omega _ { \\varepsilon } } u _ { \\varepsilon } ( x , t ) \\nabla _ { x } \\cdot v \\left ( x , \\frac { x } { \\varepsilon } \\right ) c _ { 1 } ( t ) c _ { 2 } \\left ( \\frac { t } { \\varepsilon ^ { r } } \\right ) d x d t . \\end{gather*}"}
-{"id": "9256.png", "formula": "\\begin{align*} H = \\overline { \\{ ( T \\times S ) ^ n ( e _ X , e _ Y ) : n \\in \\Z \\} } \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\varepsilon _ d = \\begin{cases} 1 & d \\equiv 1 \\bmod 4 , \\\\ i & d \\equiv 3 \\bmod 4 . \\end{cases} \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\alpha _ i ( x , v ; x ' , v ' ) : = f _ i ( x , v ; x ' , v ' ) \\big | _ { ( x , v ) \\in \\Gamma _ + } , ( x ' , v ' ) \\in \\Gamma _ - \\ , , i = 1 , 2 , 3 \\ , . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Vert f _ { t } \\Vert _ { L _ { x } ^ { 2 } } ^ { 2 } = 2 \\langle A _ { t } ^ { \\psi } f _ { t } , f _ { t } \\rangle _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} & x _ 1 ^ { p ^ n } = x _ 2 ^ { p ^ 4 } = x _ 3 ^ { p ^ 4 } = x _ 4 ^ { p ^ 2 } = 1 \\\\ & [ x _ 1 , x _ 2 ] = [ x _ 1 , x _ 3 ] = x _ 2 ^ { p ^ 2 } , \\ ; [ x _ 1 , x _ 4 ] = x _ 3 ^ { p ^ 2 } , \\\\ & [ x _ 2 , x _ 3 ] = x _ 1 ^ { p ^ { n - 2 } } , \\ ; [ x _ 2 , x _ 4 ] = x _ 3 ^ { p ^ 2 } , \\ ; [ x _ 3 , x _ 4 ] = x _ 2 ^ { p ^ 2 } . \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 g ( t ) d t = 0 \\mbox { a n d } \\int _ { - 1 } ^ 1 f ( t ) x ( t , \\rho ) d t = 0 , \\ , \\ , | \\rho | < \\rho _ 0 \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B _ t ( 0 ) } \\frac { 8 } { ( 1 + | z | ^ 2 ) ^ 2 } \\Big ( \\frac { 1 - | z | ^ 2 } { 1 + | z | ^ 2 } \\Big ) \\mathrm { d } z = \\frac { 8 \\pi t ^ 2 } { ( t ^ 2 + 1 ) ^ 2 } , \\ \\ \\textrm { a n d } \\end{aligned} \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\leq N ^ { { \\varepsilon } } \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { ( N \\eta ) ^ { \\frac 1 3 } } , \\iota = 1 , 2 \\Theta ( z , { \\frac { N ^ { 3 \\varepsilon } } { ( N \\eta ) ^ { \\frac 1 3 } } } \\ , , { \\frac { N ^ { 3 \\varepsilon } } { \\sqrt { N \\eta } } } ) \\cap \\Omega _ 3 ( z ) \\ , . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { n p } ^ 4 } \\sum ^ { p _ n } _ { \\ell = 2 } E ( z _ { n \\ell } ^ 4 ) = o ( 1 ) ~ ~ ~ \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\begin{aligned} x ' & \\approx - \\frac { 1 1 } { 2 } x ^ 2 , & y ' & \\approx - \\frac { 1 1 } { 2 } y ^ 2 , \\\\ z ' & \\approx - \\frac { 1 1 } { 3 } z ^ 2 , & w ' & \\approx - \\frac { 1 1 } { 3 } w ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\int \\langle A ^ * ( q ) \\nabla v , \\nabla \\xi \\rangle d Z = 0 , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} M ( p ) = \\int _ { - \\infty } ^ { \\infty } \\vert \\xi \\vert ^ { p } \\frac { 1 } { ( 2 \\pi ) ^ { 1 / 2 } } \\exp \\left ( - \\frac { 1 } { 2 } \\vert \\xi \\vert ^ { 2 } \\right ) d \\xi . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\mathtt { J } _ { \\nu } ( z ) = \\frac { z ^ \\nu } { 2 ^ { \\nu } \\Gamma ( \\nu + 1 ) } \\prod _ { n = 1 } ^ \\infty \\left ( 1 - \\frac { z ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } } \\right ) . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} u _ n ( z , w ) : = n ( | z + w | - 3 / 2 ) . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} \\delta ( A ) = \\lim _ { n \\rightarrow \\infty } \\frac { f _ A ( n ) } { n } = \\inf _ { n \\in \\N } \\frac { f _ A ( n ) } { n } . \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\psi _ { \\alpha } ( a b - ( - 1 ) ^ { | a | | b | } b a ) & = \\psi _ { \\alpha } ( a b ) - ( - 1 ) ^ { | a | | b | } \\psi _ { \\alpha } ( b a ) \\\\ & = a \\psi _ { \\alpha } ( b ) + ( - 1 ) ^ { | a | | b | } b \\psi _ { \\alpha } ( a ) - ( - 1 ) ^ { | a | | b | } ( b \\psi _ { \\alpha } ( a ) + ( - 1 ) ^ { | a | | b | } a \\psi _ { \\alpha } ( b ) ) \\\\ & = 0 . \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} \\lambda _ z = t ^ { - \\frac 1 2 } , \\lambda _ y = t ^ { - \\frac 1 2 } v ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} J _ { f } = \\bigcap _ { n = 0 } ^ { \\infty } f ^ { - n } \\left ( X \\right ) . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A \\hat { n } = a ^ 1 \\partial _ 1 \\eta + a ^ 2 \\partial _ 2 \\eta + b \\hat { n } . \\end{gather*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\Omega = \\left \\{ x : - a < x _ { 1 } , x _ { 2 } < a , x _ { 3 } \\in \\left ( 0 , B \\right ) \\right \\} . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} v _ 1 + & \\dots + v _ m + v ' _ 1 + \\dots + v ' _ m \\\\ [ 1 e x ] & = \\ell _ 0 \\delta - \\delta + n - m - \\ell _ 0 \\delta - \\delta ( 2 m - 1 ) - \\ell _ { 1 , 1 } \\delta - \\delta - ( m - 1 ) \\\\ [ 1 e x ] & = n - 2 m + 1 - ( 2 m + 1 ) \\delta - \\ell _ { 1 , 1 } \\delta \\\\ [ 1 e x ] & \\ge n - 2 m + \\tfrac 1 2 - ( 2 m + 1 ) \\delta > n - 2 m \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\left ( M _ r \\Phi ( V ) \\right ) _ S & = \\sum _ { T \\subseteq [ r ] } M _ r ( S , T ) ( \\Phi ( V ) ) _ T \\\\ & = \\sum _ { | T | = 0 } ^ r \\sum _ { T } M _ r ( S , T ) ( \\Phi ( V ) ) _ T \\\\ & = \\sum _ { | T | = 0 } ^ r N _ r ( | S | , | T | ) V _ T \\\\ & = ( N _ r V ) ( | S | ) = \\lambda V ( | S | ) = \\lambda ( \\Phi ( V ) ) _ S \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} u _ i ( 0 , x ) = u _ { i t } ( 0 , x ) = 0 x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} \\psi _ { \\lambda _ 0 + \\varepsilon } ( \\zeta ) = \\det M _ { \\lambda _ 0 + \\varepsilon } ( \\zeta ) = \\sigma _ 1 ( \\varepsilon , \\zeta ) \\sigma _ * ( \\varepsilon , \\zeta ) , \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} u _ i ( T , x ) = u _ { i } ^ { 0 } ( x ) \\ , , u _ { i t } ( T , x ) = u _ { i } ^ { 1 } ( x ) \\ , , x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { \\varepsilon _ n } } w _ { \\mu _ { i , n } ^ { \\prime } , \\zeta _ { i , n } ^ { \\prime } } ^ 4 \\ , z _ { i j } ^ n \\ , { \\bf z } _ { k l } ^ n = C \\ , \\delta _ { i , k } \\ , \\delta _ { j , l } + o ( 1 ) \\delta _ { i , k } = \\left \\{ \\begin{array} { l l } 1 & i = k \\\\ 0 & i \\neq k , \\end{array} \\right . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { d + 2 } \\left | \\| \\mathbf v _ 1 - \\mathbf v _ j \\| ^ 2 - 1 \\right | \\geq 3 d ^ { \\beta - 2 / 3 } . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial t } = \\log \\frac { \\omega _ { \\phi } ^ n } { \\omega _ 0 ^ n } + \\gamma \\phi + F + \\theta _ X ( \\omega _ { \\phi } ) , \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} \\Big \\| \\frac { u ^ { \\theta } } { r } \\Big \\| _ { 4 } \\leq | | u ^ { \\theta } | | _ { 4 } = \\Big \\| ( r u ^ { \\theta } ) ^ { \\frac { 1 } { 2 } } \\Big ( \\frac { u ^ { \\theta } } { r } \\Big ) ^ { \\frac { 1 } { 2 } } \\Big \\| _ { 4 } \\leq | | r u ^ { \\theta } | | _ { \\infty } ^ { \\frac { 1 } { 2 } } \\Big \\| \\frac { u ^ { \\theta } } { r } \\Big \\| _ { 2 } ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\frac { \\partial v _ { n \\dots n } } { \\partial x ^ n } = f _ { n \\dots n } . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\phi ( x ) = x ( y ^ { - 1 } x ) ^ n , \\ \\phi ( y ) = y ( x ^ { - 1 } y ) ^ n \\ \\mbox { a n d } \\ \\phi ( s ) = s \\ \\mbox { f o r a l l $ s \\in S $ } \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} { \\rm H o m } ( G r _ e N , M ) = 0 , { \\rm H o m } ( W ^ { e - 1 } N , M ) = 0 . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} a \\sum _ { n = n _ 0 } ^ { \\infty } \\frac { 1 } { n ^ { 2 \\nu } } \\le \\varepsilon \\ , . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\tilde { z } _ n ^ M = { \\mathcal { K } } _ m ( { z } _ n ^ M ) + f . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\psi { _ { s } \\left ( \\xi \\right ) = } \\frac { f { _ { s + 2 } \\left ( z \\right ) } } { f { _ { 0 } \\left ( z \\right ) } } \\ \\left ( { s = 1 , 2 , 3 , \\cdots } \\right ) , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\psi _ 1 ^ { k ' } \\prod _ { j = 2 } ^ n \\psi _ j ^ { l _ j ' } \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} J ( t , x , \\alpha , m ) : = E \\left [ \\int _ t ^ T c ( s , X _ \\alpha ^ { t , x } ( s ) , \\alpha ( s ) , m ( s ) ) d s + \\psi ( X _ \\alpha ^ { t , x } ( T ) , m ( T ) ) \\right ] . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} \\sum ^ { p _ n } _ { \\ell = 2 } E ( y _ { n \\ell } ^ 4 ) \\to 0 ~ ~ \\mbox { a n d } ~ ~ E ( \\sum _ { \\ell = 2 } ^ { p _ n } y _ { n \\ell } ^ 2 - 1 ) ^ 2 \\to 0 \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} f _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) = P ^ { + } _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) + f ^ { n e g } _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) , \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\delta _ { \\gamma _ n ( t , x ) } ( d a ) d t = [ \\widehat { \\gamma } ( t , x ) ] ( d a ) d t \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { \\infty } \\mathbb { E } \\exp \\left ( - \\frac { x W _ i } { T _ i } \\right ) & = \\int _ 0 ^ 1 \\mathbb { E } \\exp \\left ( - \\frac { x w } { T _ { i - 1 } ( 1 - w ) } \\right ) \\cdot \\theta ( 1 - w ) ^ { \\theta - 1 } d w \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\exp \\left ( - \\frac { x w } { t ( 1 - w ) } \\right ) \\cdot \\frac { \\theta } { u } \\cdot \\theta ( 1 - w ) ^ { \\theta - 1 } d t d w , \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} \\alpha ( \\bar h ) = \\overline { \\alpha ( h ) } \\quad \\quad h \\in D . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} V _ { A ( \\mathcal { O } ) } = \\oplus _ { ( r , s ) \\in \\mathcal { O } } V _ { \\mathbf { e } _ r + \\mathbf { e } _ s } . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} m _ k = \\int _ { - 1 } ^ 1 f ( t ) ( G ( t ) ) ^ k d t = 0 , ~ ~ k = 0 , 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ N { \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i + 1 ) - z ) } } \\\\ \\qquad { } = \\prod _ { i , j } { [ \\theta ( N - i ) + j - z ] _ q } \\frac { [ \\lambda _ N - z - 1 ] _ q \\cdots [ - z ] _ q } { [ \\lambda _ 1 + \\theta N - z - 1 ] _ q \\cdots [ \\theta N - z ] _ q } \\\\ \\qquad \\quad { } \\times \\frac { \\Gamma _ q ( - z ) } { \\Gamma _ q ( \\theta N - z ) } , \\end{gather*}"}
-{"id": "4766.png", "formula": "\\begin{align*} F ( a , a ) = F ( a _ 1 , a _ 2 \\cdots a _ t a _ 1 \\cdots a _ t ) + ( - 1 ) ^ { | a _ 1 | ( | a _ 2 | + \\cdots + | a _ t | ) } F ( a _ 2 \\cdots a _ t , a _ 1 a _ 1 \\cdots a _ t ) = 0 \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} a _ { 1 } > a _ { 2 } > . . . > a _ { \\ell } \\geq 0 , \\ ; 1 < p < \\frac { Q } { a _ { 1 } } , \\ ; \\textrm { a n d } \\ ; Q \\left ( \\frac { 1 } { p } - \\frac { 1 } { p _ { j } } \\right ) = a _ { j } , \\ ; j = 1 \\ldots , \\ell , \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} D ( s , S _ f ) = L ( s , f ) + \\frac { L ( s - 1 , f ) } { s + \\frac { k - 1 } { 2 } - 1 } + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\epsilon ) } L ( s - z , f ) \\zeta ( z ) \\frac { \\Gamma ( z ) \\Gamma ( s + \\frac { k - 1 } { 2 } - z ) } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\ ; d z . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} h = \\min \\{ k \\geq 2 : ( 1 , 0 ^ { k - 2 } , 1 ) { \\rm \\ i s \\ \\beta - a d m i s s i b l e } \\} . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} & 2 m _ { 1 } \\partial ( a _ { 1 } e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } ) \\wedge \\bar \\partial ( \\varphi - \\eta ) \\\\ & = 2 m _ { 1 } ( e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\partial a _ { 1 } \\wedge \\bar \\partial ( \\varphi - \\eta ) + 2 m _ { 1 } a _ { 1 } e ^ { 2 m _ { 1 } ( \\varphi - \\eta ) } \\partial ( \\varphi - \\eta ) \\wedge \\bar \\partial ( \\varphi - \\eta ) ) \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} t _ i ( ( l _ 1 ) ) = t _ { i - 1 } ( ( 0 ) : l _ 1 ) + 1 & \\le \\max \\{ t _ { i - 1 } ( K ) , t _ { i - 1 } ( ( l _ 2 ) ) , t _ { i - 2 } ( L ) \\} + 1 \\\\ & = \\max \\{ i + 2 , t _ { i - 1 } ( ( l _ 2 ) ) + 1 \\} \\\\ & = i + 2 . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\partial _ { \\nu ^ 0 , \\rm o u t } w ^ t _ { 2 1 } = \\frac 1 2 f + \\partial _ { \\nu ^ 0 , 0 } w ^ t _ { 2 1 } \\mbox { o n } \\Gamma ^ 0 \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\left [ V _ 1 , V _ 2 \\right ] = ( \\left [ Y _ 1 , Y _ 2 \\right ] , Y _ 1 ( C _ 2 ) - Y _ 2 ( C _ 2 ) - \\{ C _ 1 , C _ 2 \\} , Y _ 1 ( \\tau _ 2 ) - Y _ 2 ( \\tau _ 1 ) ) , \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} U ( x ) = ( s ( x _ 1 ) , 0 , 0 ) = s ( x _ 3 ) e _ 1 , \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\rho _ \\mathfrak { a } ^ k ( 0 , w ) & = ( * ) \\frac { L ^ k _ \\mathfrak { a } ( 4 w - 2 ) } { L ^ k _ \\mathfrak { a } ( 4 w - 1 ) } \\frac { \\Gamma ( 2 w - 1 ) } { \\Gamma ( w + \\frac { k } { 2 } ) \\Gamma ( w - \\frac { k } { 2 } ) } \\\\ \\rho _ \\mathfrak { a } ^ k ( h , w ) & = ( * ) \\lvert h \\rvert ^ { w - 1 } \\frac { 1 } { \\Gamma ( w + \\frac { \\lvert h \\rvert } { h } \\frac { k } { 2 } ) } D _ \\mathfrak { a } ^ k ( h , w ) , \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\lim _ { n \\to \\pm \\infty } \\frac { 1 } { n } \\log D ( f ^ n ( x ) ) = 0 \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} \\omega _ D ( \\{ r \\} ) = \\sum _ { z \\in X : \\ > \\pi ( x , z ) = r } \\omega _ X ( \\{ z \\} ) \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\pi _ p = 2 \\int _ 0 ^ { + \\infty } \\frac { 1 } { 1 + \\frac { 1 } { p - 1 } s ^ p } d s = 2 \\pi \\frac { ( p - 1 ) ^ { \\frac 1 p } } { p \\sin \\frac { \\pi } { p } } . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( k + 1 - 3 i ) a ^ { i } + 2 ( k - 1 ) ( k - 2 ) a ^ { k - 1 } > 0 \\ , , \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} v _ m & = \\ ( \\begin{array} { c } V _ m \\\\ V _ { m - 1 } \\end{array} \\ ) , \\\\ M _ m & = \\ ( \\begin{array} { c c } \\gamma _ m \\alpha _ m ^ { - 1 / 2 } & 0 \\\\ 1 & 0 \\end{array} \\ ) , \\\\ N _ m & = \\ ( \\begin{array} { c c } \\beta _ m \\alpha _ m ^ { - 1 / 2 } & \\alpha _ m ^ { 1 / 2 } \\alpha _ { m - 1 } ^ { - 1 / 2 } \\\\ 1 & 0 \\end{array} \\ ) , \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} ( L _ s - \\lambda ) \\Psi = 0 , \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\binom { n + k - 2 } { k } \\sigma ^ { ( k ) } + \\binom { n + k - 2 } { k - 1 } \\tau ^ { ( k - 1 ) } + \\binom { n + k - 2 } { k - 2 } h ^ { ( k - 2 ) } = 0 , \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} c = \\langle \\eta ( X + { \\mathtt p } ( X ) ) \\rangle \\ , , - { \\cal H } \\tanh ( ( \\mathtt h + c ) | D | ) { \\mathtt p } ( X ) = \\pi _ 0 ^ \\bot \\eta ( X + { \\mathtt p } ( X ) ) \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} v ^ { T } F v + ( u - v ) ^ { T } S ( u - v ) = u ^ { T } C u + z ^ { T } ( F + S ) z \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} 0 = \\phi ( c ( t ) , d ( t ) ) = \\tilde a ( c ( t ) ) - \\tilde b ( d ( t ) ) . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} \\| z _ { i } ^ { 0 } \\| ^ 2 _ { H ^ 1 _ 0 } = \\sum _ { n = 1 } ^ \\infty \\alpha ^ 2 _ { i n } n ^ 2 , \\| z _ { i } ^ { 1 } \\| ^ 2 _ { L ^ 2 } = \\sum _ { n = 1 } ^ \\infty \\rho ^ 2 _ { i n } \\ , , i = 1 , 2 . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\Sigma _ { n , j } = \\sum _ { 1 \\le i _ 1 < \\cdots < i _ j \\le n } \\mathbb { E } \\left ( \\frac { T _ n } { T _ n + P _ { i _ 1 } + \\cdots + P _ { i _ j } } \\right ) . \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} ( \\Lambda _ \\Delta ) ^ { \\otimes _ D n } = \\bigoplus _ { \\omega \\in Q _ { \\leq n } } \\left [ \\bigoplus _ { \\tau \\in T _ n ( \\omega ) } \\tau _ \\Delta \\right ] \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} \\mu _ j ( z ) = \\sum _ { n \\neq 0 } \\rho _ j ( n ) y ^ { \\frac { 1 } { 2 } } K _ { i t _ j } ( 2 \\pi \\vert n \\vert y ) e ^ { 2 \\pi i n x } . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} \\frac { d } { d \\xi } \\left \\{ { \\frac { y } { y _ { 1 } } - \\frac { y _ { 2 } } { y _ { 1 } \\left ( { y _ { 2 } / y _ { 1 } } \\right ) ^ { \\prime } } \\frac { d } { d \\xi } \\left ( { \\frac { y } { y _ { 1 } } } \\right ) } \\right \\} = 0 , \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} \\hat y _ { \\infty , x } ( r ) = Q _ { \\infty } e ^ { - r A ^ * } Q _ \\infty ^ { - 1 } x , r \\in \\ , ] - \\infty , 0 ] ; \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} D ( N _ f ) = \\{ \\xi \\in L _ 2 [ 0 , \\alpha ) : \\ , f \\xi \\in L _ 2 [ 0 , \\alpha ) \\} \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\left ( \\xi _ { , t } + \\dot { x } ^ { r } \\xi _ { , r } \\right ) \\left ( \\frac { 1 } { 2 } g _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - V \\right ) = \\frac { 1 } { 2 } \\left ( \\xi _ { , t } g _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } - 2 V \\xi _ { , t } + g _ { i j } \\xi _ { , k } \\dot { x } ^ { i } \\dot { x } ^ { j } \\dot { x } ^ { k } - 2 \\dot { x } ^ { k } \\xi _ { , k } V \\right ) \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} c \\int _ { { \\R } ^ N } \\sum _ { i = 1 } ^ n \\frac { \\varphi ^ 2 } { | x - a _ i | ^ 2 } \\ , d x \\le \\int _ { { \\R } ^ N } | \\nabla \\varphi | ^ 2 \\ , d x + \\left [ \\frac { k + ( n + 1 ) c } { r _ 0 ^ 2 } \\right ] \\int _ { \\R ^ N } \\varphi ^ 2 \\ , d x \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\mathcal { K } _ m '' ( x ) ( v _ 1 , v _ 2 ) ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\frac { \\partial ^ 2 \\kappa } { \\partial u ^ 2 } ( s , \\zeta _ i ^ j , x ( \\zeta _ i ^ j ) ) v _ 1 ( \\zeta _ i ^ j ) \\ ; v _ 2 ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\left ( Z _ { A } ^ { A ' } u , v \\right ) _ \\varphi = \\left ( u , \\delta _ { A ' } ^ A v \\right ) _ \\varphi \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\widetilde z '' ( \\omega _ \\beta ( z ) ) = \\frac { F '' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) } { F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) } \\big ( 1 - F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) \\big ) - \\frac { 1 } { ( F ' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) ) ^ 3 } F '' _ { \\mu _ \\beta } ( \\omega _ \\alpha ( z ) ) \\cdot \\big ( F ' _ { \\mu _ \\alpha } ( \\omega _ \\beta ( z ) ) \\big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} \\Sigma _ i = \\{ ( w , u ) \\in C \\ | \\ u = 1 - x _ i \\} , ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} J = - \\sin y \\partial _ { x } , \\ \\ \\ L = 1 + \\Delta ^ { - 1 } \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} A ( \\pi ) = \\{ 1 = L _ 1 < L _ 2 < \\cdots < L _ { k - 1 } = n \\} \\cup \\{ n + 1 \\} \\ , . \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} d \\sqrt { n } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } \\left ( \\frac { 1 } { 2 } - \\frac { \\alpha ^ { * } } { 2 d } \\right ) } \\epsilon _ n ^ { 1 / 2 } \\lesssim \\underbrace { n ^ { - \\frac { 2 m ( \\alpha ^ { * } ) ^ 2 - d ( m - 1 ) \\alpha ^ { * } - m d ^ 2 } { 2 d ( 2 \\alpha ^ { * } + d ) } } ( \\log { n } ) ^ { \\kappa } } _ { \\tau _ n ( m ) } \\sqrt { n \\log { n } } , \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} t _ { n p } ^ c = \\sqrt { p ( p - 1 ) } t ^ * _ { n p } + \\frac { 1 } { 2 } p ( p - 1 ) = \\sqrt { \\frac { n + 1 } { n - 2 } } ( n - 1 ) t _ { n p } + \\frac { 1 } { 2 } p ( p - 1 ) ( 1 - \\sqrt { \\frac { n + 1 } { n - 2 } } ) . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\alpha ( A ) = \\inf \\{ \\delta > 0 \\colon A \\subset \\sum _ { k = 1 } ^ { N } B _ k , d i a m ( B _ k ) \\leq \\delta \\ { \\rm a n d \\ } N < \\infty \\} . \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} P _ n ^ { ( a , b ) } \\left ( s _ 1 \\right ) = 1 , \\quad P _ n ^ { ( a , b ) } \\left ( s _ 0 \\right ) = ( 1 - b ) ^ { - n } \\quad \\quad ( n \\ge 0 ) . \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\bar { y } ( t , p ) = \\sum _ { i = 1 } ^ r \\bar { w } _ i ( t ) \\Psi _ i ( p ) \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} ( m x + r ) ^ n = \\sum _ { k = 0 } ^ n m ^ k W _ { m , r } ( n , k ) ( x ) _ k , ( \\textnormal { s e e } \\ , \\ , [ 8 ] ) . \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\phi = A ( \\phi ; \\mu ' , \\zeta ' ) \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} \\nabla _ { \\phi l } X ^ { \\beta } = \\nabla _ { 0 l } X ^ { \\beta } + X ^ k U _ { l k } ^ { \\beta } , \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} k ( Y , w ) : = \\sum _ { s \\in \\Sigma } ( \\rho _ s - 1 ) . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} u _ i ( 0 , x ) = u _ { i t } ( 0 , x ) = 0 x \\in ( 0 , \\pi ) \\ , , i = 1 , 2 , \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\phi : ( x _ { i j } ) \\mapsto ( f _ { 1 } , \\ldots , f _ { n } ) - ( x _ { i j } ) ( f _ { 1 } , \\ldots , f _ { n } ) = ( x _ { 1 1 } , \\ldots , x _ { n n } ) D \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\| u _ n ^ { * } ( \\varphi _ n ( f \\otimes a ) \\oplus & \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma _ n ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma _ n ( a ) } ^ m ) u _ n \\\\ & - \\psi _ n ( f \\otimes a ) \\oplus \\overbrace { \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma _ n ( a ) \\oplus \\cdots \\oplus \\bigoplus _ { k = 1 } ^ m f ( z _ k ) \\sigma _ n ( a ) } ^ m \\| < \\varepsilon \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} W ( \\widehat { T } ( z _ { 0 } ) \\psi , \\phi ) ( z ) = e ^ { - \\frac { i } { \\hbar } \\sigma ( z , z _ { 0 } ) } W ( \\psi , \\phi ) ( z - \\tfrac { 1 } { 2 } z _ { 0 } ) , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} g _ { k } ( x , y ) = ( 2 \\pi ) ^ { - p / 2 } \\det ( V ( x ) ) ^ { 1 / 2 } \\exp \\left [ - \\frac { 1 } { 2 } y ^ { T } V ( x ) y \\right ] , V ( x ) = \\mathrm { d i a g } \\{ \\exp ( - x ^ { ( 1 ) } ) , \\cdots , \\exp ( - x ^ { ( p ) } ) \\} , \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} \\dot { x } ( t , p ) = f ( x ( t , p ) , p ) \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } \\Gamma ( - z ) \\Gamma ( s + z ) t ^ z \\ ; d z = \\Gamma ( s ) ( 1 + t ) ^ { - s } . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} v ^ t _ 2 ( x ) = - \\int _ { \\R ^ n } d y \\Delta \\delta _ t h ^ 0 \\chi _ { \\Omega ^ t } ( y ) P ( x - y ) . \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} \\partial _ t m _ 1 ( \\lambda + 0 i ) - \\partial _ t m _ 1 ( \\lambda - 0 i ) & = \\beta \\partial _ t m _ e ( \\lambda - 0 i ) + ( \\partial _ t \\beta ) m _ e ( \\lambda - 0 i ) , \\\\ B _ u m _ 1 ( \\lambda + 0 i ) - B _ u m _ 1 ( \\lambda - 0 i ) & = \\beta B _ u m _ e ( \\lambda - 0 i ) . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\widehat { C } _ 5 ( y ) = y ^ 5 - 6 y ^ 4 + 1 9 y ^ 3 - 3 4 y ^ 2 + 3 3 y - 1 3 . \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} \\sum _ { 2 \\le q _ 1 , \\ldots , q _ { h _ 1 + h _ 2 } \\leq X } \\prod _ { i = 1 } ^ { h _ 1 + h _ 2 } \\Lambda ( q _ i ) \\prod _ { i = h _ 1 + 1 } ^ { h _ 1 + h _ 2 } \\mu ( \\omega _ { q _ i } ) \\ll ( \\log \\log x ) ^ { ( h _ 1 + 2 h _ 2 ) / 2 } ( \\log \\log \\log x ) ^ { 2 h _ 1 + h _ 2 } . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\theta ( \\xi ) \\cdot \\xi & = 0 , { \\theta } ^ { n } ( \\xi ) = 1 \\\\ \\theta ( \\xi _ { 0 } ) & = ( 0 , \\cdots , 1 ) = e _ { n } \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\underset { n \\to + \\infty } { \\lim } \\int _ { \\mathbb { R } ^ N } u ^ 2 _ n = \\int _ { \\mathbb { R } ^ N } u ^ 2 . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\Omega _ d ( x ) = \\sum _ { m = 0 } ^ d { d \\choose m } ( m + 1 ) ^ d ( x - 1 ) ^ { d - m } . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\tau ( p , \\lambda ) = \\frac { 2 - p } { 2 } ( 1 - p ) ^ { \\frac { p - 1 } { 2 - p } } \\lambda ^ { \\frac { 1 } { 2 - p } } . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\frac { 2 } { s } + \\frac { n } { r } = \\frac { n } { 2 } , \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} E _ { p , q } ^ { 1 } & = Z _ { p , q } ^ { 1 } = Z _ { p , q } ^ { r } \\oplus M _ { p , q } ^ { r - 1 } \\oplus \\cdots \\oplus M _ { p , q } ^ { 1 } \\\\ & = H _ { p , q } ^ { r } \\oplus B _ { p , q } ^ { r } \\oplus M _ { p , q } ^ { r - 1 } \\oplus \\cdots \\oplus M _ { p , q } ^ { 1 } . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} f ( b ' - a ) \\ = \\ f ( b ' - b + b - a ) \\ = \\ f ( b ' - b ) \\cdot f ( b - a ) \\ , , \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} G ( t ) = \\sqrt 2 \\left ( \\int _ 0 ^ t \\beta ( \\tau ) d \\tau + \\alpha t \\right ) ^ { \\frac 1 2 } , \\textrm { w h e r e } \\alpha = \\max \\left \\{ 0 , \\min _ { x \\in \\partial \\Omega } H ( x ) \\frac { \\partial w } { \\partial n } ( x ) \\right \\} , \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} P ( F ) = \\limsup _ { n \\to \\infty } \\bigg ( h _ { \\mu _ n } ( g ) + \\int F d \\mu _ n \\bigg ) & \\le | \\mu | ( h _ { \\mu / | \\mu | } ( g ) + \\int F d \\mu / | \\mu | ) + ( 1 - | \\mu | ) \\delta _ \\infty \\\\ & \\le | \\mu | P ( F ) + ( 1 - | \\mu | ) \\delta _ \\infty . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\varepsilon } X \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } = 0 \\quad i , j , \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} 1 - a ( p ) p ^ { - s } + \\chi ( p ) p ^ { k - 1 - 2 s } = ( 1 - \\alpha _ p p ^ { - s } ) ( 1 - \\beta _ p p ^ { - s } ) , \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{gather*} 0 \\leq \\sum _ { m = 0 } ^ k \\sum _ { n = 0 } ^ l d _ { m , n } ^ { q - 2 } S _ { m , n } ^ { q } ( r ) \\leq \\sum _ { m , n \\geq 0 } d _ { m , n } ^ { q - 2 } S _ { m , n } ^ { q } ( r ) , 0 \\leq r < 1 . \\end{gather*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ k ^ { ( i ) } \\frac { \\partial \\mathbf { h } _ i ^ * } { \\partial g _ { i k } } G \\mathbf { e } _ i \\mathbf { e } _ k ^ * X _ i G \\mathbf { e } _ i = O _ \\prec ( \\frac { 1 } { N } ) , \\frac { 1 } { N } \\sum _ k ^ { ( i ) } \\mathbf { h } _ i ^ * \\frac { \\partial G } { \\partial g _ { i k } } \\mathbf { e } _ i \\mathbf { e } _ k ^ * X _ i G \\mathbf { e } _ i = O _ \\prec ( \\Pi _ i ^ 2 ) . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} y = \\vec \\Phi ( x ) = ( \\Phi ^ 1 ( x ) , \\ldots , \\Phi ^ n ( x ) ) , ( \\det D \\vec \\Phi = 1 ) \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} \\prod _ { i , j } \\frac { 1 } { 1 - x _ i y _ j } = \\sum _ { \\lambda \\in \\mathcal { P } } s _ \\lambda ( \\mathbf { x } ) s _ \\lambda ( \\mathbf { y } ) = \\sum _ { \\mu \\in \\mathcal { P } } \\frac { p _ \\mu ( \\mathbf { x } ) p _ \\mu ( \\mathbf { y } ) } { z _ \\mu } \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} & x _ { n + 1 } = L ( r , x _ n ) : = r x _ n ( 1 - x _ n ) , \\\\ & x _ { n + 1 } = S ( r , x _ n ) : = r \\sin ( \\pi x _ n ) / 4 , \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\deg C & \\leq ( e _ j - 1 ) \\deg u \\\\ & + \\max \\{ \\deg g _ { i , e _ i } + \\deg g _ { j , e _ j - 1 } , \\deg g _ { j , e _ j } + \\deg g _ { i , e _ i - 1 } \\} - \\deg D _ { i j } . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} \\varphi \\ominus \\varphi _ 1 ( u ) = \\sup _ { v > 0 } \\{ \\varphi ( u v ) - \\varphi ( v ) \\} \\ { \\rm f o r \\ } u \\geq 0 . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} N _ { 2 d } ^ { \\mathrm { c m } } ( X , G ) = r _ d ( G ) X + O ( X Y ^ { - 1 } ) + O ( X ^ { \\alpha ' } Y ^ { d \\delta ^ { \\prime } ( 1 - \\alpha ^ { \\prime } ) + \\epsilon } ) . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} F _ { \\mu _ \\alpha } ( \\omega _ B ( z ) ) - \\omega _ A ( z ) - \\omega _ B ( z ) + z & = r _ 1 ( z ) \\ , , \\\\ F _ { \\mu _ \\beta } ( \\omega _ A ( z ) ) - \\omega _ A ( z ) - \\omega _ B ( z ) + z & = r _ 2 ( z ) \\ , , \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} k : = ( r _ 0 , r _ 1 , r _ 2 , r _ 3 , r _ 4 ) = ( 2 , 1 , 2 , 3 . 5 , 1 . 7 5 ) . \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} V = \\int _ 0 ^ 1 V ^ { \\theta } \\ , d \\theta \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} v w & = ( a \\otimes a ) ( - q c ^ * \\otimes c ) = - q a c ^ * \\otimes a c \\\\ & = q ^ 2 \\big ( - q c ^ * a \\otimes c a \\big ) \\\\ & = q ^ { 2 } w v . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} \\frak m _ { k , \\beta } ( h _ 1 , \\dots , h _ k ) = { \\rm e v } _ 0 ! ( { \\rm e v } _ 1 ^ * h _ 1 \\wedge \\dots \\wedge { \\rm e v } _ k ^ * h _ k ) \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} H _ \\infty ^ X ( \\epsilon ) = \\inf \\{ \\sup \\{ \\| x + e ^ { i \\theta } y \\| : \\ , 0 \\leq \\theta \\leq 2 \\pi \\} - 1 : \\ , \\| x \\| = 1 , \\| y \\| = \\epsilon \\} . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\omega _ R - t ^ + _ R & = \\int _ { t ^ + _ R } ^ { \\omega _ R } \\frac { \\dot r _ R ( t ) } { \\dot r _ R ( t ) } \\ , d t \\geq \\frac { 1 } { \\sqrt { 2 C _ + } } \\int _ { t ^ + _ R } ^ { \\omega _ R } \\frac { \\dot r _ R ( t ) } { r _ R ( t ) ^ { - \\alpha / 2 } } \\ , d t \\\\ & = \\frac { 1 } { \\left ( 1 + \\alpha / 2 \\right ) \\sqrt { 2 C _ + } } \\left ( R ^ { 1 + \\alpha / 2 } - K ^ { 1 + \\alpha / 2 } \\right ) , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\dfrac { \\ < \\mu ( B _ 1 ) , \\xi \\ > } { \\ < \\mu ( B _ 2 ) , \\xi \\ > } = - \\dfrac { \\ < \\mu ( B _ 1 ) , \\xi ' \\ > } { \\ < \\mu ( B _ 2 ) , \\xi ' \\ > } , \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} C _ 0 ( x ) = \\{ v = \\sum _ { i } \\alpha _ i v _ i \\in E _ x : \\sum _ { j = q + 1 } ^ n \\alpha _ j ^ 2 = \\sum _ { i = 1 } ^ q \\alpha _ i ^ 2 \\} \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\phi ( [ X , Y ] , Z ) + \\phi ( Y , [ X , Z ] ) = 0 \\forall X , Y , Z \\in \\mathfrak { s u } ( n ) \\ , . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{gather*} s _ { \\lambda } ( x _ 1 , \\dots , x _ N ) = \\frac { \\det \\big [ x _ i ^ { N + 1 - j } \\big ] _ { i , j = 1 } ^ N } { \\prod \\limits _ { 1 \\leq i < j \\leq N } { ( x _ i - x _ j ) } } . \\end{gather*}"}
-{"id": "4103.png", "formula": "\\begin{align*} h ^ + ( c _ r ) = c _ r - 2 \\ell \\int _ { 0 } ^ { 1 } \\frac { \\sin ( \\pi c _ r v ( 1 - \\delta ) ) } { \\pi v } ( 1 - v ) ^ { \\ell ^ 2 } \\ , d v + O ( 1 / \\log T ) \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} & a _ 1 ^ p = [ a _ 1 , b _ 1 ] , \\ ; a _ 2 ^ p = [ a _ 1 , b _ 1 ^ { \\lambda _ 1 } b _ 2 ^ { \\lambda _ 2 } ] , \\ ; b _ 1 ^ p = [ a _ 2 , b _ 1 b _ 2 ] , \\\\ & b _ 2 ^ p = [ a _ 2 , b _ 2 ] , \\ ; [ a _ 1 , a _ 2 ] = [ b _ 1 , b _ 2 ] = 1 . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\frac { z \\ ; \\mathtt { J } ' _ { \\nu } ( z ) } { \\mathtt { J } _ { \\nu } ( z ) } = \\nu - \\sum _ { n = 1 } ^ \\infty \\frac { 2 z ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - z ^ 2 } , \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} Q _ { k } = \\sum _ { i , j , k = 0 } ^ 3 a _ { i j k } x _ i y _ j . \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} L ^ { \\beta , \\alpha } ( t ) : = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { t } \\delta ( B ^ { \\beta , \\alpha } ( s ) - B ^ { \\beta , \\alpha } ( u ) ) \\ , d u \\ , d s . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} T = \\left \\{ \\sum _ 0 ^ n t _ j p _ j : \\sum t _ j = 1 t _ j \\geq 0 \\right \\} \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\sum _ { p _ j \\in \\widetilde X : \\pi ( p _ j ) = p } n _ { p , j } \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} & T _ w T _ { w ' } = T _ { w w ' } , \\hbox { i f } \\l ( w w ' ) = \\l ( w ) + \\l ( w ' ) , \\hbox { a n d } \\cr & ( T _ s + 1 ) ( T _ s - q ) = 0 \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} I _ t \\triangleq \\sum _ { 0 \\leq s \\leq t } \\left ( \\dot { X } _ { s + } - \\dot { X } _ { s - } \\right ) \\mathbf { 1 } _ { \\{ | X _ s | = P _ O \\} } . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} D ( X ) \\coloneqq \\sum _ { s = 0 } ^ { n - 1 } \\alpha _ s X ^ s \\left ( = 1 + X ^ { i _ 2 } + \\cdots + X ^ { i _ p } \\right ) \\in \\mathbb { F } _ { p } [ X ] . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\mathbf { X } ^ { \\left [ 1 \\right ] } L + \\frac { d \\xi } { d t } L = \\frac { d f } { d t } , \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z \\mathtt { f } ' _ { a , \\nu } ( z ) } { \\mathtt { f } _ { a , \\nu } ( z ) } \\right ) \\geq \\frac { r \\mathtt { f } ' _ { a , \\nu } ( r ) } { \\mathtt { f } _ { a , \\nu } ( r ) } = a ^ { a / 2 } - \\frac { a ^ { a / 2 } } { a \\nu - a + 1 } \\sum _ { n = 1 } ^ \\infty \\frac { 2 r ^ 2 } { \\mathtt { j } ^ 2 _ { \\nu , n } - r ^ 2 } . \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} & { { E } } \\sum _ { k = 0 } ^ { K - 1 } \\int _ { ( \\tau , \\tau + \\delta ] \\times [ 0 , 1 ] } ( K + 3 ) \\gamma ( M ) \\ell ( \\varphi ^ n _ k ( s , y ) ) \\ , d s \\ , d y + K ( K + 3 ) M \\delta + 5 { \\bar { U } } _ K \\\\ & \\le ( K + 3 ) \\gamma ( M ) M _ 0 + K ( K + 3 ) M \\delta + 5 { \\bar { U } } _ K . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{gather*} Q = Q ( \\partial _ 1 \\eta _ 1 , \\partial _ 2 \\eta _ 1 , \\partial _ 1 \\eta _ 2 , \\partial _ 2 \\eta _ 2 , \\partial _ 1 \\eta _ 3 , \\partial _ 2 \\eta _ 3 ) . \\end{gather*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\beta _ n } { n ^ t } = \\infty , t \\in \\R . \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\| f \\| _ { H [ Z ^ { \\wedge } ] } = \\| f \\| _ { Z ^ { \\wedge } } = \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| f _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in Z , n \\in \\N \\} . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} & \\Big | \\mathcal { S } \\Lambda _ \\iota + \\mathcal { T } _ \\iota \\Lambda _ \\iota ^ 2 + O ( \\Lambda _ \\iota ^ 3 ) \\Big | \\prec \\hat { \\Pi } ^ 2 , \\qquad \\iota = A , B . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\alpha _ { V , U , W } ( v \\otimes ( u \\otimes w ) ) = \\alpha ( x , y , z ) ( ( v \\otimes u ) \\otimes w ) , v \\in V _ x , \\ u \\in U _ y , \\ w \\in W _ z , \\ x , y , z \\in A , \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} \\begin{cases} v _ 1 \\partial _ y f ^ L = \\mathcal { L } [ f ^ L ] \\ , , y \\in [ 0 , \\infty ) \\ , , \\\\ f ^ L | _ { y = 0 , v _ 1 > 0 } = \\phi ^ x ( x _ 2 , x _ 3 ) \\left ( \\phi ^ v ( v ) - \\eta \\right ) \\ , , \\\\ f ^ L | _ { y = \\infty } = 0 \\ , ; \\end{cases} \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} \\ , K _ { i l } ^ { X _ 1 , 0 } & \\leq \\ , V _ { X _ 2 } ( - ( i + 1 ) A - ( l + 1 ) B ) + \\ , V _ { X _ 3 } ( - ( d - l + 1 ) B ) \\\\ & = \\ , V _ { X _ 2 } ( - ( i + 1 ) A - l B ) - 1 + \\ , V _ { X _ 3 } ( - ( d - l + 1 ) B ) \\\\ & = \\ , K _ { i l } ^ { X _ 1 , 0 } - 1 , \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\zeta ^ \\alpha _ k & : = \\langle \\zeta _ k , { \\hat { \\alpha } } \\rangle \\ \\ \\ , \\ S ^ \\alpha _ n : = \\langle S _ n , { \\hat { \\alpha } } \\rangle \\ \\ \\ , \\ x ^ \\alpha _ 0 : = \\langle x _ 0 , \\hat { \\alpha } \\rangle \\ , , \\\\ \\zeta ^ \\perp _ k & : = \\langle \\zeta _ k , { \\alpha } ^ \\perp \\rangle \\ , \\ S _ n ^ \\perp : = \\langle S _ n , { \\alpha } ^ \\perp \\rangle \\ , \\ x ^ \\perp _ 0 : = \\langle x _ 0 , \\alpha ^ \\perp \\rangle \\ , , \\\\ \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} ( \\alpha + 1 ) c _ { 1 , \\infty } ( a ^ 2 + 2 a ) + c _ { 2 , \\infty } ( b ^ 2 + 2 b ) + c _ { 3 , \\infty } ( c ^ 2 + 3 c ) = 0 . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} S _ { J , j } V ^ { , j } - \\lambda _ { 1 } S _ { J } & = 0 \\\\ \\left ( S _ { J , k } \\delta _ { j } ^ { i } + 2 S _ { J } , _ { j } \\delta _ { k } ^ { i } \\right ) V ^ { , k } + a _ { 7 } \\left ( 2 Y _ { J } ^ { ; i } { } _ { ; ~ j } - a _ { 0 } S _ { J } \\delta _ { j } ^ { i } \\right ) & = 0 \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} G _ { n , s } \\left ( \\xi \\right ) = \\sum \\limits _ { k = s } ^ { n - 1 } { F _ { k } \\left ( \\xi \\right ) F _ { s + n - k - 1 } \\left ( \\xi \\right ) . } \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\varepsilon _ d = \\begin{cases} 1 & d \\equiv 1 \\pmod 4 \\\\ i & d \\equiv 3 \\pmod 4 \\end{cases} \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} D ( a , b , c , d ) = \\sum _ { w _ { b + d } = \\max \\left \\{ t + 1 , a - c ( b - 1 ) \\right \\} } ^ { \\min \\left \\{ a - ( b - 1 ) ( t + 1 ) , c \\right \\} } D ( a - w _ { b + d } , b - 1 , c , d ) \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} A U = F - M _ { 1 } \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) U - M _ { 0 } \\partial _ { 0 } U \\in H _ { \\nu , - 1 } \\left ( \\mathbb { R } , H \\right ) \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} T \\gamma ( X ^ { \\gamma } ) = X \\circ \\gamma . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} 2 4 = \\sum _ { F \\in \\Delta [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) + \\ell ( \\Delta ) + \\ell ( \\Delta ^ * ) - \\sum _ { F \\in \\Delta [ 2 ] } \\ell ^ * ( F ) - \\sum _ { F ^ \\circ \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ^ \\circ ) - 4 \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\chi ( \\xi ) = \\begin{cases} 0 & | \\xi | \\leq \\frac 1 3 \\\\ 1 & \\ | \\xi | \\geq \\frac 2 3 \\ , , \\end{cases} \\partial _ \\xi \\chi ( \\xi ) > 0 \\forall \\xi \\in \\Big ( \\frac 1 3 , \\frac 2 3 \\Big ) \\ , . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\sum _ { i _ 1 = 1 } ^ 2 \\ldots \\sum _ { i _ k = 1 } ^ 2 \\sum _ { i _ { k + 1 } = 1 } ^ 2 g _ { i _ 1 , \\ldots , i _ { k + 1 } } ( x , s , t ) \\prod _ { l = 1 } ^ { k + 1 } ( \\zeta _ l ) _ { i _ l } , \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} x ( t ) = x _ 0 ( t - a ) ^ { \\gamma - 1 } + \\int _ { a } ^ { t } \\frac { ( t - s ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } f ( s , x ( s ) ) d s . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} K _ { \\nu } \\left ( { \\nu z } \\right ) = \\left ( { \\dfrac { \\pi } { 2 \\nu } } \\right ) ^ { 1 / 2 } \\dfrac { 1 } { \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 4 } } \\exp \\left \\{ - { \\nu \\xi + \\sum \\limits _ { s = 1 } ^ { n - 1 } } \\left ( - 1 \\right ) ^ { s } { { \\dfrac { \\tilde { { E } } _ { s } \\left ( p \\right ) } { \\nu ^ { s } } } } \\right \\} \\left \\{ { 1 + \\eta _ { n , 2 } \\left ( { \\nu , z } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} g _ { f , p } '' ( \\tau x + x _ 1 ) = g _ { f , p } '' ( x ) \\forall x \\in X . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} \\widetilde L ^ { \\rm r e s c } _ N ( v ) : = \\frac { L _ { \\widetilde A ( v ) \\to E _ N ( w ) } - ( N / \\chi + \\alpha _ 2 v N ^ { 2 / 3 } ) } { \\chi ^ { - 2 / 3 } N ^ { 1 / 3 } } \\to { \\cal A } _ 2 ( v ) - ( v - w ) ^ 2 \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} f ( x ) = g ( 1 ) = g ( 0 ) + \\int _ { 0 } ^ 1 g ' ( t ) d t = \\frac { 1 } { 2 } \\langle E x , x \\rangle + \\langle c , x \\rangle + f ( 0 ) . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i = 1 } ^ N L _ i v \\frac { v _ i } { u _ { i , \\infty } } & = \\sum _ { r = 1 } ^ R k _ r u _ { \\infty } ^ { y _ r } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } ( y _ { r , i } ' - y _ { r , i } ) y _ { r , j } \\frac { v _ i } { u _ { i , \\infty } } \\frac { v _ j } { u _ { j , \\infty } } \\\\ & = \\sum _ { r = 1 } ^ R k _ r u _ { \\infty } ^ { y _ r } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } ( y _ { r , i } ' y _ { r , j } - y _ { r , i } y _ { r , j } ) \\frac { v _ i } { u _ { i , \\infty } } \\frac { v _ j } { u _ { j , \\infty } } . \\end{aligned} \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} \\lambda _ { \\gamma } : \\sum _ { i = 1 } ^ n w ^ { \\gamma } _ i ( \\theta ) \\psi ( x _ i , \\theta ) = \\sum _ { i = 1 } ^ n \\left ( 1 + \\lambda _ \\gamma ^ T \\psi ( x _ i , \\theta ) \\right ) ^ { - \\frac { 1 } { \\gamma + 1 } } \\psi ( x _ i , \\theta ) = 0 \\ , . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} S _ { \\theta ^ k } ( n ) = \\sum _ { m \\leq n } r _ k ( m ) , \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} \\Delta _ 2 ( f ) : = \\Delta ( f ) = d i v _ { \\mathfrak { m } } ( \\nabla f ) . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} W _ { q , k } ( \\pi _ { k } ( \\mu , y _ { 0 : k } , \\cdot ) , \\pi _ { k } ( \\nu , y _ { 0 : k } , \\cdot ) ) \\leq \\exp \\left [ - \\sum _ { j = 1 } ^ { k } \\int _ { 0 } ^ { \\Delta } \\lambda ( j , y _ { j } , t ) \\mathrm { d } t \\right ] W _ { q , 0 } ( \\pi _ { 0 } ^ { \\mu } , \\pi _ { 0 } ^ { \\nu } ) , \\qquad \\forall k \\geq 1 , \\mu , \\nu \\in \\mathcal { P } _ { q } , \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} x _ { L , k } = x _ k - \\left ( L ( I _ n - \\widetilde { V } _ k \\widetilde { V } _ k ^ T ) \\right ) ^ { \\dag } L x _ k , \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} \\L = \\mathcal { M } \\ , \\Z ^ { 2 n } \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} m _ k = \\int _ { - 1 } ^ 1 f ( t ) ( G ( t ) ) ^ k d t = 0 , ~ ~ k = 0 , 1 , 2 \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\ker ( T _ 3 ) = \\begin{cases} H _ 3 / H _ 3 ^ \\prime \\simeq \\langle x y , s _ 2 \\rangle \\simeq C _ 3 \\times C _ 3 a = w = z = 0 a = 1 , \\ w = - 1 , \\\\ G ^ \\prime / H _ 3 ^ \\prime \\simeq \\langle s _ 2 \\rangle \\simeq C _ 3 . \\end{cases} \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\mathbf { S } \\left ( a _ { i } , b _ { i } , \\bar { p } , \\omega _ { i } ; \\mathbf { t } \\right ) = \\left ( \\begin{array} { c } a _ { 1 } \\cos \\omega _ { 1 } \\mathbf { t } _ { 1 } + b _ { 1 } \\sin \\omega _ { 1 } \\mathbf { t } _ { 1 } \\\\ a _ { 2 } \\cos \\omega _ { 2 } \\mathbf { t } _ { 2 } + b _ { 2 } \\sin \\omega _ { 2 } \\mathbf { t } _ { 2 } \\end{array} \\right ) + \\bar { p } \\mathbf { 1 } . \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\| \\ + F \\| \\leq \\sum _ { i < n } y _ i = ( k - 1 ) \\frac { 1 } { k - 1 } + ( n - k + 1 ) \\frac { n - k } { ( n - k + 1 ) ( k - 1 ) } = \\frac { n - 1 } { k - 1 } = 1 / p . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} n = { \\rm t r } _ { \\omega _ { \\epsilon } \\omega _ 0 } + k { \\rm t r } _ { \\omega _ { \\epsilon } } ( \\sqrt { - 1 } \\partial \\bar { \\partial } \\chi ) \\geq k \\Delta _ { \\omega _ { \\epsilon } } \\chi , \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\Theta _ u ( x _ 0 ) : = \\lim _ { r \\to 0 } W ( u , x _ 0 , r ) = \\inf _ { r > 0 } W ( u , x _ 0 , r ) = \\lim _ { r \\to 0 } W ( u _ { r , x _ 0 } ) \\ , . \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} & a _ 2 > a _ 1 > ( \\rho + 3 ) \\log 1 0 ^ { 5 / 2 } : = A _ { 1 , 2 } , & a _ 3 > 8 ( \\log 1 0 ^ 5 + 0 . 0 8 6 7 5 ( \\rho - 1 ) ) : = A _ 3 . \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} ( P _ { 1 , 1 } ( t ) , P _ { 1 , 2 } ( t ) , \\ , P _ { 2 , 1 } ( t ) , P _ { 2 , 2 } ( t ) , \\ , P _ { 3 , 1 } ( t ) , P _ { 3 , 2 } ( t ) , \\ , Q ( t ) ) = ( 0 , \\infty , \\ , 1 , - 1 , \\ , i + \\varepsilon e ^ { 2 \\pi i t } , - i , \\ , 0 ) , \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} { \\mathcal J \\left ( \\mathcal I , \\bar K \\right ) } = \\bigcup \\limits _ { i \\in \\mathcal P \\backslash \\mathcal I } { { \\mathcal J _ i } \\left ( { \\bar K } \\right ) } , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} \\Phi _ i ( u ) \\mathfrak { X } _ j ^ { \\pm } ( v , \\lambda ) \\Phi _ i ( u ) ^ { - 1 } = \\frac { \\vartheta ( u - v \\pm a ) } { \\vartheta ( u - v \\mp a ) } \\mathfrak { X } _ j ^ { \\pm } ( v , \\lambda \\pm \\hbar \\alpha _ i ) \\pm \\frac { \\vartheta ( 2 a ) \\vartheta ( u - v \\mp a - \\lambda _ j ) } { \\vartheta ( \\lambda _ j ) \\vartheta ( u - v \\mp a ) } \\mathfrak { X } _ j ^ { \\pm } ( u \\mp a , \\lambda \\pm \\hbar \\alpha _ i ) . \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\nabla _ { \\phi m } \\nabla _ { \\phi l } X ^ { \\beta } = \\nabla _ { 0 m } \\nabla _ { 0 l } X ^ { \\beta } - \\nabla _ { 0 p } X ^ { \\beta } \\cdot U _ { m l } ^ p + \\nabla _ { 0 l } X ^ p \\cdot U _ { p m } ^ { \\beta } + \\nabla _ { \\phi m } X ^ k \\cdot U _ { l k } ^ { \\beta } + X ^ k \\nabla _ { \\phi m } U _ { l k } ^ { \\beta } , \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{gather*} g _ r ( x _ 1 , \\dots , x _ N ; q , t ) = \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\oint _ C { \\frac { 1 } { y ^ { r + 1 } } \\prod _ { i = 1 } ^ N { \\frac { ( t x _ i y ; q ) _ { \\infty } } { ( x _ i y ; q ) _ { \\infty } } { \\rm d } y } } , \\end{gather*}"}
-{"id": "7928.png", "formula": "\\begin{align*} c = c ^ t = c ( t ) \\in C ^ { 2 } ( [ - 1 , 1 ] ) . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} ( \\overline { o s c } _ r f ) ( x ) : = \\sup _ { y \\in \\overline { B _ r ( x ) } \\cap D } f ( y ) - \\inf _ { y \\in \\overline { B _ r ( x ) } \\cap D } f ( y ) \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\langle { s , t , u , v , w , x , y , z } \\mid { y v = w y } , ~ z x = w z , ~ t y = z t , ~ u y = z u , ~ s v = u s , ~ v s = x v , \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} K \\subseteq D \\subseteq E = F \\subseteq K . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{gather*} F _ { 1 4 } ( a _ 1 , a _ 1 , a _ 1 , b _ 1 , b _ 2 , b _ 1 ; c _ 1 , c _ 2 , c _ 2 ; x _ 1 , x _ 2 , x _ 3 ) = \\sum _ { m , n , p = 0 } ^ \\infty \\frac { ( a _ 1 ) _ { m + n + p } ( b _ 1 ) _ { m + p } ( b _ 2 ) _ n } { ( c _ 1 ) _ m ( c _ 2 ) _ { n + p } } \\frac { x _ 1 ^ m x _ 2 ^ n x _ 3 ^ p } { m ! n ! p ! } , \\end{gather*}"}
-{"id": "8300.png", "formula": "\\begin{align*} j = \\left ( \\frac { a ( z _ 1 ) z _ 0 + b ( z _ 1 ) } { c ( z _ 1 ) z _ 0 + d ( z _ 1 ) } , \\frac { \\alpha z _ 1 + \\beta } { \\gamma z _ 1 + \\delta } \\right ) = \\left ( z _ 0 , \\frac { \\alpha z _ 1 + \\beta } { \\gamma z _ 1 + \\delta } \\right ) \\circ \\left ( \\frac { a ( z _ 1 ) z _ 0 + b ( z _ 1 ) } { c ( z _ 1 ) z _ 0 + d ( z _ 1 ) } , z _ 1 \\right ) \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} \\bigg ( \\int _ { \\delta _ 3 } \\psi _ 1 , \\int _ { \\delta _ 3 } \\psi _ 2 \\bigg ) = \\bigg ( \\dfrac { 1 } { 1 - \\omega } y _ 3 , \\dfrac { 1 } { 1 - \\omega ^ 2 } y _ 4 \\bigg ) . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} u ^ \\ast ( t ) = \\arg \\max _ { u \\in \\Omega } H ( x ^ \\ast ( t ) , u , \\lambda _ 0 , \\lambda ( t ) ) \\\\ \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\mu _ { 2 j - 1 } = \\prod _ { \\ell = 1 } ^ m \\bigg ( \\frac { L _ \\ell \\Gamma ( \\frac { L _ \\ell } { 2 } ) \\Gamma ( \\frac { L _ \\ell + 1 } { 2 } ) } { 2 \\sqrt { \\pi } } \\bigg ) ^ { 1 / 2 } \\frac { \\Gamma ( j - \\frac 1 2 ) } { \\Gamma ( \\frac { L _ \\ell } 2 + j - \\frac 1 2 ) } . \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} H _ \\theta ( t ) \\ ; = \\ ; \\left \\{ \\begin{array} { c c } 2 \\ , H ( 2 t ) \\ ; , & t \\in [ 0 , \\pi ] \\ ; , \\\\ - \\ , 2 \\ , h _ \\theta \\ ; , & t \\in ( \\pi , 2 \\pi ] \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\sum _ { j = r } ^ { d } Q _ j ( w ) = \\sum _ { j = r } ^ { d } P _ j ( w _ 1 ( \\tilde { y } + x ) + w _ 2 ( \\tilde { y } - x ) ) . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} a _ * \\Q _ \\ell = \\bigoplus _ { \\substack { d _ 1 , d _ 2 \\ge 0 \\\\ d _ 1 + d _ 2 = 2 d } } V _ { ( d _ 1 , d _ 2 ) } \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\begin{cases} \\dot w & = - i E w \\\\ \\dot \\phi _ 0 & = 1 . \\end{cases} \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} \\mathfrak { E } _ h ^ k ( s ) = \\frac { ( 2 \\pi ) ^ { \\frac { k + 1 } { 2 } } ( \\overline { \\rho _ \\infty ^ k ( h , \\frac { k + 1 } { 2 } ) + \\rho _ 0 ^ k ( h , \\frac { k + 1 } { 2 } ) } ) } { ( 2 h ) ^ { s - 1 } } \\frac { \\Gamma ( s - \\frac { k } { 2 } - \\frac { 1 } { 2 } ) } { \\Gamma ( s - \\frac { k } { 2 } ) } . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ n b _ { j , r } \\ , b _ { k , r } + \\ , b _ { j , k } \\cdot \\sum _ { \\substack { r , s = 1 \\\\ r < s } } ^ n b _ { k , r } \\ , b _ { k , s } \\equiv 0 \\bmod 2 . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\left | { \\dd ^ m \\over \\dd z ^ m } S _ { n , r } ( z ) \\Big | _ { z = 0 } \\right | \\leq C ^ m ( m - 1 ) ! r n ^ { 1 - m } . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\bar n ( P , \\langle \\rangle ) & : = P , \\\\ \\bar n ( P , \\sigma ^ \\frown a ) & : = n ( \\bar n ( P , \\sigma ) , a ) . \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ 1 = P _ t u _ 0 , \\\\ & u _ { n + 1 } = F ( u _ { n } ) = F ^ { n } ( u _ 1 ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} B _ { m , k - 1 - m } ^ { r } \\subset B _ { m , k - 1 - m } ^ { r + 1 } \\subset Z _ { m , k - 1 - m } ^ { r } = H _ { m , k - 1 - m } ^ { r } \\oplus B _ { m , k - 1 - m } ^ { r } \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} u _ i ( 0 , x ) = u _ { i t } ( 0 , x ) = 0 x \\in ( 0 , \\pi ) , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\mu ( I _ { n _ i } ( x ) ) = \\frac { 1 } { b _ i } \\leq \\frac { 1 } { c ( \\overline { \\beta } ) \\overline { \\beta } ^ { p _ i } } , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} \\sigma = \\sigma ( x ) : = \\partial _ N \\tilde u _ 0 ( x ) . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} E \\left [ g ( Y ( t ) ) \\right ] = E \\left [ g ( \\xi ) \\right ] + E \\left [ \\int _ 0 ^ t \\int _ A \\Lambda _ s ^ a g ( Y ( s ) ) [ \\widehat { \\gamma } ( s , Y ( s ) ) ] ( d a ) d s \\right ] . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z } | \\varpi | ^ { 2 k s } f _ x \\big ( ( 1 , \\varpi ^ { - k } ) \\cdot \\gamma \\big ) = | c | ^ { 2 s } = | \\epsilon | _ x ^ { - 2 s } . \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} y e _ n & = f _ n y ( \\xi ) = 0 \\xi \\perp [ e _ n ] \\textit { N o t e t h a t : } \\ , \\xi _ 0 \\perp [ e _ n ] \\\\ z e _ n & = f _ n z \\xi _ 0 = \\eta _ 0 z ( \\xi ) = 0 \\xi \\perp [ \\xi _ 0 , ( e _ n ) ] \\\\ \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} A Q _ t x = e ^ { t A } B B ^ * e ^ { t A ^ * } x - B B ^ * x - Q _ t A ^ * x \\forall x \\in D ( A ^ * ) \\forall t \\in [ 0 , + \\infty [ \\ , , \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{gather*} Q _ { \\lambda } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) = t ^ { n ( \\lambda ) } \\prod _ { s \\in \\lambda } { \\frac { 1 - q ^ { a ' ( s ) } t ^ { N - l ' ( s ) } } { 1 - q ^ { a ( s ) + 1 } t ^ { l ( s ) } } } . \\end{gather*}"}
-{"id": "3483.png", "formula": "\\begin{align*} F _ S = F _ { S , n } : = \\sum _ { \\substack { i _ 1 \\leq i _ 2 \\leq \\cdots \\leq i _ n \\\\ j \\in S \\ , \\ , \\Rightarrow \\ , \\ , i _ j < i _ { j + 1 } } } x _ { i _ 1 } \\cdots x _ { i _ n } . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} w = \\sum _ { \\gamma \\in H _ \\varepsilon } y _ \\gamma . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c c c } \\vline & \\vline & \\cdots & \\vline \\\\ p _ 1 & p _ 2 & \\cdots & p _ n \\\\ \\vline & \\vline & \\cdots & \\vline \\end{array} \\right ] \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} n ^ { - } \\left ( E ^ { s } \\oplus E ^ { u } \\right ) = \\dim E ^ { u } = n ^ { - } \\left ( L \\right ) . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} O _ P \\left ( 2 ^ { \\sum _ { l = 1 } ^ d ( j _ l / 2 + J _ { n , l } ) } \\epsilon _ n \\right ) + O \\left ( \\frac { n } { \\prod _ { l = 1 } ^ d 2 ^ { j _ l / 2 } } \\sum _ { l = 1 } ^ d 2 ^ { - \\alpha _ l J _ { n , l } } \\right ) . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} \\Sigma _ { n , j } = \\mathbb { E } \\binom { C _ n } { j } . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\frac { A \\otimes W - \\left ( \\overline { A \\otimes W } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } , \\quad \\quad \\mbox { f o r a l l $ i , j = 1 , 2 $ , } \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} | { M P } | \\leq \\sum _ j | { M v _ j v _ j ^ * } | = \\sum _ j | \\langle M v _ j , v _ j \\rangle | \\leq \\| M \\| \\sum _ j \\| v _ j \\| ^ 2 = \\| M \\| { P } . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} F ( x ) : = \\begin{cases} 0 , & x \\in \\Omega \\setminus \\omega _ h , \\\\ \\nabla \\eta , & x \\in \\omega _ h , \\end{cases} \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} S _ n : = x _ 0 + \\sum _ { k = 1 } ^ n \\zeta _ k = X _ { T _ n } \\ , \\ \\ \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\sum _ { i = 1 } ^ N \\frac { \\| v _ i \\| _ 2 ^ 2 } { u _ { i , \\infty } } + \\frac \\beta 2 \\sum _ { i = 1 } ^ { N } \\frac { \\| v _ i \\| _ 2 ^ 2 } { u _ { i , \\infty } } + \\frac \\beta 2 \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ { H ^ 1 ( \\Omega ) } ^ 2 \\leq C \\sum _ { i = 1 } ^ { N } ( \\| v _ i \\| _ { \\mu + 1 } ^ { \\mu + 1 } + \\| v _ i \\| _ { 2 + { \\delta } } ^ { 2 + { \\delta } } ) . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\| W ^ { \\ast } R _ { D } x \\| ^ { 2 } = \\| R _ { A } x \\| ^ { 2 } = \\langle A x , x \\rangle \\leq \\alpha ^ { 2 } \\langle D x , x \\rangle = \\alpha ^ { 2 } \\| R _ { D } x \\| ^ { 2 } , x \\in X , \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\nabla w ( x ) = W ' ( \\mu _ u ( u ( x ) ) ) ( \\mu _ u ) ' ( u ( x ) ) \\nabla u ( x ) f o r \\ > \\ > a . e . \\ > \\ > x \\in \\O \\ , . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} & A = c i r c s h i f t ( A , [ \\lfloor \\frac { y _ 1 } { 4 } \\rfloor ~ ~ - \\lfloor \\frac { y _ 1 } { 3 } \\rfloor ] ) , \\\\ & A = c i r c s h i f t ( A , [ \\lfloor \\frac { 2 y _ 1 } { 1 0 } \\rfloor ~ ~ \\lfloor \\frac { 2 y _ 1 } { 5 } \\rfloor ] ) . \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} s _ \\lambda = \\sum _ { \\mu \\vdash | \\lambda | } \\frac { \\chi _ { \\mu } ^ { \\lambda } p _ \\mu } { z _ \\mu } \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{gather*} \\Phi _ 0 + \\Phi _ n = 0 . \\end{gather*}"}
-{"id": "8625.png", "formula": "\\begin{align*} z ^ 4 \\ , \\textrm { s i g n } ( h ) - w z + \\frac { \\lambda } { 3 } \\ , \\textrm { s i g n } ( h ) = 0 \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} 0 \\leq f _ \\Lambda ( u ) \\leq \\frac { \\sqrt { u ^ 2 + 4 } - | u | } { 2 } = \\frac { 2 } { \\sqrt { u ^ 2 + 4 } + | u | } . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} [ \\prod _ { i = 3 } ^ k r _ q ^ { ( i ) } , a ] [ \\prod _ { i = 3 } ^ k s _ q ^ { ( i ) } , b ] \\equiv [ A , a ] [ B , b ] \\mod \\gamma _ { k + 3 } ( F ) . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\sum _ { i < n } y _ i / \\norm { F } = 1 , \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty ( | v _ i | ^ p + | v _ i | ^ { p _ i } ) = | | v | | _ p ^ p + \\sum _ { i = 1 } ^ \\infty | v _ i | ^ { p _ i } = 1 \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } V ( t , x ) = \\frac 1 2 \\| x \\| _ H ^ 2 \\forall x \\in H . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} X ^ { \\swarrow k } = X ^ { \\leftarrow k \\downarrow k } . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\frac { a } { 4 n ^ 2 - 1 } + b \\sum _ { m = n _ 0 } ^ \\infty \\frac { 1 } { 4 m ^ 2 - 1 } \\le \\varepsilon \\forall n \\ge n _ 0 \\ , . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & s _ 2 & \\cdots & s _ d & \\varphi _ { d + 1 } & \\cdots & \\varphi _ n \\\\ \\sum _ j \\lambda _ j \\Phi _ j & - \\sum _ j \\lambda _ j \\varphi _ { j 2 } & \\cdots & - \\sum _ j \\lambda _ j \\varphi _ { j d } & \\lambda _ { d + 1 } & \\cdots & \\lambda _ n \\end{pmatrix} \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\pi ^ { - \\ , ( 1 - z ) / 2 } \\ , \\Gamma ( ( 1 - z ) / 2 ) \\ , \\zeta ( 1 - z ) = \\pi ^ { - \\ , z / 2 } \\ , \\Gamma ( z / 2 ) \\ , \\zeta ( z ) \\ , . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} ( M _ s ) _ { a , b } & = \\begin{cases} \\frac { s ( v \\sigma ) } { \\sum _ { \\tau \\in \\Sigma } s ( v \\tau ) } & \\mbox { i f } ( a , b ) \\mbox { i s a n e d g e a n d } b = v \\sigma , \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\{ \\lambda _ k , \\rho _ k \\} _ { k = 1 } ^ N \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\int \\partial _ { t } ^ 2 \\partial ^ { i } ( J R \\partial _ { t } v ^ { \\beta } ) \\partial _ { t } ^ 2 \\partial _ { i } v _ { \\beta } + \\int _ { \\Omega } \\partial _ { t } ^ 2 \\partial ^ { i } ( J a ^ { \\alpha \\beta } \\partial _ { \\alpha } q ) \\partial _ { t } ^ 2 \\partial _ { i } v _ { \\beta } = 0 \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\left ( ( a _ 1 \\otimes a _ 2 \\otimes \\cdots \\otimes a _ n ) \\otimes \\sigma \\right ) \\left ( ( b _ 1 \\otimes b _ 2 \\otimes \\cdots \\otimes b _ n ) \\otimes \\rho \\right ) = ( a _ 1 b _ { \\sigma ^ { - 1 } ( 1 ) } \\otimes a _ 2 b _ { \\sigma ^ { - 1 } ( 2 ) } \\otimes \\cdots \\otimes a _ n b _ { \\sigma ^ { - 1 } ( n ) } ) \\otimes ( \\sigma \\rho ) \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\lambda _ { 1 , p } ( M , F _ k ) } { \\lambda _ { 1 , p } ( M , g ) } = 1 . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} f _ { 1 , w } ( x ) \\ = \\ f _ { 2 , w } ( x ) \\ = \\ \\dotsb \\ = \\ f _ { n , w } ( x ) \\ = \\ 0 \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\sum _ { z \\in \\Lambda ^ e _ n } \\sum _ { w \\sim z } { \\bf 1 } _ { \\{ w , z \\notin x _ 1 ^ e \\} } + \\Big ( \\sum _ { z \\in x _ 1 ^ e } \\sum _ { w \\sim z } + \\sum _ { z \\in x _ 2 ^ e } \\sum _ { w \\in x _ 1 ^ e } \\Big ) \\ ; : = \\ ; \\eqref { t e r m 1 } ( i ) \\ , + \\ , \\eqref { t e r m 1 } ( i i ) , \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{gather*} a \\left ( y ^ { n } , s ^ { m } , p ^ { k } \\right ) \\cdot p ^ { k } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) \\rightarrow a \\left ( y ^ { n } , s ^ { m } , \\nabla u + \\sum \\limits _ { j = 1 } ^ { n } \\nabla _ { y _ { j } } u _ { j } + \\delta c \\right ) \\\\ \\cdot \\left ( \\nabla u \\left ( x , t \\right ) + \\sum \\limits _ { j = 1 } ^ { n } \\nabla _ { y _ { j } } u _ { j } \\left ( x , t , y ^ { j } , s ^ { m - d _ { j } } \\right ) + \\delta c \\left ( x , t , y ^ { n } , s ^ { m } \\right ) \\right ) \\end{gather*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\sum _ { \\epsilon = ( t + 1 ) \\cdot \\max \\left \\{ K , L \\right \\} } ^ { K \\cdot L } \\frac { \\epsilon } { m ^ 2 } \\cdot A _ { K , L } \\cdot \\hat { N } _ { K , L } ^ { \\epsilon } \\cdot ( p + \\xi ) ^ \\epsilon , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} x ^ \\ast ( t ) = \\begin{pmatrix} x _ N ( \\tau _ 0 ) - a _ M b _ M k _ M x _ E ( \\tau _ 0 ) ( t - \\tau _ 0 ) \\\\ x _ M ( \\tau _ 0 ) + k _ M x _ E ( \\tau _ 0 ) ( t - \\tau _ 0 ) \\\\ x _ E ( \\tau _ 0 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\| T _ a \\chi _ k \\| _ 1 \\geq c = | \\hat a ( s ) | . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\Bigl [ \\Bigl ( \\sum _ { i = 1 } ^ k U _ i \\Bigr ) ^ 6 - \\sum _ { i = 1 } ^ k U _ i ^ 6 \\Bigr ] = \\sum _ { i = 1 } ^ k \\int _ { B _ { \\rho } ( \\zeta _ i ) } E _ i + \\sum _ { i = 1 } ^ k \\int _ { \\mathcal { O } _ \\rho } E _ i , \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} M _ \\lambda ( \\zeta ) \\mu ^ { \\frac { 1 } { 2 } } & = | \\sigma _ 1 | ^ { \\frac { 1 } { 2 } } M _ \\lambda P \\bar \\Lambda = | \\sigma _ 1 | ^ { \\frac { 1 } { 2 } } \\Bigl ( \\sigma _ 1 v _ 1 \\bar \\Lambda _ 1 + \\sum _ { l = 2 } ^ k \\bar v _ l \\bar \\Lambda _ l \\Bigr ) , \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} I \\cap J = ( x z ^ 2 + y z ^ 2 , 4 y ^ 3 z + z ^ 4 , x ^ 3 y ^ 3 - x ^ 2 y ^ 4 + x y ^ 5 - y ^ 6 - y ^ 2 z ^ 3 ) \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\chi _ j = \\lim \\limits _ { t \\to + \\infty } \\frac { 1 } { t } \\log \\Vert D X _ t ( x ) \\cdot v \\Vert , \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\begin{pmatrix} \\gamma _ 1 & 0 & \\gamma _ 2 \\\\ 0 & 1 & 0 \\\\ \\gamma _ 2 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\texttt { C } _ { \\textsf { i s } } = \\frac { \\sqrt { \\varepsilon + \\kappa \\mathtt { C } _ { P , \\Omega } ^ { 2 } } } { \\beta } . \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} T ^ F g ( x ) : = \\int _ X F ( x , y ) \\ > g ( y ) \\ > d \\omega _ X ( y ) \\quad \\quad ( x \\in X , \\ > g \\in C _ c ( X ) ) . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} = \\frac { ( a _ { 1 } - a _ { 2 } ) p q } { a _ { 1 } p q - Q ( q - p ) } \\left ( \\frac { Q ( q - p ) - a _ { 2 } p q } { a _ { 1 } p q - Q ( q - p ) } \\right ) ^ { \\frac { a _ { 2 } p q - Q ( q - p ) } { ( a _ { 1 } - a _ { 2 } ) p ^ { 2 } } } \\left ( \\frac { a _ { 1 } p q - Q ( q - p ) } { ( a _ { 1 } - a _ { 2 } ) ( q - p ) } d \\right ) ^ { \\frac { p - q } { p } } . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\left ( \\sum _ { i _ { 1 } , . . . , i _ { m } = 1 } ^ { N } \\left \\vert y _ { i _ { 1 } . . . i _ { m } } \\right \\vert ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } \\leq K _ { m , r } \\left ( { \\displaystyle \\int \\limits _ { I ^ { m } } } \\left \\vert \\sum _ { i _ { 1 } , . . . , i _ { m } = 1 } ^ { N } r _ { i _ { 1 } } \\left ( t _ { 1 } \\right ) . . . r _ { i _ { m } } \\left ( t _ { m } \\right ) y _ { i _ { 1 } . . . i _ { m } } \\right \\vert ^ { r } d t _ { 1 } . . . d t _ { m } \\right ) ^ { \\frac { 1 } { r } } \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { \\lvert S _ f ( n ) \\rvert ^ 2 } { n ^ { k - 1 } } e ^ { - n / X } = C X ^ { \\frac { 3 } { 2 } } + O _ { f , \\epsilon } ( X ^ { \\frac { 1 } { 2 } + \\epsilon } ) , \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} C ( h _ { \\rho } ) = \\{ X = ( t , x , \\tau , \\xi ) \\mid \\sigma ( X , Y ) \\leq 0 , ~ \\forall Y \\in \\Gamma ( h _ { \\rho } ) \\} \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} 2 ^ { 1 - p } ( \\| x + y \\| _ p ^ p + \\| x - y \\| _ p ^ p ) + \\sum _ { i = 1 } ^ \\infty 2 ^ { p _ i ( \\frac 1 p - 1 ) } ( | x _ i + y _ i | ^ { p _ i } + | x _ i - y _ i | ^ { p _ i } ) = 2 \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} { \\rm d e t } \\ , L _ a ( x , \\tau , \\xi , \\eta ) = \\tau ( \\tau ^ 2 - \\xi ^ 2 - x ^ 2 \\eta ^ 2 ) = h ( x , \\tau , \\xi , \\eta ) \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} A ( p ) : = \\left . \\textstyle \\frac { \\partial f } { \\partial x } \\right | _ { x = x ^ * ( p ) } = \\left . \\textstyle \\frac { \\partial \\tilde { f } } { \\partial x } \\right | _ { x = 0 } . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} a x _ { 0 } ^ { 3 } x _ { 2 } + b x _ { 0 } x _ { 2 } ^ { 3 } + c x _ { 0 } ^ { 2 } x _ { 1 } x _ { 3 } + d x _ { 1 } x _ { 2 } ^ { 2 } x _ { 3 } + x _ { 0 } x _ { 1 } ^ { 2 } x _ { 2 } + x _ { 1 } ^ { 3 } x _ { 3 } + x _ { 0 } x _ { 2 } x _ { 3 } ^ { 2 } + x _ { 1 } x _ { 3 } ^ { 3 } = 0 , \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} P _ r ( z ) = \\lim _ { t \\searrow 0 } t ^ { - r } p ( t z ) \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} R _ 1 = R _ 2 = \\frac { P ( 1 + \\rho ^ * ) } { 4 } \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} H = H _ { + } \\oplus H _ { - } , \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\P \\big ( \\Psi _ { \\zeta } = 1 \\ ; : \\ ; \\forall \\zeta \\in \\Xi _ \\delta \\big ) \\ge 1 - \\frac { C _ D } { N ^ D } \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} T B V _ { i } ^ { ( W ) } = \\dfrac { \\bigg ( q ^ { W } _ { i - 1 } ( x _ { i - \\frac { 1 } { 2 } } ) - q ^ { W } _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) \\bigg ) ^ 4 + \\bigg ( q ^ { W } _ { i + 1 } ( x _ { i + \\frac { 1 } { 2 } } ) - q ^ { W } _ { i } ( x _ { i + \\frac { 1 } { 2 } } ) \\bigg ) ^ 4 } { \\big ( \\bar { q } _ { i } - \\bar { q } _ { i - 1 } \\big ) ^ 4 + \\big ( \\bar { q } _ { i } - \\bar { q } _ { i + 1 } \\big ) ^ 4 + \\epsilon } , \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} u '' ( t ) + \\mathcal { A } u ( t ) - \\int _ 0 ^ t \\ H ( t - s ) \\mathcal { A } u ( s ) d s = 0 t \\in ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\int _ 0 ^ X ( P _ k ( x ) ) ^ 2 d x = \\delta _ { [ k = 3 ] } D ' X ^ { k - 1 } \\log X + D X ^ { k - 1 } + O _ { \\lambda } ( X ^ { k - 1 - \\lambda } ) . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} d ^ { 2 } w / d z ^ { 2 } = \\left \\{ { u ^ { 2 } f \\left ( u , z \\right ) + g \\left ( z \\right ) } \\right \\} w . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} D ^ k _ h ( s ) : = \\frac { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\langle P _ h ^ k ( \\cdot , s ) , V \\rangle . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} ( x _ { i j } , x _ { j k } ) _ { x _ i x _ j x _ k } = & x _ { i j } x _ k - x _ i x _ { j k } \\\\ = & ( x _ i x _ j - ( - 1 ) ^ { | x _ { i } | | x _ { j } | } x _ j x _ i ) x _ k - x _ i ( x _ j x _ k - ( - 1 ) ^ { | x _ { j } | | x _ { k } | } x _ k x _ j ) \\\\ = & - ( - 1 ) ^ { | x _ { i } | | x _ { j } | } x _ j x _ i x _ k + ( - 1 ) ^ { | x _ { j } | | x _ { k } | } x _ i x _ k x _ j \\\\ \\equiv & - ( - 1 ) ^ { 2 | x _ { i } | | x _ { j } | } x _ i x _ j x _ k + ( - 1 ) ^ { 2 | x _ { j } | | x _ { k } | } x _ i x _ j x _ k \\equiv 0 ~ m o d ( J ; x _ i x _ j x _ k ) . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} L a w ( Y ^ N _ i , \\alpha ^ N _ i , \\xi _ i ^ N , \\N ^ N _ i ) = L a w ( X _ { \\alpha , m } , \\alpha , \\xi , \\N ) , i \\in \\{ 1 , \\ldots , N \\} . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} L _ m ( x ) = x ^ { m - 2 } \\big ( H ( x ) - x ^ 2 C ( x ) \\big ) + ( m - 2 ) \\frac { x ^ { m + 1 } } { 1 - x } C ( x ) + \\frac { x ^ m } { 1 - x } C ( x ) F _ T ( x ) . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\kappa ^ { \\chi _ n } _ { \\ell } ( x _ { 1 } , \\dots , x _ { \\ell } ) \\ n ^ { - \\frac { \\deg x _ 1 + \\cdots + \\deg x _ \\ell - 2 ( \\ell - 1 ) } { 2 } } \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} \\norm { D v } _ { L ^ 2 ( \\Omega ) } \\lesssim \\norm { \\vec g } _ { L ^ 2 ( \\Omega ) } = \\norm { \\vec g } _ { L ^ 2 ( \\Omega ( x _ 0 , 2 R ) \\setminus B ( x _ 0 , R ) ) } . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial t } + L w - ( \\lambda + \\mu ) w = - \\frac { \\partial u } { \\partial y } \\frac { \\partial v } { \\partial y } \\ : \\mbox { i n } \\ : D _ T , w ( \\pm 1 , y ) = w ( \\pm 1 , - e y ) \\ : \\mbox { i n } \\ : D _ T ^ \\pm , w ( T ) = 0 \\ : \\mbox { i n } \\ : D \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { M _ h ( x ) } { C ^ { h / 2 } x ( \\log \\log x ) ^ { 3 h / 2 } } = s _ h = \\frac { h ! } { ( h / 2 ) ! 2 ^ { h / 2 } } , \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} d e ^ 1 = - 2 e ^ { 2 3 } , d e ^ 2 = - 2 e ^ { 1 3 } \\textrm { a n d } d e ^ 3 = 2 e ^ { 1 2 } \\ , . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} V = \\left ( v _ { \\alpha } ^ { \\beta } \\right ) _ { 1 \\leq \\alpha \\leq q N } ^ { 1 \\leq \\beta \\leq q N } \\in \\mathcal { M } _ { q N \\times q N } \\left ( \\mathbb { C } \\right ) , A = \\left ( a ^ { i j } _ { k l } \\right ) _ { 1 \\leq k , l \\leq q } ^ { 1 \\leq i , j \\leq q } \\in \\mathcal { M } _ { q ^ { 2 } \\times q ^ { 2 } } \\left ( \\mathbb { C } \\right ) , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} u ^ { L } _ { i + \\frac { 1 } { 2 } } = q _ i ( x _ { i + \\frac { 1 } { 2 } } ) \\ \\ { \\rm a n d } \\ \\ u ^ { R } _ { i - \\frac { 1 } { 2 } } = q _ { i } ( x _ { i - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} T u = \\frac { \\nabla u } { \\sqrt { 1 + | \\nabla u | ^ 2 } } \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} L ^ \\rho _ { - } ( m , n ) = L _ { ( 0 , 0 ) \\to ( 1 , 0 ) \\to ( m , n ) } , L ^ \\rho _ { | } ( m , n ) = L _ { ( 0 , 0 ) \\to ( 0 , 1 ) \\to ( m , n ) } . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} ( A Q _ t ) x = e ^ { t A } B B ^ * e ^ { t A ^ * } x - ( A Q _ t ) ^ * x - B B ^ * x \\forall x \\in D ( ( A Q _ t ) ^ * ) \\forall t \\in [ 0 , + \\infty [ \\ , . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { F _ k ^ { \\bf a } } { k ^ { j _ 0 } \\lambda _ { i _ 0 } ^ k } = c _ { { i _ 0 } { j _ 0 } } . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} \\mathcal { K } ( x ) ( s ) = \\int _ a ^ b \\kappa ( s , t , x ( t ) ) d t , \\ ; \\ ; \\ ; s \\in [ a , b ] , \\ ; x \\in X , \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{gather*} J _ { 2 1 } = J _ { 2 1 1 } + J _ { 2 1 2 } + J _ { 2 1 3 } + J _ { 2 1 4 } , \\end{gather*}"}
-{"id": "605.png", "formula": "\\begin{align*} f ( T _ 0 : T _ 1 ) = ( T _ 0 ^ 2 : T _ 1 ^ 2 + c T _ 0 ^ 2 ) \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} ( \\psi _ 2 ^ { i + 1 i } \\circ \\frak n ^ i _ k - \\frak n ^ { i + 1 } _ k \\circ \\psi _ 1 ^ { i + 1 i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } = ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak h ^ { i + 1 i } _ k + \\frak h ^ { i + 1 i } _ k \\circ \\hat d _ { 1 } ^ { i + 1 } ) _ { \\alpha ' _ 2 \\alpha _ 1 } \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\ , K _ { \\tilde { i } \\tilde { l } } = \\ , K _ { \\tilde { i } - 1 , \\tilde { l } } = \\ldots = \\ , K _ { b _ { 0 } + 1 , \\tilde { l } } = r + 1 , \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\mu _ k | _ P & = c _ 1 \\mu | _ P \\\\ \\nu _ k | _ P & = c _ 2 \\nu | _ P \\ , . \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} Q _ 0 \\big ( \\mu ( \\omega _ { q _ 0 } ) , \\dots , \\mu ( \\omega _ { q _ { \\rho ( X ) } } ) \\big ) = \\log 2 \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\omega ( y ) = \\sum _ { i = 1 } ^ k \\bigl ( 1 + \\vert y - \\zeta _ i ^ { \\prime } \\vert \\bigr ) ^ { - 1 } . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} L _ { r } ( k ) = ( - 1 ) ^ { r ' } 2 ^ { r } & \\left ( 2 \\pi \\right ) ^ { r ( r - 1 ) / 2 - k r } \\\\ & \\times D _ { \\boldsymbol { K } } ^ { ( k + 1 / 2 ) r ' - r ' ( r ' + 1 ) } \\prod _ { i = 0 } ^ { r - 1 } \\Gamma ( k - i ) \\prod _ { i = 1 } ^ { r } L ( i - k , \\chi _ { \\boldsymbol { K } } ^ { i - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } a ( p ^ { j n } ) X ^ n = \\dfrac { a ( 1 ) } { 1 - \\lambda _ j ( p ) X + p ^ { j ( k - 1 ) } \\chi ^ j ( p ) X ^ 2 } \\cdot \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} \\left ( \\boxtimes _ { i = 1 } ^ { l } \\left ( U _ i ^ { \\boxtimes | \\tau ^ { ( i ) } | } \\otimes \\mathcal { S } ^ { \\tau ^ { ( i ) } } \\right ) \\right ) \\boxtimes \\left ( \\mathbf { 1 } ^ { \\boxtimes ( n - | \\sigma | ) } \\otimes \\mathcal { S } ^ { ( n - | \\tau ^ { ( 0 ) } | - | \\sigma | , \\tau ^ { ( 0 ) } ) } \\right ) \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} ( x _ \\lambda ) \\mathbf { d } x & = \\limsup _ \\lambda x _ \\lambda \\mathbf { d } x . \\\\ x \\mathbf { d } ( x _ \\lambda ) & = \\liminf _ \\gamma x \\mathbf { d } x _ \\lambda . \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} p _ { 2 n } ( z ) = z ^ { 2 n } , p _ { 2 n + 1 } ( z ) = z ^ { 2 n + 1 } - \\left \\langle \\mathrm { T r } \\ , P _ m ^ 2 \\right \\rangle _ { 2 n \\times 2 n } z ^ { 2 n - 1 } \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} - | u ( x ) | ^ { q - 2 } u ( x ) \\overline { \\psi ( x ) } ) d x = 0 \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\hat { F } _ { i } ( \\hat { v } ) \\approx \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } \\ , F \\left ( \\sum _ { j = 1 } ^ m \\hat { v } _ j \\Phi _ j ( p ^ { ( \\ell ) } ) , p ^ { ( \\ell ) } \\right ) \\Phi _ i ( p ^ { ( \\ell ) } ) \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} \\tau ( \\phi ^ t _ X ) = \\tau + t . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\Psi _ j ^ * ( p ) = \\sum _ { i = 1 } ^ m u _ { i j } \\Phi _ i ( p ) \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} P _ { A _ 0 , \\ldots , A _ n } & ( A ) = P ( \\psi ( X _ 0 ) \\in A _ 0 , \\ldots , \\psi ( X _ n ) \\in A _ n , , \\psi ( X _ { n + 1 } ) \\in A ) \\\\ & = \\int _ D K _ l ( x _ 0 , \\psi ^ { - 1 } ( A ) ) \\ > d ( P _ { A _ 0 , \\ldots , A _ { n - 1 } } | _ { A _ n } * \\mu _ 1 ) ( l ) \\\\ & = ( P _ { A _ 0 , \\ldots , A _ { n - 1 } } | _ { A _ n } * \\mu _ 1 ) ( A ) . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\int \\chi \\cdot \\prod _ { j = 0 } ^ m S ^ j f _ j \\ , d \\nu = 0 \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} \\alpha _ X = \\lim _ { s \\to 0 } \\frac { \\log \\| D _ { 1 / s } \\| _ { X \\to X } } { \\log s } , \\beta _ X = \\lim _ { s \\to \\infty } \\frac { \\log \\| D _ { 1 / s } \\| _ { X \\to X } } { \\log s } \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} X = \\left ( d _ { 1 } t + d _ { 2 } \\right ) \\partial _ { t } \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\iota ( x ) = \\{ ( u , v ) \\in E _ \\omega : x \\in \\ell _ \\omega ( u , v ) \\} \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} u ^ g _ { k , T } = \\sum _ { t = 1 } ^ { T - 1 } \\left ( u ^ g _ { k , t + 1 } + u ^ g _ { k , t - 1 } \\right ) \\varkappa ^ T _ t . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} b _ 1 & = ( 1 , 0 ) + a e _ 1 + b e _ 2 , & b _ 2 & = ( 0 , 1 ) + c e _ 1 + d e _ 2 , \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\rho _ { ( k ) } ^ { r } ( x ) = S ( x , x ) . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} n ^ { ( d - 1 ) n } 1 _ S ^ { * d } ( f ) = n ^ { ( d - 1 ) n } \\sum _ { \\chi \\in \\hat { G } ^ n } \\hat { 1 _ S } ( \\chi ) ^ d \\chi ( f ) . \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\kappa ( x \\ , y , g ) = \\kappa ( x , y ( g ) ) \\ , \\kappa ( y , g ) \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} 2 \\ , h \\ , \\Re \\ , G ( 1 / 2 ) = - \\ , h ^ 3 \\ , \\Im \\ , G ^ { ( 2 ) } ( 1 / 2 ) \\ , . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\langle \\tilde T _ { f } g , g \\rangle _ { \\tilde X } \\ > = \\ > \\sum _ { i , j = 1 } ^ n \\overline { c _ i } \\ > c _ j \\ > \\Phi _ f ( i , j ) . \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} [ T _ { n } ( U _ 1 ) , T _ { n } ( U _ 2 ) ] = \\sum _ { U _ 3 } ( N _ { U _ 1 , U _ 2 } ^ { U _ 3 } - N _ { U _ 2 , U _ 1 } ^ { U _ 3 } ) T _ { n } ( U _ 3 ) \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\ 1 ( x ) = \\begin{cases} 1 & x > 0 \\\\ 0 & x < 0 . \\end{cases} \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} & g _ { 2 1 } = - U _ { 2 n } J _ { 2 n } g _ { 1 2 } ^ { \\tau } J _ { 2 m } U _ { 2 m } , \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\lim _ { \\substack { r \\rightarrow 0 \\\\ \\Delta \\rightarrow 0 } } \\delta _ { \\gamma ^ { \\Delta r } ( t , x ) } ( d a ) d t = [ \\widehat { \\gamma } ( t , x ) ] ( d a ) d t \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} A = \\langle e _ 2 \\rangle _ { \\overline { 0 } } \\oplus \\langle e _ 1 , e _ 3 \\rangle _ { \\overline { 1 } } . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} D _ i . D _ j = \\begin{cases} 1 & \\textnormal { i f $ v _ i $ a n d $ v _ j $ a r e n e x t t o e a c h o t h e r o n a n e d g e . } \\\\ l ^ * ( m _ { i j } ^ * ) + 1 & \\textnormal { i f $ v _ i $ a n d $ v _ j $ a r e v e r t i c e s t h a t a r e c o n n e c t e d } \\\\ & \\textnormal { b y a n e d g e $ m _ { i j } $ w h o s e d u a l i s $ m _ { i j } ^ * $ . } \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\phi ( x , \\kappa ^ i r ) \\lesssim \\phi ( x , r ) + \\norm { D u } _ { L ^ \\infty ( B ( x , \\frac 1 5 ) \\cap B ^ + _ 4 ) } \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\mathbf { A } } ( t ) } t \\ , d t + \\int _ 0 ^ { r } \\frac { \\tilde \\omega _ { \\vec g } ( t ) } t \\ , d t . \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} \\int _ { | x - y | \\leq \\delta / C _ { 0 } } e ^ { i \\lambda \\psi _ { ( 1 ) } ( x , \\beta ) } f _ { i _ { 1 } . . . i _ { m } } ( x ) C ^ { i _ { m } } ( x , \\beta ; \\lambda ) d x = \\mathcal { O } ( e ^ { - \\lambda / C } ) i _ { j } \\in \\{ 1 , \\dots , n \\} j = 1 , \\dots , ( m - 1 ) \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\int _ { B _ { r } ( x ) } \\sum _ { i , j = 1 } ^ { m } \\left ( | D u | ^ { 2 } \\delta _ { i j } - 2 \\sum _ { \\ell = 1 } ^ { Q } \\langle D _ { i } u _ { \\ell } , D _ { j } u _ { \\ell } \\rangle \\right ) D _ { i } X ^ { j } \\ , \\mathrm { d } y = 0 . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} Y ^ { ( n ) } ( t _ { n , i + 1 } ) = Y ^ { ( n ) } ( t _ { n , i } ) + \\int _ { t _ { n , i } } ^ { t _ { n , i + 1 } } g ( t _ { n , i } , W _ 0 ^ { t _ { n , i } } , Y _ 0 ^ { ( n ) , t _ { n , i } } ) d s + B ( t _ { n , i + 1 } ) - B ( t _ { n , i } ) , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} l ( y ) & < f ( z ) \\frac { y - p } { z - p } + \\frac { z - y } { z - p } \\left ( \\frac { y - p } { y - x } f ( x ) + \\frac { p - x } { y - x } f ( y ) \\right ) \\\\ & = ( z - \\alpha ) \\frac { y - p } { z - p } + \\frac { z - y } { z - p } \\left ( \\frac { y - p } { y - x } ( x - \\alpha ) + \\frac { p - x } { y - x } ( y - \\alpha ) \\right ) \\\\ & = y - \\alpha = f ( y ) , \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} L E ( x _ 0 , r ) : = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum ^ { n - 1 } _ { i = 0 } \\ln | G _ { r } ^ { \\prime } ( x _ i ) | . \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\mathbf { S } ( x ) = ( S _ 1 ( x ) , S _ 2 ( x ) ) : = ( T ( | x | ) , R ( | x | ) ) , \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} g ( v , w ) : = g _ { i j } v ^ i w ^ j , \\ \\beta ( v ) = b _ i v ^ i , \\ v = v ^ i \\frac { \\partial } { \\partial x ^ i } , \\ w = w ^ j \\frac { \\partial } { \\partial x ^ j } . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\left < Z _ { i } , \\mathcal { A } _ { k u } ^ { j } \\left ( Z _ { j } \\right ) \\right > + \\left < \\mathcal { A } _ { u k } ^ { i } \\left ( Z _ { i } \\right ) , Z _ { j } \\right > = 0 , \\quad \\quad \\quad \\quad \\mbox { f o r a l l $ k , u , i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\alpha , \\varepsilon } ( p ) = \\phi ^ { t _ { \\alpha , \\varepsilon } ( p ) } { \\varepsilon } ( p ) . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{gather*} \\overline { \\xi } \\cdot \\overline { \\xi } = - \\mu ^ 2 . \\end{gather*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\| \\widehat { U } ^ { * } ( \\Phi ^ { \\prime } ( f \\otimes a ) + \\sum _ { j = 1 } ^ m \\sum _ { k = 1 } ^ m & f ( z _ k ) \\rho ( p _ { j , k } \\otimes a ) ) \\widehat { U } \\\\ & - ( \\Psi ^ { \\prime } ( f \\otimes a ) + \\sum _ { j = 1 } ^ m \\sum _ { k = 1 } ^ m f ( z _ k ) \\rho ( p _ { j , k } \\otimes a ) ) \\| < \\frac { \\varepsilon } { 7 } \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} F ( T ) = \\lim _ { m \\to \\infty } \\sum _ { n = 0 } ^ { [ 2 ^ m T ] - 1 } \\mathbb E \\Bigl [ F \\Bigl ( \\frac { n + 1 } { 2 ^ m } \\Bigr ) - F \\Bigl ( \\frac { n } { 2 ^ m } \\Bigr ) \\Big | \\mathcal F _ { \\frac { n } { 2 ^ m } } \\Bigr ] , \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\left | \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( \\boldsymbol { X } _ i ) \\right | - \\left | \\int _ { [ 0 , 1 ] ^ d } f ( \\boldsymbol { x } ) d \\boldsymbol { x } \\right | \\right | \\leq C \\frac { 1 } { n } \\int _ { [ 0 , 1 ] ^ d } \\left | \\frac { \\partial ^ d } { \\partial x _ 1 , \\cdots \\partial x _ d } f ( \\boldsymbol { x } ) \\right | d \\boldsymbol { x } . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\Lambda _ 2 \\left [ \\left [ \\Lambda _ 3 \\Lambda _ 1 - \\partial ( \\Lambda _ 1 ) \\right ] \\varphi - n \\Lambda _ 1 ( \\mu + \\alpha ) \\right ] = n \\lambda _ 1 \\varphi N f _ s . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} \\sup _ { y \\in B _ { \\delta / \\varepsilon } ( \\zeta _ i ' ) ) } \\omega ( y ) ^ { - ( 2 + \\nu ) } | E ( y ) | \\leq C \\varepsilon ^ { \\frac { 1 } { 2 } } \\bigl | - \\mu _ i ^ { \\frac { 1 } { 2 } } g _ \\lambda ( \\zeta _ i ) + \\sum _ { j \\not = i } \\mu _ j ^ { \\frac { 1 } { 2 } } \\ , G _ { \\lambda } ( \\zeta _ i , \\zeta _ j ) \\bigr | + C \\varepsilon ^ 2 . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} A ' ( x , v ) = \\frac { v x } { 1 - v } A ( x , v ) - \\frac { x v } { 1 - v } A ( v x , 1 ) + \\frac { ( v x ^ 2 + v x - 1 ) v ^ 2 x ^ 5 } { ( 1 - 2 v x ) ( 1 - ( 1 + v ) x ) ( 1 - v x ) ^ 2 ( 1 - x ) ^ 2 } . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\sup _ { 0 \\leq t \\leq T } | | g _ { \\varepsilon } - g | | _ { p } ( t ) = 0 . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} f ^ * ( \\vartheta _ i ) = f ^ * ( \\varphi _ { n _ 1 } ) ^ { c _ 1 } \\cdots f ^ * ( \\varphi _ { n _ r } ) ^ { c _ r } = ( \\varphi _ { n _ 1 + 1 } ^ { d c _ 1 } / \\varphi ^ { c _ 1 } ) \\cdots ( \\varphi _ { n _ r + 1 } ^ { d c _ r } / \\varphi ^ { c _ r } ) , \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ t [ \\widetilde { \\varphi } ( X _ { t + s } ^ { x } ) ] = \\hat { \\mathbb { E } } [ \\widetilde { \\varphi } ( X _ { s } ^ { y } ) ] _ { y = X _ t ^ x } . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\frac { \\partial q } { \\partial t } + \\frac { \\partial f ( q ) } { \\partial x } = 0 , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} H [ X ] = \\big \\{ f \\in X \\colon \\widehat { f } ( n ) = 0 \\quad \\mbox { f o r a l l } n < 0 \\big \\} , \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\sigma _ { f } ( S ) = f | S \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\Psi ( z _ { 1 } ^ { 0 } , z _ { 1 } ^ { 1 } , z _ { 2 } ^ { 0 } , z _ { 2 } ^ { 1 } ) = ( - \\phi _ { 1 t } ( T , \\cdot ) , \\phi _ { 1 } ( T , \\cdot ) , - \\phi _ { 2 t } ( T , \\cdot ) , \\phi _ { 2 } ( T , \\cdot ) ) \\ , . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\Omega _ a = \\{ x \\in \\R ^ 3 \\ : \\ a < | x | < 1 \\} . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } L ( \\phi _ n ) & = & h _ n + \\sum _ { i , j } c _ { i j } ^ n \\ , w _ { \\mu _ { i , n } ^ { \\prime } , \\zeta _ { i , n } ^ { \\prime } } ^ 4 \\ , z _ { i j } ^ n & \\Omega _ { \\varepsilon _ n } , \\\\ \\phi _ n & = & 0 & \\partial \\Omega _ { \\varepsilon _ n } , \\\\ \\int _ { \\Omega _ { \\varepsilon _ n } } w _ { \\mu _ { i , n } ^ { \\prime } , \\zeta _ { i , n } ^ { \\prime } } ^ 4 \\ , z _ { i j } ^ n \\ , \\phi _ n & = & 0 & i , j , \\end{array} \\right . \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} g _ { 1 } ( U _ 2 , U _ 1 ) g _ { 1 } ( \\xi , Z ) = g _ { 1 } ( U _ 2 , \\phi U _ 1 ) g _ { 1 } ( H , Z ) - g _ { 1 } ( U _ 2 , U _ 1 ) g _ { 1 } ( H , \\varphi Z ) . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} d ^ { 2 } w / d z ^ { 2 } = \\left \\{ { u ^ { 2 } f _ { 0 } \\left ( z \\right ) + u f _ { 1 } \\left ( z \\right ) + g \\left ( z \\right ) } \\right \\} w . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} \\varepsilon ( \\{ a , b \\} ) & = ( - 1 ) ^ { | b _ { ( 1 ) } | | a _ { ( 2 ) } | } \\varepsilon ( \\varepsilon ( \\{ a _ { ( 1 ) } , b _ { ( 1 ) } \\} ) a _ { ( 2 ) } b _ { ( 2 ) } + \\{ a , b \\} ) \\\\ & = ( - 1 ) ^ { | b _ { ( 1 ) } | | a _ { ( 2 ) } | } ( \\varepsilon ( \\{ a _ { ( 1 ) } , b _ { ( 1 ) } \\} ) \\varepsilon ( a _ { ( 2 ) } b _ { ( 2 ) } ) + \\varepsilon ( \\{ a , b \\} ) ) \\\\ & = ( - 1 ) ^ { | b _ { ( 1 ) } | | a _ { ( 2 ) } | } ( \\varepsilon ( \\{ a , b \\} ) + \\varepsilon ( \\{ a , b \\} ) ) , \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\nabla u ( x ) = \\sum _ { i = 1 } ^ { n + 1 } \\alpha _ i ( x ) \\xi _ i ( x ) . \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{gather*} c ( N , m , \\theta ) = - \\sum _ { r = 1 } ^ m { \\sum _ { j = \\theta ( N - r + 1 ) } ^ { \\theta N - 1 } { ( \\theta ( N - r ) - j ) } } - \\sum _ { r = 1 } ^ { m } { ( \\theta r - \\theta ) ( \\theta N - 1 ) } , \\end{gather*}"}
-{"id": "7140.png", "formula": "\\begin{align*} | \\mathcal { L } _ 2 | = q + 1 + ( q ^ 2 + q ) q = q ^ 3 + q ^ 2 + q + 1 . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} b _ { \\sigma ( i ) , \\sigma ( j ) } = c _ { i , j } \\quad 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} D ^ { i j } _ { u u u ' u ' } = 0 , \\quad \\quad \\mbox { f o r a l l $ i , j , u , u ' = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} & \\int _ { | u - \\alpha _ 1 | \\leqslant \\frac { d } { M } } \\frac { \\ , \\mathrm { d } u } { \\big ( ( u - \\alpha _ 1 ) ^ 2 + ( v - \\alpha _ 2 ) ^ 2 \\big ) ^ { \\frac { p } { 2 } } } \\\\ \\leqslant { } & \\int _ { | u - \\alpha _ 1 | \\leqslant \\frac { d } { M } } \\Big ( \\frac { d } { \\sqrt { 1 + M ^ 2 } } \\Big ) ^ { - p } \\ , \\mathrm { d } u \\\\ = { } & 2 d ^ { 1 - p } M ^ { - 1 } ( 1 + M ^ 2 ) ^ { \\frac { p } { 2 } } , \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} f _ { 2 n } ( t ) = \\Upsilon _ n ( f _ { 1 n } ) ( t ) t \\ge 0 \\ , . \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\lambda ^ * \\mu = \\delta _ { \\lambda , \\mu } s ( \\lambda ) , \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} w = \\left [ \\mu _ { i _ { 1 1 } } , \\ldots , \\mu _ { i _ { 1 p _ 1 } } , \\left [ \\ldots , \\big [ \\mu _ { i _ { ( n - 1 ) 1 } } , \\ldots , \\mu _ { i _ { ( n - 1 ) p _ { n - 1 } } } , [ \\mu _ { i _ { n 1 } } , \\ldots , \\mu _ { i _ { n p _ n } } ] \\big ] \\ldots \\right ] \\right ] . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} W _ { m , r } ( n , k ) = \\sum _ { i = k } ^ n { n \\choose i } r ^ { n - i } S _ 2 ( i , k ) m ^ { i - k } . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\frac { \\Gamma ( 2 z ) } { \\Gamma ( z ) } = \\frac { \\Gamma ( z + \\tfrac { 1 } { 2 } ) 2 ^ { 2 z - 1 } } { \\sqrt \\pi } \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} A ^ { \\sharp } _ { j , \\lambda } : = \\chi _ { \\lambda } \\ast \\tilde { A } _ { j } , j = 1 , \\ , 2 . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} H ^ 0 ( X , D ) : = \\left \\{ \\phi \\in \\mathrm { R a t ( S ) } ^ { \\times } \\ , : \\ , D + ( \\phi \\otimes 1 ) _ { \\mathbb Q } \\geqslant 0 \\right \\} \\cup \\{ 0 \\} \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} r _ 0 ( \\boldsymbol { \\zeta } ( t ) ) & = \\frac { \\zeta _ 0 ( t ) } { \\zeta _ 0 ( t ) + \\sum _ { k = 1 } ^ \\infty k \\zeta _ k ( t ) } \\\\ & = \\frac { \\zeta _ 0 ( t ) } { \\zeta _ 0 ( t ) + \\sum _ { k = 1 } ^ \\infty k { \\tilde { \\zeta } } _ k ( t ) } \\ge \\frac { { \\tilde { \\zeta } } _ 0 ( t ) } { { \\tilde { \\zeta } } _ 0 ( t ) + \\sum _ { k = 1 } ^ \\infty k { \\tilde { \\zeta } } _ k ( t ) } = r _ 0 ( { \\tilde { \\boldsymbol { \\zeta } } } ( t ) ) . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} T _ 0 \\ , y _ 0 = 0 \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\omega _ a [ \\xi _ m , \\ldots , \\xi _ n ] = - \\frac { 1 } { \\psi _ { m , n } ( a ) } . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\biggr | \\frac { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 2 } \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ k - \\big ( \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } \\big ) ^ 2 } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ k } \\biggr | = \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} D _ { \\nu , \\theta } ( f ) & = - \\int _ { \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ + } A _ { \\nu , \\theta } \\big \\{ \\mathcal { M } _ { \\nu , \\theta } ( f ) - f \\big \\} \\ln f d v d I \\cr & = A _ { \\nu , \\theta } \\int _ { \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ + } \\big \\{ f - \\mathcal { M } _ { \\nu , \\theta } ( f ) \\big \\} H ^ { \\prime } ( f ) v d I \\cr & \\geq A _ { \\nu , \\theta } \\big \\{ H ( f ) - H ( \\mathcal { M } _ { \\nu , \\theta } ) \\big \\} . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} f ( T _ 0 : T _ 1 ) = ( 2 T _ 0 T _ 1 : T _ 1 ^ 2 - T _ 0 ^ 2 ) \\quad f ' ( T _ 0 : T _ 1 ) = ( 2 \\sqrt { - 1 } T _ 0 T _ 1 : T _ 1 ^ 2 - T _ 0 ^ 2 ) , \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} { \\cal D } _ \\bot : = \\begin{pmatrix} D _ \\bot & 0 \\\\ 0 & - D _ \\bot \\end{pmatrix} , D _ \\bot = { \\rm d i a g } _ { j \\in { \\mathbb S } _ 0 ^ c } \\ , \\mu _ j \\ , , \\mu _ { - j } = \\mu _ j \\ , , \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} \\lim _ { k _ 0 < k \\to \\infty } \\frac { F ^ { \\bf a } _ { k + 1 } } { F ^ { \\bf a } _ k } = \\lambda _ { i _ 0 } , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\chi ( S ; a , b ) \\cdot { V } ^ x & = ( r - 2 | T | ) \\sum _ { \\substack { T \\\\ S \\thicksim T \\\\ x \\in T } } 1 \\\\ & = \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - ( 1 - S ( x ) ) } { b } ( r - 2 | T | ) . \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 3 } \\hat n _ { \\mu } & = - g ^ { k l } \\partial _ { k } \\partial _ { t } ^ 2 v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } - \\left ( \\partial _ { t } ^ 2 ( g ^ { k l } \\partial _ { k } v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } ) - g ^ { k l } \\partial _ { k } \\partial _ { t } ^ 2 v ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } \\right ) \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} \\widetilde { \\textnormal { ( I ) } } = & - \\frac { 4 ( p - 1 ) } { p } \\int ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla ( \\eta u ^ { p / 2 } ) ) \\ ; d x \\\\ & + \\frac { 4 ( p - 2 ) } { p } \\int u ^ { p / 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x . \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\frac { 1 } { L _ 1 L _ 2 } \\int _ \\Sigma \\eta _ 0 = 0 . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} f ( z , w ) : = \\sum _ { n \\ge 1 } p _ n ( z , w ) \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} \\mathfrak { L } _ { 1 } ( u ) : = \\frac { 1 } { p } \\sum _ { j = 1 } ^ { \\ell } \\int \\limits _ { \\mathbb { G } } | \\mathcal { R } _ { j } ^ { \\frac { a _ { j } } { \\nu _ { j } } } u ( x ) | ^ { p } d x - \\frac { 1 } { q } \\int \\limits _ { \\mathbb { G } } | u ( x ) | ^ { q } d x \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } P _ k ( n ) ^ 2 = \\delta _ { [ k = 3 ] } C ' X ^ { k - 1 } \\log X + C X ^ { k - 1 } + O _ { \\lambda } ( X ^ { k - 1 - \\lambda } ) . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\int _ { [ C _ m ] } \\theta _ t = \\varepsilon \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { E } | D _ n | ^ r = \\mathbb { E } | D ^ * _ z | ^ r = \\frac { 2 } { \\mu } \\mathbb { E } \\delta _ r ( Y ) , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} s ( X ) & = \\sup \\{ p : X \\ \\} , \\\\ \\sigma ( X ) & = \\inf \\{ q : X \\ \\} . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\rho _ \\infty ^ k ( 0 , w ) = y ^ { 1 - w } 2 \\pi 4 ^ { 1 - 3 w } \\frac { \\zeta ( 2 w - 1 ) } { \\zeta ^ { ( 2 ) } ( 2 w ) } \\frac { \\Gamma ( 2 w - 1 ) } { \\Gamma ( w + \\frac { k } { 2 } ) \\Gamma ( w - \\frac { k } { 2 } ) } \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } + \\frac { \\partial v } { \\partial r } + \\frac { v } { r } + \\frac { 1 } { r } \\frac { \\partial w } { \\partial \\theta } = 0 . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\triangle _ g u - \\partial _ t u - W ( x , t ) \\cdot \\nabla u - { V } ( x , t ) u = 0 \\mbox { o n } \\ \\mathcal { M } ^ 1 , \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { + \\infty } \\textrm { t r } _ \\epsilon u _ k = \\sum _ { m \\geqslant 0 } \\lim _ { n \\to + \\infty } l _ m \\ ( \\ ( \\sum _ { k = 0 } ^ n u _ k \\ ) \\ ( x _ m \\ ) \\ ) = \\sum _ { m \\geqslant 0 } l _ m \\ ( \\ ( \\sum _ { k = 0 } ^ { + \\infty } u _ k \\ ) \\ ( x _ m \\ ) \\ ) = \\textrm { t r } _ \\epsilon \\ ( \\sum _ { k = 0 } ^ { + \\infty } u _ k \\ ) . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} U _ \\chi ^ S = 2 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma \\le T } } \\frac { Z _ { \\gamma } } { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } \\quad V _ \\chi ^ S = 2 \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi ) \\\\ \\gamma > T } } \\frac { Z _ { \\gamma } } { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} p \\lim _ { \\theta } \\left ( \\frac { \\hat { \\nu } ^ { 1 1 } } { \\hat { L } ^ { 1 1 } } \\right ) = \\frac { \\left \\{ \\textrm { a s y . v a r . o f } \\sqrt { n } \\left ( \\hat { \\theta } ^ { M } _ 1 \\right ) \\right \\} } { \\left \\{ \\textrm { a s y . v a r . o f } \\sqrt { n } \\left ( \\hat { \\theta } ^ { M L } _ 1 \\right ) \\right \\} } = \\textrm { a s y . e f f . } \\left ( \\hat { \\theta } ^ M _ 1 , F \\right ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\pi ^ { - s } \\Gamma ( s ) \\zeta ( 2 s ) E ( z , s ) = : E ^ * ( z , s ) = E ^ * ( z , 1 - s ) . \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} [ \\xi ] = ( 1 - r ) [ \\sigma ] , \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} Y : \\ ; \\R & \\longrightarrow \\R ^ n \\\\ Y ( 0 ) & = \\vec { 0 } \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} h _ { S _ { m , n } } ( t ) = \\sum _ { i = 0 } ^ { m - 1 } \\binom { m - 1 } { i } \\binom { n - 1 } { i } t ^ i \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } \\int _ { 0 } ^ { 1 } \\int _ { Y ^ { \\ast } } - u ( x , t ) v _ { 1 } ( x ) \\partial _ { t } c _ { 1 } ( t ) \\\\ + A ( y , s ) ( \\nabla u ( x , t ) + \\nabla _ { y } u _ { _ { 1 } } ( x , t , y , s ) ) \\cdot \\nabla v _ { 1 } ( x ) c _ { 1 } ( t ) d y d s d x d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega } f ( x , t ) v _ { 1 } ( x ) c _ { 1 } ( t ) d x d t \\end{gather*}"}
-{"id": "4098.png", "formula": "\\begin{align*} N ( T ) : = \\sum _ { 0 < \\gamma \\le T } 1 \\sim \\frac { T } { 2 \\pi } \\log T , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} & \\Phi _ k ^ - ( x ^ { ( k ) } ) \\star 1 ( \\mathfrak { X } _ k ^ { - } ( u , \\lambda ) , \\mathfrak { X } _ k ^ { + } ( v , \\lambda ) ) + \\mathfrak { X } _ k ^ { - } ( u , \\lambda ) \\star \\mathfrak { X } _ k ^ { + } ( v , \\lambda ) ( 1 , \\Phi _ k ( x ^ { ( k ) } ) ) \\\\ = & \\mathfrak { X } _ k ^ { + } ( v , \\lambda ) \\star \\mathfrak { X } _ k ^ { - } ( u , \\lambda ) ( 1 , \\Phi _ k ( x ^ { ( k ) } ) ) + \\Phi _ k ^ + ( x ^ { ( k ) } ) \\star 1 ( \\mathfrak { X } _ k ^ { + } ( v , \\lambda ) , \\mathfrak { X } _ k ^ { - } ( u , \\lambda ) ) . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} 1 _ { B } \\cdot u = 1 _ { B } \\cdot \\sum _ { i = 1 } ^ { l } h _ { i } = 1 _ { B } \\cdot \\sum _ { i = 1 } ^ { l } \\sum _ { j = 1 } ^ { s _ { i } } \\lambda _ { i j } u _ { i j } = \\sum _ { i = 1 } ^ { l } \\sum _ { j = 1 } ^ { s _ { i } } \\lambda _ { i j } 1 _ { B } \\cdot u _ { i j } = 0 . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\frac { q + q ' } { p } \\sum _ { j = 1 } ^ { N } w _ { j } b _ { j } = 2 \\sum _ { j = 1 } ^ { N } w _ j c _ j + \\frac { q - q ' } { p } \\sum _ { j = N + 1 } ^ { m } w _ { j } b _ { j } \\ ; . \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\begin{aligned} & - h ^ 2 \\Delta _ x \\Phi - { \\partial ^ 2 \\Phi \\over \\partial z ^ 2 } = 0 , \\ - D ( x ) < z < 0 \\cr & h ^ 2 { \\partial ^ 2 \\Phi \\over \\partial t ^ 2 } + g { \\partial \\Phi \\over \\partial z } = f ^ + ( x , t ) \\ @ \\ z = 0 \\cr & { \\partial \\Phi \\over \\partial z } + h ^ 2 \\langle \\nabla _ x D , \\nabla _ x \\Phi \\rangle = f ^ - ( x , t ) \\ @ \\ z = - D ( x ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\alpha _ e L ^ e ( M ^ e ) ^ 2 = \\alpha _ { \\tilde { e } } L ^ { \\tilde { e } } ( M ^ { \\tilde { e } } ) ^ 2 \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} C _ { \\Theta ^ { ( n , \\sigma ) } } = & \\sum _ { i , j = 1 } ^ { n } E _ { i j } \\otimes \\Theta ^ { ( n , \\sigma ) } ( E _ { i j } ) \\\\ & = P + \\sum _ { i \\in F ^ { c } \\atop i < \\sigma ( i ) } Q _ { i } . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} d = 1 + p ^ { i _ 2 } + \\cdots + p ^ { i _ p } \\ \\ \\mbox { w i t h } \\ \\ 0 \\leq i _ 2 \\leq \\cdots \\leq i _ p \\ \\ \\mbox { a n d } \\ \\ ( i _ 2 , \\dots , i _ p ) \\ne ( 0 , \\dots , 0 ) . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} ( n - 3 ) g = 0 . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} H = V ^ - ( T ) \\oplus V ^ + ( T ) \\oplus \\ker T , \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} h ^ { p , q } ( Y , V ) = \\sum _ { k } i ^ { p , k } ( H ^ { p + q } ( Y , V ) ) . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\displaystyle { \\frac { d \\gamma } { d \\theta } = - ( n - 1 ) A ( \\theta ) B ( \\theta ) \\gamma ^ 3 + [ ( n - 1 ) A ( \\theta ) - B ' ( \\theta ) ] \\gamma ^ { 2 } } , \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} X ( p , p ' ) = ( S ( p , p ' ) , T ( p , p ' ) ) \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} h _ i \\in L ^ p ( B ' ) , 1 \\le i \\le d , \\ \\ \\ \\ c , e \\in L ^ q ( B ' ) \\ \\ \\ \\ q : = \\frac { d p } { d + p } . \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\mathrm { p h \\ddot { o } l } _ \\delta ( g , \\overline { Q } ) : = \\sup \\left \\{ \\frac { | g ( x , t ) - g ( y , s ) | } { \\left ( \\| x - y \\| + \\sqrt { | t - s | } \\right ) ^ { \\delta } } : \\ ; ( x , t ) , ( y , s ) \\in \\overline { Q } , \\ ; ( x , t ) \\not = ( y , s ) \\right \\} \\in [ 0 , \\infty ] , \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} F _ 0 ( \\bar { u } ) & \\le \\liminf _ { \\varepsilon \\to 0 } F _ 0 ( u _ \\varepsilon ) \\\\ & \\le \\liminf _ { \\varepsilon \\to 0 } F _ \\varepsilon ( u _ \\varepsilon ) \\\\ & \\le \\liminf _ { \\varepsilon \\to 0 } F _ \\varepsilon ( u _ 0 ) \\\\ & = F _ 0 ( u _ 0 ) . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} ( M _ 1 ) _ { i j } & = a _ { 0 i j } x _ 0 + a _ { 1 i j } x _ 1 + a _ { 2 i j } x _ 2 a _ { 3 i j } x _ 3 , \\\\ ( M _ 2 ) _ { i j } & = a _ { i 0 j } y _ 0 + a _ { i 1 j } y _ 1 + a _ { i 2 j } y _ 2 a _ { i 3 j } y _ 3 . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U _ { 1 } , V _ { 1 } ) , \\pi _ * Z ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 1 } } \\varphi V _ { 1 } , \\mathcal { B } Z ) - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 1 } } \\varphi V _ { 1 } , \\mathcal { C } Z ) - g _ { 1 } ( V _ { 1 } , \\phi U _ { 1 } ) \\eta ( Z ) . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} K _ { h _ 2 } \\circ K _ { h _ 1 } & ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) = \\int _ { X _ 2 } K _ { h _ 1 } ( ( x _ 1 , y _ 2 ) , A _ 1 \\times A _ 2 ) \\ > K _ { h _ 2 } ^ 2 ( x _ 2 , d y _ 2 ) \\\\ & = \\int _ { X _ 2 } \\omega _ { X _ 2 } ( A _ 2 ) K _ { h _ 1 } ^ 1 ( x _ 1 , A _ 1 ) \\ > K _ { h _ 2 } ^ 2 ( x _ 2 , d y _ 2 ) = K _ { h _ 1 } ^ 1 ( x _ 1 , A _ 1 ) \\omega _ { X _ 2 } ( A _ 2 ) \\\\ & = K _ { h _ 1 } ( ( x _ 1 , x _ 2 ) , A _ 1 \\times A _ 2 ) \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} s a t ( n , k , r ) = k ( 2 r - 4 ) n + O ( 1 ) \\begin{cases} k = 2 r - 3 , \\\\ k \\ge 2 r - 2 r \\equiv 0 \\mod 2 , \\\\ k \\ge 2 r - 1 r \\equiv 2 \\mod 3 , \\end{cases} \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{gather*} J _ { j l } ( t , \\tau ) = J ^ { ( 1 ) } _ { j l } ( t , \\tau ) + J ^ { ( 2 ) } _ { j l } ( t , \\tau ) , | t | \\le t ^ { 0 0 } _ { j l } , \\\\ J ^ { ( r ) } _ { j l } ( t , \\tau ) : = t | t | \\int _ 0 ^ { \\tau } e ^ { i \\tilde { \\tau } A ( t ) ^ { 1 / 2 } } F ^ { ( r ) } _ { j l } ( t ) P _ j G _ * P _ l e ^ { - i \\tilde { \\tau } ( t ^ 2 S ) ^ { 1 / 2 } P } P \\ , d \\tilde { \\tau } , r = 1 , 2 . \\end{gather*}"}
-{"id": "4424.png", "formula": "\\begin{align*} f '' & = \\frac { a '' } { a } + 2 n \\frac { b '' } { b } \\\\ a '' & = 2 n \\left ( \\frac { a ^ 3 } { b ^ 4 } - \\frac { a ' b ' } { b } \\right ) + a ' f ' \\\\ b '' & = \\frac { 2 n + 2 } { b } - 2 \\frac { a ^ 2 } { b ^ 3 } - \\frac { a ' b ' } { a } - ( 2 n - 1 ) \\frac { ( b ' ) ^ 2 } { b } + b ' f ' , \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} b _ { i l } \\left ( Z \\right ) = 0 , \\mbox { f o r a l l $ i = 1 , \\dots , q $ a n d $ l = 1 , \\dots , p - q $ , } \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\beta _ Q ( q ^ i , p _ i , \\dot { q } ^ i , \\dot { p } _ i ) = ( q ^ i , p _ i , - \\dot { p } _ i , \\dot { q } ^ i ) . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\sqrt { \\lambda _ l ( t ) } = \\sqrt { \\gamma _ l } | t | \\left ( 1 + { \\mu _ l } ( { 2 \\gamma _ l } ) ^ { - 1 } t + \\ldots \\right ) , l = 1 , \\ldots , n , | t | \\le t _ { * * } . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\ , ( V _ 1 + V _ 2 + V _ 3 ) = & \\ , V _ 1 + \\ , V _ 2 + \\ , V _ 3 - \\ , V _ 1 \\cap V _ 2 - \\ , V _ 1 \\cap V _ 3 - \\ , V _ 2 \\cap V _ 3 \\\\ & + \\ , V _ 1 \\cap V _ 2 \\cap V _ 3 . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} g = \\tilde m _ l ( f ) : = \\frac { 1 } { W ( Q _ l ) } \\int _ { Q _ l } f W \\ ; \\ ; Q _ l , \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} 2 q \\theta _ q = z ( ( z \\theta ) _ { z z } ) = z ^ 2 \\theta _ { z z } + 2 z \\theta _ z ~ . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} v _ { i , j } ( t ) = \\ ; < x _ j ( t , \\cdot ) , \\Phi _ i ( \\cdot ) > \\mbox { a n d } w _ i ( t ) = \\ ; < y ( t , \\cdot ) , \\Phi _ i ( \\cdot ) > , \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\aligned & ( \\hat d _ { 2 } ^ { i + 1 } \\circ ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak H _ { a b } ^ { i + 1 i } - \\frak H _ { a b } ^ { i + 1 i } \\circ \\hat d _ { 1 } ^ { i } ) \\\\ & + ( \\hat d _ { 2 } ^ { i + 1 } \\circ \\frak H _ { a b } ^ { i i + 1 } - \\frak H _ { a b } ^ { i + 1 i } \\circ \\hat d _ { 1 } ^ { i } ) \\circ \\hat d _ { 1 } ^ { i } ) _ { \\alpha ' _ 2 \\alpha _ 1 } = 0 . \\endaligned \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} | | u ( t ) - u ( \\tau ) | | _ { \\tilde { L } ^ { 3 } } & \\leq | | e ^ { t A } u _ 0 - e ^ { \\tau A } u _ 0 | | _ { \\tilde { L } ^ { 3 } } + \\int _ { \\tau } ^ { t } | | e ^ { ( t - s ) A } f | | _ { \\tilde { L } ^ { 3 } } \\dd s + \\int _ { 0 } ^ { \\tau } | | e ^ { ( t - s ) A } f - e ^ { ( \\tau - s ) A } f | | _ { \\tilde { L } ^ { 3 } } \\dd s \\\\ & = : I + I I + I I I . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\frac { u _ { n + 1 } } { u _ n } & = A l ^ { \\alpha + k + 1 - \\gamma } \\frac { \\lim _ { m \\to \\infty } \\frac { m ^ { ( n + 3 ) k + ( n + 2 ) ( \\alpha + 1 - \\gamma ) + 1 } m ! } { ( ( n + 3 ) k + ( n + 2 ) ( \\alpha + 1 - \\gamma ) + 1 ) \\cdots ( ( n + 3 ) k + ( n + 2 ) ( \\alpha + 1 - \\gamma ) + m + 1 ) } } { \\lim _ { m \\to \\infty } \\frac { m ^ { ( n + 3 ) ( k + \\alpha ) + ( n + 2 ) ( 1 - \\gamma ) + 1 } m ! } { ( ( n + 3 ) ( k + \\alpha ) + ( n + 2 ) ( 1 - \\gamma ) + 1 ) \\cdots ( ( n + 3 ) ( k + \\alpha ) + ( n + 2 ) ( 1 - \\gamma ) + m + 1 ) } } \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} \\langle A \\chi _ 0 , p q \\rangle = \\langle A p ^ c , q \\rangle . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\varphi ( B _ { \\tau + \\cdot } - B _ { \\tau } ) ] = \\hat { \\mathbb { E } } [ \\varphi ( B _ { \\tau + \\cdot } - B _ { \\tau } ) ] = \\hat { \\mathbb { E } } [ \\varphi ( B _ \\cdot ) ] . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\sum ^ k _ { j = 0 } u _ { i , j } c _ j = - u _ { i , k } , i = 1 , \\ldots , k ; c _ k = 1 ; u _ { i , j } = \\big ( q , D _ { j + 1 , p + i } \\big ) , \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ t [ \\varphi ( X _ { t + t _ 1 } ^ { x } , X _ { t + t _ 2 } ^ { x } , \\cdots , X _ { t + t _ m } ^ { x } ) ] & = \\hat { \\mathbb { E } } _ t [ \\hat { \\mathbb { E } } _ { t + t _ { m - 1 } } [ \\varphi ( X _ { t + t _ 1 } ^ { x } , X _ { t + t _ 2 } ^ { x } , \\cdots , X _ { t + t _ m } ^ { x } ) ] ] \\\\ & = \\hat { \\mathbb { E } } _ t [ \\overline { \\varphi } _ { m - 1 } ( X _ { t + t _ 1 } ^ { x } , X _ { t + t _ 2 } ^ { x } , \\cdots , X _ { t + t _ { m - 1 } } ^ { x } ) ] , \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ m K _ { N , L _ i } } { k ! \\left ( ( N - k ) / 2 \\right ) ! } \\left | \\Delta \\left ( \\{ \\lambda _ l \\} _ { l = 1 } ^ k \\cup \\{ x _ j \\pm \\mathrm { i } y _ j \\} _ { j = 1 } ^ { ( N - k ) / 2 } \\right ) \\right | \\prod _ { j = 1 } ^ k w _ r ^ { ( m ) } ( \\lambda _ j ) \\prod _ { j = 1 } ^ { ( N - k ) / 2 } w _ c ^ { ( m ) } ( ( x _ j , y _ j ) ) \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ t [ \\eta X + Y ] & = \\hat { \\mathbb { E } } _ t [ \\sum _ { i = 1 } ^ n ( \\eta _ i X _ i + Y _ i ) I _ { A _ i } ] \\\\ & = \\sum _ { i = 1 } ^ n \\hat { \\mathbb { E } } _ t [ \\eta _ i X _ i + Y _ i ] I _ { A _ i } \\\\ & = \\sum _ { i = 1 } ^ n ( \\eta _ i ^ + \\hat { \\mathbb { E } } _ t [ X _ i ] + \\eta _ i ^ - \\hat { \\mathbb { E } } _ t [ - X _ i ] + Y _ i ) I _ { A _ i } \\\\ & = \\eta ^ + \\hat { \\mathbb { E } } _ t [ X ] + \\eta ^ - \\hat { \\mathbb { E } } _ t [ - X ] + Y . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\Omega _ l : = \\sum _ { i \\in D } \\omega _ i \\delta _ i \\quad \\quad \\quad \\Omega _ r : = \\Omega _ l ^ * : = \\sum _ { i \\in D } \\omega _ { \\bar i } \\delta _ i \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\sup _ { x \\in B ^ { \\R ^ N } _ { 8 / \\delta _ 1 } ( 0 ) } \\big ( | x - R e _ i | + | x + R e _ i | - 2 R \\big ) \\leq \\tilde \\eta \\qquad \\forall i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\langle E _ \\infty ^ k ( \\cdot , u ) , V \\rangle = \\int _ 0 ^ \\infty \\widetilde { V } _ 0 ( y ) y ^ { u - 1 } \\frac { d y } { y } = \\frac { \\Gamma ( s + \\frac { k + 1 } { 2 } - 1 ) } { ( 4 \\pi ) ^ { u + \\frac { k + 1 } { 2 } - 1 } } \\sum _ { n \\geq 1 } \\frac { r _ { 2 k + 1 } ( n ) r _ 1 ( n ) } { n ^ { u + \\frac { k + 1 } { 2 } - 1 } } \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\cos ( B ) = \\frac { y _ 1 ^ 2 - 2 } { 2 } , ~ ~ \\textrm { a n d } ~ ~ \\cos ( A ) = \\frac { y _ 1 } { 2 } . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} Y _ i = f ( \\boldsymbol { X } _ i ) + \\varepsilon _ i , i = 1 , \\dotsc , n , \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} \\tilde \\beta ( h ) = \\int _ S \\beta ( h ) \\ > d \\mu ( \\beta ) \\quad \\quad ( h \\in D ) \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} & ( p - 1 ) \\eta ^ 2 u _ k ^ { p - 2 } ( a \\nabla u _ k , \\nabla u _ k ) + 2 u _ k ^ { p - 1 } \\eta ( a \\nabla u _ k , \\nabla \\eta ) \\\\ & = \\frac { 4 ( p - 1 ) } { p ^ 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla ( \\eta u _ k ^ { p / 2 } ) ) \\\\ & \\ ; \\ ; \\ ; \\ ; - \\frac { 4 ( p - 2 ) } { p ^ 2 } u _ k ^ { p / 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla \\eta ) - \\frac { 4 } { p ^ 2 } u _ k ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} P & = \\sum _ t \\left [ \\binom { t } { 2 } \\binom { r - 2 } { t - s } + \\binom { r - t } { 2 } \\binom { r - 2 } { r - t } \\right ] \\\\ Q & = \\sum _ t \\left [ \\binom { t } { 2 } \\binom { r - 2 } { t } + \\binom { r - t } { 2 } \\binom { r - 2 } { r - t - 2 } \\right ] . \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\theta ( y _ { i } , 4 r ) - \\theta ( y _ i , 2 r ) < \\delta \\mbox { f o r e v e r y } i = 0 , \\dots , m - 2 , \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { P } \\left ( \\pi ( P _ { \\Lambda _ n } , P _ { \\Lambda _ \\infty } ) > \\varepsilon \\right ) = 0 . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} - \\Delta w _ j = \\lambda _ j w _ j , \\int _ { \\Omega } w _ j ^ 2 d x = 1 , \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} \\sup _ n \\sum ^ n _ { m = 1 } \\frac { w _ k ( n ) } { w _ l ( m ) } \\binom { n - 1 } { m - 1 } < \\infty . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\theta = \\left ( \\prod _ { j = 1 } ^ m \\prod _ { i = 0 } ^ { \\ell _ j } \\theta _ { j , i } \\right ) ^ { \\frac 1 { n + 1 } } \\ , . \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{gather*} \\widetilde G ( T , Y ) = Y ^ \\delta + ( \\xi _ 1 T ) Y ^ { \\delta - 1 } + \\cdots + \\xi _ \\delta T ^ \\delta , \\\\ \\widetilde F _ { i } ( T , Y ) = ( \\eta _ { i , 1 } T ^ { n _ i - \\delta + 1 } ) Y ^ { \\delta - 1 } + \\cdots + \\eta _ { i , \\delta } T ^ { n _ i } . \\end{gather*}"}
-{"id": "2489.png", "formula": "\\begin{align*} u _ 1 ( t , 0 ) = u _ 2 ( t , 0 ) = 0 \\ , , u _ 1 ( t , \\pi ) = g _ 1 ( t ) \\ , , u _ 2 ( t , \\pi ) = g _ 2 ( t ) t \\in ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{gather*} \\sum _ { n \\in \\N } \\sum _ { \\ell \\ge 0 } ( \\ell + 1 ) ^ 2 \\Big ( \\sum _ { k = 2 ^ { 2 ^ \\ell } } ^ { 2 ^ { 2 ^ { \\ell + 1 } } - 1 } | a _ k | u _ { n + k } \\Big ) ^ 2 \\le \\sum _ { n \\in \\N } \\Big ( \\sum _ { k \\ge 0 } ( \\log \\log ( k + 3 ) ) ^ 2 | a _ k | u _ { n + k } \\Big ) ^ 2 \\\\ \\le \\sum _ { n \\in \\N } \\Big ( \\sum _ { k \\ge 0 } ( \\log \\log ( n + k + 3 ) ) ^ 2 | a _ k | u _ { n + k } \\Big ) ^ 2 \\ , . \\end{gather*}"}
-{"id": "825.png", "formula": "\\begin{align*} \\| \\Pi _ { [ \\lambda , \\lambda + \\delta ] } \\| _ { L ^ 2 ( M ) \\to L ^ \\infty ( U ) } ^ 2 = \\sup _ { x \\in U } \\sum _ { \\lambda _ j \\in [ \\lambda , \\lambda + \\delta ] } | u _ { \\lambda _ j } ( x ) | ^ 2 . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\frac { ( 2 \\pi ) ^ { s + \\frac { k - 1 } { 2 } } } { \\Gamma ( s + \\frac { k - 1 } { 2 } ) } \\langle P _ h ( \\cdot , s ) , \\widetilde { V } \\rangle = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 2 k + 1 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s + \\frac { k - 1 } { 2 } } } . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} ( M _ r { V } ^ x ) _ S = \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - ( 1 - S ( x ) ) } { b } \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} [ x ] _ r = [ d _ r ] _ r r \\in ( \\{ 1 , \\ldots , t \\} \\setminus \\{ h ' , h \\} ) . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } k ^ * ( t ) e ^ { i u t } d t = \\cos ( u T ) K ^ * ( u ) \\ , , \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} w ( Y _ n ) = w ( Y _ { n - 1 } ) ( 1 + \\sum _ { i = 1 } ^ { n - 1 } c _ { i , n } x _ { n + i } ) , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} \\begin{cases} \\vect ( e ) = + 1 , & ; \\\\ \\vect ( e ) = - 1 , & \\end{cases} \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\Theta ^ { ( n , \\sigma ) } [ a ; c _ { 1 } , c _ { 2 } , \\ldots , c _ { n } ] ( X ) = \\Delta ^ { ( n , \\sigma ) } [ a ; c _ { 1 } , c _ { 2 } , \\ldots , c _ { n } ] ( X ) - X , \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} [ ( S _ 1 , a _ 1 \\xi ) , ( S _ 2 , a _ 2 \\xi ) ] = ( [ S _ 1 , S _ 2 ] , 0 ) \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} L _ q \\colon \\gg = \\mathrm T _ e G \\longrightarrow \\mathrm T _ q G . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} h = h _ { n } \\left ( { u , \\xi } \\right ) = 2 T \\varepsilon + \\frac { \\chi \\varepsilon } { u ^ { n - 1 } } - \\frac { 2 T } { u } \\frac { d \\varepsilon } { d \\xi } + \\frac { e ^ { u \\xi } \\chi } { u ^ { n - 1 } } , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} W _ { q } ( \\pi _ { k } ( \\mu , y _ { 0 : k } , \\cdot ) , \\pi _ { k } ( \\nu , y _ { 0 : k } , \\cdot ) ) \\leq \\exp \\left [ - \\sum _ { j = 1 } ^ { k } \\int _ { 0 } ^ { \\Delta } \\lambda ( j , y _ { j } , t ) \\mathrm { d } t \\right ] W _ { q } ( \\mu _ { 0 , k } , \\nu _ { 0 , k } ) , \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\mathcal { R } _ { t } ^ c ( \\alpha ) = \\left \\{ t _ { n p } \\ge \\frac { p ( p - 1 ) } { 2 ( n - 1 ) } ( \\sqrt { \\frac { n - 2 } { n + 1 } } - 1 ) + \\chi ^ 2 _ { \\alpha } ( p ( p - 1 ) / 2 ) \\sqrt { \\frac { ( n - 2 ) } { ( n - 1 ) ^ 2 ( n + 1 ) } } \\right \\} \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} p _ 3 ( X ) & = b ^ 2 ( c - a ) ^ 2 X ^ 4 - 4 b ^ 2 c ( c - a ) X ^ 3 + \\\\ & 2 b c ( 3 b c - a ^ 2 - a c - a b ) X ^ 2 - 4 b c ^ 2 ( b - a ) X + c ^ 2 ( b - a ) ^ 2 \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} K _ { h _ 1 } \\circ K _ { h _ 2 } ( x , A ) = \\int _ D K _ h ( x , A ) \\ > d ( \\delta _ { h _ 1 } * \\delta _ { h _ 2 } ) ( h ) . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} d \\sqrt { n } n ^ { 1 / 4 } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } \\left [ \\frac { 1 } { 2 } - \\frac { \\alpha ^ { * } } { 2 d } \\right ] } \\epsilon _ n ^ { 1 / 2 } \\lesssim \\underbrace { ( n / \\log { n } ) ^ { - \\frac { ( 2 \\alpha ^ { * } - 3 d ) ( \\alpha ^ { * } + d ) } { 8 d ( 2 \\alpha ^ { * } + d ) } } ( \\log { n } ) ^ { - 1 / 2 } } _ { \\tau _ n } \\sqrt { n \\log { n } } . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{gather*} t ^ { | \\lambda | } \\frac { P _ { \\lambda } \\big ( x / t , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } = \\frac { P _ { \\lambda } \\big ( x , t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , t ^ { N - 2 } , \\dots , 1 ; q , t \\big ) } = \\frac { P _ { \\lambda } \\big ( x , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } { P _ { \\lambda } \\big ( 1 , t , t ^ 2 , \\dots , t ^ { N - 1 } ; q , t \\big ) } , \\end{gather*}"}
-{"id": "8347.png", "formula": "\\begin{align*} L _ 2 = \\left ( \\begin{array} { c c c c c c } - 1 & 2 & - 1 & & & \\\\ & - 1 & 2 & - 1 & & \\\\ & & \\ddots & \\ddots & \\ddots & \\\\ & & & - 1 & 2 & - 1 \\\\ \\end{array} \\right ) \\in \\mathbb { R } ^ { ( n - 2 ) \\times n } , \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} P _ t f = T _ t f m U t > 0 . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} H ^ 0 ( \\P ( W ) , \\mathrm { S y m } ^ 2 \\Omega ^ 1 _ { \\P ( W ) } ( 2 ) ) = \\mathrm { S y m } ^ 2 \\Lambda ^ 2 W ^ { \\vee } \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\int _ { B _ { r } ( x ) } | D _ { L } u ( y ) | ^ { 2 } \\ , \\mathrm { d } y : = \\int _ { B _ { r } ( x ) } \\sum _ { i = 1 } ^ { k } | D _ { e _ i } u ( y ) | ^ { 2 } \\ , \\mathrm { d } y , \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} c _ { _ { \\rm E H } } ( K ) = c _ { _ { \\rm H Z } } ( K ) = \\min \\ , { \\cal L } ( \\partial K ) . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\partial _ t u _ t \\ , = \\ , - u _ t + g X u _ t \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} e _ { v _ 1 , v _ 2 } ^ { \\iota } = \\prod _ { i \\in I } \\prod _ { s = 1 } ^ { v _ 1 ^ i } \\prod _ { t = 1 } ^ { v _ 2 ^ i } \\vartheta ( z ^ i _ s - z ^ i _ { t + v _ 1 ^ i } + t _ 1 + t _ 2 ) , \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N } \\Vert f \\Vert _ { \\scriptstyle L ^ { p _ { i } } ( B _ { i } ) } \\leq c _ { 2 } \\Vert f \\Vert _ { \\scriptstyle L ^ { \\bar { p } ( \\cdot ) } ( \\Omega ) } . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} S I N R ^ { \\kappa } = \\frac { P _ { \\kappa } G _ 0 h _ 0 r _ 0 ^ { - \\alpha _ { \\kappa } ( r _ 0 ) } } { \\sigma ^ 2 + \\underbrace { \\sum _ { i \\in \\Phi _ c } P _ c G _ i h _ i r _ i ^ { - \\alpha _ { \\kappa } ( r _ i ) } } _ { I _ { c \\kappa } } + \\underbrace { \\sum _ { j \\in \\Phi _ d } P _ d G _ j h _ j r _ j ^ { - \\alpha _ { \\kappa } ( r _ j ) } } _ { I _ { d \\kappa } } } \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} v ^ { R ' R } ( A , B , C , E , F ) = \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\frac { d q ^ i ( t ) } { d t } & = \\dot { q } ^ i , \\frac { d \\dot { q } ^ i ( t ) } { d t } = \\xi ^ i , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} h ^ { \\mathrm { a n } } _ { ( D , g ) } = a _ 1 h ^ { \\mathrm { a n } } _ { ( E _ 1 , e _ 1 ) } + \\cdots + a _ r h ^ { \\mathrm { a n } } _ { ( E _ r , e _ r ) } \\geqslant a _ 1 C _ 1 + \\cdots + a _ r C _ r \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} | F _ x ( y ) | ~ \\leq ~ \\| F _ y \\| \\cdot \\| F _ x \\| ~ = ~ \\sqrt { F _ y ( y ) \\cdot F _ x ( x ) } \\ , . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} \\bar \\Lambda ^ 0 _ l = 0 l = 2 , \\ldots , k \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{gather*} \\operatorname { R e s } _ { z = n } { \\Gamma _ q ( - z ) } = \\frac { ( - 1 ) ^ n ( 1 - q ) ^ { n + 1 } } { \\ln { q } } \\frac { q ^ { { n + 1 \\choose 2 } } } { ( q ; q ) _ n } . \\end{gather*}"}
-{"id": "9874.png", "formula": "\\begin{align*} U _ { \\pi } ^ R = 2 \\Re \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi _ 0 ) \\\\ \\gamma \\le T } } \\frac { Z _ \\gamma } { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } \\quad V _ { \\pi } ^ R = 2 \\Re \\sum _ { \\substack { \\gamma \\in \\Gamma ^ S ( \\chi _ 0 ) \\\\ \\gamma > T } } \\frac { Z _ \\gamma } { \\sqrt { \\frac 1 4 + \\gamma ^ 2 } } . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} K ( \\Gamma _ s ) = \\mathbb { C } \\langle u _ s \\rangle ( \\Gamma _ s ) \\simeq \\mathbb { C } \\langle u _ s \\rangle ( \\tau ) = K ( \\tau ) . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{gather*} \\widetilde { e } _ { 1 } : = e _ 1 , \\widetilde { e } _ { 2 } : = e _ 2 , \\\\ \\widetilde { e } _ { 3 } : = x e _ 1 + y e _ 2 + e _ 3 \\widetilde { e } _ { 4 } : = x ' e _ 1 + y ' e _ 2 + e _ 4 . \\end{gather*}"}
-{"id": "6178.png", "formula": "\\begin{align*} u ( y , \\tau ) = v ( x , t ) \\ , ( 1 + \\tau ) ^ { - n / 2 s } , t = \\log ( 1 + \\tau ) . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} p ( \\Pi ) \\sqcup g ( \\Gamma ) = ( \\cup _ { i , j } p ( \\Pi _ { i } \\cap \\Gamma _ { j } ) ) \\sqcup ( \\cup _ { j } g ( \\Gamma _ { j } ) ) = \\cup _ { i , j } \\left ( p ( \\Pi _ { i } \\cap \\Gamma _ { j } ) \\sqcup g ( \\Gamma _ { j } ) \\right ) \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} f ^ * e _ 1 & = \\lambda e _ 1 , \\\\ f ^ * e _ 2 & = \\lambda e _ 2 + e _ 1 . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} R i c ( V ) : = \\sum _ { i = 1 } ^ { n - 1 } K ( V , e _ i ) , \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\tilde { { G } } _ { n , s } \\left ( p \\right ) = \\sum \\limits _ { k = s } ^ { n - 1 } { \\tilde { { F } } _ { k } \\left ( p \\right ) \\tilde { { F } } _ { s + n - k - 1 } \\left ( p \\right ) , } \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} t _ { i , j } ( \\alpha ' + \\alpha '' ) = \\frac { t _ { i , j } ( \\alpha ' ) { \\sigma } _ { i } ( \\alpha ' ) + t _ { i , j } ( \\alpha '' ) { \\sigma } _ { i } ( \\alpha '' ) } { { \\sigma } _ { i } ( \\alpha ' ) + { \\sigma } _ { i } ( \\alpha '' ) } , \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\vec { f } = [ f ( x _ 1 , v _ 1 ) \\ , , f ( x _ 2 , v _ 1 ) \\ , , \\cdots \\ , , f ( x _ n , v _ 1 ) \\ , , f ( x _ 1 , v _ 2 ) \\ , , \\cdots \\ , , f ( x _ { N _ x } , v _ 2 ) \\ , , \\cdots \\ , , \\cdots \\ , , f ( x _ { N _ x } , v _ { N _ v } ) ] ^ t \\ , , \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} | \\mu _ i | \\leq C \\varepsilon , i = 1 , \\ldots , k . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} | E _ { n + 1 } ' | + | E _ { n + 1 } '' | = 3 | E _ n ' | . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} r _ { t - 1 } = r _ { t - 1 } ^ { N , h } = \\int _ { - \\infty } ^ \\infty T _ t ( \\lambda ) \\ , d \\rho ^ { N , h } ( \\lambda ) , t \\in 1 , \\ldots , 2 N . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\left ( \\forall i \\in I \\right ) \\left ( \\forall j \\in I \\right ) i \\neq j \\implies \\alpha _ { i } \\alpha _ { j } = 0 \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} ( y , \\frak s ^ { \\epsilon } _ { 2 } ( y ) ) = \\tilde \\varphi _ { 2 1 } ( y , \\frak s ^ { \\epsilon } _ { 1 } ( \\pi _ { 1 2 } ( y ) ) ) ) , \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\sum _ { A , B } h _ { B A } \\overline { H _ { A B } } & = \\sum _ { A , B } h _ { A B } \\overline { H _ { A B } } + \\sum _ { A , B } ( h _ { B A } - h _ { A B } ) \\overline { H _ { A B } } \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} B i a s ( \\tilde { \\theta } ^ \\alpha _ 1 ; F ) = \\phi ( \\Phi ^ { - 1 } ( \\alpha ) ) \\Phi ^ { - 1 } ( \\alpha ) \\left ( \\sqrt { \\textrm { a s y . e f f . } \\left ( \\hat { \\theta } ^ M _ 1 , F \\right ) ^ { - 1 } } - 1 \\right ) \\ , , \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} | F _ j ( x _ j ) | & \\ll H ^ { 2 \\delta } ( \\Psi ( H ) \\ , H ^ { n - m } ) ^ { - \\frac 1 { m - 1 } } \\\\ [ 1 e x ] & \\le H ^ { 2 \\delta } ( H ^ { - ( \\ell _ 0 + 1 ) \\delta } H ^ { n - m } ) ^ { - \\frac 1 { m - 1 } } \\\\ [ 1 e x ] & = H ^ { - \\frac { n - m - \\ell _ 0 \\delta - \\delta ( 2 m - 1 ) } { m - 1 } } \\qquad ( 2 \\le j \\le m ) \\ , . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} I _ \\lambda ( \\zeta ' , \\mu ' ) = \\bar J _ \\lambda ( V + \\phi ) \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} z \\cdot F ( z , \\alpha ) = - ( z - \\underline { 0 } ) \\cdot ( \\triangledown U ( z ) - \\triangledown U ( \\underline { 0 } ) ) + z \\cdot \\triangledown U ( \\underline { 0 } ) + z \\cdot b ( z , \\alpha ) \\leq C \\left ( \\| z \\| ^ 2 + 1 \\right ) , \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} 0 = \\lambda _ 0 < \\lambda _ 1 \\leq \\lambda _ 2 \\leq \\cdots \\nearrow \\infty . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} T _ m ( b ) T _ m ( a _ 1 , a _ 2 , \\ldots , a _ n ) = T _ m ( b , a _ 1 , a _ 2 , \\ldots , a _ n ) + \\sum _ { i = 1 } ^ n T _ m ( a _ 1 , \\ldots a _ { i - 1 } , b a _ i , a _ { i + 1 } , \\ldots , a _ n ) \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} [ W , \\lambda ] ( x , y ) = \\begin{cases} W \\left ( \\frac { x } { 1 - \\lambda } , \\frac { y } { 1 - \\lambda } \\right ) & 0 \\le x \\le 1 - \\lambda 0 \\le y \\le 1 - \\lambda \\ , , \\\\ 1 & 1 - \\lambda < x \\le 1 1 - \\lambda < y \\le 1 \\ , . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\Lambda = | \\sigma _ 1 | ^ { 1 / 2 } P ( \\varepsilon , \\zeta ) \\bar { \\Lambda } . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} \\frac { d } { d t } | _ { t = 0 } \\int _ M S ( J ( t ) , f ) h f ^ { - 2 m - 1 } \\omega ^ m = 2 \\Re \\ \\int _ M ( \\nabla _ i \\nabla _ j h ) \\ v ^ { i j } \\ f ^ { - 2 m + 1 } \\ \\omega ^ m \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} n = \\prod _ { \\substack { q \\le V \\\\ q \\equiv 1 \\mod p } } q = e ^ { \\theta ( V , p , 1 ) } , \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} \\tilde { \\psi } ( ( x _ { i j } ) ^ { t } ) & = \\frac { 1 } { 2 } \\left ( \\psi ( ( x _ { j i } ) ) + \\overline { \\psi ( ( x _ { j i } ) ) } \\right ) = \\frac { 1 } { 2 } \\left ( \\sum _ { i , j = 1 } ^ { n } x _ { j i } \\psi ( E _ { i j } ) + \\sum _ { i , j = 1 } ^ { n } x _ { j i } \\overline { \\psi ( E _ { i j } ) } \\right ) \\\\ & = \\frac { 1 } { 2 } \\left ( \\sum _ { i = 1 } ^ { n } x _ { i i } ( \\psi ( E _ { i i } ) + \\psi ( E _ { i i } ) ^ { t } ) + \\sum _ { 1 \\leq i \\neq j \\leq n } x _ { j i } ( \\psi ( E _ { i j } ) + \\psi ( E _ { j i } ) ) \\right ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} y _ i - x _ i y _ n \\quad , i = 2 , \\dots , n - 1 \\quad . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\gamma ( S ) _ i = \\begin{cases} s _ j - j + 1 & i = s _ j \\\\ 0 & i \\notin S . \\end{cases} \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\Pi \\triangleq \\begin{pmatrix} \\pi ^ + ( P _ Y , 0 ^ + ) - \\pi ^ + ( P _ Y , 0 ^ - ) & \\pi ^ - ( P _ Y , 0 ^ + ) - \\pi ^ - ( P _ Y , 0 ^ - ) \\\\ \\pi ^ + ( - P _ Y , 0 ^ + ) - \\pi ^ + ( - P _ Y , 0 ^ - ) & \\pi ^ - ( - P _ Y , 0 ^ + ) - \\pi ^ - ( - P _ Y , 0 ^ - ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} d \\Omega ( J Z , J X , J Y ) = d \\Omega ( J Z , X , Y ) + d \\Omega ( Z , J X , Y ) + d \\Omega ( Z , X , J Y ) \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} M _ { d _ k } = \\{ ( \\varepsilon _ 1 , \\ldots , \\varepsilon _ { d _ k } ) \\in \\S _ { \\beta } ^ { d _ k } : \\ \\varepsilon _ 1 = 1 \\ { \\rm a n d \\ } I _ { d _ k } ( \\varepsilon _ 1 , \\ldots , \\varepsilon _ { d _ k } ) \\ { \\rm i s \\ f u l l } \\} . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} | \\zeta - \\zeta _ 0 + z | & = \\big | | \\zeta - \\zeta _ 0 | \\mathrm { e } ^ { \\mathrm { i } \\arg ( \\zeta - \\zeta _ 0 ) } - | z | \\mathrm { e } ^ { \\mathrm { i } \\theta } \\big | \\\\ & = \\big | | \\zeta - \\zeta _ 0 | - | z | \\mathrm { e } ^ { \\mathrm { i } \\theta - \\mathrm { i } \\arg ( \\zeta - \\zeta _ 0 ) } \\big | \\\\ & \\geqslant | z | \\cdot | \\sin ( \\theta - \\arg ( \\zeta - \\zeta _ 0 ) ) | \\\\ & \\geqslant | z | \\sin \\phi _ 1 . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\delta = c \\log ( n / m ) / \\log n \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\nu _ e ( \\Omega ^ \\sigma _ k ) = \\frac { ( q - 1 ) } { q } \\begin{cases} q ^ { - \\abs { k - i } } & k - i \\leq 0 , \\\\ - q ^ { - ( k - i ) + 1 } & k - i > 0 , \\end{cases} \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\sum _ { \\substack { k = 0 \\\\ k \\equiv N \\mod 2 } } ^ N \\zeta ^ k p _ { N , k } ^ { P _ m } , \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\dim H ^ 2 ( Y _ \\Delta ) = \\dim H _ c ^ 4 ( Y _ \\Delta ) = \\ell ( \\Delta ^ \\circ ) - 4 - \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) + \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} Q _ 1 [ u , v ] = \\sum \\limits _ N B [ u _ { \\ll N } , \\partial _ x ^ { - 1 } v _ N ] , Q _ 2 [ u , v ] = \\sum \\limits _ { N _ 1 \\sim N _ 2 } \\partial _ x ^ { - 1 } B [ u _ { N _ 1 } , v _ { N _ 2 } ] . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\Gamma _ { 1 2 } = \\frac { 2 \\sqrt { 2 } } { J _ 0 } - \\frac { 1 } { 2 } ( \\Gamma _ { 1 1 } + \\Gamma _ { 2 2 } ) . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} & ( M _ r { V } ^ x ) _ S = \\\\ & = \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - ( 1 - S ( x ) ) } { b } ( r - 2 | T | ) \\\\ & = \\sum _ { a , b } \\left [ \\binom { a } { 2 } + \\binom { b } { 2 } \\right ] \\binom { | S | - S ( x ) } { a - S ( x ) } \\binom { r - | S | - ( 1 - S ( x ) ) } { b } ( 2 | S | - r + 2 b - 2 a ) \\\\ & = P _ 1 ( x ) + P _ 2 ( x ) + P _ 3 ( x ) + P _ 4 ( x ) + P _ 5 ( x ) + P _ 6 ( x ) , \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} A \\cap I _ { u , n } = \\bigcup _ { j = 1 } ^ { q + 1 } ( A \\cap I _ { u , n } ^ { ( j ) } ) \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} f _ { m , p } ( x ) & = x f _ { m - 1 , p - 1 } ( x ) + x \\sum _ { i = p } ^ { m - 1 } f _ { m , i } ( x ) \\\\ & + \\frac { x ^ { p + 3 } } { ( 1 - x ) ^ 2 } \\left ( \\sum _ { i = 0 } ^ { m - 1 - p } x ^ i F _ { m - 1 - p - i } ( x ; 0 ) + \\sum _ { i = 0 } ^ { m - 2 - p } x ^ i F _ { m - 1 - p - i } ( x ; 1 ) \\right ) \\\\ & + \\frac { x ^ { p + 3 } } { ( 1 - x ) ^ 2 } \\left ( \\sum _ { i = 0 } ^ { m - 2 - p } \\frac { x ^ i } { 1 - x } F _ { m - 1 - p - i } \\left ( x ; \\frac { 1 } { 1 - x } \\right ) - \\sum _ { i = 0 } ^ { m - 2 - p } x ^ i F _ { m - 1 - p - i } ( x ; 1 ) \\right ) . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} A = \\begin{pmatrix} A _ 1 & A _ 2 \\\\ A _ 3 & A _ 4 \\end{pmatrix} , A _ i \\in O P S ^ m , i = 1 , \\ldots , 4 \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U , X ) , \\pi _ { \\ast } Y ) & = g _ { 1 } ( \\mathcal { T } _ { U } \\phi B X , Y ) + g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , \\omega \\mathcal { B } X ) , \\pi _ * Y ) - g _ { 1 } ( \\mathcal { T } _ { U } \\mathcal { C } X , \\mathcal { B } Y ) \\\\ & + g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , \\mathcal { C } X ) , \\pi _ * \\mathcal { C } Y ) + \\eta ( X ) g _ { 1 } ( Q U , \\varphi Y ) \\\\ & - U \\eta ( X ) \\eta ( Y ) - g _ { 1 } ( X , \\omega U ) \\eta ( Y ) . \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} s _ 1 + \\sigma _ 0 \\leq s _ 0 + \\mu _ 0 \\ , , \\sigma _ 0 : = \\sigma _ 0 ( \\tau , \\nu ) > 0 \\ , . \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\overline { \\partial } g ^ { - 1 } & = Q ( \\overline { \\partial } \\eta ) \\overline { \\partial } { } ^ 2 \\eta \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} \\mathrm { H i l b } ( R _ n ; q ) = ( 1 + q ) ( 1 + q + q ^ 2 ) \\cdots ( 1 + q + \\cdots + q ^ { n - 1 } ) = [ n ] ! _ q , \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} v _ { i } ^ { \\varepsilon } \\left ( 0 , x \\right ) = v _ { i } ^ { \\varepsilon , 0 } \\left ( x \\right ) \\quad \\mbox { f o r } \\ ; x \\in \\Gamma ^ { \\varepsilon } . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} v _ * ( x , 0 ) = v ^ * ( x , 0 ) = \\min ( 0 , - ( d + \\alpha ) \\log ( | x | ) ) . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\psi _ { \\lambda ( N ) / ( \\lambda ( N ) _ 1 , \\dots , \\lambda ( N ) _ { N - 1 } ) } ( q , t ) = 1 , \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} T : = R ^ { \\ast [ - 1 ] } B \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\sup _ { \\boldsymbol { \\alpha } \\in \\mathbb { A } _ { \\boldsymbol { r } } } \\sup _ { f _ 0 \\in \\mathcal { B } ^ { \\boldsymbol { \\alpha } } _ { \\infty , \\infty } ( R ) } \\mathrm { E } _ 0 \\| \\mathrm { E } ( D ^ { \\boldsymbol { r } } f | \\boldsymbol { Y } ) - D ^ { \\boldsymbol { r } } f _ 0 \\| _ \\infty \\lesssim ( n / \\log { n } ) ^ { - \\frac { \\alpha ^ { * } \\{ 1 - \\sum _ { l = 1 } ^ d ( r _ l / \\alpha _ l ) \\} } { 2 \\alpha ^ { * } + d } } . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\limsup \\limits _ { y \\uparrow 1 } \\limsup \\limits _ { x \\rightarrow \\infty } \\mathcal { J } _ 1 & \\leqslant ( 1 + \\varepsilon ) \\max \\limits _ { 1 \\leqslant k \\leqslant K \\atop k \\notin \\mathcal { K } } \\biggl \\{ \\limsup \\limits _ { y \\uparrow 1 } \\limsup \\limits _ { x \\rightarrow \\infty } \\frac { \\overline { F } _ { \\xi _ { k } } ( \\mathit { x y } ) } { \\overline { F } _ { \\xi _ { k } } ( x ) } \\biggr \\} \\\\ & = 1 + \\varepsilon , \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} ( g h ) ( \\sqrt { \\alpha _ i } ) & = g ( h ( \\sqrt { \\alpha _ i } ) ) \\\\ & = g ( x _ { h , \\sigma _ h ( i ) } \\sqrt { \\alpha _ { \\sigma _ h ( i ) } } ) \\\\ & = x _ { h , \\sigma _ h ( i ) } g ( \\sqrt { \\alpha _ { \\sigma _ h ( i ) } } ) \\\\ & = x _ { h , \\sigma _ h ( i ) } x _ { g , \\sigma _ g ( \\sigma _ { h } ( i ) ) } \\sqrt { \\alpha _ { \\sigma _ g ( \\sigma _ { h } ( i ) ) } } \\\\ & = x _ { h , \\sigma _ h ( i ) } x _ { g , \\sigma _ { g h } ( i ) } \\sqrt { \\alpha _ { \\sigma _ { g h } ( i ) } } , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} W _ 1 & = \\tilde { W } _ 1 / \\{ c \\in \\tilde { W } _ 1 : \\langle c , \\tilde { W } _ 2 \\rangle = 0 \\} \\\\ W _ 2 & = \\tilde { W } _ 2 / \\{ c \\in \\tilde { W } _ 2 : \\langle c , \\tilde { W } _ 1 \\rangle = 0 \\} , \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} E [ G ( X _ { \\rho _ 3 , m } ) ] & = E \\left [ \\left . G ( X _ { \\rho _ 3 , m } ) \\right | \\zeta = 1 \\right ] P ( \\zeta = 1 ) + E \\left [ \\left . G ( X _ { \\rho _ 3 , m } ) \\right | \\zeta = 0 \\right ] P ( \\zeta = 0 ) \\\\ & = \\theta E [ G ( X _ { \\rho _ 1 , m } ) ] + ( 1 - \\theta ) E [ G ( X _ { \\rho _ 2 , m } ) ] \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} 2 = p ^ { \\ast } \\left ( 0 , P \\right ) \\overset { } { \\longrightarrow } p ^ { \\ast } \\left ( 1 - \\frac { 1 } { C _ { P } } , P \\right ) \\overset { 1 - \\frac { 1 } { C _ { P } } \\leq t \\leq 1 } { = } p ^ { \\ast } \\left ( t , P \\right ) = P \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} \\beta ( h ) = \\int _ { \\tilde S } \\tilde \\beta ( h ) \\ > d \\mu ( \\tilde \\beta ) \\quad \\quad ( h \\in D ) \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\chi = ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { l = 1 } ^ { N } z _ { i l } \\overline { a _ { k u } ^ { i l } \\left ( Z \\right ) } + \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\overline { z } _ { j l } a _ { u k } ^ { j l } \\left ( Z \\right ) = 0 , \\mbox { f o r a l l $ i , j , k , u = 1 , \\dots , q $ w i t h $ k \\neq u $ . } \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} f ( x ^ * ( p ) , p ) = 0 \\mbox { f o r a l l } \\ ; \\ ; p \\in \\Pi . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} G _ 1 = \\varphi _ 1 + \\varphi _ 2 \\partial , \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } L ( \\phi ) & = & - N ( \\phi ) - E + \\sum _ { i , j } c _ { i j } \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } & \\Omega _ \\varepsilon , \\\\ \\phi & = & 0 & \\partial \\Omega _ \\varepsilon , \\\\ \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , \\phi & = & 0 & i , j , \\end{array} \\right . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} L u = \\sum _ { i , j = 1 } ^ n D _ i ( a ^ { i j } ( x ) D _ j u + b ^ i ( x ) u ) + \\sum _ { i = 1 } ^ n c ^ i ( x ) D _ i u + d ( x ) u , \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} \\nu \\left ( \\sum _ { d _ 1 n _ 1 + \\dots + d _ r n _ r = n } \\alpha _ { ( n _ 1 , \\dots , n _ r ) } f _ 1 ^ { n _ 1 } \\cdots f _ r ^ { n _ r } \\right ) \\geqslant \\min \\left \\{ \\sum _ { i = 1 } ^ r n _ i \\nu ( f _ i ) \\right \\} \\geqslant a n . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} { \\mathcal N } ( { \\alpha _ 1 } , { \\alpha _ 2 } ; P ) = \\mu ( \\alpha _ 2 ) - \\mu ( \\alpha _ 1 ) + \\dim R ^ 2 _ { \\alpha _ 2 } + \\dim P . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\varGamma _ { 1 } = \\left \\{ \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\left | \\omega _ { 1 } T _ { 0 } = \\left ( 2 k _ { 1 } + 1 \\right ) \\pi , \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) = \\left ( 2 k _ { 2 } + 1 \\right ) \\pi , k _ { 1 } \\in \\mathbb { Z } , k _ { 2 } \\in \\mathbb { Z } \\right . \\right \\} , \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} e ^ { ( t - s ) A } f - e ^ { ( \\tau - s ) A } f = ( e ^ { ( t - \\tau ) A } - 1 ) e ^ { \\frac { ( \\tau - s ) } { 2 } A } e ^ { \\frac { ( \\tau - s ) } { 2 } A } f , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = - g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , X ) , \\pi _ * \\omega \\phi V ) - g _ { 1 } ( \\omega V , \\mathcal { T } _ { U } \\mathcal { B } X ) \\\\ & + g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , \\mathcal { C } X ) , \\pi _ * \\omega V ) - g _ { 1 } ( V , \\phi U ) \\eta ( X ) \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} m _ 1 = 2 ^ { \\mu } m _ 1 ' , \\ , m _ 2 = \\delta _ 2 2 ^ { \\alpha } m _ 2 ' \\ , m _ 3 = \\delta _ 3 2 ^ { \\beta } m _ 3 ' , \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\Delta z + 5 \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z = 0 \\quad \\R ^ 3 . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\operatorname * { O p } \\nolimits ^ { \\mathrm { W } } ( a ) \\psi ( x ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\mathbb { R } ^ { 2 n } } a _ { \\sigma } ( z _ { 0 } ) \\widehat { T } ( z _ { 0 } ) \\psi ( x ) d ^ { 2 n } z _ { 0 } \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} \\left ( Q ^ { * } _ { n ( k , m ) } ( i ) , 0 \\le i \\le k \\right ) \\stackrel { ( d ) } { = } \\left ( Y _ m \\left ( \\log \\left ( \\frac { 1 - F _ k } { 1 - F _ i } \\right ) \\right ) , 0 \\le i \\le k \\right ) , \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\tau & = \\frac { 1 } { 2 } \\begin{pmatrix} \\sqrt { - 3 } \\eta ^ { - 1 } & - 1 \\\\ - 1 & \\sqrt { - 3 } \\eta \\\\ \\end{pmatrix} , \\\\ \\zeta & = \\frac { 1 } { 2 } \\left ( ( \\dfrac { z _ 1 } { 1 - \\omega } - \\dfrac { z _ 2 } { 1 - \\omega ^ 2 } ) \\eta ^ { - 1 } , \\dfrac { z _ 1 } { 1 - \\omega } + \\dfrac { z _ 2 } { 1 - \\omega ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} x _ m - \\mathcal { K } _ m ( x _ m ) = f . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\sum _ { q _ 1 \\ge 2 } \\sum _ { q _ 2 \\ge 2 } \\frac { \\Lambda ( q _ 1 ) \\Lambda ( q _ 2 ) } { \\phi ( q _ 1 ) \\phi ( q _ 2 ) \\phi ( [ q _ 1 , q _ 2 ] ) } = 4 A _ 0 ^ 2 + B , \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} f ( x ) = \\int _ { \\mathbb { R } ^ { 4 n } } K ( x , y ) f ( y ) e ^ { - 2 \\varphi } d V \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\big ( q , D _ { m , n } \\big ) = - \\sum ^ \\mu _ { s = 1 } \\frac { ( q , v _ s ) } { \\psi _ { m , n } ( z _ s ) } . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} \\varepsilon _ { n , 1 } \\left ( { u , \\xi } \\right ) = \\frac { 1 } { u } \\int _ { \\alpha _ { 1 } } ^ { \\xi } { \\mathsf { K } \\left ( { \\xi , t } \\right ) h _ { n , 1 } ^ { + } \\left ( { u , t } \\right ) d t , } \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} & ( p - 1 ) \\eta ^ 2 u ^ { p - 2 } ( a \\nabla u , \\nabla u ) + 2 u ^ { p - 1 } \\eta ( a \\nabla u , \\nabla \\eta ) \\\\ & = \\frac { 4 ( p - 1 ) } { p ^ 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla ( \\eta u ^ { p / 2 } ) ) \\\\ & \\ ; \\ ; \\ ; \\ ; - \\frac { 4 ( p - 2 ) } { p ^ 2 } u ^ { p / 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla \\eta ) - \\frac { 4 } { p ^ 2 } u ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\sqrt { g _ n ' } \\sqrt { g _ { n + 1 } ' } \\\\ & = \\frac { 1 } { 4 } \\sqrt { 2 - \\frac { 1 } { 2 ^ n } } \\sqrt { 2 - \\frac { 1 } { 2 ^ { n + 1 } } } + \\frac { 1 } { 4 } \\sqrt { 2 - \\frac { 1 } { 2 ^ n } } \\sqrt { \\frac { 1 } { 2 ^ { n + 1 } } } + \\frac { 1 } { 4 } \\sqrt { \\frac { 1 } { 2 ^ n } } \\sqrt { 2 - \\frac { 1 } { 2 ^ { n + 1 } } } + \\frac { 1 } { 4 } \\sqrt { \\frac { 1 } { 2 ^ n } } \\sqrt { \\frac { 1 } { 2 ^ { n + 1 } } } \\to \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\mathfrak { L } ( u ) : = \\frac { 1 } { p } \\int \\limits _ { \\mathbb { G } } | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ( x ) | ^ { p } d x + \\frac { 1 } { p } \\int \\limits _ { \\mathbb { G } } | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ( x ) | ^ { p } d x - \\frac { 1 } { q } \\int \\limits _ { \\mathbb { G } } | u ( x ) | ^ { q } d x \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} d _ \\tau ( \\pi _ 1 , \\pi _ 2 ) = d _ \\tau ( \\pi _ 1 \\vert _ A , \\pi _ 2 \\vert _ A ) + d _ \\tau ( \\pi _ 1 \\vert _ B , \\pi _ 2 \\vert _ B ) + d _ \\tau ( f ( \\pi _ 1 ) , f ( \\pi _ 2 ) ) , \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} { \\mathcal E } L ( \\phi ( { \\bf A } ^ { * } ) , { \\bf I } ) & = { \\mathcal E } [ \\mbox { t r } { \\bf D } { \\bf A } ^ { * } - \\mbox { l o g } \\mbox { d e t } { \\bf D } { \\bf A } ^ { * } - p ] \\\\ & = \\sum _ { i = 1 } ^ { p } ( n - i + 1 ) d _ { i i } - \\sum _ { i = 1 } ^ { p } \\mbox { l o g } d _ { i i } - \\sum _ { i = 1 } ^ { p } { \\mathcal E } [ \\mbox { l o g } { \\chi } ^ { 2 } _ { n - i + 1 } ] - p . \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} O \\left ( \\frac { \\left ( \\sqrt { n } \\log n \\right ) ^ { 2 } h n } { n ^ { 2 } } \\right ) = O \\left ( h \\log ^ { 2 } n \\right ) = o \\left ( \\log ^ { 3 } n \\right ) . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\kappa _ { X } ( \\zeta ) = \\int _ { 0 } ^ { \\infty } \\kappa _ { L } ( e ^ { - s } \\zeta ) d s . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} & S ( \\{ a _ { ( 1 ) } , b _ { ( 1 ) } \\} ) a _ { ( 2 ) } b _ { ( 2 ) } \\\\ = & - S ( a _ { ( 1 ) } b _ { ( 1 ) } ) \\{ a _ { ( 2 ) } , b _ { ( 2 ) } \\} = - ( - 1 ) ^ { | a _ { ( 1 ) } | | b _ { ( 1 ) } | } S ( b _ { ( 1 ) } ) S ( a _ { ( 1 ) } ) \\{ a _ { ( 2 ) } , b _ { ( 2 ) } \\} \\\\ = & ( - 1 ) ^ { | a _ { ( 1 ) } | | b _ { ( 1 ) } | + | b _ { ( 2 ) } | | a _ { ( 2 ) } | } S ( b _ { ( 1 ) } ) \\{ S ( a _ { ( 1 ) } ) , b _ { ( 2 ) } \\} a _ { ( 2 ) } = - ( - 1 ) ^ { | b _ { ( 2 ) } | | a _ { ( 2 ) } | } \\{ S ( a _ { ( 1 ) } ) , S ( b _ { ( 1 ) } ) \\} b _ { ( 2 ) } a _ { ( 2 ) } \\\\ = & - \\{ S ( a _ { ( 1 ) } ) , S ( b _ { ( 1 ) } ) \\} a _ { ( 2 ) } b _ { ( 2 ) } . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} m ( T , k - 1 ) = & m ( T - e , k - 1 ) + m ( T - u ' - v ' , k - 2 ) \\\\ = & m ( T ( a , y - 1 , c - 1 ) \\cup P _ { x - a - 1 } , k - 1 ) + m ( T ( a - 1 , y - 1 , c - 1 ) \\cup P _ { x - a - 2 } , k - 2 ) . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{gather*} \\tilde { Y } ^ { l + \\alpha , l + \\beta } = Z ( I , J \\setminus ( j _ \\alpha , j _ \\beta ) , ( j _ \\alpha , j _ \\beta ) \\sqcup K ) \\quad , \\\\ \\tilde { Y } ^ { l + 1 , l + 2 , l + 3 , l + 4 } = Z ( I , J \\setminus ( j _ 1 , j _ 2 , j _ 3 , j _ 4 ) , ( j _ 1 , j _ 2 , j _ 3 , j _ 4 ) \\sqcup K ) , \\end{gather*}"}
-{"id": "5364.png", "formula": "\\begin{align*} d ^ { 2 } W / d \\xi ^ { 2 } = \\left \\{ { u ^ { 2 } + u \\phi \\left ( \\xi \\right ) + \\psi \\left ( u , \\xi \\right ) } \\right \\} W , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\left ( 1 - \\lambda ^ T \\psi _ i + ( \\lambda ^ T \\psi _ i ) ^ 2 - ( \\lambda ^ T \\psi _ i ) ^ 3 + O _ p \\left ( ( \\lambda ^ T \\psi _ i ) ^ 4 \\right ) \\right ) = 0 \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} T _ { j , m } & ( f , g ) ( x ) = \\\\ & \\int f * \\Phi _ { a j + m } \\left ( x - \\frac { t ^ a + \\epsilon _ P ( t ) } { 2 ^ { a j } } \\right ) g * \\Phi _ { b j + m } \\left ( x - \\frac { t ^ b + \\epsilon _ Q ( t ) } { 2 ^ { b j } } \\right ) \\rho ( t ) \\ , d t , \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} S _ f ( X ) : = \\sum _ { n \\leq X } a ( n ) , S _ g ( X ) : = \\sum _ { n \\leq X } b ( n ) . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } z _ { i , j } ( y ) & = & D _ { \\zeta _ i ^ { \\prime } } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) \\cdot \\textrm { e } _ j j = 1 , 2 , 3 \\\\ z _ { i , 4 } ( y ) & = & \\frac { \\partial \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } } { \\partial \\ , \\mu _ i ^ { \\prime } } ( y ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { \\lambda \\vdash n } \\chi _ { ( n ) } ^ { \\lambda } \\chi ^ { \\lambda } _ { \\nu } = \\delta _ { \\nu , ( n ) } \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} F _ { 2 } \\left ( Z , W \\right ) = 0 . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} D \\sqrt { - 1 } \\partial \\bar { \\partial } \\dot { \\phi } = D { \\rm R i c } ( \\omega _ { \\phi } ) + D \\widetilde { \\eta } + D ( \\nabla _ { \\phi } X ^ { \\flat } ) , \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x _ N & = - v _ A - b _ M v _ M - b _ E v _ E = - a _ M b _ M v _ M - a _ E b _ E v _ E \\\\ \\dot x _ M & = v _ M \\\\ \\dot x _ E & = v _ E , \\end{aligned} \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{gather*} \\left \\{ \\frac { \\partial ^ 2 } { \\partial \\beta ^ 2 } + ( n - 1 ) \\tanh \\beta \\frac { \\partial } { \\partial \\beta } + \\rho ^ 2 + \\frac { ( n - 1 ) ^ 2 } { 4 } + \\frac { l _ 1 ( l _ 1 + n - 2 ) } { \\cosh ^ 2 \\beta } \\right \\} V ( \\beta ) = 0 , \\end{gather*}"}
-{"id": "2669.png", "formula": "\\begin{align*} A = \\inf \\{ \\eta > 0 ; \\int _ { 0 } ^ { \\infty } e ^ { - \\eta t } h ( t ) d t < \\infty \\} \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} F ( \\xi \\cdot g ) = F ( N _ \\xi ( g ) ) = N _ \\xi ( F ( g ) ) = \\xi \\cdot F ( g ) \\xi \\in L _ \\infty [ 0 , \\alpha ) , g \\in L _ 2 [ 0 , \\alpha ) . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} J _ d = x ^ { d + 1 } + \\sum _ { j = 1 } ^ d x ^ j J _ { d + 1 - j } + \\sum _ { j = 1 } ^ d \\sum _ { k = 0 } ^ { d - j } x ^ { k + j } ( J _ { d + 1 - k - j } - x J _ { d - k - j ) } ) \\ , , \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} T _ m ( a _ 1 , a _ 2 , \\ldots , a _ n ) = \\sum _ { \\sigma \\in S _ n } \\sum _ { \\lambda \\vdash n } \\frac { \\varepsilon _ \\lambda } { z _ \\lambda } T _ { m , \\lambda } ( a _ { \\sigma ( 1 ) } , a _ { \\sigma ( 2 ) } , \\ldots , a _ { \\sigma ( n ) } ) \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\rho _ t ( l _ i ) = A _ i ( t ) \\exp ( a _ i ( t ) ) A _ i ( t ) ^ { - 1 } \\qquad \\textrm { a n d } \\rho _ t ( m _ i ) = A _ i ( t ) \\exp ( b _ i ( t ) ) A _ i ( t ) ^ { - 1 } . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} d \\tilde { Z } ^ { \\alpha } _ t = B ^ { \\alpha } _ { \\beta , t } d Z ^ { \\beta } _ t , \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} { F _ { 1 } ^ { \\pm } } \\left ( \\xi \\right ) = { \\tfrac { 1 } { 2 } } \\psi \\left ( \\xi \\right ) - { \\tfrac { 1 } { 8 } } \\phi ^ { 2 } \\left ( \\xi \\right ) \\mp { \\tfrac { 1 } { 4 } } { \\phi } ^ { \\prime } \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{gather*} \\| v _ { N } \\| _ { L ^ 2 ( m ) } ^ 2 = \\sum _ { k = 0 } ^ { 2 ^ N } ( \\sum _ { n \\ge 2 ^ N + 1 } ( n + 1 ) ^ { - \\alpha } c _ { k + n } ) ^ 2 \\le C 2 ^ { N } ( \\sum _ { n \\ge 2 ^ N + 1 } ( n + 1 ) ^ { - \\alpha } c _ { n } ) ^ 2 \\le \\frac { C ' } { N ( \\log ( N + 1 ) ) ^ 3 } \\ , . \\end{gather*}"}
-{"id": "5106.png", "formula": "\\begin{align*} f ( x ) = f ^ * ( - x ) , x \\in X , \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} Q _ { j } \\phi { } _ { j , k } = Q _ { j } \\phi { } _ { j , \\infty } + Q _ { j } ( \\phi { } _ { j , k } - \\phi { } _ { j , \\infty } ) = \\varsigma _ { j - 1 } \\phi { } _ { j - 1 , \\infty } + \\varsigma _ { j - 1 } ( \\phi { } _ { j - 1 , k } - \\phi { } _ { j - 1 , \\infty } ) = \\varsigma _ { j - 1 } \\phi { } _ { j - 1 , k } , \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} 0 = \\sigma _ 0 < \\sigma _ 1 \\leq \\sigma _ 2 \\leq \\cdots \\nearrow \\infty . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\sum _ { \\nu \\ge 0 } a ( t p ^ { 2 \\nu } ) p ^ { - \\nu s } = a ( t ) \\frac { 1 - \\chi _ { t , N } ( p ) p ^ { k - 1 - s } } { 1 - \\lambda _ p p ^ { - s } + \\chi ^ 2 ( p ) p ^ { 2 k - 1 - 2 s } } , \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\log \\Phi _ { n + 1 } ^ * = \\sum _ { j = k } ^ n \\log \\left ( 1 - \\frac { \\alpha _ j z \\Phi _ j } { \\Phi _ j ^ * } \\right ) + \\log \\Phi _ k ^ * = \\sum _ { j = k } ^ n \\log \\left ( 1 - \\alpha _ j e ^ { i \\gamma _ j ( \\theta ) } \\right ) + \\log \\Phi _ k ^ * . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} L _ 2 & = \\int \\int \\tilde { K } ( x + y , y ) ( u ( x + y ) - u ( x ) ) u ( x ) d y d x \\\\ & = \\frac { 1 } { 2 } \\int \\int ( \\tilde { K } ( x , y ) - \\tilde { K } ( x + y , y ) ) ( u ( x ) - u ( x + y ) ) u ( x ) d y d x \\\\ & = \\frac { 1 } { 2 } \\int \\int ( 2 \\tilde { K } ( x , y ) - \\tilde { K } ( x + y , y ) - \\tilde { K } ( x - y , y ) ) u ( x ) ^ 2 d y d x \\\\ & \\gtrsim - \\int \\int | y | ^ 2 \\| D ^ 2 _ x K \\| _ { L ^ \\infty } | y | ^ { - d - \\alpha } u ( x ) ^ 2 d y d x . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} u _ \\infty : = \\lim _ { n \\to \\infty } u _ n = \\mu ^ { - 1 } = \\prod _ { j = 1 } ^ \\infty ( 1 - \\mathbb { P } ( X _ 1 > j ) ) , \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} X ^ { \\frac { d - 1 } { 2 } - 1 } \\cdot X ^ { \\frac { 1 } { 2 } + \\epsilon - \\lambda ( d ) } = X ^ { \\frac { d } { 2 } - 1 + \\epsilon - \\lambda ( d ) } , \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} G ( x , y ) & = \\sum _ { m \\geq 0 } G _ m ( x ) y ^ m , & & A ( x , y ) = \\sum _ { m \\geq 1 } A _ m ( x ) y ^ m , \\\\ B ( x , y ) & = \\sum _ { m \\geq 2 } B _ m ( x ) y ^ m . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{gather*} ( D f ) ( x ) = \\frac { { \\rm d } } { { \\rm d } x } f ( x ) , x \\in [ - 1 , 1 ] , \\end{gather*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\sum _ { \\overrightarrow { \\omega } \\in I } \\exp \\ ( \\sup _ { V _ { \\overrightarrow { \\omega } , t } } g _ { m , t } \\ ) = \\inf _ { J \\subseteq \\Omega ^ m , S ^ 1 = \\bigcup _ { \\overrightarrow { \\omega } \\in J } W _ { \\overrightarrow { \\omega } , t } } \\sum _ { \\overrightarrow { \\omega } \\in J } \\exp \\ ( \\sup _ { V _ { \\overrightarrow { \\omega } , t } } g _ { m , t } \\ ) . \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} 0 = \\varepsilon ( \\{ a , b \\} ) 1 _ A = S ( \\{ a , b \\} _ { ( 1 ) } ) \\{ a , b \\} _ { ( 2 ) } = ( - 1 ) ^ { | b _ { ( 1 ) } | | a _ { ( 2 ) } | } ( S ( \\{ a _ { ( 1 ) } , b _ { ( 1 ) } \\} ) a _ { ( 2 ) } b _ { ( 2 ) } + S ( a _ { ( 1 ) } b _ { ( 1 ) } ) \\{ a _ { ( 2 ) } , b _ { ( 2 ) } \\} ) . \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\tau _ n : = f _ n ( \\tau ) : = \\sum _ { i = 1 } ^ { \\infty } t ^ n _ i I _ { \\{ t ^ n _ { i - 1 } \\leq \\tau < t ^ n _ i \\} } , \\ \\ \\ \\ \\ t ^ n _ i : = \\frac { i } { 2 ^ n } , \\ i \\geq 0 . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} F ( A , q , \\iota ) = \\left ( B \\overset { \\iota } { \\longrightarrow } A \\overset { \\pi } { \\longrightarrow } \\widehat { B } \\right ) , \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , W ) & = g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Y , \\phi W ) + g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Y , \\omega W ) - \\eta ( Y ) g _ { 1 } ( X , \\omega W ) . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\phi ( a ) = a ^ 4 , \\phi ( b ) = b ^ 4 \\ \\mbox { a n d } \\ \\phi ( x ) = x , \\phi ( y ) = y \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\begin{aligned} G _ a ^ n ( z , w ) : = \\langle x , y , s _ 2 , \\ldots , s _ { n - 1 } \\mid s _ 2 = \\lbrack y , x \\rbrack , \\ ( \\forall _ { i = 3 } ^ n ) \\ s _ i = \\lbrack s _ { i - 1 } , x \\rbrack , \\ s _ n = 1 , \\ \\lbrack y , s _ 2 \\rbrack = s _ { n - 1 } ^ a , \\\\ ( \\forall _ { i = 3 } ^ { n - 1 } ) \\ \\lbrack y , s _ i \\rbrack = 1 , \\ x ^ 3 = s _ { n - 1 } ^ w , \\ y ^ 3 s _ 2 ^ 3 s _ 3 = s _ { n - 1 } ^ z , \\ ( \\forall _ { i = 2 } ^ { n - 3 } ) \\ s _ i ^ 3 s _ { i + 1 } ^ 3 s _ { i + 2 } = 1 , \\ s _ { n - 2 } ^ 3 = s _ { n - 1 } ^ 3 = 1 \\ \\rangle , \\end{aligned} \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\omega L _ { \\eta } V ^ { , i } + \\xi \\omega _ { , t } V ^ { , i } + 2 \\omega \\xi , _ { t } V ^ { , i } + \\eta ^ { i } , _ { t t } = 0 \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} f ( x ) : = E ^ { - 1 } ( x , h D ) P \\chi _ 0 q ( x , h D ) u _ { \\Sigma } + h R _ 1 \\chi _ 0 ( x , h D ) q ( x , h D ) u _ { \\Sigma } . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} = \\frac { ( a _ { 1 } - a _ { 2 } ) p q } { a _ { 1 } p q - Q ( q - p ) } \\left ( \\frac { Q ( q - p ) - a _ { 2 } p q } { a _ { 1 } p q - Q ( q - p ) } \\right ) ^ { \\frac { a _ { 2 } p q - Q ( q - p ) } { ( a _ { 1 } - a _ { 2 } ) p ^ { 2 } } } \\| \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } \\phi \\| _ { L ^ { p } ( \\mathbb { G } ) } ^ { p - q } \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\langle \\tilde { M } _ { g ^ { * } } \\ , \\phi , \\psi \\rangle _ { \\mathcal { P } } & = \\langle \\psi , \\tilde { M } _ { g ^ { * } } \\ , \\phi \\rangle _ { \\mathcal { P } } ^ { * } \\\\ & = \\big ( \\langle g ^ { * } , \\phi ^ { ( 2 ) } \\rangle _ { \\mathcal { C } } \\ , \\langle \\psi , Q \\phi ^ { ( 1 ) } \\rangle _ { \\mathcal { P } } \\big ) ^ { * } \\\\ & = \\langle \\phi ^ { ( 2 ) } , g ^ { * } \\rangle _ { \\mathcal { C } } \\ , \\langle Q \\phi ^ { ( 1 ) } , \\psi \\rangle _ { \\mathcal { P } } \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\mathcal { K } _ n ^ M ( x ) = Q _ n \\mathcal { K } ( x ) + \\mathcal { K } ( Q _ n x ) - Q _ n \\mathcal { K } ( Q _ n x ) . \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} T _ { m , \\lambda } ( a _ 1 , a _ 2 , \\ldots , a _ n ) = T _ m ( a _ 1 a _ 2 \\cdots a _ { \\lambda _ 1 } ) T _ m ( a _ { \\lambda _ 1 + 1 } a _ { \\lambda _ 1 + 2 } \\cdots a _ { \\lambda _ 1 + \\lambda _ 2 } ) \\cdots T _ m ( a _ { n - \\lambda _ { l ( \\lambda ) } + 1 } a _ { n - \\lambda _ { l ( \\lambda ) } + 1 } \\cdots a _ { n } ) \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} Q \\colon \\sum _ { i = 0 } ^ 5 a _ i t _ i ^ 2 = 0 \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{gather*} p _ a ( \\bar { C } _ 1 ) = 3 , | I | = 1 0 , p _ a ( \\bar { C } _ 2 ) = 4 , | I | = 6 , \\\\ p _ a ( \\bar { C } _ 1 ) = 5 , | I | = 2 , p _ a ( \\bar { C } _ 2 ) = 2 , | I | = 1 4 . \\end{gather*}"}
-{"id": "4088.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A ( \\sqrt { g } g ^ { i j } ) = \\sqrt { g } \\left ( \\frac { 1 } { 2 } g ^ { i j } g ^ { k l } - g ^ { l j } g ^ { i k } \\right ) \\overline { \\partial } _ A g _ { k l } , \\end{gather*}"}
-{"id": "7019.png", "formula": "\\begin{align*} H ^ i \\big ( \\operatorname { S p } ( A ) , \\mathcal { O } _ { \\operatorname { S p } ( A ) } ( < r ) \\big ) & = 0 \\enspace \\enspace i \\neq 0 \\enspace \\\\ H ^ i \\big ( \\operatorname { S p } ( A ) , G \\big ) & = 0 \\enspace \\enspace i \\neq 0 . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} K ' : = \\mathbb C ( Z _ 0 , Z _ 1 , \\dots , Z _ r ) . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , W ) & = \\cos ^ { 2 } \\theta g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , W ) - g _ { 1 } ( \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { X } Y ) , \\pi _ * ( \\omega \\phi W ) ) + g _ { 1 } ( \\mathcal { A } _ { X } \\mathcal { B } Y , \\omega W ) \\\\ & + g _ { 1 } ( \\pi _ * ( \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { C } Y ) , \\pi _ * ( \\omega W ) ) - \\eta ( Y ) g _ { 1 } ( X , \\omega W ) \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\Bigl \\{ z \\in \\mathbb C : | a _ { k k } - z | \\leq \\sum _ { 1 \\leq j \\leq n , j \\ne k } | a _ { k j } | \\Bigr \\} k = 1 , \\dots , n . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\int \\langle A ( q _ j + r _ j Z ) \\nabla v _ j , \\nabla \\xi \\rangle d Z = 0 . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} ( y + 1 ) ( y ^ n - y ^ { n - 1 } + y ^ { n - 2 } + 2 y ^ { n - 3 } - y ^ { n - 4 } ) = y ^ { n + 1 } + 3 y ^ { n - 2 } + y ^ { n - 3 } - y ^ { n - 4 } . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\begin{aligned} & - h ^ 2 \\Delta _ { x } \\Psi - { \\partial ^ 2 \\Psi \\over \\partial z ^ 2 } = f , - D ( x ) < z < 0 \\cr & { \\partial \\Psi \\over \\partial z } = \\psi ^ + ( x ) \\ @ \\ z = 0 \\cr & \\Psi = \\varphi ^ - ( x ) \\ @ \\ z = - D ( x ) \\cr \\end{aligned} \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\begin{aligned} f ( a ) = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ p _ { j } a + \\xi _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ p _ { j } a \\right \\} & = \\sum _ { j = 2 } ^ { k _ 1 } \\left ( p _ { j } a + \\xi _ { j } \\right ) + \\sum _ { j \\in \\mathcal { K } _ { 0 } ^ { + } } p _ { j } a + \\sum _ { j \\in \\mathcal { K } _ { 0 } ^ { - } } \\left ( 1 + p _ { j } a \\right ) \\\\ & = k _ { 0 } ^ { - } + \\sum _ { j = 2 } ^ { k } p _ { j } a + \\sum _ { j = 2 } ^ { k _ 1 } \\xi _ { j } , \\end{aligned} \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\mathcal { A } ( E _ 1 , E _ 2 ) = \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { \\mathcal { H } E _ 1 } \\mathcal { V } E _ 2 + \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { \\mathcal { H } E _ 1 } \\mathcal { H } E _ 2 \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\textup { V a r } ( \\Gamma _ { T } ^ 0 ( f , g ) ) = w ( x , y , 0 ) , \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i Q _ i \\varphi ( \\Gamma _ i ) \\varphi ( \\Gamma ) \\Big | \\leq N ^ { \\frac { \\varepsilon } { 4 } } \\hat { \\Pi } , \\Omega _ 2 ( z ) \\ , . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} w ^ t : = \\delta _ t v ^ 0 = \\frac 1 t \\big ( v ^ t - v \\big ) = \\frac 1 2 \\delta _ t ^ 2 \\tilde u ^ 0 . \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\prod _ { j = 1 } ^ k F _ { g _ j } ( n ) \\ll _ k x ( \\log \\log x ) ^ { ( 2 \\ell + k - 1 ) / 2 } . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} & \\sum _ { m \\in \\mathbb { Z } } r _ { d - 1 } ( m ^ 2 + h ) e ^ { - ( 2 m ^ 2 + h ) / X } \\\\ & = \\delta _ { [ d = 3 ] } \\delta _ { [ h = a ^ 2 ] } C ' X ^ { \\frac { 1 } { 2 } } \\log X + \\sum _ { 0 \\leq m < \\lceil \\frac { d } { 2 } - 1 \\rceil } C _ { m } X ^ { \\frac { d } { 2 } - 1 - \\frac { m } { 2 } } + O ( X ^ { \\frac { d } { 4 } - \\frac { 1 } { 2 } + \\epsilon } ) . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\Delta ^ \\circ = \\{ \\rho \\in \\mathrm { H o m } ( M , \\mathbb { R } ) \\cong N \\otimes \\mathbb { R } : \\rho ( x ) \\leq - 1 \\ , \\ , \\forall \\ , \\ , x \\in \\Delta \\} . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} ( 1 + \\lambda t ) ^ { \\frac { x + y } { \\lambda } } & = ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } ( 1 + \\lambda t ) ^ { \\frac { y } { \\lambda } } = \\left ( \\sum _ { l = 0 } ^ \\infty { x \\choose l } _ \\lambda t ^ l \\right ) \\left ( \\sum _ { m = 0 } ^ \\infty { y \\choose m } _ \\lambda t ^ m \\right ) \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n { y \\choose m } _ \\lambda { x \\choose n - m } _ \\lambda \\right ) t ^ n . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 } { l } + \\frac { n } { q } \\leq 1 , l \\in [ 2 , \\infty ) \\mbox { a n d } q \\in ( n , \\infty ] , \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\left ( \\frac { d ^ { 2 } S \\left ( t \\right ) } { d t ^ { 2 } } \\right ) + \\phi \\left ( t \\right ) \\frac { d S \\left ( t \\right ) } { d t } = 0 \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} { 1 \\over \\Delta t } ( u _ { 0 , m + 1 } - u _ { 0 , m } ) - { \\partial u _ 0 \\over \\partial t } ( t _ m ) = { \\partial u _ 0 \\over \\partial t } ( \\tau ) - { \\partial u _ 0 \\over \\partial t } ( t _ m ) = \\int _ { t _ m } ^ \\tau { \\partial ^ 2 u _ 0 \\over \\partial t ^ 2 } ( \\sigma ) d \\sigma , \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\Gamma _ { 1 2 } = \\Gamma _ { 2 1 } = \\frac { 2 \\sqrt { 2 } } { J _ 0 } - \\frac { 1 } { J _ 1 } - \\frac { 1 } { J _ 2 } . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} I ( M _ 1 ; Y _ 0 ^ T | M _ 2 ) & = \\frac { 1 } { 2 } E \\left [ \\int _ 0 ^ T E [ ( X _ 1 ( t ) + X _ 2 ( t ) - \\hat { X } _ 1 ( t ) - \\hat { X } _ 2 ( t ) ) ^ 2 | M _ 2 ] d t \\right ] \\\\ & = \\frac { 1 } { 2 } \\int _ 0 ^ T E [ ( X _ 1 ( t ) + X _ 2 ( t ) - \\hat { X } _ 1 ( t ) - \\hat { X } _ 2 ( t ) ) ^ 2 ] d t , \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} h ^ i { } _ k : = g _ 0 ^ { i \\bar { j } } g _ { \\phi k \\bar { j } } , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} f ( x ) = \\lim _ { t \\to \\infty } e ^ { A t } S ^ { - t } x = x - \\int _ 0 ^ \\infty e ^ { A s } \\psi ( S ^ { - s } x ) | S ^ { - s } x | ^ 2 d s , \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\mathcal E _ s ( u ; \\R ^ n _ + \\ ! \\times \\R ^ n _ - \\ ! ) = o ( 1 ) \\quad \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\sup \\{ S _ n g ( \\eta ) : \\eta \\in [ \\xi _ 1 , . . . , \\xi _ n ] \\} = S _ r g ( \\xi _ 1 , . . . , \\xi _ r , \\beta _ 1 , \\beta _ 2 , . . . ) - ( n - r ) h ( Y ) , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} d ^ n h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) = h ^ { \\mathrm { a n } } _ { ( D , g ) } ( ( f ^ { \\mathrm { a n } } ) ^ n ( \\xi ) ) = h ^ { \\mathrm { a n } } _ { ( D , g ) } ( ( f ^ { \\mathrm { a n } } ) ^ m ( \\xi ) ) = d ^ m h ^ { \\mathrm { a n } } _ { ( D , g ) } ( \\xi ) , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} T R _ 1 & = H ( M _ 1 ) \\\\ & = I ( M _ 1 ; Y _ 0 ^ T ) + H ( M _ 1 | Y _ 0 ^ T ) \\\\ & \\leq I ( M _ 1 ; Y _ 0 ^ T ) + T \\varepsilon _ T \\\\ & \\leq H ( M _ 1 ) - H ( M _ 1 | Y _ 0 ^ T ) + T \\varepsilon _ T \\\\ & \\leq H ( M _ 1 | M _ 2 ) - H ( M _ 1 | Y _ 0 ^ T , M _ 2 ) + T \\varepsilon _ T \\\\ & = I ( M _ 1 ; Y _ 0 ^ T | M _ 2 ) + T \\varepsilon _ T . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\widetilde { A } _ { { \\mathcal { G } } } \\Psi ( x ) = \\sum _ { z _ { \\lambda } \\in \\Lambda } ( ( \\Psi | \\widetilde { T } ( z _ { \\lambda } ) \\Phi ) ) \\ , \\widetilde { T } ( z _ { \\lambda } ) \\Phi , \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} g _ { 1 } ( U _ 1 , U _ 2 ) g _ { 1 } ( \\xi , Z ) = g _ 1 ( \\mathcal { T } _ { U _ 1 } \\phi U _ 2 , Z ) + g _ { 1 } ( \\mathcal { C } \\mathcal { T } _ { U _ 1 } U _ 2 , Z ) = g _ 1 ( \\mathcal { T } _ { U _ 1 } \\phi U _ 2 , Z ) - g _ { 1 } ( \\mathcal { T } _ { U _ 1 } U _ 2 , \\varphi Z ) \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\partial _ t \\phi _ j & = B _ u \\phi _ j , \\\\ \\partial _ t m _ 1 ( k ) & = B _ u m _ 1 ( k ) , \\\\ \\partial _ t m _ e ( \\lambda \\pm 0 i ) & = B _ u m _ e ( \\lambda \\pm 0 i ) - i \\lambda ^ 2 m _ e ( \\lambda \\pm 0 i ) , \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ k ^ { ( i ) } \\frac { \\partial d _ j } { \\partial g _ { i k } } \\mathbf { e } _ k ^ * X _ i G \\mathbf { e } _ i = O _ \\prec \\big ( \\Psi ^ 2 \\Pi _ i ^ 2 \\big ) \\ , , \\qquad \\frac { 1 } { N } \\sum _ k ^ { ( i ) } \\frac { \\partial d _ j } { \\partial g _ { i k } } \\mathbf { e } _ k ^ * X _ i \\mathring { \\mathbf { g } } _ i = O _ \\prec \\big ( \\Psi ^ 2 \\Pi _ i ^ 2 \\big ) \\ , , \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\kappa _ Y ( \\zeta ) = \\sum _ { m = 1 } ^ \\infty \\frac { ( i \\zeta ) ^ m } { m ! } \\kappa _ Y ^ { ( m ) } . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { \\Pi } \\Big | \\frac { \\omega ^ { \\theta } } { r } \\Big | ^ { 2 } \\dd x + 2 \\int _ { \\Pi } \\Big | \\nabla \\Big ( \\frac { \\omega ^ { \\theta } } { r } \\Big ) \\Big | ^ { 2 } \\dd x = - 2 \\int _ { \\Pi } \\Big ( \\frac { u ^ { \\theta } } { r } \\Big ) ^ { 2 } \\partial _ z \\Big ( \\frac { \\omega ^ { \\theta } } { r } \\Big ) \\dd x . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} \\norm { u } _ { x , s + 1 } = \\sum _ { n \\geq 0 } \\norm { A ^ n ( x ) u } e ^ { - \\tilde { \\lambda } n } \\leq C ( x ) \\sum _ { n \\geq 0 } e ^ { - \\varepsilon n } \\norm { u } . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\aligned & ( \\hat d ^ 2 \\circ ( \\hat d ^ 2 \\circ \\widehat \\psi - \\widehat \\psi \\circ \\hat d ^ 1 ) + ( \\hat d ^ 2 \\circ \\widehat \\psi - \\widehat \\psi \\circ \\hat d ^ 1 ) \\circ \\hat d ^ 1 ) _ { \\alpha _ + , \\alpha _ - } \\\\ & = d _ 0 \\circ b ( \\alpha _ + , \\alpha _ - ) + b ( \\alpha _ + , \\alpha _ - ) \\circ d _ 0 . \\endaligned \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\mathcal { A } : = \\left \\{ ( ( \\Omega , \\mathcal { F } , P ; \\mathbb { F } ) , \\alpha , \\xi , \\N ) \\right \\} \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\lim _ { T \\to T ^ * } \\| u \\| _ { L ^ { q _ d } ( [ 0 , T ) ; W _ x ^ { 1 , r _ d } ) } = \\infty , \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{gather*} \\begin{cases} - \\Delta v _ n + \\beta _ n ' ( w _ n ) v _ n = 0 & \\Omega , \\\\ v _ n + \\nabla w _ n \\cdot \\theta = 0 & \\partial \\Omega . \\end{cases} \\end{gather*}"}
-{"id": "3346.png", "formula": "\\begin{gather*} \\begin{pmatrix} 2 \\phi _ 0 & 0 & 0 & 0 \\\\ \\gamma ^ a { } _ 0 & \\psi ^ a { } _ b & 0 & 0 \\\\ \\gamma ^ n { } _ 0 & \\gamma ^ n { } _ a & 2 \\phi _ { n } & 0 \\\\ \\pi _ a & \\pi _ { a b } & 0 & 2 \\phi _ 0 \\delta ^ a _ b - \\psi ^ a { } _ b \\end{pmatrix} , \\end{gather*}"}
-{"id": "2329.png", "formula": "\\begin{align*} { \\rm V a r } ( X ) = \\frac { t ^ 2 } { \\mu ^ 2 } + \\frac { C _ 1 t } { \\mu } + \\frac { C _ 1 ^ 2 } { 2 } , { \\rm V a r } ( Y ) = \\frac { t ^ 2 } { \\mu ^ 2 } + \\frac { C _ 2 t } { \\mu } + \\frac { C _ 2 ^ 2 } { 2 } . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\sigma ( ( a c , b d ) ) = - , \\sigma ( ( a , 1 b ) ) = + , \\sigma ( ( a d , 1 c ) ) = - . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} h ( x + v ) = h ( x ) \\mbox { f o r e v e r y } x \\in \\R ^ { m } , \\mbox { f o r a l l } v \\in L , \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } I _ c ^ { x x } \\\\ I _ c ^ { x y } \\\\ I _ c ^ { y y } \\end{array} \\right \\} = \\int \\limits _ \\Omega \\left \\{ \\begin{array} { c } ( x - x _ c ) ^ 2 \\\\ ( x - x _ c ) ( y - y _ c ) \\\\ ( y - y _ c ) ^ 2 \\end{array} \\right \\} ~ \\mathrm { d } \\Omega \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} v _ t ^ \\epsilon + \\int \\left ( 1 - \\frac { \\exp ( v ^ \\epsilon ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( v ^ \\epsilon ( x , t ) / \\epsilon ) } \\right ) K d y = \\frac { f ( \\hat { x } | x | ^ { 1 / \\epsilon } , u ^ \\epsilon ) } { u ^ \\epsilon } ; \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} \\frac { w _ { k l } - \\overline { \\nu _ { l k } } } { 2 \\sqrt { - 1 } } = \\left < Z _ { k } , \\Xi _ { l } \\right > \\mbox { f o r a l l $ k , l = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\sum _ { n } \\mathbb E _ { \\mathcal F _ { \\frac { n } { 2 ^ m } } } \\bigl ( ( | F | ^ 2 \\star \\nu ) _ { \\frac { n + 1 } { 2 ^ m } } - ( | F | ^ 2 \\star \\nu ) _ { \\frac { n } { 2 ^ m } } \\bigr ) \\to ( | F | ^ 2 \\star \\nu ) _ T = ( | F | ^ 2 \\star \\nu ) _ { \\infty } \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} H _ k ( x ) = x H _ { k - 1 } ( x ) + \\sum _ { j \\geq 1 } H _ { m , j } ( x ) \\ , . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} M ( s , s + t ] = \\sum _ { j = 1 } ^ { N ( s ) } L _ j ( s - T _ j , s + t - T _ j ] + \\sum _ { j = N ( s ) + 1 } ^ { N ( s + t ) } L _ j ( s + t - T _ j ) \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\xi ( p ) = f _ { i } ( | p - p _ { i j } | ) ( p - p _ { i j } ) ^ { \\perp } \\ , , \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} S = \\{ ( L , v ) \\in \\beta ^ { * } ( T M ) ; v \\in L \\} \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} - \\frac { \\partial P } { \\partial x } \\left ( x , t \\right ) = L \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) + R Q \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ k e ^ { - \\alpha _ j \\eta ( t _ { j - 1 } , t _ j ] } = \\prod _ { j = 1 } ^ k \\sum _ { \\ell _ j = 0 } ^ \\infty \\frac { ( 1 - \\alpha _ j ) ^ { \\ell _ j } } { \\ell _ j ! } \\eta ( t _ { j - 1 } , t _ j ] ^ { \\ell _ j } e ^ { - \\eta ( t _ { j - 1 } , t _ j ] } a . s . , \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} d f _ \\tau = \\sum _ { \\sigma \\in \\tau ^ + } ( d f _ \\tau ) _ \\sigma = \\sum _ { \\sigma \\in \\tau _ 0 ^ + } ( d f _ \\tau ) _ \\sigma \\ + \\ \\sum _ { \\sigma \\in \\tau _ 1 ^ + } ( d f _ \\tau ) _ \\sigma + \\ \\sum _ { \\sigma \\in \\tau _ 2 ^ + } ( d f _ \\tau ) _ \\sigma . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} | z ^ * - z _ a | \\leq | \\lambda ^ * | / ( 0 . 0 3 ^ 2 + 0 . 1 5 ^ 2 ) ^ { 1 / 2 } = 5 . 2 \\ldots \\times 1 0 ^ { - 1 5 } \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} G _ m ( x ) = \\frac { x } { 1 - x } C ( x ) G _ { m - 1 } ( x ) \\ , . \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} ( \\mathcal { F } \\otimes _ { t _ 1 , t _ 2 } \\mathcal { G } ) _ { v } : = \\bigoplus _ { v _ 1 + v _ 2 = v } ( \\mathbb { S } _ { v _ 1 , v _ 2 } ) _ { * } \\Big ( ( \\mathcal { F } _ { v _ 1 } \\boxtimes \\mathcal { G } _ { v _ 1 } ) \\otimes \\mathcal L _ { v _ 1 , v _ 2 } \\Big ) , \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( i + 1 ) ( 2 k + 2 - 3 i ) a ^ { i } > 0 \\ , . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} M ^ { r , \\theta } _ { T , \\varepsilon } ( \\eta ) \\ ; = \\ ; \\frac { 1 } { 2 ( \\iota _ { + } + \\iota _ - ) q ( \\varepsilon ) \\log T } \\sum _ { \\substack { T ^ { r } - T ^ \\varepsilon \\le | z | \\le T ^ r + T ^ \\varepsilon \\\\ \\theta - q ( \\varepsilon ) \\le \\Theta ( z ) \\le \\theta + q ( \\varepsilon ) } } \\frac { \\eta ( z ) } { | z | ^ 2 } \\ ; \\cdot \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\rho _ \\mathfrak { a } ^ k ( 0 , w ) & = ( * ) \\frac { L ^ k _ \\mathfrak { a } ( 2 w - 1 ) } { L ^ k _ \\mathfrak { a } ( 2 w ) } \\frac { \\Gamma ( 2 w - 1 ) } { \\Gamma ( w + \\frac { k } { 2 } ) \\Gamma ( w - \\frac { k } { 2 } ) } \\\\ \\rho _ \\mathfrak { a } ^ k ( h , w ) & = ( * ) \\frac { \\lvert h \\rvert ^ { w - 1 } } { L ^ k _ \\mathfrak { a } ( 2 w ) } \\frac { 1 } { \\Gamma ( w + \\frac { \\lvert h \\rvert } { h } \\frac { k } { 2 } ) } D _ \\mathfrak { a } ^ k ( h , w ) \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\displaystyle { \\gamma = \\frac { r ^ { n - 1 } } { 1 + B ( \\theta ) r ^ { n - 1 } } } \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} N = \\int _ 0 ^ \\infty e ^ { - A s } \\sigma ( 0 ) d W ( s ) = \\lim _ { t \\to \\infty } M ( t ) , \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\Psi _ { \\varsigma , K } = \\frac { 1 } { \\sqrt { \\varepsilon } } \\| \\boldsymbol { \\sigma } _ K ^ { \\varsigma } \\| _ { \\boldsymbol { L } ^ { 2 } ( K ) } + \\texttt { C } _ { K } \\| \\boldsymbol { \\mathsf { o s c } } _ { K } ^ { \\varsigma } \\| _ { \\boldsymbol { L } ^ { 2 } ( K ) } . \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} \\left \\Vert \\widetilde { V } \\right \\Vert _ { H ^ { 1 } \\left ( \\Omega _ { d + c } \\right ) } ^ { 2 } \\leq C _ { 2 } \\left ( 1 + \\left \\Vert \\widetilde { V } \\right \\Vert _ { H ^ { 2 } \\left ( \\Omega \\right ) } ^ { 2 } \\right ) \\sigma ^ { 2 \\rho } , \\rho = c / \\left ( m + c \\right ) . \\square \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\lambda _ 1 ( S ) = \\nu \\lambda _ 1 ( \\Omega ) ( 1 + o ( 1 ) ) \\nu \\to \\infty v _ * \\to 0 , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} e ^ a = ( e ^ a _ 0 , ( e ^ a _ 1 , \\dots , e ^ a _ k ) ) , \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow + \\infty } { \\underline { c } _ n } ^ * _ + = \\lim \\limits _ { n \\rightarrow + \\infty } { \\overline { c } _ n } ^ * _ + = { c } ^ * _ + . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} z \\mapsto \\exp \\ ( - \\sum _ { n \\geqslant 1 } \\frac { 1 } { n } \\ ( \\sum _ { T _ \\tau ^ n x = x } \\frac { \\exp \\ ( - u \\sum _ { k = 0 } ^ { n - 1 } g \\ ( T _ \\tau ^ k x \\ ) \\ ) } { \\left | \\ ( T _ \\tau ^ n \\ ) ' \\ ( x \\ ) - 1 \\right | } \\ ) z ^ n \\ ) \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} T _ m = m _ { ( m , \\dot { 0 } ) } , m \\ge 1 . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\nu _ { \\ell } ( K ) = \\prod _ { \\chi \\in S ( K ) } \\left ( 1 - \\frac { \\chi ( \\ell ) } { \\ell } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\psi ( t ) & = \\sum _ { k = 0 } ^ { \\infty } ( k - 2 ) \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { [ 0 , r _ { k } ( \\boldsymbol { \\zeta } ( s ) ) ) } ( y ) \\ , \\varphi _ { k } ( s , y ) d s \\ , d y \\\\ \\zeta _ { k } ( t ) & = p _ { k } - \\int _ { [ 0 , t ] \\times \\lbrack 0 , 1 ] } { { 1 } } _ { [ 0 , r _ { k } ( \\boldsymbol { \\zeta } ( s ) ) ) } ( y ) \\ , \\varphi _ { k } ( s , y ) d s \\ , d y , k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} g _ { 2 } ( \\pi _ { \\ast } \\omega W , ( \\nabla \\pi _ { \\ast } ) ( Z , \\varphi U ) ) & = g _ { 1 } ( \\phi W , \\hat { \\nabla } _ { Z } \\varphi U ) \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} [ \\nabla _ i , \\nabla _ { \\bar j } ] X ^ l = R _ { i \\bar j k } \\ , ^ l X ^ k , \\quad [ \\nabla _ i , \\nabla _ { \\bar j } ] a _ k = - R _ { i \\bar j k } \\ , ^ l a _ l ; \\\\ [ \\nabla _ i , \\nabla _ { \\bar j } ] X ^ { \\bar l } = - R _ { i \\bar j } \\ , ^ { \\bar k } \\ , _ l X ^ { \\bar l } , \\quad [ \\nabla _ i , \\nabla _ { \\bar j } ] a _ k = R _ { i \\bar j } \\ , ^ { \\bar k } \\ , _ { \\bar l } a _ { \\bar k } . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\pi ^ \\rho : G _ \\bullet \\to U ( H ^ \\rho _ o ) \\ \\ , \\ \\ \\pi ^ \\rho ( g ) \\ , = \\ , g ^ \\rho \\ . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\tilde M _ \\lambda ( r ) = M _ \\lambda ( \\zeta ( r ) ) , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} J _ \\lambda \\Bigl ( \\sum _ { j = 1 } ^ k U _ j \\Bigr ) = F _ \\lambda ( \\mu , r ) + R _ \\lambda ( \\mu , r ) , \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{gather*} a _ { 1 , 0 , 0 , 2 } = a _ { 0 , 1 , 0 , 2 } = a _ { 0 , 0 , 1 , 2 } = a _ { 0 , 0 , 0 , 3 } = 0 , \\\\ a _ { 2 , 0 , 0 , 1 } a _ { 0 , 1 , 1 , 1 } ^ 2 - a _ { 1 , 0 , 1 , 1 } a _ { 1 , 1 , 0 , 1 } a _ { 0 , 1 , 1 , 1 } + a _ { 0 , 2 , 0 , 1 } a _ { 1 , 0 , 1 , 1 } ^ 2 + a _ { 0 , 0 , 2 , 1 } a _ { 1 , 1 , 0 , 1 } ^ 2 - 4 a _ { 0 , 0 , 2 , 1 } a _ { 0 , 2 , 0 , 1 } a _ { 2 , 0 , 0 , 1 } = 0 , \\end{gather*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\partial _ \\xi p \\neq 0 \\{ p = 0 \\} . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\rho \\frac { \\partial Q } { \\partial t } + \\pi a ^ { 2 } \\frac { \\partial P } { \\partial x } = 2 \\pi \\mu r \\frac { \\partial u } { \\partial r } \\mid _ { 0 } ^ { a \\left ( x , t \\right ) } . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\bar { \\eta } : = \\mathcal { P } \\eta = \\eta , \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\langle \\ ! \\langle \\Phi , \\varphi \\rangle \\ ! \\rangle _ { \\mu _ { \\beta } } = \\sum _ { n = 0 } ^ { \\infty } n ! \\langle \\Phi ^ { ( n ) } , \\varphi ^ { ( n ) } \\rangle . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} 0 = R _ { i j } u = \\sum _ { k = 1 } ^ n ( R _ { i j } u _ k ) e _ k + u _ k ( R _ { i j } e _ k ) = \\sum _ { k = 1 } ^ n ( R _ { i j } u _ k ) e _ k + u _ k ( e _ i \\delta _ { j k } - e _ j \\delta _ { i k } ) \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{gather*} \\alpha ( \\beta - \\beta \\alpha ) = \\alpha \\beta - \\alpha \\beta \\alpha = 0 , ( \\beta - \\beta \\alpha ) \\alpha = \\beta \\alpha - \\beta \\alpha ^ 2 = 0 , \\\\ ( \\beta - \\beta \\alpha ) ^ 2 = \\beta ^ 2 - \\beta ^ 2 \\alpha - \\beta \\alpha \\beta + \\beta \\alpha \\beta \\alpha = \\beta - \\beta \\alpha . \\end{gather*}"}
-{"id": "9680.png", "formula": "\\begin{align*} f ' _ { \\theta , q , 3 } ( \\mathbf { x } ^ { ( i ) } ) = \\frac { 3 ( \\theta - 1 + q ) \\mathbf { x } ^ { ( i ) } } { ( \\theta - 1 ) \\mathbf { y } ^ { ( i ) } ( \\theta \\mathbf { y } ^ { ( i ) } + 1 - \\theta - q ) } . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} { \\rm R i c } ( \\omega ) = \\gamma \\omega + ( 1 - \\beta ) [ D ] + L _ X \\omega \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} a _ { m - 1 } = \\sum _ { k = 1 } ^ { m - 1 } ( - 1 ) ^ { k } { m - 1 \\choose k } \\frac { 1 } { k } . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} G ( n ) = \\prod _ { p \\mid \\phi ( n ) } G _ p ( n ) \\quad \\log G ( n ) = \\sum _ { p \\mid \\phi ( n ) } \\log G _ p ( n ) . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\bigl \\langle ( f _ { k , m } ) _ { k = 1 , m = 0 } ^ { k = K , m = M } , ( g _ { k , m } ) _ { k = 1 , m = 0 } ^ { k = K , m = M } \\bigr \\rangle \\eqsim _ { p , q } \\bigl \\| ( f _ { k , m } ) _ { k = 1 , m = 0 } ^ { k = K , m = M } \\bigr \\| _ { Q ^ { p ' } _ { q ' } } . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\kappa _ { X ^ * } ^ { ( m ) } ( t ) = \\kappa _ X ^ { ( m ) } m I _ { m - 1 } ( t ) \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} \\int _ 2 ^ z \\frac { S ( t ) } { t ^ 2 } \\ , d t & = \\int _ 2 ^ z \\bigg ( \\frac { ( \\log \\log t ) ^ 2 } { t \\log t } + O \\bigg ( \\frac { | \\ ! \\log \\log t | } { t \\log t } \\bigg ) \\bigg ) \\ , d t \\\\ & = \\frac { ( \\log \\log t ) ^ 3 } { 3 } \\bigg | _ 2 ^ z + O \\big ( ( \\log \\log t ) ^ 2 \\big | _ 2 ^ z \\big ) = \\frac { ( \\log \\log z ) ^ 3 } 3 + O \\big ( ( \\log \\log z ) ^ 2 \\big ) \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} 0 < r _ - ^ q \\le \\dots \\le r _ - ^ 1 = r _ - \\le r _ + = r _ 1 ^ + \\le \\dots \\le r _ + ^ p . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} \\left ( \\frac { s - 1 } { s + 1 } \\right ) ^ { a ( \\chi ) } L _ F ( \\chi , s ) \\ll _ { \\epsilon , d } \\mathfrak { q } ( F , \\chi , s ) ^ { \\delta ^ { \\prime } ( 1 - \\sigma ) + \\epsilon } , 1 / 2 \\leq \\sigma : = \\textrm { R e } ( s ) \\leq 1 + \\epsilon , \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} s ^ * ( z ) \\le \\int _ { B ( z , r ) \\cap L } s ^ * \\ , d \\mu _ r = \\int _ { B ( z , r ) \\cap L } s \\ , d \\mu _ r \\le \\int _ { B ( z , r ) \\cap L } u \\ , d \\mu _ r . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ \\eta X + Y ] & = \\mathbb { L } ^ 1 \\lim _ { n \\rightarrow \\infty } ( \\eta ^ + \\hat { \\mathbb { E } } _ { \\tau _ n + } [ X ] + \\eta ^ - \\hat { \\mathbb { E } } _ { \\tau _ n + } [ - X ] ) + Y \\\\ & = \\eta ^ + \\hat { \\mathbb { E } } _ { \\tau + } [ X ] + \\eta ^ - \\hat { \\mathbb { E } } _ { \\tau + } [ - X ] + Y . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} E ( x ; q , a ) = - c ( q , a ) - \\sum _ { \\chi \\ne \\chi _ 0 } \\overline \\chi ( a ) \\sum _ { \\substack { \\gamma \\in \\R \\\\ L ( \\frac 1 2 + i \\gamma , \\chi ) = 0 } } \\frac { x ^ { i \\gamma } } { \\frac { 1 } { 2 } + i \\gamma } + O \\bigg ( \\frac 1 { \\log x } \\bigg ) , \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\left . \\partial ^ { 2 } _ { w _ { b b } w _ { a a } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot G \\left ( W , Z \\right ) \\right \\vert _ { Z = O _ { q \\times N } \\atop { W = O _ { q \\times q } } } = \\mbox { O } _ { q \\times q } , \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} u _ i ( x , t ) = \\partial _ { x _ i } P _ t ( x ) \\ , , \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} I _ \\lambda ( \\zeta ' , \\mu ' ) & = \\bar J _ \\lambda ( V ) + D \\bar J _ \\lambda ( V + \\phi ) [ \\phi ] + \\theta _ \\lambda ^ { ( 2 ) } ( \\zeta ' , \\mu ' ) , \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} ( A _ { d } ( \\delta _ { n } ) ) & \\leq ( v _ { n } \\delta _ { n } ) ^ { - \\kappa } \\int _ \\Theta c ( \\theta ) d \\Pi ( \\theta ) = o ( 1 ) \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} M [ u ] = \\int u ^ 2 \\ , d x d y . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\widehat { \\mathcal { A } } = b ( \\mathbf { D } ) ^ * g ( \\mathbf { x } ) b ( \\mathbf { D } ) \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} { \\| f \\| } _ { C _ { \\gamma } ^ { n } } = \\sum _ { k = 0 } ^ { n - 1 } { \\| f ^ { ( k ) } \\| } _ { C } + { \\| f ^ { ( n ) } \\| } _ { C _ { \\gamma } } , \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\widetilde { V } ( z ) = \\sum _ { m _ 1 , \\ldots , m _ { 2 k + 2 } \\in \\mathbb { Z } } y ^ { \\frac { k + 1 } { 2 } } e ^ { 2 \\pi i x ( m _ 1 ^ 2 + \\cdots m _ { 2 k + 1 } ^ 2 - m _ { 2 k + 2 } ^ 2 ) } e ^ { - 2 \\pi y ( m _ 1 ^ 2 + \\cdots + m _ { 2 k + 2 } ^ 2 ) } \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} ( \\alpha _ { 1 \\underline { d } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } ) ( s ) = \\alpha _ { 1 \\widetilde { \\underline { d } } } ( s ) \\in V _ { X _ 1 } ( - ( d - i ) A ) . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} w _ { \\rm d o u . } ( x ) : = \\int _ { \\Gamma _ 0 } d \\mathcal { H } ^ { n - 1 } ( y ) \\frac 1 2 \\left ( ( \\dot \\eta ^ 0 \\circ \\pi _ 1 ) ^ 2 \\frac { N \\cdot \\nu ^ 0 } { J } \\ , \\big ( J \\Delta h ^ 0 ) \\right ) ( y ) \\ , \\partial _ N P ( x - y ) , \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} V ( A ) = \\{ ( \\overline y , ( t _ 1 , \\dots , t _ k ) ) \\in V _ x \\mid \\} . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\int _ { B _ { \\mu _ i } } ( { \\bf c } \\cdot z ) \\ , U ^ 5 ( z ) \\ , d z = 0 \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\partial \\left ( \\sum a _ { i , j } \\lambda ^ i \\mu ^ j \\right ) = \\sum \\partial ( a _ { i , j } ) \\lambda ^ i \\mu ^ j , \\ , \\ , \\ , a _ { i , j } \\in K \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\iota _ { X _ { \\sigma } } \\omega _ Q = \\Theta ( \\gamma ( q ) ) . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} W \\left ( { u , \\xi } \\right ) = A \\left ( u \\right ) W _ { n , 1 } \\left ( { u , \\xi } \\right ) + B \\left ( u \\right ) W _ { n , 2 } \\left ( { u , \\xi } \\right ) + G \\left ( { u , \\xi } \\right ) , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\eqref { v a 3 } & = \\psi ( 0 ) \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ - e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } \\ , d \\xi \\\\ & \\quad + [ \\psi ( x ) - \\psi ( 0 ) ] \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ - e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } \\ , d \\xi \\\\ & + \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ - e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } [ \\psi ( \\xi ) - \\psi ( x ) ] \\ , d \\xi . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\widehat u _ + ( - 1 ) = ( - 1 , \\varepsilon / 2 ) , \\widehat u _ + ( 1 ) = ( 1 , \\varepsilon / 2 ) . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } e ^ { x t } = \\sum _ { n = 0 } ^ \\infty E _ n ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 1 , 2 ] ) . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} \\lVert A _ i ^ n ( x ) \\rVert & \\le \\lVert A _ i ^ { n - 1 } ( f ( x ) ) \\rVert \\cdot \\lVert A _ i ( x ) \\rVert \\\\ & \\le \\lVert A _ i ^ { n - 1 } ( f ( x ) ) \\rVert \\cdot \\lVert A ( x ) \\rVert . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} \\begin{array} { r c l c l } M + | L | & = & \\sum _ { j = 0 } ^ { \\infty } | q | ^ { j ( j + 1 ) / 2 - j ( k + 1 / 2 ) } & = & | q | ^ { - k ^ 2 / 2 } \\sum _ { j = 0 } ^ { \\infty } | q | ^ { ( j - k ) ^ 2 / 2 } \\\\ \\\\ & = & | q | ^ { - k ^ 2 / 2 } ( 1 + 2 \\sum _ { \\nu = 1 } ^ k | q | ^ { \\nu ^ 2 / 2 } + \\sum _ { \\nu = k + 1 } ^ { \\infty } | q | ^ { \\nu ^ 2 / 2 } ) & < & | q | ^ { - k ^ 2 / 2 } ( 1 + \\tau ( | q | ) ) ~ . \\end{array} \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log T _ n = \\log \\rho , \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha x _ { i - 1 } ( t ) + a x _ { i } ( t ) + \\beta x _ { i + 1 } ( t ) , \\ , \\ , i \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} F _ T ( x ) = 1 + x F _ T ( x ) + x \\big ( F _ T ( x ) - 1 - x F _ T ( x ) \\big ) + x \\big ( F _ T ( x ) - 1 \\big ) + \\frac { x ^ 3 ( 1 + x ) F _ T ( x ) } { 1 - 3 x + x ^ 2 } \\ , , \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\partial _ x ^ { - 1 } f = \\frac 1 2 \\int _ { - \\infty } ^ x f ( y ) \\ , d y - \\frac 1 2 \\int _ x ^ \\infty f ( y ) \\ , d y . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} B _ 1 \\cap S = \\left \\{ 0 , \\infty \\right \\} , B _ 2 \\cap S = \\left \\{ - 1 , 1 \\right \\} , B _ 3 \\cap S = \\left \\{ - i , i \\right \\} . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} H _ { b , m } ^ { 0 } ( X ) : = \\{ u \\in C ^ { \\infty } ( X ) : T u = i m u , \\bar \\partial _ { b } u = 0 \\} . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ k ( - \\lambda _ i ) = \\sum _ { i = 1 } ^ k ( - \\lambda _ i ) ^ 3 = n - k . \\end{gather*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\mathcal { G } _ { 0 u } ( y ^ n ) = \\begin{cases} M ( y ^ { n } ) & \\mbox { i f } | M ( y ^ { n } ) | = 1 \\\\ \\mathcal { E } & \\mbox { i f } | M ( y ^ { n } ) | > 1 . \\end{cases} \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} j _ n : = B _ { \\lambda r _ j } ^ { \\R ^ N } ( y _ j ) \\cap \\bigcup _ { i = 1 } ^ { n - 1 } B _ { \\lambda r _ { j _ i } } ^ { \\R ^ N } ( y _ { j _ i } ) = \\emptyset . \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\Gamma _ m ( p , n ) = \\Gamma _ m ( p , n - 1 ) \\cup \\{ \\Gamma _ m ( p - 1 , n - m ) | | { n - 1 } \\} . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\eta & = \\sum _ { i = 1 } ^ { N } \\omega _ i D _ i = \\sum _ { i = 1 } ^ { N } \\omega _ i \\sum _ { j = 1 } ^ { M } P _ { i , j } D _ { i , j } \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{gather*} b ( \\mathbf { D } ) = \\begin{pmatrix} D _ 1 & 0 \\\\ \\frac { 1 } { 2 } D _ 2 & \\frac { 1 } { 2 } D _ 1 \\\\ 0 & D _ 2 \\end{pmatrix} , g ( \\mathbf { x } ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & g _ 2 ( x _ 1 ) & 0 \\\\ 0 & 0 & g _ 3 ( x _ 1 ) \\end{pmatrix} , \\end{gather*}"}
-{"id": "7148.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\partial ^ 2 R _ 0 } { \\partial z ^ 2 } - p ^ 2 R _ 0 = 0 , \\ - D ( x ) \\leq z \\leq 0 , R _ 0 | _ { z = 0 } = 1 \\cr & \\frac { \\partial R _ 0 } { \\partial z } | _ { z = - D ( x ) } = 0 \\cr \\end{aligned} \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} P ( E ; \\Omega ) : = \\sup \\left \\{ \\int _ \\Omega \\chi _ E ( x ) \\div \\psi ( x ) \\ , d x \\ , : \\psi \\in C ^ 1 _ c ( \\Omega ; \\ , \\R ^ n ) \\ , , \\| \\psi \\| _ { \\infty } \\leq 1 \\right \\} \\ , . \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} z _ 1 ( t ) = - \\dfrac { 1 } { \\varepsilon e ^ { 2 \\pi i t } } , \\ z _ 2 ( t ) = - \\varepsilon e ^ { 2 \\pi i t } , \\ z _ 3 ( t ) = 1 , \\ z _ 4 ( t ) = - 1 , \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { r _ n ( x , \\beta ) } { \\log _ \\beta n } = 1 . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\pi _ 1 + \\pi _ 2 = \\pi _ 3 \\qquad ( \\pi _ 1 , \\pi _ 2 , \\pi _ 3 \\in S ) . \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} \\prod _ { j = k } ^ m \\frac { c _ j } { \\sqrt { n - j + 2 } } \\le \\left ( \\frac { 1 } { t ( \\alpha ) } \\ , \\sqrt { \\frac { 1 + \\epsilon } { 1 - \\epsilon } } \\right ) ^ { m - k + 1 } c _ k ^ { m - k + 1 } . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\abs { g ( 2 ^ { - r } ) - g ( 0 ) - g ' ( 0 ) \\cdot 2 ^ { - r } } & = \\abs [ \\Big ] { \\int _ { 0 } ^ { 2 ^ { - r } } s ( s + 1 ) ( 1 - t ) ^ { - s - 2 } ( 2 ^ { - r } - t ) ~ d t } \\\\ & \\leq \\abs { s } \\abs { s + 1 } 2 ^ { - r } \\int _ { 0 } ^ { 2 ^ { - r } } \\abs { 1 - t } ^ { - \\Re s - 2 } ~ d t \\\\ & \\leq \\abs { s } \\abs { s + 1 } 2 ^ { - 2 r } ( 1 - 2 ^ { - r } ) ^ { - \\Re s - 2 } , \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} e = - \\frac { 1 9 2 } { 2 4 7 } \\ , c ^ { 5 } - \\frac { 6 0 } { 2 4 7 } \\ , c ^ { 4 } - \\frac { 3 6 2 4 } { 2 4 7 } \\ , c ^ { 3 } - \\frac { 6 2 1 } { 1 9 } \\ , c ^ { 2 } - \\frac { 5 3 3 6 } { 2 4 7 } \\ , c - \\frac { 7 0 8 } { 2 4 7 } , \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\rho _ i ( x ) = \\nabla ^ 2 \\eta ( x ) ( E _ i , E _ i ) & \\geq - d ( x ) \\frac { \\sqrt { a } \\sinh \\sqrt { a } d _ 0 - \\kappa _ - \\cosh \\sqrt { a } d _ 0 } { \\cosh \\sqrt { a } d _ 0 - \\dfrac { \\kappa _ - } { \\sqrt { a } } \\sinh \\sqrt { a } d _ 0 } \\\\ & \\geq \\sqrt { a } d ( x ) \\geq 0 . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} { \\| D _ { a ^ + } ^ { \\alpha , \\beta } y _ { m } ( x ) - D _ { a ^ + } ^ { \\alpha , \\beta } y ( x ) \\| } _ { C _ { 1 - \\gamma } [ a , b ] } & = { \\| f ( x , y _ { m } ( x ) ) - f ( x , y ( x ) ) \\| } _ { C _ { 1 - \\gamma } [ a , b ] } \\\\ & \\leq A { \\| y _ { m } ( x ) - y ( x ) \\| } _ { C _ { 1 - \\gamma } [ a , b ] } . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { \\mathcal { P } ^ { * } } = \\langle f , g \\rangle _ { \\mathcal { C } } = \\langle f ^ { * } , g ^ { * } \\rangle _ { \\mathcal { C } } ^ { * } = \\langle g ^ { * } , f ^ { * } \\rangle _ { \\mathcal { P } } , \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\textnormal { ( I ) } & : = \\frac { 4 ( p - 2 ) } { p } \\int u ^ { p / 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x , \\\\ \\textnormal { ( I I ) } & : = \\int u ^ p ( \\nabla a , \\nabla ( \\eta ^ 2 ) ) \\ ; d x + ( p - 1 ) \\int u \\eta ^ 2 u ^ p \\ ; d x . \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} B ^ { \\rho _ { \\pm } } _ { n } ( u ) : = \\frac { L ^ { \\rho _ { \\pm } } ( \\gamma ^ 2 n + \\beta _ 1 u n ^ { 2 / 3 } , n ) - ( L ^ { \\rho _ { \\pm } } ( \\gamma ^ 2 n , n ) + \\frac { 1 } { 1 - \\rho _ \\pm } \\beta _ 1 u n ^ { 2 / 3 } ) } { \\beta _ 2 n ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} | F ( w ) | = | F ( \\zeta _ 0 + \\mathrm { i } \\tau ) | & \\leqslant \\int _ { \\Gamma } | F ( \\zeta ) | \\frac { C \\tau | \\mathrm { d } \\zeta | } { | \\zeta - \\zeta _ 0 | ^ 2 + \\tau ^ 2 } \\\\ & \\leqslant \\int _ { \\mathbb { R } } \\big | F \\big ( u + \\mathrm { i } a ( u ) \\big ) \\big | \\frac { C \\tau \\sqrt { 1 + M ^ 2 } \\ , \\mathrm { d } u } { | u - u _ 0 | ^ 2 + \\tau ^ 2 } , \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\Omega _ l : = \\sum _ { i \\in D } \\omega _ i \\delta _ i \\quad \\quad \\quad \\Omega _ r : = \\Omega _ l ^ * : = \\sum _ { i \\in D } \\omega _ { \\bar i } \\delta _ i \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} F | _ { T N \\cap J _ \\pm ( T N ) } = J | _ { T N \\cap J _ \\pm ( T N ) } , \\ ; F | _ { T N \\cap J ( \\nu N ) } = 0 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} u _ i ( 0 , x ) = u _ { i t } ( 0 , x ) = 0 x \\in ( 0 , \\pi ) \\ , , i = 1 , 2 , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} { \\mathcal I } _ * ( t ) = - \\frac { t ^ 3 } { \\pi } \\int _ { 0 } ^ { \\infty } \\zeta ^ { - 1 / 2 } \\left ( \\Xi ( t , \\zeta ) N _ * + N _ * \\Xi ( t , \\zeta ) - \\zeta \\ , \\Xi ( t , \\zeta ) N _ * \\Xi ( t , \\zeta ) \\right ) \\ , d \\zeta . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\ \\| f \\| _ { A ^ s ( T ) } \\leq K \\ \\ \\| f \\| _ { B ^ s ( T ) } \\leq K \\ , \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\mu _ i ^ { - \\frac { 1 } { 2 } } \\pi _ i ( x ) = - 4 \\pi \\alpha _ 3 \\ , H _ { \\lambda } ( x , \\zeta _ i ) + \\mu _ i \\ , \\mathcal { D } _ 0 \\Bigl ( \\frac { x - \\zeta _ i } { \\mu _ i } \\Bigr ) + \\mu _ i ^ { 2 - \\sigma } \\ , \\theta ( \\mu _ i , x , \\zeta _ i ) \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\partial _ { w _ { b b } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\cdot \\partial _ { w _ { a a } } \\left ( G \\left ( W , Z \\right ) \\right ) \\right ) = \\partial _ { w _ { b b } } \\left ( \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\right ) \\cdot \\partial _ { w _ { a a } } \\left ( G \\left ( W , Z \\right ) \\right ) + \\frac { 1 } { I _ { q '^ { 2 } } + R \\left ( W ' \\right ) } \\cdot \\partial ^ { 2 } _ { w _ { b b } w _ { a a } } \\left ( G \\left ( W , Z \\right ) \\right ) , \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\Phi ( z _ k ) < \\Phi ( z _ { k + 1 } ) = \\cdots = \\Phi ( z _ { k + r } ) < \\begin{cases} \\Phi ( z _ { k + r + 1 } ) & \\quad , \\\\ \\rho & \\quad \\end{cases} \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\Phi ( z _ k ) < \\Phi ( z _ { k + 1 } ) = \\cdots = \\Phi ( z _ { k + r } ) < \\Phi ( z _ { k + r + 1 } ) , \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\gamma _ { \\ell } ^ { i } = \\frac { 1 } { 2 } ( \\lambda _ { \\epsilon } \\gamma _ { \\ell - 1 } ^ { i - 1 } + \\gamma _ { \\ell - 1 } ^ { i + 1 } ) . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} u '' ( t ) + \\frac { 1 - 2 \\alpha } { t } u ' ( t ) & = A u ( t ) \\big ( t \\in ( 0 , \\infty ) \\big ) , \\\\ u ( 0 ) & = x , \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} n + p = r + s - k = l - m . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} h ^ { 1 , 0 } ( Z ) = h ^ { 2 , 0 } ( Z ) = h ^ { 3 , 0 } ( Z ) = 0 . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\zeta _ j ( r ) = ( r e ^ { 2 \\pi i \\frac { j - 1 } { k } } , 0 ) \\in \\R ^ 3 , j = 1 , \\ldots , k , \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{gather*} \\frac { \\prod \\limits _ { r = 1 } ^ m { \\prod \\limits _ { j = \\theta ( N - m + 1 ) } ^ { \\theta ( N - r + 1 ) - 1 } { \\big ( z _ r - q ^ { j - \\theta } \\big ) } } } { \\prod \\limits _ { \\substack { 1 \\leq i < j \\leq m \\\\ 0 \\leq k < \\theta } } { ( z _ i - q ^ k z _ j ) } } . \\end{gather*}"}
-{"id": "7619.png", "formula": "\\begin{align*} \\mu _ 3 = - 1 + \\sqrt { 1 + \\frac { c _ { 1 , \\infty } } { c _ { 3 , \\infty } } - ( 1 + \\mu _ 1 ) ^ 2 \\frac { c _ { 1 , \\infty } } { c _ { 3 , \\infty } } } \\geq \\sqrt { 1 + \\frac { c _ { 1 , \\infty } } { 2 c _ { 3 , \\infty } } } - 1 . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\left [ A \\omega _ { 1 } , \\omega _ { 2 } \\right ] & = \\left \\langle L J L \\omega _ { 1 } , \\omega _ { 2 } \\right \\rangle = \\left \\langle J L \\omega _ { 1 } , L \\omega _ { 2 } \\right \\rangle = - \\left \\langle L \\omega _ { 1 } , J L \\omega _ { 2 } \\right \\rangle \\\\ & = - \\left [ \\omega _ { 1 } , A \\omega _ { 2 } \\right ] , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\partial _ N w _ { \\rm i m p . } = ( N \\cdot \\nu ) \\partial _ \\nu w _ { \\rm i m p . } + \\big ( N - ( N \\cdot \\nu ) \\nu \\big ) \\cdot \\nabla w _ { \\rm i m p . } \\mbox { o n } \\Gamma ^ 0 _ { \\rm o u t } \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} g _ { 1 } ( [ X , Y ] , V ) & = g _ { 1 } ( \\varphi V , \\mathcal { A } _ { Y } \\mathcal { B } X + \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { B } Y ) - g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { C } Y ) , \\pi _ { \\ast } ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) ) \\\\ & - g _ { 1 } ( \\varphi V , \\mathcal { A } _ { X } \\mathcal { B } Y + \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { Y } \\mathcal { B } X ) - g _ { 2 } ( \\pi _ { \\ast } ( \\mathcal { C } X ) , \\pi _ { \\ast } ( \\nabla ^ { ^ { M _ 1 } } _ { Y } \\varphi V ) ) . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} S _ f ( R ) : = \\sum _ { n \\leq R } a ( n ) . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\varphi : R & \\to S = \\mathbb { Q } [ y _ 1 , \\dots , y _ 5 ] \\\\ x _ 2 + x _ 4 & \\mapsto y _ 1 \\\\ x _ 1 + x _ 3 + x _ 5 & \\mapsto y _ 2 \\\\ x _ 3 & \\mapsto y _ 3 \\\\ x _ 4 & \\mapsto y _ 4 \\\\ x _ 5 & \\mapsto y _ 5 \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\partial _ { i } ( n _ \\mu \\circ \\eta ) = - g ^ { k l } \\partial _ { i k } \\eta ^ { \\tau } \\hat n _ { \\tau } \\partial _ { l } \\eta _ { \\mu } . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\theta ( r ) = \\sup \\limits _ { z \\in \\partial A _ r } \\max \\limits _ { i , j = 1 , . . . , 2 d } | \\frac { \\partial ^ 2 H } { \\partial z _ i \\partial z _ j } | \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\big ( F _ A ' ( \\omega _ B ) - 1 \\big ) \\big ( F _ B ' ( \\omega _ A ^ c ) - 1 \\big ) - 1 = \\mathcal { S } + O ( | \\Lambda _ A | ) \\ , , F _ A ' ( \\omega _ B ^ c ) - F _ A ' ( \\omega _ B ) = O ( | \\Lambda _ B | ) \\ , . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\cos \\angle \\mathbf u _ i \\mathbf z ' \\mathbf u _ j = \\frac { \\| \\mathbf u _ i - \\mathbf z ' \\| ^ 2 + \\| \\mathbf u _ j - \\mathbf z ' \\| ^ 2 - \\| \\mathbf u _ i - \\mathbf u _ j \\| ^ 2 } { 2 \\| \\mathbf u _ i - \\mathbf z ' \\| \\| \\mathbf u _ j - \\mathbf z ' \\| } = \\frac { 2 ( r ' ) ^ 2 - 1 } { 2 ( r ' ) ^ 2 } \\leq 0 \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{gather*} B \\coloneqq \\begin{pmatrix} _ 0 ( 1 , n - 1 ) & 0 \\\\ 0 & \\pm 1 \\end{pmatrix} . \\end{gather*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\hat { \\mathcal M } _ { \\hat { \\mathfrak { a } } } = \\hat { \\mathcal M } _ { \\mathfrak { a } } = \\mathcal M . \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) ( \\ell ^ * ( F ^ \\circ ) + 1 ) + \\sum _ { F \\in \\Delta ^ \\circ [ 0 ] } \\ell ( F ) = \\ell ( \\Delta ^ \\circ ) - 1 - \\sum _ { F \\in \\Delta ^ \\circ [ 2 ] } \\ell ^ * ( F ) + \\sum _ { F \\in \\Delta ^ \\circ [ 1 ] } \\ell ^ * ( F ) \\ell ^ * ( F ^ \\circ ) . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { u ( \\cdot ) , x ( \\cdot ) } & \\ \\int \\limits _ { 0 } ^ { T } K ( x , u ) \\ ; d t \\\\ \\textnormal { s . t . } & \\dot { x } = f ( x , u ) & g ( u ) & \\geq 0 \\\\ & x ( 0 ) = x _ 0 & h ( x ) & \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} | \\mathcal { A } | + | \\mathcal { D } | = 1 + \\sum _ { i = 0 } ^ { r - 3 } \\binom { r } { i } = 2 ^ r - \\binom { r + 1 } { 2 } \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} u _ t + u _ { x x x } - 6 u u _ x + f ( x , u , u _ x , u _ { x x } , u _ { x x x } ) = 0 \\ , . \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} [ a , c , d , e ] \\cdot [ \\emptyset ] & = [ a , c ] \\cdot [ d , e ] - [ a , d ] \\cdot [ c , e ] + [ a , e ] \\cdot [ c , d ] , \\\\ [ a , b , d , e ] \\cdot [ \\emptyset ] & = [ a , b ] \\cdot [ d , e ] - [ a , d ] \\cdot [ b , e ] + [ a , e ] \\cdot [ b , d ] , \\\\ [ a , b , c , e ] \\cdot [ \\emptyset ] & = [ a , b ] \\cdot [ c , e ] - [ a , c ] \\cdot [ b , e ] + [ a , e ] \\cdot [ b , c ] , \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} 0 = \\begin{bmatrix} m _ 1 & m _ 1 ^ 2 & \\cdots & m _ 1 ^ { k - 1 } \\\\ m _ 2 & m _ 2 ^ 2 & \\cdots & m _ 2 ^ { k - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ m _ { k - 1 } & m _ { k - 1 } ^ 2 & \\cdots & m _ { k - 1 } ^ { k - 1 } \\end{bmatrix} \\begin{bmatrix} p d _ 1 \\\\ p ^ 2 d _ 2 \\\\ \\vdots \\\\ p ^ { k - 1 } d _ { k - 1 } . \\end{bmatrix} \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\left [ P , 2 v \\ , G _ 2 ( u ) - 2 u \\ , G _ 2 ( v ) - ( u - v ) \\ , G _ 2 ( u ) \\ , G _ 2 ( v ) \\right ] = 0 . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\big \\| \\dot \\Psi ^ { t + s } ( x ) - \\dot \\Psi ^ t ( x ) \\big \\| _ { C ^ { 0 } ( \\R ^ n ; \\R ^ n ) } = o ( 1 ) \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} F _ T ( x ) - 1 - G _ 1 ( x ) & = x ( F _ T ( x ) - 1 ) + \\frac { x ^ 2 } { 1 - x } ( K - 1 ) \\\\ & + \\sum _ { m \\geq 2 } J _ m ( x ) - \\frac { x } { 1 - x } J _ 1 ( x ) + \\sum _ { m \\geq 2 } x ^ m \\sum _ { j = 1 } ^ { m - 1 } \\left ( \\frac { 1 } { ( 1 - x ) ^ j - 1 } \\right ) ( K - 1 ) \\ , . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\theta _ { U _ 1 } - \\theta _ { U _ 0 } = i \\ , d \\ , l o g \\ , g _ { 0 1 } \\ , \\ , i n \\ , \\ , U _ 0 \\cap U _ 1 \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} g ( s ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { g _ F ( s ) } { d _ F ^ { 2 s } } \\textrm { a n d } h ( s ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\frac { R _ d ( F ) } { d _ F ^ { 2 s } } . \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} \\Omega ( x , r ) : = \\Omega \\cap B ( x , r ) . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\begin{pmatrix} s _ 2 & \\cdots & s _ d & \\varphi _ { d + 1 } & \\cdots & \\varphi _ n \\\\ - \\sum _ j \\lambda _ j \\varphi _ { j 2 } & \\cdots & - \\sum _ j \\lambda _ j \\varphi _ { j d } & \\lambda _ { d + 1 } & \\cdots & \\lambda _ n \\end{pmatrix} \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\frac { d w _ \\tau } { d \\tau } + \\beta ^ { \\prime } ( w _ { \\Omega } ) \\frac { d w _ \\tau } { d \\tau } = 0 , & \\Omega , \\\\ \\frac { d w _ \\tau } { d \\tau } = - \\nabla w _ \\Omega \\cdot \\theta , & \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\left | l _ m \\ ( \\ ( \\sum _ { k = 0 } ^ n u _ k \\ ) \\ ( x _ m \\ ) \\ ) \\right | \\leqslant M \\left \\| l _ m \\right \\| _ { \\ ( H ^ s \\ ) ' } \\left \\| x _ m \\right \\| _ { H ^ s } . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\mathbb E \\| F \\star \\bar { \\mu } \\| ^ 2 = \\mathbb E \\| F \\| ^ 2 \\star \\nu . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} { \\sqrt 3 } - 1 g ' = 1 + { \\sqrt 3 } < 2 . 7 4 < g . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { 2 n } H _ { \\rm B e t t i } ^ i ( X , { \\bf Q } ) \\to H _ { \\rm B e t t i } ^ d ( X , { \\bf Q } ) \\to \\bigoplus _ { i = 0 } ^ { 2 n } H _ { \\rm B e t t i } ^ i ( X , { \\bf Q } ) . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\tilde { y } _ j = \\max ( | x _ j | , y _ j ) . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha } f ( t ) = \\frac { 1 } { \\Gamma ( n - \\alpha ) } \\frac { d ^ { n } } { d t ^ { n } } \\int _ { a } ^ { t } \\frac { f ( s ) d s } { ( t - s ) ^ { \\alpha - n + 1 } } , \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} T ( z , t z ) = z t + \\sum _ { n \\geq 2 } \\sum _ { k = 1 } ^ { n - 1 } N _ { n - 1 , k } z ^ { n } t ^ { k } = \\sum _ { n \\geq 1 } z ^ { n } \\tilde { N } _ { n - 1 } ( t ) . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} z _ i = \\left ( \\frac { ( \\theta - 1 ) z _ i + \\sum _ { j = 1 } ^ { q - 1 } z _ j + 1 } { \\theta + \\sum _ { j = 1 } ^ { q - 1 } z _ j } \\right ) ^ k , 1 \\leq i \\leq q - 1 . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} j ( g h ) \\overline { \\mu ( g , h ) } = j ( g ) j ( h ) \\mu ( h ^ { - 1 } , g ^ { - 1 } ) g , h \\in \\Lambda \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} e _ n = \\sum _ { \\lambda \\vdash n } \\frac { M B _ { \\lambda } \\Pi _ { \\lambda } \\widetilde { H } _ { \\lambda } } { w _ { \\lambda } } , \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\varphi V = \\phi V + \\omega V \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha ( s ) f ( x _ { i - 1 } ( t ) ) + a ( s ) f ( x _ i ( t ) ) + \\beta ( s ) f ( x _ { i + 1 } ( t ) ) , \\ , \\ , i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} a _ 2 ' & : = 2 8 \\log ( ( 1 . 2 6 4 ) ^ 3 \\beta _ 3 ) > 6 0 \\log \\left ( 1 . 2 6 4 \\cdot \\frac { 1 } { \\sqrt [ 3 ] { 2 } } \\sqrt { c } \\right ) + 8 \\log ( ( 1 . 2 6 4 ) ^ 3 \\beta _ 3 ) \\geq \\\\ & \\geq \\varrho | \\log \\gamma _ 2 | - \\log | \\gamma _ 2 | + 8 h ( \\gamma _ 2 ) . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} d ( x , y ) = \\max _ { t \\in [ 0 , 1 ] } d ( \\pi g _ t ( x ) , \\pi g _ t ( y ) ) , \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\mathbb E \\| ( F \\star \\bar { \\mu } ) _ { \\infty } \\| _ { L ^ q ( S ) } ^ p & = \\mathbb E \\Bigl \\| \\sum _ { n \\geq 0 } ( F _ n \\star \\bar { \\mu } ) _ { \\infty } \\Bigr \\| _ { L ^ q ( S ) } ^ p = \\mathbb E \\Bigl \\| \\sum _ { n \\geq 0 } d _ n \\Bigr \\| _ { L ^ q ( S ) } ^ p \\eqsim _ { p , q } \\| ( d _ n ) \\| _ { s _ { p , q } } ^ p \\\\ & \\eqsim _ p ( \\| ( d _ n ) \\| _ { S ^ p _ q } + \\| ( d _ n ) \\| _ { D ^ p _ { q , q } } + \\| ( d _ n ) \\| _ { D ^ p _ { p , q } } ) ^ p . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} w \\ = \\ v + \\sum _ a \\beta _ a ( b ) ( 1 , a ) \\ + \\ c \\ = \\ \\sum _ s ( \\beta _ a ( b ) + \\nu ) ( 1 , a ) \\ + \\ c \\ , , \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\gamma ( A ) = \\sup \\{ \\langle A \\chi _ 0 , f \\rangle \\colon \\| f \\| _ { H [ X ] \\odot H [ Y ' ] } \\leq 1 \\} . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} \\mu ( \\omega _ q ) ^ 2 = \\frac { ( \\log \\log x ) ^ 2 } { \\phi ( q ) ^ 2 } + O \\bigg ( \\frac { \\log q } { \\phi ( q ) ^ 2 } \\log \\log x \\bigg ) . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\mathbf { t } _ { \\mathcal { N } } ^ { r _ { h } } \\sum _ { \\ell } b _ { \\ell } \\mathbf { t } _ { \\mathcal { N } } ^ { \\ell } = P ^ { + } _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) \\cdot \\prod _ { n \\in \\mathcal { N } } ( 1 - \\mathbf { t } _ { \\mathcal { N } } ^ { a _ { n } } ) + R _ { h } ( \\mathbf { t } _ { \\mathcal { N } } ) \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } u ^ \\epsilon ( x , 0 ) = \\left \\{ \\begin{array} { l l } u _ 0 ( 0 ) \\ & | x | < 1 , \\\\ u _ 0 ( x ) \\ & | x | = 1 , \\\\ 0 \\ & | x | > 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} H ^ 0 ( x ) : = \\{ \\mathbf { u } \\in \\mathbb { R } ^ d \\mid \\langle \\mathbf { u } , \\mathbf { a } \\rangle \\geq b \\} \\ , \\ , \\ , \\ , \\ , \\ , H ^ 1 ( x ) : = \\{ \\mathbf { u } \\in \\mathbb { R } ^ d \\mid \\langle \\mathbf { u } , \\mathbf { a } \\rangle \\leq b \\} = H ^ 0 ( - x ) , \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} Y _ n & = \\int _ 0 ^ T \\left [ \\varphi ( t , \\gamma _ n ( t , X _ n ( t ^ - ) ) ) - \\varphi ( t , \\gamma _ n ( t , X ( t ^ - ) ) ) \\right ] d t \\\\ Z _ n & = \\int _ 0 ^ T \\int _ A \\varphi ( t , a ) [ \\delta _ { \\gamma _ n ( t , X ( t ^ - ) ) } - \\widehat { \\gamma } ( t , X ( t ^ - ) ) ] ( d a ) d t . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { { \\bf { I I } } ( x , t ) } { A _ { d } ( b _ { d } ( t ) ) } & = \\frac { \\kappa c _ { 1 } K _ { d } } { \\alpha \\rho } ( x ^ { - \\rho / \\alpha } - 1 ) . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} & G \\left ( Z , W \\right ) : = E '^ { - 1 } \\otimes G \\left ( V ^ { - 1 } \\otimes Z , A ^ { - 1 } \\otimes W \\right ) , \\\\ & F \\left ( Z , W \\right ) : = V '^ { - 1 } \\otimes F \\left ( V ^ { - 1 } \\otimes Z , A ^ { - 1 } \\otimes W \\right ) . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} H ( X _ 1 ^ { k _ 1 } \\cdots X _ m ^ { k _ m } , \\mathfrak { c } I _ m ) = \\prod _ { j = 1 } ^ m H _ { k _ j } ( X _ j , \\mathfrak { c } ) \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} & | \\nabla \\overline { \\eta } ( t ) | \\le [ ( 1 - t ) u ^ p + t v ^ p ] ^ { ( 1 - p ) / p } [ ( 1 - t ) u ^ { p - 1 } | \\nabla u | + t v ^ { p - 1 } | \\nabla v | ] \\\\ & = [ ( 1 - t ) u ^ p + t v ^ p ] ^ { ( 1 - p ) / p } [ ( 1 - t ) ^ { 1 / p ' } u ^ { p - 1 } ( 1 - t ) ^ { 1 / p } | \\nabla u | + t ^ { 1 / p ' } v ^ { p - 1 } t ^ { 1 / p } | \\nabla v | ] \\\\ & \\le [ ( 1 - t ) u ^ p + t v ^ p ] ^ { ( 1 - p ) / p } [ ( 1 - t ) u ^ p + t v ^ p ] ^ { 1 / p ' } [ ( 1 - t ) | \\nabla u | ^ p + t | \\nabla v | ^ p ] ^ { 1 / p } ~ \\mbox { a . e . i n } ~ \\Omega \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} \\frac { u _ { k + 1 } ( z ) } { u _ { k } ( z ) } = \\frac { \\Gamma ( \\alpha k + \\beta ) z } { ( k + 1 ) \\Gamma ( \\alpha k + \\beta + \\alpha ) } \\leq \\frac { \\Gamma ( \\alpha k + \\beta ) } { \\Gamma ( \\alpha k + \\beta + \\alpha ) } . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} g ^ 0 & = | \\Omega | ^ { - 1 } \\int _ { \\Omega } \\widetilde { g } ( \\mathbf { x } ) \\ , d \\mathbf { x } , \\\\ \\widetilde { g } ( \\mathbf { x } ) & : = g ( \\mathbf { x } ) ( b ( \\mathbf { D } ) \\Lambda ( \\mathbf { x } ) + \\mathbf { 1 } _ m ) . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\tau _ 1 = T - \\frac { 2 ( k _ M b _ M - k _ E b _ E ) } { b _ M k _ M k _ E } , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} x _ m - \\mathcal { K } _ m x _ m = f \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\tilde T _ f g ( x ) = \\tilde T ^ { f \\circ \\pi } g ( x ) \\quad \\quad f \\in C _ c ( D ) , \\ > g \\in C _ c ( X ) , \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} | V _ { X ^ m } | = \\frac { q - 1 } d + 1 , \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} ( \\eta \\circ \\mu ) _ { i j } = g ^ { k l } \\left ( \\eta _ { i k } \\mu _ { j l } - \\eta _ { j k } \\mu _ { i l } \\right ) . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\frac { 1 } { B } = \\frac { A } { \\lambda _ 1 \\lambda _ 2 } A = \\frac { \\lambda _ 1 \\lambda _ 3 } { C } . \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} c = \\frac { h _ \\mu } { \\int \\rho d \\mu } \\geq \\frac { h _ { \\nu } } { \\int \\rho d \\nu } . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} G ' \\left ( t \\right ) = \\begin{cases} G ' \\left ( t - 1 \\right ) & E _ { v \\left ( t \\right ) } \\left ( t \\right ) E _ { w \\left ( t \\right ) } \\left ( t \\right ) e \\left ( t \\right ) \\nsubseteq A ; \\\\ G ' \\left ( t - 1 \\right ) + e \\left ( t \\right ) & \\end{cases} \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} \\lim _ { k _ 0 < k \\to \\infty } \\frac { F _ { k + 1 } ^ { \\bf a } } { F _ { k } ^ { \\bf a } } = \\lambda _ { i _ 0 } \\cdot \\lim _ { k _ 0 < k \\to \\infty } \\biggr ( \\frac { F _ { k + 1 } ^ { \\bf a } } { ( k + 1 ) ^ { j _ 0 } \\lambda _ { i _ 0 } ^ { k + 1 } } \\cdot \\frac { k ^ { j _ 0 } \\lambda _ { i _ 0 } ^ k } { F _ k ^ { \\bf a } } \\biggr ) = \\lambda _ { i _ 0 } . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} \\chi = \\begin{pmatrix} 5 & 1 0 & 2 5 & 1 2 \\\\ 1 5 + \\frac { 1 } { 4 } & 3 0 & 3 5 & 1 3 - \\frac { 1 } { 4 } \\\\ 2 0 + \\frac { 1 } { 4 } & 4 0 & 4 5 & 2 1 - \\frac { 1 } { 4 } \\\\ 1 2 - 2 \\cdot \\frac { 1 } { 4 } & 1 3 & 2 1 & 1 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\norm { T ^ P a } _ 1 = \\norm { T _ 0 a } _ 1 \\leq C . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} \\frac { d C } { d t } & = \\frac { A } { B } = \\frac { A ^ 2 } { \\lambda _ 1 \\lambda _ 2 } = \\frac { 1 } { \\lambda _ 1 \\lambda _ 2 } \\Bigg ( \\frac { \\lambda _ 1 ^ 2 \\lambda _ 3 ^ 2 } { C ^ 2 } \\Bigg ) , \\\\ C ^ 2 \\frac { d C } { d t } & = \\frac { \\lambda _ 1 \\lambda _ 3 ^ 2 } { \\lambda _ 2 } . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\phi _ j ' ( u ) = ( - 1 ) \\int _ { ( t _ { j - 1 } , t _ j ] \\times \\R _ + } y e ^ { - u y } \\rho ( d ( x , y ) ) \\cdot \\phi _ j ( u ) , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\int _ { \\Omega } b _ l W = \\int _ { Q _ l } b _ l W = 0 . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\left | { \\dd ^ m \\over \\dd z ^ m } S _ { n , r } ( z ) \\Big | _ { z = 0 } \\right | \\leq C ^ m ( m - 1 ) ! r n ^ { 1 - m } . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} \\rho = - \\frac { 1 } { \\lambda } \\log \\frac { | \\chi ^ { ( d ) } | } { R ( d - 1 ) ! \\lambda ^ { d - 1 } } ; \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} \\bar Y _ t & = \\int _ { t \\wedge \\tau } ^ { m \\wedge \\tau } f \\bigl ( s , Y ^ m _ s , Z ^ m _ s , V ^ m _ s \\bigr ) d s - \\int _ { t \\wedge \\tau } ^ { m \\wedge \\tau } \\bar Z _ s d W _ s \\\\ [ - 2 p t ] & - \\int _ { t \\wedge \\tau } ^ { m \\wedge \\tau } \\int _ U \\bar V _ s ( e ) \\widehat \\pi ( d u , d s ) - \\bar M _ { m \\wedge \\tau } + \\bar M _ { t \\wedge \\tau } . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} - ( \\lambda _ { n , 1 } ^ { ( 1 ) } - \\lambda _ { n , 1 } ^ { ( 2 ) } ) + \\log \\frac { \\lambda _ { n , 1 } ^ { ( 1 ) } } { \\lambda _ { n , 1 } ^ { ( 2 ) } } = \\log \\left ( 1 + O \\left ( \\sum _ { i = 1 } ^ 2 \\frac { 1 } { \\lambda _ { n , 1 } ^ { ( i ) } } \\right ) \\right ) = O \\left ( \\sum _ { i = 1 } ^ 2 \\frac { 1 } { \\lambda _ { n , 1 } ^ { ( i ) } } \\right ) . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\epsilon = ( k _ 1 - 1 ) ( k _ 2 - 1 ) + \\dim L + k _ 1 + ( i - 1 ) \\Big ( 1 + ( \\mu ( \\beta _ 2 ) + k _ 2 ) \\dim L \\Big ) \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\delta _ 0 = \\min \\{ \\delta ^ { ( l ^ 1 , l ^ 2 ) } : \\ : l ^ 1 = 1 , \\dots , m ^ 1 \\ , , \\ , l ^ 2 = 1 , \\dots , m ^ 2 \\} \\ , \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\left ( { \\rm s i n g } _ { H } ( u ) \\cap B _ { \\frac { r } { 2 } } ( 0 ) \\right ) \\setminus B _ { \\rho r } ( L ) = \\emptyset . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ p ^ p + C \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ { q } ^ p \\leq C \\sum _ { i = 1 } ^ { N } \\| v _ i \\| _ { \\mu + p - 1 } ^ { \\mu + p - 1 } + C . \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} O \\left ( ( n / \\log { n } ) ^ { - \\frac { \\alpha ^ { * } } { 2 \\alpha ^ { * } + d } \\sum _ { l = 1 } ^ d \\left ( \\frac { 1 } { d } + \\frac { 1 } { 2 \\alpha ^ { * } } - \\frac { 1 } { 2 \\alpha _ l } - \\frac { r _ l } { \\alpha _ l } \\right ) } \\right ) = O ( \\epsilon _ { n , \\boldsymbol { r } } ) . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} H [ Z ] ^ { \\wedge } = H [ Z ^ { \\wedge } ] . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\{ u \\in H ^ 1 ( \\R ) \\cap H ^ 2 ( \\R \\backslash \\{ 0 \\} ) : \\partial _ x u ( 0 + ) - \\partial _ x u ( 0 - ) = 2 q u ( 0 ) \\} , \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } \\alpha _ i F ( d _ i ) & + \\alpha _ k \\bigl ( ( 1 - \\beta _ { 1 k } ) F ( d ' ) + \\beta _ { 1 k } F ( d '' ) \\bigr ) \\\\ & \\le \\max \\left ( \\bigl ( 1 - \\sqrt { \\gamma } \\bigr ) F ( 0 ) + \\sqrt { \\gamma } F ( \\sqrt { \\gamma } ) , ( 1 - \\eta ) F ( \\eta ) + \\eta F ( 1 ) \\right ) \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} [ V ( t ) w ( t ) ] ( x ) = e ^ { - i \\lambda | W ( x ) | ^ 2 \\log t } W ( x ) + \\mathcal { O } ( t ^ { - \\frac 1 4 + \\beta } ) \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 3 } \\Phi ^ + ( f ) | v | ^ 2 d v & = \\int _ { v _ 1 > 0 } e ^ { - \\frac { 1 } { \\tau | v _ 1 | } \\int ^ x _ 0 \\rho _ f ( y ) d y } f _ { L } ( v ) | v | ^ 2 d v \\cr & + \\int _ { v _ 1 > 0 } \\int _ { 0 } ^ { x } \\frac { 1 } { \\tau | v _ 1 | } e ^ { - \\frac { \\int ^ x _ y \\rho _ f ( z ) d z } { \\tau | v _ 1 | } } \\rho _ f ( y ) \\mathcal { M } _ { \\nu } ( f ) | v | ^ 2 d y d v . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} \\xi \\left ( x , t \\right ) = \\left \\vert x \\right \\vert ^ { 2 } - \\varrho ^ { 2 } \\left ( t - T / 2 \\right ) ^ { 2 } , \\varphi _ { \\lambda } \\left ( x , t \\right ) = \\exp \\left ( \\lambda \\xi \\left ( x , t \\right ) \\right ) . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} d \\rho ^ { ( a , b ) } ( x ) = w ^ { ( a , b ) } ( x ) d x \\Bigr | _ { [ - 1 , 1 ] } + \\frac { b - a } { b } d \\delta _ { s _ 0 } \\quad { \\rm f o r } b > a \\ge 2 \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} S = \\begin{pmatrix} X ^ { 1 / 2 } & 0 \\\\ X ^ { - 1 / 2 } Y & X ^ { - 1 / 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} U ( x ) = \\frac { m _ i } { \\alpha \\vert x - c _ i \\vert ^ { \\alpha } } + U _ i ( x ) , i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} Y _ i ( t ) = a _ { i 1 } \\int _ 0 ^ t X _ 1 ( s , M _ 1 ) d s + a _ { i 2 } \\int _ 0 ^ t X _ 2 ( s , M _ 2 ) d s + \\cdots + a _ { i m } \\int _ 0 ^ t X _ m ( s , M _ m ) d s + B _ i ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\int _ { \\Omega } D u \\cdot \\vec g = \\int _ { \\Omega } \\vec b \\cdot D v = \\int _ { \\Omega ( x _ 0 , r ) } \\vec b \\cdot \\left ( D v - \\overline { D v } _ { \\Omega ( x _ 0 , r ) } \\right ) . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\mathcal { F } ( u ) = \\displaystyle u _ { \\bar { \\zeta } } + A ( u ) \\overline { u _ { \\zeta } } . \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} H _ { ( c , c ^ * ) } ( x ) = \\tilde { c } x ^ { - \\alpha } \\frac { M x ^ \\rho - 1 } { \\rho } \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\ < m , n \\ > _ C & = \\dfrac { 2 } { 3 } \\Re ( ( \\rho - \\overline { \\rho } ) \\ m U ^ t \\overline { n } ) , \\\\ \\ < m ' , n ' \\ > _ E & = \\dfrac { 2 } { 3 } \\Re ( ( \\rho ' - \\overline { \\rho ' } ) \\ m ' \\overline { n ' } ) , \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} ( \\gamma , z ) ( \\eta , w ) = ( \\gamma \\eta , \\sigma ( \\gamma , \\eta ) z w ) ( \\gamma , z ) ^ * = ( \\gamma ^ { - 1 } , \\overline { \\sigma ( \\gamma , \\gamma ^ { - 1 } ) } \\overline { z } ) . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} - \\triangle _ { H } u + u = | u | ^ { q - 2 } u , \\ ; \\ ; u \\in H ^ 1 ( \\mathbb { H } ^ { N } ) , \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} x _ { L , k } & = ( I _ n - ( L ( I _ n - \\widetilde { A } _ k ^ { \\dagger } \\widetilde { A } _ k ) ) ^ { + } L ) \\widetilde { A } _ k ^ + b \\\\ & = x _ k - ( L ( I _ n - \\widetilde { A } _ k ^ { \\dagger } \\widetilde { A } _ k ) ) ^ { + } L x _ k . \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} f ( z + 1 ) = f ( z ) , \\ , \\ f ( z + \\tau ) = e ^ { 2 \\pi i \\lambda } f ( z ) . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , & \\forall 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k } = p _ { k } \\theta , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a ^ { i j } D _ { i j } u = f \\ ; \\mbox { i n } \\ ; \\Omega , u = 0 \\ ; \\mbox { o n } \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} H ' \\geq \\frac { t } { n } \\log n ^ { 1 / 2 } = \\frac { t } { 2 n } \\log n . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} u ^ { \\theta } = e ^ { t L _ 0 } u ^ { \\theta } _ 0 - \\int _ { 0 } ^ { t } e ^ { ( t - s ) L _ 0 } \\Big ( v \\cdot \\nabla u ^ { \\theta } + \\frac { u ^ { r } } { r } u ^ { \\theta } \\Big ) \\dd s . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} F ( f ) = \\mathbb E \\sum _ { n = 1 } ^ { \\infty } \\langle f _ n , \\tilde g _ n \\rangle , \\ ; \\ ; \\ ; f = ( f _ n ) _ { n \\geq 0 } \\in Q _ q ^ p . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} x _ j - \\frac { 1 } { j } \\sum _ { k = 1 } ^ j x _ k = \\frac { 1 } { j } \\left ( ( j - 1 ) x _ j - \\sum _ { k = 1 } ^ { j - 1 } x _ k \\right ) , j \\geq 2 , \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} A _ n ^ - = A _ { n - 1 } ^ - - 2 a ( n - 2 ) + \\sum _ { j = 0 } ^ { n - 1 } a ( j ) + C _ { n - 1 } - C _ { n - 2 } + 2 ^ { n - 2 } - 2 n + 3 \\ , , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\begin{array} { l } Q ' = \\frac { 1 } { 2 } ( \\tau + 1 ) E ^ { - 1 } , \\\\ q ' = \\frac { 1 } { 2 } ( c + w ) , \\\\ \\theta ' = \\displaystyle { \\frac { \\beta } { \\tau + 1 } } + \\left \\langle \\displaystyle { \\frac { 1 - \\tau } { 4 } E ^ { - 1 } c } - \\frac { 1 } { 2 } E ^ { - 1 } w , c \\right \\rangle + \\displaystyle { \\frac { \\left \\langle E ^ { - 1 } ( w + \\tau c ) , w + \\tau c \\right \\rangle } { 4 ( \\tau + 1 ) } } . \\end{array} \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} r a ^ { a / 2 } \\mathtt { J } ' _ \\nu ( r ) - \\left ( ( \\nu - 1 ) ( 1 - a ) a ^ { a / 2 } - a ( 1 - \\nu ) + \\beta \\right ) \\mathtt { J } _ \\nu ( r ) = 0 . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} & U M _ { 4 2 } ^ { \\tau } + M _ { 1 2 } U _ n = T M _ { 3 2 } U _ n , M _ { 1 2 } ^ { \\tau } U + U _ n M _ { 4 2 } = U _ n M _ { 2 2 } T . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} \\Gamma ( \\lambda ) & = 1 + i \\int _ { \\real } u ( x ) m _ e ( x , \\lambda + 0 i ) e ^ { - i \\lambda x } ~ d x \\\\ & = \\frac { 1 } { 1 - i \\int _ { \\real } u ( x ) m _ e ( x , \\lambda - 0 i ) e ^ { - i \\lambda x } ~ d x } , \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} c ^ { \\circ } _ { j l } : = \\min \\{ c _ * , n ^ { - 1 } | \\gamma ^ { \\circ } _ l - \\gamma ^ { \\circ } _ j | \\} . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{gather*} x \\cdot x \\coloneqq - x _ { 0 } ^ { 2 } + x _ { 1 } ^ { 2 } + \\dots + x _ { n } ^ { 2 } = R ^ { 2 } , \\end{gather*}"}
-{"id": "3023.png", "formula": "\\begin{align*} _ F ^ \\textup { P R F } ( m ) = \\max _ F ^ \\textup { P R F } ( A ) , \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} P \\ = \\ \\bigcap _ { w \\in \\R ^ n } \\{ x \\in \\R ^ n \\mid w \\cdot x \\leq h _ P ( w ) \\} \\ , . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { D _ { 1 - x / t } - K _ { d } } { A _ { 1 } ( b _ { 1 } ( t ) ) } = \\lim _ { t \\to \\infty } \\left [ \\frac { { \\bf { I } } ( x , t ) } { A _ { 1 } ( b _ { 1 } ( t ) ) } - \\frac { { \\bf { I I } } ( x , t ) } { A _ { 1 } ( b _ { 1 } ( t ) ) } \\right ] = - \\frac { c _ { 1 } K _ { d } } { \\alpha \\rho _ { 1 } } ( x ^ { - \\rho _ { 1 } / \\alpha } - 1 ) . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { C _ { n , j } } { n } = \\frac { \\nu _ j } { \\mu } a . s . \\mbox { f o r } ~ j \\geq 1 , \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} \\psi \\left ( \\xi \\right ) = \\frac { 4 f \\left ( z \\right ) { f } ^ { \\prime \\prime } \\left ( z \\right ) - 5 { f } ^ { \\prime 2 } \\left ( z \\right ) } { 1 6 f ^ { 3 } \\left ( z \\right ) } + \\frac { g \\left ( z \\right ) } { f \\left ( z \\right ) } . \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} p _ 0 h _ 0 / h _ n = p _ 1 h _ 1 / h _ 0 = \\cdots = p _ n h _ n / h _ { n - 1 } \\ \\ \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} a ^ 2 = ( 2 n + 2 ) L ^ 2 > L ^ 2 = b ^ 2 \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} P ( g ) = \\sup \\left \\{ h _ \\mu + \\int g \\ , d \\mu \\right \\} , \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} Q [ \\varphi ] : = \\int _ { \\R ^ N } \\left ( | \\nabla \\varphi | ^ 2 - c V _ n ( x ) \\varphi ^ 2 \\right ) d \\mu , \\varphi \\in H ^ 1 _ \\mu . \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\log M _ { n , r } ( z ) = S _ { n , r } ( z ) + \\frac { z } { 2 } \\log ( r + 1 ) + \\frac { z r } { 2 } \\log 2 \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\frac { d v ( t ) } { d t } = ( a - 1 + \\alpha e ^ { \\mu } + \\beta e ^ { - \\mu } ) v ( t ) . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} 1 - \\sum _ { j = d + 1 } ^ n \\varphi _ j \\Phi _ j \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} D _ { \\bar \\Lambda _ 1 } E & = O ( \\varepsilon ^ { 2 - \\sigma } ) w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 4 + O ( w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 3 \\varepsilon ^ 2 ) + O ( \\varepsilon ^ 5 ) , \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} Q ( C _ i ) ^ 2 = \\frac { \\| \\delta _ i \\| ^ 2 } { C _ { i , \\infty } ^ 2 \\Bigl ( \\sqrt { \\overline { C _ i ^ 2 } } + \\overline { C _ i } \\Bigr ) ^ 2 } = \\frac { \\sqrt { \\overline { C _ i ^ 2 } } - \\overline { C _ i } } { C _ { i , \\infty } ^ 2 \\Bigl ( \\sqrt { \\overline { C _ i ^ 2 } } + \\overline { C _ i } \\Bigr ) } \\leq \\frac { 1 } { C _ { i , \\infty } ^ 2 } . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\psi _ k ( \\bar \\mu , \\zeta ) = \\frac { 1 } { 2 } ( M ( \\zeta ) \\ , \\bar \\mu ^ { \\frac { N - 2 } { 2 } } , \\bar \\mu ^ { \\frac { N - 2 } { 2 } } ) - \\frac { 1 } { 2 } \\ , B \\sum _ { i = 1 } ^ k \\bar \\mu _ i ^ 2 \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\langle B f , g \\rangle = \\langle \\chi _ { - n } A ( \\chi _ n f ) , g \\rangle \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} \\{ C ^ T \\} _ { l + 1 , m + 1 } = \\int _ { - \\infty } ^ \\infty T _ { T - l } ( \\lambda ) T _ { T - m } ( \\lambda ) \\ , d \\rho ^ { N , h } ( \\lambda ) , l , m = 0 , \\ldots , T - 1 , \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\frac { \\partial g _ { i j } } { \\partial t } = - 2 R _ { i j } \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{gather*} \\Upsilon ^ { a b } = \\theta ^ a \\circ \\omega ^ b - \\theta ^ b \\circ \\omega ^ a \\end{gather*}"}
-{"id": "6360.png", "formula": "\\begin{align*} F ( t ) = F ^ { ( 1 ) } _ { j l } ( t ) + F ^ { ( 2 ) } _ { j l } ( t ) , | t | \\le t ^ { 0 0 } _ { j l } . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { f ( n ) } { n } \\leq \\inf _ { d \\in \\N } \\frac { f ( d ) } { d } \\leq \\liminf _ { d \\rightarrow \\infty } \\frac { f ( d ) } { d } = \\liminf _ { n \\rightarrow \\infty } \\frac { f ( n ) } { n } . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} c _ { \\Omega , K , a } : = \\dfrac { \\displaystyle { \\int _ M S ( \\omega _ g , f _ { K , g , a } ) \\left ( \\frac { 1 } { f _ { K , g , a } } \\right ) ^ { 2 m + 1 } \\frac { \\omega ^ m } { m ! } } } { \\displaystyle { \\int _ M \\left ( \\frac { 1 } { f _ { K , g , a } } \\right ) ^ { 2 m + 1 } \\frac { \\omega ^ m } { m ! } } } , \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i \\Big ( G _ { i i } ( z ) - \\frac { 1 } { a _ i - \\omega _ B ( z ) } \\Big ) \\Big | \\prec \\frac { 1 } { N \\eta } \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} l a _ k y _ { k + 1 } + a _ { k - 1 } y _ { k - 1 } + b _ k y _ k = 0 , \\\\ y _ 0 = \\alpha , \\ , \\ , y _ 1 = \\beta . \\end{array} \\right . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 f ( t ) r _ k ( t ) d t = 0 , \\ , \\ , k \\geq 0 . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} E \\vert _ { S } = \\left \\{ \\left ( q ^ i , \\frac { \\partial W } { \\partial q ^ i } ; \\frac { \\partial F } { \\partial p _ i } , - \\frac { \\partial F } { \\partial q ^ i } \\right ) \\in T T ^ * Q : \\frac { \\partial F } { \\partial \\lambda ^ a } = 0 , \\frac { \\partial W } { \\partial \\mu ^ \\beta } ( q , \\mu ) = 0 \\right \\} , \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty | \\tilde { e } ^ { ( k ) } _ { n m } | = \\sum _ { m = 1 } ^ { n - 1 } \\frac { w _ k ( n ) } { w _ { k + 1 } ( m ) } | e _ { n m } | \\leq C \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n - 1 } \\frac { r _ k ^ { \\alpha _ n } } { r _ { k + 1 } ^ { \\alpha _ m } } \\frac { n ^ \\alpha } { m ^ \\alpha } . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = - g _ { 1 } ( \\phi V , \\varphi \\nabla ^ { ^ { M _ 1 } } _ { U } X ) - g _ { 1 } ( \\omega V , \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi X ) . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} \\frac { \\partial \\omega } { \\partial t } = - { \\rm R i c } ( \\omega ) + \\gamma \\omega + L _ X \\omega \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} g ( v ) = v \\ \\ \\mbox { i f a n d o n l y i f } \\ \\ v ^ { - 1 } g v \\in F _ S . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\log | { \\cal M } _ 0 | & \\le \\log | \\tilde { { \\cal M } } _ 0 | + \\log n + \\log \\log | { \\cal X } | + 2 , \\\\ \\log | { \\cal M } _ 0 | | { \\cal M } _ 1 | & \\le \\log | { \\cal T } ^ n _ { \\bar { X } } | + \\log n + \\log \\log | { \\cal X } | + 2 , \\\\ \\log | { \\cal M } _ 2 | & = \\log | \\tilde { { \\cal M } } _ 2 | , \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} | V _ { x _ { 1 } } ( a _ { 1 } ) | = C _ { 0 } > 0 . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} d \\left ( \\frac { \\Delta _ { N , 1 2 } } { \\Delta } \\right ) = \\psi ( N ) - 1 . \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} L _ m ( x ) & = x ^ m \\big ( x C ^ m F _ T ( x ) + ( C ^ m ( x ) - 1 ) ( F _ T ( x ) - 1 ) \\big ) \\ , . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} & R _ k ^ { \\rm N O M A } ( \\nu ) = \\\\ & \\left \\{ \\begin{array} { l } \\log _ 2 \\left ( 1 + p _ k ( \\nu ) g _ k ( \\nu ) \\right ) , \\ { \\rm i f } \\ g _ k ( \\nu ) > g _ { \\bar k } ( \\nu ) , \\\\ R _ { k \\rightarrow k } ^ { \\rm N O M A } ( \\nu ) , \\ { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d B } { d t } = \\frac { A } { D } = \\frac { \\lambda _ 1 \\lambda _ 2 ^ 2 \\lambda _ 3 } { \\lambda _ 4 B ^ 2 C } \\\\ \\frac { d C } { d t } = \\frac { A } { E } = \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ^ 2 } { \\lambda _ 5 B C ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A Q ( \\overline { \\partial } \\eta ) = \\widetilde { Q } ( \\overline { \\partial } \\eta ) \\overline { \\partial } _ A \\overline { \\partial } \\eta . \\end{gather*}"}
-{"id": "5511.png", "formula": "\\begin{align*} h _ i : = P _ { k , i + 1 } ^ { m _ { i + 1 } } \\cdot u _ { i + 1 } ^ { a _ { i + 1 } } \\cdots P _ { k , k } ^ { m _ k } \\cdot u _ k ^ { a _ k } \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\| x _ i \\| _ { X } ^ q \\right ) ^ { \\frac { 1 } { q } } \\le M \\left \\| \\sum _ { i = 1 } ^ n x _ i \\right \\| _ { X } . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{gather*} \\frac { g _ { n _ s + \\tau _ s ^ + - \\tau _ s ^ + } \\big ( q ^ { \\lambda _ 1 } t ^ { N - 1 } , \\dots , q ^ { \\lambda _ N } ; q , t \\big ) } { g _ { n _ s + \\tau _ s ^ + - \\tau _ s ^ - } ( t ^ { N - 1 } , \\dots , t , 1 ; q , t ) } \\\\ { } = \\frac { P _ { \\lambda } \\big ( q ^ { n _ s + \\tau _ s ^ + - \\tau _ s ^ - } t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } ( t ^ { N - 1 } , \\dots , t , 1 ; q , t ) } \\forall \\ , 1 \\leq s \\leq m . \\end{gather*}"}
-{"id": "5640.png", "formula": "\\begin{align*} X = \\left ( C ( t ) S _ { J } + D ( t ) \\right ) \\partial _ { t } + T ( t ) S _ { J } V ^ { , i } \\partial _ { i } \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n - 1 } \\frac { r _ k ^ { \\alpha _ n } } { r _ { k + 1 } ^ { \\alpha _ m } } \\frac { n ^ \\alpha } { m ^ \\alpha } \\leq \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n - 1 } \\frac { n ^ \\alpha } { m ^ \\alpha } \\leq \\frac { 1 } { n ^ { 1 - \\alpha } } \\sum _ { m = 1 } ^ { n - 1 } \\frac { 1 } { m ^ \\alpha } \\leq \\max \\left \\{ 1 , \\frac { 1 } { 1 - \\alpha } \\right \\} < \\infty \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} \\frac { d \\ell } { d t } = X _ { H _ t } \\circ \\ell . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} & \\mathcal { H } _ m = \\{ \\ , Z \\in M _ m ( \\mathbb { C } ) \\ , \\mid \\ , \\tfrac { 1 } { 2 i } ( Z - Z ^ { * } ) > 0 \\ , \\} \\ ; ( ) , \\\\ & \\Lambda _ m ( \\mathcal { O } _ { \\boldsymbol { K } } ) = \\{ \\ , H = ( h _ { j l } ) \\in M _ m ( \\boldsymbol { K } ) \\ , \\mid \\ , H ^ { * } = H , \\ h _ { j j } \\in \\mathbb { Z } , \\ , \\sqrt { - D _ { \\boldsymbol { K } } } \\ , h _ { j l } \\in \\mathcal { O } _ { \\boldsymbol { K } } \\ , \\} . \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} B \\left ( A _ j , \\frac { r _ j } { 2 M _ { \\infty } } \\right ) \\subset \\Omega ^ c \\cap B ( q _ j , r _ j ) A _ j = q _ j + r _ j A ^ - _ { \\infty } ( 0 , 1 ) . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} i ( c ^ { m _ { l _ { 2 } } + \\bar { p } L _ { i } } ) = 2 n \\bar { q } l _ { 2 } + 2 [ Q _ { 0 } ] - 2 i '' , \\ \\forall 1 \\leq i \\leq \\beta \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} m ( P _ { a - 1 } \\cup P _ { b - 1 } , k - 1 ) = & m ( P _ { x - 1 } \\cup P _ { y - 2 } \\cup P _ { b - 1 } , k - 1 ) + m ( P _ { x - 2 } \\cup P _ { y - 3 } \\cup P _ { b - 1 } , k - 2 ) . \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} & \\psi ( E _ { i j } ) = \\psi ( E _ { j i } ^ { * } ) = \\psi ( E _ { j i } ) ^ { * } = \\overline { \\psi ( E _ { j i } ) ^ { t } } = \\overline { \\psi ( E _ { j i } ) } \\quad ; \\\\ & \\psi ( E _ { i i } ) = \\psi ( E _ { i i } ) ^ { * } = \\overline { \\psi ( E _ { i i } ) ^ { t } } \\quad . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} S ^ 1 = \\bigcup _ { \\overrightarrow { \\omega } \\in I } W _ { \\overrightarrow { \\omega } , t ' } ; \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\bigg \\Vert \\sum _ { i = 1 } ^ n c _ i ( f _ i - g _ i ) \\bigg \\Vert \\leq \\lambda _ 1 \\bigg \\Vert \\sum _ { i = 1 } ^ n c _ i f _ i \\bigg \\Vert + \\lambda _ 2 \\bigg \\Vert \\sum _ { i = 1 } ^ n c _ i g _ i \\bigg \\Vert + \\mu \\bigg [ \\sum _ { i = 1 } ^ n | c _ i | ^ 2 \\bigg ] ^ { 1 / 2 } \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\eta _ { \\epsilon } : = \\lambda \\omega _ 0 + \\sum _ { i = 1 } ^ d \\sqrt { - 1 } \\tau _ i \\partial \\bar { \\partial } \\log ( | s _ i | _ { H _ i } ^ 2 + \\epsilon ^ 2 ) \\xrightarrow { \\epsilon \\to 0 } [ D ] , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} - \\frac { \\partial P } { \\partial x } \\left ( x , t \\right ) = \\mathcal { L } \\left ( x , t \\right ) \\frac { \\partial Q } { \\partial t } \\left ( x , t \\right ) + \\mathcal { R } \\left ( x , t \\right ) Q \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\left ( \\tilde { \\mathcal { K } } _ n ^ M \\right ) ' ( x ) = Q _ n \\mathcal { K } _ m ' ( x ) + ( I - Q _ n ) \\mathcal { K } _ m ' ( Q _ n x ) Q _ n . \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} I _ { a ^ + } ^ { \\alpha } f ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { t } \\frac { f ( s ) d s } { ( t - s ) ^ { 1 - \\alpha } } , t > a , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde \\sigma _ 1 } { \\partial \\lambda } ( \\lambda , r ) & = \\frac { \\partial g _ \\lambda } { \\partial \\lambda } ( \\zeta _ 1 ( r ) ) - \\sum _ { j = 1 } ^ { k - 1 } \\frac { \\partial G _ \\lambda } { \\partial \\lambda } ( \\zeta _ 1 ( r ) , \\zeta _ { j + 1 } ( r ) ) < 0 \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} J '' _ { d , e } ( x ) & = \\sum _ { t = 1 } ^ e \\frac { x ^ { d + e + 4 } } { ( 1 - x ) ^ t ( 1 - 2 x ) } C ( x ) , \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} \\lambda ^ { \\frac s 2 } = c _ s \\int _ 0 ^ \\infty t ^ { - 1 - \\frac s 2 } ( 1 - e ^ { - t \\lambda } ) d t \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} C & = \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 } { A B } \\\\ & \\approx \\frac { ( \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ) ^ { 2 / 3 } } { m } ( M t ) ^ { 3 / 7 } \\\\ & = M _ C ( M t ) ^ { 3 / 7 } . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v = f ^ \\# & \\mbox { i n } \\O ^ \\# \\ , , \\\\ v = 0 & \\mbox { o n } \\partial \\O ^ \\# \\ , . \\end{cases} \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} d _ M ( \\mu ) & : = \\inf \\{ \\dim _ M ( S ) : \\mu ( S ) = 1 \\} \\ , , \\\\ d _ H ( \\mu ) & : = \\inf \\{ \\dim _ H ( S ) : \\mu ( S ) = 1 \\} \\ , . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\gamma & = ( W ^ \\rho _ u ) ^ { - 1 } T ^ { - 1 } _ 1 \\Big ( \\sum _ i W ^ \\rho _ u ( \\gamma ) ( x _ i ) \\psi _ i \\# W ^ \\rho _ u ( u ) \\Big ) \\\\ \\gamma & = \\sum _ i \\lambda _ i \\big ( T ^ { - 1 } _ 2 W ^ \\rho _ u ( \\gamma ) \\big ) \\rho ^ * ( x _ i ) u \\\\ \\gamma & = \\sum _ i c _ i \\ , \\ , ( T ^ { - 1 } _ 3 W ^ \\rho _ u ( \\gamma ) ) \\ , \\rho ^ * ( x _ i ) u \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\| \\varphi - \\varphi _ n ^ C \\| _ \\infty = O ( h ^ r ) , \\ ; \\ ; \\ ; \\| \\varphi - \\varphi _ n ^ S \\| _ \\infty = O ( h ^ { 2 r } ) . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} \\bigg \\{ \\sum _ { j = 1 } ^ { \\infty } \\bigg ( \\int _ { \\mathbb { R } ^ n } | f _ { j } ( y ) | d y \\bigg ) ^ { r } \\bigg \\} ^ { 1 / r } \\leq C \\int _ { \\mathbb { R } ^ n } \\bigg \\{ \\sum _ { j = 1 } ^ { \\infty } | f _ { j } ( y ) | ^ { r } \\bigg \\} ^ { 1 / r } d y . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} Y ( t ) - 1 = X ( t Y ( t ) ) , \\ , ( 1 - t ) Y ( t ) = 1 + A ( t ^ 2 Y ( t ) ^ 2 ) . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w + \\beta ( w ) = { f } , & \\Omega , \\\\ w = 1 , & \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\nabla _ { P ( p , q , r ) } ( z ) & = 1 + \\frac { 1 } { 4 } \\left ( p q + q r + r p + 1 \\right ) z ^ { 2 } , \\\\ v _ { 3 } ( P ( p , q , r ) ) & = \\frac { 1 } { 6 4 } \\left ( ( p + q + r + 1 ) ( p q + q r + r p + 1 ) + ( p - 1 ) ( q - 1 ) ( r - 1 ) \\right ) \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\tilde { k } _ { \\mu , \\nu } ^ { \\lambda } = \\sum _ { \\sigma ^ { ( 1 ) } , \\sigma ^ { ( 2 ) } , \\sigma ^ { ( 3 ) } \\in \\mathcal { P } } \\sum _ { \\rho ^ { ( 1 ) } , \\rho ^ { ( 2 ) } , \\rho ^ { ( 3 ) } \\in \\mathcal { P } } k _ { \\sigma ^ { ( 2 ) } , \\sigma ^ { ( 3 ) } } ^ { \\sigma ^ { ( 1 ) } } c _ { \\sigma ^ { ( 1 ) } , \\rho ^ { ( 2 ) } , \\rho ^ { ( 3 ) } } ^ { \\lambda } c _ { \\rho ^ { ( 1 ) } , \\sigma ^ { ( 2 ) } , \\rho ^ { ( 3 ) } } ^ { \\mu } c _ { \\rho ^ { ( 1 ) } , \\rho ^ { ( 2 ) } , \\sigma ^ { ( 3 ) } } ^ { \\nu } \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\nabla \\ ! \\cdot \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , = \\ , 0 , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\widehat { T } ( S z _ { 0 } ) = \\widehat { S } \\widehat { T } ( z _ { 0 } ) \\widehat { S } ^ { - 1 } \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} P _ { \\theta _ 0 } \\left ( \\rho ( \\tilde { \\theta } _ { 1 } ^ \\alpha , x ) < \\alpha | x \\right ) = \\alpha + \\phi ( \\Phi ^ { - 1 } ( \\alpha ) ) E _ { F } \\left ( R ( \\alpha , x ) \\right ) + O ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} D ^ 2 _ x u ( x , y , \\epsilon ) & = \\frac { u ( x + \\varepsilon h , y ) - 2 u ( x , y ) + u ( x - \\varepsilon h , y ) } { \\varepsilon ^ 2 } \\\\ & = \\sum _ { i , j = 1 } ^ n u _ { x _ i x _ j } ( x , y ) h _ i h _ j + o ( 1 ) \\ , , \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( x , t \\right ) = \\frac { \\rho } { \\pi a ^ { 2 } \\left ( x , t \\right ) } , \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ m y _ j \\right \\| \\le C m ^ { 1 / p } . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} c _ 0 = - 2 \\sum _ { m = 1 } ^ { k - 1 } a _ { m + 1 } S _ m - \\sum _ { m = 0 } ^ { k - 2 } b _ { m + 1 } T _ m . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} L _ { J X _ u } J = - 2 \\nabla ^ { \\prime \\prime } _ J X _ u ^ \\prime - 2 \\nabla ^ \\prime _ J X _ u ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} S ( \\Gamma ^ 2 ) = M ( \\Gamma ^ 2 , 1 ) \\oplus M ( \\Gamma ^ 2 , \\beta ) \\oplus M ( \\Gamma ^ 2 , \\beta ^ 2 ) . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\aligned \\partial ( \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\mathcal H ^ i ) = & \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\frak H _ { ( a b ) } ^ { i + 1 } \\circ \\frak N ^ { i } _ { 1 } \\cup \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\frak H _ { ( b , 2 1 ) } ^ i \\\\ & \\cup \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\frak N ^ { i } _ { 2 } \\circ \\frak H _ { ( a b ) } ^ { i } \\cup \\frak N ^ { i + 1 } _ { 3 2 } \\circ \\frak H _ { ( a , 2 1 ) } ^ i \\endaligned \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{gather*} C ^ { 2 } \\coloneqq j ^ { 2 } - m ^ { 2 } , \\textrm { w h e r e } m ^ { 2 } \\coloneqq \\sum _ { i = 1 } ^ { n } { m _ { i 0 } } ^ { 2 } \\mathrm { a n d } j ^ { 2 } \\coloneqq \\sum _ { i < j } { m _ { i j } } ^ { 2 } . \\end{gather*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\| \\mathcal { Q } _ \\tau ( v ) \\| ^ 2 = \\| \\mathcal { Q } _ \\tau ^ 1 ( v ) \\| ^ 2 + I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} K ^ * ( u ) : = \\frac { 4 T \\pi } { \\pi ^ 2 - 4 T ^ 2 u ^ 2 } \\ , , u \\in \\C \\ , , \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} d _ 0 \\circ \\frak m ^ { 1 } _ { 1 ; \\alpha _ + , \\alpha _ - } + \\frak m ^ { 1 } _ { 1 ; \\alpha _ + , \\alpha _ - } \\circ d _ 0 + o ( \\alpha _ + , \\alpha _ - ) = 0 . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} \\left \\vert A _ { 0 } \\left ( x , \\widetilde { v } _ { i } \\right ) \\right \\vert \\leq C _ { 2 } \\left \\vert \\nabla \\widetilde { V } \\left ( x \\right ) \\right \\vert , \\forall x \\in \\Omega , i = 0 , . . . , N - 1 , \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\mathrm { E } _ 0 \\Pi \\left ( \\| D ^ { \\boldsymbol { r } } f - D ^ { \\boldsymbol { r } } f _ 0 \\| _ \\infty > M ( n / \\log { n } ) ^ { - \\frac { \\alpha ^ { * } \\{ 1 - \\sum _ { l = 1 } ^ d ( r _ l / \\alpha _ l ) \\} } { 2 \\alpha ^ { * } + d } } \\middle | \\boldsymbol { Y } \\right ) \\leq \\frac { ( \\log { n } ) ^ d } { n ^ \\xi } . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } ( | \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u | ^ { p - 2 } \\mathcal { R } _ { 1 } ^ { \\frac { a _ { 1 } } { \\nu _ { 1 } } } u ) + \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } ( | \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u | ^ { p - 2 } \\mathcal { R } _ { 2 } ^ { \\frac { a _ { 2 } } { \\nu _ { 2 } } } u ) = | u | ^ { q - 2 } u , u \\in L ^ { p } _ { a _ { 1 } , a _ { 2 } } ( \\mathbb { G } ) . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} S = \\left \\{ \\left ( q ^ i , \\frac { \\partial W } { \\partial q ^ i } ( q , \\mu ) \\right ) \\in T ^ * Q : \\frac { \\partial W } { \\partial \\mu ^ \\beta } ( q , \\mu ) = 0 \\right \\} . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} I _ { 0 ^ + } ^ { \\alpha } f ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { t } \\frac { f ( s ) d s } { ( t - s ) ^ { 1 - \\alpha } } , t > 0 , \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\Delta = \\{ \\delta _ { i k } \\mid i \\in F , k \\in F _ { i } \\} . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} t ^ 2 \\Psi '' - 2 ( 2 m - 1 ) t \\Psi ' + 2 m ( 2 m - 1 ) \\Psi = c t ^ 2 - d t - e . \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\left \\| \\mathcal { K } _ m ' ( \\varphi _ m ) ( I - Q _ n ) \\mathcal { K } _ m ' ( \\varphi _ m ) ( I - Q _ n ) \\varphi _ m \\right \\| _ { \\infty } = O ( h ^ { 4 r } ) . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\sum _ { ( a , b , c , d ) \\in B _ n ^ { b = 0 } } H ( k - k _ { c / d } ) & = \\sum _ { d \\mid n } \\sum _ { c = 0 } ^ { d - 1 } H ( k - k _ { c / d } ) = \\sum _ { d \\mid n } \\sum _ { c = 1 } ^ { d - 1 } H ( k - k _ { c / d } ) . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} \\mathcal { R } \\cap \\mathcal { L } ^ + = \\mathcal { R } ^ + \\subseteq _ { \\mathbb { R } } ( \\mathcal { R ^ + } ) \\cap \\mathcal { L } ^ + \\subseteq \\mathcal { R } \\cap \\mathcal { L } ^ + , \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} C L \\frac { \\partial ^ { 2 } p } { \\partial t ^ { 2 } } \\left ( x , t \\right ) - \\frac { C R ^ { 2 } } { 4 L } p \\left ( x , t \\right ) = \\frac { \\partial ^ { 2 } p } { \\partial x ^ { 2 } } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} p _ { \\max } : = \\frac { 1 } { \\frac { 1 } { 2 } - c \\frac { \\log \\left ( \\frac { 1 } { 1 - \\frac { \\alpha _ { \\scriptscriptstyle 0 } } { \\beta _ { \\scriptscriptstyle 0 } } } \\right ) } { \\log C _ { P } } \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { P } \\right ) } . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} L _ 1 = \\left ( \\begin{array} { c c c c c } 1 & - 1 & & & \\\\ & 1 & - 1 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & 1 & - 1 \\\\ \\end{array} \\right ) \\in \\mathbb { R } ^ { ( n - 1 ) \\times n } , \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 ^ m } \\sum _ { j = 1 } ^ { r } \\psi ^ { ( m - 1 ) } \\left ( \\frac { n - r + j } { 2 } \\right ) \\sim \\frac { 1 } { 2 ^ m } \\sum _ { k = n - r + 1 } ^ n \\ , \\frac { ( - 1 ) ^ { m - 2 } ( m - 2 ) ! } { ( k / 2 ) ^ { m - 1 } } \\\\ & = \\frac { ( - 1 ) ^ m \\ , ( m - 2 ) ! } { 2 } \\bigg [ \\sum _ { k = 1 } ^ n \\ , \\frac { 1 } { k ^ { m - 1 } } - \\sum _ { k = 1 } ^ { n - r } \\ , \\frac { 1 } { k ^ { m - 1 } } \\bigg ] \\sim \\frac { ( - 1 ) ^ { m } ( m - 3 ) ! } { 2 \\cdot n ^ { m - 2 } } \\left ( \\frac 1 { ( 1 - \\alpha ) ^ { m - 2 } } - 1 \\right ) , \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\frac { \\partial \\omega _ { \\phi } } { \\partial t } = - { \\rm R i c } ( \\omega _ { \\phi } ) + \\gamma \\omega _ { \\phi } + \\widetilde { \\eta } + L _ X \\omega _ { \\phi } , \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\delta : \\check { H } ^ 2 ( \\Delta , { } ^ 2 G ) \\to H ^ 2 ( \\Delta , G ) \\ \\ , \\ \\ \\delta [ u , g ] : = [ g ] \\ , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\sum _ { p \\le Y } \\nu _ p ( n ) \\log p = \\sum _ { n \\le x } \\sum _ { p \\le Y } \\log p \\sum _ { \\substack { j \\ge 1 \\\\ p ^ j \\mid n } } 1 & = \\sum _ { p \\le Y } \\log p \\sum _ { j \\ge 1 } \\sum _ { \\substack { n \\le x \\\\ p ^ j \\mid n } } 1 \\\\ & \\le \\sum _ { p \\le Y } \\log p \\sum _ { j \\ge 1 } \\frac x { p ^ j } = x \\sum _ { p \\le Y } \\frac { \\log p } { p - 1 } \\ll x \\log Y \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} \\omega _ { 1 } = \\omega _ { 1 } ^ { 1 } , \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{gather*} \\big | \\big ( x t ^ N ; q \\big ) _ { \\infty } \\big | \\leq \\prod _ { i = 0 } ^ { \\infty } { \\big ( 1 + | x | q ^ { N \\Re \\theta + i } \\big ) } \\leq \\prod _ { i = 0 } ^ { \\infty } { \\big ( 1 + | x | q ^ i \\big ) } , \\\\ | ( x q ; q ) _ { \\infty } | = \\prod _ { i = 1 } ^ { \\infty } { \\big | \\big ( 1 - x q ^ i \\big ) \\big | } \\geq \\prod _ { i = 1 } ^ { \\infty } { \\big ( 1 - | x | q ^ i \\big ) } , \\end{gather*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\kappa _ { n } ^ { 1 } = \\zeta _ { n } ^ { 1 } s i n \\left ( \\sqrt { C L \\left ( \\omega _ { n } ^ { 1 } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x _ { 0 } \\right ) + \\eta _ { n } ^ { 1 } c o s \\left ( \\sqrt { C L \\left ( \\omega _ { n } ^ { 1 } \\right ) ^ { 2 } - \\frac { C R ^ { 2 } } { 4 L } } x _ { 0 } \\right ) \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} A _ { i j } \\left ( Z \\right ) = 0 , \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ . } \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} g _ { f , p } ( \\tau x + E ^ { - 1 } w - E ^ { - 1 } c ) = \\tau ^ 2 g _ { f , p } ( x ) , x \\in X . \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} u _ { i } ^ { 0 } \\left ( t = 0 \\right ) = u _ { i } ^ { 0 , 0 } \\quad \\mbox { i n } \\ ; \\overline { \\Omega } , \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( | u _ 1 ( t ) | ^ 2 + | u _ 2 ( t ) | ^ 2 \\big ) \\ d t \\asymp \\sum _ { n = 1 } ^ { \\infty } \\Big ( | C _ { n } | ^ 2 + | d _ n D _ { n } | ^ 2 \\Big ) \\ , . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\mathcal { K } _ t ^ m = \\ ( \\mathcal { K } _ t ^ m \\ ) _ b + \\ ( \\mathcal { K } _ t ^ m \\ ) _ c \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} = | \\hat a ( - k _ 0 ) - \\hat a ( - k _ 0 - ( n - d ) ( k _ 1 - k _ 0 ) ) | \\geq ( 1 - \\epsilon ) c , \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} F _ { A } ( x ) = F ( A x ) . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} P ( x , t ; 1 ) = ( 4 \\pi t ) ^ { - n / 2 } \\ , e ^ { - | x | ^ 2 / 4 t } \\ , , \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} f ( x ) : = \\begin{cases} f _ * ( x ) & x \\notin B _ r ( x _ \\circ ) \\\\ \\min \\{ \\tilde f , f _ * ( x ) \\} & x \\in B _ r ( x _ \\circ ) . \\end{cases} \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\mathbb P \\times \\nu \\Big ( \\bigcup _ { i = 1 } ^ n A _ i \\times B _ i \\Big ) : = \\sum _ { i = 1 } ^ n \\mathbb E ( \\mathbf 1 _ { A _ i } \\nu ( B _ i ) ) , \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\| \\varphi - \\varphi _ m \\| _ \\infty = O \\left ( \\tilde { h } ^ { d } \\right ) . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\left ( \\sin y \\partial _ { x } - \\sin x \\partial _ { y } \\right ) \\left ( 1 + \\Delta ^ { - 1 } \\right ) = J L , \\ \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\ \\ \\ & \\frac { d } { d t } \\left ( \\int _ { \\mathbb { T } } ( | \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } - | \\partial _ { y } \\psi _ { s 2 } ^ { \\nu } | ^ { 2 } ) d x d y + \\left \\Vert \\omega _ { n 2 } ^ { \\nu } \\right \\Vert _ { X } ^ { 2 } \\right ) \\\\ & = - 2 \\nu \\left ( \\int _ { \\mathbb { T } } ( | \\partial _ { y } \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } - | \\omega _ { s 2 } ^ { \\nu } | ^ { 2 } ) d x d y + \\left \\Vert \\omega _ { n 2 } ^ { \\nu } \\right \\Vert _ { X ^ { 1 } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} { \\dot q } ^ i = \\frac { \\partial H } { \\partial p _ i } , { \\dot p } _ i = - \\frac { \\partial H } { \\partial q ^ i } . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} f _ x \\big ( ( 1 , \\varpi ^ { - k } ) \\cdot \\gamma \\cdot ( e + \\varpi ^ \\ell f ) \\big ) = \\begin{cases} 1 & \\mbox { i f } | \\varpi | ^ k = | c | \\mbox { a n d } \\ell = 0 \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\# \\{ i \\in \\{ 1 , \\dots , k ' \\} \\mid \\phi ( \\partial _ i V _ p ) \\subset \\partial ^ 0 U \\} = k . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\langle { a , b , c , r , s , t , u , v , w } \\mid { a s ^ { - 1 } v s a ^ { - 1 } = w = b r b ^ { - 1 } } , ~ c a c ^ { - 1 } = b , ~ r c r ^ { - 1 } = a , ~ w c w ^ { - 1 } = b , \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ l h _ i x _ i = y . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} L _ { \\eta } g _ { i j } & = 2 \\left ( \\frac { 1 } { 2 } \\xi _ { , t } \\right ) g _ { i j } \\\\ V _ { , k } \\eta ^ { k } + V \\xi _ { , t } + \\xi V _ { , t } & = - f _ { , t } \\\\ \\eta _ { i , t } & = f _ { , i } . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} Y ^ { ( 1 ) } ( t ) & = \\int _ 0 ^ t \\mathrm { d } \\tau \\ h _ \\vartheta ( \\tau ) , \\\\ Y ^ { ( 2 ) } ( t ) & = - \\frac { 1 } { 2 } \\int _ 0 ^ t \\mathrm { d } \\tau _ 1 \\int _ 0 ^ { \\tau _ 1 } \\mathrm { d } \\tau _ 2 \\ [ h _ \\vartheta ( \\tau _ 1 ) , h _ \\vartheta ( \\tau _ 2 ) ] , \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 = O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\ ) . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\begin{aligned} L ^ { \\rm r e s c , h } _ n ( u ) - L ^ { \\rm r e s c , h } _ n ( v ) & \\leq B ^ { \\rho _ + } _ n ( u ) - B ^ { \\rho _ + } _ n ( v ) + ( u ^ 2 - v ^ 2 ) \\\\ & + \\left ( \\frac { \\beta _ 1 } { 1 - \\rho _ + } - 2 ( 1 + \\gamma ) ^ { 5 / 3 } \\gamma ^ { 1 / 3 } \\right ) \\frac { ( u - v ) } { \\beta _ 2 } n ^ { 1 / 3 } . \\end{aligned} \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\nabla ^ { ^ { M _ 1 } } _ { X } Z = \\varphi ( \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi ) Z - \\varphi \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi Z + \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { X } Z ) \\xi \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\theta _ { \\bar { Q } } \\ominus \\theta _ Q = \\bar { p } _ { i } d \\bar { q } ^ { i } - p _ { i } d q ^ { i } = d W \\left ( q , \\bar { q } \\right ) , \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} d = 3 \\colon & p = 1 1 \\quad p \\ge 1 7 \\\\ d = 4 \\colon & p \\ge 1 9 \\\\ d = 5 \\colon & p \\ge 2 3 \\\\ d = 6 \\colon & p \\ge 2 3 \\quad p \\neq 3 7 \\\\ d = 7 \\colon & p \\ge 2 9 \\quad p \\neq 3 7 \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{gather*} \\frac { \\prod \\limits _ { s = 1 } ^ m { g _ { n _ s + \\tau _ s ^ + - \\tau _ s ^ - } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } } { Q _ { ( n _ 1 , \\dots , n _ m ) } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } \\end{gather*}"}
-{"id": "643.png", "formula": "\\begin{align*} D _ v ^ w X ( x ) : = \\left \\{ v ^ j \\frac { \\partial X ^ i } { \\partial x ^ j } ( x ) + \\Gamma ^ i _ { j k } ( w ) v ^ j X ^ k ( x ) \\right \\} \\frac { \\partial } { \\partial x ^ i } , \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\int _ { - T } ^ { T } \\Big | \\sum _ { n = - \\infty } ^ { \\infty } \\big ( C _ n e ^ { i \\omega _ n t } + R _ n e ^ { r _ n t } \\big ) \\Big | ^ 2 d t \\le c _ 2 ( T ) \\sum _ { n = - \\infty } ^ { \\infty } | C _ n | ^ 2 \\ , , \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} c ^ { \\left ( 0 , J \\right ) } _ { i j } = K \\left ( d ^ { \\left ( I ' , J ' \\right ) } _ { i , j } , \\dots \\right ) , \\quad \\mbox { w h e r e $ K \\left ( d ^ { \\left ( I ' , J ' \\right ) } _ { i , j } , \\dots \\right ) $ i s a c o n s t a n t d e f i n e d b y $ d ^ { \\left ( I ' , J ' \\right ) } _ { i , j } , \\dots $ . } \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} t = e ^ { \\frac { 1 } { \\gamma } s } . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} & A ^ + _ { j i } : = \\sum _ { s \\geq 1 } \\sum _ { j = j _ 1 < \\ldots < j _ { s + 1 } = i } ( - 1 ) ^ { s - 1 } \\tilde { e } ^ { ( 0 ) } _ { j _ 1 j _ 2 } \\cdots \\tilde { e } ^ { ( 0 ) } _ { j _ s j _ { s + 1 } } , \\\\ & A ^ - _ { i j } : = \\sum _ { s \\geq 1 } \\sum _ { j = j _ 1 < \\ldots < j _ { s + 1 } = i } ( - 1 ) ^ { s - 1 } \\tilde { f } ^ { ( 0 ) } _ { j _ { s + 1 } j _ s } \\cdots \\tilde { f } ^ { ( 0 ) } _ { j _ 2 j _ { 1 } } \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\Gamma ( a , x ) ~ : = ~ \\int _ x ^ \\infty s ^ { a - 1 } e ^ { - s } \\ , d s \\ , . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} E = \\left \\{ \\left ( q ^ i , \\frac { \\partial L } { \\partial \\dot { q } ^ i } ; \\dot { q } ^ i , \\frac { \\partial L } { \\partial q ^ i } \\right ) \\in T T ^ * Q \\right \\} \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} f ( x ) = \\tau f ^ * ( E x + c ) + \\langle w , x \\rangle + \\beta , x \\in X , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\int _ { \\Gamma } \\varphi = \\pi \\lambda ^ 2 . \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{gather*} L \\big ( x ^ 0 , \\ldots , x ^ n ; y ^ 0 , \\ldots , y ^ n \\big ) = 0 , \\end{gather*}"}
-{"id": "4515.png", "formula": "\\begin{align*} & f g \\otimes h = ( g ^ { f ^ { - 1 } } \\otimes h ^ { f ^ { - 1 } } ) ( f \\otimes h ) , \\\\ & f \\otimes g h = ( f \\otimes g ) ( f ^ { g ^ { - 1 } } \\otimes h ^ { g ^ { - 1 } } ) , \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} t ^ 3 - \\frac { 1 } { 3 } \\lambda \\ , t - \\frac { 1 } { 8 } | w | ^ 2 = 0 \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} Z ( \\mu ) = \\{ \\mu x \\mid x \\in X , r ( \\mu ) = s ( x ) \\} \\subseteq X . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} A & = G _ 0 + G _ 1 - G _ 2 - G _ \\infty \\in M _ 2 ( \\alpha _ 0 \\alpha _ 1 ) , \\\\ B & = G _ 0 - G _ 1 + G _ 2 - G _ \\infty \\in M _ 2 ( \\alpha _ 0 \\alpha _ 2 ) , \\\\ C & = G _ 0 - G _ 1 - G _ 2 + G _ \\infty \\in M _ 2 ( \\alpha _ 1 \\alpha _ 2 ) . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} D ( x ) = A ( \\log \\log x ) ^ 2 + O \\bigg ( \\frac { ( \\log \\log x ) ^ { 3 / 2 } } { ( \\log \\log \\log x ) ^ 2 } \\bigg ) , \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} I H ^ k ( Y ) \\cong \\begin{cases} H ^ k ( Y ) \\quad { } \\\\ \\Im ( \\alpha ^ * _ n ) \\quad { } \\\\ H ^ k ( U ) \\quad { } \\end{cases} \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} S _ { 1 } + S _ { 2 } + \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\left ( \\varphi _ { i l } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } \\overline { \\left ( \\varphi _ { j l } ^ { \\star \\star } \\left ( Z \\right ) \\right ) ^ { ( 2 ) } } = 0 , \\quad \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ n \\psi _ i \\left ( 1 + \\lambda ^ T \\psi _ i + \\frac { 1 } { 2 } ( \\lambda ^ T \\psi _ i ) ^ 2 + O _ p \\left ( ( \\lambda ^ T \\psi _ i ) ^ 2 \\right ) \\right ) \\ , . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} | ( \\nabla F _ x ) ( y ) | ~ = ~ | ( \\nabla ( F _ x \\circ \\iota ^ { - 1 } ) ) ( x ) | ~ = ~ | ( \\nabla F _ z ) ( x ) | \\ , . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} { \\rm R i c } ( \\omega ) = \\beta \\omega + ( 1 - \\beta ) [ D ] \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\Delta _ a & = \\mbox { P r } _ { E _ t } ( R | H _ a ) - \\mbox { P r } _ { E _ f } ( R | H _ a ) \\\\ & = ( 1 - \\delta ) - \\mbox { P r } _ { E _ f } ( R | H _ a ) \\\\ & = ( 1 - \\mbox { P r } _ { E _ f } ( R | H _ a ) ) - \\delta \\\\ & = \\mbox { P r } _ { E _ f } ( \\neg R | H _ a ) - \\delta \\\\ & = \\beta - \\delta \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\Phi \\pi _ 0 = \\pi _ 0 = \\Phi ^ { - 1 } \\pi _ 0 \\ , . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} \\alpha ( x _ 1 \\otimes x _ 2 ) = \\alpha _ 1 ( x _ 1 ) \\alpha _ 2 ( x _ 2 ) . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\lVert A _ i ^ n ( x ) \\rVert \\le \\sum _ { j = 1 } ^ t \\lvert a _ { j , n } \\rvert \\cdot \\lVert A ^ n ( x ) e _ j \\rVert \\le \\sum _ { j = 1 } ^ t \\lVert A ^ n ( x ) e _ j \\rVert . \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{gather*} \\overline { \\partial } _ A g ^ { i j } = - g ^ { l j } g ^ { i k } \\overline { \\partial } _ A g _ { k l } . \\end{gather*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\beta \\int _ 0 ^ t \\ e ^ { - \\eta ( t - s ) } u _ { 1 x x } ( s , x ) d s + a u _ 2 ( t , x ) = 0 \\ , , \\\\ \\phantom { u _ { 1 t t } ( t , x ) - u _ { 1 x x } ( t , x ) + \\int _ 0 ^ t \\ k ( t - s ) u _ { 1 x x } ( s , x ) d s + } t \\in ( 0 , T ) \\ , , \\ , \\ , \\ , x \\in ( 0 , \\pi ) \\\\ \\displaystyle u _ { 2 t t } ( t , x ) - u _ { 2 x x } ( t , x ) + b u _ 1 ( t , x ) = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\tilde { A } \\left ( W ' , Z ' \\right ) = \\frac { 1 } { \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) \\cdot \\overline { \\left ( I _ { { q ' } ^ { 2 } } + \\mathcal { L } \\left ( W ' , Z ' \\right ) \\right ) } ^ { t } } , \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} \\rho _ i ( x ) \\leq \\frac { d ( x ) } { \\kappa _ + ^ { - 1 } - d _ 0 ( x ) } = \\frac { d ( x ) } { \\bar { h } - d _ 0 ( x ) } \\leq 1 . \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\partial _ { i } ( \\sqrt { g } g ^ { i k } ) = - \\sqrt { g } g ^ { i j } g ^ { k l } \\partial _ { i j } ^ 2 \\eta ^ { \\mu } \\partial _ { l } \\eta _ { \\mu } , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} T _ j ( f , g ) ( x ) & : = \\int f ( x - P ( t ) ) g ( x - Q ( t ) ) 2 ^ j \\rho ( 2 ^ j t ) \\ , d t \\\\ & = \\iint \\hat { f } ( \\xi ) \\hat { g } ( \\eta ) e ^ { 2 \\pi i ( \\xi + \\eta ) x } m _ j ( \\xi , \\eta ) \\ , d \\xi d \\eta , \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} \\limsup _ \\delta y _ \\lambda \\mathbf { d } x _ \\delta & \\leq \\liminf _ n \\limsup _ \\delta ( y _ \\lambda \\overline { \\mathbf { d } } x _ \\lambda ^ n + x _ \\lambda ^ n \\overline { \\mathbf { d } } x _ { \\gamma _ \\lambda ^ n } + x _ { \\gamma _ \\lambda ^ n } \\mathbf { d } x _ \\delta ) \\\\ & \\leq \\liminf _ n ( y _ \\lambda \\overline { \\mathbf { d } } x _ \\lambda ^ n + s _ { \\gamma _ \\lambda ^ n } ) \\\\ & \\leq \\liminf _ n ( y _ \\lambda \\overline { \\mathbf { d } } x _ \\lambda ^ n + 2 ^ { 1 - n } t _ \\lambda ) \\\\ & = 0 . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} A ( x ) P ( x ) = P ( f ( x ) ) A ( x ) \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} & ( p - 1 ) \\eta ^ 2 u _ k ^ { p - 2 } ( a \\nabla u _ k , \\nabla u _ k ) + 2 u _ k ^ { p - 1 } \\eta ( a \\nabla u _ k , \\nabla \\eta ) \\\\ & = \\frac { 4 ( p - 1 ) } { p ^ 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla ( \\eta u _ k ^ { p / 2 } ) ) \\\\ & \\ ; \\ ; \\ ; \\ ; - \\frac { ( 2 p - 4 ) } { p } u _ k ^ { p - 1 } \\eta ( a \\nabla u _ k , \\nabla \\eta ) - \\frac { 4 ( p - 1 ) } { p ^ 2 } u _ k ^ p ( a \\nabla \\eta , \\nabla \\eta ) . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\sum _ { k = k _ 0 } ^ \\infty 3 ^ { - k n } \\norm { D u } _ { L ^ \\infty ( B ^ + _ { r _ k } ) } \\le C \\norm { D u } _ { L ^ 1 ( B ^ + _ 4 ) } + C \\int _ 0 ^ { 1 } \\frac { \\hat \\omega _ { \\vec g } ( t ) } t \\ , d t + \\sum _ { k = k _ 0 } ^ \\infty 3 ^ { - ( k + 1 ) n } \\norm { D u } _ { L ^ \\infty ( B ^ + _ { r _ { k + 1 } } ) } . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} \\langle P _ h ( \\cdot , s ) , \\mu _ j \\rangle = \\frac { \\overline { \\rho _ j ( h ) } \\sqrt { \\pi } } { ( 4 \\pi h ) ^ { s - \\frac { 1 } { 2 } } } \\frac { \\Gamma ( s - \\frac { 1 } { 2 } + i t _ j ) \\Gamma ( s - \\frac { 1 } { 2 } - i t _ j ) } { \\Gamma ( s ) } . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\begin{pmatrix} \\hat { u } \\\\ \\hat { v } \\\\ \\hat { w } \\end{pmatrix} = \\begin{pmatrix} 0 . 1 + 0 . 2 x + 0 . 2 x + 0 . 1 z + 0 . 1 5 x ^ 2 + 0 . 2 y ^ 2 + 0 . 1 z ^ 2 + 0 . 1 5 x y + 0 . 1 y z + 0 . 1 z x \\\\ 0 . 1 5 + 0 . 1 x + 0 . 1 y + 0 . 2 z + 0 . 2 x ^ 2 + 0 . 1 5 y ^ 2 + 0 . 1 z ^ 2 + 0 . 2 x y + 0 . 1 y z + 0 . 2 z x \\\\ 0 . 1 5 + 0 . 1 5 x + 0 . 2 y + 0 . 1 z + 0 . 1 5 x ^ 2 + 0 . 1 y ^ 2 + 0 . 2 z ^ 2 + 0 . 1 x y + 0 . 2 y z + 0 . 1 5 z x \\end{pmatrix} \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} D ^ { i j } _ { k k k ' u ' } = 0 , \\mbox { f o r a l l $ i , j , k , k ' , u ' = 1 , \\dots , q $ w i t h $ k ' \\neq u ' $ , } \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} m _ j ( \\xi , \\eta ) : = \\int 2 ^ j \\rho ( 2 ^ j t ) e ^ { - 2 \\pi i ( \\xi P ( t ) + \\eta Q ( t ) ) } \\ , d t . \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} v _ { n , j } ^ { ( i ) } ( y ) = \\tilde { u } _ n ^ { ( i ) } ( y ) - \\phi _ { n , j } ( y ) , \\ \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\begin{cases} \\tilde h _ 1 ( x , u , \\varphi ) & = h _ 1 ( x , x u , \\varphi ) \\\\ \\tilde h _ 2 ( x , u , \\varphi ) & = \\frac { h _ 2 ( x , x u , \\varphi ) } { h _ 1 ( x , x u , \\varphi ) } \\\\ \\tilde h _ 3 ( x , u , \\varphi ) & = h _ 3 ( x , x u , \\varphi ) \\end{cases} \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { P } ( D _ n = d ) = \\mathbb { P } ( D ^ * _ z = d ) = \\frac { 1 } { \\mu } \\mathbb { E } \\left ( \\frac { ( Y _ 1 - | d | ) _ { + } } { Y _ 1 } \\right ) \\mbox { f o r ~ } d \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\{ f , g \\} = \\omega _ Q ^ { T } ( X _ f , X _ g ) = \\frac { \\partial f } { \\partial \\dot { p } _ i } \\frac { \\partial g } { \\partial q ^ i } - \\frac { \\partial g } { \\partial \\dot { p } _ i } \\frac { \\partial f } { \\partial q ^ i } + \\frac { \\partial f } { \\partial p _ i } \\frac { \\partial g } { \\partial \\dot { q } ^ i } - \\frac { \\partial g } { \\partial p _ i } \\frac { \\partial f } { \\partial \\dot { q } ^ i } . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} \\ddot { x } ^ { i } + \\omega \\left ( t \\right ) \\frac { 1 } { n } r ^ { n } x ^ { i } = 0 ~ , ~ \\left ( n \\neq 0 , - 2 , 2 \\right ) . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} ( t ^ w ) ^ a \\ = \\ ( t ^ { w _ 1 } ) ^ { a _ 1 } \\dotsb ( t ^ { w _ n } ) ^ { a _ n } \\ = \\ t ^ { w _ 1 a _ 1 + \\dotsb + w _ n a _ n } \\ = \\ t ^ { w \\cdot a } \\ , . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\frac { z _ { k + 1 } - 2 z _ k + z _ { k - 1 } } { \\phi _ 1 ( a , h ) \\phi _ 2 ( c , h ) } + ( a c ) z _ k = 0 , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} a _ 7 = \\sqrt { a _ 5 a _ 6 } = \\sqrt { { \\cal A } ^ { - 1 } ( a _ 2 ) { \\cal A } ^ { - 1 } ( a _ 3 ) { \\cal A } ^ { - 1 } ( 1 + \\alpha _ x ) } = { \\cal A } ^ { - 1 } ( \\sqrt { a _ 2 a _ 3 } ) { \\cal A } ^ { - 1 } \\big ( \\sqrt { 1 + \\alpha _ x } \\big ) . \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\varkappa _ d ( G ) = ( \\# \\ker ( T _ { H _ i , G ^ \\prime } ) ) _ { 1 \\le i \\le p + 1 } \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} Z _ N ( \\zeta ) = \\left ( \\prod _ { i = 1 } ^ m K _ { N , L _ i } \\right ) \\mathrm { P f } \\left [ \\zeta ^ 2 \\alpha _ { j , l } + \\beta _ { j , l } \\right ] _ { j , l = 1 , \\ldots , N } \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\widetilde { U } _ k = ( \\widetilde { u } _ 1 , \\widetilde { u } _ 2 , \\ldots , \\widetilde { u } _ k ) \\in \\mathbb { R } ^ { m \\times k } , \\widetilde { V } _ k = ( \\widetilde { v } _ 1 , \\widetilde { v } _ 2 , \\ldots , \\widetilde { v } _ k ) \\in \\mathbb { R } ^ { n \\times k } , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 L _ \\varepsilon } { \\partial w _ i \\partial w _ j } ( w ) = l _ \\varepsilon ( w ) ^ { p - 2 } \\delta _ { i j } + ( p - 2 ) l _ \\varepsilon ( w ) ^ { p - 4 } w _ i w _ j , \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} S = \\left ( \\begin{array} { c c } \\alpha I & 0 \\\\ 0 & \\beta I \\end{array} \\right ) , \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} d ^ { 2 } \\varepsilon / d \\xi ^ { 2 } - u ^ { 2 } \\varepsilon = h , \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\bigl ( \\# \\{ ( c , d ) & \\in \\Z _ { > 0 } ^ 2 : d > c > 0 , \\ ; d \\mid n , \\ ; c \\equiv d k \\bmod p \\} \\\\ & \\quad { } - \\# \\{ ( c , d ) \\in \\Z _ { > 0 } ^ 2 : d > c > 0 , \\ ; d \\mid n , \\ ; c \\equiv d k _ * \\bmod p \\} \\bigr ) \\\\ & = \\tfrac { 1 } { 2 } \\bigl ( s ( k ) - s ( k _ * ) \\bigr ) \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\Xi _ p = \\big \\{ \\xi ^ { ( p ) } _ 1 , \\xi ^ { ( p ) } _ 2 , \\ldots , \\xi ^ { ( p ) } _ { p + k } \\big \\} \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 1 } ^ { N } w _ { j } b _ { j } ( q + q ' ) - 2 \\sum _ { j = 1 } ^ { N } w _ { j } c _ { j } p + \\sum _ { j = N + 1 } ^ { m } w _ { j } b _ { j } ( q ' - q ) \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\bigtriangleup ( n ) = \\begin{cases} k + k ( - 1 ) ^ { ( 2 k - 1 ) | x _ i | ^ 2 } ) , & i f ~ n = 2 k , \\\\ k + 1 + k ( - 1 ) ^ { ( 2 k - 1 ) | x _ i | ^ 2 } , & i f ~ n = 2 k + 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { k \\rightarrow + \\infty } \\displaystyle \\frac { \\tilde { C } k ^ { p _ { - } - 1 } } { e ^ { a k ( p ^ { - } - 1 - ( \\alpha ^ { + } + \\beta ^ { + } ) ) } } \\left | \\ln \\displaystyle \\frac { k } { e ^ { a k } } \\right | = 0 . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} P ( t ) = a _ { d _ 1 } t ^ { d _ 1 } + a _ { d _ 1 - 1 } t ^ { d _ 1 - 1 } + \\dots + a _ { e _ 1 } t ^ { e _ 1 } , 1 \\le e _ 1 \\le d _ 1 , a _ { d _ 1 } a _ { e _ 1 } \\neq 0 \\\\ Q ( t ) = b _ { d _ 2 } t ^ { d _ 2 } + b _ { d _ 2 - 1 } t ^ { d _ 2 - 1 } + \\dots + b _ { e _ 2 } t ^ { e _ 2 } , 1 \\le e _ 2 \\le d _ 2 , b _ { d _ 2 } b _ { e _ 2 } \\neq 0 . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\left . \\begin{matrix} x ( 0 ) = x _ 0 > 0 \\\\ y ( 0 ) = y _ 0 > 0 \\end{matrix} \\right \\} \\Longrightarrow \\begin{pmatrix} x ( t ) > 0 \\\\ y ( t ) > 0 \\end{pmatrix} , t > 0 . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} T _ { n } ^ { \\pm } \\left ( { u , \\xi } \\right ) = \\sum \\limits _ { s = 0 } ^ { n - 2 } \\left ( \\pm 1 \\right ) ^ { s } { \\frac { F _ { s + 1 } ^ { \\pm } \\left ( \\xi \\right ) } { u ^ { s } } , } \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} d \\rho ^ { ( a , b ) } ( x ) = w ^ { ( a , b ) } ( x ) d x \\Bigr | _ { [ - 1 , 1 ] } \\quad \\quad { \\rm f o r } a \\ge b \\ge 2 \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\Gamma _ \\pm = \\{ ( x , v ) : x \\in \\partial \\Omega \\ , , \\pm v \\cdot n _ x > 0 \\} \\ , , \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\mathfrak { F } ( \\varphi ^ \\hbar ) = \\{ \\L \\subset \\R ^ 2 \\mid \\delta _ * ( \\L ) > 2 \\pi \\hbar \\} . \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } L ( \\phi ) & = & - N ( \\phi ) - E + \\sum _ { i , j } c _ { i j } \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } & \\Omega _ \\varepsilon , \\\\ \\phi & = & 0 & \\partial \\Omega _ \\varepsilon , \\\\ \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , \\phi & = & 0 & i , j . \\end{array} \\right . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} M _ t ^ v : = v ( X _ t ) - v ( x ) - \\int _ 0 ^ t \\left ( \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { d } a _ { i j } \\partial _ i \\partial _ j v + \\sum _ { i = 1 } ^ { d } g _ i \\partial _ i v \\right ) ( X _ s ) \\ , d s , t \\ge 0 , \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} g ( \\lambda ) : = h _ \\rho ( f ( \\lambda ) ) , \\forall \\lambda \\in \\R ^ { \\nu + 1 } \\ , , \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\textstyle c ( z ) : = \\frac { ( a - 1 ) z - z ^ { - 1 } + ( b - 2 ) ( a - 1 ) ^ { 1 / 2 } ( b - 1 ) ^ { - 1 / 2 } } { a ( z - z ^ { - 1 } ) } . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} \\partial _ { s } u _ { 1 } ( x , t , y , s ) - \\nabla _ { y } \\cdot ( A ( y , s ) ( \\nabla u ( x , t ) + \\nabla _ { y } u _ { 1 } ( x , t , y , s ) ) ) = 0 Y ^ { ^ { \\ast } } \\times ( 0 , 1 ) \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} a _ 1 ^ 2 x ^ 4 + a _ 2 ^ 2 x ^ 2 y ^ 2 + a _ 3 ^ 2 x ^ 2 z ^ 2 + a _ 4 y ^ 4 + a _ 5 y ^ 2 z ^ 2 + a _ 6 z ^ 4 = 0 \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d t \\thinspace e ^ { - s t } \\langle T _ + \\rangle ( 0 , v , t ) = { 1 \\over s } \\thinspace \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 3 / 2 } \\psi _ { s , F } ( - v ) \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} M _ { j , m , n } ( \\xi , \\eta ) : = & m _ j ( \\xi , \\eta ) \\hat { \\Phi } \\left ( \\frac { \\xi } { 2 ^ { a j + m } } \\right ) \\hat { \\Phi } \\left ( \\frac { \\eta } { 2 ^ { b j + n } } \\right ) \\\\ = & \\int \\rho ( t ) e ^ { - 2 \\pi i \\left ( \\frac { \\xi } { 2 ^ { a j } } ( t ^ a + \\epsilon _ P ( t ) ) + \\frac { \\eta } { 2 ^ { b j } } ( t ^ b + \\epsilon _ Q ( t ) ) \\right ) } \\ , d t \\hat { \\Phi } \\left ( \\frac { \\xi } { 2 ^ { a j + m } } \\right ) \\hat { \\Phi } \\left ( \\frac { \\eta } { 2 ^ { b j + n } } \\right ) \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} G = \\{ a ^ { n _ 1 } b ^ { m _ 1 } a ^ { n _ 2 } b ^ { m _ 2 } \\dots a ^ { n _ k } b ^ { m _ k } : n _ i , m _ i \\in \\Z \\} . \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} X _ j = ( R _ j + \\varDelta _ 1 ) \\cos \\theta _ j , Y _ j = ( R _ j + \\varDelta _ 2 ) \\sin \\theta _ j , \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} T _ n = T _ 1 + \\sum _ { k = 1 } ^ { n - 1 } ( T _ { k + 1 } - T _ k ) \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\frac { d h } { h } \\wedge \\omega & = m \\ , \\psi \\wedge \\omega \\ , \\\\ d ( h \\omega ) & = \\big ( m - 2 \\big ) \\ , h \\ , \\psi \\wedge \\omega \\ . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} \\left ( D ^ 2 L ( \\nabla u ) \\nabla u _ { x _ j } \\right ) ^ i & = l ( \\nabla u ) ^ { p - 2 } u _ { x _ i x _ j } + ( p - 2 ) l ( \\nabla u ) ^ { p - 4 } \\langle \\nabla u _ { x _ j } , \\nabla u \\rangle u _ { x _ i } , \\\\ \\frac { \\partial } { \\partial x _ i } \\gamma ^ j & = l ( \\nabla u ) ^ { 2 - q } u _ { x _ i x _ j } + ( 2 - q ) l ( \\nabla u ) ^ { - q } \\langle \\nabla u _ { x _ i } , \\nabla u \\rangle u _ { x _ j } , \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathcal { F } _ G ^ { + } : = \\{ F / \\Q : ~ \\textrm { $ F $ t o t a l l y r e a l o f d e g r e e $ d $ } , ~ \\textrm { G a l } ( F ^ c / \\Q ) \\cong G \\} . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} & f ( \\{ a , b \\} ) + g ( a b ) = g ( a ) f ( b ) + f ( a ) g ( b ) , \\\\ & f ( \\{ b , a \\} ) + g ( b a ) = g ( b ) f ( a ) + f ( b ) g ( a ) . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} M ^ { \\nu } \\vert \\phi _ n ( y ) \\vert \\leq \\eta _ n ^ 3 , \\vert y - \\zeta _ { i , n } ^ \\prime \\vert = M . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} D _ { m , n } = - \\sum ^ \\mu _ { s = 1 } \\frac { v _ s } { \\psi _ { m , n } ( z _ s ) } . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\frac { d } { d t } ( A ^ 2 B C D ^ 2 ) & = \\frac { d } { d t } A ^ 2 E ^ 2 ( B ^ 2 - C ^ 2 ) = 0 \\\\ \\frac { d } { d t } ( B - C ) & = ( B - C ) ( A E - B ^ 2 - 2 B C - C ^ 2 ) . \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} V ( g ) = U ^ { - 1 } ( g \\circ \\sigma ) , g \\in S ( [ 0 , \\tau ( s ( x ) ) ) , m ) . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} b '' _ l = \\sum _ { i \\geq 1 } \\frac { 1 } { ( i - 1 ) ! } ( l - 1 ) ^ { \\underline { i - 1 } } \\sum _ { \\substack { k _ 1 , \\dots , k _ i \\geq 2 \\\\ k _ 1 + \\cdots + k _ i = l } } b ' _ { k _ 1 } a ' _ { k _ 2 } \\cdots a ' _ { k _ i } . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus N _ \\Omega } \\nabla v \\nabla \\varphi + \\int _ { \\Omega \\setminus N _ \\Omega } \\beta ' ( w ) v \\varphi = 0 . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } \\int _ { \\mathcal { Y } _ { n , m } } \\left ( - a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) + a \\left ( y ^ { n } , s ^ { m } , \\nabla u + \\sum \\limits _ { j = 1 } ^ { n } \\nabla _ { y _ { j } } u _ { j } + \\delta c \\right ) \\right ) \\\\ \\cdot \\delta c \\left ( x , t , y ^ { n } , s ^ { m } \\right ) d y ^ { n } d s ^ { m } d x d t \\geq 0 \\end{gather*}"}
-{"id": "223.png", "formula": "\\begin{align*} g J _ \\ell g ^ { - 1 } = J _ { \\sigma _ g ( \\ell ) } \\mbox { a n d } g \\Delta _ \\ell g ^ { - 1 } = \\Delta _ { \\sigma _ g ( \\ell ) } . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ { 2 n } } \\mathrm { s g n } \\left ( \\sigma \\right ) x _ { \\sigma \\left ( 1 \\right ) } x _ { \\sigma \\left ( 2 \\right ) } \\cdots x _ { \\sigma \\left ( 2 n \\right ) } = 0 \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} { } _ p \\Psi _ q \\Big [ ^ { ( a _ 1 , \\alpha _ 1 ) , . . . , ( a _ p , \\alpha _ p ) } _ { ( b _ 1 , \\beta _ 1 ) , . . . , ( b _ q , \\beta _ q ) } \\Big | z \\Big ] = \\sum _ { k = 0 } ^ \\infty \\frac { \\prod _ { i = 1 } ^ p \\Gamma ( a _ j + \\alpha _ j k ) } { \\prod _ { j = 1 } ^ q \\Gamma ( b _ j + \\beta _ j k ) } \\frac { z ^ k } { k ! } , \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} g ^ * ( x ^ * ) = \\tau h ^ * ( H x ^ * + v ) + \\langle z , x ^ * \\rangle + \\rho , x ^ * \\in X , \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\Omega ( { \\mathbf Z } , x ) = \\overline { \\big \\{ S ^ n ( x , x , \\ldots , x ) : n \\in \\Z \\big \\} } \\subset ( G / \\Gamma ) ^ { k } . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} a & b \\\\ b & c \\end{pmatrix} = 0 \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\ker Q _ t \\subseteq \\ker Q _ s \\subseteq \\ker B ^ * = \\ker B B ^ * \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} c _ { 1 } \\left ( \\nu \\right ) = \\frac { \\nu ^ { \\nu } } { e ^ { \\nu } \\Gamma \\left ( { \\nu + 1 } \\right ) } . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } = - \\frac { 1 } { 2 i } d \\omega _ { 0 } \\big | _ { T ^ { 1 , 0 } X } = \\partial \\bar \\partial \\varphi . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\mathcal K _ { j } ^ { p , q } : = \\mathcal K _ { \\lceil \\frac j 2 + 1 \\rceil , \\lfloor \\frac j 2 + 1 \\rfloor } ^ { p , q } = \\frac { \\Gamma ( \\lceil \\frac j 2 \\rceil + \\frac 1 2 ) } { \\Gamma ( p ) \\Gamma ( p + q + j + \\frac 1 2 ) } \\sum _ { \\ell = 1 } ^ { q } \\frac { \\Gamma ( j + \\ell + \\frac 1 2 ) \\Gamma ( p + q - \\ell ) } { \\Gamma ( \\lceil \\frac j 2 \\rceil + \\ell + \\frac 1 2 ) \\Gamma ( q - \\ell + 1 ) } . \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} W _ { q } ( \\pi _ { k } ( x , y _ { 0 : k } , \\cdot ) , \\pi _ { k } ( x ^ { \\prime } , y _ { 0 : k } , \\cdot ) ) \\leq \\exp \\left [ - \\sum _ { j = 1 } ^ { k } \\int _ { 0 } ^ { \\Delta } \\lambda ( j , y _ { j } , t ) \\mathrm { d } t \\right ] \\| x - x ^ { \\prime } \\| , \\qquad \\forall x , x ^ { \\prime } \\in \\mathbb { R } ^ { p } , \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} | | P _ { \\lambda ^ t } ( a ) | | _ { s - t } ^ 2 & = \\sum _ { m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { s - t } \\lambda ^ { 2 t } ( m ) | a _ m | ^ 2 \\\\ & = \\sum _ { m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ { s - t } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ t | a _ m | ^ 2 \\\\ & = \\sum _ { m } ( 1 + | m _ 1 | ^ 2 + \\cdots + | m _ n | ^ 2 ) ^ s | a _ m | ^ 2 \\\\ & = | | a | | _ s \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\mathbb { E } \\Big [ \\sum _ { t = 0 } ^ { T - 1 } Q ( t ) \\Big ] . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } \\int _ { \\mathcal { Y } _ { i , \\lambda } } - u _ { i } \\left ( x , t , y ^ { i } , s ^ { \\lambda } \\right ) v _ { 1 } \\left ( x \\right ) v _ { 2 } \\left ( y _ { 1 } \\right ) \\cdots v _ { i + 1 } \\left ( y _ { i } \\right ) \\\\ \\times c _ { 1 } \\left ( t \\right ) c _ { 2 } \\left ( s _ { 1 } \\right ) \\cdots \\partial _ { s _ { \\lambda } } c _ { \\lambda + 1 } \\left ( s _ { \\lambda } \\right ) d y ^ { i } d s ^ { \\lambda } d x d t = 0 \\end{gather*}"}
-{"id": "8440.png", "formula": "\\begin{align*} A _ { ( { \\overline 1 } , 0 ) } = \\langle e _ 1 \\rangle \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} { \\int _ M \\zeta _ n \\mathrm { d } \\mu = - b _ 0 + o ( 1 ) . } \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\nabla _ { J ( \\ell - 2 , m ) } ( z ) - \\nabla _ { J ( \\ell , m ) } ( z ) = z \\nabla _ { T ( 2 , - m ) } ( z ) , \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\varphi _ n ( y ) = \\eta _ n \\ , \\vert y - \\zeta _ { i , n } ^ { \\prime } \\vert ^ { - \\nu } \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\tilde v _ l : = v _ l / W . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\phi ( \\gamma ( l , m ) ) \\ = \\ d u ( l , m ) \\ \\ \\ , \\ \\ \\ \\forall l , m \\in \\Pi \\ . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{gather*} \\begin{cases} - \\Delta w _ n + \\beta _ n ( w _ n ) = f & \\Omega , \\\\ w _ n = 1 & \\partial \\Omega . \\end{cases} \\end{gather*}"}
-{"id": "1871.png", "formula": "\\begin{align*} f ( a x + b y ) = a f ( x ) + b f ( y ) \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} h ^ 1 ( N _ { C / S } ( - p _ 1 - \\cdots - p _ 8 - q _ 1 - \\cdots - q _ 4 ) ) & = h ^ 1 ( \\O _ C ( 2 H + L ) ( - p _ 1 - \\cdots - p _ 8 - q _ 1 - \\cdots - q _ 4 ) ) \\\\ & = h ^ 1 ( \\O _ C ( H + L ) ( p _ 9 - q _ 1 - \\cdots - q _ 4 ) ) \\\\ & = h ^ 1 ( K _ C ( p _ 9 ) ( - q _ 1 - \\cdots - q _ 4 ) ) . \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} \\max \\left ( \\bigl ( 1 - \\sqrt { \\gamma } \\bigr ) F ( 0 ) + \\sqrt { \\gamma } F ( \\sqrt { \\gamma } ) , ( 1 - \\eta ) F ( \\eta ) + \\eta F ( 1 ) \\right ) = \\tfrac 3 5 \\ , . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} ( D ^ * _ z ; ~ z \\in \\mathbb { Z } ) \\stackrel { ( d ) } { = } ( D ^ * _ { a + z } ; ~ z \\in \\mathbb { Z } ) \\mbox { f o r } ~ a \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\int _ { - \\frac { 1 } { 2 } } ^ { \\frac { 1 } { 2 } } { \\rm d } \\lambda I ( \\lambda , z ) I _ { \\ell } ( \\lambda , z ) e \\left ( u \\lambda \\right ) = z ^ { \\frac { 1 } { \\ell } } + O ( 1 ) . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} v _ { n \\dots n } ( x ) & = \\int _ { - \\infty } ^ { x ^ n } f _ { n \\dots n } ( x ^ { ' } , y ^ n ) d y ^ n \\\\ & = \\int _ { - \\infty } ^ { \\infty } f _ { n \\dots n } ( x ^ { ' } , y ^ n ) H ( x ^ n - y ^ n ) d y ^ n . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} B _ k ( \\Omega ( L ) [ 1 ] ) = \\underbrace { \\Omega ( L ) [ 1 ] \\otimes \\dots \\otimes \\Omega ( L ) [ 1 ] } _ { } . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\Im A = ( I - K ) ^ { - 1 } ( I - K K ^ * ) ( I - K ^ * ) ^ { - 1 } \\geq 0 . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ { \\ast } ) ( U _ { 1 } , V _ { 1 } ) , \\pi _ { \\ast } Z ) & = - g _ { 1 } ( \\varphi \\nabla ^ { ^ { M _ 1 } } _ { U _ { 1 } } V _ { 1 } , \\varphi Z ) - \\eta ( \\nabla ^ { ^ { M _ 1 } } _ { U _ { 1 } } V _ { 1 } ) \\eta ( Z ) . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\frac { t } { e ^ t - 1 } e ^ { x t } = \\sum _ { n = 0 } ^ \\infty B _ n ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 1 - 1 0 ] ) . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} T _ h ( { \\bf 1 } _ { \\{ r \\} } \\circ \\pi _ x ) ( y ) & = \\int _ X { \\bf 1 } _ { \\{ r \\} } ( \\pi ( x , z ) ) \\ > K _ h ( y , d z ) = K _ h ( y , \\{ z \\in X : \\ > \\pi ( x , z ) = r \\} ) \\\\ & = \\frac { 1 } { \\omega _ h } | \\{ z \\in X : \\ > \\pi ( x , z ) = r , \\ > \\pi ( y , z ) = h \\} | \\\\ & = \\frac { 1 } { \\omega _ h } | \\{ z \\in X : \\ > \\pi ( z , x ) = \\bar r , \\ > \\pi ( y , z ) = h \\} | \\\\ & = \\frac { 1 } { \\omega _ h } p _ { h , \\bar r } ^ { \\pi ( y , x ) } = \\frac { 1 } { \\omega _ h } p _ { r , \\bar h } ^ { \\pi ( x , y ) } . \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 2 } ( k + 1 - 3 i ) a ^ { i } + \\frac { ( k - 1 ) ( k - 8 ) } { 2 } a ^ { k - 1 } > 0 \\ , . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} L Q L = Q ^ { - 1 } , \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} K _ { N , L } = \\frac { \\mathrm { v o l } \\left ( O ( L ) \\right ) \\mathrm { v o l } \\left ( O ( N ) \\right ) } { \\mathrm { v o l } \\left ( O ( L + N ) \\right ) } \\left ( \\frac { ( 2 \\pi ) ^ L } { L ! } \\right ) ^ { N / 2 } = \\bigg ( \\frac { 2 ^ L } { L ! } \\bigg ) ^ { \\ ! N / 2 } \\prod _ { j = 1 } ^ N \\frac { \\Gamma ( \\frac { L + j } 2 ) } { \\Gamma ( \\frac j 2 ) } . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\frac 1 { 1 - h _ 1 - \\cdots - h _ p } \\cdot \\prod _ { i = 1 } ^ p \\frac { ( 1 - h _ i ) ^ { m _ i } } { 1 - 2 h _ i } \\quad . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\abs { x } - \\mu ( \\infty , x ) s ( x ) & = V \\mu ( \\abs { x } - \\mu ( \\infty , x ) s ( x ) ) = V ( \\mu ( x ) - \\mu ( \\infty , x ) ) \\\\ & = V \\mu ( x ) - \\mu ( \\infty , x ) V ( \\chi _ { [ 0 , \\infty ) } ) = V \\mu ( x ) - \\mu ( \\infty , x ) s ( x ) , \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} | \\lambda _ { n , j } ^ { ( 1 ) } - \\lambda _ { n , j } ^ { ( 2 ) } | = O \\left ( \\sum _ { i = 1 } ^ 2 \\frac { 1 } { \\lambda _ { n , 1 } ^ { ( i ) } } \\right ) \\quad \\textrm { f o r a l l } \\ \\ 1 \\le j \\le m , \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\gamma ( b ( F _ \\pm X , F _ \\pm Y ) - b ( X , Y ) , J _ \\pm Z ) = 0 , \\ ; \\ ; \\forall X , Y \\in P _ \\pm , Z \\in Q _ \\pm . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\widetilde { \\bf A } _ { ( k ) } & = { \\bf h } _ { k } { \\bf A } _ { ( k ) } { \\bf h ' } _ { ( k ) } = \\left [ \\begin{array} { c c } a _ { ( k ) 1 1 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf A } _ { ( k ) 2 2 : 1 } \\end{array} \\right ] , ~ ~ k = 1 , \\cdots , p . \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} \\mu ( \\omega _ q ) = \\sum _ { p \\leq x } \\frac { \\omega _ q ( p ) } { p } \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t v _ i - d _ i \\Delta v _ i & = \\nabla f _ i ( u _ { \\infty } ) \\cdot v = : L _ i v , & & x \\in \\Omega , t > 0 , \\\\ \\nabla v _ i \\cdot \\nu & = 0 , & & x \\in \\partial \\Omega , t > 0 , \\\\ v _ i ( x , 0 ) & = u _ { i , 0 } ( x ) - u _ { i , \\infty } , & & x \\in \\Omega , \\end{aligned} \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\Phi _ n ^ { - 1 } = { \\mathbb I } _ \\bot + \\check { \\Psi } _ n \\ , , \\check { \\Psi } _ n : = \\begin{pmatrix} \\check { \\Psi } _ { n , 1 } & \\check { \\Psi } _ { n , 2 } \\\\ \\overline { \\check { \\Psi } } _ { n , 2 } & \\overline { \\check { \\Psi } } _ { n , 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\begin{aligned} k = \\frac { \\lambda _ 2 } { \\lambda _ 4 } \\left ( \\lambda _ 2 \\lambda _ 4 - \\lambda _ 3 \\lambda _ 5 \\right ) = \\lambda _ 2 ^ 2 - \\frac { \\lambda _ 2 \\lambda _ 3 \\lambda _ 5 } { \\lambda _ 4 } < 0 \\\\ \\ell = \\frac { \\lambda _ 4 } { \\lambda _ 2 } \\left ( \\lambda _ 2 \\lambda _ 4 - \\lambda _ 3 \\lambda _ 5 \\right ) = \\lambda _ 4 ^ 2 - \\frac { \\lambda _ 3 \\lambda _ 4 \\lambda _ 5 } { \\lambda _ 2 } < 0 . \\end{aligned} \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\sum _ { \\substack { N ( \\varpi ) \\leq x \\\\ \\varpi \\equiv 1 \\bmod { ( 1 + i ) ^ 3 } } } \\frac { \\log N ( \\varpi ) } { N ( \\varpi ) } = \\log x + O ( \\log \\log 3 x ) . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} j _ \\alpha ( y ) : = _ 0 F _ 1 ( \\alpha + 1 ; - y ^ 2 / 4 ) \\quad \\quad ( y \\in \\mathbb C ) . \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\eta ^ { 1 2 } b _ 1 & = ( \\eta ^ { 1 2 } , 0 ) + u ( b E _ 4 ^ 2 - a E _ 6 ) e _ 1 + u ( b E _ 6 - a E _ 4 ) e _ 2 , \\\\ \\eta ^ { 1 2 } b _ 2 & = ( 0 , \\eta ^ { 1 2 } ) + u ( d E _ 4 ^ 2 - c E _ 6 ) e _ 1 + u ( d E _ 6 - c E _ 4 ) e _ 2 . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} [ h _ { j , - 1 } , \\psi ^ + _ { i , 2 } ] = 0 , \\ [ h _ { j , 1 } , \\psi ^ - _ { i , b _ i - 2 } ] = 0 . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\dim \\mathcal O _ { X ' , \\pi ( \\xi ) } = \\dim \\mathcal O _ { X , \\xi } - \\dim \\mathcal O _ { \\pi ^ { - 1 } ( \\pi ( \\xi ) ) , \\xi } = . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\left . \\frac { \\partial ^ { l } } { \\partial \\xi ^ { j _ 1 } \\cdots \\partial \\xi ^ { j _ { l } } } w ( x , \\xi ) \\right | _ { \\xi = e _ n } = 0 . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} S _ f ^ \\nu ( n ) = \\sum _ { m \\leq n } \\frac { a _ f ( m ) } { m ^ { \\frac { k - 1 } { 2 } + \\frac { 1 } { 6 } - \\epsilon } } , \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} C _ 0 ^ \\vartheta ( t , X ) & = X \\cup \\{ 0 \\} , \\\\ C _ { n + 1 } ^ \\vartheta ( t , X ) & = \\begin{aligned} [ t ] C _ { n } ^ \\vartheta ( t , X ) & \\cup \\{ s \\in \\varepsilon ( S _ { \\omega ^ \\alpha } ^ u ) \\ , | \\ , s ^ * \\in C _ { n } ^ \\vartheta ( t , X ) \\} \\\\ & \\cup \\{ \\vartheta ( s ) \\ , | \\ , s \\in C _ { n } ^ \\vartheta ( t , X ) s < t \\} \\\\ & \\cup \\{ s \\ , | \\ , s < s ' s ' \\in C _ { n } ^ \\vartheta ( t , X ) \\cap \\Omega \\} . \\end{aligned} \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} E _ { s } ^ { \\pm } \\left ( \\xi \\right ) = \\int { F _ { s } ^ { \\pm } \\left ( \\xi \\right ) d \\xi } \\ \\left ( { s = 0 , 1 , 2 , \\cdots } \\right ) , \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} \\left ( G ^ { \\star } , F ^ { \\star } \\right ) = T _ { 1 } \\circ \\left ( G , F \\right ) . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} ( \\hat d ^ 1 \\circ \\hat d ^ 1 ) _ { \\alpha _ 2 , \\alpha _ 1 } = 0 \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} B ( z , s ) = \\int _ 0 ^ \\infty \\frac { x ^ z } { ( 1 + x ) ^ { z + s } } \\ ; \\frac { d x } { x } , \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} B _ { \\eta } : = \\{ v \\in X ^ 1 _ T : \\| v \\| _ { X ^ 1 _ T } \\le \\eta \\} . \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} & L ( t , x , \\tau , \\xi ) P _ 0 ( t , x , \\xi ) \\\\ = P _ 0 ( t , x , \\xi ) & \\ , { \\rm d i a g } ( \\tau - { A } _ { 1 0 } ( t , x , \\xi ) , \\ldots , \\tau - { A } _ { l 0 } ( t , x , \\xi ) ) \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\psi ( x _ 0 , t _ 0 ) = v ^ * ( x _ 0 , t _ 0 ) > \\min ( 0 , B _ { k + 1 } t _ 0 - ( d + \\alpha ) \\log ( | x _ 0 | ) ) \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} c _ { 1 , \\infty } ( \\mu _ 1 ^ 2 + 2 \\mu _ 1 ) + c _ { 3 , \\infty } ( \\mu _ 3 ^ 2 + 2 \\mu _ 3 ) = 0 , \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} & \\hat { b } ( \\lambda ) : = \\sum _ { r = 0 } ^ { \\infty } \\star g ^ r _ { \\lambda + v \\hbar } ( \\zeta ) z ^ r \\ , \\ \\ , \\ \\hat { c } ( \\lambda ) : = \\sum _ { r = 0 } ^ { \\infty } \\star g ^ r _ { - \\lambda - ( w - v ) \\hbar } ( \\zeta ) z ^ r \\\\ & \\hat { \\Phi } : = \\star H _ { v , w } ( - z ) , \\ , \\ H _ { v , w } : = \\frac { \\prod _ { i = 1 } ^ { v } \\vartheta ( z - t _ i + \\hbar ) \\prod _ { i = v + 1 } ^ w \\vartheta ( z - t _ i - \\hbar ) } { \\prod _ { i = 1 } ^ w \\vartheta ( z - t _ i ) } \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\displaystyle { \\begin{array} { c c c } \\dot { x } & = & - y + P _ n ( x , y ) \\\\ \\dot { y } & = & x + Q _ n ( x , y ) , \\\\ \\end{array} } \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} G = S L ( V _ { 1 } ) \\times \\dotsb \\times S L ( V _ { n } ) . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\Gamma ( s ) = \\int _ 0 ^ \\infty t ^ s e ^ { - t } \\frac { d t } { t } \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\mathcal { E } _ \\varphi ( f , g ) = \\langle \\Box _ { \\varphi } f , g \\rangle _ \\varphi \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} g _ 1 ( X , Y ) \\xi - \\omega \\mathcal { A } _ { X } Y + \\mathcal { C } \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { X } Y & = \\mathcal { A } _ { X } \\mathcal { B } Y + \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { C } Y + \\eta ( Y ) X , \\\\ \\phi \\mathcal { A } _ { X } Y + \\mathcal { B } \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { X } Y & = \\mathcal { V } \\nabla ^ { ^ { M _ 1 } } _ { X } \\mathcal { B } Y + \\mathcal { A } _ { X } \\mathcal { C } Y , \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} ( g \\rtimes \\sigma ) \\cdot \\mathcal { R } _ h ( x _ 1 , \\ldots , x _ k ) : = \\mathcal { R } _ g \\left ( ( - 1 ) ^ { h _ { \\sigma ^ { - 1 } ( 1 ) } } x _ { \\sigma ^ { - 1 } ( 1 ) } , \\ldots , ( - 1 ) ^ { h _ { \\sigma ^ { - 1 } ( k ) } } x _ { \\sigma ^ { - 1 } ( k ) } \\right ) . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\overset { \\circ \\circ } S _ k ( X ' , \\widehat { \\mathcal U } ) = S _ k ( X ' ) \\cap \\mathcal R ^ { - 1 } ( \\overset { \\circ } S _ k ( X , \\widehat { \\mathcal U } ) ) , \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} K _ 0 ^ \\times \\backslash \\mathrm { I s o } ( K _ 3 , K _ 0 ) / K _ 3 ^ \\times = T _ 0 ( F ) \\backslash J ( F ) / T _ 3 ( F ) . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\sum _ { s \\in \\mathcal S } \\chi _ s { \\rm { t r } } ( P _ { U _ s \\cap M } ) \\leq \\sum _ { s \\in \\mathcal S } \\chi _ s { \\rm { t r } } ( P _ { U _ s } P _ M ) = \\chi _ 0 \\dim M . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} A _ n = \\mathbf 1 _ { [ 0 , \\tau _ n ] \\times B _ n } . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} V = \\sum _ { i = 1 } ^ k V _ i . \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} Y _ { k , r } : = \\left \\{ \\begin{array} { c c } \\omega \\in \\wedge ^ { k + \\ell } V _ d , \\ ; \\omega = \\sum \\alpha _ i \\wedge \\beta _ i \\mbox { s u c h t h a t } \\beta _ i \\in \\wedge ^ \\ell V _ d \\\\ \\mbox { a n d } \\alpha _ i \\in \\wedge ^ k U \\mbox { f o r s o m e } U \\subset V _ d \\mbox { o f d i m e n s i o n } r \\end{array} \\right \\} , \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\begin{aligned} \\log | 1 + z | & = \\frac { 1 } { 2 } \\log | 1 + z | ^ 2 \\\\ & = \\frac { 1 } { 2 } ( \\log ( 1 + 2 \\Re z + | z | ^ 2 ) \\\\ & \\leq \\frac { 1 } { 2 } ( 2 \\Re z + | z | ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} \\hat { 1 _ S } ( \\chi _ 1 , \\dots , \\chi _ m , 0 ^ { n - m } ) = \\frac { ( - 1 ) ^ { m / 2 } } { n ^ { m / 2 } } \\frac { n ! } { n ^ n } + O _ m \\ ( \\frac 1 { n ^ { m / 2 + 1 } } \\frac { n ! } { n ^ n } \\ ) . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} \\lim _ { \\ell \\rightarrow \\infty } \\gamma _ { \\ell } ^ i = \\left ( 1 - \\sqrt { 1 - \\lambda { } _ { \\epsilon } } \\right ) ^ i \\lim _ { \\ell \\rightarrow \\infty } \\beta _ { \\ell } ^ i = 1 . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} P _ { K } x = \\left ( \\max \\left \\{ \\xi _ { i } , 0 \\right \\} \\right ) _ { i \\in I } ; \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} I = \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix} \\right ) , U = \\left ( \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\dot \\eta ^ 0 ( z ) = \\left ( \\frac { \\partial _ N v } { ( N \\cdot \\nu ^ 0 ) ^ 2 \\Delta h ^ 0 } \\right ) ( z , \\eta ^ 0 ( z ) ) , \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\ddot x = \\nabla U ( x ) , x \\in \\mathbb { R } ^ 2 \\setminus \\Sigma , \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\frac { \\mathtt { g } _ { a , \\mu } ' ( 1 ) } { \\mathtt { g } _ { a , \\mu } ( 1 ) } & = a \\left ( a ^ { a / 2 } - 1 \\right ) ( \\mu - 1 ) + a ^ { a / 2 } - \\sum _ { n = 1 } ^ \\infty \\frac { 2 a ^ { a / 2 } } { \\mathtt { j } _ { \\mu , n } ^ 2 - 1 } \\\\ & \\geq a \\left ( a ^ { a / 2 } - 1 \\right ) ( \\nu - 1 ) + a ^ { a / 2 } - \\sum _ { n = 1 } ^ \\infty \\frac { 2 a ^ { a / 2 } } { \\mathtt { j } _ { \\nu , n } ^ 2 - 1 } = \\frac { \\mathtt { g } _ { a , \\nu } ' ( 1 ) } { \\mathtt { g } _ { a , \\nu } ( 1 ) } . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\max \\{ R , \\underline { \\gamma } \\} \\sum _ { ( \\boldsymbol { j } , \\boldsymbol { k } ) \\in \\mathcal { I } _ n ( \\boldsymbol { \\alpha } ) ^ c } \\prod _ { l = 1 } ^ d U _ { j _ l , n } | D ^ { \\boldsymbol { r } } \\psi _ { \\boldsymbol { j } , \\boldsymbol { k } } ( \\boldsymbol { x } ) | . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\mathbb { P } ( L _ { n , 1 } = n _ 1 , \\ldots , L _ { n , k } = n _ k \\ , | \\ , K _ n = k ) = \\frac { 1 } { ( n , k ) ^ { \\dagger } } \\prod _ { i = 1 } ^ k ( n _ i , 1 ) ^ { \\dagger } , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\sup _ { 0 \\le t \\le T } \\left | \\sum _ { k = K } ^ \\infty ( k - 2 ) { \\bar { B } } ^ n _ k ( t ) \\right | \\le \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty \\frac { k n _ k } { n } \\le K ^ { - \\varepsilon _ { \\boldsymbol { p } } } \\sup _ { n \\in \\mathbb { N } } \\sum _ { k = K } ^ \\infty \\frac { n _ k } { n } k ^ { 1 + \\varepsilon _ { \\boldsymbol { p } } } \\to 0 . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\Delta _ { h _ k e _ n } e _ n \\mid _ { q = 0 } = h _ k ( 1 , t , \\dots , t ^ { n - 1 } ) \\cdot e _ n ( 1 , t , \\dots , t ^ { n - 1 } ) \\cdot \\left [ \\frac { M B _ { ( 1 ^ n ) } \\Pi _ { ( 1 ^ n ) } \\widetilde { H } _ { ( 1 ^ n ) } } { w _ { ( 1 ^ n ) } } \\right ] _ { q = 0 } . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} d _ { w } ( x , y ) = & w ( \\varphi _ { x } ) + w ( \\varphi _ { y } ) + \\sum _ { z \\neq x , y } w ( \\varphi _ { \\{ x , z \\} } ) + \\sum _ { z \\neq x , y } w ( \\varphi _ { \\{ y , z \\} } ) \\\\ = & w ( \\varphi _ { x } ) + w ( \\varphi _ { y } ) + \\sum _ { z \\neq x } w ( \\varphi _ { \\{ x , z \\} } ) + \\sum _ { z \\neq y } w ( \\varphi _ { \\{ y , z \\} } ) - 2 w ( \\varphi _ { \\{ x , y \\} } ) = 2 a - 4 \\cdot \\frac { 1 + b _ { k } - b _ { i } } { 4 } = 2 a + b _ { i } - 1 - b _ { k } \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} c _ { 1 } ( m ) \\Lambda \\geq \\theta ( x , \\overline { \\sigma } ) \\geq \\sum _ { i = 0 } ^ { M } \\theta ( x , 2 ^ { - i } \\overline { \\sigma } ) - \\theta ( x , 2 ^ { - ( i + 1 ) } \\overline { \\sigma } ) \\geq M \\frac { 2 c _ { 1 } ( m ) \\Lambda } { - \\log _ { 2 } ( 2 \\tau ) } \\geq \\frac { 3 } { 2 } c _ { 1 } ( m ) \\Lambda , \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} D _ h ^ k ( s ) = \\sum _ { m \\in \\mathbb { Z } } \\frac { r _ { 2 k + 1 } ( m ^ 2 + h ) } { ( 2 m ^ 2 + h ) ^ { s + \\frac { k - 1 } { 2 } } } - \\mathfrak { E } _ h ^ k ( s ) \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\| f \\| _ { X \\odot Y } = \\inf \\{ \\| g \\| _ X \\| h \\| _ Y \\colon f = g h , \\ g \\in X , \\ h \\in Y \\} . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} L ( s , f ) : = \\sum _ { n \\geq 1 } \\frac { a ( n ) } { n ^ { s + \\frac { k - 1 } { 2 } } } , \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} \\hat y _ { t , x } ( r ) = Q _ { t + r } e ^ { - r A ^ * } Q _ t ^ { - 1 } x , r \\in [ - t , 0 ] ; \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} c ^ { \\circ } : = \\min _ { ( j , l ) \\in \\mathcal { Z } } c ^ { \\circ } _ { j l } \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} q ( S ) = g ( C / G ^ 0 ) . \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} \\log m = \\theta ( U ) = U + O \\bigg ( \\frac U { \\log ^ 2 U } \\bigg ) = \\frac 1 5 \\log x - \\log \\log x + O \\bigg ( \\frac { \\log x } { ( \\log \\log x ) ^ 2 } \\bigg ) , \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} g _ { m , t } \\ ( x \\ ) = \\sum _ { i = 0 } ^ { m - 1 } g _ t \\ ( T _ t ^ i \\ ( x \\ ) \\ ) . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} A ( t ) ^ { 1 / 2 } F ( t ) = \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\infty } \\zeta ^ { - 1 / 2 } ( A ( t ) + \\zeta I ) ^ { - 1 } A ( t ) F ( t ) \\ , d \\zeta , 0 < | t | \\le t ^ 0 . \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} ( T + ( f _ s ) ) ( q + ( f _ s ) ) = \\Lambda ( \\lambda ) + ( f _ s ) , \\mbox { w i t h } T = b g \\mbox { a n d } \\Lambda = a b . \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} A ( \\mathcal { O } ) = \\{ ( \\alpha _ 1 , \\ldots , \\alpha _ k ) \\mid \\alpha _ r = \\alpha _ s = 1 \\ , \\ , \\ , \\ , \\alpha _ i = 0 \\ , \\ , \\ , \\ , i \\neq r , s \\} . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathfrak { l } ^ { ( p , p ) } ( z ) \\big ] = \\mathbb { E } \\big [ O _ \\prec ( \\hat { \\Pi } ^ 2 ) \\mathfrak { l } ^ { ( p - 1 , p ) } ( z ) \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\hat { \\Pi } ^ 4 ) \\mathfrak { l } ^ { ( p - 2 , p ) } ( z ) \\big ] + \\mathbb { E } \\big [ O _ \\prec ( \\hat { \\Pi } ^ 4 ) \\mathfrak { l } ^ { ( p - 1 , p - 1 ) } ( z ) \\big ] \\ , . \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} F _ T ( x ) - 1 - x F _ T ( x ) = x C ( x ) ( F _ T ( x ) - 1 ) + \\frac { 1 } { 1 - x } H ( x ) + J ( x ) + \\frac { 1 } { 1 - x } J ' ( x ) . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} C ' _ { s } = c _ 0 + \\sum _ { r = 0 } ^ { n - 2 } \\beta ^ { \\pi ^ { r - s } } c ' _ { r } , \\ \\ \\ s = 0 , 1 , \\cdots , n - 2 . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} B _ { j , m } ( f , g ) ( x ) : = & 2 ^ { - \\frac { ( b - a ) j } { 2 } } \\int \\rho ( t ) f * \\Phi \\left ( \\frac { x } { 2 ^ { ( b - a ) j } } - 2 ^ m ( t ^ a + \\epsilon _ P ( t ) ) \\right ) \\\\ & g * \\Phi ( x - 2 ^ m ( t ^ b + \\epsilon _ Q ( t ) ) ) \\ , d t \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ s a _ j \\ge \\log | \\L _ s ( Y ) | + c \\sqrt { s } . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\omega | ^ { 2 } - | \\bigtriangledown \\psi | ^ { 2 } ) d x d y = - 2 \\nu \\int _ { \\mathbb { T } _ { \\alpha } } ( | \\bigtriangledown \\omega | ^ { 2 } - | \\omega | ^ { 2 } ) d x d y , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} ( \\varphi _ x \\circ \\Phi ) ( W _ p ) & = \\varphi _ x \\big ( b ( p ) W _ { ( x , p , x ) } \\big ) \\\\ & = b ( p ) \\sigma _ c \\big ( ( y , m + p , x ) , \\eta ^ { - 1 } \\big ) \\sigma _ c ( \\eta , ( x , p , x ) ) \\overline { \\sigma _ c ( \\eta ^ { - 1 } , \\eta ) } \\varphi _ y ( W _ { ( y , p , y ) } ) \\\\ & = \\sigma _ c \\big ( ( y , m + p , x ) , ( x , p , x ) \\big ) \\sigma _ c \\big ( \\eta , ( x , p , x ) \\big ) \\overline { \\sigma _ c ( \\eta ^ { - 1 } , \\eta ) } \\varphi _ x \\circ \\Phi ( W _ p ) . \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\sum _ { \\sigma ^ { ( n ) } \\in \\Omega _ { W _ n } } \\mu _ { \\mathbf { \\tilde { h } } } ^ { ( n ) } ( \\sigma _ { n - 1 } \\vee \\sigma ^ { ( n ) } ) = \\mu _ { \\mathbf { \\tilde { h } } } ^ { ( n - 1 ) } ( \\sigma _ { n - 1 } ) \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} k _ 1 = k _ 3 , c _ 1 = c _ 3 . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\theta _ \\lambda ^ { ( 2 ) } ( \\zeta , \\mu ) & = \\int _ 0 ^ 1 s D ^ 2 \\bar J _ \\lambda ( V + s \\phi ) [ \\phi ^ 2 ] \\ , d s . \\\\ & = \\int _ 0 ^ 1 s \\left [ \\int _ { \\Omega _ \\varepsilon } \\vert \\nabla \\phi \\vert ^ 2 - \\varepsilon ^ 2 \\lambda \\phi ^ 2 - 5 ( V + s \\phi ) ^ 4 \\phi ^ 2 \\right ] \\ , d s \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} { \\cal L } _ n = { \\mathfrak L } _ n ^ { < } + { \\cal R } _ n + { \\cal R } _ n ^ \\bot \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\lambda _ 0 ^ X ( s , X ( s ) ) = \\sum _ { k = 1 } ^ K \\lambda _ k ^ X ( s , X ( s ) ) \\lambda _ 0 ^ Z ( s , Z ( s ) ) = \\sum _ { k = 1 } ^ K \\lambda _ k ^ Z ( s , Z ( s ) ) . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\sqrt { \\tfrac { t } { 2 \\pi i } } \\int _ - e ^ { \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } \\phi ( x ) \\ , d x = \\overline { \\sqrt { \\tfrac { i t } { 2 \\pi } } \\int _ - e ^ { - \\frac { i t } { 2 } ( x - \\xi ) ^ 2 } \\bar \\phi ( x ) \\ , d x } . \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} \\Phi _ h \\bigg ( \\bigg ( \\sum _ { i = 0 } ^ \\ell r _ i x _ i \\bigg ) ^ h \\bigg ) = \\sum _ { j _ 1 = 0 } ^ \\ell \\cdots \\sum _ { j _ h = 0 } ^ \\ell r _ { j _ 1 } \\cdots r _ { j _ h } z _ { j _ 1 j _ 2 } \\cdots z _ { j _ { h - 1 } j _ h } = \\bigg ( \\sum _ { j _ 1 = 0 } ^ \\ell \\sum _ { j _ 2 = 0 } ^ \\ell r _ { j _ 1 } r _ { j _ 2 } z _ { j _ 1 j _ 2 } \\bigg ) ^ { h / 2 } , \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\lim _ k \\| \\lambda _ { n _ k } ^ { - 1 } x _ { n _ k } \\| \\neq 0 , \\ \\ \\ \\lim _ k \\| x \\pm \\lambda _ { n _ k } ^ { - 1 } x _ { n _ k } \\| = 1 , \\ \\ \\lim _ k \\| x \\pm i \\lambda _ { n _ k } ^ { - 1 } x _ { n _ k } \\| = 1 , \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } L ( \\phi ) & = & h + \\sum _ { i , j } c _ { i j } \\ , w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } & \\Omega _ \\varepsilon , \\\\ \\phi & = & 0 & \\partial \\Omega _ \\varepsilon , \\\\ \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , \\phi & = & 0 & i , j . \\end{array} \\right . \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} G ( s , t ) = | \\frac { x _ 0 } { \\Gamma ( \\gamma ) } s ^ { \\gamma - 1 } - \\frac { x _ 0 } { \\Gamma ( \\gamma ) } t ^ { \\gamma - 1 } | , ( s , t ) \\in [ 2 \\tau , \\nu ] \\times [ 2 \\tau , \\nu ] , \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} A = K [ \\ ! [ t ^ 6 , t ^ 8 , t ^ { 1 0 } , t ^ { 1 3 } ] \\ ! ] \\cong K [ \\ ! [ x , y , z , w ] \\ ! ] / ( y ^ 2 - x z , y z - x ^ 3 , z ^ 2 - x ^ 2 y , w ^ 2 - x ^ 3 y ) . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\frac { ( k - 2 ) ( k - 1 ) } { 2 } + \\sum _ { i = 1 } ^ { k - 1 } ( 3 i + 1 - 2 k ) b ^ i \\ge 0 \\ , . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} \\mu ^ \\ast ( \\omega ) = ( 0 , \\lambda ^ { - 2 } \\omega ) \\mu ^ \\ast ( \\psi ) = ( d \\log \\lambda , \\psi ) . \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\mu _ j ( z ) = \\sum _ { n \\neq 0 } \\rho _ j ( n ) W _ { \\frac { n } { \\lvert n \\rvert } \\frac { k } { 2 } , i t _ j } ( 4 \\pi \\lvert n \\rvert y ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} G _ 0 ^ 0 ( x ) = & ~ - \\frac 1 { 2 \\pi } \\int _ { - 1 } ^ x \\frac 1 { \\xi } ~ d \\xi + \\frac 1 { 2 \\pi } \\int _ 0 ^ { - 1 } \\frac { e ^ { i \\xi } - 1 } { \\xi } ~ d \\xi + \\frac { 1 } { 2 \\pi } \\int _ { - 1 } ^ { - \\infty } \\frac { e ^ { i \\xi } } { \\xi } ~ d \\xi \\\\ & - \\frac 1 { 2 \\pi } \\int _ 1 ^ { 2 } \\frac { \\chi ( \\xi ) } { \\xi } ~ d \\xi \\\\ = & ~ - \\frac 1 { 2 \\pi } \\log | x | + \\frac { c _ 2 } { 2 \\pi } . \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} | g _ i ( v ) | \\leq C \\sum _ { j = 1 } ^ N ( | v _ j | ^ { \\mu } + | v _ j | ^ { 1 + \\delta } ) v \\in \\mathbb R ^ N _ + , \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} u _ { \\infty } = 1 / \\mathbb { E } Y _ 1 \\stackrel { 1 } { = } 1 / 3 . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{gather*} a _ n ^ { ( \\theta ) } = \\frac { C _ n ^ { ( q , q ^ { \\theta } ) } \\big ( x _ 2 ^ { - 1 } x _ 1 q ^ { - \\theta } \\big ) } { \\prod \\limits _ { i = 0 } ^ { \\theta - 1 } { ( x _ 1 - q ^ i x _ 2 ) } } , 0 \\leq n \\leq \\theta . \\end{gather*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\limsup _ { t \\ , \\rightarrow \\ , \\infty } \\ ; t ^ { \\ : \\ ! \\alpha } \\ ; \\ ! \\| \\ , \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { 2 } ( \\mathbb { R } ^ { 4 } ) } } = : \\ : \\lambda _ { 0 } ( \\alpha ) \\ , < \\ , \\infty . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} d ^ 3 b + w d + w ^ 2 d ^ 2 = c ^ q . \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\frac { ( n - 4 ) r _ { i j } ^ 2 } { 1 - r _ { i j } ^ 2 } = \\frac { s _ { i , j } ^ 2 } { n - 1 } ( 1 + o _ p ( 1 ) ) , \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} & = \\frac { 1 } { \\left ( f ^ { 2 } + 4 g \\right ) ( g h ^ { 2 } - k ^ { 2 } + f h k ) } \\left | \\left [ \\begin{array} [ c ] { c c } R _ { n } \\left ( f ^ { 2 } h + 2 g h - f k \\right ) & \\ \\ g R _ { n } \\left ( 2 k - f h \\right ) \\\\ & \\\\ R _ { n } \\left ( 2 k - f h \\right ) & \\ \\ \\ R _ { n } \\left ( 2 g h + f k \\right ) \\end{array} \\right ] \\right | \\\\ & = R _ { n } ^ { 2 } . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\Phi \\cdot M = \\Phi \\cdot M ^ c + \\Phi \\cdot M ^ q + \\Phi \\cdot M ^ a . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\mathbb { P } ( L _ n ^ { * } = \\ell \\ , | \\ , K _ n = k ) & = \\sum _ { j = 1 } ^ k \\mathbb { P } ( \\mbox { p i c k } L _ { n , j } \\mbox { a n d } L _ { n , j } = \\ell \\ , | \\ , K _ n = k ) \\\\ & = k \\mathbb { P } ( \\mbox { p i c k } L _ { n , 1 } \\mbox { a n d } L _ { n , 1 } = \\ell \\ , | \\ , K _ n = k ) \\\\ & = k \\frac { \\ell } { n } \\mathbb { P } ( L _ { n , 1 } = \\ell \\ , | \\ , K _ n = k ) , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} ( z ^ 2 + t ) ^ 2 = \\bigg ( \\sqrt { 2 t } z + \\frac { | w | } { \\sqrt { 8 t } } \\bigg ) ^ 2 \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} S ^ c _ 3 ( H ) = \\sum _ { 1 \\leq a < b \\leq d _ c } S _ 3 ( \\{ h _ a , h _ b \\} ) , \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} f ^ { ( k ) } ( X ) & = f ^ { ( k - 1 ) } ( f ( X ) ) \\\\ & = b _ 1 ( a _ 1 X ^ { d _ 1 } + \\ldots + a _ s X ^ { d _ s } ) ^ { e _ 1 } + \\ldots + b _ t \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\overline { Q } \\frac { \\partial ^ 2 \\mathbf { u } _ 0 ( \\mathbf { x } , \\tau ) } { \\partial \\tau ^ 2 } = - \\mathcal { W } ^ 0 \\mathbf { u } _ 0 ( \\mathbf { x } , \\tau ) , \\\\ & \\mathbf { u } _ 0 ( \\mathbf { x } , 0 ) = \\boldsymbol { \\phi } ( \\mathbf { x } ) , \\overline { Q } \\frac { \\partial \\mathbf { u } _ 0 } { \\partial \\tau } ( \\mathbf { x } , 0 ) = \\boldsymbol { \\psi } ( \\mathbf { x } ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} \\tau ( x , y ) = \\frac { a ( x y ) } { a ( x ) a ( y ) } \\sigma ( x , y ) . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} \\bar \\Lambda ' = \\left [ \\begin{matrix} \\bar \\Lambda _ 2 \\\\ \\vdots \\\\ \\bar \\Lambda _ k \\end{matrix} \\right ] , Q ( \\varepsilon , \\zeta ) = P ' ( \\varepsilon , \\zeta ) ^ T M _ { \\lambda _ 0 + \\varepsilon } ( \\zeta ) P ' ( \\varepsilon , \\zeta ) \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\int _ U \\int _ A & f ( t , X ( t ) , u , a , m ( t ) ) [ \\widehat { \\gamma } ( t , X ( t ) ) ] ( d a ) \\nu ( d u ) \\\\ & = E \\left [ \\left . \\int _ U \\int _ A f ( t , X ( t ) , u , a , m ( t ) ) \\rho _ t ( d a ) \\nu ( d u ) \\right | X ( t ) \\right ] \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} { F _ { { \\Gamma _ { k | X \\ge Y } } } } \\left ( { { z } } \\right ) = & 1 - \\frac { \\lambda _ k + \\lambda _ { \\bar k } } { \\lambda _ { \\bar k } } { e ^ { - { \\lambda _ k } { \\varphi _ k } } } + \\frac { \\lambda _ k } { \\lambda _ { \\bar k } } e ^ { - \\left ( \\lambda _ k + \\lambda _ { \\bar k } \\right ) \\varphi _ k } , \\\\ { F _ { { \\tilde \\Gamma _ { k | X < Y } } } } \\left ( { { z } } \\right ) = & 1 - e ^ { - \\left ( \\lambda _ k + \\lambda _ { \\bar k } \\right ) \\tilde \\varphi _ k } , \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\bigl ( \\bigl ( W ( \\psi , \\phi ) | W ( \\psi ^ { \\prime } , \\phi ^ { \\prime } ) \\bigr ) \\bigr ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } ( \\psi | \\psi ^ { \\prime } ) \\overline { ( \\phi | \\phi ^ { \\prime } ) } \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} a _ 1 b _ 1 - a _ 2 - b _ 2 & = c _ 1 d _ 1 - c _ 2 - d _ 2 \\\\ a _ 1 d _ 1 - a _ 2 - d _ 2 & = c _ 1 b _ 1 - b _ 2 - b _ 2 . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\operatorname { t r a c e } ( \\rho _ u ( l ) ) = \\pm 2 \\cosh ( u / 2 ) = \\pm ( 2 + u ^ 2 / 4 + O ( u ^ 4 ) ) , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\hat d \\circ \\frak h _ { c a } ^ { i } + \\frak h _ { c a } ^ { i } \\circ \\hat d = \\psi _ { c a } ^ { i + 1 } \\circ \\frak n _ { a } ^ { i + 1 i } - \\frak n _ { c } ^ { i + 1 i } \\circ \\psi _ { c a } ^ { i } . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\| \\varphi ( f ) - ( \\varphi ^ { \\prime } ( f ) + \\sigma ( f ) ) \\| & \\leq \\| p \\varphi ( f ) - p \\varphi ( f ) p \\| + \\| ( 1 - p ) \\varphi ( f ) - \\sum _ { i = 1 } ^ l f ( z _ i ) p _ i \\| \\\\ & < \\frac { 2 \\varepsilon } { 3 } + \\frac { \\varepsilon } { 3 } = \\varepsilon \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} \\mathbf { w } _ { 1 } = \\left ( \\begin{array} { c } \\sin \\omega _ { 1 } \\mathbf { t } _ { 1 } \\\\ \\mathbf { 0 } _ { 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\theta ' _ k = \\deg \\theta \\cdot { \\rm { i d } } _ { H ^ k ( U ) } . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} \\tilde { r } _ { 0 , n } & : = \\frac { 1 } { n } \\log | \\tilde { { \\cal M } } _ 0 | + \\Delta _ n + \\frac { \\log \\log | { \\cal X } | + 2 } { n } , \\\\ \\tilde { r } _ { 2 , n } & : = \\frac { 1 } { n } \\log | \\tilde { { \\cal M } } _ 2 | + \\frac { 1 + \\log | { \\cal Y } | } { n } , \\\\ \\delta _ n & : = \\Delta _ n + \\frac { \\log \\log | { \\cal X } | + 3 + \\log | { \\cal X } | } { n } , \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} & S ( 0 ) = { \\rm I d } , \\\\ [ 1 m m ] & S ( t + s ) = S ( t ) S ( s ) , \\forall s , t \\ge 0 . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\frac 1 n \\sum _ { a _ i > 0 } \\log ( a _ i + 2 ) & \\leq \\frac { k } { n } \\log \\ ( \\frac 1 k \\sum _ { a _ i > 0 } ( a _ i + 2 ) \\ ) \\\\ & = \\frac { k } { n } \\log \\ ( \\frac { n } { k } + 2 \\ ) \\\\ & \\leq O \\ ( \\frac { H } { \\log n } \\ ( \\log \\log n + \\log H ^ { - 1 } \\ ) \\ ) \\\\ & = O \\ ( \\frac { H \\log \\log n } { \\log n } \\ ) . \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\int _ { B _ 1 ( 0 ) } D ^ 2 _ x u ( x , y , \\epsilon ) \\phi ( h ) d h & = \\sum _ { i = 1 } ^ n \\int _ { B _ 1 ( 0 ) } u _ { x _ i x _ i } ( z ) h _ i ^ 2 \\phi ( h ) d h + o ( 1 ) \\\\ & = \\frac C n \\Delta u ( x , y ) + o ( 1 ) \\ , , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ n c _ { l , t } \\ , c _ { j , t } \\ , c _ { k , t } = M _ { l j k } \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} f ( z + 1 ) = f ( z ) , \\ , \\ f ( z + \\tau ) = e ^ { 2 \\pi i \\lambda } f ( z ) . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} F _ { \\lambda } ( \\zeta , \\Lambda ) : = & \\ , k \\ , a _ 0 + a _ 1 \\sum _ { i = 1 } ^ k \\Bigl ( \\Lambda _ i ^ 2 \\ , g _ { \\lambda } ( \\zeta _ i ) - \\sum _ { j \\neq i } \\Lambda _ i \\ , \\Lambda _ j \\ , G _ { \\lambda } ( \\zeta _ i , \\zeta _ j ) \\Bigr ) + a _ 2 \\ , \\lambda \\ , \\sum _ { i = 1 } ^ k \\Lambda _ i ^ 4 \\\\ & - a _ 3 \\ , \\sum _ { i = 1 } ^ k \\Bigl ( \\Lambda _ i ^ 2 \\ , g _ { \\lambda } ( \\zeta _ i ) - \\sum _ { j \\neq i } \\Lambda _ i \\Lambda _ j \\ , G _ { \\lambda } ( \\zeta _ i , \\zeta _ j ) \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} \\rho ^ { \\alpha } ( [ 0 , t ] \\times E ) = \\int _ 0 ^ t \\rho ^ { \\alpha } _ s ( E ) d s = \\int _ 0 ^ t \\delta _ { \\alpha ( s ) } ( E ) d s . \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} t ^ { 0 0 } _ { j l } \\le ( 4 C _ 2 ) ^ { - 1 } c ^ { \\circ } _ { j l } = ( 4 \\beta _ 2 ) ^ { - 1 } \\delta ^ { 1 / 2 } \\| X _ 1 \\| ^ { - 3 } c ^ { \\circ } _ { j l } . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\eta ( x ) : = \\begin{cases} \\dfrac { 1 } { 2 } d ( x ) ^ 2 , & K _ M \\leq 0 , \\\\ 1 - \\cos \\sqrt { a } d ( x ) , & 0 < K _ M \\leq a . \\end{cases} \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} P _ m = X _ 1 X _ 2 \\cdots X _ m \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\lim _ { q \\to \\infty } R _ { q , \\ell } & = \\lim _ { q \\to \\infty } \\frac { \\sum _ { i = 3 } ^ { \\ell } ( q ^ { i - 1 } - 2 q ^ { i - 2 } + q ^ { i - 3 } ) q \\log q } { \\ell q ^ \\ell \\log q } = \\frac { 1 } { \\ell } \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\left . \\left ( \\overline { \\frac { \\partial ^ { 2 } \\left ( g ^ { \\star } _ { i j } \\left ( Z , W \\right ) \\right ) } { \\partial w _ { a a } \\partial w _ { b b } } } + \\frac { \\partial ^ { 2 } \\left ( g ^ { \\star } _ { j i } \\left ( Z , W \\right ) \\right ) } { \\partial w _ { a a } \\partial w _ { b b } } \\right ) \\right \\vert _ { Z = O _ { q \\times N } \\atop { W = O _ { q \\times q } } } = 0 , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} \\frac { \\dd ^ m } { \\dd z ^ m } \\log M _ { n , r } ( z ) = \\frac { \\dd ^ m } { \\dd z ^ m } S _ { n , r } ( z ) + { \\bf 1 } _ { \\{ m = 1 \\} } \\frac { 1 } { 2 } \\log ( r + 1 ) + { \\bf 1 } _ { \\{ m = 1 \\} } \\frac { r } { 2 } \\log 2 \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } + J \\left ( \\frac { \\partial u } { \\partial t } - X _ { H ^ { S T } _ { \\tau , t } } ( u ) \\right ) = 0 , \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} { 1 \\over s } = \\int _ 0 ^ \\infty d F \\thinspace F ^ { - 3 / 2 } \\left [ \\psi _ { s , F } ( - v ) + \\psi _ { s , F } ( v ) \\right ] \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\log _ t ( x , w ) \\geq - ( s ^ \\lambda , y ^ \\lambda ) + \\biggl ( \\lambda - \\log _ t \\Bigl ( \\frac { N } { 1 - \\theta } \\Bigr ) - \\delta ( t ) \\biggr ) e = ( x ^ \\lambda , w ^ \\lambda ) - \\biggl ( \\log _ t \\Bigl ( \\frac { N } { 1 - \\theta } \\Bigr ) + \\delta ( t ) \\biggr ) e \\ , , \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} | \\nabla h ( x ) | ^ 2 ~ \\le ~ \\lambda \\cdot ( 1 - a ^ 2 ) \\cdot \\norm { F _ x } ^ 4 ~ = ~ \\lambda \\cdot ( 1 - a ^ 2 ) \\cdot F _ x ( x ) ^ 2 \\ , . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\sum _ { m ~ \\chi } \\hat { 1 _ S } ( \\chi ) ^ 3 = \\frac { ( - 1 ) ^ { m / 2 } } { 2 ^ { m / 2 } ( m / 2 ) ! } \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 + O _ m \\ ( \\frac 1 n \\ ( \\frac { n ! } { n ^ { n } } \\ ) ^ 3 \\ ) , \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} | x _ 1 + \\dots + x _ n | = ( | x _ 1 | ^ 2 + \\dots + | x _ n | ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} v ( c ) + i _ * ( I ) - { o } _ * ( O ) = 0 \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} \\frac { F \\bigl ( t _ 0 + z ( t _ 0 ) \\bigr ) - F ( t _ 0 ) } { z ( t _ 0 ) } - \\frac { F ( t _ 0 ) - F ( 0 ) } { t _ 0 } = F ' \\bigl ( t _ 0 + z ( t _ 0 ) \\bigr ) - F ' ( t _ 0 ) \\ , . \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla ^ 2 u ( F , \\nabla u ) d v = \\frac { 1 } { 2 } \\int _ \\Sigma | \\nabla u | ^ 2 \\langle F , \\nu \\rangle d \\sigma - \\frac { 1 } { 2 } \\int _ \\Omega | \\nabla u | ^ 2 \\cdot d i v F d v . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} D = \\frac { \\deg ( D ) } { \\deg ( A ) } A + \\frac { 1 } { n _ 1 n _ 0 \\deg ( A ) } ( f ) , \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\Delta _ f ( \\widetilde { H } _ { \\lambda } ) : = f ( \\dots , q ^ { i - 1 } t ^ { j - 1 } , \\dots ) \\cdot \\widetilde { H } _ { \\lambda } , \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\mu _ 1 ( A ) : = L e b ( h _ 1 ^ { - 1 } ( A ) ) , \\quad \\mu _ 2 ( A ) : = L e b ( h _ 2 ^ { - 1 } ( A ) ) . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\mathcal { A } ( \\mathbf { k } ) & \\ge c _ * | \\mathbf { k } | ^ 2 I , \\mathbf { k } \\in \\widetilde { \\Omega } , \\\\ c _ * & = \\alpha _ 0 \\| f ^ { - 1 } \\| _ { L _ \\infty } ^ { - 2 } \\| g ^ { - 1 } \\| _ { L _ \\infty } ^ { - 1 } . \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\omega } { \\partial t } = - { \\rm R i c } ( \\omega ) + \\gamma \\omega + ( 1 - \\beta ) [ D ] + L _ X \\omega \\\\ \\omega | _ { t = 0 } = \\omega ^ { \\ast } . \\end{cases} \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\hat { r } _ i = \\frac { 1 } { m } \\sum _ { \\tau = t - m } ^ { t - 1 } r _ i ( \\tau ) \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} f | _ { \\Gamma _ - } = f _ - ( x , v ) \\ , , ( x , v ) \\in \\Gamma _ - \\ , . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\limsup _ \\lambda ( ( s - r _ \\lambda ) _ + ) = ( \\limsup _ \\lambda ( s - r _ \\lambda ) ) _ + = ( s - \\liminf _ \\lambda r _ \\lambda ) _ + , \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} ( a _ 1 + c _ 1 ) ( b _ 1 - d _ 1 ) & = 2 e _ 1 ( b _ 1 - d _ 1 ) \\\\ a _ 1 + c _ 1 & = 2 e _ 1 \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} G _ 2 ( x ; d ) & = \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ { d + 1 } } + \\frac { x ^ { d + 3 } } { ( 1 - x ) ^ { d + 2 } } + \\sum _ { s = 2 } ^ { d } \\frac { x ^ { d + 3 } } { ( 1 - x ) ^ { s + 1 } } \\\\ & + \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ { d + 1 } } ( F _ T ( x ) - 1 ) + \\sum _ { t = 1 } ^ { d } \\left ( \\frac { x ^ { d + 3 } } { ( 1 - x ) ^ { t + 2 } } \\big ( F _ T ( x ) - 1 \\big ) + \\frac { x ^ { d + 5 } } { ( 1 - x ) ^ { t + 3 } ( 1 - 2 x ) } \\right ) , \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} M _ { l + 1 } : = \\{ p \\in M _ { l } \\textrm { s u c h t h a t t h e r e e x i s t s } X _ p \\in T _ p M _ l \\\\ \\textrm { s a t i s f y i n g } i _ X \\omega _ 1 = d h _ 1 \\} . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} Q = P + \\sum _ { k = 1 } ^ m A _ j ^ * ( Q - C ) ^ { - 1 } A _ j , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { x \\in M } \\lVert A ^ n ( x ) \\rVert ^ { 1 / n } = \\sup _ { ( x , p ) \\in M \\times \\mathbb N : f ^ p ( x ) = x } \\rho ( A ^ p ( x ) ) ^ { 1 / p } . \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{gather*} B ' = \\big ( A ' A ^ { - 1 } \\big ) ' = A '' A ^ { - 1 } - A ' A ^ { - 1 } A ' A ^ { - 1 } = - \\overset { \\mathrm { s f } } R - B ^ 2 . \\end{gather*}"}
-{"id": "2403.png", "formula": "\\begin{align*} S _ 2 ( n + 1 , k | x ) = ( x + k ) S _ 2 ( n , k | x ) + S _ 2 ( n , k - 1 | x ) , \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\frac { d ^ 2 q ^ i ( t ) } { d t ^ 2 } = \\xi ^ i . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} Q = \\left ( \\begin{matrix} \\ast & \\kappa \\alpha _ A & \\alpha _ A & \\alpha _ A \\\\ \\kappa \\alpha _ G & \\ast & \\alpha _ G & \\alpha _ G \\\\ \\alpha _ C & \\alpha _ C & \\ast & \\kappa \\alpha _ C \\\\ \\alpha _ T & \\alpha _ T & \\kappa \\alpha _ T & \\ast \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} & \\sup \\ : ( T - t _ i ) K _ i < \\infty \\\\ & \\sup \\ : ( T - t _ i ) K _ i = \\infty \\\\ \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha _ n f ( x _ { i - 1 } ( t ) ) + a _ n f ( x _ { i } ( t ) ) + \\beta _ n f ( x _ { i + 1 } ( t ) ) , \\ , \\ , i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} Z _ { n , q } = \\prod _ { j = 1 } ^ n \\sum _ { i = 1 } ^ j q ^ { i - 1 } = ( 1 - q ) ^ { - n } \\prod _ { j = 1 } ^ n ( 1 - q ^ j ) \\mbox { f o r } ~ 0 < q < 1 . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ \\ell T ^ { i n } f _ i \\ ; S ^ { j n } g _ j = \\left ( \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k T ^ { i n } f _ i \\right ) \\left ( \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { n = 1 } ^ N \\prod _ { j = 1 } ^ \\ell S ^ { j n } g _ j \\right ) \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} f ( \\theta | x ^ N ) & = \\frac { p _ s ( x ^ N | \\theta ) f ( \\theta ) } { \\int p _ s ( x ^ N | \\theta ) f ( \\theta ) \\ , d \\theta } \\\\ & = \\frac { p ( x ^ N | \\theta ) f ( \\theta ) } { \\int p ( x ^ N | \\theta ) f ( \\theta ) \\ , d \\theta } \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\pi _ { k } ( x , y _ { 0 : k } , A ) \\coloneqq \\frac { \\mathbf { E } _ { x } \\left [ \\mathbf { 1 } _ { A } ( X _ { k \\Delta } ) \\prod _ { j = 0 } ^ { k } g _ { j } ( X _ { j \\Delta } , y _ { j } ) \\right ] } { \\mathbf { E } _ { x } \\left [ \\prod _ { j = 0 } ^ { k } g _ { j } ( X _ { j \\Delta } , y _ { j } ) \\right ] } , A \\in \\mathcal { B } ( \\mathbb { R } ^ { p } ) , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\pi ( f ) = \\{ \\pi _ n ( f ) \\} = \\left \\{ \\int _ { n - 1 } ^ n \\mu ( f ) \\right \\} \\in E , \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\tilde u ^ { ( k ) } = ( \\Phi _ - ) ^ { \\lambda _ 0 } \\sum _ { \\ell = 1 } ^ k \\frac { ( \\log \\Phi _ - ) ^ { k - \\ell } } { ( k - \\ell ) ! } \\tilde v ^ { ( \\ell ) } \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\left \\langle \\mathrm { T r } \\ , P _ m ^ 2 \\right \\rangle _ { 2 n \\times 2 n } = \\prod _ { i = 1 } ^ m \\frac { 2 n } { L _ i + 2 n } \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\int \\limits _ { S ^ { n - 1 } } { \\| x \\| ^ { - 1 } _ K } d \\sigma _ { n - 1 } \\leq C \\int \\limits _ { S ^ { n - 1 } } { \\| x \\| _ K } d \\sigma _ { n - 1 } = C M ( K ) . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\alpha = 2 - \\beta . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} q ( s , t ) = p ( s e _ j + t \\vec { 1 } ) = c _ 0 \\det \\left ( J + s X _ j + t \\begin{pmatrix} I & 0 \\\\ 0 & C ^ { - 1 } \\end{pmatrix} \\right ) . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\widetilde { X } ^ N _ i ( t ) = \\xi _ i ^ N + \\int _ 0 ^ t \\int _ U f ( s , \\widetilde { X } ^ N _ i ( s ^ - ) , u , [ \\alpha ^ { N , - 1 } , \\beta ] _ i ( s ) , \\widetilde { \\mu } _ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\langle \\chi _ n , f \\rangle = 0 \\ { \\rm f o r \\ e a c h \\ } n \\geq 0 , \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} ( m x + r ) ^ n = \\sum _ { k = 0 } ^ n \\left ( \\sum _ { i = k } ^ n { n \\choose i } r ^ { n - i } m ^ i S _ 2 ( i , k ) \\right ) ( x ) _ k , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} I _ g ( z ) : = \\sup _ { \\gamma \\in \\mathbb { R } } \\bigl \\{ \\gamma z - \\log g \\bigl ( e ^ \\gamma \\bigr ) \\bigr \\} . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{gather*} A = J a , \\end{gather*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\int _ { | x - y | \\leq \\delta / C _ { 0 } } e ^ { i \\lambda \\psi ( x , \\alpha ) } f _ { i _ { 1 } \\dots i _ { m } } ( x ) B ^ { i _ { 1 } \\dots i _ { m } } ( x , \\alpha ; \\lambda ) d x = \\mathcal { O } ( e ^ { - \\lambda / C } ) \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} ( F \\star \\mu ) _ { \\infty } & = \\bigl ( ( F \\circ \\tau ) \\star ( \\mu \\circ \\tau ) \\bigr ) _ { \\infty } , \\\\ ( F \\star \\nu ) _ { \\infty } & = \\bigl ( ( F \\circ \\tau ) \\star ( \\nu \\circ \\tau ) \\bigr ) _ { \\infty } , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\psi = \\sum _ { \\beta \\in G } \\psi _ { \\beta } T ^ { E ( \\beta ) } e ^ { \\mu ( \\beta ) / 2 } , \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} | d _ E | = d _ F ^ 2 \\mathcal { N } _ { F / \\Q } ( \\frak { D } _ { E / F } ) , \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\tilde { X } _ t ( \\omega ) = \\omega ( t ) , t \\ge 0 . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} a \\ , w \\ , = \\ , T _ a w \\ , + \\ , T _ w a \\ , + \\ , R ( a , w ) \\ , . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\mathrm { A u t } ( \\mathbb { C } ^ 2 ) = \\mathrm { A } _ 2 \\ast _ { \\mathrm { S } _ 2 } \\mathrm { E } _ 2 \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} f ( x ) = f ^ * ( - x ) + \\langle w , x \\rangle , x \\in X . \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} R = N _ c \\frac { F } { \\eta } . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} D = \\begin{cases} ( y / x ) & , \\\\ ( 1 ) & . \\end{cases} \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} \\lambda _ i | _ { T } = \\lambda _ { 2 n - i + 1 } | _ { T } = \\varpi _ i , \\mbox { } \\lambda _ n | _ { T } = \\lambda _ { n + 1 } | _ { T } = 2 \\varpi _ n , ~ 1 \\leqslant i \\leqslant n - 1 . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\mathcal M _ { k + 1 } ( \\beta ; [ 1 , 2 ] ) = \\bigcup _ { t \\in [ 1 , 2 ] } \\mathcal M _ { k + 1 } ( \\beta ; J _ t ) \\times \\{ t \\} . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\phi ( t , \\lambda , x ) = v ( t + \\lambda , \\phi ( t , \\lambda , x ) ) . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha , \\beta } f ( t ) = I _ { a ^ + } ^ { \\beta ( n - \\alpha ) } D ^ { n } I _ { a ^ + } ^ { ( 1 - \\beta ) ( n - \\alpha ) } f ( t ) , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\log | p ( z ) | \\leq \\Re ( \\sum _ { j = 1 } ^ { n } p _ j ( 0 ) z _ j ) + \\frac { 1 } { 2 } \\| z \\| _ { \\infty } ^ 2 ( | \\sum _ { j = 1 } ^ { n } p _ j ( 0 ) | ^ 2 - \\Re ( \\sum _ { j , k = 1 } ^ { n } p _ { j k } ( 0 ) ) ) \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\ell _ { ( a , w ) } F ( x , z ) = & F ( a ^ { - 1 } x , \\overline { w } z \\sigma ( a , a ^ { - 1 } ) \\overline { \\sigma ( a ^ { - 1 } , x ) } ) \\\\ = & F ( a ^ { - 1 } x , \\overline { w } z \\sigma ( a , a ^ { - 1 } x ) ) \\\\ = & w \\overline { z } \\ , \\overline { \\sigma ( a , a ^ { - 1 } x ) } f ( a ^ { - 1 } x ) \\\\ = & \\overline { z } w \\ , \\ell ^ \\sigma _ a f ( x ) . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\omega ^ { \\theta } ( \\eta + \\delta ) = e ^ { \\eta L _ 0 ' } \\omega ^ { \\theta } ( \\delta ) + \\int _ { 0 } ^ { \\eta } e ^ { ( \\eta - s ) L _ 0 ' } g ( \\delta + s ) \\dd s . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} K ( x , p ; E , h ) = K _ 0 + h ^ 2 \\bigl ( { K _ 0 ( E ) L _ 2 \\over L _ 0 - E } - \\widetilde C _ 0 ( E ) \\bigr ) \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} f ( x , u ) = \\mu ( x ) u - E ( x , u ) , \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} ( M ( s , s + t ] ) ^ 2 & = \\big ( \\sum _ { j = 1 } ^ { N ( s ) } L _ j ( s - T _ j , s + t - T _ j ) \\big ) ^ 2 + 2 \\sum _ { j = 1 } ^ { N ( s ) } L _ j ( s - T _ j , s + t - T _ j ] \\sum _ { j = N ( s ) + 1 } ^ { N ( s + t ) } L _ j ( s + t - T _ j ) \\\\ & + \\big ( \\sum _ { j = N ( s ) + 1 } ^ { N ( s + t ) } L _ j ( s + t - T _ j ) \\big ) ^ 2 = : \\mathrm { I } + 2 \\mathrm { I I } + \\mathrm { I I I } \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} \\nabla ^ { ^ M } _ { V } X = \\mathcal { T } _ { V } X + \\mathcal { H } ( \\nabla ^ { ^ M } _ { V } X ) ; \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} C ^ { 0 , \\beta } ( \\overline { V } ) : = \\{ f \\in C ( \\overline { V } ) : \\mathrm { h \\ddot { o } l } _ \\beta ( f , \\overline { V } ) < \\infty \\} . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\ell ( \\tau , \\xi , \\eta ) = \\tau - \\xi ^ 2 \\coth \\xi + \\xi + \\xi ^ { - 1 } \\eta ^ 2 . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { P } ( \\Pi _ n > n ) = \\mathbb { P } ( D ^ * _ z > 0 ) = \\frac { 1 } { 2 } \\left ( 1 - \\frac { 1 } { \\mu } \\right ) , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} \\langle A \\chi _ j , \\chi _ k \\rangle = a _ { k + j + 1 } \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} \\Sigma _ t = \\{ v \\in \\R : t ^ { - \\frac 1 { 1 2 } } < v < t ^ { \\frac 1 { 1 2 } } \\} , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} W ( x , v ) = \\frac { 1 } { ( 1 - x ) ( 1 - x v ) ( 1 - ( 1 + v ) x ) } . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\frac 1 2 \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } ( 2 y _ { r , i } y _ { r , i } ' - y _ { r , i } ^ 2 - y _ { r , i } '^ 2 ) = \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } ( y _ { r , i } ' y _ { r , i } - \\frac { 1 } { 2 } ( y _ { r , i } ^ 2 + y _ { r , i } '^ 2 ) ) . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\alpha = \\frac { \\gamma } { \\sqrt { 3 } } \\beta = \\frac { \\bar { \\gamma } } { \\sqrt { 3 } } . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( \\mathcal E P ) & = l _ { \\langle \\rangle } ( P ) , \\\\ o _ { \\langle \\rangle } ( \\mathcal E P ) & = \\Omega ^ { o _ { \\langle \\rangle } ( P ) } , \\\\ d ( \\mathcal E P ) & = \\max \\{ 2 , d ( P ) - 1 \\} \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\mathcal { M } _ 2 = \\{ \\alpha + \\beta : \\alpha \\in \\mathcal { M } , \\beta \\in \\N ^ n , | \\beta | \\leq 2 \\} \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{gather*} ( - 1 ) ^ { \\theta { m \\choose 2 } } \\frac { q ^ { - \\theta ^ 2 ( N - 1 ) { m \\choose 2 } } } { \\prod _ { \\substack { 1 \\leq i < j \\leq m \\\\ 0 \\leq k < \\theta } } { \\big ( x _ i - q ^ k x _ j \\big ) } } \\prod _ { i = 1 } ^ m { \\frac { 1 } { \\big ( t ^ { N - i + 1 } ; q \\big ) _ { \\theta ( i - 1 ) } } } . \\end{gather*}"}
-{"id": "1605.png", "formula": "\\begin{align*} H ^ { i j } ( \\tau , t , x ) = \\begin{cases} H ^ j ( t , x ) & , \\\\ H ^ i ( t , x ) & . \\end{cases} \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\sum _ { A ' , B ' } \\left ( h _ { A ' B ' } , { H _ { A ' B ' } } \\right ) _ { \\varphi } = \\sum _ { A ' , B ' } \\left ( h _ { A ' B ' } , { H _ { ( A ' B ' ) } } \\right ) _ { \\varphi } , \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} \\{ C ^ T _ { N , h } \\} _ { l + 1 , m + 1 } = \\int _ { - \\infty } ^ \\infty T _ { T - l } ( \\lambda ) T _ { T - m } ( \\lambda ) \\ , d \\rho ^ { N , h } ( \\lambda ) , l , m = 0 , \\ldots , T - 1 . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\begin{cases} \\partial ^ 2 _ { x } \\theta = 0 \\ , , \\\\ \\theta ( x = 0 ) = \\eta \\ , , \\quad \\theta ( x = 1 ) = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} h ( f g ) & = \\psi ( f g ) = \\sum _ { \\alpha \\in \\Lambda } \\psi _ { \\alpha } ( f g ) y _ { \\alpha } = \\sum _ { \\alpha \\in \\Lambda } f \\psi _ { \\alpha } ( g ) y _ { \\alpha } + \\sum _ { \\alpha \\in \\Lambda } ( - 1 ) ^ { | f | | g | } g \\psi _ { \\alpha } ( f ) y _ { \\alpha } \\\\ & = f \\psi ( g ) + ( - 1 ) ^ { | f | | g | } g \\psi ( f ) = m ( f ) h ( g ) + ( - 1 ) ^ { | f | | g | } m ( g ) h ( f ) . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} J _ { \\lambda , \\gamma } ^ { \\prime } \\left ( W _ { 1 } \\right ) \\left ( h \\right ) = e ^ { - 2 \\lambda \\left ( d + c \\right ) } \\displaystyle \\int \\limits _ { \\Omega } L i n \\left ( x , p , h \\right ) \\varphi _ { \\lambda } ^ { 2 } d x + 2 \\gamma \\left \\{ W , h \\right \\} . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\vert \\phi _ n ( y ) \\vert \\leq o ( 1 ) \\ , \\varepsilon _ n ^ \\nu , y \\in \\Omega _ { \\varepsilon _ n } \\setminus \\Bigl ( \\bigcup _ { i = 1 } ^ k B _ { \\frac { \\delta } { 4 \\varepsilon _ n } } ( \\zeta _ { i , n } ^ \\prime ) \\Bigr ) . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} Z _ n ( z ) = - \\frac { n ( n - 1 ) } { k ( k - 1 ) } X '' ( z ) - \\frac { n } { k - 1 } Y ' ( z ) + \\sum _ { l = 0 } ^ { k - 3 } C _ { k - l - 2 , n } \\frac { z ^ l } { l ! } , \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty u ^ h \\ , d \\sigma _ x ( u ) = \\frac { 1 } { x } \\sum _ { n \\leq x } \\bigg ( \\frac { P _ n ( x ) - D ( x ) } { \\sqrt C ( \\log \\log x ) ^ { 3 / 2 } } \\bigg ) ^ h = \\frac { M _ h ( x ) } { x C ^ { h / 2 } ( \\log \\log x ) ^ { 3 h / 2 } } . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & h _ { j - 1 } \\\\ - h _ { j - 1 } & 0 \\end{bmatrix} , h _ { j - 1 } = \\alpha _ { 2 j - 1 , 2 j } + \\beta _ { 2 j - 1 , 2 j } , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} S _ { a , b , k } = \\bigcup _ { j = 1 } ^ k R _ { b j , a ( k + 1 - j ) } = \\left \\{ ( i , j ) : \\left \\lfloor \\frac { i } { b } \\right \\rfloor + \\left \\lfloor \\frac { j } { a } \\right \\rfloor \\leq k - 1 \\right \\} . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\zeta _ 0 ( t ) & \\doteq 0 , \\zeta _ k ( t ) \\doteq p _ k ( f _ 1 ( t ) ) ^ k , k \\in \\mathbb { N } , \\\\ \\psi ( t ) & \\doteq - 2 \\int _ 0 ^ t r _ 0 ( \\boldsymbol { \\zeta } ( s ) ) \\ , d s + \\sum _ { k = 1 } ^ \\infty ( k - 2 ) ( p _ k - \\zeta _ k ( t ) ) . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} t ( H , W ) = \\int _ { [ 0 , 1 ] ^ { V ( H ) } } \\prod _ { i j \\in E ( H ) } W ( x _ i , x _ j ) \\prod _ { i \\in V ( H ) } \\mathrm { d } x _ i \\ , , \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} \\deg \\theta \\cdot \\beta ^ * _ k = \\theta ' _ k \\circ \\beta ^ * _ k = \\alpha ^ * _ k \\circ \\theta _ k \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} ( f _ s ) = I \\cap C [ \\lambda , \\mu ] = \\{ p \\in C [ \\lambda , \\mu ] \\mid p ( L _ s , A _ { 2 s + 1 } ) = 0 \\} . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} \\sigma = v \\wedge \\phi , \\omega = v \\wedge \\theta - c \\phi . \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} x _ { h , \\sigma _ { g } ^ { - 1 } ( \\sigma _ { g h } ( i ) ) } = x _ { h , \\sigma _ h ( i ) } , \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} g = d s ^ 2 + g _ { a ( s ) , b ( s ) } . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\varphi _ { \\epsilon } } { \\partial t } = \\log \\frac { \\omega _ { \\varphi _ { \\epsilon } } ^ n } { \\omega _ { \\epsilon } ^ n } + F _ { \\epsilon } + \\gamma ( k \\chi + \\varphi _ { \\epsilon } ) + \\theta _ X ( \\omega _ { \\varphi _ { \\epsilon } } ) \\\\ \\varphi _ { \\epsilon } | _ { t = 0 } = c _ { \\epsilon 0 } . \\end{cases} \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\tilde T _ h ( \\phi / \\phi _ 0 ) ( x ) = \\frac { 1 } { \\alpha _ 0 ( h ) \\phi _ 0 ( x ) } \\int _ X \\phi ( y ) K _ h ( x , d y ) = \\frac { \\alpha ( h ) \\phi ( x ) } { \\alpha _ 0 ( h ) \\phi _ 0 ( x ) } . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\psi ^ { G L M } ( y _ i , x _ i , \\beta _ j ) = \\sum _ { i = 1 } ^ n \\frac { y _ i - \\mu _ i } { \\tilde { V } ( \\mu _ i ) } \\frac { \\partial \\mu _ i } { \\partial \\beta _ j } \\ , . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\dim { \\bar Y } = \\dim Y - \\dim ( a , 3 ) & \\leq ( a ^ 2 - 1 ) + \\dim ( a - 1 , 3 ) - \\dim ( a , 3 ) \\cr & = ( a ^ 2 - 1 ) - \\dim ( a , 2 ) \\cr & = a ( a - 1 ) / 2 \\cr & = \\dim ( a - 1 , 1 , 1 ) . \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\nabla _ j R = \\nabla _ k R ^ { \\ ; k } _ j = R _ j ^ { \\ ; k } \\nabla _ k f \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} & ~ [ ( I + R _ k ) 1 ] ( x ) \\\\ = & ~ 1 + i \\int _ { - \\infty } ^ x e ^ { i k ( x - y ) } e ^ { i \\int _ y ^ x u ( t ) ~ d t } u ( y ) ~ d y \\\\ = & ~ 1 - \\frac { u ( x ) } { k } + \\frac 1 k \\int _ { - \\infty } ^ x e ^ { i k ( x - y ) } \\left ( e ^ { i \\int _ y ^ x \\int u ( t ) ~ d t } u ( y ) \\right ) _ y ~ d y \\\\ = & ~ 1 - \\frac { u ( x ) } { k } + O \\left ( \\frac 1 { | k | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} \\big ( ( \\cdot \\otimes g ) ^ { \\prime } ( \\lambda \\otimes \\omega ) \\big ) ( \\phi ) & = ( \\lambda \\otimes \\omega ) \\big ( ( \\cdot \\otimes g ) ( \\phi ) \\big ) = ( \\lambda \\otimes \\omega ) ( \\phi \\otimes g ) \\\\ & = \\lambda ( \\phi ) \\omega ( g ) = \\big ( \\omega ( g ) \\lambda \\big ) ( \\phi ) , \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\pi ^ { \\rtimes \\rho , \\upsilon } ( f ) \\ , : = \\ , \\int _ \\Pi \\pi ^ { \\rtimes \\rho } ( f ( l ) ) \\ , \\upsilon _ l \\ , d l \\ , \\in B ( H ) \\ , \\ \\ \\ f \\in C _ c ( \\Pi , A \\rtimes ^ \\alpha G ) \\ . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\left ( \\forall i \\in I \\right ) \\norm { x _ { i } } = \\beta , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\int _ M f \\ , \\omega _ g ^ m = a . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } x ' ( t ) & = v ( \\lambda _ 0 + t , x ( t ) ) \\\\ x ( \\lambda _ 0 ) & = x _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathbf { k } , \\varepsilon ) ^ { s / 2 } \\widehat { P } = \\varepsilon ^ s ( t ^ 2 + \\varepsilon ^ 2 ) ^ { - s / 2 } \\widehat { P } , s > 0 . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\Psi & = ( \\Psi ^ 1 , \\dots , \\Psi ^ { n - 1 } , \\Psi ^ n ) = ( \\hat \\Psi , \\Psi ^ n ) \\\\ X & = ( X _ 1 , \\dots , X _ { n - 1 } , X _ n ) = ( \\hat X , X _ n ) \\\\ y & = ( y _ 1 , \\dots , y _ { n - 1 } , y _ n ) = ( \\hat y , y _ n ) y \\in \\R ^ n \\ , . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { N } } \\log p _ { N , 0 } ^ { P _ 1 } = - \\frac { 1 } { \\sqrt { 2 \\pi } } \\zeta \\Big ( \\frac 3 2 \\Big ) + \\frac { C } { \\sqrt { N } } + \\cdots , \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} h _ n = \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { d ^ { i + 1 } } ( f ^ i ) ^ * ( \\lambda ) \\quad ( n \\geqslant 1 ) , \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\dot { w } = J K _ { 0 2 } ( \\widetilde \\omega t ) [ w ] + J K _ { 1 1 } ( \\widetilde \\omega t ) [ y ( 0 ) ] \\ , . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f \\right ] : = \\int _ { \\Pi } f ( p ) \\rho ( p ) \\ ; \\mbox { d } p \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} D _ { \\bar \\Lambda _ l } E & = O ( \\varepsilon ) w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 4 + O ( w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ( y ) ^ 3 \\varepsilon ^ 2 ) + O ( \\varepsilon ^ 5 ) . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\widetilde z ' ( \\omega ) = - F ' _ { \\mu _ \\alpha } ( \\omega ) + 1 + \\frac { 1 } { F ' _ { \\mu _ \\beta } \\circ F _ { \\mu _ \\beta } ^ { ( - 1 ) } \\circ F _ { \\mu _ \\alpha } ( \\omega ) } F ' _ { \\mu _ \\alpha } ( \\omega ) \\ , . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} \\Im m _ { \\mu _ \\alpha \\boxplus \\mu _ \\beta } ( z ) = \\int _ \\R \\frac { \\eta } { ( x - E ) ^ 2 + \\eta ^ 2 } \\ , { \\rm d } \\mu _ { \\alpha } \\boxplus \\mu _ \\beta ( x ) \\sim \\eta \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} & \\sum _ { i , j = 1 } ^ { n } E _ { i i } \\otimes E _ { j j } + ( n - 2 ) \\sum _ { i = 1 } ^ { n } E _ { i i } \\otimes E _ { i i } - \\sum _ { 1 \\leq i \\neq j \\leq n } E _ { i j } \\otimes E _ { i j } \\\\ = & \\sum _ { 1 \\leq i < j \\leq n } \\left ( E _ { i i } \\otimes E _ { i i } + E _ { j j } \\otimes E _ { j j } + E _ { i i } \\otimes E _ { j j } + E _ { j j } \\otimes E _ { i i } - E _ { i j } \\otimes E _ { i j } - E _ { j i } \\otimes E _ { j i } \\right ) . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\bigg \\| \\Psi ^ { t + s } ( x ) - \\Psi ^ { t } ( x ) - s \\ , \\dot \\Psi ^ t ( x ) - \\frac 1 2 s ^ 2 \\ , \\ddot \\Psi ^ t ( x ) \\bigg \\| _ { C ^ 0 ( \\R ^ n ; \\R ^ n ) } = o ( s ^ 2 ) \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} u _ { k } \\stackrel { 1 } { = } ( - 1 ) ^ { k - 1 } \\left ( 2 - \\frac { 3 } { 2 ^ k } \\right ) + \\sum _ { j = 2 } ^ { k } ( - 1 ) ^ { k - j } \\left ( 2 - \\frac { 1 } { 2 ^ { k - j } } \\right ) \\zeta ( j ) . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\alpha _ 1 \\cdots \\alpha _ m = \\prod _ { r = 1 } ^ m \\ ( ( 1 + O ( 1 / r ) ) \\frac { ( 1 - \\delta ) r } { n - r + 1 } \\ ) = e ^ { O ( \\log m ) } ( 1 - \\delta ) ^ m \\binom { n } { m } ^ { - 1 } , \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} D _ { a ^ + } ^ { \\alpha , \\beta } y ( x ) = f ( x , y ) , 0 < \\alpha < 1 , \\ , 0 \\leq \\beta \\leq 1 , \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x ^ { i } } { d s ^ { 2 } } + \\Gamma _ { j k } ^ { i } \\frac { d x ^ { j } } { d s } \\frac { d x ^ { k } } { d s } + \\frac { 1 } { \\left ( \\frac { d S \\left ( t \\right ) } { d t } \\right ) ^ { 2 } } V ^ { , i } = 0 . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} A : = \\sqrt { \\tau } U \\abs ( D ) U ^ { - 1 } . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\epsilon = ( k _ 1 - 1 ) ( k _ 2 - 1 ) + \\dim L + k _ 1 + ( i - 1 ) \\Big ( 1 + ( \\mu ( \\beta _ 2 ) + k _ 2 ) \\dim L \\Big ) . \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} K _ { \\nu } \\left ( { \\nu z } \\right ) = \\left ( { \\frac { \\pi } { 2 \\nu z } } \\right ) ^ { 1 / 2 } e ^ { - \\nu z } \\left \\{ { 1 + \\mathcal { O } \\left ( { \\frac { 1 } { z } } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\bigg ( q , \\sum ^ { k } _ { j = 0 } c _ j D _ { j + 1 , p + i } \\bigg ) = 0 , i = 1 , \\ldots , k ; c _ k = 1 , \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\P \\big ( \\cap _ { j = 1 } ^ k N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\big ) = \\prod _ { j = 1 } ^ k \\P \\big ( N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\big ) \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} \\partial _ { t } U + U \\cdot \\bigtriangledown U - \\nu \\bigtriangleup U = - \\bigtriangledown P , \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} { \\cal R } _ { [ 0 , t ] } ^ 0 : = \\left \\{ x \\in X : \\ { \\cal U } _ { [ 0 , t ] } ( 0 , x ) \\neq \\emptyset \\right \\} . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} L _ \\lambda : = \\dfrac { \\partial ^ 2 f } { \\partial x ^ 2 } ( \\l , 0 ) : H \\to H \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} \\Delta _ { ( t _ { 1 } , t _ { 2 } , \\ldots , t _ { n } ) } ( X ) = \\sum _ { i = 1 } ^ { n } t _ { i } F _ { i i } X F _ { i i } ^ { * } - \\left ( \\sum _ { i = 1 } ^ { n } F _ { i i } \\right ) X \\left ( \\sum _ { i = 1 } ^ { n } F _ { i i } \\right ) ^ { * } \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u = H u \\\\ u ( 0 ) = \\phi \\end{cases} \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\Delta = P Q ^ 2 R ^ 2 , \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} H _ v = \\left \\{ \\begin{array} { l l } \\overline { G } _ v \\simeq \\mathbb Z _ p \\rtimes \\mathbb Z _ p & \\hbox { i f $ v \\in S ^ { \\ast } $ } , \\\\ G _ v ^ { \\mathrm { c r } } \\simeq \\mathbb Z _ p \\times \\mathbb Z _ p & \\hbox { i f $ v \\in P $ } , \\end{array} \\right . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] & = 0 & [ X _ 1 , X _ 3 ] & = 0 & [ X _ 1 , X _ 4 ] & = 0 & [ X _ 1 , X _ 5 ] & = 0 & [ X _ 2 , X _ 3 ] & = 0 \\\\ [ X _ 2 , X _ 4 ] & = 0 & [ X _ 2 , X _ 5 ] & = X _ 1 & [ X _ 3 , X _ 4 ] & = X _ 1 & [ X _ 3 , X _ 5 ] & = X _ 2 & [ X _ 4 , X _ 5 ] & = X _ 3 . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{gather*} \\big ( \\Lambda ^ N _ m \\delta _ { A _ k \\lambda ( N ) } \\big ) ( \\mu ) = \\Lambda ^ N _ m ( A _ k \\lambda ( N ) , \\mu ) = \\Lambda ^ N _ m ( \\lambda ( N ) , A _ { - k } \\mu ) \\\\ \\hphantom { \\big ( \\Lambda ^ N _ m \\delta _ { A _ k \\lambda ( N ) } \\big ) ( \\mu ) } { } = \\big ( \\Lambda ^ N _ m \\delta _ { \\lambda ( N ) } \\big ) ( A _ { - k } \\mu ) \\xrightarrow { N \\rightarrow \\infty } M _ m ( A _ { - k } \\mu ) = A _ k M _ m ( \\mu ) , \\end{gather*}"}
-{"id": "3569.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X _ 1 ( s , M _ 1 , Y _ 0 ^ s ) d s + \\int _ 0 ^ t X _ 2 ( s , M _ 2 , Y _ 0 ^ s ) d s + B ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} \\frac { C } { T } \\left ( \\ln \\omega \\right ) _ { , t } + \\frac { 2 C _ { , t } } { T } + \\frac { 1 } { \\omega ( t ) } \\frac { T _ { , t t } } { T } & = 1 \\\\ T _ { , t } & = - \\omega ( t ) C \\\\ C _ { , t } & = T . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\begin{array} { c c c } S \\left ( a _ { i } , b _ { i } , \\bar { p } , \\omega _ { i } ; t \\right ) & = & \\left ( a _ { 1 } \\cos \\omega _ { 1 } t + b _ { 1 } \\sin \\omega _ { 1 } t + \\bar { p } \\right ) \\mathbf { 1 } _ { \\left [ 0 , T _ { 0 } \\right ) } \\left ( t \\right ) + \\\\ & & \\left ( a _ { 2 } \\cos \\omega _ { 2 } t + b _ { 2 } \\sin \\omega _ { 2 } t + \\bar { p } \\right ) \\mathbf { 1 } _ { \\left [ T _ { 0 } , T \\right ) } \\left ( t \\right ) , \\end{array} \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\sum _ { k , l = 1 } ^ n \\overline { g ( x _ k ) } \\ > g ( x _ l ) & \\ > \\tilde K _ { \\pi ( x _ k , x _ l ) } ( x _ k , \\{ x _ l \\} ) \\ > \\tilde \\omega _ X ( \\{ x _ k \\} ) \\cdot \\\\ & \\cdot \\tilde \\alpha ( \\pi ( x _ k , x _ l ) ) \\beta ( \\pi ( x _ k , x _ l ) ) \\tilde \\omega _ D ( \\{ \\pi ( x _ k , x _ l ) \\} ) \\ge 0 . \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} \\beta = \\sum _ { i = 0 } ^ { 2 n - 2 } { a _ i 3 ^ i } \\textup { w h e r e } a _ i = \\begin{cases} 2 & i \\textup { e v e n } \\\\ 1 & i \\textup { o d d } \\\\ \\end{cases} \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} H ^ 0 ( \\widehat P , n A ) = H ^ 0 ( P , \\nu _ * ( \\mathcal O _ { \\widehat P } ( n A ) ) ) \\subset H ^ 0 ( P , \\mathcal O _ P ( p n / p _ 0 ) ) \\subset V _ { p n } \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\{ f , g \\} = \\sum _ { m \\in \\Z } \\frac { a _ m b _ { - m } } { m ^ { n - 1 } } \\ . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} x ( t ) = u \\left ( \\frac { t } { \\omega } \\right ) , t \\in [ - \\omega , \\omega ] , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\int _ { { \\mathbb { R } } ^ { 2 n } } \\rho ( z ) d ^ { 2 n } z = 1 \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} G _ \\epsilon ( x , y ) : = \\exp \\left ( g ( \\hat { x } | x | ^ { 1 / \\epsilon } - y ) - g ( \\hat { x } | x | ^ { 1 / \\epsilon } ) \\right ) \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\mathbb E \\langle F \\star \\bar { \\mu } , G \\star \\bar { \\mu } \\rangle & = \\sum _ { k , l = 1 } ^ d \\langle x _ k , x _ l ^ * \\rangle \\mathbb E \\bigl ( F ^ k \\star \\bar { \\mu } \\cdot G ^ l \\star \\bar { \\mu } \\bigr ) = \\sum _ { k , l = 1 } ^ d \\langle x _ k , x _ l ^ * \\rangle \\mathbb E ( F ^ k G ^ l ) \\star \\nu \\\\ & = \\mathbb E \\Bigl ( \\sum _ { k , l = 1 } ^ d \\langle x _ k , x _ l ^ * \\rangle F ^ k G ^ l \\Bigr ) \\star \\nu = \\mathbb E \\langle F , G \\rangle \\star \\nu . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta { v } = f ( v ) ~ ~ & ~ \\mathbb { R } ^ N , \\\\ & v \\in H ^ 1 ( \\mathbb { R } ^ N ) , ~ ~ v \\not \\equiv 0 ~ ~ & ~ \\mathbb { R } ^ N , \\end{aligned} \\right . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} ( T _ 1 ^ 2 - T _ 0 ^ 2 ) - \\sqrt { - 1 } ( 2 T _ 0 T _ 1 ) = ( T _ 1 - \\sqrt { - 1 } T _ 0 ) ^ 2 . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\frac { d } { d t } a _ { 1 } & = - \\nu a _ { 1 } - \\int _ { \\mathbb { T } _ { \\alpha } } v _ { n } ^ { \\nu } \\omega _ { n } ^ { \\nu } \\sin y \\ d x d y , \\\\ \\frac { d } { d t } a _ { 2 } & = - \\nu a _ { 2 } + \\int _ { \\mathbb { T } _ { \\alpha } } v _ { n } ^ { \\nu } \\omega _ { n } ^ { \\nu } \\cos y \\ d x d y . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} X = & \\sum _ { k = 0 } ^ { g - 1 } ( P ^ { d } ) ^ { k + 2 } R T ^ { k } T ^ { \\pi } - P ^ { d } \\sum _ { j = 0 } ^ { n - 1 } R ^ { j + 1 } \\sum _ { h = 0 } ^ { l - 1 } ( Q ^ d ) ^ { h + j + 1 } S ^ { h } , \\\\ T ^ { d } = & P ^ \\pi \\sum _ { j = 0 } ^ { n - 1 } \\sum _ { k = 0 } ^ { j } P ^ k R ^ { j - k } \\sum _ { h = 0 } ^ { l - 1 } ( Q ^ d ) ^ { h + j + 1 } S ^ { h } , \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} \\mu \\left ( t , \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\right ) ^ { \\frac 1 p } \\right ) \\leq \\left ( \\sum _ { i = 1 } ^ n \\mu \\left ( \\frac t n , | x _ i | ^ p \\right ) \\right ) ^ { \\frac 1 p } = \\left ( \\sum _ { i = 1 } ^ n \\mu ^ p \\left ( \\frac t n , | x _ i | \\right ) \\right ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\frac { w ( \\Psi _ + , \\Psi _ - ) } { \\Psi _ + \\Psi _ - } = \\frac { \\partial ( \\Psi _ + ) } { \\Psi _ + } - \\frac { \\partial ( \\Psi _ - ) } { \\Psi _ - } = \\phi _ + - \\phi _ - = \\frac { 2 \\mu } { \\varphi } \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} v _ { i , 0 } ^ { \\varepsilon } \\left ( t , x \\right ) : = v _ { i } ^ { 0 } \\left ( t , x \\right ) . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} C [ \\Gamma _ s ] = \\frac { C [ \\lambda , \\mu ] } { ( f _ s ) } \\mbox { a n d } K [ \\Gamma _ s ] = \\frac { K [ \\lambda , \\mu ] } { ( f _ s ) } \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} \\widehat { \\Phi } _ { k , v } ( u ) : = \\prod _ { i \\in I } \\prod _ { j = 1 } ^ { v ^ { i } } \\frac { \\vartheta ( u + z _ { j } ^ { ( i ) } + ( c _ { k i } ) \\frac { \\hbar } { 2 } ) } { \\vartheta ( u + z _ { j } ^ { ( i ) } - ( c _ { k i } ) \\frac { \\hbar } { 2 } ) } . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} W ( B ( x _ 0 , 2 r ) ) \\lesssim W ( B ( x _ 0 , r ) ) , \\left ( W ( B ( x _ 0 , r ) ) : = \\int _ { B ( x _ 0 , r ) } W \\ , d x \\right ) , \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} L _ { Y } V ^ { , i } + d _ { 0 } V ^ { , i } + m Y ^ { i } = 0 . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\| A ( t ) ^ { 1 / 2 } F ( t ) - ( t ^ 2 S ) ^ { 1 / 2 } P \\| \\le C _ 5 t ^ 2 , | t | \\le t ^ 0 ; C _ 5 = \\beta _ 5 \\delta ^ { - 1 / 2 } \\bigl ( \\| X _ 1 \\| ^ 2 + c _ * ^ { - 1 / 2 } \\| X _ 1 \\| ^ 3 \\bigr ) . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} f \\# g ( x ) = \\lim _ { n \\to \\infty } \\int _ G \\psi _ n ( y ) f ( y ) \\ell _ y ^ \\sigma g ( x ) \\ , d y \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t v _ i - d _ i \\Delta v _ i & = L _ i v + f _ i ( v + u _ { \\infty } ) - \\nabla f _ i ( u _ { \\infty } ) \\cdot v = : L _ i v + g _ i ( v ) , & & x \\in \\Omega , \\\\ \\nabla v _ i \\cdot \\nu & = 0 , & & x \\in \\partial \\Omega , \\\\ v _ i ( x , 0 ) & = u _ { i , 0 } ( x ) - u _ { i , \\infty } , & & x \\in \\Omega \\end{aligned} \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\frac { 1 } { m ^ k k ! } ( 1 + \\lambda t ) ^ { \\frac { r } { \\lambda } } \\big ( ( 1 + \\lambda t ) ^ { \\frac { m } { \\lambda } } - 1 \\big ) ^ k = \\sum _ { n = k } ^ \\infty W _ { m , r } ( n , k | \\lambda ) \\frac { t ^ n } { n ! } , \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} J ( x ) = \\frac { 1 - 3 x + 3 x ^ 2 } { ( 1 - x ) ^ 2 ( 1 - 2 x ) } \\ , . \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\tau ( z ) = \\tilde { \\tau } ( z ) + 2 \\pi ( z ) \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} \\left ( - \\partial ^ 2 + u _ s ( x ) - \\chi _ 1 ( \\tau ) \\right ) \\Upsilon = 0 \\ , \\ \\left ( A _ { 2 s + 1 } - \\chi _ 2 ( \\tau ) \\right ) \\Upsilon = 0 \\ \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\| \\dot \\Psi ^ t \\| _ { C ^ { k - 1 , \\alpha } ( \\R ^ n ) } \\le C \\mbox { a n d } ( \\dot \\Psi ^ t \\circ ( \\Psi ^ t ) ^ { - 1 } ) \\cdot \\nu ^ t = \\frac { \\partial _ { \\nu ^ t } V ^ t } { \\Delta h ^ t } \\mbox { o n } \\Gamma ^ t , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} w = w _ { \\rm s o l i d } + w _ { \\rm s i n g l e } + w _ { \\rm d o u b l e } + w _ { \\rm i m p l i c i t } + \\mbox { c o n s t a n t } \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} C _ 1 & : = 2 M d ^ 2 + 2 d \\sqrt { d } M ^ d , \\\\ C _ 2 & : = 2 d \\sqrt { d } \\frac { K _ a } { \\theta } + 2 d ^ 2 K _ \\zeta , \\\\ C _ 3 & : = \\frac { 2 d ^ 2 } { \\theta } , \\\\ C _ 4 & : = K _ 2 + 2 d M _ V K _ \\zeta + 2 \\sqrt { d } M _ V \\frac { K _ a } { \\theta } + K _ 2 \\frac { K _ a } { \\theta } , \\\\ C _ 5 & : = 2 M _ V \\frac { \\sqrt { d } } { \\theta } + \\frac { K _ 2 } { \\theta } + \\sqrt { d } ( M _ \\zeta + M ) . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\frac { b } { b _ 1 } = \\frac { c } { c _ 1 } = \\frac { ( T ) } { ( T _ 1 ) } . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} | 1 | _ g = \\frac { 1 } { \\max \\{ a _ 0 , a _ 1 | z _ 1 | , \\ldots , a _ d | z _ d | \\} } , \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} T _ { - 1 } F ( x + i y ) = F ( - x + i y ) , \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\mathbb { A } _ { \\boldsymbol { r } } & = \\left \\{ \\boldsymbol { \\alpha } : \\frac { 2 ( r _ l + 1 ) \\alpha ^ { * } d } { 2 \\alpha ^ { * } + d } < \\alpha _ l < \\eta + 1 , l = 1 , \\dotsc , d \\right \\} , \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\frac { A \\otimes W - \\left ( \\overline { A \\otimes W } \\right ) ^ { t } } { 2 \\sqrt { - 1 } } = \\left ( V \\otimes Z \\right ) \\left ( \\overline { V \\otimes Z } \\right ) ^ { t } . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\Gamma _ i ' = \\lambda ( \\delta _ E \\otimes \\gamma _ i ) ( i = 1 , 2 , 3 ) , \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N G _ { i i } \\tau _ { i 1 } = 0 \\ , , \\qquad \\frac { 1 } { N } \\sum _ { i = 1 } ^ N T _ i \\tau _ { i 1 } \\Upsilon = O _ \\prec ( \\Psi \\hat { \\Upsilon } ) \\ , , \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\limsup _ \\gamma ( z \\mathbf { d } y - z \\mathbf { d } x _ \\gamma ) _ + = ( z \\mathbf { d } y - \\liminf _ \\gamma z \\mathbf { d } x _ \\gamma ) _ + . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} L ^ \\alpha [ u _ { k + 1 } ] + N _ 0 u _ { k + 1 } = f ( x , u _ k ) + N _ 0 u _ k . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\varepsilon } \\nabla \\theta ^ 0 = \\varepsilon \\nabla \\cdot ( \\mathbf { V } ^ { \\varepsilon } \\nabla \\theta ^ 0 ) - \\varepsilon \\mathbf { V } ^ { \\varepsilon } \\Delta \\theta ^ 0 - \\varepsilon ( \\nabla _ { x } \\mathbf { V } ) ^ { \\varepsilon } \\nabla \\theta ^ 0 . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} - \\lim _ { \\varepsilon \\to 0 } \\int _ { \\Omega } \\int _ { B _ 1 ( 0 ) } D ^ 2 _ x u ( x , y , \\epsilon ) & w _ \\delta ( x , y ) \\phi ( h ) d x d y d h \\\\ & = - \\int _ { \\Omega } w _ \\delta ( z ) \\Delta _ x u ( z ) \\ , d z \\ , . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} \\int _ { \\Gamma ^ 0 } ( N \\cdot \\nu ^ 0 ) ( x ) f ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) = \\int _ { \\mathcal Z } f ( z , \\eta ^ 0 ( z ) ) J ( z , \\eta ^ 0 ( z ) ) d z . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} I _ { \\nu } ( a ) = \\int _ 0 ^ { \\infty } \\frac { J _ { \\nu } ( x ) } { x ^ 2 + a ^ 2 } \\ , { \\rm d } x = \\frac { \\pi } { a \\sin ( \\pi \\nu ) } \\bigl ( { \\bar J } _ { \\nu } ( a ) - J _ { \\nu } ( a ) \\bigr ) , \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} \\mathcal K _ { j , k } ^ { p , q } = \\frac { \\Gamma ( j - \\frac 1 2 ) } { \\Gamma ( p ) \\Gamma ( p + q + j + k - \\frac 3 2 ) } \\sum _ { \\ell = 1 } ^ { q } \\frac { \\Gamma ( j + k + \\ell - \\frac 3 2 ) \\Gamma ( p + q - \\ell ) } { \\Gamma ( j + \\ell - \\frac 1 2 ) \\Gamma ( q - \\ell + 1 ) } . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} P = & \\sum _ a ( - 1 ) ^ a \\left [ \\binom { | S | - | U ' | } { 2 } \\binom { | S | - | U ' | - 2 } { a - | U ' | - 2 } \\right . \\\\ & \\left . + ( a - | U ' | ) | U ' | \\binom { | S | - | U ' | } { a - | U ' | } + \\binom { | U ' | } { 2 } \\binom { | S | - | U ' | } { a - | U ' | } \\right ] . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\int _ U \\varphi ( u _ 1 , \\ldots , u _ d ) \\nu ( d u ) = \\sum _ { y = 1 } ^ d \\int _ { U _ y } \\varphi ( 0 , \\ldots , u _ y , \\ldots , 0 ) d u _ y \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} X _ n = \\overline { \\Psi } ( X _ { n - 1 } , Z _ n , Z _ { n - 1 } ) \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} W ( \\varphi ^ \\hbar _ M , \\varphi ^ \\hbar _ { M ' } ) ( z ) = \\left ( \\tfrac { 1 } { \\pi \\hbar } \\right ) ^ { n } C _ { M , M ' } e ^ { - \\frac { 1 } { \\hbar } F z ^ { 2 } } , \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\bar { \\mathbf { A } } = \\bar { \\mathbf { A } } _ { B ^ { + } ( \\bar x , 2 r ) } , \\bar { \\vec g } = \\bar { \\vec g } _ { B ^ { + } ( \\bar x , 2 r ) } . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} x \\overline { \\mathbf { d } } z & = \\inf \\{ \\epsilon > 0 : \\forall r \\in ( 0 , \\infty ) \\ z ^ \\bullet _ r \\subseteq x ^ \\bullet _ { r + \\epsilon } \\} . \\\\ z \\underline { \\mathbf { d } } y & = \\inf \\{ \\epsilon > 0 : \\forall r \\in ( 0 , \\infty ) \\ z ^ r _ \\bullet \\subseteq y ^ { r + \\epsilon } _ \\bullet \\} . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} y = \\vec \\Phi ( x ) = ( \\Phi ^ 1 ( x ) , \\ldots , \\Phi ^ n ( x ) ) \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} \\left ( \\frac { d S ^ { - 1 } \\left ( t \\right ) } { d t } \\right ) = \\frac { \\gamma } { t } \\rightarrow S ^ { - 1 } \\left ( t \\right ) = \\gamma \\ln t = s . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} d _ 0 \\circ \\frak m ^ { 2 } _ { 1 ; \\alpha _ + , \\alpha _ - } + \\frak m ^ { 2 } _ { 1 ; \\alpha _ + , \\alpha _ - } \\circ d _ 0 + ( \\hat d ^ 2 \\circ \\hat d ^ 2 ) _ { \\alpha _ + , \\alpha _ - } = 0 , \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\left ( G ^ { \\star \\star } , F ^ { \\star \\star } \\right ) = T _ { 2 } \\circ \\left ( G , F \\right ) . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\dot { X } ( t ) = \\left [ \\begin{matrix} A - B K & B K \\\\ 0 _ n & A \\end{matrix} \\right ] X ( t ) + \\left [ \\begin{matrix} 0 _ n & 0 _ n \\\\ 0 _ n & - \\frac { 1 } { h } L C \\end{matrix} \\right ] \\int _ { t - h } ^ t X ( s ) d s . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\aleph _ 4 ( 0 , \\alpha ) : = \\alpha \\ , , \\aleph _ 4 ( { m + 1 } , \\alpha ) : = \\aleph _ 4 ( m , \\alpha + 1 ) + \\frac { m } { 2 } + 2 \\alpha + 4 \\ , , \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} \\varphi _ { h } ( x ) = h ^ { - n / 2 } \\psi \\Big ( \\frac { x - y } { h } \\Big ) . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} P \\left [ X ( t + h ) = y | X ( t ) = x \\right ] = \\lambda ( t , x , y ) \\cdot h + o ( h ) . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} 0 < \\alpha _ { \\scriptscriptstyle 0 } : = \\left \\vert \\left \\vert \\left \\vert A ^ { - 1 } _ { \\scriptscriptstyle 0 } \\right \\vert \\right \\vert \\right \\vert _ { \\infty , \\Omega } ^ { - 1 } \\leq \\left \\vert \\left \\vert \\left \\vert A _ { \\scriptscriptstyle 0 } \\right \\vert \\right \\vert \\right \\vert _ { \\infty , \\Omega } = : \\beta _ { \\scriptscriptstyle 0 } < \\infty . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} W & = \\log \\log \\log x \\\\ X & = ( \\log \\log x ) ^ { 1 / 2 } ( \\log \\log \\log x ) ^ 2 \\\\ Y & = ( \\log \\log x ) ^ 2 . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\gamma = \\gamma _ + \\cup \\gamma _ - \\cup \\{ 0 \\} , \\gamma _ \\pm = \\{ S ^ { - t } q _ \\pm : t \\in \\mathbb { R } _ + \\} , \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} U = \\mathcal { P } U + \\mathcal { Q } U \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\Phi _ { n + 1 } ^ * ( z ) = \\Phi _ k ^ * ( z ) \\prod _ { j = k } ^ n \\left ( 1 - \\frac { \\alpha _ j z \\Phi _ j } { \\Phi _ j ^ * } \\right ) . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} { \\tilde t } = t + \\epsilon \\sum _ { j = 1 } ^ d x _ j ^ 2 , \\ ; \\ ; { \\tilde x } _ j = x _ j , \\ ; \\ ; j = 1 , 2 , \\ldots , d \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z \\mathtt { g } _ { a , \\nu } ' ( z ) } { \\mathtt { g } _ { a , \\nu } ( z ) } \\right ) \\geq \\frac { r \\mathtt { g } _ { a , \\nu } ' ( r ) } { \\mathtt { g } _ { a , \\nu } ( r ) } > \\frac { \\mathtt { g } _ { a , \\nu } ' ( 1 ) } { \\mathtt { g } _ { a , \\nu } ( 1 ) } = a ( 1 - \\nu ) + a ^ { a / 2 } \\left ( a \\nu - a + 1 - \\sum _ { n = 1 } ^ \\infty \\frac { 2 } { \\mathtt { j } _ { \\nu , n } ^ 2 - 1 } \\right ) . \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} \\pi _ 1 + \\cdots + \\pi _ d = f \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\tilde \\rho ( t ) ^ * \\eta = e ^ t \\eta \\mathrm { a n d } \\tilde \\rho ( t ) ^ * \\omega = e ^ { - t } \\omega \\ , . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{gather*} F _ q ( z ; \\theta ) = x ^ z \\prod \\limits _ { i = 1 } ^ N { \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i + 1 ) - z ) } } . \\end{gather*}"}
-{"id": "3677.png", "formula": "\\begin{align*} D ( e _ 1 ) = - e _ 1 , D ( e _ 2 ) = e _ 2 \\textrm { a n d } D ( e _ 3 ) = 0 . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } V , X ) & = g _ { 1 } ( V , \\nabla ^ { ^ { M _ 1 } } _ { U } \\phi \\mathcal { B } X ) + g _ { 1 } ( V , \\nabla ^ { ^ { M _ 1 } } _ { U } \\omega \\mathcal { B } X ) + g _ { 1 } ( \\mathcal { C } X , \\mathcal { H } \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi V ) + g _ { 1 } ( V , \\varphi U ) \\eta ( X ) . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } x \\left ( | P _ n \\varphi | ^ 2 \\right ) _ x + x P _ n ( u \\varphi ) \\overline { P _ n \\varphi } = 0 . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{gather*} P _ { \\lambda } \\big ( 1 , t , t ^ 2 , \\dots , t ^ N \\big ) = \\sum _ { \\mu \\colon \\mu \\prec \\lambda } { \\psi _ { \\lambda / \\mu } ( q , t ) P _ { \\mu } \\big ( t , t ^ 2 , \\dots , t ^ N \\big ) } \\end{gather*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\Theta : = \\frac 1 2 ( N \\cdot \\nu ^ 0 ) ^ 2 \\Delta h ^ 0 \\ , ( \\ddot \\eta ^ 0 \\circ \\pi _ 1 ) . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\xi ( \\theta ) = \\xi \\left ( \\hat { \\theta } ^ M \\right ) + \\frac { 1 } { \\sqrt { n } } \\sum _ { s } \\hat { \\xi } _ s \\delta _ s + O _ p ( n ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} O _ { f _ t } = ( f _ t ) ^ * O _ { M _ t } ^ * \\otimes O _ { X } . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\sum ^ { \\infty } _ { j = N + 1 } \\abs { ( \\ , g _ j \\ , | \\ , \\chi _ { \\varepsilon _ 0 } \\ , ) _ M } ^ 2 < \\delta . \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} u \\left ( t \\right ) = - K y \\left ( t \\right ) , \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial f ^ a } { \\partial q ^ i } , \\frac { \\partial f ^ a } { \\partial \\lambda ^ b } \\right ) = k . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} f _ { 4 , \\lambda } ( x ) : = \\left \\{ \\begin{array} { l l } \\displaystyle { \\frac { \\lambda } { 2 } x ^ 2 } , & x \\in ( - \\infty , 0 ] , \\\\ \\\\ \\displaystyle { \\frac { 1 } { 2 \\lambda } x ^ 2 } , & x \\in [ 0 , \\infty ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} N _ { B ^ u _ \\sigma ( \\Lambda ) } = N ^ { s } _ { B ^ u _ \\sigma ( \\Lambda ) } \\oplus N ^ { u } _ { B ^ u _ \\sigma ( \\Lambda ) } . \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\int \\limits _ { - 1 } ^ 1 f ( t ) F ( t ) d t = \\int \\limits _ { - 1 } ^ 1 F ' ( t ) F ( t ) d t = \\frac { 1 } { 2 } \\int \\limits _ { - 1 } ^ 1 \\frac { d } { d t } F ^ 2 ( t ) d t = \\frac { 1 } { 2 } [ F ^ 2 ( 1 ) - F ^ 2 ( - 1 ) ] = 0 . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} \\nu ^ A _ y = \\int \\limits _ { v \\in S ^ { 2 n - 1 } } \\nu ^ A _ { y , v } d \\sigma _ { 2 n - 1 } ( v ) . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { i \\left ( k , B ^ { \\beta } ( t ) \\right ) } \\right ) = E _ { \\beta } \\left ( - \\frac { | k | ^ { 2 } } { 2 } t ^ { \\beta } \\right ) , k \\in \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} 0 \\le \\langle A g , g \\rangle _ { X } & = \\int _ D \\int _ X \\int _ X g ( x ) g ( y ) \\ > K _ h ( x , d y ) \\ > d \\omega _ X ( x ) \\ > \\cdot \\ > f ( h ) \\ > d \\omega _ D ( h ) \\\\ & = \\int _ { W _ e } \\int _ { U _ z } \\int _ { U _ z } g ( x ) g ( y ) \\ > K _ h ( x , d y ) \\ > d \\omega _ X ( x ) \\ > \\cdot \\ > f ( h ) \\ > d \\omega _ D ( h ) \\\\ & = : \\int _ { W _ e } \\phi ( x ) \\ > f ( x ) \\ > d \\omega _ D ( x ) \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} K _ { \\nu } \\left ( { \\nu z } \\right ) = \\frac { c _ { 2 } \\left ( \\nu \\right ) } { \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 4 } } \\exp \\left \\{ { \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\left ( { - 1 } \\right ) ^ { s } \\frac { \\tilde { { E } } _ { s } \\left ( p \\right ) } { \\nu ^ { s } } } } \\right \\} \\left \\{ { e ^ { - \\nu \\xi } + \\varepsilon _ { n , 2 } \\left ( { \\nu , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} \\inf \\{ \\| T y - R x \\| \\ : : \\ : x \\in X \\} = 0 . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\tilde { { F } } _ { 1 } \\left ( p \\right ) = { \\tfrac { 1 } { 8 } } p ^ { 2 } \\left ( { 1 - p ^ { 2 } } \\right ) \\left ( { 5 p ^ { 2 } - 1 } \\right ) , \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} I _ { a ^ { + } } ^ { 1 - \\gamma } y ( x ) { | } _ { x = a } = y _ a + { \\epsilon } , \\gamma = \\alpha + \\beta ( 1 - \\alpha ) , \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\sup _ { \\boldsymbol { x } \\in [ 0 , 1 ] ^ d } \\Sigma _ { \\boldsymbol { J } _ n } ( \\boldsymbol { x } , \\boldsymbol { x } ) \\lesssim \\| ( \\boldsymbol { \\Psi } ^ T \\boldsymbol { \\Psi } ) ^ { - 1 } \\| _ { ( 2 , 2 ) } \\sup _ { \\boldsymbol { x } \\in [ 0 , 1 ] ^ d } \\| \\boldsymbol { \\psi } _ { \\boldsymbol { J } _ n } ( \\boldsymbol { x } ) \\| ^ 2 \\leq Q _ 2 n ^ { - 1 } 2 ^ { \\sum _ { l = 1 } ^ d J _ { n , l } } \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} { n \\choose k } _ \\lambda = \\frac { ( n ) _ \\lambda ! } { k ! ( n - k \\lambda ) _ { n - k , \\lambda } } = \\frac { ( n ) _ { k , \\lambda } } { k ! } , \\ , \\ , ( n \\geq k \\geq 0 ) . \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\Phi _ 1 ( \\omega _ 1 ( z ) , \\omega _ 2 ( z ) , z ) = 0 \\ , , \\Phi _ 1 ( \\omega _ 1 ( z ) , \\omega _ 2 ( z ) , z ) = 0 \\ , , \\qquad \\forall z \\in \\mathbb { C } ^ + . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} { \\mathbb D } [ \\widehat \\phi , \\widehat y , \\widehat w , \\widehat \\alpha ] = \\begin{pmatrix} g _ 1 \\\\ g _ 2 \\\\ g _ 3 \\end{pmatrix} \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} S ( [ 0 , \\alpha ) , m ) & = \\{ f \\in L ^ 0 [ 0 , \\alpha ) : \\ , d ( f , s ) < \\infty , s \\geq 0 \\} \\\\ & = \\{ f \\in L ^ 0 [ 0 , \\alpha ) : \\ , \\exists A , m ( A ^ c ) < \\infty , f \\chi _ A \\in L _ { \\infty } [ 0 , \\alpha ) \\} . \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} U ( x ) = \\frac { m } { \\alpha \\vert x \\vert ^ { \\alpha } } + W ( x ) , \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\| a ( x , h D ) q ( x , h D ) u \\| _ { L ^ 2 } ^ 2 & = \\int | a | ^ 2 | q | ^ 2 d \\mu + o ( 1 ) , \\\\ \\| a ( x , h D ) P q ( x , h D ) u \\| _ { L ^ 2 } ^ 2 & = h ^ 2 \\int | a | ^ 2 | H _ p q | ^ 2 d \\mu + o ( h ^ 2 ) . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} V ( \\lambda ) = \\sum _ { n = 1 } ^ \\infty \\lambda _ n \\langle \\cdot , e _ n \\rangle f _ n , \\quad \\lambda = \\{ \\lambda _ n \\} \\in E . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} \\sigma _ j = 2 Q \\frac { 1 } { ( L + | j | ) ^ { \\alpha } } \\leq \\begin{cases} \\frac { C } { L ^ { \\alpha } } & | j | \\leq M L \\ , , \\\\ \\frac { C } { | j | \\alpha } & | j | > M L \\ , . \\end{cases} \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} b _ { k u } ^ { i j } \\left ( Z \\right ) = 0 , \\quad \\mbox { f o r a l l $ i , j , k , u = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\beta _ { \\ell } ^ { i } & = \\frac { 1 } { 2 } ( \\beta _ { \\ell - 1 } ^ { i - 1 } + \\beta _ { \\ell - 1 } ^ { i + 1 } ) , \\\\ \\gamma _ { \\ell } ^ { i } & = \\frac { 1 } { 2 } ( \\lambda _ { \\epsilon } \\gamma _ { \\ell - 1 } ^ { i - 1 } + \\gamma _ { \\ell - 1 } ^ { i + 1 } ) , \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\kappa ( s _ i \\ , s _ { i + 1 } \\ , s _ i , g ) = \\kappa ( s _ i , s _ { i + 1 } \\ , s _ i ( g ) ) \\ , \\kappa ( s _ { i + 1 } , s _ i ( g ) ) \\ , \\kappa ( s _ i , g ) . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty b _ n \\lambda _ n a _ n = \\sum _ { n = 1 } ^ \\infty c _ n a _ n \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { X } Y , V ) & = - g _ { 1 } ( \\mathcal { B } Y , \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) - g _ { 1 } ( \\mathcal { C } Y , \\nabla ^ { ^ { M _ 1 } } _ { X } \\varphi V ) . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} c ( x ) = u ( x ) ( x + 1 ) g ' ( x ) = x \\cdot u ( x ) g ' ( x ) + u ( x ) g ' ( x ) , \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} C _ V ( \\partial _ 0 ) = \\bigcup _ { v \\in \\partial _ 0 } W _ { k ( v ) } \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} \\nu _ \\mu ^ c ( T _ \\lambda ) = a _ \\nu \\chi _ \\mu ( \\nu ) + \\sum _ { \\xi \\in \\Lambda ^ + } { a _ \\xi \\chi _ \\mu ( \\xi ) } . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\mu ) + \\gamma _ 2 ( \\mu ) + \\ldots + \\gamma _ i ( \\mu ) = \\inf _ { n \\in \\mathbb N } \\frac 1 n \\int _ M F _ n ( q ) \\ , d \\mu ( q ) . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\| \\phi \\| _ * \\leq \\gamma = 2 C \\| E \\| _ { * * } \\leq C ( \\varepsilon ^ { \\frac { 1 } { 2 } } | M _ \\lambda ( \\zeta ) \\mu ^ { \\frac { 1 } { 2 } } | + \\varepsilon ^ 2 ) , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\int _ { \\Omega ^ { \\varepsilon } } \\left ( p ^ { \\varepsilon } - \\bar { p } \\right ) \\phi d x & = \\sum _ { k \\in \\mathbb { Z } ^ { d } } \\int _ { \\varepsilon Y _ { 1 } ^ { k } } \\left ( p ^ { \\varepsilon } - \\bar { p } \\right ) \\phi d x \\\\ & \\le C \\int _ { \\varepsilon Y _ { 1 } } \\left ( p ^ { \\varepsilon } - \\bar { p } \\right ) \\phi d x . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\rho _ s ( x ) f ( z ) = ( - ( z , b ) + \\overline { d } ) ^ { - s } f ( x ^ { - 1 } \\cdot z ) , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\ker \\theta _ k \\subseteq \\ker ( H ^ { k } ( X ) \\stackrel { \\beta ^ * _ k } { \\to } H ^ { k } ( U ) ) = \\Im ( H ^ { k } ( X , U ) \\stackrel { } { \\to } H ^ { k } ( X ) ) . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\ker ( \\sigma ) = \\{ x \\in A | \\ q ( x + y ) = q ( x ) q ( y ) \\forall y \\in A \\} = 0 \\ . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} P _ 1 = \\frac { 1 } { 6 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\Pi ^ T \\Omega ^ { - 1 } \\psi _ i - \\frac { 1 } { 2 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\bar { \\psi ^ T } \\Omega ^ { - 1 } \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\ , . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} ( \\psi _ x \\circ \\Phi ^ { - 1 } ) ( W _ u ) & = \\psi _ x \\big ( \\overline { b ( p ) } W _ { p } \\big ) \\\\ & = \\overline { b ( p ) } \\sigma _ c \\big ( ( y , m + p , x ) , \\eta ^ { - 1 } \\big ) \\sigma _ c ( \\eta , ( x , p , x ) ) \\overline { \\sigma _ c ( \\eta ^ { - 1 } , \\eta ) } \\psi _ y ( W _ { p } ) \\\\ & = \\sigma _ c \\big ( ( y , m + p , x ) , ( x , p , x ) \\big ) \\sigma _ c \\big ( \\eta , ( x , p , x ) \\big ) \\overline { \\sigma _ c ( \\eta ^ { - 1 } , \\eta ) } ( \\psi _ y \\circ \\Phi ^ { - 1 } ) ( W _ { u } ) . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} \\Phi _ h \\bigg ( \\bigg ( \\sum _ { i = 0 } ^ \\ell r _ i x _ i \\bigg ) ^ h \\bigg ) = \\frac 1 { h ! } \\sum _ { \\sigma \\in \\Sigma _ h } \\sum _ { m _ 1 = 0 } ^ \\ell \\cdots \\sum _ { m _ h = 0 } ^ \\ell r _ { m _ { \\sigma 1 } } \\cdots r _ { m _ { \\sigma h } } z _ { m _ { \\sigma 1 } m _ { \\sigma 2 } } \\cdots z _ { m _ { \\sigma ( h - 1 ) } m _ { \\sigma h } } . \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} n _ j & \\geq p _ j r _ j + q _ j - ( \\delta _ j - 1 ) + 1 + ( p _ j + 1 ) ( \\delta _ j - 1 ) \\\\ & = p _ j ( r _ j + \\delta _ j - 1 ) + q _ j + 1 \\\\ & > n _ j \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { \\tau \\in \\mathcal { T } ^ \\delta } { { E } } \\left [ \\sum _ { k = 0 } ^ \\infty ( k + 2 ) \\left | { \\bar { B } } ^ n _ k ( \\tau + \\delta ) - { \\bar { B } } ^ n _ k ( \\tau ) \\right | + \\left | { \\bar { \\eta } } ^ n ( \\tau + \\delta ) - { \\bar { \\eta } } ^ n ( \\tau ) \\right | \\right ] = 0 . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\begin{cases} u \\in K \\ , , \\\\ \\noalign { \\medskip } \\langle F ( u ) , v - u \\rangle \\geq \\langle \\varphi , v - u \\rangle & \\qquad \\forall v \\in K \\ , . \\end{cases} \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{gather*} \\prod _ { 1 \\leq i < j \\leq m } { T _ { q , x _ i } ^ { \\tau _ { i , j } } T _ { q , x _ j } ^ { 1 - \\tau _ { i , j } } } = \\prod _ { k = 1 } ^ m { T _ { q , x _ k } ^ { k - 1 + \\tau _ k ^ + - \\tau _ k ^ - } } . \\end{gather*}"}
-{"id": "1528.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i } { a \\alpha _ i + c _ i \\alpha _ { \\sigma ( i ) } } = \\sum _ { i = 1 } ^ { n } \\frac { 1 } { a + c _ { i } \\frac { \\alpha _ { \\sigma ( i ) } } { \\alpha _ i } } = \\frac { n } { a + d } \\leq 1 . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} { \\mathcal { G } } ( \\phi , \\Lambda ) = \\{ \\widehat { T } ( z _ { \\lambda } ) \\phi : z _ { \\lambda } \\in \\Lambda \\} , \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} Q _ B ( x ) : = \\sum _ { j = k + 1 } ^ d a _ j x _ j ^ 2 - \\Big ( \\sum _ { j = 1 } ^ k a _ j \\Big ) x _ d ^ 2 \\geq 0 \\ , \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\partial ^ 2 R _ 1 } { \\partial z ^ 2 } - p ^ 2 R _ 1 = 2 \\langle p , \\frac { \\partial R _ 0 } { \\partial x } \\rangle , \\ - D ( x ) \\leq z \\leq 0 \\cr & R _ 1 | _ { z = 0 } = 0 , \\frac { \\partial R _ 1 } { \\partial z } | _ { z = - D ( x ) } = \\langle \\nabla D , p \\rangle B _ 0 | _ { z = - D ( x ) } \\cr \\end{aligned} \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} A _ { j } = \\frac { i } { 2 } V _ { j } , \\mbox { a n d } q _ { j } = \\frac { 1 } { 4 } { V _ { j } ^ { 2 } } - \\frac { 1 } { 2 } \\ , \\mbox { d i v } \\ , V _ { j } , j = 1 , \\ , 2 . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\mathbf W = v ^ t r + r ^ t v - 2 \\langle \\mathbf v _ i , \\mathbf v _ j \\rangle - r ^ t r = v ^ t r + r ^ t ( v - r ) - 2 \\langle \\mathbf v _ i , \\mathbf v _ j \\rangle . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} \\mathbb { E } [ T _ i \\mathfrak { n } _ i ^ { ( p - 1 , p ) } ] = \\mathbb { E } [ \\mathring { T } _ i \\mathfrak { n } _ i ^ { ( p - 1 , p ) } ] + \\mathbb { E } [ O _ \\prec ( N ^ { - \\frac 1 2 } ) \\mathfrak { n } _ i ^ { ( p - 1 , p ) } ] \\ , . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} p ( z ) = c \\det \\left ( \\begin{pmatrix} 0 & 0 \\\\ 0 & C \\end{pmatrix} + \\sum _ { j = 1 } ^ { d } z _ j B _ j \\right ) \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} \\psi _ x ( a \\cdot \\delta _ u ) = \\| \\xi ( x ) \\| ^ { - 2 } \\big ( \\pi ( a ) \\xi ( x ) , \\xi ( x ) \\big ) \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\left ( v ^ { l k ' } _ { i k } \\left ( W ' , \\overline { W } ' \\right ) \\right ) _ { 1 \\leq k \\leq N ' } ^ { 1 \\leq k ' \\leq N ' } \\overline { \\left ( \\left ( v ^ { l ' k ' } _ { j k } \\left ( W ' , \\overline { W } ' \\right ) \\right ) _ { 1 \\leq k \\leq N ' } ^ { 1 \\leq k ' \\leq N ' } \\right ) ^ { t } } = a _ { i j } ^ { l l ' } \\left ( W ' , \\overline { W } ' \\right ) I _ { N ' } , \\quad \\mbox { f o r a l l $ i , j , l , l ' = 1 , 2 $ , } \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\ddot u ( t ) = \\omega ^ 2 \\ , \\nabla U ( u ( t ) ) , \\frac { 1 } { 2 } \\vert \\dot u ( t ) \\vert ^ 2 - \\omega ^ 2 \\ , U ( u ( t ) ) = 0 , \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\sigma _ n = \\Big ( \\prod _ { \\stackrel { i = 0 } { i \\not = 0 } } ^ { n } z _ i : \\prod _ { \\stackrel { i = 0 } { i \\not = 1 } } ^ { n } z _ i : \\ldots : \\prod _ { \\stackrel { i = 0 } { i \\not = n } } ^ { n } z _ i \\Big ) \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} r ^ { \\ast } _ \\beta ( \\mathcal { G } ) : = { \\rm s u p } \\ \\left \\{ r > 0 : \\real \\left ( \\frac { z f ' ( z ) } { f ( z ) } \\right ) > \\beta , z \\in \\mathbb { D } _ r , f \\in \\mathcal { G } \\right \\} , \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} [ h _ { i , 1 } , \\psi ^ + _ { i , 2 } ] = 0 , \\ [ h _ { i , - 1 } , \\psi ^ - _ { i , b _ i - 2 } ] = 0 . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} = \\biggr | \\frac { \\sum _ { l = 1 } ^ \\eta \\sum _ { m = l + 1 } ^ \\eta c ^ { \\bf a } _ l c ^ { \\bf a } _ m \\gamma ^ k _ l \\gamma ^ k _ m ( \\gamma _ l - \\gamma _ m ) ^ 2 } { \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ { k + 1 } \\sum _ { l = 1 } ^ \\eta c ^ { \\bf a } _ l \\gamma _ l ^ k } \\biggr | < \\epsilon . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} 0 \\leq { \\rm t r } _ { \\omega _ { \\epsilon } } ( \\sqrt { - 1 } \\partial \\bar { \\partial } F _ 0 + C _ 1 \\omega _ 0 ) \\leq \\nu ^ { - 1 } { \\rm t r } _ { \\omega _ 0 } ( \\sqrt { - 1 } \\partial \\bar { \\partial } F _ 0 + C _ 1 \\omega _ 0 ) = \\nu ^ { - 1 } ( C _ 1 n + \\Delta _ { \\omega _ 0 } F _ 0 ) . \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} F ( \\nabla f ) = F ^ * ( d f ) = \\frac { \\sqrt { ( 1 - \\| \\beta \\| ^ 2 ) | d f | ^ 2 + \\langle \\beta , d f \\rangle ^ 2 } - \\langle \\beta , d f \\rangle } { 1 - \\| \\beta \\| ^ 2 } , \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} b _ { m l } = O ( \\Vert \\phi \\Vert _ \\ast ) \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { f ( n ) } { n } \\leq \\limsup _ { n \\rightarrow \\infty } \\left ( \\frac { f ( d ) } { d } + \\frac { f ( r ) } { n } \\right ) = \\frac { f ( d ) } { d } \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} J ^ { ( \\alpha ) } _ \\lambda = \\sum _ \\pi \\theta ^ { ( \\alpha ) } _ \\pi ( \\lambda ) \\ p _ \\pi . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\rho _ i ( x ) = \\nabla ^ 2 \\eta ( x ) ( E _ i , E _ i ) \\leq - \\sqrt { a } \\sin \\sqrt { a } d ( x ) \\frac { - \\sqrt { a } \\sin \\sqrt { a } d _ 0 - \\kappa _ + \\cos \\sqrt { a } d _ 0 } { \\cos \\sqrt { a } d _ 0 - \\dfrac { \\kappa _ + } { \\sqrt { a } } \\sin \\sqrt { a } d _ 0 } . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\overline C ^ { k - 1 } _ k : = \\begin{pmatrix} \\overline c _ { 1 , 1 } & . . & . . & \\overline c _ { 1 , k - 2 } & \\overline c _ { 1 , k } \\\\ . . & . . & . . & . . \\\\ \\cdot & \\cdot & \\cdot & \\cdot \\\\ \\overline c _ { k - 1 , 1 } & . . & & \\overline c _ { k - 1 , k - 2 } & \\overline c _ { k - 1 , k } \\end{pmatrix} , \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\frac { P _ { \\lambda ( N ) } \\big ( R ^ { \\pm 1 } , t ^ { - 1 } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , \\dots , t ^ { 1 - N } \\big ) } } = \\Phi ^ { \\nu } ( R ^ { \\pm 1 } ; q , t ) , \\end{gather*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\left . \\frac { d \\boldsymbol { w } ( t ) } { d t } \\right | _ { t = k \\delta } \\approx \\lim _ { \\delta \\rightarrow 0 } \\frac { \\boldsymbol { w } _ { k + 1 } - \\boldsymbol { w } _ k } { \\delta } = - \\mu \\frac { ( \\boldsymbol { g } _ k - \\lambda _ k \\boldsymbol { b } _ k ) } { \\delta } \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} \\tilde { g } ^ { ( 1 ) } _ i = [ z ^ { - 1 } ] T ^ + _ { i i } ( z ) - \\sum _ { j < i } \\left ( [ z ^ { - 1 } ] T ^ + _ { i j } ( z ) \\right ) \\cdot A ^ + _ { j i } , \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} X ( t ) = \\xi + \\int _ 0 ^ t \\int _ U \\int _ A f ( s , X ( s ^ - ) , u , a , m ( s ) ) \\N _ { \\rho ^ { \\widehat { \\gamma } , X } } ( d s , d u , d a ) , \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} x ^ 2 = \\frac { s } { k } + \\frac { k + 1 } { 2 k } \\geq \\frac { 1 } { \\sqrt { k } } + \\frac { k + 1 } { 2 k } = \\left ( \\frac { 1 } { \\sqrt { 2 } } + \\frac { 1 } { \\sqrt { 2 k } } \\right ) ^ 2 \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} { \\rm C o r r } _ { \\frak X _ P } ( h ; \\widehat { \\frak S } ^ { \\epsilon } ) = ( f _ P , f _ t ) ! ( ( f _ P , f _ s ) ^ * h ; \\widehat { \\frak S } ^ { \\epsilon } ) . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} \\mathcal { L } u = \\triangle u - \\partial _ t u - \\tilde { V } ( x , t ) u . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} & - \\frac { t ^ 2 } { 4 \\alpha \\Gamma ( - \\alpha ) } \\int \\limits _ 0 ^ { \\infty } r ^ { - \\alpha - 2 } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } T ( r ) x \\ , d r \\ ; + \\ ; \\frac { 1 } { \\Gamma ( - \\alpha ) } \\int \\limits _ 0 ^ { \\infty } r ^ { - \\alpha - 1 } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } x \\ , d r \\\\ = \\ , & \\frac { t ^ 2 } { 4 \\alpha \\Gamma ( - \\alpha ) } \\int \\limits _ 0 ^ { \\infty } r ^ { - \\alpha - 2 } \\exp ^ { - \\frac { t ^ 2 } { 4 r } } \\big ( x - T ( r ) x \\big ) d r . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} Q _ { r , \\rho } ( x ) = \\{ z = y - t \\nu _ S ( x ) : \\ y \\in S _ \\rho ( x ) , \\ t \\in ( 0 , r ) \\} \\ , , \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} | G ^ T \\nu _ j | ^ 2 & = ( G ^ T \\nu _ j ) ^ T ( G ^ T \\nu _ j ) \\\\ & = \\nu _ j ^ T G G ^ T \\nu _ j \\\\ & = \\nu _ j ^ T \\Gamma \\nu _ j . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} \\overline { D } + t _ n \\pi ^ * ( \\zeta ) + \\widehat { ( \\theta _ n ) } = \\overline { D } + t _ n ( \\pi ^ * ( \\zeta + \\widehat { ( \\vartheta ) } ) + \\widehat { ( \\theta _ n \\pi ^ * ( \\vartheta ) ^ { - t _ n } ) } , \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} E ( r ) = \\int _ { r } ^ { \\infty } \\frac { \\sin ( \\pi w ( 1 - \\delta ) ) } { \\pi w } \\exp \\left ( - B ^ 2 w \\right ) \\ , d w . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} o ( \\alpha _ + , \\alpha _ - ) = d _ 0 \\circ ( \\frak m ^ { 2 } _ { 1 ; \\alpha _ + , \\alpha _ - } - b ( \\alpha _ + , \\alpha _ - ) ) - ( \\frak m ^ { 2 } _ { 1 ; \\alpha _ + , \\alpha _ - } - b ( \\alpha _ + , \\alpha _ - ) ) \\circ d _ 0 . \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} H ^ \\ast [ t ] = H ( x ^ \\ast ( t ) , u ^ \\ast ( t ) , \\lambda _ 0 , \\lambda ( t ) ) , \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} J ( t ) = \\int _ { 0 } ^ { \\tau } ( t - s ) ^ { \\alpha - 1 } s ^ { - \\delta } d s , t \\in [ 2 \\tau , \\nu ] . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} \\eta _ 1 ( t ) & = \\Psi _ 1 ( t \\xi _ 1 , \\ldots , t \\xi _ { k _ 1 } ) , \\\\ \\eta _ 2 ( t ) & = \\Psi _ 2 ( t \\xi _ { k _ 1 + 1 } , \\ldots , t \\xi _ k ) , \\\\ \\eta ( t ) & = \\Psi ( t \\xi _ 1 , \\ldots , t \\xi _ k ) , \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} g : & = e _ { \\theta } \\cdot \\textrm { c u r l } \\ f \\\\ & = - e _ { \\theta } \\cdot \\textrm { c u r l } \\ ( h - \\nabla \\Phi ) \\\\ & = - \\partial _ z h ^ { r } + \\partial _ r h ^ { z } \\\\ & = - v \\cdot \\nabla \\omega ^ { \\theta } + \\frac { u ^ { r } } { r } \\omega ^ { \\theta } + \\frac { \\partial _ z | u ^ { \\theta } | ^ { 2 } } { r } . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} f = a _ 1 ^ { n _ 1 - 1 } f _ 1 + a _ 2 ^ { n _ 2 } f _ 2 + \\cdots + a _ d ^ { n _ d } f _ d \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} C _ E = ( C _ F ) ^ * , \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{gather*} \\sum _ { n \\in \\Z } { M _ 1 ( n ) e ^ { i n \\theta } } = \\sum _ { n \\in \\Z } { M _ 1 ' ( n ) e ^ { i n \\theta } } . \\end{gather*}"}
-{"id": "6058.png", "formula": "\\begin{align*} G _ 2 ( x ) = \\big ( x C ( x ) - x \\big ) F _ T ( x ) \\ , . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\tilde \\eta = \\sigma ^ * ( d \\varphi + E \\pi ^ * d \\theta ) = d \\varphi + E d \\arctan u = 0 \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\left \\langle \\left ( T _ + - \\textstyle { 1 \\over 2 } t \\right ) ^ n \\right \\rangle = 0 { \\rm f o r } \\ n \\ { \\rm o d d } \\ , . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} h ( x ) = \\frac { 1 } { 2 } \\langle A x , x \\rangle , x \\in X , \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{gather*} \\lim _ { i \\rightarrow \\infty } { \\Phi ^ { \\nu ^ { ( i ) } } ( x _ 1 , \\dots , x _ m ; q , t ) } = \\Phi ^ { \\nu } ( x _ 1 , \\dots , x _ m ; q , t ) \\forall \\ , m \\in \\N \\end{gather*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\sum _ k k \\binom { n } { k } = n 2 ^ { n - 1 } \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{gather*} \\kappa _ { a b } = 0 , \\nu = 0 , \\tau _ { a b } = E _ { 0 a c b } \\theta ^ c + K _ { a b } \\pi _ n - F _ { a b } { } ^ c \\pi _ c , \\qquad \\lambda _ { a b } = 0 , \\end{gather*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\lambda : \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i ( 1 + \\lambda ^ T \\psi _ i ) ^ { - 1 } = 0 \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} L _ 1 = - \\partial ^ 2 + \\frac { 2 } { x ^ 2 } \\ \\mbox { a n d } \\ A _ 3 = \\partial ^ 3 - \\frac { 3 } { x ^ 2 } \\partial + \\frac { 3 } { x ^ 3 } \\ , \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\sum _ { h \\in L ^ { \\ast } / ( L ^ { \\ast } ) ^ { 2 } } \\left ( \\# \\right ) \\cdot \\operatorname { T r } _ { L / k } ( \\langle h \\rangle ) = 1 5 \\cdot \\langle 1 \\rangle + 1 2 \\cdot \\langle - 1 \\rangle . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} p = \\left ( { 1 + z ^ { 2 } } \\right ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { X } ^ N _ i ( t ) = & \\xi _ i ^ N + \\int _ 0 ^ t \\int _ U f ( s , \\widetilde { X } ^ N _ i ( s ^ - ) , u , \\beta ( s ) , \\widetilde { \\mu } _ N ( s ^ - ) ) \\N _ i ^ N ( d s , d u ) \\\\ \\widetilde { X } ^ N _ j ( t ) = & \\xi _ j ^ N + \\int _ 0 ^ t \\int _ U f ( s , \\widetilde { X } ^ N _ j ( s ^ - ) , u , \\gamma _ j ^ N ( s , \\widetilde { X } ^ N ( s ^ - ) ) , \\widetilde { \\mu } _ N ( s ^ - ) ) \\N _ j ^ N ( d s , d u ) \\\\ & \\mbox { i f } j \\neq i . \\end{cases} \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} y _ { 0 } ( x ) = \\frac { y _ a } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } , \\qquad \\gamma = \\alpha + \\beta - \\alpha \\beta , \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} y ' _ n ( z ) & = \\sum _ { m = 0 } ^ { n - 1 } ( - 1 ) ^ { n - m - 1 } ( m + 1 ) e _ { n - m - 1 } z ^ m , \\\\ y '' _ n ( z ) & = \\sum _ { m = 0 } ^ { n - 2 } ( - 1 ) ^ { n - m - 2 } ( m + 2 ) ( m + 1 ) e _ { n - m - 2 } z ^ m . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} h ( x , \\xi , \\lambda ) = \\lambda ^ { - \\frac { r - 1 } { r } } | x - x _ { 0 } | ^ { 2 } \\ \\Lambda _ { \\Phi _ { 0 } } . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\tau _ i ^ k = t _ { k - 1 } + q _ i \\ ; h , \\ ; \\ ; \\ ; i = 1 , \\ldots , r , \\ ; \\ ; \\ ; k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\xi _ { 0 } = e _ { n - 1 } , \\ ( 0 , 0 , \\cdots , 0 , 1 , 0 ) \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} 0 \\to ( \\sharp _ { i = 1 } ^ m M _ i ) ^ \\ast \\to ( F \\sharp ( \\sharp _ { i > 1 } M _ i ) ) ^ \\ast \\to ( G \\sharp ( \\sharp _ { i > 1 } M _ i ) ) ^ \\ast . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\Phi ( f ( z , t ) ) = ( 1 - P ) f \\Big ( \\frac { z } { ( 1 - P ) ^ { 2 } } , \\frac { z P ^ { 2 } } { ( 1 - P ) ^ { 2 } } \\Big ) . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\frac { d \\dot { \\varphi } _ { \\epsilon } } { d t } = ( \\Delta _ { \\omega _ { \\varphi _ { \\epsilon } } } + X ) \\dot { \\varphi } _ { \\epsilon } + \\gamma \\dot { \\varphi } _ { \\epsilon } . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} H = \\frac { H _ 3 ( \\mathbb { R } ) } { H _ 3 ( \\mathbb { Z } ) } \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\int _ { S ' _ { d } } \\langle w , \\varphi \\rangle _ { 0 } \\langle w , \\psi \\rangle _ { 0 } \\ , d \\mu _ { \\beta } ( w ) = \\frac { 1 } { \\Gamma ( \\beta + 1 ) } \\langle \\varphi , \\psi \\rangle _ { 0 } . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} | V _ m | & \\leq \\| v _ m \\| _ 2 \\\\ & \\leq \\prod _ { r = 2 } ^ m \\max \\ ( \\| M _ m \\| _ { 2 \\to 2 } , \\| N _ m \\| _ { 2 \\to 2 } \\ ) \\| v _ 1 \\| _ 2 \\\\ & = \\prod _ { r = 2 } ^ m \\max \\ ( \\| M _ m \\| _ { 2 \\to 2 } , \\| N _ m \\| _ { 2 \\to 2 } \\ ) \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} \\alpha ^ N [ \\gamma ^ N ] _ i ( s ) = \\gamma ^ N _ i ( s , X ^ N ( s ^ - ) ) . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\frac { x _ { k + 1 } - x _ k } { \\phi _ 1 ( a , h ) } = a x _ k , \\frac { y _ { k + 1 } - y _ k } { \\phi _ 2 ( c , h ) } = - c y _ k , \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} H _ \\epsilon ( a _ 0 , x , y ) : = G _ \\epsilon ( x , y ) \\left ( 1 - \\frac { \\exp ( a _ 0 ( \\eta ^ \\epsilon ( x , y ) , t ) / \\epsilon ) } { \\exp ( a _ 0 / \\epsilon ) } \\right ) . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\theta _ k ( \\pi ^ * _ k ( c ) ) = \\theta _ k ( 1 _ X \\cup \\pi ^ * _ k ( c ) ) = \\theta _ 0 ( 1 _ X ) \\cup c = \\deg \\theta \\cdot ( 1 _ Y \\cup c ) = \\deg \\theta \\cdot c . \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\lefteqn { H ( \\bar { X } ) + I ( \\bar { W } \\wedge \\bar { Y } | \\bar { X } ) } \\\\ & = I ( \\bar { W } \\wedge \\bar { X } , \\bar { Y } ) + H ( \\bar { X } | \\bar { W } ) \\\\ & \\le r _ { 0 , n } + r _ { 1 , n } \\\\ & \\le \\frac { 1 } { n } \\log | { \\cal T } _ { \\bar { X } } ^ n | \\\\ & ~ ~ ~ + \\frac { 3 \\log n + \\log \\log | { \\cal X } | + \\log | { \\cal X } | + 3 + | { \\cal X } | | { \\cal Y } | \\log ( n + 1 ) } { n } \\\\ & \\le H ( \\bar { X } ) + \\delta _ n . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} u ^ 5 ( t ) = ( 0 , 0 , 0 , 0 ) . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{gather*} - x _ { 0 } ^ { 2 } + x _ { 1 } ^ { 2 } + \\dots + x _ { n - 1 } ^ { 2 } = - R ^ { 2 } , \\end{gather*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\mathrm { d i v } \\left [ A ( x , 0 ) \\nabla u \\right ] - D _ s g ( x , 0 ) u = \\lambda u & \\qquad \\ , , \\\\ \\noalign { \\medskip } u = 0 & \\qquad \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\dot { \\underline x } = A \\underline x = X _ 0 ( \\underline x ) \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { | t - s | \\le \\delta } \\sum _ { k = 0 } ^ \\infty ( k + 2 ) | B ^ n _ k ( t ) - B ^ n _ k ( s ) | = 0 . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} A '' ( x , v ) & = \\frac { 1 } { 1 - x } B ( x , v ) + B ' ( x , v ) , \\\\ B ' ( x , v ) & = x ^ 2 A ( v x , 1 ) + \\frac { x ^ 2 } { v } ( A ( v x , 1 ) - 1 ) + \\frac { x } { 1 - v } ( B ' ( x , v ) - \\frac { 1 } { v ^ 2 } B ' ( v x , 1 ) ) , \\\\ B ( x , v ) & = x B ' ( x , v ) + \\frac { x ^ 4 } { ( 1 - ( 1 + v ) x ) ( 1 - x ) } . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\Delta R + 2 | R i c | ^ 2 = \\nabla ^ i R \\nabla _ i f \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\zeta \\circ \\mathcal { P } = R _ l \\circ \\zeta \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} f ( T _ 0 : \\cdots : T _ n ) = ( T _ 0 ^ d : F _ 1 ( T _ 0 , \\ldots , T _ n ) : \\cdots : F _ n ( T _ 0 , \\ldots , T _ n ) ) , \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} 2 T ^ \\ast \\sqrt { d } e ^ { T ^ \\ast C _ 1 } \\left [ C _ 2 + C _ 3 ( K _ 2 + T ^ \\ast C _ 4 ) e ^ { T ^ \\ast C _ 5 } \\right ] = 1 . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} X ^ { \\frak C \\boxplus \\tau } = \\coprod _ k B _ { \\tau } ( \\overset { \\circ \\circ } S _ k ( X ^ { \\frak C \\boxplus \\tau } ) ) \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} L _ { \\eta } \\Gamma _ { ( j k ) } ^ { i } = 2 \\xi , _ { t ( j } \\delta _ { k ) } ^ { i } \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\Sigma _ d ( Y / Y _ 3 ) = \\mathrm { K e r } ( \\eta _ d ) \\backslash Y ^ d . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} l _ { \\langle \\rangle } ( \\mathcal I _ { \\psi _ 0 \\land \\psi _ 1 , i } P ) & = ( l _ { \\langle \\rangle } ( P ) \\backslash \\{ \\psi _ 0 \\land \\psi _ 1 \\} ) \\cup \\{ \\psi _ i \\} , \\\\ o _ { \\langle \\rangle } ( \\mathcal I _ { \\psi _ 0 \\land \\psi _ 1 , i } P ) & = o _ { \\langle \\rangle } ( P ) \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} Y ( t ) = \\int _ 0 ^ t X ( s ) d s + B ( t ) , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\mathrm { G } _ n ( \\mathbb { C } ) = \\langle \\sigma _ n = \\Big ( \\prod _ { \\stackrel { i = 0 } { i \\not = 0 } } ^ { n } z _ i : \\prod _ { \\stackrel { i = 0 } { i \\not = 1 } } ^ { n } z _ i : \\ldots : \\prod _ { \\stackrel { i = 0 } { i \\not = n } } ^ { n } z _ i \\Big ) , \\ , \\mathrm { A u t } ( \\mathbb { P } ^ n _ \\mathbb { C } ) \\rangle \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\pi _ { s , P } \\circ \\widehat f _ s = \\pi _ { t , P } \\circ \\widehat f _ t . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} \\varpi _ n ( \\delta ) = \\sup _ { \\begin{smallmatrix} | u | , | v | \\leq M \\\\ | u - v | \\leq \\delta \\end{smallmatrix} } | L ^ { \\rm r e s c , h } _ n ( u ) - L ^ { \\rm r e s c , h } _ n ( v ) | . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} B ( x , y ) ( \\xi ) = \\lim _ { z \\to \\xi } d ( y , z ) - d ( x , z ) , \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} f ( X ) & = a _ 1 X ^ { d _ 1 } + \\ldots + a _ s X ^ { d _ s } ; \\\\ s & > 1 , \\ : d = d _ 1 > \\ldots > d _ s \\geq 0 , \\ : a _ 1 , \\ldots , a _ s \\in \\mathbb { F } \\setminus \\left \\lbrace 0 \\right \\rbrace , \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\imath _ V \\omega = 0 \\textrm { a n d } \\imath _ U \\omega = 0 \\ , , \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} \\chi \\left ( x ( \\nu ( \\Gamma _ d ) ) ^ { 1 / \\alpha } \\Gamma _ d \\right ) = c _ { a } x ^ { - \\alpha } \\frac { x ^ { \\rho } - 1 } { \\rho } + c _ { b } x ^ { - \\alpha } \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\sum _ { y \\in [ n ] } d ( y ) = \\sum _ { \\C \\in \\Gamma _ { a , b } } | Y ( \\C ) | = \\sum _ { \\C \\in \\Gamma _ { a , b } } | \\C | \\geq \\frac { | \\mathcal { F } _ { a , b } | } { k - 1 } , \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} H = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ \\ell T ^ { i n } f _ i S ^ { j n } g _ j \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{gather*} \\frac { \\Gamma _ q ( \\lambda _ i + \\theta ( N - i ) - z ) } { \\Gamma _ q ( \\lambda _ { i + 1 } + \\theta ( N - i ) - z ) } = \\prod _ { n = \\lambda _ { i + 1 } } ^ { \\lambda _ i - 1 } { [ n + \\theta ( N - i ) - z ] _ q } = \\prod _ { n = \\lambda _ { i + 1 } } ^ { \\lambda _ i - 1 } { \\frac { 1 - q ^ { n + \\theta ( N - i ) - z } } { 1 - q } } , \\end{gather*}"}
-{"id": "7765.png", "formula": "\\begin{align*} \\partial _ t u _ t \\ , = \\ , - u _ t + g X u _ t \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{gather*} a ( 0 , \\cdot ) = I , \\end{gather*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\P _ x \\left ( \\int _ 0 ^ t | f | ( X _ s ) d s < \\infty \\right ) = 1 , \\ \\ f \\in \\bigcup _ { r \\in [ q , \\infty ] } L ^ r ( \\R ^ d , m ) \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\omega _ { n , 1 } \\left ( { \\nu , p } \\right ) = 2 \\int _ { 1 } ^ { p } { \\left \\vert { \\frac { \\tilde { { F } } _ { n } \\left ( q \\right ) d q } { q ^ { 2 } \\left ( { 1 - q ^ { 2 } } \\right ) } } \\right \\vert } + \\sum \\limits _ { s = 1 } ^ { n - 1 } { \\frac { 1 } { \\nu ^ { s } } \\int _ { 1 } ^ { p } { \\left \\vert { \\frac { \\tilde { { G } } _ { n , s } \\left ( q \\right ) d q } { q ^ { 2 } \\left ( { 1 - q ^ { 2 } } \\right ) } } \\right \\vert , } } \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} \\begin{cases} - { \\rm R i c } ( \\omega _ 0 ) + \\omega _ 0 = \\sqrt { - 1 } \\partial \\bar { \\partial } F _ 0 \\\\ \\int _ X e ^ { - F _ 0 } \\omega _ 0 ^ n = [ \\omega _ 0 ] ^ n . \\end{cases} \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} E _ \\mathfrak { a } ^ k ( \\sigma _ \\mathfrak { b } z , w ) = \\delta _ { [ \\mathfrak { a } = \\mathfrak { b } ] } y ^ w + \\rho _ { \\mathfrak { a } , \\mathfrak { b } } ^ k ( 0 , w ) y ^ { 1 - w } + \\sum _ { h \\neq 0 } \\rho _ { \\mathfrak { a } , \\mathfrak { b } } ^ k ( h , w ) W _ { \\frac { | h | } { h } \\frac { k } { 2 } , w - \\frac { 1 } { 2 } } ( 4 \\pi \\lvert h \\rvert y ) e ^ { 2 \\pi i h x } \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\Delta _ 0 & = \\mbox { P r } _ { E _ t } ( R | H _ 0 ) - \\mbox { P r } _ { E _ f } ( R | H _ 0 ) \\\\ & = 1 / P W - \\epsilon - \\alpha \\\\ \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} T _ \\Delta x : = \\begin{cases} x + \\Delta \\frac { \\pi ( x ) - x } { | \\pi ( x ) - x | } , & \\\\ \\pi ( x ) , & . \\end{cases} \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} g _ { 2 } ( \\pi _ { \\ast } \\omega W , ( \\nabla \\pi _ { \\ast } ) ( Z , \\varphi U ) ) & = g _ { 1 } ( \\phi W , \\hat { \\nabla } _ { Z } \\varphi U ) , \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\left ( u _ t - | D _ x | ^ \\alpha u _ x + u u _ x \\right ) _ x + u _ { y y } = 0 . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\min \\{ - \\Delta u ^ t , u ^ t - h ^ t \\} = 0 \\mbox { i n } \\R ^ n , \\begin{cases} \\lim _ { | x | \\to \\infty } u ^ t ( x ) = c ^ t & n \\ge 3 \\\\ \\lim _ { | x | \\to \\infty } \\frac { u ^ t ( x ) } { - \\log | x | } = c ^ t & n = 2 . \\end{cases} \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} F ( f ) ( \\xi ) & = \\hat { f } ( \\xi ) = \\int _ { \\mathbb { R } } e ^ { - i \\xi x } f ( x ) ~ d x , \\\\ F ^ { - 1 } ( f ) ( x ) & = \\check { f } ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } e ^ { i x \\xi } f ( \\xi ) ~ d \\xi , \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} s _ i ( g ) = s _ i ^ { - 1 } ( g ) = ( g _ 1 , \\ldots g _ { i - 1 } , g _ { i + 1 } , g _ i , g _ { i + 2 } , \\ldots , g _ n ) , \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\begin{array} { l } N = \\left [ \\begin{array} { c c c c } A & 0 _ n & - I _ n & 0 _ n \\end{array} \\right ] , \\\\ \\tilde { X } = ( X , X , X , X ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } L ( \\phi ) & = & - N ( \\phi ) - E & \\Omega _ \\varepsilon , \\\\ \\phi & = & 0 & \\partial \\Omega _ \\varepsilon , \\end{array} \\right . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} P \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) = \\left \\Vert \\mathbf { Q } \\left ( \\omega _ { 1 } , \\omega _ { 2 } , b _ { 1 } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) , b _ { 2 } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) ; \\mathbf { t } \\right ) + \\bar { p } ^ { * } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\mathbf { 1 } - \\mathbf { f } \\right \\Vert _ { 2 } ^ { 2 } , \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\phi ( \\lambda x , y ) = \\phi ( x , \\lambda y ) , & \\\\ \\rho ( \\lambda x ) = \\lambda ^ { 2 / \\alpha } \\rho ( x ) , & \\\\ \\theta ( \\lambda x ) = \\lambda ^ { \\beta } \\theta ( x ) . & \\end{array} \\right . \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} u _ 1 ( t , 0 ) = u _ 2 ( t , 0 ) = 0 \\ , , u _ 1 ( t , \\pi ) = g _ 1 ( t ) \\ , , u _ 2 ( t , \\pi ) = g _ 2 ( t ) t \\in ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} F ( g _ n \\chi _ { [ i - 1 , i ) } ) \\to F ( g \\chi _ { [ i - 1 , i ) } ) F ( g _ n \\chi _ { [ i - 1 , i ) } ) = F ( \\chi _ { [ i - 1 , i ) } ) g _ n \\to F ( \\chi _ { [ i - 1 , i ) } ) g \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\frac { 1 } { i } \\partial _ x \\varphi - C _ + ( u \\varphi ) = \\lambda \\varphi , \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} \\left | \\det \\left ( A ^ { ( k ) } ( k : m , k : m ) \\right ) \\right | \\ge \\prod _ { j = k } ^ m \\frac { t ( \\alpha ) } { \\sqrt { n - j + 2 } } \\ , c _ j = t ( \\alpha ) ^ { m - k + 1 } \\ , \\prod _ { j = k } ^ m \\frac { c _ j } { \\sqrt { n - j + 2 } } . \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\bar { u } \\left ( x , t \\right ) = Q \\left ( x , t \\right ) \\left ( 2 \\pi \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } r U \\left ( r \\right ) d r \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} a _ k = \\frac { \\left ( \\det { \\overline C ^ { k + 1 } } \\right ) ^ { \\frac { 1 } { 2 } } \\left ( \\det { \\overline C ^ { k - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } } { \\det { \\overline C ^ { k } } } \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n c _ { i , j } \\equiv 0 \\bmod 2 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} L ^ \\alpha e ^ g - \\mu e ^ g = \\lambda _ 1 e ^ g . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\omega _ B ^ c = \\omega _ B + O _ \\prec ( N ^ { - \\frac { \\gamma } { 4 } } ) \\ , . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} \\sigma ( x _ { i } ) = \\begin{cases} - x _ { 1 } - x _ { 2 } - x _ { 3 } - x _ { 4 } & \\\\ x _ { \\sigma ( i ) } & \\end{cases} \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} & m ( B _ { n , 3 , 3 , 2 } ^ { ( n - 4 ) } , 1 ) = n + 1 , m ( B _ { n , 3 , 3 , 2 } ^ { ( n - 4 ) } , 2 ) = 2 n - 6 , m ( B _ { n , 3 , 3 , 2 } ^ { ( n - 4 ) } , k ) = 0 \\ f o r \\ k \\geq 3 ; \\\\ \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} N _ \\Omega = \\{ x \\in \\Omega : w _ \\Omega ( x ) = 0 \\} . \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} R = M ^ 2 _ 2 + M ^ 2 _ 3 + 2 M _ 1 M _ 2 + 2 M _ 2 M _ 3 + 2 M _ 3 M _ 1 , \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} x _ 1 & = \\frac { \\vartheta _ { ( - 1 , 0 , 0 , 1 ) / 3 } ^ 3 ( \\tau , \\zeta ) } { \\vartheta _ { ( - 2 , 0 , 0 , 2 ) / 3 } ^ 3 ( \\tau , \\zeta ) } , \\\\ x _ 2 & = \\frac { \\vartheta _ { ( - 1 , 0 , 0 , 1 ) / 3 } ^ 3 ( \\tau , \\iota ^ * ( \\zeta ) ) } { \\vartheta _ { ( - 2 , 0 , 0 , 2 ) / 3 } ^ 3 ( \\tau , \\iota ^ * ( \\zeta ) ) } , \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\left ( p ( \\Pi ) \\sqcup g ( \\Gamma ) \\right ) ^ { c } = \\left ( p ( \\Gamma ^ { c } ) \\sqcup g ( \\Gamma ^ { c } ) \\right ) \\cup p ( \\Pi ^ { c } ) = r ^ { - 1 } ( \\Gamma ^ { c } ) \\cup p ( \\Pi ^ { c } ) . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} \\aligned \\overset { \\circ } S _ A ( [ 0 , 1 ) ^ k ) & = \\{ ( t _ 1 , \\dots , t _ k ) \\in [ 0 , 1 ) ^ { k } \\mid i \\in A \\Rightarrow t _ i = 0 , \\ , \\ , i \\notin A \\Rightarrow t _ i > 0 \\} , \\\\ \\overset { \\circ } S _ A ( V _ { \\frak r } ) & = V _ { \\frak r } \\cap ( \\overline V _ { \\frak r } \\times \\overset { \\circ } S _ A ( [ 0 , 1 ) ^ k ) ) , \\\\ \\Gamma _ { \\frak r } ^ A & = \\{ \\gamma \\in \\Gamma _ { \\frak r } \\mid \\gamma A = A \\} . \\endaligned \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\frac { n } { d } - 1 = \\frac { | I | } { d } - 1 < q = \\frac { n - r } { d } \\leq \\frac { n } { d } . \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} W _ { s , a } ( f ) : = \\{ a ^ k \\pmod s : k = 1 , \\cdots , t _ s - 1 \\} \\bigcap \\{ f ( i ) \\pmod s : i \\in [ 1 , s ] \\} . \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} \\Omega _ \\varepsilon = \\{ z \\in \\R ^ 3 : 1 - \\varepsilon < | z + e _ 1 | < 1 \\} , \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} X _ H = \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial } { \\partial q ^ i } - \\frac { \\partial H } { \\partial q ^ i } \\frac { \\partial } { \\partial p _ i } \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\limsup _ { x \\to \\infty } \\sup _ { n \\geqslant \\nu } \\frac { 1 } { n \\overline { F } _ { X _ \\nu } ( x ) } \\sum \\limits _ { i = 1 } ^ { n } \\overline { F } _ { X _ i } ( x ) < \\infty . \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\max _ { G ( k ) \\leq s \\leq G ( k + 1 ) } \\left | \\log \\Phi _ s ^ * ( \\theta _ j ) \\right | \\leq \\sum _ { j \\leq k } \\lambda _ j + \\sum _ { j \\leq k } C \\left ( \\sum _ { m = G ( j ) } ^ { G ( j + 1 ) } | \\alpha _ m | ^ 2 \\right ) + C \\leq C \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} S : = ( \\P ^ 2 ) ^ { [ 3 ] } \\setminus \\P ^ 2 \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} \\chi ( J , I ) = \\big \\{ ( x _ 1 , \\ldots , x _ m ) \\in ( D ^ 2 ) ^ m \\ ; \\big | \\ ; \\big \\} . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} t ( S _ k , W ) = \\int _ 0 ^ 1 d ^ k _ W ( x ) \\ ; \\mathrm { d } x \\ , . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\aligned d _ 0 \\circ \\psi _ { \\alpha _ + , \\alpha _ - } - \\psi _ { \\alpha _ + , \\alpha _ - } \\circ d _ 0 & + \\frak m ^ { 2 } _ { 1 ; \\alpha _ + , \\alpha _ - } \\circ \\psi _ { \\alpha _ - , \\alpha _ - } \\\\ & - \\psi _ { \\alpha _ + , \\alpha _ + } \\circ \\frak m ^ { 1 } _ { 1 ; \\alpha _ + , \\alpha _ - } + b ( \\alpha _ + , \\alpha _ - ) = 0 . \\endaligned \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} & { \\cal L } _ \\omega = { \\cal L } _ \\omega ^ { < } + { \\cal R } _ \\omega + { \\cal R } _ \\omega ^ \\bot \\ , , { \\cal L } _ \\omega ^ { < } : = { \\cal V } _ n { \\mathfrak L } _ n ^ { < } { \\cal V } _ n ^ { - 1 } \\ , , { \\cal R } _ \\omega : = { \\cal V } _ n { \\cal R } _ n { \\cal V } _ n ^ { - 1 } \\ , , { \\cal R } _ \\omega ^ \\bot : = { \\cal V } _ n { \\cal R } _ n ^ \\bot { \\cal V } _ n ^ { - 1 } \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} a ^ { i j } D _ { i j } u = g - b ^ i D _ i u - c u = : g ' . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} [ M ] ^ c = \\sum _ { n = 1 } ^ { \\infty } [ M ^ n ] ^ c = \\sum _ { n = 1 } ^ { \\infty } [ M ^ { c , n } ] = [ M ^ c ] . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} a _ 0 = 1 \\textrm { a n d } a _ n = - \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } a _ j b _ { n - j } . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\Big ( \\sum _ { i , j = 1 } ^ { d } \\frac { a _ { i j } } { 2 } \\partial _ { i j } \\varphi + \\sum _ { i = 1 } ^ { d } h _ i \\partial _ i \\varphi + c \\varphi \\Big ) \\ d \\mu = \\int _ { \\R ^ d } \\varphi f \\ , d x , \\forall \\varphi \\in C _ 0 ^ { \\infty } ( \\R ^ d ) , \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi r \\frac { \\partial u } { \\partial x } d r + \\intop _ { 0 } ^ { a \\left ( x , t \\right ) } 2 \\pi \\left ( r \\frac { \\partial v } { \\partial r } + v \\right ) d r = 0 . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} U ^ e _ t ( \\phi ) = & \\int _ 0 ^ t \\sum _ { ( e , \\tilde { e } ) } \\ , 2 C ^ n _ { \\tilde { e } , e } \\big ( \\phi _ s ( x ^ e _ 1 ) - \\phi _ s ( x ^ { \\tilde { e } } _ 1 ) \\big ) \\ , \\big ( u ^ n _ { s - } ( x ^ { \\tilde { e } } _ 1 ) - u ^ n _ { s - } ( x ^ e _ 1 ) \\big ) \\ , d s \\\\ V ^ e _ t ( \\phi ) = & \\ , - \\int _ 0 ^ t \\alpha _ e \\big ( u ^ n _ { s - } ( x ^ e _ 2 ) - u ^ n _ { s - } ( x ^ e _ 1 ) \\big ) \\ , \\nabla _ L \\phi _ s ( x ^ e _ 1 ) \\ ; d s . \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\mathcal { D } _ N = \\{ Q \\in \\mathbb { D } : Q \\subset \\Lambda _ N \\} = \\Big \\{ Q \\in \\mathbb { D } : Q \\bigcap \\bigcup _ { B _ l \\in \\mathcal { F } _ N } B _ l = \\emptyset \\Big \\} . \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\partial \\Omega _ { d } = \\partial _ { 1 } \\Omega _ { d } \\cup \\partial _ { 2 } \\Omega _ { d } , \\partial _ { 1 } \\Omega _ { d } = \\xi _ { d } , \\partial _ { 2 } \\Omega _ { d } = \\Gamma _ { d } . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} [ \\phi , \\psi ] = \\int _ { \\Omega _ \\varepsilon } \\nabla \\phi \\cdot \\nabla \\psi - \\varepsilon ^ 2 \\ , \\lambda \\int _ { \\Omega _ \\varepsilon } \\phi \\ , \\psi . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} R = \\Theta ( j , 2 ^ b , A ) = \\Theta ( g ( j , 0 ) , b , A _ 0 ) \\oplus \\Theta ( g ( j , 1 ) , 2 , A _ 1 ) \\equiv _ { \\mathrm { T } } A _ 0 \\oplus A _ 1 = A . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} T _ 0 ( F ) \\backslash J ( F ) / T _ 3 ( F ) = \\tilde { T } _ 0 ( F ) \\backslash \\tilde { J } ( F ) / \\tilde { T } _ 3 ( F ) = K _ 0 ^ \\times \\backslash \\mathrm { I s o } ( K _ 3 , K _ 0 ) / K _ 3 ^ \\times , \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\frac { \\omega ^ { A _ \\Delta } ( E ) } { \\omega ^ { A _ { \\Delta } } ( \\Delta ' ) } \\sim \\dfrac { \\frac { \\omega ( E ) } { \\omega ( \\Delta ) } } { \\frac { \\omega ( \\Delta ' ) } { \\omega ( \\Delta ) } } = \\frac { \\omega ( E ) } { \\omega ( \\Delta ' ) } C \\le N ^ 4 \\frac { \\sigma ( E ) } { \\sigma ( \\Delta ' ) } . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} F \\left ( q , \\frac { \\partial W } { \\partial q } , \\lambda \\right ) = , \\frac { \\partial F } { \\partial \\lambda ^ a } \\left ( q , \\frac { \\partial W } { \\partial q } , \\lambda \\right ) = 0 . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\frac { 1 } { n + 1 } \\binom { n + 1 } { k } \\binom { n - k - 1 } { k - 1 } . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} F _ p ( p , p - 1 , b ) & = 1 + \\dim ( p - 2 , b ) + \\dim ( p - 1 , b - 1 ) + \\dim ( p + b - 3 , 1 ) \\cr & = \\dim ( p - 2 , b ) + \\dim ( p - 1 , b - 1 ) + \\dim ( p + b - 3 , 1 ) \\cr & \\ \\ \\ \\ + \\dim ( p + b - 2 ) \\cr & = \\dim ( p - 1 , b ) + \\dim ( p + b - 2 , 1 ) . \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} h & < \\frac { 3 . 4 7 5 3 } { 4 } \\left ( C \\cdot G ( a _ 1 ' ) \\cdot G ( a _ 2 ' ) ( 1 . 0 0 1 + 3 \\cdot 1 0 ^ { - 4 } C ^ { - 1 } ) + \\frac { \\log B _ 3 } { \\log 1 0 ^ 5 ( \\log 1 0 ^ { 5 / 2 } ) ^ 3 } \\right ) \\cdot \\\\ & \\cdot \\left ( h ' + \\frac { \\lambda ' } { \\sigma } \\right ) ^ 2 \\log ^ 2 \\alpha _ 2 \\log c \\\\ & = : G ( h 2 ) \\left ( h ' + \\frac { \\lambda ' } { \\sigma } \\right ) ^ 2 \\log ^ 2 \\alpha _ 2 \\log c . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\left ( F , G \\right ) _ { P } = \\tau _ { P } ^ { \\left ( F , G \\right ) } \\circ \\left ( F , G \\right ) \\circ \\sigma _ { P } ^ { 0 } = \\left ( F _ { P } , G _ { P } \\right ) , \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} ( \\alpha _ { 3 \\underline { d } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } ) ( s ) = ( \\alpha _ { 3 \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\underline { d } , \\widetilde { \\underline { d } } } \\circ \\varphi _ { \\widetilde { \\underline { d } } , \\underline { d } } ) ( s ) = ( \\alpha _ { 3 \\widetilde { \\underline { d } } } \\circ 0 ) ( s ) = 0 \\in V _ { X _ 3 } ( - ( d - l ) B ) . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\overline { F } _ { \\xi _ k } ( x ) \\leqslant c _ 2 \\varphi ( n ) \\overline { F } _ { \\xi _ \\varkappa } ( x ) , \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\left ( 1 \\pm \\gamma \\right ) \\prod _ { \\{ k , \\ell \\} \\in \\binom { [ r ] } { 2 } , \\{ k , \\ell \\} \\neq \\{ i , j \\} } d _ { k \\ell } \\prod _ { s \\in [ r ] , s \\neq i , j } | V _ s | \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} A _ { s + 1 } ( \\xi ) = - \\tfrac { 1 } { 2 } A _ { s } ^ { \\prime } ( \\xi ) + \\tfrac { 1 } { 2 } \\int \\psi ( \\xi ) A _ { s } ( \\xi ) d \\xi \\ \\left ( s = 0 , 1 , 2 , \\cdots \\right ) . \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\sigma : = \\inf \\big \\{ m \\geq 0 : \\ : \\big | S ^ 1 _ { K _ m ^ 1 } - S ^ 2 _ { K _ m ^ 2 } \\big | \\leq R ^ 1 _ { K _ m ^ 1 + 1 } + R ^ 2 _ { K _ m ^ 2 + 1 } \\big \\} \\ , . \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { Z } W , U ) & = - g _ { 1 } ( \\phi W , \\hat { \\nabla } _ { Z } \\varphi U ) + g _ { 2 } ( \\pi _ { \\ast } \\omega W , ( \\nabla \\pi _ { \\ast } ) ( Z , \\varphi U ) ) \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} { \\bar { U } } _ { K } \\doteq \\sup _ { n \\in \\mathbb { N } } { { E } } \\left \\{ \\sum _ { k = K } ^ { \\infty } \\int _ { [ 0 , T ] \\times \\lbrack 0 , 1 ] } k \\varphi _ { k } ^ { n } ( s , y ) { { 1 } } _ { [ 0 , r _ { k } ( \\bar { \\boldsymbol { X } } ^ { n } ( s ) ) ) } ( y ) \\ , d s \\ , d y \\right \\} . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} R = - 2 \\frac { a '' } { a } - 4 n \\frac { b '' } { b } - 4 n \\frac { a ' b ' } { a b } - 2 n ( 2 n - 1 ) \\left ( \\frac { b ' } { b } \\right ) ^ 2 - 2 n \\frac { a ^ 2 } { b ^ 4 } + \\frac { 2 n ( 2 n + 2 ) } { b ^ 2 } \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} T _ h \\phi ( x ) = \\int _ X \\phi ( y ) K _ h ( x , d y ) = \\phi ( x ) \\cdot \\alpha ( h ) \\quad \\quad h \\in D , \\ > x \\in X . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\mathbf { d } \\precapprox \\mathbf { e } \\qquad \\Leftrightarrow \\qquad \\lim _ { r \\rightarrow 0 } \\tfrac { \\mathbf { d } } { \\mathbf { e } } ( r ) = 0 . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\phi : \\mathrm { G a l } ( L ^ c / \\Q ) & \\longrightarrow C _ 2 \\wr G = \\{ \\pm 1 \\} ^ d \\rtimes G \\\\ g & \\longmapsto ( x _ g , \\sigma _ g ) , \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} l ( k ) & = \\frac 1 { 2 \\pi } \\int _ 0 ^ { \\infty } \\frac { \\chi ( \\xi ) } { \\xi - k } ~ d \\xi \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ 1 \\frac { 1 } { \\xi - k } ~ d \\xi + \\frac 1 { 2 \\pi } \\int _ 1 ^ 2 \\frac { \\chi ( \\xi ) } { \\xi - k } ~ d \\xi \\\\ & = \\frac 1 { 2 \\pi } \\log k + h ( k ) , \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{gather*} \\varphi = \\begin{pmatrix} \\phi _ 0 + \\phi _ n & - \\pi _ n & - \\pi _ b & - \\pi _ 0 & 0 \\\\ \\omega ^ n & \\phi _ 0 - \\phi _ n & \\gamma _ b & 0 & - \\pi _ 0 \\\\ \\omega ^ a & \\theta ^ a & - \\phi ^ a { } _ b & \\gamma ^ a & \\pi ^ a \\\\ \\omega ^ 0 & 0 & \\theta _ b & - \\phi _ 0 + \\phi _ n & - \\pi _ n \\\\ 0 & \\omega ^ 0 & - \\omega _ b & \\omega ^ n & - \\phi _ 0 - \\phi _ n \\end{pmatrix} , \\end{gather*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { A ; \\# A = \\ell } V _ { \\frak r } ^ + ( p ; A ) \\right ) / \\Gamma _ { \\frak r } . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} [ A , B ] = \\left ( \\begin{smallmatrix} x & y \\\\ 0 & 1 / x \\end{smallmatrix} \\right ) \\left ( \\begin{smallmatrix} z & w \\\\ 0 & 1 / z \\end{smallmatrix} \\right ) \\left ( \\begin{smallmatrix} 1 / x & - y \\\\ 0 & x \\end{smallmatrix} \\right ) \\left ( \\begin{smallmatrix} 1 / z & - w \\\\ 0 & z \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} 1 & x y ( 1 - z ^ 2 ) - z w ( 1 - x ^ 2 ) \\\\ 0 & 1 \\end{smallmatrix} \\right ) \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} L _ { n } \\left ( \\xi \\right ) = { \\sup _ { w \\in \\mathcal { L } \\left ( \\xi \\right ) } } \\left \\vert { G _ { n } \\left ( w \\right ) } \\right \\vert + \\frac { 1 } { 2 } \\int _ { \\mathcal { L } \\left ( \\xi \\right ) } { \\left \\vert { { G } _ { n } ^ { \\prime } \\left ( t \\right ) d t } \\right \\vert . } \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} ( \\omega _ Q ^ { T } ) = 2 \\ ( \\omega _ Q ) \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\widehat { J } \\psi ( x ) & = e ^ { - i n \\pi / 4 } \\mathcal { F } \\psi ( x ) , & & \\pi ^ { \\operatorname * { M p } } ( \\widehat { J } ) = J \\\\ \\widehat { V } _ { - P } \\psi ( x ) & = e ^ { \\frac { i } { 2 \\hbar } P x ^ { 2 } } \\psi ( x ) , & & \\pi ^ { \\operatorname * { M p } } ( \\widehat { V } _ { - P } ) = V _ { - P } \\\\ \\widehat { M } _ { L , m } \\psi ( x ) & = i ^ { m } \\sqrt { | \\det L | } \\psi ( L x ) , & & \\pi ^ { \\operatorname * { M p } } ( \\widehat { M } _ { L , m } ) = M _ { L , m } . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} W ( t ) = w _ \\Omega ( x _ 0 + t n ( x _ 0 ) ) \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} Q _ { j k } = c _ { j , k } \\cdot { N _ k \\choose 2 } \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | x - a _ i | ^ { 2 \\beta } e ^ { - \\sum _ { i = 1 } ^ n \\frac { | A ^ { \\frac { 1 } { 2 } } ( x - a _ i ) | ^ 2 } { 2 } } \\ , d x & \\le C _ 2 \\int _ { \\R ^ N } | x - a _ i | ^ { 2 \\beta } e ^ { - \\tilde \\alpha _ 1 \\frac { | x - a _ i | ^ 2 } { 2 } } \\ , d x \\\\ & = C _ 2 \\ , 2 ^ { \\beta + \\frac { N } { 2 } } \\tilde \\alpha _ 1 ^ { - \\beta - \\frac { N } { 2 } } \\int _ { \\R ^ N } | x - a _ i | ^ { 2 \\beta } e ^ { - \\frac { | x - a _ i | ^ 2 } { 2 } } \\ , d x . \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\zeta _ k ^ { \\rm O M A I I } = \\Pr \\left \\{ R _ k ^ { \\rm O M A I I } ( \\nu ) < \\bar R _ k \\right \\} . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\L ^ { - s } f = \\sum _ { j = 1 } ^ \\infty \\lambda _ j ^ { - \\frac { s } { 2 } } f _ j w _ j \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} f ! ( h ; ( \\widehat { \\frak S ^ { + \\rho _ i \\cdot } } ( i + 1 ; \\alpha _ - , \\alpha _ + ) ^ { \\epsilon } ) = f ! ( h ; ( \\widehat { \\frak S ^ { + \\rho _ i \\epsilon } } ( i + 1 ; \\alpha _ - , \\alpha _ + ) ) \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\alpha x } J _ { \\nu } ( \\beta x ) \\ , { \\rm d } x = \\frac { [ \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } - \\alpha ] ^ { \\nu } } { \\beta ^ { \\nu } \\sqrt { \\alpha ^ 2 + \\beta ^ 2 } } , \\nu > - 1 , \\alpha > 0 \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\min _ { \\boldsymbol { w } } J ( \\boldsymbol { w } _ k ) = { \\textstyle \\frac { 1 } { 2 } } \\ ! \\left ( R - \\boldsymbol { w } _ k ^ H \\boldsymbol { x } _ k \\boldsymbol { x } _ k ^ H \\boldsymbol { w } _ k \\right ) ^ 2 , ~ \\mathrm { s . t . } ~ \\| \\boldsymbol { w } _ k \\| _ p ^ p \\le c \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} ( L + 1 ) \\Gamma _ { \\varepsilon } & = ( L + 1 ) \\Gamma - \\varepsilon ( L + 1 ) ( A t + | x | ^ { 2 } ) \\\\ & = ( L + 1 ) \\Gamma - \\varepsilon ( A ( 1 + t ) + | x | ^ { 2 } + 2 b \\cdot x - 2 ) . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{gather*} \\tau _ { i , j } \\geq 0 , \\textrm { f o r a l l } 1 \\leq i < j \\leq N , \\\\ \\lambda _ n + \\sum _ { i = n + 1 } ^ N { \\tau _ { n , i } } - \\sum _ { i = 1 } ^ { n - 1 } { \\tau _ { i , n } } \\geq 0 , \\textrm { f o r a l l } n = 1 , \\dots , N . \\end{gather*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\int _ { B _ 1 ( 0 ) } \\phi ( h ) \\ , h _ i ^ 2 \\ , d h = \\frac { 1 } { n } \\int _ { B _ 1 ( 0 ) } \\phi ( h ) \\ , | h | ^ 2 \\ , d h = \\frac { C } n \\ , , \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\Delta w = D _ i ( b ^ i u ) - c u + f \\ ; \\ ; \\Omega , w = 0 \\ ; \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ R k _ r u _ { \\infty } ^ { y _ r } y _ { r , i } ^ 2 = \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } y _ { r , i } '^ 2 . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\partial _ t u \\ , = \\alpha \\ , \\Delta u + \\beta \\ , u - \\delta \\ , u ^ 2 + \\sqrt { \\gamma \\ , u } \\ , \\dot { W } \\quad \\overset { \\circ } { \\Gamma } . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} H = 2 u _ 1 + \\frac { n + 1 } { 2 } v _ 1 + \\ell L = u _ 2 - e ' v _ 2 . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} g _ { k } ( x , y _ { k } ) = \\exp \\left [ \\sum _ { i = 1 } ^ { n } \\left \\{ \\sum _ { j = 1 } ^ { p } y _ { k } ^ { ( i ) } z _ { k } ^ { ( i , j ) } x ^ { ( j ) } - \\psi \\left ( \\sum _ { j = 1 } ^ { p } z _ { k } ^ { ( i , j ) } x ^ { ( j ) } \\right ) + \\log \\phi ( y _ { k } ^ { ( i ) } ) \\right \\} \\right ] , \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\varphi X = \\mathcal { B } X + \\mathcal { C } X \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} [ A ( k _ { n ( q + 1 ) } - 1 : k _ { n q } ) ] _ { i j } \\geq \\frac { 1 } { n ^ { k _ { n ( q + 1 ) } - k _ { n q } } } = \\frac { 1 } { n ^ { l _ { q + 1 } } } . \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} S _ 2 ^ { h _ 2 } & = \\bigg ( \\sum _ { 2 \\le q \\leq X } 2 \\Lambda ( q ) \\mu ( \\omega _ { q } ) F _ { \\omega _ { q } } ( n ) \\bigg ) ^ { h _ 2 } \\ll \\sum _ { 2 \\le q _ 1 , \\ldots , q _ { h _ 2 } \\leq X } \\prod _ { i = 1 } ^ { h _ 2 } \\Lambda ( q _ i ) \\mu ( \\omega _ { q _ i } ) F _ { \\omega _ { q _ i } } ( n ) , \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\Big | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N d _ i \\Big ( G _ { i i } - \\frac { 1 } { a _ i - \\omega _ B ^ c } \\Big ) \\Big | \\prec \\Psi \\hat { \\Pi } \\ , . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} R _ h ( x _ 0 , \\dots , x _ { \\rho ( X ) } , y _ 0 , \\dots , y _ { \\rho ( X ) } ) = \\sum _ { \\beta = 1 } ^ { B _ h } r _ { h \\beta } \\prod _ { i = 1 } ^ { k _ { h \\beta } } x _ { v ( h , \\beta , i ) } \\prod _ { j = 1 } ^ { \\tilde k _ { h \\beta } } y _ { w ( h , \\beta , j ) } , \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\mathcal { G } ( u ) = u + T ( A ( u ) \\overline { u \\zeta } ) . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\rho _ i ( x ) - \\rho _ { n + 1 } ( x ) \\leq \\sum _ { i = 1 } ^ { n + 1 } \\rho _ i ( x ) ( 1 - 2 \\tilde { \\alpha } _ i ( x ) ^ 2 ) \\leq \\sum _ { i = 2 } ^ { n + 1 } \\rho _ i ( x ) - \\rho _ 1 ( x ) . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\sigma ( z , v _ 3 ) = \\sqrt { \\frac { 2 } { 1 + v _ 3 } } z \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} s u b j e c t \\ , t o \\begin{array} { c c c } a _ { 1 } \\cos \\omega _ { 1 } T _ { 0 } + b _ { 1 } \\sin \\omega _ { 1 } T _ { 0 } & = & a _ { 2 } \\cos \\omega _ { 2 } T _ { 0 } + b _ { 2 } \\sin \\omega _ { 2 } T _ { 0 } , \\\\ a _ { 1 } & = & a _ { 2 } \\cos \\omega _ { 2 } T + b _ { 2 } \\sin \\omega _ { 2 } T . \\end{array} \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} X = \\xi ( t ) \\partial _ { t } + T ( t ) V ^ { i } \\partial _ { i } ~ , ~ f = T _ { , t } S + K \\left ( t \\right ) , \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} S = \\begin{pmatrix} - a _ M b _ M & - a _ E b _ E \\\\ 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} g _ { 2 } ( ( \\nabla \\pi _ * ) ( U , X ) , \\pi _ * Y ) & = - g _ { 1 } ( \\nabla ^ { ^ { M _ 1 } } _ { U } \\varphi X , \\varphi Y ) + \\eta ( X ) g _ { 1 } ( U , \\mathcal { C } Y ) - U \\eta ( X ) \\eta ( Y ) - g _ { 1 } ( X , \\omega U ) \\eta ( Y ) . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\left \\| \\Sigma _ \\mu - \\Sigma _ 0 \\right \\| \\leq 2 4 2 \\frac { \\sigma _ 0 ^ 2 \\beta } { \\| \\Sigma _ 0 \\| m } \\leq 2 4 2 \\frac { \\sigma ^ 2 \\beta } { \\| \\Sigma _ 0 \\| m } = 2 4 2 \\| \\Sigma _ 0 \\| \\frac { \\overline { d } \\beta } { m } . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\int \\eta ^ 2 u ^ p \\ ; d x + \\frac { 4 ( p - 1 ) } { p } \\int ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla ( \\eta u ^ { p / 2 } ) ) \\ ; d x \\\\ & = \\frac { 4 ( p - 2 ) } { p } \\int u ^ { p / 2 } ( a \\nabla ( \\eta u ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x + p \\int ( u \\nabla a , \\nabla ( \\eta ^ 2 u ^ { p - 1 } ) ) \\ ; d x . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} [ ( 1 + \\mu _ 1 ) ( 1 + \\mu _ 2 ) - ( 1 + \\mu _ 3 ) ] ^ 2 = [ \\underbrace { \\mu _ 2 ( 1 + \\mu _ 1 ) } _ { \\geq 0 } + \\underbrace { \\mu _ 1 - \\mu _ 3 } _ { \\geq 0 } ] ^ 2 \\geq ( \\mu _ 1 - \\mu _ 3 ) ^ 2 \\geq \\mu _ 1 ^ 2 + \\mu _ 3 ^ 2 \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} a _ { k , k + 1 } = - \\left ( \\prod _ { j = 0 } ^ { k } a _ j \\right ) ^ { - 2 } a _ k w _ { k , k } , \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{gather*} \\begin{pmatrix} \\phi _ n - \\phi _ 0 & \\theta & 0 \\\\ { } ^ t \\ ! \\gamma & \\phi & { } ^ t \\theta \\\\ 0 & \\gamma & \\phi _ 0 - \\phi _ n \\end{pmatrix} , \\end{gather*}"}
-{"id": "4237.png", "formula": "\\begin{align*} & \\P \\big ( \\cap _ { j = 1 } ^ k \\{ T _ { s _ { j - 1 } + 1 } \\le t _ { j 1 } , \\ldots , T _ { s _ j } \\le t _ { j \\ell _ j } \\} \\mid \\cap _ { j = 1 } ^ k \\{ N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\} \\big ) \\\\ & = \\prod _ { j = 1 } ^ k \\P \\big ( T _ { s _ { j - 1 } + 1 } \\le t _ { j 1 } , \\ldots , T _ { s _ j } \\le t _ { j \\ell _ j } \\mid N ( t _ { j - 1 } , t _ j ] = \\ell _ j \\big ) . \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ { - 2 s } \\Gamma ( s ) \\Gamma ( s + k - 1 ) L ( s , f \\times \\overline g ) = : \\Lambda ( s , f \\times \\overline { g } ) = \\Lambda ( 1 - s , f \\times \\overline { g } ) , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\widehat { \\Omega } = \\left \\{ ( x _ 1 , \\ldots , x _ n ) \\in \\R ^ n \\ ; \\big | \\ ; ( | x _ 1 | , \\ldots , | x _ n | ) \\in \\Omega \\right \\} . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} f ( B x + d ) = f ^ * ( x ) + \\langle B ^ * c , x \\rangle + \\langle c , d \\rangle + \\beta , x \\in X , \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\int _ { \\Omega } F ^ { p - 1 } ( \\nabla u ) \\nabla _ { \\xi } F ( \\nabla u ) \\cdot \\nabla \\varphi \\ d x = \\Lambda \\int _ \\Omega | u | ^ { p - 2 } u \\varphi \\ d x \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{gather*} \\int _ { 0 } ^ { T } \\int _ { \\Omega _ { \\varepsilon } } - u _ { \\varepsilon } ( x , t ) v ( x ) \\partial _ { t } c ( t ) + A \\left ( \\frac { x } { \\varepsilon } , \\frac { t } { \\varepsilon ^ { 2 } } \\right ) \\nabla u _ { \\varepsilon } ( x , t ) \\cdot \\nabla v ( x ) c ( t ) d x d t \\\\ = \\int _ { 0 } ^ { T } \\int _ { \\Omega _ { \\varepsilon } } f _ { \\varepsilon } ( x , t ) v ( x ) c ( t ) d x d t , v \\in V _ { \\varepsilon } , c \\in D ( 0 , T ) \\end{gather*}"}
-{"id": "3310.png", "formula": "\\begin{align*} ( K _ { i l } ^ { X _ 2 , 0 } + K _ { i l } ^ { X _ 3 , 0 } ) = & \\ , V _ { X _ 1 } ( - ( d - i + 1 ) A ) + \\ , V _ { X _ 2 } ( - i A - ( l + 1 ) B ) \\\\ & + \\ , V _ { X _ 3 } ( - ( d - l + 1 ) B ) . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} | \\mathcal { Z } _ { K , L } ^ \\epsilon | = F ( \\epsilon , K ) \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\mathcal { K } _ m ' ( \\varphi ) v ( s ) = \\tilde { h } \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ \\rho w _ i \\ ; \\ell ( s , \\zeta _ i ^ j ) v ( \\zeta _ i ^ j ) , \\ ; \\ ; \\ ; s \\in [ a , b ] . \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} C x ^ 2 - A z ^ 2 & = 4 ( C - A ) , \\\\ C y ^ 2 - B z ^ 2 & = 4 ( C - B ) . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} x = \\sum _ { i \\in F } r _ { i } 1 _ { B _ { i } } . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\tilde D ( f ^ j ( x ) ) = a , \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} 2 \\Phi \\left ( \\frac { \\sqrt { C _ 1 n } } { 4 \\sigma } \\right ) - 1 = 1 - P \\left ( | Z | > \\frac { \\sqrt { C _ 1 n } } { 4 \\sigma } \\right ) \\geq 1 - 2 e ^ { - C _ 1 n / ( 3 2 \\sigma ^ 2 ) } \\geq 1 / \\sqrt { 2 } . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\Theta ( j , 2 ^ b , X ) = \\Theta \\bigl ( g ( j , 0 ) , b , X _ 0 \\bigr ) \\oplus \\Theta \\bigl ( g ( j , 1 ) , 2 , X _ 1 \\bigr ) , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\lim _ { E \\searrow E _ { 0 } } \\ ; \\frac { \\ln | \\ln N ( E ) | } { \\ln E } ~ = ~ - \\gamma \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} s x ( y ^ { - 1 } x ) ^ n = x t ( y ^ { - 1 } x ) ^ n = x ( y ^ { - 1 } x ) ^ n t \\ \\mbox { a n d } \\ s y ( x ^ { - 1 } y ) ^ n = y t ' ( x ^ { - 1 } y ) ^ n = y ( x ^ { - 1 } y ) ^ n t ' . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} G ( 2 ) = { \\rm S L } _ n ( 2 ) > 2 ^ { n - 1 } . { \\rm S L } _ { n - 1 } ( 2 ) > { \\rm S L } _ { n - 1 } ( 2 ) \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\int \\frac { r ^ 3 } { ( 1 + r ^ 2 ) ^ { { 2 } } } \\mathrm { d } r = \\frac { 1 } { 2 } \\Big ( \\frac { 1 } { 1 + r ^ 2 } + \\log ( 1 + r ^ 2 ) \\Big ) + C . \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} q _ 1 ( w ) = \\sum _ j p _ j ( w _ 1 ( \\tilde { y } + x ) + w _ 2 ( \\tilde { y } - x ) ) ( \\tilde { y } _ j + x _ j ) \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} | \\Lambda _ \\iota | \\leq C | \\mathcal { S } | + C \\Big ( N ^ { \\varepsilon } \\frac { | \\mathcal { S } | + \\hat { \\Lambda } } { ( N \\eta ) ^ { \\frac 1 3 } } + N ^ { - \\frac { \\gamma } { 4 } } \\hat { \\Lambda } ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\leq C N ^ { - \\varepsilon } \\hat { \\Lambda } \\ , , \\iota = A , B \\ , , \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\frac { d } { d t } | _ { t = 0 } \\Theta ( t ) = d ^ \\nabla \\nabla ( J ^ { - 1 } \\dot { J } ) . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\theta = \\frac { n } { q } - 1 , \\qquad ( 1 + \\theta ) r _ { 1 } = \\frac { 2 n } { n - 2 } , \\qquad ( 1 - \\theta ) r _ { 2 } = 2 . \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p , q } } = ( \\int _ { 0 } ^ 1 [ f ^ * ( s ) s ^ { 1 / p } ] ^ q \\frac { d s } { s } ) ^ { 1 / q } , \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\bar x = ( x ^ 1 , \\ldots , x ^ { n - 1 } , 0 ) . \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} \\mathrm { J } _ 2 = \\mathrm { P G L } ( 2 , \\mathbb { C } ( z _ 1 ) ) \\rtimes \\mathrm { P G L } ( 2 , \\mathbb { C } ) . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\int _ A f ( x ) \\ , d x = \\int _ { ( z , \\sigma ) ( A ) } f ( z , \\sigma ) \\bar J ( z , \\sigma ) \\ , d z \\ , d \\sigma . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} { \\cal E } _ n : = \\bigg \\{ P _ { \\bar { X } } : H ( \\bar { X } ) \\ge \\frac { 1 } { n } \\log | \\tilde { { \\cal M } } _ 0 | + \\frac { | { \\cal X } | \\log ( n + 1 ) } { n } \\bigg \\} . \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\prod _ { j < p } | \\lambda ( D _ { p p } ) - \\lambda ( D _ { j j } ) | = \\left | \\Delta \\left ( \\{ \\lambda _ l \\} _ { l = 1 } ^ k \\cup \\{ x _ j \\pm \\mathrm { i } y _ j \\} _ { j = 1 } ^ { ( N - k ) / 2 } \\right ) \\right | \\prod _ { j = 1 } ^ { ( N - k ) / 2 } \\frac { 1 } { 2 y _ j } \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} d & = 2 c _ i + 2 - e _ i - e _ { n + 1 } \\\\ k & = g - \\sum c _ j , \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} u ( T ) = u _ 0 \\ , , u ' ( T ) = u _ 1 \\ , . \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} & \\left ( \\frac { 1 } { 8 \\omega ^ { 3 / 2 } } \\right ) \\left ( ( \\omega ^ 2 C ^ 2 + \\omega k ) ^ { 1 / 2 } ( 2 \\omega C ^ 3 + k C ) - k ^ 2 \\ln ( \\omega C + ( \\omega ^ 2 C ^ 2 + \\omega k ) ^ { 1 / 2 } ) \\right ) \\\\ & \\qquad = \\frac { \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 ^ 2 } { \\lambda _ 5 } t + k _ 2 , \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\lambda t + c _ 3 ^ { 3 / 2 } \\lambda ^ { - 1 / 2 } ~ \\geq ~ \\begin{cases} \\lambda t & \\textrm { i f } \\lambda \\ge \\widetilde \\lambda , \\\\ c _ 3 ^ { 3 / 2 } \\lambda ^ { - 1 / 2 } & \\textrm { i f } \\lambda \\ge \\widetilde \\lambda , \\end{cases} \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} F _ 1 ( x ) = \\sum _ n \\langle y e _ n , x e _ n \\rangle = \\sum _ n \\langle f _ n , s _ n ( x ) f _ n \\rangle = \\sum _ n s _ n ( x ) = \\| x \\| _ { C _ 1 } = 1 . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} d \\mathfrak { m } _ { H T } : = \\left ( \\frac { 1 } { \\omega _ n } \\int _ { B _ x M } d e t g _ { i j } ( x , v ) d v ^ 1 \\wedge \\dots \\wedge d v ^ n \\right ) d x ^ 1 \\wedge \\dots \\wedge d x ^ n , \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} f _ { \\theta , q , 3 } ( x ) - \\mathbf { x } ^ { ( 0 ) } = \\frac { ( \\theta - 1 ) ( x - \\mathbf { x } ^ { ( 0 ) } ) } { ( x - \\mathbf { x } ^ { ( \\infty ) } ) ^ 3 } g ( x ) \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\sum _ { H ( \\chi ) \\leq \\epsilon } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) = n \\sum _ { m = 0 } ^ { 2 \\epsilon n } \\sum _ { \\substack { m ~ \\chi \\\\ H ( \\chi ) \\leq \\epsilon } } \\hat { 1 _ S } ( \\chi ) ^ 3 \\chi ( f ) . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} & m _ { t } \\coloneqq \\mathbf { E } _ { x _ { 0 } } [ X _ { t } ] = a _ { t } + e ^ { \\beta t } x _ { 0 } , \\\\ & \\Sigma _ { t } \\coloneqq \\mathbf { E } _ { x _ { 0 } } [ ( X _ { t } - m _ { t } ) ( X _ { t } - m _ { t } ) ^ { T } ] = \\sigma ^ { 2 } \\int _ { 0 } ^ { t } e ^ { \\beta ( t - s ) } ( e ^ { \\beta ( t - s ) } ) ^ { T } \\mathrm { d } s , \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} E [ e ^ { i x V ( u ) } ] = \\exp \\Big \\{ \\int _ { ( 0 , u ] \\times \\mathbb { R } } ( e ^ { i x v } - 1 ) \\ , \\rho ( d ( w , v ) ) \\Big \\} , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\eta ^ n ( 0 ) = 0 , \\eta ^ n ( t ) \\mbox { i s n o n - d e c r e a s i n g a n d } \\int _ 0 ^ T { { 1 } } _ { \\{ \\zeta ^ n _ 0 ( t ) > 0 \\} } \\ , \\eta ^ n ( d t ) = 0 . \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} _ { 2 } ( F _ j ) = 1 / 2 . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\sum _ { k = 1 } ^ K \\zeta _ k \\int _ 0 ^ t 1 _ { \\left [ \\frac { q _ { k - 1 } ( s - ) } { \\lambda _ 0 ( s - , X ( s - ) ) } , \\frac { q _ { k } ( s - ) } { \\lambda _ 0 ( s - , X ( s - ) ) } \\right ) } ( \\xi _ { R _ 0 ( s - ) } ) d R _ 0 ( s ) \\\\ R _ 0 ( s ) & = Y \\big ( \\int _ 0 ^ t \\lambda _ 0 ( s , X ( s ) ) d s \\big ) , \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} x _ * ^ { ( j ) } = x ^ { ( j ) } + \\tau _ + ( x ^ { ( i ) } , v ^ { ( i ) } ) v ^ { ( i ) } , v _ * ^ { ( j ) } = v ^ { ( j ) } \\ , . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} h - w + \\frac { \\lambda } { 4 \\sqrt { | h | } } \\textrm { s i g n } ( h ) = 0 \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} J _ { H _ 1 } J _ { H _ 2 } = J _ V J _ { H _ 2 } J _ { H _ 1 } J _ V . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} I _ { u , n } ^ { ( q + 1 ) } = [ a + q d + 1 , n ] \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\sup _ { E \\in [ - \\mathcal { K } , E _ - - N ^ { - \\frac 2 3 + 6 \\varepsilon } ] } \\Im m _ H ( E + \\mathrm { i } \\eta ) = \\sup _ { E \\in [ - \\mathcal { K } , E _ - - N ^ { - \\frac 2 3 + 6 \\varepsilon } ] } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\frac { \\eta } { ( \\lambda _ i - E ) ^ 2 + \\eta ^ 2 } \\geq \\frac { 1 } { N \\eta } \\ , , \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\alpha ( O K ) = \\sup _ { x \\in K ^ { \\circ } } \\| J ( O ) x | | _ { K } \\geq \\| J ( O ) v | | _ { K } \\geq \\sup _ { w \\in K ^ { \\circ } \\cap L } \\langle J ( O ) v , w \\rangle . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{gather*} \\Delta Y ^ { m } _ { l _ 1 , \\ldots , l _ { n - 2 } } ( \\varphi _ 1 , \\ldots , \\varphi _ { n - 2 } , \\phi ) = - l _ { n - 2 } ( l _ { n - 2 } + n - 2 ) Y ^ { m } _ { l _ 1 , \\ldots , l _ { n - 2 } } ( \\varphi _ 1 , \\ldots , \\varphi _ { n - 2 } , \\phi ) . \\end{gather*}"}
-{"id": "6177.png", "formula": "\\begin{align*} u ( x , t ) = M ( u _ 0 ) \\ , P _ t ( x ) - { \\mathcal N } _ { 1 , i } ( u _ 0 ) \\ , ( \\partial _ i P _ t ) ( x ) + o ( t ^ { - 1 / 2 s } ) \\ , P _ t ( x ) \\ , . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} A = \\{ T _ \\Delta ^ { - 1 } y \\ , | \\ , y \\in T _ \\Delta A \\} . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\Vert F \\Vert _ { L ^ { p ( \\cdot , \\cdot ) } ( B _ { i } \\times B _ { i } , \\mu ) } \\leq \\lambda = [ f ] _ { s , p ( \\cdot , \\cdot ) } ( B _ { i } ) , \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} E ^ { ( \\alpha ) } : = \\left \\{ x \\in \\R ^ n \\ , : \\ , \\theta ( E ) ( x ) = \\alpha \\right \\} \\ , . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} G _ 2 ( x ) & = \\sum _ { d \\geq 0 } \\left ( \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ { d + 1 } } F _ T ( x ) + \\sum _ { k = 1 } ^ d \\frac { x ^ { d + 2 } } { ( 1 - x ) ^ k } \\left ( F _ T ( x ) - \\frac { 1 } { 1 - x } \\right ) + x ^ { d + 2 } F _ T ( x ) \\left ( C ( x ) - \\frac { 1 } { 1 - x } \\right ) \\right ) \\\\ & = \\frac { x ^ 2 } { 1 - 2 x } F _ T ( x ) + \\frac { x ^ 3 } { ( 1 - x ) ( 1 - 2 x ) } \\left ( F _ T ( x ) - \\frac { 1 } { 1 - x } \\right ) + \\frac { x ^ 2 } { 1 - x } F _ T ( x ) \\left ( C ( x ) - \\frac { 1 } { 1 - x } \\right ) \\ , . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} G _ { s + 1 } \\left ( \\xi \\right ) = { G } _ { s } ^ { \\prime \\prime } \\left ( \\xi \\right ) - \\psi \\left ( \\xi \\right ) G _ { s } \\left ( \\xi \\right ) \\ \\left ( { s = 0 , 1 , 2 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\partial _ { \\mu _ n ' } I _ \\lambda ( \\zeta ' , \\mu ' ) & = D \\bar J _ \\lambda ( V + \\phi ) [ \\partial _ { \\mu _ n ' } V + \\partial _ { \\mu _ n ' } \\phi ] \\\\ & = - \\sum _ { i , j } c _ { i j } \\int _ { \\Omega _ \\varepsilon } w _ { \\mu _ i ^ { \\prime } , \\zeta _ i ^ { \\prime } } ^ 4 \\ , z _ { i j } \\ , ( \\partial _ { \\mu _ n ' } V + \\partial _ { \\mu _ n ' } \\phi ) . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\overline { \\partial _ { 0 } M \\left ( \\partial _ { 0 } ^ { - 1 } \\right ) + A } \\ : U = F , \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} ( M _ r { V } ) _ S = ( P Q + P ' Q ' ) ( - 1 ) ^ { r - | S | } , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} x _ d & \\equiv x _ { \\infty } \\underset { j = 1 } { \\stackrel { d - 1 } { \\textstyle { \\prod } } } x _ j ^ { \\frac { \\ell _ j - 1 } { 2 } } \\mod { [ F , F ] } , \\\\ \\eta _ i & \\equiv x _ 0 ^ { \\mathrm { l k } ( \\ell _ i , \\ell _ 0 ) } x _ { \\infty } ^ { \\mathrm { l k } ( \\ell _ i , \\ell _ d ) } \\underset { j = 1 } { \\stackrel { d - 1 } { \\textstyle { \\prod } } } x _ j ^ { \\mathrm { l k } ( \\ell _ i , \\ell _ j ) + \\frac { \\ell _ j - 1 } { 2 } \\mathrm { l k } ( \\ell _ i , \\ell _ d ) } \\mod { [ F , F ] } , \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} q n + r = p r x + q ( n - r y ) \\in \\{ p m + q n \\ , : \\ , m , n \\in \\mathbb N \\} . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} G _ { \\lambda \\pm i \\epsilon } ( x ) & = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { i x \\xi } } { \\xi - ( \\lambda \\pm i \\epsilon ) } ~ d \\xi - \\widetilde { G } _ { \\lambda \\pm i \\epsilon } ( x ) \\\\ & = \\pm i e ^ { \\mp \\epsilon x } e ^ { i \\lambda x } \\chi _ { \\real ^ { \\pm } } ( x ) - \\widetilde { G } _ { \\lambda \\pm i \\epsilon } ( x ) \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} & \\sum _ { l = 0 } ^ k a _ l z ^ l \\sum _ { m = 0 } ^ { n - 2 } ( - 1 ) ^ { n - m - 2 } ( m + 2 ) ( m + 1 ) e _ { n - m - 2 } z ^ m \\\\ & + \\sum _ { l = 0 } ^ { k - 1 } b _ l z ^ l \\sum _ { m = 0 } ^ { n - 1 } ( - 1 ) ^ { n - m - 1 } ( m + 1 ) e _ { n - m - 1 } z ^ m \\\\ & + \\sum _ { l = 0 } ^ { k - 2 } c _ l z ^ l \\sum _ { m = 0 } ^ n ( - 1 ) ^ { n - m } e _ { n - m } z ^ m = 0 . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} I _ { \\delta , \\theta } & \\geq \\frac { \\rho } { 2 } \\Big \\{ 3 ( 1 - \\theta ) ( 1 - \\nu ) \\ln T _ { t r } + ( 1 - \\theta ) \\nu \\ln \\det \\Theta + 3 \\theta \\ln T _ { \\delta } + \\delta \\ln T _ { \\theta } - ( 3 + \\delta ) \\ln T _ { \\delta } \\Big \\} \\cr & = \\frac { \\rho } { 2 } \\Big \\{ ~ ( 1 - \\theta ) \\big ( 3 ( 1 - \\nu ) \\ln T _ { t r } + \\nu \\ln \\det \\Theta - 3 \\ln T _ { \\delta } \\big ) + \\delta \\ln T _ { \\theta } - \\delta \\ln T _ { \\delta } \\Big \\} . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} & \\sum _ { 1 \\leq i _ 1 + \\cdots + i _ n } \\alpha _ { i _ 1 \\cdots i _ n } y _ 1 ^ { i _ 1 } \\cdots y _ n ^ { i _ n } + \\sum _ { 1 \\leq j _ 1 + \\cdots + j _ n , 1 \\leq l _ 1 + \\cdots + l _ n } \\beta _ { j _ 1 \\cdots j _ n l _ 1 \\cdots l _ n } y _ 1 ^ { j _ 1 } \\cdots y _ n ^ { j _ n } x _ 1 ^ { l _ 1 } \\cdots x _ n ^ { l _ n } \\\\ & + \\sum _ { 1 \\leq m _ 1 + \\cdots + m _ n } \\gamma _ { m _ 1 \\cdots m _ n } x _ 1 ^ { m _ 1 } \\cdots x _ n ^ { m _ n } + \\delta 1 = 0 , \\\\ \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} & \\left \\| f l \\left ( B ^ { ( k + 1 ) } \\right ) - B ^ { ( k + 1 ) } \\right \\| _ { 1 , 2 } \\le \\left \\| f l \\left ( B ^ { ( 1 ) } \\right ) - B ^ { ( 1 ) } \\right \\| _ { 1 , 2 } \\ , ( 1 + u ) ^ { k } \\\\ & + u \\ , \\sum _ { i = 1 } ^ { k } \\left \\| B ^ { ( i ) } \\right \\| _ { 1 , 2 } \\ , ( 1 + u ) ^ { k - i } + 3 \\ , u \\ , \\sum _ { i = 1 } ^ { k } \\left \\| B ^ { ( i ) } \\right \\| _ { 1 , 2 } \\left \\| L _ i ^ T \\right \\| _ 1 \\ , ( 1 + u ) ^ { k - i } + O ( u ^ 2 ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} | \\hat { 1 _ S } ( \\chi ) | \\leq e ^ { O ( m ^ { 3 / 2 } / n ^ { 1 / 2 } + m ^ { 1 / 2 } ) } ( 1 - \\delta ) ^ { m / 2 } \\binom { n } { m } ^ { - 1 / 2 } \\frac { n ! } { n ^ n } . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} a _ { T - 1 , T } = \\left ( \\prod _ { j = 0 } ^ { T - 1 } a _ j \\right ) ^ { - 1 } \\sum _ { k = 1 } ^ { T - 1 } b _ k \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} 0 ~ = ~ \\int _ M \\frac { F ^ H _ x ( y ) } { F ^ H _ x ( x ) } \\ , d \\mu ( y ) ~ \\ge ~ \\int _ 0 ^ D | \\partial B ( x , r ) | \\ , \\cos ( \\sqrt \\lambda _ 1 r ) \\ , d r \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} & x _ 1 ^ { p ^ 2 } = x _ 2 ^ { p ^ 2 } = x _ 3 ^ { p ^ 2 } = x _ 4 ^ { p ^ 2 } = x _ 5 ^ p = 1 , \\\\ & [ x _ 1 , x _ 2 ] = x _ 1 ^ p , \\ , [ x _ 1 , x _ 3 ] = x _ 3 ^ p , \\ , [ x _ 1 , x _ 4 ] = 1 , \\ , [ x _ 1 , x _ 5 ] = x _ 1 ^ p , \\ , [ x _ 2 , x _ 3 ] = x _ 2 ^ p , \\\\ & [ x _ 2 , x _ 4 ] = 1 , \\ , [ x _ 2 , x _ 5 ] = x _ 4 ^ p , \\ , [ x _ 3 , x _ 4 ] = 1 , \\ , [ x _ 3 , x _ 5 ] = x _ 4 ^ p , \\ , [ x _ 4 , x _ 5 ] = 1 . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\frac { P ( u + m ) } { P ( u ) } \\cdot \\frac { \\alpha - u } { \\alpha + u - l + m } = \\frac { Q ( u + m ) } { Q ( u ) } \\cdot \\frac { \\beta - u } { \\beta + u - l + m } . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\begin{aligned} k = \\frac { \\lambda _ 2 } { \\lambda _ 4 } \\left ( \\lambda _ 2 \\lambda _ 4 - \\lambda _ 3 \\lambda _ 5 \\right ) = 0 \\\\ \\ell = \\frac { \\lambda _ 4 } { \\lambda _ 2 } \\left ( \\lambda _ 2 \\lambda _ 4 - \\lambda _ 3 \\lambda _ 5 \\right ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\alpha ( g ^ H ) : = \\int _ H \\phi ( h ( g ) ) \\ > d \\omega _ H ( h ) \\quad ( g \\in G ) . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\ell = \\begin{cases} 0 \\ ; \\ ; P ( \\alpha ) \\neq 0 , \\\\ \\min _ { k \\geq 1 } \\{ P ^ { ( k - 1 ) } ( \\alpha - ( k - 1 ) m ) = 0 , \\ , P ^ { ( k ) } ( \\alpha - k m ) \\neq 0 \\} \\ ; \\end{cases} \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} f ( t ) : = \\begin{cases} \\cos \\sqrt { k } t - \\dfrac { \\theta } { \\sqrt { k } } \\sin \\sqrt { k } t , & k > 0 , \\\\ 1 - \\theta t , & k = 0 , \\\\ \\cosh \\sqrt { - k } t - \\dfrac { \\theta } { \\sqrt { - k } } \\sinh \\sqrt { - k } t , & k < 0 , \\end{cases} t \\geq 0 . \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\| P f \\| _ { Z ^ { \\wedge } } \\leq \\inf \\{ \\sum _ { k = 1 } ^ { n } \\| P f _ k \\| _ Z \\colon f = \\sum _ { k = 1 } ^ { n } f _ k , f _ k \\in Z , n \\in \\N \\} \\leq \\| P \\| _ { Z \\to Z } \\| f \\| _ { Z ^ { \\wedge } } \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\Gamma _ m = m - \\Gamma . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} f = ( x _ { 1 } x _ { 3 } + x _ { 2 } ^ 2 ) x _ { 4 } + f _ 3 ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\widehat { T } ( z _ { 0 } ) \\widehat { T } ( z _ { 1 } ) = e ^ { \\tfrac { i } { \\hbar } \\sigma ( z _ 0 , z _ 1 ) } \\widehat { T } ( z _ { 1 } ) \\widehat { T } ( z _ { 0 } ) \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} F ( a , b ) = - ( - 1 ) ^ { | a | | b | } F ( b , a ) . \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\chi ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) = 2 - h ^ 1 ( E _ { C , A } \\otimes { E _ { C , A } } ^ * ) . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} W _ m ^ { ( 1 ) } : = { \\rm I d } + w _ m ^ { ( 1 ) } ( x , D ) , w _ m ^ { ( 1 ) } ( x , \\xi ) \\in S ^ { - \\frac { m } { 2 } + \\frac 1 2 } \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} S _ 1 ( n + 1 , k ) = S _ 1 ( n , k - 1 ) - n S _ 1 ( n , k ) , \\ , \\ , ( 1 \\leq k \\leq n ) , ( \\textnormal { s e e } \\ , \\ , [ 6 ] ) . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\widehat D ( v ) = A ^ { \\rm f l a t } ( v ) - \\theta \\mathbf { e } _ \\rho , \\theta = \\delta ( k M ) ^ 2 N ^ { 1 / 3 } + \\frac { 2 ( v - k M ) N ^ { 2 / 3 } } { \\rho \\chi ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} E | \\mu ^ N ( t ) - m ( t ) | + \\frac C N \\sum _ { i = 1 } ^ N E | X ^ N _ i ( t ) - Y ^ N _ i ( t ) | \\leq \\frac { C } { \\sqrt { N } } + 2 K _ 1 \\int _ 0 ^ t e ^ { 2 K ( t - s ) } \\frac { C } { \\sqrt { N } } d s \\leq \\frac { C } { \\sqrt { N } } . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} K _ M ( u \\wedge v ) = \\frac { \\langle R ( u , v ) u , v \\rangle } { | | u \\wedge v | | ^ 2 } = \\frac { \\langle R ( u , v ) u , v \\rangle } { | | u | | ^ 2 | | v | | ^ 2 - \\langle u , v \\rangle ^ 2 } . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} S p _ 2 ( \\mathbf { Z } ) & = \\Big \\{ g \\in G L _ 4 ( \\mathbf { Z } ) \\Big | \\ ; ^ t g J g = J = \\begin{pmatrix} & - I \\\\ I & \\end{pmatrix} \\Big \\} , \\\\ \\bold H _ 2 & = \\{ \\tau \\in G L _ 2 ( \\bold C ) \\mid \\ ^ t \\tau = \\tau , \\Im ( \\tau ) > 0 \\} . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} d \\mu ( \\beta ) : = \\prod ^ \\infty _ { j = 1 } d \\mu _ { b _ j } . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d F \\thinspace \\left [ \\psi _ { s , F } ( - v ) \\psi _ { s , F } ( - v ' ) - \\psi _ { s , F } ( v ) \\psi _ { s , F } ( v ' ) \\right ] = v ^ { - 1 } \\delta ( v - v ' ) \\ , . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} & a ^ { 1 1 } _ { 1 1 } = \\overline { a ^ { 1 1 } _ { 1 1 } } , a ^ { 1 1 } _ { 1 2 } = \\overline { a ^ { 1 1 } _ { 2 1 } } , a ^ { 1 1 } _ { 2 2 } = \\overline { a ^ { 1 1 } _ { 2 2 } } , \\\\ & a ^ { 1 2 } _ { 1 1 } = \\overline { a ^ { 1 2 } _ { 1 1 } } , a ^ { 1 2 } _ { 1 2 } = \\overline { a ^ { 2 1 } _ { 2 1 } } , a ^ { 2 2 } _ { 1 1 } = \\overline { a ^ { 2 2 } _ { 1 1 } } , \\\\ & a ^ { 2 2 } _ { 1 1 } = \\overline { a ^ { 2 2 } _ { 1 1 } } , a ^ { 2 2 } _ { 1 2 } = \\overline { a ^ { 2 2 } _ { 2 1 } } , a ^ { 2 2 } _ { 2 2 } = \\overline { a ^ { 2 2 } _ { 2 2 } } . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\phi ( x ) = \\sum _ { i \\in F } r _ { i } \\phi ( 1 _ { B _ { i } } ) = \\sum _ { i \\in F } \\sum _ { k \\in F _ { i } } r _ { i } 1 _ { B _ { i k } } p _ { \\delta _ { i k } } = \\sum _ { \\delta \\in \\Delta } \\sum _ { i \\in F _ { \\delta } } r _ { i } 1 _ { B _ { i , k ( \\delta ) } } p _ { \\delta } = 0 . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\Omega ^ { - 1 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ i \\psi _ i ^ T \\Omega ^ { - 1 } \\bar { \\psi } \\bar { \\psi } ^ T \\Omega ^ { - 1 } \\psi _ i , \\textrm { o f o r d e r } O _ p ( n ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\mathbb { P } \\left ( | Z _ { T + h } | = P _ Y , | Z _ T | = P _ Y \\right ) , \\lim _ { T \\to \\infty } \\mathbb { E } Y _ T ^ 2 Y _ { T + h } ^ 2 , \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\displaystyle \\sum _ { i ' = 1 } ^ { q } z _ { i l } \\overline { a _ { k u } ^ { j l } \\left ( Z _ { i ' } \\right ) } + \\sum _ { i ' = 1 } ^ { q } \\displaystyle \\sum _ { l = 1 } ^ { p - q } \\overline { z } _ { j l } a _ { u k } ^ { i l } \\left ( Z _ { i ' } \\right ) = 0 , \\mbox { f o r a l l $ i , j , k , u = 1 , \\dots , q $ , } \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\begin{pmatrix} F _ { 1 } \\left ( Z , W \\right ) \\\\ F _ { 2 } \\left ( Z , W \\right ) \\end{pmatrix} = \\begin{pmatrix} Z \\\\ 0 \\end{pmatrix} + A \\otimes W + \\mbox { O } \\left ( 3 \\right ) , \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} \\lim _ { m , \\mathcal { V } } \\ , \\left ( \\lim _ { n , \\mathcal { U } } \\| x _ n + y _ m \\| \\right ) = \\lim _ { n , \\mathcal { U } } \\ , \\left ( \\lim _ { m , \\mathcal { V } } \\| x _ n + y _ m \\| \\right ) . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} ( ( W ( \\psi , \\phi ) | W ( \\widehat { T } ( z _ { \\lambda } ) \\phi , \\phi ) ) ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } ( \\psi | \\widehat { T } ( z _ { \\lambda } ) \\phi ) \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} f _ { 2 n } ( t ) = d _ n D _ { n } e ^ { i \\zeta _ { n } t } + \\overline { d _ n D _ { n } } e ^ { - i \\overline { \\zeta _ { n } } t } + c _ n C _ { n } e ^ { i \\omega _ { n } t } + \\overline { c _ n C _ { n } } e ^ { - i \\overline { \\omega _ { n } } t } + E _ n e ^ { - \\eta t } \\ , . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} \\delta ^ \\prime ( h ) = \\mathop { \\mathrm { t r } } \\ , [ B ( \\Psi ( q , h ) ) ] \\ , \\delta ( h ) \\ , , \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\overline { Z _ { A } ^ { A ' } } = \\sum _ { B ' = 0 ' , 1 ' } \\overline { Z _ { A B ' } } \\varepsilon ^ { B ' A ' } = - \\sum _ { B ' = 0 ' , 1 ' } Z ^ { A B ' } \\varepsilon _ { B ' A ' } = - Z ^ { A } _ { A ' } \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} \\cos \\omega _ { 1 } T _ { 0 } \\cos \\omega _ { 2 } \\left ( T - T _ { 0 } \\right ) = 1 . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\lambda _ n = - \\tfrac { 1 } { 2 } n ( n - 1 ) \\sigma '' ( z ) - n \\tau ' ( z ) , \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\kappa _ { F } ( A , \\lambda b ) = \\left \\| K \\right \\| _ 2 . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\mathcal E ^ n ( f , g ) : = \\frac { 1 } { 2 } \\sum _ { x , y \\in \\Gamma ^ n } ( f ( x ) - f ( y ) ) ( g ( x ) - g ( y ) ) \\ , C ^ n _ { x y } \\qquad \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} \\rho = \\frac { \\| D \\| _ { 1 , \\infty } } { \\| A \\| _ { 1 , \\infty } } , \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\widehat { N } _ { 0 , Q } = \\sum _ { j = 1 } ^ { p } \\mathcal { P } _ j ^ * \\widehat { N } _ Q \\mathcal { P } _ j , \\widehat { N } _ { * , Q } = \\sum _ { { 1 \\le l , j \\le p : \\ ; l \\ne j } } \\mathcal { P } _ l ^ * \\widehat { N } _ Q \\mathcal { P } _ j . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\begin{cases} \\Lambda _ 2 & \\\\ \\Lambda _ 2 \\setminus \\{ - \\infty \\} & \\end{cases} \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\varphi = \\Big ( A _ 0 \\circ \\sigma _ n \\circ A _ 0 ^ { - 1 } \\Big ) \\circ \\Big ( A _ 1 \\circ \\sigma _ n \\circ A _ 1 ^ { - 1 } \\Big ) \\circ \\ldots \\circ \\Big ( A _ k \\circ \\sigma _ n \\circ A _ k ^ { - 1 } \\Big ) \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} s _ { i , m } = & \\frac { 1 } { 4 } \\int _ 0 ^ { \\Delta t } \\left ( 1 - 2 \\left ( 1 - \\frac { | \\tau | } { \\Delta t } \\right ) ^ 2 \\right ) \\frac { \\partial ^ 3 \\nabla _ { y _ i } \\frak u _ i } { \\partial t ^ 3 } ( t _ m + \\tau ) d \\tau - \\frac { 1 } { 4 } \\int ^ 0 _ { - \\Delta t } \\left ( 1 - 2 \\left ( 1 - \\frac { | \\tau | } { \\Delta t } \\right ) ^ 2 \\right ) \\frac { \\partial ^ 3 \\nabla _ { y _ i } \\frak u _ i } { \\partial t ^ 3 } ( t _ m + \\tau ) d \\tau . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\delta \\deg h _ { n - \\mu } + ( \\deg g _ { \\mu } ) d ^ { n - \\mu } & = \\delta ( d ^ { n - \\mu } - \\delta ^ i T ^ i ) + ( d ^ { \\mu } - \\delta ) d ^ { n - \\mu } \\\\ & = d ^ n - \\delta ^ { i + 1 } T ^ i = \\deg g _ n . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{gather*} J = \\det ( \\nabla \\eta ) . \\end{gather*}"}
-{"id": "9323.png", "formula": "\\begin{align*} \\frac { d x _ i ( t ) } { d t } = - x _ i ( t ) + \\alpha f ( x _ { i - 1 } ( t ) ) + a f ( x _ { i } ( t ) ) + ( \\beta + s ) f ( x _ { i + 1 } ( t ) ) , \\ , \\ , i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} & \\Phi _ 1 ^ c = - \\Lambda _ A + ( F ' _ A ( \\omega _ B ) - 1 ) \\Lambda _ B + \\frac { 1 } { 2 } F '' _ A ( \\omega _ B ) \\Lambda _ B ^ 2 + O ( \\Lambda _ B ^ 3 ) \\ , , \\\\ & \\Phi _ 2 ^ c = - \\Lambda _ B + ( F ' _ B ( \\omega _ A ) - 1 ) \\Lambda _ A + \\frac { 1 } { 2 } F '' _ B ( \\omega _ A ) \\Lambda _ A ^ 2 + O ( \\Lambda _ A ^ 3 ) \\ , . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} H = R ( Q _ \\infty ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} \\Im m _ { \\mu _ \\alpha \\boxplus \\mu _ \\beta } ( z ) \\sim \\Im \\omega _ \\alpha ( z ) \\sim \\Im \\omega _ \\beta ( z ) \\sim \\begin{cases} \\sqrt { \\kappa + \\eta } \\ , , \\qquad & \\textrm { i f } E \\ge E _ - \\ , , \\\\ \\frac { \\eta } { \\sqrt { \\kappa + \\eta } } \\ , , & \\textrm { i f } E < E _ - \\ , , \\end{cases} \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} E A : = E D i v ( \\{ D A ^ r _ { i } , D A ^ g _ { i } , D A ^ b _ { i } \\} _ { i = 1 } ^ { 4 } ) , \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\overline \\lambda _ 0 ^ { } = \\sum _ { \\ell = 1 } ^ K \\overline \\lambda _ \\ell . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } \\epsilon ^ { - 1 / 2 } \\left ( A _ 2 ( \\epsilon x ) - A _ 2 ( 0 ) \\right ) \\stackrel { d i s t . } { = } \\sqrt { 2 } B ( x ) \\ , . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\left ( \\sum _ { r \\geq 0 } ( - 1 ) ^ r e _ r ( \\mathbf { x } ^ { ( \\mathbf { 1 } ) } ) \\right ) \\left ( \\sum _ { s \\geq 0 } ( - 1 ) ^ s e _ s ( \\mathbf { y } ^ { ( \\mathbf { 1 } ) } ) \\right ) = \\left ( \\sum _ { r \\geq 0 } ( - 1 ) ^ r e _ r ( \\mathbf { x } ^ { ( \\mathbf { 1 } ) } , \\mathbf { y } ^ { ( \\mathbf { 1 } ) } ) \\right ) \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} = \\sum _ { r _ 1 , \\dots , r _ { m - k + 1 } } \\frac { m ! } { r _ 1 ! \\cdots r _ { m - k + 1 } ! } \\left ( \\frac { x _ 1 } { 1 ! } \\right ) ^ { r _ 1 } \\cdots \\left ( \\frac { x _ { m - k + 1 } } { ( m - k + 1 ) ! } \\right ) ^ { r _ { m - k + 1 } } \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} V \\left ( \\xi \\right ) = 1 + O ( \\xi ^ { 1 / 2 - \\epsilon } ) \\ ; \\mbox { f o r } \\ ; 0 < \\xi < 1 \\mbox { a n d } V ^ { ( j ) } \\left ( \\xi \\right ) = O ( e ^ { - \\xi } ) \\ ; \\mbox { f o r } \\ ; \\xi > 0 , j \\geq 0 . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} & ( y ^ 5 - 5 y ^ 4 + 1 4 y ^ 3 - 2 3 y ^ 2 + 2 3 y - 1 1 ) ( y ^ { n - 5 } - y ^ { n - 6 } + y ^ { n - 7 } ) \\\\ & \\qquad + ( y ^ 4 - 4 y ^ 3 + 1 0 y ^ 2 - 1 3 y + 8 ) y ^ { n - 6 } \\\\ & = ( y ^ 7 - 6 y ^ 6 + 2 1 y ^ 5 - 4 6 y ^ 4 + 7 0 y ^ 3 - 7 0 y ^ 2 + 4 2 y - 1 1 ) y ^ { n - 7 } < ( y + 1 ) y ^ { n - 3 } ( y - 1 ) ( y - 2 ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} q _ 1 ( 0 ) + q _ 2 ( 0 ) = 2 \\nabla p ( 0 ) \\cdot \\tilde { y } \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} w _ t = L _ 2 ( w ) : = - ( - \\Delta ) ^ s w - \\frac 1 { 2 s } x \\cdot \\nabla w \\ , . \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\alpha _ 1 = \\rho ( \\beta _ 2 ) , \\ \\alpha _ 2 = \\rho ( \\beta _ 1 ) , \\ \\beta _ 1 , \\ \\beta _ 2 \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{gather*} \\lim _ { N \\rightarrow \\infty } { \\prod _ { i = 1 } ^ N \\prod _ { j = 0 } ^ { \\theta - 1 } { \\big ( 1 - q ^ { - z + \\lambda _ { N - i + 1 } ( N ) + \\theta ( i - 1 ) + j } \\big ) } } \\\\ \\qquad { } = \\prod _ { i = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\theta - 1 } { \\big ( 1 - q ^ { - z + \\nu _ i + \\theta ( i - 1 ) + j } \\big ) } = \\prod _ { i = 1 } ^ { \\infty } { \\frac { \\big ( q ^ { - z + \\nu _ i } t ^ { i - 1 } ; q \\big ) _ { \\infty } } { \\big ( q ^ { - z + \\nu _ i } t ^ i ; q \\big ) _ { \\infty } } } . \\end{gather*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\nu _ 0 = n ^ { - 1 / 2 } ( 1 , \\ldots , 1 ) . \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} f ( \\{ a , b \\} ) & = g ( a ) f ( b ) - ( - 1 ) ^ { ( | a | + p ) | b | } f ( b ) g ( a ) , \\\\ g ( a b ) & = f ( a ) g ( b ) + ( - 1 ) ^ { | a | | b | } f ( b ) g ( a ) , \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} G ( 1 - z ) : = K ( z ) \\ , G ( z ) \\ , , \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\left ( a + b A ^ 2 \\right ) \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 = \\int _ { \\mathbb { R } ^ N } f ( u ) u . \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ h V _ i \\subseteq [ 0 , ( h - 1 ) v ^ * ] . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\langle [ v = 0 ] , g \\rangle = \\int _ { \\{ v = 0 \\} } g . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 } S _ { 2 , \\lambda } ( n + 1 , k | x ) = ( x + k ) S _ 2 ( n , k | x ) + S _ 2 ( n , k - 1 | x ) . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} ( x y ) . v = \\alpha ( x , y | z ) x . ( y . v ) , v \\in V _ z \\ . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\hat { \\mathbb { E } } _ { \\tau + } [ X ] : = \\mathbb { L } ^ 1 \\lim _ { n \\rightarrow \\infty } \\hat { \\mathbb { E } } _ { \\tau _ n + } [ X ] . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} m ( G , k ) = & m ( G - v , k ) + \\sum \\limits _ { w \\in N ( v ) } m ( G - v - w , k - 1 ) , \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\left ( \\lambda + A \\right ) ^ { - 1 } T _ { \\alpha } x & = \\lim \\limits _ { t \\rightarrow 0 + } - t ^ { 1 - 2 \\alpha } ( \\lambda + A ) ^ { - 1 } U ' ( t ) x \\\\ & = T _ { \\alpha } \\left ( \\lambda + A \\right ) ^ { - 1 } x = c _ \\alpha J ^ \\alpha _ A \\left ( \\lambda + A \\right ) ^ { - 1 } x . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} & \\varphi : H _ { d R } ^ 2 ( \\widetilde { X } ) ( \\chi ) \\ni a \\mapsto \\varphi ( a ) = \\begin{pmatrix} \\ < \\mu ( B _ 1 ) , a \\ > \\\\ \\ < \\mu ( B _ 2 ) , a \\ > \\end{pmatrix} \\in \\bold C ^ 2 , \\\\ & \\varphi ' : H _ { d R } ^ 2 ( \\widetilde { X } ) ( \\overline { \\chi } ) \\ni a ' \\mapsto \\varphi ( a ) = \\begin{pmatrix} \\ < \\mu ( B _ 1 ) , a ' \\ > \\\\ \\ < \\mu ( B _ 2 ) , a ' \\ > \\end{pmatrix} \\in \\bold C ^ 2 . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} E = \\left \\{ \\left ( q ^ i , p _ i ; \\frac { \\partial F } { \\partial p _ i } , - \\frac { \\partial F } { \\partial q ^ i } \\right ) \\in T T ^ * Q : \\frac { \\partial F } { \\partial \\lambda ^ a } ( q , p , \\lambda ) = 0 \\right \\} . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} L _ { \\widehat D ( v ) \\to E _ N ( w ) } \\stackrel { d } { = } L _ { ( 0 , 0 ) \\to ( \\gamma ^ 2 n + \\beta _ 1 u ( v ) n ^ { 2 / 3 } , n ) } . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} u ' ( t ) \\geq \\beta \\left ( - \\frac { \\alpha } { \\beta } + u ( t ) ^ { 2 } \\right ) \\geq \\beta \\left ( - \\frac { 1 } { 4 } u ( t ) ^ { 2 } + u ( t ) ^ { 2 } \\right ) = \\frac { 3 \\beta } { 4 } u ( t ) ^ { 2 } \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} A ( \\zeta , \\widehat { \\Lambda } ) = ( A _ 0 ( \\zeta , \\widehat { \\Lambda } ) , A _ 1 ( \\zeta , \\widehat { \\Lambda } ) , \\ldots , A _ k ( \\zeta , \\widehat { \\Lambda } ) ) ) \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} R _ h ( x _ 0 , \\dots , x _ { \\rho ( X ) } , y _ 0 , \\dots , y _ { \\rho ( X ) } ) = \\big ( Q ( x _ 0 + y _ 0 , \\dots , x _ { \\rho ( X ) } + y _ { \\rho ( X ) } ) - Q ( y _ 0 , \\dots , y _ { \\rho ( X ) } ) \\big ) ^ h . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\Gamma = \\left ( \\begin{array} { c c } \\Gamma _ { 1 1 } & \\Gamma _ { 1 2 } \\\\ \\Gamma _ { 2 1 } & \\Gamma _ { 2 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} C _ 6 \\wr S _ n = H _ 0 > H _ 1 > \\cdots > H _ t = 2 . S _ n = Z . S _ n \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} X _ \\alpha ^ { t , x } ( s ) = x + \\int _ t ^ s \\int _ U f ( r ^ - , X _ \\alpha ^ { t , x } ( r ^ - ) , u , \\alpha ( r ) , m ( r ) ) \\N ( d r , d u ) \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\widehat { \\mathcal { A } } = b ( \\mathbf { D } ) ^ * g ( \\mathbf { x } ) b ( \\mathbf { D } ) . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} & Y _ { 2 h } ^ { ( a , b ) } = \\frac { b + \\sqrt { a b } } { \\sqrt { a b } } \\left ( \\frac { r + \\sqrt { a b } } { 2 } \\right ) ^ { 2 h } + \\frac { - b + \\sqrt { a b } } { \\sqrt { a b } } \\left ( \\frac { r - \\sqrt { a b } } { 2 } \\right ) ^ { 2 h } , \\\\ & Y _ { 2 m } ^ { ( b , d ) } = \\frac { \\varepsilon b + \\sqrt { b d } } { \\sqrt { b d } } \\left ( \\frac { y + \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } + \\frac { - \\varepsilon b + \\sqrt { b d } } { \\sqrt { b d } } \\left ( \\frac { y - \\sqrt { b d } } { 2 } \\right ) ^ { 2 m } . \\\\ \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} \\pi ( z ) = \\tfrac { 1 } { 2 } [ \\sigma ' ( z ) - \\tilde { \\tau } ( z ) ] \\pm \\Bigl \\{ \\tfrac { 1 } { 4 } [ \\sigma ' ( z ) - \\tilde { \\tau } ( z ) ] ^ 2 - \\tilde { \\sigma } ( z ) + g ( z ) \\sigma ( z ) \\Bigr \\} ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\tilde { A } \\left ( W ' , Z ' \\right ) \\otimes \\left ( W ' - \\overline { W ' } ^ { t } \\right ) = 2 \\sqrt { - 1 } \\left ( V \\otimes \\left ( Z ' - A \\otimes W \\right ) \\right ) \\overline { V \\otimes \\left ( Z ' - A \\otimes W \\right ) } ^ { t } , \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} \\left ( \\gamma ^ * ( \\Phi _ - ) ^ \\lambda \\mathcal { Q } _ - \\omega \\right ) _ { ( x , \\xi ) } ( \\eta _ 1 , \\dots , \\eta _ m ) = ( \\Phi _ - ) ^ \\lambda _ { ( x , \\xi ) } \\left ( ( T _ \\gamma ) ^ { \\lambda + m } U _ \\gamma ^ * \\omega \\right ) _ { B _ - ( x , \\xi ) } ( { \\tau _ - } \\eta _ 1 , \\dots , { \\tau _ - } \\eta _ m ) \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\tilde { \\phi } _ s ( x , \\tau ) = \\frac { N _ 1 ( x , \\tau ) } { N _ 2 ( x , \\tau ) } \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} \\mathcal { H } _ { d , h } : = X _ 1 ^ 2 + \\cdots + X _ { d - 1 } ^ 2 = X _ d ^ 2 + h \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\hat { \\theta } ^ M : \\sum _ { i = 1 } ^ n \\psi ( x _ i , \\theta ) = 0 , \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda } \\big ( q ^ r t ^ { N - 1 } , t ^ { N - 2 } , \\dots , t , 1 ; q , t \\big ) } { P _ { \\lambda } \\big ( t ^ { N - 1 } , \\dots , t , 1 ; q , t \\big ) } = \\frac { ( q ; q ) _ r } { \\big ( q ^ { \\theta N } ; q \\big ) _ r } \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\oint _ C { \\frac { y ^ { - ( r + 1 ) } { \\rm d } y } { \\prod \\limits _ { i = 1 } ^ N { \\prod \\limits _ { j = 0 } ^ { \\theta - 1 } { ( 1 - q ^ { \\lambda _ i + \\theta ( N - i ) + j } y ) } } } } . \\end{gather*}"}
-{"id": "169.png", "formula": "\\begin{align*} T ( z , t ) = P + \\Phi ( T ( z , t ) ) . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} v ( X ) & = a _ 1 X ^ { d _ 1 } + \\ldots + a _ t X ^ { d _ t } + b , \\\\ w _ i ( X ) & = a _ 1 ^ { r ^ { - i } } X ^ { d _ 1 } + \\ldots + a _ t ^ { r ^ { - i } } X ^ { d _ t } + b ^ { r ^ { - i } } , \\ : i \\geq 1 . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\# = 0 \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{gather*} K = \\sum _ { i = - \\mu + 2 } ^ { 3 \\mu } K ^ { ( i ) } , \\end{gather*}"}
-{"id": "4994.png", "formula": "\\begin{align*} m = \\inf _ { x \\in \\Omega _ + } ( g _ { \\varepsilon , \\gamma } ( x ) - u ( x ) ) = \\inf _ { x \\in \\partial \\Omega _ + } ( g _ { \\varepsilon , \\gamma } ( x ) - u ( x ) ) . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\Delta _ { h _ k e _ n } e _ n = \\sum _ { \\lambda \\vdash n } h _ k [ B _ { \\lambda } ] e _ n [ B _ { \\lambda } ] \\frac { M B _ { \\lambda } \\Pi _ { \\lambda } \\widetilde { H } _ { \\lambda } } { w _ { \\lambda } } . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} X = \\left ( l _ { 1 } H ^ { , i } + l _ { 2 } X _ { I J } + l _ { 3 } A ^ { i } \\right ) \\partial _ { i } . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} \\Lambda ^ { a , p } _ t g ( x ) : = \\int _ U [ g ( x + f ( t , x , u , a , p ) ) - g ( x ) ] \\nu ( d u ) \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} \\begin{aligned} Q _ { j } ( \\vec { 1 } ) & = ( 2 m ) ^ { j } P _ { j } ( \\vec { 1 } ) \\\\ \\frac { \\partial Q _ j } { \\partial w _ 1 } ( \\vec { 1 } ) & = ( 2 m ) ^ { j - 1 } \\sum _ { k = 1 } ^ { n } \\frac { \\partial P _ j } { \\partial z _ k } ( \\vec { 1 } ) ( m + x _ k ) \\\\ \\frac { \\partial Q _ j } { \\partial w _ 2 } ( \\vec { 1 } ) & = ( 2 m ) ^ { j - 1 } \\sum _ { k = 1 } ^ { n } \\frac { \\partial P _ j } { \\partial z _ k } ( \\vec { 1 } ) ( m - x _ k ) \\end{aligned} \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} ( D _ x ) ^ { \\ell } \\alpha _ x ( a ) & = ( D _ x ) ^ { \\ell } \\alpha _ x \\left ( \\sum _ m a _ m \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\right ) \\\\ & = ( D _ x ) ^ { \\ell } \\sum _ m a _ m e ^ { i x \\cdot m } \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\sum _ m a _ m m ^ { \\ell } e ^ { i x \\cdot m } \\prod _ { j = 1 } ^ n U _ j ^ { m _ j } \\\\ & = \\delta ^ { \\ell } \\alpha _ x ( a ) . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} F = { \\rm i m } \\ , \\alpha , G = { \\rm i m } \\ , \\beta . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k V _ i + \\phi , \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{gather*} E _ { a n b c } = - f _ { a b c } , \\end{gather*}"}
-{"id": "5914.png", "formula": "\\begin{align*} p \\lim _ \\theta \\frac { \\left ( \\hat { G } _ { 1 1 1 } - \\frac { 1 } { 3 } \\hat { L } _ { 1 1 1 } \\right ) } { \\hat { L } _ { 1 1 } } = 0 , n \\rightarrow \\infty \\ , , \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} u _ t ( x ) = P _ t u _ 0 ( x ) & + \\int _ 0 ^ t P _ { t - s } \\big ( \\beta \\ , u _ s ( 1 - u _ s ) \\big ) ( x ) \\ , d s \\\\ & + \\int _ { [ 0 , t ] \\times \\Gamma } p ( t - s , x , y ) \\ , \\ell ( y ) \\sqrt { \\gamma ( y ) \\ , u _ s ( y ) \\big ( 1 - u _ s ( y ) \\big ) } \\ , d W ( s , y ) \\\\ & + \\frac { 1 } { 2 } \\int _ 0 ^ t \\sum _ { v \\in V } p ( t - s , x , v ) \\ , \\ell ( v ) \\ , \\hat { \\beta } ( v ) \\ , u _ s ( v ) ( 1 - u _ s ( v ) ) \\ , d s , \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\| \\xi _ S \\| _ { \\psi _ 2 } = \\| \\xi _ { x _ 1 , y _ 1 } - \\xi _ { x _ 2 , y _ 2 } \\| _ { \\psi _ 2 } \\leq { \\frac { c } { \\sqrt { n } } } \\| S \\| _ 1 , \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { p , q , \\mu _ { \\beta } } ^ { 2 } : = \\sum _ { n = 0 } ^ { \\infty } ( n ! ) ^ { 2 } 2 ^ { n q } | \\varphi ^ { ( n ) } | _ { p } ^ { 2 } , p , q \\in \\mathbb { N } _ { 0 } , \\ ; \\varphi \\in \\mathcal { P } \\big ( S _ d ' \\big ) . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} g = d s ^ 2 + a ( s ) ^ 2 g _ { S ^ { 2 n + 1 } } \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} ( d ^ e \\otimes I + T ( d ^ e \\otimes I ) T ) \\Delta ^ e m ( a ) & = ( d ^ e \\otimes I + T ( d ^ e \\otimes I ) T ) ( m \\otimes m ) \\Delta ( a ) \\\\ & = d ^ e m ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) + ( - 1 ) ^ { | a _ { ( 1 ) } | } m ( a _ { ( 1 ) } ) \\otimes d ^ e m ( a _ { ( 2 ) } ) \\\\ & = m d ( a _ { ( 1 ) } ) \\otimes m ( a _ { ( 2 ) } ) + ( - 1 ) ^ { | a _ { ( 1 ) } | } m ( a _ { ( 1 ) } ) \\otimes m d ( a _ { ( 2 ) } ) , \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , \\ldots , x _ { 2 n - 1 } , x _ { 2 n } ) = f ^ * ( x _ 2 , - x _ 1 , \\ldots , x _ { 2 n - 1 } , - x _ { 2 n } ) , ( x _ i ) _ { i = 1 } ^ { 2 n } \\in \\R ^ { 2 n } , \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} ( \\lambda + A ) ^ { - 1 } y & = \\lim \\limits _ { n \\rightarrow \\infty } \\lim \\limits _ { t \\rightarrow 0 + } - t ^ { 1 - 2 \\alpha } ( \\lambda + A ) ^ { - 1 } U ' ( t ) x _ n \\\\ & = \\lim \\limits _ { n \\rightarrow \\infty } \\lim \\limits _ { t \\rightarrow 0 + } - t ^ { 1 - 2 \\alpha } U ' ( t ) ( \\lambda + A ) ^ { - 1 } x _ n = \\lim \\limits _ { n \\rightarrow \\infty } c _ { \\alpha } J ^ { \\alpha } _ A ( \\lambda + A ) ^ { - 1 } x _ n = 0 . \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} p = \\frac { 3 } { 2 6 } c ^ { 5 } + \\frac { 3 } { 1 3 } c ^ { 4 } + \\frac { 1 2 } { 1 3 } c ^ { 3 } + \\frac { 1 5 } { 1 3 } c ^ { 2 } + \\frac { 1 5 } { 2 6 } c + \\frac { 4 } { 1 3 } . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } L _ i v \\frac { v _ i } { u _ { i , \\infty } } = - \\frac { 1 } { 2 } \\sum _ { r = 1 } ^ { R } k _ r u _ { \\infty } ^ { y _ r } \\left ( \\sum _ { i = 1 } ^ { N } ( y _ { r , i } ' - y _ { r , i } ) \\frac { v _ i } { u _ { i , \\infty } } \\right ) ^ 2 v \\in \\mathbb R ^ N . \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\tilde { \\rho } ( U , x ) = P _ { \\tilde { \\pi } } ( \\theta _ 1 < U | x ) = \\frac { \\int ^ { U } \\ldots \\int \\tilde { L } ( \\theta | x ) \\pi ( \\theta ) d \\theta _ d \\ldots d \\theta _ 1 } { \\int \\ldots \\int \\tilde { L } ( \\theta | x ) \\pi ( \\theta ) d \\theta _ d \\ldots d \\theta _ 1 } = \\alpha \\ , . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\frac { \\mathbb { P } ( S _ { ( \\eta ) } > \\mathit { x y } ) } { \\mathbb { P } ( S _ { ( \\eta ) } > x ) } & = \\frac { \\sum _ { n = 1 } ^ { K } \\mathbb { P } ( S _ { ( n ) } > \\mathit { x y } ) \\mathbb { P } ( \\eta = n ) } { \\mathbb { P } ( S _ { ( \\eta ) } > x ) } \\\\ & + \\frac { \\sum _ { n = K + 1 } ^ { \\infty } \\mathbb { P } ( S _ { ( n ) } > \\mathit { x y } ) \\mathbb { P } ( \\eta = n ) } { \\mathbb { P } ( S _ { ( \\eta ) } > x ) } \\\\ & = : \\mathcal { I } _ { 1 } + \\mathcal { I } _ { 2 } , \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} h ^ * ( x ^ * ) = \\frac { 1 } { 2 } \\langle A ^ { - 1 } x ^ * , x ^ * \\rangle , x ^ * \\in X . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} \\pi = ( \\texttt { A A } , \\texttt { A C } , \\texttt { C A } , \\texttt { G A } , \\texttt { A G } , \\texttt { C C } , \\texttt { C G } , \\texttt { G C } , \\texttt { G G } ) , \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { h } \\oplus \\mathfrak { m } , \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} d _ \\tau ( \\pi _ 1 , \\pi _ 2 ) = \\sum _ { i = 1 } ^ { | B | } d _ \\tau ( \\pi _ 1 \\vert _ { A _ i } , \\pi _ 2 \\vert _ { A _ i } ) + q ^ 2 d _ \\tau ( c _ 1 , c _ 2 ) , \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} ( x ) _ 0 = 1 , \\ , \\ , ( x ) _ n = x ( x - 1 ) \\cdots ( x - n + 1 ) = \\sum _ { l = 0 } ^ n S _ 1 ( n , l ) x ^ l , \\ , \\ , ( n \\geq 1 ) . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} H \\left ( Z ' , Z '' \\right ) = \\tilde { U } \\otimes \\varphi _ { B } \\left ( Z ' , Z '' \\right ) , \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\frac { \\varepsilon } { 2 } \\frac { d } { d t } \\sum _ { i = 1 } ^ { N } \\left \\Vert v _ { i } ^ { \\varepsilon } - v _ { i } ^ { 0 } \\right \\Vert _ { L ^ { 2 } \\left ( \\Gamma ^ { \\varepsilon } \\right ) } ^ { 2 } \\le C \\max \\left \\{ \\varepsilon , \\varepsilon ^ { \\gamma } \\right \\} + C \\varepsilon \\sum _ { i = 1 } ^ { N } \\left \\Vert v _ { i } ^ { \\varepsilon } - v _ { i } ^ { 0 } \\right \\Vert _ { L ^ { 2 } \\left ( \\Gamma ^ { \\varepsilon } \\right ) } ^ { 2 } . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} ( F \\mathbf { d } ) Y & = \\inf _ { y \\in Y } \\sup _ { x \\in F } x \\mathbf { d } y \\\\ & \\leq \\liminf _ \\lambda \\inf _ { y \\in Y } \\sup _ { x \\in F } ( x \\mathbf { d } x _ \\lambda + x _ \\lambda \\mathbf { d } y ) \\\\ & = \\liminf _ \\lambda ( F \\mathbf { d } x _ \\lambda + x _ \\lambda \\mathbf { d } Y ) \\\\ & \\leq ( F \\mathbf { d } ) ( x _ \\lambda ) + ( x _ \\lambda ) ( \\mathbf { d } Y ) \\\\ & = 0 , \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} { \\frak P } _ { \\ell } ( u , x ) = { \\frak P } _ { \\ell } ^ { \\prime } ( u , x ) f _ { \\ell } ( u , x ) , \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} \\theta ^ { \\varepsilon } _ { 0 } \\left ( t , x \\right ) : = \\theta ^ { 0 } \\left ( t , x \\right ) , \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} ( E _ { C , A } ) = 3 , ( E _ { C , A } ) = \\mathcal { O } _ X ( C ) , c _ 2 ( E _ { C , A } ) \\\\ = d , h ^ 0 ( X , { E _ { C , A } } ^ * ) = h ^ 1 ( X , { E _ { C , A } } ^ * ) = 0 \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\lambda _ 1 ( T ) = \\gamma _ 1 \\geq 0 , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} Q _ { \\infty } & : = \\lim _ { s \\rightarrow \\infty } Q \\\\ b _ { \\infty } & : = \\lim _ { s \\rightarrow \\infty } b \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\sigma _ { 1 } ( x ) = x ' , \\ \\sigma _ { 2 } ( x ) = x ' , \\ \\sigma _ { 3 } ( x ) = x ' \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} I & = \\sum _ { j = 1 } ^ { N } \\int _ { E _ { \\tau j } } \\frac { \\big \\lvert \\Psi ' \\big ( u + \\mathrm { i } a ( u ) + \\mathrm { i } \\tau \\big ) \\big ( 1 + \\mathrm { i } a ' ( u ) \\big ) \\big \\rvert } { \\big \\lvert \\Psi \\big ( u + \\mathrm { i } a ( u ) + \\mathrm { i } \\tau \\big ) - \\alpha \\big \\rvert ^ q } \\ , \\mathrm { d } u \\\\ & = \\sum _ { j = 1 } ^ { N } I _ { \\tau j } . \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} K _ { 0 } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\kappa \\left ( y \\right ) d y , K _ { i j } : = \\frac { 1 } { \\left | Y _ { 1 } \\right | } \\int _ { Y _ { 1 } } \\kappa \\left ( y \\right ) \\frac { \\partial \\bar { \\theta } ^ { j } } { \\partial y _ { i } } d y , \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} P \\left ( \\tilde { \\omega } ^ { k } \\right ) = \\left \\Vert \\mathbf { Q } \\left ( \\tilde { \\omega } ^ { k } , b _ { 1 } ^ { * } \\left ( \\tilde { \\omega } ^ { k } \\right ) , b _ { 2 } ^ { * } \\left ( \\tilde { \\omega } ^ { k } \\right ) ; \\mathbf { t } \\right ) + \\bar { p } ^ { * } \\left ( \\tilde { \\omega } ^ { k } \\right ) \\mathbf { 1 } - \\mathbf { f } \\right \\Vert _ { 2 } ^ { 2 } \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} g ' = \\log \\max \\{ a _ 0 | w _ 0 | , \\ldots , a _ { i _ 0 - 1 } | w _ { i _ 0 - 1 } | , a _ { i _ 0 } , a _ { i _ 0 + 1 } | w _ { i _ 0 + 1 } | , \\ldots , a _ d | w _ d | \\} . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} L & = a ' ( 0 ) \\left ( a ' ( 0 ) ^ 2 - ( n + 1 ) ^ 2 \\right ) ^ { - \\frac { 1 } { 2 } } \\\\ r _ b & = - \\frac { ( n + 1 ) L } { a ' ( 0 ) } \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\hat n ^ { \\alpha } \\partial _ { t } ^ 2 v _ { \\alpha } = \\hat n ^ { \\alpha } \\hat n ^ { \\tau } \\hat n _ { \\tau } \\partial _ { t } ^ 2 v _ { \\alpha } = \\hat n ^ { \\tau } \\Pi _ { \\tau } ^ { \\alpha } \\partial _ { t } ^ 2 v _ { \\alpha } . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{gather*} M \\big ( S \\big ( \\phi ^ { ( 0 ) } \\prec \\phi ^ { ( 1 ) } \\prec \\cdots \\prec \\phi ^ { ( N - 1 ) } \\prec \\phi ^ { ( N ) } \\big ) \\big ) \\\\ \\qquad { } = \\Lambda ^ { N } _ { N - 1 } \\big ( \\phi ^ { ( N ) } , \\phi ^ { ( N - 1 ) } \\big ) \\cdots \\Lambda ^ { 1 } _ 0 \\big ( \\phi ^ { ( 1 ) } , \\phi ^ { ( 0 ) } \\big ) M _ N \\big ( \\phi ^ { ( N ) } \\big ) , \\end{gather*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\sum _ { c _ 2 \\bmod { \\varpi ^ { l - h - 1 } } } \\widetilde { e } \\left ( \\frac { c _ 2 } { \\varpi ^ { l - h - 1 } } \\right ) = 0 . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} d \\eta ( ( S _ 1 , 0 ) , ( S _ 2 , 0 ) ) = - \\eta ( [ S _ 1 , S _ 2 ] , 0 ) = 0 . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} f : ( p , r ) \\mapsto ( p , q ) = ( p , \\dfrac { ( 1 - x _ 1 p ) r } { r x _ 2 - x _ 2 + 1 - x _ 1 p } ) . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} T _ k & : = \\inf \\{ n > T _ { k - 1 } : \\ : Q _ n = q _ 0 \\} \\ , \\\\ \\zeta _ k & : = X _ { T _ k } - X _ { T _ { k - 1 } } \\ , \\\\ \\nu _ k & : = T _ k - T _ { k - 1 } \\ , \\\\ R ' _ k & : = \\sum _ { n = { T _ { k - 1 } } } ^ { T _ k - 1 } R _ { n + 1 } \\ , . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { i \\left ( k , B ^ { \\beta , 1 } ( t ) \\right ) } \\right ) = E _ { \\beta } \\left ( - \\frac { | k | ^ { 2 } } { 2 } t \\right ) , k \\in \\mathbb { R } ^ { d } . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\varepsilon _ { n } \\left ( { u , \\xi } \\right ) = \\sum \\limits _ { s = n } ^ { n + r - 1 } \\frac { G _ { s } \\left ( \\xi \\right ) } { u ^ { 2 s + 2 } } + \\varepsilon _ { n + r } \\left ( { u , \\xi } \\right ) . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} p _ { a , b } ( 0 ) = p _ { a , b } ' ( 0 ) = 0 , \\ p _ { a , b } ( 1 ) = a , \\ p ' _ { a , b } ( 1 ) = b , \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} [ ( a _ i ) , ( f _ i ) , \\sigma _ f ] [ ( b _ i ) , ( g _ i ) , \\sigma _ g ] & = ( \\sigma _ f * ( ( b _ i ) , ( g _ i ) ) + ( a _ i , f _ i ) , \\sigma _ g \\circ \\sigma _ f ) \\\\ & = [ ( ( b _ { \\sigma _ f ( i ) } ) , ( g _ { \\sigma _ f ( i ) } ) ) ( ( a _ i ) , ( f _ i ) ) , \\sigma _ g \\circ \\sigma _ f ] \\\\ & = [ ( b _ { \\sigma _ f ( i ) } + g _ { \\sigma _ f ( i ) } a _ i ) , ( g _ { \\sigma _ f ( i ) } f _ i ) , \\sigma _ g \\circ \\sigma _ f ] . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} f ( h \\tilde * \\pi _ z ( y ) ) & = \\frac { 1 } { \\alpha _ 0 ( h ) \\alpha _ 0 ( \\pi _ z ( y ) ) } \\cdot ( \\delta _ h * \\delta _ { \\pi _ z ( y ) } ) ( \\alpha _ 0 f ) \\\\ & = \\frac { 1 } { \\alpha _ 0 ( h ) \\tilde \\alpha _ 0 ( y ) } ( \\alpha _ 0 f ) _ h ( \\pi _ z ( y ) ) \\\\ & = \\frac { 1 } { \\alpha _ 0 ( h ) \\tilde \\alpha _ 0 ( y ) } T _ h ( \\alpha _ 0 f ) ( \\pi _ z ( y ) ) \\\\ & = \\int _ X f ( \\pi _ x ( w ) ) \\ > \\tilde K _ h ( y , d w ) = ( \\tilde T _ h ( f \\circ \\pi _ z ) ) ( y ) . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} V & = \\frac { ( \\log x ) ^ 2 } { ( \\log \\log x ) ^ { 2 B + 1 } } \\bigg ( 1 - \\frac 1 { \\log \\log x } \\bigg ) \\\\ Q & = \\frac { \\log x } { ( \\log \\log x ) ^ { 2 B + 1 } } ; \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} J ( \\alpha , m ) : = E \\left [ \\int _ 0 ^ T c ( s , X _ { \\alpha , m } ( s ) , \\alpha ( s ) , m ( s ) ) d s + \\psi ( X _ { \\alpha , m } ( T ) , m ( T ) ) \\right ] . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} x = \\phi \\tilde f { \\rm \\ a n d \\ } y = \\tilde g , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\widehat { M } _ { L } \\widehat { V } _ { - P } \\varphi ^ \\hbar ( x ) = \\varphi ^ \\hbar _ { P , L } ( x ) = \\left ( \\tfrac { 1 } { \\pi \\hbar } \\right ) ^ { 1 / 4 } \\sqrt { L } e ^ { - \\tfrac { 1 } { 2 \\hbar } ( L ^ 2 + i L P ) x ^ 2 } , \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} & \\frac { \\partial u } { \\partial t } + L u - \\mathbf { i } \\xi u = - g ( x , y ) , u ( T ) = 0 & \\frac { \\partial v } { \\partial t } + L v = - \\left | \\nabla u \\right | ^ 2 , v ( T ) = 0 \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\psi ( z ) = \\phi ( z ) y ( z ) , \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} ( \\mathcal { L } _ 1 , \\mathcal { L } _ 2 , \\phi ) \\otimes \\mathcal { L } = ( \\mathcal { L } _ 1 \\otimes f _ 1 ^ * \\mathcal { L } , \\mathcal { L } _ 2 \\otimes f _ 2 ^ * \\mathcal { L } , \\phi \\otimes \\mathrm { i d } ) , \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\omega ( u ^ * x u ) & = \\sum _ { i = 1 } ^ n \\langle u ^ * x u b _ i , b _ i \\rangle _ { L ^ 2 ( M , \\phi ) } = \\sum _ { i = 1 } ^ n \\langle x u b _ i , u b _ i \\rangle _ { L ^ 2 ( M , \\phi ) } = \\sum _ { i = 1 } ^ n \\langle x P _ K u b _ i , P _ K u b _ i \\rangle _ { L ^ 2 ( M , \\phi ) } \\\\ & = \\sum _ { i = 1 } ^ n \\langle P _ K x P _ K u b _ i , u b _ i \\rangle _ { K } = \\sum _ { i = 1 } ^ n \\langle P _ K x P _ K b _ i , b _ i \\rangle _ { K } = \\sum _ { i = 1 } ^ n \\langle x b _ i , b _ i \\rangle _ { L ^ 2 ( M , \\phi ) } = \\omega ( x ) . \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\ < m ' , n ' \\ > _ X & = \\ < \\lambda _ * ( \\Delta _ E \\otimes m ) , \\lambda _ * ( \\Delta _ E \\otimes n ) \\ > _ X \\\\ & = \\ < \\Delta _ E \\otimes m , \\lambda ^ * \\lambda _ * ( \\Delta _ E \\otimes n ) \\ > _ X \\\\ & = \\ < \\Delta _ E \\otimes m , ( 1 + \\hat { \\rho } + \\hat { \\rho } ^ 2 ) ( \\Delta _ E \\otimes n ) \\ > _ X \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} \\psi ( E _ { i j } ) = & z _ { j i } E _ { j i } + z _ { \\sigma ^ { - 1 } ( j ) , i } E _ { \\sigma ^ { - 1 } ( j ) , i } + z _ { j , \\sigma ^ { - 1 } ( i ) } E _ { j , \\sigma ^ { - 1 } ( i ) } \\\\ & + z _ { \\sigma ^ { - 1 } ( j ) , \\sigma ^ { - 1 } ( i ) } E _ { \\sigma ^ { - 1 } ( j ) , \\sigma ^ { - 1 } ( i ) } , \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} \\mathbb P \\otimes \\nu \\Big ( \\bigcup _ { i = 1 } ^ n A _ i \\times B _ i \\Big ) : = \\sum _ { i = 1 } ^ n \\mathbb E ( \\mathbf 1 _ { A _ i } \\nu ( B _ i ) ) , \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} - 2 T _ 0 ^ 3 T _ 2 + 4 3 T _ 0 ^ 2 T _ 1 T _ 2 + 4 8 0 T _ 0 ^ 2 T _ 2 ^ 2 + 2 T _ 0 T _ 1 ^ 3 - 3 6 T _ 0 T _ 1 ^ 2 T _ 2 - 2 4 6 0 T _ 0 T _ 1 T _ 2 ^ 2 + 8 7 0 0 T _ 0 T _ 2 ^ 3 \\\\ { } + 2 T _ 1 ^ 4 - 1 8 9 T _ 1 ^ 3 T _ 2 + 5 1 7 0 T _ 1 ^ 2 T _ 2 ^ 2 - 3 1 2 5 5 T _ 1 T _ 2 ^ 3 + 4 0 2 5 0 T _ 2 ^ 4 = 0 \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\partial _ { t } \\left \\Vert \\phi \\right \\Vert _ { L ^ { 2 } } ^ { 2 } = - \\nu \\left \\Vert \\nabla \\phi \\right \\Vert _ { L ^ { 2 } } ^ { 2 } . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} A ( p ) ^ \\top M ( p ) + M ( p ) A ( p ) + Q ( p ) = 0 \\mbox { f o r e a c h } \\ ; \\ ; p \\in \\Pi . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} d ^ { \\circ } ( f _ d ) = w _ p ( d ) \\leq n ( p - 1 ) . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} R _ \\lambda ( \\mu , r ) = O ( \\mu ^ { 3 - \\sigma } ) . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} r ^ { 2 - m } \\int _ { B _ { r } ( 0 ) } \\left | D _ { \\hat { L } } u ( z ) \\right | ^ { 2 } \\ , \\mathrm { d } z \\leq C ( m , \\rho ) \\sum _ { i = 0 } ^ { k } \\left ( \\theta ( y _ i , 4 r ) - \\theta ( y _ i , 2 r ) \\right ) . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} r ( \\tau ) = \\int _ { \\R _ + } e ^ { - \\tau \\xi } \\pi ( d \\xi ) , \\tau \\geq 0 . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\vec q _ { x , 2 ^ { - k } r } = D u ( x ) . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\hat { y } _ { 0 } ( x ) = \\frac { ( y _ a + { \\epsilon } ) } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } , \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\mu _ x ( g - \\log | s | ) = \\mu _ x ( g ) + \\mathrm { o r d } _ x ( s ) \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} W \\sim \\exp \\left \\{ \\pm u \\xi \\right \\} \\sum \\limits _ { s = 0 } ^ { \\infty } \\left ( \\pm 1 \\right ) ^ { s } { \\frac { A _ { s } \\left ( \\xi \\right ) } { u ^ { s } } , } \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\phi { } _ { j , k } - \\phi { } _ { j , k + \\ell } = \\frac { Q _ { j , k + 1 } } { \\prod _ { i = j } ^ { k } \\varsigma _ { i } } \\left ( 1 - \\frac { Q _ { k + 1 , k + \\ell + 1 } \\mathbf { 1 } _ { \\mathbb { R } ^ { p } } } { \\prod _ { i = k + 1 } ^ { k + \\ell } \\varsigma _ { i } } \\right ) . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\nu _ 2 ^ T \\Gamma \\nu _ 2 = \\Gamma _ { 2 2 } . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} N _ f ( g ) = f \\cdot g , g \\in L _ 2 [ 0 , \\alpha ) , \\quad \\eta ( N _ f ) = \\int _ 0 ^ { \\alpha } f . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} P _ { C , o l d } ( K , L , \\epsilon ) \\stackrel { \\triangle } { = } \\frac { \\epsilon } { m ^ 2 } \\cdot A _ { K , L } \\cdot \\hat { N } _ { K , L } ^ { \\epsilon } \\cdot ( p + \\xi ) ^ \\epsilon . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} \\mathcal E _ s ( u ; \\R ^ { 2 n } \\setminus ( \\R ^ n _ - ) ^ 2 ) = \\gamma _ s \\displaystyle \\int \\limits _ { \\R ^ n _ - } | x _ 1 | ^ { - 2 s } | u | ^ 2 ~ \\ ! d x ~ \\ ! , \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} Y _ { 0 } ^ { ( a , b ) } = Y _ 0 , & Y _ { 1 } ^ { ( a , b ) } = \\frac { r Y _ 0 + b X _ 0 } { 2 } , Y _ { h ' + 2 } ^ { ( a , b ) } = r Y _ { h ' + 1 } ^ { ( a , b ) } - Y _ { h ' } ^ { ( a , b ) } , \\\\ X _ { 0 } ^ { ( a , b ) } = X _ 0 , & X _ { 1 } ^ { ( a , b ) } = \\frac { r X _ 0 + a Y _ 0 } { 2 } , X _ { h ' + 2 } ^ { ( a , b ) } = r X _ { h ' + 1 } ^ { ( a , b ) } - X _ { h ' } ^ { ( a , b ) } , \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\left ( \\forall i \\in I \\right ) \\left ( \\forall j \\in I \\right ) \\alpha _ { i } \\alpha _ { j } \\norm { x _ { i } - x _ { j } } ^ { 2 } = 0 . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\Vert \\Phi _ { - T } \\vert _ { \\{ L _ T \\} \\times ( E ^ u _ \\sigma \\cap L ^ { \\perp } _ T ) } \\Vert \\leq \\Vert \\Phi ^ * _ { - T } \\vert _ { \\{ L _ T \\} \\times E ^ u _ \\sigma } \\Vert = \\Vert X _ { - T } \\vert _ { E ^ u _ \\sigma } \\Vert , \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} U _ c & = \\{ ( x _ 1 , x _ 2 ) \\in X \\mid \\Vert x _ 1 \\Vert \\leq \\vert c \\vert ^ 2 \\} \\\\ V _ c & = \\{ ( x _ 1 , x _ 2 ) \\in X \\mid \\Vert x _ 1 \\Vert \\geq \\vert c \\vert ^ 2 \\} . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} D _ t u - \\mu ( t , x , D _ x ) u = 0 , u ( t ' , x ) = \\phi ( x ) \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} W _ { \\alpha , \\beta } ^ { \\gamma , \\sigma } ( z ) \\leq \\frac { \\Gamma ( \\beta - \\alpha ) W _ { \\alpha , \\beta - \\alpha } ( z ) - 1 } { \\Gamma ( \\beta - \\alpha ) z } = W _ { \\alpha , \\beta } ^ { 1 , 2 } ( z ) . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} A = U \\Sigma V ^ T , \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\hat { y } ( t , p ) = \\sum _ { i = 1 } ^ m \\hat { w } _ i ( t ) \\Phi _ i ( p ) . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} { \\| y ( x ) - z ( x ) \\| } _ { C _ { 1 - \\gamma } [ a , x _ 1 ] } & = { \\| I _ { a ^ + } ^ { \\alpha } f ( x , y ( x ) ) - I _ { a ^ + } ^ { \\alpha } f ( x , z ( x ) ) \\| } _ { C _ { 1 - \\gamma } [ a , x _ 1 ] } \\\\ & \\leq A \\frac { ( x _ 1 - a ) ^ { \\alpha } \\Gamma ( \\gamma ) } { \\Gamma ( \\gamma + \\alpha ) } { \\| y ( x ) - z ( x ) \\| } _ { C _ { 1 - \\gamma } [ a , x _ 1 ] } \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\begin{cases} a _ { i j } : = a _ { i - j } / A _ i & \\mbox { i f $ 0 \\le j \\le i $ a n d $ A _ i > 0 $ , } \\\\ a _ { i j } : = 0 & \\mbox { i f $ j > i $ o r $ A _ i = 0 $ } . \\end{cases} \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\int _ { \\Gamma ^ 0 } ( N \\cdot \\nu ^ 0 ) ( x ) f ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) = \\lim _ { \\varepsilon \\downarrow 0 } \\frac 1 \\varepsilon \\int _ { A ^ { \\varepsilon } } f ( x ) d \\mathcal { H } ^ n ( x ) . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{gather*} \\int _ { \\Omega _ { T } } \\int _ { \\mathcal { Y } _ { n , m } } a _ { 0 } \\left ( x , t , y ^ { n } , s ^ { m } \\right ) \\\\ \\cdot v _ { 1 } \\left ( x \\right ) v _ { 2 } \\left ( y _ { 1 } \\right ) \\cdots v _ { i } \\left ( y _ { i - 1 } \\right ) \\nabla _ { y _ { i } } v _ { i + 1 } \\left ( y _ { i } \\right ) \\\\ \\times c _ { 1 } \\left ( t \\right ) c _ { 2 } \\left ( s _ { 1 } \\right ) \\cdots c _ { m - d _ { i } + 1 } \\left ( s _ { m - d _ { i } } \\right ) d y ^ { n } d s ^ { m } d x d t = 0 \\end{gather*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\frac { 1 } { ( n + m ) ^ s } = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } \\frac { \\Gamma ( z ) \\Gamma ( s - z ) } { \\Gamma ( s ) } \\frac { 1 } { n ^ { s - z } m ^ z } \\ ; d z . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} M ( s , t ) : = \\left ( \\begin{array} { c c c } - s t + t ^ 2 & s t & g ( s , t ) \\\\ s t & s ^ 2 & t ^ 2 \\\\ g ( s , t ) & t ^ 2 & h ( s , t ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } L a w ( X _ { n _ k } ) = L a w ( X _ { \\rho , m } ) , \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} B _ { x } ( t ) : = \\sqrt { L ^ e } \\ , \\dot { W } ( [ 0 , t ] \\times I _ { x } ) , \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { 1 } { x } \\# \\bigg \\{ n \\leq x \\colon \\frac { P _ n ( x ) - D ( x ) } { \\sqrt C ( \\log \\log x ) ^ { 3 / 2 } } < u \\bigg \\} = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ u e ^ { - t ^ 2 / 2 } \\ , d t . \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } D ( x ) = D _ 0 > 0 \\ \\hbox { i n } \\ C ^ 2 \\ \\hbox { t o p o l o g y } \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} s _ { \\alpha , r } = \\tau ( r \\alpha ^ \\vee ) \\circ s _ { \\alpha } . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} a ( n ; i , j ) = a ( n - 1 ; j ) - w _ { n - 5 , j - 1 } - \\cdots - w _ { i - 2 , j - 1 } - w _ { i - 3 ; j - 1 } . \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} A = \\begin{pmatrix} I & & & \\\\ L _ { 2 1 } & I & & \\\\ \\vdots & \\vdots & \\ddots & \\\\ L _ { N 1 } & L _ { N 2 } & \\cdots & I \\end{pmatrix} \\begin{pmatrix} D _ 1 & & & \\\\ & D _ 2 & & \\\\ & & \\ddots & \\\\ & & & D _ N \\end{pmatrix} \\begin{pmatrix} I & L _ { 2 1 } ^ T & \\cdots & L _ { N 1 } ^ T \\\\ & I & \\cdots & L _ { N 2 } ^ T \\\\ & & \\ddots & \\vdots \\\\ & & & I \\end{pmatrix} . \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} J _ n ( t ) = t ^ { a n } J _ n ( 1 ) . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} ( F ^ * x _ 1 ^ * ) ( x ) = z _ 1 ^ * \\left ( f _ 0 ( s _ 0 ) F _ Z x ( s _ 0 ) \\right ) = ( F _ Z ^ * z _ 1 ^ * ) ( x ( s _ 0 ) ) = z _ 1 ^ * ( x ( s _ 0 ) ) = x _ 1 ^ * ( x ) . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\Phi _ 1 ^ c = - \\frac { F _ A ( \\omega _ B ^ c ) } { ( m _ H ( z ) ) ^ 2 } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\frac { 1 } { a _ i - \\omega _ B ^ c } Q _ i \\ , , \\qquad \\Phi _ 2 ^ c = - \\frac { F _ B ( \\omega _ A ^ c ) } { ( m _ H ( z ) ) ^ 2 } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\frac { 1 } { b _ i - \\omega _ A ^ c } \\mathcal { Q } _ i \\ , , \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} \\mathcal { M } & = \\max \\{ R _ 1 + S _ 1 + 1 , S _ 1 + T _ 1 + 1 , R _ 1 + T _ 1 + 1 , \\chi \\mathcal { V } \\} , \\\\ \\mathcal { V } & = \\sqrt { ( R _ 1 + 1 ) ( S _ 1 + 1 ) ( T _ 1 + 1 ) } , \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} \\| \\Delta _ { M } e ^ { - \\Delta _ { M } t } \\| _ { L ^ { p } ( M ) \\rightarrow L ^ { p } ( M ) } \\leq \\begin{cases} c t ^ { - 1 } , 0 \\leq t < 1 , \\\\ c e ^ { - \\epsilon t } , t \\geq 1 , \\end{cases} \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} R _ { k \\rightarrow \\bar { k } } ^ { \\rm N O M A } ( \\nu ) = \\log _ 2 \\left ( 1 + \\frac { p _ { \\bar k } ( \\nu ) g _ k ( \\nu ) } { p _ k ( \\nu ) g _ k ( \\nu ) + 1 } \\right ) , \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} \\Theta ^ * ( q , z ) = \\prod _ { m = 1 } ^ { \\infty } ( 1 - q ^ m ) ( 1 + z q ^ m ) ( 1 + q ^ { m - 1 } / z ) ~ . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\widetilde { \\mathfrak { s w } } ^ { n o r m } _ { a } ( M ) : = - \\mathfrak { s w } _ { - [ h E ^ * _ { + s } ] \\ast \\sigma _ { c a n } } ( M ) - ( ( K + 2 h E ^ * _ { + s } ) ^ 2 + \\mathcal { V } ) / 8 \\ \\ \\ \\ \\ \\mbox { f o r } \\ \\ 0 \\leq h < p . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) v \\rVert & = \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) P ( x ) v \\rVert \\\\ & \\le \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) P ( x ) \\rVert \\\\ & = \\Lambda ( \\mu ) < \\lambda _ i , \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} 2 d - g _ 3 + 1 = 2 d - 2 g + 2 . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} \\mathbb { P } ( \\widehat { Q } _ 0 < \\widehat { Q } _ 1 < \\cdots \\ , | \\ , \\widehat { Q } _ 0 = m ) \\stackrel { 1 } { = } \\frac { m } { m + 1 } \\prod _ { j = { m + 1 } } ^ \\infty \\frac { j ( j + 2 ) } { ( j + 1 ) ^ 2 } = \\frac { m } { m + 2 } . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} D _ { G } ^ { \\textrm { c m } } ( s ) : = \\sum _ { F \\in \\mathcal { F } _ G ^ { + } } \\sum _ { [ E : F ] = 2 } \\frac { 1 } { | d _ E | ^ s } = \\sum _ { n = 1 } ^ { \\infty } \\frac { a ( n ) } { n ^ { s } } , \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} ( F \\star \\bar { \\mu } ) ( \\tau _ s ) = ( ( F \\circ \\tau ) \\star ( \\overline { \\mu \\circ \\tau } ) ) ( s ) , \\ ; \\ ; \\ ; s \\geq 0 . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} \\frac { \\partial W _ { n , 1 } \\left ( { u , \\xi } \\right ) } { \\partial \\xi } = u \\exp \\left \\{ { u \\xi + S _ { n } \\left ( { u , \\xi } \\right ) + \\ln \\left \\{ { 1 + u ^ { - 2 } T _ { n } \\left ( { u , \\xi } \\right ) } \\right \\} + \\eta _ { n , 1 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\Lambda ^ { s } f \\equiv ( - \\Delta ) ^ { \\frac { s } { 2 } } f : = \\sum _ { j = 1 } ^ \\infty \\lambda _ j ^ { \\frac { s } { 2 } } f _ j w _ j \\quad ~ f = \\sum _ { j = 1 } ^ \\infty f _ j w _ j , f _ j = \\int _ { \\Omega } f w _ j d x , \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{gather*} - \\theta p N m + \\frac { \\theta p m ( m + 1 ) } { 2 } = \\sum _ { i = 1 } ^ m { ( - \\theta p N + p ( i \\theta + \\tau _ i ^ + - \\tau _ i ^ - ) ) } . \\end{gather*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\mathrm { b i d e g } ( w ) & = j ( 1 , 1 ) + k ( - 1 , - 1 ) + l ( - 1 , 1 ) + m ( 1 , - 1 ) \\\\ & = ( j - k - l + m , j - k + l - m ) , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} f _ 5 = ( z _ 0 : z _ 3 - z _ 2 : z _ 3 : z _ 1 ) \\circ ( z _ 0 z _ 2 : z _ 1 z _ 3 : z _ 3 ^ 2 : z _ 2 z _ 3 ) \\circ ( z _ 0 z _ 2 : z _ 1 z _ 2 : z _ 3 ^ 2 : z _ 2 z _ 3 ) \\circ ( z _ 0 : z _ 1 : z _ 3 - z _ 2 : z _ 3 ) . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\int _ { G } f \\left ( g \\right ) d g = c \\int _ { { \\overline { \\mathfrak { a } ^ { + } } } } f \\left ( \\exp H \\right ) \\delta ( H ) d H . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} B ^ T = \\left ( \\begin{array} { c c c } a & 0 & 0 \\\\ d & ( 1 - \\epsilon ^ 2 ) ^ { 1 / 2 } & \\epsilon \\\\ \\epsilon & \\epsilon & ( 1 - \\epsilon ^ 2 ) ^ { 1 / 2 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} H ( k ) = - \\sum ^ { w - 1 } _ { i = 0 } P ( k _ i ) \\log _ 2 P ( k _ i ) , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} [ m ] _ A ( a , b ) & = q ( a , q ( q ( b , b ) , b ) ) \\\\ [ i ] _ A ( a ) & = q ( q ( a , a ) , a ) . \\\\ \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} ( N _ r V ) _ s & = \\sum _ t \\left [ \\binom { s } { 2 } \\binom { r - 2 } { r - t } + \\binom { r - s } { 2 } \\binom { r - 2 } { t } \\right ] \\left [ \\binom { t } { 2 } + \\binom { r - t } { 2 } \\right ] \\\\ & = \\binom { s } { 2 } P + \\binom { r - s } { 2 } Q , \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} \\sum _ { n \\leq R } d ( n ) = c R \\log R + c ' R + O ( R ^ { \\frac { 1 } { 2 } } ) , \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} \\rho \\ , u ^ \\perp \\ , = \\ , \\nabla \\pi \\ , , \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\begin{aligned} \\chi _ 0 ( \\xi ) = \\chi _ 0 ( - \\xi ) \\ \\ \\forall \\xi \\in \\R \\ , , \\chi _ 0 ( \\xi ) = 0 \\ \\ \\forall | \\xi | \\leq \\frac 4 5 \\ , , \\chi _ 0 ( \\xi ) = 1 \\ \\ \\forall | \\xi | \\geq \\frac 7 8 \\ , . \\end{aligned} \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\aligned & \\widehat { \\mathcal U } ( { \\rm m o r } ; 3 , 1 ; \\alpha _ 1 , \\alpha _ 3 ) ^ { \\boxplus \\tau } \\\\ & = \\bigcup _ { \\alpha _ 2 \\in \\frak A _ 2 } \\widehat { \\mathcal U } ( { \\rm m o r } ; 2 , 1 ; \\alpha _ 1 , \\alpha _ 2 ) ^ { \\boxplus \\tau } \\times _ { R _ { \\alpha _ 2 } } \\widehat { \\mathcal U } ( { \\rm m o r } ; 3 , 2 ; \\alpha _ 2 , \\alpha _ 3 ) ^ { \\boxplus \\tau } . \\endaligned \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} ( \\widetilde { B } _ { t _ 1 } , \\widetilde { B } _ { t _ 2 } - \\widetilde { B } _ { t _ { 1 } } , \\cdots , \\widetilde { B } _ { t _ n } - \\widetilde { B } _ { t _ { n - 1 } } ) \\overset { d } { = } ( B _ { t _ 1 } , { B } _ { t _ 2 } - { B } _ { t _ { 1 } } , \\cdots , B _ { t _ n } - B _ { t _ { n - 1 } } ) . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} ( y , \\frak s _ { 2 } ( y ) ) = \\tilde \\varphi _ { 2 1 } ( y , \\frak s _ { 1 } ( \\pi _ { 1 2 } ( y ) ) ) , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{gather*} C H _ { a l g , X } ^ d ( Y ) : = \\{ Z \\in C H ^ d ( Y \\times X ) : i _ y ^ * Z \\sim _ { a l g } 0 \\} , \\\\ C H _ { h o m , X } ^ d ( Y ) : = \\{ Z \\in C H ^ d ( Y \\times X ) : i _ y ^ * Z \\sim _ { h o m } 0 \\} , \\\\ { \\rm N S } _ { a l g , X } ^ d ( Y ) : = C H _ X ^ d ( Y ) / C H _ { a l g , X } ^ d ( Y ) , \\ ; { \\rm N S } _ { h o m , X } ^ d ( Y ) : = C H _ X ^ d ( Y ) / C H _ { h o m , X } ^ d ( Y ) . \\end{gather*}"}
-{"id": "8392.png", "formula": "\\begin{align*} W ( \\psi , \\phi ) ( x , p ) = \\left ( \\tfrac { 1 } { 2 \\pi \\hbar } \\right ) ^ { n } \\int _ { \\R ^ n } e ^ { - \\frac { i } { \\hbar } p y } \\psi ( x + \\tfrac { 1 } { 2 } y ) \\overline { \\phi ( x - \\tfrac { 1 } { 2 } y ) } \\ , d ^ { n } y ; \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} Q _ { k } \\phi _ { k , \\infty } = \\varsigma _ { k - 1 } \\phi _ { k - 1 , \\infty } , k \\geq 1 , \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\left ( f _ { ( j ) } \\circ { \\rm F b } _ { ( j ) } \\right ) ( x ) = \\left ( x ^ { p ^ { n - j } } \\right ) ^ { p ^ n - p ^ j - 1 } = x ^ { - 1 } = f _ { p ^ n - 2 } ( x ) , \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} C r \\left ( F , \\pi \\right ) = \\{ z \\in P : \\left \\langle d F ( z ) , V \\right \\rangle = 0 , \\forall V \\in V _ z P \\} \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} T _ k \\varphi = G _ k * ( u \\varphi ) \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} R ^ V ( X , Y ) Z : = D _ X ^ V D _ Y ^ V Z + D _ Y ^ V D _ X ^ V Z - D _ { [ X , Y ] } ^ V Z . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} I = ( y ^ 2 - x z , x ^ 3 - y z , x ^ 2 y - z ^ 2 , w ^ 2 - x ^ 3 y ) . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} x _ i = p ^ a u _ i + \\xi y _ i = p ^ a z _ i + \\xi ( 1 \\le i \\le R ) . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} c _ i = \\| \\mathbf { g } _ i \\| - h _ { i i } - \\big ( \\| \\mathbf { g } _ i \\| ^ 2 - 1 \\big ) + O _ \\prec ( \\frac { 1 } { N } ) \\ , . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} A _ { 0 } \\left ( x , v _ { x _ { 0 } } \\right ) + F _ { 2 } \\left ( x , \\nabla v , \\nabla v _ { x _ { 0 } } \\right ) = 0 , x \\in \\Omega , x _ { 0 } \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\| f \\| _ { \\hat { \\mathcal S } _ { q ' } ^ { p ' } } = \\frac 1 { \\sqrt N } \\bigl \\| ( f _ { k , m } ) _ { k = 1 , m = 0 } ^ { k = K , m = M } \\bigr \\| _ { Q _ { q ' } ^ { p ' } } . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} [ h , \\mathfrak { X } _ i ^ { \\pm } ( u , \\lambda ) ] = \\pm \\alpha _ i ( h ) \\mathfrak { X } _ i ^ { \\pm } ( u , \\lambda ) . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} ( N _ r V ) _ s & = 2 ^ { r - 2 } ( r - 1 ) \\left ( \\binom { s } { 2 } - \\binom { r - s } { 2 } \\right ) \\\\ & = 2 ^ { r - 2 } ( r - 1 ) V _ s . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{gather*} \\frac { \\Gamma _ q ( \\lambda _ N - z ) } { \\Gamma _ q ( \\lambda _ 1 + \\theta N - z ) } = \\frac { [ - z + \\lambda _ N - 1 ] _ q \\cdots [ - z + 1 ] _ q [ - z ] _ q } { [ - z + \\lambda _ 1 + \\theta N - 1 ] _ q \\cdots [ - z + \\theta N + 1 ] _ q [ - z + \\theta N ] _ q } \\frac { \\Gamma _ q ( - z ) } { \\Gamma _ q ( \\theta N - z ) } , \\end{gather*}"}
-{"id": "1224.png", "formula": "\\begin{align*} E ( \\sum _ { \\ell = 2 } ^ { p _ n } y _ { n \\ell } ^ 2 - 1 ) ^ 2 = E ( \\sum _ { \\ell = 2 } ^ { p _ n } y _ { n \\ell } ^ 4 ) + \\sum _ { 2 \\le i \\ne j \\le p _ n } E ( y _ { n i } ^ 2 y _ { n j } ^ 2 ) - 1 = O ( p _ n ^ { - 1 } ) \\to 0 , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\int _ M u _ X \\ , f ^ { - 2 m - 1 } \\omega ^ m = 0 . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} [ M ( t ) f ] ( x ) = e ^ { \\frac { i x ^ 2 } { 2 t } } f ( x ) , [ D ( t ) f ] ( x ) = ( i t ) ^ { - \\frac 1 2 } f ( \\tfrac { x } { t } ) . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} X _ t ^ i = x _ i + \\sum _ { j = 1 } ^ d \\int _ 0 ^ t \\sigma _ { i j } ( X _ s ) \\ , d W _ s ^ j + \\int ^ { t } _ { 0 } g _ i ( X _ s ) \\ , d s , 0 \\le t < \\infty , \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} H _ { \\nu , - 1 } \\left ( \\mathbb { R } , D \\left ( A ^ { * } \\right ) ^ { \\prime } \\right ) : = H _ { \\nu , 1 } \\left ( \\mathbb { R } , D \\left ( A ^ { * } \\right ) \\right ) ^ { \\prime } , \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\Delta _ V ^ \\pm = \\Gamma _ \\pm ( \\Delta _ V ) , J _ V ^ + = \\Gamma _ + ( J _ V ) \\mbox { a n d } J _ V ^ - = \\tilde Z \\Gamma _ - ( i J _ V ) . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} N \\left ( { { \\star _ { i = 1 } ^ { m - 1 } } G } , \\alpha ^ { m } \\right ) + N \\left ( { \\star _ { i = 1 } ^ { m - 2 } G } , \\alpha ^ { m - 1 } \\right ) \\leq & N \\left ( { { \\star _ { i = 1 } ^ m } G } , \\alpha ^ { m + 1 } \\right ) \\\\ \\leq & N \\left ( { { \\star _ { i = 1 } ^ { m - 1 } } G } , \\alpha ^ { m } \\right ) + N \\left ( { { \\star _ { i = 1 } ^ { m - 2 } } G } , \\alpha ^ { m - 1 } \\right ) + n \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\ddot y _ \\varepsilon = - \\sum _ { i = 1 } ^ N \\frac { m _ i ( y - \\varepsilon c _ i ) } { \\vert y - \\varepsilon c _ i \\vert ^ { \\alpha + 2 } } \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\iint e ^ { i \\lambda \\Phi ( y , x , \\xi , \\eta ) } \\tilde { \\tilde { a _ { N } } } ( x , \\xi ) f _ { i _ { 1 } . . . . i _ { m } } ( z ) { \\tilde { b } } ^ { i _ { 1 } } ( x , \\xi ) \\cdots { \\tilde { b } } ^ { i _ { m } } ( x , \\xi ) d x d \\xi = 0 . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} P _ 1 ^ { ( a , b ) } P _ n ^ { ( a , b ) } = \\frac { 1 } { a ( b - 1 ) } P _ { n - 1 } ^ { ( a , b ) } + \\frac { b - 2 } { a ( b - 1 ) } P _ n ^ { ( a , b ) } + \\frac { a - 1 } { a } P _ { n + 1 } ^ { ( a , b ) } \\quad \\quad ( n \\ge 1 ) . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\| u \\| _ s ^ { k + 1 , \\gamma } \\leq ( \\| u \\| _ { s _ 1 } ^ { k + 1 , \\gamma } ) ^ \\theta ( \\| u \\| _ { s _ 2 } ^ { k + 1 , \\gamma } ) ^ { 1 - \\theta } \\ , , s : = \\theta s _ 1 + ( 1 - \\theta ) s _ 2 \\ , . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\tilde { z } _ n ^ M = \\mathcal { K } _ m ( { z } _ n ^ M ) + f . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\# ( \\frak E \\cap ( E _ 0 , E _ 1 ] ) = 1 . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\vert q ^ - \\vert = \\vert q ^ + \\vert \\mbox { a n d } q ^ - \\neq q ^ + ; \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\varepsilon } { d \\xi ^ { 2 } } - \\frac { { \\phi } ^ { \\prime } } { 2 u + \\phi } \\frac { d \\varepsilon } { d \\xi } - \\left \\{ { u ^ { 2 } + u \\phi + \\frac { 1 } { 4 } \\phi ^ { 2 } } \\right \\} \\varepsilon = h ^ { + } , \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\sum _ { \\epsilon \\in \\{ 0 , 1 \\} ^ { m + 1 } } ( - 1 ) ^ { | \\epsilon | } \\phi ( n + \\epsilon \\cdot h ) = 0 . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} | ( \\zeta - \\zeta _ 0 ) ^ 2 + \\tau ^ 2 | & \\geqslant \\big | ( u - u _ 0 ) ^ 2 - \\big ( a ( u ) - a ( u _ 0 ) \\big ) ^ 2 + \\tau ^ 2 \\big | \\\\ & = ( u - u _ 0 ) ^ 2 - \\big ( a ( u ) - a ( u _ 0 ) \\big ) ^ 2 + \\tau ^ 2 \\\\ & > \\frac { 3 \\tau ^ 2 } { 4 } . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} L u = \\Delta u + \\frac { \\nabla \\mu } { \\mu } \\cdot \\nabla u , \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\tau = \\left [ \\begin{array} { c c c c } 1 & & & \\\\ & 1 & & \\\\ & & \\zeta & \\\\ & & & 1 \\end{array} \\right ] \\varphi = \\left [ \\begin{array} { c c c c } 1 & & & \\\\ & & & - 1 \\\\ & & 1 & \\\\ & 1 & & \\end{array} \\right ] , \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} v = \\sum _ { e \\in s ^ { - 1 } ( v ) } r ( e ) \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\phi = f | _ { \\Gamma _ + } \\quad \\phi _ i = f _ i | _ { \\Gamma _ + } \\ , , i = 1 , 2 , 3 \\ , . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} \\sigma \\circ x : = ( x _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , x _ { \\sigma ^ { - 1 } ( n ) } ) , \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ { m } } \\Big | \\sum _ { j = 1 } ^ { n } \\psi ^ { ( m - 1 ) } \\left ( \\frac { j } { 2 } \\right ) \\Big | \\le 2 ( m - 1 ) ! \\ , . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\begin{cases} i _ X \\omega = \\sqrt { - 1 } \\bar { \\partial } \\theta _ X ( \\omega ) \\\\ \\int _ M e ^ { \\theta _ X ( \\omega ) } \\omega ^ n = [ \\omega _ 0 ] ^ n . \\end{cases} \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} g _ j = a _ 0 a _ 1 ^ { \\binom j 1 } a _ 2 ^ { \\binom j 2 } \\cdots a _ s ^ { \\binom j s } , j \\in \\Z , \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} A _ t ( n ) = \\sum _ { d | n } \\chi _ { t , N } ( d ) d ^ { k - 1 } a \\left ( \\frac { n ^ 2 } { d ^ 2 } t \\right ) , \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\varphi \\left ( t , x \\right ) : = \\theta ^ { \\varepsilon } \\left ( t , x \\right ) - \\left ( \\theta ^ { \\varepsilon } _ { 0 } \\left ( t , x \\right ) + \\varepsilon m ^ { \\varepsilon } \\left ( x \\right ) \\bar { \\theta } \\left ( x , \\frac { x } { \\varepsilon } \\right ) \\cdot \\nabla _ { x } \\theta ^ { 0 } \\left ( t , x \\right ) \\right ) , \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} \\tilde X _ 0 : \\begin{cases} \\dot y _ 1 & = y _ 1 ( - u + y _ 3 ) \\\\ \\dot u & = 1 + u ^ 2 \\\\ \\dot y _ 3 & = \\frac { 1 } { 2 } ( ( 1 + y _ 3 ^ 2 ) - y _ 1 ^ 2 ( 1 + u ^ 2 ) ) . \\end{cases} \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\alpha _ \\varepsilon ^ s \\colon \\R ^ n \\to \\R ^ n , \\alpha _ \\varepsilon ^ s ( w ) = l _ \\varepsilon ( w ) ^ { s - 1 } w \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} \\xi = \\int { f ^ { 1 / 2 } \\left ( z \\right ) d z } = \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 2 } + \\ln \\left \\{ { \\frac { z } { 1 + \\left ( { 1 + z ^ { 2 } } \\right ) ^ { 1 / 2 } } } \\right \\} . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\Gamma _ V ( - 1 ) = \\chi _ V ( 0 ) + \\chi _ V ' ( 0 ) \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} f ( X _ t ) & = f ( X _ 0 ) + \\sum _ j \\int _ 0 ^ t \\partial _ j f ( X _ { s ^ - } ) d X _ { j , s } + \\frac { 1 } { 2 } \\sum _ { i , j } \\int _ 0 ^ t \\partial _ { i } \\partial _ j f ( X _ { s ^ - } ) d [ X _ i , X _ j ] _ s \\\\ & + \\sum _ { s \\leq t } \\left ( \\Delta f ( X _ s ) - \\sum _ j \\partial _ j f ( X _ { s ^ - } ) \\Delta X _ { j , s } - \\sum _ { i , j } \\frac { \\partial _ j \\partial _ i f ( X _ { s ^ - } ) } { 2 } \\Delta X _ { i , s } \\Delta X _ { j , s } \\right ) \\ ; . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} & \\ , ( e v _ { 1 } ^ { i - 1 , l } ) - ( \\ , V _ { X _ 2 } ( - ( i - 1 ) A - l B ) - \\ , V _ { X _ 2 } ( - i A - l B ) ) \\\\ & = \\ , ( e v ^ { i l } ) - ( \\ , V _ { X _ 1 } ( - ( d - i ) A ) - \\ , V _ { X _ 1 } ( - ( d - i + 1 ) A ) ) \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} \\rho ^ { 2 - m } \\int _ { B _ { \\rho } } \\abs { D v _ j } ^ { 2 } = ( \\rho r _ { j } ) ^ { 2 - m } \\int _ { B _ { \\rho r _ j } ( x _ j ) } \\abs { D u _ j } ^ { 2 } \\leq C _ { m } \\Lambda . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} N / G \\ , : = \\ , N / i ( G ) \\ \\ \\ \\ { \\mathrm { a n d } } \\ \\ \\ \\ u _ G : = u \\ , { \\mathrm { m o d } } \\ , i ( G ) \\in N / G \\ , , \\ , \\forall u \\in N \\ . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} h ( [ \\mu _ { i _ { 0 1 } } , \\ldots , \\mu _ { i _ { 0 p _ 0 } } , w ] ) = \\prod \\limits _ { k = 0 } ^ n \\left ( \\sum \\limits _ { j = 1 } ^ { p _ k } D _ { i _ { k 1 } } \\cdots D _ { i _ { k ( j - 1 ) } } S _ { i _ { k j } } D _ { i _ { k ( j + 1 ) } } \\cdots D _ { i _ { k p _ k } } \\right ) . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} I _ { 0 } & = X _ { I J } ^ { i } \\dot { x } _ { j } \\\\ I _ { 1 } & = 2 t \\left ( \\frac { 1 } { 2 } \\delta _ { i j } \\dot { x } ^ { i } \\dot { x } ^ { j } + \\frac { 1 } { n } r ^ { n } \\right ) - H ^ { i } \\dot { x } _ { i } . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\sum _ { i = 3 } ^ { n } \\binom { n } { i } \\left ( i - 1 \\right ) ! p ^ { i } \\le \\sum _ { i = 1 } ^ { \\infty } \\frac { \\left ( n p \\right ) ^ { i } } { i } = - \\log \\left ( 1 - n p \\right ) = o \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\Lambda ^ { a , p } _ t g ( x ) : = \\int _ U [ g ( x + f ( t , x , u , a , p ) ) - g ( x ) ] \\nu ( d u ) \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} b _ { \\xi ' } : = - \\min \\{ 0 , \\nu _ { \\xi ' } ( f _ 1 ) , \\dots , \\nu _ { \\xi ' } ( f _ r ) \\} \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } \\left ( \\prod _ { \\substack { \\{ k , \\ell \\} \\in S \\\\ \\{ k , \\ell \\} \\neq \\{ i , j \\} } } \\frac { \\alpha _ S } { d _ { k \\ell } } \\right ) \\lceil n / t \\rceil ^ { r - 2 } \\le 2 ( \\alpha _ S / \\delta ) ^ { \\binom { r } { 2 } - 1 } ( n / t ) ^ { r - 2 } \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} ( x _ \\lambda ) \\underline { \\mathbf { d } } y = \\sup _ { x \\in X } ( x \\mathbf { d } y - x \\mathbf { d } ( x _ \\lambda ) ) _ + . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ g p ( x _ i , y _ i , z _ i , w _ i ) + 2 y = 0 \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} W _ { n , 1 } \\left ( { u , \\xi } \\right ) = \\exp \\left \\{ { u \\xi + S _ { n } \\left ( { u , \\xi } \\right ) + \\delta _ { n , 1 } \\left ( { u , \\xi } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} Z \\ ( t , m \\ ) \\leqslant Z \\ ( t , m _ 0 \\ ) ^ q Z \\ ( t , r \\ ) \\leqslant R ^ { q m _ 0 } \\sup _ { k = 0 , \\dots , q - 1 } \\left \\| Z \\ ( . , k \\ ) \\right \\| _ { \\infty , K } \\leqslant C R ^ { m } . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{gather*} \\frac { P _ { \\lambda ( N ) } \\big ( 1 , t , \\dots , t ^ { N - 2 } , t ^ { N - 1 } z \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t , \\dots , t ^ { N - 2 } , t ^ { N - 1 } \\big ) } = \\frac { P _ { \\lambda ( N ) } \\big ( z , t ^ { - 1 } , t ^ { - 2 } , \\dots , t ^ { 1 - N } \\big ) } { P _ { \\lambda ( N ) } \\big ( 1 , t ^ { - 1 } , t ^ { - 2 } , \\dots , t ^ { 1 - N } \\big ) } = \\sum _ { n \\in \\Z } { M ^ { \\lambda ( N ) } _ 1 ( n ) z ^ n } . \\end{gather*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\mathcal { Z } _ 1 = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathfrak { d } _ { i , 1 } Q _ i + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathfrak { d } _ { i , 2 } \\mathcal { Q } _ i \\ , , \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{gather*} \\Lambda ^ { p , q } = H ^ { p , q } + \\square \\Lambda ^ { p , q } , \\end{gather*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\log ( k ( n + 1 ) ) < & ( 1 - 2 b _ n ) \\log ( p _ { n + 1 } ) \\\\ < & \\log ( p _ { n + 1 } ) + \\phi ( p _ { n + 1 } ) = u ( p _ { n + 1 } ) . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\hat { J } _ { r s t w } ( \\gamma ) = \\sum _ { j , k , l , m } \\hat { v } ^ k _ r \\hat { v } ^ k _ s \\hat { v } ^ l _ t \\hat { v } ^ m _ w \\hat { \\omega } ^ { j k } \\hat { \\omega } ^ { j l } \\hat { \\omega } ^ { k l } \\hat { \\omega } ^ { k m } \\left [ h _ 1 ( \\gamma ) \\sum _ { o , q } \\hat { \\alpha } _ { j k o } \\hat { \\omega } ^ { o q } \\hat { \\alpha } _ { l m q } - h _ 2 ( \\gamma ) \\hat { \\alpha } _ { j k l m } \\right ] \\ , . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} P _ { C , n e w } ( K , L , \\epsilon ) = \\frac { \\epsilon } { m ^ 2 } \\cdot A _ { K , L } \\cdot N _ { K , L } ^ \\epsilon \\cdot ( p + \\xi ) ^ \\epsilon . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} W _ i ( \\R ) = W _ { i , \\mathrm { E i s } } \\oplus \\left ( \\bigoplus _ \\pi W _ { i , \\pi } \\right ) , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{gather*} \\frac 1 { 2 N + 1 } \\sum _ { i = - N } ^ { N + 1 - M } \\Big ( \\sup _ { 0 \\le m \\le M } \\frac { 1 } { A _ { m } } | \\sum _ { k = 0 } ^ { m } a _ k v _ { i + k } | \\Big ) ^ p \\le \\| \\sup _ { n \\ge 0 } \\frac 1 { A _ { n } } | \\sum _ { k = 0 } ^ { n } a _ k f \\circ \\tau ^ k | \\ , \\| _ { L ^ p ( \\nu ) } ^ p \\\\ \\le C ^ p \\| f \\| _ { L ^ p ( \\nu ) } ^ p = \\frac { C ^ p } { 2 N + 1 } \\sum _ { i = - N } ^ N | v _ i | ^ p \\ , . \\end{gather*}"}
-{"id": "4968.png", "formula": "\\begin{align*} - \\dot { \\tilde { f } } ( 0 ) = \\frac { 1 } { | Y | } Y ( a - b ) = \\frac { Y a - Y b } { | Y | } . \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} f ( 0 ) \\prod _ { j = 1 } ^ { \\alpha _ 1 } p ^ { ( a _ j - b _ j ) b _ j } \\le \\prod _ { j = 1 } ^ { \\alpha _ 1 } p ^ { ( a _ j - b _ j ) b _ j } ( 1 + \\theta _ j p ^ { - 1 } ) \\le f ( 6 ) \\prod _ { j = 1 } ^ { \\alpha _ 1 } p ^ { ( a _ j - b _ j ) b _ j } . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} W _ { \\ell + n } = \\sum _ { k - j = \\ell } \\ker N ^ { k + 1 } \\cap \\mathrm { i m } \\ , N ^ j . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\displaystyle { \\frac { d r } { d \\theta } = \\frac { A ( \\theta ) r ^ n } { 1 + B ( \\theta ) r ^ { n - 1 } } } . \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } | \\nabla J _ i | ^ 2 + c \\ , J _ 3 ^ 2 \\ , V _ 2 ( x ) = \\sum _ { i = 1 , 2 } \\frac { | \\nabla J _ i | ^ 2 } { 1 - J _ i ^ 2 } + c \\ , J _ 3 ^ 2 \\ , V _ 2 ( x ) \\le \\frac { k + 2 c } { r _ 0 ^ 2 } . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\widetilde { \\textnormal { ( I ) } } = & - \\frac { 4 ( p - 1 ) } { p } \\int ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla ( \\eta u _ k ^ { p / 2 } ) ) \\ ; d x \\\\ & + \\frac { 4 ( p - 2 ) } { p } \\int u _ k ^ { p / 2 } ( a \\nabla ( \\eta u _ k ^ { p / 2 } ) , \\nabla \\eta ) \\ ; d x + \\frac { 4 } { p } \\int u _ k ^ { p } ( a \\nabla \\eta , \\nabla \\eta ) \\ ; d x . \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\Gamma ( \\chi ) = \\{ \\gamma > 0 : L ( \\tfrac 1 2 + i \\gamma , \\chi ) = 0 \\} \\qquad \\Gamma ( q ) = \\bigcup _ { \\substack { \\chi \\mod q \\\\ \\chi \\ne \\chi _ 0 } } \\Gamma ( \\chi ) . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} B _ n = \\operatorname { s p a n } _ K \\big \\{ ( \\lambda , n - d ( \\lambda ) ) ( \\mu , n - d ( \\mu ) ) ^ * \\mid \\lambda , \\mu \\in \\Lambda , s ( \\lambda ) = s ( \\mu ) \\big \\} . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} f ( u , v ) = \\int _ { \\R _ + } ( e ^ { - u x } \\cos v x - 1 ) \\nu ( d x ) + i \\int _ { \\R _ + } e ^ { - u x } \\sin v x \\nu ( d x ) = : f _ 1 ( u , v ) + i f _ 2 ( u , v ) , u \\ge 0 , \\ , v \\in \\R \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} y ( x ) = & \\frac { y _ a } { \\Gamma ( \\gamma ) } ( x - a ) ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { x _ 1 } ^ { x } ( x - t ) ^ { \\alpha - 1 } f ( t , y ( t ) ) d t \\\\ & + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x _ 1 } ( x - t ) ^ { \\alpha - 1 } f ( t , y ( t ) ) d t , x \\in [ x _ 1 , x _ 2 ] . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u = - 1 & \\mbox { i n } \\Omega , \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\\\ \\partial _ { \\nu } u = & \\mbox { o n } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\int _ { - 1 } ^ 1 [ a _ 0 + a _ 2 t ^ 2 + a _ 4 t ^ 4 ] ( t ^ n - 1 ) d t = 0 \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\Delta \\Phi = \\Phi '' + \\left ( \\frac { a ' } { a } + 2 n \\frac { b ' } { b } \\right ) \\Phi ' . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\forall t \\in \\mathfrak { h } _ n , \\ \\forall b \\in \\mathfrak { b } _ n , \\psi ( [ t , b ] ) = 0 . \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} \\inf _ { F \\in \\mathcal { F } ( M ) } \\lambda _ { 1 , p } ( M , F ) = 0 . \\ \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\prod _ { U , V \\in I ( \\mathcal { C } ) } \\left ( \\prod _ { i , j } \\frac { 1 } { 1 - x _ i ^ { ( U ) } y _ j ^ { ( V ) } } \\right ) ^ { N _ { U , V } ^ { ( \\mathbf { 1 } ) } } = \\prod _ { U , V \\in I ( \\mathcal { C } ) } \\left ( \\sum _ { \\rho \\in \\mathcal { P } } s _ \\rho ( \\mathbf { x } ^ { ( U ) } ) s _ \\rho ( \\mathbf { y } ^ { ( V ) } ) \\right ) ^ { N _ { U , V } ^ { ( \\mathbf { 1 } ) } } \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} R = 2 n \\frac { a ^ 2 } { b ^ 4 } - \\frac { 2 n ( 2 n + 2 ) } { b ^ 2 } + 4 n \\frac { a ' b ' } { a b } + 2 n ( 2 n - 1 ) \\left ( \\frac { b ' } { b } \\right ) ^ 2 - 2 f ' \\left ( \\frac { a ' } { a } + 2 n \\frac { b ' } { b } \\right ) . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c c c } Q _ 0 ( x _ 0 ) & Q _ 0 ( x _ 1 ) & \\cdots & Q _ 0 ( x _ n ) \\\\ Q _ 1 ( x _ 0 ) & Q _ 1 ( x _ 1 ) & \\cdots & Q _ 1 ( x _ n ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ Q _ n ( x _ 0 ) & Q _ n ( x _ 1 ) & \\cdots & Q _ n ( x _ n ) \\end{array} \\right | = \\left ( \\prod ^ n _ { i = 0 } a _ { i i } \\right ) V ( x _ 0 , x _ 1 , \\ldots , x _ n ) , \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} X _ 1 ^ 2 + \\cdots + X _ { 2 k + 1 } ^ 2 = X _ { 2 k + 2 } ^ 2 + h , \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty P _ k ( t ) ^ 2 e ^ { - t / X } d t = \\delta _ { [ k = 3 ] } D ' X ^ 2 \\log X + D X ^ { k - 1 } + \\delta _ { [ k = 4 ] } D '' X ^ { \\frac { 5 } { 2 } } + O ( X ^ { k - 2 + \\epsilon } ) . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = \\frac { 1 } { 2 \\pi } \\frac { 1 } { ( 1 + x _ 1 ^ 2 + x _ 2 ^ 2 ) ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} | \\mathcal R _ l ( \\zeta , \\widehat \\Lambda ) | \\leq C \\varepsilon ^ { 2 - 2 \\sigma } , l = 2 , \\ldots , k \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\frac { 1 } { l } + \\frac { \\gamma } { s } + \\frac { 2 - \\gamma } { 2 } = 1 , \\qquad \\frac { 1 } { q } + \\frac { \\gamma } { r } + \\frac { 2 - \\gamma } { 2 } = 1 . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\left ( \\partial - \\tilde { \\phi } _ s ( x , \\tau ) \\right ) \\Upsilon = 0 , \\ , \\ , \\mbox { f o r } x \\in D , \\ , \\ , \\tau \\in U . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( \\zeta _ i ) } w _ i ^ 5 \\ , U _ j \\ , d x = \\mu _ i ^ { \\frac { 1 } { 2 } } \\mu _ j ^ { \\frac { 1 } { 2 } } \\int _ { B _ { \\mu _ i } } U ^ 5 ( z ) \\ , \\mu _ j ^ { - \\frac { 1 } { 2 } } \\ , U _ j ( \\zeta _ i + \\mu _ i z ) \\ , d z . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} N _ k ^ { i + 1 } = \\left \\lceil N _ k ^ { i } \\frac { \\hat { \\Sigma } ( x ^ i ) _ { k k } \\left ( \\Phi ^ { - 1 } ( 1 - \\alpha ) \\right ) ^ 2 } { \\hat { g } ( x ^ i ) ^ 2 _ k } \\right \\rceil \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} k ( 2 r - 4 ) \\le \\alpha ( k , r ) \\le \\begin{cases} ( k - 1 ) ( 4 r - k - 6 ) & r \\le k \\le 2 r - 3 , \\\\ ( k - 1 ) ( 2 r - 3 ) & k \\ge 2 r - 3 , \\end{cases} \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\alpha _ m = \\frac { ( 1 - \\delta ) m + 2 } { n - m + 1 } , \\beta _ m = \\frac { \\delta m - 3 } { n - m + 1 } , \\gamma _ m = \\frac { m - 1 } { n - m + 1 } , \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} y '' = \\frac { c _ 1 ( s ) } { 2 y } - \\alpha \\frac { ( y ' ) ^ 2 } { y } - c _ 2 ( s ) y ' \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{gather*} J | a ^ T N | = \\sqrt { g } , \\end{gather*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\mathcal { R } _ g ( x ) : = \\cap _ { i = 1 } ^ k H ^ { g _ i } ( x _ i ) = \\cap _ { i = 1 } ^ k H ^ 0 ( ( - 1 ) ^ { g _ i } x _ i ) \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{gather*} g _ { i j } = \\partial _ i \\eta \\cdot \\partial _ j \\eta = \\partial _ i \\eta ^ \\mu \\partial _ j \\eta _ \\mu , \\end{gather*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\int _ { \\R ^ { 2 n } } \\rho ( z ) \\ , d ^ { 2 n } z = 1 , \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} H \\big ( x ; \\frac { 1 } { 1 - x } \\big ) & = H _ 1 ( x ) C ( x ) ^ 2 + E \\big ( x ; C ( x ) \\big ) - H _ { 1 , 0 } ( x ) C ( x ) ^ 2 - \\frac { x C ( x ) ^ 2 } { 1 - x } E \\big ( x ; C ( x ) \\big ) \\\\ & \\ + J \\big ( x ; C ( x ) \\big ) C ( x ) - J _ 1 ( x ) C ( x ) ^ 2 \\\\ & - \\frac { 1 } { ( 1 - x ) C ( x ) } \\left ( E \\big ( x ; C ( x ) \\big ) - ( 1 - x ) E \\big ( x ; \\frac { 1 } { 1 - x } \\big ) C ( x ) \\right ) \\\\ & = x ^ 2 C ( x ) ^ 2 F _ T ( x ) - \\frac { 2 x ^ 2 - 4 x + 1 } { 1 - x } - \\frac { x ^ 5 - 1 6 x ^ 4 + 3 6 x ^ 3 - 2 8 x ^ 2 + 9 x - 1 } { ( 1 - 2 x ) ^ 2 ( 1 - x ) } C ( x ) \\ , . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} g ( c ) = g ( \\varpi _ 1 \\cdots \\varpi _ { 2 k _ 0 } ) \\prod ^ k _ { j = 2 k _ 0 + 1 } g ( \\varpi _ j ) = g ( \\varpi _ 1 \\cdots \\varpi _ { 2 k _ 0 } ) N \\left ( \\prod ^ k _ { j = 2 k _ 0 + 1 } \\varpi _ j \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\avg { ( x , x ) , ( x , - x ) } ^ \\perp = \\avg { ( x , 0 ) , ( 0 , x ) } ^ \\perp = \\{ ( q _ 1 , q _ 2 ) : q _ 1 , q _ 2 \\perp x \\} \\end{align*}"}